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\section*{Introduction}
The singularities in General Relativity can be avoided only if the stress-energy tensor in the right hand side of Einstein's equation satisfies some particular conditions. One way to avoid them was proposed by the authors of \cite{corda2010removingBHsingularities}, who have shown that the singularities can be removed by constructing the stress-energy tensor with non-linear electrodynamics. On the other hand, Einstein's
equation leads to singularities in general conditions \cite{Pen65,Haw66i,Haw66ii,Haw67iii,HP70,HE95}, and there the time evolution breaks down. Is this a problem of the theory itself or of the way it is formulated?
This paper proposes a version of Einstein's equation which is equivalent to the standard version at the points of spacetime where the metric is non-singular. But unlike Einstein's equation, in many cases it can be extended at and beyond the singular points.
Let $(M,g)$ be a Riemannian or a {semi{-\penalty0\hskip0pt\relax}Riemannian} manifold of dimension $n$.
It is useful to recall the definition of the \textit{Kulkarni-Nomizu product} of two symmetric bilinear forms $h$ and $k$,
\begin{equation}
\label{eq_kulkarni_nomizu}
(h\circ k)_{abcd} := h_{ac}k_{bd} - h_{ad}k_{bc} + h_{bd}k_{ac} - h_{bc}k_{ad}.
\end{equation}
The Riemann curvature tensor can be decomposed algebraically as
\begin{equation}
\label{eq_ricci_decomposition}
R_{abcd} = S_{abcd} + E_{abcd} + C_{abcd}.
\end{equation}
where
\begin{equation}
\label{eq_ricci_part_S}
S_{abcd} = \dsfrac{1}{2n(n-1)}R(g\circ g)_{abcd}
\end{equation}
is the scalar part of the Riemann curvature and
\begin{equation}
\label{eq_ricci_part_E}
E_{abcd} = \dsfrac{1}{n-2}(S \circ g)_{abcd}
\end{equation}
is the \textit{semi-traceless part} of the Riemann curvature. Here
\begin{equation}
\label{eq_ricci_traceless}
S_{ab} := R_{ab} - \dsfrac{1}{n}Rg_{ab}
\end{equation}
is the traceless part of the Ricci curvature.
The \textit{Weyl curvature tensor} is defined as the \textit{traceless part} of the Riemann curvature
\begin{equation}
\label{eq_weyl_curvature}
C_{abcd} = R_{abcd} - S_{abcd} - E_{abcd}.
\end{equation}
The Einstein equation is
\begin{equation}
\label{eq_einstein}
G_{ab} + \Lambda g_{ab} = \kappa T_{ab},
\end{equation}
where $T_{ab}$ is the stress-energy tensor of the matter, the constant $\kappa$ is defined as $\kappa:=\dsfrac{8\pi \mc G}{c^4}$, where $\mc G$ and $c$ are the gravitational constant and the speed of light, and $\Lambda$ is the \textit{cosmological constant}.
The term
\begin{equation}
\label{eq_einstein_tensor}
G_{ab}:=R_{ab}-\frac 1 2 R g_{ab}
\end{equation}
is the Einstein tensor, constructed from the \textit{Ricci curvature} $R_{ab} := g^{st}R_{asbt}$ and the \textit{scalar curvature} $R := g^{st}R_{st}$.
As it is understood, the Einstein equation establishes the connection between curvature and stress-energy. The curvature contributes to the equation in the form of the Ricci tensor $R_{ab}$ and the scalar curvature. In the proposed equation, the curvature contributes in the form of the semi-traceless and scalar parts of the Riemann tensor, $E_{abcd}$ \eqref{eq_ricci_part_E} and $S_{abcd}$ \eqref{eq_ricci_part_S}, which are tensors of the same order and have the same symmetries as $R_{abcd}$.
The Ricci tensor $R_{ab}$ is obtained by contracting the tensor $E_{abcd}+S_{abcd}$, and has the same information (if the metric is {non{-\penalty0\hskip0pt\relax}degenerate}). One can move from the fourth-order tensors $E_{abcd}+S_{abcd}$ to $R_{ab}$ by contraction, and one can move back to them by taking the Kulkarni-Nomizu product \eqref{eq_kulkarni_nomizu}, but they are equivalent. Yet, if the metric $g_{ab}$ is degenerate, then $g^{ab}$ and the contraction $R_{ab}=g^{st}(E_{asbt}+S_{asbt})$ become divergent, even if $g_{ab}$, $E_{abcd}$, and $S_{abcd}$ are smooth. This suggests the possibility that $E_{abcd}$ and $S_{abcd}$ are more fundamental that the Ricci and scalar curvatures.
This suggestion is in agreement with the following observation. In the case of \textit{electrovac} solutions, where $F_{ab}$ is the electromagnetic tensor,
\begin{equation}
\label{eq_stress_energy_maxwell}
T_{ab}=\frac{1}{4\pi}\(\frac 1 4 g_{ab} F_{st}F^{st} - F_{as} F_b{}^s\)=-\frac{1}{8\pi}\(F_{ac}F_b{}^c + {}^\ast F_{ac} {}^\ast F_b{}^c\),
\end{equation}
where ${}^\ast$ is the Hodge duality operation. It can be obtained by contracting the semi-traceless part of the Riemann tensor
\begin{equation}
\label{eq_stress_energy_maxwell_expanded}
E_{abcd}=-\frac{\kappa}{8\pi}\(F_{ab}F_{cd} + {}^\ast F_{ab} {}^\ast F_{cd}\).
\end{equation}
Therefore it is natural to at least consider an equation in terms of these fourth-order tensors, rather than the Ricci and scalar curvatures.
The main advantage of this method is that there are singularities in which the new formulation of the Einstein equation is not singular (although the original Einstein equation exhibits singularities, obtained when contracting with the singular tensor $g^{ab}$). The expanded Einstein equation is written in terms of the smooth geometric objects $E_{abcd}$ and $S_{abcd}$. Because of this the solutions can be extended at singularities where the original Einstein equation diverges. This doesn't mean that the singularities are removed; for example the Kretschmann scalar $R_{abcd}R^{abcd}$ is still divergent at some of these singularities. But this is not a problem, since the Kretschmann scalar is not part of the evolution equation. It is normally used as an indicator that there is a singularity, for example to prove that the {Schwarzschild} singularity at $r=0$ cannot be removed by coordinate changes, as the event horizon singularity can. While a singularity of the Kretschmann scalar indicates the presence of a singularity of the curvature, it doesn't have implications on whether the singularity can be resolved or not. In the proposed equation we use $R_{abcd}$ which is smooth at the studied singularities, and we don't use $R^{abcd}$ which is singular and causes the singularity of the Kretschmann scalar.
A second reason to consider the expanded version of the Einstein equation and the {quasi{-\penalty0\hskip0pt\relax}regular} singularities at which it is smooth is that at these singularities the Weyl curvature tensor vanishes. The implications of this feature will be explored in \cite{Sto12c}.
It will be seen that there are some important examples of singularities which turn out to be {quasi{-\penalty0\hskip0pt\relax}regular}. While singularities still exist, our approach provides a description in terms of smooth geometric objects which remain finite at singularities. By this we hope to improve our understanding of singularities and to distinguish those to which our resolution applies.
The \textit{expanded Einstein equations} and the {quasi{-\penalty0\hskip0pt\relax}regular} spacetimes on which they hold are introduced in section \sref{s_einstein_exp_qreg}. They are obtained by taking the Kulkarni-Nomizu product between Einstein's equation and the metric tensor. In a {quasi{-\penalty0\hskip0pt\relax}regular} spacetime the metric tensor becomes degenerate at singularities in a way which cancels them and makes the equations smooth.
The situations when the new version of Einstein's equation extends at singularities include isotropic singularities (section \sref{s_qreg_examples_isotropic}) and a class of warped product singularities (section \sref{s_qreg_examples_warped}). It also contains the {Schwarzschild} singularity (section \sref{s_qreg_examples_schw}) and the {FLRW} Big Bang singularity (section \sref{s_qreg_examples_flrw}).
\section{Expanded Einstein equation and {quasi{-\penalty0\hskip0pt\relax}regular} spacetimes}
\label{s_einstein_exp_qreg}
\subsection{The expanded Einstein equation}
\label{s_einstein_exp}
An equation which is equivalent to Einstein's equation whenever the metric tensor $g_{ab}$ is {non{-\penalty0\hskip0pt\relax}degenerate}, but is valid also in a class of situations when $g_{ab}$ becomes degenerate and Einstein's tensor is not defined will be discussed in this section. Later it will be shown that the proposed version of Einstein's equation remains smooth in various important situations such as the FLRW Big-Bang singularity, isotropic singularities, and at the singularity of the {Schwarzschild} black hole.
We introduce the \textit{expanded Einstein equation}
\begin{equation}
\label{eq_einstein_expanded}
(G\circ g)_{abcd} + \Lambda (g\circ g)_{abcd} = \kappa (T\circ g)_{abcd}.
\end{equation}
If the metric is {non{-\penalty0\hskip0pt\relax}degenerate} then the Einstein equation and its expanded version are equivalent. This can be seen by contracting the expanded Einstein equation, for instance in the indices $b$ and $d$. From \eqref{eq_kulkarni_nomizu} the contraction in $b$ and $d$ of a Kulkarni-Nomizu product $(h\circ g)_{abcd}$ is
\begin{equation}
\hat h_{ac}:=(h\circ g)_{asct}g^{st} = h_{ac}g^s_s - h_{at}\delta^t_c + h^s_sg_{ac} - h_{sc}\delta^s_a = 2h_{ac} + h^s_sg_{ac}.
\end{equation}
From $\hat h_{ac}$ the original tensor $h_{ac}$ can be obtained again by
\begin{equation}
\label{eq_expanded_to_standard}
h_{ac}=\frac 1 2 \hat h_{ac} - \frac 1{12}\hat h{}^s_s g_{ac}.
\end{equation}
By this procedure the terms $G_{ab}$, $T_{ab}$, and $\Lambda g_{ab}$ can be recovered from the equation \eqref{eq_einstein_expanded}, thus obtaining the Einstein equation \eqref{eq_einstein}
. Hence, the Einstein equation and its expanded version are equivalent for a {non{-\penalty0\hskip0pt\relax}degenerate} metric.
If the metric becomes degenerate its inverse becomes singular, and in general the Riemann, Ricci, and scalar curvatures, and consequently the Einstein tensor $G_{ab}$, diverge. For certain cases the metric term from the Kulkarni-Nomizu product $G\circ g$ tends to $0$ fast enough to cancel the divergence of the Einstein tensor. The {quasi{-\penalty0\hskip0pt\relax}regular} singularities satisfy the condition that the divergence of $G$ is compensated by the degeneracy of the metric, so that $G\circ g$ is smooth.
This cancellation allows us to weaken the condition that the metric tensor is {non{-\penalty0\hskip0pt\relax}degenerate}, to some cases when it can be degenerate. It will be seen that these cases include some important singularities.
\subsection{A more explicit form of the expanded Einstein equation}
\label{s_einstein_exp_explicit}
To give a more explicit form of the expanded Einstein equation, the \textit{Ricci decomposition} of the Riemann curvature tensor is used (see \textit{e.g.} \cite{ST69,BESS87,GHLF04}).
By using the equations \eqref{eq_einstein_tensor}
and \eqref{eq_ricci_traceless} in dimension $n=4$, the Einstein tensor in terms of the traceless part of the Ricci tensor and the scalar curvature can be written:
\begin{equation}
G_{ab} = S_{ab} - \dsfrac{1}{4}R g_{ab}.
\end{equation}
This equation can be used to calculate the \textit{expanded Einstein tensor}:
\begin{equation}
\label{eq_einstein_tensor_expanded}
\begin{array}{lrl}
G_{abcd} &:=& (G\circ g)_{abcd} \\
&=& (S \circ g)_{abcd} - \dsfrac{1}{4}R (g\circ g)_{abcd}\\
&=& 2 E_{abcd} - 6 S_{abcd}.
\end{array}
\end{equation}
The expanded Einstein equation now takes the form
\begin{equation}
\label{eq_einstein_expanded_explicit}
2 E_{abcd} - 6 S_{abcd} + \Lambda (g\circ g)_{abcd} = \kappa (T\circ g)_{abcd}.
\end{equation}
\subsection{Quasi-regular spacetimes}
\label{s_qreg_spacetimes}
We are interested in singular spacetimes on which the expanded Einstein equation \eqref{eq_einstein_expanded} can be written and is smooth. From \eqref{eq_einstein_expanded_explicit} it can be seen that this requires the smoothness of the tensors $E_{abcd}$ and $S_{abcd}$.
In addition we are interested to have the nice properties of the {semi{-\penalty0\hskip0pt\relax}regular} spacetimes.
As showed in \cite{Sto11a}, the {semi{-\penalty0\hskip0pt\relax}regular} manifolds are a class of singular {semi{-\penalty0\hskip0pt\relax}Riemannian} manifolds which are nice for several reasons, one of them being that the Riemann tensor $R_{abcd}$ is smooth.
First, a contraction between covariant indices is needed. This is in general prohibited by the fact that when the metric tensor $g_{ab}$ becomes degenerate it doesn't admit a reciprocal $g^{ab}$.
Although the metric $g_{ab}$ can't induce an invariant inner product on the cotangent space $T_p^*M$, it induces one on its subspace $\flat(T_pM)$, where $\flat:T_pM\to T_p^*M$ is the vector space morphism defined by $X^\flat(Y):=\metric{X,Y}$, for any $X,Y\in T_pM$. Equivalently, $\flat(T_pM)$ is the space of $1$-forms $\omega$ on $T_pM$ so that $\omega|_{\ker\flat}=0$. The morphism $\flat$ is isomorphism if and only if $g$ is {non{-\penalty0\hskip0pt\relax}degenerate}; in this case its inverse is denoted by $\sharp$.
The inner product on $\flat(T_pM)$ is then defined by ${g}_{\bullet}(X^\flat,Y^\flat):=\metric{X,Y}$ and it is invariant.
This allows us to define a contraction between covariant slots of a tensor $T$, which vanishes when vectors from $\ker\flat$ are plugged in those slots.
This will turn out to be enough for our needs.
We denote the contractions between covariant indices of a tensor $T$ by $T(\omega_1,\ldots,\omega_r,v_1,\ldots,{{}_\bullet},\ldots,{{}_\bullet},\ldots,v_s)$.
A degenerate metric also prohibits in general the construction of a Levi-Civita connection. For vector fields we use instead of $\nabla_XY$, the \textit{Koszul form}, defined as:
\begin{equation*}
\mc K:\fivect M^3\to\mathbb{R},
\end{equation*}
\begin{equation}
\label{eq_Koszul_form}
\mc K(X,Y,Z) :=\displaystyle{\frac 1 2} \{ X \metric{Y,Z} + Y \metric{Z,X} - Z \metric{X,Y}
- \metric{X,[Y,Z]} + \metric{Y, [Z,X]} + \metric{Z, [X,Y]}\}
\end{equation}
which defines the Levi-Civita connection by $\nabla_XY=\mc K(X,Y,\_)^\sharp$ for a {non{-\penalty0\hskip0pt\relax}degenerate} metric, but not when the metric becomes degenerate.
We define now {semi{-\penalty0\hskip0pt\relax}regular} manifolds, on which we can define covariant derivatives for a large class of differential forms and tensors. We can also define a generalization of the Riemann curvature $R_{abcd}$, which turns out to be smooth and non-singular.
\begin{definition}
\label{def_semi_regular}
A singular {semi{-\penalty0\hskip0pt\relax}Riemannian} manifold satisfying the condition that $\mc K(X,Y,\_)\in\flat(T_pM)$, and that the contraction $\mc K(X,Y,{{}_\bullet})\mc K(Z,T,{{}_\bullet})$ is smooth for any local vector fields $X,Y,Z,T$, is named \textit{{semi{-\penalty0\hskip0pt\relax}regular} manifold}, and its metric is called \textit{{semi{-\penalty0\hskip0pt\relax}regular} metric}.
A $4$-dimensional {semi{-\penalty0\hskip0pt\relax}regular} manifold with metric having the signature at each point $(r,s,t)$, $s\leq 3$, $t\leq 1$, but which is {non{-\penalty0\hskip0pt\relax}degenerate} on a dense subset, is called \textit{{semi{-\penalty0\hskip0pt\relax}regular} spacetime} \cite{Sto11a}.
\end{definition}
In \cite{Sto11a} we defined the Riemann curvature $R_{abcd}$ for {semi{-\penalty0\hskip0pt\relax}regular} metrics, even for {non{-\penalty0\hskip0pt\relax}degenerate} metrics, in a way which avoids the undefined $\nabla_XY$, but relies on the defined and smooth $\mc K(X,Y,Z)$, by
\begin{equation}
\label{eq_riemann_curvature_tensor_coord}
R_{abcd}= \partial_a \Gamma_{bcd} - \partial_b \Gamma_{acd} + \Gamma_{ac{{}_\bullet}}\Gamma_{bd{{}_\bullet}} - \Gamma_{bc{{}_\bullet}}\Gamma_{ad{{}_\bullet}},
\end{equation}
where $\Gamma_{abc}=\mc K(\partial_a,\partial_b,\partial_c)$ are the Christoffel's symbols of the first kind. From Definition \ref{def_semi_regular}, $R_{abcd}$ is smooth.
More details on the {semi{-\penalty0\hskip0pt\relax}regular} manifolds can be found in \cite{Sto11a,Sto11b,Sto12e}.
In a {semi{-\penalty0\hskip0pt\relax}regular} spacetime, since $R_{abcd}$ is smooth, the densitized Einstein tensor $G_{ab}\det g$ is smooth \cite{Sto11a}, and a densitized version of the Einstein equation can be written, which is equivalent to the usual version when the metric is {non{-\penalty0\hskip0pt\relax}degenerate}:
\begin{equation}
\label{eq_einstein_idx:densitized}
G_{ab}\sqrt{-g}^W + \Lambda g_{ab}\sqrt{-g}^W = \kappa T_{ab}\sqrt{-g}^W,
\end{equation}
where it is enough to take the weight $W\leq 2$.
Although the {semi{-\penalty0\hskip0pt\relax}regular} approach is more general, here is explored the {quasi{-\penalty0\hskip0pt\relax}regular} one, which is more strict. Consequently, these results are stronger.
\begin{definition}
\label{def_quasi_regular}
We say that a {semi{-\penalty0\hskip0pt\relax}regular} manifold $(M,g_{ab})$ is \textit{{quasi{-\penalty0\hskip0pt\relax}regular}}, and that $g_{ab}$ is a \textit{{quasi{-\penalty0\hskip0pt\relax}regular} metric}, if:
\begin{enumerate}
\item
$g_{ab}$ is {non{-\penalty0\hskip0pt\relax}degenerate} on a subset dense in $M$
\item
the tensors $S_{abcd}$ and $E_{abcd}$ defined at the points where the metric is {non{-\penalty0\hskip0pt\relax}degenerate} extend smoothly to the entire manifold $M$.
\end{enumerate}
If the {quasi{-\penalty0\hskip0pt\relax}regular} manifold $M$ is a {semi{-\penalty0\hskip0pt\relax}regular} spacetime, we call it \textit{{quasi{-\penalty0\hskip0pt\relax}regular} spacetime}. Singularities of {quasi{-\penalty0\hskip0pt\relax}regular} manifolds are called {quasi{-\penalty0\hskip0pt\relax}regular}.
\end{definition}
It can be seen that on an {quasi{-\penalty0\hskip0pt\relax}regular} spacetime the expanded Einstein tensor can be extended at the points where the metric is degenerate, and the extension is smooth. This is in fact the motivation of Definition \ref{def_quasi_regular}.
\section{Examples of {quasi{-\penalty0\hskip0pt\relax}regular} spacetimes}
\label{s_qreg_examples}
The {quasi{-\penalty0\hskip0pt\relax}regular} spacetimes are more general than the regular ones (those with {non{-\penalty0\hskip0pt\relax}degenerate} metric), containing them as a particular case. The question is, are they general enough to cover the singularities which plagued General Relativity? In the following it will be seen that at least for some relevant cases the answer is positive. It will be seen that the class of {quasi{-\penalty0\hskip0pt\relax}regular} singularities contain isotropic singularities \sref{s_qreg_examples_isotropic}, singularities obtained as warped products \sref{s_qreg_examples_warped} (including the {Friedmann-Lema\^itre-Robertson-Walker} spacetime \sref{s_qreg_examples_flrw}), and even the {Schwarzschild} singularity \sref{s_qreg_examples_schw}. The existence of these examples which are extensively researched justifies the study of the more general {quasi{-\penalty0\hskip0pt\relax}regular} singularities and of the extended Einstein equations.
\subsection{Isotropic singularities}
\label{s_qreg_examples_isotropic}
\textit{Isotropic singularities} occur in conformal rescalings of {non{-\penalty0\hskip0pt\relax}degenerate} metrics, when the scaling function cancels. They were extensively studied by Tod \cite{Tod87,Tod90,Tod91,Tod92,Tod02,Tod03}, Claudel \& Newman \cite{CN98}, Anguige \& Tod \cite{AT99i,AT99ii}, in connection with cosmological models. The following theorem shows that the isotropic singularities are {quasi{-\penalty0\hskip0pt\relax}regular}.
\begin{theorem}[Isotropic singularities]
\label{thm_quasireg_example_conformal}
Let $(M,g_{ab})$ be a regular spacetime (we assume therefore that the metric $g_{ab}$ is {non{-\penalty0\hskip0pt\relax}degenerate}). Then, if $\Omega:M\to\mathbb{R}$ is a smooth function which is non-zero on a dense subset of $M$, the spacetime $(M,\widetilde g_{ab} :=\Omega^2 g_{ab})$ is {quasi{-\penalty0\hskip0pt\relax}regular}.
\end{theorem}
\begin{proof}
From \cite{Sto11a} is known that $(M,\widetilde g_{ab})$ is {semi{-\penalty0\hskip0pt\relax}regular}.
The Ricci and the scalar curvatures take the following forms (\cite{HE95}, p. 42.):
\begin{equation}
\label{eq_conformal_ricci_curv_ud}
\widetilde R^a{}_b = \Omega^{-2}R^a{}_b + 2\Omega^{-1}(\Omega^{-1})_{;bs}g^{as}-\dsfrac 1 2\Omega^{-4}(\Omega^2)_{;st}g^{st}\delta^a{}_b
\end{equation}
\begin{equation}
\label{eq_conformal_scalar_curv}
\widetilde R=\Omega^{-2}R-6\Omega^{-3}\Omega_{;st}g^{st}
\end{equation}
where the covariant derivatives correspond to the metric $g$.
From equation \eqref{eq_conformal_ricci_curv_ud} follows that
\begin{equation}
\widetilde R_{ab}=\Omega^2 g_{as} \widetilde R^s{}_b=R_{ab} + 2\Omega(\Omega^{-1})_{;ab}-\dsfrac 1 2\Omega^{-2}(\Omega^2)_{;st}g^{st}g_{ab},
\end{equation}
which tends to infinity when $\Omega\to 0$. But we are interested to prove the smoothness of the Kulkarni-Nomizu product $\widetilde\textnormal{Ric}\circ \widetilde g$. We notice that the term $\widetilde g$ contributes with a factor $\Omega^2$, and it is enough to prove the smoothness of
\begin{equation}
\Omega^2\widetilde R_{ab}=\Omega^2 R_{ab} + 2\Omega^3(\Omega^{-1})_{;ab}-\dsfrac 1 2(\Omega^2)_{;st}g^{st}g_{ab},
\end{equation}
which follows from
\begin{equation}
\begin{array}{lll}
\Omega^3(\Omega^{-1})_{;ab} &=& \Omega^3\((\Omega^{-1})_{;a}\)_{;b} = \Omega^3\(-\Omega^{-2}\Omega_{;a}\)_{;b} \\
&=& \Omega^3\(2\Omega^{-3}\Omega_{;b}\Omega_{;a} - \Omega^{-2}\Omega_{;ab}\) \\
&=& 2\Omega_{;a}\Omega_{;b} - \Omega\Omega_{;ab} \\
\end{array}
\end{equation}
Hence, the tensor $\widetilde\textnormal{Ric}\circ \widetilde g$ is smooth. The fact that $\widetilde R \widetilde g\circ \widetilde g$ is smooth follows from the observation that $\widetilde g\circ \widetilde g$ contributes with $\Omega^4$, and the least power in which $\Omega$ appears in the expression \eqref{eq_conformal_scalar_curv} of $\widetilde R$ is $-3$.
From the above follows that $\widetilde E_{abcd}$ and $\widetilde S_{abcd}$ are smooth. Hence the spacetime $(M,\widetilde g_{ab})$ is {quasi{-\penalty0\hskip0pt\relax}regular}.
\end{proof}
\subsection{{Quasi{-\penalty0\hskip0pt\relax}regular} warped products}
\label{s_qreg_examples_warped}
Another example useful in cosmology is the following, which is a generalization of the warped products. Warped products are extensively researched, since they allow the construction of {semi{-\penalty0\hskip0pt\relax}Riemannian} spacetimes, having applications to GR. But when the warping function becomes $0$, singularities occur (see \textit{e.g.} \citep{ONe83}{ 204}). Fortunately, in the cases of interest for General Relativity, these singularities are {quasi{-\penalty0\hskip0pt\relax}regular}. We will allow the warped function $f$ to become $0$ (generalizing the standard definition \cite{ONe83}, where it is not allowed to vanish because it leads to degenerate metrics), and prove that what the resulting singularities are {quasi{-\penalty0\hskip0pt\relax}regular}.
\begin{definition}
\label{def_wp}
Let $(B,\textnormal{d} s_B^2)$ and $(F,\textnormal{d} s_F^2)$ be two {semi{-\penalty0\hskip0pt\relax}Riemannian} manifolds, and $f: B\to\mathbb{R}$ a smooth function on $B$. The \textit{degenerate warped product} of $B$ and $F$ with \textit{warping function} $f$ is the manifold $B\times_f F:=\(B\times F,\textnormal{d} s_{B\times F}^2\)$, with the metric
\begin{equation}
\textnormal{d} s_{B\times F}^2 = \textnormal{d} s_B^2 + f^2\textnormal{d} s_F^2
\end{equation}
\end{definition}
\begin{theorem}[{Quasi{-\penalty0\hskip0pt\relax}regular} warped product]
\label{thm_quasireg_example_wp}
A degenerate warped product $B\times_f F$ with $\dim B=1$ is {quasi{-\penalty0\hskip0pt\relax}regular}.
\end{theorem}
\begin{proof}
From \cite{Sto11b}, $B\times_f F$ is {semi{-\penalty0\hskip0pt\relax}regular}.
Let's denote by $g_B$, $g_F$ and $g$ the metrics on $B$, $F$ and $B\times_f F$.
It is known (\cite{ONe83}, p. 211) that for horizontal vector fields $X,Y\in\fivectlift{B \times F,B}$ and vertical vector fields $V,W\in\fivectlift{B \times F,F}$,
\begin{enumerate}
\item $\tn{Ric}(X,Y) = \tn{Ric}_B(X,Y) + \dsfrac{\dim F}{f}H^f(X,Y)$
\item $\tn{Ric}(X,V) = 0$
\item $\tn{Ric}(V,W) = \tn{Ric}_F(V,W) + \(f\Delta f + (\dim F-1)g_B(\textnormal{grad } f,\textnormal{grad } f)\)g_F(V,W)$
\end{enumerate}
where $\Delta f$ is the Laplacian, $H^f$ the Hessian, and $\textnormal{grad } f$ the gradient.
It follows that $\tn{Ric}(X,V)$ and $\tn{Ric}(V,W)$ are smooth, but $\tn{Ric}(X,Y)$ in general is not, because of the term containing $f^{-1}$. But since $\dim B=1$, the only terms in the Kulkarni-Nomizu product $\textnormal{Ric}\circ g$ containing $\textnormal{Ric}(X,Y)$ are of the form
\begin{equation*}
\textnormal{Ric}(X,Y)g(V,W)=f^2\textnormal{Ric}(X,Y)g_F(V,W).
\end{equation*}
Hence, $\textnormal{Ric}\circ g$ is smooth.
From the expression of the scalar curvature
\begin{equation}
\label{eq_scalar_curv_wp}
R = R_B + \frac {R_F}{f^2} + 2\dim F\dsfrac{\Delta f}{f} + \dim F(\dim F - 1)\dsfrac{g_B(\textnormal{grad } f,\textnormal{grad } f)}{f^2}
\end{equation}
can be concluded that $S_{abcd}$ is smooth too, because $g\circ g$ contains at least one factor of $f^2$. Hence, $B\times_f F$ is {quasi{-\penalty0\hskip0pt\relax}regular}.
\end{proof}
The following example important in cosmology is a direct application of this result.
\begin{proposition}[{Semi{-\penalty0\hskip0pt\relax}regular} manifold which is not {quasi{-\penalty0\hskip0pt\relax}regular}]
Let $B=\mathbb{R}^k$, $k>1$, be an Euclidean space, with the canonical metric $g_B$, and $f:B\to\mathbb{R}$ a linear function $f\neq 0$. Let $F=\mathbb{R}^l$, $l>1$, with the canonical metric $g_F$.
Then the degenerate warped product $B\times_f F$ is {semi{-\penalty0\hskip0pt\relax}regular}, but it isn't {quasi{-\penalty0\hskip0pt\relax}regular}.
\end{proposition}
\begin{proof}
Because $f$ is linear but not constant, $\textnormal{grad } f\neq 0$ is constant, and $\Delta f=0$.
The scalar curvature \eqref{eq_scalar_curv_wp} becomes $R=l(l - 1)\dsfrac{g_B(\textnormal{grad } f,\textnormal{grad } f)}{f^2}$, which is singular at $0$. Because $k>1$, $g_B\circ g_B$ doesn't vanish, hence it doesn't cancel the denominator $f^2$ of $R$ in the term $R g_B\circ g_B$. Also, the term $R g_B\circ g_B$ is not canceled by other terms composing $S_{abcd}$, because they are all smooth, containing at least one $g_F$. Hence, $S_{abcd}$ is singular, and the degenerate warped product $B\times_f F$ isn't {quasi{-\penalty0\hskip0pt\relax}regular}. On the other hand, according to \cite{Sto11b}, because $B$ and $F$ are {non{-\penalty0\hskip0pt\relax}degenerate}, $B\times_f F$ is {semi{-\penalty0\hskip0pt\relax}regular}.
\end{proof}
\subsection{The {Friedmann-Lema\^itre-Robertson-Walker} spacetime}
\label{s_qreg_examples_flrw}
The {Friedmann-Lema\^itre-Robertson-Walker} ({FLRW}) spacetime is defined as the warped product $I\times_a \Sigma$, where
\begin{enumerate}
\item
$I\subseteq \mathbb{R}$ is an interval representing the time, which is viewed as a {semi{-\penalty0\hskip0pt\relax}Riemannian} space with the negative definite metric $-c^2\textnormal{d} t^2$.
\item
$(\Sigma,\textnormal{d}\Sigma^2)$ is a three-dimensional Riemannian space, usually one of the homogeneous spaces $S^3$, $\mathbb{R}^3$, and $H^3$ (to model the homogeneity and isotropy conditions at large scale). Then the metric on $\Sigma$ is, in spherical coordinates $(r,\theta,\phi)$,
\begin{equation}
\label{eq_flrw_sigma_metric}
\textnormal{d}\Sigma^2 = \dsfrac{\textnormal{d} r^2}{1-k r^2} + r^2\(\textnormal{d}\theta^2 + \sin^2\theta\textnormal{d}\phi^2\),
\end{equation}
where $k=1,0,-1$, for the $3$-sphere $S^3$, the Euclidean space $\mathbb{R}^3$, or hyperbolic space $H^3$ respectively.
\item
$a: I\to \mathbb{R}$ is a function of time.
\end{enumerate}
The {FLRW} metric is
\begin{equation}
\label{eq_flrw_metric}
\textnormal{d} s^2 = -c^2\textnormal{d} t^2 + a^2(t)\textnormal{d}\Sigma^2.
\end{equation}
At any moment of time $t\in I$ the space is $\Sigma_t=(\Sigma,a^2(t)g_\Sigma)$.
For a {FLRW} universe filled with a fluid with mass density $\rho(t)$ and pressure density $p(t)$, the stress-energy tensor is defined as
\begin{equation}
\label{eq_friedmann_stress_energy}
T^{ab} = \(\rho + \dsfrac{p}{c^2}\)u^a u^b + p g^{ab},
\end{equation}
where $g(u,u)=-c^2$.
From Einstein's equation with the stress-energy tensor \eqref{eq_friedmann_stress_energy} follow the \textit{Friedmann equation}
\begin{equation}
\label{eq_friedmann_density}
\rho = \kappa^{-1}\(3\dsfrac{\dot{a}^2 + kc^2}{c^2 a^2} - \Lambda \),
\end{equation}
which gives the mass density $\rho(t)$ in terms of $a(t)$, and the \textit{acceleration equation}
\begin{equation}
\label{eq_acceleration}
\dsfrac{p}{c^2} = \dsfrac{2}{\kappa c^2}\(\dsfrac{\Lambda}{3}-\dsfrac{1}{c^2} \dsfrac{\ddot{a}}{a}\) - \dsfrac \rho 3,
\end{equation}
giving the pressure density $p(t)$.
A question that may arise is what happens with the densities $\rho$ and $p$. Equations \eqref{eq_friedmann_density} and \eqref{eq_acceleration} show that $\rho$ and $p$ may diverge in most cases for $a\to 0$. As explained in \cite{Sto11h}, $\rho$ and $p$ are calculated considering orthonormal frames. If the frame is not necessarily orthonormal (because there is no orthonormal frame at the point where the metric is degenerate), then the volume element is not necessarily equal to $1$, and it has to be included in the equations. The scalars $\rho$ and $p$ are replaced by the differential $4$-forms which have the components $\rho\sqrt{-g}$ and $p\sqrt{-g}$. It can be seen by calculation that these forms are smooth.
If the metric on the manifold $\Sigma$ is denoted by $g_{\Sigma}$, then the Friedmann equation \eqref{eq_friedmann_density} becomes
\begin{equation}
\label{eq_friedmann_density_tilde}
\rho\sqrt{-g} = \dsfrac{3}{\kappa}a\(\dot a^2 + k\) \sqrt{g_{\Sigma}},
\end{equation}
and the acceleration equation \eqref{eq_acceleration} becomes
\begin{equation}
\label{eq_acceleration_tilde}
\rho\sqrt{-g} + 3p\sqrt{-g} = -\dsfrac{6}{\kappa}a^2\ddot{a} \sqrt{g_{\Sigma}},
\end{equation}
hence $\rho\sqrt{-g}$ and $p\sqrt{-g}$ are smooth.
As $a\to 0$, the metric becomes degenerate, $\rho$ and $p$ diverge, and therefore the stress-energy tensor \eqref{eq_friedmann_stress_energy} diverges too. Because of this, the Ricci tensor also diverges. But, from Theorem \ref{thm_quasireg_example_wp}, $R_{abcd}$, $E_{abcd}$, and $S_{abcd}$ are smooth. What can be said about the expanded stress-energy tensor $(T \circ g)_{abcd}$? The following corollary shows that the metric is {quasi{-\penalty0\hskip0pt\relax}regular}, hence the expanded stress-energy tensor is smooth.
\begin{corollary}
\label{thm_flrw}
The {FLRW} spacetime with smooth $a: I\to \mathbb{R}$ is {quasi{-\penalty0\hskip0pt\relax}regular}.
\end{corollary}
\begin{proof}
Since the {FLRW} spacetime is a warped product between a $1$-dimensional and a $3$-dimensional manifold with warping function $a$, this is a direct consequence of Theorem \ref{thm_quasireg_example_wp}.
\end{proof}
\begin{remark}
Corollary \ref{thm_flrw} applies not only to a {FLRW} universe filled with a fluid, but to more general ones. For this particular case a direct proof was given in \cite{Sto12a}, showing explicitly how the expected infinities of the physical fields cancel out.
\end{remark}
While the expanded Einstein equation for the {FLRW} spacetime with smooth $a$ is written in terms of smooth objects like $E_{abcd}$, $S_{abcd}$, and $T_{abcd}:=(T \circ g)_{abcd}$, a question arises, as to why use these objects, instead of $R_{ab}$, $S$, and $T_{ab}$? It is true that the expanded objects remain smooth, while the standard ones don't, but is there other, more fundamental reason? It can be said that $E_{abcd}$ and $S_{abcd}$ are more fundamental, since $R_{ab}$ and $R$ are obtained from them by contractions. But for $T_{abcd}$, unfortunately, at this time we don't know an interpretation. The stress-energy tensor $T_{ab}$ can be obtained from a Lagrangian, but we don't know yet a way to obtain directly $T_{abcd}$ from a Lagrangian. One hint that, at least for some fields, $T_{abcd}$ seems more fundamental is that, for electrovac solutions, it is given by $T_{abcd}=-\frac{1}{8\pi}\(F_{ab}F_{cd} + {}^\ast F_{ab} {}^\ast F_{cd}\)$ \eqref{eq_stress_energy_maxwell_expanded}, while $T_{ab}$ by contracting it \eqref{eq_stress_energy_maxwell}. Similar form has the stress-energy tensor for Yang-Mills fields.
Another question that may appear is what is obtained, given that the solution can be extended beyond the moment when $a(t)=0$? Say that $a(0)=0$. The extended solution will describe two universes, both originating from the same Big-Bang at the same moment $t=0$, one of them expanding toward the direction in which $t$ increases and the other one toward the direction in which $t$ decreases. The parameter $t$ is just a coordinate, and the physical laws are symmetric with respect to time reversal in General Relativity (if one wants to consider quantum fields, the combined symmetry $CPT$ should be considered instead of $T$ alone).
\subsection{{Schwarzschild} black hole}
\label{s_qreg_examples_schw}
The {Schwarzschild} solution describing a black hole of mass $m$ is given in the {Schwarzschild} coordinates by the metric tensor:
\begin{equation}
\label{eq_schw_schw}
\textnormal{d} s^2 = -\(1-\dsfrac{2m}{r}\)\textnormal{d} t^2 + \(1-\dsfrac{2m}{r}\)^{-1}\textnormal{d} r^2 + r^2\textnormal{d}\sigma^2,
\end{equation}
where
\begin{equation}
\label{eq_sphere}
\textnormal{d}\sigma^2 = \textnormal{d}\theta^2 + \sin^2\theta \textnormal{d} \phi^2
\end{equation}
is the metric of the unit sphere $S^2$. The units were chosen so that $c=1$ and $G=1$ (see \textit{e.g.} \citep{HE95}{149}).
Apparently the metric is singular at $r=2m$, on the event horizon. As it is known from the work of Eddington \cite{eddington1924comparison} and Finkelstein \cite{finkelstein1958past} appropriate coordinate changes make the metric {non{-\penalty0\hskip0pt\relax}degenerate} on the event horizon, showing that the singularity is apparent, being due to the coordinates. The coordinate change is singular, but it can be said that the proper coordinates around the event horizon are those of Eddington and Finkelstein, and the {Schwarzschild} coordinates are the singular coordinates.
Can we apply a similar method for the singularity at $r=0$? It can be checked that the Kretschmann scalar $R_{abcd}R^{abcd}$ is singular at $r=0$, and since scalars are invariant at any coordinate changes (including the singular ones), it is usually correctly concluded that the singularity at $r=0$ cannot be removed. Although it cannot be removed, it can be improved by finding coordinates making the metric analytic at $r=0$. As shown in \cite{Sto11e} the singularity $r=0$ in the {Schwarzschild} metric \eqref{eq_schw_schw} has two origins -- it is a combination of degenerate metric and singular coordinates. Firstly, the {Schwarzschild} coordinates are singular at $r=0$, but they can be desingularized by applying the coordinate transformations from equation \eqref{eq_coordinate_semireg} which necessarily have the Jacobian equal to zero at $r=0$. It is not possible to desingularize a coordinate system, by using transformations that have non-vanishing Jacobian at the singularity, because such transformations preserve the regularity of the metric. Secondly, after the transformation the singularity is not completely removed, because the metric remains degenerate. However, the metric remains {semi{-\penalty0\hskip0pt\relax}regular}, as shown in \cite{Sto11e}. Here will be shown that it is also {quasi{-\penalty0\hskip0pt\relax}regular}.
In \cite{Sto11e} we showed that the {Schwarzschild} solution can be made analytic at the singularity by a coordinate transformation of the form
\begin{equation}
\label{eq_coordinate_change}
\left\{
\begin{array}{ll}
r &= \tau^S \\
t &= \xi\tau^T \\
\end{array}
\right.
\end{equation}
As it turns out,
\begin{equation}
\label{eq_coordinate_semireg}
\left\{
\begin{array}{ll}
r &= \tau^2 \\
t &= \xi\tau^4 \\
\end{array}
\right.
\end{equation}
is the only choice which makes analytic at the singularity not only the metric, but also the Riemann curvature $R_{abcd}$. In the new coordinates the metric has the form
\begin{equation}
\label{eq_schw_semireg}
\textnormal{d} s^2 = -\dsfrac{4\tau^4}{2m-\tau^2}\textnormal{d} \tau^2 + (2m-\tau^2)\tau^4\(4\xi\textnormal{d}\tau + \tau\textnormal{d}\xi\)^2 + \tau^4\textnormal{d}\sigma^2.
\end{equation}
\begin{corollary}
\label{thm_schw_quasireg}
The {Schwarzschild} spacetime is {quasi{-\penalty0\hskip0pt\relax}regular} (in any atlas compatible with the coordinates \eqref{eq_coordinate_semireg}).
\end{corollary}
\begin{proof}
We know from \cite{Sto11e} that the {Schwarzschild} spacetime is {semi{-\penalty0\hskip0pt\relax}regular}. Since it is also Ricci flat, \textit{i.e.} $R_{ab}=0$, it follows that $S_{ab}=1$ and $R=0$, hence $S_{abcd}= \dsfrac{1}{24}R(g\circ g)_{abcd}=0$, and $E_{abcd}\dsfrac{1}{2}(S \circ g)_{abcd}=0$. Therefore, $S_{abcd}$ and $E_{abcd}$ are smooth. Consequently, the only non-vanishing part of the curvature in the Ricci decomposition \eqref{eq_ricci_decomposition} is the Weyl tensor $C_{abcd}$, which in this case is equal to $R_{abcd}$, so it is smooth too.
\end{proof}
\begin{remark}
It has been seen that even if the {Schwarzschild} metric $g_{ab}$ is singular at $r=0$ there is a coordinate system in which it becomes {quasi{-\penalty0\hskip0pt\relax}regular}. Because the metric becomes {quasi{-\penalty0\hskip0pt\relax}regular} at $r=0$, the expanded Einstein equations are valid at $r=0$ too. But also Einstein's equation can be extended at $r=0$, because in this special case it becomes $G_{ab}=0$, the {Schwarzschild} solution being a vacuum solution.
Hence, in this case we can just use the standard Einstein equations, of course in coordinates compatible with the coordinates \eqref{eq_coordinate_semireg}. Corollary \ref{thm_schw_quasireg} shows that the {Schwarzschild} singularity is {quasi{-\penalty0\hskip0pt\relax}regular} in any such coordinates. Since $S_{abcd}=E_{abcd}=0$, the only non-vanishing part of $R_{abcd}$ is the Weyl curvature $C_{abcd}=R_{abcd}$, which is smooth because $R_{abcd}$ is smooth.
\end{remark}
\begin{remark}
In the limit $m=0$, the {Schwarzschild} solution \eqref{eq_schw_schw} coincides with the Minkowski metric, which is regular at $r=0$. The event horizon singularity $r=2m$ merges with the $r=0$ singularity, and cancel one another. Because the {Schwarzschild} radius becomes $0$, the false singularity $r=0$ is not spacelike as in the case $m> 0$, but timelike.
In the case $m=0$, because there is no singularity at $r=0$, our coordinates \eqref{eq_coordinate_semireg}, rather than removing a (non-existent) singularity, introduce one. The new coordinates provide a double covering for the Minkowski spacetime, because $\tau$ extends beyond $r=0$ to negative values, in a way similar to the case described in \cite{Sto11f}.
\end{remark}
\begin{openproblem}
What can be said about the other stationary black hole solutions?
In \cite{Sto11f} and \cite{Sto11g} we showed that there are coordinate transformations which make the {Reissner-Nordstr\"om} metric and the {Kerr-Newman} metric analytic at the singularity. This is already a big step, because it allows us to foliate with Cauchy hypersurfaces these spacetimes. Is it possible to find coordinate transformations which make them {quasi{-\penalty0\hskip0pt\relax}regular} too?
\end{openproblem}
\section{Conclusions}
An important problem in General Relativity is that of singularities. At singularities some of the quantities involved in the Einstein equation become infinite. But there are other quantities which are also invariant and in addition remain finite at a large class of singularities. In this paper it has been seen that translating the Einstein equation in terms of such quantities allows it to be extended at such singularities.
The Riemann tensor is, from geometric and linear-algebraic viewpoints, more fundamental than the Ricci tensor $R_{ab}$, which is just its trace. This suggests that the scalar part $S_{abcd}$ \eqref{eq_ricci_part_S} and the Ricci part $E_{abcd}$ \eqref{eq_ricci_part_E} of the Riemann curvature may be more fundamental than the Ricci tensor. Consequently, this justifies the study of an equation equivalent to Einstein's, but in terms of $E_{abcd}$ and $S_{abcd}$, instead of $R_{ab}$ and $R$. This is the expanded Einstein equation \eqref{eq_einstein_expanded}. The idea that $E_{abcd}$ is more fundamental than $R_{ab}$ seems to be suggested also by the electrovac solution, with the expanded Einstein equation \eqref{eq_stress_energy_maxwell_expanded}, and from which the electrovac Einstein equation is obtained by contraction.
To go from Einstein's equation to its expanded version we use the Kulkarni-Nomizu product \eqref{eq_kulkarni_nomizu}. To go back, we use contraction \eqref{eq_expanded_to_standard}. When the metric is {non{-\penalty0\hskip0pt\relax}degenerate}, these operations establish an equivalence between the standard and the expanded Einstein equations.
The question of whether the Ricci part of the Riemann tensor is more fundamental than the Ricci tensor may be irrelevant, or the answer may be debatable. But an important feature is that $E_{abcd}$ and $S_{abcd}$ can be defined in more general situations than $R_{ab}$ and $R$. Hence, the expanded Einstein equation is more general than the Einstein equation -- it makes sense even when the metric is degenerate, at least for a class of singularities named {quasi{-\penalty0\hskip0pt\relax}regular}.
A brief investigation revealed that the class of {quasi{-\penalty0\hskip0pt\relax}regular} singularities is rich enough to contain some known singularities, which were already considered by researchers, but now can be understood in a unified framework. Among these there are the isotropic singularities, which are obtained by multiplying a regular metric with a scaling factor which is allowed to vanish. Another class is given by the {Friedmann-Lema\^itre-Robertson-Walker} singularities \cite{Sto12a}, and other warped product singularities. Even the {Schwarzschild} singularity (in proper coordinates which make the metric analytic \cite{Sto11e}) turns out to be quasi{-\penalty0\hskip0pt\relax}regular.
The fact that these apparently unrelated types of singularities turn out to be {quasi{-\penalty0\hskip0pt\relax}regular} suggests the following open question:
\begin{openproblem}
Are {quasi{-\penalty0\hskip0pt\relax}regular} singularities general enough to cover all possible singularities of General Relativity?
\end{openproblem}
\subsection*{Acknowledgments}
The author thanks the anonymous referees for the valuable comments and suggestions to improve the clarity and the quality of this paper.
|
{
"timestamp": "2014-01-27T02:05:38",
"yymm": "1203",
"arxiv_id": "1203.2140",
"language": "en",
"url": "https://arxiv.org/abs/1203.2140"
}
|
\section{Introduction}
It is generally believed that inflation can be a solution to the
problems of standard cosmology such as the horizon, flatness and
monopole problem. In addition to these achievements, inflation's
predictions are compatible with the large scale structure and CMB
fluctuations which is strong evidence in favour of inflation. The
idea of inflation is the existence of an exponentially expanding
universe at early times. But identifying a unique theoretical
realization of this period is challenging. Many theoretical models
are compatible with the observational data. For example, they are in
agreement with adiabatic, nearly Gaussian fluctuations in the CMB
fluctuations. To potentially discriminate between them more accurate
observations, such as PLANCK, are needed. This fact is a starting
point for a huge amount of work on studying non-Gaussianity of
primordial fluctuations. In this field the effective field theory
approach to inflation has been used to study the general possible
interaction terms in the single field models
\cite{paolo,weinberg,eftsingle} and in the multi-field context
\cite{senatore,eftmulti}.
The advantage of using effective theories can be seen in two
regimes. Sometimes a full theory exists for an energy domain of
interest. In this case the effective field theory may be performed
to simplify calculations in a special sub-domain of energy. In the
second case the full theory is not known for the energy scales of
interest. Here, by imposing the symmetries of the full theory one
can still build an effective field theory. In this situation the
most general form of the allowed theory, e.g. a general Lagrangian,
is constructed; by comparison to observations unspecified
coefficients can be fixed. Eventually the deduced effective theory
may shed some lights on the real theory which is beyond our current
understanding. In the special case of inflation in addition to the
above reasons, the effective field theory approach can be used to
justify the use of scalar fields as inflatons, as well as to provide
a systematic classification of non-Gaussianities \cite{baumann}
among other properties.
In \cite{paolo} the effective field theory has been developed for an
inflationary single field model. In their approach the Lagrangian is
determined by all spatially diffeomorphism invariant operators. Then
the broken time invariance is reproduced by a scalar field which
transforms in a definite form under diffeomorphism transformation.
This scalar field is well-known as the Goldstone boson. It is shown
that this scalar field represents the curvature perturbation in the
validity regime of the effective field theory. In \cite{senatore}
the generalization to multi-field inflation is studied. The
existence of more than one field in the early universe is not
unnatural and the extra fields may have observable consequences. For
example entropy modes (a property of multi-field models) can affect
the curvature mode which is for example in the CMB. In
\cite{weinberg} an alternative approach to \cite{paolo} has been
given for the effective field theory of single field inflation. In
this approach all the possible terms containing up to the fourth
derivative of a scalar field and the metric enter the effective
Lagrangian. The final result is in agreement with \cite{paolo},
except some additional fourth ordered contributions tracing back to
geometrical terms. In this approach, due to the presence of metric
perturbations, it is possible to study the gravitational wave
behavior which differs in the propagation of waves with different
helicities.
In this work we are going to generalize Weinberg's approach
\cite{weinberg} to multi-field models. In the following we will
avoid the scalar metric perturbations by an appropriate gauge
choice. However it is mentioned that for energy scales of interest
the existence of them has no observable effects on non-Gaussianity
\cite{baumann,paolo1}. Note that in \cite{weinberg,paolo,senatore},
the additional correction terms arise via space-time derivatives of
the perturbations. However one can extend the effective field theory
formalism to include correction terms corresponding to the potential
terms. It is mentioned in \cite{baumann-green} that they have no
significant contribution to non-Gaussianities since they are highly
restricted by the effectiveness of the inflationary era. But it is
well-known that in the context of multi-field inflation the
non-Gaussianity window becomes wider and maybe observable by the
future data \cite{gpmulti}. This fact also has been considered in
the context of effective field theory for multi-field inflation in
different aspects \cite{eftmulti}.
In the next section we briefly review the main results of
\cite{weinberg}. In the third section we generalize the idea of
\cite{weinberg} to illustrate the perturbations in the most general
multi-field model. This section is based on the first appendix where
we find the most Lagrangian for multi-field models. Then in the
fourth section we concentrate on a two-field case, studying the
evolution of adiabatic and entropy modes in details. In this section
we will discuss on the amplitude and shape of non-Gaussianity in our model
and illustrate a specific example. At the end of this section we
infer to some differences between this approach and Senatore and Zaldarriaga
\cite{senatore}. In the second appendix we
compare our results with \cite{gordon} as a check. Finally we
conclude in the last section.
\section{Briefly Review of Weinberg's Approach \cite{weinberg}}
To generalize Einstein-Hilbert action in the presence of matter
field it is possible to add the terms containing higher order
derivatives in the Lagrangian in addition to the standard second
order ones. In principle these additional terms can be important in
the larger energy scales. As discussed in \cite{weinberg} this
situation occurs naturally in the inflation era before horizon
crossing. According to the observations; the Hubble parameter, $H$,
and physical momentum, $k/a$, are equal (at horizon crossing) and
much less than $M_P$ and even unification scale. But due to
denominator of physical momentum in a period before horizon crossing
the physical momentum has had larger value. As a consequence,
considering the correction terms will help us to understand better
the inflationary predictions.
Due to the above discussions Weinberg in \cite{weinberg} has studied
the effects of the fourth order derivative terms in
the Einstein-Hilbert
Lagrangian in the presence of one scalar field. We are going to
generalize this model by adding more than one scalar field which is interesting
for its well-known observational consequences. Before that let us
review very briefly\footnote{Here we report Weinberg's idea very
quickly without any details. But in the following when we are going
to study its generalization we will do it in details in Appendix \ref{appendixA}.}
the main results of \cite{weinberg}. The starting point is the
Einstein-Hilbert Lagrangian which includes the leading term
\begin{eqnarray}\label{E-H action}
{\cal{L}}_0=\sqrt{g}\bigg[-\frac{M_P^2}{2}R-\frac{M^2}{2}g^{\mu\nu}\partial_\mu
\varphi\partial_\nu\varphi-M_P^2U(\varphi)\bigg]
\end{eqnarray}
where dimensionless $\varphi\equiv\varphi_c/M$ is defined such that
the kinetic term of $\varphi_c$ has the canonical form.
Obviously $\varphi_c$ has dimension of mass. It is now easier to define
the hierarchy of different derivative terms as an advantage of introduction
of the scale $M$ explicitly\footnote{Just remember that the ${\cal{L}}$
has dimension of $M^4$ and in natural unit $\partial_\mu$ has dimension
of $M^{-1}$.}. The leading correction terms are satisfied general
covariance and contain four derivatives. These term can be reduced to
the following form
\begin{eqnarray}\label{correction terms}
\Delta{\cal{L}}=\sqrt{g}f(\vp)\bigg(g^{\mu\nu}\vp_{,\mu}\vp_{,\nu}\bigg)^2+\sqrt{g}
h_1(\vp)C^{\mu\nu\rho\sigma}C_{\mu\nu\rho\sigma}+\sqrt{g}
h_2(\vp)\varepsilon^{\kappa\lambda\mu\nu}C_{\kappa\lambda}^{\hspace{3mm}\rho\sigma}C_{\mu\nu\rho\sigma}
\end{eqnarray}
where $f$, $h_1$ and $h_2$ are some dimensionless arbitrary
functions which are assumed to be in order
one\footnote{\label{footnote7}Actually it is the second term of an
expansion with respect to the inverse of $M^2$ i.e.
``$M^2,1,M^{-2},...$". The first term is
``$-\frac{M^2}{2}g^{\mu\nu}\partial_\mu \varphi\partial_\nu\varphi$"
in (\ref{E-H action}).} and $C_{\mu\nu\rho\sigma}$ is the Weyl
tensor. It is noteworthy that the above terms are not all the
generally covariant terms containing four derivatives. But all the
allowed terms except the above terms are transformed to these terms
by employing the equation of motion for the leading term as well as
ignoring the surface terms (for details see Appendix
\ref{appendixA}). For the scalar perturbations it is convenient to
assume a gauge in which metric scalar perturbations vanish. In this
gauge by splitting the scalar field to its background and perturbed
parts as $\vp=\vpb+\delvp$ the Lagrangian becomes
\begin{eqnarray}\label{scalar-perturbation}\nonumber
{\cal{L}}&=&\sqrt{g}\bigg[-\frac{M^2}{2}g^{\mu\nu}\partial_\mu
\varphi\partial_\nu\varphi-M_P^2U(\varphi)+f(\vp)\bigg(g^{\mu\nu}\vp_{,\mu}\vp_{,\nu}\bigg)^2\bigg]\\
&=&\bar{\cal{L}}-\frac{1}{2}a^3\bigg(M^2+4f(\vpb)\dot{\vpb}^2\bigg)\times\bigg(-\dot{\delvp}^2+a^{-2}(\vec{\nabla}\delvp)^2\bigg)\\\nonumber
&+&4 a^3
f(\vpb)\dot{\vpb}^2\bigg(\dot\delvp^2+\dot\delvp^3/\dot\vpb-a^{-2}\dot\delvp(\vec{\nabla}\delvp)^2/\dot\vpb+\frac{1}{4}\dot\delvp^4/\dot\vpb^2
-\frac{1}{2}a^{-2}\dot\delvp^2(\vec{\nabla}\delvp)^2/\dot\vpb^2+\frac{1}{4}a^{-4}(\vec{\nabla}\delvp)^4/\dot\vpb^2\bigg)
\end{eqnarray}
which reduces to the Lagrangian $(19)$ in \cite{weinberg} with
$\pi\equiv\delvp/\dot\vpb$ and $\dot
H=-\dot\vpb^2(M^2+4f(\vpb)\dot{\vpb}^2)/2M_P^2$. This result is
compatible with \cite{paolo} with a minor disagreement. This
disagreement is in the presence of quartic terms as well as
quadratic and cubic terms. Due to the above Lagrangian obviously
ignoring the correction term, i.e. setting $f(\vp)=0$, results in a
model with $c_s=1$, where $c_s$ is the speed of sound. But in the
presence of the correction term the speed of sound is not one and
may cause large non-Gaussianity. In addition the terms in the third
line of (\ref{scalar-perturbation}) infer to the possible shapes of
non-Gaussianities as well as their amplitude.
In this section we very briefly reviewed the idea of \cite{weinberg}
for scalar perturbations in the context of effective field theory
for inflation. In addition to scalar perturbation in \cite{weinberg}
the tensor perturbations have been considered. It is concluded in
\cite{weinberg} that the propagation of gravitational wave depends
on the helicity of the wave in this model. In the next section we
are going to generalize the above idea for a multi-field theory of
inflation without considering the tensor perturbations. Since
existence of multi-scalar-field has no effect on the tensor
perturbations and consequently gravitational wave. The detailed
calculations for the next section is in appendix \ref{appendixA}
which is also useful for clarifying the case of one field studying
very briefly in this section.
\section{Effective Field Theory of Multi-Field Inflation}
The corresponding Lagrangian to (\ref{scalar-perturbation}) for
multi-field inflation can be written as follow, which has been deduced in
details in the appendix \ref{appendixA},
\begin{eqnarray}\label{most-general-lagrangian-simplified}
{\cal{L}}=\sqrt{g}&\bigg\{&b_3^{IJKL}(\vec\vp)\nabla_\mu\vp_I\nabla^\mu\vp_J\nabla_\nu\vp_K\nabla^\nu\vp_L-\frac{M^2}{2}\delta^{IJ}\nabla_\mu\vp_I\nabla^\mu\vp_J
-M_P^2U(\vec\vp)\\\nonumber
&+&a_1(\vec\vp)R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}+a_2(\vec\vp)R_{\mu\nu}R^{\mu\nu}-\frac{M_P^2}{2}R\bigg\}
\end{eqnarray}
which exactly reduces to (\ref{scalar-perturbation}) for one field
case\footnote{\label{footnote8}In \cite{weinberg} instead of Riemann
and Ricci tensors in (\ref{most-general-lagrangian-simplified}),
Weyl tensor has been used.}. Now, by splitting the scalar fields to
their background and perturbed parts $\vp_I=\bar\vp_I+\delvp$ we are
going to study the Lagrangian for the perturbations as well as the
background. To do this we start with
(\ref{most-general-lagrangian-simplified}) without worrying about
the tensor perturbations. The above Lagrangian can be written as
${\cal{L}}={\cal{L}}_0+\Delta{\cal{L}}$ such that
\begin{eqnarray}\label{most-general-lagrangian-simplified+background}\nonumber
a^{-3}{\cal{L}}_0&=&b_3^{IJKL}(\vpb){\dot\vpb}_I{\dot\vpb}_J{\dot\vpb}_K{\dot{\bar\vp}}_L
+\frac{M^2}{2}\delta^{IJ}{\dot\vpb}_I{\dot\vpb}_J-M_P^2U(\vpb),
\end{eqnarray}
\begin{eqnarray}\label{most-general-lagrangian-simplified+perturbations}\nonumber
a^{-3}\Delta{\cal{L}}&=&\bigg[\sum_{n=1}\frac{1}{n!}\frac{\partial^n
b_3^{IJKL}(\vpb)}{\partial(\vpb_M)^n}(\delvp_M)^n\bigg]{\dot\vpb}_I{\dot\vpb}_J{\dot\vpb}_K{\dot{\bar\vp}}_L\\\nonumber
&-& \bigg[\sum_{n=0}\frac{1}{n!}\frac{\partial^n
b_3^{IJKL}(\vpb)}{\partial(\vpb_M)^n}(\delvp_M)^n\bigg]\dot\vpb_I\dot\vpb_J\bigg[-\dot\vpb_K\dot\delvp_L-\dot\vpb_L\dot\delvp_K-\dot\delvp_K\dot\delvp_L+
a^{-2}\partial_i\delvp_K\partial^i\delvp_L\bigg]
\\\nonumber
&-& \bigg[\sum_{n=0}\frac{1}{n!}\frac{\partial^n
b_3^{IJKL}(\vpb)}{\partial(\vpb_M)^n}(\delvp_M)^n\bigg]\bigg[-\dot\vpb_I\dot\delvp_J-\dot\vpb_J\dot\delvp_I-\dot\delvp_I\dot\delvp_J+
a^{-2}\partial_i\delvp_I\partial^i\delvp_J\bigg]\dot\vpb_K\dot\vpb_L\\\nonumber
&+&\bigg[\sum_{n=0}\frac{1}{n!}\frac{\partial^n
b_3^{IJKL}(\vpb)}{\partial(\vpb_M)^n}(\delvp_M)^n\bigg]\times\\\nonumber&&\bigg[-\dot\vpb_I\dot\delvp_J-\dot\vpb_J\dot\delvp_I-\dot\delvp_I\dot\delvp_J+
a^{-2}\partial_i\delvp_I\partial^i\delvp_J\bigg]\bigg[-\dot\vpb_K\dot\delvp_L-\dot\vpb_L\dot\delvp_K-\dot\delvp_K\dot\delvp_L+
a^{-2}\partial_i\delvp_K\partial^i\delvp_L\bigg]\\\nonumber
&-&\frac{M^2}{2}\delta^{IJ}\bigg[-2\dot\vpb_I\dot\delvp_J-\dot\delvp_I\dot\delvp_J+
a^{-2}\partial_i\delvp_I\partial^i\delvp_J\bigg]
-M_P^2\bigg[\sum_{n=1}\frac{1}{n!}\frac{\partial^n
U(\vpb)}{\partial(\vpb_M)^n}(\delvp_M)^n\bigg]
\end{eqnarray}
where $a=a(t)$ is the scale factor of the FRW metric and
$\partial_i$ are spatial derivatives. Note that the terms containing
$\delvp_I$ without any differentiations do not show themselves in
the Lagrangian effectively. Since the $n^{th}$ equation of motion
causes vanishing of the coefficients of $(n+1)^{th}$ terms without
any differentiation. Also the linear perturbation terms even
including differentiation vanish because of the same reason. So
effectively the Lagrangian for the perturbations is
\begin{eqnarray}\label{most-general-lagrangian-simplified+perturbations-effectively}\nonumber
a^{-3}\Delta{\cal{L}}&=&
b_3^{IJKL}(\vpb)\bigg\{\bigg[-\dot\vpb_I\dot\delvp_J-\dot\vpb_J\dot\delvp_I-\dot\delvp_I\dot\delvp_J+
a^{-2}\partial_i\delvp_I\partial^i\delvp_J\bigg]\bigg[-\dot\vpb_K\dot\delvp_L-\dot\vpb_L\dot\delvp_K-\dot\delvp_K\dot\delvp_L+
a^{-2}\partial_i\delvp_K\partial^i\delvp_L\bigg]\\\nonumber
&-&\dot\vpb_I\dot\vpb_J\bigg[-\dot\delvp_K\dot\delvp_L+
a^{-2}\partial_i\delvp_K\partial^i\delvp_L\bigg]-\bigg[-\dot\delvp_I\dot\delvp_J+
a^{-2}\partial_i\delvp_I\partial^i\delvp_J\bigg]\dot\vpb_K\dot\vpb_L\bigg\}\\\nonumber
&-&\frac{M^2}{2}\delta^{IJ}\bigg[-\dot\delvp_I\dot\delvp_J+
a^{-2}\partial_i\delvp_I\partial^i\delvp_J\bigg].
\end{eqnarray}
The second, third and fourth order of perturbations respectively can
be represented as follows
\begin{eqnarray}\label{most-general-lagrangian-simplified+perturbations-effectively-L2}\nonumber
a^{-3}\Delta{\cal{L}}^{(2)}&=&
b_3^{IJKL}(\vpb)\bigg\{\dot\vpb_I\dot\vpb_J\dot\delvp_K\dot\delvp_L+\dot\vpb_I\dot\vpb_K\dot\delvp_J\dot\delvp_L+
\dot\vpb_I\dot\vpb_L\dot\delvp_K\dot\delvp_J+\dot\vpb_K\dot\vpb_J\dot\delvp_I\dot\delvp_L+
\dot\vpb_L\dot\vpb_J\dot\delvp_I\dot\delvp_L+\dot\vpb_K\dot\vpb_L\dot\delvp_I\dot\delvp_J\\&-&
a^{-2}\dot\vpb_I\dot\vpb_J\partial_i\delvp_K\partial^i\delvp_L-a^{-2}\dot\vpb_K\dot\vpb_L\partial_i\delvp_I\partial^i\delvp_J
\bigg\}-\frac{M^2}{2}\delta^{IJ}\bigg[-\dot\delvp_I\dot\delvp_J+
a^{-2}\partial_i\delvp_I\partial^i\delvp_J\bigg],
\end{eqnarray}
\begin{eqnarray}\label{most-general-lagrangian-simplified+perturbations-effectively-L3}
a^{-3}\Delta{\cal{L}}^{(3)}&=&
b_3^{IJKL}(\vpb)\bigg\{\dot\vpb_I\dot\delvp_J\dot\delvp_K\dot\delvp_L+\dot\vpb_J\dot\delvp_I\dot\delvp_K\dot\delvp_L
+\dot\vpb_K\dot\delvp_L\dot\delvp_I\dot\delvp_J+\dot\vpb_L\dot\delvp_K\dot\delvp_I\dot\delvp_J\\\nonumber
&-&a^{-2}\bigg(\dot\vpb_I\dot\delvp_J\partial_i\delvp_K\partial^i\delvp_L+\dot\vpb_J\dot\delvp_I\partial_i\delvp_K\partial^i\delvp_L
+\dot\vpb_K\dot\delvp_L\partial_i\delvp_I\partial^i\delvp_J+\dot\vpb_L\dot\delvp_K\partial_i\delvp_I\partial^i\delvp_J\bigg)\bigg\},\\\nonumber&&
\end{eqnarray}
\begin{eqnarray}\label{most-general-lagrangian-simplified+perturbations-effectively-L4}
a^{-3}\Delta{\cal{L}}^{(4)}=
b_3^{IJKL}(\vpb)\bigg\{\dot\delvp_I\dot\delvp_J\dot\delvp_K\dot\delvp_L&-&a^{-2}\bigg(\dot\delvp_I\dot\delvp_J\partial_i\delvp_K\partial^i\delvp_L+
\dot\delvp_K\dot\delvp_L\partial_i\delvp_I\partial^i\delvp_J\bigg)\\\nonumber
&+&a^{-4}\partial_i\delvp_I\partial^i\delvp_J\partial_j\delvp_K\partial^j\delvp_L\bigg\}.
\end{eqnarray}
Note that the above result exactly reduces to single field result in
(\ref{scalar-perturbation}) with
$b_3^{IJKL}(\vpb)=f(\vpb)$\footnote{To do this one should set
$I=J=K=L=1$.}. It is obvious from
(\ref{most-general-lagrangian-simplified+perturbations-effectively-L2})
that the speed of sound is not one in the presence of non-vanishing
$b_3^{IJKL}(\vpb)$. Note that due to $b_3^{IJKL}(\vpb)$ the cubic
and quartic terms are appeared. This feature is in disagreement with
\cite{paolo,senatore}. In their work the coefficient which displays
$c_s$ just connects to the cubic term. But here it connects to the
fourth order term too. In the next section we restrict the model to
a two-field model. This makes it possible to study the adiabatic and
entropy perturbations in more details without loss of generality in
the main results.
\section{A Specific Case: Adiabatic versus Entropy Perturbation}
In this section we re-write the above formalism in the language of
adiabatic and entropy perturbations for a two-field model. This is
crucial in this approach since in contrast to \cite{senatore} here
the adiabatic perturbation is not initially supposed. In
\cite{senatore} the additional perturbations are added to a model
already containing the adiabatic perturbation i.e. \cite{paolo}. In
\cite{senatore} the Goldstone boson, introduced in \cite{paolo},
plays the role of the adiabatic perturbation and the additional
fields are employed as the entropy perturbations. But in our model
there is no initially difference between $\vp_I$'s and consequently
$\delvp_I$'s. So it is critical to distinguish between adiabatic and
entropy modes to manifest their different physical meanings.
\subsection{The Most General Two-Field Model}
In this subsection we re-do perturbation calculations for a two-field model.
To do this we start
with (\ref{most-general-lagrangian-simplified}) for two fields named
$\vp$ and $\chi$
\begin{eqnarray}\label{two-field-lagrangian-simplified}
{\cal{L}}=-a^3\bigg\{&-&\frac{M_1^2}{2}\partial_\mu\vp\partial^\mu\vp-\frac{M_2^2}{2}\partial_\mu\chi\partial^\mu\chi
-M_P^2U(\vp,\chi)+g_1(\vp,\chi)\big(\partial_\mu\vp\partial^\mu\vp\big)^2+g_2(\vp,\chi)\big(\partial_\mu\chi\partial^\mu\chi\big)^2\\\nonumber
&+&g_3(\vp,\chi)\big(\partial_\mu\vp\partial^\mu\vp\big)\big(\partial_\nu\vp\partial^\nu\chi\big)
+g_4(\vp,\chi)\big(\partial_\mu\chi\partial^\mu\chi\big)\big(\partial_\nu\chi\partial^\nu\vp\big)\\\nonumber&+&g_5(\vp,\chi)\big(\partial_\mu\vp\partial^\mu\vp\big)\big(\partial_\nu\chi\partial^\nu\chi\big)
+g_6(\vp,\chi)\big(\partial_\mu\vp\partial^\mu\chi\big)\big(\partial_\nu\vp\partial^\nu\chi\big)\bigg\}
\end{eqnarray}
where $a=a(t)$ is the scale factor and $g_i$'s are some arbitrary
dimensionless and order one functions as mentioned before. By assuming
$\vp=\vpb+\delvp$ and $\chi=\bar\chi+\delchi$ the above Lagrangian
reduces to
\begin{eqnarray}\label{two-field-lagrangian-simplified-order-0}
a^{-3}{\cal{L}}_0&=&\bigg\{-\frac{M_1^2}{2}\partial_\mu\vpb\partial^\mu\vpb-\frac{M_2^2}{2}\partial_\mu\chib\partial^\mu\chib
-M_P^2U(\vpb,\chib)+g_1\big(\partial_\mu\vpb\partial^\mu\vpb\big)^2+g_2\big(\partial_\mu\chib\partial^\mu\chib\big)^2
\\\nonumber&+&g_3\big(\partial_\mu\vpb\partial^\mu\vpb\big)\big(\partial_\nu\vpb\partial^\nu\chib\big)
+g_4\big(\partial_\mu\chib\partial^\mu\chib\big)\big(\partial_\nu\chib\partial^\nu\vpb\big)+
g_5\big(\partial_\mu\vpb\partial^\mu\vpb\big)\big(\partial_\nu\chib\partial^\nu\chib\big)
+g_6\big(\partial_\mu\vpb\partial^\mu\chib\big)\big(\partial_\nu\vpb\partial^\nu\chib\big)\bigg\}\\\nonumber
&=&\frac{M_1}{2}\dot\vpb^2+\frac{M_2}{2}\dot\chib^2-M_P^2U(\vpb,\chib)+g_1\dot\vpb^4+g_2\dot\chib^4+g_3\dot\vpb^3\dot\chib+g_4\dot\vpb\dot\chib^3
+(g_5+g_6)\dot\vpb^2\dot\chib^2
\end{eqnarray}
for the background part. The corresponding equations of motion for $\vpb$ reads as
\begin{eqnarray}\label{eq-mo-background}\nonumber
&&\frac{d}{dt}\bigg[M_1\dot\vpb+4g_1\dot\vpb^3+3g_3\dot\vpb^2\dot\chib+g_4\dot\chib^3
+2(g_5+g_6)\dot\vpb\dot\chib^2\bigg]+3H\bigg[M_1\dot\vpb+4g_1\dot\vpb^3+3g_3\dot\vpb^2\dot\chib+g_4\dot\chib^3
+2(g_5+g_6)\dot\vpb\dot\chib^2\bigg]\\\nonumber
&&+M_P^2U'-\bigg(g'_1\dot\vpb^4+g'_2\dot\chib^4+g'_3\dot\vpb^3\dot\chib+g'_4\dot\vpb\dot\chib^3
+(g'_5+g'_6)\dot\vpb^2\dot\chib^2\bigg)=0
\end{eqnarray}
where $'$ is the differentiation with respect to $\vpb$ and the
similar equation is true for $\chib$. It is straightforward but
messy to show that the above equation of motion (as well as
$\chib$'s) for the background causes the Lagrangian of the first
order perturbation becomes vanishing. So the non-trivial terms start
from the second order perturbations succeeding with the third and
the fourth order terms\footnote{It is obvious if one expands the
correction terms in (\ref{most-general-lagrangian-simplified}) or
(\ref{most-general-lagrangian}) for more than four derivative terms
then the higher order perturbations show themselves.} for
(\ref{two-field-lagrangian-simplified}) as the following
\begin{eqnarray}\label{two-field-lagrangian-simplified-order-2}
a^{-3}\Delta{\cal{L}}^{(2)}&=&\dot\delvp^2\big[\frac{M^2_1}{2}+6g_1\dot\vpb^2+3g_3\dot\vpb\dot\chib+(g_5+g_6)\dot\chib^2\big]
+\dot\delchi^2\big[\frac{M^2_2}{2}+6g_2\dot\chib^2+3g_4\dot\vpb\dot\chib+(g_5+g_6)\dot\vpb^2\big]\\\nonumber&+&
\dot\delvp\dot\delchi\big[3g_3\dot\vpb^2+3g_4\dot\chib^2+4(g_5+g_6)\dot\vpb\dot\chib\big]\\\nonumber
&-&a^{-2}\bigg(\partial_i\delvp\partial^i\delvp\big[\frac{M_1}{2}+2g_1\dot\vpb^2+g_3\dot\vpb\dot\chib+g_5\dot\chib^2\big]
+\partial_i\delchi\partial^i\delchi\big[\frac{M_2}{2}+2g_2\dot\chib^2+g_4\dot\vpb\dot\chib+g_5\dot\vpb^2\big]\\\nonumber
&&\hspace{1.2cm}+\partial_i\delvp\partial^i\delchi\big[g_3\dot\vpb^2+g_4\dot\chib^2+2g_6\dot\vpb\dot\chib\big]\bigg),
\end{eqnarray}
\begin{eqnarray}\label{two-field-lagrangian-simplified-order-3}
a^{-3}\Delta{\cal{L}}^{(3)}&=&\dot\delvp^3\big[4g_1\dot\vpb+g_3\dot\chib\big]+\dot\delchi^3\big[4g_2\dot\chib+g_4\dot\vpb\big]+
\dot\delvp^2\dot\delchi\big[3g_3\dot\vpb+2(g_5+g_6)\dot\chib\big]+\dot\delvp\dot\delchi^2\big[3g_4\dot\chib+2(g_5+g_6)\dot\vpb\big]\\\nonumber
&-&a^{-2}\bigg(\dot\delvp\partial_i\delvp\partial^i\delvp\big[4g_1\dot\vpb+g_3\dot\chib\big]+
\dot\delchi\partial_i\delchi\partial^i\delchi\big[4g_2\dot\chib+g_4\dot\vpb\big]+
\dot\delvp\partial_i\delchi\partial^i\delchi\big[g_4\dot\chib+2g_5\dot\vpb\big]\\\nonumber&&\hspace{1.5cm}
+\dot\delchi\partial_i\delvp\partial^i\delvp\big[g_3\dot\vpb+2g_5\dot\chib\big]
+\dot\delvp\partial_i\delvp\partial^i\delchi\big[2g_3\dot\vpb+2g_6\dot\chib\big]+
\dot\delchi\partial_i\delvp\partial^i\delchi\big[2g_4\dot\chib+2g_6\dot\vpb\big]\bigg),
\end{eqnarray}
\begin{eqnarray}\label{two-field-lagrangian-simplified-order-4}
a^{-3}\Delta{\cal{L}}^{(4)}&=&g_1\dot\delvp^4+g_2\dot\delchi^4+g_3\dot\delvp^3\dot\delchi+g_4\dot\delchi^3\dot\delvp+(g_5+g_6)\dot\delvp^2\dot\delchi^2\\\nonumber
&-&a^{-2}\bigg(2g_1\dot\delvp^2\partial_i\delvp\partial^i\delvp+2g_2\dot\delchi^2\partial_i\delchi\partial^i\delchi+
g_3\dot\delvp^2\partial_i\delvp\partial^i\delchi+g_4\dot\delchi^2\partial_i\delchi\partial^i\delvp\\\nonumber
&&\vspace{2.5cm}+g_3\dot\delvp\dot\delchi\partial_i\delvp\partial^i\delvp+g_4\dot\delchi\dot\delvp\partial_i\delchi\partial^i\delchi+
g_5(\dot\delchi^2\partial_i\delvp\partial^i\delvp+\dot\delvp^2\partial_i\delchi\partial^i\delchi)+2g_6\dot\delvp
\dot\delchi\partial_i\delvp\partial^i\delchi\bigg)\\\nonumber
&+&a^{-4}\bigg(g_1(\partial_i\delvp\partial^i\delvp)^2+g_2(\partial_i\delchi\partial^i\delchi)^2
+g_3\partial_i\delvp\partial^i\delvp\partial_j\delchi\partial^j\delvp+g_4\partial_i\delchi\partial^i\delchi\partial_j\delchi\partial^j\delvp
\\\nonumber&&\vspace{2.5cm}+g_5\partial_i\delvp\partial^i\delvp\partial_j\delchi\partial^j\delchi+g_6(\partial_i\delvp\partial^i\delchi)^2\bigg).
\end{eqnarray}
It is interesting to mention that for $\vp=\chi$, i.e. going back to one
field case, all the above relations reduce to
(\ref{scalar-perturbation}) with $f=g_1+g_2+g_3+g_4+g_5+g_6$ as it was expected.
\subsection{Adiabatic vs. Entropy Modes}
Now let us re-write the above terms in the language of adiabatic and
entropy perturbations. The adiabatic perturbation is along the classical
path and the entropy perturbation
is orthogonal to it. Due to \cite{gordon} they can be defined as
follows
\begin{eqnarray}\label{adi-ent-perturbations}
\delta\sigma\equiv\vec{T}.\vec\delta,\hspace{2cm}\delta s\equiv
\vec{N}.\vec\delta
\end{eqnarray}
where $\delta \sigma$ and $\delta s$ are the adiabatic and entropy
modes respectively and
\begin{eqnarray}\label{tangent-normal-vector}
\vec\delta\equiv\left(\delvp,\delchi\right),\hspace{2cm}\vec
T=\left(\cos\theta,\sin\theta\right)\equiv\left(\dot\vp/\dot\sigma,\dot\chi/\dot\sigma\right),\hspace{2cm}
\vec N\equiv\left(\sin\theta,-\cos\theta\right)
\end{eqnarray}
where $\dot\sigma^2=\dot\vp^2+\dot\chi^2$. One can show easily
\begin{eqnarray}\label{adi-ent-perturbations-timederivative-original-perturbations}\nonumber
\dot\delvp=\cos\theta\hspace{2mm}\vec{T}.\dot{\vec\delta}+\sin\theta\hspace{2mm}\vec{N}.\dot{\vec\delta},\hspace{2cm}
\dot\delchi=\sin\theta\hspace{2mm}\vec{T}.\dot{\vec\delta}-\cos\theta\hspace{2mm}\vec{N}.\dot{\vec\delta}
\end{eqnarray}
and it is easy to see that
\begin{eqnarray}\label{adi-ent-perturbations-timederivative}\nonumber
\dot{\delta\sigma}=\dot\theta\delta
s+\vec{T}.\dot{\vec\delta},\hspace{2cm}\dot{\delta s}=-\dot\theta
\delta\sigma+ \vec{N}.\dot{\vec\delta}
\end{eqnarray}
and due to above relations
\begin{eqnarray}\label{adi-ent-perturbations-spatialderivative-original-perturbations}\nonumber
\partial_i\delvp=\cos\theta\hspace{2mm}\partial_i\delta\sigma+\sin\theta\hspace{2mm}\partial_i\delta s,\hspace{2cm}
\partial_i\delchi=\sin\theta\hspace{2mm}\partial_i\delta\sigma-\cos\theta\hspace{2mm}\partial_i\delta
s.
\end{eqnarray}
Now by the above definitions we re-write the results of previous
sub-section by plugging $\delta\sigma$ and $\delta s$ in. Let's
start with the kinetic terms for leading order term in
(\ref{two-field-lagrangian-simplified-order-2})
\begin{eqnarray}\label{two-field-lagrangian-simplified-order-2-M1-M2-adi-ent}
&&\frac{M^2_1}{2}\dot\delvp^2+\frac{M^2_2}{2}\dot\delchi^2=\\\nonumber
&&\frac{M^2_1+M_2^2}{4}\left[\dot\delvp^2+\dot\delchi^2\right]+\frac{M_1^2-M^2_2}{4}\left[\dot\delvp^2-\dot\delchi^2\right]=\\\nonumber&&
\frac{M^2_1+M_2^2}{4}\left[(\vec T.\dot{\vec\delta})^2+(\vec
N.\dot{\vec\delta})^2\right]+\frac{M_1^2-M^2_2}{4}\left[(\cos^2\theta-\sin^2\theta)\bigg((\vec
T.\dot{\vec\delta})^2-(\vec
N.\dot{\vec\delta})^2\bigg)+4\sin\theta\cos\theta\hspace{2mm}
T.\dot{\vec\delta}\hspace{2mm}N.\dot{\vec\delta}\right]\\\nonumber&&
\end{eqnarray}
the same procedure is applicable for
$a^{-2}\left(\frac{M^2_1}{2}\partial_i\delvp\partial^i\delvp
+\frac{M^2_2}{2}\partial_i\delchi\partial^i\delchi\right)$ in
(\ref{two-field-lagrangian-simplified-order-2}). Now let us assume
$M_1=M_2=M$ to make it comparable to results in \cite{gordon}. In
this case
\begin{eqnarray}\label{two-field-lagrangian-simplified-order-2-M1=M2=1-adi-ent}
\frac{1}{M^2}a^{-3}{\cal{L}}&=&\frac{1}{2}\dot\delvp^2+\frac{1}{2}\dot\delchi^2-a^{-2}\left(\frac{1}{2}\partial_i\delvp\partial^i\delvp
+\frac{1}{2}\partial_i\delchi\partial^i\delchi\right)=\\\nonumber &&
\frac{1}{2}\left[(\vec T.\dot{\vec\delta})^2+(\vec
N.\dot{\vec\delta})^2\right]-\frac{1}{2}a^{-2}\left(\partial_i\delta\sigma\partial^i\delta\sigma
+\partial_i\delta s\partial^i\delta s\right)=\\\nonumber&&
\frac{1}{2}\left[(\dot{\delta\sigma}-\dot\theta\delta s)^2+(\dot{\delta s}+\dot\theta\delta\sigma)^2\right]
-\frac{1}{2}a^{-2}\left(\partial_i\delta\sigma\partial^i\delta\sigma
+\partial_i\delta s\partial^i\delta s\right)=\\\nonumber&&
\frac{1}{2}\left[\dot{\delta\sigma}^2+\dot\theta^2\delta s^2-2\dot\theta\delta s\dot{\delta\sigma}+\dot{\delta s}^2
+\dot\theta^2\delta\sigma^2+2\dot\theta\delta\sigma\dot{\delta s}\right]-\frac{1}{2}a^{-2}\left(\partial_i\delta\sigma\partial^i\delta\sigma
+\partial_i\delta s\partial^i\delta s\right)
\end{eqnarray}
According to the above Lagrangian the equations of motion
for $\delta\sigma$ and $\delta s$ become\footnote{The potential term in the second order of
perturbations should be added to
(\ref{two-field-lagrangian-simplified-order-2-M1=M2=1-adi-ent}) to
make our results comparable with \cite{gordon}. This term is
$-\frac{1}{2}\big(V_{\sigma\sigma}\delta\sigma^2+V_{\sigma s}\delta
\sigma \delta s+V_{ss}\delta s^2\big)$.}
\begin{eqnarray}\label{eq-mo-pert}
&&\ddot{\delta\sigma}+3H\dot{\delta\sigma}-a^{-2}\partial^i\partial_i\delta\sigma+(V_{\sigma\sigma}-\dot\theta^2)\delta\sigma=2\dot\theta\dot{\delta
s} +(\ddot\theta+3H\dot\theta-V_{\sigma s})\delta s\\\nonumber
&&\ddot{\delta s}+3H\dot{\delta s}-a^{-2}\partial^i\partial_i\delta
s+(V_{ss}-\dot\theta^2)\delta s=-2\dot\theta\dot{\delta \sigma}
-(\ddot\theta+3H\dot\theta+V_{\sigma s})\delta \sigma
\end{eqnarray}
where $V_{\sigma s}=(\cos^2\theta-\sin^2\theta)
V_{\vp\chi}+\sin\theta\cos\theta(V_{\chi\chi}-V_{\vp\vp})$. The
above results are exactly same as (47) and (48) in \cite{gordon}
when ignoring metric perturbations, see Appendix B. Now let us do
the same procedure for the second order perturbations due to the
first order correction term in
(\ref{two-field-lagrangian-simplified-order-2}). At the first the
terms containing time derivative
\begin{eqnarray}\label{adi-ent-pert-correction-order-2}\nonumber
&&\dot\delvp^2\big[6g_1\dot\vpb^2+3g_3\dot\vpb\dot\chib+(g_5+g_6)\dot\chib^2\big]+
\dot\delchi^2\big[6g_2\dot\chib^2+3g_4\dot\vpb\dot\chib+(g_5+g_6)\dot\vpb^2\big]+
\dot\delvp\dot\delchi\big[3g_3\dot\vpb^2+3g_4\dot\chib^2+4(g_5+g_6)\dot\vpb\dot\chib\big]\\\nonumber&=&
\bigg(\cos\theta\hspace{2mm}\vec{T}.\dot{\vec\delta}+\sin\theta\hspace{2mm}\vec{N}.\dot{\vec\delta}\bigg)^2
\bigg[6g_1\dot\vpb^2+3g_3\dot\vpb\dot\chib+(g_5+g_6)\dot\chib^2\bigg]+
\bigg(\sin\theta\hspace{2mm}\vec{T}.\dot{\vec\delta}-\cos\theta\hspace{2mm}\vec{N}.\dot{\vec\delta}\bigg)^2
\bigg[6g_2\dot\chib^2+3g_4\dot\vpb\dot\chib+(g_5+g_6)\dot\vpb^2\bigg]\\\nonumber&+&
\bigg(\cos\theta\hspace{2mm}\vec{T}.\dot{\vec\delta}+\sin\theta\hspace{2mm}\vec{N}.\dot{\vec\delta}\bigg)
\bigg(\sin\theta\hspace{2mm}\vec{T}.\dot{\vec\delta}-\cos\theta\hspace{2mm}\vec{N}.\dot{\vec\delta}\bigg)
\bigg[3g_3\dot\vpb^2+3g_4\dot\chib^2+4(g_5+g_6)\dot\vpb\dot\chib\bigg]\\\nonumber&=&
\big(\vec{T}.\dot{\vec\delta}\big)^2\times\bigg\{\cos^2\theta\bigg[6g_1\dot\vpb^2+3g_3\dot\vpb\dot\chib+(g_5+g_6)\dot\chib^2\bigg]
+\sin^2\theta\bigg[6g_2\dot\chib^2+3g_4\dot\vpb\dot\chib+(g_5+g_6)\dot\vpb^2\bigg]\\\nonumber&&\hspace{2cm}+\sin\theta\cos\theta
\bigg[3g_3\dot\vpb^2+3g_4\dot\chib^2+4(g_5+g_6)\dot\vpb\dot\chib\bigg]\bigg\}\\\nonumber&+&
\big(\vec{N}.\dot{\vec\delta}\big)^2\times\bigg\{\sin^2\theta\bigg[6g_1\dot\vpb^2+3g_3\dot\vpb\dot\chib+(g_5+g_6)\dot\chib^2\bigg]
+\cos^2\theta\bigg[6g_2\dot\chib^2+3g_4\dot\vpb\dot\chib+(g_5+g_6)\dot\vpb^2\bigg]\\\nonumber&&\hspace{2cm}-\sin\theta\cos\theta
\bigg[3g_3\dot\vpb^2+3g_4\dot\chib^2+4(g_5+g_6)\dot\vpb\dot\chib\bigg]\bigg\}\\\nonumber&+&
\big(\vec{T}.\dot{\vec\delta}\big)\big(\vec{N}.\dot{\vec\delta}\big)\times
\bigg\{2\cos\theta\sin\theta\bigg(\big[6g_1\dot\vpb^2+3g_3\dot\vpb\dot\chib+(g_5+g_6)\dot\chib^2\big]
-\big[6g_2\dot\chib^2+3g_4\dot\vpb\dot\chib+(g_5+g_6)\dot\vpb^2\big]\bigg)\\\nonumber&&\hspace{2.5cm}+\bigg(\sin^2\theta-\cos^2\theta\bigg)
\bigg[3g_3\dot\vpb^2+3g_4\dot\chib^2+4(g_5+g_6)\dot\vpb\dot\chib\bigg]\bigg\}
\end{eqnarray}
which can be re-written as the following\footnote{Here we do not
expand $\vec{T}.\dot{\vec\delta}$ and $\vec{N}.\dot{\vec\delta}$
since they contain no common terms to factorize. So their expansion
may cause just messy stuffs without any physical interests.}
\begin{eqnarray}\label{two-field-lagrangian-simplified-order-2-correction-terms}
&&6\dot\sigma^2\big(\vec{T}.\dot{\vec\delta}\big)^2\times\bigg[g_1\cos^4\theta+g_2\sin^4\theta+g_3\cos^3\theta\sin\theta
+g_4\cos\theta\sin^3\theta+(g_5+g_6)\cos^2\theta\sin^2\theta\bigg]\\\nonumber&+&
\dot\sigma^2\big(\vec{N}.\dot{\vec\delta}\big)^2\times\\\nonumber&&\vspace{0cm}\bigg[(g_5+g_6)\bigg(\cos^4\theta+\sin^4\theta\bigg)+3(g_4-g_3)\bigg(\cos^3\theta\sin\theta
-\cos\theta\sin^3\theta\bigg)+2\bigg(3(g_1+g_2)+2(g_5+g_6)\bigg)\cos^2\theta\sin^2\theta\bigg]\\\nonumber&+&
3\dot\sigma^2\big(\vec{T}.\dot{\vec\delta}\big)\big(\vec{N}.\dot{\vec\delta}\big)\times\\\nonumber&&
\bigg[-g_3\cos^4\theta+g_4\sin^4\theta+2\bigg(2g_1-(g_5+g_6)\bigg)\cos^3\theta\sin\theta-2\bigg(2g_2+(g_5+g_6)\bigg)\cos\theta\sin^3\theta
+3(g_3-g_4)\cos^2\theta\sin^2\theta\bigg]
\end{eqnarray}
and similarly for the spatial differentiation
\begin{eqnarray}\label{two-field-lagrangian-simplified-order-2-correction-terms-spatial}
&-&2a^{-2}\dot\sigma^2(\partial_i\delta\sigma)^2\times\bigg[g_1\cos^4\theta+g_2\sin^4\theta+g_3\cos^3\theta\sin\theta
+g_4\cos\theta\sin^3\theta+2(g_5+g_6)\cos^2\theta\sin^2\theta\bigg]\\\nonumber&-&
a^{-2}\dot\sigma^2(\partial_i\delta
s)^2\times\vspace{0cm}\bigg[g_5\bigg(\cos^4\theta+\sin^4\theta\bigg)+(g_4-g_3)\bigg(\cos^3\theta\sin\theta
-\cos\theta\sin^3\theta\bigg)+2(g_1+g_2-g_6)\cos^2\theta\sin^2\theta\bigg]\\\nonumber&-&
a^{-2}\dot\sigma^2\partial_i\delta\sigma\partial^i \delta
s\times\\\nonumber&&
\bigg[-g_3\cos^4\theta+g_4\sin^4\theta+2\bigg(2g_1-(g_5+g_6)\bigg)\cos^3\theta\sin\theta-2\bigg(2g_2+(g_5+g_6)\bigg)\cos\theta\sin^3\theta
+3(g_3-g_4)\cos^2\theta\sin^2\theta\bigg].
\end{eqnarray}
Up to now we fully considered the second order perturbation terms in
the language of adiabatic and entropy modes. In quadratic level the
speed of sound is a matter of interest hence it is noteworthy to
take a look at it. It is obvious to the above relations that $\delta
\sigma$ and $\delta s$ can have different speeds of sound generally.
To have a sense about it let's consider a special case that $\delta
\sigma$ has $c_s=1$ and $\delta s$ has $c_s\neq 1$. To see this,
assume the special case with $g_1=g_2=g_3=g_4=0$ and $g_5+g_6=0$. In
this case the coefficients of $\dot{\delta\sigma}^2$ and
$(\partial_i\delta\sigma)^2$ are same and result in $c_s=1$ for
$\delta\sigma$\footnote{Note that we employ the standard definition
of $c_s$. It means we skip the interaction terms between
$\delta\sigma$ and $\delta s$ which exist even in quadratic level.}.
But the coefficient of $\dot{\delta s}^2$ is $\frac{M^2}{2}$ and for
$(\partial_i\delta s)^2$ is
$-\frac{1}{2}a^{-2}(M^2-2g_5\dot\sigma^2)$ that means
$c_s^2=1-2\frac{g_5\dot\sigma^2}{M^2}$. Note that here we write the
$M$ explicitly to make the comparison of the terms easier. A
characteristic property of $c_s$ is its $\dot\sigma^2$-dependence
which seems interesting. However the effective field theory is valid
where the correction terms are smaller than the leading terms in
(\ref{most-general-lagrangian-simplified}) to have an acceptable
expansion i.e. $\frac{\vert
b_3^{IJKL}(\vpb)\vert\dot{\sigma}^2}{M^2}<1$. Even more,
$\frac{\vert b_3^{IJKL}(\vpb)\vert\dot{\sigma}^2}{M^2}<<1$ should be
satisfied to make skipping higher order correction terms in
(\ref{most-general-lagrangian-simplified}) acceptable. So the speed
of sound in this model is almost one. This fact shows that for the
single field model the large non-Gaussinity is not expected. But in
the following we will discuss on the case of multi-field models. In
multi-field models even with $c_s\simeq 1$ the large non-Gaussianity
can be occurred in some specific circumstances.
One can do this procedure for the higher order perturbation terms
(\ref{two-field-lagrangian-simplified-order-3}) and
(\ref{two-field-lagrangian-simplified-order-4}) which are the
fundaments of studying non-Gaussianity for multi-field models. The
results are as follows\footnote{We here just consider the terms
containing time derivatives and not any spatial derivatives. However
the procedure is same.} for $a^{-3}\Delta{\cal{L}}^{(3)}$
\begin{eqnarray}\label{adi-ent-pert-correction-order-3}\nonumber
&&4\dot\sigma\big(\vec{T}.\dot{\vec\delta}\big)^3\times\bigg[g_1\cos^4\theta+g_2\sin^4\theta+g_3\cos^3\theta\sin\theta
+g_4\cos\theta\sin^3\theta+(g_5+g_6)\cos^2\theta\sin^2\theta\bigg]\\\nonumber&+&
\dot\sigma\big(\vec{N}.\dot{\vec\delta}\big)^3\times\\\nonumber&&\vspace{0cm}
\bigg[-g_4\cos^4\theta+g_3\sin^4\theta-2\bigg(2g_2-(g_5+g_6)\bigg)\cos^3\theta\sin\theta
+2\bigg(2g_1-(g_5+g_6)\bigg)\cos\theta\sin^3\theta+3(g_4-g_3)\cos^2\theta\sin^2\theta\bigg]\\\nonumber&+&
3\dot\sigma\big(\vec{T}.\dot{\vec\delta}\big)^2\big(\vec{N}.\dot{\vec\delta}\big)\times\\\nonumber&&
\bigg[-g_3\cos^4\theta+g_4\sin^4\theta+2\bigg(2g_1-(g_5+g_6)\bigg)\cos^3\theta\sin\theta-2\bigg(2g_2-(g_5+g_6)\bigg)\cos\theta\sin^3\theta
+3(g_3-g_4)\cos^2\theta\sin^2\theta\bigg]\\\nonumber&+&
2\dot\sigma\big(\vec{T}.\dot{\vec\delta}\big)\big(\vec{N}.\dot{\vec\delta}\big)^2\times\\\nonumber&&
\bigg[(g_5+g_6)\bigg(\cos^4\theta+\sin^4\theta\bigg)+3(g_4-g_3)\bigg(\cos^3\theta\sin\theta-\cos\theta\sin^3\theta\bigg)
+\bigg(3(g_1+g_2)-2(g_5+g_6)\bigg)\cos^2\theta\sin^2\theta\bigg]
\end{eqnarray}
and for $a^{-3}\Delta{\cal{L}}^{(4)}$ it becomes
\begin{eqnarray}\label{adi-ent-pert-correction-order-4}\nonumber
&&\big(\vec{T}.\dot{\vec\delta}\big)^4\times\bigg[g_1\cos^4\theta+g_2\sin^4\theta+g_3\cos^3\theta\sin\theta
+g_4\cos\theta\sin^3\theta+(g_5+g_6)\cos^2\theta\sin^2\theta\bigg]\\\nonumber&+&
\big(\vec{N}.\dot{\vec\delta}\big)^4\times
\bigg[g_2\cos^4\theta+g_1\sin^4\theta-g_4\cos^3\theta\sin\theta
-g_3\cos\theta\sin^3\theta+(g_5+g_6)\cos^2\theta\sin^2\theta\bigg]\\\nonumber&+&
\big(\vec{T}.\dot{\vec\delta}\big)^3\big(\vec{N}.\dot{\vec\delta}\big)\times\\\nonumber&&
\bigg[-g_3\cos^4\theta+g_4\sin^4\theta+2\bigg(2g_1-(g_5+g_6)\bigg)\cos^3\theta\sin\theta-2\bigg(2g_2-(g_5+g_6)\bigg)\cos\theta\sin^3\theta
+3(g_3-g_4)\cos^2\theta\sin^2\theta\bigg]\\\nonumber&+&
\big(\vec{T}.\dot{\vec\delta}\big)\big(\vec{N}.\dot{\vec\delta}\big)^3\times\\\nonumber&&
\bigg[-g_4\cos^4\theta+g_3\sin^4\theta-2\bigg(2g_2-(g_5+g_6)\bigg)\cos^3\theta\sin\theta+2\bigg(2g_1-(g_5+g_6)\bigg)\cos\theta\sin^3\theta
+3(g_4-g_3)\cos^2\theta\sin^2\theta\bigg]\\\nonumber&+&
\big(\vec{T}.\dot{\vec\delta}\big)^2\big(\vec{N}.\dot{\vec\delta}\big)^2\times\\\nonumber&&
\bigg[(g_5+g_6)\bigg(\cos^4\theta+\sin^4\theta\bigg)+3(g_4-g_3)\bigg(\cos^3\theta\sin\theta-\cos\theta\sin^3\theta\bigg)
+2\bigg(3(g_1+g_2)-2(g_5+g_6)\bigg)\cos^2\theta\sin^2\theta\bigg].
\end{eqnarray}
Now let us just consider the terms containing
$(\vec{T}.\dot{\vec\delta}\big)$ and write them together
as\footnote{Note that the coefficient of
$\big(\vec{T}.\dot{\vec\delta}\big)^2$ in
(\ref{two-field-lagrangian-simplified-order-2-correction-terms}) is
$6$. But by comparison to
(\ref{two-field-lagrangian-simplified-order-2-correction-terms-spatial}),
$2$ of $6$ appear in definition of the speed of sound, $c_s$,
(exactly same as the second line in (\ref{scalar-perturbation})) and
what remains is $4\big(\vec{T}.\dot{\vec\delta}\big)^2$.}
\begin{eqnarray}\label{Tdelta}\nonumber
4\dot\sigma^2\bigg[g_1\cos^4\theta+g_2\sin^4\theta+g_3\cos^3\theta\sin\theta
+g_4\cos\theta\sin^3\theta+(g_5+g_6)\cos^2\theta\sin^2\theta\bigg]\times
\bigg(\big(\vec{T}.\dot{\vec\delta}\big)^2+
\big(\vec{T}.\dot{\vec\delta}\big)^3/\dot\sigma+\frac{1}{4}\big(\vec{T}.\dot{\vec\delta}\big)^4/\dot\sigma^2\bigg)
\end{eqnarray}
Comparison the above relation with the relation in
(\ref{scalar-perturbation}) manifests that $\sigma$, $[...]$ and
$\big(\vec{T}.\dot{\vec\delta}\big)$ play the role of $\vpb$,
$f(\vpb)$ and $\dot\delvp$ respectively. The significant property of
this model is the appearance of $\big(\vec{T}.\dot{\vec\delta}\big)$
and $\big(\vec{N}.\dot{\vec\delta}\big)$ or equivalently
$(\dot{\delta\sigma}-\dot\theta\delta s)$ and $(\dot{\delta
s}+\dot\theta\delta \sigma)$ respectively. This means that
$\dot{\delta\sigma}$ and $\delta s$ are always together and the same
for $\dot{\delta s}$ and $\delta\sigma$. This characteristic feature
of this model has some observational consequences which are
discussed in the following.
\subsubsection{The Amplitude of Non-Gaussianity}
Now we are going to estimate the non-Gaussianity amplitude. To do
this one procedure is comparison between the non-linear terms and
the linear ones. Mathematically, the amplitude of non-Gaussianity
$f_{NL}$, bi-spectrum, can be estimated as
$\frac{{\cal{L}}^{(3)}}{{\cal{L}}^{(2)}}\times\zeta^{-1}$ where
$\zeta$ is the curvature perturbation \cite{baumann}. Note that the
dominant amplitude of the terms containing time derivatives comes
from their amplitude at horizon crossing. At this time
$\frac{d}{dt}\sim H$ where $H$ is the Hubble constant. Hence for the
second order perturbations the Lagrangian ${\cal{L}}^{(2)}$ can be
written in an abstract form as $\{H^2, H \dot{\theta},
\dot{\theta}^2\}\times M^2 \times \delta\sigma^2$. The same analyze
for ${{\cal{L}}^{(3)}}$ results in $\{H^3, H^2 \dot{\theta}, H
\dot{\theta}^2, \dot{\theta}^3\}\times f(g_i) \times \dot\sigma
\times \delta\sigma^3$. So in an abstract form
\begin{eqnarray}\label{fNL}
\frac{{\cal{L}}^{(3)}}{{\cal{L}}^{(2)}}=\frac{\{H^3, H^2
\dot{\theta}, H \dot{\theta}^2, \dot{\theta}^3\}}{\{H^2, H
\dot{\theta}, \dot{\theta}^2\}}\times\bigg(\frac{f(g_i)}{M^2}\dot\sigma^2\bigg)\times
\frac{\delta\sigma}{\dot\sigma}=\frac{\{H^3, H^2 \dot{\theta}, H
\dot{\theta}^2, \dot{\theta}^3\}}{H\times\{H^2, H \dot{\theta},
\dot{\theta}^2\}}\times\bigg(\frac{f(g_i)}{M^2}\dot\sigma^2\bigg)\times \zeta
\end{eqnarray}
where $\zeta\sim \frac{H\delta\sigma}{\dot\sigma}$ is interpreted as
curvature perturbation. Now we consider two different regimes
$\dot{\theta}<<H$ and $\dot{\theta}>>H$. The first regime,
$\dot{\theta}<<H$, physically means that the model is a single field model effectively. In
this case the amplitude of bi-spectrum can be approximated by
\begin{eqnarray}\label{fNL-thetadot<<H}
f_{NL}\sim\frac{{\cal{L}}^{(3)}}{{\cal{L}}^{(2)}}\zeta^{-1}\sim\frac{f(g_i)}{M^2}\dot\sigma^2
\end{eqnarray}
but remember that the validity of the effective
field theory imposes $\frac{f(g_i)}{M^2}\dot\sigma^2<1$. So in this case as mentioned before
there is no significant non-Gaussianity which is in agreement with
the single field models of inflation \cite{chen}. But the other case,
$\dot{\theta}>>H$, means that the classical path in the
phase space is highly curved \cite{ana}. In other words it means the classical path
in the phase-space is far from a straight line ($\dot\theta=0$).
So the existence of the entropic field
is unavoidable. For this case the amplitude of $f_{NL}$ is
\begin{eqnarray}\label{fNL-thetadot>>H}
f_{NL}\sim\frac{{\cal{L}}^{(3)}}{{\cal{L}}^{(2)}}\zeta^{-1}
\sim\frac{\dot\theta}{H}\times\bigg(\frac{f(g_i)}{M^2}\dot\sigma^2\bigg).
\end{eqnarray}
Now the factor
$\frac{\dot\theta}{H}\times(\frac{f(g_i)}{M^2}\dot\sigma^2)$ can be
large and results in large non-Gaussianity consequently. So the
large curvature of the classical path in the phase-space results in
the large non-Gaussianity. Though this result can be compared to the
other works in the literature \cite{gpmulti} and in the effective
field theory context \cite{eftmulti} but the curvature of the
classical path is restricted due to observed scale invariant power
spectrum \cite{max}.
For a moment let us relax the constraint on the
$\frac{f(g_i)}{M^2}\dot\sigma^2$. Consequently the correction terms
in (\ref{two-field-lagrangian-simplified}) causes the large
non-Gaussinity. However the relaxation of the constraint can be
justified by assuming that our model is completely described by
(\ref{two-field-lagrangian-simplified}) without any higher order
correction terms. This needs fine tuning which is not impossible but
it is not natural. However there is another method to rationalize
this assumption. Instead of fine tuning the model automatically
shows this property via for example Vainshtein mechanism
\cite{derham,vain}.
The same is applicable for tri-spectrum by an estimation as
$\tau_{NL}\sim\frac{{\cal{L}}^{(4)}}{{\cal{L}}^{(2)}}\times\zeta^{-2}$.
The fourth order Lagrangian, ${{\cal{L}}^{(4)}}$, can be written in
the abstract form as $\{H^4,H^3\dot\theta, H^2 \dot{\theta}^2, H
\dot{\theta}^3,H \dot{\theta}^3,\dot\theta^4\}\times f(g_i) \times
\delta\sigma^4$ and then
\begin{eqnarray}\label{tNL-thetadot<<H}
\tau_{NL}\sim\frac{{\cal{L}}^{(4)}}{{\cal{L}}^{(2)}}\zeta^{-2}\sim\frac{f(g_i)}{M^2}\dot\sigma^2
\end{eqnarray}
for $\dot\theta<<H$ and
\begin{eqnarray}\label{tNL-thetadot>>H}
\tau_{NL}\sim\frac{{\cal{L}}^{(4)}}{{\cal{L}}^{(2)}}\zeta^{-2}
\sim\left(\frac{\dot\theta}{H}\right)^2\times\bigg(\frac{f(g_i)}{M^2}\dot\sigma^2\bigg).
\end{eqnarray}
for $\dot\theta>>H$.
\subsubsection{The Shape of Non-Gaussianity}
Now let us focus on the shape of possible non-Gaussinity predicted
by our model. In principle all the possible interaction terms
between $\delta\sigma$, $\delta s$ and their derivatives exist in
our model. This fact means all the non-Gaussianity shape can be
produced. However the amplitude of different shapes are different.
As a general argument it can be emphasized that for different
regimes of $\frac{\dot\theta}{H}$ different shapes are dominant. For
the case $\dot\theta<<H$ the terms containing time derivatives
become dominated. This means in this limit the equilateral shape is
the main one among the others. But it does not mean the other shapes
do not exist i.e. the ``Cosine" between different shapes are not
zero. In the other limit, $\dot\theta>>H$, the terms without
derivative become dominant and then the local shape is dominant.
This result is in agreement with the prediction for multi-field
inflation models \cite{chen}. Note that in \cite{senatore} since the
additional entropy perturbations are added by symmetry then they do
not have any term without derivative in their Lagrangian. In this
sense they do not predict a dominant local shape for their model
which is in disagreement with our result.
A characteristic feature of this model is appearance of just two
combinations of the fields i.e. $\vec{T}.\dot{\vec{\delta}}$ and
$\vec{N}.\dot{\vec{\delta}}$ in all the terms including the second,
third and fourth orders. To explain what is the physical result of
this fact let us concentrate on
$\vec{T}.\dot{\vec{\delta}}=\dot{\delta\sigma}-\dot\theta\delta s$,
as an example. The third order term of this combination is
$(\vec{T}.\dot{\vec{\delta}})^3=\dot{\delta\sigma^3}-
3\dot\theta\dot{\delta\sigma^2} \delta
s+3\dot\theta^2\dot{\delta\sigma} \delta s^2 -\dot\theta^3\delta
s^3$. Without worrying about the amplitude in this part let us focus
on the first and the last term. The definite prediction of this
model is that if any equilateral non-Gaussianity due to the first
term, i.e. $\dot{\delta\sigma^3}$, be observed then it has to be
observed a local non-Gaussianity due to the last term\footnote{ Note
that due to the first equation of motion in (\ref{eq-mo-pert}) the
$\delta s$ sources $\delta \sigma$.}, i.e. $\delta s^3$. So the
non-Gaussianity predicted by this model cannot be pure e.g. pure
equilateral shape. Hence mathematically, the ``Cosine" between two
shapes cannot be zero and more the ``Cosine" depends on the
$\dot\theta$ and is fixed by the model. This argument is true for
the other third order terms as well as fourth order ones. To
conclude, it seems that this model predicts a definite combination
of different shapes for the non-Gaussianity if the amplitude allows
to observe them.
\subsection{An Example}
In this subsection we are going to show how the general statements
mentioned before do work in a simple example. Here we assume that
all the $g_i$'s vanish except $g_1(\vp,\chi)$\footnote{Note that
except here in the rest of the paper we assumed that $g_i$'s are
constant as a matter of simplification. But here we would like to
show how the functionality of $g_i$'s may affect the final result.}
which is a generalization of the model in \cite{paolo1}. In addition
we assume there is no potential term\footnote{Note that the most
general form of the potential term can be supposed. But as mentioned
in \cite{baumann-green}, in the slow-roll regime there is no
interesting non-Gaussianity prediction for single field models.
However for multi-field models the potential term can result in
large non-Gaussianity which considered in \cite{gpmulti}. Here, we
restrict our calculations to kinetic terms.}. According to the
background Lagrangian
(\ref{two-field-lagrangian-simplified-order-0}) the equations of
motion for our special case become
\begin{eqnarray}\label{special-case-eq.of.motion}
&&\ddot{\vp}\big(1+12
\frac{g_1}{M^2}\dot\vp^2\big)+3H\dot\vp\big(1+4
\frac{g_1}{M^2}\dot\vp^2\big)+\frac{1}{M^2}\dot\vp^3\bigg(3\dot\vp\frac{\partial
g_1}{\partial \vp}+4\dot\chi\frac{\partial g_1}{\partial
\chi}\bigg)=0\\\nonumber
&&\ddot{\chi}+3H\dot\chi-\frac{1}{M^2}\dot\vp^4\frac{\partial
g_1}{\partial\chi}=0
\end{eqnarray}
where $M=M_1=M_2$ is assumed. On the other hand, what can cause the
significant non-Gaussianity is $\dot\theta$ as mentioned before. In
general due to the definition of $\theta$ in
(\ref{tangent-normal-vector}), $\dot\theta$ can be read as
\begin{eqnarray}\label{dot-theta}\nonumber
\dot\theta=\frac{-\ddot\vp\dot\chi+\dot\vp\ddot\chi}{\dot\vp^2+\dot\chi^2}
\end{eqnarray}
and in our special case by considering
(\ref{special-case-eq.of.motion}) it becomes (up to the first order
of $g_1\frac{\dot\vp^2}{M^2}$)
\begin{eqnarray}\label{dot-theta-special-case}\nonumber
\frac{\dot\theta}{H}\simeq24 \bigg(-g_1
\frac{\dot\vp^2}{M^2}\bigg)\frac{\dot\vp\dot\chi}{\dot\vp^2+\dot\chi^2}+\frac{1}{H}
\frac{1}{M^2}\frac{\dot\vp^3}{\dot\vp^2+\dot\chi^2}\bigg[3\frac{\partial
g_1}{\partial\vp}\dot\vp\dot\chi+\frac{\partial
g_1}{\partial\chi}\big(\dot\vp^2+4\dot\chi^2\big)\bigg].
\end{eqnarray}
The condition $g_1 \frac{\dot\vp^2}{M_1^2}<1$ ensures the validity
of the effective field theory. So it is not bad to estimate $24g_1
\frac{\dot\vp^2}{M_1^2}\sim 1$. Then due to the first term
$\frac{\dot\theta}{H}\sim
\frac{\dot\vp\dot\chi}{\dot\vp^2+\dot\chi^2}$ which means the
maximum of $f_{NL}$ in (\ref{fNL-thetadot>>H}) is less than one. A
successful inflation in the slow-roll regime restricts the value of
the field velocities which may restrict more the above estimation.
To discuss on the second term let us divide $g_1(\vp,\chi)$ to its
amplitude and functionality as $g_1(\vp,\chi)=\mid g_1\mid\times
f(\vp,\chi)$ such that $\mid g\mid$ is the amplitude of the
$g_1(\vp,\chi)$ and $f(\vp,\chi)$ represents its functional form. So
the second term can be estimated as (by assuming
$\dot\vp\sim\dot\chi$)
\begin{eqnarray}\label{dot-theta-special-case-second-term}\nonumber
\frac{\dot\theta}{H}\simeq \bigg(\mid
g_1\mid\frac{\dot\vp^2}{M^2}\bigg)\frac{\dot\vp}{2H}\bigg[3\frac{\partial
f}{\partial\vp}+5\frac{\partial f}{\partial\chi}\bigg],
\end{eqnarray}
where $\mid g_1\mid \frac{\dot\vp^2}{M^2}<1$ has to be satisfied. On
the other hand one of the Friedmann equations (in the absence of the
potential) is
$H^2=\frac{M^2}{2}\dot\vp^2+\frac{M^2}{2}\dot\chi^2\sim
M^2\dot\vp^2$ for the zeroth order of $g_1 \frac{\dot\vp^2}{M^2}$.
Now if $\frac{\partial f}{\partial\vp}$ or
$\frac{\partial f}{\partial\chi}$ have the significant
amplitude with respect to $M$ then a large amplitude of
non-Gaussianity would be expected. This can be realized by
assuming sharp features in the functionality of $g_1(\vp,\chi)$
maybe due to a phase transition.
\subsection{Some Clarifications on Differences with Senatore and
Zaldarriaga \cite{senatore}} The significant difference is the
existence of the terms containing the adiabatic and entropy
perturbations themselves not just their derivatives. The reason for
this difference is in how the effective field theory is constructed
in \cite{senatore}. As mentioned before in their model the adiabatic
mode is borrowed from \cite{paolo} which satisfies a shift symmetry.
Then the entropy modes are added and satisfy the shift symmetry too.
Consequently, in their formalism they have just derivative of
perturbations. But in contrast we do not start with distinguishable
fields then we do not have any difference between the perturbations
initially. So by this starting point we had to define the adiabatic
and entropy perturbations. This is what has been done in this
section in details. Now the question is that is there any special
transformation for adiabatic and entropy perturbation in our model?
Yes, it is locally rotated shift transformation i.e.
\begin{eqnarray}\label{locally-rotated-shift-symm.}
\delta\sigma\rightarrow\delta\sigma+(c_1
\cos\theta+c_2\sin\theta)\\\nonumber \delta s\rightarrow\delta
s+(c_1 \sin\theta-c_2\cos\theta)
\end{eqnarray}
where $\theta=\arctan(\dot\chi/\dot\vp)$. Note that the rotational
angle depends on the background fields time evolution. To achieve
this result, the starting point is the Lagrangian for two fields
i.e. the relations (\ref{two-field-lagrangian-simplified-order-2}),
(\ref{two-field-lagrangian-simplified-order-3}) and
(\ref{two-field-lagrangian-simplified-order-4}). By looking at these
relations it is obvious that $\delvp\rightarrow\delvp+c_1$ and
$\delchi\rightarrow\delchi+c_2$ is a symmetry of the model where
$c_1$ and $c_2$ are two independent arbitrary constants. So due to
(\ref{adi-ent-perturbations}) one can get the above relation
(\ref{locally-rotated-shift-symm.}) as the corresponding
transformation of $\delta \sigma$ and $\delta s$. According to this
relation the invariant terms corresponding to $\dot{\delvp}$ and
$\dot{\delta \chi}$ are not $\dot{\delta\sigma}$ and $\dot{\delta
s}$ but
\begin{eqnarray}\label{invariant-combination}
\dot{\delta\sigma}-\dot\theta\delta
s\rightarrow\dot{\delta\sigma}-\dot\theta\delta
s\\\nonumber\dot{\delta s}+\dot\theta\delta
\sigma\rightarrow\dot{\delta s}+\dot\theta\delta \sigma
\end{eqnarray}
which are $\vec T.\dot{\vec\delta}$ and $\vec N.\dot{\vec\delta}$
respectively. Not surprisingly, these terms construct whole
Lagrangian in adiabatic and entropy perturbations language
as seen previously.
So it seems initially supposed adiabatic perturbation in
\cite{senatore} results in lack of all possible terms in the
effective Lagrangian. Our proposition to solve this problem is based
on the discussion in this subsection. The main building blocks for
an effective field theory of multi-field inflation are not
$\dot{\delta\sigma}$ and $\dot{\delta s}$ but they are $\vec
T.\dot{\vec\delta}$ and $\vec N.\dot{\vec\delta}$. So the most
general Lagrangian for the perturbations in the multi-field context
should be written as\footnote{Note that here we just consider the
time derivative since in the discussion of this section there is no
difference between our model and \cite{senatore} for the terms
containing spatial derivatives. This is because the background is
not spatial dependent. Remember that the angle of rotation just
depends on time.}
\begin{eqnarray}\label{general-perturbation-lag.-adi-ent.}
\Delta{\cal{L}}\propto \sum_{m,n} c_{mn} \bigg(\vec
T.\dot{\vec\delta}\bigg)^m\bigg(\vec N.\dot{\vec\delta}\bigg)^n
\end{eqnarray}
for arbitrary $c_{mn}$. The above Lagrangian can be considered as
the effective field theory for the two-field inflation in the
language of \cite{senatore} but with additional terms. Note that the
above result can be generalized to multi-field inflation as
\begin{eqnarray}\label{general-perturbation-lag.-adi-ent.-multi}
\Delta{\cal{L}}\propto \sum c_{{n_0},{n_1},...,{n_N}} \bigg(\vec
T.\dot{\vec\delta}\bigg)^{n_0}\bigg(\vec
N_1.\dot{\vec\delta}\bigg)^{n_1}\bigg(\vec
N_2.\dot{\vec\delta}\bigg)^{n_2}...\bigg(\vec
N_N.\dot{\vec\delta}\bigg)^{n_N}
\end{eqnarray}
where $\vec T$ and $\vec N_i$'s are a set of orthonormal vectors for
an $(N+1)$-field model.
\section{Conclusions}
In this work the effective field theory of multi-field inflation has
been studied as a generalization of Weinberg's idea \cite{weinberg}
for a single field. In this approach the most general Lagrangian is
built by using all the covariant terms of the fields. Though
effectively the terms with higher order derivatives are interested
in the higher energy scales. In this work we restricted the model to
the first correction terms. They results in up to fourth order terms
in perturbations. Then due to the physical interests we switched to
the adiabatic and entropy formalism. It has been shown that
generally these modes can have different speeds of sound. By
considering the non-linear terms we studied the non-Gaussinity in
this model. It has been shown that the amplitude of non-Gaussianity
can be significant when the curvature of the classical path in the
phase-space becomes large. For example a sharp turn in the classical
path can realize it. However it seems that existence of the higher
order derivative terms in the Lagrangian cannot produce large
non-Gaussinity. The bottom line for this fact is the strong
constraint on the coefficients to keep the effective field theory
valid. But there is an idea that it is possible to take the higher
order correction terms under control automatically, e.g. by
Vainshtein mechanism. This relaxes the constraint on the coefficient
of the correction terms and results in large non-Gaussianity. On the
other hand the structure of the interacting terms in the Lagrangian
predicts the existence of all the shapes of non-Gaussinity with the
different amplitude for different cases. But the characteristic
feature of the model is that the non-Gaussinities are correlated.
That means if there is a local non-Gaussinity due to the entropy
mode then certainly there is a non-Gaussinity in adiabatic mode
which is equilateral. The amplitude of these different types of
non-Gaussinities are not independent.
In contrast to \cite{senatore}, the adiabatic and entropy
perturbations are not distinguishable initially. This fact results
in the existence of the perturbations as well as their derivatives.
In other words the adiabatic and entropy perturbations are not
invariant under the shift symmetry of original fields. However a
combination of them is invariant under such a symmetry. These
combinations are ``$\dot{\delta\sigma}-\dot\theta\delta s$" and
``$\dot{\delta s}+\dot\theta\delta \sigma$" or in other form $\vec
T.\dot{\vec\delta}$ and $\vec N.\dot{\vec\delta}$ respectively. This
result is important for constructing the effective field theory for
multi-field inflation and causes the additional terms with respect
to what is considered in \cite{senatore}.
\begin{acknowledgments}
We would like to thank B. A. Bassett, H. Firouzjahi, J. Fonseca, H.
R. Sepnagi, N. Sivanandam and M. M. Sheikh-Jabbari for their
comments. We are grateful of T. Battefeld for his very useful
comments and careful reading of the manuscript. We also specially
thank P. Creminelli for very fruitful discussions and comments. We
would like to thank ICTP for their warm hospitality and support when
this work was initiated.
\end{acknowledgments}
|
{
"timestamp": "2012-03-13T01:01:21",
"yymm": "1203",
"arxiv_id": "1203.2266",
"language": "en",
"url": "https://arxiv.org/abs/1203.2266"
}
|
\section{Introduction}
Periodically ordered arrays of nanoparticles, colloidal crystals,
crystalline mesophases formed from surfactant molecules or block
copolymers, etc. are all examples of complex periodic structures that
can occur in soft matter systems. Since often the interactions between
the constituent particles of these structures are to a large degree tunable, one
has the possibility of producing materials with ``tailored'' properties
which have potential applications in nanotechnological devices
\cite{1,2,3,4,5}. When seeking to provide theoretical guidance for
understanding structure-property relations in such complex soft matter
systems, a basic issue is how to judge the relative stability of
competing candidate structures, i.e. to distinguish the stable structure
(having the lowest free energy) from the metastable ones. For standard
crystals formed from atoms or small molecules, this question can be
answered by comparing ground state energies of the competing structures
(and --if necessary-- also taking entropic contributions from lattice
vibrations into account, within the framework of the harmonic
approximation). In soft matter systems, disorder in the structure and
thermally driven entropic effects rule out such an approach, and hence
there is a need for computer simulation methods that compute the free
energy of the various complex structures. However, as is well known, the
free energy of a model system is not a direct output of either
Molecular Dynamics or Monte Carlo simulations, and special techniques
have to be used \cite{6,7,8,9,10,11}.
In principle, one can obtain the absolute free energy of a structure by
linking it to some reference state of known free energy by means of
thermodynamic integration (TI)
\cite{6,7,8,9,10,11,12,13,14,15,16}. The strengths of TI are that
it is both conceptually simple and often straightforward to implement.
Its principal drawback is that the quantity of interest, namely the free
energy {\em difference} between candidate structures is typically
orders of magnitude smaller than the absolute free energies of the
individual structures which TI measures. Essentially, therefore, TI
estimates a small number by taking the difference of two large ones; As
a consequence, the precision of the method is limited and an enormous
(even sometimes wasteful) investment of computer resources may be needed
to resolve free energy difference accurately \cite{9}.
A much more elegant approach, albeit one which is not quite so easy to
implement as TI, is the ``phase switch Monte Carlo'' \cite{17a,17,18,19,20,21,21a}
technique. This method is potentially more powerful than TI because it
focuses directly on the small free energy difference between the
structures to be compared, rather than their absolute free energies. In
previous work, the precision of the method was demonstrated in the
context of measurements of the free energy difference between fcc and
hcp structures of hard spheres \cite{19}, the phase behaviour of
Lennard-Jones crystals \cite{19} and as a means of studying liquid-solid
phase transitions \cite{17}. In the latter case, simple model systems
containing only a few hundred particles could be studied, while for the
study of the fcc-hcp free energy difference \cite{17a,20} larger systems
of up to $1728$ particles could be studied by virtue of the fact that
these crystals only differ in their packing sequence of close-packed
triangular defect-free lattice planes. However, it is an open question
as to what system sizes one can attain with the phase switch method for
more general crystalline systems, including -- as in the present work --
ones which exhibit considerable structural disorder (``soliton
staircases'', see below). Furthermore, there have hitherto been no
like-for-like comparisons of the TI and phase switch methods on the same
system, so whilst their are good reasons for {\em presuming} the
superiority of phase switch (in terms of precision delivered for a given
computational investment), this has never actually been quantified.
In the present paper, we address these matters, considering as a generic
example a two-dimensional colloidal crystal in varying geometrical
confinement \cite{22,23,24,25,26}. As is well-known, two dimensional
colloidal crystals are experimentally much studied model systems
\cite{27,28,29,30,31,32,33,34,35,36,37,38} comprising, for example,
polystyrene spheres containing a superparamagnetic core adsorbed at the
air-water-interface. Applying a magnetic field oriented perpendicular to
this interface creates a repulsive interaction that scales like $r
^{-3}$, ($r$ being the particle separation), whose magnitude is
controlled by the magnetic field strength \cite{27}. Lateral
confinement of such two-dimensional crystals can be effected
mechanically or by laser fields (if the latter are also applied in the
bulk of such a crystal, one can study laser-induced melting and/or
freezing \cite{39,40,41,42}). Of course, there exist many related
problems in rather different physical contexts (``dusty plasmas''
\cite{43,44}, i.e. negatively charged SiO$_2$ fine particles with 10$\mu
m$ diameter in weakly ionized $rf$ discharges; lattices of confined
spherical block copolymer micelles \cite{45}; vortex matter in slit
channels \cite{46}, etc.). However, our study does not address a
specific system, rather we focus on the methodological aspects of
how one can study such problems by computer simulation.
The outline of the present paper is a follows. In Sec. 2, we summarize
the key facts about our model, namely strips of two-dimensional crystals
confined between two walls where structural phase transitions may occur
when the distance between the (corrugated) rigid boundaries is varied
\cite{23,24,25,26,47,48,49} (i.e., a succession of transitions in the
number of crystal rows $n$ parallel to the walls occur, $n \rightarrow
n-1 \rightarrow n-2$, with increasing compression, accompanied by the
formation of a ``soliton staircase'' at the walls \cite{23,24,25,26}). In
Sec. 3, the methods that are used are briefly described: the
thermodynamic integration method of Schmid and Schilling \cite{15,16} is
used as a baseline, while the main emphasis is on the phase switch Monte
Carlo method (implementation details of which are deferred to an
Appendix). In Sec. 4 we describe the results of the application of these
techniques to the model of Sec. 2. We show that phase switch Monte Carlo
\cite{17,18,19} can accurately locate the phase transitions despite the
need to deal with thousands of particles, and is orders of magnitude
more efficient than thermodynamic integration. Sec. 5 summarizes some
conclusions.
\section{Structural Transitions in Crystalline Strips confined by corrugated boundaries: Phenomenology}
Here we introduce the model for which our methodology is exemplified,
and recall briefly the main findings concerning the rather unconventional
transitions that have been detected \cite{23,24,25,26}, as far as they
are relevant for the present study.
We consider monodisperse colloidal particles in a strictly
two-dimensional geometry, which then are treated like point particles in
a plane interacting with a suitable effective potential $V(r)$ that
depends only on the interparticle distance $r$. In the real systems
\cite{27,29,30,31,32,33} this potential is purely repulsive, but due to
the magnetostatic dipole-dipole interaction (whose strength is
controlled by the external magnetic field) it is very slowly decaying,
$V(r) \propto r ^{-3}$. Since we here are not concerned with
quantitative comparisons with real experimental data on such systems, we
simplify the problem by adopting a computationally more efficient
$r^{-12}$ potential, in accord with previous work \cite{23,24,25,26}.
Moreover, to render it strictly short-ranged, we introduce a cutoff
$r_c$, such that $V(r \geq r_c)\equiv 0$, and employ a smoothing function
to make $V(r)$ differentiable at $r=r_c$. Thus, the model potential used
is
\begin{equation} \label{eq1}
V(r) = \varepsilon\Big[(\sigma/r)^{12} - (\sigma/r_c)^{12} \Big] \Big[\frac{(r-r_c)^4}{h^4 + (r-r_c)^4}\Big] \quad ,
\end{equation}
with parameters $r_c=2.5\sigma$ and $h=0.01 \sigma$. Henceforth, the
particle diameter $\sigma=1$ defines the length units in our model, and
for the energy scale, $\varepsilon=1$ is taken, while Boltzmann's constant
$k_B=1$. It is known that at $T=0$ the ground state of this model is a
perfect triangular lattice, with a lattice spacing $a$ related to
the choice of number density $\rho=N/V$ (with $N$ the particle number and $V$ the
(two-dimensional) ``volume'' of the system) via
\begin{equation} \label{eq2}
a^2 =2 / (\sqrt{3} \rho) \quad .
\end{equation}
Assuming the physical effect of truncating the potential can be
neglected, only the choice of the combination $X=\rho
(\varepsilon/k_BT)^{1/6}$ controls the phase behavior \cite{49a}.
Thus, following previous work in the NVT-ensemble it
suffices to choose a single density when the temperature variation is
considered \cite{23,50}. For the particular choice $\rho=1.05$, the melting
transition of this model is known to occur at about $T=T_m \approx 1.35$
\cite{50}. Note that here we are not at all concerned with the
peculiarities of melting in two dimensions \cite{51}, and hence we focus
on a temperature deep within the crystalline phase, $T=1$. Although it
is known that the density of vacancies and interstitials in $d=2$ for
any nonzero temperature is also nonzero in thermal equilibrium
\cite{51,52}, for the chosen particle number $N= 3240$ the system is
essentially defect free, since the densities of these point defects at
$T=1$ are extremely small \cite{23,50}.
\begin{figure}
\includegraphics[scale=0.28, clip=true]{fig1.eps}
\caption{\label{fig1} Sketch of the system geometry, showing the fixed
wall particles (black spheres) and the mobile particles (gray spheres).
The orientation of the coordinate axes is indicated, as well as the
lattice spacing of the triangular lattice ($a$) and the linear dimensions
$L_x,D$ of the system.}
\end{figure}
The geometry of the present system is a $D \times L_x$ slit, confined in
the y-direction and periodic in the x-direction. In the y-direction
there are $n_y=30$ rows of the triangular lattice, each containing
$n_x=108$ particles, stacked upon each other. The $x$-direction
coincides with a lattice direction so that $L_x=n_xa$. The
confining boundaries (one at the top and one at the bottom of the
system) each take the form of a double rows of particles in which the
particles are rigidly fixed at the sites of a perfect triangular lattice
(Fig.~\ref{fig1}). These rows of fixed particles represent rigid
corrugated walls, essentially acting as a periodic wall potential on the
mobile particles. While the distance of the first row at the upper wall
from the first row of mobile particles in the ideal stress-free crystal is
simply $D=n_y a \sqrt{3}/2$, in the following we are interested in the response of the
system when the walls occur at a smaller distance, caused by a misfit
$\Delta$, defined via \cite{53}
\begin{equation} \label{eq3}
D=(n_y - \Delta) a \sqrt{3} / 2 \quad .
\end{equation}
As described in the previous work \cite{23,24,25,26}, standard
Monte Carlo simulation \cite{6,7} allows one to study this model at
various values of $\Delta$, and also sample the stress
$\sigma=\sigma_{yy} - \sigma_{xx}$ ($\sigma_{\alpha \beta}$ are the
Cartesian components of the pressure tensor) applying the virial formula
\cite{6,7}. Fig.~\ref{fig2} shows that when one starts out with the
perfect crystal $(n_y=30)$ with no misfit, the crystal already shows a
small finite stress, because the rigid wall particles somewhat hinder
the vibrations of the mobile particles in their potential wells, but
this effect is of no importance here. Rather we focus on the (slightly
nonlinear) increase of the stress up to about $\Delta =\Delta_c \approx
2$, followed by the (almost) discontinuous decrease, and the subsequent
increases again with further enhancement of the misfit. A previous
structural analysis has revealed \cite{23,24,25,26} that the sudden
decrease of stress is due to a transition in the number of rows in the
crystal, $n_y \rightarrow n_y -1=29$. However, since in the NVT ensemble
the particle number is conserved, the $n_x=108$ particles of the row
that disappears must be redistributed among the remaining rows. A
closer examination of the structure revealed that none of these
particles enter the two rows adjacent to the rigid walls, instead they
all go into the $n_y-3=27$ rows of the system that are further away from the
walls. Thus, in the present case, the particle number per row becomes
$n'_x+n_x/(n_y-3)=n_x+4$, and this leads to a new lattice spacing in the
$x$-direction of $a'=a/(1+4/n_x)$, which is no longer commensurate with
the spacing between the particles forming the rigid walls (or the two immediately adjacent
layers which remain commensurate with them). While for the rows in the
center of the system (near $n_y/2)$ this compression of the lattice
spacing occurs uniformly along the $x$-direction, this is not the case
close to the walls, which provide a periodic potential (with periodicity
$a$) that acts on the row of mobile particles a little further inside
the slit. The fact that on the scale $L_x$ the effective wall potential
exhibits $n_x$ minima but that $n'_x=n_x+4$ particles need to be
accommodated, leads to the formation of a lattice of solitons close to
both walls (``soliton staircase'') \cite{54,55}, as depicted for an
idealized case in Fig.~\ref{fig3}.
\begin{figure}
\includegraphics[scale=0.32, clip=true]{fig2.eps}
\caption{\label{fig2} Stress $\sigma$ plotted vs. misfit $\Delta$, for a
system of $N=3240$ particles, and using different starting
configurations having $n_y=30$, $n_y=29$, and $n_y=28$, as indicated in
the figure. Note the huge hysteresis of the $n_y=30 \rightarrow n_y=29$
and $n_y=29 \rightarrow n_y=28$ transitions. For further explanations
see the main text.}
\end{figure}
\begin{figure}
\includegraphics[scale=0.1, clip=true]{fig3a.eps}\\
\includegraphics[scale=0.18, clip=true]{fig3b.eps}\\
\includegraphics[scale=0.18, clip=true]{fig3c.eps}
\caption{\label{fig3} a) Putting $n+1$ particles in a periodic potential with $n$ minima creates a soliton configuration,
i.e. over a range of several lattice spacings particles are displaced from the potential minima (schematic) b) Superimposed snapshot pictures of 750 configurations of the particle positions, where for a system of $n_y=30$ rows and a large misfit ($\Delta=2.6$) a transition to $n_y-1=29$ rows has occured ($n_x=108$ and $T=1.0$ were chosen). The $4$ solitons at each wall are visible due to the larger lateral displacements of the particles, leading to a darker region in the snapshot. Part (c) shows a close-up of the structure near the upper wall. Numbers shown along the axes indicate the Cartesian coordinates of the particles. Parts (b) and (c) have been adapted from Chui et al. \cite{23}.}
\end{figure}
In practice, the actual structure having $n_y-1=29$ rows that is formed
in the simulations on increasing the misfit $\Delta$ beyond the critical
value $\Delta_c$, is generally less regular than the 'idealized' one
shown in Fig.~\ref{fig3}: specifically, the relative distance between
neighboring solitons showed a considerable variation. This comes about
because (i) the solitons are formed from the stressed crystal with
$n_y=30$ rows via random defect nucleation events \cite{24}, and (ii)
the mutual interaction between neighboring solitons, which is the
thermodynamic driving force towards a regular soliton arrangement, is
very small \cite{25}. Despite this, it is nevertheless reasonable to
construct ``by hand'' the expected regular structure of $n_x/(n_y-3) \,
(=4)$ solitons near each wall as a starting configuration for a system
with $29$ rows, which can subsequently be equilibrated \cite{23}. Of
course, there is no guarantee that this guessed structure actually is
the one of lowest free energy; but it does exhibit slightly less stress
than all other structures that had been tested, for misfits in the range
$1.5 \leq \Delta \leq 3$, and hence has been used as a starting point
for studies in which $\Delta$ was varied in this range.
Starting from this idealized $29$ row structure and decreasing the
misfit one finds that the $29$ row structure is stable down to about
$\Delta'_c\approx 1.3$, at which point the soliton lattice disappears
and the system spontaneously transforms into a defect free structure
with $n_y=30$ rows again (Fig.~\ref{fig2}). This value of $\Delta$ is to
be compared with that for the reverse transition from $30$ to $29$ rows
which we recall occurs at $\Delta_c \approx 2.0$. Thus, with the standard
Monte Carlo approach there is considerable hysteresis which precludes
the accurate location of the transition point. Clearly, therefore a
method is needed from which one can locate where the transition occurs
in equilibrium.
\begin{figure}
\includegraphics[scale=0.4, clip=true]{fig4a.eps}\\
\includegraphics[scale=0.4, clip=true]{fig4b.eps}\\
\includegraphics[scale=0.4, clip=true]{fig4c.eps}\\
\includegraphics[scale=0.4, clip=true]{fig4d.eps}\\
\caption{\label{fig4} Configurations with $N=3240$ particles and $n_y-2=28$ rows, but different configurations of the solitons. In the text, they are referenced as ``configuration nr.~1,~2,~3,~4'' from top to bottom. For a clear identification of the positions of the solitons, the method described in \cite{25} was used.}
\end{figure}
Similar hysteresis is observed if one starts out from the $29$ row
structure but increases the misfit beyond $\Delta =3$ (a case that has
not been studied previously). As Fig.~\ref{fig2} shows, a transition occurs
to structures with $n_y-2=28$ rows (at about $\Delta \approx 4.1$).
Unfortunately, there seem to be no unique candidates for stable
structures having $n_y-2=28$. Fig.~\ref{fig4} displays four candidate
structures that we have identified, each of which is at least metastable
on simulation timescales. Depending on which of these $28$ row
candidates one takes, the transition from $28$ to $29$ rows on reducing
the misfit occurs at anything between $\Delta=3.2$ and $3.75$. As
regards the nature of the candidate structures, in each case $2n_x=216$
extra particles have to be distributed across the system. If we again
keep the rows adjacent to the walls free of extra particles, the
particle number per inner row becomes $n'_x=n_x + 2 n_x/(n_y-4)\approx
n_x + 8.3$, i.e. is non-integer. If we kept two rows adjacent to the wall
rows free of extra particles, we would have $9$ extra particles per row,
and thus this structure has been tried (this is configuration number $1$
in Fig.~\ref{fig4}). Another structure was obtained if we place $4$ extra
particles in the rows directly adjacent to the walls and $8$ extra
particles in each of the $26$ inner rows (configuration number $2$). By
energy minimization of a somewhat disordered structure resulting from a
transition from $29$ to $28$ rows a structure was obtained which had $9$
solitons on one wall but only $8$ on the other wall (configuration
number $3$). Finally another configuration with $8$ solitons on each
wall (configuration number $4$) was found. Note that the configurations shown in Fig.~\ref{fig4}
are not the actual structures at $T=1.0$ but the corresponding ``inherent structures''
found from the actual structures by cooling to $T=0$, to clearly display where the solitons occur.
Clearly, it again is a
problem to (i) identify which of these $4$ configurations with $28$ rows
is the stable one (at $T=1.0$), and (ii) determine at which misfit the transition to the
structure with $29$ rows occurs. As we shall demonstrate below, both
problems can be elegantly dealt with by employing the phase switch Monte Carlo
method.
\section{Free energy based simulation methodologies to locate
transitions between imperfectly ordered crystal structures}
\subsection{Thermodynamic Integration}
The general strategy of thermodynamic integration is to consider a
Hamiltonian $\mathcal{H} (\lambda)$ that depends on a parameter
$\lambda$ that can be varied from a reference state (characterized by
$\lambda_0$) whose free energy is known, to the state of interest
$(\lambda_1)$, without encountering phase transitions. The free energy
difference $\Delta F$ can then be written as
\begin{equation} \label{eq4}
\Delta F= F (\lambda_1) - F (\lambda_0) = \int\limits_{\lambda_0}^{\lambda_1} d \lambda' \langle \partial \mathcal{H}(\lambda')
/\partial \lambda' \rangle_{\lambda'} \quad .
\end{equation}
For a dense disordered system (fluid or a solid containing defects),
Schilling and Schmid \cite{15,16} proposed to take as a reference state a
configuration chosen at random from a well equilibrated simulation of
the structure of interest, at values of the external control parameters
for which one wishes to determine the free energy. Particles can be
held rigidly in the reference configuration $\{\vec{r}_i \,^ {\rm ref}\}$
by means of a suitable external potentials. (We recall that a somewhat related
thermodynamic integration scheme for disordered systems known as the
``Tethered spheres method'' has already been proposed by Speedy
\cite{55a}.) When these external potentials act, the internal
interactions can be switched off. In practice, one can use the following
pinning potential $U_{\rm ref} (\lambda)$ to create the reference state,
where $r_{\rm cut}$ is a parameter discussed below.
\begin{equation} \label{eq5}
U_{\rm ref} (\lambda)= \lambda \sum\limits_i \phi (|\vec{r}_i - \vec{r}\;^{\rm ref}_i |/r_{\rm cut}) \quad {\rm with}\,
\phi\, (x)=x-1 \quad .
\end{equation}
Here it is to be understood that particle $i$ is only pinned by well $i$ at
$\vec{r}\;^{\rm ref}_{i}$, and not by other wells. However, identity swaps
need to be carried out to ensure the indistinguishability of particles.
The free energy of this non-interacting reference system then is
\begin{equation} \label{eq6}
F_{\rm ref} (\lambda) =\ln (N/V) -\ln [1+ (V_0/V) g_\phi (\beta\lambda)]\:,
\end{equation}
where $\beta=(k_B T)^{-1}$, $V_0$ (in $d=2$ dimensions) is $V_0= \pi r^2 _{\rm cut} $ and
\begin{eqnarray} \label{eq7}
&& g_\phi (a) = \frac{2}{\lambda^2} [\exp (a) - \sum\limits_{k=0}^2 e^k / k!]\:,\nonumber\\
\end{eqnarray}
for the choice of $\phi(x)$ written in Eq.~(\ref{eq5}).
Then intermediate models $\mathcal{H}(\lambda)$ to be used in Eq.~(\ref{eq4}) are chosen as
\begin{equation} \label{eq8}
\mathcal{H}' (\lambda) = \mathcal{H}_{\rm int} + U_{\rm ref} (\lambda) \quad ,
\end{equation}
where $\mathcal{H}_{\rm int}$ describes interactions in the system,
which then are switched on (if necessary, in several steps). The free
energy contribution of switching on these interactions can easily be
determined by a Monte Carlo simulation which includes a move that
switches the interactions on and off. The logarithm of the ratio of how
many times the states with and without interactions were visited gives
the free energy contribution. The free energy difference between the
intermediate model where particle interactions are turned on and
potential wells are also turned on, and the target system with particle
interactions but without potential wells, then is computed by
thermodynamic integration, for which
\begin{equation} \label{eq9}
\langle \partial \mathcal{H}_{\rm ref} (\lambda) / \partial \lambda \rangle = \langle \sum_i \phi (
|\vec{r}_i - \vec{r}_i\;^{\rm ref}|/ r_{\rm cut}) \rangle
\end{equation}
needs to be sampled \cite{15,16}. This method has been tested for hard spheres \cite{15,16},
including also systems confined by walls from which wall excess free energies could be
sampled \cite{56}.
\subsection{Phase Switch Monte Carlo}
The phase switch method \cite{17,18,19,20,21,21a} computes directly the
relative probabilities of two phases, by switching between them and
recording the ratio of the simulation time spent in each. This ratio
directly yields their free energy difference $\Delta F$ via $\Delta F=
\ln(A^{(1)}/A^{(2)})$. Here $A^{(1)}$ and $A^{(2)}$ are the times spent
in the respective phases which are proportional to the statistical
weight of each phase \cite{9}.
\begin{figure}
\includegraphics[scale=0.32, clip=true]{fig5.eps}
\caption{\label{fig5} Schematic comparison of (a) the standard method
for linking phases via a sampling path and (b) The phase switch method.
The blobs represent the set of values of some macroscopic property (eg
order parameter or energy) associated with configurations belonging to
two distinct phases $(\alpha=1,2)$. These pure phase states (having high
probability) are separated by a ``deep valley'' in the free energy
landscape corresponding to interfacial states having a very low
probability. (a) In the standard strategy one uses extended sampling to
negotiate the valley, by climbing down into it from one side and climbing
up out of it on the other. (b) The idea of phase switch Monte Carlo is
to ``jump over the valley''.}
\end{figure}
The power of the phase switch method derives from its ability to leap
directly from configurations of one pure phase to those of another pure
phase (Fig.~\ref{fig5}), avoiding the mixed phase states which -- when
one or both phases are crystalline -- can be computationally problematic
(see appendix A). The leap is implemented as a suitable global Monte
Carlo move. One starts out by specifying for each of the two phases of
interest (labeled by index $\alpha=1,2$), a reference configuration. This
can be expressed as a set of $i=1\ldots N$ particle positions $\{
\vec{R}_i^{\,(\alpha)}\}$. Note that the specific choice of a reference
configuration for phase $\alpha$ does not matter (at least in principle, see
Appendix), it need only be a member of the set of pure phase
configurations that ``belong'' to phase $\alpha$. Thus for example in the
present case, a suitable reference configuration for the $n=30$ row
defect-free structure could simply be a typical configuration chosen
from a simulation run on this structure. However, it could equally be a
configuration in which all particles are at the lattice sites of this
structure.
Given the two reference configurations, one can express the
position vectors $\vec{r}_i^{\,(\alpha)}$ of each particle $i$ in phase $\alpha$ as
\begin{equation}
\vec{r}_i^{\,(\alpha)}= \vec{R}_i^{(\alpha)} + \vec{u}_i\:.
\end{equation}
where $\{\vec{u}_i\}$ is a set of displacement vectors which measure the
deviation of each particle from the reference site to which it is
nominally associated. Note that while there is a separate reference
configuration for each phase, the single set of displacements is
common to both phases.
Let us suppose the simulation is currently in phase $\alpha=1$. Now the phase
switch idea is to a map the current configuration $\{\vec{r}_i^{\,(1)}\}$ of this
phase on to a configuration of phase $\alpha=2$ by switching the sets of
reference sites from $\{\vec{R}_i^{\,(1)}\}$ to $\{\vec{R}_i^{\,(2)}\}$ but
keeping the set of displacements $\{\vec{u}_i\}$ {\em fixed}. This
switch can be incorporated in a global Monte Carlo move. Of
course, in general the set displacements that are typical for phase
$\alpha=1$ will not be typical displacements for phase $\alpha=2$.
As a consequence, in a naive implementation such a global move will
almost always be rejected by the Monte Carlo lottery. This problem is
circumvented by employing extended sampling methods \cite{9,10,56a} that
create a bias which enhances the occurrence of displacements
$\{\vec{u}_i\}$ for which the switch operation does have a sufficiently
high Monte Carlo acceptance probability. Such states are called
``gateway states'' \cite{17,18,19,20,21}: crucially, they do not need to
be specified beforehand - the system autonomously guides itself to them
in the course of the biased sampling.
In practice, the bias is administered with respect to an ``order
parameter'' $M$ whose instantaneous value is closely related to the
energy cost of implementing the phase switch. One then introduces a
weight function $\eta(M)$ into the sampling of the effective Hamiltonian
which enhances the probability of the system sampling configurations for
which the energy cost of the phase switch is low, thereby increasing the
switch acceptance rate. Of course, the weight function $\eta(M)$ to be
used is not known beforehand, and thus needs to be iteratively
constructed in the course of the Monte Carlo sampling. One has a choice
of ways of doing so: we have used the transition matrix Monte Carlo
method \cite{56a,57,58} (see also the Appendix for implementation
details). Alternative methods such as Wang-Landau sampling \cite{59} or
successive umbrella sampling \cite{73} could also be applied.
Once a suitable form for the weight function $\eta(M)$ has been found, a
long Monte Carlo run is performed, in the course of which both phases
are visited many times. The statistics of the switching between phases
is monitored by accumulating the histogram of $M$, which (as in all
extended sampling methods) is corrected for the imposed bias at the end
of the simulation. Doing so yields an estimate of the true equilibrium
distribution $P(M)$, which in general exhibits a double peaked form (one
peak for each phase). The free energy difference between the two phases
is simply the logarithm of the ratio of the peak weights as described at
the start of this subsection.
Of course, the above description was only intended to outline the
phase switch strategy; more extensive implementation details are given
in the appendix.
\section{Results}
\subsection{Free energy differences and computational efficiency}
Fig.~\ref{fig6} shows the absolute free energies in the NVT ensemble for
the phase with 30 rows (and no defects) and the phase with 29 rows and
the ``soliton staircases'' (Fig.~\ref{fig3}b) as a function of the
misfit $\Delta$, as obtained from the thermodynamic integration method
(Sec. III.1). One sees that these free energies are very large (note the
ordinate scale) and vary rather strongly with $\Delta$. However, the
free energy curves with these two structures are barely distinct from
each other, and hence a very substantial computational effort is needed
to locate, with meaningful accuracy, the intersection point marking the
equilibrium transition between $n=30$ and $n=29$ rows.
\begin{figure}
\includegraphics[scale=0.32, clip=true]{fig6.eps}
\caption{\label{fig6}Absolute free energy $F$ of systems of $N=3240$
particles interacting with the potential given in Eq.~(\ref{eq1}) in $L
\times D$ geometry with $L=108 a$, $a$ being the lattice spacing, and
periodic boundaries in $x$-direction, confined by two rows of fixed
particles on either side in $y$-direction (Fig.~\ref{fig1}, as a
function of the misfit $\Delta$ \(Eq.~(\ref{eq3})\). Two structures are
compared:(i) a (compressed) triangular lattice with $n_y=30$ rows
containing $n_x=108$ particles per row; (ii) a lattice with $n_y=29$
rows and corresponding soliton staircases (Fig.~\ref{fig3}b).}
\end{figure}
\begin{figure}
\includegraphics[scale=0.32, clip=true]{fig7.eps}
\caption{\label{fig7} Free energy differences between structures with 29 and 30 rows plotted versus the misfit $\Delta$. Both results obtained from thermodynamic integration and from the phase switch method are shown, as indicated.}
\end{figure}
Fig.~\ref{fig7} plots the free energy difference $\Delta F$ versus the
misfit, comparing the results from the thermodynamic integration method
(points with error bars) with the results from the phase switch method,
and focusing on the region near the transition. One can see that within
the errors the results of both methods agree very well with each other,
although for the thermodynamic integration method the error is at least
an order of magnitude larger than that of phase switch. We note that the
predicted equilibrium value of the misfit at the transition point ($\Delta_t
\approx 1.7)$ falls well within the hysteresis loop of Fig.~\ref{fig2}.
Since the absolute free energies are of the order of 20000 (for our
system with $N=3240$ particles) but, in the region of interest, free
energy differences are of order $\pm 40$ only, we have that the relative
error $\delta F/F$ is of order $1/500$. Thus for thermodynamic
integration, it would be difficult to bring the error bars down further
in Fig.~\ref{fig7}. The error bars for the phase switch simulation were
computed from the results of four independent runs for each value of the
misfit, and are hardly visible on the scale of Fig.~\ref{fig7}.
In addition to this significant difference with respect to the size of
the statistical errors, phase switch Monte Carlo also outperformed the
thermodynamic integration method with respect to the necessary
investment of computer resources. In order to obtain a suitable weight
function for our system, at a certain value of the misfit, we let the
simulation run for about 15 million steps (each step consisting of one
sweep of local moves and one attempt to switch the phases). On the ZDV
cluster of the University of Mainz, this takes about $4.5$ days on a
single core (though in hindsight we could have got away with a less
smooth weight function, further reducing the computing time of this
step). Having determined the weight function, we initiated four
production runs for every value of the misfit. These runs needed again
10 million steps each (i.e. about 3 days each) in order to perform a
sufficient number of phase switches to yield results of the desired
precision. Overall, then, computing each point of the free energy difference
curve of Fig.~\ref{fig7} by phase switch took about $16.5$ days of CPU time.
In contrast to this, the thermodynamic integration method required a
calculation not only of the free energy difference in which we are
interested, but of the free energy difference along the path of the
thermodynamic integration, gradually switching off the wells of
attraction used there, and of the free energy difference between the
state where the particle interactions were turned on and the state where
they were turned off. This needs to be done for both phases separately.
It is therefore not surprising, that considerably more CPU time was
needed: roughly $250$ days of CPU time were invested for each phase and
for each value of the misfit to obtain the absolute free energy (again
converting units to a single core). Thus, each of the 12 values of free
energy differences needed for Fig.~\ref{fig7} required 500 days (rather
than $16.5$ days), i.e. a factor of $30$ more computational effort!
However, if we were to bring the statistical errors of the
thermodynamic integration method a factor of 10 down (to make it
comparable to the phase switch method), we would need another factor of
100 in computer time; the benefit of using the (clearly much more
powerful) phase switch approach hence amounts to a gain of the
order of 10$^3$ in computational resources! Of course, this is no
surprise when we remember that the free energy differences of interest
are only of the order of (1/500) of the total free energies for the
present model system.
\subsection{Ensemble inequivalence}
\begin{figure}
\includegraphics[scale=0.25, clip=true]{fig8.eps}
\caption{\label{fig8} Schematic description of phase transitions in thin films of thickness $D$ in the conjugate NpT
(left) and NVT (right) ensembles, for the case of a vapor to liquid transition (a) and the present transition where the number of rows is reduced $(n \rightarrow n -1)$ when either the (normal) pressure $p$ increases (left) or the thickness decreases (right). Note that in the latter case two-phase coexistence is possible for the vapor-liquid transition, but
not for the transition where the number of rows parallel to the boundaries change. For further explanations cf text.}
\end{figure}
We turn now to a discussion of a puzzling aspect of the physics, namely
the fact that we treat here a first-order structural phase transition
obtained by variation of the distance $D$ between the walls formed by
the rigidly fixed particles, i.e. an {\it extensive} rather than an {\it
intensive} thermodynamic variable. If we were concerned with the study
of a vapor to liquid transition of a fluid in such a geometry, the
proper way to locate a discontinuous transition is the variation of the
intensive variable thermodynamically conjugate to $D$, which is the
normal pressure $p_N$ (force per area acting on the walls; in the
following the index $N$ will be omitted. Of course, at fixed lateral
dimensions $L$ a variation of $D$ is equivalent to a variation of the
volume $V$).
\begin{figure}[h!]
\includegraphics[scale=0.3, clip=true]{fig9a.eps}\\
\includegraphics[scale=0.3, clip=true]{fig9b.eps}\\
\includegraphics[scale=0.3, clip=true]{fig9c.eps}
\caption{\label{fig9} a) Free energy difference $\Delta F$ for the transition from $n=30$ to $n=29$ rows as a
function of pressure. (b) The distribution of the internal energy difference between the two phases $p(E_{30 rows} -E_{29 rows})$ at fixed $\{\vec{u}\}$. Curves for $4$ pressures near and at the transition pressure $p_t=22.146 \pm 0.015$ are shown, as generated via histogram reweighting. The simulation was run at a pressure of $p=22.13$. (c) System length $D$ as a function of pressure. Clearly, the curve for the stable phase exhibits a jump at the transition pressure. Statistical errors are smaller than the symbol sizes.}
\end{figure}
To fix ideas, we remind the reader about this classical vapor-liquid
problem in Fig.~\ref{fig8}a): In the NpT ensemble, we would have a jump
in volume $V=LD$ from $V_v=LD_v$ (density of the vapor $\rho_v=N/V_v)$
to $V_\ell=LD_\ell$ (density of the liquid $\rho_\ell=N/V_\ell)$ at the
transition pressure $p_t$. If we work in the conjugate NVT ensemble, of
course, the behavior simply follows from a Legendre transform, the
volume jump from $V_v$ to $V_\ell$ translates into a horizontal plateau
at $p=p_t$, and any state of this plateau is a situation of two-phase
coexistence, as schematically indicated in Fig.~\ref{fig8}a).
Of course, it is also possible to consider the present transition
between a state of $n$ rows to $n-1$ rows in the NpT ensemble
(Fig.~\ref{fig8}b and Fig.~\ref{fig9}c). Then it is clear that the transition will show up as
a jump in the thickness $D$ from $D_n(=na_n)$ to $D_{n-1}\, (=(n-1)
a_{n-1})$, where $a_n$, $a_{n-1}$ are the (average) distances between
the lattice rows (or lattice planes, in three dimensional films,
respectively). The corresponding phases of the $n$-layer state and
$(n-1)$ layer state are indicated below the isotherm in the $(p-D)$ plane
schematically.
However, one simply cannot construct a state of two-phase coexistence
out of these two ``pure phases'' at a value of $D$ intermediate between
$D_{n-1}$ and $D_n$: locally the $n$-layer state requires a thickness
$D_n$, the $(n-1)$ layer state a thickness $D_{n-1}$, so one would have
to ``break'' the walls. Of course, it is not just sufficient to have a
state with $n$ layers separated by a grain boundary from a state with
$(n-1)$ layers at the same value of $D$: these domains are {\it not} the
coexisting pure phases in the NpT ensemble!
So the phase coexistence drawn (horizontal broken curve) in
Fig.~\ref{fig8}b) is unphysical, it requires a state where the
constraining walls were broken. Requesting the integrity of the walls is
a global constraint which makes phase coexistence in the standard sense
impossible for the present transitions! Thus, the rule that the
different ensembles of statistical mechanics yield equivalent results in
the thermodynamic limit is not true for the present system; in the
transition region $D_{n-1} < D < D_n$ the NVT ensemble and the NpT
ensemble are {\it not equivalent}.
Actually this is not the first time that such an ensemble inequivalence
has been pointed out. A case much discussed in the literature is the
``escape transition'' of a single polymer chain of $N$ beads grafted at
a planar surface underneath a piston held at a distance $D$ above the
surface to compress the polymer \cite{61,62,63,64,65,66,67}. For pressures
$p<p_t$ (where the piston is at distance $D_{t,1}$) the chain is
completely confined underneath the piston (which has the cross section
of a circle in the directions parallel to the surface) while for $p >
p_t$ the chain is (partially) escaped into the region outside of where
the piston acts (the piston distance at $p_T$ jumps to a smaller value
$D_{t,2}$). When we use instead $D$ as the control variable, again a
sharp transition occurs (for $N \rightarrow \infty$) at some
intermediate value $D_t$ $(D_{t,2}< D_t <D_{t,1})$, since obviously it is
simply inconceivable to have within a single chain phase coexistence
between states ``partially escaped'' and ``fully confined'', since these
states are only defined via a global description of the whole polymer
chain.
Another case where transitions of the number $n$ of layers in layered
structures in thin films occurs is the confinement of symmetric block
copolymer melts (which may form a lamellar mesophase of period
$\lambda_0$ in the bulk) in thin films between identical walls
\cite{68,69,70,71}. When then the thickness $D$ of such films is varied,
one observes experimentally discontinuous transitions in the number $n$
of lamellae parallel to the film \cite{69,70}. However, when one
considers block copolymer films on a substrate and does not impose the
constraint of a uniform thickness but rather allows the upper surface to
be free, then indeed mixed phase configurations of a region where $n-1$
layers occur (and take a thickness $D_{n-1})$ and of a region where $n$
layers occur (and take a thickness $D_n$) are conceivable \cite{71} and
have been observed, see e.g. \cite{72}. In summary of these remarks, we
note that it is not uncommon that global geometric constraints may
destroy the possibility of phase coexistence.
In view of the above discussion, it is of interest also in the present
case to investigate the use of the (normal) pressure $p$ (instead of the
strip width $D$) as the control variable. Taking, in the spirit of the
general remarks on the phase switch method, the appropriate phase switch
energy cost as an order parameter $M$, we can sample the probability
distribution function $p(M)$ which exhibits two well separated peaks of generally different
weights. These peaks are even more clearly visible in the distribution of the energy difference $p(E_{30 rows} -E_{29 rows})$ at fixed $\{\vec{u}\}$ as the order parameter $M$ is related to this energy difference via a logarithmic function (cf. eq.~\ref{def_M}). The transition pressure $p_t$ is that for which the peaks have
equal weight (Fig.~\ref{fig9}) and can be determined accurately via histogram
reweighting. From this we estimate that $p_t=22.146
\pm 0.015$. At the transition, the measured misfit $\Delta$ jumps from
$\Delta_1=1.913 \pm 0.043$ (for $n=30$) to $\Delta_2=1.503 \pm 0.046$
(for $n=29$). Interestingly, the misfit where the transition in the NVT
ensemble occurs ($\Delta_t \approx 1.71)$ is just the average of these
two values.
\subsection{Comparison of competing candidate stable structures}
\begin{figure}[h!]
\includegraphics[scale=0.32, clip=true]{fig10.eps}
\caption{\label{fig10} Free energy differences between various structures with $n=28$ rows and the structure with $n=29$ plotted vs. the misfit $\Delta$. As configurations nr. 2 and nr. 4 turned out to be the same, their free energy curves fall on top of each other.}
\end{figure}
Returning again to the NVT ensemble, we now consider the transition from
states with 29 layers to states with 28 layers. We recall
(Fig.~\ref{fig4}) that several different candidate structures do exist,
and it is not at all clear {\em a-priori}, which of them should be
favored. Again, the phase switch Monte Carlo is a convenient tool to
solve such a problem: we utilize reference states from all four of the
candidate structures having $n=28$ (as shown in Fig.~\ref{fig4}) and
calculate the free energy difference $\Delta F$ between the (unique)
structure with $n=29$ and these four candidates.
The results (Fig.~\ref{fig10}) clearly show that configurations number
$1$ and number $3$ are metastable, because they have distinctly higher
free energy differences throughout the range of $\Delta$ than
configurations number $2$ and $4$ which practically coincide. In fact,
this coincidence between the free energies of configurations nr. $2$ and $4$ is not accidental: a closer evaluation of
their time evolution shows that they transform into each other via
sequences of ``easy'' local moves, and although the instantaneous
snapshot pictures reproduced in Fig.~\ref{fig4} were different, they do
not belong to different phases in a thermodynamic sense.
It is also interesting to note that the conclusion that structure number
2 is the stable one would not have been obtained by a simply comparison
of the internal energies of the four structures: indeed configuration
number 2 has the highest energy of all four structures.
Thus, entropy matters in soft crystals, such as those studied here.
\section{Concluding remarks}
The principle findings of our study are two-fold: (i) We have
performed a thorough test of the suitability of the phase switch Monte
Carlo method for the task of determining the relative stability of
imperfectly ordered structures of typical soft-matter systems, where one
must deal with systems which have at least one very large linear
dimension. For such a test, it is crucial to provide full information on
the model that is studied, and to give a careful description of the
method and its implementation. Moreover we have studied precisely the
same model system by a thermodynamic integration method thereby allowing
the first like-for-like comparison between the two approaches. We find
that the results from both methods are compatible, but the accuracy that
can be achieved using phase switch MC is at least an order of magnitude
better (Fig.~\ref{fig7}), despite requiring a factor of $30$ less
computational time.
The reasons for this efficiency gain can be appreciated from a glance at
Fig.~\ref{fig6}: the absolute free energies of our system of $3240$
particles vary from about $22000$ to $24000$ (in suitably scaled units),
for a misfit parameter $\Delta$ varying from $1$ to $2$, while the free
energy difference between the two states that we wish to compare vary
only from $-60$ to $+60$ in the same range. These numbers illustrate
vividly the basic concept of phase switch Monte Carlo: one does better
in focusing directly on the small free energy difference between the
states that one wishes to compare, rather than extracting them
indirectly by subtracting two measurements of large absolute free
energies. Thus (in the present context at least) phase switch Monte
Carlo seems a much more powerful approach than thermodynamic
integration. In fact, if one were to try to bring the errors of the
thermodynamic integration method down by an order of magnitude -- to make
the error bars of both methods in Fig.~\ref{fig7} comparable -- one would
have to invest a factor of 3000 more computational time. We feel that
the case of relatively small free energy differences between competing
phases and/or structures is rather typical for soft matter systems.
Indeed for many soft matter systems, such as block copolymer mesophases,
the relative magnitude of free energy differences is much less than the
factor of about $1/500$ encountered here, and hence such problems could
never be tackled successfully with thermodynamic integration methods
since the computational effort to reach the requisite accuracy would be
prohibitive.
The first problem to which phase switch Monte Carlo was applied (in the
form of the "Lattice-switch" method), evaluated the free energy
difference of perfectly ordered face-centered cubic and hexagonal close
packed crystals. Such an application might be regarded as a somewhat
special case due to the perfect long-range order in these defect-free
crystals. However, the present work shows that the method can equally be
applied to imperfectly ordered crystals. Here, due to the confinement by
structured walls together with a misfit between the distance between the
walls and the appropriate multiple of the distance between the lattice
rows, somewhat irregular long range defect structures form along the
walls (``soliton staircase''). Additionally several similarly
ill-crystallized structures can present themselves as candidates for the
optimal structure (Fig.~\ref{fig4}). It would be absolutely impossible
to identify which is the equilibrium structure and which structures are
only metastable without the phase switch Monte Carlo method
(Fig.~\ref{fig10}).
We note that the model system that we have chosen to study
(Fig.~\ref{fig1}) could also be experimentally realized in colloidal
dispersions, though with some effort: colloids coated with polymer
brushes experience a short ranged, almost hard-sphere-like, repulsive
effective potential, and bringing them to an interface where water is on
top and air is below, rather perfect two-dimensional crystals with
triangular lattice structure form. Interference of strong laser fields
can be used to create a periodic confining potential, through which the
misfit and thus the crystal structure can be manipulated. We hope that
our study will solicit some corresponding experimental studies to show
that the proposed transitions in the number of rows in these crystalline
strips actually occur.
(ii) Our second main finding is that this type of system has an
interesting physical property, namely the inequivalence between
conjugate ensembles of statistical mechanics. When we fix the distance
$D$ between the confining ``walls'', the total particle number $N$ and
the total (two-dimensional) ``volume'' $V$ of the system, we realize the
NVT ensemble. When one studies first order transitions in the bulk using
such an ensemble containing two extensive variables ($N$, $V$), a first
order transition normally shows up as a two-phase coexistence region
(e.g., at fixed $N$ the two-phase coexistence extends from $V_I$ to
$V_{II}$). However, here such a two-phase coexistence is not possible
(Fig.~\ref{fig8}), and thus one has the unusual behaviour that at the
equilibrium in the ``constant $D$''-ensemble the conjugate intensive
variable (the normal pressure $p_N$, as well as the stress $\sigma$, cf.
Fig.~\ref{fig2}) exhibit jumps (in Fig.~\ref{fig2}, we display the
hysteresis loops, but the positions of the jumps in equilibrium can be
inferred from $\Delta F=0$ in Figs.~\ref{fig7} and \ref{fig10},
respectively). When we use a ``constant $p$''-ensemble (which is
physically reasonable if the confinement of the crystal is effected
mechanically in a Surface Force Apparatus), it is the ``volume'' (i.e.,
the distance between the walls $D$) which jumps from $D_I$ to $D_{II}$
at a well-defined transition pressure, cf. Figs.~\ref{fig8},~\ref{fig9}.
One should not confuse this ensemble inequivalence with the well-known
ensemble inequivalence between NVT and NpT ensembles in systems where
$N$ is finite: in the latter case, the ensemble inequivalence is
dominated by interfacial contributions (in the NVT-ensemble, when $V_I <
V < V_{II}$, the system is in a two-phase configuration, as suggested
for $V \rightarrow \infty$ by the ``lever rule'', but for finite $V$ the
relative contribution due to the interface between the coexisting phases
dominate the finite size effects). But for $V \rightarrow \infty$ these
interfacial effects become negligible, the properties in the two
conjugate ensembles are just related by the appropriate Legendre
transformation. This equivalence between the ensembles holds also for
liquid-vapor or liquid-liquid unmixing under confinement in a thin film
geometry: when $D$ is finite and the particle number $N \rightarrow
\infty$, i.e. the lateral linear dimensions become macroscopic, we still
have ordinary two-phase coexistence in the thin films (cf.
Fig.~\ref{fig8}). The ensemble inequivalence in the present system
arises from the lack of commensurability between the thickness $D$ of
the slit and the appropriate multiple of the lattice distance. At a
transition pressure $p_t$ in the NpT ensemble we inevitably have
different distances $D_I$, $D_{II}$ between the walls for the two phases
$I$, $II$. Thus, they cannot coexist for any uniform value of $D$.
Similar phenomena (where the number of layers of a layered lamellar
structure confined between walls exhibits jump discontinuities when $D$
is varied) are already known, both experimentally and theoretically, for
block copolymer mesophases, but the aspect of ensemble inequivalence has
not been addressed, to our knowledge, in these systems studied here.
\section{Acknowledgements}
One of us (D.W.) acknowledges support from the Deutsche
Forschungsgemeinschaft (DFG) under grant number TR6/C4 and from the
Graduate School of Excellence ``Material Science in Mainz (MAINZ)''. She
is also grateful to the Department of Physics, University of Bath
(UK), for its hospitality during an extended research stay under the
auspices of the visiting postgraduate scholar scheme. We thank P.
Virnau, T. Schilling, F. Schmid and I.M. Snook for helpful discussions
and advice.
\clearpage
\section{Appendix}
|
{
"timestamp": "2012-03-09T02:02:54",
"yymm": "1203",
"arxiv_id": "1203.1794",
"language": "en",
"url": "https://arxiv.org/abs/1203.1794"
}
|
\section*{Introduction}
Let $X$ be an algebraic variety of general type, over the complex field.
The dominant rational maps of finite degree
$X \dasharrow Y$ to varieties of general type,
up to birational isomorphisms $Y \dasharrow Y'$,
form a finite set. We call this the {\em finiteness theorem for rational maps
on a variety of general type}. The proof follows from the approach of
Maehara \cite{M} joined with some recent advances in the theory of
pluricanonical maps, due to Hacon and McKernan \cite{HK}
and to Takayama \cite{Tak}, \cite{Tak2}.
In our paper \cite{GP}, motivated by the wish of some effective estimate
for the finite number of maps in the theorem, we provided some update
and refinement in the treatment of the subject.
We brought the rigidity theorem to a general form,
avoiding certain technical restrictions, we pointed out the role of
the canonical volume ${\rm vol}(K_X)$ in bounding the
rational maps in the finiteness theorem, and we proposed
a new argument leading to a refined version of the theorem.
However, something still not satisfactory was the use of a certain bunch
of subvarieties of Chow varieties as a parameter space for rational maps,
as in Maehara's approach is too. The most natural and simple parameter space
should be the space of linear projections in a suitable projective space,
already appearing for instance in the work
of Kobayashi and Ochiai \cite{KO}.
In the present paper we are able to replace the Chow parametrization
with the natural parametrization, and this leads to some new insight
into the geometry of the finiteness theorem. The main result concerns the
structure of the special birational equivalence classes of maps viewed
as unions of connected components of a certain space of linear rational maps,
see Theorem \ref{connectedcomponent}.
This has as an immediate consequence a better refined finiteness theorem,
see Theorem \ref{finiteness}.
\bigskip
\small \noindent {\em Acknowledgements.}
The first author is partially supported by:
Finanziamento Ricerca di Base 2008 Univ. Perugia.
The second author is partially supported by:
1) INdAM (GNSAGA);
2) FAR 2010 (PV):{\em ``Variet\`{a} algebriche, calcolo
algebrico, grafi orientati e topologici"}.
\normalsize
\section{Preliminary material}
\subsection*{a. Results on pluricanonical maps}
A recent achievement in the theory of pluricanonical maps is the following
theorem of uniform pluricanonical birational embedding, due to
Hacon and McKernan \cite{HK} and to Takayama \cite{Tak}.
\begin{thm} \label{HKT}
For any dimension $n$ there is some positive integer $r_n$ such that: for
every $n$-dimensional variety $V$ of general type the multicanonical divisor
$r_nK_V$ defines a birational embedding $V \dashrightarrow V' \subset
\mathbb P^M$.
\end{thm}
A basic tool is the canonical volume of a variety,
the invariant arising in the
asymptotic theory of divisors, see Lazarsfeld's book \cite{L}.
In terms of the canonical volume we have a bound
\begin{equation} \label{degvol}
\deg V' \leq {\rm vol} (r_{n}K_{V}),
\end{equation}
see \cite{HK}, Lemma 2.2.
Moreover from elementary geometry we have a bound
\begin{equation} \label{embdim}
M \leq \deg V' +n-1.
\end{equation}
Note that the embedded variety $V'$ needs not be smooth.
Intimately related to the theorem above is the following
result, proved in \cite{HK} and in \cite{Tak}.
\begin{thm} \label{HK}
For any dimension $n$ there is some positive number $\epsilon_n$
such that every $n$-dimensional variety $V$ of general type has
${\rm vol} (K_{V}) \geq \epsilon_n$.
\end{thm}
For instance, concerning the minimum $r_n$ we know from the classical theory
that $r_1=3$ and $r_2=5$, and a recent result is that $r_3 \leq 73$, while
concerning the maximum $\epsilon_n$
it is clear that $\epsilon_1 = 2$ and $\epsilon_2 = 1$ and a recent
result is $\epsilon_3 \geq 1/2660$, see J. A. Chen and M. Chen \cite{CC}.
Note that \cite{HK} and \cite{Tak} do not give
explicit bounds for $r_n$ and $\epsilon_n$ in the theorems above.
\subsection*{b. Bounds for the degree of a rational map}
Let $f: X \dasharrow Y$ be a rational map of finite degree
between varieties of general type.
Because of Theorem \ref{HKT}, taking the $r_n$-canonical
birational models $X'$ and $Y'$ in $\mathbb P^M$
(note that $Y'$ lies within the embedding space of $X'$),
the map $f$ is identified with a {\em linear rational map} $X' \dasharrow Y'$,
a rational map which is the restriction of a linear projection
$\mathbb P^M \dasharrow \mathbb P^M$.
For a linear map of finite degree the inequality
$\deg f \, \deg Y' \leq \deg X'$ holds.
Using (\ref{degvol}) it follows that
\begin{equation} \label{deg1}
\deg f \leq \deg X' \leq (r_n)^n\, {\rm vol} (K_X).
\end{equation}
A more precise estimate is as follows. For any rational map of finite degree
the inequality $\deg f \; {\rm vol} (K_Y) \leq {\rm vol} (K_X)$ holds,
see \cite{GP}, Proposition 3.2. Using Theorem \ref{HK} it follows that
\begin{equation} \label{deg2}
\deg f \leq \dfrac{1}{\epsilon_n}\, {\rm vol} (K_X).
\end{equation}
This bound is sharp for curves, and in this case it reduces
to the usual bound from the Hurwitz formula.
\subsection*{c. Families of rational maps}
Let $T$ be a smooth variety.
If $X \rightarrow T$ is a relative scheme over $T$,
we denote by $X(t)$ the scheme fibre over $t$, and
by $X_t$ the associated reduced scheme.
A {\em family of varieties}, parametrized by a smooth variety $T$,
is a surjective morphism $X \rightarrow T$, with $X$ a variety,
such that every scheme fibre $X(t)$ is: $(i)$ irreducible, $(ii)$ generically smooth
(in order to be assigned multiplicity one in the associated algebraic cycle,
see Fulton \cite{F}, Chap. 10),
and $(iii)$ of dimension equal to the relative dimension of $X$ over $T$, of course.
When the structure morphism is projective or smooth,
we speak of a family of projective varieties or a family of smooth varieties.
A {\em family of rational maps} is the datum of
a family of varieties $X \rightarrow T$ and
a relative scheme $X' \rightarrow T$, over the same smooth variety $T$,
and a rational map $f: X \dasharrow X'$, commuting with the structural
projections, which for every $t \in T$
restricts to a rational map $f_t: X_t \dasharrow X'_t$.
\subsection*{d. The rigidity theorem}
A family of rational maps {\em on a fixed variety} $X$ is the datum of a
relative scheme $Y \rightarrow T$, with $T$ smooth, and a rational map
$$f: X\times T \dasharrow Y$$
which is a family of rational maps $f_t: X \dasharrow Y_t$
in the sense of the previous definition.
A {\em trivial family} is one which is obtained as follows.
Let $h: X \dasharrow U$ be a rational map and
let $g: T \times U \dasharrow Y$ be a birational isomorphism
which is a family of birational isomorphisms $g_t: U \dasharrow Y_t$.
Then the composite map
$$T \times X \overset{1 \times h}{\dasharrow} T \times U
\overset{g}{\dasharrow} Y$$
is a trivial family, because
all maps $g_t \circ h$ are birationally equivalent.
Recall that two dominant rational maps
$f: X \dasharrow Y$ and $f': X \dasharrow Y'$,
defined on the same variety, are {\em birationally equivalent} if there is
a birational isomorphism $g:Y \dasharrow Y'$ such that $f' = g \circ f$.
For projective varieties of general type and
dominant rational maps of finite degree
there are results of rigidity.
\begin{thm} \label{rigidity}
Let $X$ be a smooth projective variety of general type.
Let $T$ be a smooth variety, let $Y \rightarrow T$ be a family of
smooth projective varieties of general type, and let
$f: X\times T \dasharrow Y$ be a family of rational maps of finite degree.
Then $f$ is a trivial family, so all maps $f_t$ are birationally equivalent.
\end{thm}
The rigidity theorem above was proved by Maehara \cite{M}
with some technical restrictions, and has been brought to the
present form in our previous paper \cite{GP}, Theorem 2.1.
More generally, if the family of image varieties is not known
to be a smooth family, one has the following.
\begin{cor} \label{weakrigidity}
Let $X$ be a projective variety of general type.
Let $T$ be a smooth variety, let $Y \rightarrow T$ be a family of
projective varieties of general type, and let
$f: X\times T \dasharrow Y$ be a family of rational maps of finite degree.
There is a nonempty open subset $T'$ of $T$ such that the restriction
$f|_{T'}: X\times T' \dasharrow Y|_{T'}$ is a trivial family.
\end{cor}
\section{Graphs and images in a family of maps}
Let $f: X \dasharrow X'$ be a family of rational maps
parametrized by a smooth variety $T$, as in \S 1.c.
Consider the relative product $X \times_{T} X'$
and call $p$ and $p'$ the projections to $X$ and $X'$.
{\em Assume now that $X \rightarrow T$ is a projective morphism}.
Thus $p'$ is a closed map. Then define: \medskip
\begin{tabular}{rl}
$\Gamma$ & the closed graph of $f$ in $X \times_{T} X'$, \\
$Y$ & the closed image of $X$ in $X'$, \\
$C$ & any closed subscheme of $X$ such that
$X \smallsetminus C \rightarrow T$ is surjective \\
& and $f$ is a regular map $X \smallsetminus C \rightarrow Y$, \\
$E$ & the inverse image of $C$ in $\Gamma$.
\end{tabular}
\medskip
\noindent Note that $p'(\Gamma) = Y$, as $p'$ is a closed map.
A natural question is whether $\Gamma \rightarrow T$
is the family of closed graphs for the given family of maps,
more precisely: whether $\Gamma \rightarrow T$ is a family of varieties,
as in \S 1.c, and every reduced fibre $\Gamma_t$
coincides with the {\em closed graph} $\Gamma(f_t)$.
A related question is whether $Y \rightarrow T$
is the family of closed images $\overline{f_t(X_t)}$, that is: whether
$Y \rightarrow T$ is a family of varieties and
every reduced fibre $Y_t$ coincides with the {\em closed image} $\overline{f_t(X_t)}$.
The following equality of reduced schemes holds:
$$\Gamma_t = \Gamma(f_t) \cup E_t$$
and from this, applying $p'$, a description of $Y_t$ follows.
\begin{prop} \label{familyofgraphs}
In the setting above, assume that $T$ is a smooth curve.
$(1)$ There is a nonempty open subset $T'$ of $T$ such that
$\Gamma|_{T'} \rightarrow T'$ is the family of closed graphs
for the restricted family $f|_{T'}$.
$(2)$ There is a nonempty open subset $T''$ of $T'$ such that moreover
$Y|_{T''} \rightarrow T''$ is the family of closed images
for the family $f|_{T''}$.
\end{prop}
\begin{proof}
We start with an easy remark.
Let $V \rightarrow T$ be a surjective morphism of varieties,
with irreducible fibres, all of the same dimension.
Then there is a nonempty open subset $T'$ of $T$ such that
the restriction $V|_{T'} \rightarrow T'$ is a family of varieties.
Now we apply this to the relative varieties $\Gamma$
and $Y$ over the curve $T$.
In order to prove the statement we only need to identify the
reduced fibres $\Gamma_t$ and $Y_t$ for sufficiently general $t$.
This is what we do in the following.
(1) First, we show that $\Gamma_t = \Gamma(f_t)$ holds for every $t$
if $E \rightarrow T$ is a flat morphism.
Recall that this happens if and only if
every irreducible component of $E$ dominates $T$.
Write $\dim X =: n+1$. We have $\Gamma_t = \Gamma(f_t) \cup E_t$. Remark that
$\dim E < n+1$. Then $\dim E_t < n$ for every $t$, because of flatness.
But all components of $\Gamma_t$ must have dimension $=n$ for every $t$.
Thus $E_t$ is not a component and $\Gamma_t = \Gamma(f_t)$, for every $t$.
In particular, every $\Gamma_t$ is irreducible of dimension $n$.
In the present situation, the statement follows from the remark in the beginning.
In the general case, by generic flatness, we have that
$E|_{T'} \rightarrow T'$ is flat for some $T'$ and then,
because of the remark, the statement follows.
(2) We know that $Y_t = p'(\Gamma_t)$, and for $t \in T'$
we have from (1) that $\Gamma_t = \Gamma(f_t)$
and hence $Y_t = \overline{f_t(X_t)}$.
In particular every such $Y_t$ is irreducible,
and necessarily of dimension $= \dim Y -1$.
Because of the remark above, the statement follows.
\end{proof}
In general, the family of graphs needs not exist for the full family of maps,
as is seen later on in Remark \ref{example}.
\section{The varieties of general type in a family}
Using the technique of extension of differentials, from a special fibre
to the total space of the family, we gave in \cite{GP}, \S 1.4, a proof
of the assertion that the property of being a variety of general type
is invariant in a 1-dimensional small deformation, where small refers
to the Zariski topology. Here we point out that the same proof shows
indeed a slightly stronger assertion, to the effect that the same property
'propagates' from a component of a fibre.
\begin{thm}[] \label{generaltype}
Let $T$ be a smooth irreducible curve,
let $Y$ be a variety and let $Y \rightarrow T$ be a
projective morphism. Assume that some fibre
$Y_a$ has an irreducible component $Z$ which
is a variety of general type, and that the restriction
$Y \smallsetminus Y_a \rightarrow T \smallsetminus \{a\}$
is a family of varieties, as in \S {\rm 1.c}.
Then there is a nonempty open subset $T'$ of $T$
such that $Y_t$ is a variety of general type for $t \in T'$.
\end{thm}
\begin{proof}
Let $V \rightarrow Y$ be a resolution of singularities
such that the strict transform $Z'$ of $Z$ is smooth.
So $Z'$ is of general type, and \
$\dim H^{0}(Z',mK_{Z'}) \geq cm^{n}$ for $m \gg 0$.
Denote by $\pi$ the composite map $V \rightarrow Y \rightarrow T$.
Since $V \rightarrow T$ is generically smooth, and since
$Y \rightarrow T$ is generically a family of varieties,
restricting to some neighborhood of $a$,
we may assume that for every $t \neq a$
the induced map $V_{t} \rightarrow Y_{t}$ is a
resolution of singularities.
As the general $V_t$ is irreducible, it follows that
every $V_t$ is connected, by the Zariski connectedness theorem.
The extension theorem of Takayama \cite{Tak2} applies,
and gives us that there is a surjective restriction homomorphism
$$\begin{array}{ccc}
\pi_{*}\mathcal O_{V}(mK_{V}) \otimes k(a)
& \longrightarrow & H^{0}(Z', mK_{Z'})
\end{array}.$$
The image $\pi_{*}\mathcal O_{V}(mK_{V})$ is a
torsion free coherent sheaf on the smooth curve $T$,
hence it is a locally free sheaf.
So the dimension of
$\pi_{*}\mathcal O_{V}(mK_{V}) \otimes k(t)$
is constant.
For $t=a$ this dimension is $\geq cm^{n}$ for $m \gg 0$,
by what we have seen above.
\newcommand{\localmentelibero}
{let $f:Y \rightarrow S$ be flat, $\mathcal F$ on $Y$ be flat over $S$.
if $f_{*} \mathcal{F} \otimes k(t) \rightarrow H^{0}(Y_{t}, \mathcal{F}_{t})$
is surjective then it is an isomorphism, and the same holds in a neighborhood of $t$.
moreover $f_{*} \mathcal{F}$ is locally free in a neighborhood of $t$
[Hartshorne, p. 290] }
For $t \neq a$, since $mK_{V}|_{V_{t}} = m K_{V_{t}}$,
one has the restriction homomorphism
$$\begin{array}{ccc}
\pi_{*}\mathcal O_{V}(mK_{V}) \otimes k(t) & \longrightarrow &
H^{0}(V_{t}, \mathcal O_{V_{t}}(mK_{V}|_{V_{t}}))
= H^{0}(V_{t}, mK_{V_{t}})
\end{array}$$
and in a smaller neighborhood of $a$ we may assume that
this is an isomorphism for $t \neq a$.
It follows that
$\dim H^{0}(V_{t},mK_{V_{t}})
\geq cm^{n}$ for $m \gg 0$, hence $Y_{t}$
is of general type. This holds for every $t$
in a neighborhood of $a$.
\end{proof}
\section{Rigidity and limits}
Another key point in our treatment is a result about
limit maps in a generically trivial family of maps.
The result that we give here is only slightly more general than the
one in our previous paper, and the proof given here is more apparent.
Let $X$ be a projective variety.
Let $T$ be a smooth irreducible curve,
let $Y \rightarrow T$ be a projective morphism,
and let $f: T \times X \dasharrow Y$ be a family of rational maps on $X$,
as in \S 1.d. Assume that for every $t \in T$ the rational map
$f_t : X \dasharrow \overline{f_t(X)}$ is of finite degree $k$.
Assume moreover that the family is {\em generically trivial},
as in Corollary \ref{weakrigidity},
i.e. that there is a nonempty open subset $T'$ of $T$ such that
the restriction $f|_{T'}$ is obtained as
$$T' \times X \overset{1 \times h}{\dasharrow} T' \times U
\overset{g}{\dasharrow} Y|_{T'}$$
where $h: X \dasharrow U$ is a fixed dominant rational map, and where $g$
is a birational isomorphism which restricts to a birational isomorphism
$g_t: U \dasharrow Y_t$ for every $t \in T'$.
Then $f_{t} = g_{t} \circ h$ for $t \in T'$,
so all these maps are birationally equivalent,
of degree $\deg(f_{t}) = k = \deg(h)$.
\begin{prop} \label{rigidityandlimits}
Assume that $f: T \times X \dasharrow Y$ is a family
of rational maps of constant degree $\deg(f_t)=k$,
and assume that the family is generically trivial, as in the setting above.
Then all maps $f_t$ are in the same birational equivalence class.
\end{prop}
\begin{proof}
Let $a \in T$ be any point, and let us
prove that $f_a$ is in the birational equivalence class
of every $f_t$ with $t \in T'$.
We may assume that $U$ is a normal variety.
Recall that for a rational map of varieties over a base curve,
from a normal variety to a variety which is proper over the base,
the exceptional locus is of codimension $\geq 2$,
by the valuative criterion of properness for instance.
It follows that $g: T \times U \dasharrow Y$
restricts to a rational map $g_a: U \dasharrow Y_a$.
Since $f = g \circ (1 \times h)$ holds as an equality of
rational maps $T \times X \dasharrow Y$
then there is equality of restrictions $f_a = g_a \circ h$.
And since $\deg(f_a) = k = \deg(h)$ then $\deg(g_a) = 1$
and $f_a$ is birationally equivalent to $h$ and to every $f_t$.
\end{proof}
\section{Linear rational maps}
Let $\mathbb P^m = {\rm P}(V^{m+1})$ and let
$X \subseteq \mathbb P^m$ be a non degenerate subvariety,
of dimension $n$. The space of linear maps
$\mathbb P^m \dasharrow \mathbb P^m$ is the projective space
\begin{center}
$\mathbb P^N = {\rm P}({\rm End(V)})$ \ with $N = (m+1)^2-1$.
\end{center}
We denote by $\alpha = \overline\ell$ a point in $\mathbb P^N$
and by $x = \overline v$ a point in $\mathbb P^m$.
The evaluation homomorphism $(\ell,v) \mapsto \ell(v)$
determines a rational map
$$\mathbb P^N \times X \dasharrow \mathbb P^m$$
and this is the family of linear rational maps
$\alpha : X \dasharrow \mathbb P^m$.
We denote by $\overline{\alpha(X)}$ the closed image
and by $\Gamma(\alpha)$ the closed graph of the map $\alpha$.
The subscheme $C \subset \mathbb P^N \times X$
defined by $\ell(v)=0$
is the exceptional locus of the rational map above.
Consider the projection $C \rightarrow \mathbb P^N$.
The fibre $C_{\alpha}$ is the trace in $X$ of the center of
the linear projection $\alpha: \mathbb P^m \dasharrow \mathbb P^m$.
\begin{rem} \label{example} \em
The subscheme $\Gamma \subset \mathbb P^N \times X \times \mathbb P^m$
defined by $\ell(v) \wedge w =0$ is the closed graph of the rational map above.
Clearly $\Gamma$ contains $C \times \mathbb P^m$.
The projection $\Gamma \rightarrow \mathbb P^N$
does not define the family of graphs. The fibre is given by
$\Gamma_{\alpha} = \Gamma(\alpha) \cup\, C_{\alpha} \times \mathbb P^m$.
It is clear, just looking at dimensions, that
$\Gamma_\alpha = \Gamma(\alpha)$ if and only if
$C_{\alpha} = \emptyset$.
\end{rem}
In $\mathbb P^N$ define the following subsets:
\begin{itemize}
\item[]
$R$ \ \ the subset of all $\alpha$ such that
$\alpha: X \dasharrow \overline{\alpha(X)}$ is of finite degree,
\item[]
$R_k$ \ the subset of all $\alpha \in R$ with $\deg(\alpha)= k$,
\end{itemize}
for every integer $k > 0$.
\begin{prop} \label{constructible}
$(1)$ $R$ is an open subset. $(2)$ $R_k$ is a constructible subset for every $k > 0$.
\end{prop}
\begin{proof}
(1) In $(\mathbb P^N \times X) \smallsetminus C$ let $U$ be the subset
of pairs $(\alpha,x)$ such that $\dim_{x} \alpha^{-1}(\overline{\alpha(X)}) = 0$.
It is an open subset. In $\mathbb P^N$ the image of $U$ coincides with $R$.
In fact, if $\alpha$ admits some point $x \in X \smallsetminus C_{\alpha}$
which is isolated in its fibre, then its general fibre is of dimension $0$.
As the projection $\mathbb P^N \times X \rightarrow \mathbb P^N$
is an open map, $R$ is open in $\mathbb P^N$.
(2) In $\mathbb P^N \times X^{\times k}$ let $U_{k}$ be the subset of sequences
$(\alpha,x_1,\ldots,x_k) =: (\alpha, \bar x)$ such that every $(\alpha, x_i)$
belongs to $U$ and $\alpha(x_1) = \cdots = \alpha(x_k)$ while in the
sequence $(x_1,\ldots,x_k)$ there is no coincidence.
For every $\alpha \in R$
denote by $U_k(\alpha)$ the fibre of $U_k$ over $\alpha$.
Let $V_k$ be the subset such that $\dim_{(\alpha, \bar x)} U_k(\alpha) = n$.
This is a locally closed subset in $\mathbb P^N \times X^{\times k}$.
In $\mathbb P^N$ the image ${V_k}'$ of $V_k$
is the locus of $\alpha \in R$ with $\deg \alpha \geq k$.
In fact, if $\alpha$ admits some sequence $(x_1,\ldots,x_k)$
such that $\dim_{(\bar x)} U_k(\alpha) = n$, as the projection
$U_k(\alpha) \rightarrow X$ has 0-dimensional fibres, then $U_k(\alpha)$
dominates $X$, and hence for a general point $x_1$ the fibre of $\alpha$
contains at least $k$ distinct points $x_1,\ldots,x_k$.
It follows that $R_k$ coincides with ${V_k}' \smallsetminus {V_{k+1}}'$.
\end{proof}
\section{Refined finiteness theorem}
Let $X$ be a smooth projective variety of general type, of dimension $n$.
Let $X' \subset \mathbb P^M$ be the image of $X$ in the $r_n$-canonical
birational embedding, see Theorem \ref{HKT}.
Here $M = h^0(X,r_nK_X) -1$ is bounded above in (\ref{embdim}).
Every rational map of finite degree
$f: X \dasharrow Y$ to a smooth projective variety of general type,
taking the $r_n$-canonical model $Y' \subset \mathbb P^M$,
gives rise to a linear rational map $\alpha: X' \dasharrow \mathbb P^M$
with $\overline{\alpha(X')} = Y'$.
In this natural way
the set of birational equivalence classes of rational maps of finite degree
from $X$ to varieties of general type is injected into
the set of birational equivalence classes
of linear rational maps of finite degree from $X'$ to $\mathbb P^M$.
Our main result is concerned
with the geometric structure of these special equivalence classes.
\begin{thm} \label{connectedcomponent}
Let $X$ be a smooth projective variety of general type.
A birational equivalence class of rational maps of degree $k$
from $X$ to smooth projective varieties of general type
forms a union of connected components of $R_k$.
\end{thm}
\begin{proof}
Let $\alpha \in R_k$ be such that $\overline{\alpha(X')}$ is of general type.
Let $T$ be a smooth irreducible curve with a morphism
$T \rightarrow R_k$, that we write as $t \mapsto \alpha_t$,
and with some point $a \in T$ such that $a \mapsto \alpha$.
We claim that all maps $\alpha_t$ are birationally isomorphic to $\alpha$.
Consider the rational map $T \times X' \dasharrow T \times \mathbb P^M$
which represents the family of maps $\alpha_t$.
Let $Y$ be its closed image in $T \times \mathbb P^M$.
There is a nonempty open subset $T'$ of $T$ such that
$Y|_{T'} \rightarrow T'$ is the family of closed images,
by Proposition \ref{familyofgraphs}.
The fibre $Y_a$ contains $\overline{\alpha(X')}$, a variety of general type.
It follows from Theorem \ref{generaltype}
that, shrinking $T'$ if necessary, we may assume that
for every $t \in T'$ the variety $\overline{\alpha_t(X')}$ is of general type.
Then it follows from Corollary \ref{weakrigidity} to the rigidity theorem
that, shrinking $T'$ again, we may assume that
the restriction $T' \times X' \dasharrow Y|_{T'}$ is a trivial family.
And then it follows from Proposition \ref{rigidityandlimits} that
all maps $\alpha_t$ with $t \in T$ are birationally equivalent, as we claimed.
So we reach the conclusion.
Every irreducible curve through $\alpha$ in $R_k$ is the image
of a smooth irreducible curve $T$ as above, and therefore
is fully contained in the birational equivalence class of $\alpha$.
Therefore every connected curve through $\alpha$ in $R_k$
is fully contained in the birational equivalence class of $\alpha$.
Since $R_k$ is constructible, by Proposition \ref{constructible},
this means that the connected component of $\alpha$ in $R_k$
is contained in the birational equivalence class of $\alpha$.
\end{proof}
The space $R$ admits the stratification
$\bigsqcup R_k$, where the degree $k$ is bounded above in $(\ref{deg1})$
in terms of the function $r_n$, or in $(\ref{deg2})$ in terms of
the function $\epsilon_n$. As an immediate consequence of the previous result
we obtain the following refined version of the finiteness theorem,
which improves our previous result \cite{GP}, Theorem 4.3.
\begin{thm} \label{finiteness}
Let $X$ be a smooth projective variety of general type.
The number of birational equivalence classes of rational maps of finite degree
from $X$ to smooth projective varieties of general type
is bounded above by the number
of connected components of strata in the stratification
$R = \bigsqcup R_k$.
\end{thm}
We showed in \cite{GP} that the finite number of classes of maps
in the finiteness theorem has an upper bound of the form $B(n,v)$
where $n = \dim (X)$ and $v = {\rm vol}(K_X)$, and that such a function
$B$ can be explicitely computed in terms of the function $r_n$.
This is obtained by means of rather cumbersome computations
with the complexity of a certain bunch of subvarieties of Chow varieties,
that was used as a parameter space for rational maps. We believe that an
analogous computation working with the much simpler parametrization
that has been established in the present paper will lead to a simpler
procedure and to a better result for the function $B$.
|
{
"timestamp": "2012-03-13T01:01:04",
"yymm": "1203",
"arxiv_id": "1203.2246",
"language": "en",
"url": "https://arxiv.org/abs/1203.2246"
}
|
\section{Introduction}
Galactic bars are believed to play a crucial role in galaxy evolution.
By reducing angular momentum, galactic bars can efficiently transport
gas from outer disk to the central kiloparsec scale
\citep{lyndenbell79,sellwood81,albada81,combes85,pfenniger91,heller94,bournaud+02,athanassoula+03, jogee+06},
as demonstrated by a number of numerical simulations
\citep[e.g.,][]{roberts79, athanassoula92,friedli93,maciejewski+02,regan+04}.
The bar-driven gas can cause a mass accumulation within the Inner Lindblad Resonance (ILR),
leading to the destruction of bars and the formation of psuedo-bulges
\citep{hasan90,pfenniger90,hasan93,norman96,das+03,shen+04,athanassoula+05,bournaud+05}.
Numerous observational studies have found the characteristics of the bar-driven gas:
i.e., inflow velocities from CO emission \citep[e.g.,][]{quillen95,benedict96}
and from $\rm{H}\alpha$ emission \citep[e.g.,][]{regan97}, higher $\rm{H}\alpha$ luminosities in barred galaxies
than in non-barred galaxies \citep[e.g.,][]{ho97}, and higher molecular gas concentrations
in the central kiloparsec region of barred galaxies \citep[e.g.,][]{sakamoto99,sheth+05}.
Because of the high efficiency of gas inflow toward the central region of galaxies,
bars are often invoked as a trigger of nuclear star formation.
Enhanced nuclear star formation has been found in the central regions of barred spiral galaxies
\citep[e.g.,][]{heckman80,hawarden86,devereux87,arsenault89,huang96, ho97,martinet97,emsellem+01,knapen+02,jogee+05,hunt+08,Ann+09}.
Some statistical studies presented high bar fractions among star-forming galaxies: e.g., 61\% in \citet{ho97},
82\%-85\% in \citet{hunt99}, and 95\% in \citet{laurikainen+04}.
Bar-driven gas inflow has been also considered as a mechanism for triggering active galactic
nucleus (AGN) activity \citep{combes+03}. For the past three decades, much effort has been devoted to understand
the connection between the presence of bars and AGN activity.
However, it is not yet clear whether bars transport gas down to the vicinity of supermassive black holes (SMBHs).
Several observational studies claimed that the fraction of barred galaxies is higher in AGN-host galaxies than
in non-AGN galaxies \citep{arsenault89,knapen+00,laine+02},
while many others found no significant excess of barred galaxies in AGN-host galaxies
\citep{moles95,mcleod95,mulchaey97,ho97,laurikainen+04,hao+09,Ann+09}.
\citet{shlosman89} suggested the ``bars within bars'' scenario as a mechanism for fueling AGNs.
In this model, large-scale stellar bars transport gas into their rotating disks of a few hundred parsec scale.
When a critical amount of gas is accumulated, the disks undergo gravitational instability,
triggering a gaseous secondary bar, which enables gas to approach closer to SMBHs \citep{mulchaey97,maciejewski97,ho97}.
Some studies suggested that nuclear spirals, instead of secondary bars, are responsible for triggering AGNs
\citep{martini99,marquez+00,martini+03}.
In a recent high-resolution smoothed particle hydrodynamics simulation, \citet{hopkins+10} showed that
disk instabilities (for $10-100$ pc scales) driven by primary bars exhibit various morphologies
as well as bar-like shapes.
Several observational studies confirmed the presence of secondary bars by detecting nuclear bars embedded
in large-scale bars, using the ISAAC/VLT spectroscopic data \citep{emsellem+01}, the Hubble Space Telescope (HST)
images \citep{malkan98,laine+02,carollo+02,erwin+02}, and the integral field spectrograph SAURON data \citep{emsellem+06}.
It is found, however, that the fraction of secondary bars in AGNs is similar to that in non-AGNs \citep{martini+03},
implying that secondary bars do not play a critical role in fueling AGNs.
Although there have been many attempts, the nature of the AGN-bar connection is still unclear.
The presence of secondary bars or nuclear spirals in non-AGN galaxies suggests that
bars are not a universal fueling mechanism \citep{marquez+00,laine+02,martini+03}.
In this paper, we investigate the connection between the presence of bars and AGN activity
using a large sample of galaxies from the Sloan Digital Sky Survey (SDSS; \citealt{york+00}).
SDSS data have been used by several previous studies in revealing the dependence of the bar fraction
either on internal galaxy properties or on environmental properties \citep{barazza+08,
aguerri+09,li+09,nair+10b,masters+11,lee+11}.
However, the AGN-bar connection has not been studied in detail using the SDSS data.
\citet{hao+09} found no excess of bars in AGN-host galaxies using SDSS data.
However, the galaxy sample in their study was relatively small and biased to blue galaxies
\citep[see][]{masters+11, lee+11}.
Therefore, it is needed to investigate the connection between bars and AGN activity
using a homogeneous and large galaxy sample.
Following our detailed study on the relation between the presence of bars and
galaxy properties \citep[][hereafter Paper I]{lee+11},
we investigate the AGN-bar connection using a homogeneous sample of late-type galaxies,
selected from the SDSS.
This paper is organized as follows. We describe the volume-limited sample and
the method for identifying bars and spectral types in Section 2.
Section 3 presents the main results including the dependence of the bar fraction on
spectral types, the dependence of the AGN fraction on the presence of bars, and
the comparison of Eddington ratio distributions between barred and non-barred AGN-host galaxies.
We discuss the implication of primary results in Section 4, and present summary and conclusions in Section 5.
\begin{figure*}
\centering
\includegraphics[scale=0.8]{fig1.eps}
\caption{(a) Classification of spectral types for 8,655 late-type galaxies in the [NII]/$\rm{H}\alpha$ versus [OIII]/$\rm{H}\beta$
diagram. There are three types of galaxies (red: AGN-host galaxies, green: composite galaxies,
and blue: star-forming galaxies) classified by two separate lines, the extreme starburst classification line
(\citealp[solid line]{kewley+01}) and the pure star formation line (\citealp[dashed line]{kauffmann+03}).
In panels (b) and (c), dots represent strong-barred (SB) and weak-barred (WB) galaxies, respectively. Contours show
the distribution for all late-type galaxies shown in panel (a).}
\label{fig1}
\end{figure*}
\section{Data and Methods}
\subsection{SDSS Galaxy Sample}
We use a volume-limited sample of 33,391 galaxies with the $r$-band absolute magnitude
$M_r \leq -19.5+5{\rm log}h$ mag
(hereafter, we drop the $+5{\rm log}h$ term in the absolute magnitude)
at redshift $0.02 \leq z \leq 0.05489$, from the SDSS Data Release 7 (DR7; \citealp{abazajian+09}).
These galaxies are extracted from the Korea Institute for Advanced Study Value-Added Galaxy Catalog
(KIAS VAGC; \citealp{choi+10}) that is based on the Large Scale Structure (LSS) sample of New York
University Value-Added Galaxy Catalog (NYU VAGC; \citealp{blanton+05}).
The rest-frame absolute magnitudes of individual galaxies are computed in fixed bandpass, shifted to $z=0.1$,
using the Galactic reddening correction of \citet{schlegel98} and $K$-corrections as described by \citet{blanton+03}.
The mean evolution correction given by \citet{tegmark+04}, $E(z)=1.6(z-0.1)$, is also applied.
The spectroscopic parameters (i.e., stellar velocity dispersions and strength of various emission lines)
are obtained from NYU VAGC and MPA/JHU DR7 VAGC \citep{tremonti+04,brinchmann+04}.
Stellar masses are also from the MPA/JHU DR7 VAGC, which are based on fits to the SDSS five-band photometry
\citep{kauffmann+03b}.
We adopt a flat $\Lambda$CDM cosmology with $\Omega_{\Lambda}=0.74$ and $\Omega_{m}=0.26$
from {\it Wilkinson Microwave Anisotropy Probe} five-year data \citep{komatsu+09}.
The detailed description of the morphological classification and
the identification of bars of the sample, and the comparison with previous
classifications \citep{rc3,nair+10a} can be found in Paper I (see Section 3).
We summarize the sample selection and classification schemes as follows.
First, after dividing all galaxies into early- and late-type galaxies
using the automated classification method \citep{park+05} and visual inspection,
we selected 19,431 late-type galaxies out of 33,391 galaxies
(see Table 1 of Paper I).
To avoid the internal extinction effects, we selected only galaxies with the minor-to-major axis ratio
$b/a>0.6$, obtaining a sample of 10,674 late-type galaxies.
Then, we classified these late-type galaxies into three groups
based on the presence (and the length) of bars:
2542 strong-barred (23.8\%), 698 weak-barred (6.5\%),
and 7434 non-barred galaxies.
When the size of bars is larger (shorter) than a quarter of the size of their host galaxies,
we classified these galaxies as strong-barred (weak-barred) galaxies.
As described in Paper I, our classification shows a good agreement with
\citet{nair+10a}'s classification.
In this study we classify the sample galaxies into AGNs and non-AGNs using the spectral features.
To determine the spectral types and to investigate
the dependence of the bar fraction on spectral types,
we select galaxies whose spectra show strong emission-lines of $\rm{H}\alpha$, $\rm{H}\beta$, ${\rm [OIII]}~\lambda5007$, and ${\rm [NII]}~\lambda6584$
with signal-to-noise ratio ${\rm S/N}\ge3$ \citep{kewley+06}.
By excluding 2019 galaxies that do not satisfy the S/N criterion,
we make a final the sample of 8655 late-type galaxies for the following analysis.
We perform the aperture correction to the velocity dispersion of the target galaxies
using the equation suggested by \citet{cappellari+06}, \begin{equation}
\sigma_{\rm corr}=\sigma_{\rm fib}\times(R_{\rm fib}/R_{\rm eff})^{(0.066\pm0.035)},
\end{equation}
where $\sigma_{\rm fib}$ is the velocity dispersion obtained from a fiber with $R_{\rm fib}=1^{\prime\prime}.5$.
$R_{\rm eff}$ is an effective radius calculated by $R_{\rm eff}=r_{\rm deV}\times(b/a)^{0.5}_{\rm deV}$
\citep{bernardi+03}, where $r_{\rm deV}$ and $(b/a)_{\rm deV}$ are, respectively,
scale radius and $b/a$ axis ratio in $i$-band in the de Vaucouleurs fit.
Hereafter, the velocity dispersion means $\sigma_{\rm corr}$, and its subscript corr will be omitted.
\subsection{Classification of Spectral Types}
\begin{deluxetable}{crrrrr}
\tablecolumns{6}
\tablewidth{0pc}
\tablecaption{Spectral Types of the Sample Galaxies}
\tablehead{
\colhead{} & \multicolumn{2}{c}{${\rm S/N}\tablenotemark{a}\geq3$} & & \multicolumn{2}{c}{${\rm S/N}\ge6$} \\
\cline{2-3} \cline{5-6}
\colhead{Spectral type} & \colhead{Number} & \colhead{Fraction} & & \colhead{Number} & \colhead{Fraction}}
\startdata
Star-forming & 4,940 & 57.1 \% & & 3,600 & 60.3 \% \\
Composite & 1,973 & 22.8 \% & & 1,411 & 23.7 \% \\
AGN-host & 1,742 & 20.1 \% & & 957 & 16.0 \% \\
\cline{1-6}
Total & 8,655 & 100 \% & & 5,968 & 100 \%
\enddata
\tablenotetext{a}{S/N for four emission lines such as $\rm{H}\alpha$, $\rm{H}\beta$, ${\rm [NII]}~\lambda6584$, and ${\rm [OIII]}~\lambda5007$}
\label{table1}
\end{deluxetable}
\begin{deluxetable*}{crrrrrrr}
\tablecolumns{8}
\tablewidth{0pc}
\tablecaption{Dependence of Bar Fraction on Spectral Types}
\tablehead{
\colhead{(1) ${\rm S/N}\tablenotemark{a}\ge3$} &&&&& \\
\cline{1-8}
\colhead{Spectral type} & \colhead{Total} & \colhead{SB\tablenotemark{b}} & \colhead{$f_{\rm SB}$ (\%)\tablenotemark{c}} &
\colhead{WB\tablenotemark{d}} & \colhead{$f_{\rm WB}$ (\%)} & \colhead{SB+WB} & \colhead{$f_{\rm SB+WB}$ (\%)}}
\startdata
Star-forming & 4,940 & 770 & $15.6\pm0.5$ & 348 & $7.0\pm0.4$ & 1,118 & $22.6\pm0.6$ \\
Composite & 1,973 & 639 & $32.4\pm1.1$ & 118 & $6.0\pm0.5$ & 757 & $38.4\pm1.1$ \\
AGN-host & 1,742 & 742 & $42.6\pm1.1$ & 109 & $6.3\pm0.6$ & 851 & $48.9\pm1.2$ \\
\cline{1-8}
\cline{1-8}
\\
(2) ${\rm S/N}\ge6$ &&&&&& \\
\cline{1-8}
Spectral type & Total & SB & $f_{\rm SB}$ (\%) & WB & $f_{\rm WB}$ (\%) & SB+WB & $f_{\rm SB+WB}$ (\%) \\
\cline{1-8}
Star-forming & 3,600 & 659 & $18.3\pm0.6$ & 265 & $7.4\pm0.4$ & 924 & $25.7\pm0.7$ \\
Composite & 1,411 & 513 & $36.4\pm1.3$ & 67 & $4.7\pm0.5$ & 580 & $41.1\pm1.3$ \\
AGN-host & 957 & 419 & $43.8\pm1.6$ & 52 & $5.4\pm0.7$ & 471 & $49.2\pm1.6$
\enddata
\tablenotetext{a}{S/N for four emission lines such as $\rm{H}\alpha$, $\rm{H}\beta$, ${\rm [NII]}~\lambda6584$, and ${\rm [OIII]}~\lambda5007$}
\tablenotetext{b}{Strong-barred galaxies}
\tablenotetext{c}{The errors of the bar fraction are obtained by calculating the standard deviation in
1,000-times-repetitive sampling method.}
\tablenotetext{d}{Weak-barred galaxies}
\label{table2}
\end{deluxetable*}
\begin{figure*}
\centering
\includegraphics[scale=0.8]{fig2.eps}
\caption{The fraction of strong-barred galaxies ($f_{\rm SB}$) in the three BPT diagnostic diagrams: (a) [NII]/$\rm{H}\alpha$ versus [OIII]/$\rm{H}\beta$,
(b) [SII]/$\rm{H}\alpha$ versus [OIII]/$\rm{H}\beta$, (c) [NII]/$\rm{H}\alpha$ versus [OIII]/$\rm{H}\beta$. Black dots and grey dots
represent strong-barred galaxies and non-SB galaxies, respectively. Contours represent constant $f_{\rm SB}$.
In panels (b) and (c) we use only 8,508 galaxies with ${\rm S/N_{[SII]}}\geq3$ and 5,622 galaxies with ${\rm S/N_{[OI]}}\geq3$.}
\label{fig2}
\end{figure*}
To determine the spectral types, we use the [NII]/$\rm{H}\alpha$ versus [OIII]/$\rm{H}\beta$ diagnostic diagram
that is known as Baldwin-Phillips-Terlevish (BPT) diagram \citep{baldwin81,veilleux87}.
We categorize the sample galaxies into three spectral types: star-forming, composite, and AGN-host galaxies.
As shown in Figure \ref{fig1}, \citet{kewley+01} drew theoretical ``maximum starburst lines''
(solid lines) to define the upper boundary of star-forming galaxies,
and \citet{kauffmann+03} added an empirical demarcation line (dashed line)
to distinguish pure star-forming galaxies from composite galaxies whose spectra are affected
by both star forming nuclei and AGNs.
We perform spectral classification using two criteria of S/N (for $\rm{H}\alpha$, $\rm{H}\beta$, [NII], and [OIII]
emission lines): ${\rm S/N}\geq3$ and ${\rm S/N}\geq6$.
When ${\rm S/N}\geq3$ is adopted, we find that the fractions of star-forming, composite, and AGN-host galaxies are
57.1\% (4940 galaxies), 22.8\% (1973 galaxies), and 20.1\% (1742 galaxies), respectively.
On the other hand, in the case of ${\rm S/N}\geq6$, the sample galaxies consist of 60.3\% (3600 galaxies) of star-forming,
23.7\% (1411 galaxies) of composite, and 16.0\% (957 galaxies) of AGN-host galaxies.
The result of spectral classification is summarized in Table \ref{table1}.
As the threshold of S/N increases from 3 to 6, the fraction of AGN-host galaxies decreases 20.1\% to 16.0\%.
This is because a significant fraction of LINERs in the low S/N sample is not included in the high S/N sample,
since LINERs tend to dominate the low S/N sample \citep{cidfernandes+10,cidfernandes+11}.
\section{Results}
\begin{figure*}
\centering
\includegraphics[scale=0.7]{fig3.eps}
\caption{(Upper) The number of galaxies as a function of (a) $u-r$ color, (b) velocity dispersion ($\sigma$), and (c) stellar mass ($M_{\rm star}$) for
strong-barred (SB), weak-barred (WB), and non-barred (NB) galaxies. (Lower) The dependence of the bar fraction on (d) $u-r$,
(e) $\sigma$, and (f) $M_{\rm star}$. Circles and diamonds represent fractions of SB and WB galaxies,
respectively. Error bars mean 1-$\sigma$ sampling errors estimated by calculating the standard deviation of the bar fraction
in 1,000-times-repetitive sampling.}
\label{fig3}
\end{figure*}
\begin{figure}
\centering
\includegraphics[scale=0.55]{fig4.eps}
\caption{The fraction of strong-barred galaxies ($f_{\rm SB}$) in (a) $u-r$ versus $M_{\rm star}$ diagram
and (b) $\sigma$ versus $M_{\rm star}$ diagram. Black dots and grey dots represent strong-barred
galaxies and non-SB galaxies, respectively. Contours represent constant strong-barred galaxy fractions.}
\label{fig4}
\end{figure}
To investigate the connection between AGN activity and the presence of bars,
first, we compare the bar fractions in AGN-host and non-AGN galaxies
in Section 3.1.
Then, we examine the AGN fraction between barred and non-barred galaxies in Section 3.2.
Finally, using AGN host galaxies, we investigate whether the Eddington ratio distribution
is different depending on the presence of bars in Section 3.3.
\subsection{Dependence of Bar Fraction on AGN activity}
\begin{figure}
\centering
\includegraphics[scale=0.65]{fig5.eps}
\caption{Distribution of $u-r$ and $\sigma$ for 8,655 late-type galaxies in [NII]/$\rm{H}\alpha$
versus [OIII]/$\rm{H}\beta$ diagnostic diagram. We divide this diagram into $20\times20$ bins and
measure median values of $u-r$ and $\sigma$ for galaxies at each bin.
After excluding bins that contain less than five galaxies, we display
representative symbols with various colors and sizes, corresponding to the median values of $u-r$ and
$\sigma$, respectively.}
\label{fig5}
\end{figure}
\begin{figure*}
\centering
\includegraphics[scale=0.7]{fig6.eps}
\caption{The dependence of the fraction of strong-barred galaxies ($f_{\rm SB}$) on spectral types for late-type galaxies
with fixed ranges of (a) $u-r$ ($\Delta(u-r)=0.3$) and velocity dispersion ($\Delta\sigma=40$ km s${}^{-1}$).
(b) Same as (a), but x-axis is log $(M_{\rm star}/M_{\odot})$ instead of $\sigma$.
Stars, diamonds, and circles with error bars represent $f_{\rm SB}$ for star-forming, composite,
and AGN-host galaxies, respectively.}
\label{fig6}
\end{figure*}
Figure \ref{fig1} shows the distributions of strong-barred (panel b) and
weak-barred galaxies (panel c) in emission-line ratio diagrams
([NII]/$\rm{H}\alpha$ versus [OIII]/$\rm{H}\beta$).
Strong-barred galaxies are widely distributed over the star-forming,
composite, and AGN-host galaxy regions, while the majority of weak-barred galaxies lie
in the star-forming galaxy region.
Some weak bars are found in the composite and AGN-host galaxy region,
but the number of those galaxies is relatively small.
We investigate how the bar fraction varies depending on the spectral types
using two S/N criteria as summarized in Table \ref{table2}.
In the sample of 8655 galaxies with ${\rm S/N}\geq3$, the fraction of strong-barred galaxies ($f_{\rm SB}$)
is $15.6\%\pm0.5\%$ in star-forming galaxies, $32.4\%\pm1.1\%$ in composite galaxies,
and $42.6\%\pm1.1\%$ in AGN-host galaxies. Among 5968 galaxies with ${\rm S/N}\geq6$, $f_{\rm SB}$ is
$18.3\%\pm0.6\%$ in star-forming galaxies, $36.4\%\pm1.3\%$ in composite galaxies, and
$43.8\%\pm1.6\%$ in AGN-host galaxies, respectively. In the low S/N case we find that $f_{\rm SB}$ is $\sim$2.5 times higher
in AGN-host galaxies than in star-forming galaxies. On the other hand, the fraction of weak-barred
galaxies ($f_{\rm WB}$) does not vary significantly with spectral types, from 6.0\% to 7.0\%.
The result for the high S/N case is not different from that for the low S/N case.
Figure \ref{fig2} presents the change of $f_{\rm SB}$ for the sample of galaxies with ${\rm S/N}\geq3$ in three BPT diagnostic diagrams
such as (a) [NII]/$\rm{H}\alpha$ versus [OIII]/$\rm{H}\beta$, (b) [SII]/$\rm{H}\alpha$ versus [OIII]/$\rm{H}\beta$, and (c) [OI]/$\rm{H}\alpha$ versus [OIII]/$\rm{H}\beta$.
In panel (b) we use 8508 galaxies with ${\rm S/N_{[SII]}}\geq3$, while only 5622 galaxies with ${\rm S/N_{[OI]}}\geq3$
are used in panel (c).
We find a noticeable trend that $f_{\rm SB}$ increases continuously from the star-forming galaxy region
(lower left) to the AGN-host galaxy region (upper right) in all diagnostic diagrams,
showing that the presence of bars is more frequent in AGN-host galaxies than non-AGN galaxies.
We check that this trend does not change significantly even when we use the high S/N sample.
In the view of the results we obtained above, it is seen that AGN activity is related to the presence of strong bars.
However, we will see below that these results do not directly indicate a connection between the presence of strong bars and AGN activity.
This is because $f_{\rm SB}$ is also a strong function of galaxy properties, i.e., $u-r$ color, velocity dispersion ($\sigma$) and stellar mass ($M_{\rm star}$).
In Paper I, we found that $u-r$ and $\sigma$ are more influential parameters in determining the bar fraction.
Therefore, we need to compare AGN-host and non-AGN galaxies with fixed $u-r$ and $\sigma$
in order to separate the effect of the two parameters on $f_{\rm SB}$.
In Figure \ref{fig3} we show the dependence of the bar fraction on three parameters: $u-r$,
$\sigma$, and $M_{\rm star}$. $M_{\rm star}$ is another important parameter affecting the bar fraction \citep[e.g.,][]{sheth+08,cameron+10,mabreu+10,nair+10b}.
The fraction of strong-barred galaxies increases significantly as $u-r$ color becomes redder,
and it has a maximum value at intermediate velocity dispersion of $\sim$130 km s$^{-1}$.
It has a constant value ($\sim$10\%) until ${\rm log}~(M/M_{\odot})=10.2$, but increases with $M_{\rm star}$ thereafter.
On the other hand, the fraction of weak-barred galaxies shows a different dependency on the three parameters.
It has a peak value at a bluer color of $u-r\simeq1.4$, and becomes larger as $\sigma$ or $M_{\rm star}$ decreases.
This result on $M_{\rm star}$ is consistent with the result of \citet{nair+10b}.
Figure \ref{fig4} shows how $f_{\rm SB}$ varies in the $u-r$ versus $M_{\rm star}$ and in the $\sigma$ versus $M_{\rm star}$
diagrams. It shows that $M_{\rm star}$ has a strong correlation with both $u-r$ and $\sigma$.
It also shows that $f_{\rm SB}$ increases with the three parameters.
However, at $\sigma\gtrsim120$ km s$^{-1}$ it decreases with $\sigma$ at a given $M_{\rm star}$ bin.
Particularly, contours are neither vertical nor horizontal,
suggesting that $f_{\rm SB}$ depends on the three parameters simultaneously.
Therefore, we conclude that $u-r$, $\sigma$, and $M_{\rm star}$ are all important parameters
in determining the bar fraction.
Because $f_{\rm SB}$ is strongly correlated with three parameters,
we need to check whether the trend of $f_{\rm SB}$ in Figure \ref{fig2} is originated from the effect of the three parameters
on $f_{\rm SB}$. To investigate the difference of $u-r$ and $\sigma$ between AGN-host
and non-AGN galaxies, we examine how $u-r$ and $\sigma$ vary
in the [NII]/$\rm{H}\alpha$ versus [OIII]/$\rm{H}\beta$ diagnostic diagram as shown in Figure \ref{fig5}.
We use $20\times20$ bins in this diagram to measure median values of $u-r$ and $\sigma$
for late-type galaxies at each bin.
After excluding bins that contain less than five galaxies,
we display representative symbols with various colors and sizes,
corresponding to the median values of $u-r$ and $\sigma$, respectively.
From the star-forming galaxy region toward the AGN-host galaxy region,
it seems obvious that $u-r$ color becomes redder.
At the same time, $\sigma$ increases along the same direction.
The median values of $u-r$ are $1.77\pm0.01$, $2.20\pm0.01$ and $2.59\pm0.01$,
respectively for star-forming, composite, and AGN-host galaxies.
In the case of $\sigma$, the median values for star-forming, composite, and AGN-host
galaxies are $66.2\pm0.5$, $95.1\pm0.7$, and $118.3\pm0.9$ km s$^{-1}$, respectively.
Considering the dependence of $f_{\rm SB}$ on $u-r$ and $\sigma$,
it is clear that the trend shown in Figure \ref{fig2} is caused by the fact that
$f_{\rm SB}$ increases with $u-r$ and $\sigma$ (or $M_{\rm star}$).
To remove the effect of $u-r$, $\sigma$, and $M_{\rm star}$, we investigate
the bar fraction in AGN-host and non-AGN galaxies at fixed $u-r$ and $\sigma$ (or $M_{\rm star}$),
as shown in Figure \ref{fig6}.
First, we divide our sample into 17 bins with fixed $u-r$ and $\sigma$ ranges.
Note that each bin contains more than fifty galaxies.
Then we measure $f_{\rm SB}$ of each spectral type in each bin.
An obvious excess of $f_{\rm SB}$ in AGN-host galaxies is found only in one bin with
$u-r=2.1-2.4$ and $\sigma=40-80$ km s$^{-1}$.
In contrast, at all other $u-r$ and $\sigma$ bins, $f_{\rm SB}$ of AGN-host galaxies is similar
to that of star-forming galaxies.
Second, we perform similar analysis at fixed $u-r$ and $M_{\rm star}$ ranges.
We do not found any clear or significant excess of $f_{\rm SB}$ in AGN-host galaxies in all bins.
These results clearly demonstrate that the $f_{\rm SB}$ excess in AGN-host galaxies shown in Figure \ref{fig2}
is caused by the fact that on average AGN-host galaxies are redder and and more massive than non-AGN galaxies.
These results suggest that the bar fraction do not depend on AGN activity.
\subsection{Dependence of AGN Fraction on Bar Presence}
\begin{deluxetable*}{crrrrrrrr}
\tablecolumns{9}
\tablewidth{0pc}
\tablecaption{Spectral Classification in Different Bar Types}
\tablehead{
\colhead{(1) ${\rm S/N}\tablenotemark{a}\geq3$} &&&&&&&& \\
\cline{1-9}
\colhead{} & \colhead{NB\tablenotemark{b}} &&& \colhead{SB\tablenotemark{c}} &&& \colhead{WB\tablenotemark{d}} & \\
\cline{2-3} \cline{5-6} \cline{8-9}
\colhead{Spectral type} & \colhead{Number} & \colhead{Fraction} && \colhead{Number} & \colhead{Fraction} &&
\colhead{Number} & \colhead{Fraction}}
\startdata
Star-forming & 3,822 & 64.5\% & & 770 & 35.8\% & & 348 & 60.5\% \\
Composite & 1,216 & 20.5\% & & 639 & 29.7\% & & 118 & 20.5\% \\
AGN-host & 891 & 15.0\% & & 742 & 34.5\% & & 109 & 19.0\% \\
\cline{1-9}
Total & 5,929 & 100.0\% & & 2,151 & 100.0\% & & 575 & 100.0\% \\
\cline{1-9}
\cline{1-9}
\\
(2) ${\rm S/N}\ge6$ &&&&&&&& \\
\cline{1-9}
& ${\rm NB~~~}$ & & & ${\rm SB~~~}$ & & & ${\rm WB~~~}$ & \\
\cline{2-3} \cline{5-6} \cline{8-9}
Spectral type & Number & Fraction && Number & Fraction && Number & Fraction \\
\cline{1-9}
Star-forming & 2,676 & 67.0\% & & 659 & 41.4\% & & 265 & 69.0\% \\
Composite & 831 & 20.8\% & & 513 & 32.2\% & & 67 & 17.4\% \\
AGN-host & 486 & 12.2\% & & 419 & 26.3\% & & 52 & 13.5\% \\
\cline{1-9}
Total & 3,993 & 100.0\% & & 1,591 & 100.0\% & & 384 & 100.0\%
\enddata
\tablenotetext{a}{S/N for four emission lines such as $\rm{H}\alpha$, $\rm{H}\beta$, ${\rm [NII]}~\lambda6584$, and ${\rm [OIII]}~\lambda5007$}
\tablenotetext{b}{Non-barred galaxies}
\tablenotetext{c}{Strong-barred galaxies}
\tablenotetext{d}{Weak-barred galaxies}
\label{table3}
\end{deluxetable*}
\begin{figure}
\centering
\includegraphics[scale=0.5]{fig7.eps}
\caption{The fraction of AGN-host (circles), composite (triangles), and star-forming galaxies (diamonds) as a function
of $u-r$ (top), $\sigma$ (middle), and $M_{\rm star}$ (bottom). Solid lines and dashed lines represent barred (strong and weak-barred)
and non-barred galaxies, respectively. Error bars mean 1-$\sigma$ sampling errors.}
\label{frac_agn_mass}
\end{figure}
\begin{figure*}
\centering
\includegraphics[scale=0.65]{fig8.eps}
\caption{The dependence of the AGN fraction ($f_{\rm AGN}$) on the presence of bars at
fixed ranges of (a) $u-r$ and $\sigma$ and (b) $u-r$ and $M_{\rm star}$. Diamonds and circles
represent $f_{\rm AGN}$ for barred (strong and weak-barred) and non-barred galaxies, respectively.
Error bars represent 1-$\sigma$ sampling errors.}
\label{agnfrac_barredness}
\end{figure*}
In this section we investigate how AGN fraction changes
depending on the presence of bars.
Among non-barred or weak-barred galaxies,
the fraction of star-forming galaxies ($>60\%$) is much larger than
that of AGN-host galaxies ($<20\%$)
while composite galaxies occupy $\sim$20\%.
In contrast, the AGN fraction increases by a factor of two
when galaxies have strong bars.
For example, among galaxies with ${\rm S/N}\geq3$
(${\rm S/N}\geq6$), the AGN fraction is $34.5\%$ ($26.3\%$).
AGN fractions in strong-barred, weak-barred, and non-barred galaxy samples
are summarized in Table \ref{table3}.
To demonstrate the dependence on galaxy properties,
we present AGN fraction of barred (combining strong and weak bars)
and non-barred galaxies,
as a function of $u-r$, $\sigma$, and $M_{\rm star}$ in Figure \ref{frac_agn_mass}.
In both barred and non-barred galaxy samples, the AGN fraction increases
with redder color, higher $\sigma$ or larger $M_{\rm star}$
while the fraction of star-forming galaxies shows an opposite trend.
At fixed $\sigma$ or $M_{\rm star}$, AGN fraction is significantly higher in barred galaxies
than in non-barred galaxies.
At fixed $u-r$ color, however, the excess of AGN fraction in barred galaxies
is marginal.
These results do not significantly change even if we exclude weak bars
from the barred galaxy sample.
The results presented in Figure \ref{frac_agn_mass} are
similar to the finding of \citet[][see their Figure 8]{oh+11},
which appeared in the literature during the review process of our manuscript.
We note that the sample size in this study is much larger than
that in Oh et al. (8655 galaxies versus 3934 galaxies), and that we classify
composite galaxies separately instead of including them in the AGN sample.
The results in Figures \ref{fig3}, \ref{fig4}, and \ref{frac_agn_mass} show that
both bar fraction and AGN fraction increase as galaxy color becomes redder.
This leads naturally to an expectation that AGN and bars may be related.
However, this does not necessarily mean that both are related. We need to investigate
whether AGN is directly connected to bars or not.
For this we investigate whether AGN fraction is different between barred and non-barred galaxies
in multi-dimensional spaces as shown in Figure \ref{agnfrac_barredness}.
When we compare galaxies at fixed ranges of $u-r$ and $\sigma$ (or $M_{\rm star}$),
the excess of AGN fraction in barred galaxies disappears or weakens.
Within the sampling errors,
there is no significant difference of AGN fraction between barred and non-barred galaxies.
These results are dramatically different from those in Figure \ref{frac_agn_mass}
due to the fact that AGN fraction is dependent of color and $\sigma$ (or $M_{\rm star}$).
Figure \ref{fig4} shows that $u-r$ has a large dispersion even when $M_{\rm star}$ is fixed.
Similarly large color dispersion is also seen when $\sigma$ is fixed (see Figure 9 in Paper I).
Therefore an excess of the AGN fraction in barred galaxies shown in Figure \ref{frac_agn_mass} is an
``apparent'' trend, which is caused by the residual dependence on $u-r$ color in $\sigma$
(or $M_{\rm star}$) bins.
Thus, in order to remove the effect of $u-r$ and $\sigma$ (or $M_{\rm star}$),
AGN fraction has to be compared using galaxies at fixed $u-r$ {\it AND} $\sigma$ (or $M_{\rm star}$).
We find no significant dependence of AGN fraction on the presence of bars,
suggesting that AGN activity is not dominated by bars.
\begin{figure*}
\centering
\includegraphics[scale=0.8]{fig9.eps}
\caption{Eddington ratio as a function of (left) $u-r$, (middle) $\sigma$,
and (right) $M_{\rm star}$ for 1,647 (upper, ${\rm S/N}\geq3$) and 893 (lower, ${\rm S/N}\geq6$)
late-type AGN-host galaxies with $\sigma>70$ km s${}^{-1}$. Solid lines, long-dashed lines, and short-dashed lines represent
median curves of Eddington ratio for strong-barred (SB), weak-barred (WB), and non-barred (NB) AGN-host galaxies,
respectively. The size of bins corresponds to 1/7 of x-axis of each panel, and only when bins contain more than
five galaxies median values of Eddington ratio are drawn. Error bars are calculated using 1,000-times resampling
method. Error bars for strong-barred and weak-barred galaxies are slightly shifted respect to ones of non-barred galaxies
in order to avoid overlap each other.}
\label{fig7}
\end{figure*}
\begin{figure}
\centering
\includegraphics[scale=0.6]{fig10.eps}
\caption{Eddington ratio versus $u-r$ for late-type AGN-host galaxies with (upper) ${\rm S/N}\geq3$, (lower)
${\rm S/N}\geq6$, (left) 70 km s$^{-1}<\sigma\leq120$ km s$^{-1}$, and (right) 120 km s$^{-1}<\sigma\leq170$ km s$^{-1}$.
Solid lines, long-dashed lines, and short-dashed lines represent median curves of
Eddington ratio for strong-barred (SB), weak-barred (WB), and non-barred (NB) galaxies, respectively.
The size of bins corresponds to 1/7 of x-axis of each panel, and only when bins contain
more than five galaxies median values of Eddington ratio are drawn. Error bars are calculated using 1,000-times resampling
method. Error bars for strong-barred and weak-barred galaxies are slightly shifted respect to ones of non-barred galaxies
in order to avoid overlap each other.}
\label{fig8}
\end{figure}
\subsection{Comparison of Eddington Ratio between Barred and Non-barred AGN-host Galaxies}
If AGN activity is triggered by bars, barred galaxies may have
higher accretion rates than non-barred galaxies.
We exclusively use AGN-host galaxies to examine whether there is any difference
in AGN power between barred and non-barred galaxies.
We use the [OIII] luminosity ($L_{\rm [OIII]}$) as a proxy for the bolometric luminosity
and infer black hole mass ($M_{\rm BH}$) from $\sigma$ using Eq. (2) as described below.
Thus, $L_{\rm [OIII]}$ to $M_{\rm BH}$ ratio can be used as an approximate
Eddington ratio indicator.
We adopt a reddening curve of $R_{V}=A_{V}/E(B-V)=3.1$ \citep{cardelli89}
and an intrinsic Balmer decrement of $\rm{H}\alpha/\rm{H}\beta=3.1$ for AGN-host galaxies \citep{osterbrock+06}
to correct for internal dust extinction.
To estimate $M_{\rm BH}$, we adopt a $M_{\rm BH}-\sigma$ relation
for late-type galaxies suggested by \citet{mcconnell+11}:
\begin{equation}
{\rm log}~(M_{\rm BH}/M_{\odot}) = 7.97+4.58\times {\rm log}~(\sigma/200 {\rm km ~s^{-1}}).
\end{equation}
We exclude galaxies with velocity dispersion values lower than the instrumental
resolution of the SDSS spectra ($\sigma<70$ km s${}^{-1}$) since these measurements
are not reliable \citep{choi+09}.
In the following analysis we use two samples: 1647 AGN-host galaxies with ${\rm S/N}\geq3$ and 893 ones with ${\rm S/N}\geq6$.
The low S/N sample contains 718 strong-barred, 97 weak-barred, and 832 non-barred galaxies. On the other hand,
the high S/N sample consists of 407 strong-barred, 46 weak-barred, and 440 non-barred galaxies.
The black hole mass of AGN-host galaxies spans $5.9 <$ log ($M_{\rm BH}/M_{\odot}$)$ < 8.3$,
while $L_{\rm [OIII]}/M_{\rm BH}$ ranges over four order of magnitude,
$10^{-2}-10^{2}$ $L_{\odot}/M_{\odot}$.
In Figure \ref{fig7} we present the Eddington ratio indicator (hereafter Eddington ratio) distributions
as a function of $u-r$, $\sigma$, and $M_{\rm star}$, respectively,
for strong-barred, weak-barred, and non-barred galaxies.
The AGN power (i.e., Eddington ratio) appears to decrease with $M_{\rm star}$ (or $\sigma$)
as previously seen by \citet{hwang+11}.
This trend can be interpreted as ``Eddington incompleteness'', which reflects the observational
selection effect that for given flux (or luminosity limits) lower Eddington ratio AGNs can be detected at higher mass scales.
Since at a fixed Eddington ratio it is harder to detect [OIII] lines for lower mass black holes,
the Eddington ratios can be distributed down to a much lower values
for higher mass black holes and galaxies as shown in Figure \ref{fig7}.
In addition, we find that the AGN power is correlated with $u-r$ of galaxies.
Blue AGN-host galaxies show significantly higher Eddington ratio than red AGN-host
galaxies, implying that gas-rich systems generally have higher Eddington ratio
than gas-poor systems. However, we note that this correlation can be also caused by
the Eddington incompleteness since bluer color galaxies have on average lower galaxy (hence black
hole) mass.
Nevertheless, we can compare the distributions of the Eddington ratio
among strong-barred, weak-barred, and non-barred galaxies.
We measure median values of log ($L_{\rm [OIII]}/M_{\rm BH}$) as a function of $u-r$, $\sigma$, and $M_{\rm star}$
for each bar class.
In panel (a) and (d), it is shown that median curves for strong-barred and weak-barred galaxies
lie slightly above those for non-barred galaxies,
although the differences between barred and non-barred
galaxies are not significant and the median Eddington ratios are consistent
within the error.
In other panels there is no difference between barred
and non-barred AGN-host galaxies over the whole ranges of $\sigma$ and $M_{\rm star}$.
To avoid the effect of Eddington incompleteness, we plot the Eddington ratio versus $u-r$ diagram
at two fixed $\sigma$ ranges in Figure \ref{fig8}.
The anti-correlation between the Eddington ratio and $u-r$ is still
present for both 70 km s$^{-1}<\sigma\le120$ km s$^{-1}$ and 120 km s$^{-1}<\sigma\le170$ km s$^{-1}$ ranges,
indicating that AGN power is correlated with the amount of cold gas in galaxies.
\citet{choi+09} also found a similar result among late-type AGN-host galaxies with $7<{\rm log}$
$M_{\rm BH}/M_{\odot}<8$.
We note that the contributions to the [OIII] lines from star formation can
systematically increase the $L_{\rm [OIII]}$/$M_{\rm BH}$ ratio in bluer galaxies.
Thus, further analysis is required to explore the connection between
the presence of gas in large scales and the Eddington ratio.
When we compare the Eddington ratio distributions of barred and non-barred
galaxies in Figure \ref{fig8}, the median values of the Eddington ratio are not significantly
different, implying that AGN power is not strongly affected by the presence of bars.
Considering the scatter in each bin, and the uncertainty and systematic errors in estimating
black hole masses from the $M_{\rm BH}-\sigma$ relation, we conclude that
there is no strong difference of the Eddington ratios between barred and non-barred
galaxies.
\section{Discussion}
\begin{figure*}
\centering
\includegraphics[scale=0.8]{fig11.eps}
\caption{Comparison of Eddington ratio as a function of $M_{\rm star}$
between strong-barred (SB), weak-barred (WB), and non-barred (NB) AGN-host galaxies with (upper)
${\rm S/N}\geq3$, (lower) ${\rm S/N}\geq6$,
when adopting $M_{\rm BH}-\sigma$ relations given by (left) \citet{gultekin+09}: $(\alpha,\beta)=(7.67,1.08)$ for barred and
(8.19,4.21) for non-barred, (middle) \citet{graham+09}: (8.03,3.94) for barred and (8.15,3.89) for non-barred, and
(right) \citet{graham+11}: (7.80,4.34) for barred and (8.25,4.57) for non-barred galaxies, respectively.}
\label{comp_Eddr_diffmsigrel}
\end{figure*}
\subsection{Do AGNs Favor Barred Galaxies?}
Over the last three decades, bars have been invoked as a mechanism for fueling SMBHs.
Although some studies claimed that AGNs are more frequently found
in barred galaxies \citep{arsenault89,knapen+00,laine+02},
no excess of bars in AGN-host galaxies has been reported by many other statistical studies
\citep{moles95,mcleod95,mulchaey97,ho97,laurikainen+04,hao+09}.
This discrepancy was at least in part caused by the small sample size, selection effect,
and contamination owing to the correlation between bars and other galaxy properties,
i.e., color and stellar mass.
In this study we find that
the fraction of strong bars is $\sim$2.5 times higher in AGN-host galaxies than
in non-AGN galaxies. This result is clearly different from the findings by \citet{hao+09},
who claimed no excess of bar fraction in AGN-host galaxies
based on a sample of 1,144 SDSS disk galaxies with $-18.5>M_{g}>-22.0$ at $0.01<z<0.03$.
The discrepancy is due to the combination of two effects.
First, by using a color cut \citep{bell+04,barazza+08} in selecting disk galaxies,
\citet{hao+09} inevitably excluded red disk galaxies,
leading to a much lower AGN fraction in their sample (11.1\%) than
that in our sample (17.0\%).
Second, they used the ellipse fitting method to identify bars.
In general the bar fraction based on the ellipse fitting \citep[e.g., $\sim$50\% in $r$-band:][]{barazza+08} is much higher than that
obtained by visual inspection (e.g., $\sim$33\% in $B$-band: \citealt{rc3}, $26\%\pm0.5\%$ in $g+r+i$ color images:
\citealt{nair+10a}, $29.4\%\pm0.5\%$ from Galaxy Zoo: \citealp{masters+10}, 30.4\%: Paper I, $36\%$: \citealt{oh+11}).
Thus, the combined effects of differences in sample selection and the
classification method between Hao et al. and ours result in different findings.
We also find that the AGN fraction is twice higher in strong-barred galaxies than in non-barred galaxies
as previous studies similarly reported \citep{arsenault89,knapen+00,laine+02}.
As shown in Figure \ref{frac_agn_mass},
the higher AGN fraction in barred systems is present over a large ranges of
$M_{\rm star}$. This result is consistent with those of recent studies \citep{coelho+11,oh+11}.
However, the excess of the bar fraction in AGN-host galaxies and
the excess of the AGN fraction in barred galaxies do not indicate
that the presence of bars and AGN activity are directly connected
since the excess of the bar fraction in AGN-host galaxies and
the excess of the AGN fraction in barred galaxies disappear
when we compare galaxies with the same $u-r$ and $\sigma$ (or $M_{\rm star}$)
(see Figure \ref{fig6} and Figure \ref{agnfrac_barredness}).
Thus, we conclude that AGN activity is not dominated by the presence of bars.
Comparing the Eddington ratios among AGN-host galaxies,
we find no significant difference between barred and
non-barred galaxies at fixed $u-r$, $\sigma$ and $M_{\rm star}$ bins
(see Figure \ref{fig7}),
indicating that AGN power is not enhanced by the presence of bars.
Among AGN host galaxies with $2.0<u-r<2.5$,
the Eddington ratios are marginally higher in strong-barred
galaxies than in non-barred galaxies (see Figure \ref{fig8}),
possibly implying that strong bars can boost AGN activity when their host galaxies
lie in green valley.
However, the number of AGN-host galaxies in our sample is not enough
to draw a clear conclusion.
\subsection{Dependence on the $M_{\rm BH}-\sigma$ Relation}
Since we estimate black hole masses from stellar velocity dispersion utilizing
the $M_{\rm BH}-\sigma$ relation, the derived Eddington ratios depend on the slope
and intercept of the $M_{\rm BH}-\sigma$ relation. Thus, it is necessary
to investigate whether the Eddington ratio difference
between barred and non-barred galaxies depends on the adopted
$M_{\rm BH}-\sigma$ relation.
Over the last decade, empirical scaling relations between $M_{\rm BH}$ and $\sigma$ have been
improved as the number of galaxies with $M_{\rm BH}$ measurements
increased \citep[e.g.,][]{ferrarese+00,gebhardt+00,tremaine+02,gultekin+09,graham+09,woo+10}.
By combining two new $M_{\rm BH}$ measurements
with the literature data published before August 2011,
a recent study by \citet{mcconnell+11} provided two $M_{\rm BH}-\sigma$ relations;
${\rm log}~(M_{\rm BH}/M_{\odot}) = \alpha+\beta~{\rm log}~(\sigma/200 {\rm km ~s^{-1}})$
with $(\alpha,\beta) = (8.38\pm0.06,4.53\pm0.40)$ for elliptical/S0 galaxies and
$(7.97\pm0.22,4.58\pm1.25)$ for spiral galaxies.
In this study we adopt the second equation for estimating $M_{\rm BH}$
since our sample is composed of late-type galaxies.
A few studies separately derived $M_{\rm BH}-\sigma$ relations for barred and non-barred galaxies
\citep{gultekin+09,graham+09,graham+11}.
For example, \citet{gultekin+09} reported $(\alpha,\beta) = (8.19\pm0.087,4.21\pm0.446)$
for non-barred and $(7.67\pm0.115,1.08\pm0.751)$ for barred galaxies
while \citet{graham+09} derived $(8.15\pm0.05,3.89\pm0.18)$ for non-barred galaxies, and
$(8.03\pm0.05,3.94\pm0.19)$ for the combined sample of barred and non-barred galaxies.
\citet{graham+11} reported steeper $M_{\rm BH}-\sigma$ relations
with $(8.25\pm0.06,4.57\pm0.35)$ for non-barred galaxies and
$(7.80\pm0.10,4.34\pm0.56)$ for barred galaxies.
To demonstrate the dependence of Eddington ratios on the adopted $M_{\rm BH}-\sigma$ relation,
we compare in Figure \ref{comp_Eddr_diffmsigrel}
Eddington ratios between barred and non-barred galaxies
using three different pairs of $M_{\rm BH}-\sigma$ relations mentioned above.
When the $M_{\rm BH}-\sigma$ relations from \citet{gultekin+09} are used,
barred and non-barred galaxies show no significant difference in Eddington ratios
as similarly shown in Figure \ref{fig7}.
In contrast, Eddington ratios tend to be higher in barred galaxies than in non-barred galaxies
when the $M_{\rm BH}-\sigma$ relations are taken from \citet{graham+09}.
The enhanced Eddington ratios in barred galaxies is particularly noticeable
when using \citet{graham+11}'s $M_{\rm BH}-\sigma$ relations. since $M_{\rm}$
in barred galaxies are significantly reduced due to the lower intercept
of the $M_{\rm BH}-\sigma$ relation.
If we adopt different $M_{\rm BH}-\sigma$ relations respectively for barred and non-barred galaxies,
AGN power appears to be enhanced by bars.
\citet{oh+11} used the $M_{\rm BH}-\sigma$ relations taken from \citet{graham+09}, and
they argued that AGN strength is enhanced by the presence bars.
However, there are several limitations in adopting two different relations
for barred and non-barred.
First, the $M_{\rm BH}-\sigma$ relation of barred galaxies is not well defined
since the relation has been derived with a small number of barred galaxies.
For example, \citet{gultekin+09} used only 8 measurements (and 11 upper limits of $M_{\rm BH}$)
of barred galaxies while
\citet{graham+11} also used only 20 barred galaxies. Second, the $M_{\rm BH}-\sigma$ relation of
non-barred galaxies
are biased to early-type galaxies since early-type galaxies are dominant in the sample.
Third, dynamical mass measurements of black holes in barred galaxies
are much more uncertain
since no stellar dynamical model truly accounts for stellar bars \citep{gultekin+09a}.
Therefore, we decide to use only one $M_{\rm BH}-\sigma$ relation for late-type galaxies
as given in \citet{mcconnell+11}, in order to avoid any systematic uncertainties of the
$M_{\rm BH}-\sigma$ relations between barred and non-barred galaxies.
It is necessary to investigate whether barred galaxies have higher Eddington ratios than non-barred
galaxies when more robust $M_{\rm BH}-\sigma$ relations of barred and non-barred late-type
galaxies become available in the future.
\subsection{What Triggers AGNs?}
Based on the statistical analysis using a large sample of $\sim$9000 late-type galaxies, we
find that AGN activity is not dominated by the presence of bars.
Then, what triggers AGNs?
Several numerical simulations suggested that interactions and mergers between galaxies are main triggers for AGN activity
\citep{noguchi87,hernquist89,barnes91,barnes92,mihos96,dimatteo+05,hopkins+06,debuhr+11}.
This scenario is supported by several observational studies. For example, \citet{sanders88}
showed that ultra-luminous infrared galaxies (ULIRG) and quasars are formed
through the strong interaction or merger between gas-rich spirals.
\citet{bahcall97} found that twenty nearby luminous quasars ($z<0.3$)
in their sample have galaxy companions that are closer than 25 kpc.
In the case of lower luminosity AGNs, i.e., Seyfert galaxies,
minor mergers between gas rich galaxies and with their satellite galaxies
are proposed as a mechanism for triggering AGN activity \citep[e.g.,][]{derobertis98}.
However, some observational studies showed conflicting results. \citet{fuentes88} found
a marginal evidence that Seyfert galaxies interact with their companions that have comparable sizes.
In addition, by investigating the environmental dependence of AGN fraction using the SDSS sample,
\citet{miller+03} claimed that the fraction of AGN-host galaxies is independent on environment
\citep[see also,][]{coziol98,shimada+00,schmitt+01}.
Recently, \citet{martinez+10} reported that AGN-host galaxies (45\%) are more frequent than composite (23\%)
or star-forming galaxies (32\%) in the Hickson compact group environment where galaxy-galaxy interactions occur violently.
\citet{hwang+11} and Choi et al. (in prep.) also found, using the SDSS data, that the AGN fraction increases
as the distance to a nearest late-type neighbor galaxy decreases in both cluster and field environments,
and concluded that AGN activity can be triggered through mergers and interactions between galaxies
when gas supply for AGN is available.
In contrast, \citet{dasilva+11} claimed that merging galaxies with signatures of recent
starburst, found in the green valley, had no detectable AGN activity.
Although it is theoretically clear that mergers and interactions between galaxies
can provide advantages for AGN activity by reducing angular momentum of interstellar medium
and by generating gas inflows to the center of galaxies, observational studies do not
clearly show the connection between galaxy interaction and AGN activity.
Additional mechanisms are expected to occur at subkiloparsec scales
in order to influence the central black hole directly.
Secondary bars residing in the nuclear region, which are
often called as nested bars, nuclear bars or inner bars,
have been a strong candidate. Some observational works
\citep{shaw95,wozniak95,friedli96,mulchaey97,jungwiert97,greusard+00,
emsellem+01,emsellem+06,laurikainen+07} found secondary bars in the central region of galaxies.
These secondary bars are predicted by the ``bars within bars'' scenario
\citep{shlosman89}. The dynamical and kinematical properties of secondary bars are investigated
by simulations \citep{englmaier+04,maciejewski+08,shen+09,shen+11} and by observations \citep{garcia98,schinnerer+06,delorenzo+08}.
Similarly, nuclear dust spirals are invoked as a means to transport material from kiloparsec scales
down to sub-kpc scales. High resolution images from HST provided a close look at dusty structures in
nuclear regions \citep{malkan98,regan99,martini99,pogge+02}.
\citet{martini+03} classified nuclear spirals into several morphological types:
grand-design, tightly wound, loosely wound, chaotic nuclear spirals. Recently, \citet{hopkins+10}
showed that gas structures formed by gravitational instabilities at several parsec
scale have diverse morphologies: spirals, rings, clumps and also bars.
So, they proposed ``stuff within stuff'' model that is a revised version of \citet{shlosman89}'s
model. However, there is another argument that the presence of nuclear spirals is not also directly connected
with current AGN activity, because the frequency of nuclear spirals in AGN-host galaxies is comparable with that in
non-AGN galaxies \citep{martini+03}. Thus, further studies are needed to investigate any relation between
nuclear spirals (or secondary bars) and activity in galactic nuclei.
\section{Summary and Conclusions}
We investigate the relation between the presence of bars and AGN activity,
using a sample of 8655 late-type galaxies with $b/a>0.6$ and $M_{r} < -19.5$
at $0.02\le z\le 0.05489$, selected from the SDSS DR7.
We divide these galaxies into three spectral types: star-forming, composite,
and AGN-host galaxies, and classified them into barred (storng \& weak) and non-barred galaxies
by visual inspection.
We summarize our main findings as follows.
1. The strong bar fraction is $\sim$2.5 times higher in AGN-host galaxies
than in star-forming galaxies. However, the excess of $f_{\rm SB}$
is caused by the fact that AGN-host galaxies have on average redder $u-r$ color
and higher $\sigma$ than non-AGN galaxies
since $f_{\rm SB}$ is higher for redder and more massive
(higher $\sigma$) galaxies.
The excess of $f_{\rm SB}$ in AGN-host galaxies disappears when
galaxies with the same $u-r$ and $\sigma$ (or $M_{\rm star}$) are compared,
indicating that $f_{\rm SB}$ do not depend on AGN activity.
2. Strong-barred galaxies have higher AGN fraction than weak-barred
or non-barred galaxies.
However, we find no difference of the AGN fraction between barred and non-barred galaxies,
when we compare galaxies with the same $u-r$ color and $\sigma$ (or $M_{\rm star}$),
indicating that AGN activity is not dominated by the presence of bars.
3. Among AGN-host galaxies, barred and non-barred systems
show similar Eddington ratio distributions as a function of $u-r$,
$\sigma$, and $M_{\rm star}$, implying that AGN power is not enhanced by bars.
In conclusion we do not find any evidence that bars trigger AGN activity.
Thus we argue that there is no direct connection between AGN activity and the presence of bars.
\vspace{1cm}
We thank the anonymous referee for his/her useful comments which improved significantly the original manuscript.
G.H.L. thank Changbom Park and Yun-Young Choi for providing help in producing the KIAS VAGC and performing
the morphology classification.
M.G.L. was supported in part by Mid-career Research Program through the NRF grant funded by the MEST (no.2010-0013875).
J.H.W. acknowledges support by the Basic Science Research Program through the NRF funded by the MEST (no.2010-0021558).
H.S.H. acknowledges the Centre National d'Etudes Spatiales (CNES) and the Smithsonian Institution for the support of
his post-doctoral fellowship.
|
{
"timestamp": "2012-03-09T02:01:16",
"yymm": "1203",
"arxiv_id": "1203.1693",
"language": "en",
"url": "https://arxiv.org/abs/1203.1693"
}
|
\section{Introduction}
\label{Introduction}
\hl{Thermodynamic measurements with both good absolute accuracy and high resolution are essential to understand fundamental properties of materials.} Demand for nanocaloric measurements is coming both from the wish to study new physics at mesoscopic scales and the need to investigate bulk behavior such as anisotropy and magnetic field dependence of novel materials that are difficult to synthesize as large crystals. A calorimetric method particularly suitable for studying sub-$\upmu$g samples is the ac steady state method \cite{Sullivan,RiouRSI,Minakov,Huth,RydhEMSAT,Garden,TagliatiCalori,Kohama}. This method usually has high resolution but rather low absolute accuracy because of the difficulties related to the choice of the correct working frequency \cite{RiouSM}. A poorly selected working frequency results in thermal disconnection of the sample, or, for small samples, uncontrolled contribution of the frequency dependent \hl{addenda} heat capacity. A detailed analysis of the behavior of the calorimetric system is thus required to know the conditions at which the most accurate results are obtained.
In the primary work by Sullivan and Seidel~\cite{Sullivan} heater and thermometer were attached directly to the sample, which was connected to the thermal bath through a suitable thermal link. Such a design is not feasible in membrane-based calorimeters, where both thermometer and heater need to be thin films directly fabricated onto a free-standing membrane to maintain the heat capacity \hl{addenda} lower than the sample heat capacity. This separation of calorimeter cell from sample is not necessarily a drawback. It simplifies the system analysis and increases the experimental reproducibility, since there is only one thermal link to the sample that may vary from experiment to experiment rather than two or more. The most ideal situation is if, furthermore, both heater and thermometer are in good thermal contact with each other and with the membrane. The thermal diagram of such a system is depicted in Fig.~\ref{Fig_Scheme}. Heater, thermometer and membrane form a central sample platform, where the internal thermal coupling between thermometer and heater is assumed ideal. The platform is weakly connected, through the membrane thermal conductance $K_\mathrm{e}$, to a thermal bath at temperature $T_\mathrm{b}$. The sample is thermally connected to the platform by a minute amount of Apiezon grease or similar. A fundamental requirement for accurate measurements is that the thermal link $K_\mathrm{i}$ between sample and platform is much greater than $K_\mathrm{e}$.
The system of Fig.~\ref{Fig_Scheme} was analyzed by Velichkov~\cite{Velichkov} for the case corresponding to a massless thermal link. Riou~\textit{et al.}~\cite{RiouSM} extended the analysis to include the thermal diffusivity of sample and membrane, following the work by Greene~\textit{et al.}~\cite{Greene}.
In this work we study the system both theoretically and experimentally. We show that the system model describes our experimental data well, provided that the frequency dependence of both the membrane \hl{addenda} and the membrane thermal link are taken into consideration for small samples. We finally suggest a criterion for the working frequency which ensures combined high resolution and good absolute accuracy.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.8\linewidth]{Fig1_ThermalSketch.pdf}%
\end{center}
\caption{Thermal diagram of the studied system. Sample with heat capacity $C_\mathrm{s}$ and temperature $T_\mathrm{s}$ is coupled through a thermal conductance $K_\mathrm{i}$ to a platform ($C_\mathrm{0}$, $T_\mathrm{0}$) which, in turn, is connected to the thermal bath through the supporting membrane. Heater and thermometer are thin films lying on top of each other. They compose the platform together with the central part of the membrane. The heat capacity $C_\mathrm{0}$ represents an always existing background contribution of the calorimetric cell. The membrane outside the platform is treated either as a massless thermal conductance $K_\mathrm{e}$ or as a distributed object with a total heat capacity $C_{\mathrm{m}}$.}
\label{Fig_Scheme}
\end{figure}
\section{AC steady state measurements}
In the ac steady state method the temperature of sample and calorimetric cell is modulated with a small amplitude, giving rise to a temperature oscillation $T_\mathrm{ac}=T_\mathrm{ac,0}\sin(\omega t-\phi)$ of the thermometer.
The power responsible for this modulation in our case is due to resistive heating, and can be expressed as $P(t)=P_0(1+\sin\omega t)$, where $P_0=R_\mathrm{h}I^{2}_\mathrm{0}$, corresponding to an ac current with amplitude $I_\mathrm{0}\sqrt{2}$ and angular frequency $\omega'=\omega/2$ flowing through a resistor $R_\mathrm{h}$. With the use of a lock-in amplifier both the temperature oscillation amplitude $T_\mathrm{ac,0}$ and phase $\phi$ are experimentally accessible.
They are given by \cite{Gmelin}
\begin{numcases}{}\label{EqTPhi}
\begin{aligned}
T_\mathrm{ac,0} &=\frac{P_\mathrm{0}}{\sqrt{(\omega C)^{2}+K^{2}}} \\
\tan\phi &=\frac{\omega C}{K}
\end{aligned}
\end{numcases
where, for the ideal case of a perfectly connected sample and massless membrane, $C=C_0+C_\mathrm{s}$ and $K=K_\mathrm{e}$. Equation~(\ref{EqTPhi}) can be reshaped to express the unknown $C$ and $K$ as a function of the measured parameters ($P_0$, $\omega$, $T_\mathrm{ac,0}$, $\phi$):
\begin{numcases}{}\label{EqCK}
\begin{aligned}
C &=\frac{P_\mathrm{0}}{\omega T_\mathrm{ac,0}}\sin\phi \\
K &=\frac{P_\mathrm{0}}{T_\mathrm{ac,0}}\cos\phi
\end{aligned}
\end{numcases
These expressions form the basics for evaluating ac steady state measurements.
At low frequency, $\omega \tau_\mathrm{e} \ll 1$, where
\begin{equation}
\tau_\mathrm{e}=(C_\mathrm{0}+C_\mathrm{s})/K_\mathrm{e}
\end{equation
is the external time constant, the phase is close to 0 and the temperature response is dominated by the thermal link. In the opposite limit the phase ideally approaches 90$^{\circ}$. For $\omega \tau_\mathrm{e} > 7$ the absolute error arising from using the simple expression $C=P_\mathrm{0}/\omega T_\mathrm{ac,0}$ is less than 1\%. However, the conditions are rarely ideal at such frequencies. To obtain absolute accuracy, a good understanding of the system and a carefully selected frequency are therefore required.
When measuring small samples, $C_\mathrm{s} \lesssim C_\mathrm{m}$, the effect of the membrane should be considered. The thermal diffusion in the membrane introduces a new frequency dependence controlled by the parameter
\begin{equation}
\tau_\mathrm{m}=C_\mathrm{m}/K_\mathrm{e}.
\end{equation
For most samples the influence of the thermal link between sample and cell is also important. A weak connection between sample and platform introduces yet another time scale, described by the internal time constant
\begin{equation}
\tau_\mathrm{i}=C_\mathrm{s}/K_\mathrm{i}.
\end{equation
We first study the effect of the membrane, assuming that the sample is well connected ($\tau_\mathrm{i}=0$). We then include the effect of a finite thermal link between sample and platform.
\section{Effect of membrane}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.75\linewidth]{Fig2a_Memb_Tac.pdf}
\\[3mm]
\includegraphics[width=0.75\linewidth]{Fig2b_Memb_TanPhi.pdf}
\end{center}
\caption{Frequency dependence of $T_\mathrm{ac,0}$ and $\tan \phi$ for an empty cell ($C_\mathrm{s}=0$) at $T=50\,\mathrm{K}$. \textbf{a)} Temperature oscillation amplitude $T_\mathrm{ac,0}$ divided by its low frequency value $T_\mathrm{dc}=P_0/K_\mathrm{e}$. Numerical simulations are from \cite{Tagliati_Sim}. \textbf{b)} Phase angle $\phi$ between power and temperature, expressed as $\tan \phi$. The model fits are given by Eq.~(\ref{EqTPhi}) with $C=C_0+C_\mathrm{m,eff}$, $K=K_\mathrm{e,eff}$, and effective frequency dependences from Eq.~(\ref{EqCeff}) and (\ref{EqKeff}). The dashed curves were obtained by using the low-frequency limits $C=C_\mathrm{0}+C_\mathrm{m}/3$ and $K=K_\mathrm{e}$. The parameters in all cases are $C_\mathrm{0}=3.18\,\mathrm{nJ/K}$, $C_\mathrm{m}=33.2\,\mathrm{nJ/K}$, and $K_\mathrm{e}=1.63\,\upmu\mathrm{W/K}$.}
\label{Fig_Memb_TacPhi}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.75\linewidth]{Fig3a_Memb_C.pdf}
\\[3mm]
\includegraphics[width=0.75\linewidth]{Fig3b_Memb_K.pdf}
\end{center}
\caption{Membrane properties. \textbf{a)} Effective membrane heat capacity. \textbf{b)} Effective thermal conductance. The experimental data were evaluated through Eq.~(\ref{EqCK}). All parameters, experimental data, and assumed frequency dependences are the same as in Fig.~\ref{Fig_Memb_TacPhi}.}
\label{Fig_Memb_CK}
\end{figure}
For small samples, the membrane can no longer be approximated as a massless thermal conductance, but should be seen as a total heat capacity $C_\mathrm{m}$ distributed over the membrane area. The generated temperature oscillation spreads in the membrane, including the metal leads of thermometer and heater, over a distance of the order of a frequency dependent thermal length $\ell_\mathrm{th}(\omega)=\sqrt{{2D}/{\omega}}$ \cite{RiouSM}, where $D=\kappa/\rho c_\mathrm{p}$ is the diffusivity, $\kappa$ the thermal conductivity, $\rho$ the density, and $c_\mathrm{p}$ the specific heat. For simplicity, we here model the membrane as a 1D rod. As shown in \ref{Freq 1D} for this model, following Greene \textit{et al.} \cite{Greene}, the diffusion practically results in an effective frequency dependent \hl{addenda} heat capacity from the membrane, given by
\begin{equation}
C_\mathrm{m,eff}=\frac{C_\mathrm{m}}{\alpha}\frac{\sinh\alpha-\sin\alpha}{\cosh\alpha-\cos\alpha},
\label{EqCeff}
\end{equation
with $\alpha=\sqrt{2\omega \tau_\mathrm{m}}$. The limits for $C_\mathrm{m,eff}$ are $C_\mathrm{m}/3$ at low frequency and $C_\mathrm{m}/\alpha$ at high frequency. With increasing frequency the effective part that is temperature modulated thus shrinks, decreasing the membrane \hl{addenda} as $\omega^{1/2}$.
However, not only $C=C_\mathrm{0}+C_\mathrm{s}+C_\mathrm{m,eff}$ becomes frequency dependent but also $K$. While the sensed heat capacity decreases, the thermal conductance increases because of a shorter effective distance of the temperature (amplitude) gradients. We show in \ref{Freq 1D} that it is natural to replace $K_\mathrm{e}$ by an effective thermal conductance
\begin{equation}
K_\mathrm{e,eff} = K_\mathrm{e} \frac{\alpha}{2}\frac{\sinh\alpha+\sin\alpha}{\cosh\alpha-\cos\alpha}.
\label{EqKeff}
\end{equation
The low-frequency limit of $K_\mathrm{e,eff}$ is again $K_\mathrm{e}$, while the high-frequency limit is given by $\alpha K_\mathrm{e}/2$.
In Fig.~\ref{Fig_Memb_TacPhi} we apply Eq.~(\ref{EqCeff}) and (\ref{EqKeff}) to describe our experimental data of a calorimeter cell without sample. \hl{The calorimeter is made of thin film heater, electrical insulation and thermometer built on top of each other. This stack covers the central $110\times110~\upmu\mathrm{m}^2$ area of a $1\times1~\mathrm{mm}^2$ and $150~\mathrm{nm}$ thick Si$_3$N$_4$ membrane, and form the platform onto which the sample is placed} \cite{Tagliati_Sim}.
The value of $K_\mathrm{e}$ is easily determined from $T_\mathrm{dc}$ at low frequency, while $C_\mathrm{m}$ is obtained from adjusting $\tau_\mathrm{m}$.
Figure~\ref{Fig_Memb_CK} shows the experimentally determined $C$ and $K$ for the empty cell, using Eq.~(\ref{EqCK}), with the corresponding model curves obtained by using the same parameter values as in Fig.~\ref{Fig_Memb_TacPhi}.
Note that $C_\mathrm{m,eff}$ is dominating over $C_\mathrm{0}$ at all frequencies. At the highest frequencies, $K$ seems to have a tendency to saturate and $C$ decreases slightly faster than expected. Whether this is due to an experimental problem, such as a phase distortion, or if a more realistic model would display a different high-frequency behavior remains an open question.
\section{Effect of thermal link to sample}
In the normal, experimental case, the effect of a non-zero internal time constant limits the upper frequency of measurements. The expressions for the stationary temperature oscillation amplitudes and phase shifts are derived in \ref{App2} for $C_\mathrm{m}=0$. Surprisingly, $T_\mathrm{ac,0}$ and $\tan\phi$ are still given by Eq.~(\ref{EqTPhi}) provided that $C$ and $K$ are defined by Eq.~(\ref{Eq_Cbar}) and Eq.~(\ref{Eq_Kbar}). Studying these equations, we see that in the low frequency limit ($\omega\tau_\mathrm{i}\ll1$), $C=C_0+C_\mathrm{s}$ and $K=K_\mathrm{e}$, i.e., the temperature is uniformly distributed in the sample-cell system. In the oppsite limit ($\omega\tau_\mathrm{i}\gg1$), the sample is thermally disconnected from the platform and just the heat capacity of the platform is probed. Nevertheless the presence of the sample is still sensed by an increase of the apparent thermal conductance, $K=K_\mathrm{e}+K_\mathrm{i}$.
From the solution (\ref{Eq_Solution}), we see that the actual sample temperature oscillation amplitude decreases faster than $T_\mathrm{ac,0}$ when the frequency increases,
\begin{equation} \label{Tac_sample}
T_\mathrm{ac,s}=T_\mathrm{ac,0}\sqrt{1-g},
\end{equation
where $g$ is defined in Eq.~(\ref{g}), going from 0 to 1 with increasing frequency. The ac signal from a thermometer in direct contact with the sample consequently goes to zero for $\omega \tau_\mathrm{i} \gg 1$, while a thermometer on the cell, as in the present case, still gives useful information such as a rough estimate of the cell \hl{addenda}.
Figure~\ref{Fig_Sample_TacPhi} shows $T_\mathrm{ac,0}$, $\tan \phi$ \hl{and the transfer function} in a typical measurement.
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.75\linewidth]{Fig4a_Sample_Tac.pdf}
\\[3mm]
\includegraphics[width=0.75\linewidth]{Fig4b_Sample_TanPhi.pdf}
\\[2mm]
\includegraphics[width=0.75\linewidth]{Fig4c_Sample_TraFunCs.pdf}
\end{center}
\caption{\textbf{a)} Temperature oscillation amplitude of cell with sample and empty cell at $T=227\,\mathrm{K}$. Dashed curves correspond to a well-connected sample ($g=0$) and no sample ($C_\mathrm{s}=0$). \textbf{b)} $\tan \phi$ for cell with sample. Dashed curves correspond to a well-connected sample ($g=0$) and a disconnected sample ($g=1$). \hl{\textbf{c)} Transfer function of cell with sample.} The arrows indicate the frequency where 1\% of $C_\mathrm{s}$ has been decoupled ($g=0.01$). The fitting parameters of cell with sample are: $C_\mathrm{s}=1.93\,\upmu\mathrm{J/K}$, $K_\mathrm{e}=3.09\,\upmu\mathrm{W/K}$, $K_\mathrm{i}=187\,\upmu\mathrm{W/K}$, $C_\mathrm{m}=280\,\mathrm{nJ/K}$, and $C_\mathrm{0}=42\,\mathrm{nJ/K}$. The parameters for the empty cell measurement are $K_\mathrm{e}=3.00\,\upmu\mathrm{W/K}$ and $C_\mathrm{0}=25\,\mathrm{nJ/K}$.}
\label{Fig_Sample_TacPhi}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.75\linewidth]{Fig5a_Sample_C.pdf}
\\[3mm]
\includegraphics[width=0.75\linewidth]{Fig5b_Sample_K.pdf}
\end{center}
\caption{\textbf{a)} Heat capacity of cell with sample. \textbf{b)} Thermal conductance. Model curves are given by Eq.~(\ref{Eq_CKtotal}) with the same parameter values as in Fig.~\ref{Fig_Sample_TacPhi}. The arrows indicate the frequency where 1\% of $C_\mathrm{s}$ has been decoupled ($g=0.01$).}
\label{Fig_Sample_CK}
\end{figure}
The sample is a $\sim12.7\,\upmu\mathrm{g}$ piece of gold attached to the platform through Apiezon-N grease. A clear signature of the decoupling of the sample is seen both in $T_\mathrm{ac,0}$, which no longer goes as $1/\omega$, and in $\tan \phi$, which displays a characteristic decrease during the process. To fit the behavior of Fig.~\ref{Fig_Sample_TacPhi}, we introduce the effect of the membrane into the solution for a weakly connected sample by replacing $C_\mathrm{0}$ in Eq.~(\ref{Eq_Cbar}) with $C_\mathrm{0}+C_\mathrm{m,eff}$ and $K_\mathrm{e}$ in Eq.~(\ref{Eq_Kbar}) with $K_\mathrm{e,eff}$. We thus have
\begin{numcases}{}
\begin{aligned}
C &= C_0+C_\mathrm{m,eff} + (1-g)C_\mathrm{s}\\
K &=K_\mathrm{e,eff}+gK_\mathrm{i}
\end{aligned}\label{Eq_CKtotal}
\end{numcases
This is a central result of the paper. The determination of the parameters is made easier by first fitting a measurement of $T_\mathrm{ac}$ for the empty cell, as shown in Fig.~\ref{Fig_Sample_TacPhi}a. This gives a well-determined value of $C_\mathrm{m}$. $K_\mathrm{e}$ is obtained from the known power and the low-frequency $T_\mathrm{ac,0}$, both for the empty cell case and for the case with sample. The values of $K_\mathrm{i}$, $C_\mathrm{s}$, and $C_\mathrm{0}$ are then obtained by fitting. A few observations can be pointed out. First, the values of $K_\mathrm{e}$ are slightly different for cell with sample and empty cell. This difference is within variations between experiments, depending on factors such as residual gas conduction and radiation. \hl{Second, the heat capacity for a Au piece with mass $m = 12.7\,\upmu\mathrm{g}$ at $T=227\,\mathrm{K}$ is about $1.59\,\upmu\mathrm{J/K}$ from literature. The difference between this value and the measured heat capacity is coming from the Apiezon grease. For good absolute accuracy, the heat capacity of the grease should thus be measured separately before attaching a sample.} Third, there is a difference between $C_\mathrm{0}$ for the empty membrane, and the residual $C_\mathrm{0}$ when the sample is disconnected. This difference can be ascribed to some Apiezon grease in good contact with the platform. Fourth, $C_\mathrm{0}$ of the empty membrane is in our case roughly 10\% of $C_\mathrm{m}$ at both discussed temperatures. The total membrane \hl{addenda} can therefore be estimated at any frequency provided that the temperature dependences of $C_\mathrm{m}$ and $K_\mathrm{e}$ are known.
In Fig.~\ref{Fig_Sample_CK}, we show $C$ and $K$ as obtained from Eq.~(\ref{EqCK}) and fitted by Eq.~(\ref{Eq_CKtotal}), with the same parameters as in Fig.~\ref{Fig_Sample_TacPhi}.
In both Fig.~\ref{Fig_Sample_TacPhi} and \ref{Fig_Sample_CK}, arrows mark the frequency above which more than 1\% of the sample heat capacity signal has been lost ($g=0.01$ corresponding to $\omega \tau_\mathrm{i} \approx 0.1$). Note that $\tan\phi$ and $K$ are showing earlier signs of the decoupling than $T_\mathrm{ac,0}$ and $C$. Apparent $1/\omega$ behavior of $T_\mathrm{ac,0}$, \hl{which corresponds to the plateau of the transfer function in Fig.}~\ref{Fig_Sample_TacPhi}c, is thus not enough for good absolute accuracy. \hl{In the middle of the plateau, where experiments are often performed, the absolute accuracy may already be worse than 1\%.}
\section{Optimizing the working frequency}
In practice, it would be too time consuming to measure the frequency dependence of $T_\mathrm{ac,0}$ and $\tan\phi$ at every temperature to determine $C_\mathrm{s}(T)$. It is therefore important to establish well-defined measurement conditions that ensure both good absolute accuracy and high resolution. This can be done by adjusting the frequency of the measurements, the working frequency $\omega_\mathrm{work}$. The best $\omega_\mathrm{work}$ depends on the requirements and limitations of the experiment: absolute accuracy, resolution, allowed dc offset $T_\mathrm{dc}$, and maximum amplitude of $T_\mathrm{ac,0}$.
The resolution of the heat capacity is obtained by differentiating $C$ \hl{in Eq.}~(\ref{EqCK}) with respect to the temperature oscillation amplitude and phase, \hl{$\Delta C = \left|\partial C/\partial T_\mathrm{ac,0}\right|\Delta T_\mathrm{ac,0}+\left|\partial C/\partial \phi\right|\Delta \phi$}. For simplicity, we assume that the sample is well-connected and that the membrane frequency dependence can be neglected. We also assume that the phase resolution is given by $\Delta \phi=\Delta T_\mathrm{ac,0}/T_\mathrm{ac,0}$, where $\Delta T_\mathrm{ac,0}$ is determined by equipment and setup. Under these conditions
\begin{equation} \label{EqDeltaC}
\frac{\Delta C}{C}\approx\frac{\Delta T_\mathrm{ac,0}}{T_\mathrm{ac,0}}\left(1+\frac{1}{\tan\phi}\right).
\end{equation
At constant $T_\mathrm{dc}$ (i.e.\ $P_0$), Eq.~(\ref{EqDeltaC}) is minimized for $T_\mathrm{ac,0}/T_\mathrm{dc}=\cos\phi=1/\sqrt{2}$ corresponding to $\phi= 45^\circ$ or $\tan\phi=1$. If we instead maintain a constant $T_\mathrm{ac,0}$, the resolution can be improved by a factor of 2 by increasing the frequency further, i.e., by decreasing the excess noise factor $1/\tan\phi$.
Considering absolute accuracy, lower frequencies are generally more accurate as long as Eq.~(\ref{EqCK}) is used. Inaccuracies (beyond experimental problems etc.) are due to a nonzero $g$ in Eq.~(\ref{Eq_CKtotal}), in turn caused by a finite $K_\mathrm{i}$. As seen in Fig.~\ref{Fig_Sample_TacPhi}b, the sample starts to decouple near the local maximum of $\tan\phi$. The key parameter that controls this maximum is the ratio
\begin{equation}
\beta \equiv K_\mathrm{i}/K_\mathrm{e}.
\end{equation
The location of the maximum is $\omega_{\max}\tau_\mathrm{e}\approx\sqrt{\beta}$, while its value is $\tan\phi_{\max}\approx\sqrt{\beta}/2$. At the maximum (still neglecting the membrane frequency dependence), the absolute error is
\begin{equation}
g_{\big|\tan \phi_{\max}}=\frac{1}{\beta+2} \approx \frac{1}{\beta}.
\end{equation
At frequencies $\omega < \omega_{\max}$, where $\tan \phi \approx \omega\tau_\mathrm{e}$, the absolute error can be estimated to
\begin{equation}\label{EqAbsError}
g_{\big| \omega < \omega_{\max}} \approx {\left({\frac{\tan\phi}{\beta}}\right)}^2.
\end{equation
With these rules of thumb, consider a few cases. To reach $\tan\phi =7$ ($\phi_{\max}\gtrsim 82^{\circ}$), as required for using the simplified relation $C=P_\mathrm{0}/\omega T_\mathrm{ac,0}$ with good absolute accuracy, a ratio $\beta \gtrsim 200$ is needed. To have less than 1\% error at the maximum of $\tan\phi$, $\beta$ must be greater than 100, corresponding to $\tan\phi_{\max} > 5$ ($\phi_{\max} \gtrsim 79^\circ$). Finally, to have less than 1\% error at $\tan\phi=1$, we only need $\beta > 10$, corresponding to $\tan\phi_{\max} > 1.58$ ($\phi_{\max} \gtrsim 58^\circ$).
From Eq.~(\ref{EqDeltaC}) and Eq.~(\ref{EqAbsError}) it is clear that the conditions are most well-controlled if $\omega$ is adjusted to maintain a constant phase $\phi_\mathrm{work}$. Higher $\tan\phi$ gives less noise, but around $\tan\phi_{\max}$ the accuracy decreases quickly and the system will become sensitive to changes in $\beta$. Looking at Eq.~(\ref{EqAbsError}), it would seem reasonable to choose $\tan\phi$ as a small fraction of $\beta$. However, such a criterion would quickly exceed $\tan\phi_{\max}$ for large $\beta$. A better choice is to take $\tan\phi$ as a fraction of $\tan\phi_{\max}$ which goes as $\sqrt{\beta}$. A suitable number is $\tan\phi= (2/3)\tan\phi_\mathrm{max}$, corresponding to
\begin{equation}
\tan\phi_\mathrm{work}\approx\sqrt{\beta}/3.
\end{equation}
This condition gives $g \approx 1/(9\beta)$ and thus less than $1\%$ error and $100\%$ excess noise as long as $\beta\gtrsim 10$. If we have $\beta=60$ as in Fig.~\ref{Fig_Sample_TacPhi}, we get $\tan\phi\approx 2.6$ ($\phi\approx 69^\circ$). The expected absolute error at this phase is $0.2\%$ and the excess noise is $38\%$, which is a fairly small number considering that a typical achievable resolution $\Delta C/C$ is 1 in $10^4$ to $10^5$.
Since $\beta$ may vary during the measurements, it is important to be able to verifying that $g$ remains small. This can be done by studying $K$. At the point where $\tan\phi\approx\sqrt{\beta}/3$ the measured value of $K$ should be $\beta g \approx 10\%$ higher than $K_\mathrm{e}$, which in turn can be obtained from calibration measurements. In practice, $\beta$ is varying rather slowly, and $\phi_\mathrm{work}$ can often be maintained at a constant value.
\section{Conclusions}
The analysis carried out in this paper illustrates the care needed to obtain absolute accuracy in ac steady-state calorimeter measurements. Two problems that cannot be avoided when studying small samples are the frequency-dependent contribution of the sample support and the thermal link between sample and support. To handle these complications, it is not enough to present a model for the temperature oscillation and phase expressed as a function of time constants. Rather, explicit expressions for the sample heat capacity and external thermal link are needed as a function of experimentally determinable parameters. Here we provide such expressions and show that they can be used to overcome the experimental obstacles. Based on the analysis, we argue that measurements are best performed at a constant phase $\phi$. With modern measurement electronics, such a condition is both feasible and practical.
\section*{Acknowledgments}
Financial support from the K.\ and A.\ Wallenberg foundation and the Swedish Research Council is gratefully acknowledged. We would like to thank V. M. Krasnov and Luca Argenti for useful discussions.
|
{
"timestamp": "2012-03-12T01:01:19",
"yymm": "1203",
"arxiv_id": "1203.2049",
"language": "en",
"url": "https://arxiv.org/abs/1203.2049"
}
|
\section{Introduction}
It is proved in \cite{TeTr} that a simplicial complex $\Delta$ is a complete intersection
if the third power $I_{\Delta}^{3}$ of its Stanley-Reisner ideal
is Cohen-Macaulay,
using a result in \cite{MiT2, Var}.
On the other hand, there is a simplicial complex $\Delta$ which is not a complete intersection
such that $I_{\Delta}^{2}$ is Cohen-Macaulay.
The simplicial complex associated with a pentagon is such an example.
Among one-dimensional simplicial complexes, the above example
is a unique one, as shown in \cite{MiT1}.
As for the two-dimensional case, such simplicial complexes are
classified in \cite{TrTu}.
In \cite{MiT2} a characterization of Cohen-Macaulayness of
the second symbolic power $I_{\Delta}^{(2)}$ is given.
\par
A main motivation of this paper is to study
the Cohen-Macaulayness of the second ordinary powers
of Stanley-Reisner ideals of any dimension.
We consider the following two questions:
\begin{enumerate}
\item What constraints does Cohen-Macaulayness of $I_{\Delta}^{2}$ impose upon a simplicial complex $\Delta$?
\item Do there exist \textit{many} simplicial complexes $\Delta$ such that $I_{\Delta}^{2}$ are Cohen-Macaulay?
\end{enumerate}
\par
As for the second question
we give two families of examples. One is a simplicial join of pentagons;
the other is a stellar subdivision of a complete intersection complex.
\par
For the first question we treat more general properties and give necessary conditions
for Cohen-Macaulayness of the square, as a result.
In each section we pick up a different condition; In Sections 2, 3, and 4 we consider
quasi-Buchsbaum property, Serre's condition $(S_2)$, and unmixedness of a (symbolic) square, respectively.
Summarizing results in these sections, we have the following theorem:
\begin{thm} \label{IntroMain}
Let $\Delta$ be a simplicial complex on $[n]=\{1,2,\ldots,n\}$.
Let $S=K[x_1,\ldots,x_n]$ be a polynomial ring.
Suppose that $S/I_{\Delta}^2$ is Cohen-Macaulay over any field $K$.
Then the following conditions are satisfied:
\begin{enumerate}
\item $\Delta$ is Gorenstein.
\item $\diam ((\link_{\Delta} F)^{(1)}) \le 2$ for any face $F \in \Delta$ with
$\dim \link_{\Delta} F \ge 1$.
\item For $F_1,F_2, F_3 \in 2^{[n]} \setminus \Delta $ there exist $G_1, G_2 \in 2^{[n]} \setminus \Delta $
such that $G_1 \cup G_2 \subset F_1 \cup F_2 \cup F_3 $ and $G_1 \cap G_2 \subset F_1 \cap F_2 \cap F_3 $.
\end{enumerate}
\end{thm}
As shown in Corollary \ref{S2thm} the condition (2)
is equivalent to Serre's condition $(S_2)$ of $S/I_{\Delta}^{(2)}$.
And as shown in Theorem \ref{SpTriangle} the condition (3)
is equivalent to the condition $I_{\Delta}^{2}=I_{\Delta}^{(2)}$.
We may ask the converse:
\begin{quest}
{\rm
Do the conditions (1), (2) and (3) imply
that $S/I_{\Delta}^2$ is Cohen-Macaulay?
}
\end{quest}
It is known that
Cohen-Macaulayness of $I_{\Delta}^{2}$ is equivalent to Cohen-Macaulayness of $I_{\Delta}^{(2)}$ and
$I_{\Delta}^{2}=I_{\Delta}^{(2)}$.
Hence the above question will be affirmative if so is the following one, which is interesting in its own right:
\begin{quest} \label{Q12}
{\rm
Do the conditions (1) and (2) imply
that $S/I_{\Delta}^{(2)}$ is Cohen-Macaulay?
}
\end{quest}
Stronger versions of the first question are as follows:
\begin{quest} \label{Q13}
{\rm
Do the conditions (1) and (3) imply
that $S/I_{\Delta}^2$ is Cohen-Macaulay?
}
\end{quest}
\begin{quest} \label{Q23}
{\rm
Do the conditions (2) and (3) imply
that $S/I_{\Delta}^2$ is Cohen-Macaulay?
}
\end{quest}
\par
By \cite{MiT1},
the above questions are true if simplicial complexes are one-dimensional.
\par
For the case that edge ideals $I(G)$ of graphs $G$ without isolated vertices
are unmixed with the condition $2\height I(G)=n$,
the above questions are also true.
If $I(G)$ is Gorenstein, then it is a complete intersection by \cite{CRT}.
Hence $I(G)^2$ is Cohen-Macaulay and Questions \ref{Q12} and \ref{Q13} are affirmative.
On the other hand, it is proved in \cite{CRTY} that
there is some face $F$ in the simplicial complex $\Delta _2$ corresponding to the polarization of the second
symbolic power $I(G)^{(2)}$ such that $\link_{\Delta _2}F$ is not strongly connected,
if $I(G)$ is not a complete intersection.
This implies that the polarization of $I(G)^{(2)}$ does not satisfy Serre's condition $(S_2)$.
By \cite{MuT}, $I(G)^{(2)}$ does not satisfy Serre's condition $(S_2)$, either.
It means that $I(G)$ is a complete intersection if $I(G)^{(2)}$ satisfies Serre's condition $(S_2)$.
Hence Question \ref{Q23} is also affirmative.
\par \vspace{2mm}
Now let us summarize the organization of the paper.
In Section 1, we fix the terminology which we need later.
\par \vspace{2mm}
In Section 2 we consider quasi-Buchsbaum property, which is weaker than Cohen-Macaulay property.
And we prove the following theorem as a main result in this section:
\par \vspace{2mm} \par \noindent
{\bf Theorem \ref{Buchsbaum}}
Let $\Delta$ be a simplicial complex on $[n]$ of dimension $d-1 \ge 2$.
Let $S=K[x_1,\ldots,x_n]$ be a polynomial ring.
Suppose that $S/I_{\Delta}^2$ is quasi-Buchsbaum over any field $K$.
Then $S/I_{\Delta}$ is Gorenstein.
\par \vspace{2mm}
Since Cohen-Macaulay property implies Serre's condition $(S_2)$,
in Section 3 we give a criterion for $I_{\Delta}^{(2)}$ to satisfy $(S_2)$, which
is a generalization of \cite[Theorem 2.3]{MiT2}; see Theorem \ref{Depth} and
Corollary \ref{S2thm}.
As an application, we show that for Reisner's complex
(a triangulation of the real projective plane) $\Delta$,
$S/I_{\Delta}^{(2)}$ satisfies $(S_2)$ but is \textit{not} Cohen-Macaulay.
\par \vspace{2mm}
In Section 4 we consider the problem
when $I^{(2)} = I^2$ holds for a Stanley-Reisner ideal $I$, which is also a necessary condition
for Cohen-Macaulayness of $I^{2}$.
It is also discussed in \cite{TrTu}.
We give a criterion for the second symbolic power
to be equal to the ordinary power for Stanley-Reisner ideals
in terms of the hypergraph of the generators; see Theorem \ref{SpTriangle}.
This generalizes a similar criterion for edge ideals.
As an application, we show that the second powers of the edge ideals
of finitely many disjoint union of pentagons are Cohen-Macaulay
as in the second symbolic power case in \cite{MiT2}.
\par \vspace{2mm}
In Section 5, we give examples of the complexes whose second powers of
the Stanley-Reisner ideals are Cohen-Macaulay.
More precisely, we prove the following theorem,
which is a generalization of a two-dimensional complex
in \cite[Theorem 3.7 (iii)]{TrTu}.
\par \vspace{2mm} \par \noindent
{\bf Theorem \ref{Subdiv}.}
Let $\Delta$ be a stellar subdivision of a non-acyclic
complete intersection complex $\Gamma$.
Then $S/I_{\Delta}^2$ is Cohen--Macaulay.
\section{Preliminaries}
In this section we recall several definitions and properties that we
will use later.
See also \cite{BH, Ma, St, SV}.
\par \vspace{2mm}
\subsection{Stanley--Reisner ideals}
Let $V=[n]$.
A nonempty subset $\Delta$ of the power set $2^V$
is called a \textit{simplicial complex} on $V$ if
(i) $F \in \Delta$, $F' \subseteq F \Longrightarrow F' \in \Delta$
and (ii) $\{v\} \in \Delta$ for all $v \in V$.
An element $F \in \Delta$ is called a \textit{face} of $\Delta$.
The dimension of $F$ is defined by $\dim F = \sharp(F)-1$, where
$\sharp(F)$ denotes the cardinality of a set $F$.
The dimension of $\Delta$, denoted by $\dim \Delta$,
is the maximum of the dimensions of all faces.
A maximal face of $\Delta$ is called a \textit{facet} of $\Delta$, and
let $\mathcal{F}(\Delta)$ denote the set of all facets of $\Delta$.
\par
In the following, let $\Delta$ be a simplicial complex with
$\dim \Delta =d-1$, and let $K$ be a field.
Then $\Delta$ is called \textit{pure}
if all the facets of $\Delta$ have the same cardinality $d$.
Put $f_i(\Delta)=\sharp\{F \in \Delta\,:\, \dim F =i\}$
for each $i=0,1,\ldots,d-1$.
For each $i$, $\widetilde{H}_i(\Delta; K)$ (resp. $\widetilde{H}^i(\Delta;K)$)
denotes the $i$th reduced simplicial homology (resp. cohomology) of $\Delta$
with values in $K$.
We omit the symbol $K$ unless otherwise specified.
The \textit{reduced Euler characteristic} of $\Delta$ is defined by
\[
\widetilde{\chi}(\Delta) = -1 + \sum_{i=0}^{d-1} f_i(\Delta) =
\sum_{i=-1}^{d-1} (-1)^i \dim_K \widetilde{H}_i (\Delta).
\]
\par
For each face $F \in \Delta$,
the $\textit{star}$ and the
\textit{link} of $F$ are defined by
\[
\star_{\Delta} F = \{H \in \Delta \;:\,H \cup F \in \Delta\}, \quad
\link_{\Delta} F = \{H \in \star_{\Delta} F \;:\, H \cap F = \emptyset\}.
\]
Note that these are also simplicial complexes.
Moreover, we note that for any subset $W \subseteq V$,
$\Delta_W = \{F \in \Delta \,:\, F \subseteq W \}$
is also a subcomplex of $\Delta$.
For any integer $k$ with $0 \le k \le d-1$, the $k$-th \textit{skeleton} of
$\Delta$ is defined by
$\Delta^{(k)} = \{F \in \Delta \,;\, \dim F \le k \}$.
Then $\Delta^{(k)}$ is a subcomplex of $\Delta$ with $\dim \Delta^{(k)} = k$.
\par
The \textit{Stanley--Reisner ideal} of $\Delta$, denoted by $I_{\Delta}$,
is the squarefree monomial ideal of $S=K[x_1,\ldots,x_n]$ generated by
\[
\{x_{i_1} x_{i_2} \cdots x_{i_p} \,:\, 1 \le i_1 < \cdots < i_p \le n,\;
\{x_{i_1},\ldots,x_{i_p}\} \notin \Delta \},
\]
and $K[\Delta]= K[x_1,\ldots,x_n]/I_{\Delta}$ is called
the \textit{Stanley--Reisner ring} of $\Delta$.
Note that the Krull dimension of $K[\Delta]$ is equal to $d$.
For any subset $\sigma$ of $V$, $x_{\sigma}$ denotes
the squarefree monomial in $K[x_1,\ldots,x_n]$ with support $\sigma$.
\par
For a simplicial complex $\Delta$ on $V$, we put
$\core V = \{x \in V\,:\, \star\{x\} \ne V \}$.
Moreover, we define the \textit{core} of $\Delta$ by
$\core \Delta = \Delta_{\core V}$.
\par
For a given face $F$ of $\Delta$ with $\dim F \ge 1$ and
a new vertex $v$, the \textit{stellar subdivision} of $\Delta$
on $F$ is the simplicial complex $\Delta_{F}$ on the vertex
set $V \cup \{v\}$ defined by
\[
\Delta_{F} = \big(\Delta \setminus
\{H \,|\, F \subseteq H \in \Delta\} \big)
\cup
\{H \cup \{v\}\,|\,
H \in \Delta,\,F \not \subseteq H,\,
F \cup H \in \Delta \}.
\]
Notice that $\Delta_{F}$ is homeomorphic to $\Delta$.
\par \vspace{2mm}
\begin{picture}(400,70)
\put(40,28){$\Delta=$}
\thicklines
\put(77,45){\circle*{5}}
\put(77,10){\circle*{5}}
\put(112,45){\circle*{5}}
\put(112,10){\circle*{5}}
\put(64,45){{\tiny $x_1$}}
\put(64,10){{\tiny $y_2$}}
\put(116,45){{\tiny $x_2$}}
\put(116,10){{\tiny $y_1$}}
\put(77,42){\line(0,-1){29}}
\put(112,42){\line(0,-1){29}}
\put(77,45){\line(1,0){35}}
\put(77,10){\line(1,0){35}}
\put(90,50){$F$}
\put(170,25){\vector(1,0){40}}
\put(160,35){{\tiny stellar subdivision}}
\put(277,45){\circle*{5}}
\put(277,10){\circle*{5}}
\put(312,45){\circle*{5}}
\put(312,10){\circle*{5}}
\put(295,62){\circle*{5}}
\put(264,45){{\tiny $x_1$}}
\put(264,10){{\tiny $y_2$}}
\put(316,45){{\tiny $x_2$}}
\put(316,10){{\tiny $y_1$}}
\put(299,64){{\tiny $v$}}
\put(277,42){\line(0,-1){29}}
\put(312,42){\line(0,-1){29}}
\put(277,10){\line(1,0){35}}
\put(277,45){\line(1,1){18}}
\put(312,45){\line(-1,1){18}}
\end{picture}
\par \vspace{2mm}
Let $G$ be a graph, which means a finite graph without loops and
multiple edges.
Let $V(G)$ (resp. $E(G)$) denote the set of vertices (resp. edges) of $G$.
Put $V(G) =[n]$.
Then the \textit{edge ideal} of $G$, denoted by $I(G)$,
is a squarefree monomial ideal
of $S=K[x_1,\ldots,x_n]$ defined by
\[
I(G) = (x_ix_j \,:\, \{i, j\} \in E(G)).
\]
\par
For an arbitrary graph $G$, the simplicial complex $\Delta(G)$
with $I(G) = I_{\Delta(G)}$ is called
the \textit{complementary simplicial complex} of $G$.
\par \vspace{2mm}
Let $G$ be a connected graph, and let $p$,$q$ be two vertices of $G$.
The \textit{distance} between $p$ and $q$, denoted by $\dist(p,q)$,
is the minimal length of paths from $p$ to $q$.
The \textit{diameter}, denoted by $\diam G$, is the maximal distance
between two vertices of $G$.
We set $\diam G = \infty$ if $G$ is a disconnected graph.
\par \vspace{2mm}
Let $\Delta$ be a simplicial complex on $V$ of dimension $1$.
Then $\Delta$ can be regarded as a graph on $V$ whose
edge set is defined by $E(\Delta) = \{F \in \Delta \,:\, \dim F=1 \}$.
\par \vspace{2mm}
\subsection{Symbolic powers}
Let $I$ be a radical ideal of $S$.
Let $\Min_S(S/I) = \{P_1,\ldots, P_r\}$ be the set of the
minimal prime ideals of $I$, and
put $W = S \setminus \bigcup_{i=1}^r P_i$.
Given an integer $\ell \ge 1$,
the \textit{$\ell$th symbolic power} of $I$
is defined to be the ideal
\[
I^{(\ell)}= I^{\ell}S_W \cap S = \bigcap_{i=1}^r P_i^{\ell}S_{P_i} \cap S.
\]
In particular, if $I=I_{\Delta}$ is the Stanley-Reisner ideal of $\Delta$,
putting $P_F=(x \in [n] \setminus F)$ for each facet $F$, then we have
\[
I_{\Delta} = \bigcap_{F \in \mathcal{F}(\Delta)} P_F
\]
and hence
\[
I_{\Delta}^{(\ell)} = \bigcap_{F \in \mathcal{F}(\Delta)} P_F^{\ell}.
\]
\par
In general, $I^{\ell} \subseteq I^{(\ell)}$ holds, but the other inclusion
does not necessarily hold.
For instance, if $I=(x_1x_2,x_2x_3,x_3x_1)$, then
\[
I^{(2)} = (x_1,x_2)^2 \cap (x_2,x_3)^2 \cap (x_1,x_3)^2
= I^2 + (x_1x_2x_3) \ne I^2.
\]
\par
Moreover, if $I$ is a unmixed squarefree monomial ideal,
then $I^{(\ell)}$ is unmixed.
Thus if $S/I^{\ell}$ is Cohen-Macaulay (or Buchsbaum),
then so is $S/I^{(\ell)}$.
\par \vspace{2mm}
\subsection{Serre's condition}
Let $S=K[x_1,\ldots,x_n]$ and $\frm = (x_1,\ldots,x_n)S$.
Let $I$ be a homogeneous ideal of $S$.
For a positive integer $k$,
$S/I$ satisfies \textit{Serre's condition} $(S_k)$
if $\depth (S/I)_P \ge \min\{\dim (S/I)_P,\, k \}$ for every $P \in \Spec S/I$.
\par
A simplicial complex $\Delta$ is called
\textit{Cohen--Macaulay} (resp. Gorenstein, (FLC) etc.)
if so is $K[\Delta]$ over any field $K$.
Moreover,
if $\Delta$ is (FLC), then $\Delta$ is pure and
$\link_{\Delta}(F)$ is Cohen-Macaulay for every nonempty face $F \in \Delta$.
\par
A homogeneous $K$-algebra $S/I$ is called \textit{quasi-Buchsbaum}
if $\frm H_{\frm}^i(S/I) =0$ for each $i=0,1,\ldots,\dim S/I-1$.
It is known that any quasi-Buchsbaum ring has (FLC) and the converse
is also true for Stanley-Reisner rings.
\par \vspace{2mm}
\subsection{Associated simplicial complex of monomial ideals}
Let $S=K[x_1,\ldots,x_n]$ be a polynomial ring
with natural $\mathbb{Z}^n$-graded structure.
Let $\frm=(x_1,\ldots,x_n)S$ be the unique homogeneous maximal
ideal of $S$.
Let $I$ be a monomial ideal of $S$, and
let $G(I)$ denote the minimal monomial generators of $I$.
For each $i$, we put $\rho_i= \max\{b_i \,:\, x^{\bf b} \in G(I)\}$,
where ${\bf b} = (b_1,\ldots,b_n) \in \mathbb{N}^n$ and $x^{\bf b} = x_1^{b_1}\cdots x_n^{b_n}$.
Then $S/I$ can be considered as a $\mathbb{Z}^n$-graded ring.
\par
Let ${\bf a} \in \mathbb{Z}^n$ be a vector.
For any $\mathbb{Z}^n$-graded $S$-module $M$,
$M_{\bf a}$ denotes the graded ${\bf a}$-component of $M$.
We put $G_{\bf a} = \{i \in [n]\,:\, a_i < 0 \}$.
As $\sqrt{I}$ is a squarefree monomial ideal,
there exists a simplicial complex $\Delta$
such that $I_{\Delta} = \sqrt{I}$. Then we define
$\Delta(I) = \Delta$.
Under this notation, a subcomplex $\Delta_{\bf a}(I)$ is defined by
\[
\Delta_{\bf a}(I) = \left\{F \in \Delta(I) \,:
\begin{array}{l}
\bullet \;\text{$F \cap G_{\bf a} = \emptyset$.}\\
\bullet \;\text{For every $x^{\bf b} \in G(I)$,
there exists an $i \in [n] \setminus (F \cup G_{\bf a})$} \\
\phantom{\bullet}\; \text{such that $b_i > a_i$}.
\end{array}
\right\}.
\]
\par
This complex plays a key role in Takayama's formula for local cohomology modules
of monomial ideals, which is known
as Hochster's formula in the case of squarefree monomial ideals.
\par \vspace{2mm}
Let $I=I_{\Delta}$ be a squarefree monomial ideal of $S$.
Then $I^{(\ell)}$ is a monomial ideal whose radical is equal to $I$.
The following lemma enables us to compute $\Delta_{\bf a}(I^{(\ell)})$ easily.
\begin{lemma}[Minh and Trung \cite{MiT1}] \label{MTrep}
Let $I$ be a squarefree monomial ideal in $S$.
Let $\ell \ge 1$ be an integer and ${\bf a} \in \mathbb{N}^n$.
Then we have
\[
\Delta_{\bf a}(I^{(\ell)})
= \langle F \in \mathcal{F}(I) \,:\,
\sum_{i \notin F} a_i \le \ell-1 \rangle.
\]
\end{lemma}
\subsection{Linkage}
Let $R$ be a Gorenstein ring, and $I$, $J$ ideals of $R$.
$I$ and $J$ said to be \textit{directly linked}, denoted by $I \sim J$,
if there exists a regular sequence
$\underline{z}=z_1,\ldots,z_h$ in $I \cap J$ such that
$J = (\underline{z}) \colon I$ and $I = (\underline{z}) \colon J$.
\par
Assume that $I$ is Cohen-Macaulay ideal of height $h$ and
$\underline{z}=z_1,\ldots,z_h$ is a regular sequence
contained in $I$.
If we set $J = (\underline{z}) \colon I$, then
$I=(\underline{z}) \colon J$ and thus $I \sim J$.
\par
Moreover, $I$ is said to be \textit{linked} to $J$
(or $I$ lies in the linkage class of $J$) if
there exists a sequence of ideals of direct links
\[
I = I_0 \sim I_1 \sim \cdots \sim I_r =J.
\]
One can easily see that $\sim$ is an equivalence relation of ideals and
any two complete intersection with the same height belongs to the same class.
In particular, $I$ is called \textit{licci} if
$I$ lies in the linkage class of a complete intersection ideal.
See e.g. \cite{Vas} for more details.
\medskip
\section{Quasi-Buchsbaumness of the second powers and Gorensteinness}
In this section we consider quasi-Buchsbaum property of the second power of
the Stanley-Reisner ideal $I_{\Delta}$.
The main purpose of this section is to prove the
following theorem$:$
\begin{thm} \label{Buchsbaum}
Let $S=K[x_1,\ldots,x_n]$ be a polynomial ring over a field $K$,
and let $\Delta$ be a simplicial complex on $V=[n]$.
Suppose that $d = \dim S/I_{\Delta} \ge 3$.
If $S/I_{\Delta}^2$ is quasi-Buchsbaum for any field $K$
then $\Delta$ is Gorenstein.
\end{thm}
\par \vspace{2mm}
We first prove the following lemma,
which is closely related to the conjecture
by Vasconcelos (see also \cite[Conjecture 3.12]{SVV}):
Let $R$ be a regular local ring and $I$ a Cohen-Macaulay
ideal of $R$. If $I$ is syzygetic and $I/I^2$ is Cohen-Macaulay,
then $I$ is a Gorenstein ideal.
The following lemma easily follows from the classification theorems for
simplicial complexes $\Delta$ such that $S/I_{\Delta}^2$ are Cohen-Macaulay
in one and two-dimensional cases.
See \cite{MiT1, TrTu}.
\begin{lemma} \label{Vasconcelos}
Let $\Delta$ be a simplicial complex on $V=[n]$, and
let $I_{\Delta} \subseteq S=K[x_1,\ldots,x_n]$ denote
the Stanley-Reisner ideal of $\Delta$.
If $S/I_{\Delta}^2$ is Cohen-Macaulay for any field $K$, then
$\Delta$ is Gorenstein.
\end{lemma}
\begin{proof}
We may assume that $\Delta = \core \Delta$.
Let $K$ be a field and fix it.
Let $F$ be a face of $\Delta$ and
put $\Gamma = \link_{\Delta} F$.
\par \vspace{2mm}
First note that $S/I_{\Gamma}^2$ and $S/I_{\Delta}$ are Cohen-Macaulay
if so is $S/I_{\Delta}^2$.
Indeed, since $S/I_{\Delta}^2$ is Cohen-Macaulay and
$I_{\Delta} = \sqrt{I_{\Delta}^2}$, we
have that $S/I_{\Delta}$ is Cohen-Macaulay; see e.g. \cite{HTT}.
On the other hand, by localizing at $x_F=\prod_{i \in F} x_i$, we get
\[
I_{\Delta} S[x_F^{-1}]
= (I_{\Gamma}, x_{i_1},\ldots,x_{i_k})S[x_F^{-1}]
\]
for some variables $x_{i_1},\ldots,x_{i_k}$.
Hence the assumption implies that
$(I_{\Gamma}, x_{i_1},\ldots,x_{i_k})^2$ is a Cohen-Macaulay ideal.
This yields that $I_{\Gamma}^2$ is also Cohen-Macaulay.
\par \vspace{2mm}
Suppose that $\dim \Gamma =0$.
Then one can take a complete graph $G$ such that $I(G)=I_{\Gamma}$.
Since $S/I(G)^2$ is Cohen-Macaulay, we have $I(G)^{(2)} = I(G)^2$.
Hence $G$ does not contain any triangle (e.g. see Corollary \ref{Triangle}).
Thus $\sharp(V(\Gamma)) = \sharp(V(G)) \le 2$.
\par \vspace{2mm}
By the above argument, $\Lambda=\link_{\Delta} F$
is a locally complete intersection complex whenever $\dim \Lambda =1$.
Moreover, since $S/I_{\Lambda}$ is Cohen-Macaulay
and thus $\Lambda$ is connected,
$\Lambda$ is an $n$-cycle or an $n$-pointed path;
see \cite[Proposition 1.11]{TY}.
On the other hand, since $\diam \Lambda \le 2$ by \cite[Theorem 2.3]{MiT1},
we get
$n \le 3$ if $\Lambda$ is an $n$-pointed path.
Hence $\Lambda = \link_{\Delta} F$ is Gorenstein.
\par \vspace{2mm}
Now suppose that $K=\mathbb{Z}/2\mathbb{Z}$.
By \cite[Chapter II, Theorem 5.1]{St}, $K[\Delta]$ is Gorenstein.
Then we get $\widetilde{\chi}(\Delta)=(-1)^{d-1}$.
\par
Let $K$ be any field.
Then $\widetilde{\chi}(\Delta)=(-1)^{d-1}$ because
$\widetilde{\chi}(\Delta)$ does not depend on $K$.
Therefore we conclude that $\Delta$ is Gorenstein over $K$
by \cite[Chapter II, Theorem 5.1]{St} again.
\end{proof}
\par
A complex $\Delta$ is called a \textit{locally Gorenstein} complex
if $\link_{\Delta}\{x\}$ is Gorenstein for every vertex $x \in V$.
Then the following corollary immediately follows from
Lemma \ref{Vasconcelos}.
\begin{cor} \label{FLC-Vas}
If $S/I_{\Delta}^2$ has $($FLC$)$ for any field $K$,
then $\Delta$ is a locally Gorenstein complex.
\end{cor}
\begin{proof}
The assumption implies that $S/I_{\link_{\Delta}\{x\}}^2$ is Cohen-Macaulay
for every vertex $x \in V$.
Then $\link_{\Delta}\{x\}$ is Gorenstein by Lemma \ref{Vasconcelos}.
\end{proof}
\begin{lemma} \label{Q-Bbm}
Suppose $d \ge 2$.
If $S/I_{\Delta}^2$ is quasi-Buchsbaum, then $S/I_{\Delta}$ is Cohen-Macaulay.
\end{lemma}
\begin{proof}
By assumption that $S/I_{\Delta}^2$ has (FLC).
Then $S/I_{\Delta}$ has (FLC) by \cite[Theorem 2.6]{HTT}
and thus it is Buchsbaum.
\par
Now suppose that $S/I_{\Delta}$ is \textit{not} Cohen-Macaulay.
Then there exists an $i$ with $0 \le i \le d-2$ such that
$H_{\frm}^{i+1}(S/I_{\Delta})_0 \cong \widetilde{H}_i(\Delta;K) \ne 0$.
Then we get the following commutative diagram (see \cite{MiN})
\begin{picture}(400,70)
\put(120,50){$H_{\frm}^{i+1}(S/I_{\Delta}^{2})_{\bf 0}$}
\put(185,55){\vector(1,0){40}}
\put(145,42){\vector(0,-1){20}}
\put(200,58){$x_1$}
\put(240,50){$H_{\frm}^{i+1}(S/I_{\Delta}^{2})_{{\bf e}_1}$}
\put(116,13){$\widetilde{H}^i(\Delta_{\bf 0}(I_{\Delta}^2))$}
\put(190,15){\vector(1,0){35}}
\put(270,42){\vector(0,-1){20}}
\put(243,13){$\widetilde{H}^i(\Delta_{{\bf e}_1}(I_{\Delta}^2))$,}
\end{picture}
\par \vspace{2mm} \par \noindent
where the bottom map is identity because
$\Delta_{\bf 0}(I^{2}) = \Delta_{{\bf e}_1}(I^2) = \Delta$ by \cite{TeTr} and
the vertical maps are isomorphism.
This yields $x_1 H_{\frm}^{i+1}(S/I_{\Delta}^2) \ne 0$.
But this contradicts the assumption.
\end{proof}
\begin{remark}
{\rm
We have an analogous result in the symbolic power case.
Namely,
if $S/I_{\Delta}^{(2)}$ is quasi-Buchsbaum, then $S/I_{\Delta}$ is Cohen-Macaulay.
The proof is almost the same since we have
$\Delta_{\bf 0}(I^{(2)}) = \Delta_{{\bf e}_1}(I^{(2)}) = \Delta$.
}
\end{remark}
\par
We are now ready to prove Theorem \ref{Buchsbaum}.
\begin{proof}[Proof of Theorem \ref{Buchsbaum}]
By assumption and Corollary \ref{FLC-Vas}, we have that
$\Delta$ is locally Gorenstein.
Moreover, $\Delta$ is Cohen-Macaulay by Lemma \ref{Q-Bbm}.
Take any face $F$ of $\Delta$ with $\dim \link_{\Delta} F =1$.
As $d \ge 3$, $\link_{\Delta} F$ is given by some
link of $\link_{\Delta} \{x\}$ for $x \in F$.
Hence such a $\link_{\Delta} F$ is also Gorenstein.
By a similar argument as in the proof of Lemma \ref{Vasconcelos},
we get the required assertion.
\end{proof}
\par \vspace{2mm}
The Gorensteinness of $S/I_{\Delta}$ does not necessarily imply
the quasi-Buchsbaumness of $S/I_{\Delta}^2$.
\par
We cannot replace the Cohen-Macaulayness of $S/I_{\Delta}^2$ with that of
$S/I_{\Delta}^{(2)}$ in Lemma \ref{Vasconcelos} as the next example shows.
\begin{exam} \label{phantomPentagon}
Let $k \ge 2$ be a given integer.
Let $I$ be the Stanley-Reisner ideal of the following simplicial complex $\Delta$,
Then since $\diam \Delta \le 2$, $S/I^{(2)}$ is Cohen-Macaulay by \cite{MiT1},
but $S/I^2$ is not.
Moreover, $S/I$ is not Gorenstein.
\par
\begin{picture}(400,70)
\thicklines
\put(195,55){\circle*{5}}
\put(170,31.5){\circle*{5}}
\put(183.5,9){\circle*{5}}
\put(206.5,9){\circle*{5}}
\put(220,31.5){\circle*{5}}
\put(170,55){\circle*{5}}
\put(220,55){\circle*{5}}
\put(200,53){$\cdots$}
\put(165,60){{\tiny $v_1$}}
\put(190,60){{\tiny $v_2$}}
\put(218,60){{\tiny $v_k$}}
\put(162,33){{\tiny $w$}}
\put(172,8){{\tiny $x$}}
\put(210,8){{\tiny $y$}}
\put(222,35){{\tiny $z$}}
\put(170,32){\line(1,1){22}}
\put(170,31.5){\line(0,1){22}}
\put(220,32){\line(-1,1){22}}
\put(220,31.5){\line(0,1){22}}
\put(182,11){\line(-3,5){10.8}}
\put(208,11){\line(3,5){10.8}}
\put(186,9){\line(1,0){18}}
\put(170,55){\line(2,-1){50}}
\put(220,55){\line(-2,-1){50}}
\end{picture}
\end{exam}
\par \vspace{2mm}
In Theorem \ref{Buchsbaum}, we cannot remove the assumption that
$\dim S/I_{\Delta} \ge 3$ as the next example shows.
\begin{exam} \label{Counter-Bbm}
Put $I_{\Delta} = (x_1x_3,x_1x_4,x_2x_4)$, the Stanley-Reisner ideal
of the $4$-pointed path $\Delta$.
Then $S/I_{\Delta}^2$ is Buchsbaum by \cite[Example 2.9]{TY} and
$S/I_{\Delta}$ is Cohen-Macaulay but not Gorenstein of dimension $2$.
\begin{picture}(400,60)
\thicklines
\put(80,25){$\Delta=$}
\put(127,45){\circle*{5}}
\put(162,45){\circle*{5}}
\put(127,10){\circle*{5}}
\put(162,10){\circle*{5}}
\put(118,45){{\tiny $1$}}
\put(118,10){{\tiny $2$}}
\put(169,45){{\tiny $4$}}
\put(169,10){{\tiny $3$}}
\put(130,10){\line(1,0){29}}
\put(127,42){\line(0,-1){29}}
\put(162,42){\line(0,-1){29}}
\end{picture}
\end{exam}
\vspace{2mm}
The following question is valid in the case that char $K=2$,
but the other cases remain open.
\begin{quest} \label{G-quest}
If $S/I_{\Delta}^2$ is Cohen-Macaulay over a fixed field $K$,
then is $\Delta$ Gorenstein over $K$?
\end{quest}
\medskip
\section{Cohen-Macaulayness versus $(S_2)$ for second symbolic powers}
\par
Throughout this section, let $S=K[x_1,\ldots,x_n]$ be a polynomial ring
over a field $K$.
Let $\frm=(x_1,\ldots,x_n)S$ be the unique graded maximal ideal of $S$
with natural graded structure.
\par
In \cite{TeTr} it is proved that for any integer $\ell \ge 3$ and
for any simplicial complex $\Delta $ on the vertex set $V=[n]$,
$S/I_{\Delta }^{(\ell)}$ is Cohen-Macaulay if and only if it satisfies Serre's condition $(S_2)$.
So it is natural to ask the following question.
\begin{quest} \label{S2-quest}
Let $I$ be the Stanley-Reisner ideal of a simplicial complex $\Delta$ on $V=[n]$.
Then $S/I^{(2)}$ is Cohen-Macaulay if and only if $S/I^{(2)}$ satisfies $(S_2)$?
\end{quest}
\par \vspace{2mm}
So the aim of this section is to give a criterion
for $S/I_{\Delta}^{(2)}$ to satisfy $(S_2)$.
In order to do that, we prove the following theorem, which is a generalization
of \cite[Theorem 2.3]{MiT1}.
Using this, we give a negative answer to the above question; see Example \ref{RealP}.
Note that in the following Theorem \ref{Depth} and Corollary \ref{S2thm}
if we replace the condition that the diameter is less than or equal to $2$
by the connectedness condition
then we have the corresponding condition for the original Stanley-Reisner ring
instead of the second symbolic power,
e.g., $\depth S/I_{\Delta} \ge 2$ is equivalent to the connectedness of $\Delta $
if $\dim \Delta \ge 1$.
\begin{thm} \label{Depth}
Let $\Delta$ be a simplicial complex with $\dim \Delta \ge 1$.
Then the following conditions are equivalent$:$
\begin{enumerate}
\item $\depth S/I_{\Delta}^{(2)} \ge 2$
$($equivalently, $\depth (S/I_{\Delta}^{(2)})_{\frm} \ge 2$$)$.
\item $\diam \Delta^{(1)} \le 2$, where $\Delta^{(1)}$ denotes the $1$-skeleton of $\Delta$.
\end{enumerate}
\end{thm}
\begin{proof}
Put $\Delta_{\bf a} :=
\langle F \in \mathcal{F}(\Delta) \,:\, \sum_{i \notin F} a_i \le 1 \rangle$.
\par \vspace{2mm} \par \noindent
(1) $\Longrightarrow (2):$
For given $r,\,s \in V=[n]$ ($r < s)$, we show that $\dist(r,s) \le 2$
in $\Delta^{(1)}$.
Put ${\bf a} = {\bf e}_r + {\bf e}_s \in \mathbb{N}^n$.
Then
$\Delta_{\bf a}
=\langle F \in \mathcal{F}(\Delta) \,:\,
r \in F \;\text{or} \; s \in F \rangle$.
Since $\depth S/I_{\Delta}^{(2)} \ge 2$,
we have that $\widetilde{H}_0(\Delta_{\bf a}) =0$ and thus
$\Delta_{\bf a}$ is connected by Takayama's formula and
Lemma \ref{MTrep}.
Hence there exists an $F \in \mathcal{F}(\Delta)$ such that $r,s \in F$
or there exist $F_r \in \mathcal{F}(\Delta)$ and
$F_s \in \mathcal{F}(\Delta)$ such that $r \in F_r$, $s \in F_s$ and
$F_r \cap F_s \ne \emptyset$.
In any case, we get $\dist(r,s) \le 2$, as required.
\par \vspace{2mm}
$(2) \Longrightarrow (1):$
Assume $\diam \Delta^{(1)} \le 2$.
By Takayama's formula, it suffices to show that $\Delta_{\bf a}$
is connected for any ${\bf a} \in \{0,1\}^n$
with $\Delta_{\bf a} \ne \emptyset$; see also \cite{MiT2}.
\begin{description}
\item[{\bf Case 1}] $\sharp(\supp {\bf a}) \le 1$.
\end{description}
Then $\Delta_{\bf a}=\Delta$ is connected by assumption.
\begin{description}
\item[{\bf Case 2}] $\sharp(\supp {\bf a}) = 2$.
\end{description}
We may assume that $a_r = a_s=1$ for some $r < s$.
Then
\[
\Delta_{\bf a} = \langle F \in \mathcal{F}(\Delta)
\,:\, r \in F \;\text{or} \;s \in F \rangle.
\]
Since $\diam \Delta^{(1)} \le 2$, we have that $\{r,s\} \in \Delta$ or
there exists a $t \in V$ such that $\{r,t\}$, $\{t,s\} \in \Delta$.
In the first case, if we choose a facet $F \in \mathcal{F}(\Delta)$ which
contains $\{r,s\}$, then $F \in \Delta_{\bf a}$ and $r,s \in F$.
In the second case, if we choose facets $F_1$, $F_2$ such that
$\{r,t\} \in F_1$ and $\{s,t\} \in F_2$.
Then $\Delta_{\bf a}$ is connected
because $F_1,\,F_2 \in \Delta_{\bf a}$.
\begin{description}
\item[{\bf Case 3}] $\sharp(\supp {\bf a}) \ge 3$.
\end{description}
We may assume that $\sharp(\mathcal{F}(\Delta_{\bf a})) \ge 2$.
Let $F_1$, $F_2 \in \mathcal{F}(\Delta_{\bf a})$.
By assumption, $\sharp(F_i \cap \supp({\bf a})) \ge \sharp(\supp({\bf a}))-1$
for each $i=1,2$.
Then we get
\[
\sharp(F_1 \cap F_2) \ge \sharp
\big(F_1 \cap \supp({\bf a})) \cap (F_2 \cap \supp({\bf a}))\big)
\ge \sharp(\supp({\bf a})) -2 \ge 1.
\]
Hence $\Delta_{\bf a}$ is connected.
\end{proof}
\begin{cor} \label{S2thm}
Let $\Delta$ be a pure simplicial complex.
Then the following conditions are equivalent$:$
\begin{enumerate}
\item $S/I_{\Delta}^{(2)}$ satisfies $(S_2)$.
\item $\diam ((\link_{\Delta} F)^{(1)}) \le 2$ for any face $F \in \Delta$ with
$\dim \link_{\Delta} F \ge 1$.
\end{enumerate}
\end{cor}
\begin{proof}
$(1) \Longrightarrow (2):$
Let $F$ be a face of $\Delta$ with $\dim \link_{\Delta} F \ge 1$.
By assumption and localization,
we obtain that $S'/I_{\link_{\Delta}(F)}^{(2)}$ satisfies $(S_2)$,
where $S'$ is a polynomial ring which corresponds to $\Gamma = \link_{\Delta}(F)$.
Then $\depth S'/I_\Gamma^{(2)} \ge 2$.
It follows from Theorem \ref{Depth} that $\diam \Gamma^{(1)} \le 2$, as required.
\par \vspace{2mm}
$(2) \Longrightarrow (1):$
The assumption (2) preserves under localization.
Hence we may assume that $S/I_{\link_{\Delta} \{x\}}^{(2)}$ satisfies $(S_2)$.
This implies that $S/I_{\link_{\Delta} \{x\}}$ also satisfies $(S_2)$ by \cite{HTT}.
Hence $(S/I_{\Delta}^{(2)})_x$ satisfies $(S_2)$ for every variable $x$.
\par
Let $P \in \Spec(S/I_{\Delta}^{(2)})$ with $\dim (S/I_{\Delta}^{(2)})_P \ge 2$.
If $P \ne \frm$, then there exists a variable $x$ such that $x \notin P$.
Then $\depth (S/I_{\Delta}^{(2)})_P \ge 2$ by the above argument.
Otherwise, $P = \frm$.
Since $\diam \Delta^{(1)} \le 2$ by assumption, we have that
$\depth (S/I_{\Delta}^{(2)})_{\frm} \ge 2$ by Theorem \ref{Depth}.
Therefore $S/I_{\Delta}^{(2)}$ satisfies $(S_2)$.
\end{proof}
\par \vspace{2mm}
The next example shows that
the $(S_2)$-ness of $I_{\Delta}^{(2)}$ does not necessarily imply its Cohen-Macaulayness.
\begin{exam}[{\bf The triangulation of the real projective plane}] \label{RealP}
Let $I=I_{\Delta}$ be the Stanley-Reisner ideal of the triangulation of
the real projective plane $\mathbb{P}^2$.
Then $I_{\Delta}$ is generated by the following monomials of degree $3$:
\[
x_1x_2x_3,\,x_1x_2x_5,\,x_1x_3x_6,\,
x_1x_4x_5,\,x_1x_4x_6,\,x_2x_3x_4,\,
x_2x_4x_6,\,x_2x_5x_6,\,x_3x_4x_5,\,x_3x_5x_6.
\]
\par \vspace{2mm}
\begin{center}
\begin{picture}(400,80)
\put(40, 30){$\Delta=$}
\thicklines
\put(150,75){\circle*{6}}
\put(100,50){\circle*{6}}
\put(100,14){\circle*{6}}
\put(150,-10){\circle*{6}}
\put(200,14){\circle*{6}}
\put(200,50){\circle*{6}}
\put(150,50){\circle*{6}}
\put(130,25){\circle*{6}}
\put(170,25){\circle*{6}}
\put(146,80){{\tiny $1$}}
\put(90,50){{\tiny $2$}}
\put(90,14){{\tiny $3$}}
\put(146,-20){{\tiny $1$}}
\put(205,14){{\tiny $2$}}
\put(205,50){{\tiny $3$}}
\put(154,54){{\tiny $4$}}
\put(127,30){{\tiny $5$}}
\put(171,30){{\tiny $6$}}
\put(147,74){\line(-2,-1){44}}
\put(153,74){\line(2,-1){44}}
\put(147,-9){\line(-2,1){44}}
\put(153,-9){\line(2,1){44}}
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\put(153,50){\line(1,0){47}}
\put(150,50){\line(-5,-6){20}}
\put(150,50){\line(5,-6){20}}
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\put(129,25){\line(-5,-2){26}}
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\put(171,25){\line(5,-2){26}}
\put(170,25){\line(6,5){27}}
\put(230,20){$\link_{\Delta}\{4\}=$}
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\put(400,25){\circle*{6}}
\put(350,25){\circle*{2}}
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\put(346,55){{\tiny $1$}}
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\put(370,0){\line(6,5){27}}
\end{picture}
\end{center}
\par \vspace{6mm}
Since
$\widetilde{\chi}(\Delta) = -1 +f_0 - f_1 + f_2 = -1 + 6 - 15 +10 =0 \ne (-1)^2$,
$K[\Delta]$ is \textit{not} Gorenstein for any field $K$.
Moreover, Reisner proved that $K[\Delta]$ is Cohen-Macaulay if and only if
$\chara K \ne 2$.
\par
The link of every vertex is a pentagon, and $\Delta^{(1)}$ is the complete $6$-graph.
Hence it follows from Corollary \ref{S2thm} that $S/I_{\Delta}^{(2)}$
has $(S_2)$.
But it is \textit{not} Cohen-Macaulay; see \cite[Example 2.8]{MiT2}.
\par
One can easily see that $x_1x_2x_3x_4x_5x_6 \in
I_{\Delta}^{(2)} \setminus I_{\Delta}^2$.
Hence $S/I_{\Delta}^2$ does not satisfy $(S_2)$.
\end{exam}
\begin{quest} \label{S2-edge}
Let $I(G)$ be the edge ideal of a graph $G$.
If $S/I(G)^{(2)}$ satisfies $(S_2)$,
then is it Cohen-Macaulay?
\end{quest}
\medskip
\section{When does $I^{(2)} = I^2$ hold}
\par
In this section, we discuss when $I^{(2)} = I^2$ holds
for any squarefree monomial ideal $I$.
First we introduce the notion of special triangles.
\begin{defn} \label{Hypergraph}
Let $I$ be a squarefree monomial ideal of $S=K[x_1,\ldots,x_n]$.
Let $G(I) = \{x^{H_1},\ldots,x^{H_\mu}\}$ be the minimal set of
monomial generators, where $x^{H} = x_{i_1}\cdots x_{i_r}$
for $H=\{i_1,\ldots,i_r\}$.
Then $\mathcal{H}(I)$ is called the \textit{associated hypergraph} of $I$
if the vertex set of $\mathcal{H}(I)$ is $V$ and
the edge set is $\{H_1,\ldots,H_\mu\}$.
\par
Then $\{i,j,k\}$ is called a \textit{special triangle} of $\mathcal{H}(I)$ if
there exist $H_i,H_j,H_k \in \mathcal{H}(I)$ such that
\[
H_i \cap \{i,j,k\} = \{j,k\},\qquad
H_j \cap \{i,j,k\} = \{i,k\},\qquad
H_k \cap \{i,j,k\} = \{i,j\}.
\]
Then we say that \lq\lq $H_i,H_j,H_k$ \textit{make a special triangle}
$\{i,j,k\}$''.
\end{defn}
\par
For instance, if $G(I)$ contains $x_1x_2L_1$, $x_2x_3L_2$, $x_3x_1L_3$
($L_1,L_2,L_3$ are monomials any of which is not divided by $x_1$,$x_2$ nor $x_3$),
then $\{1,2,3\}$ is a special triangle.
\begin{remark}
A special cycle is considered in \cite{HHTZ},
and they prove that $I^{(\ell)}=I^{\ell}$ hold for any $\ell \ge 1$ if
there exists no special odd cycle in $\mathcal{H}(I)$.
\end{remark}
\par \vspace{2mm}
The following is the main theorem in this section.
\begin{thm} \label{SpTriangle}
Let $I$ be a squarefree monomial ideal.
Then the following conditions are equivalent$:$
\begin{enumerate}
\item $I^{(2)} = I^2$ holds.
\item If there exist $\{H_1,H_2,H_3\} \subseteq \mathcal{H}(I)$ such that
$H_1,H_2,H_3$ make a special triangle,
then $x^{H_1 \cap H_2 \cap H_3} x^{H_1 \cup H_2 \cup H_3} \in I^2$.
\end{enumerate}
\end{thm}
\begin{remark}
If there exist no special triangles, then we have $I^{(2)}=I^2$.
The converse is not true.
\end{remark}
\par
The following criterion is well known; see \cite{RTY}.
\begin{cor} \label{Triangle}
Let $I(G)$ denote the edge ideal of a graph $G$.
Then $I(G)^{(2)} = I(G)^2$ holds if and only if
$G$ has no triangles $($the cycles of length $3$$)$.
\end{cor}
\par
In what follows, we prove the above theorem.
First we prove the following lemma.
\begin{lemma} \label{SpMono}
Suppose that the condition $(2)$ in Theorem $\ref{SpTriangle}$ holds.
Then $xI \cap (I^2 \colon x) \subseteq I^2$ holds for every $x \in V$.
\end{lemma}
\begin{proof}
Suppose that there exist a variable $x_1$ and a monomial $M$ such that
$M \in x_1 I \cap (I^2 \colon x) \setminus I^2$.
As $x_1M \in I^2$, we can take $N_2$, $N_3 \in G(I)$ and a monomial $L$ such that
\begin{eqnarray} \label{eq1}
x_1M = N_2N_3L.
\end{eqnarray}
On the other hand, as $M \in x_1I$, we can choose $N_1 \in G(I)$
and a monomial $L'$ such that
\begin{eqnarray} \label{eq2}
M=N_1L' \quad \text{and} \quad x_1 \,|\,L'.
\end{eqnarray}
\begin{description}
\item[Claim 1] $x_1\,|\,N_2$, $x_1\,|\,N_3$ but $x_1 \,\not |\, N_1$.
\end{description}
\par
As $M \notin I^2$, $x_1$ does not divide $L$.
By Eqs.(\ref{eq1}),(\ref{eq2}), $N_2N_3L$ is divided by $x_1^2$.
Hence $x_1$ divides both $N_2$ and $N_3$ because $N_i$
is a squarefree monomial for $i=2,3$.
By a similar reason, we have that $N_1$ is not divided by $x_1$.
\begin{description}
\item[Claim 2] $N_2\ne N_3$ and $\gcd(N_2,N_3) \,|\, L'$.
\end{description}
\par
If $N_2=N_3$, then $x_1N_1L'=N_3^2L$ is divided by $x_1N_1$
and thus $N_3L$ is divided by $x_1N_1$.
Then $M=N_1N_2(N_3L/x_1N_1) \in I^2$. This is a contradiction.
Hence $N_2 \ne N_3$.
\par
Since $x_1N_1L'=N_2N_3L$ is divided by $\gcd(N_2,N_3)^2$, $L'$ is
divided by $\gcd(N_2,N_3)$ because $x_1N_1$ is squarefree.
\begin{description}
\item[Claim 3] There exist variables $x_2$, $x_3$ such that
\[
x_2 \,\big|\, \frac{N_3}{\gcd(N_2,N_3)},\quad
x_3 \,\big|\, \frac{N_2}{\gcd(N_2,N_3)}, \quad x_2,x_3 \,|\,N_1
\]
\end{description}
\par
Note that any variable which divides $N_i$
for $i=2,3$ is a factor of $N_1$ or $L'$.
Since $L' \notin I$, $L'/\gcd(N_2,N_3)$ is not divided by $N_3/\gcd(N_2,N_3)$.
Thus there exists a variable $x_2$ such that
$x_2\,|\, N_3 /\gcd(N_2,N_3)$ and $x_2\,|\,N_1$.
The other statement follows from a similar argument.
\par \vspace{2mm}
Take $H_i \in \mathcal{H}(I)$ such that $x^{H_i} = N_i$ for each $i=1,2,3$.
\begin{description}
\item[Claim 4] $H_1,H_2,H_3$ make a special triangle $\{1,2,3\}$.
\end{description}
\par
The assertion immediately follows from Claim 1 and Claim 3.
By the Claim 4, we get a contradiction.
\par\vspace{2mm}
By assumption, we get
\[
\gcd(N_1,N_2,N_3)\sqrt{N_1N_2N_3}
= x^{H_1 \cap H_2 \cap H_3} \cdot x^{H_1 \cup H_2 \cup H_3} \in I^2,
\]
where $\sqrt{N}=x_{i_1}\cdots x_{i_r}$ for a monomial
$N=x_{i_1}^{a_{i_1}}\cdots x_{i_r}^{a_{i_r}}$ $(a_{i_j}>0)$.
Since $N_1$ divides $N_2N_3L$ and $x_1 \,|\,N_2,\;N_3$,
we have
\begin{equation} \label{eq:sqrt}
\sqrt{N_1N_2N_3}\;\big|\; \frac{N_2N_3L}{x_1}=M.
\end{equation}
On the other hand, since $x_1 \not | \gcd(N_1,N_2,N_3)$, we have
\begin{equation} \label{eq:gcd}
\gcd(N_1,N_2,N_3)^2 \;\big|\;\frac{N_2N_3}{x_1} \;\big|\;M.
\end{equation}
Hence Eqs. (\ref{eq:sqrt}), (\ref{eq:gcd}) imply
\[
\gcd(N_1,N_2,N_3)\sqrt{N_1N_2N_3}\;\big|\; M.
\]
Therefore $M \in I^2$, which contradicts the choice of $M$.
\end{proof}
\par
Now suppose that $I^{(2)}_{x}=I^2_{x}$ holds for every vertex $x \in V$.
Then $I^{(2)} = I^2$ if and only if $\frm \notin \Ass(S/I^2)$.
Hence the following lemma is useful when we use an induction.
\begin{lemma}[{\rm See the proof of \cite[Theorem 5.9]{SVV}}] \label{SVVproof}
Let $I$ be a squarefree monomial ideal of $S$ with $\dim S/I \ge 1$.
Now suppose that $xI \cap (I^2 \colon x)\subseteq I^2$ for every variable $x$.
Then $\frm \notin \Ass_S(S/I^2)$.
\end{lemma}
\begin{proof}
Since $I^2$ and $\frm$ are monomial ideals, it suffices to show
$I^2 \colon M \ne \frm$ for every variable $x$ and any monomial $M$.
\par
Now suppose that $I^2 \colon M = \frm$
for some monomial $M \notin I^2$.
Since $\frm M \subseteq I^2 \subseteq I$ and $\depth S/I > 0$, we have
$M \in I$.
So we may assume that $M=x_1\cdots x_k L$,
where $N=x_1 \cdots x_k \in G(I)$ and $L$ is a monomial.
By assumption, $x_kM = x_1(x_2 \cdots x_{k-1}x_k^2 L) \in I^2$.
Since $I$ is generated by squarefree monomials, we then have
$x_2\cdots x_{k-1}x_k^2L \in I$ and hence
$x_2\cdots x_{k-1}x_kL \in I$.
Hence $M \in x_1I \cap (I^2 \colon x_1) \subseteq I^2$.
This is a contradiction.
\end{proof}
\begin{proof}[Proof of Theorem $\ref{SpTriangle}$]
First we show $(2) \Longrightarrow (1)$.
Suppose (2).
Since this condition preserves under localization, we may assume that
$(I^{(2)})_{x}=(I^{2})_{x}$ for any variable $x$ by an induction on $\dim S/I$.
By the above two lemmata, we have $\frm \notin \Ass_S(S/I^2)$.
Hence $I^{(2)}=I^2$, as required.
\par \vspace{2mm}
Next we show $(1) \Longrightarrow (2)$.
Suppose that
there exists a subset $\{H_1,H_2,H_3\} \subseteq \mathcal{H}(I)$ such that
$H_1,H_2,H_3$ make a special triangle and
$x^{H_1\cap H_2 \cap H_3} x^{H_1 \cup H_2 \cup H_3} \notin I^2$.
Then it suffices to show $I^2 \subsetneq I^{(2)}$.
\par
Put $H= H_1 \cup H_2 \cup H_3$.
Let $I_H$ be the squarefree monomial ideal of
$K[x\,:\,x \in V \setminus H]$ such that
$I_H S + (x \in V \setminus H)=I+(x \in V \setminus H)$.
Let $P$ be any minimal prime ideal of $I_H$.
If $\height P =1$, then there exists a vertex $j \in H_1 \cap H_2 \cap H_3$ such that
$P=(x_j)$. Then $M:=x^{H_1 \cap H_2 \cap H_3} x^{H} \in (x_j^2)=P^2$.
If $\height P\ge 2$, then $P$ contains two variables $x_i,x_j$ with $i,j \in H$.
Then $x^H \in P^2$ and hence $M \in P^2$.
Therefore $M \in I_H^{(2)}$ but $M \notin I_H^2$ by the assumption that
$M \notin I^2$.
\end{proof}
\medskip
Suppose $U \cap V = \emptyset$.
Let $\Gamma$ (resp. $\Lambda$) be a simplicial complex on $U$ (resp. $V$).
Then the \textit{simplicial join} of $\Gamma$ and $\Lambda$,
denoted by $\Gamma * \Lambda$, is defined by
$\Gamma * \Lambda = \{F \cup G \,:\, F \in \Delta,\; G \in \Lambda\}$.
It is a simplicial complex on $U \cup V$.
\par
The following corollary
is probably well-known (and hence so is Corollary \ref{Disjoint}),
but we give a proof as an application of Theorem \ref{SpTriangle}.
\begin{cor} \label{Joincor}
Let $\Gamma$ be a simplicial complex on $U$
and $\Lambda$ a simplicial complex on $V$.
Let $\Delta = \Gamma * \Lambda$ denote
the simplicial join of $\Gamma$ and $\Lambda$.
Then $\Delta$ is a simplicial complex on $W =U \coprod V$.
Put $R=K[U]$, $S=K[V]$ and $T=R \otimes_K S \cong K[W]$.
Then$:$
\begin{enumerate}
\item $I_{\Delta}^{(2)}=I_{\Delta}^2$ if and only if
$I_{\Gamma}^{(2)}=I_{\Gamma}^2$ and $I_{\Lambda}^{(2)}=I_{\Lambda}^2$.
\item $T/I_{\Delta}^{2}$ is Cohen--Macaulay if and only if
so do $R/I_{\Gamma}^{2}$ and $S/I_{\Lambda}^{2}$.
\end{enumerate}
\end{cor}
\begin{proof}
(1) Note that $I_{\Delta} = I_{\Gamma}T + I_{\Lambda}T$ and
$G(I_{\Delta})$ is a disjoint union of $G(I_{\Gamma})$ and $G(I_{\Lambda})$.
Thus it immediately follows from Theorem \ref{SpTriangle}.
\par
(2) It immediately follows from (1) and \cite[Theorem 2.7]{MiT2}.
\end{proof}
\par \vspace{2mm}
A disjoint union of two graphs $G_1$ and $G_2$, denoted by
$G_1 \coprod G_2$, is the graph $G$ which satisfies
$V(G) = V(G_1) \cup V(G_2)$ and $E(G)=E(G_1) \cup E(G_2)$.
Let $G = G_1 \coprod \ldots \coprod G_r$ be a disjoint union of
graphs $G_1,\ldots,G_r$, and
let $\Delta_i$ (resp. $\Delta$) be the complementary simplicial
complex of $G_i$
for each $i=1,\ldots,r$ (resp. $G$).
Then $\Delta$ is equal to the simplicial join
$\Delta_1 * \cdots * \Delta_r$.
\begin{cor} \label{Disjoint}
Let $G=G_1 \coprod \ldots \coprod G_r$ be a disjoint union of
graphs $G_i$ for which $I(G_i)^2$ is a Cohen-Macaulay ideal.
Then $I(G)^2$ is a Cohen--Macaulay ideal.
\end{cor}
\begin{exam} \label{DisjointPentagon}
Let $G=G_1 \coprod \ldots \coprod G_r$ be a disjoint union of
the pentagons $G_i$ for $i=1,\ldots,r$.
Then $I(G)^2$ is a Cohen--Macaulay ideal.
\end{exam}
\begin{proof}[\quad Proof]
It follows that the second symbolic power of the edge ideal of the pentagon
is a Cohen--Macaulay ideal.
\end{proof}
\medskip
\section{Examples of Stanley-Reisner ideals whose square is Cohen-Macaulay}
\par
By Corollary \ref{Joincor} we know that there exists a
simplicial complex $\Delta $ with arbitrary high dimension
such that $I_{\Delta}^2$ is non-trivially Cohen-Macaulay.
We now consider the following question.
\begin{quest}
For a given integer $d\ge 2$,
is there a simplicial complex $\Delta$ with $\dim \Delta=d-1$
such that $S/I_{\Delta}^2$ is Cohen-Macaulay and such that
$\Delta$ cannot be expressed as the simplicial join of two non-empty
complexes?
\end{quest}
\par
We give two families of examples as affirmative answers, using liaison theory.
The following key proposition is due to
Buchweitz \cite{Bu}; see also Kustin and Miller \cite{KM2}.
Note that it gives a partial converse of
Theorem \ref{Buchsbaum}.
\begin{prop}[\textrm{cf. \cite[6.2.11]{Bu}, \cite[Proposition 7.1]{KM2}}]
\label{Kustin}
Let $I$ be a Gorenstein homogeneous ideal in a polynomial ring $S$.
Assume that there exist a homogeneous polynomial ring
$T=S[z_1,\ldots,z_r]$ $(\deg z_i =1)$ and a homogeneous radical ideal $L$ such that
\begin{enumerate}
\item[(a)] $S/I \cong T/(z_1,\ldots,z_r, L)$.
\item[(b)] $z_1,\ldots,z_r$ is a regular sequence on $T/L$.
\item[(c)] $L$ is in the linkage class of a complete intersection in $T$.
\end{enumerate}
Then $S/I^2$ is Cohen-Macaulay.
\end{prop}
\begin{proof}
Since $S/I^2$ is isomorphic to the ring $T/(z_1,\ldots,z_r,L^2)$, it is enough to show
that $T/L^2$ is Cohen-Macaulay.
\par
Let $\fraM$ be the unique homogeneous maximal ideal of $T$, and set
$R=\widehat{T_{\fraM}}$, the $\fraM$-adic completion of $T_{\fraM}$.
As $R/LR$ is a radical Gorenstein ideal, we can conclude that
$LR/(LR)^2$ is Cohen-Macaulay, and thus $R/(LR)^2$ is Cohen-Macaulay
by \cite[Proposition 7.1]{KM2}.
It follows from Matijevic-Roberts theorem that $T/L^2$ is Cohen-Macaulay, as required.
\end{proof}
\par
It is well-known
that any Gorenstein ideal of codimension $3$ lies in the linkage
class of a complete intersection; see \cite{BE, Wat} or
\cite[Theorem 4.15]{Vas}.
Thus we can obtain the following corollary.
\begin{cor} \label{codim3}
Let $I_{\Delta} \subseteq S$ be a Gorenstein Stanley-Reisner ideal of
codimension $3$. Then $S/I_{\Delta}^2$ is Cohen-Macaulay.
\end{cor}
\medskip
In the rest of this section
we prove the second power of the Stanley-Reisner ideal
of a stellar subdivision of any non-acyclic complete intersection complex
is Cohen-Macaulay.
In what follows,
as vertices of simplicial complexes we use indeterminates
instead of natural numbers for convenience.
Let $\Gamma$ be a non-acyclic complete intersection simplicial complex
whose Stanley-Reisner ideal is
\[
I_{\Gamma}=(x_{11}x_{12}\cdots x_{1i_{1}},
x_{21}x_{22}\cdots x_{2i_{2}}, \ldots,
x_{\mu 1}x_{\mu 2}\cdots x_{\mu i_{\mu }}).
\]
\par
Let $\mathcal{F}(\Gamma)$ be the set of all facets of $\Gamma$.
Then
\begin{eqnarray*}
\mathcal{F}(\Gamma)=
& \{ & \{x_{11}, \ldots , \widehat{x_{1k_{1}}}, \ldots, x_{1i_{1}},
x_{21}, \ldots, \widehat{x_{2k_{2}}}, \ldots, x_{2i_{2}}, \ldots,\\
&&x_{\mu 1}, \ldots , \widehat{x_{\mu k_{\mu }}}, \ldots,
x_{\mu i_{\mu }} \}\\
&&\mid
1 \le k_1 \le i_1, 1 \le k_2 \le i_2, \dots, 1 \le k_{\mu } \le i_{\mu }
\}.
\end{eqnarray*}
\par
Let $\Delta$ be the \textit{stellar subdivision} of $\Gamma$ on
\[
F=\{ x_{11}, \ldots ,x_{1j_{1}}, x_{21}, \ldots ,x_{2j_{2}}, \ldots,
x_{p 1}, \ldots , x_{p j_{p }} \},
\]
where $1 \le p \le \mu$ and $1 \le j_1 < i_1, \dots , 1 \le j_p < i_p$ and $ j_1 +\cdots +j_p \ge 2$.
\par
Let $v$ be the new added vertex.
Then
\begin{eqnarray*}
\mathcal{F}(\Delta)= &\{ & G \in \mathcal{F}(\Gamma ) \mid G \not\supset F \ \ \}
\cup \{ \{v \} \cup G \setminus \{w\} \mid G \supset F , w \in F \}\\
=& \{ & \{x_{11}, \ldots , \widehat{x_{1k_{1}}}, \dots, x_{1i_{1}},
x_{21}, \ldots, \widehat{x_{2k_{2}}}, \dots, x_{2i_{2}}, \dots,\\
&&x_{\mu 1}, \ldots , \widehat{x_{\mu k_{\mu }}}, \dots,
x_{\mu i_{\mu }} \}\\
&&\mid
1 \le k_1 \le i_1, 1 \le k_2 \le i_2, \dots, 1 \le k_{\mu } \le i_{\mu }\\
&& \mbox{ with } 1 \le k_1 \le j_1 \mbox{ or } 1 \le k_2 \le j_2 \mbox{ or }
\dots \mbox{ or } 1 \le k_{p } \le j_{p}
\}\\
\cup & \{ &
\{v, x_{11}, \dots , \widehat{x_{1k_{1}}}, \dots, x_{1i_{1}},
x_{21}, \dots, \widehat{x_{2k_{2}}}, \dots, x_{2i_{2}}, \dots,\\
&&x_{\mu 1}, \dots , \widehat{x_{\mu k_{\mu }}}, \dots,
x_{\mu i_{\mu }} \}
\setminus \{w\} \\
&&\mid
j_1+1 \le k_1 \le i_1, j_2+1 \le k_2 \le i_2, \dots, j_p+1 \le k_p \le i_p\\
&&1 \le k_{p+1 } \le i_{p+1}, \dots,
1 \le k_{\mu } \le i_{\mu }, w \in F
\}
\end{eqnarray*}
\par \noindent
and
\[
I_{\Delta}=( I_{\Gamma}, x_F, vx_{1j_1+1}\cdots
x_{1i_{1}}, vx_{2j_2+1}\cdots x_{2i_{2}}, \dots ,
vx_{pj_p+1}\cdots x_{pi_{p}})
\]
is an ideal of a polynomial ring
\[
S=k[x_{11},\ldots, x_{1i_{1}},
x_{21},\ldots, x_{2i_{2}}, \ldots,
x_{\mu 1},\ldots, x_{\mu i_{\mu }},v].
\]
\par
Applying Proposition \ref{Kustin} to this ideal $I=I_{\Delta}$, we obtain
the following theorem.
It is proved the two-dimensional case in \cite{TrTu}.
\begin{thm} \label{Subdiv}
Let $\Delta=\Gamma_F$ be the stellar subdivision of the non-acyclic
complete intersection complex $\Gamma$ as above.
Then $S/I_{\Delta}^2$ is Cohen--Macaulay.
\end{thm}
\begin{proof}
Consider the variables $\underline{z} = z_1,z_2,\ldots,z_N$, where $N = j_1+\cdots +j_p-1$
and put $Z= z_1\cdots z_N$.
Moreover, we set
\[
\begin{array}{rclcrcl}
X_1 & = & x_{1,1}\cdots x_{1,j_1},& & Y_1 & = & x_{1,j_1+1}\cdots x_{1,i_1}, \\
X_2 & = & x_{2,1} \cdots x_{2,j_2},& & Y_2 & = & x_{2,j_2+1}\cdots x_{1,i_2}, \\
&\vdots & & & & \vdots & \\
X_p & = & x_{p,1}\cdots x_{p,j_p} & & Y_p & = & x_{p,j_p+1}\cdots x_{p,i_p}, \\
& & & & Y_{p+1} & = & x_{p+1,1}\cdots x_{p+1,i_{p+1}}, \\
& & & & & \vdots & \\
& & & & Y_{\mu} & = & x_{\mu,1}\cdots x_{\mu,i_{\mu}}. \\
\end{array}
\]
and
\[
L = (I_{\Gamma},vY_1,\ldots,vY_p,vZ-x_F)
\subseteq T=S[\underline{z}].
\]
Then $I_{\Gamma} = (X_1Y_1,\ldots,X_pY_p,Y_{p+1},\ldots,Y_{\mu})$,
$I_{\Delta} = (I_{\Gamma}, x_F,vY_1,\ldots,vY_p)$ and
$S/I_{\Delta}$ is isomorphic to $T/(\underline{z},L)$.
\par
In what follows, we show that $L$ lies in the linkage class of a complete
intersection (i.e., licci).
Firstly, we can easily prove the following equality:
\begin{equation} \label{First-link}
(I_{\Gamma},Z) \colon (Y_1,\ldots,Y_{\mu},Z) = (I_{\Gamma},Z,x_F).
\end{equation}
Secondly we show the following equality:
\begin{equation} \label{Second-link}
L=(I_{\Gamma},vZ-x_F) \colon (I_{\Gamma},Z,x_F).
\end{equation}
To end this, it is enough to show the right-hand side is contained in $L$.
Let $\alpha \in (I_{\Gamma},vZ-x_F) \colon (I_{\Gamma},Z,x_F)$.
Then there exists a $\beta \in T$ such that $\alpha Z - \beta (vZ-x_F) \in I_{\Gamma}$.
Then $\beta \in (I_{\Gamma},Z) \colon x_F = (Y_1,\ldots,Y_{\mu},Z)$.
In particular, we can write $\beta = \sum_{i=1}^{\mu} \gamma_i Y_i + \delta Z$ for
some $\gamma_i$, $\delta \in T$.
It follows that
\[
Z\bigg[\alpha - \sum_{i=1}^p \gamma_i (vY_i) - \delta (vZ-x_F)\bigg]
\in I_{\Gamma}.
\]
As $Z$ is a nonzero divisor on $T/I_{\Gamma}T$, we conclude that $\alpha \in L$.
\par
In Equations (\ref{First-link}), (\ref{Second-link}),
both $(I_{\Gamma},Z)$ and $(I_{\Gamma},vZ-x_F)$
are complete intersection ideals of the same height $\mu+1$
as $(Y_1,\ldots,Y_{\mu},Z)$ or $L$.
Hence $L$ is licci.
\par
In order to prove that $S/I_{\Delta}^2$ is Cohen-Macaulay by
Proposition \ref{Kustin},
it is enough to show that $\underline{z}$ is a regular sequence on $T/L$
and that $T/L$ is reduced.
By the above proof, we have that $L$ is licci and $\dim T/L = \dim T/(Y_1,\ldots,Y_{\mu},Z)$.
In particular, $L$ is Cohen-Macaulay and $\dim T/L = i_1+\cdots + i_{\mu} - \mu +N$.
\par
On the other hand,
\[
\dim T/(\underline{z},L) = \dim S/I_{\Delta}
= \dim S/(I_{\Gamma},v) = i_1 + \cdots + i_{\mu} - \mu
= \dim T/L - N.
\]
This implies that $\underline{z}$ is a regular sequence on $T/L$.
Moreover, as $T/(\underline{z},L)$ is reduced, so is $T/L$, as required.
\end{proof}
\begin{remark}
The above Gorenstein ideals are obtained from the
so-called Herzog ideals (see \cite{He, Hu, KM1, KM2})
and $T/L$ is called the \textit{Kustin-Miller unprojection ring} (\cite{BP}).
Moreover, the assertion of Theorem \ref{Subdiv}
says that the quotient algebras
of those ideals are \textit{strongly unobstructed}.
\end{remark}
\begin{exam}[Cross Polytope] \label{cross}
Let ${\bf e}_1,\ldots,{\bf e}_d$ be the fundamental vectors of
the $d$-dimensional Euclidean space $\mathbb{R}^d$.
Then the convex hull $\mathcal{P} =
\mathrm{CONV}(\{\pm{\bf e}_1,\pm{\bf e}_2,\ldots,\pm{\bf e}_d \})$
is called the \textit{cross $d$-polytope}.
Let $\Gamma$ be the boundary complex of the cross $d$-polytope $\mathcal{P}$.
Let $W=\{x_1,\ldots,x_d,y_1,\ldots,y_d\}$.
For a sequence ${\bf i} = [i_1,\ldots,i_m]$ with $1 \le i_1 < \cdots < i_m \le d$, we assign
a subset of $W$
\[
F_{\bf i} = \big\{x_{i_1},\ldots,x_{i_m} \big\} \cup
\big\{y_j \,:\, j \in [d] \setminus \{i_1,\ldots,i_m\}\big\}.
\]
Then $\Gamma$ can be regarded as a simplicial complex on $W$ such that
\[
\mathcal{F}(\Gamma)
= \{F_{\bf i}: m=0,1,\ldots,d,\, 1 \le i_1 < \cdots < i_m \le d \},
\]
and it is a $(d-1)$-dimensional complete intersection complex
with
\[
I_{\Gamma} = (x_1y_1,x_2y_2,\ldots,x_dy_d).
\]
\par
Let $v$ be a new vertex, and choose a facet
$F_{[1,2.\ldots,d]}=\{x_1,\ldots,x_d\}$ of $\Gamma$.
Let $\Delta$ be the stellar subdivision of $\Gamma$ on $F$.
Then $\Delta$ is a $(d-1)$-dimensional Gorenstein complex on $V=W \cup\{v\}$
and its geometric realization of $\Delta$ is homeomorphic to
$\mathbb{S}^{d-1}$.
The above theorem says that
the second power of
\[
I=(x_1y_1,x_2y_2,\ldots,x_dy_d,\,vy_1,\ldots,vy_d,\,x_1x_2\cdots x_d)
\]
is Cohen-Macaulay, but the third power is not if $d \ge 2$
because the third power of the Stanley-Reisner ideal $(x_1y_1,x_2y_2,vy_1,vy_2,x_1x_2)$
of a pentagon is not.
\end{exam}
\par \vspace{2mm}
In the last of the paper, we give candidates of edge ideals $I(G)$
for which $S/I(G)^2$ is Cohen--Macaulay (but $S/I(G)^3$ is not by \cite{RTY}).
For the case that $n=2$ it is mentioned in \cite[Theorem 3.7 (iv)]{TrTu}.
\begin{conj} \label{secondpowerCM}
Let $G$ be a graph on the vertex set $V=\{x_1,x_2,\ldots,x_{3n+2}\}$ with
\[
I(G) =
\big(x_1x_2, \; \{x_{3k-1}x_{3k},x_{3k}x_{3k+1},x_{3k+1}x_{3k+2},
x_{3k+2}x_{3k-2}\}_{k=1,2,\ldots,n},\;
\{x_{3\ell-3}x_{3\ell}\}_{\ell=2,3,\ldots,n} \big).
\]
Then $S/I(G)^2$ is Cohen--Macaulay but $S/I(G)^3$ is not.
\vspace{3mm}
\begin{center}
\end{center}
\end{conj}
\par \vspace{2mm}
\begin{acknowledgement}
We would like to thank the referee for his/her advice.
Especially, Section 5 is arranged
based on the referee's advice.
Moreover,
the second author was supported by JSPS 20540047.
The third author was supported by JSPS 19340005.
\end{acknowledgement}
|
{
"timestamp": "2012-03-12T01:00:29",
"yymm": "1203",
"arxiv_id": "1203.1969",
"language": "en",
"url": "https://arxiv.org/abs/1203.1969"
}
|
\section*{Methods}
\begin{small}
\begin{bfseries}
Lasersystem for the fundamental beams.
\end{bfseries}
A schematic of the laser system to produce the three fundamental beams is shown in the lower part of Fig. \ref{Fig:1}(b). The beam at $254\,\text{nm}$ is produced by a frequency-quadrupled Yb:YAG disc laser (ELS, VersaDisk 1030-50). Frequency-quadrupling is done with two resonant enhancement cavities, the first one using a lithium triborate crystal (LBO) as nonlinear medium, the second one using a $\beta$-barium borate crystal (BBO). From $2\,\text{W}$ of infrared light at $1015\,\text{nm}$ we get up to $200\,\text{mW}$ of UV radiation. This system is capable of producing up to 750\,mW of UV light, for details see \cite{Scheid07}. The second fundamental beam at $408\,\text{nm}$ is produced by a frequency-doubled titanium:sapphire laser (Coherent, 899-21), pumped by a frequency doubled Nd:YVO$_4$ laser (Coherent, V10). The external frequency-doubling cavity uses a LBO crystal. From $1.5\,\text{W}$ of infrared (IR) light at $816\,\text{nm}$ we get up to $500\,\text{mW}$ of blue light. The third fundamental beam at $540\,\text{nm}$ is produced with a grating stabilized diode laser at 1080\,nm boosted by a fiber amplifier system (Koheras, Boostik) and frequency doubled by a modified commercial frequency-doubling cavity (Spectra Physics, Wavetrain). This system is capable of producing up to $4\,\text{W}$ of green light at 545.5\,nm \cite{Markert07}. For the present experiments we operate the IR laser at $740\,\text{mW}$, which gives $280\,\text{mW}$ of green light at 540\,nm.
\begin{bfseries}
Wavelength separation \& detection.
\end{bfseries}
The four-wave mixing region is separated from the detection region by a vacuum sealed MgF$_2$ lens which also performs the wavelength separation of the VUV light from the fundamental beams (see Fig. \ref{Fig:1}(b)). Due to the dispersion of this lens the focal length differs for the VUV wavelength ($f=21.5$\,cm at 540\,nm, $f=13$\,cm at 121\,nm). A tiny mirror is placed in the focus of the fundamental beams to reflect them to the side. The VUV beam is large at the fundamental focus ($w \approx 4.9$\,mm) and therefore the mirror just casts a shadow in the VUV beam, causing $\approx 30\%$ loss. A solar-blind photomultiplier tube is used for detection of the VUV photons. The background is suppressed by four 121\,nm filters. The overall detection efficiency due to losses in the MgF$_2$ lens, the tiny mirror, the four filters and the photomultiplier efficiency is $9 \times 10^{-7}$.
\end{small}
|
{
"timestamp": "2012-03-12T01:02:01",
"yymm": "1203",
"arxiv_id": "1203.2121",
"language": "en",
"url": "https://arxiv.org/abs/1203.2121"
}
|
\chapter*{\sc \textbf{Preface}}
\markboth{\sc \textbf{Preface}}{\sc \textbf{Preface}}
\addcontentsline{toc}{chapter}{\sc \textbf{Preface}}
\vspace{2cm}
One century has elapsed since the discovery of superconductivity
by Heike Kamerlingh Onnes, opening a new world of significant
applications in technologies ranging from electric power devices such as
motors and generators, large magnet systems such as those needed in
storage rings for particle accelerators, and electricity transmission in power
lines. As it is well-known, the technological usage of any superconducting
material is based upon its ability
to carry and maintain a current with no applied voltage whatsoever, i.e., with
an almost negligible loss of energy even in those cases when the superconductor
is subjected to strong enough applied magnetic fields. Although electrical
currents can flow with a negligible loss of energy maintaining the
superconductor in an appropriate temperature environment, superconductivity
can be destroyed by the effect of a sufficiently intense magnetic field or the
flow of a current density exceeding a critical value. Indeed, most of the
technological applications of the superconductors are directly linked to their
magnetic properties, and in particular in the way that they expell the magnetic
fields. This fact leads to the classification of superconductors in two
different kinds. On the one hand, \textit{Type-I} superconductors are mainly
characterized by a unique curve for the maximal applied magnetic field which a
superconductor is able to expell before the sudden transition to the normal
state occurs. On the other hand, \textit{Type-II} superconductors are
characterized by a new phase or ``mixed-state'' where the transition from the
superconducting state to the normal state allows the existence of bundles of
magnetic flux penetrating the sample (vortices), before reaching the sudden
transition to the normal state. This remarkable property allows to preserve the
superconducting state with the advantage of sustaining much higher magnetic
fields, and therefore carrying much higher current densities. However, this
ability is directly related to the pinning efficiency of a given material as the
motion of vortices produces a high dissipation of energy which in turn can lead
to the \textit{quench} of the superconducting state. It is worth noting that
all the superconductors, from metal-alloys to cuprates, fullerenes, $MgB_{2}$,
iron-based systems that have been discovered along the last 60 years, are
\textit{Type-II} superconductors, and consequently almost all the actual
superconducting technology is based on these kind of materials.
Thus, since the vortices must be pinned by the underlying crystallographic
structure and the presence of different kind of defects, the knowledge of the
electromagnetic properties and laws governing the pinning of vortices becomes a
crucial but not trivial issue for the understanding and developing of devices in
the framework of applied superconductivity.
In spite of significant theoretical and practical interest, from the
macroscopical point of view, the material laws and the electromagnetic
properties of type-II superconductors still deserve attention,
and currently no book exists that covers all the aspects about this
topic in full depth. This thesis attempts to contribute with some novel
numerical methods in applied superconductivity, including a comprehensive
discussion of the different mechanisms involved in the vortex dynamics.
The book has been structured in three parts with sequential chapters increasing
the level of complexity, both from the mathematical point of view and as
concerns to the underlying phenomena. On the one hand, a general critical
state theory for type-II superconductors with magnetic anisotropy, its
computation, implications, and consequently some applications for particular
problems, is what the first and second part try to convey.
On the other hand, some microscopical aspects of the superconductivity have been
also considered and the attained results have been compiled along the third part
of this thesis.
In detail, the first part of this book is devoted to the study of the
electromagnetic properties of type-II superconductors in the critical state
regime.
After an introductory Chapter 1, which reviews the classical statements of the
critical state theory and derived approaches, Chapter 2 focuses on the
variational theory for critical state problems and the establishment of a
general material law for 3D critical states with an associated magnetic
anisotropy and the underlying physics for the mechanism of flux depinning and
cutting. Then, a technical but important issue arises and is covered in Chapter
3: how to deal with large-scale nonlinear minimization problems such as those
presented in the general critical state theory for applied superconductivity,
but in a personal computer. A well defined structure for the minimization
functional, constraints, bounds, and preconditioners, based upon a set of
FORTRAN packages, solve this problem.
Hopefully, at the end of the first part, the reader will feel either attracted
or at least intrigued by the scope of our theory and methods. In
this sense, the second part of this book is devoted to sketch some of the main
results obtained along this line, i.e., we show some examples where we
have implemented our general critical state theory whose impact affects not
only the understanding of the physical properties of a superconducting system
but also at its potential applications.
In Chapter 4, the advantages of the variational method are emphasized focusing
on its numerical performance that allows to explain a wide number of physical
scenarios. In particular, we present a thorough analysis of the underlying
effects derived of the three dimensional magnetic anisotropy and different
material laws (\textit{or models}) which allow us to treat with the flux
depinning and cutting mechanisms.
Chapter 5 deals with the study of the longitudinal transport problem
(the current is applied parallel to some bias magnetic field) in type II
superconductors. In particular, for the slab geometry with three dimensional
components of the local electromagnetic quantities, the complex interaction
between shielding and transport is solved. On the one hand, based on a
simplified analytical method for 2D configurations, and on the other hand,
based on a wide set of numerical studies for general scenarios (3D), it is shown
that an outstanding inversion of the current flow in a surface layer, and the
remarkable enhancement of the current density by their compression towards the
center of the sample, are straightforwardly predicted when the physical
mechanisms of flux depinning and consumption (via line cutting) are recalled. In
addition, a number of striking collateral effects, such as local and global
paramagnetic behavior, are predicted.
Chapter 6 addresses to a comprehensive study of the electromagnetic response
of superconducting wires subjected to diverse configurations of transverse
magnetic field and/or longitudinal transport current.
In particular, we have performed a wide set of numerical experiments dealing
with the local and global effects underlying to the distribution of field and
current for a straight, infinite, type II superconducting wire, it immersed in
an oscillating magnetic field applied perpendicular to its surface
($\textbf{B}_{0}$), and the simultaneous action of an AC transport current
($I_{tr}$). Thus, in a first part we have introduced the theoretical framework
of this problem focusing on the numerical advantages of our variational method.
Likewise, we provide a thorough discussion about some of the main macroscopic
quantities which may be experimentally measured, such as the magnetization curve
and the hysteretic AC loss, as well as on the local behavior of the
electromagnetic quantities \textbf{E}, \textbf{B}, and \textbf{J}. Three
different regimes of excitation have been considered: (\textit{i}) Isolated
electromagnetic excitations, where only the action of $\textbf{B}_{0}$ or
$I_{tr}$ is considered, (\textit{ii}) Synchronous electromagnetic sources,
where the concomitant action of $\textbf{B}_{0}$ and $I_{tr}$ shows a
unique oscillating phase and frequency, and (\textit{iii}) Asynchronous
electromagnetic sources, where $\textbf{B}_{0}$ and $I_{tr}$ do not show the
same oscillating frequency and therefore are out-phase. The underlying
effects of considering premagnetized wires under the above mentioned regimes are
also considered. Thus, several striking effects as the strong localization of
the local density of power loss, a distinct low-pass filtering effect intrinsic
to the wire's magnetic response, exotic magnetization loops, increases and
decreases of the hysteretic AC loss by power supplies with double frequency
effects, and significant differences between the widely used approximate
formulae and the actual AC loss numerically calculated, have been detected
and explained.
The last part of this dissertation concerns our contribution to another aspect
of superconductivity. By means of a specific integral method applied to
spectroscopic data, we have been able to draw some conclusions on the influence
of the Electron-Phonon (E-Ph) coupling mechanism in cuprate superconductors.
More specifically, we have focused on the analysis of high-resolution angle
resolved photoemission spectroscopies (ARPES) in several families of cuprate
superconductors. Although relying on solid (and sophisticated) techniques in
the realm of quantum theory, we describe a phenomenological procedure that
allows to obtain relevant physical parameters concerning the E-Ph interaction.
Thus, in chapter~\ref{ch-7}, we introduce a novel theoretical model which allows
a quite general explanation of the so-called nodal \textit{kink effect} observed
in ARPES, for several doping levels in the cuprate
families $La_{2-x}Sr_{x}CuO_{4}$, $Bi_{2}Sr_{2}CaCu_{2}O_{8+x}$, and
$YBa_{2}Cu_{3}O_{6+x}$.
Finally, in an effort to clarify the influence of
the E-Ph coupling mechanism to the boson mechanism which causes the pair
formation in the superconducting state, chapter~\ref{ch-8} addresses the study
of the superconducting thermodynamical quantities, $T_{c}$, the ratio gap
$2\Delta_{0}/k_{B}T_{c}$, and the zero temperature gap $\Delta_{0}$, for a wide
set of natural and empirical equations.
In reading this book, we want to remark that each one of its parts have its
own introduction and concluding sections, and also the list of references to the
literature have been placed forward. In addition, a small glossary can be found
at the end of this book.
\break
Hopefully, this thesis may serve to bring a bigger community interested in
the world of superconductivity, either in the application of their macroscopical
properties or the understanding of their microscopical ground.
\vspace*{2cm}
\hspace{7cm} February 2012, Zaragoza - Spain.
\cleardoublepage
\pagenumbering{arabic}
\part{\label{Part_1}\textsc{\textbf{Electromagnetism of type II
superconductors}}}
\chapter*{}
\vspace*{-3cm}
\section*{\label{Intro-P1}\hspace*{0.55cm} \sc \textbf{Introduction}}
\markboth{\sc \textbf{Introduction}}{\hspace*{0.55cm} \sc \textbf{Introduction}}
\addcontentsline{toc}{chapter}{\hspace*{0.55cm} \sc \textbf{Introduction}}
The high interest concerning the investigation of the macroscopic
magnetic properties of type-II superconductors in the mixed state is markedly
associated with its relevance to technological and industrial applications
achieving elevated transport currents with no discernible energy dissipation.
It relates to a wide list of physical phenomena concerning the physics of
vortices, which may be basically analyzed in terms of
interactions between the flux lines themselves (lattice elasticity and line
cutting), and interactions with the underlying crystal structure averaged
by the so-called flux pinning mechanism.
In a mesoscopic description of real type-II superconductors, the distribution
of vortices may be simplified through a mean-field approach for a volume
containing a big enough number of vortices and making use of an appropriate
material law incorporating the intrinsic properties of the material. This
picture of coarse-grained fields, i.e.: magnetic induction ${\bf B}\equiv
\langle{\bf b}\rangle$, electric current density ${\bf J\equiv
\langle{\bf j}\rangle}$ and electric field ${\bf E\equiv \langle{\bf
e}\rangle}$, allows to state the problem of the driving force due to the
currents circulating in the superconducting sample and their balance with the
limiting pinning force acting on the vortex lattice so as to
prevent destabilization and the consequent propagation of dissipative states.
Per unit volume, this reads ${\bf J}\times{\bf B}={\bf F}_p$ (or $J_{\perp}B =
F_p$). The underlying concept behind this balance condition is already a
classical subject well known as the critical state model by Charles P.
Bean~\cite{P1-Bean_1964}. In this simple, but brilliant model, the response of
the superconducting sample is provided by assuming that the electrical current
density vector $\textbf{J}$ (oriented perpendicular to the direction of
the local magnetic field vector ${\bf B}$) compensates with the pinning force,
and then, it is constrained by a threshold value $\textbf{J}_{c}$ which defines
a local critical state for the array of magnetic flux lines. Thus, in
view of Faraday's law, external field variations are opposed by the maximum
current density $J_{c}$ within the material, and after the changes occur,
$J_{c}$ persists in those regions which have been affected by an electric field.
Although, such a model allows to capture the main features of the magnetic
response of superconductors with pinning at low frequencies and temperatures,
through the minimal mathematical complication, the stronger limitation
of Bean's ansatz is that one can just apply it to vortex lattices composed by
parallel flux lines perpendicular to the current flow, and unless for highly
symmetric situations \textbf{J} does not necessarily satisfy the condition
$\textbf{J}=\textbf{J}_{\perp}$. In fact, a proper theory for the critical state
must allow the coexistence of nonparallel flux lines. Thus, rotations of
\textbf{B} can lead to entanglement and recombination of neighboring flux lines
which brings a component of the current density along the local magnetic field,
$\textbf{J}_{\parallel}$. This component generates distortions which also become
unstable when a threshold value $J_{c\parallel}$ is exceeded, giving way to the
so-called flux cutting phenomenon.
When the conditions $J_{\parallel}=J_{c\parallel}$ and $J_{\perp}=J_{c\perp}$
become active, the so-called double critical state appears~\cite{P1-Clem_DCSM}.
In simple words, this upgraded theory ({\em double critical state model} or
DCSM) generalises the one-dimensional concept introduced by
Bean~\cite{P1-Bean_1964} to anisotropic scenarios for the material law in terms
of the natural concepts $E_{\parallel}(J_{\parallel})$ and
$E_{\perp}(J_{\perp})$~\cite{P1-Ruiz_PRB_2009,P1-Ruiz_SUST_2010,
P1-Ruiz_SUST_2011}.
From the mathematical point of view, the critical state problem can be
understood as finding the equilibrium distribution for the circulating
current density $\textbf{J}(\textbf{r})$ defined by the conditions
$J_{\parallel}\leq J_{c\parallel}$ and $J_{\perp}\leq J_{c\perp}$, both
consistent with the Maxwell equations in quasistationary form, and under
continuity boundary conditions that incorporate the influence of the sources.
Being a quasi-stationary approach, the critical state is customarily stated
without an explicit role for the transient electric field. Thus, Faraday's law
is implicitly used through Lenz' law by selecting the actual value $\pm
J_{c}$ or $0$ that minimizes flux variations when solving Ampere's law
$\nabla\times \textbf{B}=\mu_{0}\textbf{J}$ along the process. Customarily, one
also considers situations
where the local components of the magnetic field $\textbf{H}(\textbf{r})$ along
the superconductor (SC) are much higher than the lower critical field
$H_{c1}$ and well below $H_{c2}$ to allow the use of the linear relation
$\textbf{B}=\mu_{0}\textbf{H}$.
Within this picture of the electromagnetic problem, in this first \textit{part}
of the book we introduce the important definitions and concepts of those topics
behind the critical state theory, extending its scope for three dimensional
cases with help of numerical methods in the
framework of the variational formalism for optimal control problems. We want to
emphasize that although it is not our intent to develop a comprehensive study of
these mathematical topics, we will show common mathematical techniques which are
found to be particularly useful in applied superconductivity. The reader is
referred to the
references~\cite{P1-Badia_PRL_2001,P1-Badia_PRB_2002,P1-Jackson,P1-Mayergoyz,
P1-Arfken,P1-Pontryagin,P1-Leitmann,P1-Knowles} for a more thorough discussion
of this
material.
Chapter 1 is devoted to introduce the theoretical background that
justifies the critical state concept as a valid
constitutive law for superconducting materials. First, the critical state is
described by the classical differential formalism of the Maxwell equations, and
then, the prescribed magnetoquasistationary approximation is thoroughly
discussed.
In chapter 2, our proposed general critical state theory is developed in two
parts. Firstly, the critical state problem is posed in terms of an equivalent
optimal control problem with variational statements, i.e., the classical Maxwell
equations are translated to the variational formalism where a simpler set of
integral equations with boundary conditions is to be solved by a minimization
procedure. Is to be noticed, that despite of the fact that the reader can feel
more familiar to the differential formalism, the numerical solution of the
differential set of equations is much more cumbersome than minimizing an
integral functional. Secondly, the underlying vortex physics is posed in terms
of a quite general material law for type-II
superconductors with magnetic anisotropy, which characterizes
the conducting behavior in terms of the threshold values for the current
density and the physical mechanisms of flux depinning and cutting.
Finally, chapter 3 covers the basic facts related to the computational method
adopted for the solution of general critical state problems such
as those tackled in the following part of this book. Here, no attempt
is made to scrutinize through the FORTRAN packages for large scale nonlinear
optimization. Instead, the presentation of this chapter must be
understood as a schematic tool for dealing with a wide variety of problems in
applied superconductivity.
\chapter{\label{ch-1} \sc \textbf{General Statements Of The Critical State}}
\subsection*{\label{ch-1-1}
\hspace*{0.1cm} \textsl{1.1 The Critical State In The Maxwell Equations
Formalism}}
\vspace*{1cm}
\markboth{\hspace*{0.1cm}\textsl{1.1 The Critical State In The Maxwell
Equations Formalism}}
{\hspace*{0.1cm} \textsl{1.1 The Critical State In The Maxwell Equations
Formalism}}
\addcontentsline{toc}{chapter}
{\hspace*{0.1cm} \textsl{1.1 The Critical State In The Maxwell Equations
Formalism}}
The fundamental concept on which the critical state theory relies is that, in
many cases, the experimental conditions allow to analyze the evolution of the
system in a magnetoquasisteady (MQS) regime of the time-dependent Maxwell
equations accompanied by material constitutive laws, ${\bf H}({\bf
B})$, ${\bf D}({\bf E})$, and ${\bf J}({\bf E})$. Thus, Faraday's and Ampere's
laws represent a coupled system of time evolution field equations
\begin{eqnarray}\label{Eq.1.1}
\partial_{t}{\bf B}=-\nabla\times{\bf E}\quad , \quad
\partial_{t}{\bf D}=\nabla\times{\bf H}-{\bf J} \, ,
\end{eqnarray}
which together determine the distribution of supercurrents within the sample.
Here, the induced transient electric field is determined through an appropriate
material relation ${\bf J}({\bf E})$, and is used to update the profile of $\bf
J$.
\footnotetext[1]{If there were some average surface currents, then the actual
density of magnetic flux $\textbf{B}$ would differ from
$\mu_{0}\textbf{H}$.}
Notice that, as equilibrium magnetization is usually neglected
in the critical-state regime, one is enabled to use the
relation $\textbf{B}=\mu_{0}\textbf{H}$, so that there are no average surface
currents.\footnotemark[1] Furthermore, as the magnetic fields of interest are
some fraction of the critical transition magnetic field $H_{c2}$ that is much
greater than the penetration field $H_{c1}$, the distribution of vortices and
the corresponding supercurrents will be thermodynamically favoured to
go into the superconductor, such that a ramp in the magnetic field is induced
by the external excitation within the interval $[t,t+dt]$, see
Fig.~\ref{Figure_1_1}(a). Thus, as a consequence of a very fast diffusion
(elevated flux flow resistivity), the electric field quickly adjusts to a
constant value along the excitation interval, and once the
magnetic field ramp stops, $E$ goes back to zero again.
The {\em readjusting}
vertical bands are considered a second order effect and allow for charge
separation and recombination, according to the specific ${\bf E}({\bf J})$ model
[see Fig.~\ref{Figure_1_1}(b)]. Therefore, we are allowed to model the
flux as entering the superconductor at zero field cooling, where the electric
field arises when some {\em critical} condition for the volume current density
is reached ($J_c$ in the 1D representation). Then, corresponding to the MQS
limit, the electric field instantaneously increases to a certain value
determined by the rate of variation of the magnetic field and then goes back to
zero.
By taking divergence in both sides
of the Faraday's and Ampere's laws, and recalling integrability
(permutation of space and time derivatives) it leads to
the additional conditions
\begin{eqnarray}\label{Eq.1.2}
\partial_{t}(\nabla\cdot{\bf B})=0 \quad ,\quad
\partial_{t} (\nabla\cdot{\bf D}) + \nabla\cdot{\bf J} = 0 \, .
\end{eqnarray}
Within this picture, the remaining
Maxwell equations can be interpreted as ``\textit{spatial
initial conditions}'' for Eq.~(\ref{Eq.1.2}) which are defining the existence of
conserved electric charges, i.e.,
\begin{eqnarray}\label{Eq.1.3}
\nabla\cdot{\bf B}(t=0) = 0 \quad ,\quad \nabla\cdot {\bf D} (t=0) = \rho
(t=0)\, .
\end{eqnarray}
In this sense, the set of equations~(\ref{Eq.1.1}), upon substitution of $\bf
H$, $\bf D$ and $\bf J$ through the constitutive laws, and with appropriate
initial conditions, uniquely determine the evolution profiles ${\bf
B}({\bf r},t)$ and ${\bf E}({\bf r},t)$.
\begin{figure}[t]
\centering
\includegraphics[width=1.0\textwidth]{Figure_1_1.pdf}
\caption{\label{Figure_1_1} (a) Schematic representation the time
dependence of the electromagnetic fields within the MQS
regime. (b) Pictorial drawing of the critical state model in terms of a one
dimensional ${E}({J})$ law.}
\end{figure}
Notice that, for {\em slow} and {\em uniform} sweep rates of the external
excitations (magnetic field sources and/or transport current), the
transient variables $\bf E$, $\bf D$ and $\rho$ are small, and proportional to
$\dot{\bf B}$, whereas $\ddot {\bf B}$, $\dot {\bf E}$ and
$\dot {\rho}$ are negligible. Thus, the main hypothesis within the MQS regime is
that the {\em displacement} current densities $\partial _t{\bf D}$ are much
smaller than $\bf J$ in the bulk and vanish in a first order treatment. This
causes a crucial change in the
mathematical structure of the Maxwell equations: Ampere's law is no
longer a time evolution equation, but becomes a purely spatial
condition. It reads as
\begin{eqnarray}\label{Eq.1.4}
\nabla\times {\bf B} \simeq \mu_{0}{\bf J} \, ,
\end{eqnarray}
with approximate integrability condition $\nabla\cdot{\bf J}\simeq 0$.
In the MQS limit, Faraday's law is the unique time evolution equation. Then, one
can find the evolution profile ${\bf B}({\bf r},t)$ from
\begin{eqnarray}\label{Eq.1.5}
\partial_{t}{\bf B}=-\nabla\times{\bf
E}=-\nabla\times\left[\rho(\textbf{J})~\mu_{0}\nabla\times{\bf
B}\right] \, .
\end{eqnarray}
Here, $\rho ({\bf J})$ plays the role of a nonlinear and possibly nonscalar
resistivity
that should properly incorporate the physics of the threshold and dissipation
mechanisms associated to the flux depinning and flux cutting mechanisms.
We want to mention that, although the \textit{B-formulation} in
Eq.~(\ref{Eq.1.5}) is definitely the most extended one, the possibilities of
\textit{E-formulations}~\cite{P1-Barret_2006},
\textit{J-formulations}~\cite{P1-Wolsky_2008}, or a vector potential oriented
theory (\textit{A-formulation})~\cite{P1-Campbell_2009}, in which the dependent
variables are the fields ${\bf E}$, ${\bf J}$, or $\textbf{A}$ respectively,
have also been exploited by several authors.
\vspace*{0.5cm}
\subsection*{\label{ch-1-2}
\hspace*{0.1cm} \textsl{1.2 The Critical State Regime And The MQS
Limit}}
\vspace*{0.5cm}
\markboth{\hspace*{0.1cm}\textsl{1.2 The Critical State Regime And The MQS
Limit}}
{\hspace*{0.1cm} \textsl{1.2 The Critical State Regime And The MQS
Limit}}
\addcontentsline{toc}{chapter}
{\hspace*{0.1cm} \textsl{1.2 The Critical State Regime And The MQS
Limit}}
In spite of the seeming simplicity of the MQS approach
($\partial_{t}\textbf{D}\approxeq0$), we want to emphasize that the numerical
procedure to solve a critical state problem is closely linked to the
consequences of having assumed this limit. Below, two of the most relevant
consequences of the MQS limit are highlighted.
\begin{enumerate}
\item Notice that, making use of the conductivity law
through its inverse function ${\bf E}({\bf J})$, the successive field
penetration profiles within the superconductor
may be obtained by the finite-difference expression of Faraday's law,
\begin{eqnarray}\label{Eq.1.6}
\frac{B_{l+1}-B_{l}}{\delta t}=-\nabla\times{\bf
E}~\left(\mu_{0}\textbf{J}_{l+1}\approx\nabla\times\textbf{B}_{l+1}\right) \, .
\end{eqnarray}
Here we have assumed an evolutionary discretization scheme, where
$\textbf{B}_{l}$ stands for the local magnetic field induction at the time
layer $l\delta t$, and the current density profiles are related to some
magnetic diffusion process that takes place when the local condition for
critical state $\textbf{J}(\textbf{r})\leq \textbf{J}_{c}(\textbf{r})$ is
violated. On the other hand, the constitutive law ${\bf D}({\bf E})$ which is
not used in
Eq.~(\ref{Eq.1.6}), plays no role in the evolution of the magnetic variables
$\textbf{B}_{l+1}$ and $\textbf{J}_{l+1}$, which means that the magnetic
``\textit{sector}'' is decoupled from the charge density profile because the
coupling term (charge recombination) has disappeared. In this sense, notice that
the local profile $\textbf{B}_{l+1}$ can be solved in terms of the
previous field distribution $\textbf{B}_{l}$ and the boundary conditions at time
layer $(l+1)\delta t$.
\item As the initial conditions must fulfill the Ampere's
law $\nabla\times\textbf{B}_{l}=\mu_{0}\textbf{J}_{l}$ as well as
$\nabla\cdot\textbf{B}_{l}=0$ and $\nabla\cdot\textbf{J}_{l}=0$, only the
inductive component of $\textbf{E}$ (given by $\nabla\times {\bf E}_{\rm ind} =
- \dot {\bf B}$, $\nabla\cdot {\bf E}_{\rm ind} = 0$) determines the evolution
of $\bf B$ (Faraday's law). At this point, the conducting law in its inverse
formulation ${\bf E}({\bf J})$ seems show certain ambiguity, as far as
two different material laws related by ${\bf E}_2 ({\bf J}) = {\bf E}_1 ({\bf
J}) + \nabla\Phi ({\bf J})$ determine the same magnetic and current density
profiles. Going into some more detail, whereas for the complete Maxwell
equations statement, the potential component of the electric field
($\nabla\times{\bf E}_{\rm pot} = 0$,
$\epsilon _0 \nabla\cdot {\bf E}_{\rm pot} = \rho$), is coupled to $\bf B$ and
${\bf E}_{\rm ind}$ through the $\dot {\bf D}$
term (which contains both inductive and potential parts), within the MQS limit
it is irrelevant for the magnetic quantities. In
fact, one is enabled to include the presence of charge densities without
contradiction with the condition $\nabla\cdot {\bf J} \simeq 0$ by means of
inhomogeneity or nonlinearity in the $\textbf{E}(\textbf{J})$ relation. Then
one has that the condition $\nabla\cdot\textbf{J}=0$ does not imply
$\nabla\cdot\textbf{E}=0$. The charge density ${\rho}$ can be
understood as a parametrized charge of {\it static} character as far as $\dot
{\rho}$ is
neglected. As indicated above, once the magnetic variables are computed, one
has the freedom to modify the ``\textit{electrostatic sector}'' if necessary
by the rule ${\bf E}({\bf J})+\nabla\Phi$ while still maintaining the values
of $\bf B$ and $\bf J$. This invariance can be of practical interest as far as
the ``electrostatic'' behavior in the critical state regime is still under
discussion, it because of the inherent difficulties in
the direct measurement of transient charge
densities~\cite{P1-Ruiz_SUST_2011,P1-Joos_2006,P1-Clem_2011_PRB,Clem_2011_SUST}.
\end{enumerate}
\chapter{\label{ch-2} \sc \textbf{Variational Theory for Critical State
Problems}}
\subsection*{\label{ch-2-1}
\hspace*{0.1cm} \textsl{2.1 General Principles Of The Variational
Method}}
\vspace*{1cm}
\markboth{\hspace*{0.1cm}\textsl{2.1 General Principles Of The Variational
Method}}
{\hspace*{0.1cm} \textsl{2.1 General Principles Of The Variational
Method}}
\addcontentsline{toc}{chapter}
{\hspace*{0.1cm} \textsl{2.1 General Principles Of The Variational
Method}}
As we have mentioned before, the numerical solution of the critical state
problem from the differential formalism of the Maxwell equations may be
cumbersome. One
possibility for making the resolution of this system affordable is to find an
equivalent variational statement of Eq.~(\ref{Eq.1.6}). Then, one can avoid the
integration of these set of differential equations by \textit{inversion} of a
set of Euler-Lagrange equations
\begin{eqnarray}\label{Eq.2.1}
\mu_{0}\textbf{J}_{l+1}-\nabla\times\textbf{B}_{l+1}=0\, ,
\end{eqnarray}
and
\begin{eqnarray}\label{Eq.2.2}
\mu_{0}\nabla\times\textbf{p}_{l}+\textbf{B}_{l+1}-\textbf{B}_{l}=0\, ,
\end{eqnarray}
for arbitrary variations of the Lagrange multiplier
(i.e., $\delta\textbf{p}_{l}$), and the time-discretized local magnetic
induction field (i.e., $\delta\textbf{B}_{l+1}$).\footnotemark[1] Eventually,
the Lagrange multiplier, $\textbf{p}_{l}$, will be basically identified with the
electric field of the problem.
\footnotetext[1]{Recall that the magnetic field \textbf{H} is defined as
a modification of the induction field \textbf{B} due to magnetic fields produced
by material media. However, as in the critical state regime the use of the
linear relation $\textbf{B}=\mu_{0}\textbf{H}$ is allowed, henceforth, we
will refer to \textit{magnetic field} where either or both fields apply.}
Going into more detail, let us consider a small path step $\delta t$, from some
initial profile of the magnetic field $\textbf{B}_{l}(\textbf{r})$ to a
final profile $\textbf{B}_{l+1}(\textbf{r})$, and the
corresponding $\textbf{J}_{l}(\textbf{r})$ and $\textbf{J}_{l+1}(\textbf{r})$.
Defining $\Delta\textbf{B}=\textbf{B}_{l+1}-\textbf{B}_{l}$, both
configurations can be considered to be connected by a steady process
performing a small linear step, such that
$\textbf{B}_{l+1}=\textbf{B}_{l}+s\Delta\textbf{B}$ with $s\in[0,1]\delta t$.
Recalling that the initial condition fulfills Ampere's law
$\nabla\times\textbf{B}_{l}=\mu_{0}\textbf{J}_{l}$, as well as
$\nabla\cdot\textbf{B}_{l}=0$ and $\nabla\cdot\textbf{J}_{l}=0$, the
time-averaged Lagrange density (over whole space) is
\begin{eqnarray}\label{Eq.2.3}
{\cal L}
=\frac{1}{2}|\Delta\textbf{B}|^{2}+\textbf{E}\cdot(\nabla\times\textbf{B}_ { l+1
}
-\textbf{J}_{l+1})\delta t \, .
\end{eqnarray}
Thus, the physically admissible Lagrangian multipliers in the critical state
regime must satisfy the condition
\begin{eqnarray}\label{Eq.2.4}
\textbf{p}=\textbf{E}_{cs}\delta t \, ,
\end{eqnarray}
where the critical state electric field $(\textbf{E}_{cs})$ must be properly
defined by the imposed material law $\textbf{E}(\textbf{J})$.
However, concerning the ``{\it unknown parameter}'' $\textbf{J}_{l+1}$, as far
as it is not allowed to take arbitrary values, we cannot impose
arbitrary variations as it is customary for the typical steady condition
of the Euler-Lagrange equations. Instead, an Optimal Control-like Maximum
principle equivalent to a maximal projection rule
$\textbf{\^{E}}\cdot\textbf{J}$ must be
used (see Refs.~\cite{P1-Badia_PRL_2001,P1-Badia_PRB_1998}). For a
more comprehensive review
of
the optimal control theory which can be understood as a generalization of the
variational calculus, the interested reader is directed to see, for instance,
Refs.~\cite{P1-Pontryagin,P1-Leitmann,P1-Knowles}.
In simple terms, the optimal control concept introduces a geometrical picture of
the material law for the boundary conditions of the vector $\textbf{J}$ that may
be of much help when discussing the idea of a general
critical state theory. Summarizing, it is necessary to declare that
there must be a region $\Delta_{\textbf{r}}$ within the \textbf{J}
space (possibly oriented according to the local magnetic field $\textbf{\^{B}}$,
and/or also depending on $|\textbf{B}|$ and \textbf{r}) such that nondissipative
current flow occurs when the condition $\textbf{J}\in\Delta_{\textbf{r}}$ is
verified. Thus, the minimum of the Lagrangian
must be sought within the set of current density vectors fulfilling
${\bf J}\in\Delta_{\textbf{r}}$, i.e.: $\textbf{J}_{l+1}$ is determined by
the condition
\begin{eqnarray}\label{Eq.2.5}
{\rm Min}\{ {\cal L} \}|_{\textbf{J}\in\Delta_{\textbf{\textbf{r}}}}\equiv{\rm
Max}\{\textbf{J}\cdot\textbf{p}\}|_{\textbf{J}\in\Delta_{\textbf{\textbf{r}}}}
\, .
\end{eqnarray}
Notice that an $\textbf{E}(\textbf{J})$ law is needed in addition to
Eq.~\ref{Eq.2.3}. Thus, together with the concept of a very high dissipation
when \textbf{J} is driven outside $\Delta_{\textbf{r}}$ by some nonvanishing
electric field, Eq.~\ref{Eq.2.5} suffices to determine the relation between
the directions of \textbf{J} and \textbf{E}. Notice that the maximal shielding
condition is equivalent to the maximum projection rule, it means that the
orthogonality condition of the electric field direction with the surface of
$\Delta_{\textbf{r}}$ is recalled, and the Lagrange multiplier can be
straightforwardly identified with the electric field of the problem, i.e.,
\begin{eqnarray}\label{Eq.2.6}
{\rm
Max}\{\textbf{J}\cdot\textbf{p}\}|_{\textbf{J}\in\Delta_{\textbf{r}}}\equiv{\rm
Max}\{\textbf{E}\cdot\textbf{J}\}|_{\textbf{J}\in\Delta_{\textbf{r}}} \, .
\end{eqnarray}
Notice also that Ampere's law is imposed [Eq.~(\ref{Eq.2.1})] through the
Lagrange multiplier, while the discretized version of Faraday's law
[Eq.~(\ref{Eq.2.2})] is derived as an Euler-Lagrange equation for the
variational problem, so that absolute consistency with the Maxwell equations is
obtained. In fact, maximal global (integral) shielding is achieved through a
maximal local shielding rule [Eq.~(\ref{Eq.2.6})] that reproduces the elementary
evolution of $\partial_{t}{\bf J}$ for a perfect conductor with restricted
currents. Thus, in practice, if one explicitly introduces Ampere's
law $(\nabla \textbf{B}_{l+1}=\mu_{0}\textbf{J}_{l+1})$, minimization is made in
terms of
\begin{eqnarray}\label{Eq.2.7}
{\cal F}[\textbf{J}_{l+1}]={\rm
Min}\left\{ -\int_{\Re^{3}}\textbf{E}\cdot\textbf{J}_{l+1} \right\} \, ,
\end{eqnarray}
and the minimum is sought over $\textbf{J}_{l+1}\in\Delta_{r}$ for a fixed
$\textbf{E}$. However, as we have mentioned before, special attention must be
payed to the feasible ambiguity of the function $\textbf{E}(\textbf{J})$ as it
can lead to fake values of the variables $\textbf{J}$.
Likewise, the straightforward equivalence between the convex functionals for
Eqs.~(\ref{Eq.2.5}) and (\ref{Eq.2.7}) allows to establish an equivalent
minimization principle in terms of the general definitions
\begin{eqnarray}\label{Eq.2.8}
\textbf{B}=\nabla\times\textbf{A} \, ,
\end{eqnarray}
and
\begin{eqnarray}\label{Eq.2.9}
\textbf{E}=-\partial_{t}\textbf{A}-\nabla\varPhi \, ,
\end{eqnarray}
by imposing the material law $\textbf{E}(\textbf{J})$ through the Lagrange
multiplier $\textbf{p}_{l+1}$. Thus, the minimization problem turns to find out
the invariant gauge conditions $\nabla\varPhi_{l+1}$ and
$\textbf{J}_{l+1}\in\Delta_{\textbf{r}}$ for a given function
$\textbf{A}[\textbf{J}]$, in such manner that
\begin{eqnarray}\label{Eq.2.10}
{\cal F}[\textbf{B}_{l+1},\nabla \varPhi]
={\rm Min}\int_{\Re^{3}}&&
\frac{1}{2}|\Delta\textbf{B}|^{2}-\Delta\textbf{A}\cdot(\nabla\times\textbf{B}_{
l+1}-\textbf{J}_{l+1})\\
&&-\nabla\varPhi(\nabla\times\textbf{B}_{l+1}-\textbf{J}_{l
+1})\delta t \, . \nonumber
\end{eqnarray}
We call the readers' attention to notice that the uncoupling of the
electromagnetic potentials can be accomplished
by exploiting the arbitrariness involved in the definition of $\textbf{A}$. In
fact, since \textbf{B} is defined through Eq.~(\ref{Eq.2.8}) in terms of
\textbf{A}, the vector potential is arbitrary to the extent that the gradient of
some scalar function can be added. Therefore, the ``\textit{magnetic sector}''
could be decoupled of the ``\textit{electric sector}'' if the physical
admissible states in the time-averaged Lagrange density $L$ are invariant gauge
of the Lagrange multipliers \textbf{p}. As a consequence, if the problem is such
that there are no intrinsic electromagnetic sources, $\varPhi\equiv0$ (for
type-II superconductors it means absence of transport current), a proper choice
of \textbf{A} should satisfy the Coulomb gauge $(\nabla\cdot\textbf{A}\equiv0)$.
In this sense, by using the Laplace equation, the second term in
Eq.~(\ref{Eq.2.10}) is reduced to
$\Delta\textbf{A}\cdot\delta_{t}^{2}\textbf{A}$ meanwhile the third term
have vanished by assuming $\varPhi\equiv0$. Then, as the MQS approximation
relies in assume that the electric field quickly adjusts to a constant value
along the interval $[t+\delta t]$, for enough small time steps $\delta t$ (see
Fig.~\ref{Figure_1_1}) the action of
$\Delta\textbf{A}\cdot\delta_{t}^{2}\textbf{A}$ may be neglected, and therefore
the solution of the critical state problem can be also achieved from the
functional for the magnetic sector:
\begin{eqnarray}\label{Eq.2.11}
{\cal F}[\textbf{B}(\cdot)]={\rm Min}\int_{\Re^{3}}
\frac{1}{2}|\Delta\textbf{B}|^{2} \, .
\end{eqnarray}
Recall that, the minimization principle is based on a discretization of the path
followed by the external sources, meaning that it is an approximation to the
continuous evolution whose accuracy increases as the step diminishes.
Moreover, we must emphasize that the derived functionals
[Eqs.~(\ref{Eq.2.7}) \& (\ref{Eq.2.11})] are in matter of fact fully
equivalents, as long as the minimization procedure accomplishes the boundary
conditions imposed by the prescribed sources and the material law
$\textbf{J}\in\Delta_{\textbf{r}}$. Thus, in those cases when an intrinsic
electromagnetic source must be considered, i.e., $\nabla\varPhi\neq0$, the
global set of variables must me constrained by the prescribed conditions. For
example, if the superconductor is carrying a transport
current $I_{tr}$ flowing through the surface $s$, one has to mandatorily
consider the external constraint
\begin{eqnarray}\label{Eq.2.12}
\int_{s}\textbf{J}\cdot\hat{\textbf{n}}ds=I_{tr} \,
\end{eqnarray}
and further update the distribution of current to satisfy the physical condition
$\textbf{E}\cdot\textbf{J}=0$ (at those points where the magnetic flux does not
vary), by means the use of a \textit{calibrated} potential
$\textbf{\~{A}}$. Thus, one of the advantage
of the formulation in Eq.~(\ref{Eq.2.11}) is that the number of variables can be
reduced avoiding to include the intrinsic variables associated to $\varPhi$,
accordingly to the statement
$\textbf{E}=-\partial_{t}\textbf{A}-\nabla\varPhi\equiv\partial_{t}
\textbf{\~{A}}$
Concluding, for 3D problems, it must be emphasized that the introduced
minimization principle can be applied for any shape of the
superconducting volume $\Omega$ as well as for any general restriction (material
law) for the current density $\textbf{J}_{l+1}\in\Delta_{\textbf{r}}$.
Different possibilities for the material law are described in the following
chapter. It is
also to be noticed that the searching of the minimum for the allowed set of
current densities must fulfill the intrinsic condition
$\nabla\cdot\textbf{J}=0$ to be consistent with charge conservation
in the quasi-steady regime. Further, from the numerical point of view, the
advantage
of the variational formulation in Eqs.~(\ref{Eq.2.7})~\&~(\ref{Eq.2.11}) is
that
one can avoid the integration of the equivalent partial differential equations
and straightforwardly minimize the discretized integral by using a numerical
algorithm for constrained minimization (see Chapter~\ref{ch-3}). This fact
represents an
important advantage in the performance and power of the numerical methods
applied to the design of superconducting devices, where symmetry arguments can
allow further simplifications and correspondingly faster numerical convergence.
\vspace*{1cm}
\subsection*{\label{ch-2-2}
\hspace*{0.1cm} \textsl{2.2 The Material Law: SCs with
magnetic anisotropy}}
\vspace*{1cm}
\markboth{\hspace*{0.1cm}\textsl{2.2 The Material Law: SCs with
magnetic anisotropy}}
{\hspace*{0.1cm} \textsl{2.2 The Material Law: SCs with
magnetic anisotropy}}
\addcontentsline{toc}{chapter}
{\hspace*{0.1cm} \textsl{2.2 The Material Law: SCs with
magnetic anisotropy}}
In this section, we will continue our discussion of the critical state theory
which still needs the explicit inclusion of a material law
$\textbf{J}(\textbf{E})$ that dictates the magnetic response of a
superconducting sample for a given external excitation. For simplicity, we start
with an overview of the material law for 1D systems, that will be gradually
generalized until a 3D formulation is reached.
\vspace*{0.5cm}
\hspace*{-0.3cm}
\textbf{\textit{2.2.1 ~Onto The 1D Critical States}}
\markboth{\hspace*{0.1cm} \textsl{2.2.1 ~1D Critical States}}{\hspace*{0.1cm}
\textsl{2.2.1 ~Onto The 1D Critical States}}
\addcontentsline{toc}{chapter}{\hspace*{0.1cm} \textsl{2.2.1 ~Onto the 1D
Critical States}}\label{ch-2.2.1}
\vspace*{0.5cm}
For our purposes, it is sufficient to recall that the basic structure of the
critical state problem (Fig.~\ref{Figure_1_1}) relates to an experimental graph
within the $\{V , I \}$ plane that basically contains two regions defined by
the critical current value $I_{c}$ as follows:
\begin{enumerate}
\item $-I_c \leq I \leq I_c$ with perfect conducting behavior, i.e.:
$V=0$ and $\partial _t I = 0$.
\item For $I \gtrsim I_c$, the curve is characterized by a high
${\partial_{I} V}$ slope (and antisymmetric for $I \lesssim - I_c$). Further
steps, with $I$ increasing above the critical value $I_c$, i.e., the eventual
transition to the normal state, may be neglected for slow sweep rates of the
external sources, which produce moderate electric fields.
\end{enumerate}
Within the local description of the electromagnetic quantities involved in the
superconducting response, different models have been used for the
corresponding $E \leftrightarrow J$ graph, the most popular being
\break
\begin{enumerate}
\item The {\em power law} model~\footnotemark[2]\\
$E=\alpha\;{\rm sgn}(J)\left({|J|}/{J_c}\right) ^n$, with $\alpha$ a constant
and $n$ high.
\item The {\em piecewise continuous linear} approximation~\footnotemark[2]\\
$E=0$ for $|J|\leq J_c$, and $E=\beta\;{\rm sgn}(J)(|J|-J_c)$ for $|J|> J_c$,
$\beta$ having a high value.
\item {\em Bean's model}~\footnotemark[3]\\
$J$ constant for $E=0$, and $J={\rm
sgn}(E)J_c$ for $E\neq 0$.
\footnotetext[2]{Is to be noted that these models (1 \& 2) present a small
dependence on the sweep rate, as far as different values of $E$ give way to a
slightly different $J$. The feasibility of the selected model can only be
justified by agreement of the free parameters with the experimental results.}
\footnotetext[3]{This is the simplest model, without sweep rate dependence
because only the sign of $E$ enters the theory (see Fig.~\ref{Figure_1_1}).}
\end{enumerate}
In some treatments, the first or second models are
implemented, in order to transfer a full ${\bf E}({\bf J})$ law to the Maxwell
equations. Further, notice that Bean's model may be obtained from
the other representations through the limiting cases $n\to\infty$ and
$\beta\to\infty$ respectively.
The well known experimental evidence of a practical sweep rate independence for
magnetic moment measurements (unless for high frequency alternating sources or
at elevated temperatures) allows the use of the clearest Bean's model because
the critical state problem is no longer a time-dependent problem, but a
path-dependent one, meaning that the trajectory of the external sources
${\bf H}_{0}$ uniquely determines the magnetic evolution of the
sample. This makes an important difference when one compares to more
standard treatments, as far as Faraday's law is not completely determined from
the path~\cite{P1-Ruiz_PRB_2009}. Strictly speaking, one has
\begin{eqnarray}\label{Eq.2.13}
\Delta{\bf B} = - \nabla\times [{\bf E}\Delta t] \,\, ,
\end{eqnarray}
with $\Delta t$ (and therefore $|{\bf E}|$) depending on the external sources.
i.e., the absence of an intrinsic time constant gives way to the
arbitrariness in the time scale of the problem.
Furthermore, in the actual 1D applications of Beans's model,
Faraday's law is not strictly solved and \textbf{E} is absent from the theory.
It is just the sign rule (the {\it
vectorial} part of the material law), that is used to integrate Ampere's law.
Notice that such sign rule corresponds to a maximal shielding response against
magnetic vector variations, and thus, determines the selection of $J= \pm J_c$.
Regarding the direction of \textbf{E}, in ``1D'' problems one has ${\bf
J}\parallel {\bf E}$ and both orthogonal to $\bf B$, such that the physical
threshold related to a maximum value of the force balancing the magnetostatic
term ${\bf J}\times{\bf B}$ gives place to the material law
\begin{eqnarray}\label{Eq.2.14}
J_{\perp}={\rm sgn}(E_{\perp})J_{c\perp} \qquad{\rm for}\qquad E_{\perp}\neq
0\,\, .
\end{eqnarray}
Here, $E_{\perp}$ stands for the component of ${\bf E}$ along the direction
${\bf B}\times ({\bf J}\times{\bf B})$, and the material law falls in a ``1D''
scalar condition which describes the physical mechanism of vortex depinning.
At this point, the constitutive relation for the critical state describes the
underlying physics for the coarse-grained fields in homogeneous type-II
superconductors. However, it is well known that the coarse-grained behavior
approach straightforwardly depends on the manufacturing process of the
superconducting sample as far as inclusion of impurities, magnetic defects,
or deformation of their cristal structure imposes the local coupling of $J_{c}$
with the intrinsic variation of the magnetic field B. Thus, for practical
purposes we emphasize that the theoretical framework developed in this book is
fully general, with caution of suggest to the experimentalist the need of an
apriori measurement of the dependence $\textbf{J}_{c}[\textbf{r},\textbf{B},T]$
at least in those cases where the condition $\textbf{J}\perp \textbf{B}$ can be
asserted. Henceforth, the
implementation for a particular superconductor can be carried out.
\vspace*{0.5cm}
\hspace*{-0.3cm}
\textbf{\textit{2.2.2 ~Towards The 3D Critical States}}
\markboth{\hspace*{0.1cm} \textsl{2.2.2 ~Towards The 3D Critical
States}}{\hspace*{0.1cm} \textsl{2.2.2 ~Towards The 3D Critical States}}
\addcontentsline{toc}{chapter}{\hspace*{0.1cm} \textsl{2.2.2 ~Towards The 3D
Critical States}}\label{ch-2.2.2}
\vspace*{0.5cm}
In the case of superconductors with anisotropy of the critical current, the
description of their magnetic behavior requires the development of approaches
more sophisticated than 1D-Bean's model. The main issue is that, in general, the
parallelism of ${\bf E}$ and ${\bf J}$ and their perpendicularity to ${\bf B}$
are no longer warranted. Then, the {\em sign rule} of Eq.~(\ref{Eq.2.14})
does not suffice for determining the solution of Eq.~(\ref{Eq.2.6}), and the
optimal control condition with $\textbf{J}\in\Delta_{\textbf{r}}$
({\em a vectorial rule}) must be invoked.
The simplest assumption that translates the critical state problem to 3D
situations was already issued by Bean
in Ref.~\cite{P1-Bean_1970}. It has been called the \textbf{\textit{isotropic
critical state model}} (ICSM) and generalizes 1D Bean's law to
\begin{equation}\label{Eq.2.15}
{\bf J} =J_{c}\,\hat {\bf E}\quad{\rm if} \quad E \neq 0 \, ,
\end{equation}
i.e., the region $\Delta_{\textbf{r}}$ becomes a sphere. Noticeably, in spite
of lacking a solid physical basis, thanks to its mathematical simplicity, this
qualitative model has been widely used by several authors for reproducing a
number of experiments with rotating and crossed magnetic
fields~\cite{P1-Badia_PRL_2001,P1-Badia_PRB_2002,P1-ICSM_app}. In any case,
one could argue that statistical averaging over a system of entangled flux lines
within a random pinning structure might be responsible for the isotropization of
$\Delta_{\textbf{r}}$.
On the other hand, the general statement of the critical state in
terms of well accepted physical basis was firstly introduced by John R.
Clem~\cite{P1-Clem_DCSM}, and it is currently known as the
\textbf{\textit{double
critical
state model}} (DCSM). In particular, this theory
assumes two different critical parameters, $J_{c\parallel}$ and $J_{c\perp}$
acting as the thresholds for the components of ${\bf J}$ parallel and
perpendicular to ${\bf B}$ respectively. Notice
that, $J_{c\perp}$ relates to the flux depinning threshold induced by the
Lorentz force on flux tubes ($\textbf{J}\times\textbf{B}$),
while the additional $J_{c\parallel}$ is imposed by a maximum gradient in the
angle between adjacent vortices
($\textbf{B}\cdot\nabla\gamma=J_{\parallel}$) before mutual cutting
and recombination occurs [see Fig.~\ref{Figure_2_1}~(a)]. In brief, the
DCSM may be expressed by the statement
\begin{eqnarray}\label{Eq.2.16}
\left\{
\begin{array}{ll}
{\bf J}_{\parallel} & =J_{c\parallel}\;\hat {\bf u}
\\
{\bf J}_{\perp} & =J_{c\perp}\,\hat {\bf v}
\end{array}
\right. \, ,
\end{eqnarray}
being $\textbf{\^{u}}$ the unit vector for the direction
of \textbf{B}, and $\textbf{\^{v}}$ a unit vector in the perpendicular plane
to \textbf{B}.
Within the DCSM, the region $\Delta_{\textbf{r}}$ is a cylinder with its axis
parallel to $\bf B$, and a rectangular longitudinal section in the plane defined
by the unit vectors $\hat{\bf B},\hat{\bf J}_{\perp}$ [see
Fig.~ \ref{Figure_2_1}~(b)]. The edges of the region $\Delta_{\textbf{r}}$
introduce a criterion for classifying the CS configurations into:
\begin{enumerate}
\item T zones
or flux transport zones ($J_{\perp}=J_{c\perp}$;
$J_{\parallel}<J_{c\parallel}$) where the
flux depinning threshold has been reached (${\bf J}$ belongs to the
horizontal sides of the rectangle),
\item C-zones or flux cutting zones
($J_{\parallel}= J_{c \parallel}~;~J_{\perp}<J_{c \perp}$) where the
cutting threshold has been reached (${\bf J}$ belongs to the vertical sides
of the rectangle),
\item CT zones ($J_{\parallel} = J_{c \parallel}$ and
$J_{\perp} = J_{c \perp}$) where both $J_{\parallel}$
and $J_{\perp}$ have reached their critical values (corners of the
rectangle), and
\item O zones depicted the regions without energy dissipation (the current
density vector belongs to the interior of the rectangle).
\end{enumerate}
\begin{figure}[t]
\centering
\includegraphics[width=1.0\textwidth]{Figure_2_1.pdf}
\caption{\label{Figure_2_1} {\it Pane}~(a), Top: Schematic representation of the
local relative
orientations of {\bf B} and {\bf J}. Also sketched is the direction of
the magnetic field at some neighboring point, at an angle $\gamma$.
The vectors ${\bf B}$, ${\bf B}'$ and ${\bf J}$ do not
necessarily lie at the same plane. The current is decomposed into its parallel
and perpendicular components, i.e.: ${\bf J}={\bf
J}_{\parallel}+{\bf J}_{\perp}$.
Bottom: the {\em perfect conducting} region within the plane perpendicular to
$\textbf{B}$. An induced electric field is shown. Initially (${\bf
J}_{0}$), the high dissipation region is touched, but almost instantaneously
{\bf J} shifts along the boundary, reaching a point where the condition ${\bf
E}\perp\partial\Delta_{\textbf{r}}$ is fulfilled. Anisotropy within the plane is
allowed. {\it Pane}~(b): Geometric interpretation of the DCSM. {\bf J} is
constrained to the
boundary of a rectangular region. T, C and CT states are related to
the horizontal and vertical sides, and to the corners. Coupling between
the components $J_{c\parallel}$ and $J_{c\perp}$ is envisaged by the EDCSM
(dotted red ellipse). {\it Pane}~(c): Our generalization of the material law for
critical
state problems or SDCST. Several regions are shown from the degree of the
superelliptical functions ($n=1,2,3,4,6,10,20,40,\infty$) and
$\chi=J_{c\parallel}/J_{c\perp}=1$.}
\end{figure}
Notice that $J_{c\parallel}$ and $J_{c\perp}$ are determined from different
physical phenomena, and their values may be very different (in general
$J_{c\parallel}>J_{c\perp}$ or even $J_{c\parallel}\gg J_{c\perp}$).
Nevertheless, the coupling of parallel and perpendicular effects has been longer
recognized by the experiments~\cite{Clem_2011_SUST,P1-Boyer_80} and, for
instance,
may be included in the
theory by the condition $J_{c\parallel}=K B J_{c\perp}$ with $K$ a
material dependent constant.
Recalling that the mesoscopic parameters $\textbf{J}_{c}$
are related to averages over the flux line lattice, interacting activation
barriers for the mechanisms of flux depinning and cutting are expected and this
may give place to deformations of the boundary $\partial\Delta_{\textbf{r}}$
[see Fig.~ \ref{Figure_2_1}(b)]. Thus, validated in those cases where a good
agreement with the experiments is achieved, the theoretical scenario can be
enlarged by a number of alternative approaches that focus on different aspects
of the vast number of experimental activities in this field, e.g. one
can identify the so-called:
\begin{enumerate}
\item \textit{Isotropic critical state models
(ICSM)}~\cite{P1-Ruiz_PRB_2009,P1-Ruiz_SUST_2010,P1-ICSM_app}\\
\hspace*{3.4cm} $J^{2}=J_{\parallel}^2+J_{\perp}^2 \leq J_{c}^2$
\item \textit{Elliptical double critical state
models (EDCSM)}~\cite{P1-Ruiz_PRB_2009,P1-EDCSM_app}\\
\hspace*{3.26cm}
$J_{\parallel}^2/J_{c\parallel}^2+J_{\perp}^2/J_{c\perp}^2\leq1$
\item \textit{T critical state model
(TCSM)}~\cite{P1-Ruiz_PRB_2009,P1-Ruiz_SUST_2010,P1-Ruiz_SUST_2011,
P1-Brandt_2007,
P1-Ruiz_PRB_2011}\\
\hspace*{2.87cm} $J_{\parallel}$ unbounded $\forall$ $J_{\perp}\leq J_{c\perp}$
.
\end{enumerate}
Remarkably, the whole set of models have been recently unified by us in
Ref.~\cite{P1-Ruiz_SUST_2010} within a continuous two-parameter theory that
poses
the critical state problem in terms of geometrical concepts within the
$J_{\parallel}-J_{\perp}$ plane (see Fig.~ \ref{Figure_2_1}~(c)). To be
specific, in this framework, we have shown that by the application of our
variational statement~\cite{P1-Ruiz_PRB_2009}, one is able to specify almost any
critical state law by means of an integer index $n$, that accounts for the
smoothness of the $J_{\parallel}(J_{\perp})$ relation, and a certain {\em
bandwidth} characterizing the magnetic anisotropy ratio $\chi\equiv
J_{c\parallel}/J_{c\perp}$. This and the variational formalism introduced
above constitutes the so-called \textbf{\textit{Smooth Double
Critical State Theory}} (SDCST), which allows to elucidate the relation
between diverse physical processes and the actual material law.
Mathematically, the material law introduced in our general theory for the
critical state problem or SDCST is based upon the idea that either material or
extrinsic anisotropy can be easily incorporated by prescribing a region
$\Delta_{\textbf{r}}$ where the physically admissible states of $\textbf{J}$ are
hosted as limiting cases of a smooth expression defined
by the two-parameter family of superelliptic functions,
\begin{equation}\label{Eq.2.17}
\left(\frac{J_{\parallel}}{J_{c\parallel}}\right)^{2n}+\left(\frac{J_{\perp}}{J_
{c\perp}}\right)^{2n}\leq 1.
\end{equation}
We call the readers' attention to the fact that an index $n=1$ and a bandwidth
defined by $\chi\equiv J_{c \parallel}/J_{c \perp}=1$ correspond to the standard
ICSM~\cite{P1-ICSM_app}.
On the other hand, when one assumes enlarged bandwidth (i.e.: $\chi>1$), the
region $\Delta_{\textbf{r}}$ of the SDCST becomes the standard
EDCSM introduced by Romero-Salazar and
P\'erez-Rodr\'iguez~\cite{P1-EDCSM_app}.
When the bandwidth $\chi$ is extremely large, i.e., $J_{c\parallel}\gg J_{c
\perp}$, one recovers the so-called $T-$zones treated by Brandt and
Mikitik~\cite{P1-Brandt_2007}. Rectangular regions strictly corresponding to the
DCSM~\cite{P1-Clem_DCSM} are
obtained for the limit $n\rightarrow\infty$ and arbitrary $\chi$. Finally,
allowing $n$ to take values over the positive integers, a wide scenario
describing anisotropy effects is envisioned [Fig.~\ref{Figure_2_1}(c)]. Such
regions will be named after superelliptical and their properties can be
understood in terms of the rounding (or smoothing) of the corners for the DCSM.
\chapter{\label{ch-3} \sc \textbf{Computational Method}}
In chapter~2.1 we have mentioned that the minimization functionals
[Eq.~(\ref{Eq.2.7}) or Eq.~(\ref{Eq.2.11})] may be transformed so as to get a
practical vector potential formulation. In turn, the resulting formulation can
be expressed in terms of the so-called magnetic inductance matrices which
allows a clearest identification of the set of elements playing some role in
the minimization procedure. In this chapter, we shall discuss how to implement
the above
statements for general critical states in the
framework of the computational methods for large scale nonlinear optimization
problems.
Being more specific, in Eq.~(\ref{Eq.2.11}) the integrand $\frac{1}{2}
(\Delta{\bf B}) ^2$ can be rewritten as $\frac {1}{2} (\Delta{\bf B})\cdot
(\nabla \times \Delta{\bf A})$,
and manipulated to get $\frac {1}{2} (\Delta{\bf A})\cdot (\nabla \times
\Delta{\bf B})$ plus a divergence term, fixed by the external sources at a
distant surface. Now, the integral is restricted to the superconducting sample
volume $\Omega$, because $\nabla \times \Delta{\bf B} = \mu _0 \Delta{\bf J}$ is
only unknown within the superconductor. In addition, assuming that local
sources such as an injected transport current may be introduced as an external
constraint, and with the boundary condition that $\textbf{A}$ goes
to zero sufficiently fast as they approach infinity, the vector
potential can be expressed as:\footnotemark[1]
\footnotetext[1]{Recall that, \textbf{A} is determined by the Maxwell's
equations solution in the Lorenz gauge condition, i.e., the vector potential
must satisfy the condition $\partial_{\mu}A^{\mu}=0$ for any transformation
gauge $A^{\mu}\rightarrow A^{\mu}+\partial^{\mu}\psi$ with $\psi$ a scalar
function. Thus, as no local-sources are present into the minimization
principle of Eq.~\ref{Eq.2.11}, one is enabled to apriori assume $\psi=0$, and
thence simplify the vector potential in terms of the Coulomb gauge.}
\begin{equation}\label{Eq.3.1}
\Delta{\bf A} = \Delta{\bf A}_{0}
+ \frac {\mu _0 }{4 \pi} \int _\Omega \frac {\Delta{\bf J}}{|{\bf r} - {\bf
r}'|}
d^3 {\bf r}' \, .
\end{equation}
This transforms ${\cal F}$ into a double integral over the body of the sample,
i.e.:
\begin{eqnarray}\label{Eq.3.2}
{\cal F}[\textbf{A}(\cdot),J\in\Delta_{\textbf{r}}]= &&\frac {8\pi}{\mu_0}\int
_\Omega \Delta{\bf A}_{0}\cdot {\bf J}_{l+1}({\bf r})d^3 {\bf r}
\nonumber\\&&
+\int\!\int _{\Omega \times \Omega}
\frac {{\bf J}_{l+1}({\bf r}')
\cdot [{\bf J}_{l+1}({\bf r}) - 2 {\bf J}_{l}({\bf r})]}{|{\bf r} - {\bf
r}'|}d^3 {\bf r}d^3 {\bf r}'\;
\end{eqnarray}
As a consequence, only the unknown current components within
the superconductor $(\textbf{J}_{l+1})$
appear in the computation so reducing the number of unknown variables. At this
point let me emphasize that Eq.~(\ref{Eq.3.2}) can be applied for any shape of
the superconducting volume $\Omega$ as well as for any physical constraint
(material law $\Delta_{\textbf{r}}$) for the local current density
$\textbf{J}_{l+1}$, and further for any condition defined by the external
sources ($\textbf{A}_{0}$). Above this, minimization must ensure
the charge conservation
condition by searching the minimum for the allowed set of current densities
fulfilling $\nabla\cdot\textbf{J}=0$.
On the other hand, also it may be noticed that the double
integral in Eq.~(\ref{Eq.3.2}) can be (\textit{eventually}) identified as the
Neumann formula once it has been transformed into filamentary closed circuits. A
noteworthy fact is that regarding the superconducting volume, the coefficients
of the intrinsic inductance matrices are straightforwardly independent of time
and consequently, they appear in the root problem before going to minimize the
functional. Indeed, the proper description of the inductance coefficients
directly depends on the geometry of the superconductor and the boundary
conditions defined by the dynamics of the external electromagnetic sources,
where any symmetry of the problem allows further simplifications and
correspondingly faster numerical convergence. To be specific, upon
discretization in current elements ($I_{i}=J_{i}s_{i}$), the minimization
functional for critical state problems bears
the algebraic structure
\begin{eqnarray}\label{Eq.3.3}
{\cal
F}[I_{l+1}]=\frac{1}{2}\sum_{i,j}I_{i,l+1}M_{ij}I_{j,l+1}-\sum_{i,j}I_{i,l}M_{
ij } I_ {j,l+1}+\sum_{i}I_{i,l+1}\Delta A_{0}(M_{0}) \, ,
\end{eqnarray}
with $\{I_{i,l+1}\}$ the set of unknown currents at specific
circuits for the problem of interest, $M_{ij}$ their intrinsic {\em inductance
coupling coefficients}, and $M_{0}$ the inductance matrices associated to the
external sources $A_{0}$.
Corresponding to the critical state rule ${\bf J}\in\Delta_{\textbf{r}}$, in
order to minimize Eq.~(\ref{Eq.3.3}) each value $J_i$ must
be constrained. Thus, as it was described in chapter~\ref{ch-2}.2.2, we have
found that a number of constraints related to physically meaningful critical
state models may be expressed in the algebraic form
\begin{eqnarray}\label{Eq.3.4}
{\rm
F}_{\alpha}\left(\sum_{i}I_{i}C_{ij}^{\alpha}I_{j}\right)\leq f_{0\alpha} \,
~\forall ~ j
\end{eqnarray}
with $f_{0}$ some constant representing the physical threshold, and
${\rm F}_{\alpha}(\cdot)$ an algebraic function based upon a coupling matrix
$C_{ij}^{\alpha}$ whose elements depend on the physical model.
For example, in the simpler cases (isotropic models), the constraints
correspond to assume the matricial elements $C_{ij}=\delta_{ij}$, and the
physical threshold $f_{0}={J}_{c}^{2}$.
For simplicity, most technical procedures related to the introduction of
intricate models and either depict the minimization functional in terms of the
inductance coefficients (including those for external sources) will be left as
matter of study of the following chapters (Part~\ref{Part_2}). In return, below
we present a thorough analysis of the computational tools handled for critical
state problems at large scale.
With the purpose of obtaining a minimal understanding about how a critical state
problem can be tackled from the numerical point of view, the computational
method is sketched in the flow charts of
figures~\ref{Figure_3_1}~\&~\ref{Figure_3_2}.
\begin{figure}[t]
\centering
\includegraphics[height=13cm,width=13cm]{Figure_3_1.pdf}
\caption{\label{Figure_3_1} Flow chart describing the preparation and
management of the input elements for the objective function.}
\end{figure}
The first step is designing a grid which will allows to describe the
superconducting volume $\Omega$ as a set of elements
$\delta\Omega_{i}$, each of them characterized by a well defined current
density flowing along the coordinates $\textbf{r}_{i}$. Then, the matrices for
the intrinsic inductance coefficients between the elements
$\textbf{J}(\textbf{r}_{i})$ and $\textbf{J}(\textbf{r}_{j})$ for all the set
of possible couples $(\textbf{r}_{i},\textbf{r}_{j})\in\Omega$ must be
calculated and stored on disk. As the mesh of points
$(\textbf{r}_{i},\textbf{r}_{j})$ can be considerably large, we suggest take
advantage on the matricial formalism provided by Matlab$^{\textregistered}$
and
their own language for storage data. Once the spatial elements playing some role
into the functional have been properly defined, the temporal \textit{sector}
must be introduced by means enough small path steps of the external
electromagnetic sources, i.e., the experimental conditions must be connected by
the finite difference expressions such a
$\Delta\textbf{B}_{0}=\textbf{B}_{l+1}-\textbf{B}_{l}$, where the
associated distribution of currents $\textbf{I}_{l+1}$ plays the role of
unknown. To be specific, in those cases where the superconductor is subjected
to an external magnetic field $\textbf{B}_{0}$, additional inductance matrices
($M_{0}$)
must be introduced according to the definition
\begin{equation}\label{Eq.3.5}
\textbf{A}_{0}
(\textbf{B}_{0},\textbf{r}_{j})=\textbf{B}_{0}\times\textbf{r}_{ j } \, .
\end{equation}
On the other side, the vector potential $\textbf{A}_{0}$ not only allows to
define the contribution at the local potential $\textbf{A}$ produced by an
external magnetic field ($\textbf{A}_{{\rm B}}$), rather it also allows
consider the coupling with another materials such as ferromagnets.
Before going within the minimization procedure, it has to be noticed that those
cases
considering a transport current along the superconducting sample must
be understood as a problem where the minimization variables are required to
satisfy a set of auxiliary constraints [see Eq.~(\ref{Eq.2.12})] under the
global critical state condition $I_{tr}\leq I_{c}$. Also, as the
values for the elements $\textbf{I}_{l}(\textbf{r})$ are
assumed to be known in advance, the linear elements into the argument of the
functional (objective function) can be calculated before minimizing.
At this point, it is probably worthwhile to argue on what we mean by the
computational method for minimization of an objective function. Firstly, this
notion is clearly computer dependent, as the size of large scale problems can
require a substantial amount of memory and store. Moreover, what is large in a
personal computer can be significantly different from what is large on a super
computer. The first machine just to have a smaller memory and storage than the
second one, and therefore has more difficulty handling problems involving a
large amount of data. Secondly, the size of the objective function strongly
depends on the structure and the mathematical formulation of the problem
and exploiting it is often crucial if one wants to obtain an answer
efficiently. The complexity of this structure is often a central key in
assessing the size of a problem\footnotemark[2]. For example, for linear
objective functions
(not our case) it is possible to solve pretty large size problems (say four
million variables). However, the objective function for problems in applied
superconductivity is in general highly nonlinear and, for instance, the
quadratic terms suggest to reduce the number of variables in a root square
factor (say two thousand variables). One advisable possibility for reducing the
number of elements in the objective function is subdividing the problem into
loosely connected subsystems, i.e., all the internal operations which do not
depend of the minimization variables must
be preallocated to a well structured data (Figure.~\ref{Figure_3_1}).
Lastly, an efficient algorithm for nonlinear optimization problems
must be either invoked or built. Fortunately, nowadays there is a significant
amount of available software with standard optimization tools which allow a
faster foray in this matter~\cite{P1-Matlab,P1-Lancelot}.
\footnotetext[2]{Customarily memory access violations (segmentation faults)
appear on nonlinear large systems without a well designed structure.}
We must call reader's attention on the fact that efficient algorithms for
small-scale problems (in the sense that, assuming infinite precision,
quasi-Newton methods for unconstrained optimization are invariant under linear
transformations) do not necessarily translate into efficient algorithms
for large scale problems. Perhaps the main reason is that, in order to be able
to handle large problems with a high accuracy, the structure of the objective
function and the minimization algorithms have both of them to be enough simple
and tractable to avoid a wasting of time in the scaling of variables for the
inner iteration subproblem (the minimization itself) and the finding of an
optimum value for each one of the variables with respect to the remaining
variables sought. In this context, one of the most powerful algorithms for
large and nonlinear constrained optimization problems, known as
LANCELOT, has been developed by the professors Andrew Conn (IBM corporation,
USA), Nick Gould (Rutherford Appleton Laboratory, UK), Philippe Toint
(Facult\'es Universitaries Notre-dame de la Paix, Belgium), and Dominique Orban
(Ecole Polytechnique de Montreal, Canada)~\cite{P1-Lancelot}. The wide number of
optimization techniques provided by this package and their flexibility in
handle and storage of large amounts of variables, make this program a
clever choice for tackle highly complicated systems as those described by
Eq.~(\ref{Eq.3.2}).
\begin{figure}[t]
\centering
\includegraphics[height=13cm,width=13cm]{Figure_3_2.pdf}
\caption{\label{Figure_3_2} Flow chart describing the main structure of the
computational method implemented along this book.}
\end{figure}
A thorough study of the minimization techniques and the computational language
allocated in this package is far away of the purpose of this thesis. However,
the structure of a general problem can be understood via the flow chart in
figure~\ref{Figure_3_2}. In brief, the
minimization functional is translated into a suite of FORTRAN
procedures for minimizing an objective function, where the minimization
variables are required to satisfy a set of auxiliary constraints and possibly
internal bounds. Here, the major advantage of LANCELOT is the use of a Standard
Input Format (SIF) as a unified method for communicating numerical data and
FORTRAN subprograms with any optimization algorithm. Thus, when an optimization
problem (minimizing or maximizing a sought of variables) is specified in the SIF
decoder, one is required to write one or more files in ordered sections which
accomplish the role of introduce the set of preconditioners for the objective
function.
Once the set of input data has been structured accordingly to the number of
variables and further on the temporal dependence of the experimental conditions
(see Fig.~\ref{Figure_3_1}), one is enabled to predefine a set of input cards
allowing the knowledgeable user to specify a priori known limits on the possible
values of the objective function, as well as on the specific optimization
variables, accuracy parameters, and scaling factors (see Fig.~\ref{Figure_3_2}).
Then, the minimization functional or so-called objective function is
subdivided in a set of groups, whose purpose is twofold: On the one hand, the
linear and nonlinear (\textit{quadratic}) elements for the minimization
procedure are identified in a fore. Likewise, the specification of analytical
first derivatives is optional, but recommended whenever possible. The SIF
decoder allows also include the Hessian matrix of the objective function, if the
second-order partial derivatives of the whole set of minimization variables are
known,\footnotemark[3] otherwise the derivatives of the nonlinear element
functions can be approximated by some finite difference method.
Actually, the full Hessian matrix can be difficult to compute in practice; in
such situations, quasi-Newton algorithms\footnotemark[4] can be
straightforwardly called by LANCELOT where at least ten
different minimization algorithms have been already implemented and
coded according to the standard input format provided by the SIF
decoder~\cite{P1-Lancelot}. Notwithstanding, the solution obtained by LANCELOT
may be compromised if finite difference approximations are used, it has been our
experience that, once
understood, the programming language of SIF is in fact quite efficient for
problem specifications, in such manner that for objective functions
correctly written, and constraint functions well defined, any method can
efficiently reach to the solution sought. In this sense, additional groups may
be announced to make up the objective function by including an ``starting
point'' for the envisaged solution (if it is more or less known, or by defect
it is equals to zero) or, for introducing additional constraints (external
functions conditioning the system) as it is the case when the superconductor
implies a flow of transport current.
\footnotetext[3]{Given the real-valued function $f(x_{1},x_{2},...,x_{n}$, if
all second derivatives of $f$ exists, then the Hessian matrix of $f$ is the
matrix $H(f)_{ij}(x)=\partial_{i}\partial_{j} f(x)$. Hessian matrices are used
in large-scale optimization problems within Newton-type methods because they are
the coefficient of the quadratic term of the local Taylor expansion
$f(x+\Delta x)\approx f(x)+\breve{J}(x)\Delta x + \frac{1}{2}\Delta x^{~
\textbf{T}} H(x)\Delta x$, where $\breve{J}$ is the Jacobian matrix, which is a
vector (the gradient for scalar-valued functions)}
\footnotetext[4]{Also known as variable metric methods, are algorithms for
finding local maxima and minima of a function, with the aim of find out the
stationary point of their local Taylor expansion where the gradient is 0. Thus,
these methods are so called quasi-Newton methods, because the Hessian matrix
does not need to be a priori computed, as the Hessian is straightforwardly
updated by analyzing successive gradient vectors instead.}
It may happen that a specific problem uses variables or general constraints
whose numerical values are widely different in magnitude, causing
significant difficulties in the numerical convergence. However, LANCELOT also
gives the chance of incorporate a list of scaling factors which are applied to
the general constraints and variables separately before the optimization
commences, allowing a clearest handling of the group elements in highly
nonlinear problems as those herein considered. Thus, assuring a good
convergence, whether single or double precision,\footnotemark[5] is
implying in turn to modify the experimental time stepping and either, the
accuracy
parameters such as, the number of iterations allowed, the constraint and
gradient accuracy, the penalty parameter and the trust region for the
optimizing.
\footnotetext[5]{For large scale programming in FORTRAN based languages, one
must care about exceeding the largest positive (or negative)
floating-point number defined by the FORTRAN distribution. By default, we have
defined an architecture of double precision in 64 bits conforming to the IEEE
standard 754 for the latest versions of Intel FORTRAN Compiler (see
Ref.~\cite{P1-IEEE}).}
Finally, in applied superconductivity, a set of complementary programs have to
be developed in an effort to provide a comprehensive understanding of the
temporal evolution of the electromagnetic quantities. For example, if the
set of minimization variables corresponds to the local profiles of current
density $\textbf{J}_{i}(\textbf{r}_{i})$, additional codes must to be used for
calculating $\textbf{A}_{i}(\textbf{r})$, $\textbf{B}_{i}(\textbf{r})$,
and $\textbf{E}_{i}(\textbf{r})$ in the whole $\Re^{3}$-space. Thus,
although integrated quantities such as the magnetic moment
$\textbf{M}(\textbf{J}_{i},\textbf{r}_{i})$ may be revealing a smooth trend
despite the use of a poor numerical accuracy, it is of utter importance testing
the numerical convergence by calculating the local profiles for the
electromagnetic quantities concerning to derived quantities. In this sense, the
following part of this book is devoted to the reliable solution of some
interesting problems in applied superconductivity, where the critical state
statement falls into a large scale optimization problem.
\chapter*{}
\vspace*{-3cm}
\section*{\Huge{Conclusions I}}
\markboth{\sc \textbf{Conclusions I}}{\sc \textbf{Conclusions I}}
\addcontentsline{toc}{chapter}{\sc \textbf{Conclusions I}}
\vspace*{2cm}
In summary, in this part we have shown that the critical state theory for the
magnetic response of type-II superconductors may be built in a quite general
framework and in turn it may be solved by several means. As our interest is to
deal with highly nonlinear problems at large-scale, we
have emphasized in the performance of variational methods and computational
techniques for solving problems on personal computers.
We remark that the basic concepts underlying our generalization of the critical
state theory can be identified as follows:
\begin{enumerate}
\item The critical state theory bears a Magneto Quasi Steady (MQS) approximation
for the Maxwell equations in which $\ddot{\textbf{B}}$, $\dot{\bf E}$ and
$\dot{\rho}$
are second order quantities and consequently, the displacement current
densities $\dot{\textbf{D}}$ are much smaller than $\textbf{J}$ in the bulk and
vanish
in a first order treatment. This means that the {\em magnetic flux dynamics}
can be entirely described by the finite-difference expression of Faraday's law
\begin{eqnarray}\label{Eq.3.6}
\Delta\textbf{B}=-\nabla\times(\textbf{E}\delta t) \, ,
\end{eqnarray}
where the physically admissible states must accomplish the MQS Ampere's law,
i.e., $\nabla\times\textbf{B}=\mu_{0}\textbf{J}$. Here, the inductive
part of ${\bf E}$ may be introduced through Faraday's law, whereas the role of
electrostatic quantities is irrelevant for the magnetic sector. In other
words, {\bf E} may be modified by a gradient function (${\bf E}\to{\bf
E}+\nabla\phi$) with no effect on the magnetic response.
\item In type-II superconductors, the law that characterizes the
\textit{conducting behavior} of the material may be written in terms of
thresholds values for the current density constrained to a geometrical region
$(\textbf{J}\in\Delta_{\textbf{r}})$ which suffices to determine the relation
between the directions of \textbf{E} and \textbf{J}. Thus, \textbf{E} is no
longer an unknown variable but rather plays the role of a parameter to be
adjusted in a direct algebraic minimization, i.e.,
\begin{eqnarray}\label{Eq.3.7}
{\rm Min}\{L\}|_{\textbf{J}\in\Delta_{\textbf{\textbf{r}}}}\equiv{\rm
Max}\{\textbf{J}\cdot\textbf{p}\}|_{\textbf{J}\in\Delta_{\textbf{\textbf{r}}}}
\equiv{\rm
Max}\{\textbf{E}\cdot\textbf{J}\}|_{\textbf{J}\in\Delta_{\textbf{\textbf{r}}}}
\, .
\end{eqnarray}
In physical terms, the material ``\textit{reacts}'' with a maximal shielding
rule when electric fields are induced, and a perfect conducting behavior
characterizes the magnetostatic equilibrium when external variations cease. In
fact, is to be noticed that the above representation can be
understood as the macroscopic counterpart of the underlying vortex physics.
Thus, recalling that, in type II superconductors an incomplete isotropy for the
limitations of the current density relative to the orientation of the local
magnetic field arises from the different physical conditions of current flow
either along or across the Abrikosov vortices, one may talk about
magnetically induced anisotropy where the physical barriers of flux depinning
and cutting are customarily depicted by the
condition $\textbf{J}\leq\textbf{J}_{c}\in\Delta_{\textbf{r}}$.
The evolution
from one magnetostatic configuration to another
occurs through the local violation of this condition, i.e.: ${\bf
J}\notin\Delta_{\textbf{r}}$ ($\textbf{J}>\textbf{J}_{c}$). However, owing
to the high dissipation, an almost instantaneous response may be assumed,
represented by a {\em maximum shielding} rule in the form ${\rm Max}\{{\bf
J}\cdot\hat{\bf E}\}\left.\right|_{{\bf J}\in\Delta_{\textbf{r}}}$.
\item With the aim of offering a meaningful reduction of the number of
variables,
we have shown that the problem can be simplified by solving a minimization
functional with a underlying structure based upon inductance matrices [see
Eq.~(\ref{Eq.3.3})]. In
particular, the mutual inductance representation with ${\bf J}({\bf
r})$ as the unknown, offers two important advantages:
(i) intricate boundary
conditions and infinite domains are avoided, and
(ii) the transparency of the
numerical statement and its performance (stability) are outlined.
Then, the quantities of interest (flux penetration profiles and magnetic moment)
are obtained by integration.
\item Most popular models for critical state problems have been
generalized in our so-called \textit{smooth double critical state theory}
(SDCST) for
anisotropic material laws~\cite{P1-Ruiz_SUST_2010}. This theory relies on our
variational framework for general critical state
problems~\cite{P1-Ruiz_PRB_2009} that allows us to
incorporate the above-mentioned physical structure in the form of mathematical
restrictions for the circulating current density. Two fundamental
material-dependent quantities play key roles in this theory
$(J_{c\parallel},J_{c\perp})$ related to the flux cutting and flux depinning
thresholds. Notoriously, the boundary condition for the material law
$\textbf{J}\in\Delta_{\textbf{r}}$ and the mutual interaction between the
critical thresholds have been described in a quite general picture, based upon
the relation between the coupling parameters
$\chi\equiv J_{c\parallel}/J_{c\perp}$
and the smoothing index $n$ of the \textit{superelliptical} condition
$(J_{\parallel}/J_{c\parallel})^{2n}+(J_{\perp}/J_{c\perp})^{2n}\leq1$.
Hence, our SDCST cover a wide range of laws:
(i) the isotropic model ($\chi^{2}=1$, $n=1\Rightarrow \Delta_{\textbf{r}}$ is a
circle),
(ii) the elliptical model ($\chi^{2}>1$, $n=1\Rightarrow \Delta_{\textbf{r}}$ is
an ellipse),
(iii) the rectangular model ($\chi^{2}\geq 1$, $n\to\infty\Rightarrow
\Delta_{\textbf{r}}$ is a rectangle),
(iv) the infinite band model ($\chi^{2}\rightarrow\infty$), and
(v) else others with smooth magnetic anisotropy ($\chi^{2}\geq 1$, $n\in{\rm N}
\geqslant 1 \Rightarrow \Delta_{\textbf{r}}$ is a rectangle with smoothed
corners).
\end{enumerate}
Finally, let me emphasize that the scope of our
theory is rather beyond the actual examples treated in the following part of
this thesis. On the one side, we have shown that the critical state concept
allows arbitrariness in the presence of electrostatic charge and potential, and
one could simply upgrade the models by the rule ${\bf E}\to{\bf E}+\nabla\phi$
if necessary. For instance, a scalar function $\phi$ may be introduced if the
direction of ${\bf E}$ has to be modified respect to the maximum shielding rule
in the MQS limit. On the other side, the extension of the theory to arbitrary
sample geometries is intrinsically allowed by the mutual inductance
representation. Thus, this first part has laid necessary groundwork for
attacking general critical state problems in 3D geometry.
\renewcommand\bibname{References I}
|
{
"timestamp": "2012-03-12T01:02:34",
"yymm": "1203",
"arxiv_id": "1203.2159",
"language": "en",
"url": "https://arxiv.org/abs/1203.2159"
}
|
\section{Introduction}
In his celebrated theorem
\citet{Nas:PNASUSA1950, Nas:AM1951} used fixed point theorems to prove that any finite game admits an equilibrium in mixed strategies. The result fails in general if the strategy sets are not finite. Several authors have provided conditions under which even infinite games admit an equilibrium. Among them \citet{Deb:PNASUSA1952}, who assumed convexity and compactness of the strategy sets and continuity and quasi-concavity of the payoffs, and
\citet{Gli:PAMS1952}, who assumed compactness of the strategy sets and continuity of the payoff functions. The same year \citet{Fan:PNASUSA1952} extended Kakutani's fixed point theorem, this way permitting a generalization of Nash's existence theorem like the one in \citet{Gli:PAMS1952}.
One stream of literature considered existence theorems under various conditions that allow discontinuous payoff functions: see, e.g.
\citet{DasMas:RES1986,
Sim:RES1987,
SimZam:E1990,
Ren:E1999,
Car:IJGT2005,
Car:GEB2010,
BarSoz:Rochestermimeo2010,
BarGovWil:mimeo2012,
BicLar:mimeo2012},
papers in
\citet{Car:ET2011a}, and references therein.
\citet{Wal:AM1945} considered the case where the strategy set of either one or both players is countable and showed that the mixed extension of a game has a value if one of the strategy sets is finite, but in general it doesn't if they are both countable.
One way to overcome the lack of equilibria in some games is to enlarge the set of mixed strategy by including also finitely additive probability measures.
For the probabilistic and decision-theoretical foundations of the use of finitely additive probability measures, we refer the reader to
\citet{deF:EINAUDI1970, DeF:Wiley1972, deF:Springer2008}, and
\citet{Sav:Dover1972}.
\citet{DubSav:Dover1976} used finitely additive measures extensively in their approach to gambling.
The main issue along this road is that in general a mixed extension is not well defined. Given two measures $\mu_{1}, \mu_{2}$ on the power sets of $S_{1}$ and $S_{2}$, respectively, a product measure $\mu_{1}\otimes\mu_{2}$ is uniquely defined only on the algebra generated by the cylinders and can be extended in a non-unique way to the power set of $S_{1} \times S_{2}$. As a consequence, Fubini's theorem cannot be applied to this situation and in general the order of integration of a double integral matters.
To obviate this drawback several solutions were proposed in the framework of zero-sum two-person games. Both
\citet{Yan:TPA1970} and
\citet{Kin:JOTA1983}
defined the expected value of the payoff to be an arbitrary fixed value, whenever Fubini's theorem cannot be used, and proved this way the existence of a value for the game.
\citet{HeaSud:AMS1972} instead proved the existence of a value by selecting the product measure that corresponds to a fixed order of integration.
\citet{SheSei:BASE1996} used an approach, whose generalization we follow in our paper, that selects as product measure a convex combination of the measures obtained by interchanging the order of integration. In order to obtain the existence of a value they need the condition that, in their words, there exist some maxmin $\mu_{1}$-strategy where each good $\mu_{2}$-reply to $\mu_{1}$ is close to one of some finite collections of $\mu_{2}$.
Finitely additive mixed strategies have been used by
\citet{MaiSud:IJGT1993, MaiSud:IJGT1998} in the framework of zero-sum stochastic games.
\citet{Cot:JET1991} considered finitely additive strategies in correlated equilibria; \citet{Sti:JET2011a} showed the limitation of his approach and proposed an alternative one.
\citet{Mar:IJGT1997} proved the existence of Nash equilibria in finitely additive mixed strategies under purely measure-theoretic conditions and connected this to the existence of $\varepsilon$-equilibria with countably additive mixed strategies. His analysis needs to restrict attention to payoff functions that are measurable with respect to the algebra generated by the cylinders.
\citet{HarStiZam:GEB2005} dealt with a class of games called nearly compact and continuous and proved existence of equilibria for any game in this class via a continuous compact imbedding in a larger game. They showed the use and limitations of finitely additive mixed strategies for these games.
\citet{Sti:GEB2005} devoted his attention to games that are not nearly compact and continuous and considered several classes of equilibria for these games showing advantages of disadvantages for each of them. The use of finitely additive strategies is fundamental in his analysis.
\citet{MyeRen:mimeo2012} have recently proposed a new notion of equilibrium for infinite games using finitely additive mixed strategies that arise from suitable finite approximations.
\citet{CapMor:IJGT2012} proved existence of equilibria for a family of zero-sum two-person games on semigroups when feasible mixed strategies are restricted to a suitable subclass of the class of finitely additive probability measures.
In this paper we prove an existence result for Nash equilibria of countable games by imposing some algebraic conditions on the payoff functions. The strategy set of each player is assumed to be a countable group and the payoff functions depend on their arguments only through the group operation. No topological condition is required. We allow finitely additive mixed strategies defined on the power set of the group. As mentioned before, this requires some care since the product of finitely additive measures is not uniquely defined on the power set of the Cartesian product of the groups, but only on the algebra generated by the cylinders. Since we want to integrate payoff functions that are not measurable with respect to this algebra, we need to select a suitable extension of the product measure. We propose a natural class of extensions by considering an average over all possible orders of integration. We show that the equilibrium exists and it does not depend on the way we choose this average. We characterize the equilibrium strategies and prove that they are the invariant means over the group that solve a suitable variational problem. The equilibrium payoffs have a very simple form. To avoid drowning our result in a sea of measure-theoretic technicalities, we first develop the theory for games on countable groups; then we show how it can be extended to uncountable groups, under suitable assumptions.
The paper is organized as follows.
Section~\ref{se:Main} describes the model and states the main result.
Section~\ref{se:Variational} proves a variational principle that is of interest \emph{per se} and is used in the proof of the main result.
Section~\ref{se:Proof} contains the proof of the main result.
Section~\ref{se:Generalization} examines some interesting generalizations.
Section~\ref{se:Examples} considers several examples.
Section~\ref{se:Uncountable} deals with uncountable groups.
Section~\ref{se:Conclusions} provides some conclusive remarks.
\section{Group games}\label{se:Main}
\subsection{Finite games}
Consider the classical matching pennies game
\begin{equation}\label{eq:matchingpennies}
\begin{array}[c]{c|rr|rr|}
\multicolumn{1}{c}{} & \multicolumn{2}{c}{A} &
\multicolumn{2}{c}{B} \\
\cline{2-5}
A & -1,& 1 & 1,& -1 \\
\cline{2-5}
B & 1,& -1 & -1,& 1 \\
\cline{2-5}
\end{array}
\end{equation}
We know that the unique equilibrium of this game is the profile of mixed strategies $((1/2,1/2), (1/2,1/2))$.
Notice that the matching pennies game can be re-written as follows. Make the set $\{A,B\}$ a finite group\footnote{A set $G$ with a binary operation $*$ is called a \emph{group} if the operation is associative, it has a unit element, and every element has an inverse. If the operation is commutative, then the group is called \emph{abelian}.}
by endowing it with the binary operation $*$ defined as
\[
A*A=B*B= B, \quad A*B=B*A = A.
\]
Define $\phi : \{A,B\} \to \mathbb{R}$ as follows:
\[
\phi(x) =
\begin{cases}
1 & \text{for $x=A$}, \\
-1 & \text{for $x=B$}.
\end{cases}
\]
Consider a game played by players $1$ and $2$, where each player's pure strategy set is $\{A,B\}$ and the payoffs are
\[
u_{1}(x,y) = - u_{2}(x,y) = \phi(x*y), \quad \text{for $x,y \in \{A,B\}$}.
\]
The game that we just described is nothing else than the matching pennies game defined in \eqref{eq:matchingpennies}.
This suggests the following generalization.
Consider a finite group $(G, *)$ and $N$ functions $\phi_{1}, \dots, \phi_{N} : G \to \mathbb{R}$.
Given a set of players $P=\{1, \dots, N\}$, for $i\in P$ let $u_{i} : G^{N} \to \mathbb{R}$ be defined as
\begin{equation}\label{eq:uphifinite}
u_{i}(x_{1}, \dots, x_{N}) = \phi_{i}(x_{1} * \dots * x_{N}).
\end{equation}
For $\boldsymbol{\phi} := (\phi_{1}, \dots, \phi_{N})$, call $\mathcal{G}(P,G, \boldsymbol{\phi})$ the game where the set of players is $P$, each player's set of pure strategies is $G$, and player $i$'s payoff function is given by \eqref{eq:uphifinite}.
Call $\mathcal{P}(G)$ the set of all probability measures on $2^{G}$. A probability measure $\lambda \in \mathcal{P}(G)$ is \emph{invariant} if for all $x, y \in G$ we have $\lambda(x) = \lambda(x*y)$.
Observe that finite groups have a unique invariant
measure, that is, the uniform measure. We will see in the next sections that a countable
group may have many invariant measures.
\begin{proposition}\label{pr:mainfinite}
The game $\mathcal{G}(P,G, \boldsymbol{\phi})$ admits an equilibrium in mixed strategies $(\lambda, \dots, \lambda)$, with $\lambda$ invariant on $G$.
\end{proposition}
\begin{proof}
For $\mu_{1}, \dots, \mu_{N} \in \mathcal{P}(G)$, define
\begin{equation*}
u_{i}(\mu_{1}, \dots, \mu_{N}) = \sum_{x_{1} \in G} \dots \sum_{x_{N} \in G} u_{i}(x_{1}, \dots, x_{N}) \mu_{1}(x_{1}) \cdots \mu_{N}(x_{N}).
\end{equation*}
Then we have to prove that for all $i \in P$ and all $\mu_{i} \in \mathcal{P}(G)$ we have
\begin{equation}\label{eq:lambdamufinite}
u_{i}(\lambda, \dots, \lambda) \ge u_{i}(\lambda, \dots, \lambda, \mu_{i}, \lambda, \dots, \lambda).
\end{equation}
Notice that, by definition of $u_{i}$, for all $j \in P$, for all $x_{1}, \dots, x_{j-1}, x_{j+1}, \dots, x_{N} \in G$ we have
\begin{align*}
\sum_{x_{j} \in G} u_{i}(x_{1}, \dots, x_{j}, \dots, x_{N}) \lambda(x_{j}) &= \sum_{x_{j} \in G} \phi_{i}(x_{1} * \dots * x_{j} * \dots * x_{N}) \lambda(x_{j}) \\
&= \sum_{y_{j} \in G} \phi_{i}(y_{j}) \lambda(x_{j-1}^{-1} * \dots * x_{1}^{-1} * y_{j} * x_{N}^{-1} * \dots * x_{j+1}^{-1}) \\
&= \sum_{y_{j} \in G} \phi_{i}(y_{j}) \lambda(y_{j}),
\end{align*}
where we used the change of variable $y_{j} = x_{1} * \dots * x_{j} * \dots * x_{N}$. Hence \begin{align*}
u_{i}(\lambda, \dots, \lambda) &= \sum_{x_{1} \in G} \dots \sum_{x_{N} \in G} \phi_{i}(x_{1} * \dots * x_{N}) \lambda(x_{1}) \cdots \lambda(x_{N}) \\
&= \sum_{y_{j} \in G} \phi_{i}(y_{j}) \lambda(y_{j}) \\
&= \sum_{x_{1} \in G} \dots \sum_{x_{i} \in G} \dots \sum_{x_{N} \in G} \phi_{i}(x_{1} * \dots * x_{N}) \lambda(x_{1}) \cdots \mu_{i}(x_{i}) \cdots \lambda(x_{N}) \\
&= u_{i}(\lambda, \dots, \lambda, \mu_{i}, \lambda, \dots, \lambda),
\end{align*}
that is, \eqref{eq:lambdamufinite} holds.
\end{proof}
\citet{Mor:MM2010} proved an analogous result for zero-sum games.
In the rest of the paper we will find conditions for the existence of equilibria in countable games, that, among other things, allow to extend Proposition~\ref{pr:mainfinite} to the case of countable strategy sets.
\subsection{Countable games}
Given a set of players $P=\{1, \dots, N\}$, a countable set $S$ and bounded functions $u_i: S^{N} \to [0,1]$, $i \in P$, consider a game $\mathcal{G}= \langle P, S,(u_{i})_{i\in P}\rangle$, where $S$ is the strategy set of all players, and $u_i$ is the payoff function of player $i$.
As mentioned in the Introduction, existence of mixed equilibria may fail if only countably additive mixed strategies are allowed. Therefore we consider a mixed extension of the game $\mathcal{G}$ where the space of mixed strategies is $\mathcal{P}(S)$, the space of all finitely additive probability measures on $S$. When doing this, a selection problem immediately arises. Given $\mu_{1}, \dots, \mu_{N} \in \mathcal{P}(S)$, a product measure $\otimes_{i=1}^{N} \mu_{i}$ is uniquely defined only on the algebra generated by the cylinders $S \times \dots \times S \times A \times S \dots \times S$, for all $A \subset S$. This product measure can be (non-uniquely) extended to the power set $2^{S \times \dots \times S}$. Different extensions correspond to different values of the expected payoff $\int_{S \times \dots \times S} u \ \mathrm{d} \otimes_{i=1}^{N} \mu_{i}$. Here we consider a parametric class of possible extensions that has the advantage of being easily computable. Its simpler bivariate version has been used for zero-sum two-person games by \citet{SheSei:BASE1996}.
Call $\Sigma(P)$ the space of permutations of $P$. Let $\nu \in \mathcal{P}(\Sigma(P))$ and $\mu_{1}, \dots, \mu_{N} \in \mathcal{P}(S)$.
For $i \in P$, let $u_i : S^{N} \to [0,1]$. Define
\begin{equation}\label{eq:ualpha}
u_i^{\nu}(\mu_{1}, \dots, \mu_{N}) :=
\sum_{\pi \in \Sigma(P)} \nu(\pi) \int_{S} \dots \int_{S} u_{i}(x_{1}, \dots, x_{N}) \ \mathrm{d} \mu_{\pi(1)}(x_{\pi(1)}) \dots \ \mathrm{d} \mu_{\pi(N)}(x_{\pi(N)}).
\end{equation}
This clearly defines an extension $\mu_{1} \boxtimes_{\nu} \dots \boxtimes_{\nu} \mu_{N}$ of $\otimes_{i=1}^{N} \mu_{i}$ to $2^{S \times \dots \times S}$ as follows.
For $A \subset S \times \dots \times S$
\begin{equation}\label{eq:otimesalpha}
\mu_{1} \boxtimes_{\nu} \dots \boxtimes_{\nu} \mu_{N}(A) = \sum_{\pi \in \Sigma(P)} \nu(\pi) \int_{S} \dots \int_{S} \mathds{1}_{A}(x_{1}, \dots, x_{N}) \ \mathrm{d} \mu_{\pi(1)}(x_{\pi(1)}) \dots \ \mathrm{d} \mu_{\pi(N)}(x_{\pi(N)}),
\end{equation}
where $\mathds{1}_{A}$ is the indicator function of the set $A$.
For properties of integration with respect to finitely additive measures we refer the reader to \citet{Hil:TAMS1934},
\citet{DunSch1:Wiley1988}, \citet{DeF:Wiley1972}, and \citet{BhaBha:AcademicPress1983}. Since every bounded function on a countable set is integrable with respect to any finitely additive probability measure, we do not need any measure-theoretical assumption.
We can now state our main theorem.
Let $(G, *)$ be a countable group%
\footnote{In the whole paper all countable groups will be endowed with the discrete topology.}
and given $\phi_{1}, \dots, \phi_{N} : G \to [0,1]$, define
\begin{equation}\label{eq:uphi}
u_i(x_{1}, \dots, x_{N})=\phi_{i}(x_{1}* \dots * x_{N}).
\end{equation}
For $\boldsymbol{\phi} = (\phi_{1}, \dots, \phi_{N})$ call $\mathcal{G}(P,S, \boldsymbol{\phi}, \nu)$ the mixed extension of the game $\mathcal{G}$ when $u_{i}$ is defined as in \eqref{eq:uphi} and the product measure of the finitely additive mixed strategies is selected as in \eqref{eq:otimesalpha}.
\begin{theorem}\label{th:main}
If $(G,*)$ is a countable abelian group, then the game $\mathcal{G}(P,G, \boldsymbol{\phi}, \nu)$ admits a Nash equilibrium that does not depend on $\nu$.
\end{theorem}
A more general version of Theorem~\ref{th:main} will be proved in Section~\ref{se:Proof}.
\section{A variational principle for FC-groups}\label{se:Variational}
In this section we prove a preliminary result that has some interest \emph{per se} since it represents a new variational principle for a useful class of groups that we now define.
\begin{definition}\label{def:fcgroups}
A countable group $G$ is called an \emph{FC-group} if for all $g\in G$, the conjugacy class $\{h*g*h^{-1} : h\in G\}$ is finite.
\end{definition}
FC-groups have been introduced by
\citet{Bae:DMJ1948} and
\citet{Neu:PLMS1951}. Among others, abelian groups are FC, since every conjugacy class is a singleton.
Let $G$ be a countable group, $A\subset G$ and $g\in G$. Fix the following notation
\[
g*A=\{g*a : a\in A\}\qquad A*g=\{a*g : a\in A\}.
\]
\begin{definition}\label{def:invariantmeasure}
A finitely additive probability measure $\mu$ on the power set of $G$ is called
\begin{itemize}
\item \emph{left-invariant mean}, if $\mu(A)=\mu(g*A)$, for all $g\in G$ and $A\subset G$,
\item \emph{right-invariant mean}, if $\mu(A)=\mu(A*g)$, for all $g\in G$ and $A\subset G$,
\item \emph{invariant mean}, if it is both left- and right-invariant.
\end{itemize}
For a given countable group $G$, we call $\mathcal{L}(G)$, $\mathcal{R}(G)$ and $\mathcal{I}(G)$ the class of all left-invariant, right-invariant and invariant means on $G$, respectively.
\end{definition}
It is well-known that the existence of a left-invariant mean is equivalent to the existence of a right-invariant mean, that is equivalent to the existence of an invariant mean\footnote{Indeed, given a left-invariant mean $\lambda$, one can define a right-invariant mean $\rho$ by setting $\rho(A)=\lambda(A^{-1})$, where $A^{-1} = \{a^{-1} : a \in A\}$. Now one can define an invariant mean $\mu$ by the formula $\mu(A)=\int_G\lambda(A * g^{-1})\ \mathrm{d} \rho(g)$.}.
\begin{definition}\label{def:amenable}
A countable group is called \emph{amenable} if it admits a left-invariant mean.
\end{definition}
Amenable groups have been introduced by
\citet{vNe:FM1929} in relation to the Tarski paradox and they form a hugely studied class of groups still nowadays. Every finite group is amenable, just taking the uniform measure; abelian groups are amenable by a standard but non-trivial argument making use of the Markov-Kakutani fixed point theorem. The simplest example of a non-amenable group is the free group on two generators\footnote{The free group on two generators, say $x$ and $y$, is the group of all words in the letters $x,x^{-1},y,y^{-1}$, equipped with the operation of concatenation of words, where only the simplifications $x*x^{-1}=x^{-1}*x=y*y^{-1}=y^{-1}*y=e$ are allowed, being $e$ the empty word. It was observed by von Neumann himself that this group, usually denoted by $\mathbb F_2$, is not amenable. A celebrated example of Ol{$'$}{\v{s}}anski{\u\i} shows the existence of non-amenable groups which do not contain $\mathbb F_2$ \citep[see][]{Ols:UMN1980}.}.
We use the following theorem, which appeared in \citet[Theorem~3.2]{Pat:PJM1979}.
\begin{theorem}\label{th:Paterson}
Let $G$ be a countable amenable FC-group. Then
\[
\mathcal{R}(G)=\mathcal{L}(G)=\mathcal{I}(G).
\]
\end{theorem}
This means that we have no distinctions between left- and -right-invariant means.
Given a countable amenable group $G$, $\ell^\infty(G)$ denotes the Banach space of all bounded real-valued function on $G$. The main result of this section is the following variational principle.
\begin{theorem}\label{th:variationalprinciple}
If $G$ is a countable amenable FC-group and $f:G\rightarrow[0,1]$, then for all $\pi \in \Sigma(P)$ and all $\lambda_{2}, \dots, \lambda_{N} \in\mathcal{I}(G)$ the functional $\Psi : \mathcal{P}(G) \to \mathbb{R}$ defined as
\[
\Psi(\mu) = \int \dots \int \int f(x_{1}* \dots * x_{N}) \ \mathrm{d} \mu(x_{\pi(1)}) \ \mathrm{d} \lambda_{2}(x_{\pi(2)}) \dots \ \mathrm{d} \lambda_{N}(x_{\pi(N)})
\]
attains its maximum at some $\lambda \in\mathcal{I}(G)$.
\end{theorem}
\begin{remark}
The space $\mathcal{P}(G)$ is a closed subset of the unit ball of the dual of $\ell^{\infty}(G)$ and therefore is compact in the weak* topology by the Banach-Alaoglu theorem \citep[see, e.g.,][Theorem 5.93]{AliBor:Springer2006}
Nevertheless the existence of the maximum for the functional $\Psi$ is not automatic, since it is not continuous.
To see this take $G=(\mathbb{Z}, +)$, $N=2$, and consider the functional
\[
\Psi(\mu) := \int \int \mathds{1}_{\mathbb{N}}(x+y) \ \mathrm{d} \mu(x) \ \mathrm{d} \lambda(y),
\]
with $\lambda \in \mathcal{I}(\mathbb{Z})$ such that $\lambda(\mathbb{N})=1$. Call $\mu_{\alpha}$ a net of probability measures having finite support in $-\mathbb{N}$ and converging to some $\mu \in \mathcal{I}(\mathbb{Z})$. Observe that $\mu(\mathbb{N}) = 0$. If $\Psi$ were continuous we would have
\begin{equation}\label{eq:continuousPsi}
\Psi(\mu) = \lim_{\alpha} \Psi(\mu_{\alpha}) .
\end{equation}
But
\begin{align*}
\Psi(\mu) &= \int \int \mathds{1}_{\mathbb{N}}(x+y) \ \mathrm{d} \mu(x) \ \mathrm{d} \lambda(y) = 0, \\
\Psi(\mu_{\alpha}) &= \int \int \mathds{1}_{\mathbb{N}}(x+y) \ \mathrm{d} \mu_{\alpha}(x) \ \mathrm{d} \lambda(y) \\
&= \int \int \mathds{1}_{\mathbb{N}}(x+y) \ \mathrm{d} \lambda(y) \ \mathrm{d} \mu_{\alpha}(x) = 1 \quad \text{for all } \alpha,
\end{align*}
which contradicts \eqref{eq:continuousPsi}.
\end{remark}
Denote
\begin{equation*}
I(f)=\left\{\int f(x) \ \mathrm{d} \lambda(x) : \lambda\in\mathcal{I}(G)\right\}.
\end{equation*}
The following lemma is folklore and follows from the fact that the set $\mathcal I(G)$ is convex and weak*-compact, when seen as a subset of the dual of $\ell^\infty(G)$.
\begin{lemma}\label{lem:convex}
The set $I(f) \subset \mathbb{R}$ is convex and compact.
\end{lemma}
This lemma guarantees that the following number is well defined.
\begin{equation}\label{eq:maxima}
I(f)^{+} := \max I(f).
\end{equation}
\begin{lemma}\label{lem:minimalgain}
Let $G$ be a countable amenable FC-group and $f:G \to [0,1]$. If there exist $\mu\in\mathcal{P}(G)$ and $L\in\mathbb{R}$ such that either
\begin{equation}\label{eq:intmuL}
\int f(x*y) \ \mathrm{d} \mu(x) \geq L \quad \text{for all $y \in G$},
\end{equation}
or
\begin{equation}\label{eq:intmuL2}
\int f(y*x) \ \mathrm{d} \mu(x) \geq L \quad \text{for all $y \in G$},
\end{equation}
then there exists $\lambda \in \mathcal{I}(G)$ such that $\int f(x) \ \mathrm{d} \lambda(x)\geq L$.
\end{lemma}
Given a set $A$, its cardinality is denoted by $|A|$.
\begin{definition}
A sequence $F_{n}$ of finite subsets of $G$ is called a \emph{left-F{\o}lner sequence} for $G$ if for all $g\in G$ one has
\[
\lim_{n\rightarrow\infty}\frac{|(g*F_{n})\triangle F_{n}|}{|F_{n}|}=0
\]
and a \emph{right-F{\o}lner sequence} for $G$ if for all $g\in G$ one has
\[
\lim_{n\rightarrow\infty}\frac{|(F_{n}*g)\triangle F_{n}|}{|F_{n}|}=0,
\]
where $\triangle$ stands for the symmetric difference of sets; i.e. $A\triangle B=(A\cup B)\setminus(A\cap B) = (A \setminus B) \cup (B \setminus A)$.
\end{definition}
\citet{Fol:MS1955} proved that such sequences exist for all countable amenable groups.
\begin{proof}[Proof of Lemma~\ref{lem:minimalgain}]
Let $F_{n}$ be a left-F{\o}lner sequence for $G$.
Consider the sequence of measures $\mu_{n}$ defined by
\[
\mu_{n}(A)=\frac{1}{|F_{n}|}\sum_{g\in F_{n}}\mu(A*g)
\]
and let $\lambda$ be a weak* limit of (a subnet $\mu_{c(\alpha)}$ of) this sequence.
First we prove that $\lambda\in\mathcal{I}(G)$. Indeed, for all $A\subset G$ and for all $h\in G$, one has
\begin{align*}
|\lambda(A*h)-\lambda(A)| &= \lim_\alpha|\mu_{c(\alpha)}(A*h)-\mu_{c(\alpha)}(A)| \\
&=
\lim_\alpha\left|\sum_{g\in F_{c(\alpha)}}\frac{1}{|F_{c(\alpha)}|}\left(\mu_{c(\alpha)}(A*h*g)-\mu_{c(\alpha)}(A*g)\right)\right|.
\end{align*}
Observe that the terms that are not in $(h*F_{c(\alpha)}) \triangle F_{c(\alpha)}$ cancel out. Majorizing with $1$ each of the remaining terms, we get
\[
|\lambda(A*h)-\lambda(A)|\leq \lim_{\alpha}\frac{|(h*F_{c(\alpha)})\triangle F_{c(\alpha)}|}{|F_{c(\alpha)}|}=0,
\]
which proves that $\lambda\in\mathcal{R}(G)$. Theorem~\ref{th:Paterson} implies that $\lambda\in\mathcal{I}(G)$.
Now we prove that if \eqref{eq:intmuL} holds, then
$\int f(x) \ \mathrm{d} \lambda(x)\geq L$. Indeed, we have
\begin{align*}
\int f(x) \ \mathrm{d} \lambda(x) &= \lim_\alpha\int f(x) \ \mathrm{d} \mu_{c(\alpha)}(x) \\
&=
\lim_\alpha\int\frac{1}{|F_{c(\alpha)}|}\sum_{g\in F_{c(\alpha)}}f(x) \ \mathrm{d} \mu(x*g) \\
&=
\lim_\alpha\int\frac{1}{|F_{c(\alpha)}|}\sum_{g\in F_{c(\alpha)}}f(x*g^{-1}) \ \mathrm{d} \mu(x)\\
&=\lim_\alpha\frac{1}{|F_{c(\alpha)}|}\sum_{g\in F_{c(\alpha)}}\int f(x*g^{-1}) \ \mathrm{d} \mu(x)\\
&\geq L.
\end{align*}
where the inequality stems from the hypothesis that each of the $|F_{c(\alpha)}| $ summands is larger or equal $L$.
The proof for the case \eqref{eq:intmuL2} is similar.
\end{proof}
Given a group $G$ we call $G^{N}$ the direct product of $G$, $N$ times, endowed with the component-wise operation, still denoted by $*$, with a little abuse of notation. If $G$ is amenable, then $G^{N}$ is amenable, too \citep{Day:IJM1957}. Furthermore, a simple computation shows that if $G$ is an FC-group, then also $G^{N}$ is an FC-group.
\begin{lemma}\label{lem:product}
Under the hypotheses of Theorem~\ref{th:Paterson}
\begin{align}
\max\left\{\int f(x_{1}* \dots * x_{N})\ \mathrm{d} \sigma(x_{1}, \dots, x_{N}) : \sigma\in\mathcal{I}(G^{N})\right\}=I(f)^{+},
\end{align}
where $I(f)^{+}$ is defined as in \eqref{eq:maxima}.
\end{lemma}
\begin{proof}
For
\begin{equation}\label{eq:psif}
\psi(x_{1}, \dots, x_{N}) = f(x_{1}* \dots * x_{N})
\end{equation}
define the set
\begin{equation*}
\mathbb{R} \supset \Lambda(\psi)=\left\{\int \psi(x_{1}, \dots, x_{N}) \ \mathrm{d} \sigma(x_{1}, \dots, x_{N}) : \sigma\in\mathcal{I}(G^{N})\right\}
\end{equation*}
and call $L=\max \Lambda(\psi)$, which exists by Lemma~\ref{lem:convex}. We have to prove that $L=I(f)^{+}$.\\
\noindent
\textbf{Proof of the inequality $L\geq I(f)^{+}$.}
By Lemma~\ref{lem:convex} there exists $\lambda\in\mathcal{I}(G)$ such that $\int f(x) \ \mathrm{d} \lambda(x)=I(f)^{+}$. Let $\lambda^{\otimes N}$ denote the measure on $G^{N}$ defined by the functional
\[
\ell^\infty(G^{N}) \ni \gamma \mapsto \int \dots \int \gamma(x_{1}, \dots, x_{N})\ \mathrm{d} \lambda(x_{1}) \dots \ \mathrm{d} \lambda(x_{N}).
\]
First we show that $\lambda^{\otimes N} \in\mathcal{I}(G^{N})$. Indeed for all $\gamma \in \ell^{\infty} (G^{N})$ and for all $(g_{1}, \dots, g_{N}) \in G^{N}$ we have
\begin{align*}
&\int \gamma((g_{1}, \dots, g_{N}) * (x_{1}, \dots, x_{N})) \ \mathrm{d} \lambda^{\otimes N}(x_{1}, \dots, x_{N}) \\
&\qquad= \int \gamma(g_{1}*x_{1}, \dots, g_{N}* x_{N}) \ \mathrm{d} \lambda^{\otimes N}(x_{1}, \dots, x_{N}) \\
&\qquad=\int \dots \int \gamma(g_{1}*x_{1}, \dots, g_{N}* x_{N}) \ \mathrm{d} \lambda(x_{1}) \dots \ \mathrm{d} \lambda(x_{N}) \\
&\qquad=\int \dots \int \gamma(x_{1}, g_{2} * x_{2, },\dots, g_{N}* x_{N}) \ \mathrm{d} \lambda(x_{1}) \dots \ \mathrm{d} \lambda(x_{N}) \\
&\qquad=\int \dots \int \gamma(x_{1}, x_{2}, g_{3}* x_{3},\dots, g_{N}* x_{N}) \ \mathrm{d} \lambda(x_{1}) \dots \ \mathrm{d} \lambda(x_{N}) \\
&\qquad \qquad \vdots \\
&\qquad=\int \dots \int \gamma(x_{1},\dots, x_{N}) \ \mathrm{d} \lambda(x_{1}) \dots \ \mathrm{d} \lambda(x_{N}),
\end{align*}
where the third equality stems from the fact that $\lambda$ is a left invariant mean, therefore
\[
\int \gamma(g_{1}*x_{1}, \dots, g_{N}* x_{N}) \ \mathrm{d} \lambda(x_{1}) = \int \gamma(x_{1}, g_{2} * x_{2, }\dots, g_{N}* x_{N}) \ \mathrm{d} \lambda(x_{1}).
\]
For the forth equality define $\zeta(x_{2}, \dots, x_{N}) = \int \gamma(x_{1}, x_{2}, \dots, x_{N}) \ \mathrm{d} \lambda(x_{1})$. Then, since $\lambda$ is a left invariant mean, we have
\begin{equation*}
\int \zeta(g_{2}* x_{2}, g_{3}* x_{3}, \dots, g_{N} * x_{N}) \ \mathrm{d} \lambda(x_{2})
= \int\zeta(x_{2}, g_{3}* x_{3}, \dots, g_{N} * x_{N}) \ \mathrm{d} \lambda(x_{2}),
\end{equation*}
i.e.,
\begin{multline*}
\int \int \gamma(x_{1, }g_{2}* x_{2}, g_{3}* x_{3}, \dots, g_{N} * x_{N}) \ \mathrm{d} \lambda(x_{1}) \ \mathrm{d} \lambda(x_{2}) \\
= \int \int \gamma(x_{1}, x_{2}, g_{3}* x_{3}, \dots, g_{N} * x_{N}) \ \mathrm{d} \lambda(x_{1}) \ \mathrm{d} \lambda(x_{2}).
\end{multline*}
The remaining equalities are obtained analogously. We have then shown that $\lambda^{\otimes N} \in \mathcal{L}(G^{N})$.
By Theorem~\ref{th:Paterson} applied to the FC-group $G^{N}$ it follows that $\lambda^{\otimes N} \in \mathcal{I}(G^{N})$.
Now we show that $\int f(x_{1} * \dots * x_{N}) \ \mathrm{d} \lambda^{\otimes N}(x_{1}, \dots, x_{N}) = I(f)^{+}$. We have
\begin{align*}
\int f(x_{1} * \dots * x_{N}) \ \mathrm{d} \lambda^{\otimes N}(x_{1}, \dots, x_{N}) &= \int \dots \int f(x_{1} * \dots * x_{N}) \ \mathrm{d} \lambda(x_{1}) \dots \ \mathrm{d} \lambda(x_{N}) \\
&= \int \dots \int f(x_{1}) \ \mathrm{d} \lambda(x_{1}) \dots \ \mathrm{d} \lambda(x_{N}) \\
&= \int \dots \int I(f)^{+} \ \mathrm{d} \lambda(x_{2}) \dots \ \mathrm{d} \lambda(x_{N}) \\
&= I(f)^{+}.
\end{align*}
This shows that $I(f)^{+}\in \Lambda(\psi)$ and therefore $I(f)^{+}\leq L$.\\
\noindent
\textbf{Proof of the inequality $L\leq I(f)^{+}$.}
Let $\overline{\sigma} \in \mathcal{I}(G^{N})$ be such that
\[
\int f(x_{1} * \dots * x_{N}) \ \mathrm{d} \overline{\sigma}(x_{1},\dots,x_{N})=L
\]
(such a measure exists by Lemma~\ref{lem:convex} applied to the group $G^{N}$ and to the function $\psi(x_{1}, \dots, x_{N})=f(x_{1} * \dots * x_{N})$) and let $\sigma_{\alpha}$ be a net of countably additive probability measures on $G^{N}$ converging to $\overline{\sigma}$ in the weak* topology. Define a countably additive probability measure $\mu_{\alpha}$ on $G$ by setting for all $x_{1} \in G$
\[
\mu_{\alpha}(x_{1})=\sum_{(x_{2}, \dots, x_{N}) \in G^{N-1}}\sigma_{\alpha}(x_{1}* (x_{2} * \dots * x_{N})^{-1}, x_{2}, \dots, x_{N}).
\]
To show that this is indeed a countably additive probability measure notice that for each $x_{1} \in G$ we have $\mu_{\alpha}(x_{1})\geq 0$ and therefore it suffices to show that $\sum_{x_{1}\in G}\mu_{\alpha}(x_{1})=1$. This follows from the observation that $\mu_{\alpha}(x_{1}) = \sigma_{\alpha}(A_{x_{1}})$, where
\[
A_{x_{1}}=\left\{(x_{1}* (x_{2} * \dots * x_{N})^{-1}, x_{2}, \dots, x_{N}) : (x_{2}, \dots, x_{N}) \in G^{N-1}\right\}
\]
and the fibers $A_{x_{1}}$ form a partition of $G^{N}$.
Now, let $\mu$ be any weak* limit of a subnet, denoted by $\mu_\beta$, of the net $\mu_{\alpha}$. For any $g\in G$, one has
\begin{align*}
\int_G f(g*x_{1}) \ \mathrm{d} \mu(x_{1}) &= \lim_{\beta}\int_G f(g*x_{1}) \ \mathrm{d} \mu_{\beta}(x_{1})\\
&=\lim_{\beta}\sum_{x_{1}\in G}f(g*x_{1}) \mu_{\beta}(x_{1})\\
&=\lim_{\beta}\sum_{x_{1}\in G} \sum_{(x_{2}, \dots, x_{N}) \in G^{N-1}}f(g*x_{1}) \sigma_{\beta}(x_{1}*(x_{2} * \dots * x_{N})^{-1}, x_{2}, \dots, x_{N})\\
&=\lim_{\beta} \sum_{(x_{2}, \dots, x_{N}) \in G^{N-1}} \sum_{x_{1}\in G} f(g*x_{1}) \sigma_{\beta}(x_{1}*(x_{2} * \dots * x_{N})^{-1}, x_{2}, \dots, x_{N})\\
&= \lim_{\beta}\sum_{(x_{2}, \dots, x_{N}) \in G^{N-1}} \sum_{z\in G*(x_{2} * \dots * x_{N})^{-1}}f(g*z*x_{2} * \dots * x_{N})
\sigma_{\beta}(z, x_{2}, \dots, x_{N}),
\end{align*}
where in the last equality we put $z=x_{1}*(x_{2} * \dots * x_{N})^{-1}$.
Observe that in the fourth equality we can exchange the order of summation since the series are nonnegative and convergent. For the same reason we can now replace $G*(x_{2} * \dots * x_{N})^{-1}$ with $G$ (the mapping $x\mapsto x*(x_{2} * \dots * x_{N})^{-1}$ is a permutation). Therefore, using again \eqref{eq:psif}, we have
\begin{align*}
&\lim_{\beta}\sum_{(x_{2}, \dots, x_{N}) \in G^{N-1}} \sum_{z\in G*(x_{2} * \dots * x_{N})^{-1}}f(g*z*x_{2} * \dots * x_{N})
\sigma_{\beta}(z, x_{2}, \dots, x_{N}) \\
&\qquad=
\lim_{\beta}\sum_{(z, x_{2}, \dots, x_{N}) \in G^{N}} f(g*z*x_{2} * \dots * x_{N})
\sigma_{\beta}(z, x_{2}, \dots, x_{N}) \\
&\qquad=
\lim_{\beta}\int f(g*z*x_{2} * \dots * x_{N}) \ \mathrm{d}
\sigma_{\beta}(z, x_{2}, \dots, x_{N}) \\
&\qquad=
\lim_{\beta}\int \psi((g, 1_{G}, \dots, 1_{G}) * (z, x_{2}, \dots, x_{N})) \ \mathrm{d}
\sigma_{\beta}(z, x_{2}, \dots, x_{N}) \\
&\qquad=
\int \psi((g, 1_{G}, \dots, 1_{G}) * (z, x_{2}, \dots, x_{N})) \ \mathrm{d}
\overline{\sigma}(z, x_{2}, \dots, x_{N}) \\
&\qquad=
\int \psi(z, x_{2}, \dots, x_{N}) \ \mathrm{d}
\overline{\sigma}(z, x_{2}, \dots, x_{N}) \\
&\qquad=
\int f(z * x_{2} * \dots * x_{N}) \ \mathrm{d}
\overline{\sigma}(z, x_{2}, \dots, x_{N}) \\
&\qquad=L.
\end{align*}
We have proved that $\int_G f(g*x_{1}) \ \mathrm{d} \mu(x_{1}) = L$ for all $g\in G$.
Therefore, by Lemma \ref{lem:minimalgain}, there exists $\lambda\in\mathcal{I}(G)$ such that $\int f(x) \ \mathrm{d} \lambda(x)\geq L$. It follows that $I(f)^{+}\geq L$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{th:variationalprinciple}]
Call $\boldsymbol{\lambda} = (\lambda_{2}, \dots, \lambda_{N})$ and
\begin{align}
S_{\boldsymbol{\lambda}, \pi}=\sup_{\mu\in\mathcal{P}(G)}\left\{
\int \dots \int \int f(x_{1}* \dots * x_{N}) \ \mathrm{d} \mu(x_{\pi(1)}) \ \mathrm{d} \lambda_{2}(x_{\pi(2)}) \dots \ \mathrm{d} \lambda_{N}(x_{\pi(N)})
\right\}.
\end{align}
By Lemma \ref{lem:convex}, we know that $S_{\boldsymbol{\lambda}, \pi} \geq I(f)^{+}$, for all $\boldsymbol{\lambda}$ and $\pi$.
We recall that the value $I(f)^{+}$ is attained, basically by definition, by an invariant mean. So it suffices to show that $S_{\boldsymbol{\lambda}, \pi} = I(f)^{+}$.
Now, by contradiction, suppose that there exist $\mu\in\mathcal P(G)$ and $\lambda_{2}, \dots, \lambda_{N}\in\mathcal{I}(G)$ such that
\[
\int \dots \int \int f(x_{1}* \dots * x_{N}) \ \mathrm{d} \mu(x_{\pi(1)}) \ \mathrm{d} \lambda_{2}(x_{\pi(2)}) \dots \ \mathrm{d} \lambda_{N}(x_{\pi(N)}) =: L>I(f)^{+}.
\]
Call $\sigma$ the measure on $G^{N}$ defined by the functional
\begin{multline*}
\ell^\infty(G^{N}) \ni \gamma \mapsto \int_{G^{N}}\gamma(x_{1}, \dots, x_{N}) \ \mathrm{d} \sigma(x_{1}, \dots, x_{N}) \\
=\int \dots \int \int \gamma(x_{1}, \dots, x_{N}) \ \mathrm{d} \mu(x_{\pi(1)}) \ \mathrm{d} \lambda_{2}(x_{\pi(2)}) \dots \ \mathrm{d} \lambda_{N}(x_{\pi(N)}).
\end{multline*}
Define $\psi(x_{1}, \dots, x_{N})=f(x_{1} * \dots * x_{N})$.
We start considering the case $\pi(1) < N$. Setting $\pi(1) = j$, for any $(g_{1}, \dots, g_{N})\in G^{N}$ we have
\begin{align}
&\int \psi((x_{1}, \dots, x_{N})*(g_{1}, \dots, g_{N})) \ \mathrm{d} \sigma(x_{1}, \dots, x_{N}) \label{eq:psisigma}\\
&\qquad= \int \psi((x_{1}*g_{1}, \dots, g_{N}*x_{N}))\ \mathrm{d}\sigma(x_{1},\dots, x_{N}) \nonumber \\
&\qquad= \int \dots \int \int f(x_{1}*g_{1}* x_{2}*g_{2}\dots *x_{N}*g_{N}) \ \mathrm{d} \mu(x_{\pi(1)}) \ \mathrm{d} \lambda_{2}(x_{\pi(2)}) \dots \ \mathrm{d} \lambda_{N}(x_{\pi(N)}) \nonumber \\
&\qquad= \int \dots \int \int f(x_{1}*x_{2}*\dots *x_{N}) \ \mathrm{d} \mu(x_{\pi(1)}) \ \mathrm{d} \lambda_{2}(x_{\pi(2)}) \dots \ \mathrm{d} \lambda_{N}(x_{\pi(N)}) \nonumber \\
&\qquad=L, \nonumber
\end{align}
where the third equality can be shown by setting
\[
\xi(x_{1}, \dots, x_{j-1}, x_{j+1}, \dots, x_{N}) = \int f(x_{1} * \dots * x_{N}) \ \mathrm{d} \mu(x_{j})
\]
and noticing that
\begin{align*}
&\int \dots \int \int f(x_{1}*g_{1} * \dots * x_{j-1}*g_{j-1} * x_{j} * g_{j}* x_{j+1} * g_{j+1} * \dots * x_{N} * g_{N}) \\
& \qquad \ \mathrm{d} \mu(x_{j}) \ \mathrm{d} \lambda_{2}(x_{\pi(2)}) \dots \ \mathrm{d} \lambda_{N}(x_{\pi(N)}) \\
&\qquad = \int \dots \int \xi(x_{1}*g_{1}, \dots, x_{j-1}*g_{j-1}, g_{j}* x_{j+1} * g_{j+1}, x_{j+2} *g_{j+2}, \dots, x_{N} * g_{N}) \\
& \qquad \qquad \ \mathrm{d} \lambda_{2}(x_{\pi(2)}) \dots \ \mathrm{d} \lambda_{N}(x_{\pi(N)}) \\
&\qquad = \int \dots \int \xi(x_{1}, \dots, x_{j-1}, x_{j+1}, \dots, x_{N} ) \ \mathrm{d} \lambda_{2}(x_{\pi(2)}) \dots \ \mathrm{d} \lambda_{N}(x_{\pi(N)}) \\
&\qquad= \int \dots \int \int f(x_{1}*x_{2}*\dots *x_{N}) \ \mathrm{d} \mu(x_{\pi(1)}) \ \mathrm{d} \lambda_{2}(x_{\pi(2)}) \dots \ \mathrm{d} \lambda_{N}(x_{\pi(N)}).
\end{align*}
Therefore, we can apply Lemma \ref{lem:minimalgain} to the amenable FC-group $G^{N}$ and to the function $\psi(x_{1}, \dots, x_{N})$. This means that there exists an invariant measure $\rho'\in\mathcal{I}(G^{N})$ such that $\int f(x_{1} * \dots * x_{N}) \ \mathrm{d} \rho'(x_{1}, \dots, x_{N}) \ge L>I(f)^{+}$. This contradicts Lemma~\ref{lem:product}.
If $\pi(1) = N$, then the result can be proved replacing \eqref{eq:psisigma} with
\[
\int \psi((g_{1}, \dots, g_{N}) * (x_{1}, \dots, x_{N})) \ \mathrm{d} \sigma(x_{1}, \dots, x_{N}) \\
\]
and following the steps of the previous case.
\end{proof}
\section{Proof of the main result}\label{se:Proof}
As before, $G$ is a countable amenable FC-group and, for $i \in P$, let $\phi_i:G\rightarrow[0,1]$ and $\overline\lambda_{i}\in \mathcal{I}(G)$ be such that
\[
\int \phi_{i} \ \mathrm{d} \overline\lambda_{i}=I(\phi_{i})^{+}.
\]
Every abelian group is an amenable FC-group, hence Theorem~\ref{th:main} is a corollary of the following more general result.
\begin{theorem}\label{th:main2}
If $G$ is a countable amenable FC-group, then the profile of strategies $(\overline\lambda_{1}, \dots, \overline\lambda_{N})$ is a Nash equilibrium for the game $\mathcal{G}(P,G, \boldsymbol{\phi}, \nu)$.
\end{theorem}
\begin{proof}
We have to prove that for all $i \in P$ and all $\mu_{i} \in \mathcal{P}(G)$ we have
\begin{equation*}
u_i^{\nu}(\overline{\lambda}_{1}, \dots, \overline{\lambda}_{i}, \dots, \overline{\lambda}_{N}) \ge u_i^{\nu}(\overline{\lambda}_{1}, \dots, \overline{\lambda}_{i-1}, \mu_{i}, \overline{\lambda}_{i+1}, \dots, \overline{\lambda}_{N}).
\end{equation*}
Using \eqref{eq:ualpha}, we know that
\begin{equation*}
u_i^{\nu}(\overline{\lambda}_{1}, \dots, \overline{\lambda}_{i}, \dots, \overline{\lambda}_{N}) = \sum_{\pi \in \Sigma(P)} \nu(\pi) \int_{G} \dots \int_{G} u_{i}(x_{1}, \dots, x_{N}) \ \mathrm{d} \overline{\lambda}_{\pi(1)}(x_{\pi(1)}) \dots \ \mathrm{d} \overline{\lambda}_{\pi(N)}(x_{\pi(N)}),
\end{equation*}
where $u_{i}(x_{1}, \dots, x_{N}) = \phi_{i}(x_{1} * \dots * x_{N})$.
Since all $\overline{\lambda}_{j}$ are invariant, each of the summands of $u_i^{\nu}(\overline{\lambda}_{1}, \dots, \overline{\lambda}_{i}, \dots, \overline{\lambda}_{N})$ is equal to the corresponding summand for
\linebreak
$u_i^{\nu}(\overline{\lambda}_{1}, \dots, \overline{\lambda}_{i-1}, \mu_{i}, \overline{\lambda}_{i+1}, \dots, \overline{\lambda}_{N})$, except when $\pi(1)=i$.
If we call $\Sigma_{i}(P)$ the class of all permutations of $P$ such that $\pi(1)=i$, then all we have to prove is
\begin{multline*}
\sum_{\pi \in \Sigma_{i}(P)} \nu(\pi) \int_{G} \dots \int_{G }\int_{G} u_{i}(x_{1}, \dots, x_{N}) \ \mathrm{d} \overline{\lambda}_{i}(x_{i}) \ \mathrm{d} \overline{\lambda}_{\pi(2)}(x_{\pi(2)}) \dots \ \mathrm{d} \overline{\lambda}_{\pi(N)}(x_{\pi(N)}) \\
\ge
\sum_{\pi \in \Sigma_{i}(P)} \nu(\pi) \int_{G} \dots \int_{G} \int_{G} u_{i}(x_{1}, \dots, x_{N}) \ \mathrm{d} \mu_{i}(x_{i}) \ \mathrm{d} \overline{\lambda}_{\pi(2)}(x_{\pi(2)}) \dots \ \mathrm{d} \overline{\lambda}_{\pi(N)}(x_{\pi(N)}).
\end{multline*}
This inequality can be shown to hold summand by summand. More precisely, by Theorem~\ref{th:variationalprinciple} we know that each summand with $\mu_{i}$ is majorized by some invariant measure; but $\bar{\lambda}_{i}$ maximizes the value over all invariant measures, so the result follows.
\end{proof}
\section{Some generalizations}\label{se:Generalization}
Here we consider some generalizations of Theorem~\ref{th:main2}. The first extension allows us to show the existence of an equilibrium in Wald's game, a classical example of countable game that does not admit equilibria in $\sigma$-additive mixed strategies.
\subsection{Transformations of group operations}
Let $\eta_{1}, \dots, \eta_{N}: G \to G$ be bijections and let
\begin{equation}\label{eq:uphieta}
u^{\boldsymbol{\eta}}_{i}(x_{1}, \dots, x_{N}) = \phi_{i}(\eta_{1}(x_{1}) * \dots * \eta_{N}(x_{N})) \quad \text{for $i \in P$}.
\end{equation}
For $\boldsymbol{\eta} = (\eta_{1}, \dots, \eta_{N})$ call $\mathcal{G}(P,G, \boldsymbol{\phi}, \boldsymbol{\eta}, \nu)$ the game where the payoffs are given by the mixed extensions of \eqref{eq:uphieta}, as in \eqref{eq:ualpha}.
\begin{theorem}\label{th:generaleta}
If $G$ is a countable amenable FC-group, then the game $\mathcal{G}(P,G, \boldsymbol{\phi}, \boldsymbol{\eta}, \nu)$ admits a Nash equilibrium.
\end{theorem}
\begin{proof}
Call
\[
u_{i}(y_{1}, \dots, y_{N}) = \phi_{i}(y_{1} * \dots * y_{N}).
\]
By Theorem~\ref{th:main2} we know that the game $\mathcal{G}(P,G, \boldsymbol{\phi}, \nu)$ admits a Nash equilibrium given by $(\overline{\lambda}_{1}, \dots, \overline{\lambda}_{N})$. Therefore if we define for $i\ \in P$ a measure $\rho_{i}$ on $2^{G}$ as follows
\[
\rho_{i}(A) = \overline{\lambda}_{i}(\eta_{i}(A)),
\]
then the profile $(\rho_{1}, \dots, \rho_{N})$ is a Nash equilibrium of $\mathcal{G}(P,G, \boldsymbol{\phi}, \boldsymbol{\eta}, \nu)$.
\end{proof}
\subsection{Graph games}
Theorem~\ref{th:main} requires that the payoff of each player be a function of the group operation over the strategies of all players. This hypothesis is quite restrictive. The next theorem substantially weakens it by allowing the payoff of player $i$ to be a function of the group operation over a---possibly small---subset of players, provided it includes player $i$ herself and at least another player. This can be interpreted as a game over a graph, where the payoff function of each player depends only on her action and the actions of her neighbors.
For every $i \in P$ let $P_{i} \subset P$ be such that $i \in P_{i}$ and $|P_{i}| \ge 2$. Consider functions $\phi_{i} : G \to [0,1]$ such that
\begin{equation}\label{eq:uiNi}
u_{i}(x_{1}, \dots, x_{N}) = \phi_{i}(*_{j \in P_{i}} x_{j}),
\end{equation}
that is, the payoff of player $i$ depends only on the strategies of her neighbors.
Call $\boldsymbol{P} = (P_{1}, \dots, P_{N})$ and define the game $\mathcal{G}(P, \boldsymbol{P}, G, \boldsymbol{\phi}, \nu)$ where the payoff functions are as in \eqref{eq:uiNi}.
\begin{theorem}\label{th:generalNi}
If $G$ is a countable amenable FC-group, then the game $\mathcal{G}(P, \boldsymbol{P}, G, \boldsymbol{\phi}, \nu)$ admits a Nash equilibrium.
\end{theorem}
The proof of this theorem follows the line of the proof of Theorem~\ref{th:main2} and is therefore omitted.
\section{Examples}\label{se:Examples}
\subsection{Games on $\mathbb{Z}$}
As the next proposition shows, when the game is played on $\mathbb{Z}$, the equilibrium of Theorem~\ref{th:main2} has an interesting structure, that is, each equilibrium strategy has its mass either entirely adherent to $-\infty$ or to $+\infty$, or if it is split between the two, then it can be split in any possible way.
\begin{proposition}\label{pr:lambdapmN}
Consider a game $\mathcal{G}(P, \mathbb{Z}, \boldsymbol{\phi}, \nu)$.
Let $(\lambda_{1}, \dots, \lambda_{N})$ be an equilibrium for this game, where $\lambda_{1}, \dots, \lambda_{N} \in \mathcal{I}(\mathbb{Z})$. Then, for all $i \in P$, one of the following three possibilities is true
\begin{enumerate}[{\rm (a)}]
\item\label{it:pr:lambdapmN-a}
$\lambda_{i}(\mathbb{N})=0$,
\item\label{it:pr:lambdapmN-b}
$\lambda_{i}(\mathbb{N}) = 1$,
\item\label{it:pr:lambdapmN-c}
if $0 < \lambda_{i}(\mathbb{N}) <1$, then for any $\delta \in [0,1]$ there exists another equilibrium strategy $\lambda_{i}' \in \mathcal{I}(\mathbb{Z})$ with $\lambda_{i}'(\mathbb{N}) = \delta$.
\end{enumerate}
\end{proposition}
\begin{proof}
Assume that neither \ref{it:pr:lambdapmN-a} nor \ref{it:pr:lambdapmN-b} holds. We prove that \ref{it:pr:lambdapmN-c} must be true. Suppose that $\lambda_{i} \in\mathcal{I}(\mathbb{Z})$ is an equilibrium strategy for player $i$ and $\lambda_{i}(\mathbb{-N}) = \theta \in (0,1)$.
Call
\[
K=\int \phi_{i} \ \mathrm{d} \lambda_{i}, \quad \theta K_{1}=\int_{-\mathbb{N}} \phi_{i} \ \mathrm{d} \lambda_{i}, \quad (1-\theta) K_{2} = \int_{\mathbb{N}} \phi_{i} \ \mathrm{d} \lambda_{i}.
\]
Then
\[
K = \theta K_{1} + (1-\theta) K_{2}.
\]
Assume that $K_{2} > K_{1}$.
Consider now a measure $\lambda_{i}' \in \mathcal{I}(\mathbb{Z})$ such that $\lambda_{i}'(-\mathbb{N})=0$ and for $A \subset \mathbb{N}$ we have $\lambda_{i}'(A) = (1-\theta)^{-1} \lambda_{i}(A)$. Then
\[
\int \phi_{i} \ \mathrm{d} \lambda_{i}' = K_{2} > K,
\]
which, by Theorem~\ref{th:main2}, contradicts the fact that $\lambda_{i}$ is an equilibrium strategy. A similar argument holds if $K_{1} > K_{2}$.
It is easy to see that if $K_{1}=K_{2}$, then any measure $\lambda_{i}''$ such that for $0 < \kappa < \theta^{-1}$
\begin{align*}
\lambda_{i}''(A) &= \kappa \lambda_{i}(A) \quad \text{for $A \subset -\mathbb{N}$}, \\
\lambda_{i}''(A) &= (1-\theta\kappa) (1-\theta)^{-1} \lambda_{i}(A) \quad\text{ for $A \subset \mathbb{N}$},
\end{align*}
satisfies $\int \phi_{i} \ \mathrm{d} \lambda_{i}'' = K$ and therefore
is an equilibrium strategy. This proves part \ref{it:pr:lambdapmN-c}.
\end{proof}
\begin{example}[Matching pennies]
Consider the following countable version of matching pennies. The strategy set of each of the two players is $\mathbb{Z}$ and the payoff functions are
\[
u_{1}(x,y) = 1-u_{2}(x,y) = \mathds{1}_{2\mathbb{Z}}(x+y),
\]
where $k\mathbb{Z}$ is the set of multiples of $k$.
This game is equivalent to the one where players choose only Odd or Even and player 1 wins if both players make the same choice. Any profile of strategies $(\mu_{1}, \mu_{2})$ such that $\mu_{1}(2\mathbb{Z}) = \mu_{2}(2\mathbb{Z}) = 1/2$ is an equilibrium of the game.
The game can be generalized to $N$ players as follows. Consider a partition $A_{1}, \dots, A_{N}$ of $\mathbb{Z}$ and payoff functions
\[
u_{i}(x_{1}, \dots, x_{N}) = \mathds{1}_{A_{i}}(x_{1}+ \dots + x_{N}).
\]
If for each $i \in P$ the measure
\[
\overline{\lambda}_{i} \in \arg \max_{\lambda_{i} \in \mathcal{I}(\mathbb{Z})} \lambda_{i}(A_{i}),
\]
then, by Theorem~\ref{th:main2}, the profile $(\overline{\lambda}_{1}, \dots, \overline{\lambda}_{N})$ is a Nash equilibrium of the game.
Notice that, if all sets $A_{1}, \dots, A_{N}$ are periodic (not necessarily with the same period), then for all $\lambda, \lambda' \in \mathcal{I}(\mathbb{Z})$ we have
\[
\lambda(A_{i}) = \lambda'(A_{i}),
\]
Therefore any profile of invariant measures is an equilibrium. Let $m$ be the lowest common multiple of the periods $m_{i}$ of the $A_{i}$'s.
Consider now the $m$ congruence classes $m\mathbb{Z} + k$.
Any profile of probability measures $(\mu_{1}, \dots, \mu_{N})$ such that for $i \in P$ and $k \in \{0, \dots, m-1\}$
\[
\mu_{i}(m\mathbb{Z} + k) = \overline{\lambda}_{j}(m\mathbb{Z} + k) = 1/m
\]
is an equilibrium, too.
\end{example}
\begin{example}[Wald's game]
The following game was introduced by \citet{Wal:AM1945} as a counterexample to the existence of minmax in zero-sum two-person games when the sets of strategies for both players are infinite.
Let the strategy set be $\mathbb{Z}$ and
\[
u_{1}(x,y) = 1 - u_{2}(x,y) =
\begin{cases}
1 & \text{if $x > y$}, \\
1/2 & \text{if $x = y$}, \\
0 & \text{if $x < y$}.
\end{cases}
\]
Call $z := -y$; then the payoff function becomes
\[
u_{1}(x,z) =
\begin{cases}
1 & \text{if $x+z > 0$}, \\
1/2 & \text{if $x+z = 0$}, \\
0 & \text{if $x+z < 0$}.
\end{cases}
\]
Then $u_{1}(x,z)$ is an example of $\phi(x+z)$ with $+$ as the group operation.
Applying Theorem~\ref{th:generaleta} we obtain the equilibrium $(\lambda, \rho)$ with $\lambda, \rho \in \mathcal{I}(\mathbb{Z})$ and $\lambda(\mathbb{N}) = \rho
(-\mathbb{N}) = 1$. This shows the striking difference between countably additive and finitely additive extensions of countable games.
\end{example}
\subsection{Games on $\mathbb{Z}^{2}$}
Consider a game where the strategy set of each player is $\mathbb{Z}^{2}$ and for $i \in \{1, \dots, N\}$ the payoff function $u_{i}$ is
\begin{equation}\label{eq:uiZ2}
u_{i}(x_{1} + \dots + x_{N}) = \mathds{1}_{C_{i}}(x_{1} + \dots + x_{N}),
\end{equation}
where $C_{i}$ is some open cone, i.e., $C_{i}$ is an open subset of $\mathbb{R}^{2}$ such that if $x \in C_{i}$, then $\beta x \in C_{i}$ for all $\beta \in \mathbb{R}_{+}$.
\begin{proposition}
Let $\phi_{i} = \mathds{1}_{C_{i}}$. If $\overline{\lambda}_{i}\in \mathcal{I}(\mathbb{Z}^{2})$ and $\overline{\lambda}_{i}(C_{i})=1$ for all $i \in P$, then the profile $(\overline{\lambda}_{1}, \dots, \overline{\lambda}_{N})$ is an equilibrium of the game $\mathcal{G}(P, \mathbb{Z}^{2}, \boldsymbol{\phi}, \nu)$.\end{proposition}
\begin{proof}
By Theorem~\ref{th:main2} a measure $\overline{\lambda}_{i}$ is an equilibrium strategy if
\[
\overline{\lambda}_{i} = \arg \max_{\lambda_{i} \in \mathcal{I}(\mathbb{Z}^{2})} \lambda_{i}(C_{i}).
\]
Since $\lambda_{i}(C_{i}) \le 1$ all we need to prove is that for each open cone $C_{i}$ there exists an invariant measure $\overline{\lambda}_{i}$ such that $\overline{\lambda}_{i}(C_{i}) = 1$. This is achieved by using F{\o}lner sequences as follows.
For every open cone $C_{i}$ there exists a convex open cone $C_{i}^{*} \subset C_{i}$. We now prove the existence of $\overline{\lambda}_{i}$ such that $\overline{\lambda}_{i}(C_{i}^{*}) = 1$. Call $Q_{n} := \{-n, \dots, n\}^{2} \subset \mathbb{Z}^{2}$ and $F_{n} := Q_{n} \cap C_{i}^{*}$.
\begin{claim}\label{cl:Folner}
The sequence $F_{n}$ is a F{\o}lner sequence.
\end{claim}
\begin{proof}[Proof of Claim~\ref{cl:Folner}]
We need to prove that for all $g \in \mathbb{Z}^{2}$ we have
\[
\lim_{n \to \infty} \frac{|(g*F_{n}) \triangle F_{n}|}{|F_{n}|} = 0.
\]
There exists $\delta > 0$ such that for $n$ large enough $|F_{n}| > \delta n^{2}$. Moreover if $g=(g_{1}, g_{2})$, then $|(g*F_{n}) \triangle F_{n}| \le 4|g_{1} g_{2}| n$. This proves the Claim.
\end{proof}
Define a probability measure $\mu_{n}$ on $2^{\mathbb{Z}^{2}}$ as follows:
\[
\mu_{n}(A) = \frac{|A \cap F_{n}|}{|F_{n}|}.
\]
Call $\overline{\lambda_{i}}$ a weak* limit of a subnet $\mu_{c(\alpha)}$ of $\mu_{n}$. First we prove that $\overline{\lambda_{i}} \in \mathcal{I}(\mathbb{Z}^{2})$. For all $A \subset \mathbb{Z}^{2}$ and for all $g \in \mathbb{Z}^{2}$
\begin{align*}
|\overline{\lambda_{i}}(A+g) - \overline{\lambda_{i}}(A)| &= \lim_{\alpha} \left| \frac{|(A+g) \cap F_{c(\alpha)}|}{|F_{c(\alpha)}|} - \frac{|A \cap F_{c(\alpha)}|}{|F_{c(\alpha)}|}
\right| \\
&= \lim_{\alpha} \left| \frac{|A \cap (F_{c(\alpha)}-g)|}{|F_{c(\alpha)}|} - \frac{|A \cap F_{c(\alpha)}|}{|F_{c(\alpha)}|}
\right| \\
& \le \lim_{\alpha}\left| \frac{|A \cap ((F_{c(\alpha)}-g) \triangle F_{c(\alpha)}) |}{|F_{c(\alpha)}|}
\right| \\
&= 0,
\end{align*}
where the inequality is due to the fact that
\[
A \cap ((F_{c(\alpha)}-g) \setminus (A \cap F_{c(\alpha)})) \subset A \cap ((F_{c(\alpha)}-g) \triangle F_{c(\alpha)}).
\]
Then we prove that $\overline{\lambda}_{i}(C_{i}^{*}) = 1$. Indeed
\[
\overline{\lambda}_{i}(C_{i}^{*}) = \lim_{\alpha} \frac{|C_{i}^{*} \cap F_{c(\alpha)}|}{|F_{c(\alpha)}|}.
\]
Since $F_{c(\alpha)} \subset C_{i}^{*}$ for all $\alpha$, we have
\[
\frac{|C_{i}^{*} \cap F_{c(\alpha)}|}{|F_{c(\alpha)}|} = 1 \quad \text{for all $\alpha$}. \qedhere
\]
\end{proof}
\subsection{Games on $\mathbb{Q} \cap [0,1] \mod 1$}\label{suse:Q01}
Take $G = \mathbb{Q} \cap [0,1]$ equipped with the sum modulo $1$. Then $G$ is a countable abelian group, so it is an amenable FC-group. Observe that any invariant mean $\lambda$ on $G$ satisfies the following property.
With an abuse of language we use the symbol $[a,b]$ to denote the set $\{x \in G: a \le x \le b\}$.
\begin{proposition}\label{pr:lambda01}
For every $\lambda \in \mathcal{I}(G)$ and every $a, b \in [0,1]$, $a < b$, we have $\lambda([a,b]) = b-a$.
\end{proposition}
\begin{proof}
Observe that, by invariance, for every $n \in \mathbb{N}$
\[
\lambda\left(\left[0,\frac{1}{n}\right]\right) = \lambda\left(\left[\frac{1}{n}, \frac{2}{n}\right]\right) = \dots =\lambda\left(\left[\frac{n-1}{n}, 1\right]\right) = \frac{1}{n}.
\]
Hence, by finite additivity, if $a = k/n$ and $b = m/n$, then
\[
\lambda([a,b]) = \lambda\left(\left[\frac{k}{n}, \frac{m}{n} \right] \right) = \frac{m-k}{n}.
\]
\end{proof}
Therefore any two invariant means on $G$ coincide on the algebra generated by intervals in $G$, but they can be extended in many different ways to $2^{G}$.
\begin{proposition}
Call $\mathfrak{c}$ the cardinality of $\mathbb{R}$. Then $|\mathcal{I}(G)| \ge 2^{\mathfrak{c}}$.
\end{proposition}
\begin{proof}
If $G$ is endowed with the discrete topology, then it is a locally compact group, whose Haar measure is the counting measure. It follows that the set of essentially (with respect to the Haar measure) bounded real-valued functions on $G$ is equal to $\ell^{\infty}(G)$. Therefore \citet[Theorem on page 444]{Cho:TAMS1970} can be used to get the result.
\end{proof}
Consider now a game on $G$ where for $i \in P$,
\[
\phi_{i}(x) = \mathds{1}_{A_{i}}(x).
\]
If $A_{1}, \dots, A_{N}$ are in the algebra generated by intervals, then every profile $(\lambda_{1}, \dots, \lambda_{N})$, with $\lambda_{i} \in \mathcal{I}(G)$ is an equilibrium. Otherwise the equilibrium strategy for player $i$ is
\[
\overline{\lambda}_{i} \in \arg\max_{\lambda \in \mathcal{I}(G)} \lambda(A_{i}).
\]
\begin{example}[Rock-scissors-paper]
The classical rock-scissors-paper game is played by two players whose strategy set is $G=\{R, S, P\}$. The payoff for player $1$ is
\begin{align}\label{eq:uRSP}
u_{1}(R,S)=u_{1}(P,R)=u_{1}(S,P)&=1, \nonumber\\
u_{1}(R,R)=u_{1}(P,P)=u_{1}(S,S)&=1/2, \\
u_{1}(S,R)=u_{1}(R,P)=u_{1}(P,S)&=0, \nonumber
\end{align}
and $u_{2}(x,y) = 1- u_{1}(x,y)$.
It is well known that the unique equilibrium of this game is the profile of uniform mixed strategies for each player.
We can endow the set $G$ with an operation $*$ that makes it an abelian group, as follows:
\begin{align*}
R*R = P*S = S*P &= R \\
R*P = P*R = S*S &= P \\
R*S = S*R = P*P &= S.
\end{align*}
Note that $R$ is the unit element and
\[
R^{-1}=R, P^{-1}=S, S^{-1}=P.
\]
Actually the group $G$ can be identified with $\mathbb{Z}/3 \mathbb{Z} = \{\bar{0}, \bar{1}, \bar{2}\}$ using the following isomorphism $\Phi$:
\[
\Phi(R)=\bar{0},\quad \Phi(P)=\bar{1},\quad \Phi(S)=\bar{2}.
\]
Therefore \eqref{eq:uRSP} can be re-written as follows.
\[
u_{1}(x,y) =
\begin{cases}
1 & \text{if $y*x^{-1} = S$,}\\
1/2 & \text{if $y*x^{-1} = R$,}\\
0 & \text{if $y*x^{-1} = P$.}
\end{cases}
\]
Thus rock-scissors-paper is a game with payoffs of the form \eqref{eq:uphieta}.
The game can be generalized to a countable setting as follows.
Let the strategy set of each of the two players be the group $G$ equal to $[0,1] \cap \mathbb{Q}$ equipped with the sum $\mod 1$ and let, for $0 < \alpha < \beta < 1$, the payoff function be
\[
u_{1}(x,y) =
\begin{cases}
1 & \text{if $\beta < y-x < 1$,}\\
1/2 & \text{if $\alpha \le y-x \le \beta$, or $y-x=0$,}\\
0 & \text{if $0 < y-x < \alpha$.}
\end{cases}
\]
Combining Theorem~\ref{th:generaleta} and Proposition~\ref{pr:lambda01} we can show that every pair of invariant means is an equilibrium.
To wit, the proof of Theorem~\ref{th:generaleta} shows that $(\bar{\lambda}_{1}, \bar{\lambda}_{2})$ is an equilibrium if
\begin{enumerate}[(a)]
\item\label{it:RSP-a}
$\bar{\lambda}_{2} \in \mathcal{I}(G)$,
\item\label{it:RSP-b}
$\widetilde{\lambda}_{1} \in \mathcal{I}(G)$, where $\widetilde{\lambda}_{1}(A)=\bar{\lambda}_{1}(-A)$,
\item\label{it:RSP-c}
$\widetilde{\lambda}_{1} \in \arg \max_{\lambda \in \mathcal{I}(G)} \int \phi_{1} \ \mathrm{d} \lambda$, where
\[
\phi_{1}(x) = \mathds{1}_{(\beta, 1)}(x) + \frac{1}{2} \mathds{1}_{[\alpha,\beta]}(x) + \frac{1}{2} \mathds{1}_{\{0\}}(x),
\]
\item\label{it:RSP-d}
$\bar{\lambda}_{2} \in \arg \max_{\lambda \in \mathcal{I}(G)} \int (1-\phi_{1}) \ \mathrm{d} \lambda$.
\end{enumerate}
Proposition~\ref{pr:lambda01} shows that every invariant mean satisfies \ref{it:RSP-c} and \ref{it:RSP-d}. Furthermore, since $\widetilde{\lambda}_{1} \in \mathcal{I}(G)$ we have, $\bar{\lambda}_{1} \in \mathcal{I}(G)$, too. Therefore every pair of invariant means satisfies \ref{it:RSP-a}--\ref{it:RSP-d} and hence is an equilibrium.
\end{example}
\begin{example}[Love and hate]
This game is played by an even number of players $N=2k$. The strategy set of each player is $\mathbb{Q} \cap [0,1] \mod 1$. The payoff functions have this form for $h \in \{1, \dots, k\}$
\begin{align*}
u_{2h}(x_{1}, \dots, x_{2k}) = -d(x_{2h}, x_{2h+1}), \\
u_{2h+1}(x_{1}, \dots, x_{2k}) = d(x_{2h+1}, x_{2h+2}), \\
\end{align*}
where $N+j := j$ and
\[
d(x,y) = \min(|x-y|,1-|x-y|).
\]
In words, every even player wants to be as close as possible to the following odd player and every odd player wants to be as far as possible from the following even player.
If we define
\[
\eta_{2h}(x) = x, \quad
\eta_{2h+1}(x) = -x,
\]
then the payoffs can be written as
\begin{align*}
u_{2h}(x_{1}, \dots, x_{2k}) &= \phi_{2h}(\eta_{2h}(x_{2h})+\eta_{2h+1}(x_{2h+1})),\\
u_{2h+1}(x_{1}, \dots, x_{2k}) &= \phi_{2h+1}(\eta_{2h+1}(x_{2h+1})+\eta_{2h+2}(x_{2h+2})).
\end{align*}
Combining Theorems~\ref{th:generaleta} and \ref{th:generalNi} we obtain that there exist equilibria that are invariant means for each player.
\end{example}
\begin{remark}
Theorems~\ref{th:generaleta} and \ref{th:generalNi} and especially their combination broaden considerably the class of games for which existence of equilibria can be shown using group-theoretic arguments.
On one hand some of the group games that we have considered are the countable version of finite games that have an equilibrium in uniform strategies, see, e.g., matching pennies and rock-scissors-paper. On the other hand the finite version of some other countable group games does not have an equilibrium in uniform strategies, see, e.g., Wald's game. This shows that the class of countable group games is not a trivial extension of the class of finite group games.
\end{remark}
\section{Uncountable games}\label{se:Uncountable}
In this section we consider games $\mathcal{G}(P,G,u,\nu)$, where $G$ is an uncountable group. Some hypotheses on $G$ will be needed to generalize the results of the previous sections. In particular we will require $G$ to be a locally compact metric group, which means that the group is equipped with a locally compact metrizable topology that is compatible with the group operation.\footnote{As examples of such a group consider for instance $(\mathbb{R}, +)$ and $S^{1}$, the unit circle in $\mathbb{R}^{2}$, where the group operation is the angle sum $\mod 2 \pi$.} In this case, a classical theorem by \citet{Haa:AM1933} guarantees the existence of a unique invariant countably additive measure on $G$. This measure is finite if and only if the group $G$ is compact, therefore in general this is not a mixed strategy. Nevertheless the Haar measure was used by \citet{vNe:FM1929} to define amenability.
Denote by $L^{\infty}(G)$ the Banach space of all real-valued functions on $G$ that are essentially bounded with respect to the Haar measure. Given $g\in G$ and $f\in L^{\infty}(G)$, we denote by $L_{g}f$ and $R_{g}f$ respectively the left- and the right-translation of $f$ by $g$, i.e.,
\[
(L_{g}f)(x)=f(g*x)\quad\text{and}\quad (R_{g}f)(x)=f(x*g).
\]
\begin{definition}\label{defin:amenableuncountable}
A locally compact topological group $G$ is called amenable if there exists a linear operator $T:L^{\infty}(G)\to\mathbb R$ verifying the following properties:
\begin{itemize}
\item[] \textbf{Positivity.} If $f:G\to\mathbb R_{+}$, then $T(f)\geq0$.
\item[] \textbf{Normalization.} If $f \equiv 1$, then $T(f)=1$.
\item[] \textbf{Invariance.} For all $f\in L^\infty(G)$ and for all $g\in G$, one has
\begin{equation}\label{eq:invariantoperator}
T(L_{g}f)=T(f)=T(R_{g}f).
\end{equation}
\end{itemize}
A positive and normalized linear operator $T$ that verifies the first (second) equality in \eqref{eq:invariantoperator} is called \emph{left-invariant (right-invariant) mean}; an operator that is both left- and right-invariant is called \emph{invariant mean}.
\end{definition}
Any positive and normalized linear operator $T:L^{\infty}(G)\to\mathbb R$ defines a finitely additive probability measure $\mu$ on $\mathcal{B}$, the $\sigma$-algebra of Borel subsets of $G$, as follows
\[
\mu(A) = T(\mathds{1}_{A}), \quad\text{for all $A \in \mathcal{B}$}.
\]
Therefore, a locally compact group is amenable if and only if there exists a finitely additive probability measure on $\mathcal{B}$ which is invariant with respect to the group operation.
Now we consider a game $\mathcal{G}(P,G,u,\nu)$, where for every $i \in P$
\[
u_{i}(x_{1}, \dots, x_{N}) = \phi_{i}(x_{1}* \dots * x_{N}),
\]
for some Borel-measurable function $\phi_{i}$ which is assumed to be bounded and integrable with respect to every finitely additive probability measure on $\mathcal{B}$.
The mixed extension of the game is achieved using \eqref{eq:ualpha} as in the countable case.
In order to prove the existence of equilibria we need the following two results. The first is a more general version of F{\o}lner's theorem.
\begin{theorem}[\citet{Fol:MS1955}]\label{th:generalforlner}
A locally compact group $G$ with Haar measure $\mu$ is amenable if and only if there is a sequence of compact subsets $F_n$ of $G$ such that $\mu(F_n)\to\infty$ and
\[
\frac{\mu(F_{n} * (g\triangle F_{n}))}{\mu(F_{n})}\to 0 \quad \text{for all $g\in G$}.
\]
\end{theorem}
The following theorem is folklore, \citep[see, e.g.,][Theorem 4.3]{Mer:ZRB2000}.
\begin{theorem}\label{th:weakstar}
The set of finitely supported probability measures on a metric space is dense, with respect to the weak* topology, in the set of all finitely additive probability measures.
\end{theorem}
We can now state our general existence result for group games.
\begin{theorem}\label{th:maingeneral}
Let $G$ be an amenable, locally compact, metric group such that left-invariant and right-invariant means coincide. Then the game $\mathcal {G}(P,G,u,\nu)$ admits Nash equilibria which do not depend on $\nu$.
\end{theorem}
The proof of Theorem~\ref{th:maingeneral} follows the line of the proof of Theorem~\ref{th:main2}, using Theorems~\ref{th:generalforlner} and \ref{th:weakstar} and is therefore omitted.
The following general version of Paterson's theorem guarantees that the class of groups that satisfy the hypotheses of Theorem~\ref{th:maingeneral} is large.
\begin{theorem}[\citet{Pat:PJM1979}]
An amenable, compactly generated\footnote{A topological group is called \emph{compactly generated} if it is generated by a compact subset; namely, there is a compact subset $K$ of $G$ such that every element $g$ of $G$ can be written in the form $g=k_1*\ldots*k_n$, with $k_i\in K\cup K^{-1}$. For instance, $(\mathbb R,+)$ is compactly generated by the interval $[-1,1]$: every real number $r$ can be written in the form $r= s+\ldots+s$, where $s\in[-1,1]$.}, locally compact group $G$ has the property that right-invariant and left-invariant means coincide if and only if the closure of each conjugacy class is compact.
\end{theorem}
Wald's game can also be played on $\mathbb{R}$ and all games in Subsection~\ref{suse:Q01} can also be played on $[0,1] \mod 1$.
Theorem~\ref{th:maingeneral} guarantees existence of equilibria in this uncountable setting. All results go through with the suitable needed modifications.
\section{Conclusions}\label{se:Conclusions}
We have considered a class of games where the strategy set of each player is a group and the payoff functions depend on the strategies only through the group operation. We have shown that finitely additive equilibria exist for this class of games. In the case of countable groups we have not used any topological conditions, just the algebraic structure of the payoffs. The only measure theoretical assumption refers to the selection of the product of finitely additive mixed strategies.
Even if the algebraic condition that we use leaves out a huge set of games, it includes cases that are not covered by
\citet{Mar:IJGT1997}. In fact in general the payoff functions that we considered are not measurable with respect to the algebra generated by the cylinders. Indeed, his payoff functions satisfy Fubini's theorem \citep[Proposition~3]{Mar:IJGT1997}, whereas ours in general do not. For instance take the two-person zero-sum game on $(\mathbb{Z}, +)$ where $u_{1}(x,y)=\mathds{1}_{\mathbb{N}}(x+y)$. This function is clearly not measurable with respect to the algebra generated by the cylinders, although it is measurable with respect to the $\sigma$-algebra generated by the cylinders. Measurability assumptions on the payoff functions are crucial to prove Marinacci's theorems, whereas we based our proofs on the algebraic properties of the payoffs.
As \citep[Example~2.1]{Sti:GEB2005} shows, the games that we considered in general are not nearly compact and continuous as the ones in \citet{HarStiZam:GEB2005}.
\citet{Sti:GEB2005}
proves very general deep existence results, that are typically non-constructive. In our paper we characterize the equilibrium strategies in a simple form.
\citet{CapMor:IJGT2012} prove that invariant means are minmax strategies for zero-sum two-person games when the set of allowed strategies is restricted so that the exchange of the order of integration is possible. In our paper we do not put any restriction on the set of mixed strategies. Our results are not only more general, they also requires different tools for their proof.
\citet{CanMenOzdPar:MOR2011} show that any finite game can be decomposed into three components, a potential, a nonstrategic, and a harmonic component. The last component has the property that a profile of uniform strategies is always an equilibrium. We notice that, even if group games share the same property, they are neither a subclass nor a superclass of the class of harmonic games. For instance, if $\phi_{1}= \dots = \phi_{N}$ the group game is a potential game and therefore cannot be harmonic. On the other hand in a harmonic game the strategy sets for the different players are not necessarily the same.
We note that the class of group games is a subspace in the class of all games and that the \citet{CanMenOzdPar:MOR2011} decomposition holds in this subspace by projection.
\subsection*{Acknowledgments}
The authors thank Marco Dall'Aglio for sparking their interest in group games, Patrizia Berti and Pietro Rigo for their useful comments about finitely additive probability measures, Alain Valette for helpful discussions about amenability, Kent Morrison for pointing out a mistake in a proof, and Ozan Candogan for his insights on harmonic games.
\bibliographystyle{artbibst}
|
{
"timestamp": "2013-09-18T02:03:34",
"yymm": "1203",
"arxiv_id": "1203.2301",
"language": "en",
"url": "https://arxiv.org/abs/1203.2301"
}
|
\section{Introduction}
The first molecule discovered in space was the CH radical (Dunham 1937). At
that time, the first predictions were made regarding the abundances of
molecules in interstellar space, identifying the hydroxyl radical (OH) as a
promising candidate (Swings \& Rosenfeld 1937). It took more than two
decades until the latter was detected in the {\sc ism} (Weinreb et al.
1963), in absorption against the supernova remnant Cas A, in the $F= 1 - 1$
and $2 - 2$ transitions between the hyper-fine structure split
$\Lambda$-double levels of OH's $^2\Pi_{3/2}, J = 3/2$ rotational ground
state. Comprehensive models of the gas phase chemistry of diffuse
interstellar clouds constructed by van Dishoeck \& Black (1986) revealed
the importance of the OH radical in the network of reactions leading to the
formation of oxygen-bearing molecules.
Unfortunately, OH column densities in these objects are difficult to
determine from radio lines, due to frequently observed deviations of the
underlying level population from LTE (e.g., Neufeld et al. 2002). The highest
densities occurring in diffuse clouds amount to $\sim$10$^4\,{\rm cm}^{-3}$
(Greaves \& Williams 1994), while their mean density is
$\sim$10$^2\,{\rm cm}^{-3}$ (Snow \& McCall 2006; Cox et al. 1988 derived upper
limits of a few thousand ${\rm cm}^{-3}$).
Storey et al. (1981) first detected the $\Lambda$ doublet line from the
OH ground state ($^2\Pi_{3/2}\,J=5/2 \leftarrow 3/2$) towards Sgr~B2 near the
Galactic centre with the Kuiper Airborne Observatory, but their spectral
resolution (250~\kms) proved inadequate to separate the absorption by the
line-of-sight clouds from that occurring in Sgr~B2 itself. Here we report the
detection of one doublet line of this transition (at 2514~GHz) and of its
isotopolog $^{18}$OH (at 2495 GHz) with GREAT\footnote{GREAT is a
development by the MPI f{\"u}r Radioastronomie and the KOSMA/Universit{\"a}t
zu K{\"o}ln, in cooperation with the MPI f{\"u}r Sonnensystemforschung and the
DLR Institut f{\"u}r Planetenforschung.} onboard SOFIA, in absorption towards
the giant H{\sc ii} regions W49N and W51 and the ultracompact H{\sc ii} region
G34.26+0.15. Both lines are inaccessible for Herschel/HIFI. The three observed
lines of sight are within a $15^\circ$ wide Galactic longitude interval. Those
towards W51 and G34.26+0.15 cross the near side of the Carina-Sagittarius
spiral arm, with W51 ($\ell = 49\fdg5$) at a distance of
5.41~($+0.31,\,-0.28)$ kpc (Sato et al. 2010) and G34.26+0.15 at $\sim$2~kpc
distance (cf. measurements of G35.10-0.74, which has a comparable radial
velocity, Zhang et al. 2009).
The line of sight to W49N ($\ell = 43\fdg 2$) first crosses the near side of the Carina-Sagittarius arm, grazes the Crux-Scutum arm, and then again crosses
the Carina-Sagittarius arm on its far side, where W49N is located at a
distance of $(11.4 \pm 1.2)$\,kpc (Gwinn et al. 1992).
\section{Observations, data reduction and analysis}
The observations reported here were performed with the GREAT receiver
(Heyminck et al. 2012) onboard the SOFIA airborne observatory (Young et
al. 2012), as part of the {\it basic science} programme (flights on 2011 July 26
and November 8). On the first flight the receiver's M and L2 bands were tuned to
2514.317~GHz for the $^2\Pi_{3/2}\,J=5/2\leftarrow 3/2$ group of OH hyperfine
structure lines (in the lower sideband) and to 1837.817 GHz for the
$^2\Pi_{1/2}\,J=3/2 \leftarrow 1/2$ lines (in the upper sideband), respectively.
On the second flight, the M band was also tuned to the 2494.695~GHz frequency
of the $^2\Pi_{3/2}\,J=5/2\leftarrow 3/2$ transition of $^{18}$OH.
\begin{table}
\caption[]{List of the observed $^2\Pi_{3/2} J=5/2 \leftarrow 3/2$ $^{16}$OH and $^{18}$OH lines}
\begin{tabular}{lll}
\hline\hline
\noalign{\smallskip}
Transition & Frequency [GHz]$^{\rm (a)}$ & A$_{\rm E}$ [s$^{-1}$]$^{\rm (b)}$ \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\multicolumn{3}{c}{OH, $^2\Pi_{3/2}, J=5/2 \leftarrow 3/2$} \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
$F=2^- \leftarrow 2^+$ & 2514.298092 & 0.0137 \\
$F=3^- \leftarrow 2^+$ & 2514.316386 & 0.1368 \\
$F=2^- \leftarrow 1^+$ & 2514.353165 & 0.1231 \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\multicolumn{3}{c}{$^{18}$OH, $^2\Pi_{3/2}, J=5/2 \leftarrow 3/2$}\\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
$F=2^+ \leftarrow 2^-$ & 2494.68092 & 0.0136 \\
$F=3^+ \leftarrow 2^-$ & 2494.69507 & 0.1356 \\
$F=2^+ \leftarrow 1^-$ & 2494.73421 & 0.1221 \\
\noalign{\smallskip}
\hline
\end{tabular}
\flushleft{$^{\mathrm{(a)}}$ Varberg \& Evenson (1993,
$\sigma_{\rm rms}=30$\,kHz). The frequencies for $^{18}$OH were derived by
isotope scaling. $^{\mathrm{(b)}}$ Pickett et al. (1998).}
\label{tab:lines}
\end{table}
Typical DSB receiver temperatures were 4500~K and 2500~K for the
M and L2 bands, respectively. Total power subtraction was performed by chopping
with an amplitude of $90''$ at 1~Hz. The raw data were converted from
XFFT spectrometer (Klein et al. 2012) count rates to forward-beam
brightness temperatures with the module {\it kalibrate} (Guan et al. 2012)
as part of the {\it kosma\_software} observing software, analysing the data from
the calibration loads and the atmospheric total power and allowing us to fit
both the wet component (typically pwv=10-20~$\mu$m) and dry content of the
atmospheric emission and thus to determine the opacity correction (a few
$\times 0.1$). Further processing of the data (conversion to main-beam
brightness temperature, with a beam efficiency of 0.58, and averaging with
$1/\sigma_{\rm rms}^2$ weighting) was made with the {\sc class}
software. The overall calibration uncertainty does not exceed 20\%.
\begin{figure}
\centering
\includegraphics[angle=-90,width=7.6cm]{levelDiagram.eps}
\caption{Level diagram (not to scale) for the $^2\Pi_{3/2}$ OH ground
and first excited state. The observed $^{16}$OH transitions are
indicated by bold arrows and labelled with their corresponding
frequency. The observed $^{18}$OH transitions are drawn as grey
arrows.}
\label{fig:levelDiagram}
~\vspace{-0.5cm}
\end{figure}
\begin{table}
\caption[]{Results of the OH line profile fitting.}
\begin{tabular}{lcccc}
\hline\hline
\noalign{\smallskip}
Source & $\upsilon_0^{\rm (a)}$ & {\sc fwhm}$^{\rm (b)}$ &
$\tau_{\rm max}^{\rm (c)}$ & N$_{\rm OH}^{\rm (d)}$ \\
\noalign{\smallskip}
$(T_{\rm mb,c})$ & \multicolumn{2}{c}{\hrulefill~[\kms]~\hrulefill} & & [$10^{14}$~cm$^{-2}$] \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
W49N & $2.9 \pm 1.9 $ & $13.4 \pm 0.4 $ & 0.9 &
\multirow{2}{*}{ $8.8 \pm 4.6$ } \\
(12.3~K)
& $12.3 \pm 1.0 $ &$10.1 \pm 1.1 $ & 10.1 & \\
& $36.8 \pm 3.5 $ &$ 11.8\pm 4.4 $ & 1.8 & $ 1.6 \pm 0.9 $\\
& $39.1 \pm 1.3 $ &$ 2.8 \pm 0.5 $ & 2.3 & $ 0.5 \pm 0.1 $ \\
& $60.5 \pm 0.3 $ &$10.4 \pm 3.9 $ & 2.5 & $ 2.0 \pm 0.2 $ \\
W51 & $7.2 \pm 0.4 $ &$ 4.6\pm 2.2 $ & 2.9 & $ 0.66 \pm 0.08$ \\
(8~K) & $59.4 \pm 3.3 $ &$16.8\pm 4.5 $ & 5.2 &
\multirow{2}{*}{$5.6 \pm 2.1 $} \\
& $68.2 \pm 1.4 $ & $ 4.6 \pm 2.7 $ & 1.9 & \\
G34.26+0.15 & $12.6 \pm 0.2 $ & $ 5.7 \pm 0.8 $ & 2.5 & $ 1.12 \pm 0.21$\\
(9~K) & $28.3 \pm 0.5 $ & $ 6.0 \pm 1.6 $ & 1.4 & $ 0.67 \pm 0.09$\\
& $55.4 \pm 0.4 $ & $12.9 \pm 1.0 $ & 4.8 & $ 4.73 \pm 1.20$\\
\noalign{\smallskip}
\hline
\end{tabular}
\begin{list}{}{}
\item[$^{\mathrm{(a)}}$] {\sc lsr} velocity of component. The first and third
hyperfine component (hfc) are $+2.2$ and $-4.4$\,\kms, respectively, off the
second one.
\item[$^{\mathrm{(b)}}$] {\sc fwhm} of Gaussian absorption profile, deconvolved
from hfc split.
\item[$^{\mathrm{(c)}}$] Peak opacity in the strongest hfc. The saturated
absorption towards W49N (\vlsr $< 25$~\kms) provides only a lower opacity limit.
\item[$^{\mathrm{(d)}}$] Column density per fitted velocity component.
\end{list}
\label{tab:results}
~\vspace{-1.2cm}
\end{table}
Thanks to the high critical density of the $J=5/2 \rightarrow 3/2$ transition
($5.1\times 10^9\,{\rm cm}^{-3}$ for a 15~K gas, collision coefficients
from Dewangan et al. 1987, Einstein coefficients as given in
Tab.~\ref{tab:lines}), we can safely expect almost all OH to be in its ground
state at the density of the foreground diffuse clouds along the sight-line.
This makes the determination of column densities much more reliable
than those derived from the $\lambda$\,18~cm line. Likewise, the
$^2\Pi_{1/2}\,J = 1/2$ level (64~K above the $^2\Pi_{3/2}\,J=5/2$ level) cannot
be substantially populated either, because we failed to detect absorption in
the simultaneously observed $^2\Pi_{1/2}\,J = 3/2 \leftarrow 1/2$ transition.
The spectral profile, in absence of emission, is thus is given by
\begin{equation}
T_{\rm mb}(\upsilon) = T_{\rm mb, c}
\exp{\left ( -\sum_{i=1}^{N_{\rm vc}}\sum_{j=1}^{N_{\rm hfc}} \tau_\mathrm{ ij, \upsilon} \right) }\,,
\label{eq:radtran}
\end{equation}
where $T_{\rm mb}$ and $T_{\rm mb, c}$ are the main beam brightness
Rayleigh-Jeans temperatures of the spectral profile (here as a function of
velocity) and of the continuum (in single-sideband calibration), respectively,
and $N_{\rm vc}$ and $N_{\rm hfc}$ are the number of velocity components and
hyperfine components, respectively. While uncertainties in the calibration
temperature cancel out in the opacity determination, any residual offset of
variance $\sigma^2_{\rm rms, T{\rm c}}$ in the definition of the continuum level
leads to an additional uncertainty in the derived opacity of
$\sigma_{\rm rms, \tau}=\sigma_{\rm rms, T_c}/T_{\rm c}$. The absorption
spectra suggest $\sigma_{\rm rms, \tau} \sim$0.1, which is tolerable in view
of the substantial opacities. A simultaneous least-squares fit to the line
profiles of all velocity components (Eq.~\ref{eq:radtran}) with the opacity
\begin{equation}
\tau_\mathrm{ij, \upsilon} =
\sqrt{\frac{\ln{2}}{\pi}} \frac{A_{\mathrm E,j}c^3}{4\pi\Delta \upsilon_{\rm i}
\nu_{\rm j}^3}
\frac{g_{\rm u,j}}{g_{\rm l,j}} N_{\OH} w_\Lambda
\exp{
\left(-4\ln{2}\left(\frac{\upsilon-\upsilon_{0,ij}}
{\Delta \upsilon_i}\right)^2\right)
}
\end{equation}
yields $N_{\OH}$, the OH column density per velocity component. Here
$\Delta \upsilon_{\rm i}$ is the {\sc fwhm} of the Gaussian component
$i$, $g_{\rm u,j}$ and $g_{\rm l,j}$ are the statistical weights
($2F+1$) of the upper and lower level, respectively, of a given hyperfine
component $j$, $\upsilon_{\rm 0, ij}$ is the offset of its velocity from the
line-of-sight velocity of the source, and $w_\Lambda = 0.5$ corrects for the
fact that only one doublet line was observed (Fig.~\ref{fig:levelDiagram}).
Owing to the relatively large number of free parameters ($3N_{\rm vc}$), a
simulated annealing method (Metropolis algorithm) was used in combination with
a downhill simplex method (Press et al. 1992). The former assists the
minimisation process in escaping from a local minimum, and the latter improves
the efficiency of the convergence. The velocity structure in the
para-H$_2$O $1_{11} - 0_{00}$ spectrum towards W49N (Sonnentrucker et al. 2010)
suggests five velocity components as a strict minimum, while with more
components, the procedure would start to fit noise features. The results are
summarised in Tab.~\ref{tab:results}. The OH column density can be expressed by
the relationship
$N_\OH\,\,[\mathrm{cm}^{-2}] = 7.8\times 10^{12} \tau_{\rm max} \Delta \upsilon_{\rm fwhm}$\,
[\kms] as a function of the opacity in the strongest hyperfine component and of
the width of the absorption profile of the deconvolved spectrum. The uniqueness
of the solution was tested with a Monte Carlo study, yielding the standard
deviation of each parameter.
\section{Results}
For the spiral arm clouds there is no ambiguity in the fit results
(Fig.~\ref{fig:ohspectra}). Towards the three continuum sources the absorption
is saturated, i.e., in the $(5,20)$, $(40,80)$ and $(35,70)$~\kms~velocity
intervals for W49N, W51 and G34.26+015, respectively, where the derived main
line opacities and column densities are to be considered lower limits. For W49N,
this caveat is corroborated by the $^{18}\OH$ absorption profile
(Fig.~\ref{fig:w49n_18oh}). The $^{18}\OH/^{16}\OH$ abundance ratio is not
expected to be affected by chemical fractionation (Langer et al. 1984). The
synthesised opacity in the W49N spectrum peaks at $\tau$$=5.7$. Assuming for
the $^{18}\mathrm O/^{16}\mathrm O$ ratio the value in the 4~kpc
ring ($327 \pm 32$, Wilson \& Rood 1994; Polehampton et al. 2005 found no
evidence of an abundance gradient with galactocentric distance), the estimated
opacity in our $^{18}\OH$ detection ($\tau$$\sim$0.2) would require the main
line opacity to be higher by at least an order of magnitude with respect to
that estimated by the absorption profile fit. Unfortunately, in the
30-40\,\kms~velocity range the $^{18}\OH$ absorption is affected by a telluric
ozone feature, and a confirmation by observations of sources with a more
favourable velocity is planned to definitely rule out a baseline ripple.
In the unsaturated wings of the absorption profile, the
sensitivity of the corresponding $^{18}$OH measurement is no longer sufficient
to estimate the $^{18}\OH/\OH$ abundance ratio. A two-component fit
to the $^{18}$OH absorption (Fig.~\ref{fig:w49n_18oh}) yields a column density
of $4\times 10^{13}$\,cm$^{-2}$ for the whole absorption feature.
\begin{table*}[ht!]
\caption{OH and $\HHO$ abundances and their ratios in line-of-sight clouds
towards W49N and W51. Statistical errors are given in brackets.}
\centering
\begin{tabular}{llcccccccc}
\hline\hline
\noalign{\smallskip}
& & \multicolumn{5}{c}{\hrulefill~W49N \vlsr~intervals [\kms]}~\hrulefill &
\multicolumn{3}{c}{W51 \vlsr~intervals [\kms]} \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
& & (30,37) & (37,44) & (44,49) & (49,54) & (54,72) & ($-1$,11) & (11,16)$^{(c)}$&(43,50)\\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
$N_\OH$ & $[10^{13}\,\mathrm{cm}^{-2}]$ &
7.1(0.5) &11.3(6.3) & 1.2(0.2) & 1.5(0.3) & 18.1(1.3) & 6.4(0.9) & $-$ & 3.1(0.6) \\
$N_\HHO^{\rm (a)}$ & $[10^{13}\,\mathrm{cm}^{-2}]$ &
2.3(0.1) & 6.2(0.6) & 0.1(0.05) & 1.5(0.07) & 11.6(0.4) & 2.6(0.1) & 0.23(0.05)&2.5(0.1) \\
$N_\HH^{\rm (b)}$ &$[10^{20}\,\mathrm{cm}^{-2}]$ &
5.9(1.9) & 6.6(2.0) & 0.9(0.03)&3.2(0.1) &22.6(3.7) & 4.3(0.1) & 0.6(0.03) &
2.6(0.8) \\
$[\OH]/[\HH]$& $[10^{-8}]$ & 12.0(4.0) & 17(11) & 13.2(2.5) & 4.5(0.9) & 8.0(1.4) & 15(2.1)& $-$ & 11.9(4.3) \\
$[\HHO]/[\HH]$ & $[10^{-8}]$ & 3.9(1.3) & 9.4(3.0) & 1.1(0.6) & 4.7(0.3) & 5.1(0.9) &6.0(0.3)& 3.8(0.9) & 9.6(3.0) \\
$[\HHO]/[\OH]$ & & 0.32(0.03)& 0.55(0.31) & 0.08(0.04) & 1.03(0.20) & 0.64(0.05) &
0.40(0.06) & $-$ & 0.81(0.16) \\
\noalign{\smallskip}
\hline
\end{tabular}
\flushleft $^{\rm (a)}$ Sonnentrucker (2010), $^{\rm (b)}$ Godard et al. (2012). $^{\rm (c)}$ The sensitivity of the OH observations is not sufficient for this
velocity interval.
\label{tab:oh_h2o}
\end{table*}
Although the spiral arm clouds along the sight-line exhibit substantial
opacities on the order of unity, the absorption is not saturated and OH column
densities can be derived whose accuracy only depends on the signal-to-noise
ratio, the quality of the fitted profile,
and the assumed continuum level. For a comparison of our OH column densities
with those observed for $\HHO$ (Sonnentrucker et al. 2010) and inferred for
H$_2$ (Godard et al. 2012), the absorption profiles of the clouds towards W49N
and W51 are integrated within the velocity intervals of Tab.~\ref{tab:oh_h2o}.
With the exception of a velocity interval with an abnormally low water
abundance, probably due to a spectral baseline problem, the $\HHO/\OH$ ratios
are in the range 0.3-1.0. Plume et al. (2004) have determined the $\HHO/\OH$ ratio by a comparison of submillimeter
$\HHO$ observations with ground-based radio observations of the 18~cm
transitions within the ground rotational state of OH. They thereby
estimated a $\HHO/\OH$ ratio of 0.4 at \vlsr=68~\kms, in agreement with our
measurement of 0.6. The ratio of
$\sim$0.3 measured by Neufeld et al. (2002) towards W51, at \vlsr=6~\kms,
compares to our value of 0.4 in the \vlsr=($-1$,11)~\kms~interval. Their OH
column density is compatible with our value
($8\times10^{13}\,\mathrm{cm^{-2}}$ and $6.4\times10^{13}\,\mathrm{cm^{-2}}$,
respectively). Observations of the $^{2}\Pi_{1/2}\leftarrow$~$^{2}\Pi_{3/2}$
cross-band transitions towards Sgr~B2 (Polehampton et al. 2005) suggest a
$\HHO/\OH$ range of 0.6-1.2. Generally, discrepancies between different sets of
data may be due to {\sc nlte} effects in the radio lines, different spectral
resolutions and definitions of velocity components, and uncertainties in the
definition of the respective continuum levels.
\begin{figure}[h!]
\centering
\includegraphics[angle=-90,width=7.6cm]{w49n.eps}
\includegraphics[angle=-90,width=7.6cm]{w51.eps}
\includegraphics[angle=-90,width=7.6cm]{g34.eps}
\caption{{\bf Top:} OH absorption towards W49N. The velocities of the
line-of-sight clouds and of W49N are indicated. The red dashed
line is a least-squares fit. {\bf Middle:} Same for the OH
absorption against W51e4. The relative positions and strengths of
the hfc splitting are indicated for the \vlsr=7~\kms~component.
{\bf Bottom:} OH absorption against G34.26+0.15.}
\label{fig:ohspectra}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[angle=-90,width=7.6cm]{w49n_18oh.eps}
\caption{$^{18}$OH absorption (top, grey-shaded), a least-squares,
two-component fit to it (dashed line) and OH absorption towards
W49N. The spectra are scaled by the corresponding continuum
level, to facilitate a comparison. The insert at the top shows a
telluric ozone feature (as observed in total power), where the
calibration is more uncertain.}
\label{fig:w49n_18oh}
\end{figure}
The chemistry leading to interstellar OH and water has been
considered in many theoretical studies over the past thirty years
(e.g. Draine et al. 1983; van Dishoeck \& Black 1986;
Hollenbach et al. 2009, and references therein). Three main pathways
to OH have been identified in diffuse and translucent molecular
clouds. The first pathway involves an ion-molecule chemistry,
initiated by the cosmic-ray ionization of $\HH$ or H. The resulting $\HHplus$
and $\Hplus$ ions can lead to $\OHplus$ through the reaction sequences
$\HHplus(\HH,\H)\HHHplus(\O,\HH)\OHplus$ or
$\Hplus(\O,\H)\Oplus(\HH,\H)\OHplus$. In clouds with a low molecular fraction,
the resulting $\OHplus$ is destroyed primarily by dissociation recombination
with electrons. In clouds with a high molecular fraction, however, $\OHplus$ is
rapidly converted to $\HHHOplus$ by a series of two H atom abstraction
reactions:
$\OHplus(\HH,\H)\HHOplus(\HH,\H)\HHHOplus$. The $\HHHOplus$ ion then undergoes
dissociative recombination with electrons to form OH or $\HHO$. The branching
ratio for this process is important in determining the resultant $\OH/\HHO$
ratio and has been studied in two recent ion storage ring experiments (Jensen
et al. 2000; Neau et al. 2000): these suggest that $\sim$74\% to 83\% of
dissociative recombinations lead to OH, with almost all the remainder
leading to $\HHO$ (and less than $\sim$1\% resulting in the production of O).
In diffuse or translucent clouds, both neutral molecules are destroyed by
photodissociation, which - in the case of $\HHO$ - is an additional formation
process for OH. A second and different pathway may be important in shocks or
turbulent dissipation regions, where elevated gas temperatures can drive a
series of neutral-neutral reactions with significant energy barriers:
$\O(\HH,\H)\OH(\HH,\H)\HHO$. Finally, OH and $\HHO$ may be produced by means
of a grain-surface chemistry, in which O nuclei are hydrogenated on grain
surfaces and subsequently photodesorbed. The relative importance of these three
pathways will determine the exact $\HHO/\OH$ abundance ratio, but all three
predict a close relationship between OH and $\HHO$. This relationship is
supported by the observations reported here, which indicate a good
correspondence between the OH and $\HHO$ absorption features; detailed
modelling, which must await a larger sample of sight-lines, will be
needed to interpret observed variations in the $\HHO/\OH$ ratio.
We note, however, that the lower end of the observed range of $\HHO/\OH$ ratios
(0.3-1.0) is predicted by models for turbulent chemistry (Godard et al. 2009).
We note also that the observed distribution of OH is quite different from that
of $\OHplus$; the latter is believed to arise primarily in material with a
molecular fraction that is too low to permit the efficient
production of $\HHHOplus$, whereas the former will arise in clouds with a
substantial abundance of $\HH$.
Future data of the OH ground state transition and the relatively high precision
of the resulting column densities will not only allow us to assess the
correlation between the abundances of OH and $\HHO$, but also to
re-calibrate less accurate OH column densities derived from decades of radio
observations.
\begin{figure}
\centering
\includegraphics[angle=-90,width=7.6cm]{w49n_oh_h2o.eps}
\includegraphics[angle=-90,width=7.6cm]{w51_oh_h2o.eps}
\caption{Para-$\HHO$ absorption (dashed red line; from Sonnentrucker et
al. 2010) and OH absorption (solid line), normalised by the
respective single sideband continua, towards W49N (top) and W51
(bottom).}
\label{fig:oh_h2o}
\end{figure}
\begin{acknowledgements}
Based on observations made with the NASA/DLR Stratospheric
Observatory for Infrared Astronomy. SOFIA Science Mission Operations are
conducted jointly by the Universities Space Research Association, Inc., under
NASA contract NAS2-97001, and the Deutsches SOFIA Institut under DLR contract
50 OK 0901. We {\sc great}fully acknowledge the support by the observatory staff
and a helpful referee report.
\end{acknowledgements}
|
{
"timestamp": "2012-04-13T02:02:54",
"yymm": "1203",
"arxiv_id": "1203.1744",
"language": "en",
"url": "https://arxiv.org/abs/1203.1744"
}
|
\section{Introduction}\label{section-Introduction}
The theoretical description of strongly interacting many-particle quantum
systems in condensed matter physics is a difficult undertaking in general.
Even in those cases where it is possible to derive a Hamiltonian which
contains all relevant parts for a complete description of the many-particle
system under consideration, a complete solution of the dynamics or an exact
evaluation of the ground state is often out of reach.
However, essential features such as phase transitions or quasi-particle
excitations are already contained in drastically simplified models.
One of the most studied models which describes strongly interacting
electrons in a solid is the Fermi-Hubbard model \cite{H63,H64a,H64b}.
Although it describes only the physics of a single energy band without
long-ranged Coulomb-interactions, it is believed to exhibit various
interesting phenomena such as the metal-insulator transition,
anti-ferromagnetism, ferrimagnetism, ferromagnetism, Tomonaga-Luttinger
liquid, and superconductivity \cite{T98}.
The Fermi-Hubbard model was first established by J.~Hubbard in order to
describe correlations of electrons in narrow energy bands \cite{H63,H64a,H64b}.
It connects the Heitler-London theory of strongly interacting electrons on
one side and the band theory, valid for weak interactions, on the other side.
Apart from electrons in solids, there are also other physical systems which
can be described (within suitable approximations) by the Fermi-Hubbard model,
for example ultra-cold atoms in optical lattices.
Since optical lattice systems allow for a tuning of the model parameters
over a wide range (in contrast to most condensed matter systems), they are
predestinated for the study of the fermionic Hubbard model.
This has been experimentally realised by trapping fermionic Potassium atoms
in an optical lattice \cite{KMS05,JSG08,SHW08}.
By increasing the on-site interaction among the atoms, the transition from
a metallic phase to a Mott insulating phase was deduced from compressibility
measurements and in situ imaging \cite{SHW08}.
After replacing the fermionic by bosonic atoms, the optical lattice system
corresponds to the Bose-Hubbard model, which describes interacting bosons
in periodic potentials.
The Bose-Hubbard model was first motivated by experimental realisations
such as Helium-4 absorbed in porous media or Cooper pairs in granular
media \cite{FWGF89}.
Within the last decade, the Bose-Hubbard model has been attracting
increasing attention due to its experimental realisation with interacting
bosons, for example Rubidium atoms, in optical lattices \cite{G02,B10}.
The phase diagram, which is much better understood than in the fermionic
case, contains (for vanishing disorder) a superfluid regime and Mott
insulating phases.
The transition between the two phases is characterised by a natural order
parameter, such as the superfluid density.
This second-order quantum phase transition, which results from the
competition between kinetic energy and on-site interaction, has been
observed for bosons in optical lattices \cite{G02}.
Though at first glance seemingly very simple, the fermionic as well as the
bosonic lattice models are not exactly solvable in general, the only
exceptions being the Fermi-Hubbard model in one dimension \cite{LW68,EFGKK05}
and trivial situations, such as the case of vanishing or infinitely strong
interaction, or an empty lattice, for example.
Thus, various analytical and numerical techniques have been developed
to study their physical properties.
Widely used numerical methods, ignoring distance-dependent quantum
correlations between different lattice sites, are mean-field approaches
such as the dynamical mean-field theory (DMFT) \cite{AP98,GK96}.
Exact diagonalisation (analytical or numerical) is only possible for small
system sizes \cite{LHS88,KLA07}.
Monte Carlo methods can improve the situation \cite{SBG91}, but they can
sample only a small part of the whole Hilbert space and have problems with
general time-dependent situations.
An important and widely used analytical approach is the Gutzwiller
ansatz \cite{G63} -- for the bosonic and the fermionic case.
However, a serious drawback of this Gutzwiller mean-field approach is again
the neglect of distance-dependent quantum correlations between different
lattice sites.
This simplification leads sometimes to unphysical behaviour --
for example, the
Mott-insulator state would not react to an external force within the
Gutzwiller mean-field approach.
The main goal of the present work is to study these non-local quantum
correlations in the Bose and Fermi-Hubbard models.
To this end, we develop an analytical expansion in powers of the inverse
coordination number $Z$ (i.e., the number of tunnelling partners of a
given lattice site).
For comparison, we also calculated these non-local quantum correlations
by means of exact (numerical) diagonalisation for one and two-dimensional
lattices.
Although exact diagonalisation is possible only for small systems, the
results are in good agreement with those obtained by the analytical
expansion and the Density Matrix Renormalisation Group technique (DMRG)
\cite{CBPE12,E11}.
The paper is organised as follows.
After a brief introduction into the Bose-Hubbard model in
Sec.~\ref{section-Bose-Hubbard-Model}, we present our analytical method
for general Hubbard type Hamiltonians in Section~\ref{hierarchyofcorr}.
With this method, the ground state properties of the bosonic Mott
insulator phase are derived in Sec.~\ref{mottsection}.
Taking this Mott insulator as the initial state, the temporal evolution of
the correlations after a quantum quench to the superfluid regime as well
as within the Mott phase are studied in Sections~\ref{superquench} and
\ref{equilibration}.
In Section~\ref{tiltlattice}, we investigate particle-hole pair creation
out of the (bosonic) Mott state induced by a weak tilt of the lattice and
establish a quantitative analogy to electron-positron pair creation due to
a strong electric field (Sauter-Schwinger effect) in
Sec.~\ref{section-Analogue}.
The results of Secs.~\ref{mottsection}--\ref{section-Analogue} are derived
in first order $1/Z$ (where $Z$ is the coordination number).
In Section~\ref{Z2}, we show how to extend our analytical method to higher
orders in $1/Z$.
Subsequently, we compare our analytical results with exact numerical studies
of finite bosonic lattices in Section~\ref{numerical} and find qualitative
agreement.
In the second part of the paper, we consider similar problems for fermions.
After a brief introduction into the Fermi-Hubbard model in
Section~\ref{Fermi-Hubbard Model}, we discuss its ground state correlations
(in the fermionic Mott state) and quench dynamics Secs.~\ref{chargemodes}
and \ref{Fermi-Mott}.
Section~\ref{fermitilt} is then devoted to particle-hole pair creation
induced by a weak lattice tilt.
Finally, we address resonant tunnelling in the Bose and Fermi-Hubbard
model due to a large lattice tilt in Section \ref{restun}.
\section{Bose-Hubbard Model}\label{section-Bose-Hubbard-Model}
The Bose-Hubbard model is one of the most simple and yet non-trivial models
in condensed matter theory {\cite{FWGF89,J98,Z03}.
It describes identical bosons hopping on a lattice with the tunnelling
rate $J$.
In addition, two (or more) bosons at the same lattice site repel each other
with the interaction energy $U$.
The Hamiltonian reads
\begin{eqnarray}
\label{Bose-Hubbard-Hamiltonian}
\hat H
=
-\frac{J}{Z}\sum_{\mu\nu} T_{\mu\nu} \hat b^\dagger_\mu \hat b_\nu
+\frac{U}{2}\sum_{\mu}
\hat n_{\mu}(\hat n_{\mu}-1)
\,.
\end{eqnarray}
Here $\hat b^\dagger_\mu$ and $\hat b_\nu$ are the creation and annihilation
operators at the lattice sites $\mu$ and $\nu$, respectively, which obey
the usual commutation relations
\begin{eqnarray}
\left[\hat b_\nu,\hat b_\mu^\dagger\right]=\delta_{\mu\nu}
\;,\,
\left[\hat b_\nu^\dagger,\hat b_\mu^\dagger\right]=
\left[\hat b_\nu,\hat b_\mu\right]=0
\,.
\end{eqnarray}
The lattice structure is encoded in the adjacency matrix $T_{\mu\nu}$
which equals unity if $\mu$ and $\nu$ are tunnelling neighbours
(i.e., if a particle can hop from $\mu$ to $\nu$) and zero otherwise.
The number of tunnelling neighbours at a given site $\mu$ yields the
coordination number $Z=\sum_\nu T_{\mu\nu}$
(we assume a translationally invariant lattice).
Finally, $\hat n_{\mu}=\hat b^\dagger_\mu \hat b_\mu$ is the number
operator and we assume unit filling $\langle\hat n_{\mu}\rangle=1$
in the following.
Note that the total particle number
$\hat N=\sum_\mu \hat n_\mu$ is conserved $[\hat H,\hat N]=0$.
The Bose-Hubbard model is considered {\cite{sachdev}}
one of the prototypical examples for a quantum phase transition.
If the interaction term dominates $U\gg J$, the bosons are pinned to
their lattice sites and we have the Mott insulator state
\begin{eqnarray}
\label{Mott-state}
\ket{\Psi_{\rm Mott}}_{J=0}
=
\bigotimes\limits_\mu\ket{1}_\mu
=
\prod\limits_\mu\hat b_\mu^\dagger\ket{0}
\;\leadsto\;
\hat H\ket{\Psi_{\rm Mott}}_{J=0}=0
\,,
\end{eqnarray}
which is fully localised.
If the hopping rate dominates $U\ll J$, on the other hand, the particles
can propagate freely across the lattice and become completely delocalised
\begin{eqnarray}
\label{superfluid-state}
\ket{\Psi_{\rm superfluid}}_{U=0}
=
\frac{1}{\sqrt{N!}}
\left(\hat b_{\fk{k}=0}^\dagger\right)^N\ket{0}
=
\frac{1}{\sqrt{N! N^N}}
\left(\sum_{\mu}\hat b_\mu^\dagger\right)^N\ket{0}
\,,
\end{eqnarray}
which is the superfluid phase.
Obviously, the Mott state (\ref{Mott-state}) does not have any correlations,
for example
$\langle\hat b^\dagger_\mu \hat b_\nu\rangle_{\rm Mott}=\delta_{\mu\nu}$,
whereas the superfluid state in (\ref{superfluid-state}) shows correlations
across the whole lattice
$\langle\hat b^\dagger_\mu \hat b_\nu\rangle_{\rm superfluid}=1$.
Furthermore, the Mott insulator state is separated by a finite energy gap
from the lowest excited state, while the superfluid state possesses
sound-like modes with arbitrarily low energies
(for an infinitely large lattice $N\uparrow\infty$).
Finally, the Bose-Hubbard model can be realised experimentally
(to a very good approximation) with ultra-cold atoms in optical lattices
\cite{B05,S07,RSN97} and it was even possible to observe the aforementioned
phase transition in these systems \cite{G02}.
In spite of its simplicity, the Bose-Hubbard model
(\ref{Bose-Hubbard-Hamiltonian}) cannot be solved analytically.
Numerical simulations are limited to reduced sub-spaces or
small systems sizes, see Section \ref{numerical} below.
Analytical approaches are based on suitable approximations.
In order to control the error of these approximations, they should be
based on an expansion in term of some large or small control parameter.
For the Bose-Hubbard model (\ref{Bose-Hubbard-Hamiltonian}), one could
consider the limit of large $\langle\hat n_{\mu}\rangle\gg1$
or small $\langle\hat n_{\mu}\rangle\ll1$ filling \cite{SUXF06,FSU08},
for example, or the limit of weak coupling $U\ll J$ or strong coupling
$U\gg J$ \cite{FM94,FM96,DZ06}.
However, none of these limits is particularly well suited for studying
the Mott--superfluid phase transition.
To this end, we consider the limit $Z\gg1$ in the following and employ
an expansion into powers of $1/Z$ as small control parameter.
Note that an expansion in powers of $1/Z$ was also used to derive bosonic
dynamical mean-field equations (which were then solved numerically) in
\cite{HSH09,LBHH11,LBHH12}.
\section{Hierarchy of Correlations}\label{hierarchyofcorr}
Let us consider general Hamiltonians of the form
\begin{eqnarray}
\hat H=\frac1Z\sum_{\mu\nu}\hat H_{\mu \nu}+\sum_\mu\hat H_\mu
\,,
\end{eqnarray}
which includes the Bose-Hubbard model (\ref{Bose-Hubbard-Hamiltonian})
as a special case.
The quantum evolution of the density operator $\hat\rho$ describing the
state of the full lattice can be written as
\begin{eqnarray}
\label{Liouville}
i\partial_t\hat\rho
= \left[\hat H,\hat\rho\right]
&=&
\frac1Z\sum_{\mu\nu}\left[\hat H_{\mu \nu},\hat\rho\right]
+\sum_\mu\left[\hat H_\mu,\hat\rho\right]
\nonumber\\
&=&
\frac1Z\sum_{\mu\nu}\,{\cal L}_{\mu \nu}\hat\rho
+
\sum_\mu\,{\cal L}_\mu\hat\rho
\,,
\end{eqnarray}
where we have introduced the Liouville super-operators $\,{\cal L}_{\mu \nu}$
and $\,{\cal L}_\mu$ as short-hand notation.
As the next step, we introduce the reduced density matrices for one
or more lattice sites via averaging (tracing) over all other sites
\begin{eqnarray}
\label{reduced-density-matrices}
\hat\rho_\mu
&=&
\,{\rm Tr}_{\not\mu}\{\hat\rho\}
\nonumber\\
\hat\rho_{\mu\nu}
&=&
\,{\rm Tr}_{\not\mu\not\nu}\{\hat\rho\}
\,,
\end{eqnarray}
and so on.
Note that $\,{\rm Tr}\{\hat\rho\}=1$ implies $\,{\rm Tr}_\mu\{\hat\rho_\mu\}=1$
and $\,{\rm Tr}_{\mu\nu}\{\hat\rho_{\mu\nu}\}=1$ etc.
Since we are interested in the (quantum) correlations, we separate the
correlated and uncorrelated parts of the reduced density matrices via
\begin{eqnarray}
\label{correlated-parts}
\hat\rho_{\mu\nu}
&=&
\hat\rho_{\mu\nu}^{\rm corr}+\hat\rho_{\mu}\hat\rho_{\nu}
\nonumber\\
\hat\rho_{\mu\nu\lambda}
&=&
\hat\rho_{\mu\nu\lambda}^{\rm corr}+
\hat\rho_{\mu\nu}^{\rm corr}\hat\rho_{\lambda}+
\hat\rho_{\mu\lambda}^{\rm corr}\hat\rho_{\nu}+
\hat\rho_{\nu\lambda}^{\rm corr}\hat\rho_{\mu}+
\hat\rho_{\mu}\hat\rho_{\nu}\hat\rho_{\lambda}
\,,
\end{eqnarray}
and analogously for more lattice sites.
As a consequence, we obtain from (\ref{Liouville}) the evolution
equation for the on-site density matrix
\begin{eqnarray}
\label{one-site}
i\partial_t\hat\rho_{\mu}
=
\frac{1}{Z}
\sum_{\kappa\neq\mu}\,{\rm Tr}_{\kappa}\left\{
\,{\cal L}^S_{\mu \kappa}
(\hat\rho^{\rm corr}_{\mu \kappa}+\hat\rho_\mu \hat\rho_\kappa)\right\}
+
\,{\cal L}_\mu\hat\rho_{\mu}
\,,
\end{eqnarray}
where $\,{\cal L}_{\mu \nu}^S=\,{\cal L}_{\mu \nu}+\,{\cal L}_{\nu \mu}$ denotes the
symmetrised form.
Obviously, solving this equation exactly requires knowledge of the
two-point correlation $\hat\rho^{\rm corr}_{\mu \kappa}$.
The time-evolution of this quantity can also obtained from
(\ref{Liouville}) and reads
\begin{eqnarray}
\label{two-sites}
i \partial_t \hat\rho^{\rm corr}_{\mu \nu}
&=&
\,{\cal L}_\mu\hat\rho^{\rm corr}_{\mu\nu}
+
\frac1Z\,{\cal L}_{\mu\nu}
(\hat\rho^{\rm corr}_{\mu\nu}+\hat\rho_\mu\hat\rho_\nu)
-
\frac{\hat\rho_{\mu}}{Z}
\,{\rm Tr}_{\mu}
\left\{\,{\cal L}^S_{\mu\nu}
(\hat\rho^{\rm corr}_{\mu\nu}+\hat\rho_\mu\hat\rho_\nu)
\right\}
\nonumber\\
&&+
\frac1Z
\sum_{\kappa\not=\mu,\nu}
\,{\rm Tr}_{\kappa}
\left\{
\,{\cal L}^S_{\mu \kappa}
(\hat\rho^{\rm corr}_{\mu\nu\kappa}+
\hat\rho^{\rm corr}_{\mu\nu}\hat\rho_{\kappa}+
\hat\rho^{\rm corr}_{\nu\kappa}\hat\rho_{\mu})
\right\}
+(\mu\leftrightarrow\nu)
\,.
\end{eqnarray}
As one would expect, this equation contains the three-point
correlator $\hat\rho^{\rm corr}_{\mu\nu\kappa}$, and similarly
the evolution equation for $\hat\rho^{\rm corr}_{\mu\nu\kappa\lambda}$
contains the four-point correlator etc.
Consequently, one can never exactly solve this set of equations,
truncated at any finite order.
However, the limit $Z\gg1$ facilitates an approximate solution:
Let us imagine that we start from an initial state
$\hat\rho^{\rm in}=\bigotimes_\mu\hat\rho^{\rm in}_\mu$
without any correlations
(i.e., $\hat\rho^{\rm corr}_{\mu \nu}=0$ and
$\hat\rho^{\rm corr}_{\mu\nu\kappa}=0$, etc.)
such as the Mott state (\ref{Mott-state}).
In this case, the right-hand side of (\ref{two-sites})
is suppressed by $\,{\cal O}(1/Z)$ and thus the time evolution
creates only small correlations
$i\partial_t\hat\rho^{\rm corr}_{\mu \nu}$.
Moreover, if these correlations are small initially
$\hat\rho^{\rm corr}_{\mu \nu}=\,{\cal O}(1/Z)$, they remain small
-- at least for a finite amount of time -- because there is no
term in (\ref{two-sites}) to compensate the $\,{\cal O}(1/Z)$ suppression.
Note that the sum $\sum_\kappa$ in (\ref{two-sites}) might scale with $Z$,
but this is compensated by the $1/Z$ factor in front of it.
On the other hand, if we insert $\hat\rho^{\rm corr}_{\mu \nu}=\,{\cal O}(1/Z)$
into (\ref{one-site}), we find that we can neglect this term and arrive at
an approximate equation containing on-site density matrices only
\begin{eqnarray}
\label{one-site-approx}
i\partial_t\hat\rho_{\mu}
&=&
\frac{1}{Z}
\sum_{\kappa\neq\mu}\,{\rm Tr}_{\kappa}\left\{
\,{\cal L}^S_{\mu \kappa}
\hat\rho_\mu \hat\rho_\kappa\right\}
+
\,{\cal L}_\mu\hat\rho_{\mu}
+\,{\cal O}(1/Z)
\nonumber\\
&\approx&
\frac{1}{Z}
\sum_{\kappa\neq\mu}\,{\rm Tr}_{\kappa}\left\{
\,{\cal L}^S_{\mu \kappa}
\hat\rho_\mu \hat\rho_\kappa\right\}
+
\,{\cal L}_\mu\hat\rho_{\mu}
\,,
\end{eqnarray}
The approximate solution $\hat\rho_\mu^0$ of this self-consistent equation
is valid to lowest order in $1/Z$, i.e.,
$\hat\rho_\mu=\hat\rho_\mu^0+\,{\cal O}(1/Z)$
and reproduces the well-known Gutzwiller ansatz \cite{G63,J98,RK91}.
If we now insert this approximate solution $\hat\rho_\mu^0$ into
(\ref{two-sites}), we get an approximate evolution equation for
the two-point correlator
\begin{eqnarray}
\label{two-sites-approx}
i \partial_t \hat\rho^{\rm corr}_{\mu \nu}
&=&
\,{\cal L}_\mu\hat\rho^{\rm corr}_{\mu\nu}
+
\frac1Z\,{\cal L}_{\mu\nu}\hat\rho^0_\mu\hat\rho^0_\nu
+
\frac1Z
\sum_{\kappa\not=\mu,\nu}
\,{\rm Tr}_{\kappa}
\left\{
\,{\cal L}^S_{\mu \kappa}
(\hat\rho^{\rm corr}_{\mu\nu}\hat\rho^0_{\kappa}+
\hat\rho^{\rm corr}_{\nu\kappa}\hat\rho^0_{\mu})
\right\}
\nonumber\\
&&-
\frac{\hat\rho^0_{\mu}}{Z}
\,{\rm Tr}_{\mu}
\left\{\,{\cal L}^S_{\mu\nu}\hat\rho^0_\mu\hat\rho^0_\nu\right\}
+(\mu\leftrightarrow\nu)
+\,{\cal O}(1/Z^2)
\,.
\end{eqnarray}
Note that we have assumed that the three-point correlations
$\hat\rho^{\rm corr}_{\mu\nu\kappa}$ do not spoil this line of
arguments and are suppressed by $\,{\cal O}(1/Z^2)$ in complete analogy.
This is indeed correct and can be shown in basically the same way,
see Appendix \ref{hierarchyApp}.
More generally, we find that $\ell$-point correlations are suppressed
as $\,{\cal O}(1/Z^{\ell-1})$, i.e.,
\begin{eqnarray}
\label{hierarchy}
\hat\rho_{\mu} &=& \,{\cal O}\left(Z^0\right)
\nonumber\\
\hat\rho^{\rm corr}_{\mu\nu} &=& \,{\cal O}\left(1/Z\right)
\nonumber\\
\hat\rho^{\rm corr}_{\mu\nu\kappa} &=& \,{\cal O}\left(1/Z^2\right)
\nonumber\\
\hat\rho^{\rm corr}_{\mu\nu\kappa\lambda} &=& \,{\cal O}\left(1/Z^3\right)
\,,
\end{eqnarray}
and so on, see Appendix \ref{hierarchyApp}.
This hierarchy (\ref{hierarchy}) is related to
the quantum de~Finetti theorem \cite{CKMR07},
the generalised cumulant expansion \cite{K62},
and the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY)
hierarchy \cite{B75},
but we are considering lattice sites instead of particles.
As an example for the four-point correlator, let us consider observables
$\hat A_\mu$, $\hat B_\nu$, $\hat C_\kappa$, and $\hat D_\lambda$ at
four different lattice sites, which have vanishing on-site expectation values
$\langle\hat A_\mu\rangle=
\langle\hat B_\nu\rangle=
\langle\hat C_\kappa\rangle=
\langle\hat D_\lambda\rangle=0$.
In this case, the hierarchy (\ref{hierarchy}) implies
\begin{eqnarray}
\langle\hat A_\mu\hat B_\nu\hat C_\kappa\hat D_\lambda\rangle
&=&
\langle\hat A_\mu\hat B_\nu\rangle
\langle\hat C_\kappa\hat D_\lambda\rangle
+
\langle\hat A_\mu\hat C_\kappa\rangle
\langle\hat B_\nu\hat D_\lambda\rangle
+
\langle\hat A_\mu\hat D_\lambda\rangle
\langle\hat B_\nu\hat C_\kappa\rangle
\nonumber\\
&&
+
\,{\cal O}\left(1/Z^3\right)
\,,
\end{eqnarray}
which resembles the Wick theorem in free quantum field theory
(even though the quantum system considered here is strongly interacting).
\section{Mott Insulator State}\label{mottsection}
Now let us apply the hierarchy discussed above to the Bose-Hubbard model
(\ref{Bose-Hubbard-Hamiltonian}).
To this end, we start with the factorising Mott state (\ref{Mott-state})
at zero hopping rate $J=0$ as our initial state
\begin{eqnarray}
\hat\rho^{\rm in}
=
\bigotimes\limits_\mu\hat\rho^{\rm in}_\mu
=
\bigotimes\limits_\mu\ket{1}_\mu\!\bra{1}
\,.
\end{eqnarray}
Then we slowly switch on the hopping rate $J(t)$ until we reach its
final value.
In view of the finite energy gap, the adiabatic theorem implies that
we stay very close to the real ground state of the system if we do
this slowly enough.
Of course, we cannot cross the phase transition in this way
(i.e., adiabatically) since the energy gap vanishes at the critical point,
see Section \ref{superquench} below.
Since we have $\langle\hat b_\mu\rangle=0$ in the Mott state,
Eq.~(\ref{one-site-approx}) simplifies to
\begin{eqnarray}
\label{one-site-Mott}
i\partial_t\hat\rho_{\mu}
\approx
\frac{1}{Z}
\sum_{\kappa\neq\mu}\,{\rm Tr}_{\kappa}\left\{
\,{\cal L}^S_{\mu \kappa}
\hat\rho_\mu \hat\rho_\kappa\right\}
+
\,{\cal L}_\mu\hat\rho_{\mu}
=0
\,\leadsto\,
\hat\rho_{\mu}^0=\ket{1}_\mu\!\bra{1}
\,.
\end{eqnarray}
Thus, to zeroth order in $1/Z$ (i.e., on the Gutzwiller mean-field level),
the Mott insulator state $\hat\rho_{\mu}^0$ for finite $J$ has the same
form as for $J=0$.
To obtain the first order in $1/Z$, we insert this result into
(\ref{two-sites-approx}).
Again using $\langle\hat b_\mu\rangle=0$, we find
\begin{eqnarray}
i \partial_t \hat\rho^{\rm corr}_{\mu \nu}
&=&
\left(\,{\cal L}_\mu+\,{\cal L}_\nu\right)\hat\rho^{\rm corr}_{\mu\nu}
+
\frac1Z\,{\cal L}_{\mu\nu}^S\hat\rho_\mu^0\hat\rho_\nu^0
\nonumber\\
&&+
\frac1Z
\sum_{\kappa\not=\mu,\nu}
\,{\rm Tr}_{\kappa}
\left\{
\,{\cal L}^S_{\mu\kappa}\hat\rho^{\rm corr}_{\nu\kappa}\hat\rho_{\mu}^0
+\,{\cal L}^S_{\nu\kappa}\hat\rho^{\rm corr}_{\mu\kappa}\hat\rho_{\nu}^0
\right\}
+\,{\cal O}(1/Z^2)
\,.
\end{eqnarray}
Formally, this is an evolution equation for an infinite dimensional
matrix $\hat\rho^{\rm corr}_{\mu \nu}$.
Fortunately, however, it suffices to consider a few elements only.
If we introduce
$\hat p_\mu=\ket{1}_\mu\!\bra{2}$
and
$\hat h_\mu=\ket{0}_\mu\!\bra{1}$
as local particle and hole
operators\footnote{These excitations are sometimes \cite{CBPE12,BPCK12,KJSW90}
called doublons and holons.},
all the interesting physics can
be captured by their correlation functions (for $\mu\neq\nu$)
\begin{eqnarray}
f_{\mu\nu}^{11}
&=&
\langle\hat{h}^\dagger_\mu\hat{h}_\nu^{} \rangle
=
\,{\rm Tr}\left\{\hat{\rho}\,\hat{h}^\dagger_\mu\hat{h}_\nu^{}\right\}
=
\,{\rm Tr}_{\mu\nu}\left\{
\hat{\rho}^{\rm corr}_{\mu\nu}\hat{h}^\dagger_\mu\hat{h}_\nu^{}\right\}
\,,
\nonumber\\
f_{\mu\nu}^{12}
&=&
\langle\hat{h}^\dagger_\mu\hat{p}_\nu^{} \rangle
=
\,{\rm Tr}\left\{\hat{\rho}\,\hat{h}^\dagger_\mu\hat{p}_\nu^{}\right\}
=
\,{\rm Tr}_{\mu\nu}\left\{
\hat\rho^{\rm corr}_{\mu\nu}\hat{h}^\dagger_\mu\hat{p}_\nu^{}\right\}
\,,
\nonumber\\
f_{\mu\nu}^{21}
&=&
\langle\hat{p}^\dagger_\mu\hat{h}_\nu^{} \rangle
=
\,{\rm Tr}\left\{\hat{\rho}\,\hat{p}^\dagger_\mu\hat{h}_\nu^{}\right\}
=
\,{\rm Tr}_{\mu\nu}\left\{
\hat{\rho}^{\rm corr}_{\mu\nu}\hat{p}^\dagger_\mu\hat{h}_\nu^{}\right\}
\,,
\nonumber\\
f_{\mu\nu}^{22}
&=&
\langle\hat{p}^\dagger_\mu\hat{p}_\nu^{} \rangle
=
\,{\rm Tr}\left\{\hat{\rho}\,\hat{p}^\dagger_\mu\hat{p}_\nu^{}\right\}
=
\,{\rm Tr}_{\mu\nu}\left\{
\hat{\rho}^{\rm corr}_{\mu\nu}\hat{p}^\dagger_\mu\hat{p}_\nu^{}\right\}
\,.
\end{eqnarray}
To first order in $1/Z$, these correlation functions form a closed set
of equations
\begin{eqnarray}
\label{f12-Mott}
i\partial_t f^{12}_{\mu\nu}
&=&
-\frac{J}{Z}\sum_{\kappa\neq \mu,\nu}
\left(T_{\mu\kappa}(f^{12}_{\kappa\nu}+\sqrt{2}f^{22}_{\kappa\nu})
+
\sqrt{2}T_{\nu\kappa}(f^{11}_{\mu\kappa}+\sqrt{2}f_{\mu\kappa}^{12})\right)
\nonumber\\
&&+Uf^{12}_{\mu\nu}-\frac{J\sqrt{2}}{Z}T_{\mu\nu}
\\
i\partial_t f^{21}_{\mu\nu}
&=&
+\frac{J}{Z}\sum_{\kappa\neq \mu,\nu}
\left(T_{\nu\kappa}(f^{21}_{\kappa\mu}+\sqrt{2}f^{11}_{\kappa\mu})
+
\sqrt{2}T_{\mu\kappa}(f^{22}_{\kappa\nu}+\sqrt{2}f_{\kappa\nu}^{12})\right)
\nonumber\\
&&-Uf^{21}_{\mu\nu}+\frac{J\sqrt{2}}{Z}T_{\mu\nu}
\\
\label{f11-Mott}
i\partial_t f^{11}_{\mu\nu}
&=&
i\partial_t f^{22}_{\mu\nu}
=
-\frac{\sqrt{2}J}{Z}\sum_{\kappa\neq\mu,\nu}\left(T_{\mu\kappa}
f_{\kappa\nu}^{21}-T_{\nu\kappa}f_{\mu\kappa}^{12}\right)
\,.
\end{eqnarray}
This truncation is due to the fact that the correlation functions
$f^{mn}_{\mu\nu}$ involving higher occupation numbers $m\geq3$ or $n\geq3$
do not have any source terms of order $1/Z$ and hence do not contribute
at that level.
Exploiting translational symmetry, we may simplify these equations by a
spatial Fourier transformation with
\begin{eqnarray}
\label{T_k}
T_{\mu\nu}
&=&
\frac{Z}{N}\sum_\mathbf{k}T_{\mathbf{k}}
e^{i\mathbf{k}\cdot(\mathbf{x}_\mu-\mathbf{x}_\nu)}
\\
\label{f_k}
f^{ab}_{\mu\nu}
&=&
\frac{1}{N}\sum_\mathbf{k}f^{ab}_{\mathbf{k}}
e^{i\mathbf{k}\cdot(\mathbf{x}_\mu-\mathbf{x}_\nu)}\,,
\end{eqnarray}
where $N$ denotes the number of lattice sites
(which equals the number of particles in our case).
Formally, in order to Fourier transform equations
(\ref{f12-Mott}-\ref{f11-Mott}), one should add the summands
corresponding to $\kappa=\mu$ and $\kappa=\nu$.
Since these terms are of order $1/Z^2$, they do not spoil our first-order
analysis.
However, when going to second order $1/Z^2$ (see Section~\ref{Z2} below),
they have to be taken into account.
After the Fourier transformation (\ref{T_k}) and (\ref{f_k}),
Eqs.~(\ref{f12-Mott}-\ref{f11-Mott}) become
\begin{eqnarray}
(i\partial_t-U+3 J T_\mathbf{k})f_\mathbf{k}^{12}
&=&
-\sqrt{2}J T_\mathbf{k}(f_\mathbf{k}^{11}+f_\mathbf{k}^{22}+1)
\,,\label{diff0}
\\
(i\partial_t+U-3 J T_\mathbf{k})f_\mathbf{k}^{21}
&=&
+\sqrt{2}J T_\mathbf{k}(f_\mathbf{k}^{11}+f_\mathbf{k}^{22}+1)
\,,\label{diff1}
\\
i\partial_t f_\mathbf{k}^{11}=i\partial_t f_\mathbf{k}^{22}
&=&
\sqrt{2}JT_\mathbf{k}(f^{12}_\mathbf{k}-f^{21}_\mathbf{k})
\label{diff2}
\,.
\end{eqnarray}
From the last equation, we may infer an effective particle-hole symmetry
$f_\mathbf{k}^{11}=f_\mathbf{k}^{22}$ valid to first order in $1/Z$.
With this symmetry, any stationary state (such as the ground state)
with $\partial_tf_\mathbf{k}^{ab}=0$ must obey the condition
\begin{eqnarray}
\label{stat}
f_\mathbf{k}^{12}=f_\mathbf{k}^{21}
=
\frac{\sqrt{2}JT_\mathbf{k}(2f_\mathbf{k}^{11}+1)}{U-3 JT_\mathbf{k}}
\,.
\end{eqnarray}
The remaining unknown quantity $f_\mathbf{k}^{11}$ can be obtained in the
following way:
The evolution equations (\ref{diff0}-\ref{diff2}) leave the following
bilinear quantity invariant
\begin{eqnarray}\label{inv}
\partial_t
\left[
f_\mathbf{k}^{11}(f_\mathbf{k}^{11}+1)-f_\mathbf{k}^{12}f_\mathbf{k}^{21}
\right]
=0
\,,
\end{eqnarray}
which remains valid even for time-dependent $J(t)$.
Thus, starting in the Mott state (\ref{Mott-state}) at zero hopping rate
$J=0$ with vanishing correlations $f_\mathbf{k}^{ab}(t=0)=0$, we get the
additional condition
$f_\mathbf{k}^{11}(f_\mathbf{k}^{11}+1)=f_\mathbf{k}^{12}f_\mathbf{k}^{21}$
for all times $t>0$.
Thus, to first order in $1/Z$, the ground state correlations read
(for $\mu\neq\nu$)
\begin{eqnarray}
\langle\hat{h}^\dagger_\mu\hat{h}_\nu^{} \rangle_{\rm ground}
=
\langle\hat{p}^\dagger_\mu\hat{p}_\nu^{} \rangle_{\rm ground}
&=&
\frac{1}{N}\sum_\mathbf{k}
\frac{U-3JT_\mathbf{k}-\omega_\mathbf{k}}{2\omega_\mathbf{k}}
e^{i\mathbf{k}\cdot(\mathbf{x}_\mu-\mathbf{x}_\nu)}
\label{ground1}
\\
\langle\hat{h}^\dagger_\mu\hat{p}_\nu^{} \rangle_{\rm ground}
=
\langle\hat{p}^\dagger_\mu\hat{h}_\nu^{} \rangle_{\rm ground}
&=&
\frac{1}{N}\sum_\mathbf{k}\frac{\sqrt{2}JT_\mathbf{k}}{\omega_\mathbf{k}}
e^{i\mathbf{k}\cdot(\mathbf{x}_\mu-\mathbf{x}_\nu)}
\label{ground2}
\,.
\end{eqnarray}
Here we have used the abbreviation {\cite{KN11}}
\begin{eqnarray}
\label{eigen-frequency}
\omega_\mathbf{k}&=&\sqrt{U^2-6 J UT_\mathbf{k}+J^2 T_\mathbf{k}^2}
\,,
\end{eqnarray}
which is just the (non-zero) eigenfrequency of the evolution equations
(\ref{diff0}-\ref{diff2}) for non-stationary states and will become
important in the next Section.
The above equations (\ref{ground1}) and (\ref{ground2}) describe the
correlations and are valid for $\mu\neq\nu$ only.
The correct on-site density matrix $\rho_\mu$ can be obtained from
(\ref{one-site}) which shows that non-vanishing correlations lead
to small deviations from the lowest-order result $\rho_\mu^0$.
As one would expect, the quantum ground-state fluctuations manifest
themselves in a small depletion of the unit-filling state
$\hat\rho_{\mu}^0=\ket{1}_\mu\!\bra{1}$ given by a small but finite
probability for a particle
$f_2=\,{\rm Tr}\{\hat{\rho}_\mu|2\rangle_\mu\langle 2|\}$
or a hole
$f_0=\,{\rm Tr}\{\hat{\rho}_\mu|0\rangle_\mu\langle 0|\}$.
To first order in $1/Z$, we get from (\ref{one-site})
\begin{eqnarray}
\label{diff3}
i\partial_t f_0
=
i\partial_t f_2
=
\sum_\mathbf{k}\frac{\sqrt{2}J T_\mathbf{k}}{N}
(f_\mathbf{k}^{12}-f_\mathbf{k}^{21})
=
i \frac{1}{N}\sum_\mathbf{k} \partial_tf_\mathbf{k}^{11}
\,,
\end{eqnarray}
where we used equation (\ref{diff2}) in the last step.
This equation can be integrated easily and with the initial conditions
$f_0(t=0)=f_2(t=0)=0$ we find the $1/Z$-corrections to the one-site
density matrix
\begin{eqnarray}
\label{depletion}
\langle\hat{p}^\dagger_\mu\hat{p}_\mu^{}\rangle
=
\langle\hat{h}_\mu^{}\hat{h}^\dagger_\mu\rangle
=
f_0=f_2
=
\frac{1}{N}\sum_\mathbf{k}
\frac{U-3JT_\mathbf{k}-\omega_\mathbf{k}}{2\omega_\mathbf{k}}
\,.
\end{eqnarray}
Note that, even though the right-hand side of the above equation looks
like that of (\ref{ground1}) for $\mu=\nu$, one should be careful as they
are derived from two different equations: (\ref{one-site}) and
(\ref{two-sites}).
\section{Mott--Superfluid Quench}\label{superquench}
After having studied the ground state properties of the Mott phase,
let us consider a quantum quench from the Mott state to the superfluid
regime.
This requires a time-dependent solution of the evolution equations
(\ref{diff0}-\ref{diff2}) which crucially depends on the eigenfrequency
(\ref{eigen-frequency}).
In view of the definition (\ref{T_k}), $T_{\mathbf{k}}$ adopts its
maximum value $T_{\mathbf{k}=0}=1$ at $\mathbf{k}=0$.
Thus $\omega_{\mathbf{k}=0}=\Delta\mathcal E$ corresponds to the
energy gap of the Mott state mentioned in
Section~\ref{section-Bose-Hubbard-Model}.
For $J=0$, we have a flat dispersion relation $\omega_\mathbf{k}=U$.
If we increase $J$, the dispersion relation $\omega_\mathbf{k}$
bends down and the minimum at $\mathbf{k}=0$ approaches the axis.
Finally, at a critical value of the hopping rate \cite{NS10}
\begin{eqnarray}
J_{\rm crit}=U(3-\sqrt{8})
\,,
\end{eqnarray}
the minimum $\omega_{\mathbf{k}=0}$ touches the axis and thus
the energy gap vanishes $\Delta\mathcal E=0$.
This marks the transition to the superfluid regime and we cannot
analytically or adiabatically continue beyond this point.
However, nothing stops us from suddenly switching $J$ to a final
value $J_{\rm out}>J_{\rm crit}$ beyond this point.
Of course, this would not be adiabatic anymore and we would
no longer be close to the ground state.
For hopping rates $J$ which are a bit larger than the critical
value $J>J_{\rm crit}$, the dispersion relation dives below the
axis and the $\omega_\mathbf{k}^2$ become negative for small $\mathbf{k}$.
Thus, the eigenfrequencies $\omega_\mathbf{k}$ become imaginary
indicating an exponential growth of these modes, i.e., an instability.
This is very natural since the quantum system ``feels'' that the Mott
state is no longer the correct ground state.
If we consider even larger $J$, we find that the original minimum
of the dispersion relation $\omega_\mathbf{k}^2$ at $\mathbf{k}=0$
splits into degenerate minima at finite values of $\mathbf{k}$
when $J=3U$, while $\mathbf{k}=0$ becomes a local maximum.
This local maximum even emerges $\omega_{\mathbf{k}=0}^2>0$
on the positive side again for $J>U(3+\sqrt{8})$.
Nevertheless, there are always
unstable modes for some values of $\mathbf{k}$,
see Fig.~\ref{fig-omega} and compare \cite{S11}.
\bigskip
\begin{figure}[h]
\begin{center}
\includegraphics[width=.49\columnwidth]{frequency.eps}
\includegraphics[width=.49\columnwidth]{frequency2.eps}
\end{center}
\caption{Dispersion relation $\omega_k^2/U^2$ in one dimension for
different values of $J/U$.}
\label{fig-omega}
\end{figure}
\bigskip
After these preliminaries, let us study a quantum quench from the
Mott state to the superfluid phase.
For simplicity, we consider a sudden change of $J(t)=J\Theta(t)$
from $J=0$ to the final value of $J$
(but the calculation can easily be generalised to other scenarios).
Solving the evolution equations (\ref{diff0}-\ref{diff2}) for this case,
we find
\begin{eqnarray}
\label{quench-h+h}
\langle\hat{h}^\dagger_\mu\hat{h}_\nu^{} \rangle_{\rm quench}
&=&
\langle\hat{p}^\dagger_\mu\hat{p}_\nu^{} \rangle_{\rm quench}
=
\frac{1}{N}\sum_\mathbf{k}
4J^2 T_\mathbf{k}^2\,
\frac{1-\cos(\omega_\mathbf{k}t)}{\omega^2_\mathbf{k}}\,
e^{i\mathbf{k}\cdot(\mathbf{x}_\mu-\mathbf{x}_\nu)}
\\
\label{quench-h+p}
\langle\hat{h}^\dagger_\mu\hat{p}_\nu^{} \rangle_{\rm quench}
&=&
\frac{1}{N}\sum_\mathbf{k}
\sqrt{2}JT_\mathbf{k}(U-3JT_\mathbf{k})
\frac{1-\cos(\omega_\mathbf{k}t)}{\omega_\mathbf{k}^2}\,
e^{i\mathbf{k}\cdot(\mathbf{x}_\mu-\mathbf{x}_\nu)}
\nonumber\\
&&+
\frac{i}{N}\sum_\mathbf{k}
\sqrt{2}JT_\mathbf{k}\,
\frac{\sin(\omega_\mathbf{k}t)}{\omega_\mathbf{k}}\,
e^{i\mathbf{k}\cdot(\mathbf{x}_\mu-\mathbf{x}_\nu)}
\,.
\end{eqnarray}
The remaining correlation can simply be obtained via
$\langle\hat{p}^\dagger_\nu\hat{h}_\mu^{}\rangle=
(\langle\hat{h}^\dagger_\mu\hat{p}_\nu^{}\rangle)^*$.
The correlator in terms of the original creation and
annihilation operators $\hat b_\mu^\dagger$ and $\hat b_\nu$
is just a linear combination of these correlation functions
\begin{eqnarray}
\label{quench-b+b}
\langle\hat b_\mu^\dagger\hat b_\nu\rangle_{\rm quench}
=
\frac{1}{N}\sum_\mathbf{k}4JUT_\mathbf{k}\,
\frac{1-\cos(\omega_\mathbf{k}t)}{\omega^2_\mathbf{k}}\,
e^{i\mathbf{k}\cdot(\mathbf{x}_\mu-\mathbf{x}_\nu)}
\,.
\end{eqnarray}
Note that the momentum distribution
\begin{eqnarray}
\label{momdist}
P(\mathbf{k})=\frac{1}{N^2}\sum_\mathbf{\mu\nu}
e^{i\mathbf{k}\cdot(\mathbf{x}_\mu-\mathbf{x}_\nu)}
\langle\hat b_\mu^\dagger\hat b_\nu\rangle
\end{eqnarray}
which is basically the Fourier transform of
$\langle\hat b_\mu^\dagger\hat b_\nu\rangle$,
can be measured by time-of-flight experiments \cite{G02,G01}
The quench $J(t)$ can be realised experimentally by decreasing the
intensity of the laser field generating the optical lattice
(which lowers the potential barrier for tunnelling and thus
increases $J$).
Thus the above prediction should be testable in experiments.
As explained above, after a quench to the superfluid regime,
the frequencies $\omega_\mathbf{k}$ become imaginary for some
$\mathbf{k}$ and thus these modes grow exponentially.
As a result, the expectation value will quickly be dominated by these
fast growing modes and so most of the details of the initial state
will become unimportant.
Of course, this exponential growth cannot continue forever --
after some time, the $1/Z$-expansion breaks down since the quantum
fluctuation are too strong and the growth will saturate.
\begin{center}
\begin{figure}[h]
\includegraphics{holepart.eps}
\includegraphics{correlationsf12.eps}
\begin{center}
\includegraphics{correlationsf11.eps}
\end{center}
\caption{Time-dependence of the depletion
$\langle\hat{p}_{\mu}^\dagger\hat{p}_{\mu}\rangle$
and the nearest-neighbor correlations functions
$\langle\hat{h}_{\mu}^\dagger\hat{p}_{\nu}\rangle$
and
$\langle\hat{p}_{\mu}^\dagger\hat{p}_{\nu}\rangle$
in three dimensions after the quench within the Mott phase from
$J/U=0$ to $J/U=0.14$ in comparison to their ground state values.
After quasi-equilibration,
$\langle\hat{p}_{\mu}^\dagger\hat{p}_{\nu}\rangle_\mathrm{quench}$ and
$\langle\hat{p}_{\mu}^\dagger\hat{p}_{\nu}\rangle_\mathrm{ground}$
as well as
$\langle\hat{p}_{\mu}^\dagger\hat{p}_{\mu}\rangle_\mathrm{quench}$ and
$\langle\hat{p}_{\mu}^\dagger\hat{p}_{\mu}\rangle_\mathrm{ground}$
differ roughly by a factor of two.}
\end{figure}
\end{center}
\section{Equilibration versus Thermalisation}\label{equilibration}
Instead of a quench from the Mott to the superfluid phase, we can also study
a quench within the Mott regime.
Again, we consider a sudden change of $J$ from zero its final value for
simplicity -- but now the final value $J$ lies below the critical point
$J<J_{\rm crit}$.
In this case, all frequencies are real $\omega_\mathbf{k}\in\mathbb R$
and thus there is no exponential growth -- all modes oscillate.
Apart from this point, we can use the same solution as in
(\ref{quench-h+h}-\ref{quench-b+b}).
For an infinite (or at least extremely large) lattice, the oscillations
in (\ref{quench-h+h}-\ref{quench-b+b}) average out for sufficiently large
times $t$ and thus these observables approach a quasi-equilibrium value
\begin{eqnarray}
\label{equil-h+h}
\langle\hat{h}^\dagger_\mu\hat{h}_\nu^{} \rangle_{\rm equil}
=
\langle\hat{p}^\dagger_\mu\hat{p}_\nu^{} \rangle_{\rm equil}
&=&
\frac{1}{N}\sum_\mathbf{k}
\frac{4J^2 T_\mathbf{k}^2}{\omega^2_\mathbf{k}}\,
e^{i\mathbf{k}\cdot(\mathbf{x}_\mu-\mathbf{x}_\nu)}
\\
\label{equil-h+p}
\langle\hat{h}^\dagger_\mu\hat{p}_\nu^{} \rangle_{\rm equil}
=
\langle\hat{p}^\dagger_\mu\hat{h}_\nu^{} \rangle_{\rm equil}
&=&
\frac{1}{N}\sum_\mathbf{k}
\sqrt{2}JT_\mathbf{k}\,
\frac{U-3JT_\mathbf{k}}{\omega_\mathbf{k}^2}
e^{i\mathbf{k}\cdot(\mathbf{x}_\mu-\mathbf{x}_\nu)}
\,.
\end{eqnarray}
The quasi-equilibrium values for the local (on-site) particle or hole
probability can be derived in complete analogy to the previous case
\begin{eqnarray}\label{parthole}
\langle\hat{p}^\dagger_\mu\hat{p}_\mu^{}\rangle_{\rm equil}
=
\langle\hat{h}_\mu^{}\hat{h}^\dagger_\mu\rangle_{\rm equil}
=
\frac{1}{N}\sum_\mathbf{k}
\frac{4J^2 T_\mathbf{k}^2}{\omega^2_\mathbf{k}}
\,.
\end{eqnarray}
Having found that the observables considered above approach a
quasi-equilibrium state, it is natural to ask the question of
thermalisation.
This is one of the major unsolved questions
(or rather a set of questions) in quantum many-body theory
\cite{KWE11,GME11,EHKKMWW,KISD11,BCH11}
In one version, this question can be posed in the following way:
Given an interacting quantum many-body system
(for example the Bose-Hubbard model) on an infinite lattice
in a appropriately excited state (such as after a quench),
do all localised observables
(e.g., $\langle\hat{p}^\dagger_\mu\hat{p}_\mu^{}\rangle$
and $\langle\hat{h}^\dagger_\mu\hat{h}_\nu^{}\rangle$) settle down
to a value which is consistent with a thermal state described by
a suitable temperature?
Even though we cannot settle this question here, we can compare the
quasi-equilibrium values obtained above with a thermal state.
To this end, we derive the thermal density matrix $\hat\rho_\beta$
corresponding to a given (inverse) temperature $k_{\rm B}T=1/\beta$.
Using the grand canonical ensemble, the thermal density operator is
given by
\begin{eqnarray}
\hat\rho_\beta
=
\frac{e^{-\beta(\hat{H}-\mu \hat{N})}}
{\,{\rm Tr}\{e^{-\beta(\hat{H}-\mu \hat{N})}\}}
\,,
\end{eqnarray}
where chemical potential $\mu$ will be chosen such that the filling
is equal to unity.
Since we cannot calculate $\hat\rho_\beta$ exactly, we employ
strong-coupling perturbation theory, i.e., an expansion in powers of $J$.
It is useful to introduce the operator \cite{CFMSE08}
\begin{eqnarray}
\hat R(\beta)
=
e^{\beta\hat{H}_0}e^{-\beta(\hat{H}_0+\hat{H}_1)}
\,,
\end{eqnarray}
where $\hat{H}_0$ is the diagonal on-site part of the grand canonical
Hamiltonian $\hat{H}-\mu \hat{N}$ and $\hat{H}_1$ is the hopping term.
This operator satisfies the differential equation
\begin{eqnarray}
\partial_\beta\hat{R}(\beta)=-\hat{H}_1(\beta)\hat{R}(\beta)
\,,
\end{eqnarray}
where $\hat{H}_1(\beta)=e^{\beta\hat{H}_0} \hat{H}_1 e^{-\beta\hat{H}_0}$.
In analogy to time-dependent perturbation theory, the operator $\hat{R}$
can be calculated perturbatively by integrating this equation with respect
to $\beta$.
In first-order perturbation expansion, we have
\begin{eqnarray}
\label{thermal-perturbation}
\hat{\rho}_\beta
=
\frac{e^{-\beta\hat{H}_0}}{\,{\rm Tr}\{e^{-\beta \hat{H}_0}\}}
\left(1+\frac{J}{Z}\sum_{\mu\nu}T_{\mu\nu}\,
\frac{e^{\beta U(\hat{n}_\mu-\hat{n}_\nu-1)}-1}
{U(\hat{n}_\mu-\hat{n}_\nu-1)}\,
\hat{a}^\dagger_\mu \hat{a}^{}_\nu
+\,{\cal O}(J^2)
\right)
\,.
\end{eqnarray}
Obviously, the correction to first order in $J$ does not affect the
on-site density matrix $\hat\rho_\mu$ but the two-point correlations.
Thus, we find that the quasi-equilibrium state of the on-site density
matrix $\hat\rho_\mu$ can indeed be described by a thermal state
provided that we choose the chemical potential as $\mu=U/2$ which gives
\begin{eqnarray}\label{thermdens}
\hat\rho_\mu(\beta)
\approx
\frac{e^{-\beta U/2}}{2}\,\ket{0}_\mu\!\bra{0}
+
\left(1-e^{-\beta U/2}\right)\ket{1}_\mu\!\bra{1}
+
\frac{e^{-\beta U/2}}{2}\,\ket{2}_\mu\!\bra{2}
\,.
\end{eqnarray}
The particular value $\mu=U/2$ of the chemical potential ensures
that (in first order thermal perturbation theory) we have on average one
particle per lattice site and the particle-hole symmetry
$\langle \hat{p}_\mu^\dagger \hat{p}_\mu\rangle=
\langle \hat{h}_\mu^\dagger \hat{h}_\mu\rangle$.
To obtain the correct probabilities, we have to select the temperature
according to
\begin{eqnarray}
\label{temperature}
e^{-\beta U/2}=2\langle\hat{p}^\dagger_\mu\hat{p}_\mu^{}\rangle_{\rm equil}
=
\frac{2}{N}\sum_\mathbf{k}
\frac{4J^2 T_\mathbf{k}^2}{\omega^2_\mathbf{k}}
=\,{\cal O}(1/Z)
\,,
\end{eqnarray}
which can be deduced from (\ref{parthole}) and (\ref{thermdens}).
Since the depletion is small
$\langle\hat{p}^\dagger_\mu\hat{p}_\mu^{}\rangle=\,{\cal O}(1/Z)$,
we obtain a low effective temperature which scales as $T=\,{\cal O}(U/\ln Z)$.
Accordingly, consistent with our $1/Z$-expansion, we can neglect
higher Boltzmann factors such as $e^{-\beta U}$.
Of course, the fact that the on-site density matrix $\hat\rho_\mu$
can be described (within our limits of accuracy) by a thermal state
does not imply that the same is true for the correlations.
To study this point, let us calculate the thermal two-point correlator
from (\ref{thermal-perturbation}).
To first order in $J$ and $1/Z=\,{\cal O}(e^{-\beta U}/2)$, we find
\begin{eqnarray}
\langle\hat{h}^\dagger_\mu\hat{p}_\nu^{} \rangle_\beta
=
\langle\hat{p}^\dagger_\mu\hat{h}_\nu^{} \rangle_\beta
=
\frac{\sqrt{2}JT_{\mu\nu}}{ZU}
+\,{\cal O}(J^2)+\,{\cal O}(1/Z^2)
\,,
\end{eqnarray}
while $\langle\hat{h}^\dagger_\mu\hat{h}_\nu^{} \rangle_\beta$
and $\langle\hat{p}^\dagger_\mu\hat{p}_\nu^{} \rangle_\beta$
vanish (to first order in $J$).
If we compare this to the quasi-equilibrium value
$\langle\hat{h}^\dagger_\mu\hat{p}_\nu^{} \rangle_{\rm equil}$ in
(\ref{equil-h+p}), we find that they coincide to first order in $J$
\begin{eqnarray}
\langle\hat{h}^\dagger_\mu\hat{p}_\nu^{} \rangle_{\rm equil}
=
\langle\hat{p}^\dagger_\mu\hat{h}_\nu^{} \rangle_{\rm equil}
=
\frac{\sqrt{2}JT_{\mu\nu}}{ZU}
+\,{\cal O}(J^2)+\,{\cal O}(1/Z^2)
\,.
\end{eqnarray}
This is perhaps not too surprising since the same value can be obtained
from the ground state fluctuations
$\langle\hat{h}^\dagger_\mu\hat{p}_\nu^{} \rangle_{\rm ground}=
\langle\hat{p}^\dagger_\mu\hat{h}_\nu^{} \rangle_{\rm ground}$
in (\ref{ground2}) after expanding them to first order in $J$.
Due to the low effective temperature $T=\,{\cal O}(U/\ln Z)$, the lowest
Boltzmann factor is suppressed by $e^{-\beta U/2}=\,{\cal O}(1/Z)$.
As a consequence, because the correlations are small $\,{\cal O}(1/Z)$,
their finite-temperature corrections are even smaller $\,{\cal O}(1/Z^2)$,
and thus can be neglected.
The same is true for the other correlations
$\langle\hat{h}^\dagger_\mu\hat{h}_\nu^{} \rangle=
\langle\hat{p}^\dagger_\mu\hat{p}_\nu^{} \rangle$.
All of them: the ground state correlators
$\langle\hat{h}^\dagger_\mu\hat{h}_\nu^{} \rangle_{\rm ground}=
\langle\hat{p}^\dagger_\mu\hat{p}_\nu^{} \rangle_{\rm ground}$
in (\ref{ground1}), the quasi-equilibrium correlators
$\langle\hat{h}^\dagger_\mu\hat{h}_\nu^{} \rangle_{\rm equil}=
\langle\hat{p}^\dagger_\mu\hat{p}_\nu^{} \rangle_{\rm equil}$
in (\ref{equil-h+h}), as well as the thermal correlators
$\langle\hat{h}^\dagger_\mu\hat{h}_\nu^{} \rangle_\beta$
and $\langle\hat{p}^\dagger_\mu\hat{p}_\nu^{} \rangle_\beta$
vanish to first order in $J$.
Therefore, to first order in $J$ and $1/Z$, the thermal state
can describe the observable under consideration.
However, going to the next order in $J$, this description breaks down.
This failure can even be shown without explicitly calculating
$\hat R(\beta)$ up to second order.
If we compare the quasi-equilibrium correlators in (\ref{equil-h+h})
\begin{eqnarray}
\langle\hat{h}^\dagger_\mu\hat{h}_\nu^{} \rangle_{\rm equil}
=
\langle\hat{p}^\dagger_\mu\hat{p}_\nu^{} \rangle_{\rm equil}
=
\frac{4J^2 }{U^2Z^2}\sum_\kappa T_{\mu\kappa}T_{\kappa\nu}
+\,{\cal O}(J^3)+\,{\cal O}(1/Z^2)
\,,
\end{eqnarray}
with the ground state correlations in (\ref{ground1}),
expanded to the same order in $J$
\begin{eqnarray}
\langle\hat{h}^\dagger_\mu\hat{h}_\nu^{} \rangle_{\rm ground}
=
\langle\hat{p}^\dagger_\mu\hat{p}_\nu^{} \rangle_{\rm ground}
=
\frac{2J^2 }{U^2Z^2}\sum_\kappa T_{\mu\kappa}T_{\kappa\nu}
+\,{\cal O}(J^3)+\,{\cal O}(1/Z^2)
\,,
\end{eqnarray}
we find a discrepancy by a factor of two.~
I.e., after the quench, these correlations settle down to a value
which is twice as large as in the ground state.
This factor of two has already been found elsewhere in the context
of standard time-dependent and time-independent perturbation theory,
see also \cite{MK09}.
This is incompatible with the small Boltzmann factors
$e^{-\beta U/2}=\,{\cal O}(1/Z)$ and would require a comparably large
effective temperature $T=\,{\cal O}(U)$ instead of $T=\,{\cal O}(U/\ln Z)$.
However, such a large effective temperature $T=\,{\cal O}(U)$ is
inconsistent with the small on-site depletion (\ref{temperature}).
In summary, the considered observables settle down to a quasi-equilibrium
state -- but this state is not thermal.
Thus, real thermalisation -- if it occurs at all -- requires much longer
times scales.
This seems to be a generic feature and has been discussed for
bosonic \cite{CFMSE08,FCMSE10,CDEO08} and fermionic systems
\cite{U09,MK08,EKW10,MK10,SGJ10,EKW09}
and is sometimes called ``pre-thermalisation''.
This phenomenon can be visualised via the following intuitive picture:
The excited state generated by the quench can be viewed as a highly
coherent superposition of correlated quasi-particles.
During the subsequent quantum evolution, these quasi-particles
disperse and randomise their relative phases -- which results in
a quasi-stationary state.
However, the quasi-particles still retain their initial spectrum
(in energy and quasi-momentum), which could be approximately described
by a generalised Gibbs ensemble (i.e., a momentum-dependent temperature).
In this picture, thermalisation requires the exchange of energy and
momentum between these quasi-particles due to multiple collisions,
which changes the one-particle spectrum and takes much longer.
Ergo, one would expect a separation of time scales -- i.e., first (quasi)
equilibration and only much later thermalisation -- for many systems
in condensed matter, where the above quasi-particle picture applies.
\section{Tilted Mott Lattice}\label{tiltlattice}
In the following, we study the impact of a spatially constant but possibly
time-dependent force on the particles, which could correspond to a tilt of
the lattice, for example
\cite{M01,M02,D96,QNS11,WWMK05,SSG02,W05,KK03,KK04,K04,K03,KB03,KKG09,CN99,M12,Si11}.
This scenario can be described by a generalisation of the
Hamiltonian (\ref{Bose-Hubbard-Hamiltonian})
\begin{eqnarray}
\label{Bose-Hubbard-Tilt}
\hat H
=
-\frac{J}{Z}\sum_{\mu\nu} T_{\mu\nu} \hat b^\dagger_\mu \hat b_\nu
+\frac{U}{2}\sum_{\mu}
\hat n_{\mu}(\hat n_{\mu}-1)
+\sum_{\mu} V_\mu\hat n_{\mu}
\,.
\end{eqnarray}
The external potential $V_\mu(t)=\mathbf{x}_\mu\cdot\mathbf{E}(t)$
will be identified as an effective electric field $\mathbf{E}(t)$
and will be time-dependent in general.
If we insert this modified Hamiltonian into (\ref{one-site-Mott}),
we find that the potential $V_\mu$ has no effect to zeroth order $\,{\cal O}(Z^0)$,
i.e., the solution $\hat\rho_{\mu}^0=\ket{1}_\mu\!\bra{1}$ remains the
same.
In other words, the Gutzwiller mean field is not affected by the tilt
in the Mott state (in the superfluid phase, this would be different).
However, the next-order $\,{\cal O}(1/Z)$ quantum correlations can lead to
the creation of particle-hole pairs via tunnelling over one or more
lattice sites.
In order to study this effect, let us generalise the evolution equations
(\ref{f12-Mott}) and (\ref{f11-Mott}) in the presence of the potential
$V_\mu$
\begin{eqnarray}
\label{f12}
\left(i\partial_t+V_\mu-V_\nu-U\right)f^{12}_{\mu\nu}
&=&
-\frac{J}{Z}\sum_{\kappa\neq \mu,\nu}T_{\mu\kappa}
\left[3 f^{12}_{\kappa\nu}+\sqrt{2}f^{22}_{\kappa\nu}+
\sqrt{2}f^{11}_{\kappa\nu}\right]
\nonumber\\
& &
-\frac{J\sqrt{2}}{Z}T_{\mu\nu}
\,,
\\
\label{f11}
\left(i\partial_t+V_\mu-V_\nu\right)
f^{11}_{\mu\nu}
&=&
-\frac{\sqrt{2}J}{Z}\sum_{\kappa\neq\mu,\nu}T_{\mu\kappa}\left(
f_{\kappa\nu}^{21}-f_{\kappa\nu}^{12}\right)
\,,
\quad
\end{eqnarray}
and the same for $f^{22}_{\mu\nu}$, such that we again have an effective
particle-hole operator symmetry $f^{11}_{\mu\nu}=f^{22}_{\mu\nu}$ to
lowest order in $1/Z$.
Here we have already used translational invariance.
The tunnelling probability can now be obtained by solving the
above equations.
However, instead of solving them directly, we can simplify the problem
by effectively factorising these equations:
If we introduce the {\em effective} differential equations for
$\hat{h}_\mu$ and $\hat{p}_\mu$,
\begin{eqnarray}
\label{effective}
\left[i\partial_t-V_\mu-\frac{U}{2}\right]\hat{p}_\mu
&=&
-\frac{J}{Z}\sum_\nu T_{\mu\nu}
\left[\frac32\,\hat{p}_\nu+\sqrt{2}\,\hat{h}_\nu\right]
\,,
\nonumber\\
\left[i\partial_t-V_\mu+\frac{U}{2}\right]\hat{h}_\mu
&=&
\frac{J}{Z}\sum_\nu T_{\mu\nu}
\left[\frac32\,\hat{h}_\nu+\sqrt{2}\,\hat{p}_\nu\right]
\,,
\end{eqnarray}
and exploit the initial conditions
$\langle\hat h_\mu^\dagger\hat h_\nu\rangle_0=\delta_{\mu\nu}$
and
$\langle\hat h_\mu^\dagger\hat p_\nu\rangle_0=
\langle\hat p_\mu^\dagger\hat h_\nu\rangle_0=
\langle\hat p_\mu^\dagger\hat p_\nu\rangle_0=0$
valid in the Mott state, we exactly recover
(\ref{f12}) and (\ref{f11}) to first order in $1/Z$.
For potentials of the form $V_\mu(t)=\mathbf{x}_\mu\cdot\mathbf{E}(t)$
it is possible to apply the Peierls transformation and absorb the
potential in a phase.
After the Fourier transformations
\begin{eqnarray}
\hat{h}_\mu(t)
&=&
\exp\left\{-i\int_0^t dt' V_\mu(t')\right\}
\sum_\mathbf{k}\hat{h}_\mathbf{k}(t)\exp\{i\mathbf{k\cdot x}_\mu\}
\,,
\\
\hat{p}_\mu(t)
&=&
\exp\left\{-i\int_0^t dt' V_\mu(t')\right\}
\sum_\mathbf{k}\hat{p}_\mathbf{k}(t)\exp\{i\mathbf{k\cdot x}_\mu\}
\,,
\\
T_{\mu\nu}(t)
&=&
\frac{Z}{N}\sum_\mathbf{k}T_\mathbf{k}(t)
\exp\left\{i\mathbf{k}\cdot(\mathbf{x}_\mu-\mathbf{x}_\nu)
+i\int_0^t dt'[V_\mu(t')-V_\nu(t')]\right\}
\label{Peierls}
\,,
\end{eqnarray}
the effective evolution equations (\ref{effective}) become
\begin{eqnarray}
i\partial_t \hat{h}_\mathbf{k}
&=&
+\frac{1}{2}\left[3JT_\mathbf{k}(t)-U\right]\hat{h}_\mathbf{k}+
\sqrt{2}J T_\mathbf{k}(t)\hat{p}_\mathbf{k}
\,,
\label{holedgl}
\\
i\partial_t \hat{p}_\mathbf{k}
&=&
-\frac{1}{2}\left[3JT_\mathbf{k}(t)-U\right]\hat{p}_\mathbf{k}-
\sqrt{2}J T_\mathbf{k}(t)\hat{h}_\mathbf{k}
\label{partdgl}
\,.
\end{eqnarray}
Note that $T_\mathbf{k}(t)$ explicitly depends on time here and this
time-dependence encodes the potential $V_\mu(t)$.
Introducing the effective vector potential $\mathbf{A}(t)$ which generates
the effective electric field $\mathbf{E}(t)$ in
$V_\mu(t)=\mathbf{x}_\mu\cdot\mathbf{E}(t)$
via $\mathbf{E}(t)=\partial_t\mathbf{A}(t)$, this is equivalent to the
substitution $\mathbf{k}\to\mathbf{k}+\mathbf{A}(t)$
in complete analogy to the minimal coupling procedure
$T_\mathbf{k}(t)=T_{\mathbf{k}+\mathbf{A}(t)}$ known from
electrodynamics.
The most general solution of (\ref{holedgl}) and (\ref{partdgl})
can be written as
\begin{eqnarray}
\hat{h}_\mathbf{k}
&=&
f_\mathbf{k}^+(t)\hat{A}_\mathbf{k}+
f_\mathbf{k}^-(t)\hat{B}_\mathbf{k}
\,,
\\
\hat{p}_\mathbf{k}
&=&
g_\mathbf{k}^+(t)\hat{A}_\mathbf{k}+
g_\mathbf{k}^-(t)\hat{B}_\mathbf{k}
\,,
\end{eqnarray}
where $\hat{A}_\mathbf{k}$ and $\hat{B}_\mathbf{k}$ are time-independent
operators while $f_\mathbf{k}^\pm$ and $g_\mathbf{k}^\pm$ are time-dependent
c-number functions.
In analogy to the previous case, we assume that we start in the
Mott state (\ref{Mott-state}) with $J=V_\mu=0$.
In this case, the equations (\ref{holedgl}) and (\ref{partdgl}) decouple
and we may choose $\hat{A}_\mathbf{k}=\hat{h}_\mathbf{k}^{\rm in}$ and
$\hat{B}_\mathbf{k}=\hat{p}_\mathbf{k}^{\rm in}$ which imply
$\langle \hat{A}_\mathbf{k}^\dagger \hat{A}_\mathbf{p}\rangle_0=
\delta_{\mathbf{k},\mathbf{p}}$ and
$\langle \hat{B}_\mathbf{k}^\dagger \hat{B}_\mathbf{p}\rangle_0=0$,
as well as, $f_\mathbf{k}^+(t)=\exp(i U t/2)$ and
$g_\mathbf{k}^-(t)=\exp(-i U t/2)$ while the other two vanish.
Now we imagine the following sequence:
First we switch on $J$ adiabatically, then we apply the potential $V_\mu(t)$
for a finite period of time, and finally we switch off $J$ adiabatically.
Thus, at the very end, the equations (\ref{holedgl}) and (\ref{partdgl})
decouple again and the final particle operator $\hat{p}_\mathbf{k}^{\rm out}$
oscillates with positive frequencies $\exp(-i U t/2)$ while the final
hole operator $\hat{h}_\mathbf{k}^{\rm out}$
oscillates with negative frequencies $\exp(+i U t/2)$.
However, during the time-evolution, positive and negative frequencies will
get mixed in general by the time-dependence of
$T_\mathbf{k}(t)=T_{\mathbf{k}+\mathbf{A}(t)}$,
i.e., the potential $V_\mu(t)$.
Thus, the initial and final particle/hole-operators will be connected by
a Bogoliubov transformation
\begin{eqnarray}
\label{Bogoliubov}
\hat{p}_\mathbf{k}^{\rm out}
&=&
\alpha_\mathbf{k}\hat{p}_\mathbf{k}^{\rm in}+
\beta_\mathbf{k}\hat{h}_\mathbf{k}^{\rm in}
\,,
\nonumber\\
\hat{h}_\mathbf{k}^{\rm out}
&=&
\alpha^*_\mathbf{k}\hat{h}_\mathbf{k}^{\rm in}+
\beta^*_\mathbf{k}\hat{p}_\mathbf{k}^{\rm in}
\,,
\end{eqnarray}
where the Bogoliubov coefficients $\alpha_\mathbf{k}$ and $\beta_\mathbf{k}$
satisfy the relation $|\alpha_\mathbf{k}|^2-|\beta_\mathbf{k}|^2=1$.
In view of the initial conditions
$\langle \hat{A}_\mathbf{k}^\dagger \hat{A}_\mathbf{p}\rangle_0=
\delta_{\mathbf{k},\mathbf{p}}$ and
$\langle \hat{B}_\mathbf{k}^\dagger \hat{B}_\mathbf{p}\rangle_0=0$,
we find
\begin{eqnarray}
\langle\hat{p}^\dagger_\mathbf{k}\hat{p}_\mathbf{k}\rangle_{\rm out}
=
|\beta_\mathbf{k}|^2
\,,
\end{eqnarray}
which gives the probability to create a particle in the mode $\mathbf{k}$.
Since particles (i.e., doubly occupied lattice sites) and holes
(i.e., empty lattice sites) are always created in pairs, we get the
same probability for the holes.
Note that $\mathbf{k}$ is the canonical and not necessarily the mechanical
momentum due to the substitution $\mathbf{k}\to\mathbf{k}+\mathbf{A}(t)$
mentioned above.
\section{\label{section-Analogue}Analogue of Sauter-Schwinger Tunnelling}
The precise amount of mixing which determines the Bogoliubov coefficients
$\alpha_\mathbf{k}$ and $\beta_\mathbf{k}$ can be derived from the evolution
equations (\ref{holedgl}) and (\ref{partdgl}).
Turning these two first-order differential equations into one second-order
equation, we get for $g^+_\mathbf{k}$ and $f^+_\mathbf{k}$,
\begin{eqnarray}
\partial_t^2 f^+_\mathbf{k}-
\frac{\dot{T}_\mathbf{k}}{T_\mathbf{k}}\partial_t f^+_\mathbf{k}
+\left(
\frac{\omega^2_\mathbf{k}}{4}+i \frac{U\dot{T}_\mathbf{k}}{2T_\mathbf{k}}
\right)f^+_\mathbf{k}
&=&0
\,,
\\
\partial_t^2 g^+_\mathbf{k}-
\frac{\dot{T}_\mathbf{k}}{T_\mathbf{k}}\partial_t g^+_\mathbf{k}
+\left(
\frac{\omega^2_\mathbf{k}}{4}-i \frac{U\dot{T}_\mathbf{k}}{2T_\mathbf{k}}
\right)g^+_\mathbf{k}
&=&0
\,.
\end{eqnarray}
With the substitutions $f^+_\mathbf{k}=\sqrt{T_\mathbf{k}}u_\mathbf{k}$
and $g^+_\mathbf{k}=\sqrt{T_\mathbf{k}}v_\mathbf{k}$, we may eliminate
the first-order terms and arrive at
\begin{eqnarray}
\partial_t^2 u_\mathbf{k}+
\left(
\frac{\omega^2_\mathbf{k}}{4}+i \frac{U\dot{T}_\mathbf{k}}{2T_\mathbf{k}}+
\frac{\ddot{T}_\mathbf{k}}{2T_\mathbf{k}}-
\frac{3\dot{T}_\mathbf{k}^2}{4T_\mathbf{k}^2}
\right)u_\mathbf{k}&=&0
\,,
\label{dglexakt}
\nonumber\\
\partial_t^2 v_\mathbf{k}+
\left(
\frac{\omega^2_\mathbf{k}}{4}-i \frac{U\dot{T}_\mathbf{k}}{2T_\mathbf{k}}+
\frac{\ddot{T}_\mathbf{k}}{2T_\mathbf{k}}-
\frac{3\dot{T}_\mathbf{k}^2}{4T_\mathbf{k}^2}
\right)v_\mathbf{k}&=&0
\,.
\end{eqnarray}
Now we consider a small tilt of the lattice, corresponding to a weak
electric field $|\mathbf{E}|\ll U$.
In this case, we may approximate the above equations by neglecting the
terms with $\dot{T}_\mathbf{k}$ and $\ddot{T}_\mathbf{k}$ since
$\dot{T}_\mathbf{k}=\,{\cal O}(\mathbf{E})$.
Furthermore, for a weak tilt, the particles have to tunnel across many
lattice sites in order to gain enough energy and to be able to overcome
the energy gap before a particle-hole pair can be created.
Thus, we may consider large length scales, corresponding
to small wavenumbers $\mathbf{k}$ and Taylor expand the
${T}_\mathbf{k}(t)$
\begin{eqnarray}
T_\mathbf{k}(t)=T_{\mathbf{k}+\mathbf{A}(t)}=
1-\xi[\mathbf{k}+\mathbf{A}(t)]^2+\,{\cal O}(\mathbf{k}^4)
\,,
\end{eqnarray}
where $\xi$ is the stiffness.
With these approximations, we find that (\ref{dglexakt}) simplify to
\begin{eqnarray}
\label{klein-gordon}
\partial_t^2 \phi_\mathbf{k}+
\left(
m_{\rm eff}^2 c^4_{\rm eff}+
c^2_{\rm eff}[\mathbf{k}+\mathbf{A}(t)]^2
\right)
\phi_\mathbf{k}=0
\,,
\end{eqnarray}
which is just the Klein-Fock-Gordon equation describing charged scalar
particles in an external electromagnetic field, provided that we identify
the effective speed of light
\begin{eqnarray}
\label{c_eff}
c^2_{\rm eff}= \frac{\xi}{2}\,J(3U-J)
\,,
\end{eqnarray}
while the effective mass is given by half the energy gap $\Delta{\cal E}$
\begin{eqnarray}
\label{m_eff}
m_{\rm eff}^2 c^4_{\rm eff}=\frac{1}{4}(U^2-6 J U+J^2)
\,.
\end{eqnarray}
Consequently, there is a quantitative analogy between the tilted
Bose-Hubbard lattice and the Sauter-Schwinger effect, i.e.,
electron-positron pair creation out of the quantum vacuum due to
an external electric field, sketched in Fig.~\ref{sauter-analogy}
and the following table:
\bigskip
\begin{center}
\begin{tabular}{|c|c|}
\hline
Sauter-Schwinger effect & \;Bose-Hubbard model\; \\
\hline
electrons \& positrons & particles \& holes \\
Dirac sea & Mott state \\
\;mass of electron/positron\; & energy gap $\Delta{\cal E}$ \\
electric field $\mathbf{E}$ & lattice tilt $V_\mu$ \\
speed of light $c$ & velocity $c_{\rm eff}$ \\
\hline
\end{tabular}
\end{center}
\bigskip
\begin{figure}[h]
\begin{center}
\includegraphics[width=.8\columnwidth]{analog.eps}
\end{center}
\psfrag{test}{$\Delta\cal E$}
\caption{Sketch of the analogy:
a) Dirac sea for $E=0$,
b) Sauter-Schwinger tunneling for $E\neq0$,
c) Mott state with energy gap $\Delta\cal E$,
d) tunneling in tilted lattice.}
\label{sauter-analogy}
\end{figure}
We can now use this analogy to apply our knowledge of the
Sauter-Schwinger effect \cite{S31,S51,K65,BI70,D09,SGD08}
to particle-hole creation in the
tilted Bose-Hubbard model -- as Richard Feynman said:
{\em The same equations have the same solutions.}
For example, consider a purely time-dependent electric field of the
following form
\begin{eqnarray}
\label{sauterpulse}
\mathbf{E}=\frac{E_0\mathbf{e}_z}{\cosh^2\left(t/\tau\right)}
\,.
\end{eqnarray}
Such a profile is called Sauter pulse since F.~Sauter was the first to
realise (already in 1931) that the Dirac equation and the Klein-Fock-Gordon
equation in the presence of such a field can be solved exactly
in terms of hypergeometric functions
(although he considered the form with $t$ and $x$ interchanged).
From the exact solution of the scalar field case, one obtains
\cite{KLY08,GG96}
\begin{eqnarray}
\label{sauter}
|\beta_\mathbf{k}|^2
=
\frac{\cosh\left(\pi\tau[\omega_+-\omega_-]\right)+
\cosh\left(\pi\sqrt{4 E_0^2 c^2\tau^4-1}\right)}
{2\sinh(\pi \tau \omega_+)\sinh(\pi \tau \omega_-)}
\,,
\end{eqnarray}
with the abbreviations
\begin{eqnarray}
\omega_\pm=\sqrt{c^2(k_z\mp E_0\tau)^2+m_e^2c^4+k_\perp^2 c^2}
\,.
\end{eqnarray}
Here $k_\perp$ denotes the momentum perpendicular to the electric field
and $m_e$ is the mass of the electron.
Via the analogy established above, expression~(\ref{sauter}) yields the
momentum dependent particle-hole pair creation probability in a Mott
state subject to a time-dependent tilt according to Eq.~(\ref{sauterpulse}).
For various pulse lengths $\tau$, this result plotted in
Fig.~\ref{Pexc-N_12-L_12} and compared to numerical results for a
one-dimensional Bose-Hubbard lattice.
In the static limit $\tau\rightarrow\infty$, Eq.~(\ref{sauter}) reproduces
the well-known expression
\begin{eqnarray}
\label{sauter-inf}
|\beta_\mathbf{k}|^2
=
\exp\left(-\pi\frac{m_e^2 c^4+k_\perp^2 c^2}{E_0 c}\right)
\,.
\end{eqnarray}
As we see, the electron-positron pair creation probability is
exponentially suppressed for small electric fields $E_0$.
Inserting the translation formula (\ref{c_eff}) and (\ref{m_eff}),
we get the same exponential suppression for the particle-hole pair
creation probability via tunnelling in tilted Mott lattices.
Thus, in order to actually verify this prediction experimentally,
the tilt should not be too small.
In this case, the terms with $\dot{T}_\mathbf{k}$ and
$\ddot{T}_\mathbf{k}$ we have neglected earlier due to
$\dot{T}_\mathbf{k}=\,{\cal O}(\mathbf{E})$ might start to play a role.
Thus, let us estimate the impact of these contributions.
Including the terms involving $\dot{T}_\mathbf{k}$ and $\ddot{T}_\mathbf{k}$,
we find
\begin{eqnarray}
\partial_t^2 u_\mathbf{k}
+
\left[
m_{\rm eff}^2 c^4_{\rm eff}+k_\perp^2 c^2_{\rm eff}
+c^2_{\rm eff}(k_z-E_0 t)^2
\right]u_\mathbf{k}
&&
\nonumber\\
+\xi
\left[-E_0^2+i E_0 U(k_z-E_0 t)\right]
u_\mathbf{k}
&=&0\,,
\end{eqnarray}
where we have assumed a constant field ($\tau\rightarrow\infty$)
for simplicity.
The above differential equation can be solved in terms of parabolic
cylindrical functions from which the pair creation probability
is determined to be
\begin{eqnarray}\label{staticfield}
|\beta_\mathbf{k}|^2
\approx
\exp\left[
-\frac{\pi}{E_0c_{\rm eff}}
\left(
m_{\rm eff}^2 c^4_{\rm eff}+c^2_{\rm eff} k_\perp^2
-\xi E_0^2
+\xi^2\frac{E_0^2 U^2}{4c^2_{\rm eff}}
\right)
\right]\,.
\end{eqnarray}
In Fig.~\ref{static}, we depicted the dependence of the
particle-hole creation probability
$\langle\hat{p}^\dagger_\mu\hat{p}_\mu\rangle=
\sum_\mathbf{k}|\beta_\mathbf{k}|^2/N$ on the potential gradient.
\begin{figure}[h]
\begin{center}
\includegraphics{static.ps}
\caption{Dependence of $-\ln(\langle\hat{p}^\dagger_\mu\hat{p}_\mu\rangle)$
on the lattice tilt for $J/U=0.1$.
The black line represents the standard result (\ref{sauter-inf})
for the static Sauter-Schwinger effect while the red curve deviates
due to perturbative corrections in $E_0$ given by the lattice structure,
see equation (\ref{staticfield}).}
\label{static}
\end{center}
\end{figure}
\section{Second Order in $1/Z$}\label{Z2}
So far, we have only considered the first order in $1/Z$.
Now let us discuss the effect of higher orders by means of few examples.
Let us go back to the derivation from (\ref{two-sites}) to
(\ref{two-sites-approx}) and include $1/Z^2$ corrections.
To achieve this level of accuracy, we should not replace the exact
on-site density matrix $\hat\rho_\mu$ by it lowest-order approximation
$\hat\rho_\mu^0$ but include its first-order corrections in
(\ref{depletion}), i.e., the quantum depletion $f_0=\,{\cal O}(1/Z)$ of the unit
filling (Mott) state.
This results in a renormalisation of the eigenfrequency
\begin{eqnarray}\label{renomega}
\omega_\mathbf{k}^{\rm ren}
=
\sqrt{U^2-6JT_\mathbf{k}(1-3f_0)+J^2T_\mathbf{k}^2(1-3f_0)^2}
\,,
\end{eqnarray}
which indicates a shift of the Mott-superfluid transition to slightly
higher values of $J$,
\begin{eqnarray}
J_{\rm crit}^{\rm ren}
=
U\,\frac{3-2\sqrt{2}}{1-3f_0}>U(3-2\sqrt{2})
\,,
\end{eqnarray}
see Appendix \ref{secondorder}.
Since the net effect can roughly be understood as a reduction of the
effective hopping rate $J^{\rm ren}=J(1-3f_0)$, it is easy to visualise
that this implies also a decrease of the effective propagation velocity.
There are also other $1/Z^2$ corrections in (\ref{two-sites-approx}) such as
the three-point correlator $\hat\rho^{\rm corr}_{\mu\nu\kappa}$ but they act
as source terms and do not affect the eigenfrequency (at second order).
However, there are other quantities where these source terms are crucial.
In particular we consider correlation functions which are of the form
\begin{equation}
F_{\cal O}(\mu,\nu)
=
\langle
\hat{\cal O}_\mu
\hat{\cal O}_\nu
\rangle
-
\langle
\hat{\cal O}_\mu
\rangle
\langle
\hat{\cal O}_\nu
\rangle
\;,
\end{equation}
and vanish to first order in $1/Z$, in contrast to the off-diagonal
long-range order
$\langle\hat{a}_\mu^\dagger \hat{a}_\nu\rangle$ discussed above.
One important example is the particle-number correlation, i.e.,
$\langle \hat{n}_\mu \hat{n}_\nu\rangle-1$.
After a somewhat lengthy calculation, we find for the ground state
correlations (see Appendix \ref{secondorder})
\begin{eqnarray}\label{number}
F_{n}(\mu,\nu)
&=&
\frac{2}{N^2}\sum_{\mathbf{p},\mathbf{q}}
e^{i(\mathbf{p}+\mathbf{q})\cdot(\mathbf{x}_\mu-\mathbf{x}_\nu)}
\left(f^{11}_\mathbf{p}f^{11}_\mathbf{q}-f^{12}_\mathbf{p}f^{21}_\mathbf{q}
\right)
\,,
\end{eqnarray}
where $f^{12}_\mathbf{p},f^{21}_\mathbf{p}$ and
$f^{11}_\mathbf{p}$ are given through equations (\ref{stat}) and (\ref{inv}).
It is also possible to calculate this quantity via a perturbation expansion
into powers of $J/U$.
Note, however, that the above result is not perturbative in $J/U$, see,
for example, the non-polynomial dependence of $\omega_\mathbf{k}$ on $J$.
These predictions could be tested experimentally by site-resolved imaging,
i.e., measurements on single lattice sites \cite{B10,W09,GZHC09,S10}.
In some of these experiments, the particle number per site is not directly
measured, but only the parity -- i.e., whether the number of particles on
a given lattice site is even or odd \cite{CBPE12}.
The parity correlator reads
\begin{eqnarray}
\label{parity}
F_{(-1)^n}(\mu,\nu)
&=&
\frac{8}{N^2}\sum_{\mathbf{p},\mathbf{q}}
e^{i(\mathbf{p}+\mathbf{q})\cdot(\mathbf{x}_\mu-\mathbf{x}_\nu)}
\left(f^{11}_\mathbf{p}f^{11}_\mathbf{q}+f^{12}_\mathbf{p}f^{21}_\mathbf{q}
\right)
\,.
\end{eqnarray}
Again, this result can be compared with a perturbative expansion into
powers of $J/U$.
Assuming a hyper-cubic lattice ${\mathbb Z}^D$ in $D$ dimensions with
nearest-neighbour hopping (i.e., $Z=2D$), we obtain up to quartic order
in $J$
\begin{eqnarray}
F_{(-1)^n}(\mu,\nu)
&=&
\left(\frac{J}{ZU}\right)^2 8n(n+1)+
\nonumber\\
&&
+\left(\frac{J}{ZU}\right)^4\frac{2 n (1 + n)}{3}
\left[n(n+1)(70 - 208 Z + 48 Z^2)+
\right.
\nonumber \\
&&
\quad
\left.
+4 - 22 Z + 9 Z^2 \right]
+\,{\cal O}(J^6)
\,,
\end{eqnarray}
where $\mu$ and $\nu$ are nearest neighbours and
$n=\langle\hat{n}_\mu\rangle$ is an arbitrary (integer) filling.
Inserting $n=1$ and keeping only the lowest-order $1/Z^2$ terms, we may
compare this result with (\ref{parity}), after an expansion into powers
of $J$, and find perfect agreement.
However, there is an interesting observation regarding the above equation:
In one spatial dimension with $Z=2$ nearest neighbours, the $J^4$
contribution in the above equation is negative.
This suggests that the parity correlator assumes its maximum at a finite
value of $J$ (in the Mott phase), which can indeed be confirmed by numerical
simulations, see, e.g., \cite{E11} and Section \ref{numerical}.
In two or more spatial dimensions, the situation is different.
Even though the parity correlator should still assume its maximum at some
finite value of $J$, this value is quite close to the phase transition or
already in the superfluid regime.
Thus, this maximum is not visible in our $1/Z$-expansion starting in the
Mott state, which predicts a monotonously increasing parity correlation
in its region of applicability.
In analogy to Sections \ref{superquench} and \ref{equilibration},
we can also study the correlations after a quantum quench with
$J(t)=J\Theta(t)$.
Again, there are no contributions to the particle-number and parity
correlations in first order $1/Z$ -- but, to second order $1/Z$,
we find formally the same expressions as in the static case
(\ref{parity}) and (\ref{number})
\begin{eqnarray}
\label{numberquench}
F_{n}(\mu,\nu)
&=&
\frac{2}{N^2}\sum_{\mathbf{p},\mathbf{q}}
e^{i(\mathbf{p}+\mathbf{q})\cdot(\mathbf{x}_\mu-\mathbf{x}_\nu)}
\left(
f^{11}_\mathbf{p}(t)f^{11}_\mathbf{q}(t)-
f^{12}_\mathbf{p}(t)f^{21}_\mathbf{q}(t)
\right)
\,,
\end{eqnarray}
and
\begin{eqnarray}
\label{parityquench}
F_{(-1)^n}(\mu,\nu)
&=&
\frac{8}{N^2}\sum_{\mathbf{p},\mathbf{q}}
e^{i(\mathbf{p}+\mathbf{q})\cdot(\mathbf{x}_\mu-\mathbf{x}_\nu)}
\left(
f^{11}_\mathbf{p}(t)f^{11}_\mathbf{q}(t)
+f^{12}_\mathbf{p}(t)f^{21}_\mathbf{q}(t)
\right)
\,,
\end{eqnarray}
where $f^{12}_\mathbf{p}(t)$,$f^{21}_\mathbf{p}(t)$ and $f^{11}_\mathbf{p}(t)$
are now given by equations (\ref{quench-h+h}) and (\ref{quench-h+p}).
The parity correlations after a quench have been experimentally observed
in a one-dimensional setup \cite{CBPE12}.
Although the hierarchical expansion relies on a large coordination number,
we find qualitative agreement between the theoretical prediction
(\ref{parityquench}) for $Z=2$ and the results from \cite{CBPE12}.
For large times $t$ and distances $\mathbf{x}_\mu-\mathbf{x}_\nu$,
we may estimate the integrals over $\mathbf{p}$ and $\mathbf{q}$ in
the expressions (\ref{numberquench}) and (\ref{parityquench}) via the
stationary-phase or saddle-point approximation.
The dominant contributions stem from the momenta satisfying the
saddle-point condition
\begin{eqnarray}
\label{statphase}
\nabla_\mathbf{k}
\left[
\mathbf{k}\cdot(\mathbf{x}_\mu-\mathbf{x}_\nu)\pm\omega_\mathbf{k}t
\right]
=0
\,.
\end{eqnarray}
Thus their structure is determined by the group velocity
$\mathbf{v}_\mathbf{k}=\nabla_\mathbf{k}\omega_\mathbf{k}$.
If the equation $\mathbf{x}_\mu-\mathbf{x}_\nu=\pm\mathbf{v}_\mathbf{k}t$
has a real solution $\mathbf{k}$, i.e., if the distance
$\mathbf{x}_\mu-\mathbf{x}_\nu$ can be covered in the time $t$ with the
group velocity $\mathbf{v}_\mathbf{k}$, then we get a stationary-phase
solution -- otherwise the integral will be exponentially suppressed
(i.e., the saddle point $\mathbf{k}$ becomes complex).
For small values of $J$, the maximum group velocity is given by
$\mathbf{v}_\mathbf{k}^{\rm max}\approx3J$, which determines the
maximum propagation speed of the correlations, i.e., the effective
light-cone.
\section{Numerical Simulations for the Bose-Hubbard model}\label{numerical}
In the following we analyze the Bose-Hubbard system
(\ref{Bose-Hubbard-Hamiltonian})
numerically in one and two dimensions.
All calculations are carried out on a finite lattice with $L$
lattice sites and $N$ bosons.
\subsection{General formalism for the one-dimensional Hubbard model}
The eigenstates of lattice systems are calculated by means of
exact numerical diagonalisation of the Hamiltonian
matrix~\cite{EKO,KPS,KPPS,KS,RB,RB04,KT08,HSTR,ZD}
which can be obtained from the Hamiltonian (\ref{Bose-Hubbard-Hamiltonian})
using the basis of Fock states
\begin{equation}
\label{Fock}
|{\bf n}_\Gamma\rangle
=
\bigotimes_{\mu=1}^L
|n_{\Gamma\mu}\rangle
\;,
\quad
\Gamma=1,\dots,{\cal D}
\;,\quad
{\cal D}=\frac{(N+L-1)!}{N!(L-1)!}
\;,
\end{equation}
where $\Gamma$ labels the configuration of the bosons and the occupation
numbers
of individual lattice sites $n_{\Gamma\mu}$ satisfy the condition
\begin{equation}
N
=
\sum_{\mu=1}^L
n_{\Gamma\mu}
\;.
\end{equation}
The matrix dimension can be reduced by factor $L$ for homogeneous lattices
with periodic boundary conditions ($\hat b_{L+1}\equiv\hat b_{1}$).
In this case the Hamiltonian commutes with the unitary translation operator
$\hat{\cal T}$ which acts through cyclic permutation on the lattice bosons.
Due to the periodic boundary conditions, the operator satisfies
$\hat{{\cal T}}^L=1$.
As a basis one can use linear combinations of the Fock states~(\ref{Fock})
in the form
\begin{equation}
\label{basis}
|{\bf n}_{K\Gamma}\rangle
=
{\cal N}_\Gamma
\sum_{\mu=1}^{L}
\left(
\frac{\hat{\cal T}}{\tau_K}
\right)^{\mu-1}
|{\bf n}_\Gamma\rangle
\;,
\end{equation}
which are eigenstates of the operator $\hat{{\cal T}}$ for the eigenvalue
$
\tau_K
=
\exp
\left(
i K
\right)
$.
${\cal N}_\Gamma$ are normalisation constants chosen such that
$\langle{\bf n}_{K\Gamma}|{\bf n}_{K'\Gamma'}\rangle =
\delta_{\Gamma\Gamma'}\delta_{KK'}$.
Here the state $|{\bf n}_\Gamma\rangle$ cannot be obtained by cyclic
permutations of
$|{\bf n}_{\Gamma'}\rangle$ with $\Gamma'\ne\Gamma$.
The eigenstates of the Hamiltonian have the following form
\begin{equation}
|K\Omega\rangle
=
\sum_{\Gamma=1}^{{\cal D}_K}
C_{K\Omega\Gamma}
|{\bf n}_{K\Gamma}\rangle
\;,\quad
\Omega=1,\dots,{\cal D}_K
\;,\quad
\sum_{K=0}^{L-1}
{\cal D}_K
=
{\cal D}
\;,
\end{equation}
and the corresponding eigenenergies are denoted by $E_{K\Omega}$.
If the complete eigenvalue problem is solved, one can work out
expectation value of any operator $\hat{\cal O}$
at arbitrary temperature in a canonical ensemble as
\begin{equation}
\langle
\hat{\cal O}
\rangle_T
=
\frac{1}{{\cal Z}(T)}
\sum_{K\Omega}
\langle
K\Omega|\hat{\cal O}|K\Omega
\rangle
\exp
\left(
-\frac{E_{K\Omega}}{ T}
\right)
\end{equation}
with the partition function
\begin{equation}
{\cal Z}(T)
=
\sum_{K\Omega}
\exp
\left(
-\frac{E_{K\Omega}}{T}
\right)
\;.
\end{equation}
Using the set of the basis states (\ref{basis}) we were able to solve
numerically the complete eigenvalue problem for $N=L=9$.
In order to study zero-temperature properties of the system,
it is sufficient to calculate the ground state.
This can be done exactly with the aid
of iterative numerical solvers for sparse matrices of large dimensions
along the lines of \cite{RB}.
We were able to do this up to $N=L=14$.
\subsection{Energy spectrum}\label{energyspectrum}
In the limit of vanishing hopping, $J=0$, the basis states~(\ref{basis}) are
the eigenstates of the Hamiltonian (\ref{Bose-Hubbard-Hamiltonian}) which are,
apart from the ground state, degenerate.
The ground state has equal occupation numbers at each lattice site, that is
$n_{\Gamma\mu}\equiv n$.
It exists only at $K=0$ and has the energy
\begin{equation}
E_{01}=L\frac{U}{2}n(n-1)\,.
\end{equation}
The energy eigenvalues of the degenerated excited states are given through
\begin{equation}
E_{K\Gamma}
=
\sum_{\mu=1}^L
\frac{U}{2}
n_{\Gamma\mu}
\left(
n_{\Gamma\mu}-1
\right)
\end{equation}
and do not depend on $K$, corresponding to flat energy bands.
The lowest band contains $L(L-1)$ degenerate eigenstates with the energies
$E_{K\Gamma}=E_{01}+U$.
These states correspond to bosonic configurations with the same
occupation numbers $n$ at any site except two, one of which contains
$n-1$ boson and the other one $n+1$.
The highest band contains $L$ degenerate states with all atoms sitting at one
lattice site.
These states have the energy $E_{K\Gamma}=UN(N-1)/2$.
A finite hopping rate $J$ lifts the degeneracy,
the bands aquire finite widths and can even overlap
if the tunneling parameter is large enough.
The full energy spectrum calculated for $N=L=9$ and $J/U=0.1$,
which corresponds to the Mott-insulator state,
is shown in Fig.~\ref{sp_01}(a).
The lowest dot at $K=0$ is the ground state energy $E_{01}$,
see also Fig.~\ref{sp_01}(b).
Also the lowest excited state is located at $K=0$ and has the energy $E_{02}$.
Together with the energies $E_{K2}$, where $K\ne0$, they form the lowest
excitation branch shown by the black lines in Figs.~\ref{sp_01}(b,c).
An increase of the system size leads to more dense distribution of the
points and the solid line becomes smoother.
We see that, at small momenta $K$, the lowest excitation branch
can be approximated by a pseudo-relativistic form
\begin{equation}
\label{rel}
\omega_{\bf K}^2
\equiv
(E_{K2}-E_{01})^2
=
(\Delta{\cal E})^2
+
K^2
v_{\rm eff}^2
\;.
\end{equation}
Thus we can estimate the effective velocity $v_{\rm eff}$ using the
numerically calculated values of $E_{01}$, $E_{02}$ and $E_{11}$.
The results for different system sizes are shown in Fig.~\ref{sp_01}(d).
With the increase of $J/U$ the energy bands become broader and
the gap in the excitation spectrum $E_{02}-E_{01}$ becomes smaller.
The energy eigenvalues in the lowest band in Fig.~\ref{sp_01}
correspond to particle-hole excitations of the form
$\hat{p}^\dagger_{k_p}\hat{h}_{k_h}|\Psi_\mathrm{Mott}\rangle$
with the total momentum $k_p+k_h=K$.
When discussing the ground-state properties or the dynamics after a quench
in the previous Sections, it was sufficient to consider translationally
invariant states with $K=0$, i.e., $k_p=-k_h=k$, where $k$ corresponds to
the relative momentum.
In the discussion of the Sauter-Schwinger analogue, we considered a
spatially constant potential gradient and absorbed it into the
Fourier coefficients $T_\mathbf{k}(t)$ via a Peierls transformation,
finally arriving at the evolution equations (\ref{holedgl}) and
(\ref{partdgl}) for particle and hole operators with $k_p=-k_h=k$.
However, for arbitrary potentials $V_\mu$, this is no longer possible
in this simple form.
In order to satisfy the equations of motion (\ref{f12-Mott}-\ref{f11-Mott})
for the correlation functions for an arbitrary potential $V_\mu$,
one should employ the generalised evolution equations
\begin{eqnarray}
\left(i\partial_t-V_\mu+\frac{U}{2}\right)\hat{h}_\mu
&=&
\frac{J}{Z}\sum_\kappa T_{\mu\kappa}
\left(\hat{h}_\kappa+\sqrt{2}\hat{p}_\kappa\right)\,,
\label{holegen}\\
\left(i\partial_t-V_\mu-\frac{U}{2}\right)\hat{p}_\mu
&=&
-\frac{\sqrt{2}J}{Z}\sum_\kappa T_{\mu\kappa}
\left(\hat{h}_\kappa+\sqrt{2}\hat{p}_\kappa\right)
\label{partgen}\,.
\end{eqnarray}
Here the particle and hole-operators are fixed up to a $k_{h,p}$-independent
phase.
For the limiting case $V_\mu\to0$ we find from Eqs.~(\ref{holegen}) and
(\ref{partgen}) the following eigenfrequencies
\begin{eqnarray}
\omega^h_{k_h}&=&-\frac{1}{2}(JT_{k_h}+\omega_{k_h})\,,
\\
\omega^p_{k_p}&=&-\frac{1}{2}(JT_{k_p}-\omega_{k_p})\,,
\end{eqnarray}
where $\omega_{k}$ are the eigenfrequencies defined in
(\ref{eigen-frequency}).
These excitations define the lowest band with the energies
\begin{eqnarray}
\label{lowest-band}
E_{K\Omega}=\omega^p_{k_p}-\omega^h_{k_h}\,,
\end{eqnarray}
where $k_h+k_p=K$.
The spectrum is depicted in Fig.~\ref{partholeband} and the effective
velocity reads
\begin{eqnarray}
v_\mathrm{eff}=\frac{1}{2}\sqrt{J(3 U-J)}
\end{eqnarray}
which has for $J/U=0.1$
the numerical value $v_\mathrm{eff}/U=0.269$.
Including the second order corrections in $1/Z^2$ we have a slightly
lower value $v_\mathrm{eff}/U=0.264$, compare Fig.~\ref{sp_01}(d).
\begin{figure}[t]
\includegraphics[width=15cm]{arxiv2.eps}
\caption{
{\bf(a)} Full energy spectrum and {\bf(b)} its lowest part for $N=L=9$.
{\bf(c)} Lowest part of the spectrum for $N=L=13$.
{\bf(d)} Effective velocity in a one-dimensional lattice with $n=1$
atom per site and
$J/U=0.1$.
}
\label{sp_01}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=10cm]{band.ps}
\caption{Boundaries of the lowest energy excitation band in the continuum limit (solid lines) and
energy excitations for $L=9$ (points) from Eq.~(\ref{lowest-band}) in a one-dimensional lattice
with $n=1$ and $J/U=0.1$.}
\label{partholeband}
\end{center}
\end{figure}
\subsection{Probability distribution of the occupation numbers}
We calculate the ground state and then the probabilities $p(n_\mu)$
to have $n_\mu$ atoms at a lattice site which satisfy the normalisation
condition
\begin{equation}
\sum_{n_\mu=0}^N
p(n_\mu)
=1
\;.
\end{equation}
As in the previous Section, we consider the system with $N=L$.
From Eqs. (\ref{Mott-state}) and (\ref{superfluid-state}) it follows that
in the limit $J=0$ we have $p(n_\mu)=\delta_{n_\mu,1}$,
and in the opposite limit $U=0$ the probabilities are given by the binomial distribution
\begin{equation}
p(n_\mu)
=
\frac{N!}{(N-n_\mu)! n_\mu!}\left(\frac{1}{N}\right)^{n_\mu}\left(1-\frac{1}{N}\right)^{N-n_\mu}
\;.
\end{equation}
The result for $N=14$ at arbitrary $J/U$ and at zero temperature
is shown in Fig.~\ref{p_n}.
One can clearly see that the probability to have three particles or more at one
lattice site is very small which is consistent with the approximations
used in the $1/Z$ expansion.
The particle-number distribution at finite temperature is shown in
Fig.~\ref{p_n-T}(b).
Comparison with the zero-temperature result for the same system size
[Fig.~\ref{p_n-T}(a)]
indicates that temperature has stronger influence at smaller values of $J/U$.
\begin{figure}[t]
\psfrag{J}[c]{$J/U$}
\psfrag{p}[b]{$p(n_\mu)$}
\begin{center}
\includegraphics[width=7cm]{p1D-N_14-L_14-Z.eps}
\end{center}
\caption{Probabilities to have $n_\mu=0$ (red), $1$ (green), $2$ (blue),
$3$ (magenta)
atoms at a lattice site in a one-dimensional lattice
of $L=14$ sites with $n=1$ atom per site at zero temperature.
}
\label{p_n}
\end{figure}
\begin{figure}[t]
\psfrag{J}[c]{$J/U$}
\psfrag{p}[b]{$p(n_\mu)$}
\psfrag{a}[c]{\bf(a)}
\includegraphics[width=7cm]{p1D-N_9-L_9-Z.eps}
\hspace{1cm}
\psfrag{b}[c]{\bf(b)}
\includegraphics[width=7cm]{p1D-N_9-L_9-T_0.3-Z.eps}
\psfrag{a}[c]{\bf(c)}
\includegraphics[width=7cm]{p2D-N_9-L_9-Z.eps}
\hspace{1cm}
\psfrag{b}[c]{\bf(d)}
\includegraphics[width=7cm]{p2D-N_9-L1_3-L2_3-T_0.3-Z.eps}
\caption{Probabilities to have $n_\mu=0$ (red), $1$ (green), $2$ (blue),
$3$ (magenta)
atoms at a site in a lattice with $n=1$ atom per site.
{\bf(a)} and {\bf(b)}: one-dimensional lattice of $L=9$ sites;
{\bf(c)} and {\bf(d)}: two-dimensional lattice of $3\times 3$ sites;
{\bf(a)} and {\bf(c)}: zero temperature;
{\bf(b)} and {\bf(d)}: $T/U=0.3$.
}
\label{p_n-T}
\end{figure}
\subsection{Two-point correlation functions}
In this Subsection, we consider two-point correlation functions
which have been discussed in Section~\ref{Z2}.
Due to the translational invariance, the correlation functions depend
on the distance $s=|x_\mu-x_\nu|$ and have the property
$F_{\cal O}(\mu,\nu)=F_{\cal O}(s)=F_{\cal O}(L-s)$
in view of the periodic boundary conditions.
First we consider the parity correlation function
$F_{(-1)^n}(s)$.
Its dependence on $J/U$ as well as on the distance $s$ is shown
in Fig.~\ref{par-dd}.
This result is in a very good quantitative agreement with
the DMRG-calculations~\cite{E11}.
Note that our definition of $J$ differs by a factor $Z$ from that
used in \cite{E11}.
Fig.~\ref{par-dd} shows the number correlation function
$F_{n}(s)$.
\begin{figure}[t]
\psfrag{J}[c]{$J/U$}
\psfrag{par}[b]{$F_{(-1)^n}(s)$}
\psfrag{a}[c]{\bf(a)}
\includegraphics[width=7cm]{par1D-N_14-L_14-Z.eps}
\hspace{1cm}
\psfrag{n}[c]{$s$}
\psfrag{b}[c]{\bf(b)}
\includegraphics[width=7cm]{par1Dlog-J_0.05-N_14-L_14.eps}
\psfrag{J}[c]{$J/U$}
\psfrag{dd}[b]{$F_{n}(s)$}
\psfrag{a}[c]{\bf(c)}
\includegraphics[width=7cm]{dd1D-N_14-L_14-Z.eps}
\hspace{1cm}
\psfrag{n}[c]{$s$}
\psfrag{ddm1}[b]{$\left|F_{n}(s)\right|$}
\psfrag{b}[c]{\bf(d)}
\includegraphics[width=7cm]{dd1Dlog-J_0.05-N_14-L_14.eps}
\caption{
Parity correlation [{\bf (a)} and {\bf (b)}]
and density-density correlation [{\bf (c)} and {\bf (d)}]
in a one-dimensional lattice of $L=14$ sites with $n=1$
atom per site at zero temperature.
{\bf (a)} and {\bf (c)}: dependence on $J/U$ for
$s=1$ (red), $2$ (green), $3$ (blue), $4$ (magenta);
{\bf (b)} and {\bf (d)}: Dependence on $s$ for $J/U=0.1$.
Due to the periodic boundary conditions
correlation functions increase starting from $s=7$.
}
\label{par-dd}
\end{figure}
Fig.~\ref{obdm} shows matrix elements of the one-body density matrix
$\langle \hat b_\mu^\dagger \hat b_{\mu+s} \rangle$
as well as its momentum distribution (\ref{momdist}).
In a finite lattice the quasi-momentum takes discrete values which are integer
multiples of $2\pi/L$. These allowed values are marked in Fig.~\ref{obdm}(a)
by dots.
The momentum distribution calculated in the first order of $1/Z$ is also
shown for comparison.
We observe good quantitative agreement.
\begin{figure}[t]
\psfrag{k}[c]{$k$}
\psfrag{pdis}[b]{$P(k)$}
\psfrag{b}[c]{\bf(b)}
\psfrag{a}[c]{\bf(a)}
\includegraphics[width=7cm]{md1D-J_0.05-N_14-L_14.eps}
\hspace{1cm}
\psfrag{dm}[b]{$\langle \hat b_\mu^\dagger \hat b_{\mu+s} \rangle$}
\psfrag{n}[c]{$s$}
\includegraphics[width=7cm]{obdm1Dlog-J_0.05-N_14-L_14.eps}
\caption{
{\bf(a)}: Momentum distribution (\ref{momdist}) in a one-dimensional lattice
with $n=1$ atom per site at $J/U=0.1$ and zero temperature.
The result obtained by exact diagonalisation for a finite
lattice of $L=14$ is shown by dots. The solid line is a calculation
in the first order of $1/Z$ for an infinite lattice.
{\bf(b)}:
Correlation function $\langle\hat b_\mu^\dagger\hat b_{\mu+s}\rangle$
calculated by exact diagonalisation for a finite lattice with the
same values of parameters as in {\bf(a)}.
}
\label{obdm}
\end{figure}
\subsection{Particle-number distribution and correlation functions in 2D}
The whole procedure of exact numerical diagonalisation described for
one-dimensional
systems can be generalised to higher dimensions.
We did numerical simulations for two-dimensional square lattices of $3\times3$
lattice sites with periodic boundary conditions.
Exact calculations for square lattices of the size $4\times4$ and larger
were not
possible due to the problem with computer memory.
Due to the fact that the size of the two-dimensional system is very small
one can expect strong finite-size effects. However, as we will see later
numerical calculations give qualitatively correct predictions valid for
large systems.
The probability distribution of the occupation numbers is shown
in Figs.~\ref{p_n-T}(c) and (d).
It is very similar to the one-dimensional case.
In a lattice of $3\times 3$ sites with periodic boundary conditions
the distance $s$ takes only three values $s=0,1,\sqrt{2}$ which makes the study
of the long-range behavior of the two-point correlation functions practically
impossible. Nevertheless some useful information can be obtained in the
Mott-insulator
phase where the correlations decay exponentially.
Fig.~\ref{par-dd-obdm-2D}(a)
shows the dependence of the parity correlation function on $J/U$.
As in the one-dimensional case, $F_{(-1)^n}(s)$ has a maximum
at a finite value of $J$, which is, however,
not in the Mott phase.
The results in Fig.~\ref{par-dd-obdm-2D} are in a good agreement with those
obtained in \cite{E11} by DMRG and MPS calculations for large systems.
Correlations $\langle \hat n_\mu \hat n_\nu \rangle$
and $\langle \hat b_\mu^\dagger \hat b_\nu \rangle$
are shown in Figs.~\ref{par-dd-obdm-2D}(b),~\ref{par-dd-obdm-2D}(c),
respectively.
They become stronger for increasing values of $J/U$.
\begin{figure}[t]
\begin{center}
\psfrag{J}[c]{$J/U$}
\psfrag{par}[b]{$F_{(-1)^n}(s)$}
\psfrag{a}[c]{\bf(a)}
\includegraphics[width=7cm]{par2D-N_9-L1_3-L2_3-Z.eps}
\psfrag{J}[c]{$J/U$}
\psfrag{dd}[b]{$F_{n}(s)$}
\psfrag{b}[c]{\bf(b)}
\hspace{0.4cm}\includegraphics[width=7cm]{dd2D-N_9-L1_3-L2_3-Z.eps}
\end{center}
\psfrag{J}[c]{$J/U$}
\psfrag{dm}[b]{$\langle \hat b_\mu^\dagger \hat b_\nu \rangle$}
\psfrag{c}[c]{\bf(c)}
\begin{center}
\includegraphics[width=7cm]{obdm2D-N_9-L1_3-L2_3-Z.eps}
\end{center}
\caption{Parity correlation (a), density-density correlation (b)
and elements of the one-body density matrix (c)
in a two-dimensional lattice of $3\times 3$ sites with $n=1$
atom per site at zero temperature.
$s=1$ (red), $\sqrt{2}$ (green).
}
\label{par-dd-obdm-2D}
\end{figure}
\subsection{Time evolution after quench}
If the complete set of the eigenvalues and eigenstates of the Hamiltonian
is known, the time evolution of an arbitrary initial state $|\psi(0)\rangle$
can be calculated exactly without numerical integration,
provided that the Hamiltonian does not depend explicitly on time.
The initial state can be decomposed into the eigenstates of the Hamiltonian as
\begin{equation}
|\psi(0)\rangle
=
\sum_K
\sum_{\Omega=1}^{{\cal D}_K}
c_{K\Omega}
|K\Omega\rangle
\;,\quad
c_{K\Omega}
=
\langle
\psi(0)|K\Omega
\rangle
\;.
\end{equation}
If the parameters of the Hamiltonian do not depend on time,
the evolution of the initial state is given by
\begin{equation}
|\psi(t)\rangle
=
\sum_K
\sum_{\Gamma=1}^{{\cal D}_K}
c_{K \Gamma}(t)
|{\bf n}_{K\Gamma}\rangle
\end{equation}
with
\begin{equation}
c_{K\Gamma}(t)
=
\sum_{\Omega=1}^{{\cal D}_K}
c_{K\Omega}
C_{K\Omega\Gamma}
\exp
\left(
-i E_{K\Omega} t
\right)
\;.
\end{equation}
The time dependence of the expectation value of any operator
$\hat{\cal O}$ can be
calculated as
\begin{equation}
\langle
\hat{\cal O}
\rangle
(t)
=
\sum_{K\Gamma}
\sum_{K'\Gamma'}
\langle
{\bf n}_{K\Gamma} |\hat{\cal O}| {\bf n}_{K'\Gamma'}
\rangle
c_{K\Gamma}^*(t)
c_{K'\Gamma'}(t)
\;.
\end{equation}
We will be dealing with operators $\hat{\cal O}$ which have the property
\begin{equation}
\langle
{\bf n}_{K\Gamma} |\hat{\cal O}| {\bf n}_{K'\Gamma'}
\rangle
=
\langle
{\bf n}_{K\Gamma} |\hat{\cal O}| {\bf n}_{K\Gamma'}
\rangle
\delta_{KK'}
\;.
\end{equation}
Then the expectation value averaged over the evolution time is given by
\begin{eqnarray}
\overline
{
\langle
\hat{\cal O}
\rangle
}
&=&
\lim_{t\to\infty}
\frac{1}{t}
\int_0^t
\langle
\hat{\cal O}
\rangle(t')
dt'\nonumber\\
& =&
\sum_{K}
\sum_{\Gamma}
\sum_{\Gamma'}
\langle
{\bf n}_{K\Gamma} |\hat{\cal O}| {\bf n}_{K\Gamma'}
\rangle
\sum_{\Omega=1}^{{\cal D}_K}
\left|
c_{K\Omega}
\right|^2
C^*_{K\Omega\Gamma}
C_{K\Omega\Gamma'}
\;.
\end{eqnarray}
We study time evolution of the initial state with exactly one atom at each
lattice site which is the ground state with $K=0$ of the Bose-Hubbard
Hamiltonian
in the limit $J=0$. Since the Hamiltonian after quench preserves the
translational
invariance, the time evolution involves only states with $K=0$.
In Figs.~\ref{pn-1D-t} and \ref{pn_2D-t}
we present numerical results for the particle-number distribution
$p(n_\mu)$ and correlation function
$\langle \hat b_\mu^\dagger \hat b_\nu \rangle$ in one and two dimensions.
The purpose of this study is to address the question of (quasi) equilibration
versus thermalisation in closed quantum systems.
Time evolution of the probabilities $p(n_\mu)$ for one- and two-dimensional
systems is shown
in Figs.~\ref{pn-1D-t}(a),~\ref{pn_2D-t}(a), respectively.
On large time scales they oscillate around the averaged values shown by
straight horizontal lines.
For the chosen value of $J/U=0.1$, the behavior $p(0)$ is almost
indistinguishable from that of $p(2)$.
Figs.~\ref{pn-1D-t}(b),~\ref{pn_2D-t}(b) show the dependence of the
probabilities on the temperature.
Averaged values of the probabilities correspond to the effective
temperature which is slightly less than $0.15~U$.
\begin{figure}[t]
\psfrag{t}[c]{$tU$}
\psfrag{Pn}[b]{$p(n_\mu)$}
\psfrag{a}[c]{\bf(a)}
\includegraphics[width=7cm]{pn1Dt-J_0.05-N_9-L_9-NC_9.eps}
\hspace{1cm}
\psfrag{T}[c]{$T/U$}
\psfrag{b}[c]{\bf(b)}
\includegraphics[width=7cm]{pn1DT-J_0.05-N_9-L_9-NC_9.eps}
\psfrag{OBDM}[b]{$\langle \hat b_\mu^\dagger \hat b_{\mu+s} \rangle$}
\psfrag{t}[c]{$tU$}
\psfrag{a}[c]{\bf(c)}
\includegraphics[width=7cm]{obdm1Dt-J_0.05-N_9-L_9-NC_9.eps}
\hspace{1cm}
\psfrag{T}[c]{$T/U$}
\psfrag{b}[c]{\bf(d)}
\includegraphics[width=7cm]{obdm1DT-J_0.05-N_9-L_9-NC_9.eps}
\caption{Quench and (quasi) equilibration in a one-dimensional lattice
of $L=9$ sites
with $n=1$ atom per site.
{\bf (a)}
Time evolution of the
probabilities to have $n_\mu=0$ (red), $2$ (blue)
atoms at a lattice site after quench $J/U = 0 \to 0.1$.
{\bf (b)}
Probabilities to have $n_\mu=0$ (red), $2$ (blue)
atoms at a lattice site for $J/U=0.1$ as a function of temperature.
Straight horizontal lines in both panels show the values of probabilities
averaged over the infinite evolution time.
{\bf (c)}
Elements of the one-body density matrix with $s=1$ (red), $2$ (green),
after quench $J/U = 0 \to 0.1$.
{\bf (d)}
Elements of the one-body density matrix with $s=1$ (red), $2$ (green),
for $J/U = 0.1$ as a function of temperature.
}
\label{pn-1D-t}
\end{figure}
The time dependence of the correlation functions
$\langle \hat b_\mu^\dagger \hat b_\nu \rangle$
presented in Figs.~\ref{pn-1D-t}(c),~\ref{pn_2D-t}(c)
displays the same oscillating character.
In the one-dimensional system, the effective temperature corresponding
to the averaged values of $\langle \hat b_\mu^\dagger \hat b_\nu \rangle$
can be defined but appears to be higher than that of the probabilities
$p(n_\mu)$.
In contrast, in the two-dimensional case, the correlation functions
$\langle \hat b_\mu^\dagger \hat b_\nu \rangle$ cannot be described by
a thermal state, see Fig.~\ref{pn_2D-t}(d).
The absence of effective temperature in the two-dimensional system is
consistent with the result obtained within the $1/Z$ expansion
in Section \ref{equilibration}.
\begin{figure}[t]
\psfrag{t}[c]{$tU$}
\psfrag{Pn}[b]{$p(n_\mu)$}
\psfrag{a}[c]{\bf (a)}
\includegraphics[width=7cm]{pn2Dt-J_0.025-N_9-L1_3-L2_3-NC_9.eps}
\hspace{1cm}
\psfrag{T}[c]{$ T/U$}
\psfrag{b}[c]{\bf (b)}
\includegraphics[width=7cm]{pn2DT-J_0.025-N_9-L1_3-L2_3-NC_9.eps}
\psfrag{OBDM}[b]{$\langle \hat b_\mu^\dagger \hat b_\nu \rangle$}
\psfrag{t}[c]{$tU$}
\psfrag{a}[c]{\bf (c)}
\includegraphics[width=7cm]{obdm2Dt-J_0.025-N_9-L1_3-L2_3-NC_9.eps}
\hspace{1cm}
\psfrag{T}[c]{$T/U$}
\psfrag{b}[c]{\bf (d)}
\includegraphics[width=7cm]{obdm2DT-J_0.025-N_9-L1_3-L2_3-NC_9.eps}
\caption{Quench and (quasi) equilibration in a two-dimensional lattice
of $3\times 3$ sites with $n=1$ atom per site.
{\bf (a)}
Probabilities to have $n_\mu=0$ (red), $2$ (blue)
atoms at a lattice site after quench $J/U = 0 \to 0.1$.
{\bf (b)}
Probabilities to have $n_1=0$ (red), $2$ (blue)
atoms at a lattice site for $J/U=0.1$ as a function of temperature.
{\bf (c)}
Elements of the one-body density matrix with
$s=1$ (red), $\sqrt{2}$ (green)
after quench $J/U = 0 \to 0.1$.
{\bf (d)}
Elements of the one-body density matrix
with $s=1$ (red), $\sqrt{2}$ (green)
for $J/U=0.1$ as a function of temperature.
}
\label{pn_2D-t}
\end{figure}
\subsection{Tilt in one dimension}
In this Section, we calculate the probability to create a particle-hole
excitation
due to the time-dependent tilt. Initial state of the system $|\psi(0)\rangle$
is the ground state of the Hamiltonian (\ref{Bose-Hubbard-Hamiltonian})
for finite value of $J/U$ in the Mott-insulator phase.
During the time evolution the system is described by the Hamiltonian
(\ref{Bose-Hubbard-Tilt})
with the on-site energies
\begin{equation}
V_\mu
=
E_0
f(t/\tau)
x_\mu
\;,
\end{equation}
where function $f(s)$ has similar form as (\ref{sauterpulse})
\begin{equation}
\label{fs}
f(s)
=
\left\{
\cosh^{-2}
\left(
s-\frac{5}{2}
\right)
-
\cosh^{-2}
\left(
\frac{5}{2}
\right)
\right\}
\left[
1
-
\cosh^{-2}
\left(
\frac{5}{2}
\right)
\right]^{-1}
\,,
\end{equation}
with $0<s<5$,
such that $f(0)=f(5)=0$ and $f(5/2)=1$.
\begin{figure}[t]
\begin{center}
\includegraphics[width=.45\textwidth]{Pexcloglog.ps}
\includegraphics[width=.45\textwidth]{Decaysauterloglog2.ps}
\caption{{\bf a)} Numerical results for the excitation probability per unit time in logarithmic scales
for $N=L=12$ and $J/U=0.1$.
{\bf b)} Analytical results (\ref{sauter}) for the excitation probability per unit time where the
expressions for the effective mass
(\ref{m_eff}) and the effective velocity (\ref{c_eff}) have been used.}
\label{Pexc-N_12-L_12}
\end{center}
\end{figure}
In contrast to all the previous numerical calculation we do not impose anymore
periodic boundary conditions. Instead of that we consider the case when the
particles
cannot tunnel between the lattice sites $\mu=1$ and $\mu=L$ and
perform calculations using the basis of Fock states (\ref{Fock}).
Numerical integration of the Schr\"odinger equation
is done by means of the fourth-order Runge-Kutta method.
The accuracy of the numerical integration is controlled by the conservation
of norm of the state $|\psi(t)\rangle$.
The results for the excitation probability per unit time
\begin{equation}
\label{Pexc}
P_{\rm exc}
=
\frac
{1-\left|\langle\psi(0)|\psi(5\tau)\rangle\right|^2}
{\tau}
\;,
\end{equation}
where $5\tau$ is the total evolution time,
are shown in Fig.~\ref{Pexc-N_12-L_12}(a).
At short evolution times, the excitation probability $P_{\rm exc}$
has a power-law dependence
on $\Delta\epsilon$ which corresponds to the perturbative regime
of the pair production.
One should keep in mind that
finite-size effects start to play an important role
if the evolution time exceeds $L/v_{\rm eff}$, where $v_{\rm eff}$
is an effective
velocity
for the propagation of excitations discussed in Section \ref{energyspectrum}.
For $L=N=12$ and $J/U=0.1$ this leads to the requirement $\tau U < 7$.
At much longer times $\tau$, the dynamics of a finite-size system will
be adiabatic
and the excitation probability will tend to zero in contrast to the
infinite system,
where the excitation probability remains finite in the limit
$\tau\rightarrow\infty$ as determined by Eq.~(\ref{sauter-inf}).
In Fig.~\ref{Pexc-N_12-L_12}(b) we show the results of the same calculations
obtained
in Section~\ref{section-Analogue} in the first order of the $1/Z$-expansion
where
corrections due to time-derivatives of $T_\mathbf{k}$ have been
neglected, see Eq.~(\ref{dglexakt}).
The excitation probability~(\ref{Pexc}) and the particle-hole creation rate are related
via
\begin{eqnarray}
P_{\rm exc}
=
\frac{1-\langle\Psi_\mathrm{Mott}|\hat{\rho}(\infty)|\Psi_\mathrm{Mott}\rangle}{\tau}
\approx
\frac{1- (1-2\langle \hat{p}^\dagger_\mu\hat{p}_\mu\rangle)^N}{\tau}
\approx
2N\frac{\langle \hat{p}^\dagger_\mu\hat{p}_\mu\rangle}{\tau}
\;,
\end{eqnarray}
where
$
\langle \hat{p}^\dagger_\mu\hat{p}_\mu\rangle
=
\sum_{\bf k}
\left|
\beta_{\bf k}
\right|^2/N
$ and $\beta_\mathbf{k}$ is the Bogoliubov coefficient defined in equation
(\ref{sauter}).
We observe a very good qualitative agreement with the results of exact
numerical calculations, although the latter give somewhat
smaller values of $P_{\rm exc}$.
\section{Fermi-Hubbard Model}\label{Fermi-Hubbard Model}
Now, after having studied the bosonic case, let us investigate the
Fermi-Hubbard model \cite{H63,EFGKK05,F91}.
We shall find many similarities to the Bose-Hubbard model -- but also crucial
differences.
The Hamiltonian is given by
\begin{eqnarray}
\label{Fermi-Hubbard-Hamiltonian}
\hat H
=
-\frac{J}{Z}\sum_{\mu\nu,s}T_{\mu\nu}\hat{c}_{\mu,s}^\dagger\hat{c}_{\nu,s}
+U\sum_{\mu}\hat{n}_\mu^\uparrow\hat{n}_\mu^\downarrow
\,.
\end{eqnarray}
The nomenclature is the same as in the bosonic case
(\ref{Bose-Hubbard-Hamiltonian}) but with an additional spin label $s$
which can assume two values $s=\uparrow$ or $s=\downarrow$.
In the following, we consider the case of half-filling
$\langle\hat{n}_\mu^\uparrow+\hat{n}_\mu^\downarrow\rangle=1$
where half the particles are in the $s=\uparrow$ state and the other
have $s=\downarrow$.
Note that the total particle numbers
$\hat N^\uparrow=\sum_\mu \hat n_\mu^\uparrow$
and
$\hat N^\downarrow=\sum_\mu \hat n_\mu^\downarrow$
for each spin species are conserved separately
$[\hat H,\hat N^\uparrow]=[\hat H,\hat N^\downarrow]=0$.
The creation and annihilation operators satisfy the fermionic
anti-commutation relations
\begin{eqnarray}
\label{fermionic-commutation}
\left\{\hat c_{\nu,a},\hat c_{\mu,b}^\dagger\right\}
=\delta_{\mu\nu}\delta_{ab}
\;,\,
\left\{\hat c_{\nu,a},\hat c_{\mu,b}\right\}
=
\left\{\hat c_{\nu,a}^\dagger,\hat c_{\mu,b}^\dagger\right\}
=0
\,.
\end{eqnarray}
The fermionic nature of the particles has important consequences.
For example, let us estimate the expectation value of the hopping
Hamiltonian $\hat H_J$.
Introducing the ``coarse-grained'' operator
\begin{eqnarray}
\label{c-sigma}
\hat c_{\mu,s}^\Sigma
=
\frac{1}{\sqrt{Z}}\sum_{\nu}T_{\mu\nu}\hat{c}_{\nu,s}
\,,
\end{eqnarray}
we may write the expectation value of the tunnelling energy $\hat H_J$
per lattice site for one spin species $s$ as
$-J\langle\hat c_{\mu,s}^\dagger\hat c_{\mu,s}^\Sigma\rangle/\sqrt{Z}$.
This expectation value can be interpreted as a scalar product of the
two vectors $\hat c_{\mu,s}\ket{\Psi}$ and $\hat c_{\mu,s}^\Sigma\ket{\Psi}$
and hence it is bounded by
\begin{eqnarray}
\left|\bra{\Psi}\hat c_{\mu,s}^\dagger\hat c_{\mu,s}^\Sigma\ket{\Psi}\right|
\leq
||\hat c_{\mu,s}\ket{\Psi}||\cdot||\hat c_{\mu,s}^\Sigma\ket{\Psi}||
\,.
\end{eqnarray}
Inserting $||\hat c_{\mu,s}\ket{\Psi}||^2=
\bra{\Psi}\hat c_{\mu,s}^\dagger\hat c_{\mu,s}\ket{\Psi}=
\bra{\Psi}\hat n_{\mu,s}\ket{\Psi}$, we get the expectation value of the
number operator $\hat n_{\mu,s}$.
In contrast to the bosonic case, this operator is bounded and thus we find
$||\hat c_{\mu,s}\ket{\Psi}||\leq1$.
Furthermore, the operator $\hat c_{\mu,s}^\Sigma$ in (\ref{c-sigma}) obeys
the same anti-commutation relations (\ref{fermionic-commutation}) and thus
we find $||\hat c_{\mu,s}^\Sigma\ket{\Psi}||\leq1$ in complete analogy.
Consequently, the absolute value of the tunnelling energy per lattice site
is below $2J/\sqrt{Z}$, i.e., decreases for large $Z$.
The above result implies that the interaction term $\propto U$ always
dominates (except in the trivial case $U=0$) in the limit $Z\to\infty$
under consideration.
Hence, we are in the strongly interacting Mott regime and do not find
anything analogous to the Mott--superfluid transition as in the bosonic
case.
Note that often \cite{MV89,F99} a different $Z$-scaling is considered,
where the hopping term scales with $J/\sqrt{Z}$ instead of $J/Z$ as in
(\ref{Fermi-Hubbard-Hamiltonian}).
Using this $J/\sqrt{Z}$ scaling, one can study the transition from the
Mott state to a metallic state which is supposed to occur at a critical
value of $J$ where -- roughly speaking -- the hopping term starts to
dominate over the interaction term.
However, this transition is not as well understood as the Mott--superfluid
transition in the bosonic case.
With our $J/Z$-scaling in (\ref{Fermi-Hubbard-Hamiltonian}), we study a
different corner of the phase space where we can address question such
as tunnelling in tilted lattices and equilibration vs thermalisation etc.
\subsection{Symmetries and Degeneracy}
In addition to the usual invariances already known from the bosonic case,
the Fermi-Hubbard model has some more symmetries.
For example, the particle-hole symmetry
$\hat c_{\mu,s}^\dagger\leftrightarrow\hat c_{\mu,s}$ and thus
$\hat n_{\mu,s}=\hat c_{\mu,s}^\dagger\hat c_{\mu,s}
\leftrightarrow
\hat{\bar n}_{\mu,s}=\hat c_{\mu,s}\hat c_{\mu,s}^\dagger
=1-\hat n_{\mu,s}$
is no longer an effective approximate symmetry, but becomes exact
(for the case of half-filling considered here).
Furthermore, there is an effective $SU(2)$-symmetry corresponding to the
spin degrees of freedom.
To specify this, let us introduce the effective spin operators
\begin{eqnarray}
\label{spin-operators}
\hat S_\mu^z
=
\frac12
\sum\limits_{ab}
\hat c_{\mu,a}^\dagger\,
\sigma^z_{ab}\,
\hat c_{\mu,b}
=
\frac12
\left(\hat{n}_\mu^\uparrow-\hat{n}_\mu^\downarrow\right)
\,,
\end{eqnarray}
and analogously
$\hat S_\mu^x=\sum_{ab}\hat c_{\mu,a}^\dagger\sigma^x_{ab}\hat c_{\mu,b}/2$
as well as
$\hat S_\mu^y=\sum_{ab}\hat c_{\mu,a}^\dagger\sigma^x_{ab}\hat c_{\mu,b}/2$
where $\sigma^{x,y,z}_{ab}$ are the usual Pauli spin matrices.
These operators satisfy the usual spin, i.e., $SU(2)$, commutation relations
and the Fermi-Hubbard Hamiltonian (\ref{Fermi-Hubbard-Hamiltonian})
is invariant under global $SU(2)$ rotations generated by the total spin
operators $\hat{\mathbf S}_{\rm tot}=\sum_\mu\hat{\mathbf S}_\mu$.
In the case of zero hopping $J=0$, this global $SU(2)$ invariance even
becomes a local symmetry, i.e., we may perform a spin rotation at each
site without changing the energy.
As a result, the ground state (at half filling) is highly degenerate for
$J=0$ in contrast to the Bose-Hubbard model (at integer filling).
This degeneracy can be lifted by an additional staggered magnetic field
(see Appendix \ref{staggered}) and is related to the spin modes which become
arbitrarily soft for small $J$.
In this limit $J\ll U$, their dynamics can be described by an effective
Hamiltonian, which is basically the Heisenberg model
\begin{eqnarray}
\label{Heisenberg-Hamiltonian}
\hat H
=
\frac{2J^2}{Z^2U}
\sum_{\mu\nu}T_{\mu\nu}\,\hat{\mathbf S}_\mu\cdot\hat{\mathbf S}_\nu
\,,
\end{eqnarray}
with an effective anti-ferromagnetic coupling constant
of order $1/Z^2$.
This effective Hamiltonian describes the Fermi-Hubbard Hamiltonian
(\ref{Fermi-Hubbard-Hamiltonian}) for half-filling in the low-energy
sub-space where we have one particle per site, but with a variable spin
$\hat{\mathbf S}_\mu$.
In order to avoid complications such as frustration for the
anti-ferromagnetic Heisenberg model (\ref{Heisenberg-Hamiltonian}),
we assume a bipartite lattice -- i.e., we can divide the total lattice
into two sub-lattices $\cal A$ and $\cal B$ such that, for each site in
$\mu\in\cal A$, all the neighbouring sites $\nu$ belong to $\cal B$
and {\em vice versa}.
In this case, the ground state of the Heisenberg model
(\ref{Heisenberg-Hamiltonian}) approaches the N\'eel state for large $Z$
\begin{eqnarray}
\label{Neel}
\hat{\rho}_{\rm Neel}
=
\bigotimes_{\mu\in\cal A}
\bigotimes_{\nu\in\cal B}
\hat{n}_\mu^\downarrow\,
\hat{\bar{n}}_\mu^\uparrow\,
\hat{n}_\nu^\uparrow\,
\hat{\bar{n}}_\nu^\downarrow
\,,
\end{eqnarray}
which is just the state with exactly one particle per site, but in
alternating spin states, i.e., $s=\downarrow$ for $\mu\in\cal A$ and
$s=\uparrow$ for $\nu\in\cal B$.
Note that $\hat{n}_\mu^\downarrow$ is the projector on the
$\ket{1}_\mu^\downarrow$ state
$\hat{n}_\mu^\downarrow=\ket{1^\downarrow}_\mu\bra{1^\downarrow}$
while $\hat{\bar{n}}_\mu^\uparrow$
projects on the $\ket{0}_\mu^\uparrow$ state etc.
As usual, this state (\ref{Neel}) breaks the original symmetry group of
the Hamiltonian (\ref{Fermi-Hubbard-Hamiltonian}) containing particle-hole
symmetry, $SU(2)$ invariance, and translational symmetry, down to a
sub-group, which includes invariance under a combined spin-flip and
particle-hole exchange etc.
Let us stress that the N\'eel state (\ref{Neel}) is only the lowest-order
approximation of the real ground state of the Heisenberg model
(\ref{Heisenberg-Hamiltonian}), there are quantum spin fluctuations
of order $\,{\cal O}(1/Z)$.
These quantum spin fluctuations do not vanish in the limit $J\to 0$ since
$J$ only appears in the overall pre-factor in front of the
Heisenberg Hamiltonian (\ref{Heisenberg-Hamiltonian}) while the internal
structure remains the same.
Only after adding a suitable staggered magnetic field
(see Appendix \ref{staggered}), the N\'eel state (\ref{Neel})
is the exact unique ground state (for $J\to 0$).
Either way, in analogy to the bosonic case, we can now use this fully
factorising state (\ref{Neel}) as the starting point for our $1/Z$-expansion.
\section{Charge Modes}\label{chargemodes}
Starting with the N\'eel state (\ref{Neel}) as the zeroth order in $1/Z$,
let us now derive the first-order corrections.
To this end, let us consider the Heisenberg equations of motion
\begin{eqnarray}
i\partial_t \hat{c}_{\mu s}
&=&
-\frac{J}{Z}\sum_{\kappa\neq \mu} T_{\mu\kappa}\hat{c}_{\kappa s}
+U\hat{c}_{\mu s} \hat{n}_{\mu \bar{s}}
\label{annihilation-operator}\\
i\partial_t \hat{c}_{\mu s}^\dagger
&=&
+\frac{J}{Z}\sum_{\kappa\neq \mu} T_{\mu\kappa}\hat{c}^\dagger_{\kappa s}
-U\hat{c}^\dagger_{\mu s} \hat{n}_{\mu \bar{s}}
\label{creation-operator}\\
i\partial_t\hat{n}_{\mu s}
&=&
-i\partial_t\hat{\bar{n}}_{\mu s}
=
\frac{J}{Z}\sum_{\kappa\neq \mu}T_{\mu\kappa}
\left(
\hat{c}_{\kappa s}^\dagger \hat{c}_{\mu s}-
\hat{c}_{\mu s}^\dagger \hat{c}_{\kappa s}
\right)
\label{number-operator}
\,,
\end{eqnarray}
where $\bar s$ denotes the spin label opposite to $s$, i.e., either
$(s,\bar{s})=(\uparrow,\downarrow)$ or $(s,\bar{s})=(\downarrow,\uparrow)$.
If we now insert these evolution equations into the correlation functions
$\langle\hat{c}_{\mu a}^\dagger\hat{c}_{\nu b}
\hat{n}_{\mu\bar{a}}\hat{n}_{\nu\bar{b}}\rangle$,
$\langle\hat{c}_{\mu a}^\dagger\hat{c}_{\nu b}
\hat{\bar n}_{\mu\bar{a}}\hat{n}_{\nu\bar{b}}\rangle$,
$\langle\hat{c}_{\mu a}^\dagger\hat{c}_{\nu b}
\hat{n}_{\mu\bar{a}}\hat{\bar n}_{\nu\bar{b}}\rangle$,
and
$\langle\hat{c}_{\mu a}^\dagger\hat{c}_{\nu b}
\hat{\bar n}_{\mu\bar{a}}\hat{\bar n}_{\nu\bar{b}}\rangle$,
we find that they form a closed set of equations to first order in $1/Z$,
where we can neglect three-point correlations
\begin{eqnarray}
i\partial_t
\langle \hat{c}_{\mu a}^\dagger\hat{c}_{\nu b}
\hat{n}_{\mu\bar{a}}\hat{n}_{\nu\bar{b}}\rangle
&=&
+\frac{J}{Z}\langle\hat{n}_{\mu \bar{a}}\rangle_0
\sum_{\kappa\neq \mu,\nu}T_{\mu\kappa}
\langle\hat{c}_{\kappa a}^\dagger\hat{c}_{\nu b}
(\hat{n}_{\kappa\bar{a}}+\hat{\bar{n}}_{\kappa\bar{a}})
\hat{n}_{\nu\bar{b}}\rangle
\nonumber\\
& &
-\frac{J}{Z}\langle\hat{n}_{\nu \bar{b}}\rangle_0
\sum_{\kappa\neq \mu,\nu}T_{\nu\kappa}
\langle\hat{c}_{\mu a}^\dagger\hat{c}_{\kappa b}
\hat{n}_{\mu\bar{a}}(\hat{n}_{\kappa\bar{b}}+\hat{\bar{n}}_{\kappa\bar{b}})
\rangle
\nonumber\\
& &
+\frac{J}{Z}T_{\mu\nu}
\langle \hat{c}_{\nu a}^\dagger\hat{c}_{\nu b}
\hat{n}_{\mu\bar{a}}\hat{n}_{\nu\bar{b}}\rangle_0
-\frac{J}{Z}T_{\mu\nu}
\langle \hat{c}_{\mu a}^\dagger\hat{c}_{\mu b}
\hat{n}_{\mu\bar{a}}\hat{n}_{\nu\bar{b}}\rangle_0\label{corr1}
\,,
\end{eqnarray}
where the expectation values $\langle\hat{n}_{\mu \bar{a}}\rangle_0$ and
$\langle\hat{n}_{\nu \bar{b}}\rangle_0$ as well as those in the last line
are taken in the zeroth-order N\'eel state (\ref{Neel}).
In complete analogy, we obtain for the remaining three correlators
\begin{eqnarray}
i\partial_t
\langle \hat{c}_{\mu a}^\dagger\hat{c}_{\nu b}
\hat{n}_{\mu\bar{a}}\hat{\bar{n}}_{\nu\bar{b}}\rangle
&=&
+\frac{J}{Z}\langle\hat{n}_{\mu \bar{a}}\rangle_0
\sum_{\kappa\neq \mu,\nu}T_{\mu\kappa}
\langle\hat{c}_{\kappa a}^\dagger\hat{c}_{\nu b}
(\hat{n}_{\kappa\bar{a}}+\hat{\bar{n}}_{\kappa\bar{a}})
\hat{\bar{n}}_{\nu\bar{b}}\rangle
\nonumber\\
& &
-\frac{J}{Z}\langle\hat{\bar{n}}_{\nu \bar{b}}\rangle_0
\sum_{\kappa\neq \mu,\nu}T_{\nu\kappa}
\langle\hat{c}_{\mu a}^\dagger\hat{c}_{\kappa b}
\hat{n}_{\mu\bar{a}}(\hat{n}_{\kappa\bar{b}}+\hat{\bar{n}}_{\kappa\bar{b}})
\rangle
\nonumber\\
& &
-
\langle \hat{c}_{\mu a}^\dagger\hat{c}_{\nu b}
\hat{n}_{\mu \bar{a}}\hat{\bar{n}}_{\nu \bar{b}}\rangle
\nonumber\\
& &
+\frac{J}{Z}T_{\mu\nu}
\langle \hat{c}_{\nu a}^\dagger\hat{c}_{\nu b}
\hat{n}_{\mu\bar{a}}\hat{\bar{n}}_{\nu\bar{b}}\rangle_0
-\frac{J}{Z}T_{\mu\nu}
\langle \hat{c}_{\mu a}^\dagger\hat{c}_{\mu b}
\hat{n}_{\mu\bar{a}}\hat{\bar{n}}_{\nu\bar{b}}\rangle_0
\,,
\end{eqnarray}
as well as
\begin{eqnarray}
i\partial_t
\langle \hat{c}_{\mu a}^\dagger\hat{c}_{\nu b}
\hat{\bar{n}}_{\mu\bar{a}}\hat{n}_{\nu\bar{b}}\rangle
&=&
+\frac{J}{Z}\langle\hat{\bar{n}}_{\mu \bar{a}}\rangle_0
\sum_{\kappa\neq \mu,\nu}T_{\mu\kappa}
\langle\hat{c}_{\kappa a}^\dagger\hat{c}_{\nu b}
(\hat{n}_{\kappa\bar{a}}+\hat{\bar{n}}_{\kappa\bar{a}})
\hat{n}_{\nu\bar{b}}\rangle
\nonumber\\
& &
-\frac{J}{Z}\langle\hat{n}_{\nu \bar{b}}\rangle_0
\sum_{\kappa\neq \mu,\nu}T_{\nu\kappa}
\langle\hat{c}_{\mu a}^\dagger\hat{c}_{\kappa b}
\hat{\bar{n}}_{\mu\bar{a}}
(\hat{n}_{\kappa\bar{b}}+\hat{\bar{n}}_{\kappa\bar{b}})\rangle
\nonumber\\
& &
+
\langle \hat{c}_{\mu a}^\dagger\hat{c}_{\nu b}
\hat{\bar{n}}_{\mu \bar{a}}\hat{n}_{\nu \bar{b}}\rangle
\nonumber\\
& &
+\frac{J}{Z}T_{\mu\nu}
\langle \hat{c}_{\nu a}^\dagger\hat{c}_{\nu b}
\hat{\bar{n}}_{\mu\bar{a}}\hat{n}_{\nu\bar{b}}\rangle_0
-\frac{J}{Z}T_{\mu\nu}
\langle \hat{c}_{\mu a}^\dagger\hat{c}_{\mu b}
\hat{\bar{n}}_{\mu\bar{a}}\hat{n}_{\nu\bar{b}}\rangle_0
\,,
\end{eqnarray}
and finally
\begin{eqnarray}
i\partial_t
\langle \hat{c}_{\mu a}^\dagger\hat{c}_{\nu b}
\hat{\bar{n}}_{\mu\bar{a}}\hat{\bar{n}}_{\nu\bar{b}}\rangle
&=&
+\frac{J}{Z}\langle\hat{\bar{n}}_{\mu \bar{a}}\rangle_0
\sum_{\kappa\neq \mu,\nu}T_{\mu\kappa}
\langle\hat{c}_{\kappa a}^\dagger\hat{c}_{\nu b}
(\hat{n}_{\kappa\bar{a}}+\hat{\bar{n}}_{\kappa\bar{a}})
\hat{\bar{n}}_{\nu\bar{b}}\rangle
\nonumber\\
& &
-\frac{J}{Z}\langle\hat{\bar{n}}_{\nu \bar{b}}\rangle_0
\sum_{\kappa\neq \mu,\nu}T_{\nu\kappa}
\langle\hat{c}_{\mu a}^\dagger\hat{c}_{\kappa b}
\hat{\bar{n}}_{\mu\bar{a}}
(\hat{n}_{\kappa\bar{b}}+\hat{\bar{n}}_{\kappa\bar{b}})\rangle
\nonumber\\
& &
+\frac{J}{Z}T_{\mu\nu}
\langle \hat{c}_{\nu a}^\dagger\hat{c}_{\nu b}
\hat{\bar{n}}_{\mu\bar{a}}\hat{\bar{n}}_{\nu\bar{b}}\rangle_0
-\frac{J}{Z}T_{\mu\nu}
\langle \hat{c}_{\mu a}^\dagger\hat{c}_{\mu b}
\hat{\bar{n}}_{\mu\bar{a}}\hat{\bar{n}}_{\nu\bar{b}}\rangle_0
\label{sectors}
\,.
\end{eqnarray}
We observe that the spin structure is conserved in these equations,
i.e., the four correlators containing
$\hat{c}_{\mu\uparrow}^\dagger\hat{c}_{\nu\uparrow}$ decouple from
those with
$\hat{c}_{\mu\uparrow}^\dagger\hat{c}_{\nu\downarrow}$ etc.
Thus we can treat the four sectors separately.
Let us focus on the correlators containing
$\hat{c}_{\mu\downarrow}^\dagger\hat{c}_{\nu\downarrow}$ and introduce
the following short-hand notation:
If ${\mu}\in\cal A$ and ${\nu}\in\cal B$, we denote the correlations by
$\langle \hat{c}_{\mu \downarrow}^\dagger\hat{c}_{\nu \downarrow}
\hat{n}_{\mu\uparrow}\hat{n}_{\nu\uparrow}\rangle=f_{\mu\nu}^{1_A1_B}$,
and
$\langle \hat{c}_{\mu \downarrow}^\dagger\hat{c}_{\nu \downarrow}
\hat{\bar{n}}_{\mu\uparrow}\hat{n}_{\nu\uparrow}\rangle=f_{\mu\nu}^{0_A1_B}$,
etc.
Inserting the zeroth-order N\'eel state (\ref{Neel}),
we find four trivial equations which fully decouple
\begin{eqnarray}
i\partial_t f^{1_A0_B}_{\mu\nu}
&=&
-U f^{1_A0_B}_{\mu\nu}
\,,
\nonumber\\
i\partial_t f^{0_B1_A}_{\mu\nu}
&=&
+U f^{0_B1_A}_{\mu\nu}
\,,
\nonumber\\
i\partial_t f^{0_B0_B}_{\mu\nu}
&=&0
\,,
\nonumber\\
i\partial_t f^{1_A1_A}_{\mu\nu}
&=&0
\label{hom1}
\,.
\end{eqnarray}
Thus, if these correlations vanish initially, they remain zero
(to first order in $1/Z$).
Setting these correlations (\ref{hom1}) to zero,
we get four pairs of coupled equations
\begin{eqnarray}
i\partial_t f^{0_A0_B}_{\mu\nu}
&=&
+\frac{J}{Z}\sum_{\kappa\neq{\mu,\nu}}T_{\mu\kappa}
f^{1_B0_B}_{\kappa\nu}
\,,
\nonumber\\
i\partial_t f^{1_B0_B}_{\mu\nu}
&=&
+\frac{J}{Z}\sum_{\kappa\neq{\mu,\nu}}T_{\mu\kappa}
f^{0_A0_B}_{\kappa\nu}
-Uf^{1_B0_B}_{\mu\nu}
\,,
\label{hom2}\\
i\partial_t f^{0_B0_A}_{\mu\nu}
&=&
-\frac{J}{Z}\sum_{\kappa\neq{\mu,\nu}}T_{\kappa\nu}
f^{0_B1_B}_{\mu\kappa}
\nonumber\\
i\partial_t f^{0_B1_B}_{\mu\nu}
&=&
-\frac{J}{Z}\sum_{\kappa\neq{\mu,\nu}}T_{\kappa\nu}
f^{0_B0_A}_{\mu\kappa}
+Uf^{0_B1_B}_{\mu\nu}
\,,
\\
i\partial_t f^{1_B1_A}_{\mu\nu}
&=&
+\frac{J}{Z}\sum_{\kappa\neq{\mu,\nu}}T_{\mu\kappa}
f^{0_A1_A}_{\kappa\nu}
\nonumber\\
i\partial_t f^{0_A1_A}_{\mu\nu}
&=&
+\frac{J}{Z}\sum_{\kappa\neq{\mu,\nu}}T_{\mu\kappa}
f^{1_B1_A}_{\kappa\nu}
+Uf^{0_A1_A}_{\mu\nu}
\,,
\\
i\partial_t f^{1_A1_B}_{\mu\nu}
&=&
-\frac{J}{Z}\sum_{\kappa\neq{\mu,\nu}}T_{\kappa\nu}
f^{1_A0_A}_{\mu\kappa}
\nonumber\\
i\partial_t f^{1_A0_A}_{\mu\nu}
&=&
-\frac{J}{Z}\sum_{\kappa\neq{\mu,\nu}}T_{\kappa\nu}
f^{1_A1_B}_{\mu\kappa}
-Uf^{1_A0_A}_{\mu\nu}
\,.
\label{hom12}
\end{eqnarray}
Again, since these equations do not have any non-vanishing source terms
(to first order in $1/Z$), they can be set to zero if we start in an
initially uncorrelated state (see Appendix \ref{staggered}).
Note that they would acquire non-zero source terms if we go away from
half-filling.
The positive and negative eigenfrequencies of these modes behave as
\begin{eqnarray}\label{eigenmodes}
\omega_\mathbf{k}^\pm=\frac{U\pm\sqrt{U^2+4 J^2T_\mathbf{k}^2}}{2}
\,.
\end{eqnarray}
Thus we have soft modes which scale as
$\omega_\mathbf{k}^-\sim J^2/U$ for small $J$ and hard modes
$\omega_\mathbf{k}^+\approx U$.
These modes are important for making contact to the $t$-$J$ model
\cite{A94} which describes the low-energy excitations of the
Fermi-Hubbard Hamiltonian (\ref{Fermi-Hubbard-Hamiltonian}) for small $J$
away from half-filling.
However, at half-filling, we can set them to zero.
After doing this, we are left with four coupled equations, which do have
non-vanishing source terms
\begin{eqnarray}
i\partial_t f^{0_A0_A}_{\mu\nu}
&=&
\frac{J}{Z}\sum_{\kappa\neq{\mu,\nu}}
\left\{
T_{\mu\kappa}
f^{1_B0_A}_{\kappa\nu}
-T_{\kappa\nu}
f^{0_A1_B}_{\mu\kappa}
\right\}
\label{charge1}
\,,
\\
i\partial_t f^{0_A1_B}_{\mu\nu}
&=&
\frac{J}{Z}\sum_{\kappa\neq{\mu,\nu}}
\left\{
T_{\mu\kappa}
f^{1_B1_B}_{\kappa\nu}
-T_{\kappa\nu}
f^{0_A0_A}_{\mu\kappa}
\right\}
+U f^{0_A1_B}_{\mu\nu}-\frac{J}{Z}T_{\mu\nu}
\,,
\\
i\partial_t f^{1_B0_A}_{\mu\nu}
&=&
\frac{J}{Z}\sum_{\kappa\neq{\mu,\nu}}
\left\{
T_{\mu\kappa}
f^{0_A0_A}_{\kappa\nu}
-T_{\kappa\nu}
f^{1_B1_B}_{\mu\kappa}
\right\}
-U f^{1_B0_A}_{\mu\nu}+\frac{J}{Z}T_{\mu\nu}
\,,
\\
i\partial_t f^{1_B1_B}_{\mu\nu}
&=&
\frac{J}{Z}\sum_{\kappa\neq{\mu,\nu}}
\left\{
T_{\mu\kappa}
f^{0_A1_B}_{\kappa\nu}
-T_{\kappa\nu}
f^{1_B0_A}_{\mu\kappa}
\right\}
\,.
\label{charge4}
\end{eqnarray}
Due to the source terms $JT_{\mu\nu}/Z$, these modes will develop
correlations if we slowly (or suddenly) switch on the hopping rate $J$,
even if there are no correlations initially.
The eigenfrequencies of these modes behave as
\begin{eqnarray}
\label{omega-fermi}
\omega_\mathbf{k}=\sqrt{U^2+4 J^2T_\mathbf{k}^2}
\,.
\end{eqnarray}
A similar dispersion relation can be derived from a mean-field
approach \cite{F91}.
In contrast to the bosonic case, the origin of the Brillouin zone at
$\mathbf{k}=0$ does not have minimum but actually maximum excitation
energy $\omega_\mathbf{k}$.
The minimum is not a point but a hyper-surface where $T_\mathbf{k}=0$
(or, more generally, $T_\mathbf{k}^2$ assumes its minimum).
After Fourier transformation of (\ref{charge1})-(\ref{charge4}) we find
that the equations of motion conserve a bilinear quantity, that is
\begin{eqnarray}\label{invfermi}
\partial_t\left[
\left(f^{1_B1_B}_\mathbf{k}-1\right)f^{1_B1_B}_\mathbf{k}+
f^{0_A1_B}_\mathbf{k}f^{1_B0_A}_\mathbf{k}
\right]=0\,.
\end{eqnarray}
This relation holds, as in the bosonic case, also for time-dependent $J(t)$.
\section{Mott State}\label{Fermi-Mott}
\subsection{Ground state correlations}
In complete analogy to the bosonic case, we now imagine switching $J$
adiabatically from zero (where all the charge fluctuations vanish) to
a finite value.
Thus we find the following non-zero ground-state correlations
\begin{eqnarray}
\label{ground-11}
f^{1_B1_B}_{\mu\nu,\mathrm{ground}}
&=&
-f^{0_A0_A}_{\mu\nu,\mathrm{ground}}
=
\frac{1}{N}\sum_{\mathbf{k}}\frac{1}{2}
\left(1-\frac{U}{\omega_\mathbf{k}}\right)
e^{i(\mathbf{x}_\mu-\mathbf{x}_\nu)\cdot\mathbf{k}}
\,,
\\
f^{1_B0_A}_{\mu\nu,\mathrm{ground}}
&=&
{f}^{0_A1_B}_{\mu\nu,\mathrm{ground}}
=
\frac{1}{N}\sum_{\mathbf{k}}
\frac{JT_\mathbf{k}}{\omega_\mathbf{k}}\,
e^{i(\mathbf{x}_\mu-\mathbf{x}_\nu)\cdot\mathbf{k}}
\,.
\label{ground-10}
\end{eqnarray}
Somewhat similar to the Bose-Hubbard model, the symmetric combination
(\ref{ground-11}) scales with $J^2$ for small $J$ while the other
(\ref{ground-10}) starts linearly in $J$.
Other correlators such as
$\langle\hat{c}_{\mu\downarrow}^\dagger\hat{c}_{\nu\downarrow}\rangle$
can be obtained from these expressions.
For example, if $\mu$ and $\nu$ are in $\cal A$, we find,
using $\hat{n}_{\mu\uparrow}+\hat{\bar n}_{\mu\uparrow}=1$
and $\hat{n}_{\nu\uparrow}+\hat{\bar n}_{\nu\uparrow}=1$
\begin{eqnarray}
\langle\hat{c}_{\mu\downarrow}^\dagger\hat{c}_{\nu\downarrow}\rangle
=
f_{\mu\nu}^{1_A1_A}
+f_{\mu\nu}^{0_A1_A}
+f_{\mu\nu}^{1_A0_A}
+f_{\mu\nu}^{0_A0_A}
=
f_{\mu\nu}^{0_A0_A}
\,.
\end{eqnarray}
\subsection{Quantum depletion}
In the zeroth-order N\'eel state (\ref{Neel}), we have
$\langle\hat{n}_{\mu\uparrow}\hat{n}_{\mu\downarrow}\rangle=0$.
Thus this quantity
$\langle\hat{n}_{\mu\uparrow}\hat{n}_{\mu\downarrow}\rangle$
measures the deviation from this zeroth-order N\'eel state (\ref{Neel})
due to quantum charge fluctuations.
In order to calculate
$\langle\hat{n}_{\mu\uparrow}\hat{n}_{\mu\downarrow}\rangle$,
we also need some of the other sectors discussed after (\ref{sectors}).
Obviously, the correlators containing
$\hat{c}_{\mu\uparrow}^\dagger\hat{c}_{\nu\uparrow}$
behave in the same way as those with
$\hat{c}_{\mu\downarrow}^\dagger\hat{c}_{\nu\downarrow}$
after interchanging the sub-lattices $\cal A$ and $\cal B$.
Thus a completely analogous system of differential equations exists for
the correlations of the form
$\langle \hat{c}_{\mu \uparrow}^\dagger\hat{c}_{\mu \uparrow}
\hat{n}_{\mu\downarrow}\hat{n}_{\nu\downarrow}\rangle
=g_{\mu\nu}^{1_A1_B}$ etc.
If we insert (\ref{number-operator}) in order to calculate
$i\partial_t\langle\hat{n}_{\mu\uparrow}\hat{n}_{\mu\downarrow}\rangle$,
we find that these two sectors are enough for deriving
$\langle\hat{n}_{\mu\uparrow}\hat{n}_{\mu\downarrow}\rangle$.
Assuming $\mu\in\cal B$ for simplicity, we find
\begin{eqnarray}
i\partial_t\langle \hat{n}_{\mu s}\hat{n}_{\mu \bar{s}}\rangle
&=&
-\frac{J}{Z}\sum_{\kappa\neq\mu}T_{\kappa\mu}
\Big\{
g_{\mu\kappa}^{1_B1_A}+g_{\mu\kappa}^{1_B0_A}+
f_{\mu\kappa}^{1_B1_A}+f_{\mu\kappa}^{1_B0_A}
\nonumber\\
& &\quad
-g_{\kappa\mu}^{1_A1_B}-g_{\kappa\mu}^{0_A1_B}
-f_{\kappa\mu}^{1_A1_B}-f_{\kappa\mu}^{0_A1_B}
\Big\}
\,.
\end{eqnarray}
Setting the correlations with vanishing source terms to zero, we find
\begin{eqnarray}
i\partial_t\langle \hat{n}_{\mu s}\hat{n}_{\mu \bar{s}}\rangle
&=&
-\frac{J}{Z}\sum_{\kappa\neq\mu}T_{\kappa\mu}
\Big\{
f_{\mu\kappa}^{1_B0_A}
-f_{\kappa\mu}^{0_A1_B}
\Big\}\nonumber\\
& =&-\frac{1}{N}\sum_\mathbf{k}JT_\mathbf{k}\Big\{
f_{\mathbf{k}}^{1_B0_A}
-f_{\mathbf{k}}^{0_A1_B}
\Big\}=\frac{i}{N}\sum_\mathbf{k}\partial_t f_{\mathbf{k}}^{1_B1_B}
\,.
\end{eqnarray}
Thus, in the ground state, the quantum depletion reads
\begin{eqnarray}
\langle\hat{n}_{\mu s}\hat{n}_{\mu \bar{s}}\rangle
=
\langle\hat{\bar{n}}_{\mu s}\hat{\bar{n}}_{\mu \bar{s}}\rangle
=
\frac{1}{N}\sum_\mathbf{k}\frac{1}{2}
\left(1-\frac{U}{\omega_\mathbf{k}}\right)
\,.
\end{eqnarray}
As one would expect, this quantity scales with $J^2$ for small $J$.
\begin{center}
\begin{figure}[h]
\includegraphics{onsiteg0J05.eps}
\includegraphics{correlationsg11J05.eps}
\begin{center}
\includegraphics{correlationsg10J05.eps}
\end{center}
\caption{Time-dependence of the quantum depletion, the nearest-neighbour
correlation function $f^{1_B0_A}_{\mu\nu}$, and the next-to-nearest-neighbour
correlation function $f^{1_B1_B}_{\mu\nu}$ in three dimensions
after a quench within the Mott phase from $J/U=0$ to $J/U=0.5$
in comparison to their ground state values. }\label{quenchfermi}
\end{figure}
\end{center}
\subsection{Quench}
Now we consider a quantum quench, i.e., a sudden switch from $J=0$
to some finite value of $J$, where we find the following non-vanishing
correlations
\begin{eqnarray}
f^{1_B1_B}_{\mu\nu,\mathrm{quench}}
=
-f^{0_A0_A}_{\mu\nu,\mathrm{quench}}
&=&
\frac{1}{N}\sum_{\mathbf{k}}
2 J^2 T_{\mathbf{k}}^2\,
\frac{1-\cos(\omega_\mathbf{k} t)}{\omega_\mathbf{k}^2}\,
e^{i(\mathbf{x}_\mu-\mathbf{x}_\nu)\cdot\mathbf{k}}
\,,
\\
f^{1_B0_A}_{\mu\nu,\mathrm{quench}}
=
\left({f}^{0_A1_B}_{\mu\nu,\mathrm{quench}}\right)^*
&=&
\frac{1}{N}\sum_{\mathbf{k}}
J T_\mathbf{k}U
\frac{1 - \cos(\omega_\mathbf{k} t)}{\omega_\mathbf{k}^{2}}
e^{i(\mathbf{x}_\mu-\mathbf{x}_\nu)\cdot\mathbf{k}}
\nonumber\\
& &
-\frac{i}{N}\sum_{\mathbf{k}}
J T_\mathbf{k}\,
\frac{\sin(\omega_\mathbf{k} t)}{\omega_\mathbf{k}}\,
e^{i(\mathbf{x}_\mu-\mathbf{x}_\nu)\cdot\mathbf{k}}
\,.
\end{eqnarray}
Again, these correlations equilibrate to a quasi-stationary value,
which is, however, not thermal.
For some of these correlations, this quasi-stationary value lies
even {\em below} the ground-state correlation, see Fig.~\ref{quenchfermi}.
The probability to have two or zero particles at a site reads
\begin{eqnarray}
\langle\hat{n}_{\mu s}\hat{n}_{\mu \bar{s}}\rangle_\mathrm{quench}
=
\langle\hat{\bar{n}}_{\mu s}\hat{\bar{n}}_{\mu \bar{s}}\rangle_\mathrm{quench}
=
\frac{1}{N}\sum_\mathbf{k}
2 J^2 T_{\mathbf{k}}^2\,
\frac{1-\cos(\omega_\mathbf{k} t)}{\omega_\mathbf{k}^2}
\,.
\end{eqnarray}
This quantity also equilibrates to a quasi-stationary value of order $1/Z$.
In analogy to the bosonic case, this quasi-stationary value could be explained
by a small effective temperature -- but this small effective temperature then
does not work for the other observables, e.g., the correlations.
\subsection{Spin modes}
So far, we have considered expectations values such as
$\langle\hat{c}_{\mu a}^\dagger\hat{c}_{\nu b}
\hat{n}_{\mu\bar{a}}\hat{n}_{\nu\bar{b}}\rangle$,
where -- apart from the number operators $\hat{n}_{\mu\bar{a}}$ and
$\hat{n}_{\nu\bar{b}}$ -- one particle is annihilated at site $\nu$
and one is created at site $\mu$.
These operator combinations correspond to a change of the occupation
numbers and are thus called charge modes.
However, as already indicated in Section~\ref{Fermi-Hubbard Model},
there are also other modes which leave the total occupation number
of all lattice sites unchanged.
Examples are
$\langle\hat{c}_{\mu s}^\dagger\hat{c}_{\mu\bar{s}}
\hat{c}_{\nu\bar{s}}^\dagger\hat{c}_{\nu s}\rangle$
or
$\langle\hat{n}_{\mu a}\hat{n}_{\nu b}\rangle$
or combinations thereof.
Many of these combinations can be expressed in terms of the
effective spin operators in (\ref{spin-operators}) via
$\langle\hat{S}_{\mu}^i\hat{S}_{\nu}^j\rangle$.
As one would expect from our study of the Bose-Hubbard model,
the evolution of these spin modes vanishes to first order in $1/Z$
\begin{eqnarray}
\partial_t\langle\hat{S}_{\mu}^i\hat{S}_{\nu}^j\rangle=\,{\cal O}(1/Z^2)
\,,
\end{eqnarray}
consistent with the Heisenberg Hamiltonian (\ref{Heisenberg-Hamiltonian}).
In analogy to the $\langle\hat{n}_{\mu}\hat{n}_{\nu}\rangle$-correlator
in the bosonic case, one has to go to second order $\,{\cal O}(1/Z^2)$ in order
to calculate these quantities.
Fortunately, the charge modes discussed above do not couple to these spin
modes to first order in $1/Z$ and hence we can omit them to this level
of accuracy.
\section{Tilted Fermi-Hubbard Lattice}\label{fermitilt}
Motivated by the bosonic case, we now study particle-hole pair creation
via tunnelling in a tilted lattice.
Again, we assume a spatially constant but possibly time-dependent force
on the particles which acts on both spin species in the same way.
Accordingly, we modify our Hamiltonian via
\begin{eqnarray}
\label{Fermi-Hubbard-tilt}
\hat H
=
-\frac{J}{Z}\sum_{\mu\nu,s}T_{\mu\nu}\hat{c}_{\mu,s}^\dagger\hat{c}_{\nu,s}
+U\sum_{\mu}\hat{n}_\mu^\uparrow\hat{n}_\mu^\downarrow
+\sum_{\mu}V_\mu (\hat{n}_\mu^\uparrow+\hat{n}_\mu^\downarrow)
\,,
\end{eqnarray}
where $V_\mu(t)=\mathbf{E}(t)\cdot\mathbf{x}_\mu$ denotes the additional
potential.
Performing the same procedure as before, we find modified equations of
motion for the charge modes
\begin{eqnarray}
\label{modify-charge-mode-1}
i\partial_t f^{0_A0_A}_{\mu\nu}
&=&
+\frac{J}{Z}\sum_{\kappa\neq\mu,\nu}
\left(
T_{\mu\kappa}f^{1_B0_A}_{\kappa\nu}-T_{\kappa\nu}f^{0_A1_B}_{\mu\kappa}
\right)
\,,
\\
\label{modify-charge-mode-2}
i\partial_t f^{0_A1_B}_{\mu\nu}
&=&
+\frac{J}{Z}\sum_{\kappa\neq\mu,\nu}
\left(
T_{\mu\kappa}f^{1_B1_B}_{\kappa\nu}-T_{\kappa\nu}f^{0_A0_A}_{\mu\kappa}
\right)
\nonumber\\
& &
+(U+V_\nu-V_\mu)f^{0_A1_B}_{\mu\nu}-\frac{J}{Z}T_{\mu\nu}
\,,
\\
\label{modify-charge-mode-3}
i\partial_t f^{1_B0_A}_{\mu\nu}
&=&
-\frac{J}{Z}\sum_{\kappa\neq\mu,\nu}
\left(
T_{\kappa\nu}f^{1_B1_B}_{\mu\kappa}-T_{\mu\kappa}f^{0_A0_A}_{\kappa\nu}
\right)
\nonumber\\
& &
-(U-V_\nu+V_\mu)f^{1_B0_A}_{\mu\nu}+\frac{J}{Z}T_{\mu\nu}
\,,
\\
\label{modify-charge-mode-4}
i\partial_t f^{1_B1_B}_{\mu\nu}
&=&
+\frac{J}{Z}\sum_{\kappa\neq\mu,\nu}
\left(
T_{\mu\kappa} f^{0_A1_B}_{\kappa\nu}-T_{\kappa\nu}f^{1_B0_A}_{\mu\kappa}
\right)\,.
\end{eqnarray}
In complete analogy to the bosonic case
it is possible to factorise the differential equations
for the correlation functions.
We define {\em effective} particle and hole operators such that
we have for the correlations functions without source terms
\begin{eqnarray}
\langle \hat{p}_{\mu,A}^\dagger \hat{h}_{\nu,B}\rangle
&=&
f^{1_A0_B}_{\mu\nu},\qquad
\langle \hat{h}_{\mu,B}^\dagger \hat{p}_{\nu,A}\rangle=f^{0_B1_A}_{\mu\nu},
\\
\langle \hat{h}_{\mu,B}^\dagger \hat{h}_{\nu,B}\rangle
&=&
f^{0_B0_B}_{\mu\nu},\qquad
\langle \hat{p}_{\mu,A}^\dagger \hat{p}_{\nu,A}\rangle=f^{1_A1_A}_{\mu\nu},
\\
\langle \hat{h}_{\mu,A}^\dagger \hat{h}_{\nu,B}\rangle
&=&
f^{0_A0_B}_{\mu\nu},\qquad
\langle \hat{p}_{\mu,B}^\dagger \hat{h}_{\nu,B}\rangle=f^{1_B0_B}_{\mu\nu},
\\
\langle \hat{h}_{\mu,B}^\dagger \hat{h}_{\nu,A}\rangle
&=&
f^{0_B0_A}_{\mu\nu},\qquad
\langle \hat{h}_{\mu,B}^\dagger \hat{p}_{\nu,B}\rangle=f^{0_B1_B}_{\mu\nu},
\\
\langle \hat{p}_{\mu,B}^\dagger \hat{p}_{\nu,A}\rangle
&=&
f^{1_B1_A}_{\mu\nu},\qquad
\langle \hat{h}_{\mu,A}^\dagger \hat{p}_{\nu,A}\rangle=f^{0_A1_A}_{\mu\nu},
\\
\langle \hat{p}_{\mu,A}^\dagger \hat{p}_{\nu,B}\rangle
&=&
f^{1_A1_B}_{\mu\nu},\qquad
\langle \hat{p}_{\mu,A}^\dagger \hat{h}_{\nu,A}\rangle=f^{1_A0_A}_{\mu\nu},
\end{eqnarray}
and for the correlation functions with source terms
\begin{eqnarray}
\langle \hat{h}_{\mu,A}^\dagger \hat{p}_{\nu,B}\rangle
&=&
f^{0_A1_B}_{\mu\nu},\qquad
\langle \hat{h}_{\mu,A}^\dagger \hat{h}_{\nu,A}\rangle
=
f^{0_A0_A}_{\mu\nu}+\delta_{\mu\nu},
\\
\langle \hat{p}_{\mu,B}^\dagger \hat{h}_{\nu,A}\rangle
&=&
f^{1_B0_A}_{\mu\nu},\qquad
\langle \hat{p}_{\mu,B}^\dagger \hat{p}_{\nu,B}\rangle=f^{1_B1_B}_{\mu\nu}
\,.
\end{eqnarray}
This allows us to effectively factorise the equations for the
correlation functions
\begin{eqnarray}
i\partial_t \hat{p}_{\mu,B}
&=&
-\frac{J}{Z}\sum_{\kappa\neq\mu}T_{\mu\kappa}
\left(\hat{h}_{\kappa,A}+\hat{p}_{\kappa,A}\right)
+\left(\frac{U}{2}+V_\mu\right)\hat{p}_{\mu,B}
\\
i\partial_t \hat{h}_{\mu,A}
&=&
-\frac{J}{Z}\sum_{\kappa\neq\mu}T_{\mu\kappa}
\left(\hat{h}_{\kappa,B}+\hat{p}_{\kappa,B}\right)
+\left(-\frac{U}{2}+V_\mu\right)\hat{h}_{\mu,A}
\\
i\partial_t \hat{p}_{\mu,A}
&=&
\left(\frac{U}{2}+V_\mu\right)\hat{p}_{\mu,A}
\\
i\partial_t \hat{h}_{\mu,B}
&=&
\left(-\frac{U}{2}+V_\mu\right)\hat{h}_{\mu,B}\,.
\end{eqnarray}
Due to the initial conditions,
we can set the operators $\hat{h}_{\mu,B}$ and $\hat{p}_{\mu,A}$
to zero.
After Fourier and Peierls transformation (\ref{Peierls}),
we find the symmetric form
\begin{eqnarray}
i\partial_t \hat{p}_{\mathbf{k},B}
&=&
+\frac{U}{2}\hat{p}_{\mathbf{k},B}-JT_\mathbf{k}(t)\hat{h}_{\mathbf{k},A}\,,
\label{dirac1}
\\
i\partial_t \hat{h}_{\mathbf{k},A}
&=&
-\frac{U}{2}\hat{h}_{\mathbf{k},A}-JT_\mathbf{k}(t)\hat{p}_{\mathbf{k},B}
\label{dirac2}
\,,
\end{eqnarray}
where the $T_\mathbf{k}(t)$ are time-dependent.
Now the line of reasoning is analogous to the Bose-Hubbard model.
Initially, the operators evolve according to
\begin{eqnarray}
\hat{h}_{\mathbf{k},A}&=&e^{+iU t/2}\hat{A}_\mathbf{k}
\,,
\\
\hat{p}_{\mathbf{k},B}&=&e^{-iU t/2}\hat{B}_\mathbf{k}
\,,
\end{eqnarray}
with $\langle \hat{A}_\mathbf{k}^\dagger \hat{A}_\mathbf{p}\rangle=0$ and
$\langle\hat{B}_\mathbf{k}^\dagger\hat{B}_\mathbf{p}\rangle=
\delta_{\mathbf{k,p}}$.
At the end of the evolution, we find
\begin{eqnarray}
\hat{h}_{\mathbf{k},A}
&=&
\left(\alpha_\mathbf{k} \hat{A}_\mathbf{k}
+\beta_\mathbf{k} \hat{B}_\mathbf{k}\right)e^{+iU t/2}
\\
\hat{p}_{\mathbf{k},B}
&=&
\left(\beta_\mathbf{k}^* \hat{A}_\mathbf{k}-
\alpha_\mathbf{k}^* \hat{B}_\mathbf{k}\right)e^{-iU t/2}
\,.
\end{eqnarray}
In contrast to the bosonic case
(where $|\alpha_\mathbf{k}|^2-|\beta_\mathbf{k}|^2=1$),
we have
$|\alpha_\mathbf{k}|^2+|\beta_\mathbf{k}|^2=1$.
This difference reflects the fermionic nature of the quasi-particles
and will have consequences for the case of resonant hopping
(see below).
The number of created particle-hole pairs then yields the depletion
\begin{eqnarray}
\langle\hat{n}_{\mu s}\hat{n}_{\mu \bar{s}}\rangle
=
\langle\hat{\bar{n}}_{\mu s}\hat{\bar{n}}_{\mu \bar{s}}\rangle
=
\frac{1}{N}\sum_\mathbf{k} |\beta_\mathbf{k}|^2
\,.
\end{eqnarray}
Note that the equations (\ref{dirac1}) and (\ref{dirac2}) are analogous to
the Dirac equation in 1+1 dimensions, if we consider a small effective
electric field $\mathbf{E}$.
In this case, particle-hole pair creation will occur predominantly in the
vicinity of those points in $\mathbf{k}$-space, where $T_\mathbf{k}$
vanishes, i.e., where the energy gap $\omega_\mathbf{k}$ in
(\ref{omega-fermi}) assumes it minimum.
Inserting $\mathbf{k}=\mathbf{k}_0+\delta\mathbf{k}$ with
$T_{\mathbf{k}_0}=0$, we may approximate $T_\mathbf{k}(t)$ via
\begin{eqnarray}
T_\mathbf{k}(t)
\approx
[\delta\mathbf{k}+\mathbf{A}(t)]
\cdot
[\nabla_\mathbf{k}T_\mathbf{k}]_{\mathbf{k}_0}
\,.
\end{eqnarray}
Inserting this approximation into the equations (\ref{dirac1}) and
(\ref{dirac2}), we get
\begin{eqnarray}
i\partial_t
\left(
\begin{array}{c}
\hat{p}_{\mathbf{k},B}
\\
\hat{h}_{\mathbf{k},A}
\end{array}
\right)
=
\left(
\frac{U}{2}\,\sigma^z-
J
[\nabla_\mathbf{k}T_\mathbf{k}]_{\mathbf{k}_0}
\cdot
[\delta\mathbf{k}+\mathbf{A}(t)]
\sigma^x
\right)
\cdot
\left(
\begin{array}{c}
\hat{p}_{\mathbf{k},B}
\\
\hat{h}_{\mathbf{k},A}
\end{array}
\right)
\,.
\end{eqnarray}
This is precisely the same form as a Dirac equation in 1+1 space-time
dimensions if we identify the effective speed of light via
\begin{eqnarray}
\mathbf{c}_{\rm eff}=J[\nabla_\mathbf{k}T_\mathbf{k}]_{\mathbf{k}_0}
\,,
\end{eqnarray}
and the effective mass according to
\begin{eqnarray}
m_{\rm eff}\mathbf{c}_{\rm eff}^2=\frac{U}{2}
\,.
\end{eqnarray}
Note, however, that $\mathbf{c}_{\rm eff}$ depends on $\mathbf{k}_0$
in general, i.e., the analogy only works if we single out a specific
direction.
In contrast to the bosonic case, we do not find a full analogy valid
for all $\mathbf{k}$-directions, since the dispersion relation is not
isotropic near the minimum in the fermionic case.
Nevertheless, we may use the analogy to the 1+1 dimensional Dirac
equation in order to estimate the pair creation probability via
\begin{eqnarray}
|\beta_{\mathbf{k}\approx\mathbf{k}_0}|^2
\sim
\exp\left\{-\pi
\frac{m_{\rm eff}^2\mathbf{c}_{\rm eff}^4}{\mathbf{c}_{\rm eff}\cdot\mathbf{E}}
\right\}
=
\exp\left\{-\pi
\frac{U^2}{4J[\nabla_\mathbf{k}T_\mathbf{k}]_{\mathbf{k}_0}\cdot\mathbf{E}}
\right\}
\,,
\end{eqnarray}
where we have assumed a slowly varying field $\mathbf{E}$.
This result should be relevant for the investigations of the dielectric
break-down in the Fermi-Hubbard model, see, e.g., \cite{EOW10,OAA03,OA10}.
\section{Resonant Tunnelling}\label{restun}
In the previous Section, we have studied the case of small potential
gradients, i.e., small effective electric fields
$V_\mu(t)=\mathbf{E}(t)\cdot\mathbf{x}_\mu$, for which
we have obtained a quantitative analogy to the Sauter-Schwinger
effect, which describes tunnelling from the negative continuum
(i.e., the Dirac sea) to the positive continuum.
Now let us investigate the case of strong potential gradients.
In this case, the lattice structure becomes important and resonance
effects play a role.
For simplicity, we assume a small hopping rate $J\ll U$ where we can solve
the equations for the charge modes
(\ref{modify-charge-mode-1}-\ref{modify-charge-mode-4}) via time-dependent
perturbation theory.
In this case, Eq.~(\ref{modify-charge-mode-2}) simplifies to
\begin{eqnarray}
\left(i\partial_t-U-V_\nu+V_\mu\right)
f^{0_A1_B}_{\mu\nu}
=
-\frac{J}{Z}T_{\mu\nu}
+\,{\cal O}(J^2)
\,,
\end{eqnarray}
as the other terms
$J(T_{\mu\kappa}f^{1_B1_B}_{\kappa\nu}-T_{\kappa\nu}f^{0_A0_A}_{\mu\kappa})$
are of higher order in $J$.
We see that this equation becomes resonant if $V_\mu-V_\nu=U$, i.e.,
if the energy gained by tunnelling from site $\mu$ to site $\nu$
compensates the gap $U+\,{\cal O}(J^2)$.
In this resonance case, the correlation grows linearly with time
$f^{0_A1_B}_{\mu\nu}=iJtT_{\mu\nu}/Z+\,{\cal O}(J^2)$.
Of course, Eq.~(\ref{modify-charge-mode-3}) has the same structure, but
becomes resonant for the opposite case $V_\mu-V_\nu=-U$.
Either way, the other two correlators
\begin{eqnarray}
i\partial_t f^{0_A0_A}_{\mu\nu}
=
\frac{J}{Z}\sum_{\kappa\neq\mu,\nu}
\left(
T_{\mu\kappa}f^{1_B0_A}_{\kappa\nu}-T_{\kappa\nu}f^{0_A1_B}_{\mu\kappa}
\right)
\,,
\end{eqnarray}
and similarly $f^{1_B1_B}_{\mu\nu}$, grow quadratically
$f^{0_A0_A}_{\mu\nu}\propto J^2t^2$.
The same perturbative analysis can be done for the bosonic case,
if we start from equations (\ref{f12-Mott}-\ref{f11-Mott}).
Alternatively, we could employ the equations (\ref{diff0}-\ref{diff2})
in Fourier space
\begin{eqnarray}
\left[i\partial_t-U+3 J T_\mathbf{k}(t)\right]
f_\mathbf{k}^{12}
&=&
-\sqrt{2}J T_\mathbf{k}(t)(f_\mathbf{k}^{11}+f_\mathbf{k}^{22}+1)
\,,
\nonumber\\
\left[i\partial_t+U-3 J T_\mathbf{k}(t)\right]
f_\mathbf{k}^{21}
&=&
+\sqrt{2}J T_\mathbf{k}(t)(f_\mathbf{k}^{11}+f_\mathbf{k}^{22}+1)
\,,
\nonumber\\
i\partial_t f_\mathbf{k}^{11}
=
i\partial_t f_\mathbf{k}^{22}
&=&
\sqrt{2}JT_\mathbf{k}(t)(f^{12}_\mathbf{k}-f^{21}_\mathbf{k})
\nonumber
\,,
\end{eqnarray}
where we have inserted the time-dependent hopping matrix $T_\mathbf{k}(t)$
after the Peierls transformation (\ref{Peierls}).
Since $T_\mathbf{k}$ is periodic in $\mathbf{k}$ (the $\mathbf{k}$-space
is a periodic repetition of the Brillouin zone), the time-dependent
hopping matrices $T_\mathbf{k}(t)$ are oscillating
harmonically\footnote{For non-interacting particles, this behaviour is the
basis for the well-known Bloch oscillations.}
for constant potential gradients.
Thus the above set of equations corresponds to parametric resonance and
can be analysed with Floquet theory.
For simplicity, let us assume that the $T_\mathbf{k}(t)$ behave
after the Peierls transformation as
\begin{eqnarray}
T_\mathbf{k}(t)
=
\frac{1}{Z}
\left(e^{iE_0t}\chi_\mathbf{k}+e^{-iE_0t}\chi_\mathbf{k}^*\right)
\,.
\end{eqnarray}
In order to solve equations (\ref{diff0}-\ref{diff2}) we make the Floquet
ansatz
\begin{eqnarray}
f^{12}_\mathbf{k}
=
\sum_{n=-\infty}^{\infty}e^{i(\nu+n)E_0 t}f^{12}_n
\,,
\\
f^{11}_\mathbf{k}
=
f^{22}_\mathbf{k}
=
\sum_{n=-\infty}^{\infty}e^{i(\nu+n)E_0 t}f^{11}_n-\frac{1}{2}
\,,
\\
f^{21}_\mathbf{k}
=
\sum_{n=-\infty}^{\infty}e^{i(\nu+n)E_0 t}f^{21}_n
\,,
\end{eqnarray}
where $\nu$ denotes the Floquet exponent and the $f^{ab}_n$
are discrete Fourier coefficients
of the correlation functions $f_\mathbf{k}^{ab}$.
In order to satisfy equations (\ref{diff0}-\ref{diff2}), the
functions $f^{ab}_n$ have to fulfill the linear system of equations
\begin{eqnarray}
\label{linsys}
\hat{\mathbf{M}}\cdot\mathbf{f}=\mathbf{0}
\,,
\end{eqnarray}
where we defined the infinite-dimensional matrix
\begin{eqnarray}
\hat{\mathbf{M}}
=\left( \begin{array}{ccccccc}
....&....&....& & & & \\
&\chi_\mathbf{k}\mathbf{M}_{-1}&\mathbf{1}&\chi_\mathbf{k}^*\mathbf{M}_{-1}& & & \\
& &\chi_\mathbf{k}\mathbf{M}_{0}&\mathbf{1}&\chi_\mathbf{k}^*\mathbf{M}_{0}& & \\
& & &\chi_\mathbf{k}\mathbf{M}_{1}&\mathbf{1}&\chi_\mathbf{k}^*\mathbf{M}_{1} &\\
& & & &....&....&....
\end{array}
\right)
\end{eqnarray}
with
\begin{eqnarray}
\mathbf{M}_n=
\frac{J}{ZE_0}\left( \begin{array}{ccc}
-\frac{3}{\nu+n+{U}/{E_0}}&-\frac{2\sqrt{2}}{\nu+n+{U}/{E_0}}&0\\
\frac{2}{\nu+n}&0&-\frac{\sqrt{2}}{\nu+n}\\
0&\frac{2\sqrt{2}}{\nu+n-{U}/{E_0}}&
\frac{3}{\nu+n-{U}/{E_0}}
\end{array}\right)
\,,
\end{eqnarray}
and the vector
\begin{eqnarray}
\mathbf{f}=(...,f^{12}_{-1},f^{11}_{-1},f^{21}_{-1},f^{12}_0,f^{11}_0,f^{21}_0,f^{12}_1,f^{11}_1,f^{21}_1,...)^T
\,.
\end{eqnarray}
The set of equations (\ref{linsys}) has only nontrivial solutions
if the determinant of the infinite-dimensional matrix vanishes, that is
\begin{eqnarray}
\Delta(\nu)=\mathrm{Det}(\hat{\mathbf{M}})=0
\,.
\end{eqnarray}
The above relation determines the Floquet exponent $\nu$ up to multiples of
$2\pi$ and it can be shown that $\nu$ satisfies the equality \cite{Floquet}
\begin{eqnarray}
\sin^2(\pi\nu)=\sin^2\left(\frac{\pi U}{E_0}\right)\Delta(0)\,.
\end{eqnarray}
If the hopping rate is much smaller than the potential gradient,
that is $J\ll E_0$, we may expand $\Delta(0)$ in powers of $J/E_0$.
Using the matrix-identity
\begin{eqnarray}
\mathrm{Det}(\hat{\mathbf{M}})=\exp(\,{\rm Tr}\{\ln\hat{\mathbf{M}}\})
\,,
\end{eqnarray}
we find up to forth order in $J/E_0$
\begin{eqnarray}\label{resonance}
\sin^2\left(\pi\nu\right)
&=&
\sin^2\left(\frac{\pi U}{E_0}\right)
\Bigg[1+\frac{16 J^2|\chi_\mathbf{k}|^2\pi U}{Z^2 E_0(E_0^2 -U^2)}
\cot\left(\frac{\pi U}{E_0}\right)
\nonumber\\
& &+
\frac{8 J^4|\chi_\mathbf{k}|^4 \pi U}
{Z^4\sin^2\left(\frac{\pi U}{E_0}\right)}
\frac{1}{ E_0^2(E_0^2-U^2)^3(4E_0^2-U^2)}
\nonumber\\
& &\quad\times \Bigg\{8\pi U\left(4 E_0^4-5 E_0^2 U^2+U^4\right)
\cos\left(\frac{2\pi U}{E_0}\right)\nonumber\\
& &\quad\quad+E_0\left(-19E_0^4+76E_0^2U^2-33 U^4\right)
\sin\left(\frac{2\pi U}{E_0}\right)\Bigg\}\Bigg]
\,.
\end{eqnarray}
Two cases need to be distinguished.
In the first case, the right hand side of (\ref{resonance})
is between zero and unity,
the Floquet exponent is real and the correlation functions are bounded.
In the second case, the right hand side of (\ref{resonance})
is bigger than unity
or smaller than zero, the Floquet exponent acquires an imaginary part
and the correlation functions grow exponentially,
$f_\mathbf{k}^{ab}\sim \exp(\Im \nu t E_0) $, corresponding
to a Floquet resonance.
We find the first resonance being located at $U=E_0$ with
a width of
$\Delta U=2(\Im \nu)_\mathrm{max}E_0=4\sqrt{2}J|\chi_\mathbf{k}|/Z$.
The second resonance is located at
$U=2E_0+16J^2|\chi_\mathbf{k}|^2/(3E_0Z^2)$ and has
the width
$\Delta U=2(\Im \nu)_\mathrm{max}E_0=
12\sqrt{2}J^2|\chi_\mathbf{k}|^2/(Z^2E_0)$.
In principle, there are resonances when $E_0/U$ adopts higher integer
values but they become smaller for increasing $E_0/U$.
In contrast, the correlation functions in the Fermi-Hubbard model do not
grow exponentially.
This distinction between the bosonic and the fermionic case can already be
deduced from the difference of the conserved quantities (\ref{inv}) and
(\ref{invfermi}).
While the relation (\ref{inv}) allows in principle arbitrary large
values of the correlation functions, we find from (\ref{invfermi})
that $f_\mathbf{k}^{1_B1_B}$ cannot exceed unity and is therefore
bounded.
\section{Conclusions \& Outlook}
For the Bose and the Fermi-Hubbard model, we studied the quantum correlations
analytically by using the hierarchy of correlations obtained via a formal
expansion into powers of $1/Z$.
Starting deep in the Mott regime $J/U\to0$ with exactly one particle per
lattice site, we derived the correlations in the ground state for a finite
value of $J$ and after a quantum quench (i.e., suddenly switching on $J$).
From these correlations, we can also infer the quantum depletion, i.e.,
the probability of having zero (``holon'') or two (``doublon'') particles
at a given lattice site.
It turns out that these observables approach a quasi-equilibrium state some
time after the quench -- but this quasi-equilibrium state is {\em not}
thermal.
Furthermore, we derived the particle-hole (``doublon-holon'') pair creation
probability via tunnelling in tilted lattices and found remarkable analogies
to the Sauter-Schwinger effect (i.e., electron-positron pair creation out of
the quantum vacuum by an external electric field) in the case of weak tilts.
For strong tilts, one obtains resonant tunnelling reminiscent of the Bloch
oscillations for non-interacting particles.
For the Bose-Hubbard model, we also studied a quench from the Mott state to
the super-fluid regime and calculated the growth of phase coherence.
Going up to second order in $1/Z$, we derived the correlations of the
number and parity operators, again in the ground state and after a quench.
For the Fermi-Hubbard model, we found that the spin and charge modes
decouple to first order in $1/Z$.
The dynamics of the charge modes (particle-hole excitations) already
contributes to first order in $1/Z$ whereas the time-evolution of the
spin modes requires going up to the second order in $1/Z$, similar to the
number and parity correlations in the bosonic case.
Comparing our analytical results to numerical simulations for bosons on
finite-size lattices in one and two spatial dimensions,
we found qualitative agreement.
Thus, although our analytical approach is formally based on the limit $Z\gg1$,
we expect that our results apply -- at least qualitatively -- to lattices
in three ($Z=6$), two ($Z=4$), or even one ($Z=2$) spatial dimension.
There are only a few properties which strongly depend on the dimensionality
of the system, one example being the maximum of the parity correlator well
within the Mott regime, which occurs in one spatial dimension only.
In view of the tremendous experimental progress regarding ultra-cold atoms
in optical lattices, for example, most of our predictions should be testable
experimentally.
In this paper, we used the zero-temperature Mott phase as our initial state
-- but the presented method can easily be applied to other initial states.
For example, finite initial temperatures can be incorporated as well because
our approach is based on density matrices.
Even at zero temperature, it should be interesting to study other initial
states.
In the bosonic case, we could start with $U=0$ (instead of $J=0$),
i.e., in the super-fluid phase, where we may use
$\rho^0_\mu=\ket{\alpha}_\mu\!\bra{\alpha}$ with the coherent state
$\hat b_\mu\ket{\alpha}_\mu=\alpha\ket{\alpha}_\mu$,
see, e.g., \cite{KN11}.
In this way, an order parameter $\langle\hat b_\kappa\rangle\not=0$
is introduced at the expense of making the total particle number
ill-defined $[\hat N,\hat\rho_{\mu}]\not=0$.
As another possibility, one could assume non-integer filling
$\langle\hat n_\mu\rangle\not\in\mathbb N$, where one has a non-vanishing
super-fluid component even for arbitrarily small $J$.
For example, taking $\langle\hat n_\mu\rangle<1$, the initial state would be
$\rho^0_\mu=\ket{\psi}_\mu\!\bra{\psi}$ with
$\ket{\psi}=\alpha\ket{0}+\beta\ket{1}$.
In these cases, the time-dependence of $\rho^0_\mu$ will be non-trivial
in general.
In the fermionic case, an analogous initial state would be
$\rho^0_\mu=\ket{\psi}_\mu\!\bra{\psi}$ with
$\ket{\psi}=
\alpha\ket{0^\uparrow0^\downarrow}+\beta\ket{1^\uparrow1^\downarrow}$,
which could describe a Bose-Einstein condensate of Cooper-like pairs,
which may be stabilised by an attractive interaction $U<0$, for example.
\section*{Acknowledgements}
The authors acknowledge valuable discussions with M.~Vojta, A.~Rosch,
W.~Hofstetter, and others (e.g., several members of the SFB-TR12).
This work was supported by the DFG (SFB-TR12).
\section{Appendix: Derivation of the hierarchy}\label{hierarchyApp}
In this Appendix, we derive the hierarchical set of equations for the
correlation functions.
The quantum evolution of the one-site density matrix can be derived
by tracing von Neumann's equation (\ref{Liouville})
over all lattice sites but $\mu$ and exploiting the invariance of the
trace under cyclic permutations
\begin{eqnarray}
\label{singlesite}
i\partial_t \hat{\rho}_\mu
&=&
\frac{1}{Z}\,{\rm Tr}_{\not\mu}
\left\{\sum_{\alpha,\beta\neq\mu}\mathcal{L}_{\alpha\beta}\hat{\rho}
+\sum_{\alpha\neq \mu}\mathcal{L}^S_{\alpha\mu}\hat{\rho}\right\}
+
\,{\rm Tr}_{\not\mu}\left\{\sum_{\alpha\neq\mu}
\mathcal{L}_\alpha\hat{\rho}+\mathcal{L}_\mu\hat{\rho}\right\}
\nonumber\\
&=&
\frac{1}{Z}\sum_{\alpha\neq \mu}
\mathcal{L}^S_{\mu\alpha}\,{\rm Tr}_{\alpha}\{\hat{\rho}_{\mu\kappa}\}
+\mathcal{L}_\mu\hat{\rho}_\mu\,.
\end{eqnarray}
Using the definition of the two-point correlations given in
(\ref{correlated-parts}), we arrive at (\ref{one-site}).
Similarly, the differential equation for the two-particle density matrix can
be deduced by tracing over all lattice sites but $\mu$ and $\nu$,
\begin{eqnarray}
i\partial_t \hat{\rho}_{\mu\nu}
&=&
i\left(\partial_t \hat{\rho}_{\mu\nu}^\mathrm{corr}+
\hat{\rho}_\mu \partial_t \hat{\rho}_\nu+
\hat{\rho}_\nu \partial_t \hat{\rho}_\mu\right)
\nonumber\\
&=&
\frac{1}{Z}\sum_{\alpha\neq \mu\nu}
\,{\rm Tr}_\alpha\left\{\mathcal{L}_{\mu\kappa}^S\hat{\rho}_{\mu\nu\alpha}\right\}
+\frac{1}{Z}\sum_{\alpha\neq \mu\nu}
\,{\rm Tr}_\alpha\left\{\mathcal{L}_{\kappa\nu}^S\hat{\rho}_{\mu\nu\alpha}\right\}
\nonumber\\
&&
+\frac{1}{Z}\mathcal{L}_{\mu\nu}^S\hat{\rho}_{\mu\nu}
+\mathcal{L}_\mu \hat{\rho}_{\mu\nu}+\mathcal{L}_\nu \hat{\rho}_{\mu\nu}\,.
\end{eqnarray}
With the definitions (\ref{correlated-parts}) and the
time-evolution for the single-site density matrix (\ref{singlesite}),
we find for the two-point correlation functions (\ref{two-sites}).
The equations (\ref{one-site}) and (\ref{two-sites})
preserve the hierarchy in time if initially
$\hat{\rho}_\mu=\mathcal{O}(Z^0)$ and
$\hat{\rho}_{\mu\nu}^\mathrm{corr}=\mathcal{O}(1/Z)$ holds.
In order to derive the full hierarchy,
we define the generating functional
\begin{eqnarray}
{\cal F}(\hat\alpha)
=
{\cal F}(\{\hat\alpha_\mu\})
=
\ln\left[
\,{\rm Tr}\left\{\hat\rho\bigotimes_\mu(\mathbf{1}_\mu+\hat\alpha_\mu)\right\}
\right]
\,,
\end{eqnarray}
where $\hat\rho$ is the density matrix of the full lattice and
\begin{eqnarray}
\hat\alpha_\mu=\sum_{m,n}\alpha_\mu^{m,n}|m\rangle_\mu\langle n|
\end{eqnarray}
are arbitrary operators acting on the Hilbert spaces associated to the
lattice sites $\mu$ with the local basis $\{\ket{n}_\mu\}$.
The role of this functional is to generate all correlated density matrices
via the derivatives with respect to these operators $\hat\alpha_\mu$ which
are defined via
\begin{eqnarray}
\frac{\partial{\cal F(\{\alpha\})}}{{\partial\hat\alpha_\mu}}
=
\sum_{m,n}|n\rangle_\mu\langle m|\,
\frac{\partial \cal F(\{\alpha\})}{\partial \alpha^{m,n}_\mu}
=
\sum_{m,n}|n\rangle_\mu\langle m|\,
\frac{\partial{\cal F}(\{\alpha\})}
{\partial\, {_\mu\!\bra{m}}\hat\alpha_\mu\ket{n}_\mu}
\,.
\end{eqnarray}
If we consider an ensemble ${\cal S}=\{\mu_1, \dots ,\mu_\ell\}$
of $\ell$ different lattice sites $\mu_1\not= \dots \not=\mu_\ell$,
we obtain the correlation operators via
\begin{eqnarray}
\hat\rho^{\rm corr}_{\cal S}
=
\left.
\frac{\partial}{\partial\hat\alpha_{\mu_1}}
\frac{\partial}{\partial\hat\alpha_{\mu_2}}
\dots
\frac{\partial}{\partial\hat\alpha_{\mu_\ell}}
{\cal F}(\hat \alpha)\right|_{\hat \alpha=0}
\,.
\end{eqnarray}
These operators are related to the corresponding reduced density matrix
operator $\rho_{\cal S}$ through the relation
\begin{eqnarray}
\hat\rho_{\cal S}
=
\hat\rho_{\mu_1 \dots \mu_\ell}
=
\sum_{\cup_i{\cal P}_i={\cal S}}\prod_i\hat\rho^{\rm corr}_{{\cal P}_i}
\end{eqnarray}
where the sum runs over all possible segmentations of the subset ${\cal S}$
into partitions ${\cal P}_i$ starting from the whole subset
${\cal P}={\cal S}$ and ranging to single lattice sites
${\cal P}_i=\{\mu\}$ where
$\hat\rho^{\rm corr}_{{\cal P}_i=\{\mu\}}=\hat\rho_\mu$ is understood.
For two and three lattice sites, the above equation reproduces
Eq.~(\ref{correlated-parts}).
Our derivation is based on the following scaling hierarchy of
correlations:
\begin{eqnarray}
\label{hierarchy1}
\hat\rho_{\cal S}^{\rm c}=\,{\cal O}\left(Z^{1-|\cal S|}\right)
\end{eqnarray}
where $|\cal S|$ is the number $\ell$ of lattice sites in the set $\cal S$.
From the Liouville equation (\ref{Liouville}),
the temporal evolution of ${\cal F}$ is given by
\begin{eqnarray}
\label{eqgf}
i \partial_t
{\cal F}(\hat\alpha)
&=&
\sum_\mu\,{\rm Tr}_\mu
\left\{
\hat\alpha_\mu \,{\cal L}_\mu \frac{\partial{\cal F}}{\partial\hat\alpha_\mu}
\right\}
\\
&+&
\frac{1}{Z} \sum_{\mu,\nu} \,{\rm Tr}_{\mu\nu}
\left\{
(\hat\alpha_\mu + \hat\alpha_\nu +\hat\alpha_\mu \hat\alpha_\nu)
\,{\cal L}_{\mu \nu}
\left(
\frac{\partial^2{\cal F}}{\partial\hat\alpha_\mu\partial\hat\alpha_\nu}
+
\frac{\partial{\cal F}}{\partial\hat\alpha_\mu}
\frac{\partial{\cal F}}{\partial\hat\alpha_\nu}
\right)
\right\}\nonumber
\,.
\end{eqnarray}
By taking successive derivatives and using the generalised Leibniz rule
\begin{eqnarray}
\frac{\partial}{\partial\hat\alpha_{\mu_1}}
\frac{\partial}{\partial\hat\alpha_{\mu_2}}
\dots
\frac{\partial}{\partial\hat\alpha_{\mu_\ell}}
\left[{\cal F}(\hat \alpha)\right]^2
&=&
\sum_{{\cal P}\subseteq {\cal S}}^{{\cal P}\cup\bar{\cal P}={\cal S}}
\left[
\left(
\prod_{\mu_i \in {\cal P}}
\frac{\partial}{\partial\hat\alpha_{\mu_i}}
\right)
{\cal F}(\hat \alpha)
\right]
\times
\\
&&
\times
\left[
\left(
\prod_{\mu_j \in \bar{\cal P}}
\frac{\partial}{\partial\hat\alpha_{\mu_j}}
\right)
{\cal F}(\hat \alpha)
\right]
\,,
\end{eqnarray}
as well as the the property
\begin{eqnarray}
\frac{\partial^2{\cal F}(\hat \alpha)}{\partial\hat\alpha_{\mu}^2}
=
\frac{\partial}{\partial\hat \alpha_{\mu}}
\frac{\partial}{\partial\hat \alpha_{\mu}}
{\cal F}(\hat \alpha)
=
-
\frac{\partial{\cal F}(\hat \alpha)}{\partial\hat\alpha_{\mu}}
\frac{\partial{\cal F}(\hat \alpha)}{\partial\hat\alpha_{\mu}}
=
-
\left(\frac{\partial{\cal F}(\hat \alpha)}{\partial\hat\alpha_{\mu}}\right)^2
\,,
\end{eqnarray}
we establish the following set of
equations for the correlated density matrices:
\begin{eqnarray}
\label{general}
i \partial_t\hat\rho^{\rm corr}_{\cal S}
&=&
\sum_{\mu \in {\cal S}} \,{\cal L}_\mu \hat\rho^{\rm corr}_{\cal S}
+
\frac{1}{Z}\sum_{\mu,\nu\in{\cal S}}
\,{\cal L}_{\mu \nu}\,\hat\rho^{\rm corr}_{\cal S}
\nonumber\\
& &+
\frac{1}{Z}
\sum_{\kappa\notin{\cal S}} \sum_{\mu\in{\cal S}}
\,{\rm Tr}_{\kappa}
\Bigg[
\,{\cal L}^S_{\mu \kappa}
\hat\rho^{\rm corr}_{{\cal S}\cup {\kappa}}
+
\sum_{{\cal P}\subseteq{\cal S}\setminus\{\mu\}}
^{{\cal P}\cup\bar{\cal P}={\cal S}\setminus\{\mu\}}
\,{\cal L}^S_{\mu \kappa}
\hat\rho^{\rm corr}_{\{\mu\}\cup{\cal P}}\,
\hat\rho^{\rm corr}_{\{\kappa\}\cup\bar{\cal P}}
\Bigg]\nonumber\\
& &+
\frac{1}{Z}
\sum_{\mu,\nu\in{\cal S}}
\sum_{{\cal P}\subseteq{\cal S}\setminus\{\mu,\nu\}}
^{{\cal P}\cup\bar{\cal P}={\cal S}\setminus\{\mu,\nu\}}
\Bigg\{
\,{\cal L}_{\mu \nu}\,
\hat\rho^{\rm corr}_{\{\mu\}\cup{\cal P}}\,
\hat\rho^{\rm corr}_{\{\nu\}\cup\bar{\cal P}}
\nonumber \\
& &-\,{\rm Tr}_{\nu}\Bigg[
\,{\cal L}^S_{\mu \nu}(\hat\rho^{\rm corr}_{\{\mu,\nu\}\cup\bar{\cal P}}
+
\sum_{{\cal Q}\subseteq\bar{\cal P}}
^{{\cal Q}\cup\bar{\cal Q}=\bar{\cal P}}
\hat\rho^{\rm corr}_{\{\mu\}\cup{\cal Q}}\,
\hat\rho^{\rm corr}_{\{\nu\}\cup\bar{\cal Q}}
)
\Bigg]
\hat\rho^{\rm corr}_{\{\nu\}\cup{\cal P}}
\Bigg\}
\,.
\end{eqnarray}
For $\ell=1$ and $\ell=2$ we recover the equations
(\ref{one-site}) and (\ref{two-sites}).
A careful inspection of this set of equations shows that the hierarchy
in (\ref{hierarchy1}) is preserved in time:
Imposing the scaling
$\hat\rho^{\rm corr}_{\cal S}=\,{\cal O}(Z^{1-|\cal S|})$ on the r.h.s.\ of the
above equation, we find that the time derivative on the l.h.s.\ does also
satisfy the hierarchy (\ref{hierarchy1}).
Therefore, inserting (\ref{hierarchy1}) into (\ref{general}) and taking the
limit $Z\rightarrow\infty$, we obtain the leading-order contributions
\begin{eqnarray}
\label{general1}
i \partial_t\hat\rho^{\rm corr}_{\cal S}
&=&
\sum_{\mu \in {\cal S}} \,{\cal L}_\mu \hat\rho^{\rm corr}_{\cal S}
+
\frac{1}{Z}
\sum_{\kappa\notin{\cal S}} \sum_{\mu\in{\cal S}}
\,{\rm Tr}_{\kappa}\Bigg[
\sum_{{\cal P}\subseteq{\cal S}\setminus\{\mu\}}
^{{\cal P}\cup\bar{\cal P}={\cal S}\setminus\{\mu\}}
\,{\cal L}^S_{\mu \kappa}
\hat\rho^{\rm corr}_{\{\mu\}\cup{\cal P}}\,
\hat\rho^{\rm corr}_{\{\kappa\}\cup\bar{\cal P}}
\Bigg]
\nonumber \\
& &+\frac{1}{Z}
\sum_{\mu,\nu\in{\cal S}}
\sum_{{\cal P}\subseteq{\cal S}\setminus\{\mu,\nu\}}
^{{\cal P}\cup\bar{\cal P}={\cal S}\setminus\{\mu,\nu\}}
\Bigg\{
\,{\cal L}_{\mu \nu}\,
\hat\rho^{\rm corr}_{\{\mu\}\cup{\cal P}}\,
\hat\rho^{\rm corr}_{\{\nu\}\cup\bar{\cal P}}
\nonumber\\
& &-
\,{\rm Tr}_{\nu}\Bigg[
\,{\cal L}^S_{\mu \nu}
\sum_{{\cal Q}\subseteq\bar{\cal P}}
^{{\cal Q}\cup\bar{\cal Q}=\bar{\cal P}}
\hat\rho^{\rm corr}_{\{\mu\}\cup{\cal Q}}\,
\hat\rho^{\rm corr}_{\{\nu\}\cup\bar{\cal Q}}
\Bigg]
\hat\rho^{\rm corr}_{\{\nu\}\cup{\cal P}}
\Bigg\}
+\,{\cal O}(Z^{-|\cal S|})
\,.
\end{eqnarray}
For $\ell=1$ and $\ell=2$, we recover equations
(\ref{one-site-approx}) and (\ref{two-sites-approx}).
In contrast to the exact expression (\ref{general}), the approximated
leading-order equations (\ref{general1}) form a closed set.
The exact time evolution (\ref{general}) of the $|\cal S|$-point
correlator $\partial_t\hat\rho^{\rm corr}_{\cal S}$ also depends on the
higher-order correlation term $\hat\rho^{\rm corr}_{{\cal S}\cup{\kappa}}$
involving $|{\cal S}|+1$ points.
The approximated expression (\ref{general1}), on the other hand, only
contains correlators of the same or lower rank.
This facilitates the iterative solution of the problem sketched in
Section~\ref{hierarchyofcorr}.
First one solves the zeroth-order equation (\ref{one-site-approx})
for $\hat\rho_\mu^0$.
Inserting this result $\hat\rho_\mu^0$ into the first-order (in $1/Z$)
equation (\ref{two-sites-approx}) for $\hat\rho_{\mu\nu}^{\rm corr}$,
we obtain a first-order result for $\hat\rho_{\mu\nu}^{\rm corr}$.
This first-order result for $\hat\rho_{\mu\nu}^{\rm corr}$ can then be
inserted into the equation for $\hat\rho_{\mu\nu\lambda}^{\rm corr}$
which is of second order $1/Z^2$.
Furthermore, we may use the first-order result for
$\hat\rho_{\mu\nu}^{\rm corr}$ in order to obtain a better approximation
for the on-site density matrix $\hat\rho_\mu^1$ which is valid to first
order in $1/Z$ and contains the quantum depletion etc.
Repeating this iteration, we may successively ``climb up'' to higher and
higher orders in $1/Z$.
\section{Appendix: Staggered Magnetic Field}\label{staggered}
We assumed in Section \ref{chargemodes} that the initial state of the
Fermi-Hubbard system is given by the N\'eel state.
However, for $J=0$ we have infinitely many states with same energy and
vanishing total spin.
In order to single out the N\'eel state, we add a staggered magnetic field
to the Hubbard Hamiltonian,
\begin{eqnarray}
\hat H
&=&
-\frac{J}{Z}\sum_{\mu,\nu}T_{\mu\nu}
\left(\hat{c}_{\mu\uparrow}^\dagger\hat{c}_{\mu\uparrow}+
\hat{c}_{\mu\downarrow}^\dagger\hat{c}_{\mu\downarrow}\right)
\nonumber\\
&&+
\sum_{\mu}\left(
U\hat{n}_\mu^\uparrow\hat{n}_\mu^\downarrow
-A_{\mu\downarrow}\hat{n}_\mu^\downarrow
-A_{\mu\uparrow} \hat{n}_\mu^\uparrow
\right)
\,.
\end{eqnarray}
If we choose the magnetic field as
$A_{\mu\downarrow}(x_\mu\in \mathcal{A})=a$,
$A_{\mu\downarrow}(x_\mu\in \mathcal{B})=0$,
$A_{\mu\uparrow}(x_\mu\in \mathcal{B})=a$, and
$A_{\mu\uparrow}(x_\mu\in \mathcal{A})=0$,
the N\'eel state is the unique ground state for $J=0$ at half filling.
The Heisenberg equations (\ref{annihilation-operator})-(\ref{number-operator}) read now
\begin{eqnarray}
i\partial_t \hat{c}_{\mu s}
&=&
-\frac{J}{Z}\sum_{\kappa\neq \mu} T_{\mu\kappa}\hat{c}_{\kappa s}
+U\hat{c}_{\mu s} \hat{n}_{\mu \bar{s}}-A_{\mu s}\hat{c}_{\mu s}
\\
i\partial_t \hat{c}_{\mu s}^\dagger
&=&
\frac{J}{Z}\sum_{\kappa\neq \mu} T_{\mu\kappa}\hat{c}^\dagger_{\kappa s}
-U\hat{c}^\dagger_{\mu s} \hat{n}_{\mu \bar{s}}
+A_{\mu s}\hat{c}^\dagger_{\mu s}
\\
i\partial_t\hat{n}_{\mu s}
&=&
-i\partial_t\hat{\bar{n}}_{\mu s}
=
\frac{J}{Z}\sum_{\kappa\neq \mu}T_{\mu\kappa}\left(
\hat{c}_{\kappa s}^\dagger \hat{c}_{\mu s}
-\hat{c}_{\mu s}^\dagger \hat{c}_{\kappa s}
\right)\,,
\end{eqnarray}
To first order in $1/Z$, we find the closed set of differential equations,
cf.~Eqs.~(\ref{corr1})-(\ref{sectors}),
\begin{eqnarray}
i\partial_t \langle \hat{c}_{\mu a}^\dagger\hat{c}_{\nu b}\hat{n}_{\mu\bar{a}}\hat{n}_{\nu\bar{b}}\rangle&=&
\frac{J}{Z}\sum_{\kappa\neq \mu,\nu}T_{\mu\kappa}
\langle\hat{c}_{\kappa a}^\dagger\hat{c}_{\nu b}(\hat{n}_{\kappa\bar{a}}+\hat{\bar{n}}_{\kappa\bar{a}})
\hat{n}_{\nu\bar{b}}\rangle\langle\hat{n}_{\mu \bar{a}}\rangle_0\nonumber\\
& &-\frac{J}{Z}\sum_{\kappa\neq \mu,\nu}T_{\nu\kappa}
\langle\hat{c}_{\mu a}^\dagger\hat{c}_{\kappa b}\hat{n}_{\mu\bar{a}}(\hat{n}_{\kappa\bar{b}}+\hat{\bar{n}}_{\kappa\bar{b}})
\rangle\langle\hat{n}_{\nu \bar{b}}\rangle_0\nonumber\\
& &+(A_{\mu a}-A_{\nu b})\langle \hat{c}_{\mu a}^\dagger\hat{c}_{\nu b}\hat{n}_{\mu\bar{a}}\hat{n}_{\nu\bar{b}}\rangle\nonumber\\
& &+\frac{J}{Z}T_{\mu\nu}\langle \hat{c}_{\nu a}^\dagger\hat{c}_{\nu b}\hat{n}_{\mu\bar{a}}\hat{n}_{\nu\bar{b}}\rangle_0
-\frac{J}{Z}T_{\mu\nu}\langle \hat{c}_{\mu a}^\dagger\hat{c}_{\mu b}\hat{n}_{\mu\bar{a}}\hat{n}_{\nu\bar{b}}\rangle_0
\end{eqnarray}
\begin{eqnarray}
i\partial_t \langle \hat{c}_{\mu a}^\dagger\hat{c}_{\nu b}\hat{n}_{\mu\bar{a}}\hat{\bar{n}}_{\nu\bar{b}}\rangle&=&
\frac{J}{Z}\sum_{\kappa\neq \mu,\nu}T_{\mu\kappa}
\langle\hat{c}_{\kappa a}^\dagger\hat{c}_{\nu b}(\hat{n}_{\kappa\bar{a}}+\hat{\bar{n}}_{\kappa\bar{a}})
\hat{\bar{n}}_{\nu\bar{b}}\rangle\langle\hat{n}_{\mu \bar{a}}\rangle_0\nonumber\\
& &-\frac{J}{Z}\sum_{\kappa\neq \mu,\nu}T_{\nu\kappa}
\langle\hat{c}_{\mu a}^\dagger\hat{c}_{\kappa b}\hat{n}_{\mu\bar{a}}(\hat{n}_{\kappa\bar{b}}+\hat{\bar{n}}_{\kappa\bar{b}})
\rangle\langle\hat{\bar{n}}_{\nu \bar{b}}\rangle_0\nonumber\\
& &-(U-A_{\mu a}+A_{\nu b})\langle \hat{c}_{\mu a}^\dagger\hat{c}_{\nu b}\hat{n}_{\mu \bar{a}}\hat{\bar{n}}_{\nu \bar{b}}\rangle\nonumber\\
& &+\frac{J}{Z}T_{\mu\nu}\langle \hat{c}_{\nu a}^\dagger\hat{c}_{\nu b}\hat{n}_{\mu\bar{a}}\hat{\bar{n}}_{\nu\bar{b}}\rangle_0
-\frac{J}{Z}T_{\mu\nu}\langle \hat{c}_{\mu a}^\dagger\hat{c}_{\mu b}\hat{n}_{\mu\bar{a}}\hat{\bar{n}}_{\nu\bar{b}}\rangle_0
\end{eqnarray}
\begin{eqnarray}
i\partial_t \langle \hat{c}_{\mu a}^\dagger\hat{c}_{\nu b}\hat{\bar{n}}_{\mu\bar{a}}\hat{n}_{\nu\bar{b}}\rangle&=&
\frac{J}{Z}\sum_{\kappa\neq \mu,\nu}T_{\mu\kappa}
\langle\hat{c}_{\kappa a}^\dagger\hat{c}_{\nu b}(\hat{n}_{\kappa\bar{a}}+\hat{\bar{n}}_{\kappa\bar{a}})
\hat{n}_{\nu\bar{b}}\rangle\langle\hat{\bar{n}}_{\mu \bar{a}}\rangle_0\nonumber\\
& &-\frac{J}{Z}\sum_{\kappa\neq \mu,\nu}T_{\nu\kappa}
\langle\hat{c}_{\mu a}^\dagger\hat{c}_{\kappa b}\hat{\bar{n}}_{\mu\bar{a}}(\hat{n}_{\kappa\bar{b}}+\hat{\bar{n}}_{\kappa\bar{b}})
\rangle\langle\hat{n}_{\nu \bar{b}}\rangle_0\nonumber\\
& &+(U+A_{\mu a}-A_{\nu b})\langle \hat{c}_{\mu a}^\dagger\hat{c}_{\nu b}\hat{\bar{n}}_{\mu \bar{a}}\hat{n}_{\nu \bar{b}}\rangle\nonumber\\
& &+\frac{J}{Z}T_{\mu\nu}\langle \hat{c}_{\nu a}^\dagger\hat{c}_{\nu b}\hat{\bar{n}}_{\mu\bar{a}}\hat{n}_{\nu\bar{b}}\rangle_0
-\frac{J}{Z}T_{\mu\nu}\langle \hat{c}_{\mu a}^\dagger\hat{c}_{\mu b}\hat{\bar{n}}_{\mu\bar{a}}\hat{n}_{\nu\bar{b}}\rangle_0
\end{eqnarray}
\begin{eqnarray}
i\partial_t \langle \hat{c}_{\mu a}^\dagger\hat{c}_{\nu b}\hat{\bar{n}}_{\mu\bar{a}}\hat{\bar{n}}_{\nu\bar{b}}\rangle&=&
\frac{J}{Z}\sum_{\kappa\neq \mu,\nu}T_{\mu\kappa}
\langle\hat{c}_{\kappa a}^\dagger\hat{c}_{\nu b}(\hat{n}_{\kappa\bar{a}}+\hat{\bar{n}}_{\kappa\bar{a}})
\hat{\bar{n}}_{\nu\bar{b}}\rangle\langle\hat{\bar{n}}_{\mu \bar{a}}\rangle_0\nonumber\\
& &-\frac{J}{Z}\sum_{\kappa\neq \mu,\nu}T_{\nu\kappa}
\langle\hat{c}_{\mu a}^\dagger\hat{c}_{\kappa b}\hat{\bar{n}}_{\mu\bar{a}}(\hat{n}_{\kappa\bar{b}}+\hat{\bar{n}}_{\kappa\bar{b}})
\rangle\langle\hat{\bar{n}}_{\nu \bar{b}}\rangle_0\nonumber\\
& &+(A_{\mu a}-A_{\nu b})\langle \hat{c}_{\mu a}^\dagger\hat{c}_{\nu b}\hat{\bar{n}}_{\mu\bar{a}}\hat{\bar{n}}_{\nu\bar{b}}\rangle\nonumber\\
& &+\frac{J}{Z}T_{\mu\nu}\langle \hat{c}_{\nu a}^\dagger\hat{c}_{\nu b}\hat{\bar{n}}_{\mu\bar{a}}\hat{\bar{n}}_{\nu\bar{b}}\rangle_0
-\frac{J}{Z}T_{\mu\nu}\langle \hat{c}_{\mu a}^\dagger\hat{c}_{\mu b}\hat{\bar{n}}_{\mu\bar{a}}\hat{\bar{n}}_{\nu\bar{b}}\rangle_0\,.
\end{eqnarray}
If $x_{\mu}\in A$ and $x_{\nu}\in B$ we denote the correlations
$\langle \hat{c}_{\mu \downarrow}^\dagger\hat{c}_{\mu \downarrow}\hat{n}_{\mu\uparrow}\hat{n}_{\nu\uparrow}\rangle=f_{\mu\nu}^{1_A1_B}$,
$\langle \hat{c}_{\mu \downarrow}^\dagger\hat{c}_{\mu \downarrow}\hat{\bar{n}}_{\mu\uparrow}\hat{n}_{\nu\uparrow}\rangle=f_{\mu\nu}^{0_A1_B}$, etc.
Inserting the zeroth-order N\'eel state,
we find four equations which fully decouple, cf.~Eq.~(\ref{hom1}),
\begin{eqnarray}
i\partial_t f^{1_A0_B}_{\mu\nu}&=&-(U-a) f^{1_A0_B}_{\mu\nu}\\
i\partial_t f^{0_B1_A}_{\mu\nu}&=&(U-
a) f^{0_B1_A}_{\mu\nu}\\
i\partial_t f^{0_B0_B}_{\mu\nu}&=&0\\
i\partial_t f^{1_A1_A}_{\mu\nu}&=&0\,.
\end{eqnarray}
In general, these four correlations are sources in the following four
pairs of coupled equations, cf.~Eqs.~(\ref{hom2})-(\ref{hom12}),
\begin{eqnarray}
i\partial_t f^{0_A0_B}_{\mu\nu}&=&\frac{J}{Z}\sum_{\kappa\neq{\mu,\nu}}T_{\mu\kappa}\left(f^{1_B0_B}_{\kappa\nu}+f^{0_B0_B}_{\kappa\nu}\right)+a f^{0_A0_B}_{\mu\nu}\\
i\partial_t f^{1_B0_B}_{\mu\nu}&=&\frac{J}{Z}\sum_{\kappa\neq{\mu,\nu}}T_{\mu\kappa}\left(f^{0_A0_B}_{\kappa\nu}+f^{1_A0_B}_{\kappa\nu}\right)
-Uf^{1_B0_B}_{\mu\nu}\,,\\
i\partial_t f^{0_B0_A}_{\mu\nu}&=&-\frac{J}{Z}\sum_{\kappa\neq{\mu,\nu}}T_{\kappa\nu}\left(f^{0_B1_B}_{\mu\kappa}+f^{0_B0_B}_{\mu\kappa}\right)-af^{0_B0_A}_{\mu\nu}\\
i\partial_t f^{0_B1_B}_{\mu\nu}&=&-\frac{J}{Z}\sum_{\kappa\neq{\mu,\nu}}T_{\kappa\nu}\left(f^{0_B0_A}_{\mu\kappa}+f^{0_B1_A}_{\mu\kappa}\right)
+Uf^{0_B1_B}_{\mu\nu}\,,\\
i\partial_t f^{1_B1_A}_{\mu\nu}&=&\frac{J}{Z}\sum_{\kappa\neq{\mu,\nu}}T_{\mu\kappa}\left(f^{0_A1_A}_{\kappa\nu}+f^{1_A1_A}_{\kappa\nu}\right)-af^{1_B1_A}_{\mu\nu}\\
i\partial_t f^{0_A1_A}_{\mu\nu}&=&\frac{J}{Z}\sum_{\kappa\neq{\mu,\nu}}T_{\mu\kappa}\left(f^{0_B1_A}_{\kappa\nu}+f^{1_B1_A}_{\kappa\nu}\right)
+Uf^{0_A1_A}_{\mu\nu}\,,\\
i\partial_t f^{1_A1_B}_{\mu\nu}&=&-\frac{J}{Z}\sum_{\kappa\neq{\mu,\nu}}T_{\kappa\nu}\left(f^{1_A0_A}_{\mu\kappa}+f^{1_A1_A}_{\mu\kappa}\right)+af^{1_A1_B}_{\mu\nu}\\
i\partial_t f^{1_A0_A}_{\mu\nu}&=&-\frac{J}{Z}\sum_{\kappa\neq{\mu,\nu}}T_{\kappa\nu}\left(f^{1_A0_B}_{\mu\kappa}+f^{1_A1_B}_{\mu\kappa}\right)
-Uf^{1_A0_A}_{\mu\nu}\,.
\end{eqnarray}
The eigenfrequencies of these equations read now
\begin{eqnarray}
\label{eigenmodesstaggerd}
\omega_\mathbf{k}^\pm=\frac{U+a\pm\sqrt{4J^2T_\mathbf{k}^2+(U-a)^2}}{2}\,.
\end{eqnarray}
In contrast to the eigenmodes (\ref{eigenmodes}) we see that the modes
(\ref{eigenmodesstaggerd}) have a gap in the limit $J\rightarrow 0$.
This enables us to switch on $J$ and then switch off $a$ adiabatically
in order to have a well-defined initial state without correlations.
Furthermore we have the four coupled equations, cf.~Eqs.~(\ref{charge1})-(\ref{charge4}),
\begin{eqnarray}
i\partial_t f^{0_A0_A}_{\mu\nu}&=&\frac{J}{Z}\sum_{\kappa\neq{\mu,\nu}}\left\{T_{\mu\kappa}\left(f^{0_B0_A}_{\kappa\nu}+f^{1_B0_A}_{\kappa\nu}\right)
-T_{\kappa\nu}\left(f^{0_A0_B}_{\mu\kappa}+f^{0_A1_B}_{\mu\kappa}\right)\right\}\\
i\partial_t f^{0_A1_B}_{\mu\nu}&=&\frac{J}{Z}\sum_{\kappa\neq{\mu,\nu}}\left\{T_{\mu\kappa}\left(f^{1_B1_B}_{\kappa\nu}+f^{0_B1_B}_{\kappa\nu}\right)
-T_{\kappa\nu}\left(f^{0_A0_A}_{\mu\kappa}+f^{0_A1_A}_{\mu\kappa}\right)\right\}\nonumber\\
& &+(U+a) f^{0_A1_B}_{\mu\nu}-\frac{J}{Z}T_{\mu\nu}\\
i\partial_t f^{1_B0_A}_{\mu\nu}&=&\frac{J}{Z}\sum_{\kappa\neq{\mu,\nu}}\left\{T_{\mu\kappa}\left(f^{0_A0_A}_{\kappa\nu}+f^{1_A0_A}_{\kappa\nu}\right)
-T_{\kappa\nu}\left(f^{1_B1_B}_{\mu\kappa}+f^{1_B0_B}_{\mu\kappa}\right)\right\}\nonumber\\
& &-(U+a) f^{1_B0_A}_{\mu\nu}+\frac{J}{Z}T_{\mu\nu}\\
i\partial_t f^{1_B1_B}_{\mu\nu}&=&\frac{J}{Z}\sum_{\kappa\neq{\mu,\nu}}\left\{T_{\mu\kappa}\left(f^{0_A1_B}_{\kappa\nu}+f^{1_A1_B}_{\kappa\nu}\right)
-T_{\kappa\nu}\left(f^{1_B0_A}_{\mu\kappa}+f^{1_B1_A}_{\mu\kappa}\right)\right\}\,.
\end{eqnarray}
The eigenfrequencies of these equations read now
\begin{eqnarray}
\omega_\mathbf{k}^\pm=\pm\sqrt{4J^2T_\mathbf{k}^2+(U+a)^2}\,.
\end{eqnarray}
\section{Appendix: Second-order Equations}\label{secondorder}
The differential equation for the three-point correlator reads
\begin{eqnarray}
i\partial_t\hat{\rho}_{\alpha\beta\gamma}^\mathrm{corr}
&=&
\hat{\mathcal{L}}_\alpha\hat{\rho}_{\alpha\beta\gamma}^\mathrm{corr}
+\frac{1}{Z}\hat{\mathcal{L}}^S_{\alpha\beta}
\left(
\hat{\rho}_\alpha\hat{\rho}^\mathrm{corr}_{\beta\gamma}
+\hat{\rho}_\beta\hat{\rho}^\mathrm{corr}_{\alpha\gamma}
\right)
\nonumber\\
& &+
\frac{1}{Z}\sum_{\kappa\neq\{\alpha,\beta,\gamma\}}
\mathrm{tr}_\kappa
\left\{
\hat{\mathcal{L}}^S_{\alpha\kappa}
\left(
\hat{\rho}^\mathrm{corr}_{\alpha\beta\gamma}\hat{\rho}_{\kappa}+
\hat{\rho}^\mathrm{corr}_{\alpha\beta}\hat{\rho}^\mathrm{corr}_{\kappa\gamma}
+\hat{\rho}^\mathrm{corr}_{\alpha\gamma}
\hat{\rho}^\mathrm{corr}_{\kappa\beta}
+\hat{\rho}_{\alpha}\hat{\rho}^\mathrm{corr}_{\kappa\beta\gamma}
\right)
\right\}
\nonumber\\
& &-
\frac{1}{Z}\hat{\rho}_{\alpha\gamma}^\mathrm{corr}
\mathrm{tr}_\alpha
\left\{
\hat{\mathcal{L}}^S_{\alpha\beta}
\hat{\rho}_\beta\hat{\rho}_\alpha
\right\}
-\frac{1}{Z}\hat{\rho}_{\alpha\beta}^\mathrm{corr}
\mathrm{tr}_\alpha
\left\{
\hat{\mathcal{L}}^S_{\alpha\gamma}\hat{\rho}_\gamma\hat{\rho}_\alpha
\right\}
\nonumber\\
& &-
\frac{1}{Z}\hat{\rho}_\alpha
\mathrm{tr}_\alpha
\left\{
\hat{\mathcal{L}}_{\alpha\beta}^S
\hat{\rho}_{\beta\gamma}^\mathrm{corr}\hat{\rho}_\alpha+
\hat{\mathcal{L}}_{\alpha\beta}^S
\hat{\rho}_\beta\hat{\rho}_{\alpha\gamma}^\mathrm{corr}
\right\}
\nonumber\\
& &-
\frac{1}{Z}\hat{\rho}_\alpha
\mathrm{tr}_\alpha
\left\{
\hat{\mathcal{L}}_{\alpha\gamma}^S
\hat{\rho}_{\gamma\beta}^\mathrm{corr}\hat{\rho}_\alpha+
\hat{\mathcal{L}}_{\alpha\gamma}^S
\hat{\rho}_\gamma\hat{\rho}_{\alpha\beta}^\mathrm{corr}
\right\}
\nonumber\\
& &
+(\alpha\rightarrow\beta,\beta\rightarrow\gamma,\gamma\rightarrow\alpha)
+(\alpha\rightarrow\gamma,\gamma\rightarrow\beta,\beta\rightarrow\alpha)
\nonumber\\
& &
+\mathcal{O}(1/Z^3)
\,.
\end{eqnarray}
In the following we use the matrix elements of
$\hat{\rho}_{\mu\nu}^\mathrm{corr}$ and $\hat{\rho}_\mu$ in order $1/Z$
and define
\begin{eqnarray}
\hat{\rho}_{\alpha\beta\gamma}^\mathrm{corr}
=
\sum_{a,a',b,b',c,c'}\rho_{\alpha\beta\gamma}^{aa'bb'cc'}
|a\rangle_\alpha\langle a'|\otimes
|b\rangle_\beta \langle b'|\otimes
|c\rangle_\gamma\langle c'|
\,.
\end{eqnarray}
All three-point correlations can be deduced by permutation and
complex conjugation from the following set of differential equations
\begin{eqnarray}\label{rho001001}
i\partial_t \rho_{\alpha\beta\gamma}^{001001}
&=&
\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\beta\kappa}
\left(\rho_{\alpha\kappa\gamma}^{001001}+\sqrt{2}\rho_{\alpha\kappa\gamma}^{002101}\right)
\nonumber\\
& &-\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\gamma\kappa}
\left(
\rho_{\alpha\beta\kappa}^{001001}+\sqrt{2}\rho_{\alpha\beta\kappa}^{001012}
\right)
\nonumber\\& &
+\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\alpha\kappa}
f^{11}_{\alpha\beta}
\left(f^{11}_{\kappa\gamma}+\sqrt{2}f^{12}_{\kappa\gamma}\right)
\nonumber\\
& &-\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\alpha\kappa}
f^{11}_{\alpha\gamma}
\left(f^{11}_{\kappa\beta}+\sqrt{2}f^{21}_{\kappa\beta}\right)
\end{eqnarray}
\begin{eqnarray}
i\partial_t \rho_{\alpha\beta\gamma}^{001012}&=&-U\rho_{\alpha\beta\gamma}^{001012}
+\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\beta\kappa}
\left(\rho_{\alpha\kappa\gamma}^{001012}+\sqrt{2}\rho_{\alpha\kappa\gamma}^{002112}\right)\nonumber\\
& &+\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\gamma\kappa}
\left(\sqrt{2}\rho_{\alpha\beta\kappa}^{001001}+2\rho_{\alpha\beta\kappa}^{001012}\right)\nonumber\\& &+
\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\alpha\kappa}f^{11}_{\alpha\beta}
\left(f^{21}_{\kappa\gamma}+\sqrt{2}f^{22}_{\kappa\gamma}\right)\nonumber\\
& &-\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\alpha\kappa}f^{21}_{\alpha\gamma}
\left(f^{11}_{\kappa\beta}+\sqrt{2}f^{21}_{\kappa\beta}\right)+\frac{J}{Z}T_{\alpha\gamma}\sqrt{2}f^{11}_{\alpha\beta}
\end{eqnarray}
\begin{eqnarray}
i\partial_t \rho_{\alpha\beta\gamma}^{002101}&=&U\rho_{\alpha\beta\gamma}^{002101}-
\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\beta\kappa}
\left(\sqrt{2}\rho_{\alpha\kappa\gamma}^{001001}+2\rho_{\alpha\kappa\gamma}^{002101}\right)\nonumber\\
& &-\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\gamma\kappa}
\left(\rho_{\alpha\beta\kappa}^{002101}+\sqrt{2}\rho_{\alpha\beta\kappa}^{002112}\right)\nonumber\\& &+
\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\alpha\kappa}f^{12}_{\alpha\beta}
\left(f^{11}_{\kappa\gamma}+\sqrt{2}f^{12}_{\kappa\gamma}\right)\nonumber\\
& &-\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\alpha\kappa}f^{11}_{\alpha\gamma}
\left(f^{12}_{\kappa\beta}+\sqrt{2}f^{22}_{\kappa\beta}\right)-\frac{J}{Z}T_{\alpha\beta}\sqrt{2}f^{11}_{\alpha\gamma}
\end{eqnarray}
\begin{eqnarray}\label{rho002112}
i\partial_t \rho_{\alpha\beta\gamma}^{002112}&=&-
\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\beta\kappa}
\left(\sqrt{2}\rho_{\alpha\kappa\gamma}^{001012}+2\rho_{\alpha\kappa\gamma}^{002112}\right)\nonumber\\
& &+\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\gamma\kappa}
\left(\sqrt{2}\rho_{\alpha\beta\kappa}^{002101}+2\rho_{\alpha\beta\kappa}^{002112}\right)\nonumber\\& &+
\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\alpha\kappa}f^{12}_{\alpha\beta}
\left(f^{21}_{\kappa\gamma}+\sqrt{2}f^{22}_{\kappa\gamma}\right)\nonumber\\
& &-\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\alpha\kappa}f^{21}_{\alpha\gamma}
\left(f^{12}_{\kappa\beta}+\sqrt{2}f^{22}_{\kappa\beta}\right)\nonumber\\
& &-\frac{J}{Z}T_{\alpha\beta}\sqrt{2}f^{21}_{\alpha\gamma}+\frac{J}{Z}T_{\alpha\gamma}\sqrt{2}f^{12}_{\alpha\beta}
\end{eqnarray}
\begin{eqnarray}\label{rho221001}
i\partial_t \rho_{\alpha\beta\gamma}^{221001}&=&
\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\beta\kappa}
\left(\rho_{\alpha\kappa\gamma}^{221001}+\sqrt{2}\rho_{\alpha\kappa\gamma}^{222101}\right)\nonumber\\
& &-\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\gamma\kappa}
\left(\rho_{\alpha\beta\kappa}^{221001}+\sqrt{2}\rho_{\alpha\beta\kappa}^{221012}\right)\nonumber\\& &-
\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\alpha\kappa}f^{21}_{\alpha\beta}
\left(f^{11}_{\kappa\gamma}+\sqrt{2}f^{12}_{\kappa\gamma}\right)\nonumber\\
& &+\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\alpha\kappa}\sqrt{2}f^{12}_{\alpha\gamma}
\left(f^{11}_{\kappa\beta}+\sqrt{2}f^{21}_{\kappa\beta}\right)\nonumber\\
& &-\frac{J}{Z}T_{\alpha\gamma}\sqrt{2}f^{21}_{\alpha\beta}+\frac{J}{Z}T_{\alpha\beta}\sqrt{2}f^{12}_{\alpha\gamma}
\end{eqnarray}
\begin{eqnarray}
i\partial_t \rho_{\alpha\beta\gamma}^{221012}&=&-U\rho_{\alpha\beta\gamma}^{221012}
+\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\beta\kappa}
\left(\rho_{\alpha\kappa\gamma}^{221012}+\sqrt{2}\rho_{\alpha\kappa\gamma}^{222112}\right)\nonumber\\
& &+\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\gamma\kappa}
\left(\sqrt{2}\rho_{\alpha\beta\kappa}^{221001}+2\rho_{\alpha\beta\kappa}^{221012}\right)\nonumber\\& &-
\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\alpha\kappa}\sqrt{2}f^{21}_{\alpha\beta}
\left(f^{21}_{\kappa\gamma}+\sqrt{2}f^{22}_{\kappa\gamma}\right)\nonumber\\
& &+\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\alpha\kappa}\sqrt{2}f^{22}_{\alpha\gamma}
\left(f^{11}_{\kappa\beta}+\sqrt{2}f^{21}_{\kappa\beta}\right)+\frac{J}{Z}T_{\alpha\beta}\sqrt{2}f^{22}_{\alpha\gamma}
\end{eqnarray}
\begin{eqnarray}
i\partial_t \rho_{\alpha\beta\gamma}^{222101}&=&U\rho_{\alpha\beta\gamma}^{222101}-
\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\beta\kappa}
\left(\sqrt{2}\rho_{\alpha\kappa\gamma}^{221001}+2\rho_{\alpha\kappa\gamma}^{222101}\right)\nonumber\\
& &-\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\gamma\kappa}
\left(\rho_{\alpha\beta\kappa}^{222101}+\sqrt{2}\rho_{\alpha\beta\kappa}^{222112}\right)\nonumber\\& &-
\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\alpha\kappa}\sqrt{2}f^{22}_{\alpha\beta}
\left(f^{11}_{\kappa\gamma}+\sqrt{2}f^{12}_{\kappa\gamma}\right)\nonumber\\
& &+\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\alpha\kappa}\sqrt{2}f^{12}_{\alpha\gamma}
\left(f^{12}_{\kappa\beta}+\sqrt{2}f^{22}_{\kappa\beta}\right)-\frac{J}{Z}T_{\alpha\gamma}\sqrt{2}f^{22}_{\alpha\beta}
\end{eqnarray}
\begin{eqnarray}\label{rho222112}
i\partial_t \rho_{\alpha\beta\gamma}^{222112}&=&-
\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\beta\kappa}
\left(\sqrt{2}\rho_{\alpha\kappa\gamma}^{221012}+2\rho_{\alpha\kappa\gamma}^{222112}\right)\nonumber\\
& &+\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\gamma\kappa}
\left(\sqrt{2}\rho_{\alpha\beta\kappa}^{222101}+2\rho_{\alpha\beta\kappa}^{222112}\right)\nonumber\\& &-
\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\alpha\kappa}\sqrt{2}f^{22}_{\alpha\beta}
\left(f^{21}_{\kappa\gamma}+\sqrt{2}f^{22}_{\kappa\gamma}\right)\nonumber\\
& &+\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\alpha\kappa}\sqrt{2}f^{22}_{\alpha\gamma}
\left(f^{12}_{\kappa\beta}+\sqrt{2}f^{22}_{\kappa\beta}\right)\nonumber\\
\end{eqnarray}
\begin{eqnarray}\label{rho111001}
i\partial_t \rho_{\alpha\beta\gamma}^{111001}&=&
\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\beta\kappa}
\left(\rho_{\alpha\kappa\gamma}^{111001}+\sqrt{2}\rho_{\alpha\kappa\gamma}^{112101}\right)\nonumber\\
& &-\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\gamma\kappa}
\left(\rho_{\alpha\beta\kappa}^{111001}+\sqrt{2}\rho_{\alpha\beta\kappa}^{111012}\right)\nonumber\\
& &-\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\alpha\kappa}\left(f^{11}_{\alpha\beta}-\sqrt{2}f^{21}_{\alpha\beta}\right)
\left(f^{11}_{\kappa\gamma}+\sqrt{2}f^{12}_{\kappa\gamma}\right)\nonumber\\
& &+\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\alpha\kappa}\left(f^{11}_{\alpha\gamma}-\sqrt{2}f^{12}_{\alpha\gamma}\right)
\left(f^{11}_{\kappa\beta}+\sqrt{2}f^{21}_{\kappa\beta}\right)\nonumber\\
& &+\frac{\sqrt{2}J}{Z}T_{\alpha\gamma}f^{21}_{\alpha\beta}-\frac{\sqrt{2}J}{Z}T_{\alpha\beta}f^{12}_{\alpha\gamma}
\end{eqnarray}
\begin{eqnarray}
i\partial_t \rho_{\alpha\beta\gamma}^{111012}&=&-U\rho_{\alpha\beta\gamma}^{111012}
+\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\beta\kappa}
\left(\rho_{\alpha\kappa\gamma}^{111012}+\sqrt{2}\rho_{\alpha\kappa\gamma}^{112112}\right)\nonumber\\
& &+\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\gamma\kappa}
\left(\sqrt{2}\rho_{\alpha\beta\kappa}^{111001}+2\rho_{\alpha\beta\kappa}^{111012}\right)\nonumber\\
& &-
\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\alpha\kappa}
\left(f^{11}_{\alpha\beta}-\sqrt{2}f^{21}_{\alpha\beta}\right)
\left(f^{21}_{\kappa\gamma}+\sqrt{2}f^{22}_{\kappa\gamma}\right)\nonumber\\
& &+\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\alpha\kappa}
\left(f^{21}_{\alpha\gamma}-\sqrt{2}f^{22}_{\alpha\gamma}\right)
\left(f^{11}_{\kappa\beta}+\sqrt{2}f^{21}_{\kappa\beta}\right)\nonumber\\
& &-\frac{\sqrt{2}J}{Z}T_{\alpha\gamma}f^{11}_{\alpha\beta}-\frac{\sqrt{2}J}{Z}T_{\alpha\beta}f^{22}_{\alpha\gamma}
\end{eqnarray}
\begin{eqnarray}
i\partial_t \rho_{\alpha\beta\gamma}^{112101}&=&U\rho_{\alpha\beta\gamma}^{112101}-
\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\beta\kappa}
\left(\sqrt{2}\rho_{\alpha\kappa\gamma}^{111001}+2\rho_{\alpha\kappa\gamma}^{112101}\right)\nonumber\\
& &-\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\gamma\kappa}
\left(\rho_{\alpha\beta\kappa}^{112101}+\sqrt{2}\rho_{\alpha\beta\kappa}^{112112}\right)\nonumber\\
& &-\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\alpha\kappa}
\left(f^{11}_{\alpha\beta}-\sqrt{2}f^{22}_{\alpha\beta}\right)
\left(f^{11}_{\kappa\gamma}+\sqrt{2}f^{12}_{\kappa\gamma}\right)\nonumber\\
& &+\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\alpha\kappa}
\left(f^{11}_{\alpha\gamma}-\sqrt{2}f^{12}_{\alpha\gamma}\right)
\left(f^{12}_{\kappa\beta}+\sqrt{2}f^{22}_{\kappa\beta}\right)\nonumber\\
& &+\frac{\sqrt{2}J}{Z}T_{\alpha\gamma}f^{22}_{\alpha\beta}+\frac{\sqrt{2}J}{Z}T_{\alpha\beta}f^{11}_{\alpha\gamma}
\end{eqnarray}
\begin{eqnarray}\label{rho112112}
i\partial_t \rho_{\alpha\beta\gamma}^{112112}&=&-
\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\beta\kappa}
\left(\sqrt{2}\rho_{\alpha\kappa\gamma}^{111012}+2\rho_{\alpha\kappa\gamma}^{112112}\right)\nonumber\\
& &+\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\gamma\kappa}
\left(\sqrt{2}\rho_{\alpha\beta\kappa}^{112101}+2\rho_{\alpha\beta\kappa}^{112112}\right)\nonumber\\
& &-\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\alpha\kappa}
\left(f^{12}_{\alpha\beta}-\sqrt{2}f^{22}_{\alpha\beta}\right)
\left(f^{21}_{\kappa\gamma}+\sqrt{2}f^{22}_{\kappa\gamma}\right)\nonumber\\
& &+\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\alpha\kappa}
\left(f^{21}_{\alpha\gamma}-\sqrt{2}f^{22}_{\alpha\gamma}\right)
\left(f^{12}_{\kappa\beta}+\sqrt{2}f^{22}_{\kappa\beta}\right)\nonumber\\
& &-\frac{\sqrt{2}J}{Z}T_{\alpha\gamma}f^{12}_{\alpha\beta}+\frac{\sqrt{2}J}{Z}T_{\alpha\beta}f^{21}_{\alpha\gamma}
\end{eqnarray}
\begin{eqnarray}
i\partial_t \rho_{\alpha\beta\gamma}^{200101}&=&U \rho_{\alpha\beta\gamma}^{200101}
-\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\beta\kappa}
\left(\rho_{\alpha\kappa\gamma}^{200101}+\sqrt{2}\rho_{\alpha\kappa\gamma}^{201201}\right)\nonumber\\
& &-\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\gamma\kappa}
\left(\rho_{\alpha\beta\kappa}^{200101}+\sqrt{2}\rho_{\alpha\beta\kappa}^{200112}\right)\nonumber\\
& &+\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\alpha\kappa}
\left(f^{12}_{\alpha\beta}-\sqrt{2}f^{11}_{\alpha\beta}\right)
\left(f^{11}_{\kappa\gamma}+\sqrt{2}f^{12}_{\kappa\gamma}\right)\nonumber\\
& &+\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\alpha\kappa}
\left(f^{12}_{\alpha\gamma}-\sqrt{2}f^{11}_{\alpha\gamma}\right)
\left(f^{11}_{\kappa\beta}+\sqrt{2}f^{12}_{\kappa\beta}\right)\nonumber\\
& &-\frac{\sqrt{2}J}{Z}T_{\alpha\beta}f^{11}_{\alpha\gamma}-\frac{\sqrt{2}J}{Z}T_{\alpha\gamma}f^{11}_{\alpha\beta}
\end{eqnarray}
\begin{eqnarray}
i\partial_t \rho_{\alpha\beta\gamma}^{200112}&=&
-\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\beta\kappa}
\left(\rho_{\alpha\kappa\gamma}^{200112}+\sqrt{2}\rho_{\alpha\kappa\gamma}^{201212}\right)\nonumber\\
& &+\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\gamma\kappa}
\left(\sqrt{2}\rho_{\alpha\beta\kappa}^{200101}+2\rho_{\alpha\beta\kappa}^{200112}\right)\nonumber\\
& &+\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\alpha\kappa}
\left(f^{12}_{\alpha\beta}-\sqrt{2}f^{11}_{\alpha\beta}\right)
\left(f^{21}_{\kappa\gamma}+\sqrt{2}f^{22}_{\kappa\gamma}\right)\nonumber\\
& &+\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\alpha\kappa}
\left(f^{22}_{\alpha\gamma}-\sqrt{2}f^{21}_{\alpha\gamma}\right)
\left(f^{11}_{\kappa\beta}+\sqrt{2}f^{12}_{\kappa\beta}\right)\nonumber\\
& &+\frac{\sqrt{2}J}{Z}T_{\alpha\gamma}f^{12}_{\alpha\beta}-\frac{\sqrt{2}J}{Z}T_{\alpha\beta}f^{21}_{\alpha\gamma}
\end{eqnarray}
\begin{eqnarray}
i\partial_t \rho_{\alpha\beta\gamma}^{201201}&=&
\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\beta\kappa}
\left(\sqrt{2}\rho_{\alpha\kappa\gamma}^{200101}+2\rho_{\alpha\kappa\gamma}^{201201}\right)\nonumber\\
& &-\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\gamma\kappa}
\left(\rho_{\alpha\beta\kappa}^{201201}+\sqrt{2}\rho_{\alpha\beta\kappa}^{201212}\right)\nonumber\\
& &+\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\alpha\kappa}
\left(f^{22}_{\alpha\beta}-\sqrt{2}f^{21}_{\alpha\beta}\right)
\left(f^{11}_{\kappa\gamma}+\sqrt{2}f^{12}_{\kappa\gamma}\right)\nonumber\\
& &+\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\alpha\kappa}
\left(f^{12}_{\alpha\gamma}-\sqrt{2}f^{11}_{\alpha\gamma}\right)
\left(f^{21}_{\kappa\beta}+\sqrt{2}f^{22}_{\kappa\beta}\right)\nonumber\\
& &+\frac{\sqrt{2}J}{Z}T_{\alpha\beta}f^{12}_{\alpha\gamma}-\frac{\sqrt{2}J}{Z}T_{\alpha\gamma}f^{21}_{\alpha\beta}
\end{eqnarray}
\begin{eqnarray}
i\partial_t \rho_{\alpha\beta\gamma}^{201212}&=&-U \rho_{\alpha\beta\gamma}^{201212}
+\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\beta\kappa}
\left(\sqrt{2}\rho_{\alpha\kappa\gamma}^{200112}+2\rho_{\alpha\kappa\gamma}^{201212}\right)\nonumber\\
& &+\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\gamma\kappa}
\left(\sqrt{2}\rho_{\alpha\beta\kappa}^{201201}+2\rho_{\alpha\beta\kappa}^{201212}\right)\nonumber\\
& &+\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\alpha\kappa}
\left(f^{22}_{\alpha\beta}-\sqrt{2}f^{21}_{\alpha\beta}\right)
\left(f^{21}_{\kappa\gamma}+\sqrt{2}f^{22}_{\kappa\gamma}\right)\nonumber\\
& &+\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\alpha\kappa}
\left(f^{21}_{\alpha\gamma}-\sqrt{2}f^{22}_{\alpha\gamma}\right)
\left(f^{21}_{\kappa\beta}+\sqrt{2}f^{22}_{\kappa\beta}\right)\nonumber\\
& &+\frac{\sqrt{2}J}{Z}T_{\alpha\beta}f^{22}_{\alpha\gamma}+\frac{\sqrt{2}J}{Z}T_{\alpha\gamma}f^{22}_{\alpha\beta}
\end{eqnarray}
\begin{eqnarray}
i\partial_t \rho_{\alpha\beta\gamma}^{310101}&=&3U \rho_{\alpha\beta\gamma}^{310101}
-\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\beta\kappa}
\left(\rho_{\alpha\kappa\gamma}^{310101}+\sqrt{2}\rho_{\alpha\kappa\gamma}^{311201}\right)\nonumber\\
& &-\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\gamma\kappa}
\left(\rho_{\alpha\beta\kappa}^{310101}+\sqrt{2}\rho_{\alpha\beta\kappa}^{310112}\right)\nonumber\\
& &-\frac{\sqrt{3}J}{Z}\sum_{\kappa\neq{\alpha,\beta,\gamma}}f^{12}_{\alpha\beta}\left(f^{11}_{\kappa\gamma}+\sqrt{2}f^{12}_{\kappa\gamma}\right)\nonumber\\
& &-\frac{\sqrt{3}J}{Z}\sum_{\kappa\neq{\alpha,\beta,\gamma}}f^{12}_{\alpha\gamma}\left(f^{11}_{\kappa\beta}+\sqrt{2}f^{12}_{\kappa\beta}\right)\nonumber\\
& &-\frac{\sqrt{3}J}{Z}T_{\alpha\beta}f^{12}_{\alpha\gamma}-\frac{\sqrt{3}J}{Z}T_{\alpha\gamma}f^{12}_{\alpha\beta}
\end{eqnarray}
\begin{eqnarray}
i\partial_t \rho_{\alpha\beta\gamma}^{310112}&=&2U \rho_{\alpha\beta\gamma}^{310112}
-\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\beta\kappa}
\left(\rho_{\alpha\kappa\gamma}^{310112}+\sqrt{2}\rho_{\alpha\kappa\gamma}^{311212}\right)\nonumber\\
& &+\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\gamma\kappa}
\left(\sqrt{2}\rho_{\alpha\beta\kappa}^{310101}+2\rho_{\alpha\beta\kappa}^{310112}\right)\nonumber\\
& &-\frac{\sqrt{3}J}{Z}\sum_{\kappa\neq{\alpha,\beta,\gamma}}f^{12}_{\alpha\beta}\left(f^{21}_{\kappa\gamma}+\sqrt{2}f^{22}_{\kappa\gamma}\right)\nonumber\\
& &-\frac{\sqrt{3}J}{Z}\sum_{\kappa\neq{\alpha,\beta,\gamma}}f^{22}_{\alpha\gamma}\left(f^{11}_{\kappa\beta}+\sqrt{2}f^{12}_{\kappa\beta}\right)-\frac{\sqrt{3}J}{Z}T_{\alpha\beta}f^{22}_{\alpha\gamma}
\end{eqnarray}
\begin{eqnarray}
i\partial_t \rho_{\alpha\beta\gamma}^{311201}&=&2U \rho_{\alpha\beta\gamma}^{311201}
+\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\beta\kappa}
\left(\sqrt{2}\rho_{\alpha\kappa\gamma}^{310101}+2\rho_{\alpha\kappa\gamma}^{311201}\right)\nonumber\\
& &-\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\gamma\kappa}
\left(\rho_{\alpha\beta\kappa}^{311201}+\sqrt{2}\rho_{\alpha\beta\kappa}^{311212}\right)\nonumber\\
& &-\frac{\sqrt{3}J}{Z}\sum_{\kappa\neq{\alpha,\beta,\gamma}}f^{22}_{\alpha\beta}\left(f^{11}_{\kappa\gamma}+\sqrt{2}f^{12}_{\kappa\gamma}\right)\nonumber\\
& &-\frac{\sqrt{3}J}{Z}\sum_{\kappa\neq{\alpha,\beta,\gamma}}f^{12}_{\alpha\gamma}\left(f^{21}_{\kappa\beta}+\sqrt{2}f^{22}_{\kappa\beta}\right)-\frac{\sqrt{3}J}{Z}T_{\alpha\gamma}f^{22}_{\alpha\beta}
\end{eqnarray}
\begin{eqnarray}
i\partial_t \rho_{\alpha\beta\gamma}^{311212}&=&U \rho_{\alpha\beta\gamma}^{311212}
+\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\beta\kappa}
\left(\sqrt{2}\rho_{\alpha\kappa\gamma}^{310112}+2\rho_{\alpha\kappa\gamma}^{311212}\right)\nonumber\\
& &+\frac{J}{Z}\sum_{\kappa\neq\alpha,\beta,\gamma}T_{\gamma\kappa}
\left(\sqrt{2}\rho_{\alpha\beta\kappa}^{311201}+2\rho_{\alpha\beta\kappa}^{311212}\right)\nonumber\\
& &-\frac{\sqrt{3}J}{Z}\sum_{\kappa\neq{\alpha,\beta,\gamma}}f^{22}_{\alpha\beta}\left(f^{21}_{\kappa\gamma}+\sqrt{2}f^{22}_{\kappa\gamma}\right)\nonumber\\
& &-\frac{\sqrt{3}J}{Z}\sum_{\kappa\neq{\alpha,\beta,\gamma}}f^{22}_{\alpha\gamma}\left(f^{21}_{\kappa\beta}+\sqrt{2}f^{22}_{\kappa\beta}\right)
\end{eqnarray}
By separating the two-point correlations in terms of order $1/Z$
and of order $1/Z^2$ according to
$\hat{\rho}_{\mu\nu}^\mathrm{corr}=
\hat{\rho}_{\mu\nu}^{\mathrm{corr}(1)}+
\hat{\rho}_{\mu\nu}^{\mathrm{corr}(2)}$
we find with
\begin{eqnarray}
\hat{\rho}_{\mu\nu}^{\mathrm{corr}(2)}
=
\sum_{m,m',n,n'}\rho_{\mu\nu}^{mm'nn'}
|m\rangle_\mu\langle m'|\otimes
|n\rangle_\nu \langle n'|
\end{eqnarray}
the following set of differential equations
\begin{eqnarray}\label{rho1001}
i\partial_t \rho_{\mu\nu}^{1001}&=&\frac{J}{Z}\sum_{\kappa\neq\mu,\nu}T_{\mu\kappa}\left(\rho_{\kappa\nu}^{1001}+\sqrt{2}\rho_{\kappa\nu}^{2101}+\sqrt{3}\rho_{\kappa\nu}^{3201}\right)\nonumber\\
& &-\frac{J}{Z}\sum_{\kappa\neq\mu,\nu}T_{\nu\kappa}\left(\rho_{\mu\kappa}^{1001}+\sqrt{2}\rho_{\mu\kappa}^{1012}+\sqrt{3}\rho_{\mu\kappa}^{1023}\right)\nonumber\\
& &-\frac{J}{Z}\sum_{\kappa\neq\mu,\nu}T_{\mu\kappa}\Big(\rho^{\mu\kappa\nu}_{001001}+\sqrt{2}\rho^{\mu\kappa\nu}_{002101}-
\rho^{\mu\kappa\nu}_{111001}\nonumber\\
& &\qquad\qquad-\sqrt{2}\rho^{\mu\kappa\nu}_{112101}+\sqrt{2}\rho^{\mu\kappa\nu}_{200101}+2\rho^{\mu\kappa\nu}_{201201}\Big)\nonumber \\
& &-\frac{J}{Z}\sum_{\kappa\neq\mu,\nu}T_{\nu\kappa}\Big(\rho^{\nu\mu\kappa}_{111001}+\sqrt{2}\rho^{\nu\mu\kappa}_{111012}-
\rho^{\nu\mu\kappa}_{001001}\nonumber\\
& &\qquad\qquad-\sqrt{2}\rho^{\nu\mu\kappa}_{001012}-\sqrt{2}\rho^{\nu\mu\kappa}_{021010}-2\rho^{\nu\mu\kappa}_{021021}\Big)\nonumber\\
& &-3f_0\frac{\sqrt{2}J}{Z}\sum_{\kappa\neq\mu,\nu}\left(T_{\nu \kappa}f^{12}_{\mu\kappa}-T_{\mu\kappa}f^{21}_{\kappa\nu}\right)
\end{eqnarray}
\begin{eqnarray}
i\partial_t \rho_{\mu\nu}^{2112}&=&-\frac{J}{Z}\sum_{\kappa\neq\mu,\nu}T_{\mu\kappa}\left(\sqrt{2}\rho_{\kappa\nu}^{1012}+2\rho_{\kappa\nu}^{2112}+\sqrt{6}\rho_{\kappa\nu}^{3212}\right)\nonumber\\
& &+\frac{J}{Z}\sum_{\kappa\neq\mu,\nu}T_{\nu\kappa}\left(\sqrt{2}\rho_{\mu\kappa}^{2101}+2\rho_{\mu\kappa}^{2112}+\sqrt{6}\rho_{\mu\kappa}^{2123}\right)\nonumber\\
& &-\frac{J}{Z}\sum_{\kappa\neq\mu,\nu}T_{\mu\kappa}\bigg(\sqrt{2}\rho^{\mu\kappa\nu}_{111012}+2\rho^{\mu\kappa\nu}_{112112}-
\sqrt{2}\rho^{\mu\kappa\nu}_{221012}-2\rho^{\mu\kappa\nu}_{222112}\nonumber\\
& &\qquad\qquad+\sqrt{3}\rho^{\mu\kappa\nu}_{310112}+\sqrt{6}\rho^{\mu\kappa\nu}_{311212}-\rho^{\mu\kappa\nu}_{200112}-\sqrt{2}\rho^{\mu\kappa\nu}_{201212}\bigg) \nonumber\\
& &-\frac{J}{Z}\sum_{\kappa\neq\mu,\nu}T_{\nu\kappa}\bigg(\sqrt{2}\rho^{\nu\mu\kappa}_{222101}+2\rho^{\nu\mu\kappa}_{222112}-
\sqrt{2}\rho^{\nu\mu\kappa}_{112101}-2\rho^{\nu\mu\kappa}_{112112}\nonumber\\
& &\qquad\qquad+\rho^{\nu\mu\kappa}_{022110}+\sqrt{2}\rho^{\nu\mu\kappa}_{022121}-\sqrt{3}\rho^{\nu\mu\kappa}_{132110}-\sqrt{6}\rho^{\nu\mu\kappa}_{132121}\bigg)\nonumber\\
& &-3f_0\frac{\sqrt{2}J}{Z}\sum_{\kappa\neq\mu,\nu}\left(T_{\nu \kappa}f^{12}_{\mu\kappa}-T_{\mu\kappa}f^{21}_{\kappa\nu}\right)
\end{eqnarray}
\begin{eqnarray}
i\partial_t \rho_{\mu\nu}^{1012}&=&-U\rho_{\mu\nu}^{1012}+\frac{J}{Z}\sum_{\kappa\neq\mu,\nu}T_{\mu\kappa}\left(\rho_{\kappa\nu}^{1012}
+\sqrt{2}\rho_{\kappa\nu}^{2112}+\sqrt{3}\rho_{\kappa\nu}^{3212}\right)\nonumber\\
& &+\frac{J}{Z}\sum_{\kappa\neq\mu,\nu}T_{\nu\kappa}\left(\sqrt{2}\rho_{\mu\kappa}^{1001}+2\rho_{\mu\kappa}^{1012}+\sqrt{6}\rho_{\mu\kappa}^{1032}\right)\nonumber\\
& &-\frac{J}{Z}\sum_{\kappa\neq\mu,\nu}T_{\mu\kappa}\Big(\rho^{\mu\kappa\nu}_{001012}+\sqrt{2}\rho^{\mu\kappa\nu}_{002112}-
\rho^{\mu\kappa\nu}_{111012}\nonumber\\
& &\qquad\qquad-\sqrt{2}\rho^{\mu\kappa\nu}_{112112}+\sqrt{2}\rho^{\mu\kappa\nu}_{200112}+2\rho^{\mu\kappa\nu}_{201212}\Big) \nonumber\\
& &-\frac{J}{Z}\sum_{\kappa\neq\mu,\nu}T_{\nu\kappa}\bigg(\sqrt{2}\rho^{\nu\mu\kappa}_{221001}+2\rho^{\nu\mu\kappa}_{221012}-
\sqrt{2}\rho^{\nu\mu\kappa}_{111001}-2\rho^{\nu\mu\kappa}_{111012}\nonumber\\
& &\qquad\qquad+\rho^{\nu\mu\kappa}_{021010}+\sqrt{2}\rho^{\nu\mu\kappa}_{021021}-\sqrt{3}\rho^{\nu\mu\kappa}_{131010}-\sqrt{6}\rho^{\nu\mu\kappa}_{131021}\bigg)\nonumber \\
& &-3f_0\frac{J}{Z}\sum_{\kappa\neq\mu,\nu}\left(T_{\nu\kappa}\left(f^{21}_{\mu\kappa}+\sqrt{2}f^{22}_{\mu\kappa}\right)
+\sqrt{2}T_{\mu\kappa}\left(f^{11}_{\kappa\nu}+\sqrt{2}f^{21}_{\kappa\nu}\right)\right)\nonumber\\
& &+4f_0\frac{\sqrt{2}J}{Z}T_{\mu\nu}
\end{eqnarray}
\begin{eqnarray}\label{rho2101}
i\partial_t \rho_{\mu\nu}^{2101}&=&U\rho_{\mu\nu}^{2101}-\frac{J}{Z}\sum_{\kappa\neq\mu,\nu}T_{\nu\kappa}\left(\rho_{\mu\kappa}^{2101}
+\sqrt{2}\rho_{\mu\kappa}^{2112}+\sqrt{3}\rho_{\mu\kappa}^{2123}\right)\nonumber\\
& &-\frac{J}{Z}\sum_{\kappa\neq\mu,\nu}T_{\mu\kappa}\left(\sqrt{2}\rho_{\kappa\nu}^{1001}+2\rho_{\kappa\nu}^{2101}+
\sqrt{6}\rho_{\kappa\nu}^{3201}\right)\nonumber\\
& &+\frac{J}{Z}\sum_{\kappa\neq\mu,\nu}T_{\nu\kappa}\Big(\rho^{\nu\mu\kappa}_{002101}+\sqrt{2}\rho^{\nu\mu\kappa}_{002112}-
\rho^{\nu\mu\kappa}_{112101}\nonumber\\
& &\qquad\qquad-\sqrt{2}\rho^{\nu\mu\kappa}_{112112}+\sqrt{2}\rho^{\nu\mu\kappa}_{022110}+2\rho^{\nu\mu\kappa}_{022121}\Big) \nonumber\\
& &+\frac{J}{Z}\sum_{\kappa\neq\mu,\nu}T_{\mu\kappa}\bigg(\sqrt{2}\rho^{\mu\kappa\nu}_{221001}+2\rho^{\mu\kappa\nu}_{222101}-
\sqrt{2}\rho^{\mu\kappa\nu}_{111001}-2\rho^{\mu\kappa\nu}_{112101}\nonumber\\
& &\qquad\qquad+\rho^{\mu\kappa\nu}_{200101}+\sqrt{2}\rho^{\mu\kappa\nu}_{201201}-\sqrt{3}\rho^{\mu\kappa\nu}_{310101}-
\sqrt{6}\rho^{\mu\kappa\nu}_{311201}\bigg)\nonumber \\
& &+3f_0\frac{J}{Z}\sum_{\kappa\neq\mu,\nu}\left(T_{\mu\kappa}\left(f^{12}_{\kappa\nu}+\sqrt{2}f^{22}_{\kappa\nu}\right)
+\sqrt{2}T_{\nu\kappa}\left(f^{11}_{\mu\kappa}+\sqrt{2}f^{12}_{\mu\kappa}\right)\right)\nonumber\\
& &-4f_0\frac{\sqrt{2}J}{Z}T_{\mu\nu}
\end{eqnarray}
\begin{eqnarray}
i\partial_t \rho^{1023}_{\mu\nu}&=&-2U\rho^{1023}_{\mu\nu}+\frac{J}{Z}\sum_{\kappa\neq\mu,\nu}T_{\mu\kappa}
\left(\rho^{1023}_{\kappa\nu}+\sqrt{2}\rho^{2123}_{\kappa\nu}\right)\nonumber\\
& &-\frac{J}{Z}\sum_{\kappa\neq\mu,\nu}T_{\nu\kappa}\left(-\sqrt{3}\rho^{\nu\mu\kappa}_{221001}-\sqrt{6}\rho^{\nu\mu\kappa}_{221012}+\sqrt{2}\rho^{\nu\mu\kappa}_{131010}
+2\rho^{\nu\mu\kappa}_{131021}\right)\nonumber\\
& &+f_0\frac{\sqrt{3}J}{Z}T_{\mu\nu}+f_0\frac{\sqrt{3}J}{Z}\sum_{\kappa\neq\mu,\nu}T_{\nu\kappa}(f^{11}_{\mu\kappa}+\sqrt{2}f^{21}_{\mu\kappa})
\end{eqnarray}
\begin{eqnarray}
i\partial_t \rho^{3201}_{\mu\nu}&=&2U\rho^{3201}_{\mu\nu}-\frac{J}{Z}\sum_{\kappa\neq\mu,\nu}T_{\nu\kappa}
\left(\rho^{3201}_{\mu\kappa}+\sqrt{2}\rho^{3212}_{\mu\kappa}\right)\nonumber\\
& &-\frac{J}{Z}\sum_{\kappa\neq\mu,\nu}T_{\mu\kappa}\left(\sqrt{3}\rho^{\mu\kappa\nu}_{221001}+\sqrt{6}\rho^{\mu\kappa\nu}_{222101}-\sqrt{2}\rho^{\mu\kappa\nu}_{310101}
-2\rho^{\mu\kappa\nu}_{311201}\right)\nonumber\\
& &-f_0\frac{\sqrt{3}J}{Z}T_{\mu\nu}-f_0\frac{\sqrt{3}J}{Z}\sum_{\kappa\neq\mu,\nu}T_{\mu\kappa}(f^{11}_{\nu\kappa}+\sqrt{2}f^{12}_{\nu\kappa})
\end{eqnarray}
\begin{eqnarray}
i\partial_t \rho^{3212}_{\mu\nu}&=&U\rho^{3212}_{\mu\nu}+\frac{J}{Z}\sum_{\kappa\neq\mu,\nu}T_{\nu\kappa}
\left(\sqrt{2}\rho^{3201}_{\mu\kappa}+2\rho^{3212}_{\mu\kappa}\right)\nonumber\\
& &-\frac{J}{Z}\sum_{\kappa\neq\mu,\nu}T_{\mu\kappa}\left(\sqrt{3}\rho^{\mu\kappa\nu}_{221012}+\sqrt{6}\rho^{\mu\kappa\nu}_{222112}-\sqrt{2}\rho^{\mu\kappa\nu}_{310112}
-2\rho^{\mu\kappa\nu}_{311212}\right)\nonumber\\
& &-f_0\frac{\sqrt{3}J}{Z}\sum_{\kappa\neq\mu,\nu}T_{\mu\kappa}(f^{21}_{\nu\kappa}+\sqrt{2}f^{22}_{\nu\kappa})
\end{eqnarray}
\begin{eqnarray}
i\partial_t \rho^{2123}_{\mu\nu}&=&-U\rho^{2123}_{\mu\nu}-\frac{J}{Z}\sum_{\kappa\neq\mu,\nu}T_{\nu\kappa}
\left(\sqrt{2}\rho^{1023}_{\kappa\mu}+2\rho^{2123}_{\kappa\mu}\right)\nonumber\\
& &-\frac{J}{Z}\sum_{\kappa\neq\mu,\nu}T_{\nu\kappa}\left(-\sqrt{3}\rho^{\nu\mu\kappa}_{222101}-\sqrt{6}\rho^{\nu\mu\kappa}_{222112}+\sqrt{2}\rho^{\nu\mu\kappa}_{132110}
+2\rho^{\nu\mu\kappa}_{132121}\right)\nonumber\\
& &+f_0\frac{\sqrt{3}J}{Z}\sum_{\kappa\neq\mu,\nu}T_{\nu\kappa}(f^{12}_{\mu\kappa}+\sqrt{2}f^{22}_{\mu\kappa})
\end{eqnarray}
\begin{eqnarray}\label{1111}\label{rho1111}
i\partial_t\rho^{1111}_{\mu\nu}&=&-\frac{J}{Z}\sum_\kappa T_{\mu\kappa}\bigg(\rho^{101101}_{\kappa\nu\mu}+
\sqrt{2}\rho^{211101}_{\kappa\nu\mu}-\sqrt{2}\rho^{101112}_{\kappa\nu\mu}-2\rho^{211112}_{\kappa\nu\mu}\nonumber\\
& &\qquad\qquad-\rho^{101101}_{\mu\nu\kappa}-\sqrt{2}\rho^{101112}_{\mu\nu\kappa}+\sqrt{2}\rho^{211101}_{\mu\nu\kappa}+2\rho^{101112}_{\mu\nu\kappa}\bigg)\nonumber\\
& &-\frac{J}{Z}\sum_\kappa T_{\nu\kappa}\bigg(\rho^{111001}_{\mu\kappa\nu}+\sqrt{2}\rho^{112101}_{\mu\kappa\nu}-\sqrt{2}\rho^{111012}_{\mu\kappa\nu}-2\rho^{112112}_{\mu\kappa\nu}\nonumber\\
& &\qquad\qquad-\rho^{111001}_{\mu\nu\kappa}-\sqrt{2}\rho^{111012}_{\mu\nu\kappa}+\sqrt{2}\rho^{112101}_{\mu\nu\kappa}+2\rho^{112112}_{\mu\nu\kappa}\bigg)\nonumber\\
& &+\frac{J}{Z}2\sqrt{2}T_{\mu\nu}(f^{12}_{\mu\nu}-f^{21}_{\mu\nu})
\end{eqnarray}
\begin{eqnarray}
i\partial_t\rho^{2222}_{\mu\nu}&=&-\frac{J}{Z}\sum_{\kappa}T_{\mu\kappa}\bigg(\sqrt{2} \rho^{102212}_{\kappa\nu\mu}+
2 \rho^{212212}_{\kappa\nu\mu}-\sqrt{2}\rho^{212201}_{\mu\nu\kappa}-2\rho^{212212}_{\mu\nu\kappa}\bigg)\nonumber\\
& &-\frac{J}{Z}\sum_{\kappa}T_{\nu\kappa}\bigg(\sqrt{2} \rho^{221012}_{\mu\kappa\nu}+
2 \rho^{222112}_{\mu\kappa\nu}-\sqrt{2}\rho^{222101}_{\mu\nu\kappa}-2\rho^{222112}_{\mu\nu\kappa}\bigg)
\end{eqnarray}
\begin{eqnarray}
i\partial_t\rho^{0000}_{\mu\nu}&=&-\frac{J}{Z}\sum_{\kappa}T_{\mu\kappa}\bigg(- \rho^{100001}_{\kappa\nu\mu}-\sqrt{2}
\rho^{210001}_{\kappa\nu\mu}+\rho^{100001}_{\mu\nu\kappa}+\sqrt{2}\rho^{100012}_{\mu\nu\kappa}\bigg)\nonumber\\
& &-\frac{J}{Z}\sum_{\kappa}T_{\nu\kappa}\bigg(- \rho^{001001}_{\mu\kappa\nu}-\sqrt{2} \rho^{002101}_{\mu\kappa\nu}+
\rho^{001001}_{\mu\nu\kappa}+\sqrt{2}\rho^{001012}_{\mu\nu\kappa}\bigg)
\end{eqnarray}
\begin{eqnarray}
i\partial_t\rho^{0011}_{\mu\nu}&=&-\frac{J}{Z}\sum_{\kappa}T_{\mu\kappa}\bigg(- \rho^{101101}_{\kappa\nu\mu}-\sqrt{2}
\rho^{211101}_{\kappa\nu\mu}+\rho^{101101}_{\mu\nu\kappa}+\sqrt{2}\rho^{101112}_{\mu\nu\kappa}\bigg)\nonumber\\
& &-\frac{J}{Z}\sum_{\kappa}T_{\nu\kappa}\bigg(\rho^{001001}_{\mu\kappa\nu}+\sqrt{2} \rho^{002101}_{\mu\kappa\nu}-
\sqrt{2}\rho^{001012}_{\mu\kappa\nu}-2 \rho^{002112}_{\mu\kappa\nu}\nonumber\\
& &\qquad\qquad-
\rho^{001001}_{\mu\nu\kappa}-\sqrt{2}\rho^{001012}_{\mu\nu\kappa}+\rho^{002101}_{\mu\nu\kappa}+2\rho^{002112}_{\mu\nu\kappa}\bigg)\nonumber\\
& &-\sqrt{2}\frac{J}{Z}T_{\mu\nu}(f^{12}_{\mu\nu}-f^{21}_{\mu\nu})
\end{eqnarray}
\begin{eqnarray}
i\partial_t\rho^{1122}_{\mu\nu}&=&-\frac{J}{Z}\sum_{\kappa}T_{\mu\kappa}\bigg(\rho^{102201}_{\kappa\nu\mu}+\sqrt{2}
\rho^{212201}_{\kappa\nu\mu}-\sqrt{2}\rho^{102212}_{\kappa\nu\mu}-2\rho^{212212}_{\kappa\nu\mu}\nonumber\\
& &\qquad\qquad-\rho^{102201}_{\mu\nu\kappa}-\sqrt{2}\rho^{102212}_{\mu\nu\kappa}+\sqrt{2}\rho^{212201}_{\mu\nu\kappa}+2\rho^{212212}_{\mu\nu\kappa}\bigg)\nonumber\\
& &-\frac{J}{Z}\sum_{\kappa}T_{\nu\kappa}\bigg(\sqrt{2}\rho^{111012}_{\mu\kappa\nu}+2\rho^{112112}_{\mu\kappa\nu}
-\sqrt{2}\rho^{112101}_{\mu\nu\kappa}-2\rho^{112112}_{\mu\nu\kappa}\bigg)\nonumber\\
& &-\sqrt{2}\frac{J}{Z}T_{\mu\nu}(f^{12}_{\mu\nu}-f^{21}_{\mu\nu})
\end{eqnarray}
\begin{eqnarray}\label{rho0022}
i\partial_t\rho^{0022}_{\mu\nu}&=&-\frac{J}{Z}\sum_{\kappa}T_{\mu\kappa}\bigg(- \rho^{102201}_{\kappa\nu\mu}-\sqrt{2}
\rho^{102201}_{\kappa\nu\mu}+\rho^{102201}_{\mu\nu\kappa}+\sqrt{2}\rho^{102212}_{\mu\nu\kappa}\bigg)\nonumber\\
& &-\frac{J}{Z}\sum_{\kappa}T_{\nu\kappa}\bigg(\sqrt{2}\rho^{001012}_{\mu\kappa\nu}+2 \rho^{002112}_{\mu\kappa\nu}-
\sqrt{2}\rho^{002101}_{\mu\nu\kappa}-2 \rho^{002112}_{\mu\nu\kappa}\bigg)\nonumber\\
& &+\sqrt{2}\frac{J}{Z}T_{\mu\nu}(f^{12}_{\mu\nu}-f^{21}_{\mu\nu})
\end{eqnarray}
\subsection{Renormalised frequencies}
The two-point correlations to first order in $1/Z$ are determined by the
differential equations
\begin{eqnarray}
i\partial_tf_\mathbf{k}^{12}
&=&+(U-3 J T_\mathbf{k})f_\mathbf{k}^{12}
-\sqrt{2}J T_\mathbf{k}(f_\mathbf{k}^{11}+f_\mathbf{k}^{22})
+\mathrm{source\,term}
\,,\label{f_12}
\\
i\partial_tf_\mathbf{k}^{21}
&=&-(U-3 J T_\mathbf{k})f_\mathbf{k}^{21}
+\sqrt{2}J T_\mathbf{k}(f_\mathbf{k}^{11}+f_\mathbf{k}^{22})
+\mathrm{source\,term}
\,,\label{f_21}
\\
i\partial_t f_\mathbf{k}^{11}&=&i\partial_t f_\mathbf{k}^{22}
=
\sqrt{2}JT_\mathbf{k}(f^{12}_\mathbf{k}-f^{21}_\mathbf{k})
\label{f_22}
\,.
\end{eqnarray}
The $1/Z^2$-contribution of the correlations
$f^{12}_\mathbf{k},f^{21}_\mathbf{k},f^{11}_\mathbf{k}$
and $f^{22}_\mathbf{k}$ can be deduced from (\ref{rho1001}-\ref{rho2101}).
Defining the Fourier transform
\begin{eqnarray}
\rho_{\mu\nu}^{mm'nn'}
=
\frac{1}{N}\sum_{\mathbf{k}}\rho_{\mathbf{k}}^{mm'nn'}
e^{i\mathbf{k}\cdot(\mathbf{x}_\mu-\mathbf{x}_\nu)}\,.
\end{eqnarray}
we find from equations (\ref{rho1001}-\ref{rho2101})
\begin{eqnarray}
i\partial_t \rho^{2101}_\mathbf{k}&=&+U\rho^{2101}_\mathbf{k}-3JT_\mathbf{k}
(\rho^{2101}_\mathbf{k}-3f_0f^{12}_\mathbf{k})\nonumber\\
& &-\sqrt{2}JT_\mathbf{k}\left(
\rho_\mathbf{k}^{1001}+\rho_\mathbf{k}^{2112}
-3f_0 f^{11}_\mathbf{k}-3f_0f^{22}_\mathbf{k}\right)\nonumber\\
& &+\mathrm{source\,terms}\label{f2101}\,,\\
i\partial_t \rho^{1012}_\mathbf{k}&=&-U\rho^{1012}_\mathbf{k}+3JT_\mathbf{k}
(\rho^{1012}_\mathbf{k}-3f_0f^{21}_\mathbf{k})\nonumber\\
& &+\sqrt{2}JT_\mathbf{k}\left(
\rho_\mathbf{k}^{1001}+\rho_\mathbf{k}^{2112}
-3f_0 f^{11}_\mathbf{k}-3f_0f^{22}_\mathbf{k}\right)\nonumber\\
& &+\mathrm{source\,terms}\label{f1012}\,,\\
i\partial_t \rho^{1001}_\mathbf{k}&=&i\partial_t \rho^{2112}_\mathbf{k}=\sqrt{2}J T_p\left(\rho_\mathbf{k}^{2101}
-\rho_\mathbf{k}^{1012}-3f_0 f^{12}_\mathbf{k}+3f_0f^{21}_\mathbf{k}\right)\nonumber\\
& &+\mathrm{source\,terms}\label{f1001}\,.
\end{eqnarray}
As a next step we add equations (\ref{f_12}) and (\ref{f2101}), (\ref{f_21}) and (\ref{f1012}),
(\ref{f_22}) and (\ref{f1001}) and define
\begin{eqnarray}
\tilde{\rho}_\mathbf{k}^{2101}&=&f^{12}_\mathbf{k}+\rho_\mathbf{k}^{2101}\\
\tilde{\rho}_\mathbf{k}^{1012}&=&f^{21}_\mathbf{k}+\rho_\mathbf{k}^{1012}\\
\tilde{\rho}_\mathbf{k}^{2112}&=&f^{22}_\mathbf{k}+\rho_\mathbf{k}^{2112}\\
\tilde{\rho}_\mathbf{k}^{1001}&=&f^{11}_\mathbf{k}+\rho_\mathbf{k}^{1001}\,.
\end{eqnarray}
From this follows a system of differential equations which is valid up to $\mathcal{O}(1/Z^2)$,
\begin{eqnarray}
i\partial_t\tilde{\rho}_\mathbf{k}^{2101}
&=&+\left[U-3 J T_\mathbf{k}\left(1-3f_0\right)\right]\tilde{\rho}_\mathbf{k}^{2101}
-\sqrt{2}J T_\mathbf{k}\left(1-3f_0\right)\left(\tilde{\rho}_\mathbf{k}^{1001}+\tilde{\rho}_\mathbf{k}^{2112}\right)\nonumber\\
& &+\mathrm{source\,terms}\label{2101}
\,,
\\
i\partial_t\tilde{\rho}_\mathbf{k}^{1012}
&=&-\left[U-3 J T_\mathbf{k}(1-3f_0)\right]\tilde{\rho}_\mathbf{k}^{1012}
+\sqrt{2}J T_\mathbf{k}(1-3f_0)\left(\tilde{\rho}_\mathbf{k}^{1001}+\tilde{\rho}_\mathbf{k}^{2112}\right)\nonumber\\
& &+\mathrm{source\,terms}
\,,
\\
i\partial_t \tilde{\rho}_\mathbf{k}^{1001}&=&i\partial_t \tilde{\rho}_\mathbf{k}^{2112}
=
\sqrt{2}JT_\mathbf{k}(1-3f_0)\left(\tilde{\rho}_\mathbf{k}^{2101}-\tilde{\rho}_\mathbf{k}^{1012}\right)\nonumber\\
& &+\mathrm{source\,terms}
\,.\label{1001}
\end{eqnarray}
The homogeneous part of equations (\ref{f_12})-(\ref{f_22}) is related to the homogeneous part
of (\ref{2101})-(\ref{1001}) via the substitution $T_\mathbf{k}\rightarrow T_\mathbf{k}(1-3f_0)$ from
which follows immediately the renormalised frequency (\ref{renomega}).
\subsection{Parity-parity and particle-number correlations}
The parity-parity and the particle-number correlations are determined in
$\mathcal{O}(1/Z^2)$
by the differential equations (\ref{rho1111}-\ref{rho0022}).
Since the right-hand sides of (\ref{rho1111}-\ref{rho0022})
involve three-point correlations, we have solve
the equations (\ref{rho001001})-(\ref{rho112112}).
The calculations can be simplified by observing that it is possible
to express the
right-hand sides of (\ref{rho1111}-\ref{rho0022}) by total time-derivatives
using (\ref{rho001001}), (\ref{rho002112}), (\ref{rho221001}),
(\ref{rho222112}),
(\ref{rho111001}), (\ref{rho112112}) and (\ref{f_12})-(\ref{f_22}).
We find the exact expressions
\begin{eqnarray}\label{rho1111ex}
\rho^{1111}_{\mu\nu}&=&-\frac{1}{N^2}\sum_{\mathbf{k,p,q}}\left(\rho_{\mathbf{kpq}}^{111001}+
\rho_{\mathbf{kpq}}^{112112}\right)\left(e^{i\mathbf{q\cdot x}_\mu+i\mathbf{p\cdot x}_\mu+i\mathbf{k\cdot x}_\nu}
+e^{i\mathbf{q\cdot x}_\nu+i\mathbf{k\cdot x}_\mu+i\mathbf{p\cdot x}_\nu}\right)\nonumber\\
& &-\frac{2}{N^2}\sum_{\mathbf{p,q}}
\left(f^{11}_\mathbf{q} f^{11}_\mathbf{p}+ f^{12}_\mathbf{q} f^{21}_\mathbf{p}\right)e^{i(\mathbf{p}+\mathbf{q})\cdot
(\mathbf{x}_\mu-\mathbf{x}_\nu)}
\end{eqnarray}
\begin{eqnarray}\label{rho2222ex}
\rho^{2222}_{\mu\nu}&=&\frac{1}{N^2}\sum_{\mathbf{k,p,q}}\rho_{\mathbf{kpq}}^{222112}
\left(e^{i\mathbf{q\cdot x}_\mu+i\mathbf{p\cdot x}_\mu+i\mathbf{k\cdot x}_\nu}+
e^{i\mathbf{q\cdot x}_\nu+i\mathbf{k\cdot x}_\mu+i\mathbf{p\cdot x}_\nu}\right)\nonumber\\
& &-\frac{1}{N^2}\sum_{\mathbf{p,q}}f^{11}_\mathbf{p}f^{11}_\mathbf{q}e^{i(\mathbf{p}+\mathbf{q})\cdot
(\mathbf{x}_\mu-\mathbf{x}_\nu)}
\end{eqnarray}
\begin{eqnarray}\label{rho0000ex}
\rho^{0000}_{\mu\nu}&=&\frac{1}{N^2}\sum_{\mathbf{k,p,q}}\rho_{\mathbf{kpq}}^{001001}
\left(e^{i\mathbf{q\cdot x}_\mu+i\mathbf{p\cdot x}_\mu+i\mathbf{k\cdot x}_\nu}
+e^{i\mathbf{q\cdot x}_\nu+i\mathbf{k\cdot x}_\mu+i\mathbf{p\cdot x}_\nu}\right)\nonumber\\
& &-\frac{1}{N^2}\sum_{\mathbf{p,q}}f^{11}_\mathbf{p}f^{11}_\mathbf{q}e^{i(\mathbf{p}+\mathbf{q})\cdot
(\mathbf{x}_\mu-\mathbf{x}_\nu)}
\end{eqnarray}
\begin{eqnarray}\label{rho0011ex}
\rho^{0011}_{\mu\nu}&=&\frac{1}{N^2}\sum_{\mathbf{k,p,q}}\rho_{\mathbf{kpq}}^{111001}
e^{i\mathbf{p\cdot x}_\mu+i\mathbf{k\cdot x}_\nu+i\mathbf{q\cdot x}_\mu}
\nonumber\\& &-\frac{1}{N^2}\sum_{\mathbf{k,p,q}}\left(\rho_{\mathbf{kpq}}^{001001}+\rho_{\mathbf{kpq}}^{002112}\right)
e^{i\mathbf{q\cdot x}_\nu+i\mathbf{k\cdot x}_\mu+i\mathbf{p\cdot x}_\nu}\nonumber\\
& &+\frac{1}{N^2}\sum_{\mathbf{p,q}}\left(f^{11}_\mathbf{p}f^{11}_\mathbf{q}+f^{12}_\mathbf{p} f^{21}_\mathbf{q}\right)e^{i(\mathbf{p}+\mathbf{q})
\cdot(\mathbf{x}_\mu-\mathbf{x}_\nu)}
\end{eqnarray}
\begin{eqnarray}\label{rho1122ex}
\rho^{1122}_{\mu\nu}&=&-\frac{1}{N^2}\sum_{\mathbf{k,p,q}}\left(\rho_{\mathbf{kpq}}^{221001}+\rho_{\mathbf{kpq}}^{222112}\right)
e^{i\mathbf{p\cdot x}_\mu+i\mathbf{k\cdot x}_\nu+i\mathbf{q\cdot x}_\mu}\nonumber\\
& &+\frac{1}{N^2}\sum_{\mathbf{k,p,q}}\rho_{\mathbf{kpq}}^{112112}e^{i\mathbf{q\cdot x}_\nu+i\mathbf{k\cdot x}_\mu+i\mathbf{p\cdot x}_\nu}
\end{eqnarray}
\begin{eqnarray}\label{rho0022ex}
\rho^{0022}_{\mu\nu}&=&\frac{1}{N^2}\sum_{\mathbf{k,p,q}}\rho_{\mathbf{kpq}}^{221001}e^{i\mathbf{p\cdot x}_\mu+i\mathbf{k\cdot x}_\nu+i\mathbf{q\cdot x}_\mu}
+\frac{1}{N^2}\sum_{\mathbf{k,p,q}}\rho_{\mathbf{kpq}}^{002112}e^{i\mathbf{q\cdot x}_\nu+i\mathbf{k\cdot x}_\mu+i\mathbf{p\cdot x}_\nu}\nonumber\\
& &-\frac{1}{N^2}\sum_{\mathbf{p,q}}f^{12}_\mathbf{p} f^{21}_\mathbf{q} e^{i(\mathbf{p}+\mathbf{q})\cdot(\mathbf{x}_\mu-\mathbf{x}_\nu)}\,,
\end{eqnarray}
where we defined the Fourier transforms
\begin{eqnarray}
\rho^{aa'bb'cc'}_{\alpha\beta\gamma}=\frac{1}{N^2}\sum_{\mathbf{k,p,q}}
\rho^{aa'bb'cc'}_{\mathbf{kpq}}e^{i\mathbf{k\cdot x}_\alpha+i\mathbf{p\cdot x}_\beta+i\mathbf{q\cdot x}_\gamma}\,.
\end{eqnarray}
After solving the differential equations for the three-point-correlations
and inserting the solutions in (\ref{rho1111ex})-(\ref{rho0022ex})
we find the parity-parity and particle-number correlations which are given
in Section \ref{Z2}.
\section*{References}
|
{
"timestamp": "2012-08-02T02:05:51",
"yymm": "1203",
"arxiv_id": "1203.2164",
"language": "en",
"url": "https://arxiv.org/abs/1203.2164"
}
|
\section{Introduction}
How dark matter (DM) couples to Standard Model (SM) particles is an open
question. An interesting possibility is that the coupling of dark and
visible sectors is through a Higgs portal \cite{Patt:2006fw, Kim:2006af,
MarchRussell:2008yu, Kim:2008pp, Ahlers:2008qc, Feng:2008mu, Andreas:2008xy,
Barger:2008jx, Kadastik:2009ca, Kanemura:2010sh, Piazza:2010ye,
Arina:2010an, Low:2011kp, Djouadi:2011aa, Englert:2011yb, Kamenik:2012hn,
Gonderinger:2012rd, Lebedev:2012zw}. The operator $(H^\dagger H)$ is one of
the two lowest dimensional gauge invariant operators that one can write in
the SM (the other one being the hyper-charge gauge field strength
$B_{\mu\nu}$). Therefore, it is quite likely that also $(H^\dagger
H)$--(dark sector) will be the lowest dimension operator connecting dark and
visible sectors, and thus potentially the most important one.
Experimentally the Higgs portal is probed from two complementary
directions. On the one hand, the new generation of direct DM
detection experiments \cite{Aprile:2011hi, Ahmed:2011gh} is starting
to probe DM--nucleon scattering cross sections of roughly the size
given by a single Higgs exchange with the SM Yukawa couplings to the
quarks. On the other hand, the first hints of a SM-like Higgs boson
signal were reported by the ATLAS and CMS collaborations. The hints of
a signal are seen in several channels, pointing to a Higgs mass of
roughly $m_h\sim 124-126$ GeV \cite{:2012si,Chatrchyan:2012tx}, with
the SM Higgs boson consistent with the current data at 82\%
C.L.~\cite{Espinosa:2012ir}, see also~\cite{Azatov:2012bz,
Carmi:2012yp}. Those hints are supported by recent results from D0
and CDF. For updates from ATLAS, CMS, D0, and CDF see \cite{moriond}.
In view of these experimental developments we revisit the Higgs portal to
DM. In particular we focus on fermionic DM. The Higgs
portals for the fermionic DM and the scalar DM are qualitatively different.
For instance, if DM is a scalar, $\phi_{DM}$, then the Higgs portal operator
$(H^\dagger H) (\phi_{DM}^\dagger \phi_{DM})$ is renormalizable. The same is
true if DM is a spin-1 particle. In contrast, if DM is a fermion, $\chi$,
then the Higgs portal necessarily proceeds through non-renormalizable
interactions. The lowest dimensional Higgs portal in that case consists of
two dim~$=5$ operators
\begin{equation}\label{Qs}
Q_1=(H^\dagger H) (\bar \chi \chi) \,, \qquad Q_5=i(H^\dagger H) (\bar \chi
\gamma_5 \chi) \,,
\end{equation}
which enter the effective Hamiltonian
%
\begin{equation} \label{eq:Heff}
H_{\rm eff}=\frac{1}{\Lambda_1}Q_1+\frac{1}{\Lambda_5}Q_5 \,.
\end{equation}
The mass scales $\Lambda_{1,5}$ are roughly the masses of the mediators for
${\mathcal O}(1)$ couplings between DM and the mediators. Since the DM--Higgs
effective Hamiltonian is non-renormalizable, this means that a Higgs portal
for fermionic dark matter necessarily requires a UV completion. In this
paper we also consider situations when such UV completions are required
in order to obtain a correct description of the DM phenomenology.
We first use the effective field theory (EFT) description of the Higgs
portal \eqref{eq:Heff} and derive consequences for each of the two operators
$Q_{1,5}$. The parity conserving interaction, $Q_1$, is severely
constrained by direct detection experiments. If only $Q_1$ is present in
$H_{\rm eff}$ then one cannot obtain a small enough relic density consistent
with the bound from XENON100 for DM masses below about 2~TeV
\cite{Djouadi:2011aa}. In contrast, as we will show in
section~\ref{sec:EFT}, the parity violating operator $Q_5$ is allowed by
direct detection searches and the observed relic density can be obtained,
see also \cite{Pospelov:2011yp}. Hence, when DM interactions are mediated by
fields much heavier than $2m_{\chi}$ and $2m_h$ the EFT description is valid
and we must conclude that the Higgs portal interactions for fermionic DM
need to be parity violating (``pseudo-scalar Higgs portal''). Yet viable
scenarios with parity conserving operators can be found when EFT breaks
down. We identify two distinct options:
\begin{itemize}
\item
``resonant Higgs portal" -- where the dominant contribution is due to a
resonant annihilation either through the Higgs or the mediator,
\item
``indirect Higgs portal" -- where the DM annihilations into the mediator set
the relic density and the Higgs portal only provides the link between the visible
and dark sector thermal baths.
\end{itemize}
In section~\ref{sec:model} we illustrate both of these two possibilities
using a toy model -- the minimal extension of the SM with a DM fermion
$\chi$ and a real singlet $\phi$ see e.g.~\cite{Kim:2008pp, Baek:2011aa}.
\section{Effective field theory considerations}
\label{sec:EFT}
Let us first assume that the mediators are heavy so that they can be
integrated out. The Higgs portal is then given by
eq.~\eqref{eq:Heff}. We will be interested in the direct detection
of DM and in the annihilation of DM in the galactic halo. In both cases the
DM particles entering the process are non-relativistic, with velocities
typical of DM in the galactic halo, ${\nu} \sim 10^{-3}$. For annihilations
in the early universe, responsible for obtaining the thermal relic density,
DM is moderately relativistic.
The indirect and direct DM detection signals are given by the annihilation
of two non-relativistic DM particles and the scattering of non-relativistic DM
on the nuclei, respectively. For these two processes the two effective
operators $Q_{1,5}$ behave in exactly the opposite way. For instance, the
annihilation cross section is (for a Majorana fermion $\chi$)
\begin{equation}
\sigma_{\rm ann}=\frac{1}{4\pi}\left[\frac{\left(1-4 m_\chi^2/s\right)}{\Lambda_1^2}+\frac{1}{\Lambda_5^2}\right] \frac{f(m_\chi)}{\sqrt{1-4m_\chi^2/s}},
\end{equation}
which is in the non-relativistic limit
\begin{equation}\label{eq:ann-nonrel}
\sigma_{\rm ann}=\frac{1}{4\pi{\nu}}\left[\frac{{\nu}^2}{\Lambda_1^2}+\frac{1}{\Lambda_5^2}\right] f(m_\chi),
\end{equation}
where ${\nu}$ is the velocity of each of the DM particles in the
center-of-mass system (CMS). The contributions to annihilations from the
parity conserving operator $Q_1$ are thus velocity suppressed, while parity
violating contributions, due to $Q_5$, are unsuppressed. The function
$f(m_\chi)\equiv \sum_i f_i$ sums the available final states $i$. For
instance, for $m_\chi>m_h$ we have
\begin{equation}
f_h=\left(1+\frac{3 m_h^2}{s- m_h^2}\right)^2 \left(1-4m_h^2/s\right)^{1/2}, \qquad f_t=\frac{m_t^2}{s}\frac{(1-4m_t^2/s)^{3/2}}{(1-m_h^2/s)^2},
\end{equation}
for the decays to Higgs and top, respectively. For $m_\chi$ very heavy
$f_h\to 1$ and $f_t\to 0$.
In the
early universe, around the freeze-out temperature $T_F$ we have
${\nu}^2 \sim T_F / m_\chi \simeq 1/20$, whereas in the galactic halo we have
${\nu}^2 \sim 10^{-6}$. As a consequence, for parity conserving interactions
the annihilation cross section relevant for indirect detection signals is
significantly suppressed compared to the one relevant for
thermal freeze-out. In contrast, for parity violating interactions the
annihilation cross section is independent of the velocity.
For direct detection the situation is exactly opposite.
Integrating out the Higgs field, the scattering of DM on matter is given by
an effective Hamiltonian
\begin{equation}
H=\sum_{i=1,5}
\frac{C_i}{m_h^2} O_i,
\end{equation}
where the two operators and Wilson coefficients are
\begin{equation}
O_{1,5}=\bar \chi \{1,i \gamma_5\}\chi \sum_q \frac{m_q}{v}\bar q q,\qquad C_{1,5}=\frac{v}{\Lambda_{1,5}}.
\end{equation}
where $v=246$ GeV is the Higgs vacuum expectation value (VEV).
The cross section for $\chi$ scattering on the proton induced by the operator $O_1$
is then
\begin{equation}\label{eq:scattering}
\sigma(\chi p\to \chi p)=\frac{4}{\pi}\left(C_1 g_{Hp} \right)^2 \left(\frac{m_{\rm red}}{m_h^2}\right)^2,
\end{equation}
where $m_{\rm red} = m_p m_\chi / (m_p + m_\chi)$ is the reduced mass of the
DM--proton system and
\begin{equation}
g_{Hp}=\frac{m_{p}}{v}\left[\sum_{q=u,d,s}f^{(p)}_{q}+\frac{2}{9}\left(1-
\sum_{q=u,d,s}f^{(p)}_{q}\right)\right] \approx 1.3\times 10^{-3}\,,
\end{equation}
see e.g., \cite{Belanger:2008sj}, where also values for $f^{(p)}_q$ are
given.
The operator $O_5$, on the other hand, induces a velocity suppressed scattering
\begin{equation}
\sigma(\chi p\to \chi p)=\frac{2}{\pi}\left(C_5 g_{Hp} \right)^2 \left(\frac{m_{\rm red}}{m_h^2}\right)^2 {\nu}^2.
\end{equation}
where typically ${\nu} \sim 10^{-3}$. Hence in direct detection one obtains a
velocity suppressed scattering cross section for parity violating
interactions, but unsuppressed scattering for parity conserving ones.
\begin{figure}[tb!]
\begin{center}
\includegraphics[width=10cm]{P-DDEFT.png}
\mycaption{Proton--dark matter scattering cross
section as a function of the dark matter mass in the
effective field theory of eq.~(\ref{eq:Heff}), as predicted by
requiring that the correct relic density is obtained by thermal freeze-out.
The scattering cross section is shown for several ratios
of pseudo scalar coupling to scalar coupling $\Lambda_1/\Lambda_5$, and
compared to the limit from XENON100~\cite{Aprile:2011hi}. }
\label{fig:DDEFT}
\end{center}
\end{figure}
This means that it is possible to find regions of $C_{1,5}$ parameter
space with correct relic density but small enough direct detection
signals. In fig.~\ref{fig:DDEFT} we show the scattering cross section
for different ratios of pseudo scalar coupling to scalar coupling
$\Lambda_1/\Lambda_5$. For a fixed ratio and given DM mass we
determine the sizes of $\Lambda_i$ by requiring that the thermal relic
density is obtained in the interval $0.09<\Omega h^2<0.13$. For the
thermal freeze-out calculations we use the {\tt
micrOMEGAs}~\cite{Belanger:2008sj, Belanger:2010gh} public code, and
we assume $m_h = 125$~GeV. The curve for $\Lambda_1/\Lambda_5 = 0$
corresponds to parity conservation, i.e., pure scalar coupling. This
case was considered for example in \cite{Kim:2006af, Kanemura:2010sh,
Djouadi:2011aa} and is incompatible with the bound from
XENON100~\cite{Aprile:2011hi} for DM masses $m_\chi \lesssim
2$~TeV. However, already for $\Lambda_1/\Lambda_5 \simeq 1$ the
predicted cross section is below the XENON100 bound for $m_\chi
\gtrsim 100$~GeV, while $\Lambda_1/\Lambda_5 \simeq 10$ leads to cross
sections of about two to three orders of magnitudes below the limit.
Around $m_\chi \approx m_h/2 = 63$~GeV the effect of the $s$-channel
resonance due to Higgs exchange is clearly visible in fig.~\ref{fig:DDEFT}.
For DM masses below the resonance Higgs decays $h\to 2\chi$ become possible.
The shape of the curves below the resonance is due to the Breit-Wigner form
of the annihilation cross section. The contribution of $h\to 2\chi$ to the
Higgs decay width allows for two solutions for $\Lambda_i$ giving rise to
the correct relic density at a given DM mass above a certain minimal mass
and below the resonance. Note however, that in some cases $\Lambda_i$ may
become even smaller than $m_\chi$ and the EFT description may no longer be
valid. Moreover, typically large branching fractions of the Higgs into DM
are obtained in those cases. Therefore, the region below $m_h/2$ would be
excluded by observing a Higgs with SM-like decay branching fractions.
Let us briefly mention constraints from indirect detection. In the case of
pure scalar interactions DM annihilations are ${\nu}^2$ suppressed. In the
early universe at freeze-out ${\nu}^2 \simeq 1/20$, whereas in the galactic halo we have
${\nu}^2 \sim 10^{-6}$. This leads to a negligible signal for indirect
detection experiments. As soon as the pseudo-scalar coupling becomes
comparable to the scalar one, the annihilation cross section is dominated by
pseudo-scalar interactions with $\sigma {\nu}$ independent of the velocity,
see eq.~\eqref{eq:ann-nonrel}. Therefore, in the latter case, the annihilation cross section
will be the ``thermal'' one with $\sigma {\nu} \simeq 3\times 10^{-26}\,\rm
cm^3s^{-1}$. Current data from FERMI-LAT and BESS-Polar~II disfavour such
cross sections for $m_\chi \lesssim 30$~GeV \cite{Ackermann:2011wa,
Kappl:2011jw}. Since here we are restricted to DM masses $m_\chi \gtrsim
60$~GeV current limits from indirect detection do not constrain this model.
Let us mention that monojet searches for dark matter at colliders~\cite{Beltran:2010ww,Bai:2010hh} will apriori not constrain further this EFT model of dark matter. Limits on spin-independent interactions from recent LHC data~\cite{Fox:2011pm,cms} are much weaker than Xenon bounds in the region of interest.
\section{Beyond the EFT framework}
\label{sec:model}
\subsection{The toy model}
Now we move to situations which cannot be described by the EFT. In
order to illustrate when it is possible to have a viable fermionic DM Higgs
portal we consider a simple UV completion by introducing a real
scalar singlet that will act as mediator particle\footnote{For alternative UV completions see e.g.~\cite{Bird:2006jd,D'Eramo:2007ga}.}. For simplicity we consider a
Majorana fermion $\chi$ as DM, with $\chi = \chi_L + \chi_L^c$ in
4-component notation. (All our arguments will equally apply to the
Dirac case.) We denote the SM Higgs doublet by $H$ and the real
singlet scalar by $\varphi$. The relevant terms in the Lagrangian are
\begin{align}
\mathcal{L} &=
\frac12 \bar\chi_L(i \gamma_\mu\partial^\mu - \mu_\chi - g \varphi) \chi_L^c +
{\rm h.c.} \nonumber\\
&+
(D_\mu H)^\dagger D^\mu H +
\frac{1}{2} \partial_\mu\varphi\partial^\mu\varphi
- V(\varphi, H)\,,
\end{align}
Here $D^\mu$ is the SM gauge-covariant derivative, $V(\varphi, H)$ is the
Higgs potential, and we allow the coupling constant $g$ and the mass
parameter $\mu_\chi$ to be complex. Let us work in
unitary gauge and expand $H$ and $\varphi$ around their VEVs:
\begin{equation}
H= \frac{1}{\sqrt{2}} \left(\begin{array}{c} 0 \\ h+v_1 \end{array}\right)
\,,\qquad \varphi= \phi+v_2
\end{equation}
where $v_1 = 246$~GeV. By performing a phase transformation
$\chi_L \to e^{i\alpha/2}
\chi_L$ with $\alpha = \mathrm{Arg}(\mu_\chi + g v_2)$ we find that the
physical mass of $\chi$ corresponds to $m_\chi = |\mu_\chi + g v_2|$.
The phase of $g$ relative to the mass term determines the scalar ($S$) or
pseudo-scalar ($P$) nature of the Yukawa coupling:
\begin{equation}
g_S = \mathrm{Re}(g e^{-i\alpha}) \,,\qquad
g_P = \mathrm{Im}(g e^{-i\alpha}) \,.
\end{equation}
A non-zero value of $g_P$ violates parity.
The mass term and the
interaction terms for the Majorana fermion $\chi$ become thus:
\begin{equation}
{\cal L}_\chi=-\frac12 \left(m_\chi \bar \chi \chi+ g_S \phi \bar \chi
\chi +i g_P \phi \bar \chi \gamma_5 \chi \right) \,.
\end{equation}
As discussed in the previous section the pseudo-scalar coupling leads
to suppressed rates in direct detection. Therefore, it is always
possible to consider the situation of $g_S \ll g_P$ in order to
reconcile the annihilation cross section required for the relic
density with stringent bounds on the DM--nucleon scattering cross
section. In the following we discuss alternative ways to
achieve this goal, and therefore we assume in this section parity
conservation, $g_P = 0$, keeping always in mind the possibility of
parity violation on top of the mechanisms discussed here.
The Higgs potential is
\begin{align}
V(\varphi, H) =& -\mu_H^2 H^\dagger H + \lambda_H (H^\dagger H)^2
- \frac{\mu_\varphi^2}{2} \varphi^2
+ \frac{\lambda_\varphi}{4} \varphi^4
+ \frac{\lambda_4}{2} \varphi^2 H^\dagger H
\label{eq:V1} \\
&
+ \frac{\mu_1^3}{\sqrt{2}} \varphi
+ \frac{\mu_3}{2\sqrt{2}} \varphi^3
+ \frac{\mu}{\sqrt{2}} \varphi (H^\dagger H) \,,
\label{eq:V2}
\end{align}
where the $\lambda_4$ and $\mu$ terms provide the Higgs portal between the
dark and SM sectors. In order to keep the expressions simple we set in the following
always $\mu_1 = \mu_3 = 0$. Those terms will not introduce new physical
effects and therefore all features relevant for our discussion can be
captured within this restricted framework.
In general mixing between $h$ and $\phi$ will be induced, with physical mass
states $H_1$ and $H_2$ and a mixing angle $\alpha$ with
\begin{equation}
\tan 2\alpha = \frac{\sqrt{2} {\mu v_1} + 2 \lambda_4 v_1v_2}
{2 \lambda_H v_1^2 - 2 \lambda_\phi v_2^2 + \mu v_1^2/(2 \sqrt{2} v_2)} \,.
\end{equation}
We adopt the convention that for $\alpha \to 0$, $H_1$
corresponds to $h$. Hence, for small mixing and $m_{H_1} = 125$~GeV,
$H_1 \approx h$ becomes a SM-like Higgs.
All direct processes coupling $\chi$ to the SM are proportional to
$\sin^2 2\alpha$ and therefore the mixing angle plays a crucial role
for DM signals.
\subsection{Direct detection}
DM scattering on nuclei relevant for direct detection is mediated via
$t$-channel exchange of the Higgs mass eigenstates $H_1$ and $H_2$. Hence,
scattering is spin-independent. The elastic scattering cross section
$\sigma_{p}$ of $\chi$ off a proton $p$ is obtained as
\begin{equation}\label{eq:DD}
\sigma_{p}=\frac{g_S^2 \sin^22\alpha}{4 \pi} \, m_{\rm red}^{2}
\left(\frac{1}{m_{H_1}^2} - \frac{1}{m_{H_2}^2} \right)^{2} g_{Hp}^{2} \,.
\end{equation}
The typical size of the scattering cross sections is
\begin{equation}
\sigma_p \approx 5\times 10^{-43}\,{\rm cm}^2 \, g_S^2 \sin^22\alpha
\left(\frac{m_{\rm red}}{1 \,{\rm GeV}}\right)^2
\left(\frac{1}{m_{H_1}^2} - \frac{1}{m_{H_2}^2} \right)^{2}
(100 \,{\rm GeV})^{4} \,.
\end{equation}
This number has to be compared to the limit from XENON100, which is
$\sigma_p \lesssim 10^{-44}\,{\rm cm}^2$ for $m_\chi \simeq
50$~GeV~\cite{Aprile:2011hi}. Hence, couplings of order one and large mixing
are in tension with the bound. In eq.~\eqref{eq:DD} we take into account
only the scalar coupling $g_S$. Similar to the EFT case discussed above, for
pseudo-scalar interactions the cross section is suppressed by ${\nu}^2\sim
10^{-6}$.
\subsection{LHC Higgs signatures}
In order to define a SM-like Higgs $h$ with $m_{H_1} = 125$~GeV, we will use
the notion of signal strength reduction factor in the event number of a
specific final state of the Standard Model, $X$, in the Higgs boson decay,
see e.g.~\cite{Baek:2011aa, Englert:2011yb, Englert:2011aa}. The latter is
defined as:
\begin{equation}
r_i\equiv \frac{\sigma_{H_i} {\rm Br}_{H_i\rightarrow X}}
{\sigma_{H_i}^{\rm SM} {\rm Br}_{H_i\rightarrow X}^{\rm SM}}
\end{equation}
with $i=1,2$ and where $\sigma_{H_i}$ and
${\rm Br}_{H_i\rightarrow X}$ are the Higgs
production cross section and branching ratio of $H_i\rightarrow X$,
respectively, while $\sigma_{H_i}^{\rm SM}$ and
${\rm Br}_{H_i\rightarrow X}^{\rm SM}$ are
the same quantities for a Standard Model Higgs with $m_h=m_{H_i}$. One obtains
\begin{equation}\label{eq:r1r2}
r_1=\cos^4\alpha \, \frac{\Gamma^{\rm SM}_{H_1}}{\Gamma_{H_1}}
\quad \mbox{and} \quad
r_2=\sin^4\alpha \, \frac{\Gamma^{\rm SM}_{H_2}}{\Gamma_{H_2}}
\end{equation}
where $\alpha$ denotes the Higgs mixing angle, $\Gamma^{\rm SM}_{H_i}$ is
the total decay width of a SM Higgs of mass $m_h=m_{H_i}$ and
$\Gamma_{H_i}$ is the total decay width of $H_i$ including the decay into
$H_{j \neq i}$ and $\chi$. In order to have a SM-like Higgs we require small
mixing $\alpha$ and identify $H_1$ with the SM Higgs $h$ with $m_{H_1} =
125$~GeV. In practice, we will require that $r_1>0.9$ and $r_2<0.1$. The latter
constraint is imposed to respect the fact that no indication of a second
Higgs-like particle is seen at LHC. In the model under consideration
typically requiring $r_1 > 0.9$ automatically leads to $r_2 < 0.1$. Note
that eq.~\eqref{eq:r1r2} is independent of the Higgs decay channel $X$.
Therefore, we can compare $r_i$ directly with the ATLAS/CMS results on the
signal strength reduction factor obtained from a combination of all search
channels.
\subsection{Numerical results}
We have performed a numerical scan over the parameters of this model using
{\tt micrOMEGAs}~\cite{Belanger:2008sj, Belanger:2010gh}. We assume
$m_{H_1}=125$~GeV and set $g_P = 0$. Then we scan randomly over $m_\chi,
g_S, v_2, \mu, \lambda_4$, and $m_{H_2}$ or $\lambda_\phi$ as free
parameters. In order to ensure perturbativity, we impose that the absolute
value of the couplings $ \lambda_4, \lambda_\phi, \lambda_H$ and $g_S$ are
smaller than $\pi$. For the scalar potential to be bounded from below, we
imposed $\lambda_\phi, \lambda_H>0$ and $\lambda_4>-2
\sqrt{\lambda_\phi\lambda_H}$. We also assume that $\chi$ is the only dark
matter candidate that gives rise to a relic density $0.09<\Omega h^2<0.13$
obtained by thermal freeze-out. If not mentioned otherwise, we scanned the
following range of parameters: 5~GeV$\leq m_{H_2}, m_\chi\leq 10^4$~GeV,
$10^{-4}\,{\rm GeV}\leq|\mu|, v_2 \leq 10^4$~GeV, and
$10^{-5}\leq|\lambda_4|, |g_S|\leq \pi$. We identify two viable parity conserving
Higgs portals.
\begin{figure}[tb!]
\begin{center}
\includegraphics[width=10cm]{lastPsDDmH125-2000.png}
\mycaption{Proton--DM scattering cross section as a function of the
dark matter mass in the Higgs portal model for $m_{H_1} = 125$~GeV,
$m_{H_2} = 2$~TeV, and $g_P=0$. The green points
correspond to a SM-like $H_1$ with an LHC Higgs signal strength modifier
$r_1>0.9$, while the red points have $r_1 < 0.9$.
The points above the blue line are excluded at 95\%~CL by
the XENON100 experiment~\cite{Aprile:2011hi}. This exclusion
limit has been extended for $m_\chi>1$~TeV assuming a linear dependence in $m_\chi$.}
\label{fig:DD2000}
\end{center}
\end{figure}
\subsubsection{Resonant Higgs portal}
We first assume $m_{H_1} \ll m_{H_2}$ and fix $m_{H_2}=2000$~GeV. Requiring
the correct relic abundance we show the predicted direct detection scattering
cross section in fig.~\ref{fig:DD2000} compared to the bound from XENON100.
For DM masses $m_\chi \lesssim 500$~GeV the mediator mass is still ``heavy''
and we recover roughly the EFT behaviour from fig.~\ref{fig:DDEFT}.
However, we clearly observe the suppression of the direct detection rate
when $m_\chi \approx m_{H_{1}}/2$ or $m_{H_{2}}/2$, where there is an
$s$-channel resonance for annihilations, allowing for small coupling
constants while maintaining the correct relic abundance. The red dots in
fig.~\ref{fig:DD2000} correspond to a signal strength modifier for the Higgs
signal at LHC of $r_1 < 0.9$. Hence, those points would be excluded by
confirming a SM-like Higgs at 125~GeV, while for the green points we have
$r_1 > 0.9$, showing that close to the resonances we can easily have parity
conserving fermionic Higgs portal DM consistent with a SM-like Higgs.
\subsubsection{Indirect Higgs portal}
Let us now discuss the region $m_\chi > m_{H_2} = 2$~TeV in
fig.~\ref{fig:DD2000}. In this case annihilation of $\chi$ into the mediator
becomes kinematically allowed. There are $t$- and $u$-channel diagrams
contributing to this annihilation channel, which are independent of the
mixing angle $\alpha$ and only depend on the coupling $g_S$. Assuming pure
scalar coupling and $m_{H_{1,2}} \ll m_\chi$ we find
\begin{equation}\label{eq:t-u-chan-annih}
\sigma_{\chi\chi\to \phi\phi} = \frac{3 g_S^4 {\nu}}{32\pi m_\chi^2} \qquad
\text{($u$- and $t$-channel diagrams)}\,,
\end{equation}
where ${\nu}$ is the $\chi$ velocity in the CMS. The relic density is
obtained when the reaction $\chi\chi\leftrightarrow \phi\phi$ freezes out.
(Note that for small mixing we have $\phi \sim H_2$.) This fixes essentially
the coupling $g_S$, while leaving the Higgs mixing $\alpha$ unconstrained.
Since the direct detection cross section is proportional to
$\sin^2(2\alpha)$, essentially any value for $\sigma_p$ below the XENON100
bound can be obtained\footnote{At 1-loop DM--nucleus scattering is induced
also for zero Higgs mixing, if $\lambda_4\ne 0$, giving a lower bound on the
scattering cross section. The Wilson coefficient in
eq.~\eqref{eq:scattering} is in this case
$C_1=(\sqrt{2}g_S^2\lambda_4/16\pi^2)(m_\chi v_1/m_\phi^2)f(x)$, with
$x=m_\chi^2/m_\phi^2$ and $f(x)=1/(1-x)-x\log(x)/(1-x)^2$ so that $f(0)=1$.
Note that this means that for zero $\phi-h$ mixing the suppression scale is
$\Lambda_1\sim 16\pi^2 m_\phi^2/m_\chi$ for ${\mathcal O}(1)$ couplings.
For typical parameter choices the loop process induces tiny cross sections
below $10^{-50}\,\rm cm^2$.}, as confirmed in fig.~\ref{fig:DD2000} for
$m_\chi > 2$~TeV. We study this situation in more detail in the following.
For $m_{H_{1,2}} < m_\chi$ the exchange of light scalar
fields $H_{1,2}$ between the two annihilating dark matter particles creates
a long range attractive potential (long range compared to the Compton wavelength of $\chi$). As a result there is a {\it Sommerfeld}
enhancement of the dark matter annihilation
cross-section~\cite{sommerfeld}. This velocity dependent effect has been studied in
detail in several references, see
e.g.~\cite{Hisano:2003ec,ArkaniHamed:2008qn,MarchRussell:2008tu} (see
also~\cite{Arina:2010wv} for a very similar framework).
In the calculations of the dark matter relic density we estimate the Sommerfeld enhancement
averaged over a thermal distribution at freeze-out temperature $T_f$
following~\cite{Feng:2010zp}. We assume that the cross section determining
the dark matter relic abundance is p-wave suppressed.
The thermally averaged Sommerfeld factors $ \bar S_\Phi (x_f)$
due to $\Phi=H_1$ and $H_2$ exchanges are
functions of the couplings $\alpha_{H_1}=
\left(g_S\sin\alpha\right)^2/(4\pi)$ and
$\alpha_{H_2}= \left( g_S\cos\alpha\right)^2/(4\pi)$, respectively, and of the
dimensionless parameters $\epsilon_\Phi=m_\chi/(\alpha_\Phi m_\Phi)$
and $x_f=m_\chi/T_f$ (all in the notation of~\cite{Feng:2010zp}). Let us emphasize that in our toy model dark matter does couple to two
mediators, in which case the computation of the
exact Sommerfeld factor is more involved~\cite{McDonald:2012nc} than the results
in~\cite{Feng:2010zp}. In most of the cases considered here, only one
of the two scalars leads to a non-negligible Sommerfeld correction
$\bar S$ and the relic density is taken to be $\Omega_\chi h^2\propto
1/({\bar S \langle\sigma v\rangle}$), where the thermal averaged
annihilation cross-section $\langle\sigma v\rangle$ is obtained with
{\tt micrOMEGAs}. If both $H_1$
and $H_2$ lead to a non negligible thermally averaged Sommerfeld factor,
then $\bar S$ is taken to be the largest of the two \footnote{Notice
that in the particular framework of Ref.~\cite{McDonald:2012nc} it
was shown that the exchange of multiple mediators can increase the
Sommerfeld enhancement in the off-resonant region by $\sim 20\%$.
}.
\begin{figure}[tb!]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=7.5cm]{PgSmH125-m2lmc.png} &
\includegraphics[width=7.5cm]{lastPgSmH125-m2lmc.png}
\end{tabular}
\mycaption{Parameter choices giving rise to a relic density in the WMAP
range in the Higgs portal model with $m_{H_1} = 125$~GeV and $g_P=0$. Green
and red points correspond to $m_{H_2}<m_\chi$ with a more ($r_1>0.9$) or
less ($r_1<0.9$) SM Higgs-like $H_1$, respectively. We show the scalar coupling $g_S$ as a function of the dark
matter mass without (left) and with (right) Sommerfeld enhancement for the relic density computation. For illustration, we also show the points with $m_{H_2}>m_\chi$
(blue points). } \label{fig:mchi-g}
\end{center}
\end{figure}
For masses $m_\chi\lesssim 100$ GeV, no Sommerfeld enhancement of the
thermal averaged annihilation cross-section is observed. For
$m_\chi\gtrsim 100$ GeV, $\bar S_{H_2}$ can become larger than one and
take values up to 4.5. Above $m_\chi\sim 1$ TeV, we observe values of
$\bar S_{H_1}\geq 1$ going up to 2. The main impact of the Sommerfeld
enhancement is to allow for smaller values of the couplings $g_S$ at a
given mass $m_\chi$ in order to account for the correct relic
density. This is illustrated in fig.~\ref{fig:mchi-g} where we show
the DM coupling to the scalar singlet, $g_S$, as a function of the DM
mass with and without Sommerfeld enhancement. For the case $m_{H_2} <
m_\chi$ (red and green points) we observe a clear correlation
consistent with $g_S^2 \propto m_\chi$. This is expected when the
relic density is driven by the process $\chi\chi \leftrightarrow
\phi\phi$ according to eq.~\eqref{eq:t-u-chan-annih}. Also notice the
relative flattening of the $g_S^2 -m_\chi$ correlation for $m_\chi$ in
the right panel of fig.~\ref{fig:mchi-g}. This is due to the
presence of the Sommerfeld enhancement factor allowing for smaller
coupling at a given value $m_\chi$ in order to still be consistent
with WMAP data.
\begin{figure}[tb!]
\begin{center}
\includegraphics[width=8cm]{lastPsDDmH125-m2lmc.png}
\mycaption{Parameter choices giving rise to a relic density in the WMAP
range in the Higgs portal model with $m_{H_1} = 125$~GeV and $g_P=0$. Green
and red points correspond to $m_{H_2}<m_\chi$ with a more ($r_1>0.9$) or
less ($r_1<0.9$) SM Higgs-like $H_1$, respectively. We show the DM--proton
scattering cross section as a function of the dark matter mass for
$m_{H_2}<m_\chi$ only. The points above the blue line are excluded at 95\%
CL by the XENON100 experiment~\cite{Aprile:2011hi}. This exclusion limit
has been extended for $m_\chi>1$~TeV assuming a linear dependence in
$m_\chi$. \label{fig:mchi-SDDn}}
\end{center}
\end{figure}
In fig.~\ref{fig:mchi-SDDn}, we show the direct detection cross section requiring
$m_{H_2} < m_\chi$. We see that the $\chi$-nucleon scattering cross section can have vastly varying values. It can be well below the XENON100 bound, while still
accounting for the correct relic density. This confirms that
the size of the DM annihilation cross section (which controls the relic density) is no longer related to the strength of DM interactions with the SM. In fig.~\ref{fig:mchi-SDDn}
the green points have $r_1 > 0.9$, which means that $H_1$ will look like a SM Higgs
at the LHC, while the red points have a suppressed Higgs signal, $r_1
< 0.9$. We observe that the $r_1 > 0.9$ requirement tends to keep the
scattering cross section below the XENON100 limit.
In the relic density calculation we have assumed that the thermal bath of
$\chi$ and $\phi$ has the same temperature as the SM thermal bath. The
contact between those two sectors is provided by the Higgs portal
$\lambda_4 \varphi^2 (H^\dagger H)$, providing interactions like $\phi\phi
\leftrightarrow hh, \phi \leftrightarrow hh, \phi\phi \leftrightarrow h$. If
those interactions freeze out before $\chi$ decouples from $\phi$, in
principle the dark and visible sectors may acquire different temperatures due
to a change in the number of relativistic degrees of freedom in the visible
sector. Unless both, $\lambda_4$ and $\alpha$, are extremely tiny, this may
change the relic abundance by factors of order one compared to the situation
presented above, while maintaining the qualitative picture. Various
possibilities to obtain the relic abundance for various cases of DM and
mediator properties have been discussed recently in \cite{Chu:2011be}. Note
that as long as the scalar mixing angle $\alpha$ is not exactly zero, $H_2$
is not stable and decays via the Higgs $h$ into SM particles.
Hence, in this situation the Higgs portal acts indirectly, providing the
link between the dark and visible thermal baths in the early universe. We
call this ``indirect Higgs portal''. This situation is similar to secluded
DM models~\cite{Pospelov:2007mp}, where DM annihilations into light mediator
particles have been discussed, see also \cite{Finkbeiner:2007kk, Kim:2009ke}. A
particular version of the indirect Higgs portal has been obtained in the
model from~\cite{Lindner:2011it}. That model respects a global $U(1)$
symmetry with a complex mediator $\phi$, and the relic density may be set by
the annihilation of DM into the massless Goldstone boson from the
spontaneous breaking of the $U(1)$.
Let us mention that the inclusion of Sommerfeld enhancement in the
computation of the DM relic density does not change qualitatively the
general picture presented here. With respect to indirect detection
searches, notice that the Sommerfeld correction is a velocity
dependent effect that becomes larger when smaller velocities are
involved. The Sommerfeld enhancement that affect dark matter
annihilations in the galactic halo ($v\sim 10^{-6}$) is larger than
$\bar S$ by several orders of magnitude. The annihilation cross
sections involved in the scalar case are however always p-wave
suppressed and we have checked that they for the model under study,
they stay unconstrained by indirect detection searches.
\section{Conclusions}
Motivated by recent hints from LHC experiments for a SM-like Higgs particle
around 125~GeV we have revisited here the possibility that the operator
$(H^\dagger H)$ acts as a portal between the SM and the dark sector. We
adopt the assumption that DM is a fermion, which necessarily requires
additional degrees of freedom to couple it to the Higgs portal. We consider
configurations where those additional particles are heavy and an EFT description
is possible, as well as situations with light mediators. In the latter case
we adopt a simple renormalizable toy model where a real scalar $\phi$ plays
the role of the mediator particle. Assuming further that the DM relic
abundance is obtained by thermal freeze-out in the early universe, the most
simple realization of the fermionic Higgs portal DM is under pressure from
constraints on the DM--nucleon scattering cross section from XENON100.
We
have identified three simple ways to make thermal fermionic DM consistent
with a SM-like Higgs at 125~GeV and XENON100 bounds:
\begin{itemize}
\item
{\it Pseudo-scalar Higgs portal.} If DM couples to the Higgs portal via
$\bar\chi \gamma_5 \chi$ the direct detection cross section is suppressed by
the DM velocity ${\nu}^2 \sim 10^{-6}$, whereas the annihilation cross
section responsible for the relic abundance is unsuppressed.
\item
{\it Resonant Higgs portal.} If the DM mass $m_\chi$ is close to half of the
Higgs mass $m_h$ or the mediator mass $m_\phi$, then the annihilation cross
section is enhanced by an $s$-channel resonance, allowing for small
couplings and a suppressed direct detection cross section.
\item
{\it Indirect Higgs portal.} If the mediator $\phi$ is lighter than the DM
$\chi$, the relic density can be obtained by $\chi\chi \leftrightarrow
\phi\phi$ annihilations, where the $t$- and $u$-channel diagrams are
independent of the Higgs portal strength. The Higgs portal only acts
indirectly to provide thermal contact between the dark and the visible
sector thermal baths.
\end{itemize}
In all cases it is possible to have a SM-like Higgs, with an LHC signal
strength modifier $r_1 > 0.9$ (where $r_1 = 1$ corresponds to the SM Higgs).
This framework is sometimes called ``LHC nightmare scenario'', with no
new-physics signal at LHC apart from a SM-like Higgs. Also, by construction,
the models discussed here can have unobservably small signals in direct
detection experiments. However, in general a signal can be expected for
indirect detection. For the pseudo-scalar and the indirect Higgs portals we
predict a conventional indirect detection signal (dominated by annihilations
into $\bar b b$ or gauge bosons), with an annihilation cross section
determined by the thermal freeze-out of $\sigma {\nu} \simeq 3\times
10^{-26}\,\rm cm^3s^{-1}$. In the case of resonant Higgs portal there might
be also an enhancement of the annihilation cross section today compared to
the one in the early universe \cite{Ibe:2008ye,Feldman:2008xs} if the resonance is combined
with a pseudo-scalar coupling. However, the enhancement effect may be not
enough to overcome the velocity suppressed annihilation rate for pure scalar
couplings.
In conclusion, fermionic Higgs portal DM remains a viable option if a
SM-like Higgs should be established at the currently hinted mass of
around 125~GeV. We have outlined simple mechanisms to obtain a classic
``WIMP'' DM candidate, whose relic abundance is set by thermal freeze-out,
with no DM related signal at the LHC and highly suppressed rates in direct
detection experiments, but still potentially observable in indirect detection.
\bigskip
{\bf Acknowledgement.} We thank Joachim Kopp and Yasutaka Takanishi for
useful discussions. L.L.H and T.S. acknowledge partial support from the European Union FP7 ITN
INVISIBLES (Marie Curie Actions, PITN- GA-2011- 289442).
\bibliographystyle{my-h-physrev.bst}
|
{
"timestamp": "2012-07-10T02:06:25",
"yymm": "1203",
"arxiv_id": "1203.2064",
"language": "en",
"url": "https://arxiv.org/abs/1203.2064"
}
|
\section{Introduction}
Scalar fields are abundant in supersymmetric theories. They all
couple to each other with at least gravitational strength
interactions. Planck suppressed couplings are generically unimportant
when describing processes at low energies, but such a decoupling does
not work for inflation. This can be most easily inferred from the
slow roll parameters, which get contributions from dimension five and
six operators that are unsuppressed. Describing inflation in a
generic supergravity model is thus a challenging task, as generically
the scalar field dynamics pose a complicated multifield problem.
There are good reasons why a single--field description is
desirable. In line with Ockham's razor, it is the simplest model that
fits the data. Multifield slow--roll inflation with several (real)
light fields has been studied for over a decade
\cite{Gordon:2000hv,GrootNibbelink:2000vx,GrootNibbelink:2001qt,Wands:2002bn}
(see \cite{Malik, Peterson} and references therein),
and is very constrained by the observations, in particular through the
tight limits on isocurvature modes and non-gaussianity \cite{WMAP7}.
On the other hand, however, it is technically challenging to obtain
single-field behavior in a full multi-field set-up.
If the inflaton is the only light field in a multifield parent theory,
integrating out the heavy fields should yield an effective
single--field description that is accurate up to terms
$\mathcal{O}(\partial^2/M^2)$, with $M$ the mass of the heavy
field. Naively, if there is slow roll and a large mass hierarchy, one
would assume such terms can be ignored, but this expectation is
premature\footnote{The caveats are due to other mass scales introduced
by the changing background, whether in flat space or during
slow-roll inflation \cite{Achucarro:2010jv}. This makes the details
of decoupling during inflation different from particle
phenomenology, where the effective field theory expansion is around
a particular vacuum.}. In particular, if there are turns in the
inflationary trajectory, derivative interactions between the inflaton
and the heavy fields can become transiently strongly coupled. These
lead to features and non-gaussianity in the spectrum of primordial
perturbations that would not be inferred from the naive effective
field theory (EFT). If the heavy fields remain sufficiently massive,
the turns result in a reduced speed of sound for the adiabatic
perturbations but are otherwise completely consistent with slow-roll
\cite{Achucarro:2010jv,Achucarro:2010da,Cespedes,Shiu,Baumann}. Careful
integration of the heavy fields recovers the general low energy
effective field theory of inflation including a variable speed of
sound for the adiabatic perturbations
\cite{Cheung:2007st,Weinberg:2008hq,Tolley,Garriga,cliff1} (see
\cite{Achucarro:2012sm} for a detailed discussion).
These interactions are unavoidable whenever the potential ``valley''
provided by the multifield potential deviates from a geodesic of the
multifield sigma model metric. A corollary from the point of view of
inflationary model building is that, when it comes to precision
cosmology, the field space geometry of the ``spectator'' heavy fields
(that are supposed to sit at their adiabatic minima during slow--roll
inflation) is as important as their masses and couplings inferred from
the potential alone.
Among the many scalars in a supersymmetric theory, the sgoldstino
field stands out. The sgoldstino is the scalar partner of the
goldstino, and belongs to the chiral superfield whose non-zero F-term
breaks supersymmetry\footnote{We will not consider D-term breaking in
this work.}. It has the special property \cite{
Binetruy:2004hh, kepathesis,Achucarro:2010jv} that it decouples from all
other fields in the theory\footnote{More precisely, setting all
other superfields at the minimum of their potential is a consistent
truncation from the N=1 sugra multi--field parent theory to an
effective N=1 sugra with a single chiral superfield, the sgoldstino.
In particular, the (real, two-dimensional) sgoldstino plane is a
geodesically generated surface of the K\"ahler metric in the parent
theory, so there are no derivative interactions with the truncated
heavy fields: all turns in the inflationary trajectory are entirely
confined to the sgoldstino plane. The effects of the fluctuations of
the heavy fields are suppressed by their mass squared exactly as one
would expect from an EFT expansion.}.
This makes the sgoldstino an ideal inflaton candidate, for it allows
for a description of inflation in terms of a single complex
field. From the point of view of inflationary modeling this is still
multifield inflation (with two real fields), but this two--field model
is not a toy model, it really is the correct effective description for
the full multifield system.
If inflation is effectively driven by a single real scalar field, the
inflaton, this can be identified with a suitable linear combination of
the real and imaginary parts of the sgoldstino field. Meanwhile, the
orthogonal combination is to remain stabilized at a local minimum of
the potential during inflation. The effect of turns in the trajectory
on the spectrum of primordial perturbations have to be taken into
account when comparing to observations, but at least they can be
calculated from the two-field model (see
\cite{Chen,Achucarro:2010da,PetersonTegmark,Cremonini, Avgoustidis} for
recent discussions and references).
Needless to say, this does not mean that all other scalars in the
theory (from the susy-preserving superfields) can be completely
neglected, as they have to be stabilized in a minimum of the potential
during inflation. Even though the sgoldstino decouples from these
fields, vice versa this is not true: the masses and couplings of all
other fields generically depend on the field value of the sgoldstino
field. As during inflation the sgoldstino evolves along its
inflationary trajectory, the masses of the scalars change. If the
inflaton is the sgoldstino, they will remain at the critical points,
but they may become light or even tachyonic, triggering a
waterfall-type exit from inflation that is not seen in the two-field
model. Although it is still a complicated task to determine the
minimum of the multifield potential along the inflationary trajectory,
it is much simpler than the full multifield {\it dynamics} needed for
a generic, non-sgoldstino, inflation model.
The potential energy density driving inflation breaks susy
spontaneously \cite{Dine:1995kz, Dine:1995uk}. This source of susy
breaking in the inflaton sector is always present during inflation,
and is in principle independent of the source of vacuum susy breaking.
For sgoldstino inflation there are two possibilities. First, the same
superfield that drives inflation is also responsible for low energy
susy breaking\footnote{This possibility has been recently discussed in
\cite{minimal,minimal1,minimal2} but as we will show it is difficult
to implement in practice.}. This would be the ideal situation. Not
only does inflation decouple from all other fields in the theory, it
also links the scale of inflation to the scale of susy breaking. The
second possibility is that the two sources of susy breaking are due to
different fields. Both sources may be operative during inflation, or
alternatively, it may be that only after inflation has ended, a phase
transition takes place generating our present-day susy breaking. In
both cases the present day sgoldstino field is not the sgoldstino
during inflation.
If several sources of susy breaking are present during inflation, the
inflaton can only be approximately identified with the sgoldstino, and
only if the vacuum susy breaking scale is much below the inflationary
scale. Care should be taken in this case because, as argued in
\cite{Hardeman}, only if the lightest mass in the hidden sector
responsible for vacuum susy breaking is much larger than the inflaton
mass and if the inflaton mass is much larger than the scale of hidden
sector susy breaking, is the effect of the hidden sector on the
slow-roll dynamics of the inflaton negligible. This is far from
trivial; for example, it has been proven extremely hard to combine a
susy breaking moduli stabilization and inflation in a consistent way
\cite{Kallosh:2004yh}, even in a fine-tuned setting
\cite{DavisPostma1}.
The decoupling of the sgoldstino from the other fields fits in with
recent work on how to incorporate different fields, or sets of fields,
in a sugra set-up minimizing the coupling between them
\cite{Binetruy:2004hh,deAlwis1,deAlwis2,Kepa1,Kepa2,Kepa3,serone1,serone2,gallego,Brizi,Hardeman,Achucarro:2011yc}.
Quite commonly different sectors --- e.g. the fields and couplings
responsible for susy breaking, for inflation, for moduli
stabilization, or making up the standard model --- are combined by
simply adding their respective K\"ahler- and superpotentials. However,
following this procedure one cannot completely decouple these
sectors. Even if the K\"ahler and superpotential do not contain direct
interaction terms between fields in different sectors, the resulting
scalar potential does. There are always at least Planck suppressed
interactions between the fields, and generically the mass matrix is
not block dia\-go\-nal in the different sectors. This complicates the
analysis of the full model enormously. Sectors are affected by the
presence of others, and although they work in isolation, they may no
longer do so in the full set-up. Moreover, heavy fields generically
cannot be integrated out in a consistent supersymmetric
way.\footnote{Here, once again, approximations that are justified for
phenomenology applications where the background is static
\cite{Brizi} fail during inflation
\cite{deAlwis1,deAlwis2,Kepa3,Achucarro:2010jv}}
The cross-couplings between sectors can be minimized if instead of
adding K\"ahler and superpotentials, one adds the K\"ahler invariant
functions $G = K + \ln|W|^2$ for the two sectors \cite{Binetruy,Kepa1}. This
approach allows to integrate out fields in a susy preserving
way \cite{Binetruy:2004hh}. In Ref. \cite{Kepa1} the addition of sugra
functions was used to couple a susy breaking moduli sector (fields
$X^i$) to a susy preserving sector, for example the standard model
(fields $z_i$):
\begin{equation}
G^{\rm tot}(X^i,\bar{X^i},z_i,\bar z_i) = g(X^i,\bar{X^i}) +
G^{\rm other}(z_i,\bar z_i).
\label{G_ana}
\end{equation}
In this article we want to use the same idea to couple a susy breaking
inflationary sector (fields $X^i$) to a susy preserving sector
($z_k$)\footnote{In \cite{DavisPostma2} the separable form
\eref{G_ana} was used to combine hybrid inflation with a susy
breaking moduli sector in a successful way. In this set-up the
inflaton is not the goldstino.}. For simplicity we restrict to
effectively single field inflation, and models with a single inflaton
field $X$. As susy is broken during inflation, the inflaton is then
the sgoldstino. As it turns out, the ansatz \eref{G_ana} is actually
too strict. We can allow for explicit couplings between the inflaton
and the other fields, of the form
\begin{eqnarray}
G(X,\bar X,z_k ,\bar z_k) = g(X,\bar X) + \frac12 \sum_{i \geq j} &\bigg[&
(z_i-(z_i)_0) (z_j-(z_j)_0) f^{(ij)}(X,\bar X,z_k ,\bar z_k)
\nonumber \\
+
&&(z_i-(z_i)_0) (\bar z_j-(\bar z_j)_0) h^{(ij)}(X,\bar X,z_k ,\bar z_k)+{\rm h.c.
}\bigg]
\label{G_intro}
\end{eqnarray}
with $f,h$ arbitrary functions of its arguments. As we will show,
this is the most general ansatz consistent with X being the
sgoldstino. The explicit $X$-dependence in the second term does not
spoil the decoupling of the inflaton field, the mass matrix remains
block diagonal in the two sectors, as long as the fields $z_i$ sit at
the susy critical point $(z_i)_0$ during inflation. As we will show,
during sgoldstino inflation the K\"ahler function $G$ is well
defined, maybe except from isolated points in field space.
Single field inflation can be divided into three main classes: large
field, small field and hybrid inflation. We discuss whether and how
sgoldstino inflation might work in these three regimes. Any sugra
model of inflation has to address the $\eta$-problem; this puts
bounds on the K\"ahler geometry \cite{Covi1,Covi2,BenDayan}.
Large field sgoldstino inflation does not work, at least not for potentials that grow at most polynomial.
Hybrid inflation provides the most natural embedding for sgoldstino
inflation. Indeed, usual F-term hybrid inflation is an example of
having a sgoldstino inflaton. In this set-up susy is restored in the
vacuum, and there is no relation with low energy susy breaking. More
complicated waterfall regimes may be devised, such that susy is broken
in the minimum after inflation. However, such an analysis is
multifield, and complicated multifield dynamics enters via the back
door again.
Small field inflation offers the best possibility to link inflation to
susy breaking. Naively all that is needed is finding and tuning a
saddle or maximum in a single field potential with a susy breaking
Minkowski minimum. We only find inflection points suitable for
inflation rather than a maximum or saddle point. Inflection point
inflation yields \cite{brax, Linde:2007jn} a low spectral index $n_s \leq 0.92-0.93$ (for $N =
50-60$ efolds), which is on the verge of being ruled out by the CMB
data \cite{WMAP7}. Interestingly enough, models in which susy is
broken after inflation are much easier to embed in a multi-field
set-up than models with a susy preserving vacuum. Finally, we comment
on recent claims in the literature for small field sgoldstino
inflation \cite{minimal1,minimal2,dine} with no or very little
fine-tuning. We will explain why these models cannot work.
\section{Decoupling of the sgoldstino}
\label{s:decoupling}
In this section we will show the decoupling of the sgoldstino field
explicitly. In the first subsection we derive the mass matrix, which
is block diagonal along the sgoldstino inflation trajectory. We will
argue in subsection \ref{s:G} that the K\"ahler function for a
dynamical sgoldstino field can always be put in the form
\eref{G_intro}. In subsection \ref{s:traject} we show that this
sgoldstino trajectory is independent of the field values of all the
other fields. Vice versa that is not the case: the dynamics of the
non-sgoldstino fields does depend on the sgoldstino field. Care must
be taken so that these fields remain stabilized along the full
inflationary trajectory. Finally, in subsection \ref{s:adding} we
discuss the special limit of separable K\"ahler functions
\eref{G_ana}, in which the results of \cite{Kepa1} are retrieved.
\subsection{Mass matrix}
\label{s:mass}
We start with the general formula for the mass matrix, then specialize
to sgoldstino inflation. The scalar potential can be expressed solely
in terms of the K\"ahler function\footnote{This procedure is ill
defined for $W=0$. To cure this, one can use the variable $\phi
\equiv {\rm e}^G$ instead, which remains well defined \cite{Barbieri}.
However, in the next section we show that $W=0$ at most in isolated
points in field space.} $G = K + \ln |W|^2$:
\begin{equation}
V_F = {\rm e}^G [G_I G^{I\bar J} G_{\bar J} -3],
\label{VF}
\end{equation}
with $I,J$ running over all fields $\Phi_I$. We will be working in
Planck units $M=1$ throughout this work. The fields span the K\"ahler
manifold with complex metric $G_{I \bar J}$. The inverse metric
$G^{I\bar J}$ is such that $G_{I\bar J}\, G^{K\bar J}=\delta_I^K$ and
$G_{I\bar J}\, G^{I\bar K}=\delta_{\bar J}^{\bar K}$. The only
non-zero connection is $\Gamma_{IJ}^K = G_{IJ\bar P}G^{\bar P K}$ and
its complex conjugate. The non-zero components of the Riemann tensor
are $R_{I\bar J K \bar L} =G_{S\bar L} \partial_{\bar J} \Gamma^{
S}_{IK}$ and permutations thereof.
The mass matrix is
\begin{equation}
\mathcal{M} = \(
\begin{array}{cc}
M^I_J & M^I_{\bar J} \\
M^{\bar I}_J & M^{\bar I}_{\bar J}
\end{array} \),
\qquad
M^I_J = G^{I \bar K} \nabla_{\bar K} \nabla_J V,
\quad
M^I_{\bar J}= G^{I \bar K} \nabla_{\bar K} \nabla_{\bar J} V,
\label{M}
\end{equation}
with $\nabla_K v_L = \partial_K v_L - \Gamma_{KL}^M v_M$ the covariant
derivative of some vector $v_L$. The eigen\-va\-lues and eigenvectors of
the mass matrix correspond to the $m^2$--values and mass
eigenstates respectively. The first derivative of the potential is
\begin{equation}
V_K = G_K V + {\rm e}^G[G^I \nabla_K G_I + G_K ]
\label{dV}
\end{equation}
where we used metric compatibility $\nabla_K G_{I\bar J} = 0$,
$\nabla_K G^I = \delta^I_K$ and introduced the notation $V_K
= \partial_K V$, $G^I = G^{I\bar J} G_{\bar J}$. Stationarity is not
assumed, as the inflaton field is displaced from its minimum during
inflation. The second derivatives of the potential are
\begin{eqnarray}
\nabla_{\bar L} \nabla_K V &=& (G_{K \bar L} -G_K G_{\bar L} )V +2 G_{(K}V_{\bar L)}
+{\rm e}^G[ G^{I\bar J}(\nabla_{\bar L} G_{\bar J})(\nabla_K G_I)
- R_{I\bar J K \bar L}G^I G^{\bar J}
+G_{K\bar L}],
\nonumber \\
\nabla_{L} \nabla_K V &=& (\nabla_{(L} G_{K)} -G_{(K} G_{L)} )V +2 G_{(K}V_{L)}
+{\rm e}^G[ 2 \nabla_{(K}G_{L)} +G^I \nabla_{(L} \nabla_{K)} G_I],
\label{ddV}
\end{eqnarray}
where round brackets denote symmetrization. We used that
$[\nabla_{\bar L},\nabla_K]G_I=\nabla_{\bar L} \nabla_K G_I = -R_{K
\bar L I \bar J}G^{\bar J}$. Apart from the terms proportional to
$V_K$, which are absent for stationary situations, these equations are
the same as (2.6, 2.7) of Ref.~\cite{Covi}.
Now consider F-term breaking of susy, signaled by a non-zero $G_X \neq
0$. Here $X$ is the scalar component of the chiral superfield which
also contains the goldstino. Note that one can always make a field
redefinition such that only one linear combination of fields breaks
susy. All other fields in the theory, denoted by $z_i$ (indexed by
lower case latin letters), do not break susy. Hence, we split the
fields in $\Phi_I= \{X,z_i\}$, with
\begin{equation}
G_X|_{z_0} \neq 0, \qquad G_{i}|_{z_0} = 0
\label{condition}
\end{equation}
at some point in field space $z_0 = \{(z_1)_0, (z_2)_0, ...\}$, the so-called susy critical point.
We are interested in a cosmological situation, in which $X(t)$ is
the inflaton rolling along some trajectory with $V_X \neq 0$. While
the inflaton rolls in the $X$-direction, we want all orthogonal fields
$z_i$ to remain extremized at $z_0$. To that end we demand that
\begin{equation}
\left(\partial_{X}\right)^m\left(\partial_{\bar X}\right)^n G_i|_{z_0}
=0 ,
\qquad \forall m,n \in \mathbb N.
\label{condition2}
\end{equation}
Indeed, from \eref{dV}, we then have that
\begin{equation}
V_i|_{z_0} = G_i V + {\rm e}^G[G^P \nabla_{i} G_P + G_{i}]
= {\rm e}^G G^X \nabla_{i} G_X =0.
\end{equation}
For notational convenience we dropped the $|_{z_0}$ on the right hand
side, but the reader should keep in mind that all expressions should
be evaluated at $z = z_0$. Note that $i$ labels the $z_i$ fields, and
capital letters label $\Phi_I$ (i.e. also running over $X$). In
the first step we used \eref{condition}, in the second $\nabla_{i}
G_X|_{z_0} =0$, which is a consequence of \eref{condition2}.
Thus $z_i = (z_i)_0$ is an extremum of the potential. To see whether
this is a maximum, minimum or saddle point, we must calculate the
eigenvalues of the mass matrix, which need to be positive definite for
a stable minimum. This analysis is simplified because \eref{condition} assures
the mass matrix is in block diagonal form, i.e. $M^X_{i}|_{z_0} =
M^{\bar X}_{i}|_{z_0} =0$. To prove this last statement, it is enough
to show the block diagonal form of the second covariant derivatives,
as it follows from \eref{condition2} that the field metric $G_{I\bar
J}|_{z_0}$ is block diagonal as well. The first equation in
\eref{ddV} gives for mixed indices
\begin{eqnarray}
\nabla_{\bar i} \nabla_X V |_{z_0}&=&
(G_{X \bar i} -G_X G_{\bar i} )V +2 G_{(X}V_{\bar i)}
+{\rm e}^G[ G^{K \bar L}(\nabla_{\bar i} G_{\bar L})(\nabla_X G_K)
- R_{K \bar L X \bar i}G^K G^{\bar L}
+G_{X\bar i}]\nonumber \\
&=& -{\rm e}^G G^X G^{\bar X}R_{X \bar X X \bar i} =0.
\end{eqnarray}
In the first step we used (\ref{condition}, \ref{condition2}) and that
$\nabla_i G_X|_{z_0} =\nabla_X G_i|_{z_0} =0$; in the second step that
$R_{X \bar X X \bar i}|_{z_0} = G_{j\bar i} \partial_{\bar X}
\Gamma^{j}_{ X X} = 0$ as well, which also follows from
\eref{condition2}. The second equation in \eref{ddV} likewise
vanishes for mixed indices:
\begin{equation}
\nabla_{i} \nabla_X V|_{z_0} = (\nabla_{(i} G_{X)} -G_{(X} G_{i)} )V +2 G_{(X}V_{i)}
+{\rm e}^G[ 2 \nabla_{(X}G_{i)} +G^K \nabla_{(i} \nabla_{X)} G_K]=0.
\end{equation}
\subsection{K\"ahler invariant function for sgoldstino inflation}
\label{s:G}
Let us quickly comment on our use of the K\"ahler invariant function $G = K +
\ln|W|^2$, rather than expressing results in terms of the K\"ahler
potential and superpotential. The potential danger in using $G$ is that it becomes undefined when $W=0$. However, it is easy to show that for sgoldstino
inflation we nowhere have $W=0$, except maybe for isolated points in field
space. Therefore the K\"ahler function $G$ is well defined. To illustrate this, consider a
theory with two chiral fields --- the extension to many fields is
straightforward --- with a superpotential $W=W(X,Z)$. For sgoldstino
inflation, with $X$ the goldstino superfield, we have
\begin{equation}
D_X W|_{\{X(t),Z_0\}} \neq0, \qquad D_Z W|_{\{X(t),Z_0\}}=0,
\label{demands}
\end{equation}
with $D_XW = K_X W + W_X$ the K\"ahler covariant derivative. Setting
$W=0$ along the {\it whole} trajectory implies \begin{equation} W|_{\{X(t),Z_0\}}=0
\quad \Rightarrow \quad W_X|_{\{X(t),Z_0\}}=0 \quad \Rightarrow \quad
D_X W|_{\{X(t),Z_0\}}=0 \end{equation} in contradiction with \eref{demands}.
Therefore the superpotential can only vanish for sgoldstino inflation
at accidental zeroes at isolated points in field space (possibly on
the trajectory, but this does not change our conclusions).
As a side remark, note that when the inflaton is identified with the $Z$
field rather than the sgoldstino field $X$, as for example in the models of
Ref. \cite{rube}, it is possible to have $W=0$, $D_X W|_{\{X_0,Z(t)\}} \neq0$ and $D_Z W|_{\{X_0,Z(t)\}}=0$
along the whole trajectory $\{X_0,Z(t)\}$. In this case the K\"ahler invariant
function is not well defined, and a description in terms of $K$ and
$W$ is needed.
Expanding the K\"ahler function around the susy critical point $z^i =
z_0^i$, the most general form for sgoldstino inflation --- satisfying
\eref{condition} and \eref{condition2} --- can be written in the form
\begin{eqnarray}
G(X,\bar X,z_k ,\bar z_k) = g(X,\bar X) + \frac12 \sum_{i \geq j} &\bigg[&
(z_i-(z_i)_0) (z_j-(z_j)_0) f^{(ij)}(X,\bar X,z_k ,\bar z_k)
\nonumber\\
+
&&(z_i-(z_i)_0) (\bar z_j-(\bar z_j)_0) h^{(ij)}(X,\bar X,z_k ,\bar z_k)+{\rm h.c.
}\bigg]\nonumber\\ \label{G}
\end{eqnarray}
with $f,h,g$ arbitrary functions of its arguments.
\subsection{Inflationary trajectory}
\label{s:traject}
We have seen in subsection \ref{s:mass} that along the inflationary trajectory
all non-sgoldstino fields are extremized at $z^i = z_0^i$. Since the
mass matrix is block diagonal, we can determine the stability of the
$z_i$ extremum from the sub-block of $\mathcal{M}$ with $z_i$ indices. It can easily be shown that the inflaton trajectory itself is independent of
the field values of the other fields. Indeed, the potential along the
inflationary trajectory only depends on the function $g(X,\bar X)$ in
\eref{G}, and is thus independent of the field values of all other
fields. The height $V_0 \equiv V|_{z_0}$, slope and second
derivatives of the inflaton potential are given by
(\ref{VF},~\ref{dV},~\ref{ddV}) with $I,J$ only running over $X$ and
$G \to g$. For example we have
\begin{eqnarray}
V_0 & =&{\rm e}^g\[g_X g^{X \bar X} g_{\bar X} -3\], \\
V_X|_{z_0} &=& g_X V_0 + {\rm e}^g\[g^X \nabla_X g_X + g_X\].
\end{eqnarray}
In contrast, the mass matrix along the orthogonal directions does
depend on the inflaton field value. We find
\begin{eqnarray}
M^{i}_{j}|_{z_0} &=& G^{i \bar k} \nabla_{\bar k} \nabla_{j} V\nonumber \\
&=& G^{i \bar k}
\[ G_{{j} \bar k} V_0
+{\rm e}^G[ G^{l \bar m}(\nabla_{\bar k} G_{\bar m})(\nabla_{j} G_l)
- R_{X \bar X j \bar k}G^X G^{\bar X}
+G_{j\bar k}]\] \nonumber \\
&=& {\rm e}^g\[ \delta^i_j(b+1) + x^i_{\bar m} x^{\bar m}_j + w^i_j \],
\end{eqnarray}
and
\begin{eqnarray}
M^{\bar i}_{j}|_{z_0}
&=& G^{\bar i k} \nabla_{k} \nabla_{j} V \nonumber\\
&=& G^{\bar i k} \[
\nabla_{(k} G_{j)} V_0
+{\rm e}^G[ 2 \nabla_{(j}G_{k)} +G^X \nabla_{(k} \nabla_{j)} G_X] \]
\nonumber \\
&=& {\rm e}^g\[x^{\bar i}_j (b+2)+ y^{\bar i}_j \].
\end{eqnarray}
Here we introduced the notation
\begin{eqnarray}
b & = & V_0e^{-g} = g_{_X} g^{_X} -3
\label{b} \\
x^{\bar i}_j &= &
G^{\bar i k}\nabla_k G_m = G^{\bar i k}\nabla_m G_k
\label{x} \\
w^i_j & =& -G^{i\bar k}G^X G^{\bar X} R_{X \bar X j \bar k}
\label{w} \\
y^{\bar i}_{j} &=& G^{\bar i k}G^{X} \nabla_{(k} \nabla_{j)} G_{X}.
\label{y}
\end{eqnarray}
Note that $b = V_0/m_{3/2}^2$ gives the height of the potential in
units of the gravitino mass. During slow-roll this is approximately $b
\sim 3H^2/m_{3/2}^2$.
The functions $b,x,y,w$ can be expressed in terms of the functions
$f,g,h$ appearing in the K\"ahler function \eref{G}. In general, the
constraint that the squared masses should be positive is complicated,
but there are two situations in which it simplifies considerably. The
first one, discussed in the next section, is if the K\"ahler invariant
function is separable \cite{Kepa1,Kepa2}. In this case the matrices $y$
and $w$ vanish and the constraint involves the eigenvalues of the $x$
matrix.
The second case where the constraint simplifies is for a single $z$
field, i.e. $i=\{1\}$, such that there is only one $f$ and $h$
function. Then the matrices $x$,$y$ and $w$ become scalars
\begin{eqnarray} b &=& g_{_X} g^{_X}-3 \label{defs}
\\
x &=& h^{-1} (f -f_{_{\bar X}} g^{_{\bar X}}),
\nonumber \\
w &=& - g^{_X} g^{_{\bar X}} h^{-1} (h_{_{X \bar X}} - h_{_X} h^{-1}
h_{_{\bar X}}),
\nonumber \\
y &=& h^{-1} g^{_X}\[ f_{_X}-2 h_{_X}h^{-1}f-f_{_{\bar X}}
g_{_{X}}^{_{\bar X}} +\( f_{_{\bar X}} g_{_{XX}}^{_X} + h^{-1}h_{_X}
f_{_{\bar X}} -f_{_{X \bar X}} \) g^{_{\bar X}} \]. \nonumber \end{eqnarray}
For a canonically normalized $z$ field, $h =1, \, h_{_X} = h_{_{X\bar
X}} =0$ which implies $w=0$.
For single field inflation, or if the matrices $x,w,y$ can be
diagonalized simultaneously, the eigenvalues of the mass matrix are
given by
\begin{equation}
m_\pm^2 = {\rm e}^g \[ (1+b)+ |x|^2 +w \pm | (2+b)x +y| \].
\label{mass2}
\end{equation}
The $z$ eigenstates remain stabilized as long as the smallest mass is
positive definite $m_-^2 > 0$.
\subsection{Separable K\"ahler function}
\label{s:adding}
The results in the previous section are a generalization of the work
\cite{Kepa1,Kepa2,Kepa3}, who considered a set-up with separable
K\"ahler functions:
\begin{equation}
G(X,\bar X,z_i ,\bar z_i) = g(X,\bar X)+ \tilde g(z_i ,\bar z_i),
\label{Gsimple}
\end{equation}
which is a special limit of the more general function \eref{G}.
For the separable K\"ahler function above \eref{Gsimple} all mixed
derivatives of $G$, such as $G_{z z X}$, cancel. With this
simplification
\begin{equation}
b = g_{_X} g^{_X} -3, \qquad
x^{\bar i}_m = \tilde g^{\bar i k} \tilde g_{km}, \qquad
y^{\bar i}_j = w^i_j =0. \label{defx}
\end{equation}
We now consider the case with only one $z$ field, which turns $x^{\bar
i}_j$ into a scalar. As one can always diagonalize $x^{\bar i}_j$,
this simplification precisely gives the result along one of the
eigenvectors, and thus can be straightforwardly be generalized to
several $z$ fields. We recover the system studied in
\cite{Kepa1}\footnote{Our definition of $b$ is different from
\cite{Kepa1}, which has $b \leftrightarrow b-3$.}:
\begin{equation}
M^{z}_{z}|_{z_0} = {\rm e}^g[(b+1) + |x|^2], \qquad
M^{\bar z}_z |_{z_0}= {\rm e}^g(b+2)x,
\end{equation}
which has eigenvalues
\begin{equation} m_\pm^2 |_{z_0} = {\rm e}^g\[ 1+b +|x|^2 \pm |(2+b)x| \]
={\rm e}^g\[ \( |x|\pm\frac{1}{2}|b+2|\)^2-\frac{b^2}{4}\].
\label{msep}
\end{equation}
This result can also be obtained from the general expression for the
mass squared eigenvalues \eref{mass2}, taking the appropriate limit
$y^{\bar i}_j=w^i_j=0$. The function $b$ is bigger, equal or smaller
than zero for a dS, Minkowksi or AdS universe, respectively. Take $b
\geq 0$; in the opposite limit the masses $m_-^2$ and $m_+^2$ are
exchanged. The smallest mass eigenstate is positive $m_-^2 >0$, i.e.,
the $z$-field is stabilized along the inflationary trajectory, for
$|x| <1$ or $| x|> (1+b)$. We will put this analysis in practice for
sgoldstino inflation in subsection \ref{s:hybrid} (hybrid inflation)
and \ref{s:small} (small field inflation).
Close to the instability bounds $|x| \lessapprox 1$ or $| x|
\gtrapprox (1+b)$ the spectator field $z$ is lighter than the
gravitino mass and/or the Hubble scale, and cannot be integrated out.
In a Minkowski vacuum after inflation either $b = 0$ or $b\to\infty$;
the latter case may occur in a supersymmetric vacuum with $W \to 0$.
For $b=0$, the masses reduce to $m_\pm^2 = m_{3/2} ^2\left(1 \pm
|x|\right)^2$, with $m_{3/2}$ the gravitino mass. For $|x|>1$, the
lightest scalars from the supersymmetric sector are heavier than the
gravitino. However, for $|x|<1$ the lightest of the two eigenstates is
lighter than the gravitino and cannot be neglected from a low--energy
description. This will play an important role later. In the
supersymmetric vacuum with $b \to \infty$ we find $m_\pm^2 \approx
V_0(1\pm |x|) \to 0$, and the spectators are massless. To avoid a
plethora of massless fields in the theory, one has to either break the
supersymmetry, or else go beyond the simple separable form of the
K\"ahler function \eref{Gsimple}.
\section{Single field sgoldstino inflation}
In this paper we focus on effectively single field inflation models,
for simplicity. The inflaton, a real scalar, is identified with a
suitable linear combination of the real and imaginary parts of the
sgoldstino field; the orthogonal combination is to remain stabilized
at a local minimum of the potential during inflation. Single field
inflation can be divided into three classes: small field, large field
and hybrid inflation. In the first two cases, if the model only
contains a single chiral superfield, the inflaton is automatically the
sgoldstino. If several fields are present, as is the case for hybrid
inflation, one has to be more careful, as the sgoldstino does not have
to coincide with the inflaton direction.
As is well known any sugra model of inflation has to address the
$\eta$-problem \cite{Dine:1995kz,Lyth:2004nx,Copeland:1994vg}: the
inflaton field needs to be protected from its natural tendency to
become heavy, and obtain a mass of the order of the Hubble scale. This
is just another manifestation of the hierarchy problem that plagues
all scalars, including the standard model Higgs field. The problem is
easily spotted in the sugra context. Expand the K\"ahler potential
around $X_0$, the inflaton field value during inflation, in $\delta X
= X-X_0$; this gives $K = K_0 + K_{_{X \bar X}}\big|_{0} \, |\delta
X|^2 + ... = K_0 + |\Phi|^2 + ...$, with $|\Phi|$ the canonically
normalized complex field. The scalar potential then gives
\begin{equation}
V_F = {\rm e}^{|\Phi|^2}[V_0+ ...].
\label{eta_problem}
\end{equation}
The $\eta$-parameter measures the curvature of the potential in units
of the Hubble parameter along the inflationary trajectory: $\eta=
V_{\phi\phi}/V$, with $\phi$ the canonically normalized real inflaton
field. With the inflaton some linear combination of the real and
imaginary parts of $\Phi$, it is clear that the exponent in
\eref{eta_problem} contributes order unity: $\eta \approx 1 + ...$,
which spoils inflation.
The $\eta$-problem may be solved introducing symmetries which forbid
an inflaton mass, and thus keep the inflaton potential flat. Such a
symmetry needs to be softly broken to provide a small slope for the
inflaton potential. It is far from trivial to assure that such a
breaking does not introduce the $\eta$-problem again. Another
solution to the $\eta$-problem is to fine-tune parameters. The order
one contribution coming from the exponent in \eref{eta_problem} may be
tuned against all other contributions (from the ellipses) to obtain a
total $\eta$-parameter that is small.
In the remainder of this section we will discuss large field, small
field and hybrid sgoldstino inflation, and how the $\eta$-problem may
or may not be addressed in each case.
\subsection{Large field inflation}
In models of large field inflation \cite{Linde:1983gd}, the inflaton field traverses
super-planckian distances in field space during inflation. For a
potential dominated by a single monomial during inflation, $V \sim
\lambda \phi^n$, the slow roll parameters
\begin{equation}
\eps = \frac12 \(\frac{V_\phi}{V}\)^2, \qquad \eta =
\frac{V_{\phi\phi}}{V},
\label{sr}
\end{equation}
both scale as $\eta,\eps \sim 1/\phi^2$, and are automatically suppressed
for super-planckian field values. At first sight, no tuning of the
potential is needed. However, the problem is that for such large field
values {\it all} non-renormalizable operators are
unsuppressed. Therefore, an explicit UV completion of the model is
needed to determine whether inflation is possible.
Embedding large field inflation in sugra provides a better control
over the UV behavior of the theory. Because of the $\eta$-problem
such an embedding is far from straightforward, as the potential
\eref{eta_problem} grows exponentially rather than
polynomial. Fine-tuning $\eta$ is not an option, as $\eta$ has to be
small along the whole inflationary trajectory, which spans
super-planckian distances in field space $\Delta \phi >1$. This is in
contrast with small field inflation, discussed in subsection \ref{s:small}, where
the $\eta$-problem can be solved by tuning $\eta$ at a single point in
field space.
Instead of fine-tuning, we can try to solve the $\eta$-problem by invoking a shift
symmetry \cite{kawasaki}. Consider a K\"ahler function $G = {\cal K}(X - \bar X)$, which is
symmetric under a shift $X \to X + c$ with $c$ a real constant. Since
$G$ does not depend explicitly on $\phi \propto {\rm Re}(X)$, the
exponent in \eref{eta_problem} is independent of $\phi$ and there is
no $\eta$-problem. In fact, the potential has an exactly flat
direction. Since we want the system to end up after inflation in a
Minkowski minimum, there is no other option than to set $V =0$ along
the flat direction, which is incompatible with having inflation.
In order to get a slope for the potential and obtain inflation, the
shift symmetry needs to be weakly broken. To assure the breaking does
not reintroduce exponential terms that ruin inflation, we add a
logarithmic term $G = {\cal K}(X-\bar X) + \ln|W(X)|^2$ with $W$ not
growing faster than power law. As we want to construct a potential that is polynomial, we forbid the linear terms in the K\"ahler potential ${\cal K}={\cal K}\((X-\bar X)^2\)$. Then
the potential along the inflationary trajectory is
\begin{equation}
V_F|_{X = \bar X} = W_X G^{X\bar X} \bar W_{\bar X} - 3|W|^2\big|_{X = \bar X}.
\end{equation}
The inverse metric $G^{X\bar X}|_{X =\bar X} =-1/{\cal K}''(0)$ is a constant
along the inflationary trajectory, as it is independent of $\phi$; it
just renormalizes the field and can be absorbed by going to
canonically normalized fields: $\phi^2 = -2{\cal K}''(0) |X|^2$. If the
superpotential during inflation is dominated by a monomial term $W
\sim \lambda X^n$, we find
\begin{equation}
V_F|_{X = \bar X} \propto n^2 \phi^{2n-2} - 3\phi^{2n}
\label{VF_largefield}
\end{equation}
which goes negative for large $\phi > n/\sqrt{3}$. For field values
$\phi = {\mathcal O}(10)$ as needed for large field inflation, the
field will run off to infinity and negative potential, rather than the
Minkowksi minimum at the origin. This does not give a viable inflation
model. The instability occurs for every superpotential that does not
grow faster than power law, such that the shift symmetry is only
broken softly. Faster growing superpotentials reintroduce the $\eta$
problem.
Although we did the analysis for a single field, this
straightforwardly generalizes to the multi-field case. If the inflaton
is the sgoldstino, it decouples from the other fields, and its
potential can be analysed independently and will always be of the form
\eref{VF_largefield}. We conclude that large field sgoldstino
inflation in a sugra model does not work as it is plagued by an
instability in the scalar potential.
We note that it is certainly not impossible to have large field
inflation in sugra, only that it does not work with a single chiral
superfield. Two-field models have been constructed that avoid the
instability \cite{rube,kawasaki}, employing a shift symmetry to
address the $\eta$-problem. However, in these models the inflaton is
{\it not} the sgoldstino (rather the sgoldstino is the orthogonal
field).
\subsection{Hybrid inflation}
\label{s:hybrid}
Hybrid inflation is a multi-field model of inflation which in addition
to the inflaton contains one or more so-called waterfall fields, which
serve to end inflation \cite{HI}. During inflation the waterfall fields are
stabilized in a local minimum, and inflation is effectively single
field. If the inflaton field drops below a critical value one of the
waterfall fields becomes tachyonic, and inflation ends with a phase
transition.
Standard F-term hybrid inflation \cite{HI_susy,HI_sugra} is an example
of sgoldstino inflation. The K\"ahler function is of the separable
form \eref{Gsimple} discussed in section \ref{s:adding}.
\begin{equation}
G = g(X,\bar X) + \tilde g(\chi_1, \bar\chi_1, \chi_2, \bar \chi_2),
\end{equation}
with\footnote{To see that this setup is indeed of the general form
\eref{G}, one can move a factor of $\ln|\mu^2|^2$ from $\tilde{g}$
to $g$ and Taylor expand the remaining $\ln|\frac{\chi_1
\chi_2}{\mu^2} - 1|^2$.}
\begin{equation}
g = X \bar X + {k_s}(X \bar X)^2 +
\ln|\kappa X|^2 + ..., \qquad
\tilde g = \chi_1 \bar \chi_1
+\chi_2 \bar \chi_2 + \ln|\chi_1 \chi_2 - \mu^2|^2+...\nonumber
\label{hybridsetup}
\end{equation}
The model has an R-symmetry, which uniquely fixes the superpotential
at the normalized level, and in particular it allows for a linear term
in $X$ but forbids the quadratic and cubic terms in $W$. This kills
large contributions to the slow roll parameters, and allows for a flat
direction in the inflaton potential, which at tree level is only
lifted by higher order terms in the K\"ahler potential.
\begin{figure}[t!]
\begin{center}
\includegraphics[scale=0.75]{stabilityplot3.pdf}
\end{center}
\caption{\footnotesize (Figure adapted from \cite{Kepa1,Kepa3}.)
Stability diagram for the separable case $G = g(X,\bar X) + \tilde
g(z,\bar z)$. The variables on the axes $b,\, x$ are defined in
\protect \eref{defx}, with $x$ one of the degenerate eigenvalues
of the $x^{\bar i}_j$ matrix. The masses of the spectator fields
are positive in the shaded region, while the unstable region signals
a tachyonic mode. The black arrow represents the inflationary
trajectory for the proposed hybrid set-up, which ends when one of
the spectator fields (the waterfall fields) becomes tachyonic. Also
shown are possible inflationary trajectories for small field
inflation (red arrows).}
\label{stability}
\end{figure}
The inflaton $\phi$ is identified with the real direction via the
decomposition $X = (\phi + i\theta)/\sqrt{2}$. Inflation takes place
for large $\phi > \phi_c = \sqrt{2}\mu $, and all other fields
stabilized at zero field value. The potential along the inflationary
trajectory is
\begin{equation}
V = \kappa^2 \mu^4\( 1- 2 k_s \phi^2 +...\) + V_{1-{\rm
loop}}.
\end{equation}
The flatness of the potential is only lifted by higher order terms in
$K$, and by the one-loop Coleman-Weinberg potential $V_{1-{\rm loop}}$
\cite{Coleman:1973jx} . The $\eta$-problem is solved via a moderate
fine-tuning of $k_s \lesssim 10^{-2}$. Moreover, for the 1-loop
contribution to be sufficiently small $\sqrt{\kappa}\mu$ should be of
the grand unified scale or smaller. During inflation
$G_X=\frac{\sqrt{2}}{\phi}+\frac{\phi}{\sqrt{2}}+\frac{k_s
\phi^3}{\sqrt{2}}$ and $G_{\chi_1} = G_{\chi_2} = 0$. Hence $\phi$
is indeed the (real part of the) sgoldstino field.
The Minkowski minimum after inflation is at $X=0$, and $|\chi_1|=|
\chi_2| = \mu$. In the minimum $G_X = G_{\chi_\pm} =0$ and susy is
restored. There is no relation between inflation and low energy susy
breaking. The sgoldstino during inflation is unrelated to the
sgoldstino today.
The masses of waterfall fields along the inflationary trajectory can
be found using the results of section \ref{s:adding}. The mass
eigenstates are the linear combinations $\chi_\pm = (\chi_1 \pm
\chi_2)/\sqrt{2}$. Using these as a basis the matrix $x^{\tilde{i}}_m$
becomes diagonal during inflation. This shows that we can
restrict our attention to only one of the complex fields $\chi_\pm$,
the other field will give the same masses for its two real degrees of
freedom. Now we can directly compute the masses from \eref{msep}. The
stability region as a function of $b$ and $|x|$ is plotted in Fig
\ref{stability}. The inflationary trajectory corresponds to a
vertical trajectory in the plot, going upwards as the field rolls
down. When it irrevocably hits the instability region (i.e. when the
lower mass eigenvalue becomes negative), inflation ends.
We note that a similar stability analysis can be done for all models
of sgoldstino inflation. Whereas hybrid inflation critically makes
use of the instability regions, for any non-hybrid scenario --- being
it small or large field inflation --- the inflationary trajectory
would have to stop before reaching the instability region. This is
automatic for $|x|<1$, otherwise the stability conditions place an
upper bound on $b$ during inflation. We will return to this point
shortly when discussing small field inflation.
\subsection{Small field inflation}
\label{s:small}
Inflation in small field models \cite{{Linde:1981mu},Albrecht:1982wi}
takes place for sub-Planckian values of the inflaton field. This
allows for Taylor expanding the inflaton potential around its
Minkowski minimum. If one term in the polynomial expansion dominates
during inflation the slow roll parameters blow up: $\eps,\eta \sim
1/\phi^2$ in the small field limit, prohibiting inflation. As before,
$\phi$ is the canonically normalized inflaton field. The only way to
get around this conclusion is that several terms in the expansion
conspire together to nearly cancel, thus obtaining small slow roll
parameters.
This motivates to consider inflation near an extremum --- a maximum,
saddle point or inflection point --- of the potential. This assures
that the first slow roll parameter $\eps$ vanishes. The
$\eta$-parameter can be made small by tuning the parameters in the
potential. Since the inflaton field traverses only small,
sub-planckian distances in field space, tuning the curvature of the
potential at a single point, at the extremum, suffices. This is in
contrast with large field inflation, where $\eta$ needs to be small
along the full, super-planckian inflationary trajectory. The tuning of
the parameters in the potential is typically of the 1-permille level,
dictated by the need to get $\eta \lesssim 10^{-2}$. Note that in a
sugra the $\eta$-parameter cannot be tuned for arbitrary K\"ahler
geometry \cite{Covi1,Covi2, BenDayan}. In our example below we will assume an
(approximately) canonical K\"ahler potential, for which there are no
obstacles. Ref. \cite{BenDayan} considered modular inflation near a
maximum; we come back to this model at the end of this section.
Symmetries generically do not help in solving the $\eta$ problem in
the small field models. For example, a shift symmetry $K = K(X - \bar
X)$, so useful in large field models, does not do anything in the
small field regime. By Taylor expanding the K\"ahler potential and
performing a K\"ahler transformation, it becomes equivalent to a non
shift symmetric $K = K(X \bar X)$. R-symmetries may help in providing
a flat potential, but the R-symmetry breaking, which is necessary to
obtain a Minkowski vacuum, also tends to spoil the flatness. This is
what kills the model proposed in \cite{dine}, on which we will comment
in a bit more detail below.
We were able to construct a fine-tuned small field inflation model in
sugra containing only a single chiral field. In such a set-up the
inflaton is automatically the sgoldstino, and our example is an
existence proof for small field sgoldstino inflation. Consider a
model with\footnote{ This ansatz \eref{KWsmall} is equivalent to $
G= \sum_{n=1}\alpha_n (X \bar X)^n + \log |\sum_{n=0}\lambda_n X^n|^2$.}
\begin{equation}
K = \sum_n\alpha_n (X \bar X)^n, \qquad
W = \sum_n \lambda_n X^n. \label{KWsmall}
\end{equation}
We decompose the complex scalar $X= (\phi+i\theta)/{\sqrt{2}}$ with
$\phi$ the inflaton field. The model parameters $\lambda_n,\alpha_n$
can be tuned in such a way that the potential allows for inflation
near an inflection point which, without loss of generality, is located
at the origin $(\phi,\theta) = (0,0)$, and a Minkowski minimum at
finite field value $(\phi,\theta) = (\phi_0,0)$. In particular, we
demand
\begin{itemize}
\item{Vanishing slope and curvature of the potential at the origin
1) $V_\phi |_{(0,0)}=0$ and 2) $V_{\phi\phi}|_{(0,0)}=0$, to assure zero slow roll
parameters $\eps = \eta =0$. The condition on $\eta$ may be relaxed
to $\eta \lesssim 10^{-2}$.}
\item{The height 3) $V|_{(0,0)} \equiv V_0$ of the potential at the origin
is fixed by the COBE nor\-ma\-li\-za\-tion of the inflaton perturbations.}
\item{After inflation the inflaton settles in a local Minkowski minimum
with 4) $V|_{(\phi_0,0)} =0$ and 5) $V_\phi|_{(\phi_0,0)} =0$.
Moreover, the masses are positive definite 6) $m_i^2|_{(\phi_0,0)} >
0$.}
\item{Along the whole trajectory, from the extremum to the minimum, the
orthogonal field is stabilized 7) $V_\theta=V_{\phi\theta}=0$
and 8) $m^2_{\theta} \gtrsim H^2$.}
\end{itemize}
We consider solutions with canonical kinetic terms, i.e. we set
$\alpha_1 =1$ and $\alpha_i =0$ for $i\neq 1$. To satisfy conditions
1-5 we need at least five parameters and choose them accordingly. We
take all $\lambda_i$ real, and consider the first five in the
expansion. Tuning is required to satisfy conditions (2) and (4) ---
the smallness of $\eta$ parameter and of the cosmological constant ---
in the usual sense that large contributions should nearly cancel.
Conditions 6-8 are then checked for consistency, but do not require
any new input. Setting the minimum at $\phi_0 =1$ we find two
inflationary inflection point solutions\footnote{$\lambda_3 =0$ only
vanishes for $\phi_0 =1$, but is non-zero for other positions of the
minima.}
\begin{equation}
\{\lambda_0,\lambda_1,\lambda_2,\lambda_3,\lambda_4\}=
\sqrt{\frac{V_0}{23}}\times
\{ 3, -5\sqrt{2},3, 0,2\} ,
\label{example1}
\end{equation}
and
\begin{eqnarray}
&&\{\lambda_0,\lambda_1,\lambda_2,\lambda_3,\lambda_4\}=
\frac{\sqrt{V_0}}{19\sqrt{73}} \times
\label{example2} \\
&&
\bigg\{{3} \sqrt{39287-1464\sqrt{6}},
\sqrt{2\left(543551-19764\sqrt{6}\right)}, {3}
\sqrt{39287-1464\sqrt{6}},0,
-{2}
\sqrt{4943-1152\sqrt{6}} \bigg\},
\nonumber
\end{eqnarray}
and all other $\lambda_i$ are zero.
\begin{figure}[t!]
\begin{center}
\includegraphics[scale=0.7]{plotSF.pdf}
\end{center}
\caption{\footnotesize Scalar potential for small field inflation
corresponding to the first solution \protect \eref{example1}.}
\end{figure}
Both examples above correspond to inflection point inflation, rather
than to inflation near a maximum or saddle point. This is
unfortunate, as for inflection point inflation the spectral index is
bounded to be $n_s \lesssim 0.92$, which is on the verge of being
ruled out. We review this argument in appendix \ref{s:inflec}.
The spectral index can be larger if the cubic term is absent or
unnaturally small, as is the case for inflation at a maximum rather
than an inflection point. Then the correction to the spectral index
\eref{n_s} is set by the quartic term in the Taylor expansion around
the extremum, rather than by cubic term, with an upper bound $n_s
\lesssim 0.95$. In our set-up this would require an extra tuning
condition $V_{\phi\phi\phi} \approx 0$; without it we always find a
saddle point.
\begin{figure}[t!]
\begin{center}
\includegraphics[scale=0.75]{stab_small.pdf}
\end{center}
\caption{\footnotesize Stability plot of the spectator $z$-fields for
a separable K\"ahler function $G = g(X,\bar X)+\tilde g(z,\bar z)$.
The trajectories for small field inflation are vertical lines, going
upward (red) to infinity for solution \protect \eref{example1} which
has a susy preserving vacuum, and downward (black) to zero for
\protect \eref{example2} which has a susy breaking vacuum. Dashed
lines indicate unstable trajectories. The position on the horizontal
axis depends on the specifics of the spectator sector. Solution
\protect \eref{example1} always leads to an instability for $|x|
>1$.}
\label{f:smallfield}
\end{figure}
The first solution above \eref{example1} has a supersymmetric
Minkowksi minimum. In this scenario the susy breaking observed today
is not related to the susy breaking during inflation. The second
solution \eref{example2}, however, does end in a susy breaking
minimum, and the gravitino mass today can be related to the
inflationary scale. The gravitino mass is $m_{3/2} \sim 10^{-7}$, see
appendix \ref{s:inflec}.
There is a huge difference between the two solutions when combined
with other spectator fields. The first solution has a susy preserving
vacuum in which $W \to 0$. Although at this exact point our
description in terms of a K\"ahler function $G$ breaks down, we can
nevertheless describe the behavior of the potential as we approach this
singular limit. We find that $b \propto V_0/W_0 \to \infty$, with $b$
defined in \eref{defs}. This implies that if we draw the stability
diagram for the simplified case of separable K\"ahler functions \eref{Gsimple},
see Fig.~\ref{f:smallfield}, this inflationary model corresponds to
vertical trajectories going upwards to infinity.
The position on the horizontal axis given by $|x|$ depends on the
specifics of the spectator sector, but it is clear that for all
$|x|>1$ one of the fields becomes tachyonic as the inflaton approaches
its minimum, and the potential is unstable. Hence, solution
\eref{example1} with a susy vacuum can only be combined with different
fields if this extra sector has $|x| <1$ (for several fields the
eigenvalues of the $|x|^2$ matrix should all be less than unity). This
puts enormous limitations on the spectator sector. For $|x|<1$ the
masses of the spectator fields vanish in the vacuum, as discussed at
the end of section \eref{s:adding}. However, in a subsequent susy
breaking phase transition they may pick up a soft mass term.
This disastrous conclusion may be avoided by going to the most generic
K\"ahler function for sgoldstino inflation \eref{G} rather than
sticking to the separable case \eref{Gsimple}; it is hard to make a
general prediction as in the $b \to \infty$ limit also the other
quantities $x,w,y$ in the mass matrix \eref{mass2} may blow up.
In contrast, solution \eref{example2} has a susy breaking vacuum, and
the parameter $b = V_0/W = 0$ vanishes in the minimum. The inflaton
trajectory again corresponds to a vertical trajectory in the stability
diagram, but now going downwards. Except for a small region near $|x|
=1$ there are no instabilities in the potential, and at least for the
separable K\"ahler function \eref{Gsimple} sgoldstino inflation can
straightforwardly be combined with a spectator sector. In the region
$|x|>1$ the spectator fields are heavy in the vacuum and can be
integrated out to get a low energy EFT. In the other limit $|x| < 1$
the spectator fields are of the same order as the gravitino mass (see
the discussion at the end of section \ref{s:adding}), and are
relatively light.
Ref.~\cite{BenDayan} constructed a single-field potential with a
maximum, rather than an inflection point, suitable for inflation. As
remarked above, this set-up gives a spectral index in better agreement
with the WMAP data than our inflection point model. The flat maximum
was obtained by only allowing odd powers in the superpotential $W =
\sum \lambda_{2n+1} \Phi^{2n+1}$, and fine-tuning the lowest four
$\lambda_{2n+1}$ parameters. In the absence of a symmetry that can
guarantee this form of the superpotential, this model is more
fine-tuned than the inflection point set-up, as it also requires
tuning the even parameters $\lambda_{2n}=0$; not only the
$\eta$-parameter is tuned, but also $V_{\phi\phi\phi}$ at the extremum
should vanish. We further note that in this set-up $W \to 0$ at the
maximum, and thus $b\to \infty$. As discussed above, this puts
very strong constraints on the spectator sector, and may make it harder to
embed the inflaton model in a larger parent theory.
\subsubsection{Recent proposals for small field sgoldstino inflation}
In the recent literature there have been claims for small field
sgoldstino inflation, with no or very little fine-tuning of the
parameters in the potential. As argued in this paper, unless some
symmetry principle is invoked, this is not possible as the slow roll
parameters generically blow up in the small field limit. Indeed we
find that these proposals do not work, although the devil is sometimes
in the details.
Refs. \cite{minimal1,minimal2} propose a model of sgoldstino inflation
in a single field set-up without tuning of parameters. To address the
$\eta$ problem they add a logarithmic term to the K\"ahler potential
\begin{eqnarray}
K &=& X \bar X + a X \bar X (X + \bar X) + b (X \bar X)^2 + ... - 2
\ln(1 + X + \bar X),\nonumber\\
W &=& f X + f_n M.
\end{eqnarray}
However, in the small field regime the logarithm can simply be
expanded and does not alter the qualitative structure of the
potential. It also does not enhance the symmetry.
Taking arbitrary parameters, except for the constraint that the
minimum at the origin is stable and has zero cosmological constant,
both the epsilon and eta-parameter exceed unity throughout the whole
field space $|X| <1$. Slow roll inflation cannot happen. In
\cite{minimal1} it is actually claimed that $\epsilon<1$, but what
they calculate is $\eps_\theta =g^{\theta \theta} (V_{\theta}/V)^2$,
where we again decomposed the field $X = (\phi+i\theta)/\sqrt{2}$ and
$g_{ij}$ is the metric in field space. However, in a situation where
the potential falls much steeper in the $\phi$-direction than in the
$\theta$-direction, this is not the relevant slow roll
parameter. Instead, one should use the more general multi-field
generalization $\eps = g^{ij} V_i V_j/V^2$.
Ref. \cite{minimal2} shows inflationary
trajectories with a large number of efolds $N > 60$. However, their
trajectories are calculated in the --- non-applicable --- slow roll
approximation. For all initial points in field space proposed in
\cite{minimal1, minimal2} we have solved the full two-dimensional
field equations and the slow-roll approximations to them. In all cases
the slow roll solutions wildly diverge from the full solutions, which
can only give inflation for less than an efold, confirming once more
that this setup does not provide a slow roll regime.
The only way to get inflation in the set-up of \cite{minimal1,
minimal2} is to tune parameters near an extremum, along the lines
of our example \eref{KWsmall}.
Ref. \cite{dine} proposes a model with an approximate R-symmetry:
\begin{equation}
K = S \bar S + \alpha (S \bar S)^2, \qquad W = W_0 + \mu^2 S -
\frac{\lambda}{2(n+1)}S^{n+1}.
\end{equation}
The R-symmetry is only broken by $W_0$ and the higher order term in
the superpotential. In the absence of the constant $W_0$, this
assures that the potential is nearly flat near the origin, as there
is no quadratic and cubic term in the superpotential. The potential is only
lifted by the higher order quartic term in the K\"ahler (which must be
tuned $|\alpha| < 10^{-2}$), and the 1-loop Coleman-Weinberg correction
(which vanishes at the origin).
The set-up looks ideal for inflation. However, the $n$ degenerate
minima of the potential are all anti-de Sitter. To get a Minkowksi
minimum after inflation, the constant $W_0$ has to be turned on. And
although this is a small correction to the potential near the minimum,
it is the dominant correction to the inflationary plateau at the
origin, and gives rise to non-zero slow roll parameters $\eps$ and
$\eta$. We find that the resulting potential is too steep to generate
60 e-folds of inflation (at most a single efold is
possible). Moreover, the tilt of the classical potential (not
including the one-loop contribution, which may change this) is such
that, unless there is some initial velocity to make it roll uphill,
the inflaton will not end in the minimum which is lifted to $V=0$, but
rather in one of the other AdS minima.
For concreteness, we can choose to uplift the AdS minimum at positive
values of $\phi$ to a Minkowksi minimum (with $X =
(\phi+i\theta)/\sqrt{2}$). Moreover, just as \cite{dine}, we take the
parameters in the superpotential real, which simplifies the analysis.
The resulting potential will have a positive slope at the origin, as
argued above, which kills inflation at the origin. However, there
will always be a maximum of the potential in between the origin and
the minimum. Can we do inflation here? Although the R-symmetry has
lost all of its power here (as it can only help to keep the potential
flat near the origin), this is still a possibility. However, although
the epsilon parameter vanishes at the maximum, the $\eta$ parameter
naturally exceeds unity. Of course, $\eta$ can be tuned, but as
follows from our analysis in section \ref{s:small}, to satisfy all
constraints one needs at least five parameters. The potential of
\cite{dine} has not enough freedom to do so. Moreover, adding extra,
say, higher order terms, and trying to tune $\eta$, we find that the
maximum morphs into an inflection point (although we did by no means
an exhaustive study). This is as expected, there is no reason, no
symmetry, which assures that when expanded around the extremum as in
\eref{V_inflec}, the cubic term should vanish.
\newpage
\section{Conclusions}
Inflationary models in supergravity, where the inflaton sits in a
complex scalar superfield, necessarily involve a multifield analysis.
Any extra fields present during inflation must be integrated out to
give an effective single-field slow-roll dynamics that is consistent
with the CMB. However, even very heavy fields can leave a detectable
imprint in the spectrum of primordial perturbations, in particular
through a reduction in the speed of sound of the adiabatic
perturbations. The correct effective field theory for the adiabatic
mode has a variable speed of sound that depends on the background
trajectory. A necessary condition to recover the standard single-field
slow roll description is that the trajectory should have no turns into
the heavy directions. In this case, the speed of sound is unity, equal
to the speed of light, and integrating out the extra fields gives the
same effective action as truncating the heavy fields at their
adiabatic minima.
In supersymmetric models there is an extra complication. One
has to integrate out whole supermultiplets in order to obtain an
effective supergravity description for the remaining superfields. This
is only possible if the superfields that are being integrated out are
in configurations that do not contribute to susy breaking.
Sgoldstino inflation naturally implements these two conditions. The
full inflationary dynamics is confined to the sgoldstino plane.
Putting the scalar components of all other superfields at their minima
is a consistent truncation of the parent theory. This makes
sgoldstino inflationary models extremely attractive, because of their
simplicity and robustness.
We have analysed sgoldstino inflation scenarios exploiting the fact
that the K\"ahler in\-va\-riant function $G = K + \log |W|^2$ has a
relatively simple form \eref{G} which allows some aspects to be
analysed in a model--independent way. We derived a necessary and
sufficient condition on the K\"ahler function (\ref{mass2}) for the
stability of the susy-preserving sector, the spectator fields that are
integrated out. Figure \ref{stability} shows the constraint for a
separable K\"ahler function, in particular for hybrid F-term inflation
(which is a well studied case of sgoldstino inflation).
In the case of small field sgoldstino inflation we were able to
provide some viable fine-tuned examples around inflection points. The
spectral index is rather low, on the verge of being ruled out by the
CMB data. A higher spectral index would be possible with additional
fine--tuning. Rather surprisingly, the inflationary model can only be
straightforwardly combined with a spectator sector if the minimum
after inflation breaks susy. In our inflation example with a susy
preserving Minkowski vacuum the spectator sector is very constrained
by the condition that there should be no tachyonic modes in the
system. This is illustrated in Figure \ref{f:smallfield}. These
constraints would also affect the hilltop inflation examples in
\cite{BenDayan}.
One of the motivations for this study was the interesting suggestion,
put forward in \cite{minimal1}, that a relatively simple supergravity
model with a single chiral sgoldstino superfield could account for
both inflation and susy breaking in the vacuum. Contrary to claims in
\cite{minimal1,minimal2}, our conclusion is that this minimal scenario
is very tightly constrained and requires the usual level of
fine--tuning that is expected on general grounds. Another interesting
model was proposed in \cite{dine}, in which the flatness of the
inflationary plateau follows from an R-symmetry. However we find that the
R-symmetry breaking needed to obtain a Minkowski vacuum introduces an
unacceptable tilt in the potential, and prevents inflation. It is
possible that variations of this model may still work with some extra
fine--tuning.
\section{Acknowledgments}
We are grateful to Luis Alvarez-Gaum\'e, Michael Dine, C\'esar
G\'omez, Ra\'ul Jim\'enez and Lawrence Pack for discussions
of their models, to Jan-Willem van Holten for collaboration at the
early stages of this work, and to Lofti Boubekeur and Cristiano Germani for
discussions. PO is supported by F.O.M. (Dutch
organization for Fundamental Research in Matter). AA
acknowledges support by Basque Government grant IT559-10, Spanish
Ministry of Science and Technology grant FPA2009-10612, the
Consolider-ingenio programme CDS2007-00042. SM and MP are supported by
NWO Vidi-grant (680-47-229).
\begin{appendix}
\section{Small spectral index for inflection point inflation} \label{s:inflec}
In this appendix we derive the spectral index and power spectrum for
inflection point inflation, following the work of Refs. \cite{brax,
Linde:2007jn}. To a very good approximation the inflationary
observables only depend on the $\eta$-parameter at the extremum and on
the number of efolds.
Expanding the potential around the inflection point gives:
\begin{equation}
V = V_0(1+1/2 \eta_0 \phi^2 +C_3 \phi^3 + C_4 \phi^4 +...),
\label{V_inflec}
\end{equation}
with $\eta,C_3 <0$ so that the field rolls towards the minimum at
positive $\phi$ values. Inflation ends when the $C_3$ term becomes
important, and $\eps \approx 1$, which occurs for field values
$\phi_f^2 \sim \sqrt{2} /(3|C_3|) $. We can calculate the number of
efolds
\begin{equation}
N \approx \int_{\phi_f}^{\phi_N} \frac{V}{V'}
= \frac1{\eta} \log\[ \frac{\phi}{3C_3 \phi+\eta} \]^{\phi_N}_{\phi_f},
\end{equation}
where we used $V \approx V_0$ above. The above expression can be
inverted to obtain the value of the inflaton field $N$ efolds before
the end of inflation $\phi_N$:
\begin{equation}
\phi_N = \frac{{\rm e}^{N \eta_0} \eta_0/C3}{-3 ({\rm e}^{N \eta_0}-1)
-\eta_0/(\phi_f C_3)} \approx \frac{{\rm e}^{N \eta_0} \eta_0}{-3 C_3 ({\rm e}^{N \eta_0}-1)},
\end{equation}
where in the second step we used $\eta_0/(\phi_f |C_3|) \ll 1$. This
is a good approximation as $\eta_0 \ll 1$ is fine-tuned, whereas
$C_3$, and thus $\phi_f$, is naturally of order one\footnote{To be
precise, $C_3 = {\mathcal O}(1)$ for $\phi_0 \sim 1$. For minima
at smaller field values generically $C_3$ increases, as a sharper
turnover of the potential is needed. We do not find valid
solutions for minima for $\phi_0 \gg 1$ much larger, as then other
local minima at smaller field values appear.}. Note that in this
limit, the number of efolds is independent of the end of inflation,
as $\phi_f$ has dropped out of the equation. As a result the
inflationary observables are insensitive to the precise coefficients of the
higher order terms in \eref{V_inflec}. The spectral index is
\begin{equation}
n_s \approx 1 + 2 \eta \approx 1 + 2 \eta_0 +12 C_3 \phi_N \approx 1-2\eta_0
\frac{({\rm e}^{\eta_0 N}+1)}{({\rm e}^{\eta_0 N}-1)},
\label{n_s}
\end{equation}
where we used that $\eps \ll \eta$. For $N < 50-60$ one finds $n_s <
0.92-0.93$ for the whole range of $|\eta_0| \lesssim 10^{-2}$. The
power spectrum is
\begin{equation}
P_\zeta = \frac{V}{150\pi^2 \eps} = \frac{3C_3^2 {\rm e}^{-4N\eta_0} ({\rm e}^{N
\eta_0}-1)^4 V_0}{25\pi^2 \eta_0^4}
\end{equation}
with $P_\zeta = 4 \times 10^{-10}$ measured by WMAP.
For the first example \eref{example1} in the text $\eta_0 = 0$ and
$C_3=-2.39$. For $\eta_0 = 0$, the expressions simplify to
\begin{equation}
n_s - 1=-\frac{4}{N}, \quad
P_\zeta = \frac{3 C_3^2 N^4
V_0}{25\pi^2}, \quad
( {\rm for} \; \eta_0 = 0).
\end{equation}
Choosing $N=50$ this gives $n_s = 0.92$ and $V_0 = 9 \times 10^{-16}$.
The second example \eref{example2} has $C_3 =-3.69$, and gives the
same spectral index and similar $V_0 = 4 \times 10^{-16}$. The
gravitino mass today is related to the inflationary scale via $m_{3/2}
= {\rm e}^{K/2} W|_{\rm min} \sim 10^2 \sqrt{V_0} \sim 10^{-7}$, far above
the electroweak scale.
\end{appendix}
|
{
"timestamp": "2013-12-03T02:13:03",
"yymm": "1203",
"arxiv_id": "1203.1907",
"language": "en",
"url": "https://arxiv.org/abs/1203.1907"
}
|
\section{Introduction}
Let $G$ be a non-empty set and $n$ be a positive integer. If
$f:G^n\to G$ is an $n$-ary operation, then we use the compact
notation $f(x_1^n)$ for the elements $f(x_1, \ldots, x_n)$. In
general, if $x_i, x_{i+1}, \ldots, x_j$ is an arbitrary sequence of
elements in $G$, then we denote it as $x_i^j$. In the special case,
when all terms of this sequence are equal to a constant $x$, we show
it by $\stackrel{(t)}{x}$, where $t$ is the number of terms. During
this article, we assume that $n\geq 2$. We say that an $n$-ary
operation is {\em associative}, if for any $1\leq i<j\leq n$, the
equality
$$
f(x_1^{i-1},f(x_i^{n+i-1}),x_{n+i}^{2n-1})=
f(x_1^{j-1},f(x_j^{n+j-1}),x_{n+j}^{2n-1})
$$
holds for all $x_1,\ldots,x_{2n-1}\in G$. An $n$-ary system $(G,f)$
is called an {\em $n$-ary group} or a {\em polyadic group}, if $f$
is associative and for all $a_1,\ldots,a_n, b\in G$ and $1\leq i\leq
n$, there exists a unique element $x\in G$ such that
$$
f(a_1^{i-1},x,a_{i+1}^n)=b.
$$
It is proved that the uniqueness assumption on the solution $x$ can
be dropped, see \cite{Dud2} for details. Clearly, the case $n=2$ is
just the definition of ordinary groups.
For a review of basic notions, we introduce some materials. First of
all, there is the classical paper of E. Post \cite{Post}, which is
one of the first articles published on the subject. In this paper,
Post proves his well-known {\em Coset theorem}. Many basic
properties of polyadic groups are studied in this paper. The
articles \cite{Dor} and \cite{Kas} are among the first materials
written on the polyadic groups. Reader who knows Russian, can use
the book of Galmak \cite{Gal}, for an almost complete description of
polyadic groups and as we understand from its English abstract, some
of our results are also proved in that book by a different method.
The articles \cite{Art}, \cite{Dud1}, \cite{Gal2}, and \cite{Gle}
can be used for study of axioms of polyadic groups as well as their
varieties.
Note that an $n$-ary system $(G,f)$ of the form $\,f(x_1^n)=x_1
x_2\ldots x_nb$, where $(G,\cdot)$ is a group and $b$ a fixed
element belonging to the center of $(G,\cdot)$, is an $n$-ary group.
Such an $n$-ary group is called {\em $b$-derived} from the group
$(G,\cdot)$ and it is denoted by $der_b(G, \cdot)$. In the case when $b$ is the identity of $(G,\cdot)$ we
say that such a polyadic group is {\em reduced} to the group $(G, \cdot
)$ or {\em derived} from $(G, \cdot)$ and we use the notation $der(G, \cdot)$ for it. But for every $n>2$, there
are $n$-ary groups which are not derived from any group. An $n$-ary
group $(G,f)$ is derived from some group if and only if it contains
an element $a$ (called an {\em $n$-ary identity}) such that
$$
f(\stackrel{(i-1)}{a},x,\stackrel{(n-i)}{a})=x
$$
holds for all $x\in G$ and for all $i=1,\ldots,n$.
From the definition of an $n$-ary group $(G,f)$ we can directly see
that for every $x\in G$ there exists only one $y\in G$ satisfying
the equation
$$
f(\stackrel{(n-1)}{x},y)=x.
$$
This element is called {\em skew} to $x$ and it is denoted by
$\overline{x}$. As D\"ornte proved (see a \cite{Dor}), the
following identities hold for all $\,x,y\in G$, $2\leq i,j\leq n$
and $1\leq k\leq n$
$$
f(\stackrel{(i-2)}{x},\overline{x},\stackrel{(n-i)}{x},y)=
f(y,\stackrel{(n-j)}{x},\overline{x},\stackrel{(j-2)}{x})=y,
$$
$$
f(\stackrel{(k-1)}{x},\overline{x},\stackrel{(n-k)}{x})=x.
$$
Suppose $(G, f)$ is a polyadic group and $a\in G$ is a fixed
element. Define a binary operation
$$
x\ast y=f(x,\stackrel{(n-2)}{a},y).
$$
Then $(G, \ast)$ is an ordinary group, called the {\em retract} of
$(G, f)$ over $a$. Such a retract will be denoted by $ret_a(G,f)$. All
retracts of a polyadic group are isomorphic, see \cite{DM}. The
identity of the group $(G,\ast)$ is $\overline{a}$. One can verify
that the inverse element to $x$ has the form
$$
y=f(\overline{a},\stackrel{(n-3)}{x},\overline{x},\overline{a}).
$$
One of the most fundamental theorems of polyadic group is the
following, now known as {\em Hossz\'{u} -Gloskin's theorem}. We will use
it frequently in this article and the reader can use \cite{DG},
\cite{DM1}, \cite{Hos} and \cite{Sok} for detailed discussions.
\begin{theorem}
Let $(G,f)$ be an $n$-ary group. Then
1. on $G$ one can define an operation $\cdot$ such
that $(G,\cdot)$ is a group,
2. there exist an automorphism $\theta$ of $(G,\cdot)$ and $b\in G$, such
that $\theta(b)=b$,
3. $\theta^{n-1}(x)=b x b^{-1}$, for every $x\in
G$,
4. $f(x_1^n)=x_1\theta(x_2)\theta^2(x_3)\cdots\theta^{n-1}(x_n)b$, for
all $x_1,\ldots,x_n\in G$.
\end{theorem}
According to this theorem, we use the notation $\Gf$ for $(G,f)$
and we say that $(G,f)$ is $(\theta, b)$-derived from the group $(G,
\cdot)$. During this paper, we will assume that $(G, f)=\Gf$.
Varieties of polyadic groups and the structure of congruences on
polyadic groups are studied in \cite{Art}, \cite{Dud1} and
\cite{Usan}. It is proved that all congruences on polyadic groups
are commute and so the lattice of congruences is modular. In this
article we will give a very simple proof for this fact. Among other
issues, investigated on polyadic groups, is the representation
theory. In \cite{Dud-Shah} the representation theory of polyadic
groups is studied and \cite{Shah} contains generalizations of some
important theorems of character theory of finite groups to the case
of polyadic groups.
We established some fundamental results on the structure of
homomorphisms of polyadic groups in \cite{Khod-Shah}. We will use
the main result of that article here as a very strong tool. Suppose
$(H, \ast)$ is an ordinary group and $a\in H$. In the following
theorem, we denote the inner automorphism $x\mapsto a\ast x\ast
a^{-1}$ by $I_a$. Also $R_a$ denotes the map $x\mapsto x\ast a$.
\begin{theorem}
Suppose $(G,f)=der_{\theta, b}(G, \cdot)$ and $(H, h)=der_{\eta,
c}(H, \ast)$ are two polyadic groups. Let $\psi: (G,f)\to (H, h)$ be
a homomorphism. Then there exists $a\in H$ and an ordinary
homomorphism $\phi:
(G, \cdot)\to (H, \ast)$, such that $\psi=R_a\phi$. Further $a$ and $\phi$ satisfy the following conditions;\\
$$
h(\stackrel{(n)}{a})=\phi(b)\ast a\ \ \ and \ \ \
\phi\theta=I_{a}\eta\phi.
$$
Conversely, if $a$ and $\phi$ satisfy the above two conditions, then
$\psi=R_a\phi$ is a homomorphism $(G,f)\to (H,h)$.
\end{theorem}
Clearly, using the above theorem, we can determine when two polyadic
groups $(G,f)=der_{\theta, b}(G, \cdot)$ and $(H, h)=der_{\eta,
c}(H, \ast)$ are isomorphic, and also, it can be applied for a
complete description of polyadic representations, see
\cite{Khod-Shah} for details. Using this theorem, we will determine
the structure of polyadic subgroups in the section 4.
Before going to explanation of the motivations for the recent work,
we recall the definition of normal polyadic subgroups from
\cite{Dud-Shah}. An $n$-ary subgroup $H$ of a polyadic group $(G,f)$
is called {\em normal} if
$$
f(\overline{x},\stackrel{(n-3)}{x},h,x)\in H
$$
for all $h\in H$ and $x\in G$. If every normal subgroup of $(G, f)$
is singleton or equal to $G$, then we say that $(G, f)$ is {\em
group theoretically simple} or it is $GTS$ for short. If $H=G$ is
the only normal subgroup of $(G, f)$, then we say it is {\em
strongly simple in the group theoretic sense} or $GTS^{\ast}$ for
short. For any normal subgroup $H$ of an $n$-ary group $(G,f)$, we
define the relation $\sim_H$ on $G$, by
$$
x\sim_H y\ \Longleftrightarrow\ \exists h_1, \ldots, h_{n-1}\in H:\
y=f(x,h_1^{n-1}).
$$
Now, it is easy to see that such
defined relation is an equivalence on $G$. The equivalence class of
$G$, containing $x$ is denoted by $xH$ and is called the {\it left
coset} of $H$ with the representative $x$. On the set $G/H=\{xH: x\in
G\}$, we introduce the operation
$$
f_H(x_1H,x_2H,\ldots,x_nH)=f(x_1^n)H.
$$
Then $(G/H,f_H)$ is an $n$-ary group derived from the group
$ret_H(G/H,f_H)$, see \cite{Dud-Shah}. One of the main aims of this
article is to classify all $GTS$ polyadic groups. We will give a
necessary and sufficient condition for a polyadic group $(G, f)$ to
be $GTS$ in terms of the ordinary group $(G, \cdot)$ and the
automorphism $\theta$. It is possible to define another kind of
simpleness for polyadic groups, universal algebraically simpleness.
Note that an equivalence relation $R$ over $G$ is said to be a {\em
congruence}, if
1. $\forall i:\ x_iRy_i\Rightarrow f(x_1^n)Rf(y_1^n)$,
2. $xRy\Rightarrow \overline{x}R\overline{y}$.
For example, if $H$ is a normal polyadic subgroup of $(G, f)$, then
$R=\sim_H$ is a congruence, see \cite{Dud-Shah}. We say that $(G, f)$ is {\em universal
algebraically simple} or $UAS$ for short, if the only congruence is
the {\em equality} and $G\times G$. We also, will give a
classification of $UAS$ polyadic groups and we will prove that
$UAS\subseteq GTS$.
Our motivation to study of simple polyadic groups came from $n$-Lie
algebras (or Filippov algebras). A vector space $L$ over a field
$\mathbb{F}$ is an $n$-ary Lie algebra, if it is equipped with an
alternative $n$-linear map $[-, \cdots,-]:L^n\to L$ such that the
following {\em Jacobi identity} holds
$$
\sum_{i=1}^n[x_1, \ldots, x_{i-1}, [x_i, y_1, \ldots, y_{n-1}],
x_{i+1}, \ldots, x_n]=0.
$$
The case $n=2$ is ordinary Lie algebra. The notions such as,
subalgebra and ideal can be defined as usual, see \cite{Fil}. For
$n>2$, it is proved that there is only one simple $n$-ary Lie
algebra and the dimension of this unique simple Lie algebra is
$n+1$, see \cite{Bai} for details. This fact is a large difference
between ordinary and $n$-ary Lie algebras, because there are lots
of simple ordinary Lie algebras. The case of $n$-ary Lie algebras,
may suggest that similarly there are a few simple polyadic groups
for $n>2$. But as we shall see, this is not true and there are many
simple $n$-ary groups (both $UAS$ and $GTS$), even more than the
ordinary simple groups. This fact, suggests us that there might be a
more general definition of $n$-ary Lie algebras which is not
discovered until today. In our opinion, the recent definition of
Filippov for $n$-ary Lie algebras is only a small piece of an
unknown algebraic structure. It may be logically true way if we look
for these general notion of real $n$-Lie algebras via studying {\em
polyadic Lie groups}.
\section{Elementary notions}
We assume that our polyadic group has the form $(G, f)=\Gf$. The
identity element of $(G, \cdot)$ will be denoted by $e$. First we
express the skew element $\overline{x}$ in terms of $x$ and
$\theta$.
\begin{lemma}
We have
$$
\overline{x}=b^{-1}\theta^{n-2}(x^{-1})\cdots
\theta^2(x^{-1})\theta(x^{-1}).
$$
\end{lemma}
\begin{proof}
Suppose $y=b^{-1}\theta^{n-2}(x^{-1})\cdots
\theta^2(x^{-1})\theta(x^{-1})$. Then we have
\begin{eqnarray*}
f(y, \stackrel{(n-1)}{x})&=&y\theta(x)\theta^2(x)\cdots \theta^{n-1}(x)b\\
&=&b^{-1}\theta^{n-1}(x)b\\
&=&b^{-1}bxb^{-1}b\\
&=&x.
\end{eqnarray*}
This shows that $y=\overline{x}$.
\end{proof}
\begin{definition}
Suppose $H\leq (G, f)$ and for all $x\in G$ and $h\in H$
$$
f(\overline{x},\stackrel{(n-3)}{x}, h, x)\in H.
$$
Then $H$ is called a normal polyadic subgroup of and it is denoted
by $H\unlhd (G, f)$.
\end{definition}
\begin{lemma}
Assume that $H\leq (G, f)$. Then $H$ is normal, iff for all $x\in G$
and $h\in H$, we have $\theta^{-1}(x^{-1}h)x\in H$.
\end{lemma}
\begin{proof}
Note that
\begin{eqnarray*}
f(\overline{x},\stackrel{(n-3)}{x}, h, x)&=&b^{-1}\theta^{n-2}(x^{-1})\cdots \theta(x^{-1})\theta(x)\cdots\theta^{n-3}(x)\theta^{n-2}(h)bx\\
&=&b^{-1}\theta^{n-2}(x^{-1})\theta^{n-2}(h)bx\\
&=&b^{-1}\theta^{n-2}(x^{-1}h)bx\\
&=&b^{-1}\theta^{n-1}(\theta^{-1}(x^{-1}h))bx\\
&=&b^{-1}b\theta^{-1}(x^{-1}h)b^{-1}bx\\
&=&\theta^{-1}(x^{-1}h)x,
\end{eqnarray*}
and the lemma is proved.
\end{proof}
\begin{definition}
An equivalence relation $R$ over $G$ is said to be a {\em
congruence}, if
1. $\forall i:\ x_iRy_i\Rightarrow f(x_1^n)Rf(y_1^n)$,
2. $xRy\Rightarrow \overline{x}R\overline{y}$.
\end{definition}
We denote the set of all congruences of $(G, f)$ by $Cong(G, f)$.
The following theorem is proved in \cite{Dud-Shah}.
\begin{theorem}
Suppose $H\unlhd (G, f)$ and define $R=\sim_H$ by
$$
x\sim_Hy \Leftrightarrow \exists h_1, \ldots, h_{n-1}\in H: y=f(x,
h_1^{n-1}).
$$
Then $R$ is a congruence and if we let $xH=[x]_R$, (the equivalence
class of $x$), then the set $G/H=\{ xH:x\in G\}$ is an $n$-ary group
with the operation
$$
f_H(x_1H, \ldots, x_nH)=f(x_1^n)H.
$$
Further we have
$$
(G/H, f_H)=der(ret_H(G/H, f_H)),
$$
and so it is reduced.
\end{theorem}
\begin{definition}
A polyadic group $(G, f)$ is {\em group theoretically simple}, or
$GTS$, if
$$
H\unlhd (G, f)\Rightarrow |H|=1\ \ or\ \ H=G.
$$
We say that $(G, f)$ is {\em universal algebraically simple}, or
$UAS$, if
$$
R\in Cong(G, f)\Rightarrow R=\Delta\ \ or\ \ R=G\times G,
$$
where $\Delta$ denotes equality.
\end{definition}
\begin{proposition}
Every $UAS$ is also $GTS$.
\end{proposition}
\begin{proof}
Suppose $(G,f)$ is $UAS$ and $H\unlhd (G, f)$. Let $R=\sim_H$. So we
have $R=\Delta$ or $G\times G$. If $R=\Delta$, then the following
implication is true,
$$
(\exists h_1, \ldots, h_{n-1}\in H: y=f(x, h_1^{n-1}))\Rightarrow
x=y.
$$
Assume that $|H|>1$, so there exist distinct elements $h_1, h_2\in H$.
But then there is $h_3\in H$ such that $h_2=f(h_1, h_3,
\stackrel{(n-2)}{h_1})$. Therefore $h_1=h_2$, a contradiction. Hence
$|H|=1$. On the other hand, if $R=G\times G$, then for any $x, y\in
G$, we have
$$
\exists h_1, \ldots, h_{n-1}\in H: y=f(x, h_1^{n-1}).
$$
So, if we let $x\in H$, then for any $y\in G$, $ y=f(x,
h_1^{n-1})\in H$. This shows that $H=G$.
\end{proof}
A polyadic group $(G, f)$ is {\em strongly GTS}, if $H=G$ is its
only normal polyadic subgroup. The class of such polyadic groups
will be denoted by $GTS^{\ast}$. The next proposition shows that
$GTS^{\ast}$ is the more important part of $GTS$.
\begin{proposition}
If $(G, f)$ has a singleton normal polyadic subgroup, then it is
reduced.
\end{proposition}
\begin{proof}
Suppose $H=\{ u\}$ is a polyadic normal subgroup of $(G, f)$. For any $x\in
G$, the coset $xH$ is also singleton and so $xH=\{ x\}$. The map
$\delta:G\to G/H$ defined by $\delta(x)=\{x\}$ is then an
isomorphism and hence
\begin{eqnarray*}
(G, f)&\cong& (G/H, f_H)\\
&=& der(ret_H(G/H, f_H)),
\end{eqnarray*}
and therefore $(G, f)$ is reduced.
\end{proof}
By the above proposition, if $(G, f)$ is $GTS$ but not strong,
then $(G, f)=der(G, \cdot)$, where $(G, \cdot)$ is an ordinary
simple group. Hence it remains to determine polyadic groups which
are $GTS^{\ast}$. This will be done in the section 4.
\section{UAS and the lattice of congruences}
In this section we determine the structure of congruences of
polyadic groups and we give a necessary and sufficient condition for
a polyadic group to be $UAS$. Note that the binary group $G\times G$
has an automorphism $(x, y)\mapsto (\theta(x),\theta(y))$ which we
denote it by the same symbol $\theta$. As in the previous section,
$Cong(G, f)$ is the set of all congruences of $(G, f)$. This set is
a lattice under the operations of intersection and product
(composition). We also denote by $Eq(G)$ the set of all equivalence
relations of $G$.
\begin{theorem}
$R\in Cong(G, f)$ iff $R\in Eq(G)$ and $R$ is a $\theta$-invariant
subgroup of $G\times G$.
\end{theorem}
\begin{proof}
Let $R\in Con(G,f)$. We prove that $R$ is a $\theta$-invariant subgroup of $G\times G$. Suppose $xRy$ and
$$
x_1=x,\ y_1=y,\ x_2=y_2=e,\ \ldots, x_n=y_n=e.
$$
Now, since $f(x_1^n)Rf(y_1^n)$, we must have $xbRyb$. A similar argument with
$$
x_1=x, y_1=b^{-2},\ x_2=y_2=\cdots x_{n-1}=y_{n-1}=e,\ x_n=x,\ y_n=y,
$$
shows that $b^{-1}xRb^{-1}y$. Now, let $xRy$ and assume that
$$
x_1=y_1=e,\ x_2=x,\ y_2=y,\ x_3=y_3=e, \ldots, x_{n-1}=y_{n-1}=e,\ x_n=y_n=b.
$$
Then $f(x_1^n)Rf(y_1^n)$, which means that $\theta(x)R\theta(y)$. This shows that $\theta(R)\subseteq R$. Note that the converse is also true, i.e. if $\theta(x)R\theta(y)$, then $xRy$. This is true, since $\theta(x)R\theta(y)$ implies $\theta^{n-1}(x)R\theta^{n-1}(y)$. So we have $bxb^{-1}Rbyb^{-1}$ and therefore $xRy$ by the above argument. Summarizing what we proved above, we have
$$
xRy\Leftrightarrow \theta(x)R\theta(y).
$$
Now, assume that $xRy$ and $uRv$. Let $u^{\prime}=\theta^{-1}(u)$, and $v^{\prime}=\theta^{-1}(v)$. So $u^{\prime}Rv^{\prime}$ and again a similar argument shows that $x\theta(u^{\prime})Ry\theta(v^{\prime})$, hence $xuRyv$. This shows that $R$ is closed under the binary operation of $G\times G$. Finally we show that $xRy$ implies $x^{-1}Ry^{-1}$. Note that $xRy$ implies $\overline{x}R\overline{y}$. But
$$
\overline{x}=b^{-1}\theta^{n-2}(x^{-1})\ldots\theta^2(x^{-1})\theta(x^{-1})
$$
$$
\overline{y}=b^{-1}\theta^{n-2}(y^{-1})\ldots\theta^2(y^{-1})\theta(y^{-1}).
$$
Therefore
$$
b^{-1}\theta^{n-2}(x^{-1})\ldots\theta^2(x^{-1})\theta(x^{-1})Rb^{-1}\theta^{n-2}(y^{-1})\ldots\theta^2(y^{-1})\theta(y^{-1}),
$$
and hence
$$
\theta(\theta^{n-3}(x^{-1})\ldots\theta(x^{-1})x^{-1})R\theta(\theta^{n-3}(y^{-1})\ldots\theta(y^{-1})y^{-1}).
$$
Therefore, we conclude that
$$
\theta^{n-3}(x^{-1})\ldots\theta(x^{-1})x^{-1}R\theta^{n-3}(y^{-1})\ldots\theta(y^{-1})y^{-1}.
$$
Continuing this argument, we obtain finally $x^{-1}Ry^{-1}$. So, we proved that $R$ is a $\theta$-invariant subgroup of $G\times G$. Note that, clearly the converse is also true, so we proved the assertion.
\end{proof}
\begin{proposition}
Let $H_R=\{ x\in G: xRe\}=[e]_R$. Then $H_R$ is a $\theta$-invariant normal subgroup of $G$ and it is a normal polyadic subgroup of $(G,f)$ only in the case $bRe$. \\
\end{proposition}
\begin{proof}
Let $x, y\in H_R$. Then $xRe$ and $yRe$ and so $xy^{-1}Re$. Also if $x\in H_R$ and $a\in G$ then $axa^{-1}Re$ and therefore $H_R$ is a normal subgroup of $(G, \cdot)$. Moreover if $xRe$ then $\theta(x)Re$ and hence $H_R$ is $\theta$-invariant. Note that if $x_1, \ldots, x_n\in H_R$, then $f(x_1^n)Rb$ and hence $H_R\leq (G, f)$ iff $bRe$.
\end{proof}
Now, we are ready to give the necessary and sufficient condition for a polyadic group to be $UAS$.
\begin{theorem}
$(G,f)$ is UAS iff the only normal $\theta$-invariant subgroups of $(G,\cdot)$ are trivial subgroups.
\end{theorem}
\begin{proof}
Let $(G, \cdot)$ be $\theta$-simple (i.e. it has no non-trivial normal $\theta$-invariant subgroup) and $R\in Con(G,f)$. Since $H_R$ is a $\theta$-invariant normal subgroup of $(G,\cdot)$, so $H_R=1$ or $G$. This shows that $R=\Delta$ or $G\times G$. So $(G,f)$ is UAS. Conversely, suppose that $(G,f)$ is UAS, and let $H$ be a $\theta$-invariant normal subgroup of $(G, \cdot)$. Define
$$
R=\{ (x,y): x^{-1}y\in H\}.
$$
It is easy to see that $R\in Con(G,f)$ and so, $R=\Delta$ or $G\times G$ which implies that $H=1$ or $G$.
\end{proof}
In the previous section, we saw that $H\unlhd (G, f)$ implies that $(G/H, f_H)$ is reduced, but this is not true for $G/R$, when $R$ is an arbitrary congruence. To determine its structure, note that, since $H_R$ is $\theta$-invariant, so we can define a new automorphism
$$
\theta_R:\frac{G}{H_R}\to \frac{G}{H_R}
$$
by $\theta_R([x])=[\theta(x)]$. Let $b_R=[b]$. Then we have
\begin{eqnarray*}
f_R([x_1], \ldots,[x_n])&=&[f(x_1^n)]\\
&=&[x_1\theta(x_2)\ldots\theta^{n-1}(x_n)b]\\
&=&[x_1]\theta_R([x_2])\ldots\theta_R^{n-1}([x_n])b_R.
\end{eqnarray*}
This shows that
$$
\frac{G}{R}=der_{\theta_R, b_R}(\frac{G}{H_R}, \cdot).
$$
\begin{lemma}
Let $R\leq G\times G$. Then $R\in Eq(G)$ iff $\Delta\subseteq R$.
\end{lemma}
\begin{proof}
One side is trivial. So assume that $\Delta\subseteq R$. Let $(x,y)\in R$. We show that $(y,x)\in R$. Note that
$$
(y,x)=(x,x)(x^{-1},y^{-1})(y,y)\in R.
$$
So $R$ is symmetric. Now suppose $(x,y), (y,z)\in R$. Then
$$
(x,z)=(x,y)(y^{-1},y^{-1})(y,z)\in R.
$$
This completes the proof.
\end{proof}
\begin{corollary}
We have $Cong(G,f)=\{ R\leq_{\theta} G\times G: \Delta\subseteq R\}$.
\end{corollary}
\begin{proposition}
Let $R,Q\in Cong(G,f)$. Then the following assertions are true.
1. as subgroups of $G\times G$, we have $RQ=QR$.
2. we have $R\circ Q=RQ$.
3. we have $R\circ Q=Q\circ R$, so $Cong(G,f)$ is a modular lattice.
4. we have $H_{RQ}=H_RH_Q$ and $H_{R\cap Q}=H_R\cap H_Q$.
\end{proposition}
\begin{proof}
We know that $H_Q\unlhd G$, so $1\times H_Q\unlhd G\times G$. Let $(x,y)\in R$ and $(u,v)\in Q$. Since $R$ is a congruence, so $(xu,yu)\in R$ and also
\begin{eqnarray*}
(x,y)(u,v)&=&(x,y)(u,u)(u^{-1},u^{-1})(u,v)\\
&=&(x u,y u)(e,u^{-1}v).
\end{eqnarray*}
Now, $(e,u^{-1}v)\in Q$, so $u^{-1}v\in H_Q$ and hence $(e,u^{-1}v)\in 1\times H_Q$. But, we have $R(1\times H_Q)=(1\times H_Q)R$, so
$$
(x,y)(u,v)=(e,w)(x^{\prime},y^{\prime}),
$$
for some $(e,w)\in Q$ and $(x^{\prime},y^{\prime})\in R$. This shows that $RQ=QR$.
To prove {\em 2}, note that
$$
R\circ Q=\{ (x,y)\in G\times G: \exists u, \ (x,u)\in Q\ and\ (u,y)\in R\}.
$$
So, if $(x,y)\in R\circ Q$, then $(x,y)=(x,u)(u^{-1}, u^{-1})(u,y)\in RQ$. Hence $R\circ Q\subseteq RQ$. The converse is also true.
The proofs are {\em 3} and {\em 4} are now clear.
\end{proof}
A congruence $R\in Cong(G, f)$ is called normal if there exists a normal polyadic group $H\unlhd (G, f)$ such that $R=\sim_H$. In what follows, we determine the normal elements of $Cong(G, f)$.
\begin{proposition}
Let $R$ be a normal congruence and $H$ be the normal polyadic subgroup corresponding to $R$. Then there exists an element $a\in G$ such that $H=aH_R$. \end{proposition}
\begin{proof}
Suppose $x, y\in H$ and $h_1, \ldots, h_{n-2}\in H$ are arbitrary elements. We know that there is $h_{n-1}\in H$ such that $y=f(x, h_1^{n-1})$. Hence $x\sim_H y$ and therefore there exists $a\in G$ such that $x, y\in [a]_R$. Hence $H\subseteq aH_R$. Now, suppose $u\in aH_R$. Then there is $x\in H$ such that $u\sim_H x$, so there are $h_1, \ldots, h_{n-1}\in H$ such that $u=f(x, h_1^{n-1})\in H$. Hence $aH_R\subseteq H$.
\end{proof}
\begin{theorem}
Let $R$ be a congruence. Then $R$ is normal iff there exists $a\in G$ such that
1. $aR\overline{a}$,
2. for all $x\in G$, we have $f(\overline{x},\stackrel{(n-3)}{x}, a, x)Ra$.
\end{theorem}
\begin{proof}
Suppose $R$ satisfies {\em 1} and {\em 2}. We first show that $H=aH_R$ is a normal polyadic subgroup. Suppose $x_i\in H$ for $1\leq i\leq n$. Then for all $i$, we have $x_iRa$ and since $aR\overline{a}$, hence $f(x_1^n)Rf(\overline{a},\stackrel{(n-1)}{a})$. This shows that $f(x_1^n)Ra$ and therefore $f(x_1^n)\in H$. On the other hand, if $x\in H$, then $xRa$ and so $\overline{x}R\overline{a}$. Now, since $aR\overline{a}$, so $\overline{x}Ra$, which proves that $\overline{x}\in H$. Therefore $H$ is a polyadic subgroup. Now, let $x\in G$ and $h\in H$. Then $hRa$ and hence, $f(\overline{x},\stackrel{(n-3)}{x}, h, x)Rf(\overline{x},\stackrel{(n-3)}{x}, a, x)$. But by {\em 2}, we have $f(\overline{x},\stackrel{(n-3)}{x}, a, x)Ra$, so
$f(\overline{x}, \stackrel{(n-3)}{x}, h, x)\in H$, proving that $H$ is a normal.
Now, we prove that $R=\sim_H$. Suppose $x\sim_H y$. Then there exist $h_1, \ldots, h_{n-1}\in H$, such that $y=f(x, h_1^{n-1})$. But, then every $h_i$ is of the form $ah_i^{\prime}$, with $h_i^{\prime}\in H_R$. Remember from 3.2 that $H_R$ is a $\theta$-invariant normal subgroup of $(G, \cdot)$. So we have $y=f(x, ah_1^{\prime}, \ldots, ah_{n-1}^{\prime})=f(x, \stackrel{(n-1)}{a})h^{\prime}$ for some $h^{\prime}\in H_R$. Note that, $f(x, \stackrel{(n-1)}{a})=x(\overline{a})^{-1}a$ by 2.1. On the other hand, $(\overline{a})^{-1}a\in H_R$ by {\em 1}. Hence
$$
y=f(x, \stackrel{(n-1)}{a})h^{\prime}\in xH_R,
$$
which shows that $xRy$. Hence, we proved that $x\sim_H y$ implies $xRy$. The converse is also true, showing that $R=\sim_H$. Therefore $R$ is a normal congruence.
Now suppose $R$ is normal. So $R=\sim_H$ for some normal polyadic subgroup $H$. By 3.7, there is $a\in G$ such that $H=aH_R$. We prove that $a$ satisfies {\em 1} and {\em 2} above. Let $x_1, \ldots, x_n\in H$. Then for all $i$, we have $x_iRa$, and hence $f(x_1^n)Rf(\stackrel{(n)}{a})$. Since $H$ is a polyadic subgroup, so $f(x_1^n)Ra$ and hence $aRf(\stackrel{(n)}{a})$. Now, using 2.1, it is easy to see that $f(\stackrel{(n)}{a})=a(\overline{a})^{-1}a$. Therefore, $f(\stackrel{(n)}{a})a^{-1}\in H_R$ implies that $a(\overline{a})^{-1}aa^{-1}\in H_R$. This shows that $a(\overline{a})^{-1}\in H_R$. So $aR\overline{a}$, proving {\em 1}. The proof of {\em 2} is similar.
\end{proof}
Form the above theorem, one can deduce that if $R, Q\in Cong(G, f)$ with $R$ normal and $R\subseteq Q$, then $Q$ is also normal. We can restate the above theorem as in the following form.
\begin{corollary}
A congruence $R$ is normal iff there exists an element $a\in G$ such that
1. $aR\overline{a}$,
2. for all $x\in G$, $\theta^{-1}(x^{-1}a)xRa$.
\end{corollary}
\section{GTS and normal polyadic subgroups}
This section is devoted to $GTS$ polyadic groups. Again, we assume that $(G,f)=\Gf$ is an $n$-ary group. For $u\in G$, define a new binary operation on $G$ by $x\ast y=xu^{-1}y$. Then $(G, \ast)$ is an isomorphic copy of $(G,\cdot)$ and the isomorphism is the map $x\mapsto xu$. We denote this new group by $G_u$. Its identity is $u$ and the inverse of $x$ is $ux^{-1}u$, which we denote it by $x^{-u}$. We define an automorphism of $G_u$ by $\psi_u(x)=u\theta(x)\theta(u^{-1})$. It can be easily checked that this is actually an automorphism of $G_u$.
\begin{theorem}
We have $H\unlhd (G, f)$ iff there exists an element $u\in H$ such that
1. $H$ is a $\psi_u$-invariant normal subgroup of $G_u$,
2. for all $x\in G$, we have $\theta^{-1}(x^{-1}u)x\in H$.
\end{theorem}
\begin{proof}
We first determine the structure of polyadic subgroups, using 1.2. Suppose $H\leq (G, f)$. We denote the restriction of $f$ to $H$ by $f$, so there is a binary operation on $H$, say $\ast$, an automorphism $\psi$ and an element $c\in H$, such that
$$
(H, f)=der_{\psi, c}(H, \ast).
$$
The inclusion map $j:H\to G$ is a polyadic homomorphism, hence by 1.2, there is an element $u\in G$ and an ordinary homomorphism $\phi:(H, f)\to (G,f)$, with the properties
i-\ \ $j=R_u\phi$,
ii-\ $f(\stackrel{(n)}{u})=\phi(c)u$,
iii- $\phi\psi=I_u\theta\phi$.
From i, we deduce that for any $x\in H$, $\phi(x)=xu^{-1}$, and so by ii, we have $f(\stackrel{(n)}{u})=c$. Moreover, since $\phi$ is an ordinary homomorphism, so using $\phi(x\ast y)=\phi(x)\phi(y)$ and $\phi(x)=xu^{-1}$, we obtain $x\ast y=xu^{-1}y$. Finally, by iii, we have $\psi(x)=u\theta(x)\theta(u^{-1})$, and therefore we must have $(H, \ast)\leq G_u$. Further, $H$ is invariant under $\psi_u$ and hence $\psi=\psi_u|_H$. So, we proved that $H$ is a polyadic subgroup of $(G, f)$ iff there exists an element $u$ such that $H$ is a $\psi_u$-invariant subgroup of $G_u$. Now, suppose such an $H$ is normal in $(G, f)$. We show that $H\unlhd G_u$, equivalently
$$
x^{-u}\ast h\ast x\in H
$$
for all $x\in G_u$ and $h\in H$. Note that this last statement is equivalent to $ux^{-1}hu^{-1}x\in H$. Since $H$ is a normal polyadic subgroup, so by 2.3,
$$
\theta^{-1}(x^{-1}h)x, \theta^{-1}(x^{-1}u)x\in H.
$$
But $H$ is $\psi_u$-invariant, so
$$
\psi_u(\theta^{-1}(x^{-1}h)x), \psi_u(\theta^{-1}(x^{-1}u)x)\in H.
$$
Therefore the following element also belongs to $H$,
\begin{eqnarray*}
\psi_u(\theta^{-1}(&x^{-1}&h)x)\ast\psi_u(\theta^{-1}(x^{-1}u)x)^{-u}\\
&=&ux^{-1}h\theta(x)\theta(u^{-1})u^{-1}u(ux^{-1}u\theta(x)\theta(u^{-1}))^{-1}u\\
&=&ux^{-1}hu^{-1}x.
\end{eqnarray*}
This shows that $H\unlhd G_u$. Note that the following is also automatically holds:
$$
\forall x\in G: \theta^{-1}(x^{-1}u)x\in H.
$$
Conversely, suppose there is a $u\in G$ such that $H$ is a $\psi_u$-invariant normal subgroup of $G_u$ and
$$
\forall x\in G: \theta^{-1}(x^{-1}u)x\in H.
$$
We show that $H$ is a normal polyadic subgroup. The equality
$$
\psi_u(\theta^{-1}(x^{-1}h)x)\ast \psi_u(\theta^{-1}(x^{-1}u)x)^{-u}=x^{-u}\ast h\ast x
$$
shows that $\psi_u(\theta^{-1}(x^{-1}h)x)\in H$, and since $H$ is invariant under $\psi_u$, so $\theta^{-1}(x^{-1}h)x\in H$. Therefore, $H$ is a normal polyadic subgroup.
\end{proof}
\begin{lemma}
Suppose $u\in G$ is an arbitrary element. Then $H$ is a
$\theta$-invariant normal subgroup of $(G,\cdot)$ iff $Hu$ is
$\psi_u$-invariant normal subgroup of $G_u$.
\end{lemma}
\begin{proof}
Suppose $H\unlhd_{\theta} G$. Then it can be checked that $Hu$ is a
subgroup of $G_u$. For any $x\in G$ and $h\in H$ we have
\begin{eqnarray*}
x^{-u}\ast hu\ast x&=&(ux^{-1}u)u^{-1}(hu)u^{-1}x\\
&=&ux^{-1}hx,
\end{eqnarray*}
which is clearly an element of $Hu$. So $Hu\unlhd G_u$. Also
$$
\psi_u(hu)=u\theta(hu)\theta(u^{-1})=u\theta(u),
$$
which shows that $Hu$ is $\psi_u$-invariant. Conversely, suppose $H$
is a $\psi_u$-invariant normal subgroup of $G_u$. We show that
$Hu^{-1}$ is a $\theta$-invariant normal subgroup of $G$. We have
$$
xhu^{-1}x^{-1}=(xu)\ast h\ast (xu)^{-u}u^{-1},
$$
which belongs to $Hu^{-1}$. So $Hu^{-1}\unlhd G$. Similarly,
$\theta(hu^{-1})=u^{-1}\psi_u(h)$ belongs to $u^{-1}H=e\ast H=H\ast
e=Hu^{-1}$. Hence $Hu^{-1}$ is $\theta$-invariant.
\end{proof}
Suppose $K$ is a $\theta$-invariant normal subgroup of $(G, \cdot)$.
Then $\theta$ induces an automorphism of $G/K$ which we denote it by
$\theta_K$ in what follows.
\begin{corollary}
Let $H\unlhd(G, f)$. Then there exists an element $u$ such that
$K=H\cdot u^{-1}$ is a $\theta$-invariant normal subgroup of $G$ and
$\theta_K$ is an inner automorphism. The converse is also true.
\end{corollary}
\begin{proof}
First, we notice that $H\cdot u^{-1}$ is not a polyadic coset, but it is the set $\{ hu^{-1}: h\in H\}$.
Suppose $H\unlhd (G, f)$. By 4.1, there is an element $u\in H$ such
that $H$ is a $\psi_u$-invariant normal subgroup of $G_u$ and for
any $x\in G$, we have $\theta^{-1}(x^{-1}u)x\in H$. Let $K=H\cdot u^{-1}$.
By the above lemma $K$ is a $\theta$-invariant normal subgroup of
$G$. Now, $\theta^{-1}(x^{-1}u)x\in H$ and $H$ is
$\psi_u$-invariant, so $x^{-u}\ast \psi_u(x)\in H$. Therefore
$\psi_u(x)\ast H=x\ast H$. Since $H$ is normal in $G_u$, we have
$H\ast \psi_u(x)=H\ast x$ and this is equivalent to $K\psi_u(x)=Kx$.
Now $K$ is a normal subgroup of $G$, and hence $\psi_u(x)K=xK$. So
$\theta(xu^{-1})K=u^{-1}xK$. If we put $y=xu^{-1}$, then
$\theta(y)K=u^{-1}yuK$ and this proves that $\theta_K$ is an inner
automorphism.
Conversely, suppose $K$ is a $\theta$-invariant normal subgroup of
$G$ and $\theta_K=I_{uK}$. Then by a similar argument one can prove
that $H=Ku^{-1}\unlhd (G, f)$.
\end{proof}
We saw before that if $(G, f)$ is a non-strong $GTS$, then it is
reduced. So we determine when a polyadic group belongs to the class
$GTS^{\ast}$.
\begin{theorem}
A polyadic group $(G, f)$ is $GTS^{\ast}$ iff whenever $K$ is a
$\theta$-invariant normal subgroup of $(G, \cdot)$ with $\theta_K$ inner,
then $K=G$.
\end{theorem}
\begin{proof}
First let $(G,f)$ be $GTS^{\ast}$ and $K$ be a
$\theta$-invariant normal subgroup of $G$ with $\theta_K$ inner. Then by the above corollary, there is a $u$ such that $H=Ku^{-1}\unlhd (G, f)$. Hence $Ku^{-1}=G$ and so $K=G$. To prove the converse, suppose $H\unlhd (G, f)$. So there is a $u$ such that $K=H\cdot u^{-1}$ satisfies our hypothesis. Therefore $K=G$ and hence $H=G$, proving that $(G, f)$ is $GTS^{\ast}$.
\end{proof}
As a final remark, the reader most notice that the binary groups $G_u$ which we used in this section, are in fact retract of $(G, f)$ in the case when $u$ is the skew for some other element. Suppose $u=\overline{a}$. In the group $ret_a(G, f)$, as we said in the introduction, the identity is $\overline{a}$ and if we define $c=f(\stackrel{(n)}{\overline{a}})$ and $\phi(x)=f(\overline{a}, x, \stackrel{(n-2)}{a})$, then by \cite{Sok}, we have
$$
(G, f)=der_{\phi, c}(ret_a(G, f), \ast).
$$
It is easy to see that $x\ast y=x(\overline{a})^{-1}y$ and hence $G_u=ret_a(G, f)$. But, note that in general, it is false to say that every $u$ is equal to the skew element of some $a$; there are polyadic groups in which the skew elements of any $x$ and $y$ are equal. So in the general case $G_u$ is not equal to any retract.
|
{
"timestamp": "2012-03-12T01:02:02",
"yymm": "1203",
"arxiv_id": "1203.2125",
"language": "en",
"url": "https://arxiv.org/abs/1203.2125"
}
|
\section{Introduction}
\label{sec:intro}
Graphene, a perfect carbon monolayer $sp^2-$hybridized, has attracted a huge interest since
its discovery.\cite{Novoselov_Geim_Morozov_Jiang_Zhang_Dubonos_Grigorieva_Firsov_2004,Geim_Novoselov_2007}
Besides a pure theoretical interest, its possible applications in carbon-based electronics
represent a very exciting perspective.
Graphene displays a very peculiar electronic structure, arising from the confinement of electrons
in two dimensions and its geometrical symmetries.
Graphene is a zero gap semimetal whose specific linear electronic band dispersion near the
Brillouin zone corners (Dirac points) gives rise to electrons and holes that propagate as massless
fermions.\cite{Cast09,Dubois_Zanolli_Declerck_Charlier_2009,PhysRevB.44.13237,Wallace_1947}
Graphene nanoribbons (GNRs) are one dimensional graphene strips that are considered as
promising candidates building blocks for future electronic applications.\cite{Avouris_Chen_2007,%
Wang_Ang_Wang_Tang_Thong_Loh_2010,Wakabayashi_Fujita_Ajiki_Sigrist_1998}
Several methods allow the production of GNR, including mechanical cutting of exfoliated
graphene,\cite{Geim_Novoselov_2007} patterning of epitaxially grown
graphene,\cite{Berger_Song_Li_Wu_Brown_Naud_Mayou_Li_Hass_Marchenkov_etal_2006}
bottom-up chemical
methods,\cite{Cai_Ruffieux_Jaafar_Bieri_Braun_Blankenburg_Muoth_Seitsonen_Saleh_Feng_etal_2010}
and carbon nanotubes unzipping.\cite{Jiao_Wang_Diankov_Wang_Dai_2010}
Due to the honeycomb geometry of graphene, GNR can be patterned along two preferential
directions giving rise to armchair shaped edges graphene nanoribbon (aGNR) and zigzag
shaped edges one (zGNR).
The finite size of the GNR gives rise to a large variety of electronic behaviors that could be
relevant in transport.
Considering the case of aGNR, it has been demonstrated that its electronic properties sensibly
vary by changing the width of the ribbon.\cite{Fujita_Wakabayashi_Nakada_Kusakabe_1996,%
PhysRevB.54.17954,PhysRevB.59.8271,Loui06}
In fact, the width and the electron energy gap $\Delta$ are related to each other primarily in inverse
proportion.
In particular, there exist three different classes of $N$-aGNR (aGNR with $N$ dimer lines) for what
concerns the value of the gap $\Delta_{N}$: $\Delta_{3p+1}>\Delta_{3p}>\Delta_{3p-1}$, with
$p$ integer.\cite{Loui06}
If we restrict to each of these classes, the energy gap decreases as $p$ increases.
Transport properties of pristine GNR have been studied both for the ideal\cite{PhysRevB.73.195411,%
PhysRevLett.96.246802,PhysRevB.73.235411} and defective\cite{PhysRevB.81.245402,%
Chen_Song_Zhou_Wang_Zhou_2011,Rosales_Orellana_Barticevic_Pacheco_2007} case.
Electronic transport has also been studied, by a tight-binding approach, for junctions connecting
zGNR of different widths\cite{PhysRevB.64.125428,PhysRevLett.84.3390,PhysRevB.74.195417}
revealing the crucial role played by corner edge structures.\cite{Yamamoto_Wakabayashi_2009}
A challenging technological issue would be to exploit the high mobility properties of graphene and the
finite-size characteristics of GNR, by creating heterostructures which exploit both these advantages.
These systems may be used as interconnections to transmit signals in future pure-C-based electronic
devices.
In fact, the effects of quantum confinement in graphene nanoconstrictions have been studied showing
their analogy with optics phenomena\cite{PhysRevLett.102.136803} as well as their exploitation for
valley filter applications,\cite{Rycerz_Tworzydlo_Beenakker_2006} and recently a quantized ballistic
conductance has been measured in such structures.\cite{Tombros_2011}
Even more complex junctions have been experimentally built, with carbon nanotubes as
interconnections between graphene bilayers.\cite{Qi_Huang_Feng_Shi_Li_2011}
However, due to their flat structure, GNRs seem easier to pattern than carbon nanotubes.
Owing to the large zoology of hybrid graphene interconnections, it is of great importance to perform
a systematic investigation of their transport properties at the simplest level of configurational
complexity, in order to clarify how the basic geometrical parameters affect the conductance.
Most of the previous works investigating GNR junctions and graphene nanoconstrictions rely on
standard tight-binding calculations.
It has been shown by Louie and co-workers\cite{Loui06} that the predictions of simple tight-binding
models on GNR may lead to incorrect band structures and energy gaps, since the bonding
characteristics between atoms substantially change at the edges.
An ab-initio study of GNR junctions avoids such these difficulties, providing an accurate description
of their transport properties\cite{Roche-nn100028q}, which are directly related to the underlying
electronic band structure.
In this paper, we present a first-principles study, by means of the non-equilibrium Green's
functions (NEGF) technique, of the electronic and transport properties of systems consisting
of two semi-infinite graphene layers interconnected by an hydrogen-passivated armchair
graphene nanoribbon.
The transport properties of such junctions are predicted to strongly depend on the GNR
geometry, while are quite robust to changes of the graphene's edges geometry.
These structures combine the high mobility of graphene electrodes\cite{PhysRevLett.100.016602,%
Bolotin_Sikes_Jiang_Klima_Fudenberg_Hone_Kim_Stormer_2008,%
Katsnelson_Novoselov_Geim_2006} with the intrinsic semiconducting behavior of GNR.
We show that these semiconducting hybrid graphene-GNR junctions have a significant gap
in the transmission spectrum, which may be exploited to build logic devices which require
a large on-off ratio in the current.\cite{Qi_Huang_Feng_Shi_Li_2011}
In the following discussion, we will show the properties of aGNR junctions considering one
representative for each class according to their width: we will take into account 3-aGNR,
5-aGNR, and 7-aGNR.
In the case of 3-aGNR, we will consider ribbons of three different lengths, namely those
consisting of four (4L), six (6L), and eight (8L) zigzag lines along their axis.
The 8L 3-aGNR will be considered as an illustrative model to study the effects of the
application of a bias.
We will discuss the effect of the rotation of a phenyl ring, and show how different edges
configurations affect the transport properties.
\section{Model and Methods}
\begin{figure}
\includegraphics[width=0.48\textwidth]{Fig1.eps}
\caption{\label{fig:setup
System setup for 4L 7-aGNR.
The left and right electrodes and the extended-molecule region are highlighted.}
\end{figure}
The system setup for a 4L 7-aGNR junction is shown in \ref{fig:setup} as a representative case.
This open system is constituted by three parts: the left (L) and right (R) semi-infinite graphene
leads, and an extended-molecule (EM) region.
The junction is constructed so that the periodic replicas of the aGNR along the direction parallel
to the electrode edge are separated by 7.43~\AA.
This corresponds to three unit cells of graphene and we verified that the interactions among the
replica are negligible.
The width of the leads part included inside the EM region is chosen after converging the
transmission function: six carbon zigzag lines per side are sufficient to reach stable results.
The electronic structure calculations are carried out using the first-principles self-consistent
method implemented in SIESTA package.\cite{Sole02,Arta08}
The exchange-correlation energy and electron$-$ion interaction are described by the
Perdew-Burke-Ernzerhof (PBE)\cite{Perd96} generalized gradient approximation (GGA)
and norm-conserving pseudopotentials\cite{Troullier91} in the fully nonlocal form, respectively.
A double-$\zeta$ polarized basis set of numerical atomic orbitals is used and the energy
cutoff for real-space mesh is set to 200~Ry.\cite{Sole02}
Preliminary tests indicated that the relaxation of the carbon atoms in the leads did not affect
the transport properties of the systems under study, so in most cases we considered
non-relaxed geometries.
The edges of semi-infinite leads are saturated with one relaxed hydrogen per carbon atom.
We also verified that the relaxation of the aGNR does not affect significantly the electronic
and transport properties of the system.
For the calculation of the transmission coefficients, 60~$k$-points along the transverse
direction in the 2D first Brillouin zone are used.
Periodic images of the graphene layer are separated by 15~\AA\ along the normal direction.
The electronic transport is studied with the TranSIESTA code,\cite{Bran02} which combines
the NEGF technique with density functional theory.
The transmission function of the system can be obtained by the following equation:
\begin{equation}
T(E,V)=\mathrm{Tr}[\Gamma_{\rm{L}}(E,V)G(E,V)\Gamma_{\rm{R}}(E,V)G^{\dag}(E,V)],
\end{equation}
where the spectral density $\Gamma_{\rm{L}\rm{(R)}}$ describes the coupling between the
L (R) electrode and the EM region and it is given by the imaginary part of the electrode
self-energy: $\Gamma_{\rm L(R)}(E,V)=i(\Sigma_{\rm L(R)}-\Sigma_{\rm L(R)}^{*})/2$.
The self-energy describes the hopping across the surface separating one lead and the EM
region, and establishes the appropriate boundary conditions for the Green's function
calculation.
$G$ is the retarded Green's function of the EM region, formally given by:
\begin{equation}
G(E,V)=\left[ES-H(V)-\Sigma_{\rm{L}}(E,V)-\Sigma_{\rm{R}}(E,V)\right]^{-1},
\end{equation}
where $S$ is the overlap matrix and $H$ is the Hamiltonian of the system when a bias
voltage $V$ is applied.
The current is simply given by the following integral:
\begin{equation}
I(V)=\frac{2e^{2}}{h}\int_{-\infty}^{\infty}\mathrm{d}E\ T(E,V) \left[ f(E-\mu_{\rm{L}})-
f(E-\mu_{\rm{R}})\right],
\end{equation}
with $f$ being the occupation Fermi function, $\mu_{\rm{L}\rm{(R)}}$ the chemical
potential of the L (R) electrode, and $V=\mu_{\rm{L}}-\mu_{\rm{R}}$.
\section{Results and discussion}
\begin{figure}
\includegraphics[width=0.48\textwidth]{Fig2.eps}
\caption{\label{fig:dos-te
Electronic and transport properties of 4L 3-aGNR.
Upper panel (a): DOS (solid line) of the junction and PDOS (shaded grey area) on the ribbon
region.
Red triangles represent the eigenstates of the isolated (unconnected and hydrogenated)
4L 3-aGNR linker.
Lower panel (b): transmission function.}
\end{figure}
We first investigate in detail the electronic properties and the transmission function of the 4L
3-aGNR junction.
For the other systems, similar considerations can be done.
In \ref{fig:dos-te} we show the density of states (DOS) in the EM region and the projected
density of states (PDOS) on the ribbon region for the 4L 3-aGNR, in comparison with the
transmission function.
Around the Fermi energy ($E_{\mathrm F}$), the DOS resembles that of graphene, in fact
it goes to zero almost linearly with a deviation due to the presence of vacuum portions in
the molecular bridge region.
The signature of the molecular energy levels is clear both in the DOS and PDOS, and the
transmission function has higher intensity in correspondence to those peaks.
We also consider the isolated molecule obtained by cutting the bonds between the 4L
3-aGNR and the graphene leads, and saturating them with hydrogen atoms.
The red triangles represent the eigenvalues of the isolated molecule, which correlate well
with the position of the resonances in the DOS, PDOS, and transmission.
The (P)DOS for the other studied systems have a similar behavior, and are not reported here.
\begin{figure}
\includegraphics[width=0.48\textwidth]{Fig3.eps}
\caption{\label{fig:autofunz
4L 3-aGNR junction: isosurface plot of the wavefunctions at energies corresponding to the
LUMO and HOMO resonant transmission peaks, calculated at $k_\parallel=0$.}
\end{figure}
\begin{figure}
\includegraphics[width=0.4\textwidth,angle=-90]{Fig4.eps}
\caption{\label{fig:3aGNR-te
Transmission function of 3-aGNR junctions for three different lengths: 4L (red), 6L (green),
and 8L (blue).
The dotted line is the transmission function of the pristine infinite 3-aGNR.}
\end{figure}
Moving away from $E_{\mathrm F}$, the transmission function increses up to significant
values in correspondence to the two peaks at $-$1.2 and 1~eV; these peaks are
generated by the hybridization of the molecule's highest occupied molecular orbital (HOMO)
and the lowest unoccupied molecular orbital (LUMO) states with the leads, playing the role
of channels between the two graphene electrodes.
The corresponding wavefunctions of the interacting system (shown in \ref{fig:autofunz})
calculated at $k_\parallel=0$, have the same shape and symmetry of that of the isolated
molecule (see \ref{fig:wfiso}).
These wavefunctions represent good conducting states, as they are delocalized along the
molecule and propagate inside the electrodes with the same symmetry.
Some other states farther from $E_{\mathrm F}$ contribute to the transmission.
All these states are generated from the $\pi$ orbitals of the C atoms and they are delocalized
along the junction.
We now discuss the dependence with the length of the ribbon forming the junction.
The transmission functions for the 4L, 6L, and 8L 3-aGNR junctions, reported in
\ref{fig:3aGNR-te}, show some general trends.
The peaks become denser when the length of the junction increases, because they are
related to the discrete structure of the electronic states of the nanostructured ribbon.
Thus, the effective gap of the junction decreases as the length increases.
As expected, the transmission function of the pristine infinite 3-aGNR acts like an envelope
curve for the other curves; it has an energy gap of $\simeq$1.5~eV.\cite{Loui06}
This aspect confirms that the contacts between the graphene sheets and the nanoribbon do
not represent significant barriers to the transport of electrons, as one can naively expect due
to the chemical identity of the different subsystems.
Within the energy gap of the nanoribbon, the transmission function is very low and decreases
rapidly as the length of the ribbon increases.
This behavior is clearly due to the tunneling mechanism dominating the transmission at those
energies.
The states of the graphene lead decay exponentially along the ribbon because there are no
states in the junction supporting the conductance.
This metal-semiconductor-metal device presents an effective gap for transport, since the
conductance inside the gap is several orders of magnitude lower than outside.
\begin{figure}
\includegraphics[width=0.4\textwidth,angle=-90]{Fig5.eps}
\caption{\label{fig:width}Transmission function of 4L $N$-aGNR junctions for $N=3$, 5, and 7.
Dotted line: transmission function of the infinite pristine $N$-aGNR.}
\end{figure}
We now discuss the effects of the aGNR width in the transmission function, considering
junctions made with 4L 3-aGNR, 4L 5-aGNR, and 4L 7-aGNR.
We consider a representative aGNR for each of the three classes,
so that we can capture the main differences among them.
As illustrated in \ref{fig:width}, the aGNR junctions have different gaps in the transmission
spectrum consistently with the previously reported results.\cite{Loui06}
In miniaturized graphene-based electronic devices these structures may act both as linkers and
as active components, and it would be useful to finely tune their width in order to have different
conductive behaviors.
Within each class, we expect the energy gap to decrease as the width of the junction increases.
We notice that the intensity of the transmission function is higher for wider ribbons, as more
electronic channels are open.
Moreover, as the width of the ribbon increases the spike features of the transmission function
become less marked, since the system approaches the limit of graphene.
In this case, the transmission functions of the pristine aGNR is a good reference only for those
junctions where the aspect ratio between the length and the width is reasonably high.
This because the contribution to the conductance of the infinite aGNR comes from
$k_\parallel=0$ only.
For finite nanoribbons this restriction is relaxed and the passage of electrons with a finite
$k_\parallel$ is allowed, with $k_\parallel$ within an interval that increases as the width to length
ratio increases.
This is clearly reflected in the transmission function, which departs from an exponential decay
within the energy gap for short and wide ribbons.
This is more evident for the 4L 7-aGNR in \ref{fig:width}.
For this ribbon the conductance is linear in the gap region, reflecting the underlying electronic
structure of the graphene electrodes, and departing from a strictly tunneling behavior dominated
by the energy dependence of the effective tunneling barrier.
We now discuss the effects of the application of a bias to the junction.
We consider the junction with the 8L 3-aGNR as a representative case; this is the longest
junction considered here, thus being that in which the edges of the two graphene leads
are less interacting.
\begin{figure}
\includegraphics[width=0.48\textwidth]{Fig6.eps}
\caption{\label{fig:bias} Lower panel: current-voltage characteristics of 8L 3-aGNR junction.
Upper panels: transmission function of 8L 3-aGNR junction for four selected values of the bias
voltage, as indicated in labels. The bias window is delimited by two vertical lines.}
\end{figure}
In \ref{fig:bias} the transmission functions for different biases up to 1.0~V are shown.
For any given bias voltage, the chemical potentials of the two leads are well recognizable,
as they correspond to the left and right Dirac points where the DOS tends to zero.
Within the bias window, the intensity of the transmission function increases with the bias.
This is expected since it is determined by the number of electron tunneling from the leads, and
the DOS of graphene shows a linear energy dependence in that region.
Thus, the shape of the transmission function within the gap region can be interpreted in terms
of the product of the DOS of the left an right graphene leads and a modulation function
determined by the size of the gap of the nanoribbons.
We notice two small peaks near the Dirac points of the two electrodes, which we assign to the
edge states appearing at the two zigzag
electrodes.\cite{Wassmann_Seitsonen_Saitta_Lazzeri_Mauri_2008}
In fact, in presence of zigzag edges there is a band near $E_{\mathrm F}$ which corresponds
to a state localized at the edge.
At small applied bias, the probability for an electron lying in an edge state to be transferred to
the opposite lead is very small since the DOS at that energy is negligible.
However, at finite bias, the situation is different and we can find distinct features in the
transmission curves related to the presence of edge states.
We can also see that the resonances originated by the HOMO and LUMO of the aGNR slightly
shift as the bias is applied.
As a result, the energy gap reduces by approximatively 10\% when a bias of 1.0~V is applied.
We verified that the current remains very low (less that 0.1~$\mu$A) for biases up to 1.0~V.
For larger values, the intensity sharply increases when the two main peaks enter the bias
window, and for a bias of 3.0~V we calculated a current of 3.3~$\mu$A.
When considering junctions made of 3-aGNR, the system is actually a chain of phenyl rings.
In this case, it has been shown that the stable configuration consists of neighbouring rings lying
on different planes.\cite{Lortscher_Elbing_Tschudy_VonHanisch_Weber_Mayor_Riel_2008,Venkataraman_Klare_Nuckolls_Hybertsen_Steigerwald_2006}
We can thus take into account one more degree of freedom, that is the relative torsion angle
between the phenyl rings of the linkers.
We show the effect of the ring rotation by taking the 4L 3-aGNR as test system.
According to our calculations, the most stable configuration corresponds to a rotation of
$\simeq 45^{\circ}$.\cite{Vergniory_Granadino-Roldan_Garcia-Lekue_Wang_2010}
In fact, the central ring tends to rotate with respect to its planar configuration due to steric
repulsion between its hydrogen atoms and those of the neighboring rings.
By looking at the transmission function in \ref{fig:transm-tors} we can note a general trend.
As the ring rotates from $0^{\circ}$ to $90^{\circ}$, the peak H (HOMO) shifts towards lower
energies, while the peak L (LUMO) shifts towards higher energies.
As a result, the energy gap of the system increases.
In fact, the wavefunctions related to these peaks (see \ref{fig:wfiso}) are extended along
all the molecular backbone and they feel a distortion as the central ring is rotated, thus their
energies are expected to change.
Both HOMO and LUMO are even respect to the mirror plane that bisects the 3-aGNR molecule.
As consequence of this symmetry, when the rotation is by $90^{\circ}$ these two peaks
disappear: the linear combinations of the ${p_z}$ orbitals of the central carbon atoms are
orthogonal to those of the remaining carbon atoms, resulting in the closure of the
corresponding conduction channels.
The situation is different for HOMO$-$1 and LUMO+1, whose wavefunctions are mainly
localized on the central phenyl ring.
In fact, they are at the same energy position regardless of the torsion angle.
These considerations may be extended to the case of longer aGNR.
\begin{figure}
\includegraphics[width=0.48\textwidth]{Fig7.eps}
\caption{\label{fig:transm-tors
Transmission function at zero bias for the 4L 3-aGNR junction, under rigid torsion of the central
phenyl ring.
The curves shown here refer to rotations by $0^{\circ}, 30^{\circ}, 60^{\circ}$, and $90^{\circ}$.
The main resonances are highlighted and the labels correspond to the 3-aGNR molecular
states shown in \ref{fig:wfiso}. }
\end{figure}
\begin{figure}
\includegraphics[angle=90,width=0.24\textwidth]{Fig8.eps}
\caption{\label{fig:wfiso
Isosurface plots of the selected molecular orbitals of the isolated 3-aGNR linker: HOMO (H),
HOMO-1 (H-1), LUMO (L) and LUMO+1 (L+1).}
\end{figure}
Concerning the passivation of the graphene edges, all the results presented so far refer to
what we can call ``standard saturation''.
That is, each edge carbon atom is saturated by a single hydrogen (panel $a$ in \ref{fig:sat}).
In a real system subject to different hydrogen partial pressures, one may guess that the edge
saturation can change.\cite{Wassmann_Seitsonen_Saitta_Lazzeri_Mauri_2008}
In order to validate our results, we show the effect of four different edge saturations on the
transmission function.
The first case we consider is the formation of pentagon-heptagon reconstruction of the
graphene edges, where no H atoms are present on the edge (panel $d$).
Then, we considered the case in which each carbon edge atom is saturated by two
hydrogens (panel $b$).
One may also guess a more complicated connection between the aGNR and the leads, and
we modeled it as shown in the inset of panel $c$.
\ref{fig:sat} shows the transmission function for these four cases.
Our results are very robust with respect to these changes of the geometry and saturation of
the graphene lead edges, as we see only minor differences among these cases, mainly a
sharpening of the peaks derived from the HOMO and LUMO resonances.
This is evident in particular for the double-H and shaped edges, in which the orbitals of the
aGNR are less hybridized with the leads.
The reason is that in both cases less charge is present on the graphene layer edge.
\begin{figure}
\includegraphics[width=0.48\textwidth]{Fig9.eps}
\caption{\label{fig:sat
Zero bias transmission function of 8L 3-aGNR junctions with four different saturations of the
graphene edges.
The inset show the linking geometries.}
\end{figure}
\section{Conclusions}
Very simple prototypes of carbon-based molecular junctions have been investigated in
order to characterize their transport properties.
Carbon nanoribbons with armchair shaped edges have been considered as linkers between
between two semiinfinite graphene electrodes.
These systems present a typical metal-semiconductor-metal behavior due to the electronic
gap of the ribbons which depends on the their width.
The electronic properties of the isolated subsystems, i.e. graphene electrodes and the
nanoribbon, together with the interaction in contact regions determine the transport properties
of the junction.
For what concerns the coupling between the subsystems, this is very efficient and the contacts
do not create appreciable barrier for transport due to the same chemical species constituting
the subsystems.
Hence the transport properties are mainly determined by the shape (width and length) of the
finite ribbon included in between the graphene leads.
Larger molecules furnish more channels for convoying the electrons then the smaller ones and
consequently the conductance is generally higher.
In the energy gap region the transport occurs via electron tunneling between the electrodes
and the efficiency decays simply with the linker length.
Some additional representative configurations has been considered explicitly: thinest ribbon
a torqued geometries has been studied to evaluate the effects of the misalignment of the
$\pi$ orbitals on different phenyl rings.
The changes in transmission through HOMO and LUMO orbitals are noteworthy, while for energy
levels farther from $E_{\textrm F}$ they are negligible.
Finally, graphene edges with different saturations and reconstructions has been considered and
they do not influence significantly the calculated transport properties.
In spite of the simple model adopted in this paper, the reported results on the electronic transport
provide some parameters to control and engineer future carbon-based electronic devices.
\section{Aknowledgments}
C.M. thanks CARIPLO Foundation for its support within the
PCAM European Doctoral Programme.
DSP acknowledges support fromt from Basque Departamento de Educaci\'on,
UPV/EHU (Grant No. IT-366-07), the Spanish Ministerio de
Ciencia e Innovaci\'on (Grant No. FIS2010-19609-C02-02), and
the ETORTEK research program funded by the Basque Departamento
de Industria and the Diputaci\'on Foral de Gipuzkoa.
|
{
"timestamp": "2012-03-12T01:01:56",
"yymm": "1203",
"arxiv_id": "1203.2111",
"language": "en",
"url": "https://arxiv.org/abs/1203.2111"
}
|
\section{Introduction} \label{sect:introduction}
Let $(M^{2n+1},\xi)$ be a cooriented contact manifold with associated contact form
$\alpha$, i.e. $\xi=\ker\alpha$. This structure determines a symplectic distribution
$(\xi,d\alpha_{|\xi}) \subset TM$. Any change of the associated contact form $\alpha$ does
not change the conformal symplectic class of $d \alpha$ restricted to $\xi$.
This allows us to choose a compatible almost complex structure $J\in End(\xi).$
Thus given a cooriented contact structure we obtain in a natural way a reduction of
the structure group $Gl(2n+1,\mathbb{R})$ of the tangent bundle $TM$ to the group $U(n)\times \{1\}$,
which is unique up to homotopy. A manifold $M$ is said to be an \emph{almost contact
manifold} if the structure group of its tangent bundle can be reduced to $U(n)
\times \{ 1 \}$. In particular, cooriented contact manifolds are
almost contact manifolds and such a reduction of the structure group of the
tangent bundle of a manifold $M$ is a necessary condition for the existence of a
cooriented contact structure on $M$. It is unknown whether this condition is in general
sufficient.\\
\noindent Nevertheless there are cases in which the existence of an almost contact structure is sufficient for the manifold to admit a contact structure. For example, if the manifold $M$ is open then one can apply Gromov's
$h$--principle techniques to conclude that the condition is sufficient. See the result 10.3.2 in \cite{EM}. The scenario is quite different for closed almost contact manifolds. Using results of Lutz \cite{Lu} and Martinet \cite{Ma} one can show that every cooriented tangent $2$--plane field on a closed oriented $3$--manifold is
homotopic to a contact structure. A good account of this result from a modern perspective is given in~\cite{Ge}. For manifolds of higher dimensions there are various results
establishing the sufficiency of the condition. Important instances of these are the construction of contact structures on certain principal $\mathbb{S}^1$--bundles over closed symplectic manifolds due to Boothby and
Wang~\cite{BW}, the existence of a contact structure on the product of a contact manifold with a
surface of genus greater than zero following Bourgeois~\cite{Bo} and the existence of contact
structures on simply connected $5$--dimensional closed orientable manifolds obtained by Geiges
\cite{Ge1} and its higher dimensional analogue ~\cite{Ge2}.\\
\noindent The work in this article was presented in the Spring 2012 AIM Workshop on higher dimensional contact geometry. In its course, J. Etnyre commented on a possible alternative approach in the framework of Giroux's program using an open book decomposition. The argument has been subsequently written and it is the content of the article \cite{Et}.\\
\noindent Let us turn our attention to $5$--manifolds since the main goal of this article is to show that any orientable almost contact $5$--manifold is contact. In this case H. Geiges has been studying existence results in other situations apart from the simply connected one. In \cite{GT1} a positive result is also given for spin closed manifolds with $\pi_1=\mathbb{Z}_2$, and spin closed manifolds with finite fundamental group of odd order are studied in ~\cite{GT2}. On the other hand there is also a construction of contact structures on an orientable $5$--manifold occurring as a product of two lower dimensional manifolds by Geiges and Stipsicz~\cite{GS}. While Geiges used the topological classification of simply connected manifolds for his results in ~\cite{Ge1}, one of the ingredients in \cite{GS} is a decomposition result of a $4$--manifold into two Stein manifolds with common contact boundary ~\cite{AM}, ~\cite{Bk}.\\
Being an almost contact manifold is a purely topological condition. In fact, the reduction of the structure group can be studied via obstruction theory. For example, in the $5$--dimensional situation a manifold $M$ is almost contact if and only if the third integral Steifel--Whitney class $W_3(M)$ vanishes. Actually, using this hypothesis and the classification of simply connected manifolds due to D. Barden ~\cite{Ba}, H. Geiges deduces that any manifold with $W_3(M)=0$ can be obtained by Legendrian surgery from certain model contact manifolds. Though this approach is elegant, it seems quite difficult to extend these ideas to produce contact structures on any almost contact $5$--manifold. We therefore propose a different approach: the existence of an \emph{almost contact pencil} structure on the given almost contact manifold is the required topological property to produce a contact structure. The tools appearing in our proof use techniques from three different sources:\\
\begin{enumerate}
\item The approximately holomorphic techniques developed by Donaldson in the
symplectic setting \cite{Do,Do1} and adapted in \cite{IMP} to the contact
setting to produce the so--called \emph{quasi contact pencil}.\\
\item The generalization of the notion of overtwistedness to higher dimensions done
by Niederkr\"uger and Gromov \cite{Ni} and the generalized Lutz twist based on that defined in
\cite{EP}.\\
\item Eliashberg's classification of overtwisted $3$--dimensional manifolds \cite{El} to produce overtwisted contact structures on the fibres of the pencil.\\
\end{enumerate}
\noindent Let us state the main result.
\begin{theorem} \label{main}
Let $M$ be a closed oriented $5$--dimensional manifold. There exists a contact structure in every homotopy class of almost contact structures.
\end{theorem}
\noindent In particular closed oriented almost contact $5$--manifolds are contact. It is important to emphasize that using the techniques developed in this article, it is not possible to conclude anything about the number of distinct contact distributions that may occur in a given homotopy class of almost contact distributions. The result states that there is at least one. From the construction it will be clear that the contact structure contains a PS--structure \cite{Ni,NP} and therefore it is non--fillable. The PS--structure is used as a substitute of the overtwisted disk to introduce flexibility as Eliashberg does in the $3$--dimensional case.
\begin{remark}
The distributions are supposed to be coorientable along the course of the article. Section \ref{sec:non-coorientable} contains the corresponding results for non--coorientable distributions.
\end{remark}
\noindent The proof of Theorem \ref{main} consists of a constructive argument in which we obtain the contact condition step by step. These steps correspond to the sections of the paper as follows:\\
\begin{itemize}
\item[-] To begin with, we show that any almost contact $5$--manifold $(M, \xi)$ admits an almost contact fibration over $\mathbb{S}^2$ with singularities of some standard type. It is defined on the complement of a link. The construction of this almost contact fibration -- in fact, an almost contact pencil -- is the content of Sections \ref{sect:quasi} and \ref{sec:pencils}.\\
\item[-] In Section \ref{sect:defor_local}, we produce a first deformation of the almost contact structure $\xi$ to obtain a contact structure in some neighborhood of the singularities of the fibration and a neighborhood of the link.\\
\item[-] One of the types of singularities has the structure of a base locus of a pencil occurring in algebraic or symplectic geometry. In order to provide a Lefschetz type fibration we blow--up the base locus. This requires the notion of a contact blow--up. For the purposes of the article, it will be enough to define the contact blow--up of a contact $5$--manifold along a transverse $\mathbb{S}^1$ and the corresponding notion of contact blow--down. This is the content of Section \ref{sect:blowup}.\\
\item[-] Away from the critical points the distribution splits as $\xi= \xi_v \oplus {\mathcal{H}}$, where $\xi_v$ is the restriction of the distribution to the fibres and ${\mathcal{H}}$ is the symplectic orthogonal. Section \ref{sec:vertical} deals with a deformation of $\xi_v$ to produce a contact structure in the fibres. It strongly uses the construction of overtwisted contact manifolds due to Eliashberg \cite{El}.\\
\item[-] In Section \ref{sect:skeleton} we begin to deform the horizontal direction ${\mathcal{H}}$. This is done in two steps. Given a suitable cell decomposition of the base $\mathbb{S}^2$, we deform ${\mathcal{H}}$ in the pre--image of a neighborhood of the $1$--skeleton. Section \ref{sect:skeleton} contains this first step.\\
\item[-] The contact condition still has to be achieved in the pre--image of the $2$--cells. This is the second step. The contact structure used in order to fill the pre--image of the $2$--cells is carefully constructed in Section \ref{sec:bands}.\\
\item[-] In Section \ref{sec:end} we gather the results in the previous sections and construct the contact structure. Theorem \ref{main} is concluded.\\
\item[-] In Section \ref{sec:non-coorientable} we deal with the case of non--coorientable distributions. We introduce the suitable definitions and explain the non--coorientable version of Theorem \ref{main}.\\
\end{itemize}
\noindent It is reasonable to guess after a careful reading of this article the ingredients needed to adapt the proofs in order to work in the $7$--dimensional case. We have begun to understand this $7$--dimensional setup and it will be the goal of a forthcoming article. E. Giroux has work in progress in which he tries to prove the existence result by using an open book decomposition of the manifold \cite{Gi}.\\
\noindent{\bf Acknowledgements.} The authors are grateful to Y. Eliashberg, J. Etnyre, E. Giroux and H. Geiges for valuable conversations. The proof of Theorem \ref{thm:dishant} was outlined to us by Y. Eliashberg. The original work lacked the construction of the homotopy in the case that $2$--torsion existed in $H^2(M,\mathbb{Z})$. This case was proven after a useful discussion with J. Etnyre at the AIM Workshop.
\section{Preliminaries.} \label{sect:quasi}
\subsection{Quasi--contact structures} Let $M$ be an almost contact manifold. There exists a
choice of a symplectic distribution $(\xi, \omega) \subset TM$ for such a
manifold. Namely, we can
find a $2$--form $\eta$ on $\xi$ with the property that $\eta$ is
non--degenerate and
compatible with the almost complex structure $J$ defined on $\xi.$ By extending
$\eta$ to a form on $M$ we can find a $2$--form
$\omega$ on $M$ such that $(\xi, \omega_{|\xi})$ becomes a symplectic
vector bundle. This form $\omega$ is not necessarily closed. The triple $(M, \xi, \omega)$ is also said to be an almost contact manifold. In other words, by an almost contact structure is meant to be a triple $(\xi,J,\omega)$ for some $\omega$ as discussed. The choice of almost complex structure $J$ is homotopically unique and it might be omitted. This article only concerns codimension--$1$ distributions on $M$ which are almost complex. An almost contact manifold is subsequently described by a triple $(M,\xi,\omega)$.\\
\noindent In order to construct a contact structure out of an almost contact one, the
first step is to provide a better $2$--form on $M.$ That is, we replace $\omega$ by a closed
$2$--form.
\begin{definition}
A manifold $M^{2n+1}$ admits a {\it quasi--contact structure} if there exists
a pair $(\xi, \omega)$ such that $\xi$ is a codimension $1$--distribution and
$\omega$ is a closed $2$--form
on $M$ which is non--degenerate when restricted to $\xi.$
\end{definition}
\noindent Notice that a quasi--contact pair $(\xi,\omega)$ admits a compatible almost contact
structure, i.e. there exist a $J$ which makes $(\xi,J,\omega)$ into an almost contact
structure. These manifolds have also been called {\it
$2$--calibrated}~\cite{IbortDMT} in the literature. The following lemma justifies the appearance of the previous definition:
\begin{lemma} \label{lem:almost-quasi}
Every almost contact manifold $(M, \xi_0, \omega_0)$ admits a quasi--contact
structure $(\xi_1, \omega_1)$ homotopic to $(\xi_0, \omega_0)$ through
symplectic distributions and the class $[\omega_1]$ can be fixed to be any
prescribed cohomology class $a\in H^2(M, \mathbb{R})$.
\end{lemma}
\begin{proof}
Let $j: M \longrightarrow M \times \mathbb{R}$ be the inclusion as the zero section. We know
that we can
find a (not--necessarily closed) $2$--form $\widetilde{\omega}_0$, such that
$\omega_0= j^* \widetilde{\omega}_0$. Fix a Riemannian metric $g$ over $M$ such that
$\xi_0$ and $\ker \omega_0$ are $g$--orthogonal.\\
\noindent Apply Gromov's classification result of open symplectic manifolds
to produce a $1$--parametric family $\{ \widetilde{\omega}_t \}_{t=0}^1$ of
{\it symplectic} forms such that for $t=1$ the form is closed. See \cite{EM}, Corollary 10.2.2. Let $\pi: M \times \mathbb{R} \longrightarrow M$ be the projection and choose the cohomology class defined by $\widetilde{\omega}_1$ to be $\pi^* a$. Consider the family of $2$--forms $\omega_t= j^* \widetilde{\omega}_t$ on $M.$
Since $\widetilde{\omega_t}$ is non--degenerate on $M \times \mathbb{R}$ for each $t$,
the form $\omega_t$ has $1$--dimensional kernel $\ker \omega_t$. Define $\xi_t=
(\ker \omega_t)^{\perp g}$. Then $(\xi_t, \omega_t)$ provides the required
family.
\end{proof}
This is the farthest one can reach by the standard $h$--principle argument in
order to find contact structures on a closed manifold. One can start
with the almost contact bundle $\xi= \ker \alpha$ and find a $2$--form $d\beta$
that makes it symplectic by Lemma \ref{lem:almost-quasi}, but there is in
general no way to relate $\alpha$ and $\beta$. This is the aim of the article.
\subsection{Obstruction theory} The content of Theorem \ref{main} has two parts. The statement implies the existence of a contact structure in an almost contact manifold. This is a result in itself, regardless of the homotopy type of the resulting almost contact distribution. The construction we provide in this article also concludes that the obtained contact distribution lies in the same homotopy class of almost contact distributions as the original almost contact structure. This is achieved via the study of an obstruction class. Let us review some well--known facts. \\
\noindent Let $M$ be a smooth oriented $5$--manifold and $\pi:TM\longrightarrow M$ its tangent bundle. The projection $\pi$ is considered to be an $SO(5)$--principal frame bundle. An almost contact structure is a reduction of the structure group $G=SO(5)$ to a subgroup $H\cong U(2)\times\{1\}\cong U(2)$. The isomorphism classes of almost contact structures are parametrized by the homotopy classes of such reductions. A reduction of the structure group $G$ to a subgroup $H$ is tantamount to a section of a $G/H$--bundle over $M$. Hence the classification of almost contact structures on $M$ is reduced to the study of homotopy classes of sections of a $SO(5)/U(2)$--bundle over $M$.
\begin{lemma}
$SO(5)/U(2)\cong\mathbb{CP}^3$.
\end{lemma}
\begin{proof}
First, note that $SO(4)/U(2)\cong\mathbb{S}^2$ is given by the choice of a unitary vector in real $3$--space. The inclusion $\mathbb{S}^2\longrightarrow SO(5)/U(2)$ has homotopy cofibre $\mathbb{S}^4$ and thus we obtain a fibration $SO(5)/U(2)\longrightarrow\mathbb{S}^4$. Second, there exists an $\mathbb{S}^1$--action on the total space of the Hopf fibration $\mathbb{S}^3\longrightarrow\mathbb{S}^7\longrightarrow\mathbb{S}^4$ that restricts to the Hopf action on the fibres $\mathbb{S}^3$. Hence, the orbit space $\mathbb{CP}^3$ fibres over $\mathbb{S}^4$ with fibres $\mathbb{S}^2$. The classifying maps of both fibrations coincide as elements in $\pi_3(SO(3))$. In particular, we obtain the diffeomorphism $\mathbb{CP}^3\cong SO(5)/U(2)$.
\end{proof}
\noindent The homotopy groups $\pi_i(\mathbb{CP}^3)=0$ for $1\leq i\leq6$, $i\neq2$, hence the existence of sections of a fibre bundle with typical fibre $\mathbb{CP}^3$ over the $5$--manifold $M$ is controlled by the primary obstruction class $d=W_3(M)\in H^3(M,\pi_2(\mathbb{CP}^3))\cong H^3(M,\mathbb{Z})$. The hypothesis of Theorem \ref{main} is $d=0$.\\
\noindent Let $s_\xi$ and $s_{\xi'}$ be two sections of this $\mathbb{CP}^3$--bundle. The obstruction class dictating the existence (or the lack thereof) of a homotopy between them lies is the primary obstruction $d(\xi,\xi')\in H^2(M,\mathbb{Z})$. The obstruction theory argument can be made relative to a submanifold $A\subset M$. This implies the following
\begin{lemma}\label{lem:2sk}
Let $s_\xi$ and $s_{\xi'}$ be two sections of a $\mathbb{CP}^3$--bundle over $M$ that are homotopic over a $2$--skeleton of the pair $(M,A)$. Then $s_\xi$ and $s_{\xi'}$ are also homotopic over $(M,A)$.
\end{lemma}
\noindent Let $(M,\xi)$ be an almost contact structure, the construction of the contact structure $\xi'$ obtained in Theorem \ref{main} does not modify the homotopy class of the given section, i.e. $s_\xi\sim s_{\xi'}$. This is readily seen at the stages preceeding Section \ref{sec:bands}. In Section \ref{sec:bands} we provide a detailed account on the modification of the obstruction class $d(\xi,\xi')$ in the $2$--skeleton of certain pieces of $M$ where $\xi'$ has been constructed. This is enough to conclude that $d(\xi,\xi')=0$ once $\xi'$ is extended to $M$ in Section \ref{sec:end}.
\subsection{Homotopy of vector bundles} The argument constructing the homotopy between the initial almost contact structure and the resulting contact distribution in Theorem \ref{main} uses the following lemma. It is implicitly used in several parts of Sections \ref{sect:defor_local} to \ref{sec:end}.\\
\noindent Let $(V,\omega)$ be an oriented vector space of dimension $\dim_\mathbb{R} V=4$. Consider an splitting $V=V_0\oplus V_1$ with $V_0,V_1$ two oriented $2$--dimensional vector subspaces. Since $Sp(2,\mathbb{R})/SO(2)$ is contractible, the space of symplectic structures on $V$ such that $V_0$ and $V_1$ are symplectic orthogonal subspaces is contractible. This essentially implies the following
\begin{lemma}\label{lem:split}
Let $M$ be an almost contact $5$--manifold, $A$ an open submanifold of $M$, and $(\xi_0,\omega_0),(\xi_1,\omega_1)$ two almost contact structures on $M$ such that there exists a homotopy $\{\xi_t\}$ of oriented distributions on $(M,A)$ connecting $\xi_0$ and $\xi_1$. Suppose that there exist $L_0$ and $L_1$ two rank--$2$ symplectic subbundles of $\xi_0$ and $\xi_1$ and a homotopy $\{L_t\}\subset\{\xi_t\}$ of oriented distributions connecting $L_0$ and $L_1$ on $(M,A)$. Then $\{\xi_t\}$ is a homotopy of symplectic distributions connecting $\xi_0$ and $\xi_1$ on $(M,A)$.
\end{lemma}
\begin{proof}
Consider $J_0$ and $J_1$ two compatible complex structures on the symplectic distributions $\xi_0$ and $\xi_1$ respectively. These define two fibrewise scalar--product structures
$$g_0=\omega_0(\cdot,J_0\cdot)\mbox{ and }g_1=\omega_1(\cdot,J_1\cdot)$$
on $\xi_0$ and $\xi_1$. The space of fibrewise scalar--product structures has contractible fiber, namely $Gl^+(4,\mathbb{R})/SO(4)$, and thus is contractible. Hence, there exists a homotopy $\{g_t\}$ of fibrewise scalar--products connecting $g_0$ and $g_1$. The scalar--product $g_t$ provides an orthogonal decomposition $\xi_t=L_t\oplus L_t^{\perp_{g_t}}$. The homotopy of oriented bundles $\{L_t\}$ induces a homotopy of oriented bundles $\{L_t^{\perp_{g_t}}\}$ respecting the symplectic splitting given by $\omega_0$ and $\omega_1$ on $\xi_0$ and $\xi_1$.
\end{proof}
\subsection{Notation.} Let $\mathbb{R}^{2n}$ be Euclidean space, $B^{2n}(r)=\{p\in\mathbb{R}^{2n}:\|p\|\leq r\}$ denotes the ball of radius $r$ centered at the origin. The $2$--dimensional balls are also referred to as disks and denoted by $\mathbb{D}^2(r)$. In case the radius is omitted $B^{2n}$ and $\mathbb{D}^2$ denote the ball and disk of radius $1$ respectively.
\section{Quasi--contact pencils.} \label{sec:pencils}
Approximately holomorphic techniques have been extremely useful in symplectic
geometry. Their main application in contact geometry -- due to E. Giroux -- is
to establish the existence of a compatible open book for a contact manifold in
higher dimensions. An open book decomposition is a way of trivializing a
contact manifold by fibering it over $\mathbb{S}^1$. Such objects have also
been studied in the almost contact case, see ~\cite{DMTMPresas}.\\
\noindent There exists a construction \cite{Fran1} in the contact case analogous to the Lefschetz pencil decomposition introduced by Donaldson over a symplectic manifold \cite{Do1}. It is called a contact pencil and it allows us to express a contact manifold as a singular fibration over $\mathbb{S}^2$. It has been extended in \cite{IbortDMT2} to the quasi--contact setting. Theorem \ref{thm:exist_pencil} and Corollary \ref{coro:starting_pencil} in this Section show the existence of a quasi--contact pencil with suitable properties. Let us begin with the appropriate definitions.
\begin{definition}
An almost contact submanifold of an almost contact manifold $(M, \xi, \omega)$ is an embedded
submanifold $j: S \longrightarrow M$ such that the induced pair $(j^* \xi, j^* \omega)$ is an almost contact structure on $S$.
\end{definition}
\noindent A quasi--contact submanifold of a quasi--contact manifold is defined analogously. In particular this implies in both cases that the submanifold $S$ is transverse to the distribution $\xi$.\\
\noindent A chart $\phi :(U,p) \longrightarrow V \subset (\mathbb{C}^n \times \mathbb{R}, 0)$ of an atlas of $M$ is compatible with the almost
contact structure $(\xi,\omega)$ at a point $p \in U \subset M$ if the push--forward at $p$ of $\xi_p$ is $\mathbb{C}^n \times \{ 0\}$ and the $2$--form $\phi_*\omega(p)$ is a positive $(1,1)$--form.
\begin{definition} \label{def:almost_pencil}
An almost contact pencil on a closed almost contact manifold $(M^{2n+1}, \xi,
\omega)$ is a triple $(f,B,C)$ consisting of a codimension--$4$ almost contact
submanifold $B$, called the base locus, a finite set $C$ of smooth
transverse curves and a map $f:M\backslash B\longrightarrow \mathbb{C}\mathbb{P}^1$
conforming the following conditions:
\begin{itemize}
\item[(1)] The set $f(C)$ contains locally smooth curves with transverse
self--intersections and the map $f$ is a submersion on the complement of $C$.
\item[(2)] Each $b\in B$ has a compatible chart to $(\mathbb{C}^n \times
\mathbb{R},0)$ under which $B$ is locally cut out by $\{z_1=z_2=0\}$ and $f$ corresponds
to the projectivization of the first two coordinates, i.e. locally $\displaystyle f(z_1, \ldots, z_n,
t)= \frac{z_2}{z_1}$.
\item[(3)] At a critical point $p\in \gamma\subset M$ there exists a compatible chart $\phi_P$ such that $$(f\circ \phi_P^{-1})
(z_1,\ldots,z_n,s)=f(p)+z_1^2+\ldots+z_n^2+g(s)$$ where $g:(\mathbb{R},0)\longrightarrow (\mathbb{C},0)$ is
a submersion at the origin.
\item[(4)] The fibres $f^{-1}(p)$, for any $p\in \mathbb{CP}^1$, are almost contact
submanifolds at the regular points.
\end{itemize}
\end{definition}
The local models have to be provided through compatible charts in
the sense above. That is to say the distribution in $\xi_p$ is mapped to the horizontal
distribution $\mathbb{C}^n\times\{0\}$ and $\omega$ is a positive $(1,1)$--form when
restricted to the horizontal distribution with respect to its canonical almost
complex structure.
\begin{remark}
Quasi--contact pencils for quasi--contact manifolds and
contact pencils for contact manifolds are defined by replacing the expression almost
contact by the suitable one in each case.
\end{remark}
The generic fibres of $f$ are open almost contact submanifolds and the closures
of the fibres at the base locus are smooth. This is because the local model
$(2)$ in the Definition \ref{def:almost_pencil} is a parametrized elliptic
singularity and the fibres come in complex lines ($\displaystyle z_2=const\cdot z_1$)
joining at the origin. We refer to the compactified
fibres so constructed as the fibres of the pencil. See Figure \ref{fig:base_points}.\\
\begin{figure}[ht]
\includegraphics[scale=0.6]{elliptic.pdf}
\caption{fibres close to $B$}\label{fig:base_points}
\end{figure}
Notice that the set of critical values $\Delta= f(C)$ are no longer points, as
in the symplectic case, but immersed curves. This is because of Condition $(3)$
in the Definition \ref{def:almost_pencil}. In particular, the usual isotopy
argument between two fibres does not apply unless their images are in the same
connected component of $\mathbb{CP}^1\backslash \Delta$. This has been studied in the contact and quasi--contact cases. The set $C$ is a positive link and therefore $\Delta$ is also oriented. There is a partial order in
the complementary of $\Delta$: a connected component $P_0$ is less or equal than
a connected component $P_1$ if $P_0$ and $P_1$ can be connected by an oriented path $\gamma \subset \mathbb{CP}^1$ intersecting $\Delta$ only with positive crossings. The proposition that follows has only been proved for the contact and quasi--contact cases. An analogous statement probably remains true in the almost contact setting. It is provided to offer some geometric insight about contact and quasi--contact pencils, it is not used in the rest of the article.
\begin{proposition} [Proposition $6.1$ of ~\cite{Fran1}] \label{prop:crossing}
Let $M$ be a quasi--contact manifold equipped with a quasi--contact pencil $(f, B,
C)$. Then if two regular values of $f$, $P_0$ and $P_1$, are separated by a unique
curve of $\Delta$ then the two corresponding fibres $F_0=\overline{f^{-1}(P_0)}$
and $F_1=\overline{f^{-1}(P_1)}$ are related by an index $n-1$ surgery.\\
\noindent Suppose that the manifold and the pencil are contact, then the surgery is a
Legendrian one and it attaches a Legendrian sphere to $F_0$ if $P_0$ is
smaller than $P_1$. See Figure \ref{fig:crossing}.
\end{proposition}
\begin{figure}[ht]
\includegraphics[scale=0.6]{crossing.pdf}
\caption{The drawn orientations make $F_1=\overline{f^{-1}(P_1)}$ a Legendrian
surgery of $F_0=\overline{f^{-1}(P_0)}$.}\label{fig:crossing}
\end{figure}
\noindent In the contact case it implies that the crossing of a singular curve in
the fibration amounts to a directed Weinstein cobordism. In the quasi--contact
case no such orientation appears. For instance, the case in which the
quasi--contact distribution is a foliation --in dimension $3$ this is a taut foliation-- becomes absolutely symmetric and there is no difference in crossing one way or the other.\\
\noindent The main existence result (\cite{IbortDMT2,Fran1}) can be stated as
\begin{theorem} \label{thm:exist_pencil}
Let $(M, \xi, \omega)$ be a quasi--contact manifold. Given an integral cohomology class $a\in
H^2(M, \mathbb{Z})$, there exists a quasi--contact pencil $(f, B, C)$ such that the
fibres are Poincar\' e dual to the class $a+k[\omega]$, for some $k\in \mathbb{N}$.
\end{theorem}
The proof of this result does not work in the almost contact setting. In order to construct the pencil, the approximately holomorphic techniques are essential and
for them to work we need the closedness of the $2$--form $\omega$. In general, a quasi--contact pencil may have empty base locus. Nevertheless a pencil obtained through approximately holomorphic sections on a higher dimensional manifold does not.\\
\noindent In case the form $\omega$ of the quasi--contact structure is exact -- then called an exact quasi-contact structure -- we obtain the following
\begin{corollary} \label{coro:starting_pencil}
Let $(M,\xi,\omega)$ be an exact quasi--contact closed manifold. Then it admits a
quasi--contact pencil such that any smooth fibre $F$ satisfies
$c_1(\xi_F)=0$. Further, the base locus $B$ is non--empty if $\dim M$
is greater than $3$.
\end{corollary}
\begin{proof}
We use Theorem \ref{thm:exist_pencil} to construct a pencil such that the cohomology class of the Poincar\'e dual to the smooth fibres equals the first Chern class $c_1(\xi)$ of the
complex bundle $\xi$. This follows from the fact that for any $a \in H^2(M,\mathbb{Z})$ there is
a quasi--contact pencil with any smooth fibre Poincar\'e dual to the
the class $a + k [\omega]$. Since $\omega$ is exact
we get the pencil with the property mentioned earlier by taking
$a = c_1(\xi).$\\
\noindent Since the distribution $\xi$ is transverse to the
fibre $F$, we have that
$$ \xi_{|F} = (\xi_{|F} \cap TF) \oplus \nu(F). $$
Recall that $c_1(\nu(F))=e(\nu(F))=PD([F])$ to obtain
$$ c_1(\xi_{|F}) = c_1(\xi_{|F} \cap TF) + c_1(\nu(F))= c_1(\xi_F) + a_{|F}, $$
and so
$$ a_{|F}= c_1(\xi_F) + a_{|F}. $$
This shows that the first Chern class of the quasi--contact structure in any
smooth fibre is zero.\\
\noindent As for the non--emptiness of the set $B$, just recall that by the general theory developed in \cite{IbortDMT2,IMP}, the submanifold satisfies a
Lefschetz hyperplane theorem. It implies that whenever the dimension
of $M$ is greater than $3$, the morphism
$$ H_0(B) \longrightarrow H_0(M) $$
is surjective. Hence we conclude that $B$ is not the empty set.
\end{proof}
\noindent The triviality of the Chern class of the quasi--contact structures on the fibres and the non--emptiness of $B$ are used in the construction of the contact structure.
\section{Base locus and Critical loops.} \label{sect:defor_local}
\noindent Let $(M,\xi,\omega)$ be an exact quasi--contact $5$--manifold and $(f,B,C)$ a quasi--contact pencil on it. Assume that $B\neq\emptyset$ and $c_1(\xi_F)=0$ for a regular fibre $F$ of $f$. Such a pencil is provided in Corollary \ref{coro:starting_pencil}. A fair amount of control on the almost--contact structure can be achieved in the neighborhood of the base locus and the critical loops.
\begin{definition}
A submanifold $i:S \longrightarrow M$
of an almost contact manifold $(M, \xi, \omega)$ is said to be contact if it is an almost contact submanifold and there is a choice of adapted form $\alpha$ for $\xi$ in a
neighborhood of $S$, i.e $\xi= \ker \alpha$, such that $i^* (d\alpha)= i^*
\omega$.
\end{definition}
\noindent Note that a transverse loop is a contact submanifold. An additional property in our almost contact pencil can then be required.
\begin{definition}
An almost contact pencil $(f,B,C)$ on $(M,\xi, \omega)$ is called good if there exist neighborhoods of $B$ and $C$ which are contact submanifolds of $(M,\xi,\omega)$.
\end{definition}
\noindent The following lemma provides a perturbation achieving a suitable almost contact pencil.
\begin{lemma} \label{coro:perturb_pencil}
Let $(M,\xi,\omega)$ be a quasi--contact closed $5$--dimensional manifold and let $(f,B,C)$ be a quasi--contact pencil. There
exists a $C^0$--small perturbation $\{(\xi_t, \omega)\}$ of almost contact
structures such that:
\begin{enumerate}
\item $(\xi_t,\omega)$ is an almost contact structure $\forall t\in[0,1]$, and $(\xi_0,\omega)=(\xi,\omega)$.
\item There exist neighborhoods of $B$ and $C$ that are contact submanifolds of $(\xi_1,\omega)$.
\item $(f,B,C)$ is an almost contact pencil for $(M,\xi_1,\omega)$.
\item $c_1((\xi_1)_F)=0$ for a regular fibre $F$ of $f$.
\end{enumerate}
\end{lemma}
\noindent Fix an associated contact form $\alpha$, i.e. $\xi=\ker\alpha$. The proof of the lemma is an exercise. Indeed, in a neighborhood of the link the difference between $\omega$ and $d\alpha$ is exact and this primitive allows us to perturb the defining form until we achieve the contact condition $\omega=d\alpha'$, $\xi'=\ker\alpha'$.\\
\noindent In conclusion, we obtain the following
\begin{proposition}\label{prop:good_ac}
Let $(M,\xi,\omega)$ be an exact quasi--contact closed $5$--dimensional manifold. Then there exists an almost contact perturbation $(\xi',\omega)$ of $(\xi,\omega)$ such that $(M,\xi',\omega)$ admits a good almost contact pencil $(f,B,C)$ with $B\neq\emptyset$ and any smooth fibre $F$ satisfying $c_1(\xi'_F)=0$.
\end{proposition}
\section{Contact blow--up.} \label{sect:blowup}
Let $(f,B,C)$ be an almost contact pencil on $(M,\xi,\omega)$. The map $f$ does not define a smooth fibration on $M$ for two reasons: it is not defined on $B$ and there exist critical fibres. The former failure can be avoided if we change the domain manifold $M$, i.e. $f$ can be defined on a suitable closed manifold $\widetilde{M}$ obtained from $M$ by a specific surgery procedure.
\begin{definition}
An almost contact Lefschetz fibration is an almost contact pencil $(f,B,C)$ with $B=\emptyset$. A contact Lefschetz fibration is a contact pencil $(f,B,C)$ with $B=\emptyset$.
\end{definition}
\noindent An almost contact Lefschetz fibration can be obtained out of an almost contact Lefschetz pencil by performing a surgery along the base locus. In particular, each connected component of the link $B$ is replaced by a standard $3$--sphere $(\mathbb{S}^3,\xi_{std})$. This leads to the notion of contact blow--up. The aim of this section
is to produce such a fibration for a good almost--contact pencil on a $5$--dimensional manifold.
\begin{theorem}\label{thm:good_blowup}
Let $(M,\xi,\omega)$ be an almost contact 5--manifold and $(f,B,C)$ a good almost contact pencil with $B\neq\emptyset$ and $c_1(\xi_F)=0$ for the regular fibres $F$ of $f$. Let $(\widetilde{M},\widetilde{\xi},\widetilde{\omega})$ be an almost contact blow--up of $(M,\xi,\omega)$ along $B$. Then there exists a good almost contact Lefschetz fibration $(\widetilde{f},C)$ for $(\widetilde{M},\widetilde{\xi},\widetilde{\omega})$ with $c_1(\widetilde{\xi}_{\widetilde{F}})=0$. The Lefschetz fibration $(\widetilde{f},C)$ restricts to $(f,C)$ away from the exceptional divisors.
\end{theorem}
\noindent The necessary definitions and discussions for this statement are given in this Section. Further details on the contact blow--up procedure appear in \cite{CPP}. Theorem \ref{thm:good_blowup} is a consequence of Theorem \ref{thm:blow-up}, adapted to an almost contact pencil via Lemma \ref{lemma:expand}, and Proposition \ref{prop:choice}. The compatibility of $(\widetilde{M},\widetilde{\xi},\widetilde{\omega})$ with $(\widetilde{f},C)$ is detailed in subsection \ref{ssec:comp}.
\subsection{Contact blow--up surgery.} \label{sub:surgery}
Throughout this subsection $\dim(M)$ is not relevant. Let us describe the precise properties of the blown--up manifold $\widetilde{M}$ and show its existence. Suppose that $(M,\xi)$ is contact.
\begin{theorem} \label{thm:blow-up}
Let $(M^{2n+1}, \xi)$ be a contact manifold. Let $S\subset M$ be a smooth transverse loop in $M$. There exists a manifold $\widetilde{M}$ conforming the following conditions:
\begin{itemize}
\item[-] There exists a contact structure $\widetilde{\xi}$ on $\widetilde{M}$.
\item[-] There exists a codimension--$2$ contact submanifold $E\cong(\mathbb{S}^{2n-1},\xi_{st})$ of $(\widetilde{M},\widetilde{\xi})$ with trivial normal bundle.
\item[-] The manifolds $(M\setminus S, \xi)$ and $(\widetilde{M}\setminus E, \tilde{\xi})$ are contactomorphic.
\end{itemize}
A pair $(\widetilde{M},\widetilde{\xi})$ with the above properties is said to be a contact blow-up of $(M,\xi)$ along $S$. The submanifold $E$ is called the exceptional divisor.
\end{theorem}
\begin{proof}
By Gray's stability there is a neighborhood $U$ of $S$ and a contactomorphism
$$\Phi: \mathbb{S}^1 \times B^{2n}(\varepsilon) \longrightarrow U,\quad (\theta,r,\sigma)\longmapsto\Phi(\theta,r,\sigma),\quad \Phi^*(\xi|_S)=\ker(d\theta -r^2 \alpha_{std}),$$
where $\alpha_{std}$ is the standard contact form on $\mathbb{S}^{2n-1}$ induced by restriction of the standard Liouville form on $\mathbb{R}^{2n}$. Choose an integer $k$ satisfying that
$$ \frac{2}{\sqrt{1+4k}} <\varepsilon. $$
Then the map
\begin{eqnarray*}
\psi_k: \mathbb{S}^1 \times B^{2n}(2) & \longrightarrow & \mathbb{S}^1 \times B^{2n}\left(\varepsilon\right) \\
(\theta, r, w_1, \ldots, w_n) & \longrightarrow & \left(\theta, \frac{r}{\sqrt{1+kr^2}}, e^{ik\theta}w_1, \ldots, e^{ik\theta}w_n\right),
\end{eqnarray*}
is a contact embedding. Hence we obtain a contact embedding from a radius--$2$ neighborhood of the loop into the manifold, namely
$$\Phi_k= (\Phi \circ \psi_k): \mathbb{S}^1 \times B^{2n}(2) \longrightarrow U \subset M.$$
\noindent Consider the diffeomorphism
\begin{eqnarray*}
\phi_1: \mathbb{S}^1\times (3/2,2) \times \mathbb{S}^{2n-1} & \longrightarrow & \mathbb{S}^1\times (3/2,2) \times \mathbb{S}^{2n-1} \\
(\theta, r, w_1, \ldots, w_n) & \longrightarrow & (\theta, r, e^{i\theta}w_1, \ldots, e^{i\theta}w_n).
\end{eqnarray*}
\noindent If $V= \Phi_k(\mathbb{S}^1 \times B^{2n}(3/2))$, then $\rho= \Phi_k \circ \phi_1: \mathbb{S}^1\times (3/2,2) \times \mathbb{S}^{2n-1} \longrightarrow U\setminus V \subset M$ satisfies
$$ \rho^* \xi = \ker \left\{\alpha_{std} + \frac{r^2-1}{r^2}d\theta \right\}.$$
Note that the function
\begin{eqnarray*}
h: (3/2,2) & \longrightarrow & \mathbb{R} \\
r & \longmapsto & h(r)= \frac{r^2-1}{r^2}
\end{eqnarray*}
satisfies $h(r)>5/9$. Therefore it is possible to extend it to a smooth function $\widetilde{h}:[0,2) \longrightarrow \mathbb{R}$ satisfying the following conditions (see Figure \ref{fig:h}):
\begin{itemize}
\item[-] $\widetilde{h}(r)=r^2$, for $r\in[0,1/2]$,
\item[-] $\widetilde{h}(r)=h(r)$, for $r>3/2$,
\item[-] $\widetilde{h}(r)'>0$ for $r\in[1/2,3/2]$.
\end{itemize}
\begin{figure}[ht]
\includegraphics[scale=0.4]{h.pdf}
\caption{The function $\widetilde{h}$.}\label{fig:h}
\end{figure}
Therefore $\widetilde{\eta}= \alpha_{std}+ \widetilde{h}(r) d\theta$ is a contact form over $\mathbb{S}^1\times [0,2) \times \mathbb{S}^{2n-1} \cong B^2(2) \times \mathbb{S}^{2n-1}$. We can glue the manifold $(M\setminus V, \xi)$ and $(B^2(2) \times \mathbb{S}^{2n-1}, \ker\widetilde{\eta})$ with the gluing map $\rho$ to define a contact manifold $(\widetilde{M}, \widetilde{\xi})$. This manifold readily satisfies the statement of the theorem.
\end{proof}
\subsection{Almost contact blow--up.}
Let $(f,B,C)$ be a good almost contact pencil on $(M,\xi,\omega)$. In the contact case, the surgery performed in the proof of Theorem \ref{thm:blow-up} is local on $B$. The construction readily applies to a good almost contact pencil since $(M,\xi)$ is a contact structure on a neighborhood of $B$.\\
\noindent Concerning the almost contact structure, the condition on the parameter $k$ in the proof of Theorem \ref{thm:blow-up} is unnecessary. The size of a neighborhood of a contact submanifold for the almost contact structure can be enlarged. In precise terms:
\begin{lemma} \label{lemma:expand}
Let $(M, \xi, \omega)$ be an almost contact manifold and $(S,\xi=\ker\alpha)$ be a contact submanifold with trivial normal bundle $\nu_S\cong S\times\mathbb{R}^{2q}$. Fix a radius $R\in\mathbb{R}$. Then there exists an almost contact homotopy $(M,\xi_t,\omega)$ such that $(M,\xi_0,\omega)=(M,\xi,\omega)$ and it conforms the following conditions:
\begin{itemize}
\item[-] The homotopy is supported in an annulus around $S$, i.e. given a smooth fiberwise metric on $\nu_S$ there exist $\rho_1,\rho_2\in\mathbb{R}^+$ with $\rho_1<\rho_2$ such that
$$\xi_t|_{\mathbb{D}(\nu_S,\rho_2)\setminus \mathbb{D}(\nu_S,\rho_1)}=\xi|_{\mathbb{D}(\nu_S,\rho_2)\setminus\mathbb{D}(\nu_S,\rho_1)},$$
where $\mathbb{D}(\nu_S,r)$ is the disk bundle of radius $r$.\\
\item[-] There exists a neighborhood $U$ of $S$ and a diffeomorphism $\varphi$ such that
$$\varphi: S \times B^{2q}(R) \longrightarrow U,\quad\varphi^*\xi_1=\ker(\alpha-r^2\alpha_{std}),\quad\varphi^* \omega =d\alpha-2rdr\wedge d\alpha_{std}.$$
The $1$--form $\alpha_{std}$ is the standard contact form on $\partial B^{2q}(R)$.
\end{itemize}
\end{lemma}
\begin{proof}
This is a statement about a neighborhood $S\times B^{2q}(\varepsilon)$. Suppose that $R>\varepsilon$. In this area $(\xi,\omega)$ is a contact structure described as the kernel of the $1$--form $\eta_0= \alpha-r^2 \alpha_{std}$. Consider a function $H\in C^\infty([0,\varepsilon],\mathbb{R}^+)$ such that:
\begin{itemize}
\item[a.] $H(r)=r^2$ for $r\in [0,\varepsilon/4] \cup [3\varepsilon/4,\varepsilon]$,
\item[b.] $H'(r)>0$ for $r \in(0,\varepsilon/2)$,
\item[c.] $H(\varepsilon/2)= R^2$.
\end{itemize}
There exists a homotopy $\{H_t\}$ of functions in $C^\infty([0,\varepsilon],\mathbb{R}^+)$ with $H_0(r)=r^2$, $H_1(r)=H(r)$ and any $H_t$ satisfying properties a and b above. The homotopy of $1$--forms $\eta_t= \alpha-H_t(r)\alpha_{std}$ defines a homotopy of almost contact distributions. The diffeomorphism
\begin{eqnarray*}
\Psi: S \times B^{2q}(R) &\longrightarrow & S \times B^{2q}(\varepsilon/2) \\
(s,r, \theta) & \longmapsto & (s, \sqrt{H(r)}, \theta)
\end{eqnarray*}
satisfies $\Psi^* \eta_0 = \eta_1$ as required.
\end{proof}
\noindent The lemma does not hold for a contact structure since the contact condition is violated in the homotopy. Thus the restriction on the parameter $k$ in the contact case.\\
\noindent The surgery in the almost contact case can be performed with any given integer $k\in\mathbb{Z}$. Indeed, the construction follows the proof of Theorem \ref{thm:blow-up}. Consider a transverse embedded loop $\mathbb{S}^1\subset(M,\xi,\omega)$ and use the map $\psi_k$ to obtain a standard contact neighborhood of a certain size. Enlarge this neighborhood using Lemma \ref{lemma:expand} and apply $\phi_1$.
\subsection{Compatibility with an almost contact pencil.}\label{ssec:comp}
Let $(f,B,C)$ be a good almost contact pencil on a $5$--dimensional contact manifold $(M,\xi)$. There are several choices in the previous construction. To begin with, the map $\Phi: \mathbb{S}^1 \times B^{2n}(\varepsilon) \longrightarrow U$. This amounts to a choice of framing of the trivial normal bundle along $\mathbb{S}^1$. Since $\mathbb{S}^1\subset B$ we can use the adapted charts in Definition \ref{def:almost_pencil} and require that $\Phi$ satisfies that the map
$$f \circ \Phi: \mathbb{S}^1 \times(B^{4}(\varepsilon)\backslash\{ 0 \}) \longrightarrow \mathbb{CP}^1$$
is precisely $(f \circ \Phi)(\theta, w_1, w_2)= [w_1 :w_2]$. Therefore the compactified fibres are of the form $\mathbb{S}^1 \times L$, for any complex line $L\in \mathbb{C}^2$.\\
Let us focus on the compactification of fibres in the blow--up, i.e. the extension of $\widetilde{f}$ from $M\setminus B$ to $\widetilde{M}$. We first restrict ourselves to the transition region $\mathbb{S}^1\times(3/2,2)\times\mathbb{S}^3\subset\mathbb{S}^1\times\mathbb{C}^2$. The gluing map is $\rho=\Phi\circ\psi_k\circ\phi_1$. In order to understand the fibres of the blown--up pencil we just need to describe the map $\widetilde{f}=f\circ\Phi\circ\psi_k\circ\phi_1$. The reader can readily verify that $\widetilde{f}(\theta,r,w_1,w_2)=(f\circ\Phi)(\theta,rw_,rw_2)=[w_1:w_2]$, since $\psi_k\circ\phi_1$ acts as complex scalar multiplication in the transition area.\\
\noindent Notice that the domain of definition of $\widetilde{f}$ is $\mathbb{S}^1\times(3/2,2)\times\mathbb{S}^3$, and it is invariant with respect to the coordinates $(\theta,r)\in\mathbb{S}^1\times(3/2,2)$. Hence, the map $\widetilde{f}$ extends trivially to the model $(B^2(2)\times\mathbb{S}^3,\ker\widetilde{\eta})$. In particular, the extension of $\widetilde{f}$ restricted to the exceptional divisor $\{0\}\times\mathbb{S}^3$ consists of the Hopf fibration.\\
\noindent The fibres of the blown--up fibration are thus almost contact submanifolds. Indeed, the fibres of $\widetilde{f}$ restricted to $(B^2(2)\times\mathbb{S}^3,\ker\widetilde{\eta})$ are diffeomorphic to $B^2(2)\times\mathbb{S}^1$, the $\mathbb{S}^1$ factor being a transverse Hopf fibre. These submanifolds are certainly contact.
\subsection{Blown--up contact Lefschetz Fibration} The fibres $\widetilde{F}$ of the Lefschetz fibration $(\widetilde{f},C)$ differ from the fibres $F$ of $(f,B,C)$. Let us provide a precise description of $\widetilde{F}$ and show that the blow--up procedure can be perfomed to obtain $c_1(\widetilde{\xi}_{\widetilde{F}})=0$.\\
\noindent The trivialization of a neighborhood of a connected component $\gamma\subset B$ of the base locus provided in Definition \ref{def:almost_pencil} induces a natural framing $\nu_S\cong\mathbb{S}^1\times\mathbb{C}^2$, i.e. $\langle(1,0),(i,0),(0,1),(0,i)\rangle$. It restricts to a framing of $\mathbb{S}^1\times\mathbb{C}$ in the two fibres corresponding to the two complex axes of $\mathbb{C}^2$. Hence it induces framings in any complex line $\mathbb{S}^1\times\mathbb{C}\subset\mathbb{S}^1\times \mathbb{C}^2$: for the complex line $\{(z,w)\in\mathbb{C}^2:z-\alpha w=0\}$, we use $\langle(\alpha,1),i(\alpha,1)\rangle$. Denote by $\mathbb{F}_p(0)$ such framing of $B\subset\overline{f^{-1}(p)}$. Let $\mathbb{F}_p(n)$ be the $n$--twist of $\mathbb{F}_p(0)$ and $k_\gamma$ be the parameter used in the construction of Theorem \ref{thm:blow-up} when performing a blow--up along $\gamma$.
\begin{lemma}
Let $(f, B, C)$ be a contact pencil for the 5--manifold $(M,\xi)$. Its contact blow--up $(\widetilde{M} ,\widetilde{\xi})$ has a contact fibration
$(\tilde{f} , C)$ that coincides with $(f,B,C)$ away from $B=\gamma_1\cup\ldots\cup\gamma_s$. Its fibre over $p\in\mathbb{CP}^1$ is contactomorphic to a transverse contact $(0,1)$--surgery performed on $\overline{f^{-1}(p)}$ along $\gamma_i$ with framing $\mathbb{F}_p(-k_i-1)$, for some $k_i\in\mathbb{Z}$. The restriction of the map $f$ to each of the exceptional
divisors is given by the Hopf fibration.
\end{lemma}
\begin{proof}
The map $\psi_k$ in Theorem \ref{thm:blow-up} modifies the initial framing from $\mathbb{F}_p$ to $\mathbb{F}_p(-k_i)$, $k_i$ being the corresponding squeezing parameter $k$ in the surgery along $\gamma_i$. Using the map $\phi_1$ substracts another twist and sends the meridian to the longitude of the added solid torus. It is thus a $(p,q)=(0,1)$--Dehn surgery with respect to $\mathbb{F}_p(-k_i-1)$.
\end{proof}
\noindent Note that the coefficients $k_i$ can be arbitrarily chosen. The constructive argument will use the fact that $c_1(\widetilde{\xi}_{\widetilde{F}})=0$ for any fibre $\widetilde{F}$ of the blown--up pencil. This has been achieved for the initial fibres of the pencil. The blown--up procedure changes the almost contact manifold $(F,\xi)$ to $(\widetilde{F},\widetilde{\xi})$ and we cannot directly assume that $c_1(\widetilde{\xi}_{\widetilde{F}})=0$. This can be fixed using the following:
\begin{proposition}\label{prop:choice}
There is a choice of $(k_1,\ldots,k_s)\in \mathbb{Z}^s$, such that the first Chern class of the almost contact structure $(\widetilde{M},\widetilde{\xi},\widetilde{\omega})$ in any regular fibre $\widetilde{F}$ is zero.
\end{proposition}
\begin{proof}
Consider a connected component $\gamma\subset B$. The complex rank--2 distribution $\xi$ yields a line bundle $\displaystyle\bigwedge^2\xi$. The almost contact pencil is obtained from two generic sections of this line bundle. The determinant bundle of $\xi$ is needed because of Corollary \ref{coro:starting_pencil}. Equivalently, from a section
$$s=(s_0, s_1): M \longrightarrow \mathbb{C}^2 \otimes \bigwedge^2 \xi.$$
A point $p\in M$ maps to $[s_0(p):s_1(p)]\in\mathbb{CP}^1$. This is well--defined if $p$ is not contained in the base locus $B=\{p:s_0(p)=s_1(p)=0\}$. The construction is detailed in ~\cite{Na}.\\
\noindent Let $\widetilde{F}$ be a regular fibre of $\widetilde{f}$. Along this blown--up fibre $\widetilde{F}$ the distribution $\widetilde{\xi}$ satisfies
$$c_1(\widetilde{\xi})|_{\widetilde{F}}=c_1(\widetilde{\xi}_{\widetilde{F}})+c_1(\nu_{\widetilde{F}}).$$
The submanifold $\widetilde{F}\subset\widetilde{M}$ is a fibre of a locally trivial fibration, hence $c_1(\nu_{\widetilde{F}})=0$. The first Chern class measures the obstruction to the existence of a trivialization along the $1$--skeleton. For a transverse loop this is equivalent to a framing. Hence, it is enough to justify that the section $(s_0,s_1)$ can be lifted to a non-vanishing section $(\widetilde{s}_0, \widetilde{s}_1)$ from the blown--up manifold to the bundle $\mathbb{C}^2 \otimes \bigwedge^2 \widetilde{\xi}$. Then $c_1(\widetilde{\xi})|_{\widetilde{F}}=0$ and $c_1(\widetilde{\xi}_{\widetilde{F}})=0$.\\
\noindent A surgery procedure takes place, thus we should ensure that the relevant vector fields used in the blow--up process extend without adding zeroes. Let $X_r,X_i,X_j,X_k$ be the canonical quaternionic basis generating\footnote{Due to the choice of contact form, we should be strictly working with $\overline{\mathbb{C}}$ instead of $\mathbb{C}$.} $T\mathbb{C}^2=\mathbb{C}^2\cong\mathbb{H}^1$. The blow--up provides an interpolation between the following two bundles,
$$\xi=\ker\{{d\theta-r^2\alpha_{std}\}=\langle X_r,r^2\partial_\theta+X_i,X_j,X_k\rangle\simeq H=\ker d\theta}=\langle X_r,X_i,X_j,X_k\rangle\mbox{ for $r$ large},$$ $$\mbox{ and }\ker\{r^2d\theta+\alpha_{std}\}=\langle X_r,r^2X_i-\partial_\theta,X_j,X_k\rangle\quad \mbox{for $r$ small}.$$
Let $\varphi(r): (0, \infty) \longrightarrow [0, \pi/2]$ be a function constant equal to $0$ for small $r$ and constant equal to $\pi/2$ for large $r$. The distribution being fixed for large radii, we interpolate through $\sin \varphi\cdot X_i-\cos \varphi\cdot\partial_{\theta}$ between $X_i$ and $-\partial_\theta\simeq r^2X_i-\partial_\theta$, for $r$ small.
Suppose the original sections $(s_0,s_1)$ restrict to an $m$--twisted frame, i.e. their trivialization is $e^{m\cdot i\theta}$ times the fixed trivialization. Then the choice $k=m$ allows us to extend the field $\partial_\theta$ in the framing to the interior of the dual sphere without zeroes.\\
\noindent Consider an increasing smooth function $g:\mathbb{R}^+ \to \mathbb{R}$ such that $g(r)=\varepsilon/2$ for $r<\varepsilon/2$ and $g(r)=r$ for $r>\varepsilon$ and use it as a cut--off function to extend the sections to the whole blown--up manifold. This is the required section $\widetilde{s}=(\widetilde{s_0},\widetilde{s_1})$ extending the previous section $s=(s_0,s_1)$ away from the surgery area and restricting to the Hopf section at $r=0$. Thus the sections can be extended to the blown--up manifold in a non--vanishing manner and $c_1(\widetilde{\xi}|_{\widetilde{F}})=0$.
\end{proof}
\noindent The previous results are of a local nature and therefore they apply, accordingly modified, to good almost contact pencils. In that
case, we obtain an almost contact Lefschetz fibration that is contact in neighborhoods of the exceptional divisors and the critical loops, i.e. a good almost contact Lefschetz fibration, and the first Chern class of the distribution along the regular fibres vanishes. This concludes the proof of Theorem \ref{thm:good_blowup}.
\subsection{Contact blow--down in $5$ dimensions.}\label{sec:blow-down}
In the subsequent sections the argument uses a fibration obtained by blowing--up the base locus of a good almost contact pencil. In order to recover the initial manifold we require the inverse procedure. This is used in Section \ref{sec:end}.
\begin{lemma} \label{lem:down}
Let $(M,\xi)$ be a contact manifold and $E\cong(\mathbb{S}^3,\xi_{st})$ be a codimension--$2$ contact submanifold with trivial normal bundle. Then the submanifold $E$ can be replaced by a knot $K$ such that the resulting manifold $\overline{M}$ admits a contact structure $(\overline{M},\overline{\xi})$, $K$ is transverse and the complements $(M \setminus E,\xi)\cong(\overline{M}\setminus K,\overline{\xi})$ are contactomorphic.
\end{lemma}
\begin{proof}
The blow--up procedure can be reversed smoothly. Use $\phi_1^{-1}$ to glue instead of $\phi_1$. In this case any choice of $k$ can be made.
\end{proof}
\noindent Note that the diffeomorphism type of the blown--up manifold depends on the choice of $k\in \mathbb{Z}$. Even choices of $k$ not being diffeomorphic to odd ones.\\
\noindent The argument developed in this article to prove Theorem \ref{main} requires a smooth fibration, hence the reason for performing a blow--up. There is an alternative approach not involving a blow--up that leads to a quite complicated version of the local models used in Sections \ref{sec:vertical}, \ref{sect:skeleton} and \ref{sec:bands}. These models are essential to describe the deformation of the almost contact structure. The simpler, the better. In particular, the description in Section \ref{sec:bands} would be rather technical if the modified model was used.\\
\noindent The use of a blow--up to obtain an almost contact Lefschetz fibration eases the technicalities. In order to prove Theorem \ref{main} we perform both blow--up and blow--down processes with the same parameter $k=(k_1,\ldots,k_s)$. This preserves the homotopy type of the almost contact structure.
\section{Vertical Deformation.} \label{sec:vertical}
In Section \ref{sec:pencils} we endowed our initial $5$--dimensional closed orientable almost contact manifold $(M,\xi,\omega)$ with an almost contact pencil $(f,B,C)$ such that $B\neq0$ and $c_1(\xi_F)=0$ for the fibres $F$ of $f$. In Proposition \ref{prop:good_ac} we have obtained a contact structure in a neighborhood of the base locus $B$ and the critical curves $C$. According to Theorem \ref{thm:good_blowup} there exists an almost contact Lefschetz fibration $(\widetilde{f},C)$ in an almost contact blow--up $(\widetilde{M},\widetilde{\xi},\widetilde{\omega})$. In order to obtain a contact structure in the manifold $(M,\xi,\omega)$ we use the splitting induced by the existence of the fibration $(\widetilde{f},C)$ on $(\widetilde{M},\widetilde{\xi},\widetilde{\omega})$. Henceforth we shall consider an almost contact manifold with an almost contact Lefschetz fibration. These will be respectively denoted $(M,\xi,\omega)$ and $(f,C)$ even though in our situation they refer to the blown--up initial manifold and the blown--up pencil. This should not lead to confusion. The initial manifold is recovered in Section \ref{sec:end}.\\
\noindent Let $(M,\xi,\omega)$ be a $5$--dimensional closed orientable almost contact manifold and $(f,C)$ an almost contact Lefschetz fibration on it.
\begin{definition}
An almost contact structure $(M,\xi,\omega)$ is called vertical with respect to an almost contact
fibration $(f,C)$ if the fibres of $f$ are contact submanifolds for $(\xi,\omega)$ away from the critical points.
\end{definition}
\noindent A piece of terminology. If an almost contact Lefschetz fibration $(f,C)$ is obtained as a blown--up good almost contact pencil it contains a set $E=\{E_j\}$ of exceptional contact divisors. The data of an almost contact Lefschetz fibration $(f,C,E)$ with a non--empty set of exceptional divisors will be referred to as an almost contact exceptional fibration, and shortened to {\it ace fibration}. An ace fibration is good if the curves of $C$ and the divisors in $E$ are contact submanifolds of $(M,\xi,\omega)$.\\
\noindent The main result of this section reads:
\begin{theorem} \label{thm:vert_defor}
Let $(M,\xi,\omega)$ be an almost contact manifold and $(f,C,E)$ an associated good ace fibration. Then there exists a homotopic deformation of the almost contact structure relative to $C$ and $E$ such that the almost
contact structure becomes vertical for $(f,C)$.
\end{theorem}
The proof of the theorem relies on the existence of an overtwisted disk in each fibre, such structure allows more flexibility in handling families of distributions. Hence, it will be essential for the argument to apply that the fibres of the ace fibration are $3$--dimensional manifolds. In order to obtain a vertical fibration we need Eliashberg's classification result of overtwisted contact structures ~\cite{El}. The required details are provided.
\subsection{3--dimensional Overtwisted Structures.}
Our setup provides a fibration with a distribution on each fibre. Given such an almost contact fibration $f:M\longrightarrow\mathbb{CP}^1$, let $F_z$ denote the fibre over $z\in \mathbb{CP}^1$ and $(\xi_z, \omega_z)$ the
induced almost contact structure on $F_z$. Then the family $(F_z, \xi_z)$ can locally be viewed as a $2$--parametric family of $2$--distributions on a fixed fibre.\\
\noindent In the proof of Theorem~\ref{thm:vert_defor} we use a relative version of the following:
\begin{theorem} [Theorem 3.1.1 in \cite{El}] \label{thm:eliash}
Let $M$ be a compact closed $3$--manifold and let $G$ be a closed subset
such that $M \setminus G$ is connected. Let $K$ be a compact space and $L$ a
closed subspace of $K$. Let $\{\xi_t \}_{t \in K}$ be a family of cooriented $2$--plane
distributions on $M$ which are contact everywhere for $t\in L$ and are contact near $G$ for
$t\in K$. Suppose there exists an embedded $2$--disk ${\mathcal{D}} \subset M \setminus G$ such that
$\xi_t$ is contact near ${\mathcal{D}}$ and $({\mathcal{D}}, \xi_t)$ is equivalent to the standard
overtwisted disk for all $t\in K$. Then there exists a family $\{ \xi_t' \}_{t\in
K}$ of contact structures of $M$ such that $\xi_t'$ coincides with $\xi_t$ near $G$
for $t\in K$ and coincides with $\xi_t$ everywhere for $t\in L$. Moreover $\xi_t'$
can be connected with $\xi_t$ by a homotopy through families of distributions that
is fixed in $(G \times K) \cup (M \times L)$.
\end{theorem}
\noindent In order to allow the case of a $3$--manifold with non--empty boundary we also need:
\begin{corollary} \label{coro:eliash}
Let $M$ be a compact $3$--manifold with boundary $\partial M$ and let $G$ be a closed
subset of $M$ such that $M \setminus G $ is connected and $\partial M \subset G.$
Let $K$ be a compact space and $L$ a closed subspace of $K.$
Let $\{ \xi_t \}_{t \in K}$ be a family of cooriented $2$--plane distributions on $M$
which are contact everywhere for $t\in L$ and are contact near $G$ for $t\in K$. Suppose
there exists an embedded $2$--disk ${\mathcal{D}} \subset M\backslash G$ such that $\xi_t$ is contact
near ${\mathcal{D}}$ and $({\mathcal{D}}, \xi_t)$ is equivalent to the standard overtwisted disk for all
$t\in K$. Then there exists a family $\{ \xi_t' \}_{t\in K}$ of contact structures
of $M$ such that $\xi_t'$ coincides with $\xi_t$ near $G$ for $t\in K$ and coincides
with $\xi_t$ everywhere for $t\in L$. Moreover $\xi_t'$ can be connected with
$\xi_t$ by a homotopy through families of distributions that is fixed in $(G\times K)
\cup (M \times L)$.
\end{corollary}
\begin{proof}[Outline]
The proof for the closed case uses a suitable triangulation $P$ of the $3$--manifold having a subtriangulation $Q$ containing $G$, for which the distributions are already contact structures. Then Eliashberg's argument is of local nature, working with neighborhoods of the $0$, $1$, $2$ and $3$--skeleton of $P\backslash Q$ and assuring that no changes are made in a neighborhood of $Q$. Hence, the method is still valid since $P$ and $Q$ do exist in the case of a manifold with boundary, and only $Q$ contains the boundary.
\end{proof}
\noindent We locally treat an almost contact fibration as a $2$--parametric family of distributions over a fixed fibre, thus we may use a disk as a parameter space and the central fibre as the fixed manifold. It will be useful to be able to obtain a continuous family of distributions such that the distributions in a neighborhood of the central fibre become contact structures while the distributions near the boundary are fixed. Such a family is provided in the following
\begin{corollary} \label{coro:eliash2}
Under the same hypothesis and notation of Corollary \ref{coro:eliash}, assume that $K$ is a
topological ball, so $\SS=\partial K$ is a sphere. Let
$$\widetilde{K}=K
\cup_{\partial} (\SS \times [0,1])\quad and \quad \widetilde{L}=L \cup_{\partial} (\partial L \times [0,1])$$
where we identify $\partial K=\SS\cong\SS\times\{0\}$ and $\partial L\cong\partial L\times\{0\}$. Let the family of distributions be defined for $\{ \xi_t \}_{t\in \widetilde{K}}$, the distributions being contact near $G$ and ${\mathcal{D}}$. Then there exists a deformation $\{ \widetilde{\xi}_t?\}_{t \in \widetilde{K}}$ such that:
\begin{itemize}
\item[-] Satisfies the properties obtained in Corollary \ref{coro:eliash} when
the family of parameters is restricted to $K$. In particular, for any $t\in K$ the distributions $\xi_t$ and $\widetilde{\xi}_t$ coincide on an open
neighborhood of $G \cup {\mathcal{D}}$.
\item[-] The distributions $\xi_t$ and $\widetilde{\xi}_t$ coincide over $M$ for $t\in \partial \widetilde{K} \cong \SS \times \{ 1 \}$. Further, $\widetilde{\xi}_t$ can be connected with $\xi_t$ by a homotopy through families
of distributions fixed in $(G\times \widetilde{K}) \cup M \times (\widetilde{L} \cup
\partial \widetilde{K})$.
\end{itemize}
\end{corollary}
\begin{proof}
Corollary \ref{coro:eliash} provides the family $\widetilde{\xi}_t$ for $t\in
K$, it is left to extend the parameter over $\widetilde{K}$. Since both families must coincide at $\SS\times\{1\}$, we use the structures $\xi_t$ in the annulus $\SS\times[1/2,1]$ and start deforming to $\widetilde{\xi}_t$ at $\SS\times\{1/2\}$ until we reach $\partial K\cong\SS\times\{0\}$, where it glues with $\widetilde{\xi}_t$.\\
In more precise terms, let us denote the family $\xi_t$ restricted to $\SS \times [0,1]$ by $\{ \eta_p^s
\}_{p\in \SS}^{s\in [0,1]}$, i.e. $\xi_t = \xi_{(p,s)}= \eta_p^s$. Let $\{ \xi_t^s \}_{t\in \SS}^{s\in[0,1]}$ be the deformation of
$\xi_{(t,0)}=\eta_t^0 = \xi_t^0$ to $\widetilde{\xi}_{(t,0)}=\xi_{t}^1$, restricted to $\SS \times\{0\}
\simeq \partial K$.
Define the continuous deformation
$$ \{ \nu_t^s \}_{t \in \SS}^{s\in[0,1]} = \left\{
\begin{array}{l}
\eta_t^{1-2s}, s \in [0,1/2], \\
\xi_{t}^{2s-1}, s \in [1/2,1].
\end{array}
\right. $$
\begin{figure}[ht]
\includegraphics[scale=0.6]{interpolation.pdf}
\caption{Deformation in the boundary of the ball. $t\in \SS$ is fixed.}
\end{figure}
Finally, the following deformation extends $\{ \xi_t^s \}$ to the domain $\SS \times [0,1]$ and satisfies all the required properties:
$$ \{ (\widetilde{\xi})_{(t,r)}^s \}_{(t,r) \in \SS \times [0,1]}^{ s\in[0,1]}= \{ \nu_t^q \},
\hspace{0.5cm} q= \frac{1}{2}(1+s)(1-r). $$
See Figure $4$ for a visual realization.
\end{proof}
\begin{remark}
Notice that the assumption that $K$ is a topological ball is not necessary. However,
since we will apply Corollary~\ref{coro:eliash2} when $K$ is
a topological ball, we consider more appropriate to state it in this particular case.
\end{remark}
\noindent We need at least one overtwisted disk over each fibre in order to apply Corollary \ref{coro:eliash}. The family should behave continuously. Let us provide such a family of disks.
\subsection{Families of overtwisted disks.}\label{subsec:otdisks}
There are two basics issues to be treated: the location of the disks and their overtwistedness. The second is simply guaranteed since once a disk with a contact neighborhood is placed in each fibre we can produce overtwisted disks using Lutz twists. In order to decide the location of the disks in each fibre we need to find a section of the good ace fibration.\\
\noindent Let $(f,C,E)$ be a good ace fibration. Denote by $U(C),U(E_i)$ open neighborhoods of the critical curves $C$ and the exceptional spheres $E_i$. Consider $U(f)=U(C)\cup U(E_i)$ the union of these open neighborhoods, so in the complementary of $U(f)$ the $f$ becomes a submersion. Instead of finding a global section mapping away from $U(f)$, we shall construct two {\it disjoint} sections that will provide at least one overtwisted disk in each fibre. The global situation we achieve is described as follows:
\begin{proposition} \label{propo:disc_family}
Let $(f,C,E)$ be a good ace fibration for $(M,\xi,\omega)$. There
exists a deformation $(F_z,\widetilde{\xi}_z)_{z\in \mathbb{CP}^1}$ of the family $(F_z, \xi_z)_{z\in
\mathbb{CP}^1}$ fixed at the intersection of the set $U(f)$ with each $F_z$ satisfying:
\begin{enumerate}
\item There exist two open disks ${\mathcal{B}}_0,{\mathcal{B}}_{\infty} \subset
\mathbb{CP}^1$, containing $0$ and $\infty$ respectively such that the
intersection ${\mathcal{B}}_0\cap{\mathcal{B}}_\infty$ is an open annulus and the curves $\partial {\mathcal{B}}_i$ are disjoint from the set of curves $f(C)$.
\item There exists two disjoint families of embedded $2$--disks ${\mathcal{D}}_z^i\subset F_z$,
with $z\in{\mathcal{B}}_i$, for $i=0, 1$, not intersecting $U(f)$. Further, the structure $\widetilde{\xi}_z$ is contact in a neighborhood of such families and $({\mathcal{D}}_z^i, \widetilde{\xi}_z)$ are equivalent to standard overtwisted disks.
\end{enumerate}
\end{proposition}
\begin{remark} \label{rem:smallest}
The complement of ${\mathcal{B}}_0 \bigcap {\mathcal{B}}_{\infty}$ consists of two disjoint disks. The diameter of those disks can be chosen as small as desired. This will be needed in the subsequent sections.
\end{remark}
\noindent The fact that $\widetilde{\xi}_z$ equals $\xi_z$ in the intersection of the set $U(f)$ with $F_z$ ensures that no deformation is performed near the critical curves nor the exceptional spheres. This is mainly a global statement, involving the whole of the fibres. In order to prove the result we study the local model of a tubular neighborhood of an exceptional divisor of the good ace fibration $(f,C,E)$.\\
A good ace fibration $(f,C,E)$ is obtained by blowing--up a certain good almost contact Lefschetz pencil along its base locus $B$. Let $K$ be a knot belonging to this base locus $B$. After the blow--up procedure it is replaced by an exceptional contact divisor $E\cong(\mathbb{S}^3,\xi_{st})$. As explained in Section \ref{sect:blowup} the restriction of the fibration $f$ to $E$ is the Hopf fibration. Since the distribution $\xi$ is locally a contact structure the tubular neighborhood theorem provides a chart
\begin{equation}
\Psi: U \longrightarrow \mathbb{S}^3 \times B^2,\quad\Psi^* \xi_{st} = \xi \label{eq:chart0}
\end{equation}
where $\xi_{st}= \ker \{ \alpha_{\mathbb{S}^3} + r^2 d\theta \}$. The fibres of the induced map $f_U$ defined as
$$\xymatrix{
& \mathbb{S}^3\times B^2\ar@{->}[rd]^{f_U}\ar@{->}[r]^{\qquad\Psi^{-1}}
&U\ar@{->}[d]^{f}
\\
& & \mathbb{CP}^1
}$$
correspond to tubular neighborhoods of the Hopf fibres. That is to say, they are submanifolds $(\mathbb{S}^1\times B^2, \xi_v= \ker (d\beta + r^2
d\theta))$ for each $z\in \mathbb{CP}^1$.
Note that the variable $\beta\in \mathbb{S}^1$ parametrizing each Hopf fibre is not global since the fibration is not trivial. The differential $d\beta$ is globally well--defined since it is dual to the vector field generating the associated $\mathbb{S}^1$ action. Using an almost contact connection\footnote{If $(f, C)$ is an almost contact fibration for $(M, \xi, \omega)$, an almost contact connection $H_{\xi}$ is defined at a point $p\in M\backslash C$ as the distribution $\omega_p$--symplectic orthogonal to the distribution of the vertical fibre $\xi_p
\cap \ker df_p$ inside $\xi_p$.} the standard contact structure in $\mathbb{S}^3\times B^2$ can be expressed as the direct sum of distributions
\begin{equation}
\xi_{st} = \xi_v \oplus H, \label{eq:conn_tubular}
\end{equation}
where $\xi_v$ is the standard contact structure in $\mathbb{S}^1 \times B^2$, {\it the vertical direction}, and $H$ is the
contact\footnote{An almost contact connection is called a contact connection if the fibration and $\xi$ are, with the compatibility condition: $\exists \alpha$ a contact form such that $\omega=d\alpha$ and $\xi=\mbox{ker }\alpha$.}
connection associated to the fibration of $\mathbb{S}^3 \times B^2$ over $\mathbb{CP}^1$.\\
\noindent Topologically, the $4$--distribution $\xi_{st}$ is expressed as a direct sum of two distributions of $2$--planes. Since the $2$--form $\omega$ providing the almost contact structure is given and so is $\xi$, we may interpret $(\mathbb{S}^3 \times B^2, \xi_v)$ as a non--trivial family of contact
structures parametrized by the base, $\{\xi_q= \xi_v \}$, ${q\in \mathbb{CP}^1 }$. So far we understand the topology and contact structure of the local model of the ace fibration along an exceptional divisor. Indeed, in a neighborhood of the exceptional divisor $E$ it is precisely a piece of the blown--up fibration and the knots are the intersection of the fibres of the almost contact pencil with the exceptional sphere $E$.\\
The local model described above allows us to prove the local version of Proposition ~\ref{propo:disc_family}.
\begin{lemma} \label{lem:local_family}
There exists a deformation of the contact structures $(\mathbb{S}^3\times B^2, \xi_v)$ relative to a neighborhood of the boundary such that the deformed structures are contact and with respect to them:
\begin{enumerate}
\item There exist two continuous families of overtwisted disks $\{ {\mathcal{D}}_q^0 \}_{q \in {\mathcal{B}}^0}$ and $\{ {\mathcal{D}}_q^\infty \}_{q \in {\mathcal{B}}^\infty}$, i.e. on each point $q\in {\mathcal{B}}_0\backslash\partial{\mathcal{B}}_0$ $($resp. $q\in {\mathcal{B}}_\infty\backslash\partial{\mathcal{B}}_\infty)$ there is an overtwisted disk ${\mathcal{D}}_q^0$ $($resp. ${\mathcal{D}}_q^\infty)$.
\item The disks ${\mathcal{D}}_q^0$ and ${\mathcal{D}}_q^{\infty}$ are disjoint for $q\in {\mathcal{B}}_0 \cap {\mathcal{B}}_{\infty}$.
\end{enumerate}
Both ${\mathcal{B}}_0\backslash\partial{\mathcal{B}}_0,{\mathcal{B}}_\infty\backslash\partial{\mathcal{B}}_\infty$ can be thought as neighborhoods of the upper and lower semi--spheres.
\end{lemma}
\begin{proof}
Let $h: \mathbb{S}^3 \longrightarrow \mathbb{CP}^1$ be the Hopf fibration, extend the fibration to $h: \mathbb{S}^3
\times B^2 \longrightarrow \mathbb{CP}^1$ by projection onto the first factor. Let ${\mathcal{B}}_0$, ${\mathcal{B}}_\infty$ be two disks containing $0,\infty\in\mathbb{CP}^1$, e.g. the complements of a tubular neighborhood of $\infty$ and $0$. As explained before, the idea is to use the exceptional divisor to create a couple of sections. On the one hand, the exceptional divisor has a contact structure and we would rather not perturb around a small neighborhood of it, and on the other hand the exceptional divisor is not $\mathbb{CP}^1$ but $\mathbb{S}^3$. We use two copies of the exceptional divisor away from $\mathbb{S}^3\times\{0\}\subset \mathbb{S}^3\times B^2$ and we trivialize the base $\mathbb{CP}^1$ with the two disks ${\mathcal{B}}_0$, ${\mathcal{B}}_\infty$.\\
Let $q_0=(1/2,0)$, $q_{\infty}=(0,1/2) \in B^2$ be two fixed points and denote the two corresponding $3$--spheres
$$\mathbb{S}^3_0= \mathbb{S}^3 \times \{ q_0 \},\qquad \mathbb{S}^3_{\infty}= \mathbb{S}^3 \times \{ q_{\infty} \}.$$
The fibre of the restriction of the fibration $(\mathbb{S}^3 \times B^2, \xi_v)\longrightarrow\mathbb{CP}^1$ to the
submanifold $\mathbb{S}^3_0$ (resp. $\mathbb{S}^3_{\infty}$) is a transverse knot $K_0^p$ (resp. $K_{\infty}^p$). We will now insert two families of overtwisted disks.\\
Applying a full Lutz twist in a small neighborhood of each of those knots $K_0^p \in h^{-1}(p)$ produces a $3$--dimensional Lutz twist on each fibre, see \cite{Lu,Ge}. This family of overtwisted disks is parametrized as $\{{\mathcal{D}}^0_{t} \}_{t\in \mathbb{S}_0^3}$, thus we obtain a $\mathbb{S}^1$--family of overtwisted disks at each fibre. Perform the same procedure for the family of knots $K_{\infty}^p \in
h^{-1}(p)$ to obtain another family of disks $\{ {\mathcal{D}}_{t}^\infty \}_{t \in
\mathbb{S}_{\infty}^3}$. The two families of disks can indeed be disjoint by letting the radius in which we perform Lutz twists be small enough. This construction provides the deformed family of contact structures, coinciding with the previous distribution near the
boundary. Let us remark that the support of the pair of Lutz twists also does not intersect the exceptional divisor. See Figure \ref{fig:exceptional}.\\
\begin{figure}[ht]
\includegraphics[scale=0.4]{exceptional1.pdf}
\caption{The neighborhood of the exceptional divisor intersected with the fibre. The red cilynder is the support of the Lutz twist around $\mathbb{S}^1 \times \{ q_0 \}$, the blue one around $\mathbb{S}^1 \times \{ q_{\infty} \}$. }\label{fig:exceptional}
\end{figure}
We need the base to be the parameter space, instead of a $3$--sphere: restricted to ${\mathcal{B}}_0$ or ${\mathcal{B}}_\infty$ the Hopf fibration becomes trivial and therefore there
exist two sections $s_0: {\mathcal{B}}_0 \longrightarrow \mathbb{S}^3 \cong \mathbb{S}^3_0$ and $s_{\infty}: {\mathcal{B}}_{\infty} \longrightarrow \mathbb{S}^3 \cong \mathbb{S}_{\infty}^3$. The required families are defined as
$$ \{ {\mathcal{D}}_q^0 \} = \{ {\mathcal{D}}^0_{s_0(q)} \}, q\in {\mathcal{B}}_0, $$
$$ \{ {\mathcal{D}}_q^{\infty} \} = \{ {\mathcal{D}}^\infty_{s_{\infty}(q)} \}, q\in {\mathcal{B}}_{\infty}.$$
Note that the two families of overtwisted disks are disjoint since the two families of Lutz twists
were. Further, there exists a small neighborhood of the exceptional divisor $\mathbb{S}^3\times\{0\}$ where no deformation is performed.
\end{proof}
\noindent The global construction can be simply achieved:\\
\noindent{\em Proof of Proposition \ref{propo:disc_family}.}
Apply Lemma \ref{lem:local_family} to a neighborhood of one exceptional sphere $E_0\in E=\{E_0,E_1,\ldots,E_s\}$. The families of overtwisted disks do not meet $C$ or any $E_j$. Indeed, the two families are arbitrarily close to $E_0$ and the exceptional divisors are pairwise disjoint and none of them intersect the critical curves $C$. Thus, maybe after shrinking the neighborhood $U(E_0)$ in the construction, the families are located away from $U(f)$.\hfill $\Box$\\
\noindent Thus we obtain the families of overtwisted disks required to apply Theorem \ref{thm:eliash}. The vertical deformation is described using a suitable cell decomposition of the base $\mathbb{CP}^1$. The vertical contact condition is ensured progressively above the 0--cells, the 1--cells and the 2--cells.
\subsection{Adapted families.}
Let $(f,C)$ be an almost contact fibration. A finite set of oriented immersed
connected curves $T$ in $\mathbb{CP}^1$ will be called an adapted family for $(f,C)$ if
it satisfies the following properties:
\begin{itemize}
\item[-] The image of the set of critical values $f(C)$ is part of $T$. Let $C_i$ denote the image of each
of the components of $C$.
\item[-] Given any element $c\in T$, there exists another element of $c'\in T$ having a non--empty intersection\footnote{In case $c$ has a self--intersection, then $c'=c$ is allowed.} with $c$. Any two elements of $T$ intersect transversally.
\end{itemize}
Let $|T|\subset\mathbb{CP}^1$ be the underlying set of points of the elements of $T$. The elements of an adapted family $T$ that are not of the form $C_i$ are denoted by $F_j$ and referred to as {\it fake} components.
\begin{figure}[ht]
\includegraphics[scale=0.6]{triangula0.pdf}
\caption{Part of an adapted family $T$. The associated
subdivision consists of certain 2--cells with their boundaries being a union of parts of various
elements in the family $T$.}\label{fig:tri0}
\end{figure}
\noindent Let $N\in\mathbb{N}$ be fixed. The insertion of fake curves proves the existence of an adapted family with $\mbox{diam}_{g_0}(\mathbb{CP}^1\setminus|T|)\leq 1/N$, $g_0$ the standard round metric.\\
\noindent There is a cell decomposition of $\mathbb{CP}^1$ associated to an adapted family, the $1$--skeleton being $|T|$. See Figure~\ref{fig:tri0}. We shall first deform in a neighborhood of each vertex relative to the boundary, proceed with a neighborhood of the $1$--cells and finally obtain the vertical contact condition in the $2$--cells. To be precise in the description of the procedure, we introduce some notation.\\
Let $L_j\in T$ be a curve, $U(L_j)$ be a tubular neighborhood and denote $\partial U(L_j)=L_j^0\cup L_j^1$. Suppose that $\bigcup_{j \in J}|L^i_j|$ is isotopic to $|T|$ for both $i=0,1$; this can be achieved by taking a small enough neighborhood of each $L_j$. See Figure \ref{fig:tri3}. We use $V(L_j)$ to denote a slightly larger tubular neighborhood satisfying this same condition. Fix an intersection point $p$ of two elements $L_j,L_k\in T$. Denote by
${\mathcal{A}}_{p}$ the connected component of the intersection of $U(L_j)\cap U(L_k)$ containing $p$. Similarly, let ${\mathcal{V}} {\mathcal{A}}_{p}$ be the connected component of the
intersection of $V(L_j)\cap V(L_k)$ that contains $p$, and denote ${\mathcal{A}}\SA_p={\mathcal{V}} {\mathcal{A}}_{p}\backslash{\mathcal{A}}_p$.\\
The open connected components of $U(T)\backslash\{\bigcup{\mathcal{A}}_p\}$ are homeomorphic to rectangles ${\mathcal{B}}_i$. Here the index $p$ over the intersection points is assumed, as well as a suitable indexing for $i$. The third class of pieces constitute the
interior of the complementary in $\mathbb{CP}^1$ of the open set formed by the union of the sets ${\mathcal{A}}_{p}$ and ${\mathcal{B}}_i$; its connected components are denoted ${\mathcal{C}}_l$. Thus, neighborhoods of the $0$--cells, $1$--cells and $2$--cells are labeled ${\mathcal{A}}_p$, ${\mathcal{B}}_i$ and ${\mathcal{C}}_l$ respectively. See Figure \ref{fig:tri3}.\\
\begin{figure}[ht]
\includegraphics[scale=0.5]{triangula3.pdf}
\caption{In dark gray the sets $A_p$, in light gray the sets $B_i$, associated to
the subdivision of the figure \ref{fig:tri0}.}\label{fig:tri3}
\end{figure}
Finally, we define the sets ${\mathcal{B}}\SB_i$. Let ${\mathcal{B}}_i$ connect a couple of open sets\footnote{Both sets may be the same for the self--intersecting curves.} of the form ${\mathcal{A}}_{p}$.
There exists a curve $L_{{\mathcal{B}}_i}$ contained in ${\mathcal{B}}_i$ which is a part of a curve $L_i\in T$; so $L_{{\mathcal{B}}_i}$ is a $1$--cell in the decomposition associated to the adapted family $T$. Let $L_{{\mathcal{B}}_i}^0$ and $L_{{\mathcal{B}}_i}^\infty$ denote
the two boundary components of ${\mathcal{B}}_i$ which are part of the curves
$L_i^0$ and $L_i^1$ defined above. Then we declare ${\mathcal{B}} {\mathcal{B}}_i^0$ (resp. ${\mathcal{B}} {\mathcal{B}}_i^1$) to be the connected component of $V(L_i)\backslash{\mathcal{B}}_j$ containing the boundary curve $L_i^0$ (resp. $L_i^1$). Their union ${\mathcal{B}} {\mathcal{B}}_i^0\cup{\mathcal{B}} {\mathcal{B}}_i^1$ will be denoted ${\mathcal{B}} {\mathcal{B}}_i$. See figures
\ref{fig:tri4} and \ref{fig:tri7}.\\
\begin{figure}[ht]
\includegraphics[scale=0.4]{triangula4.pdf}
\caption{Example of two components ${\mathcal{V}} {\mathcal{A}}_{p}$ and ${\mathcal{V}} {\mathcal{A}}_{q}$ in light gray, containing ${\mathcal{A}}_{p}$ and ${\mathcal{A}}_{q}$, in dark gray.}\label{fig:tri4}
\end{figure}
\begin{figure}[ht]
\includegraphics[scale=0.45]{triangula7.pdf}
\caption{Example of the sets ${\mathcal{B}} {\mathcal{B}}_i$ for the subdivision of Figure
\ref{fig:tri4}.}\label{fig:tri7}
\end{figure}
\subsection{The vertical construction.} We prove Theorem \ref{thm:vert_defor}. The following lemma is a simple exercise in differential topology (particular case of Ehresmann's fibration theorem). It will be used in the proof of Theorem \ref{thm:vert_defor} and its idea also appears in Lemma \ref{lem:diff_top_sec7}, cf. Section \ref{sect:skeleton}. We include it for completeness.
\begin{lemma} \label{lem:trivializing}
Let $f:E\longrightarrow B^2$ be a locally trivial smooth fibration over the disk with compact
fibres. Assume that $E$ has a smooth closed boundary $\partial E$. Suppose also that there is a collar neighborhood $N$ of $\partial E$ and a
closed submanifold $S$ such that restricting $f$ to $S,N$ or $\partial E$ induces locally trivial fibrations. Let $S_0,N_0$ and $E_0$ be their fibres over $0\in B^2$.\\
\noindent Then there exists a diffeomorphism $g: E \longrightarrow E_0 \times B^2$ making the following diagram commute
$$\xymatrix{
E\ar@{->}[r]^{g}\ar@{->}[d]_\pi & E_0\times B^2\ar@{->}[d]^{\pi_0}\\
B^2\ar@{=}[r] & B^2
}$$
such that $g(N)=N_0 \times B^2$ and $g(S)=S_0 \times B^2$.
\end{lemma}
\begin{proof}
Let $g$ be Riemannian metric in $E$ such that $(TE_z)^{\perp g} \subset TS$ as well as $(TE_z)^{\perp g} \subset T(\partial E)$, for the points $z$ where the condition can be satisfied.
Let $X=\partial_r$ be the radial vector field in $B^2\backslash\{ 0 \}$ and construct the connection associated to the Riemannian fibration:
$$H_{\pi}(e)=(T_e F_{\pi(e)})^{\perp g}.$$
The condition imposed on the Riemannian metric implies that $\partial E$ and $S$ are
tangent to the horizontal connection $H_{\pi}$. Let $\widetilde{X}$ be a lift of $X$ through $H_{\pi}$ and $\phi_t(e)$ the flow of this vector field. Define
\begin{eqnarray*}
E & \stackrel{g}{\longrightarrow} & E_0 \times B^2 \\
e & \longmapsto & (\phi_{(-||\pi(e)||)}(e), \pi(e)).
\end{eqnarray*}
This map satisfies the required properties.
\end{proof}
\noindent {\bf Proof of Theorem \ref{thm:vert_defor}.}
Let $(f,C,E)$ be a good ace fibration. Note that a {\it horizontal} connection $H$ is defined away from $U(C)$ and provides the splitting specified in (\ref{eq:conn_tubular}). Fix an exceptional divisor $E_0$. Use it to apply Proposition \ref{propo:disc_family} to the family of distributions to ensure the existence of at least one overtwisted disk in each fibre. In particular we obtain ${\mathcal{B}}_0$ and ${\mathcal{B}}_\infty$.\\
Let $T$ be an adapted family to the almost contact fibration such that $\partial{\mathcal{B}}_0$ and $\partial{\mathcal{B}}_\infty$ are both contained in two different $2$--cells ${\mathcal{C}}_0$ and ${\mathcal{C}}_\infty$. In order to establish Theorem \ref{thm:vert_defor} we need to perform a deformation which is fixed in a neighborhood of $U(C)$ and leaves the distribution $H$ unchanged, i.e. it should be a strictly vertical deformation.\\
\noindent{\it Deformation at the $0$--cells}: Let $p$ be a vertex with neighborhood ${\mathcal{A}}_p$. The fibration trivializes over ${\mathcal{V}}{\mathcal{A}}_p$ and let $(F_z,\xi_z)$ be the family of fibres and distributions. In case ${\mathcal{V}}{\mathcal{A}}_p$ is small enough the manifolds with boundary ${\mathcal{F}}_z=F_z\backslash (F_z\cap U(C))$ are all diffeomorphic. Let $N_z$ be a collar neighborhood of $\partial{\mathcal{F}}_z$ in which the distribution is contact. Given an exceptional divisor $E_j\in E$ denote by $U(E_j)_z$ the intersection of $U(E_j)$ with the fibre ${\mathcal{F}}_z$. Applying the trivializing diffeomorphism provided in Lemma \ref{lem:trivializing}, we may assume ${\mathcal{F}}_z \times {\mathcal{V}}{\mathcal{A}}_p \cong{\mathcal{F}}$, $U(E_j)_z\times {\mathcal{V}}{\mathcal{A}}_p \cong U(E_j)$ and $N_z\times {\mathcal{V}}{\mathcal{A}}_p \cong N$.\\
\noindent We thus have: a manifold with boundary ${\mathcal{F}}$ with a family of distributions $\xi_z$ parametrized by the topological disk $K'={\mathcal{V}}{\mathcal{A}}_p$ containing $K={\mathcal{A}}_p$. Also a {\it good} set $G$ of submanifold that are already contact for any parameter in $K'$, $G$ consists of the union of $N$, $U(E_j)$ and a neighborhood of one of the two overtwisted disks\footnote{These disks are trivialized along with $N$ using Lemma \ref{lem:trivializing}.}. Let us say a neighborhood of ${\mathcal{D}}^\infty$. A neighborhood of this set will not be perturbed. The remaining disk ${\mathcal{D}}^0$ is contactomorphic to the standard overtwisted disk for each element of the family of distributions. This set--up satisfies the hypothesis of Corollary \ref{coro:eliash2} with $L'=\emptyset$. Since we are able to obtain a deformation relative to the boundary we may perform the deformation at each neighborhood of the $0$--cells and extend trivially to the complement of ${\mathcal{V}}{\mathcal{A}}_p$ in $\mathbb{CP}^1$.\\
\noindent{\it Deformation at the $1$--cells}: Almost the same strategy applied to the $0$--cells applies, although we should not undo the deformation in a neighborhood of the $0$--cells. Corollary \ref{coro:eliash2} allows us to perform deformations relative to a subfamily, so in this case $L'$ will be non--empty. See Figure \ref{fig:Bi}. The reader is invited to precise the remaining details.\\
\begin{figure}[ht]
\includegraphics[scale=0.6]{TheBs.pdf}
\caption{The set $L'\subset{\mathcal{V}} {\mathcal{B}}_i$ is already a contact distribution.} \label{fig:Bi}
\end{figure}
\noindent{\it Deformation at the $2$--cells}: In this situation Theorem \ref{thm:eliash} also applies after a suitable trivialization of the smooth fibration provided by Lemma \ref{lem:trivializing}. The set $L$ is a small tubular neighborhood of the boundary of the $2$--cells. Except at ${\mathcal{C}}_0$ and ${\mathcal{C}}_\infty$, we may use any of the two families of overtwisted disks to apply the result. Let it be ${\mathcal{D}}_z^0$. In the remaining family the distributions are contact and so we include the disks in the set $G$, that also contains $N$ and $U(E_j)$. At ${\mathcal{C}}_0$ we use the family ${\mathcal{D}}_z^0$, since it is the only one well defined over the whole set. Proceed analogously at ${\mathcal{C}}_\infty$. Note that this argument is possible because the deformation is relative to the boundary. Then Theorem \ref{thm:eliash} applies to the $2$--cells and we extend trivially the deformation. We obtain a vertical contact distribution $(F_z,\widetilde{\xi}_z)$ away from $U(C)$.\\
To conclude the statement, consider the direct sum $\widetilde{\xi}_z\oplus H$ to include the critical set, which has not been deformed. This is the required vertical contact structure. Notice that this construction preserves the almost contact class of the distribution since it is performed homotopically only in the vertical direction.
\hfill $\Box$\\
\section{Horizontal Deformation I} \label{sect:skeleton}
Consider the initial almost contact distribution $(M,\xi)$ and a good ace fibration $(f,C,E)$ with associated adapted family $T$. Theorem \ref{thm:vert_defor} deforms $\xi$ to a vertical contact structure with respect to $(f,C,E)$. To obtain a honest contact structure the distribution has to be suitably changed in the horizontal direction. As in the previous section, this is achieved in three stages. The content of this Section consists of the first two of these: deformation in the pre--image of a neighborhood of the $1$--cells of the adapted family $T$. The main result of this Section is the following theorem.
\begin{theorem}
\label{propo:pencil_skeleton}
Let $(M,\xi,\omega)$ be a vertical contact structure with respect to a good ace fibration $(f,C,E)$ and $T$ an adapted family. Then there exists a deformation $(\xi',\omega')$ of $(\xi,\omega)$ relative to $C$ and $E$ such that $(f,C,E)$ is a good ace fibration for $(\xi',\omega')$ and $\xi'$ is a contact structure in the pre--image of a neighborhood of $|T|$.
\end{theorem}
\noindent The resulting distribution $(\xi',\omega')$ is still vertical. In fact, the vertical distribution is fixed along the deformation. In this sense the deformation in the statement is horizontal. The blown--up fibration $(f,C)$ will not be deformed to prove this fact, just the almost contact structure.\\
\noindent Theorem \ref{propo:pencil_skeleton} follows from Proposition \ref{cor:def0} and Proposition \ref{cor:defo_curve}. To prove the statement we trivialize the vertical contact fibration over a neighborhood of the $0$--cells. Then the deformation is performed using an explicit local model. Then we proceed with the pre--image of a neighborhood of the $1$--cells. The same local model is used to deform.
\subsection{Local model} The following lemma is used to prove Proposition \ref{cor:def0} and Proposition \ref{cor:defo_curve}. It is a version of results in Section 2.3 of \cite{El} concerning deformations of a family of distributions near the $1$ and $2$--skeleta of a $3$--manifold. The connectedness condition is stated there as the vanishing of a relative fundamental group.
\begin{lemma} \label{lem:contactness_condition}
Let $\left(F, \xi_{(s,t)}\right)$ be a family of contact structures over a compact 3--manifold
$F$ parametrized by $(s,t)\in [-\varepsilon,
\varepsilon] \times [0,1]$. Let $\alpha_{(s,t)}$ be the associated contact forms. Consider the projection
$$\xymatrix{F\times [-\varepsilon,
\varepsilon] \times [0,1]\ar@[->][r]^{\qquad\pi_s}& F\times[0,1],}$$
and the distribution $\xi$ on $F\times[-\varepsilon,
\varepsilon] \times [0,1]$ defined globally by the kernel of the form
$$\alpha_H(p,s,t) = \alpha_{(s,t)} + H(p,s,t) dt,\qquad H\in C^\infty(F \times [-\varepsilon,
\varepsilon] \times [0,1]).$$
Suppose that $|H(p,s,t)|\leq c \cdot |s|$ and $\xi$ is constant along the $s$-lines, i.e. $\partial_s \alpha_{(s,t)}=0$. Assume that the $1$--form $\alpha_H$ is a contact form in a compact set $G$ such that $\pi_s^{-1}(p,s)\cap G$ is connected and contains one of the endpoints of the interval $\pi_s^{-1}(p,s)$.\\
\noindent Then, there is a small perturbation $\widetilde{H}$ of $H$ relative to $G$ such that $\alpha_{\widetilde{H}}$ defines a contact structure. In precise terms, $|\widetilde{H}-H|\leq 3c\varepsilon$ and $\widetilde{H}|_G=H|_G$.
\end{lemma}
\begin{proof}
Let us compute the contact condition on $\alpha=\alpha_H$.
$$d\alpha= d\alpha_{(s,t)} + dH \wedge dt\quad\Longrightarrow\quad (d\alpha)^n = \left(d\alpha_{(s,t)}\right)^n + \left(d\alpha_{(s,t)}\right)^{n-1} \wedge dH \wedge dt.$$
Therefore, the contact condition is described as
$$(d\alpha)^n \wedge \alpha = \left(d\alpha_{(s,t)}\right)^n \wedge Hdt +
\left(d\alpha_{(s,t)}\right)^{n-1} \wedge \alpha_{(s,t)} \wedge dH \wedge dt>0$$
Since $\partial_s\alpha_{(s,t)}=0$ the first
term of the right hand side of the equation is zero and
$$(d\alpha)^n \wedge \alpha = \left(d\alpha_{(s,t)}\right)^{n-1} \wedge \alpha_{(s,t)} \wedge(
\partial_s H\cdot ds \wedge dt).$$
Thus, the $1$--form $\alpha$ is a contact form if and only if $\partial_s H >0$.\\
\noindent Given $p\in F,t\in[0,1]$, $\pi_s^{-1}(p,t)$ is a $4$--parametric family of $1$--dimensional manifolds. The connectedness of $\pi_s^{-1}(p,t)\cap G$ and the compactness of $G$ assure that it is possible to
perturb $H$ to an $\widetilde{H}$ relative to $G$ and satisfying the contact condition. Indeed, the connectedness condition allows us to perturb at least one end of a curve in $F\times[-\varepsilon,\varepsilon]\times[0,1]$ and obtain a function $\widetilde{H}$ with $\partial_s\widetilde H>0$.
\end{proof}
\subsection{Deformation along intersection points.}\label{ssec:defpt}
In this subsection we obtain a contact structure in a neighborhood of the fibres over a neighborhood of the intersection points in $f(C)$. The precise statement reads as follows:
\begin{proposition}\label{cor:def0}
Let $(M,\xi,\omega)$ be a vertical contact structure with respect to a good ace fibration $(f,C,E)$ and $T$ an adapted family. Then there exists a deformation $(\xi',\omega')$ of $(\xi,\omega)$ relative to $C$ and $E$ such that $(f,C,E)$ is a good ace fibration for $(\xi',\omega')$ and $\xi'$ is a contact structure in the pre--image of a neighborhood of the 0--cells of $|T|$.
\end{proposition}
\noindent In order to apply \ref{lem:contactness_condition} we consider a trivialization relative to $C$ and $E$ and with a particular condition on the parallel transport. This is rather technical. The following lemmata exploit the ideas of Lemma ~\ref{lem:trivializing}. We introduce the required notation.\\
\noindent Let $z$ be a point of intersection of the adapted family $T$ and $(\phi,U)$ a sufficiently small chart centered at $z$ with polar coordinates
$(r,\theta)$. Recall the definition of the open set $U(f)=U(C)\cup U(E)$, i.e. $U(f)$ is a small open neighborhood of the critical set union the exceptional divisors. Let $N=f^{-1}(U)\backslash U(f)$. We shall also denote by $f$ the possible restrictions of the homonymous fibering map.\\
\noindent Denote by ${\mathcal{F}}\cong F\setminus (U(f)\cap F)$ the fibre of $f|_N$ over $z$. The map $f:N\longrightarrow U$ is a smooth fibration provided that the neighborhoods are small enough, and thus the fibres are diffeomorphic to ${\mathcal{F}}$. Restricting $f$ to the boundary $\partial N$ we also obtain a smooth fibration whose fibre is $\partial{\mathcal{F}}$. The collar neighborhood theorem provides a neighborhood $U_{\partial N}$ of the boundary $\partial N$ such that
$f: U_{\partial N}\subset N \longrightarrow U$ is a smooth fibration and the fibre $U_{{\mathcal{F}}}$ is diffeomorphic to $\partial{\mathcal{F}} \times
[0,\varepsilon]$. Let $\psi:U_{{\mathcal{F}}} \longrightarrow \partial {\mathcal{F}}\times
[0,\varepsilon]$ be such a diffeomorphism. Note that the almost contact structure $(\xi, \omega)$ is contact on $U_{\partial N}$, because $(f,C,E)$ is a good ace fibration.\\
\noindent The trivialization is carried out using the flow of the radial vector field lifted using a boundary preserving connection. Let us work with a slight modification of the manifold ${\mathcal{F}}$. Concretely, let $L\subset U_L$ be such that
$$\psi(L)=\partial {\mathcal{F}}\times\{\varepsilon/2\},\qquad \psi(U_L)= \partial {\mathcal{F}}\times[\varepsilon/2,3\varepsilon/4].$$
Then ${\mathcal{F}}\backslash\psi^{-1}(\partial {\mathcal{F}}\times[0,\varepsilon/2))$ is a manifold with boundary $L$ and collar neighborhood $U_L\subset U_{{\mathcal{F}}}$. To ease notation we still call this shortened manifold ${\mathcal{F}}$. In this situation, we are able to perform a trivialization respecting the vertical contact condition:
\begin{lemma}\label{lem:diff_top_sec7}
Let $(F,\Xi)$ be a contact manifold with boundary and $f:\O\longrightarrow B^2$ a fibre bundle with typical fibre $F$. Consider a vertical almost contact structure $(\O,\xi)$ restricting to $(F,\partial F,\Xi)$ in the fibre. Suppose that $(\O,\xi)$ is a contact structure in a collar neighborhood ${\mathcal{C}}_F$ of the boundary $\partial(f^{-1}(B^2))\setminus\partial(f^{-1}(\partial B^2))$. Then there exists $\varepsilon\in\mathbb{R}^+$, a flow $\varphi_t$ on $F$ and a fibre--preserving diffeomorphism
$$\xymatrix{
\tau:\tau^{-1}(F\times B^2(\varepsilon))\ar@[->][r] & F\times B^2(\varepsilon)
},\quad \tau^{-1}(F\times B^2(\varepsilon))\subset\O$$
$$p\longmapsto \tau(p)=\left(\varphi_{(-||f(p)||)}(p),f(p)\right),$$
such that $\tau_{*}(\xi)$ is still a vertical contact structure with respect to the product fibration over $B^2(\varepsilon)$, and the family $\{\xi_z = \tau_{*}(\xi)|_{f^{-1}(z)}\}_{z\in B^2(\varepsilon)}$ is constant when restricted to $\tau({\mathcal{C}}_F)$.
\end{lemma}
\begin{proof}
The idea is contained in the proof of Lemma~\ref{lem:trivializing}. We use the almost contact connection to trivialize the fibration $f:\O\longrightarrow
B^2$. Near the boundary the distribution $\xi$ is a contact structure, therefore the parallel transport is by contactomorphisms.
\noindent Let $\varphi_t$ be the flow of the lift of the radial vector field. The diffeomorphism $\varphi_t(p)$ is defined for $p\in\O\backslash\partial\O$ and a finite time depending on $p$. A uniform time $\varepsilon$ is obtained by shortening the manifold and restricting to a compact neighborhood near the boundary as before the statement of the Lemma. Define
$$\xymatrix{
\tau:\tau^{-1}(F\times B^2(\varepsilon)\ar@[->][r] & F\times B^2(\varepsilon)
},\quad\tau(p)=\left(\varphi_{(-||f(p)||)}(p),f(p)\right).$$
The splitting of $\tau_*\xi$ must now be performed using $\tau_*\omega$. The connection has been chosen in order that the vertical factor of the distribution $\tau_*\xi$ is a contact structure once restricted to the fibres $F$. Since the family of distributions was already a contact distribution close to the boundary, the parallel transport is performed along contactomorphisms and the family is indeed constant in the image of the collar neighborhood ${\mathcal{C}}_F$.
\end{proof}
\begin{remark}\label{rmk:lem_diff}
Suppose that the almost contact connection preserves the collar neighborhood $\mathcal{C}_F$. Then the radius $\varepsilon\in\mathbb{R}^+$ in Lemma \ref{lem:diff_top_sec7} can be assumed to be $\varepsilon=1$. This follows from the argument since in this case the flow $\varphi_t$ is defined for $t\in[0,1]$.
\end{remark}
\noindent Lemma \ref{lem:diff_top_sec7} applies to the situation described above once we have used the chart $(U,\phi)$. Indeed, in this situation we use $\O=N$, $F={\mathcal{F}}$ and the fibration onto $B^2$ is given by the map $\phi\circ f|_N:N\longrightarrow B^2$. Hence the fibration provided by the good ace fibration $(f,C,E)$ restricted to the pre--image of a small neighborhood of $z$ inside the manifold can be trivialized preserving the vertical contact condition.\\
\noindent In order to be able to apply Lemma ~\ref{lem:contactness_condition} we prove the existence of a deformation such that at least in one direction the parallel transport along the deformed almost contact connection is a contactomorphism. This allows us to trivialize with the almost contact connection and obtain a vertical contact distribution constant along that direction. Thus conforming the hypotheses of Lemma \ref{lem:contactness_condition}. This is the content of the subsequent lemma.\\
\noindent Let $I^2 \subset B^2$ denote a small closed rectangle with coordinates $(s,t)\in [0,1] \times [0,1]$. The appearance of both cartesian and polar coordinates can be avoided, however it is more natural to prove the previous lemma in polar coordinates and describe the following local model in cartesian coordinates.
\begin{lemma} \label{lem:first_def0}
Let $(F,\Xi)$ be a contact manifold with boundary. Consider a vertical almost contact structure $(F\times B^2,\xi)$ restricting to $(F,\Xi)$. There exists a horizontal deformation of $\xi$, supported in the pre--image of a small disk $D^2$ containing $I^2$, such that the parallel transport along the lift of $\partial_s$ through the deformed almost contact connection is a contactomorphism.
\end{lemma}
\begin{proof}
The almost contact distribution $\xi$ splits as
$$\xi=\xi_v\oplus \xi_h$$
Let us consider the vertical part $\xi_v$ restricted to the preimage of $I^2$. To ease notation, denote it by $\xi_{(s,t)}$. Perform Moser's trick to each $1$--parametric family $\xi_{(s,t)}= \ker \alpha_{(s,t)}$ where the $t$--coordinate is fixed. We obtain an $s$--family of diffeomorphisms
$$m_s^t:F\longrightarrow F\mbox{ such that }\left(m_s^t\right)_*\xi_{(s,t)}=\xi_{(0,t)}.$$
Note that these diffeomorphisms are supported away from any region where the family is constant. Moser's argument can be made parametric and the family $m_s^{t}$ can be chosen to behave smoothly with respect to the $t$--coordinate. We use this family of diffeomorphisms in the fibration to define a diffeomorphism
$$J:F \times I^2 \longrightarrow F \times I^2,\qquad (p,s,t)\longmapsto(m_{s}^{t}(p),s,t).$$
Choose a vertical distribution at each fibre constant equal to the distribution $\xi_{(0,t)}$. The condition $\left(m_s^t\right)_*\xi_{(s,t)}=\xi_{(0,t)}$ implies that $J_*\left(\xi_v\right)=\xi_{(0,t)}$.\\
\noindent We construct the appropriate distribution. Let $\tau_0=\xi_h$ and $\tau_1$ be the $2$--distribution given by $TB^2$ inside $TF\oplus TB^2$. They are isotopic through horizontal distributions. Let $\tau_l$ be such an isotopy and $\chi:D^2\longrightarrow[0,1]$ be a smooth decreasing function such that
$$\chi(s,t)=1,\mbox{ for }(s,t)\in I^2; \hspace{1cm} \chi(s,t)=0,\mbox{ for }(s,t)\in \partial D^2.$$
The required distribution in a small rectangle of the trivialization is
$$ \widetilde{\eta}\left(p,s,t\right)= \tau_{\chi(s,t)}\left(p,s,t\right).$$
The parallel transport induced by the lift of the vector field
$\frac{\partial}{\partial s}$ consists of the contactomorphisms on $I^2$ obtained through Moser's argument.
\end{proof}
\noindent Lemma \ref{lem:first_def0} allows us, up to a fibre preserving diffeomorphism, to consider the
contact structure near the central fibre of $f:N\longrightarrow U$ to be given as the pull--back of a
contact structure
$$\alpha_{(s,t)} + H(p,s,t) ds\mbox{, satisfying }\partial_s\alpha_{(s,t)} = 0.$$
\noindent Let us gather these results and conclude Proposition \ref{cor:def0}.\\
\noindent {\bf Proof of Proposition \ref{cor:def0}}: Both Lemma \ref{lem:diff_top_sec7} and Lemma \ref{lem:first_def0} provide a vertical almost contact structure $(\widetilde{\xi},\widetilde{\omega})$ that coincides with the original almost contact distribution $(\xi,\omega)$ away from a neighborhood of the fibres over the intersection points and in a neighborhood of the boundary $\partial N$. This almost contact structure is also homotopic to the original distribution on $M$ and parallel transport through a lift of at least one direction to the almost contact connection of $f:N \longrightarrow U$ consists of contactomorphisms.\\
\noindent In conclusion, there exists $\varepsilon>0$, a neighborhood $V(z)$ of $z$ and a trivializing map $\psi$ such that the following diagram commutes:
$$\xymatrix{
f^{-1}(V(z))\ar@[->][r]^{\psi} \ar@[->][d]^f & F \times B^2\left(\varepsilon\right)\ar@[->][d] \\
V(z)\ar@[->][r]^{\phi}& B^2(\varepsilon)
},$$
and the fibre of $f$ is a contact manifold $F$ with boundary. Since the map $\psi$ is defined as a flow, a smaller neighborhood might be required but such neighborhood exists because of compactness. Lemma ~\ref{lem:first_def0} has been used, hence $\psi_*\xi$ satisfies the hypothesis of Lemma~\ref{lem:contactness_condition} for some small rectangle $I^2 \subset B^2\left(\varepsilon/2\right)$. Note that the set $G$ can be taken to be a suitable neighborhood of the boundary $\partial F\times B^2(\varepsilon)$. Applying Lemma \ref{lem:contactness_condition} to the domain $F \times I^2$ and interpolating back to the almost contact structure $\left(\psi_* \widetilde{\xi}, \psi_k* \widetilde{\omega}\right)$ in $F \times
((B^2(\varepsilon)\backslash B^2(\varepsilon/2))$ we obtain an almost contact structure whose pull--back through $\psi$ has the required properties once we extend it to $M$ by $\xi$.$\hfill\Box$\\
\subsection{Deformation along curves.}
Once we have achieved the contact condition in a neighborhood of the fibres over the $0$--skeleton, we proceed with a neighborhood of the fibres over the $1$--skeleton. Let $\mathbb{T}$ denote a neighborhood of $T$.\\
\begin{proposition}\label{cor:defo_curve}
Let $(M,\xi,\omega)$ be a vertical contact structure with respect to a good ace fibration $(f,C,E)$ and $T$ an adapted family. Suppose that $(M,\xi)$ is a contact structure on a neighborhood $\mathbb{O}$ of the fibres over the $0$--cells of $T$. Then there exists a deformation $(\xi',\omega')$ of $(\xi,\omega)$ relative to $C$, $E$ and $\mathbb{O}$ over the $0$--cells of $T$ such that $(f,C,E)$ is a good ace fibration for $(\xi',\omega')$ and $\xi'$ is a contact structure in the pre--image of $\mathbb{T}$.
\end{proposition}
\noindent Let $\SS$ be a small neighborhood of the set of fibres over $\mathbb{T}\backslash\mathbb{O}$.
See Figure \ref{fig:def_domain}.
\begin{figure}[h]
\includegraphics[scale=0.6]{Neigh.pdf}
\caption{The deformation domains.}\label{fig:def_domain}
\end{figure}
The argument applied over $\mathbb{O}$ in the previous subsection works analogously when applied to $\SS$. Thus, no detailed proof is given. The only subtlety lies in the appropriate choice of the compact set $G$ when Lemma~\ref{lem:contactness_condition} is applied.\\
\noindent Let $z,w\in\mathbb{C}\mathbb{P}^1$ with corresponding neighborhood ${\mathbb{O}}_z,{\mathbb{O}}_w$; we focus on a line segment $S\subset |T|$ joining these two points. Let $\left(\phi,U\right)$ be a local chart around $S\backslash\left({\mathbb{O}}_z\cup{\mathbb{O}}_w\right)$ with cartesian coordinates $(s,t)$ such that
$$\phi(U)=[-\varepsilon,\varepsilon]\times[0,1],\qquad \phi(S)=\{0\}\times[0,1].$$
The existence of a suitable trivialization is obtained with the same arguments used in Lemma \ref{lem:first_def0}. In precise terms, we use the following pair of lemmata.
\begin{lemma}
Let $(F,\Xi)$ be a contact manifold with boundary and $f:\O\longrightarrow [-1,1]\times[0,1]$ a fibre bundle with typical fibre $F$. Consider a vertical almost contact structure $(\O,\xi)$ restricting to $(F,\partial F,\Xi)$ in the fibre. Suppose that $(\O,\xi)$ is a contact structure in a collar neighborhood ${\mathcal{C}}_F$ of the boundary $\partial(f^{-1}([-1,1]\times[0,1]))\setminus\partial(f^{-1}(\partial([-1,1]\times[0,1])))$. Then there exists $\varepsilon\in\mathbb{R}^+$, a flow $\varphi_t$ on $F$ and a fibre--preserving diffeomorphism
$$\xymatrix{
\tau:\tau^{-1}(F\times [-\varepsilon,\varepsilon]\times[0,1])\ar@[->][r] & F\times [-\varepsilon,\varepsilon]\times[0,1]
},\quad \tau^{-1}(F\times [-\varepsilon,\varepsilon]\times[0,1])\subset\O$$
$$p\longmapsto \tau(p)=\left(\varphi_{(-||f(p)||)}(p),f(p)\right),$$
such that $\tau_{*}(\xi)$ is still a vertical contact structure with respect to the product fibration over $[-\varepsilon,\varepsilon]\times[0,1]$, and the family $\{\xi_{(s,t)} = \tau_{*}(\xi)|_{f^{-1}(s,t)}\}_{(s,t)\in (-\varepsilon, \varepsilon) \times[0,1]}$ is constant when restricted to $\tau({\mathcal{C}}_F)$.\hfill$\Box$
\end{lemma}
\noindent This is proven as Lemma \ref{lem:diff_top_sec7} using the vector field $\partial_s$ instead of the radial vector field $\partial_r$.
\begin{lemma} \label{lem:first_def1}
There exist an arbitrarily small neighborhood $\mathbb{S}$ of $S$ and a horizontal deformation of the vertical almost
contact structure $(\xi,\omega)$ supported in the pre--image of $\mathbb{S}$, relative to the pre--images of $\mathbb{S}\cap\mathbb{O}_z$ and $\mathbb{S}\cap\mathbb{O}_w$, and conforming the following properties:
\begin{itemize}
\item[-] The distribution is deformed relative to $U(f)$, where it was already a contact distribution.
\item[-] There exists a trivialization $(\phi,U)$ such that the parallel transport of the associated almost contact connection along the vector field $\phi^*\partial_s$ consists of contactomorphisms.
\end{itemize}
\end{lemma}
\begin{figure}[h]
\includegraphics[scale=0.35]{convexity.pdf}
\caption{The deformation curves $\phi^*\partial_s$.}\label{fig:convex}
\end{figure}
\noindent This is proven with the same methods used in subsection \ref{ssec:defpt}.\\
\noindent{\bf Proof of Proposition \ref{cor:defo_curve}.} Trivialize the vertical contact fibration using Lemma~\ref{lem:first_def1}. Choose the coordinates in the trivialization in such a way that the curves which provide the lift of $\phi^*\partial_s$ have at most one of the ends in the fibres over the $0$--skeleton. See Figure \ref{fig:convex}. This allows us to choose a compact set $G$ containing the fibres over the two endpoints plus a neighborhood of the boundary of all the fibers such that the intersection of $G$ with any such arc is connected. There might be the need to progressively shrink the neighborhoods of the fibres over the $0$--skeleton. Apply Lemma~\ref{lem:contactness_condition} to produce a contact structure in a
neighborhood of the fibres over $1$--skeleton without perturbing the existing contact structure in
a small neighborhood of fibres over the endpoints.$\hfill\Box$
\section{Fibrations over the $2$--disk.}\label{sec:bands}
\noindent Let $(F,\xi_v)$ be a contact $3$--manifold, $\xi_v=\ker\alpha_v$ and $\mathbb{D}^2$ a 2--disk. In this Section we study contact structures on the product manifold $F\times\mathbb{D}^2$. Consider the coordinates $(p,r,\theta)\in F\times\mathbb{D}^2$. The existence part of Theorem \ref{main} can be essentially reduced to the existence of a contact structure on $F\times\mathbb{D}^2$ restricting to a prescribed contact structure on a neighborhood of the boundary $F\times\partial\mathbb{D}^2$. This is explained in Section \ref{sec:end}.\\
\noindent Fix an $\varepsilon\in(0,1)$ and consider $H\in C^\infty(F\times\mathbb{D}^2(1))$ to be a smooth function such that $\partial_r H>0$ for $r\in(1-\varepsilon,1]$. Then the $1$--form
$$\alpha=\alpha_v+H(p,r,\theta)d\theta$$
defines a distribution $\xi=\ker\alpha$. It can be endowed with the symplectic form
$$\omega=d\alpha_v+(1-\tau(r))\cdot rdr\wedge d\theta+\tau(r)dH\wedge d\theta,$$
where $\tau:[0,1]\longrightarrow[0,1]$ is an strictly increasing smooth function such that
$$\tau(x)=0\mbox{ for }x\in [0, 1-\varepsilon]\mbox{ and }\tau(x)=1\mbox{ for }x\in [1-\varepsilon/2,1].$$
Then $(\xi,\omega)$ is an almost contact structure on $F\times\mathbb{D}^2(1)$ which is a contact structure on the neighborhood $F\times(1-\varepsilon/2,1]\times\mathbb{S}^1$ of the boundary $F\times\partial\mathbb{D}^2(1)$.\\
\noindent The main result in this Section is the following:
\begin{theorem}\label{thm:band}
Let $(F,\xi_v)$ be a contact $3$--manifold with $c_1(\xi_v)=0$, $\xi_v=\ker\alpha_v$ and $L$ a transverse link. Given $\varepsilon\in(0,1)$, consider a function $H\in C^\infty(F\times\mathbb{D}^2(1))$ such that $\partial_rH>0$ in $r\in(1-\varepsilon,1]$, and the almost contact structure
$$(\xi,\omega)=(\ker(\alpha_v+H(p,r,\theta)d\theta),d\alpha_v+(1-\tau(r))\cdot rdr\wedge d\theta+\tau(r)dH\wedge d\theta),$$
where $\tau$ is the function described above.\\
\noindent Then there exists a $1$--parametric family of almost contact structures $\{(\xi_t, \omega_t)\}$, constant along the boundary $F\times\partial\mathbb{D}^2(1)$ and with $(\xi_0, \omega_0)=(\xi,\omega)$ such that:
\begin{enumerate}
\item[a.] $(\xi_1, \omega_1)=(\ker\alpha, d\alpha)$ is a contact structure for some contact form $\alpha$ on $F\times\mathbb{D}^2(1)$.
\item[b.] The submanifold $L\times\mathbb{D}^2(1)$ is a contact submanifold of $(F\times\mathbb{D}^2(1),\xi_1)$ and the induced contact structure is a full Lutz twist along $L\times\{0\}$.
\end{enumerate}
\end{theorem}
\noindent In coordinates $(z,r, \theta) \in L\times\mathbb{D}^2(1)$, the contact structure obtained by a full Lutz twist of a neighborhood of $L$ along $L$ is described as
\begin{equation*}
\xi_{|L\times\mathbb{D}^2(1)} = \ker (\cos (2\pi r) dz + r \sin (2\pi r) d\theta).
\end{equation*}
\noindent This theorem is used to conclude Theorem \ref{main} in Section \ref{sec:end}. In brief, it is used to deform the almost contact structure over the $2$--cells of the decomposition associated to an adapted family $T$ of a vertical good ace fibration $(f,C,E)$. In this description of the fibration over the $2$--cells, the part corresponding to the exceptional divisors is the submanifold $L\times\mathbb{D}^2(1)$. Although the deformation in the statement is not relative to a neighborhood of them, the resulting contact structure is described in the part b. of Theorem \ref{thm:band}.\\
\noindent{\bf Example.} Suppose that the function $H\in C^\infty(F\times\mathbb{D}^2(1))$ also satisfies
$$H(p,1, \theta)>0,\mbox{ for all }(p,\theta)\in F\times\mathbb{S}^1.$$
The contact condition for the initial form $\alpha_v+ H(p,r,\theta)d\theta$ is $\partial_rH>0$. Consider a smooth family $\{H_t\}_{t\in[0,1]}$ of functions in $F\times\mathbb{D}^2(1)$ such that
$$H_0=H,\quad H_1(p,0,\theta)=0,\quad \partial_r H_1>0\mbox{ for }r\in(0,1]\mbox{ and }H_t(p,1,\theta)=H_0(p,1,\theta).$$
Suppose that $H_1$ vanishes quadratically at the origin. Then $\alpha_t=\alpha_v+H_t(p,r,\theta)d\theta$ is a family of almost contact distributions constant along the boundary $F\times\partial\mathbb{D}^2(1)$ such that $\ker\alpha_1$ is a contact structure. The corresponding symplectic structures on $\ker\alpha_t$ is readily constructed as in the previous discussion, and an interpolation to the symplectic form $\alpha_v+dH_1\wedge d\theta$ is required to obtain the almost contact structure $(\ker\alpha,d\alpha)$. This contact structure does conform the first property in Theorem \ref{thm:band}, but not necessarily the second one. The construction in the proof differs from that in this example and satisfies both properties.\\
\noindent Theorem \ref{thm:band} also covers harder cases, such as almost contact distributions defined by functions $H(p,r,\theta)$ with positive and negative values along $F\times\partial\mathbb{D}^2(1)$.
\subsection{The model.}
In this subsection we describe the model used to obtain the contact structure in the statement of Theorem \ref{thm:band}.\\
\noindent Consider the manifold $F\times\mathbb{S}^2$. The submanifolds
$$i_0:F_0=F\times\{(1,0,0)\}\longrightarrow F\times\mathbb{S}^2\mbox{ and }i_\infty:F_\infty=F\times\{(-1,0,0)\}\longrightarrow F\times\mathbb{S}^2$$
are referred to as the fibres at zero and infinity. A construction made relative to $F_\infty$ should be thought as construction on $F\times\mathbb{D}^2(1)$ relative to the boundary. In the manifold $\mathbb{S}^1\times\mathbb{S}^2$ there exists a unique tight contact structure and a unique overtwisted contact structure isotopic to it. The latter is obtained by performing a full Lutz twist in the former along $\mathbb{S}^1\times\{0\}$. This is said to be the standard overtwisted structure on $\mathbb{S}^1\times\mathbb{S}^2$.\\
\noindent The basic geometric construction used to prove Theorem \ref{thm:band} is the content of the following result. A minor enhancement of the Proposition is also required, it is explained in Corollary \ref{coro:model}.
\begin{proposition} \label{propo:model}
Let $(F,\xi_v)$ be a contact $3$--manifold with $c_1(\xi_v)=0$, $\xi_v=\ker\alpha_v$ and $L$ a transverse link. Consider the manifold $F\times\mathbb{S}^2$, $\omega_{\mathbb{S}^2}$ the standard area form on $\mathbb{S}^2$ and the almost contact structure
$$(\xi,\omega)=(\ker\alpha_v, d\alpha_v +\omega_{\mathbb{S}^2}).$$
\noindent Then there exists a contact structure $\xi_f= \ker \alpha_f$ on $F\times\mathbb{S}^2$ conforming the properties:
\begin{enumerate}
\item[a.] The contact form $\alpha_f$ restricts to the initial contact form at the fibres $F_0$ and $F_\infty$:
$$i_0^* \alpha_f = \alpha_v\mbox{ and }i_{\infty}^* \alpha_f = \alpha_v.$$
\item[b.] Consider the inclusion $i_L: L \times S^2=\bigsqcup (\mathbb{S}^1 \times \mathbb{S}^2)\longrightarrow F \times \mathbb{S}^2$. Then the contact form $i_L^* \alpha_f$ is the standard overtwisted form on each $\mathbb{S}^1 \times \mathbb{S}^2$.\\
\item[c.] The almost contact structures $(\xi,\omega)$ and $(\ker \alpha_f, d\alpha_f)$ are homotopic relative to $F_\infty$.
\end{enumerate}
\end{proposition}
\begin{proof} This is a rather long proof. It is divided according to the construction and the verification of each of the three properties.\\
\noindent {\bf Construction.} Since $c_1(\xi_v)=0$, there exist a global framing $\{ X_1, X_2 \in \Gamma(\xi_v)\}$ of the contact distribution $\xi_v$. Denote by $X_0$ the Reeb vector field associated to the contact form $\alpha_0=\alpha_v$. Therefore $\{ X_0, X_1, X_2 \}$ is a global framing of $TF$. Let $\{ \alpha_0, \alpha_1, \alpha_2 \}$ be the dual framing. Denote the standard embedding of the $2$--sphere as $e=(e_0, e_1, e_2): \mathbb{S}^2 \longrightarrow \mathbb{R}^3$. It is a computation to verify that
$$\lambda = e_0 \cdot \alpha_0 + e_1 \cdot \alpha_1 + e_2 \cdot \alpha_2 $$
is a contact form on $F\times \mathbb{S}^2$. The important properties are that $\{ \alpha_0, \alpha_1,\alpha_2 \}$ is a framing and the map $e$ is a star--shaped embedding. \\
\noindent In spherical coordinates $(t,\theta)\in[0,1]\times[0,1]$ the embedding can be described as
\begin{eqnarray}
e_0(t,\theta) & = & \cos (\pi t), \nonumber \\
e_1(t,\theta) & = & \sin (\pi t) \cos (2\pi \theta), \nonumber \\
e_2(t,\theta) & = & \sin (\pi t) \sin(2\pi \theta). \nonumber
\end{eqnarray}
\noindent Note that $F_{\infty}=F \times (-1,0,0)$ and $F_0=F \times (1,0,0)$ are contactomorphic contact submanifolds of $(F\times\mathbb{S}^2,\ker\lambda)$ with trivial normal bundle. Consider two copies of $F\times\mathbb{S}^2$, we can perform a contact fibered sum along their $F_{\infty}$ fibres, see \cite{Ge}. This operation is done in order to obtain two fibres with the contact form $\alpha_0$. Those coming from the two zero fibres $F_0$ in the two copies of $F\times\mathbb{S}^2$. Let us provide an explicit equation for the contact form in this connected sum. \\
\noindent A tentative modification of $\lambda$ is obtained by considering the following map
\begin{eqnarray*}
\kappa_0(t,\theta) & = & \cos (2\pi t), \\
\kappa_1(t,\theta) & = & \sin (2\pi t) \cos (2\pi \theta), \\
\kappa_2(t,\theta) & = & |\sin (2\pi t)| \sin(2\pi \theta),
\end{eqnarray*}
and the $1$--form $\kappa_0\cdot\alpha_0 +\kappa_1\cdot\alpha_1+\kappa_2\cdot\alpha_2$. Due to the appearance of the absolute value this form is just continuous. Observe though that in the smooth area it is a contact form. Let us perturb it to a smooth $1$--form.\\
\noindent Define a smooth map $t:[0,1]\longrightarrow[0,1]$ such that:
$$t(0)=0,\mbox{ }t(1/2)=1/2,\mbox{ }t(1)=1, t'(v)>0\mbox{ for }v\in[0,1/2) \cup (1/2,1]\mbox{ and }t^{(k)}(1/2)=0\mbox{ }\forall k\in\mathbb{N}.$$ \\
\noindent This allows us to reparametrize the sphere with coordinates $(v,\theta)\in[0,1]\times[0,1]$. The following map is denoted by $(e_0,e_1,e_2)$ in order to ease notation. This should not lead to confusion since the map formerly referred to as $(e_0,e_1,e_2)$ is not to be considered again. Consider the smooth map
\begin{eqnarray*}
e_0(v,\theta) & = & \cos (2\pi t(v)), \\
e_1(v,\theta) & = & \sin (2\pi t(v)) \cos (2\pi \theta), \\
e_2(v,\theta) & = & |\sin (2\pi t(v))| \sin(2\pi \theta).
\end{eqnarray*}
It is indeed smooth because $t^{(k)}(1/2)=0$. This almost provides the desired $1$--form for the fibre connected sum. Define the smooth function $h(v)= v(1-v) \sin (2\pi v)$ and the $1$--form $\eta=c \cdot h(v)d\theta$, where $c$ is a small positive constant.\\
\noindent{\bf Assertion.} There exists a choice of $c\in\mathbb{R}^+$ such that the $1$--form defined as
\begin{equation}
\alpha_f = e_0 \alpha_0 + e_1 \alpha_1 + e_2 \alpha_2- \eta \label{eq:aT}
\end{equation}
is a contact form over the fibre connected sum of two copies of $F\times\mathbb{S}^2$ along the fibres $F_\infty$.\\
\noindent This concludes the construction of the contact form in the manifold $F\times\mathbb{S}^2$ obtained in the Theorem. The contact form $\alpha_f$ also conforms property a. in the statement of the Theorem.\\
\noindent{\bf Proof of Assertion.} Consider the following volume form $\nu = \sin(\pi v)dv \wedge d\theta \wedge \alpha_0 \wedge \alpha_1 \wedge \alpha_2$ on $F \times \mathbb{S}^2$ and compute the exterior differential
\begin{equation*}
d\alpha_f = de_0 \wedge \alpha_0 + de_1\wedge \alpha_1 + e_1 \wedge \alpha_2 + e_0 d\alpha_0 + e_1 d\alpha_1 + e_2 d\alpha_2 - d\eta.
\end{equation*}
\noindent The contact condition states that $\alpha_f\wedge(d\alpha_f)^2$ is a positive of $\nu$. Let us express it as
\begin{eqnarray*}
\alpha_f \wedge (d\alpha_f)^2 & = & \eta_1 + c\eta_2 + c\eta_3,
\end{eqnarray*}
where $\eta_1,\eta_2,\eta_3$ are the following $5$--forms:
\begin{eqnarray*}
\eta_1 & = & \left|
\begin{array}{ccc} e_0 & e_1 & e_2 \\
\partial_t e_0 & \partial_t e_1 & \partial_t e_2 \\
\partial_\theta e_0 & \partial_\theta e_1 & \partial_\theta e_2 \\\end{array} \right| t'(v)^2 dv \wedge d\theta \wedge \alpha_0 \wedge \alpha_1 \wedge \alpha_2 = \\[10pt] & = & 4\pi^2|\sin(2\pi t(v))|(t'(v))^2dv \wedge d\theta \wedge \alpha_0 \wedge \alpha_1 \wedge \alpha_2, \\[10pt]
\eta_2 & = & - e_0^2\cdot h'(v)\cdot\alpha_0 \wedge d\alpha_0 \wedge dv \wedge d\theta, \\[10pt]
\eta_3 & = & -\sum_{i+j\geq1} (e_i\cdot e_j\cdot h'(v))\cdot\alpha_i \wedge d \alpha_j \wedge dv \wedge d\theta + \sum_{i,j} (e_i\cdot h(v))\cdot de_j\wedge d\alpha_i \wedge \alpha_j \wedge d\theta.
\end{eqnarray*}
The indices belong to $i,j\in\{0,1,2\}$. Evaluating at $v=1/2$ we obtain:
\begin{eqnarray*}
\eta_2 (p, 1/2, \theta) & = & \frac{\pi}{2} \alpha_0 \wedge d\alpha_0 \wedge dv \wedge d\theta = \frac{\pi}{2} dv \wedge d\theta \wedge \alpha_0 \wedge \alpha_1 \wedge \alpha_2,\\
\eta_1 (p, 1/2, \theta)& = & 0,\\
\eta_3 (p, 1/2, \theta)& = & 0.
\end{eqnarray*}
\noindent Therefore, there is a small constant $\delta>0$ such that the $5$--form $\eta_2+\eta_3$ is a positive volume form in the region $F \times [1/2-\delta, 1/2 + \delta] \times [0,1]$. The function $t(v)$ is strictly increasing except at $v=1/2$. Hence, there exists a constant $B>0$ such that $t'(v) >B$ for any $v\in [0, 1/2-\delta] \cup [1/2+\delta, 1]$.\\
\noindent Let us write $\eta_1(p,v,\theta) = g_1(p, v, \theta) \nu$ and $\eta_2 +\eta_3 = g_2(p,v,\theta) \nu$. There exist constants $C,M\in\mathbb{R}^+$ such that $g_1> C>0$ for $v\in [0, 1/2-\delta] \cup [1/2+\delta, 1]$, and $|g_2| \leq M$.\\
\noindent Choose the initial constant $c\in\mathbb{R}^+$ to satisfy $cM \leq C$. Then we obtain the following bound for $v\in [0, 1/2-\delta] \cup [1/2+\delta, 1]$:
$$\alpha_f \wedge (d\alpha_f)^2 = \eta_1 + c\eta_2 + c\eta_3 = (g_1 +c g_2) \nu> C-cM\geq0.$$
Hence the form $\alpha_f$ is a contact form in this region. The following bound holds in the remaining region $v\in [1/2-\delta,1/2+\delta]$:
$$\alpha_f \wedge (d\alpha_f)^2 = \eta_1 + c\eta_2 + c\eta_3 = (g_1 +c g_2) \nu> cg_2\geq0.$$
\noindent Thus $\alpha_f$ is a contact form in the fibre connected sum $F\times\mathbb{S}^2$.\hfill$\Box$\\
\noindent {\bf Property b.} Choose the contact form $\alpha_v$ associated to $\xi_v$ such that its Reeb vector field $X_0$ is tangent to the link $L$. Restricting the contact form $\alpha_f$ in the equation (\ref{eq:aT}) to the submanifold we obtain
\begin{equation}
i_L^* (\alpha_T) = \cos(2\pi t(v)) dz - c v(1-v) \sin (2\pi v) d\theta , \label{eq:OT}
\end{equation}
where $(z,v,\theta)\in \mathbb{S}^1\times\mathbb{S}^2$. This is an equation of the standard overtwisted contact structure on each $\mathbb{S}^1 \times \mathbb{S}^2$. Indeed, consider $a(v)=\cos(2\pi t(v))$ and $b(v)=v(1-v)\sin(2\pi v)$. Then the curve parametrized by $(a(v),b(v))$ rotates once around the origin and the tangent vector field $(a'(t), b'(t))$ is transverse to the radial direction, i.e. $\partial_r$, on $(0,1)$.\\
\noindent {\bf Property c.} Let $f_F:F \longrightarrow [0,1]$ be a Morse function on the $3$--manifold $F$ with a single minimum $q\in F$. Then
$$f(p,v, \theta)= f_F(p) - (1+f_F(p))v^2: F\times\mathbb{S}^2 \longrightarrow [-1,1]$$
is a smooth Morse function on $F \times \mathbb{S}^2$ whose critical points belong to the central fibre $F_0$. Let us use the associated cell decomposition relative to the level $f^{-1}((-\infty, -1])= F_{\infty}$. It is generated by the descending manifolds associated to each critical point. It has a unique $2$--cell $\sigma_q^2= \{q \} \times (\mathbb{S}^2 \backslash \{\infty \})$, corresponding to the critical point $(q,0,0)$.\\
\noindent Due to Lemma \ref{lem:2sk}, a pair of almost contact distributions homotopic over the disk $\sigma_q^2$ relative to its boundary are homotopic on the $5$--manifold $F\times\mathbb{S}^2$. To conclude Property c. we verify that such relative homotopy exists along $\sigma_q^2$. The almost contact distribution $\xi$ in the statement of the Proposition can be written as $\xi=\ker \alpha_v \oplus T\mathbb{S}^2$. Its symplectic structure is induced by the symplectic structure on each of the factors. Note that both $\ker\alpha_v$ and $T\mathbb{S}^2$ are $\operatorname{rk}_\mathbb{R}=2$ symplectic bundles. This is tantamount to $\operatorname{rk}_\mathbb{R}=2$ oriented bundles.\\
\noindent Consider a trajectory $\gamma$ of the Reeb flow through $q$
$$\gamma:(-\varepsilon,\varepsilon)\longrightarrow F,\quad \gamma(0)=q.$$
The submanifold $(V,\xi_{ot})=(\gamma \times \mathbb{S}^2,\xi_f|_{\gamma \times \mathbb{S}^2})$ is a contact submanifold of the contact manifold $(F\times\mathbb{S}^2,\ker\alpha_f)$. A contact from is given by the equation (\ref{eq:OT}). As suggested by the notation, the contact form $\alpha_{ot}=\alpha_f|_V$ defines the standard overtwisted structure on $(-\varepsilon, \varepsilon)\times(\mathbb{S}^2\setminus\{\infty\})$.\\
\noindent Hence the two subbundles of $TV$
$$\xi_{ot}\longrightarrow\sigma_q^2,\quad T\mathbb{S}^2\longrightarrow\sigma_q^2$$
are homotopic as oriented subbundles relative to the boundary of the disk. Thus relative homotopic as symplectic bundles. This provides a homotopy in the $2$--dimensional horizontal part. Let us deal with the vertical bundle.\\
\noindent The initial vertical subbundle is $\xi_v=\ker\alpha_v$, it does satisfy the splitting
$${\xi_v}_{|\sigma_q^2} \oplus TV_{|\sigma_q^2}=T(F\times\mathbb{S}^2)_{|\sigma_q^2}.$$
The resulting vertical subbundle in the distribution $\xi_f$ can be constructed as the symplectic orthogonal subbundle $\nu_{ot}$ of $\xi_{ot}$. This yields the decomposition
$${\nu_{ot}}_{|\sigma_q^2} \oplus TV_{|\sigma_q^2}=T(F\times\mathbb{S}^2)_{|\sigma_q^2}.$$
The space of rank--2 oriented vector bundles transverse to the rank--3 vector bundle $TV$ is contractible. Hence ${\nu_{ot}}_{|\sigma_q^2}$ is homotopic to ${\xi_v}_{|\sigma_q^2}$ as rank--2 symplectic distributions.\\
\noindent On the unique 2--cell $\sigma_q^2$ both splittings $\xi=\xi_v \oplus T\mathbb{S}^2$ and $\xi_f= \nu_{ot} \oplus \xi_{ot}$ hold. Since the subbundles are pairwise homotopic as symplectic distributions, $\xi$ and $\xi_f$ are also homotopic as symplectic distributions.
\end{proof}
\noindent In the proof of Property c. of Proposition \ref{propo:model} we have only used the 2--skeleton to verify the statement. Obstruction theory ensures that this is enough. There is an alternative geometric approach to produce the homotopy. Indeed, the Reeb trajectories of $\alpha_v$ produce a foliation ${\mathcal{L}}$ on $F$. This induces a foliation ${\mathcal{L}}\times\mathbb{D}^2$ with $3$--dimensional contact leaves. The argument in the proof of Property c. can be made parametric to construct an explicit almost contact homotopy.\\
\noindent The norm of the function $H$ in the statement of Theorem \ref{thm:band} does translate into a geometric feature. This is the size of a certain neighborhood. This is explained in the subsequent subsection. Let us enhance the conclusion of Proposition \ref{propo:model} in order to obtain an arbitrarily large contact neighborhood of a fibre.\\
\noindent{\bf Property d.} Let $R\in\mathbb{R}^+$ be given. There exists a neighborhood $U_\infty$ of the fibre $F_\infty$ and a trivializing diffeomorphism $\psi: F \times \mathbb{D}^2(R) \to U_{\infty}$ such that
\begin{itemize}
\item[-] $\psi(F \times \{ 0 \})=F_{\infty}$,
\item[-] $\psi^* \alpha_f = \alpha_v + r^2d \theta$.
\end{itemize}
\noindent This property could have been included in the statement of Proposition \ref{propo:model}. It is stated apart to ease the comprehension.
\begin{corollary} \label{coro:model}
There exists a contact manifold $(F\times\mathbb{S}^2,\xi_f=\ker\alpha_f)$ conforming a. to d.
\end{corollary}
\begin{proof}
The contact structure $(F\times\mathbb{S}^2,\xi_f=\ker\alpha_f)$ obtained in Proposition \ref{propo:model} does satisfy properties a.-- c. Let us modify it in order to satisfy Property d. The contact neighborhood theorem provides a neighborhood $U_{\infty}$ of the fiber $F_{\infty}$ and a contactomorphism $\psi_\varepsilon: F \times\mathbb{D}^2(\varepsilon) \to U_{\infty}$, for some $\varepsilon\in\mathbb{R}^+$. In case $R\leq \varepsilon$ the statement follows. \\
\noindent Suppose $R\geq\varepsilon$. Let $k\in\mathbb{N}$ be an integer and consider the ramified covering
\begin{eqnarray*}
\phi_k: F \times \mathbb{S}^2 = F \times \mathbb{CP}^1 & \longrightarrow & F \times \mathbb{CP}^1 \\
(p,z) & \longmapsto & (p,z^k).
\end{eqnarray*}
The branch locus consists of the fibres $F_0$ and $F_{\infty}$. Both fibres are contact submanifolds in $(F\times\mathbb{S}^2,\ker\alpha_f)$ and we can lift the contact form to a contact form $\alpha_f^k= \phi_k^* \alpha_f$ in the domain of the covering map. Lifting the formula (\ref{eq:aT}), we obtain
\begin{equation}
\alpha_f^k = \cos(2\pi t(v)) \alpha_0 + \sin (2\pi t(v)) \cos (2\pi k\theta) \alpha_1 + |\sin (2\pi t(v))| \sin(2 \pi k\theta) \alpha_2+ k\eta \label{eq:aTk}
\end{equation}
\noindent Hence properties a.-- c. are still satisfied by the contact structure $\ker\alpha_f^k$. Regarding Property d, observe that $\psi^* \alpha_f^k= \alpha_v +kr^2 d\theta$. Consider the scaling diffeomorphism
\begin{eqnarray*}
g_k: F \times \mathbb{D}^2(\sqrt{k}\cdot\varepsilon) & \longrightarrow & F \times \mathbb{D}^2(\varepsilon) \\
(p, r, \theta) & \longmapsto & (p, r/\sqrt{k}, \theta).
\end{eqnarray*}
Then the trivializing diffeomorphism $\psi_\varepsilon\circ g_k$ satisfies $(\psi_\varepsilon \circ g_k)^* \alpha_f^k = \alpha_v + r^2 d\theta$. Choose $k\in\mathbb{N}$ such that $\sqrt{k}\cdot \varepsilon \geq R$ to conclude the statement.
\end{proof}
\noindent To ease notation, we can refer to the contact structures resulting either of Proposition \ref{propo:model} or Corollary \ref{coro:model} as $\xi_f$. Since the latter has better properties than the former, $\xi_f$ refers to that in Corollary \ref{coro:model}.
\subsection{The proof.} In this subsection we conclude the proof of \ref{thm:band}. The essential geometric ideas have been introduced in Proposition \ref{propo:model}. The necessary details to conclude are provided.\\
\noindent Let us introduce a definition. It is given in order to stress the relevance of the size in a neighborhood.
\begin{definition}
Let $(F,\xi_v=\ker \alpha_v)$ be a contact manifold. For $A \in\mathbb{R}^+$, the manifold $F \times [-A,A] \times\mathbb{S}^1$ with the contact structure $\alpha_A= \alpha_v + td\theta$ is called the $A$--standard contact band associated to $(F,\ker\alpha_v)$.
\end{definition}
\noindent The role of this definition is elucidated in the following lemma.
\begin{lemma} \label{lem:emb}
Let $(F,\xi_F)$ be a contact manifold, $\xi_F=\ker\alpha_F$. Consider a contact manifold $(F \times [0, 1] \times \mathbb{S}^1,\xi)$ with contact form $\alpha_F + Hd\theta$, $H\in C^\infty(F\times[0,1]\times\mathbb{S}^1)$.\\
\noindent Suppose that $|H|< A$, for some $A\in\mathbb{R}^+$. Then, there exists a strict contact embedding of $(F \times [0, 1] \times \mathbb{S}^1,\alpha)$ in the $A$--standard contact band associated to $(F, \alpha_F)$.
\end{lemma}
\begin{proof}
The embedding is defined as
\begin{eqnarray*}
\Psi_A: F \times [0, 1] \times \mathbb{S}^1 & \longrightarrow & F \times [-A,A] \times \mathbb{S}^1 \\
\left(p, t, \theta\right) & \longrightarrow & \left(p, H(p,t,\theta), \theta\right).
\end{eqnarray*}
\end{proof}
\noindent The remaining ingredient for the proof of Theorem \ref{thm:band} is the subsequent lemma.\\
\noindent Let $l\in\mathbb{R}^+$ be a constant. Consider a smooth function $\kappa_l:[0,2l+1]\longrightarrow[0,l]$ with
$$\kappa_l(r)=0\mbox{ for }r\in [0,l],\quad \kappa_l(r)=r-l-1\mbox{ for }r\in [2l,2l+1].$$
Consider $(r,\theta)\in\mathbb{D}^2_l$ to be polar coordinates for the $2$--disk $\mathbb{D}_l^2$ of radius $2l+1$.
\begin{lemma} \label{lemma:model}
Let $(F,\xi_v)$ be a contact $3$--manifold with $c_1(\xi_v)=0$, $\xi_v=\ker\alpha_v$, $C\in \mathbb{R}^+$ and $L$ a transverse link. Consider the standard area $\omega_{\mathbb{D}}$ on the $2$--disk $\mathbb{D}^2_l$ and the almost contact structure on $F\times\mathbb{D}^2_l$ described as
$$(\xi,\omega)=(\ker(\alpha_v+ \kappa_l(r)d\theta), d\alpha_v +\omega_{\mathbb{D}}).$$
Then there exists a contact structure $\xi_1=\ker\alpha_1$ on $F\times\mathbb{D}^2_l$ such that:
\begin{enumerate}
\item[A.] The region $F\times[1,2l+1]$ is an $l$--standard contact band for $(F,\ker\alpha_v)$:
$$\alpha_1|_{F\times[1,2l+1]} = \alpha_v + (r-l-1)d\theta.$$
\item[B.] Consider the inclusion $i_L: L \times\mathbb{D}^2_l=\bigsqcup (\mathbb{S}^1\times\mathbb{D}^2_l) \longrightarrow F \times\mathbb{D}^2_l$. Then the contact form $i_L^* \alpha_1$ is the standard overtwisted form on each $\mathbb{S}^1 \times\mathbb{D}^2_l$.\\
\item[C.] $(\xi,\omega)$ and $(\xi_1,d\alpha_1)$ are homotopic relative to the boundary $F\times\partial\mathbb{D}^2_l$.
\end{enumerate}
\end{lemma}
\begin{proof}
Consider Property d with radius $R=\sqrt{l}$. Let $(F\times\mathbb{S}^2,\xi_f=\ker\alpha_f)$ be the contact manifold obtained in Corollary \ref{coro:model}. Then there exists a contact neighborhood $U_\infty$ of the fibre $F_{\infty}$ and a trivializing diffeomorphism
\begin{equation*}
\psi: F \times [0,\sqrt{l}] \times \mathbb{S}^1 \longrightarrow U_\infty\mbox{ such that }\psi^*\alpha_f = \alpha_v + r^2 d\theta.
\end{equation*}
Note that $\psi$ also identifies $\psi:F\times(0,\sqrt{l}]\times\mathbb{S}^1\longrightarrow U_\infty\setminus F_\infty$.\\
\noindent Define the following map
$$m: F\times [-l,0) \times \mathbb{S}^1\longrightarrow F\times (0,\sqrt{l}] \times \mathbb{S}^1,\quad m(p,x,\theta)= (p, \sqrt{-x}, -\theta).$$
It satisfies $(\psi\circ m)^* \alpha_f = \alpha_v +rd\theta$. This form extends to the region $F\times [-l,l] \times \mathbb{S}^1$ with the same expression.\\
\noindent Then the manifold $F\times\mathbb{D}^2_l$ is obtained by gluing the regions $F\times[0,\sqrt{l}]\times\mathbb{S}^1$ and $F\times[-l,l]\times\mathbb{S}^1$ via the contactomorphism $\psi\circ m$. The construction implies that Property A holds. Properties B and C follow from Properties b and c in Corollary \ref{coro:model} since the manifold $(F\times\mathbb{S}^2)\setminus F_\infty$ satisfies them.
\end{proof}
\noindent {\bf Proof of Theorem \ref{thm:band}.}
The function $H$ is $C^0$--bounded on the compact manifold $F\times\mathbb{D}^2(1)$. Let $l\in\mathbb{R}^+$ be an upper bound, $\|H\|_{C^0}< l$. Consider a smooth function $h\in C^\infty(F\times[0,1]\times\mathbb{S}^1)$ such that
\begin{itemize}
\item[-] $h(p,r,\theta)=0$ for $r\in[0,1-2\varepsilon]$,
\item[-] $h(p,r,\theta)=r-l-(1-\varepsilon)$ for $r\in[1-\varepsilon,1-3\varepsilon/4]$,
\item[-] $\partial_rh>0$ for $r\in[1-3\varepsilon/4,1-\varepsilon/2]$,
\item[-] $h(p,r,\theta)=H(p,r,\theta)$ for $r\in[1-\varepsilon/2,1]$.\\
\end{itemize}
\noindent The almost contact structure $(\xi,\omega)$ is homotopic relative to the boundary to the almost contact structure defined by
$$(\xi_h,\omega_h)=(\ker(\alpha_v+h(p,r,\theta)),d\alpha_v+(1-\tau(r))\cdot rdr\wedge d\theta+\tau(r)dh\wedge d\theta).$$
\noindent The homotopy is provided by a relative homotopy between the functions $h(p,r,\theta)$ and $H(p,r,\theta)$ and Lemma \ref{lem:split}. Hence the departing the almost contact structure can be considered to be $(\xi_h,\omega_h)$ .\\
\noindent The neighborhood $F\times(1-\varepsilon,1]\times\mathbb{S}^1$ of the boundary $F\times\partial\mathbb{D}^2(1)$ is a contact manifold. By Lemma \ref{lem:emb}, it embeds in an $l$--standard contact band $F\times[-l,l]\times\mathbb{S}^1$. Denote such an embedding by $\phi$.\\
\noindent Consider the almost contact manifold $(F\times\mathbb{D}^2_l,\ker\alpha_1)$ in the statement of Lemma \ref{lemma:model}. Property A implies the existence of a contactomorphism $\iota$ embedding the $l$--standard contact band in a neighborhood of size $2l$ of the boundary of $F\times\mathbb{D}^2_l$.\\
\noindent The composition $\iota\circ\phi$ embeds a neighborhood of the boundary $F\times\{1-\varepsilon\}\times\mathbb{S}^1$ via
$$(\iota\circ\phi)(p,r,\theta)=(p,r+\varepsilon,\theta).$$
\noindent The required contact structure in the statement of Theorem \ref{thm:band} is obtained by extending the contact structure induced by $(F\times\mathbb{D}^2_l,\ker\alpha_1)$ to the area $F\times[0,1-\varepsilon]\times\mathbb{S}^1$. This is achieved with a coordinate transformation
$$F\times[0,1]\times\mathbb{S}^1\longrightarrow F\times[0,1-\varepsilon]\times\mathbb{S}^1,\quad (p,r,\theta)\longmapsto (p,c(r),\theta),$$
where $c:[0,1]\longrightarrow[0,1-\varepsilon]$ is a smooth function such that
\begin{itemize}
\item[-] $c(t)=t$ near $t=0$,
\item[-] $c(t)=t-\varepsilon$ near $t=1$,
\item[-] $c'(t)>0$ for $t\in[0,1]$.
\end{itemize}
\noindent Property B in Lemma \ref{lemma:model} implies Property b in the Theorem. Property C ensures that the obtained almost contact structure is homotopic to the initial almost contact structure relative to the boundary.\hfill $\Box$
\section{Horizontal Deformation II} \label{sec:end}
The arguments in the previous sections are gathered to conclude the proof of Theorem \ref{main}.
\subsection{Contact Structure in the fibration}
\begin{theorem} \label{thm:filling}
Let $(M,\xi,\omega)$ be an almost contact structure and $(f,C,E)$ a good ace fibration adapted to it. There exists a contact distribution $\xi'$ homotopic to $\xi$. The restriction of $\xi'$ to the exceptional $3$--spheres in $E$ induces the unique overtwisted contact structure homotopic to the standard contact structure $\xi_{std}$.
\end{theorem}
\noindent A neighborhood of the intersection of an exceptional $3$--sphere with a fibre of $f$ is diffeomorphic to $\mathbb{S}^1\times\mathbb{D}^2\times\mathbb{D}^2$. Let $(z,r,\theta,\rho,\phi)$ be coordinates for such a neighborhood, the triple $(z,\rho,\phi)$ belong to the fibre. It can be considered as a trivial fibration over the first pair of factors
$$\pi:\mathbb{S}^1\times\mathbb{D}^2\times\mathbb{D}^2\longrightarrow\mathbb{S}^1\times\mathbb{D}^2,\quad (z,r,\theta,\rho,\phi)\longmapsto (z,r,\theta).$$
There also exists a contact structure given by the contact form $\alpha=dz+r^2d\theta+\rho d\phi$ on the neighborhood. This induces a contact connection $A_{\pi}$ for the fibration $\pi$. Let $\delta\in\mathbb{R}^+$ and suppose the horizontal $2$--disk $(\rho,\phi)\in\mathbb{D}^2(\delta)$ is of radius $\delta$.
\begin{lemma} \label{lem:parallel}
Consider the contact manifold $(\mathbb{S}^1\times\mathbb{D}^2\times\mathbb{D}^2(\delta),\ker(dz+r^2d\theta+\rho d\phi))$, $\pi$ the projection onto the first pair of factors and $A_\pi$ the associated contact connection. The flow of the lift of $\partial_r$ to $A_\pi$ preserves the submanifold $\{(z,r,\theta, \rho, \phi)\in X: \rho=\delta/2 \}.$
\end{lemma}
\begin{proof}
The vector field $\partial_r$ belongs to the contact distribution. The vertical directions are generated by $\partial_\rho,\partial_\phi$ and the symplectic form pairs them via $\rho\cdot d\rho\wedge d\phi$. Hence $\partial_r$ is itself the lift to $A_\pi$. The statement follows.
\end{proof}
\noindent {\bf Proof of Theorem \ref{thm:filling}.}
Apply Theorem \ref{thm:vert_defor} to the almost contact structure and the given good ace fibration. Chosen an adapted family $T$ for $(f,C,E)$ and use Theorem \ref{propo:pencil_skeleton} to obtain a distribution which is a contact structure away from a disjoint union of pre--images of the balls $\{B_1,\ldots,B_a\}\subset \mathbb{CP}^1$. Let us still refer to this distribution as $\xi$. The distribution $\xi$ is a contact structure in the fibres close to the boundary of $\{B_i\}$, maybe after enlarging the balls if necessary. The restriction of $f$ to the preimages of each $B\in\{B_i\}$ is a smooth fibration since the critical values of $f$ lie in the complement of the set $B_1\cup\ldots\cup B_a$. In order to conclude the statement of the Theorem we produce a deformation over each ball $B\in\{B_i\}$ supported away from the boundary and resulting in a contact structure. The deformation can then be extended to a global deformation.\\
\noindent The attentive reader is probably able to conclude the proof of the statement since the results in Section \ref{sec:bands} are the essential ingredients. Nevertheless, let us precise the necessary details regarding the trivializations. Choose the ball $B_1\in\{B_1,\ldots,B_a\}$ and a trivializing diffeomorphism $\varphi:B \longrightarrow B^2(1)$. Consider $U=f^{-1}(B)$ and the map
$$g=\varphi\circ f: U \longrightarrow B^2(1).$$
For $\varepsilon>0$ a small constant, we may assume that $g^{-1}(B^2(1)\backslash B^2 (1-\varepsilon))$ is an open set where the distribution $\xi$ is a contact structure. The interior must be deformed, the boundary has already been deformed in the previous sections.\\
\noindent The only boundary contribution to this fibration is given by the neighbourhood of the exceptional divisors. Consider an exceptional divisor $E$. According to the local model used in the contact blow--up, there exists a neighbourhood $\mathcal{E}$ of $E$ and a contactomorphism
$$\varphi_E: (\mathbb{S}^3 \times\mathbb{D}^2(\delta),\alpha_{std}+ \rho^2 d\phi) \longrightarrow\mathcal{E}.$$
The composition $f \circ \varphi_E: \mathbb{S}^3 \times\mathbb{D}^2(\delta) \longrightarrow \mathbb{S}^2$ restricts to the Hopf fibration at $\mathbb{S}^3\times\{0\}$. Restricting to the region $f^{-1}(U)\cap\mathcal{E}$ we obtain a fibration
$$\varphi\circ f\circ\varphi_E:\mathbb{S}^1\times B^2(1) \times\mathbb{D}^2(\delta) \longrightarrow B^2(1)$$
over the 2--ball. Lemma \ref{lem:parallel} implies that the contact parallel transport along the neighbourhoods of the boundary is tangent to it. Therefore we can apply Lemma \ref{lem:diff_top_sec7} and Remark \ref{rmk:lem_diff} to obtain a radius--1 trivialization of the fibration $g$ respecting the vertical contact condition and preserving the boundary. In precise terms, we obtain a trivializing map
$$\tau: U \longrightarrow F \times B^2(1),\mbox{ such that }\tau_* \xi = \ker (\alpha_{(r,\theta)} + \widetilde{G}dr + \widetilde{H} d\theta)$$
using the parallel transport along the almost contact connection. The corresponding trivialization for the symplectic structure is also obtained. The trivialization is performed with the radial direction and hence $\widetilde{G}=0$. However, the fact that the contact form $\alpha_{(r,\theta)}$ depends on the point $(r,\theta)\in\mathbb{D}^2$ beclouds the vanishing of $\widetilde{G}$. Theorem \ref{thm:band} only applies to contact fibrations with constant vertical contact structure. Let us achieve this.\\
\noindent Consider a fixed angle $\theta\in\mathbb{S}^1$ and the radial family of contact structures
$$\xi_{(r, \theta)}=\ker \alpha_{(r,\theta)},\mbox{ for }r\in [0,1].$$
In a neighborhood of the boundary $\partial F$ this family has a constant contact structure. Hence Gray's stability theorem applies, relative to $\partial F$, to produce a family $m_{(r,\theta)}$ of diffeomorphisms such that
$$m_{(r,\theta)}: F\longrightarrow F,\quad (m_{(r,\theta)})_* \xi_0 = \xi_{(r, \theta)}.$$
This radial family depends smoothly in the parameter $\theta\in \mathbb{S}^1$. Consider the map
$${\mathcal{M}}: F \times B^2(1) \longrightarrow F \times B^2(1), \quad {\mathcal{M}}(p,r,\theta)= (m_{(r, \theta)}(p, r, \theta), r, \theta).$$
It trivializes the distribution $\xi$ as
$$({\mathcal{M}} \circ \tau)_* \xi= \ker \{ \alpha_0 + G(p,r,\theta) dr + H(p,r,\theta) d\theta \}. $$
In a neighborhood ${\mathcal{U}}$ of the boundary $\{r=1 \}$ the distribution $\xi$ is a contact structure and the almost contact connection is a honest contact connection in ${\mathcal{U}}$. Thus the flow obtained using Gray's stability coincides with the radial parallel transport by contactomorphisms in ${\mathcal{U}}$. In particular $G$ vanishes on ${\mathcal{U}}$ because the lift of the radial direction is contained in the contact distribution.\\
\noindent The appearance of the function $G$ in the trivialization does not ease the attainment of a contact distribution. Perform a homotopy of almost contact structures relative to the boundary, by using Lemma \ref{lem:split}, to obtain an almost contact structure $\widetilde{\xi}$ such that
$$(M\circ \tau)_* \widetilde{\xi} = \ker (\alpha_{0} + H d\theta).$$
\noindent Let us denote $\xi=\widetilde{\xi}$. This setup satisfies the hypotheses of Theorem \ref{thm:band}. It applies producing a homotopy $\xi_t$ of almost contact structures over $U$ relative to its boundary such that $\xi_0=\xi$ and $\xi_1$ is a contact structure. The exceptional divisors remain contact submanifolds and as contact submanifolds of $\xi_1$, their induced contact structure is the standard contact structure $\xi_{std}$ with a full Lutz twist performed. The construction is made relative to the pre--image of a neighborhood of the boundary of the ball $B$. The argument successively applies to the elements of $B=\{B_1,\ldots,B_a\}$. This concludes the statement.\hfill $\Box$
\subsection{Interpolation at the exceptional divisors.}
Let $(M,\xi,\omega)$ be an almost contact manifold. The argument for proving Theorem \ref{main} begins with a good almost contact pencil $(f,C,E)$. It is then blown--up to obtain a good ace fibration. The results in Section 6, 7 and 8 confer ace fibrations. These exist not on the manifold $(M,\xi,\omega)$ but in an almost contact blow--up. A contact structure has been obtained in the almost contact blow--up such that a neighborhood of the exceptional spheres has remained contact. It is left to perform an appropriate contact blow--down and obtain a contact structure in the initial manifold $M$.\\
\noindent The exceptional spheres in $(\widetilde{M},\widetilde{\xi})$ have the standard tight contact structure $(\mathbb{S}^3,\xi_{std})$ at the beginning of the argument. In the deformation performed in Section \ref{sec:bands} the exceptional spheres become overtwisted. Hence the contact blow--down procedure cannot be performed directly. This has a simple solution. We deform the contact distribution on a neighborhood of the exceptional spheres to the standard one. This is the content of the following
\begin{theorem} \label{thm:dishant}
Let $(\mathbb{S}^3 \times B^2(4),\xi_0)$ have the contact form
\begin{equation}
\eta= \alpha_{ot} + \delta\cdot r^2 d\theta, \label{eq:ot_ne}
\end{equation}
where $\delta\in\mathbb{R}^+$ is a constant and $\alpha_{ot}$ is any contact form associated to an overtwisted contact structure homotopic to the standard contact structure on $\mathbb{S}^3$.\\
\noindent Then there exists a deformation $\xi_1$ of $\xi_0$ supported in $\mathbb{S}^3 \times B^2(3)$ such that the $\xi_1$ is a contact structure and $\mathbb{S}^3 \times \{ 0 \}$ inherits the standard contact structure.
\end{theorem}
\noindent This result is a consequence of Lemma 3.2 in \cite{EP}. Let us give an alternative argument, pointed out to us by Y. Eliashberg. \\
\noindent {\bf Proof of Theorem \ref{thm:dishant}.} Let us begin with the standard contact $3$--sphere $(\mathbb{S}^3,\xi_{std})$. Performing a Lutz twist along a given transverse trivial knot $K$ produces an overtwisted contact structure $\xi^1_{ot}$ in $\mathbb{S}^3$ homotopic to $\xi_{std}$ as almost contact distribution. The contact structure $\xi^1_{ot}$ is isotopic to the contact structure $\xi^2_{to}=\ker\alpha_{ot}$. Consider both a trivial Legendrian knot $L\subset(\mathbb{S}^3,\xi_{std})$ whose positive transverse push--off is $K$, and its Legendrian push--off $L'$ with two additional zig--zags. According to \cite{DGS} a Lutz twist along $K$ is tantamount to a contact (+1)--surgery along $L$ and $L'$. Hence, given $(\mathbb{S}^3,\xi^1_{ot})$ there exists a ($-$1)--surgery on $(\mathbb{S}^3,\xi^1_{ot})$ producing $(\mathbb{S}^3,\xi_{std})$. Such surgery provides a Liouville cobordism $(W,\lambda)$ from $(\mathbb{S}^3,\xi^1_{ot})$ to $(\mathbb{S}^3,\xi_{std})$.\\
\noindent The cobordism obtained by a (+1)--surgery along $L$ and $L'$ is smoothly trivial, see \cite{DGS}. Consider $\theta\in\mathbb{S}^1$ and $\eta^1=\lambda+\mu\cdot d\theta$, for a constant $\mu\in\mathbb{R}^+$. Then the contactization $(W\times\mathbb{S}^1,\eta^1)$ of the exact symplectic manifold $(W,\lambda)\cong(\mathbb{S}^3\times[0,1],\lambda)$ is diffeomorphic to $\mathbb{S}^3\times[0,1]\times\mathbb{S}^1$. We have obtained a contact structure on the $3$--sphere times the annulus such that the inner boundary $\mathbb{S}^3\times\{0\}$ has fibres $(\mathbb{S}^3,\xi_{std})$, and $(\mathbb{S}^3,\xi^1_{ot})$ are the fibres of the outer bundary $\mathbb{S}^3\times\{1\}$. The inner part is a convex boundary and it can be filled with the contact manifold
$$(\mathbb{S}^3\times\mathbb{D}^2,\ker(\alpha_{std}+r^2d\theta))$$
in order to obtain a contact structure on $\mathbb{S}^3\times\mathbb{D}^2$ with $(\mathbb{S}^3,\xi_{std})$ as central fibre. For a choice of $\mu$ small enough, there exists a small constant $\delta\in\mathbb{R}^+$ such that in a neighborhood $\mathbb{S}^3\times(1-\varepsilon,1]\times\mathbb{S}^1$ of the outer boundary the contact structure can be expressed as
$$\eta^1=\alpha^1_{ot}+\delta\cdot r^2d\theta.$$
The contact forms $\alpha^1_{ot}$ and $\alpha^2_{ot}=\alpha_{ot}$ are isotopic via a family of contact forms $\{\alpha^r_{ot}\}$, $r\in[1,2]$. On the manifold $\mathbb{S}^3\times[1,4]\times\mathbb{S}^1$ consider the 1--form
$$\eta^2=\widetilde{\alpha}_{ot}+\delta\cdot r^2d\theta\mbox{ for }r\in[1,2]\mbox{ and } \eta^2=\alpha^2_{ot}+\delta\cdot r^2d\theta \mbox{ for }r\in[2,4]$$
where $\widetilde{\alpha}_{ot}(p,r,\theta)=\alpha^r_{ot}(p)$. The form $\eta^2$ is a contact form because the form $r^2d\theta$ does not depend on the point $p\in\mathbb{S}^3$. The gluing of the contact forms $\eta^1$ and $\eta^2$ is the required contact structure $\xi_1$ on $\mathbb{S}^3\times B^2(4)$. \hfill$\Box$\\
\noindent Notice that this deformation gives a homotopy of almost contact structures.
\subsection{Proof of Theorem \ref{main}.}
\noindent Let $(M,\xi,\omega)$ be an almost contact structure. Proposition \ref{prop:good_ac} allows us to construct a good almost contact pencil. Then Theorem \ref{thm:blow-up} provides a good ace fibration on an almost contact blow--up $(\widetilde{M},\widetilde{\xi},\widetilde{\omega})$. We apply Theorem \ref{thm:filling} to this almost contact manifold. The construction provides the standard overtwisted structure on the exceptional spheres since a sequence of full Lutz twists are performed. Let us use Theorem \ref{thm:dishant} to deform the contact structure to be standard near the exceptional spheres. Lemma \ref{lem:down} allows us to blow--down along the exceptional divisors to obtain a contact structure over the initial manifold $M$. It is only required to use the same parameter $k$ in the choice of framing in the blow--up construction and use the same $k$ in the blow--down process. This concludes the proof of the existence of a contact structure $\xi'$ in the manifold $M$.\\
\noindent Let us prove that $\xi$ and $\xi'$ are homotopic as almost contact distributions. This has been proven except in the blow--down process. That is, suppose that two distributions $\tilde{\xi}$ and $\tilde{\xi}'$ are homotopic in $\widetilde{M}$ and they coincide along the exceptional divisors $(\mathbb{S}^3,\xi_{std})$. Then the two resulting distributions are also homotopic after performing a blow--down. Indeed, the blow--down distributions corresponding to $\xi$ and $\xi'$ are homotopic over $M\setminus B$. Let us consider a cell decomposition of the manifold $M$ such that $B$ contains only $4$ and $5$--cells. Such decomposition exists due to genericity of transversality. Then Lemma \ref{lem:2sk} implies that the blow--down distributions are also homotopic over $M$. In geometric terms, the Poincar\'e dual of the obstruction class is not modified by the blow--down process. \hfill$\Box$
\subsection{Uniqueness}
The uniqueness of a contact structure in every homotopy class of almost contact structures does not hold in a $5$--fold. There are examples in the literature, for instance \cite{NK} proves that every fillable contact structure has a non--fillable contact structure in the same almost contact homotopy class.\\
\noindent The construction described in this article requires a fair amount of choices. Though, the dependence of the contact structure with respect to them may be understood. The three main ingredients are the stabilization procedure of almost contact pencils, in the same spirit than Giroux's stabilization for a contact open book decomposition ~\cite{Gi}; the addition of fake curves in the triangulation increasing the amount of holes filled with the local model and the canonicity of the contact blow--up procedure.
\section{Non--coorientable case} \label{sec:non-coorientable}
\subsection{Definitions}
Let $M$ be a ($2n+1$)--dimensional closed manifold, not necessarily orientable. In order to state the Theorem \ref{main} in the non--coorientable setting, we need to give a definition of a non--coorientable almost contact structure. This is a distribution with a suitable reduction of the structure group along with a property requiring a relation between the normal bundle and the distribution. First we introduce the Lie group $\mathfrak{A}(n)$ defined as
$$\mathfrak{A}(n)=\{A\in O(2n):\quad AJ=\pm JA\},\quad\mbox{where }
J=\left(\begin{array}{cc}
0 & Id_n\\
-Id_n & 0
\end{array}\right)$$
Notice the following properties:
\begin{itemize}
\item[1.] The group $\mathfrak{A}(n)$ has two connected components. It is homeomorphic to $U(n)\times\mathbb{Z}_2$.
\item[2.] Its group structure is isomorphic to a semidirect product $U(n)\rtimes_\rho\mathbb{Z}_2$. More precisely, let $\mathbb{I}=\left(\begin{array}{cc}
Id_n & 0\\
0 & -Id_n
\end{array}\right)$, then the action
$$\rho:\mathbb{Z}_2\longrightarrow Aut(U(n)),\quad a\longmapsto (U\longmapsto \mathbb{I}^aU\mathbb{I}^a)$$
induces the semidirect product structure in the usual way.
\item[3.] There is a natural group morphism $\mathfrak{s}:\mathfrak{A}(n)\longrightarrow\mathbb{Z}_2$ defined as
$$\mathfrak{s}(A)=tr(JAJ^{-1}A^{-1})/(2n),$$
i.e. under the previous isomorphism, $\mathfrak{s}$ is the projection onto the second factor of $U(n)\rtimes_\rho\mathbb{Z}_2$.
\end{itemize}
\noindent Let us deduce some topological implications of the existence of a contact structure. Let $\xi \subset TM$ be a possibly non--coorientable contact structure on $M$ with a fixed set $\{U_i\}$ of trivializing contractible charts. Choose $\alpha_i$ as a local equation for $\xi|_{U_i}$, then
$$\alpha_i=a_{ij}\alpha_j,\qquad\mbox{with }a_{ij}:U_i\cap U_j\longrightarrow \{\pm1\}.$$
This implies that $\{a_{ij}\}$ are the transition function of the normal line bundle $TM/\xi$. Further, $(d\alpha_i)_{|\xi}=a_{ij}(d\alpha_j)_{|\xi}$. In particular, we may choose a family of compatible complex structures $\{J_i\}$ for the bundle $\xi$ satisfying $J_i=a_{ij}J_j$.\\
\noindent First, note that there is a group injection
$$\mathfrak{A}(n)\longrightarrow O(2n+1),\quad A\longmapsto\left(\begin{array}{cc}
A & 0\\
0 & \mathfrak{s}(A)
\end{array}\right)$$
and thus the structure group of $M$ reduces to $\mathfrak{A}(n)$. And second, a $\mathfrak{A}(n)$--bundle $E$ induces via the morphism $\mathfrak{s}$ a real line bundle $\mathfrak{s}(E)$. This construction applied to $\xi$ gives the line bundle $TM/\xi$ in the case above. These two properties will be the ones required in the following:
\begin{definition}
An almost contact structure on a manifold $M$ is a codimension $1$ distribution $\xi \subset TM$ such that the structure group of $\xi$ reduces to $\mathfrak{A}(n)$ and $\mathfrak{s}(\xi)\cong TM/\xi$.
\end{definition}
\noindent Observe that the definition for a cooriented almost contact distribution coincides with the one previously given. There are some immediate topological consequences of the existence of such a $\xi$. Indeed:\\
\begin{enumerate}
\item If $n$ is an even integer, then $\mathfrak{A}(n)\subset SO(2n)$. Thus the distribution $\xi$ is oriented.
\item If $n$ is an even integer, there is an isomorphism
\begin{equation}
TM/\xi\cong det(TM). \label{eq:normdet}
\end{equation}
Hence, any almost contact structure in an orientable $5$--dimensional manifold is cooriented. Conversely, any non--orientable $5$--fold can only admit non--corientable almost contact structures.
\item If $n$ is an odd integer, then $\mathfrak{s}=det$ as morphisms from $\mathfrak{A}(n)$ to $\mathbb{Z}_2$. Therefore $M$ is orientable since
$$det(TM)\cong det(\xi\oplus (TM/\xi))\cong det(\xi)\otimes\mathfrak{s}(\xi)\cong det(\xi)^2\cong\mathbb{R}$$
\end{enumerate}
\noindent Let $M^{2n+1}$ be a non--orientable manifold with $n$ an even integer. Then there exists a canonical $2:1$ cover
$$\pi_2:M_2\longrightarrow M$$
satisfying the following properties:
\begin{itemize}
\item[1.] $M_2$ is an orientable manifold.
\item[2.] Any almost contact structure $\xi$ on $M$ lifts to an almost contact structure $\pi_2^*\xi$ on $M_2$. Moreover, such a distribution is cooriented because of equation (\ref{eq:normdet}).
\end{itemize}
\subsection{Statement of the main result}
Let us state the equivalent of Theorem \ref{main} in the non--coorientable setting:
\begin{theorem}
Let $M$ be a non--orientable closed $5$--dimensional manifold. Let $\xi$ be an almost contact structure. Then there exists a contact structure $\xi'$ homotopic to $\xi$.
\end{theorem}
\begin{proof}
Let $\pi_2:(M_2,\pi_2^*\xi)\longrightarrow (M, \xi)$ be an orientable double cover. The constructions developed in this article can be performed in a $\mathbb{Z}_2$--invariant manner. Let us discuss it:
\begin{enumerate}
\item An almost contact pencil $(f,B,C)$ can be made $\mathbb{Z}_2$--invariant. To be precise, the loci $B$ and $C$ are $\mathbb{Z}_2$--invariant subsets and $f$ is a $\mathbb{Z}_2$--invariant as a map. In particular the action preserves the fibres. This is because the approximately holomorphic techniques can be developed in that setting. See \cite{IMP} for the details of the construction in the $\mathbb{Z}_2$--invariant setting.\\
\item The deformations performed in Section \ref{sect:defor_local} can easily be done in a $\mathbb{Z}_2$--invariant way. Also, the contact blow--up along a $\mathbb{Z}_2$--invariant loop can be built to preserve that symmetry.\\
\item Subsection \ref{subsec:otdisks} is also prepared for the $\mathbb{Z}_2$--invariant setting. Instead of having a single pair of overtwisted disks, we require two pairs of overtwisted disks. Each pair in the image of the other through the $\mathbb{Z}_2$--action.\\
\item Eliashberg's construction is not $\mathbb{Z}_2$--invariant. Therefore we proceed by quotienting the whole manifold by the $\mathbb{Z}_2$--action, we then obtain an almost contact pencil over the quotient. The fibres are oriented since they are $3$--dimensional almost contact manifolds. The induced almost contact distribution on them is non--coorientable. However, there is no hypothesis on the coorientability in the results of ~\cite{El}. Once the procedure described in Section \ref{sec:vertical} is applied, we consider the orienting double cover.\\
\item Section \ref{sect:skeleton} is trivially adapted to the $\mathbb{Z}_2$--invariant setting if a serious increase of notation is allowed.\\
\item Filling the $2$--cells as in Section \ref{sec:bands} and \ref{sec:end}. We need to produce a $\mathbb{Z}_2$--invariant standard model over $M \times \mathbb{S}^2$, with $(M,\alpha_0)$ a contact manifold with a $\mathbb{Z}_2$--invariant action. The only required ingredient is to ensuring that the framing $\{\alpha_0, \alpha_1, \alpha_2 \}$ is chosen $\mathbb{Z}_2$--invariant. The rest of the proof works through up to notation details.\\
\item Blowing--down is still a $\mathbb{Z}_2$--invariant procedure if the previous choices have been done $\mathbb{Z}_2$--invariantly. Therefore, we obtain a $\mathbb{Z}_2$--invariant contact structure $\xi'_2$ on $M_2$. Its quotient produces a contact structure on $M$.
\end{enumerate}
This proves the existence part of the statement. The statement concerning the homotopy follows since the homotopies can be easily made $\mathbb{Z}_2$--invariant. This is left to the careful reader.
\end{proof}
|
{
"timestamp": "2013-11-12T02:09:32",
"yymm": "1203",
"arxiv_id": "1203.2166",
"language": "en",
"url": "https://arxiv.org/abs/1203.2166"
}
|
\section{Introduction}\label{sec_intro}
Imaging surveys provide a general tool to access the average properties
of galaxy populations. A survey data set usually consists of an
arrangement of primary images in one or several filters. These data are
often accompanied by various supplementary data. Examples for such
surveys are COMBO-17 \citep{ref_combo}, DEEP1/DEEP2
\citep{ref_groth_strip}, GOODS \citep{ref_goods}, COSMOS
\citep{ref_cosmos} or the Hubble Ultra Deep Field \citep{ref_hudf}.
Common to all imaging surveys are the specific reduction methods
involved in the data analysis. After reducing the imaging data, which
normally consists of a mosaic of many potentially (partly) overlapping
tiles, scientific sources are detected and compiled in a source
catalogue. Depending on the scientific goals, more sophisticated
methods are then applied to analyse the morphology of the sources, \ie\
quantify the structure of their light-profiles. Finally, the resulting
additional structural parameters are added to the source catalogue.
Somewhere in this process the source catalogue might (optionally) get
cleaned from duplicate source entries or other artifacts.
For the main task, source detection and extraction, the code \sex\ by
\cite{ref_sex} has been widely-used in astronomy. Based on a
simple setup script \sex\ detects sources, estimates a background sky
level, measures primary shape information, like position, position
angle and axis ratio, and even performs aperture photometry. A key
feature is the ability to properly deblend close companion sources,
while at the same time avoid breaking single large sources up into
several pieces. Other features include a neural network to separate
stars and galaxies or the option to associate the detected objects with
a given list of input positions. \sex\ is designed with minimum user
interaction, support for large images and high execution speeds in
mind.
In order to analyse galaxy light profiles quantitatively, many codes
have been developed. The ones that are most widely used employ a
two-dimensional fitting method to model ellipsoidal radial profiles,
and include convolution with a point spread function (PSF).
One of these codes is \gimtwod, which was first employed as part of an
\iraf\ pipeline to analyse survey imaging data \citep{ref_gim2d}. Based
on a Metropolis algorithm to find the minimum in $\chi$-space,
\gimtwod\ mainly uses the \sersic\ profile \citep{ref_sersic}, which is
a general expression that includes both the de~Vaucouleurs and
exponential forms (see Sec.~\ref{sec_galfit} and eq.~\ref{eq_sersic}).
The minimisation method performs a global parameter space search. As a
result, \gimtwod\ is robust, however it requires large amounts of CPU
time compared to other codes \cite[\eg][]{ref_fitting}.
Another application for modelling light profiles is \budda\
\citep{ref_budda}. \budda\ was initially developed to perform
bulge/disc decomposition. However, it has recently been updated to
include also bar and central point source modelling. Moreover, it now
also features a double exponential profile for discs.
Finally, a rather versatile and effective method was presented by
\cite{ref_galfit, ref_galfit3}: \galfit. Like the aforementioned
programmes, it is a two-dimensional fitting code to extract structural
components from galaxy images. It is designed to model galaxies in as
flexible a manner as possible, by allowing the user to fit any number of
components and functional forms. \galfit\ therefore allows for the
possibility to not only fit simple situations, but also for fitting
more complicated setups including bulge, disk, bar, halo, etc.
This freedom has the major advantage that not only may the object of
prime interest be fitted, but so may the neighbouring sources -- at the
same time, as some situation may demand. Various light profile models
are built into the code, including the ``Nuker'' law \citep{ref_nuker},
the \sersic\ profile \citep{ref_sersic}, an exponential disc, Gaussian
or Moffat functions and even a pure PSF for modelling stars. \galfit\
convolves all model profiles, except for the PSF itself by the PSF to
simulate image smearing by Earth's atmosphere and telescope optics.
Although a scientist has a multitude of options to choose from for
fitting and detecting objects, analysing a complete survey to the end
of obtaining a source catalogue with galaxy parameters, requires many
intermediate steps. For example, duplicate sources from tile overlaps
have to be differentiated; the detection and fitting codes have to be
set up; a proper local background sky value has to be estimated;
resulting source parameters have to be compiled in a catalogue. As
these steps are fairly general we have built a code that simplifies all
these steps and largely automates the entire process. Our code, \gala,
performs all the required steps from a single setup and with minimal
manual interaction provides a fitting catalogue. It runs \sex\ to
detect sources and performs an automated \sersic\ fit using \galfit.
Amongst the various codes introduced above, we opted to use \galfit\
because it outperforms \gimtwod\ both in speed and reliability
\citep{ref_fitting} and allows a much wider range of light-profile
models than \budda. Upcoming versions will include additional features
like automated multi-component fitting. The code is available freely
for public download from our website at:
\url{http://astro-staff.uibk.ac.at/~m.barden/galapagos/}.
The layout of the paper is as follows. We start by giving an overview
of the structure of the code (Sec.~\ref{sec_structure}).
Then we elaborate on the methods involved in the individual components
(Sec.~\ref{sec_components}). Next, we present some fitting results
based on simulated data and provide details concerning the reliability
of the code (Sec.~\ref{sec_quality}). Subsequently, we give estimates
on the performance of \gala\ (Sec.~\ref{sec_performance}), followed by
a summary (Sec.~\ref{sec_summary}). Upon first reading this article we
suggest to skip Sec.~\ref{sec_components},
which address mainly the frequent \gala\ user. In the course of the
paper we assume a working knowledge of \sex\ and \galfit\ and refer the
reader to the publications by \cite{ref_sex} and \cite{ref_galfit}.
\begin{figure*}
\centering\includegraphics[width=12cm]{flow-chart-timing-rev.eps}
\caption{Code structure. A yellow background indicates the four main
blocks, labelled according to the nomenclature of the \gala\ setup
file (see Fig.~\ref{fig_setupfile}). Fitting objects with \galfit\
(Block \texttt{(D)}) is a two-stage process (see
Sec.~\ref{sec_galfit}), which typically requires more than 90\%\ of
the total computation time (green background). We mark smaller tasks
by blue boxes. For further details see
Sec.~\ref{sec_structure}.}\label{fig_structure}
\end{figure*}
\section{Overview of Code Structure}\label{sec_structure}
\gala\ is divided into four main blocks, each of which is executable
independently from the others. This allows flexibility of repeating or
optimising certain segments of the analysis without re-running the
entire pipeline. These blocks are: \begin{enumerate}
\item Detect sources by running \sex\ \verb|(B)|
\item Cut out postage stamps for all detected objects \verb|(C)|
\item Estimate sky background, prep. \& run \galfit\ \verb|(D)|
\item Compile catalogue of all galaxies \verb|(F)| \end{enumerate}
Note that letters in brackets correspond to the respective sections in
the \gala\ setup file (see Sec.~\ref{sec_setup_main}). We visualise
this structure in Fig.~\ref{fig_structure}.
Also note that \gala\ does not create the PSF image, which is
required by \galfit\ in the fitting process. The user is responsible
for providing such an image. A proper PSF should have a sufficient
S/N, i.e. better than the brightest objects in the survey, in order not
to degrade the science data. Furthermore, it should contain all features
of the PSF down to the noise and it should not be truncated at the
edges. Also, it has to be background subtracted and normalised to a
total flux of 1.
\subsection{Source Detection}
In the first block \verb|(B)|, \sex\ is run to detect sources on the
individual survey images. Optionally, \gala\ features a high dynamic
range (HDR) mode for source extraction (Sec.~\ref{sec_sex}), which is
ideally suited for wide area and/or space-based, \eg\ \hst, data. After
a first pass, the user may refine this catalogue by identifying ``bad''
detections followed by re-running \sex. This may be required to fix
overly deblended sources or to remove spurious detections (see
Sec.~\ref{sec_bad_detections}). Once all tiles are analysed, \gala\
combines the individual output catalogues, rejecting duplicate sources
(see Sec.~\ref{sec_sex}) and optionally bad detections like cosmic rays
etc.~(see Sec.~\ref{sec_opt}).
\subsection{Postage Stamp Cutting}
To reduce the amount of time needed to ingest an image into \galfit\ it
is worthwhile to first extract each galaxy from the survey mosaic.
Therefore, in the second block \verb|(C)|, \gala\ estimates a size for
each object based on its \sex\ parameters. With this information it
computes the extent of a postage stamp. From the original survey
images, \gala\ then creates such a cutout for every object. It performs
the subsequent fitting with \galfit\ on these postage stamps (see
Sec.~\ref{sec_postages}). At this stage, \gala\ creates for every
survey image a ``sky-map'' containing information about the nature of
the pixel flux (either ``no flux'', ``sky'' or ``source''). It uses
this map later on to identify blank sky pixels (see
Sec.~\ref{sec_sky}).
\subsection{Sky Estimation and Fitting}\label{sec_structure_sky_fitting}
The third block \verb|(D)| performs the major fitting work. For every
object in the source catalogue it prepares and runs \galfit\ (see
Sec.~\ref{sec_galfit}). Accurate fitting analysis by \galfit\ requires
careful consideration, which includes identifying the proper sky
background, identifying neighbours and providing initial parameter
guesses to start the fitting.
\gala\ measures the sky using a flux growth curve including pixel
rejection based on the ``sky-map'', which was calculated in the
previous step (see Sec.~\ref{sec_sky}). It uses the full survey image
and not the small postage stamp to compute the sky. Note that even
though \galfit\ can fit the sky, \gala\ does not use this option to
avoid instances when neighbouring contamination makes accurate
determination infeasible, and to reduce the degree of freedom in the
fit. We provide further justification for and details on this approach
in Sec.~\ref{sec_postages} and Sec.~\ref{sec_sky}.
\subsection{Catalogue Creation}
In the last block \verb|(F)|, \gala\ reads the results of the fitting
from the headers of the \galfit\ output images and puts them into the
source catalogue (see Sec.~\ref{sec_cat}). Here, it removes a second
set of ``bad'' detections from the catalogue. Namely those that were
required in the fitting process to allow optimal results for
neighbouring objects. Usually, these are bright artefacts in close
proximity to relatively faint real sources (see
Sec.~\ref{sec_bad_detections}). Finally, \gala\ compiles the resulting
catalogue in a FITS-table.
\section{Components}\label{sec_components}
Subsequently, we describe in detail the methods involved in the
individual components of \gala. These include \sex\ and high dynamic
range (HDR) source extraction (Sec.~\ref{sec_sex}), compiling a
combined source catalogue (Sec.~\ref{sec_cat}), the cutting of postage
stamps (Sec.~\ref{sec_postages}), estimating a background sky level
robustly (Sec.~\ref{sec_sky}), and fitting with \galfit\
(Sec.~\ref{sec_galfit}). In the last part of this section we introduce
some technical mechanisms to optimise the code for robustness and speed
(Sec.~\ref{sec_opt}).
\begin{figure*}\centering\includegraphics[width=12cm]{hdr.eps}
\caption{Combining ``hot'' and ``cold'' \sex\ catalogues. The two
panels ({\it upper} and {\it lower}) show examples of a cold (left
side) and a hot (right side) SExtraction. Ellipses indicate the \sex\
Kron ellipses of the detected sources. Arrows mark objects from the hot
(red) and the cold (blue) catalogue that were incorporated into the
combined catalogue. Additional hot sources not marked with arrows (\eg\
in the upper right panel) were excluded from the combined catalogue as
described in detail in Sec.~\ref{sec_sex}.}\label{fig_hdr}
\end{figure*}
\subsection{SExtractor}\label{sec_sex}
\gala\ incorporates \sex\ to detect astronomical sources on individual
survey tiles. Details on how to operate \sex\ can be found in
\cite{ref_sex}. \sex\ uses the Kron radius to estimate the extent of a
galaxy \citep{ref_kron,ref_infante}. For both stars and galaxies, when
convolved with a Gaussian seeing, it encircles 90\% of their flux.
\gala\ applies the Kron radius \eg\ to estimate sizes of postage stamps
or to judge which pixels in an image are affected by light from
sources.
\sex\ has been used successfully with both ground- and space-based
data. Yet, recent large CCD arrays put the code to its limits due to
the wide range of object sizes and luminosities that are being observed
simultaneously. In classic pencil beam surveys, the objects of interest
are mostly faint and small. \sex\ is then fine-tuned to pick up such
sources properly at the cost of splitting up the occasional big bright
spiral galaxy into many pieces. On the other hand, wide area surveys
traditionally do not reach very deep. When fine-tuning \sex\ for these
surveys, emphasis is put on correctly deblending the larger and
brighter objects while losing some depth. In both applications one
reaches the dynamic range limit (in terms of object size and
brightness) and has to make a compromise of depth and proper
deblending.
\subsubsection*{HDR SExtraction}
Fortunately, there is a rather simple two-step approach using \sex\ to
overcome this problem. Firstly, one runs \sex\ in a so-called ``cold''
mode in which only the brightest sources are picked up and properly
deblended. As this will miss many faint sources, in a second setup
emphasis is put on depth. The second run we term ``hot'' mode.
Then one needs to combine the ``hot'' and the ``cold'' runs.
Firstly, all ``cold'' sources are imported into the output catalogue.
Then the Kron ellipses as provided by \sex\ of ``hot'' and ``cold''
sources are analysed. Every source position in the ``hot'' catalogue is
checked whether it falls inside a Kron ellipse of a ``cold'' source. If
it lies inside a Kron ellipse it is discarded and does not enter the
output catalogue; if its central position lies ``sufficiently'' outside
of all ``cold'' Kron ellipses it does enter the output catalogue.
``Sufficiently'' here refers to the possibility in \gala\ to
artificially enlarge the Kron ellipses slightly for this purpose:
parameter \texttt{B09} provides a scaling factor. Setting \texttt{B09}
to \eg\ 1.1 results in enlarging each Kron ellipse by 10\%.
In summary, it is important that the ``cold'' run properly
deblends all brighter objects, while the ``hot'' run is tuned to pick up
fainter sources. We term this mode ``High Dynamic Range (HDR)
SExtraction''.
To illustrate the process of including hot sources outside the Kron
radius of cold sources into a combined catalogue, see
Fig.\ref{fig_hdr}. In the upper left hand panel we show a ``cold'' run.
The big central spiral galaxy is deblended correctly with the fainter
galaxy below it. Also, the clumpy low surface brightness spiral in the
upper left corner is detected as a single source. All three sources are
taken over into the combined catalogue. Requiring a proper deblending
of the bright objects results in missing the faintest sources, though.
The ``hot'' run (upper right hand panel) picks those up. However, it
breaks the brighter galaxies up into many sources. In the
example, an
off-centre knot of the upper left galaxy was detected as a separate
object. Moreover, the outer regions of the central (and upper left)
galaxy are assigned separate source IDs. These ``spurious'' detections
change the effective size of the central galaxy (compare diameters of
the Kron ellipses in the left and right figures). Interestingly, the
relatively bright galaxy below the central object is not deblended
properly in the hot run. Furthermore, the size and position angle of
the upper left detection demonstrates the lower detection threshold of
the hot setup. In the hot setup a larger fraction of the low surface
brightness flux is included in the calculation of the position angle,
thus providing a much better estimate than the cold setup, which is
more heavily weighted towards the inner regions of the sources. Yet,
the values from the cold run enter the combined catalogue as deblending
is the more important source of error. Also, \galfit\ calculates
structural parameters like the position angle much more reliably. The
lower panels in Fig.~\ref{fig_hdr} show another example. Again, the
deblending in the hot run is bad, while in the cold run it is correct.
The faintest sources are only detected in the hot run. Bad deblending
in the hot run strongly affects the calculation of the position angle
of the brightest source, while in the cold run it is acceptable.
We developed and tested this method for the GEMS survey
(\citeauthor{ref_gems} \citeyear{ref_gems}; \citeauthor{ref_gems_cat}
\citeyear{ref_gems_cat}; for tests see \citeauthor{ref_fitting}
\citeyear{ref_fitting}). Subsequently, other major surveys have adopted
it as well, including COSMOS \citep{ref_cosmos_hst,ref_cosmos_lensing}
and STAGES \citep{ref_stages}. \gala\ provides the option for running
\sex\ in two-stage HDR or normal single-stage \sex\ configuration.
\begin{figure}\centering\resizebox{8cm}{!}{\includegraphics{combine_tiles.eps}}
\caption{Combining \sex\ catalogues from neighbouring tiles. Tile A
contains sources 1a, 2a and 4, while objects 1b, 2b and 3 were detected
on tile B. Ellipses show the corresponding sizes from \sex. For a
description of what source ends up in the resulting table see
Sec.~\ref{sec_cat}.}\label{fig_combine_tiles} \end{figure}
\begin{figure}\centering\resizebox{8cm}{!}{\includegraphics{comb1.eps}}
\caption{Combining \sex\ catalogues from neighbouring tiles. The left
image (blue area) extends out to the right (blue) diagonal line; the
right image (red area) extends out to the left (red) line. Shaded areas
outside of the lines corresponding to the respective image did not
receive sky flux. Pluses (blue) indicate source detections (hot and
cold already combined) from the left (blue) image; crosses (red) mark
detections from the right (red) image. Diamonds (blue) highlight
objects that are contained in the combined catalogue if they originated
from the left (blue) image; boxes (red) highlight those that were taken
from the right (red) image. Catalogue combination is based on the \sex\
ellipses (see Sec.~\ref{sec_cat}). Such an ellipse is shown in case a).
The source from the right image (red) is rejected as it lies inside an
ellipse of a detection in the left image (blue), which is further away
from the respective image boundary (blue and red lines). For the same
reason in case b) the red source is kept. The detection in case c) does
not even have a counterpart in the other catalogue.}\label{fig_comb1}
\end{figure}
\subsection{Catalogue Compilation}\label{sec_cat}
Compiling the output source catalogue is a two-stage process. \gala\
creates a first combined catalogue from the \sex\ output tables. In the
subsequent model fitting process, \gala\ fills this catalogue with the
\galfit\ output parameters.
When putting together catalogues from potentially overlapping images,
\gala\ has to take care of removing detections of the same source on
multiple images. To this end, it uses the world coordinate system of
the images to translate pixel coordinates from one to another image.
Next, \gala\ calculates the distance to the image border for each
source (not only those in the overlap area) in the corresponding image
catalogues. The area containing flux (pixels with non-zero values)
defines the image border. This is crucial in particular for
non-rectangular images (\eg\ from \hst). Now, \gala\ sorts the two
catalogues by border-distance. It starts with the source farthest
from the edge, which we assume to be on image {\it A} (source 1a;
see Fig.~\ref{fig_combine_tiles}). Then it checks whether there are
sources inside the Kron ellipse of the current object in the
neighbouring image {\it B} (sources 1b, 2b and 3). If it finds any such
targets, \gala\ removes them from the list. Note that \gala\ does not
remove objects overlapping with the source from image {\it A} from the
list (sources 2a and 4). Following this scheme it works through the
complete list, from the farthest to the closest objects to the
boundary, and constantly updates the list in the process.
A problem
arises for sources, say in image {\it A} (source 1a), extending over a
radius larger than the size of the overlap area and having overlapping
detections on image {\it B}, which are not covered by image {\it A}
(source 3 in Fig.~\ref{fig_combine_tiles}). Or put differently,
if source 1 is \eg\ deblended differently in image {\it A} than in image
{\it B}, sources might get lost in the combination process. In such a
case, \gala\ includes the main source from image {\it A} (source 1a) in
the catalogue and all overlapping sources from image {\it A} (sources 2a
and 4). Overlapping sources from image {\it B} it removes, though
(sources 2b and 3). However, in cases where source 1 was
over-deblended in image {\it B}, but not in image {\it A}, this would
result in a welcome clean-up of the catalogue by removing the spurious
source 3. Although this problem cannot be unambiguously solved, in
practice it rarely occurs. It can be avoided completely if the largest
source in the survey is smaller than the overlap between survey images.
Fig.~\ref{fig_comb1} shows an example for this procedure to remove
duplicate detections. The bright galaxy a) is just on the edge of the
red image. As only half its flux is visible on that image, the
calculated centre is far off from the real position. The blue image
fully contains this galaxy. The two central positions (in red and blue)
being so different, a normal nearest neighbour matching algorithm with
a maximum matching radius would not have been able to identify the two
detections as the same source. In the proposed scheme, though, the red
detection is not put into the combined catalogue, for being inside the
Kron ellipse of a source that is further from the image edge in the
blue image. Similarly, the red source b) is further from the image edge
than the blue source and thus, we reject the blue object. Objects
without counterpart in the other image, as in c), we do keep in the
combined catalogue.
As \gala\ performs duplicate removal before running \galfit, it does
not fit sources twice. Note that for fitting sources at the edge of an
image, \gala\ takes objects on neighbouring survey images into account
as well (see Sec.~\ref{sec_opt}).
\subsection{Postage Stamps}\label{sec_postages}
To optimise galaxy fitting with \galfit, \gala\ cuts the science images
into smaller sections centred on individual sources. The advantage of
using such postage stamps is that the total fitting time and the
demand on main memory can be reduced. Even rather deep optical surveys
contain large fractions of empty sky, which can mostly be excluded from
the fit once the information from the sky pixels is effectively used to
estimate the background (see Sec~\ref{sec_sky}; even masking cannot
totally diminish this advantage). In a typical one-orbit \hst\ survey
around a factor of 2 in the total number of pixels can be saved.
Moreover, although \galfit\ allows simultaneous fitting of multiple
sources, modelling more than a handful of objects at the same time
quickly becomes rather impractical. Thus, to optimise automated fitting
of large numbers of sources, \gala\ incorporates a postage stamp
cutting facility.
To determine the size of the postage stamps, \gala\ uses the Kron
radius. The user specifies a scale factor \verb|C03| by which the Kron
radius is enlarged. The decision for this scaling should be guided by
trying to find a compromise between maximal area, to include as much
flux of the central source (and maybe the closest neighbours) as
possible, and minimal area, to speed up computation time of \galfit.
Finding a good compromise is important as elliptical galaxies require a
larger area than spiral galaxies, owing to their extended and slowly
dimming, low surface brightness wings. For the one-orbit \hst\ surveys
GEMS and STAGES, we found a factor of 2.5 to work well. \gala\ does not
enlarge the size of the postage stamps in the presence of close
neighbouring galaxies. However, they are properly taken into account in
the fitting process (see Sec.~\ref{sec_galfit}).
We note that a disadvantage of using postage stamps for fitting {\it
with the background sky as a free fit parameter} (which we discourage
the user from in the context of \gala) is that the fit results will be
biased if the postage stamp does not contain enough empty sky pixels. In
such a case the $\chi^2$ of the fit might indicate a good fit, yet the
result would be flawed by attributing too much or too little flux to the
object. This could potentially also have a strong impact on other
structural parameters. Therefore, \gala\ does {\it not} allow a free fit
of the sky background within \galfit, but estimates a value before the
fitting. We give details on the background estimation in the following
section.
\begin{figure*}\centering\includegraphics[width=17.5cm]{sky.eps}
\caption{Sky estimation. {\it Left:} The average flux $f$ measured in
elliptical annuli (blue) centred on an object (here {\it a}) determines
the background level. In each annulus, we exclude regions surrounding
other sources from the calculation (shaded area). For the indicated
annulus, we exclude dark blue shaded regions {\it b} -- only light blue
regions {\it c} define the average background flux. {\it Right:} Flux
$f$ measured in an elliptical annulus as a function of radius $r$. {\it
a:} Starting radius. {\it b:} Slope (indicated by the diagonal lines)
turns positive for the first time, \eg\ due to galactic structure at
large radii. {\it c:} Slope turns positive for the second time. Here we
stop the iteration. {\it d:} We compute slope measurements from the
last $n$ sky estimates (here: n=5; $n$ is a user parameter). {\it e:}
The adopted background sky level. See Sec.~\ref{sec_sky} for
details.}\label{fig_sky} \end{figure*}
\begin{figure*}\centering\includegraphics[width=17.5cm]{skymap.eps}
\caption{The ``skymap'' ({\it left}): for each object that was detected
in the image ({\it right}) we calculate the Kron ellipse and scale it
up. Pixels inside a Kron ellipse get the value one. Pixel values stack,
\eg\ where two Kron ellipses overlap, the pixel value is two. Blank sky
has a value of zero; pixels without astronomical flux, as occurs after
removing image distortions \eg\ in \hst\ images, have a value of -1.
Some pixel values are indicated.}\label{fig_skymap} \end{figure*}
\subsection{Sky Estimation}\label{sec_sky}
Obtaining a precise sky level is the most critical systematic in galaxy
surface brightness profile fitting \citep[see
\eg][]{ref_dejong,ref_fitting}. To obtain a precise background
measurement \galfit\ is capable of including the sky as a free
parameter when fitting a celestial source. However, using the sky as a
free parameter requires an appropriate size of the input image, \ie\ it
has to contain {\it all} the flux of the primary source and most of the
flux of neighbouring sources that are to be fitted simultaneously and
ample sky. For estimating a proper sky background, the image should be
as large as possible. However, as detailed above, large postage stamps
become impractical once too many neighbouring sources are included.
Only a manual setup may allow using the sky as a free model parameter.
To enable automated processing of large numbers of objects, \gala\
incorporates its own subroutine to obtain an optimal sky measurement
before running \galfit\ and hence uses a fixed value during fitting.
With the proper setup, the resulting \gala\ estimate improves
significantly over values obtained from \sex.
We use a flux growth method to estimate the local sky around an object.
Calculating the average flux in elliptical annuli centred on the object
of interest while excluding other sources or image defects, we obtain
the background flux as a function of radius. Once the slope over the
last few measurements levels off, \gala\ stops and determines the sky
from those last few annuli (see Fig.~\ref{fig_sky}).
\begin{figure*}\centering\includegraphics[width=17.5cm]{galfit.eps}
\caption{Fitting \sersic\ profiles with \galfit. From left to right,
the panels show the original galaxy image, the \sersic\ model and the
residual of image and model, respectively. \gala\ excludes (masks)
areas shaded in red from the fit. In this example no bright secondary
sources were detected. The next brightest object after the primary
source is too far away to become a secondary (for details on the
definition of primary and secondary sources see Sec.~\ref{sec_galfit}).
Note that the masked region at the right edge of the image results from
the irregular shape of the \hst\ images. This area has not received any
flux and thus \gala\ masks it as well.}\label{fig_galfit} \end{figure*}
For this procedure to work, we create a ``sky-map'', \ie\ a copy of the
input images where the pixel values indicate the nature of the
contained flux. In the sky-map a pixel value of 0 stands for blank
background sky, while positive numbers indicate the presence of a
source. A value of -1 indicates no flux at all, as happens with \hst\
images that are geometrically distorted (see Fig.~\ref{fig_skymap}).
One might think that to make the decision between source or sky the
\sex\ segmentation map \citep[for a definition see][]{ref_sex} might
suffice. Unfortunately, the level out to which \sex\ detects objects is
rather limited. In particular with elliptical galaxies \sex\
underestimates the flux belonging to the object significantly. Changing
the \sex\ setup parameters cannot totally remedy this. Therefore, a
significant number of pixels still containing some source flux would be
assigned as ``sky''. To circumvent this problem, we instead use Kron
ellipses to determine the extent of an object. \gala\ regards any pixel
inside \verb|D03|$\times R_\mathrm{Kron} + $\verb|D05| as containing
source flux (for STAGES: \verb|D03| $=3$; \verb|D05| $=20$~pix). Note
that this scaling factor \verb|D03| does not have to coincide with the
scale for the size of the postage stamps \verb|C03|. Note also that
\gala\ records the total number of objects that might contribute to a
certain pixel, \ie\ when \eg\ two sources overlap, the value in the
intersection of the two Kron ellipses is also two. A weight map
(exposure time map) specified by the user defines the off-chip pixels,
which are given a value of -1. \gala\ identifies these as pixels with
zero exposure time.
\gala\ takes special care to minimise the impact of large nearby
sources on the background estimation process for the current object. To
that end, \gala\ relies on the \sex\ output catalogue to provide shape
information. Under the assumption that all sources have a \sersic\
index $n=4$ and a half-light radius
$r_e=\left(\tt{flux\_radius}\right)^{\alpha}$, with the \sex\ catalogue
parameter {\tt flux\_radius} and a user specified power $\alpha$ (we
chose $\alpha = 1.4$; \verb|D11|) to convert the \sex\ {\tt
flux\_radius} to a ``true'' half-light radius, \gala\ calculates the
flux of all catalogue objects at the position of the current source.
Any source exceeding a user specified limit \verb|D09|, \gala\ regards
as an important flux contributor for the current object. Subsequently,
we will term the sources that are selected that way ``contributors''.
Note that \verb|D09| has the units of a magnitude, \ie\ ``exceeding''
the given limit implies a number smaller than this value. As the \sex\
{\tt flux\_radius} is a rather poor proxy for the true half-light
radius and without proper estimate for the \sersic\ index, we opt for a
rather conservative limit of this flux cut.
If a proper \galfit\ fit exists for the contributors, \gala\ subtracts
their model profile from the input image temporarily, \ie\ for the time
of the current background estimation. Note that removal of a model
profile includes convolution with the telescope PSF before subtraction.
In order to optimise the profile subtraction, \gala\ processes the
\sex\ source catalogue in order of increasing magnitude. As the very
few brightest sources have a significant impact on both the sky
estimation and fitting of a large number of fainter sources, starting
the fitting process with the brightest galaxies is essential. We give
further details about the sorting process in Sec.~\ref{sec_opt}.
Normally, the Kron ellipse of the current object defines the starting
radius for the iterative measurement of the sky background in
increasing annuli. In case of the presence of potentially dominant flux
contributors, for which no \galfit\ model exists yet, and hence were
not subtracted from the input image, \gala\ increases the starting
radius to the maximal distance of all such sources from the current,
as they might potentially influence the fitting.
For each sky annulus, it estimates an average flux value excluding any
pixels that were flagged as containing an object (or that were flagged
as having a defect or no flux) in the skymap. Firstly, of the
distribution of the remaining pixels, \gala\ symmetrically clips all
$3\sigma$ outliers. Then it fits a Gaussian function to the leftover
distribution, producing a mean value for the current annulus. After
each new sky annulus measurement, \gala\ calculates a robust linear fit
to the last few estimates (\verb|D13|; in the case of STAGES 15
measurements). As long as source flux is still measurable, this slope
is negative. Once this process reaches the true background, the
estimated slope should start to randomly change its sign. When this
happens for the second time, \gala\ stops the loop and obtains the
final background value from the last \verb|D13| measurements. Stopping
the process at the first positive slope measurement often results in
suboptimal estimates as galactic inhomogeneities (like spiral arms)
might produce dips sufficient to produce a slope sign change. However,
using a much later slope change (than two) in practice is not necessary.
Note that neighbouring sources are not a problem for the termination of
this iteration as the method takes special care to take their influence
into account (as shown above). This whole process is fully
user-configurable, including options for the width of the sky annuli
\verb|D07|, their spacing \verb|D06|, the initial starting radius
\verb|D08| and the magnitude cut \verb|D09|.
\subsection{GALFIT}\label{sec_galfit}
Of the various light profiles built into \galfit, the most general one
for galaxy fitting is the \sersic\ model. It is also used by \gala:
\begin{equation}\label{eq_sersic}
\Sigma\left(R\right)=\Sigma_e\cdot\exp\left(-\kappa\left[
\left(R/R_e\right)^{1/n}-1\right]\right),
\end{equation}
where $R_e$ is the effective or half-light radius, $\Sigma_e$ is the
effective surface brightness, $\Sigma\left(R\right)$ is the surface
brightness as a function of radius $R$, $n$ is the \sersic\ index and
$\kappa=\kappa\left(n\right)$ is a normalisation constant. The \sersic\
profile is a generalisation of a de~Vaucouleurs profile with variable
\sersic\ index $n$. An exponential profile has $n=1$ while a
de~Vaucouleurs profile has $n=4$.
A simple setup script controls profile modelling with \galfit. It
contains information about input and output file locations, PSF image,
bad pixel mask, etc. A list of starting guesses defines what
light-profiles are to be fitted. Although the downhill gradient method
incorporated in \galfit\ is often speculated to be prone to converging
to a local instead of the global minimum, in practice we find it to be
extremely robust, even in comparison to global parameter space search
algorithms \citep{ref_fitting}. In application to high redshift survey
data, the other two noteworthy features are the included bad pixel mask
(\ie\ pixels that are excluded from the fitting) and the number of
simultaneously fitted objects.
We show an example of \galfit\ output in Fig.~\ref{fig_galfit}. The
left image presents the input postage stamp. In this case a single
component (one object) was fitted. We show the resulting \sersic\ model
in the middle. Note that the brightness cuts and scaling in both left
and middle panels are the same. The right panel displays the difference
image of input minus model. Bright spiral features and dark dust lanes
that strongly deviate from the smooth \sersic\ profile are very
prominent in this image. In order not to bias the fit by neighbouring
sources and the image boundaries, \gala\ excludes the shaded region
from the fit by applying a bad pixel mask (see below).
In order to define which objects do not have a high importance for the
current fit and hence may be masked instead of being fitted
simultaneously, we define the following terminology: the target for the
current fit is the {\it primary source}; any object whose expanded Kron
ellipse overlaps with that of the primary are {\it secondary sources};
objects without any overlap with the primary we term {\it tertiary
sources}. We consider tertiary sources not to be important for the
quality of the fit. As a result, we mask and exclude them from the
modelling (each pixel in a mask image is ignored during the fit by
\galfit). Secondary sources might have an impact on the outcome of the
parameters of the primary. Therefore, we fit them simultaneously
with the primary. We treat contributors (for a definition see
Sec.~\ref{sec_sky}) as secondaries. The difference between
contributors and secondaries is, that we do not require an overlap of
the Kron ellipses for contributors.
\begin{figure}\centering
\resizebox{8cm}{!}{\includegraphics{simultaneous_fitting.eps}}
\caption{Optimisation of the \galfit\ setup. Circles indicate the Kron
ellipses used for classifying the detected objects (as secondaries or
tertiaries). This example was taken from real data. However, for
clarity we do not mark the faintest detections in this image. For
details see Sec.~\ref{sec_galfit}.} \label{fig_fix} \end{figure}
\begin{figure*}\centering\includegraphics[width=17.5cm]{secondaries_contrib.eps}
\caption{\galfit\ parameter setup scheme for secondaries and
contributors. Depending on the relative position to the primary target,
\ie\ on the same postage stamp or not, with a pre-existing fit from the
same or another survey image or without a fit, we show the setup for
the \galfit\ parameters ({\it left panel}): static implies all
parameters are fixed to their initial guess (\ie\ the \sex\ estimates),
while free means that they are variable throughout the fit. In some
cases the position pos, the axis ratio $q$ or the position angle
$\theta$ take a different state than the remaining fit parameters. We
visualise the situation in the {\it right panel}: The current primary
{\it P} is located on the red survey image {\it A}, which has some
overlap (purple) with the blue image {\it B}. The solid black outline
indicates the postage stamp corresponding to {\it P}. Potential
secondaries or contributors are numbered. For sources with a black
background ({\it 3} \& {\it 4}) no prior fit exists (\sex\ values are
used as static/fixed profile parameters), while for targets shown
either in red ({\it 1}) or blue colour ({\it 1} \& {\it 2}) a fit from
the respective survey image is available. For further details see
Sec.~\ref{sec_galfit}.}\label{fig_simultaneous} \end{figure*}
\subsubsection*{Simultaneous Fits}
Often, sources are so close to each other, that they are best fitted
simultaneously. One might argue that after a simultaneous fit of two
sources, {\it A} and {\it B}, the best parameters are known for both
objects. However, in general this is not true. For example, we construct
a situation with three sources {\it A}, {\it B} and {\it C}, where {\it
C} is on the opposite side of {\it A} with {\it B} being in the middle
(see Fig.~\ref{fig_fix}). Let {\it A} be the brightest source of the
three, \ie\ {\it A} is fitted firstly (see also Sec.~\ref{sec_opt}).
The Kron ellipses of {\it A} and {\it B} and those of {\it B} and {\it
C} overlap, while the Kron ellipses of {\it A} and {\it C} do not.
Fitting of {\it A} implies fitting of {\it B} simultaneously: {\it B}
is a secondary to the primary {\it A}. {\it C} is not connected to {\it
A} and therefore following our prescription we mask it. As a tertiary
we exclude it from the fit. The resulting fit for {\it B} thus is not
optimal, as it neglects the presence of {\it C}, which is important for
fitting {\it B}, but not for fitting {\it A}. To obtain the optimal fit
for {\it B}, we have to fit {\it B} as a primary. In this case {\it A}
and {\it C} are secondaries as their Kron ellipses both overlap with
{\it B}, and are fitted simultaneously. To speed up the fitting of {\it
B}, we can now insert the known parameters for object {\it A}, thus
effectively removing one component from the fit. This example
highlights the importance of fitting all objects once as primaries,
while secondaries may be made static if a fit already exists.
Normally secondary sources are fitted simultaneously with the current
primary object (see above). Using a pre-existing fit (as in the
example) as static parameters for a secondary source, thus, is an
exception to this rule. A further complication is that the existing fit
for the secondary may have been obtained from a different survey image
as the current primary. In that case, the central position of the
secondary has to be converted via the world coordinate system
information from the original pixel coordinates to the current system
of the primary. Therefore, to allow optimal centring after such a
conversion, \gala\ fixes all parameters for the secondary, but its
central coordinates. If in the previous example, when fitting object
{\it B} with a pre-existing fit of source {\it A}, the fit for {\it A}
was performed on a different survey image than {\it B}, then the pixel
centre of {\it A} would not be static. However, a free pixel centre is
only required if the centre of the secondary {\it A} is also inside the
postage stamp of the current primary {\it B}. If the centre is off the
postage stamp, sub-pixel accuracy is not required any more for an
optimal fit, and all components of {\it A} are made static. We
visualise this situation in Fig.~\ref{fig_simultaneous} (case {\it 1}
and {\it 2}).
Furthermore, if no fit exists for a secondary, a free fit for that
source is not always the best solution. In the case that the centre of
the secondary is not on the postage stamp, a free fit results in too
many degrees of freedom. In \gala\ we opt to then fix the position,
axis ratio and position angle to the values provided by \sex\ (while
leaving the \sersic\ index $n$ and the half-light radius $R_\mathrm{e}$
as free parameters; see Fig.~\ref{fig_simultaneous} case {\it 3} and
{\it 4}). This is justified because,
on one hand, more than half the flux of the secondary cannot be seen by
\galfit, thus making it increasingly difficult to come up with precise
estimates for these parameters. On the other hand, the values given by
\sex\ usually have high enough accuracy not to bias the fit of the
primary significantly.
In addition to the ``normal'' sources (secondaries and tertiaries) in
the immediate surroundings of the current object, \gala\ has to take
bright and large contributors as defined in Sec.~\ref{sec_sky} into
account as well, although these sources may be off the current survey
image. It treats them as secondaries without the requirement of their
Kron ellipse to overlap with the Kron ellipse of the primary. In terms
of the parameter setup, \gala\ handles them exactly like other
secondaries.
\begin{figure*}\centering\includegraphics[width=12cm]{mask.eps}
\includegraphics[width=12cm]{contrib.eps} \caption{Mask creation. The
left panels show galaxy images; the right panels the corresponding bad
pixel masks. In the bad pixel masks white and red represent good and
bad pixels, respectively. {\it Upper panels:} a) and b) indicate
primary and secondary sources, respectively. c) and d) mark examples of
tertiary sources: c) is only partly masked, as it overlaps with a
secondary source; d) is masked completely for not having any overlap
with the primary or a secondary. {\it Lower panels:} The plotted area
shows a postage stamp (indicated by the solid rectangles) and some of
its surroundings. Note that the postage stamp is tilted (representation
in world and not pixel coordinates) and that the blue area is actually
not part of the postage stamp. 1) is a source that might potentially
contribute to the fit of the primary, due to its brightness, and is
included as a secondary source (with parameters fixed from a previous
fit) although its centre is off the current postage stamp. 2) is a
tertiary source without overlap with the primary and being too faint to
contribute significantly, and is therefore masked completely (red
pixels inside the postage stamp).}\label{fig_mask} \end{figure*}
\subsubsection*{Bad Pixel Masks}
\galfit\ supports so-called bad pixel masks \citep[see][]{ref_galfit}
to exclude image regions from fitting and thus speed up the fitting
process. As tertiary sources may overlap with secondaries, we take the
following approach to define the area to be masked. In general, \gala\
masks the full Kron ellipse, enlarged by a user-specified factor
\verb|D04| (which may have a different value as the one used for
computing the skymap \verb|D03|) and an additional offset \verb|D05|,
for the fitting. If the Kron ellipse of the tertiary overlaps with the
Kron ellipse of the secondary, \gala\ {\it includes} the intersection
in the fit. However, as the included area might contain significant
flux (maybe even the nucleus) of the tertiary, it {\it excludes} any
pixel marked in the \sex\ segmentation map as belonging to the
tertiary. Thus, the resulting shape of the mask may look complicated,
yet this procedure ensures having the fit of the secondary targets only
mildly affected. The primary source should not be significantly
affected at all.
To speed up the fitting process by reducing the number of simultaneous
fits, \gala\ masks secondary objects based on a magnitude criterion
\verb|D16| (for extended and \verb|D17| for point sources) in
comparison to the primary source. In the case that they are too faint
compared to the primary, \gala\ ``downgrades'' them to tertiary status
and treats them as such, \ie\ it masks their Kron ellipses completely,
but for parts which overlap with other secondaries or the primary that
are not covered by the \sex\ segmentation map.
\gala\ also masks pixels that have a value of zero in the weight map,
\ie\ an exposure time of zero. Obeying these rules results in masks as
shown in Fig.~\ref{fig_mask}.
\subsubsection*{Parameter Constraints}
\galfit\ not only applies a bad pixel mask, but also allows
fit parameters to be constrained in various ways. Examples
are keeping a parameter within an acceptable absolute range
(\eg~\sersic\ indices should satisfy $0.5<n<8$) or a relative range
depending on the given input values. Parameters might even be
constrained with respect to each other or other components. For more
details see the GALFIT homepage
\url{http://users.obs.carnegiescience.edu/peng/work/} \url{galfit/galfit.html}.
With respect to \sersic\ fitting in \gala\ providing a suitable range for
the \sersic\ index $n$ and the half-light radius $R_\mathrm{e}$ has a
stabilising effect on the procedure. To this end in \gala\ a limit
on the relative difference between \galfit\ and \sex\ magnitude is
imposed as well. \gala\ incorporates global constraints on the \sersic\
index $n$ ($0.2<n<8.0$), the half-light radius $R_\mathrm{e}$
($0.3<R_\mathrm{e}<$ \verb|E11|) and the fit magnitude $m$
(\verb|E12| $<m_\mathrm{GALFIT}-m_\mathrm{SExtractor}<$ \verb|E13|).
\subsection{Computational Optimisation}\label{sec_opt}
In the following section we will describe additional characteristics of \gala\
that increase the efficiency and robustness of the code.
\subsubsection*{Sorting and Parallel Computation}
After running \sex\ and cutting postage stamps, \galfit\ fits the
individual catalogue sources. Because efficient removal of brighter
sources is needed for accurate estimation of the sky background, an
ordered processing is required. This is extremely inefficient in terms
of total CPU time, we have developed methods to speed up this sequential
process. In the next paragraphs we will describe the mechanisms that are
incorporated into \gala\ to switch from sequential to parallel
processing and to increase the overall efficiency and robustness of the
code.
To optimise the execution time, \gala\ performs fitting in a
rank-ordered sequence starting with the brightest source in the survey
and progressing to the fainter ones. The advantage of this procedure is
twofold:\\
$\bullet$ Faint neighbours of bright sources do not have to
be included in a simultaneous fit (as a second component), as they do
not influence the resulting fit parameters of the bright object
significantly. The magnitude difference between faint and bright
neighbours is a free user parameter (\verb|D16|, \verb|D17| if the
primary is a galaxy or a star, respectively).\\
$\bullet$ When a faint
source has a brighter neighbour, which has to be included in the fit as
well, parameters for that object will already exist from a previous
fit. Hence the variables for that component can safely be held fixed to
the best values. This reduces the total number of degrees of freedom
and increases the computation speed for a large number of sources
tremendously. Another reason in favour of
sorting objects by magnitude is that the efficiency with which bright
contributors are included in the current fit is greatly enhanced.
The weakness of the sequential approach is that it voids the speed
benefits of parallel processing. To alleviate this problem, we devise
two methods:\\ a) Consecutive sources in a rank-ordered list are
usually sufficiently far apart to not affect one another (the average
object size is much smaller than their typical distance). Therefore,
\gala\ starts the next object in the sequence as a new process on
another CPU (core), given that its distance from other sources in the
queue is large enough (\verb|D20|). The extent of the brightest object
in the survey determines this distance and it should exceed the limit
out to which this object might have an influence on the fitting of
neighbours. If the next source in the queue is too close to objects
currently being executed, the code waits for these objects to finish.
b) Generally it is possible to parallelise the analysis by running the
code on one survey image only, at a time, by encapsulating the sky
fitting and \galfit\ processing. This will then enable the user to run
several instances of the code simultaneously on $n$ different
computers, thus reducing total computation time by a factor of $n$.
This is realised in \gala\ by specifying which tiles are to be
processed in a so-called ``batch file'' (\verb|E01|). The problem with
this approach is that sources may extend from one survey image onto the
next. Therefore, one might run into the situation where tile {\it A} is
fitted before tile {\it B}, with the brightest source in the two tiles
being on {\it B} and reaching into {\it A}. In this case, a fit for the
brightest object is not available for estimating the optimal sky
background for a number of galaxies on tile {\it A}. The underlying
idea of this method is that the average object size is much smaller
than the size of a survey image.
These two approaches a) and b) are implemented in the code as follows.
\sersic\ fitting with \gala\ is divided into two parts:
In the first part (see Fig.~\ref{fig_structure} upper section of block
D), \gala\ treats a fraction of all sources on all tiles in a sorted
order as laid out in method a). This assures that the brightest galaxy
from tile {\it B} is fitted before \gala\ treats tile {\it A} or {\it
B}. This part still requires sequential processing without the
possibility to run other instances of \gala\ at the same time. Also, it
produces a rather large computational overhead, as potentially with
every new source a number of large images (the complete science image,
weight image, segmentation map, etc. -- not the postage stamps) are to
be loaded into memory and processed (for fitting the sky background). A
possible working definition for the fraction of sources that have to be
fitted sequentially might encompass all sources that span an area
larger than the size of the overlaps resulting from the survey's tiling
scheme (\verb|D12|). This stage requires that all CPUs must be able to
see the whole dataset, \ie\ they have to have access to the same
harddiscs, because several threads are interacting with each other and
working on the same data.
The second part (see Fig.~\ref{fig_structure} lower section of block D)
is kept as detailed above in method b): \gala\ processes all objects
within a tile in order of decreasing brightness. Several instances of
\gala\ may be run simultaneously on {\it different} tiles. With the
sources that potentially reach into neighbouring tiles already
processed in the first part, now survey images may be treated as
individual entities, which can be processed out of order and
simultaneously. At this stage, one might think that, as the tiles are
decoupled, only a single tile is accessed at a time. However, in the
presence of big, bright sources that affect neighbouring tiles, this is
not the case any more. Therefore, even when fitting individual tiles in
parallel the whole data set must be accessible. As now only the
information for the current tile is changed, the fitting may be
distributed to different harddiscs, though (by creating identical
copies).
\begin{figure}\centering\resizebox{8cm}{!}{\includegraphics{tiling.eps}}
\caption{Definition of neighbouring tiles. For each survey image the
$n$ closest neighbours define its immediate neighbourhood. In a
checker-board configuration $n=8$ corresponds to a $3\times 3$ pattern.
At the edge of the survey more distant tiles are included. The light
green shaded areas {\it b} indicate the neighbourhoods for the central
tiles {\it a}.}\label{fig_neigh} \end{figure}
\subsubsection*{Neighbouring Tiles}
During the sky background estimation \gala\ calculates the influence of
all objects on the currently processed source. Depending on the size of
the survey, this check for contributors takes up a significant fraction
of the complete source loop computation time. However, the sources
immediately required for processing the current object are only the
ones that may have an impact on the fitting or background estimation.
Therefore, by specifying the ``reach'' of the brightest sources allows
to restrict the computation to a much smaller fraction of all sources.
This is done by providing the total number of closest tiles $n$ that
are to be included in the calculation (\verb|D18|). If the tiles are
taken on a regular grid, $n=8$ defines a ring surrounding the tile of
the current object (see Fig.~\ref{fig_neigh}). Note that in case of a
tile at the edge of the survey this ``ring'' is not cut in half, but
all tiles are selected on just one side. \gala\ {\it always} selects
the nearest $n$ neighbours. It calculates the distance between tiles
from the centres of the images.
\begin{figure}\centering\resizebox{8cm}{!}{\includegraphics{baddetections.eps}}
\caption{Correction of detection errors. Red crosses indicate
``critical'' detections; blue boxes mark ``catalogue'' detections;
green circles show ``good'' source detections. For a definition of
these terms see Sec.~\ref{sec_opt}. {\it a:} The diffraction spike of
the star was picked up as multiple individual sources. Most of them are
critical failures. Yet, one detection in each spike was kept as a
catalogue source, in order to guarantee that the fitting of nearby
objects is not biased. {\it b:} Pixel bleeding from the star. Again,
some detections were flagged critical, others as catalogue sources.
{\it c:} An over-deblended object. The excess detections are critical
errors. {\it d:} Spurious detections in the vicinity of the bright
star. All are critical failures. Note that the categorisation of
sources is an optional, subjective process, which is not performed
automatically by \gala.} \label{fig_sexerr} \end{figure}
\subsubsection*{Detection Flags}\label{sec_bad_detections}
A {\it perfect} setup for \sex\ never exists. In a small fraction of
all detections one or the other failure occurs, \eg\
(over-)~deblending, non-detection, spurious detection, etc. In
particular in the surroundings of bright stars (or even galaxies) these
errors accumulate. With respect to setting up \galfit\ properly, there
are two classes of failures: the ``critical'' and the ``catalogue''
failures. Depending on their relative brightness compared to nearby
``real'' objects they either have to be removed before the fitting
(faint sources; ``critical'') or after (bright sources; ``catalogue'').
A critical failure is a detection error that should be corrected {\it
before} running \galfit. Critical detections {\it do not affect the
fitting of neighbouring real sources}. Examples are over-deblends,
cosmic rays or a bad detection at the image edge. Critical failures
include any unwanted detection that might erroneously include
additional unnecessary components in the fitting of real objects. We
give an example for an over-deblended source in Fig.~\ref{fig_sexerr}
{\it c} and indicate several spurious detections in
Fig.~\ref{fig_sexerr} {\it d}.
In contrast, catalogue failures are detections that one has to remove
{\it after} running \galfit. They are bright in relation to
neighbouring sources and they {\it might affect the fitting of nearby
objects} if not included as separate components. Typically, they are
connected to cosmetic ``defects'' of the image. A common example for
these are diffraction spikes of stars, which may not be included in the
PSF model. Therefore, \galfit\ may not properly fit a galaxy close to
such a spike, as too much flux is in the spike compared to the source.
Common are also satellite trails or pixel column bleeding of saturated
stars. We show some examples in Fig.~\ref{fig_sexerr} {\it a} and {\it
b}.
\gala\ can optionally take care of both these failures. If the user
provides a (manually created) list of positions for critical and/or
catalogue failures (one file each), \gala\ will remove any source found
within a specified radius \verb|B16| around these positions from the
catalogues at the proper stage in the process. Otherwise, \gala\ treats
them simply as normal sources and provides \sersic\ fits for them. A
cleaner --although potentially somewhat more time consuming-- approach
would be to remove problematic regions from the data altogether (\eg\ by
replacing with white noise).
To classify unwanted detections (into one of the two categories), the
user should decide whether an object is required for obtaining a proper
fit with \galfit\ for neighbouring ``real'' sources, or not. In
principle it is save to put any detection error into the catalogue
failure list. This might lead to prolonged fitting times, though. In
practice, most detection errors are faint enough to not influence
neighbours and should therefore be put into the list of critical
failures.
Note that the definition of whether an object is a critical or
catalogue failure is subjective and depends on the user. However, the
correction of these errors is an option. \gala\ will run perfectly well
without any manual treatment. In that case, the user will have to live
with the fact that some (small) fraction of sources might be affected
by this.
\subsubsection*{Treatment of Stars}
A problem related to fitting bright saturated stars is that they are
often much brighter than the stars that one can use as a PSF model.
Because there is a limited dynamic range, the PSF cannot adequately
capture the tails seen around brighter stars, which may then
contaminate neighbouring galaxies. To deal with this situation, we fit
\sersic\ models to stars instead of the usual PSF model, because a high
\sersic\ index produces a model with extended tails. However, in so
doing, it may cause \galfit\ to not converge within a reasonable amount
of time. As the focus of \gala\ is on modelling the properties of
galaxies, no further attempt was made to apply a different, more
elaborate model (instead of the \sersic\ profile).
\begin{figure}\centering\resizebox{8cm}{!}{\includegraphics{fig_stars.eps}}
\caption{Treatment of saturated stars. Here we show source detections
from the STAGES survey in the $\log\left(\tt{fwhm\_image}\right)$
vs.~$\tt{mag\_best}$ plane. Red pluses mark objects identified as stars
\citep[see][]{ref_stages}. The line indicates the cut used to identify
saturated stars (left of the line). Black dots show other
(extragalactic) sources.}\label{fig_stars} \end{figure}
To resolve the resulting problem with the convergence of \galfit\,
\gala\ identifies saturated stars in the magnitude-size plane, which is
represented by the \sex\ parameters $\tt{mag\_best}$ and
$\log\left(\tt{fwhm\_image}\right)$ (see Fig.~\ref{fig_stars}). The
user specifies the zeropoint \verb|D15| and slope \verb|D14| of a line
below which \gala\ treats objects as saturated stars (\ie\ on the
bright and compact/small side). The reason for many of the brighter
stars to fail in the fitting is the detection of a large number of
neighbouring secondary sources (including stellar diffraction spikes),
which have to be modelled simultaneously. To reduce the number of these
secondaries the user specifies a relative magnitude cut \verb|D17|,
below which secondaries are not fitted any more and treated as
tertiaries. For the STAGES data, all objects more than two magnitudes
fainter than the primary star
($m_{\mathrm{star}}-m_{\mathrm{object}}>2$) were subject to this. Note
that for galaxies the same limitation applies, but at a much weaker
level. Again a magnitude limit \verb|D16| may be specified (\eg\
$m_{\mathrm{galaxy}}-m_{\mathrm{object}}>5$). Restricting the number of
secondaries to those objects bright enough to influence the fit and
removing the fainter ones resolves the issue.
\begin{figure*}\centering\resizebox{17.5cm}{!}{
\includegraphics{fitting_gala1b.eps}\hspace{0.5cm}
\includegraphics{fitting_gala2b.eps}} \caption{Parameter recovery as a
function of simulated mean surface brightness $\mu_\mathrm{sim}$ within
$R_\mathrm{e}$ ({\it left panel}) and simulated magnitude
$m_\mathrm{sim}$ ({\it right panel}) for two different \sersic\ index
ranges (disc-like galaxies with $n\approx1$ on the left-hand side,
early-type galaxies with $n\approx4$ on the right-hand side). Grey
levels show galaxy density, with each bin being normalised to its own
peak value. As a result, grey levels roughly resemble a mean value and
a measure for the scatter of the distribution. Due to an asymmetric
distribution and different binning, the true mean value (black line)
deviates slightly from the peak values for fainter galaxies. The
1-$\sigma$ scatter of the distributions is shown as well (dashed
lines). The light grey line indicates the ideal zero-level. Fainter
galaxies (both as function of magnitude and surface brightness) and
galaxies with higher $n$ are fitted less accurately. Also, for the
brightest galaxies in the sample, the deviation increases. Most likely
their brightness (and size) makes them the most difficult objects to
setup for fitting, because of a large number of simultaneously fitted
neighbours and because of having the highest uncertainty in their
background sky estimate.}\label{fig_param_recovery} \end{figure*}
\begin{figure*} \centering\resizebox{17.5cm}{!}{
\includegraphics{fitting_gala3b.eps}\hspace{0.5cm}
\includegraphics{fitting_gala5b.eps}} \caption{Sky recovery (flux
difference in counts) as a function of simulated galaxy \sersic\ index
({\it left panel}) and simulated magnitude ({\it right panel}).
Contours and black lines show the distribution, mean and $\sigma$ of
the estimated sky as recovered by \gala, white/grey dashed lines
indicate mean and $\sigma$ as provided by \sex. \gala\ recovers a very
accurate sky value independent of galaxy structure, whereas \sex\
overall exhibits a much larger offset, scatter and dependence on galaxy
morphology. At the brightest magnitudes a slight trend is seen in
\gala\ while \sex\ performs worse by a factor of $\sim$50. Note that
the right panel shows only objects with $3<n<5$, and thus portrays a
rather conservative scenario.}\label{fig_sky_recovery} \end{figure*}
\section{Data Quality}\label{sec_quality}
We have tested \gala\ thoroughly using simulated data as described in
more detail in \cite{ref_fitting} and \cite{ref_stages}. For the
simulations applied here, we use the same setup as for fitting the
STAGES survey. Analytical \sersic\ profiles are randomly placed on a
background image composed from patches of blank sky from real data.
The galaxy models are convolved with the same PSF as the original
STAGES data (before placing them on the background). Also Poisson
noise was added to each pixel of the galaxy models. The galaxy model
parameters randomly cover the same parameter space as the original
STAGES data with an extension to towards low fluxes and surface
brightnesses, such as to cover the full completeness space.
All in all, the simulated datasets contain around 7
million galaxies. Excluding the ones that are not recovered by \gala\
for being below the detection threshold (3 million) and the ones that
ran into any given fitting constraint ($\sim$280\,000; the following
constraints for the \sersic\ index $n$, the half-light radius
$R_\mathrm{e}$ and the magnitude $m$ were applied: $0.2<n<8$,
$0.3<R_\mathrm{e}$~[pix] $<750$,
$|m_\mathrm{GALFIT}-m_\mathrm{SExtractor}|<5$) or where the fit crashed
(293), leaves us with around 3.7 million successfully fitted galaxies.
The left panel of Fig.~\ref{fig_param_recovery} shows the deviations of
the three most important fitting parameters magnitude $m$, effective
radius $R_\mathrm{e}$ and \sersic\ index $n$ as a function of simulated
mean surface brightness $\mu_\mathrm{sim}$ within $R_\mathrm{e}$ for
two different regimes of \sersic\ index. We choose the samples such
that the completeness as a function of magnitude is roughly 90\% for
all galaxies. The low \sersic\ index sample ($m_\mathrm{sim}<24.5$)
contains $\sim$1.1 million galaxies, the high \sersic\ sample
($m_\mathrm{sim}<25.25$) contains $\sim$470\,000 galaxies. Obviously,
\gala' performance decreases at faint magnitudes and high \sersic\
indices. The right panel of Fig.~\ref{fig_param_recovery} shows the
same plot, but as a function of simulated magnitude $m_\mathrm{sim}$
rather than surface brightness to illustrate the same effects in
another commonly used parameter space. Again, we choose a cut to select
only galaxies with a surface brightness completeness exceeding 90\%.
The low \sersic\ index sample ($\mu_\mathrm{sim}<22.25$) contains
$\sim$780\,000 galaxies, the high \sersic\ sample
($\mu_\mathrm{sim}<23$) contains $\sim$295\,000 galaxies. At the faint
end, quite expectedly, the recoverability of parameters gets worse. In
both panels of Fig.~\ref{fig_param_recovery}, we see no significant
systematic trends apart from the faintest levels.
The left panel in Fig.~\ref{fig_sky_recovery} shows the deviation of
the sky value (as recovered by \gala) from the true sky value (as
derived from the empty noise image used for the galaxy simulations) as
a function of the simulated \sersic\ index $n$ of the primary object.
Obviously, the recovery of the sky in \gala\ is completely independent
of $n$. Compared to the \sex\ value for the local sky, which shows both
a much bigger offset and a larger standard deviation, the recovery is
close to ideal with very small offset and scatter. We derive the true
sky value for this plot from simple statistics on an empty noise image
used for the simulations.
\begin{figure}
\centering\resizebox{8cm}{!}{\includegraphics{fitting_gala4b.eps}}
\caption{Parameter deviations as a function of both distance ({\it left
panel}) and magnitude ({\it right panel}) of the nearest neighbouring
galaxy. The thick grey/white dashed lines indicate the zero-level;
black solid and dashed lines show the mean and $\sigma$ of the
recovered parameter, respectively. Grey contours represent the
normalised distribution of recovered parameter
values.}\label{fig_neighbour_influence} \end{figure}
Furthermore, we investigate the magnitude dependence of the sky
recoverability (see right panel of Fig.~\ref{fig_sky_recovery}). Here
we select only objects with a \sersic\ index $3<n<5$ ($\sim$300\,000
objects), which due to their extended low surface brightness wings are
hardest to fit and estimate a background value. Thus, we portray a
conservative worst case scenario. While for the large majority of all
objects there is no trend to be seen at all, at the bright end the
estimates provided by \gala\ do diverge slightly: at $m=17,18$ the mean
sky moves off by $\sim$0.04/0.03 with a scatter of $\sim$0.02,
respectively. For comparison, the values recovered by \sex\ are
$\sim$2.3/1.3 with a scatter of $\sim$0.4 at the same brightness.
To examine the influence of neighbouring galaxies in a similar fashion
as was shown in \cite{ref_fitting}, we plot parameter deviations over
both magnitude of and distance from the next neighbour. The next
neighbour we here define as the closest simulated galaxy that was found
by \sex. This does not necessarily imply that this galaxy had to be
properly deblended and simultaneously fitted when running \galfit\
(\ie\ assuming a rather conservative definition resulting in a worst
case scenario). We show these deviations in
Fig.~\ref{fig_neighbour_influence}. In contrast to the analysis in
\cite{ref_fitting}, we now have enough statistical significance to
separate both effects. We only show neighbours with
$21<m_\mathrm{sim}<23$ ($\sim$330\,000 galaxies) in the left panel and
neighbours with a distance $1<d$ [arcsec] $<1.6$ ($\sim$280\,000
galaxies) in the right panel of Fig.~\ref{fig_neighbour_influence} to
not confuse the two distinct effects: contamination by bright
neighbours and contamination by close neighbours. As one can see from
these plots, \gala\ results do not show any dependence on either of
these parameters. From this plot we conclude that the deblending and
fitting scheme applied in \gala\ works well and successful deblending
of clustered fields (as \eg\ STAGES) is possible.
\begin{figure}\centering\resizebox{8cm}{!}{\includegraphics{performance.eps}}
\caption{Performance of the galaxy modelling with \galfit. Cumulative
histogram of the fitting time per object as a fraction of the total
fitting time. The two histograms show all galaxies (black/left) and the
brightest 5\% (green/right).50\% of all sources take less than 1.25~min
to fit and more than 90\% of all objects are done within 10~min (red
lines). The brightest 5\% take about a factor of 5
longer.}\label{fig_performance} \end{figure}
\section{Performance}\label{sec_performance}
We measure the performance of \gala\ when applying it to the
single-orbit \hst\ survey STAGES \citep[see][]{ref_stages}. STAGES is a
mosaic composed of 80 tiles in the F606W filter containing
$\sim$75\,000 sources. The survey being centred on a nearby galaxy
cluster system at redshift $z\sim$0.16, it provides the ideal test case
including a high fraction of large and also peculiar objects. Large
objects serve as a test for the deblending process during the source
extraction, while peculiar objects like mergers or saturated stars with
diffraction spikes pose a challenge for the modelling with \galfit. The
total wall-clock time for processing this survey is $\sim$430~hours.
Details are given below.
The fitting process with \galfit\ is the main limitation for \gala\ '
performance. The largest amount of time was spent on fitting the
fainter 95\% of all sources in the parallel mode (second part of block
\verb|(D)| in Fig.~\ref{fig_structure}). Using eight 2.2~GHz CPU cores
in parallel, this process (\ie\ the slowest of the eight) takes
$\sim$260~hours. There is potential for further improvement by
increasing the total number of CPU cores. This would also reduce the
overhead resulting from individual pipelines not finishing at the same
time (\ie\ pipelines with fewer sources finish sooner), resulting
normally in much less than the total number of available CPUs running
simultaneously at the end of the fitting.
For the first part of block \verb|(D)|, the fitting of the brightest
5\% of all sources we used four 2.4~GHz CPU cores. This part of the fitting
takes $\sim$150~hours. Note that moving from four CPU cores to eight does
not necessarily imply halving the required computation time. The
performance increase at this stage depends on the survey geometry. A
wide area survey has a higher efficiency than a smaller survey of the
same depth, because of the higher probability that the brightest
objects in the survey are further apart from each other, thus allowing
a higher multiplicity. Fig.~\ref{fig_performance} shows a cumulative
histogram of the fitting time per object. Note that the brightest
objects take considerably longer to fit than the rest thus explaining
the necessity to find a good compromise between the time spent in the
two stages.
The remaining blocks take up an almost negligible fraction of the total
processing time. Block \verb|(B)|, the \sex\ stage, takes
$\sim$13.5~hours, including HDR mode. Cutting the postage stamps in
block \verb|(C)| requires $\sim$2.5~hours and the last block
\verb|(F)|, compilation of the output catalogue, finishes within
$\sim$0.7~hours.
Note that overheads for adjusting the setup and preparing the parallel
fitting is not taken into account in the numbers cited above. Also, for
varying survey layouts/configurations relative fractions of the total
processing times between the various stages might vary significantly.
\section{Summary}\label{sec_summary}
We present \gala, a software for automating the process of detecting
sources and modelling them with single \sersic\ profiles. \gala\
incorporates \sex\ and \galfit\ to perform these two tasks. In
addition, it provides HDR source extraction, a postage stamp cutting
facility and a robust means of estimating a local sky background. It
stores results in a combined FITS table, excluding duplicates resulting
from detections in overlapping tiles. We optimised the code for speed
and stability, making use of modern multi-core CPUs and allowing a high
degree of multiplicity. Another aim was to present the user with a
simple setup, yet enabling control over all features of the code. As a
result, \gala\ can be used on a wide variety of survey applications,
from single tile deep observations to wide area shallow surveys.
\galfit 's ability to work with any given PSF enables application of
\gala\ to both space- and ground-based data. The PSF has to be prepared
by the user before running \gala, though. This procedure is not part of
the code.
We tested \gala\ on an
extensive set of simulations and find it to be extremely robust in
terms of parameter recoverability. Note that the results of the fitting
depend on the choice of the input parameters. For example, a bad \sex\
setup will have a significant impact on the fitting procedure and thus
lower the quality of the output catalogue.
The main feature that will be implemented in \gala\ in the near future
is the option for a consistent two-component bulge-disc fitting. This
will also include an estimator providing information about whether the
increased amount of data potentially allows further insight into the
structural composition of the object or not. Based on this idea, we
will also investigate the automated fitting of bars and the application
of Fourier mode fitting, built into the most recent version of \galfit.
Another potential aspect for increasing the versatility of \gala\ could
be the implementation of a variable PSF. Currently, just one PSF is
used for convolving the \galfit\ model profiles for the whole survey.
Instead, one could allow using a different PSF depending on the
position on the tile or even varying tile by tile.
\gala\ is freely available for download from our webpage at:
\url{http://astro.uibk.ac.at/~m.barden/GALAPAGOS/}
\section*{acknowledgements}
MB was supported in part by the \emph{Austrian Science Foundation FWF} under
grant P18416. BH is grateful for support from the \emph{Science and Technology
Facilities Council (STFC)}. DHM acknowledges support from the \emph{National
Aeronautics and Space Administration (NASA)} under LTSA Grant NAG5-13102 issued
through the \emph{Office of Space Science}. CYP acknowledges support from the
Canadian NRC-HIA Plaskett Fellowship and the STScI Institute/Giacconi Fellowship
programs.
\bibliographystyle{mn2e}
|
{
"timestamp": "2012-03-09T02:03:28",
"yymm": "1203",
"arxiv_id": "1203.1831",
"language": "en",
"url": "https://arxiv.org/abs/1203.1831"
}
|
\section{Introduction}
Detailed knowledge about the statistical characteristics of the density structure is of pivotal importance for many fields in astronomy and astrophysics. Probability distribution functions (PDFs) of the density have been introduced as a simple and robust measure of the one-point statistics for many applications, ranging from cosmology, where the Press-Schechter formalism was primarily established \citep{1974PressSchechter}, to star formation and theories of the initial mass function or the core mass function (e.g., \citealp*{1982Fleck,1984Zinnecker,1997PNJ,2000KlessenBurkert}; \citealp{2004Lietal}; \citealp*{2008HennebelleChabrier,2009HennebelleChabrier,2011PN}).
In the star formation context, the relation between the width of the density PDF -- the density variance or standard deviation -- and the root-mean-square (rms) Mach number in supersonic turbulent flow is a key ingredient for analytical models of the star formation rate \citep{2005KrumholzMckee,2011PN}, and for the stellar initial mass function or the core mass function \citep{2002PN,2008HennebelleChabrier,2009HennebelleChabrier}. In this framework, supersonic turbulence plays a fundamental role in determining the density and velocity statistics of the interstellar medium \citep{2004Elmegreen,2007McKeeOstriker} and controls stellar birth \citep{2004MacLowKlessen}. Conversely, the importance of magnetic fields in the star formation process is still inconclusive, despite decades of research (\citealp{1999MouschoviasCiolek,2007McKeeOstriker}; \citealp*{2009Crutcheretal}; \citealp{2010Crutcheretal,2012Bertrametal}). Hence, the question of how magnetic fields affect the density variance--Mach number relation is still not clearly answered, despite the empirical findings of \citet{2001Ostrikeretal} and the analytical ansatz provided by \citet{2011PN}.
For purely hydrodynamical, supersonic, isothermal, turbulent gas, the relation between the density variance and Mach number has been identified and widely studied in numerical simulations (e.g., \citealp{1997PNJ,1998PassotyVS}; \citealp{2008Federrathetalb}; \citealp*{2008Federrathetala}; \citealp{2010Federrathetal}; \citealp*{2011Priceetal}). This relation is commonly assumed to be linear,
\begin{equation}\label{linear-s}
\sigma_{\rho/\rho_0}=b\mathscr{M},
\end{equation}
where $\sigma_{\rho/\rho_0}^2$ is the density variance (to emphasise the density fluctuations about the mean $\rho_0$, it makes sense to express the density in terms of the density contrast $\rho/\rho_0$), $b$ is a proportionality constant of order unity as explained in more detail below, and $\mathscr{M}$ is the rms Mach number. Usually, the density contrast is written in terms of its logarithm, $s\equiv \ln(\rho/\rho_0)$.
Several authors have noted that the PDF of the logarithm of the density contrast $s$ -- produced by supersonic turbulent flow of isothermal gas -- follows approximately a lognormal distribution (e.g. \citealp{1994Vazquez-Semadeni, 1997PNJ, 1998PassotyVS,1999NP, 2000Klessen}; \citealp*{2001Ostrikeretal}; \citealp{2003Lietal,2007Kritsuketal,2008Federrathetala, 2008Lemaster, 2009Schmidtetal,2010Gloveretal, 2010Federrathetal, 2011PN}; \citealp{2011Collinsetal, 2011Priceetal}),
\begin{equation}\label{log-norm}
p_s \dif s=\frac{1}{\sqrt{2\pi\sigma_s^2}}\exp \left[-\frac{\left(s- s_0\right)^2}{2\sigma_s^2}\right] \dif s,
\end{equation}
where the mean $ s_0$ is related to the density variance by $s_0 = -\sigma_s^2/2$, due to the constraint of mass conservation. Besides the empirical findings of \citet{1994Vazquez-Semadeni}, \citet{1997PNJ}, and \citet{1998PassotyVS}, there is no clear explanation for the shape of the PDF. From a mathematical point of view, a log-normal distribution is the result of independent random perturbations driven in a stationary system \citep{1993PopeChing} as a consequence of the central limit theorem \citep{1994Vazquez-Semadeni,1997PNJ,1999NP,2010Federrathetal}. The physical interpretation is that density fluctuations present at a given location are produced by successive passages of shocks with amplitudes independent of the local density. For a log-normal distribution, the density variance -- given by Equation~(\ref{linear-s}) -- is equivalent to
\begin{equation}\label{HD}
\sigma_s^2=\ln\left[1+{b^2\mathscr{M}^2}\right].
\end{equation}
The parameter $b$ in Equations~(\ref{linear-s}) and~(\ref{HD}) is related to the kinetic energy injection mechanism -- the forcing $\mathbf{F}$, which drives the turbulence. \citet{2008Federrathetala} found that $b=1$ for purely compressive (curl-free) forcing, $\nabla\times\mathbf{F}=0$, while $b=1/3$ for purely solenoidal (divergence-free) forcing, $\nabla\cdot\mathbf{F}=0$. In a follow-up study, \citet{2010Federrathetal} showed that $b$ increases smoothly from $1/3$ to $1$, when the amount of compressive modes, $F_\mathrm{comp}/(F_\mathrm{sol}+F_\mathrm{comp})$ is gradually increased from $0$ to $1$. For the natural mixture of modes, $F_\mathrm{comp}/(F_\mathrm{sol}+F_\mathrm{comp})=1/3$, which is also the mixture of forcing modes used in all our numerical experiments here, they found $b\approx0.4$, so we will later use that value for comparing our analytic model with numerical simulations.
When magnetic fields are included, the density variance is significantly lower than in the unmagnetised case for simulations with Mach numbers $\mathscr{M}\gtrsim 10$ \citep{2001Ostrikeretal,2011Priceetal}. Recently, \citet{2011PN} provided an analytical ansatz for the hydrodynamical density contrast in supersonic, turbulent flow, which in turn follows the approach of \citet{1980DysonWilliams} for obtaining the density contrast for strong adiabatic shocks, but extended to the magnetic case. Their $\sigma_s$--$\mathscr{M}$ relation was, however, not tested with numerical simulations.
The density PDF may or may not deviate from a log-normal form when other processes -- like heat exchange and gravitation -- are included. For example, when a non-isothermal equation of state is considered, the PDF still closely follows a log-normal distribution over a range of densities \citep[see e.g.,][]{2007GloveryMacLow}. However, depending on whether the equation of state is softer or harder than isothermal, it might acquire power-law tails either at high or low densities (\citealp{1998PassotyVS,1998Scaloetal,2001Wada}; \citealp*{2003Lietal,2007McKeeOstriker}). The density PDF also deviates from log-normal when gravity is included. In this instance, the PDF exhibits a power-law tail at high densities (\citealp{2000Klessen,2008Federrathetalb,2009Kainulainenetal,2011ChoKim}; \citealp*{2011Kritsuketal}). In addition, turbulent intermittency also leads to deviations from the log-normal PDF in the wings of the distribution \citep{2010Federrathetal}. Consequently, the accuracy of the measurement of the density variance, using Equation (\ref{log-norm}), may be compromised depending on the importance of the different processes involved in real molecular clouds.
Here, we present an analytical derivation for the $\sigma_s$--$\mathscr{M}$ relation in supersonic turbulent isothermal gas including magnetic fields. Our results are in qualitative agreement with \citet{2001Ostrikeretal} and \citet{2011Priceetal}, however, here we present quantitative predictions and tests. The present work is organised as follows: In \S\ref{ana-approach} we describe the analytical approach made for the $\sigma_s$--$\mathscr{M}$ relation. In this section, we start with the study of the density contrast of a single shock confined into a cubic box, and then we extrapolate it to the whole cloud in \S\ref{rhocontrast}. In \S\ref{sigma-s} we propose three $\sigma_s$--$\mathscr{M}$ relations given by three different assumptions of the behaviour of magnetic fields with density. We test these predictions with numerical simulations in \S\ref{test}, and conclude in \S4.
\section{Analytical derivation}\label{ana-approach}
Our basis for obtaining the density variance--Mach number relationship involves determining how the density contrast changes with the Mach number. The density variance $\sigma_{\rho/\rho_0}$ and the density contrast are related by:
\begin{equation}\label{std}
\sigma_{\rho/\rho_0}^2={1\over V}\int_V \left({\rho \over \rho_0}-1\right)^2\dif V,
\end{equation}
where $\rho$ is the local density, $\rho_0$ is the mean density in the volume, and $V$ is the volume of the cloud. The density contrast is a measure of the density fluctuations in the flow, and therefore it is useful for identifying the disturbances that originate from shock fronts and compressions.
\subsection{Density contrast in magnetohydrodynamics}\label{rhocontrast}
Supersonic turbulence in the interstellar medium generates a complex network of shock waves (or simply shocks). When the velocity of the fluid exceeds that of sound, it leads to the formation of shocks that are one of the most important distinctive effects of the compressibility of the fluid \citep[e.g.,][]{1987LandauLifshitz}.
In order to study the density contrast in a molecular cloud, we first consider the physics of the discontinuity formed by a single shock front. We then generalise the results to the ensemble of shocks confined in a cloud. Following \citet{2005Lequeux}, we describe a shock by choosing two control surfaces, one on either side of the discontinuity, and parallel to each other. Let us choose the shock surfaces as the reference frame, such that the control surfaces are stationary with respect to the shock. We also define the ``parallel'' direction as the one parallel to the flow of gas through the shock (i.e., perpendicular to the shock front). From the well known equations of fluid dynamics, it is then possible to derive equations that expresses the conservation of matter and momentum flux for a magnetised inviscid, neutral fluid:
\begin{equation}\label{masseq}
v_{\parallel,1}\rho_1=v_{\parallel,2}\rho_2,
\end{equation}
and
\begin{equation}\label{origin}
\rho_1 \left(v_{\parallel,1}^2 + {c_{s,1}^2\over \gamma_1} + {v_{A{\mathbf \perp},1}^2\over 2} \right) = \rho_2\left(v_{\parallel,2}^2+ {c_{s,2}^2\over \gamma_2} + {v_{A\perp,2}^2\over 2}\right),
\end{equation}
respectively. In these equations, the subscripts 1 and 2 indicate the pre--and post-shock conditions, respectively. The velocity of the gas into the shock is $v_\parallel$, while $c_s$ is the adiabatic sound speed, $\gamma$ is the ratio between the specific heats and $v_{A\perp}$ is the Alfv\'en velocity, defined here as $v_{A\perp}= B_\perp/(4\pi \rho)^{1/2}$, where $B_\perp$ is the magnetic field perpendicular to the flow direction. The post-shock density is described by $\rho_2$.
We now make two important approximations. First, as we wish to focus on the role of magnetic fields in determining the density variance, we assume that the gas is isothermal, deferring consideration of non-isothermal effects to future work. Our assumption of isothermality implies that $c_{s,1}=c_{s,2}=c_s$ and $\gamma_1=\gamma_2=1$. Second, as we are considering an entire molecular cloud, we approximate it as an ensemble of shocks. We assume that we can express the average pre-shock velocity in terms of the rms velocity $v_0$ -- hereafter, the subscript ``0'' indicates the volume averages -- as $v_{\parallel,0}^2=b^2v_0^2$, where the factor $b$ depends on the number of degrees of freedom available for the compressive modes \citep{2008Federrathetala}. We also assume that the typical pre-shock magnetic and thermal pressures are just those given by volume averages over the total volume, allowing us to write them in terms of the volume-averaged density $\rho_{0}$ and the rms Alf\'enic velocity $v_{A,0}$. Similarly, we assume that the typical pre-shock density is simply the volume-averaged density. Making these assumptions, and introducing the ratio of the thermal pressure to magnetic pressure
\begin{equation}\label{betafin}
\beta\equiv{P_{\rm{th}} \over P_{\rm{mag}}}={2}{c_{s}^2\over v_{A}^2},
\end{equation}
we can rewrite Equation (\ref{origin}) as
\begin{equation}\label{2-origin}
b^2\mathscr{M}^2{\rho_0\over \rho_2}\left(1-{\rho_0\over \rho_2}\right) +{\rho_0\over \rho_2}\left(1 + \beta_0^{-1}\right) = \left( 1 + \beta_2^{-1}\right),
\end{equation}
where the rms Mach number is given by $\mathscr{M}=v_0/c_{s}$.
In order to solve this equation for the characteristic density contrast associated with the shocked gas, $\rho_{2} / \rho_{0}$, it is necessary to determine $\beta_{2}$, the post-shock ratio of the thermal to magnetic pressures. The value of this will depend on the change in the magnetic field strength through the shock, which in turn depends on the orientation of the field with respect to the flow of gas through the shock. Using magnetic flux and mass conservation during compression, one can show that $B\propto\rho^\alpha$ with $0\leq\alpha\leq1$, depending on the field geometry and direction of compression. In the extreme case where the gas flows in a direction parallel to the field lines, the field strength will be the same on either side of the shock despite the jump in density, and the field strength then will be independent of density, i.e., $\alpha=0$. In the other extreme case where the field is oriented at right-angles to the gas flow, the shock jump conditions for magnetic flux freezing imply that $B \propto \rho$, i.e., $\alpha=1$. Meanwhile, compression of an isotropic field along all three spatial directions gives $B\propto\rho^{2/3}$. However, for our ``average shock'', we expect behaviour that lies somewhere between $0\lesssim \alpha \lesssim 1$. By looking at observations and existing simulations, we can get some guidance as to what this intermediate behaviour should be.
Observationally, \citet{1999Crutcher} presented a study of the magnetic field strength in molecular clouds measured with the Zeeman effect. He fitted the results with a power law $B\propto \rho^\alpha$ and found that $\alpha=0.47 \pm 0.08$. \citet*{2003Crutcheretal} provided additional support for this result. More recently, \citet{2010Crutcheretal} have presented a detailed compilation of Zeeman data based on a much larger number of measurements. They find that at number densities
$n<300$ cm$^{-3}$, the data is consistent with a field strength that is independent of density, while at higher densities they obtain $B\propto \rho^{0.65\pm 0.05}$.
From a theoretical point of view, \citet{1999PN} noted that their $B$ distributions closely match the observational scaling given by \citet{1999Crutcher} and \citet{2003Crutcheretal}, $B\propto \rho^{1/2}$, for high $B$ in their high Alfv\'enic Mach number regime. \citet*{2001Kimetal} also study the relationship between $B$ and $\rho$, and find that $\alpha \simeq 0.4$, albeit with large scatter, especially at low densities. Additionally, \citet{2009Banerjeeetal} report that the magnetic field strength appears to scale in their simulations as $B\propto \rho^{1/2}$ for number densities $10^2\lesssim n\lesssim 10^4$ cm$^{-3}$, although with significant scatter around this value. On the other hand, \citet{2000HennebellePerault} found that the magnetic field does not necessarily increase with the density. Asides from these reports, if the magnetic flux is not conserved, but increases due to turbulent dynamo amplification during compression, $\alpha$ can become larger than the values quoted above, depending on the Reynolds numbers of the gas \citep{2010Schleicheretal,2010Suretal,2011Federrathetal}. Thus, even if the gas is compressed only parallel to the field lines, turbulent tangling of the field can lead to $\alpha>0$ during compression.
Given the different possible relations between the magnetic field strength and the density, we consider three cases to include in Equation (\ref{2-origin}): the two extreme cases, where $B$ is independent of the density, and where $B \propto \rho$, and an intermediate case with $B \propto \rho^{1/2}$. We also note that if we were to take instead the relation $B\propto \rho^{0.65}$ suggested by the most recent observational data, then we would obtain results quite similar to the $B \propto \rho^{1/2}$ case.
\subsubsection{First case: $B$ independent of $\rho$}\label{1case}
We start by considering one extreme, the case where $B$ is independent of the density. In this scenario, Equation (\ref{2-origin}) becomes a second-order equation, independent of the magnetic field strength
\begin{equation*}
\left({\rho_2 \over \rho_0}\right)^2-\left(b^2\mathscr{M}^2+1\right)\left({\rho_2\over \rho_0}\right)+b^2\mathscr{M}^2=0.
\end{equation*}
This equation results in a density contrast
\begin{equation}\label{conte}
{\rho_2 \over \rho_0}=b^2\mathscr{M}^2.
\end{equation}
Equation~(\ref{conte}) matches the density contrast for the non-magnetic regime \citep[see e.g.,][]{1997PNJ}. This is not surprising, because in this case we are assuming that the gas and the magnetic field are not coupled. Therefore, amplification of the magnetic field with density is not expected under these conditions.
\subsubsection{Second case: $B\propto \rho^{1/2}$}
In the intermediate case in which $B\propto \rho^{1/2}$, we again find a second-order equation for the density contrast, but with a dependence on the magnetic field expressed in terms of $\beta_0$. From Equation (\ref{2-origin}), we obtain
\begin{equation*
\left(1+\beta_0^{-1}\right)\left({\rho_2 \over \rho_0}\right)^2-\left(b^2\mathscr{M}^2+1+\beta_0^{-1}\right)\left({\rho_2\over \rho_0}\right)+b^2\mathscr{M}^2=0.
\end{equation*}
This equation has the solution:
\begin{equation}\label{quad}
{\rho_2 \over \rho_0}=b^2\mathscr{M}^2\left({\beta_0 \over \beta_0 +1}\right).
\end{equation}
In other words, the effect of the magnetic field in this case is to reduce the density contrast by a factor $\beta_0/(\beta_0+1)$. We see from this that in the weak field limit where $\beta_0\rightarrow\infty$, we recover the hydrodynamical result, while for strong fields we have a smaller density contrast in the MHD case than in the non-magnetic case.
\subsubsection{Third case: $B\propto \rho$}
Finally, we investigate the other extreme case, where the magnetic field strength is proportional to the density. In this case, Equation (\ref{2-origin}) results in a third-order equation,
\begin{equation*
\beta_0^{-1}\left({\rho_2 \over \rho_0}\right)^3+\left({\rho_2 \over \rho_0}\right)^{2}-\left(b^2\mathscr{M}^2+1+\beta_0^{-1}\right)\left({\rho_2\over \rho_0}\right)+b^2\mathscr{M}^2=0.
\end{equation*}
The solution for the density contrast is
\begin{equation}\label{cub}
{\rho_2 \over \rho_0}={1\over 2}\left(-1-\beta_0+ \sqrt[]{\left(1+\beta_0\right)^2+4b^2\mathscr{M}^2\beta_0}\right).
\end{equation}
\subsection{Density variance--Mach number relation}\label{sigma-s}
In the previous section, we presented three different expressions for the density contrast. They correspond to three different assumptions regarding the relationship $B\propto \rho^\alpha$, with $\alpha=0,1/2$, and 1. We now determine the density variance of a fluid in which there are many shocks, for each of these three cases.
We start by noting that in a highly supersonic flow, the dominant contribution to the integral in Equation (4) will come from shocked regions, and thus we can consider this equation as a volume average over an ensemble of many shocks. We next assume that we can approximate the value of this integral with the result of integrating over a single ``average'' shock of the kind considered in the previous section. As we already know the density contrast of this representative shock, the only thing that remains to be done before we can solve Equation (4) is to determine the appropriate volume over which to integrate.
We approximate the cloud as a cubic box of side L, and consider an infinitesimal part of its volume $\dif V$ that encloses one shock. Therefore, the size of $\dif V$ depends on the size of the shock itself
\begin{equation}\label{difer}
\dif V\approx\dif V_{sh}.
\end{equation}
To define the shock volume, we make use of an approximation introduced by \citet{2011PN}, where the volume of the shock is given by the area of the box face times the shock width $\lambda$, $V_{\rm sh} = L^{2} \lambda$. However, in the absence of viscosity, it is not straightforward to define the shock width $\lambda$. Therefore, we follow \citet{2011PN} and assume that the shock width, if the compression is driven at the box scale, is given by
\begin{equation}
\lambda \simeq \theta L \rho_{0} / \rho_{2},
\end{equation}
where $\theta$ is the integral scale of the turbulence. Then, the volume of the shock $V_{\rm sh}$ is given by
\begin{equation}\label{vsh}
V_{\rm sh} \simeq \theta L^{3} \frac{\rho_{0}}{\rho_{2}}.
\end{equation}
For turbulence driven on large scales, as appears to be the case in real molecular clouds \citep{2002OssenkopfMacLow,2009Bruntetal}, we have $\theta \simeq 1$. Having made the assumption that the appropriate volume over which to average is the volume of our representative shock, and considering Equation \ref{difer}, we approximate $\dif V$ by
\begin{equation}\label{difer}
\dif V = L^3\left({\rho_0 \over \rho_2}\right)^2 \dif \left({\rho_2\over \rho_0}\right).
\end{equation}
Finally, inserting Equation (\ref{difer}) into Equation (\ref{std}), yields
\begin{equation}\label{resstd}
\sigma_{\rho/\rho_0}^2=\int_1^{\rho \over \rho_0} \left(1-{\rho_0 \over \rho_2}\right)^2 \dif \left({\rho_2\over\rho_0}\right)={\rho \over \rho_0} -{\rho_0 \over \rho} -2\ln {\left({\rho \over \rho_0}\right)}.
\end{equation}
It is important to note that in this formulation, Equation~(\ref{resstd}) is physically meaningless if the lower limit of the integral is set between $0<\rho/\rho_0<1$. It is due to the definition adopted for the shock width (Eq. \ref{vsh}), where the shock thickness is defined only for $\rho_2/\rho_0>1$. For highly supersonic turbulence, which is the regime that concerns us, the assumption $\rho\gg\rho_0$ is valid. Then, the first term in Equation (\ref{resstd}) dominates the variance and we get
\begin{equation}
\sigma_{\rho/\rho_0}^2 \approx {\rho \over \rho_0}.
\end{equation}
For practical reasons, we prefer to consider the variance of the logarithm of the density contrast, $s=\ln(\rho/\rho_0)$, instead of the variance of the linear density when we will compare this analytical model with numerical simulations. These variances are related by \citep[e.g.,][]{2008Federrathetala,2011Priceetal}
\begin{equation}\label{std2}
\sigma_{s}^2=\ln\left[1+\sigma_{\rho/\rho_0}^2\right].
\end{equation}
We now insert the three cases considered in \S\ref{rhocontrast} into Equation (\ref{std2}), in order to obtain the density variance--Mach number relation. The subscripts of the following results are chosen based on the value $\alpha=0$, $1/2$ and 1 of the $B\propto\rho^\alpha$ relationship.
\begin{itemize}
\item {\boldmath$B$} {\bf independent of} {\boldmath$\rho$}
The density variance in this case is exactly the same as for the purely hydrodynamical, isothermal case,
\begin{equation}\label{HD2}
\sigma_{s,0}^2=\ln\left[1+{b^2\mathscr{M}^2}\right].
\end{equation}
\item {\boldmath$B\propto \rho^{1/2}$}
In this case, the density variance is:
\begin{equation}\label{supreme}
\sigma_{s,1/2}^2=\ln\left[1+b^2\mathscr{M}^2\left({\beta_0 \over \beta _0+1}\right)\right].
\end{equation}
This relation is similar to Equation (\ref{HD2}) except for a correction factor due to the effects of magnetic fields, which is a function of the plasma $\beta_0$ only.
\item {\boldmath{$B\propto \rho$}}
Finally, the density variance--Mach number relation in this case is given by
\begin{equation}\label{supreme2}
\sigma_{s,1}^2=\ln\left[1+{1\over 2}\left(-1-\beta_0+ \sqrt[]{\left(1+\beta_0\right)^2+4b^2\mathscr{M}^2\beta_0}\right)\right].
\end{equation}
The density variance has a strong dependence on $\beta_0$, leaving the rms Mach number as a marginal quantity in this relation.
\end{itemize}
In the last two cases, when $\beta_0\rightarrow 0$, the Alf\'enic velocity is much higher than the sound speed, and both relations approach zero. In this scenario, the magnetic pressure is infinitely large and prevents density fluctuations from forming. The gas is ``frozen'' in the magnetic field. In the opposite limit, when $\beta_0\rightarrow \infty$, Equation (\ref{supreme}) and Equation (\ref{supreme2}) simplify to the purely hydrodynamical case, as expected. In the next section, we are going to test these cases with numerical simulations.
\section{Numerical test of the analytical model}\label{test}
\subsection{Simulations}
We have performed simulations of the evolution of the turbulent, dense, inviscid, magnetised (MHD) and unmagnetised (HD), isothermal interstellar medium using a modified version of the {\sc zeus-mp} hydrodynamical code \citep{2000Norman,2006Hayesetal}. We neglect chemical reactions in order to study the effects of magnetic fields in molecular clouds, leaving the inclusion of the effects of chemistry \citep{2010Gloveretal} for a future study.
Each of our simulations begins with an initially uniform gas distribution, with a mean hydrogen number density of $n_0=1000$ cm$^{-3}$ and a resolution of 256$^3$ cells. The initial velocity field is turbulent, with power concentrated on large scales, between wave numbers $k=1$ and 2 and with an initial rms velocity of $5$ km\,s$^{-1}$. Moreover, we drive the turbulence so as to maintain approximately the same rms velocity throughout the simulations, following the method described in \citet{1998MacLowetal} and \citet{1999MacLow}. We do not perform a Helmholtz decomposition of the force field, and thus the turbulent forcing consists of a natural mixture of solenoidal and compressive modes, i.e., $F_\mathrm{sol}/(F_\mathrm{sol}+F_\mathrm{comp})\approx2/3$. Note that \citet{2008Federrathetala,2010Federrathetal} tested the two limiting cases of purely solenoidal (divergence-free) and purely compressive (curl-free) forcing, as well as various mixtures of solenoidal and compressive modes of the turbulent forcing. They found a strong influence on the density PDF, producing a three times larger standard deviation for compressive forcing compared to solenoidal forcing. They parameterised the influence of the forcing by introducing the $b$-parameter in Equation~(\ref{HD}). Purely solenoidal forcing is characterised by $b=1/3$, while purely compressive forcing gives $b=1$. For the natural mixture, they find $b\approx0.4$. Using the present set of numerical models, we confirm that using $b = 0.4$ for the natural mixture of forcing modes used here gives the best fits with our analytically derived density variance--Mach number relation. The temperature of the gas is constant and fixed to an initial value $T_0=1062$, 170, 42 and 15 K, in order to sample a large set of Mach numbers $\langle\mathscr{M}\rangle\simeq2$, 5, 10 and 17, respectively. We adopt periodic boundary conditions for the gas using a cubical simulation volume with a side length $L=20\,$pc, such that the turbulent crossing time, $T_{\rm cross}=L/(2c_s\mathscr{M})\approx2$ Myr. We present results from $t=3\,T_{\rm cross}\approx 5.7$ Myr, sampled every $0.17\,T_{\rm cross}$, and evolved until $t=4\,T_{\rm cross}\approx 7.6$ Myr. This period of time is long enough to expect the turbulence to have reached a statistical steady state (\citealp*{2009Federrathetal}; \citealp{2010Federrathetal,2010Gloveretal,2010PriceFederrath}). This simulation time might be also short enough to obtain reliable results for the initial phase of star formation, when self-gravity did not yet have a large effect on the dynamics. In order to concentrate on turbulent compression alone, we neglect self-gravity in the present experiments.
For the MHD cases, the simulations begin with a uniform magnetic field that is initially oriented parallel to the $z$-axis of the simulation. Four of these simulations begin with an initial magnetic field strength $B_i=5.85\,\mu$G, which is our standard magnetic field strength hereafter. We also perform three MHD runs with $B_i=10$, 20 and 60$\,\mu$G, with $\mathscr{M}=10$, to check the behaviour of the results with increasing magnetic field strengths. We note that as the simulations run, dynamo amplification can lead to increased field strength, and thus we use the instantaneous magnetic field strength to compute $\beta_{0}$. Nevertheless, for simplicity we use the initial value of the magnetic field strength to label runs MHD-B2, MHD-B20 and MHD-B60.
In Table 1, we list the simulations that we have performed. In our labels, we use ``H'' to denote a hydrodynamic run and ``MHD'' to denote a magnetohydrodynamic run. Our multiple runs with fixed (or zero) magnetic field strength but different sound-speeds are labelled with an ``M", followed by the (approximate) rms Mach number of the simulation. Finally, the three runs in which we examined the effect of varying the initial magnetic field strength are labelled with a ``B'', followed by the initial field strength in $\mu$G. In Table 1, we also list the values of the quantities: $\beta_0$, the rms Alfv\'enic Mach number $\mathscr{M}_{A,0}=v_{0}/v_{A,0}$ and the sonic Mach number. They are measured in every cell and then are spatially averaged over the datacube. The brackets denote the time average over the seven snapshots, and the $1\sigma$ shows the temporal standard deviation around the mean values.
\begin{table}
\caption{List of simulations.}
\begin{tabular}{l@{ }c@{ }r@{ }c@{ }l@{ }r@{ }c@{ }l@{ }r@{ }c@{ }l@{ }r@{ }c@{ }l@{ }}
\hline
& $B_i$ &$\langle\beta_0\rangle$&$\pm$&$1\sigma$& $\langle\mathscr{M}_{A,0}\rangle$& $\pm$ & $1\sigma$&$\langle\sigma_s\rangle$&$\pm$& 1$\sigma$ &$\langle\mathscr{M}\rangle$&$\pm$ & $1\sigma$ \\
\hline
HD-M2 & 0 & &$\infty$& & & 0 & & 0.77 &$\pm$& 0.02 & 2.21 &$\pm$& 0.02 \\
HD-M5 & 0 & &$\infty$& & & 0 & & 1.3&$\pm$&0.1 & 5.4 &$\pm$& 0.1 \\
HD-M10 & 0 & &$\infty$& & & 0 & & 1.7&$\pm$&0.1& 10.6 &$\pm$& 0.2 \\
HD-M17 & 0 & &$\infty$& & & 0 & &1.92&$\pm$&0.09& 17.6 &$\pm$& 0.5 \\
MHD-M2 & 5.85 & 25 &$\pm$& 5 & 8.1 &$\pm$& 0.9& 0.69&$\pm$&0.02& 2.09 &$\pm$& 0.02 \\
MHD-M5 & 5.85 & 4.8 &$\pm$& 0.4 & 8.4 &$\pm$& 0.8& 1.18&$\pm$&0.04& 4.98 &$\pm$& 0.07 \\
MHD-M10 & 5.85 & 1.4 &$\pm$& 0.5 & 9 &$\pm$& 3 & 1.47&$\pm$&0.06& 10.2 &$\pm$& 0.3\\
MHD-M17 & 5.85 & 0.3 &$\pm$& 0.1& 7 &$\pm$& 2& 1.61&$\pm$&0.06 & 16.8 &$\pm$& 0.5 \\
MHD-B2 & 2 & 11.3 &$\pm$& 0.5 & 27 &$\pm$& 2& 1.58&$\pm$&0.09 & 10.5 &$\pm$& 0.2 \\
MHD-B20 & 20 & 0.083 &$\pm$& 0.005& 1.94 &$\pm$& 0.06& 1.48&$\pm$&0.01 & 9.9 &$\pm$& 0.2 \\
MHD-B60 & 60 & 0.030 &$\pm$& 0.001& 1.24 &$\pm$& 0.03& 1.34&$\pm$&0.01 & 10.3 &$\pm$& 0.1\\
\hline
\label{table}
\end{tabular}
$B_i$ -- initial magnetic field strength in $\mu$G. \\
$\beta_0$ -- mean thermal to instantaneous magnetic pressure ratio. \\
$\mathscr{M}_{A,0}$ -- rms Alfv\'enic Mach number. \\
$\sigma_s$ -- density variance. \\
$\mathscr{M}$ -- rms Mach number. \\
The brackets indicate the time average calculated over the snapshots after averaging over the spatial coordinates.
\end{table}
\subsection{Statistical Analysis}
In this subsection, we explain the method used to measure the density variance for every snapshot in our simulations using the PDF as a robust statistical tool for this analysis \citep{2011Priceetal}. Then, we parameterise the instantaneous $\beta_0$ in terms of $\mathscr{M}$, in the direction of testing numerically the $\sigma_{s}$--$\mathscr{M}$ relations presented in \S \ref{sigma-s}. Finally, we present the comparison between our analytical model and the simulations.
\subsubsection{Probability Density Function (PDF)}\label{1era}
In Figure \ref{PDFs}, we plot the volume-weighted dimensionless density PDFs for MHD and HD isothermal gas with the same Mach number for comparison. For these simulations, we find that all the PDFs have a log-normal shape around their peak. However, the PDFs deviate from log-normality especially in the HD simulations at low densities, being more evident for $\mathscr{M} \gtrsim 5$. The error bars in this figure show the 1$\sigma$ variations around the time average. We see that these variations cannot explain the tail at low densities. Therefore, this deviation is not explained by intermittency fluctuations, and deserves further study. However, the low-density tail does not significantly affect our $\sigma_s$ estimates, because the variance is computed from a log-normal fit in a limited interval around the peak, giving the most reliable estimates of $\sigma_s$ \citep[see][]{2011Priceetal}. In this sense, the trend of the time averages observed between MHD and HD simulations shows the magnetic field acting as a density cushion, preventing the gas from reaching very low densities during local expansion. As a consequence, there are larger parts of the volume with density $\rho \approx \rho_0$ in the MHD case than in the HD case.
\begin{figure}
\resizebox{8cm}{!}{\rotatebox{0}{\includegraphics{fig1.eps}}}
\caption{Dimensionless density PDF for magnetised and unmagnetised molecular clouds with the same initial conditions, $n_0=1000$ cm$^{-3}$, and same turbulent rms velocity, but different sound speed. The most significant features are: 1) the density variance increases with Mach number, and 2) the density variance decreases with magnetic field strength. These simulations have a ratio between thermal pressure and magnetic pressure $\beta_0\lesssim10$. All simulations have a resolution of $256^3$ zones.}
\label{PDFs}
\end{figure}
In order to avoid contamination from intermittency, numerical artefacts, etc., in the wings of the PDFs, we perform a Gaussian fitting only in a data subset selected by $s$, in each simulation. This subset consists of 60\% of the number of bins considered to calculate the density PDF which are distributed symmetrically around the mean, $s_0$. Then, we fit the Gaussian profile given by Equation (\ref{log-norm}) to obtain $\sigma_s$ in every snapshot of the simulations.
\subsubsection{Density variance--rms Mach number test}\label{sigmatest}
In the interest of comparing the density variance--Mach number relation, given by Equation (\ref{supreme}) and Equation (\ref{supreme2}), with the results obtained in the previous subsection, we parameterise the thermal-to-magnetic pressure ratio in terms of the rms Mach number for our sequence of simulations. In this sense, we rewrite Equation (\ref{betafin}) as
\begin{equation}\label{betamach}
\beta_0={2}{\mathscr{M}_{A,0}^2 \over \mathscr{M}^{2}}.
\end{equation}
Note that this parameter is calculated considering the instantaneous magnetic field strength and not the initial value.
Next, we select the four MHD simulations with different rms Mach number, but the same initial magnetic field strength, and use a linear regression considering the logarithm of Equation (\ref{betamach}): $\log_{10}\beta_0=\log_{10}C-2\log_{10}\mathscr{M}$. From the fit shown in Figure~\ref{beta}, we find $C= 111 \pm 4$. In Figure \ref{beta}, we plot $\beta_0$ as a function of the rms Mach number for the different snapshots. The triangles show $\beta_0$ for the selected simulations with $\langle\mathscr{M}\rangle\approx2$, 5, 10 and 17, while the curve shows the linear regression.
\begin{figure}
\resizebox{9cm}{!}{\rotatebox{90}{\includegraphics{fig2.eps}}}
\caption{Parameterisation of $\beta_0=P_{\rm th}/P_{0,\rm mag}$ with respect to the rms Mach number for the subset of simulations with roughly constant Alfv\'enic Mach number, $\mathscr{M}_{A,0} \approx 8$ (see Table~\ref{table}). The curve is a linear regression of the MHD simulations with $B_i=5.85\mu$G. The linear regression performed to the logarithm of Equation (\ref{betamach}) gives $\beta_0=(111\pm 4)\mathscr{M}^{-2}$.}
\label{beta}
\end{figure}
In Figure \ref{dinda}, we combine the dimensionless standard deviation $\sigma_s$, obtained from the fit over the numerical PDFs for every snapshot, and the analytical prediction for the three cases of $B\propto\rho^\alpha$ -- with $\alpha=$0,1/2, and 1 -- as a function of the rms Mach number. For the triangles around a given $\langle\mathscr{M}\rangle$, the HD simulations exhibit larger
$\sigma_s$ compared with the MHD simulations, as was expected from Figure \ref{PDFs}. For comparison, we plot the analytical prediction given by Equation (\ref{HD2}), $\sigma_{\alpha,0}$. This result matches the prediction provided by \citet{1997PNJ}. However, instead of using their proportionality parameter $b\approx 0.5$, we used the input value $b=0.4$ \citep[][dashed line]{2010Federrathetal}, which is the result of the natural mixing of solenoidal and compressive modes in the turbulent forcing field. We also plot the two extreme cases for the unmagnetised gas, $\sigma_{s,HD}$, with $b=1/3$ (lower dotted line) for purely solenoidal forcing and $b=1$ for purely compressive forcing (upper dotted line) for comparison.
In the same Figure, we superpose Equation (\ref{supreme}, light grey solid line) and Equation (\ref{supreme2}, dark grey solid line), both again with $b=0.4$. We find than the best agreement with the MHD simulations is given by Equation (\ref{supreme}), that is $\sigma_{s,1/2}$. The result obtained for the first case -- $B$ independent of density (Equation \ref{HD2}) -- may account only for low Mach number zones. This case might be appropriate for diffuse clouds \citep{2010Crutcheretal}, where the mean sound speed of the cloud may be of the same order as the rms velocity. Here, at $\mathscr{M} \sim1$, all the three cases converge to the HD result.
\begin{figure*}
\resizebox{18.cm}{!}{\rotatebox{90}{\includegraphics{fig3.eps}}}
\caption{Standard deviation of the dimensionless density contrast, plotted as a function of the rms Mach number. Circles show the purely hydrodynamical simulations that follow very well the
\citet{1997PNJ} prediction, $\sigma_{s,HD}^2=\ln(1+b^2\mathscr{M}^2)$, with $b=0.4$, expected for mixed-mode turbulent forcing \citep[][dashed line]{2010Federrathetal}. The dotted lines are for comparison with purely hydrodynamical model, assuming $b=1/3$ for purely solenoidal forcing and $b=1$ for purely compressive forcing
\citep{2008Federrathetala}. Triangles show the MHD simulations and the two formulas, Eqs. (\ref{supreme}) and (\ref{supreme2}), obtained in this work: $\sigma_{s,1/2}=\{\ln[1+b^2\mathscr{M}^2\beta_0/(\beta_0+1)]\}^{1/2}$ (light grey solid line), and $\sigma_{s,1}$ (dark grey solid line). Those curves are plotted for $b=0.4$, and using our parameterisation, $\beta_0=(111 \pm 4) \mathscr{M}^{-2}$ from Fig. \ref{beta}. Squares, stars and diamonds show the additional MHD simulations with different rms Alfv\'enic Mach number, $\mathscr{M}_{A,0}\approx 27$ ($B_i=2\,\mu$G), $\mathscr{M}_{A,0}\approx 1.9$ ($B_i=20\,\mu$G), and $\mathscr{M}_{A,0}\approx 1.2$ ($B_i=60\,\mu$G).}
\label{dinda}
\end{figure*}
Our results are qualitatively in agreement with \citet{2001Ostrikeretal} and \citet{2011Priceetal}. These authors find that the density variance in magnetised gas is significantly lower than in the HD counterparts for simulations with a Mach number $\mathscr{M}\gtrsim 10$. In addition, \citet{2003ChoLazarian} study the density contrast resulting from the Alfv\'enic waves, slow and fast magneto-sonic waves originating in different environments. The authors concluded that the three kinds of waves can coexist in those environments. In the regime that concerns us, $\beta_0\approx 1$ and $5\lesssim\mathscr{M}\lesssim10$, their density contrasts closely match ours.
To test the validity of our results for different Alfv\'enic Mach numbers, we also performed three simulations with an initial magnetic field strength different from the standard one, with $\mathscr{M}_{A,0}\approx27$, 1.9, and 1.2, at $\langle\mathscr{M}\rangle\approx 10$ (empty squares in Figure \ref{dinda}). Our model works well for $\mathscr{M}_{A,0}\gtrsim 6$, but breaks down for our test with $\mathscr{M}_{A,0}\lesssim 2$. The break occurs when the turbulence becomes trans-Alfv\'enic or sub-Alfv\'enic, i.e., when $\mathscr{M}_{A,0}\lesssim2$. This is due to anisotropies arising in this case, i.e., the turbulence is no longer isotropic, as can be seen in Figure \ref{dibujito}. This is because the back reaction of the magnetic field onto the flow is extremely strong for flows perpendicular to the magnetic field lines, if the turbulence is trans-Alfv\'enic or sub-Alfv\'enic \citep*[see e.g.,][]{2003ChoLazarian, 2010Bruntetal, 2011EsquivelLazarian}. Since our analytic derivation is based on an ensemble average (Eq.~\ref{std}), assuming statistical isotropy, the anisotropies are the most likely cause for the limitation of our model to super-Alfv\'enic turbulence. In Figure~\ref{sig-alf}, we show our prediction (Eq. \ref{supreme})\footnote{Equation (\ref{supreme}) has been written in terms of the instantaneous Alfv\'enic Mach number (Eq. \ref{betamach}), yielding the relation for the density variance: $\sigma_{s,1/2}^2=\ln[1+2b^2\mathscr{M}^2\mathscr{M}_{A,0}^2/(2\mathscr{M}_{A,0}^2+\mathscr{M}^2)]$.} for a fixed Mach number $\mathscr{M}\approx10$ and forcing parameter $b\approx 0.4$, which fits very well the data with $\mathscr{M}_{A,0}\gtrsim 6$. These simulations show high dispersion -- around the time average -- in the density variance and the rms Alfv\'enic Mach number showing the fluctuations of the gas caused by the turbulence dominating the dynamics of the flow, in contraposition of the simulations with small Alfv\'enic Mach number. In the same Figure, we also plot the model curve Eq. (\ref{supreme}) for the same sonic Mach number 10 and $b=1$. Although our turbulent forcing in the simulations is by definition mixed, and thus we expect $b\approx 0.4$ \citep{2010Federrathetal}, we find it interesting to note that $b=1$ -- corresponding to purely compressive forcing -- gives a good fit to the data with very low Alfv\'enic Mach number, $\mathscr{M}_{A,0} \lesssim 2$. We speculate that the density field for very high magnetic field strengths and thus very low Alfv\'enic Mach number starts behaving as if it was driven by purely compressive forcing. This is very different from the compression obtained with solenoidal or mixed forcing, but more similar to compressive forcing, which also directly compresses the gas \citep{2008Federrathetala}. More data at $\mathscr{M}_{A,0} \lesssim 2$ would be needed to sample this region and the transition from $b=0.4$ to 1 in detail, and we just note here that $b=1$ seems to provide a good fit for $\mathscr{M}_{A,0} \lesssim 2$, given the data at hand.
\begin{figure*}
\resizebox{18.cm}{!}{\includegraphics{fig4.eps}}
\caption{Density slices of the simulations at $t=6$ Myr. The mean magnetic field is oriented along the vertical axis. From left to right: initial magnetic field strength $B_i=2$, 5.85, 20 and 60 $\mu$G. The turbulence remains isotropic for super-Alfv\'enic gas $\mathscr{M}_{A,0}\gg1$, but when it becomes trans-Alfv\'enic or sub Alfv\'enic ($\mathscr{M}_{A,0}\lesssim3$), the turbulence becomes highly anisotropic.}
\label{dibujito}
\end{figure*}
\begin{figure}
\resizebox{9cm}{!}{\rotatebox{90}{\includegraphics{fig5.eps}}}
\caption{Standard deviation of the dimensionless density contrast, plotted as a function of the instantaneous rms Alfv\'enic Mach number at $\langle\mathscr{M}\rangle\approx 10$. The different symbols show snapshots of simulations with $\mathscr{M}_{A,0}$ time averages: $\langle \mathscr{M}_{A,0}\rangle\approx 27$ (squares), $\langle\mathscr{M}_{A,0}\rangle\approx 9$ (triangles), $\langle\mathscr{M}_{A,0}\rangle\approx 1.9$ (stars), and $\langle\mathscr{M}_{A,0}\rangle\approx 1.2$ (diamonds). When the turbulence becomes trans-Alfv\'enic or sub-Alfv\'enic, $\langle \mathscr{M}_{A,0}\rangle\lesssim 2$ (stars and diamonds), anisotropies arise in the gas, because the back reaction of the magnetic field onto the flow is extremely strong for flows perpendicular to the magnetic field lines.
The grey curve shows our prediction$^1$ using $b\approx 0.4$ that fits very well the data. Meanwhile, the black curve shows our prediction$^1$ considering $b=1$ (corresponding to purely compressive forcing). Although our turbulent forcing in the simulations is by definition mixed, and thus we expect $b\approx 0.4$ \citep{2010Federrathetal}, it is noteworthy to say that $b=1$ gives a good fit to the data with very low $\langle\mathscr{M}_{A,0}\rangle\lesssim 2$.}
\label{sig-alf}
\end{figure}
\section{Conclusions}
We presented an analytical prediction for the density variance--Mach number relation in magnetised supersonic turbulent gas. In this formulation, we considered three different cases for the relation between the magnetic field strength and density. The first case assumes that $B$ is independent of $\rho$, the second assumes that $B \propto \rho^{1/2}$, while the third is given by $B\propto \rho$. The three resulting $\sigma_s$--$\mathscr{M}$ relations were tested against numerical simulations. From this analysis we conclude that:
\begin{itemize}
\item If $B$ is independent of the density, we recover the hydrodynamical prediction of \citet{1997PNJ}. In this case, the gas and the magnetic field are not coupled. Therefore, an amplification of the magnetic field with the shock is not expected. Observationally, \citet{1999Crutcher} found that the magnetic field was independent of the density for diffuse clouds, corresponding to low rms Mach numbers, $\mathscr{M}\lesssim 2$. In this regime, all our predictions converge to the purely hydrodynamical $\sigma_s$--$\mathscr{M}$ relation.
\item For the second case, $B\propto \rho^{1/2}$, we found a one-to-one relation between $\mathscr{M}$, $\beta_0$ and the density variance. This $\sigma_s$--$\mathscr{M}$ relation (Eq.~\ref{supreme}) matches very well our numerical test considering $b=0.4$, which is the input for the natural mixture of compressive-to-solenoidal modes in the turbulent forcing field. This result is in agreement with the ones presented by \citet{2001Ostrikeretal} and \citet{2011Priceetal}, where they found lower $\sigma_s$ than in the unmagnetised case for $\mathscr{M}\gtrsim 10$. Moreover, \citet{2003ChoLazarian} presented a density contrast that closely matches our result for $\beta_0\approx 1$ and $5\lesssim\mathscr{M}\lesssim10$.
\item For the last case, $B\propto \rho$, the $\sigma_s$--$\mathscr{M}$ relation (Eq.~\ref{supreme2}) predicts a lower density variance than measured in our numerical simulations for $\mathscr{M}\geq 5$, because our simulations are closer to $B\propto\rho^{1/2}$. However, the relation given by Equation~(\ref{supreme2}) would fit better, if $B\propto\rho$.
\item The $\sigma_s$--$\mathscr{M}$ relation obtained for $B\propto \rho^{1/2}$ works very well for intermediate to high Alfv\'enic Mach number, $\mathscr{M}_{A,0}\gtrsim 6$, but breaks down for $\mathscr{M}_{A,0} \lesssim 2$ at $\langle\mathscr{M}\rangle \approx 10$. This probably occurs because in the presence of strong magnetic fields, the turbulence is no longer isotropic. This is because the back reaction of the magnetic field onto the flow is very strong for flows perpendicular to the magnetic field lines.
\end{itemize}
Magnetic fields act as a density cushion in turbulent gas, preventing the gas from reaching very low densities as well as very high densities. We conclude that magnetic fields are an important mechanism for shaping the density variance--Mach number relation, and therefore will change the quantitative predictions in models of the star formation rate, initial mass function or core mass function that depend on these quantities \citep[e.g.][]{2005KrumholzMckee,2011PN,2002PN,2008HennebelleChabrier,2009HennebelleChabrier}.
\section*{Acknowledgments}
We thanks the anonymous referee for the useful comments that helped to improve this manuscript. F.~Z.~M. thanks Paola Pinilla and Joe Ramsey for reading the manuscript and providing useful comments, as well as the International Max Planck Research School for Astronomy and Cosmic Physics at the University of Heidelberg, and the Heidelberg Graduate School of Fundamental Physics. F.~Z.~M.~received funding from the German Bundesministerium f\"ur Bildung und Forschung for funding via the ASTRONET project STAR FORMAT (grant 05A09VHA) C.F.~received funding from a Discovery Projects fellowship by the Australian Research Council (grant~DP110102191) and from the European Research Council (FP7/2007-2013 Grant Agreement no.~247060). All authors acknowledge subsidies from the Baden-W\"urttemberg-Stiftung for contract research via grant P-LS-SPII/18, and the Deutsche Forschungsgemeinschaft via SFB project 881 ``The Milky Way System'' (subprojects B1, B2, B3, and B5) as well as the SPP 1573 ``The Physics of the ISM''. The simulations described in this paper were performed using the {\em Ranger} cluster at the Texas Advanced Computing Center, using time allocated as part of Teragrid project TG-MCA99S024.
|
{
"timestamp": "2012-04-13T02:00:24",
"yymm": "1203",
"arxiv_id": "1203.2117",
"language": "en",
"url": "https://arxiv.org/abs/1203.2117"
}
|
\section{Introduction}
Macromolecules such as DNA and proteins carry an electrical charge in aqueous solution because of
ionic dissociation of molecular groups on their surface. The amount and sign of the charge depends on various
factors such as the pH of the solution. If an external electric field is applied, these macro-ions migrate along the
field with a velocity that depends on its charge, size and shape as well as on the ionic composition of
the background electrolyte. Electrophoresis is a technique of analytical chemistry for separating a mixture of
chemical compounds by exploiting their differing migration velocities in an applied electric field. It is a widely used
laboratory technique in areas such as molecular biology, forensics and medicine.
The most common format is ``gel electrophoresis,'' where the electrolyte is suspended in a porous gel. The technique of capillary
electrophoresis (CE) has developed rapidly in recent years, partly because of the possibility of integrating CE channels
on to micro-fluidic chips. In CE, the sample and suspending electrolyte
are contained in a micro-fluidic channel with width in the range of tens of microns. CE can proceed in several modes, the
simplest of which is ``zone electrophoresis''. Here a sample zone or band is injected into a micro-channel, which then moves
by electrophoresis towards the detector at the opposite end under an applied electric field. Different components arrive at different times and their arrival
is marked by a peak in some solute property (usually the UV absorbance) at the detector window \citep{czebook1,czebook2}.
The walls of the capillary, which are most commonly made of fused silica, has an electrostatic charge that is characterized by its zeta potential.
Therefore, the applied electric field results in an electro-osmotic flow along the capillary \citep{probstein}. This flow is normally
advantageous in CE because it sweeps both positive and negative ions past a single detector near the outlet and also reduces the
transit time from inlet to detector.
However, it can, under certain circumstances adversely affect the separation by causing ``anomalous dispersion''
of the sample peak \citep{ghosal_annrev06}.
The physical process of separation in CE is the result of the mutually opposed and competing processes of differential
migration of ions and diffusive spreading along the axis of separation. The resolution depends on the strength of
the applied electric field. In practice, the applied voltage is often very large, in the range of kilo volts. The highest voltage
that can be applied is limited by the ability to dissipate the considerable Joule heat that is produced in the buffer by the
electrolytic current. This is the reason why the channel width may not exceed some tens of microns. It is clearly advantageous to have an
electrolytic buffer of low conductivity in order to minimize heating. It is also advantageous to have a relatively high concentration
of sample ions, since one of the limitations of CE is the high demand placed on the sensitivity of the detector,
which must be sensitive to light attenuation over an optical path length of only a few tens of microns.
However, the ratio of sample to background ion concentration cannot be increased indefinitely; distortions due to
``electromigration dispersion'' or the ``sample overloading effect'' limits the highest ratio of
sample to background ion concentration that may be used in CE.
The underlying physical mechanism of electromigration dispersion was elucidated by \citet{mi_ev_ve_79a} and may be
roughly explained as follows: when the concentration of sample ions is comparable
to that of the carrier electrolyte, the local electrical conductivity is altered in the vicinity of the sample peak. On the other hand, since the current
through the capillary must be the same at all points along the axis,
the electric field must change locally. This follows since the product of the conductivity and electric field is the current (according to Ohm's law, neglecting
for the time being the diffusion current due to ionic concentration gradients). This
varying electric field alters the effective migration speed of the sample ions, which, in turn, alters
its concentration distribution. Thus, we have a nonlinear transport problem that must be solved in a self
consistent manner. Electromigration dispersion causes highly asymmetric concentration profiles, high rates
of dispersion and shock like structures that are reminiscent of nonlinear waves seen in many other physical systems.
These effects have been widely reported in the literature on electrophoresis \citep{bouskova_elph04}.
Simple one dimensional mathematical models of electromigration dispersion have been studied by \citet{mi_ev_ve_79a,mikkers_ac99,math_th_elph_bk}
by invoking the assumption of vanishing diffusivity which results in a single nonlinear hyperbolic equation for the concentration of sample ions.
Solutions describe the observed steepening of an intitially smooth profile leading to subsequent shock formation. Two recent reviews by \cite{gas09} and \cite{thormann09} provide more extensive
references to this early work. The restriction of zero diffusivity was removed by \cite{ghosal_chen10}
(henceforth referred to as GC)
who considered a three ion system -- the sample ion, co-ion and counter-ion -- where the diffusivities of the three ionic species were equal
but not necessarily zero.
The sample concentration was shown to obey a one dimensional nonlinear advection diffusion equation which reduced to Burgers' equation if the sample concentration was not too high
relative to that of the background ions. Since the initial value problem for Burgers' equation may be exactly solved, useful insight into the nature of electromigration dispersion
could be gained in the limits of small as well as large P\'{e}clet numbers.
A generalization of this model to the situation where the buffer is a weak electrolyte has recently been presented~\citep{EMD1}.
It was found that the evolution of the peak shape may still be described by the Burgers' equation with a slight re-interpretation
of a certain model parameter. If this parameter is taken as a fitting parameter, excellent quantitative
agreement is obtained with measured values of the peak variance in experiments.
A similar nonlinear advection diffusion equation was derived by \citet{zangle_propagation_2009}
in order to describe ``desalination shocks'' near constrictions in micro channels. However, in their problem the physical origin of the nonlinearity is related to
surface conduction effects.
In this paper, we extend the earlier work of GC by taking into account the fact that the channel wall generally has a non-zero zeta potential which
creates a bulk fluid flow in the capillary due to electro-osmosis. The electro-osmotic flow affects the transport process in the following way: the
axial variation of the electric field created by the conductivity changes results in a variable electro-osmotic slip velocity on the channel walls. This in turn results
in an induced pressure gradient and the concomitant appearance of a shear in the velocity profile across the channel due to a well known mechanism \citep{he_00,gh_02b,gh_02a,gh_02c}. The shear then enhances axial dispersion due to the
Taylor-Aris effect \citep{taylor,aris}. Our principal result is equation (\ref{generalized_GC}), which is a one dimensional
nonlinear advection diffusion equation for the sample concentration averaged over the channel cross-section.
The rest of the paper is organized as
follows. In \S~\ref{sec:formulation} the complete mathematical statement of the flow and electro-diffusion problem is presented. In \S~\ref{sec:homogenized} a systematic reduction
of the full equations leading to equation~(\ref{generalized_GC}) is achieved by introducing a number of physical approximations. The consequences of this equation are discussed in \S~\ref{sec:dispersion},
where it is shown that there is a peak in the efficiency of separation at an intermediate value of the electro-osmotic flow strength. The broad
implications of the analysis to separation efficiency in CE are discussed in \S~\ref{sec:conclusion}.
\section{Mathematical Formulation}
\label{sec:formulation}
\begin{figure}
\centerline{\includegraphics[width=6in]{figure1_geometry.eps}
\caption{Schematic diagram (not to scale) illustrating the mathematical problem of electromigration
dispersion of a sample in CE. In addition to the sample ions shown, the capillary also contains a
carrier electrolyte consisting of co- and counter- ions.}
\label{fig:geom}
\end{figure}
We consider (see Figure~1) a uniform, long
capillary of arbitrary cross-sectional shape connected to infinite reservoirs at either end. The length of the capillary is much greater than the
width; the capillary may, for most purposes, be assumed infinitely long.
The capillary contains a solute and $N$ strong (i.e. completely dissociated) electrolytes
of concentration $c_i (x,y,z,t)$ where $i=1$ to $N$. These concentrations then obey the conservation equations based on the Nernst-Planck model for ion flux~\citep{probstein}
\begin{equation}
\frac{ \partial c_i}{\partial t} + \bnabla \cdot ( {\bf u} c_i - z_i e \nu_i c_i \bnabla \phi_e - D_i \bnabla c_i ) = 0,
\label{transport}
\end{equation}
where $z_i$, $\nu_i$ and $D_i$ are the valence, mobility and diffusivity of species $i$, the electronic charge is $e$, $\phi_e$ is the electric potential and
${\bf u} (x,y,z,t)$ is the fluid flow velocity.
The potential $\phi_e$ obeys the Poisson equation (in CGS units)
\begin{equation}
\varepsilon \nabla^{2} \phi_e = - 4 \pi \rho_e = - 4 \pi \sum_{i=1}^{N} z_i e c_i (x,y,z,t)
\label{poisson}
\end{equation}
where $\varepsilon$ is the permittivity of the solvent and $\rho_e$ is the density of free charges in the solvent. Since the Reynolds number is small for flow in micro-capillaries, the fluid flow is described
by Stokes equation
\begin{equation}
- \bnabla p + \eta \nabla^{2} {\bf u} - \rho_e \bnabla \phi_e = 0,
\label{stokes}
\end{equation}
together with the incompressibility constraint
\begin{equation}
\bnabla \cdot {\bf u} = 0,
\end{equation}
where $p$ is the pressure and
$\eta$ is the dynamic viscosity of the solvent.
These equations are associated with a number of boundary conditions that
describe our situation. The concentrations $c_i$ reduce to their equilibrium distributions in a uniform capillary
very far away from the sample zone
\begin{equation}
c_{i} (x,y,z) \sim c_{i}^{eq} (y,z) \quad \mbox{if $|x| \rightarrow \infty$}.
\end{equation}
Clearly $c_{i}^{eq} (y,z) = 0$ when $i$ corresponds to the sample and is given by solutions of the two dimensional Poisson-Boltzmann model
for the other species (background electrolytes). The potential $\phi_e$ obeys Dirichlet boundary conditions
\begin{equation}
\phi_e (x,y,z) = \left\{
\begin{array}{ll}
\phi_{e}^{w} & \mbox{if $(x,y,z) \: \varepsilon \: S_w$} \\
V & \mbox{if $(x,y,z) \: \varepsilon \: S_{+}$} \\
0 & \mbox{if $(x,y,z) \: \varepsilon \: S_{-}$}
\label{dirichlet}
\end{array}
\right.
\end{equation}
where $\phi_{e}^{w}$ is the value of the potential at the wall, $V$ is the applied voltage, and, $S_w$, $S_{+}$ and $S_{-}$ are the parts of the
bounding surface of the capillary that correspond to the walls, the inlet and the outlet respectively (Figure~1). The inlet and the outlet
sections are separated by a distance $L$ and we are interested in the limit where $V$ and $L$ approach infinity but $V/L = E_0$
remains finite. The hydrodynamic flow satisfies the no-slip conditions at the wall
\begin{equation}
{\bf u} (x,y,z,t) = 0 \quad \mbox{if $(x,y,z) \: \varepsilon \: S_w$},
\label{no-slip}
\end{equation}
and,
\begin{equation}
{\bf u} (x,y,z,t) \sim {\bf u}^{eq} (y,z) \quad \mbox{if $|x| \rightarrow \infty$},
\end{equation}
where ${\bf u}^{eq} (y,z)$ is the electro-osmotic flow in the uniform capillary when no sample is present, and, when $L$ is infinite.
Finally, the ion fluxes are zero at the capillary walls ($S_w$)
\begin{equation}
( z_i e \nu_i c_i \bnabla \phi_e + D_i \bnabla c_i ) \cdot \hat{ {\bf n} } = 0 \quad \mbox{if $(x,y,z) \: \varepsilon \: S_w$}
\label{zero_flux}
\end{equation}
where $\hat{ {\bf n} }$ is the unit normal at the wall directed into the fluid.
These equations and boundary conditions form a well posed system of equations that may be integrated numerically
for given initial conditions. Such a description would, however, be unnecessarily complex. Our objective is to introduce a
series of approximations that enable a reduced description in terms of a single one dimensional evolution equation
for the cross-sectionally averaged concentrations $\bar{c}_{i} (x,t)$, since this is the quantity that is essentially measured by
the detector. Henceforth, a bar above a field variable will always denote its average over the cross-section of the capillary.
\section{One dimensional homogenized equations}
\label{sec:homogenized}
We now introduce a number of approximations that simplify the description and lead up to the one dimensional
homogenized equation for the sample ion concentration that we seek. The developments closely follow GC
except for the important difference that the possibility of a hydrodynamic flow field ${\bf u}$
is admitted.
\subsection{Thin Debye layers}
\label{ssec:TDL}
The fixed charges on the wall are shielded by a cloud of counter-ions over a length scale of the order of the Debye
length ($\lambda_D$) which is related to the equilibrium ionic concentrations~\citep{probstein}.
The Debye length is typically of the order of nanometers; much smaller than the characteristic width of the
micro-channel which is in the range of tens of microns. Under such conditions, the left hand side of (\ref{poisson})
may be set to zero~\citep{Planck}, and, as a consequence, the last term in (\ref{stokes}), representing the density of electric forces also
vanish. Electrical forces could also arise outside of Debye layers in response to variations in electrical
conductivity~\citep{chen_convective_2005,oddy_multiple-species_2005}. The contribution
from such forces may however be neglected in the current context (see Supplementary Material online).
Thus, Poisson's equation is replaced by the constraint of local electro-neutrality. The loss of the equation for $\phi_e$ is however
not catastrophic, because $\phi_e$ can be determined from the equation of current conservation
\begin{equation}
\bnabla \cdot {\bf J}_{e} = 0
\label{current_conserve}
\end{equation}
where the current density
\begin{equation}
{\bf J}_{e} = - \left( \sum_{i=1}^{N} z_i^{2} e^{2} \nu_{i} c_{i} \right) \bnabla \phi_e - \sum_{i=1}^{N} z_{i} e D_{i} \bnabla c_{i}
\label{Je}
\end{equation}
is the sum of the Ohmic conduction (first term) and a contribution due to differential diffusion (second term). Equation (\ref{current_conserve})
follows from the constraint of local electro-neutrality on summing (\ref{transport}) after multiplying by $z_i e$. The loss of the electrical forcing
term in (\ref{stokes}) is mitigated by replacing the no-slip boundary condition (\ref{no-slip}) with the Helmholtz-Smoluchowski slip boundary
condition~\citep{probstein}
\begin{equation}
{\bf u} (x,y,z) = \frac{\varepsilon \zeta \bnabla \phi_e }{4 \pi \eta} \quad \mbox{if $(x,y,z) \: \varepsilon \: S_w$}
\label{HS-slip}
\end{equation}
that takes into account the presence of a boundary layer of non-vanishing charge next to the capillary wall. Here $\zeta$, the so called
``zeta-potential'' is the difference in the values of $\phi_{e}$ between a point at the outer edge of the Debye layer and the corresponding point on the wall.
It is a material property that will be assumed constant. The boundary conditions
(\ref{dirichlet}) are replaced by
\begin{equation}
{\bf J}_{e} \cdot \hat{{\bf n}} = 0
\label{bc_wall_insulator}
\end{equation}
which follows from (\ref{zero_flux}) and is the condition of zero current flow into capillary walls. At large distances from the sample zone
we have a uniform field ($E_0$) and a uniform flow ($u_{eo}$), thus,
\begin{equation}
\phi_e (x,y,z,t) \sim - E_0 x \quad \mbox{if $|x| \rightarrow \infty$}
\end{equation}
and
\begin{equation}
{\bf u} (x,y,z,t) \sim {\bf u}^{eq} (y,z) = - \frac{\varepsilon \zeta E_0 }{4 \pi \eta} \; \hat{{\bf x}}= u_{eo} \: \hat{{\bf x}} \quad \mbox{if $|x| \rightarrow \infty$}
\end{equation}
where $\hat{{\bf x}}$ is the unit vector in the axial direction.
\subsection{Three ion model}
\label{ssec:3ion}
We introduce the Kohlrausch \citep{kohlrausch} function
\begin{equation}
K(x,y,z,t) = \sum_{i=1}^{N} \frac{1}{\nu_{i}} c_i(x,y,z,t).
\end{equation}
Then (\ref{transport}) together with the constraint of local electro-neutrality yields the following equation for $K$:
\begin{equation}
\frac{ \partial K }{\partial t} + \bnabla \cdot ( {\bf u} K ) = \nabla^{2} \left( \sum_{i=1}^{N} D_i \frac{c_i}{\nu_i} \right).
\label{transport_K}
\end{equation}
For a two ion system, equation~(\ref{transport_K}) together with the condition of local electro-neutrality leads to the
Ohmic model~\citep{Melch_Taylor_ARFM,chen_convective_2005}
where the local electrical conductivity evolves as a passive scalar with an effective diffusivity $D_{e} = 2 D_{+} D_{-} /( D_{+} + D_{-} )$; $D_{+}$
and $D_{-}$ denotes the diffusivities of the cation and anion. In our problem, we have more than two ionic species and the Ohmic model is not applicable.
However, if we assume that the ions all have the same mobilities, $\nu_i = \nu$, and therefore, on account of the Einstein relation $D_i/\nu_i = k_B T$, the
same diffusivities, $D_i = D$, then the function $K$, rather than the conductivity, is found to evolve as a passive scalar:
\begin{equation}
\frac{ \partial K }{\partial t} + \bnabla \cdot ( {\bf u} K ) = D \nabla^{2} K.
\end{equation}
We restrict ourselves to a minimal model consisting of a system of just three ions ($N=3$).
We will drop the index $i$ and instead use the suffix $p$ and $n$ to denote the positive ion and negative ion respectively,
in the background electrolyte.
The absence of a suffix will indicate the sample ion. For example, $c_p$, $c_n$ and $c$ are the concentrations of positive ions,
negative ions and sample ions respectively. Then from the local electro-neutrality constraint and the definition of $K$
we have
\begin{equation}
c_n - c_{n}^{(\infty)} = \frac{z - z_p}{z_p - z_n} c + \frac{ \nu z_p}{z_p-z_n} \, \delta K,
\label{cn}
\end{equation}
\begin{equation}
c_p + \frac{z_n}{z_p} c_{n}^{(\infty)} = - \frac{z - z_n}{z_p - z_n} c - \frac{ \nu z_n}{z_p-z_n} \, \delta K,
\label{cp}
\end{equation}
where $c_{n}^{(\infty)}$ is the concentration of negative ions in the background electrolyte far from the sample zone.
If $K_{\infty}$ denotes the value of $K$ in the far field, then the perturbation $\delta K = K - K_{\infty}$ is advected by the flow and spreads diffusively from the injection region.
Therefore (GC), after an initial transient that is small compared to the total analysis time, the sample peak,
which moves by electrophoresis in addition to being advected by the flow, migrates
into a region of space where $\delta K$ is effectively zero. Thus, in the vicinity of the sample peak, we may assume that
$\delta K = 0$, so that, the background ion concentrations $c_p$ and $c_n$
may be expressed as linear functions of the sample ion concentration $c$.
The diffusion current represented by the second term in (\ref{Je}) vanishes when $D_i = D$ on account of
local electro-neutrality, so that it reduces to Ohm's law for a homogeneous electrolyte: ${\bf J}_e = - \sigma_e \bnabla \phi_e$, where
\begin{equation}
\sigma_e = \sum_{i=1}^{N} z_i^{2} e^{2} \nu_{i} c_{i} = \sigma_{\infty} ( 1 - \alpha \phi )
\end{equation}
is the local electrolyte conductivity. The expression on the right is obtained on eliminating $c_p$ and $c_n$ using
(\ref{cn}) and (\ref{cp}) with $\delta K = 0$. Here $\sigma_{\infty}$ is the bulk electrolyte conductivity,
$\phi = c /c_{n}^{(\infty)}$ is the sample concentration relative to the background
and $\alpha$ is the parameter
\begin{equation}
\alpha = \frac{(z-z_n)(z-z_p)}{z_n (z_p - z_n)}
\end{equation}
introduced in GC to characterize the nature of the nonlinearity.
\subsection{The lubrication limit}
\label{ssec:lube}
The subsequent development is based on the premise of ``slow axial variations''
on account of the inlet to detector separation, $L$, being much larger than the characteristic channel width, $w_{0}$.
In practice, $L$ is of the order of tens of cm whereas $w_{0} \sim 10-100$ $\mu m$. Thus, $L/w_{0} \sim 10^{3} - 10^{4}$.
Therefore, as the sample moves down the micro-channel, the sample concentration
and all other quantities controlled by this concentration vary on an axial length scale $L_x \gg w_{0}$.
To see this, suppose that the initial sample concentration was a delta function in the axial direction.
Then it would spread over a distance of the order of the channel width in time $\tau_d \sim w_{0}^{2} / D$.
The total analysis time is $\tau_a \sim L / v_{0}$, where $v_{0}$ is a characteristic migration
velocity. Thus, $\tau_a / \tau_d \sim (L/w_{0}) \mbox{Pe}^{-1}$ where $\mbox{Pe} = v_{0} w_{0} /D$ is the P\'{e}clet number.
Since typically $\mbox{Pe} \sim 10 - 100$, $\tau_a / \tau_d \gg 1$ so that concentrations are
homogenized across the channel and the fluid flow and transport problems both become quasi one dimensional
a short time ($\sim \tau_d$) after sample injection.
The equation of current conservation, (\ref{current_conserve}), may then be integrated using the
boundary condition (\ref{bc_wall_insulator}) to give
\begin{equation}
{\bf E} = \hat{{\bf x}} \: E_{0}/(1 - \alpha \bar{\phi}) + \cdots,
\end{equation}
where ${\bf E} = - \bnabla \phi_e$ is the electric field and the neglected terms are asymptotically small in the lubrication limit.
Thus, the field is predominantly in the axial direction
and depends on the local sample concentration. The axial variability of the electric field affects the evolution of the sample concentration in two ways.
First, it results in a variable electrophoretic migration velocity $v_{0}/(1 - \alpha \bar{\phi})$
for the sample ions, where $v_{0} = z e \nu E_{0}$ is the migration velocity of an isolated sample ion. Second, it results in a
variable slip velocity for the electro-osmotic flow through the boundary condition (\ref{HS-slip}).
The hydrodynamic flow
field due to such a variable slip velocity may be written down using lubrication theory~\citep{gh_02c}. The velocity is predominantly
in the axial direction, ${\bf u} = u \hat{{\bf x}} + \cdots$ and the axial component $u$ may be written as the sum of a
mean $\bar{u}$ and a fluctuation about the mean, $\Delta u$, due to the induced pressure gradient. The mean flow $\bar{u}$
is a constant given by
\begin{equation}
\bar{u} = \lim_{L \rightarrow \infty} \frac{1}{L} \int_{0}^{L} u_s (x,t) \; dx = u_{eo}
\end{equation}
where $u_s (x,t)$ is the slip
velocity at the wall. In the presence of concentration gradients, the slip velocity could also have a diffusiophoretic component~\citep{prieve_motion_1984,rica_electrodiffusiophoresis:_2010}. This is however expected to be small
in the present context (see Supplementary Material online).
\begin{equation}
u_{s} (x,t) = u_{eo} / (1 - \alpha \bar{\phi} ) + \cdots,
\end{equation}
and,
\begin{equation}
\label{Deltau}
\Delta u \equiv u - \bar{u} = \left( u_s - u_{eo} \right) \left( 1 - \frac{u_p}{\bar{u}_{p}} \right) = \frac{\alpha \bar{\phi} \, u_{eo} }{1 - \alpha \bar{\phi}} \left( 1 - \frac{u_p}{\bar{u}_{p}} \right) + \cdots
\end{equation}
to leading order in the lubrication approximation.
Here $u_p$ is a function that depends solely on the cross-sectional shape and represents the flow profile due to a unit pressure gradient.
It is defined by the solution of the equation
\begin{equation}
\frac{\partial^{2} u_p}{\partial y^{2}} + \frac{\partial^{2} u_p}{\partial z^{2}} = - 1
\end{equation}
with the boundary condition
\begin{equation}
u_p (y,z) = 0 \quad \mbox{if $(x,y,z) \: \varepsilon \: S_w$}.
\end{equation}
The function $u_p$ admits analytical representation for several cross-sectional shapes.
\subsection{Taylor-Aris limit and macrotransport}
\label{ssec:taylor}
The time evolution of the concentration field of an advected scalar in the limit $\tau_a/\tau_d \gg 1$ was first described by Taylor and Aris
\citep{taylor,aris} and later applied to a wide variety of problems involving shear induced dispersion~\citep{macrotransport_processes} including dispersion problems in CE~\citep{ghosal_annrev06}.
In this limit, lateral inhomogeneities in the scalar concentration are small $c(x,y,z,t) = \bar{c} (x,t) + \cdots$, so that
the cross-sectionally averaged concentration
$\bar{c}$ is advected by the mean flow ${\bar u}$ and undergoes axial diffusion with an effective diffusivity~\citep{pof}
\begin{equation}
\label{De}
D_e = D - \frac{\overline{G u}}{D}
\end{equation}
where $G$ satisfies
\begin{equation}
\frac{\partial^{2} G}{\partial y^{2}} + \frac{\partial^{2} G}{\partial z^{2}} = \Delta u = u - \bar{u}
\end{equation}
and the conditions
\begin{equation}
\bnabla G \cdot \hat{{\bf n}} = 0 \quad \mbox{if $(x,y,z) \: \varepsilon \: S_w$}.
\end{equation}
To remove the indeterminacy of $G$ up to a constant, we further impose the condition $\bar{G}=0$.
Furthermore, the sample ions are advected with a concentration dependent velocity $v_{0}/(1 - \alpha \bar{\phi})$. Thus, the
equation satisfied by $\bar{\phi}$ is
\begin{equation}
\frac{\partial \bar{\phi} }{\partial t} + \frac{\partial}{\partial x} \left[ \left( u_{eo} + \frac{v_{0} }{1 - \alpha \bar{\phi} } \right) \bar{\phi} \right] = \frac{\partial}{\partial x} \left[
\left\{ D + \frac{k u_{eo}^{2} w_{0}^{2}}{D} \left( \frac{\alpha \bar{\phi} }{1 - \alpha \bar{\phi} }\right)^{2} \right\} \frac{\partial \bar{\phi} }{\partial x} \right]
\label{generalized_GC}
\end{equation}
where $k$ is a numerical constant depending solely on the cross-sectional shape and $w_{0}$ is a characteristic channel width.
The expression for the effective diffusivity $D_{e}$ appearing on the right hand side of (\ref{generalized_GC}) follows on
evaluating (\ref{De}) using (\ref{Deltau}).
In particular, for a planar channel of half-width $w_{0}$, a simple calculation shows that $u_p = (w_{0}^{2} - y^{2} )/2$, which leads to $k=2/105$.
Equation (\ref{generalized_GC}) is the generalization of the evolution equation derived in GC to the situation where
the channel has a non-zero electro-osmotic slip velocity. An alternative derivation of (\ref{generalized_GC}) using the
method of multiple scales is sketched in the Appendix.
If $ | \alpha | \bar{\phi} \ll 1$, the nonlinear terms in (\ref{generalized_GC}) may be expanded in
Taylor series: $(1 - \alpha \bar{\phi})^{-1} = 1 + \alpha \bar{\phi} + \alpha^{2} \bar{\phi}^{2} + \cdots$, so that, in place of Burgers' equation arrived at in GC
we get the following equation
\begin{equation}
\frac{\partial \bar{\phi} }{\partial t} + (v_{0} + u_{eo} + 2 \alpha v_{0} \bar{\phi} + 3 \alpha^{2} v_{0} \bar{\phi}^{2} ) \; \frac{\partial \bar{\phi} }{\partial x} =
\frac{\partial}{\partial x} \left\{ \left( D + \frac{k \alpha^{2} u_{eo}^{2} w_{0}^{2}}{D} {\bar{\phi}}^{2} \right) \frac{\partial \bar{\phi}}{\partial x}
\right\}
\label{generalized_weak_GC}
\end{equation}
which shows an amplitude dependent contribution to the diffusivity in addition to a correction to the term $2 \alpha v_{0} \bar{\phi} (\partial \bar{\phi}/\partial x)$ representing
nonlinear wave steepening. Equation (\ref{generalized_GC}), or, its weakly nonlinear version (\ref{generalized_weak_GC}), is our principal result.
\begin{figure}
\centerline{\includegraphics[width=6in]{figure2_VarRatioTime.eps}
\caption{Time evolution of the normalized rate of increase of variance for three different values of the dimensionless electro-osmotic
flow strength $u_{*}=u_{eo}/v_{0}$. Horizontal dotted line is the effective diffusivity predicted
by equation~(35) in GC. }
\label{fig:diffusivity}
\end{figure}
Equation (\ref{generalized_GC}) has a singularity at $\bar{\phi} = 1/\alpha$ when
$\alpha$ is positive. However, (\ref{generalized_GC}) ceases to be valid even before this singularity is reached (GC),
because, the requirement that $c_p$ and $c_n$ must be non-negative, imposes the constraint $\phi < \phi_{c}$
where $\phi_{c} = (z_p - z_n)/(z_p - z)$ when $z < 0$ and $\phi_{c} = - [ z_n (z_p - z_n) / [ z_p (z - z_n) ]$
when $z > 0$. If $\alpha > 0$, then it may be shown that $\phi_{c} < 1 / \alpha$. The physical reason for the
breakdown of (\ref{generalized_GC}) when $\phi > \phi_{c}$ is the following: under those conditions the
conductivity in the sample zone is so high as to reduce the electric field to very small values. Then the electrophoretic
motion of the sample ions become very small and there is little relative motion between
the sample peak and $\delta K$. Thus, the assumption that $\delta K$ may be set to zero in the sample zone
can no longer be employed and (\ref{generalized_GC}) is no longer valid. However, after sufficient time has passed and axial
diffusion has caused the peak value of $\bar{\phi}$ to fall below $\phi_c$, the time evolution once again
proceeds in accordance with (\ref{generalized_GC}). Numerical simulations of the full electrohydrodynamic equations
described in \S~\ref{ssec:TDL} confirm this behavior (Chen \& Ghosal, unpublished).
For the purpose of numerical integration, (\ref{generalized_weak_GC}) is a little more convenient than (\ref{generalized_GC})
since it does not exhibit the singularity
at $\bar{\phi} = 1/ \alpha$.
Nevertheless, even though solutions to (\ref{generalized_weak_GC}) may be formally calculated even for
$\bar{\phi} > \phi_c$, the solution in this regime is devoid of physical
significance, and indeed, will yield negative concentrations of background electrolytes in parts of the domain if (\ref{cn}) and (\ref{cp})
with $\delta K=0$ are employed to calculate the concentrations of the background ions.
\section{Dispersion}
\label{sec:dispersion}
Equation (\ref{generalized_GC}) or (\ref{generalized_weak_GC}) provide a compact description of electromigration dispersion that
is helpful for gaining a qualitative understanding of the underlying mechanisms. Furthermore, it
is much more amenable to numerical integration than the full three dimensional coupled problem involving fluid flow
and transport. By way of example, in this section, we use numerical integration to
illustrate a point that at first sight may appear counter-intuitive, but, is clarified by an analysis of (\ref{generalized_GC}).
Since the presence of a zeta-potential results in cross-channel variations in the flow velocity due to
induced pressure gradients, one would expect the efficiency of separation to be adversely affected. However,
in certain ranges of parameters the reverse may actually be true. To understand this, one needs to examine the roles of the
different terms in (\ref{generalized_weak_GC}). The nonlinear term on the left hand side
causes wave steepening leading to the formation of shock like structures. This is the
dominant mechanism that contributes to electromigration dispersion (GC). Taylor dispersion can actually mitigate this tendency by increasing the effective
axial diffusivity that serves to diffuse the electrokinetic shock. However, on the other hand, if the Taylor dispersion is too large, its contribution to the axial dispersion
dominates with a consequent loss of separation efficiency. Thus, there is an intermediate value of the wall zeta potential that
corresponds to the lowest peak dispersion.
\begin{figure}
\centerline{\includegraphics[width=6in]{figure3_platenumber.eps}}
\caption{The number of theoretical plates $N=L^{2}/\sigma^{2}$ as a function of dimensionless electro-osmotic
velocity $u_{*} = u_{eo}/v_{0}$ for three different detector to injection point distances ($L$) and a range of sample loading characterized by the
P\'{e}clet number $P = v_{0} \Gamma / D$. Here $N_{0}$ is the ``ideal'' value of $N$ in the
absence of electromigration dispersion.}
\label{fig:N}
\end{figure}
Equation (\ref{generalized_GC}) was integrated numerically using a finite volume method that
allows for adaptive grid refinement and variable time
steps. We used the ``ode15s'' solver in MATLAB~\citep{shampine_reichelt97}
(see Supplementary Materials online for further details).
The geometry chosen was that of a planar channel of width $2 w_{0}$.
A ``moving window'' that is advected with the mean migration velocity $v_{0} + u_{eo}$ was used to
optimize the number of grid points needed. The size of the window was chosen large enough that $\bar{\phi}$ can be set to zero at the domain boundaries
with negligible loss in accuracy. The initial concentration profile $\bar{\phi}(x,0)$ was chosen as a Gaussian of standard deviation $w_{0}$ centered on $x=0$
and with different peak strengths. The ``sample loading'' is
characterized (GC) by a P\'{e}clet number $P = v_{0} \Gamma / D$ based on the length scale
\begin{equation}
\Gamma \equiv \int_{-\infty}^{+\infty} \bar{\phi} \; dx,
\end{equation}
which is an integral of motion.
A second P\'{e}clet number that characterizes the diffusion, $\Pen = v_{0} w_{0} / D$ is held fixed at the value $200$.
The parameter $\alpha$ was set to $0.5$. These conditions are fairly typical for laboratory experiments.
Figure~\ref{fig:diffusivity} shows the evolution of the quantity $D_{t} \equiv (2D)^{-1} d \sigma^{2} / dt$, where $\sigma^{2}$ is the variance
of $\bar{\phi}(x,t)$, as a function of the
dimensionless time $v_{0} t / w_{0}$. At late times, when $\bar{\phi}$ is sufficiently small, (\ref{generalized_GC}) may be replaced by (\ref{generalized_weak_GC}).
At even larger times, the quadratic terms in $\bar{\phi}$ become vanishingly small, so that, the equation
describing the evolution of $\bar{\phi}$ essentially
reduces to Burgers' equation as discussed in GC with the minor difference that the constant part of the advection velocity
is $v_0 + u_{eo}$ and not $v_0$.
Therefore, one would expect that $D_t$, which is like an ``instantaneous'' diffusivity, should asymptote to the value given by equation (35) in GC.
Figure~\ref{fig:diffusivity} shows that indeed, this is the case for all values of the flow strength, $u_{*} = u_{eo}/v_{0}$. However, the evolution of $D_t$
to the common asymptotic value follows different trajectories depending on the strength of the electro-osmotic flow. For larger values of $u_{*}$, $D_{t}$
is initially relatively large because of the contribution of the quadratic (and higher order) terms in $\bar{\phi}$ to the effective
diffusivity in (\ref{generalized_GC}).
This is because of Taylor dispersion. At intermediate times, $D_t$ is lower than the asymptotic value predicted by equation (35) in GC,
a consequence of the fact that a larger effective axial diffusivity delays the formation of shock like structures that result from nonlinear
wave steepening. The total variance
is the initial variance ($w_{0}^{2}$) plus the area under the curve $D_t$ from $t=0$ to some time $t=t_{f}$ when the peak arrives at a detector located at $x=L$. This is shown in Figure~2 and 3 in the Supplementary Material.
\begin{figure}
\centerline{\includegraphics[width=6in]{figure4_profiles_T200.eps}}
\caption{Concentration profiles $\bar{\phi}(x,t)$ at a fixed instant of time ($v_{0} t / w_{0} = 200$) for several values of the
electro-osmotic flow strength, $u_{*}=u_{eo}/v_{0}$. The increased effective axial diffusivity due to Taylor dispersion
``softens'' the electromigration shock that tends to form at the leading edge. Here $P=50$ and $x=x_{c}$ is the location of the centroid
of the peak.}
\label{fig:profile}
\end{figure}
The efficiency of separation in CE is often characterized by the ``number of theoretical plates''
$N = L^{2} / \sigma^{2}$ which is a dimensionless measure of the peak width. In the top panel of Figure~\ref{fig:N}, we plot $N$ as a function of $u_{*}$
for a number of downstream detector positions and for sample loading ranging from $P=0.5$ (weak) to $P=50$ (strong). It is clear that increasing $P$
reduces $N$ due to electromigration dispersion. However, the dependance on $u_{*}$ is non-monotonic with $N$ increasing at first with $u_{*}$
but then decreasing or leveling off to a plateau depending on the location of the detector. Increasing the electro-osmotic flow $u_{eo}$ generally increases
$N$ due to an obvious and quite trivial reason. If the detector position is fixed, the peak reaches it faster for higher electro-osmotic flow speeds and
the accumulated variance is less simply because the peak has evolved for a shorter time period. In fact, in the absence of the nonlinearities induced by electromigration
dispersion, $\sigma^{2} \equiv \sigma_{0}^{2} = w_{0}^{2} + 2 D t_{f} = w_{0}^{2} + 2 D L / (u_{eo} + v_{0} )$. In this
ideal limit $N \equiv N_{0} = L^{2} / \sigma_{0}^{2}$. In order to eliminate this obvious effect that electro-osmotic flow has
on peak dispersion, we plot
$N/N_{0}$ as a function of $u_{*}$ in the lower panel of Figure~\ref{fig:N}, which clearly shows, that the degradation of separation efficiency by electromigration
dispersion is minimized at an intermediate value of the electro-osmotic flow $u_{*}$.
The dependence of $N$ on $L$ at different
values of $P$ is shown in Figure~4 of the Supplementary Material included in the online version.
\begin{figure}
\centerline{\includegraphics[width=6in]{figure5_at_detector.eps}
\caption{Concentration $\bar{\phi}(L,t)$ as a function of time of arrival ($t$) at a fixed detector location ($x=L$) for several values of
electro-osmotic flow strength, $u_{*}=u_{eo}/v_{0}$. It is seen that though electro-osmotic flow improves the resolution, the degree of
improvement ``saturates'' as the flow rate is increased, as one might expect from Figure~\ref{fig:N}. Here $P=50$.}
\label{fig:detector}
\end{figure}
The mechanism of this reduction in total variance when $u_{*}$ is
sufficiently small is evident from Figure~\ref{fig:profile} depicting peak shapes at a fixed time $v_{0} t / w_{0} = 200$ for a number of values of $u_{*}$.
It is seen that the increase in the effective axial diffusivity due to Taylor dispersion ``softens'' the electrokinetic shock that results from nonlinear
wave steepening due to electromigration. Thus, electro-osmotic flow has three competing effects (a) reduction in the time available for diffusion
(b) reduction in the nonlinear wave steepening on account of enhanced effective diffusivity (c) increase in axial diffusion due to the
Taylor-Aris effect. The total dispersion is determined by the relative contribution of each of (a), (b) and (c).
At low values of $u_{*}$, (a) and (b) dominate, but
at higher values, (c) is more important.
In laboratory systems, $u_{*}$ could have either sign and a magnitude that could vary between zero (e.g. in a coated capillary where electro-osmosis is suppressed) to
some number of order unity, since the electro-osmotic velocity is similar in magnitude to the electrophoretic velocity.
Figure~\ref{fig:detector} provides an alternate viewpoint that has a closer correspondence with
experiments: the signal intensity $\bar{\phi}(L,t)$ is plotted as a function of the time of arrival at a fixed detector position at distance $L$
from the injection point. It is seen that introduction of an electro-osmotic flow at first results in a significant reduction in peak width,
but with diminishing returns as the flow strength is increased.
A similar plot is shown as Figure~6 in the Supplementary Material except there $\bar{\phi}(x,t)$ is plotted as a function
of $x$ at fixed times.
\section{Conclusions}
\label{sec:conclusion}
In conclusion, we would like to make a few remarks about the relevance of the analysis presented here to laboratory practice.
The model studied here is clearly oversimplified and (\ref{generalized_GC}) cannot be expected to predict in detail
the peak shapes in a real laboratory experiment. In particular, real electrophoresis buffers contain many more than three ions
including one or more weak acids or bases that are added to maintain a stable pH. Nevertheless, it is important to understand
the qualitative effects predicted by the simplified model considered here, since the actual dispersion in a real laboratory experiment is likely to be due
to a multiplicity of causes, of which, the mechanism discussed here would be one.
In this paper, we have described the effects of electromigration dispersion by means of a small set of dimensionless
parameters: the P\'{e}clet numbers $\Pen = v w_{0} / D$, $P = v \Gamma/D$, $\alpha$, and, the dimensionless electro-osmotic
flow speed $u_{*}=u_{eo}/v_{0}$. A rough idea of typical values of these parameters in laboratory experiments
is helpful. If we take $v_{0} \sim 1$ mm/s, $w_{0} \sim 10 \mu m$, $D \sim 10^{-5}$ cm$^{2}$/s as fairly typical, we have $\Pen \sim 10$.
For large molecules, $D$ could be an order of magnitude smaller, so that $\Pen$ could be in the hundreds.
If the sample concentration is of the same order as the background electrolyte concentration and the peak width
is of the order of the channel width, then $\Gamma \sim w_{0}$.
However, such sample concentrations would be considered extremely high in CE and would show very strong
peak distortion. We may therefore take this as an upper limit for $\Gamma$. Thus, $P$ could range from
essentially zero (the linear regime) to $P = v \Gamma/D \sim v w_{0} /D = \Pen$. Since electrophoresis and electro-osmosis are generally of comparable
magnitudes, $u_{eo} \sim v$. Thus, $u_{*} \sim 1$, though $u_{*}$ could be close to zero if the channel walls
have a low zeta potential (as in a coated capillary). The magnitude of the parameter $\alpha$ is of order unity if the sample valence
and the valence of the background electrolytes are similar. However, for macro-ions $|z| \sim 10$ or larger. For such large value of $z$,
$\alpha \sim z^{2}$ could be in the hundreds, and, nonlinear effects of the kind considered in this paper would be very pronounced.
Electro-osmotic flow is generally desirable in the context of CE
as it shortens the transit time from inlet to detector and as a consequence also reduces the peak variance.
The effect of electro-osmotic flow on separation efficiency is however complicated in the presence of electromigration
dispersion since a number of competing effects are simultaneously present. This paper generalizes the analysis presented
in an earlier paper (GC) where the effect of electromigration dispersion in the absence of
any bulk flow was considered. However, capillary electrophoresis channels are usually made of fused
silica which has a large negative zeta potential and sustains strong electro-osmotic flow. Thus, the theory
presented here broadens the scope of the earlier paper making it more relevant to practical systems.
In summary, a homogenized equation for the cross-sectionally averaged concentration of sample ions was derived by exploiting the approximations
outlined in \S~\ref{sec:homogenized}, which are:
(a) the Debye length is much smaller than the typical channel width
(b) the electrolyte contains only three ionic species of equal diffusivity which are also strong electrolytes
(c) the time between injection and detection is much longer than characteristic transport times
across the capillary. The outcome of the analysis is the replacement of
the molecular diffusion coefficient in the one dimensional transport equation derived in GC by an effective axial diffusivity which is a nonlinear
function of the concentration. If the concentration is not too high, this effective diffusivity is a quadratic function of the
concentration.
The reduced one-dimensional equation (\ref{generalized_GC})
provides a compact description of electromigration dispersion that
is much more amenable to numerical integration than the full coupled three dimensional problem involving fluid flow
and transport in the capillary. In fact, since both the length to width aspect ratio of the capillary as well as the P\'{e}clet number ($\Pen$)
are usually large in experiments, such three dimensional simulations can be computationally intensive. The loss of accuracy in replacing the full equations by the one dimensional model is minimal because the requirements for its validity are well satisfied and the
asymptotic convergence of the thin Debye layer and long length scale approximations have been shown to be rapid
\citep{ghosal_annrev06,loc,pof}.
The effect of a wall zeta potential may be explained qualitatively in the following way: alteration of the ionic composition of the solute
in the sample zone results in a change of the electrical conductivity and therefore in the axial electric field. Thus, the electro-osmotic slip velocity
now has an axial variation which induces pressure gradients along the channel. This in turn induces shear in the electro-osmotic
velocity profile. The consequent shear induced (Taylor-Aris) dispersion results in a concentration dependent axial
diffusivity. Numerical integration of the homogenized equation (\ref{generalized_GC}) provided some qualitative insights into the
effect of this nonlinear diffusion on separation efficiency. It is seen that the total variance
for a fixed detector position actually decreases with the strength of the electro-osmotic flow as long as the flow is not too strong.
This is due to the fact that an increase in the effective diffusivity for axial transport counteracts the wave steepening resulting from
electromigration dispersion, and, furthermore, the transit time from injection to detection is reduced.
However, if the flow is much stronger, the dispersion caused by the enhanced axial diffusivity itself
is dominant, and this degrades any gain in resolution due to the aforemetioned causes.
Thus, if all other parameters are invariant, the peak variance is a non-monotonic
function of the electro-osmotic flow strength and is optimal at a certain intermediate value of the flow velocity.
\vspace*{0.1in}
This work was supported by the NIH under grant R01EB007596
|
{
"timestamp": "2012-03-12T01:00:19",
"yymm": "1203",
"arxiv_id": "1203.1953",
"language": "en",
"url": "https://arxiv.org/abs/1203.1953"
}
|
\section{Introduction}
Parton distributions, supplemented by factorization theorems, play a crucial role in the understanding
and exploration of QCD \cite{qcdbook}. In formulating factorization theorems it is desirable to make as little approximations
in the kinematics as possible, so as to capture more of the underlying dynamics. Frequently then one
encounters the concept of transverse-momentum-dependent (TMD), or $k_\perp$-dependent, parton distributions
which follow from TMD factorization ($k_\perp$-factorization). The TMD distributions are
important because they capture more of the parton kinematics than do the canonical integrated parton distributions,
the PDFs, and they therefore play an important role in the study of less inclusive
hadronic observables which are sensitive to the details of the parton kinematics \cite{Collins:2005uv}.
In the high energy, small-$x$, limit of QCD even inclusive
cross sections are sensitive to the TMD distributions, as the so-called Regge kinematics is dominated by
the transverse components of the momenta.
Large contributions arise from large rapidity separations, and the typical
contributing momenta are slightly off-shell, the off-shellness determined by the transverse momentum.
Much of the intuition about the TMD distributions is based on concepts directly
borrowed from the parton
model, and it is for example very frequent to find in the literature the assertion that the TMD parton distributions are field
theoretical number densities, and for example that the underlying mechanism of the phenomenon of saturation is related to the saturation of the phase space occupation number of gluons in
a hadron, thus implying that there is a upper limit for the number of partons per phase space in the hadron wave
function.
While intuitive notions may be helpful in interpreting the dynamics, what is important
is the exact formulation of TMD factorization that is a must for any proper definition of the
relevant parton distribution, and the resulting distribution may or may not have the number density interpretation.
In the small-$x$ literature we find many statements regarding factorization, yet looking
closely at these statements, we find that the necessary proofs are not always provided. We have moreover found different
meanings attached to the word ``factorization'', and we therefore take the task of illuminating what exactly
is being meant in different formalisms. We will do this in section \ref{sec:factorization} where we compare
different formalisms with each other.
We should here mention that when we do speak of factorization we shall sometimes use different names
to distinguish different formalisms. For example, we frequently use the words ``hard scattering factorization"
with which we are referring to the basic factorization of QCD processes where a hard scattering is present \cite{Collins:1981uw, Collins:1981uk, Collins:1984kg, Meng:1995yn, Collins:2003fm, qcdbook}. The hard scale sets the relevant
momentum scale by which contributions can be classified according to their power as being leading or suppressed.
The latter classification is achieved using the power
counting arguments of \cite{Sterman:1978bi, Libby:1978bx}. We will go through this factorization approach in section
\ref{sec:hardscatfact}.
We note that usually the hard scattering
factorization is referred to as the ``collinear factorization" while the small-$x$ Regge type formalisms go under
the name of ``$k_\perp$-factorization". This is rather misleading, however, since $k_\perp$-factorization (TMD factorization)
is also a central part of the hard scattering factorization approach so that it is important to realize that TMD factorization
is not only relevant for small-$x$ physics. Depending on the exact final state studied, TMD factorization is
a necessary tool for QCD studies even when $x$ is not small. We will in section \ref{sec:factorization}
also go through the Color Glass Condensate \cite{JalilianMarian:1997jx, JalilianMarian:1997gr, JalilianMarian:1997dw, Iancu:2000hn,
Iancu:2001ad, Ferreiro:2001qy} formalism which is based on a physical picture of classical color fields.
One of our main objectives will be to compare the picture of factorization that emerges from the CGC with the
hard scattering factorization approach. This is important and relevant for understanding much of the phenomenology
based on these formalisms that is currently being used.
In section \ref{sec:gluonprod} we give a detailed analysis on the validity of factorization in single inclusive
particle production at small-$x$. The main small-$x$ formula, equation \eqref{GLRfact}, or some
variation of it, has been
widely used in the applications of particle production in proton-proton ($pp$), proton-nucleus ($pA$)
and nucleus-nucleus ($AA$) collisions (see e.g. \cite{Kharzeev:2003wz, Gelis:2003vh, Blaizot:2004wu, Marquet:2004xa, Kharzeev:2004if, Fujii:2006ab, Gelis:2006yv, Gelis:2006dv, Gelis:2008rw, Gelis:2008sz, Levin:2010dw, Levin:2010zy, Albacete:2010bs, Levin:2011hr, ALbacete:2010ad, Levin:2010br, Tribedy:2011aa} and references therein).
We shall examine the foundations of the formula, the arguments given
for its validity, and we shall clarify the exact pre-factor involved in the formula (as there are variations in the literature
regarding the pre-factor). Additionally we shall examine
what exactly the definition of the corresponding TMD gluon distribution is.
The standard arguments for the validity of the $k_\perp$-factorization formula are usually based on the use of the light-cone gauge. Here, simplifications occur because the leading gluon contributions are suppressed, and Faddeev-Popov
ghosts are absent.
However, there appear severe technical difficulties
by the introduction of the unphysical singularities in the light-cone gauge propagators. One issue is that these can potentially
obstruct the contour deformations that are needed for the complete proof of factorization. Additionally,
for the TMD distributions, the singularities of the gauge propagator imply rapidity divergences starting from
one loop order, and one must then consistently regularize those divergences.
While in the moderate-$x$ region the important gluon momenta are collinear to the hadron momentum, in the
small-$x$ region one enters the Regge kinematics where actually the transverse momentum components
are dominating. If $k$ is the gluon momentum then $k^+k^- \ll k_\perp^2$.
In this case the gluons are also said to be in the Glauber region.
In light-cone gauge then, transversely
polarized gluons are no longer power-suppressed. This complicates the general treatment
because one can then have arbitrarily many transversely polarized gluons exchanged
without power-suppression.
To remove the extra gluon contributions and
establish factorization, one must then be able to perform contour deformations
on the loop momenta out of Glauber region.
It is then important that the unphysical singularities
in the gauge propagators do not block the necessary contour deformations.
In reference \cite{Kovchegov:1998bi} it is shown at least in the deep inelastic scattering of a color-singlet gauge
invariant gluon current on a hadron that the contour deformations are possible in low order graphs.
However, in \cite{Kovchegov:1998bi} specific assumptions
are made on the target state that make the application of the Ward identities simpler, at least for the low order graphs.
Going to higher order graphs, however, complications can easily arise, and a systematic treatment is therefore needed.
We will examine the applications of
axial gauge on the particle production process in sections \ref{sec:lcgauge}, \ref{sec:axialgauge}
and \ref{sec:singlehadron}, addressing in particular the ability of making the necessary contour deformations.
Apart from the technical details of the proof of factorization, another issue we address here concerns the
exact definition of the TMD gluon distribution that is associated with the factorization formula,
equation \eqref{GLRfact}.
The definitions found in the literature all center
around the so-called ``dipole gluon distribution" that is related to a (slightly modified) Fourier transform of the coordinate space dipole scattering
amplitude, see equations \eqref{dipgluedistrb} and \eqref{KTgluon}. In the arguments leading to the factorization formula, however, one makes use
of the axial gauge. In the axial gauge, one necessarily obtains
a definition for the gluon distribution that is an expectation value over the transverse gluon fields, $\langle A^iA^i\rangle$.
This is
canonically identified, not with the dipole distribution, but with the so-called small-$x$ Weizsacker-Williams (WW) distribution
which is meant to represent a number density of gluons
\cite{McLerran:1993ni, McLerran:1993ka, Kovchegov:1996ty, Kovchegov:1997pc, McLerran:1998nk, Iancu:2002xk}.
The WW distribution naturally appears also in the calculation of certain classical quantities, such as the energy density
of the so-called Glasma, see for example \cite{Lappi:2006hq}.
There is therefore a potential confusion
as to what exactly the gluon distribution is, this is for example apparent in reference \cite{Kovchegov:2001sc}.
We discuss further the form of the gluon distribution in section \ref{sec:tmdgluon}.
We should also mention here that this work is part of a larger project initiated in order to understand the connections and
differences between the various TMD factorization formalisms and the TMD gluon distributions which they give rise
to. Related points that are not covered here will therefore be discussed and addressed in two separate papers
\cite{ourpaper, mypaper2}.
This paper is somewhat long, the reason being that we cover a variety of topics
which are important for the questions regarding factorization and the correct definitions of the TMD gluon distribution,
and we do not wish to skip important and subtle points but rather try to explain and illuminate them, as
this is the goal of our project. We have also aimed at providing a coherent exposition of the various topics
that appear in different formalisms and different set of works but nevertheless all are centered around the concepts
of TMD factorization and TMD parton distributions. We have therefore decided to present all the material
in a single paper. We believe that it will be of interest for both
experimentalists and theoreticians working on related topics.
The paper is organized as follows. In section \ref{sec:unint} we analyze and explain some fundamental aspects
of unintegrated parton distributions, starting from the elementary parton model definition. We concentrate on the two type of distributions commonly found
in the small-$x$ literature. Section
\ref{sec:factorization} contains our main discussion on factorization. In section \ref{sec:hardscatfact}
we provide an analysis of the hard scattering factorization approach which leads to both collinear and
TMD factorization. Then in sections \ref{sec:bfklfact} and \ref{sec:cgcfact} we analyze the formulation of $k_\perp$-factorization
in the small-$x$ region and we compare these to the hard scattering TMD factorization.
Section \ref{sec:hybrid} gives an account
of the formalisms that combine collinear factorization with the small-$x$ formulas. Section \ref{sec:gluonprod}
gives the detailed analysis of the single inclusive particle production in the small-$x$ region as already explained
above. We have divided this section into several subsections according to the different points we cover, as
was summarized above. Finally, section \ref{sec:summary} contains a brief summary.
\section{Unintegrated parton distributions}
\label{sec:unint}
Our aim in this section is to first recall the basic
idea of parton densities. We will outline the basic definition as given by the parton model, and then shortly
discuss some of the modifications induced by the dynamics of QCD. We also examine the validity of the intuitive ideas
borrowed from the parton model in the formulation of small-$x$ QCD. We will therefore here go through the commonly used ``number density" and ``dipole" distributions from the small-$x$ literature.
The concept of parton distributions dates back to the introduction of the parton model itself by Feynman
\cite[p.\ 135]{Feynman:1972r}. In there, partons of a particular flavor are considered to have a number density in the target hadron. While for the parton model calculation in DIS it is sufficient to consider number densities in the longitudinal
momentum component $x$, the concept also naturally extends to a number density in both $x$ and $k_\perp$.
The intuitive concept of a number density of partons can be
formalized using light-front quantization and writing
\begin{equation}
\label{eq:pdf.lf.def}
f_{j/h}(x,k_\perp) = \sum_\alpha
\frac{1}{ 2x (2\pi)^3 }
\frac{\langle P,h | a_{k,\alpha,j}^{\dag} a_{k,\alpha,j} |P,h\rangle}
{\langle P,h | P,h\rangle}.
\end{equation}
Here $j$ and $h$ label parton and hadron flavor, $\alpha$ is a parton
helicity index, $|P,h\rangle$ is the target state of momentum $P$, and
$a^\dagger$ and $a$ are parton creation and annihilation operators respectively.
While intuitively clear, definition \eqref{eq:pdf.lf.def} above is not really correct in full QCD, and it cannot be used
in the exact form just given \cite{qcdbook}.
In the above formula for example, the kinematic variables $x$ and $k_\perp$ are literally the momentum
fraction and transverse momentum
of the parton probed by the electromagnetic current in DIS. Therefore the unintegrated distribution above is indeed
a simultaneous distribution of the partons in both $x$ and $k_\perp$. In QCD, however, several modifications
do occur. The variables $x$ and $k_\perp$ no longer correspond to the literal momentum fractions
of any single parton in the hadron state, and additional variables must be introduced which are connected to the
divergences that occur in loop calculations (see section \ref{sec:rapidityvariable}
below and in addition the discussions in section \ref{sec:hardscatfact}).
\subsection{The gluon ``number density"}
It is in the small-$x$ literature often implied that the TMD gluon distribution
indeed has the meaning of a phase space number density
as in the above formula. Thus we often find the statement of a certain ``number of gluons per unit phase space".
In the Color Glass Condensate (CGC) model at least, this statement is meant in the sense of the Weizsacker-Williams
method of virtual quanta. We recall that in electrodynamics this method replaces the energy density of the classical electromagnetic field created by fast moving charged particles by the equivalent field of pulse radiation.
The latter is interpreted semi-classically as
consisting of a distribution of energy quanta, that is, photons. From the average energy density of the classical field,
$\langle |E|^2 \rangle$, one can
then calculate the equivalent number of photons. This is the reason why the gluon distribution appearing in the
CGC formalism is referred to as the Weizsacker-Williams (WW) gluon distribution. In the CGC then, one solves the
classical Yang-Mills equations for the non-Abelian color field. The energy density of
the classical field then relates to the equivalent number density of energy quanta, in this case identified with the gluons.
In the light-cone gauge $A^+=0$ one defines (for a hadron with large $P^+$)
\begin{eqnarray}
f_{WW}(x,k_\perp) &=& \sum_{i,a} \frac{1}{2(2\pi)^3} \left\langle a_a^{\dagger i}(x^+\!\!, k) \, a_a^{i}(x^+\!\!, k) \right\rangle \nonumber \\
&=& \sum_{i,a} \frac{2(k^+)^2}{(2\pi)^3} \left\langle
A^i_a(x^+\!\!, k) \, A^i_a(x^+\!\!, -k) \right\rangle \nonumber \\
&=& \sum_{i,a} \frac{2}{(2\pi)^3} \left\langle
F^{i+}_a(x^+\!\!, k) \, F^{i+}_a(x^+\!\!, -k) \right\rangle
\label{numberdens}
\end{eqnarray}
where $k=(k^+,k_\perp)$, and $a$ and $a^\dagger$, as in \eqref{eq:pdf.lf.def}, denote the parton (in this case gluon)
annihilation and creation operators in the sense of light front quantization where $x^+$ plays
the role of time. Notice that $x=k^+/P^+$ should not be confused with the time variable $x^+$.
The last identity, $\langle F^{+i}F^{+i} \rangle$, can be calculated in a classical approximation, for example using
the McLerran-Venugopalan model \cite{McLerran:1993ni, McLerran:1993ka}, from which an explicit expression can be obtained for $f_{WW}$.
The definition of the WW distribution is thus essentially identical to the parton model definition \eqref{eq:pdf.lf.def}.
One trivial difference is that, by convention, the $1/x$ term in \eqref{eq:pdf.lf.def} is not included in \eqref{numberdens}.
As a less trivial difference we also note that while in \eqref{eq:pdf.lf.def} the quantum mechanical averaging is taken over the
momentum eigenstates of the target, $|P\rangle$, in the CGC definition \eqref{numberdens} one rather specifies
a classical charge density $\rho(x^-,x_\perp)$ in the transverse and longitudinal planes, and the classical averaging
is then performed with respect to the specified profile, using a classical weight functional\footnote{This functional should not be confused with our generic notation for Wilson lines which is also $W$. We therefore always explicitly indicate the $\rho$ dependence of the classical CGC functional and write $W[\rho]$.} $W[\rho]$. One is then
clearly not averaging over momentum eigenstates.
The brackets are defined such that
any function, $\mathcal{O}$, of the classical source $\rho$ has the average
\begin{eqnarray}
\langle \mathcal{O}\rangle = \int D\rho \,\mathcal{O}[\rho] W[\rho].
\label{cgcaverage}
\end{eqnarray}
This averaging is normalized to unity, so that $\langle \mathds{1} \rangle = 1$,
\emph{i.e.} the classical weight functional $W[\rho]$ is such that
\begin{eqnarray}
\int \mathcal{D}\rho \,W[\rho] = 1.
\label{cgcunity}
\end{eqnarray}
A gauge invariant version of \eqref{numberdens} can be written as
(where we now expand the $F^{+i}(x^+\!\!,k)$ in terms of $F^{+i}(x^+\!\!,x^-\!\!,x_\perp)$)
\begin{eqnarray}
f_{WW}(x,k_\perp) = \frac{2}{(2\pi)^3}
\int dx^-dy^-\!\!\int d^2x_\perp && \!\!\! d^2y_\perp e^{ixP^+ (x^- -y^-) - ik_\perp (x_\perp-y_\perp)} \nonumber \\
&&\left\langle F_a^{+i}(x) W_{ab}(x,y)F_b^{+i}(y) \right\rangle.
\label{WWdistrbadj}
\end{eqnarray}
Here $W$ denotes a Wilson line in the adjoint representation needed to make the operators
within the expectation value gauge invariant. We write down the explicit definitions of the Wilson lines
in the following sections.
\subsection{The dipole gluon distribution}
\label{sec:dipoledistrb}
The most commonly encountered ``unintegrated gluon distribution" in the small-$x$ formalism
is actually different than the above distribution and is related to the so-called dipole scattering amplitude
which itself is specified in coordinate space. The dipole scattering amplitude, and the associated ``gluon
distribution" appears as a result of the use of the dipole formalism \cite{Nikolaev:1990ja, Nikolaev:1991et, Mueller:1993rr, Mueller:1994gb, Mueller:2001fv} which canonically is applied
to DIS at small-$x$.
The basic object that enters any definition of the dipole ``gluon distribution" is the coordinate space
dipole ``scattering amplitude", $\mathcal{N}$. The standard definition of this object in
DIS, or in $\gamma^*\gamma^*$ scattering is given by (see for example \cite{Balitsky:1995ub, Balitsky:2001gj,
Kovner:2001vi, Iancu:2002xk})
\begin{eqnarray}
\mathcal{N}(x_\perp,y_\perp;y) \equiv 1 - \frac{1}{N_c}
\left\langle \mathrm{Tr} \{ W^\dagger (x_\perp)W (y_\perp)
\}\right\rangle_y,
\label{dipN}
\end{eqnarray}
where we shall freely switch between the coordinates $x_\perp$ and $y_\perp$, and
\begin{eqnarray}
r_\perp=x_\perp-y_\perp, \\
b_\perp=(x_\perp+y_\perp)/2,
\end{eqnarray}
which are respectively the dipole ``size"
and ``impact parameter" in transverse coordinate space. In \eqref{dipN}, $W$ denotes the eikonal
Wilson line given by
\begin{eqnarray}
W(x_\perp) = P \exp \left( -i g_s \int_{-\infty}^\infty d \lambda \, n \!\cdot \!A^a(x_\perp \!\!+ \!\!\lambda n )\, t_F^a
\right ).
\label{Wilsonfund}
\end{eqnarray}
Here $P$ denotes path ordering with respect to $\lambda$, and $t_F^a$ is the SU(3) color matrix in the
fundamental representation. The vector $n$ is taken along the light-like direction, and the trace in \eqref{dipN}
is meant with respect to the color matrices $t_F$. The assertion of the dipole model is that this quantity is
relevant for DIS \cite{Buchmuller:1996xw, Buchmuller:1998jv, Mueller:2001fv}, $\gamma^*\gamma^*$ scattering \cite{Balitsky:1995ub, Balitsky:2001gj}, and also for quark, or prompt photon production
in hadron-hadron collisions (see for example \cite{Gelis:2002ki, Gelis:2002nn, Gelis:2006hy}).
As for the momentum distribution referred to as the ``dipole gluon distribution" \cite{Kharzeev:2003wz, Kharzeev:2004if, Albacete:2010bs, Dominguez:2011wm}, or also very commonly
as simply the ``unintegrated gluon density" \cite{Blaizot:2004wu, Marquet:2004xa, Fujii:2006ab, Gelis:2006yv, Gelis:2006dv, Gelis:2008rw, Levin:2010dw, Levin:2010zy}, it is given by a modified Fourier transform of the dipole scattering
amplitude. Most commonly we do in the literature find the definition
\begin{eqnarray}
f_{dip}(k_\perp; y) =
\mathcal{C} \int d^2r_\perp d^2b_\perp e^{-i k_\perp \cdot r_\perp} \nabla_{r}^2 \,
\mathcal{N}(r_\perp,b_\perp; y),
\label{dipgluedistrb}
\end{eqnarray}
where now we have used the variables $r_\perp$ and $b_\perp$ instead of $x_\perp$ and $y_\perp$.
We write the pre-factor simply as $\mathcal{C}$
since there does not seem to be any universally accepted value for it, and different papers use different
pre-factors. Note also that a fully gauge invariant definition of \eqref{dipN}, and therefore also of
\eqref{dipgluedistrb}, requires that one also insert transverse gauge links at $\pm \infty$.
Formula \eqref{dipgluedistrb} is not exactly linked to the parton model definition of the unintegrated gluon distribution
in \eqref{eq:pdf.lf.def}. It is therefore also distinct from the Weizsacker-Williams distribution, and also
from the gluon distributions obtained in the TMD factorization approach that we go through in section
\ref{sec:TMD}.
We examine the derivation of the Wilson lines in the definition \eqref{dipgluedistrb} in \cite{ourpaper}.
A version of the dipole gluon distribution in the adjoint representation appears also in single inclusive
gluon production, equation \eqref{KTgluon}, which we shall examine in detail in section \ref{sec:gluonprod}.
\subsection{On the rapidity variable in the gluon distribution}
\label{sec:rapidityvariable}
It is also common to denote the rapidity dependence of the dipole distribution \eqref{dipgluedistrb}
by $x$, using $y=\ln 1/x$.
We emphasize, however, that the rapidity variable in \eqref{dipgluedistrb} is conceptually different
than the variable $x$ which appears in \eqref{eq:pdf.lf.def} and \eqref{numberdens}. In the dipole
distribution, $y=\ln 1/x$ enters as a rapidity cut-off, either as the scale in the CGC formalism where the functional $W_y[\rho]$
is evaluated, or as
the non-zero slope of the Wilson lines in the formalism by Balitsky \cite{Balitsky:1995ub, Balitsky:2001gj}. On the other hand, in \eqref{eq:pdf.lf.def},
$x =k^+/P^+$, where $k^+$ is the momentum of the parton entering the hard scattering.
Similarly in the light-cone gauge definition of the WW distribution \eqref{numberdens} it again has the meaning
of the momentum fraction of the gluon entering the hard scattering. Of course, to avoid rapidity
divergences in \eqref{numberdens} a cut-off must be inserted just as in \eqref{dipgluedistrb}. There must therefore
be present an additional variable, $\zeta$, which plays the same role as $y=\ln 1/x$ in \eqref{dipgluedistrb}.
Thus we have
\begin{eqnarray}
f_{WW}=f_{WW}(x,k_\perp;\zeta),
\end{eqnarray}
and we must generally distinguish $x$ and $\zeta$. It is customary to
choose $\zeta=x$ where for example in DIS $x$ is taken to be the Bjorken variable.
One may then naturally ask why only $y$ and $k_\perp$ appear in the definition of the dipole
distribution. The answer is that $k^+$ is actually set to 0 (this is why the Wilson line \eqref{Wilsonfund} is integrated
in $x^-$ from $-\infty$ to $+\infty$). Thus the variable $x$ which appears in $f_{WW}$
is instead set to 0 in $f_{dip}$. If therefore for example the brackets in $f_{dip}$ are evaluated
fully in the classical approximation without any effects of quantum corrections, say in the MV model,
then there is no $x$ dependence, unlike $f_{WW}$ which has a $x$ dependence even
in the classical computation.
\section{Factorization}
\label{sec:factorization}
As the word ``factorization" is often used in the literature, and as there are many formalisms which
go under the name of ``$k_\perp$-factorization", we want to examine these formalisms, to explain the similarities
and the differences among them. We believe this to be a relevant task since it is important especially for the
experimental community to have clear understanding on what exactly is meant in the different formalisms. This
is also of interest for theorists, however, and especially in the case of small-$x$ physics where many statements are
put forward, particularly regarding $k_\perp$-factorization and unintegrated parton distributions. We must
then once for all analyze these statements and the assertions made.
The original concept of factorization is to be found in the hard scattering factorization
approach \cite{Collins:1981uw, Collins:1981uk, Collins:1984kg, Meng:1995yn, Collins:2003fm}
where for a given process the contributing Feynman graphs are shown to be factorizable into
different components each of which is associated with a particular type of momentum region. The leading
momentum regions are determined by a power counting analysis that we go through in section \ref{sec:powercount}.
There is a hard part specified by the large momentum scale $Q$, and dominated by short distance, $d\sim 1/Q$, contributions.
The hard scattering factorization does not directly deal with the small-$x$ region where $\sqrt{s}$ is
asymptotically large, and where there may or may not be present in addition the hard scale $Q$.
For an up-to-date and comprehensive overview of factorization in QCD, see \cite{qcdbook}.
We go through the hard scattering factorization in section \ref{sec:hardscatfact}.
After going through the hard scattering factorization, we shall in section \ref{sec:bfklfact}
examine the basic aspects of the BFKL formalism \cite{Fadin:1975cb, Kuraev:1977fs, Balitsky:1978ic}. Here the emphasis is put on the so-called Multi-Regge-Kinematics (MRK), and ideas
borrowed from the pre-QCD Regge theory \cite{Weis:1972ir, Bartels:1974tj, Bartels:1974tk} play an important role.
Even though the methods are rather different than the hard scattering factorization,
one can actually identify a structure where different factors are associated with different momentum regions as in the hard scattering factorization (see \cite{ourpaper} for further discussions).
There is also the CCH approach \cite{Catani:1990xk, Catani:1990eg, Catani:1994sq} which is based on BFKL but is meant to build on a structure
that is closely related to the hard scattering factorization since again emphasis is put on a hard
scattering coefficient.
We will here not go through CCH since we give a detailed analysis
in \cite{ourpaper}. In \cite{ourpaper} we also go through in more detail the CCFM formalism \cite{Ciafaloni:1987ur, Catani:1989yc, Catani:1989sg} that is also
based on the CCH approach and is meant to interpolate between the small-$x$ BFKL formalism and
the collinear limit at high $Q$ encoded in the DGLAP evolution.
There is then the CGC approach \cite{JalilianMarian:1997jx, JalilianMarian:1997gr, JalilianMarian:1997dw, Iancu:2000hn, Iancu:2001ad, Ferreiro:2001qy, Iancu:2002xk, Blaizot:2004wu, Gelis:2006yv, Gelis:2006dv, Gelis:2008rw, Gelis:2008sz} which uses a very different language in terms of classical fields, $A_{cl}$, and their corresponding sources, $\rho$.
In this case emphasis is put on a power counting in $g_s \rho$ where the strong coupling $g_s$
is taken as a fixed variable which can be made as small as possible. A difference between
``dilute" and ``dense" systems is emphasized, where for dilute systems $g_s \rho \ll 1$ while
for dense systems $g_s \rho \sim 1$. The structure of the factorization formula is therefore rather different than the hard scattering factorization. We analyze factorization within the CGC formalism in section \ref{sec:cgcfact}.
We shall then in section \ref{sec:hybrid} analyze some formalisms where the ideas of collinear factorization and
the CGC are mixed.
We may also mention the dipole approach encountered above where the scattering process of parton impinging upon
a target hadron is modeled via the insertion of Wilson lines as in \eqref{Wilsonfund}, where for a quark the Wilson
line is taken in the fundamental representation while for a gluon the color matrices in \eqref{Wilsonfund} are instead
taken in the adjoint representation. The dipole formalism is easily embedded into the CGC picture because
the CGC formalism, or the MV formalism, gives an explicit way of calculating the averages of the Wilson lines
that are present in the dipole formalism. Actually factorization is more or less asserted in the dipole formalism.
In \cite{ourpaper, mypaper2} we analyze the underlying structure in more detail.
\subsection{Hard scattering factorization}
\label{sec:hardscatfact}
We now review and explain the factorization which is applied to processes where a
hard scale is present. As we shall see, however, there is a structure which does not depend
on the existence of the hard factor. It will then be important to understand the overall structure here,
since it can also be applied to the Regge
region.
We will start with the most simple case of the parton model, and then
move on to the more complicated cases in QCD, and eventually to TMD factorization which is
the main interest of this paper.
\subsubsection{Basic parton model}
\label{sec:partonmod}
In order to understand the basic idea of the hard scattering factorization, it is useful to first look at the
interpretation of DIS within the parton model. The advantage of the simple parton model is that
the intuitive ideas about the scattering and the structure of hadrons can be quantified in a mathematical
manner which then paves the way for an understanding of the more complicated case of full QCD.
The quantitive analysis of the model is simplified by the understanding of the kinematics
involved, and in DIS it is convenient to consider the frame where the target hadron
has momentum $P=(P^+\!,m^2/2P^+\!,0_\perp)$,
while the virtual photon has momentum $q=(q^+,q^-,0_\perp)$ where of course $-2q^+q^-=Q^2$.
The scattering in the parton model approximation proceeds as shown in figure \ref{partonDIS} (left graph).
The parton which is struck by the virtual photon has momentum $k$.
In the rest frame of the target all the components of $k$ are of the order of the typical hadronic
scale $m$. A large boost in the plus direction then brings the momentum of $P$ into the above form, and implies
that $k^+$ is the largest component, being of order $Q$, while $k^-$ and $k_\perp$ are of order $m^2/Q$ and $m$
respectively. This corresponds to the region where the longitudinal momentum fraction $\xi=k^+/P^+$ is not much smaller
than 1.
\begin{figure}[t]
\begin{center}
\includegraphics[angle=0, scale=0.6]{partonDIS}
\end{center}
\caption{\label{partonDIS} DIS in the simple parton model. Right: Factorized structure in the parton model.}
\end{figure}
According to the parton model one can neglect the effects of the strong interaction during the
time of the interaction with the photon, and all the effects of the long distance strong interactions is put into
the parton distribution functions. This structure is shown in figure \ref{partonDIS} (right graph). In the
upper part which contains the hard scattering, one can set
$k=\xi P$. In particular since the minus component of $P$ is power suppressed with respect to the plus component,
one can make the collinear approximation whereby only $k^+$ is kept in the calculation of the hard scattering
coefficient. We denote by $\hat{k}=(k^+\!,0^-\!,0_\perp)$ the approximated momentum.
We define the DIS hadronic tensor $W^{\mu\nu}$ as
\begin{eqnarray}
W^{\mu\nu}(q,P) &=& \frac{1}{4\pi}\int d^4z \, e^{i q\cdot z} \langle P| J^{\mu}(z)J^\nu(0) |P\rangle \nonumber \\
&=& 4\pi^3 \sum_X \delta(p_X-P-q) \langle P| J^{\mu}(0)J^\nu(0) |P\rangle.
\end{eqnarray}
A factorization formula using the basic assumptions of the parton model can then be easily obtained
for $W^{\mu\nu}$. Using the general structure of the contributing graphs shown in figure \ref{partonDIS}, we can write the hadronic tensor as
\begin{eqnarray}
W^{\mu\nu} = \sum_j \frac{e_j^2}{4\pi} \int \frac{d^4k}{(2\pi)^4} \mathrm{Tr}\, \gamma^\mu U_j(k+q)
\gamma^\nu L_j(k,P)
\end{eqnarray}
where $U$ refers to the upper part of the diagram while $L$ refers to the lower blob. The trace refers to the
Dirac trace. In the upper part only
$k^+$ is important so we replace $k$ by $\hat{k}$. Then in the lower part one can replace
$k^+ \to xP^+$ since $\xi=x(1+\mathcal{O}(m^2/Q^2))$. Thus we get
\begin{eqnarray}
W^{\mu\nu} \!=\! \sum_j \frac{e_j^2}{4\pi} \mathrm{Tr} \,\gamma^\mu \! \left [ \int \! dk^+
U_j(k^+\!\!,q^-\!\!,0_\perp)\right ] \! \gamma^\nu \! \left [ \int \! \frac{dk^-d^2k_\perp}{(2\pi)^4} L_j(xP^+\!\!,k^-\!\!,k_\perp,P)\right ]
\! + \mathrm{p.s.c.}
\label{partonmod1}
\end{eqnarray}
where ``p.s.c." stands for ``power suppressed corrections".
To finally obtain a fully factorized structure we notice that the leading contribution from the lower part
comes from the component which is enhanced by the factor $Q$ in the boost along the plus direction from the hadron rest frame.
Using Lorentz invariance, this leading component can be written as $L_{leading}= \gamma^-\tilde{L}^+= (1/4)\mathrm{Tr} \,
\gamma^+ L$. Thus the factorized structure is given by
\begin{eqnarray}
W^{\mu\nu} = \sum_j \frac{e_j^2}{4\pi} \!\! && \!\!\!\!\!\! \mathrm{Tr} \left [ \gamma^\mu\int \frac{d\xi}{\xi}
U_j(\xi P^+\!\!,q^-\!\!,0_\perp)\gamma^\nu \frac{\hat{\slashed{k}}}{2}\right ] \nonumber \\
&\times& \!\! \mathrm{Tr}\left [ \int \frac{dk^-d^2k_\perp}{(2\pi)^4} \frac{1}{2} \gamma^+ L_j(xP^+\!\!,k^-\!\!,k_\perp,P)\right ]
+ \mathrm{p.s.c.}
\label{pmfact}
\end{eqnarray}
The factor in the second row defines the unpolarized integrated quark distribution in the parton model and it can be
shown to be equivalent to \eqref{eq:pdf.lf.def}.
The unintegrated density is obtained simply by undoing the $k_\perp$ integral. Thus
\begin{eqnarray}
f_j(\xi) = \int d^2k_\perp f_j(\xi, k_\perp)
\label{intvsunintpm}
\end{eqnarray}
in the parton model.
Note that the integral is over \emph{all} $k_\perp$.
Actually as we review in detail in \cite{ourpaper}, much of the
literature on the TMD gluon distribution in small-$x$ physics uses very much the same ideas as above.
We shall also see in section \ref{sec:gluonprod} that very similar arguments are used in the
treatment of single inclusive gluon production in small-$x$ QCD.
\subsubsection{On the leading momentum regions in field theory}
\label{sec:powercount}
In trying to simplify generic graphs in a field theory, so as to extract a factorized form, it is important
to systematically classify the structure of the leading contributions. In each graph at any given order in
perturbation theory there may be many loop momenta that give rise to a rather complicated manifold of
momentum regions. It turns out, however, that there is a correspondence between divergences in massless
theories and the leading configurations in high-energy processes \cite{Sterman:1978bi, Libby:1978bx}.
These leading regions are non-UV regions that are important when the hard scale $Q$ gets large.
The UV region for momenta above $Q$ of course gives divergent contributions but these contributions are handled
by renormalization which effectively cuts off the integrals above the renormalization scale $\mu$ that
conveniently may be taken as $Q$.
If one considers the complex momentum plane, then as $Q\to \infty$, many of the momentum integrations
can be deformed away from the propagator singularities, and those give therefore vanishing contributions
at asymptotic $Q$. There may, however, be contributions which cannot be deformed away from the propagator poles.
These contributions arise from surfaces in loop momentum space which are called ``pinch-singular surfaces" (PSSs).
The PSSs therefore give important contributions which must be taken into account. To determine the strengths
of the different PSSs a power counting analysis is employed. Via the power counting one also can see the
appropriate approximations to be made in the different momentum regions, and this is highly relevant for
factorization.
The interesting regions where there might be large contributions to the graphs for any given process
are thus regions where a given loop momentum $k$ has small virtuality, $|k^2| \ll Q^2$. Consider semi-inclusive
DIS where a hadron of momentum $p_B$ is produced away from the target, \emph{i.e} the large
component of $p_B$ is its minus component. The target hadron has momentum $p_A$ which is large in the
plus direction.
\begin{figure}[t]
\begin{center}
\includegraphics[angle=0, scale=0.6]{SIDISreduced}
\includegraphics[angle=0, scale=0.6]{DYreduced}
\end{center}
\caption{\label{SIDISreduced} Left: Reduced graphs for SIDIS where a hadron with momentum
$p_B$ is detected. Right: Reduced graphs for the Drell-Yan process of lepton pair production in hadron-hadron
scattering.}
\end{figure}
We show in figure (left graph) \ref{SIDISreduced} a so-called ``reduced graph" for the important PSSs.
In obtaining a reduced graph from the full Feynman graph one contracts to points all the lines whose denominators
are not pinched. This follows from the observation that those lines in the limit $Q^2\to \infty$ carry much larger
momentum than the pinched lines and therefore in a space-time picture they would reduce to points. The regions
$H, C_A, C_B, S$ denote the different momentum regions where the momenta are large and of order $Q$ (for $H$),
collinear to $p_A$ (for $C_A$), collinear to $p_B$ (for $C_B$), and small of order $m$ (for $S$). In the asymptotic
limit, $p_A$ and $p_B$ become exactly light-like, and the exact PSSs correspond to these limits where the virtuality
vanishes. Of course in the realistic (non-asymptotic) case the momenta are not exactly light-like so the exact PSSs
form a sort of skeleton of the corresponding region (for example the PSS for $C_A$ is the skeleton where the
given momentum $k$ is exactly parallel to the light-like limit of $p_A$, while the whole region of $C_A$ also contains
momenta which are approximately collinear to $p_A$). The soft PSS corresponds to the exact
limit of $S$ where all momentum components of $k$ are 0. Thus in general, momenta belonging to $S$ have
all their component small (no component is enhanced by any factor of $Q$, and they stay fixed as $Q\to \infty$).
The soft lines can therefore connect to any other region. If $k_S$ is a soft line and is added to say $k_A$ which
is in $C_A$, then $k_S+k_A$ still belongs to $C_A$. We notice, however, that lines in $C_B$ and $C_A$ cannot
be directly added to each other because adding two light-like momenta in opposite directions gives a non-light-like
momentum far off shell, and such a line does not belong to any of the two regions (it actually belongs to
the hard region $H$).
The collinear lines can, however, be added to the hard part since the result is again a hard
momentum. Thus one finds the connections between the regions as in figure \ref{SIDISreduced}.
We also show in figure \ref{SIDISreduced} (right graph) the Drell-Yan lepton pair production where
again there are two collinear regions associated with the incoming momenta $p_A$ and $p_B$, and in
addition there is the hard part where all momenta are of order $Q$, and there is again the soft graph connecting possibly
to any of the other regions.
In a collinear pinch, say collinear to the $+$ direction, the typical scales for the momenta are $k^+ \sim Q$,
$k^- \sim m^2/Q$ and $k_\perp \sim m$. In the soft pinch on the other hand all components satisfy
$k^\mu \sim m$, while in the hard region the virtuality is large $|k^2| \sim Q^2$.
There can also be several collinear regions $C_i$ in a given process. For example in DIS we can have
several jets emerging from the hard scattering, each defining its own collinear region. Notice also that a \emph{single} Feynman graph can have
multiple leading PSSs. This is so because for any given momentum line $k$ in the original graph, we have
the possibility that $k$ is in any of the allowed regions for that graph.
Consider now in QCD gluons exchanged between the different regions. Let us assume we have a collinear-to-$A$
gluon $k$ exchanged between the hard part $H$ and $C_A$. We then have a contribution of the type
\begin{eqnarray}
H^\mu N_{\mu\nu}(k)C_A^\nu.
\end{eqnarray}
Since $C_A$ contains momenta which are large in the $+$ direction, the contribution proportional to $C_A^+$ is boosted by a factor $Q$, and we see that the leading contribution
satisfies
\begin{eqnarray}
H^\mu N_{\mu\nu}(k)C_A^\nu \approx H^-N^{+-}(k)C_A^+.
\label{HtoAapprox}
\end{eqnarray}
Similar relations hold for gluons exchanged
between $H$ and $C_B$. If, however, a gluon is exchanged between $H$ and the soft region $S$,
there is no large boost factor associated with $S$. In fact the $H$-to-$S$ couplings give power suppressed corrections and therefore
the leading power contribution does not contain any lines attaching $H$ to $S$ (see below). As a simple example consider
figure \ref{sudexample} where a time-like photon $q$
produces an exclusive pair of an anti-quark with large
minus momentum $p_B$, and a quark with large plus momentum $p_A$ (this is a two-loop contribution to the Sudakov form factor). In the Feynman graph shown in figure
\ref{sudexample}, one possibility is that the gluon $k_1$ is collinear to $p_A$, while $k_2$ is soft. It is then easily seen that
$p_A-k_1-k_2$ and $p_A-k_2$ are collinear to $p_A$, while $p_B+k_1$ and $p_B+k_1+k_2$ are hard lines (since their virtualities are of order $Q^2$). The reduced graph for this Feynman graph is shown in
figure \ref{sudreduced} (left graph).
The contribution is proportional to
\begin{eqnarray}
g_s^4\, \bar{u}(p_A) \gamma^{\mu_2}\frac{\slashed{p}_A-\slashed{k}_2}{(p_A-k_2)^2+i\epsilon}\gamma^{\mu_1}
\frac{\slashed{p}_A-\slashed{k}_1-\slashed{k}_2}{(p_A-k_1-k_2)^2+i\epsilon} \frac{N_{\mu_1\nu_1}(k_1)}
{k_1^2+i\epsilon}\gamma^\mu
\nonumber \\
\frac{\slashed{p}_B+\slashed{k}_1+\slashed{k}_2}{(p_B+k_1+k_2)^2+i\epsilon}\gamma^{\nu_2}
\frac{\slashed{p}_B+\slashed{k}_1}{(p_B+k_1)^2+i\epsilon}\gamma^{\nu_1}v(p_B)\frac{N_{\mu_2 \nu_2}(k_2)}{k_2^2+i\epsilon}.
\label{sudtwoloop}
\end{eqnarray}
To pick up the leading contributions we project out the $+$ component inside the $C_A$ part (which
consists of the factors to the left of $\gamma^\mu$). This part can then be written as
\begin{eqnarray}
\gamma^{+}\frac{2p_A^+}{-2p_A^+k_2^-+i\epsilon}
\frac{\slashed{p}_A-\slashed{k}_1}{-2(p_A^+-k_1^+)k_2^-+i\epsilon} \frac{N^{-+}(k_1)}
{k_1^2+i\epsilon} \sim \frac{Q}{Q\, \lambda_s}\frac{Q}{Q\, \lambda_s}\frac{1}{\lambda_A^2}.
\label{twoloopgraph}
\end{eqnarray}
\begin{figure}[t]
\begin{center}
\includegraphics[angle=0, scale=0.55]{sudakovexamp}
\end{center}
\caption{\label{sudexample} A two loop contribution to the Sudakov form factor. }
\end{figure}
Here we have introduced typical momentum scales for the collinear and soft regions, $\lambda_A$
and $\lambda_s$ respectively, such that for any collinear-to-$A$ ($C_A$) momentum, $k_A$, we have $k_A^2 \sim \lambda^2_A$,
while for the soft momentum, $k_s$, we have $k_s^2 \sim \lambda_s^2$. Notice that since $k_A^+\sim Q$, this
means that $k_A^- \sim \lambda_A^2/Q$. The soft region in \eqref{sudtwoloop} simply consists
of the soft propagator $1/k_2^2 \sim 1/\lambda_s^2$, and the momentum integral $\int d^4k_s \sim \int d\lambda_s
\lambda_s^3$.
The collinear-to-$B$ region, $C_B$, is elementary while the hard region power counts as
\begin{eqnarray}
\frac{\slashed{p}_B}{2p_B^-k_1^++i\epsilon}\gamma^{-}\frac{\slashed{p}_B}{2p_B^-k_1^++i\epsilon}\gamma^{-}
\sim \frac{Q}{Q^2} \frac{Q}{Q^2}.
\end{eqnarray}
The PSSs then give
\begin{eqnarray}
\int^{\sim Q}\!\! d\lambda_A \lambda_A^3 \int^{\sim Q}\!\! d\lambda_s \lambda_s^3 \frac{1}{Q^2} \frac{1}{\lambda_A^2}\frac{1}{\lambda_s^2}
\frac{Q}{Q\, \lambda_s}\frac{Q}{Q\, \lambda_s} = \int^{\sim Q}\frac{d\lambda_A}{\lambda_A} \left ( \frac{\lambda_A}{Q} \right )^2
\int^{\sim Q} \frac{d \lambda_s}{\lambda_s}.
\label{2loopsuppressed}
\end{eqnarray}
The complete result is given by multiplying \eqref{2loopsuppressed} with the LO graph.
In figure \ref{sudreduced} (right graph) we show the case where both $k_1$ and $k_2$ are soft gluons.
Here, the hard part is elementary while the soft part now
contains both gluon propagators. It is easy to see that we get in this case
\begin{eqnarray}
\int^{\sim Q} \!\!d\lambda_{s,1} \lambda_{s,1}^3 \int^{\sim Q}\!\! d\lambda_{s,2} \lambda_{s,2}^3
\frac{Q^2}{(Q\lambda_{s,1})^2} \frac{Q^2}{(Q\lambda_{s,2})^2} \frac{1}{\lambda_{s,1}^2} \frac{1}{\lambda_{s,2}^2}
= \int^{\sim Q} \frac{d\lambda_{s,1}}{\lambda_{s,1}}\int^{\sim Q} \frac{d\lambda_{s,2}}{\lambda_{s,2}}.
\label{twoloopleading}
\end{eqnarray}
The contribution from the PSS \eqref{twoloopleading} as we see has no suppression compared to the LO graph, while
\eqref{2loopsuppressed} has a power suppression. The power suppression comes from the coupling
of the soft part to the hard part.\footnote{It may seem in \eqref{2loopsuppressed} that performing the $\lambda_A$
integral gives a contribution of order unity since we integrate all the way up to $Q$. However, the integral is
completely dominated by the upper limit where the momentum is no longer collinear-to-$A$ but is instead is
hard. In the definition of the hard region there will be a subtraction of the smaller PSSs, for example $C_A$.
That subtraction will cancel the dominant contribution of the integral and ensure that \eqref{2loopsuppressed} is truly power-suppressed.}
\begin{figure}[t]
\begin{center}
\includegraphics[angle=0, scale=0.55]{sudakovreduced}
\end{center}
\caption{\label{sudreduced} Examples of reduced graphs for the two loop Sudakov form factor.}
\end{figure}
For a given amplitude, cross section or structure function to be analyzed we denote the leading
power obtained by dimensional analysis as $Q^p$, where $p=4-E_L$ with $E_L$ counting the
number of external lines. For the Sudakov form factor in figure \ref{sudexample}, $E_L=3$, so
the lowest order contribution grows as $Q$. For DIS, $E_L=4$ and the leading power is $Q^0$.
For a given PSS, we then generally have integrals of the form
\begin{eqnarray}
Q^{p_1}\int^{\sim Q} \frac{d\lambda}{\lambda}\lambda^{p_2},
\label{lambdaint}
\end{eqnarray}
where $p_1$ and $p_2$ are different powers.
Making use of dimensional analysis and Lorentz invariance, one then finds in
QCD the following results \cite{qcdbook}: For a collinear
region $C$, every line joining $C$ to $H$ gives a power $\lambda/Q$ \emph{except} for longitudinally
polarized gluons, carrying polarization $N^{+-}$,
for which there is no suppression. For the soft region, every gluon coupling $S$ to $H$
gives a factor $\lambda/Q$ (as in the example of \eqref{2loopsuppressed}) while every fermion gives $(\lambda/Q)^{3/2}$. Every fermion coupling $S$
to $C$ gives a factor $(\lambda/Q)^{1/2}$. Thus all couplings between $S$ and other regions are suppressed,
\emph{except} for longitudinally polarized gluons between $S$ and $C$ for which there is no suppression. There is
thus no penalty for coupling $C$ and $H$, and $S$ and $C$ via longitudinally polarized gluons. For more details,
see \cite{Sterman:1978bi, Libby:1978bx, qcdbook}. In the cases
where there is no suppression, the integrals \eqref{lambdaint} usually produce logarithms $\ln Q^2/m^2$
that accompany the leading power (this is due to the renormalizable nature of QCD in which the coupling
is dimensionless), as for example in \eqref{twoloopleading}.
\subsubsection{Factorization in simple theory}
\label{sec:simplefact}
\begin{figure}[t]
\begin{center}
\includegraphics[angle=0, scale=0.65]{gaugeDIS}
\end{center}
\caption{\label{gaugeDIS} Generic contribution to inclusive DIS in simplified case.}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[angle=0, scale=0.6]{gaugeDIS3}
\end{center}
\caption{\label{gaugeDIS3} Generic contribution to inclusive DIS. }
\end{figure}
The results above show that in QCD one has to take into account arbitrarily many gluon exchanges, of longitudinal
polarizations, between the different regions (except for $S$-to-$H$ couplings which are always power suppressed
regardless of polarization). The proof for factorization is then more complicated compared to the simple parton
model in figure \ref{partonDIS} where gauge bosons are not present.
Let us first, however, study a simplified situation by using the results from the power counting.
This example will be illustrative for understanding the small-$x$ calculations in section \ref{sec:gluonprod}.
In figure \ref{gaugeDIS} we show an example of inclusive DIS where arbitrarily many gluons are exchanged between the lower part $L$, which is collinear to the target
hadron $P$, and the upper part $U$, which contains the hard scattering. Of course where the final state cut goes through
$U$, the cut lines are necessarily on-shell, but the bubble will still contain internal lines that are far off-shell. In a more
complete picture one must consider instead the class of graphs shown figure \ref{gaugeDIS3}.
It can, however, be shown in the inclusive case by a sum-over-cuts argument that the momenta
in the collinear region can be deformed out to the region where it is far off-shell, effectively reducing the
leading graphs to that shown in figure \ref{gaugeDIS}. We thus treat the upper part of the diagram
as the hard region. According to the analysis in the previous section, we then see that soft gluon couplings
do not arise in the leading contributions.
\begin{figure}[t]
\begin{center}
\includegraphics[angle=0, scale=0.65]{gaugeDIS2}
\end{center}
\caption{\label{gaugeDIS2} Pure gluonic contributions to DIS. Left: The black squares indicate transversely polarized
gluons while all other gluons are longitudinally polarized. Right: Longitudinally polarized gluons only give
a super-leading contribution in the hard scattering region.}
\end{figure}
We notice that one may also consider pure gluon exchanges between the upper and lower parts. If all gluons are longitudinally polarized, \emph{i.e} contributing via $N^{-+}$, then a super-leading contribution
arises which has power $Q^2/m^2$ relative to the leading case. However, Ward identities apply for these contributions,
and a careful treatment shows that the super-leading piece actually cancels, leaving behind a remainder term that is leading
only \cite{Collins:2008sg}. A leading contribution is also obtained
when one of the gluons at each side of the cut is transversely polarized, we show this in figure
\ref{gaugeDIS2} (left graph) where we denote the transversely polarized gluons using the black squares.
Pure gluon exchange terms
are important for the analysis in the small-$x$ region which we come back to later.
The parton model result reviewed above can be exactly reproduced in a model field theory which is non-gauge
(this removes all gauge boson attachments between $L$ and $U$) and super-renormalizable (this implies that
the hard part $U$ is trivial as in figure \ref{partonDIS}). As a simplified case we instead imagine a theory which is still non-gauge but is renormalizable. This means that
the higher order corrections to the hard part are not power suppressed anymore. Moreover it means that one has
to also take into account the UV renormalization. At the same time it implies that
the gauge boson exchanges shown in figure \ref{gaugeDIS} are absent, and one obtains instead figure \ref{nongaugeDIS}. Now, another way to think of this case is to actually consider full QCD in light-cone gauge $A^+=0$. In this case the leading gluon coupling vanishes since
\begin{eqnarray}
N^{-+}(k)=g^{-+} - \frac{n^-k^++n^+k^-}{k^+n^-} = 1 - 1 = 0.
\end{eqnarray}
Therefore in figure \ref{gaugeDIS}, all gluon couplings again vanish to leading power. In figure \ref{gaugeDIS2}
it means on the other hand that only the two transversely polarized gluons remain, as shown in
figure \ref{nongaugeDIS}.
\begin{figure}[t]
\begin{center}
\includegraphics[angle=0, scale=0.65]{nongaugeDIS}
\end{center}
\caption{\label{nongaugeDIS} Leading contribution in the simplified case in non-gauge theory (only left graph) or in light-cone
gauge QCD (both graphs).}
\end{figure}
A factorization formula for figure \ref{nongaugeDIS} can now be obtained rather easily by assuming that there is a
clear separation in momenta for the exchanged line $k$, namely that it can either be hard or collinear
to $P$. We can then write the hadronic tensor as (neglecting photon indices)
\begin{eqnarray}
W = \int \frac{d^{4-2\epsilon} k}{(2\pi)^{4-2\epsilon}} U^{\{\alpha\}}(k, q) L_{\{\alpha\}}(k,P),
\end{eqnarray}
where the index $\{\alpha\}$ collectively denotes all relevant labels such as flavor, color, polarization\footnote{Of course
in a non-gauge theory we need not consider the color indices but as the analysis is also relevant for light-cone QCD
we include all quantum labels.}. We again
make the approximation of replacing $k$ in $U$ by $\hat{k}=(k^+\!,0,0_\perp)$. Thus one gets
\begin{eqnarray}
W \sim \int \frac{d k^+}{k^+} U^{\{\alpha\}}(\hat{k}, q) \,\,k^+\!\!\int \frac{dk^- d^{2-2\epsilon} k_\perp}{(2\pi)^{4-2\epsilon}}
L_{\{\alpha\}}(k,P).
\end{eqnarray}
This formula is not yet in a fully factorized form, however, since there is still the sum over the labels $\{\alpha\}$.
We note that $U$ must be diagonal in the color indices since the photon is color singlet. Consider first the
quark contribution shown in figure \ref{nongaugeDIS} (left graph). To fully factorize $F$ we can then apply
exactly the same argument as in the parton model case in going from equation \eqref{partonmod1} to \eqref{pmfact}.
We then get just as in \eqref{pmfact}
\begin{eqnarray}
W \sim \int \frac{d\xi}{\xi} \left [ \mathrm{Tr} \,
U_j(\xi P^+\!\!,q^-\!\!,0_\perp) \frac{\hat{\slashed{k}}}{2}\right ] \left [ \mathrm{Tr}
\int \frac{dk^-d^{2-2\epsilon}k_\perp}{(2\pi)^{4-2\epsilon}} \frac{1}{2} \gamma^+ L_j(k,P)\right ].
\end{eqnarray}
Summation over the color indices in $L$ is kept implicit. Corrections to the factorization formula
are power suppressed by the analysis in section \ref{sec:powercount}.
For the gluon contribution shown in the right graph of figure \ref{nongaugeDIS} we instead find
\begin{eqnarray}
W \sim \int \frac{d k^+}{k^+} U^{ij}(\hat{k}, q)\,\, k^+\!\!\int \frac{dk^- d^{2-2\epsilon} k_\perp}{(2\pi)^{4-2\epsilon}}
L^{ij}_{aa}(k,P).
\label{simplequarkfact}
\end{eqnarray}
We then notice that the upper part $U$ is diagonal in the transverse and color indices which gives the
factorized form
\begin{eqnarray}
W \sim \int \frac{d \xi}{\xi} \left [ \frac{1}{2}U^{jj}(\hat{k}, q) \right ] \left [\xi P^+\!\!\int \frac{dk^- d^{2-2\epsilon} k_\perp}{(2\pi)^{4-2\epsilon}}
L^{ii}_{aa}((\xi P^+\!\!,k^-\!,k_\perp),P)\right ].
\label{simplegluonfact}
\end{eqnarray}
The second factor here defines, preliminarily, the integrated gluon distribution. We shall see in section \ref{sec:gluonprod}
that the elementary definition
of the TMD gluon distribution in axial gauge in the small-$x$ limit is given by the very same
set of approximations.
This simple derivation of factorization cannot be strictly true, however.
Namely, the main assumption that a clear separation of scales is possible
is not generally true in a renormalizable theory like QCD. For example in the above calculation we assume that
$k_\perp \sim m$, while the case $k_\perp \sim Q$ would have instead contributed to the next-to-leading order correction
to the hard part $H$. There is, however, also an intermediate region, where
$m \lesssim k_\perp \lesssim Q$, and $k$ is neither exactly target collinear nor exactly hard, and as a consequence it is not
clear in the above formalism how to exactly handle $k$ in this case. For the assumptions above
to thus hold, it must be true that this intermediate region can be safely omitted. This is, however, not the case. In
fact, the renormalizability of QCD implies that there are in general logarithmic contributions,
\begin{eqnarray}
\int_{\sim m^2}^{\sim Q^2} \frac{dk_\perp^2}{k_\perp^2} \sim \ln Q^2/m^2.
\end{eqnarray}
\begin{figure}[t]
\begin{center}
\includegraphics[angle=0, scale=0.5]{DISsubtractedgluon}
\end{center}
\caption{\label{dissubtracted} Example of subtraction in the NLO gluon coefficient. The subtraction removes the
contribution where the loop momentum $l$ is target-collinear, indicated by $\hat{l}$ in the last graph. }
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[angle=0, scale=0.5]{DISsubtractedquark}
\end{center}
\caption{\label{dissubtracted2} Example of subtraction in the NLO quark coefficient. The subtraction removes the
contribution where the loop momentum $l$ is target-collinear, indicated by $\hat{l}$ in the last graph.}
\end{figure}
There is therefore no power suppression of the intermediate region, and in fact it is even enhanced by a
logarithm. A full treatment must therefore treat such regions correctly, and this can in general be done by a
subtractive formalism \cite{qcdbook}. This means that each PSS is defined with subtractions
of the smaller PSSs that it contains, to prevent double counting and ensure that it indeed is dominated
by the momenta associated with it. For the hard part $U$ in figure \ref{nongaugeDIS}, one should therefore
include a subtraction of the target-collinear PSS. We show examples of these subtractions
in DIS for the gluon and quark contributions in figures \ref{dissubtracted} and \ref{dissubtracted2}
respectively. If we denote by $d\Pi$ the phase
space measure for the momenta contained in $U$ then a more correct version of \eqref{simplegluonfact} reads
\begin{eqnarray}
W \sim \int \frac{d \xi}{\xi} \!\!\!\!&&\left [ \frac{1}{2}\int d\Pi \left [ U^{jj}(\hat{k}, q)
-\mathrm{subtractions} \right ] \right ] \nonumber \\
&&\times \left [\xi P^+\!\!\int \frac{dk^- d^{2-2\epsilon} k_\perp}{(2\pi)^{4-2\epsilon}}
L^{ii}_{aa}((\xi P^+\!\!,k^-\!,k_\perp),P)\right ].
\label{subtractedfact}
\end{eqnarray}
The integrated (bare) gluon distribution is thus given by
\begin{eqnarray}
f_g^{(0)}(\xi) &=& \xi P^+\!\!\int \frac{dk^- d^{2-2\epsilon} k_\perp}{(2\pi)^{4-2\epsilon}}
L^{ii}_{aa}((\xi P^+\!,k^-\!,k_\perp),P) \nonumber \\
&=& \int \frac{dx^-}{2\pi \, \xi P^+} e^{i\xi P^+ x^-} \langle P | F_{(0) \, a}^{+i}(0^+\!\!,x^-\!\!,0_\perp)F_{(0) \, a}^{+i}(0)
|P\rangle
\label{intgluonpdf}
\end{eqnarray}
where the last result holds in $A^+=0$ gauge in QCD, apart from some technical problems associated with this
gauge that we are neglecting.
As indicated in \eqref{intgluonpdf}, the basic operator definitions of the parton distributions are for the bare fields of the Lagrangian. Note that
it is these fields which have the canonical gauge transformation properties, and thus in discussing the gauge
transformation properties of the parton distributions one necessarily refers to the operator definitions constructed
out of the bare fields.
The renormalization of the bare parton distributions
is then an issue of the renormalization of non-local operators. While in the case of local
field operators, the renormalization factor can be taken as a multiplicative constant which is independent of
momenta and masses, for the non-local operators appearing in the definitions of the bare parton distributions
one instead finds that there is a convolution with a renormalization factor. Basically if we denote the bare parton
distribution for a parton of flavor $j$ as obtained from either \eqref{simplequarkfact} or \eqref{simplegluonfact}
by $f^{(0)}_j(\xi)$, and the renormalized distribution by $f_j(\xi)$, we find
\begin{eqnarray}
f_j(x; \mu) = \lim_{\epsilon \to 0} Z_{jj'}(\xi, \mu, \epsilon) \otimes_{\xi} f_{j'}^{(0)}(x/\xi; \mu, \epsilon),
\label{pdfrenorm}
\end{eqnarray}
where the convolution is an integral in $\xi$ as in \eqref{simplequarkfact} and \eqref{simplegluonfact}.
The evolution of $f_j(x; \mu)$ with respect to $\mu$ is given by the DGLAP equations.
\subsubsection{Including the gluons, and the Glauber region}
For a fully satisfactory treatment of factorization in full QCD one needs, however, to deal with the gluon
emissions. As we recall from section \ref{sec:powercount}, in QCD we can without any power suppression exchange arbitrarily many
longitudinally polarized gluons between the hard and collinear, and the soft and collinear regions respectively.
We indicated this possibility already in figures \ref{gaugeDIS3} and \ref{gaugeDIS2}. In the previous section
we argued that in the collinear factorization of inclusive DIS at least, the structure of the leading graphs can
be simplified by choosing the light-cone gauge $A^+=0$ which eliminates the leading longitudinally polarized gluons.
There is, however, a good reason to try to avoid the light-cone gauge in the generic treatment (see also
sections \ref{sec:lcgauge}, \ref{sec:axialgauge} and \ref{sec:singlehadron} below). Note from the
arguments in the previous sections that the treatment of factorization is based on first analyzing the analytic structure
of the Feynman graphs, identifying the PSSs, and then using power counting to extract the leading PSSs. To
guarantee that the power counting arguments work properly, contour deformations must be performed when necessary.
In particular, if $k$ is a momentum in the soft region, then there is the possibility that the components of $k$
do not all scale with the same power $\lambda_s$, but that the longitudinal components $k^+$ and $k^-$ might
be parametrically much smaller than $k_\perp$. This happens if $k^+$ or $k^-$ is pinched by the collinear lines it attaches to. For example, if $k$ couples to a collinear line $p_A$ then a propagator,
\begin{eqnarray}
(p_A + k)^2 + i\epsilon,
\end{eqnarray}
arises. The pole for $k^-$ is then
\begin{eqnarray}
k^- \sim \frac{m^2}{Q} - i\epsilon.
\label{kminuspole}
\end{eqnarray}
Thus $k^-$ is parametrically much smaller than $\lambda_s \sim m$. When this happens, we say the
momentum is in the Glauber region, $k^+k^- \ll k_\perp^2$. Now, if no other such pole is present,
or if all such poles lie in the same part of the imaginary plane (all below or above the real axis), then we can deform the contour away from
this pole to keep $k^- \sim \lambda_s$. If, however, another pole exists simultaneously, such that
\begin{eqnarray}
k^- \sim \frac{m^2}{Q} + i\epsilon
\label{kminuspole2}
\end{eqnarray}
then the $k^-$ contour is pinched, and cannot be deformed. It might still be possible to deform on $k^+$ but if not, then the standard power counting fails. The longitudinal polarizations
then no longer dominate and one cannot use the eikonal approximations needed to obtain factorization.
The use of the light-cone gauge implies that the analytic structure of the individual Feynman graphs is altered,
since now an additional pole $1/k^+$ is introduced with each propagator. This has obvious implications for the
factorization proofs. These poles might for example introduce pinch points that are not present in a covariant gauge.
Moreover, the gauge poles $1/k^+$ commonly give rise to integrals of the form
\begin{eqnarray}
\int_0^\infty dk^+ \frac{1}{k^+}I(k^+,k_\perp),
\label{rapdiv}
\end{eqnarray}
and these diverge as $k^+\to 0$. Notice that the divergences arise from end point singularities and can
therefore not be treated by any $i\epsilon$ prescription or principal value. In fact there exists no generalized
function which is a ``canonical regularization", in the sense described in \cite{gelfand}, of this integral.
These divergences are in fact the rapidity divergences we mentioned
in sections \ref{sec:dipoledistrb} and \ref{sec:rapidityvariable}. They also arise when the eikonal approximation
is used in a covariant gauge. In the integrated distribution, there is actually a cancellation between real and virtual
terms, which means that in \eqref{rapdiv}
\begin{eqnarray}
\int d^2k_\perp I(k^+=0,k_\perp) = 0.
\end{eqnarray}
This leads to the well-known ``plus prescription", $\left ( \frac{1}{1-z} \right )_+$. In TMD distributions, however, no cancellation occurs, since $I(0,k_\perp) \neq 0$,
and the light-cone gauge therefore introduces problems.
The light-cone gauge is moreover not useful when several different collinear directions are relevant.
The general method for factorizing the arbitrary order gluon couplings between the different regions
is based on exploiting the gauge symmetries of the leading terms, and to use Ward identities (Slavnov-Taylor-Ward identities).
The basic technique can be understood as follows. In Feynman gauge, let $k$
be a soft gluon coupling the regions $S$ and $A$. We then have a contribution of the type
\begin{eqnarray}
A^\mu(k, p_A) \, g_{\mu\nu}\, S^\nu(k).
\end{eqnarray}
Generally of course there will be many other couplings, and $A$ and $S$ will depend on additional
momenta but that does not matter for the approximation we are explaining. The leading contribution is then
\begin{eqnarray}
A^\mu(k, p_A) \, g_{\mu\nu} \, S^\nu(k) \sim A^+(\hat{k}_B, p_A)S^-(k) \nonumber \\
= A^\mu(\hat{k}_B,p_A)\frac{\hat{k}_{B,\mu} \, n_{A, \nu}}{k\cdot n_A} S^\nu(k),
\label{AtoSapprox}
\end{eqnarray}
where
\begin{eqnarray}
\hat{k}_B=(k \cdot n_A)\, n_B=(0^+\!\!, k^-\!\!, 0_\perp).
\end{eqnarray}
Here $n_A$ is a light-like vector in the direction of $p_A$,
with $n_A\cdot V = V^-$ for any $V$. Thus $\hat{k}_B\cdot n_A = k \cdot n_A$. Since now the
polarization of the gluon $k$ is multiplied by its momentum in the coupling to $A$, Ward identities
can be applied. The eikonal denominator in \eqref{AtoSapprox} gives a contribution in $S$ from a
Wilson line. The all-order gluon couplings between $A$ and $S$ can then be successively factorized
into a Wilson line contribution in $S$.
\begin{figure}[t]
\begin{center}
\includegraphics[angle=0, scale=0.65]{DISWilsonlines}
\end{center}
\caption{\label{DISwilsonlines} Factorized structure in inclusive DIS in covariant gauge. The longitudinal
gluon emissions are factorized into eikonal Wilson lines (double lines) to provide gauge invariant definitions
of the parton distributions. Left: Quark distribution. Right: Gluon distribution where the gluons with black
squares are transversely polarized gluons. }
\end{figure}
One can similarly make approximations for the $H$-to-$A$ couplings. The eikonal terms that arise are then
absorbed into $A$ to provide gauge invariant definitions of the basic parton distributions (or fragmentation
functions). An example in the case of inclusive DIS is shown in
figure \ref{DISwilsonlines} where the Wilson lines are indicated by double lines.
The procedure of using the Ward identities for extracting the gluon exchanges between the different
regions proceeds very much the same whether one is formulating collinear factorization or TMD factorization.
As we have seen, Wilson lines appear in the small-$x$ formalisms as well, both in the Weizsacker-Williams
distribution \eqref{WWdistrbadj} and the dipole distribution \eqref{dipgluedistrb}. It is then rather important
to understand the exact structure and derivation of these lines, in particular since differences appear between
the dipole definition and the TMD distributions. We analyze these points in detail in \cite{ourpaper}.
\subsubsection{TMD factorization}
\label{sec:TMD}
In the hard scattering formalism, the need for TMD factorization becomes obvious when one considers observables
which are more sensitive to the exact kinematics of the final state.
A typical example concerns the almost back-to-back production of
hadrons \cite{Collins:1981uk} in $e^+e^-$ annihilation shown in figure \ref{TMDgraphs1}. Other relevant processes
where one needs to consider TMD factorization are
single-inclusive hadron production at low $p_\perp$
in DIS (SIDIS) also shown in figure \ref{TMDgraphs1}, and Drell-Yan lepton pair production shown in figure
\ref{TMDgraphs2} where the total transverse momentum of the lepton pair
is much smaller than the hard scale. In all these cases the kinematics is sensitive to low values of the observable
transverse momentum $q_\perp$,
and one cannot therefore neglect any of the transverse momenta flowing through the regions $C_A$, $C_B$ and $S$,
as doing so would significantly change the kinematics of the observable final state products.
If on the other hand the relevant transverse momentum observables are large, of the order of the hard scale $Q$,
then the effects of the transverse momentum flowing out from the collinear regions via the soft region
is power suppressed and can be neglected. In that case one obtains the standard integrated (collinear) factorization.
\begin{figure}[t]
\begin{center}
\includegraphics[angle=0, scale=0.55]{TMDEPEM}
\includegraphics[angle=0, scale=0.55]{TMDSIDIS}
\end{center}
\caption{\label{TMDgraphs1} Processes where TMD factorization is relevant. Left: Di-hadron production in $e^+e^-$.
Right: Hadron production in SIDIS. }
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[angle=0, scale=0.55]{TMDDY}
\end{center}
\caption{\label{TMDgraphs2} Leading regions for TMD factorization in Drell-Yan lepton pair production. }
\end{figure}
Note, however, that the transverse momentum flowing directly into the hard part $H$ from the collinear regions
$C_A$ and $C_B$ can still
be neglected, since the error involved in this approximation is of order $q_\perp/Q$
which is small in the validity region of TMD factorization. As $q_\perp \to Q$ the TMD formula loses its accuracy
but then one enters the region where ordinary integrated factorization is valid. When $q_\perp \sim Q$, the
transverse momentum must be a part of the hard region, physically it corresponds to the case where several
high $q_\perp$ partons emerge from $H$. Thus
what determines the need for TMD parton distributions and fragmentation functions is the kinematics of the
final state. The momenta entering $H$ from the collinear region $C_A$ or $C_B$ can still be approximated to be on-shell,
even in the case of TMD factorization. This is somewhat different than the small-$x$ formulation
where the gluon momentum entering the hard scattering (if there is any) is off-shell, its virtuality being determined
by the transverse momentum.
The factorization formula in case of hadron pair production in $e^+e^-$ annihilation involves the transverse momentum convolution of
two fragmentation functions (since there is no hadronic initial state in this process). The factorized formula for the relevant hadronic tensor is obtained by applying the appropriate Ward identities for the longitudinally polarized gluons exchanged between leading
regions shown in figure \ref{TMDgraphs1} (left graph). If the momentum entering regions $C_A$, $C_B$ and $S$ is denoted
respectively by $k_A$, $k_B$ and $k_S$, then the factorized formula is given by (we denote $C_{A}$ by
$A$, and $C_B$ by $B$ for clarity)
\begin{eqnarray}
W^{\mu\nu} = \int d^4k_A \, d^4k_B \, d^4k_S\, A(k_A) \, B(k_B)\, S(k_S) \, H^{\mu\nu}(q)
\delta^{(4)}\!(q\!-k_A\!\!-k_B\!\!-k_S).
\end{eqnarray}
The delta function can be used to fix $k_{S,\perp}$, $k^+_A$ and $k_B^-$. One furthermore makes the
approximation of ignoring $k_A^-$ ($k_B^+$) everywhere but in $A$ ($B$), and ignoring $k_S^{\pm}$
everywhere but in $S$. These approximations are allowed since the corrections are power-suppressed
at least as $m^2/Q^2$.
The integrals over these variables can then all be short circuited and one gets
\begin{eqnarray}
W^{\mu\nu}\!\! &=& \!\! \int d^2k_{A,\perp} d^2k_{B,\perp} \! \left ( \int dk_A^-A(k_A)\right)\!\! \left ( \int dk_B^- B(k_B)\right ) \!\!\left ( \int d k_S^+ dk_S^- S(k_S) \right )\!H^{\mu\nu}(q) \nonumber \\
&=& \!\!\int d^2k_{A,\perp} d^2k_{B,\perp} \,A(z_A,k_{A,\perp})\, B(z_B,k_{B,\perp})\, S(q_\perp\!\! - k_{A,\perp}\!\! -k_{B,\perp})
H^{\mu\nu}(q).
\label{pretmdepem}
\end{eqnarray}
Each respective factor in the parentheses gives the basic operator definition of the fragmentation
functions and the soft factor.
We mentioned in sections \ref{sec:powercount} and \ref{sec:simplefact} that each given PSS contains subtractions of the smaller PSSs. Thus the collinear
factors $A$ and $B$ in \eqref{pretmdepem} contain subtractions of the soft region.
Now, the unsubtracted collinear
parts contain Wilson lines which arise from the factorized gluon couplings to the hard part $H$.
This is done by using the approximation in \eqref{HtoAapprox},
rewriting this as in \eqref{AtoSapprox} and applying the Ward identities. For the $A$-to-$H$ couplings, the approximated momenta
from \eqref{HtoAapprox} are $\hat{k}_A = (k^+,0^-,0_\perp)=(k\cdot n_B)\, n_A$ and therefore we get a Wilson line
in the direction $n_B$:
\begin{eqnarray}
W(x;n_B) = P \exp \left (-ig_s \int_0^\infty d \lambda \, A(x+n_B \lambda)\cdot n_B \right ).
\label{wilsonnb}
\end{eqnarray}
For the $B$ part we instead get a Wilson line in the direction $n_A$. In figure \ref{unsubtracted} we
graphically represent the unsubtracted collinear part, including the Wilson line \eqref{wilsonnb}
shown by double lines, for both a parton distribution (top two graphs) and a fragmentation function (bottom two graphs).
The color representation of the Wilson line \eqref{wilsonnb} is determined by the particle at the end of
the double lines in figure \ref{unsubtracted}: Fundamental for a quark (top and bottom left), adjoint
for a gluon (top and bottom right).
\begin{figure}[t]
\begin{center}
\includegraphics[angle=0, scale=0.65]{unsubtractedpdf}
\end{center}
\caption{\label{unsubtracted} Graphical representation of the unsubtracted collinear part
after the
gluon couplings to the hard part have been factorized into Wilson lines in the direction $n_B$. Left: Quark
distribution. The black squares indicate transversely polarized gluons. Top: Collinear part in a parton distribution. Bottom: Collinear part in a fragmentation
function.
}
\end{figure}
The soft gluons are similarly summed into Wilson lines using \eqref{AtoSapprox}. From the $A$ side we
see we get a line in the direction of $n_A$ while from the $B$ side we instead get a line in the direction of
$n_B$. The definition of the collinear part involves always the hadron state $|P\rangle$, either as incoming
(for a parton distribution) or as outgoing (for a fragmentation function). The soft factor on the other hand does
not contain such a hadron so it is defined as a vacuum expectation value which we represent in figure
\ref{softfigure}.
As seen from \eqref{pretmdepem}, it is convenient to make a Fourier transform into transverse coordinate $b_\perp$ to obtain
\begin{eqnarray}
W^{\mu\nu} = \int d^2b_\perp e^{-iq_\perp \cdot b_\perp} A(z_A,b_{\perp}) B(z_B,b_{\perp}) S(b_\perp)
H^{\mu\nu}(q)
\end{eqnarray}
which is simpler than the momentum convolution written above.
\begin{figure}[t]
\begin{center}
\includegraphics[angle=0, scale=0.65]{softfigure}
\end{center}
\caption{\label{softfigure} The factorized soft part. On each side of the cut, the gluons that couple
to regions $A$ and $B$ are factorized into Wilson lines in the directions $n_A$ and $n_B$ respectively.
}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[angle=0, scale=0.37]{subtractedpdf}
\end{center}
\caption{\label{subtractedpdf} The soft factor absorbed into the unsubtracted parton
distributions and fragmentation functions. In the final result, $n_A$ and $n_B$ can be taken
exactly light-like since the rapidity divergences cancel those in the unsubtracted collinear factor.
The vector $n_\zeta$ cannot be taken light-like, however.
}
\end{figure}
In the final definition, the soft factor is absorbed completely into the collinear factors to define
the final subtracted fragmentation functions given by \cite{qcdbook}
\begin{eqnarray}
D_{H_A/f}(z_A,b_{\perp}; \zeta, \mu) = D^{\mathrm{unsub}}_{H_A/f}(z_A,b_\perp;n_B)
\! \times \! \sqrt{\frac{S(b_\perp;n_A,n_\zeta)}{S(b_\perp;n_A,n_B)S(b_\perp;n_\zeta,n_B)}}
\times Z
\label{subtractedtmd}
\end{eqnarray}
Here $n_A$ and $n_B$ are taken light-like, and $Z$ is the UV renormalization factor.
The somewhat strange looking factor in the square root
is shown in figure \ref{subtractedpdf}. The precise motivation for it is described in detail in
\cite[Ch.\ 13]{qcdbook}. The final definition is free from divergences associated with Wilson line self-energy
corrections.
The vector $n_\zeta$ defining the directions of the Wilson line
in the soft factors serves as the rapidity cut-off which we indicate by the $\zeta$ dependence of the
fragmentation function. The unsubtracted factor $D^{\mathrm{unsub}}$ is given exactly by the
factors in figure \ref{unsubtracted} (bottom left graph in current example), defined in addition with integral over $k^-$
as in \eqref{pretmdepem}, and the Fourier transform from $k_\perp$ to $b_\perp$. A similar definition
applies for the second fragmentation function associated with the region $B$.
The final factorization formula then reads
\begin{eqnarray}
W^{\mu\nu} \!\!\propto \frac{z_A z_B}{Q^2}H_f^{\mu\nu}(Q; \mu)\! \int d^2 b_{\perp} e^{-iq_\perp \cdot b_\perp}
D_{H_A/f}(z_A,b_{\perp}; \zeta, \mu) \, D_{H_B/\bar{f}}(z_B,b_\perp; \zeta, \mu),
\label{tmdepem}
\end{eqnarray}
where $z_{A,B}=p_{A,B}/k_{A,B}$, and
\begin{eqnarray}
H_f^{\mu\nu} = \mathrm{Tr} \, \slashed{\hat{k}}_AH_f^\nu \slashed{\hat{k}}_B H^{\mu \dagger}_f.
\label{hardcoeff}
\end{eqnarray}
$H^\nu$ and $H^{\mu \dagger}_f$
stand for the hard blobs shown in figure \ref{TMDgraphs1}, defined to be irreducible in the collinear lines,
and containing subtractions of the collinear and soft regions, just like in \eqref{subtractedfact}.
The tensor $W^{\mu\nu}$ of course
cannot depend on the rapidity cut-off $\zeta$, and this requirement is embedded in the Collins-Soper evolution equation of the
fragmentation functions with respect to $\zeta$.
In SIDIS we instead have a convolution of one parton distribution (for the incoming target hadron) and one fragmentation function (for the final state hadron),
\begin{eqnarray}
W^{\mu\nu} \!\!\propto \frac{z}{Q^2}H_f^{\mu\nu}(Q; \mu)\! \int d^2 b_{\perp} e^{-iq_\perp \cdot b_\perp}
f_{f/H_A}(x,b_{\perp}; \zeta, \mu) \, D_{H_B/\bar{f}}(z,b_\perp; \zeta, \mu)
\label{sidis}
\end{eqnarray}
where $H_f^{\mu\nu}$ is given by the same expression as in \eqref{hardcoeff} (but of course the hard
factors $H^{\mu \dagger}$ and $H^\nu$ are different in $e^+e^-$ and DIS), and $x=k_A^+/p_A^+$
and $z=p_B^-/k_B^-$. Thus the change is that one fragmentation function is simply
exchanged for the parton distribution function of the target hadron. The parton distribution $f$ is defined
exactly as in \eqref{subtractedtmd} to include the soft factors, one simply needs to change
$D^{\mathrm{unsub}}$ to $f^{\mathrm{unsub}}$ which means (for quarks) replacing the bottom
left graph in figure \ref{unsubtracted} with the top left one.
Finally in the Drell-Yan process we instead
have two parton distributions and there is no fragmentation function since the observed final state is leptonic.
Thus
\begin{eqnarray}
W^{\mu\nu} \!\!\propto \frac{s}{Q^2}H_f^{\mu\nu}(Q; \mu)\! \int d^2 b_{\perp} e^{-iq_\perp \cdot b_\perp}
f_{f/H_A}(x_A,b_{\perp}; \zeta, \mu) \, f_{\bar{f}/H_B}(x_B,b_\perp; \zeta, \mu)
\label{drellyan}
\end{eqnarray}
where now the hard coefficient $H^{\mu\nu}$ is the tensor for the on-shell partonic reaction $f\bar{f} \to \gamma^*$.
The extra factor $s$ in front of the integral arises from the definition of the hadronic tensor for the Drell-Yan
process which reads
\begin{eqnarray}
W^{\mu\nu} = s \int d^4x \, e^{iq\cdot x} \langle p_A,p_B| J^\mu(x) J^\nu(0) |p_A,p_B\rangle.
\end{eqnarray}
In order to obtain a reliable estimate of $H^{\mu\nu}$ it is optimal to let $\mu \sim Q$ so as to avoid
large logarithms. The higher order corrections are then subleading
in factors of $\alpha_s(\mu\sim Q) \ll 1$ without any logarithmic enhancements, and thus fixed order perturbative calculations
are reliable. Notice again that in all formulas above, the hard tensor $H^{\mu\nu}$ is always outside the transverse
momentum (or coordinate) integral and the lines entering it are on-shell.
Thus we see that the TMD parton distributions or fragmentation functions, compared to the basic parton model
definitions, depend additionally on the variables $\zeta$ and $\mu$. They consequently satisfy evolution equations
with respect to both these variables. The evolution in $\mu$ is given by the standard DGLAP equations while the evolution with respect to the rapidity variable $\zeta$ is given by the (Collins-Soper) CS evolution equation \cite{qcdbook}.
The CS kernel controlling the rapidity evolution is the same for all the above reactions because it is determined by the
soft factor which is the same in all the above examples.
We have above outlined the fundamentals of factorization in QCD, in processes where
a hard scale $Q$ is present, and where the collinear directions scale with $Q$. In the small-$x$
region there may or may not be present a hard scale. The traditional process to study is small angle two-particle
elastic scattering where the momentum transfer $t$ is much smaller than the cms energy $s$, and
where the collinear momenta scale with $\sqrt{s}$. In this case the hard region, if present, has a scale
$Q$ which is fixed, and is therefore not proportional to the asymptotic variable $\sqrt{s}$. The leading regions
are therefore somewhat different than in the hard scattering factorization. We will outline the relevant
regions for the small-$x$ case in section \ref{sec:diffcases} where we examine single inclusive particle
production. We now go through the main formulations of $k_\perp$-factorization
in the BFKL and CGC formalisms, and compare these to the hard scattering case just discussed.
\subsection{Factorization in BFKL}
\label{sec:bfklfact}
``Factorization" in the BFKL formalism refers to the Regge factorization in which a given
$2\to n$ scattering amplitude is, in the asymptotic limit $s\to \infty$, written as a factorized product of effective vertices and
couplings of ``reggeized gluons". This is known as the ``multi-Regge form".
The arguments for the factorized form of the $2\to n$
amplitudes go back to the pre-QCD days of Regge theory, and the so-called ``multi-peripheral" models
\cite{Campbell:1970wy, Weis:1972ir, Bartels:1974tj, Bartels:1974tk}.
\begin{figure}[t]
\begin{center}
\includegraphics[angle=0, scale=0.55]{MultiRegge}
\end{center}
\caption{\label{multiregge} The multi-Regge-factorized form of the scattering amplitude in BFKL. }
\end{figure}
We illustrate the multi-Regge form in figure \ref{multiregge}. Here the zig-zag lines denote the
Reggeons, and each black circle denotes the Reggeon-Reggeon-gluon vertex. Figure \ref{multiregge}
is in the Regge theory valid when $s_i \to \infty$ for all $i$ \cite{Bartels:1974tj, Bartels:1974tk}.
\begin{figure}[t]
\begin{center}
\includegraphics[angle=0, scale=0.55]{Lipatovvertex4}
\end{center}
\caption{\label{lipvert1} Graphs contributing to Lipatov vertex. }
\end{figure}
In BFKL, the vertical zig-zag lines in figure \ref{multiregge} are given by gluons whose propagators (in Feynman
gauge) are obtained by
\begin{eqnarray}
P_{\mu\nu}(q_i) = \frac{-ig_{\mu\nu}}{q_i^2+i\epsilon} \to P_{\mu\nu}(q_i, s_i) = \frac{-ig_{\mu\nu}}{q_i^2+i\epsilon}
\left ( \frac{s_i}{q_{\perp,i}^2} \right )^{\omega(q_{\perp, i}^2)-1}
\label{reggegluon}
\end{eqnarray}
where
\begin{eqnarray}
\omega(q_{\perp, i}^2)-1 = \alpha_s N_c \int \frac{d^{2+2\epsilon}\kappa_\perp}{(2\pi)^{2+2\epsilon}} \frac{-q_{\perp, i}^2}{\kappa^2_\perp(\kappa_\perp-q_{\perp, i})^2},
\label{Reggefactor}
\end{eqnarray}
The function $\omega$ is called the ``gluon Regge trajectory". The vertices in figure \ref{multiregge}
are given by the Lipatov vertex which is an effective three-gluon
vertex. The Lipatov vertex is derived from the tree-level graphs of the $2\to 3$ partonic amplitude
shown in figure \ref{lipvert1}. The external partons
may be quarks or gluons, the use of the eikonal approximations implies that the vertex is
independent of the flavor of these particles (at least when the external particles are individual quarks or gluons).
The fundamental assertion of the BFKL formalism is that the multi-Regge form shown in figure \ref{multiregge}
is valid for all $2 \to n$ amplitudes. It has been argued in reference \cite{Fadin:2006bj} that the multi-Regge
result can be shown to be correct to all orders,
once it has been shown to be correct to one-loop order for \emph{all} $2\to n$ amplitudes, by essentially using the same techniques
($s$-channel unitarity relations) developed in Regge theory in \cite{Bartels:1974tj, Bartels:1974tk}. We are, however, not aware of any explicit higher order calculations in
QCD of the $2 \to n$ amplitudes for $n > 3$. For the $2\to 3$ amplitude, the multi-Regge form has been derived
in reference \cite{Fadin:1993wh} up to one-loop corrections to the graphs in figure \ref{lipvert1}.
As we saw in the previous section, factorization has to be shown to hold for all orders. In section
\ref{sec:tmdgluon} we shall show some examples of higher order corrections where TMD factorization
is know to be violated. Since the multi-Regge formula leads to a $k_\perp$-factorized form (see further \cite{ourpaper}) it is of relevance to consider such higher order graphs. As we will see, the breakdown of factorization might be hidden until
higher order corrections. For example, figures \ref{2to2gentmd}, \ref{2to2nogentmd} and \ref{2to2nogentmd2}
show that
factorization breakdown is not visible until 4 and 5 gluon exchange in the $2\to 2$ amplitudes.
If we consider only one side of the cut $2\to 2$ amplitude, then factorization breaking graphs appear
in 2 or 3 loop corrections to the $2\to 4$ amplitude. Similar factorization breaking terms might be present
in the $2\to 3$ gluon amplitudes at 2 loop corrections as well.
It may therefore very well be that one-loop corrections do not exhibit any TMD factorization breaking.
\subsection{Factorization in the CGC}
\label{sec:cgcfact}
The Color Glass Condensate (CGC) \cite{JalilianMarian:1997jx, JalilianMarian:1997gr, JalilianMarian:1997dw, Iancu:2000hn,
Iancu:2001ad, Ferreiro:2001qy} is a semi-classical approach developed to deal with the
QCD physics of ``large" objects such as heavy ions.
The set-up of the CGC formalism is rather different than the hard scattering factorization. The main assertion
here is that the color degrees of freedom of a given hadron, such as a large nucleus, can be
described by classical fields generated by
a distribution of random color sources, $\rho_a$ ($a$ being the color index), which arise due to the ``fast'' moving partons,
i.e., those partons which are in the collinear region. These then act as sources for the
softer gluons whose dynamics depend on the classical sources.
\subsubsection{Basics of CGC}
\label{sec:cgcbasic}
The classical fields generated by these sources are determined by the solutions to the classical equations
of motion
\begin{eqnarray}
D_\nu F_a^{\mu\nu}(x) = J^\mu_a,
\label{YMeqs}
\end{eqnarray}
with $D_\nu$ the usual covariant derivative.
The generic solutions to \eqref{YMeqs} give classical fields $A^a_{cl}$ that are highly non-linear in the sources $\rho_a$.
In the classical McLerran-Venugopalan (MV) model \cite{McLerran:1993ni, McLerran:1993ka}, the sources are assumed to
originate from the valence quarks of the nucleons which are randomly distributed according to some
weight functional, $W[\rho]$. This is the distribution we encountered earlier in equations \eqref{cgcaverage}
and \eqref{cgcunity}.
In the case of a single particle traveling in the plus direction, the classical current is taken as
\begin{eqnarray}
J_a^\mu(x) = \delta^{\mu +} g_s\, \rho_a(x^-\!\!,x_\perp)
\label{onecurrent}
\end{eqnarray}
where the classical source $\rho(x^-,x_\perp)$ has a very narrow support in $x^-$. In the case of
two particle scattering, with the incoming hadrons traveling along the opposite light-cones, one takes
instead
\begin{eqnarray}
J^\mu_a(x) = \delta^{\mu +}g_s\, \rho_{1,a}(x^-\!\!,x_\perp) + \delta^{\mu -}g_s\, \rho_{2,a}(x^+\!\!,x_\perp).
\label{twocurrent}
\end{eqnarray}
The model is defined at some scale $\Lambda^{\hat{\mu}}$ which sets the applicability of the classical description.
Here $\hat{\mu} = +$ or $\hat{\mu} = -$.
For a hadron with large momentum $P^\mu$ along the direction $\hat{\mu}$, this means that all
fields with $k^{\hat{\mu}} > \Lambda^{\hat{\mu}}$ are taken to be described by the classical sources $\rho$.
The distribution $W[\rho]$ is therefore specified at the scale $\Lambda^{\hat{\mu}}$.
Physical quantities of interest in the model are calculated by functional averages using the classical
distribution $W[\rho]$ as in \eqref{cgcaverage} for a single hadron, and
\begin{eqnarray}
\langle \mathcal{O} \rangle = \int D\rho_1 \,D\rho_2 \, W_{\Lambda_1^+}[\rho_1] \, W_{\Lambda_2^-}[\rho_2]
\,\mathcal{O}[\rho_1,\rho_2],
\label{twohadronav}
\end{eqnarray}
in two hadron scattering. Of course, \eqref{twohadronav} is already in a factorized form.
\subsubsection{Power counting and ``dilute" and ``dense" systems}
\label{sec:cgcpowercount}
The treatment of two particle processes is then based on a power counting argument of the
classical sources $g_s\, \rho$. A ``dilute" particle in this power counting is defined to be
one described by a source such that $|g_s \, \rho | \ll 1$ . For such a particle then, in
the calculations
only the first order dependence $(g_s\, \rho)^1$ is kept. Given a functional
$\mathcal{O}[\rho_1,\rho_2]$ which depends on both $\rho_1$ and $\rho_2$, expand it as a polynomial
\begin{eqnarray}
\mathcal{O}[\rho_1,\rho_2] = \sum_{n=1}^\infty \sum_{m=1}^\infty \mathcal{O}_{n m} \,
(g_s \, \rho_1)^n (g_s \, \rho_2)^m.
\end{eqnarray}
The definition of particle 1 being dilute then means that
\begin{eqnarray}
\mathcal{O}[\rho_1,\rho_2] \to\biggl . \mathcal{O}[\rho_1,\rho_2] \biggr \vert_{\mathrm{1,dilute}} = \sum_{m=1}^\infty \mathcal{O}_{1 m} \,
(g_s \, \rho_1) (g_s \, \rho_2)^m.
\end{eqnarray}
Conversely a particle is defined to be ``dense" if it is described by a source satisfying $|g_s \, \rho | \sim 1$.
In that case, the dependence on $g_s\, \rho$ is retained to all orders. As for real particles, a proton
or a deuteron is defined as being ``dilute", while heavy ions such as gold or lead nuclei are defined
to be ``dense".
Thus ``dilute-dilute" scattering refers essentially to $pp$ or $p\bar{p}$ scattering, while ``dilute-dense"
scattering refers to $pA$ or deuteron-Nucleus ($dA$) collisions, and finally ``dense-dense" scattering refers
to $AA$ collisions (lead-lead or gold-gold). Of course
a proton in the CGC becomes ``dense" at sufficiently high energies since the classical sources grow
as a function of energy.
In this setting, the quantum evolution is based on the logic of the leading logarithmic approximation (LLA)
where the coupling $g_s$ is fixed and small, $g_s \ll 1$. Therefore for a ``dilute" object we have
$\rho \lesssim 1$, while for a ``dense" object we have $\rho \sim 1/g_s \gg 1$. These assumptions lead to the formulation of factorization
in the CGC approach \cite{Gelis:2003vh, Blaizot:2004wu, Gelis:2006yv, Gelis:2006dv, Gelis:2008rw, Gelis:2008sz}.
We immediately notice that this power counting is rather different in logic than the power
counting described in section \ref{sec:powercount}. Here the emphasis is put on the
classical source $\rho(x)$ specified in space-time coordinates. Any correction beyond the classical approximation
is calculated to order $g_s^2$ which amounts to a one-loop calculation. For processes involving protons
then, calculations are kept at linear order in $g_s \rho$ for each proton which in a diagrammatic analogy means that
at most two gluon couplings are considered. In figure \ref{dilutefigure} we show an example of single inclusive gluon production in ``dilute-dilute" scattering. Thus in the dilute limit factorization is essentially identical to that in the parton model
we considered in section \ref{sec:partonmod}.
\begin{figure}[t]
\begin{center}
\includegraphics[angle=0, scale=0.55]{dilutesinglegluonprod}
\end{center}
\caption{\label{dilutefigure} Diagrammatic representation of particle production in ``dilute-dilute"
scattering in the language of CGC. }
\end{figure}
In general, however, the extra gluon emissions
between the different PSSs considered in section \ref{sec:powercount} all have small virtualities and
they therefore couple strongly. In particular the soft gluons have all their momentum components small,
and the QCD coupling of these gluons is therefore strong. That is, we do not have a situation where $g_s \ll 1$.
Even in the case of a weak coupling at all relevant scales, however, such as in QED, is the formalism outlined
in \ref{sec:powercount} and the factorization theorems rather useful for controlling the higher order corrections
which still might be enhanced by kinematical factors.
In the CGC higher order corrections are needed because the classical sources $\rho$
can have large values, $|\rho| \gg 1$, but $g_s$ itself is always small. Pure perturbative calculations
are thus performed when $|\rho| \lesssim 1$, which happens in the case of ``dilute" particles. In the general
treatment of factorization in QCD, or in generic field theories, however, large contributions arise from
surfaces in the multi-dimensional space of momentum integrals where the integration contours
are forced to go close to the singularities of the propagators, the pinch singular surfaces. In QCD
the momentum lines in the PSSs have large couplings. This is the reason why factorization must be
proven to all orders, and it is then convenient to employ the power counting analysis of the PSSs.
Corrections are then guaranteed to be power suppressed in the large scale $Q$. In the case of small-$x$
therefore, ideally we would want to formulate factorization (``$k_\perp$-factorization") up to power
suppressed corrections in $\sqrt{s}$.
Of course the treatment of factorization cannot be purely perturbative for the reasons just explained.
It is important to emphasize that the power counting methods of section \ref{sec:powercount} rely
generically on dimensional analysis and Lorentz invariance, and thus not exclusively on perturbation theory. The
explicit calculations are of course performed using Feynman graphs, but the structures
obtained have a meaning beyond strict perturbation theory. One can therefore apply the same
methods to the small-$x$ region where any hard scale might be absent.
\subsubsection{The LLA and basic logic of factorization}
\label{sec:cgclla}
As the LLA is important for the formulation of factorization in the CGC, we shortly outline the logic
behind it. An all order result can be obtained by
calculating the one-loop graphs using the eikonal approximation, and then exponentiating the result.
If the one loop result for a certain process is $\Gamma_1$, and
the tree level result is $\Gamma_0$, then usually one finds
\begin{eqnarray}
\Gamma_1 = g_s^2 \int^Y_0 dy \, K_s(y) \, \cdot \Gamma_0,
\label{gammaoneloop}
\end{eqnarray}
where $dy =\frac{dk^+}{k^+}$ and the limits on $y$ are determined by the kinematics of the given process.
The kernel $K_s$ is found by applying the approximations appropriate for a soft term.
We can then write the complete result up to one loop as
\begin{eqnarray}
\Gamma_0 + \Gamma_1 = \left ( 1 +g_s^2 \int^Y_0 \!dy \, K_s(y) \right ) \Gamma_0.
\end{eqnarray}
For infinitesimal change in the scale we can write this as
\begin{eqnarray}
\Gamma_{dY} = \left ( 1 +g_s^2 dY \, K_s(dY) \right )\Gamma_0,
\label{gammady}
\end{eqnarray}
so that
\begin{eqnarray}
\frac{\Gamma_{dY}-\Gamma_{0}}{dY} = g_s^2\, K_s \,\Gamma_0.
\end{eqnarray}
This gives the all order LLA result
\begin{eqnarray}
\Gamma_Y^{LLA} = \exp \left ( g_s^2 \int^Y_0 \!dy \, K_s(y) \right ) \Gamma_0.
\label{gammalla}
\end{eqnarray}
A similar construction is used in the CGC
\cite{Gelis:2003vh, Blaizot:2004wu, Gelis:2006yv, Gelis:2006dv, Gelis:2008rw, Gelis:2008sz}.
The idea is to start with a formula at the classical level, where the correlator of the classical
fields is calculated using \eqref{twohadronav}, and then to perform a one loop calculation as in \eqref{gammaoneloop}
and show that at this level the classical structure \eqref{twohadronav} still holds. The resulting
one loop formula can then be resummed as in \eqref{gammady} and \eqref{gammalla} to obtain
a final formula in the LLA.
\begin{figure}[t]
\begin{center}
\includegraphics[angle=0, scale=0.55]{CGCparticleprod}
\end{center}
\caption{\label{CGCparticleprod} Particle production by classical sources in the CGC. The crosses denote the classical field
insertions. }
\end{figure}
Take for example the single inclusive particle production in the scattering of two hadrons, described by sources
$\rho_1$ and $\rho_2$, which is studied in \cite{Gelis:2003vh, Blaizot:2004wu, Gelis:2006yv, Gelis:2006dv, Gelis:2008rw, Gelis:2008sz}. The basic classical formula which is equivalent to a tree level
calculation is given by
\begin{eqnarray}
\left\langle E_p \frac{dN}{d^3p} \right\rangle = \frac{1}{2 (2\pi)^3} \sum_\lambda \left\langle |\mathcal{M}_\lambda(p)|^2 \right\rangle
\label{classicalglueprod}
\end{eqnarray}
where
\begin{eqnarray}
\mathcal{M}_\lambda(p) =p^2 A^{cl}_\mu(p) \epsilon^{*\mu}_{(\lambda)} (p).
\end{eqnarray}
We illustrate this in figure \ref{CGCparticleprod} where the crosses denote the insertions of the
classical fields $A^{cl}(p)$. Note that $A^{cl}(p)$
is a function of both $\rho_1$ and $\rho_2$ so it contains the effects of both hadrons. At the pure classical
level, one evaluates \eqref{classicalglueprod} using \eqref{twohadronav}. This gives
\begin{eqnarray}
\langle A_\nu A_\mu \rangle_0 = \int D \rho_1 \, D \rho_2\, W_{\Lambda^+}[\rho_1] \, W_{\Lambda^-}[\rho_2] \,
(A^{cl}_\nu A^{cl}_\mu)[\rho_1,\rho_2]
\end{eqnarray}
where the subscript on the left hand side is to denote that this corresponds to the tree level calculation.
The weight functionals can at this level be fully parametrized using the MV model from which an explicit result
can be obtained for \eqref{classicalglueprod}.
The one-loop correction to the tree level result is then found to be \cite{Gelis:2008rw, Gelis:2008sz}
\begin{eqnarray}
\langle A_\nu A_\mu \rangle_1 \!=\! \int D \rho_1 \, D \rho_2 \, W_{\Lambda^+}[\rho_1] \, W_{\Lambda^-}[\rho_2]
\left [ \ln\frac{\Lambda^+}{p^+} H_1 + \ln\frac{\Lambda^-}{p^-} H_2 \right ] (A^{cl}_\nu A^{cl}_\mu)[\rho_1,\rho_2]
\label{cgconeloop}
\end{eqnarray}
where each $H_i$ corresponds to the ``JIMWLK Hamiltonian". Each $H_i$
is a Hermitian differential kernel \cite{Iancu:2002xk} (in the sense of functional differentiation) that acts on the classical fields $A^{cl}_\nu A^{cl}_\mu$ in \eqref{cgconeloop}.
We see that this result is analogous to
\eqref{gammaoneloop}. To understand the logarithmic factors in \eqref{cgconeloop}, note that if $K_s(y)$
is independent of $y$ (which it nearly always is), then the integral in \eqref{gammaoneloop} simply gives
$Y\cdot K_s$. The rapidity $Y$ exactly corresponds to the logarithmic factors in \eqref{cgconeloop} and
we see that $K_s$ corresponds\footnote{The JIMWLK Hamiltonian is of order $g_s^2$ in the quantum fluctuations, but it contains the classical sources
$g_s \, \rho$ to all orders. } to $H_i$. Using that the $H_i$ are Hermitian one can then rewrite
the complete one-loop result \eqref{cgconeloop} as \cite{Gelis:2008rw, Gelis:2008sz}
\begin{eqnarray}
\langle A_\nu A_\mu \rangle_1 \! +\! \langle A_\nu A_\mu \rangle_0 \!\!&=& \!\!
\int D \rho_1 \, D \rho_2 \left ( 1 + \ln\frac{\Lambda^+}{p^+} H_1 \right) W_{\Lambda^+} [\rho_1]
\left (1 + \ln\frac{\Lambda^-}{p^-} H_2 \right) W_{\Lambda^-}[\rho_2] A^{cl}_\nu A^{cl}_\mu \nonumber \\
&\equiv& \!\! \int D \rho_1 \, D \rho_2\, W_{p^+} [\rho_1] W_{p^-} [\rho_2] (A^{cl}_\nu A^{cl}_\mu)[\rho_1,\rho_2]
\label{cgcfact}
\end{eqnarray}
In this rewriting one uses that formally the term containing the product $H_1H_2$ in \eqref{cgcfact}
is of higher order (it is not of LLA) and thus neglected.
One then gets exactly as in \eqref{gammalla} the LLA result
\begin{eqnarray}
W_{dY}=(1+dY\, H)W_0 \to W^{LLA}_Y = \exp \left ( \int^Y_0 dy \, H(y) \right ) W_0.
\end{eqnarray}
Equation \eqref{cgcfact} is referred to as the ``high energy factorization", or ``JIMWLK factorization", formula \cite{Gelis:2003vh, Blaizot:2004wu, Gelis:2006yv, Gelis:2006dv, Gelis:2008rw, Gelis:2008sz}.
\subsubsection{Comparison to TMD factorization}
\label{sec:cgctmd}
In its derivation, \eqref{cgcfact} is rather different than the TMD factorization described in section \ref{sec:TMD}.
For example, in \eqref{cgcfact} there is a factorized product of the classical weight functionals $W[\rho_i]$
rather than a product of parton distributions and/or fragmentation functions.
Equation \eqref{cgcfact} is in the literature implied to be a generalization of ordinary TMD factorization.
In section 5 of reference \cite{Gelis:2008rw} we can for example read that \\
``\emph{JIMWLK factorization proven here is far more general and robust in
comparison to the $k_\perp$-factorization often discussed in the literature.}"
The statement on the wider generality of the CGC formula is motivated by the observation that
one can for ``dilute" systems obtain from \eqref{cgcfact} a formula which looks like a $k_\perp$-factorized formula.
Since this ``dilute" limit involves a simplified approximation within the CGC formalism, it is therefore said
that \eqref{cgcfact} is more general. For example,
for the single inclusive gluon production using \eqref{classicalglueprod} and \eqref{cgcfact} one gets
in the ``dilute" limit a formula that looks like equation \eqref{GLRfact} below which is the $k_\perp$-factorization formula canonically
used in the small-$x$ region. Moreover,
within the CGC, the TMD gluon distribution can be calculated explicitly if $W[\rho]$ is given. For example, the
WW gluon distribution can be calculated from \eqref{WWdistrbadj} once $W[\rho]$ is specified. The converse
statement on the other hand is not true: It is not enough to have an explicit formula for \eqref{WWdistrbadj}
in order to extract $W[\rho]$ uniquely.
In this sense, it can indeed be said that \eqref{cgcfact} is more general than the TMD factorization. However,
from a different perspective we find that this statement is misleading and not correct. Moreover, as we shall explain now,
the factorization explained in section \ref{sec:hardscatfact} is actually more robust.
Equation \eqref{cgcfact} is namely only derived at one loop order using the logic of the
LLA while the TMD factorization is much more general and accurate than that.
The LLA result for example
gives no hint at all as to what the higher order corrections might look like.
There are even instances where it gives the wrong result,
even qualitatively, an example being the Drell-Yan cross section at zero transverse momentum where
the LLA gives a vanishing result while the true result that can be obtained from the factorization approach
is non-zero \cite{Collins:1981va}.
Contrary to the LLA, in the factorization
approach the higher order corrections are well controlled, and even if the explicit calculations of the higher order
corrections can be difficult in practice, one can nevertheless make reliable estimates of their importance \cite{qcdbook}.
It is therefore not correct to say that the ``JIMWLK factorization" is more robust than the TMD factorization.
In fact the opposite is clearly true with regards to the accuracy of the derivation.
Moreover, when in the CGC the dilute limit is taken, the TMD gluon distribution that appears in the
factorization formula is given by \cite{Iancu:2002xk, Blaizot:2004wu, Gelis:2008rw}
\begin{eqnarray}
\bigl . f(x,k_\perp; \zeta) \bigr \vert_{\mathrm{dilute}} \!\!\!&=& \! \frac{1}{k_\perp^2}\langle \rho(k)\rho (-k) \rangle_{W_{\zeta P^+}} \!\!=
\langle F^{+i}(k)F^{+i}(-k) \rangle_{W_{\zeta P^+}} \nonumber \\
&=& \!\!\! \int d^3x \, d^3y \, e^{ixP^+ (x^-\!\! -y^-) - ik_\perp (x_\perp-y_\perp)}
\langle F_a^{+i}(x)F_a^{+i}(y)\rangle_{W_{\zeta P^+}}.
\label{dilutegluon}
\end{eqnarray}
The subscripts on the correlators imply that the averages using $W[\rho]$ are performed at the scale $\zeta P^+$.
Acting with the dilute limit of the JIMWLK Hamiltonian on the classical sources in \eqref{dilutegluon} one then recovers
the BFKL equation for the object $f(x,k_\perp; \zeta)$ (for a simple demonstration of this, see \cite{Iancu:2002xk}).
Thus the BFKL formalism can be identified with the
dilute limit of the JIMWLK formalism. Since for example the CCH formalism \cite{Catani:1990xk, Catani:1990eg} is based upon BFKL it is
indeed correct that \eqref{cgcfact} presents a generalization of the work in \cite{Catani:1990xk, Catani:1990eg}.
Moreover, as the work in \cite{Catani:1990xk, Catani:1990eg} is frequently referred to as the ``$k_\perp$-factorization" formula,
in this sense (\emph{i.e.} if `$k_\perp$-factorization" is understood to refer to \cite{Catani:1990xk, Catani:1990eg})
\eqref{cgcfact} is more general than ``$k_\perp$-factorization". The CCH formalism is, however, also based on the
LLA, and neglected terms are therefore not power-suppressed.
The argument for factorization in \cite{Catani:1990xk, Catani:1990eg} is based on the use of the
light-cone gauge (in DIS) or axial gauge (in hadron-hadron collisions).
The
final expression in \eqref{dilutegluon} actually equals the earlier light-cone gauge expression in \eqref{numberdens}.
A similar definition also appears in the factorization approach as we discussed in reference to figure
\ref{nongaugeDIS}. It is, however, important to realize that \eqref{dilutegluon} is supposed to hold in the dilute limit for \emph{any} gauge, even a covariant gauge.
This is in fact in line with the power counting we discussed in section \ref{sec:cgcpowercount}
above, where the definition of the dilute limit is that $g_s \, \rho \ll 1$. This is of course why equation \eqref{dilutegluon}
is second order only in $\rho$ (the first order term $\langle \rho \rangle$ vanishes when, as usual, the distribution
$W[\rho]$ is taken to be a Gaussian).
It is then important, however, to realize that the distribution thus obtained in \eqref{dilutegluon} is \emph{not} the
TMD gluon distribution in the TMD factorization approach. One cannot in the TMD factorization in covariant
gauge simply drop the Wilson lines because as mentioned above, the soft gluons exchanged between different
regions have strong coupling. The TMD factorization therefore does not correspond to the dilute limit of the
CGC. The factorization \eqref{cgcfact} does indeed represent a different
structure than the TMD factorization, but it cannot be said to be more general since it contains only a one-loop
calculation while the TMD factorization is valid to leading power, rather than to leading logarithm.
We want to emphasize that this point is important and not merely a technical detail. The reason is that
if we wish to establish factorization for a given process, then a possible breakdown of factorization
may not show up until higher order corrections are considered, beyond the dilute limit.
In section \ref{sec:tmdgluon} below we shall discuss this point in the context of the small-$x$ single
inclusive gluon production formula. As we explain there for example, the factorization breaking graphs
studied in \cite{Collins:2007jp, Rogers:2010dm} do not show up until one considers two gluon corrections
to the parton model graphs, see figures \ref{2to2gentmd} and \ref{2to2nogentmd}. In terms of Feynman
diagrams, the parton model graphs themselves are already at two loop order, so the factorization breaking does
not appear until 4 loop graphs. In the dilute limit considered above, or in the logic of the LLA, however, this would have been completely missed.
It is therefore difficult to discuss the validity of factorization at one loop order, or in a
``dilute" approximation in the sense described in section \ref{sec:cgcpowercount}. In that case for
example proton-proton collisions become rather trivial but the real situation is
far more complicated than that, as should be obvious from our discussion in section \ref{sec:hardscatfact}.
\subsubsection{Causality and factorization}
\begin{figure}[t]
\begin{center}
\includegraphics[angle=0, scale=0.45]{spacetime}
\end{center}
\caption{\label{spacetime} Space-time illustration of the scattering of two hadrons $H_1$ and $H_2$. In the
classical solutions of CGC, one has independent solutions in the non-casually-connected regions
$R_1$ and $R_2$. The solutions in the forward light-cone region $R_+$ are however non trivial and
give rise to the so-called Glasma \protect \cite{Lappi:2006fp}.}
\end{figure}
An argument given for the validity of \eqref{twohadronav} is based partly on causality (see
for example \cite{Gelis:2008rw}), namely that two fast moving hadrons as shown in figure \ref{spacetime}
cannot interact with each other prior to the collision. This by itself, however, does not imply that there
must be a factorized structure for the observable under study. In covariant gauge, it is true that the hadrons
cannot interact prior to the collision, and they are therefore causally disconnected before the collision. To write a factorization
formula, however, one must be able to factorize the soft emissions which can occur at late times after
the collision. Even though the hadrons are casually disconnected prior to the scattering, the scattering
might produce color entangled states which break factorization (see section \ref{sec:tmdgluon} below).
Moreover, the causality argument does not hold in ``physical gauges", such as the Coulomb gauge
or the axial gauge, where manifest Lorentz invariance is broken and faster-than-light propagation is
possible in individual graphs. The causality violating contributions should cancel in the final, physical
results, but the proofs can be very non-trivial. It was in fact early reported that \cite{Bodwin:1981fv, Bodwin:1988fs}
the faster-than-light interactions in the physical gauges would correlate the hadrons prior to the collision and
break factorization in hadron-hadron collisions such as in the Drell-Yan process.
Factorization, both collinear and TMD, in fact holds in Drell-Yan \cite{qcdbook}. The problematic gluons are
precisely the Glauber (Coulomb) gluons which complicate the proofs. However, in covariant gauge one can
consistently deform the integration contours away from the Glauber region and restore factorization. Whether
this can be done for more complicated interactions is of course the real question. We discuss this more in
section \ref{sec:gluonprod} below. What is clear, however, is that the proof of factorization is much more intricate
than what general causality arguments would suggest.
\subsection{Hybrid formalisms}
\label{sec:hybrid}
Some of the applications of the CGC model falls into a category that we shall call the ``hybrid formalisms", since they combine the CGC
treatment above with that of collinear hard scattering factorization (see e.g. \cite{Gelis:2002ki, Gelis:2002fw, Gelis:2002nn, Dumitru:2005gt, Dumitru:2005kb, Gelis:2006hy}).
These formalisms are used especially in proton-nucleus ($pA$) collisions.
Typical examples include photon production, Drell-Yan, and soft particle production in
the forward region (all in $pA$ collisions). As we shall show here, however, these formalisms do not address
the question whether there is factorization for the given process, and the validity of the proposal
to mix collinear factorization with the CGC treatment is not at all clear to us.
\begin{figure}[t]
\begin{center}
\includegraphics[angle=0, scale=0.6]{promptphoton}
\end{center}
\caption{\label{promptphoton} Examples of processes considered in the hybrid formalisms. Left: Photon
production in quark-Nucleus scattering. Right: Quark production in quark-Nucleus scattering. }
\end{figure}
We illustrate in figure \ref{promptphoton} two examples of the processes considered in this framework.
The upper incoming line refers to a quark of momentum $p$ while the lower thick line with momentum $P$
refers to the nucleus. The proton is therefore not treated explicitly. Only interactions between the active
quark and the nucleus are considered as indicated in the figure. The gluon attachments between the
lower and the upper blobs is described by a Wilson line exactly as in \eqref{Wilsonfund}.
Consider the quark production case. The incoming quark is here on-shell and has zero transverse momentum
\cite{Gelis:2002nn, Dumitru:2005gt}. Thus the transverse momentum of the ``observed" final state quark is determined by the
momentum transferred from the nucleus. This dependence is then given directly by the Fourier transform
of the Wilson line \eqref{Wilsonfund},
\begin{eqnarray}
\hat{W}(k_\perp) = \int d^2 x_\perp e^{-ik_\perp \cdot x_\perp} W(x_\perp).
\label{wilsonkt}
\end{eqnarray}
There are two possibilities, that the observed particle has low transverse momentum,
of order of the typical intrinsic transverse momentum, \emph{i.e.} $k_\perp \sim m$, or that
it has large transverse momentum, of the order of a hard scale $Q$.
The cases in figure \ref{promptphoton} suggest that the particle is produced at low
transverse momentum, since the $k_\perp$ dependence is directly
determined by \eqref{wilsonkt}. In that case, however, there is no
reason to neglect the transverse momentum from the proton side, as this could completely change the kinematics of the observed
final state particle. One must therefore formulate a TMD factorization formula, with the TMD parton distribution
and fragmentation function
of the proton taken into account.
If on the other hand the produced particle has large transverse momentum, then a hard region
must properly be included in the process. This, however, is not the case in figure \ref{promptphoton}.
The central idea of the hybrid formalisms is based on what is called the ``factorization of mass
singularities". Here an emphasis is put on the mass divergences that appear in massless on-shell
partonic reactions \cite{pinkbook}. This procedure is in fact widely found in the literature when dealing with
collinear factorization. Despite its wide use, however, it is a physically misleading procedure.
It is in fact a rather different approach than the factorization explained in section \ref{sec:hardscatfact}
above.
In this approach it is first \emph{asserted} that a hadronic cross section $\sigma_h$, or a structure
function $W_{h}$, is a convolution of the corresponding partonic cross section $\sigma_p$, or structure function
$W_p$, and a so-called ``bare parton density'', $f^{\rm bare}$:
\begin{eqnarray}
W_{h} (q,P) = W_p(q,\xi P) \otimes f^{\rm bare}(\xi).
\label{massfact1}
\end{eqnarray}
The convolution in the variable $\xi$ is here the same as in equations \eqref{simplequarkfact} and \eqref{simplegluonfact}.
In the appendix of \cite{Dumitru:2005gt} (see also \cite{Chirilli:2011km}) it is for example asserted for single
inclusive hadron production that the differential cross section is given by
\begin{eqnarray}
d \sigma_h(p,p_h;P) = f^{\rm bare}(\xi) \otimes D^{\rm bare}(z) \otimes d \sigma_p(z,\xi; P)
\label{hybridfact}
\end{eqnarray}
where $p$, $p_h$ and $P$ are the momenta of the incoming proton, the produced hadron and the
incoming nucleus respectively. For the forward particle production shown in figure \ref{promptphoton}
(right graph), the incoming quark has momentum $\xi \, p$ while the outgoing quark has momentum
$p_h/z$ and it subsequently fragments to produce the observed hadron $p_h$.
In both cases, the calculations are then
performed with massless partons and with the parton entering the scattering taken to be on-shell with zero transverse momentum. With these assumptions, collinear divergences appear in the partonic cross sections.
It has been shown in the case of \eqref{massfact1}
that the result for $W_p$ can be written as a convolution of a divergent factor, $D$ (not to be confused
with the fragmentation function),
and a finite cross section $\hat{\sigma}$ \cite{Ellis:1978ty, Curci:1980uw}.
Using the associativity of the convolution operation $\otimes$, one can then write
\begin{eqnarray}
W_h = (\hat{\sigma} \otimes D) \otimes f^{\rm bare} = \hat{\sigma} \otimes (
D \otimes f^{\rm bare}) = \hat{\sigma} \otimes f^{\rm ren}
\label{massfact2}
\end{eqnarray}
where the ``renormalized'' parton distribution is given by $f^{\rm ren} =
D \otimes f^{\rm bare}$. This final result is actually just like that in \eqref{pdfrenorm}.
The just outlined procedure is, however, problematic for several reasons.
To begin with, there is no proof for the assertion \eqref{massfact1} or \eqref{hybridfact}, which
actually \emph{is} the statement of factorization. In the hybrid
formalisms, it is simply stated that the proton side can be treated by integrated distributions. It is also in this case
not exactly clear what the ``bare parton density" is. According to the set up of the formalism,
it is supposed to represent a distribution
of on-shell and massless partons in the proton. This, however, is physically an ill-defined concept since
quarks and gluons never exist as on-shell particles inside real hadrons. Moreover, if quark masses are retained
in the calculations, there are no collinear
divergences. It is therefore dangerous to emphasize the importance of the mass divergences
since they appear only due to the approximation of using massless on-shell partons, and are therefore of
a spurious nature. The
``regularization" procedure just above is therefore conceptually different than \eqref{pdfrenorm}, and crucially, it
is not in any way related to factorization even if this might seem to be implied.
In the analysis of section \ref{sec:hardscatfact} what factorization means is that a given cross section
or structure function can be written in a factorized form where each factor is associated with a
given momentum region. For example,
in the case of DIS it means that we can factorize the hadronic tensor as
\begin{eqnarray}
W \sim \int_x^1 \frac{dz}{z} C^{(0)}_j(Q/\mu, z/x, \epsilon) \, f_j^{(0)}(z;\mu,\epsilon)
\label{baredisfact}
\end{eqnarray}
up to power-suppressed corrections. We can also write this simply as
\begin{eqnarray}
W \sim C^{(0)}_j \otimes f_j^{(0)}.
\end{eqnarray}
The meaning of the bare parton distribution is then that it is the gauge invariant
integrated or TMD parton distribution constructed out of the bare fields of the Lagrangian. An example
is the light-cone gauge definition of the bare integrated gluon distribution in \eqref{intgluonpdf}. In fact
any gauge invariant definition of a parton distribution involving suitable Wilson lines, as for example in the WW
distribution \eqref{WWdistrbadj} or the dipole distribution \eqref{dipgluedistrb}, must refer to the bare distribution, because
the gauge transformation properties are obeyed by the gauge links constructed out of the bare fields. So
strictly speaking we should have denoted all those distributions as in \eqref{intgluonpdf} and
\eqref{baredisfact}, \emph{i.e} by a
superscript $f^{(0)}$. It is important, however, to realize that this bare distribution, constructed out of the bare
fields, cannot be the same as the undefined quantities in \eqref{massfact1} and \eqref{hybridfact}.
For it is clear that it does not represent any distribution of on-shell, massless partons as is implied by
\eqref{massfact1} and \eqref{hybridfact}.
Once factorization has been proved as in \eqref{baredisfact} (or in \eqref{tmdepem}, \eqref{sidis}
and \eqref{drellyan}), which itself is a very non-trivial statement, then renormalization is a matter of removing UV divergences by a suitable redefinition of the parameters of the
Lagrangian. Order by order in perturbation theory this means adding the necessary counter terms from
the Lagrangian, for example in the $\overline{\mathrm{MS}}$ scheme. One then finds the renormalized parton
distribution via a formula as in \eqref{pdfrenorm}. For \eqref{baredisfact} we find that
\begin{eqnarray}
W \sim C^{(0)}_j \otimes f_j^{(0)} &=& C^{(0)}_{j'} \otimes \delta_{j'j} \otimes f_j^{(0)} \nonumber \\
&=& C^{(0)}_{j'} \otimes (Z^{-1} \otimes Z)_{j'j}\otimes f_j^{(0)} \nonumber \\
&=& (C^{(0)}_{j'} \otimes Z^{-1}_{j'j''}) \otimes (Z_{j'' j} \otimes f_j^{(0)} ) \nonumber \\
&=& C_{j} \otimes f_j
\label{renormdisfact}
\end{eqnarray}
where $f_j$ is the renormalized distribution given by \eqref{pdfrenorm}, and the Kronecker delta in the
first line also includes delta functions with respect to the momentum convolutions. This procedure still
applies if the quark masses are retained in which case there are no collinear divergences at all.
Now, in the factorization approach, one can indeed approximate the momentum entering the hard scattering
factor as massless and on-shell. It is crucial, however, that the hard scattering factor, $C$ in \eqref{renormdisfact},
is defined
with suitable subtractions (as we indicated in \eqref{subtractedfact} and showed in figures \ref{dissubtracted}
and \ref{dissubtracted2}) so that it genuinely describes a wide angle scattering with scale $Q$ (we also note that the UV divergences of the subtraction terms are regulated by $Z^{-1}$ in \eqref{renormdisfact}). In the TMD
factorization in section \ref{sec:TMD} for example, the errors in neglecting the transverse momenta, $q_\perp$, in the hard factor goes as $q_\perp/Q$ which indeed is small in the validity region of the formalism.
In \eqref{massfact1} and \eqref{hybridfact}, however, this is no longer the case (in particular in \eqref{hybridfact}
the partonic part still contains the scattering off the nucleus).
Moreover for particle production at low transverse momentum, the
neglected transverse momentum, from the proton side, is of the same order as the transverse momentum of the final state particle which means that the error is substantial.
What is also non-trivial is that TMD factorization is mixed into the formalism of the factorization of mass
singularities.
If in fact we want to treat the given problem using TMD distributions, then in the small-$x$ case where
the produced particle is typically soft, one must consider off-shell matrix elements, precisely because
of the reason just explained above. The off-shell matrix elements must then carefully be specified, to ensure gauge
invariance (or rather gauge-independence), and one cannot use on-shell incoming partons.
For the lowest order contributions, gauge independent off-shell scattering coefficients have been calculated
in the CCH approach \cite{Catani:1990xk, Catani:1990eg}, and an explicit all order definition in the case
of BFKL is given in \cite{Collins:1991ty}. See also \cite{Hautmann:2008vd, Deak:2009xt, Deak:2011gj, Ermolaev:2011aa}
for more recent considerations.
To summarize this section, the hybrid formalisms do not really address the question of factorization.
Factorization is in a sense assumed from the start, via equation \eqref{massfact1} or \eqref{hybridfact}.
In fact the real problem is to show a factorization like in \eqref{baredisfact} to start with.
Moreover, the procedure which is referred to as the renormalization of the parton densities is conceptually
very different from what is the case in the hard scattering factorization. It is moreover physically a misleading
procedure since the basic structures are not well-defined.
Additionally we have seen that for particles produced at low transverse momentum,
TMD distributions must be used also from the proton side, but then of course one must first formulate a valid
TMD factorization formula first, which might not be possible. We will in the coming sections analyze single particle production in the small-$x$ region.
\section{The fundamentals of single inclusive particle production}
\label{sec:gluonprod}
We will now give a comprehensive analysis of single inclusive particle production in high energy
QCD, explaining many details which are usually overlooked. We will start by going through the
basics of particle production, giving an overview of the leading regions in different kinematical
situations. We then
go on to analyze single inclusive gluon production in hadron-hadron scattering
which is a process that has been widely studied (see e.g. \cite{Gribov:1981kg, Gribov:1983fc, Gribov:1984tu, Kharzeev:2003wz, Gelis:2003vh, Blaizot:2004wu, Marquet:2004xa, Kharzeev:2004if, Gelis:2006yv, Gelis:2006dv, Gelis:2008rw, Gelis:2008sz, Levin:2010dw, Levin:2010zy, Albacete:2010bs, Levin:2011hr, ALbacete:2010ad, Levin:2010br}
and references therein) in the small-$x$ region.
We will first go through the process using the axial gauge which is essentially the gauge on which the
arguments for factorization are based, for example in \cite{Gribov:1981kg, Gribov:1983fc, Gribov:1984tu, Kovchegov:1998bi}.
We will in detail explain the technical difficulties of the axial gauge, and why after all it is not convenient for proving factorization. We will then discuss hadron production from a more complete point of view, by building upon the analysis of the leading regions for the different kinematical cases. Finally we shall address the exact form of the TMD gluon
distribution associated with this process, finishing with a discussion of the validity of factorization.
\subsection{The different cases of particle production}
\label{sec:diffcases}
\begin{figure}[t]
\begin{center}
\includegraphics[angle=0, scale=0.55]{softhadronprod}
\end{center}
\caption{\label{softhadronprod} Production of soft hadrons in the small-$x$ limit. The observed hadron $p$
is associated with the soft region. }
\end{figure}
In figures \ref{softhadronprod}, \ref{collhadronprod} and \ref{hardhadronprod} we list the possible
scenarios for single inclusive particle production at small-$x$. In this section we explain the physics
of the different cases.
Figure \ref{softhadronprod} represents a typical scenario of particle production in the Regge region,
namely that of a soft particle produced at a typical small angle scattering event. In this case there is
no hard region. All virtualities are of the typical soft scale $m^2$. The momentum $p$ of the produced particle therefore
typically scales as $|p^\mu| \sim m$. This case is relevant for soft particle production at mid-rapidity.
The inclusive charged particle spectrum at mid-rapidity,
\begin{eqnarray}
\left . \frac{dN_{ch}}{d\eta} \right \vert_{\eta=0},
\end{eqnarray}
has been measured by the different experimental groups at the LHC; ATLAS \cite{:2010ir}, CMS \cite{Khachatryan:2010nk}
and ALICE \cite{Aamodt:2010ft, Aamodt:2010pp}. This also happens to be the mostly studied case in the applications of small-$x$ physics \cite{Kharzeev:2003wz, Kharzeev:2004if, Levin:2010dw, Levin:2010zy, Albacete:2010bs, Levin:2011hr, ALbacete:2010ad, Levin:2010br, Grinyuk:2012mc}.
\begin{figure}[t]
\begin{center}
\includegraphics[angle=0, scale=0.55]{collhadronprod}
\end{center}
\caption{\label{collhadronprod} Production of hadrons in the fragmentation region of particle $B$ in the small-$x$ limit. The observed hadron $p$ has rapidity close to that of $B$. A similar graph exists for production in the opposite
direction close to $A$. These cases require the use of fracture functions rather than ordinary parton distributions
and fragmentation functions. }
\end{figure}
Next, in figure \ref{collhadronprod} we show particle production in the case where the produced particle
is close in rapidity to one of the hadron beams. This case therefore covers the forward production of particles.
At the LHC, the CMS detector can detect particles in the pseudorapidity range $|\eta| < 5$ thanks to the
hadronic forward calorimeters. Since the particles traveling in the forward region have enormous
longitudinal momentum, they must of course have high $p_\perp$ as well, since otherwise they would
have too large rapidity and escape detection via the beam pipes.
In CMS for example \cite{Chatrchyan:2012gw} forward \emph{jets} (not hadrons) in the rapidity range
$3.2 < |\eta| < 4.7$ have $p_\perp \geq 35$ GeV. One can also arrange for events where a hard di-jet
is produced at central rapidity, to accompany the forward jet. The correlations between the forward
jet and the central jets then offer important insight into the parton kinematics, see e.g. \cite{Deak:2010gk, Deak:2011gj,
Deak:2011ga}. Actually if the momentum of the produced hadron belongs to either $C_A$ or $C_B$, then one has to use so-called fracture functions
rather than ordinary fragmentation functions or parton distributions.
\begin{figure}[t]
\begin{center}
\includegraphics[angle=0, scale=0.55]{hardhadronprod}
\end{center}
\caption{\label{hardhadronprod} Production of hadron in the presence of a hard factor. The soft region
coupling the collinear regions is also present, and additional collinear factors emerging from the hard
scattering may be present as well, but for simplicity we do not show these here. }
\end{figure}
Finally in figure \ref{hardhadronprod} we show the case where the hadron is produced with large
rapidity separation to both beams (for example in the central region) and where a hard region is present.
This could for example be the case where the components $p^\mu$ are typically of order $Q \gg m$
or where we are looking at an event where a hard collision is present, that is jets with large $p_\perp$
are produced in addition to the particle we tag (we do not show these additional jets in figure \ref{hardhadronprod}).
The region decomposition here needs some explanation.
In section \ref{sec:powercount} we classified momenta according to different possible scalings.
The external scales in that case were set by $Q$ which also happens to be the hard momentum scale
in the process. Therefore such a classification is appropriate when the components of the hard momenta scale with the
longitudinal momenta of the external particles. The decomposition is thus appropriate when $x$ is not too small.
In that case we noticed that the only real possibilities for a pinch of a given momentum $k^\mu$
were as follows:
\begin{itemize}
\item{ None of $k^\mu$ scales with $Q$. Then we can characterize $k^\mu$ by the typical soft scale $m$,
\emph{i.e.} $k^\mu \sim m$. Then $k \in S$.}
\item{A longitudinal component, say $k^+$ or $k^-$ scales with $Q$. Then we have
$k^+ \sim Q$, $k^-\sim m^2/Q$, $k_\perp \sim m$ and vice versa. In this case
$k \in C_A$ (or $k \in C_B$ in opposite case).}
\item{$k_\perp \sim Q$ in which case also $k^+k^- \sim Q^2$. Thus $k^\mu \sim Q$ and in that
case $k\in H$.}
\end{itemize}
Using this classification we then saw that a power counting analysis gives that at leading
power, $C_A$ and $C_B$ can be connected to $S$ via arbitrarily many soft longitudinally
polarized gluons, while again arbitrarily many collinear gluons can be exchanged between
$H$ and the respective collinear region.
In the small-$x$ case we have a different situation. In this case the large components of the
external particles scale with $\sqrt{s}$ but the momentum transfer remains fixed as $\sqrt{s} \to \infty$.
Thus in this case there is no region in which all momentum components scale with the asymptotic
parameter $\sqrt{s}$. In the soft production case one has the possibilities that
\begin{itemize}
\item{ None of $k^\mu$ scales with $\sqrt{s}$. Then generally $k \in S$.}
\item{$k^+$ or $k^-$ scales with $\sqrt{s}$. In this case $k \in C_A$ or $k \in C_B$ respectively.}
\end{itemize}
There may, however, also be present hard collisions which give rise to jets or hadrons
of several tens of GeV. Thus we may very well have regions where $k^\mu \sim Q$. We then propose the following classification
\begin{itemize}
\item{If $k^+$ or $k^-$ scales with $\sqrt{s}$, then just as above we let $k \in C_A$ or $k \in C_B$
respectively. }
\item{ Let $|k^\mu|/\sqrt{s} \to 0$ as $\sqrt{s} \to \infty$, but such that for example $|k^+|/|k^-| \gg 1$
and $|k^+|/|k^i| \gg 1$. Then even though $k^+ \ll \sqrt{s}$, we shall let $k \in C_A$. In the opposite
case we of course let $k \in C_B$. To characterize such cases we shall let $k^{+} \sim Q \ll \sqrt{s}$
(or $k^{-} \sim Q \ll \sqrt{s}$) where $Q \gg m$.}
\item{ We define the region where $k^+k^- \sim Q^2$ to be the hard region. Thus in figure \ref{hardhadronprod}
there is momentum $k^- \sim Q$ flowing into $H$ from $C_B$, and momentum $k^+ \sim Q$ flowing
in from $C_A$. The momenta going out from $H$ to the final state is then characterized by the scale $Q$. }
\item { Momenta such that $|k^\mu| \sim m \ll Q$ are as before classified as soft. In figure \ref{hardhadronprod}
we do not explicitly draw the soft subgraph to keep the notation simple. }
\end{itemize}
With this classification we can then understand figure \ref{hardhadronprod}. Notice that the momentum
lines whose large components scale with $\sqrt{s}$, and therefore belong to one of the collinear regions,
cannot join the collinear region to the hard region $H$, since in that case a large momentum $\sqrt{s}$
would be transferred to $H$, and we would no longer be in the small-$x$ region. Thus in figure \ref{hardhadronprod}
the lines joining $C_{A,B}$ to $H$ belong to the second class above. This is a different situation then
in section \ref{sec:powercount} where any line in $C_{A,B}$ can join that region to $H$.
We shall now argue that the
power counting is essentially the same as in section \ref{sec:powercount}, despite the somewhat different
kinematics.
\begin{figure}[t]
\begin{center}
\includegraphics[angle=0, scale=0.55]{2gluonexch1}
\end{center}
\caption{\label{2gluontohard} Coupling of two gluons from $C_A$ to $H$.}
\end{figure}
In figure \ref{2gluontohard} we show an example where two gluons from $C_A$ couple to $H$ as defined above.
These two gluons have $k^+ \sim Q$, and in the lower end (not shown in the figure) they couple to collinear-to-$A$
gluons which may have momenta scaling as $\sqrt{s}$ in the plus direction. The leading contribution is then
given by
\begin{eqnarray}
\int d^4k_1 \int d^4 k \frac{\slashed{p}+\slashed{k}_1}{(p+k_1)^2}
\gamma^-\frac{\slashed{p}+\slashed{k}_1-\slashed{k}}{(p+k_1-k)^2}\gamma^-\frac{\slashed{p}}{p^2}\,
\frac{1}{k_1^2}\, \frac{1}{k^2} \, A^{++}(k_1,k,p_A)
\end{eqnarray}
where $p\in H$. We then write this expression as
\begin{eqnarray}
\int d^4k_1 \int d^4 k \frac{\slashed{p}}{2p^-k_1^+}\gamma^-\frac{\slashed{p}}{2p^-(k_1^+-k^+)}
\gamma^-\frac{\slashed{p}}{p^2}\frac{1}{k_1^2}\, \frac{1}{k^2} \, A^{++}(k_1,k,p_A).
\end{eqnarray}
Now, as in section \ref{sec:powercount} we characterize the momentum coupling $C_A$ to $H$
by a scale $\lambda_A$, such that $k^- \sim \lambda_A^2/Q$ and $k_\perp \sim \lambda_A$. When
the momentum $k$ in figure \ref{2gluontohard} couples to $A^{++}(k_1,k,p_A)$, there will be a
typical contribution of
\begin{eqnarray}
\frac{\sqrt{s}}{(p_A+k)^2} \sim \frac{\sqrt{s}}{p_A^+k^-} \sim \frac{\sqrt{s}}{\sqrt{s}\lambda_A^2/Q}
= \frac{Q}{\lambda_A^2}.
\label{Reggecount}
\end{eqnarray}
The factor $\sqrt{s}$ in the numerator comes from the large boost of $A$ in the + direction. Remember
that in the case covered in section \ref{sec:powercount} we have
\begin{eqnarray}
\frac{Q}{(p_A+k)^2} \sim \frac{Q}{Q\lambda_A^2/Q} = \frac{Q}{\lambda_A^2}.
\label{hardcount}
\end{eqnarray}
As we see \eqref{Reggecount} agrees with \eqref{hardcount}. We therefore essentially have the same situation
as before, that is arbitrarily many longitudinally polarized gluons of the second type in the classification above
can connect the collinear regions to $H$ in figure \ref{hardhadronprod}. Indeed the contribution from figure \ref{2gluontohard}
gives
\begin{eqnarray}
\left [\int d^4k_1 \frac{\slashed{p}+\slashed{k}_1}{(p+k_1)^2}\gamma^-\frac{\slashed{p}}{p^2}\frac{1}{k_1^2} A^+\right ]
\int d\lambda_A \, \lambda_A^3 \, \frac{1}{Q}\,\frac{1}{\lambda_A^2}\, \frac{Q}{\lambda_A^2}.
\end{eqnarray}
The term in the brackets corresponds to the contribution from gluon $k_1$ only. The factor outside
therefore gives the contribution from attaching the additional gluon $k$ and we see that it
gives a logarithmic contribution
\begin{eqnarray}
\int \frac{d\lambda_A}{\lambda_A}
\end{eqnarray}
so that there is no power suppression for coupling the extra gluon $k$ to $H$.
To ensure the validity of all these arguments it is again important that one can perform contour deformations
out of the Glauber region. We will in the next sections give a careful analysis of the
factorization arguments that are based on the use of axial gauge, and we will show the difficulties associated
with such arguments. We will continue the general discussion of single particle production in section \ref{sec:singlehadron}
below. Before that, however, we want in the coming sections to concentrate on the small-$x$ single inclusive gluon production
cross section that has been widely used for phenomenological applications.
\subsection{The small-$x$ formula for gluon production}
\label{sec:smallxgluon}
The most basic process for gluon production is depicted in figure \ref{glrgluonprod}
where the idea is that two gluons, $k_A$ and $k_B$, each belonging to one of the incoming hadrons, fuse to produce
a gluon of momentum $l$ which then emerges in the final state. The argument for the validity of figure \ref{glrgluonprod}
is based on the use of axial gauge. The situation is similar to that in figure \ref{nongaugeDIS} where the use of the light-cone gauge
eliminates all higher order gluon exchanges to leading power.
\begin{figure}[t]
\begin{center}
\includegraphics[angle=0, scale=0.55]{GLRgluonprod}
\end{center}
\caption{\label{glrgluonprod} Single inclusive gluon production in hadron-hadron scattering
according to equation \protect \eqref{GLRfact}. }
\end{figure}
The factorization formula being used is given by \cite{Gribov:1984tu, Kovchegov:1998bi, Kharzeev:2003wz, Kharzeev:2004if, Levin:2010dw}
\begin{eqnarray}
\frac{d\sigma}{d^2l_\perp dy} = \frac{2\alpha_s}{C_F \, l_\perp^2} \int d^2k_\perp \,
f_A(y,k_{A,\perp})\, f_B(Y\!-\!y, k_{B,\perp})
\label{GLRfact}
\end{eqnarray}
where
\begin{eqnarray}
k_A \equiv k, \,\,\,\, k_B \equiv l-k,
\end{eqnarray}
$y$ is the rapidity of the produced gluon with respect to the right moving hadron $p_A$. The functions
$f_A$ and $f_B$ represent the respective TMD gluon distributions, and we shortly write
down the definitions used.
The origin of equation \eqref{GLRfact} goes back to the GLR papers \cite{Gribov:1981kg, Gribov:1983fc, Gribov:1984tu}
where the function $f$ is ``defined'' as the derivative of the integrated gluon distribution (which is called the ``gluon structure function'' in \cite{Gribov:1981kg, Gribov:1983fc, Gribov:1984tu})
\begin{eqnarray}
f(y,k_\perp) = \frac{\partial xG(x,k_\perp)}{\partial k_\perp^2}, \,\,\,\, y = \ln 1/x .
\label{intvsunintglr}
\end{eqnarray}
We note that this relation (or rather the inverted integral version of it) is a direct application of the parton model result \eqref{intvsunintpm}, although in the parton model the integral over the unintegrated distribution is over all $k_\perp$.
There are several good reasons for why one should be very cautious with the naive application of the parton model result.
We will discuss this more in \cite{ourpaper}, and see also the comments just after equation \eqref{firstorderKTgluon} below.
As for the validity the factorization formula \eqref{GLRfact}, it is in the literature common to cite the works \cite{Kovchegov:1998bi, Kovchegov:2001sc}.
Reference \cite{Kovchegov:2001sc} makes use of the dipole formalism in studying the deep inelastic scattering
on a large nucleus, where the nucleus is taken to be described by the classical MV model. In this case
the ``unintegrated gluon distribution'' is taken to be
\begin{eqnarray}
f(k_\perp;y) = \frac{N_c}{(2\pi)^4\alpha_s}\int d^2r_\perp \int d^2 b_\perp e^{-ir_\perp \cdot k_\perp} \nabla_r^2\,
\mathcal{N}_G(r_\perp, b_\perp;y),
\label{KTgluon}
\end{eqnarray}
where $\mathcal{N}_G$ has the same meaning as $\mathcal{N}$ in \eqref{dipN} but is instead written in the
adjoint representation as
\begin{eqnarray}
\mathcal{N}_G(r_\perp, b_\perp;y) \equiv 1 - \frac{1}{N_c^2-1}
\left\langle \mathrm{Tr} \{ \tilde{W}(b_\perp \!\!+\! r_\perp/2)\tilde{W}^\dagger (b_\perp\!\!-\!r_\perp/2)
\}\right\rangle_y.
\label{dipadj}
\end{eqnarray}
The Wilson line $\tilde{W}$ has the same form as in \eqref{Wilsonfund}, but with
the replacement $t_F^a \to T^a$ where $T^a$ is the adjoint color matrix.
As can be seen we have indicated the dependence on
the rapidity variable $y$ a bit differently in \eqref{KTgluon} than in \eqref{GLRfact}. We have
in fact done this in purpose and it should later on be clear why we have done so. Notice
for now that \eqref{KTgluon} is essentially the dipole distribution \eqref{dipgluedistrb}, with
the difference that it is here written using Wilson lines in the adjoint representation.
It is important to note, however, that \eqref{KTgluon} is not directly derived from the
formalism in \cite{Kovchegov:2001sc}. Its form is rather asserted by the \emph{assumption} that the dipole formalism
used in \cite{Kovchegov:2001sc} is equivalent to the factorization formula \eqref{GLRfact}.
The results of \cite{Kovchegov:2001sc} are in turn partly based on \cite{Kovchegov:1998bi} where the light-cone gauge is employed and
it is argued that the leading regions have the structure shown in the figure \ref{glrgluonprod}. We also note that
a similar factorized formula is found in the classical DDT paper \cite{Dokshitzer:1978hw} from the early days of QCD.
We will therefore now go through the light-cone gauge calculation. First, however, we need to specify the kinematics more carefully.
\subsubsection{The kinematics}
We denote as usual the incoming momenta by $p_A$ and $p_B$. In the cms frame in the limit of very high energy
and neglecting the masses one has
\begin{eqnarray}
p_A &=& (\sqrt{s/2},0,0_\perp) \\
p_B &=& (0, \sqrt{s/2}, 0_\perp)
\end{eqnarray}
so that $s=2p_A \cdot p_B=2p_A^+p_B^-$.
\begin{figure}[t]
\begin{center}
\includegraphics[angle=0, scale=0.65]{kminuscontour}
\end{center}
\caption{\label{kmcontour} Poles in the plane of $k^-$ and possible integration contour. }
\end{figure}
We now ask which of the cases in section \ref{sec:diffcases} above that is relevant here for figure \ref{glrgluonprod}.
From figure \ref{glrgluonprod} we see that there will be a typical contribution of the type
\begin{eqnarray}
\frac{\mathrm{Numerator}}{(k_A^2+i\epsilon)(k_B^2+i\epsilon)((p_A-k_A)^2+i\epsilon)((p_B-k_B)^2+i\epsilon)}
\times (\mathrm{Rest}\,\, \mathrm{of} \,\, \mathrm{graph} ).
\end{eqnarray}
Let us now consider the $k_A$ part, and the integral over $k_A^-$. We note that if $k_A^+ < 0$ or $k_A^+ > p_A^+$
then the poles in the $k_A^-$ plane are either both in the upper or in the lower half plane respectively. In those
cases we can deform away from the poles simultaneously and we get a power suppressed contribution. Thus we have
$0 < k_A^+ < p_A^+$. In this case the pole from the $k_A$ propagator is in the lower half plane, while the pole
from the lower blob is in the upper half plane and the integration contour is therefore trapped. We show the
pole structure in figure \ref{kmcontour}.
We here simply denote the order of the magnitude of the poles, setting
$k_\perp \sim m$.
If we denote the two poles in $k_A^-$ by $k^-_1$ and $k^-_2$ we see that the distance
between them satisfies
\begin{eqnarray}
|k_1^--k_2^-| &=& \frac{k_{A,\perp}^2}{2}\left ( \frac{1}{k_A^+}+\frac{1}{p_A^+-k_A^+} \right ) \nonumber \\
&\sim & \frac{k_\perp^2}{2k_A^+}.
\end{eqnarray}
Thus when $k_\perp\to 0$ (and all masses in the theory are neglected) we get an exact pinch. As
$k_A^+ \to \sqrt{s}$ we also see that the poles are increasingly pinched and there is potentially a large
contribution (from the collinear PSS). This, however, corresponds to the non-Regge region and is
therefore not relevant for us.
Now, we can let
the integration contour pass near the $p_A-k_A$ pole in which case $|k_A^-| \sim m^2/\sqrt{s}$ (if actually
the lower blob consists of a single spectator line then this pole becomes exact because there will be
a delta function setting the spectator line on-shell).
We might, however, also ask what happens if there is a hard region as in figure \ref{hardhadronprod}.
Assume for example that $l_\perp \sim Q$.
As described in section \ref{sec:diffcases}, we must then have $k^+ \sim Q$ and $(l-k)^- \sim Q$
(we now use that $k_A=k$ and $k_B=l-k$). Then
\begin{eqnarray}
k^+ \sim Q, \,\, k^- \sim Q^2/\sqrt{s},
\end{eqnarray}
and thus
\begin{eqnarray}
k^+k^- \sim \frac{Q}{\sqrt{s}}Q^2 \ll Q^2 \sim k_\perp^2.
\end{eqnarray}
The last estimate comes from $k_\perp \sim |l_\perp-k_\perp| \sim l_\perp \sim Q$. This, however
implies $k^2 \sim Q^2$, which means that $k$ is actually not in the collinear-to-$A$ PSS. It instead belongs to $H$
and one can see that this case is suppressed. To get a leading contribution we want $k^2 \sim m^2$,
and similarly $(l-k)^2 \sim m^2$, but then it is easy to see that we cannot have $l_\perp^2 \sim Q^2$.
Thus for the graph shown in figure \ref{glrgluonprod} we do not have the situation in figure \ref{hardhadronprod}.
To have a situation with a hard region like in figure \ref{hardhadronprod} we must instead consider an
additional collinear, unobserved, jet that emerges from $H$. This, however, makes the situation rather
complicated and changes the physics involved quite a bit. We shall briefly come back to this case in
the discussions in sections \ref{sec:singlehadron} and \ref{sec:tmdgluon} below.
For analyzing the small-$x$ formula \eqref{GLRfact} we consider the situation where $k^+ \sim m$.
That is we essentially have the soft (or perhaps semi-hard) case in figure \ref{softhadronprod}.
A similar analysis as in above for the $l-k$ line implies that in this case
\begin{eqnarray}
|k^-| \!\! \sim m^2/\sqrt{s}, \,\,\,\,\, |l^+\!\!\!-\!k^+| \sim m^2/\sqrt{s}.
\label{gluonprodkin1}
\end{eqnarray}
Therefore
\begin{eqnarray}
|k| \sim (m, m^2/\sqrt{s},m),
\label{ksmallx}
\end{eqnarray}
so that
\begin{eqnarray}
k^+ = l^+ + \mathcal{O}(m^2/\sqrt{s}), \,\,\,\,\, l^- \gg |k^-|.
\label{kandlrelation}
\end{eqnarray}
Thus
\begin{eqnarray}
k^+ \!\!\sim k^i, \,\,\,\,\, (l^-\!\!\!-\!k^-)\sim l^i-k^i,
\end{eqnarray}
and
\begin{eqnarray}
|k^+k^-| &\sim& m^3/\sqrt{s} \ll m^2 \sim k_\perp^2 \\
|(l^+ \!\!\!- \!k^+)(l^-\!\!\!-\!k^-)| &\sim& m^3/\sqrt{s} \ll m^2 \sim (l_\perp\!\!-\!k_\perp)^2.
\label{gluonprodkin2}
\end{eqnarray}
Both gluons $k$ and $l-k$ are therefore in the Glauber region where the transverse momentum
components dominate. In light of what we have said earlier it would seem that we better avoid
the Glauber region. Note, however, that there is no Glauber pinch here so we can deform out of the
Glauber region if necessary.
\subsection{The use of the light-cone gauge}
\label{sec:lcgauge}
The main argument for the validity of \eqref{GLRfact} given in
\cite{Kovchegov:1998bi} is based on the use of the light-cone gauge. Since an axial gauge is also used
in \cite{Gribov:1984tu} to argue for the validity of \eqref{GLRfact}, we now go in through the derivation in these
gauges. We shall start with the light-cone gauge in this section and then in the next section give an account
based on the non-light-like axial gauge. We notice that axial or light-cone gauge is also used in establishing
the factorization formulas in the CCH \cite{Catani:1990eg} and CCFM \cite{Catani:1989sg} formalisms.
There is in fact problem with the kinematical arguments given above in the light-cone gauge.
If we choose the gauge $A^+=0$ then
the treatment of the $A$ part is as we just described. However, we do get a problem of the treatment
of the $B$ side. Similarly we do get a problem of the treatment of the $A$ side if we work in $A^-=0$
gauge. In fact the latter is the gauge on which the arguments in \cite{Kovchegov:1998bi}
are based. What we want to demonstrate in this section is that the
light-cone gauge is clearly improper for the treatment of hadron-hadron collisions (be it proton-proton,
proton-nucleus or nucleus-nucleus collisions). We will offer
several reasons for this, and we return to the just mentioned issue at the end of this section. We
will now simply push forward with the light-cone gauge and then see that it leads to severe problems.
Let us now denote the gluon propagators by
\begin{eqnarray}
P^{\mu\nu}(k) = \frac{-iN^{\mu\nu}(k)}{k^2+i\epsilon}.
\end{eqnarray}
Then in the light-cone gauge $n\cdot A=0$ we have
\begin{eqnarray}
N^{\mu\nu}(k) = g^{\mu\nu} - \frac{n^\mu k^\nu}{k\cdot n} - \frac{n^\nu k^\mu}{k\cdot n}.
\label{LCprop}
\end{eqnarray}
We shall write $N^{\mu\nu}(k)$ as
\begin{eqnarray}
N^{\mu\nu}(k) = \overrightharp{G}^{\mu\nu}(k) - \overleftharp{K}^{\mu\nu}(k),
\label{KMKG}
\end{eqnarray}
where
\begin{eqnarray}
\overrightharp{G}^{\mu\nu}(k) &\equiv& g^{\mu\nu} - \frac{n^\mu k^\nu}{k\cdot n } \\
\overleftharp{K}^{\mu\nu}(k) &\equiv& \frac{ k^\mu n^\nu}{k\cdot n}.
\end{eqnarray}
Our notation here is inspired by the so-called $K$-$G$ decomposition introduced by Grammer and Yennie \cite{Grammer:1973db}.
The directions of the harpoons indicate whether it is the left or the right Lorentz index that is carried by
the momentum $k$; $\overrightharp{G}^{\mu\nu}(k)$ (and $\overrightharp{K}^{\mu\nu}$)
contains $k^\nu$, while $ \overleftharp{K}^{\mu\nu}(k)$ (and $\overleftharp{G}^{\mu\nu}$)
contains $k^\mu$. Notice that the standard Grammer-Yennie decomposition which is applied to the
Feynman gauge propagators is in this notation given by
\begin{eqnarray}
N_{Feyn}^{\mu\nu} = g^{\mu\nu} = \overleftharp{G}^{\mu\nu}(k) + \overleftharp{K}^{\mu\nu}(k)
= \left ( g^{\mu\nu} - \frac{k^\mu n^\nu}{k\cdot n } \right ) + \frac{k^\mu n^\nu}{k\cdot n }.
\label{kgdecomp}
\end{eqnarray}
The $K$-$G$ decomposition is important in proving factorization in the hard
scattering domain since Ward identities can be applied to the $K$ terms which are the dominant
contributions. Remember from the
analysis in section \ref{sec:hardscatfact} that there can be arbitrarily many longitudinally polarized
gluons exchanged between the hard and collinear regions, $H$ and $C$, and between the collinear
and the soft regions, $C$ and $S$. These gluons precisely correspond to the $K$ terms.
If we choose $n$ such that $n\cdot A = A^+$, then for the $G$ terms we have
\begin{eqnarray}
\overrightharp{G}^{-+}(k) = g^{-+} \!\!- \frac{k^+}{k^+} = 0, \,\,\, \overleftharp{G}^{+-}(k) = g^{+-} \!\!- \frac{k^+}{k^+} = 0,
\label{Gplusminus}
\end{eqnarray}
while for the $K$ terms
\begin{eqnarray}
\overrightharp{K}^{-+}(k) = \frac{k^+}{k^+} = 1, \,\,\, \overleftharp{K}^{+-}(k)= \frac{k^+}{k^+} = 1.
\label{Kplusminus}
\end{eqnarray}
For the dominant polarization $N^{-+}$ we therefore see that only the $K$ terms contribute.
The key step to proving factorization is then to repeatedly apply the Ward identities on the $K$ terms.
If, however,
$k$ is dominated by its transverse component, then one can no longer neglect the transverse
$G$ contributions to which the Ward identities do not apply. If for example we have momentum which scales
as $l-k$ in the above example, then
\begin{eqnarray}
|\overrightharp{G}^{-i}(l-k)| = \left \vert \frac{(l-k)^i}{(l-k)^+} \right \vert \gg 1, \,\,\, |\overleftharp{K}^{i-}(l-k)|=
\left \vert \frac{(l-k)^i}{(l-k)^+}\right \vert \gg 1.
\end{eqnarray}
This means that the transverse components cannot be neglected in favor of the $+-$ components.
Moreover, even for the $K$ terms,
the application of the Ward identities leave non-factorizing remainder terms which are complicated.
These can be neglected in the collinear limit but not in the Glauber region. Therefore in all the higher order
corrections to figure \ref{glrgluonprod} we must be able make all necessary contour deformations so as
to power suppress these contributions.
\begin{figure}[t]
\begin{center}
\includegraphics[angle=0, scale=0.55]{glrgluonprod2}
\end{center}
\caption{\label{glrgluonprod2} The graphical representation of formula \protect \eqref{GLRfact}. }
\end{figure}
In the axial gauge, the singular propagators must be regularized. A canonical regularization is obtained
by treating the singularities as principal values.
Now, in \cite{Kovchegov:1998bi} the regularization is instead performed by choosing
\begin{eqnarray}
N^{\mu\nu}(k) = g^{\mu\nu} - \frac{n^\mu k^\nu}{k\cdot n - i\epsilon} - \frac{n^\nu k^\mu}{k\cdot n +i\epsilon}.
\label{KMprop}
\end{eqnarray}
Here the momentum flows from $\mu$ towards $\nu$. The vector $n$ is now chosen so that $n \cdot A = A^-$.
There is, however, a fundamental problem with this gauge, and it shows up already for the lowest
order contribution in figure \ref{glrgluonprod2}. It is related to the fact that the light-cone gauge does not
treat the hadrons symmetrically. We now demonstrate this problem by calculating the contribution in
figure \ref{glrgluonprod2}.
The polarization vector of the produced gluon $l$ is chosen in \cite{Kovchegov:1998bi} to satisfy $\epsilon^+(l) = 0$.
Since $l \cdot \epsilon(l) = 0$ one has $\epsilon^-(l) = l^i\epsilon^i/l^+$.
The contribution from the process depicted in figure \ref{glrgluonprod2} is given by
\begin{eqnarray}
-g_s\epsilon^*_\beta(l)U_a^\rho(p_B,l-k) (\overrightharp{G}_{\rho\gamma}(l-k) - \overleftharp{K}_{\rho\gamma}(l-k)
) V_{abc}^{\gamma\alpha\beta}(\overrightharp{G}_{\sigma\alpha}(k) - \overleftharp{K}_{\sigma\alpha}(k))L_b^\sigma(p_A,k)
\nonumber \\
\label{graphone}
\end{eqnarray}
where $V$ is the three-gluon vertex. The dominant component of the lower part is $L^+ \propto \sqrt{s}$,
while the dominant component of the upper part is $U^- \propto \sqrt{s}$. We notice that in the above expression,
\begin{eqnarray}
U_a^\rho(p_B,l-k) \overleftharp{K}_{\rho\gamma}(l-k) = L_b^\sigma(p_A,k)\overleftharp{K}_{\sigma\alpha}(k)
= 0
\end{eqnarray}
by the use of the Ward identity.
One is then left with
\begin{eqnarray}
-g_s\epsilon^*_\beta(l)U_a^\rho(p_B,l-k) \, \overrightharp{G}_{\rho\gamma}(l-k) \, V_{abc}^{\gamma\alpha\beta}\, \overrightharp{G}_{\sigma\alpha}(k)\, L_b^\sigma(p_A,k).
\label{graphtwo}
\end{eqnarray}
It is easily seen that the leading contributions are
\begin{eqnarray}
\overrightharp{G}^{\sigma\alpha}(k) \, L_{b,\sigma}(p_A,k) \approx \overrightharp{G}^{-\alpha}(k) \, L_{b}^+(p_A,k) =
g^{-\alpha}L_b^+(p_A,k)
\label{Lleading}
\end{eqnarray}
and
\begin{eqnarray}
U_{a,\rho}(p_B,l-k)\, \overrightharp{G}^{\rho\gamma}(l-k) \approx U_a^-(p_B,l-k)\, \overrightharp{G}^{+\gamma}(l-k).
\end{eqnarray}
For the $\gamma$ index, the $\gamma=-$ component gives zero because of the gauge $A^-=0$, while
\begin{eqnarray}
|\overrightharp{G}^{++}(l-k)| = \frac{|l^+-k^+|}{|l^--k^-|} \ll \frac{|l^i-k^i|}{|l^--k^-|} = |\overrightharp{G}^{+i}(l-k)|
\end{eqnarray}
so the leading term comes from $\gamma=i$. From \eqref{Lleading}, taking also into account the contribution
from the complex conjugate amplitude, we see that we have for the lower part (we neglect the color indices for the moment)
\begin{eqnarray}
L^{+ \dagger}(p_A,k) L^+(p_A,k)\! &=&\! \sum_X \int d^4x \, e^{ik\cdot x} \langle p_A | A^+(x)
|X,\mathrm{out}\rangle \langle X, \mathrm{out}| A^+(0) |p_A \rangle \nonumber \\
\! &=&\! \sum_X \frac{1}{(k^-)^2}\int d^4x \, e^{ik\cdot x} \langle p_A | (k\cdot A(x)+k^iA^i(x)) |X,\mathrm{out}\rangle
\nonumber \\
&& \hspace{35mm} \langle X, \mathrm{out}| (k\cdot A(0)+k^iA^i(0)) |p_A \rangle
\nonumber \\
\! &=&\! \sum_X \frac{1}{(k^-)^2}\int d^4x \, e^{ik\cdot x} \langle p_A | k^iA^i(x)|X,\mathrm{out}\rangle \langle X, \mathrm{out}| k^iA^i(0) |p_A\rangle \nonumber \\
\label{Lpdfdef}
\end{eqnarray}
where in the second equality we used the fact that $A^-=0$, while in the last equality we used the Ward identity.
For the upper part we instead have for the leading term
\begin{eqnarray}
\left (U^-\, \overrightharp{G}^{+i}\right )^\dagger \left (U^-\, \overrightharp{G}^{+i}\right )= \int d^4x \, e^{i(l-k)\cdot x}
\langle p_B | A^i(x) A^i(0) |p_B \rangle.
\label{Updfdef}
\end{eqnarray}
In the gauge $A^-=0$, the canonical definition of the TMD gluon distribution is (which directly corresponds
to the parton model definition \eqref{eq:pdf.lf.def})
\begin{eqnarray}
&k^-& \!\!\!\!\int \frac{dk^+}{(2\pi)^4} \int d^4x \, e^{ik\cdot x} \langle p_B| A^{i}(x)A^{i}(0)|p_B\rangle \nonumber \\
&=& \int \frac{dk^+}{(2\pi)^4}\frac{1}{k^-} \int d^4x \, e^{ik\cdot x} \langle p_B| F^{-i}(x)F^{-i}(0)|p_B\rangle.
\label{pdfcanonical}
\end{eqnarray}
This is for example also
the case for the Weizsacker-Williams distribution in the CGC, with the only trivial difference being that in
that case the pre-factor in the first line above is taken to be $(k^-)^2/p_B^-=\bar{x}k^-$ instead of $k^-$
(with $\bar{x} = k^-/p^-_B$).
We notice, however, that the so-called dipole gluon distribution cannot be really fully consistent
with \eqref{pdfcanonical}. The reason is that for the dipole
gluon distribution, in the corresponding derivation one must actually set $k^-$ to 0
(this is why the Wilson lines \eqref{Wilsonfund} are integrated from $-\infty$ to $+\infty$ in the
longitudinal direction).
One can therefore not multiply the definition with $k^-$ as above,
in order to obtain the canonical form \eqref{pdfcanonical}. In that case one may instead
multiply the integral by $p_B^-$.
While it is straightforward to put \eqref{Updfdef} into the proper form, this is not so with the
lower component \eqref{Lpdfdef}. Going now back to the evaluation of
the graph in figure \ref{glrgluonprod2} we thus have
\begin{eqnarray}
&&g_s\epsilon^{*\beta}(l) U_a^-(l-k)L_b^+(k) \frac{(l-k)_\perp^\gamma}{l^-\!\!-k^-}g^{-\alpha}V_{\gamma\alpha\beta}^{abc} \nonumber \\
&=& -g_s U_a^-(l-k)L_b^+(k)f^{abc}\frac{1}{l^-\!\!-k^-} [ -\epsilon^{*-}(l_\perp^2\!-\!k_\perp^2)-\epsilon^{*i} (l^i\!-\!k^i)
(k^-\!\!\!-\!2l^-) ] \nonumber \\
&\approx & - g_sU_a^-(l-k)L_b^+(k)f^{abc}\left [- \frac{\epsilon^{*i}l^i}{l^-l^+}(l_\perp^2\!-\!k_\perp^2)+ 2\epsilon^{*i} (l^i\!-\!k^i) \right ]
\end{eqnarray}
where $k^-$ has been neglected with respect to $l^-$. Using $2l^+l^-=l_\perp^2$
one then gets
\begin{eqnarray}
g_s\frac{2U_a^-(l-k)L_b^+(k)f^{abc}}{l_\perp^2} ( \epsilon^{*i}l^i (l_\perp^2\!-\!k_\perp^2) - \epsilon^{*i} (l^i\!-\!k^i)l_\perp^2).
\end{eqnarray}
Squaring and summing over polarization and color indices, and integrating over $k$, we have
\begin{eqnarray}
\frac{g_s^2}{N_c^2-1}\!\!\sum_{aa'bb'c} \int \frac{d^4k}{(2\pi)^4} 4(U_a^-U_{a'}^{-\dagger})(L_b^+L_{b'}^{+\dagger})f^{abc}f^{a'b'c} \,
\frac{k_\perp^2(l_\perp-k_\perp)^2}{l_\perp^2}.
\label{glueprodres}
\end{eqnarray}
Now, to write a factorization formula for this result we have to untangle the color flow and at the same
time make the appropriate kinematical approximations. Using \eqref{kandlrelation}, we now neglect $k^-$ with respect
to $l^-$ in the $U$ factors, and we set $k^+=l^+$ in the $L$ factors. The $k^+$ ($k^-$) integral then acts only on the
$U$ ($L$) factors.
For obtaining the differential single inclusive cross section we project the diagonal
color components in $U$ and $L$, and we find that the result can be written as
\begin{eqnarray}
\frac{d\sigma}{dy \, d^2l_\perp}=\frac{1}{2s} \frac{1}{2(2\pi)^3}\frac{4\, g_s^2\, N_c}{N_c^2-1} \frac{(2\pi)^4}{l_\perp^2}\int d^2k_\perp \left [ \int \frac{dk^+}{(2\pi)^4} \sum_a U_a^-U_{a}^{-\dagger} \,
(l_\perp\!\!-\!k_\perp)^2 \right ]_{k^-\!=0} \nonumber \\
\times \left [ \int \frac{dk^-}{(2\pi)^4} \sum_b L_b^+L_{b}^{+\dagger} \, k_\perp^2 \right]_{k^+\!=l^+}.
\label{glueprodres2}
\end{eqnarray}
We notice that up to this point the arguments have followed very closely those in section \ref{sec:simplefact}
that lead to equation \eqref{simplegluonfact}. However, as we discussed after equation \eqref{simplegluonfact},
a more careful treatment is needed since the integration over the momentum will include contributions which
are not strictly in the region where the above kinematics holds. What we saw in equation \eqref{subtractedfact}
was that this could be treated by including subtractions in the hard factor. In this case, we instead need
subtractions in the last factor of \eqref{glueprodres}. In fact one must correctly treat the gluon production
factor, the analog of the hard region, to all orders and make sure it is gauge independent. This, however,
does not affect the definition of the gluon distribution.
Now, for the first bracket
containing the upper blobs in \eqref{glueprodres2} we have from \eqref{Updfdef} (we keep the summation over the color indices
implicit)
\begin{eqnarray}
\frac{1}{p_B^-}\left [ \int \frac{dk^+}{(2\pi)^4} U_a^-U_{a}^{-\dagger} \, (l_\perp\!\!-\!k_\perp)^2 \right ]_{k^-\!\!=0} \!\!\!\!\!
= \frac{(l^-)^2}{p_B^-} \int \frac{dk^+}{(2\pi)^4} \int d^4x \, e^{i(l-k)\cdot x} \langle p_B | A_a^i(x) A_a^i(0) |p_B \rangle \nonumber \\
= \int \frac{dx^+d^2x_\perp}{ (2\pi)^3\, p_B^-}\, e^{il^- x^+\! -(l_\perp\!-k_\perp)\cdot x_\perp} \langle p_B | F_a^{-i}(x^+\!\!,0^-\!\!,x_\perp) F_a^{-i}(0) |p_B \rangle \nonumber \\
\label{Updfdef2}
\end{eqnarray}
where we have chosen to include the factor $1/p_B^-$ into the definition. For the lower blobs, however, we cannot get the standard formula because of the asymmetric
gauge choice $A^-=0$. Using \eqref{Lpdfdef} we have (again keeping summation over color indices
implicit)
\begin{eqnarray}
&&\frac{1}{p_A^+}\left [ \int \frac{dk^-}{(2\pi)^4} L_b^+L_{b}^{+\dagger}\, k_\perp^2 \right]_{k^+=l^+}
\nonumber \\
&=&\sum_X \frac{1}{p_A^+}\int \frac{dk^-}{(2\pi)^4} \frac{k_\perp^2}{(k^-)^2}\int d^4x \, e^{i k x}
\langle p_A | k^iA_b^i(x)|X,\mathrm{out}\rangle \langle X, \mathrm{out}| k^iA_b^i(0) |p_A\rangle\biggl . \biggr \vert_{k^+=l^+}.
\nonumber \\
\end{eqnarray}
This expression is clearly different than \eqref{pdfcanonical} or \eqref{Updfdef2}, and does not correspond to
any know distribution. One therefore does not obtain formula \eqref{GLRfact}.
Let us now explain the other difficulty with the light-cone gauge that we mentioned just above equation
\eqref{graphone}. As we have seen, in $A^-=0$ gauge we have a problem with the definition of the
parton distribution for particle $A$ which moves in the + direction. Similarly if we chose $A^+=0$ gauge,
then we will have a problem with the definition for particle $B$. Let $k_{A,B}$
denote momenta attached between the collinear regions $A, B$ and any other region such as $S$
or $H$. Where $k_A$ attaches to $A$, the collinear lines of $A$ will force $k_A^-$ to generally be
small as in figure \ref{kmcontour}. If we now work in the $A^-=0$ gauge it means we
additionally have the $1/k^-_A$ pole at the origin, and the combined poles from the propagator and
the collinear lines of $A$ will then generally pinch $k_A^-$ at the origin. This, however, means that the higher order terms cannot be deformed out to $k^-_A \sim Q$
to power suppress them (terms for example such as $\overleftharp{G}^{i+}$ will be large). The gauge $A^-=0$ therefore fails for the gluons attaching to $A$. A similar
argument for $B$ shows that the $A^+=0$ gauge similarly is not useful.
\subsection{Non-light-like symmetric axial gauge}
\label{sec:axialgauge}
To get a formula that looks like \eqref{GLRfact} one must instead choose a gauge that treats the two
hadrons symmetrically, this can for example be done by choosing the non light-like axial gauge $A^++A^-=0$,
\emph{i.e.} the temporal gauge $A^0=0$. Using this gauge, one can again eliminate the extra gluon
couplings to the collinear regions.
We will here use this gauge to
derive \eqref{GLRfact} and at the same time we will see what the definition of the TMD gluon distribution is.
However, in
section \ref{sec:singlehadron} we will explain the general case, and demonstrate the problem that is inherent
in this axial gauge treatment as well.
In the gauge $A^++A^-=0$, the numerator of the gluon propagator is given by
\begin{eqnarray}
N^{\mu\nu}(k) = g^{\mu\nu} - \frac{n^\mu k^\nu+n^\nu k^\mu}{n\cdot k} + \frac{k^\mu k^\nu n^2}{(n\cdot k)^2}
\label{temporalprop}
\end{eqnarray}
where $n \cdot k = k^++k^-$ for any $k$. The contribution in figure \ref{glrgluonprod2} gives
\begin{eqnarray}
-g_s\epsilon^*_\beta(l)U_a^\rho(p_B,l-k) N_{\rho\gamma}(l-k)
V_{abc}^{\gamma\alpha\beta}N_{\sigma\alpha}(k)L_b^\sigma(p_A,k).
\end{eqnarray}
The last term proportional to $n^2$ in \eqref{temporalprop} then cancels in both propagators above
when the Ward identity is applied on $U$ and $L$. One is then left with the same expression as in \eqref{graphone}
which again reduces to \eqref{graphtwo} when applying the Ward identity. As before we have that
\begin{eqnarray}
\overrightharp{G}^{\sigma\alpha}(k) \, L_{b,\sigma}(p_A,k) \approx \overrightharp{G}^{-\alpha}(k) \, L_{b}^+(p_A,k)
\end{eqnarray}
and
\begin{eqnarray}
U_{a,\rho}(p_B,l-k)\, \overrightharp{G}^{\rho\gamma}(l-k) \approx U_a^-(p_B,l-k)\, \overrightharp{G}^{+\gamma}(l-k),
\end{eqnarray}
but in this case the leading $G$ terms are different. We have
\begin{eqnarray}
&&\left \vert \overrightharp{G}^{++}(l-k) \right \vert = \left \vert \frac{l^+-k^+}{l^+-k^++l^--k^-} \right \vert
\sim \left \vert \frac{l^+-k^+}{l^-} \right \vert \sim \frac{m}{\sqrt{s}} \ll 1 \\
&& \left \vert \overrightharp{G}^{+-}(l-k) \right \vert = \left \vert 1 - \frac{l^--k^-}{l^+-k^++l^--k^-} \right \vert
\sim \left \vert \frac{k^-}{l^-} \right \vert \sim \frac{m}{\sqrt{s}}\ll 1 \\
&&\left \vert \overrightharp{G}^{+i}(l-k) \right \vert = \left \vert \frac{l^i-k^i}{l^+-k^++l^--k^-} \right \vert
\sim \left \vert \frac{l^i-k^i}{l^-} \right \vert \sim \frac{m}{m} = 1,
\end{eqnarray}
and
\begin{eqnarray}
&&\left \vert \overrightharp{G}^{--}(k) \right \vert = \left \vert \frac{k^-}{k^++k^-} \right \vert
\sim \left \vert \frac{k^-}{k^+} \right \vert \sim \frac{m}{\sqrt{s}}\ll 1 \\
&& \left \vert \overrightharp{G}^{-+}(k) \right \vert = \left \vert 1 - \frac{k^+}{k^++k^-} \right \vert
\sim \left \vert \frac{k^-}{k^+} \right \vert \sim \frac{m}{\sqrt{s}}\ll 1 \\
&&\left \vert \overrightharp{G}^{-i}(k) \right \vert = \left \vert \frac{k^i}{k^++k^-} \right \vert
\sim \left \vert \frac{k^i}{k^+} \right \vert \sim \frac{m}{m} = 1.
\end{eqnarray}
The leading contributions are therefore the transverse components in both sides.
Squaring the contribution from figure \ref{glrgluonprod2} and summing over gluon polarizations
one is then left with (we neglect for simplicity the color factors since they are exactly the same
as in the light-cone gauge calculation above)
\begin{eqnarray}
&&g_s^2 (U^-U^{-\dagger})(L^+L^{+\dagger})\frac{1}{(l^+-k^++l^--k^-)^2}\frac{1}{(k^++k^-)^2} \times
\nonumber \\
&&\sum_\lambda \left [ -2(k^+\epsilon_\lambda^-+k^-\epsilon_\lambda^+)(k_\perp^2-l_\perp\cdot k_\perp)
+\epsilon^i_{\lambda} k^i l_\perp^2 -\epsilon^i_{\lambda} l^i (-k_\perp^2
+2l_\perp \cdot k_\perp) \right ]^2.
\end{eqnarray}
We shall next choose the external polarization vector to satisfy $\epsilon^-=0$, which means
that $\epsilon^+ = \epsilon^i l^i/l^-$. Then the first term in the sum above gives
\begin{eqnarray}
-2\frac{k^-}{l^-}\epsilon^i l^i(k_\perp^2-l_\perp\cdot k_\perp)
\end{eqnarray}
which is of the order of a transverse component multiplied by $k^-/l^- \sim m/\sqrt{s} \ll 1$ and can therefore be neglected
compared to the other transverse terms. One then gets
\begin{eqnarray}
g_s^2 (U^-U^{-\dagger})(L^+L^{+\dagger})\frac{1}{(l^+-k^++l^--k^-)^2}\frac{1}{(k^++k^-)^2}
l_\perp^2 k_\perp^2 (l_\perp-k_\perp)^2.
\end{eqnarray}
Inserting now all pre-factors and color indices, we get for the gluon production cross section
\begin{eqnarray}
\frac{d\sigma}{dy \, d^2l_\perp}=\frac{1}{4} \frac{1}{2(2\pi)^3}\!\!\!\!\!&&\!\!\!\!\!\frac{(2\pi)^4\, g_s^2\, N_c}{N_c^2-1}
\label{sigmatemp1}\\
&\times& \!\!\!\int d^4 k \left [ \frac{U_a^-U_{a}^{-\dagger}(l_\perp \!\!-\!k_\perp)^2}{p_A^- (2\pi)^4} \right ]
\left [ \frac{L_b^+L_{b}^{+\dagger}k_\perp^2}{p_B^+ (2\pi)^4} \right ] \frac{l_\perp^2}{(l^+\!\!-k^+\!\!+l^-\!\!-k^-)^2
(k^+\!\!+k^-)^2}. \nonumber
\end{eqnarray}
To define the TMD gluon distribution we now notice that
\begin{eqnarray}
U_a^-U_{a}^{-\dagger}(l_\perp \!\!-\!k_\perp)^2 &=& (l^+\!\!-k^+\!\!+l^-\!\!-k^-)^2
\int d^4x e^{i(l-k)\cdot x} \langle p_A| A^i_a(x)A^i_a(0)|p_A\rangle \nonumber \\
&=& 2\int d^4x e^{i(l-k)\cdot x} \langle p_B|F_a^{0i}(x)F_a^{0i}(0)|p_B\rangle,
\label{nonlighttmdup}
\end{eqnarray}
where $F^{0i}=(1/\sqrt{2})(F^{+i}+F^{-i})$. Similarly
\begin{eqnarray}
L_b^+L_{b}^{+\dagger}k_\perp^2 = 2\int d^4x e^{ik\cdot x} \langle p_A|F_b^{0i}(x)F_b^{0i}(0)|p_A\rangle.
\label{nonlighttmddown}
\end{eqnarray}
To obtain the canonical forms of the two gluon distributions, we notice that we can drop the $F^{-i}$ contribution
in \eqref{nonlighttmddown}, since it gives rise to the contributions $k^-L^i$, $k^iL^-$ and $L^{i-}$ which
are all power-suppressed. Therefore we might as well replace $F^{0i}$ by $F^{+i}/\sqrt{2}$. Similarly for
the expression in \eqref{nonlighttmdup} we can replace $F^{0i}$ by $F^{-i}/\sqrt{2}$.
To get the factorization formula, one further needs to approximate $k^-=0$ in the upper part,
and $k^+=l^+$ lower part. Furthermore we applied the approximations from the kinematics in
\eqref{gluonprodkin1}-\eqref{gluonprodkin2} in the last factor in \eqref{sigmatemp1}
which can then be written as (up to power-suppressed corrections)
\begin{eqnarray}
\frac{l_\perp^2}{(l^+\!\!-k^+\!\!+l^-\!\!-k^-)^2(k^+\!\!+k^-)^2} \sim \frac{l_\perp^2}{(l^-)^2(l^+)^2} =
\frac{4}{l_\perp^2}.
\end{eqnarray}
Thus we find
\begin{eqnarray}
\frac{d\sigma}{dy \, d^2l_\perp}&=& \frac{2\pi^2 \, \alpha_s}{C_F \, l_\perp^2} \int d^2k_\perp
\left [ \int \frac{dk^+}{(2\pi)^4}\frac{1}{p_B^-} U_a^-U_{a}^{-\dagger}(l_\perp \!\!-\!k_\perp)^2\right ]
\left [ \int \frac{dk^-}{(2\pi)^4}\frac{1}{p_A^+} L_b^+L_{b}^{+\dagger} k_\perp^2\right ] \nonumber \\
&=& \frac{2\pi^2 \, \alpha_s}{C_F \, l_\perp^2} \int d^2k_\perp f_B(x_B,l_\perp-k_\perp) f_A(x_A,k_\perp),
\label{particleprodfact}
\end{eqnarray}
with
\begin{eqnarray}
f_A(x_A, k_\perp) = \int \frac{dx^-d^2x_\perp}{(2\pi)^3\, p_A^+}e^{ix_Ap_A^+ x^- - i k_\perp x_\perp}
\langle p_A|F_a^{+i}(0^+\!\!,x^-\!\!,x_\perp)F_a^{+i}(0)|p_A\rangle,
\label{temporalpdf1}
\end{eqnarray}
and
\begin{eqnarray}
f_B(x_B,l_\perp-k_\perp) = \int \frac{dx^+d^2x_\perp}{(2\pi)^3\, p_B^-}e^{ix_Bp_B^-x^+ - i(l-k)_\perp x_\perp}
\langle p_B|F_a^{-i}(x^+\!\!,0^-\!\!,x_\perp)F_a^{-i}(0)|p_B\rangle,
\label{temporalpdf2}
\end{eqnarray}
where $x_A=l^+/p_A^+$ and $x_B=l^-/p_B^-$.
\subsubsection{The coefficient of the formula}
As for the coefficient in front of formula \eqref{particleprodfact}, we note that different
values appear in the literature. Let us denote the coefficient in \eqref{particleprodfact} by
\begin{eqnarray}
C= \frac{2\pi^2 \, \alpha_s}{C_F}= \frac{4\pi^2 \, N_c \, \alpha_s}{N_c^2-1}.
\label{Ccoeff}
\end{eqnarray}
In the papers \cite{Kovchegov:2001sc, Kharzeev:2004if, Levin:2010dw, Levin:2010zy, Levin:2011hr, Levin:2010br}
we instead find the formula (this is the value we used in writing \eqref{GLRfact})
\begin{eqnarray}
C = \frac{2\alpha_s}{C_F} = \frac{4\, N_c \, \alpha_s}{N_c^2-1},
\label{ckt}
\end{eqnarray}
while in \cite{Gribov:1984tu} we find,
\begin{eqnarray}
C = \frac{N_c \, \alpha_s}{(2\pi)^6},
\label{cglr}
\end{eqnarray}
and in \cite{Gribov:1983fc}
\begin{eqnarray}
C = 2\pi N_c \alpha_s.
\label{cglr2}
\end{eqnarray}
Similarly we find in \cite{Albacete:2010bs}
\begin{eqnarray}
C = \frac{(2\pi)^8\, C_F\, \alpha_s^3}{\pi N_c^2},
\label{cam}
\end{eqnarray}
and in \cite{ALbacete:2010ad}
\begin{eqnarray}
C = \frac{2 \pi^2 \, K\,C_F\, \alpha_s}{N_c^2} = \frac{\pi^2 \, K\,(N_c^2-1) \alpha_s}{N_c^3}
\label{cad}
\end{eqnarray}
where $K$ is a fit parameter which is quoted to be of the numerical value 1.5-2. We see that the coefficients in \eqref{ckt}, \eqref{cglr}, \eqref{cglr2},
\eqref{cam} and \eqref{cad} are all different from each other. It appears also that none agrees with the result above,
equation \eqref{Ccoeff}. Our result \eqref{Ccoeff} on the other hand agrees with the result in \cite{Schafer:2012yx}
where it was indeed observed that an extra factor $\pi$ for each TMD distribution must be included to agree with
\eqref{ckt} above.
The numerical differences between the pre factors used in different papers are clearly rather
important. It should also further be noted that in the papers \cite{Kharzeev:2003wz, ALbacete:2010ad}
the $k_\perp$ integration is performed only up to $l_\perp$ while such a bound does not appear
in the other papers. Moreover in most of the phenomenological applications the coupling $\alpha_s$ is taken
to run with some scale which also differs from paper to paper.
\subsection{Higher order terms in axial gauge, and more complete view}
\label{sec:singlehadron}
From the contribution in figure \ref{glrgluonprod2} we have thus seen that we can in the non-light-like
axial gauge, $A^++A^-=0$, obtain the formula \eqref{GLRfact} where the TMD distributions are given by
\eqref{temporalpdf1} and \eqref{temporalpdf2}. We notice that exactly the same gauge is used in the
CCH formalism \cite{Catani:1990eg} and in the GLR paper \cite{Gribov:1984tu}.
The question is of course what happens when we include higher order corrections to figure \ref{glrgluonprod2}.
We will now in this section first prove that the axial gauge does indeed eliminate to leading
power the couplings to the collinear regions, and at the same time we will see what kinematics is necessary
for this result to hold. We will show that the kinematics is actually opposed to the usual small-$x$ kinematics.
Thus for the higher order corrections to be generally negligible we will need contour deformations to ensure
the desired kinematics. We shall then give an argument for why the needed contour deformations generally fail
in the axial gauge.
Assume now that we have a collinear region $C$ which carries momentum lines that are large in some direction $w_C$. For example this could be region $C_A$ which has large momentum in the $+$ direction. Let $\tilde{w}_C$
be the conjugate direction to $w_C$, such that $w_C \cdot \tilde{w}_C=1$. The large component of $C^\mu$ is
then given by $\tilde{w}_C \cdot C$. We now choose the axial gauge $n\cdot A = 0$ where $n$ is not necessarily
light-like. Let $V$ be any vector. We then have
\begin{eqnarray}
V\cdot C = V\cdot w_C \,\, \tilde{w}_C\cdot C + \mathrm{p.s.c.}
\end{eqnarray}
where ``p.s.c." as before stands for ``power suppressed corrections". Now we let $V=n$, and using that
we are in $n\cdot A=0$ gauge, we obtain
\begin{eqnarray}
0 = n\cdot C = n\cdot w_C \,\, \tilde{w}_C\cdot C + \mathrm{p.s.c.}
\end{eqnarray}
Assuming now that $n \cdot w_C \neq 0$, we can separately scale the gauge vector
\begin{eqnarray}
n \to \frac{n}{n\cdot w_C}
\end{eqnarray}
for each collinear region in the graph to get
\begin{eqnarray}
0 = n\cdot C = \tilde{w}_C\cdot C + \mathrm{p.s.c.}
\end{eqnarray}
Thus we conclude that the leading term vanishes in the axial gauge, and only power-suppressed
contributions remain. Notice that if $n \cdot w_C=0$ then we cannot necessarily conclude that the
leading contribution is eliminated. It might also be that, depending on the exact kinematics, several
directions of $C^\mu$ simultaneously become important. In that case the advantage of the axial
gauge vanishes. Let us illustrate these points with some examples.
Consider now a gluon $k$ coupling to region $C_A$, and
denote $\tilde{C}_A^\mu=N^{\mu\nu}(k)C_{A,\nu}$. It is actually then $\tilde{C}_A$ that corresponds to $C$
above (since $n\cdot \tilde{C}_A = 0$ but $n \cdot C_A \neq 0$).
Assume we are in the $A^-=0$ gauge.Then
\begin{eqnarray}
\tilde{C}_A^+ &\sim& N^{+-}C_A^+ \!=\left (1 - \frac{k^-}{k^-} \right ) C_A^+ = 0, \\
\tilde{C}_A^i &\sim& N^{i-}C_A^+ \!= 0, \\
\tilde{C}_A^- &\sim& N^{--} \!\!= 0.
\end{eqnarray}
Therefore only power suppressed contributions from $A$ will remain (we could have also immediately seen this
from the fact that $n\cdot C_A = 0 + \mathrm{p.s.c}$). On the other hand if we choose the gauge
$A^+=0$ then
\begin{eqnarray}
\tilde{C}_A^+ &\sim& N^{+-}C_A^+\!=\left (1 - \frac{k^+}{k^+} \right ) C_A^+ = 0, \\
\tilde{C}_A^i &\sim& N^{i-}C_A^+ \!= -\frac{k^i}{k^+}C_A^+, \\
\tilde{C}_A^- &\sim& N^{--}C_A^+ \!= -\frac{k^-}{k^+}C_A^+.
\end{eqnarray}
Here we see that $ \tilde{C}_A^i$ and $\tilde{C}_A^- $ are suppressed only if $k^+$ is the dominant
component of $k$. If not, then in the higher order terms all contributions can be important and the
situation obviously gets complicated. The gauge $A^+=0$ is useful in DIS where the
target hadron has large $P^+$. In hadron--hadron collisions, however, as we have seen, the
light-cone gauge cannot be used. There is moreover the problem with rapidity divergences
which appear in TMD distributions via integrals like \eqref{rapdiv} (the light-cone distribution \eqref{Updfdef}
for example leads to divergences and is therefore ill-defined). These divergences become visible starting
from one loop calculations. Now assume we are instead in $A^++A^-=0$ gauge. Then
\begin{eqnarray}
\tilde{C}_A^+ &\sim& N^{+-}C_A^+ \!=\left (1 - \frac{k^+\!\!+k^-}{k^+\!\!+k^-} +\frac{k^+k^-n^2}{(k^+\!\!+k^-)^2}\right ) C_A^+
= \frac{k^+k^-n^2}{(k^+\!\!+k^-)^2}C_A^+ , \\
\tilde{C}_A^i &\sim& N^{i-}C_A^+ \!=\left (- \frac{k^i}{k^+\!\!+k^-} +\frac{k^ik^-n^2}{(k^+\!\!+k^-)^2}\right ) C_A^+ , \\
\tilde{C}_A^- &\sim& N^{--}C_A^+ \!= \left ( - \frac{2k^-}{k^+\!\!+k^-} +\frac{(k^-)^2n^2}{(k^+\!\!+k^-)^2}\right ) C_A^+.
\end{eqnarray}
If for example $k$ is collinear to $C_A$, then indeed the contributions are power suppressed.
Thus for the axial gauge to be useful, the momenta emerging from $C_A$ ($C_B$) should be collinear to $C_A$ ($C_B$).
Actually none of the momentum components need to scale with $\sqrt{s}$, but the dominant component
should be $k^+$ (or $k^-$ for $C_B$). Remember indeed from our classification scheme in section \ref{sec:diffcases}
that momenta which have no components scaling with $\sqrt{s}$ but whose components along $C_A$
dominates are still classified as belonging to $C_A$.
If, however, we are in a region where for example $k_\perp$ dominates, then we see that we have a large
contribution from the transverse components. In that case we cannot neglect the higher
order corrections. This is why we must be able to always deform the contour into the region where $k^+$
(or $k^-$ for $C_B$) is the large component.
\begin{figure}[t]
\begin{center}
\includegraphics[angle=0, scale=0.55]{singlehadronprod}
\end{center}
\caption{\label{singlehadronprod} Leading regions for single inclusive hadron
production via gluon initiated jet in hadron-hadron collisions. There is an additional collinear region associated with the produced hadron $p_C$. There will generally also be additional collinear regions associated with unobserved
jets, these are not shown here for simplicity.}
\end{figure}
The analysis above and in section \ref{sec:diffcases} suggests a general
picture like in figure \ref{singlehadronprod}. We consider the case where the observed hadron, $p_C$, has
some component scaling with $Q$, the reason being that the scale $Q$ is needed to suppress the higher
order corrections as seen above.
The regions in figure \ref{singlehadronprod} are to be
understood in the classification presented in section \ref{sec:diffcases}. The momentum $Q$ is fixed
and $Q/\sqrt{s} \to 0$ asymptotically. There are actually further lines going out from the hard region
which give undetected collinear regions but we do not show them in figure \ref{singlehadronprod} for simplicity.
According to what we have just seen above, in axial gauge we generally
expect the contributions in figure \ref{singlehadronprod} to be reduced to that of figure \ref{singlehadronprodLC}.
Here the extra collinear-to-hard gluons are missing, and the remaining gluons coupling to $H$ are
transversely polarized (indicated by black squares).
\begin{figure}[t]
\begin{center}
\includegraphics[angle=0, scale=0.55]{singlehadronprodLC}
\end{center}
\caption{\label{singlehadronprodLC} Single hadron production in axial gauge where the extra collinear-to-hard
can be eliminated. The collinear regions then couple to the hard region via a single transversely polarized gluon,
indicated by the black squares, each. }
\end{figure}
Note from figure \ref{singlehadronprodLC}
that the soft region still remains. Indeed the analysis above does not directly apply to the soft region since
we needed a scale $Q$ to suppress the higher order terms. To simplify the expression completely then, one
must be able to show that the soft region can be eliminated or neglected.
\begin{figure}[t]
\begin{center}
\includegraphics[angle=0, scale=0.55]{singlehadronsoft1}
\includegraphics[angle=0, scale=0.55]{singlehadronsoft2}
\includegraphics[angle=0, scale=0.55]{singlehadronsoft3}
\end{center}
\caption{\label{singlehadronsoft} Examples of graphs in axial gauge where soft gluons are exchanged between the
collinear regions. Each of these type of emissions require contour deformations in different directions to stay
out of the Glauber region. }
\end{figure}
In figure \ref{singlehadronsoft} we show examples of soft gluons exchanged between the different regions.
In the first graph (top left) the gluon $k$ attaches to the collinear-to-$B$ gluon that goes into the hard
scattering. The momentum $k$ then runs in a loop from top to down, counterclockwise, via $H$ into $A$
and back again. The line $k_A-k$ then gives a pole (taking all $k_\perp$ to be of order $m$)
\begin{eqnarray}
k^- \sim \frac{m^2}{k_A^+} - i\epsilon \sim \frac{m^2}{Q} - i\epsilon.
\end{eqnarray}
Inside the lower blob $A$, $k$ will run along the large momentum $p_A$, and so there
will be a typical pole of the type
\begin{eqnarray}
k^- \sim \frac{m^2}{p_A^+} +i \epsilon \sim \frac{m^2}{\sqrt{s}} +i \epsilon.
\end{eqnarray}
We thus see that these poles pinch the integration contour of $k^-$.
It might also be that $k$ in the lower blob attaches to a line with plus component only
of order $Q$ instead of $\sqrt{s}$, but in any case we see that $k^-$ is at least forced to be
small as $m^2/Q$. One can still save the power counting arguments if $k^+$ can be deformed
far out so that $k^+k^- \sim k_\perp^2$.
We must now, however, exactly specify how to treat the singularities of the axial gauge propagator
\eqref{temporalprop}. The canonical regularization of these singularities is given by the
principal value prescription.
The canonical regularization is useful because the corresponding
generalized functions then obey elementary relations, such as ordinary differentiation, that are obeyed by the
corresponding regular functions
\cite{gelfand}.
The use of principal value, however, also implies that one cannot deform the contours. The variable $k\cdot n$
must therefore remain on the real axis.
As we have seen above, for the contributions in figure \ref{singlehadronsoft} we must deform
in the first graph (top left) $k^+$ but not $k^-$, in the second graph (top right) $k^-$ but not $k^+$ while
in the last graph (bottom) we must simultaneously deform $k_1^+$ and $k_2^-$ while keeping
$k_1^-$ and $k_2^+$ fixed. We then, however, see that these requirements are in contradiction
with the fact that we cannot deform on $k\cdot n$. For example, deforming in the first case $k^+$,
\emph{i.e.} letting $k^+ \to k^+ + iC$ for some large $C \sim Q$, but keeping $k^-$ fixed implies
that
\begin{eqnarray}
k\cdot n = k^++k^- \to (k^++iC)+k^- =k\cdot n +iC
\end{eqnarray}
which is not allowed. The required contour deformations therefore fail. We thus conclude
that the treatment in axial gauge is not complete.
One may also consider the possibility of using the so-called ``planar gauge" introduced
in \cite{Dokshitzer:1978hw}. In this gauge, the gauge vector $n$ is non-light-like, so that
$n^2 \neq 0$, but the last term in the axial gauge propagator \eqref{temporalprop} is eliminated
(by a clever choice of the gauge fixing term in the Lagrangian). Moreover, as shown in
\cite{Dokshitzer:1978hw}, Faddeev-Popov ghosts are still absent, just like in axial gauge. This
gauge has thus all the advantages of the axial gauge, and in addition is free from the double
pole in the propagator. It is therefore certainly much better behaved. However, the unphysical
singularity $1/k\cdot n$ still remains and must be treated via the principal value. Therefore the
above arguments still apply to this gauge. In \cite{Dokshitzer:1978hw} the authors argue that,
since the propagator poles are unphysical and have to all cancel at the end of the day, one might
as well treat $1/k\cdot n$ as a regular function, excluding this pole from loop integrals. The problem,
however, is that one still needs to perform the contour deformations to prove factorization, and
in doing this the term $1/k\cdot n$ cannot be neglected in the intermediate steps, even if the
final result should be free from unphysical poles.
It is of course possible that one chooses a regularization which is not principal value. For example,
we saw above that the choice in \cite{Kovchegov:1998bi} for the light-cone gauge is given by \eqref{KMprop}.
In any case, however, it is very hard to see how exactly a systematic procedure is developed that
is capable of treating graphs of arbitrarily high order, as is required for the full proof of factorization.
As far as we aware of, this has never been done. We leave the possibility open that
a treatment in axial gauge might work out, but it is difficult to see how this would be
achieved.
\subsection{The gluon distribution function}
\label{sec:tmdgluon}
We have systematically gone through single inclusive particle production at high energies,
and we have concentrated especially on the small-$x$ factorization formula \eqref{GLRfact}.
In this section we examine more closely
the exact definition of the TMD gluon distribution. We will moreover at the end of the section make some final
comments on the validity of factorization.
According to \eqref{KTgluon}, the gluon distribution is a (modified) Fourier transform of the dipole
scattering amplitude in the adjoint representation. The expression \eqref{KTgluon} is appropriate in
a covariant gauge, and not in an axial gauge. In the canonical definition of the parton distributions,
the direction of the Wilson lines in \eqref{dipadj} are
taken opposite to the hadron, \emph{i.e} for a hadron moving with momentum $p_A$ ($p_B$), the direction
is taken as $n_B$ ($n_A$), which is parallel to $p_B$ ($p_A$). To leading power we can also take the directions to be $n_A+n_B$ for both hadrons, and
the axial gauge $(n_A+n_B)\cdot A=0$ then sets the Wilson lines to unity. At first sight, however, this does not
seem to be strictly
correct because if in \eqref{dipadj} we set the Wilson lines to be unity then we find that \eqref{dipadj}
vanishes, $\mathcal{N}=0$, which obviously cannot be true. Part of the answer is that a fully gauge invariant definition
of \eqref{dipadj} requires that we also insert transverse gauge links at infinity, and these are non-zero
in any axial gauge. However, to match the axial gauge expressions, \eqref{temporalpdf1} and \eqref{temporalpdf2},
one must also express the distribution \eqref{dipadj} using the field tensors $F^{+i}$ and $F^{-i}$.
Let us now see how this can be done.
It is in fact a fundamental property of all gluon distributions that the field tensors $F^{\mu\nu}$ appear in the definitions.
The underlying reason for this comes from the elementary parton model definition \eqref{eq:pdf.lf.def}.
As the QCD definitions are appropriate modifications and generalizations of the parton model result, it
is then natural that the field tensors appear in the definitions of the integrated and TMD gluon distributions
\cite{qcdbook}. This is also the case in the construction scheme for the generalized TMD distributions
given in \cite{Bomhof:2004aw, Bomhof:2006dp}. It should therefore also be possible to write the dipole distribution \eqref{KTgluon} using the field tensors, if it indeed is a TMD gluon distribution as claimed.
Consider the lowest order contribution from \eqref{dipadj} where we insert a set of outgoing states
$|X,\mathrm{out}\rangle$ between the Wilson lines and then expand each Wilson line to first order in $g_s$. We will
\emph{assume} that the averaging in \eqref{dipadj} is given by an ordinary expectation value between
momentum eigenstates of the hadron, but we are not actually sure whether this is consistent with the
formalism from which \eqref{dipadj} is supposed to arise. Nevertheless, without this assumption we cannot
make any real comparison. We also neglect for the moment the regulator $y$ in \eqref{KTgluon} and
\eqref{dipadj}.
The first order expansion of \eqref{dipadj} in \eqref{KTgluon} for a hadron with momentum $p_A$
gives
\begin{eqnarray}
&&f_A^{(1)}(k_\perp) = \frac{N_c}{(2\pi)^4\alpha_s}k_\perp^2 \int d^2x_\perp\!\! \int d^2 y_\perp e^{-ik_\perp \cdot (x_\perp
-y_\perp)} \nonumber \\
&&\sum_X\frac{g_s^2N_c}{N_c^2-1} \int dx^-\! \int dy^- \frac{\langle p_A | A_a^+(x^-,x_\perp) |X,\mathrm{out} \rangle
\langle X,\mathrm{out}| A_a^+(y^-,y_\perp) |p_A\rangle}
{\langle p_A|p_A\rangle}.
\label{f1formula}
\end{eqnarray}
The argument to convert $k^i A_a^+$ into $F^{+i}_a$ can now be made as follows. In the power counting
of the contributions from the region collinear to $p_A$, the largest contribution arises from the + component as we
have seen in sections \ref{sec:powercount} and \ref{sec:diffcases}. In the $N$ gluon exchange term, the biggest contribution
therefore arises from the terms where we pick up the contribution $A^{+\cdots +}$ for all the $N$ collinear-to-$p_A$ gluons.
For every contribution where we change one of the gluon polarizations from the longitudinal index $+$ to a transverse index $i$, we lose one power of $\sqrt{s}$.
Thus one can let
\begin{eqnarray}
k^i A_a^+ \to k^i A_a^+ - k^+ A_a^i
\label{fieldtotensor}
\end{eqnarray}
since the correction produces a power suppressed term. It is important to notice that this exchange
is not permissible in the hard scattering factorization. From the power counting in section \ref{sec:powercount}
we actually see that $k^iA^+ \sim m\,Q$ and $k^+A^i \sim Q\, m$ for a collinear-to-$A$ gluon $k$. In the
small-$x$ case, however, $k^+ \ll \sqrt{s}$, so that $k^+A^i \ll \sqrt{s} \, m \sim k^iA^+$.
For the lowest order term in \eqref{f1formula}
this is enough to convert each $k^i A_a^+$ into $F^{+i}_a$ since the commutator in $F^{+i}$ contributes
at higher order.
Removing the sum over the states $X$, one can then rewrite \eqref{f1formula} as
\begin{eqnarray}
f_A^{(1)}(k_\perp) &=& \frac{N_c}{2\pi \alpha_s}\frac{g_s^2N_c}{N_c^2-1}\int \frac{dx^-d^2x_\perp}{(2\pi)^3 2p_A^+}
e^{-ik_\perp \cdot x_\perp} \langle p_A | F_a^{+i}(0^+\!,x^-\!,x_\perp) F_a^{+i}(0) |p_A\rangle \nonumber \\
&=& \frac{N_c^2}{N_c^2-1} \int \frac{dx^-d^2x_\perp}{(2\pi)^3 p_A^+}
e^{-ik_\perp \cdot x_\perp} \langle p_A | F_a^{+i}(0^+\!,x^-\!,x_\perp) F_a^{+i}(0) |p_A\rangle.
\label{firstorderKTgluon}
\end{eqnarray}
In the dipole model from which \eqref{dipadj} arises, the large $N_c$ limit is
employed which means that the coefficient $N_c^2/(N_c^2-1)$ is set to unity. The result \eqref{firstorderKTgluon}
then very strongly resembles \eqref{temporalpdf1}.
We note, however, that in \eqref{firstorderKTgluon}, there is no $x$ dependence as in \eqref{temporalpdf1}.
This is a characteristics of the dipole formalism where the longitudinal component of the total momentum
coupling to the collinear region is neglected. The rapidity dependence of the dipole distribution therefore
purely arises from the rapidity cut-off.
In \eqref{temporalpdf1}, the rapidity cut-off is
not yet included, and the $x_A$ variable which is the longitudinal momentum fraction of the gluon $k$
in figure \ref{glrgluonprod2} clearly does not play the role of a rapidity cut-off. This is also one of the reasons
why the dipole distribution \eqref{KTgluon} or \eqref{dipgluedistrb} cannot be directly related to the integrated
distribution as in \eqref{intvsunintglr}, since the meanings of the longitudinal variables in \eqref{intvsunintglr}
are completely different on the right and the left hand sides. Despite this, however, the relation
\eqref{intvsunintglr} is still widely advocated in the small-$x$ literature.
When all the gluons coupling to the collinear region contribute with their longitudinal polarizations,
however, there must be certain cancellations due to the Ward identities. In Feynman gauge the easiest
way to see this is to use the $K$-$G$ decomposition \eqref{kgdecomp}. Ward identities apply on the
$K$ terms, and these correspond to the longitudinally polarized gluons. For the region collinear to $p_A$,
we choose the vector $n$ in the $K$-$G$ decomposition \eqref{kgdecomp} to be in the opposite
direction to $p_A$, \emph{i.e.} $n=n_B$ (and the other way around for the $B$ terms). Then
as we saw in \eqref{Gplusminus} and \eqref{Kplusminus}, the longitudinal components vanish
for the $G$ terms while for the $K$ terms we get unity. The largest contribution therefore arises
from the terms where we only pick up the $K$ terms. Ward identities, however, imply that part of this
largest contribution cancel, leaving behind a reminder term which is of the same order as the contributions
where one gluon contributes as $G^{i-}$, while all the other terms contribute via the
$K^{+-}$ terms \cite{qcdbook, Collins:2008sg}. It is then the combination of the $G^{i-}$
term and the remainder term from the Ward identity cancellations that give rise to the field tensor
term $F^{+i}$ (including the commutator term)
while the sum over all the $K^{+-}$ terms give the Wilson lines. We explain this in the context of the small-$x$ calculations
in \cite{ourpaper} where we derive the TMD gluon distribution that looks like \eqref{WWdistrbadj}. That is,
a gluon distribution including the $F^{+i}$ factors is naturally constructed.
Let us now extend the above analysis to all orders.
In \cite{Bomhof:2004aw, Bomhof:2006dp} a construction scheme of TMD parton distributions was proposed.
The proposed scheme is a method of converting the collinear-to-hard gluons to Wilson lines,
thus giving the ``unsubtracted" TMD parton distributions. We now apply the scheme to the present
process.
\begin{figure}[t]
\begin{center}
\includegraphics[angle=0, scale=0.55]{2to1gluonprod}
\end{center}
\caption{\label{2to1gluon} The elementary graph for the gluon production.}
\end{figure}
The scheme starts from studying the elementary
``hard" graph for the process under consideration, that is figure \ref{2to1gluon}.
Of course here this graph does not
involve any hard momenta, but that does not really affect the structure of the Wilson lines which parametrize
the non-perturbative structure. According to \cite{Bomhof:2004aw, Bomhof:2006dp} then, the contribution from the process in
figure \ref{2to1gluon} to the TMD gluon distribution of the lower particle (with momentum $p_A$) is
\begin{eqnarray}
F_{b'}(x)\, F_b(0) \, if^{abc}\, if^{a'b'c'} (W_B^{(+)})_{cc'}(W_B^{(-)})_{aa'}
\label{gaugelinks1}
\end{eqnarray}
where
\begin{eqnarray}
W_B^{(\pm)}= W_B(0;\pm \infty^-\!\!,0_\perp)W_T(\pm \infty^-\!\!, 0_\perp;\pm \infty^-\!\!, x_\perp)W_B(\pm \infty^-\!\!,x_\perp;x^-\!\!, x_\perp),
\label{wilsonpm}
\end{eqnarray}
and
\begin{eqnarray}
\label{wilsonlong}
W_B(x;y) = P \exp \left ( -ig_s \int_x^y d z\, n_B\cdot A_a (z) T^a \right ) , \\
W_T(x,y) = P \exp \left ( -ig_s \int_x^y d z_\perp \cdot A_{\perp,a} (z) T^a \right ).
\label{wilsontrans}
\end{eqnarray}
If we instead consider the TMD distribution of the upper hadron with momentum $p_B$ then
the longitudinal direction in \eqref{wilsonpm} should be $+$ instead of $-$, and in \eqref{wilsonlong}
$n_B \to n_A$. Notice that in \eqref{gaugelinks1} the Wilson lines are in the adjoint representation
as is clear from the color subscripts. We now use $T^b_{ac}=if^{abc}$ for the adjoint representation
to rewrite \eqref{gaugelinks1} as
\begin{eqnarray}
F_{b'}(x)\, F_b(0) \, T^b_{ac}\, T^{b'}_{a'c'}\, (W_B^{(+)})_{cc'}(W_B^{(-)})_{aa'} \nonumber \\
= F_{a'c'}(x)F_{ac}(0) \, (W_B^{(+)})_{cc'}(W_B^{(-)})_{aa'}\label{gaugelinks2}
\end{eqnarray}
where we have defined
\begin{eqnarray}
F_{ac} \equiv F_b \, T^b_{ac}.
\label{adjtensor}
\end{eqnarray}
From equation \eqref{gaugelinks2} one then finds the following contribution to the
correlator in the gluon distribution
\begin{eqnarray}
\langle p_A| \left( F(x)W_B^{(+)\,\dagger} \right )_{a'c}\left ( F(0)W_B^{(-)} \right )_{ca'} |p_A\rangle \nonumber \\
= \mathrm{Tr} \langle p_A| F(x) W_B^{(+)\,\dagger} F(0)W_B^{(-)}|p_A\rangle.
\end{eqnarray}
The trace is taken with respect to the adjoint representation with the field tensor
defined as in \eqref{adjtensor}. The (unsubtracted) gluon distribution function is then given by
\begin{eqnarray}
f_A(x_A,k_\perp) = \! \int \frac{dx^-d^2x_\perp}{(2\pi)^3\,p_A^+}e^{ix_Ap_A^+x^-\!-ik_\perp\cdot x_\perp}
\mathrm{Tr} \langle p_A| F^{+i}(0^+\!\!,x^-\!\!,x_\perp) W_B^{(+)\,\dagger} F^{+i}(0)W_B^{(-)}|p_A\rangle. \nonumber \\
\label{gaugelinkpdf}
\end{eqnarray}
Actually, note that in the canonical definitions \eqref{eq:pdf.lf.def} and \eqref{pdfcanonical} we would
instead of $1/p_A^+$ insert the factor $1/k_A^+=1/(x_Ap_A^+)$. The reason we choose $1/p_A^+$ here
is that we will connect the above distribution with that of the dipole result \eqref{KTgluon} and remember
from above that the dipole result cannot be obtained if we have the factor $1/k_A^+$ (see also remarks
just below).
Strictly speaking \eqref{gaugelinkpdf} involves only the bare fields. Remember from section \ref{sec:simplefact}
that the gluon distribution has to be renormalized as in equation \eqref{pdfrenorm}. The soft region
must also properly be subtracted to cancel the rapidity divergences in \eqref{gaugelinkpdf}.
A similar definition is easily obtained for the gluon distribution associated with $p_B$
\begin{eqnarray}
f_B(x_B,k_\perp)= \! \int \frac{dx^+d^2x_\perp}{(2\pi)^3\,p_B^-}e^{ix_Bp_B^-x^+\!-ik_\perp\cdot x_\perp}
\mathrm{Tr} \langle p_B| F^{-i}(x^+\!\!,0^-\!\!,x_\perp) W_A^{(+)\,\dagger} F^{-i}(0)W_A^{(-)}|p_B\rangle. \nonumber \\
\label{gaugelinkpdfb}
\end{eqnarray}
Exchanging to leading order the Wilson line directions to $n_A+n_B$ in both cases and applying
the axial gauge $(n_A+n_B)\cdot A=0$ we then obtain \eqref{temporalpdf1} and \eqref{temporalpdf2}
respectively. There is an additional factor $N_c$ arising from the color traces in \eqref{gaugelinkpdf}
and \eqref{gaugelinkpdfb} (exactly as in \eqref{firstorderKTgluon}). Thus we can see \eqref{gaugelinkpdf}
and \eqref{gaugelinkpdfb} as possible generalizations of \eqref{temporalpdf1} and \eqref{temporalpdf2}
to arbitrary gauge.
The connection to the dipole formula \eqref{KTgluon} and \eqref{dipadj} can now be made as
follows. We consider the transverse derivatives in \eqref{KTgluon} acting on the Wilson lines in \eqref{dipadj}.
The effect of the derivative can be written as (for the hadron $p_A$)
\begin{eqnarray}
\partial_x^i \tilde{W}(x_\perp) = -i g_s\int dx^- W_B(x;\infty^-\!\!,x_\perp)\partial_x^i A^+_a(x)T^aW_B(-\infty^-\!\!,x_\perp;x)
\label{Wilsondiff}
\end{eqnarray}
where as we recall $\tilde{W}$ is given by taking \eqref{Wilsonfund} with the adjoint color matrices
while $W_B(x;\infty^-\!\!,x_\perp)$ and $W_B(-\infty^-\!\!,x_\perp;x)$ are given by \eqref{wilsonlong}.
We can again use \eqref{fieldtotensor} since the correction is power suppressed. One can also
argue that the commutator of the field tensor is subleading since at given order in $g_s$ it
contains one factor $A^i$ which replaces a factor $A^+$ from the Wilson line. In that case we could replace $ -i\partial_x^i A^+_a(x)T^a \to F^{+i}_aT^a=F^{+i}$
in \eqref{Wilsondiff}. This would imply that \eqref{KTgluon} contains the same structure as
in \eqref{gaugelinkpdf}, once we also set $x=0$ in \eqref{gaugelinkpdf} which as we remember from
above is the standard approximation in the dipole formalism.
Thus as we have seen, in a sense the formula \eqref{KTgluon} together with \eqref{dipadj}
contains the contributions from the gluon field tensors as in \eqref{gaugelinkpdf}. We motivated this
by the power counting arguments, but a word of caution is in order here. We have mentioned above
that the $K$ terms in the $K$-$G$ decomposition are subject to certain cancellations from the Ward identities. This implies actually that terms
containing one factor of $A^i$ at each side of the cut become leading. As explained above, these arise
from the $G^{i-}$ terms. Thus the transverse components
in $F^{+i}$, including the commutator, may not be automatically dropped.
The expression in
\eqref{gaugelinkpdf} is therefore more correct than \eqref{KTgluon}, assuming of course that
factorization holds. If not, then neither expression needs to be correct. Let us therefore now finish
our analysis with a discussion on the validity of factorization.
What we have thus seen is that \eqref{KTgluon} and \eqref{dipadj} can be related to the
distribution, \eqref{gaugelinkpdf} or \eqref{gaugelinkpdfb}, constructed using the scheme of \cite{Bomhof:2004aw, Bomhof:2006dp}. However, the scheme in \cite{Bomhof:2004aw, Bomhof:2006dp} by itself
does not prove whether factorization holds or not. When a TMD parton distribution
associated with a given collinear region is being constructed, one considers the attachments of the collinear-to-hard gluons
to each line of the hard graph, and replace each set of connections by a Wilson line that correctly carries
the color of the hard line. Since TMD factorization is used for two particle production in the almost back-to-back
region, as in the examples of $e^+e^-$ annihilation and Drell-Yan production in section \ref{sec:TMD},
the relevant hard graphs are usually $2 \to 2$ partonic graphs, and one can then use these basic graphs
to construct the possible gauge links for a given collinear region. An extensive list of possible gauge links
is given in \cite{Bomhof:2006dp}.
For proving factorization, however, one must consider all gluon attachments from the collinear regions
to the hard graph \emph{simultaneously}, as well as all possible soft attachments between the collinear regions.
For example, in \eqref{gaugelinks1}, following \cite{Bomhof:2004aw, Bomhof:2006dp}, the attachments from the
collinear regions $C_A$ and $C_B$ in figures \ref{softhadronprod}, \ref{hardhadronprod} or \ref{singlehadronprod},
are considered separately, and each is summed into the Wilson lines in \eqref{gaugelinks1}. Considering all possible attachments, however, as for example in the graphs in figure
\ref{singlehadronsoft}, it may very well be that the resulting structure is more complicated than
in \eqref{gaugelinks1} or that it is not even possible to identify any gauge link contributions to the TMD distributions.
At the same time, one must be able show that deformations out of the Glauber region are possible, or that
the poles producing the Glauber pinch cancel. Cancellation of the Glauber region has been demonstrated
explicitly in the case of Drell-Yan (Ch 14, \cite{qcdbook}), but difficulties may easily arise for the
more complicated processes studied in \cite{Bomhof:2004aw, Bomhof:2006dp}.
\begin{figure}[t]
\begin{center}
\includegraphics[angle=0, scale=0.6]{2to2gentmd}
\end{center}
\caption{\label{2to2gentmd} Production of two hadrons in an elementary model considered in \protect \cite{Collins:2007jp}. We indicate the hard scattering by the exchange of the zig-zag lines. The additional gluon contributions correspond to breakdown
of ordinary factorization. }
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[angle=0, scale=0.55]{4glueexchange}
\end{center}
\caption{\label{4glueexch} Examples of the type of graphs that are taken into account in the construction scheme of
equation \protect \eqref{gaugelinks1}. The solid lines indicate the spectator parts of each hadron. }
\end{figure}
In reference \cite{Collins:2007jp}, the breakdown of ordinary TMD factorization (\emph{i.e.} the TMD
factorization that is relevant for the processes in section \ref{sec:TMD}) was explicitly demonstrated in di-hadron
production in hadron-hadron collisions at the level of 2 gluon exchange between the hard part and the
collinear part. We illustrate in figure \ref{2to2gentmd} two examples of the type of graphs considered in \cite{Collins:2007jp}.
To distinguish the hard scattering we draw the hard gluons by zig-zag lines, while the collinear-to-hard
gluons are illustrated by curly lines. In the elementary model considered in \cite{Collins:2007jp}, the gluons are massive Abelian gluons, and the active lines that enter the hard scattering are scalar ``di-quarks" while the spectator lines are fermions.
The breakdown of ordinary factorization is then understood as being due to the attachments of the collinear gluons
from the lower hadron lines to the upper active ``quark" line which is of course color connected to the upper hadron.
The collinear-to-$p_A$ gluons in figure \ref{2to2gentmd} which couple to the upper active lines of the hard part
are precisely the gluons that in the scheme of \cite{Bomhof:2004aw, Bomhof:2006dp} give rise to the gauge
links of the generalized TMD distributions. The construction in \eqref{gaugelinks1} therefore contains these
contributions. We illustrate these in the single gluon production case in figure \ref{4glueexch}.
As discussed above, however, for a complete proof of factorization one must also consider the simultaneous gluon couplings
between the upper hadron and the hard part. This was
considered in reference \cite{Rogers:2010dm} which calculated in a slightly different model than \cite{Collins:2007jp}
the type of graphs shown in figure \ref{2to2nogentmd} (the zig-zag lines for example correspond to a massive
color singlet scalar boson). These graphs have an entangled color structure
which makes it impossible to factorize the color flows even in the scheme of \cite{Bomhof:2004aw, Bomhof:2006dp}.
The examples shown in figure \ref{2to2nogentmd} then break factorization for the Double Spin Asymmetry (DSA),
while in the specific model considered the contributions from figure \ref{2to2nogentmd} to the unpolarized
cross section cancel. Breakdown of factorization for the unpolarized cross section instead appears
for graphs where three additional gluons are exchanged, with at least one gluon coupling to each
hadron. We illustrate this in figure \ref{2to2nogentmd2}.
\begin{figure}[t]
\begin{center}
\includegraphics[angle=0, scale=0.6]{2to2nogentmd}
\end{center}
\caption{\label{2to2nogentmd} Examples of the class of graphs considered
in \protect \cite{Rogers:2010dm} that lead to the breakdown of TMD factorization for DSA. We indicate the hard scattering by the exchange of the zig-zag lines.}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[angle=0, scale=0.6]{2to2nogentmd2}
\end{center}
\caption{\label{2to2nogentmd2} Examples of graphs where TMD factorization is broken for the unpolarized
cross section. We indicate the hard scattering by the exchange of the zig-zag lines.}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[angle=0, scale=0.55]{5glueexchange}
\end{center}
\caption{\label{5glueexch} Examples of the type of graphs that may go beyond the construction scheme of
equation \protect \eqref{gaugelinks1} in QCD. The solid lines indicate the spectator parts of each hadron. }
\end{figure}
What this shows to us in the case of gluon production at small-$x$ is that to answer the question
of factorization one needs to consider graphs like in figure \ref{5glueexch}. These graphs have
non-trivial color flows that do not seemingly factorize into color singlet factor associated
with each collinear region. In that case one must demonstrate explicitly that such contributions cancel. Given,
however that they do not even cancel in the simple models considered in \cite{Collins:2007jp, Rogers:2010dm}
it seems rather difficult to see how they would in full QCD. Indeed we note that the results from \cite{Rogers:2010dm} have
further been systematized in \cite{Buffing:2011mj} where simultaneous couplings to different
parts are considered, generalizing the scheme in \cite{Bomhof:2004aw, Bomhof:2006dp}.
The difficulties with the color entangled contributions are there clearly demonstrated.
We mentioned earlier that the gluon production in figures \ref{glrgluonprod}, \ref{glrgluonprod2},
\ref{2to1gluon}, \ref{4glueexch} and \ref{5glueexch} corresponds to the case of soft particle production,
illustrated in figure \ref{softhadronprod}. To instead consider hard gluon (or rather hadron) production
with large transverse momentum, so that a scale $Q$ is present which can be used to suppress
transverse polarizations, we need to
take into account that the hard part contains additional jets. It can be shown that the case where
more than two jets emerge from the hard region is suppressed in the almost back-to-back region \cite{qcdbook}.
We then consider the case where two gluon jets emerge from the hard region, and where only one of them
contains the detected hadron. We illustrate this case in figure \ref{hardhadronprod2}.
\begin{figure}[t]
\begin{center}
\includegraphics[angle=0, scale=0.55]{hardhadronprod2}
\end{center}
\caption{\label{hardhadronprod2} Single hadron production where the second jet emerging from the
hadron is integrated over. Arbitrarily many gluons can be exchanged between each collinear
region and the hard region, as indicated by the dots. We do not show the soft region. }
\end{figure}
The case in figure \ref{hardhadronprod2} equals to taking di-hadron production and then
integrating over one of the hadron momenta. The $2\to 2$ hard scattering is now more intricate,
and the scheme of \cite{Bomhof:2004aw, Bomhof:2006dp} becomes rather complicated
as can be seen from table 8 in reference \cite{Bomhof:2006dp}. More importantly, however, the
results in \cite{Collins:2007jp, Rogers:2010dm, Buffing:2011mj} become highly relevant and show us that generally
factorization is broken in di-hadron production. Cancellation of the factorization breaking terms
occur for the integrated distribution, but not if we merely integrate over the momentum of one of detected
final state particles. In fact this can be seen in \cite{Buffing:2011mj} where simplifications occur
only when one integrates over \emph{all} momenta except for a single hadron. Even in that case,
however, the simplification only occurs for contributions that are termed ''tree-level".
It may be of course that the color structures simplify
in the strict large $N_c$ limit where $N_c \to \infty$. The factorization breaking graphs studied in
\cite{Rogers:2010dm}, see figure \ref{2to2nogentmd}, are for example non-leading in $N_c$. Their
effect on the production cross section may still, however, be important if there is no kinematical
suppression.
Finally we note that in more general processes like in figure \ref{hardhadronprod2} there is also the
soft factor which will now be more complicated than in standard TMD factorization. Assuming that factorization
holds, according to \cite{Bomhof:2006dp} the unsubtracted TMD gluon distribution is a highly complicated
function containing many different Wilson lines. Each light-like Wilson line produces rapidity divergences
that must be regulated. In addition to the rapidity divergences there appear divergences related to the
self energy corrections of the Wilson lines.
All these divergences are regulated by subtracting the soft factor from the collinear region,
which leads to definitions like in equation \eqref{subtractedtmd}.
In the case of the gauge link structures that appear for figure \ref{hardhadronprod2} using the scheme
of \cite{Bomhof:2006dp}, however, we dare not even ask how exactly all these issues would be dealt
with. It appears to be an immensely difficult task to obtain final definitions of the highly complicated TMD distributions which
are free from all divergences. Yet this would be extremely helpful for precise phenomenological applications.
\section{Summary}
\label{sec:summary}
Our main aim has been to provide a coherent analysis of TMD factorization
and the TMD gluon distribution, especially as used in the small-$x$ region, and to examine many
important points that usually are not well explained or are overlooked in the literature.
In section \ref{sec:factorization} we have given a unified analysis of the concept of factorization in different formalisms,
the hard scattering formalism (section \ref{sec:hardscatfact}), the BFKL formalism (section \ref{sec:bfklfact}) and the CGC formalism (section \ref{sec:cgcfact}). We also analyzed in section \ref{sec:hybrid} what we called hybrid approaches which combine collinear factorization with the use of TMD distributions.
The main point in section \ref{sec:hardscatfact} has been to explain what exactly is meant by factorization
in the hard scattering case, and what approximations and methods are built into the analysis. We have then compared
these to the small-$x$ treatments which use somewhat different methods. We emphasized in section
\ref{sec:cgctmd} the difference between
factorization which is constructed to be valid to leading power and the leading logarithmic approximation (LLA)
that is based on the one-loop calculation. As we have explained the former is of much greater accuracy and
generality which is important to understand when comparing the different treatments.
In section \ref{sec:hybrid}
we explained the idea behind the so-called factorization of mass singularities that is built into the hybrid formalisms.
Let us note here that it has been demonstrated in \cite{qcdbook} that for the simplest partonic reactions as relevant
for DIS, the method gives the same results as the hard scattering factorization for the massless limit of the hard
scattering coefficient. It is, however, not clear to us whether this still holds in the cases studied in the hybrid formalisms,
where one includes also TMD distributions, and studies proton-nucleus collisions.
We also note that the CCH and CCFM formalisms essentially base
their underlying formulas on the same approach. The use of the method in these formalisms is discussed in \cite{ourpaper}.
We have explained here why this procedure is physically misleading, and caution should be taken before
trying to move on to more complicated reactions.
In section \ref{sec:gluonprod} we have given an extensive analysis of single particle production in the small-$x$
region. We started by showing in section \ref{sec:diffcases} that one can perform a power counting analysis
very much as in section \ref{sec:powercount} to identify the leading structure. This is crucial to understand
when the higher order corrections can be neglected and how the asserted formulas can be
justified.
The main factorization formula \eqref{GLRfact} has been extensively used in phenomenological
applications of small-$x$ QCD, at both RHIC and the LHC. It is therefore crucial to understand the physics
behind it and the justifications given for its validity. We noted that
many treatments in the literature are based on the axial gauge, and we therefore examined the application of the axial
gauge in justifying the factorization formula \eqref{GLRfact}. We showed in section
\ref{sec:lcgauge} why the light-cone gauge is inappropriate for the formulation while in section \ref{sec:axialgauge}
we showed how one can obtain the standard factorization formula in a symmetric axial gauge.
Then in section \ref{sec:singlehadron} we demonstrated the technical difficulties with the use of the axial gauge
and suggested that a more complete treatment be based instead on covariant gauge.
In section \ref{sec:tmdgluon} we then
discussed the gluon distribution that is associated with \eqref{GLRfact} and how it could generally be
constructed from Feynman graphs, and we examined the graphs that are problematic for the full proof of factorization.
There have lately been many applications of TMD factorization in the small-$x$ region, in
$pp$, $pA$ and $AA$ collisions.
To fully prove factorization, however, one must show that the graphs of the type we showed in section \ref{sec:tmdgluon}
cancel.
In the case of $pA$
collisions we emphasize that the gluon couplings from the proton side cannot neglected. In particular it does not
follow that one can automatically treat the proton using integrated parton distributions and fragmentation
functions. If the observed particle is at low $p_\perp$ then the transverse momentum of the collinear region
of the proton and the soft region cannot be neglected outside of these regions, and as a consequence
TMD distributions must be used everywhere. A more complete factorization formula must then be constructed,
taking into account the difficulties outlined in sections \ref{sec:singlehadron} and \ref{sec:tmdgluon}.
Finally, a point which did not discuss much here concerns the scattering coefficient in the gluon
production formula, equation \eqref{GLRfact}. Note that this factor diverges as $l_\perp \to 0$. This is
in fact a sign that the standard treatment cannot be complete.
One should provide for the scattering factor a full definition that is valid to all orders, is gauge independent,
and which contains
necessary subtractions to remove all divergences. An example for the scattering factor in heavy
$q\bar{q}$ production is given in \cite{Collins:1991ty}.
\section*{Acknowledgements}
I would like to thank John Collins, Anna Stasto and Bo-Wen Xiao for useful discussions during
an extended period of time. This work is supported by U.S. D.O.E. grant number~DE-FG02-90-ER-40577
\bibliographystyle{JHEP}
|
{
"timestamp": "2012-04-03T02:07:30",
"yymm": "1203",
"arxiv_id": "1203.1916",
"language": "en",
"url": "https://arxiv.org/abs/1203.1916"
}
|
\section{Introduction}
The development of $\mathcal{P}\mathcal{T}$-symmetric quantum mechanics (PTQM) attracts a lot of interests during the past decade
\cite{BE, MO}. The PTQM is based on the idea that the Hermiticity condition, which is stated as an axiom of quantum mechanics,
may be replaced by a certain less mathematical but more physical condition of symmetry without losing any of the essential physical features of
quantum mechanics.
The Hamiltonians of PTQM are not self-adjoint with respect to the initial Hilbert space's inner product and their
`physical symmetries' do not depend on the choice of inner product. One of typical examples is the non-selfadjoint Hamiltonian
$$
H=-\frac{d^2}{dx^2} + x^2(ix)^\epsilon, \qquad 0\leq\epsilon<2
$$
in $L_2(\mathbb{R})$. The spectrum of $H$ is real and positive \cite{B1,D2} and $H$ has
the property of $\mathcal{P}\mathcal{T}$-symmetry $
{\mathcal P}{\mathcal
T}H=H{\mathcal P}{\mathcal T},$
where the space reflection
(parity) operator $\mathcal{P}$ and the complex conjugation operator
$\mathcal{T}$ are defined as
$({\mathcal P}f)(x)=f(-x)$ and $({\mathcal T}f)(x)=\overline{f(x)}.$
The property of $\mathcal{PT}$-symmetry depends on the choice of operators $\mathcal{P}$ and $\mathcal{T}$, which can be different for
various underlying Hilbert space $\mathfrak{H}$ and various non-selfadjoint operators $H$.
This means that Hamiltonians of PTQM may have the property of $\mathcal{PT}$-symmetry realized by different operators
$\mathcal{P}$ and $\mathcal{T}$.
However, the linear operator $\mathcal{P}$ is always a unitary involution in $\mathfrak{H}$, that is
$\mathcal{P}^2=I$, $(\mathcal{P}f,\mathcal{P}g)=(f,g)$; and the anti-linear operator $\mathcal{T}$ is a conjugation operator in $\mathfrak{H}$,
that is $\mathcal{T}^2=I$, $(\mathcal{T}f,\mathcal{T}g)=(f,g)$.
This observation allows one to develop the theory of $\mathcal{PT}$-symmetric operators in some abstract setting
keeping in mind the properties of operators $\mathcal{P}$ and $\mathcal{T}$ mentioned above (subsection 3.1).
Nowadays, scattering problems related to the Schr\"{o}dinger-type
differential expression
\begin{equation}\label{gere1}
l(\cdot)=-\frac{d^2}{dx^2}+q(x)
\end{equation}
with $\mathcal{PT}$-symmetric potential $q(x)$, i.e., $q(x)=\overline{q(-x)}$ were studied by different methods \cite{ABB,CDV,JOU1,HKS,LSZ,MOS2,ZNO1,ZNO3}. In particular, scattering on the $\mathcal{PT}$-symmetric Coulomb potential
was studied on a trajectory of the complex plane \cite{LSZ}; discretization methods were used
for getting the explicit formulae for the reflection and transmission coefficients \cite{ZNO1,ZNO3}; the relationship between $\mathcal{PT}$-symmetric Hamiltonians and reflectionless scattering systems was discussed \cite{ABB, HKS};
spectral singularities were characterized in terms of reflection and transmission coefficients \cite{MOS2}.
If the potential $q(x)$ in (\ref{gere1}) is local, that is, if its support is contained in an interval $(-\rho,\rho)$,
then the corresponding traveling wave functions have the form:
$$
f_1=\left\{\begin{array}{ll}
{e^{-i{k}x}}+R_k^{r}e^{ikx}, & x\geq\rho \\
T_k^{r}e^{-ikx}, & x\leq{-\rho}
\end{array}\right., \ \
f_2=\left\{\begin{array}{ll}
{T_k^l}e^{ikx}, & x\geq\rho \\
e^{i{k}x}+{R_k^l}e^{-ikx}, & x\leq-\rho
\end{array}\right.,
$$
where $R_k^{l}$, $R_k^{r}$ are the left and right reflection coefficients and
$T_k^{l}$, $T_k^{r}$ are the left and right transmission coefficients, respectively and $k>0$.
The $S$-matrix of (\ref{gere1}) is presented in terms of reflection/transmission coefficients and
the investigation of relationship between the $S$-matrix\footnote{or its counterparts like reflection/transmission coefficients}
and a $\mathcal{PT}$-symmetric operator $H$ generated by (\ref{gere1}) is the subject of the scattering theory.
Examples of such kind of investigations for local $\mathcal{PT}$-symmetric potentials can be found in \cite{CDV,JOU1,MOS2,ZNO1}.
In the present paper we are going to contribute to this inspiring field by studying the scattering of $\mathcal{PT}$-symmetric operators
with the use of an operator-theoretical interpretation of the Lax-Phillips approach in scattering theory \cite{LF} developed in
\cite{KU1,AlAn,Kioto}.
Our choice of the Lax-Phillips approach is explained by the fact that this approach fits the scattering on local potentials well and
it contains an algorithm for the explicit calculation of the analytic continuation of $S$-matrix
in the case of local symmetric potentials.
Another distinctive feature of the Lax-Phillips approach is its
operator-theoretical formulation that allows one to consider the
scattering of many concrete systems with locally supported perturbations from a unique point of view.
Here, the crucial role is played by the concept of $\rho$-perturbed operator (Definition \ref{did2}),
which generalizes the concept of local perturbation to the case of an abstract Hilbert space and
provides simple links to powerful mathematical methods of the extension theory of symmetric operators.
The latter leads to general formula (\ref{red1}) of the analytical continuation of $S$-matrix onto $\mathbb{C}_+$ for
the case of $\rho$-perturbed positive self-adjoint operators (subsection 2.3).
The application of (\ref{red1}) to the case of Schr\"{o}dinger-type differential expression (\ref{gere1})
with symmetric local potential gives the analytical continuation of
the known $S$-matrix, which is expressed in terms of (generalized) reflection/transmission coefficients
calculated for any $k\in\mathbb{C}_+$ (subsection 2.4).
The results of subsection 2.4 can be considered as a hint about `right definition' of $S$-matrix for
$\mathcal{PT}$-symmetric $\rho$-perturbed operators. Roughly speaking the idea is to use the general formula
(\ref{red1}) for all $k\in\mathbb{C}_+$, where this formula has sense. We found this definition useful because, for (\ref{gere1}) with
$\mathcal{PT}$-symmetric potentials, it leads to the expressions of $S$-matrices in terms of reflection/transmission coefficients
(subsection 3.3). Moreover, since the $S$-matrix is determined for some subset of $\mathbb{C}_+$ we obtain more informative relationship between reflection/transmission coefficients and $\mathcal{PT}$-symmetric $\rho$-perturbed operators that can be useful for various inverse problems studies.
Let a $\mathcal{PT}$-symmetric $\rho$-perturbed operator $H$ have the
property of $\mathcal{C}$-symmetry realized by an operator $\mathcal{C}=e^{-Q}\mathcal{P}$ (Definition \ref{neww55})
and let $H$ turn out to be a self-adjoint operator in the Hilbert space $\mathfrak{H}$ endowed with new inner product $(e^{Q}\cdot,\cdot)$.
In that case, we may expect that the corresponding $S$-matrix contains an information about the metric operator $e^Q$.
This gives rise to a problem of recovering of the metric operator $e^Q$ (and, hence the operator $\mathcal{C}$)
by the $S$-matrix of a $\mathcal{PT}$-symmetric $\rho$-perturbed operator.
In this trend, we continue the investigations of \cite{ALBKUZ} and consider the case when the metric operator $e^Q$ fits into the Lax-Phillips scattering structure (subsection 3.4). Theorems \ref{neww78}, \ref{neww144} correspond to the direct problem (metric operator $\to$ the properties of $S$-matrix) and subsection 3.5 contains an example of the successful solution of the inverse problem ($S$-matrix $\to$ recovering the metric operator $e^Q$).
Throughout the paper, $\mathcal{D}(A)$, $\mathcal{R}(A)$, and $\rho(A)$ denote the domain, the range, and the resolvent set of a linear operator $A$, respectively.
The symbol $A\upharpoonright_{\mathcal{D}}$ means the restriction of $A$ onto a set $\mathcal{D}$.
We denote by ${W^m_2}((a,b),{\mathcal N})$ the Sobolev space of vector functions on $(a,b)$ $(-\infty\leq{a}<b\leq{\infty})$
with values in an auxiliary Hilbert space ${\mathcal N}$; the set $\stackrel{0}{{W}_{2}^{m}}((a,b),{\mathcal N})$ is a subspace of ${W^m_2}((a,b),{\mathcal N})$ determined by the condition: $f\in{\stackrel{0}{{W}_{2}^{m}}((a,b),{\mathcal N})}$ if all derivatives $f^{(k)}(x)$ \ $(k=0,\ldots{m-1})$ vanish at points $x=a, x=b$
(see, e.g., \cite{GG} for detail).
\section{Elements of Lax--Phillips Scattering Theory}
\subsection{Definition of unperturbed and $\rho$-perturbed operators.}
Let $B$ be a densely defined symmetric operator in a Hilbert space $\mathfrak{H}$ with inner product $(\cdot,\cdot)$.
The defect numbers $m_\pm$ of $B$ are defined as $m_\pm=\dim\ker(B^*\pm{i}I)$, where
$B^*$ is the adjoint operator of $B$.
A symmetric operator $B$ is called \emph{simple} if it does not induce a self-adjoint
operator in any proper subspace of ${\mathfrak H}$ and $B$ is called \emph{maximal symmetric} if one of its defect numbers $m_{\pm}$
is equal to zero. In what follows, we suppose that $m_-=0$, i.e., $m=m_+=\dim\ker(B^*+{i}I)>0$.
It is known \cite{AkG1} that a simple maximal symmetric operator $B$ with nonzero defect number $m$
in $\mathfrak{H}$ \emph{is unitarily equivalent} to a simple maximal symmetric operator
\begin{equation}\label{neww45}
\mathcal{B}=i\frac{d}{dx}, \qquad \mathcal{D}(\mathcal{B})=\{u\in{W^1_2}({\mathbb R}_+,{\mathcal N}) : u(0)=0\}
\end{equation}
in the Hilbert space $L_2({\mathbb R}_+,{\mathcal N})$, where $\mathbb{R}_+=\{x\in\mathbb{R}\ : \ x\geq0\}$ and the dimension of the auxiliary Hilbert space ${\mathcal N}$ coincides with $m$. This relationship immediately leads to the following statement.
\begin{lemma}\label{l1}
The operator $B^2$ is a closed densely defined symmetric operator in $\mathfrak{H}$ and
${B^2}^{*}={B^{*}}^2.$
\end{lemma}
\begin{definition}\label{did1}
A self-adjoint extension $H$ of $B^{2}$ is called \emph{unperturbed} if
\begin{equation}\label{e5}
(Hf,f)={\Vert B^{*}f \Vert}^{2}, \qquad {f}\in{\mathcal{D}(H)}.
\end{equation}
\end{definition}
Every unperturbed operator $H$ has a purely absolutely continuous spectrum with the same multiplicity $m$ at
each point of $[0,\infty)$, where $m$ is the non-zero defect number of $B$ \cite{KU1}.
Since $B$ is unitarily equivalent to the operator $\mathcal{B}$ defined by (\ref{neww45}),
the semigroup $V(t)=e^{iBt}$ is a completely nonunitary semigroup of
isometries \cite[Theorem 9.3, Chapter III]{SNK}
Denote $\mathfrak{H}_{\rho}=V(\rho)\mathfrak{H}$ \ $(\rho\geq{0})$.
We have $\mathfrak{H}_{\rho_1}\supset\mathfrak{H}_{\rho_2}$ for $0\leq\rho_1<\rho_2$.
The restriction of $B$ on $V(\rho)\mathcal{D}(B)$ gives rise to the
simple maximal symmetric operator
\begin{equation}\label{nee3}
B_{\rho}:=B\upharpoonright_{\mathcal{D}(B_{\rho})}, \qquad \mathcal{D}(B_{\rho})=V(\rho)\mathcal{D}(B)
\end{equation}
acting in the Hilbert space $\mathfrak{H}_{\rho}$.
By Lemma \ref{l1}, $B_\rho^2$ is a densely defined symmetric operator in $\mathfrak{H}_{\rho}$.
\begin{definition}\label{did2}
A closed densely defined operator $H$ in $\mathfrak{H}$ is called \emph{$\rho$-perturbed} if
$H$ and its adjoint operator $H^*$ are extensions of $B_{\rho}^2$, i.e.,
$$
Hu={B_{\rho}^2}u \quad \mbox{and} \quad H^*u={B_{\rho}^2}u \qquad \mbox{for all} \quad u\in\mathcal{D}(B_{\rho}^2).
$$
\end{definition}
{\bf Example 1.}
Let $\mathfrak{H}=L_2(\mathbb{R})$ and let $\mathbb{R}_\pm=\{x\in\mathbb{R}\ : \ {\pm}x>0\}$. The operator
$$
B=(\textsf{sgn}\ x)i\frac{d}{dx}, \qquad \mathcal{D}(B)=\stackrel{0}{{W}_{2}^{1}}(\Bbb R_{-})\oplus\stackrel{0}{{W}_{2}^{1}}(\Bbb R_{+})
$$
is a simple maximal symmetric operator in $L_2(\mathbb{R})$ and
$$
B^2=-\frac{d^2}{dx^2}, \qquad \mathcal{D}(B^2)=\stackrel{0}{{W}_{2}^{2}}(\Bbb R_{-})\oplus\stackrel{0}{{W}_{2}^{2}}(\Bbb R_{+}).
$$
The operator
$$
H=-\frac{d^2}{dx^2}, \qquad \mathcal{D}(H)=W^2_2(\mathbb{R})
$$
is a self-adjoint extension of $B^2$ and it satisfies (\ref{e5}). Therefore, $H$ is an unperturbed operator.
Another unperturbed operator is the Friedrichs extension $H_\mu=B^*B$ of $B^2$:
\begin{equation}\label{ses121}
H_\mu=-\frac{d^2}{dx^2}, \quad \mathcal{D}(H_\mu)=\{u\in{W}^2_2(\mathbb{R}_-)\oplus{W}^2_2(\mathbb{R}_+) : u(0-)=u(0+)=0\}
\end{equation}
The isometric semigroup $V(t)=e^{iBt}$ ($t\geq0$) acts as follows
$$
V(t)f(x)=\left\{\begin{array}{ll}
f(x-t), & \quad x\geq{t}, \\
0, & \quad |x|<t, \\
f(x+t), & \quad x\leq{-t}
\end{array}\right.
$$
This means that $\mathfrak{H}_\rho=L_2(\mathbb{R}\setminus(-\rho,\rho))$,
\begin{equation}\label{ses124}
B_\rho=(\textsf{sgn}\ x)i\frac{d}{dx}, \qquad \mathcal{D}(B_\rho)=\stackrel{0}{{W}_{2}^{1}}(-\infty,-\rho)\oplus\stackrel{0}{{W}_{2}^{1}}(\rho,\infty),
\end{equation}
and
\begin{equation}\label{ses1}
B_\rho^2=-\frac{d^2}{dx^2}, \qquad \mathcal{D}(B^2_\rho)=\stackrel{0}{{W}_{2}^{2}}(-\infty,-\rho)\oplus\stackrel{0}{{W}_{2}^{2}}(\rho, \infty). \end{equation}
Consider the differential expression
\begin{equation}\label{rada1}
l(\cdot)=-\frac{d^2}{dx^2}+q(x), \qquad \textsf{supp}\ q(x)\subset(-\rho,\rho), \quad x\in\mathbb{R}
\end{equation}
and suppose that $l(\cdot)$ determines a closed densely defined operator $H$ in $L_2(\mathbb{R})$. Then, the operator
$H$ and its adjoint operator $H^*$ are extensions of the symmetric operator
$B_\rho^2$ defined by (\ref{ses1}). Therefore, due to Definition \ref{did2}, the operator $H$ is $\rho$-perturbed.
\begin{remark}
Usually \cite{LF}, the Lax-Phillips perturbed and unperturbed evolutions are defined with the use of incoming $D_-$ and outgoing $D_+$ subspaces for the unitary group of solutions $W_H(t)$ of the Cauchy problem for the operator differential equation
\begin{equation}\label{bonn60}
\frac{d^2}{dt^2}u=-Hu,
\end{equation}
where $H$ is a positive (i.e. $(Hf,f)>0$ for all nonzero $f\in\mathcal{D}(H)$) self-adjoint operator in $\mathfrak{H}$.
In particular, the existence of orthogonal subspaces $D_\pm$ of the space of Cauchy data\footnote{the Hilbert space $\mathfrak{H}_H$ is
a completion of $\mathcal{D}(H)$ with respect to the norm $\|\cdot\|_H=(H\cdot,\cdot)$} $\mathfrak{G}=\mathfrak{H}_H\oplus\mathfrak{H}$
with the properties
$$
\begin{array}{l}
(i) \quad W_H(-t)D_{-}\subset{D_{-}}, \qquad W_H(t)D_{+}\subset{D_{+}}, \qquad t\geq0; \vspace{3mm} \\
(ii) \quad \bigcap_{t\geq 0}W_H(-t)D_{-}=\bigcap_{t\geq 0}W_H(t)D_{+}=\{0\},
\end{array}
$$
characterizes the perturbed evolution. The unperturbed evolution is determined by the additional requirement
$$
(iii) \quad D_-\oplus{D_+}=\mathfrak{G}.
$$
These definitions are coordinated with Definitions \ref{did1}, \ref{did2} in the following sense:
if a positive self-adjoint operator $H$ on the right-hand side of (\ref{bonn60}) is $\rho$-perturbed or unperturbed, then
the corresponding group $W_H(t)$ of Cauchy problem solutions possesses
orthogonal subspaces $D_\pm$ with the properties (i)-(ii) or (i)-(iii), respectively \cite{AlAn}.
\end{remark}
\subsection{Definition and properties of $S$-matrix.}
It is easy to check that the relation (\ref{e5}) holds for the Friedrichs extension $H_\mu=B^*B$ of $B^2$.
Therefore, the Friedrichs extension $H_\mu$ remains an unperturbed operator for any choice of $B$.
\begin{proposition}[\cite{Kioto}]\label{pepe1}
Let $H$ be a positive self-adjoint $\rho$-perturbed operator for a given simple maximal symmetric
operator $B$ in $\mathfrak{H}$. Then the wave operators
$$
\Omega_{\pm}(H, H_\mu):=s-\lim_{t\to\pm\infty}e^{iHt}e^{-iH_{\mu}t}
$$
exist and are isometric in ${\mathfrak H}$.
\end{proposition}
The operator $S_{(H,H_\mu)}=\Omega_+^*(H, H_\mu)\Omega_-(H, H_\mu)$
is called \emph{the scattering operator}.
The properties of $S_{(H,H_\mu)}$ is feeling better in terms of the spectral representation of an unperturbed operator \cite{BM1}.
To this end we construct the spectral representation of $H_\mu$.
Since $B$ is unitarily equivalent to the operator $\mathcal{B}=i\frac{d}{dx}$ defined by (\ref{neww45}),
there exists a unitary mapping $Y : \mathfrak{H}\to{L_2}({\mathbb R}_+,{\mathcal N})$ such that
\begin{equation}\label{p21}
B=Y^{-1}{\mathcal B}Y, \quad \mathcal{D}(B)=Y^{-1}\mathcal{D}(\mathcal{B})=Y^{-1}\{u\in{W^1_2}({\mathbb R}_+,{\mathcal N}) : u(0)=0\}.
\end{equation}
Then the operator
$$
({\mathcal F}f)(\delta)=\sqrt{\frac{2}{\pi}}\int_0^\infty{\sin}{\delta{x}}(Yf)(x)dx, \qquad
{f}\in\mathfrak{H}, \ \ \delta>0
$$
isometrically maps $\mathfrak{H}$ onto ${L}_2({\mathbb R}_+,{\mathcal N})$ and
$$
({\mathcal F}B^*Bf)(\delta)=\delta^2({\mathcal F}f)(\delta), \qquad {f}\in\mathcal{D}(B^*B).
$$
The mapping $\mathcal{F}$ determines a
spectral representation $L_2({\mathbb R}_+,{\mathcal N})$ of $H_\mu=B^*B$ in which the action of $H_\mu$ corresponds to
the multiplication by the modified spectral parameter $\delta^2$.
The image ${\mathbb S}={\mathcal F}S_{(H,H_\mu)}{\mathcal F}^{-1}$ of the scattering operator
$S_{(H,H_\mu)}$ in the spectral representation ${L}_2({\mathbb R}_+,{\mathcal N})$
can be realized as the multiplication by an operator-valued
function ${\mathbb S}(\delta)$, the values of which
are bounded operators in ${\mathcal N}$ for almost all $\delta\in\mathbb{R}_+$.
Precisely,
$$
{\mathbb S}f={\mathbb S}(\delta)f(\delta), \qquad {f}\in{L_2({\mathbb R}_+,{\mathcal N})}.
$$
Let us extend the function ${\mathbb S}(\delta)$ onto the whole real axis
$$
{\mathbb S}(-\delta):={\mathbb S}^*(\delta), \qquad \delta>0,
$$
where ${\mathbb S}^*$ means the adjoint operator in ${\mathcal N}$.
The obtained operator-valued function ${\mathbb S}(\cdot)$ depends on the choice of an
auxiliary space ${\mathcal N}$ in (\ref{p21}). However, for any choice of ${\mathcal N}$,
the function ${\mathbb S}(\cdot)$ is the boundary value\footnote{in the sense of strong convergence in ${\mathcal N}$} of an analytic
function ${\mathbb S}(k)$ in $\mathbb{C}_+=\{k\in\mathbb{C} : \textsf{Im} \ k>0\}$.
The values of ${\mathbb S}(k)$ are contraction operators in ${\mathcal N}$
(see \cite[Theorems 4.1, 4.2]{AlAn}, \cite[Theorem 2.3]{Kioto} for details).
The operator-valued function ${\mathbb S}(\cdot)$ is called the $S$-{\it matrix} of
the positive self-adjoint $\rho$-perturbed operator $H$.
\subsection{An operator method for the calculation of the $S$-matrix.}
The $S$-matrix ${\mathbb S}(\cdot)$ contains information about the $\rho$-perturbed operator ${H}$
and the investigation of the relationship between
$\mathbb{S}(\cdot)$ and ${H}$ is the proper subject of Lax-Phillips scattering theory.
The common feature of unperturbed $H_\mu$ and $\rho$-perturbed ${H}$ operators is that
\emph{they are extensions of
a given symmetric operator} $B^2_\rho$. This leads to a simple recipe for finding $\mathbb{S}(\cdot)$ \cite{AlAn,Kioto}.
We begin with the following auxiliary result.
\begin{lemma}\label{meme2}
Let a closed densely defined operator $H$ be $\rho$-perturbed in the sense of Definition \ref{did2}. Then
\begin{equation}\label{aaa1}
{P_\rho}\mathcal{D}({H})\subset\mathcal{D}({B^*_\rho}^2) \quad \mbox{and} \quad P_\rho{H}{f}={B^*_\rho}^2P_\rho{f} \quad \forall{f}\in\mathcal{D}({H}),
\end{equation}
where $P_\rho$ is the orthogonal projection of ${\mathfrak{H}}$ onto $\mathfrak{H}_\rho$.
\end{lemma}
\emph{Proof.} If $H$ is $\rho$-perturbed, then $H\supset{B^2_\rho}$ and $H^*\supset{B^2_\rho}$. Hence,
\begin{eqnarray*}
(P_\rho{H}{f},u)=({H}{f},u)=(f,H^*u)=({f},B^2_{\rho}u)= & & \\
(P_\rho{f},B^2_\rho{u})=({B^2_\rho}^*P_\rho{f},{u})=({{B^*_\rho}^2}P_\rho{f},{u}) & &
\end{eqnarray*}
for all ${f}\in\mathcal{D}({H})$ and for all ${u}\in\mathcal{D}(B^2_\rho)$. The obtained relation
implies (\ref{aaa1}). \rule{2mm}{2mm}
Lemma \ref{meme2} shows that the operators
\begin{equation}\label{ada52}
H_k={B^*_\rho}^2\upharpoonright_{\mathcal{D}(H_k)}, \qquad \mathcal{D}(H_k)=P_\rho({H}-k^2{I})^{-1}\mathfrak{H}_\rho
\end{equation}
are well defined in $\mathfrak{H}_\rho$ for all $k\in\Lambda_+=\{k\in\mathbb{C}_+ : k^2\in\rho(H)\}$.
\begin{definition}\label{did56}
The set of operators $\{H_k\}_{k\in\Lambda_+}$ acting in $\mathfrak{H}_\rho$ and determined by (\ref{ada52}) is called \emph{the image set} of
a $\rho$-perturbed operator $H$.
\end{definition}
If a $\rho$-perturbed operator $H$ is a positive self-adjoint operator in $\mathfrak{H}$, then $\Lambda_+=\mathbb{C}_+$ and
operators $H_k$ from the image set are defined for all $k\in\mathbb{C}_+$.
It is useful to describe operators $H_k$ in terms of a boundary triplet
$({\mathcal H},\Gamma_{0},\Gamma_{1})$ of $B^2_\rho$ defined as follows.
Let ${\mathcal H}=\kerr({B^*_\rho}^2+I)$. Then $\mathcal{D}({B^*_\rho}^2)=\mathcal{D}(B_\rho^*B_\rho)\dot{+}{\mathcal H}$ and hence,
every function $f\in\mathcal{D}({B^*_\rho}^2)$ is uniquely decomposed:
\begin{equation}\label{bonn41}
f=u+h, \qquad u\in{D({B^*_\rho}{B_\rho})}, \quad h\in{\mathcal H}.
\end{equation}
The decomposition (\ref{bonn41}) allows to define the linear mappings $\Gamma_{0}$ and $\Gamma_{1}$ from
$\mathcal{D}({B^*_\rho}^2)$ into ${\mathcal H}$:
\begin{equation}\label{e7}
\Gamma_{0}f=\Gamma_{0}(u+h)=h, \qquad \Gamma_{1}f=\Gamma_{1}(u+h)=P_{\mathcal H}(B_\rho^*B_\rho+I)u,
\end{equation}
where $P_{\mathcal H}$ is the orthogonal projector of ${\mathfrak H}_\rho$ onto the subspace ${\mathcal H}$.
The triple $({\mathcal H},\Gamma_{0},\Gamma_{1})$ is called the positive boundary triplet (positive boundary value space) of $B_\rho^2$ \cite{GG}.
\begin{lemma}[\cite{AlAn}]\label{pepe15}
Let $H$ be a positive self-adjoint $\rho$-perturbed operator. Then
the operators $H_k$ from the image set $\{H_k\}_{k\in\mathbb{C}_+}$
are restrictions of ${B^*_\rho}^2$ onto
\begin{equation}\label{sese2}
\mathcal{D}(H_k)=\{f\in{\mathcal{D}({B^*_\rho}^2)} \ : \ T_k\Gamma_{1}f=\Gamma_{0}f\},
\end{equation}
where $T_k$ are bounded operators in the Hilbert space ${\mathcal H}=\ker({B^*_\rho}^2+I)$ and
$T_k^*=T_{-\overline{k}}$. Furthermore, the operator $T_k$ is maximal dissipative (accumulative) when $\textsf{Re} \ k>0$ ($\textsf{Re} \ k<0$) and $T_k$ is a nonnegative self-adjoint operator with $\|T_k\|\leq{1/2}$ while $\textsf{Re} \ k=0$.
\end{lemma}
It follows from (\ref{p21}) that the dimension of ${\mathcal H}$ coincides with the dimension of
the auxiliary space ${\mathcal N}$ in the definition of the $S$-matrix ${\mathbb S}(\cdot)$.
Let us identify ${\mathcal N}$ with ${\mathcal H}$ for the simplicity.
\begin{theorem}[\cite{AlAn}]\label{esse3}
Let $H$ be a positive self-adjoint $\rho$-perturbed operator. Then the
$S$-matrix ${\mathbb S}(\cdot)$ is an analytic operator-valued function
in $\mathbb{C}_+$ and
\begin{equation}\label{red1}
{\mathbb S}(k)=[I-2(1-ik)T_k][I-2(1+ik)T_k]^{-1}, \quad k\in\mathbb{C}_+.
\end{equation}
where $T_k$ are taken from (\ref{sese2}).
\end{theorem}
\begin{remark}\label{esse3d}
The properties of $T_k$ described in Lemma \ref{pepe15}
yield that the formula (\ref{red1}) determines an analytic contraction-valued function in $\mathbb{C}_+$, which satisfies the relation ${\mathbb S}(-\overline{k})={\mathbb S}^*(k)$, where ${\mathbb S}^*(k)$ is the adjoint operator of ${\mathbb S}(k)$ with respect to the inner product
$(\cdot,\cdot)$ in $\mathcal{H}$.
\end{remark}
In Theorem \ref{esse3}, the auxiliary space ${\mathcal N}$ in the spectral representation $L_2(\mathbb{R}, {\mathcal N})$
is identified with $\mathcal{H}$. It looks natural to rewrite the obtained result for the general case.
Let $Y$ be an isometric mapping of $\mathfrak{H}$ onto $L_2(\mathbb{R}_+,{\mathcal N})$ from (\ref{p21}).
Taking (\ref{nee3}) into account, we conclude that $Y$ maps $\mathcal{H}=\kerr({B_\rho^*}^2+I)$ onto the subspace
$\{e^{-x}u : {u}\in{{\mathcal N}}\}$ of ${L_2((\rho,+\infty),{\mathcal N})}$.
Identifying functions $f(x)={e^{-x}u}$ with elements $v={\alpha}u\in{{\mathcal N}}$:
\begin{equation}\label{rest1}
f(x)=e^{-x}u \leftrightarrow v \ (={\alpha}u), \qquad \alpha=\frac{1}{\sqrt{2}}e^{-\rho}
\end{equation}
where the constant $\alpha$ is chosen in such a way that
$\|f\|_{L_2}=\|v\|_{{\mathcal N}}$,
we obtain that $Y$ isometrically maps $\mathcal{H}$ onto ${\mathcal N}$.
For the operator-valued function
${\textsf S}(\cdot)=Y{\mathbb S}(\cdot)Y^{-1}$,
Theorem \ref{esse3} is reformulated as follows:
\begin{theorem}\label{esse3b}
The $S$-matrix ${\textsf S}(\cdot)$ of $H$ has the form
\begin{equation}\label{red1b}
{\textsf S}(k)=[I-2(1-ik){\textsf T}_k][I-2(1+ik){\textsf T}_k]^{-1}, \ k\in\mathbb{C}_+,
\end{equation}
where ${\textsf T}_k$ characterizes the domain $\mathcal{D}(H_k)$:
\begin{equation}\label{sese2b}
\mathcal{D}(H_k)=\{f\in{\mathcal{D}({B^*_\rho}^2)} \ : \ {\textsf T}_kY\Gamma_{1}f=Y\Gamma_{0}f\}.
\end{equation}
\end{theorem}
\subsection{Application to Schr\"{o}dinger operator with local potential.}
Let the differential expression (\ref{rada1})
determine a positive self-adjoint operator $H$ in $\mathfrak{H}=L_2(\mathbb{R})$.
Then $H$ is $\rho$-perturbed in the sense of Definition \ref{did2}, where $B_\rho^2$ is defined by (\ref{ses1})
(see Example 1).
Hence, we can find the $S$-matrix ${\textsf S}(\cdot)$ of $H$ in $\mathbb{C}_+$.
A simple calculation with the use of (\ref{ses121}) and (\ref{ses1}) leads to the conclusion that the
positive boundary triplet $({\mathcal H},\Gamma_{0},\Gamma_{1})$ of $B_\rho^2$ defined by formulas (\ref{bonn41}) and (\ref{e7})
has the following form: the space $\mathcal{H}$ coincides with the linear span of
functions
$$
\psi_-(x)=\left\{\begin{array}{ll}
0, & x\geq\rho \\
2e^{\rho+x}, & x\leq{-\rho}
\end{array}\right. \quad \mbox{and} \quad
\psi_+(x)=\left\{\begin{array}{ll}
2e^{\rho-x}, & x\geq\rho \\
0, & x\leq{-\rho}
\end{array}\right. ;
$$
the operators $\Gamma_0: \mathcal{D}({B^*_\rho}^2)\to\mathcal{H}$ act as follows
\begin{equation}\label{meme1}
\begin{array}{l}
\Gamma_0f=\frac{1}{2}(f(-\rho)\psi_-+f(\rho)\psi_+), \vspace{3mm} \\
\Gamma_1f=[f(-\rho)-f'(-\rho)]\psi_-+[f(\rho)+f'(\rho)]\psi_+,
\end{array}
\end{equation}
where $f\in\mathcal{D}({B^*_\rho}^2)={{W}_{2}^{2}}(-\infty,-\rho)\oplus{{W}_{2}^{2}}(\rho, \infty)$.
Let us choose $\mathbb{C}^2$ as the auxiliary space ${\mathcal N}$ and denote elements of its canonical basis by $e_+=(1,0)^T$ and $e_-=(0,1)^T$.
The operator $Y$ defined by the formulas:
\begin{equation}\label{deder8}
\begin{array}{ll}
Yf(x)=f(x){e_+}, \ x\in\mathbb{R}_+ & \mbox{for all} \ {f}\in{L_2(\mathbb{R}_+)} \vspace{2mm} \\
Yg(x)=g(-x){e_-}, \ x\in\mathbb{R}_- & \mbox{for all} \ {g}\in{L_2(\mathbb{R}_-)}
\end{array}
\end{equation}
maps isometrically $L_2(\mathbb{R})$ onto $L_2(\mathbb{R}_+, \mathbb{C}^2)$ and it satisfies relation (\ref{p21}).
Moreover, taking (\ref{rest1}) into account, we get $Y\psi_\pm=\sqrt{2}e_{\pm}$. The obtained relation and the explicit formulas (\ref{meme1}) for $\Gamma_j$ allow us to reinterpret (\ref{sese2b}) as follows: the domain $\mathcal{D}(H_k)$ consists of those functions $f\in{{W}_{2}^{2}}(-\infty,-\rho)\oplus{{W}_{2}^{2}}(\rho, \infty)$ for which
\begin{equation}\label{sese2c}
{\textsf T}_k\left(\begin{array}{c}
f(\rho)+f'(\rho) \\
f(-\rho)-f'(-\rho)
\end{array}\right)=\frac{1}{2}\left(\begin{array}{c}
f(\rho) \\
f(-\rho)
\end{array}\right).
\end{equation}
The matrix ${\textsf T}_k=\|t_{ij}\|_{ij}^2$ in (\ref{sese2c}) is completely determined
by two linearly independent functions $f_1, f_2\in\mathcal{D}(H_k)\setminus\mathcal{D}(B^2_\rho)$.
Precisely, we have to know values $f_j(\pm\rho)$ together with values of derivatives $f_j'(\pm\rho)$.
Since $\mathfrak{H}_\rho=\mathcal{R}(B^2_\rho-k^2I)\oplus\ker({B^*_\rho}^2-{\overline{k}}^2I)$, the second relation in
(\ref{ada52}) leads to the conclusion that
\begin{equation}\label{rest4}
\mathcal{D}(H_k)=\mathcal{D}(B^2_\rho)\dot{+}P_\rho(H-k^2I)^{-1}\ker({B^*_\rho}^2-{\overline{k}}^2I),
\end{equation}
where $P_\rho$ is the orthogonal projection onto $\mathfrak{H}_\rho=L_2(\mathbb{R}\setminus(-\rho,\rho))$.
Let us assume that $k\in\mathbb{C}_+'=\mathbb{C}_+\setminus{i\mathbb{R}_+}=\{k\in\mathbb{C}_+ : \textsf{Re} \ k\not=0\}$. Then the functions
$$
h_1=\left\{\begin{array}{l}
\beta{e^{-i\overline{k}x}}, \quad x\geq\rho \\
0, \quad x<\rho
\end{array}\right., \quad
h_2=\left\{\begin{array}{l}
0, \quad x>-\rho \\
\beta{e^{i\overline{k}x}}, \quad x\leq-\rho
\end{array}\right., \qquad \beta={\overline{k}}^2-k^2
$$
form a basis of $\ker({B^*_\rho}^2-{\overline{k}}^2I)$.
(The coefficient $\beta$ is used for the simplification of formulas below.)
Let $\tilde{f}_j\in\mathcal{D}(H)$ be solutions of equations
\begin{equation}\label{rest1963}
(H-k^2I)\tilde{f}_j=h_j, \qquad j=1,2.
\end{equation}
It follows from (\ref{rest4}) and (\ref{rest1963}) that the functions $f_j=P_\rho{\tilde{f}_j}$ belong to $\mathcal{D}(H_k)\setminus\mathcal{D}(B^2_\rho)$
and they are linearly independent. Furthermore, taking (\ref{rada1}) into account, we get
\begin{equation}\label{deder1}
f_1=\left\{\begin{array}{ll}
{e^{-i\overline{k}x}}+R_k^{r}e^{ikx}, & x\geq\rho \\
T_k^{r}e^{-ikx}, & x\leq{-\rho}
\end{array}\right., \ \
f_2=\left\{\begin{array}{ll}
{T_k^l}e^{ikx}, & x\geq\rho \\
e^{i\overline{k}x}+{R_k^l}e^{-ikx}, & x\leq-\rho
\end{array}\right.,
\end{equation}
where $R_k^{l}$, $R_k^{r}$ are the left and right reflection coefficients and
$T_k^{l}$, $T_k^{r}$ are the left and right transmission coefficients, respectively.
The formulas (\ref{deder1}) allow us to express the values of $f_j(\pm\rho)$ and $f'_j(\pm\rho)$
in terms of coefficients $R_k^{l}$, $R_k^{r}$, $T_k^{l}$ and $T_k^{r}$.
Substituting these values into (\ref{sese2c}) and making elementary calculations we determine the entries
of ${\textsf T}_k=\|t_{ij}\|_{ij}^2$. Then, the formula (\ref{red1b}) leads to the following expression
of the $S$-matrix ${\textsf S}(\cdot)$:
\begin{equation}\label{rest2}
{\textsf S}(k)=-e^{2i\rho\textsf{Re} k}\frac{k}{\textsf{Re}\ k}\left(\begin{array}{cc}
R_k^r+\displaystyle{e^{-2i\rho\textsf{Re} k}\frac{\textsf{Im}\ k}{k}} & T_k^l \vspace{4mm} \\
T_k^r & R_k^l+\displaystyle{e^{-2i\rho\textsf{Re} k}\frac{\textsf{Im}\ k}{k}}
\end{array}
\right)
\end{equation}
The formula (\ref{rest2}) is obtained for $k\in\mathbb{C}_+'$ and it can be extended onto $\mathbb{C}_+$ by the continuity.
For real $k$ the expression (\ref{rest2}) is reduced to
\begin{equation}\label{rest2b}
{\textsf S}(k)=-e^{2i\rho{k}}\left(\begin{array}{cc}
R_k^r & T_k^l \vspace{4mm} \\
T_k^r & R_k^l
\end{array}
\right).
\end{equation}
\section{Scattering for $\mathcal{PT}$-symmetric operators}
\subsection{Definition of $\mathcal{PT}$-symmetric operators}
Let $\mathfrak{H}$ be a Hilbert space with inner product
$(\cdot,\cdot)$. A linear operator $\mathcal{P}$ in $\mathfrak{H}$ is called \emph{unitary involution} if
\begin{equation}\label{bonn1}
(i) \quad \mathcal{P}^2=I, \qquad (ii) \quad (\mathcal{P}f,\mathcal{P}g)=(f,g), \quad {f,g}\in\mathfrak{H}.
\end{equation}
A modification of condition (ii) in (\ref{bonn1}) leads to the definition of the conjugation operator.
An operator $\mathcal{T}$ in $\mathfrak{H}$ is called \emph{conjugation} if
\begin{equation}\label{bonn2}
(i) \quad \mathcal{T}^2=I, \qquad (ii) \quad (\mathcal{T}f,\mathcal{T}g)=(g,f), \quad {f,g}\in\mathfrak{H}.
\end{equation}
The conjugation operator is bounded in $\mathfrak{H}$ but, in contrast to the case of an involution, $\mathcal{T}$ is \emph{anti-linear} in the sense that
\begin{equation}\label{bonn2b}
\mathcal{T}(\alpha{f}+\beta{g})=\overline{\alpha}\mathcal{T}{f}+\overline{\beta}\mathcal{T}{g}, \qquad \alpha, \beta\in\mathbb{C}, \quad f,g\in\mathfrak{H}.
\end{equation}
Let us fix a unitary involution $\mathcal{P}$ and a conjugation $\mathcal{T}$ in $\mathfrak{H}$ assuming in what follows that
$\mathcal{P}$ and $\mathcal{T}$ \emph{commute}, that is, $\mathcal{P}\mathcal{T}=\mathcal{T}\mathcal{P}$.
This means that $\mathcal{P}\mathcal{T}$ is also a conjugation.
\begin{definition}\label{ddd1}
A closed densely defined linear operator $H$ in $\mathfrak{H}$ is called $\mathcal{P}\mathcal{T}$-symmetric if the relation
\begin{equation}\label{bonn3}
\mathcal{P}\mathcal{T}Hf=H\mathcal{P}\mathcal{T}f
\end{equation}
holds for all elements $f\in\mathcal{D}(H)$.
\end{definition}
\begin{remark}
In what follows we will often use operator identities like
\begin{equation}\label{neww11}
XA=BX,
\end{equation}
where $A$ and $B$ are (possible) unbounded operators in a Hilbert space $\mathfrak{H}$ and $X$ is a bounded operator in $\mathfrak{H}$.
In that case, we \emph{always assume that (\ref{neww11}) holds on $\mathcal{D}(A)$}. This means that
$X\mathcal{D}(A)\subset\mathcal{D}(B)$ and the identity $XAu=BXu$ holds for all $u\in\mathcal{D}(A)$.
If $A$ is bounded, then (\ref{neww11}) should hold on the whole $\mathfrak{H}$.
\end{remark}
\begin{lemma}[\cite{ALBKUZ}]\label{abbb1}
If $H$ is $\mathcal{P}\mathcal{T}$-symmetric in a Hilbert space $\mathfrak{H}$, then its adjoint operator $H^*$ is also $\mathcal{P}\mathcal{T}$-symmetric.
\end{lemma}
\subsection{The $S$-matrix for $\mathcal{PT}$-symmetric $\rho$-perturbed operators.}
We begin with the following auxiliary result.
\begin{lemma}\label{neww14}
Let $B$ commute with $\mathcal{P}$
and anti-commute with $\mathcal{T}$
\begin{equation}\label{bebe3}
\mathcal{P}B=B\mathcal{P}, \qquad \mathcal{T}B=-B\mathcal{T},
\end{equation}
and let $H$ be a $\mathcal{PT}$-symmetric $\rho$-perturbed operator.
Then the operators $H_k$ from the image set $\{H_k\}_{k\in\Lambda_+}$ of $H$ (see Definition \ref{did56}) satisfy the relation
$$
\mathcal{PT}H_k=H_{-\overline{k}}\mathcal{PT}.
$$
\end{lemma}
\emph{Proof.} It follows from (\ref{bonn2b}) and (\ref{bebe3}) that $V(t)=e^{iBt}$ commutes with $\mathcal{PT}$. Hence, the subspace $\mathfrak{H}_\rho=V(\rho)\mathfrak{H}$ reduces $\mathcal{PT}$ and the orthogonal projection $P_\rho$
onto $\mathfrak{H}_\rho$ commutes with $\mathcal{PT}$. Furthermore
\begin{equation}\label{red14}
B_{\rho}{\mathcal{PT}}=-{\mathcal{PT}}B_\rho, \qquad B_\rho^*{\mathcal{PT}}=-{\mathcal{PT}}B_\rho^*,
\end{equation}
due to (\ref{nee3}) and (\ref{bebe3}).
Hence, $\mathcal{PT}{B^*_\rho}^2={B^*_\rho}^2\mathcal{PT}$ and
$$
\mathcal{PT}\mathcal{D}(H_k)=\mathcal{PT}P_\rho({H}-k^2{I})^{-1}\mathfrak{H}_\rho=P_\rho({H}-{\overline{k}}^2{I})^{-1}\mathcal{PT}\mathfrak{H}_\rho=\mathcal{D}(H_{-\overline{k}}).
$$
Combining the obtained relations with (\ref{ada52}) we complete the proof.
\rule{2mm}{2mm}
\smallskip
Let us suppose that the operators $H_k$ from the image set $\{H_k\}_{k\in\Lambda_+}$ of a $\mathcal{PT}$-symmetric
$\rho$-perturbed operator $H$ \emph{can be determined by the relation (\ref{sese2}) with
bounded operators $T_k$ in $\mathcal{H}$}. Then we can formally define
\emph{the $S$-matrix of $H$} by the formula
(\ref{red1}) for all $k\in{\Lambda_+}$ such that $0\in\rho(I-2(1+ik)T_k)$.
\smallskip
\emph{We will consider the operator-valued function ${\mathbb S}(\cdot)$ defined in such a way (as well as its image
${\textsf{S}}(\cdot)=Y{\mathbb S}(k)Y^{-1}$ in ${\mathcal N}$) as
the $S$-matrix of a $\mathcal{PT}$-symmetric $\rho$-perturbed operator $H$.}
\smallskip
Of course, in contrast to the case of positive self-adjoint $\rho$-perturbed operators considered in Section 2,
our definition is rather formal. In particular, we do not take care about the existence of wave operators and other important auxiliary things.
However, we found this definition useful because it provides an explicit relation to the image set $\{H_k\}$ of $H$ that may be important for
the inverse problem studies.
\begin{proposition}\label{neww28}
Let $B$ satisfy (\ref{bebe3}) and let \ ${\mathbb S}(\cdot)$ be the $S$-matrix of $\mathcal{PT}$-symmetric
$\rho$-perturbed operator $H$. Then
$$
\mathcal{PT}{\mathbb S}(k)={\mathbb S}(-\overline{k})\mathcal{PT}
$$
for those $k\in\Lambda_+$ that \ ${\mathbb S}(k)$ exists.
\end{proposition}
\emph{Proof.} First of all, we show that the existence of ${\mathbb S}(k)$ implies
the existence of ${\mathbb S}(-\overline{k})$.
Indeed, the existence of ${\mathbb S}(k)$ is equivalent to the following conditions:
\begin{enumerate}
\item[(i)] \quad the corresponding operator $T_k$ in (\ref{sese2}) is bounded; \vspace{1mm}
\item[(ii)] \quad $0\in\rho(I-2(1+ik)T_k)$.
\end{enumerate}
It follows from (i) and \cite[Theorem 2.2, Chapter 3]{GG} that $-1\in\rho(H_k)$ and
\begin{equation}\label{bebe45}
T_k=(H_k+I)^{-1}-(B^*_\rho{B_\rho}+I)^{-1}.
\end{equation}
The relations (\ref{red14}) imply that $\mathcal{PT}B^*_\rho{B_\rho}=B^*_\rho{B_\rho}\mathcal{PT}$. Hence,
$$
\mathcal{PT}(B^*_\rho{B_\rho}+I)^{-1}=(B^*_\rho{B_\rho}+I)^{-1}\mathcal{PT}.
$$
On the other hand, $-1\in\rho(H_{-\overline{k}})$ and $\mathcal{PT}(H_k+I)^{-1}=(H_{-\overline{k}}+I)^{-1}\mathcal{PT}$ by Lemma \ref{neww14}.
Therefore
\begin{equation}\label{dede1}
\mathcal{PT}T_k=[(H_{-\overline{k}}+I)^{-1}-(H_\mu+I)^{-1}]\mathcal{PT}=T_{-\overline{k}}\mathcal{PT},
\end{equation}
where the bounded operator $T_{-\overline{k}}=(H_{-\overline{k}}+I)^{-1}-(H_\mu+I)^{-1}$ determines $H_{-\overline{k}}$ in
(\ref{sese2}).
By virtue of (\ref{dede1}), $\mathcal{PT}[I-2(1+ik)T_k]=[I-2(1+i(-\overline{k}))T_{-\overline{k}}]\mathcal{PT}$. Hence
$0\in\rho(I-2(1+i(-\overline{k}))T_{-\overline{k}})$ and the existence of ${\mathbb S}(-\overline{k})$ is
established.
Applying the operator $\mathcal{PT}$ to the both parts of (\ref{red1})
and using (\ref{dede1}) we complete the proof.
\rule{2mm}{2mm}
\subsection{Schr\"{o}dinger operator with $\mathcal{PT}$-symmetric local potential on $\mathbb{R}$.}
Denote by $\mathcal{P}f(x)=f(-x)$ and $\mathcal{T}f(x)=\overline{f(x)}$ the unitary involution and conjugation operators in
$\mathfrak{H}=L_2(\mathbb{R})$ and consider the differential expression (\ref{rada1}) with $\mathcal{PT}$-symmetric potential
$q(x)$. In that case (\ref{rada1}) determines a $\mathcal{PT}$-symmetric $\rho$-perturbed operator
$H$.
By analogy with subsection 2.4, the image set $\{H_k\}_{k\in\Lambda_+}$ of $H$ is determined by the matrices ${\textsf T}_k=\|t_{ij}\|_{ij}^2$ in (\ref{sese2c}). Substituting the values $f_j(\pm\rho)$ and $f'_j(\pm\rho)$ of functions $f_j$ from (\ref{deder1}) into
(\ref{sese2c}) and solving the corresponding system of linear equations, we get
$$
\begin{array}{ll}
\displaystyle{t_{11}=\frac{1}{2\theta\Delta_k}[\Delta_k-e^{i\phi}(e^{i\alpha}-1)(R_k^l+e^{i(\alpha+\phi)})]}; & \displaystyle{t_{12}=\frac{T_k^l}{2\theta\Delta_k}e^{i\phi}(e^{i\alpha}-1)}; \vspace{4mm} \\
\displaystyle{t_{22}=\frac{1}{2\theta\Delta_k}[\Delta_k-e^{i\phi}(e^{i\alpha}-1)(R_k^r+e^{i(\alpha+\phi)})]}; & \displaystyle{t_{21}=\frac{T_k^r}{2\theta\Delta_k}e^{i\phi}(e^{i\alpha}-1)},
\end{array}
$$
where $\theta=1+ik$, \ $\displaystyle{e^{i\alpha}=\frac{\overline{\theta}}{\theta}}$, \ $\displaystyle{e^{i\phi}=e^{-2i\rho\textsf{Re} k}}$, \ $k\in\mathbb{C}_+'$, \ and
\begin{equation}\label{deder2}
\Delta_k=\det\left(\begin{array}{cc}
R_k^r+e^{i(\alpha+\phi)}, & T_k^r \\
T_k^l, & R_k^l+e^{i(\alpha+\phi)}
\end{array}\right).
\end{equation}
Obviously, if $\Delta_k\not=0$, then the entries $t_{ij}$ of ${\textsf T}_k$ are well defined. This means that the corresponding operator $T_k$
in $\mathcal{H}$ is bounded if and only if $\Delta_k\not=0$.
Further, $0\in\rho(I-2(1+ik)T_k)$ if and only if $\det(\sigma_0-2\theta{\textsf T}_k)\not=0$, where
$\sigma_0=\left(\begin{array}{cc} 1 & 0 \\
0 & 1 \end{array}\right)$. The explicit expressions of $t_{ij}$ obtained above lead to the conclusion that
$$
\sigma_0-2\theta{\textsf T}_k=\displaystyle{\frac{e^{i\phi}(e^{i\alpha}-1)}{\Delta_k}}\left(\begin{array}{cc}
R_k^l+e^{i(\alpha+\phi)}, & -T_k^l \\
-T_k^r, & R_k^r+e^{i(\alpha+\phi)}
\end{array}\right).
$$
Hence,
$$
\det(\sigma_0-2\theta{\textsf T}_k)=\displaystyle{\frac{e^{2i\phi}(e^{i\alpha}-1)^2}{\Delta_k}},
$$
where $e^{i\alpha}-1\not=0$ (since $k\in\mathbb{C}_+'$). Therefore, the condition $\Delta_k\not=0$
ensures that $0\in\rho(I-2(1+ik)T_k)$.
Denote
$$
\Lambda_+'=\Lambda_+\cap\mathbb{C}_+'=\{k\in\mathbb{C}_+ : \quad \textsf{Re} \ k\not=0, \quad k^2\in\rho(H)\}.
$$
Summing up the discussion above, we establish the following
\begin{theorem}\label{new345}
The $S$-matrix ${\textsf S}(\cdot)$ of a $\mathcal{PT}$-symmetric $\rho$-perturbed operator $H$
is defined by (\ref{rest2}) for all $k\in\Lambda_+'$ such that $\Delta_k\not=0$.
\end{theorem}
\subsection{The $S$-matrix for $\mathcal{PT}$-symmetric $\rho$-perturbed operators with $\mathcal{C}$-symmetry.}
The property of $\mathcal{P}\mathcal{T}$-symmetry of $H$ is significant, but we have still to show
that $H$ can serve as an Hamiltonian for quantum mechanics.
To do so one must demonstrate that $H$ is self-adjoint in a \emph{Hilbert space}.
In physical literature this problem is often solved
by finding a new symmetry represented by a linear operator $\mathcal{C}$, which commutes with $H$.
More precisely, suppose we can find an operator $\mathcal{C}=e^{-Q}\mathcal{P}$, where $Q$ is a bounded
self-adjoint operator in $\mathfrak{H}$ obeying the following algebraic equations:
\begin{equation}\label{usa6}
\mathcal{P}Q=-Q\mathcal{P}, \qquad \mathcal{T}Q=-Q\mathcal{T}, \qquad \mathcal{C}H=H\mathcal{C}.
\end{equation}
The first two relations in (\ref{usa6}) imply that $\mathcal{C}^2=I$ and $\mathcal{C}\mathcal{PT}=\mathcal{PT}\mathcal{C}$.
\begin{definition}\label{neww55}
{We say that a closed densely defined operator $H$ in $\mathfrak{H}$
\emph{has the property of $\mathcal{C}$-symmetry} if relations (\ref{usa6}) hold
for some choice of $\mathcal{C}=e^{-Q}\mathcal{P}$}.
\end{definition}
As a rule, if a physically meaningful $\mathcal{PT}$-symmetric operator $H$ has the
property of $\mathcal{C}$-symmetry realized by an operator $\mathcal{C}=e^{-Q}\mathcal{P}$,
then $H$ turns out to be a self-adjoint operator in the Hilbert space $\mathfrak{H}$ with
the new inner product $(e^{Q}\cdot,\cdot)$, which is equivalent to the initial inner product $(\cdot,\cdot)$.
In what follows we will use the notation $(\mathfrak{H}, (e^{Q}\cdot,\cdot))$ for the Hilbert space $\mathfrak{H}$ endowed with
inner product $(e^{Q}\cdot,\cdot)$.
Consider a $\mathcal{PT}$-symmetric $\rho$-perturbed operator $H$ with the
property of $\mathcal{C}$-symmetry realized by an operator $\mathcal{C}=e^{-Q}\mathcal{P}$
and assume that ${\mathbb S}(\cdot)$ is the $S$-matrix of $H$ (see subsection 3.2).
It seems natural that ${\mathbb S}(\cdot)$ will contain some information about the metric operator $e^{Q}$. We are aiming
to investigate this problem for the simplest case when the operator
$B$ has a property of $\mathcal{C}$-symmetry realized by \emph{the same} operator $\mathcal{C}=e^{-Q}{\mathcal{P}}$.
\begin{lemma}\label{newwww45}
Let $B$ satisfy (\ref{bebe3}) and let $B$ have a $\mathcal{C}$-symmetry realized by an operator $\mathcal{C}=e^{-Q}{\mathcal{P}}$.
Then $B$ keeps being a simple maximal symmetric operator in the Hilbert space
$(\mathfrak{H}, (e^{Q}\cdot,\cdot))$ and its adjoint operator in $(\mathfrak{H}, (e^{Q}\cdot,\cdot))$
coincides with the initial adjoint operator $B^*$ with respect to $(\cdot,\cdot)$.
\end{lemma}
\emph{Proof.} If $B$ has a $\mathcal{C}$-symmetry, then $B\mathcal{C}=\mathcal{C}B$, where
$\mathcal{C}=e^{-Q}\mathcal{P}$.
Then
$Be^{Q}=B{\mathcal{PC}}={\mathcal{P}}B{\mathcal{C}}={\mathcal{PC}}B=e^{Q}B$ and, hence,
$(Be^{Q})^*=e^{Q}B^*=(e^{Q}B)^*=B^*e^{Q}$. Thus, we show
\begin{equation}\label{er1}
Be^{Q}=e^{Q}B, \qquad B^*e^{Q}=e^{Q}B^*
\end{equation}
that completes the proof. \rule{2mm}{2mm}
It follows from (\ref{er1}) that $V(t)=e^{iBt}$ commutes with $e^{Q}$. Hence, the subspaces $\mathfrak{H}_\rho=V(\rho)\mathfrak{H}$ reduce
$e^{Q}$ and
\begin{equation}\label{er2}
B_{\rho}e^{Q}=e^{Q}B_\rho, \qquad B_\rho^*e^{Q}=e^{Q}B_\rho^*,
\end{equation}
where $B_\rho$ is defined by (\ref{nee3}). By analogy with Lemma \ref{newwww45} we obtain
\begin{lemma}\label{neww45b}
Let $B$ satisfy (\ref{bebe3}) and let $B$ have a $\mathcal{C}$-symmetry realized by an operator $\mathcal{C}=e^{-Q}{\mathcal{P}}$. Then
the operator $B_\rho$ is simple maximal symmetric in both of the Hilbert spaces $(\mathfrak{H}_\rho, (\cdot,\cdot))$ and $(\mathfrak{H}_\rho, (e^{Q}\cdot,\cdot))$ and its adjoint $B_\rho^*$ does not depend on the choice of inner products $(\cdot,\cdot)$ or $(e^{Q}\cdot,\cdot)$.
\end{lemma}
\begin{theorem}\label{neww78}
Let a $\mathcal{PT}$-symmetric $\rho$-perturbed operator ${H}$ have nonnegative real spectrum with $0\not\in\sigma_p(H)$
and let $H$ be self-adjoint in the Hilbert space $(\mathfrak{H}, (e^{Q}\cdot,\cdot))$, where $\mathcal{C}=e^{-Q}{\mathcal{P}}$ is an operator
of $\mathcal{C}$-symmetry of a simple maximal symmetric operator $B$ satisfying (\ref{bebe3}). Then
the $S$-matrix ${\mathbb S}(k)$ of $H$ is an analytic operator-valued function in $\mathbb{C}_+$, which is determined by (\ref{red1}), and for all $k\in\mathbb{C}_+$:
\begin{equation}\label{neww69}
\begin{array}{c}
(i) \quad e^{Q}{\mathbb S}(-\overline{k})={\mathbb S}^*(k)e^{Q}; \vspace{3mm} \\
(ii) \quad e^{Q}{\mathbb S}(-\overline{k}){\mathbb S}(k)\leq{e^{Q}} \vspace{3mm} \\
(iii) \quad \mathcal{PT}{\mathbb S}(-\overline{k})={\mathbb S}(k)\mathcal{PT}.
\end{array}
\end{equation}
\end{theorem}
\emph{Proof.} Let us consider the operators $H_\mu=B^*B$ and $H$ in the Hilbert space $(\mathfrak{H}, (e^{Q}\cdot,\cdot))$.
Due to Lemma \ref{newwww45}, the operator $H_\mu$ does not depend on the choice of $\mathcal{C}$-symmetry of $B$ and $H_\mu$ is an unperturbed operator in $(\mathfrak{H}, (e^{Q}\cdot,\cdot))$ in the sense of Definition \ref{did1}.
On the other hand, Lemma \ref{neww45b} and Definition \ref{did2} imply that $H$ is a $\rho$-perturbed self-adjoint operator
in $(\mathfrak{H}, (e^{Q}\cdot,\cdot))$ with nonnegative spectrum. Moreover, the condition $0\not\in\sigma_p(H)$ yields that
$H$ is a positive self-adjoint operator with respect to $(e^{Q}\cdot,\cdot)$.
This means that for the operator $H$ considered in $(\mathfrak{H}, (e^{Q}\cdot,\cdot))$ there exists
the $S$-matrix ${\mathbb S}(k)$ in the spectral representation $L_2({\mathbb R}_+,{\mathcal N})$ of $H_\mu$
(see subsection 2.2).
The $S$-matrix ${\mathbb S}(k)$ is defined by formula (\ref{red1}) for all $k\in\mathbb{C}_+$ and its
calculation consists of three stages, which have to be done in $(\mathfrak{H}, (e^{Q}\cdot,\cdot))$:
\begin{itemize}
\item[(A)] the determination of the image set $\{H_k\}_{k\in\mathbb{C}_+}$ of $H$;
\item[(B)] the construction of the positive boundary triplet $(\mathcal{H}, \Gamma_0, \Gamma_1)$;
\item[(C)] the determination of operators $\{T_k\}$ in (\ref{sese2}).
\end{itemize}
Let us to examine the dependence of these stages on the change of inner product: $(e^{Q}\cdot,\cdot)\to(\cdot,\cdot)$.
(A) \emph{The image set.}
Since ${H}$ is a positive self-adjoint $\rho$-perturbed operator in $(\mathfrak{H}, (e^{Q}\cdot,\cdot))$, we can define the image set
$\{H_k\}_{k\in\mathbb{C}_+}$ of $H$ by the formula (\ref{ada52}), where $P_\rho$ is the orthogonal projection onto the subspace $\mathfrak{H}_\rho$ in $(\mathfrak{H}, (e^{Q}\cdot,\cdot))$.
Since the subspace $\mathfrak{H}_\rho$ reduces $e^{Q}$,
the orthogonal decomposition $\mathfrak{H}=\mathfrak{H}_\rho\oplus\kerr{V^*(\rho)}$ with respect to $(e^{Q}\cdot,\cdot)$ remains orthogonal for inner product $(\cdot,\cdot)$. This means that $P_\rho$ is also the orthogonal projection onto $\mathfrak{H}_\rho$ in $\mathfrak{H}$ with respect to the inner product $(\cdot,\cdot)$. Therefore, the image set $\{H_k\}_{k\in\mathbb{C}_+}$ determined by (\ref{ada52})
\emph{does not depend on the choice of inner product} $(e^{Q}\cdot,\cdot)$ or $(\cdot,\cdot)$.
(B) \emph{The positive boundary triplet $(\mathcal{H}, \Gamma_0, \Gamma_1)$.} Lemma \ref{neww45b} and relations (\ref{er2}) imply that: the subspace $\mathcal{H}=\kerr({B^*_\rho}^2+I)$ does not depend on the choice $(e^{Q}\cdot,\cdot)$ or $(\cdot,\cdot)$; \ the subspace $\mathcal{H}$ reduces $e^{Q}$; \ the operator ${B^*_\rho}{B_\rho}$ commutes with $e^{Q}$. Therefore, the decomposition (\ref{bonn41}) and the operator $\Gamma_0$ in (\ref{e7}) do not depend on the choice of inner product.
Furthermore, since $\mathcal{H}$ reduces $e^{Q}$, the
orthogonal decomposition $\mathfrak{H}_\rho=\mathcal{H}\oplus\mathcal{R}(B^2_\rho+I)$ does not depend on the choice of inner product
$(e^{Q}\cdot,\cdot)$ or $(\cdot,\cdot)$. Hence, the orthogonal projection $P_{\mathcal{H}}$ does not change and the
operator $\Gamma_1$ in (\ref{e7}) does not depend on the choice of inner products. Thus, we show that
the formulas (\ref{bonn41}) and (\ref{e7}) determine the boundary triplet
$(\mathcal{H}, \Gamma_0, \Gamma_1)$ of $B_\rho^2$, which \emph{does not depend on the choice of inner product} $(e^{Q}\cdot,\cdot)$ or $(\cdot,\cdot)$.
(C) \emph{Operators $\{T_k\}$.} It follows from (A) and (B) that the operators $\{T_k\}$ describing the image set $\{H_k\}_{k\in\mathbb{C}_+}$ in (\ref{sese2}) do not depend on the choice of inner product $(e^{Q}\cdot,\cdot)$ or $(\cdot,\cdot)$.
Combining (A) - (C) we arrive at the conclusion that the $S$-matrix ${\mathbb S}(\cdot)$ defined by (\ref{red1}) \emph{does not depend on the choice of $(e^{Q}\cdot,\cdot)$ or $(\cdot,\cdot)$}.
Due to Theorem \ref{esse3} and Remark \ref{esse3d}, the $S$-matrix
${\mathbb S}(\cdot)$ is an analytic operator-valued function in $\mathbb{C}_+$
and
\begin{equation}\label{neww1}
{\mathbb S}(-\overline{k})={\mathbb S}^{[*]}(k), \qquad \forall{k}\in\mathbb{C}_+,
\end{equation}
where ${\mathbb S}^{[*]}$ is the adjoint operator in $\mathcal{H}$ with respect to $(e^{Q}\cdot,\cdot)$.
It is clear that
\begin{equation}\label{neww31}
e^{Q}{\mathbb S}^{[*]}(k)={\mathbb S}^{*}(k)e^{Q},
\end{equation}
where ${\mathbb S}^{*}$ is the adjoint operator in $\mathcal{H}$ with respect to $(\cdot,\cdot)$.
The relation (i) in (\ref{neww69}) follows from (\ref{neww1}) and (\ref{neww31}).
Further, the operators ${\mathbb S}(k)$ are contraction operators
with respect to $(e^{Q}\cdot,\cdot)$ in $\mathcal{H}$. Hence
$$
(e^{Q}{\mathbb S}(k)f,{\mathbb S}(k)f)=(e^{Q}{\mathbb S}^{[*]}(k){\mathbb S}(k)f,f)\leq(e^{Q}f,f), \qquad \forall{f}\in\mathcal{H}.
$$
The obtained inequality and (\ref{neww1}) mean that
$$
e^{Q}{\mathbb S}(-\overline{k}){\mathbb S}(k)=e^{Q}{\mathbb S}^{[*]}(k){\mathbb S}(k)\leq{e^{Q}}
$$
with respect to the inner product $(\cdot,\cdot)$. Thus the relation (ii) in (\ref{neww69})
is established. Relation (iii) follows from Proposition \ref{neww28}.
Theorem \ref{neww78} is proved. \rule{2mm}{2mm}
\smallskip
Since the operators $\mathcal{P}$, $\mathcal{PT}$, and $Q$ are reduced by the subspace $\mathcal{H}=\kerr({B_\rho^*}^2+I)$, the formulas
\begin{equation}\label{neww24}
\textsf{P}=Y\mathcal{P}Y^{-1}, \quad \textsf{T}=Y\mathcal{T}Y^{-1}, \quad \textsf{Q}=YQY^{-1},
\end{equation}
where $Y$ isometrically maps $\mathcal{H}$ onto the subspace
$\{e^{-\delta}u : {u}\in{\mathcal N}\}$ of ${L_2((\rho,+\infty),{\mathcal N})}$ with the subsequent identification (\ref{rest1}),
determine, respectively, a unitary involution $\textsf{P}$, a conjugation $\textsf{T}$, and a self-adjoint operator $\textsf{Q}$ in ${\mathcal N}$. Considering the operator-valued function
$$
{\textsf S}(k)=Y{\mathbb S}(k)Y^{-1}
$$
with values in ${\mathcal N}$ and taking (\ref{neww24}) into account we rewrite Theorem \ref{neww78} as follows.
\begin{theorem}\label{neww144}
Under conditions of Theorem \ref{neww78}, the $S$-matrix ${\textsf S}(k)$ is an analytic operator-valued function in $\mathbb{C}_+$, which is determined by (\ref{red1b}), and has the following properties:
\begin{equation}\label{neww69b}
\begin{array}{c}
(i) \quad e^{\textsf{Q}}{\textsf{S}}(-\overline{k})={\textsf{S}}^*(k)e^{\textsf{Q}}; \vspace{3mm} \\
(ii) \quad e^{\textsf{Q}}{\textsf{S}}(-\overline{k}){\textsf{S}}(k)\leq{e^{\textsf{Q}}}; \vspace{3mm} \\
(iii) \quad \textsf{PT}{\textsf S}(-\overline{k})={\textsf S}(k)\textsf{PT},
\end{array}
\end{equation}
where ${\textsf{S}}^*$ is the adjoint operator in $\mathcal{N}$.
\end{theorem}
\subsection{An example of inverse problem solution.} Relations (\ref{neww69b}) contain information about the metric operator
$e^Q$ and the corresponding inner product $(e^Q\cdot,\cdot)$ which ensures the self-adjointness of $H$.
This leads to natural question: \emph{is it possible for a given $\mathcal{PT}$-symmetric operator $H$ to recover the corresponding
metric operator $e^Q$ by the $S$-matrix ${\textsf{S}}(\cdot)$?}
A simple example considered below show that the answer is positive for certain classes of $\mathcal{PT}$-symmetric operators.
Consider the one-dimensional Schr\"{o}dinger differential expression with singular zero-range potential
\begin{equation}\label{as3}
-\frac{d^2}{dx^2}+q_\gamma(x), \quad
q_\gamma(x)=i\gamma(<\delta',\cdot>\delta(x)+
<\delta,\cdot>\delta'(x)), \ \gamma\in\mathbb{R},
\end{equation}
where $\delta(x)$ and $\delta'(x)$ are, respectively, the Dirac
$\delta$-function and its derivative (with support at $0$).
It is easy to verify that $\mathcal{PT}q_\gamma(x)=q_\gamma(x)\mathcal{PT}$,
where $\mathcal{P}$ is the space parity operator $\mathcal{P}f(x)={f(-x)}$ and $\mathcal{T}$ is the complex conjugation operator
$\mathcal{T}f(x)=\overline{f(x)}$.
The operator realization $H_\gamma$ of (\ref{as3}) in $L_2(\mathbb{R})$ is defined as
$$
H_\gamma=l_{\mathrm{reg}}\upharpoonright\mathcal{D}(H_\gamma), \quad
\mathcal{D}(H_\gamma)=\{f\in{{W_2^2}(\mathbb{R}\backslash\{0\})} \ :
\ l_{\mathrm{reg}}(f)\in{L_2(\mathbb{R})}\},
$$
where the regularization $l_{\mathrm{reg}}$ of the differential expression (\ref{as3}) onto
${W_2^2}(\mathbb{R}\backslash\{0\})$ has the form
$$
{l}_{\mathrm{reg}}(\cdot)=-\frac{d^2}{dx^2}+i\gamma(<\delta_{\mathrm{ex}}',\cdot>\delta(x)+
<\delta_{\mathrm{ex}},\cdot>\delta'(x)).
$$
Here $-{d^2}/{dx^2}$ acts on ${W_2^2}(\mathbb{R}\backslash\{0\})$ in
the distributional sense and
$$
<\delta_{\mathrm{ex}}, f>=\frac{f(+0)+f(-0)}{2}, \quad <\delta_{\mathrm{ex}}', f>=-\frac{f'(+0)+f'(-0)}{2}
$$
for all $f(x)\in{{W_2^2}(\mathbb{R}\backslash\{0\})}$.
An equivalent description of $H_\gamma$ is (see \cite[Theorem 1]{AK}):
\begin{equation}\label{neww74}
H_\gamma=-\frac{d^2}{dx^2}, \quad \mathcal{D}(H_\gamma)=\left\{f\in{{W_2^2}(\mathbb{R}\backslash\{0\})} \ : \
\begin{array}{cc}
f(0+)=e^{i\beta}f(0-) \\
f'(0+)=e^{-i\beta}f'(0-)
\end{array} \right\},
\end{equation}
where $e^{i\beta}=\frac{2+i\gamma}{2-i\gamma}$.
The operator $H_\gamma$ is $\mathcal{PT}$-symmetric in $L_2(\mathbb{R})$, its spectrum coincides with $\mathbb{R}_+$ for $|\gamma|\not=2$ and
$\sigma(H_\gamma)=\mathbb{C}$ for $|\gamma|=2$ (\cite[Theorem 2]{AK}).
It follows from (\ref{ses1}) and (\ref{neww74}) that $H_\gamma$ is $0$-perturbed. Therefore, for $|\gamma|\not=2$, we can determine
the $S$-matrix ${\textsf{S}}(\cdot)$ by the formula (\ref{rest2}).
The coefficients $T_k^l, T_k^r, R_k^l, R_k^r$ in (\ref{rest2})
are determined by the condition that functions $f_j$ in (\ref{deder1}) belong to $\mathcal{D}(H_\gamma)$ in (\ref{neww74}).
Simple calculations give
$$
T_k^l=T_k^r={\frac{\textsf{Re}\ k}{k\cos\beta}}, \quad R_k^r={i\frac{(\textsf{Re}\ k)\sin\beta-(\textsf{Im}\ k)\cos\beta}{k\cos\beta}},
$$
and $\displaystyle{R_k^l=-{i\frac{(\textsf{Re}\ k)\sin\beta+(\textsf{Im}\ k)\cos\beta}{k\cos\beta}}}$.
Substituting these quantities into (\ref{rest2}) and taking into account that $\rho=0$, we obtain
\begin{equation}\label{deder7}
{\textsf S}(k)=-\left(\begin{array}{cc}
i\tan\beta & \frac{1}{\cos\beta} \vspace{4mm} \\
\frac{1}{\cos\beta} & -i\tan\beta
\end{array}
\right).
\end{equation}
Thus, the $S$-matrix ${\textsf S}(k)$ is a constant in $\mathbb{C_+}$ and it is defined for all $|\gamma|\not=2$.
If $|\gamma|=2$, i.e., $\gamma=2$ or $\gamma=-2$, then $\beta=\frac{\pi}{2}$ or $\beta=-\frac{\pi}{2}$ and
the $S$-matrix does not exist in $k\in\mathbb{C}_+$ (the entries of ${\textsf S}(k)$ are $\infty$).
This is natural because $\sigma(H_{\pm{2}})=\mathbb{C}$.
In our case, the isometric mapping $Y:\mathcal{H}\to\mathbb{C}^2$ is defined by (\ref{deder8}). Then the image $\textsf{P}$
of $\mathcal{P}$ in (\ref{neww24}) has the form $\textsf{P}=\sigma_1=\left(\begin{array}{cc}
0 & 1 \\
1 & 0 \end{array}
\right)$. The image $\textsf{T}$ of $\mathcal{T}$ is the standard operator of conjugation in $\mathbb{C}^2$.
For such choice of $\textsf{P}$ and $\textsf{T}$ the relation (iii) of (\ref{neww69b}) holds for the $S$-matrix (\ref{deder7}).
Let us suppose that the relation (i) holds for some choice of self-adjoint operator $\textsf{Q}$ in $\mathbb{C}^2$, i.e.
\begin{equation}\label{deder9}
e^{\textsf{Q}}\left(\begin{array}{cc}
i\tan\beta & \frac{1}{\cos\beta} \vspace{4mm} \\
\frac{1}{\cos\beta} & -i\tan\beta
\end{array}
\right)=\left(\begin{array}{cc}
-i\tan\beta & \frac{1}{\cos\beta} \vspace{4mm} \\
\frac{1}{\cos\beta} & i\tan\beta
\end{array}
\right)e^{\textsf{Q}}.
\end{equation}
The operator $\textsf{Q}$ is a $2\times{2}$ matrix. Hence,
\begin{equation}\label{deder10}
\textsf{Q}=\chi_0\sigma_0+\chi_1\sigma_1+\chi_2\sigma_2+\chi_3\sigma_3, \qquad \chi_j\in\mathbb{C},
\end{equation}
where $\sigma_3=\left(\begin{array}{cc}
1 & 0 \\
0 & -1 \end{array}
\right)$ and $\sigma_2=i\sigma_1\sigma_3=\left(\begin{array}{cc}
0 & -i \\
i & 0 \end{array}
\right)$ are the Pauli matrices.
It follows from (\ref{usa6}) and (\ref{neww24}) that
$$
\textsf{Q}=\textsf{Q}^*, \quad \textsf{PQ}=-\textsf{QP}, \quad \textsf{TQ}=-\textsf{QT}.
$$
Imposing these conditions in (\ref{deder10}) we derive that $\textsf{Q}={\chi\sigma_2}$, where $\chi\in\mathbb{R}$.
Hence, $e^\textsf{Q}=e^{\chi\sigma_2}=(\cosh\chi)\sigma_0+(\sinh\chi)\sigma_2$.
Substituting this expression into (\ref{deder9}) and making elementary calculations we arrive at the conclusion that
the relation (\ref{deder9}) is true if and only if
\begin{equation}\label{deder11}
\tanh\chi=\sin\beta.
\end{equation}
Thus, we determine uniquely the operator $e^\textsf{Q}=e^{\chi\sigma_2}$ which satisfies relation (i)
in (\ref{neww69b}) with the $S$-matrix (\ref{deder7}).
Denote by $\mathcal{R}$ the operator of unitary involution $\mathcal{R}f(x)=(\textsf{sgn}\ x)f(x)$
in $L_2(\mathbb{R})$. The subspace $\mathcal{H}$ reduces $\mathcal{R}$ and its image $\textsf{R}=Y\mathcal{R}Y^{-1}$
in $\mathbb{C}^2$ coincides with $\sigma_3$. Therefore, the image of unitary involution $i\mathcal{PT}$ in $\mathbb{C}^2$
will coincide with $i\textsf{PR}=i\sigma_1\sigma_3=\sigma_2$.
Taking this relationship and Theorems \ref{neww78}, \ref{neww144} into account, it is natural to suppose that the operator $H_\gamma$ will have the property of $\mathcal{C}$-symmetry realized by the operator
\begin{equation}\label{deder12}
\mathcal{C}=e^{-i\chi\mathcal{PR}}\mathcal{P}=(\cosh\chi)\mathcal{P}+i(\sinh\chi)\mathcal{R},
\end{equation}
where $\chi$ is determined by (\ref{deder11}) and $H_\gamma$ turns out to
be a self-adjoint operator with respect to new inner product
$(e^{i\chi\mathcal{PR}}\cdot,\cdot)$ in $L_2(\mathbb{R})$.
This assumption is true due to the results \cite[Subsection 5.1]{KKK}, where was shown that the operator $H_\gamma$ has the $\mathcal{C}$-symmetry (\ref{deder12}) and $H_\gamma$ is a self-adjoint operator with respect to $(e^{i\chi\mathcal{PR}}\cdot,\cdot)$.
\section{Summary and Discussion}
In the present paper an operator-theoretical interpretation of the Lax-Phillips scattering theory \cite{LF} developed in
\cite{KU1,AlAn,Kioto} has been reshaped to define and calculate $S$-matrices
of $\mathcal{PT}$-symmetric $\rho$-perturbed operators.
The crucial role is played by the concept of $\rho$-perturbed operator (Definition \ref{did2})
that allows one to consider the scattering of many concrete systems with perturbation supported
at bounded domain from a unique point of view.
In our opinion, the advantages of the proposed approach have their origin in the
following properties:
\begin{enumerate}
\item[(i)] for typical examples, the general formula (\ref{red1}) for $S$-matrix
leads to well known expressions derived by the standard methods (see subsection 3.3 or \cite[Section 3]{AlAn});
\item[(ii)] the formula (\ref{red1}) ensure simple links to powerful mathematical methods of the extension
theory of symmetric operators.
\end{enumerate}
We believe that interplay between (i) and (ii) will provide some deeper insights into the structural subtleties of
inverse scattering problems for $\mathcal{PT}$-symmetric operators.
Let us illustrate this point by considering the case
of positive self-adjoint $\rho$-perturbed operators. In that case the corresponding $S$-matrix (\ref{red1}) uniquely determines
the image set $\{H_k\}_{k\in\mathbb{C}_+}$ by formula (\ref{sese2}). The image set $\{H_k\}_{k\in\mathbb{C}_+}$
determines a generalized resolvent of the symmetric operator $B^2_\rho$ \cite{AkG1,Sh}:
$$
R_k=(H_k-k^2)^{-1}, \qquad k\in\mathbb{C}_+.
$$
This means that
$$
(R_kf,g)=\int_{-\infty}^\infty\frac{d(F_tf,g)}{t-k}, \qquad {f,g}\in\mathfrak{H}_\rho,
$$
where $F_t$ is a spectral function of the symmetric operator $B^2_\rho$. By the Naimark theorem \cite{Naimark}, the spectral function $F_t$
uniquely determines a minimal positive self-adjoint $\rho$-perturbed operator $H$ up to unitary equivalence (see \cite[Section 4]{AlAn} for detail). Thus, we derive the solution of inverse problem in the following form:
$S$-matrix $\to$ image set $\{H_k\}_{k\in\mathbb{C}_+}$ $\to$ generalized resolvent $R_k$ $\to$ spectral function $F_t$ of symmetric operator
$B^2_\rho$ $\to$ $\rho$-perturbed operator $H$.
It seems highly interesting to extend at least part of these conclusions to the case of $\mathcal{PT}$-symmetric $\rho$-perturbed operators $H$
with $\mathcal{C}$-symmetries $\mathcal{C}=e^{-Q}\mathcal{P}$ trying to recover an information about the metric operator $e^{Q}$
from the $S$-matrix.
The methods developed in the present paper can also be applied to Dirac systems with $\rho$-perturbation type.
We hope to undertake those discussions in a forthcoming paper.
\smallskip
\noindent \textbf{Acknowledgements.}
Research supported by the Polish Ministry of Science and Higher Education: 11.11.420.04 and N201546438.
The second named author thanks JRP IZ73Z0 (28135) of SCOPES 2009-2012
for support.
|
{
"timestamp": "2012-06-25T02:03:05",
"yymm": "1203",
"arxiv_id": "1203.2110",
"language": "en",
"url": "https://arxiv.org/abs/1203.2110"
}
|
\section{Formulation of the problem}
The calculation is based on the coupled-channels AGS-Faddeev treatment of the $\bar KNN-\pi\Sigma N$ three-body system. The details of this approach have been already described in detail in several papers \cite{Nina1,Sato,Nina3}, here we shall recall them only briefly, mainly to introduce the notations. The operator AGS equations for the transition operators $U_{i1}$ read
\begin{equation}
\label{AGS}
U_{i1}=(1-\delta_{ij})G_0^{-1} +\sum_{j\neq i}T_jG_0U_{j1},
\end{equation}
where $i,j=1,2,3$ are the usual pair-spectator indices,
$$
T_j=V_j+V_jG_0T_j
$$
are the two-particle $T$-operators and $G_0(z)=(z-H_0)^{-1}$ is the free Green-operator.
The configuration space $|\textbf{x}_{i}\textbf{y}_i\nu_i\rangle$ in which these operators act, apart from the usual Jacobi momentum variables $\textbf{x}_{i}\textbf{y}_i$ contain a discrete index $\nu_i=(\alpha,\sigma_i)$, which is a combination of the particle composition index $\alpha$
$$
\alpha=\{1,2,3\}=\{\bar KN_1N_2,\pi\Sigma_1N_2,\pi N_1\Sigma_2\}
$$
and an isospin label $\sigma_i=(I_iI)$ corresponding to pair isospin $I_i$ of the particle pair $i$ and total isospin $I$:
$$
\sigma_i\sim\left[\left[t_jt_k\right]^{I_i}t_i\right]^I
$$
This choice of the isospin labels corresponds to the "isospin representation", which is useful when isospin conserving pair interactions are used. Another possibility is the equivalent "charge state" or "particle" representation, characterized by the 3rd component of the particle isospins:
$$
\sigma_0\sim\{t_{1z},t_{2z},t_{3z}\}\qquad \textrm{or for $\alpha=1$}\qquad \sigma_0\sim\{\bar K^0n_1n_2,K^-p_1n_2,K^-n_1,p_2\}
$$
These representations can be transformed into each other with the help of orthogonal matrices
\begin{equation}
\label{Bmat}
B^{\alpha}_{ij}=\langle\alpha\sigma_i|\alpha\sigma_j\rangle \qquad i,j=0,1,2,3
\end{equation}
composed from 6j symbols for $i,j=1,2,3$ and from Clebsch-Gordan coefficients for ${i=0,j=1,2,3}$.
Using isospin conserving separable interactions of the form
\begin{equation}
V^i=\sum_{\nu_i\nu_i'}
\delta_{I_iI_i'}\delta_{II'}|g^{I_i}_{\nu_i}\rangle\lambda^{I_i}_{\nu_i\nu_i'}\langle
g^{I_i}_{\nu_i'}|
\end{equation}
the $T_i(z)$ operators can be written as
\begin{equation}
T_i(z)=\sum_{\nu_i\nu_i'}|g^{I_i}_{\nu_i}\rangle\tau_{\nu_i\nu_i'}(z)\langle g^{I_i'}_{\nu_i'}|
\end{equation}
with $\tau_{\nu_i\nu_i'}(z)$ being the usual (c-number) matrix, defined as:
\begin{equation}
\label{tau}
(\tau_{\nu_i\nu_i'}(z))^{-1}=(\lambda^{I_i}_{\nu_i\nu_i'})^{-1}-\langle g^{I_i}_{\nu_i}|G_0(z)|g^{I_i'}_{\nu_i'}\rangle
\end{equation}
The matrix indices $\nu_i$ in Eq.(\ref{tau}) consist of the particle space label $\alpha$ and the isospin label $\sigma_i=(I_iI)$. Due to isospin conservation of our interactions, the coupling constant matrix $\lambda$ is diagonal in $(I_iI)$, while for particle pairs $i$ capable to change their identity $(\bar KN\leftrightarrow\pi\Sigma)$ it has non-diagonal elements in the particle labels $(\alpha\alpha')$. As for the matrix elements of $G_0$, it does not change particle identities, thus it is diagonal in $\alpha$, and if we take averaged masses for particles within an isospin multiplet, it is also diagonal in pair- and total isospin indices $(I_iI)$. However, if physical (unequal) masses are used,
$G_0$ will be diagonal only in "particle" representation, while in the $\sigma_i$ "isospin" representation it will acquire non-diagonal elements both in $I_i$ and $I$, proportional to the mass differences.
The equations (\ref{AGS}) for the transition operators take the form
\begin{equation}
U_{i1}=(1-\delta_{ij})G_0^{-1} +\sum_{j\ne i}\sum_{\nu_j'}|g_{\nu_j}\rangle \tau_{\nu_j,\nu_j'}\langle X^j_{\nu_j'}|\qquad \textrm{with}
\qquad \langle X^j_{\nu_j'}|=\langle g^{I_j}_{\nu_j'}|G_0U_{j1}
\end{equation}
Introducing the functions $X^j_{\nu_j}(\textbf{y}_j)=\langle X^j_{\nu_j}|\Phi_0\rangle$, where $\textbf{y}_i$ is the momentum of the spectator particle, corresponding to the pair $j$ and $|\Phi_0\rangle=|\varphi_dP_{K}\rangle$ is the initial state with the deuteron wave function $|\varphi_d\rangle$ and $P_K$ - the momentum of the incident kaon, we get the set of integral equations:
\begin{equation}
\label{system}
X^i_{\nu_i}(\textbf{y}_j)=(1-\delta_{i1})\langle g^{I_i}_{\nu_i}|
\Phi_0\rangle+\sum_{j\ne i}\sum_{\nu_j,\nu_j'}\int Z_{\nu_i\nu_j}(\textbf{y}_i
,\textbf{y}_j)\tau_{\nu_j\nu_j'}(z-y^2_j/2\mu_{j,ki})X^j_{\nu_j'}(\textbf{y}_j)d\textbf{y}_j
\end{equation}
with the kernel
\begin{equation}
Z_{\nu_i\nu_j}(\textbf{y}_i,\textbf{y}_j)=\langle g^{I_i}_{\nu_i}|G_0(z)|g^{I_j}_{\nu_j}\rangle .
\end{equation}
The size of the system (\ref{system}) can be reduced by introducing symmetric (antisymmetric) combinations of X-functions, with respect to interchange of baryon numbering. The baryon spins do not enter explicitly in this formalism, therefore the total baryon spin $S$ remains unchanged in the process (is a conserved quantum number). For a given $S$ value the total antisymmetry required by the Pauli principle has to be ensured by the space-isospin part. Thus for $S=0$ ($K^-pp$ system) we have to work with the symmetric combinations of $X$-s, while for $S=1$ (our $K^-d$ system) the antisymmetric combinations are needed. As a result, the labeling of the unknown functions $\nu_i=(\alpha\sigma_i)$ is changed to ${\mu_a=(a,\sigma_a)}$, where $a$ denotes a pair of interacting particles, irrespectively to which original particle composition channel they belonged and $\sigma_a$ denotes the corresponding isospin values. Thus we are left with $X_{a\sigma_a}(\textbf{y}_a)$ and $a$ can take the values $\bar KN,NN,\Sigma N$ and $\pi\Sigma$ ($\pi N$ is missing, since we neglected the $\pi N$ interaction, see next section).
The break-up transition operator $U_{01}$ can be expressed in terms of the $U_{i1}$-s as:
$$
U_{01}={1\over 2}(U_{11}+U_{21}+U_{31})
$$
and the break-up amplitude reads
\begin{equation}
\label{BU}
A_{BU}=\langle\Phi_f|U_{01}|\Phi_0\rangle .
\end{equation}
For the reaction under consideration the properly antisymmetrized final state is
$$
|\Phi_f\rangle=|\textbf{x}_{\pi\Sigma},\textbf{y}_N;\sigma_{\pi\Sigma}\rangle=
{1\over\sqrt{2}}(|\textbf{x}_{\pi\Sigma_1},\textbf{y}_{N_2};\sigma_{\pi\Sigma_1}\rangle-
|\textbf{x}_{\pi\Sigma_2},\textbf{y}_{N_1};\sigma_{\pi\Sigma_2}\rangle)
$$
The break-up amplitude can be expressed in terms of the X-functions as
\begin{eqnarray}
\label{ABU}
&A_{BU}(\textbf{x}_{\pi\Sigma},\textbf{y}_N;\sigma_{\pi\Sigma})
=
\nonumber &\\
\label{BUamp}
&-g_{\pi\Sigma}(\textbf{x}_{\pi\Sigma})
\left[\tau_{\pi\Sigma,\bar KN}(z-y^2_N/2\mu_{N,\pi\Sigma})X_{\bar KN}(\textbf{y}_N)+
\tau_{\pi\Sigma,\pi\Sigma}(z-y^2_N/2\mu_{N,\pi\Sigma})X_{\pi\Sigma}(\textbf{y}_N)\right] &\\
&-B^2_{31}g_{\Sigma N}(u\textbf{y}_N+v\textbf{x}_{\pi\Sigma})\tau_{\Sigma N,\Sigma N}
(z-|\textbf{x}_{\pi\Sigma}-w\textbf{y}_N|^2/2\mu_{\pi,\Sigma N})X_{\Sigma N}
(\textbf{x}_{\pi\Sigma}-w\textbf{y}_N)\nonumber &,\
\end{eqnarray}
where $B^2_{31}$ is an isospin recoupling matrix (see Eq.(\ref{Bmat})), $u,v\ \textrm{and}\ w$ are mass coefficients of the transformation between Jacobi momentum sets. In Eq.(\ref{BUamp}) we omitted the isospin labels, the quantities are vectors (matrices) in isospin space. The on-shell amplitude for a given neutron energy $E_n$ depends on $E_n,t$ and the isospin labels $\sigma_{\pi\Sigma}$
$$
A(E_n,t,\sigma_{\pi\Sigma})=A_{BU}(\textbf{x}_{\pi\Sigma},\textbf{y}_N;\sigma_{\pi\Sigma})
$$
with
$$
|\textbf{y}_n|=\sqrt{2E_n\mu_{N,\pi\Sigma}};\
|\textbf{x}_{\pi\Sigma}|=\sqrt{2(E_{\pi\Sigma N}-E_n)\mu_{\pi\Sigma}};\
t=\cos(\textbf{x}_{\pi\Sigma},\textbf{y}_N)
$$
The physically observable final state corresponds to a certain particle composition, not to a definite isospin state, therefore the amplitude has to be transformed into the $\sigma_0$ representation, using the suitable $B$ matrix of Eq.(\ref{Bmat}):
{$$
A(E_n,t,\sigma_0)=\sum_{\sigma_{\pi\Sigma}}\left(B^2_{03}\right)_{\sigma_0,\sigma_{\pi\Sigma}}A(E_n,t,\sigma_{\pi\Sigma}),
$$
where $\sigma_0$ can be $\{\pi^+\Sigma^-n,\pi^0\Sigma^0n,\pi^-\Sigma^+n\}$.} The neutron spectrum is proportional to the differential cross section
\begin{equation}
\label{cross}
P(E_n,t,\sigma_0)\sim{d\sigma\over d\Omega_{\textbf{x}_{\pi\Sigma}}d\Omega_{\textbf{y}_N}dE_n}=
(2\pi)^4\mu_{\pi\Sigma}\mu_{N,\pi\Sigma}\mu_{K,NN}{x_{\pi\Sigma}y_N\over P_K}|A(E_n,t,\sigma_0)|^2
\end{equation}
The inclusive neutron spectrum (when no other particles are detected) is given by
\begin{equation}
\label{PEn}
P(E_n)=\sum_{\sigma_0}\int_{-1}^{1}dtP(E_n,t,\sigma_0)
\end{equation}
The above considerations refer to the neutrons emerging from the reaction $K^-+d\rightarrow \pi+\Sigma+n$; when the energy of the incident kaon exceeds the deuteron binding energy, neutrons are also emitted from the reaction $K^-+d\rightarrow K^-+p+n$. Their spectrum can be deduced in a similar way to Eq.(\ref{BUamp}) from the $X_{\bar KN}$ and $X_{NN}$ functions. The allowed energy range for neutrons from the first reaction is $(0,E_K+E_d+\Delta)$, while for the second it is $(0,E_K+E_d)$, where $\Delta$ is the difference of the $\bar KN$ and $\pi\Sigma$ threshold energies.
\section{Details of the calculation and the input}
The main purpose of the present work is to study the possible signature(s) of the $\Lambda(1405)$ resonance in the neutron spectra from the reaction (\ref{react}). In our calculation we used the two-body interactions of \cite{Nina3} and \cite{Nina4}, which are adjusted for our three-body model. They are $s$-wave, separable isospin dependent and isospin conserving interactions with Yamaguchi type form-factors.
In particular, for the two-channel $\bar KN-\pi\Sigma$ interaction we used two variants, both having a one and a two pole version for the $\Lambda(1405)$. They both reproduce all available experimental data on the low-energy $\bar KN$ system, the first one is fitted to the KEK data on the kaonic hydrogen $1s$ level shift, while the second one reproduces the most recent SIDDHARTA data. Their pole positions are shown in Table \ref{poles}.
\begin{table}[h]
\caption{\label{poles}Pole positions of the $\bar{K}N-\pi\Sigma$ potentials (in MeV), the negative real parts correspond to distances from the $\bar{K}N$ threshold.}
\begin{center}
\begin{tabular}{c|c|c}
\hline\hline
& KEK\ & SIDDHARTA\\
\hline
1-pole\ \ &$-23.6-35.6$\ \textbf{i}\ ($1411.0-35.6$\ \textbf{i})&$-6.4-46.8$\ \textbf{i}\ ($1428.1-46.8$\ \textbf{i})\\
2-pole\ \ &$-22.2-36.3$\ \textbf{i}\ ($1412.4-36.3$\ \textbf{i})&$-14.8-57.2$\ \textbf{i}\ ($1419.8-57.2$\ \textbf{i})\\
{}&$-58.8-102.5$\ \textbf{i}\ ($1375.8-102.5$\ \textbf{i})&$-56.6-101.8$\ \textbf{i}\ ($1380.0-101.8$\ \textbf{i})\\
\hline
\end{tabular}
\end{center}
\end{table}
The numbers in Table \ref{poles} differ slightly from those given in the original papers \cite{Nina3,Nina4}. The reason is, that the above ones were calculated with averaged masses and without Coulomb interaction - as they appear in most of the 3-body calculations, - while the fitting to the experimental data was performed with physical masses and Coulomb interaction. Since the main aim of the present work is to study the appearance of subthreshold resonances of different type in a 3-body reaction, we kept both interactions, not only the more advanced one.
The triplet $NN$ interaction is a two-term one to account for the short range repulsion, with form-factors fitted to reproduce the deuteron binding energy and $s$-wave phase shifts.
The $S=1$ $\Sigma N$ interaction in the $I=1/2$ isospin state is complex, since it was deduced from a two-channel $\Sigma N-\Lambda N$ interaction, while for $I=3/2$ it is real.
The $\pi N$ interaction was neglected in our calculation due to its weak $s$-wave part.
The total angular momentum was restricted to $L=0$ since we believe, that for our $s$-wave interactions the essential dynamics can be traced in spite of this limitation. Keeping the interactions $s$-wave, the extension to higher angular momenta is straightforward, unlike the case of inclusion of $p,d,..$-wave interactions.
We considered incident kaon energies in the interval $E^{cm}_{K^-}=0-50\ MeV$ ${(P^{LAB}_K \sim 0-250\ MeV/c)}$. The system of integral equations (\ref{system}) in the case of physical masses ($I=1/2$ and $I=3/2$ mixed) consists of 12 equations, while for averaged masses - of 8.
As a numerical method we used expansion of the unknown functions on a cubic spline basis, for the distribution of nodes and collocation points the prescription of \cite{Svenne} was used with a slight modification to allow nonsymmetric intervals and distributions on the two sides of the break-up singularity. Complete convergence of the results was achieved for $\sim 20$ nodes in the non-break-up channels, while for the break-up channels $\sim 30-35$ nodes were necessary. Apart from the lower dimensionality of the matrices to be inverted, the use of spline expansion is especially advantageous when break-up amplitudes are calculated, since no interpolation of the solutions is needed.
\section{Results and discussion}
We start the presentation of our results by a "by-product": the effect of the physical versus averaged masses of the $\bar K^0$ and $K^-$ mesons on the $K^-d$ scattering length. The inclusion of the possibility of isospin mixing due to this mass difference allowed us to extend the results of \cite{Nina3,Nina4} in this respect. Our results for the two potential versions are shown in Table \ref{scatlen}. The results for averaged masses coincide with those of \cite{Nina3,Nina4}, while for physical masses they differ by a few per cent, mainly in the real part. At present level of accuracy of available information - both theoretical and experimental - on the $\bar KN$ interaction and $\bar K$-nuclear clusters this difference does not seem to be essential. However, once it might become useful to have some numerically reliable information on the order of magnitude of this effect.
\begin{table}[h]
\caption{\label{scatlen}$K^-d$ scattering lengths of the $\bar{K}N-\pi\Sigma$ potentials for physical and averaged masses of $K^-$ and $\bar{K}^0$ (in fm).}
\begin{center}
\begin{tabular}{c|c|c|c|c}
\hline
\hline
&\multicolumn{2}{c|}{KEK}&\multicolumn{2}{c}{SIDDHARTA}\\\cline{2-5}
&\ averaged\ &\ physical\ &\ averaged\ &\ physical\ \\
\hline
1-pole\ \ &$-1.49+0.97$\ \textbf{i} &$-1.52+0.98$\ \textbf{i}&$-1.47+1.22$\ \textbf{i} &$-1.50+1.23$\ \textbf{i}\\
2-pole\ \ &$-1.57+1.10$\ \textbf{i} &$-1.60+1.12$\ \textbf{i}&$-1.50+1.23$\ \textbf{i} &$-1.54+1.24$\ \textbf{i}\\
\hline
\end{tabular}
\end{center}
\end{table}
On the other hand, when calculating neutron spectra our interest was focused on qualitative signals of the $\Lambda(1405)$ in the line shapes, therefore we used averaged masses, what simplified the numerical work to some extent.
We have calculated the inclusive neutron spectra $P(E_n)$ (\ref{PEn}) for different incident kaon energies, both below and above the deuteron break-up threshold.
In order to allow a comparison with the only available experimental data in our energy range we performed a calculation for $E_K=0$ (this was also needed for the scattering lengths). The results are shown in Fig.\ref{exp}.
\begin{figure}[h]
\centering
\includegraphics[width=\textwidth]{direct0}
\caption{
Momentum distribution of the neutrons from the reaction $K^-+d\rightarrow\pi^\pm +\Sigma^\mp + n$ for stopped kaons
}
\label{exp}
\end{figure}
Unfortunately, this can be called only a "quasi-comparison", since it is not clear, what calculated theoretical quantity should be compared with the data shown on their Fig.2. If the reaction is "at rest", then probably it starts from an atomic orbit, which case necessitates a somewhat different Faddeev treatment. Our calculation can imitate the "at rest" criterion by taking $E_K=0$, but then our cross section formula (\ref{cross}) must be modified: the incoming flux normalization (division by $P_K$) has to be removed and an extra $y_N$factor must be added to get momentum distribution instead of energy spectrum. Still we believe, that the curves displayed on Fig.\ref{exp}. qualitatively correspond to the same momentum distribution, but have different absolute normalization. Therefore, to bring the curves together, the experimental ones were scaled down, as indicated in the captions. The difference of the scaling factors (and of the arbitrary units on the y-axis) corresponds to the fact, that the number of neutrons coming with $\Sigma^-$ exceeds the number of those, emitted with $\Sigma^+$ by a factor of $\sim 2.5$ (in \cite{Tan} it was estimated as $2$). The agreement can be considered as acceptable, especially having in mind the experimental uncertainties. However, due to the practical indistinguishability of the theoretical curves, from the point of view of the signature of the $\Lambda(1405)$ in this reaction, this agreement seems to be of not much help.
For kaon incident energies $E_{K^-}^{cm}=1,20,50 MeV$ the results are displayed in Figs. \ref{fig1}, \ref{fig2} and \ref{fig3}, respectively, (upper left graphs)\footnote{Since our main concern is the possible trace of the $\Lambda(1405)$ in the line shapes of the calculated spectra, the arbitrary units on the $Y$-axes of our graphs are chosen to optimize visibility.}.
\begin{figure}[h]
\centering
\includegraphics[width=\textwidth]{fig1b}
\caption{Neutron spectra for kaon incident energy $E^{cm}_{K^-}=1\ MeV$.
Upper left graph - direct neutron spectra $P(E_n)$, eq.(\ref{PEn});
lower left graph - deviation spectrum $P_{DEV}(E_n)$, eq.(\ref{DEV1});
upper right graph - single scattering deviation spectrum $P_{DEV}^{sing}(E_n)$, eq. (\ref{DEV1});
lower right graph - "original" $\Lambda(1405)$ as $\pi^0\Sigma^0$ elastic cross section.
}
\label{fig1}
\end{figure}
The overall shape of the spectra is a strong peak near the origin with no signal of the $\Lambda(1405)$ resonance. The direct $P(E_n)$ spectra are practically indistinguishable for the four considered $\bar{K}N-\pi\Sigma$ potentials. For kaon energies above the deuteron binding energy there are two modifications: the neutron spectra from the
$K^-+d\to \pi+\Sigma+n$ channel show a cusp at neutron energies $E_n=E_{th}$ when the $\bar KN$ system is at its threshold, and additional neutrons show up from the $K^-+d\to K^-+p+n$ reaction in a form of a structureless
\begin{figure}[h]
\centering
\includegraphics[width=\textwidth]{fig2b}
\caption{Neutron spectra for kaon incident energies $E^{cm}_{K^-}=20\ MeV$ The four graphs are the same as in Fig.\ref{fig1}.
\label{fig2}}
\end{figure}
bump between $E_n=0$ and $E_n=E_{th}$ (on the graphs it is scaled down to allow to draw it on the same plot with the other neutrons). The reason, why the $\Lambda(1405)$ is not seen in these spectra is essentially kinematical:
the neutron energy in the resonance region should exceed the incident energy of the kaon by the amount of energy, which separates the pole position from the $\bar KN$ threshold, while in the deuteron the neutron energy (momentum) distribution is dominated by the low energy part.
\begin{figure}[h]
\centering
\includegraphics[width=\textwidth]{fig3b}
\caption{Neutron spectra for kaon incident energies $E^{cm}_{K^-}=50\ MeV$ The four graphs are the same as in Fig.\ref{fig1}.
\label{fig3}}
\end{figure}
In order to eliminate this kinematical "inconvenience" Esmaili, Akaishi and Yamazaki (EAY) \cite{AY} propose to consider instead of $P(E_n)$ the deviation spectrum:
\begin{equation}
\label{DEV}
P_{DEV}={P(E_n)\over P_{nonres}(E_n)}
\end{equation}
In Eq.(\ref{DEV}) $P_{nonres}(E_n)$ is a non-resonant background spectrum, containing the kinematics of the reaction. Let's see, how this idea can be realized in our case.
Considering the zero-order iteration of our (symmetrized) system of integral equations (\ref{system}), the only non-vanishing $X$ will be the inhomogeneous term:
\begin{equation}
\label{sing}
X_{\bar {K}N}(y_N)=\langle g_{\bar {K}N}|\Phi_0\rangle=\langle g_{\bar {K}N}|\varphi_d P_K\rangle .
\end{equation}
This ansatz is usually called single scattering approximation. Substituting eq.(\ref{sing}) into eq.(\ref{ABU}) we get the corresponding break-up amplitude:
\begin{eqnarray}
\label{ABUsing}
A_{BU}^{sing}(x_{\pi\Sigma},y_N;\sigma_{\pi\Sigma})
&=&g(x_{\pi\Sigma})\tau_{\pi\Sigma,\bar{K}N}(z-y_N^2/2\mu_{N,\pi\Sigma})\langle g_{\bar {K}N}|\varphi_d
P_K\rangle\nonumber\\
&=&\langle x_{\pi\Sigma},y_N|T_{\pi\Sigma,\bar{K}N}|\varphi_d P_K\rangle,
\end{eqnarray}
which is the matrix element of the two-body $T$ operator between the initial and final state. It contains two-body dynamics through the $T$ operator and the kinematical input: the transformation between the Jacobi-coordinates and the deuteron wave function. This is basically the formula, which EAY used to calculate the transition amplitude from the $K^-d$ atomic state to the $\pi\Sigma n$ continuum. As for the non-resonant amplitude they suggest to replace $T_{\pi\Sigma,\bar{K}N}$ in (\ref{ABUsing}) by $V_{\pi\Sigma,\bar{K}N}$:
\begin{equation}
\label{born1}
A_{BU}^{Born}(x_{\pi\Sigma},y_N;\sigma_{\pi\Sigma})=\langle x_{\pi\Sigma},y_N|V_{\pi\Sigma,\bar{K}N}|\varphi_d P_K\rangle,
\end{equation}
that is, to use the Born-approximation, which contains all the kinematics.
Thus we have three amplitudes with the properties
\begin{itemize}
\item[-]{$A_{BU}\ \longrightarrow$ three-body dynamics + three-body kinematics,}
\item[-]{$A_{BU}^{sing}\ \longrightarrow$ two-body dynamics + three-body kinematics,}
\item[-]{$A_{BU}^{Born}\ \longrightarrow$ \qquad\qquad\qquad\qquad\qquad three-body kinematics,}
\end{itemize}
and we expect, that the DEV spectra
\begin{equation}
\label{DEV1}
P_{DEV}(E_n)=P(E_n)/P^{Born}(E_n)\ ;\ \ P_{DEV}^{sing}(E_n)=P^{sing}(E_n)/P^{Born}(E_n)
\end{equation}
will display (reveal) three- and two-body dynamics, respectively.
It is assumed, that the $E_n$ dependence of $P_{nonres}(E_n)=P^{Born}(E_n)$ is basically determined by the features of the initial and final states, while the details of the $V_{\bar KN,\pi\Sigma}$ potential (within reasonable limits) influence it only weakly. This expectation is important, if the deviation spectrum method is to be applied for extracting some information on $\Lambda(1405)$ from an experimentally measured neutron spectrum. (Hopefully, the matrix element (\ref{born1}) can be calculated in an experimental group, too.) To check this anticipated model-independence of the method, we calculated $P^{Born}(E_n)$ \emph{not} with our realistic $\bar KN-\pi\Sigma$ interactions, but with the simplest possible separable potential:
$$
\langle x_{\pi\Sigma}|V^I_{Born}|x_{\bar KN}\rangle={1\over x_{\pi\Sigma}^2+(\beta_{\pi\Sigma}^I)^2}\lambda_{\pi\Sigma,\bar KN}^I{1\over x_{\bar KN}^2+(\beta_{\bar KN}^I)^2}
$$
and took
$$
\lambda_{\pi\Sigma,\bar KN}^{I=0}=\lambda_{\pi\Sigma,\bar KN}^{I=1}=1\ , \beta_{\pi\Sigma}^{I=0}=\beta_{\pi\Sigma}^{I=1}=\beta_{\bar KN}^{I=0}=\beta_{\bar KN}^{I=1}=\beta_{Born}
$$
We calculated the $P_{DEV}(E_n)$ deviation spectra for different $\beta_{Born}$ values and incident kaon energies. Typical results are shown in Fig. \ref{born}.
\begin{figure}[h]
\centering
\includegraphics[width=\textwidth]{born}
\caption{Effect of $\beta_{Born}$ on the deviation spectra
\label{born}}
\end{figure}
The shape of the $P_{DEV}$ does not depend on the normalization of $P_{nonres}$ (see the arbitrary choice of the $\lambda_{\pi\Sigma,\bar KN}$-s) and we have normalized the $P_{DEV}$ curves to have their maxima at 1. It can be seen, that for reasonable $\pi\Sigma-\bar KN$ interaction range values the spectra show practically no dependence on $\beta_{Born}$, thus confirming the presumed (approximate) model-independence of the $P_{DEV}$ method. According to the foresaid, the rest of our calculations were done with $\beta_{Born}=3 fm^{-1}$.
Figs. 1.,2. and 3. demonstrate our main results. Apart from the direct spectra $P(E_n)$, which show no trace of the $\Lambda(1405)$, we show also the full- and single-scattering deviation spectra, too. For comparison, we plotted also the (hypothetical) $\pi^0\Sigma^0$ elastic cross sections as calculated from the $\bar{K}N -\pi\Sigma$ interaction for the corresponding $\pi\Sigma$ energy (top scale). Since this cross section "feels" only the $I=0$ part of the $\bar{K}N -\pi\Sigma$ interaction, its peak is usually identified with the $\Lambda(1405)$ resonance. Thus the similarity of the $P_{DEV}(E_n)$ and $\sigma(\pi^0\Sigma^0)$ line shapes can tell us about the reliability of extracting information about $\Lambda(1405)$ from the reaction under consideration. Obviously, the $P_{DEV}^{sing}$ spectra show more similarity with the original $\Lambda(1405)$ shape than the full $P_{DEV}(E_n)$ spectra, since their dynamical content is more or less the same. For three of the four considered potentials (KEK 1, KEK 2, SIDDHARTA 2) a clear resonant structure can be seen in the full deviation spectra, however, the shapes and positions can significantly differ from their "originals".
As for the fourth potential, SIDDHARTA 1, its deviation spectra do not show any signature of its original $\Lambda(1405)$, although there are clean maxima in the corresponding $P_{DEV}^{sing}$ and $\sigma(\pi^0\Sigma^0)$ curves. The reason might be the extreme closeness of the pole to the $\bar{K}N$ threshold combined with its large width.
How can we interpret these results?
As for the direct spectra, we can say, that they are practically indistinguishable for all four potentials, having quite different
pole structure, - at least in the considered energy range - and thus are useless for differentiating between models of $\Lambda(1405)$.
The deviation spectra in single scattering approximation show a remarkable similarity with the "original" $\sigma(\pi^0\Sigma^0)$ cross section
curve. Since the authors of \cite{AY} used this approximation for the solution of their three-body problem, this similarity lead them
to optimistic conclusions about the general and simple applicability of the DEV spectrum method.
Unfortunately, in the case of DEV spectra calculated with the true three-body operators, this similarity does not hold any more.
The most remarkable observation, concerning these spectra, however, is, that they are quite sensitive to the choice of the $\bar{K}N$
potentials, or to their pole structure. This in principle allows to distinguish between potentials leading to different pole positions.
However, this possibility does not mean, that visual observation of maxima in the DEV spectra can be used for identification of the $\Lambda(1405)$ pole positions. Even in the two-body case the connection between cross section maxima and pole positions are far from being trivial, except for very narrow resonances. This can be seen e.g. from comparison of the single-scattering DEV spectra and the $\sigma(\pi^0\Sigma^0)$ cross
sections, which are similar, proving, that they describe essentially the same resonance, however, their maxima and widths are not simply
related to the pole parameters as they are shown in Table \ref{poles}.
In the three-body case the observed
spectra can be related to the two-body characteristics of the input potentials only via reliable dynamic calculations, which in the low-energy regime mean solution of Faddeev equations. Their results, combined with the DEV method should reproduce the "observed" (experimental)
DEV spectra, and the input potentials should be tuned until this goal is reached. The situation is familiar from the history of NN
potentials, when many of the subtle details of the NN interactions were fixed from three-body experiments and calculations.
Finally, we asked the question, under which conditions could the resonance be observed in the direct, $P(E_n)$ spectra. For this purpose we modified two of the $\bar KN-\pi\Sigma$ interaction parameters of one of our potentials (KEK~1), $\lambda^{I=0}_{\bar KN,\bar KN}$ and $\lambda^{I=0}_{\pi\Sigma,\bar KN}$, in such a way, that the position of the $\Lambda(1405)$ remained at its original place, while its width could be made smaller. The results are shown in Fig.\ref{gamma}.
It is seen, that in order to show up in the direct $P(E_n)$ spectra the width of the original resonance should not exceed $10-15\ MeV$, while the deviation spectra reproduce the original resonance shape in an acceptable way. Thus the real $\Lambda(1405)$ peak, which in all models has a width of $\sim 50-100\ MeV$ has little chance to be seen directly in this reaction, at least in the considered energy region.
\begin{figure}[h]
\includegraphics[width=0.9\textwidth]{gamma}
\caption{Effect of decreasing $\Gamma_{\Lambda(1405)}$ on the neutron spectra
\label{gamma}}
\end{figure}
\section{Conclusions}
Dynamically exact Faddeev-type calculations for the $K^-+d\rightarrow \pi+\Sigma + n$ reaction were performed in the energy range $E^{cm}_{K^-}=0-50\ MeV$ in order to find the signature of the $\Lambda(1405)$ resonance in the observable neutron spectra.
Four different phenomenological $\bar KN-\pi\Sigma$ interactions were used, all well reproducing the experimental data in the two-body sector, but having rather different pole structure. It was shown, that due to strong kinematical masking effect the inclusive neutron spectra do not exhibit a peak, corresponding to the $\Lambda(1405)$ resonance and show no difference for the four potentials. These spectra are in agreement with the only available experimental data \cite{Tan}. We demonstrated, that the deviation spectrum method is able to eliminate the disturbing kinematical factors and differentiate between potentials with different pole structure. In most of the cases the "deviation" spectra show maxima, which can be related to the "original" two-body $\Lambda(1405)$ resonance. However, the shape and position of the peaks in the deviation spectra may significantly differ from those of their "original" counterparts. For one of the potentials (SIDDHARTA 1) even the deviation spectrum does not exhibit a maximum, probably due to the closeness of the original pole to the $\bar{K}N$ threshold and its large imaginary part.
\begin{acknowledgments}
The work was supported by the OTKA grant T71989.
\end{acknowledgments}
|
{
"timestamp": "2013-03-11T01:01:10",
"yymm": "1203",
"arxiv_id": "1203.1813",
"language": "en",
"url": "https://arxiv.org/abs/1203.1813"
}
|
\section{Introduction}
The modern quantization method for gauge theories is based on the BRST symmetry \cite {BRS,T}. In the framework of the canonical formalism this symmetry is generated by the BRST charge.
If the quantum BRST charge exists it is essentially determined by the corresponding classical one. The BRST charge
is defined as a solution to the master equation
with certain boundary conditions. The BRST construction in the case of reducible gauge theories was given in \cite{BF1}. The global existence of the classical BRST charge in the reducible case was proved in \cite {FHST}.
In this paper we give the general solution to the classical
master equation for the BRST charge in the case of reducible gauge theories. To this aim, we construct a new coordinate system in the extended phase space and transform the equation by changing variables. Then it can be solved by using a simple iterative procedure.
We also give a solution to the equation determining the classical BRST observables, and describe a new realization of the observable algebra.
In contract to \cite {FHST}, our approach does not require neither redefining of the constraints no modification of the reducibility functions.
In the framework of the Lagrangian formalism the master equation was solved in \cite{B1}.
The paper is organized as follows.
In section 2, we review the BRST construction and derive two auxiliary equations.
In section 3, we introduce new variables and construct a generalized inverse of the Koszul-Tate differential operator $\delta.$ With respect to the new variables, both $\delta$ and its generalized inverse take a standard form. The solution to the master equation is given in section~4.
The BRST observables are described in section 5.
In what follows Grassman parity and ghost number
of a function $X$ are denoted by $\epsilon (X)$ and $\mbox{gh}(X),$
respectively. The Poisson superbracket in phase space $\Gamma=(P_i,Q^{i}), \epsilon (P_i)=\epsilon (Q^{i}),$
is given by
\begin{eqnarray*}
\{X, Y\}= \frac {\partial X} {\partial Q^{i}} \frac
{\partial Y} {\partial P_i} - (-1)^{\epsilon (X)\epsilon (Y)}\frac
{\partial Y} {\partial Q^{i}} \frac {\partial X} {\partial P_i}.
\end{eqnarray*}
Derivatives with respect to generalized momenta $P_i$ are always understood as left-hand, and those with respect to generalized coordinates $Q^i$ as right-hand ones.
\section {Master equation for the BRST charge}
Let $\xi_{a}, {a} =1,\ldots,2m,$ be the phase space coordinates, and
let $G_{a_0},$ ${a_0 =1,\ldots,m_0
,}$ be the
first class constraints which satisfy the following Poisson brackets
\begin{eqnarray*}
\{G_{a_0},G_{b_0}\}
=U_{{a_0}{b_0}}^{c_0}G_{c_0},
\end{eqnarray*}
where $U_{a_0 b_0}^{c_0}$ are phase space functions.
The constraints are assumed to be of definite Grassmann parity $\epsilon_{a_0},$
$\epsilon(G_{a_0})=\epsilon_{a_0}.$
We shall consider a reducible theory of $L$-th order.
That is, there exist phase space functions
\begin{eqnarray*}
Z^{a_k}_{a_{k+1}},\qquad k=0,\ldots,L-1,\qquad a_k=1,\ldots,m_k,
\end{eqnarray*}
such that at each stage the $Z$'s form a complete set,
\begin{eqnarray*}
Z^{a_k}_{a_{k+1}}\lambda^{a_{k+1}}\approx 0 \Rightarrow \lambda^{a_{k+1}}\approx Z^{a_{k+1}}_{a_{k+2}}\lambda^{a_{k+2}},\qquad k=0,\ldots,L-2,
\end{eqnarray*}
\begin{eqnarray*}
Z^{a_{L-1}}_{a_L}\lambda^{a_L}\approx 0 \Rightarrow \lambda^{a_L}\approx 0.
\end{eqnarray*}
\begin{eqnarray}\label {rc}
\label{r}G_{a_0}Z^{a_0}_{a_1}=0,\qquad
Z^{a_k}_{ a_{k+1}}Z^{a_{k+1}}_{a_{k+2}}\approx 0, \qquad k=0,\ldots,L-2.
\end{eqnarray}
The weak equality $\approx$ means equality on the constraint surface
\begin{eqnarray*}\Sigma:\qquad G_{a_0}=0.
\end{eqnarray*}
Following the BRST method the ghost pairs
$({
\mathcal
P}_{a_k},c^{a_k}),k=0,\ldots,L,$ are introduced
\begin{align*}
\epsilon({\mathcal P}_{a_k})=\epsilon(c^{a_k})=\epsilon_{a_k}+k+1,\qquad
-\mbox {gh} ({\mathcal P}_{a_k}) = \mbox {gh} (c^{a_k}) =k+1.
\end{align*}
The BRST charge $\Omega $ is defined as a solution to the equations
\begin{eqnarray}
\label{1}
\{\Omega,\Omega\}
=0,
\end{eqnarray}
\begin{eqnarray} \label{o7}
\epsilon (\Omega)=1,\qquad
\mbox {gh} (\Omega)=1,
\end{eqnarray}
and the boundary conditions
\begin{eqnarray*}
\left.
\frac
{\partial \Omega}
{\partial c^{a_0}}
\right|_{c=0} =G_{a_0},\qquad
\left.\frac {\partial^2 \Omega} {\partial {\mathcal P}_{a_{k-1}}\partial c^{a_k}}\right|_{
{\mathcal P}=c=0} =Z_{a_k}^{a_{k-1}}.
\end{eqnarray*}
One can write
\begin{eqnarray} \label{us}
\Omega= \Omega^{(1)}+M,\qquad M= \sum_{n\geq 2}\Omega^{(n)},\qquad
\Omega^{(n)} \sim c^n ,
\end{eqnarray}
\begin{eqnarray}
\Omega^{(1)}= G_{a_0}c^{a_0}+\sum_{k=1}^L \bigl({\mathcal P}_{a_{k-1}}Z_
{a_k}
^{a_{k-1}}+N_{a_k}\bigr)
c^{a_k}
,
\label{sumo}
\end{eqnarray}
where $N_{a_1}=0$ and $N_{a_k},$ $k>1,$ only involves ${\mathcal P}_
{a_{s}},s\leq k-2.$
Eq. (\ref{o7}) implies
\begin{eqnarray*}
\left.N_{a_{k}}\right|_{{\mathcal P}=0}=0 ,\qquad \left. M \right|_{{\mathcal P}=0}=0.
\end{eqnarray*}
Let $V$ be the space of the formal power series in
$(\xi,{\mathcal P},c)$
which vanish on $\Sigma$ at ${\mathcal P}=0.$ For $X,Y\in V$ we have $$XY\in V,\qquad
\{X,Y\}\in V, $$ and therefore $V$ is a Poisson algebra. It is easily verified that
$\Omega\in V.$
The bracket $\{.\,,.\}$ splits as
\begin{eqnarray*}
\{X,Y\}
=
\{X,Y\}_\xi+ \{X,Y\}_\diamond -(-1)^{\epsilon (X)\epsilon (Y)}\{Y,X\}_\diamond,
\end{eqnarray*}
where $\{.\,,.\}_\xi$ refers to the Poisson bracket in the original phase space and
\begin{eqnarray*}
\{X,Y\}_\diamond= \sum_{k=0}^L \frac {\partial X} {\partial c^{a_k}}\frac {\partial Y} {\partial {\mathcal P}_{a_k}} .
\end{eqnarray*}
Let $\delta:V\to V$ be defined by
\begin{eqnarray*}
\delta =\{\Omega^{(1)},.\}_\diamond= G_{a_0}\frac {\partial }{\partial {\mathcal P}_{a_{0}}}+\sum_{k=1}^L ({\mathcal P}_{a_{k-1}}Z_{a_k}^{a_{k-1}}+N_{a_k})\frac {\partial }
{\partial {\mathcal P}_{a_{k}}}.
\end{eqnarray*}
From (\ref{sumo}) we obtain
\begin{eqnarray}
\label {chau}
\delta{\mathcal P}_{a_{0}}=G_{a_0},
\qquad
\delta{\mathcal P}_{a_{k}}=
{\mathcal P}_
{a_{k-1}}Z_{a_k}^{a_{k-1}}+N_{a_k},\qquad k=1,\ldots L.
\end{eqnarray}
Substituting (\ref {us}) in (\ref {1}) one obtains
\begin{eqnarray} \label{ru}
\delta \Omega^{(1)}=0,
\end{eqnarray}
\begin{eqnarray} \label{r}
\delta M = D,
\end{eqnarray}
where
\begin{eqnarray*}
-D=\frac 1 2 F + AM + \frac 1 2 \{M,M\},\qquad
F=\{\Omega^{(1)}, \Omega^{(1)} \}_\xi,
\end{eqnarray*}
and the operator $A$ is given by
\begin{eqnarray*}
A= \{\Omega^{(1)}, \,.\, \}_\xi+\{\,.\,,\Omega^{(1)}\}_\diamond.
\end{eqnarray*}
Eq. (\ref{ru}) is equivalent to
\begin{eqnarray*}
\delta^2=0.
\end{eqnarray*}
We shall need equations for $S=\delta \Omega^{(1)},$ and
$R=\delta M -D.$
By using the definition of $\delta,$
we get
\begin{eqnarray} \label {ff2}
\delta S=\{\Omega^{(1)},S\}_\diamond.
\end{eqnarray}
If (\ref{ru}) holds, then $R=\{\Omega,\Omega\}.$
From the Jacobi identity $\{\Omega,\{\Omega,\Omega\}\}=0$ it follows that $\{\Omega,R\}=0,$ or equivalently
\begin {eqnarray}
\label{omf}
\delta R+ AR + \{M, R\}=0.
\end {eqnarray}
\section {Generalized inversion of $\delta$}
In this section we reduce $\delta$ to a standard form and construct its generalized inverse.
For $k=L-2,$ eq. (\ref{rc}) reads
\begin{eqnarray}
\label{r1}
Z_{a'_{L-1}}^{a_{L-2}}Z_{a_L^{\phantom {\prime}}}^{a'_{L-1}}+
Z^{a_{L-2}}_{a''_{L-1}}Z^{a''_{L-1}}_
{a_L^{\phantom {\prime}}} \approx 0,
\end{eqnarray}
where
$a'_{L-1},a''_{L-1}$ are increasing index sets, such that $a'_{L-1}\cup\, a''_{L-1}= a_{L-1},$ $|a'_{L-1}|=|a_L|$ and ${\rm rank}\, Z^{a'_{L-1}}_{a_L^{\phantom{\prime}}}=|a_L|.$
For an index set $i= \{i_1,i_2,\ldots,i_n\},$ we denote $|i|=n.$
From (\ref{r1}) it follows
that ${\rm rank}\, Z^{a_{L-2}}_{a_{L-1}}=|a_{L-1}|-|a_L|=|a''_{L-1}|,$ and ${\rm rank}\, Z^{a_{L-2}}_{a''_{L-1}}=| a''_{L-1}|.$
One can split the index set $a_{L-2}$ as
$a_{L-2}=a'_{L-2}\cup\, a''_{L-2},$ such that ${|a'_{L-2}|=|a''_{L-1}|,}$ and ${\rm rank}\, Z^{a'_{L-2}}_{a''_{L-1}}=|a''_{L-1}|.$
For $k=L-3,$ eq. (\ref{rc}) implies
\begin{eqnarray*}
\label{r2}
Z_{a'_{L-2}}^{a_{L-3}}Z_{a''_{L-1}}^{a'_{L-2}}+Z^{a_{L-3}}_{a''_{L-2}}
Z^{a''_{L-2}}_{a''_{L-1}}\approx 0.
\end{eqnarray*}
From this it follows that
\begin{eqnarray*}
{\rm rank}\, Z^{a_{L-3}}_{a''_{L-2}}=
{\rm rank}\, Z^{a_{L-3}}_{a_{L-2}}=|a_{L-2}|-|a''_{L-1}|=|a''_{L-2}|.
\end{eqnarray*}
Using induction on $k,$ we obtain a set of nonsingular matrices $Z^{a'_{k-1}}_{a''_{k}},$
$k=1,\ldots,L,$
and a set of matrices $Z^{a_{k-1}}_{a''_{k}},k=1,\ldots,L,$ such that
\begin{eqnarray*}
{\rm rank}\, Z^{a_{k-1}}_{a''_{k}}= {\rm rank}\, Z^{a_{k-1}}_{a_{k}}=|a''_k|.
\end{eqnarray*}
Here $$ a'_k\cup a''_k=a_k,\qquad k=1,\ldots,L-1, \qquad a''_{L}= a_L.$$
Eq. (\ref{rc}) implies
\begin{eqnarray} \label{o2}
G_{a'_0}Z^{a'_0}_{a''_1} + G_{a''_0} Z^{a''_0}_{a''_1}= 0.
\end{eqnarray}
From this it follows that $G_{a''_0}$ are independent.
We assume that $G_{a''_0}$ satisfy the regularity conditions. It means that there are some functions $F_{\alpha}(\xi),$ ${\alpha}\cup {a''_0}=a,$ such that $(F_{\alpha},G_{a''_0})$ can be locally taken as new coordinates in the original phase space.
Let $f: a''_{k+1} \to a_{k},$ $k=~0,\ldots, L-~1,$ be an embedding, $ {f(a''_{k+1})=a''_{k+1}\in a_{k}},$
and let $\alpha_{k}$ be defined by $a_{k}= f(a''_{k+1})\cup \alpha_{k}.$ Since $|a''_{k}|=|\alpha_{k}|,$ one can write $\alpha_{k}=g(a''_{k})$ for some function $g,$ and hence
$$a_{k}= f(a''_{k+1})\cup g(a''_{k}), \qquad k=~0,\ldots, L-~1.$$
{\it Lemma.}
The nilpotent operator $\delta$ is reducible to the form
\begin{eqnarray}
\label{de9}
\delta= \xi'_{a''_0} \frac{\partial}{\partial {\mathcal P}'_{g(a''_0)}}
+\sum_{k=1}^{L} {\mathcal P}'_{f(a''_k)}\frac{\partial }{\partial {\mathcal P}'_{g(a''_k)}},
\end{eqnarray}
by the change of variables:
$(\xi_a,{\mathcal P}_{a_0}, \ldots, {\mathcal P}_{a_{L}})\to
(\xi'_a,{\mathcal P}'_{a_0}, \ldots, {\mathcal P}'_{a_{L}}),$
\begin{eqnarray*}
\xi'_{\alpha} = F_{\alpha},\qquad \xi'_{a''_0} = G_{a''_0},
\end{eqnarray*}
\begin{eqnarray}\label {cha}
{\mathcal P}'_{
f(a''_{k+1})}=\delta{\mathcal P}_{a''_{k+1}}
,\qquad {\mathcal P}'_{g(a''_{k})}={\mathcal P}_{a''_{k}},
\end{eqnarray}
\begin{eqnarray*}
{\mathcal P}'_{a_{L}} = {\mathcal P}_{a_{L}},
\end{eqnarray*}
where $k=0,\ldots,L-1,$ $g(a''_L)=a_L.$
To prove this statement we first observe that eqs. \eqref {cha} are solvable with respect to $(\xi_a,{\mathcal P}_{a_0}, \ldots, {\mathcal P}_{a_{L}}).$
The original variables can be represented as
\begin{eqnarray*}\label{cucu}
\xi_{a}= \xi_{a}(\xi'),\qquad {\mathcal P}_{a_k}={\mathcal P}_{a_k}(\xi'_{a},
{\mathcal P}'_{a_0},\ldots, {\mathcal P}'_{a_k}),\qquad k=0,\ldots,L.
\end{eqnarray*}
Here we have used the fact that the ${\mathcal P}_{a_k}$ depends
only on the functions ${\mathcal P}'_{a_s}$ with $s\leq k.$
Assume that the functions $\xi_{a}(\xi')$ have been constructed.
Then from (\ref{cha}) it follows that
\begin{eqnarray*}\label{cucu22}
{\mathcal P}_{a'_k}=
({\mathcal P}'_{f(a''_{k+1})}-
{\mathcal P}'_{g(a''_k)} Z^{ a''_{k}}_{a''_{k+1}}- N^{\prime\phantom{a_k}}_{a''_{k+1}})(Z^{(-1)})^{a''_{k+1}}_{a'_k},
\qquad {\mathcal P}_{a''_k}={\mathcal P}'_{g(a''_k)},
\end{eqnarray*}
\begin{eqnarray}\label{cucu22}
{\mathcal P}_{a_{L}}= {\mathcal P}'_{a_{L}},
\end{eqnarray}
where $k=0,\ldots,L-1,$
\begin{eqnarray}
\label{N}
N'_{a_{k+1}}(\xi', {\mathcal P}'_{a_0},\ldots, {\mathcal P}'_{a_{k-1}})=
N^{\phantom{b_k}}_{a_{k+1}}(\xi, {\mathcal P}_{a_0},\ldots,{\mathcal P}_{a_{k-1}}).
\end{eqnarray}
Therefore, the variables $$(\xi'_a,{\mathcal P}'_{a_0}, \ldots, {\mathcal P}'_{a_{L}})=({\xi'_{\alpha},\xi'_{a''_0},{\mathcal P}'_{g(a''_0)},{\mathcal P}'_{f(a''_1)},{\mathcal P}'_{g(a''_1)},\ldots,{\mathcal P}'_{f(a''_{L})},{\mathcal P}'_{g(a''_{L})}})$$ are independent.
It follows from (\ref{cha}) that
\begin{eqnarray*}
\delta \xi'_a=\delta {\mathcal P}'_{f(a''_1)}=\ldots=\delta {\mathcal P}'_{f(a''_L)}=0,
\end{eqnarray*}
\begin{eqnarray}\label{cu55}
\delta {\mathcal P}'_{g(a''_0)}=\xi'_{a''_0}, \qquad \delta {\mathcal P}'_{g(a''_k)}= {\mathcal P}'_{f(a''_{k})}, \qquad k=1,\ldots, L.
\end{eqnarray}
Eqs. (\ref {cu55}) are equivalent to (\ref {de9}).
Let $n$
be the counting operator
\begin{eqnarray*}
n = \xi'_{a''_{0}}\frac{\partial }{\partial \xi'_{a''_{0}}}+ {\mathcal P}'_{g(a''_{0})}\frac{\partial }{\partial {\mathcal P}'_{g(a''_{0})}}+ \sum_{s=1}^{L}\left({\mathcal P}'_{f(a''_{s})}\frac{\partial}{\partial {\mathcal P}'_{f(a''_{s})}}
+ {\mathcal P}'_{g(a''_{s})}\frac {\partial} {\partial {\mathcal P}'_{g(a''_{s})}}\right),
\end{eqnarray*}
and let
\begin{eqnarray*}
\sigma= {\mathcal P}'_{g(a''_0)} \frac{\partial}{\partial \xi'_{a''_0}}+
\sum_{k=1}^{L} {\mathcal P}'_{g(a''_k)}\frac{\partial }{\partial {\mathcal P}'_{f(a''_k)}}
.
\end{eqnarray*}
One can directly verify that
\begin{eqnarray} \label{us4}
\sigma^2=0,\qquad \delta\sigma+\sigma \delta=n,\qquad n\delta=\delta n,
\qquad n\sigma=\sigma n.
\end{eqnarray}
With respect to the new coordinate system the condition $X\in {V}$
becomes $$\left. X \right|_{\xi'_{a''_0}={\mathcal P}'=0}=0.$$
The space ${V}$ splits as
\begin{eqnarray}
\label {spl}
{V}= \bigoplus_{m\geq 1} {V}_{m}
\end{eqnarray}
with $
nX=mX$ for $X\in {V}_{m}.$ Hence the operator $n: {V} \to {V}$ is invertible.
It is easily verified
that $\delta^{+}=\sigma n^{(-1)}$ is a generalized inverse of $\delta$:
\begin{eqnarray} \label{mk}
\delta\delta^{+}\delta=\delta,
\qquad \delta^{+}\delta\delta^{+}=\delta^{+}.
\end{eqnarray}
From (\ref{us4}) and (\ref{spl}) it follows that for any $X
\in {V},$
\begin {eqnarray}
\label{osas}
X=
\delta^+
\delta
X+
\delta
\delta^+
X.
\end {eqnarray}
\section {Solution of the master equation}
In this section we start from eq.
(\ref {ru}). One can write
\begin{eqnarray*} \label {ff}
\delta \Omega^{(1)}=\delta N+BN+Q,
\end{eqnarray*}
where
\begin{eqnarray}
\label{No}
N=\sum_{k=2}^{L} N_{a_k}c^{a_k},\qquad Q=
\sum_{k=2}^{L} {\mathcal P}_{a_{k-2}} Z^{a_{k-2}}_{a_{k-1}}
Z^{a_{k-1}}_{a_k}c^{a_k},
\end{eqnarray}
$B:V\to V$ is defined by $B=0,$ if $L\leq 2,$ and otherwise
\begin{eqnarray*}
BX=\sum_{k=3}^{L} \frac{\partial X }{\partial c^{a_{k-1}}}
Z^{a_{k-1}}_{a_k}c^{a_k}.
\end{eqnarray*}
Then eq. (\ref{ru}) takes the form
\begin{eqnarray*}
\delta N+BN+Q =0.
\end{eqnarray*}
Changing variables
$(\xi,{\mathcal P}) \to
(\xi',{\mathcal P}'),$
we get
\begin{eqnarray} \label{f1}
\delta N'+B'N'+Q' =0.
\end{eqnarray}
Here and in what follows, for any $X(\xi, {\mathcal P},c)$ we denote by $X'$ the function
\begin{eqnarray*}
X'(\xi', {\mathcal P}',c)=X(\xi, {\mathcal P},c).
\end{eqnarray*}
We shall seek the solution to eq. (\ref{f1}) in the form of expansion in power series of variables
$${\mathcal P}'_{g(a''_0)},{\mathcal P}'_{g(a''_1)},\ldots, {\mathcal P}'_{g(a''_L)}.$$
Applying $\delta\delta^+$ to (\ref{f1}) and using (\ref{mk}) we have
$$\delta N'+\delta\delta^{+}(B'N'+Q')=0,$$ and therefore
\begin {eqnarray} \label{tor}
N'+\delta^{+}(B'N'+Q')= Y' ,
\end {eqnarray}
where
\begin {eqnarray*}
Y'=\sum_{k=2}^{L} Y'_{a_k}(\xi', {\mathcal P}')c^{a_k},\qquad
\delta Y'=0,\qquad \epsilon(Y')= 1,\qquad \mbox{\rm gh} (Y')= 1.
\end {eqnarray*}
Solving (\ref{tor}), we get
\begin {eqnarray} \label{solu}
N'=(I+\delta^{+}B')^{(-1)}(Y'-\delta^{+}Q') ,
\end {eqnarray}
where $I$ is the identity map, and
\begin{eqnarray*}
(I+\delta^+B)^{(-1)}=\sum_{m\geq 0}(-1)^m(\delta^+B)^m.
\end{eqnarray*}
It remains to show that (\ref {solu}) satisfies (\ref{f1}). We shall use the
approach of \cite{F}. With respect to the new coordinate system
eq. (\ref{ff2}) takes the form
\begin{eqnarray} \label {fy}
\delta S'= \{ \Omega^{\prime(1)},S'\}'_\diamond,
\end{eqnarray}
where
\begin{eqnarray*}
S'=\delta N'+B'N'+Q'.
\end{eqnarray*}
If $N'$ is a solution to (\ref{tor}), then
\begin {eqnarray*}
\delta^{+}N'=\delta^{+}Y',
\end {eqnarray*}
since $(\delta^+)^2=0,$ and
\begin{eqnarray}
\label {fi}
\delta^+S'=\delta^+\delta N'+\delta^+(B'N'+Q')=0.
\end{eqnarray}
Here we have used (\ref{osas}) and (\ref{tor}).
Consider eq. (\ref{fy}) and condition (\ref{fi}),
where $N'$ is the solution to (\ref{tor}).
Applying $\delta^+$ to (\ref{fy}), and using (\ref{fi}), we get
\begin {eqnarray*}
S'=\delta^{+}\{\Omega^{\prime(1)},S'\}'_
\diamond,
\end {eqnarray*}
from which by iterations it follows that $S'=0.$
The functions $N_{a_k}, k=1,\ldots,L, $ are found by substituting (\ref{cha}) in (\ref{N}), where
\begin{eqnarray*}
N'_{a_{k}}= \frac{\partial N' }{\partial c^{a_{k}}}.
\end{eqnarray*}
Recall that $N_{a_1}=0.$ Assume that $N_{a_s},$
$s\le k,$ have been constructed. It follows from (\ref{chau}) and (\ref{cha}) that
the variables $(\xi', {\mathcal P}'_{a_0},\ldots, {\mathcal P}'_{a_{k-1}})$ depends only on the functions
$N_{a_s}$ with $s\le k,$ and therefore $ N_{a_{k+1}}$ is easily computed.
Our next task is to find a solution to
eq. (\ref {r}).
Changing variables $
(\xi,{\mathcal P}) \to
(\xi',{\mathcal P}'),$
we get
\begin{eqnarray} \label{to}
\delta M'= D',
\end{eqnarray}
where
\begin{eqnarray*}
-D'= \frac 1 2 F'
+ A M'+\frac 1 2 \{M',M'\}'.
\end{eqnarray*}
Applying $\delta\delta^+$ to (\ref{to}), we have
$$\delta M'=\delta\delta^{+}D',$$ from which it follows that
\begin {eqnarray} \label{tort}
M^{\prime}= W' + \delta^{+}D^{\prime},
\end {eqnarray}
\begin {eqnarray}
\label{rssr}
W'\in {V}, \qquad
\delta W'=0,\qquad \epsilon(W')= 1,\qquad \mbox{\rm gh} (W')= 1.
\end {eqnarray}
Let $\langle .\,,. \rangle: {V}^2\to {V} $ be defined by
\begin{eqnarray*} \label {or}
\langle X_1,X_2 \rangle = -\frac 1 2 (I+\delta^+A)^{(-1)}\delta^{+}\left(\{ X_1,X_2 \}'+\{ X_2,X_1\}'\right),
\end{eqnarray*}
where $I$ is the identity map, and
$$(I+\delta^+A)^{(-1)}=\sum_{m\geq 0}(-1)^m(\delta^+A)^m.$$
One can rewrite (\ref{tort}) as
\begin {eqnarray}
\label{omee}
M' = M'_0 + \frac 1 2 \langle M',M' \rangle,
\end {eqnarray}
where
\begin {eqnarray*}
\label{mo}
M'_0 = (I+\delta^+A)^{(-1)}(W'- \frac 1 2 \delta^{+}F').
\end {eqnarray*}
Iterating eq. (\ref {omee}) we can construct the solution $M'$ in the form of a power series expansion in
$({\mathcal P}'_{g(a''_k)}, c^{a_k}, k=0,\ldots,L) $:
\begin {eqnarray}
\label{mex}
M' = M'_0 + \frac 1 2 \langle M'_0,M'_0 \rangle +\ldots.
\end {eqnarray}
Changing variables in (\ref{omf}) $
(\xi,{\mathcal P}) \to
(\xi',{\mathcal P}'),$ we get
\begin {eqnarray}
\label{oss}
\delta R'+ A R' + \{M',R'\}'=0,
\end {eqnarray}
where
\begin {eqnarray*}
R'=\delta M'+\frac 1 2 F'+AM'+\frac 1 2 \{M',M'\}'.
\end {eqnarray*}
To prove that (\ref {mex}) satisfies (\ref {to}) consider eq. (\ref{oss})
and the condition
\begin {eqnarray}
\label{ossa}
\delta^+R'=0,
\end {eqnarray}
where $M'$ is the solution to (\ref{tort}).
Applying $\delta^+$ to eq. (\ref{oss}) and using (\ref{ossa}), we get
\begin {eqnarray}
\label{ossr}
R'=-\delta^+ (A R' + \{M',R'\}').
\end {eqnarray}
From (\ref{ossr}) by iterations it follows that $R'=0.$
It remains to check \eqref {ossa}. The solution to (\ref{tort}) satisfies the condition
\begin {eqnarray*}
\delta^+M' = \delta^+W'.
\end {eqnarray*}
By using
(\ref{osas}), we have
\begin {eqnarray*}
M'=\delta^+\delta M' + \delta\delta^+W'.
\end {eqnarray*}
From this, (\ref{osas}) and (\ref{rssr}) it follows that
\begin {eqnarray}
\label{osm}
M'=\delta^+\delta M' + W'.
\end {eqnarray}
We have
\begin {eqnarray*}
\label{omt}
\delta^+R'=\delta^+\delta M'+\delta^+(\frac 1 2 F'+AM'+\frac 1 2 \{M',M'\}'),
\end {eqnarray*}
and therefore by
(\ref{osm}) and (\ref{tort}), $\delta^+R'=0.$
\section {BRST observables}
Let $P$ denote the Poisson algebra of the first class functions,
\begin{eqnarray*}
P =\{\varphi(\xi)\,|\,\left. \{\varphi, G_\alpha\}
\right|_{G=0}=0
\},
\end{eqnarray*}
and let
\begin{eqnarray*}
J =\{u(\xi)\,|\,\left. u\right|_{G=0}=0\}.
\end{eqnarray*}
Elements of $P/J$ are called classical observables.
A function $\Phi=\Phi(\xi,{\mathcal P},c)$ is called a BRST-invariant extension of $\Phi_0 \in P$ if
\begin{eqnarray*}
\Phi = \Phi_{0}+\Pi,\qquad \Pi=\sum_{n\geq 1}\Phi^{(n)},\qquad
\Phi^{(n)}\sim c^n,\qquad \mbox{\rm gh} (\Phi)=0,
\end{eqnarray*}
\begin{eqnarray}
\label{pp}
\{\Omega,\Phi\}=0.
\end{eqnarray}
Let ${\mathcal U}$ denote the space of all such extensions. The functions $\Phi_1, \Phi_2\in {\mathcal U}$ are set to be equivalent if
\begin {eqnarray}
\label {nnm}
\Phi_1-\Phi_2=\{\Omega,\Psi\}
\end {eqnarray}
for some $\Psi.$
Elements of the corresponding factorspace ${\mathcal U}/{\sim}$ are called the BRST observables. The Poisson algebras $P/J$ and ${\mathcal U}/{\sim}$ are isomorphic \cite{FHST}.
Let us consider the equation
\begin {eqnarray}
\label {linm}
\{\Omega,\Psi\}-\Lambda=0,
\end {eqnarray}
where $\Lambda$ is a given function, ${\rm gh}(\Lambda)=0,$ $\{\Omega,\Lambda\}=0,$ and $\Psi$ is an unknown one.
The equation implies that
$\Psi,\Lambda\in V,$ since ${\rm gh}(\Psi)=-1.$
Let us show that for any $\Lambda\in V$ there exist a solution to (\ref{linm}).
One can write (\ref{linm}) in the form
\begin {eqnarray}
\label{qspm}
\delta \Psi+ A \Psi + \{M, \Psi\}- \Lambda=0.
\end {eqnarray}
Changing variables from
$(\xi,{\mathcal P})$ to
$(\xi',{\mathcal P}'),$ we get
\begin {eqnarray}
\label{qsq}
\delta \Psi'+ A \Psi' + \{M', \Psi'\}'- \Lambda'=0. \end {eqnarray}
By using (\ref{mk}), one can write
\begin {eqnarray}
\label{qsqs}
\Psi'+ \delta^{+}(A \Psi' + \{M', \Psi'\}'- \Lambda')=\Upsilon',
\end {eqnarray}
where
\begin {eqnarray*}
\Upsilon'\in {V}, \qquad
\delta \Upsilon'=0,\qquad \mbox{\rm gh} (\Upsilon')=1.
\end {eqnarray*}
From (\ref{qsqs}) it follows
\begin {eqnarray}
\label{qppp}
\Psi'=(I+\delta^{+}(A+ {\rm ad}\,M') )^{(-1)}(\Upsilon'+\delta^{+}\Lambda'),
\end {eqnarray}
where $ {\rm ad}\,M'=
\{M',\,.\,\}'.$
Now, let us show that (\ref{qppp}) satisfies (\ref{qsq}).
Denote by $\Gamma$ left-hand side of (\ref{linm})
\begin {eqnarray*}
\Gamma=\{\Omega,\Psi\}-\Lambda.
\end {eqnarray*}
From the Jacoby identity $\{\Omega,\{\Omega,\Psi\}\}=0$ and the BRST invariance of $\Lambda$ it follows that
\begin {eqnarray*}
\delta \Gamma+ A\Gamma + \{M, \Gamma\}=0.
\end {eqnarray*}
Changing variables
$(\xi,{\mathcal P}) \to (\xi',{\mathcal P}'),$ we get
\begin {eqnarray}
\label{omm}
\delta \Gamma'+ A\Gamma' + \{M', \Gamma'\}'=0,
\end {eqnarray}
where
\begin {eqnarray*}
\Gamma'=\{\Omega',\Psi'\}'-\Lambda'.
\end {eqnarray*}
It is straightforward to check that if $\Psi'$ satisfies (\ref{qsqs}) then
$\delta^{+}\Psi'= \delta^{+}\Upsilon',$ and
\begin {eqnarray}
\label{sqsq}
\delta^{+}\Gamma'=0.
\end {eqnarray}
Consider (\ref{omm}) and (\ref{sqsq}), where $\Psi'$ satisfies (\ref{qsqs}). By using (\ref{osas}), we get
\begin {eqnarray}
\Gamma'=-\delta^{+}(A\Gamma' + \{M', \Gamma'\}'),
\end {eqnarray} from which it follows that $\Gamma'=0.$
From definition (\ref{nnm})
we conclude that $\Phi_1\sim \Phi_2$ if and only if $\Phi_1-\Phi_2\in {\mathcal U}\cap V.$
Let us now turn our attention to eq. (\ref{pp}).
It can be written in the form
\begin {eqnarray}
\label{qsp}
\delta \Pi+ {\{\Omega, \Phi_0\}}+ A \Pi + \{M, \Pi\}=0.
\end {eqnarray}
We note that left-hand side of (\ref{qsp}) belong to $V.$
Changing variables from
$(\xi,{\mathcal P})$ to
$(\xi',{\mathcal P}'),$ we get
\begin {eqnarray}
\label{qssp}
\delta \Pi'+ \{\Omega', \Phi'_0\}'+ A\Pi' + \{M', \Pi'\}'=0.
\end {eqnarray}
By repeating the same steps as in the case of eq. (\ref{qsq}), we obtain the general solution to (\ref{qssp})
\begin {eqnarray}
\label{q22}
\Pi'=(I+\delta^{+}(A+ {\rm ad}\,M') )^{(-1)}({X}'-\delta^{+}\{\Omega',\Phi_0'\}'),
\end {eqnarray}
\begin {eqnarray*}
{ X}'\in {V}, \qquad
\delta { X}'=0,\qquad \mbox{\rm gh} ({ X}')=0.
\end {eqnarray*}
The condition
\begin {eqnarray}
\label{r23}
\delta^+ {\Pi}'=0
\end {eqnarray}
implies ${X}'=0.$ Therefore the solution to (\ref{pp}) with boundary condition (\ref{r23}) is given by
\begin {eqnarray}
\label{q23}
\Phi'=L\Phi'_0,
\end {eqnarray}
where
\begin {eqnarray*}
L=I-(I+\delta^{+}(A+ {\rm ad}\,M') )^{(-1)}\delta^{+}{\rm ad}\,\Omega'.
\end {eqnarray*}
The operator $L$ is invertible. The inverse $L^{-1}$ is given by
\begin {eqnarray*}
L^{-1}\Phi' = \left.\Phi' \right|_{{\mathcal P}' =0}.
\end {eqnarray*}
Eq. (\ref{q23}) establishes a one-to-one correspondence between first class functions and solutions to eqs. (\ref {pp}), (\ref {r23}) .
Let us denote by $L(P)$ and $L(J)$ the images of $P$ and $J,$ respectively, under the mapping $L.$
For $\Phi'_1,\Phi'_2 \in L(P)$
\begin {eqnarray*}
\left.
\{\Phi'_1,\Phi'_2\}' \right|_{{\mathcal P} =0}'=
\{
\left.\Phi'_1\right|_{{\mathcal P}'=0},
\left.\Phi'_2 \right|_{{\mathcal P}' =0}
\}',\qquad \left.
(\Phi'_1\Phi'_2) \right|_{{\mathcal P}' =0}=
(\left.\Phi'_1\right|_{{\mathcal P}'=0})
(\left.\Phi'_2 \right|_{{\mathcal P}' =0}),
\end {eqnarray*}
from which it follows that $L(P)$ and ${P}$ are isomorphic as Poisson algebras, and
therefore $L(P)/L(J)$ gives a realization of classical observables.
|
{
"timestamp": "2014-08-13T02:08:53",
"yymm": "1203",
"arxiv_id": "1203.1937",
"language": "en",
"url": "https://arxiv.org/abs/1203.1937"
}
|
\section*{Introduction}
Throughout this note, let $X$ denote a {\em translation surface}, i.e.,
a (connected) topological surface with a translation atlas. Then $X$ is
automatically endowed with a conformal structure and a flat metric, and
so it is both a Riemann surface and a Riemannian manifold \cite{jHhM79}.
An orientation-preserving homeomorphism $\phi : X \to X$ is called
{\em affine} if it is affine in local charts. We use $\mathrm{Aff}^+(X)$ to denote
the group of affine maps of $X$. Any element $\phi$ of $\mathrm{Aff}^+(X)$ has
a well-defined global {\em derivative} $\mathrm{der}\,\phi \in \Lie{GL}_2^+(\mathbb{R})$.
The image $\Gamma(X)$ of the homomorphism
$\mathrm{der} : \mathrm{Aff}^+(X) \to \Lie{GL}_2^+(\mathbb{R})$ is called the {\em Veech group}
of $X$ \cite{wV89,yV96,cEfG97,eGcJ00}.
The existence of affine self-maps of a translation surface has applications
in the study of mapping class groups, Teichm{\"u}ller theory, algebraic
geometry, and dynamical systems (for a small sampling of such applications,
see, e.g., \cite{wpT88,pHtS00,ctM03,mM06,cLaR06,ldM11}). They measure a
kind of ``symmetry'' more general than that of isometries, which nonetheless
has consequences for such systems as geodesic flow on the surface and
geodesics in Teichm{\"u}ller space. Veech first observed the importance of
the group of derivatives of affine maps \cite{wV89}.
Let $\clos{X}$ denote the metric completion of $X$. The classical study of
translation surfaces assumes that $\clos{X}$ is itself a compact surface
and $\clos{X} \setminus X$ is finite. If these conditions are satisfied,
we will say that $X$ has {\em finite affine type}. Here we wish to consider
four other ``finiteness'' conditions that may be placed on $X$:
\begin{enumerate}
\item\label{I:an} $X$ has {\em finite analytic type} as a Riemann surface,
meaning that it is obtained from a compact Riemann surface by making
finitely many punctures.
\item\label{I:ar} $X$ has {\em finite area} as a Riemannian manifold,
meaning that the integral of the induced area form over all of $X$ is
finite.
\item\label{I:bd} $X$ is {\em bounded} as a metric space, meaning that
there exists a constant $M > 0$ such that $d_X(x,y) \le M$ for every pair
of points $x$ and $y$ in $X$.
\item\label{I:tb} $X$ is {\em totally bounded} as a metric space, meaning
that for any fixed $\varepsilon > 0$, $X$ can be covered by finitely many balls
of radius $\varepsilon$ (equivalently, $\clos{X}$ is compact).
\end{enumerate}
We will prove two main results about these conditions, one negative and
one positive.
\begin{theorem}\label{T:1}
Except for the trivial implication ``totally bounded $\implies$ bounded'',
none of the conditions \eqref{I:an}--\eqref{I:tb} on $X$ implies any of
the others. However, if $X$ has finite analytic type, then the other three
conditions are equivalent and imply that $X$ has finite affine type.
\end{theorem}
\begin{theorem}\label{T:2}
Suppose the ideal boundary of $X$ is empty. If $X$ has at least one
periodic trajectory and is totally bounded or has finite area, then its
Veech group is a discrete subgroup of $\Lie{SL}_2(\mathbb{R})$. However, there
exist bounded surfaces and surfaces of finite analytic type with
non-discrete Veech groups.
\end{theorem}
\begin{remark*}
It is likely that the condition of having a periodic trajectory follows
from the assumptions of having empty ideal boundary and being totally
bounded or of finite area, in which case it can be dropped in
Theorem~\ref{T:2}.
\end{remark*}
Translation surfaces of infinite analytic type appear, for example,
in \cite{cEfG97,rCfGnL06,pHsLsT11,pHpHbW12,fV12,jpb12}, and it is
such examples that motivated the study presented here. We will prove
Theorem~\ref{T:1} in \S\ref{S:inequiv} and Theorem~\ref{T:2} in
\S\ref{S:discrete}.
\section{Inequivalence of finiteness conditions}\label{S:inequiv}
We begin with the trivial, and only, implication among the finiteness
conditions \eqref{I:an}--\eqref{I:tb}.
\begin{proposition}\label{P:1.1}
``$X$ is totally bounded'' $\implies$ ``$X$ is bounded''.
\end{proposition}
\begin{proof}
This is a generality about metric spaces. Pick $\varepsilon > 0$, and cover $X$
with $N$ balls of radius $\varepsilon$. Then the distance between any two points
is at most $2N\varepsilon$.
\end{proof}
The rest of the first part of Theorem~\ref{T:1} is proved through a series
of examples. One general construction will be quite useful and flexible,
so we describe it first and establish some notation.
\begin{example}[An infinite ``stack of boxes'']
Let $H = \{h_n\}_{n=1}^\infty$ be a sequence of positive numbers,
and let $W = \{w_n\}_{n=1}^\infty$ be a strictly decreasing sequence
of positive numbers tending to zero. Then we construct a translation
surface $X_{H,W}$ as follows (see Figure~\ref{F:XHW}):
\begin{itemize}
\item For each $n \ge 1$, let $R_n$ be a rectangle with horizontal side
$w_n$ and vertical side $h_n$.
\item Place the sequence of rectangles in the plane $\mathbb{R}^2$, starting with
$R_1$ having its lower left corner at the origin, and with $R_{n+1}$
immediately above $R_n$ so that its left edge is along the $y$-axis.
\item Identify the right and left sides of each $R_n$ with each other via
horizontal translation, and identify the portion of the top of $R_n$ not
covered by $R_{n+1}$ (of length $w_n - w_{n+1}$) with the portion of the
bottom edge of $R_1$ directly below via vertical translation. (We omit the
vertices.)
\end{itemize}
The genus of $X_{W,H}$ is infinite, as can be seen by considering the
(pairwise non-homotopic) horizontal core curves of the $R_n$. The area of
$X_{H,W}$ is
\[
\mathrm{Area}(X_{H,W})
= \sum_{n=1}^\infty \mathrm{Area}(R_n)
= \sum_{n=1}^\infty h_n w_n.
\]
In particular, the area of $X_{H,W}$ is finite if $H$ and $W$ are sequences
in $\ell^2$, but this is not necessary. Let $\clos{X}_{H,W}$ denote the
metric completion of this surface.
\end{example}
\begin{figure}
\includegraphics{jpb-finiteness-1.eps}
\caption{A sample surface $X_{H,W}$. In addition to the identification of
vertical edges, as indicated by the arrows, each segment $A_k A_{k+1}$ is
identified with $B_k D_k$ via vertical translation.}
\label{F:XHW}
\end{figure}
\begin{lemma}
$\clos{X}_{H,W} \setminus X_{H,W}$ has only one point.
\end{lemma}
\begin{proof}
The translation structure has been defined by taking a quotient of the
union of the rectangles $R_n$ except for their vertices. The vertices are
all collapsed to a single point, as is evident in Figure~\ref{F:XHW}:
using the notation of that figure, observe that $A_0 \sim A_1 \sim B_1
\sim C_1 \sim D_1 \sim A_2 \sim B_2 \sim C_2 \sim D_2 \sim \cdots$.
\end{proof}
\begin{lemma}
$X_{H,W}$ is bounded if and only if $H$ is bounded.
\end{lemma}
\begin{proof}
Suppose $H$ is bounded by $M_H$ and $W$ by $M_W$. Then every point of
every rectangle $R_n$ is within $M = \sqrt{{M_H}^2 + {M_W}^2}$ of a corner.
Since the vertices are identified to a single point in $\clos{X}_{H,W}$,
$2M$ is an upper bound for the distance between any two points of $X_{H,W}$.
Now suppose $H$ is not bounded. Then the centers of the rectangles $R_n$
become arbitrarily far from the vertices, and so $X_{H,W}$ is not bounded.
\end{proof}
\begin{lemma}
$X_{H,W}$ is totally bounded if and only if $H$ tends to zero.
\end{lemma}
\begin{proof}
Let $s$ denote the unique point in $\clos{X}_{H,W} \setminus X_{H,W}$.
Suppose $H$ tends to zero. Then, because $W$ also tends to zero,
for every $\varepsilon > 0$ there exists $N$ such that $h_n < \sqrt{\varepsilon}$
and $w_n < \sqrt{\varepsilon}$ for all $n \ge N$. This implies that the
$\varepsilon$-neighborhood $B_\varepsilon$ of $s$ covers all $R_n$ for $n \ge N$. The
complement of $B_\varepsilon$ in $X_{H,W}$ is compact, being a finite union of
compact pieces, and can therefore be covered by finitely many $\varepsilon$-balls.
Suppose $H$ does not tend to zero. Then there exists $\varepsilon_0 > 0$ such
that $h_n > 2\varepsilon_0$ for infinitely many $n \ge 1$. Fix such an $\varepsilon_0$
and cover $\clos{X}_{H,W}$ by the following open sets:
\begin{itemize}
\item the $\varepsilon_0$-neighborhood $B_{\varepsilon_0}$ of $s$;
\item for each $R_n$ not fully covered by $B_{\varepsilon_0}$ (of which there
are infinitely many), the interior of this cylinder;
\item for each edge between $R_n$ and $R_{n+1}$, a neighborhood of radius
$(1/3) \cdot \min\{h_n, h_{n+1}\}$.
\end{itemize}
Any finite subcover of this open cover would fail to cover infinitely many
interior points of the $R_n$s, and so $\clos{X}_{H,W}$ cannot be compact.
\end{proof}
With these observations about $X_{H,W}$ in mind, we proceed to our
counterexamples.
\begin{example}[\emph{finite area $\not\Rightarrow$ finite analytic type}]
\label{Ex:aran}
Take $X_{H,W}$ with $h_n = w_n = 1/n$.
\end{example}
\begin{example}[\emph{finite area $\not\Rightarrow$ bounded}]
\label{Ex:arid}
Take $X_{H,W}$ with $h_n = n$ and $w_n = 1/n^3$.
\end{example}
\begin{example}[\emph{finite area $\not\Rightarrow$ totally bounded}]
\label{Ex:artb}
Take $X_{H,W}$ with $h_n = 1$ and $w_n = 1/n^2$.
\end{example}
\begin{example}[\emph{bounded $\not\Rightarrow$ finite analytic type}]
Take Example \ref{Ex:aran} or \ref{Ex:artb}.
\end{example}
\begin{example}[\emph{bounded $\not\Rightarrow$ finite area}]
\label{Ex:bdar}
Take $X_{H,W}$ with $h_n = 1$ and $w_n = 1/n$.
\end{example}
\begin{example}[\emph{bounded $\not\Rightarrow$ totally bounded}]
Take Example \ref{Ex:artb} or \ref{Ex:bdar}.
\end{example}
\begin{example}[\emph{totally bounded $\not\Rightarrow$ finite analytic type}]
Take Example \ref{Ex:aran}.
\end{example}
\begin{example}[\emph{totally bounded $\not\Rightarrow$ finite area}]
\label{Ex:tbar}
Take $X_{H,W}$ with $h_n = w_n = 1/\sqrt{n}$.
\end{example}
\begin{example}[\emph{finite analytic type $\not\Rightarrow$ finite area or
bounded}]
\label{Ex:anarbd}
The Riemann surface $\mathbb{C}^*$ has finite analytic type, since it is obtained
from the Riemann sphere by removing two points. However, the translation
structure given by the differential $\mathrm{d}{z}/z$ makes $\mathbb{C}^*$ isometric to an
infinite cylinder, in which case it does not have finite area, and it is
not bounded.
\end{example}
Example~\ref{Ex:anarbd} shows that the essential way a surface of finite
analytic type can fail to have finite affine type is that one could take
a {\em meromorphic} differential on a compact Riemann surface and remove
the zeroes and poles to obtain a translation surface of finite analytic
type. However, if we pair the ``finite analytic type'' condition with any
of the others, then the rest follow. This fact is likely to be well-known,
but we prove it here for completeness and to show how the analytic structure
of the translation surface plays a role.
\begin{proposition}\label{P:1.2}
If $X$ has finite analytic type and finite area, then it is totally bounded.
\end{proposition}
\begin{proof}
The translation structure is given by an abelian differential on $X$.
Let $\widetilde{X}$ denote the compact surface from which $X$ is obtained as
a Riemann surface. Because $X$ has finite area, the differential can be
extended to $\widetilde{X}$; each point of $\widetilde{X} \setminus X$ is either a
regular point or a zero of the differential. Thus $\clos{X}$ is canonically
homeomorphic to $\widetilde{X}$, so $X$ is totally bounded.
\end{proof}
\begin{proposition}\label{P:1.3}
If $X$ has finite analytic type and it is bounded, then it has finite area.
\end{proposition}
\begin{proof}
The translation structure is given by an abelian differential on $X$ that
is meromorphic on the compact Riemann surface $\widetilde{X}$ from which it is
obtained by punctures. Because $X$ is bounded, none of the punctures is at
an infinite distance from any other point of $X$. Therefore the differential
has no poles on $\widetilde{X}$, and so it has finite area.
\end{proof}
\begin{proof}[Proof of second part of Theorem~\ref{T:1}]
Immediate from Propositions \ref{P:1.1}, \ref{P:1.2}, and \ref{P:1.3}.
\end{proof}
To conclude this section, we observe that other collections of conditions
do not imply any of the remaining ones, except as trivially follows from
what has been established.
\begin{example}[\emph{finite area $+$ totally bounded $\not\Rightarrow$
finite analytic type}]
Take Example~\ref{Ex:aran}.
\end{example}
\begin{example}[\emph{finite area $+$ bounded $\not\Rightarrow$ totally bounded}]
Take Example~\ref{Ex:artb}.
\end{example}
\section{Discreteness of Veech groups}\label{S:discrete}
Recently, it has become apparent that translation surfaces of infinite
analytic type allow for Veech groups of much greater complexity than occurs
in the case of finite type. Specifically, it is well-known that the Veech
group of a translation surface of finite affine type is always a Fuchsian
(i.e., discrete) subgroup of $\Lie{SL}_2(\mathbb{R})$ and is never co-compact. In
contrast, it has been shown by direct construction that any countable
subgroup of $\Lie{SL}_2(\mathbb{R})$ (in fact, of $\Lie{GL}_2^+(\mathbb{R})$) that avoids
the set of matrices with operator norm less than $1$ can occur as the Veech
group of a translation surface whose topological type is that of a ``Loch
Ness Monster'', meaning it has infinite genus and one topological end
\cite{pPgSfV11}. Other ``naturally occurring'' examples (e.g., the surface
obtained by ``unfolding'' an irrational polygon \cite{fV12}) also
demonstrate that one cannot in general expect the Veech group of a
translation surface of infinite type to be discrete. In this section,
we show that this phenomenon of non-discreteness relies essentially on
the failure of a surface to be totally bounded or to have finite area;
i.e., it is not enough that the surface simply have infinite analytic type.
The usual proof of discreteness in the case of finite affine type is
carried out by showing that the Veech group acts on the set of holonomy
vectors of saddle connections, which is a discrete subset of $\mathbb{C}$ (see,
e.g., \cite{yV96}). For surfaces not of finite affine type, this last
clause no longer holds: in many examples, the holonomy vectors of saddle
connections do not have their lengths bounded away from zero. We find
another subset of $\mathbb{C}$ on which the Veech group acts and which, under the
conditions of Theorem~\ref{T:2}, is also discrete. Our proof holds also
for surfaces of finite affine type, and bypasses considerations of whether
the holonomy vectors of saddle connections form a discrete set or not.
Observe, first of all, that if $X$ has finite area, then any element of
$\mathrm{Aff}^+(X)$ must preserve this area, and so the condition that
$\Gamma(X) \subset \Lie{SL}_2(\mathbb{R})$ follows automatically. Similarly, we
have the following.
\begin{lemma}\label{L:2.1}
If $X$ is totally bounded, then $\Gamma(X) \subset \Lie{SL}_2(\mathbb{R})$.
\end{lemma}
\begin{proof}
We use the compactness of $\clos{X}$ to establish a kind of Poincar{\'e}
recurrence, which will permit us to define a first return map. Let
$\phi \in \mathrm{Aff}^+(X)$. For any open subset $U$ of $X$ with piecewise smooth
boundary, we observe that the images $\phi^{\circ n}(U)$ cannot all be
disjoint: for otherwise, we could take them together with one more open
subset, formed by the union of their complement and regular neighborhoods
of their boundaries, and we would have an open cover of $\clos{X}$ with no
finite subcover. Therefore, by a standard argument,
$\phi^{\circ N}(U) \cap U \ne \varnothing$ for some $N \ge 1$.
Proceeding inductively, we obtain a first return map $R_\phi$ into $U$,
defined on an open subset of $U$ whose complement has measure zero. Choose
$U$ so that it has finite area. The area of the image is
\[
\mathrm{Area}(R_\phi) = \int_U (\det DR_\phi)\,d\mathrm{Area}
\le \mathrm{Area}(U).
\]
If $\mathrm{der}\,\phi$ had determinant greater than $1$, then the Jacobian
determinant in the above integral would be greater than $1$ on the entire
domain, and the given inequality would not hold. We conclude that any
element of $\mathrm{Aff}^+(X)$ must have a derivative in $\Lie{SL}_2(\mathbb{R})$.
\end{proof}
Now we proceed to the main ideas in the proof of Theorem~\ref{T:2}.
Throughout this section, we take cylinders in $X$ to be {\em open}
subsets of $X$; that is, they do not include their boundaries.
\begin{lemma}\label{L:2.2}
Let $C_1$ and $C_2$ be two maximal cylinders in a translation surface whose
respective circumferences are $w_1$ and $w_2$ and whose respective heights
are $h_1$ and $h_2$, and suppose that they intersect but do not coincide.
Then the angle $\theta$ between the core curves of $C_1$ and $C_2$ satisfies
\[
|\tan\theta| > \min \left\{ \frac{h_1}{w_1},\frac{h_2}{w_2} \right\}.
\]
\end{lemma}
\begin{proof}
Let $\gamma_1$ and $\gamma_2$ be the core curves of $C_1$ and $C_2$,
respectively. If $\gamma_1$ and $\gamma_2$ meet at right angles, then we
are done. So suppose they do not meet at right angles. We note that each
time $\gamma_1$ crosses one boundary component of $C_2$, it must cross the
other boundary component before returning to the first, and likewise for
$\gamma_2$ crossing the boundary of $C_1$. Therefore the connected
components of $C_1 \cap C_2$ are Euclidean parallelograms whose sides are
arcs of the boundaries of $C_1$ and $C_2$. Let $P$ be one such
parallelogram (see Figure~\ref{F:C1C2}). The angles of $P$ are $\theta$
and $\pi - \theta$, so it suffices to consider the smaller of these angles.
Note that $h_1$ and $h_2$ are also the two heights of $P$. Let $l_1$ and
$l_2$ be the distances from the vertex at $\theta$ to the orthogonal
projections of the adjacent vertices onto the adjacent sides of $P$ in the
directions of $\gamma_1$ and $\gamma_2$, respectively. At least one of the
following inequalities holds: $l_1 < w_1$ or $l_2 < w_2$. But
$\tan\theta = h_1/l_1 = h_2/l_2$, from which the desired result follows.
\end{proof}
\begin{figure}
\includegraphics{jpb-finiteness-2.eps}
\caption{Setup for the proof of Lemma~\ref{L:2.2}. The length of
$\gamma_1$ is $w_1$, and the length of $\gamma_2$ is $w_2$.}
\label{F:C1C2}
\end{figure}
\begin{lemma}\label{L:2.3}
If $v_0, v \in \mathbb{C}$ satisfy $|v_0 - v| < \varepsilon < |v_0|$, then the angle
$\theta$ between $v_0$ and $v$ satisfies
\[
|\tan\theta| < \frac{\varepsilon}{\sqrt{|v_0|^2 - \varepsilon^2}}.
\]
\end{lemma}
\begin{proof}
Under the given conditions, the largest angle a vector $v$ can make with
$v_0$ is when $v$ is tangent to the circle with radius $\varepsilon$ centered at
$v_0$; the assumptions imply that this angle is strictly smaller than
$\pi/2$ in absolute value. The result now follows by direct calculation
of the tangent of the angle in this extreme case and monotonicity of the
tangent function on $(-\pi/2,\pi/2)$.
\end{proof}
The values $h_1/w_1$ and $h_2/w_2$ in Lemma~\ref{L:2.2} are, of course,
the {\em moduli} of the cylinders. The basic idea behind the next lemma
is that if two cylinders have the same area and almost the same
circumference, then their moduli are not very different; we can thus play
the two inequalities of Lemmata \ref{L:2.2} and \ref{L:2.3} against each
other.
\begin{notation*}
Given a translation surface $X$ and $A > 0$, we denote by
$\mathcal{C}(A)$ the set of maximal cylinders on $X$ with area $A$,
and by $\mathcal{V}(A) \subset \mathbb{C}$ the set of holonomy vectors of core
curves of elements of $\mathcal{C}(A)$.
\end{notation*}
\begin{lemma}\label{L:2.4}
Let $X$ be a translation surface that either is totally bounded or has
finite area, and let $A > 0$. Then $\mathcal{V}(A)$ is either empty or a
discrete subset of $\mathbb{C}$.
\end{lemma}
\begin{proof}
Let $v_0 \in \mathcal{V}(A)$. If $0 < \varepsilon < |v_0|$ and
$v \in \mathcal{V}(A)$ is any vector such that $|v - v_0| < \varepsilon$,
then the modulus of any corresponding cylinder is bounded below by
\[
f_1(\varepsilon) = \frac{A}{(|v_0| + \varepsilon)^2}.
\]
On the other hand, Lemma~\ref{L:2.3} implies that if $|v_0 - v| < \varepsilon$
and $|v_0 - w| < \varepsilon$, then the absolute value of the tangent of the
angle between $v$ and $w$ is bounded above by
\[
f_2(\varepsilon) = \frac{2\varepsilon \sqrt{|v_0|^2 - \varepsilon^2}}{|v_0|^2 - 2\varepsilon^2}.
\]
Note that, as $\varepsilon \to 0$, $f_1(\varepsilon)$ tends to $A/|v_0|^2$, while
$f_2(\varepsilon)$ tends to $0$. We can therefore choose $\varepsilon_0 > 0$ small
enough that $f_2(\varepsilon_0) < f_1(\varepsilon_0)$. Then Lemma~\ref{L:2.2} implies
that, for any pair of distinct elements $v,w \in \mathcal{V}(A)$ such
that $|v_0 - v| < \varepsilon_0$ and $|v_0 - w| < \varepsilon_0$, the corresponding
cylinders in $\mathcal{C}(A)$ must be disjoint.
If $X$ has finite area, there can only be finitely many such cylinders,
and so there can only be finitely many elements of $\mathcal{V}(A)$ within
$\varepsilon_0$ of $v_0$.
If $X$ is totally bounded, there again can be only finitely many such
cylinders; otherwise we could find infinitely many disjoint balls of some
fixed positive radius on $X$, which is impossible in a totally bounded
space.
In either case, $v_0$ is an isolated point in $\mathcal{V}(A)$; since
$v_0$ was arbitrary, $\mathcal{V}(A)$ is discrete.
\end{proof}
\begin{lemma}\label{L:2.5}
Let $X$ be a translation surface that either is totally bounded or has
finite area, and let $A > 0$. Then $\mathrm{Aff}^+(X)$ preserves $\mathcal{C}(A)$
and $\Gamma$(X) preserves $\mathcal{V}(A)$.
\end{lemma}
\begin{proof}
The affine image of a cylinder is a cylinder, and the maximality of a
cylinder is preserved because saddle connections are sent to saddle
connections by elements of $\mathrm{Aff}^+(X)$. We have already observed that
an affine self-homeomorphism of $X$ must preserve area, and so the first
claim is proved. The second follows immediately.
\end{proof}
\begin{lemma}\label{L:2.6}
Let $X$ be a translation surface without ideal boundary, and let
$C \subset X$ be a maximal cylinder of finite area. Suppose that $X$
is not a torus. Then the stabilizer $\mathrm{Stab}(C)$ of $C$ in $\mathrm{Aff}^+(X)$
is a cyclic group, hence discrete.
\end{lemma}
\begin{proof}
Because $C$ has finite area and $X$ is not a torus, $C$ has an ideal
boundary. Because the ideal boundary of $X$ is empty, $X$ does not consist
only of $C$. Therefore each boundary component of $C$ contains a saddle
connection; call these $I_1$ and $I_2$. Any element of $\mathrm{Stab}(C)$ must
also preserve the lengths of $I_1$ and $I_2$. Because the boundary of $C$
has finite length, we may assume, up to taking a finite index subgroup,
that every element in $\mathrm{Stab}(C)$ fixes $I_1$ and $I_2$. But this implies
that every element of $\mathrm{Stab}(C)$ fixes the entire boundary of $C$ and thus
is a power of a full Dehn twist in $C$. Ergo $\mathrm{Stab}(C)$ is isomorphic to a
subgroup of $\mathbb{Z}$, from which the result follows.
\end{proof}
\begin{proof}[Proof of first part of Theorem~\ref{T:2}]
The result is already known if $X$ is a torus, because then the Veech group
is (conjugate to) $\Lie{SL}_2(\mathbb{Z})$, so assume this is not the case.
Let $\gamma$ be a periodic trajectory on $X$. Then the image of $\gamma$
is contained in some maximal cylinder. This cylinder must have some finite
area $A$: this is immediate if the area of $X$ is finite, and if $X$ is
totally bounded, it follows because the points of the cylinder must be a
bounded distance apart. Thus the set $\mathcal{V}(A)$ is non-empty. By
Lemma~\ref{L:2.4}, it is therefore discrete. By Lemma~\ref{L:2.5},
$\Gamma(X)$ acts on $\mathcal{V}(A)$. So it suffices to show that the
stabilizer inside $\Gamma(X)$ of a point $v \in \mathcal{V}(A)$ is discrete
in $\Lie{SL}_2(\mathbb{R})$. To see this, we observe that, up to taking a finite
index subgroup, the stabilizer of $v$ in $\Gamma(X)$ may be identified with
the stabilizer inside $\mathrm{Aff}^+(X)$ of some cylinder in $\mathcal{C}(A)$.
Lemma~\ref{L:2.6} now implies the desired result.
\end{proof}
To prove the second part of Theorem~\ref{T:2}, we again turn to examples.
\begin{example}
Let $L$ be an ``irrational'' rhombus, meaning its angles are not rational
multiples of $\pi$. The surface $X_L$ obtained by unfolding $L$ is such
that $\clos{X}_L \setminus X_L$ consists of four points, arising from the
vertices of $L$. $X_L$ is therefore bounded. Its Veech group, however, is
an indiscrete subgroup of $\Lie{SO}(2)$, generated by rotations through the
angles of $L$.
\end{example}
\begin{example}
The infinite cylinder of Example~\ref{Ex:anarbd} has finite analytic type.
This surface has one homotopy class of periodic trajectories; these are
the images of vertical lines in the plane under the universal covering map
$\zeta \mapsto e^\zeta$ from $\mathbb{C}$ to $\mathbb{C}^*$, which is made into a
translation covering by taking the differential $\mathrm{d}\zeta$ on the domain.
The parabolic map $(x,y) \mapsto (x,y+tx)$ of $\mathbb{R}^2 \cong \mathbb{C}$ is affine
with respect to $\mathrm{d}\zeta$ for any $t \in \mathbb{R}$, and it descends to $\mathbb{C}^*$ as
an affine map with respect to $\mathrm{d}{z}/z$, acting as a Dehn twist on each
annulus $\{ 2\pi k \le t \log|z| \le 2\pi(k+1) \}$, $k \in \mathbb{Z}$. The Veech
group of $(\mathbb{C}^*,\mathrm{d}{z}/z)$ therefore contains a copy of $\mathbb{R}$, and so it is
not discrete.
\end{example}
\begin{remark*}
It is still not known whether a surface of infinite genus that has finite
area or is totally bounded can have a lattice Veech group---in particular,
whether the Veech group of such a surface can be co-compact.
\end{remark*}
\subsection*{Acknowledgements}
The author wishes to thank the Hausdorff Research Institute for
Mathematics in Bonn as well as the organizers of the trimester program
``Geometry and Dynamics of Teichm{\"u}ller Spaces'', where much of
this work was carried out. Thanks also to Pascal Hubert, Gabriela
Schmith{\"u}sen, and Ferr{\'a}n Valdez for helpful conversations
and feedback, and to the referee for useful suggestions.
\bibliographystyle{math}
|
{
"timestamp": "2012-03-09T02:04:42",
"yymm": "1203",
"arxiv_id": "1203.1903",
"language": "en",
"url": "https://arxiv.org/abs/1203.1903"
}
|
\section{Introduction}
The calculus of variations is a beautiful and useful field of mathematics
that deals with problems of determining extrema (maxima or minima)
of functionals \cite{BorisBookI,BorisBookII,book:Brunt}.
It starts with the simplest problem of finding a function extremizing
(minimizing or maximizing) an integral
\begin{equation*}
\mathcal{J}(y)=\int\limits_a^b F(t,y(t),y'(t))dt
\end{equation*}
subject to boundary conditions $y(a)=y_a$ and $y(b)=y_b$.
In the literature many generalizations of this problem were proposed, including
problems with multiple integrals, functionals containing higher-order derivatives,
and functionals depending on several functions \cite{MyID:130,MyID:203,MalinowskaTorres}.
Of our interest is an extension proposed by Riewe in 1996-1997,
where fractional derivatives (real or complex order)
are introduced in the Lagrangian \cite{CD:Riewe:1996,CD:Riewe:1997}.
During the last decade, fractional problems have increasingly attracted the attention of many
researchers. As mentioned in \cite{isi}, Science Watch of Thomson Reuters identified
the subject as an \emph{Emerging Research Front} area. Fractional derivatives
are nonlocal operators and are historically applied
in the study of nonlocal or time dependent processes \cite{book:Podlubny}.
The first and well established application of fractional calculus in Physics
was in the framework of anomalous diffusion, which is related to features observed
in many physical systems. Here we can mention the report \cite{MK} demonstrating
that fractional equations works as a complementary tool in the description of anomalous
transport processes. Within the fractional approach it is possible to include
external fields in a straightforward manner. As a consequence, in a short period of time the list
of applications expanded. Applications include chaotic dynamics \cite{Zaslavsky},
material sciences \cite{Mainardi}, mechanics of fractal and complex media \cite{Carpinteri,Li},
quantum mechanics \cite{Hilfer,Laskin}, physical kinetics \cite{Edelman},
long-range dissipation \cite{Tarasov3}, long-range interaction
\cite{Tarasov2,Tarasov1}, just to mention a few.
One of the most remarkable applications of fractional calculus appears, however,
in the fractional variational calculus, in the context of
classical mechanics. Riewe \cite{CD:Riewe:1996,CD:Riewe:1997}
shows that a Lagrangian involving fractional
time derivatives leads to an equation of motion with nonconservative forces such
as friction. It is a remarkable result since frictional and nonconservative forces
are beyond the usual macroscopic variational treatment and, consequently,
beyond the most advanced methods of classical mechanics \cite{Lanczos}. Riewe generalizes
the usual variational calculus, by considering Lagrangians that dependent on fractional derivatives,
in order to deal with nonconservative forces. Recently, several
different approaches have been developed to generalize the least action principle and the
Euler--Lagrange equations to include fractional derivatives.
Results include problems depending on Caputo fractional derivatives,
Riemann--Liouville fractional derivatives and others
\cite{Almeida:AML,MyID:182,MyID:152,MyID:179,Cresson,jmp,gastao,mal,comBasia:Frac1,comDorota,MyID:181,MyID:207,Sha}.
A more general unifying perspective to the subject is, however, possible,
by considering fractional operators depending on general kernels
\cite{OmPrakashAgrawal,book:Kiryakova,MyID:226}. In this work we follow such an approach,
developing a generalized fractional calculus of variations. We consider
very general problems, where the classical integrals are substituted by generalized fractional
integrals, and the Lagrangians depend not only on classical derivatives
but also on generalized fractional operators. Problems of the type considered here,
for particular kernels, are important in Physics \cite{Nabulsi}.
Here we obtain general necessary optimality conditions, for several types
of variational problems, which are valid for rather arbitrary operators and kernels.
By choosing particular operators and kernels, one obtains
the recent results available in the literature of Mathematical Physics
\cite{Caldirola,Nabulsi2,Nabulsi3,Nabulsi4,Nabulsi,Herrera}.
The paper is organized as follows. In Section~\ref{sec:prelim} we introduce
the generalized fractional operators and prove some of its basic properties.
Section~\ref{sec:fip} is dedicated to prove integration by parts formulas
for the generalized fractional operators. Such formulas are then used in
later sections to prove necessary optimality conditions
(Theorems~\ref{theorem:ELCaputo} and \ref{theorem:EL2}).
In Sections~\ref{sec:fp}, \ref{sec:fpfb} and \ref{sec:fp:iso}
we study three important classes of generalized variational problems:
we obtain fractional Euler--Lagrange conditions
for the fundamental (Section~\ref{sec:fp}) and generalized isoperimetric problems
(Section~\ref{sec:fp:iso}), as well as fractional natural boundary conditions
for generalized free-boundary value problems (Section~\ref{sec:fpfb}).
Finally, two illustrative examples are discussed in detail in Section~\ref{sec:ex},
while applications to Physics are given in Section~\ref{sec:appl:phys}:
in Section~\ref{sub:sec:harosc} we obtain the damped harmonic oscillator
in quantum mechanics, in Section~\ref{sub:sec:FALVA} we show how results
from FALVA Physics can be obtained. We end with Section~\ref{sec:conc}
of conclusion, pointing out an important direction of future research.
\section{Preliminaries}
\label{sec:prelim}
In this section we present definitions and properties of generalized fractional operators.
As particular cases, by choosing appropriate kernels, these operators are reduced
to standard fractional integrals and fractional derivatives.
Other nonstandard kernels can also be considered as particular cases.
For more on the subject of generalized fractional calculus and applications,
we refer the reader to the book \cite{book:Kiryakova}.
Throughout the text, $\alpha$ denotes a
real number between zero and one.
Following \cite{MyID:209}, we use round brackets for the arguments of functions,
and square brackets for the arguments of operators.
By definition, an operator receives and returns a function.
\begin{definition}[Generalized fractional integral]
The operator $K_P^\alpha$ is given by
\begin{equation*}
K_P^{\alpha}\left[f\right](x)
= K_P^{\alpha}\left[t \mapsto f(t)\right](x)
=p\int\limits_{a}^{x}k_{\alpha}(x,t)f(t)dt
+q\int\limits_{x}^{b}k_{\alpha}(t,x)f(t)dt,
\end{equation*}
where $P=\langle a,x,b,p,q\rangle$ is the \emph{parameter set} ($p$-set for brevity),
$x\in[a,b]$, $p,q$ are real numbers, and $k_{\alpha}(x,t)$
is a kernel which may depend on $\alpha$.
The operator $K_P^\alpha$ is referred as the \emph{operator $K$} ($K$-op for simplicity)
of order $\alpha$ and $p$-set $P$, while $K_P^{\alpha}[f]$ is called the
\emph{operation $K$} (or $K$-opn) of $f$ of order $\alpha$ and p-set $P$.
\end{definition}
Note that if we define
\[
G(x,t):= \left\{ \begin{array}{ll}
p k_\alpha(x,t) & \mbox{if $t < x$},\\
q k_\alpha(t,x) & \mbox{if $t \geq x$},
\end{array} \right.
\]
then the operator $K_P^\alpha$ can be written in the form
\begin{equation*}
K_P^{\alpha}\left[f\right](x)
= K_P^{\alpha}\left[ t \mapsto f(t)\right](x)
=\int_a^b G(x,t) f(t) dt.
\end{equation*}
This is a particular case of one of the oldest and most respectable class
of operators, so called Fredholm operators \cite{book:Helemskii,book:Polyanin}.
\begin{theorem}[\textrm{cf.} Example~6 of \cite{book:Helemskii}]
Let $\alpha\in(0,1)$ and $P=\langle a,x,b,p,q\rangle$. If $k_\alpha$ is a square integrable function
on the square $\Delta=[a,b]\times[a,b]$, then
$K_P^{\alpha}:L_2\left([a,b]\right)\rightarrow L_2\left([a,b]\right)$
is well defined, linear, and bounded operator.
\end{theorem}
\begin{theorem}
\label{theorem:L1}
Let $k_\alpha$ be a difference kernel, \textrm{i.e.},
let $k_\alpha\in L_1\left([a,b]\right)$ with $k_\alpha(x,t)=k_\alpha(x-t)$.
Then, $K_P^{\alpha}:L_1\left([a,b]\right)\rightarrow L_1\left([a,b]\right)$
is a well defined bounded and linear operator.
\end{theorem}
\begin{proof}
Obviously, the operator is linear. Let $\alpha\in(0,1)$,
$P=\langle a,t,b,p,q\rangle$, and $f\in L_1\left([a,b]\right)$. Define
\[
F(\tau,t):=
\left\{
\begin{array}{ll}
\left| p k_\alpha(t-\tau)\right|\cdot \left|f(\tau)\right| & \mbox{if $\tau \leq t$}\\
\left| q k_\alpha(\tau-t)\right|\cdot \left|f(\tau)\right| & \mbox{if $\tau > t$}
\end{array} \right.
\]
for all $(\tau,t)\in\Delta=[a,b]\times[a,b]$.
Since $F$ is measurable on the square $\Delta$, we have
\begin{equation*}
\begin{split}
\int_a^b \left(\int_a^b F(\tau,t)dt\right)d\tau
&=\int_a^b\left[\left|f(\tau)\right|\left(\int_{\tau}^b \left|p k_\alpha(t-\tau)\right|dt
+\int_{a}^{\tau}\left| q k_\alpha(\tau-t)\right|dt\right)\right]d\tau\\
&\leq\int_a^b\left|f(\tau)\right| \bigl| |p| - |q| \bigr| \left\|k_\alpha\right\|d\tau\\
&=\bigl| |p| - |q| \bigr| \cdot \left\|k_\alpha\right\|\cdot\left\|f\right\|.
\end{split}
\end{equation*}
It follows from Fubini's theorem that $F$
is integrable on the square $\Delta$. Moreover,
\begin{equation*}
\begin{split}
\left\|K_P^\alpha[f]\right\|
&=\int_a^b\left|p\int_{a}^{t}k_{\alpha}(t-\tau)f(\tau)d\tau
+q\int_{t}^{b}k_{\alpha}(\tau-t)f(\tau)d\tau\right|dt\\
&\leq\int_a^b\left(|p|\int_{a}^{t}\left|
k_{\alpha}(t-\tau)\right|\cdot\left|f(\tau)\right|d\tau
+|q|\int_{t}^{b}\left|k_{\alpha}(\tau-t)\right|
\cdot\left|f(\tau)\right|d\tau\right)dt\\
&=\int_a^b\left(\int_a^b F(\tau,t)d\tau\right)dt\\
&\leq \bigl| |p| - |q| \bigl| \cdot \left\|k_\alpha\right\|\cdot\left\|f\right\|.
\end{split}
\end{equation*}
Hence, $K_P^{\alpha}:L_1\left([a,b]\right)\rightarrow
L_1\left([a,b]\right)$ and $\left\|K_P^\alpha
\right\|\leq \bigl| |p| - |q| \bigr| \cdot \left\|k_\alpha\right\|$.
\end{proof}
\begin{remark}
The $K$-op reduces to the left and the right Riemann--Liouville
fractional integrals from a suitably chosen kernel
$k_{\alpha}(x,t)$ and $p$-set $P$.
Let $k_{\alpha}(x,t) = k_{\alpha}(x-t)=\frac{1}{\Gamma(\alpha)}(x-t)^{\alpha-1}$:
\begin{itemize}
\item if $P=\langle a,x,b,1,0\rangle$, then
\begin{equation*}
K_{P}^{\alpha}\left[f\right](x)
=\frac{1}{\Gamma(\alpha)}\int\limits_a^x(x-t)^{\alpha-1}f(t)dt
=: {_{a}}\textsl{I}^{\alpha}_{x} \left[f\right](x)
\end{equation*}
is the standard left Riemann--Liouville fractional integral of $f$ of order $\alpha$;
\item if $P=\langle a,x,b,0,1\rangle$, then
\begin{equation*}
K_{P}^{\alpha}\left[f\right](x)=\frac{1}{\Gamma(\alpha)}\int\limits_x^b(t-x)^{\alpha-1}f(t)dt
=: {_{x}}\textsl{I}^{\alpha}_{b} \left[f\right](x)
\end{equation*}
is the standard right Riemann--Liouville fractional integral of $f$ of order $\alpha$.
\end{itemize}
\end{remark}
\begin{corollary}
\label{corollary:Ibounded}
Operators ${_{a}}\textsl{I}^{\alpha}_{x}, {_{x}}\textsl{I}^{\alpha}_{b}:
L_1\left([a,b]\right)\rightarrow L_1\left([a,b]\right)$
are well defined, linear and bounded.
\end{corollary}
The generalized fractional derivatives $A_P^\alpha$
and $B_P^\alpha$ are defined in terms of the
generalized fractional integral $K$-op.
\begin{definition}[Generalized Riemann--Liouville fractional derivative]
\label{def:GRL}
Let $P$ be a given parameter set and $0<\alpha < 1$.
The operator $A_P^\alpha$ is defined by
$A_P^\alpha = D \circ K_P^{1-\alpha}$,
where $D$ denotes the standard derivative operator,
and is referred as the \emph{operator $A$} ($A$-op)
of order $\alpha$ and $p$-set $P$,
while $A_P^\alpha[f]$, for a function $f$ such that
$K_P^{1-\alpha}[f]\in AC\left([a,b]\right)$,
is called the \emph{operation $A$} ($A$-opn)
of $f$ of order $\alpha$ and $p$-set $P$.
\end{definition}
\begin{definition}[Generalized Caputo fractional derivative]
\label{def:GC}
Let $P$ be a given parameter set and $\alpha \in (0,1)$.
The operator $B_P^\alpha$ is defined by
$B_P^\alpha =K_P^{1-\alpha} \circ D$,
where $D$ denotes the standard derivative operator,
and is referred as the \emph{operator $B$} ($B$-op)
of order $\alpha$ and $p$-set $P$,
while $B_P^\alpha[f]$, for a function $f\in AC\left([a,b]\right)$,
is called the \emph{operation $B$} ($B$-opn)
of $f$ of order $\alpha$ and $p$-set $P$.
\end{definition}
\begin{remark}
The standard Riemann--Liouville and Caputo fractional derivatives
are easily obtained from the generalized operators
$A_P^\alpha $ and $B_P^\alpha$, respectively.
Let $k_{1-\alpha}(x,t)=k_{1-\alpha}(x-t)
=\frac{(x-t)^{-\alpha}}{\Gamma(1-\alpha)}$:
\begin{itemize}
\item if $P=\langle a,x,b,1,0\rangle$, then
\begin{equation*}
A_{P}^\alpha\left[f\right](x)=\frac{1}{\Gamma(1-\alpha)}
D\left[\xi \mapsto \int\limits_{a}^{\xi} (\xi-t)^{-\alpha}f(t)dt\right](x)
=: {_{a}}\textsl{D}^{\alpha}_{x}\left[f\right](x)
\end{equation*}
is the standard left Riemann--Liouville fractional derivative
of $f$ of order $\alpha$, while
\begin{equation*}
B_{P}^\alpha \left[f\right](x)=\frac{1}{\Gamma(1-\alpha)}
\int\limits_a^x(x-t)^{-\alpha} D[f](t)dt
=: {^{C}_{a}}\textsl{D}^{\alpha}_{x} \left[f\right](x)
\end{equation*}
is the standard left Caputo fractional derivative of $f$ of order $\alpha$;
\item if $P=\langle a,x,b,0,1\rangle$, then
\begin{equation*}
-A_{P}^\alpha \left[f\right](x)=\frac{-1}{\Gamma(1-\alpha)}
D\left[\xi \mapsto \int\limits_{\xi}^b(t-\xi)^{-\alpha}f(t)dt\right](x)
=: {_{x}}\textsl{D}^{\alpha}_{b} \left[f\right](x)
\end{equation*}
is the standard right Riemann--Liouville
fractional derivative of $f$ of order $\alpha$, while
\begin{equation*}
-B_{P}^\alpha \left[f\right](x) =\frac{-1}{\Gamma(1-\alpha)}
\int\limits_x^b(t-x)^{-\alpha} D[f](t)dt\\
=: {^{C}_{x}}\textsl{D}^{\alpha}_{b} \left[f\right](x)
\end{equation*}
is the standard right Caputo fractional derivative
of $f$ of order $\alpha$.
\end{itemize}
\end{remark}
\section{On generalized fractional integration by parts}
\label{sec:fip}
We now prove integration by parts formulas
for generalized fractional operators.
\begin{theorem}[Fractional integration by parts for the $K$-op]
\label{thm:gfip:Kop}
Let $\alpha \in (0,1)$, $P=\langle a,t,b,p,q\rangle$,
$k_{\alpha}$ be a square-integrable function
on $\Delta=[a,b]\times[a,b]$, and $f,g\in L_2\left([a,b]\right)$.
The generalized fractional integral $K_P^{\alpha}$ satisfies
the integration by parts formula
\begin{equation}
\label{eq:fracIP:K}
\int\limits_a^b g(x)K_P^{\alpha}\left[f\right](x)dx
=\int\limits_a^b f(x)K_{P^*}^{\alpha}\left[g\right](x)dx ,
\end{equation}
where $P^{*}=<a,t,b,q,p>$.
\end{theorem}
\begin{proof}
Define
\[
F(\tau,t):=
\left\{
\begin{array}{ll}
\left|pk_\alpha(t,\tau)\right|
\cdot\left|g(t)\right|\cdot \left|f(\tau)\right|
& \mbox{if $\tau \leq t$}\\
\left|qk_\alpha(\tau,t)\right|
\cdot \left|g(t)\right|\cdot\left|f(\tau)\right|
& \mbox{if $\tau > t$}
\end{array}\right.
\]
for all $(\tau,t)\in\Delta$. Applying Holder's inequality, we obtain
\begin{equation*}
\begin{split}
\int_a^b \left(\int_a^b F(\tau,t)dt\right)d\tau
&=\int_a^b\left[\left|f(\tau)\right|\left(\int_{\tau}^b\left|pk_\alpha(t,\tau)\right|
\cdot\left|g(t)\right|dt+\int_{a}^{\tau}\left|qk_\alpha(\tau,t)\right|
\cdot\left|g(t)\right|dt\right)\right]d\tau\\
&\leq \int_a^b\left[\left|f(\tau)\right|\left(
\int_{a}^b\left|pk_\alpha(t,\tau)\right|\cdot
\left|g(t)\right|dt+\int_{a}^{b}\left|qk_\alpha(\tau,t)\right|
\cdot\left|g(t)\right|dt\right)\right]d\tau\\
&\leq \int_a^b\left\{\left|f(\tau)\right|\left[\left(
\int_a^b\left|pk_\alpha(t,\tau)\right|^2dt\right)^{\frac{1}{2}}\left(
\int_a^b\left|g(t)\right|^2dt\right)^{\frac{1}{2}}\right.\right.\\
&\qquad\qquad \left.\left.+\left(\int_a^b\left|qk_\alpha(\tau,t)\right|^2
dt\right)^{\frac{1}{2}}\left(\int_a^b\left|g(t)\right|^2
dt\right)^{\frac{1}{2}}\right]\right\}d\tau.
\end{split}
\end{equation*}
By Fubini's theorem, functions $k_{\alpha,\tau}(t):=k_{\alpha}(t,\tau)$
and $\hat{k}_{\alpha,\tau}(t):=k_{\alpha}(\tau,t)$
belong to $L_2\left([a,b]\right)$ for almost all $\tau\in[a,b]$. Therefore,
\begin{equation*}
\begin{split}
\int_a^b &\left\{\left|f(\tau)\right|\left[\left(\int_a^b\left|pk_\alpha(t,\tau)\right|^2dt
\right)^{\frac{1}{2}}\left(\int_a^b\left|g(t)\right|^2dt\right)^{\frac{1}{2}}\right.\right.\\
&\qquad\left.\left.+\left(\int_a^b\left|qk_\alpha(\tau,t)\right|^2dt\right)^{\frac{1}{2}}\left(
\int_a^b\left|g(t)\right|^2dt\right)^{\frac{1}{2}}\right]\right\}d\tau\\
&=\left\|g\right\|_2\int_a^b\left[\left|f(\tau)\right|\left(\left\|pk_{\alpha,\tau}\right\|_2
+\left\|q\hat{k}_{\alpha,\tau}\right\|_2\right)\right]d\tau\\
&\leq \left\|g\right\|_2 \left(\int_a^b\left|f(\tau)\right|^2d\tau\right)^{\frac{1}{2}}\left(
\int_a^b\left|\left\|pk_{\alpha,\tau}\right\|_2+\left\|q\hat{k}_{\alpha,\tau}\right\|_2\right|^2
d\tau\right)^{\frac{1}{2}}\\
&\leq\left\|g\right\|_2\cdot\left\|f\right\|_2\left(\left\|pk_\alpha\right\|_2
+\left\|q k_\alpha\right\|_2\right) < \infty.
\end{split}
\end{equation*}
Hence, we can use again Fubini's theorem
to change the order of integration:
\begin{equation*}
\begin{split}
\int\limits_a^b g(t)K_P^{\alpha}[f](t)dt
&= p\int\limits_a^b g(t)dt\int\limits_a^t f(\tau)k_{\alpha}(t,\tau)d\tau
+q\int\limits_a^b g(t)dt\int\limits_t^b f(\tau)k_{\alpha}(\tau,t)d\tau\\
&= p\int\limits_a^b f(\tau)d\tau\int\limits_{\tau}^b g(t)k_{\alpha}(t,\tau)dt
+q\int\limits_a^b f(\tau)d\tau\int\limits_a^{\tau} g(t)k_{\alpha}(\tau,t)dt\\
&=\int\limits_a^b f(\tau)K_{P^*}^{\alpha}[g](\tau)d\tau.
\end{split}
\end{equation*}
\end{proof}
\begin{theorem}
\label{thm:IPL1}
Let $0<\alpha<1$ and $P=\langle a,x,b,p,q\rangle$. If $k_{\alpha}(x,t)=k_{\alpha}(x-t)$,
$k_{\alpha}, f\in L_1\left([a,b]\right)$, and $g\in C\left([a,b]\right)$,
then the operator $K_P^{\alpha}$ satisfies the integration
by parts formula \eqref{eq:fracIP:K}.
\end{theorem}
\begin{proof}
Define
\[
F(t,x) := \left\{
\begin{array}{ll}
\left|pk_\alpha(x-t)\right|\cdot\left|g(x)\right|\cdot \left|f(t)\right| & \mbox{if $t \leq x$}\\
\left|qk_\alpha(t-x)\right|\cdot \left|g(x)\right|\cdot\left|f(t)\right| & \mbox{if $t > x$}
\end{array}
\right.\]
for all $(t,x)\in\Delta=[a,b]\times[a,b]$. Since $g$ is a continuous function on $[a,b]$,
it is bounded on $[a,b]$, i.e., there exists a real number $C>0$ such that
$\left|g(x)\right|\leq C$ for all $x\in [a,b]$. Therefore,
\begin{equation*}
\begin{split}
\int_a^b \left(\int_a^b F(t,x)dt\right)dx
&=\int_a^b\left[\left|f(t)\right|\left(\int_{t}^b\left|pk_\alpha(x-t)\right|
\cdot\left|g(x)\right|dx+\int_{a}^{t}\left|qk_\alpha(t-x)\right|
\cdot\left|g(x)\right|dx\right)\right]dt\\
&\leq \int_a^b\left[\left|f(t)\right|\left(\int_{a}^b\left|pk_\alpha(x-t)\right|
\cdot\left|g(x)\right|dx+\int_{a}^{b}\left|qk_\alpha(t-x)\right|
\cdot\left|g(x)\right|dx\right)\right]dt\\
&\leq C\int_a^b\left[\left|f\left(t\right)\right|\left(\int_{a}^b
\left|pk_\alpha(x-t)\right|dx+\int_{a}^{b}\left|qk_\alpha(t-x)\right|dx\right)\right]dt\\
&=C\bigl|\left|p\right|-\left|q\right|\bigr|\left\|k_\alpha\right\|\left\|f\right\|<\infty.
\end{split}
\end{equation*}
Hence, we can use Fubini's theorem to change the order of integration in iterated integrals.
\end{proof}
\begin{theorem}[Generalized fractional integration by parts]
\label{thm:gfip}
Let $\alpha \in (0,1)$ and $P=\langle a,t,b,p,q\rangle$.
If functions $f,K_{P^*}^{1-\alpha}[g] \in AC([a,b])$, and we are in conditions
to use formula \eqref{eq:fracIP:K} (Theorem~\ref{thm:gfip:Kop} or Theorem~\ref{thm:IPL1}), then
\begin{equation}
\label{eq:fip:2}
\int\limits_a^b g(x) B_{P}^\alpha \left[f\right](x)dx
=\left. f(x) K_{P^*}^{1-\alpha}\left[g\right](x)\right|_a^b
-\int_a^b f(x) A_{P^*}^\alpha\left[g\right](x)dx,
\end{equation}
where $P^*=<a,t,b,q,p>$.
\end{theorem}
\begin{proof}
From Definition~\ref{def:GC} we know that
$B_P^\alpha[f](x) = K_P^{1-\alpha}\left[D[f]\right](x)$.
It follows that
$$
\int_a^b g(x) B_{P}^\alpha[f](x)dx
= \int_a^b g(x) K_{P}^{1-\alpha}\left[D[f]\right](x) dx.
$$
By relation \eqref{eq:fracIP:K}
$$
\int_a^b g(x) B_{P}^\alpha[f](x) dx
= \int_a^b D[f](x) K_{P^*}^{1-\alpha}[g](x) dx,
$$
and the standard integration by parts formula
implies \eqref{eq:fip:2}:
$$
\int_a^b g(x) B_{P}^\alpha[f](x)dx
=\left. f(x) K_{P^*}^{1-\alpha}[g](x) \right|_a^b
-\int_a^b f(x) D\left[K_{P^*}^{1-\alpha}[g]\right](x)dx.
$$
\end{proof}
\begin{corollary}[\textrm{cf.} \cite{book:Klimek}]
Let $0<\alpha<1$. If $f, {_x I_b^{1-\alpha}} \left[g\right] \in AC([a,b])$, then
\begin{equation*}
\int_{a}^{b} g(x) \, {^C_aD_x^\alpha}\left[f\right](x)dx
=\left.f(x){_x I_b^{1-\alpha}} \left[g\right](x)\right|^{x=b}_{x=a}
+\int_a^b f(x){_x D_b^\alpha}\left[g\right](x)dx.
\end{equation*}
\end{corollary}
\section{The generalized fundamental variational problem}
\label{sec:fp}
By $\partial_{i} F$ we denote the partial
derivative of a function $F$ with respect to its $i$th argument.
We consider the problem of finding a function
$y= t \mapsto y(t)$, $t\in[a,b]$,
that gives an extremum (minimum or maximum)
to the functional
\begin{equation}
\label{eq:1}
\mathcal{J}(y)=K_{P_1}^\alpha\left[t \mapsto
F\left(t,y(t),y'(t),B_{P_2}^\beta \left[y\right](t),
K_{P_3}^\gamma\left[y\right](t)\right)\right](b)
\end{equation}
when subject to the boundary conditions
\begin{equation}
\label{eq:2}
y(a)=y_a, \quad y(b)=y_b,
\end{equation}
where $\alpha,\beta, \gamma\in(0,1)$, $P_1=<a,b,b,1,0>$
and $P_j=<a,t,b,p_j,q_j>$, $j=2,3$. For simplicity of notation we introduce
the operator $\left\{ \cdot \right\}_{P_2, P_3}^{\beta,\gamma}$ defined by
\begin{equation*}
\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(t)
=\left(t,y(t),y'(t),B_{P_2}^\beta \left[\tau \mapsto y(\tau)\right](t),
K_{P_3}^\gamma \left[\tau \mapsto y(\tau)\right](t)\right).
\end{equation*}
With the new notation one can write \eqref{eq:1} simply as
$\mathcal{J}(y)=K_{P_1}^\alpha\left[F\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}\right](b)$.
The operator $K_{P_1}^\alpha$ has kernel $k_\alpha(x,t)$, and operators $B_{P_2}^\beta$
and $K_{P_3}^\gamma$ have kernels $h_{1-\beta}(t,\tau)$ and $h_\gamma(t,\tau)$, respectively.
In the sequel we assume that:
\begin{enumerate}
\item[(H1)] Lagrangian $F\in C^1\left([a,b]\times\mathbb{R}^4;\mathbb{R}\right)$;
\item[(H2)] functions
$A_{P_2^*}^\beta\left[\tau \mapsto k_\alpha(b,\tau)\partial_4F\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(\tau)\right]$,
$K_{P_3^*}^\gamma\left[\tau \mapsto k_\alpha(b,\tau)\partial_5 F\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(\tau)\right]$,\\
$D\left[t \mapsto \partial_3F\left\{y\right\}_{P_2,P_3}^{\beta,\gamma}(t)k_\alpha(b,t)\right]$ and
$t \mapsto k_\alpha(b,t)\partial_2 F \left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(t)$
are continuous on $(a,b)$;
\item[(H3)] functions $t \mapsto \partial_3F\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(t)k_\alpha(b,t)$,
$K_{P_2^*}^{1-\beta}\left[\tau \mapsto k_\alpha(b,\tau)
\partial_4 F\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(\tau)\right] \in AC([a,b])$;
\item[(H4)] kernels $k_\alpha(x,t)$, $h_{1-\beta}(t,\tau)$ and $h_\gamma(t,\tau)$
are such that we are in conditions to use Theorems~\ref{thm:gfip:Kop},
\ref{thm:IPL1} and \ref{thm:gfip}.
\end{enumerate}
\begin{definition}
A function $y\in C^1\left([a,b];\mathbb{R}\right)$ is said to be
admissible for the fractional variational problem \eqref{eq:1}--\eqref{eq:2},
if functions $B_{P_2}^\beta[y]$ and $K_{P_3}^\gamma[y]$ exist and are continuous on the
interval $[a,b]$, and $y$ satisfies the given boundary conditions \eqref{eq:2}.
\end{definition}
\begin{theorem}
\label{theorem:ELCaputo}
If $y$ is a solution to problem \eqref{eq:1}--\eqref{eq:2},
then $y$ satisfies the generalized Euler--Lagrange equation
\begin{multline}
\label{eq:eqELCaputo}
k_\alpha(b,t)\partial_2 F \left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(t)
-\frac{d}{dt}\left(\partial_3F\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(t)k_\alpha(b,t)\right)\\
-A_{P_2^*}^\beta\left[\tau \mapsto k_\alpha(b,\tau)\partial_4 F\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(\tau)\right](t)
+K_{P_3^*}^\gamma\left[\tau \mapsto k_\alpha(b,\tau)\partial_5 F\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(\tau)\right](t)=0
\end{multline}
for all $t\in(a,b)$.
\end{theorem}
\begin{proof}
Suppose that $y$ is an extremizer of $\mathcal{J}$.
Consider the value of $\mathcal{J}$ at a nearby function
$\hat{y}=y+\varepsilon\eta$,
where $\varepsilon\in\mathbb{R}$ is a small parameter,
and $\eta\in C^1\left([a,b];\mathbb{R}\right)$ is an arbitrary
function with continuous $B$-op and $K$-op.
We require that $\eta(a)=\eta(b)=0$. Let
\begin{equation*}
\begin{split}
\mathcal{J}(\hat{y})&=J(\varepsilon)=K_{P_1}^\alpha\left[t \mapsto
F\left(t,\hat{y}(t),\hat{y}'(t),B_{P_2}^\beta \left[\hat{y}\right](t),
K_{P_3}^\gamma \left[\hat{y}\right](t)\right)\right](b)\\
&=\int\limits_a^b k_\alpha(b,t)F\left(t,y(t)+\varepsilon\eta(t),\frac{d}{dt}\left(y(t)
+\varepsilon\eta(t)\right),B_{P_2}^\beta \left[y+\varepsilon\eta\right](t),
K_{P_3}^\gamma\left[y+\varepsilon\eta\right](t)\right)dt.
\end{split}
\end{equation*}
A necessary condition for $y$ to be an extremizer is given by
\begin{equation}
\label{eq:3}
\begin{split}
\left.\frac{d J}{d\varepsilon}\right|_{\varepsilon=0}=0
&\Leftrightarrow K_{P_1}^\alpha\biggl[
\partial_2 F \left\{y\right\}_{P_2, P_3}^{\beta,\gamma} \eta
+\partial_3 F \left\{y\right\}_{P_2, P_3}^{\beta,\gamma} D[\eta]\\
&\qquad \qquad + \partial_4 F \left\{y\right\}_{P_2, P_3}^{\beta,\gamma}
B_{P_2}^\beta\left[\eta\right]
+\partial_5 F\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}
K_{P_3}^\gamma \left[\eta\right]\biggr](b)=0\\
&\Leftrightarrow
\int\limits_a^b
\Biggl(\partial_2 F \left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(t)\eta(t)
+\partial_3 F \left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(t)\frac{d}{dt}\eta(t)\\
&\qquad + \partial_4 F \left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(t)
B_{P_2}^\beta\left[\eta\right](t)
+\partial_5 F\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(t)
K_{P_3}^\gamma \left[\eta\right](t)\Biggr)k_\alpha(b,t)dt = 0.
\end{split}
\end{equation}
Using classical and generalized fractional integration by parts formulas
(Theorems~\ref{thm:gfip:Kop}, \ref{thm:IPL1} and \ref{thm:gfip}),
\begin{multline*}
\int_a^b\partial_3F\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(t)k_\alpha(b,t)
\frac{d}{dt} \eta(t) dt\\
=\left.\partial_3F\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(t)
k_\alpha(b,t)\eta(t)\right|_a^b
-\int_a^b \frac{d}{dt}\left(\partial_3F\left\{y\right\}_{P_2,
P_3}^{\beta,\gamma}(t)k_\alpha(b,t)\right) \eta(t) dt,
\end{multline*}
\begin{multline*}
\int\limits_a^b\partial_4 F\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(t)k_\alpha(b,t)
B_{P_2}^\beta\left[\eta\right](t)dt\\
= \left. K_{P_2^*}^{1-\beta}\left[\tau \mapsto k_\alpha(b,\tau)
\partial_4 F\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(\tau)\right](t) \eta(t) \right|_a^b
-\int\limits_a^b A_{P_2^*}^\beta\left[\tau \mapsto k_\alpha(b,\tau)
\partial_4 F\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(\tau)\right](t) \, \eta(t) dt
\end{multline*}
and
\begin{equation*}
\int\limits_a^b k_\alpha(b,t)\partial_5 F\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(t)
K_{P_3}^\gamma\left[\eta\right](t) dt
=\int\limits_a^b K_{P_3^*}^\gamma\left[\tau \mapsto k_\alpha(b,\tau)
\partial_5 F\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(\tau)\right](t) \, \eta(t) dt,
\end{equation*}
where $P_j^*=<a,t,b,q_j,p_j>$, $j=2,3$.
Because $\eta(a)=\eta(b)=0$, \eqref{eq:3} simplifies to
\begin{multline*}
\int_a^b \Biggr\{k_\alpha(b,t)\partial_2 F \left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(t)
-\frac{d}{dt}\left(\partial_3F\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(t)k_\alpha(b,t)\right)\\
-A_{P_2^*}^\beta\left[\tau \mapsto k_\alpha(b,\tau)\partial_4 F\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(\tau)\right](t)
+K_{P_3^*}^\gamma\left[\tau \mapsto k_\alpha(b,\tau)\partial_5 F\left\{y\right\}_{P_2,P_3}^{\beta,\gamma}(\tau)\right](t)\Biggr\}
\eta(t) dt=0.
\end{multline*}
We obtain \eqref{eq:eqELCaputo} by application of the fundamental lemma
of the calculus of variations (see, \textrm{e.g.}, \cite[Section~2.2]{G:H}).
\end{proof}
The next corollary gives an extension of the main result of \cite{jmp}.
\begin{corollary}
\label{corollary:fund1}
If $y$ is a solution to the problem of minimizing or maximizing
\begin{equation}
\label{eq:4}
\mathcal{J}(y)={_{a}}\textsl{I}^{\alpha}_{b}\left[t \mapsto
F\left(t,y(t),y'(t), \, {^C_aD_t^\beta} \left[y\right](t)\right)\right](b)
\end{equation}
in the class $y\in C^1\left([a,b];\mathbb{R}\right)$ subject to the boundary conditions
\begin{equation}
\label{eq:7}
y(a)=y_a, \quad y(b)=y_b,
\end{equation}
where $\alpha,\beta\in(0,1)$, $F\in C^1\left([a,b]\times \mathbb{R}^3;\mathbb{R}\right)$
and $\tau \mapsto (b-\tau)^{\alpha-1}
\partial_4 F\left(\tau,y(\tau),y'(\tau), \, {^C_aD_\tau^\beta} \left[y\right](\tau)\right)$
has continuous Riemann-Liouville fractional derivative $_{t}D^{\beta}_{b}$, then
\begin{multline}
\label{eq:5}
\partial_2 F\left(t,y(t),y'(t), \, {^C_aD_t^\beta} \left[y\right](t)\right)\cdot(b-t)^{\alpha-1}
-\frac{d}{dt}\left\{\partial_3 F\left(t,y(t),y'(t),
\, {^C_aD_t^\beta} \left[y\right](t)\right) \cdot (b-t)^{\alpha-1}\right\}\\
+{_{t}D^{\beta}_{b}}\left[\tau \mapsto (b-\tau)^{\alpha-1}
\partial_4 F\left(\tau,y(\tau),y'(\tau), \, {^C_aD_\tau^\beta} \left[y\right](\tau)\right)\right](t)=0
\end{multline}
for all $t\in(a,b)$.
\end{corollary}
\begin{proof}
Choose $k_\alpha(x,t)=\frac{1}{\Gamma(\alpha)}(x-t)^{\alpha-1}$,
$h_{1-\beta}(t,\tau)=\frac{1}{\Gamma(1-\beta)}(t-\tau)^{-\beta}$,
and $P_2=<a,t,b,1,0>$. Then the $K$-op, the $A$-op and the $B$-op reduce
to the left fractional integral, the left Riemann-Liouville
and the left Caputo fractional derivatives, respectively.
Therefore, problem \eqref{eq:4}--\eqref{eq:7} is a particular case
of problem \eqref{eq:1}--\eqref{eq:2} and \eqref{eq:5} follows
from \eqref{eq:eqELCaputo} with $\partial_5 F=0$.
\end{proof}
The following result is the Caputo analogous to the main result of \cite{DerInt}
done for the Riemann--Liouville fractional derivative.
\begin{corollary}
\label{corollary:fund2}
Let $\beta, \gamma\in (0,1)$. If $y$ is a solution to the problem
\begin{gather*}
\int\limits_a^b F\left(t,y(t),y'(t), {_{a}^{C}}\textsl{D}_t^\beta [y](t),
{_{a}}\textsl{I}_t^\gamma [y](t)\right)dt \longrightarrow \textrm{extr}\\
y \in C^1\left([a,b]; \mathbb{R}\right)\\
y(a)=y_a, \quad y(b)=y_b,
\end{gather*}
then
\begin{multline}
\label{eq:9}
\partial_2 F\left(t,y(t),y'(t), {{_{a}^{C}}\textsl{D}}_t^\beta[y](t),
{_{a}}\textsl{I}_t^\gamma[y](t)\right)
- \frac{d}{dt}\partial_3 F\left(t,y(t),y'(t),
{_{a}^{C}}\textsl{D}_t^\beta[y](t),{_{a}}\textsl{I}_t^\gamma[y](t)\right)\\
+ {_{t}}\textsl{D}_b^\beta \left[\tau \mapsto \partial_4 F\left(\tau,y(\tau),y'(\tau),
{_{a}^{C}}\textsl{D}_\tau^\beta[y](\tau),{_{a}}\textsl{I}_\tau^\gamma[y](\tau)\right)\right](t)\\
+{_{t}}\textsl{I}_b^\beta \left[\tau \mapsto \partial_5 F\left(\tau,y(\tau),y'(\tau),
{_{a}^{C}}\textsl{D}_\tau^\beta[y](\tau),{_{a}}\textsl{I}_\tau^\gamma[y](\tau)\right)\right](t) = 0
\end{multline}
holds for all $t \in [a,b]$.
\end{corollary}
\begin{proof}
The Euler--Lagrange equation \eqref{eq:9}
follows from \eqref{eq:eqELCaputo} by choosing
$p$-sets $P_1=<a,b,b,1,0>$, $P_2=P_3=<a,t,b,1,0>$, and kernels $k_\alpha(x,t)=1$,
$h_{1-\beta}(t,\tau)=\frac{1}{\Gamma(1-\beta)}(t-\tau)^{-\beta}$,
and $h_{\gamma}(t,\tau)=\frac{1}{\Gamma(\gamma)}(t-\tau)^{\gamma-1}$.
\end{proof}
\begin{remark}
In the particular case when the Lagrangian $F$ of Corollary~\ref{corollary:fund2}
does not depend on the fractional integral and the classical derivative, one obtains
from \eqref{eq:9} the Euler--Lagrange equation of \cite{fred:tor}.
\end{remark}
\section{Generalized free-boundary variational problems}
\label{sec:fpfb}
Assume now that in problem \eqref{eq:1}--\eqref{eq:2}
the boundary conditions \eqref{eq:2} are substituted by
\begin{equation}
\label{eq:Free1}
y(a) \textnormal{ is free } \textnormal{ and } y(b)=y_b.
\end{equation}
\begin{theorem}
\label{theorem:NatBound}
If $y$ is a solution to the problem of extremizing functional \eqref{eq:1}
with \eqref{eq:Free1} as boundary conditions, then $y$ satisfies the
Euler--Lagrange equation \eqref{eq:eqELCaputo}. Moreover,
the extra natural boundary condition
\begin{equation}
\label{eq:NatBoundCond}
\partial_3 F\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(a)k_\alpha(b,a)
+K_{P_2^*}^{1-\beta}\left[\tau \mapsto \partial_4 F\left\{y\right\}_{P_2,
P_3}^{\beta,\gamma}(\tau)k_\alpha(b,\tau)\right](a)=0
\end{equation}
holds.
\end{theorem}
\begin{proof}
Under the boundary conditions \eqref{eq:Free1},
we do not require $\eta$ in the proof of Theorem~\ref{theorem:ELCaputo}
to vanish at $t=a$. Therefore, following the proof
of Theorem~\ref{theorem:ELCaputo}, we obtain
\begin{equation}
\label{eq:8}
\begin{split}
\partial_3 F&\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(a)k_\alpha(b,a)\eta(a)
+\eta(a)K_{P_2^*}^{1-\beta}\left[\tau \mapsto
\partial_4 F\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(\tau)k_\alpha(b,\tau)\right](a)\\
&+\int_a^b\eta(t)\Biggl(\partial_2 F \left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(t)k_\alpha(b,t)
-\frac{d}{dt}\left(\partial_3 F\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(t)k_\alpha(b,t)\right)\\
&-A_{P_2^*}^\beta\left[\tau \mapsto \partial_4 F\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(\tau)k_\alpha(b,\tau)\right](t)
+K_{P_3^*}^\gamma\left[\tau \mapsto
\partial_5 F\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(\tau)k_\alpha(b,\tau)\right](t)\Biggr)dt=0
\end{split}
\end{equation}
for every admissible $\eta\in C^1([a,b];\mathbb{R})$ with $\eta(b)=0$.
In particular, condition \eqref{eq:8} holds for those $\eta$ that fulfill $\eta(a)=0$.
Hence, by the fundamental lemma of the calculus of variations, equation \eqref{eq:eqELCaputo}
is satisfied. Now, let us return to \eqref{eq:8} and let $\eta$ again be arbitrary at point $t=a$.
Inserting \eqref{eq:eqELCaputo}, we obtain the natural boundary condition \eqref{eq:NatBoundCond}.
\end{proof}
\begin{corollary}
Let $\mathcal{J}$ be the functional given by
\begin{equation*}
\mathcal{J}(y)={_{a}}\textsl{I}_b^\alpha\left[t \mapsto
F\left(t,y(t),{_{a}^{C}}\textsl{D}_t^\beta[y](t)\right)\right](b).
\end{equation*}
Let $y$ be a minimizer of $\mathcal{J}$ satisfying the boundary
condition $y(b)=y_b$. Then, $y$ satisfies the Euler--Lagrange equation
\begin{equation}
\label{eq:10}
(b-t)^{\alpha-1}\partial_2 F\left(t,y(t),{_{a}^{C}}\textsl{D}_t^\alpha [y](t)\right)
+{_{t}}\textsl{D}_b^\alpha\left[\tau \mapsto (b-\tau)^{\alpha-1}
\partial_3 F\left(\tau,y(\tau),{_{a}^{C}}\textsl{D}_\tau^\beta y(\tau)\right)\right](t)=0
\end{equation}
and the natural boundary condition
\begin{equation}
\label{eq:11}
{_{a}}\textsl{I}_b^{1-\beta}\left[\tau \mapsto (b-\tau)^{\alpha-1}
\partial_3 F\left(\tau,y(\tau),{_{a}^{C}}\textsl{D}_\tau^\beta y(\tau)\right)\right](a)=0.
\end{equation}
\end{corollary}
\begin{proof}
Let functional \eqref{eq:1} be such that
it does not depend on the classical (integer) derivative
$y'(t)$ and on the $K$-op. If $P_2=<a,t,b,1,0>$,
$h_{1-\beta}(t-\tau)=\frac{1}{\Gamma(1-\beta)}(t-\tau)^{-\beta}$,
and $k_{\alpha}(x-t)=\frac{1}{\Gamma(\alpha)}(x-t)^{\alpha-1}$,
then the $B$-op reduces to the left fractional Caputo derivative and
we deduce \eqref{eq:10} and \eqref{eq:11}
from \eqref{eq:eqELCaputo} and \eqref{eq:NatBoundCond}, respectively.
\end{proof}
\begin{corollary}
Let $\mathcal{J}$ be the functional given by
\begin{equation*}
\mathcal{J}(y)=\int\limits_a^b
F\left(t,y(t),y'(t),B_{P_2}^\beta[y](t),K_{P_3}^\gamma[y](t)\right)dt.
\end{equation*}
If $y$ is a minimizer to $\mathcal{J}$ satisfying the boundary
condition $y(b)=y_b$, then $y$ satisfies the Euler--Lagrange equation
\begin{equation}
\label{eq:eqELCaputoCor}
\partial_2 F \left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(t)
-\frac{d}{dt}\partial_3 F\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(t)
-A_{P_2^*}^\beta\left[\partial_4 F\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}\right](t)
+K_{P_3^*}^\gamma\left[\partial_5 F\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}\right](t)=0
\end{equation}
and the natural boundary condition
\begin{equation}
\label{eq:NatBoundCond2}
\partial_3 F\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(a)
+K_{P_2^*}^{1-\beta}\left[\partial_4
F\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}\right](a)=0.
\end{equation}
\end{corollary}
\begin{proof}
Choose, in the problem defined by \eqref{eq:1} and \eqref{eq:Free1}, $k_\alpha(x,t) \equiv 1$.
Then, equations \eqref{eq:eqELCaputoCor} and \eqref{eq:NatBoundCond2} follow
from \eqref{eq:eqELCaputo} and \eqref{eq:NatBoundCond}, respectively.
\end{proof}
\section{Generalized isoperimetric problems}
\label{sec:fp:iso}
Let $\xi\in\mathbb{R}$. Among all functions
$y:[a,b]\rightarrow\mathbb{R}$ satisfying boundary conditions
\begin{equation}
\label{eq:IsoBound}
y(a)=y_a, \quad y(b)=y_b,
\end{equation}
and an isoperimetric constraint of the form
\begin{equation}
\label{eq:IsoConstr}
\mathcal{I}\left(y\right)=K_{P_1}^\alpha\left[G\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}\right](b)=\xi,
\end{equation}
we look for the one that extremizes (\textrm{i.e.}, minimizes or maximizes) a functional
\begin{equation}
\label{eq:IsoFunct}
\mathcal{J}\left(y\right)
=K_{P_1}^\alpha\left[F\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}\right](b).
\end{equation}
Operators $K_{P_1}^\alpha$, $B_{P_2}^\beta$ and $K_{P_3}^\gamma$,
as well as function $F$, are the same as in problem \eqref{eq:1}--\eqref{eq:2}.
Moreover, we assume that functional \eqref{eq:IsoConstr} satisfies hypotheses (H1)--(H4).
\begin{definition}
A function $y : [a,b]\to\mathbb R$
is said to be \emph{admissible} for problem
\eqref{eq:IsoBound}--\eqref{eq:IsoFunct} if functions $B_{P_2}^\beta[y]$
and $K_{P_3}^\gamma[y]$ exist and are continuous on $[a,b]$,
and $y$ satisfies the given boundary conditions \eqref{eq:IsoBound}
and the given isoperimetric constraint \eqref{eq:IsoConstr}.
\end{definition}
\begin{definition}
An admissible function $y\in C^1\left([a,b],\mathbb{R}\right)$ is said to be an \emph{extremal}
for $\mathcal{I}$ if it satisfies the Euler--Lagrange equation
\eqref{eq:eqELCaputo} associated with functional in \eqref{eq:IsoConstr}, \textrm{i.e.},
\begin{multline*}
k_\alpha(b,t)\partial_2 G \left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(t)
-\frac{d}{dt}\left(\partial_3G\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(t)k_\alpha(b,t)\right)\\
-A_{P_2^*}^\beta\left[\tau \mapsto k_\alpha(b,\tau)\partial_4 G\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(\tau)\right](t)
+K_{P_3^*}^\gamma\left[\tau \mapsto k_\alpha(b,\tau)\partial_5 G\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(\tau)\right](t)=0,
\end{multline*}
where $P_j^*=<a,t,b,q_j,p_j>$, $j=2,3$, and $t\in(a,b)$.
\end{definition}
\begin{theorem}
\label{theorem:EL2}
If $y$ is a solution to the isoperimetric problem
\eqref{eq:IsoBound}--\eqref{eq:IsoFunct} and
is not an extremal for $\mathcal{I}$, then
there exists a real constant $\lambda$ such that
\begin{multline}
\label{eq:eqEL2}
k_\alpha(b,t)\partial_2 H \left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(t)
-\frac{d}{dt}\left(\partial_3 H\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(t)k_\alpha(b,t)\right)\\
-A_{P_2^*}^\beta\left[\tau \mapsto k_\alpha(b,\tau)\partial_4 H\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(\tau)\right](t)
+K_{P_3^*}^\gamma\left[\tau \mapsto k_\alpha(b,\tau)\partial_5 H\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(\tau)\right](t)=0
\end{multline}
for all $t\in(a,b)$, where $H(t,y,u,v,w)=F(t,y,u,v,w)-\lambda G(t,y,u,v,w)$ and $P_j^*=<a,t,b,q_j,p_j>$.
\end{theorem}
\begin{proof}
Consider a two-parameter family of the form
$\hat{y}=y+\varepsilon_1\eta_1+\varepsilon_2\eta_2$,
where for each $i\in\{1,2\}$ we have $\eta_i(a)=\eta_i(b)=0$.
First we show that we can select $\varepsilon_2\eta_2$ such that
$\hat{y}$ satisfies \eqref{eq:IsoConstr}. Consider the quantity
$\mathcal{I}(\hat{y})=K_{P_1}^\alpha\left[G\left\{\hat{y}\right\}_{P_2,
P_3}^{\beta,\gamma}\right](b)$.
Looking to $\mathcal{I}(\hat{y})$ as a function
of $\varepsilon_1,\varepsilon_2$, we define
$\hat{I}(\varepsilon_1,\varepsilon_2)=\mathcal{I}(\hat{y})-\xi$.
Thus, $\hat{I}(0,0)=0$.
On the other hand, applying integration by parts formulas
(Theorems~\ref{thm:gfip:Kop}, \ref{thm:IPL1} and \ref{thm:gfip}),
we obtain that
\begin{multline*}
\left.\frac{\partial\hat{I}}{\partial\varepsilon_2}\right|_{(0,0)}
=\int\limits_a^b\eta_2(t)\Biggl(k_\alpha(b,t)
\partial_2 G \left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(t)
-\frac{d}{dt}\left(\partial_3G\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(t)
k_\alpha(b,t)\right)\\
-A_{P_2^*}^\beta\left[\tau \mapsto k_\alpha(b,\tau)
\partial_4 G\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(\tau)\right](t)
+K_{P_3^*}^\gamma\left[\tau \mapsto k_\alpha(b,\tau)
\partial_5 G\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(\tau)\right](t)\Biggr)dt,
\end{multline*}
where $P_j^*=<a,t,b,q_j,p_j>$, $j=1,2$.
We assume that $y$ is not an extremal for $\mathcal{I}$.
Hence, the fundamental lemma of the calculus of variations implies
that there exists a function $\eta_2$ such that
$\left.\frac{\partial\hat{I}}{\partial\varepsilon_2}\right|_{(0,0)}\neq 0$.
According to the implicit function theorem, there exists
a function $\varepsilon_2(\cdot)$ defined in a neighborhood of $0$ such that
$\hat{I}(\varepsilon_1,\varepsilon_2(\varepsilon_1))=0$.
Let $\hat{J}(\varepsilon_1,\varepsilon_2)=\mathcal{J}(\hat{y})$.
Function $\hat{J}$ has an extremum at $(0,0)$ subject to $\hat{I}(0,0)=0$,
and we have proved that $\nabla\hat{I}(0,0)\neq 0$. The Lagrange multiplier rule
asserts that there exists a real number $\lambda$ such that
$\nabla(\hat{J}(0,0)-\lambda\hat{I}(0,0))=0$.
Because
\begin{multline*}
\left.\frac{\partial\hat{J}}{\partial\varepsilon_1}\right|_{(0,0)}
=\int\limits_a^b\Biggl(k_\alpha(b,t)
\partial_2 F \left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(t)
-\frac{d}{dt}\left(\partial_3 F\left\{y\right\}_{P_2,
P_3}^{\beta,\gamma}(t)k_\alpha(b,t)\right)\\
-A_{P_2^*}^\beta\left[\tau \mapsto k_\alpha(b,\tau)
\partial_4 F\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(\tau)\right](t)
+K_{P_3^*}^\gamma\left[\tau \mapsto k_\alpha(b,\tau)
\partial_5 F\left\{y\right\}_{P_2, P_3}^{\beta,
\gamma}(\tau)\right](t)\Biggr) \eta_1(t)dt
\end{multline*}
and
\begin{multline*}
\left.\frac{\partial\hat{I}}{\partial\varepsilon_1}\right|_{(0,0)}
=\int\limits_a^b\Biggl(k_\alpha(b,t)\partial_2
G \left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(t)
-\frac{d}{dt}\left(\partial_3 G\left\{y\right\}_{P_2,
P_3}^{\beta,\gamma}(t)k_\alpha(b,t)\right)\\
-A_{P_2^*}^\beta\left[\tau \mapsto k_\alpha(b,\tau)\partial_4
G\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(\tau)\right](t)
+K_{P_3^*}^\gamma\left[\tau \mapsto k_\alpha(b,\tau)\partial_5
G\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(\tau)\right](t)\Biggr)\eta_1(t) dt,
\end{multline*}
one has
\begin{equation*}
\begin{split}
\int\limits_a^b & \Biggl\{k_\alpha(b,t)\partial_2
F \left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(t)
-\frac{d}{dt}\left(\partial_3 F\left\{y\right\}_{P_2,
P_3}^{\beta,\gamma}(t)k_\alpha(b,t)\right)\\
&-A_{P_2^*}^\beta\left[\tau \mapsto k_\alpha(b,\tau)\partial_4
F\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(\tau)\right](t)
+K_{P_3^*}^\gamma\left[\tau \mapsto k_\alpha(b,\tau)\partial_5
F\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(\tau)\right](t)\\
&-\lambda\biggl(k_\alpha(b,t)\partial_2
G \left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(t)
-\frac{d}{dt}\left(\partial_3 G\left\{y\right\}_{P_2,
P_3}^{\beta,\gamma}(t) k_\alpha(b,t)\right)\\
&-A_{P_2^*}^\beta\left[\tau \mapsto k_\alpha(b,\tau)\partial_4
G\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(\tau)\right](t)
+K_{P_3^*}^\gamma\left[\tau \mapsto k_\alpha(b,\tau)\partial_5
G\left\{y\right\}_{P_2, P_3}^{\beta,
\gamma}(\tau)\right](t)\biggr)\Biggr\}\eta_1 (t) dt=0.
\end{split}
\end{equation*}
From the fundamental lemma of the calculus of variations
(see, \textrm{e.g.}, \cite[Section~2.2]{G:H}) it follows
\begin{equation*}
\begin{split}
k_\alpha(b,t)&\partial_2 F\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(t)
-\frac{d}{dt}\left(\partial_3 F\left\{y\right\}_{P_2,
P_3}^{\beta,\gamma}(t)k_\alpha(b,t)\right)\\
&-A_{P_2^*}^\beta\left[\tau \mapsto k_\alpha(b,\tau)\partial_4
F\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(\tau)\right](t)
+K_{P_3^*}^\gamma\left[\tau \mapsto k_\alpha(b,\tau)\partial_5
F\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(\tau)\right](t)\\
&-\lambda\Biggl(k_\alpha(b,t)\partial_2 G
\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(t)
-\frac{d}{dt}\left(\partial_3 G\left\{y\right\}_{P_2,
P_3}^{\beta,\gamma}(t)k_\alpha(b,t)\right)\\
&-A_{P_2^*}^\beta\left[\tau \mapsto k_\alpha(b,\tau)\partial_4
G\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(\tau)\right](t)
+K_{P_3^*}^\gamma\left[\tau \mapsto k_\alpha(b,\tau)\partial_5
G\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(\tau)\right](t)\Biggr)=0,
\end{split}
\end{equation*}
that is,
\begin{multline*}
k_\alpha(b,t)\partial_2 H \left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(t)
-\frac{d}{dt}\left(\partial_3 H\left\{y\right\}_{P_2,
P_3}^{\beta,\gamma}(t)k_\alpha(b,t)\right)\\
-A_{P_2^*}^\beta\left[\tau \mapsto k_\alpha(b,\tau)\partial_4
H\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(\tau)\right](t)
+K_{P_3^*}^\gamma\left[\tau \mapsto k_\alpha(b,\tau)\partial_5
H\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(\tau)\right](t)=0
\end{multline*}
with $H=F-\lambda G$.
\end{proof}
\begin{corollary}
Let $y$ be a minimizer to the isoperimetric problem
\begin{gather}
\mathcal{J}(y)={_{a}}\textsl{I}_b^\alpha\left[t \mapsto
F\left(t,y(t),{_{a}^{C}}\textsl{D}_t^\beta[y](t)\right)
\right](b) \longrightarrow \min ,\label{eq:21}\\
\mathcal{I}(y)={_{a}}\textsl{I}_b^\alpha\left[t \mapsto G\left(t,y(t),
{_{a}^{C}}\textsl{D}_t^\beta [y](t)\right)\right](b)=\xi,\label{eq:22}\\
y(a)=y_a,\quad y(b)=y_b.\label{eq:23}
\end{gather}
If $y$ is not an extremal of $\mathcal{I}$, then there exists
a constant $\lambda$ such that $y$ satisfies
\begin{equation}
\label{eq:20}
(b-t)^{\alpha-1}\partial_2 H\left(t,y(t),
{_{a}^{C}}\textsl{D}_t^\alpha[y](t)\right)
+{_{t}}\textsl{D}_b^\beta\left[\tau \mapsto (b-\tau)^{\alpha-1}\partial_3
H\left(\tau,y(\tau),{_{a}^{C}}\textsl{D}_\tau^\beta[y](\tau)\right)\right](t)=0
\end{equation}
for all $t\in(a,b)$, where $H(t,y,v)=F(t,y,v)-\lambda G(t,y,v)$.
\end{corollary}
\begin{proof}
Let $k_{\alpha}(x,t)=\frac{1}{\Gamma(\alpha)}(x-t)^{\alpha-1}$,
$h_{1-\beta}(t,\tau)=\frac{1}{\Gamma(1-\beta)}(t-\tau)^{-\beta}$,
$P_1=<a,b,b,1,0>$ and $P_2=<a,t,b,1,0>$. Then the $K$-op and the $B$-op
reduce to the left fractional integral and the left fractional
Caputo derivative, respectively. Therefore, problem \eqref{eq:21}--\eqref{eq:23}
is a particular case of problem \eqref{eq:IsoBound}--\eqref{eq:IsoFunct},
and \eqref{eq:20} follows from \eqref{eq:eqEL2}
with $\partial_3 H=\partial_5 H=0$.
\end{proof}
\begin{corollary}
\label{IsoPro:RicDel}
Let $y$ be a minimizer to
\begin{gather*}
\mathcal{J}(y)=\int_a^b F\left(t,y(t),y'(t),\,
B_{P_2}^\beta[y](t), K_{P_3}^\gamma[y](t) \right) dt
\longrightarrow \min, \\
\mathcal{I}(y)=\int_a^b G\left(t,y(t),y'(t),\,
B_{P_2}^\beta[y](t), K_{P_3}^\gamma[y](t) \right) dt=\xi,\\
y(a)=y_a,\ y(b)=y_b.
\end{gather*}
If $y$ is not an extremal of $\mathcal{I}$,
then there exists a constant $\lambda$ such that $y$ satisfies
\begin{equation}
\label{eq:CorEL}
\partial_2 H \left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(t)
-\frac{d}{dt}\partial_3 H\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}(t)
-A_{P_2^*}^\beta\left[\partial_4 H\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}\right](t)
+K_{P_3^*}^\gamma\left[\partial_5 H\left\{y\right\}_{P_2, P_3}^{\beta,\gamma}\right](t)=0
\end{equation}
for all $t\in[a,b]$, where $H(t,y,u,v,w)=F(t,y,u,v,w)-\lambda G(t,y,u,v,w)$.
\end{corollary}
\begin{proof}
Let in problem \eqref{eq:IsoBound}--\eqref{eq:IsoFunct}
$P_1=<a,b,b,1,0>$ and kernel $k_{\alpha}(x,t) \equiv 1$. Then,
the generalized fractional integral $K_{P_1}^\alpha$
becomes the classical integral and \eqref{eq:CorEL}
follows from \eqref{eq:eqEL2}.
\end{proof}
\section{Illustrative examples}
\label{sec:ex}
We illustrate our results through two examples
with different kernels: one of a fundamental problem
\eqref{eq:1}--\eqref{eq:2} (Example~\ref{ex:1}),
the other an isoperimetric problem
\eqref{eq:IsoBound}--\eqref{eq:IsoFunct} (Example~\ref{ex:2}).
\begin{example}
\label{ex:1}
Let $\alpha,\beta \in\left(0,1\right)$, $\xi\in\mathbb{R}$,
$P_1=<0,1,1,1,0>$, and $P_2=<0,t,1,1,0>$.
Consider the following problem:
\begin{equation*}
\begin{gathered}
\mathcal{J}(y)=
K_{P_1}^\alpha\left[t \mapsto tK_{P_2}^\beta [y](t)
+\sqrt{1-\left( K_{P_2}^\beta[y](t) \right)^2}\right](1)
\longrightarrow \min,\\
y(0)=1 \, , \ y(1)=\frac{\sqrt{2}}{4}+\int_0^1 r_{\beta}(1-\tau)
\frac{1}{\left(1+\tau^2\right)^\frac{3}{2}}d\tau,
\end{gathered}
\end{equation*}
with kernel $h_{\beta}$ such that $h_\beta(t,\tau)=h_\beta(t-\tau)$ and $h_\beta(0)=1$.
Here the resolvent $r_{\beta}(t)$ is related to the kernel $h_{\beta}(t)$
by $r_{\beta}(t)=\mathcal{L}^{-1}\left[s \mapsto \frac{1}{s\widetilde{h}_{\beta}(s)}-1\right](t)$,
$\widetilde{h}_\beta(s)=\mathcal{L}\left[t \mapsto h_\beta(t)\right](s)$, where $\mathcal{L}$
and $\mathcal{L}^{-1}$ are the direct and the inverse Laplace operators, respectively.
We apply Theorem~\ref{theorem:ELCaputo} with Lagrangian $F$ given by
$F(t,y,u,v,w) =tw+\sqrt{1-w^2}$. Because
\begin{equation*}
y(t) = \frac{1}{\left(1+t^2\right)^\frac{3}{2}}
+\int_0^t r_\beta(t-\tau)\frac{1}{\left(1+\tau^2\right)^\frac{3}{2}}d\tau
\end{equation*}
is the solution to the Volterra integral equation of first kind
(see, \textrm{e.g.}, Equation~16, p.~114 of \cite{book:Polyanin})
\begin{equation*}
\textsl{K}_{P_2}^\beta [y](t)=\frac{t\sqrt{1+t^2}}{1+t^2},
\end{equation*}
it satisfies our generalized Euler--Lagrange equation
\eqref{eq:eqELCaputo}, \textrm{i.e.},
\begin{equation*}
\textsl{K}_{P_2^*}^\beta\left[\tau \mapsto k_\alpha(b,\tau)\left(
\frac{-\textsl{K}_{P_2^*}^\beta[y](\tau)}{\sqrt{1
-\left(\textsl{K}_{P_2^*}^\beta[y](\tau)\right)^2}}
+\tau\right)\right](t)=0.
\end{equation*}
In particular, for the kernel
$h_{\beta}(t-\tau)=\cosh(\beta(t-\tau))$, the boundary conditions are
$y(0)=1$ and $y(1)=1+\beta^2(1-\sqrt{2})$, and the solution is
$y(t)=\frac{1}{(1+t^2)^\frac{3}{2}}+\beta^2\left(1-\sqrt{1+t^2}\right)$
(\textrm{cf.} \cite[p.~22]{book:Polyanin}).
\end{example}
In the next example we make use of the Mittag--Leffler function
of two parameters: if $\alpha, \beta>0$, then
the Mittag--Leffler function is defined by
\begin{equation*}
E_{\alpha,\beta}(z)
=\sum_{k=0}^\infty\frac{z^k}{\Gamma(\alpha k+\beta)}\, .
\end{equation*}
This function appears naturally in the solution
of fractional differential equations,
as a generalization of the exponential function \cite{book:Kilbas}.
\begin{example}
\label{ex:2}
Let $\alpha,\beta\in\left(0,1\right)$, $\xi \in\mathbb{R}$,
and $\xi\notin\left\{\pm\frac{1}{4}\right\}$.
Consider the following problem:
\begin{equation}
\label{eq:ex}
\begin{gathered}
\mathcal{J}(y)={_{0}}\textsl{I}_1^\alpha\left[\sqrt{1+\left(y'
+\, ^{C}_{0}\textsl{D}_t^\beta [y]\right)^2}\right](1) \longrightarrow \min,\\
\mathcal{I}(y)={_{0}}\textsl{I}_1^\alpha\left[\left(y'
+ \, {^{C}_{0}}\textsl{D}_t^\beta [y]\right)^2\right](1) = \xi,\\
y(0)=0 \, , \ y(1)=\int_0^1 E_{1-\beta,1}\left(-(1-\tau)^{1-\beta}\right)
\frac{\sqrt{1-16\xi^2}}{4\xi}d\tau,
\end{gathered}
\end{equation}
which is an example of \eqref{eq:IsoBound}--\eqref{eq:IsoFunct} with
$p$-sets $P_1=<0,1,1,1,0>$ and $P_2=<0,t,1,1,0>$ and
kernels $k_\alpha(x-t)=\frac{1}{\Gamma(\alpha)}(x-t)^{\alpha-1}$
and $h_{1-\beta}(t-\tau)=\frac{1}{\Gamma(1-\beta)}(t-\tau)^{-\beta}$.
Function $H$ of Theorem~\ref{theorem:EL2} is given by
$H(t,y,u,v,w)=\sqrt{1+(u+v)^2} -\lambda (u+v)^2$.
One can easily check (see \cite[p.~324]{book:Kilbas}) that
\begin{equation}
\label{eq:y:ex}
y(t)=\int_0^t E_{1-\beta,1}\left(-(t-\tau)^{1-\beta}\right)
\frac{\sqrt{1-16\xi^2}}{4\xi}d\tau
\end{equation}
\begin{itemize}
\item is not an extremal for $\mathcal{I}$;
\item satisfies $y'+\,^{C}_{0}\textsl{D}_t^\beta [y]= \frac{\sqrt{1-16\xi^2}}{4\xi}$.
\end{itemize}
Moreover, \eqref{eq:y:ex} satisfies \eqref{eq:eqEL2} for $\lambda=2\xi$, \textrm{i.e.},
\begin{multline*}
-\frac{d}{dt}\left((1-t)^{\alpha-1}\left(y'(t)
+\,{^{C}_{0}}\textsl{D}_t^\beta[y](t)\right)\left(\frac{1}{\sqrt{1+\left(y'(t)
+\,{^{C}_{0}}\textsl{D}_t^\beta [y](t)\right)^2}}-4\xi\right)\right)\\
+ \, {_{t}}\textsl{D}_1^\beta\left[\tau \mapsto (1-\tau)^{\alpha-1}\left(y'(\tau)
+\,{^{C}_{0}}\textsl{D}_\tau^\beta [y](\tau)\right)\left(\frac{1}{\sqrt{1+\left(y'(\tau)
+\,{^{C}_{0}}\textsl{D}_\tau^\beta[y](\tau)\right)^2}}-4\xi\right)\right](t)=0
\end{multline*}
for all $t \in (0,1)$. We conclude that \eqref{eq:y:ex} is an extremal for problem \eqref{eq:ex}.
\end{example}
\section{Applications to Physics}
\label{sec:appl:phys}
If the functional \eqref{eq:1} does not depend on $B$-op and $K$-op,
then Theorem~\ref{theorem:ELCaputo} gives the following result:
if $y$ is a solution to the problem of extremizing
\begin{equation}
\label{FALVA:J}
\mathcal{J}(y)=\int_a^b F\left(t,y(t),y'(t)\right) k_\alpha(b,t) dt
\end{equation}
subject to $y(a)=y_a$ and $y(b)=y_b$, where $\alpha\in(0,1)$, then
\begin{equation}
\label{eq:FALVA}
\partial_2 F\left(t,y(t),y'(t)\right)-\frac{d}{dt}\partial_3
F\left(t,y(t),y'(t)\right)=\frac{1}{k_{\alpha}(b,t)}
\cdot \frac{d}{dt}k_{\alpha}(b,t)\partial_3 F\left(t,y(t),y'(t)\right).
\end{equation}
We recognize on the right hand side of \eqref{eq:FALVA}
the generalized weak dissipative parameter
\begin{equation*}
\delta(t)=\frac{1}{k_{\alpha}(b,t)}\cdot \frac{d}{dt}k_{\alpha}(b,t).
\end{equation*}
\subsection{Quantum mechanics of the damped harmonic oscillator}
\label{sub:sec:harosc}
As a first application, let us consider kernel
$k_\alpha(b,t)=\mathrm{e}^{\alpha (b-t)}$ and the Lagrangian
\begin{equation*}
L\left(y,\dot{y}\right)=\frac{1}{2}m \dot{y}^2-V(y),
\end{equation*}
where $V(y)$ is the potential energy and $m$ stands for mass.
The Euler--Lagrange equation \eqref{eq:FALVA} gives the following
second order ordinary differential equation:
\begin{equation}
\label{mod}
\ddot{y}(t)-\alpha \dot{y}(t) = -\frac{1}{m}V'(y(t)).
\end{equation}
Equation \eqref{mod} coincides with (14) of \cite{Herrera},
obtained by modification of Hamilton's principle.
\subsection{Fractional Action-Like Variational Approach (FALVA)}
\label{sub:sec:FALVA}
We now extend some of the recent results of \cite{Nabulsi2,Nabulsi3,Nabulsi4,Nabulsi},
where the fractional action-like variational approach (FALVA)
was proposed to model dynamical systems. FALVA functionals
are particular cases of \eqref{FALVA:J}, where the fractional time integral
introduces only one parameter $\alpha$. Let us consider the
Caldirola--Kanai Lagrangian \cite{Caldirola,Nabulsi3,Nabulsi}
\begin{equation}
\label{eq:CKLagr}
L\left(t, y, \dot{y}\right)
= m(t)\left(\frac{\dot{y}^2}{2}-\omega^2\frac{y^2}{2}\right),
\end{equation}
which describes a dynamical oscillatory system with exponentially
increasing time dependent mass, where $\omega$ is the frequency and
$m(t)=m_0 \mathrm{e}^{-\gamma b}\mathrm{e}^{\gamma t} = \bar{m}_0\mathrm{e}^{\gamma t}$,
$\bar{m}_0=m_0 \mathrm{e}^{-\gamma b}$. Using our generalized FALVA Euler--Lagrange
equation \eqref{eq:FALVA} with kernel $k_\alpha(b,t)$
to Lagrangian \eqref{eq:CKLagr}, we obtain
\begin{equation}
\label{eq:FALVAEL}
\ddot{y}(t)+\left(\delta(t)+\gamma\right)\dot{y}(t)+\omega^2y(t)=0.
\end{equation}
We study two particular kernels.
\begin{enumerate}
\item If we choose kernel
\begin{equation}
\label{eq:KuatEq}
k_\alpha (b,t)=\frac{(\rho+1)^{1-\alpha}}{\Gamma(\alpha)}\left(b^{\rho+1}
-t^{\rho+1}\right)t^\rho,
\end{equation}
defined in \cite{Katugampola}, then the Euler--Lagrange equation is
\begin{equation}
\label{eq:ELKuatugampola}
\partial_2 F\left(t,y(t),y'(t)\right)
-\frac{d}{dt}\partial_3 F\left(t,y(t),y'(t)\right)
=\left(\frac{(1-\alpha)(\rho+1)t^\rho}{b^{\rho+1}
-t^{\rho+1}}\right)\partial_3 F\left(t,y(t),y'(t)\right).
\end{equation}
In particular, when $\rho\rightarrow 0$, \eqref{eq:KuatEq}
becomes the kernel of the Riemann--Liouville fractional integral,
and equation \eqref{eq:ELKuatugampola} gives
\begin{equation*}
\partial_2 F\left(t,y(t),y'(t)\right)-\frac{d}{dt}\partial_3
F\left(t,y(t),y'(t)\right)
=\frac{1-\alpha}{b-t}\partial_3 F\left(t,y(t),y'(t)\right),
\end{equation*}
which is the Euler--Lagrange equation proved in \cite{Nabulsi3}.
For $\rho\neq 0$, we have
\begin{equation*}
\delta(t)=\frac{(1-\alpha)(\rho+1)t^\rho}{b^{\rho+1}-t^{\rho+1}}
\rightarrow 0 \text{ if } t\rightarrow\infty \text{ or } t\rightarrow 0.
\end{equation*}
Therefore, both at the very early time
and at very large time, dissipation disappears.
Moreover, if $\rho\rightarrow 0$, then
\begin{equation*}
\delta(t)=\frac{1-\alpha}{b-t}
\rightarrow
\begin{cases}
0 & \text{ if } t\rightarrow\infty \\
\frac{1-\alpha}{b} & \text{ if } t\rightarrow 0.
\end{cases}
\end{equation*}
This shows that at the origin of time, the time-dependent dissipation
becomes stationary, and that at very large time no dissipation, of any kind, exists.
\item If we choose kernel $k_{\alpha}(b,t)=\left(\cosh b- \cosh t\right)^{\alpha-1}$, then
\begin{equation}
\label{eq:extended}
\partial_2 F\left(t,y(t),y'(t)\right)
-\frac{d}{dt}\partial_3 F\left(t,y(t),y'(t)\right)
= -(\alpha-1)\frac{\sinh t}{\cosh b - \cosh t}\partial_3 F\left(t,y(t),y'(t)\right)
\end{equation}
and
\begin{equation*}
\delta(t)=-(\alpha-1)\frac{\sinh t}{\cosh b - \cosh t}
\rightarrow
\begin{cases}
\alpha-1 & \text{ if } t\rightarrow\infty\\
0 & \text{ if } t\rightarrow 0.
\end{cases}
\end{equation*}
In contrast with previous case, item 1, here
dissipation does not disappear at late-time dynamics.
\end{enumerate}
We note that there is a small inconsistence in \cite{Nabulsi3},
regarding to the coefficient of $\dot{y}(t)$ in \eqref{eq:FALVAEL},
and a small inconsistence in \cite{Nabulsi}, regarding a sign
of \eqref{eq:extended}.
\section{Conclusion}
\label{sec:conc}
In this article we unify, subsume and significantly extend
the necessary optimality conditions available in the literature
of the fractional calculus of variations.
It should be mentioned, however, that since fractional operators are nonlocal,
it can be extremely challenging to find analytical
solutions to fractional problems of the calculus of variations and,
in many cases, solutions may not exist.
In our paper we give two examples with analytic solutions, and many more can be found
borrowing different kernels from the book \cite{book:Polyanin}.
On the other hand, one can easily choose examples for which the
fractional Euler--Lagrange differential equations
are hard to solve, and in that case one needs to use numerical methods
\cite{agrawal:et:al:2012,Shakoor:01,MyID:221,MyID:225}.
The question of existence of solutions to fractional variational problems
is a complete open area of research. This needs attention.
Indeed, in the absence of existence, the necessary conditions
for extremality are vacuous: one cannot characterize an entity that does not exist in the
first place. For solving a problem of the fractional calculus of variations
one should proceed along the following three steps:
(i) first, prove that a solution to the problem exists;
(ii) second, verify the applicability of necessary optimality conditions;
(iii) finally, apply the necessary conditions which
identify the extremals (the candidates). Further elimination,
if necessary, identifies the minimizer(s) of the problem.
All three steps in the above procedure are crucial.
As mentioned by Young in \cite{young},
the calculus of variations has born from the study
of necessary optimality conditions, but any such theory is ``naive'' until
the existence of minimizers is verified. The process leading
to the existence theorems was introduced by
Leonida Tonelli in 1915 by the so-called direct method \cite{tonelli}.
During two centuries, mathematicians
were developing ``the naive approach to the calculus of variations''.
There was, of course, good reasons why the existence problem
was only solved in the beginning of XX century,
two hundred years after necessary optimality conditions began to be studied:
see \cite{cesari,torres2004} and references therein.
Similar situation happens now with the fractional
calculus of variations: the subject is only fifteen years old,
and is still in the ``naive period''. We believe
time has come to address the existence question,
and this will be considered in a forthcoming paper.
\section*{Acknowledgements}
This work was supported by {\it FEDER} funds through
{\it COMPETE} --- Operational Programme Factors of Competitiveness
(``Programa Operacional Factores de Competitividade'')
and by Portuguese funds through the
{\it Center for Research and Development
in Mathematics and Applications} (University of Aveiro)
and the Portuguese Foundation for Science and Technology
(``FCT --- Funda\c{c}\~{a}o para a Ci\^{e}ncia e a Tecnologia''),
within project PEst-C/MAT/UI4106/2011
with COMPETE number FCOMP-01-0124-FEDER-022690.
Odzijewicz was also supported by FCT through the Ph.D. fellowship
SFRH/BD/33865/2009; Malinowska by Bia{\l}ystok
University of Technology grant S/WI/02/2011;
and Torres by FCT through the Portugal--Austin (USA)
cooperation project UTAustin/MAT/0057/2008.
|
{
"timestamp": "2012-03-12T01:00:25",
"yymm": "1203",
"arxiv_id": "1203.1961",
"language": "en",
"url": "https://arxiv.org/abs/1203.1961"
}
|
\section{Introduction}
In the present paper we consider a discrete Schr\"odinger operator, i.e., a Jacobi matrix
\begin{equation}\label{J}
\mathcal J=
\left(%
\begin{array}{cccc}
b_1 & 1 & 0 & \cdots \\
1 & b_2& 1 & \cdots \\
0 & 1 & b_3 & \cdots \\
\vdots & \vdots & \vdots & \ddots \\
\end{array}%
\right),
\end{equation}
whose diagonal entries (potential) are of the form
\begin{equation}\label{b}
b_n:=\frac{c\sin(2\omega n+\delta)}n+q_n,
\end{equation}
where $c,\omega,\delta$ are real constants and $\{q_n\}_{n=1}^{\infty}$ is a real-valued sequence such that
\begin{equation}\label{conditions}
c\neq0,\omega\notin\frac{\pi\mathbb Z}2\text{ and }\{q_n\}_{n=1}^{\infty}\in l^1.
\end{equation}
The operator $\mathcal J$ is a compact perturbation of the free discrete Schr\"odinger operator and therefore, by Weyl's theorem, its essential spectrum equals the interval $[-2,2]$, cf. \cite{Birman-Solomyak-1987}. Moreover, since $\{b_n\}_{n=1}^{\infty}\in l^2$, the interval $(-2,2)$ is covered with absolutely continuous spectrum, cf. \cite{Deift-Killip-1999}. The presence of the (discrete) Wigner-von Neumann potential $\frac{c\sin(2\omega n+\delta)}n$ with frequency $\omega$ produces two critical (resonance) points, namely, the points $\pm2\cos\omega$ where $\mathcal J$ may have half-bound states or eigenvalues. Here, by a half-bound state we understand a point where a subordinate solution of the eigenfunction equation exists, which does not belong to $l^2$. The spectrum on the rest of the interval $(-2,2)$ is purely absolutely continuous, cf. \cite{Janas-Simonov-2010}.
In the present paper we study the behaviour of the spectral density of the operator $\mathcal J$ near the critical points $\pm2\cos\omega$. Our main result is Theorem \ref{thm main result}, where we show that the spectral density has a zero of the order $\frac{|c|}{2|\sin\omega|}$ at each of these points, unless an eigenvalue or a half-bound state is located there.
Vanishing of the spectral density divides the absolutely continuous spectrum into separate parts and is called pseudogap. The physical meaning of this phenomenon is that the interval of the spectrum near such point contains very few energy levels.
The proof of Theorem \ref{thm main result} is based on two ingredients. The first is a Weyl-Titchmarsh type formula taken from \cite{Janas-Simonov-2010}, which relates the value of the spectral density to the coefficient in the orthogonal polynomials asymptotics (Proposition \ref{prop Janas-Simonov}). The second is the limit behaviour of the solutions of a certain model discrete linear system (Proposition \ref{prop Naboko-Simonov}), which has been studied in \cite{Naboko-Simonov-2011}.
Operators with Wigner-von Neumann potentials \cite{Wigner-von-Neumann-1929} attracted attention of many authors \cite{Albeverio-1972, Matveev-1973, Burd-Nesterov-2010,Lukic-2011}. In \cite{Hinton-Klaus-Shaw-1991} the Weyl function behaviour near the critical points was studied for the differential Schr\"odinger operator on the half-line with an infinite sum of Wigner-von Neumann terms in the potential. The spectral density is proportional to the boundary value of imaginary part of the Weyl function. Hence the object under consideration in \cite{Hinton-Klaus-Shaw-1991} is the same as in the present paper. The questions addressed in \cite{Hinton-Klaus-Shaw-1991} were also studied by a different method in \cite{Behncke-1991-I,Behncke-1991-II,Behncke-1994}. In \cite{Klaus-1991} all the possible cases for such potentials were considered (bound state or half-bound state).
In \cite{Kurasov-Simonov-2011,Naboko-Simonov-2011} (following \cite{Kurasov-Naboko-2007}) we considered the Schr\"odinger operator with a potential which is the sum of Wigner-von Neumann part, a summable part, and a periodic background part. There we have proposed a new approach based on the study of the model discrete linear system \eqref{model system}. We consider that the main advantage of this approach is that the result can be formulated as a theorem concerning the model system. This leads to a greater universality of the method. In the present paper we show that it is applicable to the Jacobi matrix case and allows to use the same theorem as in \cite{Naboko-Simonov-2011}. We plan to consider other applications of this method, for instance, the Schr\"odinger operator on the half-line with point interactions supported by a lattice (Kronig-Penney model) with the sequence of interacting centers or strengths of interaction perturbed by a sequence of Wigner-von Neumann potential form \cite{Lotoreichik-Simonov-2012}.
The paper is organised as follows. In Section \ref{section preliminaries} we define the operator $\mathcal J$ and recall the Weyl-Titchmarsh type formula for its spectral density. In Section \ref{section reduction} we transform the eigenfunction equation to a form of the model discrete linear system \eqref{system for v-hat+}. In Section \ref{section Naboko-Simonov} we recall the results about this system which were obtained in \cite{Naboko-Simonov-2011}. In Section \ref{section final result} we prove our main result, Theorem \ref{thm main result}.
\section{Preliminaries}\label{section preliminaries}
The operator $\mathcal J$ acts in the Hilbert space $l^2$ of square summable complex-valued sequences by the rule
\begin{equation}\label{action of J}
\begin{array}{l}
(\mathcal{J}u)_1=b_1u_1+u_2, \\
(\mathcal{J}u)_n=u_{n-1}+b_nu_n+u_{n+1},\ n\ge2. \\
\end{array}
\end{equation}
on the domain
\begin{equation*}
\mathcal D(\mathcal J)=\{u\in l^2:\text{ the result of \eqref{action of J} is in }l^2\}
\end{equation*}
(maximal domain) and is self-adjoint \cite{Akhiezer-1965}. It has a matrix representation of the form \eqref{J} in the canonical base of $l^2$.
Eigenfunction equation for $\mathcal J$ is the following three-term recurrence relation:
\begin{equation}\label{spectral equation prelim}
u_{n-1}+b_nu_n+u_{n+1}=\lambda u_n,\ n\ge2,
\end{equation}
and its solutions are called generalized eigenvectors. One of these solutions is formed of polynomials $P_n(\lambda)$ which additionally satisfy the "first line" equation: $b_1u_1+u_2=\lambda u_1$, and are defined by conditions $P_1(\lambda)=1,P_2(\lambda)=\lambda-b_1$. There exists a measure $\rho$ such that polynomials $P_n(\lambda)$, $n\in\mathbb N$, form an orthogonal base in the space $L_2(\mathbb R,\rho)$. Moreover, the operator $\mathcal J$ is unitarily equivalent to the operator of multiplication by an independent variable in this space, and so the measure $\rho$ is called the spectral measure of $\mathcal J$. The derivative of the spectral measure $\rho'$ is called the spectral density and is the main object of our interest.
The spectral density of the discrete Schr\"odinger operator with summable potential can be expressed in terms of the asymptotics as $n\rightarrow\infty$ of its orthogonal polynomials by the Weyl-Titchmarsh (Kodaira) type formula. The classical Weyl-Titchmarsh formula deals with the differential Schr\"odinger operator on the half-line with summable potential \cite[Chapter 5]{Titchmarsh-1946-1}, \cite{Kodaira-1949}. For the operator $\mathcal J$ considered in this paper (and actually for a larger class of discrete Scr\"odinger operators with non-summable potentials) analogue of it was obtained in \cite{Janas-Simonov-2010} and is given by the following statement. Define the new variable $z$ as follows:
\begin{equation*}
\lambda=z+\frac1z\text{ and }z=\frac{\lambda+i\sqrt{4-\lambda^2}}2.
\end{equation*}
The interval $[-2,2]$ of the variable $\lambda$ corresponds to the upper half of the unit circle of the variable $z$.
\begin{prop}[Janas-Simonov]\label{prop Janas-Simonov}
Let $\mathcal J$ be the discrete Schr\"odinger operator with the potential $\{b_n\}_{n=1}^{\infty}$ given by \eqref{b} and let conditions \eqref{conditions} hold. Then there exists a continuous function $F:\mathbb T\backslash\{1,-1,e^{\pm i\omega},-e^{\pm i\omega}\}\rightarrow\mathbb C$ such that orthogonal polynomials associated to $\mathcal J$ have the following asymptotics for $\lambda\in(-2,2)\backslash\{\pm2\cos\omega\}$:
\begin{equation*}
P_n(\lambda)=\frac{zF(z)}{1-z^2}\cdot\frac1{z^n}+\frac{z\overline{F(z)}}{z^2-1}\cdot z^n+o(1)\text{ as }n\rightarrow\infty.
\end{equation*}
Function $F$ does not vanish on $\mathbb T\backslash\{1,-1,e^{\pm i\omega},-e^{\pm i\omega}\}$. Spectrum of $\mathcal J$ is purely absolutely continuous on $(-2;2)\backslash\{\pm2\cos\omega\}$. The spectral density of $\mathcal J$ equals:
\begin{equation}\label{Weyl-Titchmarsh type formula}
\rho'(\lambda)=\frac{\sqrt{4-\lambda^2}}{2\pi|F(z)|^2},\ \lambda\in(-2;2).
\end{equation}
\end{prop}
The Weyl-Titchmarsh type formula \eqref{Weyl-Titchmarsh type formula} will be the main tool in our analysis of the behaviour of the spectral density.
\section{Reduction of the eigenfunction equation to the model problem}\label{section reduction}
In this section we transform the eigenfunction equation for $\mathcal J$ rewriting it as a discrete linear system in $\mathbb C^2$ and reducing it to the model system of a simple form, which was studied in \cite{Naboko-Simonov-2011}. As the result we will be able to control the spectral density of $\mathcal J$ by a reformulation of Proposition \ref{prop Janas-Simonov} above in terms of a certain solution of the model system. As a byproduct we establish the asymptotic behavior of generalized eigenvectors at the critical points.
Consider for $\lambda\in(-2;2)$ the eigenfunction equation for $\mathcal J$
\begin{equation}\label{spectral equation}
u_{n-1}+b_nu_n+u_{n+1}=\lambda u_n,\ n\ge2.
\end{equation}
Write it in the vector form,
\begin{equation}\label{spectral equation, vector form}
\left(%
\begin{array}{c}
u_n \\
u_{n+1} \\
\end{array}%
\right)
=\left(%
\begin{array}{cc}
0 & 1 \\
-1 & \lambda-b_n \\
\end{array}%
\right)
\left(%
\begin{array}{c}
u_{n-1} \\
u_n \\
\end{array}%
\right),
\ n\ge 2.
\end{equation}
Consider a new parameter $\phi$ such that $\lambda=2\cos\phi$.
Variation of parameters in the form
\begin{equation*}
\left(%
\begin{array}{c}
u_n \\
u_{n+1} \\
\end{array}%
\right)
=
\left(%
\begin{array}{cc}
e^{-i\phi n} & e^{i\phi n} \\
e^{-i\phi(n+1)} & e^{i\phi(n+1)} \\
\end{array}%
\right)
v_n
\end{equation*}
in the system \eqref{spectral equation, vector form} leads to an equivalent system
\begin{equation}\label{system for v}
v_{n+1}=M_n(\phi)v_n,\ n\ge1,
\end{equation}
with the coefficient matrix
\begin{multline}\label{M}
M_n(\phi):=I+\frac{b_{n+1}}{2i\sin\phi}
\left(%
\begin{array}{cc}
1 & e^{2i\phi(n+1)} \\
-e^{-2i\phi(n+1)} & -1 \\
\end{array}%
\right)
\\
=I+V^{(1)}_n(\phi)+R^{(1)}_n(\phi),
\end{multline}
where
\begin{multline}\label{V-1}
V^{(1)}_n(\phi):=
\frac{c\sin(2\omega(n+1)+\delta)}{2i(n+1)\sin\phi}
\left(%
\begin{array}{cc}
1 & 0 \\
0 & -1 \\
\end{array}%
\right)
\\
+
\frac{c}{4(n+1)\sin\phi}
\left(%
\begin{array}{cc}
0 & e^{i(2(\phi-\omega)(n+1)-\delta)} \\
e^{-i(2(\phi-\omega)(n+1)-\delta)} & 0 \\
\end{array}%
\right)
\\
-
\frac{c}{4(n+1)\sin\phi}
\left(%
\begin{array}{cc}
0 & e^{i(2(\phi+\omega)(n+1)+\delta)} \\
e^{-i(2(\phi+\omega)(n+1)+\delta)} & 0 \\
\end{array}%
\right)
\end{multline}
and
\begin{equation}\label{R-1}
R^{(1)}_n(\phi):=\frac{q_{n+1}}{2i\sin\phi}
\left(%
\begin{array}{cc}
1 & e^{i\phi(n+1)} \\
-e^{-i\phi(n+1)} & -1 \\
\end{array}%
\right).
\end{equation}
The following theorem gives asymptotics of generalized eigenvectors of $\mathcal J$ for different values of the spectral parameter belonging to the interval $(-2,2)$.
\begin{thm}\label{thm asymptotics of GEV}
Let $\mathcal J$ be the discrete Schr\"odinger operator with the potential $\{b_n\}_{n=1}^{\infty}$ given by \eqref{b}, $\{q_n\}_{n=1}^{\infty}$ be real-valued sequence such that $\{q_n\}_{n=1}^{\infty}\in l^1$ and let the conditions \eqref{conditions} hold. Then for every $\lambda\in(-2,2)$ there exists a base $u^+(\lambda)$ and $u^-(\lambda)$ of generalized eigenvectors of $\mathcal J$ with the following asymptotics as $n\rightarrow\infty$.
\\
1. For $\lambda=2\cos\omega$
\begin{equation}\label{asymptotics at omega}
\begin{array}{l}
u^+_n(\lambda)=n^{\frac{c}{4\sin\omega}}(\cos(\omega n+\delta/2)+o(1)),
\\
u^-_n(\lambda)=n^{-\frac{c}{4\sin\omega}}(\sin(\omega n+\delta/2)+o(1)).
\end{array}
\end{equation}
2. For $\lambda=-2\cos\omega$
\begin{equation}\label{asymptotics at -omega}
\begin{array}{l}
u^+_n(\lambda)=(-1)^nn^{\frac{c}{4\sin\omega}}(\sin(\omega n+\delta/2)+o(1)),
\\
u^-_n(\lambda)=(-1)^nn^{-\frac{c}{4\sin\omega}}(\cos(\omega n+\delta/2)+o(1)).
\end{array}
\end{equation}
3. For $\lambda=2\cos\phi\in(-2;2)\backslash\{\pm2\cos\omega\}$
\begin{equation*}
\begin{array}{l}
u^+_n(\lambda)=\exp(i\phi n)+o(1),
\\
u^-_n(\lambda)=\exp(-i\phi n)+o(1).
\end{array}
\end{equation*}
\end{thm}
\begin{rem}
A similar result is obtained in, e.g., \cite{Nesterov-2010} by the method of averaging (and \cite{Janas-Simonov-2010} for the third case). Note that in $(-2,2)\backslash\{\pm2\cos\omega\}$ eigenfunction equation is elliptic (i.e., all its solutions have the same order of magnitude), while at the critical points $\pm2\cos\omega$ it is hyperbolic (the orders of two solutions are different).
\end{rem}
\begin{proof}
For $\lambda\in(-2,2)$ the transfer matrix \eqref{M} has the form
\begin{equation*}
M_n(\phi)=I+\frac{c}{4n\sin\omega}X_j+V_{j,n}(\phi),
\end{equation*}
where $\{V_{j,n}(\phi)\}_{n=1}^{\infty}$ is some conditionally summable matrix-valued sequence which belongs to $l^2$ and $X_j$ is a constant matrix, which is different in the three cases under consideration:
\\
1. $\lambda=2\cos\omega$ or $\phi=\omega$: $X_1=
\left(
\begin{array}{cc}
0 & e^{-i\delta} \\
e^{i\delta} & 0 \\
\end{array}
\right)$.
\\
2. $\lambda=-2\cos\omega$ or $\phi=\omega+\pi$: $X_2=
\left(
\begin{array}{cc}
0 & -e^{-i\delta} \\
-e^{i\delta} & 0 \\
\end{array}
\right)$.
\\
3. $\lambda\in(-2,2)\backslash\{\pm2\cos\omega\}$ or $\phi\in(0,\pi)\backslash\{\omega\mod\pi,\pi-(\omega\mod\pi)\}$: $X_3=0$.
\\
By \cite[Theorem 3.2]{Benzaid-Lutz-1987} (a version of the discrete Levinson theorem) we can neglect the term $V_n(\phi)$ (i.e., solutions of the system \eqref{system for v} have the same asymptotics as solutions of the analogous system without this term). Hence in all three cases $j=1,2,3$ the system $v_{n+1}=M_n(\phi)v_n$ has a base of solutions $v^{(1)}_n$ and $v^{(2)}_n$ with the following asymptotics as $n\rightarrow\infty$:
\begin{equation*}
v^{(1)}_n=n^{\frac{c\mu_1(X_j)}{4\sin\omega}}\left(\overrightarrow x_j^{(1)}+o(1)\right),
\end{equation*}
and
\begin{equation*}
v^{(2)}_n=n^{\frac{c\mu_2(X_j)}{4\sin\omega}}\left(\overrightarrow x_j^{(2)}+o(1)\right),
\end{equation*}
where $\mu_1(X_j),\mu_2(X_j)$ are the eigenvalues of matrices $X_j$ and $\overrightarrow x_j^{(1)},\overrightarrow x_j^{(2)}$ are the corresponding eigenvectors. Returning to the solution $u_n$ by the equality $u_n=e^{-i\phi n}(v_n)_1+e^{i\phi n}(v_n)_2$ we complete the proof (by $(v_n)_1$ and $(v_n)_2$ we denote two components of the vector $v_n\in\mathbb C^2$).
\end{proof}
From now on we will use the expression $c\sin(2\omega n+\delta)$ in the rewritten form $$c\sin(2\omega n+\delta)=|c|\sin(2\omega_1n+\delta_1)$$ with \begin{equation}\label{delta-1}
\delta_1:=\delta+\frac{\pi}2(\text{sign}\,c-1)
\end{equation}
and
\begin{equation}\label{omega-1}
\omega_1:=\omega-\pi\left\lfloor\frac{\omega}{\pi}\right\rfloor\in(0,\pi),
\end{equation}
where $\lfloor\cdot\rfloor$ denotes the standard floor function ($\lfloor x\rfloor$ is the greatest integer which is less than $x$). Now we can fix the range of the variable $\phi\in(0,\pi)$ corresponding to $\lambda\in(-2,2)$, so that $z=e^{i\phi}$.
The term $V^{(1)}_n(\phi)$ for $\phi\neq\omega_1,\pi-\omega_1$ is conditionally summable and belongs to $l^2$. As Theorem \ref{thm asymptotics of GEV} shows, this term does not affect the type of asymptotics of solutions (this leads to the preservation of the absolutely continuous spectrum and was considered in detail in \cite{Janas-Simonov-2010}). The values $\phi=\omega_1,\pi-\omega_1$ correspond to $\lambda=\pm2\cos\omega$, i.e., to the resonance points. At these points the term
$V^{(1)}_n(\phi)$ is not summable even conditionally and the type of solutions asymptotics is different.
As Proposition \ref{prop Janas-Simonov} and the forumla \eqref{system for v} suggest, the spectral density is related to the asymptotic behavior of the solution to the system $v_{n+1}=M_n(\phi)v_n$, which corresponds to orthogonal polynomials. We need to understand the dependence of asymptotics of this solution on the parameter $\lambda$ (or, equivalently, on the parameter $\phi$) near two critical points. The analysis is based upon the idea that if (for example) $\phi$ is close to $\omega_1$, then
\begin{equation*}
M_n(\phi)=I+\frac{|c|}{4n\sin\omega_1}
\left(
\begin{array}{c}
0 \qquad e^{i(2(\phi-\omega_1)n-\delta_1)} \\
e^{-i(2(\phi-\omega_1)n-\delta_1)} \qquad 0 \\
\end{array}
\right)
+\left(\begin{array}{c}
\text{some}
\\
\text{inessential part}
\end{array}
\right).
\end{equation*}
As we have seen before, the terms in matrix entries of $M_n(\phi)$ of the form $\frac{e^{i\alpha n}}n$ are "dangerous" (make effect on asymptotics) only if $\alpha\in2\pi\mathbb Z$. In the case $\alpha=0$ the term of the type $\frac Xn$, where $X$ is some constant matrix, produces a resonance (change of solutions asymptotics). Now we want to eliminate all non-resonating exponential terms from the system by a certain transformation, i.e., by substitution $v_n\mapsto w_n:=T_n(\phi)v_n$, where $\{T_n(\phi)\}_{n=1}^{\infty}$ is a sequence of invertible matrices. Such a substitution leads to the discrete linear system $w_{n+1}=T_{n+1}^{-1}M_n(\phi)T_nw_n$. Transformations that we find are local, i.e., exist and can be applied only in some neighbourhoods of the critical points. It is important to control the properties of the summable remainder to ensure that it is still uniformly summable after the transformation. Let us introduce the following notation. Let $S$ be some subset of the complex plane and $R_n(\lambda)$, $n\in\mathbb N,\lambda\in S$ be a sequence of $2\times2$ matrices depending on a parameter. We write $\{R_n(\lambda)\}_{n=1}^{\infty}\in l^1(S)$, if there exists a sequence of positive numbers $\{r_n\}_{n=1}^{\infty}\in l^1$ such that for every $\lambda\in S$ and $n\in\mathbb N$ one has $\|R_n(\lambda)\|<r_n$.
Let $U_+$ and $U_-$ be open intervals such that
\begin{equation*}
\begin{array}{rl}
\omega_1\in &U_+\subset(0,\pi)\backslash\{\pi-\omega_1\},
\\
\pi-\omega_1\in &U_-\subset(0,\pi)\backslash\{\omega_1\}.
\end{array}
\end{equation*}
Define Harris-Lutz type transformations \cite{Harris-Lutz-1975,Benzaid-Lutz-1987} as follows:
\begin{equation}\label{T-pm}
T^{\pm}_n(\phi):=-\sum_{k=n}^{\infty}
\Bigl[
V^{(1)}_k(\phi)\mp\frac{|c|}{4k\sin\omega_1}
\left(
\begin{array}{cc}
0 & e^{i(2(\phi\mp\omega_1)k\mp\delta_1)} \\
e^{-i(2(\phi\mp\omega_1)k\mp\delta_1)} & 0 \\
\end{array}
\right)
\Bigr].
\end{equation}
We will later use the following (trivial) result.
\begin{lem}\label{lem estimate of sum for Harris-Lutz}
For every real $\xi\in\mathbb R\backslash2\pi\mathbb Z$ and $n\in\mathbb N$ one has
$\l|\sum\limits_{k=n}^{\infty}\frac{e^{ik\xi}}k\r|\le\frac1{n\l|\sin\frac{\xi}2\r|}$.
\end{lem}
\begin{proof}
Straightforwardly,
\begin{multline*}
\left|(e^{i\xi}-1)\sum_{k=n}^{\infty}\frac{e^{ik\xi}}k\right|
=\left|\sum_{k=n}^{\infty}e^{i(k+1)\xi}\l(\frac1k-\frac1{k+1}\r)-\frac{e^{in\xi}}n\right|
\\
\le\sum_{k=n}^{\infty}\l(\frac1k-\frac1{k+1}\r)+\frac1n=\frac2n.
\end{multline*}
Since $|e^{i\xi}-1|=2\l|\sin\frac{\xi}2\r|$, the proof is complete.
\end{proof}
Now we are able to state the properties of the transformations $T^{\pm}$.
\begin{lem}\label{lem properties of Harris-Lutz transformation}
Sums in \eqref{T-pm}, which define $T^{\pm}(\phi)$, converge in $U_{\pm}$ and estimates
\begin{equation}\label{estimates of T-pm}
T^{\pm}_n(\phi)=O\l(\frac1n\r)\text{ as }n\rightarrow\infty
\end{equation}
hold uniformly in $U_{\pm}$, respectively. Moreover,
\begin{multline}\label{Harris-Lutz transformation T-pm}
\exp(-T^{\pm}_{n+1}(\phi))M_n(\phi)\exp(T^{\pm}_n(\phi))
\\
=I\pm\frac{|c|}{4n\sin\omega_1}
\left(%
\begin{array}{cc}
0 & e^{i(2(\phi\mp\omega_1)n\mp\delta_1)} \\
e^{-i(2(\phi\mp\omega_1)n\mp\delta_1)} & 0 \\
\end{array}%
\right)
+R_n^{\pm}(\phi),
\end{multline}
where $\{R_n^{\pm}(\phi)\}_{n=1}^{\infty}\in l^1(U_{\pm})$ and for every natural $n$ functions $R_n^{\pm}(\cdot)$ are continuous in $U_{\pm}$, respectrively.
\end{lem}
\begin{proof}
Let us prove the statement for $T^+$ (one can obtain the proof of the second statement by changing the notation). Write
\begin{equation*}
T^+_n(\phi)=\sum_{k=n}^{\infty}t^+_k(\phi),
\end{equation*}
where
\begin{multline*}
t^+_k(\phi):=
-\frac{|c|\sin(2\omega_1(k+1)+\delta_1)}{2i(k+1)\sin\phi}
\left(%
\begin{array}{cc}
1 & 0 \\
0 & -1 \\
\end{array}%
\right)
\\
-
\frac{|c|}{4(k+1)\sin\phi}
\left(%
\begin{array}{cc}
0 & e^{i(2(\phi-\omega_1)(k+1)-\delta_1)} \\
e^{-i(2(\phi-\omega_1)(k+1)-\delta_1)} & 0 \\
\end{array}%
\right)
\\
+
\frac{|c|}{4(k+1)\sin\phi}
\left(%
\begin{array}{cc}
0 & e^{i(2(\phi+\omega_1)(k+1)+\delta_1)} \\
e^{-i(2(\phi+\omega_1)(k+1)+\delta_1)} & 0 \\
\end{array}%
\right)
\\
+
\frac{|c|}{4k\sin\omega_1}
\left(%
\begin{array}{cc}
0 & e^{i(2(\phi-\omega_1)k-\delta_1)} \\
e^{-i(2(\phi-\omega_1)k-\delta_1)} & 0 \\
\end{array}%
\right).
\end{multline*}
Since the values $\xi=\pm2\omega_1,\pm2(\phi+\omega_1)$ do not belong to $\mathbb R\backslash2\pi\mathbb Z$ for $\phi\in U_+$, the sum over $k$ of first and third terms can be uniformly estimated using Lemma \ref{lem estimate of sum for Harris-Lutz}. The difference between the second term and the same expression with $k$ instead of $k+1$ in the denominator is uniformly $O(1/k^2)$, therefore the difficulty can only arise near the point $\phi=\omega_1$ when one takes the sum over $k$ of the following terms:
\begin{multline*}
\left[
\frac1{\sin\omega_1}I
-\frac1{\sin\phi}
\left(
\begin{array}{cc}
e^{2i(\phi-\omega_1)} & 0 \\
0 & e^{-2i(\phi-\omega_1)} \\
\end{array}
\right)
\right]
\\
\times
\frac{|c|}{4k}
\left(
\begin{array}{cc}
0 & e^{i(2(\phi-\omega_1)k-\delta_1)} \\
e^{-i(2(\phi-\omega_1)k-\delta_1)} & 0 \\
\end{array}
\right).
\end{multline*}
Expression in the square brackets does not depend on $k$ and is $O(\phi-\omega_1)$ as $\phi\rightarrow\omega_1$, which cancels the zero of $\xi$ when we apply Lemma \ref{lem estimate of sum for Harris-Lutz}. This gives a uniform in $U_+$ estimate $T^+_n(\phi)=O(1/n)$ as $n\rightarrow\infty$. To obtain the equality \eqref{Harris-Lutz transformation T-pm} we use the estimate $e^Y=I+Y+O(\|Y\|^2)$ as $\|Y\|\rightarrow0$ together with \eqref{estimates of T-pm}: substitute $M_n$ in the form \eqref{M},\eqref{V-1} and \eqref{R-1} and $T^+_n$ in the form \eqref{T-pm} into the expression
$(I-T^+_{n+1}+O(1/n^2))M_n(I+T^+_n+O(1/n^2))$,
open the brackets, simplify the result and leave only the terms of the order $1/n$ (the smaller terms should be included into the remainder $R^+_n$). The remainder is uniformly summable and continuous in $U_+$, which follows immediately.
\end{proof}
Transform the system further using the similarity relation
\begin{equation*}
\left(
\begin{array}{cc}
0 & e^{i\alpha} \\
e^{-i\alpha} & 0 \\
\end{array}
\right)
=
\left(
\begin{array}{cc}
1 & i \\
1 & -i \\
\end{array}
\right)
\left(
\begin{array}{cc}
\cos\alpha & \sin\alpha \\
\sin\alpha & -\cos\alpha \\
\end{array}
\right)
\left(
\begin{array}{cc}
1 & i \\
1 & -i \\
\end{array}
\right)^{-1}.
\end{equation*}
It is easy to check that by the transformation
\begin{equation*}
\hat v^+_n:=
\left(%
\begin{array}{cc}
1 & i \\
1 & -i \\
\end{array}%
\right)^{-1}
\left(%
\begin{array}{cc}
e^{i\delta_1/2} & 0 \\
0 & e^{-i\delta_1/2} \\
\end{array}%
\right)
\exp(-T^+_n(\phi))v_n
\end{equation*}
the system $v_{n+1}=M_n(\phi)v_n$ is reduced to the following one:
\begin{equation}\label{system for v-hat+}
\hat v^+_{n+1}=
\left[
I+\frac{|c|}{4n\sin\omega_1}
\left(%
\begin{array}{cc}
\cos(2(\phi-\omega_1)n) & \sin(2(\phi-\omega_1)n) \\
\sin(2(\phi-\omega_1)n) & -\cos(2(\phi-\omega_1)n) \\
\end{array}%
\right)
+\hat R_n^+(\phi)
\right]
\hat v^+_n,
\end{equation}
where $\{\hat R_n^+(\phi)\}_{n=1}^{\infty}\in l^1(U_+)$ and the function $\hat R^+_n(\cdot)$ is continuous in $U_+$ for every $n$. System \eqref{system for v-hat+} is equivalent for $\phi\in U_+$ to the eigenfunction equation \eqref{spectral equation} for the operator $\mathcal J$. Define the solution $\hat p^+(\phi)$ of \eqref{system for v-hat+} which corresponds to orthogonal polynomials:
\begin{multline}\label{p-hat+}
\hat p^+_n(\phi):=
\left(%
\begin{array}{cc}
1 & i \\
1 & -i \\
\end{array}%
\right)^{-1}
\left(%
\begin{array}{cc}
e^{i\delta_1/2} & 0 \\
0 & e^{-i\delta_1/2} \\
\end{array}%
\right)
\exp(-T^+_n(\phi))
\\
\times
\left(%
\begin{array}{cc}
e^{-i\phi n} & e^{i\phi n} \\
e^{-i\phi(n+1)} & e^{i\phi(n+1)} \\
\end{array}%
\right)^{-1}
\left(
\begin{array}{c}
P_n(2\cos\phi) \\
P_{n+1}(2\cos\phi) \\
\end{array}
\right).
\end{multline}
Now we are able to restate Proposition \ref{prop Janas-Simonov} in a form which is more convenient for our needs. The objects $\hat v^-_n,\hat R_n^-(\phi),\hat p^-_n(\phi)$ are defined in the same fashion for $\phi\in U_-$.
\begin{lem}\label{lem Weyl-Titchmarsh formula}
For every $\phi\in U_+$ the sequence $\{\hat p^+_n(\phi)\}_{n=1}^{\infty}$ given by \eqref{p-hat+} is a solution of the system \eqref{system for v-hat+}. For every $\phi\in U_+\backslash\{\omega_1\}$ it has a non-zero limit
\begin{equation*}
\lim_{n\rightarrow\infty}\hat p^+_n(\phi)=:\hat p^+_{\infty}(\phi).
\end{equation*}
The spectral density of $\mathcal J$ can be expressed in terms of this limit as
\begin{equation}\label{rho' in U_+}
\rho'(2\cos\phi)=\frac1{4\pi\sin\phi\l\|\hat p^+_{\infty}(\phi)\r\|^2},\ \phi\in U_+.
\end{equation}
Analogous statement holds true, if one replaces $\hat p^+$ by $\hat p^-$, $\hat R^+$ by $\hat R^-$, $U_+$ by $U_-$ and $\omega_1$ by $\pi-\omega_1$.
\end{lem}
\begin{proof}
The assertion of Proposition \ref{prop Janas-Simonov} for $\phi\in U_+\backslash\{\omega_1\}$ can be rewritten as
\begin{equation*}
\left(%
\begin{array}{cc}
e^{-i\phi n} & e^{i\phi n} \\
e^{-i\phi(n+1)} & e^{i\phi(n+1)} \\
\end{array}%
\right)^{-1}
\left(
\begin{array}{c}
P_n(2\cos\phi) \\
P_{n+1}(2\cos\phi) \\
\end{array}
\right)
\rightarrow
\frac1{2\sin\phi}
\left(
\begin{array}{c}
iF(e^{i\phi}) \\
\overline{iF(e^{i\phi})} \\
\end{array}
\right)
\end{equation*}
as $n\rightarrow\infty$. Together with the fact that $T^+_n(\phi)=o(1)$ this yields:
\begin{equation*}
\hat p^+_{\infty}(\phi)=
\left(%
\begin{array}{cc}
1 & i \\
1 & -i \\
\end{array}%
\right)^{-1}
\left(%
\begin{array}{cc}
e^{i\delta_1/2} & 0 \\
0 & e^{-i\delta_1/2} \\
\end{array}%
\right)
\frac1{2\sin\phi}
\left(
\begin{array}{c}
iF(e^{i\phi}) \\
\overline{iF(e^{i\phi})} \\
\end{array}
\right)
\end{equation*}
An explicit calculation shows that
\begin{equation*}
\left\|\hat p^+_{\infty}(\phi)\right\|=\frac{|F(e^{i\phi})|}{2\sin\phi}.
\end{equation*}
By Proposition \ref{prop Janas-Simonov} again,
\begin{equation*}
\rho'(2\cos\phi)=\frac{\sin\phi}{\pi|F(e^{i\phi})|^2}=\frac1{4\pi\sin\phi\l\|\hat p^+_{\infty}(\phi)\r\|^2},
\end{equation*}
which completes the proof for $\phi\in U_+$. In the second case the proof is analogous.
\end{proof}
\section{Results for the model problem}\label{section Naboko-Simonov}
In this section we formulate results concerning the model system
\begin{equation}\label{model system}
x_{n+1}=
\left[
I+\frac{\beta}n\left(%
\begin{array}{cc}
\cos(\varepsilon n) & \sin(\varepsilon n) \\
\sin(\varepsilon n) & -\cos(\varepsilon n) \\
\end{array}%
\right)
+R_n(\varepsilon)
\right]
x_n,\ n\in\mathbb N,\ \varepsilon\in U,
\end{equation}
which were obtained in \cite{Naboko-Simonov-2011}. Using these results we immediately get information about the behavior of the functions $\hat p^+_{\infty}(\phi)$ and $\hat p^-_{\infty}(\phi)$ near the points $\omega_1$ and $\pi-\omega_1$, respectively, and therefore about the behavior of the spectral density of $\mathcal J$ near the critical points, cf. \eqref{rho' in U_+}. Here $\beta$ is positive, $\varepsilon\in U$ is a small parameter, $U$ is an interval such that \begin{equation*}
0\in U\subset (-2\pi;2\pi)
\end{equation*}
and the matrices $R_n(\varepsilon)$ are supposed to be uniformly summable in $n$ with respect to $\varepsilon\in U$ and continuous in $U$ for every $n$.
Let us write the system \eqref{model system} as
\begin{equation*}
x_{n+1}=B_n(\varepsilon)x_n
\end{equation*}
with
\begin{equation}\label{B-n}
B_n(\varepsilon):
=
I+\frac{\beta}n
\left(%
\begin{array}{cc}
\cos(\varepsilon n) & \sin(\varepsilon n) \\
\sin(\varepsilon n) & -\cos(\varepsilon n) \\
\end{array}%
\right)
+R_n(\varepsilon).
\end{equation}
We parametrize different solutions by their initial conditions $f\in\mathbb C^2$ (while the system itself depends on the small parameter $\varepsilon\in U$):
\begin{equation}\label{u(f)}
\begin{array}{rl}
x_1(\varepsilon,f)&:=f,
\\
x_{n+1}(\varepsilon,f)
&:=
B_n(\varepsilon)x_n(\varepsilon,f),\,n\ge1.
\end{array}
\end{equation}
\begin{prop}[Naboko-Simonov]\label{prop Naboko-Simonov}
Assume that functions $R_n(\cdot)$ are continuous in $U$ for every $n\in\mathbb N$, the matrices $B_n(\varepsilon)$ are invertible for every $n\in\mathbb N$, $\varepsilon\in U$ and the sequence $\{R_n(\varepsilon)\}_{n=1}^{\infty}\in l^1(U)$.
Then for every $f\in\mathbb C^2$ and every $\varepsilon\in U\backslash\{0\}$ the limit
\begin{equation*}
\lim_{n\rightarrow\infty}x_n(\varepsilon,f)
\end{equation*}
exist. For $\varepsilon=0$ the limit
\begin{equation*}
\lim\limits_{n\rightarrow\infty}\frac{x_n(0,f)}{n^{\beta}}
\end{equation*}
exists for every $f$ and the linear map
\begin{equation*}
f\mapsto \lim\limits_{n\rightarrow\infty}\frac{x_n(0,f)}{n^{\beta}}
\end{equation*}
has rank one. If, moreover, $f$ is such that $\lim\limits_{n\rightarrow\infty}\frac{x_n(0,f)}{n^{\beta}}\neq0$, then there exist two one-side limits
\begin{equation*}
\lim\limits_{\varepsilon\rightarrow\pm0}|\varepsilon|^{\beta}\lim\limits_{n\rightarrow\infty}x_n(\varepsilon,f)\neq0.
\end{equation*}
\end{prop}
This result can be reformulated in terms of infinite matrix products, which will be more useful for us here. Let $R$ stand for the whole sequence $\{R_n(\varepsilon)\}_{n=1}^{\infty}$.
\begin{prop}\label{prop matrix products}
In assumptions of Proposition \ref{prop Naboko-Simonov}, the following holds. For every $\varepsilon\in U\backslash\{0\}$ there exists
\begin{equation*}
\Phi(\beta,\varepsilon,R):=\prod_{n=1}^{\infty}B_n(\varepsilon).
\end{equation*}
For $\varepsilon=0$ there exists the limit
\begin{equation*}
\Phi_0(\beta,R):=\lim_{N\rightarrow\infty}\frac1{N^{\beta}}\prod_{n=1}^NB_n(0),
\end{equation*}
which is a matrix of rank one. And finally, there exist two one-side limits
\begin{equation*}
\Phi_{\pm}(\beta,R):=
\lim_{\varepsilon\rightarrow\pm0}|\varepsilon|^{\beta}\Phi(\beta,\varepsilon,R)
\end{equation*}
such that
\begin{equation*}
\text{Ker }\Phi_0(\beta,R)=\text{Ker }\Phi_-(\beta,R)=\text{Ker }\Phi_+(\beta,R).
\end{equation*}
\end{prop}
\section{Zeroes of the spectral density}\label{section final result}
In this section we put together all the ingredients: the Weyl-Titchmarsh type formula from \cite{Janas-Simonov-2010}, the analysis of \cite{Naboko-Simonov-2011} and the transformations of Section \ref{section reduction}, to obtain the main result of the present paper.
\begin{thm}\label{thm main result}
Let $\mathcal J$ be the discrete Schr\"odinger operator with the potential
\begin{equation*}
\frac{c\sin(2\omega n+\delta)}n+q_n,
\end{equation*}
where $c,\omega,\delta$ are real constants, $\{q_n\}_{n=1}^{\infty}$ is a real-valued sequence such that
\begin{equation*}
c\neq0,\omega\notin\frac{\pi\mathbb Z}2\text{ and }\{q_n\}_{n=1}^{\infty}\in l^1.
\end{equation*}
Let $\nu_{cr}\in\{-2\cos\omega,2\cos\omega\}$. If $\nu_{cr}$ is neither an eigenvalue nor a half-bound state of $\mathcal J$, then there exist two one-side limits
\begin{equation*}
\lim_{\lambda\rightarrow\nu_{cr}\pm0}\frac{\rho'(\lambda)}
{|\lambda-\nu_{cr}|^{\frac{|c|}{2|\sin\omega|}}},
\end{equation*}
where $\rho'$ is the spectral density of $\mathcal J$.
\end{thm}
\begin{proof}
Consider the neighbourhood of the critical point $2\cos\omega_1$ and $\phi\in U_+$. Take
\begin{equation*}
\varepsilon:=2(\phi-\omega_1),
\end{equation*}
see \eqref{system for v-hat+}. Lemma \ref{lem Weyl-Titchmarsh formula} yields in the notation of Proposition \ref{prop matrix products}:
\begin{multline*}
\rho'(\lambda)=\rho'(2\cos\phi)=\rho'(2\cos(\omega_1+\varepsilon/2))
\\
=\frac1{\pi\sin(\omega_1+\varepsilon/2)}
\frac1{
\left\|
\Phi\left(\frac{|c|}{4\sin\omega_1},\varepsilon,\hat R^+\right)
\hat p^+_1(\omega_1+\varepsilon/2)
\right\|^2}.
\end{multline*}
Now the asymptotics of $\rho'(2\cos(\omega_1+\varepsilon/2))$ as $\varepsilon\rightarrow\pm0$ follow from Proposition \ref{prop matrix products} due to the continuity of $\hat p^+_1(\phi)$, if only $$\hat p^+_1(\omega_1)\notin\text{Ker}\,\Phi_0\left(\frac{|c|}{4\sin\omega_1},\hat R^+\right).$$
The latter by the definition of $\Phi_0$ means that $$\lim\limits_{n\rightarrow\infty}\frac{\hat p^+_n(\omega_1)}{n^{\frac{|c|}{4\sin\omega_1}}}\neq0.$$
Due to the relation \eqref{p-hat+} and Theorem \ref{thm asymptotics of GEV} this in turn means that orthogonal polynomials at the point $2\cos\omega_1$ are not $O\left(n^{-\frac{|c|}{4\sin\omega_1}}\right)$ as $n\rightarrow\infty$, see the formulas \eqref{asymptotics at omega} and \eqref{asymptotics at -omega}.
In the opposite case, the point $2\cos\omega_1$ is either an eigenvalue of $\mathcal J$ (if $\frac{|c|}{4\sin\omega_1}>\frac12$) or a half-bound state (if $\frac{|c|}{4\sin\omega_1}\le\frac12$). Since $\sin\omega_1=|\sin\omega|$, this proves the result for the critical point $2\cos\omega_1$. The proof for the second critical point $-2\cos\omega_1$ can be obtained by changing $\omega_1$ to $\pi-\omega_1$ and $+$ to $-$ in the notation.
\end{proof}
\subsection*{Acknowledgements}
The author expresses his deep gratitude to Prof. S.N. Naboko for his constant attention to this work and for many fruitful discussions of the subject, to Prof. Jan Janas for invaluable help in the work and to Prof. Harald Woracek for many important remarks and suggestions. The work was supported by the Chebyshev Laboratory (Department of Mathematics and Mechanics, Saint-Petersburg State University) under the grant 11.G34.31.0026 of the Government of the Russian Federation, by grants RFBR-09-01-00515-a and 11-01-90402-Ukr\_f\_a and by the Erasmus Mundus Action 2 Programme of the European Union.
|
{
"timestamp": "2012-03-12T01:00:14",
"yymm": "1203",
"arxiv_id": "1203.1935",
"language": "en",
"url": "https://arxiv.org/abs/1203.1935"
}
|
\section{Introduction}
One of the distinctive feature of the QCD phase diagram is the
possible emergence of a critical endpoint (CEP) where the first-order
chiral phase transition line at large chemical potential
terminates. Many properties of the endpoint such as, e.g., its precise
location are still unknown. However, for experiments that search for
this point the knowledge of its characteristics is inevitable. Since
the expected transition is of second-order at this point possible
signatures are based on the singular behavior of the thermodynamic
functions in its vicinity. Recently, it was pointed out that
event-by-event fluctuations of particle multiplicities and their
nonmonotonic behavior might serve as a probe to locate a possible CEP
in the phase diagram. By scanning the center of mass energy and thus
the baryochemical potential an increase and then a decrease in the
number fluctuations of, e.g., pions and protons should be seen as one
crosses the critical point. Unfortunately, in a realistic heavy-ion
collision the expected signals are washed out due to the critical
slowing down phenomenon and finite volume effects. Furthermore,
correlations can only build up for a finite time in the colliding
system and consequently the correlations length is cut off. Some
conservative estimates yield an increase of the correlation length
only by a factor 3 as the system crosses the critical
point. Therefore, more sensible quantities are needed for the analysis
of the freeze-out and critical conditions in heavy-ion collisions. For
this reason higher-order cumulants or ratios of higher-order
generalized susceptibilities have been suggested as suitable
quantities because they depend on higher powers of the correlation
length \cite{Stephanov:1999zu}.
\section{A three-flavor model analysis}
Fluctuations of conserved charges are quantified by cumulants in
statistics and are related to generalized susceptibilities
\cite{Koch:2008ia}. They are defined as derivatives of the logarithm
of the partition function with respect to the appropriate chemical
potentials. For three different chemical potentials we have
accordingly
\begin{equation}\label{eq:defchi}
\chi_{n_i,n_j,n_k} \equiv \frac{ 1}{VT} \frac{ \partial^{n_i}}{\partial (\mu_{i}/T)^{n_i}}
\frac{ \partial^{n_j}}{\partial (\mu_{j}/T)^{n_j}}
\frac{ \partial^{n_k}}{\partial (\mu_{k}/T)^{n_k}} \log Z \ ,
\end{equation}
where $n_i,\ldots$ denotes the order of the derivatives and the
indices $(i,j,k) = (u,d,s)$ the quark flavor. The generalized
susceptibilities evaluated at vanishing $\mu_f$ are the Taylor
expansion coefficients of the pressure series in powers of $\mu_f/T$.
The partition function is evaluated in a renormalized three-flavor
Polyakov-quark-meson (PQM) model with an axial $U(1)_A$ symmetry
breaking term and a logarithmic Polyakov-loop potential in mean-field
approximation where the ultraviolet divergent fermion vacuum
contribution has been taken into account \cite{Schaefer:2011ex}. Thus,
fermion fluctuations are taken into account whereas meson and
Polyakov-loop fluctuations are still ignored. The resulting phase
diagram in comparison with the one obtained in the three-flavor
quark-meson model is shown in \Fig{fig:pqmpd}
\subsection{Ratios of baryon number moments}
In the following we focus on one uniform quark chemical
potential and denote the $n$th to $m$th order moment ratio of the
quark number fluctuations $\chi_n$ as
\begin{equation}
R_{n,m}\equiv \chi_n (T,\mu)/\chi_m (T,\mu)\ .
\end{equation}
The evaluation of the ratios has been automated by using algorithmic
differentiation techniques \cite{Wagner:2009pm}. The kurtosis
$R_{4,2}$ basically measures the quark content of particles carrying
baryon number and has been calculated in the vicinity of the crossover
region of the phase diagram. In the hadron resonance gas (HRG) model
the ratio is temperature independent and all moments stay positive
since the HRG model has no singularities. Thus, any deviation from
the HRG model result might be an indicator for a real critical
phenomenon even if the lower-order moments and the thermodynamics
including particle yields are well described by the HRG
model~\cite{Andronic:2009qf}. In our model calculation the kurtosis
becomes negative and consequently a negative region around the
crossover line emerge in the phase diagram. Note that these
three-flavor results differ from a corresponding two-flavor PQM
mean-field calculation with and without the fermion vacuum term in the
grand potential. Without the vacuum term in the two-flavor case the
kurtosis $R_{4,2}$ develops a peak at zero chemical potential near the
crossover temperature which is a remnant of the first-order transition
in the chiral limit, see \cite{Nakano:2009ps} for more details. The
inclusion of the fermion vacuum term changes the transition to
second-order in the chiral limit which is consistent with universality
arguments \cite{Skokov:2010sf}. However, for three massless flavor a
first-order transition always emerges in the chiral limit with or
without the fermion vacuum term. The peak of the kurtosis for physical
pion masses is already less pronounced if the vacuum term is neglected
and finally vanishes in the three-flavor renormalized model. Thus, the
suppression of the kurtosis peak at $\mu=0$ must be caused by the
strange quarks and not by fluctuations \cite{Schaefer:2011ex,
Skokov:2010sf}.
\begin{figure}
\centering
\includegraphics[width=0.4\linewidth]{mwmf_pd_zerowuhan}
\caption{\label{fig:pqmpd}Phase diagrams for three flavor of the
renormalized PQM model with a running $T_0$ parameter in the
logarithmic Polyakov-loop potential and of the quark-meson
model. Dashed lines denote the chiral crossover, solid lines the
first-order chiral transition and dotted lines the peak in the
temperature-derivative of the Polyakov-loops. }
\end{figure}
\subsection{Sign structure of higher-order ratios close to the transition}
\begin{figure}
\centering
\subfigure[$\ $Polyakov-quark-meson model]{\label{fig:negreg}
\includegraphics[width=0.4\linewidth]{mwmf_pset_zero270Tmu_negative_Rn2_pdwuhan}}
\subfigure[$\ $Quark-meson model]{
\includegraphics[width=0.4\linewidth]{mwmf_pset_zeroqm_negative_Rn2_pdwuhan}}
\caption{\label{fig:extremapd}Three negative areas of $R_{n,2}$
ratios close to the chiral crossover line in both models.}
\end{figure}
Higher-order ratios behave in a similar way. They oscillate within a
narrow temperature interval for temperatures close to the chiral
transition. Generically, the structures of all moments at $\mu=0$ are
related to each other and the behavior including the amplitudes of
$\chi_n$ can be deduced by the temperature derivative of the preceding
$\chi_{n-2}$~\cite{Schaefer:2009st}. In contrast to the HRG model all
higher-order ratios become negative in the vicinity of the crossover
line for nonvanishing chemical potential as shown in
\Fig{fig:extremapd}. In this figure the negative regions of three even
ratios $R_{n,2}$ in the renormalized three-flavor PQM model along the
chiral crossover line are compared to a corresponding three-flavor
renormalized QM model without the Polyakov-loop. All negative regions
are closely correlated with the crossover curve and converge exactly
at the CEP. For $n>4$ they are shifted slightly in the hadronic phase.
The Polyakov loop sharpens these negative regions around the chiral
transition. In the renormalized models the CEP is pushed towards
higher chemical potentials by the inclusion of the vacuum terms but
the curvature of the crossover line seems not to be changed. However,
the crossover is washed out by fluctuations which yields larger
negative regions.
In summary the behavior of the negative regions can surely be
attributed to critical dynamics~\cite{Skokov:2011rq}. The findings
underline once more the importance of fluctuations: all regions
calculated in the renormalized models are shifted more in the hadronic
phase.
The knowledge of how the negative regions evolve towards the endpoint
might be used to construct new criteria to improve the critical
temperature estimate from a Taylor expansion around $\mu=0$. For this
reason it is instructive to define the distance $\Delta T = T_n -
T_\chi$ of the first zero in temperature direction of the ratio
$R_{n,2}$ to the crossover temperature $T_\chi$. In the left panel of
\Fig{fig:oddroots_pb} the distance $\Delta T$ of several even ratios
$R_{n,2}$ is shown as a function of $\mu/T$. Only the ratio $R_{4,2}$
remains positive away from the CEP. For all higher-order ratios the
first zero in $R_{n,2}$ is pushed into the hadronic phase and all
$\Delta T$ are negative already for $\mu/T \sim 0$. They obey a
minimum whose precise location and depth is model dependent. With the
Polyakov loop the transition is sharper and the minima are not as
deep. Remarkable is the almost linear behavior of $\Delta T$ for
intermediate $\mu/T$ ratios in the PQM and QM models for all ratios
$R_{n,2}$. The linear extrapolation of $\Delta T$ from intermediate
$\mu/T$ to larger values where $\Delta T$ vanishes might serve as an
estimator for the proximity of the thermal freeze-out to the crossover
line and the existence of a possible endpoint can be {\em ruled out}
for smaller values of $\mu/T$. This estimate could be strengthened by
considering only the difference of the subsequent roots in the odd
ratios which is independent of the knowledge of the chiral crossover
temperature. The relative temperature distance $\Delta \tau = T_{n+2}
- T_n$ for several even ratios $R_{n,2}$ is shown as a function of
$\mu/T$ in the right panel of \Fig{fig:oddroots_pb}. The curves
exhibit a similar behavior as in the previous figure except the
ordering of the curves with respect to $n$ is opposite. With
increasing $n$ the distance between subsequent ratios decreases,
signaling a possible convergence of the negative regions in the phase
diagram. An estimation of the lower bound of the CEP with a linear fit
is in between 15\% of the actual CEP model values.
\begin{figure*}
\centering
\includegraphics[width=0.4\linewidth]{mwmf_pset_zero270Tmu_deltaevenrootpb}
\includegraphics[width=0.4\linewidth]{mwmf_pset_zero270Tmu_deltaevenrootroot}
\caption{\label{fig:oddroots_pb}Distance $\Delta T = T_n - T_\chi$
(left) and
the relative distance $\Delta \tau= T_{n+2} - T_n$ (right) of the first zero
for various ratios $R_{n,2}$ for even integers $n$ as a function of
$\mu/T$ in the renormalized PQM model. $T_\chi$ denotes the chiral
critical temperature.}
\end{figure*}
\section{Conclusion}
Higher-order cumulants or generalized susceptibilities are more
sensitive on the diverging correlation length and are promising quantities for the experimental search of an endpoint in
the QCD phase diagram.
In our model analysis the higher-order moments differ significantly
from the HRG model results along the freeze-out curve due to the
existence of an endpoint. A region with negative values of the ratios
of momenta emerges close to the chiral transition line. They are
shifted towards the hadronic phase and converge at the endpoint. In
order to quantify this general trend we have introduced the distance
of the first zero in temperature direction of various ratios to the
crossover temperature and the relative distance between subsequent
roots which is independent of the insecure chiral crossover
temperature. By using linear extrapolations as estimators we could
rule out the existence of an endpoint for smaller $\mu/T$ ratios.
|
{
"timestamp": "2012-11-16T02:04:37",
"yymm": "1203",
"arxiv_id": "1203.1883",
"language": "en",
"url": "https://arxiv.org/abs/1203.1883"
}
|
\section{Introduction}
Euclid's fifth postulate (called also the eleventh or twelfth axiom) states: "If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines if produced indefinitely meet on that side on which are the angles less than two right angles." The earliest commentators found fault with this statement as being not self-evident [1]. Euclid uses terms such as ``indefinitely'' and makes logical assumptions that had not been proven or stated. Thus the parallel postulate seemed less obvious than the others.
For two thousand years, many attempts were made to prove the parallel postulate using Euclid's first four postulates. Finally the mathematical society have proven the consistency of non-euclidean plane geometry and the independence of the euclidean parallel postulate; and extend that in stating that it would be impossible to prove Euclid's Parallel Postulate from the other assumptions made by him, since this would involve the denial of the Parallel Postulate of Bolyai and Lobatschewsky [2]. This conclusions mainly derived from Beltrami's (1868) seminal papers concerning the existence of non-Euclidean objects and by the arguments offered by Jules Houel in 1860-1870 for the unprovability of the parallel postulate and for the existence of non-Euclidean geometries. The final step towards rigorous consistency proofs is taken in the 1880s by Henri Poincare.
What Houel sees in Beltrami's putative demonstration that regions of pseudospheres are complete Bolyai-Lobatschewsky planes is the presentation of an instance in which all of the properties of the euclidean plane are present except for the property of unique parallelism [3]. A trivial example will make the principle evident. If we wish to show that the property of upright posture in mammals does not imply intelligence (in this sense of 'implication'), it is sufficient to show that there are some mammals in which upright posture is present without intelligence (kangaroos), even though there may be others in which upright posture occurs together with intelligence (humans). The application of the result here to the human case, e.g. if we want to say that upright posture in humans does not imply their intelligence, turns on referring to the same properties in the case of both humans and kangaroos (Scanlan 1988).
It's obvious that the independence of the euclidean parallel postulate is well formed, but a detailed looked at the writings will show that some claims may are overstated or misinterpreted. The most striking example of a non-logical use of the word ``independent`` is in Bolyai's title of his famous appendix, one of the founding texts of hyperbolic geometry, Absolute Geometry: Independent of the Truth or Falsity of Euclid’s Axiom XI (which can never be decided a priori). Bolyai does not, of course, claim that the Parallel Postulate is independent from the rest of Euclidean geometry in the sense required to show that the Parallel Postulate is unprovable (though he does claim that it cannot be proven). He means simply that he can prove some theorems of geometry without relying on either the Parallel Postulate or its negation [4].
Sometimes, it was simply stated that the attempt to prove the Parallel Postulate had gone on too long and that repeated lack of success shows that it is impossible to prove the Parallel Postulate. On the other hand the geometers working in order to support the existence of non-Euclidean objects were in the pursue of some very impressive results. That's why Beltrami's work had a profound impact on Houel. Houel immediately after he saw it, announced that Beltrami had shown that it is impossible to prove the Parallel Postulate. Strikingly, he made this announcement in eight different journals. In reality they wanted to strongly support and consolidate the initial hypothesis of the acute angle. The geometers had made just another logical assumption, believing that by finding a contradiction to Saccheri's quadrilateral would weaken the foundations of non-Euclidean geometries.
We can also realize that some statements wasn't very well formed or may were just some logical assumptions and we have to treat them with the same way we did with the logical assumptions that Euclid did. H. S. CARSLAW (1909) states: ''How ever far the hyperbolic geometry were developed, no contradictory results could be obtained``. Stating ''How ever far'' isn't it equivalent with the term ''indefinitely'' that Euclid uses?
Finally, let us consider the following: "for every diameter of $\gamma$ let us imagine another point 'at infinity' such that all the Euclidean lines parallel in the Euclidean sense to this diameter meet in this point at infinity, just as railroad tracks appear to meet at horizon. These points at infinity will also be called ultra ideal"[5]. By imagining that the Euclidean lines meet to a point 'at infinity' we managed to show that these lines are parallel to the diameter $\gamma$, but are not parallel to each other. Nevertheless, in a specific area, by construction, we have lines that are parallel to each other due to the fact that they are parallel to a third line, something that we can not succeed without using the Euclidean sense of parallelism.
Finding a contradiction on the three hypotheses made by Saccheri does not mean that the non-euclidean geometries aren't consistent. It means that we can apply Euclid's proposition at these geometries without deconstruct them, because the independence of the Euclidean parallel postulate is bidirectional. It means that non-euclidean geometries are independent from the Euclidean parallel postulate and that the postulate, it self, is also independent from the geometries that exist in the Euclidean field. This concludes, that we can build relations without using it, but in the same time we can apply it anywhere we choose, without deconstructing our work. In reality Euclid did exactly the same thing, using the parallel axiom after the first 28 propositions. Thus was Euclid "vindicated" in an unexpected manner. Knowingly or not, the wise Greek had stated the case correctly, and only his followers had been at fault in their efforts for improvement [1].
The paper is organized in the following way, at section 2 we present the used Definitions, postulates and theorems. At section 3 we prove that the angles adjacent to the opposite sides of a quadrilateral, which are perpendicular to the base, are shorter or greater, if they are adjacent, respectively, to the greater or shorter sides. And they are equal if they adjacent to equal sides. At section 4 we prove that the bisector of the vertex angle of an isosceles triangle and the perpendicular bisector of the congruent sides of this triangle are intersecting. At section 5 we prove the rejection of the acute-angled and obtuse-angled quadrilaterals.
\section{Used definitions, postulates and theorems}
\begin{enumerate}
\item The first four Euclid's postulates, the euclidean common notions and the theorems deduced by them.
\item The plane separation postulate.
\item Theorems 1 and 2. We shall deal with them, in the course of our work.
\end{enumerate}
\section{Theorem 1}
\paragraph{}The angles adjacent to the opposite sides of a quadrilateral, which are perpendicular to the base, are shorter or greater, if they are adjacent, respectively, to the greater or shorter sides. And they are equal if they adjacent to equal sides (corresponds to the Lemma I of G. Saccheri).
\paragraph{proof:}In quadrilateral AB$\Gamma$$\Delta$ (figure 1) we are given that angles A$\Delta$$\Gamma$ and B$\Gamma$$\Delta$ are right angles and sides A$\Delta$ and B$\Gamma$ are equal. Drawing the diagonals A$\Gamma$ and B$\Delta$ two congruent triangles A$\Delta$$\Gamma$ and B$\Gamma$$\Delta$ are formed, we prove this equality easily using the Side, Angle, Side (SAS) postulate. We may therefore conclude by the corresponding parts of the above congruent triangles, that the sides A$\Gamma$=B$\Delta$ and the angles A2=B2, $\Delta$1=$\Gamma$1 . Since the angles $\Delta$2 and $\Gamma$2 are complement of equal angles $\Delta$1 and $\Gamma$1 they are themselves equal, that is, $\Delta$2=$\Gamma$2. Using again the SAS postulate we also conclude that the triangles A$\Delta$B and B$\Gamma$A are congruent, therefore by the corresponding parts of these triangles we conclude that the angles $\Delta$AB and $\Gamma$BA are equal.
\begin{figure}
\begin{center}
\includegraphics[scale=0.2]{e11.eps}
\caption{}
\end{center}
\end{figure}
\paragraph{}In quadrilateral AB$\Gamma$$\Delta$ (figure 2) we are given that angles A$\Delta$$\Gamma$ and B$\Gamma$$\Delta$ are right angles, sides A$\Delta$ and B$\Gamma$ are equal and angles $\Delta$AB and $\Gamma$BA are equal too, as it has been proved above. On the line segment B$\Gamma$ we take any point E between $\Gamma$ and B, then by joining the points A and E the quadrilateral AE$\Gamma$$\Delta$ is formed. This new quadrilateral has as above, the angles A$\Delta$$\Gamma$ and E$\Gamma$$\Delta$ right angles, but its opposite sides A$\Delta$ and E$\Gamma$ are not equal now, that is, A$\Delta$ $>$ E$\Gamma$. Since the angle AE$\Gamma$, as an exterior angle of the triangle ABE, is greater in measure than the interior remote angle ABE, the angles ABE and $\Delta$AB are equal, and the angle $\Delta$AB is greater in measure than the angle $\Delta$AE, it follows that the angle AE$\Gamma$ $>$ $\Delta$AE .
\begin{figure}
\begin{center}
\includegraphics[scale=0.2]{e22.eps}
\caption{}
\end{center}
\end{figure}
Now the converse of the above statement. The greater in measure angle of a quadrilateral, which has the consecutive angles of its base right angles, is adjacent to its shorter side and vice versa.
In the given quadrilateral AE$\Gamma$$\Delta$ (figure 2) let angle AE$\Gamma$ adjacent to the side E$\Gamma$ be greater in measure than the angle $\Delta$AE which is adjacent to the side A$\Delta$, then according to the above argument, it follows that the side A$\Delta$ $>$ E$\Gamma$, since if A$\Delta$ $<$ E$\Gamma$ or A$\Delta$=E$\Gamma$ then it must be angle\\ $\Delta$AE $>$ AE$\Gamma$ or angle $\Delta$AE=AE$\Gamma$. This last contradicts the original assumption that state angle AE$\Gamma$ $>$ $\Delta$AE .
From now on, we shall call the above mentioned two right-angled isosceles quadrilaterals, which have the angles adjacent to the opposite and perpendicular to the base sides congruent, acute-angled quadrilaterals, right-angled quadrilaterals, or obtuse-angled quadrilaterals, if the pair of the congruent angles are, respectively, acute, right or obtuse angles.
\subsection{The common property of the acute-angled, right-angled and obtuse-angled quadrilaterals (the main property of Saccheri`s quadrilaterals)}
The perpendicular bisector to the base of the acute-angled, right-angled and obtuse-angled quadrilaterals is also perpendicular bisector to the opposite the base side of them.
\begin{figure}
\begin{center}
\includegraphics[scale=0.2]{e333.eps}
\caption{}
\end{center}
\end{figure}
\paragraph{proof:}In acute-angled quadrilateral AB$\Gamma$$\Delta$ (figure 3) we are given that angles A$\Delta$$\Gamma$ and B$\Gamma$$\Delta$ are right angles, $\Delta$AB and $\Gamma$BA are equal acute angles and sides A$\Delta$=B$\Gamma$.
We draw the line segments Z$\Delta$ and Z$\Gamma$, joining the midpoint Z of the side AB, with the endpoints $\Delta$ and $\Gamma$ of the side $\Delta$$\Gamma$ of the above quadrilateral AB$\Gamma$$\Delta$. We notice that the formed triangles ZA$\Delta$ and ZB$\Gamma$ are equal, we can prove this equality using the side, angle, side postulate. We may therefore conclude by the corresponding parts of the above congruent triangles, that the angles Z3=Z4 and the sides Z$\Delta$=Z$\Gamma$. Since the triangle $\Delta$Z$\Gamma$ is isosceles it follows that the median ZE is also bisector of the vertex angle $\Delta$Z$\Gamma$ and perpendicular bisector to the base $\Delta$$\Gamma$ of the above triangle, hence the angles Z1 and Z2 are equal. Since the angles Z3, Z1, Z2 and Z4 are adjacent angles lie at the same straight line and the sums of the angles measures Z3+Z1 and Z2+Z4 are equal, it follows that the line segment ZE, which is perpendicular bisector to the base $\Delta$$\Gamma$ of the acute-angled quadrilateral AB$\Gamma$$\Delta$, is also perpendicular bisector to the opposite the base side, in which the acute angles of the quadrilateral are adjacent.
By the same argument we prove that this property holds for the right-angled and obtuse-angled quadrilaterals too.
Equivalent we can state that; in a two right-angled isosceles quadrilateral: the line segment which
joins the midpoint of the base with the midpoint of the summit is perpendicular bisector to both of
them.
\section{Theorem 2}
\paragraph{}The bisector of the vertex angle of an isosceles triangle and the perpendicular bisector of the congruent sides of this triangle are intersecting.
\paragraph{Case A:}
In the isosceles triangle AB$\Gamma$ (figure 4) we are given that sides AB and B$\Gamma$ are equal and the half of its base length A$\Gamma$ equals to the length of the bisector B$\Delta$, of the vertex angle AB$\Gamma$, that is A$\Delta$=B$\Delta$.
In this case the bisector of the vertex angle AB$\Gamma$ and the perpendicular bisector to the side AB are intersecting at the midpoint $\Delta$ of the base A$\Gamma$ of the above triangle.
\begin{figure}
\begin{center}
\includegraphics[scale=0.2]{e444.eps}
\caption{}
\end{center}
\end{figure}
\paragraph{proof:}Since the side AB of the triangle AB$\Gamma$ is now base for the isosceles triangle A$\Delta$B, it will be that the line segment E$\Delta$, which joins the midpoint of the line segment AB and the vertex $\Delta$ of the isosceles triangle A$\Delta$B, must be perpendicular bisector to the side AB .
\paragraph{Case B:}
In the isosceles triangle AB$\Gamma$ (figure 5) we are given that sides AB and B$\Gamma$ are equal and the half of its base length A$\Gamma$ is shorter than the length of the altitude B$\Delta$, that is A$\Delta$ $<$B$\Delta$.
In this case the bisector B$\Delta$ of the vertex angle AB$\Gamma$ and the perpendicular bisectors to the congruent sides AB and B$\Gamma$ are intersecting.
\begin{figure}
\begin{center}
\includegraphics[scale=0.2]{e55.eps}
\caption{}
\end{center}
\end{figure}
\paragraph{proof:}Since the side A$\Delta$ of the right triangle AB$\Delta$, is shorter in length than the side B$\Delta$, it results that the angle $\Delta$AB is greater in measure than the angle AB$\Delta$. Furthermore by the equivalence of the plane separation postulate and Pasch`s postulate [6] it follows that the perpendicular bisector $\Pi$T except with the side AB has to intersect and an other side of the triangle AB$\Delta$. We suppose that the perpendicular bisector $\Pi$T to the side AB of the triangle AB$\Delta$ also intersects and the side A$\Delta$ of the above triangle at a point M, since every point on the perpendicular bisector $\Pi$T is equidistant from the endpoints of the line segment AB, it must be that the sides AM=MB and the angles BAM and ABM are equal, but this is impossible since the angle BAM=BA$\Delta$ is greater in measure than the angle AB$\Delta$ and angle AB$\Delta$ is greater in measure than the angle ABM. Therefore the perpendicular bisector $\Pi$T except the side AB intersects always the bisector of the vertex angle AB$\Gamma$ too.
\paragraph{Case $\Gamma$:}
In the isosceles triangle AM$\Gamma$ (figure 6) we are given that sides AM and M$\Gamma$ are equal and
A$\Delta$ $>$ M$\Delta$, then the bisector of the vertex angle AM$\Gamma$ and the perpendicular bisector of the side AM are intersecting.
\begin{figure}
\begin{center}
\includegraphics[scale=0.5]{e666.eps}
\caption{}
\end{center}
\end{figure}
\paragraph{Proof:}We are given that the line segment A$\Gamma$, and the perpendicular bisector of it($\Sigma$Z) are intersecting at a point $\Delta$. Using the point $\Delta$ as a center and the line segment A$\Delta$, which is equal in length to the half of the line segment A$\Gamma$, as a radius we draw the circle($\Delta$, A$\Delta$), this circle intersects the perpendicular bisector $\Sigma$Z at the points E and H. Using the point H as a vertex and the line segment A$\Gamma$ as a base we form the isosceles triangle AH$\Gamma$, we notice that its altitude H$\Delta$ equals in length to the half of the base A$\Gamma$. Using the point of intersection E of the above circle($\Delta$, A$\Delta$)with the perpendicular bisector $\Sigma$Z as a center and the cord of the same circle($\Delta$, A$\Delta$) EA as a radius, we draw the circle(E, EA) this second circle intersects the perpendicular bisector $\Sigma$Z at the points $\Theta$ and N. Since in the circle($\Delta$, A$\Delta$) the cord EA is shorter in length than the diameter EH, it follows that the point N lies between the points H and $\Delta$, and the altitude of the formed isosceles triangle AN$\Gamma$ is shorter in length than the half of the common base A$\Gamma$. By repeating the above used procedure we can form isosceles triangles, which have the given line segment A$\Gamma$ as a base and an altitude in length, in comparison with the half of its base A$\Gamma$, as short as we want. This last takes place, since by the repetition of the above referred procedure, the arc of the given cord A$\Gamma$ continuously diminished tend to coincide with it. Since the above described isosceles triangles are inscribed in circles, it follows that the perpendicular bisectors to their equal sides intersect the common bisector of the vertex angles of them $\Sigma$Z.
\paragraph{} Let AM$\Gamma$ be any isosceles triangle in which the length of the altitude M$\Delta$ is shorter than the half of the base length A$\Gamma$.
We remind here that for any isosceles triangle AM$\Gamma$, with an altitude M$\Delta$ we can form
another inscribed isosceles triangle AN$\Gamma$ with the same base, collinear vertex angle bisector and with
an altitude in length N$\Delta$ $<$ M$\Delta$.
Then the perpendicular bisectors to its equal sides will intersect the bisector $\Sigma$Z, of the vertex angle AM$\Gamma$, in respect of the following
arguments.
Given that the line segment $\theta$M is greater in length than $\theta$N and $\theta$N is greater than $\theta$A, since
$\theta$N and $\theta$A are, respectively, diameter and cord in the same circle (E,EA) it implys that
$\theta$M $>$ $\theta$A. From this we can conclude that in the triangle A$\theta$M the angle $\theta$AM is greater
in measure than the angle $\theta$MA. This last ensure that the perpendicular bisectors to the equal
sides of the isosceles triangle AM$\Gamma$ will intersect the bisector of the vertex angle of it.
Since the opposite assumption, which states that the perpendicular bisector to the side AM will intersect
the side A$\theta$ in the triangle A$\theta$M, given the Pasch's postulate, implys that the angle $\theta$MA
is greater in measure than the angle $\theta$AM which is a contradiction.
\section{The rejection of the acute-angled and obtuse-angled quadrilaterals}
\paragraph{}In the acute-angled quadrilateral AEZ$\Delta$ (figure 7) we are given that the line segment $\Gamma$M is perpendicular bisector to the sides $\Delta$Z and AE, because it joins the midpoints of them, see the common property of the acute-angled, right-angled and obtuse-angled quadrilaterals.
\begin{figure}
\begin{center}
\includegraphics[scale=0.2]{e77.eps}
\caption{}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[scale=0.25]{e88.eps}
\caption{}
\end{center}
\end{figure}
Since the right angle AM$\Gamma$=EM$\Gamma$ is greater in measure than the congruent acute angles $\Delta$AM and ZEM it results, according to the theorem 1, that in the quadrilaterals AM$\Gamma$$\Delta$ and MEZ$\Gamma$ the sides A$\Delta$=EZ $>$ M$\Gamma$. Following, we extend the line segment $\Gamma$M, we take on it a point B, so that A$\Delta$=B$\Gamma$=EZ, we draw the line segments AB and BE and then we notice that the formed triangle ABE is an isosceles triangle, since the line segment B$\Gamma$, which now is bisector of the angle ABE, is perpendicular bisector to the side AE. By repeating the above procedure we take the line segment O$\Pi$ perpendicular bisector to the sides $\Delta$$\Gamma$ and AB of the acute-angled quadrilateral AB$\Gamma$$\Delta$, the AB$\Gamma$$\Delta$ is an acute-angled quadrilateral, since the angles A$\Delta$$\Gamma$ and B$\Gamma$$\Delta$ are right angles, the sides A$\Delta$ and B$\Gamma$ are equal, and the angles BA$\Delta$ and AB$\Gamma$ are congruent acute angles, because these are adjacent to the equal sides A$\Delta$ and B$\Gamma$, and the angle ABM=AB$\Gamma$ is a non right angle of the right triangle ABM. Similarly we take the line segment $\Sigma$T perpendicular bisector to the sides $\Gamma$Z and BE, for the same mentioned above reasons. From the above analysis we can conclude that the perpendicular bisectors O$\Pi$ and $\Sigma$T to the congruent sides of the isosceles triangle ABE, as well the bisector of the vertex angle ABE are not intersected, since they are perpendiculars to the same side $\Delta$Z, and the perpendicular to a line through a point not on the line is unique. But this conclusion contradicts to the results of the theorem 2.
\paragraph{}Using the same procedure, and taking into account that in the obtuse-angled quadrilateral AEZ$\Delta$ (figure 8) the perpendicular bisector $\Gamma$M, to the base $\Delta$Z and to the summit AE, is longer than the congruent sides A$\Delta$ and EZ, because the equal obtuse angles $\Delta$AM and ZEM are greater in measure than the right angles AM$\Gamma$ and EM$\Gamma$, we conclude again that the perpendicular bisectors to the equal sides of the isosceles triangle ABE, as well the bisector of the vertex angle ABE are not intersected. But this conclusion also contradicts to the results of the theorem 2.
\section{Conclusions}
\paragraph{}Since from the above conclusions we have to reject the hypotheses that the congruent angles adjacent to the equal opposite sides of the quadrilaterals, in which their base angles are right angles, are acute angles or obtuse angles, since these contradicts the results of the theorem 2, we have to accept that the quadrilaterals in which their base angles are right angles and their opposite sides are equal to each other and perpendicular to the base are only rectangle quadrilaterals.
|
{
"timestamp": "2012-09-11T02:01:34",
"yymm": "1203",
"arxiv_id": "1203.2243",
"language": "en",
"url": "https://arxiv.org/abs/1203.2243"
}
|
\section{Introduction}
The possibility to observe and induce valence instabilities in Eu containing intermetallic compounds has been of considerable interest and the topic of many experimental as well as theoretical investigations \cite{Bauminger1974, Felner1977,Sampathkumaran1981, Perscheid1985, Nagarajan1981, Wortmann1991, Wada1999, Ni2001, Hossain2004, Michels1995, Poettgen2000, Klauss1997, Ksenofontov2006}.
In fact, europium together with ytterbium are the only lanthanide metals which are divalent (configurations 4$f^7$; 4$f^{14}$) in their elemental metallic standard state, as well as in some alloys and intermetallic compounds \cite{Gschneidner1969}. In that case, their crystal chemistry strongly resembles that of the alkaline-earth metals, typically influenced by the large radii of the ions. Nevertheless, in many systems Eu is found to be in a trivalent (4$f^6$) state, similar to the majority of the rare-earth metals \cite{Buschow1979}. Accordingly, the physical properties of Eu are significantly different in the two states, namely Eu$^{2+}$ (S = 7/2, L = 0) carrying a high magnetic moment (J = 7/2) compared to Eu$^{3+}$ with a non-magnetic ground state (S = 3, L = 3, J = L-S = 0) and low-lying excited magnetic states (J = 1, 2,...). Many factors like the local environment (determining the crystal field splitting), the electronegativity and the concentration of alloying partners \cite{M76,B79,M83} as well as external parameters like temperature, pressure and magnetic field \cite{L81} determine and influence the valency of Eu in a compound.
In this context it has been known and controversially discussed for a considerable time that insertion of boron in cubic EuPd$_3$ (Cu$_3$Au type of structure) induces a change of the Eu valence from an essentially non-magnetic Eu$^{3+}$ ($4f^6$) state into a strongly magnetic Eu$^{2+}$ ($4f^7$) state (see Ref.~\onlinecite{gumeniuk10}). For boron contents passing a threshold value $x_c$ in the system EuPd$_3$B$_x$, a heterogeneous mixed-valence state was inferred to exist, including the stoichiometric compound EuPd$_3$B. However, a recent study of the electronic structure of intermetallic $RE$Pd$_3$B$_x$ borides covering the whole series from $RE =$ La to Lu as a function of the boron content \cite{CL07} called the stability range of EuPd$_3$B$_x$ into question. In order to clarify this issue, we reinvestigated in a joint theoretical and experimental study a large series of EuPd$_3$B$_x$ and GdPd$_3$B$_x$ compounds \cite{gumeniuk10}. By x-ray diffraction (XRD), metallography, energy and wave-length dispersive x-ray spectroscopy, as well as chemical analysis,
the homogeneity ranges for EuPd$_3$B$_x$ and GdPd$_3$B$_x$ could be established as $x \le 0.53$ and $x \le 0.42$, respectively.
Density functional (DFT) based electronic structure calculations predicted a valence change in EuPd$_3$B$_x$ above $x_c = 0.19(2)$ from a non-magnetic Eu$^{3+}$ ground state to a magnetic Eu$^{2+}$ state, which is reflected in a discontinuity of the lattice parameter. In contrast, the GdPd$_3$B$_x$ alloy system with a stable Gd$^{3+}$ state exhibits an almost linear increase of the lattice parameter following Vegard's law. Consistent with the theoretical calculations, the lattice parameter vs. $x$ indeed shows a kink for EuPd$_3$B$_x$ at $x_c = 0.22 (2)$. X-ray absorption spectroscopy (XAS) in line with magnetic susceptibility and specific heat data assign this kink to a transition into a heterogeneous mixed valence state for Eu associated with a change of the mean Eu valence from Eu$^{3+}$ ($x \le 0.2$) towards Eu$^{2.5+}$ ($x = 0.5$).
This close interplay of the Eu valence state and the discontinuity in the unit cell volume under boron insertion raises the question, if the valence transition is driven by the mere change of interatomic distances, by the B chemistry, or ruled by the valence electron count. To clarify this issue, we investigate the influence of high pressure on EuPd$_3$B$_x$ compounds with fixed B content above the valence transition ($x\ge x_c$).
In our electronic structure calculations, we find a subtle interplay between doping and volume effects on the valence state, while the disorder within the system is less relevant. However, being aware that present-day DFT calculations have still difficulties with respect to the description of strong correlation and disorder, we challenge our theoretical results by a combination of XAS and XRD experiments under high pressure. These measurements allow to follow the questions, (i) if we can observe the qualitatively predicted change of Eu valence under pressure for a fixed B content experimentally and (ii) if the transition is continuos as under B insertion. Finally, we elucidate the driving force for the onset of the transition. Our results show, that applying pressure can reverse the effect only partially, driving the magnetic Eu$^{2+}$ state towards the non-magnetic Eu$^{ 3+}$ state and thus point to an important contribution of the B chemistry or the electron count to the valence transition, ruling interatomic distance effects out.
Recently, a similar valence scenario
was observed for Eu$_{0.4}$La$_{0.6}$Pd$_3$ \cite{la_pap}.
Applying our calculational approach, we find a stable Eu$^{2+}$ state in Eu$_{0.4}$La$_{0.6}$Pd$_3$ due to the La insertion, in agreement with the reported experiments. A systematic, theoretical investigation of the whole series of Eu$_{1-x}$La$_x$Pd$_3$ ($0\le x \le1$) compounds allows to predict a critical La concentration for the transition
and supports independently the predominant role of charge doping for the Eu valence change.
The manuscript is organized as follows. All technical details concerning our experimental and theoretical methods are
shortly summarized in the next section. Then, the results of our theoretical approach (fixed-spin moment calculations) are presented (Sec.~\ref{res_calcs}) followed by high-pressure XAS (Sec.~\ref{res_xas}) and XRD (Sec.~\ref{res_xrd}) measurements to study the stability of the valence state under volume change for EuPd$_3$B$_x$.
Finally the influence of an alternative La substitution is investigated (Sec.~\ref{res_lasub}) by our computational approach allowing to distinguish between the
effects caused by B chemistry or pure valence electron change. A summary and outlook concludes the manuscript (Sec.~\ref{sum}).
\section{Methods}\label{method}
To study the influence of changes in volume and in number of valence electrons (induced by the B insertion) on the Eu valence state, electronic structure calculations have been performed using
the full potential non-orthogonal local-orbital minimum basis scheme FPLO (version: fplo5.00-19) within the local spin density approximation (LSDA).\cite{fplo1}
Within the scalar relativistic calculations the exchange and correlation potential of Perdew and Wang was chosen \cite{PW92}.
The same basis set as in Ref.~\cite{gumeniuk10} was used, treating the rare-earth $4f$ states as valence states.
The strong correlation of Eu 4$f$ electrons was considered in a mean-field way by the LSDA+$U$ approximation \cite{fplo2}
applying an on-site Coulomb repulsion $U=8$\,eV and on-site exchange $J=1$\,eV and using the atomic limit for the double counting term. The variation of $U$ from 6 to 8\,eV and $J$ from 0 to 1\,eV does not change the results qualitatively.
The disordered insertion of B for different concentrations between EuPd$_3$ and (hypothetical) EuPd$_3$B and the gradual, disordered substitution
of La from EuPd$_3$ to LaPd$_3$ was simulated by the coherent potential approximation (CPA)\cite{koep97}.
To check the accuracy of the LSDA+$U$+CPA calculations \cite{BEB} we compared them with the results for the ordered structures ($x$=0, $x$=0.5 and $x$=1).
Furthermore, to study the role of electron doping independent from the rare-earth site and from structural changes, for EuPd$_3$ the virtual crystal approximation (VCA) at the Pd site was used.
Polycrystalline samples were prepared by arc melting of the elements under Ar atmosphere and carefully characterized (for details see Ref.~\onlinecite{gumeniuk10}).
X-ray diffraction measurements (XRD) for pressures up to 30\,GPa were performed at the high-pressure beam-line ID09 of the ESRF at room temperature for two different EuPd$_3$B$_x$ samples ($x=0.32, 0.48$) above the valence transition and two GdPd$_3$B$_x$ compounds with corresponding B content as reference system. For best possible hydrostatic conditions we used a membrane diamond anvil cell (DAC) with helium as
pressure medium. The pressure was determined using the ruby fluorescence method \cite{forman72}.
The collected patterns were integrated using the program Fit2D \cite{fit2D}. After a background correction the data were refined with the fullprof package \cite{fullprof}.
Since rhombohedral and orthorhombic distortions frequently occur in compounds with perovskite- and related structures \cite{Howard2004, Aleksandrov1976}, all XRD peaks in the measured powder patterns were carefully analyzed with respect to possible splitting. In the pressure range of 15-30 GPa full-width of half maxima (FWHM) of the reflections increase slightly with increasing diffraction angle without any signs of splitting. On the other hand, in the low pressure range (0-15 GPa) some peaks in a few XRD patterns were broadened, possibly indicating a splitting (see Ref.~\onlinecite{note4}). However, the observations are not systematic and do not obey any rhombohedral, tetragonal or orthorhombic rules. Therefore we can conclude that these observations are most likely an artifact of the measurement.
Eu $L_{\mathrm{III}}$-edge (6977 eV) x-ray absorption spectroscopy (XAS) measurements were performed at the energy dispersive XAS beam-line ID24 of the ESRF, on two EuPd$_3$B$_x$ samples ($x= $0.32, 0.48).
The powder samples were pressurized up to 25\,GPa within a non-magnetic Cu-Be DAC using silicon oil as pressure transmitting medium. As in the high-pressure XRD experiment, a ruby chip was used for the pressure determination. The obtained XAS spectra were analyzed by a least-squares fitting procedure to determine the average Eu valence $\nu$ by the relative intensities of Eu$^{2+}$ and Eu$^{3+}$ components, as described in Ref.~\onlinecite{MizTsu2007}.
\begin{figure*}
\begin{center}\includegraphics[
width=15cm,
angle=0]{fig01.eps}\end{center}
\caption{\label{fsm}(color online)
Fixed spin moment calculations for four different boron contents $x$ in EuPd$_3$B$_x$ under pressure (for different volumes).
The total energies for the two ordered structures EuPd$_3$ and EuPd$_3$B are calculated by LSDA+$U$.
For the compounds with $x=0.3$ and $x=0.5$ the CPA approach was used simulating a disordered B insertion.
The total energies are given relative to the corresponding minima. Arrows indicate the global minimum for a specific B content and volume.
}\end{figure*}
\section{Results}
\subsection{Calculation}\label{res_calcs}
To elucidate the origin of the valence transition in EuPd$_3$B$_x$ we try to separate the effects caused by a change of the unit cell volume from the effects of B substitution.
Unfortunately it is not possible to simulate directly the mixed valence state observed in the experiments, as the LSDA+$U$ method favors full polarization and therefore suppresses a fractional occupation of the Eu $4f$ states. However, it is possible to stabilize Eu $4f$ occupations close to the two limiting cases of 6$\mu_B$/Eu and 7$\mu_B$/Eu in the majority spin channel: Starting from these two solutions we performed fixed-spin moment calculations for compounds with fixed B concentration and varied the volume of the unit cell \cite{note1}. As the spin component of the moment in these calculations is related to the Eu $4f$ occupation and therefore to the valence state of Eu, the interplay of volume and $4f$ charge effects can be studied.
The comparison of the total energies yields the more stable configuration for a specific
B content -- volume combination.
In Fig.~\ref{fsm} the dependence of the total energy on the spin-only moment for four compounds with different B content is depicted. While in the case of EuPd$_3$ the two branches of the energy curves regarding the two limiting Eu $4f$ occupations are clearly visible, the stabilization of the Eu 4$f^6$ phase is progressively suppressed with increasing B content.
In more detail, we find a subtle balance between the Eu$^{3+}$ and Eu$^{2+}$ state for EuPd$_3$ in its optimized equilibrium volume ($a=4.08$\AA, see left panel of Fig.~\ref{fsm}, middle graph). The energy difference between the two solutions is less than 4\,meV/Eu-atom.
Applying pressure, simulated by a reduction of the unit cell volume ($a=3.80$\AA),
clearly stabilizes the Eu$^{3+}$ state by an energy difference of about 0.25 eV/Eu-atom. If the unit cell volume is expanded ($a=4.16$\AA) compared to the equilibrium volume, the Eu$^{2+}$ state becomes energetically preferred ( arrows in Fig.~\ref{fsm}). Thus, rin our calculations, the valence state of Eu in EuPd$_3$ exhibits a clear volume effect.
In the case of EuPd$_3$B$_{0.3}$ with a B content just above the onset of the valence transition \cite{gumeniuk10}, the two solutions are still well pronounced. Although their total energy shifts against each other under pressure, the decrease of the unit cell volume does not change the position of the global energy minimum. Consequently, the Eu$^{2+}$ state remains stable in the whole volume range, in contrast to the behavior of EuPd$_3$.
This stabilizing effect of the B insertion on the Eu$^{2+}$ state is strengthened further with increasing B content. While for EuPd$_3$B$_{0.5}$ the metastable solution of Eu$^{3+}$ can be obtained only for spin-only moments below $6\mu_B/$Eu with a sizable energy difference to the Eu$^{2+}$ state, EuPd$_3$B converges always to the single solution of Eu$^{2+}$ (compare right panel of Fig.~\ref{fsm}).
To ensure that the observed effects are independent of a specific approximation to model the partial occupation of the B site, the results using the
CPA, VCA and LSDA+$U$ for ordered structures ( $x=$0, 0.5, 1) were checked and compared carefully with each other (see supplementary material \cite{supp}).
From their good agreement, we conclude that the magnetic properties of the compound depend sensitively on the B content, but not on the particular B order, in contrast to an earlier study, where the Eu valence instability was connected to an anisotropic B environment \cite{cianchi91}.
Our calculations yield a clear volume effect on the Eu valence state of the EuPd$_3$B$_x$ compounds, although a change of the preferred Eu $4f$ occupation is only observed for EuPd$_3$.
Besides this volume dependence, our calculations predict a strong influence of the B substitution
on the stability of the Eu$^{2+}$ state.
This prediction and the difficulty to describe the heterogeneous mixed-valence state of Eu by the LSDA+$U$ approach in a realistic way
require experimental support for a reliable, more quantitative picture.
\subsection{XAS under pressure}\label{res_xas}
The interplay between the occupation of the Eu 4$f$ states and the
unit cell volume suggested by our calculations proposes to tune the Eu valence state
by pressure for a fixed B concentration or even to reverse the valence transition.
As a direct probe of the Eu valence state XAS under pressure was carried out for
EuPd$_3$B$_{0.48}$ above the critical boron concentration with a mean valence $\nu=2.63(2)$.
The obtained experimental data at room temperature are depicted in Fig.~\ref{xas}. At
ambient pressure, the measured spectrum at the $L_{\mathrm{III}}$ edge exhibits a main peak centered at
6976.5(5)\,eV
corresponding to Eu$^{3+}$ states, and a shoulder at about 8\,eV lower in energy, originating from
Eu$^{2+}$ states, in good agreement with earlier measurements \cite{gumeniuk10} and the heterogeneous
mixed -valence state of Eu.
The stepwise increase of pressure reduces the intensity of the $4f^7$ shoulder significantly, though it is not suppressed totally.
For pressures up to 25\,GPa, the pure Eu$^{3+}$ state as present in the compounds EuPd$_3$B$_x$ with $x\le 0.2$, cannot be reached.
Thus, EuPd$_3$B$_{0.48}$ remains in a mixed-valence state but with a clearly enlarged mean valence $\nu=2.68(2)$.
This process is reversible by decreasing pressure (compare inset of Fig.~\ref{xas}).
For the compound EuPd$_3$B$_{0.32}$, much closer to the critical concentration for the onset of the valence transition, the two valence states should be closer
in energy according to our DFT calculations (EuPd$_3$B$_{0.3}$ exhibits a second local minimum in the energy versus moment curve
and smaller energy differences between the Eu $4f^6$ and $4f^7$ states compared to EuPd$_3$B$_{0.5}$)
and thus more sensitive to pressure.
Unfortunately, the experimental detection of any definite valence change under pressure for such small B contents was beyond the resolution of the experimental
set up. The contribution of Eu$^{2+}$ states to the XAS spectrum at ambient pressure is too small to allow a reliable observation of the transition as the small differences are of the same size as the background fluctuations.
Applying pressure drives the Eu$^{2+}$ states in EuPd$_3$B$_{0.48}$ towards Eu$^{3+}$ in a continuous way,
in analogy to the continuous valence transition upon B insertion \cite{gumeniuk10}.
However, even for pressures up to 25\,GPa the pure Eu$^{3+}$ state for EuPd$_3$B$_{0.48}$ cannot be recovered completely, raising the question whether this observation is
due to the limit of applied pressure or impeded by an intrinsic property of the compound.
\begin{figure}[tbh]
\begin{center}\includegraphics[%
clip,
width=6cm,
angle=0]{XAS_pressure_new}\end{center}
\caption{\label{xas}(color online)
Main panel: XAS under increasing pressure up to 25\,GPa for EuPd$_3$B$_{0.48}$. Increasing pressure
reduces the intensity of the $4f^7$ shoulder indicated by a dashed arrow. The effect is reversible for decreasing pressure (see inset). }
\end{figure}
\subsection{XRD under pressure}\label{res_xrd}
To estimate the pressure required to restore the equilibrium volume for the pure Eu$^{3+}$ state of the reference system EuPd$_3$,
and to ensure the stability of the
crystal structure under pressure, we applied x-ray diffraction (XRD) measurements upon pressure.
In addition, earlier XRD measurements allowed the indirect observation of the valence transition in
the series of EuPd$_3$B$_x$ compounds
by a pronounced kink in the plot of the lattice parameter {\it vs.} boron content \cite{gumeniuk10}.
For pressures up to 30\,GPa no structural changes were observed for EuPd$_3$B$_{0.32}$ and EuPd$_3$B$_{0.48}$.
The measured and evaluated data are shown in Fig.~\ref{xrd} (top, open squares). To evaluate the equation of state we fitted
the experimental volume-pressure dependence by an inverse Murnaghan-Birch equation of state (EoS)
$$V(p)=V_0 \cdot \left(\frac{B^{'}_0 \cdot p}{B_0}+1\right)^{-1/B^{'}_0}$$
with the bulk modulus of $B_0$ and its pressure derivative $B^{'}_0$.
The obtained EoS follows perfectly the experimental data in the whole pressure range with $B_0=133\pm 1$ ($B^{'}_0=4.7\pm 1$) for EuPd$_3$B$_{0.32}$ and $B_0=125\pm 3$ ($B^{'}_0=4.7\pm 1$) for EuPd$_3$B$_{0.48}$, respectively (see Fig.~\ref{xrd}, upper panel and Tab.~\ref{tab1}).
Any direct structural anomaly indicating a valence instability is absent.
However, the measured EoS allows to evaluate the volume change of the compounds in the applied pressure range. For a pressure of 30\,GPa the volume for the EuPd$_3$B$_x$ compounds decreases by about 14\%, which is more than twice the
volume difference observed under B insertion
between the boron rich EuPd$_3$B$_{0.53}$ and the boron free parent compound EuPd$_3$ \cite{note3}.
Considering the volume change as the driving force of the valence transition, a pure Eu$^{3+}$ valence state would already be expected for pressures of at most 10\,GPa, in contrast to the experimental observations.
Thus, the valence state of EuPd$_3$B$_x$ is predominantly ruled by the inserted B, while volume changes have minor influence.
These findings are in agreement with the trends obtained from DFT calculations, where in the case of EuPd$_3$ the volume has a sizable influence on the preferred valence state, while the insertion of B stabilizes the Eu$^{2+}$ state significantly (compare Fig.~\ref{fsm}).
Being aware of the experimental difficulty to resolve the small effects expected from a smooth and only partial change of the valence states, in combination
with the high-pressure method (small shear forces), we compared the results for EuPd$_3$B$_x$ with the pressure-behavior of GdPd$_3$B$_x$ ($x=0.35, 0.44$) compounds, which are used as reference systems with half filled $4f$ shell and
therefore a stable $4f$ valence state.
Evaluating the EoS for the GdPd$_3$B$_x$ compounds we obtain a significantly increased $B_0=152\pm 10$
( $B^{'}_0=4.8\pm 1$) for GdPd$_3$B$_{0.35}$
and $B_0=145\pm 1$ ( $B^{'}_0=4.8\pm 1$) for GdPd$_3$B$_{0.45}$, respectively (compare Tab.~\ref{tab1} and see \cite{note2}) in comparison to the respective EuPd$_3$B$_x$ systems.
For a more direct comparison, we normalized the derived EoS to their corresponding equilibrium volume (compare Fig.~\ref{xrd}, bottom). The normalized volume {\it vs.} pressure curves separate into two sets, namely the Eu and the Gd containing compounds,
pointing to a minor influence of the specific B content to the EoS.
Furthermore, this comparison demonstrates
clearly the smaller bulk moduli for the EuPd$_3$B$_x$ compounds compared to the Gd reference systems.
In conclusion, the "softer" pressure dependence for the EuPd$_3$B$_x$ ($x \ge 0.32$) compounds compared to the Gd reference system can be assigned to the Eu valence instability.
This trend is also found independently from a theoretical estimate of $B_0$ based on our DFT calculations, in line with the well-known problem of over-binding in LDA
the calculations result in 1\% smaller equilibrium volumes $V_0$ and about 10\% larger $B_0$, .
\begin{figure}[htb]
\begin{center}\includegraphics[%
clip,
width=8cm,
angle=0]{fig03.eps}\end{center}
\caption{\label{xrd}(color online)
Top: Measured equation of states (volume vs. pressure) for two EuPd$_3$B$_{x}$ compounds above the valence transition (squares) and
two corresponding GdPd$_3$B$_{x}$ reference systems (circles) up to 30\,GPa.
Bottom: Comparison of the EoS, normalized to the equilibrium volume $V_0$, for all four compounds.
The data separate into two sets. The EuPd$_3$B$_{x}$ compounds
show clearly a smaller bulk-module compared to the Gd-reference systems, whereas the influence of differences in the B content has only minor influence.
}
\end{figure}
\begin{table}[htb]
\begin{ruledtabular}
\begin{tabular}{l cc}
& $V_0$ & $B_0$ \\
\hline
theor. & & \\
EuPd$_3$B$_{0.3}$ & 70.92 & 139$\pm$2 \\
EuPd$_3$B$_{0.5}$ & 72.17 & 145$\pm$1\\
exptl. & & \\
EuPd$_3$B$_{0.32}$ & 71.73 & 133$\pm$1 \\
EuPd$_3$B$_{0.48}$ & 72.72 & 125$\pm$3 \\
\hline
theor. & & \\
GdPd$_3$B$_{0.35}$ & 69.2& 168$\pm$1 \\
GdPd$_3$B$_{0.45}$ & 70.45 & 168$\pm$1 \\
exptl. & & \\
GdPd$_3$B$_{0.35}$ & 70.3 & 152$\pm$10 \\
GdPd$_3$B$_{0.44}$ & 72.72 & 145$\pm$1 \\
\end{tabular}
\end{ruledtabular}
\caption{\label{tab1}Fit of equations of state to the experimental data (exptl.) and the results of
band structure calculations for comparison (theor.).}
\end{table}
The predominant role of B insertion in the valence transition (compared to a mere volume effect)
suggests two possible mechanisms influencing the system: (i)
the predominantly covalent character of the boron chemical bonding, or
(ii) an increased number of valence electrons. Stimulated by a recent report \cite{la_pap}, we try to separate these two effects by a theoretical study of the related system Eu$_{1-x}$La$_x$Pd$_3$, which allows to increase the number of additional valence electrons without the insertion of B.
\subsection{La substitution}\label{res_lasub}
Recently, a similar valence instability, as found in the system EuPd$_3$B$_x$ \cite{gumeniuk10}, was observed for the compound Eu$_{0.4}$La$_{0.6}$Pd$_3$ where the influence of chemical pressure on the Eu valence state was investigated by susceptibility and XRD measurements \cite{la_pap}.
The substitution of Eu by La changes the "non-magnetic" EuPd$_3$ into a "magnetic" Eu$_{0.4}$La$_{0.6}$Pd$_3$, comparable to the effect of B insertion in EuPd$_3$B$_x$ for $x\ge 0.2$.
Applying our calculational approach, we find a stable Eu$^{2+}$ state for Eu$_{0.4}$La$_{0.6}$Pd$_3$, in agreement with the reported experiments.
But as La and B are inserted at different crystallographic sites in the systems Eu$_{1-x}$La$_x$Pd$_3$ and EuPd$_3$B$_x$, respectively, the valence instability should be independent from changes of the structure type and the local environment of Eu.
For a more detailed analysis we simulated the gradual substitution of Eu by La using the LSDA+$U$+CPA approach, analogous to the case of B substitution.
To study the stability of the valence state, we applied the fixed spin moment method at two different volumes (the experimentally observed and a reduced volume) and varied the La content,
which is equivalent to changing the number of valence electrons in the system. The comparison of the total energies corresponding to an Eu $4f$ occupation close to $6\mu_B/$Eu (for the spin-only contribution) and $7\mu_B/$Eu are depicted in Fig.~\ref{la_fsm}.
At both volumes the La substitution strongly influences the balance between the Eu$^{2+}$ and Eu$^{3+}$ state.
For a small volume ($a=3.80$\AA), the increasing La content shifts the global energy minima from 6 to 7$\mu_B$/Eu, clearly stabilizing the Eu $4f^7$ state for a La concentration of $x\ge 0.4$.
For the experimentally observed volume of Eu$_{0.4}$La$_{0.6}$Pd$_3$ ($a=4.17$\AA) the global energy minimum around 7$\mu_B$/Eu remains stable. Nevertheless, also for the larger volume, the energy difference between the local minima and therefore the Eu $4f^6$ and Eu $4f^7$ state changes significantly depending on the La content.
Furthermore we fully optimized the whole substitution series of Eu$_{1-x}$La$_x$Pd$_3$ compounds, obtaining the volume and Eu $4f$ occupation self-consistently.
Similar to the observation for the EuPd$_3$B$_x$ series under B insertion, we found a sudden change of the optimized lattice parameters for a critical La concentration $x\approx 0.4$ (compare Fig.~\ref{lll}, top). This discontinuity of the volume is connected to a sudden change in the occupation of the Eu $4f$ majority spin channel (see Fig.~\ref{lll}, bottom).
The comparison of the critical concentration in both systems, EuPd$_3$B$_x$ and Eu$_{1-x}$La$_x$Pd$_3$, reveals the electron count as the key parameter determining the valence state of the systems.
In the case of EuPd$_3$B the system gains 3 valence electrons compared to EuPd$_3$.
Thus, for the critical B content of $x_c=0.2$, the valence transition sets in at 0.6 electrons per Eu site. Substituting La in Eu$_{1-x}$La$_x$Pd$_3$ not only increases the number of valence electrons but also reduces the number of Eu sites.
Based on one additional valence electron per substituted La atom, the critical La concentration between $x=0.35$ and $x=0.4$ (equivalent to 0.35 and 0.4 electrons, respectively) is shared by 0.6 Eu sites, resulting in about 0.54 to 0.67 additional valence electrons per Eu site.
Thus, the critical number of additional electrons per Eu site is essentially the same in both systems (compare Fig.~\ref{lll}, insets), which supports independently the predominant role of the valence electron count for the
Eu valence change.
\begin{figure}[htb]
\begin{center}\includegraphics[%
clip,
width=8cm,
angle=0]{fig04.eps}\end{center}
\caption{\label{la_fsm}(color online)
Fixed spin moment calculation for different La concentrations in Eu$_{1-x}$La$_x$Pd$_3$ at two volumes.
For a small volume ($a=3.80$\,\AA) an increasing La concentration changes the energy balance between the $4f^6$ and $4f^7$ state (left). A large volume
($a=4.17$\,\AA)
stabilizes the $4f^7$ state (right). }
\end{figure}
\begin{figure}[htb]
\begin{center}\includegraphics[%
clip,
width=8cm,
angle=0]{fig05.eps}\end{center}
\caption{\label{lll}(color online)
Optimized lattice parameter and Eu $4f$ occupation for different La concentrations in Eu$_{1-x}$La$_x$Pd$_3$. The jump
in the lattice parameter and Eu $4f$ occupancy coincides.
Insets: Comparison to the development of EuPd$_3$B$_x$ for different B contents. The data are presented with respect to the same
number of additional electrons per Eu site in the system.
}
\end{figure}
\section{summary}\label{sum}
In this joint theoretical and experimental investigation we combined DFT based electronic structure calculations
with x-ray absorption and x-ray diffraction measurements at high pressures to
elucidate the driving force of the valence transition
in the system EuPd$_3$B$_x$, which stimulated
many studies in the past.
In the series of EuPd$_3$B$_x$ compounds ($0\le x\le0.53$),
a valence transition from Eu$^{3+}$ towards Eu$^{2+}$ into a heterogeneous
mixed-valence state was observed
with increasing B content, where the onset of the transition at $x_c=0.2$ yields a
pronounced lattice anomaly. The strong interplay of the Eu valence state and the crystal structure raised the question
whether this
transition is driven by (i) the change in the volume, (ii) the specific chemical bonding of boron or by (iii) the valence electron count.
Since the substitution of B into the system influences several parameters simultaneously,
in particular the crystal structure and the number of valence electrons,
the underlying mechanism is not obvious. Furthermore, the separation of the different parameters
is impeded as the insertion of an atom usually induces at the same time disorder and changes of the local crystal field due to local distortions.
However, also theory cannot solve the problem unambiguously as a mixed-valence state cannot be simulated
in a standard DFT approach.
Thus, we applied a combination of different techniques to unravel the contributions of the
different effects on the valence transition.
To estimate the pure volume effect (i) on the Eu valence state, high-pressure experiments
for samples with a fixed B content above the critical concentration have been performed.
The results of the XAS measurements yield a sizable influence of pressure on the Eu valence,
although
the mixed-valence state can be reversed only partially for pressures up to 25\,GPa.
The critical volume for the transition (corresponding to $x=0.2$ at ambient pressure) is already reached
at about 10\,GPa, but the mean valence of EuPd$_3$B$_{0.48}$ only changes by about 10\%.
This leads to the picture that the volume change (i) is of minor importance for the transition.
However, the observed valence change of Eu is independently confirmed by XRD experiments under pressure.
The evaluation of the obtained
equations of state for two EuPd$_3$B$_x$ ($x=$ 0.32, 0.48) and two Gd-reference ($x=$ 0.35, 0.44) systems with a stable
$4f$ configuration resulted in a significantly smaller bulk modulus for the Eu compounds. This softer pressure dependence
of the Eu compounds is a fingerprint of the valence change.
The experimental results conform well with the electronic structure calculations that yield a much stronger influence of the B insertion
than a mere volume expansion of the unit cell.
The relevance of the remaining parameters ((ii) boron chemistry {\it vs.} (iii) valence electron count) could be estimated
by a systematic theoretical study of the series Eu$_{1-x}$La$_x$Pd$_3$, where a similar valence change for x=0.6 was
reported recently \cite{la_pap}.
The calculations indicate a valence transition in Eu$_{1-x}$La$_x$Pd$_3$ for $x^{\mathrm{La}}_c\ge0.4$ which corresponds
surprisingly well with the electron count per Eu
for the critical B content $x^{\mathrm{B}}_c=0.2$. Consequently, this leads to the conclusion
that the valence transition in EuPd$_3$ derived compounds is essentially of electronic origin and ruled by the number
of additional valence electrons.
A detailed experimental study of the Eu$_{1-x}$La$_x$Pd$_3$ system to challenge
the theoretical prediction is underway.
\section{acknowledgement}
We thank for the use of the computational facilities at the
IFW, Dresden and the access to the ESRF beam lines ID24 (proposals HE-3201 and IH-HC-1243) and ID09.
\bibliographystyle{apsrev4-1}
|
{
"timestamp": "2012-03-09T02:04:10",
"yymm": "1203",
"arxiv_id": "1203.1865",
"language": "en",
"url": "https://arxiv.org/abs/1203.1865"
}
|
\section{Introduction}
While our knowledge about the universe has improved over the last decades with the advent of new observational data, there are several dark sides of the universe that have not been so far described from a fundamental point of view: what caused the initial inflationary era of the universe? what is the origin of dark matter? what is the fundamental cause of the current acceleration of the universe? Even though nowadays none of the previous issues have a satisfactory answer, a parallel approach, that can shed some light on the dark sides of the universe, is a phenomenological one or a model building strategy.
The main goal of this paper is to obtain a phenomenologically consistent model for the early universe (inflationary and radiation dominated epochs) by properly modifying the generalised Chaplygin gas (GCG) \cite{chaplygin}.
A first attempt in this direction has been recently carried out in \cite{BouhmadiLopez:2009hv} where a new scenario for the early universe was proposed. Such a scenario provides a smooth transition between an early de Sitter-like phase and a subsequent radiation dominated era. Here, we give a more realistic model where the early inflationary phase of the universe is described by a ``quintessence'' inflationary phase \cite{BouhmadiLopez:2011kw}. This phase will be connected to a radiation dominated phase at later time \cite{BouhmadiLopez:2011kw}. The model can be described through a scalar field or a Chaplygin gas inspired model. We will then analyze the possible imprints of such a gas in the primordial power spectrum of scalar perturbations and the power spectrum of the stochastic background of gravitational waves.
\section{The model}
We start considering a model corresponding to an inflationary period with a ``quintessence'' like behavior (described by a power law expansion) and followed by a radiation dominated epoch. The matter content of the universe can then be modeled \textit{\`a la} Chaplygin gas as
\begin{equation}
\rho=\left(\frac{A}{a^{1+\beta}}+\frac{B}{a^{4(1+\alpha)}}\right)^{1/(1+\alpha)},
\label{rho}
\end{equation}
where $A,B,\alpha,\beta$ are constants such that $2(1+\alpha)<1+\beta<0$.
Such a model generalize the scenario presented in \cite{BouhmadiLopez:2009hv}. As this matter content is not interacting with any other fluid it fulfills the following equation of state:
\begin{equation}
p=\frac13\rho+\frac{A}{3(1+\alpha)}\frac{1+\beta-4(1+\alpha)}{a^{1+\beta}} \rho^{-\alpha}.
\label{p2}
\end{equation}
This equation shows clearly that we recover the model discussed by one of us in \cite{BouhmadiLopez:2009hv} for $\beta\rightarrow - 1$. We would like to highlight that the equation of state (\ref{p2}) has been previously analysed in \cite{Chimento:2009sh}.
The inflationary dynamics of the model presented in Eq.~(\ref{rho}) can be described through a minimally coupled scalar field, $\phi$, with a potential, $V(\phi)$, whose shape is shown in Fig.~\ref{Fig1} (please see Ref.~\cite{BouhmadiLopez:2011kw} for more details).
\begin{figure}[t]
\includegraphics[width=6cm]{phixa.eps}\hspace*{1.5cm}\includegraphics[width=6cm]{potentialVphi.eps}
\caption{The left hand side (lhs) curve corresponds to $\phi$ against $x=(B/A)a^q$ where $q=1+\beta-4(1+\alpha)$.
The rhs curve corresponds to $V(\phi)$ against $\phi$, where $V_0=A^{1/(1+\alpha)}\left(A/B\right)^{-(1+\beta)/[q(1+\alpha)]}$. The values $\phi_0,\,x_0$ and $\phi_\star,\,x_\star$ correspond to the moments when the pivot scale $k_0=0.002\, \textrm{Mpc}^{-1}$ exists the horizon and the end of inflation, respectively.}
\label{Fig1}
\end{figure}
\section{Cosmological imprints}
Inflation generates density perturbations that seeds the structure of the present universe. Those density perturbations have been constrained through observations of the cosmic microwave background (CMB). Next, we will constrain the model introduced before by using the measurements of WMAP7 \cite{Komatsu:2010fb} for the power spectrum of the comoving curvature perturbations, $P_s=2.45\times 10^{-9}$, and its index, $n_s=0.963$. These measurements correspond to a pivot scale $k_0=0.002\, {\mathrm{Mpc^{-1}}}$ \cite{Komatsu:2010fb}.
The comoving curvature perturbations is determined by the fluctuations of the scalar field $\phi$. The corresponding power spectrum for the field $\nu_k$ is \cite{Langlois:2010xc}
\begin{equation}
2\pi^2k^{-3}P_s(k)=\frac{|\nu_k|^2}{z^2}\label{powerspectrum},
\end{equation}
where $z=\frac{a}{H}\frac{d\phi}{dt}$ and $t$ is the cosmic time. The field $\nu_k$ satisfies in the Fourier space the equation \cite{Langlois:2010xc}
\begin{equation}
\frac{d^2\nu_k}{d\eta^2}+\left(k^2-\frac{1}{z}\frac{d^2z}{d\eta^2}\right)\nu_k=0,\quad \eta=\textrm{conformal time}
\label{seof}.
\end{equation}
We show in Fig.~\ref{Fig2} our results for the power spectrum of the comoving curvature perturbations and its index where we have fixed the values of the parameters of the model as follows: (i) $B$ is fixed by the current amount of radiation in the universe, (ii) for a given parameter $\alpha$, the parameter $\beta$ is fully determined by the measurement of $n_s$ and (iii) the parameter $A$ is fixed such that $P_s=2.45\times 10^{-9}$ at the pivot scale $k_0=0.002\, {\mathrm{Mpc^{-1}}}$. Finally, we have imposed that when the wavelength of a given mode $k$ is much smaller than the Hubble radius $k\gg aH$, the effect of curvature can be neglected on $\nu_k$ and, therefore, the result reduces to that of a flat Minkowski spacetime (when $k\gg aH$) \cite{Langlois:2010xc}.
\begin{figure}[t]
\includegraphics[width=6cm]{powerspectrum.eps}\hspace{1.5cm}\includegraphics[width=6cm]{ns.eps}
\caption{Primordial power spectrum $P_s(k)|_{k=aH}$ and the spectral index $n_s$ against $k$ for six different values of $\alpha$. The dashed black line is the pure power law inflation, and the vertical dashed red line locates the pivot scale $k_0=0.002\textrm{Mpc}^{-1}$. We can see that all these lines merge when small $k$; i.e. large scale, exits the horizon. The grey, violet, red, orange, green and blue curve correspond respectively to $\alpha=-1.1,-1.09,-1.08,-1.07,-1.06,-1.05.$}
\label{Fig2}
\end{figure}
Similiraly, we can obtain the spectrum of the gravitational waves (GWs) using the method of the Bogoliubov coefficients \cite{Parker:1969au}. In particular, one of these coefficients, which we will denote $\beta_k$, gives the number of gravitons $N_k$, $N_k=|\beta_k|^2$, created as the universe expands. In fact, the dimensionless relative logarithmic energy spectrum of the gravitational waves, $\Omega_{\mathrm{GW}}$, at the present time reads \cite{Sa:2008yq}:
\begin{equation}
\Omega_{GW}(\omega,\tau_0)\equiv\frac{1}{\rho_c(\tau_0)}\frac{d\rho_{GW}}{d\ln\omega}(\tau_0)=\frac{\hbar\kappa^2}{3\pi^2 c^5 H^2(\tau_0)}\omega^4\beta_k^2(\tau_0).
\label{spectrum}\end{equation}
The parameter $\rho_{\rm GW}$ is the energy density of GWs and $\omega$ the respective
angular frequency; $\rho _{\rm c}$ and $H$ are the critical density of the
universe and Hubble parameter, respectively, evaluated at the present time. Our results are shown in Fig.~\ref{Fig3}.
\begin{figure}[t]
\centering
\includegraphics[width=6cm]{betasquare}\hspace*{1.5cm}\includegraphics[width=6cm]{GWspectrum}
\caption{On the lhs figure, we show examples of $|\beta_k|^2=N_k$ for different frequencies. On the rhs figure we show $\Omega_{GW}$ against $\omega$ for different value of $\alpha$ in this GCG model: the blue line refers to $\alpha=-1.06$, the red one refers to $\alpha=-1.05$, and the grey one to $\alpha=-1.04$.}
\label{Fig3}
\end{figure}
\section{Conclusions}
We propose a phenomenological model for the early universe where there is a
smooth transition between an early ``quintessence'' phase and a radiation dominated era.
We constrain the model observationally by mapping the primordial power spectrum of the scalar perturbations to the latest data of WMAP7. We compute as well the spectrum of the primordial gravitational waves as would be measured today.
\begin{theacknowledgments}
M. B. L. is supported by the Portuguese Agency Fundac\~ao para a Ci\^{e}ncia e Tecnologia through SFRH/BPD/26542/2006 and PTDC/
FIS/111032/2009. P.C. and Y.W.L. are supported by Taiwan National Science Council under Project No. NSC 97-2112-M-002-026-MY3 and by Taiwan's National Center for Theoretical Sciences. P.C. is in addition supported by US Department of Energy under Contract No. DE-AC03-76SF00515.
\end{theacknowledgments}
\bibliographystyle{aipproc}
|
{
"timestamp": "2012-03-12T01:01:46",
"yymm": "1203",
"arxiv_id": "1203.2097",
"language": "en",
"url": "https://arxiv.org/abs/1203.2097"
}
|
\section{Introduction}
Observations of the thermal dust emission is one of the main methods
used to map dense interstellar clouds. Sub-millimetre and millimetre
emission is considered one of the most reliable ways of obtaining
information about cloud cores in the different stages of the star
formation process, from starless cores to protostellar systems
\citep{Motte1998, Andre2000, Enoch2007}. To a large extent, this
conclusion is based on the problems with the other tracers, i.e., the
difficulty of interpreting molecular line data of the very dense
clouds and the difficulty of assembling high-resolution maps using dust
extinction or scattering \citep{Lombardi2006, Goodman2009, Juvela2008,
Juvela2009}.
The main difficulties in the interpretation of dust emission are well
known. Because of the line-of-sight temperature variations, the peak
of the emission spectrum becomes wider and this results in a decrease of
the observed spectral index $\beta_{\rm Obs}$ \citep{Shetty2009a,
Malinen2011, JuvelaYsard2011b}. At the same time, because the observed
emission is dominated by the warmer dust components within the beam,
the colour temperature $T_{\rm C}$ overestimates the mass averaged
physical dust temperature. This can lead to a significant
underestimation of the dust mass \citep{Evans2001,
StamatellosWhitworth2003, Malinen2011, YsardJuvela2011b}. Furthermore,
it is difficult to estimate the intrinsic dust grain properties, such
as the spectral index, on the basis of the observed radiation.
When far-infrared and sub-millimetre observations are fitted with
modified black body spectra, the dust colour temperature, $T_{\rm C}$,
and the dust spectral index, $\beta_{\rm Obs}$, are partially
degenerate. A small increase in $T_{\rm C}$ can be compensated by a
small decrease of $\beta_{\rm Obs}$. When the observations contain
noise, the ($T_{\rm C}$, $\beta_{\rm Obs}$) values will scatter over
an elongated ellipse with a negative correlation between the two
parameters. If the noise is large enough, the points will fall on a
banana-shaped region where the $\beta_{\rm Obs}(T_{\rm C})$
relation becomes steeper at lower values of $T_{\rm C}$
\citep{Shetty2009a, Shetty2009b, Veneziani2010, Paradis2010,
Juvela2011}. The common assumption is that apart from the curvature
of the error banana, the scatter is symmetric with respect to the true
values of $T_{\rm C}$ and $\beta_{\rm Obs}$ so that the expectation
values do not show significant bias.
The $\beta_{\rm Obs}(T_{\rm C})$ relation induced by the noise is
usually steeper than the average $\beta_{\rm Obs}(T_{\rm C})$ relation
derived from observations of a large number of individual objects
\citep{Dupac2003, Desert2008, PlanckI}. If the average of the observed
($T_{\rm C}$, $\beta_{\rm Obs}$) points remains on the intrinsic
$\beta(T)$ relation and if one observes a wide range of objects with
different true temperatures, the effect of the bias can be controlled.
However, there are some indications that under certain conditions,
the error distribution is not symmetric and could behave in an even
more non-Gaussian fashion \citep{PlanckI, YsardJuvela2011b}. Under
some conditions, the $\beta_{\rm Obs}(T_{\rm C})$ can extend to low
colour temperatures and very high values of the spectral index. This could
be important for the interpretation of the observed $\beta_{\rm
Obs}(T_{\rm C})$ relations and, more generally, any attempt to measure
cloud masses and temperatures using noisy continuum data.
In this paper we investigate this question by analysing noisy modified
black body spectra, mixes of these spectra with different colour
temperature, and spectra obtained from radiative transfer modelling of
optically thick clouds. The content of the paper is the following. In
Sect.~\ref{sect:methods} we describe the methods used to produce the
spectra and to derive the colour temperature and the spectral index
estimates. The main results are presented in Sect.~\ref{sect:results}.
We start by looking at spectra calculated for optically thick
filaments and by identifying the cases where the $\chi^2$ has more
than one local minimum. In the Sect.~\ref{sect:grey} we return to
modified black body spectra in an effort to identify the basic
requirements for these anomalies. In Sect.~\ref{sect:discussion} we
discuss our results and the final conclusions are presented in
Sect.~\ref{sect:conclusions}.
\section{Methods} \label{sect:methods}
We describe below the procedures used to calculate synthetic spectra
and to derive the $T_{\rm C}$ and $\beta_{\rm Obs}$ values through
$\chi^2$ minimisation.
\subsection{Spectra from radiative transfer models}
\label{sect:model_spectra}
The first set of model spectra was obtained from the radiative transfer
models discussed in \cite{YsardJuvela2011b}. The models consist of
optically thick filaments that are represented by long cylinders with
radial density distributions following the `Plummer-like' profiles
\citep{Nutter2008, Arzoumanian2011}, which are flat at the centre of
the filaments and decrease as $R^{-2}$ in the outer regions ($R$ is
the radius of the clouds). The central extinction is varied in the
range $A_{\rm V}=$1--20$^{\rm m}$ and the analysed sub-millimetre
emission remains optically thin. The clouds are externally heated by
the standard radiation field, ISRF \citep{Mathis1983}. Because cloud cores
are often embedded in large molecular cloud complexes, this radiation
field can also be attenuated corresponding to an external layer of
dust with $A_{\rm V}^{\rm ext}=$1--5$^{\rm m}$. We used dust properties
representative of the dust in diffuse high Galactic latitude medium
(DHGL) as defined in the DustEM dust models \citep{Compiegne2011}.
The dust model consists of three dust populations: interstellar PAHs,
amorphous carbons, and amorphous silicates. In the model clouds, the
dust temperatures range from over 20\,K for the amorphous carbon at
the cloud surface to less than 10\,K for the silicate grains at the
centre of the filament, also depending on the model optical depth.
Because of the wide range of temperatures and the different intrinsic
emissivity spectral indices $\beta$ of the dust
populations\footnote{The intrinsic spectral index of the amorphous
carbon is 1.55, while it is equal to 2.11 for amorphous silicates.},
the spectra cannot be fitted precisely with a single modified black
body. For details of the calculations see \cite{YsardJuvela2011b}.
\subsection{Modified black body spectra} \label{sect:model_grey}
We additionally examined spectra that are based on modified black
bodies to which observational noise is added. The different cases are
(1) a single modified black body with a fixed value of $\beta$, (2)
the sum of two modified black bodies with the same $\beta$ but
different temperatures, and (3) a modified black body with different
$\beta$ values below and above the wavelength of 500\,$\mu$m. In the
first scenario we are testing if the noise alone can produce a tail of
solutions with high $\beta_{\rm Obs}$ and low $T_{\rm C}$ values. With
the other modifications, we tested if the probability of extreme values
is enhanced when the original spectrum cannot be described exactly as
a single modified black body.
\subsection{Analysis of the simulated spectra}
In accordance with recent observational studies, we used measurements
at a few far-infrared and sub-millimetre wavelengths. To simulate the
Planck studies, we used the wavelengths of 350, 550, and 850\,$\mu$m
complemented with the 100\,$\mu$m point that would be available from
the IRAS survey \citep[e.g.,][]{PlanckI}. As a default, we added to the
spectra observational noise that is 0.06, 0.12, 0.12, and
0.08\,MJy\,sr$^{-1}$ at the wavelengths of 100, 350, 550, and
850\,$\mu$m, respectively. For Planck the uncertainties of the
absolute surface brightness measurements are significantly smaller but
these numbers are more realistic if the flux determination includes
the separation of a background component \citep[see][]{PlanckI}. To
simulate Herschel observations, we used the wavelengths of 100, 160,
250, 350, and 500\,$\mu$m \citep[e.g.,][]{Paradis2010, Juvela2011}
with noise levels of 8.1, 3.7, 1.2, 0.85, and 0.35\,MJy\,sr$^{-1}$ per
beam. In the analysis the data are convolved to the resolution of the
500\,$\mu$m observations and this results in final uncertainties of
1.62, 1.18, 0.60, 0.60, and 0.35\,MJy\,sr$^{-1}$, respectively.
After the beam convolution, the absolute signal is lower for the
simulated Planck data (a resolution of $\sim5\arcmin$) compared
to the simulated Herschel data (a resolution of $\sim 36\arcsec$
at the wavelength of 500\,$\mu$m). This reduces the difference in the
signal-to-noise ratios (S/N) of the two data sets (see
Fig.~\ref{fig:PH_SN}).
A weighted least-squares fit of a single modified black body ($\chi^2$
minimisation) was used to estimate the colour temperature $T_{\rm C}$
and the observed spectral index $\beta_{\rm Obs}$. The wavelength
ranges fitted are 100--850\,$\mu$m for the combined data of Planck and
IRAS and 100--500\,$\mu$m for the simulated Herschel data. The
weighting was done with the actual noise in each channel although the
effect of changing the weight of the 100\,$\mu$m measurement was also
examined.
When there are temperature variations, the derived $T_{\rm C}$ and
$\beta_{\rm Obs}$ values differ from the mass weighted average dust
temperature and from the actual spectral index of the dust grains.
However, in this paper we concentrate on the effects of noise.
Therefore we concentrated on the question of how the $T_{\rm C}$ and
$\beta_{\rm Obs}$ values differ from the values that would be obtained
with a similar analysis of the noiseless spectra.
\begin{figure}
\centering
\includegraphics[width=8.5cm]{PH_SN.eps}
\caption{
Signal-to-noise ratios in the simulated observations. The solid
lines show the average and the dashed lines the minimum and the
maximum S/N ratio as the function of wavelength.
}
\label{fig:PH_SN}%
\end{figure}
\section{Results} \label{sect:results}
\subsection{Spectra from radiative transfer models} \label{sect:model}
\subsubsection{Simulated Planck and IRAS observations}
We start with a study of the spectra that were calculated in
\cite{YsardJuvela2011b} for models of cloud filaments at a
distance of $d$=100\,pc.
Figure~\ref{fig:Planck3_scatter} shows the ($T_{\rm C}$, $\beta_{\rm
Obs}$) values derived from simulated observations of 100, 350, 550,
and 850\,$\mu$m with the default noise (see
Sect~\ref{sect:model_spectra}). In the figure we include models with
$A_{\rm V}$=10--20$^{\rm m}$ and with the ISRF attenuated by an
external dust layers with $A_{\rm V}^{\rm ext}$=4--5$^{\rm m}$. For
these models the locus of the correct $T_{\rm C}$ and $\beta_{\rm
Obs}$ values (i.e., the parameters estimated in the absence of noise)
is between the $T_{\rm C}$ values of 12.0\,K and 12.8\,K and the
$\beta$ values 1.02 and 1.11.
Because of the noise the estimates scatter along a narrow
banana-shaped region. Most points cluster around the median value of 12.47\,K
and $\beta$=1.07. The figure contains 10000 points of which 217 are
below $T_{\rm C}$=8\,K.
\begin{figure}
\centering
\includegraphics[width=8.5cm]{Planck3_scatter.eps}
\caption{
Distribution of ($T_{\rm C}$, $\beta_{\rm Obs}$) values for the
modified black body fits of the spectra calculated for the filament
models. The colour scale shows the density of points per $\Delta
T_{\rm C}$=0.2\,K and $\Delta \beta_{\rm Obs}$=0.1.
The white circles (with black borders) indicate the values
obtained for the examined models in the absence of noise.
}
\label{fig:Planck3_scatter}%
\end{figure}
The histograms in Fig.~\ref{fig:NY_histo_Planck3} show that the
parameter distributions are not symmetric and this is not caused
merely by the curvature of the confidence region. The colour
temperature distribution has a tail towards low values and a secondary
peak is visible around 7--8\,K. The spectral index distribution is
correspondingly skewed in the other direction with a tail extending to
high values of $\beta$.
Altogether 4.6\% of the points have $T_{\rm C}<10$\,K. These
represent a strongly non-Gaussian part of the error distribution that,
if not properly accounted for, would negate any attempts to determine
what the real $\beta_{\rm Obs}(T_{\rm C})$ dependence would be in the absence of
noise.
\begin{figure}
\centering
\includegraphics[width=8.5cm]{NY_histo_Planck3.eps}
\caption{
Marginal distributions (i.e. the probabilities integrated
over the $\beta$ and $T_{\rm C}$ axes, respectively) of the points in
Fig.~\ref{fig:K1.000}. The upper frames show the probability density
distributions of the $T_{\rm C}$ and $\beta$ parameters and the lower
frames show a zoomed version of the upper frames. The histograms have
been normalised to represent probability distributions (area
normalised to one). The vertical dashed lines indicate the median of
the distribution and the points where the tails of the distribution
(one-sided) contain 5\% or 1\% of the data.
}
\label{fig:NY_histo_Planck3}
\end{figure}
For filament models of lower $A_{\rm V}$ and thus of lower
signal-to-noise ratio the high $\beta_{\rm Obs}$ solutions become more
common.
The $A_{\rm V}=20^{\rm m}$ models are responsible for $\sim$17\% of
all the points below $T_{\rm C}=8$\,K, while the contribution of the
clouds with a central $A_{\rm V}$ of $10^{\rm m}$ is already over 26\%.
More interestingly, the long tail to low colour temperatures is almost
exclusively produced by the models where the external radiation field
is attenuated by a dust layer of $A_{\rm V}^{\rm ext}=5^{\rm m}$. The
models with $A_{\rm V}^{\rm ext}=4^{\rm m}$ still show an asymmetry of
the $T_{\rm C}$ distribution but their contribution to the tail below
$T_{\rm C}=10.0$\,K is only a couple of percent.
Figure~\ref{fig:Planck3_scatter} also shows the ($T_{\rm C}$,
$\beta_{obs}$) values that would have been obtained from the various
models if there were no observational noise. The $A_{\rm V}^{\rm
ext}=4^{\rm m}$ and $A_{\rm V}^{\rm ext}=5^{\rm m}$ models form
separate groups, the latter being colder, as measured by the noiseless
colour temperature, by $\Delta T_{\rm C}\sim $1\,K. The higher $A_{\rm
V}^{\rm ext}$ values reduce the physical dust temperature especially
at the cloud surface. The effect is felt most strongly at 100\,$\mu$m
where the signal-to-noise ratio drops by $\sim$40\% (from $\sim$6 down
to $\sim$3.5, almost irrespective of the central $A_{\rm V}$ of the
model cloud).
Figure~\ref{fig:K1.000} shows one example of a $A_{\rm V}=5^{\rm m}$
cloud where the surface brightness values are 0.088, 1.62, 1.27, and
0.35\,MJy\,sr$^{-1}$ at the wavelengths 100, 350, 550, and
850\,$\mu$m, respectively. A fit to the original noiseless data gives
values $T_{\rm C}=$12.6\,K and $\beta_{\rm Obs}$=1.14. With this
particular noise realisation the $\chi^2$ minimum has moved to $T_{\rm
C}$=4.78\,K and $\beta_{\rm Obs}=$4.46. These values are suspicious
because the colour temperature is well below the minimum dust
temperature of the model cloud, which is always higher than 8\,K. The
spectral index $\beta_{\rm Obs}$ is also higher than the dust
intrinsic $\beta$
\citep[1.55 and 2.11 for amorphous carbons and silicates, respectively, see Fig.8 in][]{YsardJuvela2011b}
although it would be expected to be lower because of the line-of-sight
temperature variations.
In the model clouds the actual dust temperature does not decrease
below $\sim$8\,K, while the spectral indices of the dust grains only are
2.11 for astronomical silicates and 1.55 for amorphous carbon grains
\citep[see Fig. 8 in][]{YsardJuvela2011b}.
In this case the 100\,$\mu$m intensity was not much more than
1-$\sigma$ detection. However, the 100\,$\mu$m value happens to be
almost identical to the correct noiseless value. On the other hand,
the 550\,$\mu$m observation is almost 2.5$\sigma$ above the correct
value. This is the main reason for the very low temperature. The
contour plot of the $\chi^2$ values (lower frame in
Fig.~\ref{fig:K1.000}) shows that the confidence region is very
elongated. The `correct' solution (i.e., the one for the noiseless
spectrum) resides in the $\chi^2$ valley but with a $\chi^2$ value
that is four times the $\chi^2$ value of the best fit.
\begin{figure}
\centering
\includegraphics[width=7.2cm]{plot_Planck3_60065_K1.000.eps}
\caption{
Case where noise has resulted in a high $\beta$ value. In the middle
frame, the black curve is the spectrum obtained from the cloud
modelling. The red symbols are the measurements that include noise and
the red curve is the fit to these points. The uppermost frame shows
the deviations $\Delta I_{\nu}$ (the difference between the
simulated observation and the fit) in units of the assumed
uncertainty $\sigma I_{\nu}$.
The bottom frame shows the $\chi^2$ values as a function of $T_{\rm
C}$ and $\beta_{\rm Obs}$. The colour plot and the white contours show
the $\chi^2$ values for the fit to the noisy data. The contours are
drawn at 1.02, 1.05, 1.1, 1.5, 2.0, 4.0, and 8.0 times the minimum
$\chi^2$ value. In yellow are shown the corresponding contours for the
fit to the noiseless data. The two circles denote the locations of the
$\chi^2$ minima with (red circle) and without (green circle) the
noise.
}
\label{fig:K1.000}%
\end{figure}
If the observations do not perfectly fit a single modified black body,
the best fit will depend on the relative weight given to the
individual measurements. Figure~\ref{fig:example1} shows that
interesting changes take place when the assumed uncertainty of the
100\,$\mu$m point is decreased by a factor of $\sim$0.57. The measured
values (including the noise) are not changed and only the weight of
the 100\,$\mu$m point is increased in the fit. In the present example
the 100\,$\mu$m value happens to be almost correct. Therefore, the
error estimates could be decreased much more than by a factor of 0.57
without this particular noise realisation becoming improbable. In
Fig.~\ref{fig:example1} the 100\,$\mu$m noise is scaled by 0.575 in
the left hand frames and by 0.570 in the right hand frames. As the
weight of the 100\,$\mu$m point is modified, the $\chi^2$ minimum
jumps instantaneously from the high $\beta$ solution to a solution
near the value of the original noiseless spectrum. At this point
the signal-to-noise ratio of the 350\,$\mu$m data point is $\sim$18.
The $\chi^2$ values are almost identical, the change being consistent
with the effect of the very small decrease of the 100\,$\mu$m
uncertainty. The parameter space was sampled with intervals $\Delta
T$=0.1\,K and $\Delta \beta$=0.02.
\begin{figure*}
\centering
\includegraphics[width=7cm]{plot_Planck3_60065_K0.575.eps}
\includegraphics[width=7cm]{plot_Planck3_60065_K0.570.eps}
\caption{
As Fig.~\ref{fig:K1.000} but assuming in the fit a 100\,$\mu$m
uncertainty that is 0.575 times (frames on the left) or 0.570 times
(frames on the right) the original value. The intensity values are the
same as before. The $\chi^2$ plane exhibits two distinct local minima
and a small change in the weight of the 100\,$\mu$m measurement moves
the solution from a high $\beta$ solution to a low $\beta$ solution.
}
\label{fig:example1}%
\end{figure*}
The example reveals some important facts. Firstly, the solution based
on $\chi^2$ minimum can change significantly and in a non-continuous
fashion when the measured intensities or the assumed uncertainties are
slightly perturbed. Secondly, the presence of multiple local $\chi^2$
minima implies that the solution obtained by non-linear optimisation
will depend on the initial values of the optimisation. The obtained
result may correspond to a local instead of a global minimum. Thirdly,
as already indicated in Fig.~\ref{fig:NY_histo_Planck3}, the error
distributions are skewed and possibly even bimodal. This has
implications not only for the uncertainties of the $\chi^2$ approach
but more generally for the statistical models employed in other
parameter estimation methods.
In the following we call `normal' the solution that is close to the
values obtained in the absence of noise and `anomalous' the solutions
that exhibit a significantly lower value of $T_{\rm C}$ and a higher
value of $\beta_{\rm Obs}$.
Another example is shown in Appendix~\ref{sect:example2} where
the 550\,$\mu$m point is more than 2$\sigma$ above the correct
(noiseless) value. In the 217 cases of the $A_{\rm V}\ge 10^{\rm m}$
model spectra with colour temperature below 8\,K ($\sim$2\% of all
spectra), the common feature is that the 550\,$\mu$m point is high
relative to the neighbouring wavelengths and especially relative to
the 850\,$\mu$m point.
This is demonstrated in Fig.~\ref{fig:deltas} which shows the
differences between the observed and the true intensities when the
estimated $T_{\rm C}$ was below 8\,K. With a low weight of the
100\,$\mu$m measurement, a solution with a very high value of
$\beta_{\rm Obs}$ becomes possible. The anomalous solutions are seen
more frequently but not exclusively when the 100\,$\mu$m intensity is
underestimated.
One must also note that the previous colour
temperature estimates (e.g., Fig.~\ref{fig:Planck3_scatter}) were
based on $\chi^2$ optimisation where the initial values,
$T_{\rm}=$15\,K and $\beta$=1.5, favoured the high-temperature
solution.
It is time-consuming to estimate the $\chi^2$ values for the modified
black body fits with a dense grid over the whole ($T_{\rm C}$,
$\beta$) plane. Instead, to identify the cases with two $\chi^2$
minima, we started non-linear optimisation (the Powell method) at two
locations, ($T_{\rm C}$, $\beta$)=(7.0K, 4.0) and (13.0\,K, 0.9). If
two minima exist, the optimisation is likely (although not guaranteed)
to converge to different values.
With these two initial values the optimisation may still converge to
the same local minimum, missing the second one. On the other hand, if
there is only one very shallow minimum, the optimisations may
produce different results for numerical reasons.
The presence of a single $\chi^2$ minimum does not tell us whether
it corresponds to the normal or the anomalous solution. To see
whether one is close to a situation where two $\chi^2$ minima appear,
we also scaled the 100\,$\mu$m error estimates by a factor $K_{\rm
100}$ that was changed from 0.5 to 1.5 in steps of 0.1. This way one
can perturb the problem and obtain an indication whether the result of
the fit is well-defined or not. As seen in Fig.~\ref{fig:example1},
the minimum can shift very rapidly between the normal and the
anomalous solution. The second local $\chi^2$ minimum exists only for
few a $K_{\rm 100}$ values close to that point. Our emphasis is on the
non-Gaussian nature of the uncertainties (as produced by the multiple
minima). Note that for example Fig.~\ref{fig:Planck3_scatter} was
obtained using only the original weighting of the data ,
$K_{100}=$1.0.
We examined the above set of 10000 spectra from cylindrical cloud
models with $A_{\rm V}$=10--20$^{\rm m}$ and $A_{\rm V}^{\rm ext}$ of
4--5$^{\rm m}$. This revealed $\sim$680 cases where different initial
parameter estimates lead to different optimised values with $\Delta
T>0.2$ or $\Delta\beta>0.1$. This happened preferentially when the
100\,$\mu$m uncertainties were increased but could take place for any
usually small range of $K_{100}$ values. In these models the ratio of
the original intensity and the added noise was at 100\,$\mu$m higher
than 3.2, with an average value of 4.7. Of the three Planck channels
the 500\,$\mu$m band had the lowest ratio with a minimum of 15.1 and a
mean value of 18.7. The actual signal-to-noise ratios can be lower
when the observed intensity is below the expectation value of the
intensity.
The double $\chi^2$ minima are caused by the noise but their
frequency of appearance depends on the model, probably mainly through
the associated changes in the signal-to-noise ratios of the
observations. For the same model clouds with $A_{\rm V}$=10--20$^{\rm
m}$ but with the external radiation field attenuated by $A_{\rm
V}^{\rm ext}<4^{\rm m}$ rather than 4--5$^{\rm m}$, the double minima
become less frequent by a factor of ten. With $A_{\rm V}^{\rm ext}\le
2^{\rm m}$ the solutions were unique, probably because of the higher
surface brightness values of those models.
\begin{figure}
\centering
\includegraphics[width=8.0cm]{deltas_alpha.eps}
\caption{
Errors in the observed intensities in the cases where the
simulated Planck and IRAS 100\,$\mu$m observations result in colour
temperatures $T_{\rm C}<$8\,K. The errors are given as the difference
between the observed intensity and the noiseless spectrum, measured in
units of the uncertainty assumed in the modified black body fits.
}
\label{fig:deltas}%
\end{figure}
\subsubsection{Simulated Herschel observations}
The simulated Herschel observations consisted of the wavelengths of
100, 160, 250, 350, and 500\,$\mu$m. Figure~\ref{fig:Herschel_scatter}
shows the $(T_{\rm C}, \beta)$ values for the models from
\cite{YsardJuvela2011b} with the cloud central $A_{\rm
V}$=10--20$^{\rm m}$ and an external $A_{\rm V}^{\rm ext}$=4--5.
This is only a subset of all models discussed in
\cite{YsardJuvela2011b}. The original weighting of the 100\,$\mu$m
data has not been altered (i.e., $K_{\rm 100}=$1.0). The figure
includes 10000 points out of which only 83 are below $T_{\rm C}$=9\,K.
\begin{figure}
\centering
\includegraphics[width=8.0cm]{Herschel_scatter.eps}
\caption{
Distribution of $(T_{\rm C}, \beta)$ for simulated Herschel
observations. The colour scale gives the density of the points per
$\Delta T_{\rm C}$=0.2\,K and $\Delta \beta_{\rm Obs}$=0.1
The white circles (with black borders) indicate the values that would
be obtained for the included models if there were no noise.
}
\label{fig:Herschel_scatter}%
\end{figure}
The scatter of the temperature and spectral index values is more
symmetric than for the Planck+IRAS data set. This is confirmed in
Fig.~\ref{fig:NY_histo_Herschel} which shows the marginal
distributions where the $T_{\rm C}$ data have only a hint of a tail
towards higher temperatures (skewness 0.63), i.e., opposite to the
behaviour in Fig.~\ref{fig:NY_histo_Planck3}. All spectra were again
fitted using an optimisation with different initial values of $T_{\rm C}$
and $\beta$. Out of the 10000 spectra with $A_{\rm V}$=10--20$^{\rm
m}$ and $A_{\rm V}^{\rm ext}$4--5$^{\rm m}$, two $\chi^2$ minima were
inferred only in a single case. This even though the
100\,$\mu$m S/N ratios was below one and the 160\,$\mu$m ratios only a
few, the mean value being 5.0. For the longer wavelengths, the S/N
ratios were $\sim$20 or above. The $\chi^2$ plane was examined for
some of the cases with the lowest colour temperature to confirm the
presence only of a single local minimum. The other two samples, the
case with $A_{\rm V}$=10--20$^{\rm V}$ and $A_{\rm V}^{\rm ext}<4^{\rm
m}$ and the case with $A_{\rm V}$=1$^{\rm V}$ and $A_{\rm V}^{\rm
ext}<1^{\rm m}$, together contain only one additional case where two
local $\chi^2$ minima were detected.
On the basis of the previous tests the appearance of anomalous
solutions depends mainly on the noise. The comparison of the Planck
and Herschel cases shows that the set of wavelengths included in the
analysis is equally important.
The uncertainties of the simulated Herschel observations are all
higher in absolute terms but the relative uncertainty between the
short and the long wavelengths is similar to the Planck case. The
combination of five wavelengths appears to be resistant against
extreme errors that could be caused, for example, by a 2-$\sigma$ or
3-$\sigma$ error in a single channel (cf. Fig.\ref{fig:K1.000}). The
$T_{\rm C}$ and $\beta_{\rm obs}$ distributions are wide because of
the noise but there still are practically no values of $T_{\rm C}<8$\,K.
There is still the possibility that the results could depend on how
much the underlying spectrum deviates from a single modified black body.
To further examine this question, we next examined a series of models
based on modified black bodies.
\begin{figure}
\centering
\includegraphics[width=7cm]{NY_histo_Herschel.eps}
\caption{
Probability distributions of $T_{\rm C}$ and $\beta$ for the
simulated Herschel observations of Fig.~\ref{fig:Herschel_scatter}
(for further details, see the caption of
Fig.~\ref{fig:NY_histo_Planck3}).
}
\label{fig:NY_histo_Herschel}%
\end{figure}
\subsection{Modified black body spectra} \label{sect:grey}
We examined a series modified black body spectra $B_{\nu}(T)\times
\nu^{\beta}$ together with noise to look for similar anomalous cases
as shown in Fig.~\ref{fig:example1}. Several combinations of the
temperature and spectral index were investigated. We started with the
100\,$\mu$m band and the three Planck bands 350--850\,$\mu$m. When
the modified black body spectra are scaled to have a 350\,$\mu$m
signal of 2.0\,MJy\,sr$^{-1}$, the signal-to-noise ratio is comparable
to that of Fig.~\ref{fig:example1}. However, to examine the effect of
different S/N ratios, the spectra were scaled by a factor that was
varied from 0.6 to 6.0 in five logarithmic steps. To investigate the
sensitivity to the relative weighting of the frequency points in the
fit, we included the factor $K_{\rm 100}$ which was varied from 0.5 to
1.5. As before, the factor $K_{\rm 100}$ did not affect the noise,
only the relative weighting of the data points in the fit. For each
case (i.e., a combination of temperature, spectral index, the
intensity scaling, and the value of $K_{\rm 100}$), we ran 5000 noise
realisations of the spectrum. Each spectrum was fitted using the two
different initial values of the optimisation to recognise the cases
with separate $\chi^2$ minima.
\begin{figure}
\centering
\includegraphics[width=8.8cm]{scene0_Planck3_multiple.eps}
\caption{
Frequency of recognised $\chi^2$ double minima for
simulated IRAS and Planck observations. Each frame corresponds
to one combination of the temperature and the spectral index that were
inputs of the simulation. The horizontal axis gives the 350\,$\mu$m
intensity (before noise is added) and the vertical axis is the factor
$K_{\rm 100}$ (smaller numbers correspond to a larger weight of the
100\,$\mu$m point). The colour scale gives the frequency of the double
$\chi^2$ minima as a percentage of all noise realisations with the
corresponding values of $S(350\,mu{\rm m})$ and $K_{\rm 100}$.
}
\label{fig:scene0_Planck3_multiple}%
\end{figure}
Double minima are detected in $\sim$10\% of the cases but the
frequency drops close to zero by the time the S/N ratio is increased
by a factor of three (Fig.~\ref{fig:scene0_Planck3_multiple}). There
is some dependence on the model in question but this may also be
caused by the differences in the S/N ratios of the other bands. With
$T=8.0$\,K and $\beta=$1.5, the double minima are significantly more
rare when the 100\,$\mu$m data point is given a larger weight ($K_{\rm
100}<$1.0). With $T=12.0$\,K and $\beta=$2.5, the opposite is true.
Figure~\ref{fig:scene0_Herschel2_multiple} shows the corresponding
results for the Herschel data that were scaled to have the same S/N
ratios at the 350\,$\mu$m wavelength as in the previous Planck
examples (the observational uncertainties are the same as in
Sect.~\ref{sect:model}). Distinct $\chi^2$ minima are observed only
with $T_{\rm C}=8.0$\,K and $\beta=1.5$ and even in that case the
number is below 1\%. When the signal-to-noise ratio is increased by a
factor of three, no double minima are detected (probability $\sim
10^{-4}$ or less).
\begin{figure}
\centering
\includegraphics[width=8.8cm]{scene0_Herschel2_multiple.eps}
\caption{
Frequency of recognised $\chi^2$ double minima for
simulated Herschel observations. Each frame corresponds to one
modified black body with the given temperature and the spectral index.
The horizontal axis gives the 350\,$\mu$m intensity (the S/N ratios
are the same as in Fig.\ref{fig:scene0_Planck3_multiple}) and the
vertical axis is $K_{\rm 100}$, the relative weighting of the
100\,$\mu$m point. The colour scale gives the frequency of the double
$\chi^2$ minima as a percentage.
}
\label{fig:scene0_Herschel2_multiple}%
\end{figure}
If the 350\,$\mu$m signal of the simulated Herschel observations is
lowered to 2.0\,MJy\,sr$^{-1}$, the S/N ratios decrease by a factor of
seven. As shown in Fig.~\ref{fig:scene0_Herschel_multiple}, in this
case the number of double minima increases to $\sim$10\%, a level
similar to the previous Planck example.
The figures in Appendix~\ref{sect:bias} show the bias and scatter
of the $T_{\rm C}$ and $\beta_{\rm obs}$ values corresponding to the
cases in
Figs.~\ref{fig:scene0_Planck3_multiple}--\ref{fig:scene0_Herschel_multiple}.
\begin{figure}
\centering
\includegraphics[width=8.8cm]{scene0_Herschel_multiple.eps}
\caption{
Frequency of the recognised $\chi^2$ double minima for
simulated Herschel observations when the signal-to-noise ratio has
been decreased by a factor of seven compared to
Fig.~\ref{fig:scene0_Herschel2_multiple}.
}
\label{fig:scene0_Herschel_multiple}%
\end{figure}
We next examined spectra that are the sums of two modified black bodies
that have the same spectral index but different temperatures,
$I_{\nu}=(B_{\nu}(T_1) + B_{\nu}(T_2)) \times \nu^{\beta}$. The
deviations from a single modified black body could make the fit more
susceptible to ambiguity.
The results for the Planck+IRAS case are presented in
Fig.~\ref{fig:scene1_Planck3_multiple}. They look quite
similar to the single black body cases shown in
Fig.~\ref{fig:scene0_Planck3_multiple} where, of course, also the
temperatures are somewhat different. It seems that the deviations from
a single modified black body shape do not have a significant effect on
the presence of multiple $\chi^2$ minima. The conclusion is confirmed
by the Herschel simulations in
Fig.~\ref{fig:scene1_Herschel2_multiple}, which is to be compared with
Fig.~\ref{fig:scene0_Herschel2_multiple}. The differences are again
small and may be mainly caused by the temperature differences, higher
(average) temperatures leading to fewer cases of multimodal $\chi^2$
distribution.
\begin{figure}
\centering
\includegraphics[width=8.8cm]{scene1_Planck3_multiple.eps}
\caption{
Frequency of the recognised double $\chi^2$ minima for
the sum of two modified black bodies with temperatures 8 and 10\,K or
10 and 14\,K. The column density ratio of the lower temperature and
the higher temperature component is 1:2 or 2:1, as indicated in the
frames. The data correspond to simulated observations at 100\,$\mu$m
and three Planck wavelengths.
}
\label{fig:scene1_Planck3_multiple}%
\end{figure}
\begin{figure}
\centering
\includegraphics[width=8.8cm]{scene1_Herschel2_multiple.eps}
\caption{
As Fig.~\ref{fig:scene1_Planck3_multiple} but for simulated Herschel
observations.
}
\label{fig:scene1_Herschel2_multiple}%
\end{figure}
As a final deviation from single modified black bodies we examined
spectra where the spectral index is $\beta=2.0$ up to 300\,$\mu$m and
$\beta=1.5$ at the longer wavelengths.
The break in $\beta$ does not cause any significant change in the
number of double $\chi^2$ minima, either in the simulated Planck
observations (Fig.~\ref{fig:scene2_Planck3_multiple} vs.
Fig.~\ref{fig:scene0_Planck3_multiple}) or in the simulated Herschel
observations (Fig.~\ref{fig:scene2_Herschel_multiple} vs.
Fig.~\ref{fig:scene0_Herschel_multiple}).
\begin{figure}
\centering
\includegraphics[width=8.8cm]{scene2_Planck3_multiple.eps}
\caption{
Frequency of the recognised double $\chi^2$ minima for
the modified black bodies with a break in the spectral slope. The data
correspond to simulated observations at 100\,$\mu$m and three Planck
wavelengths.
}
\label{fig:scene2_Planck3_multiple}%
\end{figure}
\begin{figure}
\centering
\includegraphics[width=8.8cm]{scene2_Herschel2_multiple.eps}
\caption{
Frequency of the recognised double $\chi^2$ minima for
the modified black bodies with a break in the spectral slope. The
figure correspond to simulated Herschel data with the 350\,$\mu$m S/N
identical to that in the Planck case in
Fig.~\ref{fig:scene2_Planck3_multiple}.
}
\label{fig:scene2_Herschel_multiple}%
\end{figure}
\section{Discussion} \label{sect:discussion}
The tests with the grey body spectra showed that with four
wavelengths 100, 350, 550, and 850\,$\mu$m and signal-to-noise
ratios similar to those in Fig.~\ref{fig:example1}, the $\chi^2$ values of the
modified black body fits exhibit multiple minima in up to $\sim$10\%
of the cases. This is similar to the fraction of `anomalous' solutions
that were seen for the radiative transfer models of
cylindrical filaments ($T_{\rm C}<$10\,K) in
Figs.~\ref{fig:Planck3_scatter} and \ref{fig:NY_histo_Planck3}.
With the five wavelengths of 100, 160, 250, 350, and 500\,$\mu$m the
number of multiple $\chi^2$ minima was lower by a factor of $\sim$100
(see Fig.~\ref{fig:scene0_Herschel_multiple}) when the 350\,$\mu$m
signal-to-noise ratio was similar to that of the previous case with four
wavelengths (S/N$\sim$20). Therefore the better wavelength
sampling provided by the set of five frequencies is the main reason
for the disappearance of the double $\chi^2$ minima. With four
wavelengths, the anomalous solutions always corresponded to a
350\,$\mu$m value that was overestimated by $\sim 2\sigma$ relative to
the neighbouring wavelengths (see Fig.~\ref{fig:deltas}). With five
wavelengths, an anomalous solution might require a similar error in
two neighbouring channels (e.g., 250\,$\mu$m and 350\,$\mu$m) that,
for normally distributed errors and $2-\sigma$ deviations
should be less likely by a factor of 50. The number of anomalous
solutions increased to the 10\% level only when the S/N ratios were
decreased by a factor of six and the 350\,$\mu$m S/N ratio was
close to three.
Are the separate $\chi^2$ minima of practical importance? In the
extreme cases like the one shown in Fig.~\ref{fig:example1}, the
multimodality of the $\chi^2$ values causes clear complications. When
the minima are of similar depth, the result obtained by optimisation
methods depends on the initial values. Furthermore, an infinitesimal
change in the flux values or in their weighting can switch the global
minimum between the low- and the high-temperature solution. The
difference can be several degrees in colour temperature and
several units in spectral index. Because of the strongly
non-Gaussian behaviour of the problem the uncertainties deduced from
the {\em local} shape of the $\chi^2$ surface will radically
underestimate the true uncertainties of $T_{\rm C}$ and $\beta_{\rm
Obs}$. In principle the problem can be avoided by calculating the
$\chi^2$ values over the whole parameter plane. This is expensive but
may also not yet be a completely satisfactory solution. The examples
have shown that the shape of the $\chi^2$ surface can change rapidly
as, for example, the error estimates are changed. In
Sect.~\ref{sect:grey} we noticed that double minima could appear and
again disappear when the factor $K_{\rm 100}$ was changed at a 10\%
level. This means that also the uncertainty of the flux uncertainties
and their impact on the $\chi^2$ surface should be examined. However,
the real error estimates are rarely known to a precision of 10\%. On
the positive side, many anomalous solutions may be recognisable from
the SED plots. In Fig.~\ref{fig:example1} the low-temperature solution
would certainly be treated with some caution, in spite of it
corresponding to the lowest $\chi^2$ value. The situation becomes
less clear when the distance between the minima is shorter. One should
always take into account that for low signal-to-noise
ratio data there is a non-negligible probability, possibly even in
excess of $\sim$10\%, that the deepest $\chi^2$ minimum is not the
minimum closest to the true solution.
The problem is less tractable in large surveys of
individual sources or maps with thousands of pixels. It becomes
difficult to examine each SED fit by eye and the calculation of the
full $\chi^2$ surfaces may become impractical. If not accounted for,
even a few anomalous solutions can significantly affect
the deduced shape of the $\beta(T)$ relation. One can use Monte Carlo
methods to estimate the number and the influence of the outliers,
including the effects of the non-Gaussian statistics
\cite[e.g.][]{PlanckI}). Conversely, one can try to directly recover
the true $\beta(T)$ relation with Bayesian methods
\citep{Veneziani2010, Kelly2011}.
In those cases the multimodal $\chi^2$ distribution may not be a
significant complication because the methods are aware of the shape of
the likelihood function and that is additionally modified by the prior.
However, an accurate knowledge of the statistics of the observed
intensities is still needed.
To quantify the role of the double $\chi^2$ minima (as opposed to the
general influence of noise) we examined again the simulated
observations of Fig.~\ref{fig:scene0_Planck3_multiple}. For each
combination of $S(350\mu{\rm m})$ and $K_{\rm 100}$, we took 200
samples of ($T_{\rm C}$, $\beta_{\rm Obs}$) corresponding to the
different temperatures used in the simulation (8.0, 10.0, and 12.0\,K)
and a fixed input spectral index of $\beta=2.0$. The resulting linear
correlation coefficients between $T_{\rm C}$ and $\beta_{\rm Obs}$ are
shown in Fig.~\ref{fig:TB_fit_Planck3_scene0_2_corr}$a$. In the
absence of noise the correlation should be zero. The observed
correlation varies from -0.5 (for the data with the highest
signal-to-noise ratio) to $\sim$-0.8. Note that the correlation
coefficient does not increase monotonously to the lowest S/N ratios
because of the strong non-linearity of the $\beta_{\rm Obs}(T_{\rm
C})$ relation. The frame $b$ of
Fig.~\ref{fig:TB_fit_Planck3_scene0_2_corr} shows the correlation
coefficients excluding those cases where the $\chi^2$ shows two minima
(for that particular value of $K_{\rm 100}$). The effect of the
cases of double $\chi^2$ minima is visible at a 10\% level.
\begin{figure}
\centering
\includegraphics[width=8.8cm]{TB_fit_Planck3_scene0_2_corr.eps}
\caption{
Linear correlation coefficients between $T_{\rm C}$ and
$\beta_{\rm Obs}$ for the models of
Fig.~\ref{fig:scene0_Planck3_multiple} as functions of the source
intensity $S(350\mu{\rm m})$ and the weight given in the fit to the
100\,$\mu$m point (lower value of $K_{\rm 100}$ implies a larger
weight). Frame $b$ is the same after removing the points that
correspond to the presence of multiple local $\chi^2$ minima.
}
\label{fig:TB_fit_Planck3_scene0_2_corr}
\end{figure}
The anomalous solutions may be identified by the unrealistic values of
the colour temperature and the spectral index. However, the presence
of multiple $\chi^2$ minima also directly serves as a warning sign.
Figure~\ref{fig:scatter_12_1.5} displays the distribution of the
($T_{\rm C}$, $\beta_{\rm Obs}$) values for the modified black body
model with $T=12$\,K and $\beta$=1.5 that had particularly many
multiple minima (see
Fig.~\ref{fig:scene0_Planck3_multiple}). The points plotted in
Fig.~\ref{fig:scatter_12_1.5} corresponds to the solution obtained
from optimisation with initial values close to the true solution. The
figure shows 500 points for each value of $S(350\mu{\rm m})$, all
corresponding to the default weighting with $K_{\rm 100}$=1.0. The
blue points are the cases where two $\chi^2$ minima were detected with
any of the tested $K_{\rm 100}$ values (see Sect.~\ref{sect:grey}).
The red points are a subset where double minima were detected with the
present value of $K_{\rm 100}$=1.0. The blue points are seen to avoid
the locus of the correct solution that still contains most of the
data. Especially the red points are concentrated in the
low-temperature tail. Of all the points below the colour temperature of
8\,K, 42\% corresponded to cases with multiple $\chi^2$ minima (or
extremely shallow single minima that for numerical reasons lead to
different parameter estimates). If the test is carried out using all
$K_{\rm 100}$ factors between 0.5 and 1.5, the percentage increases to
70\%. This suggests that similar tests should be useful also for
real observations.
\begin{figure}
\centering
\includegraphics[width=8.8cm]{scatter_12_1.5.eps}
\caption{
($T_{\rm C}$, $\beta_{\rm Obs}$) values for simulated modified
black bodies with $T=12$\,K and $\beta=$1.5 (see
Sect.~\ref{sect:grey}). The data points correspond to the normal
weighting of data ($K_{\rm 100}$=1.0) and to all S/N ratios
shown in Fig.~\ref{fig:scene0_Planck3_multiple}). The red points
denote cases where double $\chi^2$ minima were detected with $K_{\rm
100}$=1.0 and the blue points the other cases where double minima were
seen with any value of $K_{\rm 100}$ between 0.5 and 1.5
}
\label{fig:scatter_12_1.5}
\end{figure}
\section{Conclusions} \label{sect:conclusions}
We have studied the behaviour of $\chi^2$ fits of SEDs that are either
sums of modified black bodies or are based on the radiative transfer
modelling of dust emission from cylindrical clouds. Using
combinations of wavelengths relevant for the current Planck and
Herschel satellite studies, we have examined the effect of noise on
the shape of the confidence regions in the ($T_{\rm C}$, $\beta_{\rm
Obs}$) plane.
The results have lead to the following conclusions:
\begin{itemize}
\item
In addition to the usual symmetric error banana, the $\chi^2$
distribution can exhibit asymmetries of varying strength. The
expectation values are close to the result obtained in the absence of
noise, but are not without bias.
\item
For low signal-to-noise data (S/N below 10) in the
100\,$\mu$m, 350\,$\mu$m, 550\,$\mu$m, and 850\,$\mu$m bands, the
noise distribution of ($T_{\rm C}$, $\beta_{\rm Obs}$) values develops
an asymmetric tail that can extend to low temperatures and very high
spectral indices.
\item
Under the same conditions, the $\chi^2$ distribution of individual
measurements can exhibit two distinct minima. A very small change in
the weighting of the frequency points or in the noise can shift the
best solution from one minimum to the other. This can correspond to a
change of several degrees in the colour temperature and a change of
several units in the spectral index.
\item
Herschel observations were simulated using five wavelengths, 100,
160, 250, 350, and 500\,$\mu$m. For the radiative transfer
models the error distributions remained relatively symmetric and very
few cases with multiple $\chi^2$ minima were detected. This although
the signal-to-noise ratios were lower than in the previously
examined four wavelength case.
\item
Investigation of pure modified black bodies (plus noise) shows that
deviations from a single modified black body, such as in the case of
line-of-sight mixing of temperatures, has no significant effect on the
appearance of double $\chi^2$ minima.
\item
The main factor behind the double $\chi^2$ minima is the noise,
but the susceptibility depends greatly on the set of wavelengths used.
Comparing the four and five wavelength cases, equal numbers of double
minima where seen when the signal-to-noise ratio of the latter were
lower by a factor of six (S/N$\sim$3).
\item
The asymmetries or the complete split of the error banana have
implications for dust studies. It can affect the interpretation of the
observations of individual targets and the reliability of the
$\beta(T)$ relations derived from low signal-to-noise data. The
probability distributions of $T_{\rm C}$ and $\beta_{\rm Obs}$ can be
non-Gaussian, strongly non-symmetric, and possibly even multimodal.
These features are sensitive to the assumptions of the flux
uncertainties and this should be taken into account, even in the
Bayesian analysis.
\end{itemize}
\begin{acknowledgements}
The authors acknowledge the support of the Academy of Finland Grants
No. 127015 and 250741. N.Y. acknowledges the support of a CNES
post-doctoral research grant.
\end{acknowledgements}
|
{
"timestamp": "2012-03-13T01:01:19",
"yymm": "1203",
"arxiv_id": "1203.2263",
"language": "en",
"url": "https://arxiv.org/abs/1203.2263"
}
|
\section{Introduction}
\setcounter{equation}{0}
\renewcommand{\theequation}{\thesection.\arabic{equation}}
In \cite{BK2}, Berenstein and Kazhdan introduced the notion of decorated
geometric crystals for reductive algebraic groups.
Geometric crystals are geometric analogue to the
Kashiwara's crystal bases (\cite{BK}). We, indeed, treated geometric
crystals in the affine/Kac-Moody settings (\cite{KNO,KNO2,N}), but
we do not need such general settings and then we shall consider the
(semi-)simple settings below.
Let $I$ be a finite index set. Associated with
a Cartan matrix $A=(a_{i,j})_{i,j\in I}$,
define the decorated geometric crystal $\cX=(\chi,f)$, which
is a pair of geometric crystal
$\chi=(X,\{e_i\}_i,\{\gamma_i\}_i,\{\vep_i\}_i)$
and a certain special rational function $f$ such that
\[
f(e_i^c(x))=f(x)+(c-1)\vp_i(x)+(c^{-1}-1)\vep_i(x),
\]
for any $i\in I$, where $e_i^c$ is the rational $\bbC^\times$ action on
$X$, and $\vp_i$ and $\vep_i=\vep_i\cdot \gamma_i$
are the rational functions on $X$.
If we apply the procedure called ``ultra-discretization''(UD) to ``positive
geometric crystals'' (see \ref{subsec-posi}),
then we would obtain certain free-crystals for
the transposed Cartan matrix (\cite{BK,N}). As for a positive decorated
geometric crystal $(\chi,f,T',\theta)$
applying UD to the function $f$ and
considering the convex polyhedral domain defined by the inequality
$UD(f)\geq0$, we get the crystal with the property
``normal''(\cite{K3}). Moreover, abstracting a connected component with
the highest weight $\lm$,
we obtain the Langlands dual
Kashiwara's crystal $B(\lm)$ with the highest weight
$\lm$.
This result makes us recall the ``polyhedral realization'' of crystal
bases (\cite{N2,NZ}) since it has very similar way to get the crystal
$B(\lm)$ from certain free-crystals, defined by the system of linear
inequalities.
Thus, one of the main aims of this article is to show that the crystals
obtained by UD from positive decorated geometric crystals and the
polyhedral realizations of crystals coincide with each other for type $A_n$.
One more aim of this article is to describe the relations between
the function $f_B$ for certain decorated geometric crystal
$(TB^-_{w_0},f_B)$
and monomial realization of crystals (\cite{K7,Nj}).
We shall propose the conjecture of their relations and
present the affirmative answer for type $A_n$.
Let us mention the statement of the conjecture:
for the function $f_B$ and certain positive structure
$T\Theta^-_{\bfi}$ on $TB^-_{w_0}$, the function
$f_B(t\Theta^-_{\bf i}(c))$ is
expressed as a sum of monomials in the crystal $\cY(p)$ with positive
coefficients (for more details, see Conjecture \ref{conj1} below.).
Observing this relation, we can deduce the refined polyhedral
realization of crystals induced from the monomial realizations.
Indeed, for the original polyhedral realizations we are forced the
condition ``ample'', which is some technical condition to guarantee the
non-emptiness of the underlying crystal (see Theorem \ref{main}).
But, if the relations among the polyhedral realizations, the UD of decorated
geometric crystals and the monomial realizations are established, it would
be possible to remove the condition ``ample'' and it would become easier
to obtain polyhedral realizations of crystals than applying
the present method.
The organization of the article is as follows:
in Sect.2, we review the theory of crystals and their polyhedral
realizations. In Sect.3, first we introduce the theory of decorated geometric
crystals following \cite{BK2}. Next, we define the decoration by using the
elementary characters and certain special positive decorated geometric
crystal on ${\bbB}_w=TB^-_w$. Finally, the ultra-discretization of $TB^-_w$ is
described explicitly. We calculate the function $f_B$ exactly for type
$A_n$ in Sect.4. In Sect.5, for the type $A_n$
the coincidence of the polyhedral
realization $\Sigma_{\io}[\lm]$ and the ultra-discretization
$B_{f_B,\Theta^-_{\bfii0}}(\lm)$ will be clarified by using the result in
Sect.4.
In the last section, we review the monomial realization of crystals
(\cite{K7,Nj})
and the function $f_B$ is expressed in terms of
the monomials in the monomial realizations of crystals for type $A_n$.
Finally, the conjecture is proposed and under the validity of the
conjecture,
we shall state the refined polyhedral realizations associated with the
monomial realizations.
The results for other simple Lie algebras are mentioned in the
forthcoming paper.
\renewcommand{\thesection}{\arabic{section}}
\section{Crystal and its polyhedral realization}
\setcounter{equation}{0}
\renewcommand{\theequation}{\thesection.\arabic{equation}}
\subsection{Notations}
We list the notations used in this paper.
Indeed, the settings below are originally Kac-Moody ones, but in the
article we do not need them and then we restrict the settings to
semi-simple ones.
Let $\ge$ be
a semi-simple Lie algebra over $\bbQ$
with a Cartan subalgebra
$\tt$, a weight lattice $P \subset \tt^*$, the set of simple roots
$\{\al_i: i\in I\} \subset \tt^*$,
and the set of coroots $\{h_i: i\in I\} \subset \tt$,
where $I$ is a finite index set.
Let $\lan h,\lm\ran=\lm(h)$ be the pairing between $\tt$ and $\tt^*$,
and $(\al, \beta)$ be an inner product on
$\tt^*$ such that $(\al_i,\al_i)\in 2\bbZ_{\geq 0}$ and
$\lan h_i,\lm\ran={{2(\al_i,\lm)}\over{(\al_i,\al_i)}}$
for $\lm\in\tt^*$ and $A:=(\lan h_i,\al_j\ran)_{i,j}$ is the associated Cartan matrix.
Let $P^*=\{h\in \tt: \lan h,P\ran\subset\ZZ\}$ and
$P_+:=\{\lm\in P:\lan h_i,\lm\ran\in\ZZ_{\geq 0}\}$.
We call an element in $P_+$ a {\it dominant integral weight}.
The quantum algebra $\uq$
is an associative
$\QQ(q)$-algebra generated by the $e_i$, $f_i \,\, (i\in I)$,
and $q^h \,\, (h\in P^*)$
satisfying the usual relations.
The algebra $\uqm$ is the subalgebra of $\uq$ generated
by the $f_i$ $(i\in I)$.
For the irreducible highest weight module of $\uq$
with the highest weight $\lm\in P_+$, we denote it by $V(\lm)$
and its {\it crystal base} we denote $(L(\lm),B(\lm))$.
Similarly, for the crystal base of the algebra $\uqm$ we denote
$(L(\ify),B(\ify))$ (see \cite{K0,K1}).
Let $\pi_{\lm}:\uqm\longrightarrow V(\lm)\cong \uqm/{\sum_i\uqm
f_i^{1+\lan h_i,\lm\ran}}$ be the canonical projection and
$\widehat \pi_{\lm}:L(\ify)/qL(\ify)\longrightarrow L(\lm)/qL(\lm)$
be the induced map from $\pi_{\lm}$. Here note that
$\widehat \pi_{\lm}(B(\ify))=B(\lm)\sqcup\{0\}$.
By the terminology {\it crystal } we mean some combinatorial object
obtained by abstracting the properties of crystal bases.
Indeed, crystal constitutes a set $B$ and the maps
$wt:B\longrightarrow P$, $\vep_i,\vp_i:B\longrightarrow \ZZ\sqcup\{-\ify\}$
and $\eit,\fit:B\sqcup\{0\}\longrightarrow B\sqcup\{0\}$
($i\in I$) satisfying several axioms (see \cite{K3},\cite{NZ},\cite{N2}).
In fact, $B(\ify)$ and $B(\lm)$ are the typical examples
of crystals.
Let $B_1$ and $B_2$ be crystals.
A {\sl strict morphism } of crystals $\psi:B_1\lar B_2$
is a map $\psi:B_1\sqcup\{0\} \lar B_2\sqcup\{0\}$
satisfying the following conditions: $\psi(0)=0$,
$wt(\psi(b)) = wt(b)$,
$\vep_i(\psi(b)) = \vep_i(b),$
$\vp_i(\psi(b)) = \vp_i(b)$
if $b\in B_1$ and $\psi(b)\in B_2,$
and the map $\psi: B_1\sqcup\{0\} \lar B_2\sqcup\{0\}$
commutes with all $\eit$ and $\fit$.
An injective strict morphism is called an {\it embedding }of crystals.
It is well-known that $\uq$ has a Hopf algebra structure.
Then the tensor product of $\uq$-modules has
a $\uq$-module structure.
The crystal bases have very nice properties for
tensor operations. Indeed, if $(L_i,B_i)$ is a crystal base of
$\uq$-module $M_i$ ($i=1,2$), $(L_1\ot_A L_2, B_1\ot B_2)$
is a crystal base of $M_1\ot_{\QQ(q)} M_2$ (\cite{K1}).
Consequently, we can consider the tensor product
of crystals and then they constitute a tensor category.
\subsection{Polyhedral Realization of $B(\ify)$}
\label{poly-uqm}
Let us recall the results in \cite{NZ}.
Consider the infinite $\bbZ$-lattice
\begin{equation}
\ZZ^{\ify}
:=\{(\cd,x_k,\cd,x_2,x_1): x_k\in\ZZ
\,\,{\rm and}\,\,x_k=0\,\,{\rm for}\,\,k\gg 0\};
\label{uni-cone}
\end{equation}
we will denote by $\ZZ^{\ify}_{\geq 0} \subset \ZZ^{\ify}$
the subsemigroup of nonnegative sequences.
To the rest of this section, we fix an infinite sequence of indices
$\io=\cd,i_k,\cd,i_2,i_1$ from $I$ such that
\begin{equation}
{\hbox{
$i_k\ne i_{k+1}$ and $\sharp\{k: i_k=i\}=\ify$ for any $i\in I$.}}
\label{seq-con}
\end{equation}
We can associate to $\io$ a crystal structure
on $\ZZ^{\ify}$ and denote it by $\ZZ^{\ify}_{\io}$
(\cite[2.4]{NZ}).
\begin{pro}[\cite{K3}, See also \cite{NZ}]
\label{emb}
There is a unique strict embedding of crystals
$($called Kashiwara embedding$)$
\begin{equation}
\Psi_{\io}:B(\ify)\hookrightarrow \ZZ^{\ify}_{\geq 0}
\subset \ZZ^{\ify}_{\io},
\label{psi}
\end{equation}
such that
$\Psi_{\io} (u_{\ify}) = (\cd,0,\cd,0,0)$, where
$u_{\ify}\in B(\ify)$ is the vector corresponding to $1\in \uqm$.
\end{pro}
Consider the infinite dimensional vector space
$$
\QQ^{\ify}:=\{{x}=
(\cd,x_k,\cd,x_2,x_1): x_k \in \QQ\,\,{\rm and }\,\,
x_k = 0\,\,{\rm for}\,\, k \gg 0\},
$$
and its dual space $(\QQ^{\ify})^*:={\rm Hom}(\QQ^{\ify},\QQ)$.
We will write a linear form $\vp \in (\QQ^{\ify})^*$ as
$\vp({x})=\sum_{k \geq 1} \vp_k x_k$ ($\vp_j\in \QQ$)
for $x\in \QQ^{\ify}$.
For the fixed infinite sequence
$\io=(i_k)$ and $k\geq1$ we set $\kp:={\rm min}\{l:l>k\,\,{\rm and }\,\,i_k=i_l\}$ and
$\km:={\rm max}\{l:l<k\,\,{\rm and }\,\,i_k=i_l\}$ if it exists,
or $\km=0$ otherwise.
We set for $x\in \QQ^{\ify}$, $\beta_0(x)=0$ and
\begin{equation}
\beta_k(x):=x_k+\sum_{k<j<\kp}\lan h_{i_k},\al_{i_j}\ran x_j+x_{\kp}
\qq(k\geq1).
\label{betak}
\end{equation}
We define the piecewise-linear operator
$S_k=S_{k,\io}$ on $(\QQ^{\ify})^*$ by
$$
S_k(\vp):=
\left\{
\begin{array}{ll}
\vp-\vp_k\beta_k & {\mbox{ if }}\vp_k>0,\\
\vp-\vp_k\beta_{\km} & {\mbox{ if }}\vp_k\leq 0.
\end{array}
\right.
$$
Here we set
\begin{eqnarray}
\Xi_{\io} &:= &\{S_{j_l}\cd S_{j_2}S_{j_1}x_{j_0}\,|\,
l\geq0,j_0,j_1,\cd,j_l\geq1\},
\label{Xi_io}\\
\Sigma_{\io} & := &
\{x\in \ZZ^{\ify}\subset \QQ^{\ify}\,|\,\vp(x)\geq0\,\,{\rm for}\,\,
{\rm any}\,\,\vp\in \Xi_{\io}\}.
\end{eqnarray}
We impose on $\io$ the following positivity assumption:
\begin{equation}
{\hbox{if $\km=0$ then $\vp_k\geq0$ for any
$\vp(x)=\sum_k\vp_kx_k\in \Xi_{\io}$}}.
\label{posi}
\end{equation}
\begin{thm}[\cite{NZ}]
Let $\io$ be a sequence of indices satisfying $(\ref{seq-con})$
and (\ref{posi}). Then we have
${\rm Im}(\Psi_{\io})(\cong B(\ify))=\Sigma_{\io}$.
\end{thm}
\subsection{Structure of $\ZZ^{\ify}_{\io}[\lm]$}
\label{ZZ-io-lm}
Let $R_{\lm}:=\{r_{\lm}\}$ be the crystal
which consists of one element $r_{\lm}$ (\cite{N2}).
Consider the crystal $\ZZ^{\ify}_{\io}\ot R_{\lm}$
and denote it by
$\ZZ^{\ify}_{\io}[\lm]$. Here note that
since the crystal $R_{\lm}$ has only one element,
as a set we can identify $\ZZ^{\ify}_{\io}[\lm]$ with
$\ZZ^{\ify}_{\io}$ but their crystal structures are different.
So we review an explicit crystal structure of
$\ZZ^{\ify}[\lm]$ in \cite{N2}.
Fix a sequence of indices $\io:=(i_k)_{k\geq 1}$ satisfying the condition
(\ref{seq-con}) and a weight $\lm\in P$
(Here we do not necessarily assume that
$\lm$ is dominant.).
$\ZZ^{\ify}[\lm]$ can be regarded
as a subset of $\QQ^{\ify}$, and then
we denote an element in $\ZZ^{\ify}[\lm]$
by $ x=(\cd,x_k,\cd,x_2,x_1)$.
For $ x=(\cd,x_k,\cd,x_2,x_1)\in \QQ^{\ify}$
we define the linear functions
\begin{eqnarray}
\sigma_k(x)&:= &x_k+\sum_{j>k}\lan h_{i_k},\al_{i_j}\ran x_j,
\q(k\geq1)
\label{sigma}\\
\sigma_0^{(i)}(x)
&:= &-\lan h_i,\lm\ran+\sum_{j\geq1}\lan h_i,\al_{i_j}\ran x_j,
\q(i\in I)
\label{sigma0}
\end{eqnarray}
Here note that
since $x_j=0$ for $j\gg0$ on $\QQ^{\ify}$,
the functions $\sigma_k$ and $\sigma^{(i)}_0$ are
well-defined.
Let $\sigma^{(i)} (x)
:= {\rm max}_{k: i_k = i}\sigma_k (x)$, and
$M^{(i)} :=
\{k: i_k = i, \sigma_k (x) = \sigma^{(i)}(x)\}.
$
Note that
$\sigma^{(i)} (x)\geq 0$, and that
$M^{(i)} = M^{(i)} (x)$ is a finite set
if and only if $\sigma^{(i)} (x) > 0$.
Now we define the maps
$\eit: \ZZ^{\ify}[\lm] \sqcup\{0\}\lar \ZZ^{\ify}[\lm] \sqcup\{0\}$
and
$\fit: \ZZ^{\ify}[\lm] \sqcup\{0\}\lar \ZZ^{\ify}[\lm] \sqcup\{0\}$
by setting $\eit(0)=\fit(0)=0$, and
\begin{equation}
(\fit(x))_k = x_k + \delta_{k,{\rm min}\,M^{(i)}}
\,\,{\rm if }\,\,\sigma^{(i)}(x)>\sigma^{(i)}_0(x);
\,\,{\rm otherwise}\,\,\fit(x)=0,
\label{action-f}
\end{equation}
\begin{equation}
(\eit(x))_k = x_k - \delta_{k,{\rm max}\,M^{(i)}} \,\, {\rm if}\,\,
\sigma^{(i)} (x) > 0\,\,
{\rm and}\,\,\sigma^{(i)}(x)\geq\sigma^{(i)}_0(x) ; \,\,
{\rm otherwise} \,\, \eit(x)=0,
\label{action-e}
\end{equation}
where $\del_{i,j}$ is the Kronecker's delta.
We also define the functions
$wt$, $\vep_i$ and $\vp_i$ on $\ZZ^{\ify}[\lm]$ by
\begin{eqnarray}
&& wt(x) :=\lm -\sum_{j=1}^{\ify} x_j \al_{i_j},
\label{wt-vep-vp-1}\\
&& \vep_i (x) := {\rm max}(\sigma^{(i)} (x),\sigma_0^{(i)}(x))
\label{wt-vep-vp}\\
&& \vp_i (x) := \lan h_i, wt(x) \ran + \vep_i(x).
\label{wt-vep-vp-3}
\end{eqnarray}
Note that
by (\ref{wt-vep-vp-1}) we have
\begin{equation}
\lan h_i,wt(x)\ran = -\sigma^{(i)}_0(x).
\label{**}
\end{equation}
\subsection{Polyhedral Realization of $B(\lm)$}
In this subsection,
we review the result in \cite{N2}.
In the rest of this section,
$\lm$ is supposed to be a dominant integral weight.
Here we define the map
\begin{equation}
\Phi_{\lm}:(B(\ify)\ot R_{\lm})\sqcup\{0\}\longrightarrow B(\lm)\sqcup\{0\},
\label{philm}
\end{equation}
by $\Phi_{\lm}(0)=0$ and $\Phi_{\lm}(b\ot r_{\lm})=\wpi(b)$ for $b\in B(\ify)$.
We set
$$
\wtil B(\lm):=
\{b\ot r_{\lm}\in B(\ify)\ot R_{\lm}\,|\,\Phi_{\lm}(b\ot r_{\lm})\ne 0\}.
$$
\begin{thm}[\cite{N2}]
\label{ify-lm}
\begin{enumerate}
\item
The map $\Phi_{\lm}$ becomes a surjective strict morphism of crystals
$B(\ify)\ot R_{\lm}\longrightarrow B(\lm)$.
\item
$\wtil B(\lm)$ is a subcrystal of $B(\ify)\ot R_{\lm}$,
and $\Phi_{\lm}$ induces the
isomorphism of crystals $\wtil B(\lm)\mapright{\sim} B(\lm)$.
\end{enumerate}
\end{thm}
By Theorem \ref{ify-lm}, we have the strict embedding of crystals
$\Omega_{\lm}:B(\lm)(\cong \wtil B(\lm))\hookrightarrow B(\ify)\ot R_{\lm}.$
Combining $\Omega_{\lm}$ and the
Kashiwara embedding $\Psi_{\io}$,
we obtain the following:
\begin{thm}[\cite{N2}]
\label{embedding}
There exists the unique strict embedding of crystals
\begin{equation}
\Psi_{\io}^{(\lm)}:B(\lm)\stackrel{\Omega_{\lm}}{\hookrightarrow}
B(\ify)\ot R_{\lm}
\stackrel{\Psi_{\io}\ot {\rm id}}{\hookrightarrow}
\ZZ^{\ify}_{\io}[\lm],
\label{Psi-lm}
\end{equation}
such that $\Psi^{(\lm)}_{\io}(u_{\lm})=(\cd,0,0,0)\ot r_{\lm}$.
\end{thm}
\vskip5pt
We fix a sequence of indices $\io$
satisfying (\ref{seq-con}) and take a dominant integral weight
$\lm\in P_+$.
For $k\geq1$ let $k^{(\pm)}$ be the ones in \ref{poly-uqm}.
Let $\beta_k^{(\pm)}(x)$ be linear functions given by
\begin{eqnarray}
&& \q
\beta_k^{(+)} (x) = \sigma_k (x) - \sigma_{\kp} (x)
= x_k+\sum_{k<j<\kp}\lan h_{i_k},\al_{i_j}\ran x_j+x_{\kp},
\label{beta}\\
&& \beta_k^{(-)} (x)
=
\left\{
\begin{array}{ll}
\sigma_{\km} (x) - \sigma_k (x)
=x_{\km}+\sum_{\km<j<k}\lan h_{i_k},\al_{i_j}\ran x_j+x_k &
\hspace{-10pt} {\mbox{ if }}\km>0,\\
\sigma_0^{(i_k)} (x) - \sigma_k (x)
=-\lan h_{i_k},\lm\ran+\sum_{1\leq j<k}\lan h_{i_k},\al_{i_j}\ran x_j+x_k
&
\hspace{-10pt}{\mbox{ if }}\km=0,
\end{array}
\right.\label{beta--}
\end{eqnarray}
Here note that
$\beta_k^{(+)}=\beta_k$ and
$\beta_k^{(-)}=\beta_{\km} {\hbox{ \,\,if\,\, $\km>0$}}$.
Using this notation, for every $k \geq 1$, we define
an operator
$\what S_k = \what S_{k,\io}$ for a linear function
$\vp(x)=c+\sum_{k\geq 1}\vp_kx_k$
$(c,\vp_k\in\QQ)$ on $\QQ^{\ify}$ by:
$$
\what S_k\,(\vp) :=\left\{
\begin{array}{ll}
\vp - \vp_k \beta_k^{(+)} & {\mbox{ if }}\vp_k > 0,\\
\vp - \vp_k \beta_k^{(-)} & {\mbox{ if }}\vp_k \leq 0.
\end{array}
\right.
$$
For the fixed sequence $\io=(i_k)$,
in case $\km=0$ for $k\geq1$, there exists unique $i\in I$ such that $i_k=i$.
We denote such $k$ by $\io^{(i)}$, namely, $\io^{(i)}$ is the first
number $k$ such that $i_k=i$.
Here for $\lm\in P_+$ and $i\in I$ we set
\begin{equation}
\lm^{(i)}(x):=
-\beta^{(-)}_{\io^{(i)}}(x)=\lan h_i,\lm\ran-\sum_{1\leq j<\io^{(i)}}
\lan h_i,\al_{i_j}\ran x_j-x_{\io^{(i)}}.
\label{lmi}
\end{equation}
For $\io$ and a dominant integral weight $\lm$,
let $\Xi_{\io}[\lm]$ be the set of all linear functions
generated by $\what S_k=\what S_{k,\io}$
from the functions $x_j$ ($j\geq1$)
and $\lm^{(i)}$ ($i\in I$), namely,
\begin{equation}
\begin{array}{ll}
\Xi_{\io}[\lm]&:=\{\what S_{j_l}\cd\what S_{j_1}x_{j_0}\,
:\,l\geq0,\,j_0,\cd,j_l\geq1\}
\\
&\cup\{\what S_{j_k}\cd \what S_{j_1}\lm^{(i)}(x)\,
:\,k\geq0,\,i\in I,\,j_1,\cd,j_k\geq1\}.
\end{array}
\label{Xi}
\end{equation}
Now we set
\begin{equation}
\Sigma_{\io}[\lm]
:=\{x\in \ZZ^{\ify}_{\io}[\lm](\subset \QQ^{\ify})\,:\,
\vp(x)\geq 0\,\,{\rm for \,\,any }\,\,\vp\in \Xi_{\io}[\lm]\}.
\label{Sigma}
\end{equation}
For a sequence $\io$ and a dominant integral weight $\lm$, a pair
$(\io,\lm)$ is called {\it ample}
if $\Sigma_{\io}[\lm]\ni\vec 0=(\cd,0,0)$.
\begin{thm}[\cite{N2}]
\label{main}
Suppose that $(\io,\lm)$ is ample.
Then we have
${\rm Im}(\plm)(\cong B(\lm))=\Sigma_{\io}[\lm]$,
where the explicit form of $\vep_i$ on $\Sigma_{\io}[\lm]$ is as follows:
\begin{equation}
\vep_i(x) =\sigma^{(i)}(x).
\label{wt-vep-vp4}
\end{equation}
The other formula for $\vp_i$, $\eit$ and $\fit$ are same as above.
\end{thm}
{\sl Proof.}
The formula (\ref{wt-vep-vp4}) slightly differs from (\ref{wt-vep-vp}).
Indeed, by (\ref{action-e}) we know that for $x\in \Sigma_{\io}[\lm]$
unless $\sigma^{(i)}(x)>0$ and $\sigma^{(i)}(x)\geq \sigma^{(i)}_0(x)$,
we find $\eit(x)=0$. Furthermore, for any $x=(\cd,x_2,x_1)\in
\Sigma_\io[\lm]$
it follows from the definition of
$\Sigma_\io[\lm]$ that $0\leq \lm^{(i)}(x)
=\sigma_{\io^{(i)}}(x)-\sigma_0^{(i)}(x)$
which implies that $\sigma^{(i)}(x)\geq \sigma_0^{(i)}(x)$ and then
we can obtain (\ref{wt-vep-vp4}).\qed
\medskip
\subsection{\bf $A_n$-case}
We shall apply the results in the previous subsection to
the case $\ge=A_n$.
Let us identify the index set $I$ with $[1,n] := \{1,2,\cd,n\}$
in the standard way; thus, the Cartan matrix
$(a_{i,j}= \lan h_i,\al_j\ran )_{1 \leq i,j \leq n}$ is given by
$a_{i,i}=2$, $a_{i,j}=-1$ for $|i-j|=1$, and
$a_{i,j}=0$ otherwise.
As the infinite sequence $\io$ let us take
the following periodic sequence
$$
\io = \cd,\underbrace{n,\cd,2,1}_{},
\cd,\underbrace{n,\cd,2,1}_{},\underbrace{n,\cd,2,1}_{}.
$$
Following \cite[Sect.5]{NZ}, we shall
change the indexing set for $\ZZ^{\ify}$
from $\ZZ_{\geq 1}$ to $\ZZ_{\geq 1} \times [1,n]$, which is given by
the bijection
$\ZZ_{\geq 1} \times [1,n] \to \ZZ_{\geq 1}$
($(j;i) \mapsto (j-1)n + i$).
According to this, we will write an element $x \in \ZZ^{\ify}$
as a doubly-indexed family $(x_{j;i})_{j \geq 1, i \in [1,n]}$.
We will adopt the convention that $x_{j;i} = 0$ unless
$j \geq 1$ and $i \in [1,n]$; in particular, $x_{j;0} = x_{j;n+1} = 0$
for all $j$.
\begin{thm}
\label{A_n}
Let $\lm=\sum_{1\leq i\leq n}\lm_i\Lm_i$ $(\lm_i\in \ZZ_{\geq0})$
be a dominant integral weight.
In the above notation, the image ${\rm Im} \,(\Psi^{(\lm)}_{\io})$
is the set of all integer families $(x_{j;i})$ such that
\beqn
&&\hspace{-30pt}\hbox{
$x_{1;i} \geq x_{2;i-1} \geq \cd \geq x_{i;1} \geq 0$
for $1 \leq i \leq n$}
\label{sl-1}\\
&&\hspace{-30pt}\hbox{
$x_{j;i} = 0$ for $i+j > n+1$, }
\label{j;i=0}\\
&&\hspace{-30pt}\hbox{
$\lm_i\geq x_{j;i-j+1}-x_{j;i-j}$ for
$1\leq j\leq i\leq n$.}
\label{sl-2}
\eeqn
\end{thm}
Observing (\ref{j;i=0}), we can rewrite the theorem in the following form:
Let $\io_0$ be one of the reduced longest words of type $A_n$:
\begin{equation}
\io_0=\underbrace{1}_{},
\underbrace{2,1}_{},\underbrace{3,2,1}_{},\cd
\underbrace{n,n-1,\cd,2,1}_{}.
\label{io0}
\end{equation}
\begin{cor}
\label{cor-A_n}
Associated with $\io_0$, we define
\begin{equation}
\bbZ_{\io_0}[\lm]:=\{(x_{j;i}|1\leq i+j\leq n+1)\in
\bbZ^{\frac{(n(n+1))}{2}}|
(x_{j;i})\text{ satisfies (\ref{sl-1}) and (\ref{sl-2}).}\}
\end{equation}
There exists the crystal structure on $\bbZ_{\io_0}[\lm]$ induced
from the one on $\bbZ_\io[\lm]$ and then the crystal $\bbZ_{\io_0}[\lm]$
is isomorphic to $B(\lm)$.
\end{cor}
\renewcommand{\thesection}{\arabic{section}}
\section{Decorated geometric crystals}
\setcounter{equation}{0}
\renewcommand{\theequation}{\thesection.\arabic{equation}}
The basic reference for this section is \cite{BK,BK2}.
\subsection{Definitions
Let
$A=(a_{ij})_{i,j\in I}$ be
an indecomposable Cartan matrix
with a finite index set $I$
(though we can consider more general Kac-Moody setting.).
Let $(\frt,\{\al_i\}_{i\in I},\{h_i\}_{i\in I})$
be the associated
root data
satisfying $\al_j(h_i)=a_{ij}$.
Let $\ge=\ge(A)=\lan \frt,e_i,f_i(i\in I)\ran$
be the simple Lie algebra associated with $A$
over $\bbC$ and $\Del=\Del_+\sqcup\Del_-$
be the root system associated with $\ge$, where $\Del_{\pm}$ is
the set of positive/negative roots.
Define the simple reflections $s_i\in{\rm Aut}(\frt)$ $(i\in I)$ by
$s_i(h)\seteq h-\al_i(h)h_i$, which generate the Weyl group $W$.
Let $G$ be the simply connected simple algebraic group
over $\bbC$ whose Lie algebra is $\ge=\frn_+\oplus \frt\oplus \frn_-$,
which is the usual triangular decomposition.
Let $U_{\al}\seteq\exp\ge_{\al}$ $(\al\in \Del)$
be the one-parameter subgroup of $G$.
The group $G$ (resp. $U^\pm$) is generated by
$\{U_{\al}|\al\in \Del\}$
(resp. $\{U_\al|\al\in\Del_{\pm}$).
Here $U^\pm$ is a unipotent radical of $G$ and
${\rm Lie}(U^\pm)=\frn_{\pm}$.
For any $i\in I$, there exists
a unique group homomorphism
$\phi_i\cl SL_2(\bbC)\rightarrow G$ such that
\[
\phi_i\left(
\left(
\begin{array}{cc}
1&t\\
0&1
\end{array}
\right)\right)=\exp(t e_i),\,\,
\phi_i\left(
\left(
\begin{array}{cc}
1&0\\
t&1
\end{array}
\right)\right)=\exp(t f_i)\qquad(t\in\bbC).
\]
Set $\al^\vee_i(c)\seteq
\phi_i\left(\left(
\begin{smallmatrix}
c&0\\
0&c^{-1}\end{smallmatrix}\right)\right)$,
$x_i(t)\seteq\exp{(t e_i)}$, $y_i(t)\seteq\exp{(t f_i)}$,
$G_i\seteq\phi_i(SL_2(\bbC))$,
$T_i\seteq \alpha_i^\vee(\bbC^\times)$
and
$N_i\seteq N_{G_i}(T_i)$. Let
$T$ be a maximal torus of $G$
which has $P$ as its weight lattice and
Lie$(T)=\frt$.
Let
$B^{\pm}(\supset T)$ be the Borel subgroup of $G$.
We have the isomorphism
$\phi:W\mapright{\sim}N/T$ defined by $\phi(s_i)=N_iT/T$.
An element $\ovl s_i:=x_i(-1)y_i(1)x_i(-1)$ is in
$N_G(T)$, which is a representative of
$s_i\in W=N_G(T)/T$.
\begin{df}
\label{def-gc}
Let $X$ be an affine algebraic variety over $\bbC$,
$\gamma_i$, $\vep_i, f$
$(i\in I)$ rational functions on $X$, and
$e_i:\bbC^\times\times X\to X$ a unital rational $\bbC^\times$-action.
A 5-tuple $\chi=(X,\{e_i\}_{i\in I},\{\gamma_i,\}_{i\in I},
\{\vep_i\}_{i\in I},f)$ is a
$G$ (or $\ge$)-{\it decorated geometric crystal}
if
\begin{enumerate}
\item
$(\{1\}\times X)\cap dom(e_i)$
is open dense in $\{1\}\times X$ for any $i\in I$, where
$dom(e_i)$ is the domain of definition of
$e_i\cl\C^\times\times X\to X$.
\item
The rational functions $\{\gamma_i\}_{i\in I}$ satisfy
$\gamma_j(e^c_i(x))=c^{a_{ij}}\gamma_j(x)$ for any $i,j\in I$.
\item
The function $f$ satisfies
\begin{equation}
f(e_i^c(x))=f(x)+{(c-1)\vp_i(x)}+{(c^{-1}-1)\vep_i(x)},
\label{f}
\end{equation}
for any $i\in I$ and $x\in X$, where $\vp_i:=\vep_i\cdot\gamma_i$.
\item
$e_i$ and $e_j$ satisfy the following relations:
\[
\begin{array}{lll}
&\hspace{-20pt}e^{c_1}_{i}e^{c_2}_{j}
=e^{c_2}_{j}e^{c_1}_{i}&
{\rm if }\,\,a_{ij}=a_{ji}=0,\\
&\hspace{-20pt} e^{c_1}_{i}e^{c_1c_2}_{j}e^{c_2}_{i}
=e^{c_2}_{j}e^{c_1c_2}_{i}e^{c_1}_{j}&
{\rm if }\,\,a_{ij}=a_{ji}=-1,\\
&\hspace{-20pt}
e^{c_1}_{i}e^{c^2_1c_2}_{j}e^{c_1c_2}_{i}e^{c_2}_{j}
=e^{c_2}_{j}e^{c_1c_2}_{i}e^{c^2_1c_2}_{j}e^{c_1}_{i}&
{\rm if }\,\,a_{ij}=-2,\,
a_{ji}=-1,\\
&\hspace{-20pt}
e^{c_1}_{i}e^{c^3_1c_2}_{j}e^{c^2_1c_2}_{i}
e^{c^3_1c^2_2}_{j}e^{c_1c_2}_{i}e^{c_2}_{j}
=e^{c_2}_{j}e^{c_1c_2}_{i}e^{c^3_1c^2_2}_{j}e^{c^2_1c_2}_{i}
e^{c^3_1c_2}_je^{c_1}_i&
{\rm if }\,\,a_{ij}=-3,\,
a_{ji}=-1.
\end{array}
\]
\item
The rational functions $\{\vep_i\}_{i\in I}$ satisfy
$\vep_i(e_i^c(x))=c^{-1}\vep_i(x)$ and
$\vep_i(e_j^c(x))=\vep_i(x)$ if $a_{i,j}=a_{j,i}=0$.
\end{enumerate}
\end{df}
We call the function $f$ in (iii) the {\it decoration} of $\chi$ and
the relations in (iv) are called
{\it Verma relations}.
If $\chi=(X,\{e_i\},\,\{\gamma_i\},\{\vep_i\})$
satisfies the conditions (i), (ii), (iv) and (v),
we call $\chi$ a {\it geometric crystal}.
{\sl Remark.}
The definitions of $\vep_i$ and $\vp_i$ are different from the ones in
e.g., \cite{BK2} since we adopt the definitions following
\cite{KNO,KNO2}. Indeed, if we flip $\vep_i\to \vep^{-1}$ and
$\vp_i\to \vp^{-1}$, they coincide with ours.
\subsection{Characters
Let $\what U:={\rm Hom}(U,\bbC)$ be the set of additive characters
of $U$.
The {\it elementary character }$\chi_i\in \what U$ and
the {\it standard regular character} $\chi^{\rm st}\in \what U$ are
defined by
\[
\chi_i(x_j(c))=\del_{i,j}\cdot c \q(c\in \bbC,\,\, i\in I),\qq
\chi^{st}=\sum_{i\in I}\chi_i.
\]
Let us define an anti-automorphism $\eta:G\to G$
by
\[
\eta(x_i(c))=x_i(c),\q \eta(y_i(c))=y_i(c),\q \eta(t)=t^{-1}\q
(c\in\bbC,\,\, t\in T),
\]
which is called the {\it positive inverse}.
The rational function $f_B$ on $G$ is defined by
\begin{equation}
f_B(g)=\chi^{st}(\pi^+(w_0^{-1}g))+\chi^{st}(\pi^+(w_0^{-1}\eta(g))),
\label{f_B}
\end{equation}
for $g\in B\ovl w_0 B$, where $\pi^+:B^-U\to U$ is the projection by
$\pi^+(bu)=u$.
For a split algebraic torus $T$ over $\bbC$, let us denote
its lattice of (multiplicative )characters(resp. co-characters) by $X^*(T)$
(resp. $X_*(T)$). By the usual way, we identify $X^*(T)$
(resp. $X_*(T)$) with the weight lattice $P$ (resp. the dual weight
lattice $P^*$).
\subsection{Positive structure and ultra-discretization
\label{subsec-posi}
In this subsection, we review the notion positive structure and
the ultra-discretization, which is called the
tropicalization in \cite{BK,BK2}.
\begin{df}
Let $T,T'$ be split algebraic tori over $\bbC$.
\begin{enumerate}
\item
A regular function $f=\sum_{\mu\in X^*(T)}c_\mu\cdot\mu$ on $T$
is {\it positive} if all coefficients $c_\mu$ are non-negative
numbers. A rational function on $T$ is said to be {\it positive} if
there exist positive regular functions $g,h$ such that
$f=\frac{g}{h}$ ($h\ne0$).
\item
Let $f:T\to T'$ be a rational map between
$T$ and $T'$. Then we say that $f$ is {\it positive} if
for any $\xi\in X^*(T')$ we have that $\xi\circ f$ is positive in the
above sense.
\end{enumerate}
\end{df}
Note that if $f,g$ are positive rational functions on $T$, then
$f\cdot g$, $f/g$ and $f+g$ are all positive.
\begin{df}
Let $\chi=(X,\{e_i\}_{i\in I},\{{\rm wt}_i\}_{i\in I},
\{\vep_i\}_{i\in I},f)$ be a decorated
geometric crystal, $T'$ an algebraic torus
and $\theta:T'\rightarrow X$
a birational map.
The birational map $\theta$ is called
{\it positive structure} on
$\chi$ if it satisfies:
\begin{enumerate}
\item For any $i\in I$ the rational functions
$\gamma_i\circ \theta, \vep_i\circ \theta, f\circ\theta:T'\rightarrow \bbC$
are all positive in the above sense.
\item
For any $i\in I$, the rational map
$e_{i,\theta}:\bbC^\tm \tm T'\rightarrow T'$ defined by
$e_{i,\theta}(c,t)
:=\theta^{-1}\circ e_i^c\circ \theta(t)$
is positive.
\end{enumerate}
\end{df}
Let $v:\bbC(c)\setminus\to\bbZ$ be a map defined by
$v(f(c)):=\deg(f(c^{-1}))$,which is different from that in e.g.,
\cite{KNO,KNO2,N,N3}. Note that this definition of the map $UD$ is
called tropicalization in \cite{BK} and much simpler than the one
in \cite{BK2} since it is sufficient in this article.
Here, we have the formula for positive rational functions $f$ and $g$:
\begin{equation}
v(f\cdot g)=v(f)+v(g),\q
v(f/g)=v(f)-v(g),\q
v(f+g)=\min(v(f),v(g)).
\label{uf-formula}
\end{equation}
Let $f\cl T\rightarrow T'$ be a
positive rational mapping
of algebraic tori $T$ and
$T'$.
We define a map $\what f\cl X_*(T)\rightarrow X_*(T')$ by
\[
\langle\chi,\what f(\xi)\rangle
=v(\chi\circ f\circ \xi),
\]
where $\chi\in X^*(T')$ and $\xi\in X_*(T)$.
Let $\cT_+$ be the category whose objects are algebraic tori over
$\bbC$ and whose morphisms are positive rational maps.
Then, we obtain the functor
\[
\begin{array}{cccc}
{\mathcal UD}:& \cT_+&\longrightarrow &{\mathfrak Set}\\
&T& \mapsto &X_*(T)\\
&(f:T\to T') &\mapsto &(\what f:X_*(T)\to X_*(T')).
\end{array}
\]
Let $\theta:T\rightarrow X$ be a positive structure on
a decorated geometric crystal $\chi=(X,\{e_i\}_{i\in I},
\{{\rm wt}_i\}_{i\in I},
\{\vep_i\}_{i\in I,f})$.
Applying the functor ${\mathcal UD}$
to positive rational morphisms
$e_{i,\theta}:\bbC^\tm \tm T'\rightarrow T'$ and
$\gamma\circ \theta:T'\ra T$
(the notations are
as above), we obtain
\begin{eqnarray*}
\til e_i&:=&{\mathcal UD}(e_{i,\theta}):
\ZZ\tm X_*(T) \rightarrow X_*(T)\\
{\rm wt}_i&:=&{\mathcal UD}(\gamma_i\circ\theta):
X_*(T')\rightarrow \bbZ,\\
\wtil\vep_i&:=&{\mathcal UD}(\vep_i\circ\theta):
X_*(T')\rightarrow \bbZ,\\
\wtil f&:=& {\mathcal UD}(f \circ\theta):
\end{eqnarray*}
Now, for given positive structure $\theta:T'\rightarrow X$
on a geometric crystal
$\chi=(X,\{e_i\}_{i\in I},\{{\rm wt}_i\}_{i\in I},
\{\vep_i\}_{i\in I})$, we associate
the quadruple $(X_*(T'),\{\til e_i\}_{i\in I},
\{{\rm wt}_i\}_{i\in I},\{\wtil\vep_i\}_{i\in I})$
with a free pre-crystal structure (see \cite[2.2]{BK})
and denote it by ${\mathcal UD}_{\theta,T'}(\chi)$.
We have the following theorem:
\begin{thm}[\cite{BK,BK2,N}]
For any geometric crystal
$\chi=(X,\{e_i\}_{i\in I},\{\gamma_i\}_{i\in I},
\{\vep_i\}_{i\in I})$ and positive structure
$\theta:T'\rightarrow X$, the associated pre-crystal
${\mathcal UD}_{\theta,T'}(\chi)=
(X_*(T'),\{e_i\}_{i\in I},\{{\rm wt}_i\}_{i\in I},
\{\wtil\vep_i\}_{i\in I})$
is a Kashiwara's crystal.
\end{thm}
{\sl Remark.}
The definition of $\wtil\vep_i$ is different from the one in
\cite[6.1.]{BK2} since our definition of $\vep_i$ corresponds to
$\vep_i^{-1}$ in \cite{BK2}.
Now, for a positive decorated geometric crystal
$\cX=((X,\{e_i\}_{i\in I},\{\gamma_i\}_{i\in I},
\{\vep_i\}_{i\in I},f),\theta,T')$, set
\begin{equation}
\wtil B_{\wtil f}:=\{\wtil x\in X_*(T')|\wtil f(\wtil x)\geq0\},
\label{btil}
\end{equation}
where $X_*(T')$ is identified with $\bbZ^{\dim(T')}$. Define
\begin{equation}
B_{f,\theta}:=(\wtil B_{\wtil f},\wt_i|_{\wtil B_{\wtil f}},
\vep_i|_{\wtil B_{\wtil f}},e_i|_{\wtil B_{\wtil f}})_{i\in I}.
\label{btheta}
\end{equation}
\begin{pro}[\cite{BK2}]
For a positive decorated geometric crystal
$\cX=((X,\{e_i\}_{i\in I},\{\gamma_i\}_{i\in I},
\{\vep_i\}_{i\in I},f),\theta,T')$, the
quadruple $B_{f,\theta}$ in (\ref{btheta}) is a normal crystal.
\end{pro}
\subsection{Decorated geometric crystal on $\bbB_w$
For a Weyl group element $w\in W$, define $B^-_w$ by
\begin{equation}
B^-_w:=B^-\cap U\ovl w U.
\label{B-w}
\end{equation}
Now, set $\bbB_w:=TB^-_w$.
Let $\gamma_i:\bbB_w\to\bbC$ be the rational function defined by
\begin{equation}
\gamma_i:\bbB_w\,\,\hookrightarrow \,\,\,
B^-\,\,\mapright{\sim}\,\,T\times U^-\,\,
\mapright{\rm proj}\,\,\, T\,\,\,\mapright{\al_i^\vee}\,\,\,\bbC.
\label{gammai}
\end{equation}
For any $i\in I$, there exists the natural projection
$pr_i:B^-\to B^-\cap \phi(SL_2)$. Hence,
for any $x\in \bbB_w$ there exists unique
$v=\begin{pmatrix}b_{11}&0\\b_{21}&b_{22}\end{pmatrix}
\in SL_2$ such that
$pr_i(x)=\phi_i(v)$. Using this fact, we define
the rational function $\vep_i$ on $\bbB_w$ by
\begin{equation}
\vep_i(x)=\frac{b_{22}}{b_{21}}\q(x\in\bbB_w).
\label{vepi}
\end{equation}
The rational $\bbC^\times$-action $e_i$ on $\bbB_w$ is defined by
\begin{equation}
e_i^c(x):=x_i\left((c-1)\vp_i(x)\right)\cdot x\cdot
x_i\left((c^{-1}-1)\vep_i(x)\right)\qq
(c\in\bbC^\times,\,\,x\in \bbB_w),
\label{ei-action}
\end{equation}
if $\vep_i(x)$ is well-defined, that is, $b_{21}\ne0$,
and $e_i^c(x)=x$ if $b_{21}=0$.\\
{\sl Remark.} The definition (\ref{vepi}) is different from the one in
\cite{BK2}. Indeed, if we take $\vep_i(x)=b_{21}/b_{22}$,
then it coincides with
the one in \cite{BK2}.
\begin{pro}[\cite{BK2}]
For any $w\in W$,
the 5-tuple $\chi:=(\bbB_w,\{e_i\}_i,\{\gamma_i\}_i,\{\vep_i\}_i,f_B)$
is a decorated geometric crystal, where
$f_B$ is in (\ref{f_B}), $\gamma_i$ is in (\ref{gammai}), $\vep_i$ is in
(\ref{vepi}) and $e_i$ is in (\ref{ei-action}).
\end{pro}
\def\ld{\ldots}
For the longest Weyl group element $w_0\in W$, let
$\bfii0=i_1\ld i_N$ be one of its reduced expressions and
define the positive structure on $B^-_{w_0}$
$\Theta^-_\bfii0:(\bbC^\times)^N\longrightarrow B^-_{w_0}$ by
\[
\Theta^-_\bfii0(c_1,\cd,c_N):=\pmby_{i_1}(c_1)\cd \fry_{i_N}(c_N),
\]
where $\pmby_i(c)=y_i(c)\al^\vee(c^{-1})$,
which is different from $Y_i(c)$ in
\cite{N,N2,KNO,KNO2}. Indeed, $Y_i(c)=\pmby_i(c^{-1})$.
We also define the positive structure on $\bbB_{w_0}$ as
$T\Theta^-_\bfii0:T\times(\bbC^\times)^N\,\,\longrightarrow\,\,\bbB_{w_0}$
by $T\Theta^-_\bfii0(t,c_1,\cd,c_N)
=t\Theta^-_\bfii0(c_1,\cd,c_N)$.
Now, for this positive structure, we describe the geometric crystal
structure on $\bbB_{w_0}=TB^-_{w_0}$ explicitly.
In fact, it is quite similar to that of the Schubert variety associated
with $w_0$ as in \cite{N} and then we obtain the following formula
by the similar method in \cite{N}.
\begin{pro}
The action $e^c_i$ on
$t\Theta^-_{\bfii0}(c_1,\cd,c_N)$ is given by
\[
e_i^c(t\Theta^-_{\bfii0}(c_1,\cd,c_N))
=t\Theta^-_{\bfii0}(c'_1,\cd,c'_N)
\]
where
\begin{equation}
c'_j\seteq
c_j\cdot \frac{\displaystyle \sum_{1\leq m< j,\,i_m=i}
{c\cdot c_1^{a_{i_1,i}}\cd c_{m-1}^{a_{i_{m-1},i}}c_m}
+\sum_{j\leq m\leq N,\,i_m=i}
{c_1^{a_{i_1,i}}\cd c_{m-1}^{a_{i_{m-1},i}}c_m}}
{\displaystyle \sum_{1\leq m\leq j,\,i_m=i}
{c\cdot c_1^{a_{i_1,i}}\cd c_{m-1}^{a_{i_{m-1},i}}c_m}+
\mathop\sum_{j< m\leq N,\,i_m=i}
{c_1^{a_{i_1,i}}\cd c_{m-1}^{a_{i_{m-1},i}}c_m}}.
\label{eici}
\end{equation}
The explicit forms of
rational functions $\vep_i$ and $\gamma_i$ are:
\begin{equation}
\vep_i(t\Theta^-_{\bfii0}(c))=
\left(\sum_{1\leq m\leq N,\,i_m=i} \frac{1}
{c_mc_{m+1}^{a_{i_{m+1},i}}\cd c_{N}^{a_{i_{N},i}}}\right)^{-1},\q
\gamma_i(t\Theta^-_{\bfii0}(c))
=\frac{\al_i(t)}{c_1^{a_{i_1,i}}\cd c_N^{a_{i_N,i}}}.
\label{th-vep-gamma}
\end{equation}
\end{pro}
{\sl Proof.}
If we rewrite $t\Theta^-_\bfii0(c)$ in the form,
\[
t\cdot \al^\vee_{i_1}(c_1^{-1}\cd \al^\vee_{i_N}(c_N^{-1})
y_{i_1}(d_1)\cd y_{i_N}(d_N),
\]
then we easily get
$d_m=c_mc_{m+1}^{a_{i_{m+1},i}}\cd c_{N}^{a_{i_{N},i}}$ for $m=1,\cd, N$.
Thus, we obtain the explicit form of $\vep_i$ as above.
To find the explicit form of the action $e_i^c$, the following formula
is crucial:
\[
x_i(a)y_j(b)=\begin{cases}
y_i(\frac{b}{1+ab})\al^\vee_i(1+ab)x_i(\frac{a}{1+ab})&\text{ if }i=j,\\
y_j(b)x_i(a)&\text{ if }i\ne j.
\end{cases}
\]
Applying this formula to (\ref{ei-action}) repeatedly,
we have the above explicit
action of $e_i^c$.\qed
\subsection{Ultra-Discretization of $\bbB_w=TB^-_w$
Applying the ultra-discretization functor to
$\bbB_w$,
we obtain the free crystal ${\mathcal UD}(\bbB_{w_0})=X_*(T)\times \bbZ^{N}$,
where $N$ is the length of the longest element $w_0$.
Then define the map $\til h:{\mathcal UD}(\bbB_{w_0})
=X_*(T)\times \bbZ^{N}\,\,\to\,\,
X_*(T)(=P^*)$ as the projection to the left component and set
\[
B_{w_0}(\lm^\vee):=\til h^{-1}(\lm^\vee),\qq
B_{f_B,\Theta^-_{\bfii0}}(\lm^\vee):=B_{w_0}(\lm^\vee)\cap
B_{{f_B},\Theta^-_{\bfii0}},
\]
for $\lm^\vee\in X_*(T)=P^*$. Set $P^*_+:=\{h\in P^*|\Lm_i(h)\geq0
\text{ for any }i\in I\}$ and for $\lm^\vee=\sum_i\lm_ih_i$, we define
$\lm=\sum_i\lm_i\Lm_i\in P_+$. Then, we have
\begin{thm}[\cite{BK2}]
The set $B_{f_B,\Theta^-_{\bfii0}}(\lm^\vee)$ is non-empty if
$\lm^\vee\in P^*_+$
and in that case, $B_{f_B,\Theta^-_{\bfii0}}(\lm^\vee)$ is isomorphic to
$B(\lm)^L$, which is the Langlands dual crystal associated with
$\ge^L$.
\end{thm}
It follows from
(\ref{eici}) and (\ref{th-vep-gamma}) that we have
\begin{thm}
\label{ud-tb}
Let $\lm^\vee\in P^*_+$.
The explicit crystal structure of $B_{f_B,\Theta^-_\bfii0}(\lm^\vee)$
is as follows:
For $x=(x_1,\cd,x_N)\in B_{f_B,\Theta^-_\bfii0}(\lm^\vee)\subset \bbZ^N$, we
have
\begin{equation}
\eit^n(x)=(x'_1,\cd,x'_N),
\end{equation}
where
\begin{eqnarray}
&& x'_j=x_j+\min\left(\min_{1\leq m<j,i_m=i}(n+\sum_{k=1 }^m
a_{i_k,i}x_k),
\min_{j\leq m\leq N,i_m=i}(\sum_{k=1 }^m a_{i_k,i}x_k)\right)
\label{eitn} \\
&&\qq -\min\left(\min_{1\leq m\leq j,i_m=i}(n+\sum_{k=1 }^m
a_{i_k,i}x_k),
\min_{j<m\leq N,i_m=i}(\sum_{k=1 }^m a_{i_k,i}x_k)\right),
\nn
\\
&&\wt_i(x)=\lm(h_i)-\sum_{k=1}^Na_{i_k,i}x_k,
\label{wt-ud}\\
&&\vep_i(x)=\max_{1\leq m\leq
N,i_m=i}(x_m+\sum_{k=m+1}^{N}a_{i_k,i}x_k),
\label{vep-ud}
\end{eqnarray}
and $x=(x_1,\cd,x_N)$ belongs to $B_{f_B,\Theta^-_\bfii0}(\lm^\vee)$ if and
only if ${\mathcal UD}(f_B)(x)\geq0$.
\end{thm}
It follows immediately from (\ref{eitn}):
\begin{lem}
\label{lem-ef}
Set $X_m:=\sum_{k=1 }^m a_{i_k,i}x_k$, ${\mathcal X}^{(i)}:=\min\{X_m|1\leq
m\leq N,i_m=i\}$ $(i\in I)$ and $M^{(i)}:=\{l|1\leq l\leq N,i_l=i,
X_l={\mathcal X}^{(i)}\}$.
Define $m_e:=\max(M^{(i)})$ and $m_f:=\min(M^{(i)})$:
for $x\in B_{f_B,\Theta^-_\bfii0}(\lm^\vee)$, we get
\begin{eqnarray}
&&\eit(x)=\begin{cases}(x_1,\cd,x_{m_e}-1,\cd,x_N)
&\text{ if }{\mathcal UD}(f_B)(x_1,\cd,x_{m_e}-1,\cd,x_N)\geq0,\\
0&\text{otherwise,}
\end{cases}
\label{th-eaction}\\
&&\fit(x)=\begin{cases}(x_1,\cd,x_{m_f}+1,\cd,x_N)
&\text{ if }{\mathcal UD}(f_B)(x_1,\cd,x_{m_f}+1,\cd,x_N)\geq0,\\
0&\text{otherwise.}
\end{cases}
\label{th-faction}
\end{eqnarray}
\end{lem}
Finally, due to the results in Sect.2 and in this section,
we obtain the following theorem
\begin{thm}
\label{coin-thm}
If we have $B_{f_B,\Theta^-_\bfii0}(\lm^\vee)
=\Sigma_{\bfii0^{-1}}[\lm]^L$ as a set.
Then they are isomorphic each other as crystals, where ${}^L$ means the
Langlands dual crystal, that is, it is defined by the transposed Cartan
matrix and $\bfii0^{-1}$ means the opposite order of $\bfii0$.
\end{thm}
{\sl Proof.}
The coincidence of the actions $\eit$ and $\fit$ are shown by comparing
(\ref{action-f}) and (\ref{action-e}) with (\ref{th-eaction}) and
(\ref{th-faction}) since the following are equivalent:\\
(a) $X_k$ is the minimum.\\
(b) $\sigma^{(i)}(x)=\sigma^{(0)}_i(x)+\lm_i-X_k$ is the maximum.\\
Similarly, comparing (\ref{wt-vep-vp-1}) with (\ref{wt-ud}) and
(\ref{wt-vep-vp4}) with (\ref{vep-ud}) respectively,
we obtain the coincidence
of $\vep_i$ and $wt_i$. \qed
\renewcommand{\thesection}{\arabic{section}}
\section{Explicit form of the decoration $f_B$ of type $A_n$}
\setcounter{equation}{0}
\renewcommand{\theequation}{\thesection.\arabic{equation}}
\subsection{Generalized Minors and the function $f_B$
For this subsection, see \cite{BFZ,BZ,BZ2}.
Let $G$ be a simply connected simple algebraic groups over $\bbC$ and
$T\subset G$ a maximal torus.
Let $X^*(T):=\Hom(T,\bbC^\times)$ and $X_*(T):=\Hom(\bbC^\times,T)$ be
the lattice of characters and co-characters respectively.
We identify $P$ (resp. $P^*$) with $X^*(T)$
(resp. $X_*(T)$) as above.
\begin{df}
For $\mu\in P_+$, the
{\it principal minor} $\Del_\mu:G\to\bbC$ is defined as
\[
\Del_\mu(u^-tu^+):=\mu(t)\q(u^\pm\in U^\pm,\,\,t\in T).
\]
Let $\gamma,\del\in P$ be extremal weights such that
$\gamma=u\mu$ and $\del=v\mu$ for some $u,v\in W$.
Then the {\it generalized minor} $\Del_{\gamma,\del}$ is defined
by
\[
\Del_{\gamma,\del}(g):=\Del_\mu(\ovl u^{-1}g\ovl v)
\q(g\in G),
\]
which is a regular function on $G$.
\end{df}
\begin{lem}[\cite{BK2}]
Suppose that $G$ is simply connected.
\begin{enumerate}
\item
For $u\in U$ and $i\in I$, we have
$\Del_{\mu,\mu}(u)=1$ and $\chi_i(u)=\Del_{\Lm_i,s_i\Lm_i}(u)$,
where $\Lm_i$ be the $i$th fundamental weight.
\item
Define the map $\pi^+:B^-\cdot U\to U$ by $\pi^+(bu)=u$ for
$b\in B^-$ and $u\in U$. For any $g\in G$, we have
\begin{equation}
\chi_i(\pi^+(g))=\frac{\Del_{\Lm_i,s_i\Lm_i}(g)}
{\Del_{\Lm_i,\Lm_i}(g)}.
\end{equation}
\end{enumerate}
\end{lem}
\begin{pro}[\cite{BK2}]
The function $f_B$ in (\ref{f_B}) is described as follows:
\begin{equation}
f_B(g)=\sum_i\frac{\Del_{w_0\Lm_i,s_i\Lm_i}(g)
+\Del_{w_0s_i\Lm_i,\Lm_i}(g)}
{\Del_{w_0\Lm_i,\Lm_i}(g)}
\end{equation}
\end{pro}
Let ${\bf i}=i_1\cd i_N$ be a reduced word for the longest Weyl
group element $w_0$.
For $t\Theta_{\bf i}^-(c)\in \bbB_{w_0}=T\cdot B^-_{w_0}$,
we get the following formula.
\begin{equation}
f_B(t\Theta_{\bf i}^-(c))
=\sum_i\Del_{w_0\Lm_i,s_i\Lm_i}(\Theta_{\bf i}^-(c))
+\al_i(t)\Del_{w_0s_i\Lm_i,\Lm_i}(\Theta_{\bf i}^-(c)).
\label{fb-th}
\end{equation}
\subsection{Bilinear Forms
Let $\omega:\ge\to\ge$ be the anti involution
\[
\omega(e_i)=f_i,\q
\omega(f_i)=e_i\,\q\omega(h)=h,
\] and extend it to $G$ by setting
$\omega(x_i(c))=y_i(c)$, $\omega(y_i(c))=x_i(c)$ and $\omega(t)=t$
$(t\in T)$.
There exists a $\ge$(or $G$)-invariant bilinear form on the
finite-dimensional irreducible
$\ge$-module $V(\lm)$ such that
\[
\lan au,v\ran=\lan u,\omega(a)v\ran,
\q\q(u,v\in V(\lm),\,\, a\in \ge(\text{or }G)).
\]
For $g\in G$,
we have the following simple fact:
\[
\Del_{\Lm_i}(g)=\lan gu_{\Lm_i},u_{\Lm_i}\ran.
\]
Hence, for $w,w'\in W$ we have
\begin{equation}
\Del_{w\Lm_i,w'\Lm_i}(g)=
\Del_{\Lm_i}({\ovl w}^{-1}g\ovl w')=
\lan {\ovl w}^{-1}g\ovl w'\cdot u_{\Lm_i},u_{\Lm_i}\ran
=\lan g\ovl w'\cdot u_{\Lm_i}\, ,\, \ovl{w}\cdot u_{\Lm_i}\ran,
\label{minor-bilin}
\end{equation}
where $u_{\Lm_i}$ is a properly normalized highest weight vector in
$V(\Lm_i)$ and note that $\omega(\ovl s_i^{\pm})=\ovl s_i^{\mp}$.
\subsection{Explicit form of $f_B(t\Theta_{\bf i}^-(c))$ of type $A_n$}
\label{fb-An}
Now, we consider the type $A_n$, that is, $G=SL_{n+1}(\bbC)$.
We fix the reduced longest word ${\bf i_0}=\underbrace{1,2,\cd,n}_{},
\underbrace{1,2,\cd,n-1}_{},\cd,\underbrace{1,2,3}_{},1,2,1$.
This is just the opposite order $\io_0=\bfii0^{-1}$ as in Sect.2.
To obtain the explicit form of $f_B(t\Theta_{\bf i_0}^-(c))$,
by (\ref{fb-th}) it suffices to know
$\Del_{w_0\Lm_j,s_j\Lm_j}(\Theta_{\bf j_0}^-(c))$ and
$\Del_{w_0s_j\Lm_j,\Lm_j}(\Theta_{\bf j_0}^-(c))$ for
\[
c=(c_{i,j}|i+j\leq n+1)=
(c_{1,1},c_{1,2},\cd,c_{1,n},c_{2,1},c_{2,2},\cd,c_{2,n-1},\cd
c_{n-1,1},c_{n-1,2},c_{n,1})\in (\bbC^\times)^N.
\]
\begin{thm}
\label{thm-a}
For $c\in (\bbC^\times)^N$ as above,
we have the following explicit forms:
\begin{eqnarray}
&&\Del_{w_0\Lm_j,s_j\Lm_j}(\Theta_\bfii0^-(c))=
c_{n-j+1,1}+\frac{c_{n-j+1,2}}{c_{n-j+2,1}}+\frac{c_{n-j+1,3}}{c_{n-j+2,2}}
+\cd+\frac{c_{n-j+1,j}}{c_{n-j+2,j-1}},
\label{del1}\\
&&\Del_{w_0s_j\Lm_j,\Lm_j}(\Theta_\bfii0^-(c))=
\frac{1}{c_{j,1}}+
\frac{c_{j-1,1}}{c_{j-1,2}}+\frac{c_{j-2,2}}{c_{j-2,3}}
+\cd+\frac{c_{1,j-1}}{c_{1,j}},\,\,
(j\in I).
\label{del2}
\end{eqnarray}
\end{thm}
The proof of this theorem will be given in the next subsection.
\subsection{Proof of Theorem \ref{thm-a}
Let $V_1:=V(\Lm_1)$ be the vector representation of
$\ssl_{n+1}(\bbC)$ with
the standard basis $\{v_1,\cd,v_{n+1}\}$, and
$\{e_i,f_i,h_i\}_{i=1,\cd,n}$ the
Chevalley generators of $\ssl_{n+1}(\bbC)$.
Their actions on the basis vectors are as follows:
\begin{equation}
e_iv_j=\begin{cases}v_{i}&\text{ if }j=i+1,\\
0&\text{otherwise},
\end{cases}
\q
f_iv_j=\begin{cases}v_{i+1}&\text{ if }j=i,\\
0&\text{otherwise},
\end{cases}
\q
h_iv_j=\begin{cases}v_{i}&\text{ if }j=i,\\
-v_{i-1}&\text{ if }j=i-1\text { and }i\ne1,\\
0&\text{otherwise},
\end{cases}
\end{equation}
By these explicit actions
we know that $e_i^2=f_i^2=0$ on $V_1$.
Thus, we can write
${\pmb x}_i(c):=\al_i^\vee(c^{-1})x_i(c)=c^{-h_i}(1+c\cdot e_i)$ and
${\pmb y}_i(c):=y_i(c)\al_i^\vee(c^{-1})=(1+c\cdot f_i)c^{-h_i}$ on $V_1$
and then
\begin{equation}
\pmbx_i(c)v_j=\begin{cases}cv_{i+1}+v_i&\text{ if }j=i+1,\\
cv_i&\text{ if }j=i,\\
v_j&\text{ otherwise, }
\end{cases}\q
\pmby_i(c)v_j=\begin{cases}c^{-1}v_i+v_{i+1}&\text{ if }j=i,\\
cv_i&\text{ if }j=i-1,\\
v_j&\text{ otherwise. }
\end{cases}
\end{equation}
For $c=(c_1,\cd,c_i)\in (\bbC^\times)^i$ set
$X^{(i)}(c):=\pmbx_i(c_i)\cd \pmbx_1(c_1)$ and its action on the basis vector is
as follows:
\begin{equation}
X^{(i)}(c)v_k=\begin{cases}
c_k^{-1}c_{k-1}v_k+v_{k-1}&\text{ if }k<i+1,\\
c_iv_{i+1}+v_i&\text{ if }k=i+1,\\
v_k&\text{ if }k>i+1.
\end{cases}
\end{equation}
Now, for $c=(c_{k,i})_{1\leq i,k\leq n,i+k\leq n+1}
\in(\bbC^\times)^{\frac{n(n+1)}{2}}$ set
$c^{(k)}:=(c_{n+1-k,k},c_{n+1-k,k-1},
\cd,c_{n+1-k,2},c_{n+1-k,1})$ and
\[
X(c):=X^{(1)}(c^{(1)})X^{(2)}(c^{(2)})\cd X^{(n-1)}(c^{(n-1)})X^{(n)}(c^{(n)})
\]
Here, note that
\begin{equation}
\omega(\Theta^-_{{\bf i}_0}(c))=X(c).
\label{omega-th-x}
\end{equation}
Writing
\[
X(c)v_i=\sum_{k=1}^i\xx(i,k)v_k,
\]
we shall get the explicit form of the coefficient $\xx(i,k)$
with the direct calculations:
For $i=1,\cd,n$ and $k=1,\cd,i$, set
\[
{}^i{\bf m}_k:=\{M|M\subset\{1,\cd,n-k+1\},\sharp M=n-i+1\}.
\]
For $M\in {}^i{\bf m}_k$, write $M=M_1\sqcup \cd\sqcup M_{i-k+1}$ where
each $M_j$ ($j=1,\cd,s:=i-k+1$) is a consecutive subsequence of $M$ satisfying
$\min(M_{l})-\max(M_j)=l-j+1$ for any $1\leq j<l\leq s$
if both $M_j$ and $M_l$ are non-empty,
which is called a segment of $M$.
For $M=M_1\sqcup\cd M_s\in {}^i{\bf m}_k$, write each segment:
\[
M_1=\{1,2,\cd,j_1-1\},\,\
M_2=\{j_1+1,j_1+2,\cd,j_2-1\},\,\,
M_s=\{j_{i-k}+1,j_{i-k}+2,\cd,n-k+1\},
\]
where $1\leq j_1<j_2<\cd<j_{i-k}\leq n-k+1$ and set
\[
c^M:=\frac{c_{1,i-1}\cd c_{j_1-1, i-1}\cd c_{j_{i-k}+1,k-1}
\cd c_{n-k+1,k-1}}
{c_{1,i}\cd c_{j_1-1,i}\cd c_{j_{i-k}+1,k}
\cd c_{n-k+1,k}}.
\]
\begin{pro}
We have the following explicit form of $\xx(i,k)$.
\begin{equation}
\xx(i,k)=\sum_{M\in{}^i{\bf m}_k}c^M.
\end{equation}
\end{pro}
This formula is obtained by direct calculations.
For the module $V(\Lm_j)$ ($j>1$), let us denote
its normalized highest (resp. lowest)weight vector by $u_{\Lm_j}$
(resp. $v_{\Lm_j}$).
Set
\begin{eqnarray*}
&&[i_1,\cd,i_j]:=v_{i_1}\wedge v_{i_2}\wedge\cd\wedge v_{i_j}
\in \bigwedge^jV_1 \\
&&I_j:=\{[i_1,i_2,\cd,i_j]\,|\,1\leq i_1<i_2<\cd<i_j\leq n+1\}.
\end{eqnarray*}
$I_j$ is a normal basis of $V(\Lm_j)$ with the weight
$\sum_{k=1}^j(\Lm_{i_k}-\Lm_{i_k-1})$.
Indeed, $u_{\Lm_j}=v_1\wedge v_2\wedge \cd\wedge v_j$ and
$v_{\Lm_j}=v_{n-k}\wedge v_{n-k+1}\wedge \cd\wedge v_{n+1}$.
The actions of $e_i$ and $f_i$ on the vector $[i_1,\cd,i_j]$ are given by
\begin{eqnarray}
&&e_i[i_1,\cd,i_j]=\begin{cases}
[i_1,\cd,i_{k-1},i,i_{k+1},\cd,i_j]&\text{ if }i_k=i+1, \,\,
i_{k-1}<i \text{ for some }k,\\
0&\text{otherwise},
\end{cases}\\
&&f_i[i_1,\cd,i_j]=\begin{cases}
[i_1,\cd,i_{k-1},i+1,i_{k+1},\cd,i_j]&\text{ if }i_k=i, \,\,
i_{k+1}>i+1 \text{ for some }k,\\
0&\text{otherwise}.
\end{cases}
\end{eqnarray}
It follows from the formula (\ref{minor-bilin}) and (\ref{omega-th-x}) that
\begin{equation}
\Del_{w_0\Lm_j,s_i\Lm_j}(\Theta_{\bf i_0}^-(c))
=\lan \Theta_{\bf i_0}^-(c)\ovl s_j\cdot u_{\Lm_j}, \ovl w_0\cdot
u_{\Lm_j}\ran
=\lan\ovl s_j\cdot u_{\Lm_j}, X(c)\ovl w_0\cdot u_{\Lm_j}\ran.
\end{equation}
Since $\ovl s_j\cdot u_{\Lm_j}=[1,2,\cd,j-1,j+1]$ and
$\ovl w_0\cdot u_{\Lm_j}=v_{\Lm_j}$, to obtain
$\Del_{w_0\Lm_j,s_i\Lm_j}(\Theta_{\bf i_0}^-(c))$ it suffices to find
the coefficient of $[1,2,\cd,j-1,j+1]$ in $X(c)v_{\Lm_j}$.
\begin{lem}
We have
\begin{equation}
X^{(j+1)}(c^{(j+1)})\cd X^{(n)}(c^{(n)})v_{\Lm_j}
=[2,3,\cd,j,j+1]+\sum_{1\leq i_1<\cd<i_j>j+1}c_{i_1,\cd,i_j}[i_1,\cd,i_j],
\end{equation}
where $c_{i_1,\cd,i_j}\in\bbC$ is the coefficient.
\end{lem}
{\sl Proof.}
First, let us see
$W_n:=X^{(n)}(c^{(n)})v_{\Lm_j}$. It is easily to see that
\begin{eqnarray}
&&W_n=\pmbx_n(c_{1,n})\cd \pmbx_1(c_{1,1})v_{\Lm_j}
\label{Xn} \\
\qq\qq &&=
[n+1-j,n+2-j,\cd,n-1,n]+
\sum_{1\leq i_1<\cd<i_{j-1}<n+1}c_{i_1,\cd,i_{j-1},n+1}[i_1,\cd,i_{j-1},n+1],
\nn
\end{eqnarray}
since the term
$c_{1,n}^{-h_n}c_{1,n}e_n\cdot
c_{1,n-1}^{-h_{n-1}}c_{1,n-1}e_{n-1}
\cd c_{1,n+1-j}^{-h_{n+1-j}}c_{1,n+1-j}e_{n+1-j}\cdot
c_{1,n-j}^{-h_{n-j}}\cd
c_{1,1}^{-h_1}$ in $X^{(n)}(c^{(n)})$
gives the leading term $[n+1-j,n+2-j,\cd,n-1,n]$ in (\ref{Xn}).
Indeed, the basis vectors appearing in
$X^{(n-1)}(c^{(n-1)})[i_1,\cd,n+1]$
are in the form $[\cd\cd,n+1]$, which means that
we may see only the vector
$X^{(n-1)}(c^{(n-1)})[n+1-j,n+2-j,,\cd,n-1,n]$ in
$X^{(n-1)}(c^{(n-1)})X^{(n)}(c^{(n)})v_{\Lm_j}$.
By considering similarly, we obtain
\[
X^{(n-1)}(c^{(n-1)})X^{(n)}(c^{(n)})v_{\Lm_j}
=[n-j,n+1-j,\cd,n-2,n-1]+
\sum_{1\leq i_1<\cd<i_{j}\geq n}c_{i_1,\cd,i_j}[i_1,\cd,i_j].
\]
Repeating this process, we get the desired result.\qed
We shall see the action of $\ovl X_j:=
X^{(1)}(c^{(1)})\cd X^{(j)}(c^{(j)})$
on the vector $[2,3,\cd,j,j+1]$.
The following lemma is shown easily.
\begin{lem}
In the expansion of
\[
\ovl X_j=c_{n,1}^{-h_n}(1+c_{n,1}e_1)\cd c_{j,1}^{-h_1}
(1+c_{j,1}e_1),
\]
the only terms $E_m:=A\cdot B_m\cdot C_m\cdot D_m$ ($m=0,\cd,j-1$)
produce the vector $[1,2,\cd,j-1,j+1]$ in
$X^{(1)}(c^{(1)})\cd X^{(j)}(c^{(j)})[2,3,\cd,j,j+1]$, where
\begin{eqnarray*}
&&A:=\ch(1,n)\ch(2,n-1)\ch(1,n-1)\cd
\ch(j-2,n-j+3)\ch(j-3,n-j+3)\cd \ch(1,n-j+3),\\
&&B_m:=\ch(j-1,n-j+2)\cc(j-1,n-j+2)e_{j-1}\cd
\ch(m+2,n-j+2)\cc(m+2,n-j+2)e_{m+2}\cdot
\ch(m+1,n-j+2)\cc(m+1,n-j+2)e_{m+1},\\
&&C_m:=\ch(m,n-j+2)\cd\ch(2,n-j+2)\ch(1,n-j+2)\ch(j,n-j+1)\ch(j-1,n-j+1)
\cd\ch(m+1,n-j+1),\\
&&D_m:=\ch(m,n-j+1)\cc(m,n-j+1)e_{m}\cdots
\ch(2,n-j+1)\cc(2,n-j+1)e_{2}\cdot
\ch(1,n-j+1)\cc(1,n-j+1)e_{1},
\end{eqnarray*}
where we understand $D_0=1$.
\end{lem}
For $m=1,2,\cd,j-1$ it is trivial that
\[
C_m\cdot D_m[2,3,\cd,j,j+1]=
\frac{\cc(m+1,n-j+1)}{\cc(m,n-j+2)}
[1,2,\cd,m,m+2,m+3,\cd,j,j+1].
\]
And then
\[
A\cdot
B_m\cdot C_m\cdot D_m[2,3,\cd,j,j+1]=
\frac{\cc(m+1,n-j+1)}{\cc(m,n-j+2)}
[1,2,3,\cd,j-1,j+1].
\]
For $m=0$, we have
\[
A\cdot B_0\cdot C_0\cdot D_0[2,3,\cd,j,j+1]=\cc(1,n-j+1)[1,2,\cd,j-1,j+1].
\]
Finally, we obtain that
the coefficient of $[1,2,\cd,j-1,j+1]$ in $X(c)v_{\Lm_j}$ is
\begin{equation}
\cc(1,n-j+1)+\sum_{m=1}^{j-1}\frac{\cc(m+1,n-j+1)}{\cc(m,n-j+2)},
\end{equation}
which is just $\Del_{w_0\Lm_j,s_i\Lm_j}(\Theta_{\bf i_0}^-(c))$
and then we have shown (\ref{del1}) in Theorem \ref{thm-a}.
The formula (\ref{del2}) would be shown by the similar way to
(\ref{del1}). \qed
Note that for $j=1$ and $k=2$, we find
$\xx(1,2)=\Del_{w_0\Lm_1,s_1\Lm_1}(\Theta^-_{{\bf i}_0}(c))$.
\renewcommand{\thesection}{\arabic{section}}
\section{Ultra-Discretization and Polyhedral Realizations of type $A_n$}
\setcounter{equation}{0}
\renewcommand{\theequation}{\thesection.\arabic{equation}}
In this section, we shall only treat the type $A_n$. Then we identify
$P$ with $P^*$ by $\lm\leftrightarrow \lm^\vee$.
Let us describe the explicit form of $B_{f_B,\Theta^-_{\bfii0}}(\lm)$ for
type $A_n$ applying the result in Theorem \ref{ud-tb} and
show the coincidence of the crystals $B_{f_B,\Theta^-_{\bfii0}}(\lm)$
and
$\Sigma_{\io_0}[\lm]$ in Sect.2 using Theorem \ref{coin-thm}.
For $\ge=\ssl_{n+1}(\bbC)$,
let $\bfii0$ be as in \ref{fb-An}. Then
we have the following:
\begin{lem}
\label{lem-coin-An}
The crystal $B_{f_B,\Theta^-_{\bfii0}}(\lm)$ is defined
by
\begin{equation}
B_{f_B,\Theta^-_{\bfii0}}(\lm):=
\left\{(x_{k,l}|k+l\leq n+1)\in \bbZ^{N}\,\,
\begin{array}{|l}
\hbox{
$x_{1,i} \geq x_{2,i-1} \geq \cd \geq x_{i,1} \geq 0$
for $1 \leq i \leq n$}
\\
\hbox{
$\lm_i\geq x_{j,i-j+1}-x_{j,i-j}$ for
$1\leq j\leq i\leq n$.}
\end{array}\right\},
\label{explicit-bfb}
\end{equation}
where $N=\frac{n(n+1)}{2}$.
\end{lem}
{\sl Proof.}
We shall see the explicit form of ${\mathcal UD}(f_B)(x)$. Indeed,
by virtue of (\ref{fb-th}), it is sufficient to know
the forms of $\Del_{w_0\Lm_j,s_j\Lm_j}(\Theta_\bfii0^-(c))$
and $\Del_{w_0s_j\Lm_j,\Lm_j}(\Theta_\bfii0^-(c))$, which are given in
(\ref{del1}) and (\ref{del2}). Thus,
we have
\[
{\mathcal UD}(f_B)(t,x)=\min_{1\leq j\leq n}
({\mathcal UD}(\Del_{w_0\Lm_j,s_j\Lm_j}(\Theta_\bfii0^-))(x),
{\mathcal UD}(\al_j(t))
+{\mathcal UD}(\Del_{w_0s_j\Lm_j,\Lm_j}(\Theta_\bfii0^-))(x))
\]
and
\begin{eqnarray}
&&{\mathcal UD}(\Del_{w_0\Lm_j,s_j\Lm_j}(\Theta_\bfii0^-))(x)
=\min_{k=1,\cd,j}(x_{n-j+1,k}-x_{n-j+2,k-1}),\\
&&{\mathcal UD}(\Del_{w_0s_j\Lm_j,\Lm_j}(\Theta_\bfii0^-))(x))
=\min_{k=1,\cd,j}(x_{j-k+1,k-1}-x_{j-k+1,k}),
\end{eqnarray}
where $x_{j,k}={\mathcal UD}(c_{j,k})$ and
we understand $x_{m,0}=0$.
Hence, if we identify ${\mathcal UD}(\al_j(t))$ with $\lm_j$, then
the condition ${\mathcal UD}(f_B)(\lm,x)\geq0$
in Theorem \ref{ud-tb}
is equivalent to the condition in
(\ref{explicit-bfb}).\qed
\begin{thm}
\label{thm-b}
For any dominant integral weight $\lm$,
there exists the following isomorphism of crystals
$B_{f_B,\Theta^-_{\bfii0}}[\lm]\cong\Sigma_{\io_0}[\lm]$
where $\Sigma_{\io_0}[\lm]$ is as in Corollary \ref{cor-A_n} and
$\io_0=\bfii0^{-1}$.
\end{thm}
{\sl Proof.} By Theorem \ref{coin-thm} it is necessarily for us to show
that $B_{f_B,\Theta^-_{\bfii0}}[\lm]=\Sigma_{\io_0}[\lm]$ as a set,
which is shown by Corollary \ref{cor-A_n} and Lemma \ref{lem-coin-An}.
\qed
\renewcommand{\thesection}{\arabic{section}}
\section{Elementary Characters and Monomial Realization of Crystals}
\setcounter{equation}{0}
\renewcommand{\theequation}{\thesection.\arabic{equation}}
We shall see the elementary characters as in Sect4 from the different point of
view, that is, the monomial realization of crystals.
Let us introduce the monomial realization of crystals (See \cite{K7,Nj}).
For variables $\{Y_{m,i}|i\in I, m\in\bbZ.\}$, define the set of monomials
\[
\cY:=\{Y=\prod_{m\in\bbZ,i\in I}
Y_{m,i}^{l_{m,i}}|l_{m,i}\in \bbZ\setminus\{0\}\text{
except for finitely many }(m,i)\}.
\]
Fix a set of integers $p=(p_{i,j})_{i,j\in I,i\ne j}$ such that
$p_{i,j}+p_{j,i}=1$. For this $p:=(p_{i,j})_{i,j\in I,i\ne j}$ and a
generalized Cartan matrix $(a_{i,j})_{i,j\in I}$, set
\[
A_{m,i}=Y_{m,i}Y_{m+1,i}\prod_{j\ne i}Y_{m+p_{j,i},j}^{a_{j,i}}.
\]
Note that
for any cyclic order $\io=\cd (i_1i_2\cd i_n)(i_1i_2\cd i_n)\cd$
s.t. $\{i_1,\cd,i_n\}=I$,
we can associate the following $(p_{i,j})$ by:
\[
p_{i_a,i_b}=\begin{cases}
1&a<b,\\0&a>b.\end{cases}
\]
For example, if we take $\io=\cd (213)(213)\cd$, then we have
$p_{2,1}=p_{1,3}=p_{2,3}=1$ and $p_{1,2}=p_{3,1}=p_{3,2}=0$.
Thus, we can identify a cyclic order $\cd(i_1\cd i_n)(i_1\cd i_n)\cd$
with such $(p_{i,j})$.
For a monomial $Y=\prod_{m,i}Y_{m,i}^{l_{m,i}}$,
set
\begin{eqnarray*}
&&\hspace{-20pt}wt(Y)=\sum_{i,m} l_{m,i}\Lm_i, \,\,
\vp_i(Y)=\operatorname{max}_{k\in\bbZ}\{\sum_{k\leq m}l_{m,i}\},\,\,
\vep_i(Y)=\vp_i(Y)-wt(Y)(h_i),\\
&&\hspace{-20pt}\fit(Y)=\begin{cases}
A_{n_f,i}^{-1}\cdot Y&\text{ if }\vp_i(Y)>0,\\
0&\text{ if }\vp_i(Y)=0,
\end{cases}\q\q
\eit(Y)=\begin{cases}
A_{n_e,i}\cdot Y&\text{ if }\vep_i(Y)>0,\\
0&\text{ if }\vep_i(Y)=0,
\end{cases}\\
&&n_f=\min\{n|\vp_i(Y)=\sum_{k\leq n}m_{k,i}\},\q
n_e=\max\{n|\vp_i(Y)=\sum_{k\leq n}m_{k,i}\}.
\end{eqnarray*}
\begin{thm}[\cite{K7,Nj}]
\begin{enumerate}
\item
In the above setting, $\cY$ is a crystal, which is denoted by $\cY(p)$.
\item
If $Y\in\cY(p)$ satisfies $\vep_i(Y)=0$ for any $i\in I$,
then the connected component containing $Y$ is isomorphic to
$B(wt(Y))$.
\end{enumerate}
\end{thm}
In the above setting, for type $A_n$ take $(p_{i,j})_{i,j\in I,i\ne j}$
such that $p_{i,j}=1$ for $i<j$, $p_{i,j}=0$
for $i>j$, which corresponds to the cyclic order
${\mathbf i}=(12\cd n)(12\cd n)\cd$.
Then we obtain
\begin{pro}
\label{mono-cry}
The crystal containing the monomial $Y_{n-i+1,1}$ (resp. $Y_{i,1}^{-1}$)
is isomorphic to $B(\Lm_1)$ (resp. $B(\Lm_n)$)
and all basis vectors are given by
\begin{eqnarray*}
&&
\til f_k\cd \til f_2\til
f_1(Y_{n-i+1,1})=\frac{Y_{n-i+1,k+1}}{Y_{n-i+2,k}}\in B(\Lm_1),\\
&&\til e_k\cd\til e_2\til e_1(Y_{i,1}^{-1})
=\frac{Y_{i-k,k}}{Y_{i-k,k+1}}\in B(\Lm_n)\q
(k=1,\cd,n).
\end{eqnarray*}
\end{pro}
{\sl Proof.}
The explicit form of $A_{m,i}$ $(m\in\bbZ,i\in I)$ is as follows:
\begin{equation}
A_{m,i}=\begin{cases}
Y_{m,1}Y_{m,2}^{-1}Y_{m+1,1}&\text{ if }i=1,\\
Y_{m,i}Y_{m,i+1}^{-1}Y_{m+1,i-1}^{-1}Y_{m+1,i}&\text{ if }i\ne1,n,\\
Y_{m,n}Y_{m+1,n-1}^{-1}Y_{m+1,n}&\text{ if }i=n.
\end{cases}
\end{equation}
Then, applying $\eit$ and $\fit$ repeatedly, we obtain the results.
For example,
\[
\til f_1(Y_{n-i+1,1})=Y_{n-i+1,1}\cdot A_{n-i+1,1}^{-1}
=\frac{Y_{n-i+1,2}}{Y_{n-i+2,1}}.
\]
\qed
Applying this results to Theorem \ref{thm-a} and
changing the variable $Y_{m,l}$ to $c_{m,l}$, we find:
\begin{pro}
\label{del-mono}
For $j=1,\cd,n$ we have
\begin{eqnarray*}
&&\chi_j(\pi^+(w_0^{-1}t\Theta_\bfi^-(c)))=
\Del_{w_0\Lm_j,s_j\Lm_j}(\Theta_\bfi^-(c))=
\sum_{k=0}^{j-1}\til f_k\cd \til f_2\til f_1(c_{n-j+1,1}),\\
&&\chi_j(\pi^+(w_0^{-1}\eta(t\Theta_\bfi^-(c))))=
\al_j(t)\Del_{w_0s_j\Lm_j,\Lm_j}(\Theta_\bfi^-(c))=
\al_j(t)
\sum_{k=0}^{j-1}\til e_k\cd\til e_2\til e_1(c_{j,1}^{-1}).
\end{eqnarray*}
\end{pro}
Note that
$\{\til f_k\cd \til f_2\til f_1(c_{n-i+1,1})|0\leq k<i\}
=B(\Lm_1){\tiny s_{k-1}\cd s_2s_1}$
is the Demazure crystal associated with the Weyl group element
$s_{k-1}\cd s_2s_1$ (\cite{K3}).
Observing Proposition \ref{del-mono}, we present the following
conjecture:
\begin{conj}
\label{conj1}
There exists certain reduced longest word
$\bfi=(i_1,\cd, i_N)$ and $p=(p_{i,j})_{i\ne j}$
such that for any $i\in I$,
there exist Demazure crystal $B^-_{w}(i)\subset B(\Lm_k)$,
Demazure crystal $B^+_{w'}(i)\subset B(\Lm_j)$ and
positive integers $\{a_b,a_{b'}|b\in B^-_w,b'\in B^+_{w'}\}$ satisfying
\begin{eqnarray*}
&&\chi_i(\pi^+(w_0^{-1}t\Theta_\bfi^-(c)))=
\Del_{w_0\Lm_i,s_i\Lm_i}(\Theta_\bfi^-(c))=
\sum_{b\in B^-_w(i)}a_b
m_b(c),\\
&&\chi_i(\pi^+(w_0^{-1}\eta(t\Theta_\bfi^-(c))))=
\al_i(t)\Del_{w_0s_i\Lm_i,\Lm_i}(\Theta_\bfi^-(c))=
\al_j(t)\sum_{b'\in B^+_{w'}(i)}a_{b'}m_{b'}(c),
\end{eqnarray*}
where $m_b(c)\in\cY(p)$ is the monomial corresponding to $b\in B(\Lm_k)$
associated with $p=(p_{i,j})_{i\ne j}$.
\end{conj}
We would see the answers to this
conjecture for other type of Lie algebras in the
forthcoming papers.
Suppose that this conjecture is right and then we can deduce the
following:
\begin{cor}
In the setting of the above conjecture, define the linear function $\til
m_b(x):={\mathcal UD}(m_b)(x)$ ($x\in \bbZ^N$) and set
\[
\wtil\Sigma_{{\bf i}^{-1}}[\lm]:=
\{x=(x_N,\cd,x_1)\in \bbZ^N_{{\bf i}^{-1}}[\lm]\,|\,
\til m_b\geq0,\,\,\lm_i+\til m_{b'}\geq0\text{ for any }b\in
B^-_w(i),b'\in B^+_{w'}(i)\,\,(i\in I)
\}.
\]
Then this is equipped with the crystal structure and
isomorphic to the crystal $B(\lm)$.
\end{cor}
{\sl Proof.} Since ${\mathcal UD}(f_B)(\lm,x)\geq0$ is equivalent to the
condition of the set $ \wtil\Sigma_{{\bf i}^{-1}}[\lm]$, we know that
the set $ \wtil\Sigma_{{\bf i}^{-1}}[\lm]$ coincides with
$B_{f_B,\Theta^-_\bfii0}(\lm^\vee)$.\qed
We call $\wtil\Sigma_{{\bf i}^{-1}}[\lm]$ the {\it refined polyhedral
realization} associated with the monomial realizations $\cY(p)$.
\bibliographystyle{amsalpha}
|
{
"timestamp": "2012-03-12T01:01:58",
"yymm": "1203",
"arxiv_id": "1203.2112",
"language": "en",
"url": "https://arxiv.org/abs/1203.2112"
}
|
\section{Introduction}
\label{sec:intro}
Let $f: \mc D \to \bb R$ be a function on a compact subset $\mc D \subseteq \bb R^d$. We would like to address the global optimization problem
\[ x_M = \argmax_{x \in \mc D} f(x). \]
Let us assume for the sake of simplicity that the objective function $f$ has a unique global maximum (although it may have many local maxima).
The space $\mc D$ might be the set of free parameters that one could feed into a time-consuming algorithm or the locations where a sensor could be deployed, and the function $f$ might be a measure of the performance of the algorithm (e.g. how long it takes to run). We refer the reader to \cite{Mockus-82,Schonlau-98,Gramacy-04,Brochu07,Lizotte-08,Martinez-Cantin-09,Garnett-10} for many practical examples of this global optimization setting. In this paper, our assumption is that once the function has been probed at point $x \in \mc D$, then the value $f(x)$ can be observed with very high precision. This is the case when the deployed sensors are very accurate or if the algorithm is deterministic. An example of this is the configuration of CPLEX parameters in mixed-integer programming \cite{Hutter2010mip}. More ambitiously, we might be interested in the \emph{simultaneous} automatic configuration of an entire system (algorithms, architectures and hardware) whose performance is deterministic in terms of several free parameters and design choices.
Global optimization is a difficult problem without any assumptions on the objective function $f$. The main complicating factor is the uncertainty over the extent of the variations of $f$, e.g. one could consider the characteristic function, which is equal to $1$ at $x_M$ and $0$ elsewhere, and none of the methods we mention here can optimize this function without exhaustively searching through every point in $\mc D$.
The way a large number of global optimization methods address this problem is by imposing some prior assumption on how fast the objective function $f$ can vary. The most explicit manifestation of this remedy is the imposition of a Lipschitz assumption on $f$, which requires the change in the value of $f(x)$, as the point $x$ moves around, to be smaller than a constant multiple of the distance traveled by $x$ \cite{LipschitzReview1992}. As pointed out in \citep[Figure 3]{Xarmed2011}, it is only important to have this kind of tight control over the function near its optimum: elsewhere in the space, we can have what they have dubbed a ``weak Lipschitz'' condition.
One way to relax these hard Lipschitz constraints is by putting a Gaussian Process (GP) prior on the function. Instead of restricting the function from oscillating too fast, a GP prior requires those fast oscillations to have low probability, cf. \citep[Theorem 5]{GhosalRoy2006}.
\begin{figure*}[t!]
\begin{center}
\includegraphics[width=\textwidth]{figs/figure1.png}
\end{center}
\caption{\capstyle{An example of the Lipschitz hypothesis being used to discard pieces of the search space when finding the maximum of a function $f$. Although $f$ is only known at the red sample points, if the derivative upper bounds (dashed lines) are below the best attained value thus far, $f(x^{+})$, the corresponding areas of the search space (shaded regions) may be discarded.
}}
\label{fig:RelevantSet}
\end{figure*}
The main point of these bounds (be they hard or soft) is to assist with the \emph{exploration-exploitation trade-off} that global optimization algorithms have to grapple with. In the absence of any assumptions of convexity on the objective function, a global optimization algorithm is forced to explore enough until it reaches a point in the process when with some degree of certainty it can localize its search space and perform local optimization (exploitation). Derivative bounds such as the ones discussed here together with the boundedness of the search space, guaranteed by the compactness assumption on $\mc D$, provide us with such certainty by producing a useful upper bound that allows us to shrink the search space.
This is illustrated in Figure \ref{fig:RelevantSet}. Suppose we know that our function is Lipschitz with constant $L$, then given sample points as shown in the figure, we can use the Lipschitz property to discard pieces of the search space. This is done by finding points in the search space where the function could not possibly be higher than the maximum value already encountered. Such points are found by placing cones at the sampled points with slope equal to $L$ and checking where those cones lie below the maximum observed value.
This crude approach is wasteful because very often the slope of the function is much smaller than $L$. As we will see below (cf. Figure \ref{fig:BB}), GPs do a better job of providing lower and upper bounds that can be used to limit the search space, by essentially choosing Lipschitz constants that vary over the search space and the algorithm run time.
We also assume that the objective function $f$ is costly to evaluate (e.g. time-wise or financially). We would like to avoid probing $f$ as much as possible and to get close to the optimum as quickly as possible. A solution to this problem is to approximate $f$ with a \emph{surrogate function} that provides a good upper bound for $f$ and which is easier to calculate and optimize. Surrogate functions can also aid with global optimization by restricting the domain of interest.
GPs enable us to construct surrogate functions, which are relatively easy to evaluate and optimize.
We refer the reader to \cite{Brochu2009at} for a general review of the literature on the various surrogate functions utilized in GP bandits in the context of Bayesian optimization.
The surrogate function that we will make extensive use of here is called the Upper Confidence Bound (UCB). It is defined to be $\mu + B\sigma$, where $\mu$ and $\sigma$ are the posterior predictive mean and standard deviation of the GP and $B$ is a constant to be chosen by the algorithm.
\begin{figure*}[t]
\begin{center}
\includegraphics[width=\textwidth]{figs/figure2.png}
\end{center}
\caption{\capstyle{An example of our branch and bound maximization algorithm with UCB surrogate $\mu+B \sigma$, where $\mu$ and $\sigma$ are the mean and standard deviation of the GP respectively. The region consisting of the points $x$ for which the upper confidence bound $\mu(x)+B \sigma(x)$ is lower that the maximum value of the lower confidence bound $\mu(x)- B \sigma(x)$
does not need to be sampled anymore. Note that the UCB surrogate function bounds $f$ from above.}}
\label{fig:BB}
\end{figure*}
This surrogate function has been studied extensively in the literature and this paper relies heavily on the ideas put forth in the paper by Srinivas et al \cite{Srinivas2010gp}, in which the algorithm consists of repeated optimization of the UCB surrogate function after each sample.
One key difference between our setting and that of \cite{Srinivas2010gp} is that, whereas we assume that the value of the function can be observed exactly, in \cite{Srinivas2010gp} it is necessary for the noise to be non-trivial (and Gaussian) because the main quantity that is used in the estimates, namely information gain, cf. \citep[Equation 3]{Srinivas2010gp}, becomes undefined when the variance of the observation noise ($\sigma^2$ in their notation) is set to $0$, cf. the expression for $\I(\mathbf{y}_A; \mathbf{f}_A)$ that was given in the paragraph following Equation (3). So, their setting is complementary to ours.
Moreover,
we show that the regret, $r(x_t) = \max_{\mc D} f - f(x_t)$, decreases according to $\mc O\left(e^{-\frac{\tau t}{\left(\ln t\right)^{d/4}}}\right)$, implying that the cumulative regret is bounded from above.
The paper whose results are most similar to ours is \cite{Munos2011soo}, but there are some key differences in the methodology, analysis and obtained rates. For instance, we are interested in cumulative regret, whereas the results of \cite{Munos2011soo} are proven for finite stop-time regret. In our case, the ideal application is the optimization of a function that is $C^2$-smooth and has an unknown non-singular Hessian at the maximum. We obtain a regret rate $\mc O\left(e^{-\frac{\tau t}{\left(\ln t\right)^{d/4}}}\right)$, whereas the DOO algorithm in \cite{Munos2011soo} has regret rate $\mc O(e^{-t})$ if the Hessian is known and the SOO algorithm has regret rate $\mc O(e^{-\sqrt{t}})$ if the Hessian is unknown. In addition, the algorithms in \cite{Munos2011soo} can handle functions that behave like $-c\|x-x_M\|^\alpha$ near the maximum (cf. Example 2 therein). This problem was also studied by
\cite{Vazquez2011ei} and \cite{Bull2011cr}, but using the
Expected Improvement surrogate instead of UCB. Our methodology and results are different, but complementary to theirs.
\section{Gaussian process bandits}
\label{sec:BG}
\subsection{Gaussian processes}
\label{sec:GP}
As in \cite{Srinivas2010gp}, the objective function is distributed according to a Gaussian process prior:
\begin{equation}
f(x) \sim \operatorname{GP}(m(\cdot), \kappa(\cdot,\cdot)).
\end{equation}
For convenience, and without loss of generality, we assume that the prior mean vanishes, i.e., $m(\cdot) = 0$.
There are many possible choices for the covariance kernel. One obvious choice is the anisotropic kernel $\kappa$ with a vector of known hyperparameters \cite{Rasmussen2006gp}:
\begin{eqnarray}
\kappa(x_i, x_j) &=& \widetilde{\kappa}\left(-(x_i-x_j)^\top\myvec{D}(x_i-x_j)\right),
\end{eqnarray}
where $\widetilde{\kappa}$ is an isotropic kernel and $\myvec{D}$ is a diagonal matrix with positive hyperparameters along the diagonal and zeros elsewhere. Our results apply to squared exponential kernels and Mat\'ern kernels with parameter $\nu \geq 2$. In this paper, we assume that the hyperparameters are fixed and known in advance.
We can sample the GP at $t$ points by choosing points $\mathbf{x}_{1:t} := \{x_1, \ldots, x_t\}$ and sampling the values of the function at these points to produce the vector $\myvec{f}_{1:t} = [f(x_1) \cdots f(x_t)]^\top$. The function values are distributed according to a multivariate Gaussian distribution $\mathcal{N}(0,\myvec{K})$, with covariance entries $ \kappa(x_i, x_j)$.
Assume that we already have several observations from previous steps, and that we want to decide what action $x_{t+1}$ should be considered next. Let us denote the value of the function at this arbitrary new point as $f_{t+1}$. Then, by the properties of GPs, $\myvec{f}_{1:t}$ and $f_{t+1}$ are jointly Gaussian:
\[
\begin{bmatrix}
\myvec{f}_{1:t} \\
f_{t+1}
\end{bmatrix}
\sim {\cal N}
\left( \mathbf{0} ,
\begin{bmatrix}
\myvec{K} & \myvec{k}^\top \\
\myvec{k} & \kappa(x_{t+1},x_{t+1})
\end{bmatrix}
\right),
\]
where
$\myvec{k} = [\kappa(x_{t+1},x_1) \cdots \kappa(x_{t+1},x_t)]^\top$.
Using the Schur complement, one arrives at an expression for the posterior predictive distribution:
\[
P(f_{t+1}|\mathbf{x}_{1:t+1}, \myvec{f}_{1:t}) = {\cal N} (\mu_t(x_{t+1}), \sigma_t^2(x_{t+1})),
\]
where
\begin{equation}
\begin{array}{l}
\mu_t(x_{t+1}) = \mathbf{k}^\top \mathbf{K}^{-1} \myvec{f}_{1:t}, \\
\sigma_t^2(x_{t+1}) = \kappa(x_{t+1},x_{t+1}) - \mathbf{k}^\top \mathbf{K}^{-1}\mathbf{k}
\end{array}
\label{eqn:Posterior}
\end{equation}
and $\myvec{f}_{1:t} = [f(x_1) \cdots f(x_t)]^\top$.
\begin{algorithm*}[t]
\caption{Branch and Bound}
\label{alg:BB}
\begin{algorithmic}
\STATE Input: A compact subset $\mc D \subseteq \bb R^d$, a discrete lattice $\mc L \subseteq \mc D$ and a function $ f: \mc D \to \bb R$.
\STATE $\mc R \gets \mc D$
\STATE $\delta \gets 1$
\REPEAT
\STATE \mbox{S\bf ample Twice as Densely:}
\STATE \qquad $\bullet$ $\delta \gets \dfrac{\delta}{2}$
\STATE \qquad $\bullet$ Sample $f$ at enough points in $\mc L$ so that every point in $\mc R$ is contained in a simplex of size $\delta$.
\STATE \mbox{\bf Shrink the Relevant Region:}
\STATE \qquad $\bullet$ Set
\[ \widetilde{\mc R} := \left\{ x \in \mc R \bigg| \mu_T(x) + \sqrt{\beta_T} \sigma_T(x) > \sup_{\mc R} \mu_T(x) - \sqrt{\beta_T} \sigma_T(x) \right\}. \]
\qquad\quad $T$ is the number points sampled so far and $\beta_T = 2\ln\left(\frac{|\mc L|T^2}{\alpha}\right) = 4\ln T + 2\ln \frac{|\mc L|}{\alpha}$ with $\alpha \in (0,1)$.
\STATE \qquad $\bullet$ Solve the following constrained optimization problem:
\[ (x_1^*,x_2^*) = \argsup_{(x_1,x_2) \in \widetilde{\mc R} \times \widetilde{\mc R}} \|x_1 - x_2\| \]
\STATE \qquad $\bullet$ $\mc R \gets B\left(\dfrac{x_1^*+x_2^*}{2}, \|x_1^*-x_2^*\|\right)$, where $B(p,r)$ is the ball of radius $r$ centred around $p$.
\UNTIL{$\mc R \cap \mc L = \varnothing$}
\end{algorithmic}
\end{algorithm*}
\subsection{Surrogates for optimization}
When it is assumed that the objective function $f$ is sampled from a GP, one can use a combination of the posterior predictive mean and variance given by Equations~(\ref{eqn:Posterior}) to construct surrogate functions, which tell us where to sample next.
Here we use the UCB combination, which is given by
\[
\mu_t(x) + B_t\sigma_t(x),
\]
where $\{B_t\}_{t=1}^\infty$ is a sequence of numbers specified by the algorithm. This surrogate trades-off exploration and exploitation since it is optimized by choosing points where the mean is high (exploitation) and where the variance is large (exploration). Since the surrogate has an analytical expression that is easy to evaluate, it is much easier to optimize than the original objective function.
Other popular surrogate functions constructed using the sufficient statistics of the GP include the Probability of Improvement, Expected Improvement and Thompson sampling.
We refer the reader to \cite{Brochu2009at,May2010ob,Hoffman-11} for details on these.
\subsection{Our algorithm}
The main idea of our algorithm (Algorithm \ref{alg:BB}) is to tighten the bound on $f$ given by the UCB surrogate function by sampling the search space more and more densely and shrinking this space as more and more of the UCB surrogate function is ``submerged'' under the maximum of the Lower Confidence Bound (LCB). Figure \ref{fig:BB} illustrates this intuition.
More specifically, the algorithm consists of two iterative stages. During the first stage, the function is sampled along a lattice of points (the red crosses in Figure \ref{fig:BB2D}). In the second stage, the search space is shrunk to discard regions where the maximum is very unlikely to reside. Such regions are obtained by finding points where the UCB is lower than the LCB (the complement of the colored region in the same panel as before). The remaining set of relevant points is denoted by $\widetilde{\mc R}$. In order to simplify the task of shrinking the search space, we simply find an enclosing ball, which is denoted by $\mc R$ in Algorithm \ref{alg:BB}.
Back to the first stage, we consider a lattice that is twice as dense as in the first stage of the previous iteration, but we only sample at points that lie within our new smaller search space.
In the second stage, the auxiliary step of approximating the relevant set $\widetilde{\mc R}$ with the ball $\mc R$ introduces inefficiencies in the algorithm, since we only need to sample inside $\widetilde{\mc R}$.
This can be easily remedied in practice to obtain an efficient algorithm. Our analysis will show that even without these improvements it is already possible to obtain very strong exponential convergence rates. Of course, practical improvement will result in better constants and ought to be considered seriously.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.8\textwidth]{figs/figBB2D2by2.png}
\end{center}
\caption{\capstyle{Branch and Bound algorithm for a 2D function. The colored region is the search space and the color-map, with red high and blue low, illustrates the value of the UCB. Four steps of the algorithm are shown; progressing from left to right and top to bottom. The green dots designate the points where the function was sampled in the previous steps, while the red crosses denote the freshly sampled points.}}
\label{fig:BB2D}
\end{figure}
\section{Analysis}
\subsection{Approximation results}
We begin our analysis by showing that, given sufficient explored
locations, the residual variance is small. More specifically, for any
point $x$ contained in the convex hull of a set of $d$ points that are
no further than $\delta$ apart from $x$, we show that the residual
is bounded by $O(\nbr{h}_\mathcal{H} \delta^2)$, where $\nbr{h}_\mathcal{H}$ is the
Hilbert Space norm of the associated function and that furthermore the
residual variance is bounded by $O(\delta^2)$.
We begin by relating residual variance, projection operators, and
interpolation in Hilbert Spaces. Lemmas 1, 2 and 3 are standard. We include their proofs in the supplementary material
for the purpose of being self-contained. Proposition 4 is our key approximation result. It plays a central role in the proof of our exponential regret bounds. Its proof, as well as the proof for the main theorem, is included in the supplementary material.
\begin{lemma}[Hilbert Space Properties]
\label{lem:rkhs}
Given a set of points $x_{1:T} := \cbr{x_1, \ldots, x_T} \in \mathcal D$ and a
Reproducing Kernel Hilbert Space (RKHS) $\mathcal{H}$ with kernel $\kappa$ the following
bounds hold:
\begin{enumerate}
\item \label{item:lipschitz}
Any $h \in \mathcal{H}$ is Lipschitz continuous with constant
$\nbr{h}_\mathcal{H} L$, where $\nbr{\cdot}_\mathcal{H}$ is the Hilbert space norm and $L$ satisfies the following:
\begin{align}
L^2 \leq \sup_{x \in \mathcal D} \partial_x \partial_{x'}
\kappa(x,x')|_{x=x'}
\end{align}
and for $\kappa(x,x') = \widetilde{\kappa}(x-x')$ we have
\[ L^2 \leq \partial_x^2 \widetilde{\kappa}(x)|_{x=0}. \]
\item \label{item:secondderivative}
Any $h \in \mathcal{H}$ has its second derivative bounded by
$\nbr{h}_\mathcal{H} Q$ where
\begin{align}
Q^2 \leq \sup_{x \in \mathcal D} \partial^2_x \partial^2_{x'}
\kappa(x,x')|_{x=x'}
\end{align}
and for $\kappa(x,x') = \widetilde{\kappa}(x-x')$ we have
\[ Q^2 \leq \partial_x^4 \widetilde{\kappa}(x)|_{x=0}. \]
\item \label{item:projection}
The projection operator $P_{1:T}$ on the subspace $\displaystyle\Span_{t = 1:T} \{\kappa(x_t,\cdot) \} \subseteq \mathcal{H}$ is given by
\begin{align}
P_{1:T}h := \myvec{k}^\top (\cdot) \myvec{K}^{-1} \left< \myvec{k}(\cdot), h \right>
\end{align}
where $\myvec{k}(\cdot) = \myvec{k}_{1:T}(\cdot) := \left[\kappa(x_1,\cdot) \cdots \kappa(x_T,\cdot) \right]^\top$ and $\myvec{K} := \left[ \kappa(x_i, x_j) \right]_{i,j = 1:T}$; moreover, we have that
\[ \left< \myvec{k}(\cdot), h \right> := \begin{bmatrix} \left< \kappa(x_1, \cdot), h \right> \\ \vdots \\ \left< \kappa(x_T, \cdot), h \right> \end{bmatrix} = \begin{bmatrix} h(x_1) \\ \vdots \\ h(x_T) \end{bmatrix}. \]
Here $P_{1:T} P_{1:T} = P_{1:T}$ and $\nbr{P_{1:T}} \leq 1$ and $\nbr{\mathbf{1} - P_{1:T}}
\leq 1$.
\item \label{item:schurcomplement}
Given sets $x_{1:T} \subseteq x_{1:T'}$ it follows that $\nbr{P_{1:T} h}_\mathcal{H} \leq
\nbr{P_{1:T'} h}_\mathcal{H} \leq \nbr{h}_\mathcal{H}$.
\item \label{item:interpolation}
Given tuples $(x_i, h_i)$ with $h_i = h(x_i)$, the minimum norm
interpolation $\bar{h}$ with $\bar{h}(x_i) = h(x_i)$ is given by
$\bar{h} = P_{1:T} h$. Consequently its residual $g := (\mathbf{1} - P_{1:T}) h$
satisfies $g(x_i) = 0$ for all $x_i \in x_{1:T}$.
\end{enumerate}
\end{lemma}
\begin{lemma}[GP Variance]\label{lem:gpvar} Under the assumptions of Lemma~\ref{lem:rkhs} it follows that
\begin{align}
\abr{h(x) - P_{1:T} h(x)} \leq \nbr{h}_\mathcal{H} \sigma_T(x),
\end{align}
where $\sigma_T^2(x) = \kappa(x,x) - \myvec{k}_{1:T}^\top(x) \myvec{K}^{-1} \myvec{k}_{1:T}(x)$
and this bound is tight. Moreover, $\sigma_T^2(x)$ is the residual variance
of a Gaussian process with the same kernel.
\end{lemma}
\begin{lemma}[Approximation Guarantees]\label{lem:approximation}~We denote by $x_{1:T} \subseteq \mathcal D$ a set of
locations and assume that $g(x_i) = 0$ for all $x_i \in x_{1:T}$.
\begin{enumerate}
\item Assume that $g$ is Lipschitz continuous with bound $L$. Then
$g(x) \leq L d(x, x_{1:T})$, where $d(x,x_{1:T})$ is the minimum distance
$\nbr{x-x_i}$ between $x$ and any $x_i \in x_{1:T}$.
\item Assume that $g$ has its second derivative bounded by
$Q^\prime$. Moreover, assume that $x$ is contained inside the convex hull
of $x_{1:T}$ such that the smallest such convex hull has a maximum
pairwise distance between vertices of $d$. Then we have
$g(x) \leq \frac{1}{4} Q^\prime d^2$.
\end{enumerate}
\end{lemma}
\begin{proposition}[Variance Bound]
\label{lem:varbound}
Let $\kappa: \bb R^d \times \bb R^d \to \bb R$ be a kernel that is four times differentiable along the diagonal $\{(x,x) \,|\, x \in \bb R^d\}$, with $Q$ defined as in Lemma \ref{lem:rkhs}.\ref{item:secondderivative}, and $f \sim \operatorname{GP}\left(0,\kappa(\cdot, \cdot)\right)$ a sample from the corresponding Gaussian Process. If $f$ is sampled at points $x_{1:T} = \{x_1, \ldots, x_T\}$ that form a $\delta$-cover of a subset $\mc D \subseteq \bb R^d$, then the resulting posterior predictive standard deviation $\sigma_T$ satisfies
\[ \sup_{\mc D} \sigma_T \leq \frac{Q\delta^2}{4}. \]
\end{proposition}
\subsection{Finiteness of regret}
Having shown that the variance vanishes according to the square of the resolution of the lattice of sampled points, we now move on to show that this estimate implies an exponential asymptotic vanishing of the regret encountered by our Branch and Bound algorithm. This is laid out in our main theorem stated below and proven in the supplementary material.
The theorem considers a function $f$, which is a sample from a GP with a kernel that is four times differentiable along its diagonal. The global maximum of $f$ can appear in the interior of the search space, with the function being twice differentiable at the maximum and with non-vanishing curvature. Alternatively,
the maximum can appear on the boundary with the function having non-vanishing gradient at the maximum.
Given a lattice that is fine enough, the theorem
asserts that the regret asymptotically decreases in exponential fashion.
The main idea of the proof of this theorem is to use the bound on $\sigma$ given by Proposition~\ref{lem:varbound} to reduce the size of the search space. The key assumption about the function that the proof utilizes is the quadratic upper bound on the objective function $f$ near its global maximum, which together with Proposition~\ref{lem:varbound} allows us to shrink the relevant region $\mc R$ in Algorithm \ref{alg:BB} rapidly. The figures in the proof give a picture of this idea.
The only complicating factor is the factor $\sqrt{\beta_t}$ in the expression for the UCB that needs to be estimated. This is dealt with by modeling the growth in the number of points sampled in each iteration with a difference equation and finding an approximate solution of that equation.
Recall that $\mc D \subseteq \bb R^d$ is assumed to be a non-empty compact subset and $f$ a sample from the Gaussian Process $\operatorname{GP}\left(0,\kappa(\cdot, \cdot)\right)$ on $\mc D$. Moreover, in what follows we will use the notation $x_M := \displaystyle\argmax_{x \in \mc D} f(x)$. Also, by convention, for any set $\mc S$, we will denote its interior by $\mc S^\circ$, its boundary by $\partial \mc S$ and if $S$ is a subset of $\bb R^d$, then $\cvx(S)$ will denote its convex hull.
The following holds true:
\begin{theorem}\label{thm:BB}
Suppose we are given:
\begin{enumerate}
\item $\alpha > 0$, a compact subset $\mc D \subseteq \bb R^d$,
and $\kappa$ a stationary kernel on $\bb R^d$ that is four times differentiable;
\item $f \sim \operatorname{GP}(0,\kappa)$ a continuous sample on $\mc D$ that has a unique global maximum $x_M$, which satisfies one of the following two conditions:
\begin{itemize}
\item[$(\dagger)$] $x_M \in \mc D^{\circ}$ and $f(x_M) - c_1 \|x-x_M\|^2 < f(x) \leq f(x_M) - c_2 \|x-x_M\|^2$ for all $x$ satisfying $x \in B(x_M, \rho_0)$ for some $\rho_0 > 0$;
\item[$(\ddagger)$] $x_M \in \partial \mc D$ and both $f$ and $\partial \mc D$ are smooth at $x_M$, with $\nabla f(x_M) \neq 0$;
\end{itemize}
\item any lattice $\mc L \subseteq \mc D$ satisfying the following two conditions
\begin{align}
\bullet\quad & 2\mc L \cap \cvx(\mc L) \subseteq \mc L \label{cdn:divby2} \\
\bullet\quad & 2^{\left\lceil -\log_2 \frac{\rho_0}{\diam(\mc D)} \right\rceil + 1}\mc L \cap \mc L \neq \varnothing \label{cdn:highresolution} \\
& \text{ if $f$ satisfies $(\dagger)$} \nonumber
\end{align}
\end{enumerate}
Then, there exist positive numbers $A$ and $\tau$ and an integer $T$ such that the points specified by the Branch and Bound algorithm, $\{x_t\}$, will satisfy the following asymptotic bound:
For all $t > T$, with probability $1-\alpha$ we have
\[ r(x_t) < Ae^{-\frac{\tau t}{\left(\ln t\right)^{d/4}}}. \]
\end{theorem}
We would like to make a few clarifying remarks about the theorem. First,
note that for a random sample $f \sim \operatorname{GP}(0,\kappa)$ one of conditions $(\dagger)$ and $(\ddagger)$ will be satisfied almost surely if $\kappa$ is a Mat\'ern kernel with $\nu > 2$ and the squared exponential kernel because the sample $f$ is twice differentiable almost surely by \citep[Theorem 1.4.2]{Adler2007rf} and \citep[\S2.6]{Stein1999gp}) and the vanishing of at least one of the eigenvalues of the Hessian is a co-dimension 1 condition in the space of all functions that are smooth at a given point, so it has zero chance of happening at the global maximum.
Second, the two conditions (\ref{cdn:divby2}) and (\ref{cdn:highresolution}) simply require that the lattice be ``divisible by 2'' and that it be fine enough so that the algorithm can sample inside the ball $B(x_M,\rho_0)$ when the maximum of the function is located in the interior of the search space $\mc D$. Finally, it is important to point out that the rate decay $\tau$ does not depend on the choice of the lattice $\mc L$, even though as stated, the statement of the theorem chooses $\tau$ only after $\mc L$ is specified. The theorem was written this way simply for the sake of readability.
Given the exponential rate of convergence we obtain in Theorem \ref{thm:BB}, we have the following finiteness conclusion for the cumulative regret accrued by our Branch and Bound algorithm:
\begin{corollary} Given $\kappa$, $f \sim \operatorname{GP}(0,\kappa)$ and $\mc L \subseteq \mc D$ as in Theorem \ref{thm:BB}, the cumulative regret is bounded from above.
\end{corollary}
\begin{remark} It is worth pointing out the trivial observation that using a simple UCB algorithm with monotonically increasing and unbounded factor $\sqrt{\beta_t}$, without any shrinking of the search space as we do here, necessarily leads to unbounded cumulative regret since eventually $\sqrt{\beta_t}$ becomes large enough so that at points $x^\prime$ far away from the maximum, $\sqrt{\beta_t}\sigma_t(x^\prime)$ becomes larger than $f(x_M)-f(x)$. In fact, eventually the UCB algorithm will sample every point in the lattice $\mc L$.
\end{remark}
\section{Discussion}
In this paper we proposed a modification of the UCB algorithm of
\cite{Srinivas2010gp} which addresses the noise free case. The key
difference is that while the original algorithm achieves an
$O(t^{-\frac{1}{2}})$ rate of convergence to the regret minimizer, we
obtain an exponential rate in the number of function evaluations. In
other words, the noise free problem is significantly easier,
statistically speaking, than the noisy case. The key difference is
that we need not invest any samples in noise reduction to determine
whether our observations deviate far from their expectation.
This allows us to discard pieces of the search space where the maximum
is very unlikely to be, when compared to \cite{Srinivas2010gp}. We
show that this additional step leads to a considerable improvement of
the regret accrued by the algorithm. In particular, the cumulative
regret obtained by our Branch and Bound algorithm is bounded from
above, whereas the cumulative regret bound obtained in the noisy
bandit algorithm is unbounded. The possibility of dispensing with
chunks of the search space can also be seen in the works involving
hierarchical partitioning, e.g. \cite{Munos2011soo}, where regions of
the space are deemed as less worthy of probing as time goes on.
Our results mirror the observation in active learning that noise free
and large margin learning of half spaces can be achieved much more
rapidly than identifying a linear separator in the noisy case
\cite{BshWat06,DasKalMon09}. This is also reflected in classical
uniform convergence results for supervised learning
\cite{AudTsy07,Vapnik98} where the achievable rate depends on the
decay of probability mass near the margin.
This suggests that the ability to extend our results to the noisy case
is somewhat limited. An indication of what might be possible can be
found in \cite{BalBeyLan09}, where regions of the version space are
eliminated once they can be excluded with sufficiently high
probability. One could model a corresponding Branch and Bound
algorithm, which dispenses with points that lie outside the current
(or perhaps the previous) relevant set when calculating the covariance
matrix $\mathbf K$ in the posterior equations
(\ref{eqn:Posterior}). Analysis of how much of an effect such a
computational cost-cutting measure would have on the regret
encountered by the algorithm is a subject of future research.
We believe that an exciting extension can be found in guarantees for
contextual bandits. Note, however, that the unpredictability of the
context introduces new difficulties in terms of speed of convergence
that need to be overcome. For instance, parameters for infrequent
contexts will be estimated slowly unless there are strong correlations among contexts.
|
{
"timestamp": "2012-03-12T01:02:48",
"yymm": "1203",
"arxiv_id": "1203.2177",
"language": "en",
"url": "https://arxiv.org/abs/1203.2177"
}
|
\section{Introduction}
Observations over the past few decades have shown that during the star formation process, mass accretion is commonly associated with mass ejection in the form of jets. The interaction between the jet and the parent cloud generates shock fronts, that form large cavities called bipolar outflows. The simultaneous presence of jets and bipolar outflows, as well as accretion disks surrounding protostellar condensations, are commonly observed \citep{Arce07}, making them the basic elements of the star formation scenario. In this picture, different excitation regimes, related to the different structures, are revealed with a diversity of observational probes, such as continuum emission and molecular transitions at different wavelengths. In the jet/outflow system, at its temperature peak the shocked material is mainly radiatively cooled through the near-IR transitions of H$_2$, whereas the low-lying CO transitions trace the lower density, lower temperature ($<$ 50 K) post-shock gas. In-between those two extreme regimes and over the 100-2000~K range, the cooling of the gas occurs mainly through the emission of atomic and molecular lines such as [OI], H$_2$O, high$-J$ CO, OH, and H$_2$ transitions, which fall in the mid- to far-infrared spectral range \citep{Nisini99}.
Traditionally, the physical conditions of the gas in protostellar outflows have been studied by means of ground-based observations of low-lying rotational transitions of the CO molecule (usually up to $J_{\rm u} = 3$, corresponding to E$_{\rm u} < 50$~K). In most cases, these low-energy transitions are more sensitive to the low excitation swept-up gas, not exclusively related with the jet and/or shock front. On the other hand, higher-lying transitions (E$_{\rm u} \gtrsim$100 K) probe the warm gas unambiguously related with the jet and/or hot spots in the shocked gas. However, transitions with relatively high$-J$ rotational quantum numbers (e.g. $J_{\rm u} \geq$ 9) fall into the window of the far-infrared regime that can not be observed from the ground. High$-J$ CO observations were pioneered with the \emph{Infrared Space Observatory} (ISO), which actually targeted several protostellar outflows \citep{Nisini99,Nisini00,Gia01}. These ISO observations revealed that the high$-J$ CO emission does indeed trace outflowing gas above 1000 K, but their poor spectral resolution did not allow us to probe either the presence of different spectral components, or the physical conditions of the gas as a function its velocity. In addition, their low spatial resolution ($\sim$ 80\arcsec~at wavelengths corresponding to its Long Wavelength Spectrometer, LWS) complicates the analysis owing to the presence of different structures within the same beam. Only recently, with \emph{Herschel} in operation and SOFIA starting operation, we had the opportunity to follow-up on these previous studies with high-resolution FIR spectroscopy and angular resolutions in the 15-20\arcsec~range.
In this letter, we present one of the first studies of two high$-J$ CO spectrally resolved lines in a protostellar outflow, as an example of the potential of SOFIA observations with GREAT in the context of protostellar jets/outflows. For this study, we selected the protostellar outflow Cep E, in the Cepheus molecular cloud, since it is associated with a relatively nearby ($\sim$730~pc) intermediate-mass class~0 source \citep[IRAS 23011+6126:][]{Chini01} of 100 L$_{\odot}$, surrounded by an envelope structure of mass $\sim$ 18 M$_{\odot}$. The jet/outflow system itself has been intensively studied at multiple wavelengths \citep{Eis96,Lefloch96,Moro01,Smith03}. The strong CO wings and the evidence of jet-like components in the spectrum of Cep E outflow make it an ideal target for the study of the high$-J$ CO emission in jet/outflow systems.
\section{Observations}
The SOFIA telescope was pointed towards the southern lobe of the Cep E outflow (Fig.~\ref{irac-2}). As seen in Fig.~\ref{irac-2}, the SOFIA observations covered the infrared knots BI and BII. The observations were performed during basic science flights in July 2011, with the German Receiver for Astronomy at Terahertz Frequencies \citep[GREAT\footnote{GREAT is a development by the MPI f\"ur Radioastronomie and the KOSMA/Universit\"at zu K\"oln, in cooperation with the MPI f\"ur Sonnensystemforschung and the DLR Institut f\"ur Planetenforschung.}:][]{Heyminck12}. The L1 band of the GREAT instrument was used to observe, in two separate flights, the CO (12--11) and (13--12) transitions at $\sim$1.4 THz and $\sim$1.5 THz, in the lower and upper sidebands, respectively. In Table~\ref{trans}, the line frequencies, diffraction-limited beam sizes, time ON source ($t_{\rm ON}$), and atmospheric opacities ($\tau_{\rm atm}$) during the observations are presented. The front-end was connected to the AFFTS back-end \citep{Klein12}, providing a total bandwidth of 1.5 GHz and a spectral resolution of 212 kHz, that at the observed frequencies correspond to $\sim$ 300 km s$^{-1}$ and $\sim$0.4 km s$^{-1}$, respectively. Single pointing observations were done in double-beam chopped mode, with a chop throw of 60\arcsec~in RA (at 1 Hz). The pointing was established with the optical guide cameras, and was stable to 4\arcsec. The calibration uncertainty is within 20\%. From observations of Jupiter and Mars, a main-beam efficiency of 0.54$\pm$0.05 in the L1 band was determined \citep{Heyminck12}, that - with a forward efficiency of 0.95 - was used to convert antenna to main beam temperatures, $T_{\rm MB}$. Subsequent analysis of the data was done with CLASS within the GILDAS software\footnote{http://www.iram.fr/IRAMFR/GILDAS}, following the standard procedures of baseline subtraction and spectra averaging. A first-order baseline was subtracted from all spectra.
Calibration was performed by carefully fitting the observed sky emission \citep{Guan12}. Unfortunately, both CO transitions suffer from residual terrestrial atmospheric features at SOFIA's flight altitude: the CO (12--11) detection bandwidth contains a strong water absorption (from the signal band) at positive velocities $>$60 km/s. This does not however affect the calibration of the pre-dominantly negative velocity emission towards the southern lobe. This is different for CO (13--12), whose calibration (and baselines) at extremely negative velocities (of lower than -~120~km s$^{-1}$) are more uncertain owing to an image band atmospheric feature. However, quite comparable integrated intensities for both lines at small S/N values supports our confidence in the adopted calibration procedure.
\begin{table}
\caption{The CO observations.}
\label{trans}
\begin{tabular}{l c c c c c c}
\hline
CO & $\nu$$_0$ & $E_{\rm u}/k$ & HPBW & rms\tablefootmark{a} & $\tau_{\rm atm}$ & $t_{\rm ON}$\\
Line & (GHz) & (K) & (\arcsec) & (mK) & & (min)\\
\hline
(13--12) & 1496.9922 & 503.1 & 20.0 & 82 & 0.03 & 15\\
(12--11) & 1381.9951 & 431.3 & 21.7 & 96 & 0.16 & 7.4\\
(2--1)\tablefootmark{b}& 230.5380 & 16.6 & 20.0 & 32 & - & - \\
\hline
\end{tabular}
\tablefoottext{a}{$T_{\rm{MB}}$ scale, at 3 km s$^{-1}$ spectral resolution.}
\tablefoottext{b}{Convolved to 20\arcsec~from original $\sim$ 11\arcsec~resolution (IRAM-30m), Lefloch et al. in prep.}
\end{table}
\begin{figure}[h!]
\centering
\includegraphics[width=8cm,angle=-90]{cepe_irac2.eps}
\caption{\emph{Spitzer}/IRAC band-two (4.5 $\mu$m) image of the Cep E protostellar outflow (retrieved from the \emph{Spitzer} archive). White circles mark the diffraction-limited SOFIA beams at the frequencies of CO (12--11) and (13-12) (decreasing beam size with increasing $J_{\rm u}$, see Table~\ref{trans}), centered at the brightest IR knot of the southern lobe, $\alpha =$23$^h$03$^m$11\fs70 $\delta =$$+$61$^{\circ}$42$'$06\farcs0 (black plus symbol). White letters label the two main knots inside our beams, BII and BI, as well as the Cep E central source (IRAS 23011: black star).
}
\label{irac-2}
\end{figure}
\section{Results}
Together with the SOFIA/GREAT spectra, in Fig.~\ref{sofia-spec} we also show a CO (2--1) line profile taken at the corresponding position from a Nyquist sampled map, obtained with the IRAM-30m telescope (Lefloch et al., in prep). The CO (2--1) spectrum was convolved to a beam of 20\arcsec ~(i.e. the SOFIA beam at the (13--12) transition, see Table~\ref{trans}). From the CO spectra shown in Fig.~\ref{sofia-spec} at least three spectral features can be distinguished: two secondary peaks, at around $-125$ km~s$^{-1}$ and $-65$ km~s$^{-1}$, and a wing-like profile (smooth decrease of intensity towards high velocities) in the range from $-50$ km s$^{-1}$ to $-20$ km~s$^{-1}$. The secondary peaks are similar to the so-called molecular \lq bullets' often observed in the spectrum from class 0 protostars and likely related to the jet component \citep[e.g. L1448:][]{Bachiller90}. These \lq bullets' were reported previously in Cep E \citep{Lefloch96}. The wing-like feature is typical of the outflow phenomenon, mostly tracing the cavity walls crated by the jet, and usually referred to as the \lq standard' high-velocity component \citep[e.g.,][]{Bachiller90}. On the basis of these spectral features, in Fig.~\ref{sofia-spec} we divide all spectra into three velocity ranges: extremely high velocity (EHV: $-140$ to $-100$ km~s$^{-1}$), intermediate-to-high velocity (IHV: $-100$ to $-50$ km~s$^{-1}$), and standard high velocity (SHV: $-50$ to $-20$ km~s$^{-1}$). In Table~\ref{tab:velint}, we present the line intensities integrated within the defined velocity ranges. From the spectra in Fig.~\ref{sofia-spec}, one can see that the bullet-like profile in the IHV range is more prominent in high$-J$ than in low-$J$ CO, with the latter still being dominated by the wing-like profile that extends into this velocity range. On the other hand, the SHV component is stronger in low$-J$ than in high$-J$ CO. From the integrated intensities in Table~\ref{tab:velint}, we found that the (2--1)/(12--11) ratio is about 0.8, 0.4, and 2.2, for EHV, IHV, and SHV, respectively, indicating different excitation conditions in the different velocity ranges. These findings underline the necessity for velocity-resolved spectroscopy in the excited CO transitions, to identify the bullets as distinct features. The significant emission of the high$-J$ CO line at high-velocity indicates that it is very likely related to the jet component, unlike the wing component, which becomes weaker in these lines. Finally, the emission at low velocities is still considerable in our high$-J$ CO lines. This implies that the \emph{a priori} assumption made in the analysis of some ISO observations, namely that the excited CO emission originates exclusively from the high-velocity gas \citep[e.g.][]{Nisini00}, could be incorrect.
\begin{figure}[h!]
\centering
\includegraphics[bb=50 80 539 717,width=8cm,height=8cm]{spectrum-cepe-rekal.ps}
\caption{SOFIA/GREAT spectra of CO (12--11) and (13--12) in Cep E, together with the low$-J$ CO (2--1) transition, taken at the position shown in Fig.~\ref{irac-2}. The CO (13--12) and (2--1) spectra are taken from data with a spatial resolution of 20\arcsec, while the spatial resolution of the CO (12--11) is 21\farcs7. The two upper spectra have been shifted along the Y-axis. The green horizontal lines show the zero level of each spectrum. The dashed vertical lines indicate the limits of each velocity range defined in Table~\ref{tab:velint}. The dotted vertical line shows the systemic velocity ($-$11 km s$^{-1}$). The spectral resolution is 3 km s$^{-1}$ for all lines.
}
\label{sofia-spec}
\end{figure}
\begin{table}
\begin{minipage}{\columnwidth}
\caption{Velocity ranges and integrated intensities.}
\label{tab:velint}
\renewcommand{\footnoterule}{}
\begin{tabular}{l c c c}
\hline
CO & \multicolumn{3}{c}{$\int T_{\rm MB}$dv (K km s$^{-1}$)\footnote{Statistical errors in parenthesis.}}\\
\cline{2-4}
Transition & EHV & IHV & SHV\\
\hline
13--12 & 4.8(0.9) & 14.9(1.0) & 16.1(0.8)\\
12--11 & 6.0(1.0) & 19.5(1.2) & 17.7(0.9)\\
2--1 & 5.1(0.2) & 8.8(0.3) & 38.9(0.2)\\
\hline
\multicolumn{4}{l}{EHV: $-140$ to $-100$ km s$^{-1}$; IHV: $-100$ to $-50$ km s$^{-1}$}\\
\multicolumn{4}{l}{SHV: $-50$ to $-20$ km s$^{-1}$}\\
\hline
\end{tabular}
\end{minipage}
\end{table}
\section{Discussion: physical conditions}
\begin{figure
\centering
\includegraphics[width=6.2cm,angle=-90]{cepe-lvg-PT-nT.eps}
\caption{LVG results produced by the RADEX code for the observed intensity ratios in the EHV, IHV, and SHV ranges (left, middle, and right panels, respectively). The results are shown as $T$ vs. $n$ and $n \times T$ (thermal pressure) vs. $T$ plots (lower and upper panels, respectively). The CO (2--1)/(12--11) ratio is indicated in green and the CO (2--1)/(13--12) ratio in blue. The observed ratio is drawn with solid lines, while the line ratio uncertainty (including statistical and 20 \% of the calibration errors) with dashed lines. $N=2\times$10$^{17}$, 5$\times$10$^{17}$, and 7$\times$10$^{17}$ cm$^{-3}$; and $\Delta \varv$ = 40, 50, and 30 km~s$^{-1}$, for EHV, IHV, and SHV, respectively.
}
\label{lvg}
\end{figure}
To constrain the physical conditions, we performed radiative transfer calculations with the RADEX code \citep{Tak07} based on the large velocity gradient (LVG) approximation and assuming a plane-parallel \lq slab' geometry for the escape probability formula. The molecular data were retrieved from the LAMDA data base\footnote{http://www.strw.leidenuniv.nl/$\sim$moldata/}. The collisional rate coefficients were adopted from \citet{Yang10}, who calculated the collisional rates between CO and H$_2$, incorporating energy levels up to $J = 40$ for kinetic temperatures of up to 3000 K.
For the purposes of our study of the physical conditions as a function of velocity, we use the integrated intensities in the EHV, IHV, and SHV ranges (Table~\ref{tab:velint}). The value that we adopt for the linewidth, $\Delta \varv$, is directly inferred from the definition of our different velocity ranges (40, 50, and 30 km s$^{-1}$, for EHV, IHV and SHV, respectively). Our background radiation field is assumed to only be produced by the CMB. Then, our procedure consists in running, for each velocity range, a grid of RADEX models to compute the integrated intensities within a three-dimensional parameter space defined by $T$ (kinetic temperature), $n$ (H$_2$ volume density), and $N$(CO) (CO column density).
The LVG results were analyzed based on the (2--1)/(12--11), (2--1)/(13--12) line ratios and the absolute integrated intensities of those three lines. We point out that owing to the use of these ratios the results are biased toward the lower excitations traced by the low$-J$ CO lines, and hence our results should be assumed to be lower limits. We use the IR size of the BI ($\sim$3\arcsec$\times$4\arcsec) and BII ($\sim$8\arcsec$\times$8\arcsec) knots to compute the brightness temperature, correcting for beam dilution effects. For the SHV component, we assumed the size of BII, while for the IHV and EHV components the size of BI was assumed. These assumptions are based on the knowledge that the EHV and IHV low$-J$ CO emission peaks around BI, while the SHV emission peaks around BII \citep{Hatchell99}. The LVG solutions were found when both the ratios and absolute integrated intensities match in the ($T$, $n$) plane. The $N$(CO) ranges constrained in this way are 4-9$\times$10$^{17}$, 4-6$\times$10$^{17}$, and 1-4$\times$10$^{17}$ cm$^{-2}$, for the SHV, IHV, and EHV components, respectively. We tested the influence of the size of the emission region on our $N$(CO) conclusions by varying the assumed size from a compact emission scenario, where the emission spreads over half the BI knot size, to a very extended case in which the filling factor is equal to one. Over this range, we found that $N$(CO) varies over up to two orders of magnitude. However, this change in $N$(CO) does not modify significantly the solution of the line ratios described below. In Fig.~\ref{lvg}, we show the RADEX results as plots of $T$ versus (vs.) $n$ and ($n \times T$) vs. $T$. In the latter, ($n \times T$) represents the thermal pressure, and is shown because in the LVG analysis the results are not as degenerate as for $T$ vs. $n$ (see Fig.~\ref{lvg}). The (2--1)/(12--11) and (2--1)/(13--12) ratios yield lower limits to the parameters of $T \gtrsim$ 100 K, $n \gtrsim$ 4.2$\times$10$^4$~cm$^{-3}$, and ($n \times T$) $\gtrsim 10^7$ K cm$^{-3}$. These lower limits indicate that the line ratios actually trace high densities and temperatures, but the remarkable result of our LVG analysis is that the thermal pressure tends to be higher in the IHV range than in both the EHV and SHV ranges. In addition, it is quiet a surprise that both EHV and SHV provide similar constraints, since it is usually found that the EHV component (probably related to the jet) has higher excitation than the SHV component (which is likely to trace the outflow cavity).
The H$_2$O mapping observations at 183 GHz of \citet{Lefloch11}, which covered the blue lobe and the central region of Cep-E with an angular resolution of $\sim$13\arcsec, also showed prominent bullet-like spectral features in the IHV range. The strongest emission was found around the central region, while weaker and no emission at all was found at BI and BII, respectively. Together with additional SiO (8--7) observations, LVG calculations made for the gas in the central region yielded the physical conditions of $T\sim$ 200 K and $n\sim$10$^6$ cm$^{-3}$, which are comparable to the ones reported here, suggesting that the SiO, H$_2$O, and high$-J$ CO emission arises from gas with similar physical conditions, despite the different frequency range and positions probed. A correlation between the physical conditions of these molecules is an expected result given that the considered formation routes of SiO and H$_2$O associate themselves with warm and dense gas. However additional observations are required to verify the spatial correlation between these molecules and to provide more accurate determinations of the physical conditions.
Previous ISO observations of CO transitions from (14--13) up to (25--26), yielded kinetic temperatures and H$_2$ densities in the range 200-1200~K and 4$\times$10$^4$-4$\times$10$^6$~cm$^{-3}$, respectively. These constraints were obtained by fitting LVG models to the total integrated emission of these spectrally unresolved high$-J$ CO lines \citep{Gia01,Moro01}. It is remarkable that our lower limits are in good agreement with the values obtained from ISO, although our results are given for three distinct velocity ranges over the same global one. However, a more complete LVG analysis is needed to strengthen our conclusions with respect to previous ISO studies, and to more clearly probe the physical conditions prevailing in the observed regions. Such an analysis should be based on additional observations of both lower- and higher-$J$ CO transitions with the GREAT spectrometer, such as the (11--10) and (16--15) ones, the latter would also allow us to make direct comparisons with ISO observations.
Spectrally resolved \emph{Herschel}/HIFI observations have been used to study the emission in different velocity ranges in low-mass outflows by measuring the ratios of low$-$ to high$-J$ CO lines, which is an approach that is similar to our analysis. The kinetic temperatures generally inferred by those studies for the outflow components are in the range 100-200 K \citep{Yild10,Bjer11}. These values are similar to our lower limits. As already pointed out, the line ratios of a few high$-J$ CO lines with low$-J$ CO lines bias the results toward lower excitation conditions, and could in the end be inappropriate owing to the different physical component possibly related to each of them, as it has been proven in a few outflows \citep{Kempen10}. In this respect, additional observations of more high$-J$ CO lines, as well low- and mid$-J$ CO transitions, are required to test the possibility of the contribution of different physical components to the CO emission in outflows. Eventually, the combination of a maximum number of CO lines will provide the optimal way to shed light on the physical processes responsible for the existence of these components \citep[e.g.][]{Gusdorf12,Visser12}.
\section{Conclusions}
The principal conclusions of the present study are:
\begin{enumerate}
\item Our analysis of SOFIA/GREAT data has demonstrated that high$-J$ CO lines are a good tracer of molecular bullets in protostellar outflows.
\item {The bullet at intermediate-to-high velocities has a higher level of excitation than the low and extremely high velocity gas, at the observed position of the Cep-E outflow.}
\item The still considerable low-velocity emission in high$-J$ CO lines should be taken into account when modeling data.
\item More and higher$-J$ CO transitions must be observed to break the degeneracy in the LVG solutions.
\end{enumerate}
\begin{acknowledgements}
Based on observations made with the NASA/DLR Stratospheric Observatory for Infrared Astronomy. SOFIA Science Mission Operations are conducted jointly by the Universities Space Research Association, Inc., under NASA contract NAS2-97001, and the Deutsches SOFIA Institut under DLR contract 50 OK 0901. We thank B. Lefloch for providing us with the CO data from IRAM-30m. A. Gusdorf acknowledges support by the grant ANR-09-BLAN-0231-01 from the French {\it Agence Nationale de la Recherche} as part of the SCHISM project.
\end{acknowledgements}
|
{
"timestamp": "2012-03-09T02:04:27",
"yymm": "1203",
"arxiv_id": "1203.1890",
"language": "en",
"url": "https://arxiv.org/abs/1203.1890"
}
|
\section*{I. Introduction}
In recent years progress has been achieved in the understanding of the heterogeneous dynamics observed in supercooled liquids and glassy systems\cite{Sillescu99, Berthier:2011p6852}.
Starting with NMR experiments\cite{SRS91, HWZS95, G13} a number of frequency-selective techniques have been developed in order to investigate the nature of the dynamic heterogeneities in the slow primary relaxation of supercooled liquids\cite{G23, Ediger00, Israeloff00, Richert02}.
Also the length scale associated with the heterogeneities could be determined in some
cases\cite{Tracht98, Reinsberg01}.
In the experimental studies the system always is monitored at more than two times via the observation of four-time correlation functions as in the quoted NMR experiments.
Alternatively, large external fields are applied giving rise to nonlinear effects as in the nonresonant hole burning studies\cite{SBLC96, G16}.
Furthermore, in computer simulations on model systems dynamic heterogeneities have been
observed via following certain trajectories\cite{Kob97, Doliwa98} or also via the calculation of four-time correlation functions\cite{Schroder03, Reichman07}.
Most of the studies on dynamic heterogeneities were concerned with systems in thermal equilibrium, but also aging glasses have been investigated\cite{Ediger02, Lunki05}.
Heterogeneous aging has also been studied theoretically in spin glasses\cite{Castillo02}, in simple spin models\cite{G48} and also in a free-energy landscape model for glassy relaxation\cite{G56}.
In recent years, both experimental techniques and theoretical tools have been refined in order to allow detailed investigations of dynamic heterogeneities.
In particular, it has been recognized that higher-order correlation functions that probe the system at different times and different locations in space can be used to observe a length scale\cite{Berthier05} and the relevant four-point correlation function $\chi_4(t)$ has been studied
theoretically\cite{Toninelli05, Berthier07a, Berthier07b}.
Earlier experimental studies used the approximative relation of $\chi_4(t)$ to a two-point correlation
function\cite{Berthier05, DalleFerrier07} in order to extract the number of cooperatively rearranging particles, $N_{\rm corr}$.
In an influential paper Bouchaud and Biroli related the nonlinear (cubic) response $\chi_3(\omega,T)$ to
$\chi_4(t)$\cite{Bouchaud05}.
The experimental determination of $\chi_3(\omega,T)$ allowed the determination $N_{\rm corr}$ more
directly\cite{CrausteThibierge10, Brun11} and the results are compatible with the earlier observations.
In particular, it was argued that the function
\be\label{X3.Def}
X(\omega,T)=\left|\chi_3(\omega,T)\right|{k_BT\over(\Delta\chi_1)^2a^3}
\ee
with $\Delta\chi_1$ denoting the static linear response, $k_B$ the Boltzmann constant and $a^3$ the molecular volume, exhibits a hump-like structure.
This behavior is assumed to be a distinctive feature of glassy correlations\cite{CrausteThibierge10}.
Additionally, the maximum of $X(\omega,T)$ is expected to decrease with increasing temperature and to be directly proportional to $N_{\rm corr}$.
If glassy correlations are absent, $X(\omega,T)$ should not be peaked and this 'trivial' behavior consists in a smooth cross-over from a low-frequency limiting value to a vanishing high-frequency limit.
In this context it has to be mentioned that Brun et al. found a hump-like shape for $X(\omega,T)$ in a calculation employing the so-called box model\cite{Brun11b}, a model devoid of spatial aspects.
Apart from the determination of $N_{\rm corr.}$ the nonlinear dielectric response has been used to investigate the nature of the heterogenous dynamics via comparison of the cubic response with the linear
response\cite{Richert06, Weinstein07} and the results were discussed in the framework of the box model.
Similar measurements were performed in order to extract the configurational heat capacity of
liquids\cite{Wang07}.
In addition also the nonlinear dielectric response of liquids due to an AC and an DC field pulse have been
recorded\cite{Rzoska12} and also dipolar glasses have been investigated\cite{Roland98}.
The present paper deals with the theory of nonlinear response functions for Markov processes, because the relaxation in complex systems often is modeled in terms of such stochastic dynamics.
For systems that follow a Hamiltonian or Langevin dynamics, nonlinear response functions have been considered quite some time ago\cite{BC59, APR76, Morita86}.
However, explicit calculations of response functions are rare and most of them relate to variants of the rotational diffusion of molecules in the presence of strong electric fields, see e.g.
refs.\cite{DD95, Dejardin00, Kalmykov01, G57}.
In addition, approximate nonlinear response theory has been investigated more generally\cite{Dyre89} and also fluctuation-dissipation relations beyond the linear regime have been discussed\cite{Eyink00, Lippiello08}.
The nonlinear response of supercooled liquids has been worked out theoretically in the framework of mode-coupling theory\cite{Tarzia10}.
Here, I will perform the calculation of the response functions in close analogy to the quantum-mechanical way of computing response functions\cite{muk95}.
Time-dependent perturbation theory for the propagator is used in order to obtain the response in the desired order in the amplitude of the external field.
I will present the results of calculations of the cubic response function for two Markovian models of relaxation.
One model describes the reorientations of dipoles in an asymmetric double well potential (ADWP) and has been used to interpret results of dielectric experiments in general\cite{Frohlich49}.
Furthermore, it has also been employed in calculations of the signals obtained in nonresonant holeburing experiments\cite{G46}.
It will be shown that $X(\omega,T)$ mainly behaves 'trivially' for this model.
Another model that will be considered is the trap model with a Gaussian density of states\cite{Dyre95, MB96}.
This model has been used in the interpretation of some features of the relaxation in simulated supercooled
liquids, both in equilibrium\cite{Denny03} and in the aging regime\cite{G64, G71}.
Here, the results for $X(\omega,T)$ are more complex and, depending on the parameters chosen, either exhibit a peak-like structure or 'trivial' behavior.
The paper is organized as follows.
In the next section, I will outline the calculation of nonlinear response functions for systems obeying a master equation.
For convenience of the reader, most of the explicit calculations are presented in the Appendix.
The sections following this theoretical part deal with a discussion of the results obtained for the two models considered and the paper closes with some concluding remarks.
\section*{II. Nonlinear response theory for Markov processes}
In this section, I will outline the general procedure to calculate the nonlinear response functions for a system that is described by a master equation (ME)\cite{vkamp81, Gardiner97}.
If one is dealing with complex systems a coarse-grained procedure may result in a description of the underlying dynamics in terms of a non-stationary Markov process.
Therefore, in order to keep the treatment general, I will treat the case of a ME with time-dependent transition rates.
In the following, $G_{kl}(t,t_0)$ denotes the conditional probability to find the system in state $k$ at time $t$ provided it was in state $l$ at time $t_0$ (Green's function, propagator) in a discrete notation.
If continuous variables are considered, all sums in the following expressions are to be replaced by the corresponding integrals.
Denoting the rates for a transition from state $k$ to state $l$ by $W_{lk}(t)$, the ME reads:
\be\label{ME.t.abh}
{\partial\over\partial t}G_{kl}(t,t_0)=
-\sum_nW_{nk}(t)G_{kl}(t,t_0)+\sum_nW_{kn}(t)G_{nl}(t,t_0)
\ee
This equation has to be solved with the initial condition $G_{kl}(t_0,t_0)\!=\!\delta_{kl}$, where $\delta_{kl}$ denotes the Kronecker symbol.
If the transition rates $W_{kl}(t)$ are time-independent the process considered is stationary.
The one-time probabilities $p_k(t)$ (the populations of the states) obey the same ME and are given by
$p_k(t)=\sum_lG_{kl}(t,t_0)p_l(t_0)$.
The $W_{kl}(t)$ can be related to the elements of the master-operator ${\cal W}(t)$ via\cite{vkamp81}:
\be\label{We.kl.t}
{\cal W}(t)_{kl}=W_{kl}(t)-\delta_{kl}\sum_nW_{nl}(t)
\ee
Here, ${\cal W}(t)_{kl}\geq 0$ holds for all $k\neq l$ and the sum rule $\sum_k{\cal W}(t)_{kl}=0$
is fulfilled for all values of $l$ as it is a general property of the transition rates for any Markov process.
At the initial time $t_0$ the system is described by a fixed set of populations,
$p_k^0\!=\!p_k(t_0)$ with $\sum_kp_k^0\!=\!1$.
If a stationary system is considered, one often starts from equilibrium populations $p_k^0=p_k^{\rm eq}$ or if one is interested in describing a situation with a certain thermal history one might choose the $p_k^0$ as the equilibrium populations at a temperature different from the working temperature.
In order to treat the system in the presence of an external field one has to specify the field-dependence of the transition rates, which is not straightforward.
In case of Hamiltonian or Langevin dynamics, the linear coupling of a variable $M(t)$ to a field $H(t)$ gives rise to an extra term $[-M(t)\cdot H(t)]$ in the Hamiltonian.
In a Fokker-Planck equation, this gives rise to a term linear in $H$\cite{Risken89}.
If one considers a ME, one choice that has been used in a number of investigations of fluctuation-disspiation relations is given by
\be\label{Wkl.HX}
W_{kl}^{(H)}(t)=W_{kl}(t)e^{\beta H[\gamma M_k-\mu M_l]}
\ee
with arbitrary $\gamma$ and $\mu$\cite{CR03, G39, G54}.
In this expression $\beta=T^{-1}$ denotes the inverse temperature with the Boltzmann constant set to unity, $k_B=1$.
If the system obeys detailed balance, one has the restriction $\gamma+\mu=1$.
In particular, for systems described by a Fokker-Planck equation, one would naturally choose $\gamma=\mu=1/2$ and a linear expansion of eq.(\ref{Wkl.HX}) gives the usual term in the Fokker-Planck operator.
However, it is obvious from eq.(\ref{Wkl.HX}) that in general one will have nonlinear contributions to the perturbation also if the coupling to the field is linear in the sense described above.
This means that couplings of a form like $[\tilde M(t)\cdot H^2]$, as it would appear for instance if the coupling to an induced dipole-moment is considered\cite{DD95}, are absent.
In order to keep the treatment general, I will formulate the response theory without fixing the field-dependence of the transition rates.
It is only assumed that it can be cast in the form:
\be\label{W.H.kl.Taylor}
W_{kl}^{(H)}(t)=\sum_{n=0}^\infty{1\over n!}W_{kl}^{(n)}(t)\cdot[\beta H(t)]^n
\quad\mbox{with}\left.\quad W_{kl}^{(n)}(t)={d^n\over d(\beta H)^n}W_{kl}^{(H)}(t)\right|_{H=0}
\ee
The elements of the propagator ${\bf G}^{(H)}(t,t_0)$ are obtained from the ME, eq.(\ref{ME.t.abh}), where the field-independent quantities are replaced by those explicitly depending on the external field, i.e.
$\dot G^{(H)}_{kl}(t,t_0)=-\sum_nW^{(H)}_{nk}(t)G^{(H)}_{kl}(t,t_0)+\sum_nW^{(H)}_{kn}(t)G^{(H)}_{nl}(t,t_0)$.
The solution of this equation is needed to calculate the response of the system to an external field applied at time $t_0$ and measured by an observable $F(t)$,
\be\label{F.expect}
\langle F(t)\rangle_{(H)}=\sum_{kl}F_kG^{(H)}_{kl}(t,t_0)p_k(t_0)
\ee
In order to be able to set up a perturbation theory for ${\bf G}^{(H)}(t,t_0)$ in terms of the corresponding 'field-free' propagator ${\bf G}(t,t_0)$, one uses the decomposition
\be\label{W.H.Vn}
{\cal W}^{(H)}(t)={\cal W}(t)+{\cal V}(t)
\quad\mbox{with}\quad
{\cal V}(t)=\sum_{n=1}^\infty{\cal V}^{(n)}(t)
\ee
where the perturbation is given according to eq.(\ref{W.H.kl.Taylor})
\be\label{Vn.def}
{\cal V}^{(n)}(t)_{kl}={[\beta H(t)]^n\over n!}\left[W_{kl}^{(n)}(t)-\delta_{kl}\sum_nW_{nl}^{(n)}(t)\right]
\ee
The theoretical treatment is very similar to the one utilized in ref.\cite{G54} and consists in performing time-dependent perturbation theory to treat ${\cal V}(t)$ in the desired order of the field.
The details of this procedure are described in Appendix A.
The explicit expressions for the response functions are given up to third order in the field and the extention to higher order is straightforward.
The main difference to the formalism utilized for Hamiltonian or Langevin dynamics with a linear coupling to the external field is that here in general the elements ${\cal V}^{(n)}(t)_{kl}$ with $n>1$ do not vanish.
This gives rise to a number of extra terms.
The situation is visualized in Fig.\ref{Plot1}, which shows the diagrams representing the interaction with the field for the third-order response.
\begin{figure}[h!]
\centering
\includegraphics[width=7.5cm]{Figure1}
\vspace{-0.5cm}
\caption{Pictorial representation of the perturbation expansion for the third-order response. The unperturbed propagators are denoted by ${\bf G}$ and the ${\cal V}^{(n)}$ are the perturbations according to
eq.(\ref{Vn.def}). }
\label{Plot1}
\end{figure}
One has the terms stemming from purely linear interactions given in the first line.
These terms also appear in a Fokker-Planck treatment of a linear coupling.
Furthermore, one has two cross terms between first-order and second-order perturbations (second and third line in Fig.\ref{Plot1}) and a term stemming from the third-order perturbation (fourth line).
For Langevin dynamics, cross-terms only appear if a quadratic coupling is considered in addition to a linear one.
While in Appendix A the general expressions for the response functions are given, in the actual model calculations I will consider only the response of systems that are in thermal equilibrium prior to the application of the external field.
Furthermore, the models treated in the present paper represent stationary Markov processes with time-independent transition rates.
The discussion will be limited to sinusoidal fields of the form
\be\label{H.cos.om.t}
H(t)=H_0\cos{(\omega t)}
\ee
For this oscillating field the linear and the cubic response for times long compared to the initial transients
can be written as:
\Be\label{Chi.om.def}
\chi^{(1)}(t)
&&\hspace{-0.6cm}=
{H_0\over2}\left[e^{-i\omega t}\chi_1(\omega)+c.c.\right]
\nonumber\\
\chi^{(3)}(t)
&&\hspace{-0.6cm}=
{H_0^3\over2}\left[e^{-i\omega t}\chi_3^{(1)}(\omega)+e^{-i3\omega t}\chi_3^{(3)}(\omega)+c.c.\right]
\Ee
where $c.c.$ denotes the complex conjugate.
In the following sections, I will mainly discuss the quantity $X(\omega,T)$ introduced in eq.(\ref{X3.Def}).
As the models that will be considered in the following are not related to any spatial aspects of dipole reorientations or relaxing units, the molecular volume will be set to unity, $a^3=1$.
Additionally, one has a separate function for each frequency-component, cf. ref.\cite{Brun11},
that can be written as ($\alpha=1, 3$):
\be\label{Xalfa.Def}
X_\alpha(\omega,T)={T\over(\Delta\chi_1)^2}\left|\chi_3^{(\alpha)}(\omega,T)\right|
\ee
This function eliminates the 'trivial' temperature dependence of $\chi_3^{(\alpha)}(\omega,T)$ because
$\Delta\chi_1\sim\beta$ , cf. eq.(\ref{Chi1}) and $\chi_3^{(\alpha)}\sim\beta^3$ according to eq.(\ref{Chi3}).
Therefore, any temperature dependence stems from the 'intrinsic' relaxation behavior of the dynamical variable considered.
\section*{III. The ADWP-model for dipole reorientations}
In this section, I will present the results for one of the simplest models for dielectric relaxation, namely the model of dipole reorientation in an asymmetric double well potential.
I will closely follow the notation used in a related investigation of the nonresonant dielectric hole burning technique\cite{SBLC96, G16, G46}.
As in ref.\cite{G46}, two dipole orientations denoted by '$1$' and '$2$', characterized by polar angles
$\theta_1=\theta$ and $\theta_2=\theta+\pi$ are assumed and the transition rates between the two are given by
$W_{12}=We^{-\beta\Delta/2}$ and $W_{21}=We^{+\beta\Delta/2}$.
Here $\Delta$ denotes the asymmetry, and $W$ is the hopping rate in the symmetric case.
For this model, the Green's functions in the field-free case are are given by:
\be\label{Gkl.ADWP}
G_{kl}(t)=p_k^{\rm eq}\left(1-e^{-t/\tau}\right)+\delta_{kl}e^{-t/\tau}
\quad\mbox{with}\quad \tau^{-1}=2W\cosh{\!(\beta\Delta/2)}
\quad\mbox{and}\quad p_k^{\rm eq}=\tau\cdot W_{kl}
\ee
The variable that couples to the field is
\[
M_k=M\cos{\!(\theta_k)}
\quad\mbox{and therefore}\quad
M_1=M\cos{\!(\theta)}
\quad;\quad
M_2=-M\cos{\!(\theta)}
\]
with $M$ denoting the static molecular dipole moment.
The field-dependent transition rates are chosen as in eq.(\ref{Wkl.HX}) with $\gamma=\mu=1/2$.
(If this restriction is relaxed all response functions depend on the sum $(\gamma+\mu)$, which equals unity in the present case.)
In the calculation of the response I assume a collection of systems characterized by an isotropic distribution of orientations and therefore an average over the angle $\theta$ is performed according to
$\langle\cos^n{\!(\theta_k)}\rangle=(n+1)^{-1}$ for $n$ even and $\langle\cos^n{\!(\theta_k)}\rangle=0$ for $n$ odd.
Using the general expressions given in Appendix A along with eq.(\ref{Gkl.ADWP}), one finds for the linear response:
\be\label{Chi1.ADWP}
\chi_1(\omega)=\Delta\chi_1{1\over1-i\omega\tau}
\quad\mbox{where}\quad
\Delta\chi_1=\beta\langle \Delta M^2\rangle
=\beta{M^2\over3}\left(1-\delta^2\right)
\ee
In this expression, I defined $\delta=\tanh{\!(\beta\Delta/2)}$. (It should be mentioned that $\Delta\chi_1$ differs by a factor $1/2$ from the definition of $\chi_{DWP}$ in ref.\cite{G46}.)
As usual, $\Delta\chi_1$ is related to the mean-square fluctuations of the dipole moment
$\langle \Delta M^2\rangle$.
Eq.(\ref{Chi1.ADWP}) follows immediately from the definition $\langle M^m\rangle=\sum_kM_k^mp_k^{\rm eq}$ and
eq.(\ref{Gkl.ADWP}) with additional isotropic average.
Note that in the ADWP-model, the static susceptibility $\Delta\chi_1$ for non-vanishing asymmetry depends on temperature due to the dependence on $\delta$ in addition to the trivial $1/T$-dependence.
This behavior for finite asymmetry is different from the model of Brownian rotational diffusion, where
$T\Delta\chi_1$ is independent of temperature\cite{Frohlich49}.
For vanishing asymmetry, the models show identical behavior (apart from irrelevant prefactors).
Without showing results here, it is mentioned that $\rm Re(\chi_1(\omega))$ decays from its low-frequency limit $\Delta\chi_1$ to zero for large frequencies and $\rm Im(\chi_1(\omega))$ shows the typical Lorentzian behavior and is peaked at $\omega\tau=1$.
The third-order response functions are calculated according to eq.(\ref{Chi.om.def}) using the general expressions given in eq.(\ref{Chi3}) in the Appendix.
In a straightforward calculation one finds:
\be\label{Chi3.ADWP}
\chi_3^{(\alpha)}(\omega)={M^4\over20}\beta^3\left(1-\delta^2\right)\times S_3^{(\alpha)}(\omega\tau)
\ee
Here, the spectral functions only depend on the product $x=\omega\tau$ and are given by:
\Be\label{S3n.ADWP}
S_3^{(1)}(x)
&&\hspace{-0.6cm}=
\delta^2{3(1+i2x)\over(1+x^2)(1+4x^2)}+{2(x^2-1)+ix(x^2-3)\over2(1+x^2)^2}
\\
S_3^{(3)}(x)
&&\hspace{-0.6cm}=
\delta^2{(1-11x^2)+i6x(1-x^2)\over(1+x^2)(1+4x^2)(1+9x^2)}
+{2(5x^2-1)+i3x(x^2-3)\over6(1+x^2)(1+9x^2)}
\nonumber
\Ee
When compared to the model of Brownian rotational diffusion, the following can be observed.
For $\Delta=0$, $\chi_3^{(\alpha)}(\omega)$ for the two models are very similar, cf. Fig.\ref{Plot2} and Figs.3,4 of ref.\cite{Dejardin00}.
For finite $\Delta$, however, the third-order response for the ADWP-model shows a characteristic temperature dependence, that is absent in the model of rotational Brownian motion.
In Fig.\ref{Plot2}, the real and the imaginary part of the 3$\omega$-component $\chi_3^{(3)}(\omega)$ are plotted
versus $\omega\tau$ for different values of the asymmetry $\Delta$ and various temperatures.
\begin{figure}[h!]
\centering
\includegraphics[width=8.0cm]{Figure2}
\vspace{-0.5cm}
\caption{Real part (red) and imaginary part (black) of the 3$\omega$-component $\chi_3^{(3)}(\omega)$ for the ADWP-model as a function of $\omega\tau$, where $\tau$ is the relaxation time according to eq.(\ref{Gkl.ADWP}). }
\label{Plot2}
\end{figure}
It is evident that the sign of both functions change as a function of frequency.
Furthermore, the shapes of $\rm Im(\chi_3^{(3)}(\omega))$ differ significantly from Lorentzians.
As mentioned above, for $\Delta=0$, $\chi_3^{(3)}(\omega)$ does not depend on temperature.
The static nonlinear susceptibilites are determined by the limiting values of the spectral dfunctions, $S_3^{(1)}(0)=(3\delta^2-1)$ and $S_3^{(3)}(0)=(3\delta^2-1)/3$, and thus are given by:
\be\label{Chi3.0.ADWP}
\chi_3^{(3)}(0)={M^4\over60}\beta^3\left(3\delta^2-1\right)\left(1-\delta^2\right)
\quad;\quad
\chi_3^{(1)}(0)=3\chi_3^{(3)}(0)
\ee
It should be mentioned, that $\chi_3^{(\alpha)}(0)$ is determined by the fourth-order cumulant,
$\kappa_4(M)=\langle M^4\rangle-4\langle M\rangle\langle M^3\rangle-3\langle M^2\rangle^2+12\langle M\rangle^2\langle M^2\rangle-6\langle M\rangle^4
=2M^4\left(3\delta^2-1\right)\left(1-\delta^2\right)$.
For finite $\Delta$, the low-frequency limit $\chi_3^{(\alpha)}(0)$ vanishes at a temperature $T_0$, at which $S_3^{(\alpha)}(0)=0$,
\[
T_0=\Delta/\ln{[(\sqrt{3}+1)/(\sqrt{3}-1)]}\simeq\Delta/1.317.
\]
For large frequencies, one always has $\chi_3^{(\alpha)}(\infty)=0$.
Instead of discussing $\chi_3^{(\alpha)}(\omega)$ further, in the following I will consider $X_\alpha(\omega,T)$ according to eq.(\ref{Xalfa.Def}).
This quantity is given by, cf. eq.(\ref{Chi1.ADWP}) and eq.(\ref{Chi3.ADWP}):
\be\label{Xalfa.ADWP}
X_\alpha(\omega,T)={9\over20}{\left|S_3^{(\alpha)}(\omega\tau)\right|\over\left(1-\delta^2\right)}
\ee
The limiting values for small and large frequencies are determined by the corresponding limits of
$S_3^{(\alpha)}(\omega\tau)$ and thus, one has for example
$X_3(0,T)=(3/20)(\left|3\delta^2-1\right|/\left(1-\delta^2\right))$.
It is evident, that $X_\alpha(\omega,T)$ will have a peak-like structure for $T\simeq T_0$.
As is shown in Fig.\ref{Plot3}, for other temperatures one has 'trivial' behavior, i.e. a continuous decay from the low-frequency limit to $X_\alpha(\omega,T)=0$ at high frequencies.
\begin{figure}[h!]
\centering
\includegraphics[width=8.0cm]{Figure3}
\vspace{-0.5cm}
\caption{$X_\alpha(\omega,T)$ for various values of the asymmetry and different temperatures.
In the uppermost panel, $X_3^{(\rm Debye)}(\omega)$\cite{Dejardin00} is shown for comparison (dashed line). }
\label{Plot3}
\end{figure}
One can see, that the behavior of the $1\omega$-component and the $3\omega$-component is very similar.
In order to further quantify the behavior of $X_\alpha(\omega,T)$ with regard to a 'hump'-like structure, in Fig.\ref{Plot4}, the ratio $X_3^{\rm max}(\omega)/X_3(0)$ is plotted versus temperature.
\begin{figure}[h!]
\centering
\includegraphics[width=8.0cm]{Figure4}
\vspace{-0.5cm}
\caption{$X_3^{\rm max}(\omega)/X_3(0)$ versus temperatures for $\Delta=1$.
The dotted line is the same with assumption of a Gaussian distribution of $\Delta$ with mean $\overline\Delta=1$ and variance $\sigma_\Delta=10$.}
\label{Plot4}
\end{figure}
For $T\ll T_0$ and also for $T\gg T_0$ trivial behavior is observed and only in the region of $T\sim T_0$ a hump develops.
This hump, however, has nothing to do with glassy correlations but is solely a consequence of the temperature dependence of the fluctuations of the dipole moments.
Finally, it is to be mentioned that the above results hardly change if one considers distributions of the hopping rate $W$ and/or the asymmetry.
In particular, the temperature-dependent change in the shape of $X_\alpha(\omega)$ is practically unaltered.
This is exemplified in Fig.\ref{Plot4}, where the dotted line represents $X_3^{\rm max}(\omega)/X_3(0)$ for the case of a broad Gaussian distribution of $\Delta$.
The reason for this is simply the steepness of the root of $S_3^{(\alpha)}(0)=0$, meaning that the overall behavior is determined by the mean value of $\Delta$.
Thus, if one considers a system with a distribution of asymmetries that is centered at $\Delta=0$, one will observe trivial behavior of $X_\alpha(\omega)$ at all temperatures.
Ladieu et al. use the ADWP-model with finite $\Delta$ and some further assumptions to fit the experimental data on supercooled liquids\cite{Ladieu12}.
\section*{IV. Trap models}
In this section, I will discuss $X_\alpha(\omega,T)$ for the trap model with a Gaussian density of states, which,
as already mentioned in the Introduction, shows some features of glassy relaxation.
It is defined by the ME for $G(\epsilon,t+t_0|\epsilon_0,t_0)=G(\epsilon,t|\epsilon_0,0)\equiv G(\epsilon,t|\epsilon_0)$, in a continuous form written as:
\be\label{ME.G}
{\dot G}(\epsilon,t|\epsilon_0)= -\kappa(\epsilon)G(\epsilon,t|\epsilon_0)+\rho(\epsilon)\!\int\!d\epsilon'\kappa(\epsilon')G(\epsilon',t|\epsilon_0)
\ee
In eq.(\ref{ME.G}), the escape rate is given by
\be\label{k.T}
\kappa(\epsilon)=\kappa_\infty e^{\beta\epsilon}
\ee
with the attempt rate $\kappa_\infty$.
Furthermore, I solely consider the model with a Gaussian DOS
\be\label{DOS.Gauss}
\rho(\epsilon)\!=\!{1\over\sqrt{2\pi}\sigma}e^{-\epsilon^2/(2\sigma^2)}
\ee
with $\sigma=1$.
From eq.(\ref{ME.G}), the equilibrium populations at a given temperature $T$ (measured in units of $\sigma$) are found to be Gaussian
$p^{\rm eq}(\epsilon)=\lim_{t\to\infty}G(\epsilon,t|\epsilon_0)={1\over\sqrt{2\pi}\sigma}e^{-(\epsilon-{\bar\epsilon})^2/(2\sigma^2)}$
with ${\bar\epsilon}=-\beta \sigma^2$.
In order to calculate the response, one further has to quantify the dependence of the dynamical variable on the trap energy $\epsilon$.
The choice of this dependence represents a further assumption of the calculation and has a strong impact on the results for the cubic response, as will be discussed below.
In order to clarify this issue, consider the linear response for the specific choice of eq.(\ref{Wkl.HX}) for the field-dependence of the transition rates.
Using eqns.(\ref{F.expect}), (\ref{Lkl.def}) and (\ref{Chi1}), one obtains the relation between the linear response and the equilibrium auto-correlation function $C_M(t)=\langle M(t)M(0)\rangle$,
$R^{(1)}_M(t)=-\beta(\gamma+\mu)[dC_M(t)/dt]$, if the system is in thermal equilibrium \cite{G54}.
In the frequency-domain, this yields eq.(\ref{Chi1.Trap}) in Appendix B, if the average over the possible realizations of the variables is performed with the following assumption:
\be\label{MkMl.mit}
\langle M(\epsilon)\rangle=0
\quad\mbox{and}\quad
\langle M(\epsilon)M(\epsilon_0)\rangle=\delta(\epsilon-\epsilon_0)\langle M(\epsilon)^2\rangle
\ee
In the calculation of the third-order response, the fourth moments of the variable are important.
For the corresponding averages I will assume a Gaussian factorization property for simplicity:
\Be\label{Mh4.mit.Gauss}
\langle M(\epsilon_1)M(\epsilon_2)M(\epsilon_3)M(\epsilon_4)\rangle
&&\hspace{-0.6cm}=
\delta(\epsilon_1-\epsilon_2)\delta(\epsilon_3-\epsilon_4)\langle M(\epsilon_1)^2\rangle\langle M(\epsilon_3)^2\rangle
\nonumber\\
&&\hspace{-0.6cm}+\;
\delta(\epsilon_1-\epsilon_3)\delta(\epsilon_2-\epsilon_4)\langle M(\epsilon_1)^2\rangle\langle M(\epsilon_2)^2\rangle
\\
&&\hspace{-0.6cm}+\;
\delta(\epsilon_1-\epsilon_4)\delta(\epsilon_2-\epsilon_3)\langle M(\epsilon_1)^2\rangle\langle M(\epsilon_2)^2\rangle
\nonumber
\Ee
In the calculation of the response, the field-dependence of the transition rates has to be fixed additionally.
I use eq.(\ref{Wkl.HX}) with arbitrary values for $\gamma$ and $\mu$.
From the physics of the model one might argue that $\mu=1$ and $\gamma=0$ is an appropriate choice because it is meaningful to assume that the activation energy of the escape is biased by the field (according to
$\epsilon\to\epsilon-M(\epsilon)\cdot H)$.
However, it is not clear that this simple argument holds in out-of-equilibrium situations and for strong fields.
Using the assumptions made, one can compute the response according to the expressions given in Appendix A.
The calculation is outlined in Appendix B and here only the results will be discussed.
In the explicit choice of the variable, I follow Fielding and Sollich\cite{FS02} and use a set of variables with an Arrhenius-like dependence on the trap energies:
\be\label{Mh2.mit.n}
\langle M(\epsilon)^2\rangle=e^{-n\beta\epsilon}
\ee
with variable $n$ and where the static value of $M^2$ has been set to unity.
For $n=0$, one has temperature-independent variables as in case of Brownian rotational diffusion.
The most important consequence of the specific choice, eq.(\ref{Mh2.mit.n}), is that it does not affect the spectral shape of the linear response.
The only quantities that strongly depend on the choice of $n$ are the static susceptibility and the
the temperature dependence of the relaxation time.
This is because one can write:
\[
\int\!d\epsilon p(\epsilon)^{\rm eq}e^{-n\beta\epsilon}{\kappa(\epsilon)\over\kappa(\epsilon)-i\omega}
=e^{{n(n+2)\over2}\beta^2\sigma^2}
\int\!d\epsilon p(\epsilon)^{\rm eq}{\kappa(\epsilon)\over\kappa(\epsilon)-i\omega_n}
\]
with
\be\label{om.n.def}
\omega_n=\omega e^{n\beta^2\sigma^2}
\ee
Thus, the susceptibility is given by:
\be\label{Chi1.n.Trap}
\chi_1(\omega)=\beta(\gamma+\mu)\int\!d\epsilon p(\epsilon)^{\rm eq}e^{-n\beta\epsilon}{\kappa(\epsilon)\over\kappa(\epsilon)-i\omega}
=\Delta\chi_1\int\!d\epsilon p(\epsilon)^{\rm eq}{\kappa(\epsilon)\over\kappa(\epsilon)-i\omega_n}
\ee
The static susceptiblity, i.e. the amplitude, $\Delta\chi_1$, strongly depends on the choice of $n$ and reads as:
\be\label{DChi1.n.Trap}
\Delta\chi_1=(\gamma+\mu)\beta \overline{\langle M^2\rangle}_T
=(\gamma+\mu)\beta e^{{n(n+2)\over2}\beta^2\sigma^2}
\ee
Here, the second moment $\overline{\langle M^2\rangle}_T$ is related to the low-frequency limit of $\chi_1(\omega)$,
$\overline{\langle M^2\rangle}_T=\int\!d\epsilon\langle M(\epsilon)^2\rangle p(\epsilon)^{\rm eq}$.
Note that $\Delta\chi_1$ is temperature independent only for $n=0$ and for $n=-2$.
In Fig.\ref{Plot5}, the imaginary part of $\chi_1(\omega)$ is shown for $n=0$ and various temperatures.
\begin{figure}[h!]
\centering
\includegraphics[width=8.0cm]{Figure5}
\vspace{-0.5cm}
\caption{Imaginary part of $T\chi_1(\omega)$, $T\chi_1''(\omega)$, for $n=0$ and various temperatures ($T/\sigma$=0.5, 0.6, 0.7, 0.8, 0.9, 1.0 as indicated by the arrow).
The dotted line represents a Lorentzian.}
\label{Plot5}
\end{figure}
The frequencies are scaled to the relaxation time of $C_M(t)$ for $n=0$,
$\tau_{\rm eq}=\int_0^\infty\!\!dtC_M(t)=\kappa_\infty^{-1}e^{{3\over2}\beta^2\sigma^2}$, cf. ref.\cite{G64}.
It is obvious that $\chi_1''(\omega)$ broadens as temperature is decreased and thus time-temperature-superposition is not obeyed.
It is stressed again, that $\chi_1(\omega)$ is basically independent of the choice of $n$.
Next, the behavior of the cubic response and its dependence on the model parameters will be discussed.
Using the limiting values of the cubic response functions given in Appendix B for small and high frequencies, one finds the following limits for $\chi_3^{(\alpha)}(\omega)$:
\be\label{Chi3n.Limit.Trap}
\chi_3^{(3)}(0)={1\over8}\beta^3(\gamma+\mu)^3(\xi_2-\xi_1)
\quad;\quad
\chi_3^{(1)}(0)=3\chi_3^{(3)}(0)
\quad\mbox{and}\quad
\chi_3^{(\alpha)}(\infty)=0
\ee
Here, I defined the averages $\xi_1=\overline{\langle M^2\rangle}_\infty\overline{\langle M^2\rangle}_T$ and
$\xi_2=\overline{\langle M^2\rangle^2}_T$, which for the Gaussian trap model are given by:
\be\label{ksi.1.2.Trap}
\xi_1=e^{n(n+1)\beta^2\sigma^2}
\quad;\quad
\xi_2=e^{2n(n+1)\beta^2\sigma^2}
\ee
With these quantities, one finds for the low-frequency limit of $X_3$:
\be\label{X3.0.Trap}
X_3(0,T)={1\over8}(\gamma+\mu){|\xi_2-\xi_1|\over\left(\overline{\langle M^2\rangle}_T\right)^2}
\ee
and similarly for $X_1(0,T)$.
It is thus clear that these low-frequency limits do strongly depend on the variable, i.e. on $n$.
Therefore, one can expect to find trivial or hump-like behavior of $X_\alpha(\omega,T)$, $\alpha=1, 3$, depending on this choice.
In Fig.\ref{Plot6} $X_3(\omega,T)$ is shown for $n=0$ and various values of $\mu$. Here, it is assumed that
$\gamma+\mu=1$.
\begin{figure}[h!]
\centering
\includegraphics[width=8.0cm]{Figure6}
\vspace{-0.5cm}
\caption{$X_3(\omega,T)$ for $n=0$ and various values of $\mu$ for $\gamma=1-\mu$ and different temperatures
($T/\sigma=0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1$) in the order indicated by the arrow.}
\label{Plot6}
\end{figure}
The main difference between the various choices for $\mu$ is the overall amplitude.
Additionally, it is clear that $X_3(\omega,T)$ exhibits a hump in all cases.
However, in contrast to the results obtained on supercooled liquids, the maximum value of $X_3$ increases as a function of temperature.
This increase is somewhat stronger for $\mu=1$ than it is for other values of $\mu$.
Next, I will consider values for $n$ different from zero, meaning that the dynamical variable that couples to field shows an explicit dependence on the trap energies.
In Fig.\ref{Plot7}a, $X_3(\omega)$ is plotted versus frequency for $n=1$ and the same values for $\mu$ as in
Fig.\ref{Plot6}.
\begin{figure}[h!]
\centering
\includegraphics[width=8.0cm]{Figure7a}
\hspace{0.5cm}
\includegraphics[width=8.0cm]{Figure7b}
\vspace{-0.5cm}
\caption{{\bf a}: (left) $X_3(\omega,T)$ for $n=1$ and different temperatures ($T/\sigma=1, 1.5, 2, 2.5, 3$).
The arrow indicates increasing temperature.
{\bf b}: (right) $X_3^{\rm max}(\omega,T)$ as a function of temperature for $n=1$.
The curves are shown for temperatures higher than the onset temperature, below which trivial behavior is observed, i.e. $\omega_{\rm max}=0$.
The dotted line is the result for $n=0$, $\mu=1$.}
\label{Plot7}
\end{figure}
It is observed that a hump is found at high temperatures, whereas trivial behavior is observed at low temperatures.
The temperature, at which a visible peak is observed depends on the value of $\mu$, i.e. on the way, the field couples to the transition rates.
This is shown in Fig.\ref{Plot7}b, where the maximum value of $X_3(\omega)$ is plotted versus temperature for temperatures higher than the onset temperature, which is defined by the first appearance of a hump in $X_3(\omega)$
indicated by the dots in Fig.\ref{Plot7}b.
In the temperature range of a hump-like shape of $X_3(\omega,T)$ its maximum, $X_3^{\rm max}(\omega,T)$, appears to be almost independent of temperature.
A similar behavior is found for other positive values of $n$.
From these model calculations it becomes apparent that the existence of a hump depends on the value of
$X_\alpha(0)$, the value of the maximum of $X_\alpha(\omega)$, and in particular their ratio.
Thus, the low-frequency limit plays an important role in determining the overall shape of $X_\alpha(\omega)$.
These considerations can be further substantiated by considering the special value of $n=-1$, because in this case one has $\xi_1=\xi_2=0$ and therefore $X_\alpha(0)=0$, cf. eq.(\ref{ksi.1.2.Trap}).
This means, a hump will be observed in this case, as is confirmed in Fig.\ref{Plot8}a, where $X_3(\omega)$ is plotted as a function of frequency for $\mu=1$.
For other values of $\mu$, the results are very similar.
On first sight, the behavior of $X_3(\omega)$ is very similar to that for $n=0$, cf. Fig.\ref{Plot6}.
However, the maximum for $n=-1$, $X_3^{\rm max}(\omega,T)$, is a decreasing function of temperature as opposed to the case of $n=0$, cf. Fig.\ref{Plot8}b.
\begin{figure}[h!]
\centering
\includegraphics[width=8.0cm]{Figure8a}
\hspace{0.5cm}
\includegraphics[width=8.0cm]{Figure8b}
\vspace{-0.5cm}
\caption{{\bf a}: (left) $X_3(\omega,T)$ for $n=-1$, $\mu=1$ and different temperatures
($T/\sigma=0.6, 0.7, 0.8, 0.9, 1$ from top to bottom).
{\bf b}: (right) $X_3^{\rm max}(\omega,T)$ as a function of temperature for $n=-1$ and $n=0$ ($\mu=1$).}
\label{Plot8}
\end{figure}
At this point, however, it has to be noted that the case $n=-1$ is somewhat special as the mean relaxation time of the linear response,
$\langle\tau\rangle=\int\!d\epsilon p(\epsilon)^{\rm eq}(e^{-\beta\epsilon}/\kappa(\epsilon))=\kappa_\infty^{-1}$, is basically temperature-independent.
Thus, although the shapes of $\chi_1(\omega)$ are identical for $n=0$ and $n=-1$ at a given temperature, the mean relaxation time for $n=-1$ does not change with temperature.
This shows that it is not straightforward to compare linear and nonlinear response functions.
\section*{V. Conclusions}
The theory of nonlinear response functions for a system obeying a master equation has been formulated in close analogy to quantum mechanical nonlinear response theory.
Time-dependent perturbation theory is used in order to compute the elements of the propagator (the Green's function or conditional probability) in the desired order of the amplitude of the applied external field.
Expressions for the response functions up to third order are given in terms of the solution of the field-free master equation for systems with arbitrary initial conditions and also for non-stationary Markov processes.
In the actual model calculations, however, only stationary systems are considered that were in thermal equilibrium prior to the application of the field.
The treatment of aging systems or other non-equilibrium situations are beyond the scope of the present paper.
For the model of dipole reorientations in an asymmetric double well potential (ADWP-model), the spectral shape of the modulus of the frequency-dependent cubic response, $X_3(\omega,T)$, shows a specific temperature dependence which strongly depends on the value of the static susceptibility, $X_3(0,T)$.
At a temperature $T_0$, which is determined by the value of the asymmetry of the potential, $X_3(0,T_0)$ vanishes.
For a narrow temperature range in the vicinity of $T_0$ a peak is observed in the modulus.
For temperatures sufficiently different from $T_0$ a monotonous decay from $X_3(0)\neq0$ to $X_3(\infty)=0$ is found.
This 'trivial' behavior is basically the same as for the model of rotational Brownian motion\cite{Dejardin00} and is at variance with experimental results obtained for supercooled glycerol\cite{CrausteThibierge10, Brun11}.
It was attributed to trivial dipole reorientations that occur independent of glassy correlations.
These correlations should give rise to a peaked behavior, i.e. the existence of a hump in $X_3(\omega)$.
If one intends to utilize the ADWP model for the dipole reorientations in supercooled liquids, it is natural to assume distributions of relaxation rates and of asymmetries.
However, as shown in Section III, such a distribution hardly affects the spectral shape of $X_3(\omega)$ (apart from the fact that a distribution of relaxation times gives rise to a broadening).
If a trap model with a Gaussian distribution of trap energies is considered, a more complex dependence of $X_3(\omega)$ on the parameters used in the calculations is observed.
In particular, the dependence of the dynamical variables that couple to the external field on the trap energies, $M(\epsilon)$, has to be fixed.
I restricted the calculations to variables that obey Gaussian statistics and depend on the trap energy in an exponential way, $M(\epsilon)=e^{-n\beta\epsilon}$, cf. eq.(\ref{Mh2.mit.n}).
This choice is particularly useful when discussing the properties of nonlinear response functions and their relation to the linear response.
This is because the exponential dependence on the trap energies has the interesting property that the spectral shape of the linear susceptibility is the same for all values of the parameter $n$.
Only the amplitude ($\Delta\chi_1$) and the temperature dependence of the relaxation time strongly depend on its specific value.
If the nonlinear response is considered, it is however found that the temperature-dependent spectral shape of $X_3(\omega)$ strongly depends on the value of $n$.
In particular, one can find a peak or 'trivial' behavior depending on both, the value of $n$ and the temperature.
Similar to the situation in the ADWP-model, the existence of a peak is related to the value of the static susceptibility.
In case of the occurrence of a hump, the temperature dependence of the peak maximum, $X_3^{\rm max}(\omega)$, can increase ($n=0$) or decrease ($n=-1$) with increasing temperature.
These results indicate that it is difficult to compare the linear and nonlinear susceptibilities.
It is left for future work to investigate the behavior of the nonlinear response in the trap model for other dynamical variables and also for non-equilibrium situations.
In the experimental determination of $X_3(\omega,T)$ in supercooled liquids\cite{CrausteThibierge10, Brun11}, the decrease of $X_3^{\rm max}(\omega)$ with increasing temperature has been used to extract the number of correlated molecules, $N_{\rm corr}$, which is a 'real space property' of the dynamical heterogenities in glasses.
Due to the mean-field nature of both models considered in the present paper, none of the results presented have any connection to real space.
Therefore, a direct comparison to experimental data is not possible.
However, the model calculations substantiate the fact observed earlier already\cite{Brun11b} that the existence of a peak in $X_3(\omega)$ does not have to be related to glassy correlations in some sense.
In conclusion, I have formulated a theory of nonlinear response for systems described by Markov processes and have presented the results of calculations for simple stochastic models.
The most important result is that the spectral shape of the nonlinear (cubic) response can vary considerably depending on the model considered.
The occurrence of a peak in the modulus of the third-order susceptibility cannot generally be attributed to glassy correlations.
Of course, this does not mean that glassy correlations do not give rise to a hump but its mere existence cannot be taken as a signature of such correlations.
Because the models considered in the present paper are of a mean-field nature, it is impossible to connect the results to a length scale of any kind.
Due to the growing interest in nonlinear responses in complex systems, calculations of the kind presented in the present paper should be performed for a variety of different models in order to gain a deeper understanding of the general features governing the shape and the temperature dependence of the corresponding
susceptibilities.
\section*{Acknowledgment}
I thank Roland B\"ohmer, Gerald Hinze, Francois Ladieu and Jeppe Dyre for fruitful discussions and
Roland B\"ohmer for helpful comments on the mansucript.
\newpage
\begin{appendix}
\section*{Appendix A: Calculation of nonlinear response functions}
\setcounter{equation}{0}
\renewcommand{\theequation}{A.\arabic{equation}}
In this appendix the calculation of the response for a system obeying the ME, eq.(\ref{ME.t.abh}), using time-dependent perturbation theory is described.
Using eq.(\ref{We.kl.t}) for the master-operator, the ME in a matrix notation reads:
\be\label{ME.matrix}
\partial_t{\bf G}(t,t_0)={\cal W}(t){\bf G}(t,t_0)
\ee
Here, the propagator has matrix-elements ${\bf G}(t,t_0)_{kl}=G_{kl}(t,t_0)$.
The solution of the ME in the absence of an external field can be written in the form:
\be\label{G.LsgME.allg}
{\bf G}(t,t_0)={\cal T}\exp{\left(\int_{t_0}^td\tau{\cal W}(\tau)\right)}{\bf G}(t_0,t_0)
\ee
where ${\cal T}$ denotes the time-ordering operator and ${\bf G}(t_0,t_0)_{kl}=\delta_{kl}$.
In the presence of the field the transition rates are given by eq.(\ref{W.H.kl.Taylor}) and the corresponding master-operator accordingly reads as ${\cal W}^{(H)}(t)_{kl}=W^{(H)}_{kl}(t)-\delta_{kl}\sum_nW^{(H)}_{nl}(t)$.
The ME is written as $\partial_t{\bf G}^{(H)}(t,t_0)={\cal W}^{(H)}(t){\bf G}^{(H)}(t,t_0)$.
In order to calculate the response of the system to an external field applied at time $t\!=\!t_0$ and measured by an observable $F(t)$, $\langle F(t)\rangle_{(H)}=\sum_{kl}F_kG^{(H)}_{kl}(t,t_0)p_k(t_0)$ as given in eq.(\ref{F.expect}), time-dependent perturbation theory is used to express the propagator as a series of the form
${\bf G}^{(H)}(t,t_0)={\bf G}(t,t_0)+\sum_{n=1}^\infty{\bf G}^{(n)}(t,t_0)$, where ${\bf G}(t,t_0)$ denotes the propagator in the field-free case.
In order to perform the calculation, one proceeds in the following way.
Starting from the Dyson-like equation
\be\label{G.Dyson}
{\bf G}^{(H)}(t,t_0)={\bf G}(t,t_0)+\int_{t_0}^t\!dt'{\bf G}(t,t'){\cal V}(t'){\bf G}^{(H)}(t',t_0)
\ee
one obtains, using eq.(\ref{W.H.Vn}), for the lowest order terms:
\Be\label{G.Reihe}
{\bf G}^{(1)}(t,t_0)
&&\hspace{-0.6cm}=
\int_{t_0}^t\!dt'{\bf G}(t,t'){\cal V}^{(1)}(t'){\bf G}(t',t_0)
\nonumber\\
{\bf G}^{(2)}(t,t_0)
&&\hspace{-0.6cm}=
\int_{t_0}^t\!dt'{\bf G}(t,t'){\cal V}^{(2)}(t'){\bf G}(t',t_0)
+\int_{t_0}^t\!dt'{\bf G}(t,t'){\cal V}^{(1)}(t'){\bf G}^{(1)}(t',t_0)
\nonumber\\
{\bf G}^{(3)}(t,t_0)
&&\hspace{-0.6cm}=
\int_{t_0}^t\!dt'{\bf G}(t,t'){\cal V}^{(3)}(t'){\bf G}(t',t_0)
+\int_{t_0}^t\!dt'{\bf G}(t,t'){\cal V}^{(1)}(t'){\bf G}^{(2)}(t',t_0)
\\
&&\hspace{4.82cm}
+\int_{t_0}^t\!dt'{\bf G}(t,t'){\cal V}^{(2)}(t'){\bf G}^{(1)}(t',t_0)
\nonumber
\Ee
In the next step, one uses the expression for the matrix elements of ${\bf G}^{(n)}(t,t_0)$, denoted by
$G^{(n)}_{kl}(t,t_0)$, in eq.(\ref{F.expect}) in order to compute the nth-order response,
$\chi^{(n)}_F(t,t_0)$.
With the definition
\be\label{Lkl.def}
L_{kj}^{(\eta)}(t_2,t_1)=\sum_m\left[G_{km}(t_2,t_1)-G_{kj}(t_2,t_1)\right]W_{mj}^{(\eta)}(t_1)
\ee
where $W_{mj}^{(n)}(t_1)$ is given in eq.(\ref{W.H.kl.Taylor}), one obtains in a straightforward calculation for the linear response:
\be\label{Chi1}
\chi^{(1)}_F(t,t_0)=\int_{t_0}^t\!dt_1H(t_1)R^{(1)}_F(t,t_1)
\quad\mbox{with}\quad
R^{(1)}_F(t,t_1)=\beta\sum_{k,l}F_kL_{kl}^{(1)}(t,t_1)p_l(t_1)
\ee
From the structure of this expression it is evident that $R$ denotes the usual response to a short field kick.
The second-order response is found to consist of two terms:
\be\label{Chi2}
\chi^{(2)}_F(t,t_0)=\chi^{(2;1)}_F(t,t_0)+\chi^{(2;2)}_F(t,t_0)
\ee
with
\Be\label{R2.def}
\chi^{(2;1)}_F(t,t_0)
&&\hspace{-0.6cm}=
\int_{t_0}^t\!dt_1H(t_1)\int_{t_0}^{t_1}\!dt_2H(t_2)R^{(2;1)}_F(t,t_1,t_2)
\nonumber\\
&&\hspace{-0.6cm}
R^{(2;1)}_F(t,t_1,t_2)=
\beta^2\sum_{k,l,m}F_kL_{km}^{(1)}(t,t_1)L_{ml}^{(1)}(t_1,t_2)p_l(t_2)
\nonumber\\
\chi^{(2;2)}_F(t,t_0)
&&\hspace{-0.6cm}=
{1\over2}\int_{t_0}^t\!dt_1H(t_1)^2R^{(2;2)}_F(t,t_1)
\\
&&\hspace{-0.6cm}
R^{(2;2)}_F(t,t_1)=
\beta^2\sum_{k,l}F_kL_{kl}^{(2)}(t,t_1)p_l(t_1)
\nonumber
\Ee
This second order response is expected to be of little relevance in most cases as it vanishes in isotropic systems.
More interesting is the third-order response because usually this is the lowest-order nonlinear contribution to the response of the system.
As can be expected from Fig.\ref{Plot1}, it has the form:
\be\label{Chi3}
\chi^{(3)}_F(t,t_0)=\chi^{(3;1)}_F(t,t_0)+\chi^{(3;2)}_F(t,t_0)+\chi^{(3;3)}_F(t,t_0)
\ee
and the individual terms are given by:
\Be\label{R31.def}
\chi^{(3;1)}_F(t,t_0)
&&\hspace{-0.6cm}=
\int_{t_0}^t\!dt_1H(t_1)\int_{t_0}^{t_1}\!dt_2H(t_2)\int_{t_0}^{t_2}\!dt_3H(t_3)
R^{(3;1)}_F(t,t_1,t_2,t_3)
\\
&&\hspace{-0.6cm}
R^{(3;1)}_F(t,t_1,t_2)=
\beta^3\!\sum_{k,l,m,n}F_kL_{km}^{(1)}(t,t_1)L_{mn}^{(1)}(t_1,t_2)L_{nl}^{(1)}(t_2,t_3)p_l(t_3)
\nonumber
\Ee
originating from the linear perturbation. This term also is found in the response theory for a Fokker-Planck equation.
The cross-terms between the first- and second-order perturbations are:
\Be\label{R32.def}
\chi^{(3;2)}_F(t,t_0)
&&\hspace{-0.6cm}=
\chi^{(3;2A)}_F(t,t_0)+\chi^{(3;2B)}_F(t,t_0)
\nonumber\\
&&\hspace{-0.6cm}
\chi^{(3;2A)}_F(t,t_0)=
{1\over2}\int_{t_0}^t\!dt_1H(t_1)\int_{t_0}^{t_1}\!dt_2H(t_2)^2R^{(3;2A)}_F(t,t_1,t_2)
\nonumber\\
&&\hspace{1.6cm}
R^{(3;2A)}_F(t,t_1,t_2)=
\beta^3\sum_{k,l,m}F_kL_{km}^{(1)}(t,t_1)L_{ml}^{(2)}(t_1,t_2)p_l(t_2)
\\
&&\hspace{-0.6cm}
\chi^{(3;2B)}_F(t,t_0)=
{1\over2}\int_{t_0}^t\!dt_1H(t_1)^2\int_{t_0}^{t_1}\!dt_2H(t_2)R^{(3;2B)}_F(t,t_1,t_2)
\nonumber\\
&&\hspace{1.6cm}
R^{(3;2B)}_F(t,t_1,t_2)=
\beta^3\sum_{k,l,m}F_kL_{km}^{(2)}(t,t_1)L_{ml}^{(1)}(t_1,t_2)p_l(t_2)
\nonumber
\Ee
Finally, the third-order contribution is:
\Be\label{R33.def}
\chi^{(3;3)}_F(t,t_0)
&&\hspace{-0.6cm}=
{1\over6}\int_{t_0}^t\!dt_1H(t_1)^3R^{(3;3)}_F(t,t_1)
\\
&&\hspace{-0.6cm}
R^{(3;3)}_F(t,t_1)=
\beta^3\sum_{k,l}F_kL_{kl}^{(3)}(t,t_1)p_l(t_1)
\nonumber
\Ee
These expressions are valid for any Markov process obeying the ME, eq.(\ref{ME.t.abh}) and arbitrary initial conditions (initial populations $p_l(t_0)$).
If the process considered is stationary, meaning that the transition rates are time-independent,
$W_{kl}(t)=W_{kl}$, the Green's functions depend only on the time-differences,
$G_{kl}(t_2,t_1)=G_{kl}(t_2-t_1)$.
Furthermore, if the system was in equilibrium initially, $p_l(t_0)=p_l^{\rm eq}$, the expressions simplify considerably.
In this case, the integrals can easily be transformed in order to find expressions for $\chi^{(n)}(t-t_0)$ that are reminiscent of the standard ones used for instance in the field of nonlinear optics\cite{muk95}.
If one is interested in the stationary response, one just starts recording $\chi^{(n)}(t-t_0)$ after times long compared to the initial transients.
Finally, the explicit choice of the field-dependence of the transition rates enters via eq.(\ref{Lkl.def}).
\section*{Appendix B: Nonlinear response functions for the trap model}
\setcounter{equation}{0}
\renewcommand{\theequation}{B.\arabic{equation}}
Using the general expressions given in Appendix A, one can calculate the response for the trap model.
In contrast to other models, the calculation is simplified by the fact that for large $N$, the number of states, one has to consider only terms of order unity and one can neglect all terms of order $1/N$.
In a discrete notation, one has for the trap model $G_{kl}(t)=\delta_{kl}e^{-\kappa_kt}+{\cal O}(1/N)$ with
$G_{kl}(t)=G(\epsilon_k,t|\epsilon_l)$ and $\kappa_k=\kappa(\epsilon_k)$.
Furthermore, the density of state, $\rho_k=\rho(\epsilon_k)$ and the equilibrium populations
$p_k^{\rm eq}=p^{\rm eq}(\epsilon_k)$ scale as $1/N$.
One thus can neglect a number of terms in the calculations.
In the actual calculations, eq.(\ref{Wkl.HX}) is used for the field-dependence of the transition rates.
With this, one has for the relevant part of $L_{kl}^{(\eta)}(t_2,t_1)=L_{kl}^{(\eta)}(t_2-t_1)$ according to
eq.(\ref{Lkl.def}):
\[
L_{kl}^{(\eta)}(t)=e^{-\kappa_kt}\kappa_l\left(\rho_kX_{kl}^\eta-\delta_{kl}\overline{X_l^\eta}\right)+{\cal O}(1/N)
\]
where $X_{kl}=\gamma M_k-\mu M_l$ and $\overline{X_l^\eta}=\sum_k\rho_kX_{kl}^\eta$.
The system is assumed to be in thermal equilibrium in the beginning and the field is assumed to be of the form $H(t)=H_0\cos{(\omega t)}$, cf. eq.(\ref{H.cos.om.t}).
The expressions given in Appendix A are used to compute the frequency-dependent response functions according to eq.(\ref{Chi.om.def}).
For the variables $M_k=M(\epsilon_k)$, the choice discussed in the text is used, cf. eqns.(\ref{MkMl.mit}) and
(\ref{Mh4.mit.Gauss}).
Furthermore, one can utilize detailed balance in the form
\be\label{Det.Bal.Trap}
\rho_k\langle\kappa\rangle=\kappa_kp_k^{\rm eq}
\quad\mbox{with}\quad
\langle\kappa\rangle=\sum_k\kappa_kp_k^{\rm eq}
\ee
For the linear response one finds according to eq.(\ref{Chi1})
\be\label{Chi1.Trap}
\chi_1(\omega)=(\gamma+\mu)\beta\sum_kp_k^{\rm eq}\langle M_k^2\rangle{\kappa_k\over\kappa_k-i\omega}
\ee
which is just the Fourier transform of the time-derivative of the correlation function $C_M(t)$.
For the third-order response, one finds for $n=1,3$:
\be\label{Chi3.om.Trap}
\chi_3^{(\alpha)}(\omega)={1\over4}\beta^3(\gamma+\mu)
\left\{
\hat\chi_{3;1}^{(\alpha)}(\omega)+\hat\chi_{3;2A}^{(\alpha)}(\omega)+\hat\chi_{3;2B}^{(\alpha)}(\omega)+\hat\chi_{3;3}^{(\alpha)}(\omega)
\right\}
\ee
where the individual terms are given by:
\Be\label{Chi3.1.Trap}
\hat\chi_{3;1}^{(\alpha)}(\omega)=
&&\hspace{-0.6cm}
3\mu^2\sum_k\rho_k\langle M_k^2\rangle^2S_{kkk}^{(\alpha)}(\omega)
-\mu^2\sum_{k,l}\rho_k\rho_l\langle M_k^2\rangle\langle M_l^2\rangle S_{kkl}^{(\alpha)}(\omega)
\\
&&\hspace{-0.6cm}
+\gamma\mu\sum_{k,l}\rho_k\rho_l\langle M_k^2\rangle\langle M_l^2\rangle S_{kll}^{(\alpha)}(\omega)
-\gamma\mu\sum_{k,l,m}\rho_k\rho_m\rho_l\langle M_k^2\rangle\langle M_l^2\rangle S_{kml}^{(\alpha)}(\omega)
\nonumber
\Ee
with
\Be\label{Sklm.n}
{\rm Re}(S_{klm}^{(1)}(\omega))
&&\hspace{-0.6cm}=
\kappa_m\kappa_l\langle\kappa\rangle{3\kappa_k\kappa_l^2\kappa_m+\omega^2(8\kappa_k\kappa_m-2\kappa_k\kappa_l-2\kappa_l\kappa_m-\kappa_l^2)
\over\kappa_l(\kappa_k^2+\omega^2)(\kappa_m^2+\omega^2)(\kappa_l^2+4\omega^2)}
\nonumber\\
{\rm Im}(S_{klm}^{(1)}(\omega))
&&\hspace{-0.6cm}=
\kappa_m\kappa_l\langle\kappa\rangle\omega{\kappa_k\kappa_l^2+2\kappa_k\kappa_m\kappa_l+3\kappa_l^2\kappa_m+2\omega^2(4\kappa_m-\kappa_l)
\over\kappa_l(\kappa_k^2+\omega^2)(\kappa_m^2+\omega^2)(\kappa_l^2+4\omega^2)}
\\
{\rm Re}(S_{klm}^{(3)}(\omega))
&&\hspace{-0.6cm}=
\kappa_m\kappa_l\langle\kappa\rangle{\kappa_k\kappa_l\kappa_m-\omega^2(2\kappa_k+3\kappa_l+6\kappa_m)
\over(\kappa_m^2+\omega^2)(\kappa_l^2+4\omega^2)(\kappa_k^2+9\omega^2)}
\nonumber\\
{\rm Im}(S_{klm}^{(3)}(\omega))
&&\hspace{-0.6cm}=
\kappa_m\kappa_l\langle\kappa\rangle\omega{\kappa_k\kappa_l+2\kappa_k\kappa_m+3\kappa_l\kappa_m-6\omega^2
\over(\kappa_m^2+\omega^2)(\kappa_l^2+4\omega^2)(\kappa_k^2+9\omega^2)}
\nonumber
\Ee
This term corresponds to the first line in Fig.\ref{Plot1}.
The second-order terms are:
\Be\label{Chi3.2A.Trap}
\hat\chi_{3;2A}^{(\alpha)}(\omega)=(\gamma-\mu)
&&\hspace{-0.6cm}
\left\{
\mu\sum_k\rho_k\langle M_k^2\rangle\left(3\langle M_k^2\rangle-\overline{\langle M^2\rangle}\right) S_{A;kk}^{(\alpha)}(\omega)
\right.
\\
&&\hspace{-0.3cm}
\left.
+\gamma\sum_{k,l}\rho_k\rho_l\langle M_k^2\rangle\left(\langle M_k^2\rangle-\overline{\langle M^2\rangle}\right) S_{A;kl}^{(\alpha)}(\omega)
\right\}
\nonumber
\Ee
where the corresponding spectral functions are:
\Be\label{SAkl.n}
{\rm Re}(S_{A;kl}^{(1)}(\omega))
&&\hspace{-0.6cm}=
\langle\kappa\rangle{1\over2}{3\kappa_k\kappa_l^2+2\omega^2(4\kappa_k-\kappa_l)\over(\kappa_k^2+\omega^2)(\kappa_l^2+4\omega^2)}
\nonumber\\
{\rm Im}(S_{A;kl}^{(1)}(\omega))
&&\hspace{-0.6cm}=
\langle\kappa\rangle{1\over2}\omega{3\kappa_l^2+2\kappa_k\kappa_l+8\omega^2\over(\kappa_k^2+\omega^2)(\kappa_l^2+4\omega^2)}
\\
{\rm Re}(S_{A;kl}^{(3)}(\omega))
&&\hspace{-0.6cm}=
\kappa_l\langle\kappa\rangle{1\over2}{\kappa_k\kappa_l-6\omega^2\over(\kappa_l^2+4\omega^2)(\kappa_k^2+9\omega^2)}
\nonumber\\
{\rm Im}(S_{A;kl}^{(3)}(\omega))
&&\hspace{-0.6cm}=
\kappa_l\langle\kappa\rangle{1\over2}\omega{2\kappa_k+3\kappa_l\over(\kappa_l^2+4\omega^2)(\kappa_k^2+9\omega^2)}
\nonumber
\Ee
Additionally, I defined
\be\label{Mh2.mit.Trap}
\overline{\langle M^2\rangle}=\sum_k\rho_k\langle M_k^2\rangle
\ee
The other second-order term is:
\Be\label{Chi3.2B.Trap}
\hat\chi_{3;2B}^{(\alpha)}(\omega)=
&&\hspace{-0.6cm}
-3\mu^2\sum_k\rho_k\langle M_k^2\rangle^2S_{B;kk}^{(\alpha)}(\omega)
-\gamma^2\overline{\langle M^2\rangle}\sum_k\rho_k\langle M_k^2\rangle S_{B;kk}^{(\alpha)}(\omega)
\\
&&\hspace{-0.6cm}
-2\gamma\mu\sum_{k,l}\rho_k\rho_l\langle M_k^2\rangle\langle M_l^2\rangle S_{B;kl}^{(\alpha)}(\omega)
\nonumber
\Ee
with
\Be\label{SBkl.n}
{\rm Re}(S_{B;kl}^{(1)}(\omega))
&&\hspace{-0.6cm}=
\kappa_l\langle\kappa\rangle{1\over2}{3\kappa_k\kappa_l-\omega^2\over(\kappa_k^2+\omega^2)(\kappa_l^2+\omega^2)}
\nonumber\\
{\rm Im}(S_{B;kl}^{(1)}(\omega))
&&\hspace{-0.6cm}=
\kappa_l\langle\kappa\rangle{1\over2}\omega{\kappa_k+3\kappa_l\over(\kappa_k^2+\omega^2)(\kappa_l^2+\omega^2)}
\\
{\rm Re}(S_{B;kl}^{(3)}(\omega))
&&\hspace{-0.6cm}=
\kappa_l\langle\kappa\rangle{1\over2}{\kappa_k\kappa_l-3\omega^2\over(\kappa_l^2+\omega^2)(\kappa_k^2+9\omega^2)}
\nonumber\\
{\rm Im}(S_{B;kl}^{(3)}(\omega))
&&\hspace{-0.6cm}=
\kappa_l\langle\kappa\rangle{1\over2}\omega{\kappa_k+3\kappa_l\over(\kappa_l^2+\omega^2)(\kappa_k^2+9\omega^2)}
\nonumber
\Ee
Finally, the term corresponding to the third-order perturbation, i. e. the last line in Fig.\ref{Plot1}, is given by:
\be\label{Chi3.3.Trap}
\hat\chi_{3;3}^{(\alpha)}(\omega)=
3(\mu^2-\gamma\mu+\gamma^2)\sum_k\rho_k\langle M_k^2\rangle^2S_k^{(\alpha)}(\omega)
+3\gamma\mu\overline{\langle M^2\rangle}\sum_k\rho_k\langle M_k^2\rangle S_k^{(\alpha)}(\omega)
\ee
where
\Be\label{Sk.n}
{\rm Re}(S_k^{(1)}(\omega))
&&\hspace{-0.6cm}=
\langle\kappa\rangle{1\over2}{\kappa_k\over(\kappa_k^2+\omega^2)}
\quad;\quad
{\rm Im}(S_k^{(1)}(\omega))
=\langle\kappa\rangle{1\over2}{\omega\over(\kappa_k^2+\omega^2)}
\\
{\rm Re}(S_k^{(3)}(\omega))
&&\hspace{-0.6cm}=
\langle\kappa\rangle{1\over6}{\kappa_k\over(\kappa_k^2+9\omega^2)}
\quad;\quad
{\rm Im}(S_k^{(3)}(\omega))
=\langle\kappa\rangle{1\over2}{\omega\over(\kappa_k^2+9\omega^2)}
\nonumber
\Ee
Here, all expressions are given in a discrete notation.
If one changes to a continuous description, one has to replace all sums by the appropriate integrals over the trap energies $\epsilon_k$.
\end{appendix}
\newpage
|
{
"timestamp": "2012-03-09T02:02:48",
"yymm": "1203",
"arxiv_id": "1203.1785",
"language": "en",
"url": "https://arxiv.org/abs/1203.1785"
}
|
\section{Introduction}
\label{section:introduction}
The more-than-ten-years-long history of the large-scale laser gravitation-wave (GW) detectors (the first one, TAMA~\cite{TAMAsite} started to operate in 1999, and the most powerful pair, the two detectors of the LIGO project~\cite{LIGOsite}, in 2001, not to forget about the two European members of the international interferometric GW detectors network, also having a pretty long history, namely, the German-British interferometer GEO\,600 \cite{GEOsite} located near Hannover, Germany, and the joint European large-scale detector Virgo \cite{VIRGOsite}, operating near Pisa, Italy) can be considered both as a great success and a complete failure, depending on the point of view. On the one hand, virtually all technical requirements for these detectors have been met, and the planned sensitivity levels have been achieved. On the other hand, no GWs have been detected thus far.
The possibility of this result had been envisaged by the community, and during the same last ten years, plans for the second-generation detectors were developed~\cite{Thorne2000, Fritschel2002, Acernese2006-2, Willke2006, AdvLIGOsite, LCGTsite}. Currently (2012), both LIGO detectors are shut down, and their upgrade to the Advanced LIGO, which should take about three years, is underway. The goal of this upgrade is to increase the detectors' sensitivity by about one order of magnitude~\cite{Smith2009}, and therefore the rate of the detectable events by three orders of magnitude, from some `half per year' (by the optimistic astrophysical predictions) of the second generation detectors to, probably, hundreds per year.
This goal will be achieved, mostly, by means of quantitative improvements (higher optical power, heavier mirrors, better seismic isolation, lower loss, both optical and mechanical) and evolutionary changes of the interferometer configurations, most notably, by introduction of the signal recycling mirror. As a result, the second-generation detectors will be \textit{quantum noise limited}. At higher GW frequencies, the main sensitivity limitation will be due to phase fluctuations of light inside the interferometer (shot noise). At lower frequencies, the random force created by the amplitude fluctuations (radiation-pressure noise) will be the main or among the major contributors to the sum noise.
It is important that these noise sources both have the same quantum origin, stemming from the fundamental quantum uncertainties of the electromagnetic field, and thus that they obey the Heisenberg uncertainty principle and can not be reduced simultaneously~\cite{81a1Ca}. In particular, the shot noise can (and will, in the second generation detectors) be reduced by means of the optical power increase. However, as a result, the radiation-pressure noise will increase. In the `naively' designed measurement schemes, built on the basis of a Michelson interferometer, kin to the first and the second generation GW detectors, but with sensitivity chiefly limited by quantum noise, the best strategy for reaching a maximal sensitivity at a given spectral frequency would be to make these noise source contributions (at this frequency) in the total noise budget equal. The corresponding sensitivity point is known as the Standard Quantum Limit (SQL)~\cite{Sov.Phys.JETP_26.831_1968, 92BookBrKh}.
This limitation is by no means an absolute one, and can be evaded using more sophisticated measurement schemes. Starting from the first pioneering works oriented on solid-state GW detectors~\cite{77a1eBrKhVo, 78a1eBrKhVo, Thorne_PRL_40_667_1978}, many methods of overcoming the SQL were proposed, including the ones suitable for practical implementation in laser-interferometer GW detectors. The primary goal of this review is to give a comprehensive introduction of these methods, as well as into the underlying theory of \emph{linear quantum measurements}, such that it remains comprehensible to a broad audience.
The paper is organized as follows. In
Section~\ref{sec:Interferometry}, we give a classical (that is,
non-quantum) treatment of the problem, with the goal to familiarize
the reader with the main components of laser GW
detectors. In Section~\ref{sec:quantum_light} we provide the necessary
basics of quantum optics. In
Section~\ref{sec:linear_quantum_measurement} we demonstrate the main
principles of linear quantum measurement theory, using simplified
toy examples of the quantum optical position meters. In
Section~\ref{sec:QN_in_GW_interferometers}, we provide the full-scale
quantum treatment of the standard Fabry--P{\'e}rot--Michelson topology
of the modern optical GW detectors. At last, in
Section~\ref{sec:sub-SQL_schemes}, we consider three methods of
overcoming the SQL, which are viewed now as the most probable
candidates for implementation in future laser GW
detectors. Concluding remarks are presented in
Section~\ref{sec:Conclusion}. Throughout the review we use the
notations and conventions presented in Table~\ref{table:notations}
below.
\ifpdf
\renewcommand{\arraystretch}{1.3}
\begin{longtable}{l p{9cm}}
\else
\begin{table}[htbp]
\fi
\caption[Notations and conventions]{Notations and conventions, used in
this review, given in alphabetical order for both, greek (first) and
latin (after greek) symbols.}
\label{table:notations}
\ifpdf\\\else
\begin{tabular}{l p{9cm}}
\fi
\toprule
\textbf{Notation and value} & \textbf{Comments} \\
\midrule
\ifpdf
\endfirsthead
\multicolumn{2}{c}{\small\textbf{\tablename~\thetable{}} -- \emph{Continued}}
\\[4mm]
\toprule
\textbf{Notation and value} & \textbf{Comments} \\
\midrule
\endhead
\fi
$\ket{\alpha}$ & coherent state of light with dimensionless complex amplitude $\alpha$\\
$\beta=\arctan\gamma/\delta$ & normalized detuning \\
$\gamma$ & interferometer half-bandwidth \\
$\varGamma=\sqrt{\gamma^2+\delta^2}$ & effective bandwidth \\
$\delta=\omega_p-\omega_0$ & optical pump detuning from the cavity resonance frequency $\omega_0$ \\
$\epsilon_d=\sqrt{\dfrac{1}{\eta_d}-1}$ & excess quantum noise due to optical losses in the detector readout system with quantum efficiency $\eta_d$ \\
$\zeta = t - x/c$ & space-time-dependent argument of the field strength of a light wave, propagating in the positive direction of the $x$-axis \\
$\eta_d$ & quantum efficiency of the readout system (e.g., of a photodetector)\\
$\theta$ & squeeze angle \\
$\vartheta,\,\varepsilon$ & some short time interval \\
$\lambda$ & optical wave length \\
$\mu$ & reduced mass \\
$\nu=\Omega-\Omega_0$ & mechanical detuning from the resonance frequency \\
$\xi=\sqrt{\dfrac{S}{S_{\mathrm{SQL}}}}$ & SQL beating factor \\
$\rho$ & signal-to-noise ratio \\
$\tau=L/c$ & miscellaneous time intervals; in particular, $L/c$ \\
$\phi_{\mathrm{LO}}$ & homodyne angle \\
$\varphi=\phi_{\mathrm{LO}}-\beta$ & \\
$\chi_{AB}(t,t') = \frac{i}{\hbar}[\hat{A}(t),\,\hat{B}(t')]$ & general linear time-domain susceptibility \\
$\chi_{xx}$ & probe body mechanical succeptibility \\
$\omega$ & optical band frequencies \\
$\omega_0$ & interferometer resonance frequency \\
$\omega_p$ & optical pumping frequency \\
$\Omega$ & mechanical band frequencies; typically, $\Omega=\omega-\omega_p$ \\
$\Omega_0$ & mechanical resonance frequency \\
$\Omega_q=\sqrt{\dfrac{2S_\mathcal{FF}}{\hbar M}}$ & quantum noise
``corner frequency'' \\
\midrule
$A$ & power absorption factor in Fabry--P{\'e}rot cavity per bounce \\
$\hat{a}(\omega),\,\hat{a}^\dag(\omega)$ & annihilation and creation operators of photons with frequency $\omega$\\
$\hat{a}_c(\Omega) = \dfrac{\hat{a}(\omega_0+\Omega)+\hat{a}^\dag(\omega_0-\Omega)}{\sqrt{2}}$ & two-photon amplitude quadrature operator\\
$\hat{a}_s(\Omega) = \dfrac{\hat{a}(\omega_0+\Omega)-\hat{a}^\dag(\omega_0-\Omega)}{i\sqrt{2}}$ & two-photon phase quadrature operator\\
$\mean{\hat{a}_i(\Omega)\circ\hat{a}_j(\Omega')}\equiv$ &
\multirow{2}{9cm}{Symmetrised (cross) correlation of the field quadrature operators ($i,j=c,s$)} \\
$\frac12\mean{\hat{a}_i(\Omega)\hat{a}_j(\Omega')+\hat{a}_j(\Omega')\hat{a}_i(\Omega)}$ & \\
$\mathcal{A}$ & light beam cross section area \\
$c$ & speed of light \\
$\mathcal{C}_0=\sqrt{\dfrac{4\pi\hbar\omega_p}{\mathcal{A}c}}$ & light quantization normalization constant\\
$\mathcal{D}=(\gamma-i\Omega)^2 + \delta^2$ & Resonance denominator of the optical cavity transfer function, defining its characteristic conjugate frequencies (``cavity poles'') \\
$E$ & electric field strength \\
$\mathcal{E}$ & classical complex amplitude of the light\\
$\mathcal{E}_{c} = \sqrt{2}\mathrm{Re}[\mathcal{E}],\,\mathcal{E}_{s} = \sqrt{2}\mathrm{Im}[\mathcal{E}]$ & classical quadrature amplitudes of the light \\
$\vb{\mathcal{E}} = \begin{bmatrix}
\mathcal{E}_c\\
\mathcal{E}_s
\end{bmatrix}$ & vector of classical quadrature amplitudes\\
$\hat{F}_{\mathrm{b.a.}}$ & back-action force of the meter \\
$G$ & signal force \\
$h$ & dimensionless GW signal (a.k.a.\ metrics variation) \\
$H=\begin{bmatrix} \cos\phi_{\mathrm{LO}} \\ \sin\phi_{\mathrm{LO}} \end{bmatrix}$ & homodyne vector \\
$\hat{\mathcal{H}}$ & Hamiltonian of a quantum system \\
$\hbar$ & Planck's constant \\
$\mathbb{I}$ & identity matrix\\
$\mathcal{I}$ & optical power \\
$\mathcal{I}_c$ & circulating optical power in a cavity \\
$\mathcal{I}_{\mathrm{arm}}$ & circulating optical power per interferometer arm cavity\\
$J=\dfrac{4\omega_0\mathcal{I}_c}{McL}$ & normalized circulating power \\
$k_p=\omega_p/c$ & optical pumping wave number \\
$K$ & rigidity, including optical rigidity \\
$\mathcal{K}=\dfrac{2J\gamma}{\Omega^2(\gamma^2+\Omega^2)}$ & Kimble's optomechanical coupling factor \\
$\mathcal{K}_{\mathrm{SM}}=\dfrac{4J\gamma}{(\gamma^2+\Omega^2)^2}$ & optomechanical coupling factor of the Sagnac speedmeter \\
$L$ & cavity length \\
$M$ & probe-body mass \\
$O$ & general linear meter readout observable \\
$\mathbb{P}[\alpha]=\begin{bmatrix}
\cos\alpha & -\sin\alpha\\
\sin\alpha & \cos\alpha
\end{bmatrix}$ & matrix of counterclockwise rotation (pivoting) by angle $\alpha$ \\
$r$ & amplitude squeezing factor ($e^r$)\\
$r_{\mathrm{dB}} = 20r\log_{10}e$ & power squeezing factor in decibels\\
$R$ & power reflectivity of a mirror \\
$\mathbb{R}(\Omega)$ & reflection matrix of the Fabry--P{\'e}rot cavity \\
$S(\Omega)$ & noise power spectral density (double-sided) \\
$S_\mathcal{XX}(\Omega)$ & measurement noise power spectral density (double-sided) \\
$S_\mathcal{FF}(\Omega)$ & back-action noise power spectral density (double-sided) \\
$S_\mathcal{XF}(\Omega)$ & cross-correlation power spectral density (double-sided) \\
$\mathbb{S}_{\vac}(\Omega) = \frac12\mathbb{I}$ & vacuum quantum state power spectral density matrix\\
$\mathbb{S}_{\sqz}(\Omega)$ & squeezed quantum state power spectral density matrix\\
$\mathbb{S}_{\sqz}[r,\theta]=\mathbb{P}[\theta]
\begin{bmatrix}
e^r & 0\\
0 & e^{-r}
\end{bmatrix}\mathbb{P}[-\theta]$ & squeezing matrix \\
$T$ & power transmissivity of a mirror \\
$\mathbb{T}$ & transmissivity matrix of the Fabry--P{\'e}rot cavity \\
$v$ & test-mass velocity \\
$\mathcal{W}$ & optical energy \\
$W_{\ket{\psi}}(X,\,Y)$ & Wigner function of the quantum state $\ket{\psi}$ \\
$x$ & test-mass position \\
$\hat{X} = \frac{\hat{a}+\hat{a}^\dag}{\sqrt{2}}$ & dimensionless oscillator (mode) displacement operator\\
$\hat{Y} = \frac{\hat{a}-\hat{a}^\dag}{i\sqrt{2}}$ & dimensionless oscillator (mode) momentum operator\\
\bottomrule
\ifpdf
\end{longtable}
\else\end{tabular}
\end{table}
\fi
\newpage
\section{Interferometry for GW Detectors: Classical Theory}
\label{sec:Interferometry}
\subsection{Interferometer as a weak force probe}
In order to have a firm basis for understanding how quantum noise
influences the sensitivity of a GW detector it would be illuminating
to give a brief description of the interferometers as weak force/tiny
displacement meters. It is by no means our intention to give a
comprehensive survey of this ample field that is certainly worthy of a
good book, which there are in abundance, but rather to provide the
reader with the wherewithal for grasping the very principles of the GW
interferometers operation as well as of other similar ultrasensitive
optomechanical gauges. The reader interested in a more detailed
description of the interferometric techniques being used in the field
of GW detectors might enjoy reading this book~\cite{91Book_Blair} or
the comprehensive Living Reviews on the subject by Freise and
Strain~\cite{lrr-2010-1} and by Pitkin et al.~\cite{lrr-2011-5}.
\subsubsection{Light phase as indicator of a weak force}
\label{sec:phasemeter}
\epubtkImage{fig1.png}{%
\begin{figure}[htb]
\centerline{\includegraphics[width=.5\textwidth]{fig1}}
\caption{Scheme of a simple weak force measurement: an external signal force $\boldsymbol{G}$ pulls the mirror from its equilibrium position $x=0$, causing displacement $\delta x$. The signal displacement is measured by monitoring the phase shift of the light beam, reflected from the mirror.}
\label{fig:phase_meas_force}
\end{figure}}
Let us, for the time being, imagine that we are capable of measuring an electromagnetic (e.g., light) wave-phase shift $\delta\phi$ with respect to some coherent reference of the same frequency. Having such a hypothetical tool, what would be the right way to use it, if one had a task to measure some tiny classical force? The simplest device one immediately conjures up is the one drawn in Figure~\ref{fig:phase_meas_force}. It consists of a movable totally-reflective mirror with mass $M$ and a coherent paraxial light beam, that impinges on the mirror and then gets reflected towards our hypothetical phase-sensitive device. The mirror acts as a \textit{probe} for an external force $G$ that one seeks to measure. The response of the mirror on the external force $G$ depends upon the details of its dynamics. For definiteness, let the mirror be a harmonic oscillator with mechanical eigenfrequency $\Omega_m = 2\pi f_m$. Then the mechanical equation of motion gives a connection between the mirror displacement $x$ and the external force $G$ in the very familiar form of the harmonic oscillator equation of motion:
\begin{equation}
\label{eq:oscillator_EOM}
M\ddot x+M\Omega_m^2 x = G(t)\,,\ \Longrightarrow\ x(t) = x_0(t)+\int_0^t dt'\, \chi_{xx}(t-t')G(t'),
\end{equation}
where $x_0(t) = x(0)\cos\Omega_m t+p(0)/(M\Omega_m)\sin\Omega_mt$ is the free motion of the mirror defined by its initial displacement $x(0)$ and momentum $p(0)$ at $t=0$ and
\begin{equation}
\label{eq:oscillator_Green's_fcn}
\chi_{xx}(t-t')=\frac{\sin\Omega_m(t-t')}{M\Omega_m}\,,\quad t\geqslant t'\,,
\end{equation}
is the oscillator Green's function. It is easy to see that the reflected light beam carries in its phase the information about the displacement $\delta x(t) = x(t)-x(0)$ induced by the external force $G$. Indeed, there is a phase shift between the incident and reflected beams, that matches the additional distance the light must propagate to the new position of the mirror and back, i.e.,
\begin{equation}
\label{eq:dphi_to_dx_relation}
\delta\phi = \frac{2\omega_0\delta x}{c} = 4\pi\frac{\delta x}{\lambda_0}\,,
\end{equation}
with $\omega_0 = 2\pi c/\lambda_0$ the incident light frequency, $c$ the speed of light and $\lambda_0$ the light wavelength. Here we implicitly assume mirror displacement to be much smaller than the light wavelength.
Apparently, the information about the signal force $G(t)$ can be obtained from the measured phase shift by \textit{post-processing} of the measurement data record $\delta\tilde{\phi}(t)\propto \delta x(t)$ by substituting it into Eq.~\eqref{eq:oscillator_EOM} instead of $x$. Thus, the estimate of the signal force $\tilde{G}$ reads:
\begin{equation}
\label{eq:GW_to_dphi_relation}
\tilde{G} = \frac{Mc}{2\omega_0}\left[\delta\ddot{\tilde{\phi}}+\Omega_m^2 \delta\tilde{\phi}\right]\,.
\end{equation}
This kind of post-processing pursues an evident goal of getting rid of any information about the eigenmotion of the test object while keeping only the signal-induced part of the total motion. The above time-domain expression can be further simplified by transforming it into a Fourier domain, since it does not depend anymore on the initial values of the mirror displacement $x(0)$ and momentum $p(0)$:
\begin{equation}
\label{eq:GW_to_dphi_spectral}
\tilde{G}(\Omega) = \frac{Mc}{2\omega_0}\left[\Omega_m^2-\Omega^2\right]\delta\tilde{\phi}(\Omega)\,,
\end{equation}
where
\begin{equation}
\label{eq:Fourier_transform}
A(\Omega) = \int_{-\infty}^\infty dt\,A(t)e^{i\Omega t}
\end{equation}
denotes a Fourier transform of an arbitrary time-domain function $A(t)$. If the expected signal spectrum occupies a frequency range that is much higher than the mirror-oscillation frequency $\Omega_m$ as is the case for ground based interferometric GW detectors, the oscillator behaves as a free mass and the term proportional to $\Omega_m^2$ in the equation of motion can be omitted yielding:
\begin{equation}
\label{eq:GW_to_dphi_free_mass_spectral}
\tilde{G}^{\mathrm{f.m.}}(\Omega) = -\frac{Mc\Omega^2}{2\omega_0}\delta\tilde{\phi}_\Omega\,.
\end{equation}
\subsubsection{Michelson interferometer}
\label{sec:MI}
Above, we assumed a direct light phase measurement with a hypothetical device in order to detect a weak external force, possibly created by a GW. However, in reality, direct phase measurement are not so easy to realize at optical frequencies. At the same time, physicists know well how to measure light intensity (amplitude) with very high precision using different kinds of photodetectors ranging from ancient--yet--die-hard reliable photographic plates to superconductive photodetectors capable of registering individual photons~\cite{goltsman:705}. How can one transform the signal, residing in the outgoing light phase, into amplitude or intensity variation? This question is rhetorical for physicists, for interference of light as well as the multitude of interferometers of various design and purpose have become common knowledge since a couple of centuries ago. Indeed, the amplitude of the superposition of two coherent waves depends on the relative phase of these two waves, thus transforming phase variation into the variation of the light amplitude.
\epubtkImage{fig2.png}{%
\begin{figure}[htb]
\centerline{\includegraphics[width=.5\textwidth]{fig2}}
\caption{Scheme of a Michelson interferometer. When the end mirrors of the interferometer arms $M_{n,e}$ are at rest the length of the arms $L$ is such that the light from the laser gets reflected back entirely (bright port), while at the dark port the reflected waves suffer destructive interference keeping it really dark. If, due to some reason, e.g., because of GWs, the lengths of the arms changed in such a way that their difference was $\delta L$, the photodetector at the dark port should measure light intensity $I_{\mathrm{dark}}(\delta L) = \frac{I_0}{2}(1-\cos4\pi\frac{\delta L}{\lambda})$.}
\label{fig:Michelson_interferometer}
\end{figure}}
For the detection of GWs, the most popular design is the Michelson interferometer~\cite{80BookeBoWo, 91Book_Blair,lrr-2010-1}, which schematic view is presented in Figure~\ref{fig:Michelson_interferometer}. Let us briefly discuss how it works. Here, the light wave from a laser source gets split by a semi-transparent mirror, called a \emph{beamsplitter}, into two waves with equal amplitudes, travelling towards two highly-reflective mirrors $M_{n,e}$\epubtkFootnote{Here, we adopt the system of labeling parts of the interferometer by the cardinal directions, they are located with respect to the interferometer central station, e.g., $M_n$ and $M_e$ in Figure~\ref{fig:Michelson_interferometer} stand for `northern' and `eastern' end mirrors, respectively.} to get reflected off them, and then recombine at the beamsplitter. The readout is performed by a photodetector, placed in the signal port. The interferometer is usually tuned in such a way as to operate at a \emph{dark fringe}, which means that by default the lengths of the arms are taken so that the optical paths for light, propagating back and forth in both arms, are equal to each other, and when they recombine at the signal port, they interfere destructively, leaving the photodetector unilluminated. On the opposite, the two waves coming back towards the laser, interfere constructively. The situation changes if the end mirrors get displaced by some external force in a differential manner, i.e., such that the difference of the arms lengths is non-zero: $\delta L = L_e-L_n \neq 0$. Let a laser send to the interferometer a monochromatic wave that, at the beamsplitter, can be written as
$$E_{\mathrm{laser}}(t) = E_0\cos(\omega_0 t)\,.$$
Hence, the waves reflected off the interferometer arms at the beamsplitter (before interacting with it for the second time) are\epubtkFootnote{Here and below we keep to a definition of the reflectivity coefficient of the mirrors that implies that the reflected wave acquires a phase shift equal to $\pi$ with respect to the incident wave if the latter impinged the reflective surface from the less optically dense medium (air or vacuum). In the opposite case, when the incident wave encounters reflective surface from inside the mirror, i.e., goes from the optically more dense medium (glass), it is assumed to acquire no phase shift upon reflection.}:
$$E^{\out}_{n,e}(t) = -\frac{E_0}{\sqrt2}\cos(\omega_0t-2\omega_0L_{n,e}/c)\,,$$
and after the beamsplitter:
\begin{eqnarray*}
E_{\mathrm{dark}}(t) &=& \dfrac{E_{n}^{\out}(t)-E_{e}^{\out}(t)}{\sqrt2} = E_0\sin\frac{\omega_0\delta L}{c}\sin\left(\omega_0t-\omega_0[L_n+L_e]/c\right)\,,\\
E_{\mathrm{bright}}(t) &=& \dfrac{E_{n}^{\out}(t)+E_{e}^{\out}(t)}{\sqrt2} = -E_0\cos\frac{\omega_0\delta L}{c}\cos\left(\omega_0t-\omega_0[L_n+L_e]/c\right)\,.
\end{eqnarray*}
And the intensity of the outgoing light in both ports can be found using a relation $\mathcal{I}\propto\overline{E^2}$ with overline meaning time-average over many oscillation periods:
\begin{equation}
\mathcal{I}_{\mathrm{dark}}(\delta L/\lambda_0) = \frac{\mathcal{I}_0}{2}\left(1-\cos4\pi\frac{\delta L}{\lambda_0}\right)\,,\quad\mbox{and}\quad \mathcal{I}_{\mathrm{bright}}(\delta L/\lambda_0) = \frac{\mathcal{I}_0}{2}\left(1+\cos4\pi\frac{\delta L}{\lambda_0}\right)\,.
\end{equation}
Apparently, for small differential displacements $\delta L\ll\lambda_0$, the Michelson interferometer tuned to operate at the dark fringe has a sensitivity to $\sim(\delta L/\lambda_0)^2$ that yields extremely weak light power on the photodetector and therefore very high levels of dark current noise. In practice, the interferometer, in the majority of cases, is slightly detuned from the dark fringe condition that can be viewed as an introduction of some constant small bias $\delta L_0$ between the arms lengths. By this simple trick experimentalists get linear response to the signal nonstationary displacement $\delta x(t)$:
\begin{eqnarray}
\label{eq:MI_out_intensity}
\mathcal{I}_{\mathrm{dark}}(\delta
x/\lambda_0) &=& \frac{\mathcal{I}_0}{2}\left(1-\cos4\pi\frac{\delta
L_0+\delta x}{\lambda_0}\right)\simeq \nonumber\\ 8\pi^2
\mathcal{I}_0\frac{\delta L_0\delta x}{\lambda_0^2} +
\mathcal{O}\left(\frac{\delta x^2}{\lambda_0^2},\,\frac{\delta
L_0^2}{\lambda_0^2}\right) &=& \mbox{const.} \times 4\pi\frac{\delta x}{\lambda_0}+ \mathcal{O}\left(\frac{\delta x^2}{\lambda_0^2},\,\frac{\delta L_0^2}{\lambda_0^2}\right)\,.
\end{eqnarray}
Comparison of this formula with Eq.~\eqref{eq:dphi_to_dx_relation}
should immediately conjure up the striking similarity between the
response of the Michelson interferometer and the single moving
mirror. The nonstationary phase difference of light beams in two
interferometer arms $\delta\phi(t) = 4\pi\delta x(t)/\lambda_0$ is
absolutely the same as in the case of a single moving mirror
(cf.\ Eq.~\eqref{eq:dphi_to_dx_relation}). It is no coincidence,
though, but a manifestation of the internal symmetry that all
Michelson-type interferometers possess with respect to coupling
between mechanical displacements of their arm mirrors and the optical
modes of the outgoing fields. In the next section \ref{sec:GW_interaction_with_IF}, we show how this symmetry displays itself in GW interferometers.
\subsubsection{Gravitational waves' interaction with interferometer}
\label{sec:GW_interaction_with_IF}
Let us see how a Michelson interferometer interacts with the GW. For this purpose we need to understand, on a very basic level, what a GW is. Following the poetic, yet precise, definition by Kip Thorne, `gravitational waves are \emph{ripples in the curvature of spacetime} that are emitted by violent astrophysical events, and that propagate out from their source with the speed of light'~\cite{04BookBlTh, 73BookeMiThWh}. A weak GW far away from its birthplace can be most easily understood from analyzing its action on the probe bodies motion in some region of spacetime. Usually, the deformation of a circular ring of free test particles is considered (see Chapter~26: Section~26.3.2 of~\cite{04BookBlTh} for more rigorous treatment) when a GW impinges it along the $z$-direction, perpendicular to the plane where the test particles are located. Each particle, having plane coordinates $(x,\,y)$ with respect to the center of the ring, undergoes displacement $\delta\boldsymbol{r}\equiv(\delta x,\,\delta y)$ from its position at rest, induced by GWs:
\begin{eqnarray}
\label{eq:GW_action_xy}
\delta x = \dfrac12 h_+x\,, & \delta y = -\dfrac12 h_+y\,,\\
\delta x = \dfrac12 h_\times y\,, & \delta y = \dfrac12 h_\times x\,.
\end{eqnarray}
Here, $h_+\equiv h_+(t-z/c)$ and $h_\times\equiv h_\times(t-z/c)$ stand for two independent polarizations of a GW that creates an acceleration field resulting in the above deformations. The above expressions comprise a solution to the equation of motion for free particles in the \emph{tidal acceleration field} created by a GW:
$$\delta\ddot{\boldsymbol{r}} = \dfrac12\left[(\ddot h_+ x+\ddot h_\times y)\boldsymbol{e}_x+(-\ddot h_+ y+\ddot h_\times x)\boldsymbol{e}_y\right]\,,$$
with $\boldsymbol{e}_{x}=\{1,0\}^{\mathsf{T}}$ and $\boldsymbol{e}_{y}=\{0,1\}^{\mathsf{T}}$ the unit vectors pointing in the $x$ and $y$ direction, respectively.
\epubtkImage{fig3.png}{%
\begin{figure}[htb]
\centerline{\includegraphics[width=\textwidth]{fig3}}
\caption{Action of the GW on a Michelson interferometer: (a) $h_+$-polarized GW periodically stretch and squeeze the interferometer arms in the $x$- and $y$-directions, (b) $h_\times$-polarized GW though have no impact on the interferometer, yet produce stretching and squeezing of the imaginary test particle ring, but along the directions, rotated by $45^\circ$ with respect to the $x$ and $y$ directions of the frame. The lower pictures feature field lines of the corresponding tidal acceleration fields $\propto\ddot h_{+,\times}$.}
\label{fig:GW_action}
\end{figure}}
For our Michelson interferometer, one can consider the end mirrors to be those test particles that lie on a circular ring with beamsplitter located in its center. One can choose arms directions to coincide with the frame $x$ and $y$ axes, then the mirrors will have coordinates $(0,\,L_n)$ and $(L_e,\,0)$, correspondingly. For this case, the action of the GW field on the mirrors is featured in Figure~\ref{fig:GW_action}. It is evident from this picture and from the above formulas that an $h_\times$-polarized component of the GW does not change the relative lengths of the Michelson interferometer arms and thus does not contribute to its output signal; at the same time, $h_+$-polarized GWs act on the end masses of the interferometer as a pair of tidal forces of the same value but opposite in direction:
$$G_n =-\dfrac12M_n\ddot h_+L_n\,,\quad G_e =\dfrac12M_e\ddot h_+L_e\,.$$
Assuming $G_e=-G_n=G$, $M_n=M_e=M$, and $L_e=L_n=L$, one can write down the equations of motion for the interferometer end mirrors that are now considered free ($\Omega_m\ll\Omega_{\mathrm{GW}}$) as:
$$M \ddot x = G\,,\quad M \ddot y = -G\,,$$
and for the differential displacement of the mirrors $\delta L = L_e-L_n = x-y$, which, we have shown above, the Michelson interferometer is sensitive to, one gets the following equation of motion:
\begin{equation}
\label{eq:GW_force_to_h_rel}
M\delta\ddot L = 2 G(t) = M\ddot h_+(t) L
\end{equation}
that is absolutely analogous to Eq.~\eqref{eq:oscillator_EOM} for a single free mirror with mass $M$. Therefore, we have proven that a Michelson interferometer has the same dynamical behavior with respect to the tidal force $G(t)=M\ddot h_+(t) L/2$ created by GWs, as the single movable mirror with mass $M$ to some external generic force $G(t)$.
The foregoing conclusion can be understood in the following way: for GWs are inherently quadruple and, when the detector's plane is orthogonal to the wave propagation direction, can only excite a differential mechanical motion of its mirrors, one can reduce a complicated dynamics of the interferometer probe masses to the dynamics of a single effective particle that is the differential motion of the mirrors in the arms. This useful observation appears to be invaluably helpful for calculation of the real complicated interferometer responses to GWs and also for estimation of its optical quantum noise, that comprises the rest of this review.
\subsection{From incident wave to outgoing light: light transformation in the GW interferometers}
To proceed with the analysis of quantum noise in GW interferometers we first need to familiarize ourselves with how a light field is transformed by an interferometer and how the ability of its mirrors to move modifies the outgoing field. In the following paragraphs, we endeavor to give a step-by-step introduction to the mathematical description of light in the interferometer and the interaction with its movable mirrors.
\subsubsection{Light propagation}
\label{sec:propagation}
We first consider how the light wave is described and how its characteristics transform, when it propagates from one point of free space to another. Yet the real light beams in the large scale interferometers have a rather complicated inhomogeneous transverse spatial structure, the approximation of a plane monochromatic wave should suffice for our purposes, since it comprises all the necessary physics and leads to right results. Inquisitive readers could find abundant material on the field structure of light in real optical resonators in particular, in the introductory book~\cite{yariv1990optical} and in the Living Review by Vinet~\cite{lrr-2009-5}.
So, consider a plane monochromatic linearly polarized light wave propagating in vacuo in the positive direction of the $x$-axis. This field can be fully characterized by the strength of its electric component $E(t-x/c)$ that should be a sinusoidal function of its argument $\zeta=t-x/c$ and can be written in three equivalent ways:
\begin{equation}
\label{eq:EMW_classic_E}
E(\zeta) = {\cal E}_0\cos\left[\omega_0\zeta-\phi_0\right]
\equiv \mathcal{E}_c\cos\omega_0\zeta + \mathcal{E}_s\sin\omega_0\zeta
\equiv \frac{\mathcal{E}e^{-i\omega_0\zeta}+\mathcal{E}^*e^{i\omega_0\zeta}}{\sqrt{2}}\,,
\end{equation}
where ${\cal E}_0$ and $\phi_0$ are called \emph{amplitude} and \emph{phase}, ${\cal E}_c$ and ${\cal E}_s$ take names of cosine and sine \emph{quadrature amplitudes}, and complex number $\mathcal{E} = |\mathcal{E}|e^{i\arg \mathcal{E}}$ is known as the \emph{complex amplitude} of the electromagnetic wave. Here, we see that our wave needs two real or one complex parameter to be fully characterized in the given location $x$ at a given time $t$. The `amplitude-phase' description is traditional for oscillations but is not very convenient since all the transformations are nonlinear in phase. Therefore, in optics, either quadrature amplitudes or complex amplitude description is applied to the analysis of wave propagation. All three descriptions are related by means of straightforward transformations:
\begin{equation}
\begin{array}{ll}
\label{eq:EMW_quadrature_def}
{\cal E}_0 = \sqrt{{\cal E}_c^2+{\cal E}_s^2} = \sqrt{2}|\mathcal{E}| & \qquad \tan\phi_0 = {\cal E}_s/{\cal E}_c = \arg \mathcal{E},\quad \phi_0\in[0,\,2\pi]\,,\\
{\cal E}_c = \frac{\mathcal{E}+\mathcal{E}^*}{\sqrt 2} = \sqrt{2}\mathrm{Re}\left[\mathcal{E}\right] = {\cal E}_0\cos\phi_0 & \qquad {\cal E}_s = \frac{\mathcal{E}-\mathcal{E}^*}{i\sqrt 2} = \sqrt{2}\mathrm{Im}\left[\mathcal{E}\right] = {\cal E}_0\sin\phi_0\,,\\
\mathcal{E} = \frac{{\cal E}_c+i{\cal E}_s}{\sqrt2} = \frac{{\cal E}_0}{\sqrt{2}}e^{i\phi_0} & \qquad \mathcal{E}^* = \frac{{\cal E}_c-i{\cal E}_s}{\sqrt2} = \frac{{\cal E}_0}{\sqrt{2}}e^{-i\phi_0}\,.
\end{array}
\end{equation}
The aforesaid means that for complete understanding of how the light field transforms in the optical device, knowing the rules of transformation for only two characteristic real numbers -- real and imaginary parts of the complex amplitude suffice. Note also that the electric field of a plane wave is, in essence, a function of a single argument $\zeta=t-x/c$ (for a forward propagating wave) and thus can be, without loss of generality, substituted by a time dependence of electric field in some fixed point, say with $x=0$, thus yielding $E(\zeta)\equiv E(t)$. We will keep to this convention throughout our review.
Now let us elaborate the way to establish a link between the wave electric field strength values taken in two spatially separated points, $x_1 = 0$ and $x_2 = L$. Obviously, if nothing obscures light propagation between these two points, the value of the electric field in the second point at time $t$ is just the same as the one in the first point, but at earlier time, i.e., at $t'=t-L/c$: $E^{(L)}(t) = E^{(0)}(t-L/c)$. This allows us to introduce a transformation that propagates EM-wave from one spatial point to another. For complex amplitude $\mathcal{E}$, the transformation is very simple:
\begin{equation}
\label{eq:EMW_free_prop_CAmp_transform}
\mathcal{E}^{(L)} = e^{i\omega_0L/c} \mathcal{E}^{(0)}\,.
\end{equation}
Basically, this transformation is just a counterclockwise rotation of a wave complex amplitude vector on a complex plane by an angle $\phi_L=\left[\frac{\omega_0 L}{c}\right]_{\mathrm{mod}\ 2\pi}$. This fact becomes even more evident if we look at the transformation for a 2-dimensional vector of quadrature amplitudes $\vb{\mathcal{E}} = \left\{{\cal E}_c,{\cal E}_s\right\}^{\mathsf{T}}$, that are:
\begin{equation}
\label{eq:EMW_free_prop_QAmp_transform}
\vb{\cal E}^{(L)} = \begin{bmatrix}
\cos\phi_L & -\sin\phi_L\\
\sin\phi_L & \cos\phi_L
\end{bmatrix}\cdot
\begin{bmatrix}
{\cal E}_c^{(0)}\\
{\cal E}_s^{(0)}
\end{bmatrix} =
\mathbb{P}\left[\phi_L\right]\vb{\cal E}^{(0)}\,,
\end{equation}
where
\begin{equation}
\label{eq:CCW_rotation_matrix}
\mathbb{P}[\theta] =
\begin{bmatrix}
\cos\theta & -\sin\theta\\
\sin\theta & \cos\theta
\end{bmatrix}
\end{equation}
stands for a standard counterclockwise rotation (pivoting) matrix on a 2D plane.
In the special case when the propagation distance is much smaller than the light wavelength $L\ll\lambda$, the above two expressions can be expanded into Taylor's series in $\phi_L=2\pi L/\lambda\ll1$ up to the first order:
\begin{equation}
\label{eq:EMW_free_prop_small_x}
E^{(L\ll\lambda)} = (1+i\phi_L)E^{(0)}\,
\end{equation}
and
\begin{equation}
\label{eq:EMW_free_prop_small_x_matrix}
\vb{\cal E}^{(L\ll\lambda)} =
\begin{bmatrix}
1 & -\phi_L\\
\phi_L & 1
\end{bmatrix}
\cdot
\begin{bmatrix}
{\cal E}_c^{(0)}\\
{\cal E}_s^{(0)}
\end{bmatrix} =
\left(\begin{bmatrix}
1 & 0\\
0 & 1
\end{bmatrix}
+
\begin{bmatrix}
0 & -\phi_L\\
\phi_L & 0
\end{bmatrix}\right)
\cdot
\begin{bmatrix}
{\cal E}_c^{(0)}\\
{\cal E}_s^{(0)}
\end{bmatrix} =
(\mathbb{I}+\delta\mathbb{P}\left[\phi_L\right])\vb{\cal E}^{(0)}\,,
\end{equation}
where $\mathbb{I}$ stands for an identity matrix and $\delta\mathbb{P}[\phi_L]$ is an \emph{infinitesimal} increment matrix that generate the difference between the field quadrature amplitudes vector $\vb{\mathcal{E}}$ after and before the propagation, respectively.
It is worthwhile to note that the quadrature amplitudes representation is used more frequently in literature devoted to quantum noise calculation in GW interferometers than the complex amplitudes formalism and there is a historical reason for this. Notwithstanding the fact that these two descriptions are absolutely equivalent, the quadrature amplitudes representation was chosen by Caves and Schumaker as a basis for their two-photon formalism for the description of quantum fluctuations of light~\cite{85a1CaSch, 85a2CaSch} that became from then on the workhorse of quantum noise calculation. More details about this extremely useful technique are given in the sections \ref{sec:2photon_formalism} and \ref{sec:light_quantum_states} of this review. Unless otherwise specified, we predominantly keep ourselves to this formalism and give all results in terms of it.
\subsubsection{Modulation of light}
\label{sec:modulation}
Above, we have seen that a GW signal displays itself in the modulation of the phase of light, passing through the interferometer. Therefore, it is illuminating to see how the modulation of the light phase and/or amplitude manifests itself in a transformation of the field complex amplitude and quadrature amplitudes.
Throughout this section we assume our carrier field is a monochromatic light wave with frequency $\omega_0$, amplitude $\mathcal{E}_0$ and initial phase $\phi_0=0$:
$$E_{\mathrm{car}}(t) = \mathcal{E}_0\cos\omega_0 t=\mathrm{Re}\left[\mathcal{E}_0e^{-i\omega_0 t}\right]\,.$$
\paragraph*{Amplitude modulation.} The modulation of light amplitude is straightforward to analyze. Let us do it for pedagogical sake: imagine one managed to modulate the carrier field amplitude slow enough compared to the carrier oscillation period, i.e., $\Omega\ll\omega_0$, then:
$$E_{\mathrm{AM}}(t) = \mathcal{E}_0(1+\epsilon_m\cos(\Omega t+\phi_m))\cos\omega_0 t\,,$$
where $\epsilon_m\ll1$ and $\phi_m$ are some constants called modulation depth and relative phase, respectively.
The complex amplitude of the modulated wave equals to
$$\mathcal{E}_{\mathrm{AM}}(t)=\frac{\mathcal{E}_0}{\sqrt{2}}\left(1+\epsilon_m\cos(\Omega t+\phi_m)\right)\,,$$
and the carrier quadrature amplitudes are, apparently, transformed as follows:
$$\mathcal{E}_{c,\mathrm{AM}}(t) = \mathcal{E}_0\left(1+\epsilon_m\cos(\Omega t+\phi_m)\right)\quad\mbox{and}\quad\mathcal{E}_{s,\mathrm{AM}}(t) = 0\,.$$
The fact that the amplitude modulation shows up only in the quadrature that is in phase with the carrier field sets forth why this quadrature is usually named \emph{amplitude quadrature} in the literature. In our review, we shall also keep to this terminology and refer to cosine quadrature as amplitude one.
Illuminating also is the calculation of the modulated light spectrum, that in our simple case of single frequency modulation is straightforward:
$$E_{\mathrm{AM}}(t) = \mathrm{Re}\left[\mathcal{E}_0e^{-i\omega_0 t}+\frac{\mathcal{E}_0\epsilon_m}{2}e^{-i\phi_m}e^{-i(\omega_0+\Omega) t}+\frac{\mathcal{E}_0\epsilon_m}{2}e^{i\phi_m}e^{-i(\omega_0-\Omega) t}\right]\,.$$
Apparently, the spectrum is discrete and comprises three components, i.e., the harmonic at carrier frequency $\omega_0$ with amplitude $A_{\omega_0}=\mathcal{E}_0$ and two satellites at frequencies $\omega_0\pm\Omega$, also referred to as \emph{modulation sidebands}, with (complex) amplitudes $A_{\omega_0\pm\Omega}=\epsilon_m\mathcal{E}_0e^{\mp i\phi_m}/2$. The graphical interpretation of the above considerations is given in the left panel of Figure~\ref{fig:modulation}. Here, carrier fields as well as sidebands are represented by rotating vectors on a complex plane. The carrier field vector has length $\mathcal{E}_0$ and rotates clockwise with the rate $\omega_0$, while sideband components participate in two rotations at a time. The sum of these three vectors yields a complex vector, whose length oscillates with time, and its projection on the real axis represents the amplitude-modulated light field.
\begin{sloppypar}
The above can be generalized to an arbitrary periodic modulation function $A(t)=\sum_{k=1}^\infty A_k\cos(k\Omega+\phi_k)$, with $E_{\mathrm{AM}}(t) = \mathcal{E}_0(1+\epsilon_mA(t))\cos\omega_0 t$. Then the spectrum of the modulated light consists again of a carrier harmonic at $\omega_0$ and an infinite discrete set of sideband harmonics at frequencies $\omega_0\pm k\Omega$ ($k=\overline{1,\infty}$):
\end{sloppypar}
\begin{equation}
\label{eq:AM_discrete}
E_{\mathrm{AM}}(t) = \mathcal{E}_0\cos\omega_0 t+\frac{\epsilon_m \mathcal{E}_0}{2}\sum_{k=1}^\infty A_k\left\{\cos[(\omega_0-k\Omega) t-\phi_k]+\cos[(\omega_0+k\Omega) t+\phi_k]\right\}
\,.
\end{equation}
\begin{sloppypar}
Further generalization to an arbitrary (real) non-periodic modulation function $A(t) = \int_{-\infty}^{\infty}\!\frac{d\omega}{2\pi}A(\Omega)e^{-i\Omega t}$ is apparent:
\end{sloppypar}
\begin{eqnarray}
\label{eq:AM_continuous}
E_{\mathrm{AM}}(t) &=& \mathrm{Re}\left[\mathcal{E}_0e^{-i\omega_0
t}+\epsilon_m \mathcal{E}_0e^{-i\omega_0
t}\int_{-\infty}^{\infty}\!\frac{d\Omega}{2\pi}A(\Omega)e^{-i\Omega t}\right] = \nonumber\\
&& \mathcal{E}_0\cos\omega_0 t+\frac{\epsilon_m \mathcal{E}_0}{2}\int_{-\infty}^{\infty}\!\frac{d\omega}{2\pi}\left\{A(\omega-\omega_0)+A(\omega+\omega_0)\right\}e^{-i\omega t}\,.
\end{eqnarray}
From the above expression, one readily sees the general structure of the modulated light spectrum, i.e., the central carrier peaks at frequencies $\pm\omega_0$ and the modulation sidebands around it, whose shape retraces the modulation function spectrum $A(\omega)$ shifted by the carrier frequency $\pm\omega_0$.
\epubtkImage{fig4.png}{%
\begin{figure}[htb]
\centerline{\includegraphics[width=\textwidth]{fig4}}
\caption{Phasor diagrams for amplitude (\emph{Left panel}) and phase (\emph{Right panel}) modulated light. Carrier field is given by a brown vector rotating clockwise with the rate $\omega_0$ around the origin of the complex plane frame. Sideband fields are depicted as blue vectors. The lower ($\omega_0-\Omega$) sideband vector origin rotates with the tip of the carrier vector, while its own tip also rotates with respect to its origin counterclockwise with the rate $\Omega$. The upper ($\omega_0+\Omega$) sideband vector origin rotates with the tip of the upper sideband vector, while its own tip also rotates with respect to its origin counterclockwise with the rate $\Omega$. Modulated oscillation is a sum of these three vectors an is given by the red vector. In the case of amplitude modulation (AM), the modulated oscillation vector is always in phase with the carrier field while its length oscillates with the modulation frequency $\Omega$. The time dependence of its projection onto the real axis that gives the AM-light electric field strength is drawn to the right of the corresponding phasor diagram. In the case of phase modulation (PM), sideband fields have a $\pi/2$ constant phase shift with respect to the carrier field (note factor $i$ in front of the corresponding terms in Eq.~\eqref{eq:PM_discrete}; therefore its sum is always orthogonal to the carrier field vector, and the resulting modulated oscillation vector (red arrow) has approximately the same length as the carrier field vector but outruns or lags behind the latter periodically with the modulation frequency $\Omega$. The resulting oscillation of the PM light electric field strength is drawn to the right of the PM phasor diagram and is the projection of the PM oscillation vector on the real axis of the complex plane.}
\label{fig:modulation}
\end{figure}}
\paragraph*{Phase modulation.} The general feature of the modulated signal that we pursued to demonstrate by this simple example is the creation of the modulation sidebands in the spectrum of the modulated light. Let us now see how it goes with a phase modulation that is more related to the topic of the current review. The simplest single-frequency phase modulation is given by the expression:
$$E_{\mathrm{PM}}(t) = \mathcal{E}_0\cos(\omega_0 t+\delta_m\cos(\Omega t+\phi_m))\,,$$
where $\Omega\ll\omega_0$, and the phase deviation $\delta_m$ is assumed to be much smaller than 1. Using Eqs.~\eqref{eq:EMW_quadrature_def}, one can write the complex amplitude of the phase-modulated light as:
$$\mathcal{E}_{\mathrm{PM}}(t)=\frac{\mathcal{E}_0}{\sqrt{2}}e^{i\delta_m\cos(\Omega t+\phi_m)}\,,$$
and quadrature amplitudes as:
$$\mathcal{E}_{c,\mathrm{PM}}(t) = \mathcal{E}_0\cos\left[\delta_m\cos(\Omega t+\phi_m)\right]\quad\mbox{and}\quad\mathcal{E}_{s,\mathrm{PM}}(t) =\mathcal{E}_0\sin\left[\delta_m\cos(\Omega t+\phi_m)\right]\,.$$
Note that in the weak modulation limit ($\delta_m\ll1$), the above equations can be approximated as:
$$\mathcal{E}_{c,\mathrm{PM}}(t) \simeq \mathcal{E}_0\quad\mbox{and}\quad\mathcal{E}_{s,\mathrm{PM}}(t) \simeq \delta_m\mathcal{E}_0\cos(\Omega t+\phi_m)\,.$$
This testifies that for a weak modulation only the sine quadrature, which is $\pi/2$ out-of-phase with respect to the carrier field, contains the modulation signal. That is why this sine quadrature is usually referred to as \emph{phase quadrature}. It is also what we will call this quadrature throughout the rest of this review.
In order to get the spectrum of the phase-modulated light it is necessary to refer to the theory of Bessel functions that provides us with the following useful relation (known as the Jacobi--Anger expansion):
\begin{equation*}
e^{i\delta_m\cos(\Omega t+\phi_m)} = \sum_{k=-\infty}^\infty i^k J_k(\delta_m)e^{ik(\Omega t+\phi_m)},
\end{equation*}
where $J_k(\delta_m)$ stands for the $k$-th Bessel function of the first kind. This looks a bit intimidating, yet for $\delta_m\ll1$ these expressions simplify dramatically, since near zero Bessel functions can be approximated as:
$$J_0(\delta_m)\simeq 1-\frac{\delta_m^2}{4}+\mathcal{O}(\delta_m^4)\,,\quad J_1(\delta_m) = \frac{\delta_m}{2}+\mathcal{O}(\delta_m^3)\,,\quad J_k(\delta_m) = \frac{1}{k!}\left(\frac{\delta_m}{2}\right)^k+\mathcal{O}(\delta_m^{k+2})\ (k\geqslant2)\,.$$
Thus, for sufficiently small $\delta_m$, we can limit ourselves only to the terms of order $\delta_m^0$ and $\delta_m^1$, which yields:
\begin{equation}
\label{eq:PM_discrete}
E_{\mathrm{PM}}(t) \simeq \mathrm{Re}\left[\mathcal{E}_0e^{i\omega_0 t}+i\frac{\delta_m \mathcal{E}_0}{2}\left(e^{i[(\omega_0+\Omega) t+\phi_m]}+e^{i[(\omega_0-\Omega) t-\phi_m]}\right)\right]\,,
\end{equation}
and we again face the situation in which modulation creates a pair of sidebands around the carrier frequency. The difference from the amplitude modulation case is in the way these sidebands behave on the complex plane. The corresponding phasor diagram for phase modulated light is drawn in Figure~\ref{fig:modulation}. In the case of PM, sideband fields have $\pi/2$ constant phase shift with respect to the carrier field (note factor $i$ in front of the corresponding terms in Eq.~\eqref{eq:PM_discrete}); therefore its sum is always orthogonal to the carrier field vector, and the resulting modulated oscillation vector has approximately the same length as the carrier field vector but outruns or lags behind the latter periodically with the modulation frequency $\Omega$. The resulting oscillation of the PM light electric field strength is drawn to the right of the PM phasor diagram and is the projection of the PM oscillation vector on the real axis of the complex plane.
Let us now generalize the obtained results to an arbitrary modulation function
$\Phi(t)$:
$$E_{\mathrm{PM}}(t)=\mathcal{E}_0\cos(\omega_0 t+\delta_m\Phi(t))\,.$$
In the most general case of arbitrary modulation index $\delta_m$, the corresponding formulas are very cumbersome and do not give much insight. Therefore, we again consider a simplified situation of sufficiently small $\delta_m\ll1$. Then one can approximate the phase-modulated oscillation as follows:
$$
E_{\mathrm{PM}}(t)=\mathrm{Re}\left[\mathcal{E}_0e^{-i\omega_0 t}e^{i\delta_m\Phi(t)}\right]\simeq\mathrm{Re}\left[\mathcal{E}_0e^{-i\omega_0 t}\left\{1+i\delta_m\Phi(t)\right\}\right]\,.
$$
When $\Phi(t)$ is a periodic function: $\Phi(t) = \sum_{k=1}^\infty\Phi_k\cos{k\Omega+\phi_k}$, and in weak modulation limit $\delta_m\ll1$, the spectrum of the PM light is apparent from the following expression:
\begin{eqnarray}
\label{eq:PM_discrete2}
E_{\mathrm{PM}}(t) &\simeq&
\mathcal{E}_0\cos\omega_0 t -
\frac{\delta_m\mathcal{E}_0}{2}\sum_{k=1}^\infty \Phi_k\left\{\sin\left[(\omega_0-k\Omega) t-\phi_k\right]+\sin\left[(\omega_0+k\Omega) t+\phi_k\right]\right\} \nonumber\\
&=& \mathrm{Re}\left[\mathcal{E}_0e^{-i\omega_0 t}+
\frac{i\delta_m\mathcal{E}_0}{2}\sum_{k=1}^\infty \Phi_k\left\{e^{-i\left[(\omega_0-k\Omega) t-\phi_k\right]}+
e^{-i\left[(\omega_0+k\Omega) t+\phi_k\right]}\right\}\right]\,,
\end{eqnarray}
while for the real non-periodic modulation function $\Phi(t) = \int_{-\infty}^{\infty}\!\frac{d\omega}{2\pi}\Phi(\Omega)e^{-i\Omega t}$ the spectrum can be obtained from the following relation:
\begin{eqnarray}
\label{eq:PM_continuous}
E_{\mathrm{PM}}(t) &\simeq& \mathrm{Re}\left[\mathcal{E}_0e^{-i\omega_0
t}+i\delta_m \mathcal{E}_0e^{-i\omega_0
t}\int_{-\infty}^{\infty}\!\frac{d\Omega}{2\pi}\Phi(\Omega)e^{-i\Omega t}\right] \nonumber\\
&=& \mathcal{E}_0\cos\omega_0 t+\frac{\delta_m \mathcal{E}_0}{2}\int_{-\infty}^{\infty}\!\frac{d\omega}{2\pi}\left\{i\Phi(\omega-\omega_0)-i\Phi(\omega+\omega_0)\right\}e^{-i\omega t}\,.
\end{eqnarray}
And again we get the same general structure of the spectrum with carrier peaks at $\pm\omega_0$ and shifted modulation spectra $i\Phi(\omega\pm\omega_0)$ as sidebands around the carrier peaks. The difference with the amplitude modulation is an additional $\pm\pi/2$ phase shifts added to the sidebands.
\subsubsection{Laser noise}
Thus far we have assumed the carrier field to be perfectly monochromatic having a single spectral component at carrier frequency $\omega_0$ fully characterized by a pair of classical quadrature amplitudes represented by a 2-vector $\vb{\mathcal{E}}$. In reality, this picture is no good at all; indeed, a real laser emits not a monochromatic light but rather some spectral line of finite width with its central frequency and intensity fluctuating. These fluctuations are usually divided into two categories: (i) \emph{quantum noise} that is associated with the spontaneous emission of photons in the gain medium, and (ii) \emph{technical noise} arising, e.g., from excess noise of the pump source, from vibrations of the laser resonator, or from temperature fluctuations and so on. It is beyond the goals of this review to discuss the details of the laser noise origin and methods of its suppression, since there is an abundance of literature on the subject that a curious reader might find interesting, e.g., the following works~\cite{Paschotta_APB_2004-79-2-153, Paschotta_APB_2004-79-2-163, Paschotta_APB_2006-82-2-265, Willke_LPR_2010-4-6-780, Harb_JOSAB_1997-14-11-2936, Heurs_APB_2006-85-1-79}.
For our purposes, the very existence of the laser noise is important as it makes us to reconsider the way we represent the carrier field. Apparently, the proper account for laser noise prescribes us to add a random time-dependent modulation of the amplitude (for intensity fluctuations) and phase (for phase and frequency fluctuations) of the carrier field~\eqref{eq:EMW_classic_E}:
$$E(t) = (\mathcal{E}_0+\hat{e}_{n}(t))\cos\left[\omega_0 t+\phi_0+\hat{\phi}_{n}(t)\right]\,,$$
where we placed hats above the noise terms on purpose, to emphasize that quantum noise is a part of laser noise and its quantum nature has to be taken into account, and that the major part of this review will be devoted to the consequences these hats lead to. However, for now, let us consider hats as some nice decoration.
Apparently, the corrections to the amplitude and phase of the carrier light due to the laser noise are small enough to enable us to use the weak modulation approximation as prescribed above. In this case one can introduce a more handy amplitude and phase quadrature description for the laser noise contribution in the following manner:
\begin{equation}
\label{eq:EMW_laser_noise}
E(t) = \left(\mathcal{E}_c + \hat{e}_c(t)\right)\cos\omega_0t + \left(\mathcal{E}_s + \hat{e}_s(t)\right)\sin\omega_0t\,,
\end{equation}
where $\hat{e}_{c,s}$ are related to $\hat{e}_{n}$ and $\hat{\phi}_{n}$ in the same manner as prescribed by Eqs.~\eqref{eq:EMW_quadrature_def}. It is convenient to represent a noisy laser field in the Fourier domain:
$$
\hat{e}_{c,s}(t) = \int_{-\infty}^{\infty}\! \frac{d\Omega}{2\pi}\hat{e}_{c,s}(\Omega)e^{-i\Omega t}\,.
$$
Worth noting is the fact that $\hat{e}_{c,s}(\Omega)$ is a spectral representation of a real quantity and thus satisfies an evident equality $\hat{e}_{c,s}^\dag(-\Omega)=\hat{e}_{c,s}(\Omega)$ (by $\dag$ we denote the Hermitian conjugate that for classical functions corresponds to taking the complex conjugate of this function). What happens if we want to know the light field of our laser with noise at some distance $L$ from our initial reference point $x=0$? For the carrier field component at $\omega_0$, nothing changes and the corresponding transform is given by Eq.~\eqref{eq:EMW_free_prop_QAmp_transform}, yet for the noise component $$\delta\hat{E}_{\mathrm{noise}}(t) = \hat{e}_c(t)\cos\omega_0t+\hat{e}_s(t)\sin\omega_0t$$
there is a slight modification. Since the field continuity relation holds for the noise field to the same extent as for the carrier field:
$$\delta\hat{E}^{(L)}_{\mathrm{noise}}(t) = \delta\hat{E}^{(0)}_{\mathrm{noise}}(t-L/c)\,,$$
the following modification applies:
\begin{equation}
\label{eq:EMW_sidebands_free_prop_matrix}
\hat{\boldsymbol{e}}^{(L)}(\Omega)=\begin{bmatrix}
\hat{e}_c^{(L)}(\Omega)\\
\hat{e}_s^{(L)}(\Omega)
\end{bmatrix} = e^{i\Omega L/c}
\begin{bmatrix}
\cos \dfrac{\omega_0L}{c} & -\sin\dfrac{\omega_0L}{c}\\
\sin\dfrac{\omega_0L}{c} & \cos\dfrac{\omega_0L}{c}
\end{bmatrix}\cdot\begin{bmatrix}
\hat{e}_c^{(0)}(\Omega)\\
\hat{e}_s^{(0)}(\Omega)
\end{bmatrix} = e^{i\Omega L/c}\mathbb{P}\left[\dfrac{\omega_0L}{c}\right]\hat{\boldsymbol{e}}^{(0)}(\Omega)\,.
\end{equation}
Therefore, for sideband field components the propagation rule shall be modified by adding a frequency-dependent phase factor $e^{i\Omega L/c}$ that describes an extra phase shift acquired by a sideband field relative to the carrier field because of the frequency difference $\Omega=\omega-\omega_0$.
\subsubsection{Light reflection from optical elements}
\label{sec:light_reflection}
So, we are one step closer to understanding how to calculate the quantum
noise of the light coming out of the GW interferometer. It is necessary
to understand what happens with light when it is reflected from such
optical elements as mirrors and beamsplitters. Let us first consider
these elements of the interferometer fixed at their positions. The
impact of mirror motion will be considered in the next subsection \ref{sec:Mirror_motion}. One
can also refer to Section~2 of the Living Review by Freise
and Strain~\cite{lrr-2010-1} for a more detailed treatment of this
topic.
\epubtkImage{fig5.png}{%
\begin{figure}[htbp]
\centerline{\includegraphics[width=\textwidth]{fig5}}
\caption{Scheme of light reflection off the coated mirror.}
\label{fig:mirror}
\end{figure}}
Mirrors of the modern interferometers are rather complicated optical systems usually consisting of a dielectric slab with both surfaces covered with multilayer dielectric coatings. These coatings are thoroughly constructed in such a way as to make one surface of the mirror highly reflective, while the other one is anti-reflective. We will not touch on the aspects of coating technology in this review and would like to refer the interested reader to an abundant literature on this topic, e.g., to the following book~\cite{2012_unpublished_Harry} and reviews and articles~\cite{lrr-2009-5, CQG.19.897_Harry, CQG.26.15.155012_2009_Martin, CQG.24.2.405_2006_Harry, CQG.20.13.2917_2003_Penn, CQG.19.5.883_2002_Crooks, PhysRevD.57.659_1998_Levin, PhysRevLett.93.250602_2004_Numata, PhysRevD.78.102003_2008_Evans}. For our purposes, assuming the reflective surface of the mirror is flat and lossless should suffice. Thus, we represent a mirror by a reflective plane with (generally speaking, complex) coefficients of reflection $r$ and $r'$ and transmission $t$ and $t'$ as drawn in Figure~\ref{fig:mirror}. Let us now see how the ingoing and outgoing light beams couple on the mirrors in the interferometer.
\paragraph*{Mirrors:}
From the general point of view, the mirror is a linear system with 2 input and 2 output ports. The way how it transforms input signals into output ones is featured by a $2\times2$ matrix that is known as the \emph{transfer matrix} of the mirror $\mathbb{M}$:
\begin{equation}
\label{eq:mirror_IOrel}
\begin{bmatrix}
E^{\out}_1(t)\\
E^{\out}_2(t)
\end{bmatrix}= \mathbb{M}\cdot
\begin{bmatrix}
E^{\in}_1(t)\\
E^{\in}_2(t)
\end{bmatrix}=
\begin{bmatrix}
r & t\\
t' & r'
\end{bmatrix}\cdot
\begin{bmatrix}
E^{\in}_1(t)\\
E^{\in}_2(t)
\end{bmatrix}\,.
\end{equation}
Since we assume no absorption in the mirror, reflection and transmission coefficients should satisfy Stokes' relations~\cite{1849a1St, 80BookeBoWo} (see also Section~12.12 of~\cite{95BookMaWo}):
\begin{equation}
\label{eq:Stokes_rel}
|r|=|r'|\,, \qquad |t|=|t'|\,, \qquad |r|^2+|t|^2 = 1\,, \qquad r^*t'+r't^* = 0\,, \qquad r^*t+r't'^{*} = 0\,,
\end{equation}
that is simply a consequence of the conservation of energy. This conservation of energy yields that the optical transfer matrix $\mathbb{M}$ must be unitary: $\mathbb{M}^\dag = \mathbb{M}^{-1}$. Stokes' relations leave some freedom in defining complex reflectivity and transmissivity coefficients. Two of the most popular variants are given by the following matrices:
\begin{equation}
\label{eq:IO_mirror_matrix}
\mathbb{M}_{\mathrm{sym}} = \begin{bmatrix}
\sqrt{R} & i\sqrt{T}\\
i\sqrt{T} & \sqrt{R}
\end{bmatrix}\,,\quad\mbox{and}\quad \mathbb{M}_{\mathrm{real}} = \begin{bmatrix}
-\sqrt{R} & \sqrt{T}\\
\sqrt{T} & \sqrt{R}
\end{bmatrix}\,,
\end{equation}
where we rewrote transfer matrices in terms of real power reflectivity and transmissivity coefficients $R=|r|^2$ and $T=|t|^2$ that will find extensive use throughout the rest of this review.
The transformation rule, or putting it another way, \emph{coupling relations} for the quadrature amplitudes can easily be obtained from Eq.~\eqref{eq:mirror_IOrel}. Now, we have two input and two output fields. Therefore, one has to deal with 4-dimensional vectors comprising of quadrature amplitudes of both input and output fields, and the transformation matrix become $4\times4$-dimensional, which can be expressed in terms of the outer product of a $2\times2$ matrix $\mathbb{M}_{\mathrm{real}}$ by a $2\times2$ identity matrix $\mathbb{I}$:
\begin{equation}
\label{eq:IO_mirror_matrix_4x4_zero}
\begin{bmatrix}
\mathcal{E}^{\out}_{1c}\\
\mathcal{E}^{\out}_{1s}\\
\mathcal{E}^{\out}_{2c}\\
\mathcal{E}^{\out}_{2s}
\end{bmatrix}=
\begin{bmatrix}
\vb{\mathcal{E}}^{\out}_{1}\\
\vb{\mathcal{E}}^{\out}_{2}
\end{bmatrix}=
\mathbb{M}_{\mathrm{real}}\otimes\mathbb{I}
\cdot
\begin{bmatrix}
\vb{\mathcal{E}}^{\in}_{1}\\
\vb{\mathcal{E}}^{\in}_{2}
\end{bmatrix}=
\begin{bmatrix}
-\sqrt{R} & 0 &\sqrt{T} & 0\\
0 & -\sqrt{R} & 0 & \sqrt{T} \\
\sqrt{T} & 0 & \sqrt{R} & 0\\
0 & \sqrt{T} & 0 & \sqrt{R}
\end{bmatrix}\cdot
\begin{bmatrix}
\mathcal{E}^{\in}_{1c}\\
\mathcal{E}^{\in}_{1s}\\
\mathcal{E}^{\in}_{2c}\\
\mathcal{E}^{\in}_{2s}
\end{bmatrix}\,.
\end{equation}
The same rules apply to the sidebands of each carrier field:
\begin{equation}
\label{eq:IO_mirror_matrix_4x4_first}
\begin{bmatrix}
\hat{\boldsymbol{e}}^{\out}_{1}(\Omega)\\
\hat{\boldsymbol{e}}^{\out}_{2}(\Omega)
\end{bmatrix}=
\mathbb{M}_{\mathrm{real}}\otimes\mathbb{I}
\cdot
\begin{bmatrix}
\hat{\boldsymbol{e}}^{\in}_{1}(\Omega)\\
\hat{\boldsymbol{e}}^{\in}_{2}(\Omega)
\end{bmatrix}=
\begin{bmatrix}
-\sqrt{R} & 0 &\sqrt{T} & 0\\
0 & -\sqrt{R} & 0 & \sqrt{T} \\
\sqrt{T} & 0 & \sqrt{R} & 0\\
0 & \sqrt{T} & 0 & \sqrt{R}
\end{bmatrix}\cdot
\begin{bmatrix}
\hat{e}^{\in}_{1c}(\Omega)\\
\hat{e}^{\in}_{1s}(\Omega)\\
\hat{e}^{\in}_{2c}(\Omega)\\
\hat{e}^{\in}_{2s}(\Omega)
\end{bmatrix}\,.
\end{equation}
\epubtkImage{fig6.png}{%
\begin{figure}[htbp]
\centerline{\includegraphics[width=\textwidth]{fig6}}
\caption{Scheme of a beamsplitter.}
\label{fig:beamsplitter}
\end{figure}}
In future, for the sake of brevity, we reduce the notation for matrices like $\mathbb{M}_{\mathrm{real}}\otimes\mathbb{I}$ to simply $\mathbb{M}_{\mathrm{real}}$.
\paragraph*{Beam splitters:}
Another optical element ubiquitous in the interferometers is a beamsplitter (see Figure~\ref{fig:beamsplitter}). In fact, it is the very same mirror considered above, but the angle of input light beams incidence is different from 0 (if measured from the normal to the mirror surface). The corresponding scheme is given in Figure~\ref{fig:beamsplitter}. In most cases, symmetric 50\%/50\% beamsplitter are used, which imply $R=T=1/2$ and the coupling matrix $\mathbb{M}_{50/50}$ then reads:
\begin{equation}
\label{eq:IO_beamsplitter_matrix}
\mathbb{M}_{50/50} =
\begin{bmatrix}
-1/\sqrt{2} & 0 &1/\sqrt{2} & 0\\
0 & -1/\sqrt{2} & 0 & 1/\sqrt{2} \\
1/\sqrt{2} & 0 & 1/\sqrt{2} & 0\\
0 & 1/\sqrt{2} & 0 & 1/\sqrt{2}
\end{bmatrix}\,.
\end{equation}
\paragraph*{Losses in optical elements:}
\label{sec:losses_in_OE}
Above, we have made one assumption that is a bit idealistic. Namely,
we assumed our mirrors and beamsplitters to be lossless, but it could
never come true in real experiments; therefore, we need some way to
describe losses within the framework of our formalism. Optical loss is a
term that comprises a very wide spectrum of physical processes, including
scattering on defects of the coating, absorption of light photons in
the mirror bulk and coating that yields heating and so on. A full
description of loss processes is rather complicated. However, the most
important features that influence the light fields, coming off the
lossy optical element, can be summarized in the following two simple
statements:
\begin{enumerate}
\item Optical loss of an optical element can be characterized by a single number (possibly, frequency dependent) $\epsilon$ (usually, $|\epsilon|\ll 1$) that is called the \emph{absorption coefficient}. $\epsilon$ is the fraction of light power being lost in the optical element:
$$E^{\out}(t)\to \sqrt{1-\epsilon}E^{\out}(t).$$
\item Due to the fundamental law of nature summarized by the Fluctuation Dissipation Theorem (FDT)~\cite{PhysRev.83.34, Landau_Lifshitz_v5}, optical loss is always accompanied by additional noise injected into the system. It means that additional noise field $\hat{n}$ uncorrelated with the original light is mixed into the outgoing light field in the proportion of $\sqrt{\epsilon}$ governed by the absorption coefficient.
\end{enumerate}
These two rules conjure up a picture of an effective system comprising of
a lossless mirror and two imaginary non-symmetric beamsplitters with
reflectivity $\sqrt{1-\epsilon}$ and transmissivity $\sqrt{\epsilon}$
that models optical loss for both input fields, as drawn in
Figure~\ref{fig:lossy_mirror}.
\epubtkImage{fig7.png}{%
\begin{figure}[htbp]
\centerline{\includegraphics[width=\textwidth]{fig7}}
\caption{Model of lossy mirror.}
\label{fig:lossy_mirror}
\end{figure}}
Using the above model, it is possible to show that for a lossy mirror the transformation of carrier fields given by Eq.~\eqref{eq:IO_mirror_matrix_4x4_zero} should be modified by simply multiplying the output fields vector by a factor $1-\epsilon$:
\begin{equation}
\label{eq:IO_lossy_mirror_relations_zero}
\begin{bmatrix}
\vb{\mathcal{E}}^{\out}_{1}\\
\vb{\mathcal{E}}^{\out}_{2}
\end{bmatrix}=
(1-\epsilon)\mathbb{M}_{\mathrm{real}}
\cdot
\begin{bmatrix}
\vb{\mathcal{E}}^{\in}_{1}\\
\vb{\mathcal{E}}^{\in}_{2}
\end{bmatrix} \simeq \mathbb{M}_{\mathrm{real}}
\cdot
\begin{bmatrix}
\vb{\mathcal{E}}^{\in}_{1}\\
\vb{\mathcal{E}}^{\in}_{2}
\end{bmatrix}
\,,
\end{equation}
where we used the fact that for low loss optics in use in GW interferometers, the absorption coefficient might be as small as $\epsilon\sim 10^{-5}$--$10^{-4}$. Therefore, the impact of optical loss on classical carrier amplitudes is negligible. Where the noise sidebands are concerned, the transformation rule given by Eq.~\eqref{eq:IO_mirror_matrix_4x4_first} changes a bit more:
\begin{eqnarray}
\label{eq:IO_lossy_mirror_relations_first}
\begin{bmatrix}
\hat{\boldsymbol{e}}^{\out}_{1}(\Omega)\nonumber\\
\hat{\boldsymbol{e}}^{\out}_{2}(\Omega)
\end{bmatrix} &=&
(1-\epsilon)\mathbb{M}_{\mathrm{real}}
\cdot
\begin{bmatrix}
\hat{\boldsymbol{e}}^{\in}_{1}(\Omega)\nonumber\\
\hat{\boldsymbol{e}}^{\in}_{2}(\Omega)
\end{bmatrix} + \sqrt{\epsilon(1-\epsilon)}\mathbb{M}_{\mathrm{real}}\cdot
\begin{bmatrix}
\hat{\boldsymbol{n}}_{1}(\Omega)\nonumber\\
\hat{\boldsymbol{n}}_{2}(\Omega)
\end{bmatrix} \nonumber\\ &\simeq&
(1-\epsilon)\mathbb{M}_{\mathrm{real}}
\cdot
\begin{bmatrix}
\hat{\boldsymbol{e}}^{\in}_{1}(\Omega)\nonumber\\
\hat{\boldsymbol{e}}^{\in}_{2}(\Omega)
\end{bmatrix} + \sqrt{\epsilon}
\begin{bmatrix}
\hat{\boldsymbol{n}}_{1}'(\Omega)\\
\hat{\boldsymbol{n}}_{2}'(\Omega)
\end{bmatrix}\,.
\end{eqnarray}
Here, we again used the smallness of $\epsilon\ll1$ and also the fact that matrix $\mathbb{M}_{\mathrm{real}}$ is unitary, i.e., we redefined the noise that enters outgoing fields due to loss as $\left\{\hat{\boldsymbol{n}}_{1}',\,\hat{\boldsymbol{n}}_{2}'\right\}^{\mathsf{T}} = \mathbb{M}\cdot\left\{\hat{\boldsymbol{n}}_{1},\,\hat{\boldsymbol{n}}_{2}\right\}^{\mathsf{T}}$, which keeps the new noise sources $\hat{n}_1'(t)$ and $\hat{n}_2'(t)$ uncorrelated: $\mean{\hat{n}_1(t)\hat{n}_2(t')}=\mean{\hat{n}_1'(t)\hat{n}_2'(t')}=0$.
\subsubsection{Light modulation by mirror motion}
\label{sec:Mirror_motion}
For full characterization of the light transformation in the GW
interferometers, one significant aspect remains untouched, i.e., the
field transformation upon reflection off the movable mirror. Above
(see Section~\ref{sec:phasemeter}), we have seen that motion
of the mirror yields phase modulation of the reflected wave. Let us
now consider this process in more detail.
\epubtkImage{fig8.png}{%
\begin{figure}[htbp]
\centerline{\includegraphics[width=\textwidth]{fig8}}
\caption{Reflection of light from the movable mirror.}
\label{fig:mov_mirr_refl}
\end{figure}}
Consider the mirror described by the matrix $\mathbb{M}_{\mathrm{real}}$, introduced above. Let us set the convention that the relations of input and output fields is written for the initial position of the movable mirror reflective surface, namely for the position where its displacement is $x=0$ as drawn in Figure~\ref{fig:mov_mirr_refl}. We assume the sway of the mirror motion to be much smaller than the optical wavelength: $x/\lambda_0\ll1$. The effect of the mirror displacement $x(t)$ on the outgoing field $E^{\out}_{1,2}(t)$ can be straightforwardly obtained from the propagation formalism. Indeed, considering the light field at a fixed spatial point, the reflected light field at any instance of time $t$ is just the result of propagation of the incident light by twice the mirror displacement taken at time of reflection and multiplied by reflectivity $\pm\sqrt{R}$\epubtkFootnote{In fact, the argument of $x$ should be written as $t_*$, that is the moment when the actual reflection takes place and is the solution to the equation: $c(t-t_*)=x(t_*)$, but since the mechanical motion is much slower than that of light one has $\delta x/c\ll 1$. This fact implies $t\simeq t_*$.}:
\begin{eqnarray}
\label{eq:IO_mirror_relations_x}
E^{\out}_1(t) = -\sqrt{R} E^{\in}_1(t-2x(t)/c)+\sqrt{T}E^{\in}_2(t)\,,\nonumber\\
E^{\out}_2(t) = \sqrt{T}E^{\in}_1(t)+\sqrt{R} E^{\in}_2(t+2x(t)/c)\,.
\end{eqnarray}
Remember now our assumption that $x\ll\lambda_0$; according to Eq.~\eqref{eq:EMW_free_prop_small_x_matrix} the mirror motion modifies the quadrature amplitudes in a way that allows one to separate this effect from the reflection. It means that the result of the light reflection from the moving mirror can be represented as a sum of two independently calculable effects, i.e., the reflection off the fixed mirror, as described above in Section~\ref{sec:light_reflection}, and the response to the mirror displacement (see Section~\ref{sec:propagation}), i.e., the signal presentable as a sideband vector $\left\{\boldsymbol{s}_1(\Omega),\boldsymbol{s}_2(\Omega)\right\}^{\mathsf{T}}$. The latter is convenient to describe in terms of the response vector $\{\boldsymbol{R}_1\,,\boldsymbol{R}_2\}^{\mathsf{T}}$ that is defined as:
\begin{eqnarray}
\boldsymbol{s}_1(\Omega)=\boldsymbol{R}_1 x(t) = -\sqrt{R}\delta\mathbb{P}\left[2\omega_0 x(t)/c\right]\cdot\vb{\mathcal{E}}^{\in}_{1} = -\dfrac{2\omega_0\sqrt{R}}{c}
\begin{bmatrix}
\mathcal{E}^{\in}_{1s}\\
-\mathcal{E}^{\in}_{1c}
\end{bmatrix} x(t) \nonumber\\
\boldsymbol{s}_2(\Omega) = \boldsymbol{R}_2 x(t) = -\sqrt{R}\delta\mathbb{P}\left[2\omega_0x(t)/c\right]\cdot\vb{\mathcal{E}}^{\in}_{2} = -\dfrac{2\omega_0\sqrt{R}}{c}
\begin{bmatrix}
\mathcal{E}^{\in}_{2s}\\
-\mathcal{E}^{\in}_{2c}
\end{bmatrix} x(t)\,.\label{eq:IO_motion_induced_sidebands}
\end{eqnarray}
Note that we did not include sideband fields $\hat{\boldsymbol{e}}^{\in}_{1,2}(\Omega)$ in the definition of the response vector. In principle, sideband fields also feel the motion induced phase shift; however, as far as it depends on the product of one very small value of $2\omega_0x(t)/c=4\pi x(t)/\lambda_0\ll1$ by a small sideband amplitude $|\hat{\boldsymbol{e}}^{\in}_{1,2}(\Omega)|\ll |\mathcal{E}^{\in}_{1,2}|$, the resulting contribution to the final response will be dwarfed by that of the classical fields. Moreover, the mirror motion induced contribution~\eqref{eq:IO_motion_induced_sidebands} is itself a quantity of the same order of magnitude as the noise sidebands, and therefore we can claim that the classical amplitudes of the carrier fields are not affected by the mirror motion and that the relations~\eqref{eq:IO_mirror_matrix_4x4_zero} hold for a moving mirror too. However, the relations for sideband amplitudes must be modified. In the case of a lossless mirror, relations~\eqref{eq:IO_mirror_matrix_4x4_first} turn:
\begin{equation}
\label{eq:IO_mirror_relations_first+x}
\begin{bmatrix}
\hat{\boldsymbol{e}}^{\out}_{1}(\Omega)\\
\hat{\boldsymbol{e}}^{\out}_{2}(\Omega)
\end{bmatrix}=
\mathbb{M}_{\mathrm{real}}
\cdot
\begin{bmatrix}
\hat{\boldsymbol{e}}^{\in}_{1}(\Omega)\\
\hat{\boldsymbol{e}}^{\in}_{2}(\Omega)
\end{bmatrix} +
\begin{bmatrix}
\boldsymbol{R}_1\\
\boldsymbol{R}_2
\end{bmatrix}x(\Omega)\,,
\end{equation}
where $x(\Omega)$ is the Fourier transform of the mirror displacement $x(t)$
$$x(\Omega) = \int_{-\infty}^{\infty}\! dt\, x(t)e^{i\Omega t}\,.$$
It is important to understand that signal sidebands characterized by a vector $\left\{\boldsymbol{s}_1(\Omega),\boldsymbol{s}_2(\Omega)\right\}^{\mathsf{T}}$, on the one hand, and the noise sidebands $\left\{\vb{\hat{e}}_1(\Omega),\,\vb{\hat{e}}_2(\Omega)\right\}^{\mathsf{T}}$, on the other hand, have the same order of magnitude in the real GW interferometers, and the main role of the advanced quantum measurement techniques we are talking about here is to either increase the former, or decrease the latter as much as possible in order to make the ratio of them, known as the \emph{signal-to-noise ratio} (SNR), as high as possible in as wide as possible a frequency range.
\subsubsection{Simple example: the reflection of light from a perfect moving mirror}
All the formulas we have derived here, though being very simple in essence, look cumbersome and not very transparent in general. In most situations, these expressions can be simplified significantly in real schemes. Let us consider a simple example for demonstration purposes, i.e., consider the reflection of a single light beam from a perfectly reflecting ($R=1$) moving mirror as drawn in Figure~\ref{fig8}. The initial phase $\phi_0$ of the incident wave does not matter and can be taken as zero. Then $\mathcal{E}^{\in}_c = \mathcal{E}_0$ and $\mathcal{E}^{\in}_s = 0$. Putting these values into Eq.~\eqref{eq:IO_mirror_matrix_4x4_zero} and accounting for $\vb{\mathcal{E}}^{\in}_2=0$, quite reasonably results in the amplitude of the carrier wave not changing upon reflection off the mirror, while the phase changes by $\pi$:
$$\mathcal{E}^{\out}_c = -\mathcal{E}^{\out}_c = -\mathcal{E}_0,\quad \mathcal{E}^{\out}_s = 0\,.$$
Since we do not have control over the laser noise, the input light has laser fluctuations in both quadratures $\hat{\boldsymbol{e}}^{\in}_1 = \{\hat{e}^{\in}_{1c},\hat{e}^{\in}_{1s}\}$ that are transformed in full accordance with Eq.~\eqref{eq:IO_mirror_matrix_4x4_first}):
$$\hat{\boldsymbol{e}}^{\out}_1(\Omega) = -\hat{\boldsymbol{e}}^{\in}_1(\Omega)\,.$$
Again, nothing surprising. Let us see what happens with a mechanical motion induced component of the reflected wave: according to Eq.~\eqref{eq:IO_mirror_relations_first+x}, the reflected light will contain a motion-induced signal in the s-quadrature:
$$\boldsymbol{s}(\Omega) = \dfrac{2 \omega_0}{c}
\begin{bmatrix}
0\\
\mathcal{E}_0
\end{bmatrix}x(\Omega)\,.
$$
This fact, i.e., that the mirror displacement that just causes phase modulation of the reflected field, enters only the s-quadrature, once again justifies why this quadrature is usually referred to as \emph{phase quadrature} (cf. section \ref{sec:modulation}).
\epubtkImage{fig9.png}{%
\begin{figure}[htbp]
\centerline{\includegraphics[width=\textwidth]{fig9}}
\caption{Schematic view of light modulation by perfectly reflecting mirror motion.
An initially monochromatic laser field $E^{\in}(t)$ with frequency $\omega_0 = 2\pi c/\lambda_0$ gets reflected from the mirror that commits slow (compared to optical oscillations) motion $x(t)$ (blue line) under the action of external force $\boldsymbol{G}$. Reflected the light wave phase is modulated by the mechanical motion so that the spectrum of the outgoing field $E^{\out}(\omega)$ contains two sidebands carrying all the information about the mirror motion. The left panel shows the spectral representation of the initial monochromatic incident light wave (upper plot), the mirror mechanical motion amplitude spectrum (middle plot) and the spectrum of the phase-modulated by the mirror motion, reflected light wave (lower plot).}
\label{fig8}
\end{figure}}
It is instructive to see the spectrum of the outgoing light
in the above considered situation. It is, expectedly, the spectrum of
a phase modulated monochromatic wave that has a central peak at the
carrier wave frequency and the two sideband peaks on either sides of
the central one, whose shape follows the spectrum of the modulation
signal, in our case, the spectrum of the mechanical displacement of
the mirror $x(t)$. The left part of Figure~\ref{fig8} illustrates
the aforesaid. As for laser noise, it enters the outgoing light in an
additive manner and the typical (though simplified) amplitude spectrum
of a noisy light reflected from a moving mirror is given in
Figure~\ref{fig9}.
\epubtkImage{fig10.png}{%
\begin{figure}[htbp]
\centerline{\includegraphics[width=\textwidth]{fig10}}
\caption{The typical spectrum (amplitude spectral density) of the light leaving the interferometer with movable mirrors. The central peak corresponds to the carrier light with frequency $\omega_0$, two smaller peaks on either side of the carrier represent the signal sidebands with the shape defined by the mechanical motion spectrum $x(\Omega)$; the noisy background represents laser noise.}
\label{fig9}
\end{figure}}
\subsection{Basics of Detection: Heterodyne and homodyne readout techniques}
\label{subsec:Detection}
Let us now address the question of how one can detect a GW signal imprinted onto the parameters of the light wave passing through the interferometer. The simple case of a Michelson interferometer considered in Section~\ref{sec:MI} where the GW signal was encoded in the phase quadrature of the light leaking out of the signal(dark) port, does not exhaust all the possibilities. In more sophisticated interferometer setups that are covered in sections \ref{sec:QN_in_GW_interferometers} and \ref{sec:sub-SQL_schemes}, a signal component might be present in both quadratures of the outgoing light and, actually, to different extent at different frequencies; therefore, a detection method that allows measurement of an arbitrary combination of amplitude and phase quadrature is required. The two main methods are in use in contemporary GW detectors: these are \emph{homodyne} and \emph{heterodyne} detection~\cite{03a1BuChMa, 06a1SoChKaMi, 08a1Wa_etal, 09a1Hi_etal}. Both are common in radio-frequency technology as methods of detection of phase- and frequency-modulated signals. The basic idea is to mix a faint signal wave with a strong \emph{local oscillator} wave, e.g., by means of a beamsplitter, and then send it to a detector with a quadratic non-linearity that shifts the spectrum of the signal to lower frequencies together with amplification by an amplitude of the local oscillator. This topic is also discussed in Section~4 of the Living Review by Freise and Strain~\cite{lrr-2010-1} with more details relevant to experimental implementation.
\subsubsection{Homodyne and DC readout}
\label{sec:homodyne}
\paragraph*{Homodyne readout.}
Homodyne detection uses local oscillator light with the same carrier frequency as the signal. Write down the signal wave as:
\begin{equation}
\label{eq:homodyne_signal_wave}
S(t) = S_c(t)\cos\omega_0t+S_s(t)\sin\omega_0t
\end{equation}
and the local oscillator wave as:
\begin{equation}
\label{eq:homodyne_LO}
L(t) = L_c(t)\cos\omega_0t+L_s(t)\sin\omega_0t\,.
\end{equation}
Signal light quadrature amplitudes $S_{c,s}(t)$ might contain GW signal $G_{c,s}(t)$ as well as quantum noise $n_{c,s}(t)$ in both quadratures:
$$S_{c,s}(t) = G_{c,s}(t)+n_{c,s}(t)\,,$$
while the local oscillator is a laser light with classical amplitudes:
$$(L^{(0)}_c,\,L^{(0)}_s)=(L_0\cos\phi_{LO},\,L_0\sin\phi_{LO})\,,$$
where we introduced a \emph{homodyne angle} $\phi_{LO}$, and laser noise $l_{c,s}(t)$:
$$L_{c,s}(t) = L^{(0)}_{c,s}+l_{c,s}(t)\,.$$
Note that the local oscillator classical amplitude $L_0$ is much larger than all other signals:
$$L_0\gg\max\left[l_{c,s},\,G_{c,s},\,n_{c,s}\right]\,.$$
Let mix these two lights at the beamsplitter as drawn in the left panel of Figure~\ref{fig:homodyne_readout_scheme} and detect the two resulting outgoing waves with two photodetectors. The two photocurrents $i_{1,2}$ will be proportional to the intensities $I_{1,2}$ of these two lights:
\begin{eqnarray*}
i_1\propto I_1\propto \frac{\tmean{(L+S)^2}}{2} =\frac{L_0^2}{2}+L_0(G_c+l_c+n_c)\cos\phi_{LO}+L_0(G_s+l_s+n_s)\sin\phi_{LO}+\mathcal{O}\left[G_{c,s}^2,l_{c,s}^2,n_{c,s}^2\right]\,,\\
i_2\propto I_2\propto \frac{\tmean{(L-S)^2}}{2} =\frac{L_0^2}{2}-L_0(G_c-l_c+n_c)\cos\phi_{LO}-L_0(G_s-l_s+n_s)\sin\phi_{LO}+\mathcal{O}\left[G_{c,s}^2,l_{c,s}^2,n_{c,s}^2\right]\,,
\end{eqnarray*}
where $\tmean{A}$ stands for time averaging of $A(t)$ over many optical oscillation periods, which reflects the inability of photodetectors to respond at optical frequencies and thus providing natural low-pass filtering for our signal. The last terms in both expressions gather all the terms quadratic in GW signal and both noise sources that are of the second order of smallness compared to the local oscillator amplitude $L_0$ and thus are omitted in further consideration.
\epubtkImage{fig11.png}{%
\begin{figure}[htb]
\centerline{\includegraphics[width=\textwidth]{fig11}}
\caption{Schematic view of homodyne readout (\emph{left panel}) and DC readout (\emph{right panel}) principle implemented by a simple Michelson interferometer.}
\label{fig:homodyne_readout_scheme}
\end{figure}}
In a classic homodyne balanced scheme, the difference current is read out that contains only a GW signal and quantum noise of the dark port:
\begin{equation}
\label{eq:homodyne_photocurrent}
i_{\mathrm{hom}}=i_1-i_2\propto 2L_0\left[(G_c+n_c)\cos\phi_{LO}+(G_s+n_s)\sin\phi_{LO}\right]\,.
\end{equation}
Whatever quadrature the GW signal is in, by proper choice of the homodyne angle $\phi_{LO}$ one can recover it with minimum additional noise. That is how homodyne detection works in theory.
However, in real interferometers, the implementation of a homodyne readout appears to be fraught with serious technical difficulties. In particular, the local oscillator frequency has to be kept extremely stable, which means its optical path length and alignment need to be actively stabilized by a low-noise control system~\cite{09a1Hi_etal}. This inflicts a significant increase in the cost of the detector, not to mention the difficulties in taming the noise of stabilising control loops, as the experience of the implementation of such stabilization in a Garching prototype interferometer has shown \cite{PhysLettA.277.3.135_Freise,PhysRevLett.81.5493_Heinzel,99pth1Heinzel}.
\paragraph*{DC readout.}
These factors provide a strong motivation to look for another way to implement homodyning. Fortunately, the search was not too long, since the suitable technique has already been used by Michelson and Morley in their seminal experiment~\cite{1887a1MiMo}. The technique is known as DC-readout and implies an introduction of a constant arms length difference, thus pulling the interferometer out of the dark fringe condition as was mentioned in Section~\ref{sec:MI}. The advantage of this method is that the local oscillator is furnished by a part of the pumping carrier light that leaks into the signal port due to arms imbalance and thus shares the optical path with the signal sidebands. It automatically solves the problem of phase-locking the local oscillator and signal lights, yet is not completely free of drawbacks. The first suggestion to use DC readout in GW interferometers belongs to Fritschel~\cite{2003LSC_Fritschel} and then got comprehensive study by the GW community~\cite{06a1SoChKaMi, 08a1Wa_etal, 09a1Hi_etal}.
Let us discuss how it works in a bit more detail. The schematic view of a Michelson interferometer with DC readout is drawn in the right panel of Figure~\ref{fig:homodyne_readout_scheme}. As already mentioned, the local oscillator light is produced by a deliberately-introduced constant difference $\delta L$ of the lengths of the interferometer arms. It is also worth noting that the component of this local oscillator created by the asymmetry in the reflectivity of the arms that is always the case in a real interferometer and attributable mostly to the difference in the absorption of the `northern' and `eastern' end mirrors as well as asymmetry of the beamsplitter. All these factors can be taken into account if one writes the carrier fields at the beamsplitter after reflection off the arms in the following symmetric form:
\begin{eqnarray*}
E^{\out}_n(t) = -\frac{E_0}{\sqrt{2}}(1-\epsilon_n)\cos\omega_0(t-2L_n/c)= -\frac{\bar{\mathcal{E}}}{\sqrt{2}}(1-\Delta\epsilon)\cos\omega_0(\tmean{t}+\Delta L/c)\,,\\
E^{\out}_e(t) = -\frac{E_0}{\sqrt{2}}(1-\epsilon_e)\cos\omega_0(t-2L_e/c)= -\frac{\bar{\mathcal{E}}}{\sqrt{2}}(1+\Delta\epsilon)\cos\omega_0(\tmean{t}-\Delta L/c)\,,
\end{eqnarray*}
where $\epsilon_{n,e}$ and $L_{n,e}$ stand for optical loss and arm lengths of the corresponding interferometer arms, $\Delta\epsilon = \frac{\epsilon_n-\epsilon_e}{2(1-\bar{\epsilon})}$ is the optical loss relative asymmetry with $\bar{\epsilon}=(\epsilon_n+\epsilon_e)/2$, $\tmean{\mathcal{E}}= E_0(1-\bar{\epsilon})$ is the mean pumping carrier amplitude at the beamsplitter, $\Delta L = L_n-L_e$ and $\bar{t} = t+\frac{L_n+L_e}{2c}$. Then the classical part of the local oscillator light in the signal (dark) port will be given by the following expression:
\begin{equation}
\label{eq:DCreadout_LO}
L^{(0)}_{\mathrm{DC}}(t) =\frac{ E^{\out}_n(t)-E^{\out}_e(t)}{\sqrt{2}} = \underbrace{\bar{\mathcal{E}}\Delta\epsilon\cos\frac{\omega_0\Delta L}{c}}_{\sqrt{2}L_c^{(0)}}\cos\omega_0\bar{t}+\underbrace{\bar{\mathcal{E}}\sin\frac{\omega_0\Delta L}{c}}_{\sqrt{2}L_s^{(0)}}\sin\omega_0\bar{t}\,,
\end{equation}
where one can define the local oscillator phase and amplitude through the apparent relations:
\begin{equation}
\label{eq:DCreadout_phiLO}
\tan\phi_{\mathrm{DC}} = \frac{1}{\Delta\epsilon}\tan\frac{\omega_0\Delta L}{c}\,\quad L_{\mathrm{DC}}^{(0)} \simeq \bar{\mathcal{E}}\sqrt{(\Delta\epsilon)^2+\left(\frac{\omega_0\Delta L}{c}\right)^2}\simeq \frac{\omega_0E_0\Delta L}{c}\,,
\end{equation}
where we have taken into account that $\omega_0\Delta L/c\ll1$ and the rather small absolute value of the optical loss coefficient $\max\left[\epsilon_n,\epsilon_e\right]\sim 10^{-4}\ll1$ available in contemporary interferometers.
One sees that were there no asymmetry in the arms optical loss, there would be no opportunity to change the local oscillator phase. At the same time, the GW signal in the considered scheme is confined to the phase quadrature since it comprises the time-dependent part of $\Delta L$ and thus the resulting photocurrent will be proportional to:
\begin{equation}
\label{eq:DCreadout_photocurrent}
i_{\mathrm{DC}}\propto\tmean{(L+S)^2}\simeq \left(L_{\mathrm{DC}}^{(0)}\right)^2+2L_{\mathrm{DC}}^{(0)}(l^{\out}_c+n_c)\cos\phi_{\mathrm{DC}}+2L_{\mathrm{DC}}^{(0)}(G_s+l^{\out}_s+n_s)\sin\phi_{\mathrm{DC}},
\end{equation}
where $l_{c,s}^{\out}$ denote the part of the input laser noise that leaked into the output port:
\begin{eqnarray}
l_c^{\out}(t) \simeq l_c^{\in}\Delta\epsilon\cos\frac{\omega_0\Delta L}{c}-l_s^{\in}\sin\frac{\omega_0\Delta L}{c}\simeq l_c^{\in}\Delta\epsilon-l_s^{\in}\frac{\omega_0\Delta L}{c}\,,\label{eq:DCreadout_laser_noise_cos}\\
l_s^{\out}(t) \simeq l_c^{\in}\sin\frac{\omega_0\Delta L}{c}+l_s^{\in}\Delta\epsilon\cos\frac{\omega_0\Delta L}{c}\simeq l_c^{\in}\frac{\omega_0\Delta L}{c}+l_s^{\in}\Delta\epsilon\label{eq:DCreadout_laser_noise_sin}\,,
\end{eqnarray}
and $n_{c,s}$ stand for the quantum noise associated with the signal sidebands and entering the interferometer from the signal port.
In the case of a small offset of the interferometer from the dark fringe condition, i.e., for $\omega_0\Delta L/c=2\pi\Delta L/\lambda_0\ll1$, the readout signal scales as local oscillator classical amplitude, which is directly proportional to the offset itself: $L_{\mathrm{DC}}^{(0)}\simeq 2\pi E_0\frac{\Delta L}{\lambda_0}$. The laser noise associated with the pumping carrier also leaks to the signal port in the same proportion, which might be considered as the main disadvantage of the DC readout as it sets rather tough requirements on the stability of the laser source, which is not necessary for the homodyne readout. However, this problem, is partly solved in more sophisticated detectors by implementing power recycling and/or Fabry--P\'erot cavities in the arms. These additional elements turn the Michelson interferometer into a resonant narrow-band cavity for a pumping carrier with effective bandwidth determined by transmissivities of the power recycling mirror (PRM) and/or input test masses (ITMs) of the arm cavities divided by the corresponding cavity length, which yields the values of bandwidths as low as $\sim$~10~Hz. Since the target GW signal occupies higher frequencies, the laser noise of the local oscillator around signal frequencies turns out to be passively filtered out by the interferometer itself.
DC readout has already been successfully tested at the LIGO 40-meter interferometer in Caltech~\cite{08a1Wa_etal} and implemented in GEO\,600~\cite{07pth1Hi, 09a1Hi_etal, 2010a1De_etal} and in Enhanced LIGO~\cite{2011LSC_Fricke, 2009LSC_TechPaper_T0900023}. It has proven a rather promising substitution for the previously ubiquitous heterodyne readout (to be considered below) and has become a baseline readout technique for future GW detectors~\cite{09a1Hi_etal}.
\subsubsection{Heterodyne readout}
\label{sec:heterodyne}
Up until recently, the only readout method used in terrestrial GW detectors has been the heterodyne readout. Yet with more and more stable lasers being available for the GW community, this technique gradually gives ground to a more promising DC readout method considered above. However, it is instructive to consider briefly how heterodyne readout works and learn some of the reasons, that it has finally given way to its homodyne adversary.
\epubtkImage{fig12.png}{%
\begin{figure}[htb]
\centerline{\includegraphics[width=.75\textwidth]{fig12}}
\caption{Schematic view of heterodyne readout principle implemented by a simple Michelson interferometer. Green lines represent modulation sidebands at radio frequency $\Omega_{\mathrm{RF}}$ and blue dotted lines feature signal sidebands}
\label{fig:heterodyne_readout_scheme}
\end{figure}}
The idea behind the heterodyne readout principle is the generalization of the homodyne readout, i.e., again, the use of strong local oscillator light to be mixed up with the faint signal light leaking out the dark port of the GW interferometer save the fact that local oscillator light frequency is shifted from the signal light carrier frequency by $\Omega_{\mathrm{RF}}\sim$ several megahertz. In GW interferometers with heterodyne readout, local oscillator light of different than $\omega_0$ frequency is produced via phase-modulation of the pumping carrier light by means of electro-optical modulator (EOM) before it enters the interferometer as drawn in Figure~\ref{fig:heterodyne_readout_scheme}. The interferometer is tuned so that the readout port is dark for the pumping carrier. At the same time, by introducing a macroscopic (several centimeters) offset $\Delta L$ of the two arms, which is known as Schnupp asymmetry~\cite{1988Conf_Schnupp}, the modulation sidebands at radio frequency $\Omega_{\mathrm{RF}}$ appear to be optimally transferred from the pumping port to the readout one. Therefore, the local oscillator at the readout port comprises two modulation sidebands, $L_{\mathrm{het}}(t) = L_+(t)+L_-(t)$, at frequencies $\omega_0+\Omega_{\mathrm{RF}}$ and $\omega_0-\Omega_{\mathrm{RF}}$, respectively. These two are detected together with the signal sidebands at the photodetector, and then the resulting photocurrent is demodulated with the RF-frequency reference signal yielding an output current proportional to GW-signal.
This method was proposed and studied in great detail in the following works~\cite{JMO.34.6.793_1987, 1988Conf_Schnupp, PhysLettA.277.3.135_Freise,PhysRevLett.81.5493_Heinzel,99pth1Heinzel,PhysRevA.44.4693, PhysRevA.43.5022} where the heterodyne technique for GW interferometers tuned in resonance with pumping carrier field was considered and, therefore, the focus was made on the detection of only phase quadrature of the outgoing GW signal light. This analysis was further generalized to detuned interferometer configurations in~\cite{03a1BuChMa, 06a1SoChKaMi} where the full analysis of quantum noise in GW dual-recycled interferometers with heterodyne readout was done.
Let us see in a bit more detail how the heterodyne readout works as exemplified by a simple Michelson interferometer drawn in Figure~\ref{fig:heterodyne_readout_scheme}. The equation of motion at the input port of the interferometer creates two phase-modulation sideband fields ($L_+(t)$ and $L_-(t)$) at frequencies $\omega_0\pm\Omega_{\mathrm{RF}}$:
\begin{eqnarray*}
L_+(t) = \left[L^{(0)}_{c+}+l_{c+}\right]\cos(\omega_0+\Omega_{\mathrm{RF}})t+\left[L^{(0)}_{s+}+l_{s+}\right]\sin(\omega_0+\Omega_{\mathrm{RF}})t\,,\\
L_-(t) = \left[L^{(0)}_{c-}+l_{c-}\right]\cos(\omega_0-\Omega_{\mathrm{RF}})t+\left[L^{(0)}_{s-}+l_{s-}\right]\sin(\omega_0-\Omega_{\mathrm{RF}})t\,,
\end{eqnarray*}
where $L^{(0)}_{(c,s)\pm}$ stand for classical quadrature amplitudes of the modulation (upper and lower) sidebands\epubtkFootnote{In the resonance-tuned case, the phase modulation of the input carrier field creates equal magnitude sideband fields as discussed in Section~\ref{sec:modulation}, and these sideband fields are transmitted to the output port thanks to Schnupp asymmetry in the same state, i.e., they remain equal in magnitude and reside in the phase quadrature. In detuned configurations of GW interferometers, the upper and lower RF-sideband fields are transformed differently, which influences both their amplitudes and phases at the readout port.} and $l_{(c,s)\pm}(t)$ represent laser noise around the corresponding modulation frequency.
Unlike the homodyne readout schemes, in the heterodyne ones, not only the quantum noise components $n_{c,s}^{\omega_0}$ falling into the GW frequency band around the carrier frequency $\omega_0$ has to be accounted for but also those rallying around twice the RF modulation frequencies $\omega_0\pm2\Omega_{\mathrm{RF}}$:
\begin{eqnarray}
S_{\mathrm{het}}(t) &=&
(G_c+n_c^{\omega_0})\cos\omega_0t+(G_s+n_s^{\omega_0})\sin\omega_0t \nonumber\\
&& + n_c^{\omega_0+2\Omega_{\mathrm{RF}}}\cos(\omega_0+2\Omega_{\mathrm{RF}})t+n_s^{\omega_0+2\Omega_{\mathrm{RF}}}\sin(\omega_0+2\Omega_{\mathrm{RF}})t \\
&& + n_c^{\omega_0-2\Omega_{\mathrm{RF}}}\cos(\omega_0-2\Omega_{\mathrm{RF}})t+n_s^{\omega_0-2\Omega_{\mathrm{RF}}}\sin(\omega_0-2\Omega_{\mathrm{RF}})t\,.
\end{eqnarray}
The analysis of the expression for the heterodyne photocurrent
\begin{equation}
\label{eq:heterodyne_photocurrent}
i_{\mathrm{het}}\propto\tmean{(L_{\mathrm{het}}+S)^2} = \tmean{S^2}+\tmean{L_+^2}+\tmean{L_-^2}+2\tmean{(L_++L_-)S}+2\tmean{L_+L_-}\,
\end{equation}
gives a clue to why these additional noise components emerge in the outgoing signal. It is easier to perform this kind of analysis if we represent the above trigonometric expressions in terms of scalar products of the vectors of the corresponding quadrature amplitudes and a special unit-length vector $\boldsymbol{H}[\phi] = \left\{\cos\phi,\,\sin\phi\right\}^{\mathsf{T}}$, e.g.:
$$S_{\mathrm{het}}\equiv(\boldsymbol{G}+\boldsymbol{n}_{\omega_0})^{\mathsf{T}}\cdot\boldsymbol{H}[\omega_0t]+\boldsymbol{n}_{\omega_0+2\Omega_{\mathrm{RF}}}^{\mathsf{T}}\cdot\boldsymbol{H}[(\omega_0+\Omega_{\mathrm{RF}})t]+\boldsymbol{n}_{\omega_0-2\Omega_{\mathrm{RF}}}^{\mathsf{T}}\cdot\boldsymbol{H}[(\omega_0-2\Omega_{\mathrm{RF}})t]$$
where $\boldsymbol{G}=\left\{G_c,\,G_s\right\}^{\mathsf{T}}$ and $\boldsymbol{n}_{\omega\alpha}=\left\{n^\omega_c,\,n_s^\omega\right\}^{\mathsf{T}}$. Another useful observation, provided that $\omega_0\gg\max[\Omega_1,\,\Omega_2]$, gives us the following relation:
$$\tmean{\boldsymbol{H}[(\omega_0+\Omega_1)t]\boldsymbol{H}^{\mathsf{T}}[(\omega_0+\Omega_2)t]}=\frac12
\begin{bmatrix}
\cos(\Omega_1-\Omega_2)t & -\sin(\Omega_1-\Omega_2)t\\
\sin(\Omega_1-\Omega_2)t & \cos(\Omega_1-\Omega_2)t
\end{bmatrix} = \frac12\mathbb{P}\left[(\Omega_1-\Omega_2)t\right] \,.
$$
Using this relation it is straightforward to see that the first three terms in Eq.~\eqref{eq:heterodyne_photocurrent} give DC components of the photocurrent, while the fifth term oscillates at double modulation frequency $2\Omega_{\mathrm{RF}}$. It is only the term $2\tmean{(L_++L_-)S}$ that is linear in GW signal and thus contains useful information:
\begin{equation*}
2\tmean{(L_++L_-)S} \simeq I_c(t)\cos\Omega_{\mathrm{RF}}t+I_s(t)\sin\Omega_{\mathrm{RF}}t+\left\{\mbox{oscillations at frequency }3\Omega_{\mathrm{RF}}\right\}
\end{equation*}
where
\begin{eqnarray*}
I_c(t) &=& (\boldsymbol{G}+\boldsymbol{n}_{\omega_0}+\boldsymbol{n}_{\omega_0+2\Omega_{\mathrm{RF}}})^{\mathsf{T}}\cdot\boldsymbol{L}^{(0)}_++(\boldsymbol{G}+\boldsymbol{n}_{\omega_0}+\boldsymbol{n}_{\omega_0-2\Omega_{\mathrm{RF}}})^{\mathsf{T}}\cdot\boldsymbol{L}^{(0)}_-\,,\\
I_s(t) &=& -i(\boldsymbol{G}+\boldsymbol{n}_{\omega_0}+\boldsymbol{n}_{\omega_0+2\Omega_{\mathrm{RF}}})^{\mathsf{T}}\cdot\sigma_y\cdot\boldsymbol{L}^{(0)}_++i(\boldsymbol{G}+\boldsymbol{n}_{\omega_0}+\boldsymbol{n}_{\omega_0-2\Omega_{\mathrm{RF}}})^{\mathsf{T}}\cdot\sigma_y\cdot\boldsymbol{L}^{(0)}_-\,,
\end{eqnarray*}
and $\sigma_y$ is the 2nd Pauli matrix:
$$\sigma_y =
\begin{bmatrix}
0 & -i\\
i & 0
\end{bmatrix}\,.
$$
In order to extract the desired signal quadrature the photodetector readout current $i_{\mathrm{het}}$ is mixed with (multiplied by) a demodulation function $D(t) = D_0\cos(\Omega_{\mathrm{RF}}t+\phi_D)$ with the resulting signal filtered by a low-pass filter with upper cut-off frequency $\Lambda\ll\Omega_{\mathrm{RF}}$ so that only components oscillating at GW frequencies $\Omega_{\mathrm{GW}}\ll\Omega_{\mathrm{RF}}$ appear in the output signal (see Figure~\ref{fig:heterodyne_readout_scheme}).
It is instructive to see what the above procedure yields in the simple case of the Michelson interferometer tuned in resonance with RF-sidebands produced by pure phase modulation: $L_{c+}^{(0)}=L_{c-}^{(0)}=0$ and $L_{s+}^{(0)}=L_{s-}^{(0)}=L_0$. The foregoing expressions simplify significantly to the following:
$$I_c(t) = 2L_0\left(G_s+n_s^{\omega_0}+\frac{n_s^{\omega_0-2\Omega_{\mathrm{RF}}}}{2}+\frac{n_s^{\omega_0+2\Omega_{\mathrm{RF}}}}{2}\right)\quad\mbox{and}\quad I_s(t) = -L_0\left(n_s^{\omega_0-2\Omega_{\mathrm{RF}}}-n_s^{\omega_0+2\Omega_{\mathrm{RF}}}\right)\,.$$
Apparently, in this simple case of equal sideband amplitudes (\emph{balanced heterodyne detection}), only single phase quadrature of the GW signal can be extracted from the output photocurrent, which is fine, because the Michelson interferometer, being equivalent to a simple movable mirror with respect to a GW tidal force as shown in Section~\ref{sec:phasemeter} and \ref{sec:MI}, is sensitive to a GW signal only in phase quadrature. Another important feature of heterodyne detection conspicuous in the above equations is the presence of additional noise from the frequency bands that are twice the RF-modulation frequency away from the carrier. As shown in~\cite{03a1BuChMa} this noise contributes to the total quantum shot noise of the interferometer and makes the high frequency sensitivity of the GW detectors with heterodyne readout 1.5 times worse compared to the ones with homodyne or DC readout.
For more realistic and thus more sophisticated optical configurations, including Fabry--P\'{e}rot cavities in the arms and additional recycling mirrors in the pumping and readout ports, the analysis of the interferometer sensitivity becomes rather complicated. Nevertheless, it is worthwhile to note that with proper optimization of the modulation sidebands and demodulation function shapes the same sensitivity as for frequency-independent homodyne readout schemes can be obtained~\cite{03a1BuChMa}. However, inherent additional frequency-independent quantum shot noise brought by the heterodyning process into the detection band hampers the simultaneous use of advanced quantum non-demolition (QND) techniques and heterodyne readout schemes significantly.
\newpage
\section{Quantum Nature of Light and Quantum Noise}
\label{sec:quantum_light}
Now is the time to remind ourselves of the word `quantum' in the title of our review. Thus far, the quantum nature of laser light being used in the GW interferometers has not been accounted for in any way. Nevertheless, quantum mechanics predicts striking differences for the variances of laser light amplitude and phase fluctuations, depending on which quantum state it is in. Squeezed vacuum~\cite{1995BookWaMi, 95BookMaWo, 97BookScZu, 81a1Ca, 02a1KiLeMaThVy} injection that has been recently implemented in the GEO\,600 detector and has pushed the high-frequency part of the total noise down by 3.5~dB~\cite{Vahlbruch_CQG_27_084027_2010,Nat.Phys.7.12.962_2011_LSC} serves as a perfect example of this. In this section, we provide a brief introduction into the quantization of light and the typical quantum states thereof that are common for the GW interferometers.
\subsection{Quantization of light: Two-photon formalism}
\label{sec:2photon_formalism}
From the point of view of quantum field theory, a freely propagating electromagnetic wave can be characterized in each spatial point with location vector $\boldsymbol{r}=(x,y,z)$ at time $t$ by a Heisenberg operator of an electric field strength $\hat{E}(\boldsymbol{r},\,t)$.\epubtkFootnote{Insofar as the light beams in the interferometer can be well approximated as paraxial beams, and the polarization of the light wave does not matter in most of the considered interferometers, we will omit the vector nature of the electric field and treat it as a scalar field with strength defined by a scalar operator-valued function $\hat{E}(x,y,z,t)$.} The electric field Heisenberg operator of a light wave traveling along the positive direction of the $z$-axis can be written as a sum of a positive- and negative-frequency parts:
\begin{equation}
\label{eq:EMW_quantum}
\hat{E}(x,y,z;t) = u(x,y,z)\left\{\hat{E}^{(+)}(t)+\hat{E}^{(-)}(t)\right\}\,,
\end{equation}
where $u(x,y,z)$ is the spatial mode shape, slowly changing on a wavelength $\lambda$ scale, and
\begin{equation}
\label{eq:EMW_expansion_in_modes}
\hat{E}^{(+)}(t) = \int_0^\infty\dfrac{d\omega}{2\pi}\sqrt{\dfrac{2\pi \hbar\omega}{\mathcal{A}c}}\hat{a}_\omega e^{-i\omega t}\,,\quad\mbox{and}\quad \hat{E}^{(-)}(t)=\left[\hat{E}^{(+)}(t)\right]^\dag\,.
\end{equation}
Here, $\mathcal{A}$ is the effective cross-section area of the light beam, and $\hat{a}_\omega$ ($\hat{a}_\omega^\dag$) is the single photon \emph{annihilation} (\emph{creation}) operator in the mode of the field with frequency $\omega$. The meaning of Eq.~\eqref{eq:EMW_expansion_in_modes} is that the travelling light wave can be represented by an expansion over the continuum of harmonic oscillators -- modes of the electromagnetic field, -- that are, essentially, independent degrees of freedom. The latter implies the commutation relations for the operators $\hat{a}_\omega$ and $\hat{a}_\omega^\dag$:
\begin{equation}
\label{eq:EMW_commutator_modes}
\left[\hat{a}_\omega,\,\hat{a}_{\omega'}^\dag\right] = 2\pi\delta(\omega-\omega')\,,\quad\mbox{and}\quad \left[\hat{a}_\omega,\,\hat{a}_{\omega'}\right] = \left[\hat{a}^\dag_\omega,\,\hat{a}^\dag_{\omega'}\right] = 0\,.
\end{equation}
In GW detectors, one deals normally with a close to monochromatic laser light with carrier frequency $\omega_0$, and a pair of modulation sidebands created by a GW signal around its frequency in the course of parametric modulation of the interferometer arm lengths. The light field coming out of the interferometer cannot be considered as the continuum of independent modes anymore. The very fact that sidebands appear in pairs implies the two-photon nature of the processes taking place in the GW interferometers, which means the modes of light at frequencies $\omega_{1,2}=\omega_0\pm\Omega$ have correlated complex amplitudes and thus the new two-photon operators and related formalism is necessary to describe quantum light field transformations in GW interferometers. This was realized in the early 1980s by Caves and Schumaker who developed the two-photon formalism~\cite{85a1CaSch, 85a2CaSch}, which is widely used in GW detectors as well as in quantum optics and optomechanics.
One starts by defining modulation sideband amplitudes as
$$\hat{a}_+ = \hat{a}_{\omega_0+\Omega}\,,\quad\hat{a}_- = \hat{a}_{\omega_0-\Omega}\,,$$
and factoring out the oscillation at carrier frequency $\omega_0$ in Eqs.~\eqref{eq:EMW_quantum}, which yields:
\begin{eqnarray} \label{eq:2photon_Eplus}
\hat{E}^{(+)}(t) = \frac{\mathcal{C}_0 e^{-i\omega_0t}}{\sqrt{2}}\int_{-\omega_0}^\infty\frac{d\Omega}{2\pi}\lambda_+(\Omega)\hat{a}_+e^{-i\Omega t}\simeq\frac{\mathcal{C}_0}{\sqrt{2}}e^{-i\omega_0t}\int_{-\infty}^{\infty}\!\frac{d\Omega}{2\pi}\lambda_+(\Omega)\hat{a}_+e^{-i\Omega t}\,,\nonumber\\
\hat{E}^{(-)}(t) = \frac{\mathcal{C}_0e^{i\omega_0t}}{\sqrt{2}}\int_{-\infty}^{\omega_0}\frac{d\Omega}{2\pi}\lambda_-(\Omega)\hat{a}^\dag_-e^{-i\Omega t}\simeq\frac{\mathcal{C}_0}{\sqrt{2}}e^{i\omega_0t}\int_{-\infty}^{\infty}\!\frac{d\Omega}{2\pi}\lambda_-(\Omega)\hat{a}^\dag_-e^{-i\Omega t}\,,
\end{eqnarray}
where we denote $\mathcal{C}_0\equiv\sqrt{\frac{4\pi\hbar\omega_0}{\mathcal{A}c}}$ and define functions $\lambda_{\pm}(\Omega)$ following~\cite{85a1CaSch} as
\begin{equation*}
\lambda_{\pm}(\Omega) = \sqrt{\frac{\omega_0\pm\Omega}{\omega_0}}\,,
\end{equation*}
and use the fact that $\omega_0\gg\Omega_{\mathrm{GW}}$ enables us to expand the limits of integrals to $\omega_0\to\infty$.
The operator expressions in front of $e^{\pm i\omega_0t}$ in the foregoing Eqs.~\eqref{eq:2photon_Eplus} are quantum analogues to the complex amplitude $\mathcal{E}$ and its complex conjugate $\mathcal{E}^*$ defined in Eqs.~\eqref{eq:EMW_quadrature_def}:
\begin{equation*}
\hat{\mathcal{E}}(t)=\frac{\mathcal{C}_0}{\sqrt{2}}\hat{a}(t)\equiv\frac{\mathcal{C}_0}{\sqrt{2}}\int_{-\infty}^{\infty}\!\frac{d\Omega}{2\pi}\lambda_+(\Omega)\hat{a}_+e^{-i\Omega
t}\,,\ \mbox{and}\ \hat{\mathcal{E}}^\dag(t)=\frac{\mathcal{C}_0}{\sqrt{2}}\hat{a}^\dag(t)\equiv\frac{\mathcal{C}_0}{\sqrt{2}}\int_{-\infty}^{\infty}\!\frac{d\Omega}{2\pi}\lambda_-(\Omega)\hat{a}^\dag_-e^{-i\Omega t}\,.
\end{equation*}
Again using Eqs.~\eqref{eq:EMW_quadrature_def}, we can define two-photon quadrature amplitudes as:
\begin{eqnarray}
\label{eq:2photon_quadratures}
\hat{\mathcal{E}}_c(t) = \frac{\hat{\mathcal{E}}(t)+\hat{\mathcal{E}}^\dag(t)}{\sqrt{2}} = \frac{\mathcal{C}_0}{\sqrt{2}}\int_{-\infty}^{\infty}\!\frac{d\Omega}{2\pi}\frac{\lambda_+\hat{a}_++\lambda_-\hat{a}^\dag_-}{\sqrt{2}}e^{-i\Omega t} \equiv \frac{\mathcal{C}_0}{\sqrt{2}}\int_{-\infty}^{\infty}\!\frac{d\Omega}{2\pi}\hat{a}_c(\Omega)e^{-i\Omega t} \nonumber\\
\hat{\mathcal{E}}_s(t) = \frac{\hat{\mathcal{E}}(t)-\hat{\mathcal{E}}^\dag(t)}{i\sqrt{2}} = \frac{\mathcal{C}_0}{\sqrt{2}}\int_{-\infty}^{\infty}\!\frac{d\Omega}{2\pi}\frac{\lambda_+\hat{a}_+-\lambda_-\hat{a}^\dag_-}{i\sqrt{2}}e^{-i\Omega t} \equiv \frac{\mathcal{C}_0}{\sqrt{2}}\int_{-\infty}^{\infty}\!\frac{d\Omega}{2\pi}\hat{a}_s(\Omega)e^{-i\Omega t}.
\end{eqnarray}
Note that so-introduced operators of two-photon quadrature amplitudes $\hat{a}_{c,s}(t)$ are Hermitian and thus their frequency domain counterparts satisfy the relations for the spectra of Hermitian operator:
\begin{equation*}
\hat{a}^\dag_{c,s}(t) = \hat{a}_{c,s}(t)\ \Longrightarrow\ \hat{a}^\dag_{c,s}(\Omega) = \hat{a}_{c,s}(-\Omega)\,.
\end{equation*}
Now we are able to write down commutation relations for the quadrature operators, which can be derived from Eq.~\eqref{eq:EMW_commutator_modes}:
\begin{eqnarray}
\left[\hat{a}_c(\Omega),\,\hat{a}^\dag_c(\Omega')\right] &=& \left[\hat{a}_s(\Omega),\,\hat{a}^\dag_s(\Omega')\right] = 2\pi\frac{\Omega}{\omega_0}\delta(\Omega-\Omega')\,,\label{eq:2photon_commutator1}\\
\left[\hat{a}_c(\Omega),\,\hat{a}^\dag_s(\Omega')\right] &=& \left[\hat{a}^\dag_c(\Omega),\,\hat{a}_s(\Omega')\right] = 2\pi i\delta(\Omega-\Omega')\,,\label{eq:2photon_commutator2}
\end{eqnarray}
The commutation relations represented by
Eqs.~\eqref{eq:2photon_commutator1} indicate that quadrature
amplitudes do not commute at different times,
i.e., $[\hat{a}_c(t),\,\hat{a}_c(t')]=[\hat{a}_s(t),\,\hat{a}_s(t')]\neq0$,
which imply they could not be considered for proper output observables
of the detector, for a nonzero commutator, as we would see later,
means an additional quantum noise inevitably contributes to the final
measurement result. The detailed explanation of why it is so can be
found in many works devoted to continuous linear quantum measurement
theory, in particular, in Chapter~6 of~\cite{92BookBrKh}, Appendix~2.7
of~\cite{03pth1Ch} or in~\cite{03a1BrGoKhMaThVy}. Where GW
detection is concerned, all the authors are agreed on the point that
the values of GW frequencies $\Omega$
$(1\mathrm{\ Hz} \leqslant\Omega/2\pi\leqslant 10^{3}\mathrm{\ Hz})$, being much smaller than
optical frequencies $\omega_0/2\pi\sim10^{15}\mathrm{\ Hz}$, allow one to neglect
such weak commutators as those of Eqs.~\eqref{eq:2photon_commutator1}
in all calculations related to GW detectors output quantum noise. This
statement has gotten an additional ground in the calculation conducted in
Appendix~2.7 of~\cite{03pth1Ch} where the value of the additional
quantum noise arising due to the nonzero value of commutators
\eqref{eq:2photon_commutator1} has been derived and its extreme
minuteness compared to other quantum noise sources has been
proven. Braginsky et~al. argued in~\cite{03a1BrGoKhMaThVy} that the
two-photon quadrature amplitudes defined by
Eqs.~\eqref{eq:2photon_quadratures} are not the real measured
observables at the output of the interferometer, since the
photodetectors actually measure not the energy flux
\begin{equation}
\hat{\mathcal{I}}(t)=\int\int_0^\infty\frac{d\omega d\omega'}{(2\pi)^2}\,\hbar\sqrt{\omega\omega'}\hat{a}^\dag_\omega\hat{a}_{\omega'}e^{i(\omega-\omega')t}
\end{equation}
but rather the photon number flux:
\begin{equation}
\hat{\mathcal{N}}(t)=\int\int_0^\infty\frac{d\omega d\omega'}{(2\pi)^2}\,\hat{a}^\dag_\omega\hat{a}_{\omega'}e^{i(\omega-\omega')t}\,.
\end{equation}
The former does not commute with itself: $[\hat{\mathcal{I}}(t),\,\hat{\mathcal{I}}(t')]\neq0$, while the latter apparently does $[\hat{\mathcal{N}}(t),\,\hat{\mathcal{N}}(t')]=0$ and therefore
is the right observable for a self-consistent quantum description of the GW interferometer output signal.
In the course of our review, we shall adhere to the approximate quadrature amplitude operators that can be obtained from the exact ones given by Eqs.~\eqref{eq:2photon_quadratures} by setting $\lambda_\pm(\Omega)\to1$, i.e.,
\begin{eqnarray}
\label{eq:2photon_quads_def}
\hat{a}_c(t) &=& \int_{-\infty}^{\infty}\!\frac{d\Omega}{2\pi}\frac{\hat{a}_++\hat{a}_-^\dag}{\sqrt{2}}e^{-i\Omega t}\ \Longleftrightarrow\ \hat{a}_c(\Omega) = \frac{\hat{a}_++\hat{a}_-^\dag}{\sqrt{2}}\,,\nonumber\\
\hat{a}_s(t) &=& \int_{-\infty}^{\infty}\!\frac{d\Omega}{2\pi}\frac{\hat{a}_+-\hat{a}_-^\dag}{i\sqrt{2}}e^{-i\Omega t}\ \Longleftrightarrow\ \hat{a}_s(\Omega) = \frac{\hat{a}_+-\hat{a}_-^\dag}{i\sqrt{2}}\,.
\end{eqnarray}
The new approximate two-photon quadrature operators satisfy the following commutation relations in the frequency domain:
\begin{equation}
\label{eq:2photon_quads_commutator_spectral}
\left[\hat{a}_c(\Omega),\,\hat{a}_s(\Omega')\right] = 2\pi i\delta(\Omega+\Omega')\,,\quad\mbox{and}\quad \left[\hat{a}_c(\Omega),\,\hat{a}_c(\Omega')\right] = \left[\hat{a}_s(\Omega),\,\hat{a}_s(\Omega')\right] = 0\,,
\end{equation}
and in the time domain:
\begin{equation}
\label{eq:2photon_quads_commutator_time}
\left[\hat{a}_c(t),\,\hat{a}_s(t')\right] = i\delta(t-t')\,,\quad\mbox{and}\quad \left[\hat{a}_c(t),\,\hat{a}_c(t')\right] = \left[\hat{a}_s(t),\,\hat{a}_s(t')\right] = 0\,
\end{equation}
Then the electric field strength operator~\eqref{eq:EMW_quantum} can be rewritten in terms of the two-photon quadrature operators as:
\begin{equation}
\label{eq:2photon_EMW_quadratures}
\hat{E}(x,y,z;t) = u(x,y,z)\mathcal{C}_0\left[\hat{a}_c(t)\cos\omega_0t+\hat{a}_s(t)\sin\omega_0t\right].
\end{equation}
Hereafter, we will omit the spatial mode factor $u(x,y,z)$ since it does not influence the final result for quantum noise spectral densities. Moreover, in order to comply with the already introduced division of the optical field into classical carrier field and to the 1st order corrections to it comprising of laser noise and signal induced sidebands, we adopt the same division for the quantum fields, i.e., we detach the mean values of the corresponding quadrature operators via the following redefinition
$\hat{a}^{\mathrm{old}}_{c_s}\to A_{c,s}+\hat{a}^{\mathrm{new}}_{c,s}$ with $A_{c,s}\equiv\mean{\hat{a}^{\mathrm{old}}_{c,s}}$. Here, by $\mean{\hat{a}^{\mathrm{old}}_{c,s}}$ we denote an ensemble average over the quantum state $\ket{\psi}$ of the light wave: $\mean{\hat{A}}\equiv\bra{\psi}\hat{A}\ket{\psi}$. Thus, the electric field strength operator for a plain electromagnetic wave will have the following form:
\begin{equation}
\label{eq:2photon_E_strain}
\hat{E}(x,y;t) = \mathcal{C}_0\left[(A_c+\hat{a}_c(t))\cos\omega_0t+(A_s+\hat{a}_s(t))\sin\omega_0t\right]\,.
\end{equation}
Further, we combine the two-photon quadratures into vectors in the same manner as we used to do for classical fields:
$$\boldsymbol{A}\equiv
\begin{bmatrix}
A_c\\
A_s
\end{bmatrix}\,,\quad\mbox{and}\quad
\hat{\boldsymbol{a}}\equiv\begin{bmatrix}
\hat{a}_c\\
\hat{a}_s
\end{bmatrix}\,.
$$
Now, when we have defined a quantum Heisenberg operator of the electric field of a light wave, and introduced quantum operators of two-photon quadratures, the last obstacle on our way towards the description of quantum noise in GW interferometers is that we do not know the quantum state the light field finds itself in. Since it is the quantum state that defines the magnitude and mutual correlations of the amplitude and phase fluctuations of the outgoing light, and through it the total level of quantum noise setting the limit on the future GW detectors' sensitivity. In what follows, we shall consider vacuum and coherent states of the light, and also squeezed states, for they comprise the vast majority of possible states one could encounter in GW interferometers.
\subsection{Quantum states of light}
\label{sec:light_quantum_states}
\subsubsection{Vacuum state}
\label{sec:vacuum_state}
The quantum state of the travelling wave is a subtle structure, for the system it describes comprises a continuum of modes. However, each of these modes can be viewed at as a quantum oscillator with its own generalized coordinate $\hat{X}_\omega = (\hat{a}_\omega+\hat{a}^\dag_\omega)/\sqrt{2}$ and momentum $\hat{Y}_\omega=(\hat{a}_\omega-\hat{a}^\dag_\omega)/i\sqrt{2}$. The ground state of this system, known as a \emph{vacuum} state $\ket{\vac}$, is straightforward and is simply the direct product of the ground states $\ket{0}_\omega$ of all modes over all frequencies $\omega$:
\begin{equation}\label{eq:field_vac_state}
\ket{\vac}\equiv\bigotimes\limits_\omega\ket{0}_\omega\,.
\end{equation}
By definition, the ground state of a mode with frequency $\omega$ is the state with minimum energy $E_{\vac}=\hbar\omega/2$ and no excitation:
\begin{equation}
\label{eq:annihilation_operator}
\hat{a}_\omega\ket{0}_\omega = 0\quad \mbox{and}\quad \bra{0}_\omega\hat{a}^\dag_\omega = 0\,.
\end{equation}
Consider now statistical properties of the vacuum state. The mean values of annihilation and creation operators as well as any linear combination thereof that includes quadrature amplitudes, are zero:
$$\bra{\vac}\hat{a}_\omega\ket{\vac}\equiv\mean{\hat{a}_\omega}=\mean{\hat{a}_\omega^\dag}=0\ \Rightarrow\ \mean{\hat{a}_c(\Omega)}=\mean{\hat{a}_s(\Omega)}=0\,.$$
Apparently, this also holds for time domain operators:
$$\bra{\vac}\hat{a}(t)\ket{\vac}\equiv\mean{\hat{a}(t)}=\mean{\hat{a}^\dag(t)}=0\ \Rightarrow\ \mean{\hat{a}_c(t)}=\mean{\hat{a}_s(t)}=0\,.$$
That the ground state of the oscillator is Gaussian is evident from its q-representation~\cite{97BookScZu}, namely
$$\psi_{\vac}(X_\omega) \equiv \braket{X_\omega}{0}_\omega = \dfrac{1}{\sqrt[4]{\pi}}\exp\left\{-\dfrac{X_\omega^2}{2}\right\}.$$
It means that knowing the second moments of quadrature amplitudes suffices for full characterization of the state $\ket{\vac}$. For this purpose, let us introduce a quadrature amplitudes matrix of spectral densities\epubtkFootnote{Herein, we make use of a double-sided power spectral density defined on a whole range of frequencies, both negative and positive, that yields the following connection to the variance of an arbitrary observable $\hat{o}(t)$:
$$\mathrm{Var}\left[\hat{o}(t)\right]\equiv\mean{\hat{o}^2(t)}-\mean{\hat{o}(t)}^2 = \int_{-\infty}^{\infty}\!\dfrac{d\Omega}{2\pi}S_o(\Omega)\,,$$
It is worth noting that in the GW community, the sensitivity of GW detectors as well as the individual noise sousces are usually characterized by a single-sided power spectral density $S^+_o(\Omega)$, that is simply defined on positive frequencies $\Omega\geqslant0$. The connection between these two is straightforward: $S^+_o(\Omega)=2S_o(\Omega)$ for $\Omega\geqslant0$ and $0$ otherwise.
} $\mathbb{S}(\Omega)$ according to the rule:
\begin{equation}
\label{eq:SpDens_matrix_def}
\smatrix{\mean{\hat{a}_c(\Omega)\circ\hat{a}_c(\Omega')}}
{\mean{\hat{a}_c(\Omega)\circ\hat{a}_s(\Omega')}}
{\mean{\hat{a}_s(\Omega)\circ\hat{a}_c(\Omega')}}
{\mean{\hat{a}_s(\Omega)\circ\hat{a}_s(\Omega')}}
= 2\pi\delta(\Omega+\Omega')\smatrix{S_{cc}(\Omega)}{S_{cs}(\Omega)}{S_{sc}(\Omega)}{S_{ss}(\Omega)}
= 2\pi\mathbb{S}\delta(\Omega+\Omega') \,.
\end{equation}
where $S_{ij}(\Omega)$ ($i,j=c,s$) denote (cross) power spectral densities of the corresponding quadrature amplitudes ${\mean{\hat{a}_i(\Omega)\circ\hat{a}_j(\Omega')}}$ standing for the symmetrized product of the corresponding quadrature operators, i.e.:
$$\mean{\hat{a}_i(\Omega)\circ\hat{a}_j(\Omega')}\equiv\frac12\mean{\hat{a}_i(\Omega)\hat{a}_j(\Omega')+\hat{a}_j(\Omega')\hat{a}_i(\Omega)} \equiv 2\pi S_{ij}(\Omega)\delta(\Omega-\Omega')\,.$$
For a vacuum state, this matrix of spectral densities can easily be obtained from the commutation relations~\eqref{eq:2photon_quads_commutator_spectral} and equals to:
\begin{equation}
\label{eq:SpDens_matrix_vac}
\mathbb{S}_{\vac}(\Omega) = \smatrix{1/2}{0}{0}{1/2}\,,
\end{equation}
which implies that the (double-sided) power spectral densities of the quadrature amplitudes as well as their cross-spectral density are equal to:
$$S_{cc}(\Omega) = S_{ss}(\Omega) = \frac{1}{2}\qquad\mbox{and}\qquad S_{cs}(\Omega) = 0\,.$$
In time domain, the corresponding matrix of second moments, known as a covariance matrix with elements defined as $\mathbb{V}_{ij}\delta(t-t') = \mean{\hat{a}_i(t)\circ\hat{a}_j(t')}$, is absolutely the same as $\mathbb{S}_{\vac}(\Omega)$ :
\begin{equation}
\label{eq:Cov_matrix_vac}
\mathbb{V}_{\vac} = \smatrix{1/2}{0}{0}{1/2}\,.
\end{equation}
It is instructive to discuss the meaning of these matrices, $\mathbb{S}$ and $\mathbb{V}$, and of the values they comprise. To do so, let us think of the light wave as a sequence of very short square-wave light pulses with infinitesimally small duration $\varepsilon\to0$. The delta function of time in Eq.~\eqref{eq:Cov_matrix_vac} tells us that the noise levels at different times, i.e., the amplitudes of the different square waves, are statistically independent. To put it another way, this noise is Markovian. It is also evident from Eq.~\eqref{eq:SpDens_matrix_vac} that quadrature amplitudes' fluctuations are stationary, and it is this stationarity, as noted in~\cite{85a1CaSch} that makes quadrature amplitudes such a convenient language for describing the quantum noise of light in parametric systems exemplified by GW interferometers.
It is instructive to pay some attention to a pictorial representation of the quantum noise described by the covariance and spectral density matrices $\mathbb{V}$ and $\mathbb{S}$. With this end in view let us introduce quadrature operators for each short light pulse as follows:
\begin{equation}
\label{eq:X_and_Y_quadrature_def}
\hat{X}_\varepsilon(t)\equiv\frac{1}{\sqrt{\varepsilon}}\int_{t-\varepsilon/2}^{t+\varepsilon/2}d\tau\,\hat{a}_c(\tau)\,,
\qquad\mbox{and}\qquad
\hat{Y}_\varepsilon(t)\equiv\frac{1}{\sqrt{\varepsilon}}\int_{t-\varepsilon/2}^{t+\varepsilon/2}d\tau\,\hat{a}_s(\tau)\,.
\end{equation}
These operators $\hat{X}_\varepsilon(t)$ and $\hat{Y}_\varepsilon(t)$ are nothing else than dimensionless displacement and momentum of the corresponding mode (called quadratures in quantum optics), normalized by zero point fluctuation amplitudes $X_0$ and $P_0$: $\hat{X}_\varepsilon(t)\equiv\hat{x}_\varepsilon/X_0$ and $\hat{X}_\varepsilon(t)\equiv\hat{p}_\varepsilon/P_0$.
This fact is also justified by the value of their commutator:
$$\left[\hat{X}_\varepsilon(t),\,\hat{Y}_\varepsilon(t)\right]=i\,.$$
There is no difficulty in showing that diagonal elements of the covariance matrix $\mathbb{V}_{ii}$ are equal to the variances of the corresponding mode displacement $\hat{X}_\varepsilon$ and momentum $\hat{Y}_\varepsilon$:
$$\mathbb{V}_{cc} = \mean{\hat{X}^2_\varepsilon(t)} = 1/2\,,\quad \mbox{and} \quad \mathbb{V}_{ss} = \mean{\hat{Y}^2_\varepsilon(t)} = 1/2\,, $$
while non-diagonal terms represent correlations between these operators (zero in our case):
$\mathbb{V}_{cs} = \mathbb{V}_{sc} = \mean{\hat{X}_\varepsilon(t)\circ\hat{Y}_\varepsilon(t)} = 0$.
At the same time, we see that there is no correlation between the pulses, justifying the Markovianity of the quantum noise of light in vacuum state: $$\mean{\hat{X}_\varepsilon(t)\hat{X}_\varepsilon(t')} = \mean{\hat{Y}_\varepsilon(t)\hat{Y}_\varepsilon(t')} = \mean{\hat{X}_\varepsilon(t)\circ\hat{Y}_\varepsilon(t')}=0\,,\ t\neq t'\,.$$
An attempt to measure the light field amplitude as a function of time will
give the result depicted in Figure~\ref{fig:vac_meas_result}.
\epubtkImage{fig13.png}{%
\begin{figure}[htbp]
\centerline{\includegraphics[width=\textwidth,]{fig13}}
\caption{Light field in a vacuum quantum state $\ket{\vac}$. \emph{Left panel (a)} features a typical result one could get measuring the (normalized) electric field strength of the light wave in a vacuum state as a function of time. \emph{Right panel (b)} represents a phase space picture of the results of measurement. A red dashed circle displays the error ellipse for the state $\ket{\vac}$ that encircles the area of single standard deviation for a two-dimensional random vector $\hat{\boldsymbol{a}}$ of measured light quadrature amplitudes. The principal radii of the error ellipse (equal in vacuum state case) are equal to square roots of the covariance matrix $\mathbb{V}_{\vac}$ eigenvalues, i.e., to $1/\sqrt{2}$.}
\label{fig:vac_meas_result}
\end{figure}}
\epubtkImage{fig14.png}{%
\begin{figure}[htbp]
\centerline{\includegraphics[width=.5\textwidth]{fig14a}\hfill\includegraphics[width=.33\textwidth]{fig14b}}
\caption{Wigner function $W_{\ket{\vac}}(X_\varepsilon,\,Y_\varepsilon)$ of a ground state of harmonic oscillator (\emph{left panel}) and its representation in terms of the noise ellipse (\emph{right panel}).}
\label{fig:vac_Wigner_fcn}
\end{figure}}
The measurement outcome at each instance of time will be a random variable with zero mean and variance defined by a covariance matrix $\mathbb{V}_{\vac}$ of Eq.~\eqref{eq:Cov_matrix_vac}:
$$\mathrm{Var}[\hat{E}(t)] = \{\cos\omega_0t,\,\sin\omega_0t\}\mathbb{V}_{\vac}\{\cos\omega_0t,\,\sin\omega_0t\}^{\mathsf{T}} = \frac12\,.$$
In quantum mechanics, it is convenient to describe a quantum state in terms of a \emph{Wigner function}, a quantum version of joint (quasi) probability distribution for particle displacement and momentum ($X_\varepsilon$ and $Y_\varepsilon$ in our case):
\begin{eqnarray}
\label{eq:WignerFcn_vac}
W_{\ket{\vac}}(X_\varepsilon,\,Y_\varepsilon) &=& \int_{-\infty}^{\infty}\!\frac{d\xi}{2\pi}\exp\left\{-i\xi Y_\varepsilon\right\}\braket{X_\varepsilon+\xi/2}{\vac}\braket{\vac}{X_\varepsilon-\xi/2}\nonumber\\
&=& \dfrac{1}{2\pi\sqrt{\det\mathbb{V}_{\vac}}}\exp\left\{-\frac12\{X_\varepsilon,\,Y_\varepsilon\}^{\mathsf{T}}\mathbb{V}_{\vac}^{-1}\{X_\varepsilon,\,Y_\varepsilon\}\right\} = \frac{1}{\pi}\exp\left\{-(X_\varepsilon^2+Y_\varepsilon^2)\right\},
\end{eqnarray}
where $\xi$ is simply the variable of integration.
The above Wigner function describes a Gaussian state, which is simply the ground state of a harmonic oscillator represented by a mode with displacement $\hat{X}_\varepsilon$ and momentum $\hat{Y}_\varepsilon$. The corresponding plot is given in the left panel of Figure~\ref{fig:vac_Wigner_fcn}. Gaussian states are traditionally pictured by error ellipses on a phase plane, as drawn in the right panel of Figure~\ref{fig:vac_Wigner_fcn} (cf.\ right panel of Figure~\ref{fig:vac_meas_result}). Here as well as in Figure~\ref{fig:vac_meas_result}, a red line in both plots circumscribes all the values of $X_\varepsilon$ and $Y_\varepsilon$ that fall inside the standard deviation region of the Wigner function, i.e., the region where all pertinent points are within 1 standard deviation from the center of the distribution. For a vacuum state, such a region is a circle with radius $\sqrt{\mathbb{V}_{cc}}=\sqrt{\mathbb{V}_{ss}}=1/\sqrt{2}$. The area of this circle, equal to $1/2$ in dimensionless units and to $\hbar/2$ in case of dimensional displacement and momentum, is the smallest area a physical quantum state can occupy in a phase space. This fact yields from a very general physical principle, the Heisenberg uncertainty relation, that limits the minimal uncertainty product for canonically conjugate observables (displacement $X_\varepsilon$ and momentum $Y_\varepsilon$, in our case) to be less than $1/2$ in $\hbar$-units:
$$\left(\mathrm{Var}[\hat{X}_\varepsilon]\right)^{1/2}\left(\mathrm{Var}[\hat{Y}_\varepsilon]\right)^{1/2}\geqslant\frac12\,.$$
The fact that for a ground state this area is exactly equal to $1/2$ is due to the fact that it is a pure quantum state, i.e., the state of the particle that can be described by a wave function $\ket{\psi}$, rather than by a density operator $\hat{\rho}$. For more sophisticated Gaussian states with a non-diagonal covariance matrix $\mathbb{V}$, the Heisenberg uncertainty relation reads:
\begin{equation}
\label{eq:CovMat_HUP}
\det\mathbb{V}\geqslant\frac14\,,
\end{equation}
and noise ellipse major semi-axes are given by the square root of the matrix $\mathbb{V}$ eigenvalues.
Note the difference between Figures~\ref{fig:vac_meas_result} and \ref{fig:vac_Wigner_fcn}; the former features the result of measurement of an ensemble of oscillators (subsequent light pulses with infinitesimally short duration $\varepsilon$), while the latter
gives the probability density function for a single oscillator displacement and momentum.
\subsubsection{Coherent state}
\label{sec:coherent_state}
Another important state of light is a \emph{coherent state} (see, e.g.,~\cite{1995BookWaMi, 97BookScZu, 95BookMaWo, 01BookSchleich}). It is straightforward to introduce a coherent state $\ket{\alpha}$ of a single mode or a harmonic oscillator as a result of its ground state $\ket{0}$ shift on a complex plane by the distance and in the direction governed by a complex number $\alpha=|\alpha|e^{i\arg(\alpha)}$. This can be caused, e.g., by the action of a classical effective force on the oscillator. Such a shift can be described by a unitary operator called a displacement operator, since its action on a ground state $\ket{0}$ inflicts its shift in a phase plane yielding a state that is called a coherent state:
$$\ket{\alpha} = \hat{D}[\alpha]\ket{0} \equiv e^{\alpha\hat{a}^\dag-\alpha^*\hat{a}}\ket{0}\,,$$
or, more vividly, in q-representation of a corresponding mode of the field~\cite{01BookSchleich}
$$\psi_{coh}(X_\omega) \equiv \braket{X_\omega}{\alpha} = \frac{1}{\sqrt[4]{\pi}}\exp\left\{-\frac{(X_\omega-\sqrt{2}\alpha)^2}{2}\right\}\,.$$
The shift described by $\hat{D}[\alpha]$ is even more apparent if one writes down its action on an annihilation (creation) operator:
$$\hat{D}^\dag[\alpha]\hat{a}\hat{D}[\alpha] = \hat{a}+\alpha\,, \quad\left(\hat{D}^\dag[\alpha]\hat{a}^\dag\hat{D}[\alpha]= \hat{a}^\dag+\alpha^*\right)\,.$$
Moreover, a coherent state is an eigenstate of the annihilation operator:
\begin{equation*}
\hat{a}\ket{\alpha} = \alpha\ket{\alpha}\,.
\end{equation*}
Using the definitions of the mode quadrature operators $\hat{X}_\omega\equiv\hat{X}$ and $\hat{Y}_\omega\equiv\hat{Y}$ (dimensionless oscillator displacement and momentum normalized by zero-point oscillations amplitude) given above, one immediately obtains for their mean values in a coherent state:
\begin{equation*}
\bra{\alpha}\hat{X}\ket{\alpha} = \bra{0}\hat{D}^\dag[\alpha]\hat{X}\hat{D}[\alpha]\ket{0} = \sqrt{2}\mathrm{Re}[\alpha]\,,\qquad
\bra{\alpha}\hat{Y}\ket{\alpha} = \bra{0}\hat{D}^\dag[\alpha]\hat{Y}\hat{D}[\alpha]\ket{0} = \sqrt{2}\mathrm{Im}[\alpha]\,.
\end{equation*}
Further calculation shows that quadratures variances:
\begin{equation*}
\mathrm{Var}[\hat{X}] = \bra{\alpha}\hat{X}^2\ket{\alpha}-\left(\bra{\alpha}\hat{X}\ket{\alpha}\right)^2 =\frac12\,,\qquad \mathrm{Var}[\hat{Y}] = \bra{\alpha}\hat{Y}^2\ket{\alpha}-\left(\bra{\alpha}\hat{Y}\ket{\alpha}\right)^2 =\frac12
\end{equation*}
have the same values as those for a ground state. These two facts unequivocally testify in favour of the statement that a coherent state is just the ground state shifted from the origin of the phase plane to the point with coordinates $(\mean{\hat{X}}_\alpha,\,\mean{\hat{Y}}_\alpha) = \sqrt{2}(\mathrm{Re}[\alpha]\,,\mathrm{Re}[\alpha])$. It is instructive to calculate a Wigner function for the coherent state using a definition of Eq.~\eqref{eq:WignerFcn_vac}:
\begin{eqnarray*}
W_{\ket{\alpha}}(X,\,Y) = \int_{-\infty}^{\infty}\!\frac{d\xi}{2\pi}\exp\left\{-i\xi Y\right\}\braket{X+\xi/2}{\alpha}\braket{\alpha}{X-\xi/2}=\\
\dfrac{1}{2\pi\sqrt{\det\mathbb{V}_{\vac}}}\exp\left\{-\frac12\{X-\sqrt{2}\mathrm{Re}[\alpha],\,Y-\sqrt{2}\mathrm{Im}[\alpha]\}^{\mathsf{T}}\mathbb{V}_{\vac}^{-1}\{X-\sqrt{2}\mathrm{Re}[\alpha],\,Y-\sqrt{2}\mathrm{Im}[\alpha]\}\right\} =\\ \frac{1}{\pi}\exp\left\{-\left[(X-\sqrt{2}\mathrm{Re}[\alpha])^2+(Y-\sqrt{2}\mathrm{Im}[\alpha])^2\right]\right\}\,,
\end{eqnarray*}
which once again demonstrates the correctness of the former statement.
Generalization to the case of continuum of modes comprising a light
wave is straightforward~\cite{PhysRevA.42.4102_1990} and goes along
the same lines as the definition of the field vacuum state, namely
(see Eq.~\eqref{eq:field_vac_state}):
\begin{equation}
\ket{\alpha(\omega)} \equiv \bigotimes\limits_\omega\ket{\alpha}_\omega = \bigotimes\limits_\omega\hat{D}[\alpha(\omega)]\ket{0}_\omega =
\exp\left\{\int_{-\infty}^{\infty}\!\frac{d\omega}{2\pi}(\alpha(\omega)\hat{a}_\omega^\dag-\alpha^*(\omega)\hat{a}_\omega)\right\}\ket{\vac}\,,
\end{equation}
where $\ket{\alpha}_\omega$ is the coherent state that the mode of the
field with frequency $\omega$ is in, and $\alpha(\omega)$ is the
distribution of complex amplitudes $\alpha$ over frequencies
$\omega$. Basically, $\alpha(\omega)$ is the spectrum of normalized
complex amplitudes of the field, i.e., $\alpha(\omega)\propto
\mathcal{E}(\omega)$. For example, the state of a free light wave
emitted by a perfectly monochromatic laser with emission frequency
$\omega_p$ and mean optical power $\mathcal{I}_0$ will be defined by
$\alpha(\omega) =
\pi\sqrt{\frac{2\mathcal{I}_0}{\hbar\omega_p}}\delta(\omega-\omega_p)$,
which implies that only the mode at frequency $\omega_p$ will be in a
coherent state, while all other modes of the field will be in their
ground states.
\epubtkImage{fig15.png}{%
\begin{figure}[htbp]
\centerline{\includegraphics[width=\textwidth]{fig15}}
\caption{Light field in a coherent quantum state $\ket{\alpha(\omega)}$. \emph{Left panel a)} features a typical result one could get measuring the (normalized) electric field strength of the light wave in a coherent state as a function of time. \emph{Right panel b)} represents a phase space picture of the results of measurement. The red dashed line in the left panel marks the mean value $\mean{\hat{E}(t)}$. The red arrow in the right panel features the vector of the mean values of quadrature amplitudes, i.e., $\boldsymbol{A}$, while the red dashed circle displays the error ellipse for the state $\ket{\alpha(\omega)}$ that encircles the area of single standard deviation for a two-dimensional random vector $\hat{\boldsymbol{a}}$ of quadrature amplitudes. The principle radii of the error ellipse (equal in the coherent state case) are equal to square roots of the covariance matrix $\mathbb{V}_{\mathrm{coh}}$, i.e., to $1/\sqrt{2}$.}
\label{fig:coh_meas_result}
\end{figure}}
Operator $\hat{D}[\alpha]$ is unitary, i.e., $\hat{D}^\dag[\alpha]\hat{D}[\alpha]=\hat{D}[\alpha]\hat{D}^\dag[\alpha]=\hat{I}$ with $\hat{I}$ the identity operator, while the physical meaning is in the translation and rotation of the Hilbert space that keeps all the physical processes unchanged. Therefore, one can simply use vacuum states instead of coherent states and subtract the mean values from the corresponding operators in the same way we have done previously for the light wave classical amplitudes, just below Eq.~\eqref{eq:2photon_EMW_quadratures}.
The covariance matrix and the matrix of power spectral densities for the quantum noise of light in a coherent state is thus the same as that of a vacuum state case.
The typical result one can get measuring the electric field strength of light emitted by the aforementioned ideal laser is drawn in the left panel of Figure~\ref{fig:coh_meas_result}.
\subsubsection{Squeezed state}
\label{sec:squeezed_state}
One more quantum state of light that is worth consideration is a squeezed state. To put it in simple words, it is a state where one of the oscillator quadratures variance appears decreased by some factor compared to that in a vacuum or coherent state, while the conjugate quadrature variance finds itself swollen by the same factor, so that their product still remains Heisenberg-limited. Squeezed states of light are usually obtained as a result of a parametric down conversion (PDC) process~\cite{67a1Kl, 69a1Kl} in optically nonlinear crystals. This is the most robust and experimentally elaborated way of generating squeezed states of light for various applications, e.g., for GW detectors~\cite{PhysRevLett.95.211102_2005_Vahlbruch, PhysRevLett.100.033602_2008, Takeno:07}, or for quantum communications and computation purposes~\cite{RevModPhys.77.513_2005}. However, there is another way to generate squeezed light by means of a ponderomotive nonlinearity inherent in such optomechanical devices as GW detectors. This method, first proposed by Corbitt et~al.~\cite{PhysRevA.73.023801_2006}, utilizes the parametric coupling between the resonance frequencies of the optical modes in the Fabry--P\'{e}rot cavity and the mechanical motion of its mirrors arising from the quantum radiation pressure fluctuations inflicting random mechanical motion on the cavity mirrors. Further, we will see that the light leaving the signal port of a GW interferometer finds itself in a ponderomotively squeezed state (see, e.g.,~\cite{02a1KiLeMaThVy} for details). A dedicated reader might find it illuminating to read the following review articles on this topic~\cite{Schnabel2010, LPOR:LPOR201000034}.
Worth noting is the fact that generation of squeezed states of light is the process that inherently invokes two modes of the field and thus naturally calls for usage of the two-photon formalism contrived by Caves and Schumaker~\cite{85a1CaSch, 85a2CaSch}. To demonstrate this let us consider the physics of a squeezed state generation in a nonlinear crystal. Here photons of a pump light with frequency $\omega_p=2\omega_0$ give birth to pairs of correlated photons with frequencies $\omega_1$ and $\omega_2$ (traditionally called \emph{signal} and \emph{idler}) by means of the nonlinear dependence of polarization in a birefringent crystal on electric field. Such a process can be described by the following Hamiltonian, provided that the pump field is in a coherent state $\ket{\alpha}_{\omega_p}$ with strong classical amplitude $|\alpha_p|\gg1$ (see, e.g., Section~5.2 of~\cite{1995BookWaMi} for details):
\begin{equation}
\label{eq:PDC_hamiltonian}
\frac{\hat{H}_{\mathrm{PDC}}}{\hbar} = \omega_1\hat{a}^\dag_1\hat{a}_1 + \omega_2\hat{a}^\dag_2\hat{a}_2 + i\left(\chi\hat{a}^\dag_1\hat{a}^\dag_2e^{-2i\omega_0t}-\chi^*\hat{a}_1\hat{a}_2e^{2i\omega_0t}\right)\,,
\end{equation}
where $\hat{a}_{1,2}$ describe annihilation operators for the photons of the signal and idler modes and $\chi=\rho e^{2i\phi}$ is the complex coupling constant that is proportional to the second-order susceptibility of the crystal and to the pump complex amplitude. Worth noting is the meaning of $t$ in this Hamiltonian: it is a parameter that describes the duration of a pump light interaction with the nonlinear crystal, which, in the simplest situation, is either the length of the crystal divided by the speed of light $c$, or, if the crystal is placed between the mirrors of the optical cavity, the same as the above but multiplied by an average number of bounces of the photon inside this cavity, which is, in turn, proportional to the cavity finesse $\mathcal{F}$. It is straightforward to obtain the evolution of the two modes in the interaction picture (leaving apart the obvious free evolution time dependence $e^{-i\omega_{s,i}t}$) solving the Heisenberg equations:
\begin{equation}
\label{eq:PDC_HEqs_solution}
\hat{a}_1(t) = \hat{a}_1\cosh \rho t+\hat{a}^\dag_2e^{2i\phi}\sinh \rho t\,,\qquad
\hat{a}_2(t) = \hat{a}_2\cosh \rho t+\hat{a}^\dag_1e^{2i\phi}\sinh \rho t\,.
\end{equation}
Let us then assume the signal and idler mode frequencies symmetric with respect to the half of pump frequency $\omega_0=\omega_p/2$: $\omega_1\to\omega_+=\omega_0+\Omega$ and $\omega_2\to\omega_-=\omega_0-\Omega$ ($\hat{a}_1\to\hat{a}_+$ and $\hat{a}_2\to\hat{a}_-$). Then the electric field of a two-mode state going out of the nonlinear crystal will be written as (we did not include the pump field here assuming it can be ruled out by an appropriate filter):
\begin{equation*}
\hat{E}(t) = \mathcal{C}_0\left[\hat{X}(t)\cos\omega_0t+\hat{Y}(t)\sin\omega_0t\right],
\end{equation*}
where two-mode quadrature amplitudes $\hat{X}(t)$ and $\hat{Y}(t)$ are defined along the lines of Eqs.~\eqref{eq:2photon_quads_def}, keeping in mind that only idler and signal components at the frequencies $\omega_\pm-\omega_0 = \pm\Omega$ should be kept in the integral, which yields:
\begin{eqnarray*}
\hat{X}(t) &=& \frac{1}{\sqrt{2}}\left[\hat{a}_+(t)e^{i\Omega t}+\hat{a}^\dag_+(t)e^{-i\Omega t}+\hat{a}_-(t)e^{-i\Omega t}+\hat{a}^\dag_-(t)e^{i\Omega t}\right] = \hat{a}_c^{\sqz} e^{-i\Omega t}+\hat{a}_c^{\sqz\dag} e^{i\Omega t}\,,\\
\hat{Y}(t) &=& \frac{1}{i\sqrt{2}}\left[\hat{a}_+(t)e^{i\Omega t}-\hat{a}^\dag_+(t)e^{-i\Omega t}+\hat{a}_-(t)e^{-i\Omega t}-\hat{a}^\dag_-(t)e^{i\Omega t}\right] = \hat{a}_s^{\sqz} e^{-i\Omega t}+\hat{a}_s^{\sqz\dag} e^{i\Omega t}\,,
\end{eqnarray*}
where $\hat{a}_c^{\sqz} = (\hat{a}_+(t)+\hat{a}^\dag_-(t))/\sqrt{2}$ and $\hat{a}_s^{\sqz} = (\hat{a}_+(t)-\hat{a}^\dag_-(t))/(i\sqrt{2})$ are the spectral quadrature amplitudes of the two-mode field at sideband frequency $\Omega$ (cf.\ Eqs.~\eqref{eq:2photon_quadratures}) after it leaves the nonlinear crystal.
Substituting Eqs.~\eqref{eq:PDC_HEqs_solution} into the above expressions yields transformation rules for quadrature amplitudes:
\begin{eqnarray}
\label{eq:SQZ_quad_transform}
\hat{a}_c^{\sqz} &=& \hat{a}_c\left(\cosh\rho t+\cos2\phi\sinh\rho t\right)+\hat{a}_s\sin2\phi\sinh\rho t\,,\nonumber\\
\hat{a}_s^{\sqz} &=& \hat{a}_c\sin2\phi\sinh\rho t+\hat{a}_s\left(\cosh\rho t-\cos2\phi\sinh\rho t\right)\,,
\end{eqnarray}
where $\hat{a}_c=(\hat{a}_++\hat{a}^\dag_-)/\sqrt{2}$ and $\hat{a}_s=(\hat{a}_+-\hat{a}^\dag_-)/(i\sqrt{2})$ stand for initial values of spectral quadrature amplitudes of the two-mode light wave created in the PDC process. A close look at these transformations written in the matrix form reveals that it can be represented as the following sequence:
\begin{equation}
\label{eq:SQZ_matrix_transform}
\hat{\boldsymbol{a}}^{\sqz} = \begin{pmatrix}
\hat{a}_c^{\sqz}\\
\hat{a}_s^{\sqz}
\end{pmatrix} = \mathbb{S}_{\sqz}[\rho t,\phi]\hat{\boldsymbol{a}} = \mathbb{P}[\phi]\mathbb{S}_{\sqz}[\rho t,0]\mathbb{P}[-\phi]\hat{\boldsymbol{a}}
\end{equation}
where
\begin{equation}
\label{eq:SQZ_matrix_transform_def}
\mathbb{S}_{\sqz}[\rho t,\phi]\equiv
\begin{bmatrix}
\cosh\rho t+\cos2\phi\sinh\rho t & \sin2\phi\sinh\rho t\\
\sin2\phi\sinh\rho t & \cosh\rho t-\cos2\phi\sinh\rho t
\end{bmatrix}\ \Longrightarrow\ \mathbb{S}_{\sqz}[\rho t,0] =
\begin{bmatrix}
e^{\rho t} & 0\\
0 & e^{-\rho t}
\end{bmatrix}
\end{equation}
are squeezing matrices in general and in special ($\phi=0$) case, while $\mathbb{P}[\phi]$ stands for a counterclockwise 2D-rotation matrix by angle $\phi$ defined by~\eqref{eq:CCW_rotation_matrix}.
Therefore, the evolution of a two-mode light quadrature amplitude vector $\hat{\boldsymbol{a}}$ in a PDC process described by the Hamiltonian~\eqref{eq:PDC_hamiltonian} consists of a clockwise rotation by an angle $\phi$ followed by a deformation along the main axes (stretching along the $a_c$-axis and proportional squeezing along the $a_s$-axis) and rotation back by the same angle. It is straightforward to show that vector $\hat{\boldsymbol{X}}^{\sqz} = \left\{\hat{X}(t),\,\hat{Y}(t)\right\}^{\mathsf{T}} = \hat{\boldsymbol{a}}^{\sqz}e^{-i\Omega t}+\hat{\boldsymbol{a}}^{\sqz*}e^{i\Omega t}$ transforms similarly (here $\hat{\boldsymbol{a}}^{\sqz*}=\left\{\hat{a}_c^{\sqz\dag},\,\hat{a}_s^{\sqz\dag}\right\}^{\mathsf{T}}$ ).
This geometric representation is rather useful, particularly for the characterization of a squeezed state. If the initial state of the two-mode field is a vacuum state then the outgoing field will be in a squeezed vacuum state. One can define it as a result of action of a special squeezing operator $\hat{S}[\rho t,\phi]$ on the vacuum state
\begin{equation}
\label{eq:SQZ_operator_action}
\ket{\sqz_0(\rho t,\phi)} = \hat{S}[\rho t,\phi]\ket{\vac}\,.
\end{equation}
This operator is no more and no less than the evolution operator for the PDC process in the interaction picture, i.e.,
\begin{equation}
\label{eq:SQZ_operator_def}
\hat{S}[\rho t,\phi] \equiv \exp\left\{\rho t(\hat{a}_+\hat{a}_-e^{-2i\phi}-\hat{a}_+^\dag\hat{a}_-^\dag e^{2i\phi})\right\}\,.
\end{equation}
Action of this operator on the two-photon quadrature amplitudes is fully described by Eqs.~\eqref{eq:SQZ_operator_action}:
\begin{equation*}
\hat{\boldsymbol{a}}^{\sqz}=\hat{S}^\dag[\rho t,\phi]\hat{\boldsymbol{a}}\hat{S}[\rho t,\phi] = \mathbb{P}[\phi]\mathbb{S}_{\sqz}[\rho t,0]\mathbb{P}[-\phi]\hat{\boldsymbol{a}}\,,
\end{equation*}
while annihilation operators of the corresponding modes $\hat{a}_\pm$
are transformed in accordance with Eqs.~\eqref{eq:PDC_HEqs_solution}.
\epubtkImage{fig16.png}{%
\begin{figure}[htb]
\centerline{\includegraphics[width=.6\textwidth]{fig16}}
\caption{Schematic plot of a vacuum state transformation under the action of the squeezing operator $\hat{S}[r,\phi]$. Eqs.~\eqref{eq:SQZ_matrix_transform} demonstrate the equivalence of the general squeezing operator $\hat{S}[r,\phi]$ to a sequence of phase plane counterclockwise rotation by an angle $\phi$ (transition from a) to b)), phase plane squeezing and stretching by a factor $e^r$ (transition from b) to c)) and rotation back by the same angle $\phi$ (transition from c) to d)). Point $P'$ tracks how transformations change the initial state marked with point $P$.}
\label{fig:sqz_op_action_on_vac}
\end{figure}}
The linearity of the squeezing transformations implies that the squeezed vacuum state is Gaussian since it is obtained from the Gaussian vacuum state and therefore can be fully characterized by the expectation values of operators $\hat{X}$ and $\hat{Y}$ and their covariance matrix $\mathbb{V}_{\sqz}$. Let us calculate these values:
\begin{equation*}
\mean{\hat{X}}_{\sqz} = \bra{\sqz_0(r,\phi)}\hat{X}\ket{\sqz_0(r,\phi)} = \bra{\vac}\hat{S}^\dag[r,\phi]\hat{X}\hat{S}[r,\phi]\ket{\vac} =\bra{\vac}\hat{X}(t)\ket{\vac} = 0\,,\quad \mean{\hat{Y}}_{\sqz} = 0\,,
\end{equation*}
and for a covariance matrix one can get the following expression:
\begin{eqnarray}
\label{eq:SQZ_V}
\mathbb{V}_{\sqz} = \bra{\sqz_0(r,\phi)}\hat{\boldsymbol{X}}\circ\hat{\boldsymbol{X}}^{\mathsf{T}}\ket{\sqz_0(r,\phi)} = \mathbb{P}[-\phi]\mathbb{S}_{\sqz}[r,0]\mathbb{V}_{\vac}\mathbb{S}_{\sqz}[r,0]\mathbb{P}[\phi] =\nonumber\\ \frac12\mathbb{P}[-\phi]\mathbb{S}_{\sqz}[2r,0]\mathbb{P}[\phi]=
\frac12\begin{bmatrix}
\cos\phi & \sin\phi\\
-\sin\phi & \cos\phi
\end{bmatrix}
\begin{bmatrix}
e^{2r} & 0\\
0 & e^{-2r}
\end{bmatrix}
\begin{bmatrix}
\cos\phi & -\sin\phi\\
\sin\phi & \cos\phi
\end{bmatrix}\,,
\end{eqnarray}
where we introduced squeezing parameter $r\equiv\rho t$ and used a short notation for the symmetrized outer product of vector $\hat{\boldsymbol{X}}$ with itself:
$$
\hat{\boldsymbol{X}}\circ\hat{\boldsymbol{X}}^{\mathsf{T}}\equiv
\begin{bmatrix}
\hat{X}\circ\hat{X} & \hat{X}\circ\hat{Y}\\
\hat{Y}\circ\hat{X} & \hat{Y}\circ\hat{Y}
\end{bmatrix}\,.
$$
The squeezing parameter $r$ is the quantity reflecting the strength of the squeezing. This way of characterizing the squeezing strength, though convenient enough for calculations, is not very ostensive. Conventionally, squeezing strength is measured in decibels (dB) that are related to the squeezing parameter $r$ through the following simple formula:
\begin{equation}
\label{eq:SQZ_factor_dB_def}
r_{\mathrm{dB}} = 10\log_{10}e^{2r}=20r\log_{10}e\ \Longleftrightarrow r = r_{\mathrm{dB}}/(20\log_{10}e)\,.
\end{equation}
For example, 10~dB squeezing corresponds to $r\simeq1.15$.
The covariance matrix~\eqref{eq:SQZ_V} refers to a unique error ellipse on a phase plane with semi-major axis $e^r/\sqrt{2}$ and semi-minor axis $e^{-r}/\sqrt{2}$ rotated by angle $\phi$ clockwise as featured in Figure~\ref{fig:sqz_op_action_on_vac}.
It would be a wise guess to make, that a squeezed vacuum Wigner function can be obtained from that of a vacuum state, using these simple geometric considerations. Indeed, for a squeezed vacuum state it reads:
\begin{equation}
\label{eq:SQZ_WignerFcn}
W_{\ket{\sqz}}(X,\,Y) = \dfrac{1}{2\pi\sqrt{\det\mathbb{V}_{\sqz}}}\exp\left\{-\frac12\{X,\,Y\}^{\mathsf{T}}\mathbb{V}_{\sqz}^{-1}\{X,\,Y\}\right\}\,,
\end{equation}
where the error ellipse refers to the level where the Wigner function
value falls to $1/\sqrt{e}$ of the maximum. The corresponding plot and
phase plane picture of the squeezed vacuum Wigner function are
featured in Figure~\ref{fig:sqz_Wigner_fcn}.
\epubtkImage{fig17.png}{%
\begin{figure}[htbp]
\centerline{\includegraphics[width=.47\textwidth]{fig17a}\hfill\includegraphics[width=.33\textwidth]{fig17b}}
\caption{\emph{Left panel}: Wigner function of a squeezed vacuum state with squeeze parameter $r=0.5$ (5~dB) and rotation angle $\phi=-\pi/4$. \emph{Right panel:} Error ellipse corresponding to that Wigner function.}
\label{fig:sqz_Wigner_fcn}
\end{figure}}
Another important state that arises in GW detectors is the displaced squeezed state $\ket{\sqz_\alpha(r,\phi)}$ that is obtained from the squeezed vacuum state in the same manner as the coherent state yields from the vacuum state, i.e., by the application of the displacement operator (equivalent to the action of a classical force):
\begin{equation}
\ket{\sqz_\alpha(r,\phi)} = \hat{D}[\alpha]\ket{\sqz_0(r,\phi)} = \hat{D}[\alpha]\hat{S}[r,\phi]\ket{\vac}\,.
\end{equation}
The light leaving a GW interferometer from the signal port finds itself in such a state, if a classical GW-like force changes the difference of the arm lengths, thus displacing a ponderomotively squeezed vacuum state in phase quadrature $Y$ by an amount proportional to the magnitude of the signal force. Such a displacement has no other consequence than simply to shift the mean values of $\hat{X}$ and $\hat{Y}$ by some constant values dependent on shift complex amplitude $\alpha$:
\begin{equation*}
\mean{\hat{X}}_{\sqz}=\sqrt{2}\mathrm{Re}[\alpha]\,, \qquad \mean{\hat{Y}}_{\sqz}=\sqrt{2}\mathrm{Im}[\alpha]\,.
\end{equation*}
\epubtkImage{fig18.png}{%
\begin{figure}[htb]
\centerline{\includegraphics[width=\textwidth]{fig18}}
\caption{Light field in a squeezed state $\ket{\sqz_\alpha(r,\phi)}$. \emph{Upper row} features time dependence of the electric field strength $E(t)$ in three different squeezed states (10~dB squeezing assumed for all): a) squeezed vacuum state with squeezing angle $\phi=\pi/4$; b) displaced squeezed state with classical amplitude $A_c=5$ (mean field strength oscillations $\mean{\hat{E}(t)}_{\sqz}$ are given by red dashed line) and amplitude squeezing ($\phi=\pi/2$); c) displaced squeezed state with classical amplitude $A_c=5$ and phase squeezing ($\phi=0$). \emph{Lower row} features error ellipses (red dashed lines) for the corresponding plots in the upper row.}
\label{fig:sqz_meas_result}
\end{figure}}
Let us now generalize the results of a two-mode consideration to a continuous spectrum case. Apparently, quadrature operators $\hat{X}(t)$ and $\hat{Y}(t)$ are similar to $\hat{a}_c(t)$ and $\hat{a}_s(t)$ for the traveling wave case. Utilizing this similarity, let us define a squeezing operator for the continuum of modes as:
\begin{equation}
\hat{S}[r(\Omega),\phi(\Omega)]\equiv\exp\left\{\int_{-\infty}^{\infty}\!\frac{d\Omega}{2\pi}r(\Omega)\left[\hat{a}_+\hat{a}_-e^{-2i\phi(\Omega)}-\hat{a}^\dag_+\hat{a}^\dag_-e^{2i\phi(\Omega)}\right]\right\}\,,
\end{equation}
where $r(\Omega)$ and $\phi(\Omega)$ are frequency-dependent squeezing factor and angle, respectively. Acting with this operator on a vacuum state of the travelling wave yields a squeezed vacuum state of a continuum of modes in the very same manner as in Eq.~\eqref{eq:SQZ_operator_action}. The result one could get in the measurement of the electric field amplitude of light in a squeezed state as a function of time is presented in Figure~\ref{fig:sqz_meas_result}. Quadrature amplitudes for each frequency $\Omega$ transform in accordance with Eqs.~\eqref{eq:SQZ_quad_transform}. Thus, we are free to use these formulas for calculation of the power spectral density matrix for a traveling wave squeezed vacuum state. Indeed, substituting $\hat{a}_{c,s}(\Omega)\to\hat{a}^{\sqz}_{c,s}(\Omega)$ in Eq.~\eqref{eq:SpDens_matrix_def} and using Eq.~\eqref{eq:SQZ_matrix_transform} one immediately gets:
\begin{equation}
\label{eq:SQZ_SpDens_matrix}
\mathbb{S}_{\sqz}(\Omega) = \mathbb{P}[-\phi(\Omega)]\mathbb{S}_{\sqz}[r(\Omega),0]\mathbb{S}_{\vac}(\Omega)\mathbb{S}_{\sqz}[r(\Omega),0]\mathbb{P}[\phi(\Omega)] = \mathbb{S}_{\sqz}(r(\Omega),\phi(\Omega))\,.
\end{equation}
Note that entries of $\mathbb{S}_{\sqz}(\Omega)$ might be frequency dependent if squeezing parameter $r(\Omega)$ and squeezing angle $\phi(\Omega)$ are frequency dependent as is the case in all physical situations. This indicates that quantum noise in a squeezed state of light is not Markovian and this can easily be shown by calculating the the covariance matrix, which is simply a Fourier transform of $\mathbb{S}(\Omega)$ according to the Wiener--Khinchin theorem:
\begin{equation}
\mathbb{V}_{\sqz}(t-t') = \int_{-\infty}^{\infty}\!\frac{d\Omega}{2\pi}\mathbb{S}_{\sqz}(\Omega)e^{-i\Omega(t-t')}\,.
\end{equation}
Of course, the exact shape of $\mathbb{V}_{\sqz}(t-t')$ could be obtained only if we specify $r(\Omega)$ and $\phi(\Omega)$. Note that the noise described by $\mathbb{V}_{\sqz}(t-t')$ is stationary since all the entries of the covariance matrix (correlation functions) depend on the difference of times $t-t'$.
The spectral density matrix allows for pictorial representation of a
multimode squeezed state where an error ellipse is assigned to each
sideband frequency $\Omega$. This effectively adds one more dimension
to a phase plane picture already used by us for the characterization
of a two-mode squeezed states. Figure~\ref{fig:sqz_cont_3D_diagram}
exemplifies the state of a ponderomotively squeezed light that would
leave the speedmeter type of the interferometer (see Section~\ref{sec:speedmeter}).
\epubtkImage{fig19.png}{%
\begin{figure}[htbp]
\centerline{\includegraphics[width=\textwidth]{fig19}}
\caption{Example of a squeezed state of the continuum of modes: output state of a speedmeter interferometer. \emph{Left panel} shows the plots of squeezing parameter $r_{\mathrm{dB}}(\Omega)$ and squeezing angle $\phi(\Omega)$ versus normalized sideband frequency $\Omega/\gamma$ (here $\gamma$ is the interferometer half-bandwidth). \emph{Right panel} features a family of error ellipses for different sideband frequencies $\Omega$ that illustrates the squeezed state defined by $r_{\mathrm{dB}}(\Omega)$ and $\phi(\Omega)$ drawn in the left panel.}
\label{fig:sqz_cont_3D_diagram}
\end{figure}}
\subsection{How to calculate spectral densities of quantum noise in linear optical measurement?}
In this section, we give a brief introduction to calculation of the power spectral densities of quantum noise one usually encounters in linear optical measurement. In optomechanical sensors, as we have discussed earlier, the outgoing light carries the information about the measured quantity (e.g., the displacement due to GW tidal forces) in its phase and (sometimes) amplitude quadratures. The general transformation from the input light characterized by a vector of quadrature amplitudes $\hat{\boldsymbol{a}}(\Omega) = \left\{\hat{a}_c(\Omega),\,\hat{a}_s(\Omega)\right\}^{\mathsf{T}}$ to the readout quantity of a meter is linear and can be written in spectral form as:
\begin{equation}
\label{eq:generic_Y}
\hat{Y}(\Omega) = \vb{\mathcal{Y}}^\dag(\Omega)\hat{\boldsymbol{a}}(\Omega) + G(\Omega) = \mathcal{Y}_c^*(\Omega)\hat{a}_c(\Omega) + \mathcal{Y}_s^*(\Omega)\hat{a}_s(\Omega) + G(\Omega),
\end{equation}
where $G(\Omega)$ is the spectrum of the measured quantity, $\mathcal{Y}_{c,s}(\Omega)$ are some complex-valued functions of $\Omega$ that characterize how the light is transformed by the device. Quantum noise is represented by the terms of the above expression not dependent on the measured quantity $G$, i.e.,
\begin{equation}
\label{eq:generic_N_Y}
\hat{N}_Y(\Omega) = \vb{\mathcal{Y}}^\dag(\Omega)\hat{\boldsymbol{a}}(\Omega) =
\begin{pmatrix}
\mathcal{Y}^*_{c}(\Omega) & \mathcal{Y}^*_{s}(\Omega)
\end{pmatrix}
\cdot
\begin{pmatrix}
\hat{a}_c(\Omega)\\
\hat{a}_s(\Omega)
\end{pmatrix}.
\end{equation}
The measure of quantum noise is the power spectral density $S_Y(\Omega)$ that is defined by the following expression:
\begin{equation}
\label{eq:generic_S_Y}
2\pi S_Y(\Omega)\delta(\Omega-\Omega') = \bra{\psi}\hat{N}_Y(\Omega)\circ\hat{N}^\dag_Y(\Omega')\ket{\psi} = \mean{\hat{N}_Y(\Omega)\circ\hat{N}_Y^\dag(\Omega')}\,.
\end{equation}
Here $\ket{\psi}$ is the quantum state of the light wave.
In our review, we will encounter two types of quantum states that we have described above, i.e., vacuum $\ket{\vac}$ and squeezed vacuum $\ket{\sqz_0(r,\phi)}$ states. Let us show how to calculate the power (double-sided) spectral density of a generic quantity $\hat{Y}(\Omega)$ in a vacuum state. To do so, one should substitute Eq.~\eqref{eq:generic_N_Y} into Eq.~\eqref{eq:generic_S_Y} and obtain that:
\begin{eqnarray}
\label{eq:S_Y_vac}
S^{\vac}_Y(\Omega) &=& \vb{\mathcal{Y}}^\dag(\Omega)\mean{\hat{\boldsymbol{a}}(\Omega)\circ\hat{\boldsymbol{a}}^\dag(\Omega)}_{\vac}\vb{\mathcal{Y}}(\Omega) = \vb{\mathcal{Y}}^\dag(\Omega)\mathbb{S}_{\vac}(\Omega)\vb{\mathcal{Y}}(\Omega) \nonumber\\
&=& \frac{\vb{\mathcal{Y}}^\dag(\Omega)\vb{\mathcal{Y}}(\Omega)}{2} = \frac{|\mathcal{Y}_c(\Omega)|^2+|\mathcal{Y}_s(\Omega)|^2}{2}\,,
\end{eqnarray}
where we used the definition of the power spectral density matrix of light in a vacuum state~\eqref{eq:SpDens_matrix_vac}\epubtkFootnote{Hereafter we will omit, for the sake of brevity, the factor $2\pi\delta(\Omega-\Omega')$ in equations that define the power (double-sided) spectral densities of relevant quantum observables, as well as assume $\Omega=\Omega'$, though keeping in mind that a mathematically rigorous definition should be written in the form of Eq.~\eqref{eq:generic_S_Y}.}.
Similarly, one can calculate the spectral density of quantum noise if the light is in a squeezed state $\ket{\sqz_0(r,\phi)}$, utilizing the definition of the squeezed state density matrix given in Eq.~\eqref{eq:SQZ_SpDens_matrix}:
\begin{eqnarray}
\label{eq:S_Y_sqz}
S^{\sqz}_Y(\Omega) &=&
\vb{\mathcal{Y}}^\dag(\Omega)\mean{\hat{\boldsymbol{a}}(\Omega)\circ\hat{\boldsymbol{a}}^\dag(\Omega)}_{\sqz}\vb{\mathcal{Y}}(\Omega) = \vb{\mathcal{Y}}^\dag(\Omega)\mathbb{S}_{\sqz}(\Omega)\vb{\mathcal{Y}}(\Omega) = \frac12\vb{\mathcal{Y}}^\dag(\Omega)\mathbb{P}[-\phi]\mathbb{S}_{\sqz}[2r,0]\mathbb{P}[\phi]\vb{\mathcal{Y}}(\Omega) \nonumber\\ &=&
\frac{|\mathcal{Y}_c(\Omega)|^2}{2}(\cosh2r+\sinh2r\cos2\phi)+\frac{|\mathcal{Y}_s(\Omega)|^2}{2}(\cosh2r-\sinh2r\cos2\phi)\nonumber\\ && -\mathrm{Re}\left[\mathcal{Y}_c(\Omega)\mathcal{Y}^*_s(\Omega)\right]\sinh2r\sin2\phi
\,.
\end{eqnarray}
It might also be necessary to calculate also cross-correlation spectral density $S_{YZ}(\Omega)$ of $\hat{Y}(\Omega)$ with some other quantity $\hat{Z}(\Omega)$ with quantum noise defined as:
\begin{equation*}
\hat{N}_Z(\Omega) = \vb{\mathcal{Z}}(\Omega)^\dag\hat{\boldsymbol{a}}(\Omega) = \mathcal{Z}_c^*(\Omega)\hat{a}_c(\Omega) + \mathcal{Z}_s^*(\Omega)\hat{a}_s(\Omega)\,.
\end{equation*}
Using the definition of cross-spectral density $S_{YZ}(\Omega)$ similar to~\eqref{eq:generic_S_Y}:
\begin{equation}
2\pi S_{YZ}(\Omega)\delta(\Omega-\Omega') = \bra{\psi}\hat{N}_Y(\Omega)\circ\hat{N}^\dag_Z(\Omega')\ket{\psi} = \mean{\hat{N}_Y(\Omega)\circ\hat{N}_Z^\dag(\Omega')}\,,
\end{equation}
one can get the following expressions for spectral densities in both cases of the vacuum state:
\begin{eqnarray}
\label{eq:S_YZ_vac}
S^{\vac}_{YZ}(\Omega) &=& \vb{\mathcal{Y}}^\dag(\Omega)\mean{\hat{\boldsymbol{a}}(\Omega)\circ\hat{\boldsymbol{a}}^\dag(\Omega)}_{\vac}\vb{\mathcal{Z}}(\Omega) = \vb{\mathcal{Y}}^\dag(\Omega)\mathbb{S}_{\vac}(\Omega)\vb{\mathcal{Z}}(\Omega) \nonumber\\
&=& \frac{\vb{\mathcal{Y}}^\dag(\Omega)\vb{\mathcal{Z}}(\Omega)}{2} = \frac{\mathcal{Y}^*_c(\Omega)\mathcal{Z}_c(\Omega)+\mathcal{Y}^*_s(\Omega)\mathcal{Z}_s(\Omega)}{2}\,,
\end{eqnarray}
and the squeezed state:
\begin{eqnarray}
\label{eq:S_YZ_sqz}
S^{\sqz}_{YZ}(\Omega) &=& \vb{\mathcal{Y}}^\dag(\Omega)\mean{\hat{\boldsymbol{a}}(\Omega)\circ\hat{\boldsymbol{a}}^\dag(\Omega)}_{\sqz}\vb{\mathcal{Z}}(\Omega) = \vb{\mathcal{Y}}^\dag(\Omega)\mathbb{S}_{\sqz}(\Omega)\vb{\mathcal{Z}}(\Omega)\nonumber\\ &=& \frac12\vb{\mathcal{Y}}^\dag(\Omega)\mathbb{P}[-\phi]\mathbb{S}_{\sqz}[2r,0]\mathbb{P}[\phi]\vb{\mathcal{Z}}(\Omega) \nonumber\\ &=& \frac{\mathcal{Y}^*_c(\Omega)\mathcal{Z}_c(\Omega)}{2}(\cosh2r+\sinh2r\cos2\phi)+\frac{\mathcal{Y}^*_s(\Omega)\mathcal{Z}_s(\Omega)}{2}(\cosh2r-\sinh2r\cos2\phi) \nonumber\\&&- \frac{\mathcal{Y}_s^*(\Omega)\mathcal{Z}_c(\Omega)+\mathcal{Y}^*_c(\Omega)\mathcal{Z}_s(\Omega)}{2}\sinh2r\sin2\phi
\,.
\end{eqnarray}
Note that since the observables $\hat{Y}(t)$ and $\hat{Z}(t)$ that one calculates spectral densities for are Hermitian, it is compulsory, as is well known, for any operator to represent a physical quantity, then the following relation holds for their spectral coefficients $\mathcal{Y}_{c,s}(\Omega)$ and $\mathcal{Z}_{c,s}(\Omega)$:
\begin{equation}
\label{eq:Y_Z_hermiticity_cont}
\mathcal{Y}^*_{c,s}(\Omega) = \mathcal{Y}_{c,s}(-\Omega)\,,\qquad\mbox{and}\qquad\mathcal{Z}^*_{c,s}(\Omega) = \mathcal{Z}_{c,s}(-\Omega)\,.
\end{equation}
This leads to an interesting observation that the coefficients $\mathcal{Y}_{c,s}(\Omega)$ and $\mathcal{Z}_{c,s}(\Omega)$ should be real-valued functions of variable $s = i\Omega$.
Now we can make further generalizations and consider multiple light and vacuum fields comprising the quantity of interest:
\begin{equation}
\hat{Y}(\Omega)\ \to\ \hat{N}_Y(\Omega) = \sum\limits_{i=1}^N \vb{\mathcal{Y}}^\dag_i(\Omega)\hat{\boldsymbol{a}}_i(\Omega)\,,
\end{equation}
where $\hat{\boldsymbol{a}}_i(\Omega)$ stand for quadrature amplitude vectors of $N$ independent electromagnetic fields, and $\vb{\mathcal{Y}}^\dag_i(\Omega)$ are the corresponding complex-valued coefficient functions indicating how these fields are transmitted to the output. In reality, the readout observable of a GW detector is always a combination of the input light field and vacuum fields that mix into the output optical train as a result of optical loss of various origin. This statement can be exemplified by a single lossy mirror I/O-relations given by Eq.~\eqref{eq:IO_lossy_mirror_relations_first} of Section~\ref{sec:losses_in_OE}.
Thus, to calculate the spectral density for such an observable one needs to know the initial state of all light fields under consideration. Since we assume $\hat{\boldsymbol{a}}_i(\Omega)$ independent from each other, the initial state will simply be a direct product of the initial states for each of the fields:
\begin{equation*}
\ket{\Psi} = \bigotimes_{i=1}^N\ket{\psi_i}\,,
\end{equation*}
and the formula for the corresponding power (double-sided) spectral density reads:
\begin{equation}
S_Y(\Omega) = \sum\limits_{i=1}^N \vb{\mathcal{Y}}^\dag_i(\Omega)\bra{\psi_i}\hat{\boldsymbol{a}}_i(\Omega)\circ\hat{\boldsymbol{a}}^\dag_i(\Omega)\ket{\psi_i}\vb{\mathcal{Y}}_i(\Omega) = \sum\limits_{i=1}^N\vb{\mathcal{Y}}^\dag_i(\Omega)\mathbb{S}_i(\Omega)\vb{\mathcal{Y}}_i(\Omega) = \sum\limits_{i=1}^N S_{Y_i}(\Omega)\,,
\end{equation}
with $\mathbb{S}_i(\Omega)$ standing for the i-th input field spectral density matrix.
Hence, the total spectral density is just a sum of spectral densities of each of the fields. The cross-spectral density for two observables $\hat{Y}(\Omega)$ and $\hat{Z}(\Omega)$ can be built by analogy and we leave this task to the reader.
\newpage
\section{Linear Quantum Measurement}
\label{sec:linear_quantum_measurement}
In Section~\ref{sec:quantum_light}, we discussed the quantum nature of light and fluctuations of the light field observables like phase and amplitude that stem thereof and yield what is usually called the quantum noise of optical measurement. In GW detection applications, where a sensitivity of the phase measurement is essential, as discussed in Section~\ref{sec:GW_interaction_with_IF}, the natural question arises: is there a limit to the measurement precision imposed by quantum mechanics? A seemingly simple answer would be that such a limit is set by the quantum fluctuations of the outgoing light phase quadrature, which are, in turn, governed by the quantum state the outgoing light finds itself in. The difficult part is that on its way through the interferometer, the light wave inflicts an additional back-action noise that adds up to the phase fluctuations of the incident wave and contaminates the output of the interferometer. The origin of this back action is in amplitude fluctuations of the incident light, giving rise to a random radiation pressure force that acts on the interferometer mirrors along with the signal GW force, thus effectively mimicking it. And it is the fundamental principle of quantum mechanics, the Heisenberg uncertainty principle, that sets a limit on the product of the phase and amplitude uncertainties (since these are complementary observables), thus leading up to the lower bound of the achievable precision of phase measurement. This limit appears to be a general feature for a very broad class of measurement known as linear measurement and is referred to as the SQL~\cite{Sov.Phys.JETP_26.831_1968, 92BookBrKh}.
In this section, we try to give a brief introduction to quantum measurement theory, starting from rather basic examples with discrete measurement and then passing to a general theory of continuous linear measurement. We introduce the concept of the SQL and derive it for special cases of probe bodies. We also discuss briefly possible ways to overcome this limit by contriving smarter ways of weak force measurement then direct coordinate monitoring.
\subsection{Quantum measurement of a classical force}
\subsubsection{Discrete position measurement}
\label{sec:linear_toy}
Let us consider a very simple measurement scheme, which, nevertheless,
embodies all key features of a general position measurement. In the
scheme shown in Figure~\ref{fig:toy_0}, a sequence of very short light
pulses are used to monitor the displacement of a probe body $M$. The
position $x$ of $M$ is probed periodically with time interval
$\vartheta$. In order to make our model more realistic, we suppose
that each pulse reflects from the test mass $\digamma>1$ times, thus
increasing the optomechanical coupling and thereby the information of
the measured quantity contained in each reflected pulse. We also
assume mass $M$ large enough to neglect the displacement inflicted by
the pulses radiation pressure in the course of the measurement
process.
\epubtkImage{fig20.png}{%
\begin{figure}[htbp]
\centerline{\includegraphics[width=.85\textwidth]{fig20}}
\caption{Toy example of a linear optical position measurement.}
\label{fig:toy_0}
\end{figure}}
Then each $j$-th pulse, when reflected, carries a phase shift proportional to the value of the test-mass position $x(t_j)$ at the moment of reflection:
\begin{equation}
\label{eq:phi_refl}
\hat{\phi}_j^{\mathrm{refl}} = \hat{\phi}_j - 2\digamma k_p\hat{x}(t_j) \,,
\end{equation}
where $k_p=\omega_p/c$, $\omega_p$ is the light frequency, $j=\dots,-1,0,1,\dots$ is the pulse number and $\hat{\phi}_j$ is the initial (random) phase of the $j$-th pulse. We assume that the mean value of all these phases is equal to zero, $\mean{\hat{\phi}_j}=0$, and their root mean square (RMS) uncertainty $\mean{(\hat{\phi^2}}-\mean{\hat{\phi}}^2)^{1/2}$ is equal to $\Delta\phi$.
The reflected pulses are detected by a phase-sensitive device (the phase detector). The implementation of an optical phase detector is considered in detail in Section~\ref{sec:homodyne}. Here we suppose only that the phase $\hat{\phi}_j^{\mathrm{refl}}$ measurement error introduced by the detector is much smaller than the initial uncertainty of the phases $\Delta\phi$. In this case, the initial uncertainty will be the only source of the position measurement error:
\begin{equation}
\label{eq:Delta_x_meas}
\Delta x_{\mathrm{meas}} = \frac{\Delta\phi}{2\digamma k_p} \,.
\end{equation}
For convenience, we renormalize Eq.~\eqref{eq:phi_refl} as the equivalent test-mass displacement:
\begin{equation}
\label{eq:tilde_x_j}
\tilde{x}_j \equiv -\frac{\hat{\phi}_j^{\mathrm{refl}}}{2\digamma k_p}
= \hat{x}(t_j) + \hat{x}_{\mathrm{fl}}(t_j) \,,
\end{equation}
where
\begin{equation}
\label{eq:x_j_fl}
\hat{x}_{\mathrm{fl}}(t_j) = -\frac{\hat{\phi}_j}{2\digamma k_p}
\end{equation}
are the independent random values with the RMS uncertainties given by Eq.~\eqref{eq:Delta_x_meas}.
Upon reflection, each light pulse kicks the test mass, transferring to it a back-action momentum equal to
\begin{equation}
\label{eq:p_j}
\hat{p}_j^{\mathrm{after}} - \hat{p}_j^{\mathrm{before}} = \hat{p}_j^{\mathrm{b.a.}}
= \frac{2\digamma}{c}\hat{\mathcal{W}}_j \,,
\end{equation}
where $\hat{p}_j^{\mathrm{before}}$ and $\hat{p}_j^{\mathrm{after}}$ are the test-mass momentum values just before and just after the light pulse reflection, and $\mathcal{W}_j$ is the energy of the $j$-th pulse. The major part of this perturbation is contributed by classical radiation pressure:
\begin{equation}
\mean{\hat{p}_j^{\mathrm{b.a.}}} = \frac{2\digamma}{c}\mathcal{W} \,,
\end{equation}
with $\mathcal{W}$ the mean energy of the pulses. Therefore, one could neglect its effect, for it could be either subtracted from the measurement result or compensated by an actuator. The random part, which cannot be compensated, is proportional to the deviation of the pulse energy:
\begin{equation}
\label{eq:p_j_fl}
\hat{p}^{\mathrm{b.a.}}(t_j) = \hat{p}_j^{\mathrm{b.a.}} - \mean{\hat{p}_j^{\mathrm{b.a.}}}
= \frac{2\digamma}{c}\bigl(\hat{\mathcal{W}}_j - \mathcal{W}\bigr) \,,
\end{equation}
and its RMS uncertainly is equal to
\begin{equation}
\label{eq:Delta_p_pert}
\Delta p_{\mathrm{b.a.}} = \frac{2\digamma\Delta\mathcal{W}}{c} \,,
\end{equation}
with $\Delta\mathcal{W}$ the RMS uncertainty of the pulse energy.
The energy and the phase of each pulse are canonically conjugate observables and thus obey the following uncertainty relation:
\begin{equation}
\label{eq:dE_dphi}
\Delta\mathcal{W}\Delta\phi \ge \frac{\hbar\omega_p}{2} \,.
\end{equation}
Therefore, it follows from Eqs.~(\ref{eq:Delta_x_meas} and \ref{eq:Delta_p_pert}) that the position measurement error $\Delta x_{\mathrm{meas}}$ and the momentum perturbation $\Delta p_{\mathrm{b.a.}}$ due to back action also satisfy the uncertainty relation:
\begin{equation}
\label{eq:Delta_x_Delta_p}
\Delta x_{\mathrm{meas}}\Delta p_{\mathrm{b.a.}} \ge \frac{\hbar}{2} \,.
\end{equation}
This example represents a simple particular case of a \emph{linear measurement}. This class of measurement schemes can be fully described by two linear equations of the form~\eqref{eq:tilde_x_j} and \eqref{eq:p_j}, provided that both the measurement uncertainty and the object back-action perturbation ($\hat{x}_{\mathrm{fl}}(t_j)$ and $\hat{p}^{\mathrm{b.a.}}(t_j)$ in this case) are statistically independent of the test object initial quantum state and satisfy the same uncertainty relation as the measured observable and its canonically conjugate counterpart (the object position and momentum in this case).
\subsubsection{From discrete to continuous measurement}
\label{sec:disc2cont}
Suppose the test mass to be heavy enough for a single pulse to either perturb its momentum noticeably, or measure its position with the required precision (which is a perfectly realistic assumption for the kilogram-scale test masses of GW interferometers). In this case, many pulses should be used to accumulate the measurement precisions; at the same time, the test-mass momentum perturbation will be accumulated as well. Choose now such a time interval $T$, which, on the one hand, is long enough to comprise a large number of individual pulses:
\begin{equation}
N = \frac{T}{\vartheta} \gg 1 \,,
\end{equation}
and, on the other hand, is sufficiently short for the test-mass position $x$ not to change considerably during this time due to the test-mass self-evolution. Then one can use all the $N$ measurement results to refine the precision of the test-mass position $x$ estimate, thus getting $\sqrt{N}$ times smaller uncertainty
\begin{equation}
\label{eq:Dx_T}
\Delta x_T = \frac{\Delta x_{\mathrm{meas}}}{\sqrt{N}}
= \Delta x_{\mathrm{meas}}\sqrt{\frac{\vartheta}{T}} \,.
\end{equation}
At the same time, the accumulated random kicks the object received from each of the pulses random kicks, see Eq.~\eqref{eq:p_j_fl}, result in random change of the object's momentum similar to that of Brownian motion, and thus increasing in the same diffusive manner:
\begin{equation}
\label{eq:Dp_T}
\Delta p_T = \Delta p_{\mathrm{b.a.}}\sqrt{N} = \Delta p_{\mathrm{b.a.}}\sqrt{\frac{T}{\vartheta}} \,.
\end{equation}
If we now assume the interval between the measurements to be infinitesimally small ($\vartheta\to0$), keeping at the same time each single measurement strength infinitesimally weak:
\begin{equation*}
\Delta x_{\mathrm{meas}} \to \infty \Leftrightarrow \Delta p_{\mathrm{b.a.}} \to 0 \,,
\end{equation*}
then we get a \emph{continuous} measurement of the test-mass position $\hat{x}(t)$ as a result.
We need more adequate parameters to characterize its `strength' than $\Delta x_{\mathrm{meas}}$ and $\Delta p_{\mathrm{b.a.}}$. For continuous measurement we introduce the following parameters instead:
\begin{equation}
\label{eq:S_xS_F_lim}
S_x = \lim_{\vartheta\to0}(\Delta x_{\mathrm{meas}})^2\vartheta = \frac{S_\phi}{4\digamma^2k_p^2}\,,\qquad
S_F = \lim_{\vartheta\to0}\frac{(\Delta p_{\mathrm{b.a.}})^2}{\vartheta} = \frac{4\digamma^2S_\mathcal{I}}{c^2}
\,,
\end{equation}
with
\begin{equation}
\label{S_phiS_I_lim}
S_\phi = \lim_{\vartheta\to0}(\Delta \phi)^2\vartheta \,,\qquad
S_\mathcal{I} = \lim_{\vartheta\to0}\frac{(\Delta\mathcal{W})^2}{\vartheta} \,.
\end{equation}
This allows us to rewrite Eqs.~\eqref{eq:Dx_T} and \eqref{eq:Dp_T} in a form that does not contain the time interval $\vartheta$:
\begin{equation}
\label{dx_dp_cont}
\Delta x_T = \sqrt{\frac{S_x}{T}} \,,\qquad \Delta p_T = \sqrt{S_FT} \,.
\end{equation}
To clarify the physical meaning of the quantities $S_x$ and $S_\phi$ let us rewrite Eq.~\eqref{eq:tilde_x_j} in the continuous limit:
\begin{equation}
\label{eq:tilde_x}
\tilde{x}(t) = \hat{x}(t) + \hat{x}_{\mathrm{fl}} (t) \,,
\end{equation}
where
\begin{equation}
\label{eq:x_fl_cont}
\hat{x}_{\mathrm{fl}}(t) = -\frac{\hat{\phi}(t)}{2\digamma k_p}
\end{equation}
stands for \emph{measurement noise}, proportional to the phase $\hat{\phi}(t)$ of the light beam (in the continuous limit the sequence of individual pulses transforms into a continuous beam). Then there is no difficulty in seeing that $S_x$ is a power (double-sided) spectral density of this noise, and $S_\phi$ is a power double-sided spectral density of $\hat{\phi}(t)$.
If we turn to Eq.~\eqref{eq:p_j}, which describes the meter back action, and rewrite it in a continuous limit we will get the following differential equation for the object momentum:
\begin{equation}
\label{eq:F_fl_def}
\frac{d\hat{p}(t)}{dt} = \hat{F}_{\mathrm{fl}}(t) + \dots
\end{equation}
where $\hat{F}_{\mathrm{fl}}(t)$ is a continuous Markovian random force, defined as a limiting case of the following discrete Markov process:
\begin{equation}
\label{eq:F_fl_cont}
\hat{F}_{\mathrm{fl}}(t_j) = \lim_{\vartheta\to0}\frac{\hat{p}^{\mathrm{b.a.}}(t_j)}{\vartheta}
= \frac{2\digamma}{c}
\lim_{\vartheta\to0}\frac{\hat{\mathcal{W}}_j-\mathcal{W}}{\vartheta}
= \frac{2\digamma}{c}[\hat{\mathcal{I}}(t_j)-\mathcal{I}_0] \,,
\end{equation}
with $\hat{\mathcal{I}}(t)$ the optical power, $\mathcal{I}_0$ its mean value, and `$\dots$' here meaning all forces (if any), acting on the object but having nothing to do with the meter (light, in our case). Double-sided power spectral density of $\hat{F}_{\mathrm{b.a.}}$ is equal to $S_F$, and double-sided power spectral density of $\hat{\mathcal{I}}$ is $S_\mathcal{I}$.
We have just built a simple model of a \emph{continuous linear measurement}, which nevertheless comprises the main features of a more general theory, i.e., it contains equations for the calculation of measurement noise~\eqref{eq:tilde_x} and also for back action~\eqref{eq:F_fl_def}. The precision of this measurement and the object back action in this case are described by the spectral densities $S_x$ and $S_F$ of the two meter noise sources, which are assumed to not be correlated in our simple model, and thus satisfy the following relation (cf.\ Eqs.~\eqref{eq:S_xS_F_lim}):
\begin{equation}
\label{eq:S_xS_F_simple}
S_xS_F = \frac{S_\phi S_\mathcal{I}}{\omega_p^2} \ge \frac{\hbar^2}{4} \,.
\end{equation}
This relation (as well as its more general version to be discussed later) for continuous linear measurements plays the same role as the uncertainty relation~\eqref{eq:Delta_x_Delta_p} for discrete measurements, establishing a universal connection between the accuracy of the monitoring and the perturbation of the monitored object.
\paragraph*{Simple case: light in a coherent state.}
Recall now that scheme of representing the quantized light wave as a sequence of short statistically-independent pulses with duration $\varepsilon\equiv\vartheta$ we referred to in Section~\ref{sec:light_quantum_states}. It is the very concept we used here, and thus we can use it to calculate the spectral densities of the measurement and back-action noise sources for our simple device featured in Figure~\ref{fig:toy_0} assuming the light to be in a coherent state with classical amplitude $A_c=\sqrt{2\mathcal{I}_0/(\hbar\omega_p)}$ (we chose $A_s=0$ thus making the mean phase of light $\mean{\hat{\phi}}=0$). To do so we need to express phase $\hat{\phi}$ and energy $\hat{\mathcal{W}}$ in the pulse in terms of the quadrature amplitudes $\hat{a}_{c,s}(t)$. This can be done if we refer to Eq.~\eqref{eq:2photon_E_strain} and make use of the following definition of the mean electromagnetic energy of the light wave contained in the volume $v_\vartheta\equiv\mathcal{A}c\vartheta$ (here, $\mathcal{A}$ is the effective cross-sectional area of the light beam):
\begin{equation}
\hat{\mathcal{W}} = \frac{v_\vartheta}{4\pi}\overline{\hat{E}^2(t)} = \frac{v_\vartheta}{4\pi \vartheta}\int_{-\vartheta/2}^{\vartheta/2}d\tau\,\hat{E}^2(\tau) = \mathcal{W}+\delta\hat{\mathcal{W}}\,,
\end{equation}
where $\mathcal{W}=v_\vartheta \mathcal{C}_0^2 A_c^2/(8\pi) = \mathcal{I}_0\vartheta$ is the mean pulse energy, and
\begin{equation}
\delta\hat{\mathcal{W}} \simeq \frac{\mathcal{A}c\mathcal{C}_0^2}{4\pi}2A_c\int_{-\vartheta/2}^{\vartheta/2}d\tau\,\hat{a}_c(\tau) = \sqrt{2\hbar\omega_p\mathcal{I}_0\vartheta}\hat{X}_\vartheta(t) = \sqrt{2\hbar\omega_p\mathcal{W}}\hat{X}_\vartheta(t)\,
\end{equation}
is a fluctuating part of the pulse energy\epubtkFootnote{Here, we
omitted the terms of $\delta\hat{\mathcal{W}}$ proportional to
$\hat{a}^2_{c,s}(t)$ since their contribution to the integral is of
the second order of smallness in $\hat{a}_{c,s}/A_c$ compared to the
one for the first order term.}. We used here the definition of the
mean pulse quadrature amplitude operators introduced in
Eqs.~\eqref{eq:X_and_Y_quadrature_def}. In the same manner, one can
define a phase for each pulse using Eqs.~\eqref{eq:EMW_quadrature_def}
and with the assumption of small phase fluctuations ($\Delta\phi\ll1$) one
can get:
\begin{equation}
\hat{\phi} \simeq \frac{1}{A_c \vartheta}\int_{-\vartheta/2}^{\vartheta/2}d\tau \hat{a}_s(\tau) = \sqrt{\frac{\hbar\omega_p}{2\mathcal{I}_0\vartheta}}\hat{Y}_\vartheta= \sqrt{\frac{\hbar\omega_p}{2\mathcal{W}}}\hat{Y}_\vartheta\,.
\end{equation}
Thus, since in a coherent state $\Delta\hat{X}^2_\vartheta = \Delta\hat{Y}^2_\vartheta = 1/2$ the phase and energy uncertainties are equal to
\begin{equation}
\label{DphiDE_coh}
\Delta\phi = \frac{1}{2}\sqrt{\frac{\hbar\omega_p}{\mathcal{W}}} \,, \qquad
\Delta\mathcal{W} = \sqrt{\hbar\omega_p\mathcal{W}} \,,
\end{equation}
and hence
\begin{equation}
\label{DxDp_coh}
\Delta x_{\mathrm{meas}} = \frac{c}{4\digamma}\sqrt{\frac{\hbar}{\omega_p\mathcal{W}}} \,, \qquad
\Delta p_{\mathrm{b.a.}} = \frac{2\digamma}{c}\sqrt{\hbar\omega_p\mathcal{W}} \,.
\end{equation}
Substituting these expressions into Eqs.~(\ref{S_phiS_I_lim}, \ref{eq:S_xS_F_lim}), we get the following expressions for the power (double-sided) spectral densities of the measurement and back-action noise sources:
\begin{equation}
\label{eq:S_phiS_I_toy}
S_\phi = \frac{\hbar\omega_p}{4\mathcal{I}_0} \,, \qquad S_\mathcal{I} = \hbar\omega_p\mathcal{I}_0 \,,
\end{equation}
and
\begin{equation}
\label{S_xS_F_toy}
S_x = \frac{\hbar c^2}{16\omega_p \mathcal{I}_0\digamma^2} \,, \qquad
S_F = \frac{4\hbar\omega_p\mathcal{I}_0\digamma^2}{c^2} \,.
\end{equation}
We should emphasize that this simple measurement model and the corresponding uncertainty relation~\eqref{eq:S_xS_F_simple} are by no means general. We have made several rather strong assumptions in the course of derivation, i.e., we assumed:
\begin{enumerate}
\item energy and phase fluctuations in each of the light pulses uncorrelated: $\mean{\hat{\mathcal{W}}(t_j)\hat{\phi}(t_j)}=0$;
\item all pulses to have the same energy and phase uncertainties $\Delta\mathcal{W}$ and $\Delta\phi$, respectively;
\item the pulses statistically independent from each other, particularly taking $\mean{\hat{\mathcal{W}}(t_i)\hat{\mathcal{W}}(t_j)}=\mean{\hat{\phi}(t_i)\hat{\phi}(t_j)}=\mean{\hat{\mathcal{W}}(t_i)\hat{\phi}(t_j)}=0$ with $t_i\neq t_j$.
\end{enumerate}
These assumptions can be mapped to the following features of the fluctuations $\hat{x}_{\mathrm{fl}}(t)$ and $\hat{F}_{\mathrm{b.a.}}(t)$ in the continuous case:
\begin{enumerate}
\item these noise sources are mutually not correlated;
\item they are stationary (invariant to the time shift) and, therefore, can be described by spectral densities $S_x$ and $S_F$;
\item they are Markovian (white) with constant (frequency-independent) spectral densities.
\end{enumerate}
The features 1 and 2, in turn, lead to characteristic fundamentally-looking sensitivity limitations, the SQL. We will call linear quantum meters, which obey these limitations (that is, with mutually non-correlated and stationary noises $\hat{x}_{\mathrm{fl}}$ and $\hat{F}_{\mathrm{b.a.}}$), \emph{Simple Quantum Meters} (SQM).
\subsection{General linear measurement}
\label{sec:gen_linear_measurement}
In this Section, we generalize the concept of linear quantum
measurement discussed above and give a comprehensive overview of the
formalism introduced in~\cite{92BookBrKh} and further elaborated
in~\cite{PhysRevD.65.042001, 03pth1Ch}. This formalism can be applied
to any system that performs a transformation from some unknown
classical observable (e.g., GW tidal force in GW interferometers) into
another classical observable of a measurement device that can be
measured with (ideally) arbitrarily high precision (e.g., in GW
detectors, the readout photocurrent serves such an observable) and its
value depends on the value of unknown observable linearly. For
definiteness, let us keep closer to GW detectors and assume the
continuous measurement of a classical force.
\epubtkImage{fig21.png}{%
\begin{figure}[htbp]
\centerline{\includegraphics[width=0.5\textwidth]{fig21}}
\caption{General scheme of the continuous linear measurement with $G$ standing for measured classical force, $\hat{\mathcal{X}}$ the measurement noise, $\hat{\mathcal{F}}$ the back-action noise, $\hat{O}$ the meter readout observable, $\hat{x}$ the actual probe's displacement.}
\label{fig:gen_pos}
\end{figure}}
The abstract scheme of such a device is drawn in Figure~\ref{fig:gen_pos}. It consists of a probe $\mathcal{P}$ that is exposed to the action of a classical force $G(t)$, and the meter. The action of this force on the probe causes its displacement $\hat{x}$ that is monitored by the meter (e.g., light, circulating in the interferometer). The output observable of the meter $\hat{O}$ is monitored by some arbitrary classical device that makes a measurement record $o(t)$. The quantum nature of the probe--meter interaction is reflected by the back-action force $\hat{F}$ that randomly kicks the probe on the part of the meter (e.g., radiation pressure fluctuations). At the same time, the meter itself is the source of additional quantum noise $\hat{O}_{\mathrm{fl}}(t)$ in the readout signal. Quantum mechanically, this system can be described by the following Hamiltonian:
\begin{equation}
\label{eq:linear_system_H}
\hat{\mathcal{H}} = \hat{\mathcal{H}}^{(0)}_{\mathrm{probe}} + \hat{\mathcal{H}}^{(0)}_{\mathrm{meter}} + \hat{V}(t) \,,
\end{equation}
where $\hat{\mathcal{H}}^{(0)}_{\mathrm{probe}}$ and $\hat{\mathcal{H}}^{(0)}_{\mathrm{meter}}$ are the Hamiltonians describing the free evolution of the probe and the meter, respectively, i.e., when there is no coupling between these systems, and $\hat{V}(t) = - \hat{x}(G(t)+\hat{F})$ is the interaction Hamiltonian. The evolution of this system can be found, in general, by solving Heisenberg equations for all of the system observables. However, it is convenient to rewrite it first in the interaction picture factoring out the free evolution of the probe and the meter (see Appendix 3.7 of~\cite{03pth1Ch} for detailed derivation):
\begin{equation}
\label{eq:linear_system_H_IP}
\hat{\mathcal{H}}_I(t)
= \exp\left\{\frac{i}{\hbar}\hat{\mathcal{H}}^{(0)}(t-t_0)\right\}
\hat{V}(t)\exp\left\{-\frac{i}{\hbar}\hat{\mathcal{H}}^{(0)}(t-t_0)\right\}
= -\hat{x}^{(0)}(t)(G(t)+\hat{F}^{(0)}(t))\,,
\end{equation}
where $\hat{\mathcal{H}}^{(0)} = \hat{\mathcal{H}}^{(0)}_{\mathrm{probe}} + \hat{\mathcal{H}}^{(0)}_{\mathrm{meter}}$, and $\hat{x}^{(0)}(t)$ and $\hat{F}^{(0)}(t)$ are the Heisenberg operators of the probe's displacement and the meter back-action force, respectively, in the case of no coupling between these systems, i.e., the solution to the following system of independent Heisenberg equations:
\begin{equation*}
\frac{d\hat{x}^{(0)}(t)}{dt} = \frac{i}{\hbar}\left[\hat{\mathcal{H}}^{(0)}_{\mathrm{probe}},\hat{x}^{(0)}(t)\right]\,, \qquad \frac{d\hat{F}^{(0)}(t)}{dt} = \frac{i}{\hbar}\left[\hat{\mathcal{H}}^{(0)}_{\mathrm{meter}},\hat{F}^{(0)}(t)\right]\,,
\end{equation*}
and $t_0$ is the arbitrary initial moment of time that can be set to $-\infty$ without loss of generality.
The following statement can be proven (see~\cite{Kubo1956}, Section~VI of~\cite{92BookBrKh}, and Theorems~3 and 4 in Appendix~3.7 of~\cite{03pth1Ch} for proof):
\textit{For a linear system with Hamiltonian~\eqref{eq:linear_system_H}, for any linear observable $\hat{A}$ of the probe and for any linear observable $\hat{B}$ of the meter, their full Heisenberg evolutions are given by:
\begin{eqnarray}
\label{eq:linear_system_AB_evolution}
\hat{A}(t) &=& \hat{A}^{(0)}(t) + \int_{t_0}^t dt'\,\chi_{Ax}(t,t')[\hat{F}(t')+G(t')]\,,\nonumber\\
\hat{B}(t) &=& \hat{B}^{(0)}(t) + \int_{t_0}^t dt'\,\chi_{BF}(t,t')\hat{x}(t')\,,
\end{eqnarray}
where $\hat{A}^{(0)}(t)$ and $\hat{B}^{(0)}(t)$ stand for the free Heisenberg evolutions in the case of no coupling, and the functions $\chi_{Ax}(t,t')$ and $\chi_{BF}(t,t')$ are called (time-domain) susceptibilities and defined as:}
\begin{eqnarray}
\label{eq:linear_system_AB_chi}
\chi_{Ax}(t,t') &\equiv& \begin{cases}
\dfrac{i}{\hbar}\left[\hat{A}^{(0)}(t),\hat{x}^{(0)}(t')\right]\,, &
t\geqslant t' \\
0 \,, \quad t<t'
\end{cases}
\nonumber \\
\chi_{BF}(t,t') &\equiv& \begin{cases}
\dfrac{i}{\hbar}\left[\hat{B}^{(0)}(t),\hat{F}^{(0)}(t')\right]\,, &
t\geqslant t' \\
0 \,, \quad t<t'
\end{cases}\,.
\end{eqnarray}
The second clauses in these equations maintain the causality principle.
For time independent Hamiltonian $\hat{\mathcal{H}}^{(0)}$ and operator $\hat{F}$ (in the Schr\"odinger picture), the susceptibilities are invariant to time shifts, i.e., $\chi(t,t')=\chi(t+\tau,t'+\tau)$, therefore they depend only on the difference of times: $\chi(t,t')\to\chi(t-t')$. In this case, one can rewrite Eqs.~\eqref{eq:linear_system_AB_evolution} in frequency domain as:
\begin{equation}
\label{eq:linear_system_AB_Fourier}
\hat{A}(\Omega) = \hat{A}^{(0)}(\Omega) + \chi_{Ax}(\Omega)[\hat{F}(\Omega)+G(\Omega)]\,, \qquad \hat{B}(\Omega) = \hat{B}^{(0)}(\Omega) + \chi_{BF}(\Omega)\hat{x}(\Omega)\,,
\end{equation}
where the Fourier transforms of all of the observables are defined in accordance with Eq.~\eqref{eq:Fourier_transform}.
Let us now use these theorems to find the full set of equations of motion for the system of linear observables $\hat{x}$, $\hat{F}$ and $\hat{O}$ that fully characterize our linear measurement process in the scheme featured in Figure~\ref{fig:gen_pos}:
\begin{eqnarray}
\label{eq:gen_linear_meas_EOM}
\hat{O}(t) &=& \hat{O}^{(0)}(t) + \int_{t_0}^t dt'\,\chi_{OF}(t-t')\hat{x}(t')\,,\nonumber \\
\hat{F}(t) &=& \hat{F}^{(0)}(t) + \int_{t_0}^t dt'\,\chi_{FF}(t-t')\hat{x}(t')\,,\nonumber\\
\hat{x}(t) &=& \hat{x}^{(0)}(t) + \int_{t_0}^t dt'\,\chi_{xx}(t-t')\left[G(t')+\hat{F}(t')\right]\,,
\end{eqnarray}
where time-domain susceptibilities are defined as
\begin{eqnarray}
\label{eq:gen_linear_meas_succep}
\chi_{OF}(t-t') &=& \frac{i}{\hbar}\left[\hat{O}^{(0)}(t),\hat{F}^{(0)}(t')\right]\,, \nonumber\\
\chi_{FF}(t-t') &=& \frac{i}{\hbar}\left[\hat{F}^{(0)}(t),\hat{F}^{(0)}(t')\right]\,, \nonumber\\
\chi_{xx}(t-t') &=& \frac{i}{\hbar}\left[\hat{x}^{(0)}(t),\hat{x}^{(0)}(t')\right]\,.
\end{eqnarray}
The meaning of the above equations is worth discussing. The first of Eqs.~\eqref{eq:gen_linear_meas_EOM} describes how the readout observable $\hat{O}(t)$ of the meter, say the particular quadrature of the outgoing light field measured by the homodyne detector (cf.\ Eq.~\eqref{eq:homodyne_photocurrent}), depends on the actual displacement $\hat{x}(t)$ of the probe, and the corresponding susceptibility $\chi_{OF}(t-t')$ is the transfer function for the meter from $\hat{x}$ to $\hat{O}$. The term $\hat{O}^{(0)}(t)$ stands for the free evolution of the readout observable, provided that there was no coupling between the probe and the meter. In the case of the GW detector, this is just a pure quantum noise of the outgoing light that would have come out were all of the interferometer test masses fixed. It was shown explicitly in~\cite{02a1KiLeMaThVy} and we will demonstrate below that this noise is fully equivalent to that of the input light except for the insignificant phase shift acquired by the light in the course of propagation through the interferometer.
The following important remark should be made concerning the meter's output observable $\hat{O}(t)$. As we have mentioned already, the output observable in the linear measurement process should be precisely measurable at any instance of time. This implies a \emph{simultaneous measurability condition}~\cite{Sci.209.4456.547_1980_BrVoTh, 80a1CaThDrSaZi, PhysRevD.19.2888, 92BookBrKh, 03pth1Ch, PhysRevD.65.042001} on the observable $\hat{O}(t)$ requiring that it should commute with itself at any moment of time:
\begin{equation}
\label{eq:measurability_cond}
\left[\hat{O}(t),\,\hat{O}(t')\right]=0\,,\quad \forall t,\,t'\,.
\end{equation}
Initially, this condition was introduced as the definition of the quantum non-demolition (QND) observables by Braginsky et al.~\cite{Sov.Phys.Uspekhi.17.5.644_1975_BrVo, 77a1eBrKhVo}. In our case it means that the measurement of $\hat{O}(t_1)$ at some moment of time $t_1$ shall not disturb the measurement result at any other moments of time and therefore the sample data $\left\{\hat{O}(t_1),\hat{O}(t_1),\ldots,\hat{O}(t_n)\right\}$ can be stored directly as bits of classical data in a classical storage medium, and any noise from subsequent processing of the signal can be made arbitrarily small. It means that all noise sources of quantum origin are already included in the quantum fluctuations of $\hat{O}(t)$~\cite{03pth1Ch, PhysRevD.65.042001}. And the fact that due to~\eqref{eq:measurability_cond} this susceptibility turns out to be zero reflects the fact that $\hat{O}(t)$ should be a classical observable.
The second equation in~\eqref{eq:gen_linear_meas_EOM} describes how the back-action force exerted by the meter on the probe system evolves in time and how it depends on the probe's displacement. The first term, $\hat{F}^{(0)}(t)$, meaning is rather obvious. In GW interferometer, it is the radiation pressure force that the light exerts on the mirrors while reflecting off them. It depends only on the mean value and quantum fluctuations of the amplitude of the incident light and does not depend on the mirror motion. The second term here stands for a \emph{dynamical back-action} of the meter and since, by construction, it is the part of the back-action force that depends, in a linear way, from the probe's displacement, the meaning of the susceptibility $\chi_{FF}(t-t')$ becomes apparent: it is the generalized rigidity that the meter introduces, effectively modifying the dynamics of the probe. We will see later how this effective rigidity can be used to improve the sensitivity of the GW interferometers without introducing additional noise and thus enhancing the SNR of the GW detection process.
The third equation of~\eqref{eq:gen_linear_meas_EOM} concerns the evolution of the probe's displacement in time. Three distinct parts comprise this evolution. Let us start with the second and the third ones:
\begin{equation}
x_s(t) = \int_{t_0}^t dt'\,\chi_{xx}(t-t')G(t')\,,\quad\mbox{and}\quad \hat{x}_{\mathrm{b.a.}}(t) = \int_{t_0}^t dt'\,\chi_{xx}(t-t')\hat{F}(t')\,.
\end{equation}
Here $x_s(t)$ is the probe's response on the signal force $G(t)$ and is, actually, the part we are mostly interested in. This expression also unravels the role of susceptibility $\chi_{xx}(t-t')$: it is just the Green's function of the equation of motion governing the probe's bare dynamics (also known as impulse response function) that can be shown to be a solution of the following initial value problem:
\begin{eqnarray*}
\mathbf{D}\chi_{xx}(t-t') = \delta(t-t')\,,\quad \chi_{xx}(t-t')|_{t\to t'} = 0\,,\quad \ldots\\ \ldots\ \frac{\partial^{n-2}\chi_{xx}(t-t')}{\partial t^{n-2}}\bigr|_{t\to t'+0} = 0\,,\quad \frac{\partial^{n-1}\chi_{xx}(t-t')}{\partial t^{n-1}}\bigr|_{t\to t'+0} = \frac{1}{a_n}\,,
\end{eqnarray*}
where $\mathbf{D} = \sum_{k=0}^n a_k\frac{d^k}{dt^k}$ is the linear differential operator that is governed by the dynamics of the probe, e.g., it is equal to $\mathbf{D}_{\mathrm{f.m.}} =M\frac{d^2}{dt^2}$ for a free mass $M$ and to $\mathbf{D}_{\mathrm{osc}} =M\frac{d^2}{dt^2}+M\Omega^2_m$
for a harmonic oscillator with eigenfrequency $\Omega_m$. Apparently, operator $\mathbf{D}$ is an inverse of the integral operator $\vb{\chi}_{xx}$ whose kernel is $\chi_{xx}(t-t')$:
\begin{equation*}
x_s(t) = \mathbf{D}^{-1}G(t) = \vb{\chi}_{xx}G(t) = \int_{t_0}^t dt'\,\chi_{xx}(t-t')G(t')\,.
\end{equation*}
The second value, $\hat{x}_{\mathrm{b.a.}}(t)$, is the displacement of the probe due to the back-action force exerted by the meter on the probe. Since it enters the probe's response in the very same way the signal does, it is the most problematic part of the quantum noise that, as we demonstrate later, imposes the SQL~\cite{Sov.Phys.JETP_26.831_1968, 92BookBrKh}.
And finally, $\hat{x}^{(0)}(t)$ simply features a free evolution of the probe in accordance with its equations of motion and thus depends on the initial values of the probe's displacement $\hat{x}^{(0)}(t_0)$, momentum $\hat{p}^{(0)}(t_0)$, and, possibly, on higher order time derivatives of $\hat{x}^{(0)}(t)$ taken at $t_0$, as per the structure of the operator $\mathbf{D}$ governing the probe's dynamics. It is this part of the actual displacement that bears quantum uncertainties imposed by the initial quantum state of the probe. One could argue that these uncertainties might become a source of additional quantum noise obstructing the detection of GWs, augmenting the noise of the meter. This is not the case as was shown explicitly in~\cite{03a1BrGoKhMaThVy}, since our primary interest is in the detection of a classical force rather than the probe's displacement. Therefore, performing over the measured data record $o(t)$ the linear transformation corresponding to first applying the operator $\mathrm{\vb{\chi}}_{OF}^{-1}$ on the readout quantity that results in expressing $o(t)$ in terms of the probe's displacement:
\begin{equation*}
\tilde{x}(t)=\mathrm{\vb{\chi}}_{OF}^{-1} O(t) = x_{\mathrm{fl}}(t)+x^{(0)}(t)+x_s(t)+x_{\mathrm{b.a.}}(t)\,,
\end{equation*}
with $x_{\mathrm{fl}}(t) = \mathrm{\vb{\chi}}_{OF}^{-1} O^{(0)}(t)$ standing for the meter's own quantum noise (measurement uncertainty), and then applying a probe dynamics operator $\mathbf{D}$ that yields a force signal equivalent to the readout quantity $o(t)$:
\begin{equation*}
\tilde{F}(t) = \mathbf{D}\tilde{x}(t) = \mathbf{D}x_{\mathrm{fl}}(t)+F_{\mathrm{b.a.}}(t)+G(t)\,.
\end{equation*}
The term $\mathbf{D}x^{(0)}(t)$ vanishes since $x^{(0)}(t)$ is the solution of a free-evolution equation of motion. Thus, we see that the result of measurement contains two noise sources, $\hat{x}_{\mathrm{fl}}(t)$ and $\hat{F}_{\mathrm{b.a.}}(t)$, which comprise the sum noise masking the signal force $G(t)$.
Since we can remove initial quantum uncertainties associated with the state of the probe, it would be beneficial to turn to the Fourier domain and rewrite Eqs.~\eqref{eq:gen_linear_meas_EOM} in the spectral form:
\begin{eqnarray}
\label{eq:gen_linear_meas_EOM_Fourier}
\hat{O}(\Omega) &=& \hat{O}^{(0)}(\Omega) + \chi_{OF}(\Omega)\hat{x}(\Omega)\,,\nonumber\\
\hat{F}(\Omega) &=& \hat{F}^{(0)}(\Omega) + \chi_{FF}(\Omega)\hat{x}(\Omega)\,,\nonumber\\
\hat{x}(\Omega) &=& \chi_{xx}(\Omega)[\hat{F}(\Omega)+G(\Omega)]\,,
\end{eqnarray}
where spectral susceptibilities are defined as:
\begin{equation}
\chi_{AB}(\Omega) = \int_0^{\infty}d\tau\,\chi_{AB}(\tau)e^{i\Omega\tau},
\end{equation}
with $(A,B) \Rightarrow (O,F,x)$, and we omit the term $\hat{x}^{(0)}(\Omega)$ in the last equation for the reasons discussed above.
The solution of these equations is straightforward to get and reads:
\begin{eqnarray}
\hat{O}(\Omega) &=& \hat{O}^{(0)}(\Omega) +\frac{\chi_{xx}(\Omega)\chi_{OF}(\Omega)}{1-\chi_{xx}(\Omega)\chi_{FF}(\Omega)}\left[G(\Omega)+\hat{F}^{(0)}(\Omega)\right]\,,\label{eq:gen_lin_meas_output_spectral}\\
\hat{F}(\Omega) &=& \frac{1}{1-\chi_{xx}(\Omega)\chi_{FF}(\Omega)}\left[\hat{F}^{(0)}(\Omega)+\chi_{FF}(\Omega)\chi_{xx}(\Omega)G(\Omega)\right]\,,\label{eq:gen_lin_meas_F_ba_spectral}\\
\hat{x}(\Omega) &=& \frac{\chi_{xx}(\Omega)}{1-\chi_{xx}(\Omega)\chi_{FF}(\Omega)}\left[G(\Omega)+\hat{F}^{(0)}(\Omega)\right]\label{eq:gen_lin_meas_x_spectral}\,.
\end{eqnarray}
It is common to normalize the output quantity of the meter $\hat{O}(\Omega)$ to unit signal. In GW interferometers, two such normalizations are popular. The first one tends to consider the tidal force $G$ as a signal and thus set to 1 the coefficient in front of $G(\Omega)$ in Eq.~\eqref{eq:gen_lin_meas_output_spectral}. The other one takes GW spectral amplitude $h(\Omega)$ as a signal and sets the corresponding coefficient in $\hat{O}(\Omega)$ to unity. Basically, these normalizations are equivalent by virtue of Eq.~\eqref{eq:GW_force_to_h_rel} as:
\begin{equation}
\label{eq:gen_lin_F_to_h_transform}
-M\Omega^2 x_h(\Omega)\equiv-M\Omega^2 \frac{Lh(\Omega)}{2} = G(\Omega)\quad\Rightarrow\quad h(\Omega) = -2 G(\Omega)/(ML\Omega^2)\,.
\end{equation}
In both cases, the renormalized output quantities can be considered as a sum of the noise and signal constituents:
\begin{equation}
\label{eq:gen_out_noise_def}
\hat{O}^{F} = \hat{\mathcal{N}}^F+G \qquad\mbox{or}\qquad \hat{O}^{h} = \hat{\mathcal{N}}^h+h(\Omega)\,.
\end{equation}
And it is the noise term in both cases that we are seeking to calculate to determine the sensitivity of the GW detector. Let us rewrite $\hat{O}(\Omega)$ in force normalization:
\begin{eqnarray}
\label{eq:gen_force_noise_def}
\hat{O}^F(\Omega) &=& \frac{1-\chi_{FF}(\Omega)\chi_{xx}(\Omega)}{\chi_{OF}(\Omega)\chi_{xx}(\Omega)} \hat{O}(\Omega) =
\frac{\hat{O}^{(0)}(\Omega)}{\chi_{OF}(\Omega)\chi_{xx}(\Omega)}+\left(\hat{F}^{(0)}(\Omega)-\frac{\chi_{FF}(\Omega)}{\chi_{OF}(\Omega)}\hat{O}^{(0)}(\Omega)\right)+G(\Omega)\nonumber\\&\equiv&
\frac{\hat{\mathcal{X}}(\Omega)}{\chi_{xx}(\Omega)}+\hat{\mathcal{F}}(\Omega)+G(\Omega)\,,
\end{eqnarray}
where we introduce two new linear observables $\hat{\mathcal{X}}$ and $\hat{\mathcal{F}}$ of the meter defined as:
\begin{equation}
\label{eq:gen_noise_def}
\hat{\mathcal{X}}(\Omega) \equiv \frac{\hat{O}^{(0)}(\Omega)}{\chi_{OF}(\Omega)}\,,\qquad \hat{\mathcal{F}}(\Omega) \equiv \hat{F}^{(0)}(\Omega)-\frac{\chi_{FF}(\Omega)}{\chi_{OF}(\Omega)}\hat{O}^{(0)}(\Omega)\,,
\end{equation}
that have the following meaning:
\begin{itemize}
\item $\hat{\mathcal{X}}$ is the effective output fluctuation of the meter not dependent on the probe. Henceforth, we will refer to it as the \emph{effective measurement noise} (shot noise, in the GW interferometer common terminology);
\item $\hat{\mathcal{F}}$ is the effective response of the output at time $t$ to the meter's back-action force at earlier times $t<t'$. In the following we will refer to $\hat{\mathcal{F}}$ as the \emph{effective back-action noise} (radiation-pressure noise, in the GW interferometer common terminology).
\end{itemize}
These two new observables that embody the two types of noise inherent in any linear measurement satisfy the following commutation relations:
\begin{eqnarray}
\label{eq:gen_lin_commutator}
\left[\hat{\mathcal{X}}(\Omega),\hat{\mathcal{X}}^\dag(\Omega')\right]=\left[\hat{\mathcal{F}}(\Omega),\hat{\mathcal{F}}^\dag(\Omega')\right] = 0\,,\quad &\Longleftrightarrow\quad &\left[\hat{\mathcal{X}}(t),\hat{\mathcal{X}}(t')\right]=\left[\hat{\mathcal{F}}(t),\hat{\mathcal{F}}(t')\right] = 0\,,\\
\left[\hat{\mathcal{X}}(\Omega),\hat{\mathcal{F}}^\dag(\Omega')\right]=-2\pi i\hbar\delta(\Omega-\Omega')\,,\quad &\Longleftrightarrow\quad &\left[\hat{\mathcal{X}}(t),\hat{\mathcal{F}}^\dag(t')\right]=-i\hbar\delta(t-t')\,,
\end{eqnarray}
that can be interpreted in the way that $\hat{\mathcal{X}}(t)$ and $\hat{\mathcal{F}}(t)$ can be seen at each instance of time as the canonical momentum and the coordinate of different effective linear measuring devices (meter+probe), thus defining an infinite set of subsequent measurements similar to the successive independent monitors of von Neumann's model~\cite{1996Book_vonNeumann}. In this case, however, the monitors described by $\hat{\mathcal{X}}(t)$ and $\hat{\mathcal{F}}(t)$ are not, generally speaking, independent. In GW detectors, these monitors appear correlated due to the internal dynamics of the detector, i.e., the noise processes they describe are non-Markovian.
In particular, this can be seen when one calculates the power (double-sided) spectral density of the sum noise $\hat{\mathcal{N}}^{F}(t)$:
\begin{equation}
\label{eq:gen_spdens}
S^F(\Omega) = \int_{-\infty}^{\infty}\! dt\,\mean{\hat{\mathcal{N}}^{F}(t)\circ\hat{\mathcal{N}}^{F}(t')}e^{i\Omega(t-t')} = \frac{S_{\mathcal{X}\mathcal{X}}(\Omega)}{|\chi_{xx}(\Omega)|^2}+S_{\mathcal{F}\mathcal{F}}(\Omega)+2\mathrm{Re}\left[\frac{S_{\mathcal{X}\mathcal{F}}(\Omega)}{\chi_{xx}(\Omega)}\right]\,,
\end{equation}
where spectral densities:
\begin{eqnarray}
S_{\mathcal{X}\mathcal{X}}(\Omega) &=& \int_{-\infty}^{\infty}\! dt\,\mean{\hat{\mathcal{X}}(t)\circ\hat{\mathcal{X}}(t')}e^{i\Omega(t-t')}\,,\\
S_{\mathcal{F}\mathcal{F}}(\Omega) &=& \int_{-\infty}^{\infty}\! dt\,\mean{\hat{\mathcal{F}}(t)\circ\hat{\mathcal{F}}(t')}e^{i\Omega(t-t')}\,,\\
S_{\mathcal{X}\mathcal{F}}(\Omega) &=& \int_{-\infty}^{\infty}\! dt\,\mean{\hat{\mathcal{X}}(t)\circ\hat{\mathcal{F}}(t')}e^{i\Omega(t-t')}\,,
\end{eqnarray}
are not necessarily constant and, thus, describe non-Markovian random processes. It can also be shown that since $\hat{\mathcal{X}}(t)$ and $\hat{\mathcal{F}}(t)$ satisfy commutation relations~\eqref{eq:gen_lin_commutator}, their spectral densities shall satisfy the \emph{Schr{\"o}dinger--Robertson} uncertainty relation:
\begin{equation}
\label{eq:gen_spdens_uncert_rel}
S_{\mathcal{X}\mathcal{X}}(\Omega)S_{\mathcal{F}\mathcal{F}}(\Omega)-|S_{\mathcal{X}\mathcal{F}}(\Omega)|^2\geqslant \frac{\hbar^2}{4}\,
\end{equation}
that is the generalization of a Heisenberg uncertainty relation in the case of correlated observables.
The general structure of quantum noise in the linear measurement process, comprising two types of noise sources whose spectral densities are bound by the uncertainty relation~\eqref{eq:gen_spdens_uncert_rel}, gives a clue to several rather important corollaries. One of the most important is the emergence of the SQL, which we consider in detail below.
\subsection{Standard Quantum Limit}
\label{sec:SQL}
Recall the SQM in Section~\ref{sec:disc2cont}.
The SQM has non-correlated effective measurement and back-action noises that results in $S_{\mathcal{X}\mathcal{F}}(\Omega)=0$. Apparently, under these conditions $\hat{\mathcal{X}}$ and $\hat{\mathcal{F}}$ turn into $\hat{x}_{\mathrm{fl}}$ and $\hat{F}_{\mathrm{fl}}$ of Eqs.~\eqref{eq:tilde_x} and \eqref{eq:F_fl_def}, respectively. Hence, we will use $S_x(\Omega)$ instead of $S_{\mathcal{X}\mathcal{X}}(\Omega)$ and $S_F(\Omega)$ instead of $S_{\mathcal{F}\mathcal{F}}(\Omega)$ Then the uncertainty relation~\eqref{eq:gen_spdens_uncert_rel} transforms into:
\begin{equation}
\label{eq:simple_spdens_uncert_rel}
S_x(\Omega)S_F(\Omega)\geqslant\frac{\hbar^2}{4}\,.
\end{equation}
The SQL is the name for an ultimate lower bound of a sum noise spectral density the SQM can, in principle, have \emph{at any given frequency $\Omega$}. To derive this limit we assume noise sources $x_{\mathrm{fl}}$ and $F_{\mathrm{b.a.}}$ to have minimal values allowed by quantum mechanics, i.e.
\begin{equation}
\label{eq:minimal_qnoise_req}
S_x(\Omega)S_F(\Omega)=\frac{\hbar^2}{4}\,.
\end{equation}
Then, using this condition, one can minimize SQM's sum noise:
\begin{equation*}
S^F(\Omega) = \frac{S_x(\Omega)}{|\chi_{xx}(\Omega)|^2}+S_F(\Omega) = \frac{S_x(\Omega)}{|\chi_{xx}(\Omega)|^2}+\frac{\hbar^2}{4S_x(\Omega)}
\end{equation*}
to yield:
\begin{equation}
\label{eq:SQL_force}
S^F_{\mathrm{SQL}}(\Omega) = \frac{\hbar}{|\chi_{xx}(\Omega)|}
\end{equation},
that is \emph{achieved when contributions of measurement noise and back-action noise to the sum noise are equal to each other}, i.e., when
\begin{equation}
\label{eq:SQL_optimization}
S_x(\Omega) = \frac{\hbar}{2}|\chi_{xx}(\Omega)|\,,\qquad\Longleftrightarrow\qquad S_F(\Omega) = \frac{\hbar}{2|\chi_{xx}(\Omega)|}\,.
\end{equation}
It is instructive to cite the forms of the SQL in other normalizations, i.e., for $h$-normalization and for $x$-normalization. The former is obtained from~\eqref{eq:SQL_force} via multiplication by $4/(M^2L^2\Omega^4)$:
\begin{equation}
\label{eq:SQL_h}
S^h_{\mathrm{SQL}}(\Omega) = \frac{4 S^F_{\mathrm{SQL}}(\Omega)}{M^2L^2\Omega^4} = \frac{4\hbar}{M^2L^2\Omega^4|\chi_{xx}(\Omega)|}\,.
\end{equation}
The latter is obtained fromEq.~\eqref{eq:SQL_force} using the obvious connection between force and displacement $x(\Omega) = \chi_{xx}(\Omega)F(\Omega)$:
\begin{equation}
\label{eq:SQL_x}
S^x_{\mathrm{SQL}}(\Omega) = |\chi_{xx}(\Omega)|^2 S^F_{\mathrm{SQL}}(\Omega) = \hbar|\chi_{xx}(\Omega)|\,.
\end{equation}
These limits look fundamental. There are no parameters of the meter (only $\hbar$ as a reminder of the uncertainty relation~\eqref{eq:S_xS_F_simple}), and only the probe's dynamics is in there. Nevertheless, this is not the case and, actually, this limit can be beaten by more sophisticated, but still linear, position meters.
At the same time, the SQL represents an important landmark beyond which the ordinary brute-force methods of sensitivity improving cease working, and methods that allow one to blot out the back-action noise $\hat{\mathcal{F}}(t)$ from the meter output signal have to be used instead. Due to this reason, the SQL, and especially the SQL for the simplest test object -- free mass -- is usually considered as a borderline between the classical and the quantum domains.
\subsubsection{Free mass SQL}
\label{sec:SQL_fm}
In the rest of this section, we consider in more detail the SQLs for a free mass and for a harmonic oscillator. We also assume the minimal quantum noise requirement~\eqref{eq:minimal_qnoise_req} to hold.
The free mass is not only the simplest model for the probe's dynamics,
but also the most important class of test objects for GW detection. Test
masses of GW detectors must be isolated as much as possible from
the noisy environment. To this end, the design of GW interferometers
implies suspension of the test masses on thin fibers. The real
suspensions are rather sophisticated and comprise several stages slung
one over another, with mechanical eigenfrequencies $f_m$ in $\lesssim
1\mathrm{Hz}$ range. The sufficient degree of isolation is provided at
frequencies much higher than $f_m$, where the dynamics of test masses
can be approximated with good precision by that of a free mass.
Let us introduce the following convenient measure of measurement strength (precision) in the first place:
\begin{equation}
\label{eq:Omega_q_def}
\Omega_q = \left(\frac{S_F}{M^2S_x}\right)^{1/4} .
\end{equation}
Using the uncertainty relation~\eqref{eq:minimal_qnoise_req}, the noise spectral densities $S_x$ and $S_F$ can be expressed through $\Omega_q$ as follows:
\begin{equation}
\label{eq:Omega_q}
S_x = \frac{\hbar}{2M\Omega_q^2} \,, \qquad S_F = \frac{\hbar M\Omega_q^2}{2} \,.
\end{equation}
Therefore, the larger $\Omega_q$ is, the smaller $S_x$ is (the higher is the measurement precision), and the larger $S_F$ is (the stronger the meter back action is).
In the case of interferometers, $\Omega_q^2$ is proportional to the circulating optical power. For example, for the toy optical meter considered above,
\begin{equation}
\label{eq:Omega_q_toy}
\Omega_q = \sqrt{\frac{8\omega_p\mathcal{I}_0\digamma^2}{Mc^2}} \,,
\end{equation}
see Eqs.~\eqref{S_xS_F_toy}.
Using this notation, and taking into account that for a free mass $M$,
\begin{equation}
\label{D_fm}
\chi_{xx}^{\mathrm{f.m.}}(\Omega) = -\frac{1}{M\Omega^2} \,,
\end{equation}
the sum quantum noise power (double-sided) spectral density can be written as follows:
\begin{equation}
\label{S_F_sum_fm}
S^F_{\mathrm{f.m.}}(\Omega) = M^2\Omega^4S_x + S_F
= \frac{\hbar M\Omega_q^2}{2}\left(\frac{\Omega^4}{\Omega_q^4} + 1\right) \,.
\end{equation}
The SQL optimization~\eqref{eq:SQL_optimization} takes the following simple form in this case:
\begin{equation}
\label{eq:SQL_fm_cond}
\Omega_q = \Omega \,,
\end{equation}
giving:
\begin{equation}
\label{eq:S_F_SQL_fm}
S^F_{\mathrm{SQL\,f.m.}}(\Omega) = \hbar M\Omega^2 \,.
\end{equation}
This consideration is illustrated by Figure~\ref{fig:S_F_SQL} (left),
where power (double-sided) spectral density~\eqref{S_F_sum_fm} is plotted for three
different values of $\Omega_q$. It is easy to see that these plots
never dive under the SQL line~\eqref{eq:S_F_SQL_fm}, which embodies a
common envelope for them. Due to this reason, the sensitivities area
above this line is typically considered as the `classical domain', and
below it -- as the `quantum domain'.
\epubtkImage{fig22.png}{%
\begin{figure}[htbp]
\centerline{
\includegraphics[width=.45\textwidth]{fig22a}\hfill
\includegraphics[width=.45\textwidth]{fig22b}
}
\caption{Sum quantum noise power (double-sided) spectral densities of the Simple Quantum Meter for different values of measurement strength $\Omega_q(\mathrm{red})<\Omega_q(\mathrm{green})<\Omega_q(\mathrm{blue})$. Thin black line: SQL. \emph{Left:} free mass. \emph{Right:} harmonic oscillator}
\label{fig:S_F_SQL}
\end{figure}}
\subsubsection{Harmonic oscillator SQL}
\label{sec:SQL_osc}
The simplest way to overcome the limit~\eqref{eq:S_F_SQL_fm}, which does not require any quantum tricks with the meter, is to use a harmonic oscillator as a test object, instead of the free mass. It is easy to see from Eq.\eqref{eq:SQL_force} that the more responsive the test object is at some given frequency $\Omega$ (that is, the bigger $\chi_{xx}(\Omega)$) is, the smaller its force SQL at this frequency is. In the harmonic oscillator case,
\begin{equation}
\chi_{xx}^{\mathrm{osc}}(\Omega) = \frac{1}{M(\Omega_0^2-\Omega^2)} \,,
\end{equation}
with $\Omega_0$ standing for the oscillator mechanical eigenfrequency, and the sum quantum noise power (double-sided) spectral density equal to
\begin{equation}
\label{S_F_sum_osc}
S^F_{\mathrm{osc}}(\Omega) = M^2(\Omega_0^2-\Omega^2)^2S_x + S_F
= \frac{\hbar M\Omega_q^2}{2}\left[\frac{(\Omega_0^2-\Omega^2)^2}{\Omega_q^4} + 1\right].
\end{equation}
Due to a strong response of the harmonic oscillator near resonance, the first (measurement noise) term in Eq.~\eqref{S_F_sum_fm} goes to zero in the vicinity of $\Omega_0$. Therefore, reducing the value of $\Omega_q$, that is, using weaker measurement, it is possible to increase the sensitivity in a narrow band around $\Omega_0$. At the same time, the smaller $\Omega_q$ is, the more narrow the bandwidth is where this sensitivity is achieved, as can be seen from the plots drawn in Figure~\ref{fig:S_F_SQL} (right).
Consider, in particular, the following minimax optimization of the narrow-band sensitivity. Let $\nu=\Omega-\Omega_0$ be the detuning from the resonance frequency. Suppose also
\begin{equation}
|\nu| \ll \Omega_0 \,.
\end{equation}
In this case, the sum noise power (double-sided) spectral density~\eqref{S_F_sum_osc} can be approximated as follows:
\begin{equation}
\label{eq:S_F_osc_nb}
S^F_{\mathrm{osc}}(\Omega_0+\nu)
= \frac{\hbar M\Omega_q^2}{2}\left(\frac{4\Omega_0^2\nu^2}{\Omega_q^4} + 1\right) .
\end{equation}
Then require the maximum of $S^F_{\mathrm{osc}}$ in a given frequency range $\Delta\Omega$ be as small as possible.
It is evident that this frequency range has to be centered around the resonance frequency $\Omega_0$, with the maximums at its edges, $\nu=\pm\Delta\Omega/2$. The sum noise power (double-sided) spectral density at these points is equal to
\begin{equation}
S^F_{\mathrm{osc}}(\Omega_0\pm\Delta\Omega/2)
= \frac{\hbar M\Omega_q^2}{2}\left(\frac{\Omega_0^2\Delta\Omega^2}{\Omega_q^4} + 1\right) .
\end{equation}
The minimum of this expression is provided by
\begin{equation}
\label{eq:Omega_q_osc}
\Omega_q = \sqrt{\Omega_0\Delta\Omega} \,.
\end{equation}
Substitution of this value back into Eq.~\eqref{eq:S_F_osc_nb} gives the following optimized power (double-sided) spectral density:
\begin{equation}
\label{eq:S_F_osc_nb_opt}
S^F_{\mathrm{osc}}(\Omega_0+\nu)
= \frac{\hbar M\Omega_0}{2}\left(\frac{4\nu^2}{\Delta\Omega} + \Delta\Omega\right) ,
\end{equation}
with
\begin{equation}
S^F_{\mathrm{osc}}(\Omega_0\pm\Delta\Omega/2)= \hbar M\Omega_0\Delta\Omega \,.
\end{equation}
Therefore, the harmonic oscillator can provide a narrow-band sensitivity gain, compared to the \emph{free mass} SQL~\eqref{eq:S_F_SQL_fm}, which reads
\begin{equation}
\label{eq:xi2_osc}
\xi^2_{\mathrm{osc}} = \frac{S^F_{\mathrm{osc}}(\Omega_0+\nu)}{S^F_{\mathrm{SQL\,f.m.}}(\Omega_0)} \simeq \frac12\left(\frac{4 \nu^2}{\Omega_q^2}+\frac{\Omega_q^2}{\Omega_0^2}\right)\,,
\end{equation}
and can be further written accounting for the above optimization as:
\begin{equation}
\label{eq:osc_fm_SQL}
\frac{S^F_{\mathrm{osc}}(\Omega_0+\nu)}
{S^F_{\mathrm{SQL\,f.m.}}(\Omega_0)}\biggr|_{|\nu|\le\Delta\Omega/2}
\le \frac{S^F_{\mathrm{osc}}(\Omega_0\pm\Delta\Omega/2)}{S^F_{\mathrm{SQL\,f.m.}}(\Omega_0)}
= \frac{\Delta\Omega}{\Omega_0} \,.
\end{equation}
Of course, the \emph{oscillator} SQL, equal to
\begin{equation}
\label{eq:S_F_SQL_osc}
S^F_{\mathrm{SQL\,osc}} = \hbar M|\Omega_0^2-\Omega^2| \approx 2\hbar M\Omega_0|\nu|
\end{equation}
cannot be beaten is this way, and the question of whether the sensitivity~\eqref{eq:osc_fm_SQL} is the `true' beating of the SQL or not, is the question to answer (and the subject of many discussions).
\subsubsection{Sensitivity in different normalizations. Free mass and harmonic oscillator}
Above, we have discussed, in brief, different normalizations of the sum noise spectral density and derived the general expressions for the SQL in these normalizations (cf.\ Eqs.~\eqref{eq:SQL_h} and \eqref{eq:SQL_x}). Let us consider how these expressions look for the free mass and harmonic oscillator and how the sensitivity curves transform when changing to different normalizations.
\paragraph*{$h$-normalization:}
The noise spectral density in $h$-normalization can be obtained using Eq.~\eqref{eq:GW_force_to_h_rel}. Where the SQM is concerned, the sum noise in $h$-normalization reads
\begin{equation*}
h_{\mathrm{sum}}(\Omega)\equiv\hat{\mathcal{N}}^h(\Omega) = -\frac{2}{ML\Omega^2}\left[\frac{\hat{x}_{\mathrm{fl}}(\Omega)}{\chi_{xx}(\Omega)}+\hat{F}_{\mathrm{fl}}(\Omega)\right]\,.
\end{equation*}
In the case of a free mass with $\chi_{xx}(\Omega) = -1/(M\Omega^2)$ the above expression transforms as:
\begin{equation*}
h^{\mathrm{f.m.}}_{\mathrm{sum}}(\Omega) = \frac{2\hat{x}_{\mathrm{fl}}(\Omega)}{L} - \frac{2\hat{F}_{\mathrm{fl}}}{ML\Omega^2}
\end{equation*}
and that results in the following power (double-sided) spectral density formula:
\begin{equation}\label{eq:S_h_fm}
S^h_{\mathrm{f.m.}}(\Omega) = \frac{4}{L^2}\left[S_x+\frac{S_F}{M^2\Omega^4}\right] = \frac{2\hbar}{ML^2\Omega_q^2}\left(1+\frac{\Omega_q^4}{\Omega^4}\right)
\end{equation}
and results in the following formula for free mass SQL in $h$-normalization:
\begin{equation}\label{eq:S_h_SQL_fm}
S^h_{\mathrm{SQL\,f.m.}}(\Omega) = \frac{4\hbar}{ML^2\Omega^2}\,.
\end{equation}
The plots of these spectral densities at different values of
$\Omega_q$ are given in the left panel of Figure~\ref{fig:S_X_SQL}.
\epubtkImage{fig23.png}{%
\begin{figure}[htbp]
\centerline{
\includegraphics[width=.45\textwidth]{fig23a}\hfill
\includegraphics[width=.45\textwidth]{fig23b}
}
\caption{Sum quantum noise power (double-sided) spectral densities of the Simple Quantum Meter in the $h$-normalization for different values of measurement strength: $\Omega_q(\mathrm{red})<\Omega_q(\mathrm{green})<\Omega_q(\mathrm{blue})$. Thin black line: SQL. \emph{Left:} free mass. \emph{Right:} harmonic oscillator.}
\label{fig:S_X_SQL}
\end{figure}}
As for the harmonic oscillator, similar formulas can be obtained taking into account that $\chi_{xx}^{\mathrm{osc}}(\Omega) = 1/(M(\Omega_0^2-\Omega^2))$. Thus, one has:
\begin{equation*}
h^{\mathrm{osc}}_{\mathrm{sum}}(\Omega) = -\frac{2(\Omega_0^2-\Omega^2)\hat{x}_{\mathrm{fl}}(\Omega)}{L\Omega^2} - \frac{2\hat{F}_{\mathrm{fl}}}{ML\Omega^2}
\end{equation*}
that results in the following power (double-sided) spectral density formula:
\begin{equation}
\label{eq:S_h_sum_osc}
S^h_{\mathrm{osc}}(\Omega) = \frac{4}{L^2}\left[\left(1-\frac{\Omega_0^2}{\Omega^2}\right)^2S_x+\frac{S_F}{M^2\Omega^4}\right] = \frac{2\hbar}{ML^2\Omega_q^2}\left[\frac{\Omega_q^4}{\Omega^4}\left(1+\frac{(\Omega_0^2-\Omega^2)^2}{\Omega_q^4}\right)\right]
\end{equation}
and results in the following formula for free mass SQL in $h$-normalization:
\begin{equation}\label{eq:S_h__SQL_osc}
S^h_{\mathrm{SQL\,osc}}(\Omega) = \frac{4\hbar|\Omega_0^2-\Omega^2|}{ML^2\Omega^4}\,.
\end{equation}
The corresponding plots are drawn in the right panel of Figure~\ref{fig:S_X_SQL}. Despite a quite different look, in essence, these spectral densities are the same \emph{force} spectral densities as those drawn in Figure~\ref{fig:S_F_SQL}, yet tilted rightwards by virtue of factor $1/\Omega^4$. In particular, they are characterized by the same minimum at the resonance frequency, created by the strong response of the harmonic oscillator on a near-resonance force, as the corresponding force-normalized spectral densities (\ref{S_F_sum_osc}, \ref{eq:S_F_SQL_osc}).
\paragraph*{$x$-normalization:}
Another normalization that is worth considering is the actual probe displacement, or $x$-normalization. In this normalization, the sum noise spectrum is obtained by multiplying noise term $\hat{\mathcal{N}}^F(\Omega)$ in Eq.~\eqref{eq:gen_out_noise_def} by the probe's susceptibility
\begin{equation}
\label{x_sum}
\hat{x}_{\mathrm{sum}}(\Omega) = \hat{x}_{\mathrm{fl}}(\Omega) + \chi_{xx}(\Omega)\hat{F}_{\mathrm{fl}}(\Omega)\,.
\end{equation}
It looks rather natural at a first glance; however, as we have shown below, it is less heuristic than the force normalization and could even be misleading. Nevertheless, for completeness, we consider this normalization here.
Spectral density of $\hat{x}_{\mathrm{sum}}(\Omega)$ and the corresponding SQL are equal to
\begin{eqnarray}
S^x(\Omega) &=& S_x(\Omega) + |\chi_{xx}(\Omega)|^2S_F(\Omega) \,, \\
S^x_{\mathrm{SQL}}(\Omega) &=& \hbar|\chi_{xx}(\Omega)| \,.
\end{eqnarray}
In the free mass case, the formulas are the same as in $h$-normalization except for the multiplication by $4/L^2$:
\begin{equation}
\label{eq:S_x_fm}
S^x_{\mathrm{f.m.}}(\Omega) = \left[S_x+\frac{S_F}{M^2\Omega^4}\right] = \frac{\hbar}{2M\Omega_q^2}\left(1+\frac{\Omega_q^4}{\Omega^2}\right)
\end{equation}
with SQL equal to:
\begin{equation}
\label{eq:SQL_x_fm}
S^x_{\mathrm{SQL\, f.m.}}(\Omega) = \frac{\hbar}{M\Omega^2}\,.
\end{equation}
In the harmonic oscillator case, these equations have the following form:
\begin{eqnarray}
S^x_{\mathrm{osc}}(\Omega) = S_x + \frac{S_F}{M^2(\Omega_0^2-\Omega^2)^2}
&=& \frac{\hbar}{2M\Omega_q^2}\left[\frac{\Omega_q^4}{\Omega^4}\left(1+\frac{(\Omega_0^2-\Omega^2)^2}{\Omega_q^4}\right)\right]
\,, \\
S^x_{\mathrm{SQL\,osc}}(\Omega) &=& \frac{\hbar}{m|\Omega_0^2-\Omega^2|} \,.
\end{eqnarray}
The corresponding plot of the harmonic oscillator power (double-sided) spectral density in
$x$-normalization is given in Figure~\ref{fig:S_x_SQL}.
\epubtkImage{fig24.png}{%
\begin{figure}[htbp]
\centerline{\includegraphics{fig24}}
\caption{Sum quantum noise power (double-sided) spectral densities of Simple Quantum Meter and harmonic oscillator in displacement normalization for different values of measurement strength: $\Omega_q(\mathrm{red})<\Omega_q(\mathrm{green})<\Omega_q(\mathrm{blue})$. Thin black line: SQL.}
\label{fig:S_x_SQL}
\end{figure}}
Note that the curves display a sharp upsurge of noise around the resonance frequencies. However, the resonance growth of the displacement due to signal force $G$ have a long start over this seeming noise outburst, as we have shown already, leads to the substantial sensitivity gain for a near-resonance force. This sensitivity increase is clearly visible in the force and equivalent displacement normalization, see Figures~\ref{fig:S_F_SQL} and \ref{fig:S_X_SQL}, but completely masked in Figure~\ref{fig:S_x_SQL}.
\subsection{Beating the SQL by means of noise cancellation}
\label{sec:toy_FI_correlation}
The SQL is not a fundamental limitation as we have mentioned already, and the clue to how to overcome it can be devised from the expression for the general linear measurement sum noise spectral density~\eqref{eq:gen_spdens}. One can see that a properly constructed cross-correlation between the measurement noise $\hat{\mathcal{X}} = \hat{O}^{(0)}/\chi_{OF}(\Omega)$ and back-action noise $\hat{F}^{(0)}$, i.e., the right choice of $\chi_{FF}(\Omega)$ that should be at any frequency equal to:
\begin{equation}
\label{eq:sub_SQL_corr}
\chi_{FF}(\Omega) = S_{\mathcal{X}F}(\Omega)/S_{\mathcal{X}\mathcal{X}}(\Omega)
\end{equation}
can compensate the back-action term and leave only the measurement noise-related contribution to the final sum quantum noise:
\begin{equation*}
S^F(\Omega) = S_{\mathcal{X}\mathcal{X}}(\Omega)/|\chi_{xx}(\Omega)|^2\,.
\end{equation*}
This spectral density could be arbitrarily small, providing the unbound measurement strength. However, there is a significant obstacle on the way towards back-action free measurement: the optimal correlation should be frequency dependent in the right and rather peculiar way. Another drawback of such back-action evasion via noise cancellation resides in the dissipation that is always present in real measurement setups and, according to Fluctuation-Dissipation Theorem~\cite{PhysRev.83.34, Landau_Lifshitz_v5} is a source of additional noise that undermines any quantum correlations that might be built in the ideal system.
The simplest way is to make the relation~\eqref{eq:sub_SQL_corr} hold at some fixed frequency, which can always be done either (i) by preparing the meter in some special initial quantum state that has measurement and back-action fluctuations correlated (Unruh~\cite{Unruh1982, PhysRevD.19.2888} proposed to prepare input light in a squeezed state to achieve such correlations), or (ii) by monitoring a linear combination of the probe's displacement and momentum~\cite{94a1VyZuMa, 96a2eVyMa, 96a1eVyMa, 94a1VyZu, 95a1VyZu, Phys.Lett.A.300.547_2002_Danilishin, Phys.Lett.A.278.3.123_2000_Danilishin} that can be accomplished, e.g., via homodyne detection, as we demonstrate below.
We consider the basic principles of the schemes, utilizing the noise cancellation via building cross-correlations between the measurement and back-action noise. We start from the very toy example discussed in Section~\ref{sec:linear_toy}.
\epubtkImage{fig25.png}{%
\begin{figure}[htbp]
\centerline{\includegraphics[width=.85\textwidth]{fig25}}
\caption{Toy example of a linear optical position measurement.}
\label{fig:toy_1}
\end{figure}}
The advanced version of that example is shown in Figure~\ref{fig:toy_1}. The only difference between this scheme and the initial one (see Figure~\ref{fig:toy_0}) is that here the detector measures not the phase of light pulses, but linear combination of the phase and energy, parametrized by the \emph{homodyne angle} $\phi_{\mathrm{LO}}$ (cf.\ Eq.~\eqref{eq:homodyne_photocurrent}):
\begin{equation}
\label{i_j}
\hat{O}(t_j) = -\hat{\phi}_j^{\mathrm{refl}}\sin\phi_{\mathrm{LO}}
+ \frac{\hat{\mathcal{W}}_j - \mathcal{W}}{2\mathcal{W}}\cos\phi_{\mathrm{LO}} \,,
\end{equation}
(we subtracted the regular term proportional to the mean energy $\mathcal{W}$ from the output signal $\hat{O}(t_j)$).
Similar to Eq.~\eqref{eq:tilde_x_j}, renormalize this output signal as the equivalent test object displacement:
\begin{equation}
\label{tilde_x_j_corr}
\tilde{x}(t_j) \equiv \frac{\hat{O}_j}{2\digamma k_p\sin\phi_{\mathrm{LO}}}
= \hat{x}(t_j) + \hat{x}_{\mathrm{fl}}(t_j) \,,
\end{equation}
where the noise term has in this case the following form:
\begin{equation}
\label{x_j_corr}
\hat{x}_{\mathrm{fl}}(t_j) = \frac{1}{2\digamma k_p}\biggl(
-\hat{\phi}(t_j) + \frac{\hat{\mathcal{W}}_j - \mathcal{W}}{2\mathcal{W}}\cot\phi_{\mathrm{LO}}
\biggr).
\end{equation}
RMS uncertainty of this value (the measurement error) is equal to
\begin{equation}
\label{Delta_x_corr}
\Delta x_{\mathrm{meas}} = \frac{1}{2\digamma k_p}
\sqrt{(\Delta\phi)^2 + \frac{(\Delta\mathcal{W})^2}{4\mathcal{W}^2}\cot^2\phi_{\mathrm{LO}}} \,.
\end{equation}
At first glance, it seems like we just obtained the increased measurement error, for the same value of the test object perturbation, which is still described by Eq.~\eqref{eq:Delta_p_pert}. However, the additional term in Eq.~\eqref{i_j} can be viewed not only as the additional noise, but as the source of information about the test object perturbation. It can be used to subtract, at least in part, the terms induced by this perturbation from the subsequent measurement results. Quantitatively, this information is characterized by the cross-correlation of the measurement error and the back action:
\begin{equation}
\label{eq:Delta_xp}
\Delta(xp) = \mean{\hat{x}_{\mathrm{fl}}(t_j)\circ\hat{p}^{\mathrm{b.a.}}(t_j)}
= \frac{(\Delta\mathcal{W})^2}{2\omega_p\mathcal{W}}\cot\phi_{\mathrm{LO}} \,.
\end{equation}
It is easy to see that the uncertainties (\ref{eq:Delta_p_pert}, \ref{Delta_x_corr}, \ref{eq:Delta_xp}) satisfy the following Schr\"odinger--Robertson uncertainty relation:
\begin{equation}
\label{eq:DxDp_corr}
(\Delta x_{\mathrm{meas}})^2(\Delta p_{\mathrm{b.a.}})^2 - [\Delta(xp)]^2
= \frac{(\Delta\phi)^2(\Delta\mathcal{W})^2}{\omega_p^2} \ge \frac{\hbar^2}{4} \,.
\end{equation}
Now we can perform the transition to the continuous measurement limit as we did in Section~\ref{sec:disc2cont}:
\begin{eqnarray}
S_x &=& \lim_{\vartheta\to0}(\Delta x_{\mathrm{meas}})^2\vartheta
= \frac{1}{4\digamma^2k_p^2}\left(S_\phi + \frac{S_\mathcal{I}}{4\mathcal{I}^2}\cot^2\phi_{\mathrm{LO}}\right)\,,\nonumber \\
S_F &=& \lim_{\vartheta\to0}\frac{(\Delta p_{\mathrm{b.a.}})^2}{\vartheta}
= \frac{4\digamma^2 S_\mathcal{I}}{c^2}\,, \nonumber\\
S_{xF} &=& \lim_{\vartheta\to0}\Delta(xp) = \frac{S_\mathcal{I}}{2\omega_p\mathcal{I}}\cot\phi_{\mathrm{LO}} \,.\label{eq:S_xS_F_corr_lim},
\end{eqnarray}
which transform inequality~\eqref{eq:DxDp_corr} to the Sch\"odinger--Robertson uncertainty relation for continuous measurements:
\begin{equation}
S_xS_F - S_{xF}^2 = \frac{S_\phi S_\mathcal{I}}{\omega_p^2} \ge \frac{\hbar^2}{4} \,.
\end{equation}
In the particular case of the light pulses in a coherent quantum state~\eqref{DphiDE_coh}, the measurement error~\eqref{Delta_x_corr}, the momentum perturbation~\eqref{eq:Delta_p_pert}, and the cross-correlation term~\eqref{eq:Delta_xp} are equal to:
\begin{equation}
\label{eq:DxDp_corr_coh}
\Delta x_{\mathrm{meas}}
= \frac{c}{4\digamma}\sqrt{\frac{\hbar}{\omega_p\mathcal{W}\sin^2\phi_{\mathrm{LO}}}}\,, \qquad
\Delta p_{\mathrm{b.a.}} = \frac{2\digamma}{c}\sqrt{\hbar\omega_p\mathcal{W}} \,, \qquad
\Delta(xp) = \frac{\hbar}{2}\cot\phi_{\mathrm{LO}}
\end{equation}
(the momentum perturbation $\Delta p_{\mathrm{b.a.}}$ evidently remains the same as in the uncorrelated case, and we provided its value here only for convenience), which gives the exact equality in the Schr\"odinger--Robertson uncertainty relation:
\begin{equation}
(\Delta x_{\mathrm{meas}})^2(\Delta p_{\mathrm{b.a.}})^2 - [\Delta(xp)]^2 = \frac{\hbar^2}{4} \,.
\end{equation}
Correspondingly, substituting the coherent quantum state power (double-sided) spectral densities~\eqref{eq:S_phiS_I_toy} into Eqs.~\eqref{eq:S_xS_F_corr_lim}, we obtain:
\begin{equation}
\label{eq:S_x_S_F_corr}
S_x = \frac{\hbar c^2}{16\omega_p \mathcal{I}_0\digamma^2\sin^2\phi_{\mathrm{LO}}} \,, \qquad
S_F = \frac{4\hbar\omega_p\mathcal{I}_0\digamma^2}{c^2} \qquad,
S_{xF} = \frac{\hbar}{2}\cot\phi_{\mathrm{LO}} \,,
\end{equation}
with
\begin{equation}
S_xS_F - S_{xF}^2 = \frac{\hbar^2}{4}
\end{equation}
[compare with Eqs.~\eqref{S_xS_F_toy}].
The cross-correlation between the measurement and back-action fluctuations is equivalent to the \emph{virtual rigidity} $\chi^{\mathrm{virt}}_{FF}(\Omega)\equiv-K_{\mathrm{virt}}=\mathrm{const.}$ as one can conclude looking at Eqs.~\eqref{eq:gen_noise_def}. Indeed,
\begin{equation*}
\hat{x}_{\mathrm{fl}}(t) = \hat{\mathcal{X}}(t)\,,\qquad\mbox{and}\qquad\hat{F}^{(0)} = \hat{\mathcal{F}}(t)+K_{\mathrm{virt}}\hat{x}_{\mathrm{fl}}(t)\,.
\end{equation*}
The sum noise does not change under the above transformation and can be written as:
\begin{equation*}
\hat{F}_{\mathrm{sum}}(t) = \mathbf{D}\hat{x}_{\mathrm{fl}}(t) + \hat{F}^{(0)}_{\mathrm{fl}}(t)
= \mathbf{D}_{\mathrm{eff}}\hat{x}_{\mathrm{fl}}(t) + \hat{\mathcal{F}}(t) \,,
\end{equation*}
where the new effective dynamics that correspond to the new noise are governed by the following differential operator
\begin{equation}
\label{D_new}
\mathbf{D}_{\mathrm{eff}} = \mathbf{D} + K_{\mathrm{virt}} \,.
\end{equation}
The above explains why we refer to $K_{\mathrm{virt}}$ as `virtual rigidity'.
To see how virtual rigidity created by cross-correlation of noise sources can help beat the SQL consider a free mass as a probe body in the above considered toy example. The modified dynamics:
\begin{equation}
\mathbf{D}_{\mathrm{eff}} = M\frac{d^2}{dt^2} + K_{\mathrm{virt}} \,, \qquad S_{xF} = K_{\mathrm{virt}} S_x \ne 0 \,,
\end{equation}
correspond to a harmonic oscillator with eigenfrequency $\Omega_0^2=K_{\mathrm{virt}}/M$, and as we have demonstrated in Eq.~\eqref{eq:osc_fm_SQL} provide a narrow-band sensitivity gain versus a free mass SQL near the resonance frequency $\Omega_0$ .
However, there is a drawback of virtual rigidity compared to the real one: it requires higher measurement strength, and therefore higher power, to reach the same gain in sensitivity as provided by a harmonic oscillator. This becomes evident if one weighs the back-action spectral density $S_F$, which is a good measure of measurement strength according to Eqs.~\eqref{eq:Omega_q}, for the virtual rigidity against the real one.
For the latter, to overcome the free mass SQL by a factor
\begin{equation}
\label{xi2_nb}
\xi^2 = \frac{\Delta\Omega}{\Omega_0}
\end{equation}
(see Eq.~\eqref{eq:osc_fm_SQL}) at a given frequency $\Omega_0$, the back-action noise spectral density has to be \emph{reduced} by this factor:
\begin{equation}
S_F = \frac{\hbar M \Omega_q^2}{2} = \xi^2S_F^{\mathrm{opt\,f.m.}} \,,
\end{equation}
see Eqs.~(\ref{eq:Omega_q}, \ref{eq:SQL_fm_cond}, \ref{eq:Omega_q_osc}). Here, $S_F^{\mathrm{opt\,f.m.}} = \hbar M \Omega_0/2$ is the back-action noise spectral density, which allows one to reach the free mass SQL~\eqref{eq:S_F_SQL_fm} at frequency $\Omega_0$. Such a sensitivity gain is achieved at the expense of proportionally reduced bandwidth:
\begin{equation}
\Delta\Omega = \xi^2\Omega_0 \,.
\end{equation}
For the virtual rigidity, the optimal value of $S_F$ results from Eq.~\eqref{eq:Omega_q_osc}:
\begin{equation}
S_F = \frac{\hbar^2/4 + S_{xF}^2}{S_x}
= \frac{\hbar M}{2}\left(\Omega_q^2 + \frac{\Omega_0^4}{\Omega_q^2}\right)
= S_F^{\mathrm{opt\,f.m.}}\left(\xi^2 + \frac{1}{\xi^2}\right) .
\end{equation}
Hence, the better the sensitivity (the smaller $\xi^2$), the larger $S_F$ must be and, therefore, measurement strength.
Another evident flaw of the virtual rigidity, which it shares with the real one, is the narrow-band character of the sensitivity gain it provides around $\Omega_0$ and that this band shrinks as the sensitivity gain rises (cf.\ Eq.~\eqref{xi2_nb}). In order to provide a broadband enhancement in sensitivity, either the real rigidity $K=M\Omega_0^2$, or the virtual one $K_{\mathrm{virt}}=S_{xF}/S_F$ should depend on frequency in such a way as to be proportional (if only approximately) to $\Omega^2$ in the frequency band of interest. Of all the proposed solutions providing frequency dependent virtual rigidity, the most well known are the \emph{quantum speedmeter}~\cite{90a1BrKh} and the \emph{filter cavities}~\cite{02a1KiLeMaThVy} schemes. Section~\ref{sec:toy_speedmeter}, we consider the basic principles of the former scheme. Then, in Section~\ref{sec:sub-SQL_schemes} we provide a detailed treatment of both of them.
\subsection{Quantum speedmeter}
\label{sec:toy_speedmeter}
\subsubsection{The idea of the quantum speedmeter}
The toy scheme that demonstrates a bare idea of the quantum speedmeter is shown in Figure~\ref{fig:speedmeter}. The main difference of this scheme from the position meters considered above (see Figures~\ref{fig:toy_0}, \ref{fig:toy_1}) is that each light pulse reflects from the test mass twice: first from the front and then from the rear face after passing the delay line with delay time $\tau$. An outgoing pulse acquires a phase shift proportional to the difference of the test-object positions at time moments separated by $\tau$, which is proportional to the test-mass average velocity $\hat{\bar{v}}(t_j)$ in this time interval ($t_j$ indicates the time moment after the second reflection):
\begin{equation}
\label{phi_sm_refl}
\hat{\phi}^{\mathrm{refl}}(t_j) = \hat{\phi}(t_j) + 2\digamma k_p\tau\hat{\bar{v}}(t_j) \,,
\end{equation}
where
\begin{equation}
\label{V_j}
\hat{\bar{v}}(t_j) = \frac{\hat{x}(t_j)-\hat{x}(t_j-\tau)}{\tau} \,.
\end{equation}
\epubtkImage{fig26.png}{
\begin{figure}
\centerline{\includegraphics[width=\textwidth]{fig26}}
\caption{Toy example of the quantum speedmeter scheme.}
\label{fig:speedmeter}
\end{figure}}
We omit here mathematical details of the transition to the continuous measurement limit as they are essentially the same as in the position measurement case, see Section~\ref{sec:disc2cont}, and start directly with the continuous time equations. The output signal of the homodyne detector in the speedmeter case is described by the following equations:
\begin{eqnarray}
\hat{O}(t) &=& -\hat{\phi}^{\mathrm{refl}}(t)\sin\phi_{\mathrm{LO}} + \frac{\hat{\mathcal{I}}(t) - \mathcal{I}_0}{2\mathcal{I}_0}\,\cos\phi_{\mathrm{LO}}
\,,\\
\hat{\phi}^{\mathrm{refl}}(t)
&=& \hat{\phi}(t) + 2\digamma k_p[\hat{x}(t)-\hat{x}(t-\tau)] \,.
\end{eqnarray}
In spectral representation these equations yield:
\begin{equation}
\tilde x(\Omega) \equiv -\frac{\hat{O}(\Omega)}{2\digamma k_p\sin\phi_{\mathrm{LO}}}
= \hat{x}(\Omega) + \hat{x}_{\mathrm{fl}}(\Omega) \,,
\end{equation}
where
\begin{equation}
\label{eq:sm_x_fl_raw}
\hat{x}_{\mathrm{fl}}(\Omega) = \frac{1}{2\digamma k_p(1-e^{i\Omega\tau})}
\left[\hat{\phi}(\Omega) - \frac{\hat{\mathcal{I}}(\Omega) - \mathcal{I}_0}{2\mathcal{I}_0}\,\cot\phi_{\mathrm{LO}}\right]
\end{equation}
is the equivalent displacement measurement noise.
The back-action force with account for the two subsequent light reflections off the faces of the probe, can be written as:
\begin{equation}
\label{sm_F_pert_raw}
\hat{F}_{\mathrm{b.a.}}(t) = \frac{2\digamma}{c}[\hat{\mathcal{I}}(t+\tau)-\hat{\mathcal{I}}(t)] \,
\end{equation}
and in spectral form as:
\begin{equation}
\hat{F}_{\mathrm{b.a.}}(\Omega)
= \frac{2\digamma}{c}\hat{\mathcal{I}}(\Omega)(e^{-i\Omega\tau}-1) .
\end{equation}
Then one can make a reasonable assumption that the time between the two reflections $\tau$ is much smaller than the signal force variation characteristic time ($\sim\Omega^{-1}$) that spills over into the following condition:
\begin{equation}
\Omega\tau \ll 1 \,,
\end{equation}
and allows one to expand the exponents in Eqs.~(\ref{eq:sm_x_fl_raw}, \ref{sm_F_pert_raw}) into a Taylor series:
\begin{equation}
\hat{x}_{\mathrm{fl}}(\Omega) = \frac{\hat{v}_{\mathrm{fl}}(\Omega)}{-i\Omega} \,, \qquad
\hat{F}_{\mathrm{b.a.}}(t) = -i\Omega\hat{p}_{\mathrm{b.a.}}(\Omega) \,,
\end{equation}
where
\begin{eqnarray}
\hat{v}_{\mathrm{fl}}(\Omega) &=& \frac{1}{2\digamma k_p\tau}
\left[\hat{\phi}(\Omega) - \frac{\hat{\mathcal{I}}(\Omega) - \mathcal{I}}{2\mathcal{I}}\,\cot\phi_{\mathrm{LO}}\right] , \\
\hat{p}_{\mathrm{b.a.}}(\Omega) &=& \frac{\hat{F}_{\mathrm{b.a.}}(\Omega)}{-i\Omega}= \frac{2\digamma\tau\hat{\mathcal{I}}(\Omega)}{c} \,.
\end{eqnarray}
Spectral densities of theses noises are equal to
\begin{equation}
\label{eq:sm_S_xS_F}
S_x(\Omega) = \frac{S_v}{\Omega^2} \,, \qquad S_F(\Omega) = \Omega^2 S_p(\Omega) \,, \qquad
S_{xF}(\Omega) = -S_{vp}(\Omega) \,,
\end{equation}
where
\begin{eqnarray}
\label{eq:SM_toy_spdens}
S_v &=& \frac{1}{4\digamma^2k_p^2\tau^2}
\left(S_\phi + \frac{S_\mathcal{I}}{4\mathcal{I}^2}\cot^2\phi_{\mathrm{LO}}\right)\,,\nonumber \\
S_p &=& \frac{4\digamma^2\tau^2S_\mathcal{I}}{c^2}\,, \nonumber\\
S_{vp} &=& -\frac{S_\mathcal{I}}{2\omega_p\mathcal{I}}\cot\phi_{\mathrm{LO}}
\end{eqnarray}
Note also that
\begin{equation}
\label{eq:SvSp-Svp2_uncert_rel}
S_x(\Omega)S_F(\Omega) - S_{xF}^2(\Omega) = S_vS_p-S_{vp}^2
= \frac{S_\phi S_\mathcal{I}}{\omega_p^2} \ge \frac{\hbar^2}{4} \,.
\end{equation}
The apparent difference of the spectral densities presented in
Eq.~\eqref{eq:sm_S_xS_F} from the ones describing the `ordinary'
position meter (see Eqs.~\eqref{eq:S_xS_F_corr_lim}) is that they now
have rather special frequency dependence. It is this frequency
dependence that together with the cross-correlation of the measurement
and back-action fluctuations, $S_{xF}\ne0$, allows the reduction of
the sum noise spectral density to arbitrarily small values. One can
easily see it after the substitution of Eq.~\eqref{eq:sm_S_xS_F} into
Eq.~\eqref{eq:gen_spdens} with a free mass
$\chi_{xx}(\Omega)=-1/(M\Omega^2)$ in mind:
\begin{equation}
\label{eq:SM_F_sum_spdens}
S^F = M^2\Omega^4S_x(\Omega) + S_F(\Omega) - 2M\Omega^2S_{xF}(\Omega) = \Omega^2(M^2 S_v + 2MS_{vp} + S_p) \,.
\end{equation}
If there was no correlation between the back-action and measurement fluctuations, i.e., $S_{vp} = 0$, then by virtue of the uncertainty relation, the sum sensitivity appeared limited by the SQL~\eqref{eq:S_F_SQL_fm}:
\begin{equation}
\label{eq:SM_F_sum_spdens_uncorr}
S^F = \Omega^2\left(\frac{\hbar^2M^2}{4S_p} + S_p\right)\geqslant \hbar M\Omega^2\,.
\end{equation}
One might wonder, what is the reason to implement such a complicated measurement strategy to find ourselves at the same point as in the case of a simple coordinate measurement? However, recall that in the position measurement case, a constant cross-correlation $S_{xF}\propto\cot\phi_{\mathrm{LO}}$ allows one to get only a narrow-band sub-SQL sensitivity akin to that of a harmonic oscillator. This effect we called \emph{virtual rigidity}, and showed that for position measurement this rigidity $K_{\mathrm{virt}} = S_{xF}/S_x$ is constant. In the speedmeter case, the situation is totally different; it is clearly seen if one calculates virtual rigidity for a speedmeter:
\begin{equation}
K_{\mathrm{virt}}^{\mathrm{SM}} = \frac{S_{xF}}{S_x(\Omega)} = -\Omega^2\frac{S_{vp}}{S_v}\,.
\end{equation}
It turns out to be frequency dependent exactly in the way that is necessary to compensate the free mass dynamical response to the back-action fluctuations. Indeed, in order to minimize the sum noise spectral density~\eqref{eq:SM_F_sum_spdens} conditioned on uncertainty relation~\eqref{eq:SvSp-Svp2_uncert_rel}, one needs to set
\begin{equation}
S_{xF} = -S_{vp} = \frac{S_{F}}{M\Omega^2} = \frac{S_{p}}{M} = \mathrm{ const.},
\end{equation}
which allows one to overcome the SQL simply by choosing the right fixed homodyne angle:
\begin{equation}
\cot\phi_{\mathrm{LO}} = \frac{8\digamma^2\tau^2\omega_p\mathcal{I}_0}{Mc^2} \,.
\end{equation}
Then the sum noise is equal to
\begin{equation}
\label{cont_opt}
S^F(\Omega) = \frac{S_\phi S_\mathcal{I}}{\omega_p^2}\,\frac{M^2\Omega^2}{S_p}
= \frac{M^2\Omega^2}{4\digamma^2k_p^2\tau^2}S_\phi \,.
\end{equation}
and, in principle, can be made arbitrarily small, if a sufficient value of $S_p$ is provided; that is, ifthe optomechanical coupling is sufficiently strong.
\paragraph*{Simple case: light in a coherent state.}
Let us consider how the spectral density of a speedmeter will appear if the light field is in a coherent state. The spectral densities of phase and power fluctuations are given in Eqs.~\eqref{eq:S_phiS_I_toy}, hence the sum noise power (double-sided) spectral density for the speedmeter takes the following form:
\begin{equation}
S^F(\Omega) = \frac{\hbar M^2c^2\Omega^2}{16\omega_p\mathcal{I}_0\digamma^2\tau^2} = \frac{\hbar M}{2\tau^2}\left(\frac{\Omega}{\Omega_q}\right)^2 = \frac{S^F_{\mathrm{SQL\,f.m.}}}{2\Omega_q^2\tau^2}\,,
\end{equation}
where $\Omega_q$ for our scheme is defined in Eq.~\eqref{eq:Omega_q_toy}. This formula indicates the ability of a speedmeter to have a sub-SQL sensitivity in all frequencies provided high enough optical power and no optical loss.
\subsubsection{QND measurement of a free mass velocity}
The initial motivation to consider speed measurement rested on the assumption that a velocity $\hat{v}$ of a free mass is directly proportional to its momentum $\hat{p}=M\hat{v}$. And the momentum in turn is, as an integral of motion, a QND-observable, i.e., it satisfies the simultaneous measurability condition~\eqref{eq:measurability_cond}:
\begin{equation*}
\left[\hat{p}(t),\,\hat{p}(t')\right] = 0\quad \forall t,\,t'.
\end{equation*}
But this connection between $\hat{p}$ and $\hat{v}$ holds only if one considers an isolated free mass not coupled to a meter. As the measurement starts, the velocity value gets perturbed by the meter and it is not proportional to the momentum anymore. Let us illustrate this statement by our simple velocity measurement scheme. The distinctive feature of this example is that the meter probes the test position $\hat{x}$ twice, with opposite signs of the coupling factor. Therefore, the Lagrangian of this scheme can be written as:
\begin{equation}
\label{sm_L_orig}
\hat{\mathcal{L}} = \frac{M\hat{v}^2}{2} + \beta(t)\hat{x}\hat{\mathcal{N}} + \hat{\mathcal{L}}_{\mathrm{meter}} \,,
\end{equation}
where
\begin{equation}
\label{sm_v}
\hat{v} = \frac{d\hat{x}}{dt}
\end{equation}
is the test-mass velocity, $\mathcal{N}$ stands for the meter's observable, which provides coupling to the test mass, $\mathcal{L}_{\mathrm{meter}}$ is the self-Lagrangian of the meter, and $\beta(t)$ is the coupling factor, which has the form of two short pulses with the opposite signs, separated by the time $\tau$, see Figure~\ref{fig:sm_alpha_beta}. We suppose for simplicity that the evolution of the meter observable $\mathcal{N}$ can be neglected during the measurement (this is a reasonable assumption, for in real schemes of the speedmeter and in the gedanken experiment considered above, this observable is proportional to the number of optical quanta, which does not change during the measurement). This assumption allows one to omit the term $\mathcal{L}_{\mathrm{meter}}$ from consideration.
\epubtkImage{fig27.png}{
\begin{figure}
\centerline{\includegraphics{fig27}}
\caption{Real ($\beta(t)$) and effective ($\alpha(t)$) coupling constants in the speedmeter scheme.}
\label{fig:sm_alpha_beta}
\end{figure}}
This Lagrangian does not satisfy the most well-known sufficient (but not necessary!) condition of the QND measurement, namely the commutator of the interaction Hamiltonian $\hat{\mathcal{H}}_{\mathrm{int}}=-\beta(t)\hat{x}\hat{\mathcal{N}}$ with the operator of measured observable $\hat{x}$ does not vanish~\cite{77a1eBrKhVo}. However, it can be shown that a more general condition is satisfied:
\begin{equation*}
[\hat{U}(0,\tau),\hat{x}]=0
\end{equation*}
where $\hat{U}(0,\tau)$ is the evolution operator of probe-meter dynamics from the initial moment $t_{\mathrm{start}}=0$ when the measurement starts till the final moment $t_{\mathrm{end}}=\tau$ when it ends. Basically, the latter condition guarantees that the value of $\hat{x}$ before the measurement will be equal to that after the measurement, but does not say what it should be in between (see Section~4.4 of~\cite{92BookBrKh} for details).
Moreover, using the following nice trick~\epubtkFootnote{Personal communication with Yanbei Chen.}, the Lagrangian~\eqref{sm_L_orig} can be converted to the form, satisfying the simple condition of~\cite{77a1eBrKhVo}:
\begin{equation}
\hat{\mathcal{L}}' = \hat{\mathcal{L}} - \frac{d\alpha(t)\hat{x}}{dt}\hat{\mathcal{N}} \,,
\end{equation}
where
\begin{equation}
\alpha(t) = \int_{-\infty}^t\beta(t')\,dt' \,,
\end{equation}
see Figure~\ref{fig:sm_alpha_beta}.
It is known that two Lagrangians are equivalent if they differ only by a full time derivative of and arbitrary function of the generalized coordinates. Lagrangian equations of motion for the coordinates of the system are invariant to such a transformation.
The new Lagrangian has the required form with the interaction term proportional to the test-mass velocity:
\begin{equation}
\hat{\mathcal{L}} = \frac{M\hat{v}^2}{2} - \alpha(t)\hat{v}\hat{\mathcal{N}} \,.
\end{equation}
Note that the antisymmetric shape of the function $\beta(t)$ guarantees that the coupling factor $\alpha(t)$ becomes equal to zero when the measurement ends. The canonical momentum of the mass $M$ is equal to
\begin{equation}
p = \partdiff{\mathcal{L}}{v} = Mv - \alpha(t)\mathcal{N} \,,
\end{equation}
and the Hamiltonian of the system reads
\begin{equation}
\label{sm_gen_H}
\hat{\mathcal{H}} = pv - \mathcal{L} = \frac{[p+\alpha(t)\mathcal{N}]^2}{2M}\,.
\end{equation}
The complete set of observables describing our system includes in addition apart to $\hat{x}$, $\hat{p}$, and $\hat{\mathcal{N}}$, the observable $\hat{\Phi}$ canonically conjugated to $\hat{\mathcal{N}}$:
\begin{equation}
\label{sm_gen_Phi_N}
[\hat{\Phi},\hat{\mathcal{N}}] = i\hbar
\end{equation}
(if $\hat{\mathcal{N}}$ is proportional to the number of quanta in the light pulse then $\hat{\Phi}$ is proportional to its phase), which represents the output signal $\hat{O}$ of the meter. The Heisenberg equations of motion for these observables are the following:
\begin{equation}
\begin{array}{ll}
\label{eq:sm_gen_eqs}
\displaystyle
\frac{d\hat{x}(t)}{dt} = \hat{v}(t) = \frac{\hat{p} + \alpha(t)\hat{\mathcal{N}}}{M} \,, & \qquad
\displaystyle
\frac{d\hat{p}(t)}{dt} = 0 \,,\\[1em]
\displaystyle
\frac{d\hat\Phi(t)}{dt} = \frac{\alpha(t)[\hat{p} + \alpha(t)\hat{\mathcal{N}}]}{M}
= \alpha(t)\hat{v}(t) \,, & \qquad
\displaystyle
\frac{d\hat{\mathcal{N}}(t)}{dt} = 0 \,.
\end{array}
\end{equation}
These equations show clearly that (i) a canonical momentum $\hat{p}$ is preserved by this measurement scheme while (ii) the test-mass velocity $\hat{v}$, as well as its kinematic momentum $M\hat{v}$, are perturbed during the measurement (that is, while $\alpha\ne0$), yet restore their initial values after the measurement, and (iii) the output signal of the meter $\hat{\Phi}$ carries the information about this perturbed value of the velocity.
Specify $\alpha(t)$ to be of a simple rectangular shape:
\begin{equation*}
\alpha(t) = \begin{cases}
1 \,, & \mbox{if}\ 0\le t<\tau \\
0 \,, & \mbox{if}\ t<0\ \mbox{or}\ t\ge\tau \,.
\end{cases}
\end{equation*}
This assumption does not affect our main results, but simplifies the calculation. In this case, the solution of Eqs.~\eqref{eq:sm_gen_eqs} reads:
\begin{subequations}
\begin{eqnarray}
\hat{x}(\tau) &=& \hat{x}(0) + \hat{\bar{v}}\tau \,, \\
\hat{\Phi}(\tau) &=& \hat{\Phi}(0) + \hat{\bar{v}}\tau \,,
\end{eqnarray}
\end{subequations}
with
\begin{equation}
\label{sm_gen_bar_v}
\hat{\bar{v}} = \frac{\hat{p}+\hat{\mathcal{N}}}{M}
\end{equation}
the test-mass velocity during the measurement [compare with Eqs.~(\ref{phi_sm_refl}, \ref{V_j})].
Therefore, by detecting the variable $\Phi(\tau)$, the \emph{perturbed} value of velocity $\bar v$ is measured with an imprecision
\begin{equation}
\label{sm_gen_v_meas}
\Delta v_{\mathrm{meas}} = \frac{\Delta\Phi}{\tau} \,,
\end{equation}
where $\Delta\Phi$ is the initial uncertainty of $\Phi$. The test-mass position perturbation after this measurement is proportional to the initial uncertainty of $\mathcal{N}$:
\begin{equation}
\label{sm_gen_pert}
\Delta x_{\mathrm{b.a.}} = \Delta v_{\mathrm{b.a.}}\tau = \frac{\Delta\mathcal{N}\tau}{M} \,.
\end{equation}
It follows from the last two equations that
\begin{equation}
\Delta v_{\mathrm{meas}}\Delta x_{\mathrm{b.a.}}
= \frac{\Delta\Phi\Delta\mathcal{N}}{M} \ge \frac{\hbar}{2M} \,.
\end{equation}
The sum error of the initial velocity estimate $v_{\mathrm{init}} = p/m$ yielding from this measurement is thus equal to:
\begin{equation}
\Delta v_{\mathrm{sum}}
= \sqrt{\left(\frac{\Delta\Phi}{\tau}\right)^2 + \left(\frac{\Delta\mathcal{N}}{M}\right)^2}
\ge \sqrt{\frac{\hbar}{M\tau}} \,.
\end{equation}
As we see, it is limited by a value of the \emph{velocity measurement} SQL:
\begin{equation}
\label{eq:v_SQL}
\Delta v_{\mathrm{SQL}} = \sqrt{\frac{\hbar}{M\tau}}\,.
\end{equation}
To overcome this SQL one has to use cross-correlation between the measurement error and back-action. Then it becomes possible to measure $v_{\mathrm{init}}$ with arbitrarily high precision. Such a cross-correlation can be achieved by measuring the following combination of the meter observables
\begin{equation}
\hat{\Phi}(\tau)-\hat{\mathcal{N}}\tau/m = \hat{\Phi} (0) + \frac{\hat{p}\tau}{M}
\end{equation}
instead of $\hat{\Phi}(\tau)$, which gives a sum measurement uncertainty for the initial velocity $\hat{p}/m$, proportional to the initial uncertainty of $\hat{\Phi}$ only:
\begin{equation}
\Delta v_{\mathrm{sum}} = \frac{\Delta\Phi}{\tau} \,,
\end{equation}
and hence not limited by the SQL.
\newpage
\section{Quantum Noise in Conventional GW Interferometers}
\label{sec:QN_in_GW_interferometers}
FY{In Section~\ref{sec:linear_quantum_measurement}}, we have talked about the quantum measurement, the general structure of quantum noise implied by the quantum mechanics and the restrictions on the achievable sensitivity it imposes. In this section, we turn to the application of these general and lofty principles to real life, i.e., we are going to calculate quantum noise for several types of the schemes of GW interferometers and consider the advantages and drawbacks they possess.
To grasp the main features of quantum noise in an advanced GW interferometer it would be elucidating to consider first two elementary examples: (i) a single movable mirror coupled to a free optical field, reflecting from it, and (ii) a Fabry--P{\'e}rot cavity comprising two movable mirrors and pumped from both sides. These two systems embody all the main features and phenomena that also mold the advanced and more complicated interferometers' quantum noise. Should one encounter these phenomena in real-life GW detectors, knowledge of how they manifest themselves in these simple situations would be of much help in successfully discerning them.
\subsection{Movable mirror}
The scheme of the mirror is drawn in
Figure~\ref{fig:Single_Mirror}. It is illuminated from both sides by
the two independent laser sources with frequency $\omega_p$, and mean
power values $\mathcal{I}_1$ and $\mathcal{I}_2$. In
terms of the general linear measurement theory of
Section~\ref{sec:gen_linear_measurement} we have two meters
represented by these two incident light waves. The two arbitrary
quadratures of the reflected waves are deemed as measured quantities
$\hat{O}_1$ and $\hat{O}_2$. Measurement can be performed, e.g., by
means of two independent homodyne detectors. Let us analyze quantum
noise in such a model keeping to the scheme given by
Eqs.~\eqref{eq:gen_linear_meas_EOM_Fourier}.
\epubtkImage{fig28.png}{
\begin{figure}[htbp]
\centerline{\includegraphics[width=.3\textwidth]{fig28}}
\caption{Scheme of light reflection off the single movable mirror of mass $M$ pulled by an external force $G$.}
\label{fig:Single_Mirror}
\end{figure}}
\subsubsection{Optical transfer matrix of the movable mirror}
The first one of Eqs.~\eqref{eq:gen_linear_meas_EOM_Fourier} in our simple scheme is represented by the input-output coupling equations (\ref{eq:IO_mirror_matrix_4x4_zero}, \ref{eq:IO_mirror_relations_first+x}) of light on a movable mirror derived in Section~\ref{sec:light_reflection}. We choose the transfer matrix of the mirror to be real:
\begin{equation}
\label{eq:mirror_Mreal}
\mathbb{M}_{\mathrm{real}} =
\begin{bmatrix}
-\sqrt{R} & \sqrt{T}\\
\sqrt{T} & \sqrt{R}
\end{bmatrix}
\end{equation}
according to Eq.~\eqref{eq:IO_mirror_matrix}. Then we can write down the coupling equations, substituting electric field strength amplitudes $\vb{\mathcal{E}}_{1,2}$ and $\hat{\boldsymbol{e}}_{1,2}(\Omega)$ by their dimensionless counterparts as introduced by Eq.~\eqref{eq:2photon_E_strain} of Section~\ref{sec:2photon_formalism}:
\begin{equation}
\label{eq:IO_mirror_relations+x_quant}
\begin{bmatrix}
\hat{\mathbf{B}}_{1}(\Omega)\\
\hat{\mathbf{B}}_{2}(\Omega)
\end{bmatrix}=
\mathbb{M}_{\mathrm{real}}
\cdot
\begin{bmatrix}
\hat{\mathbf{A}}_{1}(\Omega)\\
\hat{\mathbf{A}}_{2}(\Omega)
\end{bmatrix}\,,\qquad\mbox{and}\qquad
\begin{bmatrix}
\hat{\boldsymbol{b}}_{1}(\Omega)\\
\hat{\boldsymbol{b}}_{2}(\Omega)
\end{bmatrix}=
\mathbb{M}_{\mathrm{real}}
\cdot
\begin{bmatrix}
\hat{\boldsymbol{a}}_{1}(\Omega)\\
\hat{\boldsymbol{a}}_{2}(\Omega)
\end{bmatrix} +
\begin{bmatrix}
\boldsymbol{R}_1\\
\boldsymbol{R}_2
\end{bmatrix}\hat{x}(\Omega)\,,
\end{equation}
where
\begin{equation}
\boldsymbol{R}_1 = 2\sqrt{R}\frac{\omega_p}{c}
\begin{bmatrix}
\mathrm{A}_{1s}\\
-\mathrm{A}_{1c}
\end{bmatrix}\,,\qquad\mbox{and}\qquad
\boldsymbol{R}_2 = 2\sqrt{R}\frac{\omega_p}{c}
\begin{bmatrix}
\mathrm{A}_{2s}\\
-\mathrm{A}_{2c}
\end{bmatrix}\,.
\end{equation}
Without any loss of generality one can choose the phase of the light incident from the left to be such that $\mathrm{A}_{1s} = 0$ and $\mathrm{A}_{1c} = \sqrt{2\mathcal{I}_1/(\hbar\omega_p)}$. Then factoring in the constant phase difference between the left and the right beams equal to $\Phi_0$, one would obtain for the left light $\{A_{2c},A_{2s}\} = \sqrt{2\mathcal{I}_2/(\hbar\omega_p)}\{\cos\Phi_0,\sin\Phi_0\}$.
The two output measured quantities will then be given by the two homodyne photocurrents:
\begin{eqnarray}
\label{eq:mirror_readout_quant}
\hat{O}_1(\Omega) &=& \boldsymbol{H}^{\mathsf{T}}[\phi_1]\hat{\boldsymbol{b}}_1 = \hat{b}_{1c}(\Omega)\cos\phi_1 + \hat{b}_{1s}(\Omega)\sin\phi_1 \,,\\
\hat{O}_2(\Omega) &=& \boldsymbol{H}^{\mathsf{T}}[\phi_2]\hat{\boldsymbol{b}}_2 = \hat{b}_{2c}(\Omega)\cos\phi_2 + \hat{b}_{2s}(\Omega)\sin\phi_2\,,
\end{eqnarray}
where vector $\boldsymbol{H}[\phi]$ was first introduced in Section~\ref{sec:heterodyne} after Eq.~\eqref{eq:heterodyne_photocurrent} as:
\begin{equation}
\boldsymbol{H}[\phi] \equiv \begin{bmatrix}
\cos\phi\\
\sin\phi
\end{bmatrix}\,.
\end{equation}
Then, after substitution of~\eqref{eq:mirror_Mreal} into~\eqref{eq:IO_mirror_relations+x_quant}, and then into~\eqref{eq:mirror_readout_quant}, one gets
\begin{eqnarray}
\hat{O}^{(0)}_1(\Omega) &=& \boldsymbol{H}^{\mathsf{T}}[\phi_1](-\sqrt{R}\hat{\boldsymbol{a}_1}+\sqrt{T}\hat{\boldsymbol{a}_2})\,,\\
\hat{O}^{(0)}_2(\Omega) &=& \boldsymbol{H}^{\mathsf{T}}[\phi_2](\sqrt{T}\hat{\boldsymbol{a}_1}+\sqrt{R}\hat{\boldsymbol{a}_2})\,,
\end{eqnarray}
and
\begin{eqnarray}
\chi_{O_1F}(\Omega) &=& \boldsymbol{H}^{\mathsf{T}}[\phi_1]\boldsymbol{R}_1 = -\frac{2\sqrt{2R\hbar\omega_p\mathcal{I}_1}}{\hbar c}\sin\phi_1\,,\\
\chi_{O_2F}(\Omega) &=& \boldsymbol{H}^{\mathsf{T}}[\phi_2]\boldsymbol{R}_2 = -\frac{2\sqrt{2R\hbar\omega_p\mathcal{I}_2}}{\hbar c}\sin(\phi_2-\Phi_0)\,.
\end{eqnarray}
\subsubsection{Probe's dynamics: radiation pressure force and ponderomotive rigidity}
Now we can write down the equation of motion for the mirror assuming it is pulled by a GW tidal force $G$:
\begin{equation}
\label{eq:Mirror_EOMs}
m\ddot{\hat{x}}(t) = \hat{F}_{\mathrm{r.p.}}(t) + G(t)\qquad\Longrightarrow\qquad -M\Omega^2\hat{x}(\Omega) = \hat{F}_{\mathrm{r.p.}}(\Omega) + G(\Omega)\,,
\end{equation}
that gives us the probe's dynamics equation (the third one) in~\eqref{eq:gen_linear_meas_EOM_Fourier}:
\begin{equation}
\hat{x}(\Omega) = \chi_{xx}(\Omega)[\hat{F}_{\mathrm{b.a.}}(\Omega) + G(\Omega)] = -\frac{1}{M\Omega^2}[\hat{F}_{\mathrm{r.p.}}(\Omega) + G(\Omega)]\,,
\end{equation}
where the free mirror mechanical susceptibility $\chi_{xx}(\Omega) = -1/(M\Omega^2)$.
The term $\hat{F}_{\mathrm{b.a.}}(t)$ stands for the radiation pressure force imposed by the light that can be calculated as
\begin{eqnarray}
&&\hat{F}_{\mathrm{r.p.}}(\Omega) = F_0 + \hat{F}_{\mathrm{b.a.}}(\Omega) = \frac{\hat{\mathcal{I}}_{a_1}(\Omega)+\hat{\mathcal{I}}_{b_1}(\Omega)-\hat{\mathcal{I}}_{a_2}(\Omega)-\hat{\mathcal{I}}_{b_2}(\Omega)}{c} \simeq\nonumber\\ &&\frac{\hbar k_p}{2}\left[({\mathbf{A}}_1^{\mathsf{T}}{\mathbf{A}}_1+{\mathbf{B}}_1^{\mathsf{T}}{\mathbf{B}}_1-{\mathbf{A}}_2^{\mathsf{T}}{\mathbf{A}}_2-{\mathbf{B}}_2^{\mathsf{T}}{\mathbf{B}}_2)+ 2({\mathbf{A}}_1^{\mathsf{T}}\hat{\boldsymbol{a}}_1(t)+{\mathbf{B}}_1^{\mathsf{T}}\hat{\boldsymbol{b}}_1(t)-{\mathbf{A}}_2^{\mathsf{T}}\hat{\boldsymbol{a}}_2(t)-{\mathbf{B}}_2^{\mathsf{T}}\hat{\boldsymbol{b}}_2(t))\right]
\end{eqnarray}
where $k_p\equiv\omega_p/c$
\begin{equation}
F_0 = \frac{\hbar k_p}{2}({\mathbf{A}}_1^{\mathsf{T}}{\mathbf{A}}_1+{\mathbf{B}}_1^{\mathsf{T}}{\mathbf{B}}_1-{\mathbf{A}}_2^{\mathsf{T}}{\mathbf{A}}_2-{\mathbf{B}}_2^{\mathsf{T}}{\mathbf{B}}_2) = \frac{2 R}{c}(\mathcal{I}_1-\mathcal{I}_2)-\frac{4\sqrt{RT}}{c}\sqrt{\mathcal{I}_1\mathcal{I}_2}\cos\Phi_0\,,
\end{equation}
is the regular part of the radiation pressure force\epubtkFootnote{Note the second term proportional to $\sqrt{\mathcal{I}_1\mathcal{I}_2}\cos\Phi_0$, which owes its existence to the interference of the two traveling waves running in opposite directions. An interesting consequence of this is that the radiation pressure does not vanish even if the two waves have equal powers, i.e., $\mathcal{I}_1=\mathcal{I}_2$, that is, in order to compensate for the radiation pressure force of one field on the semi-transparent mirror, the other one should not only have the right intensity but also the right phase with respect to the former one:
$$\cos\Phi_0 = \frac{1}{2}\sqrt{\frac{R}{T}}\frac{\mathcal{I}_1-\mathcal{I}_2}{\sqrt{\mathcal{I}_1\mathcal{I}_2}}\,.$$
}. It is constant and thus can be compensated by applying a fixed restoring force of the same magnitude but with opposite direction, which can be done either by employing an actuator, or by turning the mirror into a low-frequency pendulum with $\omega_m\ll\Omega_{\mathrm{GW}}$ by suspending it on thin fibers, as is the case for the GW interferometers, that provides a necessary gravity restoring force in a natural way. However, it does not change the quantum noise and thus can be omitted from further consideration.
The latter term represents a quantum correction to the former one
\begin{equation}
\label{eq:Mirror_F_ba}
\hat{F}_{\mathrm{b.a.}}(\Omega) \simeq \hbar k_p({\mathbf{A}}_1^{\mathsf{T}}\hat{\boldsymbol{a}}_1(\Omega)+{\mathbf{B}}_1^{\mathsf{T}}\hat{\boldsymbol{b}}_1(\Omega)-{\mathbf{A}}_2^{\mathsf{T}}\hat{\boldsymbol{a}}_2(\Omega)-{\mathbf{B}}_2^{\mathsf{T}}\hat{\boldsymbol{b}}_2(\Omega)) = \boldsymbol{F}_1^{\mathsf{T}}\hat{\boldsymbol{a}}_1(\Omega)+\boldsymbol{F}_2^{\mathsf{T}}\hat{\boldsymbol{a}}_2(\Omega)-K\hat{x}(\Omega)
\end{equation}
where $\hat{F}_{\mathrm{b.a.}}^{(0)} \equiv \boldsymbol{F}_1^{\mathsf{T}}\hat{\boldsymbol{a}}_1(\Omega)+\boldsymbol{F}_2^{\mathsf{T}}\hat{\boldsymbol{a}}_2(\Omega)$ is the random part of the radiation pressure that depends on the input light quantum fluctuations described by quantum quadrature amplitudes vectors $\hat{\boldsymbol{a}}_1(\Omega)$ and $\hat{\boldsymbol{a}}_2(\Omega)$ with coefficients given by vectors:
\begin{equation*}
\boldsymbol{F}_1 = \frac{2\sqrt{2\hbar\omega_p R}}{c}
\begin{bmatrix}
\sqrt{R \mathcal{I}_1}-\sqrt{T\mathcal{I}_2}\cos \Phi_0\\
-\sqrt{T\mathcal{I}_2}\sin\Phi_0
\end{bmatrix}\,,\quad\mbox{and}\quad
\boldsymbol{F}_2 = -\frac{2\sqrt{2\hbar\omega_p R}}{c}
\begin{bmatrix}
\sqrt{T \mathcal{I}_1}+\sqrt{R\mathcal{I}_2}\cos \Phi_0\\
\sqrt{R\mathcal{I}_2}\sin\Phi_0
\end{bmatrix}\,,
\end{equation*}
and the term $-K\hat{x}(\Omega)$ represents the dynamical back action with
\begin{equation}
K = \frac{8\omega_p\sqrt{RT\mathcal{I}_1\mathcal{I}_2}\sin\Phi_0}{c^2}
\end{equation}
being a ponderomotive rigidity that arises in the potential created by the two counter propagating light waves. Eq.~\eqref{eq:Mirror_F_ba} gives us the second of Eqs.~\eqref{eq:gen_linear_meas_EOM_Fourier}. Here $\chi_{FF}(\Omega) = -K$.
\subsubsection{Spectral densities}
We can reduce both our readout quantities to the units of the signal force $G$ according to Eq.~\eqref{eq:gen_force_noise_def}:
\begin{equation}
\label{eq:mirror_sum_noises}
\hat{O}^F_1 = \frac{\hat{\mathcal{X}_1}(\Omega)}{\chi_{xx}^{\mathrm{eff}}(\Omega)} + \hat{\mathcal{F}}(\Omega) +G\,,
\qquad
\hat{O}^F_2 = \frac{\hat{\mathcal{X}_2}(\Omega)}{\chi_{xx}^{\mathrm{eff}}(\Omega)} + \hat{\mathcal{F}}(\Omega) +G
\end{equation}
and define the two effective measurement noise sources as
\begin{equation}
\hat{\mathcal{X}}_1(\Omega) = \frac{\hat{O}_1^{(0)}}{\chi_{O_1F}(\Omega)}\,,\quad\mbox{and}\quad\hat{\mathcal{X}}_2(\Omega) = \frac{\hat{O}_2^{(0)}}{\chi_{O_2F}(\Omega)}
\end{equation}
and an effective force noise as
\begin{equation}
\hat{\mathcal{F}}(\Omega) = \hat{F}_{\mathrm{b.a.}}^{(0)}\,,
\end{equation}
absorbing optical rigidity $K$ into the effective mechanical susceptibility:
\begin{equation}
\chi_{xx}^{\mathrm{eff}}(\Omega) = \frac{\chi_{xx}(\Omega)}{1+K\chi_{xx}(\Omega)} = \frac{1}{K-M\Omega^2}\,.
\end{equation}
One can then easily calculate their power (double-sided) spectral densities according to Eq.~\eqref{eq:gen_spdens}:
\begin{eqnarray}
\label{eq:mirror_spdens_sum}
S^F_{11} = \frac{S_{\mathcal{X}_1\mathcal{X}_1}(\Omega)}{|\chi^{\mathrm{eff}}_{xx}(\Omega)|^2}&+&S_{\mathcal{F}\mathcal{F}}(\Omega)+2\mathrm{Re}\left[\frac{S_{\mathcal{X}_1\mathcal{F}}(\Omega)}{\chi^{\mathrm{eff}}_{xx}(\Omega)}\right]\,,\nonumber\\
S^F_{22} = \frac{S_{\mathcal{X}_2\mathcal{X}_2}(\Omega)}{|\chi^{\mathrm{eff}}_{xx}(\Omega)|^2}&+&S_{\mathcal{F}\mathcal{F}}(\Omega)+2\mathrm{Re}\left[\frac{S_{\mathcal{X}_2\mathcal{F}}(\Omega)}{\chi^{\mathrm{eff}}_{xx}(\Omega)}\right]\,,\nonumber\\
S^F_{12} = S^{F*}_{21} = \frac{S_{\mathcal{X}_1\mathcal{X}_2}(\Omega)}{|\chi^{\mathrm{eff}}_{xx}(\Omega)|^2}&+&S_{\mathcal{F}\mathcal{F}}(\Omega)+\left[\frac{S_{\mathcal{X}_1\mathcal{F}}(\Omega)}{\chi^{\mathrm{eff}}_{xx}(\Omega)}+\frac{S^*_{\mathcal{X}_2\mathcal{F}}(\Omega)}{\chi^{\mathrm{eff*}}_{xx}(\Omega)}\right],
\end{eqnarray}
where, if both lights are in coherent states that implies
$$\mean{\hat{\boldsymbol{a}}_i(\Omega)\circ\hat{\boldsymbol{a}}^\dag_j(\Omega')} = 2\pi\mathbb{S}_{\vac}\delta_{ij}\delta(\Omega-\Omega')\,,\qquad (i,j)=(1,2)$$,
one can get:
\begin{eqnarray}
\label{eq:mirror_spdens_partial}
&&S_{\mathcal{X}_1\mathcal{X}_1}(\Omega) = \frac{\hbar c^2}{16\omega_p\mathcal{I}_1R\sin^2\phi_1}\,,\qquad S_{\mathcal{X}_2\mathcal{X}_2}(\Omega) = \frac{\hbar c^2}{16\omega_p\mathcal{I}_2R\sin^2(\phi_2-\Phi_0)}\,,\qquad S_{\mathcal{X}_1\mathcal{X}_2}(\Omega) = 0\,,\nonumber\\
&&S_{\mathcal{F}\mathcal{F}}(\Omega) = \frac{4\hbar\omega_p R(\mathcal{I}_1+\mathcal{I}_2)}{c^2}\,,\qquad S_{\mathcal{X}_1\mathcal{F}}(\Omega) = \frac{\hbar}{2}\cot\phi_1\,,\qquad S_{\mathcal{X}_2\mathcal{F}}(\Omega) = \frac{\hbar}{2}\cot(\phi_2-\Phi_0)\,.
\end{eqnarray}
Comparison of these expressions with the Eqs.~\eqref{eq:S_x_S_F_corr} shows that we have obtained the results similar to that of the toy example in Section~\ref{sec:toy_FI_correlation}. If we switched one of the pumping carriers off, say the right one, the resulting spectral densities for $\hat{O}^F_1(\Omega)$ would be exactly the same as in the simple case of Eqs.~\eqref{eq:S_x_S_F_corr}, except for the substitution of $\digamma^2\mathcal{I}_0\to R\mathcal{I}_1$ and $\phi_1\to\phi_{\mathrm{LO}}$.
\subsubsection{Full transfer matrix approach to the calculation of quantum noise spectral densities}
\label{sec:mirror_full_transfer_mat}
It was easy to calculate the above spectral densities by parts, distinguishing the effective measurement and back-action noise sources and making separate calculations for them. Had we considered a bit more complicated situation with the incident fields in the squeezed states with arbitrary squeezing angles, the calculation of all six of the above individual spectral densities~\eqref{eq:mirror_spdens_partial} and subsequent substitution to the sum spectral densities expressions~\eqref{eq:mirror_spdens_sum} would be more difficult. Thus, it would be beneficial to have a tool to do all these operations at once numerically.
It is achievable if we build a \emph{full transfer matrix} of our system. To do so, let us first consider the readout observables separately. We start with $\hat{O}_1$ and rewrite it as follows:
\begin{eqnarray}
\hat{O}_1 &=&
\boldsymbol{H}^{\mathsf{T}}[\phi_1]\left(-\sqrt{R}\hat{\boldsymbol{a}}_1+\chi_{xx}^{\mathrm{eff}}\boldsymbol{R}_1\boldsymbol{F}^{\mathsf{T}}_1\hat{\boldsymbol{a}}_1+\sqrt{T}\hat{\boldsymbol{a}}_2+\chi_{xx}^{\mathrm{eff}}\boldsymbol{R}_1\boldsymbol{F}^{\mathsf{T}}_2\hat{\boldsymbol{a}}_2\right)+\chi_{xx}^{\mathrm{eff}}\boldsymbol{H}^{\mathsf{T}}[\phi_1]\boldsymbol{R}_1G \nonumber\\ &=& \boldsymbol{H}^{\mathsf{T}}[\phi_1]\cdot\mathbb{M}_1\cdot
\begin{bmatrix}
\hat{\boldsymbol{a}}_1\\
\hat{\boldsymbol{a}}_2
\end{bmatrix}
+\chi_{xx}^{\mathrm{eff}}\boldsymbol{H}^{\mathsf{T}}[\phi_1]\boldsymbol{R}_1G
\,,
\end{eqnarray}
where we omitted the frequency dependence of the constituents for the sake of brevity and introduced a $2\times4$ full transfer matrix $\mathbb{M}_1$ for the first readout observable defined as
\begin{equation}
\mathbb{M}_1 =
\begin{bmatrix}
-\sqrt{R}\mathbb{I} + \chi_{xx}^{\mathrm{eff}}\boldsymbol{R}_1\boldsymbol{F}^{\mathsf{T}}_1 & \sqrt{T}\mathbb{I} + \chi_{xx}^{\mathrm{eff}}\boldsymbol{R}_1\boldsymbol{F}^{\mathsf{T}}_2
\end{bmatrix}
\end{equation}
with outer product of two arbitrary vectors $\vb{\alpha} = \{\alpha_1,\alpha_2\}^{\mathsf{T}}$ and $\vb{\beta} = \{\beta_1,\beta_2\}^{\mathsf{T}}$ written in short notation as:
\begin{equation}
\vb{\alpha}\vb{\beta}^{\mathsf{T}} \equiv
\begin{bmatrix}
\alpha_1\beta_1 & \alpha_1\beta_2\\
\alpha_2\beta_1 & \alpha_2\beta_2
\end{bmatrix}\,.
\end{equation}
In a similar manner, the full transfer matrix for the second readout can be defined as:
\begin{equation}
\mathbb{M}_2 =
\begin{bmatrix}
\sqrt{T}\mathbb{I} + \chi_{xx}^{\mathrm{eff}}\boldsymbol{R}_2\boldsymbol{F}^{\mathsf{T}}_1 & \sqrt{R}\mathbb{I} + \chi_{xx}^{\mathrm{eff}}\boldsymbol{R}_2\boldsymbol{F}^{\mathsf{T}}_2
\end{bmatrix}\,.
\end{equation}
Having accomplished this, one is prepared to calculate all the spectral densities~\eqref{eq:mirror_spdens_sum} at once using the following matrix formulas:
\begin{equation}
\label{eq:mirror_gen_spdens_lossless}
S^F_{ij}(\Omega) =\frac{1}{2|\chi_{xx}^{\mathrm{eff}}|^2} \frac{\boldsymbol{H}^{\mathsf{T}}[\phi_i]\cdot\mathbb{M}_i\mathbb{S}_{\in}\mathbb{M}^\dag_j\cdot\boldsymbol{H}[\phi_j]+\boldsymbol{H}^{\mathsf{T}}[\phi_i]\cdot\mathbb{M}^*_j\mathbb{S}_{\in}\mathbb{M}^{\mathsf{T}}_i\cdot\boldsymbol{H}[\phi_j]}{\boldsymbol{H}^{\mathsf{T}}[\phi_i]\boldsymbol{R}_i\boldsymbol{R}^\dag_j\boldsymbol{H}[\phi_j]}\,,\qquad (i,j)=(1,2)
\end{equation}
where $\mathbb{M}^*\equiv (\mathbb{M}^\dag)^{\mathsf{T}}$ and $\mathbb{S}_{\in}$ is the $4\times4$-matrix of spectral densities for the two input fields:
\begin{equation}
\mathbb{S}_{\in} =
\begin{bmatrix}
\mathbb{S}_{\sqz}[r_1,\theta_1] & 0\\
0 & \mathbb{S}_{\sqz}[r_2,\theta_2]
\end{bmatrix}
\end{equation}
with $\mathbb{S}_{\sqz}[r_i,\theta_i]$ defined by Eq.~\eqref{eq:SQZ_SpDens_matrix}.
Thus, we obtain the formula that can be (and, actually, is) used for the calculation of quantum noise spectral densities of any, however complicated, interferometer given the full transfer matrix of this interferometer.
\subsubsection{Losses in a readout train}
Thus far we have assumed that there is no dissipation in the transition from the outgoing light to the readout photocurrent of the homodyne detector. This is, unfortunately, not the case since any real photodetector has the finite quantum efficiency $\eta_d<1$ that indicates how many photons absorbed by the detector give birth to photoelectrons, i.e., it is the measure of the probability for the photon to be transformed into the photoelectron. As with any other dissipation, this loss of photons gives rise to an additional noise according to the FDT that we should factor in. We have shown in Section~\ref{sec:losses_in_OE} that this kind of loss can be taken into account by means of a virtual asymmetric beamsplitter with transmission coefficients $\sqrt{\eta_d}$ and $\sqrt{1-\eta_d}$ for the signal light and for the additional noise, respectively. This beamsplitter is set into the readout optical train as shown in Figure~\ref{fig:mov_mirr_refl} and the $i$-th readout quantity needs to be modified in the following way:
\begin{equation}
\hat{O}_i^{\mathrm{loss}}(\Omega) = \sqrt{\eta_d}\hat{O}_i(\Omega)+\sqrt{1-\eta_d}\hat{n}_i(\Omega)\,,
\end{equation}
where
$$\hat{n}_i(\Omega) = \boldsymbol{H}^{\mathsf{T}}[\phi_i]\hat{\boldsymbol{n}}_i(\Omega) = \hat{n}_{i,c}(\Omega)\cos\phi_i+\hat{n}_{i,s}(\Omega)\sin\phi_i$$
is the additional noise that is assumed to be in a vacuum state. Since the overall factor in front of the readout quantity does not matter for the final noise spectral density, one can redefine $\hat{O}_i^{\mathrm{loss}}(\Omega)$ in the following way:
\begin{equation}
\label{eq:homodyne_lossy}
\hat{O}_i^{\mathrm{loss}}(\Omega) = \hat{O}_i(\Omega)+\epsilon_d\hat{n}_i(\Omega)\,,\qquad\mbox{where}\qquad\epsilon_d\equiv\sqrt{\frac{1}{\eta_d}-1}\,.
\end{equation}
The influence of this loss on the final sum spectral densities~\eqref{eq:mirror_gen_spdens_lossless} is straightforward to calculate if one assumes the additional noise sources in different readout trains to be uncorrelated. If it is so, then Eq.~\eqref{eq:mirror_gen_spdens_lossless} modifies as follows:
\begin{eqnarray}
\label{eq:mirror_gen_spdens_lossy}
S^{F,\mathrm{loss}}_{ij}(\Omega) =\frac{1}{2|\chi_{xx}^{\mathrm{eff}}|^2} \left\{\frac{\boldsymbol{H}^{\mathsf{T}}[\phi_i]\cdot\mathbb{M}_i\mathbb{S}_{\in}\mathbb{M}^\dag_j\cdot\boldsymbol{H}[\phi_j]+\boldsymbol{H}^{\mathsf{T}}[\phi_i]\cdot\mathbb{M}^*_j\mathbb{S}_{\in}\mathbb{M}^{\mathsf{T}}_i\cdot\boldsymbol{H}[\phi_j]}{\boldsymbol{H}^{\mathsf{T}}[\phi_i]\boldsymbol{R}_i\boldsymbol{R}^\dag_j\boldsymbol{H}[\phi_j]} +\right.\nonumber\\\left. \frac{\epsilon_d^2\delta_{ij}}{\boldsymbol{H}^{\mathsf{T}}[\phi_i]\boldsymbol{R}_i\boldsymbol{R}^\dag_i\boldsymbol{H}[\phi_i]}\right\}\,,\qquad (i,j)=(1,2)\,.
\end{eqnarray}
Now, when we have considered all the stages of the quantum noise spectral densities calculation on a simple example of a single movable mirror, we are ready to consider more complicated systems. Our next target is a Fabry--P{\'e}rot cavity.
\subsection{Fabry--P{\'e}rot cavity}
\label{sec:Fabry-Perot}
The schematic view of a Fabry--P{\'e}rot cavity with two movable mirrors is drawn in Figure~\ref{fig:Fabry-Perot}. This simple scheme is important for at least two reasons: (i) it is the most common element for more sophisticated interferometer configurations, which are considered below; and (ii) the analysis of real-life high-sensitivity interferometers devoted, in particular, to detection of GWs, can be reduced to a single Fabry--P{\'e}rot cavity by virtue of the `scaling law' theorem~\cite{Buonanno2003}, see Section~\ref{sec:advligo}.
\epubtkImage{fig29.png}{
\begin{figure}
\centerline{\includegraphics[width=.5\textwidth]{fig29}}
\caption{Fabry--P{\'e}rot cavity}
\label{fig:Fabry-Perot}
\end{figure}}
A Fabry--P{\'e}rot cavity consists of two movable mirrors that are separated by a distance $L+x_1+x_2$, where $L=c\tau$ is the distance at rest with $\tau$ standing for a single pass light travel time, and $x_1$ and $x_2$ are the small deviations of the mirrors' position from the equilibrium. Each of the mirrors is described by the transfer matrix $\mathbb{M}_{1,2}$ with real coefficients of reflection $\sqrt{R_{1,2}}$ and transmission $\sqrt{T_{1,2}}$ according to Eq.~\eqref{eq:mirror_Mreal}.
As indicated on the scheme, the outer faces of the mirrors are assumed to have negative reflectivities. While the intermediate equations depend on this choice, the final results are invariant to it. The cavity is pumped from both sides by two laser sources with the same optical frequency $\omega_p$.
The coupling equations for the ingoing and outgoing fields at each of the mirrors are absolutely the same as in Section~\ref{sec:Mirror_motion}. The only new thing is the free propagation of light between the mirrors that adds two more field continuity equations to the full set, describing the transformation of light in the Fabry--P{\'e}rot cavity. It is illuminating to write down input-output relations first in the time domain:
\begin{equation}
\begin{array}{ll}
\label{eq:FP_I/O_time}
\hat{E}_{b_1}(t) = -\sqrt{R_1}\hat{E}_{a_1}(t+2x_1/c)+\sqrt{T_1}\hat{E}_{e_1}(t)\,, & \qquad \hat{E}_{b_2}(t) = -\sqrt{R_2}\hat{E}_{a_2}(t+2x_2/c)+\sqrt{T_2}\hat{E}_{e_2}(t)\,,\\
\hat{E}_{f_1}(t) = \sqrt{R_1}\hat{E}_{e_1}(t-2x_1/c)+\sqrt{T_1}\hat{E}_{a_1}(t)\,, & \qquad \hat{E}_{f_2}(t) = \sqrt{R_2}\hat{E}_{a_2}(t-2x_2/c)+\sqrt{T_2}\hat{E}_{e_2}(t)\,,\\
\hat{E}_{e_1}(t) = \hat{E}_{f_2}(t-L/c)\,, & \qquad \hat{E}_{e_2}(t) =
\hat{E}_{f_1}(t-L/c)\,.
\end{array}
\end{equation}
Further we use notation $\tau=L/c$ for the light travel time between the mirrors.
The frequency domain version of the above equations and their solutions are derived in Appendix~\ref{app:FP_I/O-relations}. We write these I/O-relations given in Eqs.~\eqref{eq:FP_1} in terms of complex amplitudes instead of 2 photon amplitudes, for the expressions look much more compact in that representation. However, one can simplify them even more using the \emph{single-mode approximation}.
\paragraph*{Single-mode approximation.}
We note that:
(i) in GW detection, rather high-finesse cavities are used, which implies low transmittance coefficients for the mirrors
\begin{equation}
T_{1,2}\ll 1 \,;
\end{equation}
(ii) the cavities are relatively short, so their Free Spectral Range (FSR) $f_{\mathrm{FSR}}=(2\tau)^{-1}$ is much larger than the characteristic frequencies of the mirrors' motion:
\begin{equation}
|\Omega|\tau\ll 1 \,;
\end{equation}
and (iii) the detuning of the pump frequency from one of the cavity eigenfrequencies:
\begin{equation}
\delta = \omega_p - \frac{\pi n}{\tau} \quad (n\ \mbox{is an integer})
\end{equation}
is also small in comparison with the FSR:
\begin{equation}
|\delta|\tau \ll 1 \,.
\end{equation}
In this case, only this mode of the cavity can be taken into account, and the cavity can be treated as a single-mode lumped resonator.
Note also that, while our intermediate equations below depend on whether $n$ is even or odd, the final results do not. Therefore, we assume for simplicity that $n$ is even.
Expanding the numerators and denominators of Eqs.~(\ref{eq:FP_0}, \ref{eq:FP_1}) into Taylor series in $\tau$ and keeping only the first non-vanishing terms, we obtain that
\begin{eqnarray}
\label{eq:FP_0_short}
\mathrm{B}_{1,2} &=& \mathcal{R}_{1,2}(0)\mathrm{A}_{1,2} + \mathcal{T}(0)\mathrm{A}_{2,1}\,,\nonumber\\
\mathrm{E}_{1,2} &=& \mathrm{F}_{1,2} = \mathrm{E}
= \frac{\sqrt{\gamma_1}\mathrm{A}_1 + \sqrt{\gamma_2}\mathrm{A}_2}{\ell(0)\sqrt{\tau}} \,,
\end{eqnarray}
and
\begin{eqnarray}
\hat{\mathrm{b}}_{1,2}(\omega) &=& \mathcal{R}_{1,2}(\Omega)\hat{\mathrm{a}}_{1,2}(\omega)
+ \mathcal{T}(\Omega)\hat{\mathrm{a}}_{2,1}(\omega)
+ \frac{2\sqrt{\gamma_{1,2}}X(\Omega)}{\ell(\Omega)} \,,\label{eq:FP_1_short}\\
\hat{\mathrm{e}}_{1,2}(\omega) &=& \hat{\mathrm{f}}_{1,2}(\omega) = \hat{\mathrm{e}}(\omega)
= \frac{
\sqrt{\gamma_1}\hat{\mathrm{a}}_1(\omega) + \sqrt{\gamma_2}\hat{\mathrm{a}}_2(\omega)
+ \hat{X}(\Omega)
}{\ell(\Omega)\sqrt{\tau}} \,, \label{FP_1_short_e}
\end{eqnarray}
where
\begin{eqnarray}
\gamma_{1,2} = \frac{T_{1,2}}{4\tau} \,, \\
\gamma = \gamma_1 + \gamma_2
\end{eqnarray}
is the cavity half-bandwidth,
\begin{equation}
\ell(\Omega) = \gamma - i(\delta+\Omega) \,,
\end{equation}
\begin{equation}
\label{eq:FP_RT}
\mathcal{R}_{1,2}(\Omega) = \frac{2\gamma_{1,2}}{\ell(\Omega)} - 1 \,, \qquad
\mathcal{T}(\Omega) = \frac{2\sqrt{\gamma_1\gamma_2}}{\ell(\Omega)}
\end{equation}
are the cavity left and right reflectivities and its transmittance,
\begin{equation}
\hat{X}(\Omega) = \frac{ik_p\mathrm{E}\hat{x}(\Omega)}{\sqrt{\tau}} \,,
\end{equation}
and
\begin{equation}
\hat{x} = \hat{x}_1 + \hat{x}_2
\end{equation}
is the sum variation of the cavity length.
\subsubsection{Optical transfer matrix for a Fabry--P{\'e}rot cavity}
\label{sec:FP_cav_OTMat}
The above optical I/O-relations are obtained in terms of the complex amplitudes. In order to transform them to two-photon quadrature notations, one needs to employ the following linear transformations:
\begin{enumerate}
\item change frequency $\omega\to\omega_p\pm\Omega$ and rewrite the relations between the input $\hat{\alpha}(\omega)$ and output operators $\hat{\beta}(\omega)$ in the form:
\begin{eqnarray}
\hat{\beta}(\omega) =
f(\Omega)\hat{\alpha}(\omega)\ \to\ \hat{\beta}_+&\equiv&\hat{\beta}(\omega_p+\Omega)
= f(\omega_p+\Omega)\hat{\alpha}(\omega_p+\Omega) \equiv
f_+\hat{\alpha}_+ \ \mbox{and} \nonumber\\
\hat{\beta}_-^\dag&\equiv&\hat{\beta}^\dag(\omega_p-\Omega) = f^*(\omega_p-\Omega)\hat{\alpha}^\dag(\omega_p-\Omega) \equiv f^*_-\hat{\alpha}^\dag_-\,;
\end{eqnarray}
where $f(\Omega)$ is an arbitrary complex-valued function of sideband frequency $\Omega$;
\item use the definition~\eqref{eq:2photon_quads_def} to get the following relations for two-photon quadrature operators:
\begin{equation}
\label{eq:sideband_to_2photon}
\begin{bmatrix}
\hat{\beta}_c(\Omega)\\
\hat{\beta}_s(\Omega)
\end{bmatrix} = \frac12
\begin{bmatrix}
(f_++f^*_-) & i(f_+-f^*_-)\\
-i(f_+-f^*_-) & (f_++f^*_-)
\end{bmatrix}\cdot
\begin{bmatrix}
\hat{\alpha}_c(\Omega)\\
\hat{\alpha}_s(\Omega)\,
\end{bmatrix}.
\end{equation}
\end{enumerate}
Applying transformations~\eqref{eq:sideband_to_2photon} to Eqs.~\eqref{eq:FP_1_short}, we rewrite the I/O-relations for a Fabry--P{\'e}rot cavity in the two-photon quadratures notations:
\begin{eqnarray}
\hat{\boldsymbol{b}}_{1,2}(\Omega) &=& \mathbb{R}_{1,2}(\Omega)\hat{\boldsymbol{a}}_{1,2}(\Omega)
+ \mathbb{T}(\Omega)\hat{\boldsymbol{a}}_{2,1}(\Omega)
+ 2\sqrt{\gamma_{1,2}}\mathbb{L}(\Omega)\hat{\mathbf{X}}(\Omega) \,,
\label{FP_1_quad(a)} \\
\hat{\boldsymbol{e}}(\Omega) &=& \frac{1}{\sqrt{\tau}}\mathbb{L}(\Omega)\left[
\sqrt{\gamma_{1,2}}\hat{\boldsymbol{a}}_{1,2}(\Omega)
+ \sqrt{\gamma_{2,1}}\hat{\boldsymbol{a}}_{2,1}(\Omega)
+ \hat{\mathbf{X}}(\Omega)
\right] , \label{FP_1_quad(b)}
\end{eqnarray}
where
\begin{eqnarray}
\hat{\mathbf{X}}(\Omega)
= \svector{-\mathrm{E}_s}{\mathrm{E}_c}\frac{k_p\hat{x}(\Omega)}{\sqrt{\tau}} \,, \\
\mathrm{E}_c = \sqrt{2}\Re\mathrm{E}\,,\qquad \mathrm{E}_s = \sqrt{2}\Im\mathrm{E}\,,
\end{eqnarray}
\begin{eqnarray}
\mathbb{L}(\Omega) = \frac{1}{\mathcal{D}(\Omega)}
\smatrix{\gamma-i\Omega}{-\delta}{\delta}{\gamma-i\Omega} , \label{FP_bbL} \\
\mathcal{D}(\Omega) = \ell(\Omega)\ell^*(-\Omega) = (\gamma-i\Omega)^2 + \delta^2 \,,
\end{eqnarray}
\begin{equation}
\label{FP_bbRT}
\mathbb{R}_{1,2}(\Omega) = 2\gamma_{1,2}\mathbb{L}(\Omega) - \mathbb{I} \,, \qquad
\mathbb{T}(\Omega) = 2\sqrt{\gamma_1\gamma_2}\mathbb{L}(\Omega) \,.
\end{equation}
Therefore, the I/O-relations in standard form read:
\begin{equation}
\begin{bmatrix}
\hat{\boldsymbol{b}}_1(\Omega)\\
\hat{\boldsymbol{b}}_2(\Omega)
\end{bmatrix}=
\mathbb{M}^{(0)}_{\mathrm{FP}}\cdot
\begin{bmatrix}
\hat{\boldsymbol{a}}_1(\Omega)\\
\hat{\boldsymbol{a}}_2(\Omega)
\end{bmatrix}+
\begin{bmatrix}
\boldsymbol{R}^{\mathrm{FP}}_1(\Omega)\\
\boldsymbol{R}^{\mathrm{FP}}_2(\Omega)
\end{bmatrix}\hat{x}(\Omega)
\end{equation}
with optical transfer matrix defined as:
\begin{equation}
\label{eq:FP_OTMat}
\mathbb{M}^{(0)}_{\mathrm{FP}}(\Omega) =
\begin{bmatrix}
\mathbb{R}_1(\Omega) & \mathbb{T}(\Omega)\\
\mathbb{T}(\Omega) & \mathbb{R}_2(\Omega)
\end{bmatrix}
\end{equation}
and the response to the cavity elongation $\hat{x}(\Omega)$ defined as:
\begin{equation}
\boldsymbol{R}^{\mathrm{FP}}_1(\Omega)= 2k_p\sqrt{\frac{\gamma_1}{\tau}}\mathbb{L}(\Omega)\cdot
\begin{bmatrix}
-\mathrm{E}_s\\
\mathrm{E}_c
\end{bmatrix}\quad\mbox{and}\quad
\boldsymbol{R}^{\mathrm{FP}}_2(\Omega)=2k_p\sqrt{\frac{\gamma_2}{\tau}}\mathbb{L}(\Omega)\cdot
\begin{bmatrix}
-\mathrm{E}_s\\
\mathrm{E}_c
\end{bmatrix}\,.
\end{equation}
Note that due to the fact that $(\mathbb{M}^{(0)}_{\mathrm{FP}})^\dag = (\mathbb{M}^{(0)}_{\mathrm{FP}})^{-1}$ the reflectivity and the transmission matrices $\mathbb{R}_{1,2}$ and $\mathbb{T}$ satisfy the following unitarity relations:
\begin{equation}
\label{eq:FP_bbRT_u}
\mathbb{R}_1\mathbb{R}_1^\dagger + \mathbb{T}\mathbb{T}^\dagger
= \mathbb{R}_2\mathbb{R}_2^\dagger + \mathbb{T}\mathbb{T}^\dagger
= \mathbb{I} \,, \qquad
\mathbb{R}_1\mathbb{T}^\dagger + \mathbb{T}\mathbb{R}_2^\dagger = 0 \,.
\end{equation}
\subsubsection{Mirror dynamics, radiation pressure forces and ponderomotive rigidity}
The mechanical equations of motion of the Fabry--P{\'e}rot cavity mirrors, in spectral representation, are the following:
\begin{equation}
\label{eq:x_12}\mathrm{f}
\chi_{xx,i}^{-1}(\Omega)\hat{x}_{i}(\Omega)
= \hat{F}_{i}(\Omega) + G_i(\Omega)\qquad i=1,2\,,
\end{equation}
where $\chi_{xx,i}$ are the mechanical susceptibilities of the mirrors, $G_{i}$ stand for any external classical forces that could act on the mirrors (for example, a signal force to be detected),
\begin{equation}
\hat{F}_{i} = \frac{
\hat{\mathcal{I}}_{\mathrm{e}\,i} + \hat{\mathcal{I}}_{\mathrm{f}\,i}
- \hat{\mathcal{I}}_{\mathrm{a}\,i} - \hat{\mathcal{I}}_{\mathrm{b}\,i}
}{c}
\end{equation}
are the radiation pressure forces acting on the mirrors, and $\hat{\mathcal{I}}_{\mathrm{e}\,i}$, $\hat{\mathcal{I}}_{\mathrm{f}\,i}$, $\hat{\mathcal{I}}_{\mathrm{a}\,i}$, $\hat{\mathcal{I}}_{\mathrm{b}\,i}$ are the powers of the waves $\mathrm{e}_{i}$, $\mathrm{f}_{i}$, etc. The signs for all forces are chosen in such a way that the positive forces are oriented outwards from the cavity, increasing the corresponding mirror displacement $x_{1,2}$.
In the spectral representation, using the quadrature amplitudes notation, the radiation pressure forces read:
\begin{eqnarray}
\hat{F}_{1,2}(\Omega) &=& \frac{\hbar k_p}{2}\left(
\mathbf{E}_{1,2}^{\mathsf{T}}\mathbf{E}_{1,2} + \mathbf{F}_{1,2}^{\mathsf{T}}\mathbf{F}_{1,2}
- \mathbf{A}_{1,2}^{\mathsf{T}}\mathbf{A}_{1,2} - \mathbf{B}_{1,2}^{\mathsf{T}}\mathbf{B}_{1,2}
\right) \nonumber\\
&& + \hbar k_p\left[
\mathbf{E}_{1,2}^{\mathsf{T}}\hat{\boldsymbol{e}}_{1,2}(\Omega)
+ \mathbf{F}_{1,2}^{\mathsf{T}}\hat{\boldsymbol{f}}_{1,2}(\Omega)
- \mathbf{A}_{1,2}^{\mathsf{T}}\hat{\boldsymbol{a}}_{1,2}(\Omega)
- \mathbf{B}_{1,2}^{\mathsf{T}}\hat{\boldsymbol{b}}_{1,2}(\Omega)
\right] .
\end{eqnarray}
The first group, as we have already seen, describes the regular constant force; therefore, we omit it henceforth.
In the single-mode approximation, the radiation pressure forces acting on both mirrors are equal to each other:
\begin{equation}
\label{eq:FP_F_RP}
\hat{F}_{1,2}(\Omega) \equiv \hat{F}_{\mathrm{b.a.}}(\Omega)
= 2\hbar k_p\mathbf{E}^{\mathsf{T}}\hat{\boldsymbol{e}}(\Omega) \,,
\end{equation}
and the optical field in the cavity is sensitive only to the elongation mechanical mode described by the coordinate $x$. Therefore, combining Eqs.~\eqref{eq:x_12}, we obtain for this mode:
\begin{equation}
\label{eq_x_raw}
\chi_{xx}^{-1}(\Omega)\hat{x}(\Omega) = \hat{F}_{\mathrm{b.a.}}(\Omega) + G(\Omega) \,,
\end{equation}
where
\begin{equation}
\chi_{xx}(\Omega) = \left[\chi_{xx,1}(\Omega)+\chi_{xx,2}(\Omega)\right]
\end{equation}
is the reduced mechanical susceptibility and
\begin{equation}
G(\Omega) = \frac{
\chi_{xx,1}(\Omega)G_1(\Omega) + \chi_{xx,2}(\Omega)G_2(\Omega)
}{\chi_{xx}(\Omega)}
\end{equation}
is the effective external force.
In the simplest and at the same time the most important particular case of free mirrors:
\begin{equation}
\chi_{xx,i}(\Omega) = -\frac{1}{m_{i}\Omega^2}\qquad i=1,2\,,
\end{equation}
the reduced mechanical susceptibility and the effective external force are equal to
\begin{equation}
\chi_{xx}(\Omega) = -\frac{1}{\mu\Omega^2} \,,
\end{equation}
and
\begin{equation}
G(\Omega)
= \mu\left[\frac{G_1(\Omega)}{m_1} + \frac{G_2(\Omega)}{m_2}\right] ,
\end{equation}
where
\begin{equation}
\mu = \left(\frac{1}{m_1}+\frac{1}{m_2}\right)^{-1}
\end{equation}
is the effective mass of the elongation mechanical mode.
It follows from Eqs.~(\ref{FP_1_quad(b)}) and~(\ref{eq:FP_F_RP}) that the radiation pressure force can be written as a sum of the random and dynamical back-action terms, similarly to the single mirror case:
\begin{equation}
\label{eq:FP_F_fl_K}
\hat{F}_{\mathrm{b.a.}}(\Omega) = \hat{F}^{(0)}_{\mathrm{b.a.}}(\Omega) - K(\Omega)\hat{x}(\Omega) \,,
\end{equation}
with the random component equal to
\begin{equation}
\label{eq:FP_F_fl}
\hat{F}_{\mathrm{b.a.}}^{(0)}(\Omega) = \frac{2\hbar k_p\mathbf{E}^{\mathsf{T}}}{\sqrt{\tau}}
\mathbb{L}(\Omega)\left[
\sqrt{\gamma_1}\hat{\boldsymbol{a}}_1(\Omega) + \sqrt{\gamma_2}\hat{\boldsymbol{a}}_2(\Omega)
\right]
\end{equation}
and the ponderomotive rigidity that reads
\begin{equation}
\label{eq:FP_K}
K(\Omega) = \frac{MJ\delta}{\mathcal{D}(\Omega)}\,.
\end{equation}
We introduced here the normalized optical power
\begin{equation}
\label{eq:FP_J}
J = \frac{4\hbar k_p^2|E|^2}{M\tau} = \frac{4\omega_p\mathcal{I}_c}{McL}
\end{equation}
with
\begin{equation}
\mathcal{I}_c = \hbar\omega_p|E|^2
\end{equation}
standing for the mean optical power circulating inside the cavity, and $M$ is some (in general, arbitrary) mass. Typically, it is convenient to make it equal to the reduced mass $\mu$.
Substitution of the force~\eqref{eq:FP_F_fl_K} into Eq.~\eqref{eq_x_raw} gives the following final equation of motion:
\begin{equation}
\label{eq:FP_mech_eq}
[\chi_{xx}^{-1}(\Omega)+K(\Omega)]\hat{x}(\Omega)
= \hat{F}_{\mathrm{b.a.}}^{(0)}(\Omega) + G(\Omega) \,.
\end{equation}
Thus, the effective mechanical susceptibility $\chi_{xx}^{\mathrm{eff,\,FP}}(\Omega)$ for a Fabry--P{\'e}rot cavity reads:
\begin{equation}
\label{eq:chi_xx_eff_FP}
\chi_{xx}^{\mathrm{eff,\,FP}}(\Omega) = \bigl(\chi_{xx}^{-1}(\Omega)+K(\Omega)\bigr)^{-1} = \frac{1}{K(\Omega)-\mu\Omega^2}\,.
\end{equation}
\subsection{Fabry--P{\'e}rot--Michelson interferometer}
\label{sec:advligo}
In real-life high-precision experiments with mechanical test objects,
interferometer schemes that are much more sophisticated than the
simple Fabry--P{\'e}rot cavity are used. In particular, the best
sensitivity in mechanical displacement measurements is achieved by the
laser GW detectors. The typical scheme of such a
detector is shown in Figure~\ref{fig:advligo}. It is this scheme
which is planned to be used for the next generation Advanced
LIGO~\cite{Thorne2000, Fritschel2002, Smith2009}, Advanced
VIRGO~\cite{Acernese2006-2}, and LCGT~\cite{LCGTsite} GW detectors, and its simplified versions are (or were) in use in
the contemporary first generation detectors: Initial
LIGO~\cite{Abramovici1992, LIGOsite}, VIRGO~\cite{VIRGOsite},
GEO\,600~\cite{Willke2002, GEOsite}, and TAMA~\cite{Ando2001,
TAMAsite}.
\epubtkImage{fig30.png}{
\begin{figure}[htbp]
\centerline{\includegraphics[width=.8\textwidth]{fig30}}
\caption{Power- and signal-recycled Fabry--P{\'e}rot--Michelson interferometer.}
\label{fig:advligo}
\end{figure}}
This scheme works similar to the ordinary Michelson interferometer considered briefly in Section~\ref{sec:MI}. The beamsplitter \textsf{BS} distributes the pump power from the laser evenly between the arms. The beams, reflected off the Fabry--P{\'e}rot cavities are recombined on the beamsplitter in such a way that, in the ideal case of perfect symmetry of the arms, all the light goes back to the laser, i.e., keeping the signal (`south') port dark. Any imbalance of the interferometer arms, caused by signal forces acting on the end test masses (ETMs) makes part of the pumping light leak into the dark port where it is monitored by a photodetector.
The Fabry--P{\'e}rot cavities in the arms, formed by the input test masses (ITMs) and the end test masses, provide the increase of the optomechanical coupling, thus making photons bounce many times in the cavity and therefore carry away a proportionally-amplified mirror displacement signal in their phase (cf.\ with the $\digamma$ factor in the toy systems considered in Section~\ref{sec:linear_quantum_measurement}). The two auxiliary \emph{recycling} mirrors: the PRM and the signal recycling (SRM) allow one to increase the power, circulating inside the Fabry--P{\'e}rot cavities, for a given laser power, and provide the means for fine-tuning of the quantum noise spectral density~\cite{PhysRevD.38.2317_1988_Meers, PhysRevD.38.433_1988_Vinet}, respectively.
It was shown in~\cite{Buonanno2003} that quantum noise of this dual (power and signal) recycled interferometer is equivalent to that of a single Fabry--P{\'e}rot cavity with some effective parameters (the analysis in that paper was based on earlier works~\cite{Mizuno1995, 00pth1Ra}, where the classical regime had been considered). Here we reproduce this \emph{scaling law} theorem, extending it in two aspects: (i) we factor in optical losses in the arm cavities by virtue of modeling it by the finite transmissivity of the ETMs, and (ii) we do not assume the arm cavities tuned in resonance (the detuned arm cavities could be used, in particular, to create optical rigidity in non-signal-recycled configurations).
\subsubsection{Optical I/O-relations}
We start with Eqs.~(\eqref{eq:FP_0_short}) and~(\eqref{eq:FP_1_short}) for the arm cavities. The notation for the field amplitudes is shown in Figure~\ref{fig:advligo}. The fields referring to the interferometer arms are marked with the subscripts $N$ (`northern') and $E$ (`eastern') following the convention of labeling the GW interferometer parts in accordance with the cardinal directions they are located at with respect to the drawing (up-direction coincides with north). In order to avoid subscripts, we rename some of the field amplitudes as follows:
\begin{equation}
\mathrm{a}_{1\,N,E} \to \mathrm{a}_{N,E} \,, \qquad \mathrm{b}_{1\,N,E} \to \mathrm{b}_{N,E} \,, \qquad \mathrm{a}_{2\,N,E} \to \mathrm{g}_{N,E} \,;
\end{equation}
see also Figure~\ref{fig:advligo}. Note that the fields $\mathrm{g}_{N,E}$ now describe the noise sources due to optical losses in the arm cavities.
Rewrite those Eqs.~(\ref{eq:FP_0_short}), (\ref{eq:FP_1_short}), (\ref{FP_1_short_e}) that are relevant to our consideration, in these notations:
\begin{eqnarray}
\label{eq:sl_cav_0}
\mathrm{B}_{N,E} &=& \mathcal{R}_{\mathrm{arm}}(0)\mathrm{A}_{N,E} \,,\nonumber \\
\mathrm{E}_{N,E} &=& \frac{1}{\ell_{\mathrm{arm}}(0)}\sqrt{\frac{\gamma_{1\mathrm{arm}}}{\tau}}
\mathrm{A}_{N,E} \,,
\end{eqnarray}
\begin{eqnarray}
\label{eq:sl_cav_1}
\hat{\mathrm{b}}_{N,E}(\omega) &=& \mathcal{R}_{\mathrm{arm}}(\Omega)\hat{\mathrm{a}}_{N,E}(\omega)
+ \mathcal{T}_{\mathrm{arm}}(\Omega)\hat{\mathrm{g}}_{N,E}(\omega)
+ \frac{2\sqrt{\gamma_{1\mathrm{arm}}}\hat{X}_{N,E}(\Omega)}{\ell_{\mathrm{arm}}(\Omega)} \,,\nonumber \\
\hat{\mathrm{e}}_{N,E}(\omega) &=& \frac{
\sqrt{\gamma_{1\mathrm{arm}}}\hat{\mathrm{a}}_{N,E}(\omega)
+ \sqrt{\gamma_2}\hat{\mathrm{g}}_{N,E}(\omega)
+ \hat{X}_{N,E}(\Omega)
}{\ell_{\mathrm{arm}}(\Omega)\sqrt{\tau}} \,,
\end{eqnarray}
where
\begin{equation}
\label{sl_gammas}
\gamma_{1\mathrm{arm}} = \frac{T_{\mathrm{arm}}}{4\tau} \,, \qquad
\gamma_2 = \frac{A_{\mathrm{arm}}}{4\tau} \,,
\end{equation}
$T_{\mathrm{arm}}$ is the input mirrors power transmittance, $A_{\mathrm{arm}}$ is the arm cavities power losses per bounce,
\begin{equation}
\gamma_{\mathrm{arm}} = \gamma_{1\mathrm{arm}} + \gamma_2
\end{equation}
is the arm cavities half-bandwidth,
\begin{equation}
\ell_{\mathrm{arm}}(\Omega) = \gamma_{\mathrm{arm}} - i(\delta_{\mathrm{arm}}+\Omega) \,,
\end{equation}
$\delta_{\mathrm{arm}}$ is the arm cavities detuning,
\begin{equation}
\label{eq:FP_rt}
\mathcal{R}_{\mathrm{arm}}(\Omega)
= \frac{2\gamma_{1\mathrm{arm}}}{\ell_{\mathrm{arm}}(\Omega)} - 1 \,, \qquad
\mathcal{T}_{\mathrm{arm}}(\Omega)
= \frac{2\sqrt{\gamma_{1\mathrm{arm}}\gamma_2}}{\ell_{\mathrm{arm}}(\Omega)}
\end{equation}
and
\begin{equation}
\hat{X}_{N,E} = \frac{ik_p\mathrm{E}_{N,E}\hat{x}_{N,E}(\Omega)}{\sqrt{\tau}} \,.
\end{equation}
Assume then that the beamsplitter is described by the matrix~\eqref{eq:IO_beamsplitter_matrix}, with $R=T=1/2$. Let $l_W=c\tau_W$ be the power recycling cavity length (the optical distance between the power recycling mirror and the input test masses) and $l_S=c\tau_S$ -- power recycling cavity length (the optical distance between the signal recycling mirror and the input test masses). In this case, the light propagation between the recycling mirrors and the input test masses is described by the following equations for the classical field amplitudes:
\begin{eqnarray}
\label{eq:sl_bs_0}
\mathrm{A}_{N,E} &=& \frac{\mathrm{D}_We^{i\phi_W} \pm \mathrm{D}_Se^{i\phi_S}}{\sqrt{2}} \,,\nonumber \\
\mathrm{C}_{W,S} &=& \frac{\mathrm{B}_N \pm \mathrm{B}_E}{\sqrt{2}}e^{i\phi_{W,S}} \,,
\end{eqnarray}
where
\begin{equation}
\phi_{W,S} = \omega_p\tau_{W,S} \,,
\end{equation}
are the phase shifts gained by the carrier light with frequency $\omega_p$ passing through the power and signal recycling cavities, and the similar equations:
\begin{eqnarray}\label{eq:sl_bs_1}
\hat{\mathrm{a}}_{N,E}(\omega) &=& \frac{
\hat{\mathrm{d}}_W(\omega)e^{i\omega\tau_W} \pm \hat{\mathrm{d}}_S(\omega)e^{i\omega\tau_S}
}{\sqrt{2}} \,, \nonumber\\
\hat{\mathrm{c}}_{W,S}(\omega)
&=& \frac{\hat{\mathrm{b}}_N(\omega) \pm \hat{\mathrm{b}}_E(\omega)}{\sqrt{2}}e^{i\omega\tau_{W,S}}
\end{eqnarray}
appling to the quantum fields' amplitudes.
The last group of equations that completes our equations set is for the coupling of the light fields at the recycling mirrors:
\begin{equation}
\begin{array}{ll}
\label{eq:sl_rec}
\hat{\mathrm{b}}_W = -\sqrt{R_W}\hat{\mathrm{a}}_W + \sqrt{T_{\mathrm{W}}}\hat{\mathrm{c}}_W \,, & \qquad
\hat{\mathrm{d}}_W = \sqrt{T_W}\hat{\mathrm{a}}_W + \sqrt{R_{\mathrm{W}}}\hat{\mathrm{c}}_W \,, \\
\hat{\mathrm{b}}_S = -\sqrt{R_S}\hat{\mathrm{a}}_S + \sqrt{T_{\mathrm{S}}}\hat{\mathrm{c}}_S \,, & \qquad
\hat{\mathrm{d}}_S = \sqrt{T_S}\hat{\mathrm{a}}_S + \sqrt{R_{\mathrm{W}}}\hat{\mathrm{c}}_S \,,
\end{array}
\end{equation}
where $R_W$, $T_W$ and $R_S$, $T_S$ are the reflectivities and transmissivities of the power and signal recycling mirrors, respectively. These equations, being linear and frequency independent, are valid both for the zeroth-order classical amplitudes and for the first-order quantum ones.
\subsubsection{Common and differential optical modes}
The striking symmetry of the above equations suggests that the convenient way to describe this system is to decompose all the optical fields in the interferometer arms into the superposition of the symmetric (common) and antisymmetric (differential) modes, which we shall denote by the subscripts $+$ and $-$, respectively:
\begin{equation}
\label{eq:ne2pm}
\hat{\mathrm{a}}_\pm = \frac{\hat{\mathrm{a}}_N\pm\hat{\mathrm{a}}_E}{\sqrt{2}} \,, \qquad
\hat{\mathrm{b}}_\pm = \frac{\hat{\mathrm{b}}_N\pm\hat{\mathrm{b}}_E}{\sqrt{2}} \,, \qquad
\hat{\mathrm{e}}_\pm = \frac{\hat{\mathrm{e}}_N\pm\hat{\mathrm{e}}_E}{\sqrt{2}} \,, \qquad
\hat{\mathrm{g}}_\pm = \frac{\hat{\mathrm{g}}_N\pm\hat{\mathrm{g}}_E}{\sqrt{2}} \,.
\end{equation}
It follows from Eqs.~\eqref{eq:sl_bs_1} that the symmetric mode is coupled solely to the `western' (bright) port, while the antisymmetric one couples exclusively to the `southern' (dark) port of the interferometer.
It is easy to see that the classical field amplitudes of the antisymmetric mode are equal to zero. For the common mode, combining Eqs.~(\ref{eq:sl_cav_0}), (\ref{eq:sl_bs_0}), (\ref{eq:sl_rec}), (\ref{eq:ne2pm}), it is easy to obtain the following set of equation:
\begin{eqnarray}
\label{eq:sl_pm_eq_0}
\mathrm{B}_+ &=& \mathcal{R}_{\mathrm{arm}}(0)\mathrm{A}_+ \,,\nonumber \\
\mathrm{E}_+ &=& \frac{1}{\ell_{\mathrm{arm}}(0)}\sqrt{\frac{\gamma_{1\mathrm{arm}}}{\tau}}\mathrm{A}_+ \,,\nonumber \\
\mathrm{A}_+ &=& \mathrm{D}_We^{i\phi_W} \,,\nonumber \\
\mathrm{C}_W &=& \mathrm{B}_+e^{i\phi_W} \,, \nonumber\\
\mathrm{B}_W &=& -\sqrt{R_W}\mathrm{A}_W + \sqrt{T_W}\mathrm{C}_W \,, \nonumber\\
\mathrm{D}_W &=& \sqrt{T_W}\mathrm{A}_W + \sqrt{R_W}\mathrm{C}_W \,.
\end{eqnarray}
Its solution is equal to (only those amplitudes are provided that we shall need below):
\begin{subequations}
\label{sl_pm_raw_0}
\begin{eqnarray}
\mathrm{B}_W &=& \frac{\mathcal{R}_{\mathrm{arm}}(0)e^{2i\phi_W} - \sqrt{R_W}}
{1 - \mathcal{R}_{\mathrm{arm}}(0)\sqrt{R_W}e^{2i\phi_W}}\,\mathrm{A}_W \,, \\
\mathrm{E}_+ &=& \frac{1}{\ell_{\mathrm{arm}}(0)}\sqrt{\frac{\gamma_{1\mathrm{arm}}}{\tau}}
\frac{\sqrt{T_W}e^{i\phi_W}}{1 - \mathcal{R}_{\mathrm{arm}}(0)\sqrt{R_W}e^{2i\phi_W}}\,\mathrm{A}_W \,.
\end{eqnarray}
\end{subequations}
In the differential mode, the first non-vanishing terms are the first-order quantum-field amplitudes. In this case, using Eqs.~(\ref{eq:sl_cav_1}), (\ref{eq:sl_bs_1}), (\ref{eq:sl_rec}), (\ref{eq:ne2pm}), and taking into account that
\begin{equation}
\label{E_plus}
\mathrm{E}_N = \mathrm{E}_E = \frac{\mathrm{E}_+}{\sqrt{2}} \,,
\end{equation}
we obtain:
\begin{eqnarray}
\label{eq:sl_pm_eq_1}
\hat{\mathrm{b}}_-(\omega) &=& \mathcal{R}_{1\mathrm{arm}}(\Omega)\hat{\mathrm{a}}_-(\omega)
+ \mathcal{T}_{\mathrm{arm}}(\Omega)\hat{\mathrm{g}}_-(\omega)
+ \frac{2\sqrt{\gamma_{1\mathrm{arm}}}\hat{X}_-(\Omega)}{\ell_{\mathrm{arm}}(\Omega)} \,, \nonumber\\
\hat{\mathrm{e}}_-(\omega) &=& \frac{
\sqrt{\gamma_{1\mathrm{arm}}}\hat{\mathrm{a}}_-(\omega)
+ \sqrt{\gamma_2}\hat{\mathrm{g}}_-(\omega)
+ \hat{X}_-(\Omega)
}{\ell_{\mathrm{arm}}(\Omega)\sqrt{\tau}} \,, \\
\hat{\mathrm{a}}_-(\omega) &=& \hat{\mathrm{d}}_S(\omega)e^{i\omega\tau_S} \,, \nonumber\\
\hat{\mathrm{c}}_S(\omega) &=& \hat{\mathrm{b}}_-(\omega)e^{i\omega\tau_S} \,, \nonumber\\
\hat{\mathrm{b}}_S(\omega) &=& -\sqrt{R_S}\hat{\mathrm{a}}_S + \sqrt{T_S}\hat{\mathrm{c}}_S(\omega) \,, \nonumber\\
\hat{\mathrm{d}}_S(\omega) &=& \sqrt{T_S}\hat{\mathrm{a}}_S + \sqrt{R_S}\hat{\mathrm{c}}_S(\omega) \,,
\end{eqnarray}
where
\begin{eqnarray}
\hat{X}_-(\Omega) &=& \frac{ik_p\mathrm{E}_+\hat{x}_-(\Omega)}{\sqrt{\tau}} \,, \\
\hat{x}_- &=& \frac{\hat{x}_N - \hat{x}_E}{2} \,. \label{eq:x_minus}
\end{eqnarray}
The solution of this equation set is the following:
\begin{eqnarray}
\label{eq:sl_pm_raw_1}
\hat{\mathrm{b}}_S(\omega) &=&
\frac{
[\mathcal{R}_{\mathrm{arm}}(\Omega)e^{2i\omega\tau_S} - \sqrt{R_S}]\hat{\mathrm{a}}_S(\Omega)
+ \sqrt{T_S}e^{i\omega\tau_S}\biggl[
\mathcal{T}_{\mathrm{arm}}(\Omega)\hat{\mathrm{g}}_-(\omega)
+ \dfrac{2\sqrt{\gamma_{1\mathrm{arm}}}\hat{X}_-(\Omega)}{\ell_{\mathrm{arm}}(\Omega)}
\biggr]
}{1 - \mathcal{R}_{\mathrm{arm}}(\Omega)\sqrt{R_S}e^{2i\omega\tau_S}} \,,\nonumber \\
\hat{\mathrm{e}}_-(\omega) &=& \frac{
\sqrt{T_S}e^{i\omega\tau_S}\sqrt{\gamma_{1\mathrm{arm}}}\hat{\mathrm{a}}_S(\Omega)
+ [1+\sqrt{R_S}e^{2i\omega\tau_S}]
[\sqrt{\gamma_2}\hat{\mathrm{g}}_-(\omega) +
\hat{X}_-(\Omega)]}
{\ell_{\mathrm{arm}}(\Omega)\sqrt{\tau}[1 - \mathcal{R}_{\mathrm{arm}}(\Omega)\sqrt{R_S}e^{2i\omega\tau_S}]}
\,.
\end{eqnarray}
Eqs.~\eqref{eq:sl_pm_eq_0} and \eqref{sl_pm_raw_0}, on the one hand,
and \eqref{eq:sl_pm_eq_1} and \eqref{eq:sl_pm_raw_1}, on the other,
describe two almost independent optical configurations each consisting
of the two coupled Fabry--P{\'e}rot cavities as featured in
Figure~\ref{fig:pm_modes_1}. `Almost independent' means that they do
not couple in an ordinary linear way (and, thus, indeed represent
two optical modes). However, any variation of the differential
mechanical coordinate $x_-$ makes part of the pumping carrier energy
stored in the common mode pour into the differential mode, which
means a non-linear parametric coupling between these modes.
\epubtkImage{fig34.png}{
\begin{figure}
\centerline{\includegraphics[width=.6\textwidth]{fig31}}
\caption{Effective model of the dual-recycled Fabry--P{\'e}rot--Michelson interferometer, consisting of the common (a) and the differential (b) modes, coupled only through the mirrors displacements.}
\label{fig:pm_modes_1}
\end{figure}}
\subsubsection{Interferometer dynamics: mechanical equations of motion, radiation pressure forces and ponderomotive rigidity}
The mechanical elongation modes of the two Fabry--P{\'e}rot cavities are described by the following equations of motion [see Eq.~\eqref{eq_x_raw}]:
\begin{equation}
\label{eq:x_NE_eq}
-\mu\Omega^2\hat{x}_{N,E}(\Omega)
= 2\hbar k_p\mathbf{E}_{N,E}^{\mathsf{T}}\hat{\boldsymbol{e}}_{N,E}(\Omega)
+ \frac{G_{N,E}(\Omega)}{2} \,,
\end{equation}
where $\mu=M/2$ is the effective mass of these modes and $G_{N,E}$ are the external classical forces acting on the cavities end mirrors. The differential mechanical mode equation of motion~\eqref{eq:x_minus} taking into account Eq.~\eqref{E_plus} reads:
\begin{equation}
\label{eq:x_minus_eq_raw}
-2\mu\Omega^2\hat{x}_-(\Omega) = \hat{F}_-^{\mathrm{r.p.}}(\Omega) + \frac{G_-(\Omega)}{2}
\,,
\end{equation}
where
\begin{equation}
\label{F_minus_RP}
F_-^{\mathrm{r.p.}}(\Omega) = 2\hbar k_p\mathbf{E}_+^{\mathsf{T}}\hat{\boldsymbol{e}}_-(\Omega)
\end{equation}
is the differential radiation-pressure force and
\begin{equation}
G_- = G_N-G_E
\end{equation}
is the differential external force.
Equations~\eqref{eq:sl_pm_raw_1} and \eqref{eq:x_minus_eq_raw} together form a complete set of equations describing the differential optomechanical mode of the interferometer featured in Figure~\ref{fig:pm_modes_1}(b). Eq.~\eqref{eq:x_minus_eq_raw} implies that the effective mass of the differential mechanical degree of freedom coincides with the single mirror mass:
\begin{equation}
\label{sl_M_eff}
2\mu=M,
\end{equation}
which prescribes the mirrors of the effective cavity to be twice as heavy as the real mirrors, i.e., $2M$. For the same reason Eq.~\eqref{E_plus} implies for the effective optical power a value twice as high as the power of light, circulating in the arm cavities:
\begin{equation}
\mathcal{I}_c = \hbar\omega_p\mathrm{E}_+^2 = 2\mathcal{I}_{\mathrm{arm}} = 2\hbar\omega_p\mathrm{E}_{N,E}^2 \,.
\end{equation}
\subsubsection{Scaling law theorem}
\label{sec:scaling_low}
Return to Eqs.~\eqref{sl_pm_raw_0} for the common mode. Introduce the following notations:
\begin{eqnarray}
\gamma_{1W} = \gamma_{1\mathrm{arm}}\Re\frac{1-\sqrt{R_W}e^{2i\phi_W}}{1+\sqrt{R_W}e^{2i\phi_W}}
&=& \frac{\gamma_{1\mathrm{arm}}T_W}{1 + 2\sqrt{R_W}\cos2\phi_W + R_W} \,, \\
\delta_W = \delta_{\mathrm{arm}} - \gamma_{1\mathrm{arm}}\Im\frac{1-\sqrt{R_W}e^{2i\phi_W}}{1+\sqrt{R_W}e^{2i\phi_W}}
&=& \delta_{\mathrm{arm}}
+ \frac{2\gamma_{1\mathrm{arm}}\sqrt{R_W}\sin2\phi_W}{1 + 2\sqrt{R_W}\cos2\phi_W + R_W} \,, \\
\gamma_W &=& \gamma_{1W} + \gamma_2 \,, \\
\ell_W(0) &=& \gamma_W - i\delta_W \,.
\end{eqnarray}
In these notation, Eqs.~\eqref{sl_pm_raw_0} have the following form:
\begin{eqnarray}
\mathrm{B}_W &=& \mathcal{R}_W\mathrm{A}_We^{2i\alpha_W} \,, \nonumber \\
\mathrm{E}_+ &=& \frac{\sqrt{\gamma_{1W}}}{\ell_W(0)\sqrt{\tau}}\mathrm{A}_We^{i\alpha_W} \,,
\label{eq:BE_SL_raw}
\end{eqnarray}
where
\begin{eqnarray}
\mathcal{R}_W &=& \frac{2\gamma_{1W}}{\ell_W(0)} - 1 \,, \\
\alpha_W &=& \arg\frac{e^{i\phi_W}}{1+\sqrt{R_W}e^{2i\phi_W}} \,.
\end{eqnarray}
It is easy to see that these equations have the same form as Eqs.~\eqref{eq:FP_0_short} for the single Fabry--P{\'e}rot cavity, with the evident replacement of $\gamma_1$ and $\delta$ with the effective parameters $\gamma_{1W}$ and $\delta_W$. The only difference is an additional phase shift $\alpha_W$. Thus, we have shown that the power recycling cavity formed by the PRM and the ITMs can be treated as a single mirror with some effective parameters defined implicitly by Eqs.~\eqref{eq:BE_SL_raw}, complemented by light propagation over length $\alpha_W/k_p$. Note also that the phase shift $\alpha_W$ can be absorbed into the field amplitudes simply by renaming
\begin{equation}
\label{eq:sl_noalpha_0}
\mathrm{A}_We^{i\alpha_W} \to \mathrm{A}_W \,, \qquad \mathrm{B}_We^{-i\alpha_W} \to \mathrm{B}_W \,,
\end{equation}
which yields:
\begin{eqnarray}
\label{eq:sl_pm_0}
\mathrm{B}_W &=& \mathcal{R}_W\mathrm{A}_W \,,\nonumber \\
\mathrm{E}_+ &=& \frac{\sqrt{\gamma_{1W}}}{\ell_W(0)\sqrt{\tau}}\mathrm{A}_W \,.
\end{eqnarray}
The corresponding equivalent model of the common mode is shown in
Figure~\ref{fig:pm_modes_2}(a).
\epubtkImage{fig32.png}{
\begin{figure}
\centerline{\includegraphics[width=.5\textwidth]{fig32}}
\caption{Common (top) and differential (bottom) modes of the dual-recycled Fabry--P{\'e}rot--Michelson interferometer, reduced to the single cavities using the scaling law model.}
\label{fig:pm_modes_2}
\end{figure}}
Taking into account that the main goal of power recycling is the increase of the power $\mathcal{I}_c=\hbar\omega_p|E_+|^2$ circulating in the arm cavities, for a given laser power $\mathcal{I}_0=\hbar\omega_p|A_W|^2$, the optimal tuning of the power recycling cavity corresponds to the critical coupling of the common mode with the laser:
\begin{equation}
\gamma_{1W} = \gamma_2 \,, \qquad \delta_W = 0 \,.
\end{equation}
In this case,
\begin{equation}
\mathrm{E}_+ = \frac{\mathrm{A}_W}{2\sqrt{\gamma_2\tau}} \Longrightarrow
\mathcal{I}_c = 2\mathcal{I}_{\mathrm{arm}} = \frac{\mathcal{I}_0}{4\gamma_2\tau} \,.
\end{equation}
Note that this regime can be achieved even with the detuned arm cavities, $\delta\ne0$.
Consider now the differential mode quantum field amplitudes as given in Eqs.~\eqref{eq:sl_pm_raw_1}. Note a factor $e^{i\omega\tau_S}$ that describes a frequency-dependent phase shift the sideband fields acquire on their pass through the signal recycling cavity. It is due to this frequency-dependent phase shift that the differential mode cannot be reduced, strictly speaking, to a single effective cavity mode, and a more complicated two-cavity model of Figure~\ref{fig:pm_modes_1} should be used instead. The reduction to a single mode is nevertheless possible in the special case of a short signal-recycling cavity, i.e., such that:
\begin{equation}
|\Omega|\tau_S \ll 1 \,.
\end{equation}
The above condition is satisfied in a vast majority of the proposed schemes of advanced GW interferometers and in all current interferometers that make use of the recycling techniques~\cite{VIRGOsite,GEOsite}.
In this case, the phase shift $\phi_S$ can be approximated by the frequency-independent value:
\begin{equation}
\omega\tau_S \approx \omega_p\tau_S \equiv \phi_S \,.
\end{equation}
It allows one to introduce the following effective parameters:
\begin{eqnarray}
\label{sl_gd}
\gamma_1 = \gamma_{1\mathrm{arm}}\Re\frac{1-\sqrt{R_S}e^{2i\phi_S}}{1+\sqrt{R_S}e^{2i\phi_S}}
&=& \frac{\gamma_{1\mathrm{arm}}T_S}{1 + 2\sqrt{R_S}\cos2\phi_S + R_S} \,,\nonumber \\
\delta = \delta_{\mathrm{arm}}
- \gamma_{1\mathrm{arm}}\Im\frac{1-\sqrt{R_S}e^{2i\phi_S}}{1+\sqrt{R_S}e^{2i\phi_S}}
&=& \delta_{\mathrm{arm}}
+ \frac{2\gamma_{1\mathrm{arm}}\sqrt{R_S}\sin2\phi_S}{1 + 2\sqrt{R_S}\cos2\phi_S + R_S} \,, \nonumber \\
\gamma &=& \gamma_1 + \gamma_2 \,, \nonumber \\
\ell(\Omega) &=& \gamma - i(\delta+\Omega) \,
\end{eqnarray}
and to rewrite Eqs.~\eqref{eq:sl_pm_raw_1} as follows:
\begin{eqnarray}
\hat{\mathrm{b}}_S(\omega) &=&
\left[\mathcal{R}_1(\Omega)\hat{\mathrm{a}}_S(\omega)e^{i\alpha_S}
+ \mathcal{T}(\Omega)\hat{\mathrm{g}}_-(\omega)
+ \frac{2\sqrt{\gamma_1}\hat{X}_-(\omega)}{\ell(\Omega)}\right]e^{i\alpha_S} \,,
\label{eq:sl_pm_1_b} \\
\hat{\mathrm{e}}_-(\omega) &=& \frac{
\sqrt{\gamma_1}\hat{\mathrm{a}}_S(\omega)e^{i\alpha_S}
+ \sqrt{\gamma_2}\hat{\mathrm{g}}_-(\omega)
+ \hat{X}_-(\omega)
}{\ell(\Omega)\sqrt{\tau}} \,, \label{eq:sl_pm_1_e}
\end{eqnarray}
where the reflectivity and transmittance of the equivalent Fabry--P{\'e}rot cavity are still defined by the Eqs.~\eqref{eq:FP_RT}, but with new effective parameters~\eqref{sl_gd}, and
\begin{equation}
\alpha_S = \arg\frac{e^{i\phi_S}}{1+\sqrt{R_S}e^{2i\phi_S}}
\end{equation}
is the phase shift introduced by the signal recycled cavity. Along similar lines as in the common mode case, we make the following change of variables
\begin{equation}
\hat{\mathrm{a}}_Se^{i\alpha_S} \to \hat{\mathrm{a}}_S \,, \qquad
\hat{\mathrm{b}}_Se^{-i\alpha_S} \to \hat{\mathrm{b}}_S \,.
\end{equation}
Eqs.~\eqref{eq:sl_pm_1_b} and \eqref{eq:sl_pm_1_e} have exactly the same form as the corresponding equations for the Fabry--P{\'e}rot cavity~\eqref{eq:FP_1_short}. Thus, we have successfully built a single cavity model for the differential mode, see Figure~\ref{fig:pm_modes_2}(b).
The mechanical equations of motion for the effective cavity are absolutely the same as for an ordinary Fabry--P{\'e}rot cavity considered in Section~\ref{sec:Fabry-Perot} except for the new values of the effective mirrors' mass $2M$ and effective circulating power $\mathcal{I}_c=2\mathcal{I}_{\mathrm{arm}}$. Bearing this in mind, we can procede to the quantum noise spectral density calculation for this interferometer.
\subsubsection{Spectral densities for the Fabry--P{\'e}rot--Michelson interferometer}
\label{sec:sl_noises}
The scaling law we have derived above enables us to calculate spectral
densities of quantum noise for a dual-recycled
Fabry--P{\'e}rot--Michelson featured in Figure~\ref{fig:advligo} as if
it were a bare Fabry--P{\'e}rot cavity with movable mirrors pumped
from one side, similar to that shown in Figure~\ref{fig:pm_modes_2}.
\epubtkImage{fig33.png}{
\begin{figure}
\centerline{\includegraphics[width=.55\textwidth]{fig33}}
\caption{The differential mode of the dual-recycled Fabry--P{\'e}rot--Michelson interferometer in simplified notation~\eqref{eq:simple_notations}.}
\label{fig:model}
\end{figure}}
We remove some of the subscripts in our notations, for the sake of notational brevity:
\begin{equation}
\label{eq:simple_notations}
\begin{array}{llll}
\hat{\mathrm{a}}_S \to \hat{\mathrm{a}} \,, & \qquad \hat{\mathrm{b}}_S \to \hat{\mathrm{b}} \,, & \qquad \hat{\mathrm{e}}_- \to \hat{\mathrm{e}} \,, & \qquad \mathrm{E}_- \to \mathrm{E} \,,\\
x_- \to x \,, & \qquad F_-^{\mathrm{r.p.}} \to F_{\mathrm{r.p.}} \,, & \qquad
G_- \to G \,, & \qquad F_-^{\mathrm{b.a.}} \to F_{\mathrm{b.a.}},
\end{array}
\end{equation}
(compare Figures~\ref{fig:pm_modes_2} and \ref{fig:model}). We also choose the phase of the classical field $\mathrm{E}$ amplitude inside the arm cavities to be zero:
\begin{equation}
\Im\mathrm{E} = 0 \Longrightarrow \mathbf{E} = \sqrt{2}\mathrm{E}\genfrac{(}{)}{0pt}{}{1}{0}
\end{equation}
that obviously does not limit the generality of our consideration, yet sets the reference point for all the classical and quantum fields' phases.
With this in mind, we rewrite I/O-relations~\eqref{eq:sl_pm_1_b} and \eqref{eq:sl_pm_1_e} in the two-photon quadratures notation:
\begin{eqnarray}
\hat{\boldsymbol{b}}(\Omega) &=& \mathbb{R}_1(\Omega)\hat{\boldsymbol{a}}(\Omega)
+ \mathbb{T}(\Omega)\hat{\boldsymbol{g}}(\Omega)
+ \sqrt{\frac{2MJ\gamma_1}{\hbar}}\,
\frac{\mathbf{D}(\Omega)\hat{x}(\Omega)}{\mathcal{D}(\Omega)}
\,, \label{eq:sl_quad_b} \\
\hat{\boldsymbol{e}}(\Omega) &=& \frac{1}{\sqrt{\tau}}\biggl\{
\mathbb{L}(\Omega)
[\sqrt{\gamma_1}\hat{\boldsymbol{a}}(\Omega) + \sqrt{\gamma_2}\hat{\boldsymbol{g}}(\Omega)]
+ \sqrt{\frac{MJ}{2\hbar}}\,
\frac{\mathbf{D}(\Omega)\hat{x}(\Omega)}{\mathcal{D}(\Omega)}
\biggr\} , \label{eq:sl_quad_e}
\end{eqnarray}
where the matrices $\mathbb{L}$, $\mathbb{R}_1$, $\mathbb{T}$ are defined by Eqs.~\eqref{FP_bbL} and \eqref{FP_bbRT},
\begin{equation}
\mathbf{D}(\Omega) = \mathcal{D}(\Omega)\mathbb{L}(\Omega)\genfrac{(}{)}{0pt}{}{0}{1}
= \svector{-\delta}{\gamma-i\Omega}
\end{equation}
and
\begin{equation}
\label{eq:FPMI_J_def}
J = \frac{4\hbar k_p^2\mathrm{E}^2}{M\tau} = \frac{4\omega_p\mathcal{I}_c}{McL}
= \frac{8\omega_p\mathcal{I}_{\mathrm{arm}}}{McL}
\end{equation}
is the normalized optical power, circulating in the interferometer arms.
Suppose that the output beam is registered by the homodyne detector; see Section~\ref{sec:homodyne}. Combining Eqs.~\eqref{eq:sl_quad_b} and \eqref{eq:homodyne_lossy}, we obtain for the homodyne detector a readout expressed in units of signal force:
\begin{equation}
\label{ord_i_meas}
\hat{O}^{F,\mathrm{loss}}(\Omega) = \frac{\hat{\mathcal{X}}^{\mathrm{loss}}}{\chi_{xx}^{\mathrm{eff,\,FP}}(\Omega)} + \hat{\mathcal{F}}(\Omega)+\frac{G(\Omega)}{2} \,,
\end{equation}
where
\begin{equation}
\label{eq:ord_x_meas}
\hat{\mathcal{X}}^{\mathrm{loss}}(\Omega) = \frac{\hat{O}^{(0),\,\mathrm{loss}}(\Omega)}{\chi_{OF}(\Omega)} = \sqrt{\frac{\hbar}{2MJ\gamma_1}}\,
\frac{\mathcal{D}(\Omega)}{\boldsymbol{H}^{\mathsf{T}}[\phi_{\mathrm{LO}}]\mathbf{D}(\Omega)}
\boldsymbol{H}^{\mathsf{T}}[\phi_{\mathrm{LO}}][
\mathbb{R}_1(\Omega)\hat{\boldsymbol{a}}(\Omega) + \mathbb{T}(\Omega)\hat{\boldsymbol{g}}(\Omega)
+ \epsilon_d\hat{\boldsymbol{n}}(\Omega)
]
\end{equation}
stands for the measurement noise, which is typically referred to as shot noise in optomechanical measurement with the interferometer opto-mechanical response function defined as
\begin{equation}
\chi_{OF}(\Omega) = \sqrt{\frac{2MJ\gamma_1}{\hbar}}\frac{\boldsymbol{H}^{\mathsf{T}}[\phi_{\mathrm{LO}}]\mathbf{D}(\Omega)}{\mathcal{D}(\Omega)}
\end{equation}
and the back-action noise caused by the radiation pressure fluctuations equal to
\begin{equation}
\label{eq:ord_F_pert}
\hat{\mathcal{F}}(\Omega)\equiv\hat{F}_{\mathrm{b.a.}}(\Omega) = \sqrt{2\hbar MJ}\begin{bmatrix}1\\0\end{bmatrix}^{\mathsf{T}}\mathbb{L}(\Omega)
[\sqrt{\gamma_1}\hat{\boldsymbol{a}}(\Omega) + \sqrt{\gamma_2}\hat{\boldsymbol{g}}(\Omega)]
\end{equation}
The dynamics of the interferometer is described by the effective susceptibility $\chi_{xx}^{\mathrm{eff,\,FP}}(\Omega)$ that is the same as the one given by Eq.~\eqref{eq:chi_xx_eff_FP} where
\begin{equation}
\label{eq:sl_K}
K(\Omega) = \frac{MJ\delta}{\mathcal{D}(\Omega)}
\end{equation}
is the frequency-dependent optical rigidity that has absolutely the same form as that of a single Fabry--P{\'e}rot cavity given by Eq.~\eqref{eq:FP_K}.
Suppose then that the input field of the interferometer is in the squeezed quantum state that is equivalent to the following transformation of the input fields:
\begin{equation}
\label{a_sqz}
\hat{\boldsymbol{a}} = \mathbb{S}_{\sqz}[r,\theta]\hat{\boldsymbol{a}}^{\vac} \,,
\end{equation}
where the squeezing matrix is defined by Eq.~\eqref{eq:SQZ_matrix_transform}, and the quadrature vector $\hat{\boldsymbol{a}}^{\vac}$ corresponds to the vacuum state.
Using the rules of spectral densities computation given in Eqs.~\eqref{eq:S_Y_sqz} and \eqref{eq:S_YZ_sqz}, taking into account unitarity conditions~\eqref{eq:FP_bbRT_u}, one can get the following expressions for the power (double-sided) spectral densities of the dual-recycled Fabry--P{\'e}rot--Michelson interferometer measurement and back-action noise sources as well as their cross-correlation spectral density:
\begin{eqnarray}
S_{\mathcal{X}\mathcal{X}}(\Omega) &=& \frac{\hbar}{4MJ\gamma_1}
\frac{|\mathcal{D}(\Omega)|^2}{\left|\boldsymbol{H}^{\mathsf{T}}[\phi_{\mathrm{LO}}]\mathbf{D}(\Omega)\right|^2}\nonumber\\
&&\times \bigl\{
\boldsymbol{H}^{\mathsf{T}}(\phi_{\mathrm{LO}})\mathbb{R}_1(\Omega)
[\mathbb{S}_{\sqz}(2r,\theta) - \mathbb{I}]
\mathbb{R}_1^\dagger(\Omega)\boldsymbol{H}[\phi_{\mathrm{LO}}]
+ 1 + \epsilon_d^2
\bigr\} ,\label{eq:FPMI_Sx} \\
S_{\mathcal{F}\mathcal{F}}(\Omega) &=& \hbar MJ\,\begin{bmatrix}1\\0\end{bmatrix}^{\mathsf{T}}\mathbb{L}(\Omega)
[\gamma_1\mathbb{S}_{\sqz}(2r,\theta) + \gamma_2]\mathbb{L}^\dagger(\Omega)\genfrac{(}{)}{0pt}{}{1}{0} \,,\label{eq:FPMI_SF} \\
S_{\mathcal{X}\mathcal{F}}(\Omega) &=& \frac{\hbar}{2}
\frac{\mathcal{D}(\Omega)}{\boldsymbol{H}^{\mathsf{T}}[\phi_{\mathrm{LO}}]\mathbf{D}(\Omega)}
\boldsymbol{H}^{\mathsf{T}}(\phi_{\mathrm{LO}})\bigl[
\mathbb{R}_1(\Omega)\mathbb{S}_{\sqz}(2r,\theta)
+ \sqrt{\gamma_2/\gamma_1}\mathbb{T}(\Omega)
\bigr]\mathbb{L}^\dagger(\Omega)\genfrac{(}{)}{0pt}{}{1}{0} \,. \label{eq:FPMI_SxF}
\end{eqnarray}
These spectral densities satisfy the Sch\"rodinger--Robertson uncertainty relation:
\begin{equation}
\label{SxSFSxf}
S_{\mathcal{X}\mathcal{X}}(\Omega)S_{\mathcal{F}\mathcal{F}}(\Omega) - |S_{\mathcal{X}\mathcal{F}}(\Omega)|^2 \ge \frac{\hbar^2}{4}
\end{equation}
of the same form as in the general linear measurement case considered in Section~\ref{sec:gen_linear_measurement}, see Eq.~\eqref{eq:gen_spdens_uncert_rel}, with the exact equality in the ideal lossless case:
\begin{equation}
\label{no_loss}
\gamma_2 = 0 \,, \qquad \eta_d = 1 \,.
\end{equation}
see Appendix~\ref{app:SxSFSxf}
\subsubsection{Full transfer matrix approach to calculation of the Fabry--P{\'e}rot--Michelson interferometer quantum noise}
In order to compute the sum quantum noise spectral density one has to first calculate $S_{\mathcal{X}\mathcal{X}}(\Omega)$, $S_{\mathcal{F}\mathcal{F}}(\Omega)$ and $S_{\mathcal{X}\mathcal{F}}(\Omega)$ using Eqs.~\eqref{eq:FPMI_Sx}, \eqref{eq:FPMI_SF}, and \eqref{eq:FPMI_SxF} and then insert them into the general formula~\eqref{eq:gen_spdens}.
However, there is another option that is more convenient from the computational point of view. One can compute the full quantum noise transfer matrix of the Fabry--P{\'e}rot--Michelson interferometer in the same manner as for a single mirror in Section~\ref{sec:mirror_full_transfer_mat}. The procedure is rather straightforward. Write down the readout observable of the homodyne detector in units of signal force:
\begin{eqnarray}
\label{eq:X_sum_BC}
\hat{O}^F(\Omega) =
\frac{\hat{\mathcal{X}}^{\mathrm{loss}}(\Omega)}{\chi_{xx}^{\mathrm{eff,\,FP}}(\Omega)}+\hat{\mathcal{F}}(\Omega)+\frac{G(\Omega)}{2} = \frac{G(\Omega)}{2}+
\sqrt{\frac{\hbar}{2MJ\gamma_1}}
\frac{-M\Omega^2}{\boldsymbol{H}^{\mathsf{T}}[\phi_{\mathrm{LO}}]\mathbf{D}(\Omega)}
\boldsymbol{H}^{\mathsf{T}}[\phi_{\mathrm{LO}}] \nonumber\\ \times\biggl\{
\mathbb{C}_1(\Omega)\mathbb{S}_{\sqz}[r,\theta]\hat{\boldsymbol{a}}^{\vac}(\Omega)
+ \mathbb{C}_2(\Omega)\hat{\boldsymbol{g}}(\Omega)
+ \left[\mathcal{D}(\Omega) - \frac{J\delta}{\Omega^2}\right]
\epsilon_d\hat{\boldsymbol{n}}(\Omega)
\biggr\}\,,
\end{eqnarray}
where matrices $\mathbb{C}_{1,2}(\Omega)$ can be computed using the fact that
\begin{equation*}
[\delta + \mathbf{D}(\Omega)\begin{bmatrix}1\\0\end{bmatrix}^{\mathsf{T}}]\mathbb{L}(\Omega) = \smatrix{0}{0}{1}{0} ,
\end{equation*}
which yields:
\begin{eqnarray}
\mathbb{C}_1(\Omega)
&=& \smatrix{2\gamma_1(\gamma-i\Omega) - \mathcal{D}(\Omega) + J\delta/\Omega^2}
{-2\gamma_1\delta}{2\gamma_1\delta-2J\gamma_1/\Omega^2}
{2\gamma_1(\gamma-i\Omega) - \mathcal{D}(\Omega) + J\delta/\Omega^2} , \\
\mathbb{C}_2(\Omega) &=& 2\sqrt{\gamma_1\gamma_2}
\smatrix{\gamma-i\Omega}{-\delta}{\delta-J/\Omega^2}{\gamma-i\Omega} .
\end{eqnarray}
In the GW community, it is more common to normalize the signal of the interferometer in units of GW amplitude spectrum $h(\Omega)$. This can easily be done using the simple rule given in Eq.~\eqref{eq:gen_lin_F_to_h_transform} and taking into account that in our case $G(\Omega)\to G(\Omega)/2$:
\begin{eqnarray}
\hat{O}^h(\Omega) = h_{\mathrm{GW}}(\Omega)+\hat{h}_n(\Omega) = h_{\mathrm{GW}}(\Omega)+
\frac{2}{L}\sqrt{\frac{\hbar}{2MJ\gamma_1}}
\frac{1}{\boldsymbol{H}^{\mathsf{T}}[\phi_{\mathrm{LO}}]\mathbf{D}(\Omega)}
\boldsymbol{H}^{\mathsf{T}}[\phi_{\mathrm{LO}}] \nonumber\\ \times\biggl\{
\mathbb{C}_1(\Omega)\mathbb{S}_{\sqz}[r,\theta]\hat{\boldsymbol{a}}^{\vac}(\Omega)
+ \mathbb{C}_2(\Omega)\hat{\boldsymbol{g}}(\Omega)
+ \left[\mathcal{D}(\Omega) - \frac{J\delta}{\Omega^2}\right]
\epsilon_d\hat{\boldsymbol{n}}(\Omega)
\biggr\}\,,
\end{eqnarray}
where $h_{\mathrm{GW}}(\Omega)$ is the spectrum of the GW signal and the second term $\hat{h}_n(\Omega)$ stands for the sum quantum noise expressed in terms of metrics variation spectrum units, i.e., in $\mathrm{Hz}^{-1/2}$.
The power (double-sided) spectral density of the sum quantum noise then reads:
\begin{eqnarray}
\label{eq:S_sum_BC}
S^h(\Omega) = \frac{4 S^F(\Omega)}{M^2L^2\Omega^4} &=& \frac{\hbar}{MJ\gamma_1L^2}
\frac{1}{|\boldsymbol{H}^{\mathsf{T}}[\phi_{\mathrm{LO}}]\mathbf{D}(\Omega)|^2} \nonumber\\ \times\biggl\{
\boldsymbol{H}^{\mathsf{T}}[\phi_{\mathrm{LO}}]\bigl[
\mathbb{C}_1(\Omega)\mathbb{S}_{\sqz}[2r,\theta]\mathbb{C}_1^\dagger(\Omega)
&+& \mathbb{C}_2(\Omega)\mathbb{C}_2^\dagger(\Omega)
\bigr]\boldsymbol{H}[\phi_{\mathrm{LO}}]
+ \left|\mathcal{D}(\Omega) - \frac{J\delta}{\Omega^2}\right|^2\epsilon_d^2
\biggr\} .
\end{eqnarray}
In conclusion, we should say that the quantum noise of the Fabry--P{\'e}rot--Michelson interferometer has been calculated in many papers, starting from the seminal work by Kimble et al.~\cite{02a1KiLeMaThVy} where a resonance-tuned case with $\delta=0$ was analyzed, and then by Buonanno and Chen in~\cite{Buonanno2001, Buonanno2003}, who considered a more general detuned case.
Thus, treading their steps, we have shown that the quantum noise of the Fabry--P{\'e}rot--Michelson interferometer (as well as the single cavity Fabry--P{\'e}rot one) has the following distinctive features:
\begin{itemize}
\item It comprises two effective noise sources as in any quantum
linear measurement device. These are measurement noise
$\hat{\mathcal{X}}^{\mathrm{loss}}$, more frequently called \emph{quantum
shot noise} in the GW community, and the back-action noise $\hat{\mathcal{F}}$, often referred to as \emph{quantum radiation-pressure noise}.
\item These noise sources are correlated and this correlation depends not only on the homodyne angle $\phi_{\mathrm{LO}}$ or the correlations in the input light (e.g., squeezing angle $\theta$ in case of squeezed input), but also on the interferometer effective detuning $\delta$, which, according to the scaling law theorem, can be changed by varying signal-recycling cavity parameters.
\item The scaling law theorem also shows that changing the arm cavities' detuning is equivalent to the modification of the signal recycling cavity parameters in terms of effective detuning and bandwidth of the interferometer.
\item Another important corollary of the scaling law is that the effective bandwidths and detunings for the common and differential optical modes can be chosen independently, thus making it possible to tune the former in resonance with the pumping laser to keep as high a value of the circulating optical power in the arms as possible, and to detune the latter one to modify the test masses dynamical response by virtue of the introduction of optical rigidity that arises in the detuned cavity as we have shown.
\end{itemize}
All of these features can be used to decrease the quantum noise of the interferometer and reach a sensitivity below the SQL in a decent range of frequencies as we show in Section~\ref{sec:sub-SQL_schemes}.
\newpage
\section{Schemes of GW Interferometers with Sub-SQL Sensitivity}
\label{sec:sub-SQL_schemes}
\subsection{Noise cancellation by means of cross-correlation}
\label{sec:noise_cancellation}
\subsubsection{Introduction}
In this section, we consider the interferometer configurations that use the idea of the cross-correlation of the shot and the radiation pressure noise sources discussed in Section~\ref{sec:toy_FI_correlation}. This cross-correlation allows the measurement and the back-action noise to partially cancel each other out and thus effectively reduce the sum quantum noise to below the SQL.
As we noted above, Eq.~\eqref{eq:FPMI_SxF} tells us that this cross-correlation can be created by tuning either the homodyne angle $\phi_{\mathrm{LO}}$, the squeezing angle $\theta$, or the detuning $\delta$. In Section~\ref{sec:toy_FI_correlation}, the simplest particular case of the frequency-independent correlation created by means of measurement of linear combination of the phase and amplitude quadratures, that is, by using the homodyne angle $\phi_{\mathrm{LO}}\ne\pi/2$, has been considered. We were able to obtain a narrow-band sensitivity gain at some given frequency that was similar to the one achievable by introducing a constant rigidity to the system, therefore such correlation was called effective rigidity.
However, the broadband gain requires a frequency-dependent correlation, as it was first demonstrated for optical interferometric position meters~\cite{Unruh1982}, and then for general position measurement case~\cite{87a1eKh}. Later, this idea was developed in different contexts by several authors~\cite{JaekelReynaud1990, Pace1993, 96a2eVyMa, 02a1KiLeMaThVy, PhysRevD.68.042001_2003_Harms, 06pth1Ha, PhysRevLett.95.211102_2005_Vahlbruch, Arcizet2006}. In particular, in~\cite{02a1KiLeMaThVy}, a practical method of creation of the frequency-dependent correlation was proposed, based on the use of additional \emph{filter cavities}, which were proposed to be placed either between the squeeze light source and the main interferometer, creating the frequency-dependent squeezing angle (called \emph{pre-filtering}), or between the main interferometer and the homodyne detector, creating the effective frequency-dependent squeezing angle (\emph{post-filtering}). As we show below, in principle, both pre- and post-filtering can be used together, providing some additional sensitivity gain.
It is necessary to note also an interesting method of noise cancellation proposed by Tsang and Caves recently~\cite{PhysRevLett.105.123601_2010_Tsang}. The idea was to use \emph{matched squeezing}; that is, to place an additional cavity inside the main interferometer and couple the light inside this additional cavity with the differential mode of the interferometer by means of an optical parametric amplifier (OPA). The squeezed light created by the OPA should compensate for the ponderomotive squeezing created by back-action at all frequencies and thus decrease the quantum noise below the SQL at a very broad frequency band. However, the thorough analysis of the optical losses influence, that as we show later, are ruinous for the subtle quantum correlations this scheme is based on, was not performed.
Coming back to the filter-cavities--based interferometer topologies, we limit ourselves here by the case of the resonance-tuned interferometer, $\delta=0$. This assumption simplifies all the equations considerably, and allows one to clearly separate the sensitivity gain provided by the quantum noise cancellation due to cross-correlation from the one provided by the optical rigidity, which will be considered in Section~\ref{sec:optical_rigidity}.
We also neglect optical losses \emph{inside} the interferometer, assuming that $\gamma_2=0$. In broadband interferometer configurations considered here, with typical values of $\gamma\gtrsim10^3\mathrm{\ s}^{-1}$, the influence of these losses is negligible compared to those of the photodetector inefficiency and the losses in the filter cavities. Indeed, taking into account the fact that with modern high-reflectivity mirrors, the losses per bounce do not exceed $A_{\mathrm{arm}}\lesssim10^{-4}$, and the arms lengths of the large-scale GW detectors are equal to several kilometers, the values of $\gamma_2\lesssim1\mathrm{\ s}^{-1}$, and, correspondingly, $\gamma_2/\gamma\lesssim10^{-3}$, are feasible. At the same time, the value of photodetector quantum inefficiency $\epsilon_d^2\approx1-\eta_d\approx0.05$ (factoring in the losses in the interferometer output optical elements as well) is considered quite optimistic. Note, however, that in narrow-band regimes considered in Section~\ref{sec:optical_rigidity}, the bandwidth $\gamma$ can be much smaller and influence of $\gamma_2$ could be significant; therefore, we take these losses into account in Section~\ref{sec:optical_rigidity}.
Using these assumptions, the quantum noises power (double-sided) spectral densities~\eqref{eq:FPMI_Sx}, \eqref{eq:FPMI_SF} and \eqref{eq:FPMI_SxF} can be rewritten in the following explicit form:
\begin{eqnarray}
S_{\mathcal{X}\mathcal{X}}(\Omega) &=& \frac{\hbar}{2M\Omega^2\mathcal{K}(\Omega)\sin^2\phi_{\mathrm{LO}}}
\bigl[\cosh2r + \sinh2r\cos2(\theta-\phi_{\mathrm{LO}}) + \epsilon_d^2\bigr] , \label{eq:vr_noises_x} \\
S_{\mathcal{F}\mathcal{F}}(\Omega)
&=& \frac{\hbar M\Omega^2\mathcal{K}(\Omega)}{2}(\cosh2r + \sinh2r\cos2\theta) , \label{eq:vr_noises_F}\\
S_{\mathcal{X}\mathcal{F}}(\Omega)
&=& \frac{\hbar}{2\sin\phi_{\mathrm{LO}}}[\cosh2r\cos\phi_{\mathrm{LO}} + \sinh2r\cos(2\theta-\phi_{\mathrm{LO}})]
\label{eq:vr_noises_xF} ,
\end{eqnarray}
where
\begin{equation}
\label{eq:varK}
\mathcal{K}(\Omega) = \frac{2J\gamma}{\Omega^2(\gamma^2+\Omega^2)}
\end{equation}
is the convenient optomechanical coupling factor introduced in~\cite{02a1KiLeMaThVy}.
Eq.~\eqref{eq:X_sum_BC} and \eqref{eq:S_sum_BC} for the sum quantum noise and its power (double-sided) spectral density in this case can also be simplified significantly:
\begin{eqnarray}
\hat{h}_{n}(\Omega) &=& \frac{2}{L}\sqrt{\frac{\hbar}{2MJ\gamma}}\,
\frac{1}{\sin\phi_{\mathrm{LO}}}
\boldsymbol{H}^{\mathsf{T}}[\phi_{\mathrm{LO}}]\bigl[
(\gamma+i\Omega)\mathbb{K}(\Omega)\mathbb{S}_{\sqz}[r,\theta]\hat{\boldsymbol{a}}^{\vac}(\Omega)
+ (\gamma-i\Omega)\epsilon_d\hat{\boldsymbol{n}}(\Omega)
\bigr] , \label{eq:vr_X_sum} \\
S^h(\Omega) &=& \frac{2\hbar}{ML^2\Omega^2\mathcal{K}(\Omega)\sin^2\phi_{\mathrm{LO}}}
\bigl[
\boldsymbol{H}^{\mathsf{T}}[\phi_{\mathrm{LO}}]\mathbb{K}(\Omega)\mathbb{S}_{\sqz}[2r,\theta]
\mathbb{K}^\dagger(\Omega)\boldsymbol{H}[\phi_{\mathrm{LO}}]
+ \epsilon_d^2
\bigr] , \label{eq:vr_sum_noise}
\end{eqnarray}
where
\begin{equation}
\mathbb{K}(\Omega) = \smatrix{1}{0}{-\mathcal{K}(\Omega)}{1} .
\end{equation}
In Section~\ref{sec:corr_ideal} we consider the optimization of the spectral density~\eqref{eq:vr_sum_noise}, assuming that the arbitrary frequency dependence of the homodyne and/or squeezing angles can be implemented. As we see below, this case corresponds to the ideal lossless filter cavities. In Section~\ref{sec:corr_real}, we consider two realistic schemes, taking into account the losses in the filter cavities.
\subsubsection{Frequency-dependent homodyne and/or squeezing angles}
\label{sec:corr_ideal}
\paragraph*{Classical optimization.}
As a reference point, consider first the simplest case of frequency \emph{in}dependent homodyne and squeezing angles. We choose the specific values of these parameters following the \emph{classical optimization}, which minimizes the shot noise~\eqref{eq:vr_noises_x} without taking into account the back action. Because, the shot noise dominates at high frequencies, therefore, this optimization gives a smooth broadband shape of the sum noise spectral density.
It is evident that this minimum is provided by
\begin{equation}
\phi_{\mathrm{LO}} = \frac{\pi}{2} \,, \qquad \theta = 0 \,.
\end{equation}
In this case, the sum quantum noise power (double-sided) spectral density is equal to
\begin{equation}
\label{eq:S_X_Caves}
S^h(\Omega) = \frac{2\hbar}{ML^2\Omega^2}\left[
\frac{e^{-2r} + \epsilon_d^2}{\mathcal{K}(\Omega)} + \mathcal{K}(\Omega)e^{2r}
\right] .
\end{equation}
It is easy to note the similarity of this spectral density with the ones of the toy position meter considered above, see Eq.~\eqref{eq:S_h_fm}. The only significant differences introduced here are the optical losses and the decrease of the optomechanical coupling at high frequencies due to the finite bandwidth $\gamma$ of the interferometer. If $\epsilon_d=0$ and $\gamma\gg\Omega$, then Eqs.~\eqref{eq:S_h_fm} and \eqref{eq:S_X_Caves} become identical, with the evident correspondence
\begin{equation}
\mathcal{I}_0\digamma^2 \leftrightarrow \frac{\mathcal{I}_c}{\gamma\tau} \,.
\end{equation}
In particular, the spectral density~\eqref{eq:S_X_Caves} can never be smaller than the free mass SQL $S^h_{\mathrm{SQL\,f.m.}}(\Omega)$ (see~\eqref{eq:S_h_SQL_fm}). Indeed, it can be minimized at any given frequency $\Omega$ by setting
\begin{equation}
\mathcal{K}(\Omega) = e^{-r}\sqrt{e^{-2r} + \epsilon_d^2} \,,
\end{equation}
and in this case,
\begin{equation}
\xi^2(\Omega) \equiv \frac{S^h(\Omega)}{S^h_{\mathrm{SQL\,f.m.}}(\Omega)}
= \sqrt{1 + \epsilon_d^2e^{2r}} \ge 1 \,.
\end{equation}
The spectral density~\eqref{eq:S_X_Caves} was first calculated in the pioneering work of~\cite{81a1Ca}, where the existence of two kinds of quantum noise in optical interferometric devices, namely the measurement (shot) noise and the back action (radiation pressure) noise, were identified for the first time, and it was shown that the injection of squeezed light with $\theta=0$ into the interferometer dark port is equivalent to the increase of the optical pumping power. However, it should be noted that in the presence of optical losses this equivalence holds unless squeezing is not too strong, $e^{-r}>\epsilon_d$.
\epubtkImage{fig34.png}{
\begin{figure}[htb]
\centerline{\includegraphics[width=.8\textwidth]{fig34}}
\caption{Examples of the sum quantum noise spectral densities of the classically-optimized ($\phi_{\mathrm{LO}}=\pi/2$, $\theta=0$) resonance-tuned interferometer. `Ordinary': $J=J_{\mathrm{aLIGO}}$, no squeezing. `Increased power': $J=10J_{\mathrm{aLIGO}}$, no squeezing. `Squeezed': $J=J_{\mathrm{aLIGO}}$, 10~dB squeezing. For all plots, $\gamma=2\pi\times500\mathrm{\ s}^{-1}$ and $\eta_d=0.95$.}
\label{fig:LIGO_res}
\end{figure}}
The noise spectral density curves for the resonance-tuned interferometer are drawn in Figure~\ref{fig:LIGO_res}. The default parameters for this and all subsequent similar plots are chosen to be close to those planned for the Advanced LIGO interferometer: the value of $J = J_{\mathrm{aLIGO}} \equiv (2\pi\times100)^3\mathrm{\ s}^{-3}$ corresponds to the circulating power of $\mathcal{I}_{\mathrm{arm}} = 840\mathrm{\ kW}$, $L=4\mathrm{\ km}$, and $M=40\mathrm{\ kg}$; the interferometer bandwidth $\gamma=2\pi\times500\mathrm{\ s}^{-1}$ is close to the one providing the best sensitivity for Advanced LIGO in the presence of technical noise~\cite{08a1KoSiKhDa}; 10~dB squeezing ($e^{2r}=10$), which corresponds to the best squeezing available at the moment (2011) in the low-frequency band~\cite{McKenzie2004, Vahlbruch2006, Vahlbruch_CQG_27_084027_2010}); $\eta_d=0.95$ can be considered a reasonably optimistic estimate for the real interferometer quantum efficiency.
Noteworthy is the proximity of the plots for the interferometer with 10~dB input squeezing and the one with 10-fold increased optical power. The noticeable gap at higher frequencies is due to optical loss.
\paragraph*{Frequency dependent squeezing angle.}
\label{sec:var_sqz}
Now suppose that the homodyne angle can be frequency dependent, and calculate the corresponding minimum of the sum noise spectral density~\eqref{eq:vr_sum_noise}. The first term in square brackets in this equation can be rewritten as:
\begin{equation}
\label{vr_sqz_1}
\boldsymbol{V}^{\mathsf{T}}\mathbb{P}[\theta]\mathbb{S}_{\sqz}[2r,0]\mathbb{P}^\dagger[\theta]\boldsymbol{V}
= e^{2r}(V_c\cos\theta + V_s\sin\theta)^2 + e^{-2r}(-V_c\sin\theta + V_s\cos\theta)^2
\,,
\end{equation}
where
\begin{equation}
\mathbf{V} \equiv \svector{V_c}{V_s} = \mathbb{K}^\dagger(\Omega)\boldsymbol{H}[\phi_{\mathrm{LO}}] \,.
\end{equation}
It is evident that the minimum of~\eqref{vr_sqz_1} is provided by
\begin{equation}
\label{eq:vr_preopt_theta}
\tan\theta = -\frac{V_c}{V_s} = -\cot\phi_{\mathrm{LO}} + \mathcal{K}(\Omega)
\end{equation}
and is equal to $\boldsymbol{V}^{\mathsf{T}}\boldsymbol{V}e^{-2r}$. Therefore,
\begin{eqnarray}
\label{eq:vr_opt_2}
S^h(\Omega) &=& \frac{2\hbar}{ML^2\Omega^2\mathcal{K}(\Omega)\sin^2\phi_{\mathrm{LO}}}
\Bigl[
\boldsymbol{H}^{\mathsf{T}}[\phi_{\mathrm{LO}}]\mathbb{K}(\Omega)
\mathbb{K}^\dagger(\Omega)\boldsymbol{H}[\phi_{\mathrm{LO}}]e^{-2r}
+ \epsilon_d^2
\Bigr] \nonumber\\
&=& \frac{2\hbar}{ML^2\Omega^2}\left\{
\biggl[
\frac{1}{\mathcal{K}(\Omega)\sin^2\phi_{\mathrm{LO}}} - 2\cot\phi_{\mathrm{LO}} + \mathcal{K}(\Omega)
\biggr]e^{-2r}
+ \frac{\epsilon_d^2}{\mathcal{K}(\Omega)\sin^2\phi_{\mathrm{LO}}}
\right\} \,.
\end{eqnarray}
Thus, we obtaine a well-known result~\cite{02a1KiLeMaThVy} that, using an optimal squeezing angle, the quantum noise spectral density can be reduced by the squeezing factor $e^{-2r}$ in comparison with the vacuum input case. Note, however, that the noise contribution due to optical losses remains unchanged. Concerning the homodyne angle $\phi_{\mathrm{LO}}$, we use again the classical optimization, setting
\begin{equation}
\label{vr_sqz_zeta}
\phi_{\mathrm{LO}} = \frac{\pi}{2} \,.
\end{equation}
In this case, the sum noise power (double-sided) spectral density and the optimal squeezing angle are equal to
\begin{equation}
\label{eq:vr_sqz_3}
S^h(\Omega) = \frac{2\hbar}{ML^2\Omega^2}\left[
\frac{e^{-2r} + \epsilon_d^2}{\mathcal{K}(\Omega)} + \mathcal{K}(\Omega)e^{-2r}
\right]
\end{equation}
and
\begin{equation}
\label{eq:vr_sqz_theta}
\tan\theta = \mathcal{K}(\Omega) \,.
\end{equation}
The sum quantum noise power (double-sided) spectral density~\eqref{eq:vr_sqz_3} is plotted
in Figure~\ref{fig:LIGO_var} for the ideal lossless case and for
$\eta_d=0.95$ (dotted line). In both cases, the optical power and the squeezing
factor are equal to $J = J_{\mathrm{aLIGO}}$ and $e^{2r}=10$, respectively.
\epubtkImage{fig35.png}{
\begin{figure}[htb]
\includegraphics[width=.48\textwidth]{fig35a}\hfill\includegraphics[width=.48\textwidth]{fig35b}
\caption{Examples of the sum quantum noise power (double-sided) spectral densities of the resonance-tuned interferometers with frequency-dependent squeezing and/or homodyne angles. Left: no optical losses, right: with optical losses, $\eta_d=0.95$. `Ordinary': no squeezing, $\phi_{\mathrm{LO}}=\pi/2$. `Squeezed': 10~dB squeezing, $\theta=0$, $\phi_{\mathrm{LO}}=\pi/2$ (these two plots are provided for comparison). Dots [pre-filtering, Eq.~\eqref{eq:vr_sqz_3}]: 10~dB squeezing, $\phi_{\mathrm{LO}}=\pi/2$, frequency-dependent squeezing angle. Dashes [post-filtering, Eq.~\eqref{eq:vr_post_opt}]: 10~dB squeezing, $\theta=0$, frequency-dependent homodyne angle. Dash-dots [pre- and post-filtering, Eq.~\eqref{eq:vr_opt_3}]: 10~dB squeezing, frequency-dependent squeeze and homodyne angles. For all plots, $J=J_{\mathrm{aLIGO}}$ and $\gamma=2\pi\times500\mathrm{\ s}^{-1}$.}
\label{fig:LIGO_var}
\end{figure}}
\paragraph*{Frequency dependent homodyne angle.}
\label{sec:var_post}
Suppose now that the squeezing angle corresponds to the classical optimization:
\begin{equation}
\label{eq:vr_post_theta}
\theta = 0 \,
\end{equation}
and minimize the resulting sum noise spectral density:
\begin{equation}
S^h(\Omega) = \frac{2\hbar}{ML^2\Omega^2}\left[
\frac{\cosh2r + \sinh2r\cos2\phi_{\mathrm{LO}} + \epsilon_d^2}{\mathcal{K}(\Omega)\sin^2\phi_{\mathrm{LO}}}
- 2e^{2r}\cot\phi_{\mathrm{LO}} + \mathcal{K}(\Omega)e^{2r}
\right]
\end{equation}
with respect to $\phi_{\mathrm{LO}}$. The minimum is provided by the following dependence
\begin{equation}
\label{eq:vr_post_zeta}
\cot\phi_{\mathrm{LO}} = \frac{\mathcal{K}(\Omega)}{1+\epsilon_d^2e^{-2r}}\,,
\end{equation}
and is equal to
\begin{equation}
\label{eq:vr_post_opt}
S^h(\Omega) = \frac{2\hbar}{ML^2\Omega^2}\left[
\frac{e^{-2r} + \epsilon_d^2}{\mathcal{K}(\Omega)}
+ \frac{\epsilon_d^2}{1 + \epsilon_d^2e^{-2r}}\,\mathcal{K}(\Omega)
\right] .
\end{equation}
The sum quantum noise spectral density~\eqref{eq:vr_post_opt} is plotted in Figure~\ref{fig:LIGO_var} for the ideal lossless case and for $\eta_d=0.95$ (dashed lines).
Compare this spectral density with the one for the frequency-dependent squeezing angle (pre-filtering) case, see Eq.~\eqref{eq:vr_sqz_3}. The shot noise components in both cases are exactly equal to each other. Concerning the residual back-action noise, in the pre-filtering case it is limited by the available squeezing, while in the post-filtering case -- by the optical losses. In the latter case, were there no optical losses, the back-action noise could be removed completely, as shown in Figure~\ref{fig:LIGO_var}\,(left). For the parameters of the noise curves presented in Figure~\ref{fig:LIGO_var}\,(right), the post-filtering still has some advantage of about 40\% in the back-action noise amplitude $\sqrt{S}$.
Note that the required frequency dependences~\eqref{eq:vr_sqz_theta} and \eqref{eq:vr_post_zeta} in both cases are similar to each other (and become exactly equal to each other in the lossless case $\epsilon_d=0$). Therefore, similar setups can be used in both cases in order to create the necessary frequency dependences with about the same implementation cost. From this simple consideration, it is possible to conclude that pre-filtering is preferable if good squeezing is available, and the optical losses are relatively large, and vice versa. In particular, post-filtering can be used even without squeezing, $r=0$.
\paragraph*{Frequency dependent homodyne and squeezing angles.}
\label{sec:var_both}
And, finally, consider the most sophisticated configuration: double-filtering with both the homodyne angle $\phi_{\mathrm{LO}}$ and the squeezing angle $\theta$ being frequency dependent.
Concerning the squeezing angle, we can reuse Eqs.~(\ref{eq:vr_preopt_theta}) and (\ref{eq:vr_opt_2}). The minimum of the spectral density~\eqref{eq:vr_opt_2} in $\phi_{\mathrm{LO}}$ corresponds to
\begin{equation}
\label{vr_opt_zeta}
\cot\phi_{\mathrm{LO}} = \frac{\mathcal{K}(\Omega)}{1 + \epsilon_d^2e^{2r}}
\,,
\end{equation}
and is equal to
\begin{equation}
\label{eq:vr_opt_3}
S^h(\Omega) = \frac{2\hbar}{ML^2\Omega^2}\left[
\frac{e^{-2r} + \epsilon_d^2}{\mathcal{K}(\Omega)}
+ \frac{\epsilon_d^2}{1 + \epsilon_d^2e^{2r}}\,\mathcal{K}(\Omega)
\right] .
\end{equation}
It also follows from Eqs.(\ref{eq:vr_preopt_theta}) and (\ref{vr_opt_zeta}) that the optimal squeezing angle in this case is given by
\begin{equation}
\label{eq:vr_opt_theta}
\tan\theta = \frac{\epsilon_d^2}{e^{-2r} + \epsilon_d^2}\,\mathcal{K}(\Omega) \,.
\end{equation}
It is easy to see that in the ideal lossless case the double-filtering configuration reduces to a post-filtering one. Really, if $\epsilon_d=0$, the spectral density~\eqref{eq:vr_opt_3} becomes exactly equal to that for the post-filtering case~\eqref{eq:vr_post_opt}, and the frequency dependent squeezing angle~\eqref{eq:vr_opt_theta} degenerates into a constant value~\eqref{eq:vr_post_theta}. However, if $\epsilon_d>0$, then the additional pre-filtering allows one to decrease more the residual back-action term. For example, if $e^{2r}=10$ and $\eta_d=0.95$ then the gain in the back-action noise amplitude $\sqrt{S}$ is equal to about 25\%.
We have plotted the sum quantum noise spectral density~\eqref{eq:vr_opt_3} in Figure~\ref{fig:LIGO_res}, right (dash-dots). This plot demonstrates the best sensitivity gain of about 3 in signal amplitude, which can be provided employing squeezing and filter cavities at the contemporary technological level.
Due to the presence of the residual back-action term in the spectral density~\eqref{eq:vr_opt_3}, there exists an optimal value of the coupling factor $\mathcal{K}(\Omega)$ (that is, the optical power) which provides the minimum to the sum quantum noise spectral density at any given frequency $\Omega$:
\begin{equation}
\label{vr_K_opt}
\mathcal{K}(\Omega) = \frac{1}{\epsilon_de^r} + \epsilon_de^r \,,
\end{equation}
The minimum is equal to
\begin{equation}
\label{vr_loss_lim}
S^h_{\mathrm{min}} = \frac{4\hbar}{ML^2\Omega^2}\,\epsilon_de^{-r} \,.
\end{equation}
This limitation is severe. The reasonably optimistic value of quantum efficiency $\eta_d=0.95$ that we use for our estimates corresponds to $\epsilon_d\approx0.23$. It means that without squeezing ($r=0$) one is only able to beat the SQL in amplitude by
\begin{equation}
\xi_{\mathrm{min}} \equiv \sqrt{\frac{S^h_{\mathrm{min}}}{S^h_{\mathrm{SQL}}}}
= \sqrt{\epsilon_d} \approx 0.5 \,.
\end{equation}
The gain can be improved using squeezing and, if $r\to\infty$ then, in principle, arbitrarily high sensitivity can be reached. But $\xi$ depends on $r$ only as $e^{-r/2}$, and for the 10~dB squeezing, only a modest value of
\begin{equation}
\xi_{\mathrm{min}} \approx 0.27
\end{equation}
can be obtained.
In our particular case, the fact that the additional noise associated with the photodetector quantum inefficiency $\epsilon_d>0$ does not correlate with the quantum fluctuations of the light in the interferometer gives rise to this limit. This effect is universal for any kind of optical loss in the system, impairing the cross-correlation of the measurement and back-action noises and thus limiting the performance of the quantum measurement schemes, which rely on this cross-correlation.
Noteworthy is that Eq.~\eqref{eq:vr_opt_3} does not take into account optical losses in the filter cavities. As we shall see below, the sensitivity degradation thereby depends on the ratio of the light absorption per bounce to the filter cavities length, $A_f/L_f$. Therefore, this method calls for long filter cavities. In particular, in the original paper~\cite{02a1KiLeMaThVy}, filter cavities with the same length as the main interferometer arm cavities (4\,km), placed side by side with them in the same vacuum tubes, were proposed. For such long and expensive filter cavities, the influence of their losses indeed can be small. However, as we show below, in Section~\ref{sec:corr_real}, for the more practical short (up to tens of meters) filter cavities, optical losses thereof could be the main limiting factor in terms of sensitivity.
\paragraph*{Virtual rigidity for prototype interferometers.}
\label{sec:vr_short}
The optimization performed above can be viewed also in a different way, namely, as the minimization of the sum quantum noise spectral density of an ordinary interferometer with frequency-independent homodyne and squeezing angles, yet \emph{at some given frequency} $\Omega_0$. In Section~\ref{sec:toy_FI_correlation}, this kind of optimization was considered for a simple lossless system. It was shown capable of the narrow-band gain in sensitivity, similar to the one provided by the harmonic oscillator (thus the term `virtual rigidity').
This narrow-band gain could be more interesting not for the full-scale
GW detectors (where broadband optimization of the
sensitivity is required in most cases) but for smaller devices like the
10-m Hannover prototype interferometer~\cite{10m_site}, designed for
the development of the measurement methods with sub-SQL
sensitivity. Due to shorter arm length, the bandwidth $\gamma$ in
those devices is typically much larger than the mechanical frequencies
$\Omega$. If one takes, e.g., the power transmissivity value of
$T\gtrsim10^{-2}$ for the ITMs and length of arms equal to
$L\sim10\mathrm{\ m}$, then $\gamma\gtrsim10^5\mathrm{\ s}^{-1}$, which is
above the typical working frequencies band of such devices. In the
literature, this particular case is usually referred to as a \emph{bad
cavity approximation}.
In this case, the coupling factor $\mathcal{K}(\Omega)$ can be approximated as:
\begin{equation}
\label{eq:short_calK}
\mathcal{K}(\Omega) \approx \frac{\Omega_q^2}{\Omega^2} \,,
\end{equation}
where
\begin{equation}
\label{eq:short_Omega_q}
\Omega_q^2 = \frac{2J}{\gamma} \,.
\end{equation}
Note that in this approximation, the noise spectral densities~\eqref{eq:vr_noises_x}, \eqref{eq:vr_noises_F} and \eqref{eq:vr_noises_xF} turn out to be frequency independent.
\epubtkImage{fig36.png}{
\begin{figure}[htbp]
\centerline{
\includegraphics[width=.48\textwidth]{fig36a}\hfill\includegraphics[width=.48\textwidth]{fig36b}
}
\centerline{
\includegraphics[width=.48\textwidth]{fig36c}\hfill\includegraphics[width=.48\textwidth]{fig36d}
}
\caption{Plots of the locally-optimized SQL beating factor $\xi(\Omega)$~\eqref{eq:xi2} of the interferometer with cross-correlated noises for the ``bad cavity'' case $\Omega_0\ll\gamma$, for several different values of the optimization frequency $\Omega_0$ within the range $0.1\times\Omega_q\le\Omega_0\le\sqrt{10}\times\Omega_q$. Thick solid lines: the common envelopes of these plots; see Eq.~\eqref{vr_xi2_opt}. Left column: $\eta_d=1$; right column: $\eta_d=0.95$. Top row: no squeezing, $r=0$; bottom row: 10~dB squeezing, $e^{2r}=10$.}
\label{fig:short_vr_noises}
\end{figure}
}
In Figure~\ref{fig:short_vr_noises}, the SQL beating factor
\begin{equation}
\label{eq:xi2}
\xi(\Omega) = \sqrt{\frac{S^h(\Omega)}{S^h_{\mathrm{SQL\,f.m.}}(\Omega)}}
\end{equation}
is plotted for the sum quantum noise spectral density $S^h(\Omega)$ with the following values of homodyne and squeezing angles
\begin{equation}
\cot\phi_{\mathrm{LO}} = \frac{\mathcal{K}(\Omega_0)}{1 + \epsilon_d^2e^{2r}} \,, \qquad
\tan\theta = \frac{\epsilon_d^2}{e^{-2r} + \epsilon_d^2}\,\mathcal{K}(\Omega_0) \,,
\end{equation}
and factoring in the ``bad cavity''
condition~\eqref{eq:short_calK}. The four panes correspond to the
following four combinations: (upper left) no losses ($\eta_d=1$) and
no squeezing ($r=0$); (lower left) no losses ($\eta_d=1$) and 10~dB
squeezing ($e^{2r}=10$); (upper right) with losses ($\eta_d=0.95$) and
no squeezing ($r=0$); (lower right) with losses ($\eta_d=0.95$) and
10~dB squeezing($e^{2r}=10$). In each pane, the family of plots is
shown that corresponds to different values of the ratio
$\Omega_0/\Omega_q$, ranging from 0.1 to $\sqrt{10}$.
The minima of these plots form the common envelope, given by Eqs.~\eqref{eq:vr_opt_3} and \eqref{eq:short_calK}:
\begin{equation}
\label{vr_xi2_opt}
\xi^2_{\mathrm{min}}(\Omega_0) = \frac{1}{2}\left[
(e^{-2r} + \epsilon_d^2)\frac{\Omega_0^2}{\Omega_q^2}
+ \frac{\epsilon_d^2}{1 + \epsilon_d^2e^{2r}}\,\frac{\Omega_q^2}{\Omega_0^2}
\right] ,
\end{equation}
which is also plotted in Figure~\ref{fig:short_vr_noises}. It is easy to see that in the ideal case of $\epsilon_d=0$, there is no limitation on the SQL beating factor, provided a sufficiently small ratio of $\Omega_0/\Omega_q$:
\begin{equation}
\label{vr_xi2_min0}
\xi^2_{\mathrm{min}}(\Omega) = \frac{e^{-2r}}{2}\,\frac{\Omega_0^2}{\Omega_q^2} \,.
\end{equation}
However, if $\epsilon_d>0$, then function~\eqref{vr_xi2_opt} has a minimum in $\Omega_q$ at
\begin{equation}
\Omega_q = \Omega_0\sqrt{\frac{1}{\epsilon_de^{r}} + \epsilon_de^{r}} \,,
\end{equation}
[compare with Eq.~\eqref{vr_K_opt}], equal to~\eqref{vr_loss_lim}.
\subsubsection{Filter cavities in GW interferometers}
\label{sec:corr_real}
\paragraph*{Input/output relations for the filter cavity.}
In essence, a filter cavity is an ordinary Fabry--P{\'e}rot cavity with one partly transparent input/output mirror. The technical problem of how to spatially separate the input and output beam can be solved in different ways. In the original paper~\cite{02a1KiLeMaThVy} the triangular cavities were considered. However, in this case, an additional mirror in each cavity is required, which adds to the optical loss per bounce. Another option is an ordinary linear cavity with additional optical circulator, which can be implemented, for example, by means of the polarization beamsplitter and Faraday rotator (note that while the typical polarization optics elements have much higher losses than the modern high-quality mirrors, the mirrors losses appear in the final expressions inflated by the filter cavity finesse).
In both cases, the filter cavity can be described by the input/output relation, which can be easily obtained from Eqs.~\eqref{FP_1_quad(a)} and \eqref{FP_1_quad(b)} by setting $\mathrm{E}_{c,s}=0$ (there is no classical pumping in the filter cavity and, therefore, there is no displacement sensitivity) and by some changes in the notations:
\begin{equation}
\label{eq:filter_io}
\hat{\mathbf{o}}(\Omega)
= \mathbb{R}_f(\Omega)\hat{\mathbf{i}}(\Omega) + \mathbb{T}_f(\Omega)\hat{\mathbf{q}}(\Omega)\,,
\end{equation}
where $\hat{\mathbf{i}}$ and $\hat{\mathbf{o}}$ are the two-photon quadrature amplitude vectors of the input and output beams, $\hat{\mathbf{q}}$ stands for noise fields entering the cavity due to optical losses (which are assumed to be in a vacuum state),
\begin{eqnarray}
\mathbb{R}_f(\Omega) &=& \frac{1}{\mathcal{D}_f(\Omega)}
\smatrix{\gamma_{f1}^2 - \gamma_{f2}^2 - \delta_f^2 + \Omega^2 + 2i\Omega\gamma_{f2}}
{-2\gamma_{f1}\delta_f}{2\gamma_{f1}\delta_f}
{\gamma_{f1}^2 - \gamma_{f2}^2 - \delta_f^2 + \Omega^2 + 2i\Omega\gamma_{f2}} ,
\label{filter_bbR} \\\noalign{\smallskip}
\mathbb{T}_f(\Omega) &=& \frac{2\sqrt{\gamma_{f1}\gamma_{f2}}}{\mathcal{D}_f(\Omega)}
\smatrix{\gamma_f-i\Omega}{-\delta_f}{\delta_f}{\gamma_f-i\Omega} ,
\label{filter_bbT} \\\noalign{\smallskip}
\mathcal{D}_f(\Omega) &=& (\gamma_f-i\Omega)^2 + \delta_f^2 \,, \\
\gamma_{f1} &=& \frac{cT_f}{4L_f} \\
\gamma_{f2} &=& \frac{cA_f}{4L_f} \,,
\end{eqnarray}
where $T_f$ is the power transmittance of the input/output mirror, $A_f$ is the factor of power loss per bounce, $L_f$ is the filter cavity length,
\begin{equation}
\gamma_f = \gamma_{f1} + \gamma_{f2}
\end{equation}
is its half-bandwidth, and $\delta_f$ is its detuning.
In order to demonstrate how the filter cavity works, consider the particular case of the lossless cavity. In this case,
\begin{equation}
\label{filter_io_0}
\hat{\mathbf{o}}(\Omega) = \mathbb{R}_f(\Omega)\hat{\mathbf{i}}(\Omega) \,,
\end{equation}
and the reflection matrix describes field amplitude rotations with the frequency-dependent rotation angle:
\begin{equation}
\label{eq:filter_bbR0}
\mathbb{R}_f(\Omega)
= \mathbb{P}\bigl[\theta_f(\Omega)\bigr]e^{i\beta_f(\Omega)} \,,
\end{equation}
where
\begin{eqnarray}
\theta_f(\Omega) &=& \arctan\frac{2\gamma_f\delta_f}{\gamma_f^2 - \delta_f^2 + \Omega^2}
\,, \label{theta_f} \\
e^{i\beta_f(\Omega)} &=& \frac{|\mathcal{D}_f(\Omega)|}{\mathcal{D}_f(\Omega)} \,.
\end{eqnarray}
The phase factor $e^{i\beta_f(\Omega)}$ is irrelevant, for it does not appear in the final equations for the spectral densities.
Let us now analyze the influence of the filter cavities on the interferometer sensitivity in post- and pre-filtering variational schemes.
Start with the latter one. Suppose that the light, entering the interferometer from the signal port is in the squeezed state with fixed squeezing angle $\theta$ and squeezing factor $r$ and thus can be described by the following two-photon quadrature vector
\begin{equation}
\hat{\boldsymbol{a}} = \mathbb{S}_{\sqz}[r,\theta]\hat{\boldsymbol{a}}^{\vac} \,,
\end{equation}
where the quadratures vector $\hat{\boldsymbol{a}}^{\vac}$ describes the vacuum state. After reflecting off the filter cavity, this light will be described with the following expression
\begin{equation}
\hat{\mathbf{o}}(\Omega)
= \mathbb{P}\bigl[\theta_f(\Omega)\bigr]\mathbb{S}_{\sqz}[r,\theta]\hat{\boldsymbol{a}}^{\vac}
e^{i\beta_f(\Omega)}
= \mathbb{S}_{\sqz}\bigl[r,\theta + \theta_f(\Omega)\bigr]\hat{\boldsymbol{a}}^{\vac'} \,,
\end{equation}
[see Eq.~\eqref{eq:SQZ_matrix_transform_def}], where $\hat{\boldsymbol{a}}^{vac'} = \mathbb{P}\bigl[\theta_f(\Omega)\bigr]\hat{\boldsymbol{a}}^{\vac}e^{i\beta_f(\Omega)}$ also describes the light field in a vacuum state. Thus, the pre-filtering indeed rotates the squeezing angle by a frequency-dependent angle $\theta_f(\Omega)$.
In a similar manner, we can consider the post-filtering schemes. Consider a homodyne detection scheme with losses, described by Eq.~\eqref{eq:homodyne_lossy}. Suppose that prior to detection, the light described by the quadrature vector $\hat{\boldsymbol{b}}$, reflects from the filter cavity. In this case, the photocurrent (in Fourier representation) is proportional to
\begin{eqnarray}
\label{HD_i_var}
i_-(\Omega) &\propto& \boldsymbol{H}^{\mathsf{T}}[\phi_{\mathrm{LO}}]\bigl[
\mathbb{P}\bigl[\theta_f(\Omega)\bigr]\hat{\boldsymbol{b}}(\Omega)
e^{i\beta_f(\Omega)}
+ \epsilon_d\hat{\boldsymbol{n}}(\Omega)
\bigr] \nonumber\\
&=& \boldsymbol{H}^{\mathsf{T}}\bigl[\phi_{\mathrm{LO}} - \theta_f(\Omega)\bigr]
\bigl[\hat{\boldsymbol{b}}(\Omega) + \epsilon_d\hat{\boldsymbol{n}}'(\Omega)\bigr]e^{i\beta_f(\Omega)}
\,,
\end{eqnarray}
where $\hat{{\boldsymbol{n}}}' = \mathbb{P}\bigl[-\theta_f(\Omega)\bigr]\hat{{\boldsymbol{n}}}e^{-i\beta_f(\Omega)}$ again describes some new vacuum field. This formula demonstrates that post-filtering is equivalent to the introduction of a frequency-dependent shift of the homodyne angle by $-\theta_f(\Omega)$.
It is easy to see that the necessary frequency dependencies of the homodyne and squeezing angles~\eqref{eq:vr_sqz_theta} or \eqref{eq:vr_post_zeta} (with the second-order polynomials in $\Omega^2$ in the r.h.s.\ denominators) cannot be implemented by the rotation angle~\eqref{theta_f} (with its first order in $\Omega^2$ polynomial in the r.h.s.\ denominator). As was shown in the paper~\cite{02a1KiLeMaThVy}, two filter cavities are required in both these cases. In the double pre- and post-filtering case, the total number of the filter cavities increases to four. Later it was also shown that, in principle, arbitrary frequency dependence of the homodyne and/or squeezing angle can be implemented, providing a sufficient number of filter cavities~\cite{Buonanno2004}.
However, in most cases, a more simple setup consisting of a single filter cavity might suffice. Really, the goal of the filter cavities is to compensate the back-action noise, which contributes significantly in the sum quantum noise only at low frequencies $\Omega\lesssim\Omega_q=\sqrt{2J/\gamma}$. However, when
\begin{equation}
\label{large_gamma}
\gamma > J^{1/3},
\end{equation}
which is actually the case for the planned second and third generation GW detectors, the factor $\mathcal{K}(\Omega)$ can be well approximated by Eq.~\eqref{eq:short_calK} in the low-frequency region. In such a case the single filter cavity can provide the necessary frequency dependence. Moreover, the second filter cavity could actually degrade sensitivity due to the additional optical losses it superinduces to the system.
Following this reasoning, we consider below two schemes, each based on a single filter cavity that realize pre-filtering and post-filtering, respectively.
\paragraph*{Single-filter cavity-based schemes.}
\label{sec:single_filter}
The schemes under consideration are shown in
Figure~\ref{fig:filter}. In the pre-filtering scheme drawn in the left
panel of Figure~\ref{fig:filter}, a squeezed light source emits
frequency-independent squeezed vacuum towards the filter cavity, where
it gets reflected, gaining a frequency-dependent phase shift
$\theta_f(\Omega)$, and then enters the dark port of the main
interferometer. The light going out of the dark port is detected by
the homodyne detector with fixed homodyne angle $\phi_{\mathrm{LO}}$ in the
usual way.
\epubtkImage{fig37.png}{
\begin{figure}[htb]
\centerline{\includegraphics[width=\textwidth]{fig37}}
\caption{Schemes of interferometer with the single filter cavity. \emph{Left:} In the pre-filtering scheme, squeezed vacuum from the squeezor is injected into the signal port of the interferometer after the reflection from the filter cavity; \emph{right:} in the post-filtering scheme, a squeezed vacuum first passes through the interferometer and, coming out, gets reflected from the filter cavity. In both cases the readout is performed by an ordinary homodyne detector with frequency independent homodyne angle $\phi_{\mathrm{LO}}$.}
\label{fig:filter}
\end{figure}}
Following the prescriptions of Section~\ref{sec:var_sqz}, we suppose the homodyne angle defined by Eq.~\eqref{vr_sqz_zeta}. The optimal squeezing angle should then be equal to zero at higher frequencies, see~\eqref{eq:vr_sqz_theta}. Taking into account that the phase shift introduced by the filter cavity goes to zero at high frequencies, we obtain that the squeezing angle $\theta$ of the input squeezed vacuum must be zero. Combining Eqs.~\eqref{eq:vr_X_sum} and \eqref{eq:filter_io} taking these assumptions into account, we obtain the following equation for the sum quantum noise of the pre-filtering scheme:
\begin{equation}
\label{eq:pre_X_sum}
\hat{h}_{\mathrm{sum}}(\Omega) = -\frac{2}{L}\sqrt{\frac{\hbar}{2MJ\gamma}}\,
\boldsymbol{H}^{\mathsf{T}}[\pi/2]\bigl\{
(\gamma+i\Omega)\mathbb{K}(\Omega)\bigl[
\mathbb{R}_f(\Omega)\mathbb{S}_{\sqz}[r,0]\hat{\boldsymbol{a}}^{\vac}(\Omega)
+ \mathbb{T}_f(\Omega)\hat{\mathbf{q}}(\Omega)
\bigr]
+ (\gamma-i\Omega)\epsilon_d\hat{\boldsymbol{n}}(\Omega)
\bigr\}\,,
\end{equation}
which yields the following expression for a power (double-sided) spectral density
\begin{equation}
\label{eq:pre_sum_noise}
S^h(\Omega) = \frac{2\hbar}{ML^2\Omega^2\mathcal{K}(\Omega)}\bigl\{
\boldsymbol{H}^{\mathsf{T}}[\pi/2]\mathbb{K}(\Omega)\bigl[
\mathbb{R}_f(\Omega)\mathbb{S}_{\sqz}[2r,0]\mathbb{R}_f^\dagger(\Omega)
+ \mathbb{T}_f(\Omega)\mathbb{T}_f^\dagger(\Omega)
\bigr]\mathbb{K}^\dagger(\Omega)\boldsymbol{H}[\pi/2]
+ \epsilon_d^2
\bigr\} .
\end{equation}
In the ideal lossless filter cavity case, taking into account Eq.~\eqref{eq:filter_bbR0}, this spectral density can be simplified as follows:
\begin{equation}
\label{eq:pre_sum_noise0}
S^h(\Omega) = \frac{2\hbar}{ML^2\Omega^2\mathcal{K}(\Omega)}\bigl[
\boldsymbol{H}^{\mathsf{T}}[\pi/2]\mathbb{K}(\Omega)\mathbb{S}\bigl(2r,\theta_f(\Omega)\bigr)
\mathbb{K}^\dagger(\Omega)\boldsymbol{H}[\pi/2]
+ \epsilon_d^2
\bigr]
\end{equation}
[compare with Eq.~\eqref{eq:vr_sum_noise}]. In this case, the necessary frequency dependence of the squeezing angle~\eqref{eq:vr_sqz_theta} can be implemented by the following filter cavity parameters:
\begin{equation}
\label{filter0_pre}
\gamma_f = \delta_f = \gamma_{f0} \,,
\end{equation}
where
\begin{equation}
\label{filter_g_f0}
\gamma_{f0} = \sqrt{J/\gamma} \,.
\end{equation}
Along similar lines, the post-filtering scheme drawn in the right panel of Figure~\ref{fig:filter} can be considered. Here, the squeezed-vacuum produced by the squeezor first passes through the interferometer and then, coming out, gets reflected from the filter cavity, gaining a frequency-dependent phase shift, which is equivalent to introducing a frequency dependence into the homodyne angle, and then goes to the fixed angle homodyne detector. Taking into account that this equivalent homodyne angle at high frequencies has to be $\pi/2$, and that the phase shift introduced by the filter cavity goes to zero at high frequencies, we obtain that the real homodyne angle must also be $\pi/2$. Assuming that the squeezing angle is defined by Eq.~\eqref{eq:vr_post_theta} and again using Eqs.~\eqref{eq:vr_X_sum} and \eqref{eq:filter_io}, we obtain that the sum quantum noise and its power (double-sided) spectral density are equal to
\begin{eqnarray}
\label{eq:post_X_sum}
\hat{h}_{\mathrm{sum}}(\Omega) &=& -\frac{2}{L}\sqrt{\frac{\hbar}{2MJ\gamma}}\,
\frac{1}{\boldsymbol{H}^{\mathsf{T}}[\pi/2]\mathbb{R}_f(\Omega)\binom{0}{1}} \nonumber\\ \times
\boldsymbol{H}^{\mathsf{T}}[\pi/2]\bigl\{
(\gamma&+&i\Omega)
\mathbb{R}_f(\Omega)\mathbb{K}(\Omega)\mathbb{S}_{\sqz}[r,0]\hat{\boldsymbol{a}}^{\vac}(\Omega)
+ (\gamma-i\Omega)\bigl[
\mathbb{T}_f(\Omega)\hat{\mathbf{q}}(\Omega) + \epsilon_d\hat{\boldsymbol{n}}(\Omega)
\bigr]
\bigr\}
\end{eqnarray}
and
\begin{eqnarray}
\label{eq:post_sum_noise}
S^h(\Omega) &=& \frac{2\hbar}{ML^2\Omega^2\mathcal{K}(\Omega)}
\frac{1}{\left|\boldsymbol{H}^{\mathsf{T}}[\pi/2]\mathbb{R}_f(\Omega)\binom{0}{1}\right|^2}
\nonumber\\ && \times
\bigl\{
\boldsymbol{H}^{\mathsf{T}}[\pi/2]\bigl[
\mathbb{R}_f(\Omega)\mathbb{K}(\Omega)\mathbb{S}_{\sqz}[2r,0]\mathbb{K}^\dagger(\Omega)
\mathbb{R}_f^\dagger(\Omega)
+ \mathbb{T}_f(\Omega)\mathbb{T}_f^\dagger(\Omega)
\bigr]\boldsymbol{H}[\pi/2]
+ \epsilon_d^2
\bigr\} .
\end{eqnarray}
In the ideal lossless filter cavity case, factoring in Eq.~\eqref{eq:filter_bbR0}, this spectral density takes a form similar to~\eqref{eq:vr_sum_noise}, but with the frequency-dependent homodyne angle:
\begin{equation}
\label{post_sum_noise0}
S^h(\Omega) = \frac{2\hbar}{ML^2\Omega^2\mathcal{K}(\Omega)\sin^2\phi_{\mathrm{LO}}(\Omega)}
\bigl[
\boldsymbol{H}^{\mathsf{T}}\bigl[\phi_{\mathrm{LO}}(\Omega)\bigr]\mathbb{K}(\Omega)\mathbb{S}_{\sqz}[2r,0]
\mathbb{K}^\dagger(\Omega)\boldsymbol{H}^{\mathsf{T}}\bigl[\phi_{\mathrm{LO}}(\Omega)\bigr]
+ \epsilon_d^2
\bigr] ,
\end{equation}
where
\begin{equation}
\phi_{\mathrm{LO}}(\Omega) = \pi/2-\theta_f(\Omega).
\end{equation}
The necessary frequency dependence~\eqref{eq:vr_post_zeta} of this effective homodyne angle can be implemented by the following parameters of the filter cavity:
\begin{equation}
\label{filter0_post}
\gamma_f = \delta_f = \frac{\gamma_{f0}}{\sqrt{1+\epsilon_d^2e^{-2r}}} \,.
\end{equation}
Note that for reasonable values of loss and squeezing factors, these parameters differ only by a few percents from the ones for the pre-filtering.
It is easy to show that substitution of the conditions~\eqref{filter0_pre} and \eqref{filter0_post} into Eqs.~\eqref{eq:pre_sum_noise0} and \eqref{post_sum_noise0}, respectively, taking Condition~\eqref{large_gamma} into account, results in spectral densities for the ideal frequency dependent squeezing and homodyne angle, see Eqs.~\eqref{eq:vr_sqz_3} and \eqref{eq:vr_post_opt}.
In the general case of lossy filter cavities, the conditions (\ref{eq:vr_sqz_theta}) and~(\ref{eq:vr_post_zeta}) cannot be satisfied exactly by a single filter cavity at all frequencies. Therefore, the optimal filter cavity parameters should be determined using some integral sensitivity criterion, which will be considered at the end of this section.
However, it would be a reasonable assumption that the above consideration holds with good precision, if losses in the filter cavity are low compared to other optical losses in the system:
\begin{equation}
\label{filter_lim_1}
\frac{\gamma_{f2}}{\gamma_f} \approx \frac{\gamma_{f2}}{\gamma_{f0}} \ll \epsilon_d^2 \,.
\end{equation}
This inequality can be rewritten as the following condition for the filter cavity specific losses:
\begin{equation}
\label{filter_lim_1_num}
\frac{A_f}{L_f} \lesssim \frac{4\gamma_{f0}}{c}\,\epsilon_d^2 \,.
\end{equation}
In particular, for our standard parameters used for numerical estimates ($J=J_{\mathrm{aLIGO}}$, $\gamma=2\pi\times500\mathrm{\ s}^{-1}$, $\eta_d=0.95$), we obtain $\gamma_{f0}\approx280\mathrm{\ s}^{-1}$, and there should be
\begin{equation}
\frac{A_f}{L_f} \lesssim 2\times10^{-7}\mathrm{\ m}^{-1},
\end{equation}
(the r.h.s.\ corresponds, in particular, to a 50~m filter cavity with the losses per bounce $A_f=10^{-5}$).
Another more crude limitation can be obtained from the condition that $\gamma_{f2}$ should be small compared to the filter cavity bandwidth $\gamma_f$:
\begin{equation}
\label{filter_lim_2}
\frac{A_f}{L_f} \lesssim \frac{4\gamma_{f0}}{c} \,.
\end{equation}
Apparently, were it not the case, the filter cavity would just cease to work properly. For the same numerical values of $J$ and $\gamma$ as above, we obtain:
\begin{equation}
\label{filter_lim_2_num}
\frac{A_f}{L_f} \lesssim 4\times10^{-6}\mathrm{\ m}^{-1},
\end{equation}
(for example, a very short 2.5~m filter cavity with $A_f=10^{-5}$ or 25~m cavity with $A_f=10^{-4}$).
\paragraph*{Numerical optimization of filter cavities.}
\label{sec:filter_snr_opt}
In the experiments devoted to detection of small forces, and, in particular, in the GW detection experiments, the main integral sensitivity measure is the probability to detect some calibrated signal. This probability, in turn, depends on the matched filtered SNR defined as
\begin{equation}
\label{6:snr}
\rho^2= \int_{-\infty}^{\infty}\!\frac{|h_{s}(\Omega)|^2}{S^h(\Omega)}\,\frac{d\Omega}{2\pi}
\end{equation}
with $h_{s}(\Omega)$ the spectrum of this calibrated signal.
In the low and medium frequency range, where back-action noise dominates, and wherein our interest is focused, the most probable source of signal is the gravitational radiation of the inspiraling binary systems of compact objects such as neutron stars and/or black holes~\cite{lrr-2009-2,lrr-2006-6}. In this case, the SNR is equal to (see~\cite{Flanagan1998})
\begin{equation}
\label{snr_nsns}
\rho^2 = k_0\int_0^{2\pi f_{\mathrm{max}}}\frac{\Omega^{-7/3}}{S^h(\Omega)}\,
\frac{d\Omega}{2\pi} \,,
\end{equation}
where $k_0$ is a factor that does not depend on the interferometer parameters, and $f_{\mathrm{max}}$ is the cut-off frequency that depends on the binary system components' masses. In particular, for neutron stars with masses equal to 1.4 solar mass, $f_{\mathrm{max}}\approx1.5\mathrm{\ kHz}$.
Since our goal here is not the maximal value of the SNR itself, but rather the relative sensitivity gain offered by the filter cavity, and the corresponding optimal parameters $\gamma_{f1}$ and $\delta_f$, providing this gain, we choose to normalize the SNR by the value corresponding to the ordinary interferometer (without the filter cavities):
\begin{equation}
\label{filter_rho_0}
\rho_0^2 = k_0\int_0^{2\pi f_{\mathrm{max}}}\frac{\Omega^{-7/3}}{S_0^h(\Omega)}\,
\frac{d\Omega}{2\pi} \,,
\end{equation}
with power (double-sided) spectral density
\begin{equation}
S_0^h(\Omega) = \frac{2\hbar}{ML^2\Omega^2}
\left[\frac{1+\epsilon_d^2}{\mathcal{K}(\Omega)} + \mathcal{K}(\Omega)\right]
\end{equation}
[see Eq.~\eqref{eq:S_X_Caves}].
We optimized numerically the ratio $\rho^2/\rho_0^2$, with filter cavity half-bandwidth $\gamma_{f1}$ and detuning $\delta_f$ as the optimization parameters, for the values of the specific loss factor $A_f/L_f$ ranging from $10^{-9}$ (e.g., very long 10~km filter cavity with $A_f=10^{-5}$) to $10^{-5}$ (e.g., 10~m filter cavity with $A_f=10^{-4}$). Concerning the main interferometer parameters, we used the same values as in all our previous examples, namely, $J=J_{\mathrm{aLIGO}}$, $\gamma=2\pi\times500\mathrm{\ s}^{-1}$, and $\eta_d=0.95$.
The results of the optimization are shown in
Figure~\ref{fig:filter_opt}. In the left pane, the optimal values of
the filter cavity parameters $\gamma_{f1}$ and $\delta_f$ are plotted,
and in the right one the corresponding optimized values of the
SNR are. It follows from these plots that the optimal values of
$\gamma_{f1}$ and $\delta_f$ are virtually the same as $\gamma_{f0}$,
while the specific loss factor $A_f/L_f$ satisfies the condition
\eqref{filter_lim_1}, and starts to deviate sensibly from
$\gamma_{f0}$ only when $A_f/L_f$ approaches the limit
\eqref{filter_lim_2}. Actually, for such high values of specific
losses, the filter cavities only degrade the sensitivity, and the
optimization algorithm effectively turns them off, switching to the
ordinary frequency-independent squeezing regime (see the right-most
part of the right pane).
\epubtkImage{fig38.png}{
\begin{figure}[htbp]
\includegraphics[width=.48\textwidth]{fig38a}\hfill\includegraphics[width=.48\textwidth]{fig38b}
\caption{Left: Numerically-optimized filter-cavity parameters for a single cavity based pre- and post-filtering schemes: half-bandwidth $\gamma_{f1}$ (solid lines) and detuning $\delta_f$ (dashed lines), normalized by $\gamma_{f0}$ [see Eq.~\eqref{filter_g_f0}], as functions of the filter cavity specific losses $A_f/L_f$. Right: the corresponding optimal SNRs, normalized by the SNR for the ordinary interferometer [see Eq.~\eqref{filter_rho_0}]. Dashed lines: the normalized SNRs for the ideal frequency-dependent squeeze and homodyne angle cases, see Eqs.~\eqref{eq:vr_sqz_3} and \eqref{eq:vr_post_opt}. `Ordinary squeezing': frequency-independent 10~dB squeezing with $\theta=0$. In all cases, $J=J_{\mathrm{aLIGO}}$, $\gamma=2\pi\times500\mathrm{\ s}^{-1}$, and $\eta_d=0.95$.}
\label{fig:filter_opt}
\end{figure}}
It also follows from these plots that post-filtering provides slightly better sensitivity, if the optical losses in the filter cavity are low, while the pre-filtering has some advantage in the high-losses scenario. This difference can be explained in the following way~\cite{10a1Kh}. The post-filtration effectively rotates the homodyne angle from $\phi_{\mathrm{LO}}=\pi/2$ (phase quadrature) at high frequencies to $\phi_{\mathrm{LO}}\to0$ (amplitude quadrature) at low frequencies, in order to measure the back-action noise, which dominates the low frequencies. As a result, the optomechanical transfer function reduces at low frequencies, emphasizing all noises introduced after the interferometer [see the factor $\sin^2\phi_{\mathrm{LO}}(\Omega)$ in the denominator of Eq.~\eqref{post_sum_noise0}]. In the pre-filtering case there is no such effect, for the value of $\phi_{\mathrm{LO}}=\pi/2$, corresponding to the maximum of the optomechanical transfer function, holds for all frequencies (the squeezing angle got rotated instead).
The optimized sum quantum noise power (double-sided) spectral densities are plotted in
Figure~\ref{fig:LIGO_filter} for several typical values of the
specific loss factor, and for the same values of the rest of the
parameters, as in Figure~\ref{fig:filter_opt}. For comparison, the
spectral densities for the ideal frequency-dependent squeezing angle
Eqs.~\eqref{eq:vr_sqz_3} and homodyne angle~\eqref{eq:vr_post_opt} are
also shown. These plots clearly demonstrate that providing
sufficiently-low optical losses (say, $A_f/L_f\lesssim10^{-8}$), the
single filter cavity based schemes can provide virtually the same
result as the abstract ones with the ideal frequency dependence for
squeezing or homodyne angles.
\epubtkImage{fig39.png}{
\begin{figure}[htb]
\includegraphics[width=.48\textwidth]{fig39a}\hfill\includegraphics[width=.48\textwidth]{fig39b}
\caption{Examples of the sum quantum noise power (double-sided)
spectral densities of the resonance-tuned interferometers with the
single filter cavity based pre- and post-filtering. \emph{Left:}
pre-filtering, see Figure~\ref{fig:filter}\,(left); dashes --
10~dB squeezing, $\phi_{\mathrm{LO}}=\pi/2$, ideal
frequency-dependent squeezing angle~\eqref{eq:vr_sqz_theta}; thin
solid -- 10~dB squeezing, $\phi_{\mathrm{LO}}=\pi/2$, numerically-optimized lossy pre-filtering cavity with $A_f/L_f =
10^{-9},\ 10^{-7}\,\ 10^{-6.5}\ \mbox{and}\ 10^{-6}$. \emph{Right:} post-filtering, see Figure~\ref{fig:filter}\,(right); dashes: 10~dB squeezing, $\theta=0$, ideal frequency-dependent homodyne angle~\eqref{eq:vr_post_zeta}; thin solid -- 10~dB squeezing, $\theta=0$, numerically optimized lossy post-filtering cavity with $A_f/L_f = 10^{-9},\ 10^{-8}\ \mbox{and}\ 10^{-7}$. In both panes (for the comparison): `Ordinary' -- no squeezing, $\phi_{\mathrm{LO}}=\pi/2$; `Squeezed': 10~dB squeezing, $\theta=0$, $\phi_{\mathrm{LO}}=\pi/2$}
\label{fig:LIGO_filter}
\end{figure}}
\subsection{Quantum speedmeter}
\label{sec:speedmeter}
\subsubsection{Quantum speedmeter topologies}
A quantum speedmeter epitomizes the approach to the broadband SQL beating, in some sense, opposite to the one based on the quantum noises cross-correlation tailoring with filter cavities, considered above. Here, instead of fitting the quantum noise spectral dependence to the Fabry--P{\'e}rot--Michelson interferometer optomechanical coupling factor~\eqref{eq:varK}, the interferometer topology is modified in such a way as to mold the new optomechanical coupling factor $\mathcal{K}_{\mathrm{SM}}(\Omega)$ so that it turns out frequency-independent in the low- and medium-frequency range, thus making the frequency-dependent cross-correlation not necessary.
The general approach to speed measurement is to use pairs of position measurements separated by a time delay $\tau\lesssim 1/\Omega$, where $\Omega$ is the characteristic signal frequency (cf.\ the simplified consideration in Section~\ref{sec:toy_speedmeter}). Ideally, the successive measurements should be coherent, i.e., they should be performed by the same photons. In effect, the velocity $v$ of the test mass is measured in this way, which gives the necessary frequency dependence of the $\mathcal{K}_{\mathrm{SM}}(\Omega)$.
In Section~\ref{sec:toy_speedmeter}, we have considered the simplest
toy scheme that implements this principle and which was first
proposed by Braginsky and Khalili in~\cite{90a1BrKh}. Also in this
paper, a modified version of this scheme, called the \emph{sloshing-cavity
speedmeter}, was proposed. This version uses two coupled resonators
(e.g., microwave ones), as shown in Figure~\ref{figSM2}\,(left), one
of which (2), the \emph{sloshing cavity}, is pumped on resonance
through the input waveguide, so that another one (1) becomes excited
at its eigenfrequency $\omega_e$. The eigenfrequency of resonator~1
is modulated by the position $x$ of the test mass and puts a voltage
signal proportional to position $x$ into resonator~2, and a voltage
signal proportional to velocity $dx/dt$ into resonator~1. The velocity
signal flows from resonator~1 into an output waveguide, from which it
is monitored. One can understand the production of this velocity
signal as follows. The coupling between the resonators causes voltage
signals to slosh periodically from one resonator to the other at
frequency $\Omega$. After each cycle of sloshing, the sign of the
signal is reversed, so the net signal in resonator~1 is proportional
to the difference of the position at times $t$ and $t+2\pi/\Omega$,
thus implementing the same principle of the double position
measurement.
\epubtkImage{fig40.png}{
\begin{figure}[htbp]
\centerline{
\includegraphics[width=0.38\textwidth]{fig40a}\hfill
\includegraphics[width=0.58\textwidth]{fig40b}
}
\caption{\emph{Left:} schematic diagram of the microwave speedmeter on coupled cavities as given in~\cite{90a1BrKh}. \emph{Right:} optical version of coupled-cavities speedmeter proposed in~\cite{Purdue2002}.}
\label{figSM2}
\end{figure}}
Later, the optical version of the sloshing-cavity speedmeter scheme suitable for large-scale laser GW detectors was developed~\cite{00a1BrGoKhTh, Purdue2001, Purdue2002}. The most elaborated variant proposed in~\cite{Purdue2002} is shown in Figure~\ref{figSM2}\,(right). Here, the differential mode of a Michelson interferometer serves as the resonator 1 of the initial scheme of~\cite{90a1BrKh}, and an additional kilometer-scale Fabry--P{\'e}rot cavity -- as the resonator 2, thus making a practical interferometer configuration.
\epubtkImage{fig41.png}{
\begin{figure}[htbp]
\centerline{\includegraphics[width=\textwidth]{fig41}}
\caption{Two possible optical realizations of zero area Sagnac speedmeter.
\emph{Left panel:} The ring cavities can be used to spatially separate the ingoing fields from the outgoing ones, in order to redirect output light from one arm to another~\cite{Chen2002}.
\emph{Right panel:} The same goal can be achieved using an optical circulator consisting of the polarization beamsplitter (PBS) and two $\lambda/4$-plates~\cite{02a2Kh, 04a1Da}.
}
\label{fig:Sagnacfig}
\end{figure}}
In parallel, it was realized by Chen and Khalili~\cite{Chen2002, 02a2Kh} that the zero area Sagnac interferometer~\cite{Byer1996, Byer1999, Byer2000} actually implements the initial double-measurement variant of the quantum speedmeter, shown in Figure~\ref{fig:speedmeter}. Further analysis with account for optical losses was performed in~\cite{04a1Da} and with detuned signal-recycling in~\cite{MuellerEbhardt2009}. Suggested configurations are pictured in Figure~\ref{fig:Sagnacfig}. The core idea is that light from the laser gets split by the beamsplitter (BS) and directed to Fabry--P\'erot cavities in the arms, exactly as in conventional Fabry--P{\'e}rot--Michelson interferometers. However, after it leaves the cavity, it does not go back to the beamsplitter, but rather enters the cavity in the other arm, and only afterwards returns to the beamsplitter, and finally to the photo detector at the dark port. The scheme of~\cite{Chen2002} uses ring Fabry--P{\'e}rot cavities in the arms to spatially separate ingoing and outgoing light beams to redirect the light leaving the first arm to the second one evading the output beamsplitter. The variant analyzed in~\cite{02a2Kh, 04a1Da} uses polarized optics for the same purposes: light beams after ordinary beamsplitter, having linear (e.g., vertical) polarization, pass through the polarized beamsplitter (PBS), then meet the $\lambda/4$ plates that transform their linear polarization into a circular one, and then enter the Fabry--P{\'e}rot cavity. After reflection from the Fabry--P{\'e}rot cavity, light passes through a $\lambda/4$-plate again, changing its polarization again to linear, but orthogonal to the initial one. As a result, the PBS reflects it and redirects to another arm of the interferometer where it passes through the same stages, restoring finally the initial polarization and comes out of the interferometer. With the exception of the implementation method for this round-robin pass of the light through the interferometer, both schemes have the same performance, and the same appellation \emph{Sagnac speedmeter} will be used for them below.
Visiting both arms, counter propagating light beams acquire phase shifts proportional to a sum of end mirrors displacements of both cavities taken with time delay equal to average single cavity storage time $\tau_{\mathrm{arm}}$:
\begin{equation}
\delta\phi_R \propto x_N(t)+x_E(t+\tau_{\mathrm{arm}})\,,\quad
\delta\phi_L \propto x_E(t)+x_N(t+\tau_{\mathrm{arm}})\,.
\end{equation}
After recombining at the beamsplitter and photo detection the output signal will be proportional to the phase difference of clockwise (R) and counter clockwise (L) propagating light beams:
\begin{equation}
\label{phi_speedmeter}
\delta\phi_R - \delta\phi_L \propto [x_N(t)-x_N(t+\tau_{\mathrm{arm}})]
-[x_E(t)-x_E(t+\tau_{\mathrm{arm}})]
\propto \dot x_N(t) - \dot x_E(t) + O(\tau_{\mathrm{arm}})
\end{equation}
that, for frequencies $\ll\tau_{\mathrm{arm}}^{-1}$, are proportional to the relative velocity of the interferometer end test masses.
Both versions of the optical speedmeter, the sloshing cavity and the Sagnac ones, promise about the same sensitivity, and the choice between them depends mostly on the relative implementation cost of these schemes. Below we consider in more detail the Sagnac speedmeter, which does not require the additional long sloshing cavity.
We will not present here the full analysis of the Sagnac topology similar to the one we have provided for the Fabry--P{\'e}rot--Michelson one. The reader can find it in~\cite{Chen2002, 04a1Da}. We limit ourselves by the particular case of the resonance tuned interferometer (that is, no signal recycling and resonance tuned arm cavities). It seems that the detuned Sagnac interferometer can provide a quite interesting regime, in particular, the \emph{negative inertia} one~\cite{MuellerEbhardt2009}. However, for now (2011) the exhaustive analysis of these regimes is yet to be done. We assume that the squeezed light can be injected into the interferometer dark port, but consider only the particular case of the classical optimization, $\theta=0$, which gives the best broadband sensitivity for a given optical power.
\subsubsection{Speedmeter sensitivity, no optical losses}
\label{sec:sm_noloss}
In order to reveal the main properties of the quantum speedmeter, start with the simplified case of the lossless interferometer and the ideal photodetector. In this case, the sum quantum noise power (double-sided) spectral density of the speedmeter can be written in a form similar to the one for the Fabry--P{\'e}rot--Michelson interferometer [see, e.g., Eqs.~\eqref{eq:vr_noises_x}, \eqref{eq:vr_noises_F} and \eqref{eq:vr_noises_xF}]:
\begin{equation}
\label{S_SM}
S^h(\Omega) = \frac{S^h_{\mathrm{SQL}}(\Omega)}{2}\left[
\frac{e^{-2r} + e^{2r}\cot^2\phi_{\mathrm{LO}}}{\mathcal{K}_{\mathrm{SM}}(\Omega)}
- 2e^{2r}\cot\phi_{\mathrm{LO}} + \mathcal{K}_{\mathrm{SM}}(\Omega)e^{2r}
\right] ,
\end{equation}
but with a different form of the optomechanical coupling factor, see~\cite{Chen2002}:
\begin{equation}
\label{KSM}
\mathcal{K}_{\mathrm{SM}}(\Omega) = \frac{4J\gamma}{(\gamma^2+\Omega^2)^2} \,.
\end{equation}
The factor $J$ here is still defined by Eq.~\eqref{eq:FPMI_J_def}, but the circulating power is now twice as high as that of the position meter, for the given input power, because after leaving the beamsplitter, here each of the ``north'' and ``east'' beams visit both arms sequentially.
The key advantage of speedmeters over position meters is that at low frequencies, $\Omega<\gamma$,
$\mathcal{K}_{\mathrm{SM}}$ is approximately constant and reaches the maximum there:
\begin{equation}
\mathcal{K}_{\mathrm{SM}}(\Omega\ll\gamma) \approx \mathop{{\mathrm{arccot}}\nolimits\mathcal{K}_{\mathrm{SM}}(0)
= \mathop{{\mathrm{arccot}}\nolimits\frac{4J}{\gamma^3} \,.
\end{equation}
As a consequence, a \emph{frequency-independent} readout quadrature optimized for low frequencies can be used:
\begin{equation}
\phi_{\mathrm{LO}} = \mathop{{\mathrm{arccot}}\nolimits\mathcal{K}_{\mathrm{SM}}(0) = \mathop{{\mathrm{arccot}}\nolimits\frac{4J}{\gamma^3} \,,
\end{equation}
which gives the following power (double-sided) spectral density
\begin{equation}
\label{S_SM_LF}
S^h(\Omega) = \frac{S^h_{\mathrm{SQL}}(\Omega)}{2}\left[
\frac{e^{-2r}}{\mathcal{K}_{\mathrm{SM}}(\Omega)}
+ \frac{\Omega^4(2\gamma^2+\Omega^2)^2}{\gamma^8}\mathcal{K}_{\mathrm{SM}}(\Omega)e^{2r}
\right] .
\end{equation}
Here, the radiation-pressure noise (the second term in brackets) is
significantly suppressed in low frequencies ($\Omega
\stackrel{<}{_\sim}\gamma$), and $ S^h_{\mathrm{SM\,LF}}$ can beat the
SQL in a broad frequency band.
This spectral density is plotted in Figure~\ref{fig:LIGO_sm}\,(left). For comparison, spectral densities for the lossless ordinary Fabry--P{\'e}rot--Michelson interferometer without and with squeezing, as well as for the ideal post-filtering configuration [see Eq.~\eqref{eq:vr_post_opt}] are also given. One might conclude from these plots that the Fabry--P{\'e}rot--Michelson interferometer with the additional filter cavities is clearly better than the speedmeter. However, below we demonstrate that optical losses change this picture significantly.
\epubtkImage{fig42.png}{
\begin{figure}[htbp]
\centerline{\includegraphics[width=.48\textwidth]{fig42a}\hfill\includegraphics[width=.48\textwidth]{fig42b}}
\caption{Examples of the sum quantum noise power (double-sided) spectral densities of the Sagnac speedmeter interferometer (thick solid line) in comparison with the Fabry--P{\'e}rot--Michelson based topologies considerd above (dashed lines). Left: no optical losses, right: with optical losses, $\eta_d=0.95$, the losses part of the bandwidth $\gamma_2=1.875\mathrm{\ s}^{-1}$ (which corresponds to the losses $A_{\mathrm{arm}}=10^{-4}$ per bounce in the 4~km length arms). ``Ordinary'': no squeezing, $\phi_{\mathrm{LO}}=\pi/2$. ``Squeezed'': 10~dB squeezing, $\theta=0$, $\phi_{\mathrm{LO}}=\pi/2$. ``Post-filtering'': 10~dB squeezing, $\theta=0$, ideal frequency-dependent homodyne angle [see Eq.~\eqref{eq:vr_post_opt}]. For the Fabry--P{\'e}rot--Michelson-based topologies, $J=J_{\mathrm{aLIGO}}$ and $\gamma=2\pi\times500\mathrm{\ s}^{-1}$. In the speedmeter case, $J=2J_{\mathrm{aLIGO}}$ and the bandwidth is set to provide the same high-frequency noise as in the other plots ($\gamma=2\pi\times385\mathrm{\ s}^{-1}$ in the lossless case and $\gamma=2\pi\times360\mathrm{\ s}^{-1}$ in the lossy one).}
\label{fig:LIGO_sm}
\end{figure}}
\subsubsection{Optical losses in speedmeters}
\label{sec:sm_loss}
In speedmeters, optical losses in the arm cavities could noticeably affect the sum noise at low frequencies, even if
\begin{equation}
\epsilon_{\mathrm{arm}}^2 \equiv \frac{\gamma_2}{\gamma_1} \ll \epsilon_d^2 \,,
\end{equation}
because the radiation pressure noise component created by the arm cavity losses has a frequency dependence similar to the one for position meters (remember that $\mathcal{K}/\mathcal{K}_{\mathrm{SM}}\to\infty$ if $\Omega\to0$; see Eqs.~\eqref{eq:varK}, \eqref{KSM}). In this paper, we will use the following expression for the lossy speedmeter sum noise, which takes these losses into account (more detailed treatment of the lossy speedmeter can be found in papers~\cite{Purdue2002, 04a1Da}):
\begin{eqnarray}
\label{S_SM_LF_loss}
S^h(\Omega) = \frac{S^h_{\mathrm{SQL}}(\Omega)}{2}\biggl\{
\frac{1}{\mathcal{K}_{\mathrm{SM}}(\Omega)}
\left[e^{-2r} + e^{2r}\cot^2\phi_{\mathrm{LO}} +
\frac{\epsilon_d^2}{\sin^2\phi_{\mathrm{LO}}}\right] \nonumber\\
- 2e^{2r}\cot\phi_{\mathrm{LO}} + \mathcal{K}_{\mathrm{SM}}(\Omega)e^{2r}
+ \epsilon_{\mathrm{arm}}^2\mathcal{K}(\Omega)
\biggr\} .
\end{eqnarray}
The low-frequency optimized detection angle, in presence of loss, is
\begin{equation}
\phi_{\mathrm{LO}} = \mathop{{\mathrm{arccot}}\nolimits\frac{\mathcal{K}_{\mathrm{SM}}(0)}{1 + \epsilon_d^2e^{-2r}}
= \mathop{{\mathrm{arccot}}\nolimits\frac{4J/\gamma^3}{1 + \epsilon_d^2e^{-2r}} \,,
\end{equation}
which gives
\begin{equation}
\label{S_SM_LF_loss_opt}
S^h(\Omega) = \frac{S^h_{\mathrm{SQL}}(\Omega)}{2}\biggl\{
\frac{e^{-2r} + \epsilon_d^2}{\mathcal{K}_{\mathrm{SM}}(\Omega)}
+ \frac{\mathcal{K}_{\mathrm{SM}}(\Omega)}{1 + \epsilon_d^2e^{-2r}}\left[
\frac{\Omega^4(2\gamma^2 + \Omega^2)^2}{\gamma^8}\,e^{2r} + \epsilon_d^2
\right]
+ \epsilon_{\mathrm{arm}}^2\mathcal{K}(\Omega)
\biggr\}
\end{equation}
[compare with Eq.~\eqref{S_SM_LF} and note the additional residual back-action term similar to one in Eq.~\eqref{eq:vr_post_opt}].
This spectral density is plotted in Figure~\ref{fig:LIGO_sm}\,(right), together with the lossy variants of the same configurations as in Figure~\ref{fig:LIGO_sm}\,(left), for the same moderately optimistic value of $\eta_d=0.95$, the losses part of the bandwidth and for $\gamma_2=1.875\mathrm{\ s}^{-1}$ [which corresponds to the losses $A_{\mathrm{arm}}=10^{-4}$ per bounce in the 4~km length arms, see Eq.~\eqref{sl_gammas}].
These plots demonstrate that the speedmeter in more robust with respect to optical losses than the filter cavities based configuration and is able to provide better sensitivity at very low frequencies.
It should also be noted that we have not taken into account here optical losses in the filter cavity. Comparison of Figure~\ref{fig:LIGO_sm} with Figure~\ref{fig:LIGO_filter}, where the noise spectral density for the more realistic lossy--filter-cavity cases are plotted, shows that the speedmeter has advantage over, at least, the short and medium length (tens or hundred of meters) filter cavities. In the choice between very long (and hence expensive) kilometer scale filter cavities and the speedmeter, the decision depends, probably, on the implementation costs of both configurations.
\subsection{Optical rigidity}
\label{sec:optical_rigidity}
\subsubsection{Introduction}
\label{sec:opt_rig_intro}
We have seen in Section~\ref{sec:SQL} that the harmonic oscillator, due to its strong response on near-resonance force, is characterized by the reduced values of the effective quantum noise and, therefore, by the SQL around the resonance frequency, see Eqs.~(\ref{eq:S_F_osc_nb}, \ref{eq:S_F_SQL_osc}) and Figure~\ref{fig:S_F_SQL}. However, practical implementation of this gain is limited by the following two shortcomings: (i) the stronger the sensitivity gain, the more narrow the frequency band in which it is achieved; see Eq.~\eqref{eq:osc_fm_SQL}; (ii) in many cases, and, in particular, in a GW detection scenario with its low signal frequencies and heavy test masses separated by the kilometers-scale distances, ordinary solid-state springs cannot be used due to unacceptably high levels of mechanical loss and the associated thermal noise.
At the same time, in detuned Fabry--P{\'e}rot cavities, as well as in the detuned configurations of the Fabry--P{\'e}rot--Michelson interferometer, the radiation pressure force depends on the mirror displacement (see Eqs.~\eqref{eq:FP_F_fl_K}), which is equivalent to the additional rigidity, called the \emph{optical spring}, inserted between the cavity mirrors. It does not introduce any additional thermal noise, except for the radiation pressure noise $\hat{F}_{\mathrm{b.a.}}$, and, therefore, is free from the latter of the above mentioned shortcomings. Moreover, as we shall show below, spectral dependence of the optical rigidity $K(\Omega)$ alleviates, to some extent, the former shortcoming of the `ordinary' rigidity and provides some limited sensitivity gain in a relatively broad band.
The electromagnetic rigidity was first discovered experimentally in radio-frequency systems~\cite{64a1eBrMi}. Then its existence was predicted for the optical Fabry--P{\'e}rot cavities~\cite{67a1eBrMa}. Much later it was shown that the excellent noise properties of the optical rigidity allows its use in quantum experiments with macroscopic mechanical objects~\cite{97a1BrGoKh, 99a1BrKh, 01a1BrKhVo}. The frequency dependence of the optical rigidity was explored in papers~\cite{Buonanno2001, 01a2Kh, PhysRevD.65.042001}. It was shown that depending on the interferometer tuning, either two resonances can exist in the system, \emph{mechanical} and \emph{optical} ones, or a single broader second-order resonance will exist.
In the last decade, the optical rigidity has been observed experimentally both in the table-top setup~\cite{Corbitt2007} and in the larger prototype interferometer~\cite{Miyakawa_PRD_74_022001_2006}.
\subsubsection{The optical noise redefinition}
\label{sec:opt_rig_noise_redef}
In detuned interferometer configurations, where the optical rigidity arises, the phase shifts between the input and output fields, as well as between the input fields and the field, circulating inside the interferometer, depend in sophisticated way on the frequency $\Omega$. Therefore, in order to draw full advantage from the squeezing, the squeezing angle of the input field should follow this frequency dependence, which is problematic from the implementation point of view. Due to this reason, considering the optical-rigidity--based regimes, we limit ourselves to the vacuum-input case only, setting $\mathbb{S}_{\sqz}[r,\theta]=\mathbb{I}$ in Eq.~\eqref{a_sqz}.
In this case, it is convenient to redefine the input noise operators as follows:
\begin{eqnarray}
\label{eq:new_noises}
\sqrt{\gamma}\hat{\boldsymbol{a}}_{\mathrm{new}}
&=& \sqrt{\gamma_1}\hat{\boldsymbol{a}} + \sqrt{\gamma_2}\hat{\boldsymbol{g}} \,,\nonumber \\
\sqrt{\gamma}\hat{\boldsymbol{g}}_{\mathrm{new}}
&=& \sqrt{\gamma_1}\hat{\boldsymbol{g}} - \sqrt{\gamma_2}\hat{\boldsymbol{a}} \,, \nonumber\\
\epsilon\hat{\boldsymbol{n}}_{\mathrm{new}}
&=& \sqrt{\frac{\gamma_2}{\gamma_1}}\,\hat{\boldsymbol{g}}
+ \sqrt{\frac{\gamma}{\gamma_1}}\epsilon_d\hat{\boldsymbol{n}} \,,
\end{eqnarray}
where
\begin{equation}
\label{vac_eta}
\epsilon = \sqrt{\frac{1}{\eta}-1} \qquad\mbox{and}\qquad
\eta = \frac{\gamma_1}{\gamma}\eta_d
\end{equation}
is the \emph{unified quantum efficiency}, which accounts for optical losses both in the interferometer and in the homodyne detector.
Note that if the operators $\hat{\boldsymbol{a}}$, $\hat{\boldsymbol{g}}$, and $\hat{\boldsymbol{n}}$ describe mutually-uncorrelated vacuum noises, then the same is valid for the new $\hat{\boldsymbol{a}}_{\mathrm{new}}$, $\hat{\boldsymbol{g}}_{\mathrm{new}}$, and $\hat{\boldsymbol{n}}_{\mathrm{new}}$. Expressing Eqs.~(\ref{eq:ord_x_meas} and~\ref{eq:ord_F_pert}) in terms of new noises~\eqref{eq:new_noises} and renaming them, for brevity,
\begin{equation}
\hat{\boldsymbol{a}}_{\mathrm{new}} \to \hat{\boldsymbol{a}} \,, \qquad
\hat{\boldsymbol{n}}_{\mathrm{new}} \to \hat{\boldsymbol{n}} \,,
\end{equation}
we obtain:
\begin{eqnarray}
\hat{\mathcal{X}}_{\mathrm{meas}}(\Omega) &=& \sqrt{\frac{\hbar}{2MJ\gamma}}\,
\frac{\mathcal{D}(\Omega)}
{\mathbf{H}^{\mathsf{T}}(\phi_{\mathrm{LO}})\mathbf{D}(\Omega)}
\mathbf{H}^{\mathsf{T}}(\phi_{\mathrm{LO}})
[\mathbb{R}(\Omega)\hat{\boldsymbol{a}}(\Omega) + \epsilon\hat{\boldsymbol{n}}(\Omega)] \,,
\label{eq:ord0_x_meas} \\
\hat{\mathcal{F}}_{\mathrm{b.a.}}(\Omega)
&=& \sqrt{2\hbar MJ\gamma}\begin{bmatrix}1\\0\end{bmatrix}^{\mathsf{T}}\mathbb{L}(\Omega)\hat{\boldsymbol{a}}(\Omega) \,,
\label{eq:ord0_F_pert}
\end{eqnarray}
where $\mathbb{R}(\Omega)$ is the lossless cavity reflection factor; see Eq.~\eqref{R_lossles}.
Thus, we have effectively reduced our lossy interferometer to the equivalent lossless one, but with less effective homodyne detector, described by the unified quantum efficiency $\eta<\eta_d$. Now we can write down explicit expressions for the interferometer quantum noises~\eqref{eq:FPMI_Sx}, \eqref{eq:FPMI_SF} and \eqref{eq:FPMI_SxF}, which can be calculated using Eqs.~\eqref{RL_props}:
\begin{eqnarray}
\label{eq:sl_noises0}
S_{\mathcal{X}\mathcal{X}}(\Omega) &=& \frac{\hbar}{4MJ\gamma\eta}\,
\frac{|\mathcal{D}(\Omega)|^2}{\varGamma^2\sin^2\varphi + \Omega^2\sin^2\phi_{\mathrm{LO}}}\,,\nonumber\\
S_{\mathcal{F}\mathcal{F}}(\Omega) &=& \frac{\hbar MJ\gamma(\varGamma^2 + \Omega^2)}{|\mathcal{D}(\Omega)|^2} \,, \nonumber\\
S_{\mathcal{X}\mathcal{F}}(\Omega) &=& \frac{\hbar}{2}\,\frac{\varGamma\cos\varphi - i\Omega\cos\phi_{\mathrm{LO}}}
{\varGamma\sin\varphi - i\Omega\sin\phi_{\mathrm{LO}}} \,,
\end{eqnarray}
where
\begin{equation}
\varGamma = \sqrt{\gamma^2+\delta^2} \,, \qquad
\varphi = \phi_{\mathrm{LO}} - \beta \,, \qquad
\beta = \arctan\frac{\delta}{\gamma}\,.
\end{equation}
\subsubsection{Bad cavities approximation}
\label{sec:opt_rig_short}
We start our treatment of the optical rigidity with the ``bad cavity'' approximation, discussed in Section~\ref{sec:vr_short} for the resonance-tuned interferometer case. This approximation, in addition to its importance for the smaller-scale prototype interferometers, provides a bridge between our idealized harmonic oscillator consideration of Section~\ref{sec:SQL_osc} and the frequency-dependent rigidity case specific to the large-scale GW detectors, which will be considered below, in Section~\ref{sec:fd_rigidity}.
In the ``bad cavity'' approximation $\varGamma\gg\Omega$, the Eqs.~\eqref{eq:sl_noises0} for the interferometer quantum noises, as well as the expression~\eqref{eq:sl_K} for the optical rigidity can be significantly simplified:
\begin{equation}
\label{eq:SxSFK_short}
S_{\mathcal{X}\mathcal{X}} = \frac{\hbar\varGamma^2}{4MJ\gamma\eta\sin^2\varphi} \,, \qquad
S_{\mathcal{F}\mathcal{F}} =\frac{\hbar MJ\gamma}{\varGamma^2} \,, \qquad
S_{\mathcal{X}\mathcal{F}} = \frac{\hbar}{2}\,\cot\varphi \,,
\end{equation}
and
\begin{equation}
\label{eq:short_K}
K = \frac{MJ\delta}{\varGamma^2} \,.
\end{equation}
Substituting these equations into the equation for the sum quantum noise (cf.\ Eq.~\eqref{eq:gen_spdens}):
\begin{eqnarray}
S^F(\Omega) &=& |K-M\Omega^2|^2S_{\mathcal{X}\mathcal{X}}
+ 2\Re\bigl\{[K-M\Omega^2]S_{\mathcal{X}\mathcal{F}}\bigr\}
+ S_{\mathcal{F}\mathcal{F}} \nonumber\\
&=& |K_{\mathrm{eff}}-M\Omega^2|^2S_{\mathcal{X}\mathcal{X}}+S_{\mathcal{F}\mathcal{F}}^{\mathrm{eff}}\,,
\end{eqnarray}
where
\begin{equation}
S_{\mathcal{F}\mathcal{F}}^{\mathrm{eff}} = \frac{
S_{\mathcal{X}\mathcal{X}}S_{\mathcal{F}\mathcal{F}}
- |S_{\mathcal{X}\mathcal{F}}|^2
}{S_{\mathcal{X}\mathcal{X}}}
\qquad\mbox{and}\qquad
K_{\mathrm{eff}} = K - \frac{S_{\mathcal{X}\mathcal{F}}}{S_{\mathcal{X}\mathcal{X}}}
\end{equation}
stand for the effective back-action noise and effective optical rigidity, respectively,
and dividing by $S^F_{\mathrm{SQL,\,f.m.}}$ defined by Eq.~\eqref{eq:S_F_SQL_fm}, we obtain the SQL beating factor~\eqref{eq:xi2}:
\begin{equation}
\label{short_gen}
\xi^2(\Omega) = \frac{1}{\Omega^2}\left[
(\Omega_m^2-\Omega^2)^2\frac{\varGamma^2}{4J\gamma\eta\sin^2\varphi}
+ \frac{J\gamma}{\varGamma^2}(1-\eta\cos^2\varphi) \right] ,
\end{equation}
where
\begin{equation}
\Omega_m^2 = \frac{K_{\mathrm{eff}}}{M}
= \frac{J}{\varGamma^2}(\delta-\gamma\eta\sin2\varphi)
\end{equation}
is the effective resonance frequency (which takes into account both real and virtual parts of the effective rigidity $K_{\mathrm{eff}}$). Following the reasoning of Section~\ref{sec:SQL_osc}, it is easy to see that this spectral density allows for narrow-band sensitivity gain equal to
\begin{equation}
\xi^2(\Omega_m+\nu) \le \xi^2(\Omega_m\pm\Delta\Omega/2)
\approx \frac{2J\gamma}{\varGamma^2\Omega_m^2}(1-\eta\cos^2\varphi)
\end{equation}
within the bandwidth
\begin{equation}
\frac{\Delta\Omega}{\Omega_m}
= \frac{2J\gamma}{\varGamma^2\Omega_m^2}\sqrt{\eta\sin^2\varphi(1-\eta\cos^2\phi)}
= \xi^2(\Omega_m\pm\Delta\Omega/2)
\sqrt{\frac{\eta\sin^2\varphi}{1-\eta\cos^2\varphi}} \,.
\end{equation}
In the ideal lossless case ($\eta=1$),
\begin{equation}
\frac{\Delta\Omega}{\Omega_m} = \xi^2(\Omega_m\pm\Delta\Omega/2) \,,
\end{equation}
in accord with Eq.\eqref{eq:osc_fm_SQL}. However, if $\eta<1$, then the bandwidth, for a given $\xi$ lessens gradually as the homodyne angle $\varphi$ goes down. Therefore, the optimal case of the broadest bandwidth, for a given $\xi$, corresponds to $\varphi=\pi/2$, and, therefore, to $S_{xF}=0$ [see Eqs.~\eqref{eq:SxSFK_short}], that is, to the pure `real' rigidity case with non-correlated radiation-pressure and shot noises. This result naturally follows from the above conclusion concerning the amenability of the quantum noise sources cross-correlation to the influence of optical loss.
Therefore, setting $\varphi=\pi/2$ in Eq.~\eqref{short_gen} and taking into account that
\begin{equation}
\frac{J\gamma}{\varGamma^2} = \frac{\Omega_q^2}{2}\cos^2\beta \,, \qquad
\frac{K}{M} = \frac{J\delta}{\varGamma^2} = \frac{\Omega_q^2}{4}\sin2\beta \,,
\end{equation}
where $\Omega_q^2$ is the normalized optical power defined in Eq.~\eqref{eq:short_Omega_q},
we obtain that
\begin{equation}
\label{eq:short_or_sigma}
\xi^2(\Omega) = \frac{1}{2\Omega^2}\biggl[
\left(\frac{\Omega_q^2}{4}\sin2\beta - \Omega^2\right)^2
\frac{1}{\Omega_q^2\eta\cos^2\beta}
+ \Omega_q^2\cos^2\beta
\biggr] .
\end{equation}
Consider now the local minimization of this function at some given frequency $\Omega_0$, similar to one discussed in Section~\ref{sec:vr_short}. Now, the optimization parameter is $\beta$, that is, the detuning $\delta$ of the interferometer. It is easy to show that the optimal $\beta$ is given by the following equation:
\begin{equation}
\label{eq:or_short_env}
\frac{4\Omega_0^2}{\Omega_q^2} - \frac{2}{\tan\beta}
- \frac{\Omega_q^2}{\Omega_0^2}(4\eta-1)\cos^4\beta = 0 \,.
\end{equation}
This fifth-order equation for $\tan\beta$ cannot be solved in radicals. However, in the most interesting case of $\Omega_0\ll\Omega_q$, the following asymptotic solution can easily be obtained:
\begin{equation}
\beta \approx \frac{\pi}{2} - \frac{2\Omega_0^2}{\Omega_q^2} \,,
\end{equation}
thus yielding
\begin{equation}
\label{eq:short_or_sigma_app}
\xi^2(\Omega) \approx \frac{1}{2}\biggl[
\left(1 - \frac{\Omega^2}{\Omega_0^2}\right)^2\frac{\Omega_q^2}{\Omega_0^2\eta}
+ \frac{4\Omega_0^2}{\Omega_q^2}
\biggr] .
\end{equation}
\epubtkImage{fig43.png}{
\begin{figure}
\centerline{\includegraphics[width=.5\textwidth]{fig43}}
\caption{Plots of the SQL beating factor~\eqref{eq:short_or_sigma} of the detuned interferometer, for different values of the normalized detuning: $0\le\beta\equiv\arctan(\delta/\gamma)<\pi/2$, and for unified quantum efficiency $\eta=0.95$. \emph{Thick solid line:} the common envelope of these plots. \emph{Dashed lines:} the common envelopes~\eqref{vr_xi2_opt} of the SQL beating factors for the virtual rigidity case, without squeezing, $r=0$, and with 10~dB squeezing, $e^{2r}=10$ (for comparison).}
\label{fig:short_or_noises}
\end{figure}}
The function~\eqref{eq:short_or_sigma}, with optimal values of $\beta$ defined by the condition~\eqref{eq:or_short_env}, is plotted in Figure~\ref{fig:short_or_noises} for several values of the normalized detuning. We assumed in these plots that the unified quantum efficiency is equal to $\eta=0.95$. In the ideal lossless case $\eta=1$, the corresponding curves do not differ noticeably from the plotted ones. It means that in the real rigidity case, contrary to the virtual one, the sensitivity is not affected significantly by optical loss. This conclusion can also be derived directly from Eqs.~(\ref{eq:short_or_sigma}) and (\ref{eq:short_or_sigma_app}). It stems from the fact that quantum noise sources cross-correlation, amenable to the optical loss, has not been used here. Instead, the sensitivity gain is obtained by means of signal amplification using the resonance character of the effective harmonic oscillator response, provided by the optical rigidity.
The common envelope of these plots (that is, the optimal SQL-beating factor), defined implicitly by Eqs.~\eqref{eq:short_or_sigma} and \eqref{eq:or_short_env}, is also shown in Figure~\ref{fig:short_or_noises}. Note that at low frequencies, $\Omega\ll\Omega_q$, it can be approximated as follows:
\begin{equation}
\label{or_short_env_app}
\xi^2_{\mathrm{env}}(\Omega) = \frac{2\Omega^2}{\Omega_q^2}
\approx \frac{\gamma}{\delta}\,,
\end{equation}
(actually, this approximation works well starting from $\Omega\lesssim0.3\Omega_q$). It follows from this equation that in order to obtain a sensitivity significantly better than the SQL level, the interferometer should be detuned far from the resonance, $\delta\gg\gamma$.
For comparison, we reproduce here the common envelopes of the plots of $\xi^2(\Omega)$ for the virtual rigidity case with $\eta=0.95$; see Figure~\ref{fig:short_vr_noises} (the dashed lines). It follows from Eqs.~\eqref{or_short_env_app} and \eqref{vr_xi2_opt} that in absence of the optical loss, the sensitivity of the real rigidity case is inferior to that of the virtual rigidity one. However, even a very modest optical loss value changes the situation drastically. The noise cancellation (virtual rigidity) method proves to be advantageous only for rather moderate values of the SQL beating factor of $\xi\gtrsim0.5$ in the absence of squeezing and $\xi\gtrsim0.3$ with 10~dB squeezing. The conclusion is forced upon you that in order to dive really deep under the SQL, the use of real rather than virtual rigidity is inevitable.
Noteworthy, however, is the fact that optical rigidity has an inherent feature that can complicate its experimental implementation. It is dynamically unstable. Really, the expansion of the optical rigidity~\eqref{eq:sl_K} into a Taylor series in $\Omega$ gives
\begin{equation}
\label{eq:short_K_dump}
K(\Omega) \approx \frac{MJ\delta}{\varGamma^2}
+ \frac{2MJ\gamma\delta}{\varGamma^4}i\Omega + \dots
\end{equation}
The second term, proportional to $i\Omega$, describes an optical friction, and the positive sign of this term (if $\delta>0$) means that this friction is negative.
The corresponding characteristic instability time is equal to
\begin{equation}
\tau_{\mathrm{inst}} = \frac{\varGamma^4}{2J\gamma\delta} \,.
\end{equation}
In principle, this instability can be damped by some feedback control system as analyzed in~\cite{Buonanno2001, 03pth1Ch}. However, it can be done without significantly affecting the system dynamics, only if the instability is slow in the timescale of the mechanical oscillations frequency:
\begin{equation}
\Omega_m\tau_{\mathrm{inst}} = \frac{\varGamma^2}{2\gamma\Omega_m}
\approx \frac{J}{2\Omega_m^3\xi^2} \gg 1\,.
\end{equation}
Taking into account that in real-life experiments the normalized optical power $J$ is limited for technological reasons, the only way to get a more stable configuration is to decrease $\xi$, that is, to \emph{improve} the sensitivity by means of increasing the detuning. Another way to vanquish the instability is to create a stable optical spring by employing the second pumping carrier light with opposite detuning as proposed in~\cite{Corbitt2007, PhysRevD.78.062003_2008_Rehbein}. The parameters of the second carrier should be chosen so that the total optical rigidity must have both positive real and imaginary parts in Eq.~\eqref{eq:short_K_dump}:
\begin{equation}
K_{\mathrm{sum}}(\Omega) = K_1(\Omega) + K_2(\Omega) \approx \left(\frac{MJ_1\delta_1}{\varGamma_1^2}+\frac{MJ_2\delta_2}{\varGamma_2^2}\right)
+ i\Omega\left(\frac{2MJ_1\gamma_1\delta_1}{\varGamma_1^4}+\frac{2MJ_2\gamma_2\delta_2}{\varGamma_2^4}\right) + \dots
\end{equation}
that can always be achieved by a proper choice of the parameters $J_{1,2}$, $\gamma_{1,2}$ and $\delta_{1,2}$ ($\delta_1\delta_2<0$).
\subsubsection{General case}
\label{sec:fd_rigidity}
\paragraph*{Frequency-dependent rigidity.}
In the large-scale laser GW detectors with kilometer-scale arm cavities, the interferometer bandwidth can easily be made comparable or smaller than the GW signal frequency $\Omega$. In this case, frequency dependences of the quantum noise spectral densities~\eqref{eq:FPMI_Sx}, \eqref{eq:FPMI_SF} and \eqref{eq:FPMI_SxF} and of the optical rigidity~\eqref{eq:sl_K} influence the shape of the sum quantum noise and, therefore, the detector sensitivity.
Most quantum noise spectral density is affected by the effective mechanical dynamics of the probe bodies, established by the frequency-dependent optical rigidity~\eqref{eq:sl_K}. Consider the characteristic equation for this system:
\begin{equation}
\label{eq:long_cheq}
-\Omega^2[(\gamma-i\Omega)^2+\delta^2] + J\delta = 0 \,.
\end{equation}
In the asymptotic case of $\gamma=0$, the roots of this equation are equal to
\begin{equation}
\label{long_roots0}
\Omega_m^{(0)} = \sqrt{\frac{\delta^2}{2}-\sqrt{\frac{\delta^4}{4}-J\delta}} \,, \qquad
\Omega_o^{(0)} = \sqrt{\frac{\delta^2}{2}+\sqrt{\frac{\delta^4}{4}-J\delta}},
\end{equation}
(hereafter we omit the roots with negative-valued real parts). The corresponding maxima of the effective mechanical susceptibility:
\begin{equation}
\label{mech_Green}
\chi_{xx}^{\mathrm{eff}}(\Omega) = \frac{1}{K(\Omega)-M\Omega^2}
\end{equation}
are, respectively, called the \emph{mechanical resonance} ($\Omega_m$) and the \emph{optical resonance} ($\Omega_o$) of the interferometer~\cite{Buonanno2001}. In order to clarify their origin, consider an asymptotic case of the weak optomechanical coupling, $J\ll\delta^3$. In this case,
\begin{equation}
\Omega_m \approx \frac{J}{\delta^3} = \sqrt{\frac{K(0)}{M}}\,, \qquad
\Omega_o \approx \delta \,.
\end{equation}
It is easy to see that $\Omega_m$ originates from the ordinary resonance of the mechanical oscillator consisting of the test mass $M$ and the optical spring $K$ [compare with Eq.~\eqref{eq:short_K}]. At the same time, $\Omega_o$, in this approximation, does not depend on the optomechanical coupling and, therefore, has a pure optical origin -- namely, sloshing of the optical energy between the carrier power and the differential optical mode of the interferometer, detuned from the carrier frequency by $\delta$.
In the realistic general case of $\gamma\ne0$, the characteristic equation roots are complex. For small values of $\gamma$, keeping only linear in $\gamma$ terms, they can be approximated as follows:
\begin{equation}
\label{eq:long_roots1}
\Omega_m = \Omega_m^{(0)}
\biggl[1 + \frac{i\gamma\Omega_m^{(0)}}{\sqrt{\delta^4-4J\gamma}}\biggr] , \qquad
\Omega_o = \Omega_o^{(0)}
\biggl[1 - \frac{i\gamma\Omega_o^{(0)}}{\sqrt{\delta^4-4J\gamma}}\biggr] .
\end{equation}
Note that the signs of the imaginary parts correspond to a positive dumping for the optical resonance, and to a negative one (that is, to instability) for the mechanical resonance (compare with Eq.~\eqref{eq:short_K_dump}).
\epubtkImage{fig44.png}{
\begin{figure}
\centerline{\includegraphics[width=.8\textwidth]{fig44}}
\caption{Roots of the characteristic equation~\eqref{eq:long_cheq} as functions of the optical power, for $\gamma/\delta=0.03$. Solid lines: numerical solution. Dashed lines: approximate solution, see Eqs.~\eqref{eq:long_roots1}}
\label{fig:long_roots}
\end{figure}}
In Figure~\ref{fig:long_roots}, the numerically-calculated roots of Eq.~\eqref{eq:long_cheq} are plotted as a function of the normalized optical power $J/\delta^3$, together with the analytical approximate solution~\eqref{eq:long_roots1}, for the particular case of $\gamma/\delta=0.03$. These plots demonstrate the peculiar feature of the parametric optomechanical interaction, namely, the \emph{decrease} of the separation between the eigenfrequencies of the system as the optomechanical coupling strength goes up. This behavior is opposite to that of the ordinary coupled linear oscillators, where the separation between the eigenfrequencies increases as the coupling strength grows (the well-known avoided crossing feature).
As a result, if the optomechanical coupling reaches the critical value:
\begin{equation}
\label{long_J_crit}
J = \frac{\delta^3}{4} \,,
\end{equation}
then, in the asymptotic case of $\gamma=0$, the eigenfrequencies become equal to each other:
\begin{equation}
\label{long_Omega_0}
\Omega_m^{(0)} = \Omega_o^{(0)} = \Omega_0 \equiv \frac{\delta}{\sqrt{2}} \,.
\end{equation}
If $\gamma>0$, then some separation retains, but it gets smaller than $2\gamma$, which means that the corresponding resonance curves effectively merge, forming a single, broader resonance. This \emph{second-order pole regime}, described for the first time in~\cite{01a2Kh}, promises some significant advantages for high-precision mechanical measurements, and we shall consider it in more detail below.
If $J<J_{\mathrm{crit}}$, then two resonances yield two more or less
separated minima of the sum quantum noise spectral density, whose
location on the frequency axis mostly depends on the detuning
$\delta$, and their depth (inversely proportional to their width)
hinges on the bandwidth $\gamma$.The choice of the preferable
configuration depends on the criterion of the optimization, and also
on the level of the technical (non-quantum) noise in the
interferometer.
\epubtkImage{fig45.png}{
\begin{figure}[htb]
\centerline{\includegraphics[width=.8\textwidth]{fig45}}
\caption{Examples of the sum noise power (double-sided) spectral densities of the detuned interferometer. `Broadband': double optimization of the Advanced LIGO interferometer for NS-NS inspiraling and burst sources in presence of the classical noises~\cite{08a1KoSiKhDa} ($J=J_{\mathrm{aLIGO}}\equiv(2\pi\times100)^3\mathrm{\ s}^{-3}$, $\varGamma=3100\mathrm{\ s}^{-1}$, $\beta=0.80$, $\phi_{\mathrm{LO}}=\pi/2-0.44$). `High-frequency': low-power configuration suitable for detection of the GW signals from the millisecond pulsars, similar to one planned for GEO HF~\cite{Willke2006} [$J=0.1J_{\mathrm{aLIGO}}$, $\varGamma=2\pi\times1000\mathrm{\ s}^{-1}$, $\beta=\pi/2-0.01$, $\phi_{\mathrm{LO}}=0$]. `Second-order pole': the regime close to the second-order pole one, which provides a maximum of the SNR for the GW burst sources given that technical noise is smaller than the SQL [$S_{\mathrm{tech}}=0.1S_{\mathrm{SQL}}$, $J=J_{\mathrm{aLIGO}}$, $\varGamma=1050\mathrm{\ s}^{-1}$, $\beta=\pi/2-0.040$, $\phi_{\mathrm{LO}}=0.91$].
In all cases, $\eta_d=0.95$ and the losses part of the bandwidth $\gamma_2=1.875\mathrm{\ s}^{-1}$ (which corresponds to the losses $A_{\mathrm{arm}}=10^{-4}$ per bounce in the 4~km long arms).}
\label{fig:LIGO_det}
\end{figure}}
Two opposite examples are drawn in Figure~\ref{fig:LIGO_det}. The first one features the sensitivity of a broadband configuration, which provides the best SNR for the GW radiation from the inspiraling neutron-star--neutron-star binary and, at the same time, for broadband radiation from the GW burst sources, for the parameters planned for the Advanced LIGO interferometer (in particular, the circulating optical power $\mathcal{I}_{\mathrm{arm}}=840\mathrm{\ kW}$, $L=4\mathrm{\ km}$, and $M=40\mathrm{\ kg}$, which translates to $J=J_{\mathrm{aLIGO}}\equiv(2\pi\times100)^3\mathrm{\ s}^{-3}$, and the planned technical noise). The optimization performed in~\cite{08a1KoSiKhDa} gave the quantum noise spectral density, labeled as `Broadband' in Figure~\ref{fig:LIGO_det}. It is easy to notice two (yet not discernible) minima on this plot, which correspond to the mechanical and the optical resonances.
Another example is the configuration suitable for detection of the narrow-band GW radiation from millisecond pulsars. Apparently, one of two resonances should coincide with the signal frequency in this case. It is well to bear in mind that in order to create an optical spring with mechanical resonance in a kHz region in contemporary and planned GW detectors, an enormous amount of optical power might be required. This is why the optical resonance, whose frequency depends mostly on the detuning $\delta$, should be used for this purpose. This is, actually, the idea behind the GEO HF project~\cite{Willke2006}. The example of this regime is represented by the curve labeled as `High-frequency' in Figure~\ref{fig:LIGO_det}. Here, despite one order of magnitude less optical power used ($J=0.1J_{\mathrm{aLIGO}}$), several times better sensitivity at frequency 1~kHz, than in the `Broadband' regime, can be obtained. Note that the mechanical resonance in this case corresponds to 10~Hz only and therefore is virtually useless.
\paragraph*{The second-order pole regime.}
\label{sec:double_pole}
In order to clarify the main properties of the second-order pole regime, start with the asymptotic case of $\gamma\to0$. In this case, the optical rigidity and the mechanical susceptibility~\eqref{mech_Green} read
\begin{eqnarray}
K(\Omega) &=& \frac{MJ\delta}{\delta^2-\Omega^2} \,, \\
\chi^{\mathrm{eff,\,dbl}}_{xx}(\Omega) &=& \frac{1}{K(\Omega)-M\Omega^2}
= \frac{1}{M}\frac{\delta^2-\Omega^2}{J\delta - \delta^2\Omega^2 + \Omega^4} \,.
\end{eqnarray}
If condition~\eqref{long_J_crit} is satisfied, then in the close vicinity of the frequency $\Omega_0$ (see Eq.~\eqref{long_Omega_0}):
\begin{equation}
|\Omega-\Omega_0| \ll \Omega_0 \,,
\end{equation}
the susceptibility can be approximated as follows:
\begin{equation}
\chi^{\mathrm{eff,\,dbl}}_{xx}(\Omega) \approx \frac{\Omega_0^2}{M(\Omega_0^2-\Omega^2)^2} \,.
\end{equation}
Note that this susceptibility is proportional to the square of the susceptibility of the ordinary oscillator,
\begin{equation}
\chi_{xx}^{\mathrm{osc}}(\Omega) = \frac{1}{M(\Omega_0^2-\Omega^2)} \,,
\end{equation}
and has the second-order pole at the frequency $\Omega_0$ (thus the name of this regime).
This `double-resonance' feature creates a stronger response to the external forces with spectra concentrated near the frequency $\Omega_0$, than in the ordinary harmonic oscillator case. Consider, for example, the resonance force $F_0\sin\Omega_0t$. The response of the ordinary harmonic oscillator with eigenfrequency $\Omega_0$ on this force increases linearly with time:
\begin{equation}
x_{\mathrm{osc}}(t) = \frac{F_0t}{2M\Omega_0}\sin\Omega_0t \,,
\end{equation}
while that of the second-order pole object grows quadratically:
\begin{equation}
x_{\mathrm{dbl}}(t)
= -\frac{F_0}{8M}\left(t^2\cos\Omega_0t - \frac{\sin\Omega_0t}{\Omega_0}\right) .
\end{equation}
It follows from Eq.~\eqref{eq:SQL_force} that due to this strong response, the second-order pole test object has a reduced value of the SQL around $\Omega_0$ by contrast to the harmonic oscillator.
Consider the quantum noise of the system, consisting of this test object and the SQM (that is, the Heisenberg's-uncertainty-relation--limited quantum meter with frequency-independent and non-correlated measurement and back-action noises; see Section~\ref{sec:linear_toy}), which monitors its position. Below we show that the real-life long-arm interferometer, under some assumptions, can be approximated by this model.
The sum quantum noise power (double-sided) spectral density of this system is equal to
\begin{equation}
S^h(\Omega) = \frac{4}{M^2L^2\Omega^4}(|\chi^{\mathrm{eff,\,dbl}}_{xx}(\Omega)|^2S_{\mathcal{X}\mathcal{X}} + S_{\mathcal{F}\mathcal{F}}) \,.
\end{equation}
If the frequency $\Omega$ is close to $\Omega_0$:
\begin{equation}
\Omega = \Omega_0+\nu \,, \qquad |\nu|\ll\Omega_0 \,,
\end{equation}
then
\begin{equation}
\label{eq:S_X_dbl}
S^h(\Omega_0+\nu)
\approx \frac{2\hbar}{ML^2\Omega_0^4}\left(\frac{16\nu^4}{\Omega_q^2} + \Omega_q^2\right),
\end{equation}
and
\begin{equation}
\label{eq:xi2_dbl}
\xi^2(\Omega_0+\nu)
\equiv \frac{S^h(\Omega_0+\nu)}{S^h_{\mathrm{SQL\,f.m.}}(\Omega_0+\nu)}
\approx
\frac{1}{2\Omega_0^2}\left(\frac{16\nu^4}{\Omega_q^2} + \Omega_q^2\right),
\end{equation}
(compare with Eq.~\eqref{eq:S_F_osc_nb}), where the frequency $\Omega_q$ is defined by Eq.~\eqref{eq:Omega_q_def}.
The same minimax optimization as performed in Section~\ref{sec:SQL_osc} for the harmonic oscillator case gives that the optimal value of $\Omega_q$ is equal to
\begin{equation}
\Omega_q = \Delta\Omega \,,
\end{equation}
and in this case,
\begin{equation}
\label{eq:dbl_fm_SQL}
\xi^2(\Omega_0+\nu)\Bigr|_{|\nu|\le\Delta\Omega/2}
\le \xi^2(\Omega_0\pm\Delta\Omega/2) = \left(\frac{\Delta\Omega}{\Omega_0}\right)^2.
\end{equation}
Comparison of Eqs.~\eqref{eq:dbl_fm_SQL} and \eqref{eq:osc_fm_SQL} shows that for a given SQL-beating bandwidth $\Delta\Omega$, the second-order pole system can provide a much stronger sensitivity gain (i.e., much smaller value of $\xi^2$), than the harmonic oscillator, or, alternatively, much broader bandwidth $\Delta\Omega$ for a given value of $\xi^2$. It is noteworthy that the factor~\eqref{eq:dbl_fm_SQL} can be made smaller than the normalized oscillator SQL $2|\nu|/\Omega_0$ (see Eq.~\eqref{eq:osc_fm_SQL}), which means beating not only the free mass SQL, but also the harmonic oscillator one.
\epubtkImage{fig44.png}{%
\begin{figure}
\centerline{\includegraphics[width=.48\textwidth]{fig46a}\hfill\includegraphics[width=.48\textwidth]{fig46b}}
\caption{\emph{Left panel:} the SQL beating factors $\xi^2$ for $\Omega_q/\Omega_0=0.1$. Thick solid: the second-order pole system~\eqref{eq:xi2_dbl}; dots: the two-pole system with optimal separation between the poles~\eqref{eq:dbl_sigma2}, \eqref{dbl_optimal}; dashes: the harmonic oscillator~\eqref{eq:xi2_osc}; thin solid -- SQL of the harmonic oscillator~\eqref{eq:osc_fm_SQL}. \emph{Right:} the normalized SNR~\eqref{eq:dbl_rho2_tech}. Solid line: analytical optimization, Eq.~\eqref{eq:dbl_rho2_opt}; pluses: numerical optimization of the spectral density~\eqref{eq:sl_sum0} in the lossless case ($\eta=1$); diamonds: the same for the interferometer with $J=(2\pi\times100)^3\mathrm{\ s}^{-3}$, $\eta_d=0.95$ and the losses part of the bandwidth $\gamma_2=1.875\mathrm{\ s}^{-1}$ (which corresponds to the losses $A_{\mathrm{arm}}=10^{-4}$ per bounce in the 4~km long arms).}
\label{fig:sigma2_dbl}
\end{figure}}
This consideration is illustrated by the left panel of Figure~\ref{fig:sigma2_dbl}, where the factors $\xi^2$ for the harmonic oscillator~\eqref{eq:xi2_osc} and of the second-order pole system~\eqref{eq:xi2_dbl} are plotted for the same value of the normalized back-action noise spectral density $(\Omega_q/\Omega_0)^2=0.01$, as well as the normalized oscillator SQL~\eqref{eq:osc_fm_SQL}.
Now return to the quantum noise of a real interferometer. With account of the noises redefinition~\eqref{eq:new_noises}, Eq.~\eqref{eq:S_sum_BC} for the sum quantum noise power (double-sided) spectral density takes the following form:
\begin{eqnarray}
\label{eq:sl_sum0}
S^h(\Omega) = \frac{\hbar}{ML^2J\gamma}
\frac{1}{\Omega^2\sin^2\phi_{\mathrm{LO}} + \varGamma^2\sin^2\varphi}
\biggl\{
\left[
\gamma^2 - \delta^2 + \Omega^2 +\frac{J}{\Omega^2}(\delta-\gamma\sin2\phi_{\mathrm{LO}})
\right]^2 \nonumber\\
+ 4\gamma^2\left(\delta - \frac{J}{\Omega^2}\sin^2\phi_{\mathrm{LO}}\right)^2
+ \epsilon^2\left|\mathcal{D}(\Omega) - \frac{J\delta}{\Omega^2}\right|^2
\biggr\} .
\end{eqnarray}
Suppose that the interferometer parameters satisfy approximately the second-order pole conditions. Namely, introduce a new parameter $\Lambda$ defined by the following equation:
\begin{equation}
J(\delta-\gamma\sin2\phi_{\mathrm{LO}}) = \Omega_0^4 - \Omega_0^2\Lambda^2 \,,
\end{equation}
where the frequency $\Omega_0$ is defined by Eq.~\eqref{long_Omega_0}, and assume that
\begin{equation}
\label{dbl_conds}
\nu^2 \sim \Lambda^2 \sim \Omega_0\gamma \ll \Omega_0^2 \,.
\end{equation}
Keeping only the first non-vanishing terms in $\nu^2$, $\Lambda^2$, and $\gamma$ in Eq.~\eqref{eq:sl_sum0}, we obtain that
\begin{equation}
\label{eq:S_X_nb}
S^h(\Omega_0+\nu) \approx \frac{2\hbar}{ML^2\Omega_0^4}\Biggl(
\frac{1}{\Omega_q^2}\biggl\{
(4\nu^2 - \Lambda^2)^2
+ \epsilon^2\biggl[
\left(4\nu^2-\Lambda^2+\frac{\Omega_0\gamma}{\sqrt{2}}\sin2\phi_{\mathrm{LO}}\right)^2
+ 4\gamma^2\Omega_0^2
\biggr]
\biggr\}
+ \Omega_q^2
\Biggr) ,
\end{equation}
where
\begin{equation}
\label{eq:Omega_q_nb}
\Omega_q^2 = \sqrt{2}\gamma\Omega_0(1+\cos^2\phi_{\mathrm{LO}}) \,.
\end{equation}
It follows from Eq.~\eqref{eq:S_X_nb} that the parameter $\Lambda$ is equal to the separation between the two poles of the susceptibility $\chi_{xx}^{\mathrm{eff,\,dbl}}$.
It is evident that the spectral density~\eqref{eq:S_X_nb} represents a direct generalization of Eq.~\eqref{eq:S_X_dbl} in two aspects. First, it factors in optical losses in the interferometer. Second, it includes the case of $\Lambda\ne0$. We show below that a small yet non-zero value of $\Lambda$ allows one to further increase the sensitivity.
\paragraph*{Optimization of the signal-to-noise ratio.}
The peculiar feature of the second-order pole regime is that, while being, in essence, narrow-band, it can provide an arbitrarily-high SNR for the broadband signals, limited only by the level of the additional noise of non-quantum (technical) origin. At the same time, in the ordinary harmonic oscillator case, the SNR is fundamentally limited.
In both the harmonic oscillator and the second-order pole test object cases, the quantum noise spectral density has a deep and narrow minimum, which makes the major part of the SNR integral. If the bandwidth of the signal force exceeds the width of this minimum (which is typically the case in GW experiments, save to the narrow-band signals from pulsars), then the SNR integral~\eqref{6:snr} can be approximated as follows:
\begin{equation}
\rho^2 = \frac{|h_{\mathrm{s}}(\Omega_0)|^2}{\pi}
\int_{-\infty}^{\infty}\!\frac{d\nu}{S^h(\Omega_0+\nu)} \,.
\end{equation}
It is convenient to normalize both the signal force and the noise spectral density by the corresponding SQL values, which gives:
\begin{equation}
\rho^2 = \frac{ML^2\Omega_0^3}{4\hbar}\,|h_{\mathrm{s}}(\Omega_0)|^2\sigma^2 \,,
\end{equation}
where
\begin{equation}
\label{eq:snr_rho2}
\sigma^2 = \frac{1}{\pi\Omega_0}\int_{-\infty}^{\infty}\!\frac{d\nu}{\xi^2(\Omega_0+\nu)}
\end{equation}
is the dimensionless integral sensitivity measure, which we shall use here, and
\begin{equation}
\xi^2(\Omega_0+\nu) = \frac{ML^2\Omega_0^2}{4\hbar}\,S^h(\Omega_0+\nu) \,.
\end{equation}
For a harmonic oscillator, using Eq.~\eqref{eq:xi2_osc}, we obtain
\begin{equation}
\sigma^2 = 1 \,.
\end{equation}
This result is natural, since the depth of the sum quantum-noise spectral-density minimum (which makes the dominating part of the integral~\eqref{eq:snr_rho2}) in the harmonic oscillator case is inversely proportional to its width $\Delta\Omega$, see Eq.~\eqref{eq:osc_fm_SQL}. As a result, the integral does not depend on how small the minimal value of $\xi^2$ is.
The situation is different for the second-order pole-test object. Here, the minimal value of $\xi^2$ is proportional to $(\Delta\Omega)^{-2}$ (see Eq.~\eqref{eq:dbl_fm_SQL}) and, therefore, it is possible to expect that the SNR will be proportional to
\begin{equation}
\sigma^2 \propto \frac{1}{(\Delta\Omega)^2}\times\Delta\Omega
\propto \frac{1}{\Delta\Omega} \propto \frac{1}{\xi} \,.
\end{equation}
Indeed, after substitution of Eq.~\eqref{eq:xi2_dbl} into~\eqref{eq:snr_rho2}, we obtain:
\begin{equation}
\sigma^2 = \frac{1}{\sqrt{2}}\frac{\Omega_0}{\Omega_q} = \frac{1}{2\xi(\Omega_0)} \,.
\end{equation}
Therefore, decreasing the width of the dip in the sum quantum noise spectral density and increasing its depth, it is possible, in principle, to obtain an arbitrarily high value of the SNR.
Of course, it is possible only if there are no other noise sources in the interferometer except for the quantum noise. Consider, though, a more realistic situation. Let there be an additional (technical) noise in the system with the spectral density $S_{\mathrm{tech}}(\Omega)$. Suppose also that this spectral density does not vary much within our frequency band of interest $\Delta\Omega$. Then the factor $\sigma^2$ can be approximated as follows:
\begin{equation}
\label{eq:dbl_rho2_tech}
\sigma^2 = \frac{1}{\pi\Omega_0}
\int_{-\infty}^{\infty}\!\frac{d\nu}{\xi^2(\Omega_0+\nu) + \xi^2_{\mathrm{tech}}} \,,
\end{equation}
where
\begin{equation}
\label{dbl_sigma2_tech}
\xi^2_{\mathrm{tech}} = \frac{S_{\mathrm{tech}}(\Omega_0)}{S_{\mathrm{SQL\,f.m.}}(\Omega_0)} \,.
\end{equation}
Concerning quantum noise, we consider the regime close (but not necessarily
exactly equal) to that yielding the second-order pole, that is, we suppose $0\le\Lambda\ll\Omega_0$. In order to simplify our calculations, we neglect the contribution from optical loss into the sum spectral density (we show below that it does not affect the final sensitivity much). Thus, as follows from Eq.~\eqref{eq:S_X_nb}, one gets
\begin{equation}
\label{eq:dbl_sigma2}
\xi^2(\Omega_0+\nu) = \frac{1}{2\Omega_0^2}
\biggl[\frac{(4\nu^2 - \Lambda^2)^2}{\Omega_q^2} + \Omega_q^2\biggr] .
\end{equation}
In the Appendix~\ref{app:SOP}, we calculate integral Eq.~\eqref{eq:dbl_rho2_tech} and optimize it over $\Lambda$ and $\Omega_q$. The optimization gives the best sensitivity, for a given value of $\xi^2_{\mathrm{tech}}$, is provided by
\begin{equation}
\label{dbl_optimal}
\Lambda = \Omega_q = \Omega_0\xi_{\mathrm{tech}} \,.
\end{equation}
In this case,
\begin{equation}
\label{eq:dbl_rho2_opt}
\sigma^2 = \frac{1}{2\sqrt{2}\xi_{\mathrm{tech}}} \,.
\end{equation}
The pure second-order pole regime ($\Lambda=0$), with the same optimal value of $\Omega_q$, provides slightly worse sensitivity:
\begin{equation}
\label{eq:dbl_rho2_sub}
\sigma^2 = \frac{1}{\sqrt{6\sqrt{3}}\xi_{\mathrm{tech}}} \,.
\end{equation}
The optimized function~\eqref{eq:dbl_sigma2} is shown in Figure~\ref{fig:sigma2_dbl}\,(left) for the particular case of $\Lambda=\Omega_q=0.1\Omega_0$. In Figure~\ref{fig:sigma2_dbl}\,(right), the optimal SNR~\eqref{eq:dbl_rho2_opt} is plotted as a function of the normalized technical noise $\xi_{\mathrm{tech}}^2$.
In order to verify our narrow-band model, we optimized numerically the general normalized SNR for the broadband burst-type signals:
\begin{equation}
\label{eq:dbl_snr_burst}
\sigma^2_{\mathrm{burst}} = \frac{2\hbar}{\pi ML^2\Omega_0^2}
\int_{-\infty}^{\infty}\!\frac{d\Omega/\Omega}{S^h(\Omega) + S^h_{\mathrm{tech}}} \,,
\end{equation}
where $S^h$ is the sum quantum noise of the interferometer defined by Eq.~\eqref{eq:sl_sum0}. The only assumption we have made here is that the technical noise power (double-sided) spectral density
\begin{equation}
S^h_{\mathrm{tech}} = \frac{2\hbar}{ML^2\Omega_0^2}\,\xi^2_{\mathrm{tech}}
\end{equation}
does not depend on frequency, which is reasonable, since only the narrow frequency region around $\Omega_0$ contributes noticeably to the integral~\eqref{eq:dbl_snr_burst}. The result is shown in Figure~\ref{fig:sigma2_dbl}\,(right) for two particular cases: the ideal (no loss) case with $\eta=1$, and the realistic case of the interferometer with $J=(2\pi\times100)^3\mathrm{\ s}^{-3}$, $\eta_d=0.95$ and $\gamma_2=1.875\mathrm{\ s}^{-1}$ (which corresponds to the loss factor of $A_{\mathrm{arm}}=10^{-4}$ per bounce in the $4\mathrm{\ km}$ long arms; see Eq.~\eqref{sl_gammas}). The typical optimized quantum noise spectral density (for the particular case of $\xi_{\mathrm{tech}}^2=0.1$) is plotted in Figure~\ref{fig:LIGO_det}.
It is easy to see that the approximations~\eqref{eq:dbl_sigma2} work very well, even if $\xi_{\mathrm{tech}}\sim1$ and, therefore, the assumptions~\eqref{dbl_conds} cease to be valid. One can conclude, looking at these plots, that optical losses do not significantly affect the sensitivity of the interferometer, working in the second-order pole regime. The reason behind it is apparent. In the optical rigidity based systems, the origin of the sensitivity gain is simply the resonance increase of the probe object dynamical response to the signal force, which is, evidently, immune to the optical loss.
The only noticeable discrepancy between the analytical model and the numerical calculations for the lossless case, on the one hand, and the numerical calculations for the lossy case, on the other hand, appears only for very small values of $\xi_{\mathrm{tech}}^2\sim0.01$. It follows from Eqs.~\eqref{eq:Omega_q_nb} and \eqref{dbl_optimal} that this case corresponds to the proportionally reduced bandwidth of the interferometer,
\begin{equation}
\gamma \sim \Omega_0\xi_{\mathrm{tech}}^2 \sim 10\mathrm{\ s}^{-1}
\end{equation}
(for a typical value of $\Omega_0\sim10^3\mathrm{\ s}^{-1}$). Therefore, the loss-induced part of the total bandwidth $\gamma_2$, which has no noticeable effect on the unified quantum efficiency $\eta$ [see Eq.~\eqref{vac_eta}] for the `normal' broadband values of $\gamma\sim10^3\mathrm{\ s}^{-1}$, degrades it in this narrow-band case. However, it has to be emphasized that the degradation of $\sigma^2$, for the reasonable values of $\xi_{\mathrm{tech}}^2$, is only about a few percent, and even for the quite unrealistic case of $\xi_{\mathrm{tech}}^2 = 0.01$, does not exceed $\sim$~25\%.
\newpage
\section{Conclusion and Future Directions}
\label{sec:Conclusion}
In this review, our primary goal was to tell in a clear and understandable way what is meant by quantum measurement in GW detectors. It was conceived as a comprehensive introduction to the quantum noise calculation techniques that are employed currently for the development of advanced interferometric detectors. The target audience are the young researchers, students and postdocs, who have just started their way in this field and need a guide that provides a step-by-step tutorial into the techniques and covers all the current achievements in the field. At the same time, we tried to make this manuscript interesting to all our colleagues from the GW community and, perhaps, from other branches of physics, who might be interested in getting themselves familiar with this area, not necessarily close to their own research field.
However, the reality is crude and such a lofty ambition is always a pot of gold at the end of the rainbow. Thus, we could not claim this review to be a complete and comprehensive description of the field of quantum measurement. We present here a pretty detailed analysis of the quantum noise features in the first and second generation of GW interferometers, contemplating the techniques considered robust and established. However, many hot topics, related to the planned third generation of GW interferometers~\cite{GRG.43.2.671_2011_Chen, CQG.27.19.194002_2010_Punturo, CQG.28.9.094013_2011_Hild} remained uncovered. Here are only some of them: (i) xylophone configurations~\cite{CQG.27.1.015003_2010_Hild}, (ii) multiple-carrier detectors~\cite{PhysRevD.76.062002_2007_Rehbein, PhysRevD.78.062003_2008_Rehbein}, (iii) negative optical inertia~\cite{PhysRevD.83.062003_2011_Khalili}, (iv) intracavity detection schemes~\cite{98a1BrGoKh, 97a1BrGoKh, Phys.Lett.A.298.5.308_2002_Khalili, Phys.Lett.A.317.3.169_2003_Khalili, PhysRevD.73.022002_2006_Danilishin}, etc. It is our determined intention to enjoy the great advantages of the format of \emph{living} reviews and include those topics in future revisions of this review.
We would like to conclude our review by pointing out how the new swiftly-developing areas of modern science and technology, not directly related to GW astronomy and detector science, turn out to be deeply rooted in the quantum measurement theory developed by the GW community.
It is amazing how sinuous the ways of scientific progress are. The history of how GW detection and quantum-measurement theory developed and interwove might serve as an example thereof. Indeed, from the very first steps towards the experimental observation of GWs made by Weber in the early 1960s~\cite{weber1961general, PRL.18.498_1967_Weber}, it was realized that the extreme weakness of interaction between the ripples of space-time and matter appeals for unprecedentedly precise measurement. And almost at the same time, Braginsky realized that the expected amplitude of the GW-induced oscillations of the bar detector signal mode would be on the order of the zero point oscillations of this mode, as predicted by quantum mechanics; that is, in order to observe GWs, one has to treat a detector quantum-mechanically and as a consequence there will be a quantum back action, setting a limitation on the achievable sensitivity, the SQL~\cite{Sov.Phys.JETP_26.831_1968}.
This serendipity had a powerful impact on the quantum measurement theory development, for it set an objective to contrive some ways to overcome this limitation. For decades up to this point, it was a purely theoretical discipline having little in common with experimental science, and, fancy that, become a vital necessity for GW astronomy. And again, for several decades, GW detection has been perhaps the only field where the results of quantum measurement theory were applied, mainly in the struggle with quantum noise, considered as a hindrance towards the noble goal of the detection of GWs. And only recently, the same optomechanical interaction, begetting quantum noise and the SQL in the interferometric GW detectors, has aroused a keen interest among wide circles of researchers studying the quantum behavior of macroscopic objects and testing the very foundations of quantum mechanics in the macroscopic world~\cite{Sci.321.5893.1172_2008_Kippenberg, JOSAB.27.6.A189_2010_Aspelmeyer}.
All the techniques and concepts developed in the GW community turn out to
be highly sought by this new
field~\cite{RevModPhys.82.1155_2010_Clerk}. Such methods, initially
developed for future GW detectors, as back-action evasion via properly
constructed cross correlation between the measurement and back-action
noise sources~\cite{94a1VyZuMa, 96a2eVyMa, 96a1eVyMa, 94a1VyZu,
95a1VyZu, Phys.Lett.A.300.547_2002_Danilishin,
Phys.Lett.A.278.3.123_2000_Danilishin}, find a use in the
optomechanical experiments with micro- and nanoscale mechanical
oscillators~\cite{NJP.10.9.095010_2008_Clerk,
PhysRevA.80.043802_2009_Mueller-Ebhardt,
PhysRevA.81.012114_2010_Miao, PhysRevA.81.052307_2010_Miao,
PhysRevLett.103.100402_2009_Miao,
PhysRevA.81.033849_2010_Yamamoto,arXiv:1104.3251_2011_Friedrich}. It
turns out that GW detectors themselves fit extremely well for testing the
fundamental principles of quantum mechanics just for the record low
values of the noise, having non-quantum origin, that owes to the
ingenuity, patience and dedication by an entire generation of
experimental physicists~\cite{lrr-2009-2}. The very fact that the
mechanical differential mode of the km-scale LIGO detector has been
cooled down to $T_{\mathrm{eff}} = 1.4\mathrm{\ \mu\,K}$ without
any special arrangement, just by modifying the standard feedback
kernel of the actuators to provide a virtual rigidity, shifting the
10-kg suspended mirrors oscillation frequency from $\Omega_m/2\pi =
0.74\mathrm{\ Hz}$ to $150\mathrm{\ Hz}$, where the GW detector is most
sensitive~\cite{NJP.11.7.073032_2009_Abbott}, tells its own
tale. Noteworthy also is the experiment on cooling a several-ton
AURIGA bar detector mechanical oscillation mode to $T_{\mathrm{eff}}
= 0.17\mathrm{\ mK}$~\cite{PhysRevLett.101.033601_2008_Vinante}. In
principle, some dedicated efforts might yield even cooling to ground
state of these definitely macroscopic
oscillators~\cite{arXiv:0809.2024_2008_Danilishin,
NJP.12.8.083032_2010_Miao}.
One might foresee even more striking, really quantum phenomena, to be demonstrated experimentally by future GW detectors, whose sensitivity will be governed by quantum noise and not limited by the SQL. It is possible, e.g., to prepare the mechanical degree of freedom of the interferometer in a close-to-pure squeezed quantum state~\cite{PhysRevA.80.043802_2009_Mueller-Ebhardt}, entangle the differential and common motion of the kg-scale mirrors in the EPR-like fashion~\cite{PhysRev.47.777_1935_Einstein, PhysRevLett.100.013601_2008_Mueller-Ebhardt}, or even prepare it in a highly non-classical Schr\"odinger-cat state~\cite{NatWiss.23.48.807_1935_Schroedinger, PhysRevLett.105.070403_2010_Miao}.
\newpage
\section{Acknowledgements}
\label{sec:Acknowledgements}
This review owes its existence to the wholehearted support and sound
advice of our colleagues and friends. We would like to express our
special thanks to our friends, Yanbei Chen and Haixing Miao for
enlightening discussions, helpful suggestions and encouragement we
enjoyed in the course of writing this review. Also we are greatly thankful to the referees and to our younger colleagues, Mikhail Korobko and Nikita Voronchev, who went to the trouble of thoroughly reading the manuscript and pointing out many imperfections, typos, and misprints. We greatly acknowledge as well
our fellow researchers from LIGO-Virgo Scientific Collaboration for
all the invaluable experience and knowledge they shared with us over
the years. Especially, we want to say thank you to Gregg Harry,
Innocenzo Pinto and Roman Schnabel for consulting with us on literature in
the areas of their expertise. And finally, we would like to thank
\textit{Living Reviews in Relativity} and especially Bala Iyer for the
rewarding opportunity to prepare this manuscript.
\newpage
|
{
"timestamp": "2012-05-10T02:02:59",
"yymm": "1203",
"arxiv_id": "1203.1706",
"language": "en",
"url": "https://arxiv.org/abs/1203.1706"
}
|
\section{Introduction}\label{sect:intro}
Game theory is the theory of competitive interactions between decision makers having different interests. Its primary purpose is to further understand such real-world interactions through mathematical modelling, so it usually studies games that involve many players and many possible outcomes, which are meant to describe faithfully the many shades of situations involving many stakeholders. Apart from some earlier related works, the field of game theory is usually said to be born in the first part of the 20th century, especially thanks to von Neumann~\cite{NM44}, but also Borel~\cite{Borel21} and some others. Since then, it has been applied to many concrete areas such as economics, political science, evolutionary biology, computer science, \textit{etc}. Conversely, specific problems in these concrete areas have been triggering new general questions and have thus been helpful in developing game theory.
Surprisingly, game theory has also provided useful point of view and terminology to abstract areas such as logic, descriptive set theory, and theoretical informatics. For instance, Martin~\cite{Martin90} describes the (quasi-)Borel sets \textit{via} the guaranteed existence of winning strategies in some games that are built using these (quasi-)Borel sets. More specifically, these games involve only two players and two outcomes saying who wins, \textit{i.e.} they are win-lose games, and they are built on infinite trees. Similarly, \cite{BL69}, \cite{GH82},\cite{EJ91}, \cite{Mostowski91}, \textit{etc}, relate some statements about logic to the existence/computation of simple winning strategies in some two-player win-lose games that are built on finite graphs, namely Muller games and parity games. In such frameworks, existence of a winning strategy (of some sort) is called determinacy, and the games that enjoy it are said to be determined.
There are substantial differences between the two above-mentioned types of game theory: games for logic are simple in terms of players, outcomes, and preferences, but the underlying structures are complex (since they pertain to the objects under study), whereas it is the other way round with games for economics, where the underlying structures involve, \textit{e.g.}, finiteness and continuous functions. Of course one would wish to get the best of the two worlds and to understand multi-player, multi-outcome games that are built on complex structures.
The above wish may partly come true. Indeed for some games of specific structures, determinacy results have already been generalised by considering the same structures for many player, many outcomes and the usual generalisation of the notion of winning strategy, namely the notion of Nash equilibrium: for instance, quasi-Borel determinacy was generalised in~\cite{SLR13} for (infinitely) many players and (infinitely) many outcomes; similarly, finite-memory determinacy of Muller games was generalised in~\cite{PS09}; and a similar question was asked in~\cite{Ummels05} for parity games. Said otherwise, for specific structures such as infinite trees or finite graphs, existence of winning strategies for all two-player win-lose games that are built on these structures might be transferred, through ad-hoc proofs, into existence of Nash equilibrium for all multi-player multi-outcome games that are built on these same structures.
This article shows through a single, uniform proof that such a transfer of equilibrium from win-lose to multi-outcome setting holds irrespective of the structure of the game, \textit{i.e.} it need not be a tree, or a graph, \textit{etc.} However, a universal equilibrium-transfer theorem seems to hold for two-player games only, and to fail for three players already, although a clear-cut general counterexample is still missing. Furthermore, the general order-theoretic condition of the theorem, \textit{i.e.} that the preference orders over the outcomes be of (uniformly) finite height, can be relaxed to "inverted" well-foundness when strategy sets are countable. The case in between, where only one strategy set is countable and the inverse relation of preferences are well-founded yet with chains of arbitrary length, is still open.
Note that the Nash equilibria that are considered in this article are all pure since the concept need not be weakened through probabilistic means.
Section~\ref{sect:scgc} introduces the main result of this article intuitively, using an example; Section~\ref{sect:def} defines the required concepts and makes two straightforward remarks; Section~\ref{sect:mr-psc} states the main result of this article, \textit{i.e.} the equilibrium-transfer theorem, and provides a proof in a very simple yet informative case. Section~\ref{sect:tt} presents the equilibrium-transfer theorem and details the algorithmic content of the proof when the outcomes are finitely many; Section~\ref{sect:app-et} invokes the equilibrium-transfer theorem to generalise Martin's theorem on Borel determinacy, positional determinacy of parity games (with infinitely many priorities), and finite-memory determinacy of Muller games; Section~\ref{sect:ltt} gives counterexamples to reasonable candidates to generalise the theorem; and Section~\ref{sect:c} concludes and shows in passing that, generally speaking in game theory, linearly ordered preferences do not account for partially ordered preferences.
\subsection{From simple example to general idea}\label{sect:scgc}
Let us exemplify what the main result of this article, \textit{i.e.} the equilibrium-transfer theorem, can actually do. A finite real-valued two-player game in extensive form is an object built on a finite rooted tree; each internal node is owned by exactly one player, and each leaf encloses a real-valued payoff function, \textit{i.e.} an ordered pair assigning one real number to each player. The leftmost two objects below are such games, the second game being even a win-lose game. Intuitively, the left-most game is played as follows: player $b$ at the root chooses left or right. Right yields payoff $1$ for $a$ and $0$ for $b$, left requires a choice from player $a$, and so on until a leaf is reached.
\begin{tabular}{ccccc}
\begin{tikzpicture}[level distance=7mm]
\node{b}[sibling distance=8mm]
child{node{a}[sibling distance=8mm]
child{node{b}[sibling distance=8mm]
child{node{$4,2$}}
child{node{$1,0$}}
}
child{node{$3,3$}}
}
child{node{$1,0$}};
\end{tikzpicture}
&
\begin{tikzpicture}[level distance=7mm]
\node{b}[sibling distance=8mm]
child{node{a}[sibling distance=8mm]
child{node{b}[sibling distance=8mm]
child{node{$1,0$}}
child{node{$0,1$}}
}
child{node{$1,0$}}
}
child{node{$0,1$}};
\end{tikzpicture}
&
\begin{tikzpicture}[level distance=7mm]
\node{b}[sibling distance=8mm]
child{node{a}[sibling distance=8mm]
child{node{b}[sibling distance=8mm] edge from parent[double]
child{node{$4,2$}}
child{node{$1,0$} edge from parent[double]}
}
child{node{$3,3$}}
}
child{node{$1,0$}edge from parent[double]};
\end{tikzpicture}
&
\begin{tikzpicture}[level distance=7mm]
\node{b}[sibling distance=8mm]
child{node{a}[sibling distance=8mm]
child{node{b}[sibling distance=8mm] edge from parent[double]
child{node{$1,0$}}
child{node{$0,1$} edge from parent[double]}
}
child{node{$0,1$}}
}
child{node{$0,1$} edge from parent[double]};
\end{tikzpicture}
&
\begin{tikzpicture}[level distance=7mm]
\node{b}[sibling distance=8mm]
child{node{a}[sibling distance=8mm]
child{node{b}[sibling distance=8mm]
child{node{$X$}}
child{node{$Y$}}
}
child{node{$Z$}}
}
child{node{$Y$}};
\end{tikzpicture}
\end{tabular}
A strategy profile is an ordered pair enclosing one strategy per player, where a strategy of a player is a complete collection of the (unique) choices that the player would make at each node (MODIFY that he/she owns) if the play ever reached this node. The third (resp. fourth) object from the left above represents a strategy profile for the first (resp. second) game, where the double lines represent the strategical choices. A strategy profile induces one unique payoff function, by following the unique choices from the root to the leaves. When assumed that the players prefer greater payoffs for themselves, the third object above is not a Nash equilibrium, because at least one of the player, namely $b$, can improve upon his/her payoff by changing his/her strategy from "right-right" to "left-left" and obtain $2$ instead of $0$. The fourth object above is a Nash equilibrium because no player can improve upon what they already have. Said otherwise, "right-right" is a winning strategy for player $b$.
The rightmost object above, with variables $X$, $Y$ and $Z$, is an abstraction of the leftmost game: it represents the structure of the game. Especially, repetition of the variable $Y$ captures equality between payoff functions in the original game. There are $2^3$ possibilities to instantiate the variables of the game structure with $(1,0)$ and $(0,1)$, one of them being the second game above. There are infinitely many possibilities to instantiate the variables of the game structure with real-valued payoff functions, one of them being the leftmost game above. The two strategy profiles above suggest that there are many more Nash equilibria in win-lose games than in games with real-valued payoff functions. This is actually true, but since all the $2^3$ win-lose games that are derived from the game structure above have a Nash equilibrium (equivalently, a winning strategy), which is easy to check, the equilibrium-transfer theorem, stated in Section~\ref{sect:mr-psc}, implies that all the games with real-valued payoff functions that are derived from the same game structure also have a Nash equilibrium, which is more difficult to check! (Albeit already proved in~\cite{Kuhn53}.)
In the same way, the equilibrium-transfer theorem turns essentially all determinacy results into existence of Nash equilibrium in a two-player, multi-outcome setting. It is applied to Borel determinacy, finite-memory determinacy of Muller game, and positional determinacy of parity games in Section~\ref{sect:app-et}. Apart from the generalisation of positional determinacy of parity games, which is new, the two other obtained results are weaker than existing results; but the key point here, however, is that the three applications are almost effortless, and that the same would hold for further applications.
The equilibrium-transfer theorem relates to all two-player win-lose games where the notion of winning strategy makes sense, namely strategy-based games. Since on the one hand, all strategy-based games are faithfully embeddable into games in normal form, as far as existence of Nash equilibrium is concerned, and since on the other hand, games with abstract outcomes and preferences are much more general than real-valued games, the theorem will be stated for abstract games in normal form.
\subsection{Definitions}\label{sect:def}
Unlike traditional games in normal form, the definition below involves abstract outcomes (instead of mere real-valued payoff functions) and preferences that are arbitrary (instead of transitive, reflexive, total, \textit{etc.}). It is important since there is no reason why games, \textit{e.g.}, with real-valued payoff functions should account for all possible games. For instance, Section~\ref{sect:c} defines a simple preference that has a game-theoretic property relating to Nash equilibrium but whose every linear extension fails to have the same game-theoretic property.
\begin{definition}[Games in normal form]\label{defn:gnf}
They are tuples $\langle A,(S_a)_{a\in A},O,v,(\prec_a)_{a\in A}\rangle$ satisfying the following:
\begin{itemize}
\item $A$ is a non-empty set (of players, or agents),
\item $\prod_{a\in A}S_a$ is a non-empty Cartesian product (whose elements are the strategy profiles and where $S_a$ represents the strategies available to player $a$),
\item $O$ is a non-empty set (of possible outcomes),
\item $v:\prod_{a\in A} S_a\to O$ (the outcome function that values the strategy profiles),
\item Each $\prec_a$ is a binary relation over $O$ (modelling the preference of player $a$).
\end{itemize}
\end{definition}
The traditional notion of Nash equilibrium is rephrased below in the abstract setting with a subtle semantic change (but remains the same in extension): each binary relation $\prec_a$, which I call preference, is the complement of the inverse of what is traditionally called preference.
\begin{definition}[Nash equilibrium]\label{defn:ne}
Let $g=\langle A,(S_a)_{a\in A} ,O,v,(\prec_a)_{a\in A}\rangle$ be a game in normal form. A strategy profile $s$ in $S:=\prod_{a\in A} S_a$ is a Nash equilibrium if it makes every player $a$ stable, \textit{i.e.} $v(s)\not\prec_a v(s')$ for all $s'\in S$ that differ from $s$ at most at the $a$-component.
\[NE(s)\quad:=\quad\forall a\in A,\forall s'\in S,\quad\neg(v(s)\prec_a v(s')\,\wedge\,\forall b\in A-\{a\},\,s_b= s'_b)\]
\end{definition}
Three games in normal form are represented below as arrays. They all involve two players, say $a$ and $b$, two strategies for $a$ (resp. $b$), namely $a_l$ and $a_r$ (resp. $b_l$ and $b_r$), and outcomes in $\mathbb{R}^2$. In this specific case, an outcome is usually called a payoff function because it assigns one payoff to every player. Player $a$ (resp. $b$) prefers payoff functions with greater first (resp. second) component. In the first game, if player $a$ picks the strategy $a_l$ and player $b$ picks $b_l$, the strategy profile $(a_l,b_l)$ then yields payoff $1$ for $a$ and $0$ for $b$. This profile is not a Nash equilibrium because, by changing strategies, player $a$ can convert the profile $(a_l,b_l)$ into $(a_r,b_l)$ and obtain payoff $2$, which is greater than $1$. The game has two Nash equilibria, namely profiles $(a_r,b_l)$ and $(a_l,b_r)$. The second game has no Nash equilibrium and the third game, which enjoys some symmetry, has two Nash equilibria. These last two games suggest that the notion of Nash equilibrium cannot, at least directly, and even by using probabilities, lead to a notion of best move, \textit{i.e.} of recommendations on how to play.
\begin{displaymath}
\begin{array}{c@{\hspace{1cm}}c@{\hspace{1cm}}c}
\begin{array}{c|c@{,\;}c@{\;\vline\;}c@{,\;}c|}
\multicolumn{1}{c}{}&
\multicolumn{2}{c}{b_l}&
\multicolumn{2}{c}{b_r}\\
\cline{2-5}
a_l & 1 & 0 & 5 & 0 \\
\cline{2-5}
a_r & 2 & 4 & 5 & 3\\
\cline{2-5}
\end{array}
&
\begin{array}{c|c@{,\;}c@{\;\vline\;}c@{,\;}c|}
\multicolumn{1}{c}{}&
\multicolumn{2}{c}{b_l}&
\multicolumn{2}{c}{b_r}\\
\cline{2-5}
a_l & 0 & 1 & 1 & 0 \\
\cline{2-5}
a_r & 1 & 0 & 0 & 1\\
\cline{2-5}
\end{array}
&
\begin{array}{c|c@{,\;}c@{\;\vline\;}c@{,\;}c|}
\multicolumn{1}{c}{}&
\multicolumn{2}{c}{b_l}&
\multicolumn{2}{c}{b_r}\\
\cline{2-5}
a_l & 2 & 1 & 0 & 0 \\
\cline{2-5}
a_r & 0 & 0 & 1 & 2\\
\cline{2-5}
\end{array}
\end{array}
\end{displaymath}
The two-player win-lose games in normal form, defined below, are special cases of games in normal form.
\begin{definition}[Win-lose games in normal form, winning strategies, and determinacy]\label{def:win-lose}
\begin{itemize}
\item A win-lose game is a game where $A=\{1,2\}$ and $O=\{(1,0),(0,1)\}$ and the preferences are defined by $(0,1)\prec_1(1,0)$ and $(1,0)\prec_2(0,1)$, so all these usually may remain implicit.
\item A winning strategy for player $1$ is a strategy $s_1\in S_1$ such that $v(s_1,s_2)=(1,0)$ for all $s_2\in S_2$. A winning strategy for player $2$ is a strategy $s_2\in S_2$ such that $v(s_1,s_2)=(0,1)$ for all $s_1\in S_1$.
\item A win-lose game such that one player has a winning strategy is said to be determined.
\end{itemize}
\end{definition}
Three win-lose games are represented below. For the first game, player $a$ has the winning strategy $a_r$; for the second game, player $b$ has the winning strategy $b_r$; there is no winning strategy for the third game.
\begin{displaymath}
\begin{array}{c@{\hspace{1cm}}c@{\hspace{1cm}}c}
\begin{array}{c|c@{,\;}c@{\;\vline\;}c@{,\;}c|}
\multicolumn{1}{c}{}&
\multicolumn{2}{c}{b_l}&
\multicolumn{2}{c}{b_r}\\
\cline{2-5}
a_l & 0 & 1 & 0 & 1 \\
\cline{2-5}
a_r & 1 & 0 & 1 & 0\\
\cline{2-5}
\end{array}
&
\begin{array}{c|c@{,\;}c@{\;\vline\;}c@{,\;}c|}
\multicolumn{1}{c}{}&
\multicolumn{2}{c}{b_l}&
\multicolumn{2}{c}{b_r}\\
\cline{2-5}
a_l & 1 & 0 & 0 & 1 \\
\cline{2-5}
a_r & 1 & 0 & 0 & 1\\
\cline{2-5}
\end{array}
&
\begin{array}{c|c@{,\;}c@{\;\vline\;}c@{,\;}c|}
\multicolumn{1}{c}{}&
\multicolumn{2}{c}{b_l}&
\multicolumn{2}{c}{b_r}\\
\cline{2-5}
a_l & 0 & 1 & 1 & 0 \\
\cline{2-5}
a_r & 1 & 0 & 0 & 1\\
\cline{2-5}
\end{array}
\end{array}
\end{displaymath}
The notion of winning strategy is relevant in win-lose games only, but the following remark clarifies why the transfer from winning strategy to multi-outcome Nash equilibrium is a process of generalisation.
\begin{remark}\label{rmk:ws-ne}
A win-lose game has a winning strategies iff it has a Nash equilibrium.
\end{remark}
\begin{proof}
On the one hand, the strategy profile made of a winning strategy of a player and any strategy of his or her opponent constitutes a Nash equilibrium; conversely, the $X$-component of a Nash equilibrium where player $X$ wins is a winning strategy for $X$.
\end{proof}
Section~\ref{sect:scgc} has already exemplified that both win-lose games and abstract games can be derived from a game structure. This is formalised below.
\begin{definition}[Induced structures, derived games, determined structures, enforcement]\label{defn:is-dg-ds}\hfill
Let $\langle \{1,2\},S_1,S_2,O,v,\{\prec_1,\prec_2\}\rangle$ be a two-player game.
\begin{itemize}
\item $\langle \{1,2\},S_1,S_2,O,v\rangle$ is the structure induced by the game, and conversely, the game is said to be derived from the structure.
\item Let $wl$ be a function from $O$ to $\{(1,0),(0,1)\}$, the win-lose game $\langle S_1,S_2,wl\circ v\rangle$ is also said to be derived from the structure.
\item Let $R_1\subseteq S_1$ and $R_2\subseteq S_2$. If all win-lose games derived from a structure are determined (\textit{via} strategies in $R_1$ or $R_2$), the structure is also said to be determined (\textit{via} strategies in $R_1$ or $R_2$).
\item Let $\langle \{1,2\},S_1,S_2,O,v\rangle$ be a game structure, let $P\subseteq O$, and let $s_1\in S_1$ such that $v(s_1,S_2):=\{v(s_1,s_2)\,\mid\,s_2\in S_2\}\subseteq P$. The strategy $s_1$ is said to enforce $P$ and exclude $O\backslash P$.
\end{itemize}
\end{definition}
The subsets $R_i$ from Definition~\ref{defn:is-dg-ds} represent strategies of special interest. For instance, as already mentioned, Muller games are determined through strategies that can be described by finite automata, and parity games are determined through strategies that are called positional.
The leftmost game structure below is not determined, \textit{e.g.}, because instantiating $X$ with $(1,0)$ and $Y$ with $(0,1)$ yields a game without Nash equilibrium, or equivalently without winning strategy. The other two game structures are determined. (To see this for the rightmost one, it suffices to make a case distinction on how $Y$ is instantiated.)
\begin{displaymath}
\begin{array}{c@{\hspace{1cm}}c@{\hspace{1cm}}c}
\begin{array}{c|c@{\;\vline\;}c|}
\multicolumn{1}{c}{}&
\multicolumn{1}{c}{b_{l}}&
\multicolumn{1}{c}{b_{r}}\\
\cline{2-3}
a_{l} & X & Y \\
\cline{2-3}
a_{r} & Y & X\\
\cline{2-3}
\end{array}
&
\begin{array}{c|c@{\;\vline\;}c|}
\multicolumn{1}{c}{}&
\multicolumn{1}{c}{b_{l}}&
\multicolumn{1}{c}{b_{r}}\\
\cline{2-3}
a_l & X & Z\\
\cline{2-3}
a_r & Y & Y\\
\cline{2-3}
\end{array}
&
\begin{array}{c|c@{\;\vline\;}c@{\;\vline\;}c|}
\multicolumn{1}{c}{}&
\multicolumn{1}{c}{b_{l}}&
\multicolumn{1}{c}{b_{m}}&
\multicolumn{1}{c}{b_{r}}\\
\cline{2-4}
a_l & X & Z &Y \\
\cline{2-4}
a_r & Y & Y &Y\\
\cline{2-4}
\end{array}
\end{array}
\end{displaymath}
The following remark holds since deriving a win-lose game from a structure amounts to choosing the characteristic function of a subset of the outcomes.
\begin{remark} \label{rmk:d-e}
A game structure is determined iff each subset of the outcomes can be either enforced by player $1$ or excluded by player $2$.
\end{remark}
\subsection{Main result and proof in the simplest case}\label{sect:mr-psc}
Put simply, the equilibrium-transfer theorem reads as follows: if a game structure is determined through nice strategies, all (reasonable) abstract games derived from the structure have a nice Nash equilibrium; and the converse is of course true! (In the sequel, if $\prec$ is a binary relation, its inverse is defined by $x \prec^{-1} y$ iff $y \prec x$.)
\begin{theorem}[Equilibrium transfer]\label{thm:intro-et}
Let $\langle \{1,2\},S_1,S_2,O,v,\{\prec_1,\prec_2\}\rangle$ be a two-player game whose induced structure is determined through strategies in $R_1\subseteq S_1$ and $R_2\subseteq S_2$, and assume either of the following conditions:
\begin{enumerate}
\item the preferences $\prec_1$ (resp. $\prec_2$) has (uniformly) finite height,\\
\textit{i.e.} there is $n\in\mathbb{N}$ such that there is no $o_0\prec_1 o_1\prec_1 \dots\prec_1 o_n$ (resp. $o_0\prec_2 o_1\prec_2 \dots\prec_2 o_n$).
\item the strategy sets $S_1$ and $S_2$ are countable and $\prec_1^{-1}$ and $\prec_2^{-1}$ are well-founded,\\
\textit{i.e.} there is no infinite ascending sequence $o_0\prec_1 o_1\prec_1 \dots$ (resp. $o_0\prec_2 o_1\prec_2 \dots$)
\end{enumerate}
Then $\langle \{1,2\},S_1,S_2,O,v,\{\prec_1,\prec_2\}\rangle$ has a Nash equilibrium in $R_1\times R_2$.
\end{theorem}
Following up on Section~\ref{sect:scgc}: to prove that all finite real-valued two-player games in extensive form have a Nash equilibrium, it suffices, firstly, to show that the structures induced by these games are determined, secondly, to argue that the preferences of each given real-valued game have finite height (since they are acyclic and the game involves only finitely many payoff functions), and finally, to invoke the equilibrium-transfer theorem. Note that theses games are not derived from the same game structure, but rather from infinitely many structures.
The equilibrium-transfer theorem should be of interest to both game theorists and mathematicians dealing with games. To game theorists, it provides an economical way of proving existence of multi-outcome Nash equilibrium for all games in a class that is derived from a set of two-player game structures; to mathematicians, it provides an easy way to generalise their determinacy results, which are dedicated to logic in the first place, and thus advertise their work to game theorists.
Proposition~\ref{prop:mmt} below is a very simple version of the equilibrium-transfer theorem, but the very basic idea is already there. It uses only an abstraction of zero-sum games, \textit{i.e.} two-player games where the two preference relations are inverses of each other. It is named after von Neumann's Minimax Theorem (see, \textit{e.g.}, \cite{NM44}) to hint at the similarities, although it is neither a generalisation nor a special case of it. (For instance, the Minimax Theorem involves infinitely many mixed strategies whereas Proposition~\ref{prop:mmt} involves finitely many pure strategies only.)
\begin{proposition}[Minimax transfer]\label{prop:mmt}
Let $\langle \{1,2\}, S_1,S_2,O,v,\{<_1,<_1^{-1}\}\rangle$ be a two-player game in normal form whose induced structure is determined, and assume that $<_1$ is a strict linear order over the finite domain $O$. Then the game has a Nash equilibrium (and all Nash equilibria yield the same outcome).
\end{proposition}
\begin{proof}
Let $P$ be the smallest (for inclusion) $<_1$-terminal interval (\textit{i.e.} $x\in P\wedge x<_1 y\Rightarrow y\in P$) that player $1$ can enforce \textit{via} some strategy $s_1$, and let $m$ be the $<_1$-minimum (and the $<_1^{-1}$-maximum!) of $P$. Since player $1$ cannot enforce $P-\{m\}$, by remark~\ref{rmk:d-e} player $2$ can enforce $(O\backslash P)\cup\{m\}$ \textit{via} some strategy $s_2$. Therefore $(s_1,s_2)$ is a Nash equilibrium yielding outcome $m$, and any strategy profile that does not yield $m$ may be improved upon by one player $i$ \textit{via} $s_i$.
\end{proof}
All the conditions of application of Theorem~\ref{thm:intro-et} are important, as mentioned below and further detailed in Section~\ref{sect:ltt}. First, when the inverse of some preference is not well-founded, it is easy to build a (one-player) game without Nash equilibrium. Second, this article defines a two-player game without Nash equilibrium although its induced game structure is determined, while the strategies of one player only are countably many, one preference only has finite height, and the inverse of the other preference is still well-founded. Third, a natural three-player version of the equilibrium-transfer theorem may sound as follows: "given a three-player game with preferences of finite height, if replacing the actual preferences by preferences of smaller height always yields a game with a Nash equilibrium, and/or if merging two players (into a super-player) or slicing the induced structure (\textit{i.e} fixing one player's strategy) always yields a determined structure, then the given three-player game has a Nash equilibrium." Counter-examples show that simpler versions of the statement above do not hold, but the general case is still open.
\section{The equilibrium-transfer theorem}\label{sect:tt}
This section proves the theorem by transfinite induction on the preferences. The three main ingredients of the proof are: an equilibrium-reflecting reduction that shrinks games in terms of preferences, a property on functions from $\mathbb{N}^2$ to $\mathbb{N}$ that enables a diagonal argument when shrinking games is not possible, and a finite-case version of the theorem, which itself relies on lifting binary relations to the power set of their domains. This lift, defined below, is the key idea of the equilibrium transfer: especially, it overcomes the difficulty that the proof of the minimax transfer does not scale up for preferences that are not inverses of each other. Note that a simpler and sufficient (to prove equilibrium transfer) version of the lift is mentioned in Remark~\ref{rem:finite-linear} afterwards.
\begin{definition}\label{defn:lift}
A binary relation $\prec$ on a set $S$ may be lifted to the power set of $S$ as below.
\[\forall A,B\subseteq S,\quad A\prec^{\mathcal{P}} B\,:=\,\exists a\in A\backslash B,\forall b\in B\backslash A,\,a\prec b\]
\end{definition}
\begin{lemma}\label{lem:lrs-lp}
Let $\prec$ be a binary relation on a set $S$. If $\prec$ is a strict linear order, $\prec^{\mathcal{P}}$ is a strict partial order.
\end{lemma}
\begin{proof}
A strict partial order is a transitive and irreflexive binary relation. A strict linear order is a strict partial order such that any two distinct elements are comparable. Assume that $\prec$ is as strict linear order. Since $\prec^{\mathcal{P}}$ is irreflexive by definition, it suffices to show that $\prec^{\mathcal{P}}$ is transitive. Assume that $A\prec^{\mathcal{P}}B$ and $B\prec^{\mathcal{P}}C$ with respective witnesses $a\in A\backslash B$ and $b\in B\backslash C$. First note that $a\neq b$ since $a\notin B$ and $b\in B$. Now let us case-split to show that $A\prec^{\mathcal{P}}C$.
\begin{itemize}
\item Assume that $a\prec b$, so $\neg(b\prec a)$ by transitivity and irreflexivity assumptions, so $a\notin C\backslash B$ since $b$ is a witness for $B\prec^{\mathcal{P}}C$. Together with $a\notin B$ it yields $a\notin C$, so $a\in A\backslash C$. Now let $x$ be in $C\backslash A$. If $x\in B$, then $x\in B\backslash A$, and $a\prec x$ since $a$ is a witness for $A\prec^{\mathcal{P}}B$. If $x\notin B$, then $x\in C\backslash B$, and $b\prec x$ since $b$ is a witness for $B\prec^{\mathcal{P}}C$, so $a\prec x$ by transitivity. Therefore $A\prec^{\mathcal{P}}C$ is witnessed by $a$.
\item Assume that $b\prec a$, so $\neg(a\prec b)$ by transitivity and irreflexivity assumptions, so $b\notin B\backslash A$ since $a$ is a witness for $A\prec^{\mathcal{P}}B$. Together with $b\in B$ it yields $b\in A$, so $b\in A\backslash C$. Now let $x$ be in $C\backslash A$. If $x\notin B$, then $x\in C\backslash B$, and $b\prec x$ since $b$ is a witness for $B\prec^{\mathcal{P}}C$. If $x\in B$, then $x\in B\backslash A$, and $a\prec x$ since $a$ is a witness for $A\prec^{\mathcal{P}}B$, so $b\prec x$ by transitivity. Therefore $A\prec^{\mathcal{P}}C$ is witnessed by $b$.
\end{itemize}
\end{proof}
\begin{remark}\label{rem:finite-linear}
For finite linear order, Definition~\ref{defn:lift} can be rephrased as $A<^{\mathcal{P}} B\,:= A\neq B\,\wedge\, \mathrm{min}_{<}(A\backslash B \cup B\backslash A)\in A$, and Lemma~\ref{lem:lrs-lp} can be strengthen into "$<^{\mathcal{P}}$ is also a strict linear order". This $<^{\mathcal{P}}$ is isomorphic to the lexicographic order induced by $<$ on the characteristic functions of the complements of the subsets. For example let $o_1 < o_2 < o_3 < o_4 < o_5$, then
$\{o_2,o_3,o_4,o_5\} <^{\mathcal{P}} \{o_2,o_4\}$ corresponds to $10000 <_{lex} 10101$.
\end{remark}
The lemma below states a bit more than a mere finitely-many-outcome version of the forthcoming theorem; it sounds a bit less natural too, due to Condition~\ref{cond:lem-et3} (which is obviously fulfilled when the outcomes are finitely many), but it is very useful in the proof of the equilibrium-transfer theorem. In the statement of the lemma, $\prec_i^*$ denotes the reflexive and transitive closure of $\prec_i$.
\begin{lemma}[Finitary equilibrium transfer]\label{lem:et}
Let $\langle \{1,2\}, S_1,S_2,O,v,\{\prec_1,\prec_2\}\rangle$ be a two-player game in normal form, let $R_1\subseteq S_1$ and $R_2\subseteq S_2$, and let us assume the following:
\begin{enumerate}
\item\label{cond:lem-et1} the game structure is determined \textit{via} strategies in $R_1$ and $R_2$.
\item\label{cond:lem-et2} both preferences $\prec_1$ and $\prec_2$ are acyclic.
\item\label{cond:lem-et3} $\exists s_1\in S_1,\,|\{o\in O\,\mid\,\exists s_2\in S_2,\,v(s_1,s_2)\,\prec_1^*\, o\}|<\infty\quad\vee$\\
$\exists s_2\in S_2,\,|\{o\in O\,\mid\,\exists s_1\in S_1,\,v(s_1,s_2)\,\prec_2^*\, o\}|<\infty$\\
That is, one player $i$ can enforce a subset of outcomes whose $\prec_i$-upward-generated cone is finite.
\end{enumerate}
Then the game $\langle \{1,2\}, S_1,S_2,O,v,\{\prec_1,\prec_2\}\rangle$ has a Nash equilibrium in $R_1\times R_2$.
\end{lemma}
\begin{proof}
First note that if player $i$ can enforce a subset of outcome (\textit{via} some strategy in $S_i$), he or she can enforce it \textit{via} some strategy in $R_i$, since the opponent cannot exclude it and by determinacy assumption together with Remark~\ref{rmk:d-e}. Assume that, \textit{e.g.}, player $1$ can enforce a finite $\prec_1$-upward-generated cone $C$. Since $\prec_1$ is acyclic, so is its restriction $\prec_1\mid_C$ to $C$; let $<$ be a strict linear extension of $\prec_1\mid_C$, so $<^{\mathcal{P}}$ is a strict linear order too, by Remark~\ref{rem:finite-linear}. Let $M$ be the $<^{\mathcal{P}}$-greatest subset of $C$ that player $1$ can enforce and let $s_1\in R_1$ be a strategy enforcing $M$.
Since $M$ is finite and non-empty and since $\prec_2$ is acyclic, let $m$ be $\prec_2\mid_M$-maximal and let $X:=\{x\in M\,\mid\, x<m\}\cup\{x\in C\,\mid\,m<x\}$. Since $M<^{\mathcal{P}}X$ by Remark~\ref{rem:finite-linear} (or by Definition~\ref{defn:lift}), and since $X\subseteq C$ by definition of $M$ and $X$, player $1$ cannot enforce $X$ by definition of $M$, so player $2$ can enforce $O\backslash X$ by determinacy assumption and Remark~\ref{rmk:d-e}. Let $s_2\in R_2$ be a strategy enforcing $O\backslash X$, so that $v(s_1,s_2)\in M\cap (O\backslash X)=\{m\}$.
First, the strategy profile $(s_1,s_2)$ makes Player $2$ stable, since $m$ is $\prec_2$-maximal among $M$, which is enforced by $s_1$. Second, let $o\in O$ be such that $m\prec_1 o$, so $o\in C$ by definition of $C$, so $o\in X$ by definition of $<$ and $X$, so $m$ is $\prec_1$-maximal among $O\backslash X$, which is enforced by $s_2$. Therefore $(s_1,s_2)\in R_1\times R_2$ is a Nash equilibrium.
\end{proof}
There is a straightforward algorithmic consequence of the proof of Lemma~\ref{lem:et}. Namely, finding a suitable Nash equilibrium in a two-player game that involves $n$ outcomes requires at most $n$ (resp. 2) calls to the function expecting a win-lose game and returning the winning player (resp. a suitable winning strategy). To justify this, let us fix a game $\langle \{1,2\}, S_1,S_2,\{o_1,\dots,o_n\},v,\{\prec_1,\prec_2\}\rangle$ and let $<$ be a linear extension of $\prec_1$ and assume that $o_1 < \dots < o_n$ up to renaming. Let us represent every subset $O'$ of $\{o_1,\dots,o_n\}$ via a characteristic word $u$ over $\{0,1\}$ (which may be seen as characteristic functions), where $u_i=1$ iff $o_i\in O'$. For all $u \in \{0,1\}^n$ let $w_a(u)$ be the winner of the derived win-lose game $\langle S_1,S_2, u\circ v\rangle$ (see Definition~\ref{def:win-lose}) and $w_s(u) \in R_1 \sqcup R_2$ be a winning strategy for the winner. When called with the arguments $(n,\epsilon)$, the function $g$ defined below returns within $n$ recursive calls the characteristic word of the $<$-maximum subset that player $1$ can enforce, where each recursive step calls $w_a$ only once.
\begin{itemize}
\item $g(0,u) := u$
\item $g(k+1,u):= g(k, u\cdot b)$ where $b := 0$ if $w_a(u\cdot 0\cdot 1^{k}) = 1$ and $b := 1$ otherwise.
\end{itemize}
Now let $o_j$ be $\prec_2$-maximal in the set represented by $g(n,\epsilon)$. The strategy profile $(w_s\circ g(n,\epsilon),w_s(g(n,\epsilon)_{<j}\cdot 0\cdot 1^{n-j})$ is the Nash equilibrium from the proof of Lemma~\ref{lem:et}.
When considering infinitely many outcomes, there may not exist a maximal subset that a given player can enforce. Nonetheless, the lemma above and the following two lemmas will be combined to prove the theorem by transfinite induction on (the order types of the inverses of) the preferences.
Lemma~\ref{lem:reflect-eq} below relies on the remark that if a player can exclude a lower/downward interval of least-preferred outcomes, no Nash equilibrium will ever yield such outcomes. So, the excludable least-preferred outcomes may just be merged into one single worst outcome of the player, and become the best outcome of the opponent: indeed, this reduction does not create any Nash equilibrium but yields in many cases a smaller game in terms of outcomes and especially of preferences, thus enabling a step in the transfinite induction. Lemma~\ref{lem:reflect-eq} is named after the well-known elimination of dominated strategies (see, \textit{e.g.}, \cite{LR57}), which simplifies a game through its set of strategies only. The two procedures have nothing much in common, but the naming is meant to suggest that they may complement each other nicely (although not in this article).
\begin{lemma}[Elimination of dominated outcomes]\label{lem:reflect-eq}
Let $g=\langle \{1,2\}, S_1,S_2,O,v,\{<_1,<_2\}\rangle$ be a two-player game in normal form with strict linear preferences. Let $e\in S_1$ and $o\in O$ and assume that $o<_1v(e,s_2)$ for all $s_2\in S_2$. Let $g^\prime:=\langle \{1,2\}, S_1,S_2,O^\prime,v^\prime,\{<^\prime_1,<^\prime_2\}\rangle$, where
\begin{itemize}
\item $O^\prime :=\{x\in O\,\mid\, o\leq_1 x\}$
\item $v^\prime(s):=v(s)$ if $v(s)\in O^\prime$ and $v^\prime(s):=o$ otherwise.
\item $<^\prime_1$ is the restrictions of $<_1$ to $O^\prime$.
\item $x<^\prime_2 y:= x\neq o\,\wedge\, (x<_2 y\,\vee\, y=o)$.
\end{itemize}
Then every Nash equilibrium of $g^\prime$ is also Nash equilibrium of $g$.
Moreover, if the inverse relations of $<_1$ and $<_2$ are well-orders, the order types of $(<'_1)^{-1}$ and $(<'_2)^{-1}$ are not greater than those of $(<_1)^{-1}$ and $(<_2)^{-1}$ respectively. Furthermore, if $o$ is not the $<_1$-least element of $O$, the order type of $(<'_1)^{-1}$ is less than that of $(<_1)^{-1}$.
\end{lemma}
\begin{proof}
Let $s$ be a Nash equilibrium of $g^\prime$. Since $o<_1 v(e,s_2)$ by assumption about $e$, the outcome $v(e,s_2)$ is in $O^\prime-\{o\}$ by definition of $O^\prime$, so $o<^\prime_1v^\prime(e,s_2)$ by definitions of $v^\prime$ and $<^\prime_1$. Since $v^\prime(e,s_2)\leq^\prime_1 v^\prime(s)$ by definition of NE and since $<^\prime_1$ is also strict linear, we have $o<^\prime_1v^\prime(s)$, so $v^\prime(s)\in O^\prime-\{o\}$ and $v(s)=v^\prime(s)$ by definitions of $O^\prime$ and $v^\prime$.
Now let us prove by contradiction that both players are stable w.r.t. $s$ and $g$. If $v(s)<_1v(x,s_2)$ for some $x\in S_1$, then $v(x,s_2)\in O^\prime$ since $v(s)\in O^\prime$ and by definition of $O^\prime$, so $v^\prime(s)<^\prime_1v^\prime(x,s_2)$ by definitions of $v^\prime$ and $<^\prime_1$, which contradicts $s$ being an NE of $g^\prime$. If $v(s)<_2v(s_1,y)$ for some $y\in S_2$, then $v^\prime(s)=o$ since $v^\prime(s_1,y)\leq^\prime_2 v^\prime(s)$ (by definition of NE) and by definition of $<^\prime_2$, which is also a contradiction.
\end{proof}
The transfinite-inductive step of the proof of Theorem~\ref{thm:et} will be justified by Lemma~\ref{lem:reflect-eq} above; but in cases where no player is able to exclude a non-trivial downward interval of outcomes, existence of sets $A$ and $B$ will be proved to feed Lemma~\ref{lem:countable-finite} below, which will subsequently yield the existence of a set $C$ that contradicts the the determinacy assumption.
\begin{lemma}\label{lem:countable-finite}
Let $f:\mathbb{N}^2\to\mathbb{N}$. The following two propositions are equivalent.
\begin{enumerate}
\item There exists a subset of the naturals that intersects each $f(n,\mathbb{N})$ and whose complement intersects each $f(\mathbb{N},n)$, where $f(n,\mathbb{N}):=\{f(n,k)\,\mid\,k\in\mathbb{N}\}$.
\item There exist $A$ and $B$ disjoint subsets of the naturals such that either $f(n,\mathbb{N})$ and $A$ overlap or $f(n,\mathbb{N})\backslash(A\cup B)$ is infinite, and likewise, either $f(\mathbb{N},n)$ and $B$ overlap or $f(\mathbb{N},n)\backslash(A\cup B)$ is infinite.
\end{enumerate}
The statement is formalised below.
\[\begin{array}{c}
\forall f:\mathbb{N}^2\to\mathbb{N},\\
\exists C\subseteq\mathbb{N},\forall n\in\mathbb{N},\quad f(n,\mathbb{N})\cap C\neq\emptyset\quad\wedge\quad f(\mathbb{N},n)\cap\mathbb{N}\backslash C\neq\emptyset\\
\Updownarrow\\
\exists A,B\subseteq\mathbb{N}, A\cap B=\emptyset\quad\wedge\quad\forall n\in\mathbb{N},\quad (f(n,\mathbb{N})\cap A\neq\emptyset\quad\vee\quad|f(n,\mathbb{N})\backslash(A\cup B)|=\aleph_0)\quad\wedge\\
\phantom{\exists A,B\subseteq\mathbb{N}, A\cap B=\emptyset\quad\wedge\quad\forall n\in\mathbb{N},\quad} (f(\mathbb{N},n)\cap B\neq\emptyset\quad\vee\quad|f(\mathbb{N},n)\backslash(A\cup B)|=\aleph_0)\phantom{\quad\wedge}
\end{array}\]
\end{lemma}
\begin{proof}
For the top-bottom implication $A:=C$ and $B:=\mathbb{N}\backslash C$ witness the claim. To prove the bottom-top implication let us define two sequences of subsets of $\mathbb{N}$ as follows, by mutual induction.
\begin{eqnarray*}
X_0 &:=& A\\
Y_0 &:=& B\\
X_{n+1} &:=& X_n\cup\{min (f(n,\mathbb{N})\backslash(X_n\cup Y_n))\}\mbox{ if } f(n,\mathbb{N})\cap A=\emptyset\mbox{, otherwise } X_{n+1} := X_n \\
Y_{n+1} &:=& Y_n\cup\{min (f(\mathbb{N},n)\backslash(X_{n+1}\cup Y_n))\}\mbox{ if } f(\mathbb{N},n)\cap B=\emptyset\mbox{, otherwise } Y_{n+1} := Y_n \\
\end{eqnarray*}
The inductive steps above are well-defined by the assumed disjunctions and since the $X_n\backslash A$ and $Y_n\backslash B$ are finite by construction. It is provable by induction on $n$ that $X_n$ and $Y_n$ are disjoint for all $n$, and so are $X:=\bigcup_{n=0}^{\infty}X_n$ and $Y:=\bigcup_{n=0}^{\infty}Y_n$. Now note that that $C:=X$ witnesses the claim since $X_{n+1}$ (resp. $Y_{n+1}$) intersects $f(n,\mathbb{N})$ (resp. $f(\mathbb{N},n)$) by construction.
\end{proof}
The proof of Theorem~\ref{thm:et} starts with a case distinction on Condition~\ref{cond:thm-et2}. The first case is exactly Lemma~\ref{lem:et}; the second case is reduced to Lemma~\ref{lem:et}; and the third case is proved by transfinite induction on the preferences: the base step invokes Lemma~\ref{lem:et} again and the inductive step performs a case distinction, invoking Lemma~\ref{lem:reflect-eq} when possible and proving by Lemma~\ref{lem:countable-finite} that impossibility would contradict Condition~\ref{cond:thm-et1}. (Note that Theorem~\ref{thm:et} proves slightly more than what was promised by Theorem~\ref{thm:intro-et}.)
\begin{theorem}[Equilibrium transfer]\label{thm:et}
Let $\langle \{1,2\}, S_1,S_2,O,v,\{\prec_1,\prec_2\}\rangle$ be a two-player game in normal form, let $R_1\subseteq S_1$ and $R_2\subseteq S_2$, and let us assume the following:
\begin{enumerate}
\item\label{cond:thm-et1} the induced game structure is determined \textit{via} strategies in $R_1$ and $R_2$.
\item\label{cond:thm-et2} one of the following assertions holds:
\begin{itemize}
\item the preferences are acyclic and one player $i$ can enforce a finite $\prec_i$-upward cone.
\item the preferences have (uniformly) finite height.
\item $S_1$ and $S_2$ are countably many and the inverses of the preferences are well-founded.
\end{itemize}
\end{enumerate}
Then the game $\langle \{1,2\}, S_1,S_2,O,v,\{\prec_1,\prec_2\}\rangle$ has a Nash equilibrium in $R_1\times R_2$.
\end{theorem}
\begin{proof}
The first case is proved by Lemma~\ref{lem:et}. For the second case, let $n\in\mathbb{N}$ bound the height of both $\prec_1$ and $\prec_2$, and let $\rho_1:O\to\{0,\dots,n-1\}$ and $\rho_2:O\to\{0,\dots,n-1\}$ be corresponding rank functions, that is, $x\prec_i y$ implies $\rho_i(x)<\rho_i(y)$. Consider the game $\langle \{1,2\}, S_1,S_2,\{0,\dots,n-1\}^2,(\rho_1\circ v,\rho_2\circ v),\{\prec'_1,\prec'_2\}\rangle$ where $(i,j)\prec'_1(k,l)$ iff $i<k$ and $(i,j)\prec'_2(k,l)$ iff $j<l$. By Lemma~\ref{lem:et} the derived game has a Nash equilibrium in $R_1\times R_2$, which happens to be a Nash equilibrium for the original game too, by property of $\rho_i$.
As for the third case, it suffices to prove the statement for well-ordered preferences, by linear extension and preservation of Nash equilibrium by (set-theoretic) inclusion. Without loss of generality, let us also assume that $S_1$ and $S_2$ are infinite, otherwise let us duplicate strategies, and that $O$ is countable, otherwise let us replace it with $v(S_1,S_2)$. Note that if player $i$ can enforce a subset of outcome (\textit{via} some strategy in $S_i$), he or she can enforce it \textit{via} some strategy in $R_i$, since the opponent cannot exclude it and by determinacy assumption together with Remark~\ref{rmk:d-e}. Let us define a well-founded binary relation over pairs of ordinals, as follows: $(\alpha,\beta)\prec(\gamma,\delta):=(\alpha<\gamma\,\wedge\,\beta\leq\delta)\,\vee\,(\alpha\leq\gamma\,\wedge\,\beta<\delta)$, where $<$ is the usual well-order over ordinals, and let us proceed with the proof by induction (w.r.t $\prec$) on the pairs of order types that correspond to the inverses of the preferences.
If one order type is finite, it suffices to invoke Lemma~\ref{lem:et}, so let us deal with the case where both order types are infinite. If one player can exclude two or more of his/her least-preferred outcomes, Lemma~\ref{lem:reflect-eq} and the induction hypothesis prove the claim, so let us deal with the case where no player can exclude two or more of his/her least-preferred outcomes. Let $A$ be the set containing (when they exist) the $\prec_1$-least and second-$\prec_1$-least outcomes, so that $A$ is empty if the order-type for player $a$ is a limit ordinal and $A$ is a singleton when the order type is a limit ordinal plus one. Since player $a$ cannot exclude any non-trivial downward interval, every finite set of outcomes that he/she can enforce must intersect $A$. Let us define $B$ likewise and invoke the bottom-to-top implication of Lemma~\ref{lem:countable-finite} instantiated with, up to bijection, $A$ and $B$ defined above and $f:=v$. It implies the existence of a subset of the outcomes ($C$ in Lemma~\ref{lem:countable-finite}) that player $b$ cannot enforce and that player $a$ cannot exclude, which contradicts the determinacy assumption.
\end{proof}
\section{Applications of the equilibrium-transfer theorem}\label{sect:app-et}
Section~\ref{sect:gmt} generalises Martin's Theorem on Borel determinacy, from descriptive set theory, and Section~\ref{sect:pg-mg} generalises two determinacy results from theoretical informatics, namely positional determinacy of parity games and finite-memory determinacy of Muller games. Note that in each case the corresponding class of win-lose games is derived from infinitely many game structures (rather than from a single game structure), as similarly mentioned in Section~\ref{sect:mr-psc}.
\subsection{Generalisation of Borel Determinacy}\label{sect:gmt}
An infinite two-player alternate game consists of two players that play alternately and infinitely many times. In addition, the same non-empty set of choices $C$ is available at each stage, so the first player picks an element in $C$, then the second player picks an element in $C$, then the first player picks an element in $C$ again, and so on. The underlying structure of such a game is a leafless and uniform rooted tree. Moreover, each infinite sequence of choices is mapped to some outcome and both players have preferences over the outcomes.
\begin{definition}[Infinite two-player alternate games and strategies]
An infinite 2-player alternate game is an object $\langle C,O,v,\{\prec_1,\prec_2\}\rangle$ complying with the following:
\begin{itemize}
\item $C$ is a non-empty set (of choices).
\item $O$ is a non-empty set (of possible outcomes of the game).
\item $v:C^\omega\to O$ (uses outcomes to value the infinite sequences of choice).
\item $\prec_1$ and $\prec_2$ are binary relations over $O$ (called the preferences of player $1$ and $2$, respectively).
\end{itemize}
A function of type $C^{2*}\to C$ (resp. $C^{2*+1}\to C$) is a strategy for player $1$ (resp. $2$).
\end{definition}
A strategy of a player tells what he/she would play at each node of the game. Since the tree has a uniform structure, it is convenient to represent a strategy of the first (resp. second) player by a function of type $C^{2*}\to C$ (resp. $C^{2*+1}\to C$), where $C^{2*}$ represents the finite sequences on $C$ of even (resp. odd) length. When both players have chosen their individual strategies, their combination induces a unique play, \textit{i.e.} a unique infinite sequence of choices.
\begin{definition}[Induced play]
Given a game $g=\langle C,W\rangle$, and $s_1:C^{2*}\to C$, and $s_2:C^{2*+1}\to C$, let us define $p(s_1,s_2)$ through its prefixes, inductively as below, where $p_{<n}$ is the prefix of $p$ of length $n$ and the symbol $\cdot$ represents concatenation.
\begin{itemize}
\item $p(s_1,s_2)_{< 2n+1}:=p(s_1,s_2)_{< 2n}\cdot s_1(p(s_1,s_2)_{< 2n})$
\item $p(s_1,s_2)_{< 2n+2}:=p(s_1,s_2)_{< 2n+1}\cdot s_2(p(s_1,s_2)_{< 2n+1})$
\end{itemize}
\end{definition}
An infinite two-player alternate game $\langle C,O,v,\{\prec_1,\prec_2\}\rangle$ may be translated into a game in normal form $\langle\{1,2\}, (C^{2*}\to C)\times (C^{2*+1}\to C),O,v\circ p,\{\prec_1,\prec_2\}\rangle$, which provides the infinite two-player alternate game framework with a natural notion of Nash equilibrium.
Before stating Borel determinacy below, let us recall that a subset of a topological space $X$ is called Borel if it belongs to the smallest collection of subsets of $X$ which contains all the open sets and is closed under complementation and countable union.
\begin{theorem}[Martin~\cite{Martin75},~\cite{Martin85}]
Let $C$ be a non-empty set and $v:C^\omega\to\{(1,0),(0,1)\}$ be such that $v^{-1}\{(1,0)\}$ is a Borel set of $C^\omega$ (which is endowed with for the product topology of the discrete topology on $C$). Then the win-lose game $\langle (C^{2*}\to C)\times (C^{2*+1}\to C),v\circ p\rangle$ is determined.
\end{theorem}
The generalisation of Martin's theorem below is a straightforward corollary of both Martin's Theorem itself and the equilibrium transfer theorem.
\begin{corollary}\label{cor:gmt}
Let $\langle C,O,v,\{\prec_1,\prec_2\}\rangle$ be an infinite 2-player alternate game and assume the following three conditions.
\begin{itemize}
\item $O$ is countable.
\item $\prec_1$ and $\prec_2$ have (uniformly) finite height.
\item For all $o\in O$, the pre-image $v^{-1}\{o\}$ is Borel.
\end{itemize}
Then the game $\langle C,O,v,\{\prec_1,\prec_2\}\rangle$ has a Nash equilibrium.
\end{corollary}
\begin{proof}
Thanks to the above-mentioned embedding (of alternate games into games in normal form) and the (uniformly) finite height assumption, it suffices to check Condition~\ref{cond:thm-et1} from Theorem~\ref{thm:et}. This is done along Definition~\ref{defn:is-dg-ds}, so let $wl:O\to\{(1,0),(0,1)\}$. The set $(wl\circ v)^{-1}\{(1,0)\}$ is Borel since it equals $\cup\{v^{-1}\{o\}\,\mid\,v(o)=(1,0)\}$, a countable union of set that are Borel by assumtion. So, by Borel determinacy, the win-lose game $\langle (C^{2*}\to C)\times (C^{2*+1}\to C),w\circ v\circ p\rangle$ is determined.
\end{proof}
\subsection{Generalisations on parity games and Muller games}\label{sect:pg-mg}
These infinite two-player games are played on graphs. Unfolding these graphs yields infinite trees, so Borel determinacy may imply determinacy for these games. However, Borel determinacy does not say whether there exist simple winning strategies, or more generally winning strategies satisfying some predicate. So, the results that are generalised in Section~\ref{sect:pg-mg} are not mere corollaries of Borel determinacy. The definitions below rephrase, \textit{e.g.}, \cite{GW06}.
\begin{definition}[Arena and strategy]
An arena is an object $\langle V, V',E,C,\gamma\rangle$ complying with the following:
\begin{itemize}
\item $V$ is a non-empty set of vertexes.
\item $V'\subseteq V$ are the vertexes owned by player $1$.
\item $E\subseteq V\times V$ are the edges of a sink-free graph, \textit{i.e.} $\forall x\in V,xE:=\{y\in V\,\mid\,xEy\}\neq\emptyset$.
\item $C$ is a non-empty set of colours.
\item $\gamma:V\to C$ assigns a colour to each of the vertexes.
\end{itemize}
A strategy of player $1$ (resp. $2$) is a function of (dependent) type $V^*\to\forall v\in V',\, vE$ (resp. $V^*\to\forall v\in V\backslash V',\, vE$). A strategy profile is a function of (dependent) type $V^*\to\forall v\in V,\, vE$. The combination $(s_1,s_2)$ of a strategy $s_1$ for players $1$ and $s_2$ for player $2$ amounts to a strategy profile, which, when starting from a given vertex, induces a unique infinite sequence of colours $\Gamma(s_1,s_2)\in C^{\mathbb{N}}$.
\end{definition}
\begin{definition}[multi-outcome priority/Muller games]
A multi-outcome priority (resp. Muller) game is an object $\langle\mathcal{G},O,r,\{\prec_1,\prec_2\}\rangle$ complying with the following:
\begin{itemize}
\item $\mathcal{G}$ is an arena where $C=\mathbb{N}$ (resp. $\mathcal{G}$ a is finite arena) as defined above.
\item $O$ is a non empty set of outcomes.
\item $r:\mathbb{N}\cup\{\bot\}\to O$ (resp. $r:\mathcal{P}(C)\to O$)
\item $\prec_1$ and $\prec_2$ are binary relations over $O$, the preferences.
\end{itemize}
For every infinite sequence of colours $\Gamma$, let $cl(\Gamma)$ be its cluster set, \textit{i.e.} the set of the colours occurring infinitely often in $\Gamma$. The outcome that is induced by a sequence of colours $\Gamma\in C^{\mathbb{N}}$, \textit{i.e.} by a strategy profile and a starting vertex, is:
\begin{itemize}
\item For Muller games, $r\circ cl(\Gamma)$.
\item For priority games, $r\circ min\circ cl(\Gamma)$ if $cl(\Gamma)\neq\emptyset$, otherwise $r(\bot)$.
\end{itemize}
\end{definition}
Note that setting $O:=\{(1,0),(0,1)\}$ and $(0,1)\prec_1 (1,0)$ and $(1,0)\prec_2 (0,1)$ (plus $r(2n):=r(\bot):=(1,0)$ and $r(2n+1):=(0,1)$ for a multi-outcome priority game) in the definition above yields a parity (resp. Muller) game, up to isomorphism.
It was proved in~\cite{GH82} that Muller games are determined through finite-memory strategies and in~\cite{GW06} that parity games with priorities in $\mathbb{N}$ are positionally determined. Since these are determinacy results, let us extend them to multi-outcome settings below. (Note that one need not know what positional or finite-memory means.)
\begin{corollary}\label{cor:mo-mg}
Every multi-outcome Muller game (initiated with a starting vertex) with acyclic preferences has a finite-memory Nash equilibrium.
\end{corollary}
\begin{proof}
Let $\langle V, V',E,C,\gamma,O,r,\{\prec_1,\prec_2\}\rangle$ be a multi-outcome Muller game with acyclic preferences, and let $v_0\in V$ be the starting vertex. Since it is naturally embedded into a game in normal form (as far as NE are concerned), it suffices to invoke Lemma~\ref{lem:et} to prove the claim, where Conditions~\ref{cond:lem-et2} and \ref{cond:lem-et3} are fulfilled by assumption and finiteness of the game, respectively. Finally, Condition~\ref{cond:lem-et1} is also fulfilled: indeed, for every $wl:O\to\{(1,0),(0,1)\}$, the derived win-lose game $\langle V, V',E,C,\gamma,wl\circ r\rangle$ is a Muller game, so by \cite{GH82} it is determined \textit{via} finite-memory strategies.
\end{proof}
\begin{corollary}\label{cor:mo-pg}
Every multi-outcome priority game where preferences have finite height has a positional Nash equilibrium.
\end{corollary}
\begin{proof}
The proof is similar to the proof of Corollary~\ref{cor:mo-mg}, up to one point: let $wl:O\to\{(1,0),(0,1)\}$, the derived win-lose game $\langle V, V',E,C,\gamma,wl\circ r\rangle$ is not obviously isomorphic to a parity game, because the function $wl\circ r$ may not map even numbers and $\bot$ to $(1,0)$, and odd numbers to $(0,1)$. This is easily overcome by renaming the colours: let $\gamma'(v):=2\cdot\gamma(v)$ if $wl\circ r\circ\gamma(v)=wl\circ r(\bot)$ and $\gamma'(v):=2\cdot\gamma(v)+1$ otherwise. By \cite{GW06} the parity game $\langle V, V',E,C,\gamma'\rangle$ is determined, and so is $\langle V, V',E,C,\gamma,wl\circ r\rangle$.
\end{proof}
Note that, in the area of graph games for program verification, \cite{Ummels05} has already investigated extensions of determinacy in various directions, namely for subgame perfect equilibrium (a stronger notion of Nash equilibrium), for $n$-player games instead of two-player games, or for payoff functions in $\{0,1\}^n$ instead of $\{(0,1),(1,0)\}$. For instance, Theorem 4.19. in \cite{Ummels05} states that any initialised two-player parity game has a positional subgame perfect equilibrium and Theorem 4.20. states that any initialised finite multiplayer parity game has a finite-state subgame perfect equilibrium.
\section{Limitations of transfer possibilities}\label{sect:ltt}
Section~\ref{sect:lim-order-set} below suggests that the order and set-theoretic assumptions of Theorem~\ref{thm:et} are tight, although the case where exactly one strategy set is countable and the preferences are well-founded yet with chains of arbitrary length is still open; Section~\ref{sect:3player} afterwards suggests that the two-player assumptions of Theorem~\ref{thm:et} is tight, although there is still room for a three-player version of the equilibrium-transfer theorem since I failed to find a counterexample in the most general case.
\subsection{Order and set-theoretic limitations}\label{sect:lim-order-set}
Proposition~\ref{prop:lim-uncountable} below shows that the countability condition of Theorem~\ref{thm:et} is difficult to weaken in general.
\begin{proposition}\label{prop:lim-uncountable}
There exists a game satisfying the following:
\begin{itemize}
\item player $1$ has countably many strategies,
\item the preference of player $1$ has no infinite ascending chain,
\item the preference of player $2$ has one maximum and the other outcomes are minimal,
\item the underlying game structure is determined,
\item the game has no Nash equilibrium.
\end{itemize}
\end{proposition}
\begin{proof}
Let $I$ (resp. $C$) be the infinite (resp. cofinite) subsets of the naturals. Consider the two-player game structure $\langle \{1,2\}, (C\cup\{\alpha,\beta\})\times(I\cup\{\alpha,\beta\}),\mathbb{N}\cup\{a,b\},v\rangle$ where the unions are disjoints and where $v$ is as below and $min$ refers to the usual order over $\mathbb{N}$.
\[\begin{array}{cllll}
v: & (C\cup\{\alpha,\beta\})\times(I\cup\{\alpha,\beta\}) &\to& \mathbb{N}\cup\{a,b\}\\
& (X,Y)&\mapsto & a &\mbox{ if } X=Y\in\{\alpha,\beta\}\\
&&& b &\mbox{ if } \{X\}\cup\{Y\}=\{\alpha,\beta\}\\
&&& min(X) &\mbox{ if } (X,Y)\in C\times\{\alpha,\beta\}\\
&&& min(Y) &\mbox{ if } (X,Y)\in\{\alpha,\beta\}\times I\\
&&& min(X\cap Y-\{min\, X\cap Y\}) &\mbox{ if } X,Y\notin\{\alpha,\beta\}\\
\end{array}\]
Let us first show the determinacy of the game structure. Let $P\subseteq \mathbb{N}\cup\{a,b\}$. If $P\cap\mathbb{N}$ is cofinite, player $1$ can enforce it (by playing it) since $v(X,Y)$ is in $X$ for all $X\in C$ and $Y\in I\cup\{\alpha,\beta\}$, by definition of $v$; so a fortiori player $1$ can enforce $P$. If $P\cap\mathbb{N}$ is not cofinite, $\mathbb{N}\backslash P$ is infinite, so player $2$ can enforce it, and a fortiori $(\mathbb{N}\cup\{a,b\})\backslash P$.
Second, let us define some preferences: for all $x\in\mathbb{N}\cup\{a\}$, set $x\prec_2b$. Set $b\prec_1 a$ and $n+k+1\prec_1n\prec_1a$ for all $n,k\in\mathbb{N}$. Let us now show that the game has no Nash equilibrium.
\begin{itemize}
\item $v(\alpha,\alpha)<_2v(\alpha,\beta)$ and $v(\beta,\beta)<_2v(\beta,\alpha)$.
\item $v(\alpha,\beta)<_1v(\beta,\beta)$ and $v(\beta,\alpha)<_1v(\alpha,\alpha)$.
\item If $X\in C$ and $Y\in\{\alpha,\beta\}$ then $v(X,Y)<_1v(Y,Y)=a$.
\item If $X\in\{\alpha,\beta\}$ and $Y\in I$ then $v(X,Y)<_2v(X,X')=b$, where $\{X,X'\}=\{\alpha,\beta\}$.
\item If $X,Y\notin\{\alpha,\beta\}$ then $v(X,Y)<_1v(\alpha,Y)$ since $min(X\cap Y-\{min X\cap Y\})<min(Y)$.
\end{itemize}
\end{proof}
Note that replacing "player $1$ has countably many strategies" with "player $2$ has countably many strategies" in Proposition~\ref{prop:lim-uncountable} yields a correct statement. In the proof, indeed, it suffices to swap the sets $I$ and $C$ in the definition of the witness game.
Proposition~\ref{prop:lim-bounded-height} below shows that the (uniform) finite height condition of Theorem~\ref{thm:et} is difficult to weaken in general. The proof idea is similar to that of Proposition~\ref{prop:lim-uncountable}, albeit slightly more complex.
\begin{proposition}\label{prop:lim-bounded-height}
There exists a game satisfying the following:
\begin{itemize}
\item the preference of player $1$ has no infinite chain,
\item the preference of player $2$ has one maximum and the other outcomes are minimal,
\item the underlying game structure is determined,
\item the game has no Nash equilibrium.
\end{itemize}
\end{proposition}
\begin{proof}
For every $n\in\mathbb{N}$ let $A_n:=\{k\in\mathbb{N}\,\mid\,n^2\leq k<(n+1)^2\}$, and let $I:=\{X\subseteq\mathbb{N}\,\mid\,\forall n\in\mathbb{N},\exists m\in\mathbb{N},n\leq|X\cap A_m|\}$, and let $C:=\{X\subseteq\mathbb{N}\,\mid\,\exists n\in\mathbb{N},\forall m\in\mathbb{N},|A_m\backslash X|\leq n\}$. Consider the two-player game structure that is defined by its outcome function below, where $S_1:=C\cup\{\alpha_0,\alpha_1,\dots\}$ and $S_2:=I\cup\{\alpha_0,\alpha_1,\dots\}$, where the unions are disjoints, and where $min$ refers to the usual order over $\mathbb{N}$. .
\[\begin{array}{cllll}
v: & S_1\times S_2 &\to& \mathbb{N}\cup\{a,b\}\\
& (X,Y)&\mapsto & a &\mbox{ if } X=Y=\alpha_n\\
&&& b &\mbox{ if } (X,Y)=(\alpha_i,\alpha_j) \mbox{ with } i\neq j\\
&&& min(X) &\mbox{ if } X\in C \mbox{ and } Y=\alpha_n\\
&&& min(Y\cap \{n^2,n^2+1,\dots\}) &\mbox{ if } X=\alpha_n \mbox{ and } Y\in I\\
&&& min(X\cap Y\cap A_n-\{min\, X\cap Y\cap A_n\}) &\mbox{ if } (X,Y)\in C\times I,\mbox{ where }\\
&&& &n:=min\{k\,\mid\,2\leq|X\cap Y\cap A_k|\}
\end{array}\]
Note that in the last line of the definition above, $\{k\,\mid\,2\leq|X\cap Y\cap A_k|\}$ is non-empty by definition of $C$ and $I$.
Let us first show the determinacy of the game structure. Let $P\subseteq \mathbb{N}\cup\{a,b\}$. If $P\cap\mathbb{N}$ is in $C$, player $1$ can enforce it (by playing it) since $v(X,Y)$ is in $X$ for all $X\in C$ and $Y\in I\cup\{\alpha_0,\alpha_1,\dots\}$, by definition of $v$; so a fortiori player $1$ can enforce $P$. If $P\cap\mathbb{N}$ is not in $C$, $\mathbb{N}\backslash P$ is in $I$, by definition of $C$ and $I$, so player $2$ can enforce it, and a fortiori $(\mathbb{N}\cup\{a,b\})\backslash P$.
Second, let us define some preferences: set $x\prec_2 b$ for all $x\in\mathbb{N}\cup\{a\}$; set $x\prec_1 a$ for all $x\in\mathbb{N}\cup\{b\}$ and $i\prec_1j$ whenever $n^2\leq j<i<(n+1)^2$ for $i,j,n\in\mathbb{N}$. The game thus defined has no Nash equilibrium, as shown below.
\begin{itemize}
\item $v(\alpha_n,\alpha_n)<_2v(\alpha_n,\alpha_{n+1})$ and $v(\alpha_i,\alpha_j)<_1v(\alpha_j,\alpha_j)$ for $i\neq j$.
\item If $X\in C$ and $Y=\alpha_n$ then $v(X,Y)<_1v(Y,Y)=a$.
\item If $X=\alpha_n$ and $Y\in I$ then $v(X,Y)<_2v(X,\alpha_{n+1})=b$.
\item If $(X,Y)\in C\times I$ then $v(X,Y)<_1v(\alpha_n,Y)$, where $n:=min\{k\,\mid\,2\leq|X\cap Y\cap A_k|\}$.
\end{itemize}
\end{proof}
\subsection{Three-player limitations}\label{sect:3player}
The equilibrium-transfer theorem considers two-player games only, which raises the issue of the existence of a three-player version of the theorem. However, the condition of application of such a version can no longer state a mere determinacy of the game structure, because the very notion of determinacy makes little sense for three players. Instead, the condition may require that some specific games that are derived from and are simpler than the original game all have Nash equilibria. (Note that in the two-player case, the determinacy of the game structure falls into this type of condition.) The counter-examples in this section suggest that there is no general three-player version, without giving a definitive answer, though.
Theorem~\ref{thm:et} can be rephrased as follows: if equipping a given two-player game structure with simple preferences (\textit{i.e.} free of three-outcome chains) always yields a game with a Nash equilibrium, so does equipping the same structure with more complex preferences. The example below shows that one cannot just replace "two-player" with "three-player" in the above statement, since preferences with three-outcome chains may already be problematic.
\begin{remark}\label{prop:3aet-simple}
Let $a$, $b$ and $c$ be three players, let $l$ and $r$ be two strategies available to each player, let $x$, $y$ and $z$ be three possible outcomes, let us define three transitive preferences by $z<_ay<_ax$ and $x<_by<_bz$ and $<_c:=<_b$, and define an outcome function as follows: $v(l,l,l):=v(l,r,l):=v(r,r,l):=y$ and $v(r,l,l):=v(l,l,r):=v(l,r,r):=z$ and $v(r,l,r):=v(r,r,r):=x$. See the graphical representation below, where players $a$, $b$, and $c$ choosing $l$ yields top rows, left columns, and left array, respectively.
\[\begin{array}{c@{\hspace{1cm}}c}
\begin{array}{|c|c|}
\hline y & y\\
\hline z & y\\
\hline
\end{array}
&
\begin{array}{|c|c|}
\hline z & z\\
\hline x & x\\
\hline
\end{array}
\end{array}
\]
\begin{itemize}
\item The game $\langle\{a,b,c\},(\{l,r\})_{d\in\{a,b,c\}},\{x,y,z\},v,(<_d)_{d\in\{a,b,c\}}\rangle$ has no Nash equilibrium.
\item Let $\{0,1\}^3$ be an alternative set of outcomes, for $i,j,k,n\in\{0,1\}$ let $(0,i,j)\prec_a(1,k,n)$ and $(i,0,j)\prec_b(k,1,n)$ and $(i,j,0)\prec_c(k,n,1)$. Then for all $wl:\{x,y,z\}\to\{0,1\}^3$ the game\\
\noindent $\langle\{a,b,c\},(\{l,r\})_{d\in\{a,b,c\}},\{0,1\}^3,wl\circ v,(\prec_d)_{d\in\{a,b,c\}}\rangle$ has a Nash equilibrium.
\end{itemize}
\end{remark}
\begin{proof}
The original game has no Nash equilibrium, by construction. Let $wl:\{x,y,z\}\to\{0,1\}^3$ and from now on let us consider the modified game only, assume that it has no Nash equilibrium, and draw a contradiction. Both players $a$ and $b$ are stable \textit{w.r.t.} the strategy profile $(l,r,l)$ since $v(l,l,l)=v(l,r,l)=v(r,r,l)=y$ by construction, so $wl(y)=wl\circ v(l,r,l)\prec_cwl\circ v(l,r,r)=wl(z)$. So player $c$ is stable \textit{w.r.t.} the strategy profile $(l,r,r)$, and so is player $b$ since $v(l,l,r)=v(l,r,r)=z$, therefore $wl(z)=wl\circ v(l,r,r)\prec_a wl\circ v(r,r,r)=wl(x)$. So player $a$ is stable \textit{w.r.t.} the strategy profile $(r,r,r)$, and so is player $b$ since $v(r,l,r)=v(r,r,r)=x$, so $wl(x)=wl\circ v(r,r,r)\prec_c wl\circ v(r,r,l)=wl(y)$. Therefore $wl(x)\prec_c wl(y)\prec_c wl(z)$, contradiction since $\prec_c$ has no three-outcome chain.
\end{proof}
A weaker three-player version of Theorem~\ref{thm:et} might obtain by using a large-enough natural number $n$ as follows: "if equipping a given three-player game structure with preferences free of $(n+1)$-outcome chains always yields a game with a Nash equilibrium, so does equipping the same structure with more complex preferences." Proposition~\ref{prop:3aet} below shows that preferences with $(n+1)$-outcome chains may already be problematic. (Note that the case $n=2$ corresponds to Remark~\ref{prop:3aet-simple}.)
\begin{proposition}\label{prop:3aet}
For every natural $2\leq n$ there exists a finite three-player game in normal form that complies with the following:
\begin{itemize}
\item The preferences are linear orders over $\{0,1,\dots,n\}$.
\item The game has no Nash equilibrium.
\item Replacing the preferences with preferences that have no chain of length $n+1$ yields a game with a Nash equilibrium.
\end{itemize}
\end{proposition}
\begin{proof}
Let $2\leq n$ be a natural, let $<_b$ and $<_c$ be the restrictions of the usual order over the naturals to $\{0,\dots,n\}$, and let $<_a:=<_b^{-1}$, that is, $n<_an-1<_a\dots<_a1<_a0$. Let us define the outcome function $v:\{1,\dots,n\}^3\to\{0,\dots,n\}$ below.
\begin{itemize}
\item For $1\leq i\leq n$ let $v(n,i,n):=0$.
\item For $1\leq i<n$ let $v(i,i,n):=i+1$.
\item Otherwise let $v(\cdot,\cdot,n)$ return $n$.
\item For $1\leq i<n$ let $v(n,1,i):=n$.
\item For $1<i<n$ and $1\leq j\leq n$ let $v(i,j,i):=v(j,i,i):=i$.
\item Otherwise let $v$ return $1$.
\end{itemize}
For example, the game $\langle\{a,b,c\},(\{1,2,3,4\})_{d\in\{a,b,c\}},\{0,1,2,3,4\},v,(<_d)_{d\in\{a,b,c\}}\rangle$ is represented below, where player $a$ chooses the row, $b$ the column, and $c$ the array.
\[\begin{array}{c@{\hspace{1cm}}c@{\hspace{1cm}}c@{\hspace{1cm}}c}
\begin{array}{|c|c|c|c|}
\hline 1 & 1 & 1 & 1\\
\hline 1 & 1 & 1 & 1\\
\hline 1 & 1 & 1 & 1\\
\hline 4 & 1 & 1 & 1\\
\hline
\end{array}
&
\begin{array}{|c|c|c|c|}
\hline 1 & 2 & 1 & 1\\
\hline 2 & 2 & 2 & 2\\
\hline 1 & 2 & 1 & 1\\
\hline 4 & 2 & 1 & 1\\
\hline
\end{array}
&
\begin{array}{|c|c|c|c|}
\hline 1 & 1 & 3 & 1\\
\hline 1 & 1 & 3 & 1\\
\hline 3 & 3 & 3 & 3\\
\hline 4 & 1 & 3 & 1\\
\hline
\end{array}
&
\begin{array}{|c|c|c|c|}
\hline 2 & 4 & 4 & 4\\
\hline 4 & 3 & 4 & 4\\
\hline 4 & 4 & 4 & 4\\
\hline 0 & 0 & 0 & 0\\
\hline
\end{array}
\end{array}
\]
Let us show that the game $\langle\{a,b,c\},(\{1,\dots,n\})_{d\in\{a,b,c\}},\{0,\dots,n\},v,(<_d)_{d\in\{a,b,c\}}\rangle$ witnesses the claim. First, the preferences are linear orders indeed. Second, let us show that there is no Nash equilibrium by case-splitting below.
\begin{itemize}
\item If $i,k\neq n$, then $v(i,j,k)<_cv(i,j,n)$.
\item If $i\neq n$, then $v(i,j,n)<_a0=v(n,j,n)$.
\item $v(n,j,n)=0<_cv(n,j,1)$.
\item $v(n,1,1)=n\prec_a 1=v(1,1,1)$ and if $j\neq 1$, then $v(n,j,1)=1\prec_b n=v(n,1,1)$.
\item If $j\neq 1$ and $1<k<n$, then $v(n,j,k)<_bn=v(n,1,k)$.
\item If $1<k<n$, then $v(n,1,k)=n<_a1=v(1,1,k)$.
\end{itemize}
Third, by contraposition, let $\prec_a$, $\prec_b$ and $\prec_c$ be arbitrary acyclic preferences, assume that the game\\
$\langle\{a,b,c\},(\{1,\dots,n\})_{d\in\{a,b,c\}},\{0,\dots,n\},v,(\prec_d)_{d\in\{a,b,c\}}\rangle$ has no Nash equilibrium, and let us prove that $\prec_c=<_c$, thus contradicting the assumption on the chains. If $i\neq n$, by construction $v(i,i,i)=i=v(j,i,i)=v(i,j,i)$ and $v(i,i,j)\in\{1,i,i+1\}$. By assumption there is no Nash equilibrium and the preferences are acyclic, so $i\prec_c1$ or $i\prec_ci+1$ if $1\leq i< n$, so $1\prec_c 2$, and it is provable by induction that $i\prec_ci+1$ for all $1\leq i< n$. Now it suffices to prove $0\prec_c 1$ to conclude. By assumption the strategy profile $(n-1,n,n)$ is not a Nash equilibrium, so $v(n-1,n,n)=n\prec_a 0=v(n,n,n)$ since $v(i,n,n)=v(n-1,j,n)=n$ for $i\neq n$, since $v(n-1,n,k)\in\{1,n-1,n\}$, and by assumption of acyclic preferences. Now, the profile $(n,n,n)$ is not a Nash equilibrium either. Since $v(n,j,n)=0$, since $v(i,n,n)=n$ if $i\neq n$, and since $n\prec_a 0$, the players $a$ and $b$ are sable. Since $v(n,n,k)\in\{0,1\}$, we must have $v(n,n,n)=0\prec_c 1$.
\end{proof}
Note that in the statement of Proposition~\ref{prop:3aet}, one may also modify the games without Nash equilibrium so that they are zero-sum, by defining $z(n):=(-2n,n,n)$ and using the outcome function $z\circ v$ instead of $v$.
Let us now try to find an alternative equilibrium-transfer theorem that still states existence of Nash equilibrium in three-player game. As a lead, notice that many two-player game structures may be derived from a given three-player game structure: first, by slicing the game cuboid, \textit{i.e.} by fixing the strategy of one player; second, by merging any two players into a super player. We may hope that if all thus-derived two-player game structures are determined (and therefore also have Nash equilibria for more complex preferences), then the original three-player game structure has a Nash equilibrium when equipped with simple preferences. However, Proposition~\ref{prop:3sm1} contradicts it.
\begin{proposition}\label{prop:3sm1}
There exists a finite game structure $G=\langle\{a,b,c\},S_a,S_b,S_c,\{X,Y,Z\},v\rangle$ that satisfies the following:
\begin{itemize}
\item for all $s_c\in S_c$ the game structure $\langle\{a,b\},S_a,S_b,\{X,Y,Z\},v(\cdot,\cdot,s_c)\rangle$, which is obtained by slicing $G$ along $s_c$, is determined. (And similarly by slicing along some $s_a\in S_a$ or $s_b\in S_b$.)
\item the game structure $\langle\{a\times b,c\},S_a\times S_b,S_c,\{X,Y,Z\},v')\rangle$, where $v'((s_a,s_b),s_c):=v(s_a,s_b,s_c)$, which is obtained by merging players $a$ and $b$, is determined. (And similarly by merging $c$ with $a$ or $b$.)
\item instantiating $G$ with $X:=(1,0,0)$, $Y:=(0,1,0)$, and $Z:=(0,0,1)$ yields a game without Nash equilibrium.
\end{itemize}
\end{proposition}
\begin{proof}
Let us define a three-player game structure, each player having six strategies. Informally, the $6\times 6\times 6$ empty cube is filled with three $3\times 3\times 6$ cylinders and two $3\times 3\times 3$ cylinders. The cross section of each of these cylinders looks like this:
\[\begin{array}{|c|c|c|}
\hline X & Y & Z\\
\hline Y & Z & X\\
\hline Z & X & Y\\
\hline
\end{array}\]
Let us define the outcome function of the game formally below, where each of the five blocks of three lines corresponds to a cylinder.
\[\begin{array}{l@{\hspace{2cm}}l}
v(1,1,\cdot):=v(2,3,\cdot):=v(3,2,\cdot):=X & v(\cdot,4,1):=v(\cdot,5,3):=v(\cdot,6,2):=X\\
v(1,2,\cdot):=v(2,1,\cdot):=v(3,3,\cdot):=Y & v(\cdot,4,2):=v(\cdot,5,1):=v(\cdot,6,3):=Y\\
v(1,3,\cdot):=v(2,2,\cdot):=v(3,1,\cdot):=Z & v(\cdot,4,3):=v(\cdot,5,2):=v(\cdot,6,1):=Z\\
\\
v(4,\cdot,4):=v(5,\cdot,6):=v(6,\cdot,5):=X\\
v(4,\cdot,5):=v(5,\cdot,4):=v(6,\cdot,6):=Y\\
v(4,\cdot,6):=v(5,\cdot,5):=v(6,\cdot,4):=Z\\
\end{array}\]
For $1\leq i\leq 3$, set the following:
\[\begin{array}{l@{\hspace{2cm}}l}
v(4,1,i):=v(5,3,i):=v(6,2,i):=X & v(4,i,4):=v(5,i,6):=v(6,i,5):=X\\
v(4,2,i):=v(5,1,i):=v(6,3,i):=Y & v(4,i,5):=v(5,i,4):=v(6,i,6):=Y\\
v(4,3,i):=v(5,2,i):=v(6,1,i):=Z & v(4,i,6):=v(5,i,5):=v(6,i,4):=Z\\
\end{array}\]
A typical section of the whole game structure, actually $v(\cdot,\cdot,1)$, looks as below: the cross sections of two cylinders on the left-hand side, the "outer face" of another cylinder on the right-hand side:
\[\begin{array}{|c|c|c|c|c|c|}
\hline X & Y & Z & X & Y & Z\\
\hline Y & Z & X& X & Y & Z\\
\hline Z & X & Y& X & Y & Z\\
\hline X & Y & Z & X & Y & Z\\
\hline Y & Z & X& X & Y & Z\\
\hline Z & X & Y& X & Y & Z\\
\hline
\end{array}\]
Instantiating the game structure with $X:=(1,0,0)$, $Y:=(0,1,0)$, and $Z:=(0,0,1)$ yields a game without Nash equilibrium, because none of the cylinders constituting the whole game has a Nash equilibrium.
Nonetheless, all the two-player game structures that are derived from the three-player game structure by slicing or merging have a Nash equilibrium, as partially justified below.
Slicing: fix the strategy of c between $1$ and $3$ (resp. $4$ and $6$), then player $b$ (resp. $a$) can choose the outcome he/she wants. For instance fixing the strategy of $c$ to $1$ yields the game represented graphically above. Similar situations arise when fixing strategies of $a$ or $b$.
Merging: The three $3\times 3\times 6$ cylinders ensure that any two players can collectively enforce any outcome.
\end{proof}
The next and last idea is to combine the two previous attempts to get even stronger assumptions: given a game structure, we may hope that if slicing or merging it always yields determined game structures, and if equipping it with simple preferences always yields games with Nash equilibrium, then so does equipping it with more complex preferences. However, the following does not sound very promising.
\begin{proposition}\label{prop:3sm}
There exists a finite game structure $G=\langle\{a,b,c\},S_a,S_b,S_c,\{X,Y,Z\},v\rangle$ that satisfies the following:
\begin{itemize}
\item for all $s_c\in S_c$ the game structure $\langle\{a,b\},S_a,S_b,\{X,Y,Z\},v(\cdot,\cdot,s_c)\rangle$, which is obtained by slicing $G$ along $s_c$, is determined. (And similarly by slicing along some $s_a\in S_a$ or $s_b\in S_b$.)
\item the game structure $\langle\{a\times b,c\},S_a\times S_b,S_c,\{X,Y,Z\},v')\rangle$, where $v'((s_a,s_b),s_c):=v(s_a,s_b,s_c)$, which is obtained by merging players $a$ and $b$, is determined. (And similarly by merging $c$ with $a$ or $b$.)
\item Equipping $G$ with preferences that are free of three-outcome chains yields a game a Nash equilibrium.
\item Equipping $G$ with $Z<_aY<_aX$ and $X<_bZ<_bY$ and $Y<_cX<_cZ$ yields a game without Nash equilibrium.
\end{itemize}
\end{proposition}
\begin{proof}
Let us define a three-player game structure that is not as symmetric as the one from Proposition~\ref{prop:3sm1}: Player $a$ has four strategies and players $b$ and $c$ have seven strategies each. Informally, the $4\times 7\times 7$ empty cuboid is filled with one $4\times 3\times 3$ cylinder like in Proposition~\ref{prop:3sm1}, four thinner $2\times 2\times 7$ cylinders, and four shorter versions of these, namely $2\times 2\times 3$ cylinders. The different sections of the thinner cylinders look like these:
\[\begin{array}{ccc}
\begin{array}{|c|c|}
\hline X & Y \\
\hline Y & X \\
\hline
\end{array}
&
\begin{array}{|c|c|}
\hline X & Z \\
\hline Z & X \\
\hline
\end{array}
&
\begin{array}{|c|c|}
\hline Y & Z \\
\hline Z & Y \\
\hline
\end{array}
\end{array}
\]
Let us define the outcome function of the game formally below, where each of the five blocks of three lines corresponds to a cylinder.
\[\begin{array}{l}
v(\cdot,5,1):=v(\cdot,6,3):=v(\cdot,7,2):=X\\
v(\cdot,5,2):=v(\cdot,6,1):=v(\cdot,7,3):=Y\\
v(\cdot,5,3):=v(\cdot,6,2):=v(\cdot,7,1):=Z\\
\\
v(1,1,\cdot):=v(2,2,\cdot):=v(1,3,\cdot):=v(2,4,\cdot):=X\\
v(1,2,\cdot):=v(2,1,\cdot):=Y\\
v(1,4,\cdot):=v(2,3,\cdot):=Z\\
\\
v(3,\cdot,5):=v(4,\cdot,4):=X\\
v(3,\cdot,7):=v(4,\cdot,6):=Y\\
v(3,\cdot,4):=v(4,\cdot,5):=v(3,\cdot,6):=v(4,\cdot,7):=Z\\
\end{array}\]
For $1\leq i\leq 3$, set the following:
\[\begin{array}{l}
v(3,1,i):=v(4,2,i):=v(3,3,i):=v(4,4,i):=X \\
v(3,2,i):=v(4,1,i):=Y \\
v(3,4,i):=v(4,3,i):=Z \\
\\
v(1,i+4,5):=v(2,i+4,4):=X\\
v(1,i+4,7):=v(2,i+4,6):=Y\\
v(1,i+4,4):=v(2,i+4,5):=v(1,i+4,6):=v(2,i+4,7):=Z\\
\end{array}\]
Typical cross sections of the whole game structure look like the two below, $v(\cdot,\cdot,1)$ on the left-hand side and $v(\cdot,\cdot,7)$ on the right-hand side.
\[\begin{array}{c@{\hspace{2cm}}c}
\begin{array}{|c|c|c|c|c|c|c|}
\hline X & Y & X & Z & X &Y & Z\\
\hline Y & X & Z & X & X & Y & Z\\
\hline X & Y & X & Z & X & Y & Z\\
\hline Y & X & Z & X & X & Y & Z\\
\hline
\end{array}
&
\begin{array}{|c|c|c|c|c|c|c|}
\hline X & Y & X & Z & Y & Y & Y\\
\hline Y & X & Z & X & Z & Z & Z\\
\hline Y & Y & Y & Y & Y & Y & Y\\
\hline Z & Z & Z & Z & Z & Z & Z\\
\hline
\end{array}
\end{array}\]
Equipping the game structure with the preferences $Z<_aY<_aX$ and $X,Z<_bY$ and $X,Y<_cZ$ yields a game without Nash equilibrium.
Nonetheless, slicing the game structure or merging any two of its players yields a determined game structure. It is proved either by similar arguments as in Proposition~\ref{prop:3sm1}, or as follows to show that the cross section $v(\cdot,\cdot,7)$ above to the right is determined: if $Y$ or $Z$ makes player $a$ win, player $a$ wins for sure by playing one of the last two rows; if $Y$ and $Z$ make player $a$ lose, player $b$ wins for sure by playing the last column.
Let us now show that equipping the game structure with preferences that are free of three-outcome chains yields a game with a Nash equilibrium: If two players share a preferred outcome, they can collectively enforce it, which yields a Nash equilibrium, so now let us assume that the three players prefer distinct outcomes. The following array points to one Nash equilibrium for each of the six permutations of $(X,Y,Z)$ as preferred outcomes:
\[\begin{array}{|c|c|c|c|}
\cline{1-4}
a & b & c & \mbox{Nash equilibrium}\\
\cline{1-4}
Z & X & Y & 1,1,1\\
\cline{1-4}
Z & Y & X & 1,2,1\\
\cline{1-4}
Y & X & Z & 1,3,1\\
\cline{1-4}
Y & Z & X & 1,4,1\\
\cline{1-4}
X & Y & Z & 4,7,7\\
\cline{1-4}
X & Z & Y & 4,7,6\\
\cline{1-4}
\end{array}\]
\end{proof}
Proposition~\ref{prop:3sm} does not fully dash the hope for a three-player version of the equilibrium-transfer theorem, though. Indeed, an alternative, even weaker statement could be as follows for a given natural number $n$: "If merging or slicing a given three-player game always yields determined structures, and if replacing the preferences of the game with preferences of height at most $n$ always yields games with a Nash equilibrium, then the original game also has a Nash equilibrium." The case $n=2$ is disproved by Proposition~\ref{prop:3sm}, but the cases $3\leq n$ are still open. If there are counterexamples too, building them and proving their combinatorial property might be rather complex, though.
\section{Conclusion}\label{sect:c}
This article has shown that every determinacy result over a given two-player game structure is transferable into existence of multi-outcome Nash equilibrium over the same game structure. Moreover, when the outcomes are finitely many, the proof provides an algorithm that computes a Nash equilibrium without significant complexity loss compare to the win-lose case.
Contrary to most game-theoretic results, which state that every game of a given class of games has some property, this result is a higher-order theorem: it states that every class of games that is derived from any game structure has itself some property.
If the heights of the preferences of the two players are finite, the equilibrium transfer holds regardless of the game structure; furthermore, if the structure has countably many strategies, the finite-height condition can be relaxed and phrased as absence of infinite ascending sequences, which was the hardest to prove in this article.
Although counterexamples from Section~\ref{sect:lim-order-set} show that these conditions are useful, there is still room for fine-tuning. In particular, it is still open whether the following is a sufficient condition for equilibrium transfer: "the strategy set of one player is countable and the preferences of the players have no infinite sequences (even descending)".
Section~\ref{sect:app-et} gave three examples of applications of the equilibrium-transfer theorem. Apart from the generalisation of positional determinacy of parity games, which is new, the two other obtained results are weaker than existing results; but the key point here is, however, that the three applications are almost effortless, and that the same would hold for further applications.
The two above-mentioned existing generalisations of Borel determinacy and finite-memory determinacy of Muller games are indeed more general than what can be obtained by application of the equilibrium-transfer theorem, because they hold in a multi-player setting. The proofs are ad hoc, though, which raises the question of a uniform equilibrium-transfer theorem for games with three or more players. A natural attempt is to replace the determinacy condition with existence of Nash equilibrium in some simpler derived games. Counterexamples in Section~\ref{sect:3player} disproved some simple variants of such an attempt, but existence of a slightly more complex variant is still an open question. Alternatively, one may try to add strong conditions on the structure, \textit{e.g.}, after noticing that the players play sequentially in both Muller games and the games used for Borel determinacy.
\medskip
Unexpectedly, the finite-height condition of Theorem~\ref{thm:et} leads to an interesting general phenomenon about preferences in game theory: contrary to a widespread belief, linear orders do not account for partial orders. Indeed the remark below considers two finite-height preferences, therefore fulfilling Condition~\ref{cond:thm-et2} of Theorem~\ref{thm:et}, yet for all possible linear extensions of these preferences, equilibrium transfer does not hold!
\begin{remark}\label{rem:total}
Let $\prec_1$ and $\prec_2$ be two binary relations over $\mathbb{N}$ that are defined by $2n\prec_1 2n+1$ and $\prec_2:=\prec_1^{-1}$. For all $<_1$ and $<_2$ linear extensions of $\prec_1$ and $\prec_2$ respectively, there exists a game satisfying Condition~\ref{cond:thm-et1} of Theorem~\ref{thm:et}, but without Nash equilibrium.
\end{remark}
\begin{proof}
If the inverse of $<_1$ is not a well-order, the game $\langle \{1\},\mathbb{N},\mathbb{N},id,\{<_1\}\rangle$ has no Nash equilibrium although the induced structure is determined, so let us assume that the inverse of $<_1$ is a well-order. Since the sequence $(2n)_{n\in\mathbb{N}}$ has no $<_1$-increasing subsequence, it has a $<_1$-decreasing subsequence $(2\phi(n))_{n\in\mathbb{N}}$ (as a consequence of Ramsey Theorem). Setting $a:=2\phi(0)+1$ and $b:=2\phi(0)$ and $x_n=2\phi(n+1)$ embeds the preferences from Proposition~\ref{prop:lim-uncountable} into $<_1$ and $<_2$, respectively, so the witness game from Proposition~\ref{prop:lim-uncountable} also witnesses the remark at hand.
\end{proof}
Nonetheless, it is often very convenient to consider linearly ordered preferences only, when actually done without loss of generality. Remark~\ref{rem:total} above just exemplifies that one ought to be very cautious because a loss of generality may actually occur.
\section*{Acknowledgement}
I thank Achim Blumensath and Michael Ummels for discussions on parity games and the like, Alexander Kreuzer for explanations on Ramsey's theorem, an anonymous referee in particular for a remark that helped shorten and clarify the proof of Theorem~\ref{thm:et}, and I am especially grateful to Vassilios Gregoriades for useful discussions and advice, and his determined and determinative help with Proposition~\ref{prop:lim-uncountable}.
\bibliographystyle{plain}
|
{
"timestamp": "2014-05-09T02:09:38",
"yymm": "1203",
"arxiv_id": "1203.1866",
"language": "en",
"url": "https://arxiv.org/abs/1203.1866"
}
|
\section{Introduction}
The Standard Model (SM) has proven to be an incredibly successful theory over the past decades.
However successful, it is an effective theory that must break down above a certain
energy scale, and there are strong theoretical arguments to believe that it breaks down
at the electroweak scale. For a theoretical review of this subject, see \cite{EWTheoryGautam}
and the contribution in this conference from the same author.
In 2011, operating at a center-of-mass energy of 7~TeV in $pp$ collisions, the LHC has been able
to deliver several fb$^{-1}$ of data to both ATLAS~\cite{ATLAS} and CMS~\cite{CMS} detectors within a few months,
allowing to extend the reach of searches for phenomena beyond the Standard Model well beyond the ones
carried by the TeVatron.
This article presents some of the searches carried by ATLAS and CMS using up to 1.6~fb$^{-1}$ of data
on supersymmetry and exotic signatures.
I will start with a summary of the searches for supersymmetry, followed by an overview of some
exotic searches, divided somewhat arbitrarily in three section: search for heavy resonances,
search for strong gravity at the TeV-scale, and search for long-lived particles.
Unfortunately no deviation from the SM expectation is observed, but limits on many theories beyond
the SM are improved significantly.
Searches related to Higgs boson, top-antitop resonance and fourth generation quarks are described in
other contributions of this conference~\cite{HiggsTalk, TopTalk, QCDTalk}.
Only a selection of results is shown here; all results can be found on the ATLAS~\cite{AtlasWeb}
and CMS~\cite{CMSWeb} web pages.
\section{Supersymmetry}
During the past decades, supersymmetry~\cite{SUSY1, SUSY2} has been considered the most promising extension of the SM.
The phenomenology of supersymmetry is very diverse, which requires a search strategy following
several classes of models and covering many signatures.
In its most hoped for incarnation, supersymmetry is expected to be discovered at the LHC through
pair production of supersymmetric particles decaying in a cascade of supersymmetric and SM particles.
If R-parity is conserved, the lightest supersymmetric particle (LSP) is stable and neutral, and
the cascade ends with the production of LSP's, which escape the detector, producing
missing transverse momentum.
In $pp$ collisions, strongly coupled particles are much more likely to be produced, thus the production
of squarks and gluinos is expected to dominate, leading predominantly to a final state with
jets and missing transverse momentum. The ``workhorse'' of supersymmetry searches at the LHC is thus
the channel with large missing transverse momentum and jets of high transverse momentum.
No excess above the expected SM background is observed and limits are set on supersymmetric models.
Figures~\ref{fig:ATLASSUSYS0lepton} and~\ref{fig:CMSSUSYSummary} show the limits from
ATLAS~\cite{ATLAS0lepton} and CMS~\cite{CMS0lepton}.
In addition to setting limits on the CMSSM/MSUGRA model, ATLAS also presents a limit for a simplified
model assuming only squark and gluino production, and a cascade involving only quarks and gluons,
and the LSP. For equal masses of squarks and gluinos, a limit of about 1~TeV is set at 95\% CL.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.45\columnwidth]{Plots/ATLAS_fig_02a.png}
\includegraphics[width=0.45\columnwidth]{Plots/ATLAS_fig_02b.png}
\caption{Limits on supersymmetric models from the 0-lepton channel at ATLAS~\cite{ATLAS0lepton}.
Left: simplified model assuming only squark and gluino production, and a light LSP.
Right: CMSSM/MSUGRA model.}
\label{fig:ATLASSUSYS0lepton}
\end{center}
\end{figure}
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\columnwidth]{Plots/CMS_SUSY_2011Limits_tanb10.pdf}
\caption{Summary of the searches for supersymmetry at CMS on CMSSM/MSUGRA in the 0-lepton,
1-lepton, and 2-lepton channel.}
\label{fig:CMSSUSYSummary}
\end{center}
\end{figure}
The cascade can also produce leptons through the decay of sleptons, charginos, or W/Z bosons.
Due to the smaller branching ratio, channels containing one~\cite{ATLAS1lepton, CMS1lepton}
or more electron or muon
are less sensitive to squark and gluino strong production, but are complementary
to the fully hadronic channel, as shown in Figure~\ref{fig:CMSSUSYSummary} for the CMS results.
In the di-lepton channel, several strategies are employed: opposite-sign~\cite{CMSOSlepton}
or same-sign~\cite{CMSSSlepton}, flavor subtraction~\cite{ATLAS2lepton}
to remove the flavor-correlated background, or explicit reconstruction of a Z produced in the
cascade and decaying to a pair of muons or electrons~\cite{CMSZMET}.
Of particular interest are scenarios in which the third generation of supersymmetric particles is
much lighter than the others. The current luminosity allows to test such scenarios only through
production of gluinos decaying to stop or sbottom, leading to a final state of top and/or bottom quarks.
Assuming that the stop is the only light squark,
gluino pair production leads to a complex final state containing top and bottom quarks.
Figure~\ref{fig:ATLASSUSY3G} (top) shows that this scenario is excluded for gluino masses
up to 500~GeV in the channel with one lepton and at least four jets, one of which identified as
a b-jet~\cite{ATLAS1lepton3G}.
Alternatively, if the only light squark is a sbottom, gluino pair production leads
to a final state with four b-jets and two LPS's; in this case, in the channel with at least 3 jets,
at least two of which identified as b-jets, gluino masses
are excluded up to 700~GeV, as shown on Figure~\ref{fig:ATLASSUSY3G} (bottom)~\cite{ATLAS0lepton3G}.
Additional luminosity will allow to search for direct production of third generation quarks and
gauginos.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\columnwidth]{Plots/ATLAS_SUSY_fig_06_bjets1lepton.png}
\includegraphics[width=0.7\columnwidth]{Plots/ATLAS_SUSY_fig_04_bjets.png}
\caption{Search for light third generation supersymmetric models.
Top: 1-lepton with at least one b-jet~\cite{ATLAS1lepton3G}.
Bottom: 0-lepton with b-jets~\cite{ATLAS0lepton3G}.}
\label{fig:ATLASSUSY3G}
\end{center}
\end{figure}
In gauge-mediated supersymmetry breaking (GMSB) models~\cite{GMSB}, the LSP is the gravitino and the next lightest
supersymmetric particle (NLSP) is a neutralino or a chargino. This leads to a cascade ending with
photons and missing transverse momentum in the final state.
CMS has looked for both single-photon and di-photon final states~\cite{CMSDiphotonMET}.
Results are shown in Figure~\ref{fig:CMSGMSB}. In the di-photon channel,
the result is also interpreted for the scenario of wino-like NLSP (neutralino
and chargino nearly degenerate in mass).
Universal Extra-Dimensions (UED) models~\cite{UED} predict cascades that are very similar to supersymmetry,
which allows to interpret the same analysis in both models~\cite{ATLASDiphotonMET}.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.53\columnwidth]{Plots/CMS_SUSY_Diphoton.png}
\includegraphics[width=0.45\columnwidth]{Plots/CMS_SUSY_SinglePhoton_Wino.png}
\caption{Search for GMSB in the single-photon plus missing transverse momentum and diphoton
plus missing transverse momentum final states~\cite{CMSDiphotonMET}. Left: di-photon, bino-like interpretation.
Right: single photon, wino-like interpretation.}
\label{fig:CMSGMSB}
\end{center}
\end{figure}
Supersymmetric signatures involving long-lived particles are discussed in the last section.
\section{Heavy resonances}
Heavy resonances are predicted by many extensions of the SM.
Some Grand Unified Theories~\cite{GUTE6} predict the existence of
additional gauge bosons while Randall-Sundrum models with warped
extra-dimensions~\cite{RS1, RS2} predict Kaluza-Klein excitations of the graviton.
Both lead to a narrow resonance decaying to a pair of fermions or bosons
with branching ratios varying widely depending on the model considered.
In the di-lepton channel (di-electron or di-muon)~\cite{ATLASZp, CMSZp},
a neutral gauge boson with the same couplings as the SM $Z^0$ (Sequential Standard Model Z'~\cite{SSM})
is excluded up to a mass of 1.9~TeV at 95\% CL. A Randall-Sundrum Kaluza-Klein graviton
with a coupling of $k/M_{Pl} = 0.1$ is excluded up to 1.8~TeV at 95\% CL combining the di-electron
and the di-muon channel,
and up to 1.7~TeV in the diphoton channel alone~\cite{CMSDiphoton}.
Figure~\ref{fig:zp} and figure~\ref{fig:diph-wp} (left) show the di-leptons and the di-photon mass spectra,
respectively.
A charged gauge boson ($W'$) is searched for in the $e\nu$ and $\mu\nu$ channels by reconstructing the transverse mass of
the lepton transverse momentum and the event missing transverse momentum.
Figure~\ref{fig:diph-wp} (right) shows the ATLAS transverse mass in $\mu\nu$ events.
A $W'$ with the same couplings as the SM $W$ (Sequential Standard Model W') is
excluded up to a mass of 2.3~TeV at 95\% CL when combining $e\nu$ and $\mu\nu$
channels~\cite{ATLASWp, CMSWp}.
A W' is also expected to decay to $WZ$, which is also a channel of interest for
Technicolor~\cite{TC} searches; CMS has looked for a narrow resonance in the final state
$WZ \rightarrow l\nu ll$ and excludes an SSM W' up a mass of 784~GeV and a techni-rho
up to a mass of 436~GeV in the parameter space $m_{\rho_{TC}} < m_{\pi_{TC}} + m_W$~\cite{CMSWpWZ}.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.4\columnwidth]{Plots/CMS_zprime_MuonsPlusMuonsMinus_log.pdf}
\includegraphics[width=0.55\columnwidth]{Plots/ATLAS_fig_01a_zprime.png}
\caption{Search for heavy resonances in the di-lepton channel.
Left: reconstructed di-muon mass spectrum (CMS)~\cite{CMSZp}.
Right: reconstructed di-electron mass spectrum (ATLAS)~\cite{ATLASZp}.}
\label{fig:zp}
\end{center}
\end{figure}
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.4\columnwidth]{Plots/CMS_Diphoton_invMass.png}
\includegraphics[width=0.54\columnwidth]{Plots/ATLAS_fig_01f_wprime.png}
\caption{
Left: Di-photon reconstructed mass spectrum (CMS)~\cite{CMSDiphoton}.
Right: reconstructed transverse mass in events with one muon and missing transverse momentum (ATLAS)~\cite{ATLASWp}.}
\label{fig:diph-wp}
\end{center}
\end{figure}
A narrow resonance decaying to a pair of jets is also predicted by numerous models.
Considering the excited quark model ($q^*$)~\cite{qstar} as a benchmark, no narrow resonance
in the di-jet system is observed up to 2.9~TeV at 95\% CL~\cite{ATLASdijet, CMSdijet}.
Figure~\ref{fig:dijet} shows the ATLAS model-independent limit on the cross-section for several hypotheses
of the resonance width, and the CMS limits on several models depending on the nature of the jet (quark jet or gluon jet).
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.4\columnwidth]{Plots/ATLAS_fig_03_dijet.png}
\includegraphics[width=0.4\columnwidth]{Plots/CMS_Dijets_c_xs_sys.png}
\caption{95\% CL upper limits on the production cross-section times acceptance of resonances decaying to
a pair of jets. Left: the limit is presented in a model-independent way as a function of
the full width (both physical and experimental) of the resonance (ATLAS)~\cite{ATLASdijet}.
Right: limits on narrow resonances of type gluon-gluon, gluon-quark, and quark-quark are compared to
various theoretical predictions (CMS)~\cite{CMSdijet}.}
\label{fig:dijet}
\end{center}
\end{figure}
A heavy particle decaying to a pair of charged leptons of same-sign, such as a doubly-charged Higgs,
would be a striking signature of physics beyond the SM. More generally, final states including a pair of charged leptons of same-sign are predicted
by many BSM models (including supersymmetry, same-sign top production, fourth generation b', heavy
Majorana neutrino, etc...) and enjoy a very small SM background.
Thus an inclusive search for same-sign di-lepton pair is very sensitive to a wide range of models,
and thanks to the small background is almost as sensitive as a search optimized for a particular model.
With 1.6~fb$^{-1}$ of integrated luminosity, ATLAS sets a
model-independent limit on the fiducial cross-section of isolated pairs of same-sign muons as
a function of the di-lepton pair mass~\cite{ATLASinclSS} as shown on Figure~\ref{fig:ATLASSameSign}.
The same mass spectrum is used to search for a narrow resonance,
allowing to exclude doubly-charged Higgs pair production up to a mass of 375~GeV in the
left-handed coupling triplet model~\cite{ATLASHpp}.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.57\columnwidth]{Plots/ATLAS_SSincl_fig_06.png}
\includegraphics[width=0.42\columnwidth]{Plots/ATLAS_SS_fig_01.png}
\caption{Left: model-independent fiducial cross section
limit on the production of same-sign di-muon pairs~\cite{ATLASinclSS}.
Right: reconstructed mass of same-sign muon pairs in ATLAS and expected
doubly-charged Higgs signal of various masses~\cite{ATLASHpp}.}
\label{fig:ATLASSameSign}
\end{center}
\end{figure}
\section{Strong gravity}
Theories of extra-dimension are a possible answer to the hierarchy problem.
In the large extra-dimension ADD model~\cite{ADD}, gravity is allowed to propagate into extra-dimensions,
thus appearing weak at (spatial) scales much larger than the scale of the extra-dimensions,
but possibly becoming strong at a scale of 1/TeV. The fundamental mass scale $M_D$
at which gravity becomes strong is related to the Planck scale via
$m^2_{Pl} = m^{2+n}_D R^n$ where $n$ is the number of extra-dimensions, and $R$ is the size of the
extra-dimension, and can indeed be close to the TeV scale for well-chosen values of $n$ and $R$.
A promising signature at colliders is the production of a single graviton
escaping the detector and recoiling against a jet or a photon, leading
to mono-jet~\cite{ATLASmonojet, CMSmonojet} or mono-photon~\cite{CMSmonophoton}
final states with large missing transverse momentum.
Figure~\ref{fig:mono} shows the missing transverse momentum spectrum in the ATLAS mono-jet (left)
and CMS mono-photon (right) analyses.
Thanks to a larger cross-section the mono-jet channel gives the most stringent limits,
excluding $M_D$ up to 3.7~TeV for $n=2$ and 2.3~TeV $n=6$ (conservatively assuming LO cross-sections).
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.56\columnwidth]{Plots/ATLAS_fig_06b_monojet.png}
\includegraphics[width=0.43\columnwidth]{Plots/CMS_monophoton_met_plot_pergev.png}
\caption{Missing transverse momentum in ATLAS mono-jet (left) and CMS mono-photon (right) analyses~\cite{ATLASmonojet, CMSmonojet}.}
\label{fig:mono}
\end{center}
\end{figure}
Another signature of ADD extra-dimensions is a non-resonant enhancement of expected di-lepton and di-photon events
at high invariant mass through virtual graviton exchange. CMS has searched for deviations in the
di-muon~\cite{CMSADDdimuon} and di-photon~\cite{CMSADDdiphoton} spectra, with a sensitivity similar to the monojet channel.
Finally, if gravity becomes strong at the TeV scale, microscopic black-holes may be produced
at the LHC. Due to our lack of understanding of quantum gravity, it is impossible to make precise predictions
of such phenomena. However one can expect such objects to decay democratically and isotropically,
leading to a final state with a large multiplicity of high-momentum particles, and a high content of leptons.
Several channels have been considered: multi-jet~\cite{ATLASblackholeMJ},
same-sign di-muon with a high track multiplicity~\cite{ATLASblackholeSS},
and multi-object~\cite{CMSblackhole} (where an object refers to an electron, a muon, a photon, or a jet, and
a large number of objects is required in the event).
In the latter case, CMS sets limits on black-hole masses up to 4-5~TeV for some classes of models.
Figure~\ref{fig:CMSbh} shows the $S_T$ variable, defined as the scalar sum of the transverse momentum of all objects
in the event, for events with at least six objects (left), and the limits achieved on the black-hole mass (right).
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.45\columnwidth]{Plots/CMS_bh_ST_Mul6up.pdf}
\includegraphics[width=0.45\columnwidth]{Plots/CMS_bh_CombinedMassLimits.pdf}
\caption{Multi-object search for microscopic black-holes at CMS~\cite{CMSblackhole}.
Left: Scalar sum of all objects transverse momentum in events with at least six objects.
Right: Limit on black-hole mass as a function of $M_D$, number of extra-dimensions,
for two black-hole models.}
\label{fig:CMSbh}
\end{center}
\end{figure}
\section{Long-lived particles}
Several extensions of the SM, including Hidden Valley models~\cite{HV}, and supersymmetry in some scenarios~\cite{split},
predict the existence of long-lived heavy particles.
In the case of supersymmetry, a long-lived gluino or squark hadronizes into hadronic states called R-hadrons.
The experimental signature depends strongly on the property of the particle, and in particular its life-time.
If the life-time is short (between 1 ps and several ns) the particle decays within the detector in time with the
collision that produced it; in this case it is possible to identify the decay thanks to dedicated vertexing~\cite{ATLASdisplaced}.
If the particle life-time is much longer than 1~ns there is no hope to see it decay in the detector. If the particle
is charged, it is possible to take advantage of the properties of a slow heavy particle and identify it thanks to
high energy loss in the tracking detectors and long time-of-flight~\cite{CMSslow}.
Alternatively, for a life-time up to about 1 month, if the particle is stopped within the detector, it is possible
to observe its decay long after the collision that produced it occurred~\cite{CMSoutoftime}.
Figure~\ref{fig:CMSLongLived} shows the limit on the production of long-lived scalar top stopping in the detector
and decaying out-of-time; the analysis is sensitive over 13 orders of magnitudes, from 100 ns to 1 month.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\columnwidth]{Plots/CMS_stopLifetime_stopped.png}
\caption{Search for out-of-time decay of heavy long-lived particles stopped in the detector:
limit on the cross section times branching ratio as a function of the stop life-time~\cite{CMSoutoftime}.}
\label{fig:CMSLongLived}
\end{center}
\end{figure}
\section{Conclusion}
The LHC has performed exceptionally well and has provided ATLAS and CMS with more luminosity than
expected. Many searches for physics beyond the SM have been conducted with up to 1.6~fb$^{-1}$, covering
a wide range of signatures. Unfortunately no deviation from the SM has been observed so far.
Supersymmetry in its most hoped-for incarnation is starting to be pushed
to the boarder of fine-tuning: in the framework of the CMSSM, supersymmetry is excluded up to a mass of 1~TeV in the
(optimistic) scenario of equal squark-gluino mass. This opens the field to variations of supersymmetry that require
more luminosity or new search strategies. Heavy gauge bosons are excluded up to masses of about 2~TeV, while
quark compositeness is tested up to 3~TeV.
\acknowledgments
I want to acknowledge the immense work of the LHC, ATLAS, and CMS collaborations that was
required to achieve the results presented here.
I am very grateful to the organizers for a very fruitful conference and for an unforgettable stay in Mumbai.
\bibliographystyle{pramana}
|
{
"timestamp": "2012-03-12T01:01:30",
"yymm": "1203",
"arxiv_id": "1203.2069",
"language": "en",
"url": "https://arxiv.org/abs/1203.2069"
}
|
\section{Introduction}
A typical manifestation of the reflection-asymmetric quadrupole-octupole
deformation in the energy spectra of even-even atomic nuclei is the formation
of level sequences with alternating parities \cite{BN96}. Usually the levels
with opposite parity are related through enhanced electric E1 and/or E3
transitions. The negative-parity sequence is shifted up with respect to the
positive-parity sequence due to a tunneling of the system between the two
opposite orientations along the principal symmetry axis. The magnitude of the
energy shift corresponds to the softness of the shape with respect to the
octupole deformation. The typical alternating-parity band is formed by the
members of the ground-state ($g$) band and the levels of the lowest
negative-parity sequence with odd angular momenta. In the relatively narrow
region of the light actinide nuclei Rn, Ra and Th these two sequences merge
into a single rotation band also called ``octupole band''
\cite{Th226,Th224,Cocks97}. The octupole band develops in the higher angular
momenta and indicates the appearance of a quite stiff octupole deformation.
Away from the light actinide region both sequences diverge and do not form a
single rotation band in the conventional meaning. Nevertheless, in some heavier
actinides, like U and Pu and some rare-earth isotopes like Nd, Sm, Gd and Dy,
they still remain related by E1 and E3 transitions, which indicates the
presence of a soft octupole mode in the collective motion. In this case the
term ``alternating-parity band (or spectrum)'' does not have the same strict
meaning as in the light actinide nuclei but simply refers to sequences of
levels with opposite parities which could be connected (coupled) through
electric transitions.
Various theoretical models have been developed over the years to explain and
describe the formation of alternating-parity (or octupole) bands in the stiff
and soft octupole regimes of coupling between the $g$-band and the lowest
negative-parity sequences in different nuclear regions
\cite{Krappe69}--\cite{Buck08}. Particularly, a collective model assuming
coherent quadrupole-octupole vibrations and rotations \cite{b2b3mod} was
applied to the nuclei $^{150}$Nd, $^{152}$Sm, $^{154}$Gd and $^{156}$Dy with
the presence of a soft octupole collectivity. Although the $g$-band and the
lowest negative-parity bands in these nuclei were successfully described as
members of an yrast alternating-parity band together with the attendant B(E1)
and B(E2) transition probabilities, a question arises about the validity of
such a consideration with respect to the higher-energy (non-yrast) part of the
spectrum.
The purpose of the present work is to clarify the above question within the
model of Coherent Quadrupole-Octupole Motion (CQOM) \cite{b2b3mod} by
examining the possible formation of non-yrast alternating-parity structures in
addition to the yrast band. For this reason the model scheme is extended by
assuming that the excited $\beta$-bands can be connected to higher
negative-parity sequences with odd angular momenta. Therefore, it is supposed
that the quadrupole-octupole structure of the spectrum develops along the
non-yrast regions of the energy spectrum. Such a study provides not only a test
of the model in the higher energy parts of the spectra, but also gives an
interpretation of a larger number of data that may guide the experimental
search for similar level structures in other nuclear regions. In principle, the
systematic analysis of the non-yrast levels with alternating parity may favour
different band-coupling schemes in the different nuclear regions allowing one
to compare the capabilities of various theoretical models. For example, an
extended study of non-yrast energy sequences with different parities has been
implemented within the Extended Coherent States Model \cite{RRR06} by
considering a coupling of the $\beta$ and $\gamma$ bands with respective bands
possessing the same spins but opposite parities, as well as a coupling between
$K^{\pi}=1^{+}$ and $K^{\pi}=1^{-}$ energy sequences. In the model scheme of
the present work the positive-parity $\beta$-band appears connected to a
negative-parity non-yrast sequence with odd angular momenta in the same way as
in the yrast alternating-parity configuration. This is a consequence of the
assumed mechanism of coupling between the quadrupole and octupole vibration
modes. Therefore, the present work suggests a different band-coupling scheme
and supposes a persistent role of the quadrupole-octupole motion in the forming
of the higher-energy (non-yrast) part of the spectrum. Of course, by developing
such an approach one should keep in mind the non-conventional meaning of the
term ``alternating-parity band'' mentioned above. Also, presently the CQOM
model is limited to excitations associated with the axial quadrupole and
octupole degrees of freedom. Therefore, the study is focused on the related
part of the collective spectrum, while other kinds of excitation modes as the
$\gamma$-vibrations remain beyond the present consideration.
The paper is organized as follows. In Sec. 2 the CQOM model is presented and
the model mechanism for the appearance of non-yrast alternating parity bands is
shown. Model expressions for reduced B(E1), B(E2) and B(E3) transitions in the
non-yrast spectra are given in Sec. 3. In Sec. 4 numerical results and
discussion on the application of the model to the nuclei of different regions
are given. Sec. 5 contains concluding remarks.
\section{Model of Coherent Quadrupole--Octupole Motion}
The CQOM model \cite{b2b3mod} is a particular realization of the more general
geometric concept of collective nuclear motion characterized by the
quadrupole-octupole shape deformations \cite{BN96}. The expansion of the
surface radius $R(\theta,\varphi)$, in polar coordinates, with respect to
spherical harmonics up to multipolarity $\lambda =3$ is given by
\begin{eqnarray}
R(\theta ,\varphi)=R_{0}\left[ 1+\sum_{\lambda
=2}^{3}\sum^{\lambda}_{\mu = -\lambda}\alpha_{\lambda \mu}
Y_{\lambda\mu}^{\ast}(\theta,\varphi)\right],
\end{eqnarray}
where $R_0$ is the spherical radius and $\alpha_{\lambda \mu}$ are the twelve
quadrupole and octupole collective coordinates in the laboratory frame. The
collective coordinates are transformed into a body-fixed frame
\begin{equation}
a_{\lambda\nu}=\sum_{\mu}\alpha_{\lambda\mu}D^{\lambda}_{\mu\nu}(\hat{\theta}),
\end{equation}
determined by the ``canonical'' quadrupole coordinates $a_0=a_{20}$ and
$a_2=a_{22}=a_{2-2}$ and the three Euler angles
$\hat{\theta}=(\theta_{1},\theta_{2},\theta_{3})$. The remaining seven octupole
coordinates $a_{3\mu}$ ($\mu =-3,...,3$) together with $a_0$ and $a_2$
determine the quadrupole-octupole shape of the nucleus. In the particular case
of axial symmetry the quadrupole-octupole deformation represents a pear-like
shape determined by the only non-zero coordinates $\beta_2\equiv a_0$ and
$\beta_3\equiv a_{30}$. The respective physical states of the nucleus in the
intrinsic (body-fixed) frame are characterized by the symmetrization group
D$_{\infty}$ which consists of arbitrary (infinite number) rotations about the
intrinsic $z$-axis and rotations about the axes perpendicular to $z$ through
the angle $\pi$. In principal the symmetrization group of the nucleus in the
intrinsic frame is determined by the rotations $g$ satisfying a set of
equations in the form $D^{\lambda}_{\mu\nu}(g)=0$, which in the case of axial
symmetry is $D^{\lambda}_{\mu 0}(g)=0$ for all $\mu \neq 0$ \cite{GSDD11}.
In the CQOM model \cite{b2b3mod} the geometric concept is implemented in the
limits of the axial symmetry. It is considered that the even--even nucleus can
oscillate with respect to the quadrupole $\beta_2$ and octupole $\beta_3$ axial
deformation variables, which are mixed through a centrifugal
(rotation-vibration) interaction. The collective Hamiltonian of the nucleus is
then taken in the form
\begin{eqnarray}
H_{qo}&=&-\frac{\hbar^2}{2B_2}\frac{\partial^2}{\partial\beta_2^2}
-\frac{\hbar^2}{2B_3}\frac{\partial^2}{\partial\beta_3^2}+
U(\beta_2,\beta_3,I) \ , \label{Hqo}
\end{eqnarray}
where
\begin{equation}
U(\beta_2,\beta_3, I)=\frac{1}{2}C_2{\beta_2}^{2}+
\frac{1}{2}C_3{\beta_3}^{2} +
\frac{X(I)}{d_2\beta_2^2+d_3\beta_3^2}\ ,
\label{Ub2b3I}
\end{equation}
with $X(I)=[d_0+I(I+1)]/2$. $B_2$ and $B_3$ are effective quadrupole and
octupole mass parameters and $C_2$ and $C_3$ are stiffness parameters for the
respective oscillation modes. The quantity ${\mathcal{J}}^{(\mbox{\scriptsize
quad+oct})}=(d_2\beta_2^2+d_3\beta_3^2)$ can be associated to the moment of
inertia of an axially symmetric quadrupole-octupole deformed shape
\cite{JPDav68} with $d_2$ and $d_3$ being inertia parameters. The energy
potential (\ref{Ub2b3I}) represents a two-dimensional surface determined by the
variables $\beta_2$ and $\beta_3$ with an angular-momentum-dependent repulsive
core at zero deformation (see Fig. 1 in \cite{b2b3mod}). The parameter $d_0$ in
the centrifugal factor $X(I)$ characterizes the repulsive core at $I=0$ and
determines the overall energy scale for the rotation part of the energy.
The model Hamiltonian (\ref{Hqo}) represents a D$_{\infty}$ invariant. Also, it
is important to remark that (\ref{Hqo}) corresponds to a class of
axial-symmetric Hamiltonians \cite{DD93}, \cite{DD95}, \cite{AQOA} whose
kinetic vibration parts are derived by ignoring the non-axial degrees of
freedom (e.g. $\gamma$-vibrations) in a way similar to the approach of Davidov
and Chaban \cite{DCh60}. The scalar product in the space of the wave functions
(e.g. see Eqs. (2) and (4) in \cite{AQOA}) corresponding to the particular form
of the $\beta_2$- and $\beta_3$-derivatives in (\ref{Hqo}) is characterized by
a unit weight factor, i.e. $\langle\Phi_2|\Phi_1\rangle=\int\int
d\beta_2d\beta_3 \Phi_2^\ast (\beta_2 ,\beta_3) \Phi_1(\beta_2 ,\beta_3)$.
If a condition for the simultaneous presence of nonzero coordinates
$(\beta_{2}^{\mbox{\scriptsize min}}, \beta_{3}^{\mbox{\scriptsize min}})$ of
the potential minimum is imposed, the stiffness and inertial parameters are
correlated as $d_2/C_2=d_3/C_3$ (see Eqs. (3)--(6) in \cite{b2b3mod}). In this
case the potential bottom represents an ellipse in the space of $\beta_2$ and
$\beta_3$ which surrounds the infinite zero-deformation core (see Fig. 3 in
\cite{b2b3mod}). If prolate quadrupole deformations $\beta_2>0$ are considered,
the system is characterized by oscillations between positive and negative
$\beta_3$-values along the ellipse surrounding the potential core. By
introducing polar-type of curvilinear or, more precise, ellipsoidal variables
\begin{eqnarray} \eta=\left[
\frac{2(d_2\beta_2^2+d_3\beta_3^2)}{d_2+d_3}\right]^{\frac{1}{2}} \qquad
\mbox{and}\qquad \phi=\arctan\left( {\frac{\beta_3}{\beta_2}
\sqrt{\frac{d_3}{d_2}}}\right )\ , \nonumber
\end{eqnarray}
such that
\begin{eqnarray}
\beta_{2}=p\eta\cos\phi, \qquad \beta_{3}=q\eta\sin\phi,
\label{polar}
\end{eqnarray}
with
\begin{eqnarray}
p=\sqrt{\frac{d}{d_{2}}},\qquad q=\sqrt{\frac{d}{d_{3}}}\qquad \mathrm{and}
\qquad d=\frac{1}{2}(d_{2}+d_{3}),
\label{pqd}
\end{eqnarray}
the potential (\ref{Ub2b3I}) appears in the form
\begin{eqnarray}
U_{I}(\eta)=\frac{1}{2}C\eta^2+\frac{X(I)}{d\eta^2}\ ,
\label{cpotenc}
\end{eqnarray}
where $C=(d/d_2)C_2=(d/d_3)C_3$.
Further, it is assumed that the quadrupole and octupole modes are represented
in the collective motion with the same oscillation frequencies
$\omega_{2}=\omega_{3}=\omega$, with
\begin{eqnarray}
\omega=\sqrt{\frac{C_2}{B_2}}=\sqrt{\frac{C_3}{B_3}}\equiv \sqrt{\frac{C}{B}}.
\label{om}
\end{eqnarray}
The condition (\ref{om}) imposes certain correlations between the mass,
stiffness and inertia parameters of the model Hamiltonian (\ref{Hqo}),
corresponding to a coherent quadrupole-octupole motion of the system. Note that
here the term ``coherent'' is used in the context of the mixing between the
quadrupole and octupole degrees of freedom, which is different from the meaning
of the same term used in \cite{RRR06}. In this case the Hamiltonian is obtained
in a simple form
\begin{eqnarray}
H_{qo}&=&-\frac{\hbar^2}{2B}\left[\frac{\partial^2}{\partial\eta^2}+
\frac{1}{\eta }\frac{\partial}{\partial \eta }+ \frac{1}{\eta
^2}\frac{\partial^2}{\partial \phi^2} \right] +U_{I}(\eta
) \ . \label{Hqo02}
\end{eqnarray}
It allows an exact separation of variables in the wave function $\Phi(\eta
,\phi)=\psi(\eta )\varphi(\phi)$ with the subsequent equations for $\psi(\eta
)$ and $\varphi(\phi)$
\begin{eqnarray}
\frac{\partial^2}{\partial \eta ^2}\psi(\eta )+ \frac{1}{\eta
}\frac{\partial}{\partial \eta }\psi(\eta )
+\frac{2B}{\hbar^2}\left[ E-\frac{\hbar^2} {2B}\frac{k^2}{\eta
^2}-U_{I}(\eta )\right] \psi (\eta )&=&0 \ ; \label{Hqo03} \\
\frac{\partial^2}{\partial
\phi^2}\varphi(\phi)+k^2\varphi(\phi)&=&0 \ , \label{wphi}
\end{eqnarray}
where $k$ is the separation quantum number. Eq. (\ref{Hqo03}) with the
potential (\ref{cpotenc}) is similar to the equation for the Davidson potential
\cite{Dav32} and has the following analytic solution for the energy spectrum
\cite{b2b3mod}
\begin{equation}
E_{n,k}(I) =\hbar\omega \left[ 2n+1+\sqrt{k^2+bX(I)}\right],
\label{enspect}
\end{equation}
where $\omega$ is defined in (\ref{om}), $n=0,1,2,...$ and $b=2B/(\hbar^2 d)$.
The quantum numbers $n$ and $k$ have the meaning of ``radial'' and ``angular''
oscillation quantum numbers, respectively. The normalized ``radial''
eigenfunctions $\psi(\eta )$ are obtained in terms of the generalized Laguerre
polynomials
\begin{equation}
\psi^I_{n,k}(\eta )=\sqrt
{\frac {2c\Gamma(n+1)}{\Gamma(n+2s+1)}}
e^{-c\eta^2/2}(c\eta^{2})^sL^{2s}_n(c\eta^2)\ , \label{psieta1}
\end{equation}
with $c=\sqrt{BC}/\hbar$ and $s=(1/2)\sqrt{k^2+bX(I)}$. Eq.~(\ref{wphi}) in the
``angular'' variable $\phi$ is solved under the boundary condition
$\varphi(-\pi /2)=\varphi(\pi /2)=0$. This is equivalent to the consideration
of an infinite potential wall at $\beta_2=0$ (or $\varphi =\pm\pi /2$). Then
one has two identical solutions for $\beta_2>0$ and $\beta_2<0$. As mentioned
above the physical space of the model is taken in the prolate $\beta_2>0$ half
of the $(\beta_2,\beta_3)$-plane. (See Figs. 4 and 5 in \cite{b2b3mod} and the
related text in that reference). Within this half-plane Eq.~(\ref{wphi}) has
two different solutions with positive and negative parities, $\pi =(+)$ and
$\pi=(-)$, respectively
\begin{eqnarray}
\varphi_{k}^{+}(\phi)&=& \sqrt{2/\pi}\cos (k\phi ) \ , \qquad k=1, 3, 5,
...\ ,\label{parplus} \\
\varphi_{k}^{-}(\phi)&=& \sqrt{2/\pi}\sin (k\phi ) \ , \qquad k=2, 4, 6,
...\ . \label{parminus}
\end{eqnarray}
Note that the square root term in the wave function $\psi^I_{n,k}(\eta )$,
Eq.~(\ref{psieta1}), differs from the respective term used in Eq.~(24) of
Ref.~\cite{b2b3mod} by the factor $c$ which is newly included in the numerator.
(In \cite{b2b3mod} the quantity $c$ is denoted by `$a$' which in the case of
odd nuclei leads to confusion with the notation for the decoupling factor.) One
can easily check that this factor is necessary to normalize $\psi^I_{n,k}(\eta
)$ to unity. The results for the transition probabilities obtained in
\cite{b2b3mod} are not affected by the missing factor $c$ due to the use of
overall scaling factors in Eqs.~(46) and (47) of \cite{b2b3mod}.
Since the consideration is restricted to axial deformations only, the
projection $K$ of the collective angular momentum on the principal symmetry
axis is taken zero. Then the total wave function of the coherent
quadrupole-octuple vibration and collective rotation of an even-even nucleus
has the form
\begin{eqnarray}
\Psi^{\pi}_{nkIM0}(\eta ,\phi)= \sqrt{\frac{2I+1}{8\pi^2}} D^{I}_{M\,0}(\theta )
\Phi^{\pi}_{nkI} (\eta,\phi)
\ , \label{wftot}
\end{eqnarray}
where $D^{I}_{M\,0}(\theta )$ is the Wigner function defined according to the
phase convention in \cite{BM75}. Note that due to a different phase convention
in some other works, e.g. in \cite{EG87} and \cite{RS80}, the complex
conjugated $D$-function appears in the rotation part. The relations between the
different definitions of the $D$-function are given in Table 4.2 in
\cite{VMK88}. The quadrupole-octupole vibration part of (\ref{wftot}) is
\begin{eqnarray}
\Phi^{\pi}_{nkI} (\eta,\phi)= \psi_{nk}^{I}(\eta )\varphi^{\pi}_{k}(\phi)\ .
\label{wvib}
\end{eqnarray}
The quantum numbers of the quadrupole-octupole vibration function (\ref{wvib})
are determined by the requirement for a conservation of the
$\mathcal{R}\mathcal{P}$-symmetry of the total wave function (\ref{wftot}).
($\mathcal{P}$ is the parity operator and $\mathcal{R}$ represents a rotation
by an angle $\pi$ about an axis perpendicular to the intrinsic $z$-axis) The
$\mathcal{R}$-symmetry of the rotation function $D^{I}_{M\,0}(\theta )$ is
characterized by the factor $(-1)^I$, while the action of $\mathcal{P}$ on
$\Phi^{\pi}_{nkI} (\eta,\phi)$ gives the factor $\pi =\pm$. Then the
conservation of the $\mathcal{R}\mathcal{P}$-symmetry is equivalent to the
conservation of the so called simplex quantum number $simplex=\pi (-1)^I=1$.
This condition imposes a positive parity for the states with even angular
momentum, and negative parity for the odd angular momentum states, i.e. one has
\begin{eqnarray}
\Phi^{+}_{nkI} (\eta,\phi)&=& \psi_{nk}^{I}(\eta )\varphi^{+}_{k}(\phi)\
\mbox{for}\ I=\mbox{even}\nonumber\\
\Phi^{-}_{nkI} (\eta,\phi)&=& \psi_{nk}^{I}(\eta )\varphi^{-}_{k}(\phi)\
\mbox{for}\ I=\mbox{odd}\nonumber .
\end{eqnarray}
It should be noted that the above conditions are in conjunction with the
transformation properties of the variables $\eta$ and $\phi$ in (\ref{wvib})
under the rotation $\mathcal{R}$ ($\eta$ is invariant, while $\phi$ changes in
sign) so that together with the simplex conservation condition the total wave
function (\ref{wftot}) appears to be an D$_{\infty}$ invariant as it should be
due to the axial symmetry.
The structure of the energy spectrum is determined by the oscillator quantum
numbers $n$ (``radial'') and $k$ (``angular'') in Eq.~(\ref{enspect}). Since
according to Eqs.~(\ref{parplus}) and (\ref{parminus}) $k$ obtains different
values for the states with opposite parity the energy sequences with even and
odd angular momenta corresponding to a given $n$ appear shifted to each other,
i.e. a parity shift effect is observed. In \cite{b2b3mod} it was supposed that
the $g$-band and the lowest negative-parity band belong to a $n=0$ set with
$k=k^{(+)}=1$ for $g$ and $k=k^{(-)}=2$ for the negative-parity band. In the
present work the model scheme is extended through the following three
suppositions.
i) The energy spectrum determined by the coherent axial quadrupole-octupole
vibrations and rotations consists of couples of level-sequences with opposite
parity. The sequences in each couple are characterized by the same value of the
quantum number $n=0,1,2,...$ and by different values of $k$,
$k=k_n^{(+)}=1\,\mbox{or}\,3\,\mbox{or}\,5\,\mbox{or}\,...$ for the even-$I$
sequence, and $k=k_n^{(-)}=2\,\mbox{or}\,4\,\mbox{or}\,6\,\mbox{or}\,...$ for
the odd-$I$ sequence.
ii) The lowest values of the ``radial'' quantum number $n$ correspond to the
lowest alternating parity bands, with $n=0$ being the yrast band, $n=1$
corresponding to the next non-yrast alternating parity structure and so on. The
values of the ``angular'' quantum number $k$ are not restricted and should only
satisfy the parity condition in i). The particular values of $k_n^{(+)}$ and
$k_n^{(-)}$ can be determined so as to reproduce the experimentally observed
parity shift in the set of levels with a given $n$.
iii) Due to the coherent interplay between the $\beta_2$ and $\beta_3$
variables in the oscillation motion, the excited $\beta$-bands in even-even
nuclei can be interpreted as the positive-parity counterparts of higher
negative-parity sequences, or as the members of non-yrast alternating-parity
bands.
Based on the above assumptions the extended alternating-parity spectrum of an
even-even nucleus can be considered in the following form.
\smallskip
\noindent Yrast alternating-parity set $(n=0)$: unites the $g$-band
$(k=k_{0}^{(+)})$ $I_{\nu}^{\pi}=0_{1}^{+},2_{1}^{+},4_{1}^{+}, 6_{1}^{+},...$
with the first negative-parity band denoted here as $n1$ $(k=k_{0}^{(-)})$
$I_{\nu}^{\pi}=1_{1}^{-},3_{1}^{-},5_{1}^{-},...;$
\smallskip
\noindent First non-yrast set $(n=1)$: unites the first $\beta$-band denoted by
$b1$ $(k=k_{1}^{(+)})$ $I_{\nu}^{\pi}=0_{2}^{+},2_{2}^{+},4_{2}^{+},...$ with
the second negative-parity band denoted by $n2$ $(k=k_{1}^{(-)})$
$I_{\nu}^{\pi}=1_{2}^{-},3_{2}^{-},5_{2}^{-},...;$
\smallskip
\noindent Second non-yrast set $(n=2)$: unites the second $\beta$-band $b2$
$(k=k_{2}^{(+)})$ $I_{\nu}^{\pi}=0_{3}^{+},2_{3}^{+},4_{3}^{+},...$ with the
third negative-parity band $n3$ $(k=k_{2}^{(-)})$
$I_{\nu}^{\pi}=1_{3}^{-},3_{3}^{-},5_{3}^{-},...$, and so on, higher non-yrast
sequences, where $\nu=1,2,3,...$ is the consequent number of the appearance of
a state with a given angular momentum. Also, it is convenient to use the band
labels introduced above to denote the different excited states as for example
$2_{g}^{+}$, $1_{n1}^{-}$, $0_{b1}^{+}$, $1_{n2}^{-}$ etc.
Obviously the above model scheme makes no claim to exhaust the entire
collective spectrum but rather provides a tool to identify the extent to which
the considered quadrupole-octupole motion can influence the excited band
structures in even-even nuclei. In the end of this section, it should be
remarked that the extension of the model to higher energy levels, together with
assumption ii), which releases $k$ from the fixed values $k^{(+)}=1$ and
$k^{(-)}=2$ (originally imposed in \cite{b2b3mod} for the yrast case), now
requires a new readjustment of the model parameters.
\section{Transition probabilities in the non-yrast quadrupole-octupole states}
As the B(E1) and B(E3) reduced transition probabilities are known to provide a
sensitive test for the structure of the alternating-parity sequences it is of
special importance to examine their behaviour in the non-yrast part of the
spectrum. The basic CQOM concept for the electromagnetic transitions has been
given in \cite{b2b3mod}. Here the formalism is modified so as to describe
B(E1), B(E2) and B(E3) reduced transition probabilities in the higher lying
alternating-parity bands along with the extended treatment of the model energy
quantum numbers. A more essential modification is related to a generalization
of the angular part of the electric transition operators dictated by the
complicated quadrupole-octupole shape density distribution inherent for the
coherent motion mode (see below). In addition the E1-E3 charge factors are
treated explicitly and the model parameters $p$ and $q$ (\ref{pqd}) providing
information about the potential shape are considered without including them
into scaling constants.
The reduced transition probability for an electric transition with a given
multipolarity $\lambda$ between model states (\ref{wftot}) with $n=n_i$,
$k=k_i$, $I=I_i$ and $n=n_f$, $k=k_f$, $I=I_f$ is
\begin{eqnarray}
B(E\lambda;n_i k_i I_{i}\rightarrow n_f k_f I_{f}) =
\frac{1}{2I_{i}+1}\sum_{M_{i}M_{f}\mu}\left| \left\langle
\Psi^{\pi_f}_{n_fk_fI_fM_f0}(\eta ,\phi)|\mathcal{M}_{\mu}(E\lambda)
|\Psi^{\pi_i}_{n_ik_iI_iM_i0}(\eta ,\phi) \right\rangle \right |^{2}.
\label{betrangen}
\end{eqnarray}
The operators for electric E1, E2 and E3 transitions have the following general
form
\begin{eqnarray}
\mathcal{M}_{\mu}(E\lambda)=\sqrt{\frac{2\lambda +1}
{4\pi (4-3\delta_{\lambda,1})}} \hat{Q}_{\lambda 0} D^{\lambda}_{0\mu},
\ \ \lambda=1,2,3,\ \ \mu=0,\pm 1, ..., \pm\lambda.
\end{eqnarray}
The vibration parts of these operators are given up to the first order of
$\beta_{2}$ and $\beta_{3}$, for E2 and E3, and in second order, for E1, as
\begin{eqnarray}
\hat{Q}_{1 0}&=&M_{1}\beta_{2}\beta_{3}
\label{q10} \\
\hat{Q}_{\lambda 0} &=& M_{\lambda}\beta_{\lambda}, \ \ \ \
\lambda = 2,3 \label{qL0}.
\end{eqnarray}
The electric charge factors $M_{\lambda}$ $(\lambda = 2,3)$ are taken as
\cite{LC88}
\begin{eqnarray}
M_{\lambda}=\frac{3}{\sqrt{(2\lambda +1)\pi}}ZeR_{0}^{\lambda},
\qquad \lambda = 2,3 \ ,
\label{multcrg}
\end{eqnarray}
where $R_{0}=r_0A^{1/3}$, $r_0\approx 1.2$ fm, $Z$ is the proton number, and
$e$ is the electric charge of the proton. The charge factor $M_{1}$ is taken
according to the droplet model concept \cite{MS74}-\cite{DMS86} in the form
\cite{DD95}
\begin{eqnarray}
M_{1}=\frac{9AZe^{3}}{56\sqrt{35}\pi}
\left( \frac{1}{J}+\frac{15}{8QA^{\frac{1}{3}}}\right ),
\label{m1d}
\end{eqnarray}
where the quantities $J$ and $Q$ are related to the volume and surface symmetry
energy, respectively and their values are assumed in the limits $25\lesssim
J\lesssim 44$ MeV and $17\lesssim Q\lesssim 70$ MeV \cite{BN91} (see also the
values below Eq.~(79) in \cite{DD95}). In the present work fixed average values
of these quantities $ J=35$ MeV and $Q=45$ MeV are used for all considered
nuclei. One should remark that so far there is no unique approach to estimate
the factor $M_{1}$. Therefore, here in (\ref{m1d}) the proton charge $e$ is
replaced by an effective charge $e^{1}_{eff}$ which is considered as an
adjustable parameter and can be different from one. Note that to obtain the
B(E1) transition probabilities in the units $e^2\cdot \mbox{fm}^2$, and
subsequently in Weisskopf units one has to take into account that
$e^2=1.4399764\, \mbox{MeV}\cdot \mbox{fm}$ [or
$e^6/\mbox{MeV}^2=(1.4399764)^2e^2\cdot \mbox{fm}^2$] which leads to an
additional multiplication factor 1.4399764 in (\ref{m1d}) when numerical values
are produced.
In the space of the ellipsoidal coordinates (\ref{polar}), (\ref{pqd}) one has
\begin{eqnarray}
\hat{Q}_{1 0}&=&M_{1}pq\eta^{2}\cos\phi\sin\phi \label{q1cs}\\
\hat{Q}_{2 0}&=&M_{2}p\eta\cos\phi \label{q2c}\\
\hat{Q}_{3 0}&=&M_{3}q\eta\sin\phi \label{q3s}.
\end{eqnarray}
The definitions of the operators (\ref{q10})--(\ref{qL0}) and
(\ref{q1cs})--(\ref{q3s}) originally correspond to a situation in which the
nuclear shape is characterized by fixed values of the deformation parameters
$\beta_{2}$ and $\beta_{3}$. In this case the density distribution of the
collective state is characterized by a single maximum in the space of
$\beta_{2}$ and $\beta_{3}$. In the case of the model potential (\ref{Ub2b3I})
taken with an elliptic bottom the density distribution can be characterized by
more than one maximum. Indeed, the density of the model state (\ref{wvib}) is
characterized by a different number of maxima depending on the quantum number
$k$. This feature is a result of the assumed soft quadrupole--octupole mode. It
is illustrated graphically in Appendix A, where the density distribution of the
state (\ref{wvib}) in the space of the quadrupole-octupole deformations is
plotted for different $k$-values at $n=0$ after transforming the wave function
$\Phi^{\pi}_{nkI}$ in the ($\beta_{2}$,$\beta_{3}$) variables. It is seen that
for $\beta_{2}>0$ the number of maxima is equal to $k$ and by analogy with the
acoustics may be interpreted as the number of ``overtones'' which characterize
the coherent collective oscillations of the system. Thus, it appears that the
transition operators should connect states with different numbers of maxima (or
overtones). In the space of ellipsoidal variables the positions of the maxima
are determined by the angular variable $\phi$. On the other hand the original
operators (\ref{q1cs})--(\ref{q3s}) do not take into account the presence of
multiple maxima in the shape density distributions of the different states. One
particular effect due to this circumstance is that the integrals over the
angular part of (\ref{q3s}), $\sin\phi$, vanish when the difference between the
$k$ numbers of the initial and final states is larger than a unit and the
respective B(E3) transition probabilities vanish too. This limitation is
removed if the operators are generalized appropriately. The most general forms
of the angular parts of the operators corresponding to the first orders of
$\beta_{2}$ and $\beta_{3}$ according to (\ref{q2c}) and (\ref{q3s}) can be
sought in terms of a Fourier expansion with respect to $\phi$ through the
replacements
\begin{eqnarray}
\cos\phi\rightarrow A_{2 0}(\phi)\equiv \sum_{k=1}^\infty a^{(k)}_{2 0}
\cos (k\phi),\qquad \sin\phi\rightarrow A_{3 0}(\phi)\equiv
\sum_{k=1}^\infty a^{(k)}_{3 0} \sin (k\phi ),
\end{eqnarray}
where the expansion coefficients should be chosen so as to provide an
appropriate convergence. A choice made here for both type of coefficients is
$a^{(k)}=1/k$ for which the above expansions can be obtained in analytic form
\begin{eqnarray}
A_{2 0}(\phi)&=&\sum_{k=1}^\infty \frac {\cos (k\phi )}{k}
=-\frac{1}{2}[\ln 2 +\ln(1-\cos \phi)] \label{a20}\\
A_{3 0}(\phi)&=&\sum_{k=1}^\infty \frac {\sin (k\phi )}{k}
=\frac{\pi- \phi}{2} + \pi \mbox{Floor}\left(\frac{\phi}{2\pi}\right),
\label{a30}
\end{eqnarray}
where the Floor function maps a real number to the largest previous integer.
Then the angular part of the second order operator (\ref{q1cs}) can be
generalized in an obvious way
\begin{eqnarray}
\cos\phi\sin\phi\rightarrow A_{1 0}(\phi)\equiv
A_{2 0}(\phi)A_{3 0}(\phi)=
\sum_{m=1}^\infty\sum_{n=1}^\infty \frac {\cos (m\phi )}{m}
\frac{\sin (n\phi )}{n}.
\label{a10}
\end{eqnarray}
Note that the first terms of the above expansions represent the original
angular parts in (\ref{q1cs})--(\ref{q3s}). So, the new angular operators
(\ref{a20})--(\ref{a10}), which are extensions of the old ones, provide a
connection between states whose ``dynamical'' deformations (i.e. the
probability distribution in the deformation space) are characterized by the
co-existence of a large number of maxima. These specific shape properties of
the system are due to the assumed coupling between quadrupole and octupole
degrees of freedom. Now the operators (\ref{q1cs})--(\ref{q3s}) are redefined
as
\begin{eqnarray}
\hat{Q}_{1 0}(\eta ,\phi)&=& M_{1}pq\eta^{2} A_{1 0}(\phi) \label{q1gen}\\
\hat{Q}_{2 0}(\eta ,\phi)&=& M_{2}p\eta A_{2 0}(\phi)\label{q2gen}\\
\hat{Q}_{3 0}(\eta ,\phi)&=& M_{3}q\eta A_{3 0}(\phi)\label{q3gen}.
\end{eqnarray}
After carrying out the integration over the rotation part in (\ref{betrangen})
one obtains
\begin{eqnarray}
B(E\lambda;n_i k_i I_{i}\rightarrow n_f k_f I_{f}) =
\frac{2\lambda +1}{4\pi (4-3\delta_{\lambda,1})}
\langle I_i0\lambda 0|I_f0\rangle^2
R_{\lambda}^{2}(n_i k_i I_{i}\rightarrow n_f k_f I_{f}),
\end{eqnarray}
which involves the squares of the Clebsch-Gordan coefficient and the matrix
element of the electric multipole operators (\ref{q1gen})-(\ref{q3gen}) between
the quadrupole-octupole vibration wave functions (\ref{wvib})
\begin{eqnarray}
R_{\lambda}(n_i k_i I_{i}\rightarrow n_f k_f I_{f})
=\left\langle \Phi^{\pi_f}_{n_f k_f I_f} (\eta,\phi) |\hat{Q}_{\lambda 0} |
\Phi^{\pi_i}_{n_i k_i I_i}(\eta,\phi) \right\rangle.
\label{rlam}
\end{eqnarray}
By further separating the integrations over the ``radial'' variable $\eta$ and
the ``angular'' variable $\phi$ in (\ref{rlam}) according to (\ref{wvib}) one
obtains
\begin{eqnarray}
R_{1}(n_i k_i I_{i}\rightarrow n_f k_f I_{f})
&=&M_{1}pqS_{2}(n_i,I_i;n_f,I_f)I_{1}^{\pi_i,\pi_f}(k_i,k_f) \label{rlam1}\\
R_{2}(n_i k_i I_{i}\rightarrow n_f k_f I_{f})
&=& M_{2}pS_{1}(n_i,I_i;n_f,I_f)I_{2}^{\pi_i,\pi_f}(k_i,k_f) \label{rlam2}\\
R_{3}(n_i k_i I_{i}\rightarrow n_f k_f I_{f})
&=& M_{3}qS_{1}(n_i,I_i;n_f,I_f)I_{3}^{\pi_i,\pi_f}(k_i,k_f),
\label{rlam3}
\end{eqnarray}
where
\begin{eqnarray}
S_{1}(n_i,I_i;n_f,I_f)&=&\int_0^{\infty} d\eta
\psi_{n_f}^{I_f}(\eta)\eta^{2} \psi_{n_i}^{I_i}(\eta)
\label{s1} \\
S_{2}(n_i,I_i;n_f,I_f)&=&\int_0^{\infty} d\eta
\psi_{n_f}^{I_f}(\eta)\eta^3 \psi_{n_i}^{I_i}(\eta),
\label{s2}
\end{eqnarray}
and
\begin{eqnarray}
I_{\lambda}^{\pi_i,\pi_f}(k_i,k_f)&=&\frac{2}{\pi}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}
A_{\lambda 0}(\phi)\varphi^{\pi_f}_{k_f}(\phi)\varphi^{\pi_i}_{k_i}(\phi)d\phi ,
\ \ \ \lambda =1,2,3.
\label{Ilam}
\end{eqnarray}
The integrals over $\eta$, (\ref{s1}) and (\ref{s2}), involve the ``radial''
wave functions (\ref{psieta1}). Analytic expressions for these integrals are
given in Appendix B. The integrals over $\phi$ (\ref{Ilam}) involve the
``angular'' wave functions (\ref{parplus}) and/or (\ref{parminus}). The
explicit forms of these integrals with the relevant parities $\pi_{i}$ and
$\pi_{f}$ are given in Appendix C.
From the generalized definitions (\ref{q1gen})-(\ref{q3gen}) of the operators
$\hat{Q}_{\lambda 0}$ it is seen that the inertial factors $p$, $q$ and their
product $pq$ defined through Eq.~(\ref{pqd}) are not included in any scaling
factors, as done in Ref.~\cite{b2b3mod} and can be considered as model
parameters. Actually, $p$ and $q$ are not independent. From (\ref{pqd}) it can
be easily seen that $1/p^{2}+1/q^{2}=2$. Then $q$ can be expressed by $p$ as
\begin{eqnarray}
q=\frac{p}{\sqrt{2p^{2}-1}}, \qquad \mathrm{with}
\qquad p>\frac{1}{\sqrt{2}}\approx 0.7071.
\label{q}
\end{eqnarray}
The inequality in (\ref{q}) corresponds to the condition $d_2<2d$. Analogically
one can express $p$ by $q$ with the condition $d_3<2d$. (Note that for $p=1 $
one has $q=1$ and $pq=1$.) Here the adjustable parameter is chosen to be $p$.
It should be noted that with the involvement of the new parameter $p$ the
scaling factors in Eqs.~(46) and (47) of Ref.~\cite{b2b3mod} are not considered
anymore and the charge factors $M_2$ and $M_3$ are directly calculated in
(\ref{multcrg}). Also the charge factor $M_1$ is directly calculated in
(\ref{m1d}), but with the effective charge $e^{1}_{eff}$ being adjusted to
determine the correct scale of the B(E1) transition probabilities with respect
to B(E2). From another side the parameter $p$ determines the relative scale
between B(E2) and B(E3). It is interesting to remark that $p$ does not play any
role if the model energy levels are fitted without taking into account
transition probabilities. However, in this case there is an ambiguity in the
choice of the inertial parameters $d_2$ and $d_3$. This is seen by the
following relations between the parameters of the potential (\ref{Ub2b3I}) and
the fitting parameters $\omega$ and $b$, imposed by the coherent condition
(\ref{om})
\begin{eqnarray}
C_2&=& \frac{\omega^{2}b}{2}d_2, \qquad C_3= \frac{\omega^{2}b}{2}d_3
\label{C2C3}\\
B_2&=& \frac{b}{2}d_2, \qquad\ \ \ B_3= \frac{b}{2}d_3,
\label{B2B3C2C3}
\end{eqnarray}
in which $d_2$ and $d_3$ are not determined. (The parameter $d_0$ does not
enter these relations.) This means that for a given set of parameters $\omega$,
$b$ and $d_0$ the energy spectrum corresponds to an infinite number of
potential shapes with different eccentricities of the ellipsoidal bottom. Now,
after determining the parameters $p$ and $c$ with respect to transition data
one gets
\begin{eqnarray}
d_2=\frac{d}{p^{2}}, \qquad d_3=(2p^{2}-1)\frac{d}{p^{2}},
\qquad \mathrm{with} \qquad d=\frac{2c}{\omega b}.
\label{d2d3}
\end{eqnarray}
Thus for given values of the parameters $\omega$, $b$, $c$ and $p$ the original
parameters $B_2$, $B_3$, $C_2$, $C_3$, $d_2$ and $d_3$ of potential
(\ref{Ub2b3I}) are fixed, and given additionally $d_0$, its form is
unambiguously determined.
\section{Numerical results and discussion}
The extended CQOM formalism was applied to several nuclei, namely
$^{152,154}$Sm, $^{154,156,158}$Gd, $^{236}$U and $^{100}$Mo, in which one or
two non-yrast alternating-parity bands can be constructed by the experimentally
observed $\beta$ and higher-lying negative-parity levels. In these nuclei a
number of data on E1 and/or E3 transitions are available providing the
possibility to test the complete model scheme. In all selected nuclei the
experimental data \cite{ensdf} provide well determined yrast and first
non-yrast alternating-parity bands except for $^{100}$Mo where the structure of
the non-yrast band is proposed here on the basis of the model analysis (see
below). In three of the nuclei, $^{154}$Sm, $^{154}$Gd and $^{158}$Gd second
excited (non-yrast) alternating-parity bands are additionally considered. The
structure of these bands is not clearly determined in the experimental data.
Therefore, the model description and prediction provides a possible
interpretation of the respective experimental levels. In this meaning the
present description not only provides a test for the CQOM model scheme, but
also suggests a possible classification of some highly non-yrast excited states
whose interpretation in the experimental data bases is not unambiguous.
The model description is obtained by taking the theoretical energy levels
$\tilde{E}_{n,k}(I) =E_{n,k}(I)-E_{0,k_{0}^{(+)}}(0)$ from Eq.~(\ref{enspect}).
The parameters $\omega$, $b$, $d_{0}$, $c$, $p$ and the effective charge
$e^{1}_{eff}$ have been adjusted by simultaneously taking into account
experimental data on the energy bands and the available B(E1)-B(E3) transition
probabilities. The parameter values obtained in the considered nuclei are given
in Table 1. The resulting values of the original Hamiltonian parameters in
(\ref{Hqo}) and (\ref{Ub2b3I}) are given in Table~2. For each nucleus the
calculations are performed in a net over the values of the ``angular'' quantum
numbers $k$ with appropriate parity in the limits $1\leq k\leq 20$. In all
nuclei sets of values for the $k$-quantum numbers providing the best model
description of both, energies and reduced transition probabilities, are
obtained. These values are given in Figures 1--7 where the theoretical and
experimental energy levels of the considered nuclei are compared. The
theoretical and experimental values of the B(E1), B(E2) and B(E3) transition
probabilities are compared in Table 3. Model predictions for some not yet
observed transitions are also given there.
The results in Figs. 1--7 show that the model scheme correctly reproduces the
structure of the alternating-parity spectra in the considered nuclei with a
reasonably good agreement between the theoretical and experimental energy
levels. The correct reproduction of the mutual displacement of the different
positive and negative parity sequence is related to the involvement of $k$
quantum number values larger than 1 and 2. On the other hand the determination
of the $k$-values is strongly dictated by the interband transitions between the
positive- and negative- parity levels as well as by the transitions between the
different alternating-parity sequences. The above remarks explain why for the
nuclei $^{152}$Sm and $^{154}$Gd new sets of $k$-quantum numbers appear
together with renormalized values of the fitting parameters compared to the
previous descriptions limited to the yrast bands \cite{b2b3mod} (see below).
One should remark that at the same time the main (radial) oscillator quantum
number $n$ is uniquely determined, $n=0$ for the yrast sequence, $n=1$ for the
first excited alternating parity band and so on, as explained in the end of
Sec. 2.
In $^{152}$Sm the yrast band is described together with the first excited band
(see Fig.~1). The calculations provide two identical couples of $k$ values
$(k^{(+)}=1,k^{(-)}=8)$ for each band. Thus it is seen that $k^{(-)}$ obtains a
value larger than the lowest even value $2$ considered in \cite{b2b3mod}. From
Table~3 one can see that with this configuration of $k$-numbers the model
fairly good reproduces the data \cite{nudat2} on the B(E2) intraband transition
probabilities in the ground-state band ($g$) and on the B(E1) probabilities for
transitions between the $g$- and the first negative-parity band ($n1$). Some
interband $E2$ transitions between the $g$- and the first $\beta$-band ($b1$),
as $2_{b1}^{+}\rightarrow 2_{g}^{+}$ and $4_{b1}^{+}\rightarrow 4_{g}^{+}$ are
also well described, while others like $4_{b1}^{+}\rightarrow 6_{g}^{+}$ are
overestimated. The calculated intraband transitions in the $b1$-band are in
rough agreement with the experimental data, while the $E1$ intraband transition
$1_{n1}^{-}\rightarrow 2_{b1}^{+}$ is overestimated by an order. The E3
transition probability B$(E3;3_{n1}^{-}\rightarrow 0_{g}^{+})=14$ W.u.
\cite{K02} is exactly reproduced due to the adjustable parameter $p$ which
determines the factor $q$ in (\ref{rlam3}) according to Eq.~(\ref{q}). This
allows one to predict other E3 transitions like $1_{n1}\rightarrow 4_{g}$ and
other similar transitions between the $b1$-band and the second negative-parity
band ($n2$) as shown in Table~3. Although not all theoretical transition
probabilities are in strict agreement with the experimental data it is seen
that the model scheme correctly takes into account the different scales of the
various kinds of probabilities. A similar behaviour of transition probabilities
is observed in the other considered nuclei.
In $^{154}$Sm totally three alternating-parity bands are considered as seen
from Fig.~2. The positive-parity states of the second excited band are
interpreted in \cite{ensdf} as members of a second $K^{\pi}=0^+$ band, or of a
second $\beta$-band $(b2)$. The respective negative-parity levels are selected
in the present work among levels for which there is no interpretation given in
\cite{ensdf}. Here they form a third negative-parity band $(n3)$. From Table~3
it is seen that the intraband B(E2) transition probabilities in the $g$-band of
this nucleus are reasonably well described up to $I=10$, while the
B$(E2;12_{g}^{+}\rightarrow 10_{g}^{+})$ value is considerably overestimated.
The B(E1) probabilities between the $n1$- and $g$-bands are also well
described. The theoretical interband transition value
B$(E2;0_{b1}^{+}\rightarrow 2_{g}^{+})$ is by an order smaller than the
experimental one. For the other similar E2 transitions like
$2_{b1}^{+}\rightarrow 0_{g}^{+}$ the theoretical values are obtained below
the upper limits given for the respective experimental data \cite{nudat2}.
In $^{154}$Gd, again, three alternating-parity bands are considered. The model
description is given in Fig.~3. Here, the second non-yrast band is constructed
by the second excited $K^{\pi}=0^+$ band and a $3^{-}$ state with energy
1796.96 keV \cite{ensdf}. Although the latter is interpreted in \cite{ensdf} as
a member of a $K^{\pi}=2^-$ octupole band it reasonably fits the present scheme
as a member of an $n3$-sequence. In this nucleus the B(E1)--B(E3) transition
probabilities are also reasonably described with the largest discrepancies
between the theory and experiment, about a factor of 2, being observed for the
E2 transitions $2_{b1}^{+}\rightarrow 0_{b1}^{+}$ and $0_{b1}^{+}\rightarrow
2_{g}^{+}$ (see Table~3). Note that here the theoretical B(E1) value for the
interband transition B$(E1;1_{n1}^{-}\rightarrow 2_{b1}^{+})=0.0064$ W.u. is
obtained close to the experimental one, $0.0099$ W.u.
In $^{156}$Gd two alternating-parity bands, the yrast and first excited, are
considered (see Fig.~4). The B(E2) and B(E1) transition probabilities between
the members of the $g$-, $b1$- and $n1$-bands are well described with a few
exceptions as in the transitions $4_{b1}^{+}\rightarrow 2_{b1}^{+}$ and
$4_{b1}^{+}\rightarrow 2_{g}^{+}$ for which the B(E2)-values are
underestimated with respect to the experiment by a factor of about two and an
order, respectively (see Table 3). On the other hand the model predictions for
the B(E1) transition probabilities between the second negative-parity band $n2$
and the $g$-band suggest $2$-$3$ orders of magnitude in suppression compared to
experimental data.
In $^{158}$Gd, three alternating-parity bands are considered (see Fig.~5).
Similarly to $^{154}$Gd the $1^{-}$ and $3^{-}$ states included in the second
excited band enter the present model scheme as $n3$-members, while in
\cite{ensdf} they are interpreted as members of a $K^{\pi}=1^-$ octupole band.
For this nucleus, quite a large number of data on B(E1) and B(E2) transition
probabilities are available \cite{nudat2}. One should remark that compared to
the other considered rare-earth nuclei $^{158}$Gd is closer situated to the
region of pronounced rotation collectivity. From Table~3 it is seen that the
theoretical intraband B(E2) probabilities in the $g$-band of $^{158}$Gd faster
increase with the angular momentum compared to the experimental data. On the
other hand six experimental B(E1) values for the transitions between the $g$-
and the $n1$-bands are described quite well. It is remarkable that an
experimental estimation for a E1 transition between the $n2$- and $b1$-band is
available with B$(E1;3_{n2}^{-}\rightarrow 2_{b1}^{+})>0.00035$ W.u. This
circumstance is in a conjunction with the model assumption about the
quadrupole-octupole coupling of both bands. The model description predicts for
this probability a smaller value of 0.00011 W.u., which is of the same order as
the B(E1) values connecting the $g$- and $n1$-bands. Further, model prediction
values for similar B(E1) transition probabilities as
B$(E1;1_{n2}^{-}\rightarrow 0_{b1}^{+})=8\times 10^{-5}$ W.u. and
B$(E1;1_{n2}^{-}\rightarrow 2_{b1}^{+})=0.0002$ W.u. are given in Table~3. Also
there one can find available experimental estimations for intraband transition
probabilities like B$(E2;5_{n1}^{-}\rightarrow 3_{n1}^{-})=369$ W.u. and
B$(E2;3_{n2}^{-}\rightarrow 1_{n2}^{-})>1600$ W.u. which are underestimated by
the theory. In addition, a number of B(E1) transition probabilities from $b1$-
to $n1$-, from $n2$- to $g$- and from $b2$- to $n1$- and $n2$-bands are
generally underestimated by one or two orders of magnitude.
In $^{236}$U two alternating-parity bands, the yrast and first excited, are
considered (see Fig.~6). This nucleus was selected because of the possibility
to examine two observed reduced probabilities for E3 transitions, namely
B$(E3;1_{n1}^{-}\rightarrow 4_{g})=62$ W.u. \cite{nudat2} and
B$(E3;3_{n1}\rightarrow 0_{g})=22.9$ W.u. \cite{K02}. From Table~3 it is seen
that the first one is exactly reproduced. The second one is underestimated by
the theoretical value, $15$ W.u., which is still reasonably close to the
experiment. For the B(E2) transition probabilities within the $g$-band the
description is good as overall up to a quite high angular momentum $I=26$. The
experimental B(E1) value for the transition $1_{n1}^{-}\rightarrow 0_{g}^{+}$
is exactly reproduced because of the use of the effective charge. This allows
one to predict other B(E1) transition probabilities in the described spectrum
which are given in Table~3.
In $^{100}$Mo the experimentally observed $2_{3}^{+}$ state with energy
$1463.9$ keV is considered in \cite{ensdf} as a possible member of a
$\beta$-band. However, the present scheme suggests that the $2^{+}$ state
belonging to this band should lie essentially lower. The calculations show that
the experimental $2_{2}^{+}$ state with energy $1063.78$ keV considered in
\cite{ensdf} as a possible member of a $\gamma$-band is more appropriate as a
$\beta$-band member. The result in Fig.~7 shows that if this state is included
in the $b1$-band (in the present notations) a non-yrast alternating-parity
sequence can be constructed and reasonably well described by taking three
additional states, namely $1^-$ at $2156$ keV, $3^-$ at $2369.6$ keV and $4^+$
at $1771.4$ keV from the set of available but not interpreted data for
$^{100}$Mo \cite{ensdf}. The observed B(E1)-B(E3) transition probabilities are
reasonably described as seen from Table~3. The main discrepancy between the
theory and the experiment, a factor of 5, is obtained for the E2 intraband
transition $2_{b1}^{+}\rightarrow 0_{b1}^{+}$.
The following comments on the model results can be made here. The parameters of
the fits shown in Table~1 reflect the common collective structure of the
various energy sequences ($g$, $b1$, $b2$, $n1$, $n2$, $n3$) in a given
nucleus, while the sets of $k$ values given in Figs 1-7 reflect their mutual
dispositions. Note that the parameters for $^{152}$Sm and $^{154}$Gd are
essentially renormalized compared to the fits of the yrast band only
\cite{b2b3mod}. As seen from Table~1 the parameters $\omega$ and $b$, which are
responsible for the rotation-vibration behaviour of the different sequences,
vary relatively smoothly between the different nuclei. The parameter $d_0$,
which is responsible for the shape of the potential at zero angular momentum,
shows more pronounced differences in its values, especially for the nuclei from
different regions as $^{236}$U and $^{100}$Mo. Also, the parameter $c$, which
determines the overall scale for the transition probabilities in the ``radial''
integrals, considerably varies, while the parameter $p$ which is related to the
quadrupole and octupole contributions to the moment of inertia changes quite
smoothly. It is remarkable that in three nuclei, $^{152}$Sm, $^{154}$Sm and
$^{154}$Gd, the effective charge for the E1 transitions is practically unit
which means that there is no need for this parameter to describe them. In
$^{156}$Gd it is still close to 1, while in the other three nuclei its need for
the model description is already essential.
By using the relations (\ref{B2B3C2C3}) and (\ref{d2d3}) between the model
parameters in ellipsoidal coordinates and the parameters of the original
Hamiltonian, (\ref{Hqo}) with (\ref{Ub2b3I}), one can obtain the latters from
the values given in Table~1. Subsequently one can obtain the semi-axes (sa)
$\beta_{2}^{\mbox{\scriptsize sa}}$ and $\beta_{3}^{\mbox{\scriptsize sa}}$ of
the ellipsoidal potential bottom in the space of the quadrupole--octupole
variables given by
\begin{eqnarray}
\beta_{\lambda}^{\mbox{\scriptsize sa}}(I)=[2X(I)/d_{\lambda}C_{\lambda}]^{1/4},
\qquad \lambda=2,3.
\label{semiax}
\end{eqnarray}
(For more details see the text after Eqs. (3) and (4) of \cite{b2b3mod}.) The
resulting values of the parameters $B_2$, $B_3$, $C_2$, $C_3$, $d_2$, $d_3$ and
the semi-axes are given in Table~2. Note that in the present work they are {\em
not} directly adjusted, but obtained as a result of the adjustment of the
parameters $\omega$, $b$, $d_0$, $c$, $p$, and $e^{1}_{eff}$. As such they
only give a rough estimation about the order of the potential parameters and
its shape. One can see that for $^{152,154}$Sm and $^{154,156}$Gd these
parameters vary relatively smoothly, while for the remaining three nuclei they
show some essential fluctuations. The values of the
$\beta_{2}^{\mbox{\scriptsize sa}}$ semi-axis are obtained close to the known
values of the static quadrupole deformations in these nuclei while the values
of the octupole semi-axis $\beta_{3}^{\mbox{\scriptsize sa}}$ appear
considerably larger. This result is correlated with the larger values of the
quadrupole stiffness parameters $C_2$ compared to the values of $C_3$. Hence
the present parameters correspond to a vibration motion with a larger softness
of the system with respect to the octupole mode compared to the quadrupole one.
A closer look on the formalism shows that the ratio between both semi-axes is
related to the matrix elements of the quadrupole and octupole electric
multipole operators (\ref{q2gen}) and (\ref{q3gen}). By using (\ref{C2C3}),
(\ref{d2d3}) and (\ref{q}) in (\ref{semiax}) one finds that
\begin{eqnarray}
\frac{\beta_{3}^{\mbox{\scriptsize sa}}}{\beta_{2}^{\mbox{\scriptsize sa}}}
=\frac{q}{p}=\frac{1}{\sqrt{2p^{2}-1}}.
\label{semrat}
\end{eqnarray}
It is seen that the ratio $\beta_{3}^{\mbox{\scriptsize
sa}}/\beta_{2}^{\mbox{\scriptsize sa}}$ depends on the inertia factors $p$ and
$q$, Eq~(\ref{pqd}), which determine the strength of the E2 and E3 transitions,
respectively. This ratio is less than 1 for $p>1$ $(q<1)$. It can be easily
checked that to obtain $\beta_{3}^{\mbox{\scriptsize
sa}}/\beta_{2}^{\mbox{\scriptsize sa}}<1$ one has to introduce an additional
scaling constant, $c_3$, having the meaning of an effective charge for the
octupole mode. Then the octupole charge factor is renormalized as
$M_{3}'=c_3M_{3}$. The numerical analysis shows that if $c_3$ is chosen in the
limits $2\leq c_3\leq4$ the parameter $p$ is renormalized so that $q\rightarrow
q/3$ and the same theoretical levels and transition probabilities are obtained
with $\beta_{3}^{\mbox{\scriptsize sa}}<\beta_{2}^{\mbox{\scriptsize sa}}$ in
correspondence to the usually observed values of the deformation parameters
$\beta_{2}$ and $\beta_{3}$. For example if $c_3=4$ one obtains the following
set of renormalized parameters for $^{154}$Gd, $c'=269.6$, $p'=1.197$,
${e^{1}_{eff}}'=1.512$, while the parameters $\omega$, $b$ and $d_{0}$ remain
unchanged compared to the values given in Table~1. Compared to the values in
Table~2 the renormalized parameters for $^{154}$Gd are ${B_3}'=1146$
$\hbar^{2}/$MeV, ${C_3}'=108$ MeV, ${d_3}'=777$ $\hbar^{2}/$MeV and
${\beta_{3}^{\mbox{\scriptsize sa}}}'=0.192$, while the other parameters
referring to the quadrupole deformation remain unchanged. It is seen that now
the length of the potential bottom semi-axis in the $\beta_{3}$-direction
corresponds to a more realistic octupole deformation. This result is equivalent
to the involvement of a renormalized octupole operator $\hat{Q}_{3 0}'(\eta
,\phi)=c_3\hat{Q}_{3 0}(\eta ,\phi)$. Since the use of such an effective charge
does not change the model description but only leads to the renormalization of
the parameters it is not considered in the present work.
Further, it is important to comment the obtained configurations of quantum
numbers $k_n^{(+)}$ and $k_n^{(-)}$ which characterize the energy shifts in
the described alternating-parity spectra. From Figs.~1--7 it is seen that the
relevant energy shift in the excited level sequences is obtained through a jump
of $k$ over several lower values. In this way certain low-lying states
available in the scheme do not enter the considered spectrum, while others
lying at higher energy are used to obtain the model description. This result is
a consequence of the fact that the same oscillation frequency $\omega$ is
imposed to all alternating-parity bands. Actually, the non-yrast bandheads and
the energy shifts could be reproduced through the lowest possible
$k$-configurations [$k_n^{(+)}=1,k_n^{(-)}=2$] if separate vibration
frequencies are considered in the different bands. Speaking about $k$ as a
number of angular oscillation quanta (phonons) it appears that the restricted
freedom of the frequency imposed by the coherent condition is compensated in
the model description by the presence of a larger number of quanta on which the
rotation bands are built. Since the eventual consideration of different
oscillation frequencies would correspond to the introduction of parameters
external for the model the larger numbers of quanta are retained in the present
work. The obtained pairs of values $k_n^{(+)}$ and $k_n^{(-)}$ for the quantum
number $k$ provide a detailed systematic information about the mutual
disposition of the positive- and negative-parity bands in the different nuclei,
and subsequently, about the evolution of the quadrupole-octupole spectra in a
given nuclear region. It should be noted that the involvement of the extended
transition operators (\ref{q1gen})-(\ref{q3gen}) in the present CQOM
development is related to the appearance of larger $k$-values and the
subsequent large $k$-differences taken into account in the electric transition
probabilities. These features of the model can change if it is applied beyond
the coherent-mode assumption. In this case the unrestricted Hamiltonian
(\ref{Hqo}) can be diagonalized by using the present analytic solution as a
basis. Then the parameters in (\ref{Hqo}) can be directly adjusted to describe
the spectrum without restriction of the quadrupole and octupole oscillator
frequencies. This could allow one to construct the spectrum by always choosing
the lowest possible eigenvalues, while the structure of the spectrum obtained
in the present analytic solution could only guide the construction of non-yrast
bands. Work in this direction is in progress.
Finally, it should be noted that the present model descriptions are obtained
within some natural limits of the applied formalism with respect to
experimental data. It is well known that rotation terms like the one entering
the model potential can only describe smooth changes of the rotation spectra
with increasing angular momentum, as for example the so called ``centrifugal
stretching''. The treatment of angular momentum regions where sharper changes
in the rotation spectrum due to changes in the intrinsic structure like
backbending effects occur, needs a special development which is not the subject
of the present work. That is why in some of the considered nuclei descriptions
and/or predictions of rotation levels with very high angular momenta are
avoided, especially in the cases where the negative-parity levels are not
observed. An exception is done for $^{236}$U (Fig.~6), where higher-spin
negative-parity levels were predicted in accordance to the last observed state
with even angular momentum. This prediction should be meaningful since in the
actinide region the rotation spectra exhibit more regular rotation motion in
the high-spin regions. On the other hand the prediction of missing low-spin
states, like the $1^{-}_{n3}$ level in $^{154}$Gd and the $6^{+}_{b1}$ and
$5^{-}_{n2}$ levels in $^{100}$Mo, as well as, a number of not observed
transition probabilities shown in Table~3 should be also reasonable in the
present framework. In this meaning the applied CQOM model scheme rather
describes the ``horizontal'' evolution of the alternating-parity spectra beyond
the yrast line than the high-spin properties of individual rotation bands.
\section{Concluding remarks}
The present work provides a model description and respective classification of
the yrast and non-yrast alternating-parity spectra and the attendant B(E1),
B(E2) and B(E3) transition probabilities in several rare-earth nuclei, one U
and one Mo nucleus within the collective model of Coherent Quadrupole and
Octupole Motion (CQOM). The theoretical formalism and the obtained model
descriptions outline a possible way for the development of nuclear
alternating-parity spectra toward the highly non-yrast region of collective
excitations. In the considered scheme the different negative parity
level-sequences appear in couples together with the ground-state band and the
excited $\beta$-bands. On this basis the model predicts possible E(1) and E(3)
transitions between states with opposite parity within various
alternating-parity bands. The presence of experimentally observed E(1)
transitions between such states in the non-yrast part of the spectrum is
noticed. Further experimental measurements of electric transition probabilities
would be very useful to check the possible coupling of non-yrast energy
sequences with opposite parities. It was demonstrated that the considered
scheme can be used for the interpretation of data on excitation energies whose
place in the structure of the collective spectrum has not yet been determined.
The approach was applied to selected nuclei for which a relatively large number
of data on B(E1)-B(E3) transitional probabilities are available, but it can be
easily extended to wider ranges of nuclei especially in the rare-earth and
actinide regions. Further, the formalism takes into account the complex-shape
effects in the motion of the system and in addition provides estimations about
the shape of the quadrupole--octupole potential which governs the collective
properties of the considered nuclei. More refined model descriptions and
realistic estimations about the potential shape can be obtained beyond the
limits of the present coherent-mode assumption. Work in this direction is in
progress.
\section*{Acknowledgements}
\noindent We thank Professor Jerzy Dudek for valuable discussions and comments.
This work is supported by DFG and by the Bulgarian National Science Fund
(contract DID-02/16-17.12.2009).
\section*{\bf Appendix A: CQOM shape-density distributions}
The density distribution of the CQOM vibration state in the space of the
quadrupole-octupole shapes is given by the square of the wave function
(\ref{wvib}), $\rho_{nkI}(\beta_{2},\beta_{3})=|\Phi^{\pi}_{nkI}
(\beta_{2},\beta_{3})|^{2}$, after a transformation from the ellipsoidal
coordinates $(\eta ,\phi)$ to the deformation coordinates $(\beta_{2}
,\beta_{3})$. In Fig.~8 three-dimensional plots of $\rho_{nkI}$ are given for
the lowest $k=1$ and $k=2$ states for $n=0$ and for the schematic parameter
values $\omega =0.3\,$MeV/$\hbar$, $b=3\,\hbar^{-2}$, $d_0=100\,\hbar^{2}$,
$d_2=300\,\hbar^{2}$/MeV, $d_3=500\,\hbar^{2}$/MeV. Note that according to the
discussion in the end of Sec. 3 the shape of the potential is determined
unambiguously when the values of the inertia parameters $d_2$ and $d_3$ are
given. In Fig.~9 two-dimensional plots showing the maxima of $\rho_{nkI}$ for
$k=1-4$ are given together with contours showing the ellipsoidal potential
bottom for the above set of schematic parameters.
\section*{\bf Appendix B: Explicit form of the integrals over $\eta$}
The integrals over $\eta$, (\ref{s1}) and (\ref{s2}), can be written in the
following common form after taking into account the explicit expression for the
radial wave functions (\ref{psieta1})
\begin{eqnarray}
S_{l}(n_i,I_i;n_f,I_f)&=&\int_{0}^{\infty}d \eta
\psi_{n_{f}}^{I_{f}}(\eta)\eta^{l+1}\psi_{n_{i}}^{I_{i}}(\eta)
\nonumber \\
&=&N \int_{0}^{\infty}
e^{-c\eta^{2}}c^{s_{f}}\eta^{2s_{f}}L_{n_{f}}^{2s_{f}}(c\eta^{2})\eta^{l+1}
c^{s_{i}}\eta^{2s_{i}}L_{n_{i}}^{2s_{i}}(c\eta^{2})d \eta,
\label{Sleta}
\end{eqnarray}
where $l=1,2$, $s_{i}=(1/2)\sqrt{k_{i}^{2}+bX(I_i)}$,\ \
$s_{f}=(1/2)\sqrt{ k_{f}^{2}+bX(I_f)}$ and
\begin{eqnarray}
N=N_{n_i,n_f}(c,s_i,s_f)=2c\left[\frac{\Gamma(n_{f}+1)\Gamma(n_{i}+1)}
{\Gamma(n_{f}+2s_{f}+1)\Gamma(n_{i}+2s_{i}+1)}\right]^
{\frac{1}{2}} .
\end{eqnarray}
To derive an explicit expression for the integral (\ref{Sleta}) one can apply
the substitution $c\eta^{2}=x$ with $dx=2c\eta d\eta$, such that
\begin{eqnarray}
\eta^{l+1}d\eta =\frac{1}{2c^{1+l/2}}x^{l/2}dx.
\end{eqnarray}
Then Eq.~(\ref{Sleta}) reads as
\begin{eqnarray}
S_{l}(n_i,I_i;n_f,I_f)
=\frac{N_{n_i,n_f}(c,s_i,s_f)}{2c^{1+l/2}}\int_{0}^{\infty}
e^{-x}x^{s_{i}+s_{f}+\frac{l}{2}}L_{n_{f}}^{2s_{f}}(x)
L_{n_{i}}^{2s_{i}}(x)dx.
\label{Slx}
\end{eqnarray}
By using known formulas for integration of two generalized Laguerre polynomials
with different real ranks \cite{Prudnikov}, \cite{Wlagint} one obtains
(\ref{Slx}) in the following explicit form
\begin{eqnarray}
& &S_{l}(n_i,I_i;n_f,I_f)
\nonumber \\
&=&\frac{N_{n_i,n_f}(c,s_i,s_f)}{2c^{1+l/2}}
\frac{\Gamma (n_{f}+2s_{f}+1)}{\Gamma(1+2s_{f})}
\frac{\Gamma(n_{i}+s_{i}-s_{f}-\frac{l}{2})}{\Gamma(s_{i}-s_{f}-1)}
\frac{\Gamma(s_{i}+s_{f}+\frac{l}{2}+1)} {n_{i}!n_{f}!} \label{sgen} \\
&\times&
_{3}F_{2}\left(-n_{f},s_{i}+s_{f}+\frac{l}{2}+1,s_{f}-s_{i}+\frac{l}{2}+1;
2s_{f}+1,s_{f}-s_{i}+\frac{l}{2}+1-n_{i};1\right),
\nonumber
\end{eqnarray}
where $_{3}F_{2}$ denotes a generalized hypergeometric function \cite{3F2}. The
generalized hypergeometric function $_{3}F_{2}$ is calculated numerically
through a summation of its series representation for which a Fortran code is
available \cite{genhyp93}. It can be easily checked that if the first argument
of $_{3}F_{2}$ in (\ref{sgen}) is zero, $n_f=0$, one has $_{3}F_{2}=1$. In this
case Eq.~(\ref{sgen}) reduces to the following simpler expression
\begin{eqnarray}
S_{l}(n_i,I_i;0,I_f)
&=&\frac{1}{c^{\,l/2}}\frac{\Gamma(s_{i}+s_{f}+\frac{l}{2}+1)
\Gamma(n_{i}+s_{i}-s_{f}-\frac{l}{2})}
{\sqrt{n_{i}!\Gamma(2s_{f}+1)\Gamma(n_{i}+2s_{i}+1)}
\Gamma(s_{i}-s_{f}-\frac{l}{2})}.
\label{snf0}
\end{eqnarray}
This corresponds to a transition from a non-yrast to an yrast state. The
integrals for the yrast intraband transitions, Eqs. (50) and (51) in
\cite{b2b3mod}, are directly obtained from Eq.~(\ref{snf0}) when $n_i=0$.
Simple explicit forms of the $S_l$ integrals for interband and intraband
transitions in the particular cases up to $n=2$, which are of practical
interest, are given below
\begin{eqnarray}
& &S_{l}(1,I_i;1,I_f)\nonumber \\
&=& \frac{1}{c^{\,l/2}}\Bigl[(2s_{i}+1)(2s_{f}+1)-
(s_{i}+s_{f}-\frac{l}{2})(s_{i}+s_{f}+\frac{l}{2}+1)\Bigr] \nonumber \\
&\times&\frac{\Gamma(s_{i}+s_{f}+\frac{l}{2}+1)}
{\sqrt{\Gamma(2s_{i}+2)\Gamma(2s_{f}+2)}},
\end{eqnarray}
\begin{eqnarray}
& &S_{l}(2,I_i;1,I_f)\nonumber \\
&=& \frac{\sqrt{2}}{2c^{\,l/2}}\biggl\{2(s_{i}+1)(2s_{i}+1)(2s_{f}+1)-
(s_{i}+s_{f}+\frac{l}{2}+1)\biggr. \nonumber \\
&\times&\biggl. \Bigl[2(s_{i}+1)(2s_{i}+4s_{f}+3) -(s_{i}+s_{f}+\frac{l}{2}+2)
(3s_{i}+s_{f}-\frac{l}{2}+2) \Bigr] \biggr\} \nonumber \\
&\times&\frac{\Gamma(s_{i}+s_{f}+\frac{l}{2}+1)}
{\sqrt{\Gamma(2s_{i}+3)\Gamma(2s_{f}+2)}}.
\end{eqnarray}
\begin{eqnarray}
& &S_{l}(2,I_i;2,I_f)\nonumber \\
&=&\frac{1}{2c^{\,l/2}} \Biggl\{ 4(s_{i}+1)(2s_{i}+1)(s_{f}+1)(2s_{f}+1)\Biggr.
\nonumber \\
&-&(s_{i}+s_{f}+\frac{l}{2}+1)\biggl[ 16(s_{i}+1)(s_{f}+1)(s_{i}+s_{f}+1)
\biggr.
\nonumber \\
&-&(s_{i}+s_{f}+\frac{l}{2}+2)\Bigl\{2(s_{i}+1)(2s_{i}+1)+2(s_{f}+1)(2s_{f}+1)
+16(s_{i}+1)(s_{f}+1) \Bigr.
\nonumber \\
&-&\Biggl.\biggl.\Bigl.(s_{i}+s_{f}+\frac{l}{2}+3)(3s_{i}+3s_{f}-\frac{l}{2}+4)
\Bigr\}\biggr] \Biggr\}
\frac{\Gamma(s_{i}+s_{f}+\frac{l}{2}+1)}{\sqrt{\Gamma(2s_{i}+3)\Gamma(2s_{f}+3)}}.
\end{eqnarray}
\section*{\bf Appendix C: Explicit form of the integrals over $\phi$}
The integrals over the angular variable $\phi$, (\ref{Ilam}), with the relevant
parities $\pi_{i}$ and $\pi_{f}$ can be obtained in the following explicit
forms. For $\lambda =2$ the integral $I_{2}^{\pm\pm}$ with
$k_1=k_2=k=\mbox{odd}$ (++) or even $(--)$ is
\begin{eqnarray}
I_{2}^{\pm\pm}(k)
=\frac{2}{\pi}\mbox{Cat}+\frac{(-1)^{k+1}}{4k}\left[1+\frac{4}{\pi}
\sum_{m=1}^{2k-1}\frac{\sin (m\pi/2)}{m}\right],
\end{eqnarray}
where $\mbox{Cat}=\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)^{2}}\approx 0.915
965 594 177...$ is the Catalan constant. In the case of $k_1\neq k_2$, both odd
or even, the integral is
\begin{eqnarray}
I_{2}^{\pm\pm}(k_1,k_2)&=&
\frac{1}{2|k_2-k_1|}\left[1+\frac{4}{\pi}\sum_{m=1}^{|k_2-k_1|-1}\frac{\sin
(m\pi/2)}{m} \right]\\
\nonumber
&+&\frac{(-1)^{k_{1}+1}}{2(k_2+k_1)}\left[1+\frac{4}{\pi}\sum_{m=1}^{k_2+k_1-1}
\frac{\sin (m\pi/2)}{m} \right].
\end{eqnarray}
For $\lambda =3$ one has
\begin{eqnarray}
I_{3}^{+-}(k_1,k_2)=\frac{2k_2}{k_2^{2}-k_1^{2}}-
\frac{1}{\pi}\left[\frac{(-1)^{(k_2-k_1-1)/2}}
{(k_2-k_1)^2}+\frac{(-1)^{(k_2+k_1-1)/2}}{(k_2+k_1)^2}
\right],
\end{eqnarray}
where $ k_1= 1, 3, 5, \dots \ ,\ \ k_2= 2, 4, 6,
\dots$
For $\lambda =1$ the integral is obtained in the form of an infinite,
but reasonably converging series
\begin{eqnarray}
I_{1}^{+-}&=&\frac{1}{2\pi}\sum_{m=\pm1}^{\pm\infty}
\sum_{n=\pm1}^{\pm\infty} \sum_{\nu =\pm 1}
\frac{\mbox{sign}(-n)}{|mn|}\nonumber \\
&\times&\left[(1-\delta_{k_2+\nu k_1,-m-n})
\frac{\sin[(k_2+\nu k_1+m+n)\frac{\pi}{2}]}{(k_2+\nu k_1+m+n)}+
\frac{\pi}{2}\delta_{k_2+\nu k_1,-m-n}\right],
\end{eqnarray}
where $ k_1= 1, 3, 5, \dots \ ,\ \ k_2= 2, 4, 6,
\dots$
|
{
"timestamp": "2012-03-09T02:04:14",
"yymm": "1203",
"arxiv_id": "1203.1873",
"language": "en",
"url": "https://arxiv.org/abs/1203.1873"
}
|
\section{Introduction}
The CMS ridge effect is a two-particle angular correlation effect observed by the Compact Muon Solenoid (CMS) experiment in Large Hadron Collider (LHC) proton-proton collisions at high charged track multiplicities ($N_{ch} > 110$) and in a specific transverse momentum range ($p_T = 1-3$~GeV). This effect, together with other visible structures in the $R\left(\Delta\eta,\Delta\phi\right)$ correlation distribution, was described in \cite{cmsridge}. \vskip 1em \noindent
Two-particle correlation function $R\left(\Delta\eta,\Delta\phi\right)$ is defined as:
\begin{equation}
R\left(\Delta\eta,\Delta\phi\right) = \left\langle \left(\langle N\rangle - 1 \right) \left( \frac{S_N\left(\Delta\eta,\Delta\phi\right)}{B_N\left(\Delta\eta,\Delta\phi\right)}-1\right)\right\rangle_{bins}
\end{equation}
Data is binned according to charged track multiplicity $N_{ch}$. The signal $S_N(\Delta\eta,\Delta\phi)$ consists of the charged two-particle density, while the background $B_N(\Delta\eta,\Delta\phi)$ is given by the distribution of uncorrelated particle pairs -- the product of two single-particle distributions. Finally, the data is averaged, weighted with bin multiplicity, over all bins. The analysis is repeated for four sets of data. On one hand two minimum bias sets (all $N_{ch}$), one including all particles with transverse momenta above $0.1$~GeV and the other including all particles with transverse momenta between $1$ and $3$~GeV. On the other hand two high-multiplicity sets ($N_{ch} > 110$), again with the same two transerverse momentum selections. \vskip 1em \noindent
Some of the effects reported in \cite{cmsridge}, including the near-side peak at $\left(\Delta\eta,\Delta\phi\right) = \left(0,0\right)$, the away-side ridge at $\left(\Delta\eta,\Delta\phi\right) = \left(\Delta\eta,\pi\right)$ and the Gaussian ridge at $\left(\Delta\eta,\Delta\phi\right) = \left(0,\Delta\phi\right)$ can be explained with single two-to-two partonic interactions. The first two are visible in all sets of data, while the third one is most clear in the minimum bias $p_T > 0.1$~GeV case. A fourth effect, the near-side ridge, a long-range azimuthal correlation at $\left(\Delta\eta,\Delta\phi\right) = \left(\Delta\eta,0\right)$ only visible in high-multiplicity events at moderate $p_T$, requires further study. It is for this last effect that we propose a model. \vskip 1em \noindent
For our study we observe the effect of our modification of the \textsc{Pythia6}~\cite{pythia6} Monte Carlo (MC) event generator on select observables and consider changes in a few existing \textsc{Pythia6} parameters to counteract the side-effects of our modification. This latter step can be considered a re-tuning to CMS data. Note that we only use a limited set of CMS data and start from the existing \textsc{Pythia6} tune Z2~\cite{cmsue}. More global tuning including other experiments' data was not within the scope of this study, but may be added later.
\section{The azimuthal alignment model}
For large enough impact parameter $b$ (figure \ref{fig_pprofile}), the multiparton interactions in proton-proton collisions tend to lie in the collision plane of the hardest interaction and the final state particles will have similar azimuthal angle $\phi$ -- this results in near-side effects. Furthermore, an explanation for the ridge effect with multiparton interactions would require enough such interactions to be taking place, which leads to high-multiplicity events. At the same time we require that the multiparton interactions are semi-hard, and thus yield moderate-$p_T$ particles. Finally, we are dealing with incoming partons with very different $x_{bj}$ and as such will have interactions in a broad pseudo-rapidity range $\eta$ -- this gives rise to long-range effects. So far, everything is still consistent with the observations made by CMS.
\begin{figure}[h]
\centering
\includegraphics[width=0.35\textwidth]{profile}
\caption{\label{fig_pprofile}Protons separated by impact parameter $b$.}
\end{figure}
\vskip 0em \noindent
What is still a problem, is that high-multiplicity events are generally central collisions which have an impact parameter $b \sim 0$, while the definition of the collision plane of the hardest interaction requires large $b$. In light of this issue, we study whether a small upward fluctuation in the amount of multiparton interactions, for the case of moderate impact parameter, suffices to explain the CMS ridge effect. \vskip 1em \noindent
The modification we introduce goes on top of the most recent multiparton interaction model currently in \textsc{Pythia6}~\cite{pythia6mpi}. In this existing model, the amount of multiparton interactions, a measure for the activity, is inversely proportional to impact parameter $b$ (VINT(139), rescaled to $b_{avg}=1$ for the minimum bias case). The azimuthal angle $\hat{\phi}$ (VINT(24)) is chosen randomly. This last point makes that angular correlations -- also the long-range, near-side ones -- would be missing in events generated with \textsc{Pythia6}. \vskip 1em \noindent
We propose sampling random points $\left(x_i,y_i\right)$ in Gaussian proton profiles (figure \ref{fig_sample}), these protons being separated by impact parameter $b$, and using trigonometry to calculate the $\phi$-offset from the hardest interaction. To allow for some tuning freedom we add a scaling parameter $\alpha$ to the impact parameter $b$. Ideally, the scaling parameter would be one. This results in:
\begin{equation}
\phi_i = \phi_{hardest} + \mbox{arctan}\left(\frac{y_2-y_1}{(x_2+\alpha\cdot b/b_{avg}) - x_1}\right)
\end{equation}
We implement the modification for two different modes of the multiparton interaction model of \textsc{Pythia6} which both make use of hadronic overlap according to Gaussian distributions. In those cases, the above $\phi$-definition makes sense. In our tuning activity reported in section \ref{sec_tuning}, we focus on the mode which uses double-gaussian matter profiles (MSTP(82) = 4).
\begin{figure}[h]
\centering
\includegraphics[width=0.31\textwidth]{sample}
\caption{\label{fig_sample}Sampling of random points in Gaussian proton profiles, separated by impact parameter $b$, and introduction of scaling parameter $\alpha$ to allow some tuning freedom.}
\end{figure}
\vskip 0em \noindent
The modification has several implications. We study two sets of data: CMS underlying event (UE) data~\cite{cmsue}, showing the charged multiplicity $N_{ch}$ and transverse momentum sum $\sum p_T$ in the region transverse to a jet or hard interaction (figure \ref{fig_tta}), and CMS minimum bias (MB) data~\cite{cmsmb}, showing the charged multiplicity $N_{ch}$ integrated over azimuthal angle $\phi$. By introducing the modification, we generate interactions with an azimuthal separation from the hardest interaction smaller than would be the case with the previous uniform azimuthal distribution. The interactions get shifted to the toward/away regions and the plateau for $N_{ch}^{transverse}$ drops (figure \ref{fig_sens2}, top). Re-raising this plateau to describe the data requires a re-tune, modifying the $p_T$-cutoff and by proxy the activity, $N_{ch}$. The $p_T$-cutoff in \textsc{Pythia6} is given by:
\begin{equation}
p_T^{min}\left(E_{CM}\right) = p_T^0 \cdot \left(\frac{E_{CM}}{E_{REF}}\right)^{\gamma} = PARP(82)\cdot \left(\frac{E_{CM}}{E_{REF}}\right)^{PARP(90)}
\end{equation}
\vskip 1em \noindent
In contrast with the clear effect on UE results, we expect little or no sensitivity to the modification for the MB results, which are integrated over azimuth $\phi$ (figure \ref{fig_sens2}, bottom). Possibly this diffence in sensitivity also allows to lift some of the tension which exists between the UE and MB descriptions.
\begin{figure}[!h]
\centering
\includegraphics[width=0.25\textwidth]{ntt}
\caption{\label{fig_tta}Areas in $\Delta\phi$ with respect to the leading track jet.}
\end{figure}
\begin{figure}[ht]
\centering
\vspace{-0.5em}
\hfill\includegraphics[width=0.302\textwidth]{UE03_alpha}\hspace{1cm}
\includegraphics[width=0.302\textwidth]{UE01_alpha}\hfill\mbox{\ }\vskip 0em
\hfill\includegraphics[width=0.302\textwidth]{MB06_alpha}\hspace{1cm}
\includegraphics[width=0.302\textwidth]{MB16_alpha}\hfill\mbox{\ }\vspace{-0.8em}
\caption{\label{fig_sens2}Overview of the sensitivity of $N_{ch}$ observables in CMS UE (top) and MB (bottom) data to changes in scaling parameter \textcolor{magenta}{$\alpha$ [purple]}, with Z2 [black] as a reference.}
\end{figure}
\section{Tuning}\label{sec_tuning}
In this second part we report the results of our small-scale automized three-parameter re-tuning to the two earlier described CMS data sets. We start off with a review of the sensitivity of the observables to the three parameters and end with two tunes, one simple tune to just four $N_{ch}$ distributions (transverse \& total $N_{ch}$ and 0.9 \& 7.0~TeV) and one two-step tune to all MB and UE observables in the data sets. The first tune allows us to get a feeling of the parameter space in play, while the second one aims to reach a more solid result fixing the $p_T$-cutoff based on the MB observables (insensitive to scaling parameter $\alpha$) and then using the UE observables to fix $\alpha$. For the actual tuning, we make use of the \textsc{Professor} package~\cite{prof}, which takes care of automated tuning based on \textsc{Rivet} plots~\cite{rivet} of observables with reference data. For the interpretation of the tune result we make the comparison with existing tune Z2*
rather than with tune Z2. Z2* is a \textsc{Professor} re-tuning of Z2 to CMS UE data, for parameters PARP(82) and PARP(90).
\subsection{Sensitivity}\label{tuneSens}
The first observation (figure~\ref{fig_sens1}) is that \textsc{Pythia6} $p_T^0$-reference PARP(82) and energy-scaling parameter PARP(90) affect all activity, both transverse and total activity at both $0.9$ and $7.0$~TeV. For variations in the $p_T^0$-reference (red/blue solid) the effect is the same at both energies, while for variations in the energy-scaling parameter (green/orange dashed) the effect is opposite at the two energies. This is to be expected since $0.9$ and $7.0$~TeV lie on both sides of reference energy $1.8$~TeV used in the $p_T$-cutoff formula. On the part of $\alpha$ (figure~\ref{fig_sens2}), we can see a clear effect in the transverse region (UE dataset) and little to no effect in the $\phi$-integrated case (MB dataset).
\definecolor{orange}{rgb}{1,0.5,0}
\begin{figure}[ht]
\centering
\hfill\includegraphics[width=0.33\textwidth]{UE11_P82P90}\hspace{1cm}
\includegraphics[width=0.33\textwidth]{UE08_P82P90}\hfill\mbox{\ }\vskip 0.2em
\hfill\includegraphics[width=0.33\textwidth]{MB06_P82P90}\hspace{1cm}
\includegraphics[width=0.33\textwidth]{MB16_P82P90}\hfill\mbox{\ }
\caption{\label{fig_sens1}Overview of the sensitivity of $N_{ch}$ observables in CMS UE (top) and MB (bottom) data to changes in PARP(82) [\textcolor{blue}{blue}, \textcolor{red}{red} solid] and PARP(90) [\textcolor{green}{green}, \textcolor{orange}{orange} dashed], with Z2 [black] as a reference. In brackets the values (PARP(82), PARP(90), $\alpha$) are marked.}
\end{figure}
\subsection{Tune one}\label{tuneOne}
This first crude tune, which we call Z2R, is made with the \textsc{Professor} package to just four observables (two UE and two MB) and confirms what is also more or less visible by eye. We want a moderate $\alpha$, so as not to destroy the match with data so much that it cannot be restored (either in MB, UE or both), while still having enough power to introduce the intended long-range near-side effect. Next, we need a slight lowering in the $p_T^0$-reference (compared to optimal tune Z2*) to re-raise the $N_{ch}$ plateau in the transverse region. The energy-dependence will be of less importance. We find exactly this in our Z2R \textsc{Professor} tune (table \ref{tab_Z2R}), for which we used the cubic interpolation mode. In general we find that, to begin with, the match with data for the four observables to which we tuned, is of the same quality as in the case of Z2*. Furthermore, also for the observables in the dataset which we did not include in the tuning, the match remains acceptable. We show the graphical result of tune Z2R (red solid) in figure \ref{fig_Z2RZ2Rp}, with tunes Z2 (black dotted) and Z2* (blue dashed) as reference.
\begin{table}[ht]
\centering\footnotesize
\begin{tabular}{|c|c|c|c|}
\hline
& PARP(82) & PARP(90) & $\alpha$ \\ \hline\hline
Z2 & 1.83 & 0.28 & 0.00 \\ \hline
Z2* & 1.93 & 0.23 & 0.00 \\ \hline\hline
Z2R & 1.87 & 0.23 & 4.15 \\ \hline
\end{tabular}
\caption{Result of the 1-step 3-parameter tune to 4 observables.}
\label{tab_Z2R}
\end{table}
\subsection{Tune two}\label{tuneTwo}
The second tune we consider is made in two-steps, we call it Z2R'. This time using all the observables in the same UE and MB CMS datasets, we again perform an automized \textsc{Professor} tune with cubic interpolation. In the first step, the $p_T$-cutoff (both PARP(82) and PARP(90)) is fixed to the MB data, disregarding any match with UE data. In the second step, $\alpha$ is tuned to the UE data. After the first step, the match with data is good for MB, but less so for UE. After the second step, also the match with UE is restored to an acceptable level, comparable to the Z2* tune. Quantitatively, we again find the tune to be insensitive to PARP(90), while PARP(82) and $\alpha$ settle on values inbetween those of Z2* and Z2R (table \ref{tab_Z2Rp}, figure \ref{fig_Z2RZ2Rp}).
\begin{table}[h]
\centering\footnotesize
\begin{tabular}{|c|c|c|c|}
\hline
& PARP(82) & PARP(90) & $\alpha$ \\ \hline\hline
Z2 & 1.83 & 0.28 & 0.00 \\ \hline
Z2* & 1.93 & 0.23 & 0.00 \\ \hline\hline
Z2R & 1.87 & 0.23 & 4.15 \\ \hline\hline
Z2R' & 1.90 & 0.23 & 2.67 \\ \hline
\end{tabular}
\caption{Result of the 2-step 3-parameter tune to all UE and MB observables.}
\label{tab_Z2Rp}
\end{table}
\begin{figure}[p]
\centering
\hfill\includegraphics[width=0.33\textwidth]{UE03_T2}\hspace{1cm}
\includegraphics[width=0.33\textwidth]{UE01_T2}\hfill\mbox{\ }\vskip 0.2em
\hfill\includegraphics[width=0.33\textwidth]{UE11_T2}\hspace{1cm}
\includegraphics[width=0.33\textwidth]{UE08_T2}\hfill\mbox{\ }\vskip 0.2em
\hfill\includegraphics[width=0.33\textwidth]{MB06_T2}\hspace{1cm}
\includegraphics[width=0.33\textwidth]{MB16_T2}\hfill\mbox{\ }\vskip 0.2em
\hfill\includegraphics[width=0.33\textwidth]{MB20_T2}\hspace{1cm}
\includegraphics[width=0.33\textwidth]{MB22_T2}\hfill\mbox{\ }
\definecolor{darkgreen}{rgb}{0.,0.65,0.}
\caption{\label{fig_Z2RZ2Rp}Performance of the \textcolor{red}{Z2R [red solid]} and the \textcolor{darkgreen}{Z2R' [green dashed} (intermediate), \textcolor{darkgreen}{solid} (final)\textcolor{darkgreen}{]} tunes, compared to tunes Z2 [black dashdotted] and \textcolor{blue}{Z2* [blue dashed]}, for select observables in the full UE (rows 1-2) and MB (rows 3-4) data set. The four observables used for tune Z2R are given in rows 1 and 3.}
\end{figure}
\subsection{The CMS ridge}
Finally we consider correlation function $R\left(\Delta\eta,\Delta\phi\right)$, comparing results using tunes Z2R and Z2R' with those from the original paper (figure~\ref{ridge} (top row)). In the middle row, the results for Z2R are shown. It is clear that for high-multiplicity moderate-$p_T$ events (middle, left), the long-range near-side ridge is visible, fully in agreement with the CMS results. In the same row (middle, centre), one can see that also for moderate-multiplicity events a ridge is visible, denoting that perhaps the effect of the modification is too strong. In the bottom row, the same plots are shown for tune Z2R'. Here, the effect is not strong enough at high-multiplicity (bottom, left), as no near-side ridge is visible, while it is still too strong at moderate-multiplicity (bottom, centre), where an unwanted ridge is visible. For high-multiplicity events, including all $p_T > 0.1$~GeV (middle/bottom, right), both tunes show similar effects. There is no near-side ridge and some broadening around $\Delta\eta = 0$ is visible, both in agreement with CMS data, but there is an unexplained additional peak at $\left(\Delta\eta,\Delta\phi\right) = \left(0,\pi\right)$.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.295\textwidth]{HM_MpT}\hfill
\includegraphics[width=0.295\textwidth]{MB_MpT}\hfill
\includegraphics[width=0.295\textwidth]{HM_LpT}\vskip -0.2em
\includegraphics[width=0.295\textwidth]{1869_0229_415}\hfill
\includegraphics[width=0.295\textwidth]{1869_0229_415_lowN}\hfill
\includegraphics[width=0.295\textwidth]{1869_0229_415_allpT}\vskip -0.8em
\includegraphics[width=0.295\textwidth]{1899_0231_267}\hfill
\includegraphics[width=0.295\textwidth]{1899_0231_267_lowN}\hfill
\includegraphics[width=0.295\textwidth]{1899_0231_267_allpT}
\caption{\label{ridge}Results for $R\left(\Delta\eta,\Delta\phi\right)$: original (top row), Z2R (middle row) and Z2R' (bottom row); (left) high multiplicity, moderate $p_T$; (centre) minimum bias, moderate $p_T$; (right) high multiplicity, all $p_T > 0.1$~GeV.}
\end{figure}
\section{Conclusions}
We proposed a modification of \textsc{Pythia6}, explaining the ridge effect with multiparton interactions. The model introduces a correlation between the azimuth of the event planes of individual multiparton interactions and the event plane of the hardest interaction. This correlation can be naturally explained in a physical picture based on the impact parameter between the protons. The two main implications of this modification are the appearance of the near-side ridge in high-multiplicity moderate-$p_T$ events and a shift in the activity in the transverse region. This latter effect can be counteracted by a re-tune of the $p_T$-cutoff parameters to underlying event data. In a slightly broader picture, minimum bias data can be included in the re-tuning. Implementing this with the \textsc{Professor} package, we found tunes Z2R and Z2R'.
\begin{footnotesize}
|
{
"timestamp": "2012-05-17T02:02:49",
"yymm": "1203",
"arxiv_id": "1203.2048",
"language": "en",
"url": "https://arxiv.org/abs/1203.2048"
}
|
\section{Introduction}
\label{sec:intro}
Electronic transport properties, the motion of electrons, in a
random potential are closely related to the phenomenon of
Anderson localization\cite{PAnderson1958}.
The phenomena of Anderson localization have been studied
in various fields including photonics\cite{caoh2010},
cold atoms \cite{AdLagendijk2009}, circuits \cite{lazo2011},
and DNA molecules \cite{zhangw2004,zhangw2010}.
Many accurate numerical approaches have been developed, by
the quantum transfer matrix renormalization group method
for finite temperature systems \cite{lpyang2009},
the density matrix renormalization group method
for interacting systems \cite{pshmitteckert1998}, and the integral
equation method for systems in the thermodynamic limit
\cite{kkang2010,kkang2011}, respectively. In this work we will
study the zero energy behavior for the one-dimensional model
with correlated weak diagonal disorder. We first extend the
numerical method we developed earlier in Ref.~\cite{kkang2010}
for uncorrelated disorder to correlated system. Our numerical method
was an application of the transfer matrix method \cite{jbpendry1994}
in localized phase in the thermodynamic limit.
In one-dimensional Anderson model\cite{PAnderson1958}
with diagonal disorder is described by,
\begin{equation}
\psi_{i-1}+\psi_{i+1}=(E-\epsilon_i)\psi_i, \label{eq:se}
\end{equation}
where hopping term is set to unity and
$\psi_i$ is the electron wavefunction at site-$i$.
${\epsilon_i}$ is the on-site energy with a certain type of
random distribution which satisfying an exponential correlation:
$\langle\epsilon_i^2\rangle=\sigma^2$ and
$\langle\epsilon_i \epsilon_j\rangle=\sigma^2\exp[-|i-j|/l_{cor}]$
for different sites.
$\sigma^2$ and $l_{cor}$ are the strength and correlation length
for the disordered on-site energy, respectively. Uncorrelated
disorder is given by $l_{cor}\to 0$.
Recently the anomaly around the band edge $E=\pm 2$ has been carefully
investigated. \cite{gurevich2011} In the following we focus on
the zero energy anomaly with exponentially correlated diagonal disorder.
All the eigenstates are exponentially localized for one-dimensional
uncorrelated disordered systems. \cite{EAbrahams1979} The Lyapunov
exponent $\gamma$ is the inverse of the localization length.
It is well known that for the zero energy anomaly of the uncorrelated disorder system,
the Lyapunov exponent $\gamma$ is singular at $E=\sigma=0$.
\cite{HSchomerus2002,AStone1983}.
The physical picture behind was also clear\cite{BAltshuler2003}.
For a box distribution of uncorrelated disorder with width $W$
and height $1/W$, the perturbation result revealed that the Lyapunov
exponent depends only on energy $E$ and disorder strength $W$
\cite{DThouless1979}
\begin{equation}
\gamma=\frac{W^2}{96(1-E^2/4)}.\label{eq:sp}
\end{equation}
At the band center, another perturbation yielded
\cite{MKappus1981,BDerrida1984,FIzrailev1998}
\begin{equation}
\gamma=\frac{W^2}{105.045\cdots}.\label{eq:c}
\end{equation}
The standard variance of the disorder is $\sigma^2=W^2/12$.
In uncorrelated systems, order by order
perturbation expansion in $\sigma^2$ and $E/\sigma^2$
has been demonstrated \cite{kkang2011}.
For exponentially correlated disorder, the formula for the Lyapunov
exponent at finite energy and in the weak disorder strength limit
is given by\cite{FIzrailev2001}
\begin{equation}
\gamma = {\frac {\sigma^2 } {8 ({1-{\frac{E^2}{4}}})}} \cdot
{\frac { \sinh{\frac{1}{ l_{cor}}}}
{1+\cosh{\frac{1}{ l_{cor}}}-{\frac{E^2}{2}}}}
\label{eq:gmcor}.
\end{equation}
It is straight forward to take the uncorrelated limit $l_{cor}\to 0$ of
formula Eq.~(\ref{eq:gmcor}), then obtain $\gamma/\sigma^2=1/8$ when $E$
approaches to $0$, i.e.
${\displaystyle
\lim_{E\to 0}\lim_{\sigma^2\to 0} {\frac \gamma {\sigma^2}}=1/8 }$.
On the other hand, if we stay at $E=0$, we should have
$\gamma/\sigma^2=1/8.754$ in the uncorrelated limit in accordance to Eq.~(\ref{eq:c}),
which implies
${\displaystyle
\lim_{\sigma^2\to 0} \lim_{E\to 0} {\frac \gamma {\sigma^2}}=1/8.754 }$.
Therefore, we found that the order of the limiting
processes for $E\to 0$ and $\sigma^2 \to 0$ can not be interchanged.
It means that the point $E=\sigma=0$ remains singular for perturbation
expansions in $\sigma^2$ and $E$ for correlated disorders.
The existence of strong anomalies phenomena in a correlated disorder
system was pointed by Titov and Schomerus\cite{HSchomerus2005}.
In this work we study the anomaly at $E=0$.
\section{Parametrization method}
In the transfer matrix method, Eq.~(\ref{eq:se}) can be written as
\begin{equation}
\Psi_{i+1}=\left(\begin{array}{c} \psi_{i+1} \\ \psi_i
\end{array}\right)= \left(\begin{array}{cc} {E-\epsilon_i} & -1 \\ 1 & 0
\end{array}\right) \left(\begin{array}{c} \psi_i \\ \psi_{i-1}
\end{array}\right)=\mathbf{T}_i\Psi_i, \label{eq:tm}
\end{equation}
where $\mathbf{T}_i$ is the transfer matrix.
Using a parametrization method of the transfer matrix proposed in our
previous work \cite{kkang2010,kkang2011}, we will calculate the Lyapunov
exponent in the thermodynamic limit within the localization regime.
Let $\mathbf{M}_L=\mathbf{T}_L\mathbf{T}_{L-1}\cdots\mathbf{T}_1$.
Then we parameterize $\mathbf{M}\mathbf{M}^t$ as follows
\begin{equation}
\mathbf{U}(\theta_L)\mathbf{M}_L\mathbf{M}^t_L \mathbf{U}(-\theta_L)
=\left(\begin{array}{cc} e^{2\lambda_L} & \\ & e^{-2\lambda_L}
\end{array} \right),
\end{equation}
where $\mathbf{M}^t$ is the transpose of $\mathbf{M}$ and
\begin{equation}
\mathbf{U}(\theta_L)=\left(\begin{array}{cc} \cos\theta_L & -\sin\theta_L
\\ \sin\theta_L & \cos\theta_L \end{array} \right).
\end{equation}
The recursion relation of $\theta$ in the large $L$ limit is
\begin{equation}
\tan\theta_{L+1}=\frac{1}{E-\epsilon_{L+1}-\tan\theta_L}. \label{eq:theta}
\end{equation}
We introduce the correlations between $\epsilon_i$ through the
transformation of a group of independent random variables $\eta_l$
in terms of an identical Gaussian density distribution,
\begin{equation}
p_\eta (\eta)=(1/\sqrt{2\pi}\sigma)\exp[-\eta^2/2\sigma^2].
\end{equation}
Let
$q=e^{-1/l_{cor}}$,
the exponentially correlated variable
is generated implicitly by
$\epsilon_i={\displaystyle {\sqrt{1 - q^2}} \sum_{l=0}^{\infty} }
\eta_{i-l}q^l$, or equivalently in the following recursive form,
\begin{equation}\epsilon_L={\sqrt{1 - q^2}\eta_L} + q \epsilon_{L-1}.
\end{equation}
The three parameters $\lambda$, $\theta$, and $\epsilon$ at a step $L$
are what we need in order to calculate new parameters for the next step
$L+1$.
In the localized region, the equation we obtained for the
density distribution function $p(\theta,\epsilon)$ is
\begin{equation}
p(\theta,\epsilon)=\frac{1}{\sin^2\theta}
\int\mathrm{d}\eta \mathrm{d}\epsilon' \mathrm{d}\theta'
p_\eta(\eta) p(\theta',\epsilon')
\delta(\epsilon-{\sqrt{1 - q^2}\eta} - q \epsilon')
\delta({\frac 1 {\tan\theta}} + \tan\theta' -E +\epsilon).
\label{eq:theta_distri0}
\end{equation}
After we numerically solve this equation, the
Lyapunov exponent $\gamma$ can be calculated through the following
formula,
\begin{equation}
\gamma={\frac 1 2 }
\int\mathrm{d}\eta \mathrm{d}\epsilon \mathrm{d}\theta
p_\eta(\eta) p(\theta,\epsilon)
\ln [1-(E-{\sqrt{1 - q^2}\eta} - q \epsilon)\sin 2\theta
+(E-{\sqrt{1 - q^2}\eta} - q \epsilon)^2\cos^2 \theta ].
\label{eq:length0}
\end{equation}
By defining the distribution function $p(\theta) $, which is
similar to the one in the uncorrelated diagonal disorder case,
\begin{equation}
p(\theta)=\int p(\theta,\epsilon)\mathrm{d}\epsilon,
\end{equation}
we obtain the same simple relationship between $p(\theta) $
and the Lyapunov exponent,
\begin{equation}
\gamma=-\int p(\theta)\ln |\tan \theta |\mathrm{d}\theta.
\label{eq:length}
\end{equation}
If we take the limit of $l_{cor} \to 0$ in the present correlated disorder situation,
the equations of $p(\theta)$ in the uncorrelated disorder case will
be recovered\cite{kkang2010,kkang2011}. However, $p(\theta,\epsilon)$
is not exactly the product of $p(\theta)p_\eta(\epsilon)$ in this limit.
\begin{figure}
\includegraphics[width=12cm]{fig01.eps}
\caption{Distribution $p(\theta,\epsilon)$ for $E=l_{cor}=\sigma=1$.
The forty lines in $\theta$ direction are evenly spaced in the region
$[-\pi/2,\pi/2]$. The forty lines in $\epsilon$ direction are scaled
to display a better global view.
} \label{fig:figure1}
\end{figure}
\begin{figure}
\includegraphics[width=12cm]{fig02.eps}
\caption{Distributions $p(\theta)$ at $E=1$. The full line is
for $l_{cor}=\sigma=1$; and dotted line for $l_{cor}=\sigma=0.01$.
Each point on the dotted line is differed
from the analytical curve $\displaystyle {\frac { \sqrt{3} }{\pi(2-\sin 2 \theta)}}$
within no more than a relative error $10^{-4}$.
} \label{fig:figure2}
\end{figure}
We use the Gaussian distribution $p_\eta$ to solve
Eq.~(\ref{eq:theta_distri0}) and to calculate $\gamma$
numerically. This method is very efficient to yield high precision results for
various disorder correlation length $l_{cor}$, disorder
strength $\sigma^2$, and energy $E$ in the thermodynamic limit.
Fig.~\ref{fig:figure1} and Fig.~\ref{fig:figure2} are shown
the calculated distribution functions $p(\theta,\epsilon)$
and $p(\theta)$, respectively. In these calculations we have set a relative
precision $10^{-10}$ for $p(\theta,\epsilon)$. Similar
distributions were calculated recently \cite{tkaya2009}
in the dichotomous correlated disorder case.
The structure of the joint density distribution of $\theta$
and $\epsilon$ is demonstrated by the $p(\theta,\epsilon)$
of $E=l_{cor}=\sigma=1$ in Fig.~\ref{fig:figure1}. The
distribution is not so complicated to perceive, but it can not
be decomposed into a direct product of a density distribution
for $\theta$ and a density distribution for $\epsilon$. Two
curves for $p(\theta)$ of $E=1$ are given in Fig.~\ref{fig:figure2}.
One of the curve with $l_{cor}=\sigma=0.01$ has very small disorder
strength $\sigma$ and very small disorder correlation $l_{cor}$.
This curve can be approximated very well by the expression for
distribution $p(\theta)$ of uncorrelated disorder at a finite $E$ in
the weak disorder limit \cite{MKappus1981,FIzrailev1998,kkang2011},
\begin{equation}
p(\theta)=\frac{\sin\mu}{\pi(1-\cos\mu\sin 2\theta)}, \label{eq:uni},
\end{equation}
where $\cos \mu = E/2$.
Another curve with $l_{cor}=\sigma=1$ is not in the case for small disorder
strength or small disorder correlation, which is different from
the curve in small disorder strength or small disorder correlation.
The Lyapunov exponent $\gamma(l_{cor},\sigma)$ is then calculated
by using the two curves shown in Fig.~\ref{fig:figure2} at $E=1$. We obtain
$\gamma(1,1)=0.1252$ and $\gamma(0.01,0.01)=0.00001667$.
The direct calculated results from formula Eq.~(\ref{eq:gmcor}) yield
$\gamma(1,1)=0.09587$
and $\gamma(0.01,0.01)=0.00001667$. We see the numerically
calculated Lyapunov exponent for the finite energy and in the weak
disorder strength limit is well predicted by formula Eq.~(\ref{eq:gmcor}).
The situation for zero energy is different compared to that for the
finite energy. There is no analytical result obtained so far for the zero
energy anomaly in the presence of correlated disorder in the weak disorder limit;
nor the formula predicting the Lyapunov exponent for a finite
correlation length. Our method is a good choice to perform calculation in these
situations.
\section{Anomaly at $E=0$}
We will investigate how the localization length changes
as the correlation of the disorder varies at $E=0$ in the weak
disorder limit. At finite energy, the disorder strength $\sigma$
and the correlation $\l_{cor}$ are decoupled
in function $\gamma(l_{cor},\sigma)$ in Eq.~(\ref{eq:gmcor}).
Since Eq.~(\ref{eq:gmcor}) was derived without any limitation on
the magnitude of the correlation length, with the help of the
Lyapunov exponent $\gamma(0,\sigma)$ for uncorrelated disorder,
the ratio
$\gamma(l_{cor},\sigma)/\gamma(0,\sigma)=\tanh{\frac{1}{2 l_{cor}}}$
might be exactly held for any $l_{cor}$.
At $E=0$ anomaly, even if we keep only $\sigma^2$ term in the weak
disorder limit, it is not known whether higher order terms from
correlation exists beyond the perturbation result.
To answer this question, we compare the numerically calculated result
with the perturbation one given by Eq.~(\ref{eq:gmcor}) at $E=0$:
\begin{equation}
\gamma_p = {\frac {\sigma^2 } {8 }}\tanh{\frac{1}{2 l_{cor}} }
\label{eq:gmcor0}.
\end{equation}
The small quantity related to correlation in $\gamma_p$ can be
considered in two limit cases. In the short correlation length limit
$l_{cor}\to 0$, i.e. $\tanh{\frac{1}{2 l_{cor}}} \to
1$, the small quantity for expansion is $1-\tanh{\frac{1}{2 l_{cor}}}
\sim 2e^{-1/l_{cor}}$; whereas in the large correlation length limit
$l_{cor}\to \infty$, the small quantity for expansion is
$\tanh{\frac{1}{2 l_{cor}}}$ itself, $\tanh{\frac{1}{2 l_{cor}}}
\sim {\frac{1}{2 l_{cor}}}$. Therefore, we will calculate for a
group of different correlations with $\tanh{\frac{1}{2 l_{cor}}}$
close to zero as well as to one. In order to neglect the contribution from the
higher order terms of $\sigma$ in our calculation in the weak
disorder limit, we will calculate only for small $\sigma$. It is sufficient to
keep three significant digits for the Lyapunov exponent $\gamma(
l_{cor},\sigma)$.
\begin{figure}
\includegraphics[width=12cm]{fig03.eps}
\caption{Distribution $p(\theta,\epsilon)$ for $E=0$,
$\tanh{\frac{1}{2 l_{cor}}}=0.475$, and $\sigma=0.1$.
The forty lines in $\theta$ direction are evenly spaced in between
$[-\pi/2,\pi/2]$. The forty lines in $\epsilon$ direction are scaled
to give a better global view.
}
\label{fig:figure3}
\end{figure}
To demonstrate the anomalous behavior at $E=0$, we plot
in Fig.~\ref{fig:figure3} the distribution $p(\theta,\epsilon)$
for $E=0$, $\tanh{\frac{1}{2 l_{cor}}}=0.475$, and $\sigma=0.1$;
and in Fig.~\ref{fig:figure4} the distributions $p(\theta)$ for $E=0$
and $\sigma=0.1$ with $x=1-\tanh{\frac{1}{2 l_{cor}}}=$ $0.025$,
$0.075$, $0.125$, $\ldots$, $0.925$, and $0.975$, respectively.
It shows clearly in Fig.~\ref{fig:figure3} that the joint distribution
for $\theta$ and $\epsilon$ has some inner structure.
We have observed the flattening of the distribution
$p(\theta,\epsilon)$ when increasing the correlation length $l_{cor}$
in the weak disorder limit. The flattening will not be presented
in Fig.~\ref{fig:figure3}.
The flattening of $p(\theta)$ can be seen in Fig.~\ref{fig:figure4}.
In the figure, when correlation length is small, the
distribution $p(\theta)$ turns out to be similar to the distribution for the
uncorrelated disorder\cite{CBarnes1990,FIzrailev1998,kkang2011}:
\begin{equation}
p(\theta)=\frac{1}{K(1/2)\sqrt{3+\cos4\theta}},
\end{equation}
where $K$ is the complete elliptic integral of the first kind.
As the correlation increases, we see that the distribution $p(\theta)$
flattened towards $1/\pi$. Let's take $x=1-\tanh{\frac{1}{2 l_{cor}}}$
as a new parameter of the correlation in disorder.
In the limit of $x \to 0$, which corresponds to the
uncorrelated limit $l_{cor} \to 0$, both the anomalous distribution
$1/\sqrt{3+\cos4\theta}$ and the anomalous Lyapunov exponent
$\gamma=\sigma^2/8.754$ of the uncorrelated disorder will be recovered.
In the limit of $x \to 1$, which is equivalent to the large correlation limit
$l_{cor}\to \infty$, $p(\theta)=1/\pi$ will correctly give
a zero Lyapunov exponent.
\begin{figure}
\includegraphics[width=12cm]{fig04.eps}
\caption{Distributions $p(\theta)$ for $E=0$ and $\sigma=0.1$.
The forty lines in $\theta$ direction are evenly spaced in between
$[-\pi/2,\pi/2]$. The twenty lines in $x$ direction are for
$x=1-\tanh{\frac{1}{2 l_{cor}}}=$ $0.025$, $0.075$, $0.125$, $\ldots$,
$0.925$, $0.975$, respectively.
} \label{fig:figure4}
\end{figure}
\section{High order terms of correlation at $E=0$}
Now we analyze the contribution from higher order terms of the
correlation in the weak disorder limit. In Ref.~\cite{HSchomerus2003}
the authors gave analyses, which cover not only the localization length, but also
all the higher moments of the distribution of the Lyapunov exponent for
uncorrelated finite systems. For correlated systems we expect
the deviation from Eq.~(\ref{eq:gmcor0}) comes from higher order
terms of correlation too.
In the perturbation result $\gamma_p$ in Eq.~(\ref{eq:gmcor0}),
by using variable $x=1-\tanh{\frac{1}{2 l_{cor}}}$ to denote the correlation,
we see that $\gamma_p$ included the first order correction of small
$x$ when $x\to 0$ and also the first order correction of small $1-x$
when $x\to 1$. $\gamma_p$ has included only the
first order term. From the discussion on $E=0$ anomaly in the previous section
we know that $\gamma/(\sigma^2\tanh{\frac{1}{2 l_{cor}}}) = 1/8.754$
for $x\to 0$, while $\gamma/(\sigma^2\tanh{\frac{1}{2 l_{cor}}}) = 1/8$
is predicted by perturbation result for $E\to0$. The question on how
$\gamma$ really behaves at $E=0$ is still not answered: whether
$\gamma/(\sigma^2\tanh{\frac{1}{2 l_{cor}}}) = 1/8.754$ always holds, or
there is a crossing to $\gamma/(\sigma^2\tanh{\frac{1}{2 l_{cor}}}) = 1/8$
as $ l_{cor}$ increases. We plot Fig.~\ref{fig:figure5} to answer
this question.
\begin{figure}
\includegraphics[width=12cm]{fig05.eps}
\caption{The Lyapunov exponent $\gamma$ for $E=0$ and $\sigma=0.1$.
The variable $x$ used for different correlations is
$x=1-\tanh{\frac{1}{2 l_{cor}}}$. The function is
$y=\gamma/(\sigma^2 \tanh{\frac{1}{2 l_{cor}}})$.
When $x$ is close to zero, $y$ is close to $1/8.754$;
and when $x$ is close to one, $y$ is close to $1/8$.
} \label{fig:figure5}
\end{figure}
In Fig.~\ref{fig:figure5}, the Lyapunov exponent $\gamma$ for $E=0$
and $\sigma=0.1$ are presented. We plot for different correlations by
using the parameter $x=1-\tanh{\frac{1}{2 l_{cor}}}$, and we plot
$y=\gamma/(\sigma^2 \tanh{\frac{1}{2 l_{cor}}})$ as the function of $x$.
When $x$ is close to zero, $y$ is close to $1/8.754$; and when $x$
is close to one, $y$ is close to $1/8$. We see a crossover between
the anomalous value $1/8.754$ and the perturbation result $1/8$.
In the weak disorder limit, besides the term $\tanh{\frac{1}{2 l_{cor}}}$,
there are higher order terms in $x$ or $1-x$ from the correlation.
The higher order terms connect smoothly the anomalous $1/8.754$
at zero correlation with the perturbation result $1/8$ for large
correlation length.
The physical picture is rich behind a finite magnitude of $\sigma$
and a large correlation length. The $\sigma$ in Fig.~\ref{fig:figure5}
is not a small enough disorder strength. The higher order terms
in $\sigma^2$ contributes when $x$ approaches one in
Fig.~\ref{fig:figure5}. We have calculated
for much smaller $\sigma$ and confirmed that the contribution of higher
order terms in $\sigma^2$ goes to zero in the weak disorder limit. Our
observation suggests further perturbation investigations.
To numerically provide the next leading term of the correlation
closed to the uncorrelated limit, we fit $\gamma$ for
$x$ close to zero in Fig.~\ref{fig:figure6}. In Fig.~\ref{fig:figure6}
the Lyapunov exponent $\gamma$ for $E=0$ and $\sigma=0.01$ is plotted.
$y$ represents the difference between the Lyapunov exponent for a
finite correlation and for zero correlation:
$y=\gamma(l_{cor},\sigma)/(\sigma^2 \tanh{\frac{1}{2 l_{cor}}})-1/8.754$.
The variable $x$ used for different correlations is
$x=1-\tanh{\frac{1}{2 l_{cor}}}$. We obtain a fitting line
$y = 0.01533 x$. Therefore the perturbation expansion of $\gamma$
to the sub-leading order of the correlation in power of $x$ is obtained,
\begin{equation}
\gamma = ( 1 + 0.1342 x )
{\frac {\sigma^2 } {8.754 }}
\tanh{\frac{1}{2 l_{cor}} }
\label{eq:gm fit}.
\end{equation}
In Fig.~\ref{fig:figure5} it is clear that higher order terms contributes
when $x$ is even bigger. In the weak disorder limit,
when $1-x$ close zero, the next order term in the correction
factor used to multiply to $\gamma_p$ in Eq.~(\ref{eq:gmcor0}) is $(1-x)^2$.
\begin{figure}
\includegraphics[width=12cm]{fig06.eps}
\caption{Fitting of the Lyapunov exponent $\gamma$ for
$E=0$ and $\sigma=0.01$. $y$ is certain the difference between the
Lyapunov exponent for finite correlation and zero correlation:
$y=\gamma(l_{cor},\sigma)/\sigma^2/\tanh{\frac{1}{2 l_{cor}}}-1/8.754$.
The variable $x$ used for different correlations is
$x=1-\tanh{\frac{1}{2 l_{cor}}}$. The fitting line is $y = 0.01533 x$.
} \label{fig:figure6}
\end{figure}
\section{Conclusion}
In summary, we calculated the inverse localization length in
one-dimensional Anderson model with correlated diagonal disorder.
We obtained numerically the curve of the inverse localization length
for correlations at zero energy in the case of weak disorder. A
nonsingular curve was obtained for different correlation lengths in
the weak disorder limit at zero energy.
The variable used to plot the unifying curve is
$\tanh{\frac{1}{2 l_{cor}}}$, which has correspondence to
the Poisson process of the phase accumulation. The inverse
localization length will be singular as the function of other
variables as $l_{cor}$, $1/l_{cor}$, or $e^{-1/l_{cor}}$.
We suggest further studies on the inverse localization length in
perturbation expansions or functional expansions
with the parameter $\tanh{\frac{1}{2 l_{cor}}}$.
We have obtained numerically in this work the next leading term
for comparison.
We also saw rich behavior for finite disorder strength and large
correlation length. A unifying description of the band center
anomaly and the correlated disorder will be very interesting.
\begin{acknowledgments}
This work was supported by National Natural Science Foundation of
China No.~10374093,
and the Knowledge Innovation Project of Chinese Academy of Sciences.
\end{acknowledgments}
|
{
"timestamp": "2012-03-09T02:02:15",
"yymm": "1203",
"arxiv_id": "1203.1746",
"language": "en",
"url": "https://arxiv.org/abs/1203.1746"
}
|
\section{Introduction}
Let ${\rm PG}(2,q)$, $q=p^h$, $p$ a prime, denote the Desarguesian projective plane
of order $q$. A {\em maximal arc} of {\em degree} $n$ is a set of points of
${\rm PG}(2,q)$ meeting every line in either $0$ or $n\le q$ points. For example, a
point or the complement of a line are maximal arcs; these are called {\em
trivial} maximal arcs. In \cite{bbm} it was proved that no non-trivial
maximal arc exists in ${\rm PG}(2,q)$, with $q$ odd. Instead, in \cite{d},
Denniston constructed maximal arcs in ${\rm PG}(2,q)$, $q$ even, each of which
is the union of
irreducible conics from a partial pencil plus their common nucleus \cite{d}.
In \cite{t1,t2} J.A. Thas constructed two classes of maximal arcs of
${\rm PG}(2,q)$, $q$ even. In \cite{hp} it was proved that some of the maximal arcs
in the first class as well as all maximal arcs of the second class are of
Denniston type. Many years later, Mathon studied the following problem:
{\em Do there exist other maximal arcs in ${\rm PG}(2,q)$, each of which is
the union of conics plus their common nucleus?}
In his paper \cite{m} he gave a positive answer by constructing the
{\em Mathon maximal arcs}.
In this paper we deal with a similar problem about unitals of ${\rm PG}(2,q^2)$.
A {\em unital} in ${\rm PG}(2,q^2)$ is a set $\mathcal U$ of $q^3+1$ points such that each
line meets $\mathcal U$ in 1 or $q+1$ points. A line of ${\rm PG}(2,q^2)$ is a {\em
tangent} or {\em secant} line to $\mathcal U$ according if it contains 1 or $q+1$
points of $\mathcal U$. Through each point of $\mathcal U$, there is exactly one tangent and
$q^2$ secants to $\mathcal U$, while through each point not in $\mathcal U$, there are $q+1$
tangents and $q^2-q$ secant lines.
An example of a unital is given by the set of absolute points of a
non-degenerate unitary polarity of ${\rm PG}(2,q^2)$. This is a {\em
classical} or {\em Hermitian unital}.
In \cite{b,m} Buekenhout and Metz constructed non-classical unitals by using
the Andr\`e/Bruck--Bose representation of ${\rm PG}(2,q^2)$ in ${\rm PG}(4,q)$ for
$q>2$. These unitals are {\em Buekenhout--Metz unitals}. In \cite{be,hs} a
nice geometric description in ${\rm PG}(2,q^2)$, $q$ odd, was given for some of
these unitals. For $a\in{\rm GF}(q^2)$, consider the conic $\mathcal C_a$ with equation
$2yz-x^2+az^2=0$. The set $\{\mathcal C_a: a\in {\rm GF}(q^2)\}$ is a hyperosculating
pencil with base point $(0,1,0)$. Let $t$ be a fixed non-square of ${\rm GF}(q^2)$.
Then the set
\[
\mathcal U=\bigcup_{a\in t{\rm GF}(q)}{\mathcal C_a}
\]
turns out to be a
Buekenhout--Metz unital that we call of {\em
Baker--Ebert--Hirschfeld--Sz\H{o}nyi type} or {\em BEHS-type} for short. The
following question arises:
{\em Do there exist other unitals of ${\rm PG}(2,q^2)$ which are unions of conics?}
The answer is negative.
\begin{theorem}\label{th_2}
Let $\mathcal U$ be a unital of ${\rm PG}(2,q^2)$ and suppose that $\mathcal U$ is a union of
conics. Then $q$ is odd and $\mathcal U$ is a Buekenhout--Metz unital of BEHS-type.
\end{theorem}
\section{Proof of Theorem \ref{th_2}}
Let $\mathcal U$ be an unital of $PG(2,q^2)$ and let $\mathcal C$ be an irreducible conic
contained in $\mathcal U$. For every point $P$ of $\mathcal C$, the tangent at $P$ to $\mathcal C$
coincide with the tangent at $P$ to $\mathcal U$ .
For $q$ even, the tangents to $\mathcal C$ all contain a common
point $\mathcal N$, the nucleus of $\mathcal C$, \cite[Chapter 7]{h}. Thus there would be $q^2+1$
tangents to $\mathcal U$ on $\mathcal N$, a contradiction. Hence, we conclude that if $\mathcal U$
contains an irreducible conic, then $q$ must be odd.
{}From now on, $q$ is an odd prime power and $\mathcal U$ a union of irreducible
conics.
In \cite{pr} Penttila and Royle gave a complete classification of
two-intersection sets in the projective planes of order $9$. From this
classification, the Buekenhout--Metz unitals of BEHS-type are the only unitals
in ${\rm PG}(2,9)$ containing conics. Thus we may assume $q>3$.
In ${\rm PG}(2,q)$, equipped with the homogeneous coordinates $(x,y,z)$, any conic
$\mathcal C$ is defined by the equation
\begin{equation}\label{eq_1}
f(x,y,z)=a_{11}x^2 +a_{22}y^2 + a_{33}z^2+ 2a_{12}xy + 2a_{13}xz
+ 2a_{23}yz = 0
\end{equation}
and the associated symmetric matrix is
\[
A(\mathcal C)=\begin{pmatrix}
a_{11} & a_{12}& a_{13} \\
a_{12} & a_{22}& a_{23} \\
a_{13} & a_{23}& a_{33}
\end{pmatrix}.
\]
The {\em rank} of $\mathcal C$ is the rank of the matrix $A(\mathcal C)$. Conics of rank 3
are said to be {\em irreducible} or {\em non-singular}. Singular conics are of
two types: a pair of distinct lines (when the associated matrix has rank 2)
and a repeated line (when the associated matrix has rank 1).
If $\mathcal C$ is irreducible, the points that are not in $\mathcal C$ split in two sets:
the set $\mathfrak E(\mathcal C)$ of {\em external} points, lying on two tangents to $\mathcal C$
and the set $\mathfrak I(\mathcal C)$ of {\em internal} points, lying on no tangent to $\mathcal C$.
\begin{theorem}\label{th_3}
{\rm\cite{s}}
Let $\mathcal C: f(x,y,z)=0$ be a irreducible conic of ${\rm PG}(2,q)$, $q$ odd. Then a point $(x,y,z)$ is in $\mathfrak E(\mathcal C)$ if and only if $-{\rm det}(A(\mathcal C)) \cdot f(x,y,z)$ is a non-zero square in $
{\rm GF}(q)$.
\end{theorem}
Fix an irreducible conic $\mathcal C$ in ${\rm PG}(2,q)$. In \cite{afkl}, irreducible
conics such that the points not in $\mathcal C$ are all in $\mathfrak I(\mathcal C)$ are described.
More precisely, the following theorem is proved.
\begin{theorem}\label{th_1}{\rm\cite{afkl}}
Let $\mathcal C$ and $\mathcal D$ be two irreducible conics of ${\rm PG}(2,q),\, q$ odd,
$q\ge 17$, such that $\mathcal D\setminus\mathcal C$ has empty intersection with $\mathfrak E(\mathcal C)$. Then the points of $\mathcal C\setminus\mathcal D$ consists entirely of internal points of $\mathcal D$ and one of the following cases occur:
\begin{enumerate}[\rm(i)]
\item $\mathcal C\cap \mathcal D=\{P,Q\}$, $\mathcal C$ and $\mathcal D$ being two conics of a
bitangent pencil at $P$ and at $Q;$
\item $\mathcal C \cap \mathcal D =\emptyset$, $\mathcal C$ and $\mathcal D$ being two conics of a
bitangent pencil at $P$ and at $Q$, the two common points of $\mathcal C$ and $\mathcal D$
in the quadratic extension ${\rm PG}(2,q^2)$ of ${\rm PG}(2,q);$
\item $\mathcal C\cap \mathcal D=\{P\}$, $\mathcal C$ and $\mathcal D$ being two conics of a
hyperosculating pencil at $P$.
\end{enumerate}
\end{theorem}
It is worth pointing out that all the above pencils contain a conic of rank
1.
The stabilizer of $\mathcal C$ in the group ${\rm PGL}(3,q)$ of the
linear collineations of ${\rm PG}(2,q)$ has three orbits on points of ${\rm PG}(2,q)$,
namely, $\mathcal C$ itself, $\mathfrak E(\mathcal C)$ and $\mathfrak I(\mathcal C)$. Dually, there are three
orbits on lines, namely, the tangent lines to $\mathcal C$, the secant lines to
$\mathcal C$ and the external lines to $\mathcal C$. Since every line of ${\rm PG}(2,q)$ can be
viewed as a conic of rank 1, we can fix a projective frame such that the
conic $\mathcal C$ and the pencils in Theorem \ref{th_1} have the following forms.
\begin{enumerate}[\rm(i)]
\item $\mathcal C$ is the hyperbola $2xy=z^2$ and the pencil consists of the conics in the family $2xy=kz^2$, $k\in{\rm GF}(q)$, plus the repeated line $z^2=0$. The points $P$ and $Q$ are the points at infinity of $\mathcal C$.
\item $\mathcal C$ is the circle $x^2-\alpha y^2=z^2$ where $\alpha$ is a fixed non-square of ${\rm GF}(q)$ and the pencil consists of the conics in the family $x^2-\alpha y^2=kz^2$, $k\in{\rm GF}(q)$, plus the repeated line $z^2=0$. The points $P$ and $Q$ are the points at infinity of $\mathcal C$ in ${\rm PG}(2,q^2)$.
\item $\mathcal C$ is the parabola $2yz=x^2$ and the pencil consists of the conics in the family $2yz=x^2+kz^2$, $k\in{\rm GF}(q)$, plus the repeated line $z^2=0$. The point $P$ is the point at infinity of $\mathcal C$.
\end{enumerate}
Assume now that $\mathcal C$ is contained in $\mathcal U$. Since every tangent to $\mathcal C$ is
also a tangent to $\mathcal U$ we see that $\mathcal U\setminus \mathcal C$ is contained in
$\mathfrak I(\mathcal C)$.
In what follows we will use the representation of conics of ${\rm PG}(2,q)$ as
points of ${\rm PG}(5,q)$. We also recall some relevant properties of the Veronese
surface of ${\rm PG}(5,q)$. For a fuller treatment we refer the reader to
\cite[Chapter 25]{ht}.
If the 5-dimensional projective space ${\rm PG}(5,q)$ is equipped with the
homogeneous coordinates $(a_{11},a_{22}, a_{33}, a_{12}, a_{13},a_{23})$, the
conic $\mathcal C$ with equation (\ref{eq_1}) defines the point
$P(\mathcal C)=(a_{11},a_{22}, a_{33}, a_{12}, a_{13},a_{23})$ of ${\rm PG}(5,q)$, and
conversely. Under this 1-1 correspondence, the set of singular conics defines
the hypersurface with equation ${\rm det}(A)=0$ of ${\rm PG}(5,q)$, where $A$ is the
matrix associated with the generic conic $\mathcal C$, and the set of rank 1 conics
defines the Veronese surface
\[
\mathcal V=\{(a^2,b^2,c^2,ab, ac,bc):a,b,c\in{\rm GF}(q),
(a,b,c)\neq (0,0,0)\}.
\]
It is also easy to check that the representations in
${\rm PG}(5,q)$ of the pencils of conics of type (i), (ii), (iii), are lines
intersecting the Veronese surface $\mathcal V$ at $P=(0,0,1,0,0,0)$. Further, every
conic $\mathcal C$ with rank $>1$ determines the cone $\Gamma(\mathcal C)$ projecting $\mathcal V$
from $P(\mathcal C)$.
If $\mathcal D$ is a second conic in $\mathcal U$, then $\mathcal D\setminus\mathcal C$ has empty
intersection with $\mathfrak E(\mathcal C)$. This implies that the pencil determined by $\mathcal C$
and $\mathcal D$ in ${\rm PG}(2,q^2)$ is one of those described in Theorem \ref{th_1}. We
also observe the symmetric relationship between the conics $\mathcal C$ and $\mathcal D$: if
all points of $\mathcal D\setminus\mathcal C$ are in $\mathfrak I(\mathcal C)$ then all points of
$\mathcal C\setminus\mathcal D$ are in $\mathfrak I(\mathcal D)$.
It is clear that the cones $\Gamma(\mathcal C)$ and $\Gamma(\mathcal D)$ share the line $P(\mathcal C)P(\mathcal D)$ and $\mathcal V$.
By Theorem \ref{th_1}, for every other conic $\mathcal E$ contained in $\mathcal U$ the point $P(\mathcal E)$ is contained in the intersection of the cones $\Gamma(\mathcal C)$ and $\Gamma(\mathcal D)$.
Since $\mathcal U$ does not contain lines of ${\rm PG}(2,q^2)$, then no point of $\mathcal V$ represents a conic contained in $\mathcal U$.
Hence we are reduced to studying which point in $(\Gamma(\mathcal C)\cap\Gamma(\mathcal D))\setminus \mathcal V$ represents a conic in $\mathcal U$. We will do this by considering the above three cases for the pencil defined by $\mathcal C$ and $\mathcal D$.
{\bf Case 1.} {\em $\mathcal C:2xy=z^2$ and $\mathcal D:2xy=kz^2$ for some
$k\in{\rm GF}(q^2)\setminus\{0,1\}$}.
The intersection points between $\mathcal C$ and $\mathcal D$ are $P=(1,0,0)$ and
$Q=(0,1,0)$. Further, we have ${\rm det}(A(\mathcal C))=1$ and ${\rm det}(A(\mathcal D))=k$. By
Theorem \ref{th_3}, every point of $\mathcal D\setminus\mathcal C$ is in $\mathfrak I(\mathcal C)$ if and
only if $k-1$ is a non-square in ${\rm GF}(q^2)$. By the symmetric relationship
between the conics $\mathcal C$ and $\mathcal D$, we have that $k(k-1)$ is a non-square of
${\rm GF}(q^2)$. Hence, $k$ is a non-zero square of ${\rm GF}(q^2)$.
We first consider the irreducible conics of the pencil defined by $\mathcal C$ and
$\mathcal D$. If $\mathcal E: 2xy=hz^2$, $h\neq 1,k,$ is contained in $\mathcal U$, then $h$ is a
non-zero square in ${\rm GF}(q^2)$ and $h-1$, $h-k$ are non-squares in ${\rm GF}(q^2)$.
Hence such a set of conics determines a subset $X$ of ${\rm GF}(q^2)$ such that
$1\in X$, all elements of $X$ are non-zero squares and for any $h,k\in X$,
$h-k$ is a non-square.
\begin{lemma} \label{lem_1}
Let $X$ be a subset of ${\rm GF}(q^2)$ of non-zero squares with the property that
the difference of any two elements is always a non-square. Then $X$ has at
most $(q+1)/2$ elements.
\end{lemma}
\begin{proof}
As usual we represent ${\rm GF}(q^2)$ as the affine plane ${\rm AG}(2,q)$. The lines of
this plane are subsets of ${\rm GF}(q^2)$ with the property that the difference of
two elements is either always a square, or always a non-square, depending only
on slope of the line. Thus the lines are partitioned into two classes, square
type $S$ and non-square type $N$. Through each point of ${\rm AG}(2,q)$ there pass
$(q+1)/2$ lines of $S$ and $(q+1)/2$ lines of $N$. Hence, on an arbitrary line
$L$ of $S$ not passing through the origin $O$, there are $(q+1)/2$ non-squares,
since the line parallel to $L$ containing the origin is also in $S$.
Let $A$ and $B$ two distinct points of $X$ collinear with the origin. Then
$A-B$ is always a square, a contradiction. This implies that on each line of
type $S$ on $O$ there is at most one point of $X$. Then $X$ contains at most
$(q+1)/2$ points.
\end{proof}
A consequence of this lemma is that the conics of the pencil defined by $\mathcal C$
and $\mathcal D$ cover at most $2+(q^2-1)(q+1)/2$ points of $\mathcal U$. Since $q>3$, in
order to cover the remaining points of $\mathcal U$ we need more then one conic not
in the pencil defined by $\mathcal C$ and $\mathcal D$. So we investigate $\Gamma(\mathcal C)\cap
\Gamma(\mathcal D) \setminus (P(\mathcal C) P(\mathcal D) \cup \mathcal V)$.
It is easily seen that the points of $\Gamma(\mathcal C)$ and $\Gamma(\mathcal D)$ not in
the surface $\mathcal V$ are
\begin{eqnarray*}
&&\hspace*{-8mm}\{(sa^2,sb^2,1+sc^2,-1+sab,sac,sbc):a,b,c,s\in{\rm GF}(q^2),
(a,b,c)\neq(0,0,0)\},\\
&&\hspace*{-8mm}\{(ta'^2,tb'^2,k+tc'^2,-1+ta'b',ta'c',tb'c'):a',b',c',t\in{\rm GF}(q^2),
(a',b',c')\neq(0,0,0)\}.
\end{eqnarray*}
It is worth pointing out that the points of the line
$P(\mathcal C)P(\mathcal D)$ are those for which $(a,b,c)=(a',b',c')=(0,0,1)$. Further, we
get $P(\mathcal C)$ and $P(\mathcal D)$ also for $s=0$ and $t=0$, respectively. In the
following we assume $(a,b,c)\neq(0,0,1)\neq(a',b',c')$ and $st\neq0$.
Then the points in $(\Gamma(\mathcal C)\cap\Gamma(\mathcal D))\setminus(P(\mathcal C)P(\mathcal D)\cup\mathcal V)$ satisfy the following equations:
\begin{equation}\label{eq_2}
\begin{split}
sa^2 &=\rho ta'^2 \\
sb^2 &=\rho tb'^2 \\
1+sc^2&=\rho(k+ tc'^2) \\
-1+sab&=\rho(-1+ ta'b')\\
sac &=\rho ta'c'\\
sbc &=\rho tb'c'
\end{split}
\end{equation}
{}for some $\rho\in{\rm GF}(q^2)^*$.
{}First we consider $abc\neq0$. {}From Equations (\ref{eq_2}), we get
$a'b'c' \ne 0$ and
\[
\frac{s}{t} = \frac{\rho a'^2}{a^2} =
\frac{\rho a'c'}{ac}= \frac{\rho b'c'}{bc},\ {\rm i.e.}\
\frac{a'}{a} = \frac{b'}{b} =\frac{c'}{c}.
\]
This implies that the generator with base point $(a,b,c)$ of $\Gamma(\mathcal C)$ meets $\Gamma(\mathcal D)$ on $\mathcal V$. Hence there are no points in $(\Gamma(\mathcal C)\cap\Gamma(\mathcal D))\setminus(P(\mathcal C)P(\mathcal D)\cup\mathcal V)$ with $abc\neq0$.
Now suppose that $a=0$ and $bc\neq0$. Equations (\ref{eq_2}) reduce to
\begin{equation}
\begin{split}
0 &= a'^2 \\
sb^2 &=\rho tb'^2 \\
1+sc^2&=\rho(k+ tc'^2) \\
-1&=\rho(-1+ ta'b')\\
0 &= a'c'\\
sbc &=\rho tb'c'.
\end{split}
\end{equation}
It follows immediately that $a'=0$ and $b'c'\neq0$; hence we get the same
conclusion as before.
The same reasoning applies to the case $b=0$ and $ac\neq0$.
If $c=0$ and $ab\neq0$, Equations (\ref{eq_2}) reduce to
\begin{equation}\label{eq_3}
\begin{split}
sa^2 &=\rho ta'^2 \\
sb^2 &=\rho tb'^2 \\
1&=\rho(k+ tc'^2) \\
-1+sab&=\rho(-1+ ta'b')\\
0 &= a'c'\\
0 &= b'c'.
\end{split}
\end{equation}
As $ab\neq0$ we have $a'b'\neq0$, $c'=0$ and $\rho=k^{-1}$. We can assume that $a=1=a'$. Thus Equations (\ref{eq_3}) reduce to
\begin{equation}\label{eq_5}
\begin{split}
s&=k^{-1} t \\
b^2&=b'^2 \\
-1+sb&=-k^{-1}+k^{-1}tb'.
\end{split}
\end{equation}
Since $b^2=b'^2$, we have either $b=b'$ or $b=-b'$. If $b=b'$ we get $k=1$, a contradiction. Hence $b=-b'$. Then $t=(1-k)/2b'$ and it is easy to check that the cones $\Gamma(\mathcal C)$ and $\Gamma(\mathcal D)$ share the points $P_{\mathcal E_{b'}}=(1-k,(1-k)b'^2,2kb',-(k+1)b',0,0))$, $b'\in{\rm GF}(q^2)$. We note that the conic \begin{equation}\label{eq_4}
\mathcal E_{b'}:(1-k)x^2+(1-k)b'^{2}y^2+2kb'z^2-2(1+k)b'xy=0,
\end{equation}
has rank 3 for all $b'$.
We proceed by considering separately the cases $b'$ a non-square and $b'$ a
non-zero square.
Let $b'$ be a non-square of ${\rm GF}(q^2)$. As $k(k-1)$ is a non-square, we see
that the line $x=0$ intersects $\mathcal E_{b'}$ in $(0,\bar y,1)$, with $\bar
y=\sqrt{\frac{2k}{(k-1)b'}}$. By Theorem \ref{th_3}, we have that $(0,\bar
y,1)$ is in $\mathfrak E(\mathcal C)$. Then $\mathcal E_{b'}$ cannot be contained in $\mathcal U$.
We now turn to the case $b'$ a non-zero square. Assume that $\mathcal E_{b'}$ is
contained in $\mathcal U$. As $q>3$, $\mathcal U$ contains another conic $\tilde\mathcal E$. By
applying the same reasoning to $\tilde\mathcal E$, we get
$$
\tilde\mathcal E=\mathcal E_{b''}:(1-k)x^2+(1-k)b''^{2}y^2+2k b''z^2-2(1+k)b''xy=0.
$$
Since $\mathcal E_{b'}$ and $\mathcal E_{b''}$ are contained in $\mathcal U$, they define one of
the pencils in Theorem \ref{th_1}. This implies that the line defined by
$P(\mathcal E_{b'})$ and $P(\mathcal E_{b''})$ in ${\rm PG}(5,q)$ should intersect the surface
$\mathcal V$. But we will see that this is not the case.
To simplify calculations, we apply the collineation
\begin{equation}\label{eq_7}
\sigma:\left\{
\begin{array}{rl}
x' &=x\\
y' &=b' y \\
z' &=\sqrt{b'}z
\end{array}\right.
\end{equation}
of ${\rm PG}(2,q^2)$.
Then $\sigma$ takes $\mathcal E_{b'}$ to $\mathcal E_{1}$ and $\mathcal E_{b''}$ to
and $\mathcal E_{\beta}$, with $\beta=b''/b'\neq1$. The line of ${\rm PG}(5,q)$
defined by
\begin{eqnarray*}
P(\mathcal E_{1}) & = &(1-k,1-k,2k,-(1+k),0,0)\\
P(\mathcal E_{\beta})& = &(1-k,(1-k)\beta^2,2k\beta,-\beta(1+k),0,0)
\end{eqnarray*}
intersects $\mathcal V$ if and only if
\begin{equation} \label{eq_6}
\begin{split}
(1-k)(1+s) &=\rho l^2 \\
(1-k)(1+s\beta^2) &=\rho m^2 \\
1+s\beta &=\rho n^2 \\
-(1+k)(1+s\beta)&=\rho lm\\
0 &= ln\\
0 &= mn
\end{split}
\end{equation}
{}for some $l,m,n\in{\rm GF}(q^2)$, $(l,m,n)\neq(0,0,0)$, and $s,\rho\in{\rm GF}(q^2)^*$; we recall that $k\neq1$.
In the following we use Equations (\ref{eq_6}).
Assume $n=0$ and $lm\neq0$. Then $1+s\beta=0$ and this implies that $\rho
lm=0$, a contradiction. Assume $n=0=l$. Without loss of generality we may
assume $m=1$. Then $s=-1$ and $\beta=1$, a contradiction. Assume $n=0=m$.
Without loss of generality we may assume $l=1$. Then $1+s\beta=0$. Hence
$s\beta=-1$. These forces $\beta=1$, a contradiction. Assume $l=0=m$. Without
loss of generality we may assume $n=1$. Then $s=-1$ and $\beta=-1$. This
forces $k=-1$ which contradicts the fact that $k-1$ has to be a non-square in
${\rm GF}(q^2)$. This proves that $\mathcal U$ cannot contain the conic $\mathcal E_{b''}$, a
contradiction.
We leave it to the reader to verify that, when $(a,b,c)$ is either $(0,1,0)$
or $(1,0,0)$, there are no points in
$(\Gamma(\mathcal C)\cap\Gamma(\mathcal D))\setminus(P(\mathcal C)P(\mathcal D)\cup\mathcal V)$.
\comment{
\item $a=b=0, c=1$ or $a=c=0, b=1$ or $b=c=0, a=1$.
If $a=b=0$ it follows that $a'=0, b'=0$ hence the generators with base point $Q(s,0,0,1)$ and $Q'(t,0,0,1)$ meet just in ${\cal V}$.
If $a=c=0$ it follows $a'=0$ and $b'c'=0$. Since $sb^2=\rho(tb'^2)$ from $b\ne 0$ we have $b'\ne 0$ and hence $c'=0$.
So again the generators with base point $Q(s,0,1,0)$ and $Q'(t,0,1,0)$ meet just in ${\cal V}$.
If $b=c=0$ and $a=1$ then $b'=0$ and $a'c'=0$ but since $a\ne 0$ it follows $a'\ne 0$ and hence $c'=0$ and again the generators meet just in ${\cal V}$.
}
Hence, we have proved that there is at most one conic $\mathcal E$ not in the pencil
defined by $\mathcal C$ and $\mathcal D$ that can be contained in $\mathcal U$. From Lemma
\ref{lem_1}, we get that does not exist a unital $\mathcal U$ which is union of
irreducible conics with two conics defining a pencil of type (i).
{\bf Case 2.} {\em $\mathcal C:x^2-\alpha y^2=z^2$ and $\mathcal D:x^2-\alpha y^2=kz^2$ for a fixed non-square $\alpha\in{\rm GF}(q^2)$ and some $k\in{\rm GF}(q^2)\setminus\{0,1\}$}.
The conics $\mathcal C$ and $\mathcal D$ have empty intersection in ${\rm PG}(2,q^2)$. Further, we have ${\rm det}(A(\mathcal C))=\alpha$ and ${\rm det}(A(\mathcal D))=\alpha k$. By Theorem \ref{th_3}, every point of $\mathcal D\setminus\mathcal C$ is in $\mathfrak I(\mathcal C)$ if and only if $k-1$ is a non-zero square in ${\rm GF}(q^2)$. By the symmetric relationship between the conics $\mathcal C$ and $\mathcal D$, we have that $k(k-1)$ is non-zero square in ${\rm GF}(q^2)$. Hence $k$ must be a non-zero square of ${\rm GF}(q^2)$.
We now proceed similarly to the previous case. We first consider the
irreducible conics of the pencil defined by $\mathcal C$ and $\mathcal D$. We point out that
these conics are disjoint in ${\rm PG}(2,q^2)$. As $q^2+1$ does not divide
$q^3+1$ we see that $\mathcal U$ must contain a further conic not in the pencil. So
we investigate $\Gamma(\mathcal C)\cap \Gamma(\mathcal D) \setminus (P(\mathcal C) P(\mathcal D) \cup
\mathcal V)$.
It is easily seen that the points of $\Gamma(\mathcal C)$ and $\Gamma(\mathcal D)$ not in
the surface $\mathcal V$ are
\begin{eqnarray*}
&&\hspace*{-7mm}\{(1+sa^2,-\alpha+sb^2,-1+sc^2,sab,sac,sbc): a,b,c,s\in{\rm GF}(q^2),
(a,b,c)\neq(0,0,0)\},\\
&&\hspace*{-7mm}\left\{(1+ta'^2,-\alpha+tb'^2,-k+tc'^2,ta'b',ta'c',tb'c'):
a',b',c',t\in{\rm GF}(q^2),\right.\\
&&\hspace*{10cm}\left.(a',b',c')\neq(0,0,0)\right\}.
\end{eqnarray*}
As in the previous case, we have
$(a,b,c)\neq(0,0,1)\neq(a',b',c')$ and $st\neq0$.
Then, the points in $\Gamma(\mathcal C)\cap\Gamma(\mathcal D)\setminus(P(\mathcal C)P(\mathcal D)\cup\mathcal V)$
satisfy the following equations:
\begin{equation}\label{eq_8}
\begin{split}
1+sa^2 &=\rho (1+ta'^2) \\
-\alpha+sb^2 &=\rho (-\alpha+tb'^2) \\
-1+sc^2&=\rho(-k+ tc'^2) \\
sab&=\rho ta'b'\\
sac &=\rho ta'c'\\
sbc &=\rho tb'c'
\end{split}
\end{equation}
{}for some $\rho\in{\rm GF}(q^2)^*$.
{}First we consider $abc\neq0$. {}As in Case 1, from the above equations, we
get that there are no points in
$\Gamma(\mathcal C)\cap\Gamma(\mathcal D)\setminus(P(\mathcal C)P(\mathcal D)\cup\mathcal V)$ with $abc\neq0$.
Now suppose that $a=0$ and $bc\neq0$. Equations (\ref{eq_8}) reduce to
\begin{equation*}
\begin{split}
1 &=\rho (1+ta'^2) \\
-\alpha+sb^2 &=\rho (-\alpha+tb'^2) \\
-1+sc^2&=\rho(-k+ tc'^2) \\
0&=a'b'\\
0 &= a'c'\\
sbc &= \rho tb'c'.
\end{split}
\end{equation*}
It follows immediately that $a'=0$ and $b'c'\neq0$; hence we get the same
conclusion as before. The same reasoning applies to the case $b=0$ and
$ac\neq0$.
If $c=0$ and $ab\neq0$, Equations (\ref{eq_8}) reduce to
\begin{equation}\label{eq_9}
\begin{split}
1+sa^2 &=\rho (1+ta'^2) \\
-\alpha+sb^2 &=\rho (-\alpha+tb'^2) \\
-1&=\rho(-k+ tc'^2) \\
sab&=\rho ta'b'\\
0 &= a'c'\\
0 &= b'c'.
\end{split}
\end{equation}
As $ab\neq0$ we have $a'b'\neq0$, $c'=0$ and $\rho=k^{-1}$. We can assume that $a=1=a'$. Thus Equations (\ref{eq_9}) reduce to
\begin{equation}\label{eq_10}
\begin{split}
1+s &=k^{-1} (1+t) \\
-\alpha+sb^2 &=k^{-1} (-\alpha+tb'^2) \\
sb&=k^{-1} tb'.
\end{split}
\end{equation}
{}From the first and third equation of (\ref{eq_10}) we get
\[
\begin{split}
t&=k(1+s)-1\\
b'&=\frac{sbk}{t}.
\end{split}
\]
By substituting these expressions into the second equation of $(\ref{eq_10}),$ we get
\begin{equation}\label{eq_11}
s=\frac{\alpha(1-k)}{k(\alpha-b^2)}.
\end{equation}
By substituting $a=1$, $c=0$, $\rho=k^{-1}$ and (\ref{eq_11}) into Equations
(\ref{eq_9}), we get that
$\Gamma(\mathcal C)\cap\Gamma(\mathcal D)\setminus(P(\mathcal C)P(\mathcal D)\cup\mathcal V)$ consists of the
points
\[
P(\mathcal E_{b,k})=(\alpha-kb^2,\alpha(b^2-\alpha k),k(b^2-\alpha),\alpha
b(1-k),0,0),
\]
with $b\in{\rm GF}(q^2)^*$.
In order for $\mathcal E_{b,k}$ to be a conic in $\mathcal U$, the sets $\mathcal C\setminus\mathcal E_{b,k}$ and
$\mathcal D\setminus\mathcal E_{b,k}$ should be both contained in $\mathfrak I(\mathcal E_{b,k})$. By using
Theorem \ref{th_3}, with straightforward calculations we obtain that the point
$(1,0,1)$ of $\mathcal C$ is in $\mathfrak I(\mathcal E_{b,k})$ if and only if $b^2-\alpha$ is a
non-square of ${\rm GF}(q^2)$ and the point $(\sqrt k,0,1)$ of $\mathcal D$ is in
$\mathfrak I(\mathcal E_{b,k})$ if and only if $b^2-\alpha$ is a non-zero square of
${\rm GF}(q^2)$, a contradiction. Thus we can conclude that no conic $\mathcal E_{b,k}$ is
contained in $\mathcal U$.
Assume $a=c=0$, so we can suppose $b=1$. Equations (\ref{eq_8}) reduce to
\begin{equation}\label{eq_12}
\begin{split}
1 &=\rho (1+ta'^2) \\
-\alpha+s &=\rho (-\alpha+tb'^2) \\
-1&=\rho(-k+ tc'^2) \\
0 &= a'b'\\
0 &= a'c'\\
0 &= b'c'.
\end{split}
\end{equation}
If $a'=0$ it is easily seen that there are no points in
$\Gamma(\mathcal C)\cap \Gamma(\mathcal D) \setminus (P(\mathcal C) P(\mathcal D) \cup \mathcal V)$.
Hence $a'\ne 0$ and $b'=0=c'$. Equations (\ref{eq_12}) reduce to
\begin{equation*}\label{eq_13}
\begin{split}
1 &=\rho (1+t) \\
-\alpha+s &=-\rho\alpha \\
-1&=-\rho k.
\end{split}
\end{equation*}
Hence, $\rho=k^{-1}$, $k=1+t$ and $s=\alpha (1-k^{-1})$ and we get the unique
common point $P(\mathcal E)=(k,-\alpha,-k,0,0,0)$. In order for $\mathcal E$ to be a conic
in $\mathcal U$, we should have that the line $P(\mathcal E)P(\mathcal G)$ intersects the surface
$\mathcal V$ in exactly one point, for every conic $\mathcal G$ of the pencil defined by
$\mathcal C$ and $\mathcal D$ and contained in $\mathcal U$. But this happens if and only if $\mathcal G$
coincides with either $\mathcal C$ or $\mathcal D$. Since $q>3$, the conics $\mathcal C$, $\mathcal D$ and
$\mathcal E$ don't cover all points of $\mathcal U$.
Assume $b=c=0$; so we can suppose $a=1$. Equations (\ref{eq_8}) reduce to
\begin{equation}\label{eq_14}
\begin{split}
1 +s &=\rho (1+ta'^2) \\
-\alpha &=\rho (-\alpha+tb'^2) \\
-1&=\rho(-k+ tc'^2) \\
0 &= a'b'\\
0 &= a'c'\\
0 &= b'c'.
\end{split}
\end{equation}
If $b'=0$ it is easily seen that there are no points in $\Gamma(\mathcal C)\cap \Gamma(\mathcal D) \setminus (P(\mathcal C) P(\mathcal D) \cup \mathcal V)$. Hence $b'\ne 0$ and $a'=0=c'$. Equations (\ref{eq_14}) reduce to
\begin{equation*}
\begin{split}
1+s &=\rho \\
-\alpha &=\rho (-\alpha+t) \\
-1&=-\rho k.
\end{split}
\end{equation*}
Analysis similar to the above case shows that the conics contained in $\mathcal U$
are precisely $\mathcal C$, $\mathcal D$ and $\mathcal E:x^2-\alpha k y^2 =kz^2$, a contradiction.
Finally, we conclude that there does not exist a unital $\mathcal U$ which union of
irreducible conics with two conics defining a pencil of type (ii).
{\bf Case 3.} {\em $\mathcal C:2yz=x^2$ and $\mathcal D:2yz=x^2+kz^2$ for some
$k\in{\rm GF}(q^2)\setminus\{0\}$}.
The intersection between $\mathcal C$ and $\mathcal D$ is the point $P(0,1,0)$. Further, we
have ${\rm det}(A(\mathcal C))={\rm det}(A(\mathcal D))=-1$. By Theorem \ref{th_3}, every point of
$\mathcal D\setminus\mathcal C$ is in $\mathfrak I(\mathcal C)$ if and only if $k$ is a non-square in
${\rm GF}(q^2)$.
We first consider the irreducible conics in the pencil defined by $\mathcal C$ and
$\mathcal D$. If a conic $\mathcal E: 2yz=x^2+hz^2$, $h\ne 1,k$, is contained in $\mathcal U$ then
$h$, $h-k$ are non-square in ${\rm GF}(q^2)$.
Hence such a set of conics determines a subset $X$ of ${\rm GF}(q^2)$ such that
$1\in X$, all elements of $X$ are non-squares and for any $h,k\in X$, $h-k$ is
a non-square. To obtain a unital $X$ must have size $q$.
\begin{lemma} \label{lem_2}
{\rm\cite{bl} }
Let $X$ be a subset of ${\rm GF}(q^2)$ of non-squares such that the
difference of any two elements is always a non-square. If $|X|=q$, then $X=t
{\rm GF}(q)$ for some non-square $t\in{\rm GF}(q^2)$.
\end{lemma}
It follows that such a set $X$ gives a Buekenhout--Metz unital of BEHS-type.
In order to investigate if there are further unitals union of conics we need,
also in this case, to study $\Gamma(\mathcal C)\cap \Gamma(\mathcal D) \setminus (P(\mathcal C)
P(\mathcal D) \cup \mathcal V)$:
\begin{equation}\label{eq_30}
\begin{split}
1+sa^2 &=\rho (1+ta'^2) \\
sb^2 &=\rho tb'^2 \\
sc^2&=\rho(-k+ tc'^2) \\
sab&=\rho ta'b'\\
sac &=\rho ta'c'\\
-1+sbc &=\rho (-1+tb'c')
\end{split}
\end{equation}
for some $\rho\in{\rm GF}(q)^*$.
Also in this case we have $(a,b,c)\neq(0,0,1)\neq(a',b',c')$ and $st\neq0$.
If either $abc\ne 0$ or $a=0$ or $b=0$, it is easy to check that there are no points in
$\Gamma(\mathcal C)\cap\Gamma(\mathcal D)\setminus(P(\mathcal C)P(\mathcal D)\cup\mathcal V)$.
If $c=0$ and $ab\ne 0$, then Equations (\ref{eq_30}) become
\begin{equation*}
\begin{split}
1+sa^2 &= \rho(1+ta'^2) \\
sb^2 &=\rho t b'^2 \\
0 &=-k+tc'^2 \\
sab &= \rho ta'b' \\
0 &=a'c' \\
-1 &=\rho(-1+tb'c').
\end{split}
\end{equation*}
If $a'=0$, then $ab=0$, a contradiction. If $c'=0$, then $k=0$, a
contradiction.
If $(a,b,c)$ iss either $(0,1,0)$ or $(1,0,0)$, we leave to the reader to
verify that there are no points in $\Gamma(\mathcal C)\cap \Gamma(\mathcal D) \setminus
(P(\mathcal C) P(\mathcal D) \cup \mathcal V)$.
This concludes the proof of Theorem \ref{th_2}.
|
{
"timestamp": "2012-03-09T02:02:28",
"yymm": "1203",
"arxiv_id": "1203.1766",
"language": "en",
"url": "https://arxiv.org/abs/1203.1766"
}
|
\section{Introduction}
The large-$N$ limit plays an important role in our understanding of gauge and spin models. It is also crucial for
the justification of the so-called AdS/CFT correspondence. In this context, a very interesting connection between instabilities of the
de Sitter space on the gravitational side and Dyson's instability on the gauge theory side has been suggested \cite{AM2008199}.
In a more general context, the compatibility of Dyson's instability with perturbative series having a finite radius of convergence in the large-$N$ limit
often leads to animated
discussions. To the best of our knowledge, this has never been
discussed systematically in the literature. The goal of this article is to provide such discussion together with an example where explicit calculations are possible.
The following observations lead to an apparent paradox.
There are known examples of perturbative series in the 't Hooft coupling $\lambda ^t$ with a finite radius of convergence \cite{PhysRevD.7.2911,Brezin:1977sv}.
The fact that the series converges for some negative values of $\lambda ^t$ is in apparent conflict with Dyson's argument \cite{Dyson52} which generically
suggests a zero radius of convergence due to the instability at negative coupling. This apparent conflict is called ``Dyson's large-$N$ paradox''
hereafter.
There are two possible points of view. On one hand, one could say that the large-$N$ limit provides a regularization of the divergence of
perturbative series. On the other hand, this regularization can only be obtained at the price of a ``mutilation'' \cite{Brezin:1977sv} of the
large field contributions in the functional integral. In the following, we provide a model calculation where the two points of views
can be discussed quantitatively.
We consider the {\it linear} $\sigma$-model with a $\lambda (\phi^2)^2$ interaction in three dimension (3D) and a sharp momentum cutoff \cite{PhysRevD.10.2491,Arefeva:1977bt,Arefeva:1979bd,david84,david85}. For the linear $\sigma$ model, the fields can take arbitrarily large values and, at least at large but finite $N$, a general argument \cite{convpert,optim03} can be used to infer that the radius of convergence of the perturbative series should be zero. For the nonlinear version,
the fields belong to a unit sphere of dimension $N-1$ and the situation is more complicated, as discussed in Ref. \cite{Meurice:2009bq} for a lattice regularization. Only the linear $\sigma$-model is discussed hereafter.
The paper is organized as follows.
In Sec. \ref{sec:paradox}, we state Dyson's large-$N$ paradox more precisely and review the scant literature on the subject.
The large-$N$ limit of the linear $\sigma$-model is reviewed in Sec. \ref{sec:model}. A perturbative series for the renormalized mass in the symmetric phase is constructed in Sec. \ref{sec:pert}.
A numerical analysis of the ratios of successive coefficients provides strong evidence for a finite radius of convergence $|\lambda ^t_c|$ and a square-root singularity.
This can be explained by the disappearance of saddle points for $\lambda ^t<-|\lambda ^t_c|$.
The effective potential is calculated in Sec. \ref{sec:effpot}. It is shown that for for $-|\lambda ^t_c|< \lambda ^t <0 $, the definition of the effective theory requires a large field cutoff denoted $\phi^2_{max}(\lambda ^t)$ which
provides a measure of the mutilation mentioned above and is described as a function of $\lambda ^t$ in Sec. \ref{sec:resol}.
The implications of these findings and possible improvements are discussed in the conclusions.
\section{Statement of the paradox}
\label{sec:paradox}
In this section, we state the nature of Dyson's large-$N$ paradox.
For the general discussion, we use the generic notation $\lambda ^t$ for the 't Hooft coupling which is $\lambda N$ (or $g^2 N$) for scalar (for gauge, respectively) coupling and is kept constant in the large-$N$ limit. For scalar models, it is well-known \cite{PhysRevD.7.2911,PhysRevA.7.2172,Brezin:1977sv}, that at leading order in the large-$N$ limit, the 1-Particle Irreducible functions can be expressed
as sums of bubble diagrams. This results
is series with a finite radius of convergence in $\lambda ^t$ and a singularity on the negative real axis. Explicit determination of such singularities can be found, for instance in Ref. \cite{Brezin:1977sv}.
It was noticed by Wilson \cite{PhysRevD.7.2911} that when the quartic coupling is negative ``one expects the theory to have no ground state $ [ \dots ] $. This difficulty does not seem to show up, however, when we sum only bubble graphs." This sentence could suggest that the use of the large-$N$ limit provides a regularization of the divergence of the
perturbative series, however, the use of ``seem'' indicates that this is not the end of the story.
On the other hand, the fact that we obtain a converging series for values of the coupling where the theory does not make sense indicates some limitation of the approximation. Some researchers may be acquainted with this fact, however the only mention we found in the literature is in the introduction of Ref. \cite{Brezin:1977sv}, where it is mentioned that the analyticity near the origin "reveals that the large field
region of the Feynman path integral has been drastically mutilated". In Secs. \ref{sec:effpot} and \ref{sec:resol}, we will provide a quantitative measure of this mutilation for the specific model discussed in the next section.
\section{The model and its saddle points for $\lambda ^t>0$}
\label{sec:model}
The two points of view presented in the previous section will be discussed for the 3D linear $O(N)$ $\sigma$-model with a $\lambda (\phi^2)^2$ interaction and a sharp momentum cutoff.
The partition function reads:
\begin{equation}\nonumber
Z=\int \mathcal{D}\phi {\rm e}^{-\int d^3x[(1/2)(\partial \phi)^2+(1/2)m_B^2\phi^2+\lambda (\phi^2)^2-\sqrt{N}\vec{J}\vec{\phi}]}\ .
\end{equation}
Most vector products are implicit ($\phi^2\equiv \vec{\phi}.\vec{\phi}$ etc ...), $ \mathcal{D}\phi$ refers to path-integration of
$\vec\phi$ over $\mathbb{R}^N$ rather than over the unit sphere (as in the nonlinear version) and the source $\vec{J}$ is $x$-independent.
We now follow closely the notations of \cite{david84}.
We introduce auxiliary fields $M^2(x)$ enforcing the condition $\phi^2(x)=NX(x)$ and then integrate over $\phi$. Using cutoff units,
introducing the 't Hooft coupling $\lambda ^t =\lambda N$ and dropping the
non-zero modes (which is justified in the large-$N$ limit), the action $\mathcal{A}$ per volume and number of fields reads:
\begin{eqnarray}
\label{eq:mxaction}
\mathcal{A}=&(1/2)&\int_{|k|\leq 1}d^3k\ {\rm ln}(k^2+M^2)\\
&+&(1/2)(m_B^2-M^2)X +\lambda ^t X^2-(1/2)J^2/M^2 \nonumber \ .
\end{eqnarray}
The saddle point equations obtained by varying $X$ and $M^2$ read:
\begin{eqnarray}
\label{eq:spm}
M^2&=&m^2_B+4\lambda ^t X, \\
\label{eq:spx}
X&=&F(M^2) +J^2/(M^2)^2,
\end{eqnarray}
with the cutoff one-loop function
\begin{equation}
F(M^2) \equiv \int_{|k|\leq 1}\frac{d^3k}{(2\pi)^D}\frac{1}{k^2+M^2}.
\end{equation}
When $M^2<0$, it is possible to analytically continue $F(M^2)$. When $-1<M^2<0$, $F(M^2)$ picks up an imaginary part. Its sign depends on the way the path goes around the pole.
The situation is displayed in Fig. \ref{fig:F}, where we have used the prescription $M^2-i\epsilon$.
\begin{figure}
\includegraphics[width=2.3in,angle=0]{F.eps}
\caption{Real part of $F(M^2)$ (red online) and imaginary part of $F(M^2-i\epsilon)$ (non-zero for $-1<M^2<0$, blue online).
\label{fig:F}}
\end{figure}
The solutions of the saddle point equations (\ref{eq:spm}) and (\ref{eq:spx}), and the phase diagram for $\lambda ^t>0$ can be obtained with the help of Fig. \ref{fig:SPSB}. When $m_B^2>0$ and $\lambda ^t>0$, the line
\begin{equation}
\label{eq:line}
X(M^2)=(M^2-m_B^2)/(4\lambda ^t)\ ,
\end{equation}
coming from Eq. (\ref{eq:spm}) crosses $F(M^2)$ at positive $M^2$ and there is exactly one solution of Eq. (\ref{eq:spx}) when
the source goes to zero. This situation corresponds to the symmetric phase.
When $m_B^2<0$ and $\lambda ^t>0$, the line
$X(M^2)$ does not cross $F(M^2)$ at positive $M^2$. We can nevertheless obtain a solution of Eq. (\ref{eq:spx}) with $M^2
\rightarrow 0$ and $J^2\rightarrow 0$ with $J^2/(M^2)^2=\phi^2$ kept at the constant value $X(0)-F(0)$. This gap at $M^2=0$ is illustrated on Fig. \ref{fig:SPSB}. This corresponds to the broken symmetry phase. The solution for $\phi^2=0$ with $M^2<0$ is briefly discussed in Sec. \ref{sec:effpot}.
\begin{figure}
\includegraphics[width=2.3in,angle=0]{SPSB.eps}
\caption{Real part of $F(M^2)$ (red online) and the lines of $X(M^2)$ from Eq. (\ref{eq:line}) with $m_B^2=-0.5$ and $\lambda ^t$=2 (left) and $m_B^2=1$ and $\lambda ^t$=2 (right).
\label{fig:SPSB}}
\end{figure}
\section{Perturbative series with a finite radius of convergence}
\label{sec:pert}
In this section, we construct explicitly a perturbative series in $\lambda ^t$ for the renormalized mass in the symmetric phase.
From the action Eq. (\ref{eq:mxaction}), we see that
in the symmetric phase, the saddle point solution for $M^2$ in the limit of zero source, is the inverse of the zero-momentum 2-point function and we call this value of $M^2$ the renormalized mass $M^2_R$.
Using Eqs. (\ref{eq:spm}) and (\ref{eq:spx}) in the zero source limit, we obtain the ``self-consistent" integral equation for $M^2$:
\begin{equation}
\label{eq:selfcons}
M^2_R=m_B^2+4\lambda ^t F(M^2_R)\ .
\end{equation}
This equation can be solved perturbatively by plugging the expansion
\begin{equation}
M^2_R=m_B^2+\sum_{n=1}^{\infty}c_n(m^2 _B)(\lambda ^t ) ^n \ ,
\end{equation}
in Eq. (\ref{eq:selfcons}). One finds $c_1(m^2 _B)=4F(m_B^2)$ and so on.
The series has been calculated numerically up to order 30 for $m^2_B=1$. The ratio of successive coefficients is showed in Fig. \ref{fig:rat}. The last 25 ratios have been fitted as follows:
\begin{equation}
c_n/c_{n+1}\simeq -8.34356-12.3454/n -8.73165/n^2 \ .
\end{equation}
We can compare this behavior with the corresponding ratios for the series of $(x-x_c)^\alpha$ which read
\begin{equation}
c_n/c_{n+1}\simeq x_c(1+(1+\alpha)/n+\mathcal{O}(1/n^2)) \ .
\end{equation}
We conclude that the series for $M_R^2(\lambda ^t)$ has a singularity at $\lambda ^t \simeq -8.344$ and a power behavior
$\alpha\simeq 0.493$. These two approximate numbers have a simple interpretation.
\begin{figure}
\includegraphics[width=2.3in,angle=0]{rat.eps}
\caption{
\label{fig:rat}
Ratios $c_{n}/c_{n+1}$ and the fit described in the text.}
\end{figure}
The singularity near -8.344 can be explained in terms of the saddle points properties when $\lambda ^t <0$. The solutions of the
zero source, symmetric phase, saddle point equations can be read from Fig. \ref{fig:SP} where we have displayed the lines $X(M^2)$ for $m_B^2=1$ and a certain number of values of $\lambda ^t$ ranging from 10 to -10. For positive
$\lambda ^t$ there is only one intersection between the lines and $F(M^2)$. As $\lambda ^t$ becomes negative but not too negative, this property persists. When $\lambda ^t$ is decreased to approximately -4.935, a second solution appears at lower (but positive) $M^2$.
As $\lambda ^t$ further decreases, the two solutions get closer and finally coalesce for $\lambda ^t = -8.3419 ...$ and disappear for more negative values. This provides strong numerical evidence that the singularity of the perturbative series is due to the absence of saddle point beyond this critical value.
\begin{figure}
\includegraphics[width=2.3in,angle=0]{SP.eps}
\caption{Real part of $F(M^2)$ (red online) and the lines of $X(M^2)$ from Eq. (\ref{eq:spx}) with $m_B^2=1$ and $\lambda ^t$=
10, 1, 0.01, -1, -4, -7, -8.3 and -10 (couterclockwise).
\label{fig:SP}}
\end{figure}
The nature of the singularity can be further understood using the strong coupling limit of the self-consistent Eq. (\ref{eq:selfcons}). By inspection, it is clear that when $\lambda ^t$ becomes large so does $M^2$. In this limit, we can neglect $m_B^2$ and
the $k^2$ at the denominator of $F(M^2)$. This implies
\begin{equation}
M^2_R\simeq \sqrt{2\lambda ^t/(3\pi^2)} .
\end{equation}
This square root behavior is in very good agreement with the estimate of the power of the singularity $\alpha \simeq 0.493$.
\section{The effective potential}
\label{sec:effpot}
The effective potential $V_{eff}(\vec{\phi})$ can be constructed by a Legendre transformation \cite{PhysRevD.10.2491,Arefeva:1977bt,david84}. The source can be eliminated through the relation
$\vec{J}=M^2 \vec{\phi}$
and we obtain
\begin{eqnarray}
V_{eff}(\vec{\phi})=(&1/2&) (m_B^2-M^2)X+\lambda ^t X^2\\ \nonumber &+&(1/2)M^2\phi^2+\int_{|k|\leq 1}d^3k \ {\rm ln}(k^2+M^2)\ .
\end{eqnarray}
The saddle point equations are Eq. (\ref{eq:spm}) and
\begin{equation}
\label{eq:spXphi}
X=F(M^2) +\phi^2 \ .
\end{equation}
The values of $V_{eff}(\vec{\phi})$ can be constructed by using $M^2$ as a parameter. For fixed values of $m_B^2$ and $\lambda ^t$, choosing a value of $M^2$ determines
$X(M^2)$ using Eq. (\ref{eq:line}) as before. Plugging this expression in Eq. (\ref{eq:spXphi}), we obtain $\phi^2(M^2)$ which is positive provided that $X(M^2)>F(M^2)$. For $m^2_B>0$ and $\lambda ^t >0$ (symmetric phase), this implies $M^2>M^2_R$ as discussed in Sec. \ref{sec:model} and shown in Fig. \ref{fig:SPSB}. The important point is that in this case all positive values of $\phi^2$ are allowed. On the other hand, for $m^2_B<0$ and $\lambda ^t >0$ (broken symmetry phase), $M^2$ can take any positive value but $\phi^2\geq X(0)-F(0)>0$ as illustrated in Fig. \ref{fig:SPSB}. Smaller values of
$\phi^2$ could in principle be reached by using negative values of $M^2$, but then an imaginary part appears signaling an instability \cite{PhysRevD.10.2491}. This can be understood with a simple double-well potential example $(\phi^2-v^2)^2$: if a source is introduced the absolute
minimum is always for $\phi^2>v^2$ and the extrema for $\phi^2<v^2$ correspond to subdominant saddles. For this reason we only consider the case $M^2\geq 0$ in the following.
We now restrict the discussion to $m^2_B>0$ and vary $\lambda ^t$ from positive to negative values. We use the same numerical values as in Sec. \ref{sec:pert}. The curves $\phi^2(M^2)$ are shown in Fig. \ref{fig:phiofm}. The three curves on the right correspond to $\lambda ^t >0$ and illustrate
that in this case, $\phi^2$ can take arbitrarily large positive values. The five curves on the left correspond to $\lambda ^t<0$ and have a maximal value
of $\phi^2$. For $\lambda ^t =-1$ this maximum is reached outside of the figure. For $\lambda ^t$ not too negative, $\phi^2$ remains positive when $M^2$ goes to zero. If $\lambda ^t$ is decreased below the approximate value -4.935, $\phi^2$ goes back to zero before $M^2$ is zero. The correct range of $M^2$ is then
selected by requiring the positivity of $\phi^2$ as illustrated in Fig. \ref{fig:msbound}. We can now vary $M^2$ within this correct range and construct a parametric representation of $V_{eff}(\vec{\phi})$.
Fig. \ref{fig:phiofm} shows that as $M^2$ is varied, some values of $\phi^2$ will appear twice. When this is the case, we need to pick the value that has the smallest $V_{eff}(\vec{\phi})$. The effective potential corresponding to the set of values of $\lambda ^t$ discussed above, is shown in Fig. \ref{fig:effpot} where the two possible solutions, whenever present, have been kept. It should be understood that when $\phi^2$ reaches the value where there are two solutions, $V_{eff}(\vec{\phi})$ drops discontinuously to the lowest solution.
\begin{figure}
\includegraphics[width=2.3in,angle=0]{phiofm.eps}
\caption{$\phi^2(M^2)$ for $\lambda ^t$=
10, 1, 0.01, -1, -4, -7, -8.3 and -10 (conterclockwise). The correct range of $M^2$ is selected by requiring the positivity of $\phi^2$.
\label{fig:phiofm}}
\end{figure}
\begin{figure}
\includegraphics[width=2.3in,angle=0]{msbound.eps}
\caption{Boundary of the values of $M^2$ insuring $\phi^2>0$ as a function of $\lambda ^t$. The lower right corner is inside the boundary.
\label{fig:msbound}}
\end{figure}
\begin{figure}[t]
\vskip5pt
\includegraphics[width=2.3in,angle=0]{effpot.eps}
\caption{$V_{eff}(\vec{\phi})$ for $\lambda ^t$=
10, 1, 0.01, -1, -4, -7, and, barely visible, -8.3 (clockwise). The two solutions were kept whenever present.
\label{fig:effpot}}
\end{figure}
\section{Partial resolution of the paradox}
\label{sec:resol}
It is clear from Eq. (\ref{eq:spXphi}) and Figs. \ref{fig:SP} and \ref{fig:effpot} that when $-|\lambda ^t_c|<\lambda ^t<0$, $\phi^2$ cannot take arbitrary large values.
In the following, we call the maximal value $\phi^2_{max} (\lambda ^t)$. For $m_B^2$ and $\lambda ^t$ fixed, $\phi^2_{max}(\lambda ^t)$ can be obtained by maximizing $\phi^2$ in Eq. (\ref{eq:spXphi}).
This yields
\begin{equation}
F'(M^2)=1/(4\lambda ^t).
\end{equation}
For negative value of $\lambda ^t$ with a small absolute value, the function $X(M^2)$ is almost vertical and the matching with the slope of $F(M^2)$ requires
a small value of $M^2$, namely $M^2 \simeq (\lambda ^t)^2 /({4\pi^2})$. Keeping only the leading term in $X-F$, we obtain that in this limit,
\begin{equation}\phi^2_{max}\simeq m_B^2/(4|\lambda ^t|) .
\end{equation}
On the other hand, when $\lambda ^t$ approaches $\lambda ^t_c$ from above, $\phi^2_{max}(\lambda ^t)$ goes to zero linearly. These two behaviors are illustrated in Fig. \ref{fig:phimax}.
\begin{figure}
\includegraphics[width=2.3in,angle=0]{phisqmax.eps}
\caption{Values of $\phi^2_{max}(\lambda ^t)$ versus $\lambda ^t$.
\label{fig:phimax}}
\end{figure}
An effective theory for the zero mode of the field can in principle be defined at negative $\lambda ^t$ provided that we discard the large field region
$\phi^2> \phi^2_{max}(\lambda ^t )$. Under these circumstances, it is possible to set bounds on the perturbative coefficients \cite{convpert,optim03} of the
partition function that guarantees that the series converges and the perturbative series have a finite radius of convergence determined
by the zeros of the partition function. In the example considered here, the introduction of a large field cutoff represents a departure from the original model.
Consequently, the function $\phi^2_{max}(\lambda ^t )$ can be interpreted as a measure of the mutilation of the large field contributions introduced in Sec. \ref{sec:paradox}.
\section{Conclusions}
In summary, we have discussed quantitatively the two aspects of Dyson large-$N$ paradox for the 3D linear $O(N)$ sigma model.
At leading order in the large-$N$ expansion, the perturbative series in $\lambda ^t$ for the renormalized mass shows very good evidence for
a singular behavior of the form $(\lambda ^t +|\lambda ^t_c|)^{1/2}$, where $\lambda ^t_c$ is the smallest (negative) value for which a saddle point exists.
The convergence of the series for $-|\lambda ^t_c|< \lambda ^t <0 $ can be explained by the fact that for this range of $\lambda ^t$, the effective theory can only be defined
for $\phi^2<\phi^2_{max}(\lambda ^t )$.
The explicit construction of the effective potential
for $\lambda ^t$ negative but not too negative, shows that $\phi^2=0$ remains an absolute minimum of the effective potential in the range where it can be defined. This saddle point can be used in the large-$N$ limit despite the pathologies that the effective potential develops at larger $\phi^2$, namely
a discontinuity and a finite range of definition. If $\lambda ^t$ is lowered below $\lambda ^t_c$, the two solutions of the saddle point equation coalesce and disappear completely, and so does the effective potential.
This mechanism seems to be generic and it should be possible to observe similar disappearances of large-$N$ saddle points in other models.
More work is necessary in order to understand
how the $1/N$ corrections affect the large field behavior of the effective potential at positive $\lambda ^t$. Effects that only appear at the $1/N$ level are particularly interesting with this respect. In the context of QCD-like theories, the breaking of the axial $U(1)$ \cite{Witten:1979vv,Veneziano:1979ec} is an example of such phenomenon.
We also expect that a better understanding of the connection between de Sitter and anti de Sitter spaces could help us understand complex renormalization group flows \cite{Denbleyker:2010sv,PhysRevD.83.056009,Liu:2011zzh} in gauge theories and $\sigma$-models and find improved weak coupling expansions.
\begin{acknowledgments}
This article was written while attending the KITP workshop ``Novel Numerical Methods for Strongly Coupled Quantum Field Theory and Quantum Gravity". We had many interesting conversations with many participants and especially
L. Pando Zayas, J. Polchinski and P. Damgaard regarding Dyson and de Sitter instabilities.
This
research was supported in part by the Department of Energy
under Contract No. FG02-91ER40664 and by the National Science Foundation under Grant No. PHY11-25915. \end{acknowledgments}
|
{
"timestamp": "2012-05-10T02:04:28",
"yymm": "1203",
"arxiv_id": "1203.2256",
"language": "en",
"url": "https://arxiv.org/abs/1203.2256"
}
|
\section{Introduction}\label{sec:intro}
Some of the main applications of modern algebraic topology, including
the development of structured ring spectra
\cite{ekmm,hovey-shipley-smith}, have been to the subject of
algebraic $K$-theory. These new foundations introduce strictly
associative and commutative ring objects in the category of spectra,
together with their categories of modules. These provide a large
library of new objects whose algebraic $K$-theory can be calculated
and studied. These illuminate general phenomena that bear on old
calculations of the algebraic $K$-theory of rings, of simplicial
rings, and of spaces.
Based on computational studies in algebraic $K$-theory, Ausoni and
Rognes \cite[Introduction]{ausoni-rognes} now expect the
existence of a redshift phenomenon (generalizing the Bott-periodic
phenomena appearing in the algebraic $K$-theory of fields) and
initiated a long-term program to study the relationship between
algebraic $K$-theory and the chromatic filtration. These conjectures
have been supported by their work showing that the algebraic
$K$-theory of complex $K$-theory supports information at chromatic
level $2$.
The next computational steps in such a research program would
involve study of the algebraic $K$-theory of objects at chromatic
level $2$. Ongoing work of Bruner and Rognes aims to compute the
algebraic $K$-theory of the topological modular forms spectrum
$K_*(\mathit{tmf}_{(2)})$ \cite{john-slide} and of related spectra such as
$\BPP{2}$. These computations take place using the machinery of
topological cyclic homology.
This computational work is greatly assisted by the use of higher
multiplicative structures. For an associative object $R$, the
algebraic $K$-theory $K(R)$ and topological cyclic homology $TC(R)$
are spectra connected by a cyclotomic trace $K(R) \to TC(R)$.
However, if the category of $R$-modules has a symmetric monoidal
structure analogous to the tensor product of modules over a
commutative ring, the algebraic $K$-theory and topological cyclic
homology inherit the structure of ring objects themselves
\cite{elmendorf-mandell}.
Computations in topological
cyclic homology involve numerous spectral sequence calculations, and
these are greatly assisted by the existence of ring structures (or the
data of power operations \cite{bruner-rognes}) and by naturality
arguments.
In addition, many of these computations begin with the B\"okstedt
spectral sequence, which requires information about the mod-$p$
homology of the spectrum in question.
Previous work, based on a study of the moduli of elliptic curves with
level $\Gamma_1(3)$-structures, showed the following result.
\begin{thm}[{\cite[Theorem~1.1]{lawsonnaumann}}]
\label{thm:reminder}
There exists a $2$-local complex oriented $\cal E_\infty$-ring
spectrum $\mathit{tmf}_1(3)_{(2)}$ such that the composite map of graded rings
\[
\mb Z_{(2)}[v_1, v_2] \subset BP_* \to MU_{(2),*} \to \mathit{tmf}_1(3)_{(2),*}
\]
is an isomorphism.
\end{thm}
(Here we say that a multiplicative cohomology theory is ``complex
oriented'' if it is given compatible choices of orientation for all
complex vector bundles; we employ the fact that this is equivalent to
the choice of a map of ring spectra $MU \to R$.)
However, this result was obtained by means of obstruction theory, and
only used the modular interpretation of $\mathit{tmf}_1(3)_{(2)}$ in a
superficial way. The goal of the current paper is to gain a better
understanding of the larger context inhabited by the spectrum
$\mathit{tmf}_1(3)_{(2)}$; this is closely related to the study made by
Mahowald and Rezk in~\cite{mahowald-rezk}. With the above
$K$-theoretic applications in mind, another goal is to exhibit the
mod-$2$ cohomology of $\mathit{tmf}_1(3)_{(2)}$. (Forthcoming work of Hill and
the first author should recover a $C_2$-action and a connective
spectrum $\mathit{tmf}_0(3)$.)
There is a map of moduli stacks of generalized elliptic curves
\[
\overline{\cal M}_1(3)\to \overline{\cal M}
\]
(\cite[Theorem 4.1.1, (1) with $N=3$, $n=1$]{conrad}). This is the
unique map extending the map that takes a smooth elliptic curve with a
$3$-torsion point and forgets the point. This map ramified at exactly
one of the two cusps of $\overline{\cal M}_1(3)$ but is log-\'etale.
The modular interpretation of $\mathit{tmf}$ suggests that this map should
have a topological realization. In fact, we would like to construct a
$2$-local commutative diagram of ${\cal E}_\infty$-ring spectra
corresponding to (the connective covers of) the global sections of
sheaves of ${\cal E}_\infty$-ring spectra in the following diagram:
\[
\xymatrix{ \overline{\cal M} & & [{\rm Spec}(\mb Z_{(2)})/\!/ \{ \pm 1\}] \ar[ll]_(.6){\mathrm{\small cusp}}\\
\overline{\cal M}_1(3)\ar[u] & & {\rm Spec}(\mb
Z_{(2)})\ar[ll]_{\mathrm{\small ramified\,\, cusp}} \ar[u] }
\]
A realization of this diagram is achieved by the following main
result of this note.
To state it, we recall that, for each $n$, the mod-$2$ Steenrod algebra
${\cal A}^*$ contains exterior subalgebras $E(n)$ generated by the Milnor
primitives $Q^0, \ldots, Q^n$, and larger subalgebras ${\cal
A}(n)$ generated by $Sq^1, \ldots, Sq^{2^{n+1}}$.
\begin{thm}\label{thm:main}
There is a commutative diagram of connective ${\cal E}_\infty$-ring
spectra as follows:
\begin{equation}
\label{eq:main-diagram}
\xymatrix{ \mathit{tmf}_{(2)} \ar[r]^{c}\ar[d]^{o} & ko_{(2)}\ar[d]^{\iota} \\
\mathit{tmf}_1(3)_{(2)} \ar[r]^{\tilde{c}} & ku_{(2)}}
\end{equation}
Here $\iota$ is the complexification map, $o$ is a
$\mathit{tmf}_{(2)}$-orientation of $\mathit{tmf}_1(3)$, $c$ corresponds to the
cusp on the moduli space of elliptic curves, and $\tilde{c}$
corresponds to the unique ramified cusp on the moduli space of
elliptic curves with level $\Gamma_1(3)$-structure.
In mod-$2$ cohomology, this induces the
following canonical diagram of modules over the mod-$2$ Steenrod
algebra ${\cal A}^*$:
\[ \xymatrix{ {\cal A}^*/\!/{\cal A}(2) & {\cal A}^*/\!/{\cal A}(1)\ar[l]\\
{\cal A}^*/\!/E(2)\ar[u]& {\cal A}^*/\!/E(1).\ar[l]\ar[u]}
\]
There exists a complex orientation of $\mathit{tmf}_1(3)_{(2)}$ such that
in homotopy, $\tilde{c}$ induces a map sending the Hazewinkel
generators $v_1$ to $v_1$ and $v_2$ to zero, and there is a cofiber
sequence of $\mathit{tmf}_1(3)_{(2)}$-modules
\[
\Sigma^6 \mathit{tmf}_1(3)_{(2)}\stackrel{\cdot v_2}{\longrightarrow} \mathit{tmf}_1(3)_{(2)}
\stackrel{\tilde{c}}{\longrightarrow} ku_{(2)}.
\]
\end{thm}
We note that the restriction to the ramified cusp of $\overline{{\cal M}}_1(3)$
is not material in the above discussion. The unramified cusp
also gives rise to a commutative diagram of the same form and with
similar properties. However, the spectrum in the lower-right corner
is no longer the connective $K$-theory spectrum $ku_{(2)}$ if
multiplicative structure is taken into account, but instead
corresponds to a Galois twist of the multiplicative formal group law
which is defined over $\mb Z[1/3]$. We discuss the modifications
necessary to use this form of $K$-theory in an appendix. Its periodic
version is most easily described as a homotopy fixed point spectrum:
\[
KU^\tau = (KU \smsh{} \mb S_{(2)}[\omega])^{hC_2}
\]
Here $\mb S_{(2)}[\omega]$ is obtained by adjoining third roots
of unity to the $2$-local sphere spectrum. The generator of the
cyclic group $C_2$ acts by $\psi^{-1} \smsh{} \sigma$, where
$\psi^{-1}$ is the Adams operation associated to complex conjugation
and $\sigma$ is complex conjugation acting on the roots of unity.
We conclude with a short overview of the content. In
Section~\ref{sec:elliptic-curve} we collect basic facts about the
moduli of generalized elliptic curves equipped with a
$\Gamma_1(3)$-structure. In Section~\ref{sec:constructmaps} we
construct the ${\cal E}_\infty$-maps in Theorem~\ref{thm:main} using
Goerss-Hopkins obstruction theory and chromatic fracture squares.
While familiar to the experts, we found it worthwhile to spell out the
details of the $K(1)$-local obstruction theory. Section
\ref{sec:cohomology} shows $H^*(\mathit{tmf}_1(3)_{(2)},\mb F_2)\cong {\cal
A}^*/\!/ E(2)$ by showing more generally that every generalized
$\BPP{n}$ has the same cohomology as the standard $\BPP{n}$
(Definition \ref{def:generalbpn} and Theorem \ref{thm:coh-comp}).
Section \ref{sec:proof} collects these results into a proof of Theorem
\ref{thm:main}. The appendix discusses forms of $K$-theory and the
changes necessary to realize diagram~(\ref{eq:main-diagram}) using the
unramified cusp rather than the ramified one.
\begin{conv}
With the exception of the appendix, throughout this paper we will
work in the category of $2$-local spectra. In particular, the
names $\mathit{tmf}$, $\mathit{tmf}_1(3)$, $ko$, $ku$, and the like denote
$2$-localizations.
\end{conv}
The authors would like to thank several people: Gerd Laures for
help with the proof of Proposition \ref{prop:TMFtoKO}; Matthew
Ando and Paul Goerss for discussion relating to the
$\mathit{tmf}$-orientations of $KO$ and $KO\pow{q}$; Andrew Baker for
discussions relating to uniqueness of $\BPP{n}$; Andrew Baker and
Justin Noel for indicating a generalization, and simplification, of
the argument for Theorem~\ref{thm:coh-comp}; John Rognes and
Robert Bruner for motivation and discussions relating to the material
of this paper; and referee for several helpful suggestions.
\section{The $2$-localized moduli $\overline{{\mathcal M}}_1(3)_{(2)}$}\label{sec:elliptic-curve}
This section collects basic facts about the moduli of generalized
elliptic curves with a $\Gamma_1(3)$-structure. It provides
background for the somewhat ad-hoc construction of \cite[Section
8.2]{lawsonnaumann}, where it was shown that a certain formal
group law over $\mb Z_{(2)}[A,B]$, to be recalled presently, defines a
$\BPP{2}$-realization problem at the prime $2$ which can be
solved. We will focus here on determining the ordinary and
supersingular loci of the moduli stack, as those dictate the
chromatic properties of $\mathit{tmf}_1(3)$.
Consider the curve ${\cal E}\subseteq {\mathbb
P}^2_{\mathbb{Z}_{(2)}[A,B]}$ defined by the affine Weierstrass equation
\begin{equation}\label{eq:Weierstrass}
y^2 + Axy + By = x^3
\end{equation}
over the graded ring $\mb Z_{(2)}[A,B]$, where $|A| = 1$, $|B| = 3$.
There is a $3$-torsion point at the origin $(0,0)$, and conversely
this Weierstrass form is forced by the requirements that $(0,0)$ is a
$3$-torsion point whose tangent line is horizontal; the elliptic curve
${\cal E}$ is the universal generalized elliptic curve with a choice
of $3$-torsion point (namely $[0:0:1]$) and a choice of invariant
differential $dy/3x^2$ by \cite[Proposition 3.2]{mahowald-rezk}. The
grading, which can be interpreted as acting on the invariant
differential, is reflected in the action of the multiplicative group
$\mb G_m = {\rm Spec}(\mb Z[\lambda^{\pm 1}])$ determined by
\[
A \mapsto \lambda A, B \mapsto \lambda^3 B.
\]
This lifts to a $\mb G_m$-action on ${\cal
E}/{\rm Spec}(\mathbb{Z}_{(2)}[A,B])$ via
\[
(x,y) \mapsto (\lambda^2 x,\lambda^3 y).
\]
The curve ${\cal E}$ has additive reduction only at $(A,B) = (0,0)$.
\begin{prop}
\label{prop:reduction}
The restriction of ${\cal E}$ to $\mb A^2_{\mb Z_{(2)}}\setminus\{ (0,0)\}$
is a generalized elliptic curve with irreducible geometric fibers
as follows:
\begin{enumerate}[i)]
\item A nodal curve of arithmetic genus one if $A^3=27B$ or $B=0$.
\item A supersingular elliptic curve if $A=0$ and $2=0$.
\item An ordinary elliptic curve otherwise.
\end{enumerate}
\end{prop}
This, in particular, expresses $A$ as a lift of the Hasse invariant
$v_1$, which is well-defined mod $2$; the element $B$ is a lift of
$v_2$, which is well-defined mod $(2,v_1)$.
\begin{proof}
To say that ${\cal E}/(\mb A^2_{\mb Z_{(2)}}\setminus\{ (0,0)\})$
is a generalized elliptic curve with irreducible fibers
in the sense of \cite[Chapitre I, D\'efinition 1.12]{deligne-rapoport} means that the modular
quantities $c_4({\cal E})$ and $\Delta({\cal E})$ have no common zero
on $\mb A^2_{\mb Z_{(2)}} \setminus\{ (0,0)\}$. This results from the
following computations:
\begin{eqnarray*}
c_4({\cal E})&=& A(A^3-24B)\\
\Delta({\cal E})&=&B^3(A^3-27B)\\
j({\cal E}[\Delta^{-1}])&=&\frac{A^3(A^3-24B)^3}{B^3(A^3-27B)}\\
\end{eqnarray*}
A geometric fiber is a nodal curve if and only if $\Delta=0$, so i)
is clear and ii) and iii) follow by recalling that the only
supersingular $j$-invariant in characteristic $2$ is $j=0$.
\end{proof}
The $\mb G_m$-action on ${\cal E}$ over $\mb A^2_{\mb Z_{(2)}} \setminus
\{ (0,0)\}$ descends to determine a generalized elliptic curve over
the quotient stack
\[
\overline{{\cal M}}_1(3)_{(2)} \cong \left[ (\mb A^2_{\mb Z_{(2)}}
\setminus \{ (0,0)\})\ /\!/\ \mathbb{G}_m\right].
\]
The
stack $\overline{{\cal M}}_1(3)_{(2)}$ is a stack with coarse moduli
isomorphic to the weighted projective space ${\rm Proj}(\mb Z_{(2)}[A,B])$.
Since the zero section of a generalized elliptic curve lies in the
smooth locus, we have associated with ${\cal E}$ a 1-dimensional
formal group $\widehat{\cal E}/\overline{{\cal M}}_1(3)_{(2)}$.
The function $-x/y$ is a coordinate in a neighborhood of $\infty$ on
${\cal E}$, and this gives an isomorphism between the pullback of the
relative cotangent bundle $\omega$ of $\widehat{\cal E}/\overline{\cal
M}_1(3)$ along the zero section and the tautological line bundle
${\cal O}(1)$ on $\overline{\cal M}_1(3)$. Compatibility with the
grading implies that the formal group $\widehat{\cal E}$ comes from a
graded formal group law, and is induced by a map of graded rings $MU_*
\to \mb Z_{(2)}[A,B]$. Here we follow the standard convention that
elements in algebraic grading $k$ lie in topological grading $2k$.
The elements $A$ and $B$ can then be interpreted as global sections:
\begin{eqnarray*}
A &\in& H^0(\overline{\cal M}_1(3), {\cal O}(1))\\
B &\in& H^0(\overline{\cal M}_1(3), {\cal O}(3))
\end{eqnarray*}
Let $\mb Z_{(2)}[A,B] \to \mb Z_{(2)}[b]$ be the ungraded map given by
$A \mapsto 1$, $B \mapsto b$. The composite map
\[
{\rm Spec}(\mb Z_{(2)}[b]) \to \mb A^2_{\mb Z_{(2)}} \setminus \{ (0,0)\}
\to \overline{\cal M}_1(3)
\]
is an open immersion and thus determines an affine coordinate chart
$V$ of $\overline{\cal M}_1(3)$. On this chart the elliptic curve
$y^2 + xy + by = x^3$ has no supersingular fibers by Proposition
\ref{prop:reduction},
ii).
\begin{prop}
\label{prop:restriction}
The restriction of $\widehat{\cal E}$ to $V={\rm Spec}(\mb Z_{(2)}[b])$
is a formal $2$-divisible group whose mod-$2$ reduction has
constant height $1$.
\end{prop}
\begin{proof}
By Proposition~\ref{prop:reduction} the restriction
$\widehat{\cal E}|_V$ is a $1$-dimensional formal group of constant
height $1$, so it is $2$-divisible.
\end{proof}
Observe that $V\subseteq \overline{\mathcal M}_1(3)$ is the maximal
open substack over which ${\mathcal E}$ is ordinary.
Let $\mb Z_{(2)}[A,B] \to \mb Z_{(2)}[a]$ be the map given by $A
\mapsto a$, $B \mapsto 1$. The composite map
\[
{\rm Spec}(\mb Z_{(2)}[a]) \to \mb A^2_{\mb Z_{(2)}} \setminus \{ (0,0)\}
\to \overline{\cal M}_1(3)
\]
is \'etale. The stack-theoretic image is the quotient of
${\rm Spec}(\mb Z_{(2)}[a])$ by the action of the group $\mu_3$ of
third roots of unity, given by $\omega \cdot a = \omega a$ for
$\omega$ any third root of unity. The induced map
\[
W= \left[ {\rm Spec}(\mb Z_{(2)}[a]) /\!/ \mu_3 \right] \to \overline{\cal
M}_1(3)
\]
is an open immersion, and so ${\rm Spec}(\mb Z_{(2)}[a])$ determines an
\'etale coordinate chart for the stack near $a=0$. On this chart the
elliptic curve is defined by the Weierstrass equation $y^2 + axy
+ y = x^3$, with $\mu_3$-action given by $\omega \cdot (x,y)
= (\omega^2 x, y)$.
Let $U$ be the formal scheme ${\rm Spf}(\mb Z_{2}\pow{a})$, with formally
\'etale map $U \to \overline{\cal M}_1(3)$. By Proposition
\ref{prop:reduction}, the pullback of ${\cal E}$ to $U$ has special
fiber a supersingular elliptic curve. We denote by $\mb G/U$ the
$2$-divisible group of ${\cal E}|_U$.
\begin{prop}
\label{prop:universal-def}
The $2$-divisible group $\mb G/U$ has height $2$
and is a universal deformation of its special fiber.
\end{prop}
\begin{proof}
This is a restatement of \cite[Proposition 8.2 and Remark 4.2]{lawsonnaumann}.
\end{proof}
The common overlap of the coordinate charts $V$ and $W$ is determined by the
identity $a^3 b = 1$. The Mayer-Vietoris sequence for this weighted
projective space, using these affine coordinate charts, allows us to
compute the cohomology of $\overline{\cal M}_1(3)_{(2)}$ with
coefficients in ${\cal O}(*)$.
\begin{cor}
\label{cor:moduli-cohomology}
The cohomology of $\overline{\cal M}_1(3)$ with coefficients in
${\cal O}(*)$ vanishes above degree $1$.
The cohomology groups $H^0(\overline{\cal M}_1(3), {\cal O}(*))$
form the graded ring $\mb Z_{(2)}[A,B]$.
The cohomology $H^1(\overline{\cal M}_1(3), {\cal O}(*))$ is the
module $\mb Z_{(2)}[A,B]/(A^\infty, B^\infty)$ of elements $A^{-n}
B^{-m} D$, where $D$ is a duality class in $H^1(\overline{\cal
M}_1(3), {\cal O}(-4))$ annihilated by $A$ and $B$.
\end{cor}
\section{Constructing the maps}\label{sec:constructmaps}
The goal of this section is to construct the ${\cal E}_\infty$-maps of
Theorem~\ref{thm:main}, diagram~(\ref{eq:main-diagram}),
which we reproduce here:
\[
\xymatrix{ \mathit{tmf} \ar[r]^{c}\ar[d]^{o} & ko\ar[d]^{\iota} \\
\mathit{tmf}_1(3) \ar[r]^{\tilde{c}} & ku}
\]
All spectra appearing in this diagram are the connective covers of
their $K(0)\vee K(1)\vee K(2)$-localizations. Accordingly, we will
construct the required ${\cal E}_\infty$-maps using two chromatic
fracture squares, followed by taking connective covers.
\subsection{The $K(2)$-local maps}\label{subsec:k2-local-maps}
We identify the $K(2)$-localizations of the connective spectra in
Theorem \ref{thm:main} as follows. We have that $L_{K(2)}KU\simeq *$,
and as $K(n)$-localization does not distinguish between a spectrum and
its connective cover we have $L_{K(2)}ku \simeq *$ as well. From the
familiar fibration
\begin{equation}\label{eq:eta}
\Sigma ko\stackrel{\eta}{\to} ko\to ku,
\end{equation}
it follows that the nilpotent map $\eta$ induces an equivalence on
$L_{K(2)}ko$, hence $L_{K(2)}ko\cong *$ as well.
Let $E$ denote the Lubin-Tate spectrum associated with the formal
group of the supersingular elliptic curve
\[
C\colon\thinspace y^2+y=x^3
\]
over $\mb F_4$ \cite[Section 7]{goerss-hopkins}. The group
$G_{48}=Aut(C/\mathbb{F}_4) \rtimes Gal(\mb F_4/\mb F_2)$ acts on $E$
and $L_{K(2)}\mathit{tmf}\simeq E^{hG_{48}}$ \cite{hopkins-mahowald}. The
subgroup $\langle\omega\rangle\subseteq Aut(C/\mb F_{4})$ fixing the
point at infinity on $C$ (whose generator sends $(x,y)$ to $(\omega^2
x, y)$) is cyclic of order $3$ and defines a subgroup
\begin{equation}\label{subgroup}
S_3 \cong \langle\omega\rangle \rtimes Gal(\mb F_4/\mb F_2)\subseteq G_{48}.
\end{equation}
We have $L_{K(2)}\mathit{tmf}_1(3)\simeq E^{hS_3}$ by the
construction of $\mathit{tmf}_1(3)$ \cite[proof of Theorem
4.4]{lawsonnaumann}.
We define the $K(2)$-localizations of the maps from
diagram~(\ref{eq:main-diagram}) as follows:
\[
\xymatrix{ E^{hG_{48}} \ar[r] \ar[d]_{o_{K(2)}} & \ast \ar[d]\\
E^{hS_3}\ar[r] & \ast }
\]
The map $o_{K(2)}$ is defined to be the canonical map of homotopy
fixed point spectra associated with the inclusion of
equation~(\ref{subgroup}).
\subsection{The $K(1)$-local maps}\label{subsec:K1-local-maps}
We refer the reader to
\cite{laures,hopkins,ando-hopkins-strickland}, as well as
\cite[Sections 5 and 6]{lawsonnaumann} and references therein,
for an account of basic results about $K(1)$-local ${\cal
E}_\infty$-ring spectra which we will use freely.
To ease reading, in this subsection only we will abbreviate
\begin{equation}
\label{eq:k1-local-spectra}
\begin{split}
{\rm TMF} &=L_{K(1)}\mathit{tmf},\\
{\rm TMF}_1(3) &=L_{K(1)}\mathit{tmf}_1(3),\\
KO &=L_{K(1)}ko, \text{ and}\\
K &=L_{K(1)}ku.
\end{split}
\end{equation}
Furthermore, all smash products will implicitly be $K(1)$-localized
and all abelian groups implicitly $2$-completed. We use $K^\vee_*(-)$
to denote ($K(1)$-localized) $K$-homology, so that
\[
K^\vee_*(-)=\pi_*L_{K(1)}(K\wedge -).
\]
In order to construct the required maps between the ${\cal
E}_\infty$-ring spectra in equation~(\ref{eq:k1-local-spectra}),
we first construct maps between the $\psi\mbox{-}\theta$-algebras given by
their $K^\vee$-homology.
\begin{prop}
All spectra in equation~(\ref{eq:k1-local-spectra}) have $K^\vee_*$
concentrated in even degrees, and there are isomorphisms of
$\psi\mbox{-}\theta$-algebras as follows.
\begin{align}
K^\vee_0({\rm TMF}) &\cong V \label{eq:khomoftmf} \\
K^\vee_0(KO) &\cong \ensuremath{{\rm Hom}}_c(\mb Z_2^\times/\{ \pm 1\}, K_0)\label{eq:khomofko}\\
\label{eq:k_homology_of_k} K^\vee_0(K) &\cong \ensuremath{{\rm Hom}}_c(\mb Z_2^\times, K_0)
\end{align}
Here $V$ is (the level $1$-analogue of) Katz's ring of generalized
modular functions \cite[(1.4.9.1)]{katzhigher}, where it is denoted
$V_{\infty,\infty}$.
These fit into a commutative diagram of $\psi\mbox{-}\theta$-algebras as follows:
\begin{equation}\label{diag:psthdiagram}
\xymatrix{
K_0^\vee({\rm TMF})\cong V \ar[r]^-a \ar[d]^d &
Hom_c(\mb Z_2^\times/\{ \pm 1\}, K_0)\cong K_0^\vee(KO) \ar[d]^b \\
K_0^\vee({\rm TMF}_1(3)) \ar[r]^-c &
Hom_c(\mb Z_2^\times, K_0)\cong K_0^\vee(K)}
\end{equation}
\end{prop}
\begin{proof}
First, we review the structure of these $\psi\mbox{-}\theta$-algebras.
According to \cite[Lemma 1]{hopkins} and \cite[Proposition
3.4]{lauressplitting}, or \cite[Proposition 9.2]{ando-hopkins-rezk},
we have isomorphisms of $\psi\mbox{-}\theta$-algebras as follows:
\begin{equation}\label{eq:structure-K-theory}
K^\vee_*K\cong Hom_c(\mb Z_2^\times, K_*)
\end{equation}
\begin{equation}\label{eq:structure-K-KO}
K^\vee_*KO\cong Hom_c(\mb Z_2^\times/\{ \pm 1\}, K_*)
\end{equation}
\begin{equation}\label{eq:structure-KO-theory}
KO^\vee_*KO\cong Hom_c(\mb Z_2^\times/\{ \pm 1\}, KO_*)
\end{equation}
Here we have obvious $\mb Z_2^\times$-actions and trivial
$\theta$, in the sense that the ring homomorphism $\psi^2$ is the
identity.
Furthermore, the inclusion of the constant functions $K_*\subseteq
K_*^\vee K$ (resp. $K_*\subseteq K_*^\vee KO$), as the ring
of $\mb Z_2^\times$-invariants, is a split $\mb Z_2^\times$- (resp.
$\mb Z_2^\times/\{ \pm 1 \}$-) Galois extension.
Next, we consider $K_0^\vee({\rm TMF}_1(3))$. We have a generalized elliptic
curve
\begin{equation}
\label{eq:ordinary-curve}
E\colon\thinspace y^2+xy+by=x^3
\end{equation}
over $\comp{\mb Z_2[b]}_2$. By construction \cite[Section
6.2]{lawsonnaumann}, the Hurewicz map
\[
\pi_0{\rm TMF}_1(3) \to K^\vee_0{\rm TMF}_1(3)
\]
has domain $\comp{\mb Z_2[b]}_2$, and is the $\mb Z_2^\times$-Galois
extension classifying isomorphisms $\widehat{\mb G}_m \to
\widehat{E}$. We will refer to such an isomorphism as a
trivialization of the ordinary elliptic curve $E/\pi_0({\rm TMF}_1(3))$.
The operation $\theta\colon\thinspace K^\vee_0({\rm TMF}_1(3))\to K^\vee_0({\rm TMF}_1(3))$ is
determined by the canonical subgroup of $E$ \cite[page 35]{gouvea} and
can be computed explicitly \cite[proof of Proposition
8.5]{lawsonnaumann}. Still by construction, we also have $K_1^\vee
{\rm TMF}_1(3)=0$.
The $\psi\mbox{-}\theta$-algebra structure on $V$ is determined similarly. The
ring $V$ carries a universal isomorphism class of trivialization of
its elliptic curve. It has a continuous action of $\mb Z_2^\times$
(acting on the universal trivialization) and a canonical lift of
Frobenius $\psi^2$, which induces the natural transformation on $V$
determined by the quotient by the canonical subgroup. Since $V$ is
torsion free, there is a unique self-map $\theta$ of $V$ such that
$\psi^2(x)=x^2+2\theta(x)$ for all $x\in V$. This gives the structure
of a $\psi\mbox{-}\theta$-algebra to $V$.
We will now establish the isomorphism of
equation~(\ref{eq:khomoftmf}). From \cite[Theorem 3 and
Proposition 1]{laures}, we know $KO^\vee_*{\rm TMF}\cong KO_*\otimes
KO^\vee_0 {\rm TMF}$ and $V\simeq KO^\vee_0 {\rm TMF}$ as $\psi\mbox{-}\theta$-algebras.
Since ${\rm TMF}$ is equivalent to a ($K(1)$-local) wedge of copies of
$KO$ \cite[Corollary 3]{laures}, we find that
\begin{equation}\label{eq:khomsplitoftmf}
K^\vee_* {\rm TMF} \cong \widehat\oplus K^\vee_* KO.
\end{equation}
Therefore, by equation~(\ref{eq:structure-K-KO}) we find that
$K^\vee_* {\rm TMF}$ is concentrated in even degrees. Moreover,
by equation~(\ref{eq:structure-KO-theory}) we have that the map $V
\cong KO^\vee_0 {\rm TMF} \to K^\vee_0 {\rm TMF}$ is an isomorphism.
We next construct maps between these $\psi\mbox{-}\theta$-algebras as required in diagram~(\ref{diag:psthdiagram}).
\begin{description}
\item{Construction of the map $b$.}
The map $b$ is determined by equations
(\ref{eq:structure-K-theory}), (\ref{eq:structure-K-KO}), and
pull-back along the canonical projection $\mb Z_2^\times\to\mb
Z_2^\times/\{ \pm 1\}$, and is clearly a map of $\psi\mbox{-}\theta$-algebras.
\item{Construction of the map $d$.}
By construction there is a trivialization of the elliptic curve
$E$ over $K^\vee_0{\rm TMF}_1(3)$. As $V$ carries the universal
example of a trivialized elliptic curve over a $2$-adically
complete ring, this determines a map $d\colon\thinspace V=K^\vee_0{\rm TMF}\to
K^\vee_0{\rm TMF}_1(3)$. A $\Gamma_1(3)$ structure on an elliptic curve
$E$ determines a unique compatible structure on the quotient by the
canonical subgroup, as $2$ and $3$ are relatively prime. This
implies that the induced map of classifying rings is a map of
$\psi\mbox{-}\theta$-algebras.
\item{Construction of the map $a$.}
The map $a$ is constructed in the same manner as $d$: the ring
$K^\vee_0 KO$ carries the Tate curve $y^2 + xy = x^3$ as universal among
isomorphism classes of nodal elliptic curve equipped with a choice
of trivialization, and the map $a\colon\thinspace V \to K^\vee_0 KO$ classifies it.
\item{Construction of the map $c$.} The map
\[
\pi_0{\rm TMF}_1(3)\cong\comp{\mb Z_2[b]}_2\to K^\vee_0\cong\mb
Z_2,
\]
determined by sending $b\mapsto 0$ specializes the elliptic curve
$E$ of equation~(\ref{eq:ordinary-curve}) over $\pi_0{\rm TMF}_1(3)$ to
the Tate curve $T\colon\thinspace y^2+xy=x^3$. We fix an isomorphism of formal
groups $\widehat{T}\cong\widehat{\mb G}_m$, so that the pullback of
the elliptic curve $E$ under the composite map $\pi_0 {\rm TMF}_1(3)\to
K^\vee_0K$ has a trivialization. By the universal property of
$K^\vee_0 {\rm TMF}_1(3)$, this trivialization determines a map $c\colon\thinspace
K^\vee_0{\rm TMF}_1(3)\to K^\vee_0K$, and we need to show that it
commutes with $\psi^\alpha$ ($\alpha\in\mb Z_2^\times$) and
$\psi^2$.
Let $(E, f\colon\thinspace \widehat{E}\stackrel{\cong}{\to}\widehat{\mb G}_m)$ be the
universal trivialization of $E$ over $K^\vee_0({\rm TMF}_1(3))$. Then
\[
(\psi^\alpha)_*(E,f)=(E,[\alpha]\circ f)
\]
for any $\alpha\in\mb Z_2^\times=\mathit{Aut}(\widehat{\mb G}_m)$. Hence
\[
(c\circ\psi^\alpha)_*(E,f)=
(T,c_*([\alpha])\circ c_*(f)) =
(T,[\alpha]\circ c_*(f)).
\]
On the other hand,
\[
(\psi^\alpha\circ c)_*(E,f)=\psi_*^\alpha(T,c_*(f))=(T,[\alpha]\circ
c_*(f)),
\]
so $\psi^\alpha\circ c=c\circ\psi^\alpha$, making $c$ compatible
with the $\mb Z_2^\times$-action.
Recall \cite[page 35]{gouvea} the canonical subgroup $C\subseteq E$ is
defined so that there is the following diagram
of formal groups over $K_0^\vee ({\rm TMF}_1(3))$ with exact columns:
\begin{equation}\label{eq:psipcheck}
\xymatrix{ C \ar[rr]^{\cong} \ar@{^(->}[d] & & \mu_2 \ar@{^(->}[d]\\
\widehat{E} \ar[d]\ar[rr]^f_\cong & & \widehat{\mb G}_m \ar[d]_{[2]} \\
\widehat{E/C} \ar[rr]^{\bar{f}}_\cong & & \widehat{\mb G}_m}
\end{equation}
From this, we find that $c_*(\bar{f})=\overline{c_*(f)}$.
By construction of $\psi^2$, we have
\[ (\psi^2)_*(E,f)=(E/C,\bar{f}),\]
By the functoriality of the canonical subgroup and
(\ref{eq:psipcheck}), we therefore find that
\[ (c\circ\psi^2)_*(E,f)=(c_*(E/C),c_*(\bar{f}))=(T/C,\overline{c_*(f)}),\]
On the other hand,
\[ (\psi^2\circ c)_*(E,f)=\psi_*^2(T,c_*(f))=(T/C,\overline{c_*(f)}).\]
Hence $\psi^2\circ c=c\circ\psi^2$, and $c$ is indeed a map of
$\psi\mbox{-}\theta$-algebras.
\end{description}
To see that diagram~(\ref{diag:psthdiagram}) commutes, it
suffices to remark that both composites $b\circ a$ and $c\circ d$
classify the same trivialized generalized elliptic curve over
$K^\vee_0K$, and this is true by construction.
\end{proof}
We now start to realize diagram~(\ref{diag:psthdiagram}) as the
$K^\vee$-homology of a commutative diagram of $K(1)$-local ${\cal
E}_\infty$-ring spectra. The authors have not been able to locate a
complete proof in the literature for the following result, though it
is known to the experts and a proof sketch can be found in
\cite[Remark 2.2]{davis-mahowald-connective-versions}.
\begin{prop}\label{prop:TMFtoKO}
There is an ${\cal E}_\infty$-map $c_{K(1)}\colon\thinspace {\rm TMF}\to KO$ such that
$K^\vee_0(c_{K(1)})=a$ as in diagram~(\ref{diag:psthdiagram}).
\end{prop}
\begin{proof}
We remind the reader of the presentation of $KO$ and $TMF$ as finite
cell $L_{K(1)}\mb S$-algebras. We will write $\mb PX$ for the free
$K(1)$-local $E_\infty$-ring spectrum on a $K(1)$-local spectrum $X$.
There is a generator $\zeta\in\pi_{-1}(L_{K(1)}S^0)$, and we define
$T_\zeta$ to be $\mb S \cup_\zeta e^0$: the pushout of the diagram
\[
\mb S \mathop\leftarrow^{0} \mb PS^{-1} \mathop\rightarrow^{\zeta} \mb S
\]
in the category of $K(1)$-local ${\cal E}_\infty$-ring spectra.
We refer to this as the ${\cal E}_\infty$-cone over $\zeta$.
There are elements $y$ and $f$ in $\pi_0 T_\zeta$. We refer
the reader to the discussion surrounding \cite[Proposition 5]{laures}
for the definitions of these elements, and to \cite[end of
appendix]{laures} for the existence of a factorization of the
attaching maps
\[
\xymatrix{
y\colon\thinspace \mb PS^0 \ar[rr]^{\theta(x)-h(x)} &&
\mb PS^0 \ar[r]^{f} & T_\zeta.
}
\]
The spectrum $KO$ is $T_\zeta\cup_f e^1$, the ${\cal
E}_\infty$-cone on $f$ \cite[Proposition 13]{hopkins}, and
${\rm TMF}\simeq T_\zeta\cup_y e^1$ \cite[Convention on page 390]{laures}
as ${\cal E}_\infty$-ring spectra. Therefore, there is an ${\cal
E}_\infty$-map $c_{K(1)}\colon\thinspace {\rm TMF}\to KO$ factoring the given
attaching maps.
It remains to see that $K^\vee_0(c_{K(1)})=a$ and to this end, we
first consider the effect of $c_{K(1)}$ in homotopy. Remembering that
everything is implicitly $2$-completed, we know that
\begin{equation}\label{eq:homotopyofk1localtmf} \pi_0{\rm TMF} =
\mathbb{Z}_2[f]= \mb Z_2[j^{-1}]
\end{equation}
by \cite[Proposition 6 and Lemma 9]{laures}.
By construction $f$ maps to zero under $c_{K(1)}$, and this
implies that $j^{-1}$ also maps to zero by the following
computation.
The element $j^{-1}$ is a $2$-adically convergent power series in $f$:
\begin{equation}\label{eq:expansion}
j^{-1}=\sum\limits_{n=0}^\infty a_n f^n
\end{equation}
Clearly $c_{K(1)}$ sends $j^{-1}$ to $a_0$.
We now pass to $q$-expansions. It is classical that
$j^{-1}(q)=q+O(q^2)$. Since $f$ is of the form $f=\psi(b)-b$ for a
suitable $2$-adic modular function $b$ and $\psi$ the Frobenius
operator \cite[Equation (33)]{laures}, we learn that the $q$-expansion
\[
f(q)=\psi(b)(q)-b(q)=b(q^2)-b(q)
\]
has constant term $0$. Hence, taking $q$-expansions of
equation~(\ref{eq:expansion}) and setting $q=0$ yields $0=a_0$, as
desired.
Finally, knowing that $c_{K(1)}$ sends $j^{-1}$ to $0$ implies that
$K_0^\vee (c_{K(1)})=a$, because $K_0^\vee (c_{K(1)})$ is the map induced
on the Igusa towers (\cite[Definition 5.6]{lawsonnaumann}) by the map
$\pi_0(c_{K(1)})$.
\end{proof}
All other maps of $K(1)$-local ${\mathcal E}_\infty$-ring spectra we
require will be constructed by obstruction theory. (The reason
the map $c_{K(1)}$ cannot be thus constructed is that $K^\vee_0(KO)$
is not an induced $\mb Z_2^\times$-module
(Equation~\ref{eq:khomofko}), which is a
manifestation of the fact that $KO$ is not complex orientable.)
We recall, for a graded $\psi\mbox{-}\theta$-algebra $B_*$ over $(\comp{K}_p)_*$,
$\Omega^t B_*$ is the kernel of the map of augmented $\psi\mbox{-}\theta$-algebras
\[
B_* \otimes_{(\comp{K}_p)_*}(\comp{K}_p)^* S^t \to B_*.
\]
\begin{prop}\label{prop:mappings}
Let $p$ be a prime and suppose $X$ and $Y$ are $K(1)$-local
${\mathcal E}_\infty$-ring spectra with the following properties.
\begin{enumerate}[i)]
\item $K_0^\vee (X)$ and $K^\vee _0(Y)$ are $p$-adically complete and
$K^\vee _1(X)=K^\vee _1(Y)=0$.
\item The inclusion $\left(K^\vee _0(X)\right)^{\mb
Z_p^\times}\subseteq K^\vee _0(X)$ is the $p$-adic completion of
an ind-\'etale extension.
\item The ring $\left( K^\vee _0(X)\right)^{\mb Z_p^\times}$ is the
$p$-adic completion of a smooth $\mb Z_p$-algebra.
\item For all $s>0$ we have $H^s_c(\mb Z_p^\times, K_*^\vee(Y))=0.$
\end{enumerate}
Then there is an isomorphism
\[
f \mapsto K^\vee_0(f)\colon\thinspace \pi_0 {\cal E}_\infty(X,Y)\stackrel{\cong}{\to}
Hom_{\psi\mbox{-}\theta}(K_0^\vee X, K_0^\vee Y),
\]
given by the canonical map evaluating on $K_0^\vee$, from the set
of connected components of the derived ${\mathcal E}_\infty$-mapping
space to the set of $\psi\mbox{-}\theta$-algebra maps.
\end{prop}
\begin{proof} The first assumption implies that $A_*:=K^\vee_*(X)$ and
$B_*:=K_*^\vee (Y)$ are graded, $p$-adic, even-periodic
$\psi\mbox{-}\theta$-algebras. The remaining conditions are exactly those
of \cite[Lemma 5.14]{lawsonnaumann}, application of which implies
that for all $s\ge 2$ or $t\in\mb Z$ odd we have vanishing of
the $\psi\mbox{-}\theta$-algebra cohomology groups
\[
H^s_{\psi\mbox{-}\theta}(A_*/(\comp{K}_p)_*,\Omega^t B_*).
\]
The claim now follows from Goerss-Hopkins obstruction theory as
in \cite[Theorem 5.13, 3]{lawsonnaumann}.
\end{proof}
We will make use of the following particular
instances of this result.
\begin{prop}\label{prop:mappingspaces}
For each dotted arrow between $K(1)$-local ${\cal
E}_\infty$-ring spectra $X$ and $Y$ in the diagram
\[
\xymatrix{
&
{\rm TMF} \ar@{.>}[d] \ar@{.>}[dr] \ar@{.>}[dl] &
KO \ar@{.>}[d]\\
L_{K(1)} L_{K(2)} \mathit{tmf}_1(3) &
{\rm TMF}_1(3) \ar@{.>}[r] &
K,
}
\]
there is an isomorphism
\[
f \mapsto K^\vee_0(f)\colon\thinspace \pi_0 {\cal E}_\infty(X,Y) \stackrel{\cong}{\to}
Hom_{\psi\mbox{-}\theta}(K_0^\vee X, K_0^\vee Y)
\]
given by evaluation on $K_0^\vee$.
\end{prop}
\begin{proof}
In order to deduce this from Proposition \ref{prop:mappings}, we
need to know certain properties of the $K^\vee$-homology of the
spectra involved, the local references for which we summarize in
the following table.
\begin{tabular*}{1.0\linewidth}{lccccc}
&$KO$ & $K$ & ${\rm TMF}_1(3)$ & $L_{K(1)} L_{K(2)}{\rm TMF}_1(3)$ & ${\rm TMF}$ \\
\text{$p$-adic, even}&(\ref{eq:structure-K-KO}) & (\ref{eq:structure-K-theory}) & \cite[5.4]{lawsonnaumann} & \cite[5.4]{lawsonnaumann} & (\ref{eq:khomoftmf})+(\ref{eq:structure-K-KO})\\
\text{ind-\'etale $K^\vee_0$}&(\ref{eq:structure-K-KO}) & (\ref{eq:structure-K-theory}) & \cite[5.8]{lawsonnaumann} & \cite[5.8]{lawsonnaumann} & (\ref{eq:khomoftmf})\\
\text{smooth subring}&(\ref{eq:structure-K-KO}) & (\ref{eq:structure-K-theory}) & \cite[6.1]{lawsonnaumann} & \cite[3.5]{lawsonnaumann} & (\ref{eq:homotopyofk1localtmf}) \\
\text{no cohomology}&* & (\ref{eq:structure-K-theory}) & \cite[5.8,3]{lawsonnaumann} & \cite[5.8,3]{lawsonnaumann} &*
\end{tabular*}
Here, the rows correspond to the itemized conditions in Proposition
\ref{prop:mappings} and the columns to the spectra under
consideration. Note an entry means that the given spectrum satisfies
any assertions about either the domain spectrum $X$ or the target $Y$.
The statements labeled with an asterisk are actually false:
$K^\vee_0(KO)$ and $K^\vee _0({\rm TMF})$ are not cohomologically trivial
$\mb Z_2^\times$-modules. These statements are not needed,
because this proposition does not make any assertions about maps {\em
into} $KO$ or ${\rm TMF}$.
\end{proof}
\begin{cor}\label{cor:k1-diagram}
There is a diagram of ${\cal E}_\infty$-maps
\begin{equation}\label{eq:K1-diag}
\xymatrix{
{\rm TMF}=L_{K(1)}\mathit{tmf} \ar[r]^-{c_{K(1)}} \ar[d]_{o_{K(1)}} & KO=L_{K(1)}ko\ar[d]^{\iota_{K(1)}} \\
{\rm TMF}_1(3)=L_{K(1)}\mathit{tmf}_1(3) \ar[r]^-{\tilde{c}_{K(1)}} & K=L_{K(1)}ku,
}
\end{equation}
with diagram (\ref{diag:psthdiagram}) being realized by the
$K_0^\vee$-homology of diagram (\ref{eq:K1-diag}) and diagram
(\ref{eq:K1-diag}) commuting up to homotopy in the category of
${\cal E}_\infty$-ring spectra.
\end{cor}
\begin{proof}
Applying Proposition \ref{prop:mappingspaces}, we obtain
${\mathcal E}_\infty$-maps $o_{K(1)}$, $\tilde{c}_{K(1)}$, and
$\iota_{K(1)}$ that are characterized up to homotopy by
satisfying $K_0^\vee (o_{K(1)})=d$, $K_0^\vee (\tilde{c}_{K(1)})=c$, and
$K_0^\vee (\iota_{K(1)})=b$. From Proposition \ref{prop:TMFtoKO} we
already have the ${\mathcal E_\infty}$-map $c_{K(1)}$ satisfying
$K_0(c_{K(1)})=a$. Note that we do not need to know
whether $a$ is characterized by its effect in $K^\vee$-homology.
Proposition \ref{prop:mappingspaces} then reduces the homotopy
commutativity of diagram (\ref{eq:K1-diag}) to the previously
established commutativity of diagram (\ref{diag:psthdiagram}).
\end{proof}
\subsection{Chromatic gluing of maps}\label{subsec:che-glue-maps}
We briefly remind the reader of chromatic pullbacks in stable
homotopy, referring to the introduction of
\cite{goerss-henn-mahowald-rezk} for more details and references.
Fixing a prime $p$, every $p$-local spectrum $X$ maps canonically
to a tower of Bousfield localizations
\[
X\longrightarrow\left( \cdots L_n X\longrightarrow
L_{n-1}X\longrightarrow\cdots \longrightarrow
L_0X=X\otimes\mathbb{Q}\right),
\]
and the various stages of this tower are determined by canonical
homotopy pullbacks, called chromatic fracture squares
\cite{hovey-strickland-localisation} or \cite[Lecture 23, Proposition 5]{luriechromatic}:
\begin{equation}\label{diag:lntolnplus1}
\xymatrix{ L_n X\ar[r]\ar[d] & L_{K(n)}X\ar[d]\\
L_{n-1}X\ar[r] & L_{n-1}(L_{K(n)}X)}
\end{equation}
Here, $K(n)$ denotes any Morava $K$-theory of height $n$ at the prime
$p$ and the localization functor $L_n$ is naturally equivalent to
$L_{K(0) \vee \cdots \vee K(n)}$. We will write $L_{K(1)} L_{K(2)} X$
for the iterated localization $L_{K(1)} L_{K(2)} X$, and similarly for
other iterates.
We will use other canonical homotopy pullbacks similar to
(\ref{diag:lntolnplus1}), such as the following:
\begin{equation}\label{diag:k1andk2glueing}
\xymatrix{ L_{K(1)\vee K(2)} Y\ar[r]\ar[d] & L_{K(2)}Y\ar[d]\\
L_{K(1)}Y\ar[r] & L_{K(1)} L_{K(2)}Y}
\end{equation}
\begin{lem}\label{lem:chrom-glue-maps}
Assume $f_{K(i)}\colon\thinspace L_{K(i)}X\to L_{K(i)}Y$ ($i=1,2$) are ${\cal
E}_\infty$-maps such that the diagram
\[
\xymatrix{
L_{K(1)}X \ar[r] \ar[d]^{f_{K(1)}} & L_{K(1)} L_{K(2)}X\ar[d]^{L_{K(1)}(f_{K(2)})} \\
L_{K(1)} Y\ar[r] & L_{K(1)} L_{K(2)} Y
}
\]
commutes up to homotopy in the category of ${\cal E}_\infty$-ring spectra.
Then there is an ${\cal E}_\infty$-map $f\colon\thinspace L_{K(1)\vee K(2)} X\to
L_{K(1)\vee K(2)}Y$, not necessarily unique, such that the diagrams
\[
\xymatrix{
L_{K(1)\vee K(2)} X\ar[r]^{f} \ar[d] & L_{K(1)\vee K(2)} Y\ar[d]
& & L_{K(1)\vee K(2)} X\ar[r]^{f} \ar[d] & L_{K(1)\vee
K(2)} Y\ar[d] \\
L_{K(1)}X\ar[r]^{f_{K(1)}} & L_{K(1)} Y & & L_{K(2)}X
\ar[r]^{f_{K(2)}} & L_{K(2)} Y
}
\]
commute up to homotopy.
\end{lem}
\begin{proof}
Apply the derived mapping-space functor ${\cal E}_\infty(L_{K(1)\vee
K(2)} X, -)$ to the chromatic fracture square
(\ref{diag:k1andk2glueing}).
\end{proof}
\begin{cor}\label{cor:o-k1-k2-local}
There exists an ${\mathcal E}_\infty$-map
\[ o_{K(1)\vee K(2)}\colon\thinspace L_{K(1)\vee K(2)}\mathit{tmf}\to L_{K(1)\vee K(2)}\mathit{tmf}_1(3)\]
such that the diagrams of ${\mathcal E}_\infty$-maps
\[
\xymatrix{
L_{K(1)\vee K(2)} \mathit{tmf}\ar[r]_-{o_{K(1)\vee K(2)}} \ar[d] & L_{K(1)\vee
K(2)} \mathit{tmf}_1(3)\ar[d] \\
L_{K(1)}\mathit{tmf}\ar[r]_-{o_{K(1)}} & L_{K(1)} \mathit{tmf}_1(3)}
\hspace{1pc}
\xymatrix{
L_{K(1)\vee K(2)} \mathit{tmf}\ar[r]_-{o_{K(1)\vee K(2)}} \ar[d] & L_{K(1)\vee
K(2)} \mathit{tmf}_1(3)\ar[d] \\
L_{K(2)}\mathit{tmf} \ar[r]_-{o_{K(2)}} & L_{K(2)} \mathit{tmf}_1(3)
}
\]
both commute up to homotopy.
\end{cor}
\begin{proof}
This follows from Lemma \ref{lem:chrom-glue-maps}, provided we can
establish the commutativity up to homotopy of the following diagram of
${\mathcal E}_\infty$-ring spectra:
\begin{equation}\label{diag:k1}
\xymatrix{ L_{K(1)}\mathit{tmf}\ar[rr]^-{o_{K(1)}} \ar[d]& &
L_{K(1)}\mathit{tmf}_1(3)\ar[d]\\
L_{K(1)} L_{K(2)}\mathit{tmf}\ar[rr]^-{L_{K(1)}(o_{K(2)})} & &
L_{K(1)} L_{K(2)} \mathit{tmf}_1(3)}
\end{equation}
The initial and terminal objects in this diagram appear in
Proposition \ref{prop:mappingspaces}, and so it suffices to see
that the induced diagram in $K^\vee$-homology commutes:
\[
\xymatrix{V\ar[r]\ar[d] & K_0^\vee L_{K(1)}\mathit{tmf}_1(3)\ar[d]\\
K_0^\vee L_{K(1)} L_{K(2)}\mathit{tmf}\ar[r] & K_0^\vee L_{K(1)} L_{K(2)} \mathit{tmf}_1(3)}
\]
This holds true because both composites classify isomorphic
trivializations of the elliptic curves $y^2 + xy + a^{-3}y = x^3$
and $y^2 + axy + y = x^3$ over $\comp{\mb Z\laur{a}}_2$
$\cong\pi_0L_{K(1)} L_{K(2)}\mathit{tmf}_1(3)$ (see
Section~\ref{sec:elliptic-curve}).
\end{proof}
\begin{prop}\label{prop:k1-k2-diag}
There exists a diagram of ${\cal E}_\infty$-ring spectra which
commutes up to homotopy as follows:
\[
\xymatrix{
L_{K(1)\vee K(2)}\mathit{tmf} \ar[rr]^-{c_{K(1)\vee K(2)}} \ar[d]^{o_{K(1)\vee
K(2)}} & &
L_{K(1) \vee K(2)}ko\ar[d]^{\iota_{K(1)\vee K(2)}} \\
L_{K(1)\vee K(2)}\mathit{tmf}_1(3)\ar[rr]^-{\tilde{c}_{K(1)\vee K(2)}} & &
L_{K(1) \vee K(2)}ku
}
\]
\end{prop}
\begin{proof}
We have the following diagram of ${\cal E}_\infty$-ring spectra
which commutes up to homotopy:
\begin{equation}\label{eq:K1-K2-diag}
\xymatrix{
L_{K(1)\vee K(2)}\mathit{tmf} \ar[r] \ar[d]^{o_{K(1)\vee
K(2)}} &
L_{K(1)} \mathit{tmf} \ar[r]^-{c_{K(1)}} \ar[d]^{o_{K(1)}}&
L_{K(1)}ko\ar[d]^{\iota_{K(1)\vee K(2)}} \\
L_{K(1)\vee K(2)}\mathit{tmf}_1(3)\ar[r] &
L_{K(1)}\mathit{tmf}_1(3) \ar[r]^-{\tilde c_{K(1)}} &
L_{K(1)}ku}
\end{equation}
Here, the left square is from
Corollary~\ref{cor:o-k1-k2-local}, and the right one is from
Corollary~\ref{cor:k1-diagram}. Since $L_{K(1) \vee K(2)} ko
\simeq L_{K(1)} ko$ and similarly for $ku$, we can define the upper
and lower horizontal composites to be $c_{K(1) \vee K(2)}$ and $\tilde
c_{K(1) \vee K(2)}$ respectively.
\end{proof}
\subsection{The rational maps}\label{subsec:K0-local-maps}
We first note the following about rational ${\cal
E}_\infty$-ring spectra.
\begin{lem}\label{lem:rationalmaps}
Suppose $X$ and $Y$ are ${\cal E}_\infty$-ring spectra such that
$\pi_* X \otimes \mb Q$ is a free graded-commutative $\mb Q$-algebra
on generators in even nonnegative degrees, and $\pi_* Y$ is rational
with homotopy in nonnegative odd degrees. Then the natural map
\[
\pi_0 {\cal E}_\infty(X,Y) \to \ensuremath{{\rm Hom}}_{\text{graded rings}}(X_*,Y_*)
\]
is bijective, and all path components of the derived mapping space
${\cal E}_\infty(X,Y)$ are simply connected.
\end{lem}
\begin{proof}
Since $Y$ is rational, the natural map
\[
{\cal E}_\infty(X \otimes \mb Q, Y) \to {\cal E}_\infty(X,Y)
\]
is a weak equivalence. As $X \otimes \mb Q$ is equivalent to a free $H\mb
Q$-algebra on some family of cells $x_i\colon\thinspace S^{2n_i} \to X$,
evaluation on the generators gives a weak equivalence, natural in
$Y$, of the form
\[
{\cal E}_\infty(X \otimes \mb Q, Y) \to \prod \Omega^{\infty+{2n_i}} Y.
\]
The result follows by considering $\pi_0$ and $\pi_1$ of the
right-hand side.
\end{proof}
\begin{thm}
\label{thm:rationalcube}
There exists a strictly commutative diagram in the category of
rational ${\cal E}_\infty$-ring spectra as follows:
\begin{equation}
\label{eq:rationalcube}
\xymatrix{
L_{K(0)}\mathit{tmf} \ar@{.>}[rrr] \ar@{.>}[d] \ar[ddr]& & &
L_{K(0)}ko \ar@{.>}[d] \ar[ddr] \\
L_{K(0)}\mathit{tmf}_1(3) \ar@{.>}[rrr] \ar[ddr] & & &
L_{K(0)}ku \ar[ddr] \\
&L_{K(0)} L_{K(1) \vee K(2)} \mathit{tmf}\ar[rrr] \ar[d] & & &
L_{K(0)} L_{K(1) \vee K(2)} ko \ar[d] \\
&L_{K(0)} L_{K(1)\vee K(2)} \mathit{tmf}_1(3) \ar[rrr] & & &
L_{K(0)} L_{K(1)\vee K(2)} ku
}
\end{equation}
In this diagram, the bottom square is the rationalization of the
diagram displayed in Proposition \ref{prop:k1-k2-diag} and the
diagonal maps are arithmetic attaching maps.
\end{thm}
\begin{proof}
To construct diagram~(\ref{eq:rationalcube}) we must first construct
the maps in the top square of the cube as to render the entire diagram
homotopy commutative.
Recall:
\begin{align*}
\pi_*\mathit{tmf}\otimes\mb Q&\cong \mb Q[c_4,c_6] &\text{where } &|c_i| = 2i\\
\pi_* \mathit{tmf}_1(3)\otimes \mb Q&\cong \mb Q[A,B] && |A| = 2, |B| = 6
\text{ \cite[proof of Theorem 1.1]{lawsonnaumann}}\\
\pi_*ku\otimes\mb Q&\cong \mb Q[\beta] && |\beta| = 2 \\
\pi_*ko\otimes\mb Q&\cong \mb Q[\beta^2] &&
\end{align*}
In nonnegative degrees, the diagonal maps in
diagram~(\ref{eq:rationalcube}) are given on homotopy groups by
extension of scalars from $\mb Q$ to $\mb Q_2$.
Evaluating the modular forms $c_4$ and $c_6$ on
\[
y^2 + Axy + By = x^3,
\]
the universal elliptic curve with $\Gamma_1(3)$-structure used to
construct $\mathit{tmf}_1(3)$, we find that
\[
c_4\mapsto A^4-24\, AB\, ,\, c_6\mapsto -A^6+36\, A^3B-216\, B^2.
\]
Similarly, evaluating at the Tate curve $y^2 + \beta xy = x^3$, we
find that
\begin{equation}
\label{eq:cusp-orientation}
A\mapsto\beta,\, B\mapsto 0,
\end{equation}
and
\[
c_4\mapsto \beta^4,\,c_6\mapsto -\beta^6.
\]
These formulas make diagram~(\ref{eq:rationalcube})
commutative on homotopy groups.
The homotopy groups of the spectra in the upper square of
diagram~(\ref{eq:rationalcube}) form polynomial algebras, and all
spectra in the diagram have zero homotopy in positive odd degrees.
Lemma~\ref{lem:rationalmaps} thus implies that constructing the maps
in this diagram is equivalent to defining the maps on homotopy
groups, and that the homotopy-commutativity of each square
subdiagram is equivalent to the commutativity of the square on
homotopy groups. This shows that the cubical diagram commutes in the
homotopy category.
Finally, the obstruction to lifting a homotopy commutative
cubical diagram to an honestly commutative, homotopy equivalent cubical
diagram lies in $\pi_1 {\cal E}_\infty(L_{K(0)} \mathit{tmf}, L_{K(0), K(1)
\vee K(2)} ku)$, which is the zero group (again by Lemma
\ref{lem:rationalmaps}).
\end{proof}
\begin{cor}
There is a commutative square of ${\cal E}_\infty$-ring
spectra as follows:
\begin{equation}\label{eq:K012-diag}
\xymatrix{ L_{K(0)\vee K(1)\vee K(2)}\mathit{tmf}\ar[rrr]^{c_{K(0)\vee K(1)\vee K(2)}}\ar[d]_{o_{K(0)\vee K(1)\vee K(2)}} & & & L_{K(0)\vee K(1)\vee K(2)} ko\ar[d]^{\iota_{K(0)\vee K(1)\vee K(2)}} \\
L_{K(0)\vee K(1)\vee K(2)} \mathit{tmf}_1(3) \ar[rrr]^{\tilde{c}_{K(0)\vee K(1)\vee K(2)}} & & & L_{K(0)\vee K(1)\vee K(2)} ku}
\end{equation}
\end{cor}
\begin{proof}
For the $K(0) \vee K(1) \vee K(2)$-local spectra under
consideration, $L_{K(1) \vee K(2)}$ is $p$-adic completion. We have
canonical arithmetic squares
\[
\xymatrix{ L_{K(0)\vee K(1)\vee K(2)} Y\ar[rr]\ar[d]
& & L_{K(1)\vee K(2)}Y\ar[d]\\
L_{K(0)}Y\ar[rr]
& & L_{K(0)} L_{K(1)\vee K(2)} Y.}
\]
We can then take levelwise homotopy pullbacks of the maps
\[
L_{K(0)}Y \to L_{K(0)} L_{K(1)\vee K(2)} Y \leftarrow L_{K(1)\vee K(2)} Y
\]
from the diagonals of diagram~(\ref{eq:rationalcube}) and obtain
the desired commutative square.
\end{proof}
\section{The cohomology computation}\label{sec:cohomology}
The techniques used in this section are very similar to those employed
in \cite{rezk-tmfnotes} to calculate $H^*(\mathit{tmf})$.
Let $p$ be a prime, abbreviate $H:=H\mb F_p$ and recall that the
dual Steenrod algebra $A_* = H_* H$ takes the form
\[
A_* \cong \begin{cases}
P\left({\bar\xi}_1, {\bar\xi}_2, \ldots,\right)
&\text{if }p=2,\\
P({\bar\xi}_1, {\bar\xi}_2, \ldots )\otimes E({\bar\tau}_0, {\bar\tau}_1\ldots)
&\text{if }p \neq 2.
\end{cases}
\]
Suppose a $p$-local spectrum $X$ is connective and of finite
type, with a map $X \to H$, such that the mod-$p$ homology maps
isomorphically to the sub-Hopf-algebra of $A_*$ given by
\[
H_* X \cong \begin{cases}
P\left({\bar\xi}_1^2,\ldots,{\bar\xi}_{n+1}^2,{\bar\xi}_{n+2},\ldots\right)
&\text{if }p=2,\\
P({\bar\xi}_1,\ldots )\otimes E(\overline{\tau}_{n+1},\ldots)
&\text{if }p \neq 2.
\end{cases}
\]
For example, this is true when $X = \BPP{n}$. In these
circumstances the Adams spectral sequence degenerates, and we find
that
\[
\pi_*X\cong\mb Z_{(p)}[v_1,\ldots,v_n]
\]
where $|v_i|=2(p^i-1)$ \cite[Chapter 4, Section 2, page
111]{ravenel}. (This is only necessarily an isomorphism as
graded abelian groups unless $X$ is a homotopy commutative and
associative ring spectrum.)
We will establish a converse to this isomorphism under the
assumption that $X$ is a ring spectrum in Theorem
\ref{thm:coh-comp}.
\begin{defn}\label{def:generalbpn}
A $p$-local ring spectrum $R$ is a {\em generalized $\BPP{n}$} if it
admits a complex orientation such that the resulting composite map
\[
\mb Z_{(p)}[v_1,\ldots,v_n]\subseteq \pi_*BP \to \pi_*MU_{(p)} \to\pi_* R
\]
is an isomorphism.
\end{defn}
We remark that as the element $v_i$ is an invariant of the formal
group modulo $(p,v_1,\ldots,v_{i-1})$, the property of a $p$-local,
homotopy commutative, complex orientable ring spectrum being a
generalized $\BPP{n}$ depends only on the ring structure and is
independent of the choice of complex orientation. In particular,
it does not depend on the choice of Hazewinkel, Araki, or arbitrary
other $p$-typical $v_i$-classes.
The following fact served as the basis for the construction of
$\mathit{tmf}_1(3)$ by chromatic fracture.
\begin{prop}
If $n > 0$, any generalized $\BPP{n}$ is the connective cover of
its $L_n$-localization (or equivalently its $L_{K(0) \vee K(1) \vee
\cdots \vee K(n)}$-localization).
\end{prop}
\begin{proof}
Let $L_n^f$ denote the finite localization of Miller
\cite{miller-finitelocalizations}. The fiber of the map $L_n^f \to
L_n$ is $BP$-acyclic. As both localizations are smashing, this is
also true for all (homotopy) $BP$-modules, including $p$-local complex
orientable ring spectra.
Therefore, the fiber of the localization map $\BPP{n} \to L_n \BPP{n}$
is equivalent to the finite colocalization $C^f \BPP{n}$. By
\cite[Proposition~7.10(a)]{hovey-strickland-localisation}, this finite
colocalization is an appropriate homotopy colimit of function spectra
\[
\ensuremath{\mathop{\rm hocolim}} F(M(p^{i_0}, v_1^{i_1}, \ldots, v_n^{i_n}), \BPP{n})
\]
out of a tower of generalized Moore spectra. The homotopy groups of
this function spectrum are
\[
\Sigma^{-(\sum i_k |v_k|)-n-1} \BPP{n}_*/(p^{i_0}, v_1^{i_1}, \ldots, v_n^{i_n}),
\]
and the colimit is ultimately
\[
\Sigma^{-n-1} \BPP{n}_*/(p^\infty, v_1^\infty, \ldots, v_n^\infty)
\]
whose top homotopy group is in degree $-n-1-\Sigma_{k=0}^n (2p^k-2)$.
As $n > 0$, the map $\BPP{n} \to L_n^f \BPP{n}$ is then a model
for the connective cover.
\end{proof}
There does not appear to be an easier method to prove this than direct
calculation. There are many closely related spectra where the
connective cover is not the correct tool (such as those associated to
moduli problems where the underlying curve has positive genus).
The authors are not aware of any results establishing uniqueness
of $\BPP{n}$ when $n \geq 2$. It is not clear when, after
forgetting the ring spectrum structure, two generalized $\BPP{n}$ with
nonisomorphic formal groups might have the same underlying homotopy
type. It is also not clear, for any particular formal group of the
correct form, how many weak equivalence classes of generalized
$\BPP{n}$ might exist realizing this formal group. (There exist
results if one assumes additional structure, such as that of an
$MU$-module or $MU$-algebra; see, for example,
\cite{jeanneret-wuthrich-quadratic}.)
The following is a consequence of Definition \ref{def:generalbpn} and
of the existence of the Quillen idempotent splitting $MU_{(p)} \to
BP$.
\begin{lem}
\label{lem:ringmaps}
Suppose $R$ is a generalized $\BPP{n}$. Then there are maps of
ring spectra $BP \to R \to H$, with the former map $(2p^{n+1} -
2)$-connected and the latter unique up to homotopy. If $R$ is an
${\cal A}_\infty$-ring spectrum or an ${\cal E}_\infty$-ring spectrum,
then the map $R \to H$ is a map of ${\cal A}_\infty$-ring spectra or
${\cal E}_\infty$-ring spectra accordingly.
\end{lem}
\begin{thm}
\label{thm:coh-comp}
Suppose $R$ is a generalized $\BPP{n}$. Then the map $R\to H$
induces an isomorphism of $H^* R$ with the left ${\cal A}^*$-module
${\cal A}^*/\!/E(n)$, and of $H_* R$ with the subalgebra
$B_*$ of the dual Steenrod algebra given as follows:
\[
B_* =
\begin{cases}
P({\bar\xi}_1^2, \ldots, {\bar\xi}_{n+1}^2, {\bar\xi}_{n+2}, \ldots) &\text{if }p=2,\\
P({\bar\xi}_1^2, \ldots) \otimes E({\bar\tau}_{n+1}, \ldots) &\text{if }p\neq 2.
\end{cases}
\]
\end{thm}
\begin{proof}
We recall from \cite[Theorem~3.4]{baker-jeanneret} that the
Brown-Peterson spectrum $BP$ admits an ${\cal A}_\infty$ ring
structure (in fact, it admits many). In the following we
choose one for definiteness. Smashing the maps from
Lemma~\ref{lem:ringmaps} on the right with $BP$ gives a sequence of
maps
\[
BP \smsh{} BP \to R \smsh{} BP \to H \smsh{} BP.
\]
Complex orientability of $BP$, $R$, and $H$ implies that on
homotopy groups, this becomes the sequence of maps of polynomial
algebras
\[
BP_*[t_i] \to R_*[t_i] \to \mb F_p[t_i],
\]
with the polynomial generators $t_i$ mapped identically.
We then apply the natural equivalence $(- \smsh{} BP) \smsh{BP} H
\simeq (-) \smsh{} H$, together with the K\"unneth spectral sequence
\cite[Theorem IV.4.1]{ekmm}, to obtain a natural map of spectral
sequences:
\begin{equation}
\label{eq:kunneth-sseqs}
\ensuremath{{\rm Tor}}^{BP_*}_{**}(R_*[t_i], \mb F_p) \to
\ensuremath{{\rm Tor}}^{BP_*}_{**}(\mb F_p[t_i], \mb F_p),
\end{equation}
which strongly converges to the map $R_*H\to H_* H$.
Here the action of the generators $v_i \in BP_*$ is through their
images under the right unit $BP_* \stackrel{\eta_R}{\to} BP_*[t_i]$.
Modulo $(p,v_1,\ldots,v_{k-1})$, the image of $v_k$ under the right
unit is equal to $v_k$.
Writing $S_* = BP_*/(p,v_1,\ldots,v_n)$, we have an identification
of derived tensor products
\[
(- \otimes^{\mb L}_{BP_*} S_*) \otimes^{\mb L}_{S_*} \mb F_p \cong (-)
\otimes^{\mb L}_{BP_*} \mb F_p.
\]
This shows that the map of equation~(\ref{eq:kunneth-sseqs}) is the
abutment of a map of Cartan-Eilenberg spectral sequences:
\begin{equation}
\label{eq:cartan-sseqs}
\ensuremath{{\rm Tor}}^{S_*}_{**}(\ensuremath{{\rm Tor}}^{BP_*}_{**}(R_*[t_i], S_*), \mb F_p) \to
\ensuremath{{\rm Tor}}^{S_*}_{**}(\ensuremath{{\rm Tor}}^{BP_*}_{**}(\mb F_p[t_i], S_*), \mb F_p)
\end{equation}
The elements $(p,v_1,\ldots,v_n)$ form a regular sequence in
$R_*$. The image of the regular sequence $(p,v_1,\ldots ,v_n)\in
BP_*$ under the map
\[ BP_*\stackrel{\eta_R}{\to} BP_*BP\to R_*BP\simeq R_*[t_i]\]
is therefore a regular sequence, by induction, because
every $v_k$ is invariant modulo $(p,\ldots,v_{k-1})$ and because
of the assumed properties of the map $BP_*\to R_*$.
This shows that the higher Tor-groups in
\[
\ensuremath{{\rm Tor}}^{BP_*}_{**}(R_*[t_i], S_*)
\]
are zero, with the zero'th term given by the tensor product $R_*[t_i]
\otimes_{BP_*} S_* \cong \mb F_p[t_i]$.
By contrast, the image of $v_k$ in $\mb F_p[t_i]$ under the right unit
is zero for all $k$, and hence the Tor-algebras are exterior
algebras.
Therefore, the map of equation~(\ref{eq:cartan-sseqs}) degenerates to
an edge inclusion:
\[
\mb F_p[t_i] \otimes \Lambda[x_{n+1}, x_{n+2}, \ldots] \to \mb
F_p[t_i] \otimes \Lambda[x_1, \ldots, x_n] \otimes \Lambda[x_{n+1}, x_{n+2},
\ldots]
\]
Here $t_i$ is in total degree $2p^i - 2$ and $x_i$ is in total degree
$2p^i - 1$. For the right-hand term, the associated graded vector
space already has the same dimension in each total degree as the dual
Steenrod algebra. Therefore, both the Cartan-Eilenberg and K\"unneth
spectral sequences must degenerate, as the final target is the dual
Steenrod algebra and any non-trivial differentials would result in a
graded vector space with strictly smaller dimension in some degree.
We find that the map $R_* H \to H_* H$ is an inclusion of right
comodules over the dual Steenrod algebra, and the image below degree
$2p^{n+1}-1$ consists only of terms in even degrees. On cohomology,
this implies that the map ${\cal A}^* \to H^* R$ is a surjection of
left ${\cal A}^*$-modules, and the image of the generator $1 \in {\cal
A}^*$ is acted on trivially by the odd-degree Milnor primitives
$Q^0,\ldots, Q^n$. The induced map ${\cal A}^*/\!/E(n) \to H^* R$ is
still a surjection and both sides are graded vector spaces of the
same, levelwise finite, dimensions over $\mb F_p$.
This shows that $H^* R$ has the desired form. The statement for
homology follows by dualizing the cohomology description.
\end{proof}
\section{Proof of Theorem \ref{thm:main}}\label{sec:proof}
We have now assembled all the preliminaries needed to give the proof
of Theorem~\ref{thm:main}, For ease of reference, we recall the
statements we need to prove.
\begin{enumerate}[i)]
\item There is a commutative diagram of connective ${\cal E}_\infty$-ring
spectra as follows:
\[
\xymatrix{ \mathit{tmf}_{(2)} \ar[r]^{c}\ar[d]^{o} & ko_{(2)}\ar[d]^{\iota} \\
\mathit{tmf}_1(3)_{(2)} \ar[r]^{\tilde{c}} & ku_{(2)}}
\]
\item In mod-$2$ cohomology, this induces the
following canonical diagram of modules over the mod $2$ Steenrod
algebra ${\cal A}^*$:
\[ \xymatrix{ {\cal A}^*/\!/{\cal A}(2) & {\cal A}^*/\!/{\cal A}(1)\ar[l]\\
{\cal A}^*/\!/E(2)\ar[u]& {\cal A}^*/\!/E(1).\ar[l]\ar[u]}\]
\item There exists a complex orientation of $\mathit{tmf}_1(3)_{(2)}$ such that in
homotopy $\tilde{c}$ sends the Hazewinkel generators $v_1$ to $v_1$
and $v_2$ to zero.
\item There is a cofiber sequence of $\mathit{tmf}_1(3)_{(2)}$-modules
\[
\Sigma^6 \mathit{tmf}_1(3)_{(2)}\stackrel{\cdot v_2}{\longrightarrow} \mathit{tmf}_1(3)_{(2)}
\stackrel{\tilde{c}}{\longrightarrow} ku_{(2)}.
\]
\end{enumerate}
\begin{proof}
The existence of the desired commutative diagram of ${\mathcal
E}_\infty$-ring spectra is established by taking connective
covers of diagram~(\ref{eq:K012-diag}).
It is well known that there are isomorphisms of ${\cal A}^*$-modules
$H^*(ko)\cong {\cal A}^*/\!/{\cal A}(1)$ and $H^*(ku)\cong {\cal
A}^*/\!/E(1)$. Theorem \ref{thm:reminder}, together with Theorem
\ref{thm:coh-comp}, implies $H_*\mathit{tmf}_1(3)\cong
P({\bar\xi}_1^2,{\bar\xi}_2^2,{\bar\xi}_3^2, {\bar\xi}_4,\ldots)$ and
$H^*\mathit{tmf}_1(3)\cong {\cal A}^*/\!/E(2)$. A well-known result, based on
work on $\mathit{tmf}$ initiated by Hopkins, Mahowald and Miller, is that
$H^*(\mathit{tmf})\cong {\cal A}/\!/{\cal A}(2)$ as a module over the Steenrod
algebra \cite[Theorem 9.2]{hopkins-mahowald}. To the best of the
authors' knowledge, this result still awaits official documentation.
A sketch based on the characterization of \cite[Theorem
14.5]{rezk-tmfnotes} can be found in \cite[Section 21]{rezk-tmfnotes}.
The diagram of ${\cal A}^*$-modules in the statement of the theorem
commutes because all appearing ${\cal A}^*$-modules are cyclic,
generated by $1$.
To address the remaining statements, note that the map
$\pi_*(\tilde{c})\colon\thinspace\pi_*\mathit{tmf}_1(3)_{(2)} \to\pi_*ku_{(2)}$ is determined by
its rationalization
\[
\mb Q [A,B] \to \mb Q [\beta],
\]
where we have $A\mapsto\beta$ and $B\mapsto 0$ by construction (see
Theorem~\ref{thm:rationalcube}). Now recall that there exists an
orientation $BP\to \mathit{tmf}_1(3)_{(2)}$ which maps $v_1$ to $A$ and $v_2$
to $B$ by \cite[Proposition 8.2]{lawsonnaumann}.
The composite $\mathit{tmf}_1(3)_{(2)}$-module map $\tilde{c}\circ(\cdot
v_2)\colon\thinspace\Sigma^6 \mathit{tmf}_1(3)_{(2)} \to ku_{(2)}$ sends the generator to
$0=\pi_*(\tilde{c})(v_2)\in\pi_6(ku_{(2)})$, and hence there is a
factorization in the category of $\mathit{tmf}_1(3)_{(2)}$-modules
\[
\xymatrix{
\Sigma^6\mathit{tmf}_1(3)_{(2)} \ar[rr]^{\cdot v_2} \ar[drr]^{0} & & \mathit{tmf}_1(3)_{(2)}
\ar[rr]\ar[d]^{\tilde c} & & \mathrm{cof}((\cdot v_2))\ar@{.>}[dll]\\
&&
ku_{(2)}
}
\]
Examining homotopy groups, we get an induced equivalence between
$ku_{(2)}$ and the cofiber of $v_2$ as spectra, and hence as
$\mathit{tmf}_1(3)_{(2)}$-modules.
\end{proof}
|
{
"timestamp": "2013-01-16T02:01:17",
"yymm": "1203",
"arxiv_id": "1203.1696",
"language": "en",
"url": "https://arxiv.org/abs/1203.1696"
}
|
\section{Introduction}
In recent years, the real variable theory of Hardy spaces, which originates from the work of Fefferman and Stein \cite{fs}, has been extend to a variety of new settings.
These developments involve replacing the euclidean Laplacian with a different semigroup generator $L$, and the space ${\mathbb R}^{n}$ endowed with the Borel algebra and the Lebesgue measure with a different metric measure space $(M,d,\mu)$. Prominent examples include Hofmann and Mayboroda's work \cite{hm} on the euclidean space with $\Delta$ replaced by a more general divergence form second order elliptic differential operator with bounded measurable coefficients, and Auscher-McIntosh-Russ's Hardy spaces of differential forms associated with the Hodge Laplacian on a Riemannian manifold \cite{amr}.
These results rely heavily on two assumptions. At the level of the metric measure space, one requires the doubling property:
there exists $C>0$ such that for all $x \in M$ and all $r>0$:
$$
\mu(B(x,2r)) \leq C \mu(B(x,r)).
$$
At the level of the semigroup $(e^{tL})_{t \geq 0}$, one requires some heat kernel estimates or, at least, some appropriate $L^{2}$ off-diagonal decay of the form
$$
\|1_{E}e^{tL}(1_{F}u)\|_{2} \leq C (1+\frac{d(E,F)^{2}}{t})^{-k}\|1_{F}u\|_{2},
$$
where $E,F$ are Borel sets, $1_{E},1_{F}$ denote the corresponding characteristic functions, $u \in L^{2}$, $k>0$, $t>0$ and $C$ is independent of $E,F,t$ and $u$.
This paper is concerned with the gaussian case: the metric measure space is ${\mathbb R}^{n}$ with the gaussian measure $d\gamma(x) = \pi^{-\frac{n}{2}}e^{-|x|^{2}}dx$ and the operator is the Ornstein-Uhlenbeck operator defined by
$$
Lf(x) := \frac{1}{2}\Delta f(x)-x.\nabla f(x), \quad x \in {\mathbb R}^{n}.
$$
This setting is motivated by stochastic analysis and has a long history (see the survey \cite{survey}).
Hardy spaces in this context were first introduced by Mauceri and Meda in \cite{mm}. Their work is striking because the gaussian measure is not doubling,
and the Ornstein-Uhlenbeck semigroup does not satisfy the kernel bounds required to apply the non-doubling theory of Tolsa \cite{t}.
While \cite{mm} contains highly interesting results, it does not provide a fully satisfying theory. This is due to the fact that Mauceri-Meda's Hardy spaces
$h^{1}_{\text{at}}(\gamma)$ are defined via an atomic decomposition that may not relate to the Ornstein-Uhlenbeck operator as well as classical Hardy spaces relate to the usual Laplacian (see \cite{fs}). In particular, the fact, proven in \cite{mms}, that some associated Riesz transforms are not bounded from $h^{1}_{\text{at}}(\gamma)$ to $L^{1}(\gamma)$ in dimension greater than $1$ is problematic. More generally, Mauceri-Meda's $h^{1}_{\text{at}}(\gamma)$ spaces provide a good endpoint to the $L^p$ scale from the interpolation point of view, but their theory does not contain all the machinery that makes Fefferman-Stein \cite{fs} so outstanding, and has proven useful in a range of applications, especially to partial differential equations.\\
In \cite{mnp,mnp2}, Jan Maas, Jan van Neerven, and the author have started the development of such a complete theory.
This involves adequate dyadic cubes, covering lemmas of Whitney type, related tent spaces and their atomic decomposition, and techniques to estimate the following non-tangential maximal functions and conical square functions:
\begin{equation*}
\begin{split}
T^{*}_{a}u(x):=\underset{(y,t)\in \Gamma^{a}_{x}(\gamma)}{\sup}|e^{t^{2}L}u(y)|,\\
S_{a}u(x) =
\Big(\int_{\Gamma^{a} _x(\gamma)} \frac1{\gamma(B(y,t))}|t\nabla
e^{t^{2}L}u(y)|^{2}\,d\gamma(y)\,\frac{dt}{t}\Big)^\frac12,
\end{split}
\end{equation*}
where
$$\Gamma ^{a}_{x}(\gamma) := \Big\{(y,t) \in {\mathbb R}^{n}\times(0,\infty)\colon
|y-x|<t<a m(x)\Big\}$$
is the {\em admissible cone} based at the point
$x\in{\mathbb R}^n$, $m(x):=\min\big\{1, \frac1{|x|}\big\}$ is the corresponding admissibility function, and $a$ the admissibility parameter.
From the point of view of Hardy space theory, one defines $h^{1}_{\text{max},a}(\gamma)$ as the completion of the space of smooth compactly supported functions
$C^{\infty}_{c}({\mathbb R}^{n})$ with respect to $$\|u\|_{h^{1}_{\text{max},a}(\gamma)}:=\|T_{a} ^{*}u\|_{L^{1}(\gamma)},$$ and $h^{1}_{\text{quad},a}(\gamma)$ as the completion of $C^{\infty}_{c}({\mathbb R}^{n})$
with respect to $$\|u\|_{h^{1}_{\text{quad},a}(\gamma)}:=\|S_{a}u\|_{L^{1}(\gamma)}+\|u\|_{L^{1}(\gamma)}.$$
A key result should be that these two norms are equivalent for some choice of $a$. However, \cite{mnp2} only gives one inequality:
$\|S_{a}u\|_{1} \leq C\|T_{a'}^{*}u\|_{1}$, for some $C,a'>0$ independent of $u$ (actually \cite{mnp2} gives a slightly stronger inequality involving an averaged version of $T^{*}_{a}u$).
The purpose of this paper is to prove the reverse inequality to establish the following result.
\begin{theorem}
\label{thm:main}
Given $a>0$, there exists $a'>0$ such that
$h^{1}_{\text{quad},a}(\gamma)=h^{1}_{\text{max},a'}(\gamma)$.
\end{theorem}
Since $h^{1}_{\text{quad},a} = h^{1}_{\text{quad},1}$ for all $a>1$ (as a consequence of \cite[Theorem 3.8]{mnp}), we then call $h^{1}(\gamma):=h^{1}_{\text{quad},2}$ the Gaussian Hardy space.
In the final section, the techniques used in the proof of the above reverse inequality are used again to prove that the Riesz transforms associated with $L$ are bounded on $h^{1}(\gamma)$.
The proof is based on a version of Calder\'on reproducing formula:
$$
u = C \int \limits _{0} ^{\infty} (t^{2}L)^{N+1}e^{\frac{(1+a^{2})t^{2}}{\alpha}L}u \frac{dt}{t}+\int \limits _{{\mathbb R}^{n}}u d\gamma,
$$
for $u \in L^{2}$ and some suitable constants $N,C$ and $\alpha$. The part
$$J_{1}u(x):=\int \limits _{0} ^{m(x)} (t^{2}L)^{N+1}e^{\frac{(1+a^{2})t^{2}}{\alpha}L}u(x) \frac{dt}{t}$$
is treated via the atomic decomposition of tent spaces established in \cite{mnp}, leading to the estimate
$\|J_{1}u\|_{h^{1}_{\text{max},a'}(\gamma)} \leq C'(\|u\|_{h^{1}_{\text{quad},a}(\gamma)}+\|u\|_{L^{1}(\gamma)})$.
The remainder term
$$J_{\infty}u(x):=\int \limits _{m(x)} ^{\infty} (t^{2}L)^{N+1}e^{\frac{(1+a^{2})t^{2}}{\alpha}L}u(x) \frac{dt}{t}$$
is a priori problematic, as the boundedness of the square function norm $\|S_{a}u\|_{1}$ does not give information about it.
It turns out, however, that properties of the kernel of the Ornstein-Uhlenbeck semigroup give the estimate
$\|J_{\infty}u\|_{h^{1}_{\text{max},a'}(\gamma)} \leq C''\|u\|_{L^{1}(\gamma)}$. This phenomenon is typical of local Hardy spaces, as can be seen, for instance, in \cite{art} and \cite{jyz}.\\
The paper is organised as follows. In Section 2, we recall the necessary definitions and known results, and set up the proof,
decomposing $J_{1}u$ into a main term and two remainder terms similar to $J_{\infty}u$.
In Section 3, we prove the relevant kernel estimates, and deduce appropriate off-diagonal bounds.
In Section 4, we show that the main term can be decomposed as a sum of molecules,
and estimate the $h^{1}_{\text{max}}$ norm of molecules.
In Section 5, we estimate $J_{\infty}u$ and the remainder terms, and thus conclude the proof.
In Section 6, we use the same techniques to prove that the Riesz transforms associated with $L$ are bounded on $h^{1}(\gamma)$.
\subsection*{Acknowledgement}
This work completes the first part of a larger project, started in \cite{mnp,mnp2} in collaboration with Jan Maas and Jan van Neerven. It owes a great deal to our discussions.
Thanks to Adam Sikora, for correcting an incorrect comment in an earlier version of the introduction.
Many thanks also go to Jonas Teuwen, who read a draft of this paper as part of his master thesis, under the supervision of Jan van Neerven at the TU Delft. Numerous misprints and omissions have been fixed thanks to his careful reading.
\section{Preliminaries}
We start by recalling some basic properties of the Ornstein-Uhlenbeck operator $L$ (details can be found in the survey paper \cite{survey}).
On $L^{2}(\gamma)$, $L$ generates a semigroup for which the Hermite polynomials $(H_{\alpha})_{\alpha \in {\mathbb Z}_{+}^{n}}$ form an orthonormal basis of eigenfunctions. Using this chaos decomposition, we have:
$$
e^{tL}(\sum \limits _{\beta \in {\mathbb Z}_{+}^{n}} c_{\beta}H_{\beta}) = \sum \limits _{\beta \in {\mathbb Z}_{+}^{n}} e^{-t|\beta|}c_{\beta}H_{\beta},
$$
for $c_{\beta} \in {\mathbb C}$ and $|\beta|:= \sum \limits _{j=1} ^{n} \beta_{j}$.
As a direct consequence, we have the following Calder\'on reproducing formula.
\begin{lemma}
\label{lem:repro}
For all $N \in {\mathbb N}$ and $a,\alpha>0$, there exists $C>0$ such that for all $u \in L^{2}(\gamma)$
$$
u = C \int \limits _{0} ^{\infty} (t^{2}L)^{N+1}e^{\frac{(1+a^{2})t^{2}}{\alpha}L}u \frac{dt}{t}+\int \limits _{{\mathbb R}^{n}}u d\gamma.
$$
\end{lemma}
On $L^{p}({\mathbb R}^{n},\gamma)$, for $1\leq p<\infty$, $L$ generates the semigroup defined by
$$
e^{tL}f(x) := \int \limits _{{\mathbb R}^{n}} M_{t}(x,y)f(y)dy,
$$
where $f \in L^{p}(\gamma)$, $x \in {\mathbb R}^{n}$, and $M_{t}$ denotes the Mehler kernel
$$
M_{t}(x,y) := \pi^{-\frac{n}{2}}(1-e^{-2t})^{-\frac{n}{2}}\exp(-\frac{|e^{-t}x-y|^{2}}{1-e^{-2t}}).
$$
A well know technique in gaussian harmonic analysis, going back to \cite{m}, consists in splitting kernels such as the Mehler kernel into a local and a global part, the idea being that the local part behaves like a Caldero\'n-Zygmund operator, and the global part has some specific decay properties. The local region is defined as
$$
N_{a} := \{(x,y) \in {\mathbb R}^{2n} \;;\; |x-y| \leq a m(x)\},
$$
where $a>0$ and $m(x):=\min\big\{1, \frac1{|x|}\big\}$.
A typical result obtained by this technique is the weak-type 1-1 of the local part of the Hardy-Littlewood maximal operator and the strong type 1-1 of its global part, proven by Harboure, Torrea, and Vivani in \cite[Theorem 2.7]{htv}.
In this paper, we will use the corresponding result for the non-tangential maximal function.
Before stating this result, we recall \cite[Lemma 2.3]{mnp}, and introduce some notation.
\begin{lemma}
\label{lem:mnp1}
Let $a>0$, and $x,y \in {\mathbb R}^{n}$.
If $|x-y|<am(x)$, then $m(x)\le (1+a)m(y)$ and $m(y)\le (2+2a)m(x)$.
\end{lemma}
Given $A,a>0$, we define
$$\Gamma^{(A,a)}_{x}(\gamma) := \Big\{(y,t) \in {\mathbb R}^{n}\times(0,\infty)\colon
|y-x|<At, \;\text{and}\; t\leq am(x)\Big\},$$
and call $\Gamma^{(A,a)}_{x}(\gamma)$ the admissible cone with aperture $A$ and admissibility parameter $a$ based at the point $x$.
To simplify notation we write $\Gamma_{x}(\gamma):=\Gamma^{(1,1)}_{x}(\gamma)$ and $\Gamma^{a}_{x}(\gamma):=\Gamma^{(1,a)}_{x}(\gamma)$.
Non-tangential maximal functions are pointwise dominated by the Hardy-Littlewood maximal function. This is the following lemma, proven by Pineda and Urbina in \cite[Lemma 1.1]{pu} (for the particular choice $(A,a)=(1,\frac{1}{2})$, but the proof carries over to different apertures and admissibility parameters).
\begin{lemma}
\label{lem:pu}
Let $A,a>0$. There exists $C>0$ such that for all $x \in {\mathbb R}^{n}$ and all $f \in L^{2}(\gamma)$
$$
\underset{(y,t) \in \Gamma_{x} ^{(A,a)}(\gamma)}{\sup} |e^{t^{2}L}f(y)| \leq C \underset{r>0}{\sup} \frac{1}{\gamma(B(x,r))} \int \limits _{B(x,r)} |f(z)|d\gamma(z).
$$
\end{lemma}
Using \cite[Theorem 2.7]{htv}, we get the $L^{2}$ boundedness of non-tangential maximal functions, and the $L^{1}$ boundedness of their global parts.
\begin{proposition}
\label{prop:glob}
Let $A,a>$ and set $\tau := \frac{(1+aA)(1+2aA)}{2}$. Then, for $f \in C_{c}^{\infty}({\mathbb R}^{n})$,
\begin{enumerate}[(i)]
\item
$$
\|T^{*}_{glob,a,A}f:x \mapsto \underset{(y,t) \in \Gamma_{x} ^{(A,a)}(\gamma)}{\sup} \int \limits _{{\mathbb R}^{n}} M_{t^{2}}(y,z)1_{N_{\tau}^{c}}(y,z)|f(z)|dz\|_{1} \lesssim \|f\|_{1}.
$$
\item
$$
\|x \mapsto \underset{(y,t) \in \Gamma_{x} ^{(A,a)}(\gamma)}{\sup} \int \limits _{{\mathbb R}^{n}} M_{t^{2}}(y,z)|f(z)|dz\|_{2} \lesssim \|f\|_{2}.
$$
\end{enumerate}
\end{proposition}
Here, $\|x \mapsto \underset{(y,t) \in \Gamma_{x} ^{(A,a)}(\gamma)}{\sup} \int \limits _{{\mathbb R}^{n}} M_{t^{2}}(y,z)1_{N_{\tau}^{c}}(y,z)|f(z)|dz\|_{1} \lesssim \|f\|_{1}$ means
$$\|x \mapsto \underset{(y,t) \in \Gamma_{x} ^{(A,a)}(\gamma)}{\sup} \int \limits _{{\mathbb R}^{n}} M_{t^{2}}(y,z)1_{N_{\tau}^{c}}(y,z)|f(z)|dz\|_{1} \leq C \|f\|_{1}$$ for some $C>0$ independent of $f$. We will use this notation throughout the paper.
\begin{proof}
For $x\in {\mathbb R}^{n}$, $(y,z) \in N_{\tau} ^{c}$, and $(y,t) \in \Gamma_{x} ^{(A,a)}(\gamma)$, we have that
$$
|x-z| \geq \tau m(y) -aAm(x) \geq (\frac{\tau}{1+aA}-aA)m(x)=\frac{1}{2}m(x).
$$
Therefore
$$
\|x \mapsto \underset{(y,t) \in \Gamma_{x} ^{(A,a)}(\gamma)}{\sup} \int \limits _{{\mathbb R}^{n}} M_{t^{2}}(y,z)1_{N_{\tau}^{c}}(y,z)|f(z)|dz\|_{1} \leq
\|x \mapsto \underset{(y,t) \in \Gamma_{x} ^{(A,a)}(\gamma)}{\sup} \int \limits _{{\mathbb R}^{n}} M_{t^{2}}(y,z)g_{x}(z)dz\|_{1},
$$
where $g_{x}(z):=1_{N_{\frac{1}{2}}^{c}}(x,z)|f(z)|$.
Lemma \ref{lem:pu}, combined with \cite[Theorem 2.7]{htv} thus gives
$$
\|x \mapsto \underset{(y,t) \in \Gamma_{x} ^{(A,a)}(\gamma)}{\sup} \int \limits _{{\mathbb R}^{n}} M_{t^{2}}(y,z)1_{N_{\tau}^{c}}(y,z)|f(z)|dz\|_{1} \lesssim
\int \limits _{{\mathbb R}^{n}} \underset{r>0}{\sup} \frac{1}{\gamma(B(x,r))} \int \limits _{B(x,r)}1_{N_{\frac{1}{2}}^{c}}(x,z) |f(z)|d\gamma(z) \lesssim \|f\|_{1}.
$$
To prove (ii), we apply Lemma \ref{lem:pu} and Lemma \ref{lem:mnp1} to obtain, for $x\in {\mathbb R}^{n}$,
$$
\underset{(y,t) \in \Gamma_{x} ^{(A,a)}(\gamma)}{\sup} \int \limits _{{\mathbb R}^{n}} 1_{N_{\tau}}(y,z)M_{t^{2}}(y,z)|f(z)|dz \lesssim
\underset{r \in (0,\tau' m(x))}{\sup} \frac{1}{\gamma(B(x,r))} \int \limits _{B(x,r)} |f(z)| d\gamma(z),
$$
for $\tau' = aA+\tau(2+2aA)$ and an implicit constant independent of $x$.
The weak type $1-1$ of this local part is proven, for instance, in \cite[Lemma 3.2]{mnp}. Combined with (i), this gives the weak type $1-1$ of
$$x \mapsto \underset{(y,t) \in \Gamma_{x} ^{(A,a)}(\gamma)}{\sup} \int \limits _{{\mathbb R}^{n}} M_{t^{2}}(y,z)|f(z)|dz.$$
Given the (obvious) $L^{\infty}$ boundedness of the Hardy-Littlewood maximal function
(and thus of the non-tangential maximal function by Lemma \ref{lem:pu}), the proof follows by interpolation.
\end{proof}
A geometric version of the local/global dichotomy is given by the key notion of admissible balls, introduced in \cite{mm}.
Defining
$$
\mathcal{B}_{a} := \{B(x,r) \;;\; x \in {\mathbb R}^{n}, \quad 0<r \leq am(x)\},
$$
we say that a ball $B \in \mathcal{B}_{a}$ is admissible at scale $a$.
The gaussian measure acts as a doubling measure on admissible balls, as Mauceri and Meda have pointed out in \cite[Proposition 2.1]{mm}.
We recall here a version of their result.
\begin{lemma}
\label{lem:mm}
There exists $C>0$ such that for all $a,b\geq 1$ and all $B(x,r) \in \mathcal{B}_{a}$ we have
$$
\gamma(B(x,br)) \leq e^{2a^{2}(2b+1)^{2}}\gamma(B(x,r)).
$$
\end{lemma}
This led Jan Maas, Jan van Neerven and the author to introduce gaussian tent spaces, in \cite{mnp}, as follows.
Let $D:= \{(t,x)\in (0,\infty) \times {\mathbb R}^{n} \;;\; t<m(x)\}$. Then $t^{1,2}(\gamma)$ is the completion of $C_{c}(D)$ with respect to the norm
$$
\|F\|_{t^{1,2}(\gamma)}:= \int \limits _{{\mathbb R}^{n}} \Big(\int_{\Gamma_x(\gamma)} \frac1{\gamma(B(y,t))}|F(t,y)|^{2}\,d\gamma(y)\,\frac{dt}{t}\Big)^\frac12 d\gamma(x).
$$
Compared to \cite{mnp}, we are using here the notation $t^{1,2}(\gamma)$ rather than $T^{1,2}(\gamma)$ to emphasise the local nature of this space.
Theorem 3.4 in \cite{mnp} gives an atomic decomposition of $t^{1,2}(\gamma)$. Given $a>0$, a function $F:D\to{\mathbb C}$ is called a $t^{1,2}(\gamma)$ a-atom if
there exists a ball $B \in \mathcal{B}_{a}$ such that $supp(F) \subset \{(t,y)\in (0,\infty) \times {\mathbb R}^{n} \;;\; t\leq \min(d(y,B^{c}),m(y))\}$ and
$$
\int \limits _{{\mathbb R}^{n}} \int \limits _{0} ^{\infty} |F(t,y)|^{2} \frac{dydt}{t} \leq \gamma(B)^{-1}.
$$
\begin{theorem}
\label{thm:atomic}
For all $f \in t^{1,2}(\gamma)$ and $a>1$, there exists a sequence
$(\lambda_{n})_{n\geq 1}
\in \ell_{1}$ and a sequence of $t^{1,2}(\gamma)$ $a$-atoms $(F_{n})_{n\geq 1}$
such that
\begin{enumerate}
\item[\rm(i)] $f = \sum _{n\geq 1}\lambda_{n}F_{n}$;
\item[\rm(ii)]
$\sum _{n\geq 1}|\lambda_{n}| \lesssim
\|f\|_{t^{1,2}(\gamma)}$.
\end{enumerate}
\end{theorem}
To simplify notation we will simply call atoms the $t^{1,2}(\gamma)$ 2-atoms.
Combining the atomic decomposition of $t^{1,2}(\gamma)$ and Lemma \ref{lem:repro} we get the following decomposition, which is the basis of the proof of Theorem \ref{thm:main}.
\begin{corollary}
\label{cor:dec}
For all $N \in {\mathbb N}$, $a>1$, $b>0$, and $\alpha>a^{2}$, there exists $C>0$ and $n$ sequences of atoms $(F_{m,j})_{m\in {\mathbb N}}$ and complex numbers
$(\lambda_{m,j})_{m\in{\mathbb N}}$ for $j=1,...n$, such that for all
$u \in C_{c}^{\infty}({\mathbb R}^{n})$ and $x\in {\mathbb R}^{n}$:
\begin{equation*}
\begin{split}
u(x) = \int \limits _{{\mathbb R}^{n}} u d\gamma
&- C \sum \limits _{j=1} ^{n} \sum \limits _{m=1} ^{\infty} \lambda_{m,j} \int \limits _{0} ^{2} (t^{2}L)^{N}e^{\frac{t^{2}}{\alpha}L}t\partial_{x_{j}}^{*}F_{m,j}(t,x)\frac{dt}{t}\\
&+C \sum \limits _{j=1} ^{n}\sum \limits _{m=1} ^{\infty} \lambda_{m,j} \int \limits _{0}^{2} 1_{[\frac{m(x)}{b},2]}(t) (t^{2}L)^{N}e^{\frac{t^{2}}{\alpha}L}t\partial_{x_{j}}^{*}F_{m,j}(t,x)\frac{dt}{t}
\\&-C \sum \limits _{j=1} ^{n} \int \limits _{0} ^{\frac{m(x)}{b}} (t^{2}L)^{N}e^{\frac{t^{2}}{\alpha}L}t\partial_{x_{j}}^{*}(1_{D^{c}}(t,.)t\partial_{x_{j}}e^{\frac{a^{2}t^{2}}{\alpha}L})u(x)\frac{dt}{t}- C \int \limits _{\frac{m(x)}{b}} ^{\infty} (t^{2}L)^{N+1}e^{\frac{(1+a^{2})t^{2}}{\alpha}L}u(x) \frac{dt}{t},
\end{split}
\end{equation*}
and $ \sum \limits _{j=1} ^{n} \sum \limits _{m=1} ^{\infty} |\lambda_{m,j}| \lesssim \|u\|_{h^{1}_{\text{quad},a}}$.
\end{corollary}
Here $\partial_{x_{j}}^{*}$ denotes the adjoint of $\partial_{x_{j}}$ in $L^{2}(\gamma)$.
\begin{proof}
We first remark that
$$
(t^{2}L)^{N+1}e^{\frac{(1+a^{2})t^{2}}{\alpha}L} = -\frac{1}{2} \sum \limits _{j=1} ^{n} (t^{2}L)^{N}e^{\frac{t^{2}}{\alpha}L}
t\partial_{x_{j}}^{*}((1_{D}(t,.)+1_{D^{c}}(t,.))t\partial_{x_{j}}e^{\frac{a^{2}t^{2}}{\alpha}L}u).
$$
It remains to check that the terms $1_{D}(t,.)t\partial_{x_{j}}e^{\frac{a^{2}t^{2}}{\alpha}L}u$, for $j\in \{1,...,n\}$, belong to $t^{1,2}(\gamma)$.
Using Lemma \ref{lem:mm} we have
\begin{equation*}
\begin{split}
\|(t,x) \mapsto 1_{D}(t,x)t\partial_{x_{j}}e^{\frac{a^{2}t^{2}}{\alpha}L}u(x)\|_{t^{1,2}(\gamma)}
&\lesssim \int \limits_{{\mathbb R}^{n}} ( \int \limits _{0} ^{\frac{m(x)}{\sqrt{\alpha}}} \int \limits _{B(x,\sqrt{\alpha}s)}
\frac{1_{D}(\sqrt{\alpha}s,y)}{\gamma(B(y,\sqrt{\alpha}s))} |s\nabla e^{a^{2}s^{2}L}u(y)|^{2} d\gamma(y)\frac{ds}{s})^{\frac{1}{2}}d\gamma(x)\\
&\lesssim \int \limits_{{\mathbb R}^{n}} ( \int \limits _{0} ^{m(x)} \int \limits _{B(x,\sqrt{\alpha}s)}
\frac{1_{D}(as,y)}{\gamma(B(y,s))} |s\nabla e^{a^{2}s^{2}L}u(y)|^{2} d\gamma(y)\frac{ds}{s})^{\frac{1}{2}}d\gamma(x).
\end{split}
\end{equation*}
By \cite[Theorem 3.8]{mnp}, we thus have
\begin{equation*}
\begin{split}
\|(t,x) \mapsto &1_{D}(t,x)t\partial_{x_{j}}e^{\frac{a^{2}t^{2}}{\alpha}L}u(x)\|_{t^{1,2}(\gamma)}
\lesssim \int \limits_{{\mathbb R}^{n}} ( \int \limits _{0} ^{m(x)} \int \limits _{B(x,as)}
\frac{1_{D}(as,y)}{\gamma(B(y,s))} |s\nabla e^{a^{2}s^{2}L}u(y)|^{2} d\gamma(y)\frac{ds}{s})^{\frac{1}{2}}d\gamma(x)\\
&\lesssim \int \limits_{{\mathbb R}^{n}} ( \int \limits _{0} ^{am(x)} \int \limits _{B(x,t)}
\frac{1_{D}(t,y)}{\gamma(B(y,t))} |t\nabla e^{t^{2}L}u(y)|^{2} d\gamma(y)\frac{dt}{t})^{\frac{1}{2}}d\gamma(x)
= \|u\|_{h^{1}_{\text{quad},a}}.
\end{split}
\end{equation*}
\end{proof}
Theorem \ref{thm:main} is then proven by combining the results in the next sections as follows.\\
{\em Proof of Theorem \ref{thm:main}:}\\
For $a>0$, \cite[Theorem 1.1]{mnp2} gives that there exists $a'>0$ such that $h^{1}_{\text{max},a'}(\gamma) \subset h^{1}_{\text{quad},a}(\gamma)$.
Let us fix this $a'$ and pick $\alpha>\max(2^{38},32e^{4},4\sqrt{a}e^{2a^{2}})$, $b \geq \max(2e,\sqrt{\frac{32e^{4}}{(\alpha-32e^{4})(1-e^{-2\frac{a^{2}}{\alpha}})}})$,
and $N> \frac{n}{4}$.
Let $u \in C_{c}^{\infty}({\mathbb R}^{n})$ and apply Corollary \ref{cor:dec}.
We have that
\begin{equation*}
\begin{split}
\|u\|_{h^{1}_{\text{max},a'}(\gamma)}
\lesssim
\|T^{*}_{a'}(\int \limits _{{\mathbb R}^{n}} u d\gamma)\|_{1}
&+ C \sum \limits _{j=1} ^{n} \sum \limits _{m=1} ^{\infty} |\lambda_{m,j}|
\| \int \limits _{0} ^{2} (t^{2}L)^{N}e^{\frac{t^{2}}{\alpha}L}t\partial_{x_{j}}^{*}F_{m,j}(t,.)\frac{dt}{t}\|_{h^{1}_{\text{max},a'}(\gamma)}\\
&+C \sum \limits _{j=1} ^{n}\sum \limits _{m=1} ^{\infty} |\lambda_{m,j}|
\| \int \limits _{0}^{2} 1_{[\frac{m(.)}{b},2]}(t) (t^{2}L)^{N}e^{\frac{t^{2}}{\alpha}L}t\partial_{x_{j}}^{*}F_{n,j}(t,.)\frac{dt}{t}\|_{h^{1}_{\text{max},a'}(\gamma)}
\\&+C \sum \limits _{j=1} ^{n}
\|\int \limits _{0} ^{\frac{m(.)}{b}} (t^{2}L)^{N}e^{\frac{t^{2}}{\alpha}L}t\partial_{x_{j}}^{*}(1_{D^{c}}(t,.)t\partial_{x_{j}}e^{\frac{a^{2}t^{2}}{\alpha}L})u\frac{dt}{t}\|_{h^{1}_{\text{max},a'}(\gamma)}
\\&+ C \|\int \limits _{\frac{m(.)}{b}} ^{\infty} (t^{2}L)^{N+1}e^{\frac{(1+a^{2})t^{2}}{\alpha}L}u \frac{dt}{t}\|_{h^{1}_{\text{max},a'}(\gamma)} + \|u\|_{L^{1}(\gamma)}.
\end{split}
\end{equation*}
Since $e^{sL}1=1$ for all $s \geq0$, we have
$$
\|T^{*}_{a'}(\int ud\gamma)\|_{1} \leq \|u\|_{1} \leq \|u\|_{h^{1}_{\text{quad},a}(\gamma)}.
$$
Proposition \ref{prop:jinf} gives that
$$
\|\int \limits _{\frac{m(.)}{b}} ^{\infty} (t^{2}L)^{N+1}e^{\frac{(1+a^{2})t^{2}}{\alpha}L}u \frac{dt}{t}\|_{h^{1}_{\text{max},a'}(\gamma)} \lesssim \|u\|_{1} \leq \|u\|_{h^{1}_{\text{quad},a}(\gamma)}.
$$
For $j \in \{1,...,n\}$, Proposition \ref{prop:Dcomp} then gives
$$
\|\int \limits _{0} ^{\frac{m(.)}{b}} (t^{2}L)^{N}e^{\frac{t^{2}}{\alpha}L}t\partial_{x_{j}}^{*}(1_{D^{c}}(t,.)t\partial_{x_{j}}e^{\frac{a^{2}t^{2}}{\alpha}L})u\frac{dt}{t}\|_{h^{1}_{\text{max},a'}(\gamma)}\lesssim \|u\|_{1} \leq \|u\|_{h^{1}_{\text{quad},a}(\gamma)}.
$$
Proposition \ref{prop:r1} gives that
$$
\| \int \limits _{0}^{2} 1_{[\frac{m(.)}{b},2]}(t) (t^{2}L)^{N}e^{\frac{t^{2}}{\alpha}L}t\partial_{x_{j}}^{*}F_{n,j}(t,.)\frac{dt}{t}\|_{h^{1}_{\text{max},a'}}
\lesssim 1,
$$
while Proposition \ref{prop:mol} combined with Theorem \ref{thm:mol} gives
$$
\| \int \limits _{0} ^{2} (t^{2}L)^{N}e^{\frac{t^{2}}{\alpha}L}t\partial_{x_{j}}^{*}F_{n,j}(t,.)\frac{dt}{t}\|_{h^{1}_{\text{max},a'}(\gamma)}\lesssim 1.$$
Therefore
$$
\|u\|_{h^{1}_{\text{max},a'}(\gamma)} \lesssim \|u\|_{h^{1}_{\text{quad},a}(\gamma)} + \sum \limits _{j=1} ^{n} \sum \limits _{m=1} ^{\infty} |\lambda_{m,j}|
\lesssim \|u\|_{h^{1}_{\text{quad},a}(\gamma)}.
$$
\section{Kernel estimates}
In this section, we establish some properties of the Mehler kernel, and use them to prove the following off-diagonal decay result.
Given $a>0$, $B=B(c_{B},r_{B}) \in \mathcal{B}_{a}$ and $k \in {\mathbb Z}_{+}$ we consider the following sets.
$$
C_{k}(B):= \begin{cases} B(c_{B},2r_{B}) \; \text{if} \; k=0, \\ B(c_{B},2^{k+1}r_{B}) \backslash B(c_{B},2^{k}r_{B}) \; \text{otherwise}.\end{cases}
$$
\begin{lemma}[Off-diagonal estimates]
Let $N \in {\mathbb Z}_{+}$, $a>0$, $j\in \{1,...,n\}$, $B \in \mathcal{B}_{a}$, $\alpha\geq 4e^{2a^{2}}$, and $k\in {\mathbb N}$. Then for all $u \in L^{2}(\gamma)$
$$
\|1_{C_{k}(B)}1_{(0,r_{B})}(t)(t^{2N+1}L^{N}e^{\frac{t^{2}}{\alpha}L}\partial_{x_{j}}^{*})1_{B}u\|_{2} \lesssim
\exp(-\frac{\alpha}{2^{6}e^{2a^{2}}}4^{k}(\frac{r_{B}}{t})^{2})\|u\|_{2},
$$
with implied constant depending only on $\alpha$ and $N$.
\end{lemma}
The above lemma plays a key role in the next sections, and could be deduced from more general methods giving $L^2$ off-diagonal bounds
(see \cite{cs} or \cite{mcm}). We prove it through direct kernel estimates which are used in various parts of the paper. In the next sections, it will become clear that one needs off-diagonal decay of the form $\exp(-c4^{k})$ with $c$ large enough to compensate for the growth in Lemma \ref{lem:mm}. This is the reason why we use $e^{\frac{(1+a^{2})t^{2}}{\alpha}L}$ in the reproducing formula and pick $\alpha$ large enough.\\
Given $t,\alpha>0$, $j \in \{1,...,n\}$, and $N \in {\mathbb Z}_{+}$, we denote by $K_{t^{2},N,\alpha}$ and $\tilde{K}_{t^{2},N,\alpha,j}$ the relevant kernels defined, given $u \in L^{2}(\gamma)$, by
\begin{equation*}
\begin{split}
\int \limits _{{\mathbb R}^{n}} K_{t^{2},N,\alpha}(x,y)u(y)dy &= (t^{2}L)^{N}e^{\frac{t^{2}}{\alpha}L}u(x),\\
\int \limits _{{\mathbb R}^{n}} \tilde{K}_{t^{2},N,\alpha,j}(x,y)u(y)dy &= (t^{2}L)^{N}e^{\frac{t^{2}}{\alpha}L}t\partial_{x_{j}}^{*}u(x).
\end{split}
\end{equation*}
Note that $K_{t^{2},N,\alpha}(x,y) = t^{2N}\partial_{s}^{N}M_{s}(x,y)_{|s=\frac{t^{2}}{\alpha}}$, and that, by duality
$$\tilde{K}_{t^{2},N,\alpha,j}(x,y) = t^{2N+1}\partial_{y_{j}}\partial_{s}^{N}M_{s}(y,x)_{|s=\frac{t^{2}}{\alpha}}\exp(|x|^{2}-|y|^{2}).$$
To prove Lemma \ref{lem:od}, we need preparatory lemmas of independent interest.
\begin{lemma}
\label{lem:K}
Let $N \in {\mathbb Z}_{+}$. There exists $C_{N}\in {\mathbb N}$ and a polynomial of $2n+1$ variables $P_{N}$ of degree $C_{N}$ such that
for all $x,y \in {\mathbb R}^{n}$ and $s>0$:
$$
\partial_{s}^{N}M_{s}(x,y) = (1-e^{-2s})^{-N}P_{N}(e^{-s},(\frac{e^{-s}x_{j}-y_{j}}{\sqrt{1-e^{-2s}}})_{j=1,...,n} , (\sqrt{1-e^{-2s}}x_{j})_{j=1,...,n})M_{s}(x,y).
$$
\end{lemma}
\begin{proof}
Let $j\in\{1,...,n\}$, $s>0$, $x,y \in {\mathbb R}^{n}$. We have the following.
\begin{equation*}
\begin{split}
\partial_{s}(\frac{e^{-s}x_{j}-y_{j}}{\sqrt{1-e^{-2s}}}) &= -(1-e^{-2s})^{-1}(e^{-s}x_{j}\sqrt{1-e^{-2s}}+e^{-2s}\frac{e^{-s}x_{j}-y_{j}}{\sqrt{1-e^{-2s}}}).\\
\partial_{s}(\sqrt{1-e^{-2s}}x_{j}) &= (1-e^{-2s})^{-1}(e^{-2s}\sqrt{1-e^{-2s}}x_{j}).\\
\partial_{s}M_{s}(x,y) &= -(1-e^{-2s})^{-1}ne^{-2s}M_{s}(x,y)-M_{s}(x,y)\partial_{s}(\frac{|e^{-s}x-y|^{2}}{1-e^{-2s}}).\\
\partial_{s}(\frac{(e^{-s}x_{j}-y_{j})^{2}}{1-e^{-2s}}) &= -(1-e^{-2s})^{-1}((2e^{-s}\sqrt{1-e^{-2s}}x_{j})(\frac{e^{-s}x_{j}-y_{j}}{\sqrt{1-e^{-2s}}})
+(\frac{e^{-s}x_{j}-y_{j}}{\sqrt{1-e^{-2s}}})^{2}2e^{-2s}).
\end{split}
\end{equation*}
The proof thus follows by induction.
\end{proof}
Computing partial derivatives in $x_{j}$ one obtains in the same way:
\begin{corollary}
\label{cor:Ktilde}
Let $N \in {\mathbb Z}_{+}$ and $j\in\{1,...,n\}$. There exists $C_{N}\in {\mathbb N}$ and a polynomial of $2n+1$ variables $Q_{N}$ of degree $C_{N}$ such that
for all $x,y \in {\mathbb R}^{n}$ and $s>0$:
$$
\partial _{x_{j}}\partial_{s}^{N}M_{s}(x,y) = (1-e^{-2s})^{-(N+\frac{1}{2})}Q_{N}(e^{-s},(\frac{e^{-s}x_{j}-y_{j}}{\sqrt{1-e^{-2s}}})_{j=1,...,n} , (\sqrt{1-e^{-2s}}x_{j})_{j=1,...,n})M_{s}(x,y).
$$
\end{corollary}
\begin{lemma} \label{lem:slow2}
For $a,C>0$, $\alpha >1,$ $t \in (0,a],$ and $x,y \in {\mathbb R}^n$ we have
\begin{enumerate}[(i)]
\item
$\exp(-C\frac{|e^{-\frac{t^{2}}{\alpha}}x-y|^{2}}{1-e^{-2\frac{t^{2}}{\alpha}}})
\leq \exp(-C\frac{\alpha}{2e^{2a^{2}}}\frac{|e^{-t^{2}}x-y|^{2}}{1-e^{-2t^{2}}})\exp(C\frac{t^{4}|x|^{2}}{1-e^{-2\frac{t^{2}}{\alpha}}}).$
\item
$\exp(-C\frac{|e^{-\frac{t^{2}}{\alpha}}x-y|^{2}}{1-e^{-2\frac{t^{2}}{\alpha}}})
\leq \exp(-C\frac{\alpha}{2e^{2a^{2}}}\frac{|e^{-t^{2}}x-y|^{2}}{1-e^{-2t^{2}}})\exp(C\frac{t^{4}|y|^{2}}{1-e^{-2\frac{t^{2}}{\alpha}}}).$
\end{enumerate}
\end{lemma}
\begin{proof}
Let $t\in (0,a]$ and $\alpha>1 $. Applying the mean value theorem to $f(\xi) = \xi^\alpha$, we have
\begin{align*}
\frac{1-e^{-2t^2}}{1-e^{-\frac{2t^2}{\alpha}}}
= \alpha \hat\xi^{\alpha-1}
\end{align*}
for some $\hat \xi \in [e^{-2t^2/\alpha} ,1].$ Therefore,
\begin{align*}
\alpha e^{-2a^{2}}
\leq \alpha e^{-\frac{2t^2(\alpha-1)}{\alpha}}
\leq \frac{1-e^{-2t^2}}{1-e^{-\frac{2t^2}{\alpha}}}
\leq \alpha.
\end{align*}
To prove (i), we notice that
$$
|e^{-\frac{t^{2}}{\alpha}}x-y| \geq |e^{-t^{2}}x-y| - |e^{-t^{2}}-e^{-\frac{t^{2}}{\alpha}}||x|
\geq |e^{-t^{2}}x-y| -t^{2}|x|,
$$
and thus, by Cauchy-Schwarz:
$$
|e^{-\frac{t^{2}}{\alpha}}x-y|^{2} \geq \frac{|e^{-t^{2}}x-y|^{2}}{2} - t^{4}|x|^{2}.
$$
This gives
\begin{equation*}
\begin{split}
\exp(-C\frac{|e^{-\frac{t^{2}}{\alpha}}x-y|^{2}}{1-e^{-2\frac{t^{2}}{\alpha}}})
&\leq \exp(-\frac{C}{2}(\frac{1-e^{-2t^{2}}}{1-e^{-2\frac{t^{2}}{\alpha}}})\frac{|e^{-t^{2}}x-y|^{2}}{1-e^{-2t^{2}}})\exp(C\frac{t^{4}|x|^{2}}{1-e^{-2\frac{t^{2}}{\alpha}}})
\\&\leq \exp(-C\frac{\alpha}{2e^{2a^{2}}}\frac{|e^{-t^{2}}x-y|^{2}}{1-e^{-2t^{2}}})\exp(C\frac{t^{4}|x|^{2}}{1-e^{-2\frac{t^{2}}{\alpha}}}).
\end{split}
\end{equation*}
The estimate (ii) is proven in the same way, noticing that
$$
|e^{-\frac{t^{2}}{\alpha}}x-y| \geq e^{(\frac{\alpha-1}{\alpha})t^{2}}|e^{-t^{2}}x-e^{-(\frac{\alpha-1}{\alpha})t^{2}}y|
\geq |e^{-t^{2}}x-y|- |1-e^{-(\frac{\alpha-1}{\alpha})t^{2}}||y|
\geq |e^{-t^{2}}x-y| -t^{2}|y|.
$$
\end{proof}
\begin{lemma}
\label{lem:est}
Let $N \in {\mathbb Z}_{+}$, $j \in \{1,...,n\}$, $a>0$ and $\alpha \geq 4e^{2a^{2}}$. Let $x,y \in {\mathbb R}^{n}$ and $t\in (0,a]$.
\begin{enumerate}[(i)]
\item If $t \lesssim m(y)$ then $M_{\frac{t^{2}}{\alpha}}(x,y) \lesssim \exp(-\frac{\alpha}{2e^{2a^{2}}}\frac{|e^{-t^{2}}x-y|^{2}}{1-e^{-2t^{2}}})M_{t^{2}}(x,y)$.
\item If $t \lesssim m(x)$ then $|K_{t^{2},N,\alpha}(x,y)|\lesssim
\exp(-\frac{\alpha}{4e^{2a^{2}}}\frac{|e^{-t^{2}}x-y|^{2}}{1-e^{-2t^{2}}})M_{t^{2}}(x,y)$.
\item If $t \lesssim m(y)$ then $|\tilde{K}_{t^{2},N,\alpha,j}(x,y)|\lesssim \exp(-\frac{\alpha}{4e^{2a^{2}}}\frac{|e^{-t^{2}}y-x|^{2}}{1-e^{-2t^{2}}})M_{t^{2}}(x,y)$.
\end{enumerate}
\end{lemma}
\begin{proof}
(i) follows from Lemma \ref{lem:slow2}.\\
(ii) follows from Lemma \ref{lem:K} and Lemma \ref{lem:slow2} using that $\underset{w>0}{\sup}w^{k}e^{-Cw^{2}}<\infty$ for all $k\geq0$ and $C>0$.\\
(iii) follows from Corollary \ref{cor:Ktilde} and Lemma \ref{lem:slow2} in the same way, using that $$M_{t^{2}}(y,x)\exp(|x|^{2}-|y|^{2}) = M_{t^{2}}(x,y).$$
\end{proof}
We can now prove our main lemma.
\begin{lemma}[Off-diagonal estimates]
\label{lem:od}
Let $N \in {\mathbb Z}_{+}$, $a>0$, $j \in \{1,...,n\}$, $B \in \mathcal{B}_{a}$, $\alpha>4e^{2a^{2}}$, and $k \in {\mathbb N}$. Then for all $u \in L^{2}(\gamma)$
$$
\|1_{C_{k}(B)}1_{(0,r_{B})}(t)(t^{2N+1}L^{N}e^{\frac{t^{2}}{\alpha}L}\partial_{x_{j}}^{*})1_{B}u\|_{2} \lesssim
\exp(-\frac{\alpha}{2^{6}e^{2a^{2}}}4^{k}(\frac{r_{B}}{t})^{2})\|u\|_{2},
$$
with implied constant depending only on $\alpha$, $a$ and $N$.
\end{lemma}
\begin{proof}
For $t\leq r_{B} \leq am(c_{B})$ and $y \in B$, we have $t \leq a(1+a)m(y)$ by Lemma \ref{lem:mnp1}.
Given $x\in {\mathbb R}^{n}$, we also have, using Cauchy-Schwarz, $|y-x|^{2} \leq 2(|e^{-t^{2}}y-x|^{2}+(1-e^{-t^{2}})^{2}|y|^{2})$, and thus
\begin{equation*}
\begin{split}
\exp(-\frac{\alpha}{2^{3}e^{2a^{2}}}\frac{|e^{-t^{2}}y-x|^{2}}{t^{2}})&\leq
\exp(-\frac{\alpha}{2^{4}e^{2a^{2}}}\frac{|y-x|^{2}}{t^{2}})\exp(\frac{\alpha}{2^{3}e^{2a^{2}}}(t|y|)^{2}) \\&\lesssim \exp(-\frac{\alpha}{2^{4}e^{2a^{2}}}\frac{|y-x|^{2}}{t^{2}}).
\end{split}
\end{equation*}
Therefore, using Lemma \ref{lem:est}, we have the following estimates.
\begin{equation*}
\begin{split}
\int \limits _{C_{k}(B)} &\left( \int \limits _{B} |\tilde{K}_{t^{2},N,\alpha,j}(x,y)|1_{(0,r_{B})}(t)|u(y)|dy \right)^{2} d\gamma(x)\\
& \lesssim \int \limits _{C_{k}(B)} \left( \int \limits _{B} \exp(-\frac{\alpha}{2^{3}e^{2a^{2}}}\frac{|e^{-t^{2}}y-x|^{2}}{t^{2}})M_{t^{2}}(x,y)1_{(0,r_{B})}(t)|u(y)|dy \right)^{2} d\gamma(x) \\
& \lesssim \exp(-\frac{\alpha}{2^{6}e^{2a^{2}}}4^{k}(\frac{r_{B}}{t})^{2})\|e^{t^{2}L}|u|\|_{2}
\lesssim \exp(-\frac{\alpha}{2^{6}e^{2a^{2}}}4^{k}(\frac{r_{B}}{t})^{2})\|u\|_{2} ^{2}.
\end{split}
\end{equation*}
\end{proof}
We conclude this section with a property of the sets $C_{k}(B)$ in the local region
$N_{\tau}(B):= \{x \in {\mathbb R}^{n} \;;\; |x-c_{B}| \leq \tau m(c_{B})\}$, which will be helpful when off-diagonal estimates fail.
\begin{lemma}\label{lem:region}
Let $a,\tau>0$ and $B=B(c_{B},r_{B}) \in \mathcal{B}_{a}$. There exists $C>0$ such that for all $k \in {\mathbb Z}_{+}$
$$
\gamma(C_{k}(B)\cap N_{\tau}(B)) \leq C 2^{kn}\gamma(B).
$$
\end{lemma}
\begin{proof}
Let $k \in {\mathbb Z}_{+}$ and $x\in C_{k}(B) \cap N_{\tau}(B)$.
We have $|x-c_{B}| \leq \tau m(c_{B}) \leq \tau(1+\tau)m(x)$, by Lemma \ref{lem:mnp1}.
Therefore
\begin{equation*}
\begin{split}
|x|^{2} \geq |c_{B}|^{2}-2\tau m(c_{B})|c_{B}| \\
|c_{B}|^{2} \geq |x|^{2}-2\tau(1+\tau) m(x)|x|,
\end{split}
\end{equation*}
and thus $e^{-|x|^{2}}\sim e^{-|c_{B}|^{2}}$ for all $x\in C_{k}(B) \cap N_{\tau}(B)$, with implicit constants independent of $k,B$ and $x$.
In particular, for $k=0$, we have
$$
\gamma(B) \sim e^{-|c_{B}|^{2}}\int \limits _{B}dx \sim r_{B}^{n}e^{-|c_{B}|^{2}}.
$$
For $k \in {\mathbb Z}_{+}$, this gives
$$
\gamma(C_{k}(B)\cap N_{\tau}(B)) \lesssim \int \limits _{2^{k+1}B}e^{-|c_{B}|^{2}}dx
\lesssim (2^{k}r_{B})^{n}e^{-|c_{B}|^{2}} \lesssim 2^{kn} \gamma(B).
$$
\end{proof}
\section{Molecules}
In this section, we show that, given a $t^{1,2}(\gamma)$ atom $F$ associated with a ball $B=B(c_{B},r_{B})\in \mathcal{B}_{2}$, the function
$$
\int \limits _{0} ^{r_{B}} (t^{2}L)^{N}e^{\frac{t^{2}}{\alpha}L}t\partial_{x_{j}}^{*}F(t,.)\frac{dt}{t}
$$
is a $(2,N,2^{-23}\alpha)$-molecule in the following sense.
\begin{definition}
Let $N \in {\mathbb N}$, $a>0$, and $C>0$. A function $f \in L^{2}(\gamma)$ is called a $(a,N,C)$-molecule if there exist $B=B(c_{B},r_{B})\in \mathcal{B}_{a}$
and $\tilde{f}$ in $L^{2}(\gamma)$ such that the following holds:
\begin{enumerate}[(i)]
\item
$\|1_{C_{k}(B)}f\|_{2} \leq e^{-C4^{k}}\gamma(B)^{-\frac{1}{2}} \quad \forall k \in {\mathbb Z}_{+}$,
\item
$f = L^{N}\tilde{f}$,
\item
$\|1_{C_{k}(B)}\tilde{f}\|_{2} \leq r_{B}^{2N}e^{-C4^{k}}\gamma(B)^{-\frac{1}{2}} \quad \forall k \in {\mathbb Z}_{+}$.
\end{enumerate}
\end{definition}
We then show that there exists $M>0$ depending only on $(a,N,C)$, such that $\|f\|_{h^{1}_{\text{max}}}\leq M$ for all $(a,N,C)$-molecules.
\begin{proposition}
\label{prop:mol}
Let $N \in {\mathbb N}$, $j\in \{1,...,n\}$ and $\alpha>0$. Let $B=B(c_{B},r_{B})\in \mathcal{B}_{2}$ and $F$ be a $t^{1,2}(\gamma)$ atom $F$ associated with $B$. The function
$$
\int \limits _{0} ^{r_{B}} (t^{2}L)^{N}e^{\frac{t^{2}}{\alpha}L}t\partial_{x_{j}}^{*}F(t,.)\frac{dt}{t}
$$
is a $(2,N,2^{-23}\alpha)$-molecule.
\end{proposition}
\begin{proof}
Let us treat the case $k=0$ first. Let $g = \sum \limits _{\beta \in {\mathbb Z}^{n}_{+}} c_{\beta}H_{\beta} \in L^{2}({\mathbb R}^{n},\gamma)$ be such that
$ \sum \limits _{\beta \in {\mathbb Z}^{n}_{+}} |c_{\beta}|^{2} \leq 1$. We need to estimate
$$
\int \limits _{0} ^{r_{B}} \int \limits _{{\mathbb R}^{n}} |(t^{2}L)^{N}e^{\frac{t^{2}}{\alpha}L}t\partial_{x_{j}}^{*}F(t,x)g(x)| d\gamma(x)\frac{dt}{t}.
$$
By duality, and the $L^2$ boundedness of the Riesz transforms, we have that
\begin{equation*}
\begin{split}
\int \limits _{0} ^{r_{B}} \int \limits _{{\mathbb R}^{n}} |(t^{2}L)^{N}e^{\frac{t^{2}}{\alpha}L}t\partial_{x_{j}}^{*}F(t,x)g(x)| d\gamma(x)\frac{dt}{t}
& \lesssim (\int \limits _{0} ^{r_{B}} \int \limits _{{\mathbb R}^{n}} |F(t,x)|^{2} d\gamma(x)\frac{dt}{t})^{\frac{1}{2}}
(\int \limits _{0} ^{r_{B}} \sum \limits _{\beta \in {\mathbb Z}^{n}_{+}} |(t^{2}|\beta|)^{N+\frac{1}{2}}e^{-\frac{t^{2}}{\alpha}|\beta|}c_{\beta}|^{2} \frac{dt}{t})^{\frac{1}{2}} \\
&\lesssim \gamma(B)^{-\frac{1}{2}} (\sum \limits _{\beta \in {\mathbb Z}^{n}_{+}} |c_{\beta}|^{2} \int \limits _{0} ^{\infty}
(t^{2}|\beta|)^{2N+1}e^{-\frac{2t^{2}}{\alpha}|\beta|} \frac{dt}{t})^{\frac{1}{2}} \lesssim \gamma(B)^{-\frac{1}{2}}.
\end{split}
\end{equation*}
Moreover $\int \limits _{0} ^{r_{B}} (t^{2}L)^{N}e^{\frac{t^{2}}{\alpha}L}t\partial_{x_{j}}^{*}F(t,.)\frac{dt}{t} = L^{N}\tilde{f}$ for
$\tilde{f}:=\int \limits _{0} ^{r_{B}} t^{2N+1}e^{\frac{t^{2}}{\alpha}L}\partial_{x_{j}}^{*}F(t,.)\frac{dt}{t}$.
The same argument thus gives
$$
\|\tilde{f}\|_{2} \lesssim r_{B}^{2N}\gamma(B)^{-\frac{1}{2}} (\int \limits _{0} ^{\infty}
t^{2}|\beta|e^{-\frac{2t^{2}}{\alpha}|\beta|} \frac{dt}{t})^{\frac{1}{2}} \lesssim r_{B}^{2N}\gamma(B)^{-\frac{1}{2}}.
$$
Now let $k \in {\mathbb Z}_{+}$ be such that $k \neq 0$. By Lemma \ref{lem:od}, we have the following.
\begin{equation*}
\begin{split}
\|1_{C_{k}(B)}\int \limits _{0} ^{r_{B}} (t^{2}L)^{N}e^{\frac{t^{2}}{\alpha}L}t\partial_{x_{j}}^{*}F(t,.)\frac{dt}{t}\|_{2}
&\lesssim \int \limits _{0} ^{r_{B}} \exp(-\frac{\alpha}{2^{6}e^{8}}4^{k}(\frac{r_{B}}{t})^{2})\|F(t,.)\|_{2} \frac{dt}{t}\\
&\lesssim \exp(-\frac{\alpha}{2^{23}}4^{k})
(\int \limits _{0} ^{1} \exp(-\frac{\alpha}{2^{22}}(\frac{1}{t})^{2})\frac{dt}{t})^{\frac{1}{2}} (\int \limits _{0} ^{r_{B}} \|F(t,.)\|_{2} ^{2} \frac{dt}{t})^{\frac{1}{2}}\\
& \lesssim \exp(-\frac{\alpha}{2^{23}}4^{k}) \gamma(B)^{-\frac{1}{2}}.
\end{split}
\end{equation*}
Since
$$
\|1_{C_{k}(B)}\tilde{f}\|_{2} \leq r_{B}^{2N} \int \limits _{0} ^{r_{B}} \|1_{C_{k}(B)}1_{(0,r_{B})}(t)e^{\frac{t^{2}}{\alpha}L}t\partial_{x_{j}}^{*}F(t,.)\|_{2}
\frac{dt}{t},
$$
the proof is concluded as above, using Lemma \ref{lem:od} with $N$ replaced by $0$.
\end{proof}
\begin{theorem}
\label{thm:mol}
Let $a>0$, and $f$ a $(2,N,C)$-molecule with $N>\frac{n}{4}$ and $C>2^{11}$.
Then $f \in h^{1}_{\text{max},a}$ and $\|f\|_{h^{1}_{\text{max},a}} \leq M$ for some $M$ independent of $f$.
\end{theorem}
\begin{proof}
Let $B=B(c_{B},r_{B}) \in \mathcal{B}_{2}$ be the ball associated with $f$.
Pick $\alpha>2^{31}$, and let $C_{a}:=(4+4a)\tau+2a$ where $\tau:=\frac{(1+a)(1+2a)}{2}$ as in Proposition \ref{prop:glob}.
We use the following decomposition:
$$
\|f\|_{h^{1}_{\text{max},a}} \leq I + \sum \limits _{k=0} ^{\infty} \sum \limits _{l=0} ^{\infty} I'_{k,l} + \sum \limits _{k=0} ^{\infty} \sum \limits _{l=0} ^{\infty} I''_{k,l}, $$
where
\begin{equation*}
\begin{split}
I&:= \int \limits _{{\mathbb R}^{n}} \sup\{ |e^{s^{2}L}f(y)| \;;\; (y,s) \in \Gamma_{x} ^{a}(\gamma),
s\leq \frac{r_{B}}{\sqrt{\alpha}}\}d\gamma(x),\\
I'_{k,l}&:= \int \limits _{C_{k}(B)} \sup\{ |e^{s^{2}L}(1_{C_{l}(B)}f)(y)| \;;\; (y,s) \in \Gamma_{x} ^{a}(\gamma),
s\geq \frac{r_{B}}{\sqrt{\alpha}}\}1_{(0,\frac{2^{k}r_{B}}{C_{a}})}(m(x))d\gamma(x),\\
I''_{k,l}&:= \int \limits _{C_{k}(B)} \sup\{ |L^{N}e^{s^{2}L}(1_{C_{l}(B)}\tilde{f})(y)| \;;\; (y,s) \in \Gamma_{x} ^{a}(\gamma),
s\geq \frac{r_{B}}{\sqrt{\alpha}}\}1_{[\frac{2^{k}r_{B}}{C_{a}},1]}(m(x))d\gamma(x).
\end{split}
\end{equation*}
{\em Estimating $I$:}
Decomposing into a local and global part and using Proposition \ref{prop:glob}, we have that
$$
I \lesssim \|f\|_{1} + \sum \limits _{k=0} ^{\infty} \sum \limits _{l=0} ^{\infty} I^{loc}_{k,l},
$$
where
$$
I ^{loc} _{k,l}:= \int \limits _{C_{k}(B)} \sup\{ \int \limits _{C_{l}(B)} M_{s^{2}}(z,w)1_{N_{\tau}}(z,w)|f(w)|dw \;;\; (z,s) \in \Gamma_{x} ^{a}(\gamma), s\leq \frac{r_{B}}{\sqrt{\alpha}}\}d\gamma(x).
$$
By Lemma \ref{lem:mm} we also have that
$$
\|f\|_{1} \leq \sum \limits _{k=0} ^{\infty} \sqrt{\gamma(2^{k+1}B)} \|1_{C_{k}(B)}f\|_{2}
\leq \sum \limits _{k=0} ^{\infty} e^{8(2^{k+2}+1)^{2}}e^{-C4^{k}} \lesssim 1,
$$
since $C>2^{9}$.\\
{\em Estimating $I_{k,l}^{loc}$ for $k< l+2$:}\\
By Lemma \ref{lem:mm} and Proposition \ref{prop:glob} we have that
$$
I_{k,l}^{loc} \leq \sqrt{\gamma(2^{k+1}B)} \|x\mapsto \sup\{e^{s^{2}L}|1_{C_{l}(B)}f|(y) \;;\; (y,s) \in \Gamma_{x}^{a}\}\|_{2}
\lesssim e^{2^{9}.4^{k}}\sqrt{\gamma(B)}\|1_{C_{l}(B)}f\|_{2} \leq e^{2^{9}.4^{k}} e^{-C.4^{l}},
$$
and thus:
$$
\sum \limits _{l=0} ^{\infty} \sum \limits _{k=0} ^{l+1} I^{loc}_{k,l} \leq \sum \limits _{l=0} ^{\infty} (l+2)e^{-(C-2^{11})4^{l}} \lesssim 1.
$$
{\em Estimating $I_{k,l}^{loc}$ for $k\geq l+2$:}\\
We use Lemma \ref{lem:est} as follows:
\begin{equation*}
\begin{split}
I_{k,l} ^{loc} &= \int \limits _{C_{k}(B)} \sup\{ \int \limits _{C_{l}(B)} M_{\frac{t^{2}}{\alpha}}(z,w)1_{N_{\tau}}(z,w)|f(w)|dw \;;\; (z,t) \in
\Gamma_{x} ^{(\frac{1}{\sqrt{\alpha}},a\sqrt{\alpha})}(\gamma), t\leq r_{B}\}d\gamma(x)\\
&\lesssim \int \limits _{C_{k}(B)} \sup\{ \int \limits _{C_{l}(B)} M_{t^{2}}(z,w)\exp(-\frac{\alpha}{2^{17}}\frac{|e^{-t^{2}}z-w|^{2}}{1-e^{-2t^{2}}})1_{N_{\tau}}(z,w)|f(w)|dw \;;\; (z,t) \in
\Gamma_{x} ^{(\frac{1}{\sqrt{\alpha}},a\sqrt{\alpha})}(\gamma), t\leq r_{B}\}d\gamma(x),
\end{split}
\end{equation*}
where we have used Lemma \ref{lem:mnp1} to see that
\begin{equation*}
\begin{split}
|z-x| \leq am(x) &\implies m(x) \leq (1+a)m(z),\\
|z-w| \leq \tau m(z) &\implies m(z) \leq (1+\tau)m(w),\\
t \leq a\sqrt{\alpha}m(x) &\implies t \leq a\sqrt{\alpha}(1+a)(1+\tau)m(w).
\end{split}
\end{equation*}
Now, for $x \in C_{k}(B)$, $w \in C_{l}(B)$, $t \leq \min(r_{B}, a\sqrt{\alpha}(1+a)m(z))$, and $z \in B(x,\frac{t}{\sqrt{\alpha}})$, we have
$$
|e^{-t^{2}}z-w| \geq |x-w|-|x-z|-(1-e^{-t^{2}})|z|
\geq (2^{k-1}-\frac{2}{\sqrt{\alpha}}-2a\sqrt{\alpha}(1+a))r_{B}.
$$
Let $M_{a,\alpha} \in {\mathbb N}$ be such that $\frac{2}{\sqrt{\alpha}}+2a\sqrt{\alpha}(1+a)\leq 2^{M_{a,\alpha}}$.
Then, for $k \geq \max(l,M_{a,\alpha})+2$ we have the following.
\begin{equation*}
\begin{split}
I_{k,l}^{loc} &\lesssim \exp(-\frac{\alpha}{2^{18}}(2^{k-2})^{2})
\int \limits _{C_{k}(B)} \sup\{ e^{t^{2}L}|1_{C_{l}(B)}f|(z)\;;\; (z,t) \in
\Gamma_{x} ^{(\frac{1}{\sqrt{\alpha}},\sqrt{\alpha}a)}(\gamma)\}d\gamma(x)\\
&\lesssim \exp(-\frac{\alpha}{2^{22}}4^{k}) \sqrt{\gamma(2^{k+1}B)}\|1_{C_{l}(B)}f\|_{2}
\leq \exp(-\frac{\alpha}{2^{22}}4^{k})\exp(2^{9}.4^{k})\exp(-C4^{l}),
\end{split}
\end{equation*}
where we have used Proposition \ref{prop:glob} and Lemma \ref{lem:mm}. Noticing that
$$
\sum \limits _{k=0} ^{M_{a,\alpha}+2} \sum \limits _{l=0} ^{M_{a,\alpha}} I_{k,l} ^{loc}
\lesssim \sum \limits _{k=0} ^{M_{a,\alpha}+2} \sum \limits _{l=0} ^{M_{a,\alpha}} \sqrt{\gamma(2^{k+1}B)}\|f\|_{2}
\leq \sum \limits _{k=0} ^{M_{a,\alpha}+2} \sum \limits _{l=0} ^{M_{a,\alpha}} \exp(2^{9}.4^{k}) \lesssim 1,
$$
and using the fact that $\alpha>2^{31}$,
we get that $\sum \limits _{l=0} ^{\infty}\sum \limits _{k=l+2} ^{\infty} I_{k,l}^{loc} \lesssim 1$ and thus that $I \lesssim 1$.\\
{\em Estimating $I'_{k,l}$ for $k < l+2$:}\\
Reasoning as above, using Proposition \ref{prop:glob} and Lemma \ref{lem:mm}, we have that
$$
I'_{k,l} \lesssim \exp(2^{9}.4^{k}) \sqrt{\gamma(B)} \|1_{C_{l}(B)}f\|_{2} \lesssim \exp(2^{9}.4^{k}-C4^{l}),
$$
and thus $$
\sum \limits _{l=0} ^{\infty} \sum \limits _{k=0} ^{l+1} I'_{k,l} \leq \sum \limits _{l=0} ^{\infty} (l+2)\exp(-(C-2^{13})4^{l}) \lesssim 1.
$$
{\em Estimating $I'_{k,l}$ for $k \geq l+2$:}\\
Given $x \in C_{k}(B)$ such that $m(x) \leq \frac{2^{k}r_{B}}{C_{a}}$, $s\leq am(x)$,
$y \in B(x,s)$, and $w \in C_{l}(B)$, we have, using Lemma \ref{lem:mnp1}:
$$
|y-w| \geq |x-w|-|x-y| \geq 2^{k-1}r_{B}(2-2^{l+2-k})-am(x) \geq (\frac{C_{a}}{2}-a)m(x) \geq \frac{1}{2+2a}(\frac{C_{a}}{2}-a)m(y)= \tau m(y).
$$
By Proposition \ref{prop:glob}, we thus have
$$
\sum \limits _{l=0} ^{\infty} \sum \limits _{k=l+2} ^{\infty} I'_{k,l}\leq \sum \limits _{l=0} ^{\infty} \|T^{*}_{glob,a,1}|1_{C_{l}(B)}f|\|_{1}
\lesssim \|f\|_{1} \lesssim 1.
$$
{\em Estimating $I''_{k,l}$:}\\
For $x \in {\mathbb R}^{n}$, $t\leq a\sqrt{\alpha}m(x)$, $y \in B(x,\frac{t}{\sqrt{\alpha}})$,
we have $t \lesssim m(y)$ by Lemma \ref{lem:mnp1} and thus
$$
|L^{N}e^{\frac{t^{2}}{\alpha}L}(1_{C_{l}(B)}\tilde{f})(y)| \lesssim t^{-2N} \int \limits _{C_{l}(B)}
|K_{t^{2},N,\alpha}(y,w)||\tilde{f}(w)|dw \lesssim t^{-2N} \int \limits _{C_{l}(B)} M_{t^{2}}(y,w)|\tilde{f}(w)|dw,
$$
by Lemma \ref{lem:est}.
Therefore
$$
I''_{k,l} \lesssim \int \limits _{C_{k}(B)} \sup \{t^{-2N}e^{t^{2}L}|1_{C_{l}(B)}\tilde{f}|(z) \;;\; (z,t) \in
\Gamma_{x}^{(\frac{1}{\sqrt{\alpha}},a\sqrt{\alpha})}(\gamma), t \geq r_{B}\}1_{[\frac{2^{k}r_{B}}{C_{a}},1]}(m(x))d\gamma(x)
\lesssim r_{B}^{-2N}J_{k,l}^{glob}+J_{k,l}^{loc},
$$
where
\begin{equation*}
\begin{split}
J_{k,l}^{glob} &:= \int \limits _{C_{k}(B)} \sup \{\int \limits _{C_{l}(B)} M_{t^{2}}(z,w)1_{N_{\tau} ^{c}}(z,w) |\tilde{f}|(w)dw \;;\; (z,t) \in
\Gamma_{x}^{(\frac{1}{\sqrt{\alpha}},a\sqrt{\alpha})}(\gamma)\}d\gamma(x),\\
J_{k,l}^{loc} &:= \int \limits _{C_{k}(B)} \sup \{t^{-2N}\int \limits _{C_{l}(B)} M_{t^{2}}(z,w)1_{N_{\tau}}(z,w) |\tilde{f}|(w)dw \;;\; (z,t) \in
\Gamma_{x}^{(\frac{1}{\sqrt{\alpha}},a\sqrt{\alpha})}(\gamma), t \geq r_{B}\}1_{[\frac{2^{k}r_{B}}{C_{a}},1]}(m(x))d\gamma(x),
\end{split}
\end{equation*}
and $\tau$ is defined as in Proposition \ref{prop:glob} for the parameters $(\frac{1}{\sqrt{\alpha}},a\sqrt{\alpha})$.
Proposition \ref{prop:glob} then gives that
$$
\sum \limits _{l=0} ^{\infty} \sum \limits _{k=0} ^{\infty}J_{k,l} ^{glob} \lesssim \sum \limits _{l=0} ^{\infty} \|1_{C_{l}(B)}\tilde{f}\|_{1} \lesssim r_{B}^{2N}.
$$
For $x \in C_{k}(B)$ and $m(x) \geq \frac{2^{k}r_{B}}{C_{a}}$ we have
$$
|x-c_{B}|\leq 2^{k+1}r_{B} \leq 2C_{a}m(x) \leq 2C_{a}(1+2C_{a})m(c_{B})=:\tau'm(c_{B}).
$$
Therefore
$$
J_{k,l}^{loc} \leq \int \limits _{C_{k}(B)\cap N_{\tau'}(B)} \sup \{t^{-2N}\int \limits _{C_{l}(B)} M_{t^{2}}(z,w)1_{N_{\tau}}(z,w) |\tilde{f}|(w)dw \;;\; (z,t) \in
\Gamma_{x}^{(\frac{1}{\sqrt{\alpha}},a\sqrt{\alpha})}(\gamma), t \geq r_{B}\}d\gamma(x).
$$
{\em Estimating $J_{k,l}^{loc}$ for $k < l+2$:}\\
Using Proposition \ref{prop:glob} and Lemma \ref{lem:region}, we have
$$
\sum \limits _{l=0} ^{\infty} \sum \limits _{k=0} ^{l+1} J_{k,l}^{loc}
\lesssim r_{B}^{-2N} \sum \limits _{l=0} ^{\infty} \sum \limits _{k=0} ^{l+1} \sqrt{\gamma(C_{k}(B)\cap N_{\tau'}(B))}\|1_{C_{l}(B)}\tilde{f}\|_{2}
\lesssim \sum \limits _{l=0} ^{\infty} \exp(-C4^{l})\sum \limits _{k=0} ^{l+1} 2^{k\frac{n}{2}} \lesssim 1.
$$
{\em Estimating $J_{k,l}^{loc}$ for $k \geq l+2$:}\\
For $x \in {\mathbb R}^{n}$, $s\leq a\alpha m(x)$, $z \in B(x,am(x))$, and $(z,w) \in N_{\tau}$, we have $m(w)\sim m(z) \sim m(x)$ and thus $s \lesssim m(w)$.
Therefore, using Lemma \ref{lem:est} we have
\begin{equation*}
\begin{split}
J_{k,l} ^{loc} &\lesssim \int \limits _{C_{k}(B)\cap N_{\tau'}(B)} \sup \{s^{-2N} \int \limits _{C_{l}(B)}
M_{\frac{s^{2}}{\alpha}}(z,w)1_{N_{\tau}}(z,w)|\tilde{f}(w)|dw \;;\; (z,s) \in \Gamma_{x}^{(\frac{1}{\alpha},a\alpha)}(\gamma), s\geq \sqrt{\alpha}r_{B}\}d\gamma(x)\\
&\lesssim \int \limits _{C_{k}(B)\cap N_{\tau'}(B)} \sup \{s^{-2N} \int \limits _{C_{l}(B)}
M_{s^{2}}(z,w) \exp(-\frac{\alpha}{2^{17}}\frac{|e^{-s^{2}}z-w|^{2}}{1-e^{-2s^{2}}})|\tilde{f}(w)|dw \;;\; (z,s) \in \Gamma_{x}^{(\frac{1}{\alpha},a\alpha)}(\gamma)\}d\gamma(x).
\end{split}
\end{equation*}
For $x \in C_{k}(B)$, $w \in C_{l}(B)$, $s \leq \alpha am(x)$, and $z \in B(x,\frac{1}{\alpha}s)$ we have
$$
|e^{-s^{2}}z-w| \geq |x-w| -|x-z| - (1-e^{-s^{2}})|z| \geq 2^{k-1}r_{B}-(\frac{1}{\alpha}+\alpha(a+2a^{2}))s.
$$
Therefore, there exists $C_{\alpha}>0$ such that
\begin{equation*}
\begin{split}
J_{k,l}^{loc} & \lesssim \int \limits _{C_{k}(B)\cap N_{\tau'}(B)} \sup \{s^{-2N} \exp(-C_{\alpha}4^{k}(\frac{r_{B}}{s})^{2})
\int \limits _{C_{l}(B)} M_{s^{2}}(z,w)|\tilde{f}(w)|dw \;;\; (z,s) \in \Gamma_{x}^{(\frac{1}{\alpha},a\alpha)}(\gamma)\}d\gamma(x)\\
& \lesssim (2^{k}r_{B})^{-2N}\int \limits _{C_{k}(B)\cap N_{\tau'}(B)} \sup \{
\int \limits _{C_{l}(B)} M_{s^{2}}(z,w)|\tilde{f}(w)|dw \;;\; (z,s) \in \Gamma_{x}^{(\frac{1}{\alpha},a\alpha)}(\gamma)\}d\gamma(x)\\
& \lesssim (2^{k}r_{B})^{-2N}\sqrt{\gamma(C_{k}(B)\cap N_{\tau'}(B))}\|1_{C_{l}(B)}\tilde{f}\|_{2}
\lesssim 4^{-kN}\exp(-C4^{l})2^{k\frac{n}{2}},
\end{split}
\end{equation*}
where we have used Proposition \ref{prop:glob} and Lemma \ref{lem:region}.
This gives
$$
\sum \limits _{l=0} ^{\infty} \sum \limits _{k=0} ^{l+2} J^{loc}_{k,l}
\lesssim \sum \limits _{l=0} ^{\infty} \sum \limits _{k=0} ^{l+2} 4^{-k(N-\frac{n}{4})}\exp(-C4^{l}) \lesssim 1,
$$
which concludes the proof.
\end{proof}
\section{Remainder terms}
In this section, we handle the remainder terms
\begin{enumerate}
\item
$
\int \limits _{0} ^{2} 1_{[\frac{m(.)}{b},2]}(t)t^{2N+1}L^{N}e^{\frac{t^{2}}{\alpha}L}\partial_{x_{j}}^{*}F(t,.) \frac{dt}{t}$,
\item
$
\int \limits _{0} ^{\frac{m(.)}{b}} t^{2N+1}L^{N}e^{\frac{(1+a^{2})t^{2}}{\alpha}L}\partial_{x_{j}}^{*}(1_{D^{c}}(t,.)t\partial_{x_{j}}e^{\frac{a^{2}t^{2}}{\alpha}L})u \frac{dt}{t}$,
\item
$ \int \limits _{\frac{m(.)}{b}} ^{\infty} t^{2N+2}L^{N}e^{\frac{(1+a^{2})t^{2}}{\alpha}L}u \frac{dt}{t}$,
\end{enumerate}
where $u \in L^{1}(\gamma)$ and $F$ is a $t^{1,2}(\gamma)$ atom.
\begin{lemma}
Let $N \in {\mathbb Z}_{+}$, $j \in \{1,...,n\}$, $b>0$ and $\alpha>2^{32}$.
Let $F$ be a $t^{1,2}(\gamma)$ atom associated with the ball $B=B(c_{B},r_{B}) \in \mathcal{B}_{2}$.
Then
$$
\|\int \limits _{0} ^{r_{B}} 1_{[\frac{m(.)}{b},2]}(t)t^{2N+1}L^{N}e^{\frac{t^{2}}{\alpha}L}\partial_{x_{j}}^{*}F(t,.) \frac{dt}{t}\|_{L^{1}} \lesssim 1.
$$
\end{lemma}
\begin{proof}
By Lemma \ref{lem:mnp1}, we have $m(y)\sim m(c_{B})$ for $y \in B$. Therefore, by Lemma \ref{lem:est}, and reasoning as in Proposition \ref{prop:mol}, we have
\begin{equation*}
\begin{split}
\|\int \limits _{0} ^{r_{B}} & 1_{[\frac{m(.)}{b},2]}(t)t^{2N+1}L^{N}e^{\frac{t^{2}}{\alpha}L}\partial_{x_{j}}^{*}F(t,.) \frac{dt}{t}\|_{L^{1}}
\lesssim \sum \limits _{k=0} ^{\infty}
\int \limits _{C_{k}(B)} \int \limits _{0} ^{r_{B}} \int \limits _{B} |\tilde{K}_{t^{2},N,\alpha,j}(x,y)||F(t,y)| dy \frac{dt}{t} d\gamma(x)\\
&\lesssim 1+\sum \limits _{k=1} ^{\infty} \int \limits _{0} ^{r_{B}} \exp(-\frac{\alpha}{2^{22}}4^{k}(\frac{r_{B}}{t})^{2}) \sqrt{\gamma(2^{k+1}B)}\|F(t,.)\|_{2} \frac{dt}{t}\\
&\lesssim 1+\sum \limits _{k=1} ^{\infty} \exp(2^{9}.4^{k})\sqrt{\gamma(B)}\exp(-\frac{\alpha}{2^{23}}4^{k})
(\int \limits _{0} ^{r_{B}} \exp(-\frac{\alpha}{2^{22}}4^{k}(\frac{r_{B}}{t})^{2})\frac{dt}{t})^{\frac{1}{2}}\gamma(B)^{-\frac{1}{2}}\\
&\lesssim 1+\sum \limits _{k=1} ^{\infty} \exp(-(\frac{\alpha}{2^{23}}-2^{9})4^{k}) \lesssim 1.
\end{split}
\end{equation*}
\end{proof}
Combined with Proposition \ref{prop:glob}, this gives
\begin{corollary}
Let $a,b>0$, $N \in {\mathbb Z}_{+}$, $\{j=1,...,n\}$, and $\alpha>2^{32}$.
Let $F$ be a $t^{1,2}(\gamma)$ atom associated with the ball $B=B(c_{B},r_{B}) \in \mathcal{B}_{2}$.
Then
$$
\|T^{*}_{\text{glob},a}(\int \limits _{0} ^{r_{B}} 1_{[\frac{m(.)}{b},2]}(t)t^{2N+1}L^{N}e^{\frac{t^{2}}{\alpha}L}\partial_{x_{j}}^{*}F(t,.) \frac{dt}{t})\|_{1} \lesssim 1.
$$
\end{corollary}
\begin{proposition}
\label{prop:r1}
Let $a>0$, $N \in {\mathbb Z}_{+}$, $\{j=1,...,n\}$, and $\alpha>2^{38}$.
Let $F$ be a $t^{1,2}(\gamma)$ atom associated with the ball $B=B(c_{B},r_{B}) \in \mathcal{B}_{2}$.
Then
$$
\|\int \limits _{0} ^{r_{B}} 1_{[\frac{m(.)}{b},2]}(t)t^{2N+1}L^{N}e^{\frac{t^{2}}{\alpha}L}\partial_{x_{j}}^{*}F(t,.) \frac{dt}{t}\|_{h^{1}_{\text{max},a}} \lesssim 1.
$$
\end{proposition}
\begin{proof}
Given the above Corollary, and $\tau$ as in Proposition \ref{prop:glob}, we only have to estimate
$$
I = \int \limits _{{\mathbb R}^{n}} \sup \{ \int \limits _{{\mathbb R}^{n}} M_{s^{2}}(y,z)1_{N_{\tau}}(y,z)
\int \limits _{0} ^{r_{B}} \int \limits _{{\mathbb R}^{n}}1_{[\frac{m(z)}{b},2]}(t)|\tilde{K}_{t^{2},N,\alpha,j}(z,w)||F(t,w)|dw \frac{dt}{t} dz \;;\; (y,s) \in \Gamma_{x}^{a}(\gamma)\} d\gamma(x).
$$
For $w \in B$ and $t\leq r_{B}$, we have $t \lesssim m(w)$ by Lemma \ref{lem:mnp1}. Therefore, by Lemma \ref{lem:est}
\begin{equation*}
\begin{split}
I &\lesssim \int \limits _{{\mathbb R}^{n}} \underset{(y,s) \in \Gamma_{x}^{a}(\gamma)}{\sup} \int \limits _{{\mathbb R}^{n}} M_{s^{2}}(y,z)1_{N_{\tau}}(y,z)
\int \limits _{0} ^{r_{B}} \int \limits _{{\mathbb R}^{n}}1_{[\frac{m(z)}{b},2]}(t) \exp(-\frac{\alpha}{2^{23}}\frac{|e^{-t^{2}}w-z|^{2}}{1-e^{-2t^{2}}})M_{t^{2}}(z,w)
|F(t,w)|dw \frac{dt}{t} dz d\gamma(x)\\
&\lesssim I_{loc}+\sum \limits _{k=0} ^{\infty} I^{glob} _{k},
\end{split}
\end{equation*}
where
\begin{equation*}
\begin{split}
I^{glob}_{k}:=&\int \limits _{C_{k}(B)} \underset{(y,s) \in \Gamma_{x}^{a}(\gamma)}{\sup} \int \limits _{{\mathbb R}^{n}} M_{s^{2}}(y,z)1_{N_{\tau}}(y,z)
\int \limits _{0} ^{r_{B}} \int \limits _{{\mathbb R}^{n}} 1_{[\frac{m(z)}{b},2]}(t) e^{-\frac{\alpha}{2^{23}}\frac{|e^{-t^{2}}w-z|^{2}}{1-e^{-2t^{2}}}}1_{N_{1} ^{c}}(z,w)M_{t^{2}}(z,w)
|F(t,w)|dw \frac{dt}{t} dz d\gamma(x),\\
I_{loc}:=&\int \limits _{{\mathbb R}^{n}} \underset{(y,s) \in \Gamma_{x}^{a}(\gamma)}{\sup} \int \limits _{{\mathbb R}^{n}} M_{s^{2}}(y,z)1_{N_{\tau}}(y,z)
\int \limits _{0} ^{r_{B}} \int \limits _{{\mathbb R}^{n}}1_{[\frac{m(z)}{b},2]}(t) e^{-\frac{\alpha}{2^{23}}\frac{|e^{-t^{2}}w-z|^{2}}{1-e^{-2t^{2}}}}1_{N_{1}}(z,w)M_{t^{2}}(z,w)
|F(t,w)|dw \frac{dt}{t} dz d\gamma(x).
\end{split}
\end{equation*}
{\em Estimating $I^{glob}_{k}$:}\\
For $w \in B$, $x \in C_{k}(B)$, $y \in B(x,am(x))$, $z \in B(y,\tau m(y))$, $t \leq r_{B}$, and $m(z)\leq br_{B}$,
Lemma \ref{lem:mnp1}, gives that $t \lesssim m(w)$, $|x-z| \leq (a+2\tau(1+a))m(x)$ and $m(x) \leq (1+a+2\tau(1+a))m(z) \leq b(1+a+2\tau(1+a))r_{B}$.
Therefore
$$
|e^{-t^{2}}w-z| \geq |w-x|-|x-z|-(1-e^{-t^{2}})|w| \geq 2^{k-1}r_{B}-C_{a,b}r_{B},
$$
for some $C_{a,b}>0$.
Let $M_{a,b} \in {\mathbb N}$ be such that $C_{a,b} \leq 2^{M_{a,b}}$.
We first notice that, for $k \leq M_{a,b}+1$, $ x \in C_{k}(B)$, and $z \in B(x,(a+2\tau(1+a))m(x))$ Lemma \ref{lem:mnp1} gives
$m(z) \sim m(x) \sim m(c_{B})$ with implicit constant depending only on $a$ and $b$. In particular $\frac{m(z)}{b} \geq \kappa_{a,b}m(c_{B})$ for some
$\kappa_{a,b}>0$.
Therefore
\begin{equation*}
\begin{split}
\sum \limits _{k=0} ^{M_{a,b}+1} I^{glob}_{k} &\lesssim
\sum \limits _{k=0} ^{M_{a,b}+1} \sqrt{\gamma(2^{k+1}B)} \int \limits _{\kappa_{a,b}m(c_{B})} ^{2m(c_{B})}
\|T^{*}_{a}(e^{t^{2}L}|F(t,.)|)\|_{2} \frac{dt}{t} \\
&\lesssim
\sum \limits _{k=0} ^{M_{a,b}+1} \sqrt{\gamma(B)}\exp(2^{9}.4^{k})
(\int \limits _{\kappa_{a,b}m(c_{B})} ^{2m(c_{B})} \frac{dt}{t})^{\frac{1}{2}}
(\int \limits _{0} ^{r_{B}}\|F(t,.)\|_{2} ^{2}\frac{dt}{t})^{\frac{1}{2}} \\
&\lesssim \sum \limits _{k=0} ^{M_{a,b}+1} \exp(2^{9}.4^{k}) \lesssim 1.
\end{split}
\end{equation*}
For $k \geq M_{a,b}+2$ we estimate as follows, using Lemma \ref{lem:est},
\begin{equation*}
\begin{split}
\sum \limits _{k=M_{a,b}+2} ^{\infty} I^{glob}_{k} &\lesssim
\sum \limits _{k=M_{a,b}+2} ^{\infty} \sqrt{\gamma(2^{k+1}B)} \int \limits _{0} ^{r_{B}} \exp(-\frac{\alpha}{2^{28}}4^{k}(\frac{r_{B}}{t})^{2})
\|T^{*}_{a}(e^{t^{2}L}|F(t,.)|)\|_{2} \frac{dt}{t} \\
&\lesssim
\sum \limits _{k=M_{a,b}+2} ^{\infty} \sqrt{\gamma(B)}\exp(2^{9}.4^{k})\exp(-\frac{\alpha}{2^{29}}4^{k})
(\int \limits _{0} ^{r_{B}} \exp(-\frac{\alpha}{2^{28}}(\frac{2^{k}r_{B}}{t})^{2}) \frac{dt}{t})^{\frac{1}{2}}
(\int \limits _{0} ^{r_{B}}\|F(t,.)\|_{2} ^{2}\frac{dt}{t})^{\frac{1}{2}} \\
&\lesssim \sum _{k=M_{a,b}+2} ^{\infty}\exp(2^{9}.4^{k})\exp(-\frac{\alpha}{2^{29}}4^{k}) \lesssim 1.
\end{split}
\end{equation*}
{\em Estimating $I_{loc}$:}\\
We have
$$
I_{loc} \lesssim \int \limits _{{\mathbb R}^{n}} \underset{(y,s) \in \Gamma_{x}^{a}(\gamma)}{\sup} \int \limits _{{\mathbb R}^{n}} M_{s^{2}}(y,z)1_{N_{\tau}}(y,z)
\int \limits _{0} ^{r_{B}} \int \limits _{{\mathbb R}^{n}}1_{[\frac{m(z)}{b},2]}(t) 1_{N_{1}}(z,w)m(z)^{-n}|F(t,w)|dw \frac{dt}{t} dz d\gamma(x).
$$
For $w \in B$, $(z,w) \in N_{1}$, $(y,z) \in N_{\tau}$, and $(x,y) \in N_{a}$, we have that $m(x)\sim m(y) \sim m(z) \sim m(w) \sim m(c_{B})$.
Moreover $|x-c_{B}|\leq am(x)+\tau m(y)+m(z)+m(c_{B}) \lesssim m(c_{B})$, $|x-w| \lesssim m(w)$, and $e^{-|w|^{2}}\sim e^{-|x|^{2}}$.
Let $\kappa, \lambda$ be such that $\frac{m(z)}{b} \geq \kappa m(c_{B})$ and $|x-c_{B}| \leq \lambda m(c_{B})$.
Using the positivity of $(e^{tL})_{t>0}$, and the fact that $e^{L}1=1$, we have that
\begin{equation*}
I_{loc} \lesssim \int \limits _{\kappa m(c_{B})} ^{r_{B}} m(c_{B})^{-n}\int \limits _{B(c_{B},\lambda m(c_{B}))} \|F(t,.)\|_{1} dx \frac{dt}{t}
\lesssim (\int \limits _{\kappa m(c_{B})} ^{2m(c_{B})} \frac{dt}{t})^{\frac{1}{2}}\sqrt{\gamma(B)} (\int \limits _{0} ^{r_{B}} \|F(t,.)\|_{2} \frac{dt}{t})^{\frac{1}{2}}
\lesssim 1.
\end{equation*}
\end{proof}
\begin{proposition}
\label{prop:Dcomp}
Let $a,a'>0$, $N \in {\mathbb Z}_{+}$, $j \in \{1,...,n\}$ and $\alpha > \max(32e^{4},4\sqrt{a}e^{2a^{2}})$. Let $b \geq
\max(2e,\sqrt{\frac{32e^{4}}{(\alpha-32e^{4})(1-e^{-2\frac{a^{2}}{\alpha}})}})$.
Then $$
\|\int \limits _{0} ^{\frac{m(.)}{b}} t^{2N+1}L^{N}e^{\frac{t^{2}}{\alpha}L}\partial_{x_{j}}^{*}(1_{D^{c}}(t,.)t\partial_{x_{j}}e^{\frac{a^{2}t^{2}}{\alpha}L})u \frac{dt}{t}\|_{h^{1}_{\text{max},a'}} \lesssim \|u\|_{L^{1}(\gamma)}. $$
\end{proposition}
\begin{proof}
We claim that
$$
\|\int \limits _{0} ^{\frac{m(.)}{b}} t^{2N+1}L^{N}e^{\frac{t^{2}}{\alpha}L}\partial_{x_{j}}^{*}(1_{D^{c}}(t,.)t\partial_{x_{j}}e^{\frac{a^{2}t^{2}}{\alpha}L})u \frac{dt}{t}\|_{\infty} \lesssim \|u\|_{1}. $$
The result then follows from the fact that $e^{sL}1=1$ for all $s>0$ and the positivity of $e^{sL}$.
To prove the claim, fix $x \in {\mathbb R}^{n}$, and consider $t\geq 0$ and $y \in {\mathbb R}^{n}$ such that $m(y) \leq t \leq \frac{m(x)}{b}$.
Then $|y| \geq 1$ and $|y| \geq b|x|\geq 2e|x|$. Therefore $|e^{-t^{2}}y-x|\geq \frac{|y|}{2e}+\frac{|y|}{2e}-|x|\geq\frac{|y|}{2e}$
and $t^{-1} \leq |y|$. Using Corollary \ref{cor:Ktilde} and Lemma \ref{lem:est}, this gives, for some $M>0$
\begin{equation*}
\begin{split}
t^{-1}|\tilde{K}_{t^{2},N,\alpha,j}(x,y)| &\lesssim |y|^{M}\exp(-\frac{\alpha}{2e^{2}}\frac{|e^{-t^{2}}y-x|^{2}}{1-e^{-2t^{2}}})M_{t^{2}}(x,y)\\
&\lesssim |y|^{M+n}\exp(-\frac{\alpha}{16e^{4}}|y|^{2}) \lesssim \exp(-\frac{\alpha}{32e^{4}}|y|^{2}).
\end{split}
\end{equation*}
Using Lemma \ref{lem:slow2}, and the fact that $t\mapsto \frac{t^{2}}{1-e^{-\frac{2a^{2}t^{2}}{\alpha}}}$ is increasing on $(0,1)$, we then have
\begin{equation*}
\begin{split}
\|\int \limits _{0} ^{\frac{m(.)}{b}} &t^{2N+1}L^{N}e^{\frac{t^{2}}{\alpha}L}\partial_{x_{j}}^{*}(1_{D^{c}}(t,.)t\partial_{x_{j}}e^{\frac{a^{2}t^{2}}{\alpha}L})u \frac{dt}{t}\|_{\infty}
\lesssim \int \limits _{0} ^{\frac{1}{b}} \int \limits _{{\mathbb R}^{n}} \int \limits _{{\mathbb R}^{n}} \frac{|e^{-\frac{a^{2}t^{2}}{\alpha}}y_{j}-z_{j}|}{\sqrt{1-e^{-\frac{2a^{2}t^{2}}{\alpha}}}}
M_{\frac{a^{2}t^{2}}{\alpha}}(y,z)\exp(-\frac{\alpha}{32e^{4}}|y|^{2})|u(z)|dzdydt\\
&\lesssim \int \limits _{0} ^{\frac{1}{b}} \int \limits _{{\mathbb R}^{n}} \int \limits _{{\mathbb R}^{n}}
t^{-n}\exp(-\frac{\alpha}{4\sqrt{a}e^{2a^{2}}}\frac{|e^{-t^{2}}y-z|^{2}}{1-e^{-2t^{2}}})\exp(\frac{t^{2}}{1-e^{-\frac{2a^{2}t^{2}}{\alpha}}}\frac{1}{2b^{2}}|y|^{2})
\exp(-\frac{\alpha}{32e^{4}}|y|^{2})|u(z)|dzdydt\\
&\lesssim \int \limits _{0} ^{\frac{1}{b}} \int \limits _{{\mathbb R}^{n}} \int \limits _{{\mathbb R}^{n}}
M_{t^{2}}(y,z)\exp(\frac{1}{2b^{2}(1-e^{-\frac{2a^{2}}{\alpha}})}|y|^{2})
\exp(-\frac{\alpha}{32e^{4}}|y|^{2})|u(z)|dzdydt\\
&\lesssim \int \limits _{0} ^{\frac{1}{b}} \int \limits _{{\mathbb R}^{n}} e^{t^{2}L}|u(y)| d\gamma(y)dt \lesssim \|u\|_{1}.
\end{split}
\end{equation*}
\end{proof}
\begin{proposition}
\label{prop:jinf}
Let $N \in {\mathbb Z}_{+}$, $a,a',b>0$, and $\alpha>8e^{2a^{2}}$. For all $u\in C_{c}^{\infty}({\mathbb R}^{n})$, we have
$$
\| \int\limits _{\frac{m(.)}{b}} ^{\infty} (t^{2}L)^{N+1}e^{\frac{(1+a^{2})t^{2}}{\alpha}L}u \frac{dt}{t}\|_{h^{1}_{\text{max},a'}} \lesssim \|u\|_{1}.
$$
\end{proposition}
\begin{proof}
Let $M>1$ and $x \in {\mathbb R}^{n}$. Without loss of generality we assume that $\int u d\gamma =0$ (since $L1=0$).
\begin{equation*}
\begin{split}
|\int \limits _{\frac{m(x)}{b}} ^{M} (t^{2}L)^{N+1}e^{(1+a^{2})\frac{t^{2}}{\alpha}L}u(x) \frac{dt}{t}|
&\lesssim |\int \limits _{\frac{(1+a^{2})m(x)^{2}}{b^{2}\alpha}} ^{\frac{(1+a^{2})M^{2}}{\alpha}} s^{N+1}\partial_{s} ^{N+1}e^{sL}u(x) \frac{ds}{s}|\\
&\lesssim \sum \limits _{k=0} ^{N} \int \limits _{{\mathbb R}^{n}} |K_{(1+a^{2})b^{-2}m(x)^{2},k,\alpha}(x,y)||u(y)|dy
+ \sum \limits _{k=0} ^{N} |(M^{2}L)^{k}e^{\frac{(1+a^{2})M^{2}}{\alpha}L}u(x)|.
\end{split}
\end{equation*}
Given $k \in \{0,...,N\}$ we have, using chaos decomposition and Proposition \ref{prop:glob}:
$$
\|(M^{2}L)^{k}e^{\frac{(1+a^{2})M^{2}}{\alpha}L}u\|_{h^{1}_{\text{max},a'}} \leq \|T^{*}_{a'}(M^{2}L)^{k}e^{\frac{(1+a^{2})M^{2}}{\alpha}L}u\|_{2}
\lesssim \|(M^{2}L)^{k}e^{\frac{(1+a^{2})M^{2}}{\alpha}L}u\|_{2} \leq M^{2k}e^{-\frac{(1+a^{2})M^{2}}{\alpha}}\|u\|_{2} \underset{M\to \infty}{\to} 0.
$$
It thus remains to prove that, given $k \in \{0,...,N\}$,
$$
\|T^{*}_{a'}(\int \limits _{{\mathbb R}^{n}} |K_{(1+a^{2})b^{-2}m(.)^{2},k,\alpha}(x,y)||u(y)|dy)\|_{1} \lesssim \|u\|_{1}.
$$
Using Lemma \ref{lem:est}, the positivity of $(e^{tL})_{t\geq 0}$, and the fact that $e^{L}1=1$, this further reduces to proving
$$
\|T^{*}_{a'}(\int \limits _{{\mathbb R}^{n}} M_{(1+a^{2})b^{-2}m(.)^{2}}(x,y)|u(y)|dy)\|_{1} \lesssim \|u\|_{1}.
$$
We first use Proposition \ref{prop:glob} to obtain
$$
\|T^{*}_{glob,a',1}(\int \limits _{{\mathbb R}^{n}} M_{(1+a^{2})b^{-2}m(.)^{2}}(x,y)|u(y)|dy)\|_{1} \lesssim \int \limits _{{\mathbb R}^{n}} \int \limits _{{\mathbb R}^{n}} M_{(1+a^{2})b^{-2}m(x)^{2}}(x,y)|u(y)|dy d\gamma(x).
$$
We decompose the right hand side into a local and a global part. Let $\tau:= \frac{1}{2}(1+b^{-1}\sqrt{1+a^{2}})(1+2b^{-1}\sqrt{1+a^{2}})$
and $\overline{\tau}=2(1+\sqrt{1+a^{2}}b^{-1})\tau+\sqrt{1+a^{2}}b^{-1}$. For $x,y,z \in {\mathbb R}^{n}$ such that $|x-y| \geq \overline{\tau} m(x)$ and
$|z-x| \leq \frac{\sqrt{1+a^{2}}}{b} m(x)$, we have that $|z-y|\geq \tau m(z)$. Therefore
$$
\int \limits _{{\mathbb R}^{n}} \int \limits _{{\mathbb R}^{n}} M_{(1+a^{2})b^{-2}m(x)^{2}}(x,y)1_{N_{\overline{\tau}}^{c}}(x,y)|u(y)|dy d\gamma(x)
\lesssim \int \limits _{{\mathbb R}^{n}} \underset{(z,t) \in \Gamma ^{b^{-1}\sqrt{1+a^{2}}} _{x}(\gamma)}{\sup}\int \limits _{{\mathbb R}^{n}} M_{t^{2}}(z,y)1_{N_{\tau}^{c}}(z,y)|u(y)|dy d\gamma(x)
\lesssim \|u\|_{1},
$$
by Proposition \ref{prop:glob}.
Now, for $(x,y) \in N_{\overline{\tau}}$, we have $m(x)\sim m(y)$ by Lemma \ref{lem:mnp1}. Therefore
$$
\int \limits _{{\mathbb R}^{n}} \int \limits _{{\mathbb R}^{n}} M_{(1+a^{2})b^{-2}m(x)^{2}}(x,y)1_{N_{\overline{\tau}}}(x,y)|u(y)|dy d\gamma(x)
\lesssim \int \limits _{{\mathbb R}^{n}} m(x)^{-n}\int \limits _{B(x,\overline{\tau} m(x))}|u(y)|dy d\gamma(x).
$$
For $(x,y) \in N_{\overline{\tau}}$, we also have $e^{-|x|^{2}}\sim e^{-|y|^{2}}$, therefore
$$
\int \limits _{{\mathbb R}^{n}} m(x)^{-n}\int \limits _{B(x,\overline{\tau} m(x))}|u(y)|dy d\gamma(x)
\lesssim \int \limits _{{\mathbb R}^{n}} |u(y)|m(y)^{-n}\int \limits _{B(y,\overline{\tau}(1+\overline{\tau})m(y))}d\gamma(x) dy
\lesssim \int \limits _{{\mathbb R}^{n}} |u(y)| e^{-|y|^{2}}dy \lesssim \|u\|_{1}.
$$
The proof will be completed once we have estimated the two following terms.
\begin{equation*}
\begin{split}
J_{glob}:=& \int \limits \underset{(y,t)\in \Gamma_{x} ^{a}}{\sup} \int \limits _{{\mathbb R}^{n}} M_{t^{2}}(y,z) 1_{N_{\tau'}}(y,z)
\int \limits _{{\mathbb R}^{n}} M_{(1+a^{2})b^{-2}m(z)^{2}}(z,w)1_{N_{\tau''}^{c}}(z,w)|u(w)|dwdzd\gamma(x),\\
J_{loc}:=& \int \limits \underset{(y,t)\in \Gamma_{x} ^{a}}{\sup} \int \limits _{{\mathbb R}^{n}} M_{t^{2}}(y,z) 1_{N_{\tau'}}(y,z)
\int \limits _{{\mathbb R}^{n}} M_{(1+a^{2})b^{-2}m(z)^{2}}(z,w)1_{N_{\tau''}}(z,w)|u(w)|dwdzd\gamma(x),
\end{split}
\end{equation*}
where $\tau'$ is defined in Proposition \ref{prop:glob} for the parameters $(1,a')$, and $\tau''$ is defined as follows.
For $(x,y) \in N_{a}$ and $(y,z) \in N_{\tau'}$, we have $m(x)\sim m(y) \sim m(z)$ by Lemma \ref{lem:mnp1}.
Let $\lambda>0$ be such that $\lambda^{-1} m(x) \leq m(z) \leq \lambda m(x)$, and fix
$\tau''$ as in Proposition \ref{prop:glob}, for the parameters $(\tilde{A},\tilde{a}) = ((2\tau'(1+a)+a)b/(\lambda\sqrt{1+a^{2}}),\sqrt{1+a^{2}}b^{-1}\lambda)$.
Using Proposition \ref{prop:glob}, the positivity of $(e^{tL})_{t\geq 0}$, and the fact that $e^{L}1=1$, we have that
\begin{equation*}
\begin{split}
J_{glob} &\lesssim \int \limits \underset{(y,t)\in \Gamma_{x} ^{a}}{\sup} \int \limits _{{\mathbb R}^{n}} M_{t^{2}}(y,z) 1_{N_{\tau'}}(y,z)
\underset{(\eta,s)\in \Gamma_{x} ^{(\tilde{A},\tilde{a})}(\gamma)}{\sup}\int \limits _{{\mathbb R}^{n}} M_{s^{2}}(\eta,w)1_{N_{\tau''}^{c}}(\eta,w)|u(w)|dwdzd\gamma(x)\\
&\lesssim \int \limits \underset{(\eta,s)\in \Gamma_{x} ^{(\tilde{A},\tilde{a})}(\gamma)}{\sup}\int \limits _{{\mathbb R}^{n}} M_{s^{2}}(\eta,w)1_{N_{\tau''}^{c}}(\eta,w)|u(w)|dwd\gamma(x)\\
&\lesssim \|u\|_{1},
\end{split}
\end{equation*}
Finally, for $(x,y) \in N_{a}$, $(y,z) \in N_{\tau'}$, and $(z,w) \in N_{\tau''}$, we have $m(x)\sim m(y) \sim m(z) \sim m(w)$,
$|w-x| \leq \lambda m(x)$
for some numerical constant $\lambda>0$ by Lemma \ref{lem:mnp1}, and $e^{-|w|^{2}}\sim e^{-|x|^{2}}$. Let $C>0$ be such that $m(x) \leq Cm(w)$.
Using the positivity of $(e^{tL})_{t\geq 0}$, and the fact that $e^{L}1=1$, we have that
\begin{equation*}
\begin{split}
J_{loc}\lesssim& \int \limits \underset{(y,t)\in \Gamma_{x} ^{a}}{\sup} \int \limits _{{\mathbb R}^{n}} M_{t^{2}}(y,z) 1_{N_{\tau'}}(y,z)
m(x)^{-n}\int \limits _{B(x,\lambda m(x))} |u(w)|dwdzd\gamma(x) \\
&\lesssim \int \limits m(x)^{-n}\int \limits _{B(x,\lambda m(x))} |u(w)|dwd\gamma(x)
\lesssim \int \limits |u(w)| m(w)^{-n}\int \limits _{B(w,C\lambda m(w))}d\gamma(x)dw\\
&\lesssim \int \limits |u(w)| e^{-|w|^{2}}dw \lesssim \|u\|_{1}.
\end{split}
\end{equation*}
\end{proof}
\section{Riesz transforms}
\label{sect:riesz}
In this section, we prove the following boundedness result for the Riesz transforms associated with $L$.
Let $M: L^{2}({\mathbb R}^{n},d\gamma) \to L^{2}({\mathbb R}^{n},d\gamma)$ be defined by $MH_{\alpha} = |\alpha|^{-\frac{1}{2}}H_{\alpha}$
for all $\alpha \in {\mathbb Z}^{n}_{+} \backslash \{0\}$, and $MH_{0} = 0$.
\begin{theorem}
\label{thm:riesz}
For all $k=1,..,n$, the Riesz transforms
$$
R_{k} = \partial_{x_{k}} M, \quad S_{k} = \partial^{*} _{x_{k}}M,
$$
extend to bounded operators from $h^{1}(\gamma)$ to $L^{1}(\gamma)$.
\end{theorem}
Recall that $h^{1}(\gamma):=h^{1}_{\text{quad},2}(\gamma)$.
The proof of this theorem follows the approach of the preceding sections. We start with an appropriate Calder\'on reproducing formula, which can be established through chaos expansion.
\begin{lemma}
For all $N \in {\mathbb N}$, $k \in \{1,...,n\}$, and $a,\alpha>0$, there exists $C>0$ such that for all $u \in L^{2}(\gamma)$
\begin{equation*}
\begin{split}
u &= C \int \limits _{0} ^{\infty} (t^{2}L)^{N+\frac{3}{2}}e^{\frac{5t^{2}}{\alpha}L}u \frac{dt}{t},\\
R_{k}u &= C \int \limits _{0} ^{\infty} t\partial_{x_{k}}(t^{2}L)^{N+1}e^{\frac{5t^{2}}{\alpha}L}u \frac{dt}{t},
\quad S_{k}u = C \int \limits _{0} ^{\infty} t\partial_{x_{k}}^{*}(t^{2}L)^{N+1}e^{\frac{5t^{2}}{\alpha}L}u \frac{dt}{t}.
\end{split}
\end{equation*}
\end{lemma}
In what follows, $k \in \{1,...,n\}$ is fixed. With the same proof as Corollary \ref{cor:dec}, we get the following.
\begin{corollary}
For all $N \in {\mathbb N}$, $b>0$, and $\alpha>4$, there exists $C>0$ and $n$ sequences of atoms $(F_{m,j})_{m\in {\mathbb N}}$ and complex numbers
$(\lambda_{m,j})_{m\in{\mathbb N}}$ for $j=1,...n$, such that for all
$u \in C_{c}^{\infty}({\mathbb R}^{n})$ and $x\in {\mathbb R}^{n}$:
\begin{equation*}
\begin{split}
-R_{k}u(x) =
& C \sum \limits _{j=1} ^{n} \sum \limits _{m=1} ^{\infty} \lambda_{m,j} \int \limits _{0} ^{\frac{m(x)}{b}}
t\partial_{k}(t^{2}L)^{N}e^{\frac{t^{2}}{\alpha}L}t\partial_{x_{j}}^{*}F_{m,j}(t,x)\frac{dt}{t}
\\&+C \sum \limits _{j=1} ^{n} \int \limits _{0} ^{\frac{m(x)}{b}} t\partial_{k}(t^{2}L)^{N}e^{\frac{t^{2}}{\alpha}L}t\partial_{x_{j}}^{*}(1_{D^{c}}(t,.)t\partial_{x_{j}}e^{\frac{4t^{2}}{\alpha}L})u(x)\frac{dt}{t}+ C \int \limits _{\frac{m(x)}{b}} ^{\infty} t\partial_{k}(t^{2}L)^{N+1}e^{\frac{5t^{2}}{\alpha}L}u(x) \frac{dt}{t},
\end{split}
\end{equation*}
and $ \sum \limits _{j=1} ^{n} \sum \limits _{m=1} ^{\infty} |\lambda_{m,j}| \lesssim \|u\|_{h^{1}_{\text{quad},2}}$.
\end{corollary}
The same result holds for $S_{k}u$ (replacing $\partial_{x_{k}}$ by its adjoint).
Theorem \ref{thm:riesz} will be proven, once we have obtained the following three estimates (and their analogues for $\partial_{x_{k}}^{*}$ instead of $\partial_{x_{k}}$).
$$\|\int \limits _{0} ^{\frac{m(.)}{b}}
t\partial_{k}(t^{2}L)^{N}e^{\frac{t^{2}}{\alpha}L}t\partial_{x_{j}}^{*}F(t,.)\frac{dt}{t}\|_{L^{1}(\gamma)}
\lesssim 1,$$
for all $t^{1,2}(\gamma)$ atoms $F$.
$$
\|\int \limits _{0} ^{\frac{m(.)}{b}} t\partial_{k}(t^{2}L)^{N}e^{\frac{t^{2}}{\alpha}L}t\partial_{x_{j}}^{*}(1_{D^{c}}(t,.)t\partial_{x_{j}}e^{\frac{4t^{2}}{\alpha}L})u\frac{dt}{t}\|_{L^{1}(\gamma)} \lesssim \|u\|_{L^{1}(\gamma)}.
$$
$$
\|\int \limits _{\frac{m(.)}{b}} ^{\infty} t\partial_{k}(t^{2}L)^{N+1}e^{\frac{5t^{2}}{\alpha}L}u\frac{dt}{t}
\|_{L^{1}(\gamma)} \lesssim \|u\|_{L^{1}(\gamma)}.
$$
We start with the relevant kernel estimate.
\begin{lemma}
\label{lem:RTest}
Let $N \in {\mathbb Z}_{+}$, $j \in \{1,...,n\}$, and $\alpha \geq 4e^{8}$. Let $x,y \in {\mathbb R}^{n}$ and $t\in (0,a]$.
If $t \lesssim m(y)$ then $|t\partial_{x_{k}}\tilde{K}_{t^{2},N,\alpha,j}(x,y)|\lesssim (1+t|x|)
\exp(-\frac{\alpha}{4e^{8}}\frac{|e^{-t^{2}}y-x|^{2}}{1-e^{-2t^{2}}})M_{t^{2}}(x,y)$.
\end{lemma}
\begin{proof}
As in Corollary \ref{cor:Ktilde}, there exists $C_{N}\in {\mathbb N}$ and two polynomials of $2n$ variables $Q_{N},\tilde{Q}_{N}$ of degree $C_{N}$ such that for all $x,y \in {\mathbb R}^{n}$ and $t>0$:
\begin{equation*}
\begin{split}
t\partial _{x_{k}}&\tilde{K}_{t^{2},N,\alpha,j}(x,y) = \\
&t^{2N+2}(1-e^{-\frac{2t^{2}}{\alpha}})^{-(N+1)}\tilde{Q}_{N}((\frac{e^{-\frac{t^{2}}{\alpha}}y_{j}-x_{j}}{\sqrt{1-e^{-\frac{2t^{2}}{\alpha}}}})_{j=1,...,n} , (\sqrt{1-e^{-\frac{2t^{2}}{\alpha}}}y_{j})_{j=1,...,n})M_{\frac{t^{2}}{\alpha}}(y,x)
exp(|x|^{2}-|y|^{2})\\
+&t^{2N+2}x_{k}(1-e^{-\frac{2t^{2}}{\alpha}})^{-(N+\frac{1}{2})}Q_{N}((\frac{e^{-\frac{t^{2}}{\alpha}}y_{j}-x_{j}}{\sqrt{1-e^{-\frac{2t^{2}}{\alpha}}}})_{j=1,...,n} , (\sqrt{1-e^{-\frac{2t^{2}}{\alpha}}}y_{j})_{j=1,...,n})M_{\frac{t^{2}}{\alpha}}(y,x)
exp(|x|^{2}-|y|^{2}).
\end{split}
\end{equation*}
Therefore
$$
|t\partial _{x_{k}}\tilde{K}_{t^{2},N,\alpha,j}(x,y)| \lesssim (1+t|x|)exp(-\frac{1}{2}
\frac{|e^{-\frac{t^{2}}{\alpha}}y-x|^{2}}{1-e^{-\frac{2t^{2}}{\alpha}}})exp(|x|^{2}-|y|^{2}).
$$
Using Lemma \ref{lem:slow2}, and the fact that $t \lesssim m(y)$, we have that
\begin{equation*}
\begin{split}
|t\partial _{x_{k}}\tilde{K}_{t^{2},N,\alpha,j}(x,y)| &\lesssim (1+t|x|) \exp(-\frac{\alpha}{4e^{8}}\frac{|e^{-t^{2}}y-x|^{2}}{1-e^{-2t^{2}}})M_{t^{2}}(y,x)exp(|x|^{2}-|y|^{2})\\&= (1+t|x|)
\exp(-\frac{\alpha}{4e^{8}}\frac{|e^{-t^{2}}y-x|^{2}}{1-e^{-2t^{2}}})M_{t^{2}}(x,y).
\end{split}
\end{equation*}
\end{proof}
\begin{proposition}
Let $N \in {\mathbb N}$, $j\in \{1,...,n\}$ and $\alpha>2^{32}$. Let $B=B(c_{B},r_{B})\in \mathcal{B}_{2}$ and $F$ be a $t^{1,2}(\gamma)$ atom $F$ associated with $B$.
\begin{enumerate}[(i)]
\item
$
\|\int \limits _{0} ^{r_{B}} |t\partial_{x_{k}}(t^{2}L)^{N}e^{\frac{t^{2}}{\alpha}L}t\partial_{x_{j}}^{*}F(t,.)|\frac{dt}{t}\|_{L^{1}(\gamma)} \lesssim 1,
$
\item
$
\|\int \limits _{0} ^{r_{B}} |t\partial_{x_{k}}^{*}(t^{2}L)^{N}e^{\frac{t^{2}}{\alpha}L}t\partial_{x_{j}}^{*}F(t,.)|\frac{dt}{t}\|_{L^{1}(\gamma)} \lesssim 1.
$
\end{enumerate}
\end{proposition}
\begin{proof}
For $l \in {\mathbb Z}_{+}$, we have, using Lemma \ref{lem:mm}:
\begin{equation*}
\begin{split}
\|1_{C_{l}(B)}\int \limits _{0} ^{r_{B}}|t\partial_{x_{k}}(t^{2}L)^{N}e^{\frac{t^{2}}{\alpha}L}t\partial_{x_{j}}^{*}F(t,.)|\frac{dt}{t}\|_{L^{1}(\gamma)} &\lesssim \sqrt{\gamma(2^{l+1}B)}\|1_{C_{l}(B)}\int \limits _{0} ^{r_{B}}|t\partial_{x_{k}}(t^{2}L)^{N}e^{\frac{t^{2}}{\alpha}L}t\partial_{x_{j}}^{*}F(t,.)|\frac{dt}{t}\|_{L^{2}(\gamma)}\\
&\lesssim 2^{2^{9}.4^{l}}\sqrt{\gamma(B)}\|1_{C_{l}(B)}\int \limits _{0} ^{r_{B}}|t\partial_{x_{k}}(t^{2}L)^{N}e^{\frac{t^{2}}{\alpha}L}t\partial_{x_{j}}^{*}F(t,.)|\frac{dt}{t}\|_{L^{2}(\gamma)}.
\end{split}
\end{equation*}
For $l=0$, we use the $L^2$ boundedness of $R_{j}$, and duality.
\begin{equation*}
\begin{split}
\|1_{C_{0}(B)}\int \limits _{0} ^{r_{B}}|t\partial_{x_{k}}(t^{2}L)^{N}e^{\frac{t^{2}}{\alpha}L}t\partial_{x_{j}}^{*}F(t,.)|\frac{dt}{t}\|_{L^{2}(\gamma)}
&\lesssim (\int \limits _{0} ^{r_{B}} \int \limits _{B} |F(t,x)|^{2} d\gamma(x)\frac{dt}{t})^{\frac{1}{2}}
\underset{\|g\|_{2}\leq 1}{\sup} (\int \limits _{0} ^{r_{B}}
\|(t^{2}L)^{N+1}e^{\frac{t^{2}}{\alpha}L}R_{k}^{*}g\|_{L^{2}(\gamma)}^{2} \frac{dt}{t} )^{\frac{1}{2}} \\
&\lesssim \gamma(B)^{-\frac{1}{2}} \underset{\|g\|_{2}\leq 1}{\sup} \|R_{k}^{*}g\|_{L^{2}(\gamma)} \lesssim \gamma(B)^{-\frac{1}{2}},
\end{split}
\end{equation*}
where we have used chaos decomposition (or the $L^{2}$ functional calculus of $L$) as in the proof of Proposition \ref{prop:mol}.
For $l>0$, we use off-diagonal estimates, obtained from Lemma \ref{lem:RTest} as in Lemma \ref{lem:od}, and the fact that $|r_{B}x| \lesssim r_{B}|x-c_{B}|+1 \lesssim 2^{l}$ for all $x \in C_{l}(B)$.
\begin{equation*}
\begin{split}
\|1_{C_{l}(B)}\int \limits _{0} ^{r_{B}}|t\partial_{x_{k}}(t^{2}L)^{N}e^{\frac{t^{2}}{\alpha}L}t\partial_{x_{j}}^{*}F(t,.)|\frac{dt}{t}\|_{L^{2}(\gamma)}
&\lesssim 2^{l}\int \limits _{0} ^{r_{B}} exp(-\frac{\alpha}{2^{6}e^{8}}4^{l}(\frac{r_{B}}{t})^{2})
\|F(t,.)\|_{L^{2}(\gamma)} \frac{dt}{t}\\
& \lesssim 2^{l} exp(-\frac{\alpha}{2^{23}}4^{l})
(\int \limits _{0} ^{1} exp(-\frac{\alpha}{2^{22}}4^{l}(\frac{1}{t})^{2})
\frac{dt}{t})^{\frac{1}{2}} \gamma(B)^{-\frac{1}{2}}\\
& \lesssim 2^{l} exp(-\frac{\alpha}{2^{23}}4^{l})\gamma(B)^{-\frac{1}{2}}.
\end{split}
\end{equation*}
Summing in $l$ gives (i).\\
The same argument also gives
$
\|x \mapsto \int \limits _{0} ^{r_{B}} |tx(t^{2}L)^{N}e^{\frac{t^{2}}{\alpha}L}t\partial_{x_{j}}^{*}F(t,.)|\frac{dt}{t}\|_{L^{1}(\gamma)} \lesssim 1,
$ and thus (ii).
\end{proof}
We now turn to the remainder terms. With exactly the same proof as Proposition \ref{prop:Dcomp} ,we get the following.
\begin{proposition}
Let $N \in {\mathbb Z}_{+}$, $j \in \{1,...,n\}$ and $\alpha > \max(32e^{4},8e^{8})$. Let $b \geq
\max(2e,\sqrt{\frac{32e^{4}}{(\alpha-32e^{4})(1-e^{-\frac{8}{\alpha}})}})$.
Then
\begin{enumerate}[(i)]
\item
$
\|\int \limits _{0} ^{\frac{m(.)}{b}} t\partial_{x_{k}}t^{2N+1}L^{N}e^{\frac{t^{2}}{\alpha}L}\partial_{x_{j}}^{*}(1_{D^{c}}(t,.)t\partial_{x_{j}}e^{\frac{4t^{2}}{\alpha}L})u \frac{dt}{t}\|_{L^{1}(\gamma)} \lesssim \|u\|_{L^{1}(\gamma)}. $
\item
$
\|\int \limits _{0} ^{\frac{m(.)}{b}} t\partial_{x_{k}}^{*}t^{2N+1}L^{N}e^{\frac{t^{2}}{\alpha}L}\partial_{x_{j}}^{*}(1_{D^{c}}(t,.)t\partial_{x_{j}}e^{\frac{4t^{2}}{\alpha}L})u \frac{dt}{t}\|_{L^{1}(\gamma)} \lesssim \|u\|_{L^{1}(\gamma)}. $
\end{enumerate}
\end{proposition}
The final estimate is obtained as in Proposition \ref{prop:jinf}.
\begin{proposition}
Let $N \in {\mathbb Z}_{+}$, $b>0$, and $\alpha>4e^{8}$. For all $u\in C_{c}^{\infty}({\mathbb R}^{n})$, we have
\begin{enumerate}[(i)]
\item
$
\| \int\limits _{\frac{m(.)}{b}} ^{\infty} t\partial_{x_{k}}(t^{2}L)^{N+1}e^{\frac{5t^{2}}{\alpha}L}u \frac{dt}{t}\|_{L^{1}(\gamma)} \lesssim \|u\|_{L^{1}(\gamma)}.
$
\item
$
\| \int\limits _{\frac{m(.)}{b}} ^{\infty} t\partial_{x_{k}}^{*}(t^{2}L)^{N+1}e^{\frac{5t^{2}}{\alpha}L}u \frac{dt}{t}\|_{L^{1}(\gamma)} \lesssim \|u\|_{L^{1}(\gamma)}.
$
\end{enumerate}
\end{proposition}
\begin{proof}
Let $M>0$ and $x \in {\mathbb R}^{n}$. Using Corollary \ref{cor:Ktilde} and Lemma \ref{lem:slow2}, we have that
\begin{equation*}
\begin{split}
|\int\limits _{\frac{m(x)}{b}} ^{M} & t\partial_{x_{k}}(t^{2}L)^{N+1}e^{\frac{5t^{2}}{\alpha}L}u \frac{dt}{t}|
\lesssim
|\int\limits _{\frac{5m(x)^{2}}{b^{2}\alpha}} ^{\frac{5M^{2}}{\alpha}} s^{N+\frac{1}{2}}
\int \limits _{{\mathbb R}^{n}} \partial_{x_{k}}\partial_{s}^{N+1}M_{s}(x,y)u(y)dyds| \\
& \lesssim
\sum \limits _{l=0} ^{N}
\int \limits _{{\mathbb R}^{n}} Q_{l}(1,
(\frac{e^{-\frac{5m(x)^{2}}{b^{2}\alpha}}x_{j}-y_{j}}{\sqrt{1-e^{-2\frac{5m(x)^{2}}{b^{2}\alpha}}}})_{j=1,...,n} , (\sqrt{1-e^{-2\frac{5m(x)^{2}}{b^{2}\alpha}}}x_{j})_{j=1,...,n})
M_{\frac{5m(x)^{2}}{b^{2}\alpha}}(x,y)u(y)|dy \\
& \qquad + \sum \limits _{l=0} ^{N} |M^{2l+1}\partial_{x_{k}}L^{l}e^{\frac{5}{\alpha}M^{2}L}u(x)|\\
& \lesssim
\int \limits _{{\mathbb R}^{n}} exp(-\frac{\alpha}{4e^{8}}
\frac{|e^{-\frac{5m(x)^{2}}{b^{2}}}x-y|^{2}}{1-e^{-2\frac{5m(x)^{2}}{b^{2}}}})
|u(y)|dy
+\sum \limits _{l=0} ^{N} |M^{2l+1}\partial_{x_{k}}L^{l}e^{\frac{5}{\alpha}M^{2}L}u(x)|\\
&\lesssim \int \limits _{{\mathbb R}^{n}} M_{\frac{5m(x)^{2}}{b^{2}}}(x,y)|u(y)|dy
+\sum \limits _{l=0} ^{N} |M^{2l+1}\partial_{x_{k}}L^{l}e^{\frac{5}{\alpha}M^{2}L}u(x)|.
\end{split}
\end{equation*}
Using chaos decomposition, this gives
$$
\| \int\limits _{\frac{m(.)}{b}} ^{M} t\partial_{x_{k}}(t^{2}L)^{N+1}e^{\frac{5t^{2}}{\alpha}L}u \frac{dt}{t}\|_{L^{1}(\gamma)} \lesssim
\int \limits_{{\mathbb R}^{n}} \int \limits _{{\mathbb R}^{n}} M_{\frac{5m(x)^{2}}{b^{2}}}(x,y)|u(y)|dy d\gamma(x)
+\sum \limits _{l=0} ^{N} M^{2l+1}e^{-\frac{5}{\alpha}M^{2}} \|u\|_{L^{2}(\gamma)},
$$
and thus, letting $M$ go to infinty
$$
\| \int\limits _{\frac{m(.)}{b}} ^{\infty} t\partial_{x_{k}}(t^{2}L)^{N+1}e^{\frac{5t^{2}}{\alpha}L}u \frac{dt}{t}\|_{L^{1}(\gamma)} \lesssim
\int \limits_{{\mathbb R}^{n}} \int \limits _{{\mathbb R}^{n}} M_{\frac{5m(x)^{2}}{b^{2}}}(x,y)|u(y)|dy d\gamma(x).
$$
The proof of \ref{prop:jinf} gives
$$
\int \limits_{{\mathbb R}^{n}} \int \limits _{{\mathbb R}^{n}} M_{\frac{5m(x)^{2}}{b^{2}}}(x,y)|u(y)|dy d\gamma(x)
\lesssim \|u\|_{L^{1}(\gamma)},
$$
which concludes the proof of (i). The same proof also gives (ii), using that $|xm(x)| \leq 1$ for all $x \in {\mathbb R}^{n}$.
\end{proof}
|
{
"timestamp": "2012-05-31T02:02:09",
"yymm": "1203",
"arxiv_id": "1203.1998",
"language": "en",
"url": "https://arxiv.org/abs/1203.1998"
}
|
\section{Introduction}
``Alternate quantization'' was first studied by Breitenlohner and
Freedman\cite{Freedman1} in the context of compactifications of
supergravity theories to anti-de Sitter space. In the wake of
developments in AdS/CFT correspondence, there has been a renewed
interest in it. Klebanov and Witten\cite{Witten1} first discussed it
in AdS/CFT, providing interesting boundary conformal field theories
generated by the alternate scheme\cite{Witten3,Amit1}. In a certain
window of the conformal dimensions of the boundary composite operators
corresponding the bulk excitations in $AdS_{d+1}$, there are two
possible quantization schemes for their boundary $CFT$s.
For massive scalar fields in $AdS_{d+1}$, in the window of their
masses as $ -\frac{d^2}{4}\leq m^2 \leq -\frac{d^2}{4}+1$, there are
two possible quantization schemes, so two possible boundary conformal
field theories, which are called $\Delta_+$-theory and
$\Delta_-$-theory, where
$\Delta_{\pm}=\frac{d}{2}\pm\sqrt{\frac{d^2}{4}+m^2}$, which is
conformal dimension of the boundary operator in each theory
\cite{Witten1}. In this window, both $CFT$s are above the
unitarity bound, $\Delta_\pm \geq \frac{d}{2}-1$, so their two point
correlators are positive definite in the position space. In
$\Delta_+$-theory, near $AdS$ boundary expansion of the bulk scalar
fields is $\phi(r,x^\mu)|_{r \rightarrow
0}=\phi_{0}(x^\mu)r^{d-\Delta_{+}}+A(x^\mu)r^{\Delta_+}$. It is
well known that $A(x^\mu)$ corresponds to certain composite operator
in the $CFT$ and $\phi_{0}(x^\mu)$ corresponds to the external source
coupled to it. In fact, $\Delta_+$-theory is not independent from
$\Delta_-$-theory. They are related to each other by Legendre
transform. This Legendre transform switches the role of $A(x^\mu)$
and $\phi_{0}(x^\mu)$ in $\Delta_-$-theory, because they are
canonical conjugates of each other.
In the dual gravity theory, two possible boundary $CFT$s can be
obtained by imposing different boundary conditions. Boundary
conditions for bulk fields and corresponding boundary terms in the
AdS/CFT context has been studied by various authors in the past. For
Dirac fields it was studied by Henneaux \cite{Henneaux:1998ch},
whereas for Rarita-Schwinger fields it was analysed in
\cite{Rashkov:1999ji}. This issue in case of inequivalent
quantization was addressed by \cite{Witten3,Mueck:1999kk} and in
Lorentzian AdS/CFT case was dealt with in \cite{Marolf:2004fy}. For
$\Delta_+$-theory, the corresponding boundary condition is the
Dirichlet boundary condition as $\delta \phi_0(x^\mu)=0$. Dirichlet
boundary condition is the usual boundary condition in AdS/CFT context.
Since the $\Delta_+$-theory is unitary even when $m^2 \geq
-\frac{d^2}{4}+1$, Dirichlet boundary condition is always a possible
boundary condition. The $\Delta_-$-theory can be obtained by imposing
Neumann boundary condition, $\delta A(x^\mu)=0$. This Neumann boundary
condition is obtained by adding boundary term $S_{bdy}\sim\int
\phi(x^\mu)A(x^\mu) d^dx$ at $r=0$. Adding such a term in turn
generates the same effect as performing the Legendre transform of the
$\Delta_+$-theory, therefore such a boundary term takes the
$\Delta_+$-theory to the $\Delta_-$-theory.
The Neumann boundary condition can be generalized by deforming the
boundary $CFT$ by adding a general form of $S_{bdy}$. Such a boundary
action can be an arbitrary function of $\phi_0(x^\mu)$ and $A(x^\mu)$.
By adding the boundary action, one can obtain the ``on-shell action''
$I_{os}=S_{bulk}+S_{bdy}$, where $S_{bulk}$ is the boundary
contributions from the bulk action. The boundary condition is
obtained by performing functional variation of the on-shell action and
setting it to zero, $\delta I_{os}=0$. This corresponds to saddle
point of the on-shell action, which is the classical vacuum of the
boundary theory.
An interesting example with general deformations is the conformally
coupled scalar field theory in $AdS_4$\cite{Sebastian2}. In
\cite{Sebastian2} authors consider a massless scalar field theory with
$\lambda\phi^4$ and $\frac{1}{6}R\phi^2$ interactions, where $\phi$ is
the scalar field, $\lambda$ is the quartic self-coupling of the scalar
field and $R$ is curvature scalar of $AdS_4$. The boundary theory
corresponding to the conformally coupled bulk scalar field contains a
triple trace deformation term $S_{bdy} \sim \alpha \int d^3x
\phi^3_{0}(x^\mu)$, where $\alpha$ is a numerical parameter and
$\phi_{0}$ is the boundary value of the scalar field $\phi$. Under
the field redefinition $\phi_0=\varphi^2$ and truncation up to the
second order in small derivative expansion, the boundary on-shell
action takes the canonical form with $\varphi^6$ coupling\cite{Sebastian3}.
$\Delta_\pm$-theories for $U(1)$ vector fields in $AdS_4$ are
well-defined \cite{Witten2,Donald1,Sebastian1} in which $\Delta_+=2$
and $\Delta_-=1$. The unitarity bound for the vector like local
observables in $d$-dimensional $CFT$ is $\Delta \geq
d-1$\cite{Minwalla1}, so $\Delta_+=2$ theory is unitary when $d=3$.
It also would imply that the $\Delta_-=1$ theory does not satisfy the
unitarity bound. One way to interpret the dual operator with
conformal dimension `1' (the conjugate of source term from the bulk
action) is as the $U(1)$ vector gauge field in the boundary theory.
Clearly there is an ambiguity in defining this operator due to
non-invariance of it under gauge transformations but instead one
interprets it as an observable which is not local\cite{Leigh1}. The
field strength constructed out of this gauge fields is in fact a local
observable. It also resolves the apparent contradiction with the
unitarity bound because the field strength has conformal dimension 2,
which satisfies the unitarity bound. There are many possible boundary
deformations which provide interesting on-shell actions. In
\cite{Sebastian1}, the authors consider ``massive deformation'' from
which they derive the on-shell action to be the massive gauge field
action. In the case that self-duality condition together with massive
deformation, one obtain massive Chern Simons boundary on-shell
action\cite{Townsend1}.
In this paper, we extend the discussion to non-abelian gauge theory in
$AdS_4$. Before summarizing our main result, let us briefly explain
our motivations. The motivations are three folds. First, Dirichlet
and Neumann boundary conditions in abelian gauge field theory in the
bulk correspond to free $CFT$. The most natural way to introduce
interactions is to consider Yang-Mills theory. As will be shown, this
will give non-trivial momentum dependent interactions in the boundary
action for Dirichlet and Neumann boundary conditions. We will discuss
generalizations of other deformations which in the abelian context
were considered in \cite{Sebastian1}.
Second, it is well-known that abelian gauge field theory action in 4-dimension is manifestly invariant under electric-magnetic duality,
and it is also successfully embodied in $AdS$ space\cite{Sebastian1}.
We want to extend this duality to Yang-Mills. In \cite{Deser2}, it is reported that when one retains cubic order interactions only, one can implement
electric-magnetic duality in Yang-Mills system upto that order, even if that is not possible to construct with quartic interaction terms.
In fact, this symmetry is not manifest
electric-magnetic duality since it turns out that the variation of electric field is not proportional to magnetic field, but it is the most natural extension of
the abelian duality. We will discuss how this symmetry is embodied in $AdS$ space.
Finally, Yang-Mills theory on
$AdS_4$($U(1)$ gauge theory too) should be the same with that in the
half of 4-dimension flat space through Weyl scaling of the $AdS_4$
metric. Therefore, $SU(2)$ Yang-Mills instanton solution in $\mathbb R^4$ can
easily be adapted for $AdS_4$ as well. As $r \rightarrow 0$,
Yang-Mills instanton has non-trivial boundary value whereas near
Poincare horizon $r\to\infty$ it becomes a pure gauge
solution. Therefore, Yang-Mills instanton definitely changes the
boundary condition on the $AdS_4$ boundary and exploring implications
of these boundary condition is interesting.
In our study, we develop boundary deformations with certain reasonable
boundary conditions derived from perturbative and non-perturbative
bulk solutions. We briefly list the results here. For perturbative
approach, we solve bulk Yang-Mills equations of motion in power
expansion order by order in small amplitudes of Yang-Mills fields. To
retain the leading interactions in Yang-Mills coupling $g$, we obtain
the bulk solutions up to first subleading order corrections in the
small amplitudes. Up to this order, we can account for cubic
interactions in the boundary on-shell actions(also its dual $CFT$
actions).
In the Dirichlet boundary condition case(in which case, the boundary source
becomes $A^{a(0)}_i$, the boundary value of Yang-Mills fields), the
boundary on-shell action $I^{D}_{os}$ gives rise to the
propagator which is proportional to absolute value of the 3-momenta,
$|q|$, and exotic 3-momenta dependent cubic interactions as
\begin{equation}
\Delta_{ijk}^{D,abc}(q,l,p) \sim ig\epsilon^{abc}\delta^{3}(q+p+l)
\frac{(l-q)_k\delta_{ij}+(p-l)_i\delta_{jk}+(q-p)_j\delta_{ik}}{2(|q|+|p|+|l|)},
\end{equation}
where $\Delta^{D,abc}_{ijk}(q,l,p)$ are 3-point function on the
boundary $CFT$ $a$,$b$,$c$ are $SU(2)$ gauge indices and $q_i$, $l_i$
and $p_i$ are 3-momenta along the boundary directions with the
boundary spacetime indices $i,j,k=1,2,3$. For the Neumann boundary
condition(the source becomes $-A^{a(1)}_i$, which is the boundary
value of the canonical momentum of Yang-Mills fields), the
propagator is proportional to $\frac{1}{|q|}$ and the 3-point function
is given by
\begin{equation}
\Delta_{ijk}^{N,abc}(q,l,p) \sim \frac{ \Delta_{ijk}^{D,abc}(q,l,p)}{|q||p||l|} .
\end{equation}
The most interesting cases are massive and self-dual boundary
conditions. The massive boundary condition is written as
\begin{equation}
(\sqrt{-\nabla^2}-m)A^{a(0)}_i(x^i)=0,
\end{equation}
where $m$ is a mass dimension 1 constant. Once one applies this
boundary condition to the bulk fields, then the boundary theory
becomes massive gauge field theory. In case of the self-dual boundary
condition we do small $r$ expansion of the self-duality condition,
\begin{equation}
F^{a}_{MN}=\frac{1}{2}\epsilon_{MNPQ}F^a_{PQ},
\end{equation}
which is given by
\begin{equation}
A^{a(1)}_{i}=\frac{1}{2}\epsilon_{ijk}F^{a}_{jk}.
\end{equation}
Imposing these two different boundary conditions on the bulk fields,
we get non-abelian massive Chern Simons gauge theory on the
3-dimensional boundary.
We also explore the non-perturbative solutions from the bulk
theory. We consider Yang-Mills instanton with its winding number 1. It
turns out that one of the possible boundary conditions that we can
impose results in the boundary term consisting of the Chern Simon
action with some non-local deformations. In this non-local
deformation contains a line integration of the form
\begin{equation}
\sim e^{\int^z_0\epsilon^{ai}_{\ \ j} A^{a(0)}_i(\tilde z) d\tilde z^j}.
\end{equation}
This kind of non-local interactions never comes out from any
perturbative deformations, which can be the genuine properties from
the bulk instanton backgrounds.
This note is organized as follows. In Sec.\ref{Bulk Solutions}, the
bulk equations of motion of Yang-Mills fields are solved and its
perturbative solutions are obtained up to the first subleading order
corrections in the small amplitudes of Yang-Mills fields. In
Sec.\ref{Boundary Conditions and the Effective Action}, we perform
various boundary deformations of the on-shell actions and obtain some
interesting boundary actions. In Sec.\ref{em-duality}, we explore
implications of approximate electric-magnetic duality in the context
of $SU(2)$ Yang-Mills theory in $AdS_4$. In Sec.\ref{Yang-Mills
Instanton}, we explore Yang-Mills instanton solutions and their
boundary conditions. In the conclusion section we summarise our
results and discuss their implications. Some technical details
are presented in appendices.
\section{$SU(2)$ Yang-Mills on $AdS_4$ and its Solution}
\label{Bulk Solutions}\setcounter{equation}{0}
We start with the $SU(2)$ Yang-Mills(Euclidean) action in $AdS_4$
space-time background as
\begin{equation}
S[A]=\frac{1}{4}\int d^4x \sqrt{G} F^a_{MN}F^{aMN},
\end{equation}
where the space-time indices $M,N$ run from 1 to 4 and the gauge
indices $a$ does from 1 to 3. The background metric, $G$ is
\begin{equation}
ds^2=G_{MN}dx^Mdx^N=\frac{dr^2+\sum_{i=1}^{3}dx^i dx^i}{r^2},
\end{equation}
where we define the radial coordinate $r$ as $r \equiv x^4$, the
indices $i,j..$ are defined boundary space-time coordinate, which run
from 1 to 3. Yang-Mills field strength is given by
\begin{equation}
F^{a}_{MN}=\partial_{M} A^a_{N}-\partial_{N} A^a_{M}-g\epsilon^{abc}A^b_{M} A^c_N.
\end{equation}
One interesting feature of this system is that the Weyl rescaling of
background metric, $ds^2 \rightarrow r^2 ds^2$, maps the Yang-Mills
theory in $AdS_4$ to that defined in 4-dimensional flat space-time.
The space-time is a half of $\mathbb R^4$, because the radial
coordinate in $AdS$ space runs from $0$ to $\infty$. Therefore, the
Yang-Mills action becomes
\begin{equation}
\label{flat-space-action}
S[A]=\frac{1}{4}\int_{\mathbb R^4_{+}} d^4x F^a_{MN}F^{aMN},
\end{equation}
where the space-time indices are contracted with $\delta_{MN}$ and
$\mathbb R^4_{+}$ denotes a half of the 4-dimensional flat space.
In this section, we evaluate bulk equations of motion and obtain their
solutions with a power series expansion in small amplitude of
Yang-Mills fields(This expansion is effectively the same as the
Yang-Mills coupling $g$ expansion). We would solve Yang-Mills
equations up to the first sub-leading order corrections to take into
account effects of interaction terms in Yang-Mills action. Up to this
order, only cubic interactions terms participate. The equations of
motion are given by
\begin{equation}
\label{equation-of-motion}
0=\mathcal D_M F^{a}_{MN}=\partial_M F^a_{MN}+g\epsilon^{abc} F^{b}_{MN}A^{c}_{M},
\end{equation}
where $\mathcal D_{M}$ is the gauge covariant derivative. To evaluate
the perturbative equations of motion, we expand Yang-Mills field as
\begin{equation}
A^{a}_M=\varepsilon \bar A^a_M+\varepsilon^2 \tilde A^a_M+O(\varepsilon^3).
\end{equation}
where $\varepsilon$ is a book keeping parameter for the expansion,
which is a dimensionless small number. The equations of motion are
evaluated for each order in $\varepsilon$ as
\begin{eqnarray}
\label{first-order-EOM}
{\rm First\ Order\ :\ }0&=&\partial_{M}(\partial_{M}\bar{A}^a_N-
\partial_{N}\bar{A}^a_M), \\
\label{scond-order-EOM}
{\rm Second\ Order\ :\ }0&=&\partial_{M}(\partial_{M}\tilde{A}^a_N-
\partial_{N}\tilde{A}^a_M)-g\epsilon^{abc}\left( \partial_M(\bar A^b_M
\bar A^c_N) \right. \\ \nonumber
&&-\left. (\partial_M\bar{A}^b_N-\partial_N\bar A^b_M)\bar A^c_M \right),
\end{eqnarray}
and so on.
We start with the first order equations in $O(\varepsilon)$ are given by
\begin{eqnarray}
0&=&\nabla^2 \bar A^a_{r} -\partial_r \partial_i \bar A^a_i, \\
0&=&(\partial^2_r +\nabla^2)\bar A^a_i-\partial_{i}(\partial_r
\bar A^a_{r}+\partial_j \bar A^a_j),
\end{eqnarray}
where we split the indices $M,N..$ into $r$ and $i,j..$ and $\nabla^2
\equiv \sum_{j=1}^3 \partial_j \partial_j$. At this order, equations
of motion are identical to 3 copies of $U(1)$ gauge theory equations.
Solutions to these equations has already been obtained in
\cite{Sebastian1} (See Sec.2 and Appendix.B in it). We briefly list
the leading order solution $\bar A^a_M$, in momentum space as
\begin{eqnarray}
\label{first order solution}
\bar A^a_{i,q}(r)&=&\bar A^{aT}_{i,q}(r) -iq_i \bar\phi^a_q(r) {, \ \
} \bar A^{a}_{r,q}(r)=\partial_r \bar\phi^a_q(r){, \ \ } q_i
\bar A^{aT}_{i,q}(r)=0, \\ \nonumber
{\rm \ and \ } \bar A^{aT}_{i,q}(r)&=&\bar A^{aT(0)}_{i,q}\cosh(|q|r)
+\frac{1}{|q|}\bar A^{aT(1)}_{i,q}\sinh(|q|r),
\end{eqnarray}
where $q_i$ are three momenta along the boundary direction and
the solution is obtained by using Fourier transform of the
position space representation defined as
\begin{eqnarray}
\label{Fourier Transform}
\Phi^a_M (x,r)&=&\int^{\infty}_{-\infty} d^3 p
e^{-ip_{i}x_{i}}\Phi^a_{Mp}(r), \\ \nonumber
\Phi^a_{Mp}(r)&=&\frac{1}{(2\pi)^3}\int^{\infty}_{-\infty} d^3 x
e^{ip_{i}x_{i}}\Phi^a_M (x,r),
\end{eqnarray}
where $\Phi$ denotes any fields appearing in the bulk theory. $\bar
A^{aT}_{i,q}$ is the transverse part of the gauge field, which is given by
\begin{equation}
\bar A^{aT}_{i,q} = P_{ij}(q) \bar A^{a}_{j,q},
\end{equation}
where we define a projection operator,
\begin{equation}
P_{ij}(q)=\delta_{ij}-\frac{q_i q_j}{q^2}.
\end{equation}
If $\bar A^a_{i,q}(r)$ and $\bar A^a_{r,q}(r)$ are solutions then
\begin{equation}
\bar A^{a\prime}_{i,q}(r)=\bar A^a_{i,q}(r)-iq_i \bar\phi^a_q(r) {\rm
\ and \ } \bar A^{a\prime}_{r,q}(r)=\partial_r \bar\phi^a_q(r)
\end{equation}
also solve the equations of motion. $\bar\phi^a$ is a gauge freedom
which is not completely determined by equations of motion.
To proceed further we will use the radial gauge, namely $\bar
A^{a}_{r,q}(r)=0$. In the radial gauge, the residual gauge freedom is
obtained by restricting the gauge parameter $\bar\phi^a_{r,q}(r)$ to be
independent of $r$,
\begin{equation}
\label{the radial gauge}
\bar \phi^a_q(r) \rightarrow \bar\phi^a_q.
\end{equation}
Then by definition, $\bar A^{aT}_{i,q}(r)$ is gauge invariant under this
residual gauge transformation.
For the regularity of the solutions at the Poincare horizon, at
$r=\infty$, we require that
\begin{equation}
\label{regularity-condition}
\bar A^{aT(0)}_{ip}+\frac{1}{|p|}\bar A^{aT(1)}_{ip}=0.
\end{equation}
This removes the term proportional to $e^{|p|r}$ near the Poincare horizon.
Using this regularity condition we can write the solution in the following form
\begin{equation}
\label{first order bulk solution}
\bar A^{aT}_{i,p}(r)
=\bar A^{aT(0)}_{ip}e^{-|p|r}.
\end{equation}
After having obtained leading order solution, we will now solve the
equation second order in $\varepsilon$. The
precise procedure is given in Appendix.\ref{Evaluation of the Second
order Bulk Solution}, here we briefly discuss the equations and
their solutions. The equation for $N=r$ and $N=i$ from
eq.(\ref{scond-order-EOM}) become
\begin{eqnarray}
\label{N=r the second order Bulk equation}
0&=& \left( \nabla^2\tilde A^a_{r}-\partial_r \partial_j
\tilde A^a_j \right) -g\epsilon^{abc}\left( \partial_i(\bar A^b_i
\bar A^c_r)-\bar A^c_i
(\partial_i \bar A^b_r-\partial_r \bar A^b_i) \right), \\
\label{N=i the second order Bulk equation}
0&=&(\partial_r^2+\nabla^2)\tilde A^a_i -\partial_i
(\partial_r \tilde A^a_r+\partial_j \tilde A^a_j)-g\epsilon^{abc}
\left( \partial_r (\bar A^b_r \bar A^c_i)
+\partial_j (\bar A^b_j \bar A^c_i)\right. \\ \nonumber
&-&\left.(\partial_j \bar A^b_i - \partial_i \bar A^b_j)\bar A^c_j
-(\partial_r \bar A^b_i-\partial_i \bar A^b_r) \bar A^c_r \right),
\end{eqnarray}
respectively.
The first order solution in $\varepsilon$, $\bar A^a_{i,q}$ appear in
the form of source terms in the second order equations in
$\varepsilon$, (\ref{N=r the second order Bulk equation}) and
(\ref{N=i the second order Bulk equation}). Using this we divide up
the second order solutions into the homogeneous part $\tilde
A^{a}_{i,q(H)}$ and inhomogeneous part $\tilde A^{a}_{i,q(I)}$.
The homogeneous solution has the same form with $\bar A^a_M$ as in
eq.(\ref{first order solution}). This is because the homogeneous
equations are again linear and are identical to
eq.(\ref{first-order-EOM}). With the regularity condition in the
interior, we get $\tilde A^{a}_{i,p(H)}(r)=\tilde A^{a(0)}_{i,p(H) }e^{-|p|r}$ and by
imposing the radial gauge, the solution takes the form
\begin{equation}
\label{the second order homogeneous solution}
\tilde A^{a}_{i,p(H)}(r)=\tilde A^{aT(0)}_{ip(H)}e^{-|p|r}-ip_i \tilde \phi^a_{p(H)},
\end{equation}
where as argued previously, $\tilde A^{aT(0)}_{ip(H)}e^{-|p|r}$ is
gauge invariant part of the solutions and $\tilde \phi^a_{p(H)}$ is
gauge parameter, which also does not depend on $r$ like $\bar
\phi^a_{p}$ in the radial gauge.
With the projection operator, one can split the inhomogeneous part
$\tilde A^a_{i,q(I)}$ into two pieces as
\begin{equation}
\tilde A^{a}_{i,q(I)}=\tilde A^{aT}_{i,q(I)}+\tilde A^{aL}_{i,q(I)},
\end{equation}
where $\tilde A^{aT}_{i,q(I)}=P_{ij}(q)\tilde A^{a}_{j,q(I)}$ is the
transverse part of the gauge field and $\tilde
A^{aL}_{i,q(I)}=\frac{q_iq_j}{q^2}\tilde A^{a}_{j,q(I)}$ is the
longitudinal part of the inhomogeneous
solutions.
The equations are also separated into the longitudinal and the transverse part,
which are given by
\begin{equation}
\label{the final form of longitudinal}
-iq_j \partial_r \tilde A^{aL} _{j,q(I)}(r)
=g\epsilon^{abc}\int^{\infty}_{-\infty} d^3 p |q-p| \left(
\bar A^{cT}_{j,p}(r)\bar A^{bT}_{j,{q-p}}(r)
-iq_j \bar \phi^c_p \partial_r A^{bT}_{j,q-p}(r)\right),
\end{equation}
\begin{eqnarray}
\label{the final form of TRAnsverse}
(\partial^2_r-q^2)\tilde A^{aT} _{i,q(I)}&=&
g \epsilon^{abc}\int^{\infty}_{-\infty}d^3p
\bar A^{cT}_{k,p}(r) \alpha_{ijk}(p,q) \bar A^{bT}_{j,q-p}(r)
\\ \nonumber
&+&g\epsilon^{abc}P_{ij}(q)\int^\infty_{-\infty}d^3 p ((q-p)^2-q^2)
\bar \phi^b_p \bar A^{cT}_{j,q-p}(r)\\ \nonumber
&-&\frac{i}{2}g\epsilon^{abc}q^2P_{ij}(q)\int^\infty_{-\infty}d^3 p
p_j\bar \phi^b_p \bar \phi^c_{q-p}
\end{eqnarray}
in the radial gauge,
where
\begin{equation}
\label{definition of alpha}
\alpha_{ijk}(p,q) \equiv \left( \frac{iq_i}{q^2}(q-p)^2-i(q-p)_i
\right)\delta_{jk}+iq_k\delta_{ij}-iq_j \delta_{ik}.
\end{equation}
Solutions to these equations in the momentum space are given by
\begin{eqnarray}
\label{the longitudinal the secon order}
\tilde
A^{aL}_{i,q(I)}(r)&=&-ig\epsilon^{abc}\frac{q_i}{q^2}\int^\infty_{-\infty}
d^3p \bar A^{cT(0)}_{j,p} \bar
A^{bT(0)}_{j,q-p}\frac{|q-p|}{|p|+|q-p|}e^{-(|p|+|q-p|)r} \\ \nonumber
&-&g\epsilon^{abc}\frac{q_i q_j}{q^2}\int^\infty_{-\infty}d^3p
\bar \phi^c_p \bar A^{bT(0)}_{j,q-p}e^{-|q-p|r} + q_i f^a_q, \\
\label{the transverse the secon order}
\tilde A^{aT}_{i,q(I)}&=&g\epsilon^{abc}\int^\infty_{-\infty}d^3p
\bar A^{cT(0)}_{k,p} \bar
A^{bT(0)}_{j,q-p}\frac{\alpha_{ijk}(p,q)}{(|p|+|q-p|)^2
-q^2}e^{-(|p|+|q-p|)r} \\ \nonumber
&+&\frac{i}{2}g\epsilon^{abc}P_{ij}(q)\int^\infty_{-\infty} d^3 p p_j
\bar \phi^b_p \bar \phi^c_{q-p} + g\epsilon^{abc}
P_{ij}(q)\int^\infty_{-\infty}
d^3 p \bar \phi^b_p \bar A^{cT(0)}_{j,q-p}e^{-|q-p|r},
\end{eqnarray}
where $f^a_q$ is an integration constant. Therefore, the total
solution up to $O(\varepsilon^2)$ is given by
\begin{equation}
\label{total solutions}
A^{a}_{i,p}(r)=\varepsilon \bar A^{a}_{i,p}(r) + \varepsilon^2
(\tilde A^{a}_{i,p(H)}(r) + \tilde A^{aT}_{i,p(I)}(r)+ \tilde
A^{aL}_{i,p(I)}(r) )+O(\varepsilon^3),
\end{equation}
under radial gauge. This total solution can be sorted out into gauge
invariant parts and gauge parameter dependent parts. The transverse
parts in $\bar A^a_{i,q}(r)$ and $\tilde A^a_{i,q(H)}(r)$ are gauge
invariant. In eq.(\ref{the longitudinal the secon order}) and
eq.(\ref{the transverse the secon order}), the first term in each
equation is gauge invariant because it is comprised of $\bar
A^{aT}_{i,q}(r)$ only.
One can choose the integration constant $f^a_q$ as
\begin{equation}
f^a_q=\frac{i}{2}g\epsilon^{abc}\frac{q_j}{q^2}\int^\infty_{-\infty}
d^3p p_j\bar \phi^b_p \bar \phi^c_{q-p} + f^{a\prime}_q
\end{equation}
with another arbitrary function $f^{\prime}_q$ and it can absorbed
into $\tilde \phi^a_{q(H)}$ by a redefinition
\begin{equation}
\tilde \phi^a_{q(H)} \rightarrow \tilde \phi^a_{q(H)} -if^{a\prime}_q.
\end{equation}
At this point it is worth pointing out that under such choice, gauge
parameters dependent parts of the total solution(\ref{total
solutions}) has exactly the same form as the gauge
transformation(\ref{GAUGE TRANSFORM}) except the fact that the gauge
parameters are $r$ independent(See
eq.(\ref{first-order-gauge-transform}) for the gauge transformation in
$O(\epsilon)$ and eq.(\ref{second-order-gauge-transform}) in
$O(\epsilon^2)$). Since the bulk action is manifestly gauge invariant
under the residual gauge transformation, if we plug in this
solution into the bulk action, the gauge parameter dependent
parts drop out and the bulk on-shell action is written purely in terms
of gauge invariant parts of the total solution.
It has already been noted in the past
\cite{Witten2,Leigh1,Yee1,Donald1} that imposition of the Neumann
boundary condition on the $AdS$ boundary, leads to an ambiguity in the
computation of correlation functions of the dual operators. This
ambiguity is associated with the residual gauge symmetry surviving at
the boundary. However, we want to look at the boundary on-shell action,
and this ambiguity appears as a total derivative term in the boundary action as long as the
current coupled to the boundary value of the Yang-Mills field is covariantly conserved, $\mathcal D_i F^a_{ri}=0$.
\section{Boundary Conditions and the Effective Action}
\label{Boundary Conditions and the Effective Action}\setcounter{equation}{0}
In the previous section, we have discussed the bulk solution in the
radial gauge $A^a_r=0$. In this section, we would like to discuss
boundary deformations due to the bulk
solutions that we obtained in the previous section.
Before we get into the detailed discussion, we briefly discuss bulk
action. Up to equations of motion, the bulk
action(\ref{flat-space-action}) can be written as
\begin{equation}
\label{bulk action up equation of motionN}
S[A]=\frac{1}{2}\int d^4 x \left(\partial_M(A^a_N F^a_{MN})+
\frac{1}{2}g\epsilon^{abc}A^a_M F^b_{NM}A^c_N\right).
\end{equation}
We do not need to add any counter
terms\cite{Balasubramanian1,Clifford1} since there are no divergences
in the $r\to 0$ limit, which is manifest from the bulk solutions obtained in the previous section\footnote{There is another way of adding
counterterm action subtracting all the terms in the boundary action
at any finite $r$ slice as \cite{Dileep1} which is indeed cut-off
independent action.}.
We are interested in studying small $r$ behaviour (equivalently
behaviour near the $AdS$ boundary).
Both terms in the action contain radial derivatives and can be written
as total derivative with respect to $r$ which would result in boundary
contribution. However, once we choose the radial gauge, the action
becomes
\begin{equation}
\label{radial gauge action}
S[A]=\frac{1}{2}\int d^4 x \left(\partial_r(A^a_i F^a_{ri})+
\partial_j(A^a_i F^a_{ji})+\frac{1}{2}g\epsilon^{abc}A^a_i
F^b_{ji}A^c_j\right).
\end{equation}
The second term in eq.(\ref{radial gauge action}) then
becomes independent of $r$ derivatives and the only place where
$r$-derivative appears in the first term, and that too as total
derivative. The third term also contributes to small $r$ boundary even if it is not total derivative with respect to $r$.
In general, it is non-trivial to extract its boundary contributions out but by using our perturbative solutions, we can evaluate those upto cubic order in small
amplitude expansion(The precise expresssion will be given in Sec.\ref{Boundary Deformation in the Second Order in}).
As a result contribution of the bulk Yang-Mills action up
to the bulk equations of motion to small $r$-boundary is given by
\begin{equation}
\label{boundary action from bulk}
S_{bulk}\equiv\frac{1}{2}\int d^3 x A^a_i(r,x) F^a_{ri}(r,x)+\frac{1}{4}\int d^3xdr g\epsilon^{abc}A^a_i
F^b_{ji}A^c_j.
\end{equation}
{}From now on we will call eq.(\ref{boundary action from bulk}) the
bulk action, although it is a contribution of bulk theory to the
boundary action.
We will mostly work in momentum space. Therefore, we perform a Fourier transform of bulk action(\ref{boundary action from bulk}) using eq.(\ref{Fourier Transform}) and we define a new
bulk action as
\begin{equation}
\hat{S}_{bulk}\equiv\frac{S_{bulk}}{(2\pi)^3},
\end{equation}
where $S_{bulk}$ is a momentum space expression of the bulk action. We define $\hat{S}_{bulk}$ to remove $(2\pi)^3$ factor from $S_{bulk}$ and
$\hat{S}_{bulk}$ will be mostly used for the construction of boundary action .
One
can define the boundary value of bulk canonical momentum, $\partial_r
A^{a}_{i,q}(r)$ of Yang-Mills field $A^{a(0)}_{i,q}(r)$ as
\begin{equation}
\hat \Pi^a_{i,q}\equiv\frac{\delta \hat{S}_{bulk}}{\delta A^{a(0)}_{i,q}
\end{equation}
The boundary on-shell action, $I_{os}$ can be defined by choosing
specific boundary conditions. To fix the boundary condition, we
add the boundary action, $S_{bdy}$ to the bulk action as
\begin{equation}
\label{DEFINITION of ON-SHELL action}
I_{os}=S_{bulk}+S_{bdy},
\end{equation}
where we want $S_{bdy}$ is composed of the boundary value
of the gauge invariant part of the total solution(\ref{total solutions}) and
that of its conjugate momentum only.
Then, the on-shell action is a functional of $A^{a}_{i}$ and its canonical
momentum $\Pi^a_{i}$.
After adding $S_{bdy}$, the generating functional for the boundary
$CFT$ will have two integration measures with $A^{a(0)}_{i}$ and
$\Pi^{a}_{i}$ as
\begin{equation}
Z[J]=e^{-W[J(A^{a(0)}_{i},\Pi^{a}_{i})]}=\int D[ A^{a(0)}_{i},
\Pi^{a}_{i}]exp\left( -S_{bulk}(A^{a(0)}_{i}) - S_{bdy}
( A^{a(0)}_{i}, \Pi^{a}_{i}) \right).
\end{equation}
The generating functional, $W[J]$ with a source $J$ is defined as
\begin{equation}
\label{on-shell action and the generating functionals}
W[J(A^{a}_{i},\Pi^a_{i})]\equiv I_{os}[A^{a}_{i},\Pi^a_{i}],
\end{equation}
where the source $J$ is again a non-trivial function of $A^{a(0)}_{i}$
and $\Pi^{a}_{i}$ in general. The boundary conditions are given at
the saddle point of the on-shell action:
\begin{equation}
\label{saddle point on on-shell action}
\frac{\delta I_{os}[A^{a(0)}_{i},\Pi^a_{i}]}{\delta A^{a(0)}_{i}}=0
{\rm \ and \ }\frac{\delta I_{os}[A^{a(0)}_{i},\Pi^a_{i}]}{\delta \Pi^{a}_{i}}=0,
\end{equation}
and in terms of the generating functional, which is given by
\begin{equation}
\frac{\delta W[J(A^{a(0)}_{i},\Pi^a_{i})]}{\delta
J[A^{a(0)}_{i},\Pi^a_{i}]}
\frac{\delta J(A^{a(0)}_{i},\Pi^a_{i})}{\delta A^{a(0)}_i}=0 {\rm\ and \ }
\frac{\delta W[J(A^{a(0)}_{i},\Pi^a_{i})]}{\delta
J[A^{a(0)}_{i},\Pi^a_{i}]}
\frac{\delta J(A^{a(0)}_{i},\Pi^a_{i})}{\delta \Pi^{a}_i}=0.
\end{equation}
This corresponds to the vacuum states of the boundary $CFT$.
eq.(\ref{saddle point on on-shell action}) provides a relation between
$A^{a(0)}_{i}$ and $\Pi^a_{i}$. Using this, one can re-write the
on-shell action in terms of $A^{a(0)}_{i}$ as saddle point
approximation.
The boundary effective action can be obtained by
Legendre transform defined as
\begin{equation}
\label{Legendre transform}
\Gamma[\sigma]=-\int J\sigma+W[J],
\end{equation}
where $\Gamma$ is the boundary effective action and $\sigma$ is the
vacuum expectation value of certain boundary operators. From this
relation, one gets
\begin{equation}
\sigma=\frac{\delta W[J]}{\delta J} {\rm\ \ and \ \ } J=-\frac{\delta
\Gamma[\sigma]}{\delta \sigma}.
\end{equation}
Now, let us suppose that for a certain boundary deformation, $S_{bdy}$,
the effective action changes in the following way
\begin{equation}
\label{deformed Gamma}
\tilde\Gamma[\sigma]=\Gamma[\sigma]+\int d^d x f(\sigma(x)),
\end{equation}
where $\Gamma$ denotes the effective action before the deformation and
$\tilde \Gamma$ denotes that after the deformation. $f$ is a function
of the vacuum expectation value $\sigma$. The relation between $f$
and $S_{bdy}$ will become clear momentarily. Varying both sides of
(\ref{deformed Gamma}), one obtains the expression for the deformed
source $\tilde J \equiv -\frac{\delta \tilde\Gamma[\sigma]}{\delta
\sigma}$ as
\begin{equation}
\tilde J=J-\frac{d f(\sigma)}{d \sigma}.
\end{equation}
Finally, the deformed generating functional $\tilde W[\sigma] = \tilde
\Gamma[\sigma]+\int \tilde J \sigma$ can be written as
\begin{equation}
\tilde W[\tilde J]=W[J]+\int d^d x \left(f(\sigma)-\sigma f(\sigma)\right).
\end{equation}
It is now clear from the definition of the on-shell
action(\ref{DEFINITION of ON-SHELL action}) and (\ref{on-shell action
and the generating functionals}) that
\begin{equation}
S_{bdy}=\int d^d x \left(f\left(\frac{\delta W[J]}{\delta J}\right)-
\frac{\delta W[J]}{\delta J} f\left(\frac{\delta W[J]}{\delta J}\right)\right).
\end{equation}
In next section, we use these relations to derive $I_{os}$, $W[J]$ and
$\Gamma[\sigma]$ for the various deformations from $SU(2)$ Yang-Mills
theory in $AdS_4$. Before going on, we note that the effective action
of $\Delta_+$ theory has the same form as the on-shell action of
$\Delta_-$ theory. In the case of $S_{bdy}=0$, the only possible
boundary condition is the Dirichlet boundary condition, which gives
us the $\Delta_+=2$ theory. As we will see, to obtain the Neumann
boundary condition, we will have to set $S_{bdy}=-\int d^d x \Pi^a_i
A^{a(0)}_i$. Since $\Pi^a_i$ is canonically conjugate of
$A^{a(0)}_i$, adding this boundary term results in Legendre transform
from $\Delta_+$ theory to $\Delta_-$ theory. Imposition of the
Neumann boundary condition therefore results in the $\Delta_-=1$
theory. Thus we have argued that Legendre transform of the generating
functional $W[J]$ gives us the classical effective action
$\Gamma[\sigma]$. Therefore, the effective action of $\Delta_+$
theory should be the same with the on-shell action of $\Delta_-$
theory.
\subsection{Boundary Deformations in the First Order in $\varepsilon$}
As a warm up, we start with bulk solutions with truncations up to
$O(\varepsilon)$ and derive their on-shell actions, generating
functionals and boundary effective actions. Since, we are considering
the non-abelian gauge theory case, we explicitly write the gauge group
indices, however, up to $O(\varepsilon)$, the precess is almost the
same with the abelian gauge theory on $AdS_4$\cite{Sebastian1}. The
only difference is that we have 3 copies of them. Therefore, the
genuine properties of the boundary effective action from $SU(2)$
Yang-Mills on $AdS_4$ will appear from the second order in
$\varepsilon$ onwards, which would be discussed in the next subsection.
The bulk solution in the first order in $\varepsilon$ in momentum
space would be expanded near $AdS$ boundary as
\begin{equation}
\label{small-r-expansion-in-momentum-space}
A^{a}_{i,q}=A^{a(0)}_{i,q}+r A^{a(1)}_{i,q}+O(r^2),
\end{equation}
where
\begin{equation}
\label{boundary expansion supple}
A^{a(0)}_{i,q}=\varepsilon \bar A^{aT(0)}_{i,q} {\rm \ and \ }
A^{a(1)}_{i,q}=\varepsilon \bar A^{a(1)}_{i,q}=-\varepsilon|q|
\bar A^{aT(0)}_{i,q}.
\end{equation}
In eq.(\ref{boundary expansion supple}), we have used the regularity
condition(\ref{regularity-condition}) for the last equality. As
discussed in the last section, we only deal with the gauge invariant
parts of the solutions. The bulk action up to the bulk EOM is given
by
\begin{equation}
\label{D and N on-shell}
\hat S_{bulk}=\frac{1}{2}\varepsilon^2\int d^3 p \bar A^{a(0)}_{i,p}
\bar A^{a(1)}_{i,-p} =-\frac{1}{2}\varepsilon^2\int d^3 p |p|
\bar A^{a(0)}_{i,p} \bar A^{a(0)}_{i,-p}.
\end{equation}
With this expression, one can find the canonical momentum of
the boundary Yang-Mills field $A^{a(0)}_{i,q}$, which is given by
\begin{equation}
\hat \Pi^a_{i,q}=\frac{\delta \hat{S}_{bulk}}{\delta A^{a(0)}_{i,q}}
=\frac{\delta \hat{S}_{bulk}}{\delta \varepsilon \bar A^{a(0)}_{i,q}}
=-\varepsilon |q|\bar A^{aT(0)}_{i,-q}= -|q|A^{a(0)}_{i,-q}=A^{a(1)}_{i,-q}.
\end{equation}
Variation of the bulk action with respect to the boundary
field $A^{a(0)}_{i,q}$ is then given by
\begin{equation}
\delta \hat S_{bulk} = \int d^3 p |p| \delta A^{a(0)}_{i,p}
A^{a(0)}_{i,-p}= \int d^3 p \delta A^{a(0)}_{i,p} \hat \Pi^a_{i,p}.
\end{equation}
{\em Dirichlet\ and\ Neumann\ Boundary\ Conditions}: For the case that
$\hat S_{bdy}=0$, a possible boundary condition is the Dirichlet boundary
condition, $\delta A^{a(0)}_{i,q}=0$. In this case, the on-shell
action(also the generating functional) is the same as $\hat
S_{bulk}$ and the source $J$ and the corresponding vacuum expectation
value, $\sigma$ in the generating functional are
\begin{equation}
J_D= A^{a(0)}_{i,q} {\rm \ \ and \ \ }\sigma_D \equiv\frac{\delta W[J_D]}{\delta J_D}
=\frac{\delta \hat S_{bulk}}{\delta A^{a(0)}_{i,q}}=\hat \Pi^a_{i,q}=-|q| A^{a(0)}_{i,-q},
\end{equation}
respectively, where the subscript $D$ denotes ``Dirichlet''. The
boundary effective action can be obtained by Legendre transform
defined in eq.(\ref{Legendre transform}).
We apply the Legendre transform
for the Dirichlet case, then the effective action is given by
\begin{equation}
\label{Dirichlet boundary effective order varepsilon}
\Gamma^D[\hat \Pi^a_{i,q}]=\frac{1}{2} \int^\infty_{-\infty}
\frac{d^3 p}{|p|}\hat\Pi^a_{i,p}\hat\Pi^a_{i,-p}.
\end{equation}
Neumann boundary condition can be obtained by considering that $\hat
S^N_{bdy}=-\int d^3 p A^{a(0)}_{i,p} \hat\Pi^a_{i,p}$, where the
superscript $N$ denotes ``Neumann''. To find out stationary points,
we vary $I^N_{os}[ A^{a(0)}_{i,q},\Pi^a_{i,q}]$ as
\begin{equation}
\delta I^N_{os}[A^{a(0)}_{i,q},\hat\Pi^a_{i,q}] = \int d^3 p
\delta A^{a(0)}_{i,p} \Pi^a_{i,p} + \delta S_{bdy}
= -\int d^3 p A^{a(0)}_{i,p} \delta \hat\Pi^a_{i,p}=0,
\end{equation}
so we get Neumann boundary condition: $\delta \hat\Pi^a_{i,q}=0$. For
Neumann case, the role of the source $J$ and the vacuum expectation
value $\sigma$ are interchanged with respect to the Dirichlet case. This is because
adding $\hat S^N_{bdy} = \int J\sigma$ is effectively performing
Legendre transform of $\hat S_{bulk}$. As a result, the
boundary effective action is obtained from Legendre transformation of
eq.(\ref{Dirichlet boundary effective order varepsilon}):
\begin{equation}
\Gamma^N[A^{a(0)}_{i,p}]=-\frac{1}{2}\int d^3 p |p| A^{a(0)}_{i,p}
A^{a(0)}_{i,-p},{\ \ }J_N=\hat\Pi^a_{i,q}
{\rm \ \ and\ \ }\sigma_N=A^{a(0)}_{i,p}.
\end{equation}
{\em Massive\ Deformation}: One can also discuss generalized Neumann
boundary conditions, for example, the {\em Massive\ Deformation}.
At the first order in $\varepsilon$, the massive deformation leads to
a boundary condition given by
\begin{equation}
\label{Massive deformation}
\bar A^{a(0)T}_{ip}+\frac{1}{m}\bar A^{a(1)}_{ip}=0.
\end{equation}
To obtain this boundary condition, we introduce the boundary action
\begin{equation}
\hat S^M_{bdy}=-\int d^3 p \left( A^{a(0)}_{i,p} \hat\Pi^a_{i,p}+
\frac{1}{2m}\hat\Pi^a_{i,p}\hat\Pi^a_{i,-p}\right).
\end{equation}
By varying the on-shell action with above boundary action, we end up
with
\begin{equation}
\label{delta I^N_os}
\delta I^M_{os}[A^{a(0)}_{i,q},\Pi^a_{i,q}]=-\int d^3 p
\delta A^{a(0)}_{i,p}\left( |p| A^{a(0)T}_{i,-p}+\hat\Pi^a_{i,p}\right)
-\int d^3 p \delta \Pi^a_{i,-p}\left( A^{aT(0)}_{i,-p}+
\frac{1}{m}\Pi^a_{i,p} \right)=0,
\end{equation}
where the first integral gives the regularity
condition.
Rather than imposing Neumann boundary condition for the second
integration, if we set the quantity inside the parenthesis to zero,
then the canonical momentum becomes
\begin{equation}
\label{massive canonical momentum constraint}
\hat\Pi^a_{i,-q}=-m A^{a(0)T}_{i,q}.
\end{equation}
For the consistency with the regularity
condition (\ref{regularity-condition}), it is demanded that $|p|=m$.
Therefore, the boundary field $A^{a(0)}_{i,q}$ becomes on-shell and
massive under such a condition.
We rewrite the on-shell action $I^M_{os}$ with replacing every
$\Pi^a_{i,q}$ by $A^{a(0)}_{i,q}$ using eq.(\ref{massive canonical
momentum constraint})
as
\begin{equation}
\label{massive effective action in order epsilon}
I^M_{os}[A^{a(0)}_{i,q}]=-\frac{1}{2}\int d^3 p
\left( |p|-m \right)A^{a(0)T}_{i,p}A^{a(0)T}_{i,-p}.
\end{equation}
The fact that this procedure is
justified can be seen by varying $I^M_{os}[A^{a(0)}_{i,q}]$ with
respect to $A^{a(0)}_{i,q}$ and noticing that it produces the correct
boundary condition
\begin{equation}
\frac{\delta I^M_{os}[A^{a(0)}_{i,q}]}{\delta A^{a(0)}_{i,q}}=-
\left( |p|-m \right)A^{a(0)}_{i,-p}=0.
\end{equation}
The final step for the massive deformation case is to obtain the dual
$CFT$ (or effective) action.
Unfortunately, one cannot easily figure out what is the deformed
source $J$ in above expression and therefore cannot perform Legendre
transform either. However, there is another way to deal with this
situation where one writes down an expected form of the dual $CFT$
action. Let us consider the following form:
\begin{equation}
\Gamma^M[A^{a(0)}_{i,q}]=\frac{1}{2}\int d^3 p
\alpha(p)A^{a(0)}_{i,p}A^{a(0)}_{i,-p},
\end{equation}
where $\alpha$ is an arbitrary momentum dependent function and we
assume that vacuum expectation value $\sigma$ is still
$A^{a(0)}_{i,q}$ under any deformation\cite{Ioannis1}. Using this for
the effective action, one can derive the expression of the source
\begin{equation}
J_M[A^{a(0)}_{i,q}]=-\frac{\delta \Gamma^M[J(A^{a(0)}_{i,q})]}{\delta
A^{a(0)}_{i,q}}=-\alpha(q)A^{a(0)}_{i,-q}.
\end{equation}
We can then use this source term $J$ to perform inverse Legendre
transform from $\Gamma$ to obtain the generating functional $W$
using eq.(\ref{Legendre transform}). We then demand that this inverse
transformation reproduce the correct generating functional $W$, which
imposes a constraint on $\alpha$, and also determines expression of
the source term,
\begin{equation}
\label{alp-sou}
\alpha=(|p|-m){\ , \ }J_M=-(|p|-m)A^{a(0)}_{i,-q}{\rm\ and\ }
\Gamma^M=\frac{1}{2}\int d^3 p (|p|-m)A^{a(0)}_{i,p}A^{a(0)}_{i,-p}.
\end{equation}
The generating functional $W$ is usually expressed as the functional
of source $J_M$, which is done by using eq.(\ref{alp-sou}),
\begin{equation}
W^M[J^M_{i,q}]=\frac{1}{2}\int d^3 p \frac{J^{M}_{i,p}J^{M}_{i,-p}}{ |p|-m}.
\end{equation}
{\em Self-Dual\ Boundary\ Condition\ and\ Massive\ Deformation}: The
most interesting case is the self-dual boundary condition, together
with the massive deformation. Self-duality condition in four dimension is
given by
\begin{equation}
\label{self-dual-condition}
F^a_{MN}=\frac{1}{2}\epsilon_{MNPQ}F^a_{PQ}.
\end{equation}
To study self-dual boundary condition, we expand Yang-Mills
field near the $AdS$ boundary, {\em i.e.}, around $r=0$ as in
eq.(\ref{small-r-expansion-in-momentum-space}).
Once we choose the index $M=r$ in eq.(\ref{self-dual-condition}), the
boundary condition derived from it becomes
\begin{equation}
\label{self-dual-in-boundary}
A^{a(1)}_i=\mathcal D_iA^{a(0)}_r+\frac{1}{2}\epsilon_{ijk}F^{a(0)}_{jk},
\end{equation}
where $\mathcal
D_iA^{a}_r=\partial_iA^{a}_r-g\epsilon^{abc}A^{b}_iA^{c}_r$.
Since we have used the radial gauge $A^a_r=0$ for our bulk solutions,
$\mathcal D_iA^{a}_r=0$ in eq.(\ref{self-dual-in-boundary}).
Up to the leading order in $\varepsilon$, the self dual
boundary condition is given by
\begin{equation}
\label{Self=duality}
A^{a(1)}_i(x)= \epsilon_{ijk}\partial_j A^{a(0)}_{k}(x), {\rm \ in
\ momentum\ space\ }
A^{a(1)}_{i,q}=\hat \Pi^a_{i,-q}= \epsilon_{ijk}(-iq_j)A^{a(0)}_{k,q}.
\end{equation}
In addition to this, if we impose the on-shell condition,
$(|p|-m)A^{a(0)}_{i,p}=0$, it gives rise to massive deformation of the
boundary on-shell action. That is, eq.(\ref{Massive deformation})
together with eq.(\ref{Self=duality}), gives rise to the boundary
condition
\begin{equation}
\label{massive-sef-duality-CONdITion}
0=mA^{a(0)}_{i,p}+\epsilon_{ijk}(-ip_j)A^{a(0)}_{k,p}.
\end{equation}
This boundary condition can be incorporated in boundary on-shell
action in the following way,
\begin{equation}
\hat S^{MS}_{bdy}=\int d^3 p \left[ \beta \left( A^{a(0)}_{i,p} \hat\Pi^a_{i,p}
+\frac{1}{2m}\hat\Pi^a_{i,p}\hat\Pi^a_{i,-p} \right)
-\frac{\beta+1}{2}\epsilon_{ijk}A^{a(0)}_{i,p}(ip_j)A^{a(0)}_{k,-p}\right],
\end{equation}
where $\beta$ is a numerical parameter. Variation of the on-shell
action, $I^{MS}_{os}=\hat S_{bulk}+\hat S^{MS}_{bdy}$, provides
\begin{eqnarray}
\label{sd-massive}
\delta I^{MS}_{os}[A^{a(0)}_{i,q},\hat\Pi^a_{i,q}]&=& \int d^3 p
\beta\delta \hat\Pi^a_{i,p}\left( A^{a(0)}_{i,p}+\frac{1}{m}
\hat\Pi^a_{i,-p} \right) \\ \nonumber
&-& \int d^3 p \delta A^{a(0)}_{i,p}\left( |p|A^{a(0)}_{i,-p}-
\beta\hat\Pi^a_{i,p}+(\beta+1)\epsilon_{ijk}(ip_j)A^{a(0)}_{k,-p}\right).
\end{eqnarray}
The first line in above equation (\ref{sd-massive}) can be set to zero
by considering the massive deformation
\begin{equation}
A^{a(0)}_{i,q}+\frac{1}{m}\hat\Pi^a_{i,-q} =0
\end{equation}
rather than imposing the Neumann boundary condition, $\delta
\Pi^a_{i,q}=0$. For consistency with the regularity condition, we
demand $(|p|-m)A^{a(0)}_{i,p}=0$ and the massive deformation implies
the canonical momentum is given by $\Pi^a_{i,-q}=-m A^{a(0)}_{i,q}$.
For the second line in equation (\ref{sd-massive}), rather than
imposing Dirichlet boundary condition $ \delta A^{a(0)}_{i,q}=0$, we
equate the expression inside the parenthesis to zero. This choice
corresponds to the self dual boundary condition in momentum
space,
\begin{equation}
\hat\Pi^a_{i,q}=-|q|A^{a(0)}_{i,-q}=\epsilon_{ijk}(-iq_j)A^{a(0)}_{k,-q}.
\end{equation}
This condition together with on-shell condition, is exactly the same
with eq.(\ref{massive-sef-duality-CONdITion}).
When we substitute this relation into
$I^{MS}_{os}[A^{a(0)}_{i,q},\hat\Pi^a_{i,q}]$ along with the
regularity condition
we can eliminate $\hat\Pi^a_{i,q}$ by expressing it in terms of
$A^{a(0)}_{i,-q}$ to get
\begin{equation}
I^{MS}_{os}[A^{a(0)}_{i,q}]=-\frac{1}{2}(1+\beta)\int d^3p
\left(
mA^{a(0)T}_{i,p}A^{a(0)T}_{i,-p}+\epsilon_{ijk}A^{a(0)}_{i,p}(ip_j)
A^{a(0)}_{k,-p}\right),
\end{equation}
which is abelian massive Chern-Simons
action\cite{Townsend1,Deser1}. We also obtain the deformed source
and dual $CFT$ action by the same method in the previous
discussion with massive deformation. They are given by
\begin{eqnarray}
J_{MS}&=&-(1+\beta)\left( mA^{a(0)}_{i,-q}+\epsilon_{ijk}(iq_j)
A^{a(0)}_{k,-q}\right), \\
\Gamma^{MS}[A^{a(0)}_{i,q}]&=&\frac{1}{2}(1+\beta)\int d^3p\left(
mA^{a(0)T}_{i,p}A^{a(0)T}_{i,-p}+\epsilon_{ijk}A^{a(0)}_{i,p}(ip_j)A^{a(0)}_{k,-p}\right).
\end{eqnarray}
\subsection{Boundary Deformation in the Second Order in $\varepsilon$}
\label{Boundary Deformation in the Second Order in}
A way of imposing boundary conditions for the second order solution in
$\varepsilon$ is in principle the same with previous discussion.
There are some technical difficulties due to appearance of quadratic
terms involving the first order solutions. However, this nonlinearity
in the equation involves lower order solutions only, which are already
derived using the small amplitude expansion.
For evaluating the boundary on-shell action, we would like to choose a gauge for boundary gauge fields,
$A^{a(0)}_{i,q}$, in fact, we will
set $\phi^a_{i,q}=\varepsilon \bar \phi^a_{i,q} +\varepsilon^2 \tilde \phi^a_{i,q}=0$.
Since the bulk action is manifestly gauge invaraint, choosing a particular gauge is not a problem.
With such choice of gauge degree of freedom, the boundary gauge field appearing on the boundary on-shell action will be effectively transverse.
Therefore, in the following, we only deal with gauge parameter independent
parts of the solutions for the construction of the boundary theory.
We start with a general
discussion of the solution(\ref{total solutions}). The near $AdS$
boundary expansion is given by
\begin{equation}
A^{a}_{i,q}(r)|_{r\rightarrow 0}=A^{a(0)}_{i,q}+rA^{a(1)}_{i,q}+O(r^2),
\end{equation}
where
\begin{equation}
A^{a(1)}_{i,-q}=
-|q|A^{a(0)}_{i,-q}-g\epsilon^{abc}\int^{\infty}_{-\infty}d^3 p
A^{c(0)}_{k,p} A^{b(0)}_{j,-q-p}\Delta_{ijk}(p,-q)+O(\varepsilon^3)
\end{equation}
and
\begin{equation}
\label{definition of Delta}
\Delta_{ijk}(p,q)=\frac{\alpha_{ijk}(p,q)}{|p|+|q-p|+|q|}-\frac{iq_i
\delta_{jk}|q-p|(|q-p|+|p|-|q|)}{q^2(|p|+|q-p|)}.
\end{equation}
$A^{a(0)}_{i,q}$ is the boundary value of the full solution
$A^{a}_{i,p}(r)$ defined in eq.(\ref{total solutions})(See also
eq.(\ref{the transverse the secon order}), eq.(\ref{the longitudinal the
secon order}) and eq.(\ref{the second order homogeneous solution})),
which is given by
\begin{eqnarray}
\label{A0exp}
A^{a(0)}_{i,q}&=&\varepsilon \bar A^{aT(0)}_{i,p}
+ \varepsilon^2 \tilde A^{aT(0)}_{i,p(H)}
-\varepsilon^2
ig\epsilon^{abc}\frac{q_i}{q^2}\int^\infty_{-\infty}d^3p
\bar A^{cT(0)}_{j,p} \bar A^{bT(0)}_{j,q-p}\frac{|q-p|}{|p|+|q-p|} \\ \nonumber
&+&\varepsilon^2 g\epsilon^{abc}\int^\infty_{-\infty}d^3p
\bar A^{cT(0)}_{k,p} \bar A^{bT(0)}_{j,q-p}\frac{\alpha_{ijk}(p,q)}
{(|p|+|q-p|)^2-q^2}+O(\varepsilon^3).
\end{eqnarray}
Now, we evaluate the bulk action(\ref{boundary action from bulk})
explicitly by substituting the bulk solution and keeping terms upto
the leading interaction terms,
\begin{equation}
\label{bulk-second-sol}
S_{bulk}\equiv\frac{1}{2}\int d^3 q A^{a(0)}_{i,q} A^{a(1)}_{i,-q}+\frac{1}{4}\int d^3q d^3l d^3 p dr g\epsilon^{abc}A^a_{i,q}
F^b_{ji,l}A^c_{j,p}\delta^3(q+l+p),
\end{equation}
where gauge fields in the second term contains the first order
solutions only, which means that
\begin{equation}
A^a_i=\varepsilon \bar A^{a(0)}_{i,q}e^{-|q|r} + O(\varepsilon^2)=A^{a(0)}_{i,q}e^{-|q|r}+ O(\varepsilon^2).
\end{equation}
Therefore, the second term in eq.(\ref{bulk-second-sol}) becomes
\begin{eqnarray}
S^{2nd \ term}_{bulk}\!\!\!\!\!
&=&\frac{1}{2}\int d^3q d^3l d^3 p \left.\varepsilon^3 \bar A^{a(0)T}_{i,q}
\bar A^{b(0)T}_{j,l}\bar A^{c(0)T}_{k,p}\frac{il_k \delta_{ij}}{|q|+|l|+|p|}e^{-(|q|+|l|+|p|)r}\delta^3(q+l+p)\right|^{r=0}_{r=\infty}\\ \nonumber
&=&\frac{1}{2}\int d^3q d^3l d^3 p A^{a(0)}_{i,q}
A^{b(0)}_{j,l} A^{c(0)}_{k,p}\frac{il_k \delta_{ij}}{|q|+|l|+|p|}\delta^3(q+l+p)+O((A^{a(0)}_{i,q})^4)
\end{eqnarray}
With this, one can construct the bulk action as
\begin{eqnarray}
\label{Dirichlet action}
\hat S_{bulk}&=&-\frac{1}{2} \int^\infty_{-\infty} d^3 q
|q|A^{a(0)}_{i,q} A^{a(0)}_{i,-q}
-\frac{1}{2}g\epsilon^{abc}\int^\infty_{-\infty} d^3 q d^3 p
A^{a(0)}_{i,q} A^{b(0)}_{j,-q-p} A^{c(0)}_{k,p}\Delta_{ijk}(p,-q) \\ \nonumber
&+&\frac{1}{2}\int d^3q d^3l d^3 p A^{a(0)}_{i,q}
A^{b(0)}_{j,l} A^{c(0)}_{k,p}\frac{il_k \delta_{ij}}{|q|+|l|+|p|}\delta^3(q+l+p),
\end{eqnarray}
upto cubic interactions.
Notice that $\Delta_{ijk}$ and $\frac{il_k \delta_{ij}}{|q|+|l|+|p|}$, in order to be non-vanishing, should be
fully anti-symmetric in indices, $i$, $j$ and $k$ together with
appropriate momentum exchange due to $\epsilon^{abc}$. The second
term is then written as
\begin{equation}
\label{the second term}
\hat S_{2nd\ term} = -\frac{1}{2}g \epsilon^{abc}
\int^{\infty}_{-\infty}d^3 q d^3 p d^3 l A^{a(0)}_{i,q} A^{b(0)}_{j,l}
A^{c(0)}_{k,p}\delta^3(q+p+l)\tilde\Delta_{ijk}(q,l,p),
\end{equation}
where
\begin{equation}
\tilde\Delta_{ijk}(q,l,p)=\tilde\Delta^T_{ijk}(q,l,p)+
\tilde\Delta^L_{ijk}(q,l,p).
\end{equation}
$\tilde\Delta^T_{ijk}(q,l,p)$ and $\tilde\Delta^L_{ijk}(q,l,p)$ are
given by
\begin{equation}
\label{DeltaT}
\tilde\Delta^T_{ijk}(q,l,p)=\frac{i(l-q)_k \delta_{ij}+i(p-l)_i
\delta_{jk}+i(q-p)_j \delta_{ik}}{2(|q|+|l|+|p|)}.
\end{equation}
\begin{eqnarray}
\label{DeltaL}
\tilde\Delta^L_{ijk}(q,l,p)&=&\frac{iq_i\delta_{jk}(|l|-|p|)
(|p|+|l|-|q|)}{6q^2(|p|+|l|)}
+\frac{il_j\delta_{ki}(|p|-|q|)(|q|+|p|-|l|)}{6l^2(|q|+|p|)}\\ \nonumber
&+&\frac{ip_k\delta_{ij}(|q|-|l|)(|l|+|q|-|p|)}{6p^2(|l|+|q|)},
\end{eqnarray}
(eq.(\ref{DeltaT}) and eq.(\ref{DeltaL}) can be obtained from
eq.(\ref{definition of alpha}) and eq.(\ref{definition of Delta})
after some computation using the fact that $\bar A^{aT(0)}_{i,q}$ is
transverse). In fact, $\tilde\Delta^L_{ijk}(q,l,p)$ does not
contribute to the bulk action, since the fields multiplying it in the
action are effectively transverse\footnote{The bulk solution of
Yang-Mills fields up to second order in $\varepsilon$, requires
terms only up to cubic in $\varepsilon$ in $\hat S_{bulk}$. Using
the expansion (\ref{A0exp}), of the boundary value of the Yang-Mills
field $A^{a(0)}_{i,q}$ in the cubic interaction in
eq.(\ref{Dirichlet action}), it is easy to see that up to
$O(\varepsilon^3)$ this term is effectively transverse
\begin{equation}
A^{a(0)}_{i,q} A^{b(0)}_{j,-q-p} A^{c(0)}_{k,p} = \varepsilon^3 \bar
A^{aT(0)}_{i,q} \bar A^{bT(0)}_{j,-q-p} \bar A^{cT(0)}_{k,p}+O(\varepsilon^4).
\end{equation}
As a result, at this order, $\tilde \Delta^L_{ijk}(q,l,p)$ disappears
from the boundary on-shell action.}.
The third term in eq.(\ref{Dirichlet action}) is given by
\begin{equation}
\hat S^{3rd\ term}=\frac{1}{6}\int d^3q d^3l d^3 p A^{a(0)}_{i,q}
A^{b(0)}_{j,l} A^{c(0)}_{k,p}\tilde \Delta^T_{ijk}(q,l,p)\delta^3(q+l+p),
\end{equation}
it also has the same anti-symmetrization.
The canonical momentum of the source $A^a_{i,q}$ is given by
\begin{equation}
\label{the second order momentum definietion}
\hat\Pi^a_{i,q}=\frac{\delta \hat{S}_{bulk}}{\delta A^{a(0)}_{i,q}}
=-|q|A^{a(0)}_{i,-q}-g\epsilon^{abc}\int^{\infty}_{-\infty}
d^3 p A^{b(0)}_{j,-q-p}A^{c(0)}_{k,p}\tilde\Delta_{ijk}(q,-q-p,p)
\end{equation}
{\em Dirichlet\ Boundary\ Condition}: Without adding any boundary
action, the on-shell action, $I^D_{os}$ is given by
\begin{equation}
I^{D}_{os}(A^{a(0)}_{i,q})=
-\frac{1}{2} \int^\infty_{-\infty} d^3 q |q|A^{a(0)}_{i,q} A^{a(0)}_{i,-q}
-\frac{1}{3}g\epsilon^{abc}\int^\infty_{-\infty} d^3 q d^3 p
A^{a(0)}_{i,q} A^{b(0)}_{j,-q-p} A^{c(0)}_{k,p}
\tilde\Delta^T_{ijk}(q,-q-p,p).
\end{equation}
The Legendre transform of $I^D_{os}$ becomes the boundary effective action
in terms of dual operator $\Pi^{a}_{i,q}$, which is given by
\begin{equation}
\label{Dirichlet-epsilon-square-action}
\Gamma^{D}(\Pi^a_{i,q})=\frac{1}{2}\int^{\infty}_{-\infty}
\frac{d^3 q}{|q|}\hat\Pi^a_{i,q}\hat\Pi^a_{i,-q}
+\frac{1}{3}g\epsilon^{abc}\int^{\infty}_{-\infty}d^3 q d^3 p
\frac{\tilde \Delta_{ijk}(q,-q-p,p)}{|q||q+p||p|}
\hat\Pi^a_{i,q}\hat\Pi^b_{j,-q-p}\hat\Pi^c_{k,p}.
\end{equation}
This action has exotic momentum dependent cubic interaction,
which is classically marginal. Up to this order, we can evaluate
2-point and 3-point functions of the boundary $CFT$ and the dual
$CFT$.
{\em Neumann\ Boundary\ Condition}: The effective action in Neumann
boundary condition can be obtained by Legendre transform
of (\ref{Dirichlet-epsilon-square-action}), which becomes
\begin{equation}
\Gamma^{N}(A^{a(0)}_{i,q})=
-\frac{1}{2} \int^\infty_{-\infty} d^3 q |q|A^{a(0)}_{i,q} A^{a(0)}_{i,-q}
-\frac{1}{3}g\epsilon^{abc}\int^\infty_{-\infty} d^3 q d^3 p
A^{a(0)}_{i,q} A^{b(0)}_{j,-q-p} A^{c(0)}_{k,p}\tilde\Delta_{ijk}(q,-q-p,p),
\end{equation}
with $J^N=\hat\Pi^a_{i,q}$. The generating functionals for each
boundary condition are given by
\begin{eqnarray}
I^{D}_{os}(A^{a(0)}_{i,q})=W^D[A^{a(0)}_{i,q}]=\Gamma^{N}[A^{a(0)}_{i,q}]
{\rm \ \ and \ \ } I^{N}_{os}(\Pi^{a}_{i,q})=W^{N}(\Pi^a_{i,q})=
\Gamma^{D}[\Pi^a_{i,q}].
\end{eqnarray}
{\em Massive\ and\ Self-Dual\ Boundary\ Condition}: To impose
self-dual and massive deformation as a boundary condition, we
add the following boundary action to $\hat S_{bulk}$:
\begin{eqnarray}
\hat S_{bdy}&=&\int^\infty_{-\infty} d^3 q \left[ -\beta
\left(A^{a(0)}_{i,q}\Pi^a_{i,q}+\frac{1}{2m}\Pi^a_{i,q}\Pi^a_{i,-q}
\right) \right.\\ \nonumber
&+&\frac{3}{2m}\alpha \beta g\epsilon^{abc}\Pi^a_{i,q}
\int^{\infty}_{-\infty}d^3 p d^3 l A^{b(0)}_{j,l}A^{c(0)}_{k,p}
\delta^3(-q+l+p)\tilde \Delta_{ijk}(-q,l,p)
\\ \nonumber
&+& \eta\epsilon_{ijk}\left(A^{a(0)}_{i,q}(iq_j)
A^{a(0)}_{k,-q}-\frac{1}{3}g\epsilon^{abc}\int^{\infty}_{-\infty}d^3 p
d^3 l A^{a(0)}_{i,q}A^{b(0)}_{j,l}A^{c(0)}_{k,p}\delta^3(q+l+p)\right)\\ \nonumber
&+&\left.\frac{\gamma}{3} g\epsilon^{abc}\int^{\infty}_{-\infty}d^3p
d^3l A^{a(0)}_{i,q}A^{b(0)}_{j,l}A^{c(0)}_{k,p}\delta^3(q+l+p)
\tilde \Delta_{ijk}(q,l,p) \right],
\end{eqnarray}
where $\alpha$, $\beta$, $\gamma$ and $\eta$ are numerical constants which
would be determined by imposing right boundary condition. Variation
of $I^{MS}_{os}[A^{a(0)}_{i,q}]=\hat S_{bulk}+\hat S_{bdy}$ with
respect to $A^{a(0)}_{i,q}$ and $\hat\Pi^a_{i,q}$ provides the
following boundary conditions:
\begin{eqnarray}
\label{massive-self-dual condition-1}
\hat\Pi^a_{i,q}&=&-mA^{a(0)}_{i,-q}+\frac{3}{2}\alpha g \epsilon^{abc}
\int^{\infty}_{-\infty}d^3 p d^3 l A^{b(0)}_{j,l}A^{c(0)}_{k,p}
\delta^3(q+l+p)\tilde \Delta_{ijk}(q,l,p), \\
\label{massive-self-dual condition-2}
(\beta-1)\Pi^{a}_{i,q}&=&g\epsilon^{abc}\int^{\infty}_{-\infty}d^3 p
d^3 l A^{b(0)}_{j,l}A^{c(0)}_{k,p}\delta^3(q+l+p)
(\gamma-\frac{3\alpha\beta(|l|+|p|)}{2m})\tilde\Delta_{ijk}(q,l,p) \\ \nonumber
&+&\eta\epsilon_{ijk}F^{a}_{jk,-q},
\end{eqnarray}
where
\begin{equation}
F^{a}_{ij,q}=-iq_i A^a_{j,q}+iq_j A^a_{i,q}
-g\epsilon^{abc}\int^{\infty}_{-\infty}d^3 p d^3 l A^{b(0)}_{i,l}
A^{c(0)}_{j,p}\delta^3(-q+l+p),
\end{equation}
is Yang-Mills field strength in momentum space. For the consistency
between canonical momentum(\ref{the second order momentum
definietion}) and boundary condition(\ref{massive-self-dual
condition-1}), one requires
\begin{equation}
\alpha=-\frac{2}{3} {\rm \ and \ } A^{a(0)}_{iq}\rightarrow A^{a(0)}_{iq}|_{|q|=m}.
\end{equation}
We can use the second boundary condition(\ref{massive-self-dual
condition-2}) to impose non-abelian version of massive self-dual
boundary condition\cite{Townsend1}, which is given by
\begin{equation}
A^{a(0)}_{i,q}=-\frac{1}{2m}\epsilon_{ijk}F^{a}_{jk,q}.
\end{equation}
To do this, we plug eq.(\ref{massive-self-dual condition-1}) into
eq.(\ref{massive-self-dual condition-2}). Then the massive self-dual
boundary condition can be obtained if we impose the condition,
\begin{equation}
\gamma=1-3\beta {\rm\ \ and\ \ } \eta=\frac{\beta-1}{2}.
\end{equation}
The on-shell action, dual $CFT$ action and the source term can then be
derived using this massive self-dual condition as,
\begin{eqnarray}
\label{MASSIVE SELFDUAL ONSHELL ACTION}
I^{MS}_{os}[A^{a(0)}_{i,q}]&=&\frac{1}{2}(\beta-1)\int d^3p
\left[mA^{a(0)}_{i,p}A^{a(0)}_{i,-p}+\epsilon_{ijk}\left(A^{a(0)}_{i,p}
(ip_j)A^{a(0)}_{k,-p} \right.\right.\\ \nonumber
&-&\left.\left.\frac{1}{3}g\epsilon^{abc}\int d^3 l d^3 p
A^{a(0)}_{i,q}A^{b(0)}_{j,l}A^{c(0)}_{k,p}\delta^{3}(q+l+p) \right)\right], \\
\Gamma^{MS}[A^{a(0)}_{i,q}]&=&-\frac{1}{2}(\beta-1)\int d^3p\left[m
A^{a(0)}_{i,p}A^{a(0)}_{i,-p}+\epsilon_{ijk}\left(A^{a(0)}_{i,p}(ip_j)
A^{a(0)}_{k,-p} \right.\right.\\ \nonumber
&-&\left.\left.\frac{1}{6}g\epsilon^{abc}\int d^3 l d^3 p
A^{a(0)}_{i,q}A^{b(0)}_{j,l}A^{c(0)}_{k,p}\delta^{3}(q+l+p) \right)\right]. \\
{\rm and\ }J_{MS}&=&(\beta-1)\left[mA^{a(0)}_{i,-p}
+\epsilon_{ijk}\left(ip_jA^{a(0)}_{k,-p}
\right.\right.\\ \nonumber &-&\frac{1}{4}\left.\left.
g\epsilon^{abc}\int d^3 l d^3 p A^{b(0)}_{j,l}A^{c(0)}_{k,p}
\delta^{3}(q+l+p) \right)\right],
\end{eqnarray}
respectively. The on-shell effective action (\ref{MASSIVE SELFDUAL
ONSHELL ACTION}) turns out to be proportional to the non-abelian
Chern-Simon action.
\section{``Approximate'' Electric-Magnetic Duality in SU(2)
Yang-Mills in $AdS_4$}
\label{em-duality}\setcounter{equation}{0}
It is well-known fact that explicit electric-magnetic duality cannot
be demonstrated for non-abelian gauge field theory, pure U(1) gauge
theory equations of motion, on the other hand, are manifestly invariant.
Exchanging electric and magnetic fields is possible even for
Yang-Mills, but such a transformation is not a canonical
transformation\footnote{There is, in fact, a no-go theorem for this
duality. At least in a particular gauge this has been demonstrated
in \cite{Deser3}.}.
There is, however, an attempt to construct a canonical transformation
in SU(2) Yang-Mills, which is gives rise to an {\it approximate}
electric-magnetic duality transformation, if one restrict to
Yang-Mills action truncated upto cubic order interactions in weak
field expansion\cite{Deser2}.
To see this more clearly, let us explain the meaning of ``{\it
approximate} '' electric-magnetic duality. The authors in
\cite{Deser2} construct an infinitesimal canonical transformation
which is a natural extension of U(1) electric-magnetic duality to
SU(2) Yang-Mills, which is manifest symmetry in Yang-Mills action when
the action only retains cubic order interactions in small amplitudes
of gauge fields in it(i.e. they do not keep quartic order
interactions). Therefore, if Yang-Mills coupling vanishes, then this
symmetry becomes the usual duality in U(1). However, this is not
precisely electric-magnetic duality in SU(2) Yang-Mills since the
variation of electric field is not proportional to magnetic field even
upto such a truncation. Therefore by ``approximate'' duality we mean that
there exists a canonical transformation which is the most natural
generalization of electric-magnetic duality in U(1).
It is worth mentioning at this point that this has been demonstrated
in a particular gauge for the Yang-Mills fields, namely the transverse
gauge. In this gauge, components of gauge fields
surviving in the action are all transverse. In any other gauge, the
transformation may be difficult to implement.
In this section, we will discuss ``{\it approximate} ''
electric-magnetic duality transformation for our system. The
difference between flat space and $AdS$ space here only comes from
their boundaries. In general, electric-magnetic duality is not a
manifest symmetry of the Lagrangian but it is a symmetry of equations
of motion. The total derivative terms in the Lagrangian which inhibit
this manifestation disappear in the flat space, if we suppose that all the
fields die off sufficiently fast at infinite boundary. However, Weyl transformed
action(\ref{flat-space-action}) from $AdS_4$ has conformal boundary
at $x^4\equiv r=0$ and gauge fields do not die off fast enough at this
boundary. Therefore, we need to keep total derivative terms with
respect to `$r$'.
Now let us apply the canonical transformation of \cite{Deser2} to our
case. Yang-Mills action (\ref{flat-space-action}) can be written in terms
of the Legendre transform of the Hamiltonian as
\begin{equation}
\label{primitive-em-h-action}
S[A^a_i,E^a_i,A^a_r]= \int d^4 x \left(-E^a_i \partial_r A^a_i -\mathcal H[\Pi^a_i,A^a_i]\right),
\end{equation}
where the canonical momentum $\Pi^a_i=-E^a_i$, the electric field, and the
Hamiltonian density $\mathcal H$ is given by
\begin{equation}
\mathcal H=\frac{1}{2}\left(E^a_i E^a_i -B^a_i B^a_i\right) + A^a_r\left( \partial_i E^a_i + \epsilon^{abc}E^b_iA^c_i \right),
\end{equation}
where we set Yang-Mills coupling $g=1$ for convenience, $B^a_i=\frac{1}{2}\epsilon_{ijk}F^a_{jk}$, magnetic field, and negative sign in front of the magnetic field square in the
Hamiltonian density appears since we are working in the Euclidean
space . Notice the
Legendre transform is taken with respect to the radial coordinate.
The gauge field component $A^a_r$ has no dynamics and in fact, it is a
Lagrange multiplier, which gives rise to the Gauss law constraint. By
imposing the Gauss law constraint, $(D_i E_i)^a=\partial_i E^a_i +
\epsilon^{abc}E^b_iA^c_i =0$, we can remove the terms which are proportional to $A^a_r$ from the action.
Another important point is the gauge choice. In \cite{Deser2}, authors
point out that it is very crucial to choose transverse gauge. Under
such a choice, longitudinal parts of electric fields becomes quartic
order in the weak field expansion in the action, so at the cubic
approximation we will not need to worry about those terms. With all
these conditions, the action can be expressed as
\begin{equation}
S[E^a_i,A^a_i]=\int d^4 x \left[-E^{a,T}_i \partial_r A^a_i -\frac{1}{2}\left(E^{a,T}_i E^{a,T}_i -\bar B^a_i \bar B^a_i\right)
- \frac{1}{2}\bar B^a_i \epsilon^{abc}\epsilon_{ijk} A^b_j A^c_k + O((A^a_i)^\alpha (E^a_i)^\beta)\right],
\end{equation}
where $\alpha$ and $\beta$ are positive integers which satisfy an
inequality $\alpha + \beta \geq 4$ and $\bar
B^a_i\equiv\epsilon_{ijk}\partial_j A^a_k$.
It turns out that this action is invariant upto cubic order
in small amplitude expansion of the fields under the following
infinitesimal transformation:
\begin{eqnarray}
E^{a,T}_i &\rightarrow& E^{a,T}_i +\eta \left(\bar B^a_i -\frac{3}{2}
\epsilon^{abc}\epsilon_{ijk} (A^b_j A^c_k)^T\right) + {\rm \ higher\
order,}\\
\label{longitudinal-em-transform}
E^{a,L}_i &\rightarrow& E^{a,L}_i
-\frac{1}{2}\eta\epsilon^{abc}\epsilon_{ijk}\left( A^b_jA^c_k -
E^{b,T}_j\frac{1}{\nabla^2}E^{c,T}_k\right)^L+ {\rm \ higher\
order,}\\
A^{a}_i &\rightarrow& A^{a}_i - \eta \frac{1}{\nabla^2}\epsilon_{ijk}\partial_j E^{a,T}_k + {\rm \ higher\ order,} \\
{\rm and\ } A^a_r &\rightarrow& A^a_r,
\end{eqnarray}
where `higher order' denotes cubic or higher than cubic order in weak fields expansion, $\eta$ is the
infinitesimal duality rotation angle and the superscripts $T$ and $L$
mean that only transverse and longitudinal parts of the terms would be kept
respectively.
Under such transformation, the action changes as
\begin{equation}
S[E^a_i,A^a_i] \rightarrow S[E^a_i,A^a_i]
+ \frac{1}{2}\eta \int_{r=0} d^3 x\left( A^a \cdot \nabla \times A^a+\epsilon^{abc}A^a \cdot A^b \times A^c +E^a \cdot \frac{1}{\nabla^2}(\nabla \times E^a)\right),
\end{equation}
The action is invariant upto the boundary terms. These boundary terms
will be treated as an infinitesimal boundary deformations. While the
last term in the boundary action is non-local, first two terms are
similar to the Chern-Simons term. Since the duality transformation is
approximate, we are not able to get the relative factors correctly.
One may wonder if transformation (\ref{longitudinal-em-transform}) of
the longitudinal part of the electric field will not be necessary since they appear at the quartic order and our transformations
are applicable only up to cubic part of the action. However, the term
proportional to $A^a_r$ is eliminated from the action
(\ref{primitive-em-h-action}) by imposing the Gauss law constraint.
To ensure that the Gauss law constraint is not affected up to this
order requires the transformation (\ref{longitudinal-em-transform}).
\section{Yang-Mills Instanton}
\label{Yang-Mills Instanton}\setcounter{equation}{0}
In the previous section, we have developed various kinds of
deformations to obtain the corresponding boundary actions.
While there are many reasonable deformations, most of them are obtained
by doing small amplitude expansion about perturbative classical
solution.
In this section, we will consider a nonperturbative solution in the
bulk, namely, the instanton solution, and construct the boundary
action corresponding to this Yang-Mills instanton solution in the
bulk. The instanton solution in the flat space is known for a long
time and we use the same solution\cite{Belavin1,Rajaraman1,Stefan1}.
The reason is that in four dimensions Yang-Mills theory is classically
conformally invariant and the four dimensional anti-de Sitter space,
$AdS_4$, is conformally flat. As a result the instanton solution to
Yang-Mills theory in the Euclidean $AdS_4$ has same form as that in
the $\mathbb R^4$. There is a crucial difference between these two
cases because the Euclidean $AdS_4$ is conformally equivalent to
$\mathbb R^4_+$ because of the semi-infinite range of the radial
coordinate. This fact plays an important role in determining the
boundary action. We start our discussion with the 't Hooft
instanton\cite{'tHooft:1976fv} with winding number 1 which is a
solution to the self-duality condition (\ref{self-dual-condition}),
and is given by
\begin{equation}
\label{the instanton solution}
A^{a}_{M}(x,x_0,\rho)=-\frac{2}{g}\frac{\eta^{a}_{MN}(x-x_{0})^N}
{(x-x_0)^2+\rho^2},
\end{equation}
where, $\rho$ is a real parameter which is size of the instanton and
$x^M_0$ indicates its position. For simplicity, we choose $x^4_0=0$, then
our instanton solution is located on $AdS$ boundary. The gauge condition is chosen as
$\partial_{M} A^{aM}=0$ and $\eta^a_{ij}=\epsilon^{aij}$,
$\eta^a_{ir}=-\eta^a_{ri}=\delta^a_{i}$ for $i=1,2,3$. Even if
equations of motion are the same under the Weyl rescaling defined at
the beginning in the Sec.\ref{Bulk Solutions}, the gauge condition
does not. Lorentz gauge condition $\partial_{M} A^{aM}=0$ in flat
space is different from that in $AdS$ space, which is $\partial_{N}
(\sqrt{-G}G^{NM}(r)A^{a}_M)=0$. However, the radial gauge is the same
in both cases. It is therefore convenient to work in the radial
gauge (For details of gauge transformation and the radial gauge
solution of Yang-Mills instanton, See Appendix.\ref{Yang-Mills
Instantons in Radial Gauge}).
The field strength of 't Hooft instanton solution is given by
\begin{equation}
\label{the instanton field strength}
F^a_{MN}=\frac{4}{g}\eta^{a}_{MN}\frac{\rho^2}{((x-x_0)^2+\rho^2)^2},
\end{equation}
and the action has a finite value as
$S[A_{instanton}]=\frac{8\pi^2}{g^2}$.
In flat $\mathbb{R}^4_+$, the instanton solution(\ref{the instanton
solution}) approaches pure gauge solution and the field
strength(\ref{the instanton field strength}) becomes zero at $x_4=r
\rightarrow \infty$. This region, however, gets mapped to the
Poincare horizon in $AdS_4$ space under the Weyl scaling (See the
beginning in Sec.\ref{Bulk Solutions}). Therefore, Yang-Mills
instanton solutions do not change the boundary conditions at the
horizon. Interestingly, the instanton solution does not become pure gauge
solution at $r=0$ and the field strength has the finite value. As
shown in Appendix \ref{Yang-Mills Instantons in Radial Gauge}, the
Fefferman-Graham expansion of Yang-Mills instanton in the radial gauge
near $AdS$ boundary is given by
\begin{equation}
\label{Fefferman-Graham expansion of instantons}
A^a_{i}=A^{a(0)}_{i}+r A^{a(1)}_{i}+O(r^2),
\end{equation}
with,
\begin{eqnarray}
\label{boundary Yang-Mills A1}
A^{a(0)}_{i}&=&-\frac{2}{g}\frac{\eta^a_{ij}(y-y_0)_j}{{(y-y_0)^2+\rho^2}}, \\
\label{boundary Yang-Mills A2}
A^{a(1)}_{i}&=&-\frac{4}{g}\frac{\delta^a_i\rho^2}{((y-y_0)^2+\rho^2)^2},
\end{eqnarray}
where we have defined a boundary coordinate $y^i \equiv x^i$, and
$y^2=\sum_{i=1}^3y^iy^i$.
$A^{a(1)}_{i}$ is related to $A^{a(0)}_{i}$ by the small $r$ limit of
the self-duality condition eq.(\ref{self-dual-in-boundary}), which in radial
gauge becomes
\begin{equation}
\label{self-dual-boundary-in-radial-gauge}
A^{a(1)}_i=\frac{1}{2}\epsilon_{ijk}F^{a(0)}_{jk}.
\end{equation}
It is easy to see that the boundary values of Yang-Mills instanton
solution (\ref{boundary Yang-Mills A1}) and (\ref{boundary Yang-Mills
A2}) satisfy the boundary condition
(\ref{self-dual-boundary-in-radial-gauge}). We want to
write down this boundary condition in terms of the boundary field
$A^{a(0)}_i$ only.
Although there are various ways of expressing this boundary condition,
we find representation of the Yang-Mills
gauge field corresponding to the 't Hooft instanton solution in terms
of a scalar function $\lambda(y)$ is most convenient for writing the
boundary term.
The usual ansatz for Yang-Mills instanton solution is given by
\begin{equation}
A^{a(0)}_i=\frac{1}{g}\eta^a_{ij}\partial_jln(\lambda(y)), {\rm \ where \ }
\lambda(y)=\frac{\rho^2}{(y-y_0)^2+\rho^2}.
\end{equation}
This equation can be inverted to write $\lambda(y)$ as a non-local
function of $A^{a(0)}_i$,
\begin{equation}
\lambda(y)=e^{\frac{g}{2}\epsilon^{ai}_{\ \ j}\int^y_{y_0}A^{a(0)}_i(z) dz^j}.
\end{equation}
The $A^{a(1)}_i$ can be written in terms of $\lambda(y)$ as
\begin{equation}
A^{a(1)}_{i}= -\frac{4}{g}\delta^a_i \frac{\lambda^2(y)}{\rho^2}.
\end{equation}
With these, we can rewrite the self-dual boundary
condition (\ref{self-dual-boundary-in-radial-gauge}) in terms of
$A^{a(0)}_{i}$ only as
\begin{equation}
\label{Self-dual-boundary-with_A}
\frac{1}{2}\epsilon_{ijk}F^{a(0)}_{jk}(z)
= -\frac{4}{g\rho^2}\delta^a_i e^{g\int^z_{0}\epsilon^a_{ij}
A^{a(0)}_i(\tilde z)d\tilde z^j},
\end{equation}
where $z \equiv y-y_0$.
We can now ask if it is possible to write down the boundary on-shell
action such that the boundary condition
(\ref{Self-dual-boundary-with_A}) is the equation of motion of this
boundary action.
It is easy to seen that the left hand side of
eq.(\ref{Self-dual-boundary-with_A}) comes from the non-abelian Chern
Simons action. The right hand side contains line integration in the
exponent. This line integral resembles the Wilson line, but it, in
fact, corresponds a non-local interaction. Such a term in the
deformed boundary conditions cannot be obtained within the
perturbative approach. Moreover, in the case of multi-instanton
solution\cite{'tHooft:1976fv} in the bulk, the corresponding boundary
condition would continue to have this type of non-local, although its
precise form is different from eq.(\ref{Self-dual-boundary-with_A}).
To discuss this boundary condition in general, we promote
eq.(\ref{Self-dual-boundary-with_A}) to a general boundary condition
and treat $\rho$ as a parameter in the corresponding boundary
theory. The boundary on-shell action providing the boundary condition
then takes the following form:
\begin{equation}
I_{os}=a \int^{\infty}_{-\infty} d^3z\left[
\epsilon_{ijk}\left(A^{a(0)}_{i}(z)\partial_jA^{a(0)}_{k}(z)
-\frac{1}{3}g\epsilon^{abc}A^{a(0)}_{i}(z)A^{b(0)}_{j}(z)A^{c(0)}_{k}
(z)\right)+ \frac{1}{\rho^2}\mathcal {L}_{NL}\right],
\end{equation}
where $a$ is a real constant and $\mathcal {L}_{NL}$ is a Lagrangian
providing the non-local term in eq.(\ref{Self-dual-boundary-with_A})
and we have pulled out $\rho$ dependence explicitly. We now note that the
coupling $\frac{1}{\rho^2}$ explicitly breaks the scaling symmetry as
$z \rightarrow Lz$ and $A^{a(0)}_{i} \rightarrow \frac{1}{L}
A^{a(0)}_{i}$ and its scaling property shows that it is a relevant coupling.
Since, we promote $\rho$ as a parameter in the boundary on-shell
action, we can take a limit as $\rho \rightarrow \infty$. In this
limit, $\mathcal{L}_{NL}$ will relatively suppressed, and
the boundary theory becomes approximately pure Chern Simons theory.
\section{Conclusion}
\label{sec:conclusion}
We studied various boundary conditions for $SU(2)$ Yang-Mills theory
in $AdS_4$ background. One of the motivation was to introduce
interactions in the boundary CFT. Momentum dependent cubic
interactions in Yang-Mills theory lead to non-trivial interaction
terms in the boundary action both in cases of Dirichlet and Neumann
boundary conditions. We computed bulk Yang-Mills solutions to the first
subleading order to incorporate the leading effects of Yang-Mills interaction.
We found that in case of the Dirichlet boundary condition the boundary
propagator is proportional to $|\vec q|$, where $\vec q$ is three
dimensional momentum. The cubic interaction has the form
\begin{equation}
\Delta_{ijk}^{D,abc}(q,l,p) \sim ig\epsilon^{abc}\delta^{3}(q+p+l)
\frac{(l-q)_k\delta_{ij}+(p-l)_i\delta_{jk}+(q-p)_j\delta_{ik}}{2(|q|+|p|+|l|)}.
\end{equation}
The Neumann boundary condition on the other hand has the propagator
proportional to $1/|q|$ and the cubic interaction is
\begin{equation}
\Delta_{ijk}^{N,abc}(q,l,p) \sim \frac{ \Delta_{ijk}^{D,abc}(q,l,p)}{|q||p||l|} .
\end{equation}
Another motivation was to study more interesting boundary conditions
like massive and self dual boundary conditions in the context of
non-abelian gauge theory. While the massive boundary condition gives
rise to massive gauge theory on the boundary, the self dual boundary
condition takes the form of Bogomolnyi equation in the small $r$
expansion around the boundary. The combined massive and self dual
condition gives massive Chern Simons gauge theory action on the
boundary. Equations of motion derived from this action were studied
as self-duality conditions in odd dimensions\cite{Townsend1}.
We studied the effect of approximate electric-magnetic duality on
$SU(2)$ Yang-Mills theory defined on $AdS_4$ and resulting boundary
contribution. Although the symmetry is not exact it seems to point
towards a Chern-Simon like term on the boundary in addition to a
non-local piece. It would be interesting to explore effects of
duality on boundary conditions in the $AdS$ space, particularly in the
context of supersymmetric gauge theories.
We also studied instanton solution in $AdS_4$ with unit charge. While
it was a straightforward generalization of the solution in $\mathbb
R^4$ due to conformal invariance of classical action and self-duality
condition, implication of the solution are quite interesting. In
contrast to what happens in $AdS_5/CFT_4$ correspondence, where
D-instantons in $AdS_5$ do not modify the boundary condition, in
$AdS_4$ case, the Yang-Mills instanton becomes pure gauge on the
Poincare horizon and modifies the boundary condition on $AdS$
boundary. We showed that the boundary action is the Chern Simons
action with a non-local deformation. It would be interesting to
understand this non-local deformation better.
In this paper we concentrated only on
the gauge field sector, it would be interesting to combine it with
analysis of fermion boundary conditions \cite{Henneaux:1998ch}. In
particular, it would be interesting to classify supersymmetric
boundary conditions\footnote{For some work along these lines see
\cite{Sebastian1}.}. This analysis however is beyond the scope of this
work but we will address some of these issues in future.
\subsection*{Acknowledgments} {We would like to thank Stanley Deser,
Ashoke Sen, Rajesh Gopakumar, Suvrat Raju, Sudhakar Panda for useful
discussion. Especially, Jae-Hyuk Oh thanks his ${\mathcal W.J.}$ We
would also like to thank anonymous referee for useful suggestions
and providing pointers in those directions. This work is partially
supported by 11-R$\&$D-HRI-5.02-0304 at Harish-Chandra Research
Institute, India. }
\section*{Appendix}
|
{
"timestamp": "2012-10-08T02:02:39",
"yymm": "1203",
"arxiv_id": "1203.2106",
"language": "en",
"url": "https://arxiv.org/abs/1203.2106"
}
|
\section{INTRODUCTION}
Solar flares have been understood as the result of magnetic reconnection in the solar corona \citep[see recent review by][]{Hudson2011SSRv..158....5H}.
Although magnetic field evolution in solar photosphere plays important roles in building energy and triggering eruption, most models of flares imply that photospheric magnetic fields do not have rapid, irreversible changes associated with the eruptions. The traditional picture is that the solar surface, where the coronal magnetic fields are anchored, has much higher density and gas pressure than the corona. In recent years, however, rapid and irreversible changes of photospheric magnetic field have been found to be closely associated with flares \citep[e.g.,][]{WangH1992SoPh..140...85W, Wang+etal1994ApJ...424..436W, Kosovichev+Zharkova2001ApJ...550L.105K, Spirock+etal2002ApJ...572.1072S, Wang+etal2002ApJ...576..497W, Yurchyshyn+etal2004ApJ...605..546Y, Wang+etal2004ApJ...605..931W, Sudol+Harvey2005ApJ...635..647S, WangH2006ApJ...649..490W, Wang+Liu2010ApJ...716L.195W, Petrie+Sudol2010ApJ...724.1218P, WangS+etal2011arXiv1103.0027W}. A flare-associated change of sunspot white-light structure was also discovered
\citep{WangH+etal2004ApJ...601L.195W, Deng+etal2005ApJ...623.1195D, LiuC+etal2005ApJ...622..722L, ChenW+etal2007ChJAA...7..733C}. I.e., Pieces of peripheral sunspot penumbra can suddenly disappear right after a flare. In particular, \citet{LiuC+etal2005ApJ...622..722L} discussed the peripheral penumbral decay and central penumbral darkening of $\delta$ spots seen in a number of X-class flares, and proposed a reconnection picture where the central (i.e., near flaring polarity inversion line (PIL)) magnetic fields collapse inward resulting in a more horizontal configuration and the peripheral fields turn toward the flaring PIL resulting in a more vertical configuration after flares. These observational results have been summarized recently by \citet{Wang+Liu2010ApJ...716L.195W} in the context of the conjecture of \citet{Hudson+Fisher+Welsch2008ASPC..383..221H} and \citet{Fisher+etal2010arXiv1006.5247F}. These authors quantitatively assessed the back reaction on the solar surface and interior resulting from drastic coronal field evolution required to release energy, and made the prediction that after flares, the coronal magnetic fields should collapse toward the flaring PIL. In addition, flare induced sudden lateral motions of penumbrae were also found \citep{Gosain09,Mattews11}.
On a different front, sunspot penumbrae have been suggested to exhibit an ``uncombed'' structure \citep{Solanki+Montavon1993A&A...275..283S}. The basic idea is that the penumbral fields have two intermingled magnetic components: a more vertical magnetic background called ``spines'' and a more horizontal magnetic component called ``intraspines'' \citep{lites+etal1993}. There is a broad consensus that the intraspines carry most of Evershed flows throughout the entire penumbra \citep[e.g.,][]{BellotRubio+etal2003A&A...403L..47B, BellotRubio+etal2004A&A...427..319B, Borrero+etal2005A&A...436..333B, Ichimoto+etal2007PASJ...59S.593I, Borrero+Solanki2008ApJ...687..668B, Deng+etal2010ApJ...719..385D}. Reflecting in photospheric intensity images, the intraspines are manifested as (1) bright grains or filaments (a.k.a. bright heads) in the inner and middle penumbra representing locations of hot Evershed upflows \citep[e.g.,][]{rimmele+marino2006, Ichimoto2010mcia.conf..186I}, and (2) dark fibrils (a.k.a. dark tails) following those bright grains/filaments and extending to the middle and outer penumbra corresponding to cooled horizontal Evershed flows that are guided by nearly horizontal magnetic field \citep[e.g.,][]{stanchfield+thomas+lites1997, schlichenmaier+jahn+schmidt1998b, Rempel2011ApJ...729....5R}. These are evidenced by recent observations and simulations that the Evershed flows measured by Dopplergrams correlate with brighter features in the inner penumbra and with darker features in the outer penumbra \citep{Schlichenmaier+etal2005AN....326..301S, BellotRubio+etal2006A&A...453.1117B, Ichimoto+etal2007PASJ...59S.593I, Rempel2011ApJ...729....5R}. It should be noted that in the outer penumbra, the spines could appear relatively brighter than the horizontal-flow-carrying intraspines seen as dark fibrils. Thus, in short, the bright grains or filaments in the inner and middle penumbra are most likely associated with the inner portion of intraspines, while bright filaments in the outer penumbra may also be associated with spines. It is also worth mentioning that the outer ends of intraspines especially in the inner penumbra do not necessarily dip into the photosphere but may tilt upward \citep[see Figure 19 of][]{Rempel2011ApJ...729....5R}, and that intraspines in the outer penumbra are more likely to turn back into the photosphere. The uncombed penumbral structure has been corroborated by recent advanced space- and ground-based spectro-polarimetric observations as well as radiative magnetohydrodynamic (MHD) modeling \citep[e.g.,][]{Ichimoto+etal2007PASJ...59S.593I,Rempel+etal2009Sci...325..171R, Jurcak+BellotRubio2008A&A...481L..17J,BeckC2011A&A...525A.133B,Rempel2011ApJ...729....5R,Borrero+Ichimoto2011LRSP....8....4B}.
We note that besides the uncombed penumbra model, ``gappy penumbra model'' \citep{Spruit+Scharmer2006A&A...447..343S, Scharmer+Spruit2006A&A...460..605S} explains the penumbra in a different scenario for high-resolution observations in the order of 0.1$^{\prime\prime}$, in which the dark cores in penumbral grains can be resolved. In this model, the penumbral filamentary structure is interpreted as due to convection in radially aligned field-free gaps beneath a nearly potential magnetic field. The gappy model can also explain many aspects of penumbral structure however, at present the angular resolution of the spectropolarimetric observations is insufficient to verify it.
To facilitate our analysis and discussion, we first present an investigation on the uncombed penumbral structure (Section~\ref{structure}), and then base the interpretation of our observations (Section~\ref{flare}) on the framework of the uncombed penumbra model depicted using schematic cartoons (Section~\ref{discussion}). We mainly concern ourselves with a natural question that when sunspot penumbra suddenly disappears, what would happen with respect to the uncombed magnetic structure? The high-resolution observation from Hinode covering major flares provides an excellent opportunity to answer this question. We demonstrate evidences that at the site of the penumbral decay, dark fibrils completely disappear while bright grains/filaments evolve into faculae that are signatures of vertical magnetic flux tubes. These observations suggest that the intraspines are straightened upward to become more vertical. Therefore, the original uncombed magnetic structure seems to be re-organized toward the vertical direction as a consequence of the field restructuring due to flares, which results in the disappearance of penumbrae.
\section{OBSERVATIONS AND RESULTS}
\subsection{The ``Umcombed'' Penumbral Structure} \label{structure}
We study the relationship between intensity and magnetic inclination of the detailed penumbral structure using Hinode Spectro-Polarimeter (SP) vector magnetograms\footnote{http://sot.lmsal.com/data/sot/level2d/} obtained for a $\beta$ sunspot near disk center in the NOAA Active Region (AR) 10923 on 2006 November 14 07:15:04~UT, the results of which are presented in Figure~1. The vector data were retrieved using High Altitude Observatory Milne-Eddington inversion \citep[e.g.,][]{Lites+etal2008ApJ...672.1237L}, and we further transformed the observed fields to the local Cartesian coordinates so that the inclination angle could be measured with respect to the local surface normal (i.e., 0$^{\circ}$ is vertical and 90$^{\circ}$ is horizontal). We applied the threshold method around the median value to individually identify the bright and dark penumbral fibrils (enclosed by the yellow and purple contours, respectively, in Figure~1, left column). Besides analyzing the overall intensity and magnetic inclination distribution for the entire penumbra (Figure~1, middle column), we also trace the shape of the spot (red contours in Figure~1, left column) to examine the azimuthal averages of intensity and inclination angle at different normalized radius (Figure~1, right column). The results of histograms and radial profiles unambiguously indicate that the dark penumbral fibrils generally exhibit a more horizontal orientation than the bright fibrils, with larger difference nearer the outer penumbra. This is consistent with the results derived based on the ground-based high-spatial resolution observations of line-of-sight (LOS) magnetograms \citep{Langhans+etal2005A&A...436.1087L}, and is thus in line with the uncombed penumbral model with two distinct but related magnetic components. We note that in the uncombed model, the dark fibrils that we identified could be the outer portion of intraspines, while the bright fibrils could be either the inner portion of intraspines or spines.
\subsection{Rapid Restructuring of Penumbrae during Flares} \label{flare}
The high spatial and temporal resolution observations obtained with Hinode/Solar Optical Telescope \citep[SOT;][]{Tsuneta+etal2008SoPh..249..167T} on 2006 December~6 and 2007 June~4 provide useful data sets to study the evolution of penumbral fine structure associated with major flares.
An X6.5 flare occurred at 18:29~UT on 2006 December 6 in AR 10930 (S06E60) and an M8.9 flare occurred at 05:06~UT on 2007 June 4 in AR 10960 (S06E51). The main data source is the continuous G-band images that have a nominal cadence of 1 or 2 minutes and pixel scale of 0.11$^{\prime\prime}$ covering the flares.
Figure 2 shows selected images exhibiting the rapid disappearance of a penumbral section (marked by the red box) associated with the X6.5 flare on 2006 December 6. The lower row images clearly show the transitional decaying phases of the penumbra within an hour after the onset of the flare. As reference, a stable penumbral area and a facular area are selected and marked by green and yellow boxes, respectively. Figure 3 shows images before and after the M8.9 flare on 2007 June 4. Two mpeg movies for the two events are provided as online material to depict more dynamical detail. Here are the few points that we noted by studying the time sequence of images.
(1) Morphologically, before the flares, the rapidly changing areas appear as regular penumbral structure, consisting of bright grains/filaments and dark fibrils lying at the periphery of sunspots. After the flares, the morphology is similar to the area consisting of faculae.
(2) As these two regions are far away from disk center, the faculae
(G-band bright points) appear much more obviously comparing to the regions close to disk center, such as on 2006 December 13 when an X3.4 flare occurred \citep{Tan+etal2009ApJ...690.1820T}. This is likely due to the ``hot wall''
effect --- the faculae contrast enhances when ${\rm cos}\theta$ is between 0.5 and 0.1,
where $\theta$ is the heliocentric angle \citep{spruit1976}. This gives us the opportunity to observe the structure and evolution of bright points that correspond to magnetic elements. Unfortunately, high-resolution magnetograms from SOT/SP were not available to cover these two flares. Therefore, we use the faculae as a proxy for vertical flux tubes.
(3) Based on the analysis of movies, it is clear that the bright points appear in-situ immediately after the penumbral decay, i.e., they were not transported from other areas.
(4) We can not trace all the bright points continuously due to data gaps, but it is obvious that some penumbral bright grains/filaments evolve into faculae. Statistically, the number of post-flare faculae is similar to that of pre-flare penumbral bright grains/filaments in the same area. For example, we identified 10 local maxima in the form of penumbral bright grains/filaments inside the red box before the flare on 2006 December 6. 10 local maxima are also identified in the form of faculae in the same box after the flare. For the 2007 June 4 flare, 8 local maxima are identified in the form of penumbral bright grains/filaments before the flare, while 9 local maxima are identified in the form of faculae after the flare.
(5) The penumbrae started to decay immediately after the flare brightening swept across them. Within an hour after the onset of the flare, the penumbral dark fibrils gradually lose their filamentary structure and progressively disappear from the outer part toward the inner part, which leads to a reduction of penumbral area and an increase of the corresponding brightness. Finally, the dark fibrils completely disappear, leaving bright facular points at the locations where the original penumbral bright grains/filaments are located.
(6) After the penumbrae decayed, the area of the parent umbrae apparently increased.
To quantitatively demonstrate the evolution of the decaying penumbrae, in Figures 4 and 5 we construct the time sequence of intensity histogram for the outlined areas shown in Figures 2 and 3. Each figure has three panels. The center panel is for the rapidly decaying penumbral area. The left and the right panels are for the referential stable penumbral and facular regions, respectively. In each panel, the X-axis marks the time and the Y-axis marks the normalized intensity. The colors along the Y-direction represent the intensity distribution (histogram) of all pixels inside each outlined area, i.e., the brightness represents the fraction of those pixels falling in the individual intensity range. Soft X-ray and hard X-ray flare light curves are overplotted. It is obvious that only the decaying penumbral areas have a sudden change of the intensity distribution closely associated with the flares. Specifically, their intensity distribution in the pre-flare state well resembles that of the regular penumbra, while that in the post-flare state becomes like the facular region. The change of the penumbral structure associated with the X6.5 flare is more abrupt (noticeable changes took place within a few minutes and reached the stable level in about a half hour), while the intensity distribution of the 2007 June 4 region associated with the M8.9 flare evolves gradually for about 40 minutes before the data gap.
To further explore the change of the magnetic structure, in Figures 6 and 7 we use LOS magnetograms from the Michelson Doppler Imager (MDI) and Global Oscillation Network Group (GONG) to study the magnetic flux change in the decaying penumbral areas. It can be clearly seen that for both events, the flux has a rapid stepwise change temporally correlated with the flares, with a significant decreases for AR 10930 while a pronounced increases for AR 10960. We draw schematic pictures to explain the different behavior of the flux change in these two active regions close to the limb. For the X6.5 flare occurred in AR 10930, the decaying penumbra is located at the disk-ward side of the flaring PIL, in which case the flux would decrease as the field lines turn from horizontal to vertical (i.e., away from the LOS direction). On the contrary, for the M8.9 flare occurred in AR 10960, the decaying penumbra is located at the limb-ward side of the flaring PIL, in which case the flux would increase when the field lines turn from horizontal to vertical (i.e., toward the LOS direction). Therefore, the rapid flux variation in the two events supports a common change of the field lines turning from horizontal to vertical direction in the peripheral, decaying penumbral region.
\section{SUMMARY AND DISCUSSION} \label{discussion}
Based on the G-band observations presented above, we conclude that the rapid decay of penumbra has two distinct aspects: the dark penumbral fibrils disappear completely, while bright penumbral grains/filaments evolve to faculae. The magnetic fluxes measured in the decaying penumbrae show stepwise changes temporally correlated with the flares, suggesting that those magnetic fields turn from horizontal to vertical.
To accommodate the observation of penumbral decay in the uncombed penumbral model, we propose a scenario as illustrated in Figure 8 highlighting the changes of field structure in the penumbra region associated with the flare occurrence. The upper panel depicts a $\delta$ spot in the preflare state with uncombed penumbral structure as described in Section~1, which consists of spines (blue lines) and intraspines (red lines). A majority of Evershed flows (red arrows) are carried by the intraspines. The bright grains/filaments (yellow spots) indicate the inner portion of intraspines in the inner and middle penumbra or spines in the outer penumbra. In the postflare state as shown in the lower panel of Figure 8, the horizontal segment of intraspines (the black shaded portion) could turn vertical hence leading to the disappearance of dark fibrils. At the locations of the original bright penumbral grains/filaments manifesting the inner portion (footpoints) of the intraspines, the straightened flux tubes naturally appear as faculae subsequently. Meanwhile, the spines may also become more vertical following the overall trend, so that the corresponding bright filaments evolve to faculae as well. As a result, the original uncombed magnetic structure of penumbra seems to be combed toward the vertical direction as the consequence of flares, which results in the transition of penumbrae to faculae. For the field lines near umbra, they may merge into the umbral fields as they become more vertical, which can then explain the observed increase of the umbral area.
We further discuss the observed rapid changes of penumbral structure and the possible interpretation and implication as follows.
(1) In a $\delta$ configuration (see Figure 8), part of the penumbrae is located at the central region in between the two umbrae with opposite polarities and usually exhibits highly sheared configuration along the PIL, while other portions of penumbrae lie at the periphery of the $\delta$ spot and is separated from the PIL by its parent umbra. We must clarify that all the rapid penumbral decays associated with flares that we have found so far \citep[][and in this paper]{WangH+etal2004ApJ...601L.195W, Deng+etal2005ApJ...623.1195D, LiuC+etal2005ApJ...622..722L, ChenW+etal2007ChJAA...7..733C} occur at the peripheral penumbra region. As a comparison, we found that the penumbrae at the central region are usually enhanced after flares (i.e., the intensity becomes lower and the horizontal Evershed flow increases) \citep[see e.g.,][]{LiuC+etal2005ApJ...622..722L, DengN+etal2011ApJ...733L..14D}, even though they may also be swept by flare ribbons. This also explains that in the cases studied by Li \& Zhang (2009), the sudden penmubral decay was not observed while the flare ribbons swept sunspots.
(2) In order that the horizontal segment of the intraspine flux tubes could easily turn vertical, the dark fibrils carrying horizontal Evershed flows may be relatively shallow while bright grains may be deeply anchored. This is supported by observations of helioseismic and spectropolarimetric inversions \citep{gizon+duvall+larsen2000, Borrero+etal2006A&A...450..383B, BeckC2011A&A...525A.133B}. The magneto-convection with similar properties exist in both quiet region and penumbra in the forms of granulation and anisotropic granulation, respectively \citep{Rempel2011ApJ...729....5R}. This might also be supported by the finding of \citet{KuboM+etal2008ApJ...681.1677K} that when the penumbral boundary moves inward, the granules appear in the outer boundary. The observation of sudden disappearance of penumbra in the 2006 December 13 X3.4 event close to disk center demonstrates the similar pattern, in which the regular granules appear immediately after the penumbra disappear \citep{Tan+etal2009ApJ...690.1820T}.
(3) The rapid penumbral decay is temporally and spatially correlated with flares, which indicates the crucial role of the field restructuring due to flares in such a rapid decay process different from the general sunspot evolution. Although the decayed penumbrae are swept across by flare ribbons, the penumbral decay is not due to transient heating or brightening induced by flares, as the decay is permanent and non-reversible indicating that the original uncombed magnetic fields must undergo some fundamental changes most probably toward the vertical direction. Although we believe that it is the flare-induced field change that leads to the permanent penumbral decay, it is not clear whether the flare heating plays a role in the mass flow within penumbral flux tubes to help the horizontal magnetic fields to turn vertical more easily.
(4) As mentioned previously, the peripheral penumbral decay could be accompanied by the enhancement (darkening) of the central penumbrae \citep[e.g.,][]{LiuC+etal2005ApJ...622..722L, DengN+etal2011ApJ...733L..14D} as the field lines at the flaring PIL become more horizontal after flares \citep{Wang+Liu2010ApJ...716L.195W}, which is consistent with the ``implosion'' picture of \citet{Hudson2000ApJ...531L..75H} \citep[also see][]{LiuR+WangH2009ApJ...703L..23L}. Recently, this was further discussed in terms of back reaction of coronal field restructuring on the solar surface, as the implosion would result in a downward Lorentz force \citep{Hudson+Fisher+Welsch2008ASPC..383..221H, Fisher+etal2010arXiv1006.5247F}. We speculate that when the magnetic energy in the flaring core region is decreased, a negative pressure core would form above the flaring PIL attracting ambient fields to fill the void. Subsequently, the central magnetic fields collapse downward to become more horizontal near the PIL, while the fields in the peripheral region turn more vertical toward the central region (see Figure 8). As reflected in white-light observations, the central penumbrae could then be enhanced (darkened), and portions of the peripheral penumbrae may decay depending on the flare magnitude, the distance of the peripheral penumbrae to the PIL, and whether being swept by flare ribbons (i.e., in direct association with the flaring process). The analysis of the three-dimensional MHD simulations of an emerging twisted flux tube \citep{FanY2010ApJ...719..728F} also evidences the enhanced and more inclined field at the central flaring PIL together with the decreased and more vertical peripheral fields after the eruption and the associated flares \citep{LiY+etal2011ApJ...727L..19L}. As related, \citet{KuboM+etal2011ApJ...731...84K} elaborated that the decay of penumbra must be closely associated with the increase of field inclination.
(5) As \citet{BellotRubio+etal2008ApJ...676..698B} observed, sunspot penumbra may decay slowly in the course of a few days leaving the structure of ``naked'' sunspots (i.e., spots without penumbrae; \citealt{Liggett+Zirin1983SoPh...84....3L}). When dark fibrils disappear, some bright structures (they name as ``fingers'') appear and carry upward flows in the order of 1--2~km~s$^{-1}$. The ``fingers'' may be similar flux tubes as the faculae discussed in the current paper. In this sense, flares may accelerate the decay process of sunspot penumbra in the outskirt to within an hour instead of days. Interestingly, \citet{RempelM2011SPD....42.1001R} found that the top boundary conditions determine the radial extent of penumbra in the simulated sunspots, and that the penumbra forms or destructs within 0.5 hours after the top boundary changes. This is very similar to our observations where the penumbra disappears rapidly within an hour after the onset of the flares, which suggest a reconfiguration of coronal fields accompanied with violent changes in the upper atmosphere that resembles the change of the top boundary in the simulation. Therefore, our observations strongly support the possible impact by the coronal transients on the photospheric magnetic structure of sunspots.
\acknowledgments
The authors thank Drs. K. Ichimoto, M. Rempel, and R. Liu for reading the manuscript and very helpful comments to improve the paper. We also thank the referee for valuable comments that help us to improve the paper, and Yuan Yuan for help on image processing. 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).
This work is supported by NSF grants AGS-0839216, AGS-0819662, and AGS-0849453, NASA grants NNX08AQ90G, NNX11AQ55G and NNX11AO70G. ND is supported by NASA grant NNX08AQ32G.
|
{
"timestamp": "2012-03-13T01:01:22",
"yymm": "1203",
"arxiv_id": "1203.2267",
"language": "en",
"url": "https://arxiv.org/abs/1203.2267"
}
|
\section{Introduction}
New generation supercapacitors are used for a broad range of applications in nanoscopic scale technologies. In water purification technology, capacitive desalination is an efficient candidate that might replace the current leading technics such as reverse osmosis, a membrane based purification method known to suffer from the membrane fouling phenomenon~\cite{rev1}. Supercapacitors are also used as low cost and long life energy storage devices with considerably higher energy densities than conventional electrolytic capacitors~\cite{rev2}. A through understanding of the double layer structure of these devices is thus necessary to optimize their efficiency.
The understanding of the double layer structure was limited for several decades to the Gouy-Chapman-Stern model~\cite{rev3}. This model was later completed by considering additional effects specific to electrolyte solutions, to name but a few, the steric layer associated with the size of solvent molecules as well as the dipolar alignment close to the interface~\cite{Blum}, non-local effects in electrolytes at metallic interfaces~\cite{Kor1}, ionic crowding~\cite{Kor2} and overscreening~\cite{Kor3}.
Supercapacitors are commonly fabricated from carbon based materials with a dielectric permittivity $\varepsilon_m\approx 2-5$ much lower than the permittivity of the water solvent $\varepsilon_w=78$. The polarization of the interface resulting from this large dielectric discontinuity can drastically change the physics of the double layer. Image dipole interactions were considered in Ref.~\cite{mac} for a metallic interface. However, the work accounted exclusively for the effect of image interactions on the dipolar orientation without considering their role on the interfacial dipole density. Furthermore, it was recently shown in Ref.~\cite{prlnetz} that the Gouy-Chapman (GC) capacitance largely overestimates the experimental data obtained for carbon based materials. The failure of the GC capacitance was explained by the unability of the Poisson Boltzmann formalism to account for non-local dielectric effects.
In order to gain insight about the contribution of surface polarization effects on the capacitance of low dielectric substrates, we introduce in this work a first microscopic modeling of solvent molecules beyond the MF level approximation. Namely, we derive an extended dipolar PB (EDPB) equation that can self-consistently take into account the interfacial solvent depletion. This depletion results from the interaction of solvent molecules (modeled as dipoles) with their electrostatic image, an effect absent in the mean-field level DPB equation~\cite{dundip,prlorl}. The prediction of the EDPB equation is shown to agree well with experimental data for the differential capacitance of carbon based materials. However, it is also shown that the DPB formalism yields the same result as the PB equation, that is, it overestimates the experimental data by one order of magnitude. These observations strongly suggests that the dielectric discontinuity between the substrate and the solvent can solely explain the observed low values of the differential capacitance of carbon based materials, unlike the conclusion of Ref.~\cite{prlnetz} where it was argued that the surface polarity does not play a major role in the differential capacitance. Our results are also in agreement with the experimental work in Ref.~\cite{exp}, where the surface hydrophobicity was actually shown to strongly reduce the capacitance of carbone nanotubes.
\section{Extended Dipolar Poisson Boltzmann (EDPB) equation}
We will present in this part the derivation of an extended dipolar Poisson-Boltzmann formalism. The field-theoretic partition function of ions immersed in a dipolar liquid was derived in Ref.~\cite{dundip} as a functional integral over a fluctuating electrostatic potential $\phi(\mathbf{r})$ in the form $\mathcal{Z}=\int \mathcal{D}\phi\;e^{-H[\phi]}$, where the Hamiltonian functional is given by
\begin{eqnarray}\label{HamFunc2}
H[\phi]&=&\int\mathrm{d}\mathbf{r}\left[\frac{\left[\nabla\phi(\mathbf{r})\right]^2}{8\pi\ell_B(\mathbf{r})}-i\sigma(\mathbf{r})\phi(\mathbf{r})\right]\\
&&-\int\frac{\mathrm{d}\mathbf{r}\mathrm{d}\mathbf{\Omega}}{4\pi}\lambda_de^{E_d-V_w(\mathbf{r})+i\left(\mathbf{p}\cdot\nabla\phi\right)}\nonumber\\
&&-\sum_i\lambda_i \int\mathrm{d}\mathbf{r} e^{E_i-V_w(\mathbf{r})+i \left[q_i\phi(\mathbf{r})\right]}.\nonumber
\end{eqnarray}
The first integral term of the Hamiltonian~(\ref{HamFunc2}) is composed of the Maxwell tensor associated with a freely propagating electric field $\nabla\phi(\mathbf{r})$ in the air, and a second part that couples the corresponding potential $\phi(\mathbf{r})$ to a fixed surface charge distribution $\sigma(\mathbf{r})$. The second and third integrals respectively account for the presence of solvent molecules (point dipoles) and ions of different species denoted by the index $i$. Moreover, $\mathbf{r}=(x,y,z)$ is the configurational space and $\mathbf{\Omega}=(\theta,\varphi)$ stands for the solid angle characterizing the orientation of solvent molecules, with $\theta$ the angle between the dipole and the $z$ axis. We note that the external wall potential $V_w(\mathbf{r})$ in Eq.~(\ref{HamFunc2}) restricts the space accessible to the particles, and in the case of the single dielectric interface located at $z=0$, it is of the form $V_w(z<0)=\infty$ and $V_w(z>0)=0$. Furthermore, $\lambda_d$ and $\lambda_i$ are respectively the fugacity of dipoles and ions, $\mathbf{p}$ the dipole moment vector, and $q_i$ stands for the valency of ions for the species $i$. The heterogeneous Bjerrum length is defined as $\ell_B(\mathbf{r})=e^2/\left[4\pi\varepsilon(\mathbf{r})k_BT\right]$, where $e$ is the elementary charge, $T=300$ K is the ambient temperature, and $\varepsilon(\mathbf{r})=\varepsilon_0\theta(z)+\varepsilon_m\theta(-z)$ is the dielectric permittivity of the medium in the absence of solvent molecules for the same single planar interface geometry. More precisely, $\varepsilon_0$ and $\varepsilon_m$ denote respectively the dielectric permittivity of the air (the subspace in $z>0$) and the low dielectric subbstrate located at $z<0$. From now on, the dielectric permittivities will be expressed in units of $\varepsilon_0$. We also note that the Bjerrum length in the air is $\ell_B\approx 55$ nm. Finally, the self energy of ions and polar molecules that are substracted from the potential and electrostatic field respectively read $E_i=\frac{q_i^2}{2}v_c^b(\mathbf{r}-\mathbf{r}')|_{\mathbf{r}=\mathbf{r}'}$ and $E_d=\frac{1}{2}(\mathbf{p}\cdot\nabla_\mathbf{r})(\mathbf{p}\cdot\nabla_{\mathbf{r}'})v_c^b(\mathbf{r}-\mathbf{r}')$, where the Coulomb operator in the air is defined as ${v^b_c}^{-1}(\mathbf{r},\mathbf{r}')=-\frac{k_BT\varepsilon_0}{e^2}\Delta\delta(\mathbf{r}-\mathbf{r}')$.
In this work, we aim at investigating the model of Eq.~(\ref{HamFunc2}) beyond the MF approximation where surface polarization effects are absent~\cite{dundip, prlorl}. One way to progress consists in opting for a variational minimization procedure that aims at finding the upper boundary for the dimensionless Grand potential of the system $\Omega=-\ln Z$ by minimizing the variational Grand potential defined as $\Omega_v=\Omega_0+\left\langle H-H_0\right\rangle_0$, where the reference Hamiltonian is a Gaussian functional of the form
\begin{equation}
H_0=\frac{1}{2}\int_{\mathbf{r},\mathbf{r}'}\left[\phi(\mathbf{r})-i\phi_0(\mathbf{r})\right]
v^{-1}_0(\mathbf{r},\mathbf{r}')\left[\phi(\mathbf{r}')-i\phi_0(\mathbf{r}')\right].
\label{H0phi}
\end{equation}
Furthermore, $\phi_0(z)$ is a variational external potential and the electrostatic trial kernel is chosen in the same form as in Refs.~\cite{pre,prl,jcp},
\begin{equation}\label{DH1}
v_0^{-1}(\mathbf{r},\mathbf{r}')=\frac{k_BT}{e^2}\left[-\nabla(\varepsilon_v(\mathbf{r})\nabla)+\varepsilon_v(\mathbf{r})\kappa_c^2(\mathbf{r})\right]\delta(\mathbf{r}-\mathbf{r}'),
\end{equation}
where the piecewise variational dielectric permittivity is defined as $\varepsilon_v(\mathbf{r})=\varepsilon_w\theta(z)+\varepsilon_m\theta(-z)$ and the trial screening length is given by $\kappa_c(\mathbf{r})=\kappa_c\theta(z)$. After performing the functional integrals over $\phi(\mathbf{r})$, one gets
\begin{eqnarray}
\label{vargrpot}
\Omega_v&=&\Omega_{0}+\frac{k_BT}{2e^2}\int\mathrm{d}\mathbf{r}\mathrm{d}\mathbf{r}'\delta(\mathbf{r}-\mathbf{r}')\nonumber\\
&&\hspace{9mm}\times\left\{\left[\varepsilon(\mathbf{r})-\varepsilon_v(\mathbf{r})\right]\nabla_\mathbf{r}\cdot\nabla_{\mathbf{r}'}-\varepsilon_v(\mathbf{r})\kappa_c^2(\mathbf{r})\right\}v_0(\mathbf{r},\mathbf{r}')\nonumber\\
&&+\int\mathrm{d}\mathbf{r}\left\{\sigma(\mathbf{r})\phi_0(\mathbf{r})-\frac{k_BT}{2e^2}\varepsilon(\mathbf{r})\left[\nabla\phi_0(\mathbf{r})\right]^2\right\}\\
&&-\sum_i\int\mathrm{d}\mathbf{r}\rho_i(\mathbf{r})-\int\frac{\mathrm{d}\mathbf{r}\mathrm{d}\mathbf{\Omega}}{4\pi}\bar\rho_d(\mathbf{r},\mathbf{\Omega}),\nonumber
\end{eqnarray}
where the gaussian contribution reads $\Omega_0=-\ln\int \mathcal{D}\phi\;e^{-H_0[\phi]}$. We also defined above the local ion density
\begin{equation}\label{deni1}
\rho_i(\mathbf{r})=\lambda_ie^{E_i-V_w(\mathbf{r})}e^{-q_i\phi_0(\mathbf{r})-\frac{q_i^2}{2}v_0(\mathbf{r},\mathbf{r})}
\end{equation}
and the local density of dipoles with orientation $\mathbf{\Omega}$
\begin{eqnarray}\label{dend1}
\bar\rho_d(\mathbf{r},\mathbf{\Omega})&=&\lambda_de^{E_d-V_w(\mathbf{r})}\\
&&\times e^{-\mathbf{p}\cdot\nabla\phi_0(\mathbf{r})-\frac{1}{2}(\mathbf{p}\cdot\nabla_\mathbf{r})(\mathbf{p}\cdot\nabla_{\mathbf{r}'})v_0(\mathbf{r},\mathbf{r}')|_{\mathbf{r}'=\mathbf{r}}}.\nonumber
\end{eqnarray}
By taking the derivative of the variational Grand potential Eq.~(\ref{vargrpot}) with respect to $\kappa_c$ and $\varepsilon_v$, one gets $\kappa_c^2=4\pi\ell_w\sum_i\rho_{b,i}q_i^2$ and $\varepsilon_v=\ell_B/\ell_w=\varepsilon_w=1+\frac{4\pi}{3}\ell_Bp_0^2\rho_{bd}$. These two relations respectively introduce the Debye-Huckel screening parameter and the Debye-Langevin form for the bulk dielectric permittivity of the water medium $\varepsilon_w$. The additional variational equation for $\phi_0(z)$, i.e. $\delta\Omega_v/\delta\phi_0(\mathbf{r})=0$ yields
\begin{equation}
\label{varel}
\frac{\partial}{\partial z}\tilde\varepsilon(z)\frac{\partial\phi_0(z)}{\partial z}+4\pi\ell_B\sigma(z)+4\pi\ell_B\sum\rho_i(z)q_i=0,
\end{equation}
where we took into account the translational symmetry of the electrostatic potential within the $(x,y)$ plane. We note that the variational minimization left us in Eq.~(\ref{varel}) with a spatially varying dielectric permittivity of the form
\begin{equation}\label{per1}
\tilde\varepsilon(z)=1-\frac{4\pi\ell_B}{\phi'_0(z)}\int\frac{\mathrm{d}\mathbf{\Omega}}{4\pi}\bar\rho_d(z,\mathbf{\Omega})p_z,
\end{equation}
where $p_z=p_0\cos\theta$ stands for the component of the dipolar moment vector $\mathbf{p}$ in the $z$ direction. We will call Eq.~(\ref{varel}) the Extended Dipolar Poisson Boltzmann (EDPB) equation.
The fugacity of dipoles and ions can be related to their bulk density in the limit $z\to\infty$ of the equations~(\ref{deni1}) and~(\ref{dend1}).
By injecting the obtained relations for the fugacities with the inverse of the kernel Eq.~(\ref{DH1})~\cite{pre} into Eqs.~(\ref{deni1}) and~(\ref{dend1}), the local density functions take the form
\begin{eqnarray}\label{deni2}
&&\rho_i(z)=\rho_{b,i}e^{-V_w(z)}e^{-q_i\phi_0(z)-V_c(z)}\\
\label{dend2}
&&\bar\rho_d(z,\mathbf{\Omega})=\rho_{bd}e^{-V_w(z)}e^{-\mathbf{p}\cdot\nabla\phi_0(z)-V_d(z,\mathbf{\Omega})},
\end{eqnarray}
where we defined the following ionic and dipolar potentials,
\begin{eqnarray}
\label{ion3}
&&V_c(z)=\frac{q^2\ell_w}{2}\int_0^\infty\frac{\mathrm{d}kk}{\rho_c}\Delta e^{-2\rho_cz}\\
\label{dip3}
&&V_d(z,\mathbf{\Omega})=U_d(z)+T_d(z)\cos^2\theta,
\end{eqnarray}
with the functions
\begin{eqnarray}
\label{funu}
&&U_d(z)=\frac{\ell_wp_0^2}{4}\int\frac{\mathrm{d}kk^3}{\rho_c}\Delta e^{-2\rho_cz}\\
\label{funt}
&&T_d(z)=\frac{\ell_wp_0^2}{4}\int\frac{\mathrm{d}kk}{\rho_c}(2\rho_c^2-k^2)\Delta e^{-2\rho_cz}.
\end{eqnarray}
and $\Delta=(\rho_c-\eta k)/(\rho_c+\eta k)$, $\eta=\varepsilon_m/\varepsilon_w$, and $\rho_c=\sqrt{\kappa_c^2+k^2}$.
Carrying out the integral over $\theta$ in Eq.~(\ref{per1}) with the dipole density Eq.~(\ref{dend2}) and the potential Eq.~(\ref{dip3}), the local dielectric permittivity takes the form
\begin{equation}\label{per2}
\tilde\varepsilon(z)=1+\frac{4\pi}{3}\ell_Bp_0^2\rho_{db}e^{-V_w(z)}e^{-U_d(z)}J(z),
\end{equation}
where we defined the function
\begin{eqnarray}
J(z)&=&\frac{3\sqrt\pi}{8T_d^{3/2}(z)}e^{\frac{p_0^2\phi'^2_0(z)}{4T_d(z)}}\left\{\mathrm{Erf}\left[\Psi_+(z)\right]+
\mathrm{Erf}\left[\Psi_-(z)\right]\right\}.\nonumber\\
&&-\frac{3e^{-T_d(z)}}{2T_d(z)}\frac{\sinh\left[p_0\phi'_0(z)\right]}{p_0\phi'_0(z)},
\end{eqnarray}
and the potentials
\begin{eqnarray}
\Psi_\pm(z)=\frac{2T_d(z)\pm p_0\phi'_0(z)}{2\sqrt{T_d(z)}}.
\end{eqnarray}
The EDPB Eq.~(\ref{varel}) has to be solved numerically with the ionic density profiles of Eq.~(\ref{deni2}) and the dielectric permittivity profile of Eq.~(\ref{per2}).
The second order differential equation~(\ref{varel}) should be solved with the boundary conditions $\phi_0(z\to\infty)=0$ and $\phi'_0(z\to 0^+)=2\varepsilon_w/\mu$, where the second boundary condition valid over the parameter domain $0\leq\varepsilon_m\leq\varepsilon_w$ follows by integrating Eq.~(\ref{varel}) in the close neighborhood of the interface, and noting that according to the dipolar potentials of Eqs.~(\ref{funu}) and~(\ref{funt}), one has $\rho_d(0)=0$ and $\tilde{\epsilon}(0)=1$ on the boundary. We also note that in the limit where the potentials $V_c(z)$, $U_d(z)$, and $T_d(z)$ vanish, EDPB equation.~(\ref{varel}) reduces to the mean field DPB equation of Refs.~\cite{dundip,prlorl}.
We finally note that the orientation averaged density of solvent molecules is obtained according to $\rho_d(\mathbf{r})=\int\frac{\mathrm{d}\mathbf{\Omega}}{4\pi}\bar\rho_d(\mathbf{r},\mathbf{\Omega})$. Evaluating the integral over $\theta$, the solvent density takes the form
\begin{eqnarray}
\label{dend3}
\rho_d(z)&=&\rho_{bd}\frac{\sqrt\pi}{4\sqrt T_d(z)}e^{-V_w(z)}e^{-U_d(z)}e^{\frac{p_0^2\phi'^2_0(z)}{4T_d(z)}}\\
&&\times\left\{\mathrm{Erf}\left[\Psi_+(z)\right]+\mathrm{Erf}\left[\Psi_-(z)\right]\right\}.\nonumber
\end{eqnarray}
\section{Numerical results}
We will investigate in this part the EDPB Eq.~(\ref{varel}) for a symmetric electrolyte composed of two ion species of bulk densities $\rho_{b,i}=\rho_{bi}$ and valency $q_i=q$. All numerical results will be derived for monovalent ions ($q=1$) in contact with a negatively charged planar surface, i.e. $\sigma(z)=-\sigma_s\delta(z)$ with $\sigma_s>0$. We also note that within the convention adopted in this article, the surface charge $\sigma_s$ is expressed in units of the elementary charge $e$. Moreover, the model parameters $\rho_{db}$ and $\varepsilon_w$ are taken the same as in Ref.~\cite{prlnetz}. Namely, the bulk density of solvent molecules is $\rho_{db}=50.8$ M, which yields with the dipole moment $p_0=1$ {\AA} the bulk dielectric permittivity $\varepsilon_w=71$.
The potential profile obtained from the numerical solution of the EDPB Eq.~(\ref{varel}) for the parameters $\rho_{bi}=0.1$ M, $\varepsilon_m=1$ and $\sigma_s=0.01$ $\mbox{nm}^{-2}$ is reported in Fig.~\ref{den1}.a. One notices that the potential profile is composed of three regions, namely two successive layers close to the interface where $\phi_0(z)$ behaves as a linear function of $z$, and a third layer over which $\phi_0(z)$ exponentially decays.
\begin{figure}
(a)\includegraphics[width=0.9\linewidth]{fig1.pdf}
(b)\includegraphics[width=0.9\linewidth]{fig2.pdf}
\caption{(Color online) (a) Electrostatic potential profile ($\sigma_s=0.01$ $\mbox{nm}^{-2}$) and (b) renormalized density and dielectric permittivity profiles for $\varepsilon_w=71$ and $\rho_{bi}=0.1$ M. The red line in (a) is from the restricted variational ansatz Eq.~(\ref{phires}) and the dashed black line corresponds to the solution of the EDPB equation.}
\label{den1}
\end{figure}
In order to understand the form of the potential profile, we illustrate in Fig.~\ref{den1} the form of the dielectric permittivity $\tilde{\epsilon}(z)$ and the screening parameter $\kappa^2(z)=\kappa_{c}^2e^{-V_c(z)}$ in the vanishing surface charge limit of Eq.~(\ref{varel}). It is seen that with increasing distance from the surface, the dielectric permittivity increases from the air permittivity $\tilde{\epsilon}(z)=1$ to the bulk permittivity $\tilde{\epsilon}(z)=\varepsilon_w$ over a distance $h\approx 2$ {\AA}. We note that this dipolar exclusion effect is mainly due to the interaction of solvent molecules with their electrostatic images. Then, one sees that this solvent depletion regime is followed by an ionic depletion regime of thickness $d\approx 6$ {\AA}, an effect known to originate from image charge interactions~\cite{pre}.
Inspired by the behaviour of $\tilde{\epsilon}(z)$ and $\kappa(z)$ that results from the interfacial depletion of solvent molecules and ions, we will introduce a restricted variational ansatz based on a piecewise trial solution for the electrostatic potential. We assume that $\phi_0(z)$ is the solution of Eq.~(\ref{varel}) in the linear limit of weak surface charge, with $\tilde{\epsilon}(z)=\theta(h-z)+\varepsilon_w\theta(z-h)$ and $\kappa(z)=\kappa_c\theta(z-d)$, where the dipolar and ionic depletion lengths $h$ and $d$ are trial parameters that will be obtained from a numerical optimization procedure of the Grand potential of Eq.~(\ref{vargrpot}). The solution of Eq.~(\ref{varel}) with the above piecewise dielectric permittivity and ion density profiles, and satisfying the continuity of the potential $\phi_0(z)$ and the displacement field $D(z)=\tilde{\epsilon}(z)\phi'_0(z)$ at $z=0$, $z=h$, and $z=d$, reads
\begin{eqnarray}\label{phires}
\phi_0(z)&=& -\frac{2}{\mu\kappa_c}\left[1+\kappa_c(d-h)\right]+\frac{2\varepsilon_w}{\mu}(z-h),\quad 0<z\leq h\nonumber\\
\phi_0(z)&=&-\frac{2}{\mu\kappa_c}+\frac{2}{\mu}(z-d),\quad h\leq z\leq d\\
\phi_0(z)&=&-\frac{2}{\mu\kappa_c}e^{-\kappa_c(z-d)},\quad z\geq d\nonumber.
\end{eqnarray}
Numerical optimization yields $h=0.6$ {\AA} and $d=2.3$ {\AA}. We note that due to the piecewise nature of the trial potential in Eqs.~(\ref{phires}), these values correspond approximately to half saturation densities. Figure~\ref{den1} shows that the potential profile obtained from the numerical optimization agrees very well with the general form obtained from the numerical solution of the EDPB equation. In Eqs.~(\ref{phires}), the first linear regime at $0<z\leq h$ corresponds to the solvent depletion layer resulting from image dipole interactions. This layer associated with dielectric screening deficiency is responsible for an amplification of the PB prediction of the surface potential by a factor of five. The second and third intervals correspond respectively to the usual ionic depletion and diffuse layers~\cite{pre}. The contribution of these layers to the differential capacitance will be investigated below.
\begin{figure}
(a)\includegraphics[width=0.9\linewidth]{fig3.pdf}
(b)\includegraphics[width=0.9\linewidth]{fig4.pdf}
\caption{(Color online) (a) Differential capacitance against the bulk ion concentration for $\sigma_s=0$, $\varepsilon_m=1$, and $\varepsilon_w=71$. The black circles are the experimental data, the red solid curve is the result of the EDPB equation, the red squares are from Eq.~(\ref{cap2}), the dashed blue line is the MPB result, the dotted black line is the GC capacitance, and the black squares correspond to the prediction of the DPB equation. (b) The same plot as in (a) for various $\varepsilon_m$. The inset displays the evolution of the dipolar depletion length $h$ (solid curve) and $l_d$ (dashed curve) as a function of $\varepsilon_m$ for $\rho_{bi}=0.1$ M.}
\label{den2}
\end{figure}
The differential capacitance of the double layer is defined as
\begin{equation}\label{cap1}
C_d=\frac{qe^2}{k_BT}\left|\frac{\partial\sigma_s}{\partial\phi_s}\right|,
\end{equation}
where $\phi_s=\phi_0(z=0)$ is the surface potential. Fig.~\ref{den2}.a compares the differential capacitance computed with Eq.~(\ref{varel}) in the vanishing surface charge limit with experimental data obtained for several types of monovalent electrolytes at various concentrations (for details see Ref.~\cite{prlnetz} where the data were taken from). We also report in this figure the prediction of various formalisms. As stressed in Ref.~\cite{prlnetz}, the PB result largely overestimates the experimental data. Furthermore, the result of the modified PB (MPB) equation (see Ref.~\cite{pre}) that can exclusively take into account the ionic depletion effect brings a very small correction to the PB result. However, the EDPB result that additionally contains the surface depletion effect of solvent molecules exhibits a good agreement with the experimental data. We finally note that in the vanishing surface charge limit considered in this part, the DPB equation yields the same result as the PB one (see black squares in Fig.~\ref{den2}.a).
The physics of the EDPB prediction for the capacitance can be understood within the restricted self-consistent scheme of Eq.~(\ref{phires}), where the differential capacitance in Eq.~(\ref{cap1}) takes in the limit $\sigma_s\to0$ the simple form
\begin{equation}\label{cap2}
C_d=\frac{\varepsilon_w\kappa_c}{1+\kappa_c(d-h)+\varepsilon_w\kappa_c h}.
\end{equation}
We note that the prediction of this equation reported in Fig.~\ref{den2} (red squares) fits very well the numerical result of the EDPB equation. The inverse capacitance of Eq.~(\ref{cap2}) is composed of three parts. The first contribution from the diffuse layer is the inverse GC capacitance $C_{d1}^{-1}=(\varepsilon_w\kappa_c)^{-1}$ corresponding to the PB result in Fig.~\ref{den2}.a. The second part $C_{d2}^{-1}=(d-h)/\varepsilon_w$ associated with the ionic depletion layer is shown to drop the differential capacitance to the MPB curve. Finally, the third contribution from the solvent depletion layer $C_{d3}^{-1}=h$ characterized by the dielectric screening deficiency brings the most important correction to the total capacitance by dropping the latter to the correct order of magnitude.
In order to estimate the order of magnitude of the dipolar depletion length $h$, we will compute the asymptotic limit of $\tilde{\epsilon}(z)$ far from the interface, where the dipolar potentials in Eqs.~(\ref{funu}) and~(\ref{funt}) become weak, by expanding Eq.~(\ref{per2}) in $U_d(z)$, $T_d(z)$, and $p_0\phi'_0(z)$. Furthermore, we note that in the limit $\varepsilon_m=0$, the dipolar potentials are given by the closed form expressions
\begin{eqnarray}
\label{ud}
U_d(z)&=&\ell_wp_0^2\frac{1+2\kappa_cz}{16z^3}e^{-2\kappa_cz}\\
\label{td}
T_d(z)&=&\ell_wp_0^2\frac{1+2\kappa_cz(1+2\kappa_cz)}{16z^3}e^{-2\kappa_cz}.
\end{eqnarray}
Renormalizing all lengths by the length scale $l_d=(\ell_wp_0^2/10)^{1/3}$ according to $\bar\kappa_c=\kappa_cl_d$ and $\bar z=z/l_d$, and taking into account that $\varepsilon_w\gg1$, the asymptotic form of Eq.~(\ref{per2}) far from the dielectric interface reads
\begin{equation}
\label{diel4}
\frac{\tilde{\epsilon}(z)}{\varepsilon_w}\approx1-\frac{1+2\bar\kappa_c\bar z+3\bar\kappa_c^2\bar z^2/2}{\bar z^3}e^{-2\bar\kappa_c\bar z}.
\end{equation}
We now note that for a bulk permittivity $\varepsilon_w=71$ and ionic concentration $\rho_{bi}=10^{-1}$ M, one has $\kappa_cl_d=8.10^{-2}$. Hence, in the regime $z\sim l_d=0.9$ {\AA}, the terms in Eq.~(\ref{diel4}) that depend on the screening length become negligible. This simple calculation fixes $l_d$ as the characteristic length over which the local permittivity tends to its bulk value according to an inverse cubic power law, i.e. $\tilde{\epsilon}(z)/\varepsilon_w\approx1-l_d^3/z^3$. We note that an inverse cubic law for the dielectric permittivity profile was derived in Ref.~\cite{hansenepl} in the strict limit of a single dipole (i.e. $\varepsilon_w=1$) and without salt. In the dilute salt limit $\kappa_c\to 0$, one can actually extend the estimation of $h$ to finite values of $\varepsilon_m$ by noting that the dipolar potentials possed the close form expression $U_d(z)=T_d(z)=\ell_wp_0^2\Delta_0/(16z^3)$, where $\Delta_0=(\varepsilon_w-\varepsilon_m)/(\varepsilon_w+\varepsilon_m)$. Following the same steps as above, one obtains for the characteristic dipolar depletion length the more general expression $l_d=(\Delta_0\ell_wp_0^2/10)^{1/3}$. Hence, for biological solvent concentrations and dilute electrolytes with bulk density $\rho_{bi}\lesssim 0.1$ M, the dielectric screening is mainly responsible for the decay of the image dipole interactions and solely determines the region over which a reduced dielectric permittivity is observed. For larger ion concentrations, Eq.~(\ref{diel4}) shows that the screening of image dipole interactions by surrounding ions positively adds to the dielectric screening of these forces.
We display in the inset of Fig.~\ref{den2}.b the evolution of $h$ as a function of $\varepsilon_m$ together with $l_d$, while the main plot shows $C_d$ for various values of the membrane permittivity from $\varepsilon_m=1$ to $\varepsilon_m=\varepsilon_w$. One notices that within the range $1\leq\varepsilon_m\leq60$, $h$ exhibits a slow linear decrease with increasing $\varepsilon_m$ while $C_d$ remains within the same order of magnitude as the experimental capacitance data. However, with an increase of $\varepsilon_m$ from 60 to $\varepsilon_w$, the dipolar depletion length quickly drops to zero and consequently, $C_d$ approaches the PB result. One also sees in the inset that although $l_d$ is slightly higher than the dipolar depletion length $h$, it can reproduce the correct trend of the latter as a function of $\varepsilon_m$. These observations suggest that image dipole interactions are mainly responsible for the low values of the experimental capacitance data in Fig.~\ref{den2}.a. We emphasize that this result is in agreement with the experimental observation of a strong reduction of the double layer capacitance with increasing surface hydrophobicity~\cite{exp}.
In addition to the dipolar depletion effect, the orientation is also expected to play some role in the form of the interfacial dielectric permittivity profile. The measure of the dipolar orientation is defined in the literature as $\mu_m(z)=\left\langle p_z^2\right\rangle/[p_0^2\rho_d(z)]$, where $\left\langle p_z^2 \right\rangle=\int\frac{\mathrm{d}\mathbf{\Omega}}{4\pi}\bar\rho_d(z,\mathbf{\Omega})p_z^2$. The function $\mu_m$ was studied in Ref.~\cite{Kanduc} for multipolar ions and it was found invariably below the free dipole value 1/3 in the SC limit (i.e. dipolar alignment parallel to the wall) and above this value in the WC limit (alignment along the electrostatic field). We show in Fig.~\ref{den3} that for a neutral interface, one has $\mu_m(z)<1/3$ as in the SC limit, i.e. the solvent molecules exhibit a tendency to align parallel to the wall over a distance $\approx 2$ {\AA}, that is, until image dipole forces vanish. We now define an effective dielectric permittivity function of the form $\tilde\varepsilon_{eff}(z)=1+4\pi\ell_Bp_0^2\rho_d(z)/3$ that solely accounts for the dipolar depletion. The comparison of $\tilde\varepsilon_{eff}(z)$ in Fig.~\ref{den1}.a with $\tilde\varepsilon(z)$ shows that the main effect of the dipolar alignment close to the interface is a slight reduction of the local dielectric permittivity. However, it is seen that this effect is largely dominated by the solvent depletion.
\begin{figure}
(a)\includegraphics[width=0.95\linewidth]{fig5.pdf}
\caption{(Color online) (a) Dipolar orientation profile evaluated for various $\sigma_s$ and the same model parameters as in Fig.~\ref{den2}. The black dashed reference line marks the freely rotating dipole case $\mu_m(z)=1/3$.}
\label{den3}
\end{figure}
In the presence of a finite surface charge, Fig.~(\ref{den3}) shows that interestingly, the function $\mu_m(z)$ exhibits a non-monotonous behavior. Namely, in the close vicinity of the surface, strong image dipole effects lead to a net dipolar alignment parallel to the wall. However, above a characteristic distance from the dielectric interface where the surface charge induced electric field $p_0\phi'_0(z)$ dominates the image dipole potential, $\mu(z)$ exceeds 1/3 and reaches a peak where the dipoles exhibit the maximum tendency to align in the direction of the field, i.e. perpendicular to the dielectric wall. With increasing distance, one notices a reversal of this behavior where the image dipole potential dominates for a second time the electrostatic field. As a result, the solvent molecules exhibit again some tendency to align again parallel to the interface, but this regime gradually disappears with increasing surface charge.
\section{Conclusions}
We have considered the dielectric discontinuity effects on the differential capacitance of low dielectric substrates. To this aim, we derived an extended DPB equation that can explicitly account for the interactions of solvent molecules with their electrostatic image. Within this approach, we showed that the overestimation of the experimental data by the GC capacitance is due to the inability of the latter to account for the solvent depletion effect driven by image dipole interactions. The prediction of the EDPB equation for the differential capacitance of monovalent electrolytes was compared with experimental data and good agreement was found.
The EDPB formalism is a first order theoretical approach in the explicit modeling of solvent molecules beyond the MF level, and it has its limitations. Excluded volume~\cite{jcp} and non-local dielectric effects that lead in MD simulations to interfacial structure formation~\cite{prlnetz} are absent. The theory could be extended by using more general trial kernels, but the analytical solution of the Debye-Huckel equation with a local dielectric permittivity and screening parameter is still an open problem. Furthermore, the present formalism does not include multipolar moments, which are known to enhance the interfacial dielectric exclusion\cite{Kanduc}. Hence, multipolar contributions are expected to further lower the capacitance curves in Fig.~\ref{den2}.
\acknowledgements This work has been supported in part by The Academy of Finland through its COMP CoE and NanoFluid grants.
\smallskip
|
{
"timestamp": "2012-03-13T01:01:34",
"yymm": "1203",
"arxiv_id": "1203.2285",
"language": "en",
"url": "https://arxiv.org/abs/1203.2285"
}
|
\section{\label{Intro}Introduction}
The beautiful classification of $2$-bridge links by rational numbers has not
yet been generalized to 3-bridge links. One of the goals of this paper is to
introduce a tool that eventually might lead to a generalization of this classification.
It is well known \cite{seifert} that every closed, orientable $3$-manifold can
be obtained by pasting pairs of faces of a polygonization of the boundary
$S^{2}$ of a closed $3$-cell $\mathbf{B}^{3}$.
Thurston's construction of the borromean rings, \cite{Th1} and \cite{Th2}, is a nice
example that we generalize for all links in this paper, Fig. \ref{borro}. In
this example we notice that the cube is actually a closed $3$-cell
$\mathbf{B}^{3},$ with twelve faces on its boundary that are identified by
reflections along some axes (double arrows). Moreover, pasting the faces of
the cube we obtain $S^{3}$ and the set of axes become the borromean rings.
These reflections resemble the way a butterfly closes its wings, and we will
say that the borromean rings have a $6$-butterfly representation, and the six
faces of the real cube are the six butterflies involved.
\begin{figure}
[h]
\begin{center}
\includegraphics[
height=1.8325in,
width=4.1926in
{graphics/figure1.eps
\caption{Borromean rings.}
\label{borro}
\end{center}
\end{figure}
Similarly to the borromean rings, the $2$-bridge knots or rational links $p/q$
can be obtained by pasting the northern and southern hemispheres of $S^{2}$
with themselves by reflections through half meridians separated apart $2\pi
q/p.$ For instance, Fig. \ref{fig16} depicts this construction for $p/q=3/1$,
the trefoil knot. As in Thurston's example, $S^{3}$ is obtained by pasting the faces. We say that the rational
link $p/q$ has a $2$-butterfly representation, and the northern and southern
hemispheres of $S^{2}$ are the two butterflies involved.
This butterfly representation of $p/q$ has two main advantages. First, it is a
pure $2$-dimensional diagram, and secondly, it exhibits directly the rational
number $p/q$ that classifies the knot or link.
With these two properties in mind, we wondered if all knots and links have a
similar structure, allowing two or more butterflies on the boundary $S^{2}$ of
$\mathbf{B}^{3}.$ One such structure with three butterflies is depicted in
Fig. \ref{fig2}b.
It turns out that every knot or link admits such a representation. We prove
this fact here. In Sections \ref{ButLink} and \ref{LinkBut} we give algorithms
to pass back and forth from a link to a butterfly representation of it.
We define accordingly the butterfly number of a knot or link and we prove that
it coincides with its bridge number (Section \ref{number}). To obtain this
last result we need to reduce the number of butterflies of a particular
butterfly representation of a link. This involves the definition of a move
that does precisely this. See Section \ref{Move}.
As each $m$-bridge link diagram has an $m$-butterfly representation, a natural
question arises: \textit{Is it possible to associate a set of rational numbers
to describe this butterfly}? In the case $m=3$ this assignation can, in fact,
be made \cite{HMTT4}, where a triple of rational numbers is associated to each
3-butterfly. In this paper we show some examples of $3$-butterflies and its
corresponding set of rational numbers.
We give many examples and in particular two different $3$-butterfly
representations of the same knot $8_{20}.$ This raises the problem of relating
$3$-butterfly representations by a set of potential moves. This is left as an
open problem. Using the concept of $3$-butterfly, we hope to obtain a
classification of $3$-bridge links, similar to the Schubert classification of
$2$-bridge links.
In Section \ref{Def} we present a technical definition of an $m$-butterfly
even though in the rest of the paper, for simplicity, we speak more
intuitively about $m$-butterflies.
In the last decade, Kauffman \cite{Ka} has been developing the theory of
virtual knots. This theory has several applications. The technical definition
of an $m$-butterfly is used intensively in \cite{HMTT5} where we prove that
any virtual knot also admits a representation by a generalized $(n,g)
-butterfly, that is a handlebody of genus $g$ with $2n$ faces on its boundary
that are identified by reflections along some axis.
As we have remarked above, pasting the faces of an $m$-butterfly gives the $3
$-sphere $S^{3}$. Section \ref{S3} is devoted to showing this fact. In
general, this result is not true for generalized $(n,g)$-butterflies that
represent virtual knots.
\section{\label{Def}Butterflies: Definitions and Examples}
Intuitively, an $m$-butterfly is a $3$-ball $\mathbf{B}^{3}$ with $m$ $>0$
polygonal faces on its boundary $S^{2}=\partial\mathbf{B}^{3},$ such that each
face $P$\ is subdivided by an arc $t_{P}$\ in two subfaces (that have the same
number of vertices) that are identified by a ''reflection'' along this arc
$t_{P}.$
In order to formalize this concept,\ we give some technical definitions.
Let $R$ be a connected graph embedded in $S^{2}=\partial\mathbf{B}^{3},$ where
$\mathbf{B}^{3}$ is a closed $3$-cell, so that $S^{2}-R$ is a disjoint union
of open 2-cells. For our purposes we assume that $\mathbf{B}^{3}$ is the half
ball $x^{2}+y^{2}+z^{2}\leq r^{2};z\leq0,$ and that the graph $R$ and later
the graph $R\cup T$, when $T$ has been defined, is contained in the planar
part of $\mathbf{B}^{3},\mathbb{R}^{2}\times\{0\}.$ The edges in $R$ and in
$T$ are simple arcs. However, by \cite{F1}, for any such graph $R\cup T$ there
is an autohomeomorphism of $S^{2}$ such that the images of the edges are
straight planar line segments. We shall assume, in the proofs of theorems that
follow, but not in the drawn figures, that the edges of $R\cup T$ are straight
planar line segments. We denote each open $2$-cell generically by $P.$ We
would like to parameterize each $2$-cell $P$.
For any $n\in\mathbb{N}$, let $P_{2n}$ be the regular polygon that is the
closed convex hull of the $2n^{th}\ $roots of unity. We define a
parameterization of $P$ to be a function $f$ from $P_{2n}$ to the closure
$\overline{P}$ of $P,$ with the following properties:
a) The restriction of $f$ to interior $P_{2n}$ is a homeomorphism from
interior $P_{2n}$ to $P.$
b) The restriction of $f$ to an edge of $P_{2n}$ is a piecewise linear
homeomorphism from that edge to an edge in the graph $R.$
c) $f$ as a map from the edges of $\partial P_{2n}$ to the edges of $\partial
P$ is at most 2 to 1.
The existence of a parameterization of $P$ places restrictions on $P$ and on
$R.$ We will assume that $R$ is such that each $P$ has a parameterization
$f:P_{2n}\rightarrow\overline{P},$ and we fix a parameterization $f_{P}$ for
each $P.$
Complex conjugation, $z\rightarrow\overline{z},$ restricted to $P_{2n}$ or to
boundary of $P_{2n}$ defines an involution and an equivalence relation on the
edges and vertices of $P_{2n}$, and this in turn, induces an equivalence
relation on the edges and vertices of $\overline{P},$ and on the points of $P
$ as well. That is to say for $A$ and $B$ points of $\overline{P}$, $A\sim B$
if $f_{P}^{-1}\left( A\right) =f_{P}^{-1}\left( B\right) $ or $f_{P
^{-1}\left( A\right) =\overline{f_{P}^{-1}\left( B\right) },$ where
$\overline{f_{P}^{-1}\left( B\right) }=\left\{ \overline{z}/z\in f_{P
^{-1}\left( B\right) \right\} .$
The equivalence relation on the edges and vertices of each $\overline{P}$
induces an equivalence relation on the graph $R.$ That is $x\simeq y$ if and
only if there exists a finite sequence $x=x_{1},\cdots,x_{l}=y$ with
$x_{i}\sim x_{i+1}$ for $i=1,\cdots,l-1.$ Equivalence classes of points of $P
$ contain two points except for those points in $f\left( \left[ -1,1\right]
\right) $ where there is only one point. Note that if $x$ is a vertex of $R,$
its complete class under the equivalence relation $\simeq$ is composed
entirely of vertices.
Figures \ref{fig1a} and \ref{fig1b} illustrate two different
parameterizations. In Fig. \ref{fig1a} \ we have $f(1)=f(5)$ and $f(2)=f(4);$
and in Fig. \ref{fig1b} we have $f\left( 0\right) =f\left( 6\right)
,f\left( 1\right) =f\left( 5\right) $ and $f\left( 2\right) =f\left(
4\right) .
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=1.701in,
width=4.4952in
{graphics/figure2.eps
\caption{$f$ parameterizes a pair $(P,t)$.
\label{fig1a
\end{center}
\end{figure}
\begin{figure}
[h]
\begin{center}
\includegraphics[
height=1.9298in,
width=4.5798in
{graphics/figure3.eps
\caption{$f$ parametrizes a $1$-butterfly.
\label{fig1b
\end{center}
\end{figure}
Each $P_{2n}$ contains the line segment $\left[ -1,1\right] $ which is the
fixed point set of complex conjugation restricted to $P_{2n}.$ The image of
this line segment $f_{p}\left( \left[ -1,1\right] \right) $ is called the
\textit{trunk} $t$. A pair $\left( P,t\right) $ will be called a
\textit{butterfly with trunk} $t.$ The \textit{wings} $W$ and $W^{\prime}$ are
just $f_{P}\left( P_{2n}\cap upper\ half\ plane\right) $ and $f_{P}\left(
P_{2n}\cap lower\ half\ plane\right) $ and $W\cap W^{\prime}=t.$ Each time
that we consider a trunk $t$ we are implicitly considering the equivalence
relation described above. We denote by $T$ the collection of all trunks $t$
(over all $P).$ Notice that the boundaries of the $n$ butterflies form a graph
$R$ on $S^{2}=\partial\mathbf{B}^{3}.$ As before, (See \cite{F1}), we can
assume the edges in the graph $R\cup T$ as straight line segments.
Let us denote by $M(R,T)$ the space $\mathbf{B}^{3}/\simeq$ with the topology
of the identification map $p:\mathbf{B}^{3}\rightarrow M(R,T)$.
As in Thurston's example, we would like that the image of $T,p(T),$ became a knot or link. In
order to guarantee this fact, we distinguish three types of vertices on $R.$
A member of $R\cap T$ will be called an $A$\emph{-vertex}. A member of
$p^{-1}\left( p\left( v\right) \right) ,$ $v\in R\cap T,$ which is not an
$A$-vertex will be called an $E$\emph{-vertex}. A vertex of $R$ which is not
an $A$-vertex nor an $E$-vertex will be called a $B$\emph{-vertex }iff
$p^{-1}\left( p\left( v\right) \right) $ contains at least one
non-bivalent vertex of $R$.
We do not give an explicit name for those vertices that are neither $A,B$ nor
$E$-vertices. Of course it is possible to construct $3$-balls with polygonal
faces on their boundaries with those kind of vertices but for our purposes (we
want to represent knots or links) it is enough to consider graphs without
them. There are also interesting examples in which there are $E$-vertices that
are not bivalent, as the one shown in Fig. \ref{exemike}, but for our purpose
we do not consider them as $m$-butterflies. In further research we will
consider some generalization of our construction
\begin{figure}
[h]
\begin{center}
\includegraphics[
height=1.6164in,
width=2.334in
{graphics/figure4.eps
\caption{In this poligonization, the vertices marked with $\blacklozenge$ are
trivalent $E$-vertices of $R$.
\label{exemike
\end{center}
\end{figure}
With these definitions we formalize our intuitive definition of $m$-butterfly,
given at the beginning of this section.
\begin{definition}
For $m\geq1$, an $m$-\textit{butterfly} is a $3$-ball $\mathbf{B}^{3}$ with
$m$ butterflies $(P,t)$ on its boundary $S^{2}=\partial\mathbf{B}^{3},$ such
that (i) the graph $R$ has only $A$-vertices, $E$-vertices and $B$-vertices;
(ii) the $A$- and $E$-vertices are bivalent in $R,$ and (iii) $T$ has $m$ components.
\end{definition}
Moreover, an $m$-butterfly can be represented by a planar graph (or by an
$m$-\textit{butterfly diagram}), denoted by a pair $\left( R,T\right) ,$
such that conditions (i), (ii), and (iii) are satisfied. The $m$-butterfly
represented by the diagram $(R,T)$ is also denoted by $(R,T).$
\begin{example}
Figure \ref{fig2} depicts three different butterfly diagrams. Fig \ref{fig2}b
represents a $3$-butterfly. The full equivalence class of the two trivalent
vertices $0$ and $\infty$ on it are $B$-vertices. Fig. \ref{fig2}a shows a
$2$-butterfly that has only $A$ or $E$-vertices, while the $1$-butterfly given
in \ref{fig2}c has only two $A$-vertices and three $B$-vertices.
\end{example}
In the examples of Fig. \ref{fig2} we will assume that $\mathbf{B}^{3}$ is the
closed $3$-cell that lies over the paper in $\mathbb{R}^{3}+\infty$. The
members of $T$ will be displayed as thick lines. The $B$-vertices are depicted
by *. See \ref{fig2}b and c. The other vertices of the diagram are either
boundaries of members of $T$ ($A$-vertices) or $E$-vertices
\begin{figure}
[h]
\begin{center}
\includegraphics[
height=1.5904in,
width=5.0713in
{graphics/figure5.eps
\caption{Representing butterflies with planar graphs.
\label{fig2
\end{center}
\end{figure}
\section{\label{S3}The Quotient Space $M(R,T)$ is $S^{3}$}
In this section we are going to prove that under our definitions, the space
$M(R,T)$ is $S^{3}$ and that the image of $T$ under the identification map $p$
is a knot (or link). So we are sure to obtain a knot (or link) inside $S^{3}$
when we make the identifications by the equivalence relation$.$
\begin{theorem}
\label{teos3}For any $m$-butterfly $(R,T),$ the space $M\left( R,T\right) $
is homeomorphic to $S^{3}$ and $p\left( T\right) $ is a knot or a link,
where $p:\mathbf{B}^{3}\rightarrow M(R,T)$ is the identification map.
\end{theorem}
\begin{proof}
Set\thinspace$M=M\left( R,T\right) \ $for shortness. Let $R^{\ast}=p(R)$,
$T^{\ast}=p(T)$ and $V^{\ast}=p(V)$, where $V$ is the set of vertices of $R. $
Let $U(V^{\ast})$ be a regular neighbourhood of $V^{\ast}$ in the space
$M=M\left( R,T\right) .$ Then $U(V^{\ast})$ is a disjoint union of regular
neighbourhoods (we choose $U(V^{\ast})$ as small as we need) of the vertices
of $V^{\ast}$. Let $v^{\ast}\in V^{\ast}$ be one of these vertices. Of course
any regular neighbourhood of $v^{\ast}$ is the cone over an orientable surface
$\Sigma_{v^{\ast}}.$
\textbf{Claim 1: }The surface $\Sigma_{v^{\ast}}$ is connected.
\textbf{Proof: }Consider the subset $p^{-1}\left( v^{\ast}\right) $ of the
set $V.$ Let $v\in p^{-1}\left( v^{\ast}\right) .$ A regular neighbourhood
of $v$ in $\mathbf{B}^{3}$ is a cone from $v$ over a $2$-disk $\Delta_{v}$
properly embedded in $\mathbf{B}^{3}$. Denote this cone by $C\left(
v,\Delta_{v}\right) .$ It is possible to select the regular neighbourhood of
members of $p^{-1}\left( v^{\ast}\right) $ so that
\[
\Sigma_{v^{\ast}}=
{\textstyle\bigcup_{v\in p^{-1}\left( v^{\ast}\right) }}p\left( \Delta_{v}\right).
\]
Now, if $v_{1},v_{2}\in p^{-1}\left( v^{\ast}\right) $ then $v_{1}\simeq
v_{2},$ so there exits a finite sequence of vertices of $p^{-1}\left(
v^{\ast}\right) $ say $u_{1}=v_{1},u_{2},\cdots,u_{k}=v_{2}$ such that
$u_{i}\sim u_{i+1},i=1,\cdots,k-1$. If we assume that $u_{i},u_{i+1}$ belong
to some $\overline{P},$ where $(P,t)$ is the corresponding butterfly, then the
boundary of $\Delta_{u_{i}}\cap P$ and $\Delta_{u_{i+1}}\cap P$ are also
identified and it follows that $p\left( \Delta_{u_{i}}\right) \cup p\left(
\Delta_{u_{i+1}}\right) $ is a connected set. From this, the claim follows easily.
We continue with the proof of the theorem. The closure of $M\smallsetminus
U\left( V^{\ast}\right) $ is clearly a compact, connected $3$-manifold
$M^{\ast}$ with boundary $\partial M^{\ast}=
{\textstyle\bigcup_{v^{\ast}\in V^{\ast}}}\Sigma_{v^{\ast}}.$
The closure in $M^{\ast}$ of the set $R^{\ast
}\smallsetminus U\left( V^{\ast}\right) $ (resp. $T^{\ast}\smallsetminus
U\left( V^{\ast}\right) $) is a set of disjoint, properly embedded arcs in
$M^{\ast}$ that will be denoted by $R^{\ast\ast}$ (resp. $T^{\ast\ast}$).
Now drill from $M^{\ast}$ a regular neighbourhood $U(R^{\ast\ast})\cup
U\left( T^{\ast\ast}\right) $ of $R^{\ast\ast}\cup T^{\ast\ast}$ and take
the closure $M^{\ast\ast}$ of the result. Then $M^{\ast\ast}$ is the image
under $p$ of the ball $C=\mathbf{B}^{3}\smallsetminus U\left( R\cup T\right)
,$ where $U\left( R\cup T\right) $ is a suitable regular neighbourhood of
$R\cup T.$ The set $\partial\mathbf{B}^{3}\smallsetminus U\left( R\cup
T\right) $ is a system $\left\{ \check{W}_{1},\check{W}_{1}^{^{\prime
},\cdots,\check{W}_{m},\check{W}_{m}^{^{\prime}}\right\} $ of $2m$ disks in
$\partial C.$ Here $\check{W}_{i},\check{W}_{i}^{^{\prime}} $ are contained in
the wings $W_{i},W_{i}^{^{\prime}}$ of the butterfly $P_{i}\ $with trunk
$t_{i}$ and $p$ identifies $\check{W}_{i},\check{W}_{i}^{^{\prime}}$. Thus
$M^{\ast\ast}$ is a handlebody. Therefore $M=M^{\ast\ast}\cup U\left(
T^{\ast\ast}\right) \cup U(R^{\ast\ast})\cup U(V^{\ast}).$ The set $U\left(
T^{\ast\ast}\right) $ is a set of $m $ $2$-handles that are attached to the
handlebody $M^{\ast\ast}.$ The attaching spheres for these $2$-handles are
meridians $\mu_{1},\cdots,\mu_{m}$ of $p\left( t_{1}\right) ,\cdots,p\left(
t_{m}\right) $. Then $\mu_{i}$ cuts $p\left( \check{W}_{i}\right) =p\left(
\check{W}_{i}^{^{\prime}}\right) $ transversely in just one point. Therefore
$M^{\ast\ast}\cup U\left( T^{\ast\ast}\right) $ is a $3$-ball $C^{3}$. Thus
\[
M=C^{3}\cup U(R^{\ast\ast})\cup U(V^{\ast}).
\]
Since $U(R^{\ast\ast})$ are $2$-handles attached to $C^{3}$ it follows that
$C^{3}\cup U(R^{\ast\ast})$ is a punctured $3$-ball. Since the boundary of
$C^{3}\cup U(R^{\ast\ast})$ and $U(V^{\ast})$ coincide, it follows that
$\partial U(V^{\ast})$ is a disjoint union of spheres. From the above claim,
it follows that $U(V^{\ast})$ is a disjoint union of cones over spheres. That
is, $U(V^{\ast})$ is a disjoint union of balls. Then $M$ is homeomorphic to
$S^{3}.$
To prove that $p\left( T\right) $ is a knot or a link, it is enough to show
that $p^{-1}(p(v)),$ for every $A$-vertex $v$, contains exactly two
$A$-vertices. To prove this we construct the following graph $\Gamma.$
Assume that the $3$-cell $\mathbf{B}^{3}$ is the upper half space
$\mathbb{R}_{+}^{3}$ of $\mathbb{R}^{3}+\infty$, and that the graph $R$ lies
in its boundary $\mathbb{R}^{2}\times\left\{ 0\right\} .$
Let $(P,t)\ $be a butterfly of $(R,T)$ and let $f_{P}:P_{2k}\rightarrow
\overline{P}$ be its fixed parameterization. Let $w_{1},w_{2},\cdots,w_{2r}$
be the vertices of $P_{2k}$ and let $v_{j}=f_{P}(w_{j})$ be the vertices of
$\partial P$. For a vertex $w_{j}=\cos(k\pi/r)\pm i\sin(k\pi/r),$
$k=1,2,...,r-1,$ let $L(w_{j})$ be the open vertical line segment $(\cos
(k\pi/r)+i\sin(k\pi/r),\cos(k\pi/r)-i\sin(k\pi/r)).$ For each $A$-vertex in
$\partial P$ not in $\partial t\ $and each $E$-vertex $v_{j}=f_{P}(w_{j})$ in
$\partial P,$ take the arc $Q_{v_{j}}=f_{p}(L(w_{j})). $
Denote by $\Gamma$ the union of all possible $Q_{v}$'s for any $v\in R$ that
is an $A$- or $E$-vertex.
\noindent\textbf{Claim 2:} $\Gamma$ is a disjoint union of arcs bounded by $A$-vertices.
\textbf{Proof: \ }(1) Noting that if $v$ is an $A$-vertex then any other
vertex, related to it, is an $A$- or $E$-vertex and it follows that the
vertices of $\Gamma$ are all $A$- or $E$-vertices.
(2) Since by definition the $A$-vertices are bivalent in $R$ and they are end
points of some trunk it follows that they are monovalent vertices of $\Gamma.$
(3) Since by definition the $E$-vertices are bivalent in $R$ and they are not
end points of a trunk it follows that they are bivalent vertices of $\Gamma.$
Thus, each component of the graph $\Gamma$ is linear and it is bounded by two
$A$ vertices.
To finish the proof of the theorem we observe that if $\Gamma_{0}$ is a
component of $\Gamma,$ the set of vertices of $\Gamma_{0}$ form a complete
equivalence class under $\simeq.$ Therefore, $p^{-1}\left( p\left( v\right)
\right) $ for every $A$-vertex $v$ contains exactly two $A$-vertices. Hence
the graph $p\left( T\right) $ is a knot or a link.
\end{proof}
\begin{definition}
The knot or link $p(T)$ defined by the $m$-butterfly $\left( R,T\right) $
will be denoted by $L(R,T)$, and we say that $L(R,T)$ has the butterfly
representation $\left( R,T\right) $ with butterfly number $m,$ or that the
$m $-butterfly diagram $\left( R,T\right) $ represents $L(R,T)$.
\end{definition}
\section{\label{ButLink}From the Butterfly to the Link}
In this section we show how to construct the link $L(R,T)$ from an
$m$-butterfly $(R,T).$
Recall, \cite{Mu}, that a regular diagram $D_{L}$ of a link $L$ is an
$m$-\textit{bridge diagram} for the link $L$ if we can divide up $D_{L}$ into
two sets of polygonal curves $O=\left\{ o_{1},o_{2},\cdots,o_{m}\right\} $
and $U=\left\{ u_{1},u_{2},\cdots,u_{m}\right\} $ $(m>0)$ such that:
i. $D_{L}=o_{1}\cup o_{2}\cup\cdots\cup o_{m}\cup u_{1}\cup u_{2}\cup
\cdots\cup u_{m}$,
ii. $o_{1},o_{2},\cdots,o_{m}$ are mutually disjoint simple curves,
iii. $u_{1},u_{2},\cdots,u_{m}$ are mutually disjoint simple curves,
iv. At the crossing points of $D_{L},$ $o_{1},o_{2},\cdots,o_{m}$ are segments
that pass \textit{over }at least one crossing\textit{\ }point, while
$u_{1},u_{2},\cdots,u_{m}$ are segments that pass \textit{under }at least one
crossing point\textit{. }
The arcs $o_{1},o_{2},\cdots,o_{m}$ are called \textit{bridges or overarcs}.
We use the notation $D_{L}=\left( O,U\right) $ when we want to describe
explicitly the bridge presentation of the link $L.$
Note that, by condition iv., there are link diagrams that are not bridge
diagrams. For instance, a simple closed curve is not a \textit{bridge diagram}
for the trivial knot. In this paper, we follow \cite{ChLi} and we differ from
\cite{NeOk}, where it is considered the trivial knot with no crossing as
having an $m$-bridge diagram, for all $m\in\mathbb{N}.$ When a link $L$ has
unknotted components, we need to take some care about them, in order to obtain
an $m$-bridge diagram of $L$ because no component can be expresed as a union
of only $o's$ or $u's$.
Actually, we have to make at least one kink to the trivial knot to obtain
a bridge diagram for it.
\begin{definition}
Given a link $L,$ the\textsl{\ }\emph{bridge number of} $L$ is the minimum
number $m\,$among of all possible $m$-bridge diagrams of the link $L.$ It is
denoted by $b(L).$
\end{definition}
For example, the trivial knot has bridge number $1$ (see Fig. \ref{fig5}c).
\begin{lemma}
\label{diagram}Given a link $L$, there exists an $m$-bridge diagram $D_{L}$
for $L,$ such that $D_{L}\ $is connected and has no closed curves.
\end{lemma}
\begin{proof}
If the diagram has a closed circle that splits or if it is not connected,
apply the moves shown in Figures \ref{fig19a} and \ref{fig19b}.
\begin{figure}
[h]
\begin{center}
\includegraphics[
height=1.3351in,
width=3.9707in
]
{graphics/figure6.eps
\caption{Eliminating closed curves.
\label{fig19a
\end{center}
\end{figure}
\begin{figure}
[ptbh]
\begin{center}
\includegraphics[
height=1.1149in,
width=3.88in
]
{graphics/figure7.eps}
\caption{Connecting the diagram.}
\label{fig19b}
\end{center}
\end{figure}
\end{proof}
Now, given an $m$-butterfly diagram $\left( R,T\right) $ we will describe an
algorithm (the \textit{butterfly-link algorithm}) to construct the link
$L=L(R,T)$. Moreover, we will produce an $m$-bridge diagram for the link
$L\left( R,T\right) $.
First of all, consider the following link $K^{\ast}$ of $\mathbb{R}_{+}^{3}$
\[
K^{\ast}=\left( \Gamma\times\left\{ 1/2\right\} \right) \cup\left(
T\times\left\{ 1\right\} \right) \cup\left( \partial T\times\left[
1/2,1\right] \right) ,
\]
where $\Gamma$ is the graph defined in the proof of Theorem \ref{teos3}. By
the second claim in the proof of Theorem \ref{teos3}, $\Gamma\times\left\{
1/2\right\} $ is a disjoint union of arcs lying in $\mathbb{R}^{2
\times\left\{ 1/2\right\} .$ Therefore $(\Gamma\times\left\{ 1/2\right\}
,T\times\left\{ 1\right\} \cup\left( \partial T\times\left[ 1/2,1\right]
\right) )$ is an $m$-bridge presentation of the knot (or link) $K^{\ast}$.
This proves the second part of Theorem \ref{teobridge}.
In Fig. \ref{fig3} we illustrate a portion of $K^{\ast}.$ On plane
$\mathbb{R}^{2}\times\{0\}$ we see a component of $\Gamma,\Gamma_{1},$ that is
bounded by two components of $T$ (denoted generically by $T),$ whose
intersection with that $\Gamma_{1}$ is composed of two $A$-vertices (denoted
generically by $A$) and that passes through two $E$-vertices (denoted by $E$).
The points $f,g$ and $h$ are intersections of some components of $T$ with
$\Gamma_{1}$ (we do not depict those components but they are transversal to
$\Gamma_{1}$)$.
\begin{figure}
[h]
\begin{center}
\includegraphics[
height=2.0815in,
width=2.6593in
{graphics/figure8.eps
\caption{$K^{\ast}=\Gamma\times\left\{ 1/2\right\} \cup T\times\left\{
1\right\} \cup\partial T\times\left[ 1/2,1\right] $}
\label{fig3}
\end{center}
\end{figure}
\begin{theorem}
\label{teobridge} Given an $m$-butterfly diagram $(R,T)$ the link $L(R,T)$ is
isotopic to $K^{\ast}$. Moreover $(\Gamma\times\left\{ 1/2\right\}
,T\times\left\{ 1\right\} \cup\left( \partial T\times\left[ 1/2,1\right]
\right) )$ is an $m$-bridge presentation of $L(R,T)$.
\end{theorem}
\begin{proof}
Consider a component $\Gamma_{1}$ of $\Gamma.$ It is linear and bounded by two
$A$-vertices. Call $\partial\Gamma_{1}$ the set of these two $A$-vertices. \
Consider the subset $\Gamma_{1}\times\left[ 0,1/2\right] $ of$\ \mathbb{\ R
_{+}^{3}$. Then $p\left( \Gamma_{1}\times\left[ 0,1/2\right] \right) $ is
a cone $C\left( w,p\left( \Gamma_{1}\times\left\{ 1/2\right\} \right)
\right) $ from the point $w=p\left( \Gamma_{1}^{\left( 0\right)
\times\left\{ 0\right\} \right) $ over $p\left( \Gamma_{1}\times\left\{
1/2\right\} \right) $ (compare Figures \ref{fig3} and \ref{fig4}) where
$\Gamma_{1}^{\left( 0\right) }$ is the set of vertices of $\Gamma_{1}.$ We
push $p\left( \Gamma_{1}\times\left\{ 1/2\right\} \right) $ along the cone
$C\left( w,p\left( \partial\Gamma_{1}\times\left\{ 1/2\right\} \right)
\right) .$ This we do, as shown in Fig. \ref{fig4}, by an isotopy $H_{i}$
whose final image is just $p\left( \partial\Gamma_{1}\times\left[
0,1/2\right] \right) .$
\begin{figure}
[h]
\begin{center}
\includegraphics[
height=1.9426in,
width=2.1357in
]
{graphics/figure9.eps}
\caption{Isotopy}
\label{fig4}
\end{center}
\end{figure}
Combining these isotopies $H_{i}$ for all components $\Gamma_{i}$ of $\Gamma$
we obtain an isotopy $H$ sending $K^{\ast}$ onto the set
\[
p\left( \left( T\times\left\{ 1\right\} \right) \cup\left( \partial
T\times\left[ 0,1\right] \right) \right) .
\]
But there is certainly an isotopy $H^{\prime}$ sending $p\left( \left(
T\times\left\{ 1\right\} \right) \cup\left( \partial T\times\left[
0,1\right] \right) \right) $ onto $p\left( T\times\left\{ 0\right\}
\right) =K. $ This finishes the first part of the proof.
\end{proof}
\textbf{Algorithm (Butterfly-Link algorithm).}
Finally we have:
\begin{itemize}
\item Start with an $m$-butterfly diagram on the plane $\mathbb{R}^{2
\times\{0\}$. We want to construct the link $L(R,T)$.
\item Construct the graph $\Gamma\subset$ $\mathbb{R}^{2}\times\{0\}$ as in
the proof of the Theorem \ref{teos3}. See the dotted lines in Fig. \ref{fig5}.
\item Then the link $L(R,T)$ is $\left( \Gamma\times\left\{ 0\right\}
\right) \cup\left( T\times\left\{ 1\right\} \right) \cup\left( \partial
T\times\left[ 0,1\right] \right) $.
\item And $(\Gamma\times\left\{ 0\right\} ,T\times\left\{ 1\right\}
\cup\left( \partial T\times\left[ 0,1\right] \right) )$ is an $m$-bridge
diagram of $L(R,T)$.
\end{itemize}
\begin{example}
Applying the butterfly-link algorithm found in the proof of Theorem
\ref{teobridge} to the three butterfly diagrams of Fig. \ref{fig2} we obtain
the knots of Fig. \ref{fig5}. The knot of Fig. \ref{fig5}a is the knot
$4_{1},$ the knot of Fig. \ref{fig5}b is the knot $8_{20}$ and the knot in
\ref{fig5}c is the trivial knot.
\end{example}
\begin{figure}
[h]
\begin{center}
\includegraphics[
height=1.7513in,
width=5.1102in
]
{graphics/figure10.eps
\caption{Examples of knots produced by the butterfly-link algorithm.
\label{fig5}
\end{center}
\end{figure}
\section{\label{LinkBut}From Links to Butterflies}
Now, in the other direction, we explain how to obtain a butterfly from a given link.
\begin{theorem}
\label{linkbut}Every knot or link can be represented by an $m$-butterfly
diagram, for some $m>0.$ Moreover the $m$-butterfly can be chosen with no $E$-vertices.
\end{theorem}
\begin{proof}
Given a link $L$, let $D_{L}$ be an $m$-bridge diagram of $L,$ connected. See
Fig. \ref{Bvertices}. Usually, in the theory of knots, we do not draw the
dotted lines. We assume that they are under the plane $\mathbb{R}^{2
\times\{0\}$ and so the diagram can be seen as a finite collection $T=\left\{
t_{1},\cdots,t_{m}\right\} $ of disjoint arcs (\textit{no closed curves}) in
the plane $\mathbb{R}^{2}\times\{0\}$. Select\textit{\ }a point $B_{i}$ in
each one of the \textit{regions} of the complement of $D_{L}$ in
$\mathbb{R}^{2}\times\{0\}$. For the unbounded component, set $B_{0}=\infty$
\begin{figure}
[h]
\begin{center}
\includegraphics[
height=2.0612in,
width=3.7292in
{graphics/figure11.eps}
\caption{Regions of $\mathbb{R}^{2}\backslash D_{L}$.
\label{Bvertices
\end{center}
\end{figure}
The boundary points of the arcs $t_{i}$ of the link-diagram $D_{L}$ will be
the $A$-vertices of our $m$-butterfly diagram.
Each $A$-vertex belongs to the boundary of two regions. The vertices denoted
by $B$ (and selected before) in these two regions will be called
the\textit{\ neighboring }$B$\textit{\'{}s} \textit{of the }$A$\textit{-vertex}. (In Fig. \ref{fig6}, the neighboring
$B$\'{}s of the $A$-vertex $A_{1}$ are $B_{1}$ and $B_{4}$.)
The diagram $D_{L}$ contains also \textit{crossings}. A crossing involves an
overarc and two adjacent \textit{arcs. }
We now proceed to construct an $m$-butterfly diagram $\left( R,T\right) .$
Joint every $A$-vertex of $D_{L}$ with its two neighboring $B$\'{}s
by arcs lying in the regions to which these two belong. Thus we obtain a set
of arcs $R$ and we assume that these arcs have mutually disjoint interiors
among themselves and with the arcs of $T$
\begin{figure}
[h]
\begin{center}
\includegraphics[
height=2.0484in,
width=3.5268in
]
{graphics/figure12.eps
\caption{$m$-butterfly from a link-diagram $D_{L}.$}
\label{fig6}
\end{center}
\end{figure}
Then $\left( R,T\right) $ is an $m$-butterfly diagram, where $m$ is the
number of arcs in $T$. The graph $R$ is connected because the diagram $D_{L}$
is connected. Moreover, $S^{2}\backslash R$ is a disjoint union of open
$2$-cells, namely, open neighbourhoods of the arcs $t_{i}$ of the diagram.
Finally the $A$-vertices are bivalent in $R.$ Note that there are no
$E$-vertices in $R$. The set of $B$-vertices of the $m$-butterfly diagram is
the set of $B$\'{}s.
Applying the butterfly-link algorithm found in the proof of Theorem
\ref{teobridge} to $\left( R,T\right) $ (here the graph $\Gamma$ is the set
of dotted lines), it is easy to see that $L=L(R,T)$, see Fig. \ref{fig7}.
\begin{figure}
[h]
\begin{center}
\includegraphics[
height=2.0315in,
width=3.4844in
{graphics/figure13.eps
\caption{Link $L\left( R,T\right) $ from an $m$-butterfly diagram.
\label{fig7}
\end{center}
\end{figure}
\end{proof}
We will refer to the algorithm described in the proof of Theorem \ref{linkbut}
as the \textit{link-butterfly algorithm.}
\begin{definition}
The minimum $m$ among all possible $m$-butterfly diagrams of a given link $L$
is called the\emph{\ butterfly number} of $L$ and it is denoted by $m(L).$
\end{definition}
For example, the butterfly number of the trivial knot is 1, see Fig.
\ref{fig2} c; the butterfly number of any rational knot is 2, see the
Introduction and Fig. \ref{fig16}; and the butterfly number of the borromean
rings is 3, see Fig. \ref{fig21}.
\section{\label{Move}Trunk-reducing Move}
Our goal in the next two sections is to prove that the butterfly and bridge
number of knots and links coincide. To achieve this we need to know how to
reduce the number of trunks obtained by the link-butterfly algorithm described
in Section \ref{LinkBut}.
Let $L$ be a link and $(R,T)$ be an $m$-butterfly diagram of $L$ found by the
link-butterfly algorithm. We observed that it does not produce $E$-vertices.
Actually it produces only two types of butterflies. The butterflies, coming
from trunks that are overarcs, have more than two $A$-vertices, as illustrated
in Fig. \ref{fig8}a. The butterflies coming from trunks that are not overarcs
(simple arcs) have only two $A$-vertices. We call this last kind of
butterflies \textit{simple butterflies. }They have the shape illustrated in
Fig. \ref{fig8}b
\begin{figure}
[h]
\begin{center}
\includegraphics[
height=1.2785in,
width=4.1594in
{graphics/figure14.eps
\caption{a. A non simple butterfly. \ \ \ \ \ \ \ \ \ b. A simple butterfly.
\ \ \ }
\label{fig8}
\end{center}
\end{figure}
We also notice that the value of $m$ in the $m$-butterfly diagram $(R,T)$ is
just the number of all arcs in the chosen link diagram.
So given a connected $m$-bridge diagram of a link $L$, together with the
$m$-butterfly diagram $(R,T)$ representation of $L$ produced using the
link-butterfly algorithm, a natural question arises:
Is it possible to make some \textit{moves} on the $m$-butterfly diagram
$(R,T),$ in such a way, that we find a different $l$-butterfly diagram
$(R^{\prime},T^{\prime})$ of $L$ but with $l<m$? We will see that we can do
this, but at the expense of producing $E$-vertices.
Now we will show how to decrease the number of butterflies in a given
$m$-butterfly. More specifically, trunks of simple butterflies will be
converted into $E$-vertices.
Let $P$ be the simple butterfly of $(R,T)$ shown in Fig. \ref{fig9}, where the
vertex labeled by $D$ at the rightmost part of the Figure is an $A$- or
$E$-vertex and the vertex labeled by $C$ at the leftmost part of the Figure is
an $A$-vertex.
\begin{figure}
[ptbh]
\begin{center}
\includegraphics[
height=1.192in,
width=3.636in
]
{graphics/figure15.eps
\caption{Simple butterfly
\label{fig9}
\end{center}
\end{figure}
For simplicity, we will assume here that the closed $3$-cell of $(R,T)$ is
below the paper. Consider the notations given in Fig. \ref{fig9}. On both
sides of the trunk $t^{\prime}$ we draw the arcs $C^{\prime}c$ and $C^{\prime
}d$ (See Fig. \ref{fig10}). We use the same notation on both sides, to
indicate that they match by the "reflection" along $t^{\prime}.$ Inside the
$3$-cell we trace an arc $C^{\prime}D$ getting two triangles $C^{\prime}cD$
and $C^{\prime}dD$ that have only two edges on the boundary of $(R,T).$ These
triangles together with the wings $CcD$ and $CDd$ of the simple buttterfly on
the boundary of $\partial\mathbf{B}$ can be considered as the boundary of a
pyramid with quadrilateral base $CcC^{\prime}d$ and apex $D.
\begin{figure}
[h]
\begin{center}
\includegraphics[
height=1.4631in,
width=4.0308in
]
{graphics/figure16.eps}
\caption{First step}
\label{fig10}
\end{center}
\end{figure}
Now we cut the pyramid $CcC^{\prime}dD$ out of the ball $(R,T)$ (Fig.
\ref{fig11}) and glue it on the other side of $t^{\prime}$ to the
corresponding base $CcC^{\prime}d,$ thus obtaining finally Fig. \ref{fig12}
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=1.4631in,
width=4.0427in
]
{graphics/figure17.eps}
\caption{Cutting off $CcC^{\prime}dD$
\label{fig11}
\end{center}
\end{figure}
\begin{figure}
[h]
\begin{center}
\includegraphics[
height=1.4631in,
width=4.0291in
]
{graphics/figure18.eps
\caption{Gluing $CcC^{\prime}dD$
\label{fig12}
\end{center}
\end{figure}
\begin{figure}
[h]
\begin{center}
\includegraphics[
height=1.6859in,
width=3.2006in
]
{graphics/figure19.eps
\caption{A new $E$-vertex
\label{fig13}
\end{center}
\end{figure}
In this way the simple butterfly has been substituted by two edges $cD$ and
$dD$ and a $E$-vertex $D$ (see Fig. \ref{fig13}). In this process the graph
$R$ becomes a connected graph $R_{1}$ such that $S^{2}\backslash R_{1
=S^{2}\backslash\left( R\cup\bar{P}\right) ,$ where $P$ is the simple
butterfly of $(R,T)$ shown in Fig. \ref{fig9}. Hence $S^{2}\backslash R_{1}$
consists of a disjoint union of open $2$-cells. Therefore $R_{1}$ together
with the new collection of trunks $T_{1}$ is in fact a butterfly diagram.
Moreover, note that the new $E$-vertex $D$ is bivalent in $R$. See the center
part of Fig. \ref{fig13}. The point $C$ is not any more an $A$-vertex
(actually, it is now a point in the interior of a trunk$,$ (See the leftmost
part of Fig. \ref{fig13}), and notice that the valence of the $B$-vertices of
the simple butterfly $P$ decreases by one. Recall that a vertex of $R$ is a
$B$-vertex iff $p^{-1}\left( p\left( v\right) \right) $ contains at least
one non-bivalent vertex, where $p:\mathbf{B}^{3}\rightarrow M(R,T)$ is the
identification map. So, it is possible that some of the $B$-vertices are not
any more $B$-vertices but it is not a problem since they can be considered as
any other point in $R_{1}$ that is not a vertex.
The transition from Fig. \ref{fig9} to Fig. \ref{fig13} will be referred to as
a \textit{\textquotedblleft trunk-reducing move\textquotedblright.}
We have proved the following theorem
\begin{theorem}
A trunk-reducing move converts an $m$-butterfly diagram of a link $L$ into an
$\left( m-1\right) $-butterfly diagram of the same link $L$. The new diagram
gets a new $E$-vertex in place of a simple butterfly.
\end{theorem}
\begin{proof}
It is enough to apply the butterfly-link algorithm to both butterfly diagrams.
Apply it to Figures \ref{fig9} and \ref{fig13}.
\end{proof}
\begin{example}
\label{EjemploTrebol}Let us apply trunk-reducing moves to the $4$-butterfly
diagram of the trefoil knot illustrated in Fig. \ref{fig14}. There, we have
four trunks: $t_{1},t_{2},t_{3},t_{4},$ and six $B$-vertices; $a,b,c,d,e,f$
corresponding to each region of the diagram of the knot. For simplicity, we do
not draw the edges joining $A$- and $B$-vertices of the corresponding
butterfly $(R,T).$
\begin{figure}
[h]
\begin{center}
\includegraphics[
height=1.9527in,
width=1.4707in
]
{graphics/figure20.eps}
\caption{A 4-butterfly representation of the trefoil knot.}
\label{fig14}
\end{center}
\end{figure}
The arcs $t_{1}$ and $t_{4}$ correspond to simple butterflies. Therefore,
performing two trunk-reducing moves in $t_{1}$ and $t_{4}$ (in this order)$,$
the trunks $t_{1}$ and $t_{4}$ are reduced to the $E$-vertices labeled by
$E_{1}$ and $E_{4}$, respectively (see Fig. \ref{fig15}). The diagram of Fig.
\ref{fig15} is not yet a butterfly diagram because it contains too many
vertices. Indeed, under the application of the trunk-reducing moves the
$B$-vertices of the original diagram become bivalent vertices of the new
diagram that are not $A$-vertices nor $E$-vertices. Therefore we can delete
them, thus obtaining the $2$-butterfly diagram of Fig. \ref{fig16}.
\end{example}
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=1.9654in,
width=1.4207in
]
{graphics/figure21.eps}
\caption{Diagram with new $E$-vertices. }
\label{fig15}
\end{center}
\end{figure}
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=1.8239in,
width=2.9659in
]
{graphics/figure22.eps
\caption{A 2-butterfly representation of the trefoil knot.
\label{fig16}
\end{center}
\end{figure}
A $4$-butterfly diagram for the trivial link with two components is depicted
in Fig. \ref{fig17}.
\begin{figure}
[h]
\begin{center}
\includegraphics[
height=2.6228in,
width=3.0032in
]
{graphics/figure23.eps
\caption{A $4$-butterfly for the trivial link with two components.
\label{fig17}
\end{center}
\end{figure}
Applying one trunk-reducing move we get Fig. \ref{fig1}.
\begin{figure}
[h]
\begin{center}
\includegraphics[
height=1.9459in,
width=2.4517in
]
{graphics/figure24.eps}
\caption{A $3$-butterfly for the trivial link with two components, obtained by
a trunk-reducing move.}
\label{fig1}
\end{center}
\end{figure}
A second trunk-reducing move produces the $2$-butterfly diagram representing
the trivial link with two components shown in Fig. \ref{fig18}a. In Fig.
\ref{fig18}b we apply the butterfly-link algorithm to the $2$-butterfly to
recover the link.
\begin{figure}
[ptbh]
\begin{center}
\includegraphics[
height=1.7689in,
width=3.6513in
]
{graphics/figure25.eps
\caption{ A $2$-butterfly diagram representing the trivial link with two
components.}
\label{fig18}
\end{center}
\end{figure}
\begin{remark}
\label{inverso} The inverse of a trunk-reducing move can certainly be applied
to any $E$-vertex in an $m$-butterfly diagram to increase the number of
trunks. In this way it is always possible to obtain a butterfly diagram
without $E$-vertices from any given butterfly diagram of a link.
\end{remark}
\section{\label{number}The Bridge Number and the Butterfly Number}
Let us remark that the knot-diagram of the trefoil knot given in Example
\ref{EjemploTrebol} corresponds to a $2$-bridge presentation of it and by
applying trunk-reducing moves we obtained a $2$-butterfly diagram of the
trefoil knot. Actually this is a general result, and we want to show that for
any link $L,$ the butterfly number equals the bridge number, i.e., $m(L)=b(L)$.
\begin{theorem}
\label{TeoremaBridgeigualmariposa}For any link $L,$ $b(L)=m(L).$
\end{theorem}
\begin{proof}
The fact that $b(L)\leq m(L)$ is a corollary of Theorem \ref{teobridge}.
Now we will show that $m(L)\leq b(L)$ for any link $L.$
Let $D_{L}$ be a link-diagram of $L$, such that it satisfies the conditions of
Lemma \ref{diagram} and\ the number of bridges (or overarcs) is $b(L)$.
We can apply the link-butterfly algorithm to $D_{L}$ to obtain an
$m$-butterfly diagram $\left( R,T\right) $ without $E$-vertices, where $m$
is the number of arcs of $D_{L}$ (Theorem \ref{linkbut}).
Next apply trunk-reducing moves to $\left( R,T\right) $ in order to trade
simple butterflies by pairs of edges and $E$-vertices. We have to be careful
because we cannot apply the trunk-reducing moves at random. (Remember that to
be able to apply a trunk-reducing move we need that one of the two
neighbouring vertices be an $A$-vertex.) To have a consistent order of
application for a component $L_{i}$ of $L$, we start with an overarc of the
projection of $L_{i}$ (granted by Proposition \ref{diagram}) and we tour
$L_{i},$ following some orientation, performing trunk-reducing moves to the
simple butterflies in the same order that they are found. In this way we
eliminate all the simple arcs belonging to $L_{i}$ and convert them into
$E$-points. We do this for every component of $L.$ Therefore all simple
butterflies disappear (converted into $E$-vertices) and there remains only the
trunks coming from overarcs. Since the number of overarcs of $D_{L}$ is $b(L)$
the new butterfly is a $b(L)$-butterfly diagram. Then $m(L)\leq b(L).$
\end{proof}
\begin{example}
Consider the 3-bridge presentation of the borromean rings given in Fig.
\ref{fig20}.
\begin{figure}
[ptbh]
\begin{center}
\includegraphics[
height=2.4251in,
width=2.4408in
]
{graphics/figure26.eps}
\caption{ A 3-bridge presentation.}
\label{fig20}
\end{center}
\end{figure}
Make trunk-reducing moves first to the sequence $t_{2},t_{3},t_{4}$. Next to
the sequence $t_{6},t_{7,}t_{8},$ and finally to the sequence $t_{10
,t_{11},t_{12}.$ You will get the 3-butterfly diagram of Fig. \ref{fig21},
where those trunks have been exchanged by the $E$-vertices $A,B,C,D,E,F,G,H,$
and $I$, respectively. The vertices $o$, $\infty$, $1$, $2 $, $3$, $4$, $5$,
$6$, $7$, $8$, $9$,$10$, $11$, and $12$ are $B$-vertices and all of them
belong to the orbit of $\left\{ o\right\} $, under the equivalence relation
$\simeq.$
\begin{figure}
[ptbh]
\begin{center}
\includegraphics[
height=2.411in,
width=2.6482in
]
{graphics/figure27.eps}
\caption{ A 3-butterfly.}
\label{fig21}
\end{center}
\end{figure}
Another way to visualize this 3-butterfly diagram is shown in Fig.
\ref{fig22}, where, for simplicity, we do not mark the $B$-vertices, except
$o$ and $\infty.$
\end{example}
\begin{figure}
[ptbh]
\begin{center}
\includegraphics[
height=2.399in,
width=2.2485in
]
{graphics/figure28.eps
\caption{A 3-butterfly diagram for de borromean rings (without some
B-vertices).}
\label{fig22}
\end{center}
\end{figure}
\section{Conclusions}
We have proved that any link can be represented as an $m$-butterfly. We
defined the butterfly number of a link and we proved that the butterfly number
equals the bridge number of a link. Therefore it is feasible to study the
$m$-bridge links via $m$-butterflies. For each $2$-bridge link the associated
$2$-butterfly allow us to visualize the corresponding rational number. For
example, in Fig. \ref{fig2}a we have a 2-butterfly that represents the
rational knot $5/2.$ For the 3-bridge links, as we have announced in the
introduction, it is possible to associate a set of 3 rational numbers to each
3-butterfly. For more details about the way to assign a set of three rational
numbers to a $3$-butterfly diagram see \cite{To}, \cite{HMTT4}.
For example, in Fig. \ref{fig23} we show the diagrams of two 3-butterflies,
$(R_{1},T_{1})$ and $(R_{2},T_{2}),$ with the associated set of rational
numbers.
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=2.3903in,
width=3.9729in
{graphics/figure29.eps}
\caption{Two butterfly diagrams for the knot $8_{20}$, with the associate
rational numbers.}
\label{fig23
\end{center}
\end{figure}
The two diagrams are different, however $L\left( R_{1},T_{1}\right) $ and
$L\left( R_{2},T_{2}\right) $ are equivalent $3$-bridge presentation of the
knot $8_{20}$ with bridge (and butterfly) number $3$. To exhibit the
equivalence between the bridge presentations $L\left( R_{1},T_{1}\right) $
and $L\left( R_{2},T_{2}\right) $ we modify the presentation of the two
$3$-butterfly diagrams $(R_{1},T_{1})$ and $(R_{2},T_{2})$, as shown in Fig.
\ref{fig28}, on the left. In the center we have the link diagrams obtained
when we close the $3$-butterflies.
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=3.426in,
width=4.545in
]
{graphics/figure30.eps
\caption{Two non equivalent 3-butterfly diagram with equivalent 3-bridge
presentation.
\label{fig28
\end{center}
\end{figure}
Then we move the dotted arc as shown in each diagram.
This raises the problem \textit{of finding a set of moves} in a butterfly
diagram connecting diagrams representing the same link. This is left as an
open problem.
|
{
"timestamp": "2012-03-12T01:01:16",
"yymm": "1203",
"arxiv_id": "1203.2045",
"language": "en",
"url": "https://arxiv.org/abs/1203.2045"
}
|
\section{Introduction}
Ultracold Fermi gases offer a unique possibility to investigate many-body
phenomena in a controlled environment
\cite{Giorgini:2008,Ketterle:2008,Lee:2008fa,Chin:2010aa}.
In dilute systems of two-component
fermions, the interactions are characterized by the S-wave scattering length.
Close to a Feshbach resonance, the scattering length
can be tuned experimentally by varying an external magnetic field.
In particular, the crossover from the
BCS limit of weakly interacting fermions to a BEC of bosonic dimers
by tuning through a resonance has been studied in great detail
\cite{Chin:2010aa}.
The behavior of such a system is constrained by universal relations
that involve the so-called contact, which measures the number of pairs of
fermions with different spins that have small separations
\cite{Tan:2005xx,Braaten:2010if}.
More recently, ultracold gases of three-component fermions
have also been investigated. The interest in such systems has various
motivations.
First, the manifestation of the Efimov effect \cite{Efimov:1970zz}
has been studied
in systems consisting of three hyperfine states of fermionic
$^6$Li atoms. A resonant enhancement of the recombination rates
at certain values of the scattering lengths was observed
in experiment \cite{Ottenstein:2008,Huckans:2008}.
These observations were analyzed theoretically and
traced back to the appearance of
an Efimov trimer close to the three-atom threshold
\cite{Braaten:2008wd,schmidt:2008fz,Naidon:2009,Braaten:2009ey}.
Subsequently, the direct
association of Efimov trimers was also achieved~\cite{Lompe:10,Nakajima:11}.
A second line of research has focused on the
phase structure of such systems
\cite{pair-three-comp,super-phases,Breached-Pairing,Catelani:2008}.
In these theoretical studies, two components are typically paired
while the third one remains unpaired.
This mechanism can be
regarded as a generalization of the BCS case. Moreover, the BEC-BCS
crossover has also been investigated in a three component system.
In Ref.~\cite{Crossover-3komp}, the dynamics of such a system
was analyzed on time scales long enough
to see two-body physics but short enough to be able to
neglect Efimov states or three-body collisions.
For three-component fermions in an optical lattice, the formation of a superfluid phase at weak coupling
and a \lq\lq trion'' phase of three-fermion
bound states at strong coupling has been predicted
\cite{Rapp:2006rx}.
In this work, we combine both lines of research and investigate
three-body correlations in the medium.
We investigate the effect of Pauli blocking induced by the presence
of a Fermi sphere on universal two- and
three-body states in the medium. Their corresponding energies are
extracted from the poles of two- and three-body scattering amplitudes
in the medium.
A similar study was carried out in Ref.~\cite{P-Block-Ef} for the case
of a fermion immersed in a Fermi sea interacting with two heavy bosons.
The Born-Oppenheimer approximation was used to map the system
to an effective two-body problem and calculate the dependence of the
universal spectrum of Efimov trimers on the Fermi density in that case.
In Ref.~\cite{Nygaard:2011aa}, the modification of the Efimov
spectrum for three equal-mass fermions when one of the fermions
is embedded in a Fermi sea was calculated numerically and the
modification of the universal scaling
behavior by the background density of fermionic particles was
investigated.
Here, we investigate the medium
modifications for three equal-mass fermions all of which are
embedded in a Fermi sphere. We solve the two- and
three-body scattering equations
for this system (cf. Ref.~\cite{Braaten:2008wd}) in the medium and present
a detailed study of the poles of the in-medium scattering amplitude.
In particular, we study the emergence of positive energy
three-body poles analog to
the Cooper pairs in the two-body system.
A similar analysis was carried out in Refs.~\cite{Schuck-BF} for the
in-medium scattering amplitude of a boson and a fermion. In these
studies, the boson-fermion Cooper pairs were found to persist for
vanishing attraction.
We consider three distinguishable
non-relativistic particles of equal mass with
resonant interactions in a Fermi sea at zero temperature.
The system is assumed to be dilute, i.e. $k_F R\ll 1$, where $R$
is the range of the interaction and $k_F$ the Fermi momentum.
In this case, the two-body interactions of the particles are determined
by their scattering length $a$. We assume the two-body interactions to be
resonant, i.e. $|a| \gg R$.
Effective range corrections are suppressed and can be treated
in perturbation theory. Because we are at zero temperature, all states up to
$k_F$ are occupied. For $k_F a \ll 1$, a perturbative low-density expansion
can be derived~\cite{Hammer:2000xg}, but for $k_F a \sim 1$ an infinite
class of diagrams has to be resummed and one has to resort to Monte Carlo
simulations or additional expansions~\cite{Lee:2008fa,Furnstahl:2008df}.
In this study, we include only the interaction of the particles
with the Fermi sea via Pauli blocking. These effects dominate in an
expansion in the inverse number of dimensions
\cite{Steele:2000qt,Schafer:2005kg} and determine the qualitative
behavior of the system. Other effects of the medium,
such as the excitation of particles out of the Fermi sea through
scattering processes, are neglected.
Our theoretical framework is based on
an effective Lagrangian for the fermion fields
$\Psi_i$, $i=0,1,2$:
\begin{equation}
\mathcal{L} = \sum_{i=0}^2 \Psi_i^{\dagger}\left( i \partial_t +
\frac{ \vec{\nabla}^2 }{2m}\right)\Psi_i -\sum_{k=0}^{2} \frac{g_{k}}{2}
\Psi_i^{\dagger} \Psi_j^{\dagger} \Psi_i \Psi_j + h\, \Psi_0^{\dagger}\Psi_1^{\dagger}\Psi_2^{\dagger}\Psi_0\Psi_1\Psi_2\:,
\label{Lagrangedichte1}
\end{equation}
where the coupling constant ${g_{k}}$ with $i\neq j\neq k$ parametrizes the
interaction of fermions $i$ and $j$.
The term proportional to $h$ is a contact
three-body interaction of all three fermions. It determines
the spectrum of three-body Efimov states in the vacuum.
The explicit form of this
term will not be required for our study, since the dependence on the
three-body term can be traded for a dependence on the cutoff in leading order
calculations \cite{Hammer:2000nf}. For
practical calculations, it is convenient to introduce auxiliary dimer
fields $d_k$ and rewrite the Lagrangian in the form:
\begin{equation}
\mathcal{L} = \sum_{i=0}^2 \Psi_i^{\dagger}\left( i \partial_t +
\frac{ \vec{\nabla}^2 }{2m}\right)\Psi_i +\sum_{k=0}^2 \left(
\Delta_k d_k^{\dagger} d_k -\frac{g_k}{2} \left( d_k^{\dagger}
\Psi_i \Psi_j + \Psi_i^{\dagger} \Psi_j^{\dagger} d_k \right) \right)
+ h\, \Psi_0^{\dagger}\Psi_1^{\dagger}\Psi_2^{\dagger}\Psi_0\Psi_1\Psi_2\:.
\label{Lagrangedichte2}
\end{equation}
The dimer field $d_k$ describes two interacting particles $i$ and $j$ with
$i \neq j \neq k$. Using the classical equations of motion, the equivalence
of equation (\ref{Lagrangedichte1})
and (\ref{Lagrangedichte2}) can be demonstrated. This framework has been
widely used to describe the universal properties of few-body systems close
to the universal limit \cite{hw-review}. It has also been used as
the basis for studies of the Efimov effect in systems three-component fermions
\cite{Braaten:2008wd,Braaten:2009ey}.
\section{Two-body sector}
\subsection{Vacuum case}
We are now in the position to investigate the effect of Pauli blocking
on universal two- and
three-body states in the medium.
We start by briefly reviewing the vacuum case.
More details can be found in Ref.~\cite{hw-review}.
For convenience, we set $\hbar=m=1$ from now on.
The bare dimer propagator derived from the Lagrangian (\ref{Lagrangedichte2})
is simply a constant, $i/\Delta_k$.
The full, interacting dimer propagator is given by dressing the bare
propagator with fermion bubbles, see Fig.~\ref{VollerDimer}.
It represents the exact solution of the vacuum two-body problem
for the Lagrangian (\ref{Lagrangedichte2}).
\begin{figure}[ht]
\centering
\includegraphics[clip,width=15cm,angle=0]{dimerbubbles.eps}
\caption{Bubble sum for the full interacting dimer propagator (thick line).
The double lines correspond to the bare dimer propagator and the single
lines indicate particle propagators.}
\label{VollerDimer}
\end{figure}
The diagrams constitute a geometric series,
which can easily be summed. The result can be written as
\begin{align}
iD_k(P_0,\mathbf{P}) = \frac{i}{\Delta_k\left(1-\frac{g_k^2}{4\Delta_k}I(P_0,\mathbf{P}) \right)} \:,
\end{align}
where $I(P_0,\mathbf{P})$ is the loop function for
the two-fermion bubble in Fig.~\ref{VollerDimer}.
In the vacuum, the loop function is
\begin{eqnarray}
iI(P_0,\mathbf{P}) &=& \int_{|\mathbf{q}|<\Lambda} \frac{d^4q}{\left( 2 \pi \right)^4} \frac{i }{\frac{P_0}{2}+q_0 -\frac{1}{2}\left( \frac{\mathbf{P}}{2} +\mathbf{q} \right)^2 +i\epsilon}
\frac{i }{\frac{P_0}{2}-q_0 -\frac{1}{2}\left( \frac{\mathbf{P}}{2} -\mathbf{q} \right)^2 +i \epsilon} \nonumber\\
&=& \frac{i}{4\pi}\left( -\frac{2\Lambda}{\pi} +\sqrt{-P_0+P^2/4-i \epsilon}\right)\:,
\label{Loop-Vak}
\end{eqnarray}
where $P \equiv |\mathbf{P}|$ and
the UV divergence of the loop integral
has been regulated by a momentum cutoff $\Lambda$. The cutoff dependence is
absorbed into the coupling constant $g_k$,
such that all observable quantities are independent of $\Lambda$.
The two-body scattering amplitude is obtained by multiplying the
full dimer propagator with the square of the dimer-fermion coupling, $(-ig_k/2)^2$.
Matching to the amplitude for scattering of particles $i$ and $j$
in the center of mass at energy $E=p^2$,
\begin{equation}
T_k(p^2)= \frac{4\pi}{-1/a_k -ip} \stackrel{!}{=} -\frac{g_k^2}{4}
D_k(p^2,0)\:,
\end{equation}
we obtain
\begin{align}
&\frac{ g_k^2} {\Delta_k}=\frac{16\pi a_k}{1-2a_k\Lambda/\pi} \:.
\label{ak-matching}
\end{align}
Note that $ g_k$ and $\Delta_k$ are not independent at this order
and all observables depend on the combination $g_k^2/\Delta_k$. The renormalized dimer propagator
in the vacuum can thus be written as
\begin{equation}
iD_k(P_0,\mathbf{P})_{vak}=i \frac{16\pi}{g_k^2} \left[ 1/a_k -\sqrt{-P_0+P^2/4-i \epsilon} \right]^{-1} \: .
\label{eq:vacprop}
\end{equation}
The propagator has a bound state pole at $P_0= -1/a_k^2 + P^2/4$ if $a_k>0$.
The energy at the pole is composed of the binding energy $-1/a_k^2$
and the kinetic energy of the dimer $P^2/4$. The total
mass is $2m=2$, as expected for a dimer state. For negative scattering length, the pole is on the
unphysical sheet and represents a virtual state.
\subsection{Medium case}
We now move on to medium case.
In the presence of a Fermi sphere, the loop integral changes to
\begin{equation}
iI(P_0,\mathbf{P}) = \int_{|\mathbf{q}|<\Lambda} \frac{d^4q}{\left( 2 \pi \right)^4} \frac{i\Theta \left(| \frac{ \mathbf{P}}{2} +\mathbf{q}|
-k_F \right) }{\frac{P_0}{2}+q_0 -\frac{1}{2}\left( \frac{\mathbf{P}}{2} +\mathbf{q} \right)^2 +i\epsilon}
\frac{i\Theta \left(| \frac{ \mathbf{P}}{2} -\mathbf{q}| -k_F \right) }{\frac{P_0}{2}-q_0
-\frac{1}{2}\left( \frac{\mathbf{P}}{2} -\mathbf{q} \right)^2 +i \epsilon} \:,
\end{equation}
where the theta functions encode the Pauli blocking.
They ensure that the intermediate particles can not scatter into occupied
states in the Fermi sea. This introduces boundary conditions for the loop integrals at small
momenta. Different cases must be considered. A summary of the calculation
is given in Appendix~\ref{sec:medium-integ}. Here we focus on the results.
For vanishing total momentum $P$ the boundary conditions become simple. In this case, the argument of both
theta functions is $|\mathbf{q}|-k_F$. Consequently, the integration over $|\mathbf{q}|$
starts at $k_F$ and ends at $\Lambda$. It is evident that only the infrared behavior of the integrals
is modified by the Fermi sea. The renormalization of UV divergences is the same as in the vacuum.
The in-medium dimer propagator then has the form
\begin{align}
iD_k(P_0,P)=i \frac{16\pi}{g_k^2}\left[ \frac{1}{a_k} -\frac{1}{\pi}L(P_0,P) \right]^{-1} \: ,
\end{align}
with
\begin{equation}
L(P_0,P=0)=2k_F + \sqrt{P_0+ i \epsilon} \left[ \ln(k_F-\sqrt{P_0+ i\epsilon})
-\ln(k_F+\sqrt{P_0+ i \epsilon}) \right] \: .
\end{equation}
The poles of the propagator are determined by solving
\begin{equation}
\frac{1}{a_k}= \frac{2k_F}{\pi} + \frac{\sqrt{P_0+ i \epsilon}}{\pi} \left( \ln
\left( k_F-\sqrt{P_0+i \epsilon}\right)-\ln \left(k_F+\sqrt{P_0+ i \epsilon} \right) \right) \:
\end{equation}
for $P_0$. If $P_0$ is negative, this equation can be written as
\begin{equation}
\frac{1}{a_k}=\frac{2k_F}{\pi}+\frac{2}{\pi} \sqrt{|P_0|}\arctan \left( \frac{\sqrt{|P_0|}}{k_F} \right) \:,
\end{equation}
where the $i \epsilon$ has been omitted.
This equation is formally similar to Eq.~(3) of Ref.~\cite{P-Block-Ef} for the binding energy of a light fermion
immersed in a Fermi sea interacting with two heavy bosons. In this case, the Born-Oppenheimer approximation
may be used and the three-body problem reduces to an effective two-body problem.
In the general case, the boundary conditions are more complex (cf. Appendix \ref{sec:medium-integ}).
Two cases have to be distinguished:
$P<2k_F$ and $P > 2k_F$. The result for the general in-medium loop function $L(P_0,P)$ is:
\begin{itemize}
\item[(a)] $P<2k_F$:
\begin{align}
L(P_0,P)=&P/2+k_F+\sqrt{\sigma}\left[ \ln\left(P/2+k_F-\sqrt{\sigma}\right)
-\ln \left(P/2+k_F+\sqrt{\sigma}\right) \right] \notag\\
&+\frac{k_F^2-P_0 -i \epsilon}{P} \bigg [ \ln \left( P/2+k_F-\sqrt{\sigma}\right)+
\ln \left( P/2+k_F+\sqrt{\sigma} \right) \notag \\
& - \ln \left( \sqrt{k_F^2-\tfrac{1}{4}P^2}-\sqrt{\sigma} \right) -
\ln \left( \sqrt{k_F^2-\tfrac{1}{4} P^2}+\sqrt{\sigma} \right) \bigg ]\,,
\label{eq:medprop1}
\end{align}
\item[(b)] $P>2k_F$:
\begin{align}
L(P_0,P)=&2k_F + \pi \sqrt{-\sigma}
+ \sqrt{\sigma} \bigg[\ln \left(P/2-k_F+\sqrt{\sigma} \right)
+ \ln \left( P/2 +k_F-\sqrt{\sigma}\right) \notag\\
&-\ln \left( P/2-k_F-\sqrt{\sigma}\right)
- \ln \left( P/2 +k_F+\sqrt{\sigma} \right) \bigg ] \notag \\
&+\frac{-k_F^2+P_0+i\epsilon}{P} \bigg[ \ln \left( P/2-k_F-\sqrt{\sigma} \right)
+ \ln \left( P/2 -k_F+\sqrt{\sigma}\right) \notag \\
&- \ln \left(P/2+k_F-\sqrt{\sigma} \right) - \ln \left( P/2 + k_F + \sqrt{\sigma} \right) \bigg ]\,,
\label{eq:medprop2}
\end{align}
\end{itemize}
with $\sqrt{\sigma}=\sqrt{P_0-P^2/4 + i \epsilon}$\:.
We now discuss the poles of the dimer propagator in the medium. Our aim is to recover the known two-body physics
from the viewpoint of the pole structure and then use the same strategy to understand the three-body sector.
First, we specify our units. Since there is one free length scale $l_0$ in the calculations,
we express all dimensionful quantities in units of $l_0$: the energy has the unit $[1/l_0^2]$,
scattering lengths $[l_0]$ and momenta $[1/l_0]$.
\begin{figure}[t]
\centering
\includegraphics[clip,width=10cm,angle=0]{bild1av2.eps}
\caption{(Color online) The energy $E$ of the dimer pole at $P=0$
plotted against the inverse scattering length
$1/a_k$ for a Fermi momentum $k_F=0.7/l_0$ (solid line) and $k_F=1/l_0$ (dash-dotted line).
In addition, four selected points are marked I, II, III, and IV
on the solid line. For comparison, the vacuum pole
energy is shown by the dashed line. In the inset, the dimer pole energy is displayed
as a function of the total momentrum $P$ for $k_F=0.7/l_0$ and $a=l_0$. Curves are as
above.
}
\label{fig:Streulaenge1}
\end{figure}
We never find more than one pole on the physical sheet in the in-medium
dimer propagator. The physical conditions under which this pole can
disappear are discussed below.
In Fig.~\ref{fig:Streulaenge1}, the energy of the pole, $E$, is plotted against the inverse
scattering length $1/a_k$ at vanishing momentum $P=0$ for $k_F l_0 =0.7$ (solid line)
and $1$ (dash-dotted line), respectively. The dashed curve represents
the dimer energy in the vacuum case. For positive scattering length, the energy of the vacuum
pole is $ E=-1/a_k^2$. There is no vacuum pole on the physical
sheet if the scattering length is negative.
For non-vanishing Fermi momentum, a pole with positive energy appears in the negative
scattering length region. In the limit $1/a_k \to -\infty$, this pole asymptotically approaches
the values $(k_F l_0)^2=0.49$ and $1$ for $k_F l_0 =0.7$ and $1$, respectively.
In the positive scattering length region, the pole behaves like a vacuum pole if the
scattering length is sufficiently small. However, the corresponding binding energy is reduced
by medium effects. Additionally four selected points are marked on the solid line: I, II, III, and IV.
To gain deeper insight into the nature of the pole in the in-medium dimer propagator,
these points will be further investigated below.
The parameters $k_F$ and $a$ are kept fixed while the momentum $P$ will be varied.
In the inset of Fig.~\ref{fig:Streulaenge1}, which corresponds to point I, the dependence of
the pole energy on the total momentum $P$ is shown
for $a_k=l_0$ and $ k_F=0.7/l_0$ (solid line). The dashed line shows the vacuum pole energy as before.
Medium and vacuum poles have a similar behavior as function of the total momentum: with increasing
momentum $P$, the pole energy is increased and the binding is reduced.
For the vaccuum pole, this is evident from Eq.~(\ref{eq:vacprop}).
In the medium, it follows from the dominant functional dependence of
the in-medium dimer propagator on $\sigma=P_0-P^2/4$
(cf.~Eqs.~(\ref{eq:medprop1}, \ref{eq:medprop2})). Moreover, medium effects are again seen to lower
the pole energies.
\begin{figure}[t]
\centering
\includegraphics[clip,width=12cm,angle=0]{dim-kf-a.eps}
\caption{(Color online)
(a) The energy $E$ depicted as function of the total momentum $P$
for $a_k=-3l_0$ and $k_F=0.7/l_0$ (solid line). The vertical dotted line
gives $E=k_F^2$. (b) Energy $E$ plotted against the Fermi momentum
$k_F$ for $a_k$=$2l_0$ (solid), $4l_0$ (dashed), $6l_0$ (dashed dotted); $P=0$.
}
\label{fig:dimer-kf2}
\end{figure}
Next, we examine the positive energy poles more thoroughly. In Fig. \ref{fig:dimer-kf2}~(a), the energy
is plotted against the total momentum for a negative scattering length $a_k=-3 l_0$,
which corresponds to point II in Fig.~\ref{fig:Streulaenge1}. The energy of
the pole is positive and continuously rises as the momentum $P$ is increased
until the energy reaches the value $k_F^2 = 0.49/l_0^2$,
where the pole disappears.
This positive energy pole can be associated with Cooper pairs~\cite{Fetter}.
With this interpretation, their peculiar behaviour can be understood. Assume that the two particles are inside the Fermi
sphere.
If there is no interaction, the energy of the particles is just
their kinetic energy. In the presence of attractive interactions, the energy of the two particles, given by the
pole energy, is lowered.
Consequently, the energy gain $\Delta E$ is the difference of the kinetic energy and the pole energy. Because the
maximum kinetic energy of two particles inside the Fermi sea is $k_F^2/2 + k_F^2/2$,
the maximum energy of the pole is also $k_F^2$.
When the total momentum of the two particles becomes too large, the pole disappears.
This property is compatible with the intepretation as Cooper pairs,
whose total momentum is commonly assumed to be zero. Remember that the energy threshold already appeared in
Fig.~\ref{fig:Streulaenge1}. But in this instance a different limit was considered.
The energy of the pole approaches the threshold $k_F^2$ asymptotically
in the limit $1/a_k \rightarrow - \infty$. The energy gain $\Delta E$, hence, decreases
in this limit. But the poles never disappear and Cooper pairs can always be formed in this region.
We now turn to the dependence of the poles on the Fermi momentum in the positive scattering length region.
In Fig. \ref{fig:dimer-kf2}~(b), the pole energy is plotted against the Fermi momentum. The total momentum $P$
is set to zero and the scattering lengths are $a_k=2 l_0,\, 4 l_0$ and $6 l_0$. As expected, the energy is negative
at $k_F =0$ and the pole corresponds to a bound state. With increasing Fermi momentum, the medium effects
become stronger and the binding is reduced. For small $k_F$ the energy rises only slowly but at larger
Fermi momentum the energy changes rapidly, crosses zero, and becomes positive.
Hence, we observe a continuous crossover from bound states to positive energy poles as the Fermi momentum
is increased.
So far, we could associate the left and right regions in Fig.~\ref{fig:Streulaenge1} to Cooper pairs (cf. II) and bound
states (cf. I), respectively.
In between lies the crossover region. We will investigate this region further at the two remaining points III and IV.
Here the nature of the poles changes as a function of momentum.
\begin{figure}[t]
\centering
\includegraphics[clip,width=12cm,angle=0]{crossover.eps}
\caption{(Color online)
The energy of the poles plotted as a function of the total momentum $P$ for
$a=2l_0$ [panel (a)] and $a=3l_0$ [panel (b)] and $k_F=0.7l_0$
(solid line). The dashed line shows
the vacuum poles and the dash-dotted line gives the kinetic energy $P^2/4$.
The horizontal dotted line in (b) gives $k_F^2$ and the vertical dotted line gives $2k_F$. }
\label{fig:crossover}
\end{figure}
In Fig.~\ref{fig:crossover}~(a) the energy of the pole is plotted against the total momentum (solid line) with
$a= 2l_0$ corresponding to point III in Fig.~\ref{fig:Streulaenge1}.
By comparision, the dashed line shows the vacuum pole and the dash-dotted line the kinetic energy $P^2/4$.
For vanishing momentum the energy of the pole is extremely reduced compared to the vaccuum
but still negative. However, in the region around
$P=l_0$ the energy becomes bigger than the kinetic energy. Hence, this pole can not correspond to a bound state
in this region. For larger momenta the energy again drops below the kinetic energy and the poles behave
similiar to vacuum poles.
We now turn to point IV in the crossover region. In Fig.~\ref{fig:crossover}~(b) an entirely different behaviour can
be observerd. Again the vacuum pole (dashed line) and the kinetic energy (dash-dotted line) are shown. Striking
are the three qualitatively different regions in this graph. For momenta $P<2k_F$ the pole seems to correspond to
a Cooper pair: at $P=0$ the energy of the poles is positive, in this whole region the energy is larger than the
kinetic energy, and the pole disappears when the energy approaches $k_F^2$. In a region around $P\approx 2k_F$
there is no pole at all. For slightly larger momenta, the pole reappears. At first the energy is very close
to the kinetic energy, but for larger momentum it approaches the vacuum energy, as expected.
These pole now behaves like a bound state.
In summary, we have related the positive energy poles at $P=0$ to Cooper pairs and the negative energy poles to bound
states. A finite momentum $P$ leads to an increase in the pole energy. In the vacuum, the additional energy is simply
the kinetic energy $P^2/4$. In the medium, the pole energy also increases but the dependence on $P$ is more complicated.
In particular, the poles can vanish and change their character.
As the Fermi momentum $k_F$ is increased, e.g.,
the binding energy is reduced by medium effects.
We identified the two extremes I and II in Fig.~\ref{fig:Streulaenge1} with the BCS and BEC domains,
respectively. In between there is a crossover region. In this region the poles change their character
as a function of the momentum $P$ and they
can not be uniquely related to one of the two cases. Equipped with this qualitative understanding of
in-medium two-body physics, we move on to the three-body amplitude.
\section{Three-body sector}
\subsection{Vaccuum case}
We start by briefly reviewing the physics issues of the vacuum case and then move on
to the medium. In the three-body system with resonant interactions, there is a
universal spectrum of three-body bound states with an accumulation point at zero
energy, called Efimov states \cite{Efimov:1970zz}.
The spectrum is given by Efimov's universal equation
\begin{eqnarray}
E_B^{(n)} + {1\over a^2} =
\left(e^{-2 \pi/ s_0} \right)^{n-n_*}
\exp \left[ \Delta ( \xi )/s_0 \right] \kappa_*^2 \,,
\label{eq:bind}
\end{eqnarray}
where the angle $\xi$ is defined by
\begin{eqnarray}
\label{xin-def}
\tan \xi = - a\sqrt{E_B^{(n)}} \,,
\end{eqnarray}
$s_0\approx 1.00624$ is a transcendental
number, and $\kappa_*$ is the binding wave number of the state labelled $n_*$.
The function $\Delta ( \xi )$ was first calculated in Ref.~\cite{Braaten:2002sr}
and satisfies $\Delta( -{1\over2}\pi )=0$.
In the unitary limit of infinite scattering length, the spectrum thus becomes
geometric.
The qualitative features of this spectrum are determined by the scattering length $a$, but
the exact energies depend on the three-body interaction in Eq.~(\ref{Lagrangedichte2})
which fixes the value of $\kappa_*$ \cite{Bedaque:1998kg}.
The spectrum exhibits a discrete scaling symmetry which is evident in Eq.~(\ref{eq:bind}):
if the scattering length $a$ and the energies $E_B$ are rescaled by the discrete
scaling factor $\lambda=\exp(\pi/s_0)$ and $\lambda^{-2}$, respectively, but $\kappa_*$ remains fixed
the spectrum is mapped onto itself. If the scattering length dependence of one state is known, thus all other can be
obtained from the scaling transformation. A detailed discussion
of these issues can be found in Ref.~\cite{hw-review}. Here, we focus on the modification of
this spectrum in the medium and on possible positive energy poles in the three-body amplitude
similar to the two-body case discussed above.
As discussed above, we set the three-body interaction to zero in our calculation.
Thus $\kappa_*$ is proportional to the momentum cutoff $\Lambda$. The exact proportionality factor
is not required for our purpose.
A detailed study of the Efimov spectrum and the universal scaling relations
in the presence of one Fermi sphere was carried out in Ref.~\cite{Nygaard:2011aa}.
We go beyond this study by considering three Fermi spheres and explicitly focusing
on the emergence positive energy poles in the three-body amplitude.
Preliminary results of our study were already presented in~\cite{PatrickDip}.
\subsection{Medium case}
The three-particle scattering amplitude in the medium
can be calculated by solving an integral equation.
In order to simplify the boundary conditions given by the Pauli blocking, we will constrain
the total momentum of the three particles to be zero. We note that the Fermi sea provides a special
reference frame and a non-zero momentum can not be obtained from a simple Galilei
transformation. However, we have seen in the two-body case that a non-zero
momentum essentially increases the pole energy. Outside of the
crossover region, the qualitative behavior remains unchanged (cf. inset of Fig.~\ref{fig:Streulaenge1}).
We expect the same to be true in the three-body case.
The Feynman diagrams for three-body scattering amplitude are depicted in Fig.~\ref{Fig:Streuamplitude}.
\begin{figure}[t]
\centering
\includegraphics[clip,width=14cm,angle=0]{streuamplitude.eps}
\caption{Feynman diagrams for the fermion-dimer scattering amplitude for zero
total momentum. Momenta $\mathbf{p},\,\mathbf{q},\,\mathbf{k}$
and fermion indices $i,j,r,r'$ are assigned as in Eq.~(\ref{eq:3bfeyn}).}
\label{Fig:Streuamplitude}
\end{figure}
Since we are interested in the three-body singularities of the amplitude, it is sufficient to consider the
fermion-dimer scattering amplitude where the external dimer propagators are amputated.
Because the three particles are distinguishable,
we have one inhomogeneous and two homogeneous contributions
to the amplitude as the intermediate dimer can be formed in two ways \cite{Braaten:2008wd}.
The integral equation for the amplitude $\mathcal{A}_{ij}$
can be written as
\begin{align}
i\mathcal{A}_{ij}(\mathbf{p},\mathbf{k},E,E_i,E_j) =& -\frac{g_i g_j}{4}
\frac{i\theta(|\mathbf{p}+\mathbf{k}|-k_F)}{E-E_i-E_j-\frac{(\mathbf{p}+\mathbf{k})^2}{2}
+ i \epsilon} \cdot (1-\delta_{ij}) \notag \\
&+ \sum_{r=0}^2 -\frac{g_i g_r}{4} \int_{|\mathbf{q}|<\Lambda} \frac{d^4q}{\left( 2 \pi \right)^4} \frac{i\theta(q-k_F)}{q_0-\frac{1}{2}q^2+
i \epsilon} \cdot \frac{i\theta(|\mathbf{p}+\mathbf{q}|-k_F)}{E-E_i -q_0 - \frac{1}{2}
(\mathbf{p}+\mathbf{q})^2+ i \epsilon} \notag \\
& \times iD_r (E-q_0,q)\cdot(1-\delta_{ir}) \cdot i \mathcal{A}_{rj}(\mathbf{q},\mathbf{k},E,q_0,E_j) \:,
\label{eq:3bfeyn}
\end{align}
where the momenta and particle indices are assigned as in Fig.~\ref{Fig:Streuamplitude} and $E$ is
the total energy.
After setting the energies of the incoming and outgoing particles, $E_i$ and $E_j$, on shell,
the bare coupling constants are removed by defining a renormalized amplitude:
\begin{equation}
\mathcal{A}_{ij}^R(\mathbf{p},\mathbf{k},E)=\sqrt{|Z_i||Z_j|}\mathcal{A}_{ij}(\mathbf{p},\mathbf{k},E) \:,
\end{equation}
where $Z_i$ is the residue of the dimer pole $i$ in the vacuum,
\begin{equation}
Z_i=\frac{32 \pi}{g_i^2 a_i} \:.
\label{Res-faktoren}
\end{equation}
This renormalized amplitude has the same poles in the three-body sector as the
three-particle scattering amplitude. We now expand the fermion-dimer amplitude in partial waves as
\begin{equation}
\mathcal{A}_{ij}^R(\mathbf{p},\mathbf{k},E) = \sum_{l=0}^\infty (2l+1)\,(\mathcal{A}^R_{ij})_l[p,k,E]
P_l(\cos\theta_k)\:,
\label{partial waves}
\end{equation}
where $\cos \theta_k = \mathbf{p}\cdot \mathbf{k}/(pk)$ and $P_l$ is a Legendre
polynomial. The different partial waves decouple and the integral equation for
the $l$th partial wave amplitudes is
\begin{align}
\label{eq:ampliudeeq}
i(\mathcal{A}^R_{ij})_l[p,k,E] =& \frac{1}{2} \frac{-8\pi i}{\sqrt{|a_i||a_j|}} \int_{-1}^1 d
\cos \theta_k P_l( \cos \theta_k)
\, t_{ij}(p,k, \theta_k,E) \notag \\
&+ i\sum_{r=0}^2 4 \pi \frac{\sqrt{|a_r|}}{\sqrt{|a_i|}} \int_{k_F}^{\Lambda} \frac{dq}{(2\pi)^2}q^2
\int_{-1}^1 d\cos \theta_q \, P_l(\cos \theta_q) \,t_{ir}(p,q,\theta_q,E) \, \notag \\
& \times \overline{D}_r (q,E) \mathcal({A}_{rj}^R)_l[q,k,E] \:,
\end{align}
where
\begin{align}
t_{ij}(p,k,\theta_{k},E):=&\frac{\theta (|\mathbf{p}+\mathbf{k}|-k_F)(1-\delta_{ij})}{E-p^2 -k^2 -pk
\cos \theta_{k} + i \epsilon} \:,
\end{align}
and
\begin{equation}
\overline{D}_r(q,E):= \left[\frac{1}{a_r}-\frac{1}{\pi}L(E- \tfrac{1}{2} q^2,q)\right]^{-1} \:
\end{equation}
is the dimer propagator without prefactors. In the vacuum only the S-wave amplitude has bound state
poles. This remains true in the medium and we thus focus on the poles of the S-wave in-medium
amplitude $(\mathcal{A}^R_{ij})_0[p,k,E]$. The technical details of the implementation of the boundary
conditions from the Pauli blocking are discussed in Appendix \ref{sec:medium-integ3}. In the next
section, we present our results for the pole structure of $(\mathcal{A}^R_{ij})_0[p,k,E]$.
\subsection{Results}
In this section we discuss the poles of the the S-wave amplitude $(\mathcal{A}^R_{ij})_0$,
in general for three different scattering lengths.
In Eq.~(\ref{eq:ampliudeeq}) the three-body force dependence was traded for the cutoff dependence,
so that the cutoff determines the three-body energy in the vacuum for given scattering lengths.
A spectrum of two states as a function of the Fermi momentum is shown in Fig. \ref{fig:spektrum}.
Similar to the two-body case, the binding energy of each state decreases with rising
Fermi momentum due to medium effects. Remarkable is the difference of the energy loss with increasing Fermi
momentum for shallow and deep states. The
less bound state disappears through the threshold while the more deeply bound state
looses only about 5\% of its binding energy as $k_F l_0$ is increased from 0 to 1.
This behaviour of the
three-body spectra is generic and was always observed in our calculations.
\begin{figure}[t]
\centering
\includegraphics[clip,width=8cm,angle=0]{spektrum-plus.eps}
\caption{(Color online)
Binding energy $E_B$ of two states depicted in dependence of $k_F$:
$a_k=l_0$ for $k=0,1,2$ and $\Lambda=250/l_0$.}
\label{fig:spektrum}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[clip,width=12cm]{coopertripel.eps}
\caption{(Color online)
(a) Energy of three-body poles plotted against $k_F$ for $a_0=-1.0l_0$ (dashed), $a_0=-1.25l_0$
(dash-dotted) and $a_0=-1.5l_0$ (solid); $\Lambda=150/l_0$ and the remaining scattering lengths are
$a_1=-2l_0$ and $a_2=-2.5 l_0$.
(b) The pole energy is shown as a function of $1/a_0$ for $\Lambda= 150/l_0$ (dash-dotted), $\Lambda=160/l_0$
(solid) and $\Lambda= 170/l_0$ (dashed); $k_F=0.7/l_0$ and the remaining scattering lengths are
$a_1=-l_0$ and $a_2=-0.99l_0$. The horizontal line gives $E=1.5 \, k_F^2$.}
\label{fig:coopertripel}
\end{figure}
In Fig. \ref{fig:coopertripel}~(a) the energy
of a generic three-body pole is plotted against the Fermi momentum for three negative scattering lengths.
As in the previous case, the binding energy reduces with increasing Fermi momentum. Indeed,
the energy goes to zero and continuously rises to positive values. Hence, we have found poles with
positive energy. Since the total momentum is zero, they can not correspond to bound states.
Note the resemblance between this figure and Fig.~\ref{fig:dimer-kf2} which shows dimer poles.
To get a better understanding of these positive energy poles, we have
varied one of the three negative scattering lengths while keeping the other two constant, see
Fig. \ref{fig:coopertripel}~(b).
The energy rises with decreasing $1/a$, but vanishes when
the value of the energy becomes $1.5 \,k_F^2$. For different configurations of the Fermi momenta,
scattering lengths, and the cutoff, we have always found this threshold. The accuracy of the location
of this threshold reaches to the third (fourth) decimal place for cutoffs of the order 100 (10) $l_0$.
In order to explain this observation, we draw an analogy with the positive energy poles in the two-body case.
There, the energy gain $\Delta E$ is the kinetic energy minus the energy of the pole. Hence, the energy of
the pole can not be larger than the maximum kinetic energy.
\begin{figure}[ht]
\centering
\includegraphics[clip,width=10cm,angle=0]{skizze-fermikugel.eps}
\caption{(Color online)
Configurations in the Fermi sphere for two (left) and three particles (right) with total momentum $P=0$.}
\label{fig:skizze-fermikugel}
\end{figure}
In Fig.~\ref{fig:skizze-fermikugel} configurations of two and three particles inside the Fermi
sphere are shown for total momentum $P=0$. As discussed in the previous section,
the maximum two-body pole energy is $k_F^2$.
In the case of three particles the magnitude of each momentum can be
$k_F$, whereas the total momentum remains zero. So the maximum kinetic energy of three particles
inside the Fermi sphere is $3 \times k_F^2/2=1.5 k_F^2$.
We conjecture that the three-particle poles
belong to a state similiar to a Cooper pair, but built out of three particles, which we call a \lq\lq Cooper triple''.
In contrast to Cooper pairs, these Cooper triples are fermions.
If the three pair scattering lengths are equal, the Cooper triples
are SU(3) singlets. However, for different scattering lengths, the
SU(3) symmetry is broken.
Cooper triples also appear if one scattering length is
positive. Since the energy of the triples is continuous in $1/a_i$ ($i=0,1,2$), the region
of three negative scattering lengths merges into the region of one positive
and two negative scattering lengths at the point $1/a_i=0$ (the other two
scattering lengths are considered constant). Therefore the pole energy has
to remain positive in the limit $1/a_i \rightarrow 0^-$ to obtain Cooper
triples for one positive scattering length. For this scenario, the Fermi
momentum must be sufficiently large. The actual value depends on the two constant scattering
lengths. Hence, Cooper triples also occur in this region. An analogous argument holds if two
or three scattering lengths are positive. In all three cases, we have observed Cooper triples
in our calculations.
However, it remains to be verified that the Fermi spheres assumed in
our calculation persist in this region.
\begin{figure}[t]
\centering
\includegraphics[clip,width=8cm,angle=0]{deltaEvergleich2.eps}
\caption{(Color online)
$\Delta E$ is plotted against the inverse scattering length $1/a_0$ of three-body (solid line) and
two-body (dashed line) poles; $\Lambda=160/l_0$, $k_F=0.7/l_0$ and the constant scattering lengths are
$a_1=-l_0$ and $a_2=-0.99l_0$. The
horizontal dotted line shows $\Delta E$ for the constant scattering length $a_2=-0.99l_0$.
}
\label{fig:deltaE}
\end{figure}
Next, we examine which state is energetically favorable. If Cooper pairs are built in a three component
Fermi gas, the pairs are typically formed between two components while the residual component remains unpaired.
Therefore, we compare the energy gain of a Cooper triple, $1.5 \,k_F^2$
minus pole energy, with the energy gain of a Cooper pair, $k_F^2$ minus the pole energy.
The energy gain $\Delta E$ of a Cooper pair and a Cooper triple are compared as a function of one
variable scattering length in Fig. \ref{fig:deltaE}. The remaining parameters stay the same as in
Fig.~\ref{fig:coopertripel}~(b). Since $\Delta E$ of the Cooper pair depends on the scattering length,
it can be energetically favorable to build a different Cooper pair connected to one of the constant scattering
lengths. To account for this, we have also plotted the energy gain of the larger constant scattering length.
We find that the energy gain of the three-particle poles is much larger (note the logarithmic axis). Only near the
threshold for the triple, the three-particle $\Delta E$ rapidly falls off and drops below the energy gain of both Cooper
pairs. This suggests that Cooper triples could play an important role in three-component
Fermi gases in the continuum.
In principle, it should always be possible to find these positive energy poles for three negative scattering
lengths. In contrast to the two-body case, the poles do not newly emerge in the medium. Primarily,
the poles were bound states in the
vacuum which became modified by the medium, see Fig.~\ref{fig:coopertripel}~(a). Thus, the Fermi momentum
must be large enough to obtain positive energy poles. This is most easily achieved for
Cooper triples emerging from rather shallow three-body bound states in vacuum.
\section{Conclusion and Outlook}
In this paper, we have examined the influence of Pauli blocking on universal two- and three-body states.
First the poles of the two-body scattering amplitude in the medium were regarded. We were able to
recover the physics of Cooper pairs and BEC-BCS crossover from the pole structure of the amplitude.
In particular, we found that the binding energy of bound states decreases with rising Fermi
momentum due to medium effects.
In the negative scattering length region, positive energy poles emerge which can be
identified with Cooper pairs.
In the crossover region, the poles show a different behaviour and their nature changes
with the total momentum. They can not be uniquely identified as bound states or
Cooper pairs.
We have used the same strategy to investigate the pole structure of the three-body
scattering amplitude. We found that the medium effects reduce the binding
of three-body states compared to the vacuum. This is in agreement with the findings
of Ref.~\cite{Nygaard:2011aa}, where the modification of the Efimov
spectrum for three equal-mass fermions with one of the fermions
embedded in a Fermi sea was calculated. Moreover, we found
three-body poles with positive energy. As in the two-body sector,
we observed a continuous crossover from
negative energy poles to positive energy poles as the Fermi momentum is varied.
The maximum energy of the poles was found to be $1.5 k_F^2$. In analogy to
the connection between positive energy poles in the two-body sector and
Cooper pairs, we
have interpreted
this as evidence for the formation of Cooper triples
composed out of three particles.
These Cooper triples are fermions and thus can not Bose condense.
The energy gain of such a triple was found to be larger than the energy gain of the corresponding
Cooper pair over a large region of scattering lengths. Consequently, it appears to be energetically favorable
to form a triple instead of a pair and an unpaired third atom in this region.
In the case of equal pair scattering lengths, the Cooper triples are
SU(3) singlets. For different scattering lengths, however, the SU(3)
symmetry is broken (cf. Fig.~\ref{fig:coopertripel}).
If the three scattering lengths are large, the
SU(3) breaking is small since the leading corrections to the SU(3) limit are
proportional to the inverse scattering lengths \cite{Mehen:1999qs}.
How these three-body correlations affect a many-body system is an open question.
It would be interesting to extend previous studies of the phase structure
of three-component Fermi gases
\cite{pair-three-comp,super-phases,Breached-Pairing} to include the triples
and investigate their influence.
A qualitative picture of the many-body
structure in the SU(3) symmetric limit was given
by Floerchinger and collaborators \cite{Floerchinger:2009fp}.
They argue that at small density, if the scattering length is varied from
large negative to large positive values, the BCS and BEC phases are separated
by a trion phase of three-body bound states. At small densities, our Cooper
triples must reduce to the trions of Ref.~\cite{Floerchinger:2009fp}.
A related study for three-component fermions in an optical lattice was
carried out in Ref.~\cite{Rapp:2006rx}.
Within a Hubbard model with SU(3) symmetry, a trion
phase of three-fermion bound states
has been predicted at strong coupling and a parallel to the
baryonic phase of QCD was drawn.
There may also be a connection to Ref.~\cite{Schuck-BF}, where Boson-Fermion
(BF) interactions were regarded in a similiar analysis. BF pairs
at positive energies were found and the ground state was assumed to be a
Fermi Gas of BF-Cooper pairs, since the pairs are still fermions. The
interaction of three distinguishable fermions in our case could also be
regarded as the interaction of a Cooper pair (Boson) and a fermion of the
remaining type, if the scattering lengths are negative (BCS region) and at
least two scattering lengths are different. In this case, we can conjecture
that the ground state of the system is a Fermi gas of Cooper triples, which
are composites of a Cooper pair and a unpaired fermion.
Since we have only included the Pauli blocking effects from the medium,
further theoretical study
is required. This could for example be achieved by performing Monte
Carlo simulations of such
systems similar to the two-flavour case \cite{Lee:2008fa}. Such a
calculation would allow for more quantitative
predictions of the effect. In analogy to Ref.~\cite{Schuck-BF},
the triples might also lead to a new type
of superfluidity in three-component Fermi systems which
could be observed in ultracold atomic gases.
For this purpose, it would be useful to calculate the interactions
of the triples.
If their interactions are attractive, they could again form Cooper
pairs and Bose condense.
Much insight would be gained if one could calculate
the energy of such a condensate
and compare it with a BCS condensate. This would allow to determine
under which conditions such a new type of superfluidity might occur.
An experimental test of this scenario could be carried out
with $^6$Li atoms where mixtures of three different hyperfine
states with tunable interactions are already available
\cite{Ottenstein:2008,Huckans:2008,Lompe:10,Nakajima:11}.
\begin{acknowledgments}
We thank R.J.~Furnstahl for discussions and Dean Lee for comments on the manuscript.
Partial financial support from the Deutsche Forschungsgemeinschaft (SFB/TR 16),
and BMBF~(grant 06BN9006) are acknowledged. This work was further supported by the
EU HadronPhysics3 project ``Study of strongly interacting matter''.
\end{acknowledgments}
|
{
"timestamp": "2012-08-16T02:03:37",
"yymm": "1203",
"arxiv_id": "1203.1824",
"language": "en",
"url": "https://arxiv.org/abs/1203.1824"
}
|
\section{Introduction}\label{s:introduction}
A Requirement Engineering Method (\xc{rem}) can be thought of as a combination of a formalism for the representation and analysis of requirements, and of processes to support and guide the user of the formalism through Requirements Engineering (\xc{re}) tasks, such as requirements elicitation, representation, validation, verification, and evolution. Examples of \xc{rem}s are \xc{rml} \cite{Greenspan:1984:PHD}, \xc{erae} \cite{Dubois+:1988:PJR}, \xc{nfr} \cite{Mylopoulos+:1992:TSE}, \xc{kaos} \cite{Dardenne+:1993:SCP}, i* \cite{Yu+:1994:ICSE}, Viewpoints \cite{Finkelstein+:1994:TSE}, Labeled Quasi-Classical Logic \cite{Hunter+:1998:TOSEM}, Tropos \cite{Castro+:2002:IS}, Formal Tropos \cite{Fuxman+:2004:REJ}, \xc{carl} \cite{Gervasi+:2005:TOSEM}, Techne \cite{Jureta+:2010:RE}, and many others.
It is difficult to estimate the number of \xc{rem}s and of the publications on \xc{rem}s. \zi{Method} informally means a procedure for doing something, so really \zi{any} publication which proposes \zi{how to do something} within the scope of \xc{re} proposes \zi{a method for} \xc{re}, regardless of what that method's intended scope (coverage) and depth (level of detail) may be. Figure \ref{f:cumulative-publications} gives a very rough estimate of the cumulative number of publications related to \xc{rem}s since 1991. \xc{rem}s are also of interest outside \xc{re}, such as in Business Analysis \cite{HP:2007,BABOK:2009}.
\begin{figure}[b!]
\vspace{-8mm}
\centering
\includegraphics[width=7.5cm]{cumulative-publication-data}
\vspace{-3mm}
\caption{Cumulative number of publications since 1991 that cite in the title, abstract or main text the exact phrase ``requirements engineering'' together with one or more of the terms ``method'', ``approach'', ``framework'', ``methodology''. Data from Google Scholar.}
\label{f:cumulative-publications}
\end{figure}
Despite the interest in \xc{rem}s, it is still unclear how to answer basic questions about them. Which components does an \xc{rem} have? Which components must it have and why? When is an \xc{rem} domain-specific? Given two \xc{rem}s, how can we compare them? How to systematically make an \xc{rem}? How to know if some contribution in \xc{re} is an \xc{rem}, an \xc{rem} component, or something else? How does \xc{rem} research relate to formal methods, logics, ontology engineering? What to include in a course on \xc{rem} design?
This paper does not answer all of these questions. Instead, the proposal is a classification framework for \xc{rem}s that distinguishes \xc{rem}s based on the properties of their components. A component is a set of concepts, relations, rules, or other tools having a well-delimited role within \xc{rem}. The framework uses five components, Requirements Problem and Solution, Ontology, Formalism, Organization Mechanism, and Guidelines. The classification framework is intended to help (i) the analysis, teaching and extension of existing \xc{rem}s, (ii) the engineering and validation of new \xc{rem}s, and (iii) the identification and organization of research challenges in \xc{rem}s design and validation. Classification dimensions other than components are certainly relevant, but stay outside the scope of this paper (e.g., classification by scope, by domain, by results from use, etc.).
The framework is first introduced and illustrated (\S\ref{s:components}), research challenges (\S\ref{s:challenges}) and limitations (\S\ref{s:discussion}) are discussed, and conclusions are summarized (\S\ref{s:conclusions}).
\section{Classification Framework}\label{s:components}
The classification framework uses specific terminology, capitalized hereafter and italicized when introduced first.
A \zi{Component} in the classification framework is a set of concepts, relations, rules, or otherwise, which together serve a specific purpose within an \xc{rem}, such as, e.g., to categorize requirements, to visualize requirements, etc. Each Component comes with \zi{Component Properties}. Component Properties are \zi{domain-independent}, which in this paper means that a Component Property is independent from a ``paradigm'' followed in the design of an \xc{rem}, so that there is nothing in the framework to make it specific to goal-oriented \xc{rem}s, or others.
The classification framework uses five Components:
\begin{enumerate}
\item{\xb{Requirements Problem and Solution} that the \xc{rem} should help, respectively, define and find;}
\item{\xb{Ontology}, defining categories of information input to, used and output by an \xc{rem};}
\item{\xb{Formalism}, for representation and analysis instances of the concepts and relations in the Ontology;}
\item{\xb{Organization Mechanism} for the organization of representations made with the Formalism;}
\item{\xb{Guidelines}, advising how to use the Ontology, Formalism and Organization Mechanisms to define the Requirements Problem and find Solutions to it.}
\end{enumerate}
The overall idea behind the framework is simple. The purpose of an \xc{rem} needs to be explicit, and is conveyed through the Requirements Problem that it should help solve, and the Solution it should help produce. The Ontology identifies the information that the \xc{rem} will manipulate. That information ought to be represented in some structured way, to help answer questions about requirements. Because there can be a considerable amount of information to manipulate, there are Organization Mechanisms, to help decompose and manage representations. Finally, Guidelines will say why and how to use the conceptual tools, namely the Ontology, Formalism, and Organization Mechanisms to instantiate the Requirements Problem to a specific system-to-be, and find and describe Solution instances.
The rest of this section discusses Components and Component Properties. Each Component is presented in the same way, with (i) a definition of the Component, (ii) the purpose of the Component in an \xc{rem}, (iii) Component Properties, (iv) examples that illustrate Component Properties in existing \xc{rem}s, and (v) discussion of Component Properties in relation to the hypothetical \xc{rem} in the case study.
\subsection*{Case Study}
Suppose that the aim is to design an \xc{rem}, call it X, that should help solve the requirements problem as Zave \& Jackson \cite{Zave+:1997:TOSEM} defined it. The problem says that, given a set of requirements that must be satisfied together (denote the set $R$) and domain assumptions which must be satisfied together ($K$), we should find a specification of properties and behaviors of the system-to-be ($S$) such that $K, S \vdash R$, where $\vdash$ is the consequence relation of classical logic.
\subsection{Requirements Problem and Solution}\label{s:components:requirements-problem-and-solution}
\subsubsection{Definition} The Requirements Problem concept defines the undesirable properties of the situation at the start of \xc{re}, which the \xc{rem} should help solve. The Solution concept defines desirable properties of the result that the requirements engineer aims to make with the \xc{rem}.
\subsubsection{Purpose of Req. Problem and Solution in \xc{rem}}
\begin{itemize}
\item{Define the purpose of \xc{rem}.}
\item{Define the desired result of applying the \xc{rem}.}
\item{Force the \xc{rem} designer to clearly state what her \xc{rem} is intended to do within \xc{re}, thereby forcing her to state the scope and depth at which she views the problem that the \xc{rem} should help solve.}
\item{Compare \xc{rem}s, in that the Problem and Solution statements help us evaluate if an \xc{rem} is focusing on the same problem as another, if it focuses on a more specific, or a more general problem.}
\item{Justify design choices in the \xc{rem}: Decisions to include some concepts and relations in the Ontology, support specific rules of reasoning in the Formalism, and include processes in the Guidelines can be justified through their relevance to the description of the Requirements Problem, and finding and description of Solution instances.}
\end{itemize}
\subsubsection{Component Properties}
\begin{itemize}
\item{\zi{Implicit Definition}: No explicit statement of the Requirements Problem and Solution concepts is given, but can be inferred from other Components of the \xc{rem}.}
\item{\zi{Informal Definition}: Component Property is satisfied if the Requirements Problem and Solution concepts are defined in natural language.}
\item{\zi{Formal Definition}: The Requirements Problem and Solution are defined as expressions in a Formalism.}
\end{itemize}
\subsubsection{Examples}
As there are no explicit definitions of Requirements Problem and Solution statements, all \xc{rem}s cited up to this point, except for Techne have Implicit Definitions. For example, it is clear in Tropos and \xc{kaos} that the aim is to find operationalizations of high-level goals, but the explicit statement of this problem, such as the one given by Zave \& Jackson, is absent. Techne is an example where both Informal and Formal Definition are used.
\subsubsection{Case Study}
X is intended to solve the Zave \& Jackson statement of the Requirements Problem, and this problem has both an Informal and a Formal Definition, given in the Case Study. The Solution concept is implicit in the definition of the Requirements Problem, as it is a specification $S$ which is consistent with $K$ and $R$ (otherwise it cannot be that $K, S \vdash R$), and which together with domain assumptions is enough to derive requirements, i.e., $K, S \vdash R$.
\subsection{Ontology}\label{s:components:ontology}
\subsubsection{Definition} An Ontology in an \xc{rem} is an explicit specification of concepts and relations, whose instances are input, used and output by the \xc{rem}.
\subsubsection{Purposes of an Ontology in \xc{rem}}
\begin{itemize}
\item{Scoping: By including some and excluding other concepts and relations, the Ontology identifies the categories of information judged relevant to achieve the purpose of the \xc{rem}.}
\item{Communication: Concepts and relations of the Ontology give the starting stable set of terms to use in communication about requirements.}
\item{Documentation: Categories of information in the Ontology should be captured by artifacts that document requirements. Ontology helps structure these artifacts.}
\item{Focusing: For the engineer/user of the \xc{rem}, the Ontology acts as a checklist information to focus on when applying the \xc{rem}.}
\end{itemize}
\subsubsection{Component Properties}
\begin{itemize}
\item{\zi{Implicit Definition}: Concepts and relations are not explicitly defined, but can be inferred from other Components of the \xc{rem}. In such cases, it appears as if there is no Ontology; it can, however, be determined by looking at the kinds of information used by other Components.}
\item{\zi{Informal Definition}: Informal interpretation for concepts and relations is given via definitions in natural language, the choice of concepts and relations is justified with regards to the purpose of the \xc{rem}, and a discussion given of the ontological commitments, i.e., the assumptions for choosing particular concepts and relations rather than others, and for defining them exactly as proposed.}
\item{\zi{Structured Definition}: Concepts and relations are represented as a graph, where concepts are nodes and relations are edges of the graph. This is the case when, e.g., an Entity-Relationship model is used to describe the Ontology of the \xc{rem}.}
\item{\zi{Formal Definition}: Concepts and relations are defined using expressions of a formal logic.}
\end{itemize}
\subsubsection{Examples}
In \xc{kaos}, the conceptual meta-model provides the Structured Definition of the Ontology. Informal definitions are given in intensional form, listing properties of concepts (e.g., properties of the goal concept) and properties of relations (e.g., minimality and consistency for the goal refinement relation). Similar combination is applied in Tropos, Formal Tropos, i*. To the best of my knowledge, there are no \xc{rem}s with Formal Definitions for their complete Ontology. The Core Ontology for Requirements, subsequently used in Techne, accomplishes this only in part and indirectly, by mapping its concepts to a foundational ontology which has a Formal Definition (see Appendix B of \cite{Jureta+:2009:AO}). For illustration of Formal Definition of Ontology, but unrelated to \xc{re}, see \xc{dolce} \cite{Masolo+:2003}. \xc{carl} is a case of Informal Definition. Implicit Definition occurs in the original presentation of both Viewpoints and Labeled Quasi Classical Logic. In both cases, concepts and relations are implicit in the Formalisms and Guidelines, but explicit and separate definitions are not given.
\subsubsection{Case Study}
The Ontology for X should include the Requirement, Domain assumption, and Specification concepts. Assume for simplicity that all are top-level concepts. The consequence relation $\vdash$, since it is from classical logic, tells us the Ontology should include a Satisfaction relation from the Domain assumption and Specification concepts to the Requirements concept. Note that $\vdash$ is about derivability, not satisfaction, but classical logic is sound and complete so we can talk about Satisfaction as it fits the intuition that requirements are there to be satisfied.
Informal Definition in X consists of giving (at least) natural language definitions for the K, S, R concepts and all relations, then justifying why there are the three concepts and not more or less, and why they are top-level (i.e., one is not a specialization of another). An Informal Definition of the Requirement concept is that it is an optative statement about the environment of, and/or about the system-to-be. A Structured Definition for X can be an Entity-Relationship diagram, showing the three concepts as nodes, and relations as links. A Formal Definition can consist of defining the concepts using predicates of a more abstract ontology, such as writing that $\zi{Optative}(\phi) \rightarrow \zi{Requirement}(\phi)$, to say that $\phi$ is a requirement if it is optative, whereby the predicate $\zi{Optative}$ would be defined in the more abstract ontology.
\subsection{Formalism}\label{s:components:formalism}
\subsubsection{Definition} A Formalism serves for the representation of, and reasoning about instances of concepts and relations of the Ontology. A Formalism in \xc{rem} defines (i) symbols, (ii) rules for combining symbols into expressions, whereby the expressions refer to instances of the concepts and relations of the Ontology, (iii) semantic domain and semantic mapping function, to assign values within a domain of interest to expressions, and (iv) rules and algorithms for making deductions from, and/or checking properties of, expressions.
\subsubsection{Purposes of a Formalism in \xc{rem}}
\begin{itemize}
\item{Representation/Modeling: Expressions written in the Formalism are models of information elicited for, and used in \xc{rem}. Modeling helps reflection on requirements and helps highlight the relationships between requirements.}
\item{Communication and Learning: Models facilitate communication between stakeholders and help new project participants to learn about the requirements of the system-to-be.}
\item{Analysis: If the Formalism has the necessary features, models can automatically be checked for properties of interest, such as consistency, completeness, presence of solutions to the requirements problem that the model defines.}
\item{Prediction: If the Formalism has the necessary features, simulations can be performed on models, to evaluate, e.g., the probability of failure of a requirement, given a particular way to operationalize it.}
\item{Traceability: Provided that the Formalism allows the distinction of model versions and for capturing the rationale for version changes, the Formalism can help document traces and aid traceability.}
\end{itemize}
\subsubsection{Component Properties}
\begin{itemize}
\item{\zi{Multi-Formalism}: \xc{rem} uses more than one Formalism.}
\item{\zi{Syntax Properties}: Properties of alphabet and of grammar (i.e., rules for combining symbols in the \xc{rem}):
\begin{itemize}
\item{\zi{Symbolic Syntax}: Models are well formed formulas as in a formal logic; it is relevant then to look into properties such as the presence of labels on formulas (to indicate sorts, or to keep track of formulas in deductions, as in labeled deduction), of predicates, quantifiers, and so on.}
\item{\zi{Graphical Syntax}: Models are drawn as diagrams; it is relevant then to look into the properties of graphs that these diagrams define, how the diagrams evaluate on cognitive effectiveness criteria \cite{Moody+:2010:REJ}, etc.}
\item{\zi{Syntax Maps}: Presence of rules to map expressions written in one syntax to expressions written in another syntax, in order to indicate that the expressions refer to the same (or aspects of the same) instances of concepts and relation (i.e., that the expressions aim to state the same information). This Component Property applies to \xc{rem}s which include two or more ways to represent the same information (e.g., symbolic and graphical syntax)}
\end{itemize}}
\item{\zi{Deductive System Properties}: Properties of the set of rules capturing correct inferences from a given set of expressions; properties of interest include:
\begin{itemize}
\item{\zi{Classicality}: How the Deductive System in the \xc{rem} relates to that of classical logic; this can be established by verifying which of Gabbay's \cite{Gabbay:1985} 13 properties the \xc{rem}'s Deductive System satisfies.}
\item{\zi{Paraconsistency}: Whether the Deduction System allows deriving any formula from an inconsistent set of formulas; property relevant for inconsistency handling, as a Paraconsistent Deductive System allows drawing useful conclusions from an inconsistent set of formulas.}
\end{itemize}}
\item{\zi{Model Theory Properties}: Presence of a semantic domain and a function mapping expressions to elements of the semantic domain; properties of interest include:
\begin{itemize}
\item{\zi{Truth Valuation System}: The number of, and relationships between truth values (e.g., four truth values with two order relations, as in Belnap's four valued logic \cite{Belnap:1977}).}
\item{\zi{Inconsistency Valuation}: If an expression can be evaluated as both true and false, and what aggregate truth value such expressions obtain.}
\item{\zi{Incompleteness Valuation}: If the truth value of an expression can be undetermined, and what truth value such expressions then obtain.}
\item{\zi{Utility Valuation}: If the truth, falsity, or another valuation of an expression is interpreted as being (and how) valuable with regards to a purpose; e.g., truth of a requirement can be informally interpreted, in an \xc{rem}, as that the requirement will be satisfied by the system-to-be if it is designed according to the Solution of the \xc{rem}, so we see some truth values are being more desirable than others in the resolution of the Requirements Problem.}
\end{itemize}}
\end{itemize}
\subsubsection{Examples}
\xc{kaos}, Formal Tropos and Techne are Multi-Formalism \xc{rem}s, as each includes a Symbolic and a Graphic Syntax: linear-temporal logic and goal trees in \xc{kaos}, linear temporal logic and i* diagrams in Formal Tropos, a custom symbolic syntax with a deductive system and corresponding graphs in Techne. Viewpoints can be a Multi-formalism \xc{rem} if different Viewpoints use different Formalisms. Techne has Syntax Maps which ensure that all represented in Symbolic Syntax can be translated into Grphical Syntax, and back. This is not the case in \xc{kaos} and Formal Tropos: e.g., temporal relations that can be captured in linear temporal logic have no corresponding representation in Graphical Syntax in either of these \xc{rem}s. Neither \xc{kaos} nor Formal Tropos have a Deductive System, while Techne does. In contrast, Techne has no Model Theory, while both \xc{kaos} and Formal Tropos do. Deductive Systems of both Labeled Quasi-Classical Logic and Techne fail Classicality, both being Paraconsistent; however, they are not Paraconsistent in the same way, meaning that they would not derive the same conclusions given a same set of inconsistent formulas. i* has Graphical Syntax, no Deductive System and no Model Theory. Truth Valuation System in \xc{kaos} and Formal Tropos is that of linear temporal logic, so it can be understood as involving two truth values (if we say ``true'' for a formula which is satisfied, false otherwise), so that there are no interesting Inconsistency and Incompleteness Valuations. Utility Valuation in \xc{kaos} and Formal Tropos is not developed beyond the idea that if a formula representing a requirement is true/satisfied, then this is seen as beneficial.
\subsubsection{Case Study}
Since $\vdash$ in the Requirements Problem is the consequence relation of classical logic, the Formalism of X must be either classical logic, or another formalism which ensures that we can check derivability or satisfaction (e.g., linear temporal logic will work) and has a notion of inconsistency, so that we can check if some given set $R$, $K$, and/or $S$ is consistent (as the formulation $K, S \vdash R$ requires that $K \cup S \cup R$ is consistent). If we add a Graphical Syntax, then X will be a Multi-Formalism. If so, then Syntax Maps define how formulas from the Symbolic Syntax map to (combinations of) primitives in Graphical Syntax. If we keep classical logic as one of the two Formalisms, then it will fail Paraconsistency. The Truth Valuation System properties in that case are also straightforward. Note that there is no Incompleteness and Utility Valuation.
\subsection{Organization Mechanism}\label{s:components:organization-mechanism}
\subsubsection{Definition} Organization Mechanisms are intended to facilitate the creation and manipulation of expressions written in the Formalism of the \xc{rem}. An Organization Mechanism will include rules enabling, e.g., to reuse and combine model fragments, to highlight relations between model fragments that the Formalism cannot or is not intended to show.
\subsubsection{Purposes of Organization Mechanisms in \xc{rem}}
\begin{itemize}
\item{Modularity: A model can be split into pieces and each piece presented individually, perhaps accompanied with comments helping the reader of the model.}
\item{Problem decomposition: Different pieces of the model can focus on different aspects of functionality of the system-to-be. The Organization Mechanism can allow an aspect to be considered while hiding others, and showing only its relationships with others. This can help distribute work among those involved in modeling.}
\item{Reuse: A piece of a model may represent requirements that need to be satisfied by different features of the system-to-be. The Organization Mechanism can allow inclusion of pieces by referencing them, thus avoiding repetition and enabling reuse.}
\end{itemize}
\subsubsection{Component Properties}
\begin{itemize}
\item{\zi{Reference}: Presence of tools to reference, without reproducing, pieces of a model.}
\item{\zi{Structure}: Presence of part-of and is-a relations between model pieces, to capture, respectively, (i) that a piece is an aggregate of other pieces, that the latter are parts of the former, and (ii) that a piece is a generalization of other pieces, i.e., that the latter are specializations of the former.}
\item{\zi{Interface}: Presence of tools to describe how a model piece depends on another model piece, without describing the internals of either, and thus, how model pieces depend on contents of other model pieces, or of operations defined in other model pieces.}
\item{\zi{View}: Presence of tools to group pieces of a model according to interests of different stakeholder groups, such as clients or suppliers, managers or engineers, etc.}
\item{\zi{Constraint}: Ability to define constraints on relationships between pieces of models, such as conditions which should be satisfied for a set of pieces to be parts of another piece, or that some pieces together are a refinement of another piece, etc.}
\item{\zi{Verification}: availability of algorithms to automatically verify whether constraints on relationships between pieces of models are satisfied.}
\end{itemize}
\subsubsection{Examples}
i* allows Structuring via actor boundaries, to indicate that some model pieces belong to the same actor (stakeholder, user, or otherwise). Tropos and Formal Tropos inherit this mechanism. Formal Tropos includes templates, each template being associated to an instance of the ontology in i*. This means that i* models act as-if they are the Organizing Mechanism for a specification in linear temporal logic. In other words, goals, tasks, and other notions in i* models play the same role as schemas and schema relations play in the Z notation: they are used to organize pieces of a model. This idea is in \xc{kaos}, which uses the goal concept and goal trees as an Organization Mechanism for formulas in linear temporal logic. Some relations in \xc{kaos} are defined in a way which allows Verification: e.g., goal refinement is defined via properties between formulas of linear temporal logic in the goals participating in the refinement (consistency of the goals in refinement, minimality of the refining goal set, etc.) so that Verification of such properties is feasible. Labeled Quasi-Classical Logic and Techne have no Organization Mechanism. Viewpoints themselves are an Organization Mechanism of model pieces. \xc{carl} uses its Ontology as an Organizing Mechanism, taking sets of formulas to be extensions of concepts in its Ontology. These examples raise the issue of what interplay there can be between Ontology, Formalism, and Organization Mechanisms in an \xc{rem} (see, \S\ref{s:challenges:integration}).
\subsubsection{Case Study}
For example, to have views in X, we can adopt the ideas from Viewpoints. In this case, we can define meta-level rules for, e.g., solving inconsistencies between viewpoints, including a viewpoint into another viewpoint, and for referencing viewpoints in one another. We can thus use Viewpoints to satisfy Component Properties such as Reference, Structure, View, and Constraint. If meta-level rules are themselves defined in a logic, and there are means for, say, model checking for that logic, X can satisfy Verification.
\subsection{Guidelines}\label{s:components:guidelines}
\subsubsection{Definition}
Guidelines include all recommendations given on how to instantiate the concepts and relations of the Ontology, make models using the Formalism and manage models using Organization Mechanisms in an \xc{rem}.
\subsubsection{Purposes of Guidelines in \xc{rem}}
Guidelines suggest how to use the components of an \xc{rem} to accomplish activities in \xc{re}, such as elicitation, modeling, analysis, negotiation, validation, or otherwise.
\subsubsection{Component Properties}
\begin{itemize}
\item{\zi{Design Guidelines}: Presence of rules and steps in which to apply rules to structure the problem space. Design Guidelines are present if it is explained how to refine and operationalize requirements, and identify/define alternative refinements and operationalizations. A refinement relates requirements at different levels of detail; operationalization relates a requirement to resources and processes to use and apply to satisfy the requirement.}
\item{\zi{Decision Making Guidelines}: Presence of rules and steps for defining criteria for ranking alternative refinements and operationalizations of requirements, and select the refinements and operationalizations ranking highest according to most important criteria.}
\item{\zi{Inconsistency Handling Guidelines}: Presence of rules and steps for knowing what model pieces are inconsistent and what to do about them, such as whether to tolerate or resolve the inconsistencies, which of them to resolve when (as soon as detected, or later), etc.}
\item{\zi{Tool Support}: availability of software designed to facilitate the modeling or other applications of the \xc{rem}.}
\end{itemize}
\subsubsection{Examples}
Any \xc{rem} that includes the refinement (or decomposition) and operationalization (or means-ends) relations allows the structuring of the requirements problem and solution space, which includes all \xc{rem}s being able to formalize such relations. This does not mean that they all include Design Guidelines. \xc{kaos}, i*, Tropos, Formal Tropos include guidelines on how to refine/decompose and operationalize requirements. Viewpoints comes with instructions on how to relate and combine viewpoints, and in this sense also provide Design Guidelines. Techne, Labeled Quasi-Classical Logic and \xc{carl} have the necessary concepts and relations, but are not explicit on steps to follow. Decision Making Guidelines are less common. Notions such as quality criteria and nonfunctional requirements are present in Tropos, i*, Techne, but methods on how to rank alternatives are explicit only in \xc{nfr} and in \xc{kaos} in relation to uncertainty \cite{Letier+:2004:FSE}. Labeled Quasi-Classical Logic, \xc{carl} and Viewpoints do not include the necessary concepts, relations, and guidelines. Inconsistency Handling Guidelines are given in Viewpoints, \xc{nfr}, \xc{kaos}, Tropos (as in \xc{nfr}). Techne is paraconsistent, but does not provide explicit Guidelines on what to do, when inconsistency is deduced. \xc{kaos}, i*, \xc{nfr}, Tropos, Viewpoints, \xc{carl} all have software tools to support Guidelines.
\subsubsection{Case Study}
To make X into a Design Method, we need at least to ensure that it has the refinement relation. To have Decision Making Guidelines in X, we need to add at least one preference relation, rules for aggregating preferences, and a decision rule to rank alternatives. An alternative can amount to a consistent specification $S$, which also satisfies the condition that $K, S \vdash R$. Preference relations would indicate relative desirability of each specification. The rule for aggregating preferences and for ranking alternatives should let us define a total order over all specifications. The idea is then, that we would select the highest-ranking specification. If the Formalism is classical logic and Viewpoints are used as the Organization Mechanism, then Inconsistency Handling Method can be defined using meta-level rules. Finally, Tool Support will be satisfied if there is software that helps make and do reasoning on models made with X.
\section{Research Challenges}\label{s:challenges}
Research challenges can be summarized in the following questions:
\begin{itemize}
\item{How to design \xc{rem} in a systematic way?}
\item{How to validate \xc{rem} in a systematic way?}
\item{How to teach design and validation of \xc{rem}?}
\end{itemize}
These questions become more specific when considered for each Component and Component Property.
\subsection{What criteria should we use to evaluate the relevance of ontological commitments?}
That is, how to make and justify assumptions and decisions that led to define an \xc{rem}'s Ontology in a particular way, and specifically why the Ontology has the given scope and depth? In the case study, this means explaining why there are three and not more top-level concepts in X, why some of the three are not specializations of others, whether some or all of top level concepts should be specialized in the Ontology, and if yes, then to what depth.
There are different complementary methods for answering these questions. Justification for the three concepts can be given in terms of arguments against ontologies in existing \xc{rem}, by deriving the Ontology from a more general body of knowledge (e.g., claiming that concepts should cover specific grammatical moods), deriving the Ontology from a higher-level (e.g., a foundational) ontology, or justifying concepts by the presence/absence of some specific information in many experience reports and case studies. Relative merits and limitations of these different approaches are not clear enough, making it difficult to say how one should approach Ontology engineering for an \xc{rem}.
\subsection{How can we inform the design of Ontology and Formalism, through empirical research into categories of information and reasoning rules that engineers tend to recurrently use or disregard during \xc{re}?}\label{s:challenges:validation}
Leaving aside the application of \xc{rem} to case studies as a form of validation, empirical research can be done to inform Ontology and Formalism design for \xc{rem}. Empirical research on human nonmonotonic reasoning suggests a direction. Namely, just as factors influencing human nonmonotonic reasoning have been studied (cf., e.g., \cite{Ford+2000:CI}), so can factors influencing reasoning about requirements problems be studied. In the former, data collection consists of asking subjects to choose among predefined answers to problems requiring nonmonotonic reasoning as formalized in, say, default logic. If such an approach is applied to evaluate \xc{rem}s, then observing systematic departure in answers people give to a specific problem of modeling or reasoning about requirements, from answers that an \xc{rem} would provide, can suggest that \xc{rem} helps reduce error in that specific modeling and/or reasoning task.
\subsection{Can there be a core ontology applicable across \xc{rem}s?} An ontology is a core ontology if it is minimal with regards to a purpose, i.e., includes only non-overlapping concepts and relations that are necessary and sufficient for satisfying a purpose. In the case study, the minimal ontology includes the Requirement, Specification, and Domain assumption concepts, along with only those relations necessary and sufficient to capture the complex relationship $K, S \vdash R$ (complex, because to define $\vdash$, one uses connectives over formulas in $K$, $S$, $R$ and relations between premises and conclusions in proof rules of classical logic). To have a core ontology applicable across \xc{rem}s requires recognizing and successfully arguing that there are concepts and relations without which a proposal for an \xc{rem} fails. This then leads to questions such as, Can there be an \xc{rem} which has Design Guidelines, but which cannot model the refinement relation?, or Can an \xc{rem} support decision making by modeling alternative solutions to a requirements problem, while not having some form of the preference relation between requirements? So if one claims an \xc{rem} must have Design Guidelines (for how would it otherwise solve a requirements problem which assumes unclear and incomplete requirements are what we start with in \xc{re}?), then one needs also to choose whether refinement is a core relation. Not only this, but one also has to determine if there are relations from which refinement can be defined (e.g., as in Techne), so that refinement, while perhaps necessary in any \xc{rem}, really is a derived relation, not a primitive one.
\subsection{Can we further clarify the role that concepts and relations have in reasoning about requirements?}\label{s:challenges:meaning-is-use}
There is an important difference between \xc{rem}s such as \xc{kaos}, Tropos, Techne, i* and Formal Methods such as Z and Larch. In Z, the concept of \zi{Schema} groups definitions and expressions. In the mentioned \xc{rem}s, the Goal concept (and other concepts) are intended to do more than organize formulas. The Goal concept is particularly illustrative, in that it says the conditions stated in the formulas (the linear temporal logic formulas in \xc{kaos} and Formal Tropos) it ``includes'' \zi{are desired}, which is something that these formulas alone do not convey (as there is no sort, modality, or otherwise in linear temporal logic which refers to desirability). \zi{But} these same \xc{rem}s do not take the next step, one analogous to what modal logics do with regards to classical logic: the \xc{rem}s do not study how to introduce these modalities into the semantics of the Formalism in the \xc{rem}. The very specific question is, for example, does the Goal concept give a sort on formulas in a Formalism, and if yes, then does this sort merely label formulas, or does the sort of the formula influence the role this formula has in proof theory of that Formalism?
To make the point here clearer, consider the case study again. If we take classical logic, and make its language sorted, with the three sorts $R$, $K$, and $S$, but at the same time, we keep the semantics of classical logic, we have only introduced labels on formulas: reasoning is still that of classical logic, and so we have failed to capture, \zi{in the Formalism}, the intuitive ideas on what it means for a formula to be a Requirement, while another one is a Domain assumption.
If we follow Wittgenstein's aphorism that ``meaning is use'', then \xc{rem}s such as \xc{kaos}, Tropos, Formal Tropos, Techne, along with all those cited in this paper are limited, since they use pre-existing logics which disregard these sorts defined by the Ontology of the \xc{rem}.
\zi{If we wanted conceptual tools designed to the specific purpose of the \xc{rem}}, then our intuitions about the difference \zi{in use and during} \xc{re} of $R$, $K$, and $S$ expressions, should be reflected in the rules used to draw conclusions \zi{in the Formalism}. So if we focus on proof theory, we would want to ``embed'' whatever meaning we have in mind for $R$, $K$, and $S$ \zi{by the proof theory itself}, precisely in order to make sure that the conclusions are drawn in a way that satisfies the intended meaning.
This brings me back to the question of how we could further clarify the role of concepts and relations in reasoning about requirements. We can do this -- namely, embed the intended interpretation of concepts and relation of an \xc{rem}'s Ontology into its Formalism -- \zi{only if we have clarified the role that these concepts and relations have when reasoning about requirements in that \xc{rem}}, i.e., only if we are very clear on their use, and from there, of their meaning.
To put it plainly: it is relatively easy to say that some formula $\phi$ is an instance of the Requirement concept; what is harder is explaining how this additional information -- that $\phi$ is a Requirement, not a Domain assumption -- influences the conclusions we will draw about the satisfaction of $\phi$, or about inconsistency between $\phi$ and some other formulas.
To illustrate this, suppose that we want to have a knowledge base which includes all formulas with K, S, and R labels. If that knowledge base makes deduction using the classical $\vdash$, then the question ``Is a requirement $\phi$ satisfied?'' gets an irrelevant answer in at least two cases:
\begin{enumerate}
\item{If the knowledge base is inconsistent, then $\vdash$ will derive $\phi$ regardless of what actually is in that knowledge base (just as it will derive \zi{any} other formula, because of \zi{ex falso quodlibet}).}
\item{Because $\vdash$ is reflexive, meaning that any formula on its left-hand side is always deduced, we will conclude that any formula on the left-hand side is satisfied (formally, for any set $X$ of formulas, and any formula $\phi$, we have that $X \cup \{ \phi \} \vdash \phi$).}
\end{enumerate}
Yet in both cases, it makes no sense to say that $\phi$ is satisfied, for the simple reason that in both cases, we have said nothing about if $\phi$ is operationalized, refined, or otherwise.
Now, observe that the knowledge base will give wrong answers \zi{not} because $\vdash$ is somehow deficient by itself, \zi{but because the proof theory defining $\vdash$ sees no difference between formulas that are requirements, domain assumptions, and specifications (or any other category one deems relevant)}. We can respond to this in two ways:
\begin{enumerate}
\item{We can make tools that are \zi{outside} the knowledge base, and which filter, \zi{after} deduction, the results of deduction by applying some rules. This is what happens in \xc{kaos} for example (although it is not about deduction, but model checking, but that makes no difference here), as it requires first that conflicts and obstacles be eliminated to repair consistency, and only then can questions, such as whether a requirement is satisfied, be asked.}
\item{We can make a Formalism which is attentive to which formulas instantiate which concepts and relations from the Ontology. But this requires a considerable change in how \xc{rem}s are made, as the following question suggests.}
\end{enumerate}
\subsection{Integration of Components}\label{s:challenges:integration}
It is without doubt good for a representation of requirements to be modular. But it is not clear whether it is good for an \xc{rem} to be modular. All \xc{rem}s mentioned in this paper are modular in the following sense: \zi{\xc{rem} Components are designed to a considerable extent independently from the Formalism component}. This is a strong claim, but one not difficult to argue for.
Take Tropos as an example. It uses the Ontology of i*. One of its two Formalisms is the graphical language of i*. The other is linear temporal logic. The language of i* is obviously defined to fit the Ontology of i*. But linear temporal logic is independent from the i* Ontology and from the i* Formalism. Yet it is via linear temporal logic that one can do automated reasoning in Tropos. Just as linear temporal logic ignores that there are actors, goals, tasks in i* models, so it ignores that there are goals, agents, refinements in \xc{kaos} models.
For further illustration, take the \xc{rem} X in the case study. As mentioned earlier (cf., \S\ref{s:challenges:meaning-is-use}), if X has classical logic as its Formalism, then asking questions to the knowledge base of X will give misleading answers. That is, the answers are likely to violate the rules and processes stated in the Guidelines of the \xc{rem}. One such rule is that a requirement must be satisfied by functionality described in the specification $S$ and which is consistent with the domain assumptions $K$. Yet, we would still get the answer from the knowledge base that a requirement $\phi$ is satisfied, even when $S$ is inconsistent and includes no descriptions of functionality for performing tasks which satisfy $\phi$.
If one prefers to think in terms of goals and tasks that satisfy the goals, then suppose $\phi$ is a goal (i.e., $\phi$ is an instance of the \zi{Goal} concept in the Ontology of the \xc{rem}). Suppose that the guideline in the \xc{rem} is this: a goal is satisfied if there are tasks that operationalize it, meaning that if these tasks are satisfied, then the goal is satisfied as well. If \xc{rem} uses classical logic as its Formalism, then deducing $\phi$ does not mean it is satisfied in the said sense, because (as mentioned above -- cf., \S\ref{s:challenges:meaning-is-use}) it can be deduced in cases when there are no tasks which operationalize $\phi$.
\section{Discussion}\label{s:discussion}
The Research Challenges section (cf., \S\ref{s:challenges}) started by asking three questions, namely, how to (i) systematically design \xc{rem}, (ii) validate them, and (iii) teach design and validation.
It should be clear that the proposed classification framework has a limited use in answering these questions. For systematic design, the framework gives a checklist of ingredients of \xc{rem} that a \xc{rem} designer will in one way or another need to think about. This checklist itself suggests what knowledge one will need to apply when designing an \xc{rem} -- ontology engineering, formal logic, process design, etc., \zi{in addition to} her understanding of \xc{re}. The framework does not say what concepts and relations are more relevant than others, what reasoning rules to use, etc. For validation, the framework suggests how validation methods already known for specific Components and Component Properties can be used for validation of an \xc{rem} (cf., \S\ref{s:challenges:validation}). For teaching, the framework suggests the topics to cover with students and researchers interested in the application, engineering, extension, and validation of \xc{rem}s.
The rest of this section discusses the relationship between the concept of \xc{rem} and concepts of Requirements Modeling Language and Formal Method.
\subsection{Requirements Modeling Language}
The emphasis in a Requirements Modeling Language is on \zi{language}, i.e., a conceptual tool for representation of, and reasoning about requirements. In an \xc{rem}, this conceptual tool would amount to the combination of Ontology, Formalism, and Organization Mechanism components. In Requirements Modeling Languages such as \xc{rml} and i*, guidelines for the use of the language are usually treated separately, as are the Requirements Problem and Solution concepts. An \xc{rem} can be viewed as including an \xc{rml}, if we take an \xc{rml} to include the Ontology, Formalism, and Organization Mechanism components.
\subsection{Formal Methods}
I take Wing's definition of Formal Methods \cite{Wing:1990:C} (\xc{fm}s hereafter) as the definition of \xc{fm}s.
In terms of Components, \xc{fm}s are combinations of Formalism, Organization Mechanisms, and Guidelines. Ontology is not developed in the same sense as in \xc{rem}s, and at best can amount to syntactic sugar, to make specifications readable by customers, in addition to specifiers and implementors. The classification framework suggests that Requirements Problem and Solution, Ontology, and Guidelines in an \xc{rem} are not merely syntactic sugar, but, by influencing the conceptualization of the requirements problem and the process of its resolution, influence how one designs, or should design, the Formalism and Organization Mechanisms.
We can, so to speak, hack an \xc{fm} by adding an Ontology to it as syntactic sugar, and so make it look like an \xc{rem}. The limitation of doing so is that the hacked artifact -- the \xc{fm} -- shows its limitations as soon as we start using it to \xc{re}-specific tasks, which were not of interest to the designer of the \xc{fm}. An example is to make knowledge bases using the Formalism in the \xc{rem}, and to ask questions about which requirements are satisfied. If Ontology is only syntactic sugar and Guidelines are merely text alongside the \xc{fm}, we are likely to get wrong answers (for reasons stated earlier -- cf., \S\ref{s:challenges:meaning-is-use}). This is not the problem of the underlying \xc{fm}, but of the fact we are using it despite knowing that it ignores Ontology added on top of it and the Guidelines for its use, so that it cannot make sure its answers reflect the knowledge that the Ontology and Guidelines capture. In a summary, \xc{rem}s are \zi{not} specializations of \xc{fm}s.
\section{Conclusions}\label{s:conclusions}
This paper suggests a classification framework for Requirements Engineering Methods (\xc{rem}s). The framework categorizes \xc{rem}s by the properties of \xc{rem} components. The framework is intended to help the analysis, teaching, and extension of existing \xc{rem}s, and the engineering and validation of new \xc{rem}s. The paper discusses research challenges highlighted by the framework. The framework clarifies the relations between the concept of \xc{rem} and other concepts of interest in and to \xc{re}, and in particular, Requirements Problem and Solution, Requirements Modeling Language, and Formal Method. As noted in the Introduction, this classification framework focuses on one dimension only -- the components of \xc{rem} -- while other dimensions of classification are not discussed here.
The classification framework identifies, through Components and Component Properties, the knowledge applicable when designing \xc{rem}s. In doing so, the framework suggests fragments for a body of knowledge of a research methodology proper to the design of \xc{rem}s. To the extent that \xc{rem} design and validation are important activities in \xc{re} research, the framework contributes to forming a research methodology specific to \xc{re}.
\subsection*{Acknowledgments}
Since 2008, I had discussed the ideas behind this classification framework with many colleagues at University of Trento, University of Toronto, Fondazione Bruno Kessler, and University of Namur. I am indebted to John Mylopoulos and Alexander Borgida for introducing me to requirements modeling languages. I thank St\'{e}phane Faulkner, Neil Ernst, Sotirios Liaskos, Alexei Lapouchnian, Angelo Susi, and Anna Perini for discussions on topics related to \xc{rem}s. This does not mean that they agree with me on this classification framework.
\bibliographystyle{plain}
|
{
"timestamp": "2012-03-09T02:01:41",
"yymm": "1203",
"arxiv_id": "1203.1717",
"language": "en",
"url": "https://arxiv.org/abs/1203.1717"
}
|
\section{Introduction}
The physics of cold dense quark matter is governed by the theory of strong interactions, the Quantum Chromodynamics (QCD). Unfortunately, it becomes strongly coupled in the phenomenologically interesting range of densities. At present, there is no analytic method in the market that would be able to perform reliable first-principle calculations in this regime, although first attempts have already been made~\cite{Kurkela:2009gj}. Likewise, lattice simulations at high density are out of reach of the standard Monte-Carlo techniques due to the sign problem.
As a consequence, one usually has to resort to simplified calculations using models which more or less imitate the full QCD. Since the technical difficulty of QCD stems from the strong gauge interaction, these models invariably replace the color gauge symmetry with a global one, the prototype being the Nambu--Jona-Lasinio (NJL) model (see Ref.~\cite{Vogl:1991qt,*Klevansky:1992qe,*Hatsuda:1994pi,*Buballa:2003qv} for extensive reviews). One then has to deal with various model artifacts that are particularly severe at high baryon density where cold quark matter is expected to behave as a color superconductor (see Ref.~\cite{Alford:2007xm,*Wang:2009xf,*Huang:2010nn,*Fukushima:2010bq} for recent reviews). A condensate of quark Cooper pairs breaks color symmetry and (in most color-superconducting phases) induces nonzero color charge density. This cannot be physical since, due to the long-range nature of gauge interactions, it would give rise to nonextensive energy density and an ill-defined thermodynamic limit~\cite{Alford:2002kj}. In QCD, it is compensated by an induced gluon condensate so that the system as a whole is color neutral~\cite{Gerhold:2003js,*Dietrich:2003nu}. On the other hand, in models with global color symmetry color neutrality has to be imposed as a thermodynamic constraint.
Arranging for global color neutrality by introducing one or more chemical potentials associated with the color charge(s) costs energy as compared to the unconstrained equilibrium state. The neutrality constraint then has to be defined carefully in order to avoid spurious instabilities. Initially, only neutrality with respect to the three colors of fundamental quarks was required in literature on color superconductivity, leading to the existence of seemingly neutral and energetically preferred states~\cite{He:2005jq,*Blaschke:2005km}. As was pointed out in Ref.~\cite{Buballa:2005bv}, such a restricted neutrality requirement is sufficient only for special orientations of the diquark condensate in the color space. In general, neutrality with respect to all eight generators of color $\gr{SU(3)}$ has to be imposed, and the full set of eight color chemical potentials then have to be introduced~\footnote{This is in no contradiction with the common knowledge that only mutually commuting charges can be simultaneously fixed in a grand canonical ensemble, giving rise to just two independent chemical potentials for $\gr{SU(3)}$, as dictated by its rank. In fact, one can always choose a basis in the Lie algebra of $\gr{SU(3)}$ such that only two color charges are nonzero. In such a basis, two chemical potentials are sufficient to make the system neutral. However, even in this case, the other chemical potentials will be important in our discussion of order parameter fluctuations.}.
Once the chemical potentials are fixed to make the equilibrium (``ground state'') neutral, color symmetry is apparently broken explicitly. This observation led to the conclusion that the Nambu--Goldstone (NG) bosons of the spontaneously broken global color symmetry have small, yet nonzero masses, proportional to the color chemical potential(s)~\cite{He:2005mp,*Ebert:2005fi,*Ebert:2006bq}. The goal of the present paper is to show that this conclusion is premature. The physical picture behind our claim is as follows. The introduction of chemical potentials is enforced by the diquark condensate, which itself is generated dynamically as a non-perturbative solution to the equations of motion. The color chemical potentials therefore are not mere external fields, but stem from the dynamics of the system. In full QCD, this is indeed the case~\cite{Gerhold:2003js,*Dietrich:2003nu}. After all, none of the color symmetry is broken explicitly once the dynamics in the gauge sector is properly taken into account. In the NJL model, the color chemical potentials mimic the role of the gluon condensate in the full QCD.
One should note that the chemical potentials commonly used in literature are demanded to make the \emph{ground state} neutral. However, the NG bosons constitute its excitations. One therefore cannot use the very same values of the chemical potentials when dealing with such non-equilibrium field configurations, and the thermodynamic constraint of color neutrality needs to be extended properly. We show that this can be technically achieved by treating the chemical potentials as dynamical variables, or in other words, secondary order parameters in the gauge sector induced by the primary order parameter, that is, the diquark condensate. Once this is done, the NG bosons remain exactly massless, as predicted by the Goldstone theorem.
The plan of the paper is as follows. In Section~\ref{Sec:toymodel} we explain the central idea of the paper using a very simple toy model. While this is rather trivial, it is intended to demonstrate the conceptual simplicity of our strategy, which might otherwise be concealed by unimportant technical details of the NJL calculation to follow. Section~\ref{Sec:NGmasses} provides a correction of the calculation of the NG boson masses of Ref.~\cite{He:2005mp,*Ebert:2005fi,*Ebert:2006bq}, showing that they are exactly zero once color neutrality is properly imposed. In Section~\ref{Sec:NGdispersions} we extend the calculation and determine the full dispersion relations of the NG bosons. We employ the high-density approximation~\cite{Evans:1998ek,*Hong:1998tn,*Hong:1999ru} (see Ref.~\cite{Nardulli:2002ma} for a review and further references), which both simplifies the calculation and makes the results model independent, for it is known to capture the leading order of the high-density asymptotic behavior in full QCD. Finally, in Section~\ref{Sec:conclusions} we summarize and conclude.
\section{Neutrality in a scalar toy model}
\label{Sec:toymodel}
Consider the scalar theory with a global $\gr{SU(2)\times U(1)}$ symmetry, defined by the (Minkowski space) Lagrangian
\begin{equation}
\mathcal L=\mathcal D_\mu\he\phi \mathcal D^{\mu}\phi-M^2\he\phi\phi-\lambda(\he\phi\phi)^2,
\label{Lagrangian}
\end{equation}
where $\phi$ is a complex doublet field. This model was investigated many times before~\cite{Miransky:2001tw,*Schaefer:2001bq,*Andersen:2005yk} and it was shown that when the symmetry is spontaneously broken in presence of a chemical potential associated with the $\gr{U(1)}$ subgroup, the three broken generators give rise to one type-I and one type-II NG boson whose dispersion relations at low momentum are linear and quadratic, respectively~\footnote{This conclusion does not depend on whether the symmetry is actually broken by the chemical potential itself, or rather by $M^2<0$ as in the Higgs model, and the chemical potential only introduces medium effects.}. This is in accordance with the general Nielsen--Chadha counting rule for the number of NG bosons as well as the fact that in the ground state, isospin acquires nonzero density~\cite{Nielsen:1975hm,*Watanabe:2011ec,*Watanabe:2012,*Hidaka:2012}.
Here we wish to investigate the effect of enforcing neutrality with respect to the isospin $\gr{SU(2)}$ group. This may be regarded as a simple toy model for understanding charge neutrality in non-Abelian gauge theories where the global symmetry (after gauge fixing) is spontaneously broken such as in color superconductors. We will explain that using the values of chemical potentials obtained from the neutrality constraint on the ground state to determine the excitation spectrum leads to a spurious instability. We will then use two different approaches to deal with it. First, we shall demand, and justify, that all uniform field configurations be $\gr{SU(2)}$ neutral, whether they correspond to the thermodynamic equilibrium or not. A more model-independent argument will be given afterwards, showing that the $\gr{SU(2)}$ chemical potentials can be treated as induced, secondary order parameters. These two approaches will be later used respectively in Sections~\ref{Sec:NGmasses} and~\ref{Sec:NGdispersions}. Here we just remark that while the former has the advantage of being conceptually more straightforward, the latter is more powerful and allows us in particular to determine the full dispersion relations of the NG bosons.
\subsection{Spurious instability and its cure}
\label{Subsec:instability}
We start by introducing independent chemical potentials for all generators of the symmetry group, $\mu$ for $\gr{U(1)}$ and $\vec\mu$ for $\gr{SU(2)}$. The covariant derivative in Eq.~\eqref{Lagrangian} then reads
\begin{equation}
\mathcal D_\mu\phi=[\partial_\mu-i\delta_{\mu0}(\mu+\vec\tau\cdot\vec\mu)]\phi,
\end{equation}
where $\vec\tau$ is the vector of Pauli matrices. The presence of chemical potentials for all charges allows us to easily find the associated charge density operators (here for the isospin charges),
\begin{equation}
\vec n=\frac{\partial\mathcal L}{\partial\vec\mu}=i\left(\he\phi\vec\tau\partial_0\phi-\partial_0\he\phi\vec\tau\phi\right)+2\he\phi(\mu\vec\tau+\vec\mu)\phi.
\label{charge_density}
\end{equation}
Using this expression, the Lagrangian \eqref{Lagrangian} becomes
\begin{equation}
\begin{split}
\mathcal L=&\partial_\mu\he\phi\partial^\mu\phi+i\mu(\he\phi\partial_0\phi-\partial_0\he\phi\phi)+\vec\mu\cdot\vec n\\
&-\he\phi(M^2-\mu^2+\vec\mu^2)\phi-\lambda(\he\phi\phi)^2.
\end{split}
\label{Lagrangian_expanded}
\end{equation}
All matrix structure is now hidden in the density $\vec n$.
Let us assume that the scalar field develops nonzero vacuum expectation value (the conditions for this to happen will be discussed below) and choose the ground state as usual as $\phi_0=(0,v)^T$. The complex doublet field will thus be parameterized as $\phi_1=\varphi$, $\phi_2=v+H+i\theta$, where $\varphi$ is a complex field, whereas $H$ and $\theta$ are real. For constant field configurations, the charge densities~\eqref{charge_density} are then expressed as
\begin{equation}
\begin{split}
\frac12n_1&=\mu_1(v^2+2vH)+2\mu v\,\mathrm{Re}\,\varphi+\text{second-order terms},\\
\frac12n_2&=\mu_2(v^2+2vH)-2\mu v\,\mathrm{Im}\,\varphi+\text{second-order terms},\\
\frac12n_3&=(\mu_3-\mu)(v^2+2vH+H^2+\theta^2)+(\mu_3+\mu)\he\varphi\varphi.
\end{split}
\end{equation}
Observe that when $\vec\mu=\vec0$, the isospin density, $n_3=-2\mu v^2$, appears as soon as the field condenses. For $M^2>0$, this happens once the chemical potential $\mu$ exceeds the mass $M$. This is the Bose--Einstein condensation.
If we now want to make the ground state $\gr{SU(2)}$-neutral, the first terms in the expressions for charge densities lead to the conditions
\begin{equation}
\mu_1=\mu_2=0,\qquad
\mu_3=\mu\quad\text{(neutral ground state)}.
\label{vacuum_chempot}
\end{equation}
The static bilinear (mass) part of the Lagrangian~\eqref{Lagrangian_expanded} reads, up to an overall minus sign,
\begin{align}
V_{\text{bilin}}(\phi)=&2v(M^2+2\lambda v^2)H+(M^2+6\lambda v^2)H^2\\
\notag
&+(M^2+2\lambda v^2)\theta^2+(M^2+2\lambda v^2-4\mu^2)\he\varphi\varphi.
\end{align}
From the linear term we see that when charge neutrality is imposed, spontaneous symmetry breaking can only occur for $M^2<0$, that is, when it already appears in the vacuum. This is natural: normally, Bose--Einstein condensation (at zero temperature) sets when charge density starts to be nonzero, but here we keep the isospin density equal to zero, which prevents the field from condensing.
Substituting the vacuum expectation value, $v^2=-M^2/2\lambda$, we find that the Higgs mode has the mass term $4\lambda v^2H^2$, while the phase fluctuation of the condensate, $\theta$, becomes massless. However, the same is not true for $\varphi$. This observation led in the context of color superconductivity to the conclusion that the chemical potential needed to render the so-called 2SC phase color-neutral breaks some of the generators explicitly and the associated NG bosons thus naturally acquire nonzero mass~\cite{He:2005mp,*Ebert:2005fi,*Ebert:2006bq}. In our case this corresponds to the $\mu_3$ chemical potential ``explicitly breaking'' the $\tau_{1,2}$ generators, i.e., giving mass to the $\varphi$ excitation. However, the situation is even worse: the mass squared of $\varphi$ is negative! A similar problem appears in the 2SC phase, as will be shown in the following section. In fact, the presence of terms in the Lagrangian~\eqref{Lagrangian_expanded} with a single time derivative makes the discussion slightly more complicated than just concluding that $\varphi$ has a negative mass squared. One finds that the (anti)particle mode annihilated by $\varphi$ ($\he\varphi$) has the dispersion $E_{\vek k}=|\vek k|{\mp}2\mu$. Consequently, the particle mode with momentum smaller than $2\mu$ is ``tachyonic''. To conclude, fixing the chemical potential to make the ground state neutral as we just did obviously leads to an unphysical result.
The roots of the problem lie in the fact that we fixed the chemical potentials in the ground state once for all. The fluctuations of the order parameter then drive the system off the neutrality, thus naturally lowering the energy. This feature was already observed in Ref.~\cite{He:2005jq,*Blaschke:2005km}, where it was erroneously interpreted as an instability of the 2SC ground state. Buballa and Shovkovy~\cite{Buballa:2005bv} pointed out that the instability disappears when the chemical potentials necessary to make the ground state neutral are transformed simultaneously with the ground state itself. It should be stressed that our approach is technically very close to that of Ref.~\cite{Buballa:2005bv}: after all, the NG collective modes in the infinite-wavelength limit correspond merely to a change of the ground state. However, we make one step further by demanding charge neutrality also for certain non-equilibrium field configurations, corresponding to collective modes in the infinite-wavelength limit. In other words, we will require that for every uniform field configuration the chemical potentials acquire such values that the system has zero $\gr{SU(2)}$ charge. This is reasonable since otherwise there would be a uniform charge distribution, leading to an ill-defined thermodynamic limit. Such considerations are sufficient to determine the mass spectrum of the theory, and we will now demonstrate that the NG bosons are indeed exactly massless as they should. A formalism how to deal with nonuniform field fluctuations, that allows us to determine the NG boson dispersion relations, will be developed below.
Let us therefore assume constant fields $H,\theta,\varphi$ and demand that the $\gr{SU(2)}$ charge densities~\eqref{charge_density} are zero for all values of the fields. This gives the chemical potentials the following values,
\begin{equation}
\begin{split}
\mu_1&=-\frac{2\mu}v\,\mathrm{Re}\,\varphi,\qquad\mu_2=+\frac{2\mu}v\,\mathrm{Im}\,\varphi,\\
\mu_3&=\mu-\frac{2\mu}{v^2}\he\varphi\varphi\qquad\text{(neutral excitations)},
\end{split}
\end{equation}
to lowest nontrivial order in the fields. Substituting this back into Eq.~\eqref{Lagrangian_expanded}, the term $\vec\mu\cdot\vec n$ vanishes by definition of charge neutrality, while the rest reduces to
\begin{equation}
\begin{split}
V_{\text{bilin}}(\phi)=&2v(M^2+2\lambda v^2)H+(M^2+6\lambda v^2)H^2\\
&+(M^2+2\lambda v^2)\theta^2+(M^2+2\lambda v^2)\he\varphi\varphi.
\end{split}
\end{equation}
Upon minimizing the potential to make the linear term vanish, the field $\varphi$ becomes exactly massless as it should.
\subsection{Effective field theory approach}
\label{Subsec:EFT}
The masslessness of both $\theta$ and $\varphi$ can be achieved naturally once the chemical potentials are treated as dynamical fields and the local symmetry transformation which generates the NG excitations is accompanied by a corresponding transformation of the chemical potentials. The validity of the Goldstone theorem is then an immediate consequence of the exact symmetry of the action. Let us explain on our simple example the line of reasoning. In the underlying gauge theory the charge neutrality is maintained by a gluon condensate which compensates for the charge of the primary order parameter, here the vacuum expectation value $\phi_0$. Under a symmetry transformation, the primary order parameter and the gluon condensate transform simultaneously. In order to correctly capture the symmetry properties of the gauge theory in a model with a global symmetry, we must allow the chemical potentials to transform in the same way the gluon condensate would.
Technically this means that we deal with a theory with a global symmetry and two order parameters: the primary $\phi_0$, and the secondary one, that is, the chemical potentials. A low-energy effective Lagrangian for the NG bosons is then constructed as usual by performing a spacetime-dependent symmetry transformation on the order parameter(s). In our specific example, let us parameterize the scalar field as $\phi=\mathcal U(\pi)\phi_0$, where $\mathcal U(\pi)=\exp(\frac iv\vec\tau\cdot\vec\pi)$. Analogously, the temporal background gauge field $\mathcal A$ that contains the chemical potentials and enters via the covariant derivative, $\mathcal D_\mu\phi=(\partial_\mu-i\delta_{\mu0}\mathcal A)\phi$, is parameterized as
\begin{equation}
\mathcal A=\mathcal U(\pi)\mathcal A_0\he{\mathcal U(\pi)},
\label{adjoint}
\end{equation}
where $\mathcal A_0=\mu+\vec\tau\cdot\vec\mu$ with the ground state values of the chemical potentials determined by Eq.~\eqref{vacuum_chempot}. Note that this parameterization abandons the amplitude mode $H$, that is, it defines the nonlinear sigma model.
Inserting the parameterization into the Lagrangian~\eqref{Lagrangian}, one finds
\begin{equation}
\begin{split}
\mathcal L=&\he\phi_0(\partial_\mu\he{\mathcal U})(\partial^\mu\mathcal U)\phi_0+i\he\phi_0\mathcal A_0(\he{\mathcal U}\partial_0\mathcal U)\phi_0\\
&-i\he\phi_0(\partial_0\he{\mathcal U}\,\mathcal U)\mathcal A_0\phi_0+\he\phi_0\mathcal A_0\mathcal A_0\phi_0\\
&-M^2\he\phi_0\phi_0-\lambda(\he\phi_0\phi_0)^2.
\end{split}
\end{equation}
It is easy to see that the second, third, and fourth term drops because $\mathcal A_0\phi_0=0$, while the last two terms do not include the NG fields $\vec\pi$. The Lagrangian therefore reduces to $\mathcal L=\he\phi_0(\partial_\mu\he{\mathcal U})(\partial^\mu\mathcal U)\phi_0$ up to a constant. Interestingly, it does not depend on the chemical potentials at all. In particular we can see that all NG bosons $\vec\pi$ are exactly massless and have the usual Lorentz-invariant dispersion relation. This is due to the fact that despite the chemical potentials, the densities of all conserved charges, including the $\gr{U(1)}$ charge with respect to which neutrality is not required, are zero in the ground state~\footnote{This is an artifact of the scalar toy model we consider here. The ground state is automatically neutral under an unbroken $\gr{U(1)}'$ group generated by a linear combination of the $\gr{U(1)}$ charge and one of the $\gr{SU(2)}$ charges. Together with the imposed $\gr{SU(2)}$ neutrality, this is already sufficient to guarantee vanishing of all four conserved charges of the $\gr{SU(2)\times U(1)}$ group in the ground state.}.
\section{NG boson masses in NJL model}
\label{Sec:NGmasses}
The aim of this section is to show that when color neutrality is implemented using the strategy outlined above, the NG bosons become exactly massless in the NJL model description of color-superconducting quark matter. Our plan is as follows. We first introduce the model and essentially repeat the calculation of Ref.~\cite{He:2005mp,*Ebert:2005fi,*Ebert:2006bq} for fixed color chemical potentials. Next, we adapt the approach introduced in Section~\ref{Subsec:instability} for the present purposes. Finally, in Section~\ref{Subsec:corrections} we present some details of the computation of the corrected mass spectrum of the NG bosons.
For the sake of simplicity, we will use a version of the model considered in Ref.~\cite{He:2005mp,*Ebert:2005fi,*Ebert:2006bq} which does not take into account dynamically generated constituent quark masses. This cannot change the conclusions qualitatively. Moreover, in color superconductors the constituent masses of $u$ and $d$ quarks are usually small. Since the calculation is rather technical, we will omit details, yet providing the key steps and definitions necessary for the reader who might desire to reproduce our results.
The model is defined by the Lagrangian in Minkowski space,
\begin{equation}
\mathcal L=\bar q(i\slashed\partial+\gamma_0\mu_\alpha T_\alpha-m)q+
\frac G4\sum_a({\bar q}^{\mathcal C}P_aq)(\bar q\bar P_aq^{\mathcal C}),
\end{equation}
where $q^{\mathcal C}\equiv C\bar q^T$ is the charge conjugated Dirac spinor and the matrices $P_a$ specify the structure of Cooper pairs. In the 2SC phase, they read $(P_a)^{ij}_{bc}=\gamma_5\epsilon^{ij}\epsilon_{abc}$, where $i,j$ are flavor and $a,b,c$ fundamental color indices, respectively. Also, $\bar P_a=\gamma_0\he P_a\gamma_0$. We introduced nine chemical potentials $\mu_\alpha$, $\alpha=0,\dotsc,8$, associated with the symmetry generators $T_\alpha$ in the color space, normalized by $\tr(T_\alpha T_\beta)=\frac12\delta_{\alpha\beta}$. They include the quark number chemical potential $\mu$, related to $\mu_0$ by $\mu_0=\mu\Sqrt6$ since $T_0=\openone/\Sqrt6$.
In the mean-field approximation, the thermodynamic potential $\Omega$ of the model is given by
\begin{equation}
\beta\Omega=\beta V\frac{\Delta_a\Delta^*_a}G-\frac12\mathrm{Tr}\log S^{-1},
\label{NJL_TD}
\end{equation}
where $V$ is the space volume and $\Delta_a$ is the collective field that stands for the composite operator $\frac G2{\bar q}^{\mathcal C}P_aq$. Its ground state expectation value is determined by the minimization of the thermodynamic potential. Furthermore, the quark propagator in the Nambu space, $\Psi\equiv(q,q^{\mathcal C})^T$, reads
\begin{equation}
S^{-1}(k)=\begin{pmatrix}
\slashed k+\gamma_0\mu_\alpha T_\alpha-m & \Delta_a\bar P_a\\
\Delta^*_aP_a & \slashed k-\gamma_0\mu_\alpha T_\alpha^T-m
\end{pmatrix}.
\label{Nambu_propagator}
\end{equation}
The symbol ``$\mathrm{Tr}$'' in Eq.~\eqref{NJL_TD} denotes a trace in the operator sense.
Let us introduce some further notation. In the following, we will not need to work with the general orientation of the condensate $\Delta_a$ in the color space as well as with all the chemical potentials $\mu_\alpha$. In fact, all integrals will be evaluated with only the $\Delta_3\equiv\Delta$ and $\mu_{0,8}$ components nonzero. The quasiquark spectrum is then easily determined analytically, and the dispersion relations of the individual fundamental colors, denoted as $r,g,b$, are $E^e_{\vek k(r,g)}=\Sqrt{(\xi^e_{\vek k(r,g)})^2+\Delta^2}$, $E^e_{\vek k(b)}=|\xi^e_{\vek k(b)}|$, $e=\pm$, with
\begin{equation}
\begin{split}
\xi^e_{\vek k(r)}&=\xi^e_{\vek k(g)}=\epsilon_{\vek k}+e\Bigl(\mu+\frac{\mu_8}{2\Sqrt3}\Bigr),\\
\xi^e_{\vek k(b)}&=\epsilon_{\vek k}+e\Bigl(\mu-\frac{\mu_8}{\Sqrt3}\Bigr),\qquad
\epsilon_{\vek k}=\Sqrt{\vek k^2+m^2}.
\end{split}
\end{equation}
At zero temperature to which we will from now on limit our discussion, the density of an individual (fundamental) quark color $a$ is
\begin{equation}
n_{a}=2\sum_{e=\pm}\int\frac{d^3\vek k}{(2\pi)^3} \frac{e\xi^e_{\vek
k(a)}}{E^e_{\vek k(a)}}.
\label{color_density}
\end{equation}
\subsection{Calculation for fixed chemical potentials}
\label{Subsec:fixedmu}
With the particular orientation of the condensate that we chose, only the $T_8$ generator of the color $\gr{SU(3)}$ develops nonzero density, $n_8=(n_{r}+n_{g}-2n_{b})/(2\Sqrt3)$. It can be made to vanish by tuning $\mu_8$ appropriately. Let us assume that this has been done and calculate the propagator of the four modes that couple to $\Delta_{1,2}$. These form a complex doublet under the unbroken $\gr{SU(2)}$ subgroup and, had we not imposed the neutrality constraint, they would give rise to two type-II NG bosons with quadratic dispersion relation at low momentum~\cite{Nielsen:1975hm,Blaschke:2004cs}. A necessary condition for these to appear is nonzero density of some of the color charges (see Sch\"afer \emph{et al.} in Ref.~\cite{Schaefer:2001bq}). That is why one expects to recover four usual (type-I) NG bosons once the system is made neutral.
Instead, we will show that once the color chemical potential is adjusted in order to make the system color neutral, the propagator of $\Delta_{1,2}$ has a double pole at the frequency $\omega_0=\mu_8\Sqrt3/2$. This was in Ref.~\cite{He:2005mp,*Ebert:2005fi,*Ebert:2006bq} misinterpreted as a manifestation of two pseudo-NG states with degenerate masses. On the contrary, it actually means that the system exhibits an instability of the same type as revealed in the scalar toy model in Section~\ref{Subsec:instability}. In order to understand this, one should note that the 2SC phase possesses an unbroken $\gr{U(1)}_{\tilde{B}}$ symmetry, corresponding to the blue quark number and generated by $\tilde{B}=B-2T_8/\Sqrt{3}$ with $B$ the baryon number. In the space of $\Delta_a$, which transforms in the antitriplet representation of $\gr{SU(3)}$, this symmetry is generated by the matrix $\mathrm{diag}(1,1,0)$. The two degrees of freedom contained in the complex field $\Delta_1$ (and equivalently $\Delta_2$) carry opposite charges, $\tilde{B}=\pm1$; they are a particle--antiparticle pair. Recalling the Umezawa--Kamefuchi--K\"all\'en--Lehmann spectral representation, the particle pole should show up at positive frequencies, while the antiparticle one at negative frequencies. Since the chemical potential required to make the 2SC state neutral is typically negative, the double pole at $\omega_0$ thus actually describes an antiparticle with mass $|\omega_0|$ and a particle with a negative mass, $-|\omega_0|$. This is yet another manifestation of the seeming instability of the color-neutral 2SC state with respect to fluctuations that generate off-diagonal color charges~\cite{He:2005jq,*Blaschke:2005km}. In the following, we will show how this problem can be fixed in a way that renders the theory stable and the NG bosons exactly massless.
Let us now proceed to the proof of the existence of the double pole in the $\Delta_{1}$ propagator. This is defined as usual by $\Pi_{11}^{-1}(x-y)=-i\langle0|T\{\Delta_1(x)\Delta_1^*(y)\}|0\rangle$, where ``$T$'' denotes time ordering. The inverse propagator, or polarization function, $\Pi_{11}$ is most conveniently evaluated in the random phase approximation. Upon performing Fourier transformation to momentum space and setting the momentum to zero, one finds
\begin{equation}
\begin{split}
\Pi_{11}(\omega)=&\frac1G-2\sum_{e=\pm}\int\frac{d^3\vek k}{(2\pi)^3}\frac1{E^e_{\vek k(r)}}\Biggl[\theta(e\xi^e_{\vek k(b)})\\
&\times\frac{E^e_{\vek k(r)}+e\xi^e_{\vek k(r)}}{\omega+E^e_{\vek k(r)}+|\xi^e_{\vek k(b)}|}\\
&-\theta(-e\xi^e_{\vek k(r)})\frac{E^e_{\vek k(r)}-e\xi^e_{\vek k(r)}}{\omega-E^e_{\vek k(r)}-|\xi^e_{\vek k(b)}|}\Biggr].
\end{split}
\label{invprop}
\end{equation}
The polarization function $\Pi_{22}$ is identical. Using the gap equation at zero temperature,
\begin{equation}
\frac1G=2\sum_{e=\pm}\int\frac{d^3\vek k}{(2\pi)^3}\frac1{E^e_{\vek k(r)}},
\end{equation}
the inverse propagator~\eqref{invprop} is easily brought to the form
\begin{equation}
\begin{split}
&\Pi_{11}(\omega)=(2\omega-\mu_8\Sqrt3)\sum_{e=\pm}\int\frac{d^3\vek k}{(2\pi)^3}\frac1{E^e_{\vek k(r)}}\\
&\times\left[\frac{\theta(e\xi^e_{\vek k(b)})}{\omega+E^e_{\vek k(r)}+|\xi^e_{\vek k(b)}|}+\frac{\theta(-e\xi^e_{\vek k(b)})}{\omega-E^e_{\vek k(r)}-|\xi^e_{\vek k(b)}|}\right].
\end{split}
\label{doublepole}
\end{equation}
The prefactor in Eq.~\eqref{doublepole} immediately tells us that there is a pole at $\omega=\omega_0$. Since this pole becomes exactly massless in the limit of $\mu_8=0$, thereby representing a NG boson, it is the particle pole. To find the antiparticle pole requires some further manipulation. Evaluating the expression in brackets at $\omega=\omega_0$ yields
\begin{equation}
\begin{split}
&\frac{\theta(e\xi^e_{\vek k(b)})}{\frac{\mu_8\Sqrt3}2+E^e_{\vek k(r)}+e\xi^e_{\vek k(b)}}+\frac{\theta(-e\xi^e_{\vek k(b)})}{\frac{\mu_8\Sqrt3}2-E^e_{\vek k(r)}+e\xi^e_{\vek k(b)}}\\
&=\frac{\theta(e\xi^e_{\vek k(b)})}{E^e_{\vek k(r)}+e\xi^e_{\vek k(r)}}+\frac{\theta(-e\xi^e_{\vek k(b)})}{-E^e_{\vek k(r)}+e\xi^e_{\vek k(r)}}\\
&=\frac{E^e_{\vek k(r)}\,\mathrm{sgn}(e\xi^e_{\vek k(b)})-e\xi^e_{\vek k(r)}}{\Delta^2}.
\end{split}
\label{manipulation}
\end{equation}
Using Eq.~\eqref{color_density}, one then arrives at the conclusion that the integral (including the sum over $e$) in Eq.~\eqref{doublepole} equals $-n_8\Sqrt3/(2\Delta^2)$. In fact, it was already observed in Ref.~\cite{Brauner:2008td} that the coefficient of the term linear in $\omega$ in the expansion of the inverse propagator around the particle pole is proportional to the density $n_8$. The present result is just a generalization to nonzero values of $\mu_8$. We therefore conclude that when the chemical potential $\mu_8$ is tuned so that $n_8=0$, the propagator indeed has a double pole at $\omega=\omega_0$.
We note in passing that, as the second line of Eq.~\eqref{manipulation} clearly shows, the double pole occurs outside the two-body continuum, so the implied instability cannot be alleviated by decay processes.
\subsection{Induced fluctuations of chemical potentials}
\label{Subsec:induced}
We would now like to apply the same strategy as in Section~\ref{Subsec:instability} to remove the instability revealed above, and to show that all NG modes in the 2SC phase are exactly massless as they should. However, due to the complicated form of the mean-field thermodynamic potential in the NJL model, it is not possible to solve analytically for the color chemical potentials as a function of the (uniform) collective fields. Fortunately, this is not really necessary if we are only interested in the mass spectrum of the collective excitations.
Let us consider generally a system possessing a set of (real) order parameters, $\Delta_a$, constrained by the requirement of zero densities, $n_\alpha$, of a set of conserved charges. Introducing the associated chemical potentials, $\mu_\alpha$, its thermodynamics is governed by the grand canonical potential, $\Omega(\Delta_a,\mu_\alpha)$. The values of the order parameters and the chemical potentials in the ground state are determined by the set of gap equations, $\partial\Omega/\partial\Delta_a=0$, together with the constraints,
\begin{equation}
\frac{\partial\Omega}{\partial\mu_\alpha}=0.
\label{constraint}
\end{equation}
In order to study the fluctuations in a neutral system, one uses this equation to eliminate the chemical potentials in favor of the order parameters, and thereby arrives at a thermodynamic potential as a function of $\Delta_a$ solely, $\tilde\Omega\bigl(\Delta_a,\mu_\alpha(\Delta_a)\bigr)$. Such a thermodynamic potential has a minimum that coincides with the simultaneous solution of the gap and constraint equations for $\Omega(\Delta_a,\mu_\alpha)$. In an unconstrained system, the mass matrix of the collective modes is, up to an irrelevant factor, proportional to $\partial^2\Omega/\partial\Delta_a\partial\Delta_b$. Once the constraint~\eqref{constraint} is imposed, this must be obviously replaced with the total second derivative, $d^2\tilde\Omega/d\Delta_ad\Delta_b$. Differentiating Eq.~\eqref{constraint} with respect to $\Delta_a$ (this builds in the requirement that the constraint be satisfied for all values of the order parameter, at least in the vicinity of the equilibrium), one arrives at
\begin{align}
\label{total_mass}
\frac{d^2\tilde\Omega}{d\Delta_ad\Delta_b}&=\frac{\partial^2\Omega}{\partial\Delta_a\partial\Delta_b}-\frac{\partial^2\Omega}{\partial\mu_\alpha\partial\mu_\beta}\frac{\partial\mu_\alpha}{\partial\Delta_a}\frac{\partial\mu_\beta}{\partial\Delta_b}\\
\notag
&=\frac{\partial^2\Omega}{\partial\Delta_a\partial\Delta_b}-\frac{\partial^2\Omega}{\partial\Delta_a\partial\mu_\alpha}\left(\frac{\partial^2\Omega}{\partial\mu\partial\mu}\right)^{-1}_{\alpha\beta}\frac{\partial^2\Omega}{\partial\Delta_b\partial\mu_\beta}.
\end{align}
All partial derivatives are to be evaluated at the ground state values of the order parameters and chemical potentials.
In the next subsection it will be demonstrated explicitly that this prescription results in exactly massless NG bosons in the color-neutral 2SC phase. Here we just note that this is very natural: the potential $\Omega(\Delta_a,\mu_\alpha)$ is invariant under simultaneous transformations of the order parameters and the chemical potentials. Therefore, the potential $\tilde\Omega\bigl(\Delta_a,\mu_\alpha(\Delta_a)\bigr)$ is invariant under the symmetry transformation of $\Delta_a$, and the masslessness of NG bosons follows immediately from the Goldstone theorem.
Before proceeding to the explicit NJL-model calculation, we would like to point out one subtlety hidden in Eq.~\eqref{total_mass}. The matrix $\partial^2\Omega/\partial\mu_\alpha\partial\mu_\beta$ in Eq.~\eqref{total_mass} represents minus the density--density correlator (or the color number susceptibility matrix) in the ground state so that it is negative-semidefinite. The charges of the unbroken symmetry have by definition sharp values in the ground state, and thus give rise to zero modes of the correlator. The matrix of second partial derivatives is therefore not invertible. There is a simple remedy: one adds a term $-\zeta^2\mu_\alpha\mu_\alpha$ to the thermodynamic potential which makes all expressions well defined. In the end, the limit $\zeta\to0$ is performed. Since the unbroken generators do not couple to NG bosons, this subtlety does not affect the calculation of their mass spectrum.
\subsection{Induced corrections to NG boson masses}
\label{Subsec:corrections}
We will again focus on the complex doublet of NG bosons that couple to $\Delta_{1,2}$~\footnote{There is one more NG boson annihilated by $\Delta_3$, associated with the spontaneous breaking of the $T_8$ generator. Its masslessness is not questioned because $T_8$ is not ``explicitly broken'' by the chemical potential $\mu_8$.}. Recall that these fields are complex and we are interested in the derivative $d^2\tilde\Omega/d\Delta_1^*d\Delta_1$ which yields the static part of the inverse propagator~\eqref{invprop}.
We need the three second partial derivatives of the thermodynamic potential, $\partial^2\Omega/\partial\Delta^*_1\partial\Delta_1,$ $\partial^2\Omega/\partial\mu_\alpha\partial\mu_\beta$, and $\partial^2\Omega/\partial\Delta^*_1\partial\mu_\alpha$. The first one is already contained in Eq.~\eqref{doublepole},
\begin{equation}
\begin{split}
\frac1V\frac{\partial^2\Omega}{\partial\Delta^*_1\partial\Delta_1}&=-\mu_8\Sqrt3\,X,\\
X&=\sum_{e=\pm}\int\frac{d^3\vek k}{(2\pi)^3}\frac1{E^e_{\vek k(r)}} \frac{\mathrm{sgn}(e\xi^e_{\vek k(b)})}{E^e_{\vek k(r)}+|\xi^e_{\vek k(b)}|}.
\end{split}
\label{piece1}
\end{equation}
The density--density correlator is given by two one-loop diagrams, with normal and anomalous components of the quark propagator. Carrying out the partial derivative of the thermodynamic potential~\eqref{NJL_TD}, we obtain
\begin{equation}
\beta\frac{\partial^2\Omega}{\partial\mu_\alpha\partial\mu_\beta}=\mathrm{Tr}\left(\gamma_0T_\alpha S_{q\bar q}\gamma_0T_\beta S_{q\bar q}-\gamma_0T_\alpha S_{qq}\gamma_0T_\beta^TS_{\bar q\bar q}\right).
\label{auxeq0}
\end{equation}
The subscripts of the propagator denote matrix elements in the Nambu space. Inserting the quark propagator~\eqref{Nambu_propagator} and carrying out the frequency integration, the first term evaluates to
\begin{equation}
\begin{split}
&-\frac{\beta V}2\Delta^2\tr(T_\alpha\mathcal P_{12}T_\beta\mathcal P_{12})Y\\
&-\beta V\left[\tr(T_\alpha\mathcal P_{12}T_\beta\mathcal P_3)+\tr(T_\alpha\mathcal P_3T_\beta\mathcal P_{12})\right]Z,
\end{split}
\label{auxeq}
\end{equation}
where
\begin{align}
Y&=\sum_{e=\pm}\int\frac{d^3\vek k}{(2\pi)^3}\frac1{(E^e_{\vek k(r)})^3},\\
\notag
Z&=\sum_{e=\pm}\int\frac{d^3\vek k}{(2\pi)^3}\frac1{E^e_{\vek k(r)}|\xi^e_{\vek k(b)}|}\frac{E^e_{\vek k(r)}|\xi^e_{\vek k(b)}|-\xi^e_{\vek k(r)}\xi^e_{\vek k(b)}}{E^e_{\vek k(r)}+|\xi^e_{\vek k(b)}|},
\end{align}
and $\mathcal P_{12}$ and $\mathcal P_3$ are projectors on the subspace of first two colors and the third color, respectively. The symbol ``$\tr$'' stands for a trace over color and flavor indices here. The second term in Eq.~\eqref{auxeq0} reduces to $-(\beta V/2)\tr(T_\alpha MT_\beta^T\he M)Y$, where $M$ is a matrix in color--flavor space defined by $M=\gamma_5P_3$. Putting all the pieces together, we obtain
\begin{equation}
\begin{split}
&\frac1V\frac{\partial^2\Omega}{\partial\mu_\alpha\partial\mu_\beta}\\
&=-\frac13\begin{pmatrix}
2 & \Sqrt2\\
\Sqrt2 & 1
\end{pmatrix}\Delta^2Y\quad\text{in the $(T_0,T_8)$ sector},\\
&=-Z\delta_{\alpha\beta}\quad\text{for $\alpha=4,5,6,7$},
\end{split}
\label{piece2}
\end{equation}
and zero otherwise. Finally, the mixed partial derivative of $\Omega$ is given by a one-loop diagram with one normal and one anomalous propagator,
\begin{equation}
\beta\frac{\partial^2\Omega}{\partial\Delta_a^*\partial\mu_\alpha}=\mathrm{Tr}\left(\gamma_0T_\alpha S_{qq}P_aS_{q\bar q}\right),
\end{equation}
which, after appropriate manipulations, yields
\begin{align}
\notag
\frac1V\frac{\partial^2\Omega}{\partial\Delta_a^*\partial\mu_\alpha}=&\tr(T_\alpha M\gamma_5P_a\mathcal P_3)\,X+\frac12\tr(T_\alpha M\gamma_5P_a\mathcal P_{12})\\
&\times\sum_{e=\pm} \int\frac{d^3\vek k}{(2\pi)^3}\frac{e\xi^e_{\vek k(r)}}{(E^e_{\vek k(r)})^3}.
\label{piece3}
\end{align}
For the first component of the diquark field, the color--flavor traces are $\tr(T_\alpha M\gamma_5P_1\mathcal P_{12})=0$ and $\tr(T_\alpha M\gamma_5P_1\mathcal P_3)=(0,0,0,-\Delta,-i\Delta,0,0,0)$ for $\alpha=1,\dotsc,8$.
Finally, we combine Eqs.~\eqref{total_mass}, \eqref{piece1}, \eqref{piece2}, and \eqref{piece3} to arrive at the simple result
\begin{equation}
\frac1V\frac{d^2\Omega}{d\Delta_1^*d\Delta_1}=-\Sqrt3\,\frac XZ\biggl(\mu_8Z
-\frac2{\Sqrt3}\Delta^2X\biggr).
\end{equation}
A slight manipulation with the integrals reveals that the expression in the parentheses equals the charge density $n_8$. This concludes the argument that once Eq.~\eqref{total_mass} is used instead of a simple second partial derivative to calculate the mass spectrum, the NG bosons are rendered exactly massless, as required by the Goldstone theorem.
\section{Dispersion relations of NG bosons in high-density approximation}
\label{Sec:NGdispersions}
The approach taken in the previous section was straightforward, yet intrinsically limited to uniform order parameter fluctuations, so demonstrating the masslessness of the NG modes was the best it could do for us. Here we will follow the path outlined in Section~\ref{Subsec:EFT} and construct the effective action for the NG modes in the 2SC phase. We will use the high-density effective theory (HDET, see Ref.~\cite{Nardulli:2002ma} for a review), providing model-independent results in the limit of high baryon density.
The HDET Lagrangian is constructed using quark fields and their coupling to colored diquarks. For the 2SC superconductor, it takes the following form,
\begin{align}
\notag
\mathcal L_{\text{HDET}}=&\frac{1}{2}\sum_{\vek{v}}\Bigl[\he{q_+}(iV\cdot\partial+\hat{\mu}) q_++\he{q_-}(i\tilde{V}\cdot\partial+\hat{\mu})q_-\\
\label{HDETlag}
&+\Delta_{a}^*q_-^T{\tau}_2{\vek{\epsilon}}_a q_++\Delta_{a}\he{q_+}{\tau}_2\vek{\epsilon}_aq_-^*\Bigr]\\
\notag
&+(\text{L}\to\text{R},\Delta_{a}\to-\Delta_{a}).
\end{align}
Here $q$ stands for the left-handed positive-energy-projected quark field and the subscript $\pm$ denotes the (conjugated) velocity. The two four-velocities appearing in the Lagrangian are defined as $V^\mu=(1,\vek{v})$ and $\tilde{V}^\mu=(1,-\vek{v})$; the sum is taken over pairs of patches on the Fermi surface characterized by velocities $\pm\vek v$. Furthermore, ${\tau}_2$ is the Pauli matrix in the flavor space and $\hat{\mu}$ is the chemical potential matrix in the color space; we consider the form $\hat{\mu}=\mu\openone+\sum_{\alpha=1}^8\mu_\alpha T_\alpha$ ignoring the electric charge neutrality. Also $\vek{\epsilon}_a=2T_7,2T_5,2T_2$ for $a=1,2,3$ are the color antisymmetric generators, and accordingly the $\Delta_{a}$'s stand for the background quark pair fields. The subscript $a$ represents the color antitriplet indices which can be thought of as color-antisymmetric pairs, $[g,b],[b,r],[r,g]$, respectively. Finally, the ``$\mathrm{L\to R}$'' in the last line of Eq.~\eqref{HDETlag} is to remind us that an equivalent Lagrangian for the right-handed quarks has to be added.
By introducing the Nambu doublet field, $\Psi\equiv(q_+,-q_-^*)^T$, the Lagrangian~\eqref{HDETlag} becomes
\begin{equation}
\begin{split}
\mathcal L_{\text{HDET}}=&\frac{1}{2}\sum_{\vek{v}}\he\Psi
\begin{pmatrix}
iV\cdot\partial+\hat{\mu} & -\Delta_a\tau_2\vek\epsilon_a\\
-\Delta^*_a\tau_2\vek\epsilon_a & i\tilde{V}\cdot\partial-\hat{\mu}^*
\end{pmatrix}\Psi\\
&+(\text{L}\to\text{R},\Delta_a\to-\Delta_a).
\end{split}
\label{HDETefflag}
\end{equation}
Let us construct the effective Lagrangian for the NG modes. Following the logic explained in Section~\ref{Sec:toymodel} we have to consider fluctuations of both the pairing field $\Delta_a$ and the color chemical potential matrix $\hat\mu$, which is treated as a secondary order parameter induced by the color neutrality constraint. Without loss of generality, we adopt the convention that the ground state is characterized by the nonzero condensate $\Delta_3=\Delta_3^*\equiv\Delta$. Accordingly, only the color chemical potential in the $T_8$-direction is nonzero in the ground state. The actual values of $\Delta$ and $\mu_8$ are set by the gap equation and the neutrality constraint. Having all this in mind, we parameterize the NG fields $\Delta_a$ and chemical potentials $\mu_\alpha$ in Eq.~\eqref{HDETefflag} as
\begin{equation}
\begin{split}
\Delta_a\vek\epsilon_a&=\mathcal U(\vek{\pi})2\Delta T_2\mathcal U(\vek{\pi})^T,\\
\hat\mu&=\mu\openone+\mathcal U(\vek{\pi})\mu_8T_8\he{\mathcal U(\vek{\pi})},
\end{split}
\label{eq:parametrization}
\end{equation}
where $\mathcal U$ is expressed in terms of the NG fields $\vek{\pi}$ as
\begin{equation}
\mathcal U(\vek{\pi})=\exp\biggl[\frac i{f_\pi}\sum_{\alpha=4}^8\pi_\alpha(x)T_\alpha\biggr],
\label{eq:calU}
\end{equation}
and we introduced the decay constant $f_\pi$. In principle, an
independent decay constant should be used for every real irreducible representation of the unbroken global symmetry, in this case one for $\alpha=4,5,6,7$ and one for $\alpha=8$. We just use the same symbol for them in order to simplify the notation.
It is now easy to obtain an effective action for the NG modes in a gradient expansion. Following essentially the same steps as in Section~\ref{Sec:NGmasses}, we integrate out the quark fields to obtain
\begin{equation}
S_{\text{eff}}=-\mathrm{Tr}\log S^{-1},
\end{equation}
where $S$ is the quark propagator in Nambu space in presence of the fluctuating order parameters. This formula is to be contrasted to Eq.~\eqref{NJL_TD}. The mass term $\Delta_a\Delta_a^*/G$ is missing here and consequently the NG nature of the order parameter fluctuations is made manifest.
We find the following explicit expression for the propagator up to the second order in the NG fields,
\begin{equation}
\begin{split}
S^{-1}(x,y)=&\begin{pmatrix}
iV\cdot\partial+\hat{\mu} & -2\Delta\tau_2T_2\\
-2\Delta\tau_2T_2 & i\tilde{V}\cdot\partial-\hat{\mu}
\end{pmatrix}
\delta^4(x-y)\\
&+\frac{\pi_\alpha(x)}{f_\pi}\Sigma^{(1)}_{\alpha}\delta^4(x-y)\\
&-\frac{\pi_\alpha(x)\pi_\beta(x)}{f_\pi^2}\Sigma^{(2)}_{\alpha\beta}\delta^4(x-y),
\label{propagator}
\end{split}
\end{equation}
where
\begin{equation}
\begin{split}
\Sigma^{(1)}_\alpha&=\begin{pmatrix}
-\mu_8\Xi_\alpha&\Delta\tau_2\Gamma_\alpha\\
\Delta\tau_2\he{\Gamma_\alpha}&+\mu_8\Xi_{\alpha}^*
\end{pmatrix},\\
\Sigma^{(2)}_{\alpha\beta}&=\begin{pmatrix}
-\mu_8\Xi_{\alpha\beta}&\Delta\tau_2\Gamma_{\alpha\beta}\\
\Delta\tau_2\he{\Gamma_{\alpha\beta}}&+\mu_8\Xi_{\alpha\beta}^*
\end{pmatrix},
\end{split}
\end{equation}
and $\Sigma_{\alpha}$, $\Gamma_\alpha$, $\Sigma_{\alpha\beta}$, and $\Xi_{\alpha\beta}$ are matrices in the fundamental color space. The explicit forms of these matrices are found by expanding Eq.~\eqref{eq:parametrization} up to second order in the NG fields. We note that the diagonal entries proportional to $\mu_8$ take into account the effect of fluctuations of the chemical potentials that was missed in previous analyses~\cite{Blaschke:2004cs,He:2005mp}. The gradient expansion of the action thus reads
\begin{align}
\notag
S_{\text{eff}}=&-\mathrm{Tr}\log S^{-1}_0\\
\notag
&+\int d^4x\frac{\pi_\alpha(x)\pi_\beta(x)}{f_\pi^2}\tr\left[S_0(x,x)\Sigma^{(2)}_{\alpha\beta}\right]\\
&+\int d^4x\,d^4y\frac{\pi_\alpha(x)\pi_\beta(y)}{2f_\pi^2}\\
\notag
&\times\tr\left[S_0(x,y)\Sigma^{(1)}_\alpha S_0(y,x)\Sigma^{(1)}_\beta\right],
\end{align}
with $S_0^{-1}$ being the first term of Eq.~\eqref{propagator}. The first term above is proportional to the thermodynamic potential in the equilibrium. The second term gives the tadpole contribution stemming from the fact that we employed a non-linear parameterization of the collective fields. Omitting the part of zeroth order in the NG fields and working at nonzero temperature within the Matsubara formalism, the bilinear part of the effective action becomes, in momentum space,
\begin{equation}
\begin{split}
S_{\text{eff}}=&\frac{T}{2}\sum_{N}\int\frac{d^3\vek{P}}{(2\pi)^3}\pi_\alpha(-i\Omega_N,-\vek{P})\\
&\times\Pi_{\alpha\beta}(i\Omega_N,\vek{P})\pi_{\beta}(i\Omega_N,\vek{P}),
\end{split}
\end{equation}
where $\Omega_N=2N\pi T$ and $N$ is an integer. Reality of the NG fields in the coordinate space implies the constraint on their Fourier components, $\pi_{\alpha}(-i\Omega_N,-\vek{P})=\pi_{\alpha}(i\Omega_N,\vek{P})$. The inverse propagator of the NG fields is given by
\begin{align}
\notag
{\Pi}_{\alpha\beta}(i\Omega_N,\vek{P})=&\frac{T}{f_\pi^2}\sum_{n}\int\frac{d^3\vek{q}}{(2\pi)^3}\tr\left[2S_0(i\omega_n,\vek{q})\Sigma_{\alpha\beta}^{(2)}\right]\\
\label{eq:selfenergy3}
&+\frac{T}{f_\pi^2}\sum_{n}\int\frac{d^3\vek{q}}{(2\pi)^3}\tr\Bigl[S_0(i\omega_n,\vek{q})\\
\notag
&\times\Sigma_{\alpha}^{(1)}S_0(i\omega_n+i\Omega_N,\vek{q}+\vek{P})\Sigma_{\beta}^{(1)}\Bigr],
\end{align}
with $\omega_n=(2n+1)\pi T$ being the fermionic Matsubara frequency. In the Nambu space, the fermion propagator takes the form
\begin{equation}
S_0(i\omega_n,\vek{q})=\begin{pmatrix}
S_{q\bar{q}}(i\omega_n,\vek{q})& S_{qq}(i\omega_n,\vek{q})\\
S_{\bar{q}\bar{q}}(i\omega_n,\vek{q})&S_{\bar{q}q}(i\omega_n,\vek{q})
\end{pmatrix},
\end{equation}
where the individual elements are given explicitly by
\begin{equation}
\begin{split}
S_{q\bar{q}}(i\omega_n,\vek{q})&= (\openone-\tilde{B})S_{q\bar{q}}^{(r,g)}+\tilde{B}S_{q\bar{q}}^{(b)},\\
S_{qq}(i\omega_n,\vek{q})&=2\tau_2T_2S_{qq}^{(r,g)},\\
S_{q\bar{q}}^{(r,g)}&=-\frac{i\omega_n+\vek{v}\cdot\vek{q}-\mu_r}{\omega_n^2+(\vek{v}\cdot\vek{q}-\mu_r)^2+\Delta^2},\\
S_{q\bar{q}}^{(b)}&=\frac{1}{i\omega_n-(\vek{v}\cdot\vek{q}-\mu_b)},\\
S_{qq}^{(r,g)}&=-\frac{\Delta}{\omega_n^2+(\vek{v}\cdot\vek{q}-\mu_r)^2+\Delta^2}.\\
\end{split}
\end{equation}
As before, $\tilde{B}=\frac{1}{3}\openone-2T_8/\Sqrt3=\mathrm{diag}(0,0,1)$ is the projector to the blue quark space, or in other words, the blue quark number in the fundamental color space. The chemical potentials for the red, green, and blue quarks are defined by $\mu_{r,g}=\mu+\mu_8/(2\Sqrt{3})$ and $\mu_b=\mu-\mu_8/\Sqrt{3}$.
Upon carrying out the trace in the color and flavor spaces and performing the Matsubara summation, ${\Pi}_{\alpha\beta}$ turns out to have the following block-diagonal structure in the adjoint color space, dictated by the unbroken symmetry,
\begin{equation}
\begin{split}
\Pi_{\alpha\beta}=&\begin{pmatrix}
\Pi_{\text{tad}}^{\text{ch}}+\Pi^{\text{ch}} & \Pi_{\text{mix}}\\
-\Pi_{\text{mix}} & \Pi_{\text{tad}}^{\text{ch}}+\Pi^{\text{ch}}\\
\end{pmatrix}_{(\pi_4,\pi_5)}\\
&\oplus
\begin{pmatrix}
\Pi_{\text{tad}}^{\text{ch}}+\Pi^{\text{ch}} & \Pi_{\text{mix}}\\
-\Pi_{\text{mix}} & \Pi_{\text{tad}}^{\text{ch}}+\Pi^{\text{ch}}\\
\end{pmatrix}_{(\pi_6,\pi_7)}\\
&\oplus
(\Pi^{\rm n}_{\text{tad}}+\Pi^{\rm n})_{(\pi_8)}.
\end{split}
\label{eq:polarization}
\end{equation}
The superscripts ``n'' and ``ch'' distinguish modes that are respectively neutral and charged with respect to the unbroken quantum number $\tilde B$. In accord with what we already observed in Section~\ref{Sec:NGmasses}, there should be four charged modes that form a complex doublet of the unbroken global color $\gr{SU(2)}$ symmetry, and one neutral NG mode. Using the unbroken symmetry, the propagator in the charged sector can be diagonalized in the basis $(\pi_4\pm i\pi_5)/\Sqrt2$, $(\pi_6\pm i\pi_7)/\Sqrt2$. The various contributions to the propagator are labelled by subscripts: ``tad'' refers to the tadpole contribution coming from the first term in Eq.~\eqref{eq:selfenergy3}, whereas ``mix'' refers to terms that mix different components $\pi_\alpha$, thereby giving rise to mass splitting of modes with opposite values of $\tilde B$. As we will see below, the spacetime dependence of the propagator at long-wavelengths can be extracted analytically within HDET.
\subsection{NG mode in the neutral sector}
Let us first have a look at the neutral NG mode, $\pi_8$, which corresponds to mere phase fluctuations of the order parameter and hence, based on the toy model analyzed in Section~\ref{Sec:toymodel}, should not suffer from artifacts associated with the color neutrality constraint. Its dispersion relation can be extracted from the gradient expansion of $\Pi^{\rm n}_{\text{tad}}+\Pi^{\rm n}(i\Omega_N,\vek{P})$. Explicit computation yields
\begin{equation}
\Pi^{\rm n}_{\text{tad}}=\frac{4N_{\rm f}}{3f_\pi^2}\Delta\phi(\Delta,\mu_r),
\label{eq:tadneutral}
\end{equation}
with $\phi$ being the anomalous pair density defined by $\langle q_{-}^{ai}q_{+}^{bj}\rangle=-(\tau_2)_{ij}(2T_{2})_{ab}\phi(\Delta,\mu_r)$; $N_{\rm f}=2$ is the number of flavors. In terms of the quark propagator, it is expressed as
\begin{equation}
\phi(\Delta,\mu_r)=-T\sum_n\int\frac{d^3\vek{q}} {(2\pi)^3}S_{qq}^{(r,g)}(i\omega_n,\vek{q}).
\end{equation}
On the other hand, the gradient expansion of $\Pi^{\rm n}(i\Omega_N,\vek P)$ becomes
\begin{equation}
\Pi^{\rm n}(i\Omega_N,\vek{P})=\Pi^{\rm n}(0,\vek0)-A(i\Omega_N)^2+B\vek{P}^2+\dotsb,
\end{equation}
with $\Pi^{\rm n}(0,\vek0)=-\Pi^{\rm n}_{\text{tad}}$ which guarantees that the mode is gapless in accord with the Goldstone theorem. The coefficients $A$ and $B$ are evaluated within HDET in the limit of vanishing temperature and at the leading order in $\Delta/\mu,\mu_8/\mu$ as
\begin{equation}
A=\frac{N_{\rm f}N_0}{3f_\pi^2},\qquad
B=\frac{N_{\rm f}N_0}{9f_\pi^2},
\end{equation}
where $N_0\equiv{\mu^2}/{2\pi^2}$ is the density of states at the Fermi surface. In order for the NG mode effective action to take the canonical form with $A=1$, we set the decay constant to $f_\pi^2=N_{\rm f}N_0/3$. The phase velocity of $\pi_8$ becomes
\begin{equation}
v_\pi=\Sqrt{\frac{B}{A}}=\Sqrt{\frac{1}{3}},
\end{equation}
which is the usual result reflecting the number of space dimensions~\cite{Nardulli:2002ma}. Since $\pi_8$ is a singlet under the unbroken global color $\gr{SU(2)}$ symmetry and carries zero $\tilde B$ charge, it is not much affected by the color neutrality constraint. The above conclusion therefore remains correct irrespective of the value of the background color charge density $n_8$ as long as $\mu_8/\mu$ is small. In the following, we will concentrate on the $\tilde{B}$-charged sector of the NG effective action.
\subsection{NG modes in the charged sector}
We are now ready to analyze in detail the excitation spectrum of the charged NG modes in the $(\pi_4,\pi_5)$ sector. [The spectrum in the $(\pi_6,\pi_7)$ sector is identical as a consequence of the unbroken global color $\gr{SU(2)}$ symmetry.] Before going into details, let us briefly pause to overview the general properties of the charged sector. Following the discussion below Eq.~\eqref{eq:polarization}, we define the combinations $\pi_\pm\equiv(\pi_4\pm i\pi_5)/\Sqrt2$ which carry the charges $\tilde B=\pm1$. The low-energy behavior of these can be extracted from the polarization function in the charged $(\pi_4,\pi_5)$-sector in Eq.~\eqref{eq:polarization}. The components of the propagator satisfy the complex conjugation properties $\Pi^{\text{ch}}(-i\Omega_N,\vek{P})=\Pi^{\text{ch}}(i\Omega_N,\vek{P})$ and $\Pi_{\text{mix}}(-i\Omega_N,\vek{P})=-\Pi_{\text{mix}}(i\Omega_N,\vek{P})$. In the basis $(\pi_+,\pi_-)$, the polarization matrix becomes diagonal with the entries $\Pi_\pm=\Pi_{\text{tad}}^{\text{ch}}+\Pi^{\text{ch}}\mp i\Pi_{\text{mix}}$. The charge conjugation then implies that $\Pi_-(i\Omega_N,\vek{P})=\Pi_{+}(-i\Omega_N,\vek{P})$.
In order to clarify the role of the fluctuations of the chemical potentials and the coupling between charge density fluctuations and the NG modes, we decompose the functions $\Pi^{\text{ch}}$ and $\Pi_{\text{mix}}$ as
\begin{equation}
\begin{split}
\Pi^{\text{ch}}&=\Pi_{\Delta\Delta}^{\text{ch}}+\Pi_{\Delta\mu}^{\text{ch}}+\Pi_{\mu\mu}^{\text{ch}},\\
\Pi_{\text{mix}}&=\Pi_{\text{mix}}^{\Delta\Delta}+\Pi_{\text{mix}}^{\Delta\mu}+\Pi_{\text{mix}}^{\mu\mu},
\end{split}
\end{equation}
where the first, second, and third terms are proportional to $\Delta^2$, $\Delta\mu_8$, and $\mu_8^2$, respectively. Diagrammatically, they are expressed as the quark one-loop diagrams containing two anomalous propagators, one normal and one anomalous propagator, and two normal propagators. $\Pi_{\text{tad}}^{\text{ch}}$ is computed explicitly as
\begin{equation}
\Pi_{\text{tad}}^{\text{ch}}=\frac{N_{\text{f}}}{f_\pi^2}\Delta\phi(\Delta,\mu_r)+\frac{3}{4f_\pi^2}\mu_8n_8(\Delta,\hat{\mu}),
\label{eq:tad}
\end{equation}
where $n_8(\Delta,\hat{\mu})\equiv(n_r+n_g-2n_b)/2\Sqrt{3}$ is the color charge density at an arbitrary $\mu_8$. In contrast to the tadpole contribution~\eqref{eq:tadneutral} in the neutral sector, there is an additional term proportional to $\mu_8n_8$ which comes from the tadpole diagram with a quark loop containing the normal propagator. This is because the charged NG mode is sensitive to the secondary order parameter, that is, $\mu_8$. This additional term is absent if we ignore the fluctuations in color chemical potentials.
We remark that $\pi_+,\pi_-$ correspond to $\Delta_1,\Delta_1^*$ in the notation of Section~\ref{Sec:NGmasses}. Therefore, the curvature mass matrix calculated in Section~\ref{Subsec:corrections} can be obtained as the long-wavelength limit of the polarization function,
\begin{equation}
\begin{split}
\frac{1}{V}\frac{d^2\Omega}{d\Delta_1^*d\Delta_1}&=\lim_{\vek{P}\to\vek{0}}\Pi_\pm(0,\vek{P})\\
&=\Pi_{\text{tad}}^{\text{ch}}(0,\vek{0})+\Pi^{\text{ch}}(0,\vek{0}).
\end{split}
\end{equation}
We shall now calculate in detail the polarization functions $\Pi^\pm$, and hence the dispersion relations of the $\pi_\pm$ modes. In order to elucidate the role of the color neutrality and the necessity to implement it carefully, we will analyze three different scenarios:
\begin{itemize}
\item[(i)] The case with $\mu_8=0$. In this case the primary order parameter, that is $\Delta$, induce nonzero color charge density in the system. We expect the spectrum to contain type-II NG bosons.
\item[(ii)] The case of color neutrality ensured by ``hard'' (fixed) chemical potential(s). The chemical potential $\mu_8$ is tuned so that there is no color charge in the ground state, but we ignore the chemical potential fluctuations; this is analogous to the discussion in Section~\ref{Subsec:fixedmu}. Technically this means that we discard the specific contributions to the polarization function which couple to color density such as $\Pi^{\text{ch}}_{\Delta\mu}$, $\Pi^{\text{ch}}_{\mu\mu}$ etc. In this case one may naively expect four NG bosons to acquire nonzero masses since the hard external background $\mu_8$ serves as an explicit symmetry breaking source in the quark sector~\cite{He:2005mp,*Ebert:2005fi,*Ebert:2006bq}.
\item[(iii)] The full analysis of NG bosons in the neutral system. We take into account fluctuations in the chemical potentials and their coupling to NG bosons; this is analogous to the discussion in Section~\ref{Subsec:corrections}. Technically this means including all contributions to the polarization function. We expect in this case to recover five massless type-I NG bosons.
\end{itemize}
\subsubsection{Analysis for the case (i)}
We set $\mu_8=0$ and $\mu_r=\mu_g=\mu_b=\mu$. After analytically continuing the polarization function to real frequencies, $i\Omega_N\to\omega+i\delta$, and expanding in powers of $\omega$ and $\vek{P}$, one finds, up to the second order,
\begin{equation}
\Pi^{\text{ch}}(\omega,\vek{P})=\Pi^{\text{ch}}(0,\vek{0})-A\omega^2+B\vek{P}^2+\dotsb,
\end{equation}
where $\Pi^{\text{ch}}(0,\vek{0})=-\frac{N_{\rm f}}{f_\pi^2}\Delta\phi(\Delta,\mu)$ which exactly cancels the tadpole contribution in Eq.~\eqref{eq:tad} with $\mu_8=0$, reflecting the Goldstone theorem. The coefficients $A$ and $B$ can be evaluated by employing the high-density approximation (at zero temperature), which consists in the replacement $\int\frac{d^3\vek{q}}{(2\pi)^3}\to N_0\int_{-\infty}^\infty d\ell$ with $\ell\equiv|\vek{q}|-\mu$ being the momentum measured with respect to the Fermi surface. This yields the result~\footnote{We note that to this order of approximation, the subtle problem associated with the momentum assignment to the two fermion propagators in the loop integral (see Ref.~\cite{Brauner:2008td}) is absent. This is because the one-dimensional $\ell$-integral is finite; any two momentum assignments differ by a shift which can be absorbed into a redefinition of the integration variable $\ell$.}
\begin{equation}
A=\frac{N_{\rm f}N_0}{2f_\pi^2},\qquad
B=\frac{N_{\rm f}N_0}{6f_\pi^2}.
\label{eq:coeff}
\end{equation}
In order for the NG boson Lagrangian to have the canonical form, we adjust the decay constant as $f_\pi^2=N_{\text{f}}N_0/2$.
The knowledge of the diagonal elements of the polarization matrix is not sufficient to determine the dispersion relation or even to conclude that there are two massless modes. To that end, we need to evaluate the offdiagonal component for which we obtain the result at the leading order in the gradient expansion,
\begin{equation}
\Pi_{\text{mix}}(\omega,\vek{P})=-iC\omega,
\end{equation}
where $C=\Sqrt{3}n_8/(2f_\pi^2)$. In HDET, the color charge density takes the value $n_8=N_{\rm f}N_0\Delta^2/(2\Sqrt{3}\mu)$. This is suppressed by the small ratio $\Delta/\mu$ and thus belongs to the next-to-leading order in HDET. Nevertheless, we keep the term $C$ since the charge density $n_8$ is finite and can be evaluated without the high-density approximation. The polarization matrix in the $(\pi_4,\pi_5)$ sector then becomes, in the gradient expansion,
\begin{equation}
\begin{pmatrix}
-A\omega^2+B\vek{P}^2& -iC\omega\\
+iC\omega&-A\omega^2+B\vek{P}^2
\end{pmatrix}.
\label{casei}
\end{equation}
Upon diagonalization in the $(\pi_+,\pi_-)$ basis, this leads to $\Pi_\pm(\omega,\vek{P})=-A\omega^2+B\vek{P}^2\mp C\omega$, resulting in the dispersion relations
\begin{equation}
\omega_{\pi_+}=\frac{B}{C}\vek{P}^2,\qquad
\omega_{\pi_-}=\frac{C}{A}+\frac{B}{C}\vek{P}^2
\end{equation}
to second order in momentum. We can see that $\pi_+$, having the like charge as the medium, becomes a type-II NG boson with a quadratic dispersion relation, while $\pi_-$ with the opposite charge acquires a gap proportional to the color density $n_8$. This is in agreement with the general theorems concerning NG bosons in systems lacking Lorentz invariance~\cite{Nielsen:1975hm,Blaschke:2004cs,Brauner:2008td}. Also, it is clear that the system is stable at least at the quadratic order in the derivative expansion.
\subsubsection{Analysis for the case (ii)}
The calculation is basically the same as in the case (i), but we need to take into account the chemical potential differences, $\mu_r=\mu_g\ne\mu_b$. As before, we use the basis $(\pi_+,\pi_-)$, and thus need to analyze the polarization function $\Pi_+(\omega,\vek{P})=\Pi^{\text{ch}}_{\text{tad}}+\Pi^{\text{ch}}(\omega,\vek{P})-i\Pi_{\text{mix}}(\omega,\vek{P})$. It is easy to evaluate the long-wavelength limit of $\Pi^{\text{ch}}(\omega,\vek{P})[=\Pi^{\text{ch}}_{\Delta\Delta}(\omega,\vek{P})]$,
\begin{align}
\notag
\Pi_{\Delta\Delta}^{\text{ch}}(0,\vek{0})=&-\frac{N_{\rm f}}{f_\pi^2}\Delta\phi(\Delta,\mu_r)\\
\notag
&+\frac{\Sqrt{3}N_{\rm f}}{4f_\pi^2}\mu_8\Delta^2\int\frac{d^3\vek{q}}{(2\pi)^3}\Biggl[\frac{\tanh\left(\frac{\ell_b}{2T}\right)}{\ell^2+\Delta^2-\ell_b^2}\\
&-\frac{\ell_b\tanh\Bigl(\frac{\Sqrt{\ell^2+\Delta^2}}{2T}\Bigr)}{\Sqrt{\ell^2+\Delta^2}(\ell^2+\Delta^2-\ell_b^2)}\Biggr],
\label{eq:DD}
\end{align}
where $\ell=|\vek{q}|-\mu-\mu_8/(2\Sqrt{3})$ and $\ell_b=\ell+\Sqrt{3}\mu_8/2$ are the momenta for red/green quarks and for blue quarks measured from the Fermi surface. The first term in $\Pi_{\Delta\Delta}^{\text{ch}}(0,\vek{0})$ is cancelled by the tadpole contribution $\Pi^{\text{ch}}_{\text{tad}}$, but the second term survives. Therefore, in this case the curvature mass remains finite.
Let us take a slightly different approach to the problem. An explicit computation reveals that the function $\Pi_+(\omega,\vek{P})$ depends on $\omega$ only through the combination $\omega+\mu_b-\mu_r=\omega-\Sqrt{3}\mu_8/2$. This fact suggests that it would be more natural to perform the expansion about $\omega=\Sqrt{3}\mu_8/2$ for $\Pi_+(\omega,\vek{P})$ and similarly about $\omega=-\Sqrt{3}\mu_8/2$ for $\Pi_-(\omega,\vek{P})$. In fact, we can easily show that the offset of $\Pi^{\text{ch}}(\omega,\vek{P})\mp i\Pi_{\text{mix}}(\omega,\vek{P})$ at $\omega=\pm\Sqrt{3}\mu_8/2$ completely cancels the tadpole contribution so that
\begin{equation}
\Pi_{+}(\omega=\Sqrt{3}\mu_8/2,\vek{0})=\Pi_{-}(\omega=-\Sqrt{3}\mu_8/2,\vek{0})=0.
\end{equation}
Therefore, we here perform the gradient expansion of $\Pi_{+}$ about $\omega=\Sqrt{3}\mu_8/2$. Up to second order in $\omega$ and $\vek{P}$ we find
\begin{equation}
\begin{split}
\Pi_{+}(\omega,\vek{P})=&A(\omega-\Sqrt{3}\mu_8/2)\\
&-B(\omega-\Sqrt{3}\mu_8/2)^2+C\vek{P}^2,
\end{split}
\end{equation}
where $A$ is proportional to the color density,
\begin{equation}
A=\frac{\Sqrt{3}}{4f_\pi^2}n_8(\Delta,\hat{\mu})=0.
\end{equation}
This is nothing but the neutrality condition which determines the value of $\mu_8$. In the high-density approximation, this can be calculated explicitly, $\mu_8=-\Delta^2/(2\Sqrt3\mu)$. (This is actually a next-to-leading order result, for it is suppressed as compared to $\Delta$ by the factor $\Delta/\mu$.) One can also provide explicit integral formulas for $B$ and $C$; since they are rather complicated, we again use the high-density approximation with the result
\begin{equation}
\begin{split}
B&=\frac{N_{0r}N_{\rm f}}{2f_\pi^2}\frac{\Delta^2+2\mu_8^2/3}{\Delta^2}\simeq\frac{N_0N_{\rm f}}{2f_\pi^2},\\
C&=\frac{N_{0r}}{2f_\pi^2}\frac{N_{\rm f}}{3}\simeq\frac{N_0N_{\rm f}}{6f_\pi^2},
\end{split}
\end{equation}
where $N_{0r}\equiv{\mu_r^2}/{2\pi^2}$. Setting $f_\pi=\Sqrt{N_0N_{\rm f}/2}$ and the phase velocity $v_\pi=1/\Sqrt{3}$, the polarization functions acquire the final form
\begin{equation}
\Pi_\pm\simeq-(\omega\mp\Sqrt{3}\mu_8/2)^2+v_\pi^2\vek{P}^2,
\label{caseii}
\end{equation}
near $\omega=\pm\Sqrt{3}\mu_8/2$, $\vek{P}=\vek{0}$.
We can see that under the color neutrality condition, the propagator of the charged NG modes acquires a double pole at $\omega=\pm\Sqrt{3}\mu_8/2$ which clearly indicates some kind of instability as already explained in Section~\ref{Subsec:fixedmu}. These shifted poles were recognized as the ``mass'' of the pseudo-Goldstone modes due to the``explicit symmetry breaking'' by $\mu_8$~\cite{Blaschke:2004cs,He:2005mp} since $\det{\Pi}(\omega,\vek{0})\propto(\omega^2-3\mu_8^2/4)^2$. Finally, we remark that, in accord with the discussion in Section~\ref{Subsec:instability}, the instability is also clearly seen in the curvature mass squared,
\begin{equation}
\frac{1}{V}\frac{\partial^2\Omega}{\partial\Delta_{1}^*\partial\Delta_{1}}=\Pi_\pm(0,\vek{0})=-\frac{3\mu_8^2}{4}<0.
\end{equation}
\subsubsection{Analysis for the case (iii)}
We finally consider the most general case with an arbitrary value of $\mu_8$ and the corresponding color density $n_8$, but with a proper account of the fluctuations in the color chemical potentials.
Let us start with the diagonal element of the polarization matrix, $\Pi_{\text{tad}}+\Pi^{\text{ch}}(\omega,\vek{P})$. In this case we need to take into account the contributions from $\Pi^{\text{ch}}_{\Delta\mu}$ and $\Pi^{\text{ch}}_{{\mu\mu}}$ in addition to $\Pi_{\Delta\Delta}^{\text{ch}}$ whose long-wavelength limit is given in Eq.~\eqref{eq:DD}. Computing all the contributions using the explicit expressions for quark propagators, we obtain, in the long-wavelength limit
\begin{multline}
\Pi_{\Delta\Delta}^{\text{ch}}(0,\vek{0})+\Pi_{\mu\Delta}^{\text{ch}}(0,\vek{0})+\Pi_{\mu\mu}^{\text{ch}}(0,\vek{0})\\
=-\frac{N_{\text{f}}}{f_\pi^2}\Delta\phi(\Delta,\mu_r)-\frac{3}{4f_\pi^2}\mu_8 n_8(\Delta,\hat{\mu}).
\end{multline}
Note that the above offset is completely cancelled by the tadpole contribution in Eq.~\eqref{eq:tad}. Thus, the diagonal entry of the polarization matrix can be expanded up to second order in $\omega$ and $\vek P$ as
\begin{equation}
\Pi_{\text{tad}}+\Pi^{\text{ch}}=-B\omega^2+C\vek{P}^2,
\end{equation}
where $B$ and $C$ are given by the same expression as in Eq.~\eqref{eq:coeff} up to the leading order in $\mu_8/\mu$. This is true regardless of the value of $\mu_8$ as long as the color chemical potentials fluctuate with the primary order parameter according to Eq.~\eqref{eq:parametrization}. In this case the curvature mass squared always vanishes,
\begin{equation}
\frac{1}{V}\frac{d^2\Omega}{d\Delta_1^*d\Delta_1}=\Pi_{\text{tad}}(0,\vek{0})+\Pi^{\text{ch}}(0,\vek{0})=0.
\end{equation}
In order to understand the low-energy behavior of the NG modes, we still need to evaluate the off-diagonal part of the polarization matrix, $\Pi_{\text{mix}}$; this consists of three parts $\Pi_{\text{mix}}^{\Delta\Delta}$, $\Pi_{\text{mix}}^{\Delta\mu}$, and $\Pi_{\text{mix}}^{\mu\mu}$. Each of the contributions takes quite a complicated form, but surprisingly putting them all together gives rise to the following simple formula at the lowest nontrivial order in the gradient expansion,
\begin{multline}
\Pi^{\Delta\Delta}_{\text{mix}}(\omega,\vek{P})+\Pi^{\mu\Delta}_{\text{mix}}(\omega,\vek{P})+\Pi^{\mu\mu}_{\text{mix}}(\omega,\vek{P})\\
=-i\frac{\Sqrt{3}}{2f_\pi^2}\omega n_8(\Delta,\hat{\mu})+{\mathcal O}(\omega^3,\omega P^2).
\end{multline}
This only depends on $\mu_8$ through the color density $n_8$. To summarize, the polarization matrix in the charged sector behaves at long wavelength as
\begin{equation}
\begin{pmatrix}
-B\omega^2+C\vek{P}^2 & -i\frac{\Sqrt{3}}{2f_\pi^2}n_8(\Delta,\hat{\mu})\omega\\
i\frac{\Sqrt{3}}{2f_\pi^2}n_8(\Delta,\hat{\mu})\omega & -B\omega^2+C\vek{P}^2
\end{pmatrix}.
\label{caseiii}
\end{equation}
It is now obvious that only when $\mu_8$ is tuned such that the color density $n_8$ vanishes, the full set of five type-I NG bosons are recovered.
In Fig.~\ref{fig:ngmasses} we show the schematic plot of the excitation gaps (masses) of the NG modes as a function of~$\mu_8$. There is one neutral NG boson which is always type-I irrespective of the value of $\mu_8$ and the charge density of the system. The four charged modes with $\tilde{B}=\pm1$ are, on the contrary, sensitive to $\mu_8$. When $\mu_8>-\Delta^2/(2\Sqrt{3}\mu)$, the system is positively charged with color $n_8$ and the two type-II NG modes appear in the $\tilde B=+1$ sector; their antiparticles have the gap $\omega=\Sqrt{3}n_8/(N_{\rm f}N_0)$. On the other hand, if $\mu_8<-\Delta^2/(2\Sqrt{3}\mu)$, the system is negatively charged, and the quantum numbers of type-II NG modes and those of their massive partners are interchanged. At the neutrality point, $\mu_8=-\Delta^2/(2\Sqrt{3}\mu)$, the type-II NG modes change smoothly to the type-I NG modes.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{NGmasses}
\caption{(color online). The schematic picture of the NG boson masses as a function of the color chemical potential $\mu_8$. The neutral NG mode is always type-I and it is always gapless to the lowest order of approximation.}
\label{fig:ngmasses}
\end{figure}
\subsection{Effective Lagrangian for the charged NG modes}
The results for the dispersion relations of the charged NG modes can be conveniently encoded in an effective Lagrangian. Generally, the nonlinear effective Lagrangian can be constructed using the coset field $\mathcal U(\vek\pi)$ introduced in Eq.~\eqref{eq:calU}. A symmetry transformation $g\in\gr{SU(3)\times U(1)}$ acts on this field as $\mathcal U(\vek\pi)\xrightarrow{g}\mathcal U(\vek\pi')=g\,\mathcal U(\vek\pi)h^{-1}(\vek\pi,g)$, where $h(\vek\pi,g)$ is a suitable element of the unbroken subgroup $\gr{SU(2)\times U(1)}_{\tilde B}$.
A convenient way to ensure invariance of the effective Lagrangian is to introduce two matrix-valued fields
\begin{equation}
\mathcal T\equiv\mathcal UT_2\mathcal U^{T},\qquad
\mathcal A\equiv\mathcal UT_8\he{\mathcal U}.
\end{equation}
These embody the existence of two order parameters, transforming in the antisymmetric rank-2 tensor representation and the adjoint representation, respectively. In the context of HDET, they were introduced in Eq.~\eqref{eq:parametrization}, see also Eq.~\eqref{adjoint} for the analogous construction within the toy model of Section~\ref{Sec:toymodel}. Under the action of $g$, these fields transform homogeneously as $\mathcal T\xrightarrow{g}g\mathcal Tg^T$ and $\mathcal A\xrightarrow{g}g\mathcal A\he g$.
As long as we are interested only in the dispersion relations of the charged NG modes, we can use any of the fields $\mathcal T,\mathcal A$ to construct the effective Lagrangian. (They lead to kinetic terms which are equivalent up to a redefinition of the parameters of the Lagrangian.) We choose $\mathcal A$ without loss of generality, and write down the most general $\gr{SU(3)\times U(1)}$-invariant Euclidean effective Lagrangian up to second order in the fields,
\begin{equation}
\mathcal L_{\text{eff},0}=\frac{4f_\pi^2}{3}\tr\left(\partial_\tau\mathcal A\partial_\tau\mathcal A+v_\pi^2\nabla\mathcal A\cdot\nabla\mathcal A\right),
\end{equation}
where the prefactor has been chosen just for convenience. Since there is an unbroken $\gr{U(1)}_{\tilde B}$ symmetry, we should allow for the possibility that it is endowed with a chemical potential. This is done by the replacement
\begin{equation}
\partial_\tau\mathcal A\to\partial_\tau\mathcal A-\mu_{\tilde{B}}[\tilde B,\mathcal A].
\end{equation}
Finally, the background $n_8$ charge density breaks the global color symmetry ``explicitly''. Since the charge density transforms in the adjoint representation, we have to add a term
\begin{equation}
\delta{\mathcal L}_{\text{eff}}=-\frac{8}{3}M_{\pi}^2f_{\pi}^2\tr\left(\mathcal AT_8\right),
\end{equation}
where the parameter $M_\pi^2$ measures the amount of explicit symmetry breaking, very much like the pion mass in the chiral perturbation theory of QCD. Expanding the action up to second order in $\vek{\pi}$ and discarding irrelevant terms, we arrive at the lowest-order action for the NG modes in the $(\pi_4,\pi_5)$ sector,
\begin{align}
\notag
\mathcal L_{\text{eff}}=&\tr\Bigl[\bigl(\partial_\tau\vek{\pi}-\mu_{\tilde{B}}[\tilde{B},\vek{\pi}]\bigr)^2+v_\pi^2(\nabla\vek{\pi})^2+M_\pi^2\vek{\pi^2}\Bigr]\\
\notag
=&[(\partial_\tau+\mu_{\tilde B})\pi_-][(\partial_\tau-\mu_{\tilde B})\pi_+]+v_\pi^2\nabla\pi_-\cdot\nabla\pi_+\\
&+M_\pi^2\pi_-\pi_+,
\label{eq:lag}
\end{align}
where now $\vek{\pi}\equiv \pi_4T_4+\pi_5 T_5$. The low-energy couplings $\mu_{\tilde{B}}$, $v_\pi^2$, and $M_\pi^2$ have to be computed in some microscopic model, as we have done above using HDET. Let us emphasize that $\mu_{\tilde B}$ is to be interpreted as an \emph{effective} chemical potential; it is not equal to the chemical potential of the blue quarks even though $\tilde B$ represents the operator of blue quark number up to an overall factor.
The Lagrangian~\eqref{eq:lag} implies the dispersion relations for the $\pi_\pm$ modes,
\begin{equation}
\omega_{\pi_\pm}=\sqrt{v_\pi^2\vek P^2+M_\pi^2}\mp\mu_{\tilde B}.
\end{equation}
Concretely, in the case (ii) where the fluctuations of color chemical potentials are ignored, we obtained $\mu_{\tilde{B}}=-\Sqrt{3}\mu_8/2$, $v_\pi^2=1/3$, and $M_\pi^2=0$, see Eq.~\eqref{caseii}. The instability of the $\pi_+$ mode, akin to the instability revealed in Section~\ref{Subsec:instability}, is now manifest.
The cases (i) and (iii) can be treated together; the expressions valid for the former can be obtained by setting $\mu_8=0$ in those for the latter. We obtained $\mu_{\tilde{B}}=\Sqrt3n_8(\Delta,\hat{\mu})/(4f_\pi^2)=\Sqrt3n_8(\Delta,\hat{\mu})/(2N_{\rm f }N_0)$, $v_\pi^2=1/3$, and $M_\pi=|\mu_{\tilde{B}}|$, see Eqs.~\eqref{casei} and \eqref{caseiii}. In the case (i) where color neutrality is not imposed, both $\mu_{\tilde B}$ and $M_\pi$ are nonzero and we find one type-II NG mode and one massive mode with the gap $2M_\pi$. On the other hand, in the case (iii) both parameters are zero and we find two type-I NG bosons with the phase velocity $v_\pi$.
On the general note, the above result can be interpreted as the background charge density in the fermion sector acting as a chemical potential for the modified baryon number. This provides some intuitive understanding of how the propagation of the NG modes is affected by the background charge. Putting together the collective modes in all (charged as well as neutral) sectors, the conclusion for the two physical cases (i) and (iii) is as follows. For $n_8\ne0$, there are three NG modes, two of which are type-II with $\tilde{B}=1$ and one type-I with $\tilde{B}=0$. When $n_8=0$, two gapped modes with $\tilde B=-1$, the antiparticles of the $\tilde B=1$ NG bosons, become gapless, and five type-I NG modes are correctly recovered.
\section{Conclusions}
\label{Sec:conclusions}
In this paper we analyzed in detail the issue of color neutrality in the 2SC phase of dense quark matter with particular attention to the spectrum of fluctuations of the order parameter. We showed explicitly that once color neutrality is imposed properly, the spurious instability as well as explicit breaking of the global color symmetry reported previously in literature disappear. To avoid confusion, we first addressed the problem in Section~\ref{Sec:NGmasses} using the same setting as in the previous publications. Only then, in Section~\ref{Sec:NGdispersions}, we provided a model-independent approach to the dispersion relations of the NG modes. While we used the 2SC phase as a specific example, it is obvious that our conclusions are valid more generally. Apart from other color-superconducting phases, they apply equally well to all systems with a set of conserved charges, some of them being demanded to be zero in the equilibrium.
It should be pointed out that the problem investigated in this paper is to some extent artificial. In the full QCD, color neutrality is automatically guaranteed by the gauge dynamics. Moreover, the collective modes discussed here are all colored and thus do not correspond to physical states in the spectrum; they will eventually be absorbed by the gluons by means of the Higgs--Anderson mechanism. It was shown that the collective excitations, the NG modes, play an important role in establishing gauge invariance of the Meissner effect~\cite{Nambu:1960tm}. We thus believe that our work provides a first step towards the proof of gauge invariance of the color Meissner effect in color superconductors, which has not been demonstrated satisfactorily so far. We leave a deeper consideration of this issue to future work.
\begin{acknowledgments}
Our current understanding of the problem, as presented in this paper, has been shaped by discussions with numerous colleagues. We would like to express our gratitude to those who influenced it most strongly, namely J.~O.~Andersen, M.~Buballa, X.-g.~Huang, K.~Fukushima, D.~H.~Rischke, and especially I.~A.~Shovkovy. The research of T.B.~was supported by the Sofja Kovalevskaja program of the Alexander von Humboldt Foundation.
\end{acknowledgments}
|
{
"timestamp": "2012-03-09T02:01:24",
"yymm": "1203",
"arxiv_id": "1203.1705",
"language": "en",
"url": "https://arxiv.org/abs/1203.1705"
}
|
{
"timestamp": "2012-03-12T01:02:42",
"yymm": "1203",
"arxiv_id": "1203.2168",
"language": "en",
"url": "https://arxiv.org/abs/1203.2168"
}
|
|
\section{Introduction}
Fix integers $L\ge 2$ and $N\ge 1$.
We consider following time-dependent Schr\"odinger system
\begin{equation}\label{eq Sch Intro}
\kappa \frac{\partial}{\partial z_i}\Psi({\bf q,z})=H_i\left({\bf q, \frac{\partial}{\partial q},z}\right)\Psi({\bf q,z})\quad
(1\le i\le N)
\end{equation}
where $\kappa\in \mathbb{C}$ and $\Psi({\bf q}, {\bf z})$ is an unknown function of
\begin{equation*}
{\bf q}=\left(q_1^{(1)},\ldots, q_{L-1}^{(1)}, q_1^{(2)},\ldots, q_{L-1}^{(2)},\ldots, q_1^{(N)},\ldots, q_{L-1}^{(N)}\right)
\end{equation*}
and ${\bf z}=(z_1,\ldots,z_N)$. The Hamiltonians $H_i$ are defined in Definition \ref{def Hamiltonian}.
The Schr\"odinger system \eqref{eq Sch Intro} is a quantization of the classical Hamiltonian system $\mathcal{H}_{L,N}$ obtained from a similarity reduction of the Drinfeld-Sokolov hierarchy by
K.~Fuji and T.~Suzuki ($L=3, N=1$) \cite{FS}, T.~Suzuki ($L\ge 2, N=1$) \cite{S1}, and
a similarity reduction of the UC hierarchy by
T.~Tsuda
($L\ge 2, N\ge 1$) \cite{T2}, independently. In \cite{T2},
T.~Tsuda
showed that the classical Hamiltonian system $\mathcal{H}_{L,N}$ is
equivalent to a Schlesinger system governing
isomonodromic deformation for a certain Fuchsian system.
On the other hand,
Y.~Yamada conjectured in the context of the so-called AGT relation that
the instanton partition function, in the presence of the full surface operator in $\mathcal{N} = 2$ $SU(L)$ gauge theory, is determined by
the Schr\"odinger system \eqref{eq Sch Intro} for $N=1$ \cite{Y}.
In the case of $L=2$, the Schr\"odinger system \eqref{eq Sch Intro} is a quantization of the Garnier systems \cite{Garnier},
\cite{KO}, which has been appeared in the conformal field theory \cite{Teschner}.
In this paper, we present a family of hypergeometric integrals as particular solutions
to the Sch\"rodinger system \eqref{eq Sch Intro}. These solutions are polynomials in ${\bf q}$
with the degree $M\in {\mathbb Z}_{\ge 1}$ and the coefficients are
integral representations of hypergeometric type.
A key to find special solutions to quantum isomonodromic systems is to observe
special solutions to the corresponding classical isomonodromic systems. For example,
both the classical and quantum sixth Painlev\'e equation has
a particular solution expressed
in terms of
the Gauss hypergeometric function \cite{N HGS}.
It is known that
the classical Hamiltonian system $\mathcal{H}_{L,N}$ has a particular solution
expressed in terms of a generalization of the Gauss hypergeometric function by
T.~Suzuki ($L\ge 2, N=1$) \cite{S2}, T.~Tsuda
($L\ge 2, N\ge 1$)
\cite{T1}.
Observing the linear Pfaffian system derived from this generalization of the Gauss hypergeometric function, we see
indeed that hypergeometric integrals given in \cite{T1} yield a particular
solution to the Schr\"odinger system \eqref{eq Sch Intro}:
\begin{thm}
The integral formula
\begin{equation}\label{eq IF Intro M=1}
\int_\Delta\prod_{n=1}^{L-1}t_n^{\alpha_n/\kappa}
\prod_{i=1}^N\left(
1-z_it_{L-1}
\right)^{-\beta_i/\kappa}\prod_{n=1}^{L-1}\left(
t_{n-1}-t_n
\right)^{-\gamma_n/\kappa}
\left(\varphi_0(t)-\sum_{i=1}^N\sum_{n=1}^{L-1}\varphi_n^{(i)}(t)q_n^{(i)}\right),
\end{equation}
which is a polynomial in ${\bf q}$ with the degree $1$, is a particular solution to the
Schr\"odinger system \eqref{eq Sch Intro}.
Here $\Delta$ is a twist cycle and $\varphi_0(t)$, $\varphi_n^{(i)}(t)$ are certain
rational ($L-1$)-forms defined in \eqref{eq basis phi}.
\end{thm}
(see Theorem \ref{thm M=1})
\medskip
In order to generalize the integral formula as particular solution to the case of
polynomials in ${\bf q}$ with the degree $M\in{\mathbb Z}_{\ge 2}$, let us recall equivalence between
the Knizhnik-Zamolodchikov equation of
the
conformal field theory and a quantization of a Schlesinger system \cite{Har}, \cite{R}.
The KZ equations for the simple Lie algebra $\operatorname{\mathfrak{g}}$, have integral representations as
solutions taking values in tensor products of Verma modules of $\operatorname{\mathfrak{g}}$ (see, for example, \cite{ATY},
\cite{SV}).
From the point of view that the integral formula \eqref{eq IF Intro M=1} may be a solution to
the Knizhnik-Zamolodchikov equation, it should be viewed that the integral variables are corresponding to the simple roots of
$\frak{sl}_L$. While, for the case of $L=2$ and $N=1$, it is known that the Schr\"odinger system \eqref{eq Sch Intro}, the quantum sixth
Painlev\'e equation, has hypergeometric solutions \cite{N HGS}:
\begin{equation*}
\int_\Delta \prod_{1\le a<b\le M}(t^{(a)}-t^{(b)})^{2/\kappa}\prod_{a=1}^M(t^{(a)})^{\alpha/\kappa}(1-zt^{(a)})^{-\beta/\kappa}
(1-t^{(a)})^{-\gamma/\kappa}\left(
\varphi_0(t^{(a)})-\varphi_1^{(1)}(t^{(a)})q_1^{(1)}
\right).
\end{equation*}
Note that the integrand above consists of $M$-copies of the integrand of \eqref{eq IF Intro M=1} multiplied by the
coupled term $\prod_{1\le a<b\le M}(t^{(a)}-t^{(b)})^{2/\kappa}$.
Considering upon these, we arrive at
\begin{thm}
The integral formula
\begin{align*}
\int_\Delta& \prod_{1\le a<b\le M, \atop 1\le n\le L-1}\left(t_n^{(a)}-t_n^{(b)}\right)^{2/\kappa}
\prod_{ 1\le a, b\le M,\atop 1\le n\le L-2 }
\left(t_n^{(a)}-t_{n+1}^{(b)}\right)^{-1/\kappa}
\\
&\times \prod_{a=1}^M\left\{\prod_{n=1}^{L-1}\left(
t_n^{(a)}
\right)^{\alpha_n/\kappa}
\prod_{i=1}^N\left(1-z_it^{(a)}_{L-1}\right)^{-\beta_i/\kappa}\left(
1-t_1^{(a)}
\right)^{-\gamma/\kappa}
\left(\varphi_0(t^{(a)})-\sum_{i=1}^N\sum_{n=1}^{L-1}\varphi_n^{(i)}(t^{(a)})q_n^{(i)}\right)
\right\},
\end{align*}
which is a polynomial in ${\bf q}$ with the degree $M$, is a particular solution to the Schr\"odinger system
\eqref{eq Sch Intro}. Here $\Delta$ is a skew-symmetric twist cycle.
\end{thm}
(see Theorem \ref{thm IF})
\medskip
The remainder of this paper is organized as follows.
In section 2, we introduce quantizations of the classical Hamiltonians of
$\mathcal{H}_{L,N}$ and show that those quantum Hamiltonians are mutually
commutative. In section 3, we introduce our Schr\"odinger systems and discuss properties of
them. In section 4,
we give integral formulas for solutions.
\begin{re} As mentioned above,
the classical Hamiltonian system $\mathcal{H}_{L,N}$ describes isomonodromic
deformation for an $L\times L$ Fuchsian system
\begin{equation*}
\frac{\partial}{\partial u}\Phi(u) =\sum_{i=0}^{N+1}\frac{A_i}{u-u_i}\Phi(u),
\end{equation*}
where $u_0=1$, $u_i=1/z_i$ ($1\le i\le N$), and $u_{N+1}=0$,
whose spectral type is given by the ($N+3$)-tuple
\begin{equation*}
(1,1,\ldots,1),(1,1,\ldots,1),(L-1,1),\ldots,(L-1,1)
\end{equation*}
of partitions of $L$. A spectral type defines multiplicities of the eigenvalues of
each residue matrix $A_i$.
Consequently, $L-1$ parameters are associated with
singular points $0$ and $\infty$, and one parameter is associated with
each singular point $u_i$ for $i=0,\ldots, N$. Notice that in the integrand given in the Theorems above, $L-1$ parameters are associated with
the singular point $0$, and one parameter is associated with each singular point
$1$, $1/z_i$ ($1\le i\le N$).
\end{re}
\section{Hamiltonian}
Let us define a non-commutative associative algebra $W_{L,N}$ over $\mathbb{C}$ with generators
\begin{align*}
&q_m^{(i)}, p_m^{(i)} \quad (1\le m\le L-1, \ 1\le i\le N),
\\
&e_n, \kappa_n, \theta_j, \hbar \quad (0\le n\le L-1, \ 0\le j\le N)
\end{align*}
and commutation relations
\begin{equation}\label{eq com rel}
\left[
p_m^{(j)}, q_n^{(i)}
\right]=\delta_{n,m}\delta_{i,j}\hbar\quad (1\le n, m\le L-1,\ 1\le i, j\le N),
\end{equation}
where $\delta_{i,j}$ is Kronecker's delta,
and the other commutation relations are zero, and relations
\begin{equation*}
\sum_{m=0}^{L-1}e_{m}=\frac{L-1}{2},\quad \sum_{m=0}^{L-1}\kappa_{m}=\sum_{i=0}^{N}\theta_{i}.
\end{equation*}
The non-commutative associative algebra $W_{L,N}$ is an Ore domain, so that
we can define its skew field $\mathcal{K}_{L,N}$ (see, for example, \cite{bjork}, Chapter 1, Section 8).
\medskip
\begin{dfn}\label{def Hamiltonian}
\begin{em}
We introduce Hamiltonians $H_i$ ($i=1,\ldots, N$) in the rational function field $W_{L,N}(z_{1},\ldots, z_{N})$
in variables $z_{1}$, \ldots, $z_{N}$
by
\begin{align}
z_iH_i
=&\sum_{n=0}^{L-1}e_nq_n^{(i)}p_n^{(i)}
+\sum_{j=0}^N\sum_{0\le m<n\le L-1}q_m^{(i)}p_m^{(j)}q_n^{(j)}p_n^{(i)}
+\frac{1}{z_i-1}\sum_{m,n=0}^{L-1}q_m^{(i)}p_m^{(0)}q_n^{(0)}p_n^{(i)} \nonumber
\\
&+\sum_{j=1\atop j\neq i}^N\frac{z_j}{z_i-z_j}\sum_{m,n=0}^{L-1}
q_m^{(i)}p_n^{(i)}q_n^{(j)}p_m^{(j)}
+\theta_i\left(
e_0+\kappa_0-\sum_{j=1}^N\theta_j-\sum_{j=1,\atop j\neq i}^N\frac{\theta_j z_j}{z_i-z_j}
\right)
, \label{eq def Hamiltonian}
\end{align}
where
\begin{align*}
&q_0^{(i)}=\theta_i+\sum_{m=1}^{L-1}q_m^{(i)}p_m^{(i)},\quad p_0^{(i)}=-1 \quad (1\le i\le N),
\\
&q_m^{(0)}=-1,\quad p_m^{(0)}=\kappa_m+\sum_{i=1}^Nq_m^{(i)}p_m^{(i)},\quad (1\le m\le L-1),
\\
& q_0^{(0)}=\kappa_{0}-\sum_{i=1}^{N}\theta_{i}-\sum_{i=1}^{N}\sum_{m=1}^{L-1}q_m^{(i)}p_m^{(i)},\quad p_0^{(0)}=-1.
\end{align*}
\end{em}
\end{dfn}
The Hamiltonians $H_i$ ($i=1,\ldots, N$) are canonical quantization of the polynomial Hamiltonians in \cite{T2}, Appendix A. What we mean by canonical quantization
is, to replace the Poisson bracket with the commutator.
Since the canonical variables in the classical Hamiltonians are not
separated,
quantization of the Hamiltonians is not
unique. In the following, we show that the Hamiltonians $H_i$ are mutually commutative and the Schr\"odinger equations
associated with the Hamiltonians $H_i$ have integral formulas.
\begin{exmp}
We give an example of the Hamiltonians $H_{i}$ in the case of $L=2$.
Set $\left(q_{i}, p_{i}\right)=\left(q_{1}^{(i)}, p_{1}^{(i)}\right)$.
The Hamiltonian $H_{i}$ is expressed as follows:
\begin{align*}
z_{i}(z_{i}-1)H_{i}=&q_{i}\left(
\kappa_{1}-\theta_{0}+\sum_{j=1}^{N}q_{j}p_{j}
\right)\left(
\kappa_{1}+\sum_{j=1}^{N}q_{j}p_{j}
\right)+z_{i}\left(
\theta_{i}+q_{i}p_{i}
\right)p_{i}
\\
&-\sum_{j=1,\atop j\neq i}^{N}\frac{z_{j}}{z_{i}-z_{j}}\left(
\theta_{j}+q_{j}p_{j}
\right)q_{i}p_{j}
-\sum_{j=1,\atop j\neq i}^{N}\frac{z_{i}}{z_{i}-z_{j}}\left(
\theta_{i}+q_{i}p_{i}
\right)q_{j}p_{i}
\\
&-\sum_{j=1,\atop j\neq i}^{N}\frac{z_{i}(z_{j}-1)}{z_{j}-z_{i}}\left(
\theta_{i}+q_{i}p_{i}
\right)q_{j}p_{j}
-\sum_{j=1,\atop j\neq i}^{N}\frac{z_{i}(z_{j}-1)}{z_{j}-z_{i}}\left(
\theta_{j}+q_{j}p_{j}
\right)q_{i}p_{i}
\\
&-(z_{i}+1)\left(
\theta_{i}+q_{i}p_{i}
\right)q_{i}p_{i}-
\left(
(e_{1}-e_{0})z_{i}+e_{0}-e_{1}-\hbar+\kappa_{1}-\kappa_{0}
\right)q_{i}p_{i}
\end{align*}
plus some function in only $(z_{1},\ldots, z_{N})$. These Hamiltonians are quantizations
of the polynomial Hamiltonians for the Garnier system \cite{KO}.
\end{exmp}
\begin{exmp}\label{ex FS}
We give an example of the Hamiltonians $H_1$ in the case of $N=1$.
Set $\left(q_{m}, p_{m}\right)=\left(q_{m}^{(1)}, p_{m}^{(1)}\right)$, $H=H_1$, and $z=z_1$.
The Hamiltonian $H$ is written in a {\it coupled} form as follows:
\begin{align*}
z(z-1)H=&\sum_{m=1}^{L-1}H_{\mathrm{VI}}\left(
\sum_{n=0}^{L-1}\alpha_{2n+1}-\alpha_{2m-1}-\eta, \sum_{n=0}^{m-1}\alpha_{2n}, \sum_{n=m}^{L-1}\alpha_{2n},
\alpha_{2n-1}\eta; q_m, p_m
\right)
\\
&+\frac{1}{4}\sum_{1\le m< n\le L-1}\left(
\left(
\left(
q_m-1
\right)p_mq_m+q_mp_m\left( q_m-1\right)+2\alpha_{2m-1}\left(q_m-1\right)
\right)
\left(
p_n\left(
q_n-z
\right)+\left(
q_n-z
\right)p_n
\right)\right.
\\
&+\left.
\left(
\left(
q_n-z
\right)p_nq_n+q_np_n\left(
q_n-z
\right)
+2\alpha_{2n-1}\left(q_n-z\right)
\right)
\left(
p_m\left(
q_m-1
\right)
+\left(
q_m-1
\right)p_m
\right)
\right)
\end{align*}
plus some function in only $(z_{1},\ldots, z_{N})$,
where
\begin{align*}
H_{\mathrm{VI}}\left(
a_0, a_1, a_z, a; q,p
\right)=&
\frac{1}{6}\left(qp(q-1)p(q-z)
+(q-1)p(q-z)pq
+(q-z)pqp(q-1)+\right.
\\
&\left.+(q-z)p(q-1)pq
+(q-1)pqp(q-z)
+qp(q-z)p(q-1)\right)\nonumber
\\
&-\frac{1}{2}\left(a_0((q-1)p(q-z)
+(q-z)p(q-1))
+a_1(qp(q-z)+(q-z)pq)
\right.\nonumber
\\
&\left.+
(a_z-1)(qp(q-1)
+(q-1)pq)
\right)+a q.
\nonumber
\end{align*}
Here, we let
\begin{align*}
&\alpha_{2m-1}=\kappa_n-\hbar\quad (1\le m\le L-1), \quad
\alpha_{2m}=e_{m}-e_{m+1}-\kappa_m+\hbar \quad (1\le m\le L-2),
\\
&\alpha_0=e_0-e_1,\quad \alpha_{2L-1}=-\kappa_0+(L-2)\hbar,\quad \sum_{m=0}^{2L-1}\alpha_m=\kappa,
\quad \eta=-\kappa_0+\theta_1-\frac{L-2}{2}\hbar.
\end{align*}
The Hamiltonian $H$ is a quantization of the Hamiltonian obtained by Fuji-Suzuki ($L=3$) \cite{FS} and
Suzuki ($L\ge 3$) \cite{S1}. The Hamiltonian $H_{\mathrm{VI}}$ is of the sixth quantum Painlev\'e equation with the affine Weyl group symmetry of type $D_4^{(1)}$
introduced in \cite{N quantum PVI}.
\end{exmp}
\subsection{Commutativity}
The Hamiltonians $H_{i}$ ($i=1,\ldots, N$) are expressed in the following forms
\begin{equation*}
-z_{i}^{2}H_{i}=\sum_{j=0,\atop j\neq i}^{N+1}\frac{\Omega_{i,j}}{u_{i}-u_{j}},
\end{equation*}
where $\Omega_{i,j}$ are elements in $W_{L,N}$ and $u_0=1$, $u_{i}=1/z_{i}$ ($i= 1,\ldots, n$) and $u_{N+1}=0$.
For $i, j=1,\ldots, N$, the forms $\Omega_{i,j}$ read as
\begin{equation}\label{eq Omega}
\Omega_{i,j}=\frac{1}{2}\operatorname{tr} \left(
\widehat{A}^{(i)}\widehat{A}^{(j)}
\right),
\end{equation}
where $\widehat{A}^{(i)}$ is a $L\times L$ matrix defined as
\begin{equation}\label{eq A qp}
\left(\widehat{A}^{(i)}\right)_{m,n}=q_{m}^{(i)}p_{n}^{(i)}
\end{equation}
for $m,n=0,1,\ldots, L-1$, where $\left(\widehat{A}^{(i)}\right)_{m,n}$ is the ($m,n$) entry of the matrix $\widehat{A}^{(i)}$.
The entries of $\widehat{A}^{(i)}$ satisfy the following commutation relations.
\begin{lem}\label{lem A com}
It holds that
\begin{equation}\label{eq A com}
\frac{1}{\hbar}
\left[\left(\widehat{A}^{i}\right)_{m,n}, \left(\widehat{A}^{j}\right)_{m',n'}\right]=\delta_{i,j}\left(
\delta_{n,m'}\left(\widehat{A}^{i}\right)_{m,n'}-\delta_{n',m}\left(\widehat{A}^{i}\right)_{m',n}\right)
\end{equation}
for $0\le m, n, m', n'\le L-1$ and $1\le i,j\le N$.
\end{lem}
A proof is given by a straightforward calculation.
Recall the definition of the Gaudin Hamiltonians (see, for example, \cite{Har}, Section 2).
The Gaudin Hamiltonians $G_i$ ($i=1,\ldots, N$) for $\frak{gl}_L$ are defined as
\begin{equation*}
G_i=\frac{1}{2}\sum_{j=1,\atop j\neq i}^{N}\frac{\operatorname{tr} \left(
{B}^{(i)}{B}^{(j)}
\right)}{u_{i}-u_{j}},
\end{equation*}
where $B^{(i)}$ ($i=1,\ldots, N$) are $L\times L$ matrices whose entries satisfy the commutation relations \eqref{eq A com}.
Since the commutativity of Gaudin Hamiltonians equals the so-called {\it infinitesimal braid relations},
Lemma \ref{lem A com} yields:
\begin{align}
&\left[\Omega_{i,j}, \Omega_{k,l} \right]=0\quad (i,j,k,l \ \text{are distinct}),
\\
&\left[\Omega_{i,j}, \Omega_{i,k}+\Omega_{k,j}\right]=0\quad (i,j,k \ \text{are distinct})
\end{align}
for $i, j,k,l=1,\ldots, N$.
The other elements $\Omega_{i, 0}$ and $\Omega_{i,N+1}$ ($i=1,\ldots, N$) are not expressed in a similar way as
\eqref{eq Omega} and \eqref{eq A qp}. However, we can check
by a straightforward calculation
that the {\it infinitesimal braid relations} above hold even if $i,j,k,l=0,1,\ldots, N+1$.
Therefore, we have
\begin{prop}\label{prop commutativity}
Hamiltonians $H_{i}$ ($i=1,\ldots, N$) are mutually commutative.
\end{prop}
\section{Schr\"odinger system}
Denote by
\begin{equation}\label{eq hamiltonian sch}
H_{i}\left(
{\bf q, \frac{\partial}{\partial q},z}
\right)
\end{equation}
for $i=1,\ldots, N$, the Hamiltonians obtained by substituting $q_m^{(i)}$ and $\partial /\partial q_m^{(i)}$ into
$q_m^{(i)}$ and $p_m^{(i)}$, resepectively,
of the Hamiltonians $H_i$ defined in Definition \ref{def Hamiltonian}.
We consider the following Schr\"odinger system:
\begin{equation}\label{eq Schrodinger}
\kappa \frac{\partial }{\partial z_{i}}\Psi({\bf q}, {\bf z})=H_{i}\left(
{\bf q}, \frac{\partial}{\partial {\bf q}}, {\bf z}
\right)\Psi({\bf q}, {\bf z})\qua
\end{equation}
where $\kappa\in \mathbb{C}$, $\Psi({\bf q}, {\bf z})$ is an unknown function of
\begin{equation*}
{\bf q}=\left(q_1^{(1)},\ldots, q_{L-1}^{(1)}, q_1^{(2)},\ldots, q_{L-1}^{(2)},\ldots, q_1^{(N)},\ldots, q_{L-1}^{(N)}\right)
\end{equation*}
and ${\bf z}=(z_1,\ldots,z_N)$. Here, we regard $e_n$, $\kappa_n$, $\theta_i$ as complex parameters.
\begin{prop}
The Schr\"odinger system \eqref{eq Schrodinger} is completely integrable in the sense of Frobenius, that is,
it holds
\begin{equation*}
\left[
\kappa \frac{\partial }{\partial z_{i}}-H_{i}\left(
{\bf q}, \frac{\partial}{\partial {\bf q}}, {\bf z}
\right),
\kappa \frac{\partial }{\partial z_{j}}-H_{j}\left(
{\bf q}, \frac{\partial}{\partial {\bf q}}, {\bf z}
\right)
\right]=0,
\end{equation*}
for $i,j=1,\ldots, N$.
\end{prop}
\begin{proof}
Thanks to Proposition \ref{prop commutativity}, we have only to show
\begin{equation*}
\frac{\partial }{\partial z_{i}}H_{j}\left(
{\bf q}, \frac{\partial}{\partial {\bf q}}, {\bf z}
\right)=\frac{\partial }{\partial z_{j}}H_{i}\left(
{\bf q}, \frac{\partial}{\partial {\bf q}}, {\bf z}
\right).
\end{equation*}
It is easily calculated as follows. For $i\neq j$, we have
\begin{align*}
\frac{\partial }{\partial z_{i}}H_{j}\left(
{\bf q}, \frac{\partial}{\partial {\bf q}}, {\bf z}
\right)=&\frac{\partial }{\partial z_{i}}\left(
\frac{z_i}{z_j(z_j-z_i)}\left(\sum_{m,n=0}^{L-1}
q_m^{(j)}p_n^{(j)}q_n^{(i)}p_m^{(i)}-\theta_i\theta_j\right)
\right)
\\
=&\frac{1}{(z_i-z_j)^2}\left(\sum_{m,n=0}^{L-1}
q_m^{(j)}p_n^{(j)}q_n^{(i)}p_m^{(i)}-\theta_i\theta_j\right).
\end{align*}
From Lemma \ref{lem A com}, the last line is symmetrical with respect to $i$ and $j$.
Thus, we finished the proof.
\end{proof}
In the most simplest case, namely, the case of $L=2$ and $N=1$, the Schr\"odinger system \eqref{eq Schrodinger}
is the quantum sixth Painlev\'e equation. In the previous work \cite{N HGS}, we showed that the quantum
sixth Painlev\'e equation has polynomial solutions in terms of $q$.
In the general case, the Schr\"odinger system \eqref{eq Schrodinger} also has polynomial solutions
in terms of ${\bf q}$ due to the following proposition.
For a $L-1\times N$ matrix $A$ whose entries are non-negative integers,
let $q^A$ be the monomial defined by
\begin{equation*}
q^A=\prod_{m=1}^{L-1}\prod_{i=1}^N \left(
q_m^{(i)}
\right)^{A_{m,i}},
\end{equation*}
where $A_{m,i}$ is the ($m,i$) entry of the matrix $A$. Set $d(A)=\sum_{m=1}^{L-1}\sum_{i=1}^N A_{m,i}$.
Let $V(M)$ ($M\in {\mathbb Z}_{\ge 0}$) be the subspace of the polynomial ring $\mathbb{C}[{\bf q}]$ such that
the degree of elements in $V(M)$ is less than $M$, namely, $V(M)=\bigoplus_{A}\mathbb{C} q^A$, where
the summation is taken over all $L-1\times N$ matrices $A$ such that $d(A)\le M$.
\begin{prop}\label{prop action H}
For each $i=1,\ldots, N$,
the Hamiltonian $H_i({\bf q}, \partial /\partial {\bf q}, {\bf z})$ acts on $V(M)$ if
$\kappa_0-\sum_{i=1}^N\theta_i=M$.
\end{prop}
\begin{proof}
We compute the action of the Hamiltonian $H_i({\bf q}, \partial /\partial {\bf q}, {\bf z})$ on $q^A$ such that $d(A)=M$
as follows.
\begin{align}
z_i(z_i-1)H_{i}\left(
{\bf q}, \frac{\partial}{\partial {\bf q}}, {\bf z}
\right) q^A=&-\sum_{n=1}^{L-1}q_n^{(i)}q_0^{(0)}p_n^{(0)} q^A+f({\bf q})
\nonumber
\\
=&-\sum_{n=1}^{L-1}\left(
\kappa_0-\sum_{i=1}^N\theta_i-M
\right)\left(
\kappa_n+\sum_{i=1}^N A_{n,i}
\right)q_n^{(i)}q^A+f({\bf q}). \label{eq action H}
\end{align}
Here $f({\bf q})$ is a polynomial whose degree is equal to or less than $M$.
Hence,
if $\kappa_0-\sum_{i=1}^N\theta_i=M$, then the first term of \eqref{eq action H} vanishes, which finishes the proof.
\end{proof}
By virtue of Proposition \ref{prop action H}, for the Schr\"odinger equation \eqref{eq Schrodinger},
we can consider polynomial solutions
\begin{equation*}
\Psi({\bf q}, {\bf z})=\sum_{A\in\mathcal{A}_M}c_A({\bf z}) q^A,
\end{equation*}
where
\begin{equation}\label{def matrix A}
\mathcal{A}_M=\left\{
A=\left(A_{m,i}\right)\big| A_{m,i}\in\mathbb{Z}_{\ge 0},\ d(A)\le M
\right\}
\end{equation}
and $c_A({\bf z})$ is a function of ${\bf z}$.
In the next section, we present integral formulas taking values in $V(M)$ and show that
they are solutions to the Schr\"odinger system \eqref{eq Schrodinger}.
The Hamiltonians $H_i$ act on another subspaces of the polynomial ring $\mathbb{C}[{\bf q}]$.
Let $F( T_{1},\ldots,T_{L-1})$ ($T_{1},\ldots, T_{L-1} \in{\mathbb Z}_{\ge 0}$) be the subspace
of the polynomial ring $\mathbb{C}[{\bf q}]$
defined as
$
F(T_1,\ldots, T_{L-1})=\bigoplus_{A}\mathbb{C} q^A,
$
where the summation is taken over all $L-1\times N$ matrices $A$ such that the entries of $A$ are
non-negative integers and
$\sum_{i=1}^N A_{m,i}\le T_m$ ($m=1,\ldots, L-1$). Set $d_m(A)=\sum_{i=1}^N A_{m,i}$.
\begin{prop}
For each $i=1,\ldots, N$,
the Hamiltonian $H_i({\bf q}, \partial /\partial {\bf q}, {\bf z})$ acts on $F(T_1,\ldots, T_{L-1})$ if
$\kappa_m=-T_m$.
\end{prop}
\begin{proof}
Take a $L-1\times N$ matrix $A$ such that the entries of $A$ are
non-negative integers and $d_m(A)=T_m$ for any $m\in\{1,\ldots, L-1\}$.
The Hamiltonian $H_i({\bf q}, \partial /\partial {\bf q}, {\bf z})$ acts on $q^A$
as follows:
\begin{align}
z_i(z_i-1)H_{i}\left(
{\bf q}, \frac{\partial}{\partial {\bf q}}, {\bf z}
\right) q^A=&-\left(q_m^{(i)}q_0^{(0)}p_m^{(0)}
+\sum_{n=m+1}^{L-1}q_m^{(i)}p_m^{(0)}p_n^{(i)}
+\sum_{n=1,\atop n\neq m}^{L-1}q_m^{(i)}p_m^{(0)}p_n^{(i)}\right)q^A
+f({\bf q})
\nonumber
\\
=&-\left(
\kappa_m+T_m
\right)
\left\{
\kappa_0-\sum_{j=1}^N\theta_j-d(A)
+\sum_{n=m+1}^{L-1}p_n^{(i)}
+\sum_{n=1,\atop n\neq m}^{L-1}p_n^{(i)} \right\}q_m^{(i)}q^A
+f({\bf q}). \label{eq action H 2}
\end{align}
Here, $f({\bf q})$ is a polynomial such that as a polynomial in terms of $q_m^{(1)},\ldots, q_m^{(N)}$,
the degree of $f({\bf q})$ is equal to or less than $d_m(A)$. Thus, if $\kappa_m=-T_m$, then
the first term of \eqref{eq action H 2} vanishes, which finishes the proof.
\end{proof}
Consequently, we can also consider polynomial solutions taking values in $F(T_1,\ldots, T_{L-1})$.
\section{Integral formula}
In this section, we construct integral formulas for the Schr\"odinger systems \eqref{eq Schrodinger},
as particular solutions.
Recall that the Gauss hypergeometric function is a particular solution to both the classical and
quantum sixth Painlev\'e equation \cite{N HGS}.
Hypergeometric solutions to the classical Hamiltonian systems $\mathcal{H}_{L,N}$ were given by
T.~Suzuki ($L\ge 2$, $N=1$) \cite{S2} and T.~Tsuda ($L\ge 2$, $N\ge 1$) \cite{T1}
independently, under the condition $\kappa_0-\sum_{i=1}^N\theta_i=0$.
These hypergeometric solutions are the generalized hypergeometric functions
(Thomae's hypergeometric function) $_LF_{L-1}$ in the case of ($L\ge 2$, $N=1$) and
their generalizations in the case of ($L\ge 2$, $N\ge 1$).
We expect that these generalized hypergeometric functions are also solutions to a quantization
of the classical Hamiltonian systems $\mathcal{H}_{L,N}$,
the Schr\"odinger systems \eqref{eq Schrodinger}.
Indeed, this is true if we consider polynomial solutions to the Schr\"odinger systems \eqref{eq Schrodinger} with
$ \kappa_0-\sum_{i=1}^N\theta_i=1$.
Set $\kappa_0-\sum_{i=1}^N\theta_i=M\in\mathbb{Z}_{\ge 0}$.
We begin with the case $M=1$ and later we deal with general case.
\subsection{The case of $M=1$}
Consider a multivalued function
\begin{equation*}
U(t)=\prod_{n=1}^{L-1}t_n^{\alpha_n/\kappa}
\prod_{i=1}^N\left(
1-z_it_{L-1}
\right)^{-\beta_i/\kappa}\prod_{n=1}^{L-1}\left(
t_{n-1}-t_n
\right)^{-\gamma_n/\kappa}
\end{equation*}
with $t_0=1$ defined on the complement $T\in\mathbb{C}^{(L-1)}$ of singular locus $D$ given by
\begin{equation*}
D=
\bigcup_{ 1\le n\le L-1}\left\{
t_{n-1}=t_{n}
\right\}
\cup
\bigcup_{ 1\le n\le L-1}
\left\{
t_n= 0
\right\}
\cup
\bigcup_{ 1\le i\le N}\left\{
t_{L-1}= 1/z_i
\right\}
.
\end{equation*}
Let $\mathcal{S}$ be the rank one local system determined by $U(t)$ and
$\mathcal{S}^*$, the dual local system of $\mathcal{S}$.
The hypergeometric paring between the twisted homology group and twisted de Rham cohomology group
is
\begin{align*}
H_{L-1}(T, \mathcal{S}^*)\times H^{L-1}(T,\nabla)&\longrightarrow \mathbb{C}
\\
\left(
\Delta, \varphi
\right)&\longmapsto \int_\Delta U(t)\varphi,
\end{align*}
where $\varphi$ is a rational ($L-1$)-form holomorphic outside $D$ and $\nabla$
is the covariant differential operator given by
$\nabla=d+d\log(U(t))\wedge$.
According to
\cite{T2}, the following rational ($L-1$)-forms
\begin{equation}\label{eq basis phi}
\varphi_0(t)=\frac{dt_1\wedge\cdots\wedge dt_{L-1}}{t_{L-1}}
\prod_{n=1}^{L-1}\frac{1}{t_{n-1}-t_n},
\quad \varphi^{(i)}_n(t)=\frac{dt_1\wedge\cdots\wedge dt_{L-1}}{(1-z_it_{L-1})t_{L-1}}\prod_{m=1,\atop m\neq n}^{L-1}
\frac{1}{t_{m-1}-t_m}
\end{equation}
represent a basis of $H^{L-1}(T,\nabla)$.
Define the integral formula $\Psi_1({\bf q,z})$ by
\begin{equation*}
\Psi_1({\bf q,z})=\int_\Delta U(t)\left(\varphi_0(t)-\sum_{i=1}^N\sum_{n=1}^{L-1}\varphi_n^{(i)}(t)q_n^{(i)}\right)
\end{equation*}
with $\Delta\in H_{L-1}(T, \mathcal{S}^*)$. From Proposition \ref{prop action H}, when $\kappa_0-\sum_{i=1}^N\theta_i=1$, the action of Hamiltonian
$H_i$ ($i=1,\ldots, N$) on the integral formula,
$H_i\Psi_1({\bf q,z})$, is also
a polynomial of degree equal to or less than $1$ and then the constant term and the coefficient of $q_n^{(j)}$ ($1\le n\le L-1$, $1\le j\le N$) of $H_i\Psi_1({\bf q,z})$
are
linear combinations of $\int_\Delta U(t)\varphi_0(t)$ and $\int_\Delta U(t)\varphi_n^{(j)}(t)$ ($1\le n\le L-1$, $1\le j\le N$).
Remarkably they coincide with $\kappa \partial \varphi_0(t)/ \partial z_i$ and $\kappa \partial \varphi_n^{(j)}(t)/\partial z_i$
with appropriate correspondence between parameters.
Namely, we have
\begin{thm}\label{thm M=1}
If $\kappa_0-\sum_{i=1}^N\theta_i=1$, then the integral formula $\Psi_1({\bf q,z})$ is a solution to the
Schr\"odinger system \eqref{eq Schrodinger}, with
\begin{equation*}
\alpha_n=e_{n+1}-e_n+\kappa_{n+1},\quad \beta_i=-\theta_i,
\quad \gamma_n=\kappa_n,
\end{equation*}
for $1\le n\le L-1$ and $1\le i\le N$, where $e_L=e_0$ and $\kappa_{L}=1$.
\end{thm}
\subsection{The case of $M\ge 2$}
Fix $M\in{\mathbb Z}_{\ge 2}$.
We consider a multivalued function
\begin{align*}
U(t)=& \prod_{1\le a<b\le M, \atop 1\le n\le L-1}\left(t_n^{(a)}-t_n^{(b)}\right)^{2/\kappa}
\prod_{ 1\le a, b\le M,\atop 1\le n\le L-2 }
\left(t_n^{(a)}-t_{n+1}^{(b)}\right)^{-1/\kappa}
\\
&\times \prod_{a=1}^M\left\{\prod_{n=1}^{L-1}\left(
t_n^{(a)}
\right)^{\alpha_n/\kappa}
\prod_{i=1}^N\left(1-z_it^{(a)}_{L-1}\right)^{-\beta_i/\kappa}\left(
1-t_1^{(a)}
\right)^{-\gamma/\kappa}\right\}
\end{align*}
defined on the complement $T\in\mathbb{C}^{(L-1)M}$ of singular locus $D$ given by
\begin{equation*}
D=\bigcup_{1\le a<b\le M,\atop 1\le n\le L-1} \left\{t^{(a)}_n= t^{(b)}_n\right\}
\cup
\bigcup_{1\le a, b\le M,\atop 1\le n\le L-2}\left\{
t^{(a)}_n=t^{(b)}_{n+1}
\right\}
\cup
\bigcup_{1\le a\le M,\atop 1\le n\le L-1}
\left\{
t^{(a)}_n= 0
\right\}
\cup
\bigcup_{1\le a\le M,\atop 1\le i\le N}\left\{
t^{(a)}_{L-1}= 1/z_i
\right\}
\cup
\bigcup_{1\le a\le M}
\left\{
t^{(a)}_{1}= 1
\right\} .
\end{equation*}
Let $\mathcal{S}$ be the rank one local system determined by $U(t)$ and
$\mathcal{S}^*$, the dual local system of $\mathcal{S}$.
The hypergeometric paring between the twisted homology group and twisted de Rham cohomology group
is
\begin{align*}
H_{(L-1)M}(T, \mathcal{S}^*)\times H^{(L-1)M}(T,\nabla)&\longrightarrow \mathbb{C}
\\
\left(
\Delta, \varphi
\right)&\longmapsto \int_\Delta U(t)\varphi,
\end{align*}
where $\varphi$ is a rational $(L-1)M$-form holomorphic outside $D$ and $\nabla$
is the covariant differential operator given by
$\nabla=d+d\log(U(t))\wedge$.
Denote by $\frak{S}_M^{L-1}$, ($L-1$)-th products of the symmetric group with the degree $M$.
Let the action of $\frak{S}_M^{L-1}$ on a rational function $f(t)$ of variables $t=(t_1^{(1)},\ldots,t_{L-1}^{(1)},\ldots,
t_1^{(M)},\ldots,t_{L-1}^{(M)})$ be defined by
\begin{equation}\label{eq sigma}
\sigma(f(t))=f(t_1^{(\sigma_1(1))},\ldots,t_{L-1}^{(\sigma_{L-1}(1))},
\ldots, t_1^{(\sigma_1(M))},\ldots,t_{L-1}^{(\sigma_{L-1}(M))})
\end{equation}
for $\sigma=(\sigma_1,\ldots,\sigma_{L-1})\in\frak{S}_M^{L-1}$. Let $\mathrm{Sym}[f(t)]$ be the symmetrization
of $f(t)$, given by $\mathrm{Sym}[f(t)]=\sum_{\sigma\in\frak{S}_M^{L-1}}\sigma(f(t))$.
\begin{dfn}\label{def integral formula}
For $M\in\mathbb{Z}_{\ge 2}$,
we define an integral formula by
\begin{equation*}
\Psi_M\left(
{\bf q,z}
\right)=
\int_\Delta U(t)\cdot\mathrm{Sym}\left[
\prod_{a=1}^M
\left(f_0(t^{(a)})-\sum_{i=1}^N\sum_{n=1}^{L-1}
f_n^{(i)}(t^{(a)})q_n^{(i)}\right)\right]dt,
\end{equation*}
where $\Delta\in H_{(L-1)M}(T, \mathcal{S}^*)$ and
\begin{align*}
&f_0(t^{(a)})=\prod_{m=1}^{L-1}\frac{1}{t^{(a)}_{m-1}-t^{(a)}_m}, \quad f^{(i)}_n(t^{(a)})=\frac{1}{1-z_it^{(a)}_{L-1}}\prod_{m=1,\atop m\neq n}^{L-1}\frac{1}{t^{(a)}_{m-1}-t^{(a)}_m},
\quad t_0^{(a)}=1
\\
&dt=dt^{(1)}_1\wedge\cdots \wedge dt^{(1)}_{L-1}\wedge dt^{(2)}_{1}\wedge\cdots \wedge dt^{(2)}_{L-1}\wedge
\cdots\wedge dt^{(m)}_{1}\wedge \cdots \wedge
dt^{(m)}_{L-1}.
\end{align*}
\end{dfn}
\begin{thm}\label{thm IF}
If $\kappa_0-\sum_{i=1}^N\theta_i=M$ and $\kappa_n=1$ ($2\le n\le L-1$), then
the integral formula $\Psi_M({\bf q,z})$ is a solution to the Schr\"odinger system \eqref{eq Schrodinger}, with
\begin{equation*}
\alpha_n=e_{n+1}-e_n+1,\quad
\beta_i=-\theta_i, \quad \gamma=\kappa_1+M-1,
\end{equation*}
for $1\le n\le L-1$ and $1\le i\le N$, where $e_L=e_0$.
\end{thm}
For $A\in\mathcal{A}_M$,
let $\varphi_A(t)$ be the rational $(L-1)M$-form holomorphic outside $D$
defined by
\begin{equation*}
\varphi_{A}(t)
=\mathrm{Sym}\left[
(-1)^{M-A_0}\begin{pmatrix}
M\\
A
\end{pmatrix}
\prod_{i=1}^N\prod_{n=1}^{L-1}
\prod_{a=S_{n-1}^{(i)}+1}^{S_n^{(i)}}
f^{(i)}_n\left(
t^{(a)}
\right)\prod_{a=M-A_0+1}^Mf_0\left(
t^{(a)}
\right)\right]dt,
\end{equation*}
where
\begin{align*}
&A_0=M-\sum_{1\le i\le N, \atop 1\le n\le L-1}A_{n,i}, \quad \begin{pmatrix}
M\\
A
\end{pmatrix}
=\frac{M!}{A_0!\prod_{1\le i\le N, \atop 1\le n\le L-1}A_{n,i}!},
\quad
S_n^{(i)}=\sum_{j=1}^{i-1}\sum_{m=1}^{L-1}A_{m,j}+\sum_{m=1}^nA_{m,i}.
\end{align*}
Then,
the integral formula is expressed as
\begin{equation*}
\Psi_M\left(
{\bf q,z}
\right)=\sum_{A\in\mathcal{A}_M}q^A\int_\Delta U(t) \varphi_A(t).
\end{equation*}
Since, in general, it holds for an $(L-1)M$-form $\varphi$ that
\begin{equation*}
\frac{\partial}{\partial z_i}\int_\Delta U \varphi=\int_\Delta U \left(
\frac{1}{U}\frac{\partial U}{\partial z_i}\varphi+\frac{\partial \varphi}{\partial z_i}
\right),
\end{equation*}
let a linear operator $\nabla_i$ ($i=1,\ldots,N$) acting on
$\varphi$ be defined as
\begin{equation*}
\nabla_i\varphi=\frac{1}{U}\frac{\partial U}{\partial z_i}\varphi+\frac{\partial \varphi}{\partial z_i}.
\end{equation*}
Let us explain our proof of Theorem \ref{thm IF} briefly.
We compute $\kappa\nabla_i\varphi_A(t)$ and obtain the
linear Pfaffian system for $\{\int_\Delta U \varphi_A(t)|A\in\mathcal{A}_M\}$.
While we compute the action of the Hamiltonians $H_i$ on $q^A$
and obtain the coefficient of $q^A$ of $H_i\Psi_M({\bf q,z})$
as a linear combination of elements of $\{\int_\Delta U \varphi_A(t)|A\in\mathcal{A}_M\}$.
Finally, comparing
both results, we obtain Theorem \ref{thm IF}.
{\bf A proof of Theorem \ref{thm IF}
Fix $i\in\{1,\ldots, N\}$ and $A\in\mathcal{A}_M$. We compute $\nabla_i \varphi_{A}(t)$ as follows. First,
we have
\begin{equation}\label{first eq}
\kappa\nabla_i \varphi_{A}(t)=\mathrm{Sym}\left[
\left(
\beta_i\sum_{j=1,\atop j\neq i}^N\sum_{n=1}^{L-1}\frac{A_{n,j}t_{L-1}^{\left(
S_n^{(j)}
\right)}}{1-z_it_{L-1}^{\left(
S_n^{(j)}
\right)}}
+\beta_i\frac{A_0t_{L-1}^{\left(
M-A_0+1
\right)}}{1-z_it_{L-1}^{\left(M-A_0+1
\right)}}
+ \left(\beta_i+ \kappa\right)\sum_{n=1}^{L-1}\frac{A_{n,i}t_{L-1}^{\left(
S_n^{(i)}
\right)}}{1-z_it_{L-1}^{\left(S_n^{(i)}\right)}}
\right)\bar{\varphi}_A(t)\right]dt,
\end{equation}
where $\bar{\varphi}_A(t)$ is defined by
\begin{equation*}
\bar{\varphi}_A(t)=(-1)^{M-A_0}\begin{pmatrix}
M\\
A
\end{pmatrix}
\prod_{i=1}^N\prod_{n=1}^{L-1}
\prod_{a=S_{n-1}^{(i)}+1}^{S_n^{(i)}}
f^{(i)}_n\left(
t^{(a)}
\right)\prod_{a=M-A_0+1}^Mf_0\left(
t^{(a)}
\right).
\end{equation*}
Using a relation
\begin{equation}\label{Jacobi identity}
\frac{t}{1-z_it}=\frac{1-z_jt}{z_i-z_j}\left(
\frac{1}{1-z_it}-\frac{1}{1-z_jt}
\right),
\end{equation}
we get
\begin{equation*}
\text{the first term of \eqref{first eq}}
=\beta_i\sum_{j=1,\atop j\neq i}^N\sum_{n=1}^{L-1}\frac{1}{z_i-z_j}
\left((A_{n,i}+1) \varphi_{\left(A_{n,j}-1, A_{n,i}+1\right)}(t)
-A_{n,j}
\varphi_{A}(t)\right),
\end{equation*}
where $\varphi_{\left(A_{n,j}-1, A_{n,i}+1\right)}(t)$ is the rational $(L-1)M$-form defined for
the matrix in $\mathcal{A}_M$ whose $(n,j)$ entry is $A_{n,j}-1$ and $(n,i)$ entry is $A_{n,i}+1$, and the other $(m,k)$ entries
are $A_{m,k}$.
As for the second term of \eqref{first eq},
using a relation
\begin{equation}\label{eq f0}
\frac{t_{L-1}}{1-z_it_{L-1}}=\frac{1}{(z_i-1)f_0(t)}\left(
-f_0(t)+\sum_{n=1}^{L-1}f^{(i)}_n(t)
\right)
\end{equation}
we obtain
\begin{equation*}
\text{the second term of \eqref{first eq} }
=\frac{-\beta_i}{z_i-1}\left(A_0
\varphi_{A}(t)
+\sum_{n=1}^{L-1}
\left(
A_{n,i}+1
\right)
\varphi_{\left( A_{n,i}+1\right)}(t)
\right),
\end{equation*}
where $\varphi_{\left(A_{n,i}+1\right)}(t)$ is the rational $(L-1)M$-form defined for
the matrix in $\mathcal{A}_M$ whose $(n,i)$ entry is $A_{n,i}+1$, and the other $(m,k)$ entries
are $A_{m,k}$.
In order to calculate the third term of \eqref{first eq},
we compute coboundaries $X_n^{(i)}$ ($n=1,\ldots, L-1$) defined by
\begin{equation}\label{eq coboundary}
X_n^{(i)}=\kappa\sum_{m=n}^{L-1}\nabla\left(\sum_{\sigma\in\frak{S}_M^{L-1}}\sigma\left(
t_{m}^{( (S^{(i)}_n)}
\bar{\varphi}_A(t)\right) \
{*dt}_m^{(\sigma_m(S_n^{(i)}))}\right),
\end{equation}
where $*dt_m^{(a)}$ is defined by
\begin{equation*}
*dt_m^{(a)}=(-1)^{(L-1)(a-1)+m-1}dt_1^{(1)}\wedge
\cdots \wedge \widehat{dt_m^{(a)}}\wedge \cdots \wedge t_{L-1}^{(M)},
\end{equation*}
so that $dt_m^{(a)}\wedge *dt_m^{(a)}=dt$.
For $m\neq n$, denote by
$\varphi_{\left( A_{n,i}-1, A_{m,i}+1\right)}(t)$, the rational $(L-1)M$-form defined for
the matrix in $\mathcal{A}_M$ whose $(n,i)$ entry is $A_{n,i}-1$ and $(m,i)$ entry is $A_{m,i}+1$,
and the other $(l,k)$ entries
are $A_{l,k}$, and denote by $\varphi_{\left(A_{n,i}-1\right)}(t)$, the rational $(L-1)M$-form defined for
the matrix in $\mathcal{A}_M$ whose $(n,i)$ entry is $A_{n,i}-1$, and the other $(l,k)$ entries
are $A_{l,k}$.
Using
the relations \eqref{Jacobi identity} and \eqref{eq f0},
we obtain by straightforward calculations
\begin{align*}
X_n^{(i)}=&\mathrm{Sym}\left[\left(\beta_i+ \kappa\right)z_i \frac{t_{L-1}^{(S_n^{(i)})}}{1-z_it_{L-1}^{(S_n^{(i)})}}
\bar{\varphi}_A(t)\right]dt
\\
&+\left(\sum_{m=n}^{L-1}\alpha_m-(L-1-n)\right)\varphi_{A}(t)
+\left(1+\delta_{n,1}(\gamma-1)\right)\sum_{m=n+1}^{L-1}\frac{A_{m,i}+1}{A_{n,i}}
\varphi_{\left(A_{m,i}+1, A_{n,i}-1 \right)}(t)
\\
&+\left(1+\delta_{n,1}(\gamma-1)\right){1\over z_i-1}\left(\frac{A_0+1}{A_{n,i}}
\varphi_{\left(
A_{n,i}-1
\right)}(t)
+\sum_{m=1,\atop m\neq n}^{L-1}\frac{A_{m,i}+1}{A_{n,i}}
\varphi_{\left(A_{m,i}+1,
A_{n,i}-1
\right)}(t)+ \varphi_{A}(t)
\right)
\\
&+\sum_{j=1,\atop j\neq i}^N\frac{\beta_jz_j}{z_i-z_j}\left(
\varphi_{A}(t)-
\frac{A_{n,j}+1}{A_{n,i}}
\varphi_{\left(
A_{n,i}-1, A_{n,j}+1
\right)}(t)
\right)
+Y_{n}^{(i)},
\end{align*}
where
\begin{equation}\label{eq Y Z}
Y_{n}^{(i)}=\mathrm{Sym}\left[\sum_{m=n}^{L-1}t_{m}^{\left(
S_n^{(i)}
\right)}W_{n,m}^{(i)}\bar{\varphi}_A(t)\right]dt,
\end{equation}
with
\begin{equation*}
W^{(i)}_{n,m}=\sum_{a=1,\atop a\neq
S_n^{(i)}}^M\left(
\frac{-1}{t_m^{\left(
S_n^{(i)}
\right)}
-
t_{m-1}^{(a)}}
+
\frac{2}{t_m^{\left(
S_n^{(i)}
\right)}
-
t_m^{(a)}}
+\frac{-1}{t_m^{\left(
S_n^{(i)}\right)
}
-
t_{m+1}^{(a)}}+\delta_{m,1}\frac{1}{t_{1}^{\left(
S_n^{(i)}
\right)}-1}\right).
\end{equation*}
We compute $Y_n^{(i)}$ in Lemmas \ref{lem l<n Y}, \ref{lem n<l Y}, \ref{lem n=l Y} and \ref{lem l=0 Y}. Owing to those lemmas,
we obtain
\begin{align}
&\kappa z_i\nabla_i \varphi_A(t)-\sum_{n=1}^{L-1}A_{n,i}X_n^{(i)}=\left\{
-\sum_{n=1}^{L-1}A_{n,i}\left(
\sum_{m=n}^{L-1}\alpha_m+L-n-\beta_i+\sum_{m=1}^nA_{m,i}
\right)\right.\nonumber
\\
&+\frac{1}{z_i-1}\left(
A_0\left(
\sum_{n=1}^{L-1}A_{n,i}-\beta_i
\right)
-\sum_{j=1}^N\sum_{n=1}^{L-1}A_{n,i}A_{n,j}+A_{1,i}(M-\gamma)
\right)\nonumber
\\
&\left.+\sum_{j=1,\atop j\neq i}^N\frac{z_j}{z_i-z_j}\sum_{n=1}^{L-1}\left(A_{n,i}\left(
\sum_{m=1}^{L-1}A_{m,j}+A_{n,j}-\beta_j
\right)
-\beta_iA_{n,j}
\right)\right\}\varphi_A(t)\nonumber
\\
&-\frac{A_0+1}{z_i-1}
\sum_{n=1}^{L-1}\left(\sum_{j=1}^NA_{n,j}+\delta_{n,1}(\gamma-M)\right)\varphi_{(A_{n,i}-1)}(t)
+\frac{z_i}{z_i-1}\left(\sum_{m=1}^{L-1}A_{m,i}-\beta_i\right)\sum_{n=1}^{L-1}(A_{n,i}+1)\varphi_{(A_{n,i}+1)}(t)\nonumber
\\
&-\frac{1}{z_i-1}\sum_{n=1}^{L-1}\left(\sum_{j=1}^NA_{n,j}+\delta_{n,1}(\gamma-M)\right)
\left(\sum_{m=1}^{n-1}
(A_{m,i}+1)
\varphi_{\left(A_{m,i}+1, A_{n,i}-1 \right)}(t)
+z_i\sum_{m=n+1}^{L-1}
(A_{m,i}+1)
\varphi_{\left(A_{m,i}+1, A_{n,i}-1 \right)}(t)\right)\nonumber
\\
&+\sum_{j=1,\atop j\neq i}^{N}\frac{A_{n,j}+1}{z_i-z_j}
\left(z_j\sum_{m=1}^{n-1}
(A_{m,i}+1)
\varphi_{\left(A_{m,j}-1, A_{m,i}+1, A_{n,i}-1, A_{n,j}+1 \right)}(t)
+z_i\sum_{m=n+1}^{L-1}
(A_{m,i}+1)
\varphi_{\left(A_{m,j}-1, A_{m,i}+1, A_{n,i}-1,A_{n,j}+1 \right)}(t)
\right) \nonumber
\\
&+\sum_{j=1,\atop j\neq i}^{N}\frac{z_i}{z_i-z_j}\left(\beta_i-\sum_{m=1}^{L-1}A_{m,i}\right)
\sum_{n=1}^{L-1}(A_{n,i}+1)\varphi_{(A_{n,j}-1, A_{n,i}+1)}(t)\nonumber
\\
&+\sum_{j=1,\atop j\neq i}^{N}\frac{z_j}{z_i-z_j}\left(\beta_j-\sum_{m=1}^{L-1}A_{m,j}\right)
\sum_{n=1}^{L-1}(A_{n,j}+1)\varphi_{(A_{n,i}-1, A_{n,j}+1)}(t),
\label{eq thm final result}
\end{align}
where the rational $(L-1)M$-form $\varphi_{\left(A_{m,j}-1, A_{m,i}+1, A_{n,i}-1, A_{n,j}+1 \right)}(t)$ be defined for
the matrix in $\mathcal{A}_M$ whose $(m,j)$, $(m,i)$, $(n,i)$, $(n,j)$ entries are $A_{m,j}-1$, $A_{m,i}+1$, $A_{n,i}-1$,
and $A_{n,j}+1$, respectively, and the other $(l,k)$ entries
are $A_{l,k}$.
Hence, for $A\in\mathcal{A}_M$, as an element in the twisted de Rham cohomology group $H^{(L-1)M}(T, \nabla)$,
$\kappa\nabla_i\varphi_A(t)$ is expressed in terms of elements of $\{\varphi_B(t)|B\in \mathcal{A}_M\}$.
On the other hand, computations of the action of the Hamiltonian $H_i$ on $q^A$ for $A\in\mathcal{A}_M$
is straightforward and it is easy to see that the coefficient of $q^A$ of $H_i \Psi_M({\bf q,z})$ is equal to
the hypergeometric pairing between the cycle $\Delta\in H_{(L-1)M}(T,\mathcal{S}^*)$ and the right hand
side of \eqref{eq thm final result}. Therefore, we complete our proof.
\qed
\subsection{Lemmas}
Through lemmas below, fix $1\le n\le L-1$, $1\le i\le N$ and $A\in\mathcal{A}_M$.
For a triple $(n,i,A)$, the coboundary $X_n^{(i)}$ is defined by \eqref{eq coboundary}
and expressed as
a linear combination of elements in $\{
\varphi_B(t) |B\in\mathcal{A}_M\}$, and $Y_n^{(i)}$. In this subsection,
we compute $Y_n^{(i)}$, so that we show that they are also expressed as
a linear combination of elements in
$\{\varphi_B(t) |B\in\mathcal{A}_M\}$.
We divide $Y_n^{(i)}$ as
\begin{equation*}
Y_n^{(i)}=\sum_{1\le j\le N, \atop 1\le l \le L-1 }\left(Y_n^{(i)}\right)_{l,j}+\left(Y_n^{(i)}\right)_0
\end{equation*}
and we compute $\left(Y_n^{(i)}\right)_{l,j}$ and $\left(Y_n^{(i)}\right)_0$,
where for $l\neq n$ or $j\neq i$,
\begin{align*}
&\left(Y_n^{(i)}\right)_{l,j}=\mathrm{Sym}\left[A_{l,j}C\left(n, S_n^{(i)}, S_l^{(j)}\right)
\bar{\varphi}_A(t)\right]dt,
\\
&\left(Y_n^{(i)}\right)_{n,i}=\mathrm{Sym}\left[(A_{n,i}-1)C\left(n, S_n^{(i)}, S_n^{(i)}-1\right)\bar{\varphi}_A(t)\right]dt,
\\
&\left(Y_n^{(i)}\right)_0=\mathrm{Sym}\left[A_0C\left(n, S_n^{(i)}, M-A_0+1\right)\bar{\varphi}_A(t)\right]dt.
\end{align*}
Here, for $1\le a\neq b\le M$,
\begin{equation*}
C(n,a,b)=\sum_{m=n}^{L-1}t_m^{(a)}\left(
\frac{-1}{t_{m}^{(a)}-t_{m-1}^{(b)}}+\frac{2}{t_{m}^{(a)}-t_{m}^{(b)}}
+\frac{-1}{t_{m}^{(a)}-t_{m+1}^{(b)}}\right)+\delta_{n,1}\frac{t_{1}^{(a)}}{t_{1}^{(a)}-1}.
\end{equation*}
Let the rational functions $f_{l,m}^{(j)}(t^{(a)})$ be defined by
\begin{equation*}
f_{l,m}^{(j)}(t^{(a)})={1\over 1-z_jt_{L-1}^{(a)}}\sum_{k=1,\atop k\neq l,m}^{L-1}\frac{1}{t_{k-1}^{(a)}-t_k^{(a)}}.
\end{equation*}
\begin{lem}\label{lem l<n Y}
When $1\le l< n$, for $1\le j\neq i\le N$, we have
\begin{align*}
\left(Y_n^{(i)}\right)_{l,j}=&\left(A_{l,i}+1\right) \varphi_{\left(A_{l,j}-1, A_{l,i}+1 \right)}(t)
+\frac{z_j}{z_i-z_j}\left(\left(
A_{l,i}+1
\right)
\varphi_{\left(A_{l,j}-1, A_{l,i}+1 \right)}(t)
\right.
\\
&-A_{l,j} \varphi_{A }(t)
-\frac{\left(
A_{l,i}+1
\right)
\left(
A_{n,j}+1
\right)}{A_{n,i}}
\varphi_{\left(A_{l,j}-1, A_{l,i}+1, A_{n,i}-1, A_{n,j}+1 \right)}(t)
\left.+\frac{\left(
A_{n,j}+1
\right)A_{l,j}}{A_{n,i}}
\varphi_{\left(A_{n,j}+1, A_{n,i}-1 \right)}(t)
\right),
\end{align*}
and
for $j=i$, we have
\begin{equation*}
\left(Y_n^{(i)}\right)_{l,i}=A_{l,i} \varphi_{A}(t) .
\end{equation*}
\end{lem}
\begin{proof}
It suffices to show that
\begin{align}
&\mathrm{Sym}\left[
C(n,1,2){1\over t_{L-1}^{(1)}}f_n^{(i)}(t^{(1)}){1\over t_{L-1}^{(2)}}f_l^{(j)}(t^{(2)})
\right]\nonumber
\\
&=\mathrm{Sym}\left[
{1\over t_{L-1}^{(1)}}f_n^{(i)}(t^{(1)}){1\over t_{L-1}^{(2)}}f_l^{(i)}(t^{(2)})
\right]\nonumber
\\
&+{z_j\over z_i-z_j}\mathrm{Sym}\left[
{1\over t_{L-1}^{(1)}}{1\over t_{L-1}^{(2)}}\left(f_n^{(i)}(t^{(1)})-f_n^{(j)}(t^{(1)})\right)
\left(
f_l^{(i)}(t^{(2)})-f_l^{(j)}(t^{(2)})\right)
\right], \label{eq l<n Y}
\end{align}
where the symmetrization $\mathrm{Sym}[f(t)]$ stands for $\sum_{\sigma\in\frak{S}_2^{L-1}}\sigma(f(t))$
(see \eqref{eq sigma}), the rational functions $f_n^{(i)}(t^{(a)})$ are defined in Definition \ref{def integral formula}, and
if $j=i$, then
we understand that the second line of the right hand side of \eqref{eq l<n Y} is vanished.
Firstly, we claim that for $n\le k \le L-2$, we have
\begin{align}
&\mathrm{Sym}\left[
\sum_{m=n}^k t_m^{(1)}\left(
\frac{-1}{t_{m}^{(1)}-t_{m-1}^{(2)}}+\frac{2}{t_{m}^{(1)}-t_{m}^{(2)}}
+\frac{-1}{t_{m}^{(1)}-t_{m+1}^{(2)}}\right){1\over t_{L-1}^{(1)}}f_n^{(i)}(t^{(1)}){1\over t_{L-1}^{(2)}}f_l^{(j)}(t^{(2)})
\right]\nonumber
\\
&=\mathrm{Sym}\left[
{1\over \left(t_k^{(1)}-t_{k+1}^{(2)}\right)}{ 1\over \left( t_{k+1}^{(1)}-t_{k}^{(2)} \right)}{t_{k+1}^{(1)}\over t_{L-1}^{(1)}}
f_n^{(i)}(t^{(1)}){1\over t_{L-1}^{(2)}}f_{l,k+1}^{(j)}(t^{(2)})
\right]. \label{eq l<n claim}
\end{align}
We show \eqref{eq l<n claim} by induction. Let $k=n$. then, we have
\begin{align}
&\mathrm{Sym}\left[\left(\frac{-1}{t_{n}^{(1)}-t_{n-1}^{(2)}}+\frac{1}{t_{n}^{(1)}-t_{n}^{(2)}}
\right){t_{n}^{(1)}\over t_{L-1}^{(1)}}f_n^{(i)}(t^{(1)}){1\over t_{L-1}^{(2)}}f_l^{(j)}(t^{(2)})
\right]\nonumber
\\
&=\mathrm{Sym}\left[
\frac{1}{t_{n-1}^{(2)}-t_{n}^{(1)}}\frac{1}{t_{n}^{(1)}-t_{n}^{(2)}}{t_{n}^{(1)}\over t_{L-1}^{(1)}}f_n^{(i)}(t^{(1)}){1\over t_{L-1}^{(2)}}f_{l,n}^{(j)}(t^{(2)})
\right]\nonumber
\\
&=\mathrm{Sym}\left[
\frac{-1}{t_{n}^{(1)}-t_{n}^{(2)}}\frac{1}{t_{n}^{(2)}-t_{n+1}^{(1)}}\frac{1}{t_{n}^{(1)}-t_{n+1}^{(2)}}{t_{n}^{(2)}\over t_{L-1}^{(1)}}
f_{n,n+1}^{(i)}(t^{(1)}){1\over t_{L-1}^{(2)}}f_{l,n+1}^{(j)}(t^{(2)})
\right],\label{eq l<n n1}
\end{align}
where in the last line, we interchange $t_n^{(1)}$ with $t_n^{(2)}$ and
\begin{align}
&\mathrm{Sym}\left[\left(\frac{1}{t_{n}^{(1)}-t_{n}^{(2)}}+\frac{-1}{t_{n}^{(1)}-t_{n+1}^{(2)}}
\right){t_{n}^{(1)}\over t_{L-1}^{(1)}}f_n^{(i)}(t^{(1)}){1\over t_{L-1}^{(2)}}f_l^{(j)}(t^{(2)})
\right]\nonumber
\\
&=\mathrm{Sym}\left[\frac{1}{t_{n}^{(1)}-t_{n}^{(2)}}\frac{1}{t_{n}^{(1)}-t_{n+1}^{(2)}}
{t_{n}^{(1)}\over t_{L-1}^{(1)}}f_n^{(i)}(t^{(1)}){1\over t_{L-1}^{(2)}}f_{l,n+1}^{(j)}(t^{(2)})
\right]. \label{eq l<n n2}
\end{align}
Thus, the left hand side of \eqref{eq l<n claim} for $k=n$, that is, \eqref{eq l<n n1} plus \eqref{eq l<n n2},
becomes the right hand side of \eqref{eq l<n claim} for $k=n$.
Suppose \eqref{eq l<n claim} holds for $k-1$, then
\begin{align*}
&\mathrm{Sym}\left[\left(t_{k}^{(1)}\left(\frac{-1}{t_{k}^{(1)}-t_{k-1}^{(2)}}+\frac{1}{t_{k}^{(1)}-t_{k}^{(2)}}\right)+
\sum_{m=n}^{k-1} t_{m}^{(1)}\left(\frac{-1}{t_{m}^{(1)}-t_{m-1}^{(2)}}+\frac{2}{t_{m}^{(1)}-t_{m}^{(2)}}
+\frac{-1}{t_{m}^{(1)}-t_{m+1}^{(2)}}\right)\right){1\over t_{L-1}^{(1)}}f_n^{(i)}(t^{(1)}){1\over t_{L-1}^{(2)}}f_l^{(j)}(t^{(2)})
\right]
\nonumber
\\
&=\mathrm{Sym}\left[
\frac{1}{t_{k}^{(1)}-t_{k}^{(2)}}\frac{1}{t_{k-1}^{(2)}-t_{k}^{(1)}}\frac{1}{t_{k-1}^{(1)}-t_{k}^{(2)}}{t_{k}^{(1)}\over t_{L-1}^{(1)}}
f_{n,k}^{(i)}(t^{(1)}){1\over t_{L-1}^{(2)}}f_{l,k}^{(j)}(t^{(2)})
\right]
\\
&=\mathrm{Sym}\left[
\frac{-1}{t_{k}^{(1)}-t_{k}^{(2)}}\frac{1}{t_{k}^{(2)}-t_{k+1}^{(1)}}\frac{1}{t_{k}^{(1)}-t_{k+1}^{(2)}}{t_{k}^{(2)}\over t_{L-1}^{(1)}}
f_{n,k+1}^{(i)}(t^{(1)}){1\over t_{L-1}^{(2)}}f_{l,k+1}^{(j)}(t^{(2)})
\right],
\end{align*}
where in the last line, we interchange $t_k^{(1)}$ with $t_k^{(2)}$. Since
\begin{align*}
&\mathrm{Sym}\left[\left(\frac{1}{t_{k}^{(1)}-t_{k}^{(2)}}+\frac{-1}{t_{k}^{(1)}-t_{k+1}^{(2)}}
\right){t_{k}^{(1)}\over t_{L-1}^{(1)}}f_n^{(i)}(t^{(1)}){1\over t_{L-1}^{(2)}}f_l^{(j)}(t^{(2)})
\right]\nonumber
\\
&=\mathrm{Sym}\left[\frac{1}{t_{k}^{(1)}-t_{k}^{(2)}}\frac{1}{t_{k}^{(1)}-t_{k+1}^{(2)}}{t_{k}^{(1)}\over t_{L-1}^{(1)}}
f_n^{(i)}(t^{(1)}){1\over t_{L-1}^{(2)}}f_{l,k+1}^{(j)}(t^{(2)})
\right],
\end{align*}
the left hand side of \eqref{eq l<n claim} for $k$ becomes the right hand side of \eqref{eq l<n claim} for $k$.
Secondly, using \eqref{eq l<n claim} for $k=L-2$, we have
\begin{align*}
&\mathrm{Sym}\left[\left(t_{L-1}^{(1)}\left(\frac{-1}{t_{L-1}^{(1)}-t_{L-2}^{(2)}}+\frac{1}{t_{L-1}^{(1)}-t_{L-1}^{(2)}}\right)+
\sum_{m=n}^{L-2}t_{m}^{(1)} \left(\frac{-1}{t_{m}^{(1)}-t_{m-1}^{(2)}}+\frac{2}{t_{m}^{(1)}-t_{m}^{(2)}}
+\frac{-1}{t_{m}^{(1)}-t_{m+1}^{(2)}}\right)\right){1\over t_{L-1}^{(1)}}f_n^{(i)}(t^{(1)}){1\over t_{L-1}^{(2)}}f_l^{(j)}(t^{(2)})
\right]
\nonumber
\\
&=\mathrm{Sym}\left[
\frac{1}{t_{L-1}^{(1)}-t_{L-1}^{(2)}}\frac{1}{t_{L-2}^{(2)}-t_{L-1}^{(1)}}\frac{1}{t_{L-2}^{(1)}-t_{L-1}^{(2)}}
f_{n,L-1}^{(i)}(t^{(1)}){1\over t_{L-1}^{(2)}}f_{l,L-1}^{(j)}(t^{(2)})
\right]
\\
&=\mathrm{Sym}\left[
\frac{-1}{t_{L-1}^{(1)}-t_{L-1}^{(2)}}
f_{n}^{(j)}(t^{(1)}){1\over t_{L-1}^{(1)}}f_{l}^{(i)}(t^{(2)})
\right],
\end{align*}
where in the last line, we interchange $t_{L-1}^{(1)}$ with $t_{L-1}^{(2)}$. Hence, the left hand side of \eqref{eq l<n Y}
is equal to
\begin{align*}
&\mathrm{Sym}\left[
\frac{-1}{t_{L-1}^{(1)}-t_{L-1}^{(2)}}
f_{n}^{(j)}(t^{(1)}){1\over t_{L-1}^{(1)}}f_{l}^{(i)}(t^{(2)})+\frac{1}{t_{L-1}^{(1)}-t_{L-1}^{(2)}}
f_n^{(i)}(t^{(1)}){1\over t_{L-1}^{(2)}}f_{l}^{(j)}(t^{(2)})
\right]\nonumber
\\
&=\mathrm{Sym}\left[{1\over t_{L-1}^{(1)}}{1\over t_{L-1}^{(2)}}
f_{n}^{(i)}(t^{(1)})f_{l}^{(i)}(t^{(2)})\frac{1-z_j(t_{L-1}^{(1)}+t_{L-1}^{(2)})+z_iz_jt_{L-1}^{(1)}t_{L-1}^{(2)}}{(1-z_jt_{L-1}^{(1)})(1-z_jt_{L-1}^{(2)})}
\right].
\end{align*}
Therefore, the relation \eqref{eq l<n Y} holds.
\end{proof}
\begin{lem}\label{lem n<l Y}
When $n < l\le L-1$, for $1\le j\neq i\le N$, we have
\begin{align*}
\left(Y_n^{(i)}\right)_{l,j}=&-A_{l,j}\left( \frac{\delta_{n,1}}{z_i-1}+ \frac{z_j}{z_i-z_j} \right) \varphi_{A}(t)
+
\left(
A_{l,i}+1
\right)\frac{z_i}{z_i-z_j}
\varphi_{\left(A_{l,j}-1, A_{l,i}+1 \right)}(t)
\\
&
-\frac{\left(
A_{l,i}+1
\right)
\left(
A_{n,j}+1
\right)}{A_{n,i}}\frac{z_i}{z_i-z_j}
\varphi_{\left(A_{l,j}-1, A_{l,i}+1, A_{n,i}-1, A_{n,j}+1\right)}(t)
+
\frac{\left(
A_{n,j}+1
\right)A_{l,j}}{A_{n,i}} \frac{z_j}{z_i-z_j} \varphi_{\left(A_{n,j}+1, A_{n,i}-1 \right)}(t)
\\
&-\frac{A_{l,j}}{A_{n,i}}\frac{\delta_{n,1}}{(z_i-1)}\left(\left(
A_0+1
\right)
\varphi_{\left( A_{n,i}-1 \right)}(t)
+z_i\sum_{m=2}^{L-1}\left(
A_{m,i}+1
\right) \varphi_{\left(A_{m,i}+1, A_{1,i}-1 \right)}(t)
\right),
\end{align*}
and
for $j=i$, we have
\begin{align*}
\left(Y_n^{(i)}\right)_{l,i}=&
-\frac{\delta_{n,1}A_{l,i}}{(z_i-1)}\left( \varphi_{A}(t)+{\left(
A_0+1
\right)\over A_{1,i}}
\varphi_{\left( A_{1,i}-1 \right)}(t)
+z_i\sum_{m=2}^{L-1}{\left(
A_{m,i}+1
\right) \over A_{1,i} }\varphi_{\left(A_{m,i}+1, A_{1,i}-1 \right)}(t)
\right)
\end{align*}
\end{lem}
\begin{proof}
It suffices to show that for $n\ge 2$,
\begin{align}
&\mathrm{Sym}\left[
C(n,1,2){1\over t_{L-1}^{(1)}}f_n^{(i)}(t^{(1)}){1\over t_{L-1}^{(2)}}f_l^{(j)}(t^{(2)})
\right]\nonumber
\\
&=\frac{1}{z_i-z_j}\mathrm{Sym}\left[{1\over t_{L-1}^{(1)}}{1\over t_{L-1}^{(2)}}
\left(f_n^{(i)}(t^{(1)})-f_n^{(j)}(t^{(1)})\right)\left(z_if_l^{(i)}(t^{(2)})-z_jf_l^{(j)}(t^{(2)})\right)
\right], \label{eq n<l Y}
\end{align}
and for $n=1$,
\begin{align}
&\mathrm{Sym}\left[
C(1,1,2){1\over t_{L-1}^{(1)}}f_1^{(i)}(t^{(1)}){1\over t_{L-1}^{(2)}}f_l^{(j)}(t^{(2)})
\right]\nonumber
\\
&=\frac{1}{z_i-z_j}\mathrm{Sym}\left[{1\over t_{L-1}^{(1)}}{1\over t_{L-1}^{(2)}}
\left(f_1^{(i)}(t^{(1)})-f_1^{(j)}(t^{(1)})\right)\left(z_if_l^{(i)}(t^{(2)})-z_jf_l^{(j)}(t^{(2)})\right)
\right],\nonumber
\\
&+\frac{1}{z_i-1}\mathrm{Sym}\left[{1\over t_{L-1}^{(1)}}\left(
f_0(t^{(1)})-f_1^{(i)}(t^{(1)})-z_i\sum_{m=2}^{L-1}f_m^{(i)}(t^{(1)})
\right)
{1\over t_{L-1}^{(2)}}f_l^{(j)}(t^{(2)})
\right], \label{eq 1<l Y}
\end{align}
where the symmetrization $\mathrm{Sym}[f(t)]$ stands for $\sum_{\sigma\in\frak{S}_2^{L-1}}\sigma(f(t))$
(see \eqref{eq sigma}), and
if $j=i$, then
we understand that the right hand side of \eqref{eq n<l Y} and the first line of the right hand side of
\eqref{eq 1<l Y} is zero.
We shall show \eqref{eq n<l Y}. Firstly, using \eqref{eq l<n claim} for $k=l-2$, we have
\begin{align}
&\mathrm{Sym}\left[\left(
\sum_{m=n}^{l-2}t_m^{(1)}\left(
\frac{-1}{t_{m}^{(1)}-t_{m-1}^{(2)}}+\frac{2}{t_{m}^{(1)}-t_{m}^{(2)}}
+\frac{-1}{t_{m}^{(1)}-t_{m+1}^{(2)}}\right)+t_{l-1}^{(1)}\left(\frac{-1}{t_{l-1}^{(1)}-t_{l-2}^{(1)}}+\frac{1}{t_{l-1}^{(1)}-t_{l-1}^{(2)}}\right)
\right)
{1\over t_{L-1}^{(1)}}f_n^{(i)}(t^{(1)}){1\over t_{L-1}^{(2)}}f_l^{(j)}(t^{(2)})
\right]\nonumber
\\
&=\mathrm{Sym}\left[
\frac{1}{t_{l-1}^{(1)}-t_{l-1}^{(2)}}\frac{1}{t_{l-2}^{(1)}-t_{l-1}^{(2)}}\frac{1}{t_{l-2}^{(2)}-t_{l-1}^{(1)}}
{t_{l-1}^{(1)}\over t_{L-1}^{(1)}}f_{n,l-1}^{(i)}(t^{(1)}){1\over t_{L-1}^{(2)}}f_{l-1,l}^{(j)}(t^{(2)})
\right]\nonumber
\\
&=\mathrm{Sym}\left[
\frac{-1}{t_{l-1}^{(1)}-t_{l-1}^{(2)}}\frac{1}{t_{l-1}^{(2)}-t_{l}^{(1)}}
{t_{l-1}^{(2)}\over t_{L-1}^{(1)}}f_{n,l}^{(i)}(t^{(1)}){1\over t_{L-1}^{(2)}}f_{l}^{(j)}(t^{(2)})
\right],\label{eq n<l Y 1}
\end{align}
where in the last line, we interchange $t_{l-1}^{(1)}$ with $t_{l-1}^{(2)}$. Thus, \eqref{eq n<l Y 1} is equal to
\begin{equation*}
-\mathrm{Sym}\left[
\left( \frac{t_{l-1}^{(1)}}{t_{l-1}^{(1)}-t_{l-1}^{(2)}}+\frac{-t_{l}^{(1)}}{t_{l}^{(1)}-t_{l-1}^{(2)}} \right)
{1\over t_{L-1}^{(1)}}f_n^{(i)}(t^{(1)}){1\over t_{L-1}^{(2)}}f_l^{(j)}(t^{(2)})
\right].
\end{equation*}
Secondly, we claim that for $l\le k\le L-2$, we have
\begin{align}
&\mathrm{Sym}\left[
\sum_{m=l}^k \left(\frac{-t_{m+1}^{(1)}}{t_{m+1}^{(1)}-t_{m}^{(2)}}+\frac{2t_{m}^{(1)}}{t_{m}^{(1)}-t_{m}^{(2)}}
+\frac{-t_{m-1}^{(1)}}{t_{m-1}^{(1)}-t_{m}^{(2)}}\right){1\over t_{L-1}^{(1)}}f_n^{(i)}(t^{(1)}){1\over t_{L-1}^{(2)}}f_l^{(j)}(t^{(2)})
\right]\nonumber
\\
&=\mathrm{Sym}\left[
{1\over \left(t_k^{(1)}-t_{k+1}^{(2)}\right)}{ 1\over \left( t_{k}^{(2)}-t_{k+1}^{(1)} \right)}{1\over t_{L-1}^{(1)}}
f_{n,k+1}^{(i)}(t^{(1)}){t_{k+1}^{(2)}\over t_{L-1}^{(2)}}f_{l}^{(j)}(t^{(2)})
\right]. \label{eq n<l claim}
\end{align}
We can prove \eqref{eq n<l claim} by induction and omit the proof of this claim.
Using \eqref{eq n<l claim} for $k=L-2$, we have
\begin{align*}
&\mathrm{Sym}\left[\left(\frac{t_{L-1}^{(1)}}{t_{L-1}^{(1)}-t_{L-1}^{(2)}}+\frac{-t_{L-2}^{(1)}}{t_{L-2}^{(1)}-t_{L-1}^{(2)}}+
\sum_{m=l}^{L-2} \left(\frac{-t_{m+1}^{(1)}}{t_{m+1}^{(1)}-t_{m}^{(2)}}+\frac{2t_{m}^{(1)}}{t_{m}^{(1)}-t_{m}^{(2)}}
+\frac{-t_{m-1}^{(1)}}{t_{m-1}^{(1)}-t_{m}^{(2)}}\right)\right){1\over t_{L-1}^{(1)}}f_n^{(i)}(t^{(1)}){1\over t_{L-1}^{(2)}}f_l^{(j)}(t^{(2)})
\right]\nonumber
\\
&=\mathrm{Sym}\left[
\frac{1}{t_{L-1}^{(1)}-t_{L-1}^{(2)}}\frac{1}{t_{L-2}^{(1)}-t_{L-1}^{(2)}}\frac{1}{t_{L-2}^{(2)}-t_{L-1}^{(1)}}
{1\over t_{L-1}^{(1)}}f_{n,L-1}^{(i)}(t^{(1)})f_{l,L-1}^{(j)}(t^{(2)})
\right]
\\
&=\mathrm{Sym}\left[
\frac{-1}{t_{L-1}^{(1)}-t_{L-1}^{(2)}}{1\over t_{L-1}^{(2)}}f_{n}^{(j)}(t^{(1)})f_{l}^{(i)}(t^{(2)})
\right],
\end{align*}
where in the last line, we interchange $t_{L-1}^{(1)}$ with $t_{L-1}^{(2)}$. Hence, the left hand side of \eqref{eq n<l Y} is
equal to
\begin{align*}
&\mathrm{Sym}\left[
\frac{-1}{t_{L-1}^{(1)}-t_{L-1}^{(2)}}
f_{n}^{(j)}(t^{(1)}){1\over t_{L-1}^{(2)}}f_{l}^{(i)}(t^{(2)})+\frac{1}{t_{L-1}^{(1)}-t_{L-1}^{(2)}}
f_n^{(i)}(t^{(1)}){1\over t_{L-1}^{(2)}}f_{l}^{(j)}(t^{(2)})
\right]\nonumber
\\
&=(z_i-z_j)\mathrm{Sym}\left[\frac{1}{1-z_jt_{L-1}^{(1)}}f_n^{(i)}(t^{(1)})\frac{1}{t_{L-1}^{(2)}(1-z_it_{L-1}^{(2)})}
f_l^{(j)}
\right].
\end{align*}
Therefore, the relation \eqref{eq n<l Y} holds.
We shall show \eqref{eq 1<l Y}. We compute the left hand side of \eqref{eq 1<l Y} as follows.
\begin{align*}
\mathrm{L.H.S.\ of \ \eqref{eq 1<l Y}}=&\mathrm{Sym}\left[\left(\frac{-t_1^{(1)}}{t_1^{(1)}-1}+
C(1,1,2)\right){1\over t_{L-1}^{(1)}}f_1^{(i)}(t^{(1)}){1\over t_{L-1}^{(2)}}f_l^{(j)}(t^{(2)})
\right]
\\
&+\mathrm{Sym}\left[{t_{1}^{(1)}\over t_1^{(1)}-1}{1\over t_{L-1}^{(1)}}f_1^{(i)}(t^{(1)}){1\over t_{L-1}^{(2)}}f_l(t^{(2)})
\right].
\end{align*}
The first line of the right hand side of the relation above becomes the first line of the right hand side of \eqref{eq 1<l Y}
in the same way of the proof of \eqref{eq n<l Y}. While, we have
\begin{align*}
&\mathrm{Sym}\left[{1\over t_1^{(1)}-1}{t_{1}^{(1)}\over t_{L-1}^{(1)}}f_1^{(i)}(t^{(1)}){1\over t_{L-1}^{(2)}}f_l(t^{(2)})
\right]
\\
&=\mathrm{Sym}\left[{t_{L-1}^{(1)}+\sum_{m=2}^{L-1}(t_{m-1}^{(1)}-t_m^{(1)})
\over t_1^{(1)}-1}{1\over t_{L-1}^{(1)}}f_1^{(i)}(t^{(1)}){1\over t_{L-1}^{(2)}}f_l(t^{(2)})
\right]
\\
&=\mathrm{Sym}\left[{1\over z_i-1}{1\over t_{L-1}^{(1)}}\left(f_0(t^{(1)})-f_1(t^{(1)})-z_i\sum_{m=2}^{L-1}f_m^{(i)}(t^{(1)})\right){1\over t_{L-1}^{(2)}}f_l(t^{(2)})
\right].
\end{align*}
Therefore, the relation \eqref{eq 1<l Y} holds.
\end{proof}
\begin{lem}\label{lem n=l Y}
For $1\le j\neq i\le L-1$, we have
\begin{align*}
\left(Y_n^{(i)}\right)_{n,j}=&
A_{n,j} \left( \frac{1-\delta_{n,1}}{z_i-1}-\frac{2z_j}{z_i-z_j} \right) \varphi_{A }(t)+
\frac{1-\delta_{n,1}}{z_i-1}\frac{\left(A_0+1\right)A_{n,j}}{A_{n,i}}
\varphi_{\left(A_{n,i}-1 \right)}(t)
\\
&+\frac{1-\delta_{n,1}}{z_i-1}\frac{A_{n,j}}{A_{n,i}}\left(\sum_{m=1}^{n-1}
(A_{m,i}+1)
\varphi_{\left(A_{m,i}+1, A_{n,i}-1 \right)}(t)
+z_i\sum_{m=n+1}^{L-1}
(A_{m,i}+1)
\varphi_{\left(A_{m,i}+1, A_{n,i}-1 \right)}(t)\right)
\\
&+\left(
A_{n,i}+1
\right) \frac{z_i}{z_i-z_j} \varphi_{\left(A_{n,j}-1, A_{n,i}+1 \right)}(t)
+\frac{\left(
A_{n,j}+1
\right)A_{n,j}}{A_{n,i}}\frac{z_j}{z_i-z_j}\varphi_{\left(A_{n,j}+1, A_{n,i}-1 \right)}(t),
\end{align*}
and
for $j=i$, we have
\begin{align*}
\left(Y_n^{(i)}\right)_{n,i}=&
\left(
A_{n,i}-1
\right)\left(\frac{ z_i-\delta_{n,1}}{z_i-1}\right)
\varphi_{A}(t)+
\frac{ 1-\delta_{n,1}}{z_i-1}\frac{\left(A_0+1\right)\left(A_{n,i}-1\right)}{A_{n,i}}
\varphi_{\left(A_{n,i}-1 \right)}(t)
\\
&+\frac{1-\delta_{n,1}}{z_i-1}\frac{A_{n,i}-1}{A_{n,i}}\left(\sum_{m=1}^{n-1}
(A_{m,i}+1)
\varphi_{\left(A_{m,i}+1, A_{n,i}-1 \right)}(t)
+z_i\sum_{m=n+1}^{L-1}
(A_{m,i}+1)
\varphi_{\left(A_{m,i}+1, A_{n,i}-1 \right)}(t)\right).
\end{align*}
\end{lem}
\begin{proof}
It suffices to show that
\begin{align}
&\mathrm{Sym}\left[
C(n,1,2){1\over t_{L-1}^{(1)}}f_n^{(i)}(t^{(1)}){1\over t_{L-1}^{(2)}}f_n^{(j)}(t^{(2)})
\right]\nonumber
\\
&=\frac{1-\delta_{n,1}}{z_i-1}
\mathrm{Sym}\left[{1\over t_{L-1}^{(1)}}\left(-f_0(t^{(1)})+\sum_{m=1}^{n}f_m^{(i)}(t^{(1)})+z_i\sum_{m=n+1}^{L-1}f_m^{(i)}(t^{(1)})\right){1\over t_{L-1}^{(2)}}f_n^{(j)}(t^{(2)})
\right]\nonumber
\\
&+\mathrm{Sym}\left[
{1\over t_{L-1}^{(1)}}f_n^{(i)}(t^{(1)}){1\over t_{L-1}^{(2)}}f_n^{(i)}(t^{(2)})
\right]\nonumber
\\
&+{z_j\over z_i-z_j}\mathrm{Sym}\left[
{1\over t_{L-1}^{(1)}}{1\over t_{L-1}^{(2)}}\left(f_n^{(i)}(t^{(1)})-f_n^{(j)}(t^{(1)})\right)
\left(
f_n^{(i)}(t^{(2)})-f_n^{(j)}(t^{(2)})\right)
\right],
\label{eq l=n Y}
\end{align}
where the symmetrization $\mathrm{Sym}[f(t)]$ stands for $\sum_{\sigma\in\frak{S}_2^{L-1}}\sigma(f(t))$
(see \eqref{eq sigma}), and
if $j=i$, then
we understand that the third line of the right hand side of \eqref{eq l=n Y} is vanished.
Firstly, we have
\begin{align*}
&\mathrm{Sym}\left[\frac{-t_n^{(1)}}{t_n^{(1)}-t_{n-1}^{(2)}}
{1\over t_{L-1}^{(1)}}f_n^{(i)}(t^{(1)}){1\over t_{L-1}^{(2)}}f_n^{(j)}(t^{(2)})
\right]
\\
&=\mathrm{Sym}\left[{t_{L-1}^{(1)}+\sum_{m=n+1}^{L-1}(t_{m-1}^{(1)}-t_m^{(1)})
\over t_n^{(1)}-t_{n-1}^{(1)}}{1\over t_{L-1}^{(1)}}f_n^{(i)}(t^{(1)}){1\over t_{L-1}^{(2)}}f_n^{(j)}(t^{(2)})
\right]
\\
&={1\over z_i-1}\mathrm{Sym}\left[{1\over t_{L-1}^{(1)}}\left(-f_0(t^{(1)})+\sum_{m=1}^{n}f_m^{(i)}(t^{(1)})+z_i\sum_{m=n+1}^{L-1}f_m^{(i)}(t^{(1)})\right){1\over t_{L-1}^{(2)}}f_n^{(j)}(t^{(2)})
\right].
\end{align*}
Secondly, we notice that for $n\le m\le L-2$, we have
\begin{equation*}
\mathrm{Sym}\left[
\left(\frac{-t_{m+1}^{(1)}}{t_{m+1}^{(1)}-t_{m}^{(2)}}+\frac{2t_{m}^{(1)}}{t_{m}^{(1)}-t_{m}^{(2)}}
+\frac{-t_{m-1}^{(1)}}{t_{m-1}^{(1)}-t_{m}^{(2)}}\right){1\over t_{L-1}^{(1)}}f_n^{(i)}(t^{(1)}){1\over t_{L-1}^{(2)}}f_n^{(j)}(t^{(2)})
\right]=0.
\end{equation*}
Thirdly, we compute the remaining term as follows.
\begin{align*}
&\mathrm{Sym}\left[\frac{2}{t_{L-1}^{(1)}-t_{L-1}^{(2)}}
f_n^{(i)}(t^{(1)}){1\over t_{L-1}^{(2)}}f_n^{(j)}(t^{(2)})\right]
\\
&=\mathrm{Sym}\left[
\frac{-1}{t_{L-1}^{(1)}-t_{L-1}^{(2)}}
f_{n}^{(j)}(t^{(1)}){1\over t_{L-1}^{(1)}}f_{n}^{(i)}(t^{(2)})+\frac{1}{t_{L-1}^{(1)}-t_{L-1}^{(2)}}
f_n^{(i)}(t^{(1)}){1\over t_{L-1}^{(2)}}f_{n}^{(j)}(t^{(2)})
\right]\nonumber
\\
&=\mathrm{Sym}\left[{1\over t_{L-1}^{(1)}}{1\over t_{L-1}^{(2)}}
f_{n}^{(i)}(t^{(1)})f_{n}^{(i)}(t^{(2)})\frac{1-z_j(t_{L-1}^{(1)}+t_{L-1}^{(2)})+z_iz_jt_{L-1}^{(1)}t_{L-1}^{(2)}}{(1-z_jt_{L-1}^{(1)})(1-z_jt_{L-1}^{(2)})}
\right].
\end{align*}
Therefore, the relation \eqref{eq l=n Y} holds.
\end{proof}
\begin{lem}\label{lem l=0 Y}
We have
\begin{align*}
\left(Y_n^{(i)}\right)_0=&
-\frac{1+\delta_{n,1}}{z_i-1}A_0 \varphi_{A}(t)
+\frac{z_i}{z_i-1}
\sum_{m=1}^{L-1}\left(
A_{m,i}+1
\right) \varphi_{\left(A_{m,i}+1 \right)}(t)
\\
&-\delta_{n,1}\frac{1}{(z_i-1)}{A_0\left(
A_0+1
\right)\over A_{1,i}}
\varphi_{\left(A_{1,i}-1 \right)}(t)
-\delta_{n,1}\frac{z_i}{z_i-1}\sum_{m=2}^{L-1}{A_0\left(
A_{m,i}+1
\right) \over A_{1,i}}\varphi_{\left(A_{m,i}+1, A_{1,i}-1 \right)}(t).
\end{align*}
\end{lem}
\begin{proof}
It suffices to show that
\begin{align}
&\mathrm{Sym}\left[
C(n,1,2){1\over t_{L-1}^{(1)}}f_n^{(i)}(t^{(1)}){1\over t_{L-1}^{(2)}}f_0(t^{(2)})
\right]\nonumber
\\
&=\mathrm{Sym}\left[{1\over (z_i-1)t_{L-1}^{(1)} t_{L-1}^{(2)}}f_n^{(i)}(t^{(1)})\left(-(1+\delta_{n,1})
f_0(t^{(2)})
+z_i
\sum_{m=1}^{L-1}f_m^{(i)}(t^{(2)})
\right)
\right]\nonumber
\\
&+\delta_{n,1}\mathrm{Sym}\left[
{1\over(z_i-1) t_{L-1}^{(1)} t_{L-1}^{(2)}}f_0(t^{(2)})\left(f_0(t^{(1)})-z_i\sum_{m=2}^{L-1}f_m^{(i)}(t^{(1)})\right)
\right], \label{eq l=0}
\end{align}
where the symmetrization $\mathrm{Sym}[f(t)]$ stands for $\sum_{\sigma\in\frak{S}_2^{L-1}}\sigma(f(t))$ and
the rational functions $f_0(t^{(a)})$ are defined in Definition \ref{def integral formula}.
Firstly, using \eqref{eq l<n claim} for $l=0$ and $k=L-2$, we have
\begin{align*}
&\mathrm{Sym}\left[\left(C(n,1,2)+\delta_{n,1}{-1\over t_1^{(1)}-1}{t_{1}^{(1)}\over t_{L-1}^{(1)}}\right)
{1\over t_{L-1}^{(1)}}f_n^{(i)}(t^{(1)}){1\over t_{L-1}^{(2)}}f_0(t^{(2)})
\right]
\\
&=\mathrm{Sym}\left[
\frac{-1}{t_{L-1}^{(1)}-t_{L-1}^{(2)}}\frac{1-z_it_{L-1}^{(1)}}{1-z_it_{L-1}^{(2)}}
f_{n}^{(i)}(t^{(1)}){1\over t_{L-1}^{(1)}}f_{0}(t^{(2)})+\frac{1}{t_{L-1}^{(1)}-t_{L-1}^{(2)}}
f_n^{(i)}(t^{(1)}){1\over t_{L-1}^{(2)}}f_{0}(t^{(2)})
\right]
\\
&=
\mathrm{Sym}\left[{1\over t_{L-1}^{(1)}}f_n^{(i)}(t^{(1)})\frac{1}{t_{L-1}^{(2)}(1-z_it_{L-1}^{(2)})}f_{0}(t^{(2)})
\right].
\end{align*}
Secondly, we have
\begin{align*}
&\mathrm{Sym}\left[{1\over t_1^{(1)}-1}{t_{1}^{(1)}\over t_{L-1}^{(1)}}f_1^{(i)}(t^{(1)}){1\over t_{L-1}^{(2)}}f_0(t^{(2)})
\right]
\\
&=\mathrm{Sym}\left[{t_{L-1}^{(1)}+\sum_{m=2}^{L-1}(t_{m-1}^{(1)}-t_m^{(1)})
\over t_1^{(1)}-1}{1\over t_{L-1}^{(1)}}f_1^{(i)}(t^{(1)}){1\over t_{L-1}^{(2)}}f_0(t^{(2)})
\right]
\\
&=\mathrm{Sym}\left[{1\over z_i-1}{1\over t_{L-1}^{(1)}}\left(f_0(t^{(1)})-f_1(t^{(1)})-z_i\sum_{m=2}^{L-1}f_m^{(i)}(t^{(1)})\right){1\over t_{L-1}^{(2)}}f_0(t^{(2)})
\right].
\end{align*}
Therefore, the relation \eqref{eq l=0} holds.
\end{proof}
\bigskip
{\bf Acknowledgements.}
The author is grateful to
T.~Tsuda
and Y.~Yamada
for helpful discussions.
This work was partially supported by Grant-in-Aid for Japan Society for the Promotion Science Fellows 22-2255.
|
{
"timestamp": "2012-03-12T01:00:52",
"yymm": "1203",
"arxiv_id": "1203.2009",
"language": "en",
"url": "https://arxiv.org/abs/1203.2009"
}
|
\section{Introduction} The wave-particle interaction has long been considered a dominant energy-momentum exchange mechanism in space and astrophysical plasmas. Beyond the dense and internal boundaries of stars and planetary magnetospheres, space and astrophysical plasmas are predominantly collisionless, and populated by distributions functions inconsistent with collisional equilibrium conditions. These plasmas are also believed to be turbulent systems described by a conventional energy cascade from large scales to small scales where dissipation takes place. While fluid theories provide satisfactory descriptions of macroscopic quantities at large scales, they are not equipped to explain the plasma physics at smaller kinetic scales, and one needs to include nonlinear kinetic processes (wave-particle interactions and wave-wave interactions) for a correct description of these turbulent and collisionless plasmas \cite{Kulsrud}.\\
Studies of wave-particle interactions in space and astrophysical turbulent plasmas have commonly fallen under the scope of quasi-linear theory \cite{Kennel66, Roux70}. Quasi-linear theory departs from linear theory in conserving energy and momentum through the account of the wave-particle interaction, resulting in a diffusion process for an ensemble averaged distribution function solution to a Fokker-Planck equation. However, quasi-linear theory is constrained by a number of severe limitations making it inapplicable for plasmas containing large-amplitude quasi-monochromatic electromagnetic and/or electrostatic waves. Indeed, the first assumption for quasi-linear theory consists in constraining particle orbits to their unperturbed components. A second assumption consists in assuming a wave spectrum sufficiently dense so that interference between modes is destroyed by phase-mixing. Hence, quasi-linear theory is valid only when the bandwidth is broad enough to enable resonances to be maintained as particles are scattered, and non-linear trapping effects by individual waves are to weak to be taken into consideration. \\%Paragraph on the Radiation belts
Due to the inherent difficulties of strong turbulence theories, test-particle methods have become a favored tool for the study of wave-particle interaction beyond the constraints of quasi-linear theory. Numerous methods have been developed in the context of the radiation belts alone, ranging from guiding-center approximation \cite{Omura82}, to resonance-averaged Hamiltonian \cite{Albert02}, gyroresonance averaged equations \cite{Bortnik08} and computer simulations taking into account approximate and exact dipolar fields [see review \cite{Sh08} and associated references]. However, the general case of oblique has often been avoided in favor of the more tractable case of parallel of propagation \cite{Tao10}. A strong case can be made for the neglect of oblique propagation and nonlinear effects for small amplitude waves since the appearance of a small parallel electric field can not trap orbits \cite{Albert10}. However, if the electric field component of the wave becomes sufficiently large, such that nonlinear effects can be triggered, then a rich class of orbits can result, and the parallel approximation becomes invalid.\\
Recent observations of the weakly collisional plasma of the radiation belts are suggesting that obliquely propagating waves with Poynting flux two orders of magnitude larger than previously observed whistlers waves are commonly generated in the radiation belts and appear correlated with relativistic electron microbursts \cite{Wilson11}. These large amplitude waves can propagate at large propagation angles ($\theta \leq 70^o$), and possess amplitudes capable of energizing electrons on time scales of the order of the milliseconds \cite{Yoon11}. If these large-amplitudes waves are shown to be a common observational signature in the radiation belts, the conventional models used to describe the wave-particle interaction using a quasi-linear formalism will have to be revisited. It is not only inaccurate to assume that a particle will execute a random walk in pitch angle during the course of one bounce period, but as demonstrated below, a particle can be irreversibly accelerated to relativistic energies in less than one bounce period.\\
In this report, we investigate the exact nonlinear wave-particle interaction in the relativistic regime. The inclusion of relativistic effects is a \textit{sine qua non} condition for any attempt at solving the outstanding problems which have emerged in radiation belts dynamics as well as galactic cosmic ray. Our goal is to provide a general framework for the wave-particle interaction by using a dynamical systems' approach. Such approach, although lacking the level of self-consistency found in numerical simulations, can facilitate the understanding of complex systems such as cosmic and space plasmas and therefore provide for an intuitive as well as quantitative leap between theoretical models and simulations.\\
The report is written as follows. In section 2 we derive the dynamical system as well as its fixed points and invariants. In section 3 we treat the special cases of parallel and purely perpendicular propagation. In section 4 we study the general case of oblique propagation for the cyclotron-resonance case as well as the Landau resonance. Section 5 contains a discussion of the general framework for the understanding of the wave-particle interaction and the effect of oblique propagation in collisionless plasmas such as the radiation belts. In section 6 we conclude and discuss studies currently underway to address limitations of the dynamical system approach.
\section{Dynamical System.}
\subsection{Equation of motion for the general case.}
Our study begins with the equation of motion of a particle in an electromagnetic field, as described by the Lorentz equation. The equations of motion can be written as
\begin{equation}
\frac{d\textbf{p}}{dt}= e\bigg{[}\textbf{E}(\textbf{x},t)+\frac{\textbf{p}}{m\gamma c} \times \textbf{B}(\textbf{x},t)\bigg{]},
\end{equation}
for a particle of charge $e$, momentum $\textbf{p}=m\gamma\textbf{v}$ and rest mass $m$. The Lorentz factor, $\gamma$, is defined in terms of the relativistic momentum as follows :
\begin{equation}
\label{eq:constraint}
\gamma=\sqrt{1+\frac{p^2}{m^2c^2}}.
\end{equation}
The electromagnetic field configuration consists of a background magnetic field $B_0$ to which is superposed an electromagnetic wave given by $(\delta \textbf{E}, \delta \textbf{B})$ :
\begin{equation}
\textbf{E}(\textbf{x},t)=\delta \textbf{E}(\textbf{x},t)
\end{equation}
\begin{equation}
\textbf{B}(\textbf{x}.t)=\textbf{B}_0+\delta \textbf{B}(\textbf{x},t)
\end{equation}
The electromagnetic wave vector $\mathbf{k}$ is chosen to point in the $\hat{z}$ direction, obliquely to the background magnetic field lying in the y-z plane :
\begin{equation}
\textbf{k} \cdot \textbf{B}_0=kB_0\cos(\theta)
\end{equation}
\begin{equation}
\left\{
\begin{array}{l l}
\delta\textbf{E} = \delta E_x \hat{\textbf{x}} + \delta E_y \hat{\textbf{y}}\\
\delta\textbf{B} = \delta B_x \hat{\textbf{x}} + \delta B_y \hat{\textbf{y}}\\
\end{array} \right.
\end{equation}
with the wave magnetic field components written as
\begin{equation}
\left\{
\begin{array}{l l}
\delta B_x= \delta B \sin(kz-\omega t)\\
\delta B_y=\delta B \cos(kz-\omega t)\\
\end{array} \right.
\end{equation}
and Faraday's law, expressed in terms of the Fourier components, providing for the components of the electric field\begin{equation}
c\textbf{k} \times \delta \textbf{E} (\textbf{k},\omega) =\omega \delta \textbf{B}(\textbf{k},\omega)
\end{equation}
We can therefore express the dynamical system in terms of the phase velocity $v_\phi=\omega/k$, and the variables $p_\Phi=m\gamma v_\Phi$ ; $\Omega_1=e\delta B/mc\gamma$ ; $\Omega_0=e B_0/mc\gamma$, resulting in the following coupled ordinary differential equations :
\begin{equation}
\left\{
\begin{array}{l l}
\dot{p}_x=p_y\Omega_0\cos(\theta)+(p_\Phi-p_z)\Omega_1\cos(kz-\omega t) +p_z\Omega_0\sin(\theta)\\
\dot{p}_y=-p_x\Omega_0\cos(\theta)+(p_z-p_\Phi)\Omega_1\sin(kz-\omega t)\\
\dot{p}_z=-p_x\Omega_0 \sin(\theta)+p_x\Omega_1\cos(kz-\omega t) -p_y\Omega_1\sin(kz-\omega t)\\
\dot{z}=p_z v_\Phi/p_\Phi
\end{array}
\right.
\end{equation}
In the classical case, the dynamical system is composed of four equations, the three components of the velocity plus the position coordinate along $k$. However, in the relativistic case the expression for the Lorentz factor must be obeyed and constitutes a constraint on the particle's trajectory. We can keep track of this constraint by adding an equation in the expression of a dynamical gyrofrequency :
\begin{eqnarray}
\label{eq:omega_dot}
\dot{\Omega}_0 & = & \frac{d}{dt}\bigg{(}\frac{eB_0}{mc\gamma}\bigg{)} \nonumber \\
&=&-\Omega_0 \frac{pc^2}{m^2c^4+p^2c^2}\dot{p}\nonumber\\
&=&-\frac{\Omega_0\Omega_1p_\Phi}{m^2\gamma^2c^2}\bigg{(}p_x\cos(kz-\omega t)-p_y\sin(kz-\omega t)\bigg{)}
\end{eqnarray}
If we define the constant $\delta_1=\Omega_1/\Omega_0$, it is straightforward to see that $\dot{\Omega}_1=\delta_1 \dot{\Omega}_0$, and similarly, since $p_\Phi=p_\Phi(\gamma)$, the time evolution of this quantity can be written as :
\begin{equation}
\label{eq:p_phi}
\dot{p}_\Phi=-mv_\Phi\gamma\frac{\dot{\Omega}_0}{\Omega_0}.
\end{equation}
In order to simplify the dynamical system, we can eliminate the explicit time dependence of the equations by making the following mathematical transformation :
\begin{equation}
p_x'=p_x, \hspace{2mm}p_y'=p_y, \hspace{2mm} p_z'=\gamma_w(p_z-p_\phi), \hspace{2mm} z'=\gamma_w(z-v_\phi t)
\end{equation}
for the Lorentz factor :
\begin{equation}
\gamma_w=\frac{1}{\sqrt{1-\frac{v_\Phi^2}{c^2}}}.
\end{equation}
We can therefore write the equations of motion as follow :
\begin{equation}
\left\{
\begin{array}{l l}
\dot{p}_x'=\Omega_0 p_y'\cos(\theta)-\Omega_1p_z' \cos(kz'/\gamma_w)/\gamma_w +\Omega_0 (p_z'/\gamma_w+p_\phi) \sin(\theta)\\
\dot{p}_y'=-\Omega_0p_x'\cos(\theta)+\Omega_1p_z' \sin(kz'/\gamma_w)/\gamma_w\\
\dot{p}_z'/\gamma_w=-\Omega_0p_x'\sin(\theta)+\Omega_1p_x'\cos(kz'/\gamma_w) -\Omega_1p_y'\sin(kz'/\gamma_w)-\dot{p}_\Phi\\
\dot{z}'=p_z'v_\Phi/p_\Phi\\
\end{array}
\right.
\end{equation}
If we absorb the Lorentz factor $\gamma_w$ into $p_z'$ and $k$, that is we write $p_z' \to p_z'/\gamma_w$ and $k \to k/\gamma_w$, and write $\dot{p}_\Phi$ in terms of $(p_x',p_y',p_z',z',\Omega_0)$, we can express the dynamical system as :
\begin{equation}
\label{eq:ds_in_ps}
\left\{
\begin{array}{l l}
\dot{p}_x'=\Omega_0 p_y'\cos(\theta)-\Omega_1p_z' \cos(kz') +\Omega_0 (p_z'+p_\phi) \sin(\theta)\\
\dot{p}_y'=-\Omega_0p_x'\cos(\theta)+\Omega_1p_z' \sin(kz')\\
\dot{p}_z'=-\Omega_0p_x'\sin(\theta)+\Omega_1(\frac{n^2-1}{n^2})(p_x'\cos(kz') -p_y'\sin(kz'))\\
\dot{z}'=p_z'v_\Phi/p_\Phi\\
\end{array}
\right.
\end{equation}
where the refractive index is represented as $n^2=c^2/v_\Phi^2$. The magnitude of the momentum is now written as $p'=\sqrt{p_x^{'2}+p_y^{'2}+(p_z^{'}/\gamma_w)^2}$, hence the Lorentz contraction factor also transforms from $\gamma(p) \to \gamma(p')$. The dynamical system for the classical case can be recovered by setting $\gamma = 1$ and $1/n^2\rightarrow 0$. Indeed, the equations are equivalent to the classical case under the following transformations : $p \to u$ and $\Omega \to \Omega/\gamma$ . Hence the main difference lies in the time dependence of the Larmor frequencies and the extra term that goes as $1/n^2$ in the $\dot{p}_z'$ equation.
\subsection{Representation in terms of ($P$, $\alpha$, $\Phi$, $z'$).}
It is convenient to express the relativistic momentum in spherical coordinates, that is in terms of a magnitude $p'$ and phase angles $(\alpha, \Phi)$. This can be achieved by introducing the following scalar and vector variables :
\begin{equation}
\left\{
\begin{array}{l l}
p_{\parallel} = \textbf{p}\cdot \hat{\textbf{b}}_0\\
\textbf{p}_\perp=\hat{\textbf{p}}_0 \times (\textbf{p} \times \hat{\textbf{p}}_0) = \textbf{p}-p_{\parallel}\hat{\textbf{b}}_0\\
\end{array}
\right.
\end{equation}
where $\hat{\textbf{b}}_0 = \textbf{B}_0/B_0$. Using these definitions, we can rewrite the momentum $\textbf{p}'=(p_x',p_y',p_z')$ in terms of the pitch angle $\alpha$ and the dynamical gyrophase $\Phi$, both defined as
\begin{equation}
\left\{
\begin{array}{l l}
\label{eq:tan_alpha}
\tan (\alpha) = {\displaystyle\frac{p'_\perp}{p'_{\parallel}}}\\
\tan (\Phi) ={\displaystyle\frac{p'_{\perp 1}}{p'_{\perp 2}}=\frac{p_x'}{p_y'\cos(\theta)+p_z'\sin(\theta)}}
\end{array}
\right.
\end{equation}
Hence, all three momentum components in the wave frame are written as :
\begin{equation}
\label{eq:spherical}
\left\{
\begin{array}{l l}
p_x'=p'\sin(\alpha)\cos(\Phi)\\
p_y'=p'\sin(\alpha)\sin(\Phi)\cos(\theta)-p'\cos(\alpha)\sin(\theta)\\
p_z'=p'\sin(\alpha)\sin(\Phi)\sin(\theta)+p'\cos(\alpha)\cos(\theta).\\
\end{array}
\right.
\end{equation}
Using the definitions (\ref{eq:tan_alpha}) and the representation of the momentum in (\ref{eq:spherical}), we can proceed to write the dynamical system (\ref{eq:ds_in_ps}) in terms of the normalized variables :
\begin{equation}
\label{eq:norm}
P=\frac{kp'}{m\omega},\hspace{1mm}\delta_1=\frac{\Omega_1}{\Omega_0},\hspace{1mm}\delta_2=\frac{m\omega c}{eB_0}, \hspace{1mm}\delta_3=\frac{1}{\delta_2\gamma}, \hspace{1mm}n^2=\frac{c^2}{v_\Phi^2},\hspace{1mm} Z=kz', \tau=\omega t
\end{equation}
and the function $F(\alpha, \Phi, Z)$, as follow :
\begin{equation}
\label{eq:DS}
\left\{
\begin{array}{l l}
{\displaystyle\frac{dP}{d\tau}}=\sin(\alpha)\cos(\Phi)\sin(\theta)/\delta_2 \\
\hspace{8mm}- {\displaystyle\frac{\delta_1\delta_3 P}{n^2}}\bigg{(}\sin(\alpha)\sin(\Phi)\sin(\theta)+\cos(\alpha)\cos(\theta)\bigg{)}F(\alpha, \Phi, Z)
\\
\\
{\displaystyle\frac{d\alpha}{d\tau}} = {\displaystyle\frac{1}{P\delta_2}}\cos(\alpha)\cos(\Phi)\sin(\theta) \\
\hspace{8mm} -\delta_1\delta_3\bigg{(}\cos^2 {\displaystyle(\frac{\theta}{2})}\cos(\Phi+Z)-\sin^2 {\displaystyle(\frac{\theta}{2})}\cos(\Phi-Z)\bigg{)}\\
\hspace{8mm}+ {\displaystyle\frac{\delta_1\delta_3}{n^2}}\bigg{(}\cos(\theta)\sin(\alpha)-\cos(\alpha)\sin(\Phi)\sin(\theta)\bigg{)}F(\alpha,\Phi,Z)
\\
\\
{\displaystyle\frac{d\Phi}{d\tau}} = -\delta_3 + \sin(\theta)\bigg{(}\delta_1\delta_3\cos(Z)- {\displaystyle\frac{\sin(\Phi)}{\delta_2P\sin(\alpha)}}\bigg{)} \\
\hspace{8mm}+ {\displaystyle\frac{\delta_1\delta_3}{\tan(\alpha)}}\bigg{(}\cos^2 {\displaystyle(\frac{\theta}{2})}\sin(\Phi+Z)+\sin^2 {\displaystyle(\frac{\theta}{2})}\sin(Z-\Phi)\bigg{)}\\
\hspace{8mm}- {\displaystyle\frac{\delta_1\delta_3}{n^2}\frac{\cos(\Phi)\sin(\theta)}{\sin(\alpha)}}F(\alpha, \Phi, Z)\\
\\
{\displaystyle\frac{dZ}{d\tau}}=\delta_2\delta_3 P\bigg{(}\sin(\alpha)\sin(\Phi)\sin(\theta)+\cos(\alpha)\cos(\theta)\bigg{)}\\
\\
{\displaystyle\frac{d\delta_3}{d\tau}}=- {\displaystyle\frac{\delta_1\delta_2\delta_3^3 P}{n^2}}F(\alpha, \Phi, Z)\\
\\
F(\alpha,\Phi,Z)=\sin(\alpha)\sin^2 {\displaystyle(\frac{\theta}{2})}\cos(\Phi-Z)\\
\hspace{22mm}+\sin(\alpha)\cos^2 {\displaystyle(\frac{\theta}{2})}\cos(\Phi+Z) +\cos(\alpha)\sin(\theta)\sin(Z).
\end{array}
\right.
\end{equation}
It is easy to observe that we can recover the classical regime by setting $\dot{\delta}_3=\dot{\gamma}/\gamma\delta_2=0$ or $F(\alpha, \Phi, Z)=0$. We now proceed to study some of the properties of the dynamical system.
\subsection{Fixed Points.}
A common first step in the study of dynamical systems is to find and investigate the properties of fixed (stationary) points. The fixed points of the dynamical system (\ref{eq:DS}) are defined as the values in $(P, \alpha, \Phi, Z, \gamma)$, for which $(\dot{P}=\dot{\alpha}=\dot{\Phi}=\dot{Z}=\dot{\gamma}=0)$. It can be demonstrated (see Appendix 1) that the dynamical system, for $-\pi/2<\theta<0$, possesses the following values for the fixed points :
\begin{eqnarray}
\label{eq:FP77}
P=-\gamma \tan(\theta); \hspace{10mm} \alpha =\pm\theta\pm\frac{\pi}{2}; \nonumber \\
\Phi=\pm \frac{\pi}{2}; \hspace{15mm} Z=0,\pi ; \\
\gamma=\frac{1}{\sqrt{1-{\displaystyle\frac{v_\Phi^2}{c^2}}\bigg{(}1+\tan^2(\theta)\bigg{)}}}.\nonumber
\end{eqnarray}
It is already evident from (\ref{eq:FP77}) that in the case of parallel propagation ($\theta=0$), the only fixed point is that for the trivial case $P=0$. The fixed point for the relativistic regime appears therefore similar to the classical one for parallel and oblique propagation \cite{Hamza06}. The fixed point for the relativistic regime will translate into properties found in the non-relativistic regime, but also results in different types of structures in their vicinity. For non-zero propagation angles, fixed points identify volumes of phase-space composed of physically trapped orbits. The trapped orbits could give rise to kinetic distortions in the distribution functions, such as beams and temperature anisotropies, as was revealed in the classical non-relativistic case \cite{Osmane10}. However, because the relativistic equations possess a constraint in the form of the Lorentz factor $\gamma$, different effects are shown to arise.
\subsection{Invariants.}
The dynamical system (\ref{eq:DS}) also possesses a number of invariants valid for the general case of oblique propagation. Knowledge of these invariants is used to construct pseudo-potential structures. In turn, these structures provide information on trapped and quasi-trapped orbits.
\subsubsection{First invariant : $I_1$.}
Using equations (\ref{eq:norm}) to normalize equation (\ref{eq:omega_dot}), the equation describing the evolution of the gyrofrequency can be written as follow :
\begin{equation}
\dot{\delta_3}=-\delta_3 \frac{1}{1+\frac{n^2}{\Gamma^2}}\frac{\dot{\Gamma}}{\Gamma}
\end{equation}
for $\Gamma=kp/m\omega$. Hence, this equation has an exact solution, providing the following constant of the motion :
\begin{equation}
I_1=\delta_3\sqrt{\Gamma^2+n^2}.
\end{equation}
We can write this invariant in terms of the variables $P, \alpha, \Phi, Z, $ and $\delta_3$ as follow:
\begin{eqnarray}
I_1&=&\sqrt{\delta_3^2P^2+2\delta_3P_z/\delta_2+\delta_3^2n^2}
\end{eqnarray}
The conservation of this quantity will indicate the degree to which the constraint for the Lorentz factor (\ref{eq:constraint}) is respected in a numerical scheme.
\subsubsection{Second invariant : $I_2$.}
A second general invariant can be found and expressed in terms of the normalized variables as follow :
\begin{equation}
I_2=\delta_2(n^2-1)\gamma\cos(\theta)-\delta_2P\cos(\alpha)+\delta_1\sin(\theta)\cos(Z)
\end{equation}
This invariant underlies a fundamental property of oblique propagation. One can indeed rewrite the invariant in the form $E=m\gamma c^2\sim P_\parallel$, which means that one needs a change in the parallel momentum to change the energy. This is a well-known statement resulting from the Maxwell-Lorentz invariant quantity $\mathbf{E}\cdot\mathbf{B}=0$, since a parallel component of the electric field can not be eliminated by any Lorentz translation, while the physics in a frame with $E_\parallel=0$, such as in the case of parallel propagation, is no different, therefore, than the physics in a frame where $E_\perp=E_\parallel=0$ for which energy is a constant of the motion.
\section{Special Cases.}
\subsection{Parallel Propagation : $\theta=0$.}
The wave-particle interaction problem has overwhelmingly been treated for the special case of parallel propagation. Even though we do not present any new result in this section, we find it useful to briefly discuss the parallel case as a means of comparison to the general oblique case. Setting $\theta=0$ in (\ref{eq:DS}), we recover the following dynamical system :
\begin{equation}
\label{eq:DS_0}
\left\{
\begin{array}{l l}
{\displaystyle\frac{dP}{d\tau}}= - {\displaystyle\frac{\delta_1\delta_3 P}{n^2}}\cos(\alpha)F(\alpha, \Phi, Z)
\\
\\
{\displaystyle\frac{d\alpha}{d\tau}} = -\delta_1\delta_3\cos(\Phi+Z)+{\displaystyle\frac{\delta_1\delta_3}{n^2}}\sin(\alpha)F(\alpha,\Phi,Z)
\\
\\
{\displaystyle\frac{d\Phi}{d\tau}} = -\delta_3+{\displaystyle\frac{\delta_1\delta_3}{\tan(\alpha)}}\sin(\Phi+Z)\\
\\
{\displaystyle\frac{dZ}{d\tau}}=\delta_2\delta_3 P\cos(\alpha)\\
\\
{\displaystyle\frac{d\delta_3}{d\tau}}=- {\displaystyle\frac{\delta_1\delta_2\delta_3^3 P}{n^2}}F(\alpha, \Phi, Z)\\
\\
F(\alpha,\Phi,Z)=\sin(\alpha)\cos(\Phi+Z).
\end{array}
\right.
\end{equation}
In addition to the two invariants $I_1$ and $I_2$, equations (\ref{eq:DS_0}) also possesses the following constant of motion\footnote[22]{This invariant is the relativistic equivalent found in previous studies, i.e., \cite{Sudan66} and \cite{Hamza06}.} for $n^2\ne 1$ :
\begin{equation}
\label{eq:parallel_invariant}
(\delta_2P\cos(\alpha)-1)^2=2\delta_1\delta_2\frac{n^2-1}{n^2}P\sin(\alpha)\sin(\Phi+Z).
\end{equation}\\
Moreover, the existence of physically trapped orbits for $\theta=0$ requires that $\cos(\alpha)=\cos(\Phi+Z)=0$, hence, $\alpha=\Phi+Z=\pi/2$. However, this conditions results in $\dot{\Phi}\ne 0$. Aside from the trivial case of $P=0$, no fixed point exists and the parallel propagation has the particular distinction, with respect to oblique propagation, to not possess solutions for which a particle could be trapped in $Z$.\\
The parallel case has been studied in both the classical and relativistic regime. The classical treatment covered by \textit{Matsumoto}\citep[][]{Matsumoto85} and \textit{Hamza et al.}\citep[][]{Hamza06}, have shown that one can find exact solutions in terms of elliptical integrals. \textit{Lutomirski and Sudan}\citep{Sudan66} have studied the relativistic case showing that similar solutions were also possible. \textit{Roberts and Buchsbaum} \citep[][]{Roberts64} have also treated the relativistic case with a special focus on the case $n^2=1$, for which a cyclotron-resonant particle was shown to gain energy indefinitely, while for $n^2\ne 1$, the particle simply becomes phase trapped at cyclotron-resonance with no net gain in energy on average. Using the invariant in equation (\ref{eq:parallel_invariant}) as well as $I_1$ and $I_2$, those results can be expressed in terms of a pseudo-potential equation in the parallel component of the momentum that we write as $y=\delta_2P\cos(\alpha)=\delta_2 P_\parallel$ :
\begin{eqnarray}
\label{eq:potential_parallel}
\frac{\dot{y}^2}{2}&=&-V(y; \delta_1,\delta_2, n^2, I_2)\nonumber\\
&=&\frac{\delta_1^2}{2}\Bigg{[}\frac{n^2-1}{n^2}\Bigg{]}^2\Bigg{[}\frac{n^2-1}{\sigma(y)^2}-2\frac{y}{\sigma(y)}-\bigg{(}y^2+n^2\delta_2\bigg{)}\Bigg{]}-\frac{1}{8}\sigma(y)^2\bigg{(}y-1\bigg{)}^4
\end{eqnarray}
for the function $\sigma(y)=(n^2-1)/(I_2+y)$. Solutions to equation (\ref{eq:potential_parallel}) for $V(y ; \delta_1, \delta_2, n^2, I_2) < 0$ are bound states of the system for as far as parallel momentum is concerned. However, this only holds for $n^2\ne 1$. In the case of $n^2=1$, we can easily recover the unlimited acceleration found by \textit{Roberts and Buchsbaum}\citep[][]{Roberts64} from the invariants of the motion. Setting $\theta=0$ and $n^2=1$ for $I_2$, one finds that $P_\parallel$ is constant. That is, if a particle is at cyclotron-resonance, it will remain so forever (or until the wave damps), and gain energy indefinitely. It is demonstrated in the remainder of the report, that unlimited acceleration is also possible for oblique propagation and that it underlies a specific property of the fixed points.\\
\subsection{Perpendicular Propagation : $\theta=-\pi/2$.}
We now investigate the purely perpendicular case, as its treatment will be useful to characterize the dynamics for propagation angles that increase towards $|\pi/2|$. The dynamical system is written in the following form :
\begin{equation}
\label{eq:DS_90}
\left\{
\begin{array}{l l}
{\displaystyle\frac{dP}{d\tau}}= -{\displaystyle\frac{\sin(\alpha)\cos(\Phi)}{\delta_2}}+{\displaystyle\frac{\delta_1\delta_3 P}{n^2}}\sin(\alpha)\sin(\Phi)F(\alpha, \Phi, Z)
\\
\\
{\displaystyle\frac{d\alpha}{d\tau}} = -{\displaystyle\frac{1}{\delta_2P}}\cos(\alpha)\cos(\Phi)+\delta_1\delta_3\sin(\Phi)\sin(Z)\\
\hspace{6mm}+{\displaystyle\frac{\delta_1\delta_3}{n^2}}\cos(\alpha)\sin(\Phi)F(\alpha,\Phi,Z)
\\
\\
{\displaystyle\frac{d\Phi}{d\tau}} = -\delta_3-\delta_1\delta_3\cos(Z)+{\displaystyle\frac{\sin(\Phi)}{\delta_2P\sin(\alpha)}}\\
\hspace{6mm}+{\displaystyle\frac{\delta_1\delta_3}{\tan(\alpha)}}\cos(\Phi+Z)+{\displaystyle\frac{\delta_1\delta_3\cos(\Phi)}{n^2\sin(\alpha)}}\cos(\Phi+Z)F(\alpha, \Phi, Z)\\
\\
{\displaystyle\frac{dZ}{d\tau}}=-\delta_2\delta_3 P\sin(\alpha)\sin(\Phi)\\
\\
{\displaystyle\frac{d\delta_3}{d\tau}}=- {\displaystyle\frac{\delta_1\delta_2\delta_3^3 P}{n^2}}F(\alpha, \Phi, Z)\\
\\
F(\alpha,\Phi,Z)=\sin(\alpha)\cos(\Phi)\cos(Z)-\cos(\alpha)\sin(Z).
\end{array}
\right.
\end{equation}
Similarly to the parallel case, the dynamical system (\ref{eq:DS_90}) possesses its own set of invariants written as :
\begin{equation}
\left\{
\begin{array}{l l}
I_4=\delta_1\cos(Z)+\delta_2P\cos(\alpha) \nonumber\\
I_5=\delta_2P\cos(\Phi)\sin(\alpha)+\delta_1\sin(Z)+Z+\tau . \nonumber
\end{array}
\right.
\end{equation}
With the fixed point analysis for this particular case showing that no fixed points exists, i.e., a particle can not be physically trapped, we make the assumption that the solution for $Z$ takes the form of a linear relationship in time : $Z=Z_0+\beta \tau$, with $Z_0$ as the initial condition and $\beta$ as a constant. Replacing the solution for $Z$ in the invariant $I_5$ results in the following expression :
\begin{equation}
I_5=\delta_2P\cos(\Phi)\sin(\alpha)+\delta_1\sin(Z_0+\beta \tau)+Z_0+\beta \tau+\tau
\end{equation}
It is therefore evident that for $\dot{I_5}=0$ to be true, the term $(\beta+1)\tau$ must be either zero, or compensated by the momentum in $\hat{x}$, $P_x=P\sin(\alpha)\cos(\Phi)$, to grow to minus infinity as $\tau$ goes to infinity. In the absence of accessible Landau and cyclotron resonances, the latter solution does not appear acceptable. We can qualitatively demonstrate this assumption by noting that for $\tau \gg \delta_1$, the following approximation must be respected : $\frac{\gamma}{\tan(\Phi)}\simeq\frac{1-\beta}{\delta_2\beta}\tau$. Hence, either a) $\gamma\rightarrow\infty$, or b) $\Phi\rightarrow 0$.In the first case, if $\gamma\rightarrow\infty$, then $P\rightarrow\infty$ as well. Hence for $I_4$ to be constant, stationary solutions giving $P_y=P\cos(\alpha)\sim constant$ are required. Such solutions would necessitate $\dot{P_y}\sim0$. Such constraint means that either $P_z=0$ or $Z=0$. But both solutions are unacceptable since they would imply the existence of a fixed point, which has been demonstrated to not exist for the special case of perpendicular propagation. In the second case, the requirement that $\Phi\rightarrow 0$ means that since $\delta_3P\simeq v\leq c$ is bounded, $\dot{Z}\rightarrow 0$, which is in contradiction with the evidence that $Z$ must be linear in time because of a zero parallel electric field. We are therefore left with the assumption that $\beta\sim-1$, an assumption that can indeed be verified by numerical integration. \\
Without any loss of generality, we set $Z_0=0$, resulting in the solution $Z=-\tau$. Using $I_4$ we find the following solutions for $P_\parallel$ :
\begin{equation}
\delta_2P_\parallel=I_4-\delta_1\cos(\tau).
\end{equation}
Similarly, the solution for $P_x$ can be directly found from $I_5$ :
\begin{equation}
\delta_2P_x=I_5+\delta_1\sin(\tau).
\end{equation}
Using those two solutions we can find the exact differential for $\delta_3$ :
\begin{equation}
\frac{d\delta_3}{\delta_3^3}=-\frac{\delta_1}{n^2}[I_5\cos(\tau)+I_4\sin(\tau)]d\tau.
\end{equation}
Hence, the following solutions for $\delta_3$ :
\begin{equation}
\label{eq:gamma_90}
\frac{1}{\delta_3^2}-{\frac{1}{\delta_3^2(0)}=\frac{2\delta_1}{n^2}[I_5\sin(\tau)+I_4-I_4\cos(\tau)]}.
\end{equation}
We can finally find the exact solution for the last variable in terms of $\tau$ from the dynamical system equation in $Z$, that is :
\begin{equation}
\delta_2P_z=\sqrt{\frac{1}{\delta_3^2(0)}+\frac{2\delta_1}{n^2}[I_5\sin(\tau)+I_4-I_4\cos(\tau)]}.
\end{equation}
We have therefore derived exact solutions for the perpendicular case, based on the existence of two invariants and the nonexistence of fixed points. Two limiting cases can be deduced from these solutions. If the wave is sustained for long periods, such that the time of interaction with the particles $\tau_{int}\sim 1/\epsilon\omega$ for $\epsilon \ll 1$, the perpendicular propagation results in phase trapped orbits with no net gain of energy on average. In the opposite case where the interaction would be short-lived such that $\tau_{int}\sim\epsilon/\omega$, we can calculate the average increment in energy during the time of interaction. If we write (\ref{eq:gamma_90}) in terms of $E=m\gamma c^2$, and assume large amplitude, low-frequency waves such that $\delta_1/\delta_2 \sim 1$, then $E/E_0=\sqrt{1+2\delta_1^2/\delta_2^2n^2\gamma_0^2}\sim 1+v_\Phi^2/c^2$ and a particle gains energy of the order of $\Delta E/E\sim v_\Phi^2/c^2$ for every interaction. Given a prescription in the probability of interaction $P(v_\Phi, \Delta t)$ with an electromagnetic wave of phase-speed $v_\Phi$, one could build a map to describe the nonlinear interaction of a particle in a relativistic turbulent plasma composed of highly oblique electromagnetic waves. This qualitative analysis for purely perpendicular wave applies for particles that do not belong to Landau or cyclotron resonance.
\section{Cyclotron and Landau Resonances.}
\subsection{Stochastic acceleration at Cyclotron resonance : $\omega-k_\parallel v_\parallel=\pm s\Omega_0/\gamma$.}
The most commonly studied problems of wave-particle interactions have been addressed in the context of cyclotron-resonance.
However, we demonstrate below that the case of cyclotron-resonance contains further intricacies when the general case of oblique propagation and nonlinear interaction is treated in the relativistic limit. In order to do so, we construct a pseudo-potential function for a particle crossing resonances.\\
The resonance condition is written in terms of the normalized variables as :
\begin{equation}
\gamma\sin^2(\theta)-P\cos(\alpha)\cos(\theta)=\pm s/\delta_2.
\end{equation}
Using the resonance condition to replace the expression of $P\cos(\alpha)$ in $I_2$, we find the following expression :
\begin{equation}
\label{eq:invariant_cyclotron}
I_2\cos(\theta)\mp s=\delta_2\gamma(n^2\cos^2(\theta)-1)+\delta_1\cos(Z)\sin(\theta)\cos(\theta).
\end{equation}
If $\gamma$ and $Z$ do not have singularities in their derivatives when the resonance condition is respected, the following relationship must be satisfied :
\begin{equation}
\frac{d\gamma}{d\tau}=\frac{\delta_1\sin(\theta)\cos(\theta)}{\delta_2(n^2\cos^2(\theta)-1)}\sin(Z)\frac{dZ}{d\tau}.
\end{equation}
We can find an expression between $\dot{Z}$ and $\gamma$ from the invariant $I_1$. In order to do so, we write the invariant quantity in the following form :
\begin{eqnarray}
\label{eq:approxZ}
\frac{dZ}{d\tau}-\frac{n^2-1}{2} & = &-\frac{P^2+n^2}{2\gamma^2} \nonumber \\
&=&-\frac{n^2}{2\gamma^2}\Bigg{[}\bigg{(}\frac{P}{n}+1\bigg{)}^2-2\frac{P}{n}\Bigg{]}\nonumber \\
&\simeq&-\frac{n^2}{2\gamma^2}; \hspace{49mm}{if \hspace{2mm}\frac{P}{n}\ll 1}.
\end{eqnarray}
Hence, using equation (\ref{eq:approxZ}) in addition to (\ref{eq:invariant_cyclotron}) we can replace $\dot{Z}$ and $\sin(Z)$ and find a pseudo-potential equation in $\gamma$ of the form :
\begin{equation}
\label{eq:potential_gamma}
\frac{\dot{\gamma}^2}{2}+V(\gamma ; \delta_1, \delta_2, \theta)=0
\end{equation}
for a pseudo-potential written as :
\begin{eqnarray}
V(\gamma ; \delta_1, \delta_2, \theta)&=&-\frac{1}{2}\Bigg{[}\frac{n^2-1}{2}-\frac{n^2}{2\gamma^2}\Bigg{]}^2\nonumber \\
&\times& \Bigg{[}\Bigg{(}\frac{\delta_1}{\delta_2}\frac{\sin(\theta)\cos(\theta)}{n^2\cos^2(\theta)-1}\Bigg{)}^2-\Bigg{(}\frac{I_2\cos(\theta)\mp s}{\delta_2n^2\cos^2(\theta)-\delta_2}-\gamma}{\Bigg{)}^2\Bigg{]} \nonumber\\
&=&-\frac{1}{8}\Bigg{[}\beta_1-\frac{\beta_1+1}{\gamma^2}\Bigg{]}^2\Bigg{[}\beta_2^2-\Bigg{(}\beta_3-\gamma\Bigg{)}^2\Bigg{]},
\end{eqnarray}
for the set of constants $\beta_1, \beta_2, \beta_3$ defined as follows :
\begin{equation}
\beta_1=n^2-1\nonumber
\end{equation}
\begin{equation}
\beta_2=\frac{\delta_1}{\delta_2}\frac{\sin(\theta)\cos(\theta)}{n^2\cos^2(\theta)-1}\nonumber
\end{equation}
\begin{equation}
\beta_3=\frac{I_2\cos(\theta)\mp s}{\delta_2 n^2\cos^2(\theta)-\delta_2}
\end{equation}
If we set the initial conditions $Z_0=0$ and $\gamma_0=1$, we can write $\beta_3=\beta_2+1$. Taking the second derivative of (\ref{eq:potential_gamma}), we find the following expression :
\begin{eqnarray}
\ddot{\gamma}&=&\frac{1}{4}\beta_1^2\gamma-\frac{1}{4}\beta_2\beta_1^2+\frac{1}{2}\frac{\beta_3\beta_1(\beta_1+1)}{\gamma^2}+\frac{1}{2}\frac{(\beta_1+1)^2+(\beta_3^2-\beta_2^2)(\beta_1+1)}{\gamma^3}\nonumber \\
&+&\frac{3}{4}\frac{\beta_3(\beta_1+1)^2}{\gamma^4}-\frac{1}{2}\frac{(\beta_2^2-\beta_3^2)(\beta_1+1)^2}{\gamma^5}.
\end{eqnarray}
This equation can be used to treat the cyclotron-resonance for different limits. We hereafter focus on the relativistic low energy case for which $\gamma=\gamma_0+\delta\gamma$, with $\delta\gamma\ll\gamma_0$. Using Newton's approximation to express $\gamma^{-n}\simeq\gamma_0^{-n}(1-n\delta\gamma/\gamma_0)$ and setting $\gamma_0=1$, we find the following forced oscillator equation :
\begin{equation}
\ddot{\delta\gamma}+\Theta^2\delta\gamma=\Lambda(\beta_1, \beta_2),
\end{equation}
for the frequency squared :
\begin{eqnarray}
\Theta^2 &=&-\frac{1}{4}\beta_1^2-\beta_3\beta_1(\beta_1+1)+\frac{3}{2}[(\beta_1+1)^2+(\beta_3^2-\beta_2^2)(\beta_1+1)]\nonumber \\
&+& 3\beta_3(\beta_1+1)^2-\frac{5}{2}[(\beta_2^2-\beta_3^2)(\beta_1+1)^2],
\end{eqnarray}
and the constant forcing term
\begin{eqnarray}
\Lambda&=&-\frac{1}{4}\beta_1^2(\beta_3-1)+\frac{1}{4}\beta_3(\beta_1+1)(\beta_1+3)\nonumber \\
&+& \frac{1}{2}[(\beta_1+1^2)+(\beta_3^2-\beta_2^2)(\beta_1+1)]-\frac{1}{2}(\beta_2^2-\beta_3^2)(\beta_1+1)^2.
\end{eqnarray}
Figure \ref{fig1} represents the dependence of $\Theta$ as a function of $\theta$ for fixed values of $\delta_1$. It is clear that for the range of chosen parameters ($v_\Phi/c\sim .70, \delta_2=1$), the oscillations in $\delta\gamma$ can evolve from harmonic solutions to hyperbolic solutions as the amplitude of the wave increases. As a result of a large wave-amplitude, that is $\delta_1$ growing, a wide range of propagation angles will result in hyperbolic perturbations for a relativistic particle in cyclotron resonance. Figure \ref{fig2} represents the transition from $\Theta^2>0$ to $\Theta^2<0$. As the wave-amplitude increases, the particle transit from trapped-orbits to quasi-trapped orbits in phase-space. If the amplitude is further increased, the orbit becomes stochastic. Quasi-trapped and stochastic orbits are resulting from the wandering of the particle from one cyclotron harmonic to another. Hence, the particle gains energy stochastically. This result is an extension of the overlapping resonances studied by \textit{Smith and Kaufman}\citep[][]{Smith78} for classical regimes. Using a restrictive choice of parameters, they found that wave amplitudes of the order of $\delta_1=\delta B/B_0\geq 15$ were necessary to have overlapping resonances. However, our analysis shows that there is a window in parameter space belonging to the relativistic regime that allows for the overlapping of resonances for amplitudes two orders of magnitude smaller. Similarly to the classical case, large-amplitudes translate into a broadening of the phase-trapping cell. Trapping cells are also largest for propagations at $\theta=45^o$. Plotted in figures (\ref{FIG6}) and (\ref{FIG7}) are Arnold tongues, that is regions of parameter space ($n^2, \theta, \delta_1, \delta_2$) leading to stochastic orbits. It is evident from the Arnold tongues that even though the effect described by our analysis is purely relativistic, a wide range of parameters can result in stochastic orbits.\\
It should be noted that even though the equations presented in this section also apply to the case of Landau resonance, the parameter space, in which unstable orbits and overlapping can operate, belongs to velocities that must go beyond the speed of light. Therefore, the aforementioned result applies specifically to the case of cyclotron-resonances.
\begin{figure}[ht]
\centering
\includegraphics[width=0.50\textwidth]{FIG1
\caption{Squared frequency $\Theta^2$ as a function of the propagation angle $\theta$ for a relativistic particle in cyclotron resonance. Each curves are for different values in the wave-amplitude parameter spanning $0.01\le\delta_1\le1$. The four bold red lines correspond, from top to bottom, to $\delta_1=(0.05, 0.09, 0.1, 0.3)$}
\label{fig1}
\end{figure}
\begin{figure}[p]
\centering
\includegraphics[width=0.5\textwidth]{FIG2
\caption{a)Pitch angle $\alpha$ vs dynamical gyrophase $\Phi$ for $\Theta^2>0$. The particle is phase-trapped. b)Pitch angle $\alpha$ vs dynamical gyrophase $\Phi$ for $\Theta^2<0$. The particle is quasi-trapped in phase-space c) Lorentz factor $\gamma$ vs Z for $\delta_1=(0.05, 0.09, 0.1, 0.3)$. When $\Theta^2<0$, the orbit is unstable in $\gamma$ and depart from the forced harmonic oscillation observed for $\Theta^2>0$. d)Resonance condition quantified by $s(P, \gamma, \alpha ; \theta, \delta_2)$ for the case of $\Theta^2<0$. The particle travels through multiple resonances as it gains energy through repeated kicks.}
\label{fig2}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width=0.50\textwidth]{FIG6
\caption{Arnold tongue in the parameter space $(\theta, n=c/v_\Phi$), for $\delta_1=0.3$, $\delta_2=1$.}
\label{FIG6}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width=0.50\textwidth]{FIG7
\caption{Arnold tongue in the parameter space $(n, \delta_2$), for $\delta_1=0.5$, $\theta=45^o$. }
\label{FIG7}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width=0.50\textwidth]{FIG3_1
\caption{Case $\theta<\theta_c=\arctan{\sqrt{n^2-1}}$. Particle orbit for parameters $\delta_1=0.1$, $\delta_2=0.0696$, $n^2=4$, $\theta=\theta_c-1^o$ and initial conditions $v_{x0}'=0$, $v_{y0}'=-v_\Phi\tan(\theta)-1.6v_\Phi$, $v_{z0}'=-v_\Phi$, $Z_0=0$. The particle is physically and phase trapped. It bounces back and forth in the potential well with no net gain in energy.}
\label{fig3}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width=0.50\textwidth]{FIG4_2
\caption{ Case $\theta=\theta_c$. Three particle orbits seeded with different initial conditions show that the attractor is periodic. Parameters $\delta_1=0.1$, $\delta_2=0.0696$, $n^2=4$. The orbit is locked in pitch-angle $\alpha$ and dynamical gyrophase $\Phi$, trapped along $Z$ and accelerated uniformly.}
\label{fig4_2}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width=0.50\textwidth]{FIG4_1
\caption{ Case $\theta=\theta_c$. Parameters $\delta_1=0.1$, $\delta_2=0.0696$, $n^2=4$. The orbit is locked in pitch-angle $\alpha$ and dynamical gyrophase $\Phi$, trapped along $Z$ and accelerated uniformly.}
\label{fig4_1}
\end{figure}
\subsection{Hopf bifurcation at Landau resonance : $\omega=k_\parallel v_\parallel$.}
A fundamental property of a given dynamical system can be deduced by investigating whether the phase-space density and volume is conserved or not. That is, whether or not Liouville's theorem applies \cite{Regev}. The validity of Liouville's theorem provides the possibility to construct distribution functions and follow their evolution in time. Non-conservation of phase-space density, either locally or globally, stems from the existence of either attractors or non-bounded orbits. Making use of the invariant $I_2$, we compute the divergence of the flow in phase-space as follow :
\begin{eqnarray}
\frac{1}{V}\frac{dV}{dt}&=&\vec{\nabla}\cdot \frac{d\vec{\xi}}{dt}\nonumber \\
&=&\frac{\partial \dot{P}_x}{\partial P_x} +\frac{\partial \dot{P}_y}{\partial P_y} +\frac{\partial \dot{P}_z}{\partial P_z} +\frac{\partial \dot{Z}}{\partial Z} \nonumber \\
&=&-\frac{\dot{\gamma}}{\gamma},
\label{Liouville}
\end{eqnarray}
for the volume in phase-space $V$ and the phase-space vector coordinate $\vec{\xi}=(P_x, P_y, P_z, Z)$. Hence, equation (\ref{Liouville}) hints at the existence of an attractor if the volume in phase-space shrinks as $\gamma \rightarrow \infty$. In the case where the particle's energy oscillates back and forth such as for a volume of physically trapped orbits, we can consider Liouville's theorem to apply. But it can be shown that such an attractor does exist. A recently published Letter has shown that the attractor arises from a change in parameters that results in the bifurcation of the orbits around the fixed points \cite{Osmane12}. Indeed, the stability analysis in Appendix B demonstrates that to every fixed points, combined values in $(\theta, n^2)$, satisfying the condition $n^2-1=\tan^2(\theta)$, correspond to a bifurcation in stability \footnote[11]{The fixed point at $Z=\pi$ is unstable. Hence, no physical trapping is possible for $\theta<\theta_c$ and no uniform acceleration can arise for $\theta=\theta_c$.}. That is, an orbit, close to the fixed point will experience a transition from a (marginally) stable orbit to an unstable orbit. We observe that when the condition in parameter space is respected, and for a large enough amplitude of the wave magnetic field, the real part of one of the eigenvalues becomes positive. This type of bifurcation for pairs of complex conjugate eigenvalues crossing through the imaginary axis, is the well-known Hopf bifurcations\cite{Gucken}. \\
Represented in Figures (\ref{fig3}), (\ref{fig4_2}), (\ref{fig4_1}), (\ref{fig5_1}) and (\ref{fig5_2}), are typical families of orbits for parameters below, equal to, and above the propagation angle at the Hopf bifurcations ($\theta_c=\arctan{\sqrt{n^2-1}}$) for a given refractive index $n$, respectively. The wave parameters are chosen for a large-amplitude ($\delta_1=0.1$), low-frequency wave ($\delta_2=0.0696$), but similar results also apply to frequencies of the order of the gyrofrequency as long as the wave-amplitude is sufficiently large to allow physical trapping. We can observe that when $\theta<\theta_c$, the particle becomes physically and phase-trapped in the phase-space region centered at the fixed point. The particle eventually closes unto itself with no net gain on average in energy. For $\theta=\theta_c$, the particles belonging to the basin of attraction centered around the fixed point becomes locked in pitch-angle $\alpha$ and dynamical gyrophase $\Phi$, and trapped along $Z$. This locking effect results in the divergence of the momentum to infinity under a uniform acceleration. This mechanism is similar to the surfatron process commonly studied in the physics of lasers and in the problem of wave-particle acceleration in astrophysical shocks\cite{Katsouleas, Karimabadi90, Chernikov92}. Such an effect is purely relativistic and requires the presence of the Lorentz-invariant parallel electric field. The violation of Liouville's theorem belongs to volumes composed of these surfing and trapped orbits. However, since the surfing acceleration is so efficient, a wave would be expected to damp away before considerations for self-consistency and collisions are deemed necessary. The case of $\theta>\theta_c$ manifests itself through the loss of stability of the fixed point and the evolution of the attractor into two-dimensional tori. The particle is initially trapped in the $\alpha$, $\Phi$ and $Z$ plane, but eventually becomes untrapped in $Z$ while its orbit never closes. Such a regime in parameter space can as well result in the acceleration of particles. Figure \ref{fig4_1} shows that despite the incapacity to trap physically the orbits, the particle can be accelerated to relativistic levels. It is therefore clear from the above examples that the fixed points manifest themselves differently as a function of the wave obliquity and that the propagation angle is a critical parameter for relativistic orbits in the presence of large-amplitude waves.
\begin{figure}[ht]
\centering
\includegraphics[width=0.50\textwidth]{FIG5_1
\caption{ Case $\theta>\theta_c$.Particle orbits for parameters $\delta_1=0.1$, $\delta_2=0.0696$, $n^2=9$. The attractor is lost and can give rise to quasi-trapped orbits in the dynamical phase angles.}
\label{fig5_1}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width=0.50\textwidth]{FIG5_2
\caption{ Case $\theta>\theta_c$.Particle orbits for parameters $\delta_1=0.1$, $\delta_2=0.0696$, $n^2=9$. Despite the lose of the attractor particles can still be accelerated to relativistic energy levels.}
\label{fig5_2}
\end{figure}
\section{Discussion.}
\subsection{General framework}
A general framework for understanding the wave-particle interaction for a monochromatic wave can be drawn from the previous theoretical analysis of the dynamical system presented in this paper. When the propagation is parallel, that is $\theta=0$, the electric field can be eliminated by making a transformation to the wave-frame, resulting in the particle's dynamics being resolved entirely. The particle can be phase-trapped but never physically trapped. When the propagation angle increases, the obliquity becomes manifest through the appearance of a Lorentz-invariant parallel electric field. This electric field physically traps orbits and can result in the creation of a beam parallel to the background magnetic field as well as anisotropies in temperature. Indeed, the oblique propagation can provide an explanation for kinetic distortions of distribution functions for relativistic energies, in a similar manner that it does for the classical case. As the propagation angle increases, the stable fixed point ($\theta<\theta_c$), responsible for trapped orbits and kinetic distortions of distribution functions, goes as $P=\tan(\theta)$ and therefore shifts trapped cells to higher parallel velocities. If the stable fixed point is too distant from the tail of the distribution, no particles will be trapped. This transition from physically trapped to untrapped orbits is singularized by the treatment of the purely perpendicular case. For $\theta=-\pi/2$, as well as for particles that do not belong to the basin of the stable fixed point for $\theta\neq -\pi/2$, the dynamics of the orbit can be simplified to a back and forth slushing on the wave with no net gain in energy on average. \\
On the other hand, if the propagation angle reaches the critical value $\theta=\theta_c$, at which the stability of the fixed point is destroyed by the Hopf bifurcation, the particle belonging to the basin of attraction will be accelerated uniformly to relativistic energies. For $\theta>\theta_c$, a particle initially belonging to the basin of attraction now becomes chaotic and physical trapping is lost. \\
In-between the regions of phase-space composed of physically trapped and surfing orbits, resides one further source of particle energization. The acceleration in this case originates in the cyclotron-resonance and results in stochastic trajectories. The inclusion of obliquity as well as the preservation of nonlinearities and relativistic effects, reveal that for a given propagation angle, there is a window in parameter space for which a particle can be accelerated in a diffusive manner primarily along the perpendicular direction. This stochastic process is similar to that of the overlapping resonances for the classical case for obliquely propagating electrostatic waves \cite{Smith78}. The results described in the section above consist indeed of an overlapping of resonances, but does operate for wave-amplitudes about two orders of magnitude lower than those previously assumed. The explanation for this discrepancy with the classical regime is that as the particle gains energy, the dynamical gyrofrequency $\Omega_0=eB_0/m\gamma c$ decreases sufficiently to allow the particle to wander from one resonance to another.\\
The acceleration mechanisms described above both have the important and interesting particularity to operate on short kinetic time scales. The difference is that one operates stochastically and energizes particles primarily along the perpendicular direction, while the second results in a locking in pitch angle and gyrophase, and accelerates particles coherently and primarily along the parallel direction. \\
\subsection{Applications to planetary radiation belts.}
The most recent waveforms measured in the radiation belts have revealed an unexpected discovery. Large-amplitudes, monochromatic, obliquely propagating, and bursty waveforms were not only repeatedly measured in the radiation belts \cite{Catell08, Kellog10, Kersten11, Wilson11}, but appeared correlated with electron energization\cite{Wilson11} as well as relativistic microbursts events \cite{Kersten11}. The correlation between chorus waves and electron energization in the radiation belts is not recent, but it is suspected that if such waveforms were more commonly present in the radiation belts that they could be the dominant trigger responsible for the energization of electrons on short time-scales. A study by Yoon\cite{Yoon11} has shown that if one solves the plasma equations self-consistently, that such waveforms were indeed capable to accelerate electrons on kinetic time scales consistently with the observations. Even though our study lacks the levels of self-consistency provided by the numerical method developed by Yoon\cite{Yoon11}, we arrive to similar conclusions if we choose parameters consistent with the radiation belts measured waveforms. If we integrate the dynamical system for a few wave-periods, and with low-frequency $\delta_2=0.1$, large amplitude $\delta_1\sim0.06$ and for propagation angles obeying the Hopf bifurcations, we find that keV electrons commonly found in the radiation belts could be accelerated on the order of the milliseconds to MeV energies.\\
However, despite encouraging results, we would like to leave a few words of caution. We can not rule that such a mechanism is at play in the radiation belts and the reasons are as follows. 1) There is no clear understanding of the origin of the observed large-amplitude oblique waveforms in the radiation belts. Before we can pinpoint their origin, it is impossible to attempt any self-consistent approach to the current problem. 2) The observations of these waveforms are plagued by uncertainties large enough to seriously undermine any attempt to determine precisely one or multiple acceleration processes. In the very case of the surfatron at Landau resonance, one would need good resolution for the electric and magnetic field components of the waves to obtain propagation angles and wave vectors. 3) Finally, the wave forms are observed with an electrostatic component and the analysis above needs to be conducted with the addition of this compressive electric component. Even though it can be shown that the addition of the electrostatic field with the same phase as the electromagnetic components of the fields would result in the same condition for the surfatron process, a difference in phase would shift the Hopf bifurcation and have non-trivial effects that needs to be scrutinized.\\
In such context, we can not claim that such a mechanism is at play in the radiation belts, but we do suggest that since electrons with keV energies can be accelerated to MeV energies on kinetic timescales, that such mechanism could possibly arise in the radiation belts and other space and cosmic plasmas who are suspected to be permeated by equivalent large-amplitude waveforms \footnote[16]{The stochastic process for the cyclotron resonance case is unlikely to be relevant to the Earth's radiation belts since the amplitudes and phase-speed of the waves are much larger than those observed, but could be of interest in the context of cosmic rays where growing evidence of a structured spectrum suggests multiple acceleration mechanisms.}.
\section{Conclusion.}
We have developed a dynamical system to model the interaction of an ion with an obliquely propagating electromagnetic wave in the relativistic limit. We have given a particular focus on the effect of the obliquity on the particle dynamics. It was demonstrated that physical trapping of Landau resonant particles could be identified by the fixed points analysis. Perhaps the main conclusion of our study is that the wave-particle interaction of a single wave demonstrates a rich diversity of mechanisms (acceleration, surfing, stochasticity, trapping) for which the propagation angle is an important and critical parameter. Indeed, the most telling observation, is that the physics at one propagation angle $\theta$ can be significantly altered for an angle $\theta\pm\epsilon$. \\
Even though the prime difference between oblique propagations with parallel and perpendicular propagations, resides in the inclusion of a region of phase-space for which particles are physically trapped, we have shown that the relativistic treatment also translates in coupled values in $(\theta, v_\Phi)$ for which particles are accelerated to relativistic energies on kinetic time-scales $\Omega_0\tau\leq1$. Such a mechanism, even though requiring specific wave-properties, can be efficient since it operates on short-time scales, and the volume encompassed by the attractor is large enough to affect a non-negligible portion of a distribution function.\\
Furthermore, it was shown that relativistic effects enhance the cyclotron-resonant stochastic acceleration. As a result of the overlapping in resonances, particles can wander through multiple resonances resulting in a stochastic increase in energy. This relativistic effect is of interest, since it provides acceleration for wave-amplitudes lower than those required for classical regimes of overlapping cyclotron-resonance. Such mechanism could pertain and be more spread than initially assumed in weakly collisional plasmas where particles can be confined for long-time scales.\\
It should finally be pointed out that the model we used is not self-consistent, and will therefore require corrections in order to take into account the complexity of space and astrophysical plasmas. Among these necessary corrections, the departure from a monochromatic spectrum to one composed of a bandwidth, appears today as the most fundamental of them all. Even though some of the large-amplitude waves recently measured in the radiation belt show a significant degree of monochromaticity, the cosmic and space plasmas are mostly turbulent, and the inclusion of additional waves to confirm or infirm the nature of the processes responsible for the acceleration of particles is an inevitable step. However, the dynamical system approach offers numerous advantages and the endeavors for greater self-consistency can be achieved accordingly. Indeed, the dynamical system for the general case, once families of solutions have been found, can be used as a background nonlinear solution to the wave-particle interaction, upon which corrections, such as addition of waves, changing polarization, dispersion effects, inhomogeneous background magnetic field, etc., can all be treated as perturbations to the family of solutions of the "nonlinear homogeneous" system. Such method could be investigated theoretically and numerically, in the similar methodological fashion and with comparable tools that Hamiltonian systems were constructed to investigate the impact of nonlinear perturbations.
\acknowledgments
We thank K. Meziane and L.B. Wilson III for helpful discussions. This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). One of the authors, A. M. H., wishes to acknowledge CSA (Canadian Space Agency) support. Computational facilities are provided by ACEnet, the regional high performance computing consortium for universities in Atlantic Canada. \\
|
{
"timestamp": "2012-03-13T01:01:47",
"yymm": "1203",
"arxiv_id": "1203.2302",
"language": "en",
"url": "https://arxiv.org/abs/1203.2302"
}
|
\section{Introduction}
Consider the generalized continuous-time algebraic Lyapunov equation (GCALE)
\begin{equation}\label{GCALE}
A^*XB + B^*XA =-Q
\end{equation}
with given matrices $Q, A, B$ and unknown matrix $X$. Such equations play an important role
in stability theory \cite{13, 38}, optimal control problems \cite{33, 37} and balanced model reduction \cite{36}.
In \cite{39}, it is proved that Eq. (\ref{GCALE}) has a unique Hermitian, positive definite solution $X$ for every Hermitian
positive definite matrix $Q$ if and only if all eigenvalues of the pencil $\lambda A -B$ are finite and lie
in the open left half-plane. When $A=I$ (the identity matrix), Eq. (\ref{GCALE}) becomes the standard Lyapunov equation
\begin{equation}\label{SLE}
XB + B^*X =-Q.
\end{equation}
The classical numerical methods to solve Eq. (\ref{SLE}) are the Bartels-Stewart method \cite{2}, the Hammarling method \cite{19} and the Hessenberg-Schur method \cite{18}. An extension of these methods to solve Eq. (\ref{GCALE}) with the assumption $A$ is nonsingular, is given in \cite{9, 14, 15, 18, 39}.
Other approaches to solve Eq. (\ref{GCALE}) are the sign function method \cite{6, 30, 35}, the ADI method \cite{ADI1, ADI2, ADI3}.
In this paper, we consider Eq. (\ref{GCALE}), where $Q$ is an $N\times N$ Hermitian positive definite matrix and $A,B$ are arbitrary $N\times N$ matrices.
Using Bhaskar-Lakshmikantham coupled fixed point theorem \cite{Bhaskar2006}, we provide a sufficient condition that assures the existence and uniqueness of a Hermitian positive definite solution to Eq. (\ref{GCALE}). Moreover, we present an algorithm to solve this equation. Numerical experiments are given to illustrate our theoretical result.
\section{Notations and preliminaries}
We shall use the following notations: $M(N)$ denotes the set of all $N\times N$ matrices, $H(N)\subset M(N)$ the set of all $N\times N$ Hermitian matrices and
$P(N) \subset H(N)$ is the set of all $N\times N$ positive definite matrices. Instead of $X \in P(N)$, we will also write $X >0$.
Furthermore, $X\geq 0$ means that $X$ is positive semidefinite.
As a different notation for $X-Y \geq 0$ ($X-Y > 0$) we will use $X \geq Y$ ($X > Y$). If $X, Y \in H(N)$ such
that $X \leq Y$, then $[X,Y]$ will be the set of all $Z \in H(N)$ satisfying $X \leq Z \leq Y$. If $X, Y \in H(N)$ such
that $X < Y$, then $]X,Y[$ will be the set of all $Z \in H(N)$ satisfying $X < Z < Y$.
We denote by $\|\cdot\|$ the spectral norm, i.e., $\|A\|=\sqrt{\lambda^+(A^*A)}$, where $\lambda^+(A^*A)$ is the
largest eigenvalue of $A^*A$. The $N\times N$ identity matrix will be written as $I$.
It turns out that it is convenient here to use the metric induced by the trace norm $\|\cdot\|_1$.
Recall that this norm is given by $\|A\|_1=\sum_{j=1}^n s_j(A)$, where $s_j(A)$, $j = 1,\dots,n$ are the singular values of
$A$. In fact, we shall use a slight modification of this norm. For $Q \in P(N)$, we define $\|A\|_{1,Q} =\|Q^{1/2}AQ^{1/2}\|_1$.
For any $U\in M(N)$, we denote by $\mbox{Sp}(U)$ the spectrum of $U$, that is, the set of its eigenvalues.
The following lemmas will be useful later.
\begin{lemma}[See \cite{Ran2004}]\label{Lem1}
Let $A \geq 0$ and $B \geq 0$ be $N \times N$ matrices, then $0 \leq tr(AB) \leq \left\| A\right\| \cdot tr(B)$.
\end{lemma}
\begin{lemma}[See \cite{lem}]\label{Lem2}
Let $A \in H(n)$ satisfying $-I<A < I$, then $\|A\| < 1$.
\end{lemma}
\begin{definition}
Let $(\Delta,\le)$ be partially ordered set. We say that a mapping $F:\Delta \times \Delta\to \Delta$ has the mixed monotone property if for all $X,Y,J,K \in \Delta$,
$$
X \le J, Y \ge K \Longrightarrow F(X,Y) \le F(J,K).
$$
\end{definition}
The proof of our main result is based on the following fixed point theorems.
\begin{theorem}[Bhaskar and Lakshmikantham \cite{Bhaskar2006}]\label{TBL}
Let $(\Delta,\leq)$ be a partially ordered set and $d$ be a metric on $X$ such that $(\Delta, d)$ is a complete
metric space. Let $F : \Delta \times \Delta \rightarrow \Delta$ be a continuous mapping having the mixed monotone
property on $\Delta$. Assume that there exists a $\delta \in [0, 1)$ with
$$
d(F(x,y),F(u,v))\leq \frac{\delta}{2}[d(x,u)+d(y,v)]
$$
for all $x\geq u$ and $y\leq v$.
Suppose also that
\begin{itemize}
\item[(i)] there exist $x_0, y_0 \in \Delta$ such that $x_0\leq F(x_0,y_0)$ and $y_0\geq F(y_0,x_0)$;
\item[(ii)] every pair of elements has either a lower bound or an upper bound, that is, for every $(x,y)\in \Delta\times \Delta$, there exists a $z\in \Delta$ such that
$x\leq z$ and $y\leq z$.
\end{itemize}
Then, there exists a unique $\overline{x}\in \Delta$ such that $\overline{x}=F(\overline{x},\overline{x})$. Moreover, the sequences $\{x_n\}$ and $\{y_n\}$ defined by $x_{n+1}=F(x_n,y_n)$ and $y_{n+1}=F(y_n,x_n)$ converge to $\overline{x}$, with the following estimate
$$
\max\{d(x_{n},\overline{x}),d(y_{n},\overline{x})\}\leq \frac{\delta^n}{1-\delta} \max\{d(x_0,x_1),d(y_0,y_1)\},\, \mbox{ for all }\,n.
$$
\end{theorem}
\begin{theorem}[Schauder Fixed point theorem]\label{SC}
Let $S$ be a nonempty, compact, convex subset of a normed vector space. Every continuous function $f : S \rightarrow S$ mapping $S$ into itself has a fixed point.
\end{theorem}
\section{Main result}
Our main result is the following.
\begin{theorem}\label{T1}
Suppose that there exists $(\widetilde{Q},M)\in P(N)\times P(N)$ such that
\begin{enumerate}[{\rm (a)}]
\item $2(U\widetilde{Q}U^*+B\widetilde{Q}B^*)<\widetilde{Q}$;
\item $2(V\widetilde{Q}V^*+W\widetilde{Q}W^*)<\widetilde{Q}$;
\item $U^*MU+B^*MB<M-(V^*MV+W^*MW)$;
\item $Q\in ]U^*MU+B^*MB, M-(V^*MV+W^*MW)[$,
\end{enumerate}
where
$$
U=\frac{A-B+I}{\sqrt 2},\quad V=\frac{A+B+I}{\sqrt 2}\quad\mbox{and} \quad W=B-I.
$$
Then,
\begin{enumerate}[{\rm (i)}]
\item Eq. (\ref{GCALE}) has one and only one solution $\widehat{X}\in P(N)$.
\item $\widehat{X}\in [Q-(U^*MU+B^*MB),Q+(V^*MV+W^*MW)]$.
\item Let $(X_n)$ and $(Y_n)$ the sequences defined by $X_0= 0$, $Y_0=M$, and
\begin{eqnarray}\label{seqxy}
\left\{\begin{array}{lll}
X_{n+1}&=&Q +(V^*X_nV+W^*X_nW)-(U^*Y_nU+B^*Y_nB)\\
Y_{n+1}&=&Q +(V^*Y_nV+W^*Y_nW)-(U^*X_nU+B^*X_nB)
\end{array}.\right.
\end{eqnarray}
We have
\begin{equation}\label{lim}
\lim_{n\rightarrow \infty}\|X_n-\widehat{X}\|=\lim_{n\rightarrow \infty}\|Y_n-\widehat{X}\|=0,
\end{equation}
and the error estimation is given by
\begin{equation}\label{Eerr}
\max\left\{\|X_{n}-\widehat{X}\|,\|Y_{n}-\widehat{X}\|\right\}\leq \frac{\delta^n}{1-\delta}\max\left\{\|X_1-X_0\|,\|Y_1-Y_0\|\right\},
\end{equation}
for all $n$, where $0<\delta<1$.
\end{enumerate}
\end{theorem}
\noindent{\bf Proof.}
It is easy to show that Eq. (\ref{GCALE}) is equivalent to
\begin{equation}\label{GCALE2}
X=Q +(V^*XV+W^*XW)-(U^*XU+B^*XB).
\end{equation}
Consider the continuous mapping $F: H(N)\times H(N)\rightarrow H(N)$ defined by
\begin{equation}\label{F}
F(X,Y)=Q +(V^*XV+W^*XW)-(U^*YU+B^*YB),\quad \mbox{for all}\quad X,Y\in H(N).
\end{equation}
Clearly, Eq. (\ref{GCALE2}) is equivalent to
\begin{equation}\label{GCALE3}
X=F(X,X).
\end{equation}
Let $X,Y,J,K\in H(N)$ such that $X\leq J$ and $Y\geq K$. Then,
\begin{eqnarray*}
F(X,Y)&=&Q +(V^*XV+W^*XW)-(U^*YU+B^*YB)\\
&\leq & Q +(V^*JV+W^*JW)-(U^*KU+B^*KB)\\
&=&F(J,K).
\end{eqnarray*}
This implies that $F$ is a mixed monotone mapping.
Let $X,Y,J,K \in H(N)$ such that $X \geq J$ and $Y \leq K$. We have
\begin{align*}
&\|F(X,Y)-F(J,K)\|_{1,\Q}= \tr\left( \Q^{1/2}(F(X,Y)-F(J,K))\Q^{1/2} \right)\\
&=\tr\left( \Q^{1/2}(V^*(X-J)V+W^*(X-J)W+U^*(K-Y)U+B^*(K-Y)B)\Q^{1/2} \right)\\
&=\tr\left( \Q^{1/2}(V^*(X-J)V + W^*(X-J)W)\Q^{1/2} \right)+\tr\left( \Q^{1/2}(U^*(K-Y)U+B^*(K-Y)B)\Q^{1/2} \right).
\end{align*}
On the other hand, using Lemma \ref{Lem1}, we have
\begin{align*}
&\tr\left( \Q^{1/2}(V^*(X-J)V + W^*(X-J)W)\Q^{1/2} \right) =
\tr\left(V\Q V^*(X-J)+W\Q W^*(X-J) \right)\\
&=\tr\left(V\Q V^*(X-J)\Q ^{1/2}\Q ^{-1/2}+W\Q W^*(X-J)\Q ^{1/2}\Q ^{-1/2}\right)\\
&=\tr\left(\Q ^{-1/2}V\Q V^*(X-J)\Q ^{1/2}+\Q ^{-1/2}W\Q W^*(X-J)\Q ^{1/2}\right)\\
&=\tr\left(\Q ^{-1/2}V\Q V^*\Q ^{-1/2}\Q ^{1/2}(X-J)\Q ^{1/2}+\Q ^{-1/2}W\Q W^*\Q ^{-1/2}\Q ^{1/2}(X-J)\Q ^{1/2}\right)\\
&\le \left\|\Q ^{-1/2}V\Q V^*\Q ^{-1/2}+\Q ^{-1/2}W\Q W^*\Q ^{-1/2}\right\| \tr\left(\Q ^{1/2}(X-J)\Q ^{1/2}\right) \\
&= \left\|\Q ^{-1/2}(V\Q V^*+W\Q W^*)\Q ^{-1/2}\right\| \tr\left(\Q ^{1/2}(X-J)\Q ^{1/2}\right) \\
&=\left\|\Q ^{-1/2}(V\Q V^*+W\Q W^*)\Q ^{-1/2}\right\|\|X-J\|_{1,\Q }.
\end{align*}
Thus, we have
\begin{equation}\label{in1}
\tr\left( \Q^{1/2}(V^*(X-J)V + W^*(X-J)W)\Q^{1/2} \right) \leq \left\|\Q^{-1/2}(V\Q V^*+W\Q W^*)\Q^{-1/2}\right\|\|X-J\|_{1,\Q}.
\end{equation}
Similarly, we have
\begin{equation}\label{in2}
\tr\left( \Q^{1/2}(U^*(K-Y)U+B^*(K-Y)B)\Q^{1/2} \right) \leq \left\|\Q^{-1/2}(U\Q U^*+B\Q B^*)\Q^{-1/2}\right\|\|K-Y\|_{1,\Q}.
\end{equation}
Now, using (\ref{in1}) and (\ref{in2}), we get
$$
\|F(X,Y)-F(J,K)\|_{1,\Q }\leq \frac{\delta }{2} \left(\|X-J\|_{1,\Q}+\|K-Y\|_{1,\Q}\right),
$$
where
$$
\delta=
2\max\left\{\left\|\Q^{-1/2}(V\Q V^*+W\Q W^*)\Q^{-1/2}\right\|,\left\|\Q^{-1/2}(U\Q U^*+B\Q B^*)\Q^{-1/2}\right\|\right\}.
$$
Now, from Lemma \ref{Lem2}, (a) and (b), we have $\delta\in (0,1)$.
Taking $X_0=0$ and $Y_0=M$, from (c) and (d), it follows that
\begin{equation}\label{MM}
X_0< F(X_0,Y_0)\quad\mbox{and}\quad Y_0> F(Y_0,X_0).
\end{equation}
Since all the hypotheses of Theorem \ref{TBL} are satisfied, we deduce that there exists a unique $\widehat{X}\in H(N)$ solution to Eq. (\ref{GCALE3}).
This implies that Eq. (\ref{GCALE}) has a unique solution $\widehat{X}\in H(N)$.
Now, to establish (i), we need to prove that $\widehat{X}\in P(N)$. The Schauder fixed point theorem will be useful in this step.
Define the mapping $G: [F(0,M),F(M,0)]\rightarrow H(N)$ by
$$
G(X)=F(X,X),\quad\mbox{for all}\quad X\in [F(0,M),F(M,0)].
$$
Note that from (\ref{MM}) and the mixed monotone property of $F$, we have $F(0,M)\leq F(M,0)$.
We shall prove that $G( [F(0,M),F(M,0)])\subseteq [F(0,M),F(M,0)]$. Let $X\in [F(0,M),F(M,0)]$, that is,
$$
F(0,M)\leq X\leq F(M,0).
$$
Using the mixed monotone property of $F$, we get
$$
F(F(0,M),F(M,0))\leq F(X,X)=G(X) \leq F(F(M,0),F(0,M)).
$$
On the other hand, from (\ref{MM}), we have
$$
0<F(0,M)\quad\mbox{and}\quad M> F(M,0).
$$
Again, using the mixed monotone property of $F$, we get
$$
F(F(M,0),F(0,M))\leq F(M,0)\quad\mbox{and}\quad F(F(0,M),F(M,0))\geq F(0,M).
$$
Then,
$$
F(0,M) \leq G(X) \leq F(M,0).
$$
Thus we proved that $G( [F(0,M),F(M,0)])\subseteq [F(0,M),F(M,0)]$.
Now, $G$ maps the compact convex set $[F(0,M),F(M,0)]$ into itself. Since $G$ is
continuous, it follows from Schauder's fixed point theorem (see Theorem \ref{SC}) that $G$ has at least one
fixed point in this set. However, fixed points of G are solutions of (\ref{GCALE}), and we
proved already that (\ref{GCALE}) has a unique Hermitian solution. Thus this solution must
be in the set $[F(0,M),F(M,0)]$, that is,
$$
\widehat{X}\in [Q-(U^*MU+B^*MB),Q+(V^*MV+W^*MW)]\subset P(N).
$$
Thus, we proved (i) and (ii). The proof of (iii) follows immediately from Theorem \ref{TBL}. \hfill $\square$\\
Now, we present some consequences following from Theorem \ref{T1}, when $A,B$ are Hermitian matrices.
\begin{corollary}\label{CR1}
Suppose that
\begin{enumerate}[{\rm (a)}]
\item $A, B\in H(N)$;
\item $2(UQU+BQB)<Q$;
\item $2(VQV+WQW)<Q$,
\end{enumerate}
where
$$
U=\frac{A-B+I}{\sqrt 2},\quad V=\frac{A+B+I}{\sqrt 2}\quad\mbox{and} \quad W=B-I.
$$
Then,
\begin{enumerate}[{\rm (i)}]
\item Eq. (\ref{GCALE}) has one and only one solution $\widehat{X}\in P(N)$.
\item $\widehat{X}\in [Q-2(UQU+BQB),Q+2(VQV+WQW)]$.
\item Let $(X_n)$ and $(Y_n)$ the sequences defined by $X_0= 0$, $Y_0=2Q$, and
\begin{eqnarray*}
\left\{\begin{array}{lll}
X_{n+1}&=&Q +(VX_nV+WX_nW)-(UY_nU+BY_nB)\\
Y_{n+1}&=&Q +(VY_nV+WY_nW)-(UX_nU+BX_nB)
\end{array}.\right.
\end{eqnarray*}
We have
\begin{equation*}
\lim_{n\rightarrow \infty}\|X_n-\widehat{X}\|=\lim_{n\rightarrow \infty}\|Y_n-\widehat{X}\|=0,
\end{equation*}
and the error estimation is given by
\begin{equation*}
\max\left\{\|X_{n}-\widehat{X}\|,\|Y_{n}-\widehat{X}\|\right\}\leq \frac{\delta^n}{1-\delta}\max\left\{\|X_1-X_0\|,\|Y_1-Y_0\|\right\},
\end{equation*}
for all $n$, where $0<\delta<1$.
\end{enumerate}
\end{corollary}
\noindent{\bf Proof.} It follows from Theorem \ref{T1} by taking $\Q = Q$ and $M=2Q$. \hfill $\square$\\
\begin{corollary}\label{CR2}
Suppose that
\begin{enumerate}[{\rm (a)}]
\item $A,B\in H(N)$;
\item $2(U^2+B^2)<I,\quad 2(V^2+W^2)<I$;
\item $U^2+B^2<Q<I-(V^2+W^2)$,
\end{enumerate}
where
$$
U=\frac{A-B+I}{\sqrt 2},\quad V=\frac{A+B+I}{\sqrt 2}\quad\mbox{and} \quad W=B-I.
$$
Then,
\begin{enumerate}[{\rm (i)}]
\item Eq. (\ref{GCALE}) has one and only one solution $\widehat{X}\in P(N)$.
\item $\widehat{X}\in [Q-(U^2+B^2),Q+(V^2+W^2)]$.
\item Let $(X_n)$ and $(Y_n)$ the sequences defined by $X_0= 0$, $Y_0=I$, and
\begin{eqnarray*}
\left\{\begin{array}{lll}
X_{n+1}&=&Q +(VX_nV+WX_nW)-(UY_nU+BY_nB)\\
Y_{n+1}&=&Q +(VY_nV+WY_nW)-(UX_nU+BX_nB)
\end{array}.\right.
\end{eqnarray*}
We have
$$
\lim_{n\rightarrow \infty}\|X_n-\widehat{X}\|=\lim_{n\rightarrow \infty}\|Y_n-\widehat{X}\|=0,
$$
and the error estimation is given by
$$
\max\left\{\|X_{n}-\widehat{X}\|,\|Y_{n}-\widehat{X}\|\right\}\leq \frac{\delta^n}{1-\delta}\max\left\{\|X_1-X_0\|,\|Y_1-Y_0\|\right\},
$$
for all $n$, where $0<\delta<1$.
\end{enumerate}
\end{corollary}
\noindent{\bf Proof.} It follows from Theorem \ref{T1} by taking $\Q = M=I$. \hfill $\square$
\section{Numerical experiments}
In this section, we present some numerical experiments to check the convergence of the proposed algorithm (\ref{seqxy}).
We take $X_0=0$ and $Y_0=M\in P(N)$. For each iteration $i$, we consider the residual errors
$$
E_i(X)=\|X_i-(Q +(V^*X_iV+W^*X_iW)-(U^*X_iU+B^*X_iB))\|,
$$
$$
E_i(Y)=\|Y_i-(Q +(V^*Y_iV+W^*Y_iW)-(U^*Y_iU+B^*Y_iB))\|
$$
and
$$
E_i=\max\{E_i(X),E_i(Y)\}.
$$
All programs are written in MATLAB version 7.1.
\begin{example}
We consider Eq. (\ref{GCALE}) with
\begin{eqnarray*}
Q=\left(
\begin{array}{ccc}
2 & 0.02 & 0.05 \\
0.02 & 2 & 0.02 \\
0.05 & 0.02 & 2 \\
\end{array}
\right),\quad
A=
\left(
\begin{array}{ccc}
-0.95 & 0.001 & 0.001 \\
0.001 & -0.95 & 0.001 \\
0.001 & 0.001 & -0.95 \\
\end{array}
\right),\quad
B=\left(
\begin{array}{ccc}
0.54 & -0.002 & -0.002 \\
-0.002 & 0.54 & -0.002 \\
-0.002 & -0.002 & 0.54 \\
\end{array}
\right).
\end{eqnarray*}
In this case, we have $A,B\in H(3)$,
$$
\mbox{Sp}\bigg(Q -2(UQU+BQB)\bigg)=\bigg\{0.3345, 0.3379, 0.3880\bigg\},\,\,\mbox{Sp}\bigg(Q -2(VQV+WQW)\bigg)=\bigg\{0.4532, 0.4573, 0.4612\bigg\},
$$
which imply that conditions (a)-(c) of Corollary \ref{CR1} are satisfied.
Considering the iterative method (\ref{seqxy}) with $X_0= 0$ and $Y_0=2Q$, after 100 iterations one gets an approximation to the positive definite solution $\widehat{X}$ and it is
\begin{eqnarray*}
\widehat{X}\approx X_{100}=Y_{100}=\left(
\begin{array}{ccc}
1.9495 & 0.0288 & 0.0142 \\
0.0288 & 1.9496 & 0.0288 \\
0.0142 & 0.0288 & 1.9495 \\
\end{array}
\right)
\end{eqnarray*}
and $E_{100}=1.9215\times 10^{-13}$.
\end{example}
\begin{example}
We consider Eq. (\ref{GCALE}) with
\begin{eqnarray*}
Q=\left(
\begin{array}{ccccc}
0.4 & 0.01 & 0.02 & 0.03 & 0.04 \\
0.01 & 0.4 & 0.01 & 0.02 & 0.03 \\
0.02 & 0.01 & 0.4 & 0.01 & 0.02 \\
0.03 & 0.02 & 0.01 & 0.4 & 0.01 \\
0.04 & 0.03 & 0.02 & 0.01 & 0.4
\end{array}
\right),\quad
A=
\left(
\begin{array}{ccccc}
-0.95 & 0.001 & 0.001 & 0.001 & 0.001 \\
0.001 & -0.95 & 0.001 & 0.001 & 0.001 \\
0.001 & 0.001 & -0.95 & 0.001 & 0.001 \\
0.001 & 0.001 & 0.001 & -0.95 & 0.001 \\
0.001 & 0.001 & 0.001 & 0.001 & -0.95
\end{array}
\right)
\end{eqnarray*}
and
\begin{eqnarray*}
B=
\left(
\begin{array}{ccccc}
0.44 & -0.02 & -0.02 & -0.02 & -0.02 \\
-0.02 & 0.44 & -0.02 & -0.02 & -0.02 \\
-0.02 & -0.02 & 0.44 & -0.02 & -0.02 \\
-0.02 & -0.02 & -0.02 & 0.44 & -0.02 \\
-0.02 & -0.02 & -0.02 & -0.02 & 0.44
\end{array}
\right).
\end{eqnarray*}
In this case, we have $A,B\in H(5)$,
$$
\mbox{Sp}\bigg(I -2(U^2+B^2)\bigg)=\bigg\{0.4078, 0.4078, 0.4078, 0.4078, 0.64716\bigg\},
$$
$$
\mbox{Sp}\bigg(I -2(V^2+W^2)\bigg)=\bigg\{0.0094, 0.1577, 0.1577, 0.1577, 0.1577\bigg\},
$$
$$
\mbox{Sp}\bigg(Q -(U^2+B^2)\bigg)=\bigg\{0.0516, 0.0882, 0.0963, 0.0984, 0.3049\bigg\},
$$
and
$$
\mbox{Sp}\bigg(I -(V^2+W^2)-Q\bigg)=\bigg\{0.0231, 0.1844, 0.1865, 0.1949, 0.2312\bigg\},
$$
which imply that conditions (a)-(c) of Corollary \ref{CR2} are satisfied.
Considering the iterative method (\ref{seqxy}) with $X_0= 0$ and $Y_0=I$, after 82 iterations one gets an approximation to the positive definite solution $\widehat{X}$ and it is
\begin{eqnarray*}
\widehat{X}\approx X_{82}=Y_{82}=
\left(
\begin{array}{ccccc}
0.4895 & 0.0429 & 0.0541 & 0.0658 & 0.0781 \\
0.0429 & 0.4878 & 0.0418 & 0.0535 & 0.0658 \\
0.0541 & 0.0418 & 0.4873 & 0.0418 & 0.0541 \\
0.0658 & 0.0535 & 0.0418 & 0.4878 & 0.0429 \\
0.0781 & 0.0658 & 0.0541& 0.0429 & 0.4895
\end{array}
\right)
\end{eqnarray*}
and $E_{82}=7.0549\times 10^{-16}$.
\end{example}
|
{
"timestamp": "2012-03-09T02:03:15",
"yymm": "1203",
"arxiv_id": "1203.1821",
"language": "en",
"url": "https://arxiv.org/abs/1203.1821"
}
|
\section{Introduction}
Blue compact dwarf galaxies (BCDs) are known by their intense processes of star-formation.
These make that their optical spectra were dominated by the blue light from massive young
stars and the bright emission lines from the ionised gas, so they are also called
H {\sc ii} galaxies.
Their high specific star-formation rates and their low gas-phase metallicities were taken as
evidences that these galaxies were undergoing their first bursts of star formation
\citep{sargent70}. In fact, one of the most important aspects about BCDs is that they
constitute an important link to the high redshift universe and the early epoch of galaxy
formation \citep{bergvall02}. During the last decade, however, deep imaging of BCDs showed
that in addition to bright young stars from the present starburst most of them have an underlying
older stellar population \citep{papaderos96a,aloisi99,ostlin00}.
One of the challenges of the BCDs is to reconcile their low--observed metallicity with the
relatively high star formation rate (SFR) in these galaxies. \cite{matteucci83} proposed
three possible mechanisms to explain this fact: 1) variations in the initial mass function
(IMF); 2) accretion of metal-poor gas; and 3) galactic winds powered by supernovae explosions.
According to numerical models, it seems that the last mechanism is the most simple to reproduce
the observed properties (see \citealt{tolstoy09} and references therein for a detailed
explanation).
The star formation history (SFH) and the metal content of BCDs must be figured out in order
to shed some light on these questions. In the case of SFH, although some advances have been
done by means of the fitting of the optical spectrum with stellar populations synthesis
\citep{perez-montero10}, one of the most reliable method consists of the analysis of
colour-magnitude diagrams (CMDs). However, this is limited to the closest objects whose stellar
population can be resolved. Another important contribution comes from the study of the ionic
chemical abundances of elements with a different nucleosynthetic origin, such as
oxygen and nitrogen \citep{molla06}. These chemical abundances can be derived very accurately
in the metal-poor gas phase of BCDs, where the cooling rate is not efficient, increasing the
electron temperature of the ionised gas and enhancing the emissivity of the collisional lines
necessary to derive the ionic abundances following the method based on the determination of
the electron temperature \citep{pmontero03,hagele06}.
NGC\,6789 is, according to \cite{drozdovsky01}, the closest BCD to our Galaxy, with a distance
modulus of (m - M) = 27.80, or a distance of 3.6 Mpc. This value yields a linear scale of
17.5 pc arcsec$^{-1}$. \cite{drozdovsky00} imaged NGC 6789 from the ground. They found that
it belongs to the iE subtype (following \citealt{loose86} classification), exhibiting the
morphology most characteristic of the vast majority of BCDs. NGC 6789 presents several H$\alpha$
emission knots which show evidence of actual star formation activity. It also shows a high
surface brightness in its central region and a small radial velocity
\citep[V$_0$ = -141 km s$^{-1}$,][]{karachentsev98}. NGC\, 6789's closeness, spatial isolation
and morphology offer the prospect of studying the structure, SFH, and metallicity of different
star forming knots within the same BCD.
To investigate these issues, we obtained simultaneous blue and red long-slit observations
with the ISIS double-arm spectrograph at the 4.2m William Herschel Telescope (WHT),
of the brightest five knots of NGC 6789. We also used archival data from HST/WFPC2,
GALEX and H$\alpha$ narrow filter, and B and R broad filters. These images were used to
derive the SFH and to give an observational input to derive the properties of the
ionising populations with the aid of photoionisation models. These were
also used to provide more accurate chemical abundances in each one of the five
studied knots of this galaxy and to better understand their differential
chemical evolution.
In the following section the long-slit WHT observations and reduction are described
and the results from the analysis are presented. In section \ref{phot} the optical and UV
photometry are described together with the resolved stellar photometry. Finally, we discuss
all these results in Section \ref{discussion} and conclusions are presented in
Section \ref{conclusions}.
\section{Long-slit spectroscopy}
\label{spec}
\subsection{Observations and reduction}
The long-slit spectrophotometric observations of NGC~6789 were obtained using the
ISIS double-beam spectrograph mounted on the 4.2 m William Herschel Telescope (WHT) of the
Isaac Newton group (ING) at the Roque de los Muchachos Observatory on the Spanish
island of La Palma. They were acquired on 2005 July 8 during one single night observing run
and under photometric conditions, with an average seeing of 0.7 arcsec. The EEV12 and
Marconi2 detectors were attached to the blue and red arms of the spectrograph, respectively.
The R600B grating was used in the blue covering the wavelength range 3670 - 5070 \AA\
(centred at $\lambda_c$ = 4370 \AA), giving a spectral dispersion of 0.45 \AA\ pixel$^{-1}$.
On the red arm, the R316R grating was mounted in two different central wavelengths providing
a spectral range from 5500 to 7800 \AA\ ($\lambda_c$ = 6650 \AA) and
from 7600 to 9900 \AA\ ($\lambda_c$ = 8750 \AA) with a spectral dispersion
of 0.86 \AA\ pixel$^{-1}$. To reduce the readout noise of our images,
the observations were taken with the ``SLOW'' CCD speed. The
pixel size for this set-up is 0.2 arcsec for both spectral
ranges. The slit width was 1 arcsec, which, combined with
the spectral dispersions, yields spectral resolutions of about 1.0
and 3.5 \AA\ FWHM in the blue and red arms, respectively.
The instrumental configuration and other details on the exposures
are given in the journal of observations in Table \ref{journal}.
\begin{table*}
\centering
\caption[]{WHT instrumental configuration}
\label{journal}
\begin{tabular} {l c c c c c c}
\hline
Slit position & Spectral range & Disp. & FWHM & Spatial res. & Exposure Time \\
& (\AA) & (\AA\,px$^{-1}$) & (\AA) & (\arcsec\,px$^{-1}$) & s \\
\hline
S1 & 3670-5070 & 0.45 & 1.0 & 0.2 & 4 $\times$ 900 \\
S1 & 5500-7800 & 0.86 & 3.5 & 0.2 & 2 $\times$ 900, 1 $\times$ 300 \\
S1 & 7600-9900 & 0.86 & 3.5 & 0.2 & 2 $\times$ 900 \\
S2 & 3670-5070 & 0.45 & 1.0 & 0.2 & 5 $\times$ 900 \\
S2 & 5500-7800 & 0.86 & 3.5 & 0.2 & 2 $\times$ 900, 1 $\times$ 300 \\
S2 & 7600-9900 & 0.86 & 3.5 & 0.2 & 2 $\times$ 900 \\
\hline
\end{tabular}
\end{table*}
\begin{figure*}
\includegraphics[width=\linewidth,clip=]{ngc6789-reg-slit}
\caption[]{H$\alpha$ image of NGC 6789 from \cite{gildepaz03} with the identification of the
observed knots (labelled from A to E). We show the regions for which the H$\alpha$ flux was
measured and the position of the two slits for the observations described in the text. The
R-band contours show the position of the host galaxy. Finally, we also plot part of the rectangle
encompassing the field-of-view of the PC chip of the WFPC2 data used to obtain the resolved stellar
photometry described in Section \ref{rphot}. North is up and east is towards the left-hand side.}
\label{regslit}
\end{figure*}
Several bias and sky flat field frames were taken at the beginning and at the end
of the night in both arms. In addition, two lamp flat fields and one calibration
lamp exposure were taken for each telescope position. The calibration lamp
used was +CuAr.
The images were processed and analysed with
IRAF\footnote{IRAF: the Image Reduction and Analysis Facility is distributed by
the National Optical Astronomy Observatories, which is operated by the
Association of Universities for Research in Astronomy, In. (AURA) under
cooperative agreement with the National Science Foundation (NSF).} routines in
the usual manner. The procedure included cosmic rays removal, bias subtraction,
division by a normalised flat field and wavelength calibration. Typical wavelength
fits to second to third order polynomials were performed using around 40
lines in the blue and 20-25 lines in the red. These fits were done at 100 different
locations along the slit in both arms (beam size of 10 pixels) obtaining rms residuals
between $\sim$0.1 and $\sim$0.2\,pix. In the last step, the spectra were corrected for
atmospheric extinction and flux calibrated. For both arms, BD+254655 standard star
observations were used, allowing a good spectrophotometric calibration with an estimated
accuracy of about 5\%.
Unfortunately, the sky subtraction in the spectral range from 7600 to 9900 \AA\ left
strong residuals from night-sky emission lines and telluric absorption, leaving this
part of the spectra unusable and therefore, we were not able to measure with
enough accuracy the [S{\sc iii}]\,$\lambda\lambda$\,9069,9532\,\AA\ lines in any of
the five knots.
Figure \ref{regslit} shows the H$\alpha$ image and continuum contours in the R-band
from \cite{gildepaz03} in order to illustrate the position of the slits in relation to the
position of the bursts of star formation and the host galaxy. We also show elliptical
regions in H$\alpha$ taken to measure the total H$\alpha$ and GALEX flux of the knots
(see Section \ref{Halfa} below). The five knots are labelled with the first alphabet letters.
Knots A and C were taken at PA = 138$^o$, which in average is close to the parallactic angle.
In the case of knots B, D, and E the slit was at PA = 76$^o$, which is in average
$\approx$ 40$^o$ off the parallactic angle.
Therefore, the observations in this second position can be affected by a
certain differential atmospheric refraction (DAR). Nevertheless, taking into
account the curves given by \cite{filippenko82} for the mean air mass during
the observations ($\approx$ 1.30), we calculated that the angular deviation
between [O{\sc ii}] and [S{\sc ii}] is not larger than the slit width at this P.A.
Besides, we checked that the emission line-ratios in the spatial position
where both slit positions intersect do not vary in more than the quoted
errors. Nevertheless, the line measurements errors of the second
slit position (B, D and E knots) might be larger due to this effect.
For the sake of comparison, we also plot in Figure \ref{regslit} part of the rectangle
encompassing the field-of-view (FOV) of the PC chip of the WFPC2 data used to obtain the
resolved stellar photometry (see Section \ref{rphot})
\subsection{Line intensities and reddening correction}
\label{line}
The optical calibrated spectra of the five observed knots with some relevant identified emission
lines are shown in Figure \ref{spectra}. The spectrum of each knot is split into two panels.
\begin{figure*}
\includegraphics[width=\textwidth,clip=]{ngc6789-knotA}
\includegraphics[width=\textwidth,clip=]{ngc6789-knotB}
\includegraphics[width=\textwidth,clip=]{ngc6789-knotC}
\includegraphics[width=\textwidth,clip=]{ngc6789-knotD}
\caption{Blue and red spectra for the knots A, B, C, D, and E of NGC~6789.}
\label{spectra}
\end{figure*}
\addtocounter{figure}{-1}
\begin{figure*}
\centering
\includegraphics[width=\textwidth,clip=]{ngc6789-knotE}
\caption{-- {\it continued}}
\end{figure*}
Underlying stellar population in star forming galaxies have several effects in the measure of the
emission lines produced by the ionised gas. Balmer and Paschen emission lines are depressed by the
presence of absorption wings of stellar origin and does not allow the measure of their fluxes with
acceptable accuracy \citep{diaz88}. All the properties derived from ratios that involve these
lines, like reddening or ionic abundances, will be affected.
We subtracted from the observed spectra the spectral energy distribution of the underlying stellar
population found by the spectral synthesis code STARLIGHT\footnote{The STARLIGHT project is
supported by the Brazilian agencies CNPq, CAPES and FAPESP and by the France-Brazil CAPES/Cofecub
program.} \citep{cidfernandes04,cidfernandes05,mateus06}. STARLIGHT fits an observed continuum
spectral energy distribution using a combination of the synthesis spectra of different single
stellar populations (SSPs; also known as instantaneous burst) using a $\chi^2$ minimisation
procedure. We chose for our analysis the SSP spectra from \cite{bruzual03}, based on the STELIB
library of \cite{leborgne03}, Padova 1994 evolutionary tracks, and a \cite{chabrier03} IMF between
0.1 and 100 M$_{\odot}$. The metallicity of the libraries was strictly constrained following the
results derived from Section \ref{ssfh}. Thus, we fixed the metallicity of the stellar populations to
Z = 0.004 ($\approx$ 3/10 Z$_{\odot}$\footnote{Z$_\odot$ = 0.0122}) for the interval log(age) =
[8.1,8.5] and Z = 0.001 ($\approx$ 1/12 Z$_{\odot}$) for log(age) = \{6.0, 6.4, 6.6, 6.8, 7.0,
7.1, 7.2, 7.9, 8.0, 8.1, 8.2, 8.3, 9.1, 9.2, 9.3, 9.7, 9.8, 9.9, 10.0, 10.1\} shown in the SFH solution
(see Figure \ref{sfh}). As explained in section \ref{ssfh}, although both metallicities appear in
the youngest burst ($<$ 12 Myr) SFH solution, we assigned the lower metallicity to this event
according to the abundances derived in section \ref{abundances}.
The STARLIGHT code solves simultaneously the ages and relative contributions of the different
SSPs and the average reddening. The reddening law from \cite{cardelli89} with $R_{V}$ = 3.1 was
used. Prior to the fitting procedure, the spectra were shifted to the rest frame and re-sampled
to a wavelength interval of 1 \AA\ in the entire wavelength range by interpolation, as required by
the program. Bad pixels and emission lines were excluded from the final fits.
It should be noted that while emission lines can be masked out, this is not possible for the
nebular continuum emission. We tested the contribution of this emission using
Starburst99 libraries \citep[SB99;][]{leitherer99} in the same STARLIGHT models and we checked that
the nebular continuum does not affect significantly neither the subtraction of the underlying continuum
nor the determination of the stellar mass, consistently with the weak gas emission in this galaxy.
The H{\sc i} series (emitted as a consequence of recombination) were used to determine the extinction,
comparing the observed line ratios with the expected theoretical values. Case B (optically thick in
all the Lyman lines) is the best simple approximation to describe the physical conditions in the
ionisation of the gas. This method takes advantage of the fact that the ratio between the emissivities
of two hydrogen recombination lines, which depends on electron temperature and density, is almost
constant. As an example, the ratio between the emissivity of H$\alpha$ and H$\beta$ is 2.86 for
the case B with $n_{e} = 100$ $cm^{-3}$ and $T_{e} = 10000$ K, and this value varies less than
10\% in the range of interest of temperatures and densities for an H{\sc ii} region.
We used an iterative method to estimate them, taking as starting values those derived from the
measured [S{\sc ii}] $\lambda\lambda$ 6717,6731 \AA\ and
[O{\sc iii}] $\lambda\lambda$ 4363,4959,5007 \AA. A least square fit of the measured decrements to
the theoretical ones, computed based on the data by \cite{storey95}, was performed that provides
the reddening coefficient, c(H$\beta$), and adopting the extinction law given by \cite{cardelli89}
with R$_{V}$ = 3.1. Due to the large error introduced by the presence of the underlying population,
only the strongest Balmer emission lines (H$\alpha$, H$\beta$, H$\gamma$, and H$\delta$) were used.
Line fluxes for the most relevant emission lines were measured using the \texttt{splot} task
in {\sc IRAF}. Balmer lines were obtained from the STARLIGHT residual spectra, while the rest
of the lines were measured on the original spectra. We checked in all spectra that the
Starlight models properly reproduce the observed Balmer absorption profiles\footnote{From several
fitting experiments using other libraries it is found that the resolution of the SSP spectra is a
critical factor for Starlight in order to fit the absorption lines.}. The statistical errors
associated with the observed emission line fluxes were calculated using the expression:
\begin{equation}
\sigma_{l} = \sigma_{c}\sqrt{N\left(1 + \frac{EW}{N\Delta}\right)}
\end{equation}
\noindent where $\sigma_{l}$ is the error in the observed line flux, $\sigma_{c}$ represents the
standard deviation in a box near the measured emission line and stands for the error in the
continuum placement, N is the number of pixels used in the measure of the line flux, EW is the line
equivalent width, and $\Delta$ is the wavelength dispersion in \AA\ per pixel
\citep{gdelgado94}. This expression takes into account the error in the continuum and the photon
count statistics of the emission line.
Table \ref{lines} gives the equivalent widths and the emission-line fluxes relative to
1000$\cdot$F(H$\beta$), before and after reddening correction, in the optical spectra of the
five observed knots, together with the reddening constants and their errors, their corresponding
A(V) value\footnote{c(H$\beta$) = 0.4656*A(V), using \cite{cardelli89} and R$_{V}$ = 3.1.} and the
extinction-corrected H$\beta$ flux. We also provide the adopted reddening curve, $f(\lambda)$ normalised
to H$\beta$. The errors in the emission-line ratios were obtained by propagating in quadrature the
observational errors in the emission-line fluxes and the reddening constant uncertainties.
\begin{table*}
\caption{Observed and reddening corrected relative line intensities [F(H$\beta)$=I(H$\beta)$=1000]
with their corresponding errors for the five knots. The adopted reddening curve, $f(\lambda)$
(normalised to H$\beta$), the equivalent width of the emission lines, the extinction-corrected
H$\beta$ intensity, the reddening constant c(H$\beta$) and the corresponding A(V) are also given.}
\label{lines}
\begin{tabular}{lc@{\hspace{3em}}ccc@{\hspace{3em}}ccc}
\hline
\multicolumn{1}{l}{} & \multicolumn{1}{l}{} & \multicolumn{3}{c}{\hspace{-3em}\large NGC 6789-A} & \multicolumn{3}{c}{\large NGC 6789-B} \\
\multicolumn{1}{c}{$\lambda$ ({\AA})} & f($\lambda$) & F($\lambda$) & EW(\AA) & I($\lambda$) & F($\lambda$) & EW(\AA) & I($\lambda$) \\
\hline
3726 [O{\sc ii}] & 0.322 & 896 $\pm$ 19 & -15.0 & 979 $\pm$ 63 & 1186 $\pm$ 17 & -34.9 & 1277 $\pm$ 80 \\
3729 [O{\sc ii}] & 0.322 & 1241 $\pm$ 16 & -20.7 & 1357 $\pm$ 84 & 1744 $\pm$ 16 & -51.9 & 1878 $\pm$ 116 \\
3835 H9 & 0.299 & 37 $\pm$ 10 & -0.6 & 40 $\pm$ 11 & 80 $\pm$ 4 & -3.5 & 85 $\pm$ 7 \\
3868 [Ne{\sc iii}] & 0.291 & 302 $\pm$ 17 & -2.8 & 327 $\pm$ 26 & 153 $\pm$ 14 & -3.7 & 164 $\pm$ 17 \\
3889 He{\sc i}+H8 & 0.286 & 153 $\pm$ 6 & -2.6 & 166 $\pm$ 12 & 146 $\pm$ 10 & -4.5 & 156 $\pm$ 14 \\
3968 [Ne{\sc iii}]+H7 & 0.266 & 81 $\pm$ 8 & -1.3 & 88 $\pm$ 10 & $\cdots$ & $\cdots$ & $\cdots$ \\
4102 H$\delta$ & 0.229 & 247 $\pm$ 5 & -3.9 & 263 $\pm$ 16 & 248 $\pm$ 16 & -6.7 & 261 $\pm$ 22 \\
4340 H$\gamma$ & 0.157 & 454 $\pm$ 11 & -7.2 & 474 $\pm$ 28 & 448 $\pm$ 7 & -12.4 & 465 $\pm$ 26 \\
4363 [O{\sc iii}] & 0.149 & 63 $\pm$ 3 & -0.7 & 66 $\pm$ 5 & 23 $\pm$ 4 & -0.4 & 23 $\pm$ 4 \\
4471 He{\sc i} & 0.115 & 36 $\pm$ 4 & -0.4 & 37 $\pm$ 5 & 37 $\pm$ 6 & -0.7 & 39 $\pm$ 7 \\
4861 H$\beta$ & 0.000 & 1000 $\pm$ 10 & -17.3 & 1000 $\pm$ 47 & 1000 $\pm$ 12 & -29.1 & 1000 $\pm$ 48 \\
4959 [O{\sc iii}] & -0.026 & 1240 $\pm$ 12 & -16.0 & 1231 $\pm$ 57 & 512 $\pm$ 12 & -11.4 & 509 $\pm$ 26 \\
5007 [O{\sc iii}] & -0.038 & 3801 $\pm$ 13 & -50.1 & 3762 $\pm$ 167 & 1492 $\pm$ 9 & -33.0 & 1479 $\pm$ 66 \\
5876 He{\sc i} & -0.203 & 102 $\pm$ 14 & -1.6 & 97 $\pm$ 14 & 96 $\pm$ 16 & -2.3 & 92 $\pm$ 16 \\
6312 [S{\sc iii}] & -0.264 & 11 $\pm$ 2 & -0.2 & 10 $\pm$ 2 & 16 $\pm$ 5 & -0.5 & 16 $\pm$ 5 \\
6548 [N{\sc ii}] & -0.296 & 28 $\pm$ 5 & -0.5 & 26 $\pm$ 5 & 61 $\pm$ 9 & -1.8 & 57 $\pm$ 8 \\
6563 H$\alpha$ & -0.298 & 3038 $\pm$ 29 & -57.0 & 2797 $\pm$ 38 & 3003 $\pm$ 15 & -89.3 & 2807 $\pm$ 20 \\
6584 [N{\sc ii}] & -0.300 & 136 $\pm$ 10 & -2.5 & 126 $\pm$ 9 & 186 $\pm$ 11 & -5.6 & 174 $\pm$ 10 \\
6717 [S{\sc ii}] & -0.318 & 277 $\pm$ 10 & -5.0 & 254 $\pm$ 10 & 314 $\pm$ 9 & -9.5 & 292 $\pm$ 8 \\
6731 [S{\sc ii}] & -0.320 & 188 $\pm$ 10 & -3.4 & 172 $\pm$ 9 & 217 $\pm$ 10 & -6.6 & 202 $\pm$ 9 \\
7136 [Ar{\sc iii}] & -0.374 & 112 $\pm$ 7 & -2.4 & 101 $\pm$ 7 & 76 $\pm$ 7 & -2.7 & 70 $\pm$ 7 \\
7319 [O{\sc ii}] & -0.398 & $\cdots$ & $\cdots$ & $\cdots$ & 46 $\pm$ 6 & -1.6 & 42 $\pm$ 6 \\
7330 [O{\sc ii}] & -0.400 & $\cdots$ & $\cdots$ & $\cdots$ & 35 $\pm$ 4 & -1.2 & 32 $\pm$ 4 \\
\hline
I(H$\beta$)(erg\,seg$^{-1}$\,cm$^{-2}$) & & \multicolumn{3}{c}{ 2.70e-15} & \multicolumn{3}{c}{ 3.19e-15} \\
c(H$\beta$) & & \multicolumn{3}{c}{0.12 $\pm$ 0.02} & \multicolumn{3}{c}{0.10 $\pm$ 0.02} \\
A(V) & & \multicolumn{3}{c}{0.26 $\pm$ 0.04} & \multicolumn{3}{c}{0.21 $\pm$ 0.04} \\
\hline
\multicolumn{1}{l}{} & \multicolumn{1}{l}{} & \multicolumn{3}{c}{\hspace{-3em}\large NGC 6789-C} & \multicolumn{3}{c}{\large NGC 6789-D} \\
\multicolumn{1}{c}{$\lambda$ ({\AA})} & f($\lambda$) & F($\lambda$) & EW(\AA) & I($\lambda$) & F($\lambda$) & EW(\AA) & I($\lambda$) \\
\hline
3726 [O{\sc ii}] & 0.322 & 940 $\pm$ 13 & -29.1 & 1020 $\pm$ 64 & 1159 $\pm$ 26 & -15.4 & 1267 $\pm$ 82 \\
3729 [O{\sc ii}] & 0.322 & 1377 $\pm$ 12 & -42.4 & 1494 $\pm$ 92 & 1712 $\pm$ 21 & -22.5 & 1871 $\pm$ 116 \\
3835 H9 & 0.299 & 47 $\pm$ 9 & -1.4 & 50 $\pm$ 10 & $\cdots$ & $\cdots$ & $\cdots$ \\
3868 [Ne{\sc iii}] & 0.291 & 300 $\pm$ 12 & -5.2 & 323 $\pm$ 23 & 105 $\pm$ 9 & -0.9 & 114 $\pm$ 12 \\
3889 He{\sc i}+H8 & 0.286 & 169 $\pm$ 12 & -5.5 & 182 $\pm$ 17 & 167 $\pm$ 20 & -2.4 & 180 $\pm$ 24 \\
3968 [Ne{\sc iii}]+H7 & 0.266 & 74 $\pm$ 9 & -2.3 & 79 $\pm$ 11 & $\cdots$ & $\cdots$ & $\cdots$ \\
4102 H$\delta$ & 0.229 & 246 $\pm$ 16 & -5.9 & 261 $\pm$ 22 & 244 $\pm$ 7 & -3.2 & 260 $\pm$ 16 \\
4340 H$\gamma$ & 0.157 & 446 $\pm$ 7 & -12.4 & 464 $\pm$ 26 & 444 $\pm$ 10 & -5.3 & 464 $\pm$ 27 \\
4363 [O{\sc iii}] & 0.149 & 59 $\pm$ 3 & -1.3 & 61 $\pm$ 5 & 24 $\pm$ 8 & -0.2 & 25 $\pm$ 8 \\
4471 He{\sc i} & 0.115 & 37 $\pm$ 5 & -0.7 & 38 $\pm$ 6 & 36 $\pm$ 6 & -0.3 & 37 $\pm$ 7 \\
4686 He{\sc ii} & 0.050 & 26 $\pm$ 5 & -0.5 & 26 $\pm$ 5 & $\cdots$ & $\cdots$ & $\cdots$ \\
4861 H$\beta$ & 0.000 & 1000 $\pm$ 8 & -27.1 & 1000 $\pm$ 47 & 1000 $\pm$ 10 & -12.1 & 1000 $\pm$ 47 \\
4959 [O{\sc iii}] & -0.026 & 1170 $\pm$ 9 & -26.9 & 1163 $\pm$ 53 & 607 $\pm$ 11 & -6.2 & 602 $\pm$ 29 \\
5007 [O{\sc iii}] & -0.038 & 3471 $\pm$ 11 & -75.0 & 3438 $\pm$ 153 & 1750 $\pm$ 12 & -16.7 & 1732 $\pm$ 78 \\
5876 He{\sc i} & -0.203 & 96 $\pm$ 14 & -2.8 & 91 $\pm$ 14 & 104 $\pm$ 20 & -1.3 & 98 $\pm$ 19 \\
6312 [S{\sc iii}] & -0.264 & 19 $\pm$ 2 & -0.6 & 18 $\pm$ 2 & 22 $\pm$ 6 & -0.3 & 20 $\pm$ 5 \\
6548 [N{\sc ii}] & -0.296 & 51 $\pm$ 11 & -1.6 & 47 $\pm$ 10 & 51 $\pm$ 12 & -0.8 & 48 $\pm$ 11 \\
6563 H$\alpha$ & -0.298 & 3000 $\pm$ 23 & -92.8 & 2797 $\pm$ 30 & 3026 $\pm$ 16 & -46.7 & 2814 $\pm$ 21 \\
6584 [N{\sc ii}] & -0.300 & 145 $\pm$ 11 & -4.5 & 135 $\pm$ 10 & 149 $\pm$ 6 & -2.2 & 138 $\pm$ 6 \\
6678 He{\sc i} & -0.313 & 27 $\pm$ 5 & -0.8 & 25 $\pm$ 4 & $\cdots$ & $\cdots$ & $\cdots$ \\
6717 [S{\sc ii}] & -0.318 & 307 $\pm$ 7 & -9.4 & 285 $\pm$ 7 & 300 $\pm$ 17 & -4.4 & 277 $\pm$ 16 \\
6731 [S{\sc ii}] & -0.320 & 236 $\pm$ 8 & -7.2 & 219 $\pm$ 8 & 206 $\pm$ 14 & -3.0 & 191 $\pm$ 13 \\
7065 He{\sc i} & -0.364 & 23 $\pm$ 5 & -0.7 & 21 $\pm$ 4 & $\cdots$ & $\cdots$ & $\cdots$ \\
7136 [Ar{\sc iii}] & -0.374 & 87 $\pm$ 11 & -2.7 & 80 $\pm$ 10 & 80 $\pm$ 16 & -1.3 & 73 $\pm$ 15 \\
7319 [O{\sc ii}] & -0.398 & 44 $\pm$ 4 & -1.4 & 40 $\pm$ 4 & $\cdots$ & $\cdots$ & $\cdots$ \\
7330 [O{\sc ii}] & -0.400 & 42 $\pm$ 3 & -1.3 & 38 $\pm$ 2 & $\cdots$ & $\cdots$ & $\cdots$ \\
\hline
I(H$\beta$)(erg\,s$^{-1}$\,cm$^{-2}$) & & \multicolumn{3}{c}{ 3.93e-15} & \multicolumn{3}{c}{ 2.61e-15} \\
c(H$\beta$) & & \multicolumn{3}{c}{0.11 $\pm$ 0.02} & \multicolumn{3}{c}{0.12 $\pm$ 0.02} \\
A(V) & & \multicolumn{3}{c}{0.24 $\pm$ 0.04} & \multicolumn{3}{c}{0.26 $\pm$ 0.04} \\
\hline
\end{tabular}
\end{table*}
\addtocounter{table}{-1}
\begin{table*}
\caption{-- {\it Continued}}
\begin{tabular}{lc@{\hspace{3em}}ccc}
\hline
\multicolumn{1}{l}{} & \multicolumn{1}{l}{} & \multicolumn{3}{c}{\hspace{-3em}\large NGC 6789-E} \\
\multicolumn{1}{c}{$\lambda$ ({\AA})} & f($\lambda$) & F($\lambda$) & EW(\AA) & I($\lambda$) \\
\hline
3726 [O{\sc ii}] & 0.322 & 347 $\pm$ 51 & -30.5 & 492 $\pm$ 94 \\
3729 [O{\sc ii}] & 0.322 & 398 $\pm$ 25 & -28.3 & 563 $\pm$ 77 \\
3868 [Ne{\sc iii}] & 0.291 & 62 $\pm$ 11 & -2.2 & 86 $\pm$ 18 \\
4102 H$\delta$ & 0.229 & 145 $\pm$ 10 & -7.6 & 186 $\pm$ 25 \\
4340 H$\gamma$ & 0.157 & 374 $\pm$ 9 & -11.9 & 443 $\pm$ 48 \\
4363 [O{\sc iii}] & 0.149 & 20 $\pm$ 9 & -0.6 & 23 $\pm$ 11 \\
4471 He{\sc i} & 0.115 & 36 $\pm$ 5 & -1.7 & 41 $\pm$ 7 \\
4861 H$\beta$ & 0.000 & 1000 $\pm$ 13 & -35.7 & 1000 $\pm$ 93 \\
4959 [O{\sc iii}] & -0.026 & 842 $\pm$ 15 & -22.9 & 818 $\pm$ 75 \\
5007 [O{\sc iii}] & -0.038 & 2358 $\pm$ 17 & -60.0 & 2263 $\pm$ 201 \\
5876 He{\sc i} & -0.203 & 124 $\pm$ 23 & -4.3 & 100 $\pm$ 20 \\
6548 [N{\sc ii}] & -0.296 & 42 $\pm$ 14 & -1.3 & 39 $\pm$ 13 \\
6563 H$\alpha$ & -0.298 & 3090 $\pm$ 32 & -98.9 & 2836 $\pm$ 42 \\
6584 [N{\sc ii}] & -0.300 & 69 $\pm$ 12 & -2.1 & 64 $\pm$ 11 \\
6717 [S{\sc ii}] & -0.318 & 225 $\pm$ 23 & -8.4 & 202 $\pm$ 21 \\
6731 [S{\sc ii}] & -0.320 & 207 $\pm$ 18 & -8.4 & 186 $\pm$ 17 \\
7136 [Ar{\sc iii}] & -0.374 & 80 $\pm$ 8 & -2.8 & 68 $\pm$ 6 \\
7319 [O{\sc ii}] & -0.398 & 27 $\pm$ 4 & -1.2 & 22 $\pm$ 3 \\
7330 [O{\sc ii}] & -0.400 & 20 $\pm$ 8 & -1.0 & 17 $\pm$ 6 \\
\hline
I(H$\beta$)(erg\,s$^{-1}$\,cm$^{-2}$) & & \multicolumn{3}{c}{ 2.94e-15}\\
c(H$\beta$) & & \multicolumn{3}{c}{0.47 $\pm$ 0.06}\\
A(V) & & \multicolumn{3}{c}{1.00 $\pm$ 0.10} \\
\hline
\end{tabular}
\end{table*}
\subsection{Electron densities and temperatures}
\label{elden}
\begin{table*}
\caption{Electron densities and temperatures for the five knots. Densities units are in
cm$^{-3}$ and temperatures in 10$^{4}$ K.}
\label{temden}
\begin{tabular}{l c c c c c }
\hline
& NGC~6789-A & NGC~6789-B & NGC~6789-C & NGC~6789-D & NGC~6789-E \\
\hline
n([S{\sc ii}]) & 80: & 60: & 60: & 90: & 300: \\
n([O{\sc ii}]) & 60: & 70: & 60: & 70: & 250: \\
t([O{\sc iii}]) & 1.44 $\pm$ 0.05& 1.36 $\pm$ 0.10& 1.44 $\pm$ 0.05& 1.30 $\pm$ 0.15& 1.16 $\pm$ 0.18\\
t([O{\sc ii}]) & 1.35 $\pm$ 0.06$^{a}$& 1.18 $\pm$ 0.10& 1.41 $\pm$ 0.08& 1.21 $\pm$ 0.10$^{a}$& 1.33 $\pm$ 0.25\\
t([S{\sc iii}])$^{b}$ & 1.38 $\pm$ 0.16& 1.30 $\pm$ 0.19& 1.38 $\pm$ 0.16& 1.22 $\pm$ 0.24& 1.06 $\pm$ 0.25\\
\hline
\end{tabular}
\begin{flushleft}
$^{a}$ From a relation with T([O{\sc iii}]) based on photoionisation models\\
$^{b}$ From an empirical relation with T([O{\sc iii}])
\end{flushleft}
\end{table*}
The electron density and temperatures of the ionised gas were derived from the emission line
data using the same procedures as in \cite{pmontero03}, based on the five-level statistical
equilibrium atom approximation in the IRAF task \texttt{temden} \citep{derobertis87,shaw95}.
See \cite{hagele08} and Appendix B of \cite{garcia-benito09} for a description of the
temperature and densities relations, respectively. We took as sources of error the
uncertainties associated with the reddening-corrected emission-line fluxes and
we propagated them through our calculations.
Electron densities were derived from the [S{\sc ii}] $\lambda\lambda$ 6717,6731 \AA\ and
[O{\sc ii}] $\lambda\lambda$ 3726,3729 \AA\ line ratios, which are representative of the
low-excitation zone of the ionised gas. In all cases they provide upper limits, which are
remarkably similar for both ions. In the case of the oxygen lines, the spectral dispersion
does not allow the total resolution of the lines, and they were deblended by a multi-Gaussian fit.
The upper limits for the electron density are lower in all cases than the critical value for
collisional de-excitation.
We computed three electron temperatures, T([O{\sc ii}]), T([O{\sc iii}]) and T([S{\sc iii}]), for
each of the five knots. The auroral [O{\sc iii}] $4363$ was detected in all the five knots with
sufficient signal-to-noise, and the [O{\sc iii}] electron temperature was derived directly from
the ratio (I(4959 \AA) + I(5007 \AA))/ I(4363 \AA).
The auroral [O{\sc ii}] $7319$ and [O{\sc ii}] $7330$ lines from the 2F
multiplet are emission-line doublets. Due to the spectral resolution, we report here the sum of
each doublet. Using the calculated [O{\sc iii}] electron temperatures, we checked that the
contribution by direct recombination is negligible for these lines \citep{liu00}. For knot A and
D, the intensity of the [O{\sc ii}] $\lambda\lambda$ 7319,7330 \AA\ did not allow an accurate
measure, therefore we derived their [O{\sc ii}] temperature from T([O{\sc iii}]) using the relation
based on the photoionisation models described in \cite{pmontero03}, which takes into account
explicitly the dependence of T([O{\sc ii}]) on the electron density.
Although we could measure the [S{\sc iii}] $\lambda$ 6312 \AA\ in some of the knots, we were not
able to get the [S{\sc iii}] $\lambda$ 9069,9532 \AA\ lines, so we estimated the [S{\sc iii}]
temperature from the empirical relation
\begin{equation}
t([\textrm{{S\sc iii}}]) = (1.19 \pm 0.08) t([\textrm{{O\sc iii}}]) - (0.32 \pm 0.10)
\end{equation}
\noindent found by \cite{hagele06}.
The electron densities and temperatures derived in the five knots of NGC~6789 are listed in
Table \ref{temden} along with their corresponding errors.
\begin{table*}
\caption{Ionic chemical abundances for helium.}
\label{helium}
\begin{tabular}{l c c c c c }
\hline
& NGC 6789-A & NGC 6789-B & NGC 6789-C & NGC 6789-D & NGC 6789-E \\
\hline
He$^{+}$/H$^{+}$ (4471) & 0.077 $\pm$ 0.011& 0.080 $\pm$ 0.015& 0.079 $\pm$ 0.013& 0.076 $\pm$ 0.014& 0.082 $\pm$ 0.015 \\
He$^{+}$/H$^{+}$ (5876) & 0.075 $\pm$ 0.011& 0.071 $\pm$ 0.013& 0.072 $\pm$ 0.012& 0.076 $\pm$ 0.015& 0.073 $\pm$ 0.016\\
He$^{+}$/H$^{+}$ (6678) & $\cdots$ & $\cdots$ & 0.071 $\pm$ 0.013& $\cdots$ & $\cdots$ \\
He$^{+}$/H$^{+}$ (7065) & $\cdots$ & $\cdots$ & 0.074 $\pm$ 0.016& $\cdots$ & $\cdots$ \\
He$^{+}$/H$^{+}$ & 0.076 $\pm$ 0.010& 0.075 $\pm$ 0.012& 0.074 $\pm$ 0.009& 0.076 $\pm$ 0.011& 0.077 $\pm$ 0.011\\
He$^{2+}$/H$^{+}$ (4686) & $\cdots$ & $\cdots$ & 0.0023 $\pm$ 0.0006& $\cdots$ & $\cdots$ \\
He/H & $\cdots$ & $\cdots$ & 0.076 $\pm$ 0.009& $\cdots$ & $\cdots$ \\
\hline
\end{tabular}
\end{table*}
\subsection{Ionic abundances}
\label{abundances}
We derived the ionic abundances of the different chemical species using the brightest available
emission lines detected in the optical spectra using the task \texttt{ionic} of the STSDAS package
in {\sc iraf} \citep[see][]{hagele08}. The total abundances were derived by taking into account,
when required, the unseen ionisation stages of each element, resorting to the ionisation correction
factors (ICFs) for each species derived from the photoionisation models described in Section
\ref{stion} below.
\begin{equation}
\frac{X}{H} = ICF(X^{i})\frac{X^{+i}}{H^{+}}
\end{equation}
\subsubsection{Helium abundance}
Four of the strongest helium emission lines, He{\sc i} $\lambda\lambda$ 4471, 5876, 6678, and 7065
\AA\ were used to calculate He$^{}+$/H$^{}+$. The last two lines were only detected with enough
signal-to-noise in knot C. Also in this knot, He {\sc ii} $\lambda$ 4686 was measured, allowing the
calculation of twice ionised He.
Helium lines arise mainly from pure recombination; however, they could have some contribution from
collisional excitation as well as being affected by self-absorption and, if present, by underlying
stellar absorptions \citep[see][for a complete treatment]{olive01}, but we checked
that these absorptions produce negligible variations in the measure of the
lines in the STARLIGHT subtracted spectra.
We took the electron temperature of [O{\sc iii}] as representative of the zone where the He
emission arises and we used the equations given by \cite{olive01} to derive the He$^{+}$/H$^{+}$,
using the theoretical emissivities scaled to H$\beta$ from \cite{smits96} and the expressions
for the collisional correction factors from \cite{kingdon95}. To calculate the helium twice
ionised, we used the equation found by \cite{kunth83}. We did not take into account, however,
the corrections for fluorescence since the involved helium lines have negligible dependence
on optical-depth effects and the observed knots have low densities.
The results obtained for each line and their corresponding errors are presented in Table
\ref{helium}. For knot C, the total abundance of He was calculated by adding directly the
two ionic abundances, He/H = (He$^{+}$ + He$^{2+}$)/H$^{+}$. Also given in the table is
the adopted value for He$^{+}$/H$^{+}$, a weighted average of the values, using the error
of each line as weight.
\subsubsection{Ionic abundances from forbidden lines}
The oxygen ionic abundance ratios, O$^{+}$/H$^{+}$ and O$^{2+}$/H$^{+}$, were derived from
the [O{\sc ii}]\, $\lambda\lambda$\,3726,3729\,\AA\ and
[O{\sc iii}] $\lambda\lambda$\,4959, 5007\,\AA\ lines, respectively using the appropriate
electron temperature for each ion.
The ionic abundance of nitrogen, N$^{+}$/H$^{+}$, was derived from the intensities of the
[N{\sc ii}]\, $\lambda\lambda$\,6548,6584\,\AA\ lines assuming that T([N{\sc ii}]) $\approx$
T([O{\sc ii}]).
For sulphur, we derived S$^{+}$/H$^{+}$ abundances from the fluxes of the [S{\sc ii}] emission lines
at $\lambda\lambda$ 6717,6731 \AA\ assuming that T([S{\sc ii}]) $\approx$ T([O{\sc ii}]).
Since we were not able to measure the near infrared [S{\sc iii}] $\lambda\lambda$ 9069,9532 \AA\,
we derived the S$^{2+}$/H$^{+}$ abundances using T([S{\sc iii}]) and [S{\sc iii}] $\lambda$ 6312 \AA\
by means of the equation
\begin{align}
12\,+\,log(S^{2+}/H^{+})\,&=&\,log\frac{I(6312)}{I(H\beta)}\,+\,6.74 \nonumber \\
& & +\,\frac{1.672}{t}\,-\,0.595\,log\,t
\end{align}
Neon ionic abundance was estimated from the [Ne {\sc iii}] emission line at $\lambda$ 3869 \AA.
For this ion, we took the electron temperature of [O{\sc iii}] as representative of the high
excitation zone \citep{peimbert69}.
[Ar {\sc iii}] $\lambda$ 7136 \AA\ was the only argon line detected in the spectra and
the abundance of Ar$^{2+}$ was calculated assuming that T([Ar{\sc iii}]) $\approx$
T([S{\sc iii}]) \citep{garnett92}.
The ionic abundances -and their corresponding errors- for each observed element for
the five knots are given in Table \ref{ionic}.
\begin{table*}
\caption{Ionic chemical abundances derived from forbidden emission lines, ICFs$^{a}$ and total
chemical abundances for elements heavier than helium.}
\label{ionic}
\begin{tabular}{l c c c c c }
\hline
& NGC~6789-A & NGC~6789-B & NGC~6789-C & NGC~6789-D & NGC~6789-E \\
\hline
12 + log(O$^{+}$/H$^{+}$) & 7.43 $\pm$ 0.04& 7.82 $\pm$ 0.11& 7.42 $\pm$ 0.07& 7.73 $\pm$ 0.11& 7.16 $\pm$ 0.23\\
12 + log(O$^{2+}$/H$^{+}$) & 7.63 $\pm$ 0.04& 7.28 $\pm$ 0.07& 7.59 $\pm$ 0.04& 7.41 $\pm$ 0.13& 7.68 $\pm$ 0.17\\
\textbf{12 + log(O/H)} & 7.84 $\pm$ 0.04& 7.93 $\pm$ 0.10& 7.81 $\pm$ 0.05& 7.90 $\pm$ 0.12& 7.80 $\pm$ 0.18\\
12 + log(N$^{+}$/H$^{+}$) & 6.03 $\pm$ 0.04& 6.37 $\pm$ 0.07& 6.08 $\pm$ 0.06& 6.22 $\pm$ 0.07& 5.85 $\pm$ 0.16\\
ICF(N$^{+}$) & 3.57 & 1.57 & 4.57 & 1.97 & 4.09 \\
\textbf{12 + log(N/H)} & 6.58 $\pm$ 0.04& 6.57 $\pm$ 0.07& 6.74 $\pm$ 0.06& 6.51 $\pm$ 0.07& 6.50 $\pm$ 0.16\\
log(N/O) & -1.26 $\pm$ 0.06& -1.37 $\pm$ 0.12& -1.07 $\pm$ 0.08& -1.38 $\pm$ 0.14& -1.30 $\pm$ 0.24\\
12 + log(S$^{+}$/H$^{+}$) & 5.70 $\pm$ 0.03& 5.91 $\pm$ 0.07& 5.75 $\pm$ 0.05& 5.83 $\pm$ 0.07& 5.70 $\pm$ 0.14\\
12 + log(S$^{2+}$/H$^{+}$) & 5.88 $\pm$ 0.16& 6.14 $\pm$ 0.23& 6.09 $\pm$ 0.16& 6.35 $\pm$ 0.30& $\cdots$ \\
ICF(S$^{+}$ + S$^{2+}$) & 1.11 & 1.01 & 1.24 & 1.01 & 7.64$^{b}$ \\
\textbf{12 + log(S/H)} & 6.20 $\pm$ 0.11& 6.35 $\pm$ 0.18& 6.41 $\pm$ 0.11& 6.48 $\pm$ 0.25& 6.58 $\pm$ 0.14\\
log(S/O) & -1.64 $\pm$ 0.12& -1.58 $\pm$ 0.20& -1.41 $\pm$ 0.12& -1.41 $\pm$ 0.27& -1.22 $\pm$ -0.25\\
12 + log(Ne$^{2+}$/H$^{+}$) & 7.00 $\pm$ 0.06& 6.77 $\pm$ 0.10& 7.00 $\pm$ 0.06& 6.68 $\pm$ 0.17& 6.75 $\pm$ 0.20\\
ICF(Ne$^{2+}$) & 1.39 & 1.61 & 1.28 & 2.07 & 1.22 \\
\textbf{12 + log(Ne/H)} & 7.14 $\pm$ 0.06& 6.98 $\pm$ 0.10& 7.11 $\pm$ 0.06& 7.00 $\pm$ 0.17& 6.82 $\pm$ 0.20\\
log(Ne/O) & -0.70 $\pm$ 0.07& -0.95 $\pm$ 0.14& -0.70 $\pm$ 0.08& -0.90 $\pm$ 0.21& -0.96 $\pm$ 0.27\\
12 + log(Ar$^{2+}$/H$^{+}$) & 5.67 $\pm$ 0.08& 5.56 $\pm$ 0.12& 5.56 $\pm$ 0.09& 5.63 $\pm$ 0.17& 5.70 $\pm$ 0.19\\
ICF(Ar$^{2+}$) & 1.12 & 1.21 & 1.15 & 1.14 & 1.10 \\
\textbf{12 + log(Ar/H)} & 5.72 $\pm$ 0.08& 5.64 $\pm$ 0.12& 5.62 $\pm$ 0.09& 5.68 $\pm$ 0.17& 5.74 $\pm$ 0.19\\
log(Ar/O) & -2.12 $\pm$ 0.09& -2.28 $\pm$ 0.16& -2.19 $\pm$ 0.10& -2.22 $\pm$ 0.21& -2.06 $\pm$ 0.22\\
\hline
\end{tabular}
\begin{flushleft}
$^{a}$ ICFs estimated from tailored photoionisation models (see Section \ref{stion})\\
$^{b}$ ICF(S$^+$)
\end{flushleft}
\end{table*}
\section{Optical and UV photometry}
\label{phot}
\subsection{H$\alpha$ and GALEX photometry}
\label{Halfa}
We analysed the H$\alpha$ image of the galaxy, retrieved from the Palomar/Las Campanas atlas
of blue compact dwarf galaxies \citep{gildepaz03}. We defined elliptical apertures on this image
for each of the five knots extracted in the spectroscopic observations and we measured all the
flux inside the elliptical apertures up to the isophote corresponding to the 50\% of the peak
of the intensity for each knot (see Figure \ref{regslit}). The total fluxes and the
corresponding luminosities at the adopted distance are listed in Table \ref{hafot} with their
corresponding errors. The observed H$\alpha$ fluxes were corrected in each knot for dust extinction
using the values of c(H$\beta$), given in Table \ref{lines}. We also list in the same table the size
of each knot, as the radius of the circular aperture whose area is equal to that encompassed by the
elliptical aperture. As can be seen in Figure \ref{regslit}, NGC 6789 has an H$\alpha$ major
axis of 740 pc and 560 pc for its minor axis, measured using an elliptical aperture that
encompasses the isophote at 3$\sigma$ over the level of the sky background noise. The total flux
inside this isophote (not extinction-corrected)
is (2.0 $\pm$ 0.1) $\times$ 10$^{-12}$ erg s$^{-1}$ cm$^{-2}$.
\begin{table*}
\caption{Properties of the individual knots as measured in the H$\alpha$ and GALEX photometry.}
\label{hafot}
\begin{tabular}{l c c c c c }
\hline
& NGC 6789-A & NGC 6789-B & NGC 6789-C & NGC 6789-D & NGC 6789-E \\
\hline
log F(H$\alpha$) (erg/s/cm$^2$) & -13.96 $\pm$ 0.03 & -13.85 $\pm$ 0.03 & -13.69 $\pm$ 0.03 & -14.03 $\pm$ 0.03 & -14.17 $\pm$ 0.03 \\
log L(H$\alpha$) (erg/s) & 37.23 $\pm$ 0.02 & 37.34 $\pm$ 0.02 & 37.50 $\pm$ 0.03 & 37.17 $\pm$ 0.02 & 37.02 $\pm$ 0.01\\
Radius (pc) & 30 & 34 & 37 & 36 & 21 \\
FUV (mag) & 20.00 $\pm$ 0.14 & 20.00 $\pm$ 0.16 & 20.18 $\pm$ 0.20 & 19.71 $\pm$ 0.15 & 21.24 $\pm$ 0.04 \\
NUV (mag) & 19.86 $\pm$ 0.08 & 19.69 $\pm$ 0.09 & 19.86 $\pm$ 0.10 & 19.68 $\pm$ 0.09 & 21.18 $\pm$ 0.01 \\
FUV - NUV (mag) & 0.14 $\pm$ 0.16 & 0.28 $\pm$ 0.19 & 0.31 $\pm$ 0.22 & 0.03 $\pm$ 0.18 & 0.06 $\pm$ 0.04 \\
\hline
\end{tabular}
\end{table*}
The \textit{Galaxy Evolution Explorer} \citep[GALEX;][]{martin05,morrissey05} far-ultraviolet
(FUV; $\lambda_{ref}$ = 1530 \AA, $\Delta\lambda$ = 400 \AA) and near-ultraviolet
(NUV; $\lambda_{ref}$ = 2310 \AA, $\Delta\lambda$ = 1000 \AA) images of NGC 6789 were retrieved
from the Nearby Galaxies Survey (NGS). We used the same elliptical apertures defined in the
H$\alpha$ image to obtain the FUV and NUV fluxes in each knot. In Table \ref{hafot} we show for each
knot both the FUV and NUV magnitudes once extinction corrected together with the colour index
FUV-NUV, all of them in AB mags.
As typical sizes of the studied star-forming knots are lower than the GALEX point-spread function
(PSF), some aperture effects to the measured UV fluxes that cannot be quantified may exist.
However, the characterization of the knots by the corresponding colour indices are
less affected by this effect than the total flux measurements in these regions,
as we checked by taking different aperture sizes around the position of
the knots.
As can be seen, the four brightest knots (from A to D) have similar
UV luminosities both in FUV and NUV, being the knot D the brightest one, although this is not
the brightest knot in H$\alpha$. Regarding colours, knots B and C present redder UV colours
as compared with the other three knots.
\subsection{WFPC2 Photometry \label{rphot}}
\textit{Hubble Space Telescope} (HST) Wide Field Planetary Camera 2 (WFPC2)
images of NGC~6789 were retrieved from the HST archive. We discuss photometry
based on the images taken in 2000 July-September (GO-8122) through two continuum
filters: F555W (V) and F814W (I). Each camera images onto a
Loral 800$\times$800 CCD which gives a plate scale of 0\farcs046 pixel$^{-1}$
for the PC camera and 0\farcs10 pixel$^{-1}$ for the three WF cameras, with a
readout noise of $\sim$5 e$^{-}$ and a gain of 7 e$^{-}/DN$ for this observations.
The central star-forming region of NGC~6789 was centered in the PC camera in all
images using three different pointings. The scale of the PC CCD at NGC~6789, for which we
assumed a distance modulus of $(m - M) = 27.80$ \citep{drozdovsky01}, is $0.80$ pc pixel$^{-1}$.
Table \ref{log} lists the details concerning the WFPC2 data. Figure \ref{hstgalex}
shows the HST WFPC2 images of NGC 6789 in F555W and F814W filters. The contours are
drawn from FUV and NUV GALEX images, respectively. The aperture of knot E defined in the
H$\alpha$ image and the rectangle encompassing the FOV of the PC chip of the WFPC2 data are
also shown for reference.
\begin{table*}
\caption{Journal of HST/WFPC2 observations of NGC~6789. The images were
obtained in 2000 July-September for the cycle 8 program GO-8122, with
Regina Schulte-Ladbeck as PI. The object of study is centered in the PC chip.}
\label{log}
\begin{tabular}{l c c c c c }
\hline
Filter & RA & Dec & Exposure & Observation ID\\
& (J2000) & (J2000) & (s) & \\
\hline
F555W & 19:16:37.77 & +63:58:37.2 & 2 $\times$ 1300 & U5BF0301R, U5BF0302R \\
& 19:16:37.73 & +63:58:37.4 & 2 $\times$ 1400 & U5BF0303R, U5BF0304R \\
& 19:16:37.68 & +63:58:37.6 & 2 $\times$ 1400 & U5BF0305R, U5BF0306R \\
F814W & 19:16:37.77 & +63:58:37.2 & 2 $\times$ 1300 & U5BF0401R, U5BF0402R \\
& 19:16:37.73 & +63:58:37.4 & 2 $\times$ 1400 & U5BF0403R, U5BF0404R \\
& 19:16:37.68 & +63:58:37.6 & 2 $\times$ 1400 & U5BF0405R, U5BF0406R \\
\hline
\end{tabular}
\end{table*}
\begin{figure*}
\begin{minipage}{\textwidth}
\centering
\includegraphics[width=0.49\textwidth,clip=]{f555w-fuv}
\includegraphics[width=0.49\textwidth,clip=]{f814w-nuv}
\end{minipage}
\caption{HST WFPC2 images of NGC 6789 in continuum filters F555W and F814W. The solid contours
are drawn from FUV and NUV GALEX images, respectively. The dashed contour corresponds to the
isophote at 3$\sigma$ over the level of the sky background of the Palomar/Las Campanas H$\alpha$
image. The aperture of knot E defined in the H$\alpha$ image and the rectangle encompassing the
FOV of the PC chip of the WFPC2 data are also shown for reference. North is up and east is towards
the left-hand side.}
\label{hstgalex}
\end{figure*}
The stellar photometric analysis was performed with the HSTphot package \citep{dolphin00a}.
This package is specifically designed for use with HST WFPC2 images and uses a library of
Tiny Tim \citep{krist95} undersampled PSFs for different locations
of the star on the camera and of the star within the pixel, to centre the star and
to find its magnitude, given in the flight system magnitude. The first step was to run
the \texttt{mask} task using the HST data quality file (\texttt{c1f}) to mask out the bad
pixels and other image defects. The next step was to run \texttt{crmask} for cosmic ray
removal. We used a registration factor of 0.5 and $\sigma$-threshold of 3. It has the
capability of cleaning images that are not perfectly aligned, and it can handle images
from different filters. After cosmic-ray rejection, sets of images of each filter at a common
pointing were combined into a single image, using the routine \texttt{coadd}. Since
there are two images per pointing, we ended up with 3 images per filter. The sky
computation is made by \texttt{getsky}, which takes all pixels in an annulus around each
pixel, determines the sky value and calculates the sky background map. These sky values
are used only as a starting guess in the HSTphot photometry. The final step requires the
use of the \texttt{hotpixels} procedure on each combined image, which uses the result from
\texttt{getsky} and tries to locate and remove all hot pixels. This is an important step,
since hot pixels can create false detections and also, can throw off the PSF solutions.
The main \texttt{hstphot} routine was run on the images in the F555W and F814W bands.
This task performs stellar PSF photometry on multiple images from different
filters and pointings (providing the dithering pattern, it analyses the dithering image
as a whole), including alignment and aperture corrections, as well as PSF modifications
to correct for errors of geometric distortion via the \citet{holtzman95} distortion
correction equations and the 34th row error, noted by \citet{shaklan95} (see also
\citealt{anderson99}), and correction for charge transfer inefficiency \citep{dolphin00b}.
We enabled the determination of a ``local sky'' value (option 2).
We selected ``good stars'' from the \texttt{hstphot} output. Object types were classified
as good star, possible unresolved binary, bad star, single-pixel cosmic ray or hot pixel,
and extended object. To ensure selecting high fidelity point sources, we also used
the ``sharpness'' parameter (absolute value to be $\leq$ 0.35), a measure of the quality
of the fit ($\chi^{2}$ $\leq$ 2.5) and a minimum signal-to-noise ratio of 10 to reject
false star detections in regions with structured nebulosity or artifacts. The final number
of stars detected in the PC chip with these parameters in both filters were 7410.
To quantify the completeness and systematic uncertainty of the photometry, a grid of
artificial stars was generated on a 2-dimensional CMD and distributed according to the
flux of the images with an artificial star routine provided by HSTphot.
The parameters of the routine are the minimum and maximum of the measured colour and
magnitude. The magnitude steps used were multiple of 0.5, while colour steps are by 0.25.
The artificial stars were distributed on the CMD in accordance with the number observed.
Approximately 60,000 fake stars were added in each image (at different trials, in order
to leave the crowding conditions unaltered) and were given random magnitudes and colours
in the observed range. The 50\% completeness of the F555W filter is reached at 26.9
magnitude, while for the F814W filter is 25.6 mag.
The final CMD of NGC~6789 with typical photometric errors per magnitude bin is shown in
Figure \ref{cmd}.
\begin{figure}
\includegraphics[width=0.5\textwidth,clip=]{cmd-er-ngc6789}
\caption{CMD of NGC~6789 with average photometric uncertainties per
magnitude bin.}
\label{cmd}
\end{figure}
\section{Discussion}
\label{discussion}
\subsection{Star formation history and stellar populations \label{ssfh}}
Studies to obtain information on the star formation of composite stellar systems
have been proven to be very successful (see e.g., \citealt{aparicio97};
\citealt{hernandez00}; \citealt{dolphin02}). Any SFH recovery method relies on
the assumption that a composite stellar population can be considered simply as the
combination of SSPs, assigning a certain relative weight to each SSP. The observed
CMD is compared with theoretical ones created via Monte-Carlo methods for a variety
of IMFs, binary fractions, star-formation laws, etc., extracting the stellar information
from isochrones or stellar evolution tracks. Observed and theoretical CMDs are divided
into boxes and converted into two-dimensional histograms of stellar density as function
of colour and magnitude (Hess diagrams) and compared using statistical methods.
For the analysis of this work, we used the StarFISH code\footnote{Available at
\texttt{http://www.noao.edu/staff/jharris/SFH/}} developed by \cite{harris01}. This code
was successfully used in a number of cases (e.g. \citealt{harris04};
\citealt{brown06}; \citealt{williams07}; \citealt{harris09}; \citealt{garcia-benito11}).
Using determinations of the interstellar extinction, photometric errors, and distance
moduli, it uses minimisation of a chi-squared-like statistics technique to find the linear
combination of single-component stellar population models that best fit an observed CMD.
The ages and metallicities of the underlying stellar population are characterized by
the ages and metallicities of the CMDs included in the best, while SFR at each age is
provided by the weights given to the CMDs.
The theoretical isochrones chosen for analysis were those of \cite{marigo08}\footnote{Available at
\texttt{http://stev.oapd.inaf.it/cgi-bin/cmd/}} for ages in the range 1 Myr-14Gyr. This data set
includes models with age bins spaced logarithmically since the CMD changes much more rapidly at
young ages than at old ones. We chose the two metallicity values closest to the observed ones (see
Section \ref{abundances}), namely Z = \{0.001, 0.004\}. \cite{marigo08} provide metallicities
in the range 0.0001 $\leq$ Z $\leq$ 0.03. Nevertheless, for the sake of consistency we selected
only the same values that were available for the STARLIGHT libraries \citep{bruzual03}. The
photometric error and completeness estimates were taken directly from the results of the
artificial star experiments described in Section \ref{rphot}. We adopted a Salpeter IMF with a
spectral index of -1.35 from 0.1-100 M$_{\odot}$ \cite{salpeter55}. This assumption is likely
to be valid since our CMD does not contain stars with masses $<$ 1 M$_{\odot}$. The binary
fraction was set to a value of 0.5.
Our reference value for NGC 6789 distance modulus is (m - M) = 27.80 $\pm$ 0.13 $\pm$ 0.18, value
reported by \cite{drozdovsky01} using the tip of the red giant branch distance method. The Galactic
extinction is A$_{V}$ = 0.212, value provided by \cite{schlegel98} extinction maps for an area
with radius equal to 5 arcmin around NGC 6789, remarkably close to the reddening value obtained
from our spectroscopic data (Section \ref{line}). Nevertheless, we built a set of models to
explore the space of parameters. The SFH recovery is repeated for each point in the grid
(m - M) vs A$_{V}$, and then we built a final $\chi^{2}_{min}$ map for the solutions.
We explored the space of parameters in steps of 0.01 in distance modulus and extinction. To
evaluate the errors of the recovered solutions, we generated a series of synthetic CMDs
using the best-fitting SFR and find a correspondence between the $\chi^{2}_{min}$
and the confidence level of significance.
Although it is tempting to perform a SFH analysis for each knot, the resulting individual CMDs,
taken to be as the stars included inside the defined ellipses (see Figure \ref{regslit}
and subsection \ref{Halfa}), do not contain enough stars\footnote{Specially Knot E, part of
which falls outside the PC FOV.} and, therefore, the large errors associated with each individual
SFH prevent us to draw any conclusion. At any rate, the distribution of stars in each individual
CMD is very similar for all the knots. Thus, we use the entire CMD of the galaxy to derive the
global SFH.
Figure \ref{sfh} shows the overall best-fitting StarFISH solution for the SFH of NGC 6789, located
at (m - M) = 27.83 $\pm$ 0.06 and A$_{V}$ = 0.64 $\pm$ 0.08, where the errors stand for the
1$\sigma$ confidence level. The errors in the star formation diagram only allow the consideration
of a few main bursts in the SFH. Regarding the lower metallicity (Z = 0.001),
the solution shows a few bursty events around 1.6 Gyr and 6-10 Gyr, followed by a more recent
burst around 100 Myr and a very young one during the last 12 Myr. As for the higher metallicity
(Z = 0.004), only one event between 150-300 Myr is clearly seen. Although for the very young
stars ($<$ 12 Myr) both metallicities present a significant error, we take Z = 0.001 as the
metallicity for the youngest event, as derived from the nebular analysis (see section
\ref{abundances}). The SFR, according to StarFISH's results, was stronger around 100 Myr,
with a peak of 0.015 M$_{\odot}$/yr.
\begin{figure}
\includegraphics[width=0.5\textwidth,clip=]{ngc6789-sfh}
\caption{Best StarFISH SFH fit derived from the HST optical observations of the resolved
stellar populations for NGC 6789. The best distance modulus and extinction values are also
shown.}
\label{sfh}
\end{figure}
\begin{figure*}
\begin{minipage}{\textwidth}
\centering
\includegraphics[width=0.49\textwidth,clip=]{ngc6789-sfh-sl}
\includegraphics[width=0.49\textwidth,clip=]{ngc6789-sfh-slfree}
\end{minipage}
\caption{Histograms of the distribution (light fraction) of the best fit derived by STARLIGHT
for the spectra of the five knots of NGC 6789. The left panel corresponds to the fits
using the constrains obtained by the CMD analysis, while the right panel
shows the fits using non-constrained age-metallicity libraries.}
\label{sfhsl}
\end{figure*}
In the light of these results, we used the age-metallicity solution derived from the CMD
as a prior to constrain the libraries of the STARLIGHT code, used to fit the spectra of
the five knots (see Section \ref{line} for more details). In left panel of Figure
\ref{sfhsl} we show the age distribution of the light fraction obtained from STARLIGHT for
each knot under the constrains mentioned above. Another fitting was carried out without any
specific age-metallicity constrain in order to compare it with the results of the first fitting.
These are shown in the right panel of Figure \ref{sfhsl}. As can be seen, all knots present
a very young stellar population, with ages younger than 10 Myr, responsible for the ionisation
of the gas, combined with other older populations. The presence of these
old populations is confirmed with the detection of absorption metal lines of
Ca {\sc ii} K $\lambda$ 3933 \AA\ and Ca {\sc ii} H $\lambda$ 3968 \AA\, characteristic of
old stellar populations, except for knot E, in which these features are marginally
detected. The combination of these two different stellar population is also obtained by
\cite{perez-montero10} in a sample of 10 BCDs using the same technique. In terms of mass,
more than the 80\% comes from the older population. The estimated total stellar mass and
the fraction of the mass of the stellar population with an age younger than 10 Myr with
respect to the total mass are given in Table \ref{slprop}. The stellar masses were corrected
for aperture effects, with factors calculated using the ratio between the extinction-corrected
H$\alpha$ fluxes measured in the elliptical regions of the photometry and those measured in
the extracted long-slit observations for each knot. In the same table we show the EW(H$\beta$)
values calculated once the contribution of the older stellar populations was removed, in order
to use EW(H$\beta$) to derive the properties of the ionising stellar population (see
Section \ref{stion}). The inner extinctions calculated by STARLIGHT for each fit are also
listed in the same table. Finally, the results for the non-constrained case are given in the
second row for each knot. The values from Table \ref{slprop} show that those fits obtained
from non-constrained libraries do not significantly alter some of the properties of
the stellar populations, such as the total stellar mass and the correction
factor for EW(H$\beta$), even that some relevant differences appear in the
age and metallicity distribution of the resulting SFHs. Therefore, we relied on the general
chemical evolution solution derived by StarFISH to constrain the age-metallicity libraries of
STARLIGHT. At any rate, the general distribution of the light is similar in both cases. We
recall that the CMD SFH (STARFISH) is derived for the whole galaxy, while the spectra SFH
(STARLIGHT) is derived for each individual knot.
Less significant is the difference of the fitted spectra between both cases. For these two
particular metallicities, the variation of the intensity of the first four Balmer lines on the
residual spectra for each knot is well within the observational errors when using one or the
other case. The change of age-metallicity distribution in the libraries only affects significantly
in these two particular cases to the SFH in the assignments of the metallicity of
the stellar populations, but the overall derived age distribution is very similar.
Regarding the extinction, we find some discrepancies between the values found by
STARLIGHT and those derived using Balmer decrements. However, the average extinction value
for all knots (0.52 $\pm$ 0.16) is very close to the CMD best fitting value given by StarFish
(0.64 $\pm$ 0.08). It should be noted, though, that in principle it is not expected to find
the same values in the gas and the stellar population.
\begin{table}
\caption{Properties of the stellar populations as obtained with STARLIGHT for
each knot of NGC 6789, including the inner extinction, the aperture corrected
total stellar mass, the mass of stars younger than 10 Myr (M$_{ion *}$), and the
EW(H$\beta$) corrected for the contribution of the older population. The first row
for each knot corresponds to the constrained age-metallicity libraries, while the
second row is for the unconstrained case.}
\label{slprop}
\begin{tabular}{l c c c c}
\hline
ID & A(V) & log M$_*$ & log M$_{ion *}$ & EW(H$\beta$)$_c$ \\
& (mag) & & & (\AA) \\
\hline
\hline
Knot A & 0.4 & 5.34 & 3.69 & 23 \\
& 0.4 & 5.35 & 3.81 & 19 \\
Knot B & 0.3 & 5.26 & 3.16 & 30 \\
& 0.7 & 5.22 & 3.68 & 30 \\
Knot C & 0.7 & 6.00 & 4.03 & 49 \\
& 0.6 & 5.97 & 3.95 & 51 \\
Knot D & 0.5 & 5.42 & 3.79 & 24 \\
& 0.5 & 5.51 & 3.66 & 24 \\
Knot E & 0.7 & 5.16 & 2.71 & 84 \\
& 0.5 & 5.21 & 2.47 & 95 \\
\hline
\end{tabular}
\end{table}
\begin{table*}
\caption{Properties of the individual knots derived from tailored photoionisation models compared
with the observed values.}
\label{model}
\begin{tabular}{l | c c | c c | c c | c c | c c }
\hline
& \multicolumn{2}{c|}{NGC 6789-A} & \multicolumn{2}{c|}{NGC 6789-B} & \multicolumn{2}{c|}{NGC 6789-C} & \multicolumn{2}{c|}{NGC 6789-D} & \multicolumn{2}{c}{NGC 6789-E} \\
& Mod. & Obs. & Mod. & Obs. & Mod. & Obs. & Mod. & Obs. & Mod. & Obs. \\
\hline
\hline
log L(H$\alpha$) (erg/s) & 37.32 & 37.23 $\pm$ 0.02 & 37.34 & 37.34 $\pm$ 0.02 & 37.51 & 37.50 $\pm$ 0.03 & 37.18 & 37.17 $\pm$ 0.02 & 37.03 & 37.02 $\pm$ 0.01\\
Radius (pc) & 7 & 30 & 33 & 34 & 22 & 37 & 10 & 36 & 5 & 21 \\
FUV-NUV (mag) & 0.17 & 0.14 $\pm$ 0.16 & 0.28 & 0.28 $\pm$ 0.19 & 0.30 & 0.31 $\pm$ 0.22 & 0.22 & 0.03 $\pm$ 0.18 & 0.10 & 0.06 $\pm$ 0.04 \\
I([O{\sc ii}]/I(H$\beta$) & 2.30 & 2.34 $\pm$ 0.15 & 3.33 & 3.16 $\pm$ 0.20 & 2.35 & 2.51 $\pm$ 0.16 &
3.05 & 3.14 $\pm$ 0.20 & 1.09 & 1.06 $\pm$ 0.04 \\
I([O{\sc iii}]/I(H$\beta$) & 3.82 & 3.76 $\pm$ 0.17 & 1.47 & 1.48 $\pm$ 0.07 & 3.45 & 3.44 $\pm$ 0.15 &
1.72 & 1.73 $\pm$ 0.08 & 2.30 & 2.26 $\pm$ 0.20 \\
12+log(O/H) & 7.93 & 7.84 $\pm$ 0.04 & 7.82 & 7.93 $\pm$ 0.10 & 7.79 & 7.81 $\pm$ 0.05 & 7.77 & 7.90 $\pm$ 0.12 &
7.64 & 7.80 $\pm$ 0.18 \\
-EW(H$\beta$ (\AA) & 21 & 23 & 30 & 30 & 48 & 49 & 29 & 24 & 80 & 84 \\
Age (Myr) & 5.4 & ... & 4.2 & ... & 3.5 & ... & 5.2 & ... & 3.9 & ... \\
Dust-to-gas ratio & 0.026 & ... & 0.34 & ... & 0.12 & ... & 0.15 & ... & 0.015 & ... \\
Abs. factor ($f_d$) & 5.34 & .... & 7.28 & ... & 8.21 & ... & 4.37 & ... & 2.23 & ... \\
\hline
\end{tabular}
\end{table*}
\subsection{Ionising stellar populations and photoionisation models}
\label{stion}
Among the properties of the young ionising populations that can be derived from
optical spectroscopy are their stellar masses and ages. The former can be derived from
the H$\alpha$ luminosity and EW(H$\beta$) ({\em e.g.} \citealt{diaz98}).
H$\alpha$ luminosity can also be used to estimate the SFR and the mass of ionised hydrogen.
The age can be estimated by using the EW(H$\beta$) as compared to sequences of evolutionary
synthesis models. In our case, the contamination of the older underlying population to
the H$\beta$ continuum was removed by using the analysis made with STARLIGHT. As can be seen
in Table \ref{slprop}, these corrections are quite different between the knots, going from a
3\% in the case of knot B to a 42\% in the case of knot E.
Nevertheless, as already shown by \cite{pmd07} and later in \cite{perez-montero10}
for other samples of BCDs, the corrected EW(H$\beta$) are still much lower than the
expected values for the ionisation of stellar clusters younger than 10 Myr. The
H{\sc i} mass can be estimated using the CO observations of this object reported
by \cite{leroy05} (marginally detected and thus, an upper limit)
and using the $X_{CO}$ factor at the appropriate metallicity proposed by \cite{magrini11} to
derive the H{\sc i} column density. For the adopted distance of NGC~6789 this gives a total
mass of log(M$_{HI}$/M$_{\odot}$) $\sim$ 6. Thus, this implies that dust absorption
could be the main responsible of the disagreement between expected and observed EW(H$\beta$).
Thus, both L(H$\alpha$) and EW(H$\beta$) must be corrected using the dust absorption factor ($f_d$)
before the derivation of the properties of the ionising cluster. This correction must be done even
after the extinction correction, because it corresponds to the fraction of ionising photons
absorbed by the dust and, therefore, not absorbed and re-emitted by the gas:
\begin{equation}
Q(H) = f_d \times Q_{obs}(H)
\end{equation}
To estimate both $f_d$ and the ages of the ionising cluster for each knot, we made tailored
photoionisation models using the code CLOUDY v.08.00 \citep{ferland98}. Our models assume a
one-dimensional structure, with the central ionising cluster and a gas and dust geometry, and
compute the emergent spectrum. For each knot we took as input the Starburst99 libraries
\citep{leitherer99,vazquez05}, based on stellar model atmospheres from \cite{smith02},
Geneva evolutionary tracks with high stellar mass loss \citep{meynet94}, a Kroupa IMF
\citep{kroupa02} in two intervals (0.1-0.5 and 0.5-100 M$_\odot$) with different exponents
(1.3 and 2.3 respectively), the theoretical wind model \citep{leitherer92}
and a supernova cut-off of 8 M$_\odot$. We fixed the metallicity of the stellar
populations to Z = 0.001 ($\approx$ 1/12 Z$_\odot$), which corresponds to the closest total
oxygen abundance measured using the direct method in all the knots. We also fixed the density
of particles to match the value derived in each knot from the [S{\sc ii}] emission-line ratio
and the chemical abundances in the gas (except oxygen) to match those derived using the direct
method. Those species that were not measured were put to scale with the solar proportions
measured by \cite{asplund09}.
Although not all species are expected to follow solar proportions, the deviation of this
assumption does not affect the results of our models of all the other studied species.
In each knot, we used an iterative method to find the values
of the free parameters (oxygen abundance, dust-to-gas ratio, number of
ionising photons, internal radius, filling factor, and age of the ionising cluster)
that better fit the observed properties (3727 {\AA} [O{\sc ii}]/H$\beta$,
5007 {\AA} [O{\sc iii}]/H$\beta$, 12+log(O/H), L(H$\alpha$), and FUV-NUV colour
index). The fitted properties along with the derived cluster age and $f_d$ in the
corresponding model for each knot are shown in Table \ref{model}.
The FUV-NUV colours were obtained by convolving the model output emergent spectrum for
each knot with the shape of the corresponding GALEX filter, performing
a direct measure of the resulting total flux, and finally calculating the
corresponding colour in AB magnitudes.
The assumptions taken for the models described above may be an
oversimplification as they do not take into account the three-dimensional structure
of the ionized gas and the dust. Besides, we cannot take as input parameter the
dust-to-gas ratio of this galaxy and possible uncertainties related with the adopted
synthesis stellar atmospheres can affect the final result.
As can be seen, the observed properties are generally well fitted by the models,
by assuming different geometries and dust-to-gas ratios without noticeably changing the
derived metallicity of the gas.
According to our results, $f_d$ correlates better with the FUV-NUV index
that with the reddening constant derived from the Balmer decrement in the optical spectra.
For instance, knot C has the highest $f_d$ with the lowest c(H$\beta$) and, in contrast, knots
A and E have the lowest $f_d$ and the highest c(H$\beta$). This could be indicative of a
complex inner dust structure which is not well traced by the reddening constant derived from
the Balmer decrement. Regarding the cluster ages, our results indicate that all knots have
ionising cluster in a range of ages between 3.5 Myr for knot C and 5.4 Myr for knot A. The
youngest cluster appears in knot C, although knot E, which has a very low $f_d$ correction,
has the lowest EW(H$\beta$). The age sequence in the studied knots does not correspond to any
specific spatial order.
\begin{table*}
\caption{Properties of ionising stellar populations in the knots as derived using the
Q(H$^{0}$) value obtained from the photoionisation models.}
\label{haprops}
\begin{tabular}{l c c c c c c c}
\hline
ID & log Q(H$^0$) & log M$_{burst}$ & log M(H{\sc ii}) & log SFS & SFR \\
& (erg/s) & (M$_{\odot}$) & (M$_{\odot}$) & (M$_{\odot}$) & (10$^{-3}$ M$_{\odot}$/yr) \\
\hline
\hline
Knot A & 49.94 & 3.82 & 3.45 & 4.34 & 1.04 \\
Knot B & 50.09 & 3.65 & 3.60 & 4.32 & 1.47 \\
Knot C & 50.31 & 3.69 & 3.82 & 4.54 & 2.44 \\
Knot D & 49.69 & 3.52 & 3.20 & 4.09 & 0.59 \\
Knot E & 49.25 & 2.73 & 2.76 & 3.48 & 0.21 \\
NGC 6789$^{a}$ & 51.49 & $\cdots$ & 5.00 & 5.89 & 36.97 \\
\hline
\end{tabular}
\begin{center}
$^{a}$ Values derived from the H$\alpha$ image, not corrected for extinction.
\end{center}
\end{table*}
The total masses of the ionising clusters were also derived by using the number of ionising
photons and the ages obtained from the respective CLOUDY models and the Starburst99
libraries of the corresponding synthesis cluster atmospheres. These are listed in
Table \ref{haprops}. As can be seen, the stellar masses of the bursts in the four first
knots are quite similar, while knot E has the lowest value. These masses are in agreement
within 0.3 dex with the masses derived for the younger stellar population using the fitting
to the optical spectrum made with STARLIGHT (Table \ref{slprop}).
Once the $f_d$ were estimated for the five knots, it was possible to derive the
properties that depend on the corrected total L(H$\alpha$), including
the mass of ionised hydrogen, the Star Formation Strengths (SFS, defined as the total
mass of gas transformed into stars during the burst) and the SFR. These are listed in
Table \ref{haprops}, along with the values for the whole galaxy.
The total ionised masses of the gas clouds in each knot were calculated
using \cite{oster89}:
\begin{equation}
M_{H^+} = Q(H^{0}) \frac{m_{p}}{n_{e}\alpha_{B}}
\end{equation}
\noindent assuming case B, n$_{e}$ = 100 cm$^{-3}$, and
$\alpha_{B} \sim 2.59 \times 10^{-13}$ cm$^{-3}$ s$^{-1}$.
The individual knots show masses of ionised hydrogen of the order of hundreds of solar masses
for knot E and up to $\sim$ 6600 M$_{\odot}$ in knot C.
In the case of the whole galaxy, the total flux described in Section \ref{Halfa} can be
converted into (1.0 $\pm$ 0.1) $\times$ 10$^{5}$ M$_{\odot}$ of ionised hydrogen, comparable to
knot A of the Giant Extragalactic H{\sc ii} Region NGC 5471 in M101 \citep{garcia-benito11} or
to the whole NGC 604 giant H{\sc ii} region in M33 \citep{relano09}.
It is known that the SFR determined from H$\alpha$ is sensitive to several uncertainties,
as those related with the extinction and the assumed IMF. Besides, not always
all the ionising photons are absorbed, being the escape fraction of ionising radiation from
individual H{\sc ii} regions (in nearby galaxies) between 15\% and 50\%
\citep{kennicutt98}.
We applied the calibration by
\cite{oti-floranes10}\footnote{Their web tool can be found at\\
\texttt{http://www.laeff.cab.inta-csic.es/research/sfr/}}, which makes a distinction
between instantaneous (IB) and extended burst (EB). We used our extinction and dust absorption
corrected H$\alpha$ luminosity (except for the value of the whole galaxy) and a Salpeter IMF in
the range 0.1-100 M$_{\odot}$ as input. For the IB case, the value given by the
calibration is the SFS. The SFS values in Table \ref{haprops} were calculated using the age
provided by the calibration, namely 4, 5, and 6 Myr, closest to the age estimated from the
photoionisation models. According to this calibration, the mass of gas transformed in stars
during the actual burst in each knot is of the order of tens of thousands of solar masses,
except for knot E, which is one order of magnitude lower.
We provide as well the SFR, for the EB case (continuous star formation), as a reference value.
Regarding the SFR of the whole galaxy, the value estimated by means of the H$\alpha$ flux is
slightly higher than the peak obtained from the CMD (see Figure \ref{sfh}).
The SFR for BCDs spans from a few 10$^{-3}$ to several times 10$^{1}$ M$_{\odot}$/yr
\citep{hopkins02,zhao11}, showing NGC 6789 not a particularly high value.
Although our models are able to reproduce much of the observational
information both in the optical and the UV, some of their limitations must be
taken into account. It is possible that the EW(H$\beta$) could be modified owing to great
fractions of leaked photons. However, comparing the mass of ionised gas in this galaxy
[log(M/M$_{\odot}$) $\sim$ 5, see Table \ref{haprops}] with the mass of neutral hydrogen
does not support this scenario, but taking into account the oversimplified geometry of our
models it cannot be ruled out completely. In this sense, as the long-slit
does not cover entirely the gas-emission from the knots, the
EW(H$\beta$) are lower limits. Nonetheless, the aperture factors
derived from the direct comparison with the H$\alpha$ image do not
lead to a satisfactory solution.
Larger dust-to-gas ratios and $f_d$ than typical values for H{\sc ii} regions
with larger metallicity ({\em e.g.} \citealt{inoue01}) were derived by the models
in all knots in order to fit the observed EW(H$\beta$) and FUV-NUV colours.
However, these values cannot be taken as representative of the whole galaxy
but only for the analysed star-forming knots, considering that the fraction of
ionised-to-neutral gas is quite small, as mentioned above, and that this
galaxy does not present a very high IR luminosity ({\em i.e.} it does not
appear in the IRAS catalogue). We have run other set of models with
standard dust-to-gas ration values, but they do not fit the observables mentioned
above. The dust-to-gas ratios could be considerably decreased by assuming
different geometries of the gas and dust inside the H{\sc ii} regions, but
this approach is beyond the scope of this work.
Finally, our results about the masses and ages of the ionising clusters must be used
only as a guiding signpost, as the involved masses range in a regime
($\sim$ 10$^4$ M$_{\odot}$) where stochastic fluctuations in the IMF can be important
\citep{cervino03,cervino06}. In this sense, IMF sampling effects
\citep{pflamm-altenburg07,pflamm-altenburg09} can also alter our results, as there can be
a lower number of massive stars than expected for the mass range of these knots.
\subsection{Densities, temperatures and chemical abundances}
The derived densities, both for [S{\sc ii}] and [O{\sc ii}], are very similar for the first
four knots, being lower than 100 cm$^{-3}$. Interestingly, knot E, located in the outskirts
of the galaxy, shows a density twice higher than the knots located in the central part of
the galaxy. At any rate, these densities are typical of low density environments found in BCDs.
Regarding electron temperatures, T([O{\sc iii}]) was measured in the five knots, with errors
which depend on the quality of the spectrum in each knot going from 3\% in knot A up to 13\% in
knot E. The first four star-forming regions show very similar [O{\sc iii}] temperatures, all
within a relatively narrow range, between 13000 and 14300 K, while knot E has a lower
temperature by about 2000 K. To our knowledge, there is no previous report on T([O{\sc iii}])
for this galaxy in the literature.
T([O{\sc ii}]) was also measured in three of the knots (B, C, and E) and it was estimated in
the other two knots by means of relations between these temperatures and the measured
value of T([O{\sc iii}]). The knot C,
which is the brightest in H$\alpha$, has in average higher temperatures.
We derived oxygen, nitrogen, sulphur, neon and argon total abundances in all the
studied knots by taking the ionic abundances calculated in Section
\ref{abundances} and the ICFs estimated in our tailored photoionisation models.
As in the case of temperatures, no other direct abundance determination exists in
the literature for this galaxy. In average, NGC 6789 shows an oxygen abundance
characteristic of the low values found in strong line BCDs, with values in all knots
in the range of 12 + log(O/H) = 7.80-7.93, what is compatible with a similar abundance
in all knots taking into account the errors. This behaviour, i.e. showing
different star forming knots within a BCD close abundance values (i.e., $<$ 0.2 dex),
is also found in other objects using both long-slit spectroscopy
\citep{papaderos06,cairos09,perez-montero09a,hagele11}
and integral field spectroscopy \citep{kehrig08,perez-montero11}, or
in H{\sc ii} complexes in spiral galaxies with a similar spatial scale
\citep{kennicutt03,garcia-benito10}.
Since nitrogen and oxygen have different nucleosynthetic origins, their ratio is related to the
chemical history of galaxies. Primary nitrogen synthesis is thought to occur in intermediate-mass
stars in the CNO process during hydrogen burning being hence independent of the initial
heavy-element abundances, while secondary nitrogen production is expected to be produced in
stars of all masses \citep{vilacostas93}. At low metallicity, most part of nitrogen has
a primary origin and a constant log(N/O) ratio is observed. However, the N/O values found
in the studied knots of NGC 6789 range in an interval (from -1.38 to -1.26) is sensibly
higher than the observed values for other BCDs or low-metallicity dwarf irregular
galaxies \citep{izotov99a,vilchez03,vanzee06,pmc09}, which is around log(N/O)
$\approx$ -1.6. In the case of knot C, this ratio is even higher, with
a value of -1.07.
The cause of these relatively high N/O can be partially found
in the corresponding ICFs derived by the models as
the total nitrogen abundances derived from the assumption N/O $\approx$ N$^+$/O$^+$
give lower N/O values in knots A [log(N/O) = -1.41] and C
[log(N/O) = -1.34], although still higher within the errors than the
typical N/O value for BCDs.
On the other hand, several causes are cited in the literature to explain this
overabundance of nitrogen in low-metallicity environments ({\em e.g.} the pollution
of the ISM by Wolf-Rayet stellar winds), but it is especially indicative in this case that
we found this overabundance in the five knots, which present different evolutionary
and excitation properties. Therefore, the high N/O is apparently related with some other process
affecting all the ISM of the galaxy. The fall of pristine gas in the galaxy could explain
the N/O overabundance \citep{koppen05} and, at same time, the triggering of the star
formation in different places of the galaxy with very short time intervals between them.
The ratio of the alpha elements, sulphur, neon, and argon, to oxygen should be constant and show
no dependence on the oxygen abundance, since all are products of $\alpha$-processes in the
same massive stars that make oxygen. The derived log(S/O) ratios range between
-1.64 and -1.41 in the four first knots and it is sensibly higher in knot E, but
in this knot the derivation is much more uncertain as no [S{\sc iii}] was measured.
Although the ratio seems to increase from knot A to D, the average error ($\sim$ 0.18)
prevents us from drawing any conclusion about the homogeneity.
Regarding the logarithmic Ne/O ratio, knots A and C show the same value, -0.70, while for
the rest of the knots the mean value is -0.94, with an average error of 0.07 dex for A and C
and 0.20 dex for B, D, and E.
The Ar/O ratios found, in the range -2.28 to -2.06, show a very similar value for
all knots, taking 0.17 dex as the average error (ranging from 0.09 up to 0.22 dex).
Finally, the derived helium abundances are the same for all five knots of NGC 6789 within
observational errors, and similar to the values found for other BCDs \citep{izotov04c,hagele08}.
\section{Conclusions}
\label{conclusions}
In this paper, we present resolved stellar, H$\alpha$, and GALEX photometry,
from different data archives, and WHT-ISIS optical spectroscopy of the five brightest
star-forming knots of the nearest BCD galaxy NGC 6789 in order to study their SFH and
metallicity.
The spectroscopic observations of NGC 6789 were taken using ISIS double-arm spectrograph
attached to the 4.2m WHT, which allowed the simultaneous analysis of the spectra, covering
from 3650 up to 7500 \AA. From the long-slit spectra, we extracted and performed a detailed
analysis of the five main star forming knots of the galaxy. Thanks to the measure of the
[O{\sc iii}] electron temperature in all knots and the measure of T([O{\sc ii}]) in three
of them, we provide ``direct method'' abundances for oxygen, nitrogen, sulphur, neon,
and argon, not reported previously in the literature. Our analysis indicates that this
galaxy is metal-poor (12+log(O/H) in the range 7.80 - 7.93) and chemically homogeneous,
with quite similar values of the studied species in all knots. At same time, all the
knots present values of N/O which are higher than expected for the metal regime of this
galaxy.
We used optical data obtained from the HST archive to derive the SFH of NGC 6789 by
means of the classical method of CMD reconstruction.
We used the derived SFH in the program STARLIGHT to fit the optical spectrum
and we compared these results with another non-constrained case.
We corrected the emission-line measures from absorption and we derived the
total stellar masses for each knot. Although the SFH obtained from
the CMD is not foreseen by STARLIGHT in the non-constrained case,
we checked that the total stellar mass and the ratio of older and younger
stellar populations are not noticeably affected by the assumed SFH.
Anyway, whenever possible, the extension of this type of
studies for objects for which simultaneous resolved stellar photometry
and optical spectroscopy are available will provide a much more accurate
constrain to the age-metallicity-extinction degeneracy of these galaxies, as
well as the comparison between nebular and stellar properties.
Finally, the ages and masses of the bursts of star formation in each knot were derived
using CLOUDY tailored photoionisation models to fit the optical and UV photometric
properties and the observed emission-line ratios and corrected H$\beta$ equivalent widths.
We found that dust absorption factors correlate much better with the GALEX FUV-NUV
colour index than with the reddening constants derived from the Balmer decrement,
indicating a very complex inner dust extinction structure. The models predict for all
knots instantaneous bursts with ages in the range between 3 and 6 Myr. These ages do
not follow any spatial trend in the galaxy image, so they are possibly related with
the distribution of the galaxy in the line of sight. The dust absorption-corrected
H$\alpha$ fluxes were used to derive accurate SFRs for the individual knots.
The combination of several observational and model techniques lead to a better and auto-consistent
study of NGC 6789. The derivation of a non-typical metallicity evolution using a CMD allows the use
of the SFH to the subtraction of the older stellar population using spectral fitting to the optical
spectrum with STARLIGHT. This information, together with the derivation of accurate physical properties
and ionic chemical abundances pointing to similar low O/H and high N/O ratios in all knots allow the
application of photoionisation models that predict dust-absorption factors fitting the GALEX colour
indices and the derivation of ages (IBs in the range 3-6 Myr).
\section*{Acknowledgements}
R.G.B. acknowledges support from the China National Postdoc Fund Grant No. 20100480144 and
MICINN AYA2010-15081. This work has been partially supported by DGICYT grant AYA2007-67965-C03,
AYA2007-67965-C03-02, AYA2007-64712, and Junta de Andaluc\'ia TIC 114.
The WHT is operated on the island of La Palma by the ING in the Spanish
Observatorio del Roque de los Muchachos of the Instituto de Astrof\'isica de
Canarias. We thank the Spanish allocation committee (CAT) for awarding
observing time. We thanks We would like to thank to Armando Gil de Paz, for allowing the
study of B, R, and H$\alpha$ images of NGC~6789 in the Palomar/Las Campanas Atlas
of Blue Compact Galaxies. We thank Roberto Cid Fernandes and the people of the
STARLIGHT Project Team (UFSC, Brazil), for making the STARLIGHT code available.
We acknowledge fruitful discussions with Enrique P\'erez and Ricardo Amor\'in. We
thank an anonymous referee for very useful comments that improved
the presentation of the paper.
\input{ngc6789.bbl}
\end{document}
|
{
"timestamp": "2012-03-13T01:00:11",
"yymm": "1203",
"arxiv_id": "1203.2186",
"language": "en",
"url": "https://arxiv.org/abs/1203.2186"
}
|
\section{Introduction}
The evolution of a nucleus from a single ground-state shape into two
separated fragments in nuclear fission has, since its discovery
\cite{hahn39:a}, been described in terms of potential-energy surfaces
that are functions of suitable shape coordinates
\cite{meitner39:a,bohr39:a}. Originally the potential energy was
modeled in terms of a macroscopic liquid-drop model
\cite{meitner39:a,bohr39:a,frankel47:a,hill53:a}. Subsequently it
became clear that the liquid-drop model cannot explain many features of
fission such as the fission-fragment mass yields, fission-barrier
structure, and actinide fission half-lives
\cite{hill53:a,swiatecki55:a,strutinsky67:a,strutinsky67:b,strutinsky68:a,moller70:a,moller01:a,moller09:a},
because microscopic shell effects significantly perturb the energy
surface given by the liquid-drop model. Although the energy release in
fission, that is the potential-energy change between the ground state of
a single system and well-separated fragments, is more than 200 MeV,
microscopic effects in the narrow range of zero to ten MeV can affect
half-lives by more than ten orders of magnitude and change
fission-fragment mass yields from symmetric to significantly asymmetric.
Experimental observations are that fission-fragment mass distributions
are asymmetric in low-energy fission of typical actinide nuclei for
nucleon number $A$ in the range $228 \lnsim A \lnsim 258$ and proton
number $Z$ in the range $90\lnsim Z \lnsim 100$. In those nuclei, it
has been established that the heavy-mass peak in the yield distribution
is close to $A=140$, independently of fissioning system, see for example
\cite{wilets64:a,vandenbosch73:a}. This was thought to originate from
the strong spherical shell effects present in fragments near the doubly
magic nucleus $^{132}_{\phantom{0}50}$Sn$_{82}$, although we now know
that an analysis of high-dimensional potential-energy surfaces
\cite{moller01:a,moller09:a}, coupled
with a dynamical description is required to robustly establish this
connection \cite{randrup11:a}. In particular, we
now know that ``fragment-shell'' arguments or saddle-point properties
cannot by themselves reliably predict the degree of asymmetry; rather,
the character of the entire potential-energy surface between the
ground-state and separated fragments must be considered
\cite{andreyev10:a,randrup11:a}.
A large-scale experiment studying fission of nuclei in the region
$205\leq A \leq 234$ showed that a transition to symmetric fission
occurred just below the actinide region and that fission remained
symmetric at least down to proton number $Z=85$ and nucleon number
$A=205$. The dividing line between asymmetric and symmetric fission was
found to approximately follow constant nucleon number, $A=226$
\cite{schmidt01:a}. The position of this transition line
was predicted to within about 2 neutrons in a
simple static calculation in 1972 \cite{moller72:a}.
For slightly lighter systems
\cite{itkis90:a,itkis91:a} near $Z=82$ and $A=200$, a hint of asymmetric
fission was observed for energies up to about 10 MeV above the
saddle-point energy. Itkis referred to this as ``asymmetry of symmetric
fission'' \cite{itkis90:a}, so it is unclear whether or not he viewed
his results as a clear indication of the onset of a new region of
asymmetric fission. Despite this intriguing result, it has often been
assumed that fission mass distributions for systems below the actinide
region would be symmetric because, based on the proton and neutron
numbers of possible compound systems, division into fragments with $Z$
and $N$ sufficiently close to $^{132}$Sn (or to much lighter doubly
magic nuclides) so as to exhibit strong shell effects appeared not
possible for almost all compound systems below $A\approx 200$.
Surprisingly, a recent experiment showed \cite{andreyev10:a} that
fission of $^{180}$Hg following electron capture by $^{180}$Tl is
asymmetric.
It was earlier argued that the asymmetric fission of $^{180}$Hg was a
new type of asymmetric fission with its origins in the local structure
of the fission potential-energy surface near the fission saddle point
\cite{andreyev10:a}. Moreover, it was argued that these observations
showed that consideration of ``fragment shells'' does not offer a
general method of predicting or explaining asymmetry in fission.
To illustrate the contrasting origins of asymmetric fission in the Hg
and actinide regions, we calculate and analyze the structure of
five-dimensional fission potential-energy surfaces for even Hg isotopes
in the range $ 178 \leq A \leq 200 $ and compare them to a typical
actinide potential-energy surface, namely that of $^{236}$U.
\begin{figure}[t]
\includegraphics[keepaspectratio,width=\linewidth]{fig01.eps}
\caption{
(Color online) Potential-energy curves, minima, saddles, and ridges for
$^{180}$Hg versus $q_{\rm 2}$ from oblate shapes to very deformed
configurations. The solid line denotes the optimum fission path leading
to a mass-asymmetric split. The gray (green) dashed line denotes a
symmetric valley in the potential-energy surface, corresponding to a
compact fusion valley with zero-radius neck shapes along the entire
valley. The solid (red) line with superimposed triangles is the ridge
separating those two channels.} \label{Hg180-1d}
\end{figure}
\begin{figure}[t]
\includegraphics[keepaspectratio,width=\linewidth]{fig02.eps}\\%
\caption{(Color online) Potential-energy curves, minima, saddles and
ridges for $^{198}$Hg versus $q_{\rm 2}$ from oblate shapes to very
deformed configurations. The dashed line is a symmetric valley
corresponding to a fusion valley with deformed shapes connected by a
conicoidal neck region. The symbols are the same as
Fig.~\ref{Hg180-1d}.} \label{Hg198-1d}
\end{figure}
\section{Model}
The potential energy is calculated in the FRLDM
\cite{moller95:b,moller09:a}, with the 2002 parameter set for
the macroscopic model \cite{moller04:a}.
We use two shape parameterizations. For more elongated
shapes somewhat beyond the ground state
we use the three-quadratic-surface (3QS)
parametrization~\cite{nix68:a,nix69:a} to describe nuclear shapes in a
five-dimensional deformation space. The shape degrees of freedom are a
quadrupole-moment parameter $q_2$, a neck-related parameter $\eta$,
heavier- and lighter-fragment deformation parameters $\epsilon_{\rm H}$ and
$\epsilon_{\rm L}$, and a mass-asymmetry parameter $\alpha_{\rm g}$.
The parameter $\eta$ is related to the curvature of the middle body.
The parameter $q_2$ is the dimensionless quadrupole moment in units of
$3ZR_{0}^2/4\pi(e^2b)$, where $Z$ is the proton number and $R_0$ is the
nuclear radius. The parameter $\epsilon$ is the Nilsson
perturbed-spheroid parameter. The mass-asymmetry parameter is
$\alpha_{\rm g}=(M_{\rm H} - M_{\rm L})/(M_{\rm H} + M_{\rm L})$, where
$M_{\rm H}$ and $M_{\rm L}$ are the masses of the heavier and lighter nascent
fragments, respectively. For finite neck radii these masses are defined
as discussed in \cite{moller01:a}. The microscopic single-particle
potential is calculated by folding a Yukawa function over the shape of a
``sharp-surface generating volume'' \cite{bolsterli72:a}.
We calculate the adiabatic potential-energy surfaces in this
five-dimensional deformation space for the 12 even isotopes in the range
$^{178-200}$Hg and for $^{236}$U and analyze their structure using the
immersion method \cite{moller09:a}. The potential energies are
determined at $41\times15\times15\times15\times35$ grid points for $q_2
\times\eta\times\epsilon_{\rm H}\times\epsilon_{\rm L}\times\alpha_{\rm
g}$. For $q_2$ and $\eta$ we use similar, and for fragment deformations
and asymmetry $\alpha_{\rm g}$, exactly the same points as in
Ref.~\cite{moller09:a}. We take into account the shape-dependent Wigner
and $A^{0}$ terms in our calculations~\cite{moller09:a}.
Near the
ground states where $q_2 \leq 0.5$, we also perform complementary
constrained-multipole ($\beta_2$) calculations, which better describe
compact shapes for small deformations \cite{moller95:b}. We identify
the minima and potential valleys under the condition that their depths
are deeper than 0.05 and 0.2 MeV, respectively. In our static studies
we can make realistic determinations of major features in the
potential-energy surfaces, such as minima, saddles, valleys, and ridges
between valleys, because in our model we 1) calculate the energy in
millions of grid points for the five most essential shape degrees of
freedom and 2) use an immersion method to extract structure features
\cite{moller09:a}. In contrast, in self-consistent methods in which {\it
constraints} are imposed, the inferred saddle points and ridges may be
overestimated by amounts that can be quite large. Moreover, the
magnitude of this overestimation is impossible to determine, see
Ref. \cite{moller09:a} for a detailed discussion.
\section{Calculated structure of potential-energy surfaces}
In the early days of theoretical fission studies based on the
macroscopic-microscopic method, most or all investigations calculated
the fission potential-energy surface in terms of only two independent
shape variables, for example variables related to elongation and neck
radius \cite{strutinsky67:a,nilsson69:a} or elongation and fragment mass
asymmetries \cite{moller70:a}. Complete results from such calculations
could be faithfully displayed in terms of two-dimensional contour
diagrams. In contrast, it is impossible to show all essential features
of five-dimensional potential surfaces by reducing them to
two-dimensional contour plots. To identify major features of the
5D spaces we start by locating all minima, saddles, ridges and valleys
by use of the immersion technique; for details see
Ref.~\cite{moller09:a}. We then show features identified to be of
special interest in one-dimensional plots versus quadrupole moment. For
example, we show the energies along specific one-dimensional paths,
such as valleys and ridges,
embedded in the full 5D space and relevant
minima and saddles. To more clearly visualize the substantial
differences of asymmetric fission in the neutron-deficient Hg region and
actinide region we will also plot 2D surfaces embedded in the full 5D
deformation space.
\begin{figure}[t]
\includegraphics[keepaspectratio,width=\linewidth]{fig03.eps}\\%
\caption{(Color online) Saddle and ridge locations for a range of Hg
isotopes. An extended ridge is only present for isotopes in the interval
$180\leq A \leq190$. For $A=192$ we could not clearly interpret the
ridge features so data are omitted for this isotope. } \label{q2dep}
\end{figure}
\begin{figure}[t]
\includegraphics[keepaspectratio,width=\linewidth]{fig04.eps}\\%
\caption{(Color online) Ground-state energy, saddle energy
and maximum ridge height, all with respect to the spherical macroscopic
energy. The difference between the saddle energy and the ground-state
energy is the barrier height, as indicated with an arrow for $^{192}$Hg.
It is only for isotopes in the interval $178\leq A \leq190$ that a ridge
rises above the saddle.} \label{sadhgt}
\end{figure}
\begin{figure}[t]
\includegraphics[keepaspectratio,width=\linewidth]{fig05.eps}\\%
\caption{Mass number (top), proton number (second), neutron number
(third), and deformation (bottom) of the heavy and light nascent fission
fragments at the vanishing point of the separating ridge determined from
wave-function densities in the two fragments by methods described in
\cite{ichikawa09:a}. } \label{wavedens}
\end{figure}
\begin{figure}[t]
\includegraphics[keepaspectratio,width=\linewidth]{fig06.eps}\\%
\caption{(Color online) Potential-energy curves, minima, saddles, and
ridges for $^{236}$U from a spherical shape to very deformed
configurations. Here the symmetric valley is well separated from the
asymmetric valley by a ridge that is about 5 MeV high along the entire
deformation range between the saddle and the asymmetric scission
configuration.} \label{U236-1d}
\end{figure}
Figures \ref{Hg180-1d} and \ref{Hg198-1d} show calculated ``optimal''
one-dimensional potential-energy curves or ``fission barriers'',
embedded in the five-dimensional space, as functions of $q_2$ (solid
line) for $^{180}$Hg and $^{198}$Hg. In this study, all potential
energies are measured from the spherical macroscopic energy.
Minima and saddle points are
indicated by open squares and triangles, respectively. Shapes of the
nuclear macroscopic densities at several saddle points and minima are
also displayed.
In both systems the ground-state shapes are slightly oblate. However,
the density evolutions from the ground state to the fission saddle
points differ substantially. For $^{180}$Hg, mass asymmetry has
developed already near the local energy minimum at $q_2=4.0$, although
no distinct fragments have yet emerged, cf. Fig.~\ref{Hg180-1d}.
Subsequently the neck develops, while the degree of mass asymmetry is
retained. At the fission saddle point $E_{\rm sad}=11.35$ MeV, and its
shape corresponds to $q_2=7.84$, $\epsilon_{\rm H}= 0.275$,
$\epsilon_{\rm L}= 0.30$, and $\alpha_{\rm g}=0.14$ or equivalently
$A_{\rm H}/A_{\rm L}= 102.6/77.4$.
On the other hand, the shape for $^{198}$Hg remains symmetric up to the
local energy minimum at $q_2=7.5$, although the neck is well developed
there. Beyond this local minimum, the mass asymmetry of the fissioning
nuclei develops in tandem with neck formation. At the fission saddle
point $E_{\rm sad}=15.47$ MeV, $q_2=10.08$, $\epsilon_{\rm H}= 0.35$,
$\epsilon_{\rm L}=0.10$, and $\alpha_{\rm g}=0.12$, or equivalently
$A_{\rm H}/A_{\rm L}= 110.9/87.1$.
In the outer saddle region additional valleys appear in the two
potential-energy surfaces. For each of the two systems we show only one
of these valleys, namely the one corresponding to symmetric shapes as
dashed (green) lines. To leave the figures uncluttered we do not show an
asymmetric valley which is also present. Often these valleys are
referred to as {\it fusion} valleys because along the entire curve the
neck radius is zero. In a more general treatment allowing for a family
of shapes of separated nuclei, the fragments, or equivalently, the two
colliding heavy ions would be separated along this curve until they have
approached sufficiently close that they touch. Separated fragments are
inaccessible in the 3QS parameterization in its current implementation.
Instead these configurations are represented as two spheroidal nascent
fragments connected by a conicoidal neck \cite{nix68:a,nix69:a}. This
limitation does not affect our study here, since we only follow the
shape evolution until just before zero neck radius (in a more general
treatment, separation) occurs. What we wish to establish here is the
structure of the potential-energy surface from outside the saddle point
to just before separation. Is it possible to determine if it favors
evolution towards the symmetric valley or the asymmetric valley? And
when is the final fragment asymmetry established? Clearly it will be
frozen in prior to reaching the bottom of any of the valleys, since
zero-neck-radius shapes occur already above the valley floors. For
$^{180}$Hg, the shape configuration in the symmetric fusion path/valley
is two spherical shapes with $^{90}$Zr + $^{90}$Zr, which exists because
in the macroscopic model symmetric separated fragments are energetically
favored over asymmetric fragments, and the $N=50$ shell favors spherical
fragments. The nascent fragment shapes in the symmetric fusion valley
for $^{198}$Hg are fairly deformed with $\epsilon=0.275$, because the
fragment neutron numbers are $N=59$, corresponding to onset of
deformation in separated nuclei.
An important feature for $^{180}$Hg is that the optimal potential-energy
curve from the ground state across the saddle and somewhat beyond and
the symmetric fusion valley are well separated by the potential ridge,
which initially is 8 MeV above the saddle region. On the other hand,
the height of the corresponding ridge for $^{198}$Hg is much lower
(initially only 2 MeV high) and only persists for a narrow range in
$q_2$, suggesting that a change from the asymmetric shapes along the
initial fission path to different final fragment mass asymmetries is
less hindered in $^{198}$Hg than in $^{180}$Hg.
The separating ridge for $^{180}$Hg vanishes at $q_2=10.31$,
$\epsilon_{\rm H}=0.30$, $\epsilon_{\rm L}=0.15$, and $\alpha_{\rm
g}=0.20$ corresponding to $A_{\rm H}/A_{\rm L}=108.0/72.0$. For
$^{198}$Hg, the ridge vanishes at $q_2=13.47$, $\epsilon_{\rm H}=0.40$,
$\epsilon_{\rm L}=0.0$, and $\alpha_{\rm g}=0.18$, corresponding to
$A_{\rm H}/A_{\rm L}=115.82/81.82$. At the point where the separating
ridge vanishes, no ``obvious'' valley connects this location to a
scission configuration. Instead we are on a rather flat
potential-energy surface which in the full 5D space gently slopes in
many directions. An analogy is being just below the top of a gently
sloping hill. Therefore we cannot determine a plausible optimum fission
path by a static analysis alone. However, when the neck is quite well
developed where the ridge disappears, it was suggested that the mass
asymmetry here might to a significant extent be preserved in the separated
fission fragments \cite{andreyev10:a}.
\begin{figure}[t]
\includegraphics[keepaspectratio,width=\linewidth]{fig07.eps}\\%
\caption{(Color online) Two-dimensional potential-energy surface for
$^{180}$Hg which shows some essential features of the full 5D
potential-energy surface. Two crossed (red) lines show the location of some
saddle points. Note in particular that the valley across the asymmetric
saddle disappears slightly beyond $q_2=10$} \label{Hg180-3D}
\end{figure}
\begin{figure}[t]
\includegraphics[keepaspectratio,width=\linewidth]{fig08.eps}\\%
\caption{(Color online) Two-dimensional potential-energy surface for
$^{236}$U which shows some essential features of the full 5D
potential-energy surface. Two crossed (red) lines show the location of some
saddle points. Note in particular that the valley across the asymmetric
saddle continues to the largest $q_2$ shown. It also continues beyond to
a point where the nucleus separates into two fragments. This is very
much in contrast to the potential-energy surface for $^{180}$Hg.}
\label{U236-3D}
\end{figure}
\begin{figure*}[t]
\includegraphics[keepaspectratio,width=0.8\linewidth]{fig09.eps}\\%
\caption{Total and macroscopic energies along the asymmetric fission
paths for $^{180}$Hg, and $^{236}$U are shown in the top two frames. The
microscopic energies along these paths are shown in the lower two
frames. There are very significant differences between the microscopic energy
of the two nuclei. } \label{sshellasympath}
\end{figure*}
\section{Saddle features and fission-fragment mass
asymmetry in mercury isotopes} The fragment mass asymmetry in fission is
affected by the saddles, ridges and valleys in the fission
potential-energy surface that appear beyond the fission isomeric
minimum. We have identified these features using the immersion method. The
results are summarized in Figs.~\ref{q2dep} and \ref{sadhgt}. We pay
particular attention to the point where the ridge between the optimal
fission path and the fusion valleys disappears, which for $^{180}$Hg
occurs at $q_2=10.31$. For the nuclei we study, it is not possible to
identify a clear mass-asymmetric fission path, because the ``fission
valley'' that takes us across the saddle point disappears at elongations
slightly beyond the saddle. That is, there is no continuous asymmetric
valley from the region of the saddle point to scission, very much in
contrast to the situation in the actinide region.
The mass-symmetric fusion path shown in Fig.~\ref{Hg180-1d} corresponds
to compact, nearly spherical fragment shapes. This type of fusion valley
is only present in Hg nuclei from $A=178$ to $A=190$. The ridges
separating the compact mass-symmetric fusion path become very low,
almost non-existent, at $A=190$, and this compact symmetric fusion
valley vanishes at $A=192$. Instead, for somewhat heavier isotopes a
mass-symmetric fusion path with large nascent-fragment deformations
appears. To summarize, some general trends in the structure of the
potential-energy surfaces along the Hg isotope chain are:
\begin{itemize}
\item
With increasing $A$ the barrier height increases,
partly due to a lowering of the ground state as
$N=126$ is approached, and also to a decrease of fissility.
\item
The saddle shapes are more elongated (larger $q_2$)
for the heavier Hg isotopes.
\item
For low $A$ the ridges are prominent;
for higher $A$,
they almost disappear.
\end{itemize}
In the specific case of electron-capture-delayed fission of $^{180}$Hg
the shape asymmetry where the ridge vanishes could be related to the
observed fission-fragment mass asymmetry \cite{andreyev10:a}. In
Fig.~\ref{wavedens} we show the asymmetry at this vanishing point for
the entire range of isotopes. We calculate the asymmetry from the
wave-function densities (top three panels), cf.\
Ref.~\cite{ichikawa09:a} for details.
In the bottom panel we show the
nascent-fragment shape-deformation parameters at this point. These
features stand out:
\begin{enumerate}
\item The proton number of the light-(heavy)-mass fragment is close to $Z=34(46)$
in all the Hg isotopes (see the second panel).
However, no strong shell effect is present
in the ground states of these fragments. The ground-state shapes
for all the $Z=34(46)$ fragments are well deformed
with uniformly positive microscopic corrections~\cite{moller95:b}.
\item The neutron number of the light-mass fragment is close to $N=50$
for $A > 190$ and the deformation
of those light fragments is spherical (cf. the third panel).
\item At the vanishing point of the ridge
the degree of fragment mass asymmetry becomes smaller with
increasing mass number. But for the
heavier isotopes the ridge is very short and low in energy so the
asymmetry at the vanishing point might not be closely related to
the final fragment mass asymmetry.
\end{enumerate}
\section{Two types of asymmetric fission}
Asymmetric fission in the actinide region has since its discovery been
``explained'' in terms of strong ``shells'' in the heavy fragment
related to its proximity to doubly magic $^{132}$Sn. But it should be
observed that in fission of actinides the heavy fragment is not exactly
$^{132}$Sn and just small changes in $Z$ and $N$ from the doubly-magic
nucleus drastically decrease the extra binding due to proximity to a
doubly closed shell. For example, the most probable heavy/light mass
split of $^{240}$Pu is $M_{\rm H}/M_{\rm L}= 140/100$. This corresponds
to the heavy fragment $^{140}_{\phantom{0}55}$Cs$_{\rm 85}$ with a
ground-state microscopic correction $-2.96$ MeV \cite{moller95:b}, which is
not even close to the $^{132}$Sn ground-state microscopic correction of
$-11.55$ MeV. But, when the nascent fragments start to emerge, they have
not absorbed some nucleons in the neck regions. Thus, the partially
formed heavy fragment in the case of $^{240}$Pu is closer in size and
shape to $^{132}$Sn than it is to $^{140}_{\phantom{0}55}$Cs$_{\rm 85}$,
which could significantly affect the microscopic correction. For example, just
removing one proton and one neutron from
$^{140}_{\phantom{0}55}$Cs$_{\rm 85}$ leads to
$^{138}_{\phantom{0}54}$Xe$_{\rm 84}$, with a ground-state microscopic
correction of $-5.35$ MeV \cite{moller95:b}.
Clearly, one should only invoke such hand-waving arguments related to
fragment properties as a starting point for understanding the
mass-asymmetric fission-fragment division in the actinide region. A more
complete understanding should involve the potential energy from the
ground-state shape to separated fragments in terms of a sufficiently
large number of shape degrees of freedom \cite{moller01:a}. It has
indeed been shown that a deep asymmetric valley separated from a
symmetric fission valley for most actinides extends from the saddle
region to scission configurations \cite{moller01:a,moller09:a}. As an
example, we show in Fig.~\ref{U236-1d} calculated energies along symmetric and asymmetric
optimal fission paths and the separating ridge for $^{236}$U. Here we
note an asymmetric valley extending from the outer saddle region to
scission. It is shielded from the symmetric valley by an about 5
MeV-high ridge along its entire path. This contrasts very much with the
situation in the Hg region.
To illustrate more clearly the differences between Hg and actinides in
the fission potential-energy surfaces and the presence and absence of
``fragment'' shell effects in the potential-energy surfaces of the
compound system, we plot in Fig.~\ref{Hg180-3D} a by necessity somewhat
schematic two-dimensional representation of the most important features
of the full 5D potential-energy surface for $^{180}$Hg. In the left part
of Fig.~\ref{sshellasympath} we show the total energy, macroscopic
energy, and microscopic energy along a section of the asymmetric fission
path of $^{180}$Hg. In Fig.~\ref{U236-3D} and the right part of
\ref{sshellasympath} we show the corresponding quantities for $^{236}$U.
These figures illustrate visually the different origins of asymmetric
fission in the Hg and actinide regions. For $^{236}$U the asymmetric
valley extends from the outer saddle point to scission-like shapes. It
is a plausible assumption that the mean asymmetry in thermal
neutron-induced fission is close to the asymmetry of the shapes at the
bottom of the asymmetric valley. This correlation was indeed verified in
the investigation of Ref.~\cite{moller01:a} in which the calculated
asymmetry of the asymmetric valley bottom agreed with observed
fission-fragment mass asymmetries for 25 even-even actinide nuclides
with a mean deviation of only 3.0 nucleons. The large negative microscopic
energy $E_{\rm sh} =-12$ MeV at scission where $q_2=9$,
cf. Fig.~\ref{sshellasympath}, remains almost constant for more compact
shapes; it is still very substantial, $E_{\rm sh} =-6$ MeV at the
saddle-point deformation $q_2 = 5$.
In contrast, for $^{180}$Hg there is no valley extending from the saddle
region towards scission. Rather, for elongations only moderately beyond
the saddle the ridge separating the saddle region and the symmetric
fusion valley disappears. From static considerations alone it is not
obvious what trajectory towards separated fragments the nucleus will
follow. Thus, as stated in Ref.~\cite{andreyev10:a}, the asymmetric
fission in $^{180}$Hg is a new type of asymmetric fission with its
origins in the local fission potential-energy-surface structure in the
saddle region, whereas in the actinide region a deep, persistent
asymmetric valley extends over the entire range from saddle-point shapes to
separated fragments. Figure \ref{sshellasympath} shows that there is no
significant fragment-related microscopic effect in the saddle region or
beyond for $^{180}$Hg; this microscopic energy is very low, fluctuating between $\pm 2$
MeV along the trajectory shown.
\section{Summary Discussion}
The recent observation of mass asymmetry in electron-capture delayed
fission of $^{180}$Hg \cite{andreyev10:a} has stimulated renewed
interest in fission since some simple ``fragment-shell'' type arguments
had anticipated that the most probable division would be into two
symmetric $^{90}$Zr fragments, because these exhibit two instances of
the spherical $N=50$ magic number and two instances of the spherical
$Z=40$ subshell. It was proposed that a new type of asymmetric fission
had been observed, with its origins in the {\it local} structure in the
outer saddle-point region. Currently, the experimental data in this
neutron-deficient region in terms of energy range and number of nuclides
are extremely sparse, in particular in comparison with the data
available for heavier nuclei \cite{vandenbosch73:a,schmidt00:a} We have
calculated potential-energy surfaces of 12 even Hg isotopes in this
neutron-deficient region to establish the systematics of significant
structures. The most important finding is that it is only for nuclei in
the range $180\lnsim A \lnsim 190$ that the saddle region is somewhat
shielded from the symmetric fusion valley by a moderately high ridge
that also has some moderate extension in the elongation direction. In
the $^{180}$Hg experiment the compound-nucleus excitation was limited to
about 1 MeV above the saddle point. This constraint and the ridge
structure allowed some qualitative conclusions about the expected
fragment asymmetries in this experiment \cite{andreyev10:a}.
In the actinide region numerous models have been proposed to describe
the observed fission mass asymmetries, for example
Refs.~\cite{fong56:a,wilkins76:a,brosa90:a,benlliure98:a,schmidt00:a,goutte05:a,randrup11:a}.
Often encouraging results are presented. We have shown here and
elsewhere \cite{moller01:a,moller09:a} that in calculated, realistic 5D
potential-energy surfaces, very strongly expressed, deep asymmetric
valleys are present. These valleys usually
appear also in more approximate calculations, so that when the
respective model parameters are adjusted to experimental yields the
model results agree to varying degrees of accuracy with the experimental
data. However none of these models have been applied to $^{180}$Hg,
with the exception of the Brownian shape-motion model
\cite{randrup11:a,moller12:a}, in which no parameter is adjusted. Although the statistics of the
$^{180}$Hg experiment are limited, the Brownian shape-motion model may
be less accurate in this case than in the actinide region since for
$^{180}$Hg the result was $M_{\rm H}/M_{\rm L} = 104.4/75.6$ whereas the
experimental result was given as $M_{\rm H}/M_{\rm L} = 100/80$. More
striking is that the calculated FWHM width \cite{moller12:a} is about
twice the experimental result of 9 mass units~\cite{andreyev10:a}. In the actinide region the
calculated widths agreed very well with the experimental data
\cite{randrup11:a} with no obvious deviations except in the tails of the
yield distributions at very large asymmetries. A possible explanation of
these results is that in the actinide region the confining influence of
the steep walls of the asymmetric valley defines the width of the yield
distributions, and this feature is realistically described in the
calculations. In the Hg region, where there are no confining ``fission
valley'' walls, the yield distribution is determined on the downslope of
a steep, smooth mountain side, cf. Fig.~\ref{Hg180-3D}. Here the fine
details of the dynamical part of the model may be more important than in
the actinide region. The models in the other
Refs. \cite{fong56:a,wilkins76:a,brosa90:a,benlliure98:a,schmidt00:a,goutte05:a}
have not yet been tested in this mass region.
Clearly, it will be a challenge to fission theories to reproduce
experimental data both in the Hg region and across the entire actinide
region without arbitrary model parametrizations which differ from region
to region. Since we have now shown the different issues presented to
theory by fission in the Hg and actinide regions, we strongly encourage
efforts to obtain a more extensive set of fission data in the region
$180 \leq A \leq 200$ be undertaken, both in terms of excitation-energy
range and number of nuclides. Such experiments would present new and
highly useful challenges to fission theories.
\begin{acknowledgments}
A part of this research has been funded by MEXT HPCI
STRATEGIC PROGRAM.
P.M. and A.J.S. acknowledge that this work was carried out under
the auspices of the National Nuclear Security Administration of the US
Department of Energy at Los Alamos National Laboratory under Contract
DE-AC52-06NA25396, and the US Department of Energy through the LANL/LDRD
Program. P.M. was also supported by a travel grant to JUSTIPEN
(Japan-US Theory Institute for Physics with Exotic Nuclei) under Grant
DE-FG02-06ER41407 (U. Tennessee). The numerical calculations were
carried out on SR16000 at YITP at Kyoto University.
\end{acknowledgments}
|
{
"timestamp": "2012-06-18T02:03:02",
"yymm": "1203",
"arxiv_id": "1203.2011",
"language": "en",
"url": "https://arxiv.org/abs/1203.2011"
}
|
\section{Introduction}
One of the most interesting problems of geometric mechanics is related
to the integrability conditions of the inverse problem of the calculus
of variations for time-dependent second-order ordinary dif\/ferential
equations (SODE). The inverse problem can be formulated as follows. Given
a~time-dependent system of SODE
\[
\frac{d^{2}x^{i}}{dt^{2}}+2G^{i}\left(t,x,\frac{dx}{dt}\right)=0,\qquad i\in \{1,\dots,n\},
\]
under what conditions this system can be made equivalent, using a
multiplier matrix $g_{ij}$, with the system of Euler--Lagrange equations
of a regular Lagrangian \[
g_{ij}\left(t,x,\frac{dx}{dt}\right)\left(\frac{d^{2}x^{i}}{dt^{2}}+2G^{i}\left(t,x,\frac{dx}{dt}\right)\right)=\frac{d}{dt}\left(\frac{\partial L}{\partial y^{i}}\right)-\frac{\partial L}{\partial x^{i}}\,\,?
\]
In this case such a system is called variational. The necessary and
suf\/f\/icient conditions under which such a system is variational are
known as the \emph{Helmholtz conditions}.
This inverse problem was solved for the case $n=1$ by Darboux~\cite{Dar},
and for $n=2$ by Douglas~\cite{douglas41}. Douglas's approach consists
in an application of the Riquier theory of systems of partial dif\/ferential
equations~\cite{Rhi}, to a certain associated linear dif\/ferential
system. The generalization of its results in the higher dimensional
case is a very dif\/f\/icult problem because the system provided by the
Helmholtz conditions is extremely over-determined. Some of the f\/irst
studies of the inverse problem in spaces of arbitrary dimension are
those of Davis~\cite{davis29} and Kosambi~\cite{kosambi35}.
There are dif\/ferent attempts to solve this problem. First, there are
some reformulations of the Helmholtz conditions in better geometric
forms, which are close enough to the f\/irst analytical formulations
\cite{douglas41,santilli78,Sarlet81,sarlet82}, but undercover more
of the geometry behind them \cite{carinena89,crampin81,crampin84a,crampin94,deleon88,krupkova08,massa94}.
The system of SODE is identif\/ied with a semispray on the f\/irst jet
bundle of a f\/ibred manifold over~$\mathbb{R}$. The most important
geometric tools induced by a semispray are nonlinear connection, Jacobi
endomorphism, dynamical covariant derivative, linear connections and
their curvatures. Some reformulations of the Helmholtz conditions
are using either the special derivations along the tangent bundle
projection introduced in \cite{sarlet95}, or the semi-basic 1-forms~\cite{B-C-2010} and the Fr\"olicher--Nijenhuis theory of derivations
on the algebra of vector-valued forms~\cite{frolicher56}.
Anderson and Thompson \cite{anderso92} analyzed the inverse problem
based on the exterior dif\/ferential system approach \cite{BCG91}.
Using the variational bicomplex associated to a system of arbitrary
order ordinary dif\/ferential equations, they derived the fundamental
system of equations for the va\-riatio\-nal multiplier and proved their
suf\/f\/iciency. They made a detailed study of two dimensional sprays
and they proved, for general degrees of freedom, that all isotropic
semisprays are va\-ria\-tional. It means semisprays that have the associated
Jacobi endomorphism a multiple of the identity. This correspond to
the \emph{Case~I} of Douglas's classif\/ication. This approach is continued
in~\cite{Ald-Pr-Sa-TH}, where the case of $\Phi$ diagonalizable,
with distinct eigenfunctions, is exposed in detail. The same Case
I was proved to be variational also in~\cite{Sar-Cram-Mart-integr}.
This paper uses Riquier theory, but in a more geometric way. The process
of repeated dif\/ferentiations of equations and searching for new nontrivial
relations is realized by intrinsic operations.
\looseness=1
Another subcase of Douglas's case II is discussed in \cite{Cr-Pr-Sa-Th-sep}:
separable systems of SODE. Any systems of SODE from this subcase is
variational. They showed that any system of SODE in~\emph{Case II1}
with $n$ degrees of freedom can be separated into n separate systems
of two f\/irst-order equations. They also proved that there are systems
separable in the above sense but not separable into single independent
second-order equations. This case was treated in~\cite{cantrijn96}.
In~\cite{sarlet02} the authors reinvestigated the case $n=2$ with
their more intrinsic version of the Riquier algorithm. Their approach
is based on the same underlying methodology as the analytical work
of Douglas.
Another method of studying the integrability conditions of the inverse
problem of the calculus of variation is the Spencer--Goldsmchmidt theory
of formal integrability of partial dif\/ferential operators, using two
suf\/f\/icient conditions provided by Cartan--K\"ahler theorem~\cite{Cartan,Gold,Spencer}.
This method was applied for autonomous SODE in~\cite{grifone00},
using the Fr\"olicher--Nijenhuis theory of derivations of vector-valued
dif\/ferential forms. Grifone and Muzsnay gave the f\/irst obstructions
so that a \emph{spray} (homogeneous semispray) is variational, for
general degrees of freedom. In order to obtain a complete classif\/ication
of variational sprays, they restricted their work to some particular
cases. The Spencer theory is fully applied to the two dimensional
case, corresponding to Douglas's paper. For the general n-dimensional
case, it is proved only that isotropic sprays are variational. It
is important to notice that Grifone and Muzsnay's analysis starts
from the Euler--Lagrange partial dif\/ferential operator, and not from
the Helmholtz conditions.
For time independent, homogeneous SODE, the inverse problem is known
as the projective metrizability problem. This problem and its formal
integrability is studied in~\cite{B-M_arxiv} using Spencer theory.
It was shown that there exists only one f\/irst obstruction for the
formal integrability of the projective metrizability operator, expressed
in terms of the curvature tensor of the nonlinear connection induced
by the spray. This obstruction correspond to second obstruction for
the formal integrability of the Euler--Lagrange operator.
An interesting and new approach regarding variational PDE's is the
one of A.~Pr\'asta\-ro~\mbox{\cite{Prast.99,Prast.06}}. Using suitable cohomologies
and integral bordism groups, the author characterizes variational
systems constrained by means of PDE's of submanifolds of f\/iber bundles.
He presents a~new algebraic topological characterization of global
solutions of variational problems.
In this paper we address the integrability conditions of the inverse
problem of the calculus of variations for time-dependent SODE using
also the Spencer version of the Cartan--K\"ahler theorem. The proper
setting is the f\/irst jet bundle $J^{1}\pi$ of an $(n+1)$ manifold~$M$ f\/ibred over~$\mathbb{R}$. In \cite{B-C-2010} it is proved that
a time-dependent semispray is Lagrangian if and only if there exists
a~semi-basic 1-form $\theta$ on~$J^{1}\pi$, that satisf\/ies a dif\/ferential
system. This gives rise to a linear partial dif\/ferential operator~$P$.
We study the formal integrability of~$P$ using two suf\/f\/icient conditions
provided by Cartan--K\"ahler theorem. We prove that the symbol $\sigma^{1}(P)$
is involutive (Theorem~\ref{thm:-involutivity}) and hence there is
only one obstruction for the formal integrability of the ope\-ra\-tor~$P$, which is due to curvature tensor~$R$ (Theorem~\ref{thm:A-first-order}).
Based on this result, we recover some of the classes of Lagrangian
semisprays: f\/lat semisprays, isotropic semisprays and arbitrary semisprays
on $2$-dimensional jet spaces ($n=1$).
The motivation for this article is double-folded. So far all the results
about the inverse problem of the calculus of variations were obtained
separately, in the autonomous and non\-auto\-nomous settings. This is
due to the dif\/ferent frameworks involved: the tangent bundle~$TM$
(a~vector bundle) and respectively the f\/irst jet bundle~$J^{1}\pi$
(an af\/f\/ine bundle). The geometric tools are usually constructed in
dif\/ferent ways, and special attention was given to the time-depending
situation. This paper follows the line of~\cite{B-M_arxiv} but naturally
the proofs of the main theorems have some particularities due to the
dif\/ferent setting.
Secondly, there are similarities between the formulation of the Helmholtz
conditions for sprays in the autonomous setting and respectively for
semisprays in the nonautonomous one \cite{B-C-2010, bucataru09}. This
is natural because $J^{1}\pi$ can be embedded in $\widetilde{TM}$
(the tangent bundle with the zero section removed). Due to this embedding
one can associate to any regular Lagrangian on the velocity-phase
space $J^{1}\pi$ a homogeneous degenerate Lagrangian on the extended
phase space $\widetilde{TM}$, such that the action def\/ined by a curve
in the jet formalism coincides with the action def\/ined by the corresponding
curve in the extended formalism. There are correspondences between
the main geometric objects associated to these Lagrangians: Poincar\'e
1-~and~2-forms, energies, canonical semisprays-sprays
\cite{Ant-B,CantrijnHom86,CarinenaHom2006,Klein}. Therefore,
due to this homogeneous formalism, it is natural to expect such kind
of similarities between the results corresponding to homogeneous structures
on~$\widetilde{TM}$ and nonhomogeneous one on~$J^{1}\pi$.
The paper is organized as follows. In Section~\ref{sec:prelim} we introduce the principal
geometric tools induced by a time-dependent semispray on $J^{1}\pi$
and characterize Lagrangian vector f\/ields with respect to semi-basic
1-forms. Section~\ref{section3} is dedicated to the application of the
Spencer theory to the study of formal integrability of the partial dif\/ferential operator (PDO) $P=(d_{J},d_{h})$.
The most important results are Theorems \ref{thm:-involutivity} and~\ref{thm:A-first-order}. Section~\ref{section3.3} presents classes
of semisprays for which the obstruction in Theorem~\ref{thm:A-first-order}
is automatically satisf\/ied. For these classes, the PDO~$P$ is formally
integrable, and hence these semisprays will be Lagrangian SODE.
\section{Preliminaries} \label{sec:prelim}
\subsection[The first-order jet bundle $J^{1}\pi$]{The f\/irst-order jet bundle $\boldsymbol{J^{1}\pi}$} \label{subsec:fn}
The appropriate geometric setting for the study of time-dependent
SODE is the af\/f\/ine jet bundle $(J^{1}\pi,\pi_{10},M)$~\cite{saunders89}.
We consider an $(n+1)$-dimensional, real, smooth manifold $M$, which
is f\/ibred over $\mathbb{R}$, $\pi:M\rightarrow\mathbb{R}$, and represents
the space-time. The f\/irst jet bundle of $\pi$ is denoted by $\pi_{10}:J^{1}\pi\to M$,
$\pi_{10}(j_{t}^{1}\phi)=\phi(t),$ for $\phi$ a local section of
$\pi$ and $j_{t}^{1}\phi$ the f\/irst-order jet of~$\phi$ at~$t$.
A local coordinate system $(t,x^{i})_{i\in \{1,\dots,n\}}$ on $M$
induces a local coordinate system on $J^{1}\pi$, denoted by $(t,x^{i},y^{i})$.
Submersion $\pi_{10}$ induces a \emph{natural foliation} on $J^{1}\pi$
such that~$(t,x^{i})$ are transverse coordinates for this foliation,
while~$(y^{i})$ are coordinates for the leaves of the foliation.
Throughout the paper we consider Latin indices $i\in \{1,\dots,n\}$
and Greek indices $\alpha\in \{0,\dots,n\}$, using the notation~$\left(x^{\alpha}\right)=(t=x^{0},x^{i})$.
In this article we use the Fr\"olicher--Nijenhuis theory \cite{frolicher56,grifone00,KMS93}
of derivations of vector-valued dif\/ferential forms on the f\/irst jet
bundle $J^{1}\pi$. We adopt the following notations: $C^{\infty}(J^{1}\pi)$
for the ring of smooth functions on~$J^{1}\pi$, $\mathfrak{X}(J^{1}\pi)$
for the~$C^{\infty}$ module of vector f\/ields on $J^{1}\pi$ and $\Lambda^{k}(J^{1}\pi)$
for the~$C^{\infty}$ module of $k$-forms on~$J^{1}\pi$. The $C^{\infty}$
module of $(r,s)$-type vector f\/ields on~$J^{1}\pi$ is denoted by
$\mathcal{T}_{s}^{r}(J^{1}\pi)$ and the tensor algebra on~$J^{1}\pi$
is denoted by $\mathcal{T}(J^{1}\pi)$. The graded algebra of dif\/ferential
forms on $J^{1}\pi$ is written as $\Lambda(J^{1}\pi)=\bigoplus_{k\in \{1,\dots,2n+1\}}\Lambda^{k}(J^{1}\pi)$.
We denote by~$S^{k}(J^{1}\pi)$ the space of symmetric $(0,k)$ tensors
on~$J^{1}\pi$ and by $\Psi(J^{1}\pi)=\bigoplus_{k\in \{1,\dots,2n+1\}}\Psi^{k}(J^{1}\pi)$
the graded algebra of vector-valued dif\/ferential forms on $J^{1}\pi$.
Throughout the paper we assume that all objects are $C^{\infty}$-smooth
where def\/ined.
A parametrized curve on $M$ is a section of $\pi$: $\gamma:\mathbb{R}\rightarrow M$,
$\gamma(t)=(t,x^{i}(t))$. Its \emph{first-order jet prolongation}
$J^{1}\gamma:t\in\mathbb{R}\rightarrow J^{1}\gamma(t)=\left(t,x^{i}(t),dx^{i}/dt\right)\in J^{1}\pi$
is a section of the f\/ibration $\pi_{1}:=\pi\circ\pi_{10}: J^{1}\pi\rightarrow\mathbb{R}$.
Let $VJ^{1}\pi$ be the \emph{vertical subbundle} of $TJ^{1}\pi,$
$VJ^{1}\pi=\{\xi\in TJ^{1}\pi,\, D\pi_{10}(\xi)=0\}\subset TJ^{1}\pi$.
The f\/ibers $V_{u}J^{1}\pi=\operatorname{Ker} D_{u}\pi_{10}$, $u\in J^{1}\pi$ determine
a regular, $n$-dimensional, integrable vertical distribution. Remark
that $V_{u}J^{1}\pi=\operatorname{spann}\{\partial/\partial y^{i}\}$
and its annihilators are the \emph{contact $1$-forms} $\delta x^{i}=dx^{i}-y^{i}dt$, $i\in \{1,\dots,n\}$
and \emph{basic $1$-forms} $\lambda dt$, $\lambda\in C^{\infty}(J^{1}\pi)$.
The \emph{vertical endomorphism} $J=\frac{\partial}{\partial y^{i}}\otimes\delta x^{i}$
is a vector-valued 1-form on~$J^{1}\pi$, with $\operatorname{Im}J=V(J^{1}\pi)$,
$V(J^{1}\pi)\subset\operatorname{Ker}J$ and $J^{2}=0$.
Its Fr\"olicher--Nijenhuis tensor is given by
\begin{gather*}
N_{J}=\frac{1}{2}[J,J]=-\frac{\partial}{\partial y^{i}}\otimes\delta x^{i}\wedge dt=-J\wedge dt.
\end{gather*}
Consequently, $d_{J}^{2}=d_{N_{J}}=-d_{J\wedge dt}\neq0$ and therefore
$d_{J}$-exact forms on $J^{1}\pi$ may not be $d_{J}$-closed. Here
$d_{J}$ is the \emph{exterior derivative} with respect to the vertical
endomorphisms.
\begin{remark}
For $A\in\Psi^{1}(J^{1}\pi)$ a vector-valued 1-form, the exterior
derivative with respect to $A$ is a derivation of degree 1 given
by $d_{A}=i_{A}\circ d-d\circ i_{A}.$
\end{remark}
A $k$-form $\omega$ on $J^{1}\pi$, $k\geq1$, is called \emph{semi-basic}
if it vanishes whenever one of the arguments is vertical.
A vector-valued $k$-form $A$ on $J^{1}\pi$ is called semi-basic
if it takes values in the vertical bundle and it vanishes whenever
one of the arguments is vertical.
A semi-basic $k$-form satisf\/ies the relation $i_{J}\theta=0$ and
locally can be expressed as $\theta=\theta_{0}dt+\theta_{i}\delta x^{i}$.
For example, contact 1-forms $\delta x^{i}$ are semi-basic 1-forms.
If a vector-valued $k$-form $A$ is semi-basic, then $J\circ A=0$
and $i_{J}A=0$. The vertical endomorphism $J$ is a vector-valued,
semi-basic $1$ -form.
Locally, a semi-basic $k$-form $\theta$ has the next form \begin{eqnarray*}
\theta=\frac{1}{k!}\theta_{i_{1}\dots i_{k}}(x^{\alpha},y^{j})\delta x^{i_{1}}\wedge\cdots\wedge\delta x^{i_{k}}+\frac{1}{(k-1)!}\widetilde{\theta}_{i_{1}\dots i_{k-1}}(x^{\alpha},y^{j})\delta x^{i_{1}}\wedge\cdots\wedge\delta x^{i_{k-1}}\wedge dt.\end{eqnarray*}
For simplicity, we denote by $T^{*}$ the vector bundle of $1$-forms
on $J^{1}\pi$, by $T_{v}^{*}$ the vector bundle of semi-basic $1$-forms
on $J^{1}\pi$ and by $\Lambda^{k}T_{v}^{*}$ the vector bundle of
semi-basic $k$-forms on $J^{1}\pi$. We also denote by $\Lambda_{v}^{k}=\operatorname{Sec}\left(\Lambda^{k}T_{v}^{*}\right)$
the $C^{\infty}(J^{1}\pi)$-module of sections of $\Lambda^{k}T_{v}^{*}$
and by $S^{k}T^{*}$ the vector bundle of symmetric tensors of $(0,k)$-type
on $J^{1}\pi$. $S^{1}T^{*}$ will be identif\/ied with $T^{*}$.
A \emph{semispray} is a globally def\/ined vector f\/ield $S$ on $J^{1}\pi$
such that \begin{gather*}
J(S)=0\qquad\textrm{and}\qquad dt(S)=1
\end{gather*}
The integral curves of a semispray are f\/irst-order jet prolongations
of sections of $\pi\circ\pi_{10}:\! J^{1}\pi\!\rightarrow\!\mathbb{R}$.
Locally, a semispray has the form{\samepage \begin{gather}
S=\frac{\partial}{\partial t}+y^{i}\frac{\partial}{\partial x^{i}}-2G^{i}(x^{\alpha},y^{j})\frac{\partial}{\partial y^{i}},\label{eq:5}
\end{gather}
where functions $G^{i}$, called the semispray coef\/f\/icients, are
locally def\/ined on~$J^{1}\pi$.}
A parametrized curve $\gamma: I\rightarrow M$ is a \emph{geodesic}
of $S$ if $S\circ J^{1}\gamma=\frac{d}{dt}(J^{1}\gamma).$
In local coordinates, $\gamma(t)=(t,x^{i}(t))$ is a geodesic of the
semispray $S$ given by \eqref{eq:5} if and only if it satisf\/ies
the system of SODE \begin{gather}
\frac{d^{2}x^{i}}{dt^{2}}+2G^{i}\left(t,x,\frac{dx}{dt}\right)=0.\label{sode}
\end{gather}
Therefore such a system of time-dependent SODE can be identif\/ied with
a semispray on $J^{1}\pi$.
{\bf Canonical nonlinear connection.}
A~\emph{nonlinear connection} on $J^{1}\pi$ is an $(n+1)$-dimensional
distribution $H:u\in J^{1}\pi\mapsto H_{u}\subset T_{u}J^{1}\pi$,
supplementary to $VJ^{1}\pi$: $\forall\, u\in J^{1}\pi$, $T_{u}J^{1}\pi=H_{u}\oplus V_{u}$.
A semispray $S$ induces a nonlinear connection on $J^{1}\pi$, given
by the \emph{almost product structure} $\Gamma=-\mathcal{L}_{S}J+S\otimes dt$, $\Gamma^{2}={\rm Id}$.
The \emph{horizontal projector} that corresponds to this almost product
structure is $h=\frac{1}{2}\left({\rm Id}-\mathcal{L}_{S}J+S\otimes dt\right)$
and the \emph{vertical projector} is $v={\rm Id}-h.$
The horizontal subspace is spanned by $S$ and by $\frac{\delta}{\delta x^{i}}:=\frac{\partial}{\partial x^{i}}-N_{i}^{j}\frac{\partial}{\partial y^{j}}$, where $N_{j}^{i}=\frac{\partial G^{i}}{\partial y^{j}}$.
In this paper we prefer to work with the following adapted basis and
cobasis:
\begin{gather}
\left\{ S,\frac{\delta}{\delta x^{i}},\frac{\partial}{\partial y^{i}}\right\} ,\qquad\{dt,\delta x^{i},\delta y^{i}\},\label{eq:11}
\end{gather}
with $\delta x^{i}$ the \emph{contact $1$-forms} and $\delta y^{i}=dy^{i}+N_{\alpha}^{i}dx^{\alpha}$, $N_{0}^{i}=2G^{i}-N_{j}^{i}y^{j}$.
Functions $N_{j}^{i}$ and $N_{0}^{i}$ are the coef\/f\/icients of the
nonlinear connection induced by the semispray~$S$.
With respect to basis and cobasis \eqref{eq:11}, the horizontal and
vertical projectors are locally expressed as $h=S\otimes dt+\frac{\delta}{\delta x^{i}}\otimes\delta x^{i}$, $v=\frac{\partial}{\partial y^{i}}\otimes\delta y^{i}$.
We consider the $(1,1)$-type tensor f\/ield $\mathbb{F}=h\circ\mathcal{L}_{S}h-J$,
which corresponds to the almost complex structure in the autonomous
case. It satisf\/ies $\mathbb{F}^{3}+\mathbb{F}=0,$ which means that
it is an~$f(3,1)$ structure. It can be expressed locally as $\mathbb{F}=\frac{\delta}{\delta x^{i}}\otimes\delta y^{i}-\frac{\partial}{\partial y^{j}}\otimes\delta x^{i}$.
{\bf Curvature.}
The following properties for the torsion and curvature of the nonlinear
connection induced by the semispray are proved in~\cite{B-C-2010}.
The weak torsion tensor f\/ield of the nonlinear connection $\Gamma$
vanishes: $[J,h]=0,$ which is equivalent also with $[J,\Gamma]=0$.
The curvature tensor $R=N_{h}$ of the nonlinear connection $\Gamma$
is a vector-valued semi-basic 2-form, locally given by
\begin{gather}
R=\frac{1}{2}[h,h]=\frac{1}{2}R_{ij}^{k}\frac{\partial}{\partial y^{k}}\otimes\delta x^{i}\wedge\delta x^{j}+R_{i}^{j}\frac{\partial}{\partial y^{j}}\otimes dt\wedge\delta x^{i},\label{eq:19}
\end{gather}
where \begin{gather*}
R_{jk}^{i}=\frac{\delta N_{j}^{i}}{\delta x^{k}}-\frac{\delta N_{k}^{i}}{\delta x^{j}
\end{gather*}
and \begin{gather}
R_{j}^{i}=2\frac{\partial G^{i}}{\partial x^{j}}-\frac{\partial G^{i}}{\partial y^{k}}\frac{\partial G^{k}}{\partial y^{j}}-S\left(\frac{\partial G^{i}}{\partial y^{j}}\right).\label{eq:local-Jacobi}
\end{gather}
The \emph{Jacobi endomorphism} is def\/ined as \begin{gather}
\Phi=v\circ\mathcal{L}_{S}h=\mathcal{L}_{S}h-\mathbb{F}-J.\label{eq:22}\end{gather}
Jacobi endomorphism $\Phi$ is a semi-basic, vector-valued 1-form
and satisf\/ies $\Phi^{2}=0$. Locally, can be expressed as $\Phi=R_{i}^{j}\frac{\partial}{\partial y^{j}}\otimes\delta x^{i},$
where $R_{j}^{i}$ are given by~(\ref{eq:local-Jacobi}).
The Jacobi endomorphism and the curvature of the nonlinear connection
are related by the following formulae:
\begin{gather}
\Phi = i_{S}R,\label{eq:23'}\\
\left[J,\Phi\right] = 3R+\Phi\wedge dt.\label{eq:24'}
\end{gather}
Remark that $R=0$ if and only if $\Phi=0$.
\begin{definition}
A semispray $S$ is called \emph{isotropic} if its Jacobi endomorphism
has the form \begin{gather}
\Phi=\lambda J,
\label{eq:phi_iso}
\end{gather}
where $\lambda\in C^{\infty}(J^{1}\pi)$.
\end{definition}
Next we express the isotropy condition \eqref{eq:phi_iso} for a semispray
in terms of the curvature tensor~$R$.
\begin{proposition}
\label{prop:iso} A semispray $S$ is isotropic if and only if its
curvature tensor $R$ has the form
\begin{gather*}
R=\alpha\wedge J
\end{gather*}
where $\alpha$ is a semi-basic $1$-form on $J^{1}\pi$.
\end{proposition}
\begin{proof}
Suppose that $S$ is an isotropic SODE. Then there exists $\lambda\in C^{\infty}(J^{1}\pi)$
such that $\Phi=\lambda J$. From (\ref{eq:24'}) it results
\begin{gather*}
3R = [J,\lambda J]-\Phi\wedge dt,\\
{}[J,\lambda J] = \left(d_{J}\lambda\right)\wedge J-d\lambda\wedge J^{2}+\lambda[J,J] \ \Rightarrow \
R = \frac{1}{3}\left(d_{J}\lambda\right)\wedge J-\lambda J\wedge dt=\alpha\wedge J,
\end{gather*}
with $\alpha=\frac{1}{3}d_{J}\lambda+\lambda dt\in T_{v}^{*}$.
In the above calculus we used the formula \cite{grifone00}\[
[K,gL]= (d_{K}g )\wedge L-dg\wedge KL+g[K,L],\]
for $K$, $L$ vector-valued one-forms on $J^{1}\pi$ and $g\in C^{\infty}(J^{1}\pi)$.
For the converse, suppose that $R=\alpha\wedge J$, with $\alpha\in T_{v}^{*}$.
Formula (\ref{eq:23'}) implies $\Phi=i_{S} (\alpha\wedge J )= (i_{S}\alpha )J-\alpha\wedge i_{S}J= (i_{S}\alpha )J$.
\end{proof}
\subsection{Lagrangian semisprays}
In this subsection we recall some basic notions about Lagrangian semisprays.
\begin{definition}\null \qquad
1) A smooth function $L\in C^{\infty}(J^{1}\pi)$ is called a \emph{Lagrangian
function}.
2) The Lagrangian $L$ is regular if the $(0,2)$ type tensor with
local components
\begin{gather*}
g_{ij}\big(x^{\alpha},y^{k}\big)=\frac{\partial^{2}L}{\partial y^{i}\partial y^{j}
\end{gather*}
has rank $n$ on $J^{1}\pi$. The tensor $g=g_{ij}\delta x^{i}\otimes\delta x^{j}$
is called the \emph{metric tensor} of the Lagrangian~$L$.
\end{definition}
\begin{remark}
More exactly, \cite{saunders89}, a function $L\in C^{\infty}(J^{1}\pi)$
is called a Lagrangian density on~$\pi$. If~$\Omega$ is a volume
form on $\mathbb{R}$, the corresponding Lagrangian is the semi-basic
1-form $L\pi_{1}^{*}\Omega$ on~$J^{1}\pi$. Using a f\/ixed volume form
on~$\mathbb{R}$, for example~$dt$, it is natural to consider the
function~$L$ as a~(f\/irst-order) Lagrangian.
\end{remark}
For the particular choice of $dt$ as volume form on $\mathbb{R}$,
the \emph{Poincar\'e--Cartan $1$-form} of the Lagrangian $L$ is
$\theta_{L}:=Ldt+d_{J}L$. The Lagrangian $L$ is regular if and only
if the \emph{Poincar\'e--Cartan $2$-form} $d\theta_{L}$ has maximal
rank $2n$ on $J^{1}\pi$.
For a detailed exposition on the regularity conditions for Lagrangians
see~\cite{Krupkova}.
The \emph{geodesics} of a semispray $S$, given by the system of SODE
(\ref{sode}), coincide with the solutions of the \emph{Euler--Lagrange
equations} \begin{gather*}
\frac{d}{dt}\left(\frac{\partial L}{\partial y^{i}}\right)-\frac{\partial L}{\partial x^{i}}=
\end{gather*}
if and only if \begin{gather}
g_{ij}\left(t,x,\frac{dx}{dt}\right)\left(\frac{d^{2}x^{i}}{dt^{2}}+2G^{i}\left(t,x,\frac{dx}{dt}\right)\right)=\frac{d}{dt}\left(\frac{\partial L}{\partial y^{i}}\right)-\frac{\partial L}{\partial x^{i}}.\label{eq:SEL}\end{gather}
Therefore, for a semispray $S,$ there exists a Lagrangian function
$L$ such that (\ref{eq:SEL}) holds true if and only if $S\left(\frac{\partial L}{\partial y^{i}}\right)-\frac{\partial L}{\partial x^{i}}=0,$
which can be further expressed as
\begin{gather}
\mathcal{L}_{S}\theta_{L}=dL \ \Leftrightarrow \ i_{S}d\theta_{L}=0.\label{eq:LSTL}
\end{gather}
\begin{definition}
A semispray $S$ is called a \emph{Lagrangian semispray} (or a Lagrangian
vector f\/ield) if there exists a Lagrangian function~$L$,
locally def\/ined on~$J^{1}\pi$, that satisf\/ies~(\ref{eq:LSTL}).
\end{definition}
In \cite{B-C-2010} it has been shown that a semispray $S$ is a Lagrangian
semispray if and only if there exists a semi-basic 1-form $\theta\in\Lambda_{v}^{1}$
with $\operatorname{rank}(d\theta)=2n$ on $J^{1}\pi$, such that
$\mathcal{L}_{S}\theta$ is closed. This represents a reformulation,
in terms of semi basic $1$-forms, of the result in terms of $2$-forms
obtained by Crampin et al.\ in~\cite{crampin84a}. The characterization
of Lagrangian higher order semisprays in terms of a closed 2-form
appears also in~\cite{anderso92}.
Based on this result we can obtain the following reformulation in
terms of semi-basic 1-forms of the known Helmholtz conditions \cite[Lemma~4.2, Lemma~4.3, Theorem~4.5, Theorem~5.1]{B-C-2010}.
\begin{theorem}
A semispray $S$ is a Lagrangian vector field if and only if there
exists a semi-basic $1$-form $\theta\in\Lambda_{v}^{1}$, with $\operatorname{rank}(d\theta)=2n$
on $J^{1}\pi$, such that \begin{gather}
d_{J}\theta=0,\qquad d_{h}\theta=0.\label{eq:P}
\end{gather}
\end{theorem}
\begin{proof}
In order to make this paper self contained, we give a direct proof
of this theorem.
Suppose that $S$ is a Lagrangian semispray. It results that there
exists a regular Lagrangian~$L$ on~$J^{1}\pi$ with $\mathcal{L}_{S}\theta_{L}=dL$,
or equivalently $i_{S}d\theta_{L}=0$, where $\theta_{L}=Ldt+d_{J}L$
is its Poincar\'e 1-form. Evidently $\theta_{L}$ is a semi-basic 1-form
with $\operatorname{rank}(d\theta_{L})=2n$ on $J^{1}\pi$. We will
prove that $d_{J}\theta_{L}=d_{h}\theta_{L}=0$.
Indeed, $d_{J}\theta_{L}=d_{J}L\wedge dt+Ld_{J}dt+d_{J}^{2}L=i_{J}dL\wedge dt-d_{J\wedge dt}L=i_{J}dL\wedge dt-i_{J\wedge dt}dL=0$.
From the formula $i_{J}\mathcal{L}_{S}-\mathcal{L}_{S}i_{J}=i_{\Gamma-S\otimes dt}$
and $i_{J}\mathcal{L}_{S}d\theta_{L}=0$ we obtain $\mathcal{L}_{S}d_{J}\theta_{L}+i_{\Gamma}d\theta_{L}-i_{S\otimes dt}d\theta_{L}=0$.
We also compute $i_{\Gamma}d\theta_{L}=i_{2h-\operatorname{Id}}d\theta_{L}=2i_{h}d\theta_{L}-2d\theta_{L}=2d_{h}\theta_{L}$.
Therefore $2d_{h}\theta_{L}=i_{S\otimes dt}d\theta_{L}=-i_{S}d\theta_{L}\wedge dt=0$.
Conversely, suppose that there exists a semi-basic $1$-form $\theta\in\Lambda_{v}^{1}$,
with $\operatorname{rank}(d\theta)=2n$ on~$J^{1}\pi$, such that
$d_{J}\theta=0$, $d_{h}\theta=0$. In order to prove that $S$ is
a Lagrangian vector f\/ield, we will f\/irst show that~$\mathcal{L}_{S}\theta=d(i_{S}\theta)$.
The hypothesis $d_{J}\theta=0$ implies $\theta=(i_{S}\theta)dt+d_{J}(i_{S}\theta).$
Indeed, $d_{J}i_{S}+i_{S}d_{J}=\mathcal{L}_{JS}-i_{[S,J]}=i_{h-S\otimes dt-v}\Rightarrow d_{J}(i_{S}\theta)=i_{h}\theta-(i_{S}\theta)dt-i_{v}\theta=\theta-(i_{S}\theta)dt.$
Next, from $d_{h}i_{S}+i_{S}d_{h}=\mathcal{L}_{hS}-i_{[S,h]}$ and
$d_{h}\theta=0$ it results that $d_{h}i_{S}\theta=\mathcal{L}_{S}\theta-i_{\mathbb{F}+J+\Phi}\theta=\mathcal{L}_{S}\theta-i_{\mathbb{F}}\theta$.
{\sloppy From $i_{\mathbb{F}}\theta=i_{\mathbb{F}}\left((i_{S}\theta)dt+d_{J}(i_{S}\theta)\right)=i_{\mathbb{F}}\left(d_{J}(i_{S}\theta)\right)$
and $i_{\mathbb{F}}d_{J}-d_{J}i_{\mathbb{F}}=d_{J\circ\mathbb{F}}-i_{[\mathbb{F},J]}\Rightarrow $ $ i_{\mathbb{F}}\left(d_{J}(i_{S}\theta)\right)=d_{v}(i_{S}\theta)$.
It results that $\mathcal{L}_{S}\theta=d_{h}(i_{S}\theta)+d_{v}(i_{S}\theta)=d(i_{S}\theta)$.
}
Consider $L=i_{S}\theta$. Then $\theta$ is the Poincar\'e--Cartan 1-form
of $L$ and $\mathcal{L}_{S}\theta_{L}=dL$. From $\operatorname{rank}(d\theta)=2n$
on $J^{1}\pi$ it results that $L$ is a regular Lagrangian and $S$
is a Lagrangian vector f\/ield.
\end{proof}
In the next section we discuss the formal integrability of these Helmholtz
conditions using two suf\/f\/icient conditions provided by Cartan--K\"ahler
theorem.
\section{Formal integrability for the nonautonomus inverse problem\\ of the
calculus of variations}\label{section3}
In order to study the integrability conditions of the set of dif\/ferential
equations \eqref{eq:P}, we associate to it a linear partial dif\/ferential
operator and study its formal integrability, using Spencer's technique.
The approach in this work follows the one developed in \cite{B-M_arxiv}
for studying the projective metrizability problem for autonomous sprays.
For the basic notions of formal integrability theory of linear partial
dif\/ferential operators see \cite{B-M_arxiv,grifone00}.
Consider $T_{v}^{*}$ the vector bundle of semi-basic $1$-forms on
$J^{1}\pi$ and $\Lambda_{v}^{1}$ the module of sections of~$T_{v}^{*}$.
For $\theta\in\Lambda_{v}^{1}$ and $k\geq1$ we denote by $j_{u}^{k}\theta$
the $k$th order jet of $\theta$ at the base point $u$ in $J^{1}\pi$.
The bundle of $k$th order jets of sections of $T_{v}^{*}$ is denoted
by $J^{k}T_{v}^{*}$. The projection $\pi_{0}:J^{k}T_{v}^{*}\rightarrow J^{1}\pi$
is def\/ined by $\pi_{0}(j_{u}^{k}\theta)=u$. If $l>k$, one def\/ines
the projections $\pi_{k}$ as follows: $\pi_{k}(j_{u}^{l}\theta)=j_{u}^{k}\theta$
and $J^{l}T_{v}^{*}$ is also a f\/ibred manifold over $J^{k}T_{v}^{*}$.
If $f_{1},\dots,f_{k}\in C^{\infty}(J^{1}\pi)$ are functions vanishing
at $u\in J^{1}\pi$ and $\theta\in\Lambda_{v}^{1}$, we def\/ine $\epsilon: \mathcal{S}^{k}T^{*}\otimes T_{v}^{*}\overset{}{\longrightarrow}J^{k}T_{v}^{*}$
by $\epsilon(df_{1}\odot\cdots\odot df_{k}\otimes\theta)_{u}=j_{u}^{k}(f_{1}\cdots f_{k}\theta)$,
where $\odot$ is the symmetric product. Then the sequence
\[
0\overset{}{\longrightarrow}\mathcal{S}^{k}T^{*}\otimes T_{v}^{*}\overset{\epsilon}{\longrightarrow}J^{k}T_{v}^{*}\overset{\pi_{k-1}}{\longrightarrow}J^{k-1}T_{v}^{*}\overset{}{\longrightarrow}0\]
is exact.
Consider the \emph{linear partial differential operator} of order
one\begin{gather}
P: \ \Lambda_{v}^{1} \rightarrow \Lambda_{v}^{2}\oplus\Lambda_{v}^{2},\qquad
P = \left(d_{J}, d_{h}\right).\label{eq:PDE}
\end{gather}
Remark that $P(\theta)$ can be expressed in terms of f\/irst-order
jets of $\theta$, for any $\theta\in\Lambda_{v}^{1}$, and therefore
it induces a morphism between vector bundles:
\begin{gather*}
p^{0}(P): \ J^{1}T_{v}^{*} \rightarrow \Lambda^{2}T_{v}^{*}\oplus\Lambda^{2}T_{v}^{*},\qquad
p^{0}(P)(j_{u}^{1}\theta) = P(\theta)_{u},\quad\forall\,\theta\in\Lambda_{v}^{1}.
\end{gather*}
We also consider the \emph{$l$th order jet prolongations} of the
dif\/ferential operator $P$, $l\geq1$, which will be identif\/ied with
the morphisms of vector bundles over $M$,
\begin{gather*}
p^{l}(P): \ J^{l+1}T_{v}^{*} \rightarrow J^{l}\left(\Lambda^{2}T_{v}^{*}\oplus\Lambda^{2}T_{v}^{*}\right),\qquad
p^{l}(P)\big(j_{u}^{l+1}\theta\big) = j_{u}^{l}\left(P(\theta)\right),\quad \forall\,\theta\in\Lambda_{v}^{1}.
\end{gather*}
Remark that for a semi-basic 1-form $\theta=\theta_{\alpha}\delta x^{\alpha}$,
its f\/irst-order jet $j^{1}\theta=\frac{\delta\theta_{\alpha}}{\delta x^{\beta}}\delta x^{\beta}\otimes\delta x^{\alpha}+\frac{\partial\theta_{\alpha}}{\partial y^{i}}\delta y^{i}\otimes\delta x^{\alpha}$
determines the local coordinates $\left(x^{\alpha},y^{i},\theta_{\alpha},\theta_{\alpha\beta},\theta_{\alpha\underline{i}}\right)$
on $J^{1}T_{v}^{*}$. In this work all contravariant or covariant
indices, related to vertical components of tensor f\/ields will be underlined.
Consider $\theta=\theta_{\alpha}\delta x^{\alpha}$, a semi-basic
$1$-form on $J^{1}\pi$. Then
\begin{gather*}
d\theta = \left(\frac{\partial\theta_{i}}{\partial t}-\theta_{j}N_{i}^{j}-\frac{\delta\theta_{0}}{\delta x^{i}}\right)\delta x^{0}\wedge\delta x^{i}+\left(\theta_{i}-\frac{\partial\theta_{0}}{\partial y^{i}}\right)\delta x^{0}\wedge\delta y^{i}\\
\hphantom{d\theta =}{}
+\frac{1}{2}\left(\frac{\delta\theta_{j}}{\delta x^{i}}-\frac{\delta\theta_{i}}{\delta x^{j}}\right)\delta x^{i}\wedge\delta x^{j}
+\left(\frac{\partial\theta_{j}}{\partial y^{i}}\right)\delta y^{i}\wedge\delta x^{j},\\
d_{J}\theta = \left(\theta_{i}-\frac{\partial\theta_{0}}{\partial y^{i}}\right)\delta x^{0}\wedge\delta y^{i}+\frac{1}{2}\left(\frac{\partial\theta_{i}}{\partial y^{j}}-\frac{\partial\theta_{j}}{\partial y^{i}}\right)\delta x^{j}\wedge\delta x^{i},\\
d_{h}\theta = \left(\frac{\partial\theta_{i}}{\partial t}-\theta_{j}N_{i}^{j}-\frac{\delta\theta_{0}}{\delta x^{i}}\right)\delta x^{0}\wedge\delta x^{i}+\frac{1}{2}\left(\frac{\delta\theta_{i}}{\delta x^{j}}-\frac{\delta\theta_{j}}{\delta x^{i}}\right)\delta x^{j}\wedge\delta x^{i}.
\end{gather*}
Using these formulae we obtain
\begin{gather*}
p^{0}(P)\left(j^{1}\theta\right)=
\left(\left(\theta_{i}-\frac{\partial\theta_{0}}{\partial y^{i}}\right)\delta x^{0}\wedge\delta y^{i}+\frac{1}{2}\left(\frac{\partial\theta_{i}}{\partial y^{j}}-\frac{\partial\theta_{j}}{\partial y^{i}}\right)\delta x^{j}\wedge\delta x^{i} , \right.\\
\left.\phantom{p^{0}(P)\left(j^{1}\theta\right)= }{}
\left(\frac{\partial\theta_{i}}{\partial t}-\theta_{j}N_{i}^{j}-\frac{\delta\theta_{0}}{\delta x^{i}}\right)\delta x^{0}\wedge\delta x^{i}+\frac{1}{2}\left(\frac{\delta\theta_{i}}{\delta x^{j}}-\frac{\delta\theta_{j}}{\delta x^{i}}\right)\delta x^{j}\wedge\delta x^{i}\right).
\end{gather*}
The \emph{symbol} of $P$ is the vector bundle morphism $\sigma^{1}(P): T^{*}\otimes T_{v}^{*}\rightarrow\Lambda^{2}T_{v}^{*}\oplus\Lambda^{2}T_{v}^{*}$
def\/ined by the f\/irst-order terms of $p^{0}(P)$. More exactly, $\sigma^{1}(P)=p^{0}(P)\circ\epsilon$.
For $A\in T^{*}\otimes T_{v}^{*}$, $A=A_{\alpha\beta}\delta x^{\alpha}\otimes\delta x^{\beta}+A_{\underline{i}\beta}\delta y^{i}\otimes\delta x^{\beta}$,
we compute
\begin{gather*}
\sigma^{1}(d_{J})A = -A_{\underline{i}0}\delta x^{0}\wedge\delta x^{i}+\frac{1}{2}\big(A_{\underline{j}i}-A_{\underline{i}j}\big)\delta x^{j}\wedge\delta x^{i}
\\
\sigma^{1}(d_{h})A = \left(A_{0i}-A_{i0}\right)\delta x^{0}\wedge\delta x^{i}+\frac{1}{2}\left(A_{ji}-A_{ij}\right)\delta x^{j}\wedge\delta x^{i}
\end{gather*}
and hence \begin{gather*}
\sigma^{1}(P)A = \left(\tau_{J}A , \tau_{h}A\right),\qquad
\left(\tau_{J}A\right)(X,Y) = A(JX,Y)-A(JY,X), \\
\left(\tau_{h}A\right)(X,Y) = A(hX,Y)-A(hY,X),
\end{gather*}
for $X,Y\in\mathfrak{X}(J^{1}\pi)$. In the above formulae $\tau_{J}$, $\tau_{L}$
are \emph{alternating operators}~\cite{B-M_arxiv}.
\begin{remark}
The alternating operators are def\/ined in general as follows. For $K\in\Psi^{k}(J^{1}\pi)$,
a~vector-valued $k$-form, we consider $\tau_{K}:\Psi^{1}(J^{1}\pi)\otimes\Psi^{l}(J^{1}\pi)\to\Psi^{l+k}(J^{1}\pi)$,
\begin{gather}
(\tau_{K}B)(X_{1},\dots ,X_{l+k})=
\frac{1}{l!k!}\sum_{\sigma\in S_{l+k}}\varepsilon(\sigma)B(K(X_{\sigma(1)},\dots ,X_{\sigma(k)}),X_{\sigma(k+1)},\dots ,X_{\sigma(k+l)}),\label{taull} \end{gather}
where $X_{1},\dots ,X_{l+k}\in{\mathfrak{X}}(J^{1}\pi)$ and $S_{l+k}$
is the permutation group of $\{1,\dots ,l+k\}$. The restriction of $\tau_{K}$
to $\Psi^{l+1}(J^{1}\pi)$, is a derivation of degree $(k-1)$ and
it coincides with the \emph{inner product} $i_{K}$.
\end{remark}
The \emph{first-order prolongation of the symbol} of $P$ is the vector
bundle morphism
$\sigma^{2}(P): S^{2}T^{*}\otimes T_{v}^{*}\rightarrow T^{*}\otimes\left(\Lambda^{2}T_{v}^{*}\oplus\Lambda^{2}T_{v}^{*}\right)$
that verif\/ies \[
i_{X}\left(\sigma^{2}(P)B\right)=\sigma^{1}(P)\left(i_{X}B\right),\qquad \forall\, B\in S^{2}T^{*}\otimes T_{v}^{*},\quad \forall\, X\in\mathfrak{X}(J^{1}\pi).
\]
Therefore \begin{gather*}
\sigma^{2}(P)B=\left(\sigma^{2}(d_{J})B , \sigma^{2}(d_{h})B\right),\\
\left(\sigma^{2}(d_{J})B\right)(X,Y,Z) = B(X,JY,Z)-B(X,JZ,Y), \\
\left(\sigma^{2}(d_{h})B\right)(X,Y,Z) = B(X,hY,Z)-B(X,hZ,Y).
\end{gather*}
In local coordinates we obtain the following formulae.
If $B\in S^{2}T^{*}\otimes T_{v}^{*}$, then it has the local decomposition
\begin{gather}
B = B_{\alpha\beta\gamma}\delta x^{\alpha}\otimes\delta x^{\beta}\otimes\delta x^{\gamma}+B_{\underline{i}\alpha\beta}\delta y^{i}\otimes\delta x^{\alpha}\otimes\delta x^{\beta}\nonumber \\
\hphantom{B =}{} +B_{\alpha\underline{i}\beta}\delta x^{\alpha}\otimes\delta y^{i}\otimes\delta x^{\beta}+B_{\underline{ij}\alpha}\delta y^{i}\otimes\delta y^{j}\otimes\delta x^{\alpha},\label{eq:localB}
\end{gather}
with \begin{gather}
B_{\alpha\beta\gamma}=B_{\beta\alpha\gamma},\qquad B_{\underline{i}j\alpha}=B_{i\underline{j}\alpha},\qquad B_{\underline{i}0\alpha}=B_{0\underline{i}\alpha},\qquad B_{\underline{ij}\alpha}=B_{\underline{ji}\alpha}.\label{eq:condB}
\end{gather}
The f\/irst-order prolongation of the symbol of $P$ is given by
\begin{gather*}
\sigma^{2}(d_{J})B = B_{\alpha\underline{i}0}\delta x^{\alpha}\otimes\delta x^{i}\wedge\delta x^{0}+B_{\underline{ij}0}\delta y^{i}\otimes\delta x^{j}\wedge\delta x^{0}\nonumber \\
\hphantom{\sigma^{2}(d_{J})B =}{}
+\frac{1}{2}\big(B_{\alpha\underline{i}j}-B_{\alpha\underline{j}i}\big)\delta x^{\alpha}\otimes\delta x^{i}\wedge\delta x^{j}+\frac{1}{2}\big(B_{\underline{ij}k}-B_{\underline{ik}j}\big)\delta y^{i}\otimes\delta x^{j}\wedge\delta x^{k},\\
\sigma^{2}(d_{h})B = \frac{1}{2}\left(B_{\alpha\beta\gamma}-B_{\alpha\gamma\beta}\right)\delta x^{\alpha}\otimes\delta x^{\beta}\wedge\delta x^{\gamma}+\frac{1}{2}\left(B_{\underline{i}\alpha\beta}-B_{\underline{i}\beta\alpha}\right)\delta y^{i}\otimes\delta x^{\alpha}\wedge\delta x^{\beta}. \end{gather*}
For each $u\in J^{1}\pi$, we consider
\begin{gather*}
g_{u}^{k}(P) = \operatorname{Ker}\sigma_{u}^{k}(P),\qquad k\in\{1,2\},\\
g_{u}^{1}(P)_{e_{1}\dots e_{j}} = \{A\in g_{u}^{1}(P)|i_{e_{1}}A=\cdots=i_{e_{j}}A=0\},\qquad j\in\{1,\dots ,n\},
\end{gather*}
where $\{e_{1},\dots ,e_{n}\}$ is a basis of $T_{u}(J^{1}\pi)$. Such
a basis is called \emph{quasi-regular} if it satisf\/ies
\begin{gather*}
\dim g_{u}^{2}(P)=\dim g_{u}^{1}(P)+\sum_{j=1}^{n}\dim g_{u}^{1}(P)_{e_{1}\dots e_{j}}
\end{gather*}
\begin{definition}
The symbol $\sigma^{1}(\mathcal{P})$ is called \emph{involutive}
at $u$ in $J^{1}\pi$ if there exists a quasi-regular basis of $T_{u}J^{1}\pi$.
\end{definition}
A f\/irst-order jet $j_{u}^{1}\theta\in J^{1}T_{v}^{*}$ is \emph{a~first-order formal solution of~$P$ at~$u$} in $J^{1}\pi$ if $p^{0}(P)(\theta)_{u}=0$.
For $l\geq1$, a $(1+l)$th order jet $j_{u}^{1+l}\theta\in J_{u}^{1+l}T_{v}^{*}$
is \emph{a $(1+l)$th order formal solution of $P$ at $u$} in $J^{1}\pi$
if $p^{l}(P)(\theta)_{u}=0$.
For any $l\geq0$, consider $R_{u}^{1+l}(P)=\ker p_{u}^{l}(P)$ the
space of \emph{$(1+l)$th order formal solutions} of~$P$ at~$u$.
We denote also $\bar{\pi}_{l,u} : R_{u}^{1+l}(P)\rightarrow R_{u}^{l}(P)$
the restriction of $\pi_{l,u} : J_{u}^{1+l}(T_{v}^{*})\rightarrow J_{u}^{l}(T_{v}^{*})$
to $R_{u}^{1+l}(P)$.
\begin{definition}
The partial dif\/ferential operator $P$ is called \emph{formally integrable}
at $u$ in $J^{1}\pi$ if $R^{1+l}(P)=\bigcup_{u\in J^{1}\pi}R_{u}^{1+l}(P)$
is a vector bundle over $J^{1}\pi$, for all $l\geq0$, and the map
$\bar{\pi}_{l,u}:R_{u}^{1+l}(P)\rightarrow R_{u}^{l}(P)$ is onto
for all $l\geq1$.
\end{definition}
The f\/ibred submanifold $R^{1}(P)$ of $\pi_{0}:J_{u}^{1}(T_{v}^{*})\rightarrow J^{1}\pi$
is called \emph{the partial differential equation corresponding to
the first-order PDO}~$P$. A solution of the operator $P$ on an open
set $U\subset J^{1}\pi$ is a section $\theta\in\Lambda_{v}^{1}$
def\/ined on $U$ such that $P\theta=0\Leftrightarrow p^{0}(P)(j_{u}^{1}\theta)=0$, $\forall\, u\in U$.
The Cartan--K\"ahler theorem \cite{grifone00} takes the following form
for the particular case of f\/irst-order PDO.
\begin{theorem}
Let $P$ be a first-order linear partial differential operator with
$g^{2}(P)$ a vector bundle over $R^{1}(P)$. If $\overline{\pi}_{1}:R^{2}(P)\to R^{1}(P)$
is onto and the symbol $\sigma^{1}(P)$ is involutive, then $P$ is
formally integrable.
\end{theorem}
\subsection[The involutivity of the symbol of $P$]{The involutivity of the symbol of $\boldsymbol{P}$}
In this subsection we prove that the operator $P$ satisf\/ies one of
the two suf\/f\/icient conditions for formal integrability, provided by
Cartan--K\"ahler theorem: the involutivity of the symbol~$\sigma^{1}(P)$.
\begin{theorem}\label{thm:-involutivity}
The symbol $\sigma^{1}(P)$ of the PDO
$P=(d_{J},d_{h})$ is involutive.
\end{theorem}
\begin{proof}
First we determine $g^{1}(P)=\left\{ A\in T^{*}\otimes T_{v}^{*}\,\vert\,\sigma^{1}(P)A=0\right\} $,
and compute the dimension of its f\/ibers. We obtain
\begin{gather*}
g_{u}^{1}(P) = \big\{ A=A_{\alpha\beta}\delta x^{\alpha}\otimes\delta x^{\beta}+A_{\underline{i}\beta}\delta y^{i}\otimes\delta x^{\beta}\,\,\vert\,\, A_{\underline{i}0}=0,\, A_{\underline{j}i}=A_{\underline{i}j},\, A_{\alpha\beta}=A_{\beta\alpha}\big\} .
\end{gather*}
From $A_{\underline{i}0}=0$ and $A_{\underline{j}i}=A_{\underline{i}j}$
it results that $A_{\underline{i}j}$ contribute with $n(n+1)/2$
components to the dimension of $g_{u}^{1}(P)$, and from $A_{\alpha\beta}=A_{\beta\alpha}$
it follows that $A_{\alpha\beta}$ contribute with $(n+1)(n+2)/2$
components to the dimension of $g_{u}^{1}(P)$. So
\[
\dim g_{u}^{1}(P)=\frac{n(n+1)}{2}+\frac{(n+1)(n+2)}{2}=(n+1)^{2}.
\]
Next we determine $g^{2}(P)=\left\{ B\in S^{2}T^{*}\otimes T_{v}^{*}\,\vert\,\sigma^{2}(P)B=0\right\} $.
If $B\in S^{2}T^{*}\otimes T_{v}^{*}$ has the local components (\ref{eq:localB}),
then $B\in g^{2}(P)$ if and only if the following relations are satisf\/ied:
\begin{gather}
B_{\alpha\underline{i}0}=0, \qquad B_{\underline{ij}0}=0, \qquad
B_{\alpha\underline{i}j}=B_{\alpha\underline{j}i}, \nonumber \\
B_{\underline{ij}k}=B_{\underline{ik}j},\qquad
B_{\alpha\beta\gamma}=B_{\alpha\gamma\beta}, \qquad B_{\underline{i}\alpha\beta}=B_{\underline{i}\beta\alpha}.\label{eq:eq:cond_B_in_g2}
\end{gather}
From the relations \eqref{eq:condB} and \eqref{eq:eq:cond_B_in_g2}
it results that $B\in g^{2}(P)$ if and only if its local components
$ B_{\alpha\beta\gamma}$, $B_{\underline{i}jk}$, $B_{i\underline{j}k}$, $B_{\underline{ij}k}$
are totally symmetric and the rest are vanishing. Therefore $B_{\alpha\beta\gamma}$
contribute with $(n+1)(n+2)(n+3)/6$ components to the dimension of
$g_{u}^{2}(P)$, and $B_{\underline{i}jk}$, $B_{\underline{ij}k}$
with $n(n+1)(n+2)/6$ components each of them. It results
\[
\dim g_{u}^{2}(P)=\frac{(n+1)(n+2)(n+3)}{6}+2\frac{n(n+1)(n+2)}{6}=\frac{(n+1)^{2}(n+2)}{2}.
\]
Consider \[
\mathcal{B}=\left\{ h_{0}=S,\, h_{1},\,\dots,h_{n},\, v_{1}=Jh_{1},\,\dots,\, v_{n}=Jh_{n}\right\} \]
a basis in $T_{u}J^{1}\pi$ with $h_{0}=S$, $h_{1},\dots,h_{n}$
horizontal vector f\/ields. For any $A\in g^{1}(P)$, we denote \[
A(h_{\alpha},h_{\beta})=a_{\alpha\beta},\qquad A(v_{i},h_{\alpha})=b_{\underline{i}\alpha}.\]
Because $A$ is semi-basic in the second argument it follows that
these are the only components of $A$.
Since $A\in\operatorname{Ker}\sigma^{1}(d_{J})$ it follows that $A_{\underline{i}0}=0$, $A_{\underline{i}j}=A_{\underline{j}i}$ $\Rightarrow$ $b_{\underline{i}0}=0$, $b_{\underline{i}j}=b_{\underline{j}i}$.
Because $A\in\operatorname{Ker}\sigma^{1}(d_{h})$ it results that
$A_{0i}=A_{i0}$, $A_{ij}=A_{ji}$ and hence $a_{0i}=a_{i0}$, $a_{ij}=a_{ji}$.
Consider $j\in \{1,\dots, n\} $ arbitrarily f\/ixed and
\begin{gather*}
\mathcal{\tilde{B}}=\big\{ e_{0}=S+h_{j}+v_{n},\, e_{1}=h_{1},\, e_{2}=h_{2}+v_{1}, \, \dots,\\
\hphantom{\mathcal{\tilde{B}}=\big\{}{} e_{i}=h_{i}+v_{i-1},\, \dots,\, e_{n}=h_{n}+v_{n-1},\, v_{1},\, \dots,\, v_{n}\big\}
\end{gather*}
a new basis in $T_{u}J^{1}\pi$. If we denote \[
A(e_{\alpha},e_{\beta})=\tilde{a}_{\alpha\beta},\qquad A(v_{i},e_{\alpha})=\tilde{b}_{\underline{i}\alpha},\]
a simple computation and the fact that $A$ is semi-basic in the
second argument determine \begin{gather*}
\tilde{a}_{00} = a_{00}+2a_{0j}+a_{jj}+b_{\underline{n}j},\\
\tilde{a}_{ik} = a_{ik}+b_{\underline{i-1},k}\neq\tilde{a}_{ki}=a_{ki}+b_{\underline{k-1},i},\\
\tilde{a}_{i0} = a_{i0}+a_{ij}+b_{\underline{i-1},j}\neq\tilde{a}_{0i}=a_{0i}+a_{ji}+b_{\underline{n}i},\\
\tilde{b}_{\underline{i}k} =b_{\underline{i}k} =\tilde{b}_{\underline{k}i},\qquad
\tilde{b}_{\underline{i}0} = b_{\underline{i}j}.
\end{gather*}
It can be seen that all the independent components of $A$ in the
basis $\mathcal{B}$ can be obtained from the components of $A$ in
the basis $\mathcal{\tilde{B}}$, and hence we can use the later for
determining the dimensions of $(g_{u}^{1})_{e_{0} e_{1}\dots e_{k}}$.
If $A\in(g_{u}^{1})_{e_{0}}$ it results $\tilde{a}_{0\alpha}=0$,
so using this new basis we impose $n+1$ supplementary independent
restrictions. It follows that $\dim(g_{u}^{1})_{e_{0}}=(n+1)^{2}-(n+1)=(n+1)n$.
If $A\in(g_{u}^{1})_{e_{0} e_{1}}$ it results that together with
the previous restrictions we impose also $\tilde{a}_{1\alpha}=0$,
so another independent $n+1$ restrictions. Hence $\dim (g_{u}^{1})_{e_{0} e_{1}}=(n+1)n-(n+1)=(n+1)(n-1)$.
In general $\dim (g_{u}^{1})_{e_{0} e_{1} \dots e_{k}}=(n+1)(n-k)$, $\forall\, k\in \{1,\dots,n\}$.
Hence $\dim (g_{u}^{1})_{e_{0} e_{1} \dots e_{n}}=0\Rightarrow\dim (g_{u}^{1})_{e_{0} \dots e_{n} v_{1} \dots v_{k}}=0$, $\forall\, k\in \{1,\dots,n\}$,
\begin{gather*}
\dim (g_{u}^{1})+\sum_{k=0}^{n}\dim (g_{u}^{1})_{e_{0} e_{1} \dots e_{k}}
= (n+1)^{2}+(n+1)[n+n-1+\cdots+1]\\
\hphantom{\dim (g_{u}^{1})+\sum_{k=0}^{n}\dim (g_{u}^{1})_{e_{0} e_{1} \dots e_{k}}}{} = \frac{(n+1)^{2}(n+2)}{2}=\dim g_{u}^{2}(P).
\end{gather*}
We proved that $\tilde{B}$ is a a quasi-regular basis, hence the
symbol $\sigma^{1}(P)$ is involutive.
\end{proof}
\subsection{First obstruction to the inverse problem}
In this subsection we determine necessary and suf\/f\/icient conditions
for $\bar{\pi}_{1}$ to be onto. We will obtain only one obstruction
for the integrability of the operator $P$. The obstruction is due
to the curvature tensor of the nonlinear connection induced by the
semispray.
\begin{theorem}
\label{thm:A-first-order}A first-order formal solution $\theta\in\Lambda_{v}^{1}$
of the system $d_{J}\theta=0$, $d_{h}\theta=0$ can be lifted into
a second-order solution, which means that $\bar{\pi}_{1} : R^{2}(P)\rightarrow R^{1}(P)$
is onto, if and only if \[
d_{R}\theta=0,\]
where $R$ is the curvature tensor \eqref{eq:19}.
\end{theorem}
\begin{proof}
We use a known result from \cite[Proposition~1.1]{grifone00}.
If $\mathcal{K}$ is the cokernel of $\sigma^{2}(P)$,
\begin{gather*}
\mathcal{K}=\frac{T^{*}\otimes\left(\Lambda^{2}T_{v}^{*}\oplus\Lambda^{2}T_{v}^{*}\right)}{{\normalcolor \operatorname{Im}}\sigma^{2}(P)}
\end{gather*}
there exists a morphism $\varphi :R^{1}(P)\rightarrow\mathcal{K}$
such that the sequence \[
R^{2}(P)\overset{\bar{\pi}_{1}}{\longrightarrow}R^{1}(P)\overset{\varphi}{\longrightarrow}\mathcal{K}\]
is exact. In particular $\bar{\pi}_{1}$ is onto if and only if $\varphi=0$.
After def\/ining $\varphi$, we will prove that for $\theta\in\Lambda_{v}^{1}$,
with $j_{u}^{1}\theta\in R_{u}^{1}(P)$, a f\/irst-order formal solution
of $P$ at $u\in J^{1}\pi$, we have that $\varphi_{u}\theta=0$ if
and only if $\left(d_{R}\theta\right)_{u}=0$.
The construction of the morphism $\varphi$ is represented in the
next diagram by dashed arrows. We denote $F=\Lambda^{2}T_{v}^{*}\oplus\Lambda^{2}T_{v}^{*}$.
$$
\includegraphics{Diagram}
$$
Remark that $\dim T^{*}=2n+1$, $\dim T_{v}^{*}=n+1$, $\dim\Lambda^{2}T_{v}^{*}=\frac{(n+1)n}{2}$, $\dim\mathcal{S}^{2}T^{*}=\frac{(2n+1)(2n+2)}{2}$,
$\dim T^{*}\otimes\left(\Lambda^{2}T_{v}^{*}\oplus\Lambda^{2}T_{v}^{*}\right)=(2n+1)n(n+1)$.
Therefore
\begin{gather*}
\dim\mathcal{K} = \dim\left[T^{*}\otimes\left(\Lambda^{2}T_{v}^{*}\oplus\Lambda^{2}T_{v}^{*}\right)\right]-\dim\left(\operatorname{Im}\sigma^{2}(P)\right)\\
\hphantom{\dim\mathcal{K}}{} = \dim\left[T^{*}\otimes\left(\Lambda^{2}T_{v}^{*}\oplus\Lambda^{2}T_{v}^{*}\right)\right]-\left[\dim\left(\mathcal{S}^{2}T^{*}\otimes T_{v}^{*}\right)-\dim\left(\ker\sigma^{2}(P)\right)\right]\\
\hphantom{\dim\mathcal{K}}{}
= \frac{(n-1)n(n+1)}{2}=3 \begin{pmatrix}
n+1\\
3\end{pmatrix}.\end{gather*}
It results from this that \[
\mathcal{K}\simeq\oplus^{(3)}\Lambda^{3}T_{v}^{*}.\]
Next we def\/ine $\tau: T^{*}\otimes\left(\Lambda^{2}T_{v}^{*}\oplus\Lambda^{2}T_{v}^{*}\right)\rightarrow\oplus^{(3)}\Lambda^{3}T_{v}^{*}$
such as the next sequence is exact:
\begin{gather}
0\rightarrow g^{2}(P)\overset{i}{\longrightarrow}\mathcal{S}^{2}T^{*}\otimes T_{v}^{*}\overset{\sigma^{2}(P)}{\longrightarrow}T^{*}
\otimes\left(\Lambda^{2}T_{v}^{*}\oplus\Lambda^{2}T_{v}^{*}\right)\overset{\tau}{\longrightarrow}\oplus^{(3)}\Lambda^{3}T_{v}^{*}\rightarrow0.
\label{eq:exact_tau}\end{gather}
For $B_{1}, B_{2}\in T^{*}\otimes\Lambda^{2}T_{v}^{*}$, we def\/ine
$\tau(B_{1},B_{2})=\left(\tau_{1}(B_{1},B_{2}) , \tau_{2}(B_{1},B_{2}) , \tau_{3}(B_{1},B_{2})\right)$,
where $\tau_{i}: T^{*}\otimes\left(\Lambda^{2}T_{v}^{*}\oplus\Lambda^{2}T_{v}^{*}\right)\rightarrow\Lambda^{3}T_{v}^{*}$, $ i\in \{1,2,3\}$,
are given by
\begin{gather*}
\tau_{1}(B_{1},B_{2})=\tau_{J}B_{1},\qquad
\tau_{2}(B_{1},B_{2})=\tau_{h}B_{2},\qquad \tau_{3}(B_{1},B_{2})=\tau_{h}B_{1}+\tau_{J}B_{2}
\end{gather*}
Using the def\/inition (\ref{taull}) of the alternating operators $\tau_{J}$, $\tau_{h}$,
we prove that $\tau\circ\sigma^{2}(P)=0$.
Indeed, using that any $B\in\mathcal{S}^{2}T^{*}\otimes T_{v}^{*}$
is symmetric in the f\/irst two arguments, it follows that $\left(\tau\circ\sigma^{2}(P)\right)(B)=\tau\left(\sigma^{2}(d_{J})B, \sigma^{2}(d_{h})B\right)=\left(\tau_{J}\sigma^{2}(d_{J})B, \tau_{h}\sigma^{2}(d_{h})B, \tau_{h}\sigma^{2}(d_{J})B+\tau_{J}\sigma^{2}(d_{h})B\right)=0$, $\forall\, B\in\mathcal{S}^{2}T^{*}\otimes T_{v}^{*}$.
For example,\begin{gather*}
\tau_{J}\left(\sigma^{2}(d_{J})B\right)(X,Y,Z) \\
\qquad{} = \left[\sigma^{2}(d_{J})B\left(JX,Y,Z\right)-\sigma^{2}(d_{J})B\left(JY,X,Z\right)+\sigma^{2}(d_{J})B\left(JZ,X,Y\right)\right]\\
\qquad{}{} = [B(JX,JY,Z)-B(JX,JZ,Y)-B(JY,JX,Z)+B(JY,JZ,X)\\
\qquad\quad{}{} +B(JZ,JX,Y)-B(JZ,JY,X)]=0,\qquad \forall\, X,Y,Z\in\mathfrak{X}(J^{1}\pi).
\end{gather*}
The relation $\tau\circ\sigma^{2}(P)=0$ implies that $\operatorname{Im}(\sigma^{2}(P))\subseteq\operatorname{Ker}\tau$.
Using that $\tau$ is onto ($\tau_{J}$,~$\tau_{h}$~are both onto)
it results that $\dim\left[\operatorname{Im}(\sigma^{2}(P))\right]=\dim\left(\operatorname{Ker}\tau\right)$
and hence $\operatorname{Im}(\sigma^{2}(P))=\operatorname{Ker}\tau$
and the sequence~(\ref{eq:exact_tau}) is exact.
The last step before def\/ining $\varphi :R^{1}(P)\rightarrow\mathcal{K}$
is to consider a linear connection $\nabla$ on $J^{1}\pi$ such that
$\nabla J=0$. It means that $\nabla$ preserve semi-basic forms and
$\nabla$ can be considered as a~connection in the f\/iber bundle $\Lambda^{2}T_{v}^{*}\oplus\Lambda^{2}T_{v}^{*}\rightarrow J^{1}\pi$.
As a f\/irst-order PDO we can identify $\nabla$ with the bundle morphism
$p^{0}(\nabla): J^{1}\left(\Lambda^{2}T_{v}^{*}\oplus\Lambda^{2}T_{v}^{*}\right)\rightarrow T^{*}\otimes\left(\Lambda^{2}T_{v}^{*}\oplus\Lambda^{2}T_{v}^{*}\right)$.
We will also use two derivations of degree~1 introduced in~\cite{B-M_arxiv},
def\/ined by $\mathcal{D}_{J}=\tau_{J}\nabla$, $\mathcal{D}_{h}=\tau_{h}\nabla.$
Both derivations $\mathcal{D}_{J}$, $\mathcal{D}_{h}$ preserve semi-basic
forms and $d_{J}-\mathcal{D}_{J}$, $ d_{h}-\mathcal{D}_{h}$ are algebraic
derivations. It means that if $\omega\in\Lambda^{k}\left(J^{1}\pi\right)$
vanishes at some point $u\in J^{1}\pi$, then $\left(\mathcal{D}_{J}\omega\right)_{u}=\left(d_{J}\omega\right)_{u}$
and $\left(\mathcal{D}_{h}\omega\right)_{u}=\left(d_{h}\omega\right)_{u}$
\cite[Lemma~2.1]{B-M_arxiv}.
Now we are able to def\/ine $\varphi :R^{1}(P)\rightarrow\mathcal{K}$
such that the sequence \[
R^{2}(P)\overset{\bar{\pi}_{1}}{\longrightarrow}R^{1}(P)\overset{\varphi}{\longrightarrow}\mathcal{K}\]
is exact.
Let $\theta\in\Lambda_{v}^{1}$ such that $j_{u}^{1}\theta\in R_{u}^{1}(P)\subset J_{u}^{1}T_{v}^{*}$
is a f\/irst-order formal solution of $P$ at $u\in J^{1}\pi$, which
means that $\left(d_{J}\theta\right)_{u}=\left(d_{h}\theta\right)_{u}=0$.
Consider \[
\varphi_{u}\theta=\tau_{u}\nabla P\theta=\tau_{u}\left(\nabla d_{J}\theta, \nabla d_{h}\theta\right).\]
Using the fact that $d_{J}-\tau_{J}\nabla$ and $d_{h}-\tau_{h}\nabla$
are algebraic derivations and $\left(d_{J}\theta\right)_{u}=\left(d_{h}\theta\right)_{u}=0$
it results $d_{J}\left(d_{J}\theta\right)_{u}=\tau_{J}\nabla\left(d_{J}\theta\right)_{u}$
and $d_{h}\left(d_{h}\theta\right)_{u}=\tau_{h}\nabla\left(d_{h}\theta\right)_{u}$.
We will compute the three components of the map $\varphi$. It follows
that \begin{gather*}
\tau_{1}\left(\nabla P\theta\right)_{u} = \tau_{1}\left(\nabla d_{J}\theta, \nabla d_{h}\theta\right)_{u}=\tau_{J}\left(\nabla d_{J}\theta\right)_{u} \\
\hphantom{\tau_{1}\left(\nabla P\theta\right)_{u}}{}
=\left(d_{J}^{2}\theta\right)_{u}=\frac{1}{2}\left(d_{[J,J]}\theta\right)_{u}=-\left(d_{J\wedge dt}\theta\right)_{u}=-\left(d_{J}\theta\right)_{u}\wedge dt=0,
\\
\tau_{2}\left(\nabla P\theta\right)_{u} = \tau_{2}\left(\nabla d_{J}\theta, \nabla d_{h}\theta\right)_{u}=\tau_{h}\left(\nabla d_{h}\theta\right)_{u}
= \left(d_{h}^{2}\theta\right)_{u}=\frac{1}{2}\left(d_{[h,h]}\theta\right)_{u}=\left(d_{R}\theta\right)_{u}
\end{gather*}
where $R$ is given by (\ref{eq:19}),
\begin{gather*}
\tau_{3}\left(\nabla P\theta\right)_{u} = \tau_{3}\left(\nabla d_{J}\theta, \nabla d_{h}\theta\right)_{u}=\tau_{h}\left(\nabla d_{J}\theta\right)_{u}+\tau_{J}\left(\nabla d_{h}\theta\right)_{u}\nonumber \\
\hphantom{\tau_{3}\left(\nabla P\theta\right)_{u}}{}
= \left(d_{h}d_{J}\theta\right)_{u}+\left(d_{J}d_{h}\theta\right)_{u}=\left(d_{[h,J]}\theta\right)_{u}=0
\end{gather*}
Hence $\varphi=0$ if and only if $d_{R}\theta=0$.
\end{proof}
\begin{remark}
Locally, $d_{R}\theta$ has the following form:
\begin{gather*}
R=\underbrace{\frac{1}{2}R_{ij}^{k}\frac{\partial}{\partial y^{k}}\otimes\delta x^{i}\wedge\delta x^{j}}_{\widetilde{R}=\frac{1}{3}[J,\Phi]}-\Phi\wedge dt \ \Rightarrow \ d_{R}\theta=d_{\widetilde{R}}\theta-d_{\Phi}\theta\wedge dt,
\\
d_{\Phi}\theta = R_{i}^{j}\left(\theta_{j}-\frac{\partial\theta_{0}}{\partial y^{j}}\right)dt\wedge\delta x^{i}+\frac{1}{2!}\left(\frac{\partial\theta_{j}}{\partial y^{k}}R_{i}^{k}-\frac{\partial\theta_{i}}{\partial y^{k}}R_{j}^{k}\right)\delta x^{j}\wedge\delta x^{i} \quad \Rightarrow\\
d_{R}\theta = \frac{1}{3!}\left(a_{il}R_{jk}^{l}+a_{jl}R_{ki}^{l}+a_{kl}R_{ij}^{l}\right)\delta x^{i}\wedge\delta x^{j}\wedge\delta x^{k}+\frac{1}{2!}\left(a_{jk}R_{i}^{k}-a_{ik}R_{j}^{k}\right)dt\wedge\delta x^{i}\wedge\delta x^{j},
\end{gather*}
where we denoted $a_{ij}=\frac{\partial\theta_{i}}{\partial y^{j}}$.
Hence $d_{R}\theta=0$ if and only if $a_{il}R_{jk}^{l}+a_{jl}R_{ki}^{l}+a_{kl}R_{ij}^{l}=0$
and $a_{jk}R_{i}^{k}-a_{ik}R_{j}^{k}=0$. The f\/irst identity represents
the algebraic Bianchi identity for the curvatures of the nonlinear
connection. The second identity is one of the classical Helmholtz
condition for the multiplier matrix~$a_{ij}$. These obstructions
appear also in~\cite{anderso92}.
It can be seen that for $n=2$ the formula of $d_{R}\theta$ becomes
\[
d_{R}\theta=\frac{1}{2!}\big(a_{jk}R_{i}^{k}-a_{ik}R_{j}^{k}\big)dt\wedge\delta x^{i}\wedge\delta x^{j}=-d_{\Phi}\theta\wedge dt.\]
Therefore, for $n=2$, the obstruction is
equivalent with $d_{\Phi}\theta\wedge dt=0$.
\end{remark}
\subsection{Classes of Lagrangian time-dependent SODE}\label{section3.3}
We present now some classes of semisprays for which the obstruction
in Theorem~\ref{thm:A-first-order} is auto\-ma\-ti\-cally satisf\/ied. Therefore
the PDO $P$ is formally integrable, and hence these semisprays will
be Lagrangians SODEs. These classes of semisprays are:
\begin{itemize}\itemsep=0pt
\item f\/lat semisprays, $R=0\Leftrightarrow\Phi=0$;
\item arbitrary semisprays on 2-dimensional manifolds;
\item isotropic semisprays, $\Phi=\lambda J$, for $\lambda$ a smooth function
on $J^{1}\pi$.
\end{itemize}
All these classes of semisprays were already studied in the articles
cited in the introduction.
In the f\/lat case, the obstruction is automatically satisf\/ied.
If $\dim M=1$ then for a semi-basic $1$-form $\theta$ on $J^{1}\pi$,
$d_{R}\theta$ is a semi-basic $3$-form on $J^{1}\pi$. Because $\dim\Lambda^{3}\left(T_{v}^{*}\right)=(n+1)n(n-1)/6$
and it is zero if $n=1$, $d_{R}\theta$ will necessarily vanish.
We consider now the last case, of isotropic semisprays.
\begin{proposition}
Any isotropic semispray is a Lagrangian second-order vector field.
\end{proposition}
\begin{proof}
Assume now that $S$ is an isotropic SODE and $\theta$ a semi-basic
1-form on $J^{1}\pi$ such that $\left(d_{J}\theta\right)_{u}=\left(d_{h}\theta\right)_{u}=0$,
for some $u\in J^{1}\pi$, \begin{gather*}
\left(d_{R}\theta\right)_{u} = d_{\alpha\wedge J}\theta=\alpha\wedge d_{J}\theta+(-1)^{2}d\alpha\wedge i_{J}\theta=0.
\end{gather*}
We used that $i_{J}\theta=0$, $\left(d_{J}\theta\right)_{u}=0$ and
the formula~\cite{grifone00}
\[
d_{\omega\wedge K}\pi=\omega\wedge d_{K}\pi+(-1)^{q+k}d\omega\wedge i_{K}\pi,
\]
for $\omega$ a $q$-form on $J^{1}\pi$ and $K$ a vector-valued
$k$-form on $J^{1}\pi$.
Since $\left(d_{R}\theta\right)_{u}$ vanishes, $S$ is a Lagrangian
semispray.
\end{proof}
Next we give some simple examples of Lagrangian semisprays, corresponding
to the above general classes.
We start with the semispray expressed by the SODE
\[
\frac{d^{2}x^{1}}{dt^{2}}+f\left(t,\frac{dx^{2}}{dt}\right) =0,\qquad
\frac{d^{2}x^{2}}{dt^{2}}+g(t) =0, \]
with $f$ an arbitrary smooth function depending only on $t$ and
$y^{2}=\frac{dx^{2}}{dt}$, and~$g$ an arbitrary smooth function
depending only on $t$. The only possible nonvanishing local component
of the Jacobi endomorphism is
\[
R_{2}^{1}=-S\big(N_{2}^{1}\big)=-\frac{1}{2}\frac{\partial^{2}f}{\partial t\partial y^{2}}+g(t)\frac{\partial^{2}f}{\partial(y^{2})^{2}}.
\]
Hence, if $\frac{\partial^{2}f}{\partial t\partial y^{2}}=2g(t)\frac{\partial^{2}f}{\partial(y^{2})^{2}}$
the semispray is f\/lat ($\Phi=0$). This example is a generalization
of the one given by Douglas \cite[(8.14)]{douglas41}.
If $\frac{\partial^{2}f}{\partial t\partial y^{2}}\neq2g(t)\frac{\partial^{2}f}{\partial(y^{2})^{2}}$,
then the semispray is isotropic \[
\Phi= \begin{pmatrix}
0 & -\dfrac{1}{2}\dfrac{\partial^{2}f}{\partial t\partial y^{2}}+g(t)\dfrac{\partial^{2}f}{\partial(y^{2})^{2}}\vspace{1mm}\\
0 & 0\end{pmatrix} =\left(-\frac{1}{2}\frac{\partial^{2}f}{\partial t\partial y^{2}}+g(t)\frac{\partial^{2}f}{\partial(y^{2})^{2}}\right)J.\]
Another example of isotropic (or f\/lat) semispray is the one given
by the SODE
\[
\frac{d^{2}x^{1}}{dt^{2}}+f\left(t,x^{2}\right) =0,\qquad
\frac{d^{2}x^{2}}{dt^{2}}+g(t) =0, \]
with $f$ an arbitrary smooth function depending only on~$t$ and~$x^{2}$, and~$g$ an arbitrary smooth function depending only on~$t$. All the local coef\/f\/icients of the associated nonlinear connection
are vanishing. Evidently \[
\Phi= \begin{pmatrix}
0 & \dfrac{\partial f}{\partial x^{2}}\vspace{1mm}\\
0 & 0\end{pmatrix} =\left(\frac{\partial f}{\partial x^{2}}\right)J.\]
This example was treated in \cite[(6.1)]{anderso92} and \cite[(15.4)]{douglas41}
for $f(x^{2})=-x^{2}$ and $g=0$. The f\/irst paper also presents all
the Lagrangians corresponding to the given SODE.
Consider also the semispray given by the SODE \[
\frac{d^{2}x^{1}}{dt^{2}}+2\frac{dx^{2}}{dt} =0,\qquad
\frac{d^{2}x^{2}}{dt^{2}}-\left(\frac{dx^{2}}{dt}\right)^{2} =0. \]
Evidently \[
\Phi= \begin{pmatrix}
0 & y^{2}\\
0 & 0\end{pmatrix},\]
hence the semispray is isotropic.
\subsection*{Acknowledgements}
The author express his thanks to Ioan Bucataru for the many interesting
discussions about the paper.
\pdfbookmark[1]{References}{ref}
|
{
"timestamp": "2012-09-07T02:01:57",
"yymm": "1203",
"arxiv_id": "1203.1716",
"language": "en",
"url": "https://arxiv.org/abs/1203.1716"
}
|
\section{ Introduction}
\label{sec-1}
\IAENGPARstart{T}{he} goal of this paper is to investigate a geometric flow of closed plane curves $\Gamma^t, t\ge0$, minimizing the anisoperimetric ratio. We will show that the normal velocity $\beta$ for such a geometric flow is a function of the anisotropic curvature, the total interfacial energy and enclosed area of an evolved curve,
\begin{equation}\label{geomrovnonloc}
\beta = \delta(\nu) k + {\cal F}_\Gamma,
\end{equation}
where $k$ is the curvature and $\delta(\nu)>0$ is a strictly positive coefficient depending on the tangent angle $\nu$ at a point $\bm{x}\in \Gamma^t$. Here ${\cal F}_\Gamma$ is a nonlocal part of the normal velocity depending on the entire shape of the curve $\Gamma=\Gamma^t$ and the term $\delta(\nu) k$ represents the anisotropic curvature. In typical situations, the nonlocal part is a function of the enclosed area $A$ and the interfacial energy $L_\sigma = \int_\Gamma \sigma \mathrm{d}s$, i.e. ${\cal F}_\Gamma = {\cal F}(A,L_\sigma)$.
As an example one can consider
\[
\beta = k - \frac{2\pi}{L},
\]
where $L \equiv L_1$ is the length of an evolved closed curve $\Gamma$. It is well known that such a flow represents the area preserving geometric evolution of closed embedded plane curves investigated by Gage \cite{Gage1986}. Among other geometric flows with nonlocal normal velocity we mention the curvature driven length preserving flow in which $\beta = k - \frac{1}{2\pi} \int_\Gamma k^2 \mathrm{d}s$
studied by Ma and Zhu \cite{Ma2012} and the inverse curvature driven flow preserving the length $\beta = -k^{-1} + \frac{L}{2\pi}$ studied by Pan and Yang \cite{Pan2008}.
The isoperimetric ratio gradient flow with $\beta = k - L/(2A)$ has been proposed and investigated by Jiang and Zhu \cite{Jiang2008} for convex curves and by the authors in \cite{SY2011} for general closed Jordan curves evolving in the plane.
Recently, a classical nonlocal curvature flow preserving the enclosed area was reinvestigated by Xiao \emph{et al.} in \cite{Xiao2013}. They proved uniform upper bound and lower bound on the curvature. Furthermore, Mao \emph{et al.} \cite{Mao2012} showed that such a nonlocal flow will decrease the perimeter of the evolving curve and make the curve more and more circular during the evolution process. Applying inequalities of Andrews and Green-Osher type, Lin and Tsai \cite{Lin2012} showed that the evolving curves will converge to a round circle, provided that the curvature is a-priori bounded. However, most of those fine results for area preserving flow still have to be extended to the case of a class of non-local flows minimizing the isoperimetric and/or anisoperimetric ratio.
The main goal of this paper is twofold. First we derive the normal velocity $\beta$ corresponding to the anisoperimetric ratio gradient flow. It turns out that $\beta = k_\sigma - L_\sigma/(2A)$ where $k_\sigma$ is the anisotropic curvature, i.e. $\beta$ has the form of (\ref{geomrovnonloc}). We derive and analyze several important properties of such a geometric flow. In contrast to the isoperimetric ratio gradient flow
(c.f. Jiang and Zhu \cite{Jiang2008}, \cite{SY2011}),
we show that the anisoperimetric ratio gradient flow may initially increase the total length and, conversely, decrease the enclosed area of evolved curves. In order to verify such striking phenomena, an accurate numerical discretization scheme for fine approximation of the geometric flow has to be proposed. This is the second principal goal of the paper. We derive a numerical scheme based on the method of flowing finite volumes with combination of asymptotically uniform tangential redistribution of grid points.
The idea of a uniform tangential redistribution has been proposed by How \emph{et al} in \cite{HouLS1994} and further analyzed by Mikula and \v{S}ev\v{c}ovi\v{c} in \cite{MikulaS2001}. The asymptotically uniform tangential redistribution has been analyzed in \cite{MikulaS2004b,MikulaS2004a}.
The scheme is tested on the area-decrease and length-increase phenomena as well as on various other examples of evolution of initial curves having large variations in the curvature.
The paper is organized as follows. In the next section we recall the system of governing PDEs describing the evolution of all relevant geometric quantities. In section 3 we recall basic properties the anisotropic curvature and Wulff shape. We prove an important duality identity between total interfacial energies corresponding to different anisotropies. In section 4 we investigate a gradient flow for the anisoperimetric ratio. It turns out that the flow of plane minimizing the anisoperimetric ratio has the normal velocity locally depending on the anisotropic curvature and nonlocally depending on the total interfacial energy and the enclosed area of the evolved curve. Section 5 is devoted to the proof of a mixed anisoperimetric inequality for the product of two total interfacial energies corresponding to two anisotropy functions. In section 6 we investigate properties of the enclosed area for the anisoperimetric gradient flow. In contrast to a gradient flow for the isoperimetric ratio, we will show that there are initial convex curves for which the enclosed area is strictly decreasing. Finally, in section 7 we construct a counterexample to a comparison principle showing that there initial noninteresting curves such that they intersect each other immediately when evolved in the normal direction by the anisoperimetric ratio gradient flow. In section 8 we derive a numerical scheme for solving curvature driven flows with normal velocity depending on no-local terms. The scheme is based on a flowing finite volume method combined with a precise scheme for approximation of non-local terms. We present several numerical examples illustrating theoretical results and interesting phenomena for the gradient flow for anisoperimetric ratio.
\section{System of governing equations and curvature adjusted tangential redistribution}
\label{sec:GE}
In this section we recall description and basic properties of geometric evolution of a closed plane Jordan curve $\Gamma$ which can be parameterized by a smooth function $\bm{x}:\ [0, 1]\to\mathbb{R}^2$ such that $\Gamma=\mathop\mathrm{Image}(\bm{x})=\{\bm{x}(u);\ u\in[0, 1]\}$ and $|\partial_u\bm{x}|>0$. We identify the interval $[0,1]$ with the quotient space $\mathbb{R}/\mathbb{Z}$ by imposing periodic boundary conditions for $\bm{x}(u)$ at $u=0,1$. We denote $\partial_\xi{\sf F}=\partial{\sf F}/\partial\xi$, and
$|\bm{a}|=\sqrt{\bm{a} \cdot \bm{a}}$ where $\bm{a} \cdot \bm{b}$ is the Euclidean inner product between vectors $\bm{a}$ and $\bm{b}$. The unit tangent vector is given by $\bm{T}=\partial_u\bm{x}/|\partial_u\bm{x}|=\partial_s\bm{x}$, where $s$ is the arc-length parameter $\mathrm{d}s=|\partial_u\bm{x}|\mathrm{d}u$. The unit inward normal vector is defined in such a way that $\det(\bm{T}, \bm{N})=1$. Then the signed curvature $k$ in the direction $\bm{N}$ is given by $k=\det(\partial_s\bm{x}, \partial_s^2\bm{x})$. Let $\nu$ be a tangent angle, i.e., $\bm{T}=(\cos\nu, \sin\nu)^{\mathrm{T}}$ and $\bm{N}=(-\sin\nu, \cos\nu)^{\mathrm{T}}$. From the Fren\'et formulae $\partial_s \bm{T} = k \bm{N}$ and $\partial_s \bm{N} = -k \bm{T}$ we deduce that $\partial_s\nu =k$.
Geometric evolution problem can be formulated as follows: for a given initial curve $\Gamma^0=\mathop\mathrm{Image}(\bm{x}^0)=\Gamma$, find a family of curve $\{\Gamma^t\}_{t\geq 0}$, $\Gamma^t=\{\bm{x}(u, t);\ u\in[0, 1]\}$ starting from $\Gamma^0$ and evolving in the normal direction with the velocity $\beta$. In this paper we follow the so-called direct approach in which evolution of the position vector $\bm{x} = \bm{x}(u,t)$ is governed by the equation:
\begin{equation}
\partial_t\bm{x}=\beta\bm{N}+\alpha\bm{T}, \quad \bm{x}(\cdot, 0)=\bm{x}^0(\cdot).
\label{eq:direct}
\end{equation}
Here $\alpha$ is the tangential component of the velocity vector. Note that $\alpha$ has no effect on the shape of evolving closed curves,
and the shape is determined by the value of the normal velocity $\beta$ only.
Therefore, one can take take $\alpha\equiv0$ when analyzing analytical properties of the geometric flow driven by (\ref{eq:direct}). On the other hand, the impact of a suitable choice of a tangential velocity $\alpha$ on construction of robust and stable numerical schemes has been pointed out by many authors (see \cite{SY2008,SY2011} and references therein).
In what follows, we shall assume that $\beta=\delta(\nu) k + {\cal F}_\Gamma$ where $\delta(\nu)>0$ is a strictly positive $2\pi$-periodic smooth function of the tangent angle $\nu$ and ${\cal F}_\Gamma$ is a nonlocal part of the normal velocity depending on the entire shape of the curve $\Gamma$. According to \cite{MikulaS2004a} (see also \cite{MikulaS2001, MikulaS2004b}) the system of PDEs governing evolution of plane curves evolving in the normal and tangential directions with velocities $\beta$ and $\alpha$ reads as follows:
\begin{align}
& \partial_t k=\partial_s^2 \beta+\alpha\partial_sk + k^2\beta,
\label{eq:equation-k} \\
& \partial_t\nu= \partial_s \beta + \alpha k,
\label{eq:equation-nu} \\
& \partial_t g=\left(-k\beta+\partial_s\alpha\right)g,
\label{eq:equation-g} \\
& \partial_t\bm{x}=\delta(\nu) \partial_s^2\bm{x} + \alpha\partial_s\bm{x} + {\cal F}_\Gamma \bm{N},
\label{eq:equation-x}
\end{align}
for $u\in[0, 1]$ and $t>0$. Here $g=|\partial_u\bm{x}|$ is the so-called local length (c.f. \cite{MikulaS2001}). A solution to (\ref{eq:equation-k})--(\ref{eq:equation-x}) is subject to periodic boundary conditions for $g, k, \bm{x}$ at $u=0,1$, $\nu(0,t)\equiv\nu(1,t)$ mod($2\pi$) and the initial condition
$k(\cdot,0)=k_0(\cdot), \nu(\cdot,0)=\nu_0(\cdot), g(\cdot,0)=g_0(\cdot), \bm{x}(\cdot,0)=\bm{x}_0(\cdot)$ corresponding to the initial curve $\Gamma^0=\mathop\mathrm{Image}(\bm{x}^0)$.
Local existence and continuation of a classical smooth solution to system (\ref{eq:equation-k})--(\ref{eq:equation-x}) has been investigated by the authors in \cite{SY2008,SY2011}. In this paper we therefore take for granted that classical solutions to (\ref{eq:equation-k})--(\ref{eq:equation-x}) exists on some maximal time interval $[0,T_{max})$ (c.f. \cite{SY2011,MikulaS2004b}).
\section{The Wulff shape and interfacial energy functional}
The anisotropic curvature driven flow of embedded closed plane curves is associated with the so-called interfacial energy density (anisotropy) function $\sigma$ defined on $\Gamma$. It is assumed that $\sigma=\sigma(\nu)$ is a strictly positive function depending on the tangent angle $\nu$ only. With this notation we can introduce the total interfacial energy
\[
L_{\sigma}(\Gamma)=\int_{\Gamma}\sigma(\nu)\,\mathrm{d}s
\]
associated with a given anisotropy density function $\sigma$. If $\sigma\equiv1$ then $L_1(\Gamma)$ is just the total length $L(\Gamma)$ of a curve $\Gamma$. The Wulff shape is defined as an intersection of hyperplanes:
\[
W_{\sigma}=\bigcap_{\nu\in S^1}\biggl\{\bm{x}=(x_1,x_2)^{\mathrm{T}};\ -\bm{x} \cdot \bm{N} \leq \sigma(\nu)\biggr\}.
\]
If the boundary $\partial W_\sigma$ of the Wulff shape is smooth and it is parameterized by
$\partial W_\sigma =\{ \bm{x}=-\sigma(\nu)\bm{N}+a(\nu)\bm{T}, \nu\in[0,2\pi] \}$, then, it follows from the relation $\partial_s\nu=k$ that
\[
\bm{T}
=\partial_s\bm{x}
=(-\sigma'(\nu)+a(\nu))k\bm{N}+(\sigma(\nu)+a'(\nu))k\bm{T}.
\]
Hence $a(\nu)=\sigma'(\nu)$ and $(\sigma(\nu)+\sigma''(\nu))k=1$ holds and the boundary $\partial W_\sigma$ can be parameterized as follows:
\[
\partial W_{\sigma}=\left\{\bm{x};\ \bm{x}=-\sigma(\nu)\bm{N}+\sigma'(\nu)\bm{T}, \ \nu\in [0,2\pi] \right\},
\]
and its curvature is given by $k=(\sigma(\nu)+\sigma''(\nu))^{-1}$.
Let us denote by $k_\sigma$ the anisotropic curvature defined by $k_\sigma:= (\sigma(\nu)+\sigma''(\nu)) k$. It means that the anisotropic curvature $k_\sigma$ of the boundary $\partial W_\sigma$ of the Wulff shape $W_\sigma$ is constant, $k_\sigma \equiv 1$.
Moreover, the area $|W_{\sigma}| = A(\partial W_\sigma)$ of the Wulff shape satisfies:
\begin{eqnarray*}
|W_{\sigma}|
&=&-\frac{1}{2}\int_{\partial W_{\sigma}}\bm{x} \cdot \bm{N}\,\mathrm{d}s
=\frac{1}{2}\int_{\partial W_{\sigma}}\sigma(\nu)\,\mathrm{d}s
\\
&=&\frac{1}{2}L_{\sigma}(\partial W_{\sigma}).
\end{eqnarray*}
Clearly, $|W_{1}|=\pi$ for the case $\sigma\equiv1$. If we consider the anisotropy density function $\sigma(\nu) = 1 + \varepsilon \cos(m \nu)$ for $m=2, 3, \cdots, \ \varepsilon (m^2-1)<1$ then the area of $W_\sigma$ can be easily calculated:
\begin{eqnarray}
|W_\sigma|
&=&\frac{1}{2}\int_{\partial W_{\sigma}}\sigma(\nu)\,\mathrm{d}s
=\frac{1}{2}\int_0^{2\pi}\sigma(\sigma''+\sigma)\mathrm{d}\nu
\nonumber \\
&=&\frac{\pi}{2}(2-\varepsilon^2(m^2-1)).
\label{eq:areaW}
\end{eqnarray}
In Fig~\ref{fig:Wulff} we plot shapes of $\partial W_\sigma$ for various degrees $m$.
\begin{figure}[ht]
\begin{center}
\begin{tabular}{@{}ccc@{}}
\scalebox{0.45}{\includegraphics{figures/Wulff-sigma-peak2.eps}} &
\scalebox{0.45}{\includegraphics{figures/Wulff-sigma-peak3.eps}} &
\scalebox{0.45}{\includegraphics{figures/Wulff-sigma-peak4.eps}} \\
$m=2$ & $m=3$ & $m=4$
\end{tabular}
\begin{tabular}{@{}cc@{}}
\scalebox{0.45}{\includegraphics{figures/Wulff-sigma-peak5.eps}} &
\scalebox{0.45}{\includegraphics{figures/Wulff-sigma-peak6.eps}}\\
$m=5$ & $m=6$
\end{tabular}
\end{center}
\caption{%
The Wulff shapes $W_\sigma$ for $m=2, \cdots, 6$ and $\varepsilon=0.99/(m^2-1)$.
}
\label{fig:Wulff}
\end{figure}
Since the global quantities evaluated over the closed curve $\Gamma$ do not depend on the tangential velocity $\alpha$
we may take $\alpha\equiv0$. Hence $\partial_t g = -k\beta g$ and $\partial_t \nu = \partial_s\beta$.
These identities follow from (\ref{eq:equation-nu}) and (\ref{eq:equation-g}) with $\alpha\equiv0$.
Recall that $\partial_s \nu = k$.
Therefore $\partial_s \sigma'(\nu) = \sigma''(\nu)\partial_s\nu = \sigma''(\nu)k$ and so $\int_\Gamma \sigma''(\nu)k \mathrm{d}s = 0$.
Hence
\[
\int_\Gamma k_\sigma \mathrm{d}s = \int_\Gamma \sigma k \mathrm{d}s
\]
holds. For the time derivative of $\int_\Gamma k_\sigma \mathrm{d}s$ we obtain
\begin{align*}
\frac{\mathrm{d}}{\mathrm{d}t} \int_\Gamma k_\sigma \mathrm{d}s
&= \frac{\mathrm{d}}{\mathrm{d}t} \int_0^1 \sigma k g \mathrm{d}u
= \int_0^1[ \partial_t(\sigma k) g + \sigma k \partial_t g] \mathrm{d}u
\\
&= \int_\Gamma[ \partial_t(\sigma k) - \sigma k^2 \beta] \mathrm{d}s
\\
&= \int_\Gamma[ k \partial_t\sigma(\nu) + \sigma(\nu)\partial_t k - \sigma(\nu) k^2 \beta] \mathrm{d}s
\\
&= \int_\Gamma[ k \sigma'(\nu) \partial_t\nu + \sigma(\nu)(\partial_t k - k^2 \beta)] \mathrm{d}s
\\
&= \int_\Gamma[ k \sigma'(\nu) \partial_s\beta + \sigma(\nu) \partial_s^2 \beta] \mathrm{d}s= 0,
\end{align*}
because $\partial_t k - k^2 \beta = \partial_s^2 \beta$ and
$k \sigma'(\nu) = \sigma'(\nu)\partial_s\nu = \partial_s\sigma(\nu)$.
From the previous equality we can deduce the following identity:
\begin{equation}
\int_{\Gamma^t} k_\sigma \mathrm{d}s = \int_{\Gamma^0} k_\sigma \mathrm{d}s, \quad \hbox{for any }\ 0\le t < T_{max},
\label{conservation}
\end{equation}
where the family of planar embedded closed curves $\Gamma^t, t\in [0,T_{max})$, evolves in the normal direction with the velocity $\beta$.
Now, let us consider an evolving family of plane embedded closed curves $\Gamma^t, t\in[0,T]$, homotopicaly connecting a given curve $\Gamma=\Gamma^0$ and the boundary $\Gamma^T=\partial W_\sigma$ of the Wulff shape $W_\sigma$. The homotopy can be realized by taking a suitable normal velocity $\beta$ (eventually depending on the position vector $\bm{x}$). Using such a normal velocity we deduce the identity:
\begin{equation}
\int_\Gamma k_\sigma \mathrm{d}s = \int_{\partial W_\sigma} k_\sigma \mathrm{d}s
= L(\partial W_\sigma).
\label{identity}
\end{equation}
It means that $\int_\Gamma k_\sigma \mathrm{d}s$ is equal to the length of the boundary $\partial W_\sigma$ of the Wulff shape. The same result has been recently obtained by Barrett \emph{et al.} in \cite[Lemma 2.1]{BGN}.
We can say that identity (\ref{identity}) is a generalization of the rotation number:
$\frac{1}{2\pi}\int_\Gamma k\mathrm{d}s=1$, since $2\pi=L(\partial W_1)$.
\begin{remark}
Identity (\ref{identity}) can be easily shown for convex curves. Indeed, if $\Gamma$ is convex then its arc-length parameterization $s$ can be reparameterized by the tangent angle $\nu\in[0,2\pi]$.
We have $\partial_s \nu = k >0$ and therefore $\mathrm{d}s = k^{-1} \mathrm{d}\nu$. Hence
\[
\int_\Gamma k_\sigma \mathrm{d}s
= \int_\Gamma \sigma k \mathrm{d}s
= \int_0^{2\pi} \sigma(\nu) \mathrm{d}\nu.
\]
For the length $L(\partial W_\sigma)$ of the boundary of a convex Wulff shape we obtain
\begin{eqnarray*}
L(\partial W_\sigma)
&=& \int_{\partial W_\sigma} \mathrm{d}s
= \int_0^{2\pi} \frac{1}{k} \mathrm{d}\nu
\\
&=& \int_0^{2\pi} [\sigma(\nu) + \sigma''(\nu)] \mathrm{d}\nu
= \int_0^{2\pi} \sigma(\nu) \mathrm{d}\nu.
\end{eqnarray*}
Therefore $\int_\Gamma k_\sigma \mathrm{d}s = L(\partial W_\sigma)$ because $\int_0^{2\pi} \sigma''(\nu) \mathrm{d}\nu = 0$ and $k=[\sigma(\nu) + \sigma''(\nu)]^{-1}$ on $\partial W_\sigma$. If $\Gamma$ is not convex we can apply the famous Grayson's theorem \cite{Gr}. We let it evolve according to the normal velocity $\beta=k$ until a time $t=T$ when $\Gamma^T$ becomes convex. Using (\ref{conservation}) and previous argument we again obtain identity (\ref{identity}).
\end{remark}
Let us denote by $L_1$ the total interfacial energy corresponding to $\sigma\equiv 1$, i.e. $L_1\equiv L$. Let $\Gamma=\partial W_1$ be the unit circle.
Then, by applying identity (\ref{identity}), we deduce
\begin{equation}
L_1(\partial W_\sigma) = L_\sigma(\partial W_1).
\label{L1Lsigma}
\end{equation}
Latter identity can be rephrased as follows: the length of the boundary $\partial W_\sigma$ of the Wulff shape equals to the total interfacial energy of the unit circle.
It can be easily generalized to the case of arbitrary two anisotropies $\sigma(\nu)$ and $\mu(\nu)$. We have the following proposition:
\begin{theorem}\label{prop:duality}
Let $\sigma$ and $\mu$ be two smooth anisotropy functions satisfying $\sigma(\nu)+\sigma''(\nu)>0, \mu(\nu)+\mu''(\nu)>0$. Then the duality
\begin{equation}
L_\mu(\partial W_\sigma) = L_\sigma(\partial W_\mu)
\label{LmuLsigma}
\end{equation}
between total interfacial energies of boundaries $\partial W_\sigma$ and $\partial W_\mu$ of Wulff shapes holds.
\end{theorem}
\noindent{P r o o f.}
Notice that the Wulff shapes $W_\sigma$ and $W_\mu$ are convex sets because
$\sigma(\nu)+\sigma''(\nu)>0$ and $\mu(\nu)+\mu''(\nu)>0$ hold.
For the curvature $k$ at the boundary $\partial W_\sigma$ we have $k =[\sigma(\nu)+\sigma''(\nu)]^{-1}$ and so
\begin{align}
\label{LmuLsigmaProof}
L_\mu(\partial W_\sigma)
&= \int_{\partial W_\sigma} \mu(\nu) \mathrm{d}s
\\
&= \int_0^{2\pi} \mu(\nu) \frac{1}{k} \mathrm{d}\nu
= \int_0^{2\pi} \mu(\nu) (\sigma(\nu)+\sigma''(\nu)) \mathrm{d}\nu \nonumber
\\
&= \int_0^{2\pi}[ \mu(\nu) \sigma(\nu)- \sigma'(\nu) \mu'(\nu)] \mathrm{d}\nu
= L_\sigma(\partial W_\mu),
\end{align}
arguing vice versa. \hfill $\diamondsuit$
\section{Gradient flow for the anisoperimetric ratio.}
\label{sec:nonlocal}
Recall that for the enclosed area $A=A(\Gamma^t)$ and the total length $L=L(\Gamma^t)$ for a flow of embedded closed plane curves driven in normal direction by the velocity $\beta$ we have
\begin{equation}
\frac{\mathrm{d}}{\mathrm{d}t} A + \int_{\Gamma^t} \beta \mathrm{d}s = 0,\quad
\frac{\mathrm{d}}{\mathrm{d}t} L + \int_{\Gamma^t} k \beta \mathrm{d}s = 0,
\label{eq:area}
\end{equation}
(c.f. \cite{MikulaS2001}). Using governing equations (\ref{eq:equation-k})--(\ref{eq:equation-x}), for the total interfacial energy $L_\sigma = L_\sigma(\Gamma^t)$ of a curve $\Gamma^t$, we obtain
\begin{align}
\label{eq:lengthaniso}
\frac{\mathrm{d}}{\mathrm{d}t} L_{\sigma}
&= \frac{\mathrm{d}}{\mathrm{d}t} \int_{\Gamma}\sigma(\nu)\mathrm{d}s
= \frac{\mathrm{d}}{\mathrm{d}t} \int_0^1\sigma(\nu) g \mathrm{d}u
\\
&= \int_0^1 [ \sigma'(\nu)\partial_t\nu g + \sigma(\nu) \partial_tg ] \mathrm{d}u
\nonumber
\\
&= \int_{\Gamma} [ \sigma'(\nu)\partial_s \beta - \sigma(\nu) k\beta ] \mathrm{d}s
\\
&= -\int_{\Gamma} [ \sigma''(\nu)\partial_s \nu \beta + \sigma(\nu) k\beta ] \mathrm{d}s
\\
&= - \int_{\Gamma} [\sigma''(\nu) + \sigma(\nu) ] k\beta \mathrm{d}s
= - \int_{\Gamma} k_\sigma \beta \mathrm{d}s.
\nonumber
\end{align}
Here we have used the governing equations (\ref{eq:equation-g}) and (\ref{eq:equation-nu}) (with $\alpha\equiv 0$) and the identity $\partial_s\nu = k$. For the anisoperimetric ratio
\[
\Pi_{\sigma}(\Gamma)=\frac{L_{\sigma}(\Gamma)^2}{4|W_{\sigma}|A(\Gamma)},
\]
we have $\Pi_{\sigma}(\Gamma) \ge 1$ and, in particular, $\Pi_{\sigma}(\partial W_{\sigma})=1$ (see Remark~\ref{remaniso}).
Taking into account identities (\ref{eq:lengthaniso}) and (\ref{eq:area}) we obtain
\begin{eqnarray*}
\frac{\mathrm{d}}{\mathrm{d}t}\Pi_\sigma
&=& \frac{L_\sigma \partial_t L_\sigma }{2|W_{\sigma}| A} - \frac{L_\sigma^2 \partial_t A}{4 |W_{\sigma}| A^2}
\\
&=& - \frac{L_\sigma}{2 |W_{\sigma}| A} \int_{\Gamma}\left(k_\sigma - \frac{L_\sigma }{2 A}\right) \beta \mathrm{d}s.
\end{eqnarray*}
Hence, the flow driven in the normal direction by the non-locally dependent velocity
\begin{equation}
\beta = k_\sigma - \frac{L_\sigma}{2A}
\label{eq:isoperimnormalaniso}
\end{equation}
represents a gradient flow for the anisoperimetric ratio $\Pi_\sigma$ with the property $\partial_t \Pi_\sigma <0$ for $\beta\not\equiv0$. Notice that $\beta\equiv0$ on $\Gamma$ if and only if $\Gamma \propto \partial W_\sigma$, i.e. $\Gamma$ is homotheticaly similar to $\partial W_\sigma$.
In the case $\sigma\equiv 1$ the isoperimetric ratio gradient flow has been analyzed by Jiang and Zhu in \cite{Jiang2008} and by the authors in \cite{SY2008}. In this case the normal velocity has the form:
$\beta = k - L/(2 A)$.
\section{A mixed anisoperimetric inequality}
The aim of this section is to prove a mixed anisoperimetric inequality of the form
\begin{equation}
\frac{L_\sigma(\Gamma) L_\mu(\Gamma)}{A(\Gamma)} \ge K_{\sigma,\mu},
\label{mixedanisoperim}
\end{equation}
which holds for any $C^2$ smooth Jordan curve $\Gamma$ in the plane.
Here $K_{\sigma,\mu}>0$ is a constant depending only on the anisotropy functions $\sigma$ and $\mu$ such that
$\sigma(\nu)+\sigma''(\nu)>0$ and $\mu(\nu)+\mu''(\nu)>0$ hold for any $\nu$. The existence of a minimizer of the mixed anisoperimetric ratio is discussed in Remark~\ref{minimizer}. The idea of the proof of the inequality (\ref{mixedanisoperim}) is rather simple and consists in solving the constrained minimization problem:
\begin{equation}
\min_{\Gamma} L_\sigma(\Gamma), \qquad \hbox{s.t.}\ \ L_\mu(\Gamma) = c A(\Gamma),
\label{minimproblem}
\end{equation}
where $c>0$ is a given constant. To this end, let us assume that a curve $\Gamma=\Gamma(\bm{x})$ is parameterized by a $C^2$ smooth function $\bm{x}: S^1\to \mathbb{R}^2$. If we denote $g\equiv g(\bm{x}) = |\partial_u \bm{x}|$ the local length then, for the derivative of $g$ in the direction $\bm{y}:S^1\to\mathbb{R}^2$,
we obtain $2 g(\bm{x}) g'(\bm{x}) \bm{y} = 2 (\partial_u \bm{x} \cdot \partial_u \bm{y})$ and so
\begin{equation}
g'(\bm{x}) \bm{y} = (\bm{T} \cdot \partial_s \bm{y}) g.
\label{derg}
\end{equation}
Here and here after, for scalar-valued function $f(\bm{x})$ and vector-valued function $\bm{f}(\bm{x})=(f_1(\bm{x}), f_2(\bm{x}))^{\mathrm{T}}$
we denote their derivatives in the direction $\bm{y}$ by
\[
f'(\bm{x})\bm{y}:=\nabla f(\bm{x})\cdot\bm{y}=\lim_{\varepsilon\to 0}\frac{f(\bm{x}+\varepsilon\bm{y})-f(\bm{x})}{\varepsilon},
\]
\[
\bm{f}'(\bm{x})\bm{y}:=\begin{pmatrix}f_1'(\bm{x})\bm{y}\\ f_2'(\bm{x})\bm{y}\end{pmatrix},
\]
respectively.
As for the tangent vector $\bm{T}=\bm{T}(\bm{x})= (\cos\nu, \sin\nu)^{\mathrm{T}}$ we have $\bm{T}(\bm{x}) = g^{-1}\partial_u\bm{x}$ and so
$\bm{T}'(\bm{x}) \bm{y}
= g^{-1} \partial_u \bm{y} - g^{-2} \partial_u\bm{x} \, g'(\bm{x}) \bm{y}
= \partial_s\bm{y} -(\bm{T} \cdot \partial_s \bm{y}) \bm{T}
= (\bm{N} \cdot \partial_s \bm{y}) \bm{N}$.
As $\bm{N}= (-\sin\nu, \cos\nu)^{\mathrm{T}}$, for the derivative of the tangent angle $\nu=\nu(\bm{x})$, we obtain
\begin{equation}
\nu'(\bm{x}) \bm{y} = \bm{N} \cdot \partial_s \bm{y}.
\label{dernu}
\end{equation}
Recall that $k_\sigma :=(\sigma(\nu)+ \sigma''(\nu)) k$ and $\partial_s\nu = k$. Since $L_\sigma(\Gamma)= \int_\Gamma \sigma \mathrm{d}s = \int_0^1 \sigma(\nu) g \mathrm{d}u$
we obtain
\begin{align*}
L_\sigma^\prime (\Gamma(\bm{x})) \bm{y}
& = \int_0^1
\left[
\sigma'(\nu) \nu'(\bm{x}) \bm{y} g + \sigma (\nu) g'(\bm{x}) \bm{y}
\right] \mathrm{d}u
\\
& = \int_\Gamma
\left[
\sigma'(\nu) (\bm{N} \cdot \partial_s\bm{y}) + \sigma(\nu) (\bm{T} \cdot \partial_s\bm{y})
\right]\mathrm{d}s
\\
& = - \int_\Gamma
\left[
\sigma''(\nu)\partial_s\nu (\bm{N} \cdot \bm{y})
- \sigma'(\nu) k (\bm{T} \cdot \bm{y})\right.
\\
& \left. \hskip 1truecm + \sigma'(\nu) \partial_s\nu (\bm{T} \cdot \bm{y})
+ \sigma(\nu) k (\bm{N} \cdot \bm{y})
\right]\mathrm{d}s
\\
&
= - \int_\Gamma
\left[
\sigma(\nu)+ \sigma''(\nu) \right] k (\bm{N} \cdot \bm{y}) \mathrm{d}s
\\
&= - \int_\Gamma k_\sigma (\bm{N} \cdot \bm{y}) \mathrm{d}s .
\end{align*}
Hence
\begin{eqnarray}
L_\sigma^\prime(\Gamma(\bm{x})) \bm{y}
&=& - \int_\Gamma k_\sigma (\bm{N} \cdot \bm{y}) \mathrm{d}s, \quad
\nonumber \\
L_\mu^\prime(\Gamma(\bm{x})) \bm{y} &=& - \int_\Gamma k_\mu (\bm{N} \cdot \bm{y}) \mathrm{d}s.
\label{derL}
\end{eqnarray}
For the area $A=A(\Gamma)$ enclosed by a Jordan curve $\Gamma=\Gamma(\bm{x})$ we have
$A = \frac12 \int_0^1 \det(\bm{x}, \partial_u\bm{x} ) \mathrm{d}u$. Therefore
\begin{align*}
A'(\Gamma(\bm{x}))\bm{y} &= \frac12 \int_0^1 \det( \bm{y}, \partial_u\bm{x} ) + \det(\bm{x}, \partial_u\bm{y} ) \mathrm{d}u
\\
&= \int_0^1 \det(\bm{y}, \partial_u\bm{x} ) \mathrm{d}u= \int_\Gamma \det(\bm{y}, \bm{T})\mathrm{d}s.
\end{align*}
Since $\det(\bm{y}, \bm{T}) = - \bm{y} \cdot \bm{N}$ we obtain
\begin{equation}
A'(\Gamma(\bm{x})) \bm{y} = - \int_\Gamma \bm{N} \cdot \bm{y} \mathrm{d}s.
\label{derA}
\end{equation}
In order to solve the constrained minimization problem (\ref{minimproblem}) we introduce the Lagrange function
${\mathcal L}(\bm{x},\lambda) = L_\sigma (\Gamma(\bm{x})) + \lambda (L_\mu (\Gamma(\bm{x})) - c A(\Gamma(\bm{x})))$ with $\lambda>0$.
Then the first order condition for $\bar\Gamma =\Gamma(\bar\bm{x})$ to be a minimizer of (\ref{minimproblem}) reads as follows:
$ 0= {\mathcal L}^\prime_{\bm{x}}(\bm{x},\lambda) \bm{y} \equiv L_\sigma^\prime(\Gamma(\bm{x})) \bm{y} + \lambda (L_\mu^\prime(\Gamma(\bm{x})) \bm{y} -c A'(\Gamma(\bm{x})) \bm{y})$ at $\bm{x}=\bar\bm{x}$. Latter equality has to be satisfied for any smooth function $\bm{y}:S^1\to \mathbb{R}^2$. Taking into account (\ref{derL}) and (\ref{derA}) we obtain
\[
k_\sigma + \lambda k_\mu = \lambda c, \quad\hbox{on}\ \ \bar\Gamma.
\]
It means that
\begin{equation}
k_{\bar\sigma} = \lambda c, \quad\hbox{on}\ \ \bar\Gamma, \quad \hbox{where}\ \
\bar\sigma = \sigma + \lambda\mu.
\label{neccondk}
\end{equation}
In other words, $\bar\Gamma = \frac{1}{\lambda c} \partial W_{\bar\sigma}$ (up to an affine translation in the plane $\mathbb{R}^2$).
The Lagrange multiplier $\lambda\in\mathbb{R}$ can be computed from the constraint $L_\mu(\bar\Gamma) = c A(\bar\Gamma)$.
It follows from duality (\ref{LmuLsigma}) (see Proposition~\ref{prop:duality}) that
\begin{eqnarray*}
L_\mu(\partial W_{\bar\sigma})
&=& L_{\bar\sigma}(\partial W_\mu)
= L_\sigma (\partial W_\mu) + \lambda L_\mu (\partial W_\mu)
\\
&=& L_\sigma (\partial W_\mu) + 2 \lambda A(\partial W_\mu).
\end{eqnarray*}
To calculate the enclosed area $A(\bar\Gamma) = \frac{1}{\lambda^2 c^2} A(\partial W_{\bar\sigma})$ we make use of the identity $A(\partial W_{\bar\sigma}) = \frac12 L_{\bar\sigma}(\partial W_{\bar\sigma})$. Clearly, as $\bar\sigma=\sigma+\lambda\mu$ we obtain
\begin{align*}
L_{\bar\sigma}(\partial W_{\bar\sigma})
& = L_\sigma(\partial W_{\bar\sigma}) + \lambda L_\mu(\partial W_{\bar\sigma})
\\
& = L_{\bar\sigma}(\partial W_\sigma) + \lambda L_{\bar\sigma}(\partial W_\mu)
\\
&
= L_\sigma(\partial W_\sigma) + \lambda L_\mu(\partial W_\sigma)
+ \lambda L_\sigma(\partial W_\mu)
\\
& + \lambda^2 L_\mu(\partial W_\mu)
\\
& = 2 A(\partial W_\sigma) + 2 \lambda L_\sigma(\partial W_\mu) + 2 \lambda^2 A(\partial W_\mu).
\end{align*}
Since $\frac{1}{\lambda c} L_\mu(\partial W_{\bar\sigma}) = L_\mu(\bar\Gamma) = c A(\bar\Gamma) = \frac{c}{\lambda^2 c^2} A(\partial W_{\bar\sigma})$ we end up with the identity
\begin{eqnarray*}
&&\frac{1}{\lambda c}
\left(
L_\sigma(\partial W_\mu) + 2\lambda A(\partial W_\mu)
\right)
\\
&=&
\frac{c}{\lambda^2 c^2}
\left(
A(\partial W_\sigma) + \lambda L_\sigma(\partial W_\mu) + \lambda^2 A(\partial W_\mu)
\right).
\end{eqnarray*}
Since the Lagrange multiplier $\lambda>0$ it is given by $\lambda = \sqrt{A(\partial W_\sigma)/A(\partial W_\mu)}$. Furthermore,
\begin{eqnarray*}
L_\sigma(\partial W_{\bar\sigma})
&=& L_{\bar\sigma}(\partial W_\sigma)
= L_\sigma (\partial W_\sigma) + \lambda L_\mu (\partial W_\sigma)
\\
&=& 2 A(\partial W_\sigma) + \lambda L_\sigma (\partial W_\mu).
\end{eqnarray*}
Now, let $\Gamma$ be an arbitrary $C^2$ smooth Jordan curve in the plane. Set $c= L_\mu(\Gamma)/A(\Gamma)$. Then
\begin{align*}
\frac{L_\sigma(\Gamma) L_\mu(\Gamma)}{A(\Gamma)}
&= c L_\sigma(\Gamma) \ge c L_\sigma(\bar\Gamma) =\frac{c}{\lambda c} L_\sigma(\partial W_{\bar\sigma})
\\
&= 2 \sqrt{A(\partial W_\sigma) A(\partial W_\mu) } + L_\sigma (\partial W_\mu).
\end{align*}
\begin{remark}\label{minimizer}
The proof of existence of a minimizer of the mixed anisoperimetric ratio $L_\sigma(\Gamma) L_\mu(\Gamma)/A(\Gamma)$ is as follows: let $\Gamma^n=\Gamma(\bm{x}^n)$ be a sequence of Jordan curves minimizing this ratio. As $L_\sigma(\gamma\Gamma)=\gamma L_\sigma(\Gamma)$, and $A(\gamma\Gamma)=\gamma^2A(\Gamma)$ for each $\gamma>0$, without lost of generality, we may assume $L(\Gamma^n)=1$ for all $n\in\mathbb{N}$. We can also fix the barycenter of $\Gamma^n$ at the origin.
Since $c_0 L(\Gamma)\le L_\sigma(\Gamma)\le c_1 L(\Gamma)$ where $0<c_0=\min_\Gamma\sigma\le c_1=\max_\Gamma\sigma<\infty$, then, by the isoperimetric inequality, the value of the infimum is positive. Moreover, the parameterization $\bm{x}^n$ of $\Gamma^n$ can be chosen in such a way that $|\partial_u \bm{x}^n|=L(\Gamma^n)=1$. As a consequence, the position vectors $\{\bm{x}^n(u), u\in[0,1]\}$ are uniformly bounded. By the Arzel\`a-Ascoli theorem there is a convergent subsequence converging to some function $\{\bm{x}(u), u\in[0,1]\}$ which is the minimizer of the mixed anisoperimetric ratio.
\end{remark}
In summary, we have shown the following mixed anisoperimetric inequality:
\begin{theorem}\label{anisoineq}
Let $\Gamma$ be a $C^2$ smooth Jordan curve in the plane. Then
\begin{equation}
\frac{L_\sigma(\Gamma) L_\mu(\Gamma)}{A(\Gamma)} \ge K_{\sigma,\mu},
\label{mixedanisoperim2}
\end{equation}
where $K_{\sigma,\mu} = 2 \sqrt{|W_\sigma| |W_\mu| } + L_\sigma (\partial W_\mu)$.The equality in (\ref{mixedanisoperim2}) holds if and only if the curve $\Gamma$ is homothetically similar to the boundary $\partial W_{\widetilde{\sigma}}$ of a Wulff shape corresponding to the mixed anisotropy function $\widetilde{\sigma} = \sqrt{|W_\mu|}\, \sigma + \sqrt{|W_\sigma|} \,\mu$.
\end{theorem}
\medskip
\begin{remark} \label{remaniso}
If $\sigma=\mu\equiv1$ we obtain the well known isoperimetric inequality $L(\Gamma)^2/A(\Gamma) \ge K_{1,1} \equiv 2 \sqrt{\pi^2} + L(W_1) = 4\pi$.
If $\sigma=\mu$ we obtain the anisoperimetric inequality
$L_\sigma(\Gamma)^2/A(\Gamma) \ge K_{\sigma,\sigma} = 2 \sqrt{|W_\sigma|^2} + L_\sigma(\partial W_\sigma) = 4 |W_\sigma|$.
Finally, if $\mu\equiv 1$ we obtain the mixed anisoperimetric inequality
\[
\frac{L_\sigma(\Gamma) L(\Gamma)}{A(\Gamma)} \ge K_{\sigma,1} \equiv 2 \sqrt{\pi |W_\sigma|}
+ L(\partial W_\sigma).
\]
\end{remark}
\begin{remark} \label{remaniso2}
In the case $\mu=\sigma$, the anisoperimetric inequality in the plane has been stated in a paper by G. Wulff \cite{Wulff1901} from 1901. Later, it was proved by Dinghas in \cite{Dinghas1944} for a special class of polytopes. Recently, Fonseca and M\"uller \cite{Fonseca1991} proved the anisotropic inequality in the plane. Later Fusco \emph{et al.} \cite{Fusco2005} proved it in arbitrary dimension. Giga in \cite{Giga2003} pointed out that the anisotropic inequality where $\mu=\sigma$ are $\pi$-periodic function is the isoperimetric inequality in a suitable Minkowski metric. It is a useful tool in the proof of anisotropic version of the so-called Gage's inequality (c.f. \cite[Corollary 4.3]{Gage1993}).
However, in all aforementioned proofs, the surface energy was associated with a functional $L_\Phi(\Gamma)= \int_\Gamma \Phi(\bm{N}) \mathrm{d}s$ where $\Phi:\mathbb{R}^2 \to\mathbb{R}$ is an absolute homogeneous anisotropy function of degree one, i.e. $\Phi(t\bm{x}) = |t|\Phi(\bm{x})$ for any $t\in\mathbb{R}, \bm{x}\in\mathbb{R}^2$. The relation between our description of anisotropy and the latter one is:
$\sigma(\nu) = \Phi(-\sin\nu, \cos\nu)$ and, conversely, $\Phi(\bm{x})= \sigma(\nu)$ where $\bm{x}/\Vert\bm{x}\Vert = (-\sin\nu, \cos\nu)$. Since we do not require $\pi$-periodicity of $\sigma$, in our approach of description of anisotropy we therefore allow for non-symmetric anisotropies, like e.g. functions $\sigma$ with odd degree $m$ (see Fig \ref{fig:Wulff}) corresponding thus to anisotropy function $\Phi$ which are positive homogeneous only, i.e. $\Phi(t\bm{x}) = t\Phi(\bm{x})$ for any $t\ge 0, \bm{x}\in\mathbb{R}^2$.
In the case of general anisotropy functions $\mu\not\equiv\sigma$, the mixed anisoperimetric inequality derived in Theorem~\ref{anisoineq} is, to our best knowledge, new even in the case of symmetric ($\pi$-periodic) anisotropy functions.
\end{remark}
\section{Convexity preservation. Temporal area and length behavior}
In this section we analyze behavior of the enclosed area $A(\Gamma^t)$ of a curve $\Gamma^t$ evolved in the normal direction by the anisoperimetric ratio gradient flow, i.e. $\beta = k_\sigma - L_\sigma/(2 A)$.
First we prove the preservation of convexity result stating that the anisoperimetric ratio gradient flow preserves convexity of evolved curves.
In the case of the isoperimetric ratio gradient flow of convex curves with $\beta = k - L/(2 A)$, the convexity preservation has been shown by Jiang and Pan in \cite{Jiang2008}. However, similarly as Mu and Zhu in \cite{Ma2012}, they utilized the Gauss parameterization of the curvature equation (\ref{eq:equation-k}) by the tangent angle $\nu$ and this is why their results are applicable to evolution of convex curves only. In our paper we first prove convexity preservation based on the analysis of the curvature equation (\ref{eq:equation-k}) with arc-length parameterization. Moreover, we show the anisoperimetric ratio gradient flow may initially increase the total length and decrease the enclosed area. This phenomenon cannot be found in the isoperimetric ratio gradient flow (c.f. \cite{Jiang2008,SY2008}).
\begin{theorem}\label{th:convexity} Let $\Gamma^t, t\in[0,T_{max})$, be the anisoperimetric ratio gradient flow of smooth Jordan curves in the plane evolving in the normal direction by the velocity $\beta=k_\sigma - \frac{L_\sigma}{2 A}$. If the curve $\Gamma^{t_0}$ is convex at some time $t_0\in[0,T_{max})$ then $\Gamma^{t}$ remains convex for any $t\in [t_0,T_{max})$.
\end{theorem}
\noindent{P r o o f.}
Since $\partial_t k = \partial_s^2\beta + k^2 \beta$, $\partial_t\nu = \partial_s\beta = \partial_s k_\sigma$ and $\beta = k_\sigma +{\cal F}_\Gamma$ we have
\begin{align*}
\partial_t k_\sigma
&= \delta(\nu) \partial_t k + \delta^\prime(\nu) k \partial_t \nu
\\
&= \delta(\nu) \partial_s^2 k_\sigma + \delta(\nu) k^2 \beta + \delta^\prime(\nu) k \partial_s k_\sigma \\
&= \delta(\nu) \partial_s^2 k_\sigma + \frac{1}{\delta(\nu)} k_\sigma^2 \beta + \delta^\prime(\nu) k \partial_s k_\sigma,
\end{align*}
where $\delta(\nu) := \sigma(\nu) + \sigma^{\prime\prime}(\nu)>0$.
Let us denote by $K(t) = \min_{\Gamma^t} k_\sigma(.,t)$ the minimum of the anisotropic curvature $k_\sigma = \delta(\nu) k$ over the curve $\Gamma^t$. Denote by $s^*(t)\in [0, L^t]$ the argument of the minimum of $k_\sigma$, i.e. $K(t) = k_\sigma(s^*(t), t)$. Then $\partial_s k_\sigma (s^*(t), t)=0$ and
$\partial_s^2 k_\sigma (s^*(t), t)\ge 0$. Hence
\[
K'(t) \ge \frac{1}{\Delta(t)}K(t)^2 (K(t) + {\cal F}_\Gamma^t),
\]
where $\Delta(t) = \delta(\nu(s^*(t), t))$, and ${\cal F}_\Gamma^t=-L^t_\sigma/(2A^t)$.
Notice that $\Delta(t)\ge \Delta_{min}:=\min_{\nu} \delta(\nu)>0$ for all $t\in[0,T_{max})$. Suppose that $K$ is a solution to this ordinary differential inequality existing on some interval $[t_0, T_{max})$ and such that $K(t_0)>0$. Then, it should be obvious that $K(t)>0$ for $t\in[t_0, T_{max})$ provided that
\begin{equation}
\inf_{t_0\le t \le t^*}{\cal F}_\Gamma^t > -\infty,
\label{Fbound}
\end{equation}
for every $0<t^*<T_{max}$. In order to prove convexity preservation for the anisoperimetric ratio gradient flow it is therefore sufficient to verify that the nonlocal part
${\cal F}_\Gamma^t=-L^t_\sigma/(2A^t)$ remains bounded from below for $t\ge t_0$. To prove boundedness of ${\cal F}_\Gamma^t$ from below we utilize a property of the anisoperimetric ratio. Indeed, as $\beta = k_\sigma -L_\sigma/(2A)$ represents gradient flow for the anisoperimetric ratio
$\Pi^t_\sigma = (L_\sigma^t)^2/(4|W_\sigma| A^t)$ we have $1\le \Pi^t_\sigma \le \Pi^0_\sigma$ for all $t\in [0,T_{max})$. Thus
\[
{\cal F}_\Gamma^t=-\frac{L^t_\sigma}{2 A^t} \ge - \frac{(L^0_\sigma)^2}{2 A^0} \frac{1}{L^t_\sigma}.
\]
Now, since the classical solution exists on the time interval $[0,T_{max})$ then $\inf_{0\le t\le t^*} L^t_\sigma >0$ for each $0<t^*<T_{max}$ and the estimate (\ref{Fbound}) follows. \hfill $\diamondsuit$
\bigskip
In what follows, we shall investigate the enclosed area and length behavior of curves evolved by the normal velocity $\beta =k_\sigma - L_\sigma/(2A)$ representing thus a gradient flow for the anisoperimetric ratio.
Using the area equation (\ref{eq:area}) we obtain
\begin{eqnarray}
\frac{\mathrm{d}}{\mathrm{d}t} A
&=& - \int_{\Gamma} \beta \mathrm{d}s
= - \int_{\Gamma}\left(k_\sigma - \frac{L_\sigma}{2 A}\right)\mathrm{d}s
\nonumber
\\
&=& - L (\partial W_\sigma) + \frac{L L_\sigma}{2 A}.
\label{der1A}
\end{eqnarray}
By applying the isoperimetric inequality (see Remark~\ref{remaniso}) for the case $\sigma\equiv 1$ and any curve $\Gamma=\Gamma^t$, the following inequality:
\[
\frac{\mathrm{d}}{\mathrm{d}t} A = - 2\pi + \frac{L(\Gamma)^2}{2 A(\Gamma)} \ge 0
\]
holds. It means that the gradient flow for the isoperimetric ratio does not decrease the enclosed area. On the other hand, if anisotropy density function $\sigma\not\equiv const$, then for a curve $\Gamma=\Gamma^t \propto \partial W_{\bar\sigma}$ corresponding to the Wulff shape
$W_{\bar\sigma}$ with the anisotropy function $\bar\sigma = \sqrt{\pi}\, \sigma + \sqrt{|W_\sigma|}$ we obtain
\begin{eqnarray}
\frac{\mathrm{d}}{\mathrm{d}t} A
&=& - L (\partial W_\sigma) + \frac{L L_\sigma }{2 A}
= - L (\partial W_\sigma) + \frac{1}{2} K_{\sigma,1}
\nonumber
\\
&=& \sqrt{\pi |W_\sigma|}
- \frac12 L(\partial W_\sigma) < 0
\label{derAneg}
\end{eqnarray}
due to the isoperimetric inequality $L(\partial W_\sigma)^2 \ge 4 \pi A(\partial W_\sigma)=4 \pi |W_\sigma|$.
It means that the gradient flow for the anisoperimetric ratio may initially decrease the enclosed area for special initial curves.
Next we recall the isoperimetric inequality by Gage. According to \cite{Gage1983} the following inequality holds:
\begin{equation}
\int_\Gamma k^2 \mathrm{d}s
\ge
\pi \frac{L}{A}
\label{Gageaniso}
\end{equation}
for any convex $C^2$ smooth Jordan curve in the plane. The equality in (\ref{Gageaniso}) holds iff $\Gamma$ is a circle. Therefore, in the case of isoperimetric gradient flow with $\sigma\equiv 1$ and the convex curve $\Gamma^t$, we have
\begin{equation}
\frac{\mathrm{d}}{\mathrm{d}t} L^t= - \int k\beta \mathrm{d}s = - \int_\Gamma k^2 \mathrm{d}s + \pi\frac{L}{A} \le 0.
\label{length-decrease}
\end{equation}
However, if $\sigma\not\equiv const$ is a smooth nonconstant anisotropy density function such that $\sigma +\sigma^{\prime\prime}>0$, there exists an initial curve $\Gamma^0$ such that the length $L^t$ may initially increase, i.e. $\frac{\mathrm{d}}{\mathrm{d}t} L^t>0$ at $t=0$. Indeed, let $\Gamma^0$ be an initial curve which homothetically similar to the boundary $\partial W_{\bar\sigma}$ of the Wulff shape corresponding to the mixed anisotropy function $\bar\sigma = a \sigma + b$ where $a, b>0$ are constants. Then
\[
k_\sigma = k_{\frac{\bar\sigma -b}{a}} = \frac{1}{a} k_{\bar\sigma} - \frac{b}{a} k.
\]
Hence $k_\sigma = \frac{1}{a} - \frac{b}{a} k $ on the Wulff shape $\Gamma=\Gamma^0 = \partial W_{\bar\sigma}$ because $k_{\bar\sigma} \equiv 1$ on $\partial W_{\bar\sigma}$.
Using (\ref{eq:area}), we have
\begin{eqnarray*}
\frac{\mathrm{d}}{\mathrm{d}t} L^t
&=& - \int_\Gamma k k_\sigma \mathrm{d}s + \pi \frac{L_\sigma(\partial W_{\bar\sigma})}{A(\partial W_{\bar\sigma})}
\\
&=&
\frac{b}{a} \int_\Gamma k^2 \mathrm{d}s - \frac{2\pi}{a} + \pi\frac{L_\sigma(\partial W_{\bar\sigma} )}{A(\partial W_{\bar\sigma})}
\end{eqnarray*}
at $t=0$. Since $\sigma=(\bar\sigma -b)/a$ we have
\begin{align*}
L_\sigma(\partial W_{\bar\sigma}) &= \int_{\partial W_{\bar\sigma}} \frac{\bar\sigma -b}{a} \mathrm{d}s
\\
&= \frac{1}{a} L_{\bar\sigma}(\partial W_{\bar\sigma}) - \frac{b}{a} L(\partial W_{\bar\sigma})
\\
&= \frac{2}{a} A(\partial W_{\bar\sigma}) - \frac{b}{a} L(\partial W_{\bar\sigma}).
\end{align*}
Thus
\[
\frac{\mathrm{d}}{\mathrm{d}t} L^t= \frac{b}{a} \left(
\int_{\partial W_{\bar\sigma}} k^2 \mathrm{d}s - \pi\frac{L(\partial W_{\bar\sigma})}{A(\partial W_{\bar\sigma})} \right) >0 \quad\hbox{at}\ t=0,
\]
due to inequality (\ref{Gageaniso}) and the fact that $\partial W_{\bar\sigma}$ is a convex curve different from a circle for $\bar\sigma\not\equiv const$.
\medskip
In summary, we have shown the following result.
\begin{theorem}\label{th:areadecrasing}
If $\sigma\equiv 1$ then the isoperimetric ratio gradient flow with the normal velocity $\beta=k-L/(2A)$ is area nondecreasing and length nonincreasing flow of smooth Jordan curves $\Gamma^t, t\in[0,T_{max})$ in the plane provided that $\Gamma^0$ is a convex curve.
Assume the anisotropy function $\sigma$ is not constant and such that $\sigma + \sigma^{\prime\prime}>0$. Let $\Gamma^{0}$ be an initial curve which is homothetically similar to the boundary $\partial W_{\bar\sigma}$ of a Wulff shape with the modified anisotropy density function $\bar\sigma = a \sigma + b$ where $a,b$ are constants, $a,b>0$. Then, for the anisoperimetric ratio gradient flow $\Gamma^t, t\in[0,T_{max})$, evolving in the normal direction by the velocity $\beta=k_\sigma - \frac{L_\sigma}{2 A}$, we have
\begin{enumerate}
\item $\frac{\mathrm{d}}{\mathrm{d}t} L(\Gamma^t) > 0$ at $t=0$.
\item If, moreover, $a/b = \sqrt{\pi}/\sqrt{|W_\sigma|}$ then $\frac{\mathrm{d}}{\mathrm{d}t} A(\Gamma^t) < 0$ at $t=0$.
\end{enumerate}
\end{theorem}
\section{A counterexample to the comparison principle}
The aim of this section is to demonstrate that the comparison property does not hold under the anisoperimetric gradient flow, which is quite in a contrast to the total-length gradient flow $\beta=k$. It is a well-known fact that the comparison argument plays a key role in the proof of the famous Gage-Hamilton-Grayson theorem
for the curvature driven flow $\beta=k$, and it states that two smooth curves, one of them included in the closure of the interior of the second one, evolved by the normal velocity $\beta=k$ never intersects each other \cite{GH1986,Gr}. The aim of this section is to show that the analogous comparison property does not hold for the anisoperimetric gradient flow. As the flow $\beta=k_\sigma - L_\sigma/(2A)$ is nonlocal, violation of comparison principle can be expected. Nevertheless, we provide an explicit construction of a counterexample in this section.
Clearly, any curve $\Gamma$ homotheticaly similar to the Wulff shape $\partial W_\sigma$ is a stationary curve, i.e. $\beta\equiv 0$ on $\Gamma$. Indeed, $k_\sigma \equiv 1$ on $\partial W_\sigma$ and ${\cal F}_{\partial W_\sigma} = - L_\sigma(\partial W_\sigma)/(2 A(\partial W_\sigma)) = -1$ and therefore $\beta=k_\sigma + {\cal F}_{\partial W_\sigma} \equiv 0$ on $\partial W_\sigma$.
In what follows, we shall construct a smooth initial curve $\tilde\Gamma^0$ containing in its interior the Wulff shape $\partial W_\sigma$ and such that $\tilde\Gamma^t$ intersects the stationary Wulff shape $\partial W_\sigma$ for all sufficiently small times $0<t\ll 1$. The construction is as follows. First, we shall construct a nonsmooth curve $\hat\Gamma$ as the union $\hat\Gamma = \partial W_\sigma \cup r\cdot \partial W_1$ of the Wulff shape $\partial W_\sigma$ and the circle $ r\cdot \partial W_1$ of a radius $r>0$ touching the Wulff shape from outside at a point $\bm{y}$ (see Fig~\ref{fig:failurecomparison} (left)). For such a curve we have
\begin{eqnarray*}
{\cal F}_{\hat\Gamma}
&=& - \frac{L_\sigma(\hat\Gamma)}{2 A(\hat\Gamma)}
= - \frac{L_\sigma(\partial W_\sigma)+L_\sigma(r\cdot\partial W_1)}{2 (A(\partial W_\sigma)+A(r\cdot \partial W_1) )}
\\
&=& - \frac{L_\sigma(\partial W_\sigma)+r L_\sigma(\partial W_1)}{L_\sigma(\partial W_\sigma)+ 2\pi r^2}
\end{eqnarray*}
because $A(\partial W_\sigma) = L_\sigma(\hat\Gamma)/2$. Hence
${\cal F}_{\hat\Gamma} > -1 $ provided that the radius $r$ is sufficiently large, $r> L_\sigma(\partial W_1)/2\pi$. For instance, if $\sigma(\nu) = 1 +\varepsilon\cos(m\nu)$, then $r>1$ because $L_\sigma(\partial W_1) =2\pi$ (see section 3).
Let $\tilde\bm{x}^0\in\hat\Gamma\cap \partial W_\sigma$ be a point different from $\bm{y}$ and belonging to a part of the curve representing the Wulff shape. Now, let us construct an initial smooth curve $\tilde\Gamma^0$ which is a continuous perturbation of $\hat\Gamma$, it contains the Wulff shape in the closure of its interior, and such that $\tilde\Gamma^0 \equiv \hat \Gamma$ in some neighborhood ${\mathcal O}(\tilde\bm{x})$ of $\tilde\bm{x}$ (see Fig~\ref{fig:failurecomparison} (right)). The anisoperimetric ratio gradient flow $\tilde\Gamma^t,t\ge 0,$ starting from $\tilde\Gamma^0$ intersects the stationary Wulff shape $\partial W_\sigma$ in the neighborhood ${\mathcal O}(\tilde\bm{x})$ for any time $0<t\ll 1$ ($\tilde\Gamma^t$ is plotted by a dashed curve in Fig~\ref{fig:failurecomparison} (right)). This is a consequence of the fact that the normal velocity $\beta$ at $\tilde\bm{x}^0$ is strictly positive for $t=0$ because $\beta = k_\sigma +{\cal F}_{\tilde\Gamma^0} = 1 +{\cal F}_{\tilde\Gamma^0} >0$ at $\tilde\bm{x}^0$, and $\tilde\Gamma^0 \cap {\mathcal O}(\tilde\bm{x}) = \partial W_\sigma \cap {\mathcal O}(\tilde\bm{x})$, i.e. $k_\sigma =1$ at $\bm{x}^0$.
\begin{figure}[ht]
\centering
\includegraphics[width=1.5in]{figures/counterexample-sharp.eps}
\includegraphics[width=1.5in]{figures/counterexample-smooth.eps}
\caption{%
An initial nonsmooth curve $\hat\Gamma$ (left) and its smooth perturbation $\tilde\Gamma^0$ (right). Failure of a comparison principle occurs at the point $\tilde\bm{x}^0$ and $0<t\ll 1$.
}
\label{fig:failurecomparison}
\end{figure}
\section{Numerical experiments}
In this section we present several examples of evolution of plane curves minimizing their anisoperimetric ratio. Our scheme belongs to a class of boundary tracking methods taking into account tangential redistribution. In construction of the scheme we employed flowing finite volume discretization in space with a nontrivial tangential velocity, and semi-implicit discretization in time. The advantage of applying a nontrivial tangential velocity consists in its capability to overcome various numerical instabilities of a flow of plane curves like swallow tails and/or merging of numerical grid points leading to break-up of the numerical scheme known for the case when the numerical scheme is constructed with no tangential redistribution. One can find recent progress in \cite{SY2008, SY2011}. Such a scheme is simple and fast, but even if the original problem has a variational structure, it is unclear that the discretized problem has variational structure. On the other hand, in \cite{BenesKimuraYazaki2009}, the authors proposed a semi-discrete scheme with variational structure in a discrete sense. Their scheme has the second order of accuracy in time. However, discretized polygonal curves are restricted to a certain class of curves which is analogous to the admissible class
in crystalline curvature flows or crystalline algorithm. In what follows, we propose a hybrid scheme taking into account advantages from both aforementioned schemes.
{\bf Discretization scheme.}\
For a given initial $N$-sided polygonal curve ${\cal P}^0=\bigcup_{i=1}^N{\cal S}_i^0$,
we will find a family of $N$-sided polygonal curves $\{{\cal P}^j\}_{j=1, 2, \cdots}$, ${\cal P}^j=\bigcup_{i=1}^N{\cal S}_i^j$, where ${\cal S}_i^j=[\bm{x}_{i-1}^j, \bm{x}_{i}^j]$ is the $i$-th edge with $\bm{x}_{0}^j=\bm{x}_N^j$ for $j=0, 1, 2, \cdots$. The initial polygon ${\cal P}^0$ is an approximation of $\Gamma^0$ satisfying $\{\bm{x}_i^0\}_{i=1}^N\subset{\cal P}^0\cap\Gamma^0$,
and ${\cal P}^j$ is an approximation of $\Gamma^t$ at the time $t=t_j$,
where $t_j=j\tau$ is the $j$-th discrete time ($j=0, 1, 2, \cdots$) if we use a fixed time increment $\tau>0$, or
$t_j=\sum_{l=0}^{j-1}\tau_l$ ($j=1, 2, \cdots$; $t_0=0$) if we use
adaptive time increments $\tau_l>0, l=0, \cdots, j-1 $.
The updated curve ${\cal P}^{j+1}$ is determined from the data for ${\cal P}^j$ at the previous time step by using discretization in space and time.
Our two steps scheme will be constructed as follows: in the first step we construct moving polygonal curves which is continuous in time and discrete in space. In the second step we make use of the semi-implicit time discretization scheme for moving polygonal curves.
{\bf Step 1: Moving polygonal curves.}\
Let ${\cal P}(t)=\bigcup_{i=1}^N{\cal S}_i(t)$ be an $N$-sided polygonal curve continuously in time with ${\cal P}(0)={\cal P}^0$, where
${\cal S}_i(t)=[\bm{x}_{i-1}(t), \bm{x}_{i}(t)]$ is the $i$-th edge and $\bm{x}_i(t)$ is the $i$-th vertex
($i=1, 2, \cdots, N$; $\bm{x}_0(t)=\bm{x}_N(t)$).
The length of ${\cal S}_i$ is denoted by $r_i=|\bm{x}_{i}-\bm{x}_{i-1}|$.
The $i$-th unit tangent vector $\bm{T}_i$ can be defined as $\bm{T}_i=(\bm{x}_{i}-\bm{x}_{i-1})/r_i$, and
the $i$-th unit inward normal vector $\bm{N}_i=\bm{T}_i^{\bot}$, where $(a, b)^{\bot}=(-b, a)$.
Then the $i$-th unit tangent angle $\nu_i$ is obtained from
$\bm{T}_i=(\cos\nu_i, \sin\nu_i)^{\mathrm{T}}$ in the following way:
Firstly, from $\bm{T}_1=(T_{11}, T_{12})^{\mathrm{T}}$,
we obtain $\nu_1=-\arccos(T_{11})$ if $T_{12}<0$; $\nu_1=\arccos(T_{11})$ if $T_{12}\geq 0$.
Secondly, for $i=1, 2, \cdots, N$ we successively compute $\nu_{i+1}$ from $\nu_{i}$:
\[
\nu_{i+1}=\left\{\begin{array}{@{}ll}
\nu_i-\arccos(I), & \mbox{if $D<0$}, \\
\nu_i+\arccos(I), & \mbox{if $D>0$}, \\
\nu_i, & \mbox{otherwise},
\end{array}\right.
\]
where $D=\det(\bm{T}_i, \bm{T}_{i+1}),
I=\bm{T}_i\cdot\bm{T}_{i+1}$. Finally, we obtain
$\nu_0=\nu_1-(\nu_{N+1}-\nu_{N})$.
Then the $i$-th unit inward normal vector $\bm{N}_i$ is $\bm{N}_i=(-\sin\nu_i, \cos\nu_i)^{\mathrm{T}}$.
Let us introduce the ``dual'' volume
${\cal S}_i^*=[\bm{x}_{i}^*, \bm{x}_{i}]\cup[\bm{x}_{i}, \bm{x}_{i+1}^*]$ of ${\cal S}_i$,
where $\bm{x}_{i}^*=(\bm{x}_{i-1}+\bm{x}_{i})/2$ is the mid point of the $i$-th edge ${\cal S}_i$
($i=1, 2, \cdots, N$; $\bm{x}_{N+1}^*=\bm{x}_1^*$).
The length of ${\cal S}_i^*$ is $r_i^*=(r_i+r_{i+1})/2$.
Then the total length of ${\cal P}$ is
$L=\sum_{i=1}^Nr_i=\sum_{i=1}^Nr_i^*$, and
the enclosed area of ${\cal P}$ is
$A=-\sum_{i=1}^N(\bm{x}_i\cdot\bm{N}_i)r_i/2=\sum_{i=1}^N\bm{x}_{i-1}^{\bot}\cdot\bm{x}_i/2$.
We define the $i$-th unit tangent angle of ${\cal S}_i^*$ by $\nu_i^*=(\nu_i+\nu_{i+1})/2=\nu_i+\phi_i/2$, where
$\phi_i=\nu_{i+1}-\nu_i$ is the angle between the adjacent two edges.
Then the $i$-th tangent vector at the vertex $\bm{x}_i$ is
$\bm{T}_i^*=(\cos\nu_i^*, \sin\nu_i^*)^{\mathrm{T}}$ and the inward normal vector
$\bm{N}_i^*=(-\sin\nu_i^*, \cos\nu_i^*)^{\mathrm{T}}$.
Hereafter we will use the following abbreviations:
\[
c_i = \cos\frac{\phi_i}{2}, \quad
s_i = \sin\frac{\phi_i}{2} \quad (i=1, 2, \cdots, N).
\]
Then it is easy to check that
\[
\bm{T}_i^*=c_i\bm{T}_i+s_i\bm{N}_i, \quad
\bm{N}_i^*=c_i\bm{N}_i-s_i\bm{T}_i \quad (i=1, 2, \cdots, N).
\]
The evolution equations of ${\cal P}(t)$ read as follows:
\begin{equation} \label{eq:dot{vecx}_i}
\dot{\bm{x}}_i=\alpha_i\bm{T}_i^*+\beta_i\bm{N}_i^* \quad (i=1, 2, \cdots, N),
\end{equation}
where $\alpha_i$ and $\beta_i$ are quantities defined on ${\cal S}_i^*$.
Here and hereafter, we denote $\dot{u}=\mathrm{d}u/\mathrm{d}t$.
The tangential velocities $\{\alpha_i\}$ are defined below and
the $i$-th normal velocity $\beta_i$ is defined such as
\begin{equation} \label{eq:beta_i}
\beta_i=\frac{\beta_i^*+\beta_{i+1}^*}{2c_i} \quad (i=1, 2, \cdots, N),
\end{equation}
where the $i$-th normal velocity $\beta_i^*$ is defined on ${\cal S}_i$. It is an approximation of (\ref{geomrovnonloc}) such as
\[
\beta_i^*=\delta_ik_i + {\cal F}_\Gamma \quad (i=1, 2, \cdots, N).
\]
Here $k_i$ is the $i$-th curvature and the constant value on ${\cal S}_i$ defined as
\[
k_i=\frac{\tan(\phi_i/2)+\tan(\phi_{i-1}/2)}{r_i} \quad (i=1, 2, \cdots, N),
\]
which is the same as the polygonal curvature in \cite{BenesKimuraYazaki2009}, and
$\delta_i$ is an approximation of $\delta(\nu_i)$ defined later.
Then we obtain the time evolution of the total length of ${\cal P}(t)$:
\[
\dot{L}
=-2\sum_{i=1}^N\beta_is_i
=-\sum_{i=1}^Nk_i\beta_i^*r_i,
\]
and the time evolution of the enclosed area of ${\cal P}(t)$:
\begin{align}
\dot{A}
&=-\sum_{i=1}^N\beta_ic_ir_i^*+\sum_{i=1}^N\alpha_is_i\frac{r_{i+1}-r_i}{2}
\\
&=-\sum_{i=1}^N\beta_i^*r_i+\mathrm{err}_A ,
\label{eq:semi-discrete-dA/dt} \\
\mathrm{err}_A
&= -\sum_{i=1}^N\beta_i^*\frac{r_{i+1}-2r_i+r_{i-1}}{4}+\sum_{i=1}^N\alpha_is_i\frac{r_{i+1}-r_i}{2}.
\nonumber
\end{align}
These identities represent a discrete version of equations (\ref{eq:area}) provided that the distribution $r_i\equiv L/N, i=1, 2, \cdots, N,$ is uniform because the error term $\mathrm{err}_A=0$ is vanishing.
To realize this uniform distribution asymptotically, we assume that
\[
r_i-\frac{L}{N}=\eta_ie^{-f(t)}
\]
\[
\left(\mbox{$\sum_{i=1}^N\eta_i=0$, $f(t)\to\infty$ as $t\to T_{max}\leq \infty$}\right).
\]
By using a relaxation term $\omega(t)=f'(t)$ we obtain
\begin{equation}
\dot{r}_i-\frac{\dot{L}}{N}
=\left(\frac{L}{N}-r_i\right)\omega(t), \quad \int_0^{T_{max}}\omega(t)\,\mathrm{d}t=\infty
\label{eq:relaxation-omega}
\end{equation}
$(i=1, 2, \cdots, N)$. Taking into account the relations:
\begin{align*}
\dot{r}_i
&= (\dot{\bm{x}}_i-\dot{\bm{x}}_{i-1})\cdot\bm{T}_i
\\
&= -\beta_is_i-\beta_{i-1}s_{i-1}+c_i\alpha_i-c_{i-1}\alpha_{i-1}
\\
&= \frac{\dot{L}}{N}+\left(\frac{L}{N}-r_i\right)\omega(t),
\end{align*}
we deduce $N-1$ equations for tangential velocities $\alpha_i$ ($i=2, 3, \cdots, N$):
\begin{align*}
& \alpha_i=\frac{\Psi_i}{c_i}+\frac{c_1}{c_i}\alpha_1 \quad (i=2, 3, \cdots, N), \\
& \Psi_i=\psi_2+\psi_3+\cdots+\psi_i \quad (i=2, 3, \cdots, N), \\
& \psi_i=\beta_is_i+\beta_{i-1}s_{i-1}-\frac{2}{N}\sum_{i=1}^N\beta_is_i+\left(\frac{L}{N}-r_i\right)\omega(t).
\end{align*}
To determine $\alpha_1$, we add one more linear equation of the form $\sum_{i=1}^N\alpha_ip_i=P$,
which is independent of the above $N-1$ equations. Since
\begin{equation} \label{eq:RQ}
R=c_1\sum_{i=1}^N\frac{p_i}{c_i}, \quad Q=\sum_{i=2}^N\frac{p_i}{c_i}\Psi_i,
\end{equation}
we obtain $\alpha_1=(P-Q)/R$.
Next, we propose three candidates for each $\{p_i\}$ and $P$, and choose one of them in the following way:
{\it Candidate 1.}\ We put
\[
p_i=s_i\frac{r_{i+1}-r_i}{2} \quad (i=1, 2, \cdots, N),
\]
\[
P=\sum_{i=1}^N\beta_i^*\frac{r_{i+1}-2r_i+r_{i-1}}{4},
\]
and from (\ref{eq:RQ}) we calculate $R$ and $Q$. We denote this $R$ by $R_1$.
If the above equation holds, then $\mathrm{err}_A=0$ and $\dot{A}=-\sum_{i=1}^N\beta_i^*r_i$ hold in (\ref{eq:semi-discrete-dA/dt}).
However, if distribution of grid points are almost uniform, then the above equation is almost nothing.
Therefore we need another candidate.
{\it Candidate 2.}\
For the $i$-th quantities ${\sf F}_i$ defined on ${\cal S}_i$ and ${\sf G}_i$ defined on ${\cal S}_i^*$,
we define the average along ${\cal P}$ such as
\[
\langle{\sf F}\rangle = \frac{1}{L}\sum_{i=1}^N{\sf F}_ir_i, \quad
\langle{\sf G}\rangle^* = \frac{1}{L}\sum_{i=1}^N{\sf G}_ir_i^*.
\]
Since $L=\sum_{i=1}^Nr_i=\sum_{i=1}^Nr_i^*$, we have $\langle 1\rangle=\langle 1\rangle^*=1$.
Moreover, for $\alpha_i^*=(\alpha_i+\alpha_{i-1})/2$ defined on ${\cal S}_i$,
the relation $\langle \alpha\rangle^*=\langle \alpha^*\rangle$ holds.
The second candidate of linear equation is the zero-average $\langle \alpha \rangle^*=0$,
that is, $p_i=r_i^*$ for $i=1, 2, \cdots, N$ and $P=0$.
From this and (\ref{eq:RQ}) we calculate $R$ and $Q$. We denote this $R$ by $R_2$.
The purpose of this section is to present numerical simulations of the geometric flow evolving according to the evolution equation (\ref{eq:isoperimnormalaniso}).
Before we introduce the third candidate,
we calculate the discrete version of (\ref{eq:isoperimnormalaniso}).
Let the total interfacial energy be
\[
L_\sigma(t)=\sum_{i=1}^N\sigma(\nu_i(t))r_i(t).
\]
The time derivative of $\bm{T}_i=(\cos\nu_i, \sin\nu_i)^{\mathrm{T}}$ is
$\dot{\bm{T}}_i=\dot{\nu}_i\bm{N}_i$.
Then we have $r_i\dot{\nu}_i=(\dot{\bm{x}}_i-\dot{\bm{x}}_{i-1})\cdot\bm{N}_i$,
from which it follows that
\begin{align*}
\dot{L}_\sigma
&= \sum_{i=1}^N(\sigma'_i\dot{\nu}_ir_i+\sigma_i\dot{r}_i)
= -\sum_{i=1}^N{k_{\sigma}}_i\beta_i^*r_i+\sum_{i=1}^N{k_{\sigma}}_i\tilde{p}_i\alpha_i, \\
\tilde{p}_i
&=(\sigma'_i+\sigma'_{i+1})s_i+(\sigma_i-\sigma_{i+1})c_i.
\end{align*}
Here $\sigma_i=\sigma(\nu_i)$, $\sigma'_i=\sigma'(\nu_i)$, and
${k_{\sigma}}_i$ is discrete version of the weighted curvature in the following sense:
\[
{k_{\sigma}}_i
=\delta_ik_i,
\]
\[
\delta_i=\frac{\sigma'_{i+1}-\sigma'_{i-1}}{2(t_i+t_{i-1})}
+\frac{\sigma_{i+1}t_i
+\sigma_{i}(t_i+t_{i-1})
+\sigma_{i-1}t_{i-1}}{2(t_i+t_{i-1})},
\]
\[
t_i=\tan\frac{\phi_i}{2},
\]
and $\delta_i$ is discrete weight of $\delta(\nu)=\sigma''(\nu)+\sigma(\nu)$ at $\nu=\nu_i$.
Note that $\delta_i\to\delta(\nu_i)$ holds as $\phi_i, \phi_{i-1}\to0$ formally, and
even the case where $t_i+t_{i-1}=0$,
${k_{\sigma}}_i$ is well-defined,
since $k_i=(t_i+t_{i-1})/r_i$ and then denominator of ${k_{\sigma}}_i$ is $2r_i$.
We obtain
\begin{align*}
\frac{\mathrm{d}}{\mathrm{d}t}\frac{L_\sigma^2}{A}
&= \frac{2L_\sigma}{A}\left(\dot{L}_\sigma-\frac{L_\sigma}{2A}\dot{A}\right),
\\
\dot{L}_\sigma-\frac{L_\sigma}{2A}\dot{A}
&= -\sum_{i=1}^N\left({k_{\sigma}}_i-\frac{L_\sigma}{2A}\right)\beta_i^*r_i
+\mathrm{err}_{ratio}, \\
\mathrm{err}_{ratio}
&= \sum_{i=1}^N\tilde{p}_i\alpha_i
\\
&+\frac{L_\sigma}{2A}\biggl(
\sum_{i=1}^N\beta_i^*\frac{r_{i+1}-2r_i+r_{i-1}}{4}
\\
&-\sum_{i=1}^N\alpha_is_i\frac{r_{i+1}-r_i}{2}
\biggr).
\end{align*}
{\it Candidate 3.}\ We put
\[
p_i=s_i\frac{r_{i+1}-r_i}{2}-\frac{2A}{L_\sigma}\tilde{p}_i \quad (i=1, 2, \cdots, N),
\]
\[
P=\sum_{i=1}^N\beta_i^*\frac{r_{i+1}-2r_i+r_{i-1}}{4},
\]
and from (\ref{eq:RQ}) we calculate $R$ and $Q$. We denote this $R$ by $R_3$.
Let the $i$-th normal velocity defined on ${\cal S}_i$ be
\[
\beta_i^*={k_{\sigma}}_i-\frac{L_\sigma}{2A} \quad (i=1, 2, \cdots, N),
\]
which is discrete version of (\ref{eq:isoperimnormalaniso}).
If we choose the candidate 3 and its hold exactly,
then $\mathrm{err}_{ratio}=0$ holds and we obtain
\[
\frac{\mathrm{d}}{\mathrm{d}t}\frac{L_\sigma^2}{A}
=-\frac{2L_\sigma}{A}
\sum_{i=1}^N\left({k_{\sigma}}_i-\frac{L_\sigma}{2A}\right)^2r_i<0.
\]
{\it Choice one from three candidates.}\
We choose candidate number $l$ satisfying
\[
|R_l|=\max\{|R_1|, |R_2|, |R_3|\}.
\]
\begin{remark}
If, for definition of the normal velocity defined on ${\cal S}_i^*$, we use
\[
\beta_i=\frac{\beta_i^*r_i+\beta_{i+1}^*r_{i+1}}{2c_ir_i^*}
\]
instead of (\ref{eq:beta_i}),
then ${k_{\sigma}}_i$ can not be divided into weighted part $w_i$ and the curvature $k_i$.
Moreover, if we use
\[
\beta_i=\frac{\beta_i^*+\beta_{i+1}^*}{2}
\]
instead of (\ref{eq:beta_i}),
then ${k_{\sigma}}_i$ can be divided into weighted part and the curvature.
However, we have $\mathrm{err}_A \neq 0$, that is $\dot{A}=-\sum_{i=1}^N\beta_i^*r_i$ does not hold
even if uniform distribution $r_i\equiv L/N$ holds for all $i=1, 2, \cdots, N$.
\end{remark}
{\bf Step 2: Discretization in time.}\
We use semi-implicit scheme for discretization of (\ref{eq:dot{vecx}_i}).
Next we develop expression (\ref{eq:dot{vecx}_i}) as follows:
\begin{eqnarray}
\dot{\bm{x}}_i
&=& \alpha_i\bm{T}_i^*+\beta_i\bm{N}_i^* \nonumber \\
&=& \frac{1}{2}\left(\frac{\alpha_i}{c_i}-\beta_is_i\right)\bm{T}_{i}
+ \frac{1}{2}\left(\frac{\alpha_i}{c_i}+\beta_is_i\right)\bm{T}_{i+1}
\nonumber \\
&&+ \beta_ic_i\frac{\bm{N}_{i}+\bm{N}_{i+1}}{2} \label{eq:dot{vecx}3}
\label{eq:dot{vecx}4}\\
&=& \frac{1}{2}\left(\frac{\alpha_i}{c_i}-\frac{\beta_i}{s_i}\right)\bm{T}_{i}
+ \frac{1}{2}\left(\frac{\alpha_i}{c_i}+\frac{\beta_i}{s_i}\right)\bm{T}_{i+1}. \nonumber
\end{eqnarray}
Here we have used the relation
\begin{eqnarray*}
\bm{T}_i^*
&=&\frac{\bm{T}_{i+1}+\bm{T}_{i}}{2c_i}, \quad
\\
\bm{N}_i^*
&=&s_i\frac{\bm{T}_{i+1}-\bm{T}_{i}}{2}
+c_i\frac{\bm{N}_{i+1}+\bm{N}_{i}}{2}
=\frac{\bm{T}_{i+1}-\bm{T}_{i}}{2s_i}.
\end{eqnarray*}
Let $\mu$ be a parameter satisfying $\mu=0$ if $\min_{1\leq i\leq N}|s_i|=0$, otherwise $\mu\in (0, 1]$.
For the parameter $\mu\in[0, 1]$ we use the linear interpolation of (\ref{eq:dot{vecx}3}) and (\ref{eq:dot{vecx}4}),
since we can not use (\ref{eq:dot{vecx}4}) if $s_i=0$.
Put $b_i=(1-\mu)s_i+\mu/s_i$ for $\mu\in (0, 1]$ and $b_i=s_i$ for $\mu=0$.
Then the evolution equation instead of (\ref{eq:dot{vecx}_i}) will be
\begin{eqnarray}
\dot{\bm{x}}_i
&=&\frac{1}{2}\left(\frac{\alpha_i}{c_i}-\beta_ib_i\right)\bm{T}_{i}
+\frac{1}{2}\left(\frac{\alpha_i}{c_i}+\beta_ib_i\right)\bm{T}_{i+1}
\nonumber \\
&& +\beta_ic_i(1-\mu)\frac{\bm{N}_{i}+\bm{N}_{i+1}}{2}.
\label{eq:dot{vecx}5}
\end{eqnarray}
From $\bm{T}_i=(\bm{x}_i-\bm{x}_{i-1})/r_i$,
we discretize (\ref{eq:dot{vecx}5}) in time and obtain the following tridiagonal linear system under
the periodic boundary condition:
\begin{eqnarray*}
\frac{\bm{x}_i^{j+1}-\bm{x}_i^{j}}{\tau_j}
&=&-a_-\bm{x}_{i-1}^{j+1}+a_0\bm{x}_i^{j+1}-a_+\bm{x}_{i+1}^{j+1}
\\
&&+\beta_i^jc_i^j(1-\mu)\frac{\bm{N}_{i}^j+\bm{N}_{i+1}^j}{2}.
\end{eqnarray*}
Here $a_0=a_-+a_+$ and
\[
a_-=\frac{1}{2r_i^j}\left(\frac{\alpha_i^j}{c_i^j}-\beta_i^jb_i^j\right),\]
\[
a_+=-\frac{1}{2r_{i+1}^j}\left(\frac{\alpha_i^j}{c_i^j}+\beta_i^jb_i^j\right)
\]
for $i=1, 2, \cdots, N$ and $j=0, 1, 2, \cdots$.
For the choice of $\mu$ we use
\[
\mu=\frac{\min_i|s_i^j|}{\max_i|s_i^j|}\in [0, 1].
\]
Here and hereafter,
$\min_i$ and $\max_i$ mean $\min_{1\leq i\leq N}$ and $\max_{1\leq i\leq N}$, respectively.
Note that $\max_i|s_i^j|>0$ holds for closed curves.
In order to ensure solvability of the above linear system, we require a simple condition on the diagonal dominance. Adopting such a condition the adaptive time step $\tau_j$ satisfies
\[
\tau_j=\frac{\min_i r_i^j}
{2(1+\lambda)(\max_i|\alpha_i^j/c_i^j|+\max_i|\beta_i^jb_i^j|)} \quad
(\lambda>0).
\]
{\bf Simulation.}\
In following all figures, for the prescribed $\widehat{\tau}>0$, we plot every $\mu\widehat{\tau}$ discrete
time step using discrete points representing the evolving curve.
In every $3\mu\widehat{\tau}$ time step, we plot a polygonal curve connecting those points,
where $\mu=[[T/\widehat{\tau}]/100]$ ($[x]$ is the integer part of $x$),
and $T=1.5$ is the final computational time.
We use $\omega\equiv 1000$ as the relaxation term in (\ref{eq:relaxation-omega}).
Note that if we use small $\omega$,
then asymptotic speed for uniform distribution becomes slow, and
for some choice of $\sigma$ area-decreasing phenomena do not hold near the initial time (cf. Fig~\ref{fig:Wulff-sigma-bar-peak6-AL-vs-time}~(a)).
{\bf Wulff shapes and area-decreasing phenomenon.}\
If $\sigma(\nu) =1 + \varepsilon \cos(m \nu)$ is the anisotropy density function of the degree $m$ then, by using (\ref{eq:areaW}), we obtain the explicit expression for the mixed anisotropy function:
\[
\bar{\sigma} = \sqrt{\pi}\, \sigma + \sqrt{|W_\sigma|}
\]
\[
=\sqrt{\pi}\left(1+\varepsilon\cos m\nu+\sqrt{1-\varepsilon^2\frac{m^2-1}{2}}\right).
\]
In order to verify the area-decreasing and the length-increasing phenomenon at the initial time as in Theorem~\ref{th:areadecrasing},
we use $\partial W_{\bar{\sigma}}$ as the initial curve.
Fig~\ref{fig:Wulff-sigma-bar-peak6}~(a) indicates $\partial W_{\bar{\sigma}}$ with $m=6$.
Its discretization is given by the uniform $N$-division of the $u$-range $[0, 1]$.
Fig~\ref{fig:Wulff-sigma-bar-peak6}~(b) indicates the same Wulff shape, but the grid points are distributed uniformly.
Fig~\ref{fig:Wulff-sigma-bar-peak6}~(c) indicates the time evolution starting from (b).
\begin{figure}[ht]
\begin{center}
\includegraphics[width=1.5in]{figures/Wulff-sigma-bar-peak6.eps}
\quad
\includegraphics[width=1.5in]{figures/Wulff-sigma-bar-peak6-uniform.eps}
\\
(a) \hskip 1in (b)
\\
\includegraphics[width=1.5in]{figures/Wulff-sigma-bar-peak6-allcurves.eps}
\\
(c)
\end{center}
\caption{%
(a) The Wulff shape $\partial W_{\bar{\sigma}}$ with $N=120$ points,
(b) its uniform parameterization and its time evolution (c).
}
\label{fig:Wulff-sigma-bar-peak6}
\end{figure}
\begin{figure}[ht]
\begin{center}
\begin{tabular}{@{}c@{}}
\scalebox{0.65}{\includegraphics{figures/Wulff-sigma-bar-peak6-AL-vs-time.eps}}
\end{tabular}
\end{center}
\caption{%
Initial decrease of the enclosed area $A$ and increase of the total length $L$.
}
\label{fig:Wulff-sigma-bar-peak6-AL-vs-time}
\end{figure}
\begin{figure}[ht]
\begin{center}
\includegraphics[width=1.65in]{figures/ellipse-peak2-allcurves.eps}
\\
(a)
\\ \
\\
\includegraphics[width=1.65in]{figures/dumbbell-peak3-allcurves.eps}
\\
(b)
\\ \
\\
\includegraphics[width=1.65in]{figures/ms1-peak4-allcurves.eps}
\\
(c)
\\ \
\\
\includegraphics[width=1.65in]{figures/ms2-peak5-allcurves.eps}
\\
(d)
\\ \
\\
\includegraphics[width=1.65in]{figures/thindumbbell-peak6-allcurves.eps}
\\
(e)
\\
\end{center}
\caption{%
Evolution of curves starting from the initial curves with various choice of peak of $\sigma$.
}
\label{fig:evolution-of-curves}
\end{figure}
Although deformation of $\partial W_{\bar{\sigma}}$ is very small (see Fig~\ref{fig:Wulff-sigma-bar-peak6}~(c)), the area-decreasing and the length-increasing phenomenon can numerically verified by using the aforementioned numerical discretization scheme. The behavior of the enclosed area and total length of curves evolved from the initial Wulff shape $\partial W_{\bar\sigma}$ with the mixed anisotropy density function is shown in Fig~\ref{fig:Wulff-sigma-bar-peak6-AL-vs-time}.
{\bf Initial test curves.}\
As initial test examples we use the boundary $\partial W_{\bar{\sigma}}$ of the Wulff shape as well as
the following initial curves $\bm{x}(u, 0)=(x_1(u), x_2(u))^{\mathrm{T}}$ ($u\in[0, 1]$) parameterized by
\begin{align*}
&\mathrm{(a)}&
& \mbox{ellipse}:
x_1(u)=a\cos 2\pi u, \
x_2(u)=b\sin 2\pi u, \\
&\mathrm{(b)}&
& \mbox{dumbbell shape}: z = 2\pi u, \\
& & &
x_1(u)=\cos z, \
x_2(u)=2.0\sin z - 1.99\sin^3z, \
\\
&\mathrm{(c)}&\
& x_1(u)=\cos z, \ x_2(u)=0.7\sin z+\sin x_1+x_3^2, \
\\
& & &
x_3=\sin(3z)\sin z, \ z=2\pi u, \\
&\mathrm{(d)}&
&
x_1(u)=1.5\cos z, \
\\
& & &
x_2(u)=1.5(0.6\sin z+0.5x_3^2+0.4\sin x_4+0.1\sin x_5), \\
&&&
x_3=\sin(3z)\sin z, \ x_4=2x_1^2, \ x_5=3e^{-x_1}, \ z=2\pi u, \\
&\mathrm{(e)}&
& \mbox{thin-dumbbell shape}: \\
&&&
(x_1(u), x_2(u))=\left\{\begin{array}{@{}ll}
(\bar{x}_1(u), \bar{x}_2(u)) & \\ \qquad \hbox{for}\ 0\leq u<0.5 & \\
-(\bar{x}_1(u-0.5), \bar{x}_2(u-0.5)) \\ \qquad \hbox{for}\ 0.5\leq u\leq 1 &
\end{array}\right., \\
&&&
(\bar{x}_1(u), \bar{x}_2(u))=\left\{\begin{array}{@{}ll}
(\hat{x}_1(u), \hat{x}_2(u)) \\ \qquad \hbox{for}\ 0\leq u<0.25 & \\
(-\hat{x}_1(0.5-u), \hat{x}_2(0.5-u)) \\ \qquad \hbox{for}\ 0.25\leq u\leq 0.5 &
\end{array}\right., \\
&&&
\hat{x}_1(u)=\left\{\begin{array}{@{}ll}
beam+rad(1+\cos(8(\pi-\theta_\varepsilon)u)) \\ \quad \hbox{for}\ 0\leq u<0.125& \\
2(beam+rad(1-\cos\theta_\varepsilon))(1-4u) \\ \qquad \hbox{for}\ 0.125\leq u\leq 0.25 &
\end{array}\right., \\
&&&
\hat{x}_2(u)=\left\{\begin{array}{@{}ll}
rad\sin(8(\pi-\theta_\varepsilon)u)) \\ \qquad \hbox{for}\ 0\leq u<0.125 & \\
\varepsilon \\ \qquad \hbox{for}\ 0.125\leq u\leq 0.25 &
\end{array}\right.,
\end{align*}
where $beam>0$ and $0<\varepsilon<rad$ are parameters and $\theta_\varepsilon=\arcsin(\varepsilon/rad)$.
In all examples, the initial discretization is given by the uniform $N$-division of the $u$-range $[0, 1]$. Fig~\ref{fig:evolution-of-curves} indicates numerical simulation with the initial curves. We choose several peaks of $\sigma$ such as (a) $m=2$, (b) $m=3$, (c) $m=4$, (d) $m=5$, and (e) $m=6$.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=1.8in]{figures/cnvbreak-init-sharp.eps}
\\
(a)
\\ \
\\
\includegraphics[width=1.8in]{figures/cnvbreak-allcurves.eps}
\\
(b)
\\ \
\\
\includegraphics[width=1.8in]{figures/cnvbreak-init-final-curves.eps}
\\
(c)
\end{center}
\caption{%
Anisoperimetric ratio gradient flow breaking comparison principle.
}
\label{fig:comparison-breaking}
\end{figure}
{\bf Breaking of a comparison principle.}\
In Fig~\ref{fig:comparison-breaking} we plot the initial curve consisting of the union of the boundary of a Wulff shape $\partial W_\sigma$ (with $\sigma$ having degree $m=3$) touched by a circle with a sufficiently large circle with a radius $r>1$ (see section 7). As soon as it evolved by the anisoperimetric ratio gradient flow it intersects the stationary Wulff shape $\partial W_\sigma$.
Numerically computed examples displayed in Fig~\ref{fig:comparison-breaking} (a) and (c) correspond to those of the conceptual Fig~\ref{fig:failurecomparison}.
\section{Conclusions}
In this paper, we have derived and analyzed a gradient flow of closed planar curves minimizing the anisoperimetric ratio. A geometric law for normal velocity is a function of the anisotropic curvature and it depends on the total interfacial energy and enclosed area of the curve. We also derived a new mixed anisoperimetric inequality for the product of total interfacial energies corresponding to different anisotropy functions. Interestingly enough, there exist initial curves for which the enclosed area is a decreasing function with respect to time. This is in contrast to the known property of a gradient flow minimizing isoperimetric ratio. We also derived a stable numerical scheme based on the flowing finite volumes method. Theoretical results have been illustrated be several computational examples.
\IAENGpeerreviewmaketitle
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
|
{
"timestamp": "2013-06-06T02:00:30",
"yymm": "1203",
"arxiv_id": "1203.2238",
"language": "en",
"url": "https://arxiv.org/abs/1203.2238"
}
|
\section{Introduction}
For a long time spicules \citep{Beckers1968} have been examined as structures that intermittently
couple the chromosphere and the corona through a continuous ejection of mass flux and thereby
causing heating. However, the coronal counterparts of these jets were never found \citep{Withbroe1983,
Mariska1992}. Thus the idea of their direct contribution to coronal heating was shelved
\citep{Withbroe1983} until recently. \citet{DePontieu2007} revived this idea when they used off-limb
coronal hole data to show that high velocity spicules (so called type II spicules) existing in the solar chromosphere
exhibit only upward motions. \cite{RouppeVanderVoort2009} correlated the off-limb type~II spicules
to on-disk Rapid Blueshifted Excursions (RBEs) and established that these features occur
ubiquitously on the Sun. Their disappearance from the Ca~{\sc ii}~H passband images taken with the \textit{Solar Optical Telescope} (SOT)
on-board the \textit{Hinode} spacecraft has been interpreted as heating which causes the singly ionised calcium to become
at least doubly ionised. This subject received further excitement when \citet{DePontieu2011} correlated the SOT Ca~{\sc ii}~H spicules
with their coronal equivalent as seen in 171~\AA\ and 211~\AA\ bandpass images of the \textit{Atmospheric Imaging Assembly} (AIA) on-board
\textit{Solar Dynamics Observatory} (SDO). Recently, \citet{Madjarska2011} analysed three large spicules seen in SOT Ca~{\sc ii}~H
images and concluded that ``these spicules although very large and dynamic, are not present in spectral lines formed at
temperatures above 300\,000~K'' observed with the \textit{Extreme-ultraviolet Imaging Spectrometer} (EIS) on-board \textit{Hinode}. In addition,
their preliminary analysis of solar prominences, which have very similar plasma parameters as spicules with respect to temperatures and
densities, showed that prominences are seen in the AIA 171~\AA\ images while EIS observations for one of these prominences
revealed an emission not higher than 400\,000~K (Fe~{\sc viii}, log$T~(K)~\sim~5.6$). The authors suggested, therefore,
that the recent observations of spicules by \cite{DePontieu2011} in AIA/SDO 171~\AA\ and 211~\AA\ channels may come
from the existence of transition region emission in these passbands.
The AIA \citep{Lemen2012} on-board SDO \citep{Pesnell2012} has revolutionised our view of the Sun
with its unprecedented cadence, high spatial resolution and large field-of-view. AIA has 10 passbands
which cover a wide range of temperatures thus enabling the study of the solar atmosphere from the
chromosphere to the corona. Unfortunately, due to the relatively large spectral widths of the AIA
passbands, the resultant images are not spectrally pure. Although the emission from these filters
are dominated by the intended primary ion, initial analysis show that the contribution from
several adjacent spectral lines cannot be ruled out \citep{ODwyer2010}. For their analysis they used typical Differential Emission
Measure (DEM) distributions for a coronal hole, quiet Sun, active region and flaring plasma. By forward modeling the emission
in the AIA filters, \cite{Martinez-Sykora2011} concluded that the 131\AA, 171~\AA\ and 304~\AA\ passbands
have a negligible contribution from the non-dominant ions. The authors used 3D MHD numerical simulations of the solar atmosphere
for regions representing ``coronal hole, and quiet Sun with hot emerging regions and with hot corona" to calculate the intensity (using CHIANTI)
of spectral lines within the AIA passbands, and investigate the importance of non-dominant lines for the various solar coronal conditions.
Important issues related to the AIA passbands have been highlighted recently such as the presence of unidentified transition region and coronal
lines in the 211~\AA\ channel \citep{DelZanna2011}. From their study of active region loop footpoints they also suggest that since the
171~\AA\ channel has a significant contribution from cool plasma (log$T~(K)~<~5.7$), observations in this channel should be treated
with extreme caution.
Similar studies have been conducted for some of the earlier observations by \citet{Brooks2006} where they investigated the response functions
for the 171~\AA\ and 195~\AA\ channels of \textit{Extreme-ultraviolet Imaging Telescope} (EIT) on-board
the \textit{Solar and Heliospheric Observatory} (SoHO) and the \textit{Transition Region and Coronal Explorer} (TRACE)
using coordinated \textit{Coronal Diagnostic Spectrometer} (CDS) observations of the quiet Sun. The TRACE response functions were also studied by
\citet{DelZanna2003} and \citet{Cirtain2005} for active regions and flares, respectively.
The overview given above shows that until now there has been no attempt made to understand the thermal response of the filters on
different telescopes to spicule observations. The time is now ripe to study this problem as the current high spatial and time resolution
observations make it possible to look at features like spicules, bright points, prominences \textit{etc.} in great detail. In the present paper we
aim to obtain for the first time a DEM distribution to study the thermal properties of emission registered through the AIA 171~\AA\
passband during off-limb activity above a coronal hole, dominated by spicules. The results obtained here could be applied to
observations of any phenomenon with emission at chromospheric and/or transition region temperatures with the 171~\AA\ channel of
AIA/SDO. The data used here are described in Section 2. The DEM analysis is given in Section 3. In Section 4 we describe and
discuss our results, and in Section 5 we state our conclusions.
\section{Observations}
The \textit{Solar Ultraviolet Measurements of Emitted Radiation} \citep[SUMER;][]{Wilhelm1995} instrument on-board SoHO \citep{Domingo1995}
is a telescope and a spectrometer which uses two detectors to cover a large spectral range from 465~\AA\ to 1610~\AA. While detector A is
sensitive to wavelengths 780~\AA\ to 1610~\AA, detector B works from 660~\AA\ to 1500~\AA. From the SUMER archive we selected suitable
data which will enable us to study transient events like spicules. The data were taken on 13~June~1998 in and above a coronal hole at the
North Pole. Coronal hole off-limb data are the most suitable for this study as the dominant features observed there are spicules.
\cite{RouppeVanderVoort2009} calculated that there are between 1.5 and 3 Ca~{\sc ii}~H type~II spicules per linear arcsec along the
limb in a coronal hole. Hence, our off-limb region of study is intermixed with both type~I and type~II spicules. In Figure~\ref{eit_sumer}
we show context images in Fe~{\sc ix}~171~\AA\ (top, left) and He~{\sc ii}~304~\AA\ (bottom, left) taken with the EIT with the
position of the SUMER slit over-plotted. We also show in the Appendix, Figure~\ref{aia_eit}, a comparison between EIT and AIA 304~\AA\ images in order
to demonstrate that we are able to determine with EIT images various phenomena in coronal holes off-limb data, e.g. spicules and bright points, despite the instrumental limitations.
In Figure~\ref{eit_sumer}, we also show examples of the SUMER spectral data (top/bottom, right) taken co-temporally with the EIT images.
We used these
images to verify that the emission under the SUMER slit comes only from spicular material and no other features like, for instance, coronal
bright points, are present. The spectra were recorded on detector B with a slit of 0.3$''$~$\times$~120$''$\ and 300~s exposure time.
The SUMER data reduction was done with the standard software. The spectral lines were identified with the help of the SUMER spectral atlas
\citep{Curdt2001}. A list of all the lines used, their corresponding formation temperatures and radiances are given in Table~\ref{T1}. We
selected all lines with formation temperatures between log$T~(K)~=~4.1$ and log$T~(K)~=~5.8$ which have a sufficient signal-to-noise ratio (above 10).
The Ne~{\sc viii} line was chosen as the upper limit since it is the best SUMER line which has high formation temperature (log$T~(K)~=~5.8$)
close to the formation temperature of Fe~{\sc ix}. All other SUMER lines with higher formation temperatures have in general very poor
signal-to-noise ratio. Furthermore, from the analysis of \cite{Madjarska2011} we know that spicules do not have any signal at temperatures higher than
0.3~MK which makes the use of only SUMER data for this study a reasonable approach. The alignment of the SUMER slit with the EIT images
was done by first selecting spectral lines in the SUMER data which have similar formation temperatures as the EIT channels (Fe {\sc ix} and
He {\sc ii}) and then by comparing them.
\begin{figure}[ht]
\centering
\includegraphics[width=10cm]{images_used/eit_sumer.ps}
\caption{\textbf{Top left}: EIT Fe~{\sc ix/x}~171~\AA\ context image with the SUMER slit over-plotted.
\textbf{Bottom left}: EIT 304~\AA\ image of the same region showing spicular emission. The
plotted solid line represents the position of the SUMER slit. \textbf{Top right}: Ne~{\sc viii}~770.42~\AA\
emission along the SUMER slit taken at 19:02:36 UT. \textbf{Bottom right}: O~{\sc iv}~787.72~\AA\ along the SUMER
slit taken at 19:02:36 UT. The horizontal line in both images denotes the limb position.}
\label{eit_sumer}
\end{figure}
\begin{table}
\caption{Details of the SUMER spectral lines used for the construction of the DEM}
\label{T1}
\begin{tabular}{c c c c}
\hline\hline
Wavelength (\AA) & Ion & log~$T_{max}~(K)$ & Radiance $\times 10^{3} $\\
& & & $(ergs~cm^{-2} s^{-1} sr^{-1}$)\\
\hline
765.150 & N~{\sc iv} & 5.1 & 10.30\\
770.420 & Ne~{\sc viii} & 5.8 & 6.32\\
780.300 & Ne~{\sc viii} & 5.8 & 3.58\\
786.470 & S~{\sc v} & 5.2 & 4.82\\
787.720 & O~{\sc iv} & 5.2 & 8.89\\
977.030 & C~{\sc iii} & 4.8 & 179.98\\
1031.93 & O~{\sc vi} & 5.5 & 54.95\\
1238.82 & N~{\sc v} & 5.3 & 20.88\\
1242.80 & N~{\sc v} & 5.3 & 10.68\\
1253.80 & S~{\sc ii} & 4.2 & 2.52\\
1298.96 & Si~{\sc iii} & 4.7 & 3.50\\
1334.53 & C~{\sc ii} & 4.4 & 155.64\\
\hline
\end{tabular}
\end{table}
In order to obtain the off-limb radiances it is important to accurately determine the position
of the solar limb. For this purpose we first selected the part of the SUMER spectrum dominated by
continuum emission around 1230~\AA\ and the emission in the C~{\sc i} line at 1252.21~\AA\
(log$T~(K)~=~4.0$). A tilt in the orientation of the grating with respect to the detector causes
the emission along the SUMER slit to be shifted with the dispersion. This vertical displacement was
accounted for by using the program delta\_pixel.pro. We cut a further two pixels above
the identified limb in order to avoid contamination from disk emission. After having subtracted the
background from our data we fitted each spectral line with a single Gaussian to determine the
total off-limb flux contribution by that line. The Poisson noise for these radiances were
also calculated.
\section{DEM analysis}
A DEM distribution gives information about the plasma distribution as a function of temperature along a
given line-of-sight (LOS). A DEM, $\phi(T)$, is defined as
\begin{equation}
\phi(T) = n_{e}^2~\frac{dh}{dT}
\end{equation}
where $n_{e}$ is the electron density at position $h$ along the LOS at temperature $T$. For an
optically thin line in ionisation equilibrium, the DEM and the observed intensity ($I_{obs}$) are
associated to each other by the equation
\begin{equation}
I_{obs} = \int A(x)~G(N,T)~\phi(T)~dT
\end{equation}
where $A(x)$ is the elemental abundance with respect to hydrogen and $G(N, T)$ is the contribution
function for a given spectral line. This relationship can be
used to derive the DEM from an observed spectrum \citep{Withbroe1975} using inversion techniques.
\begin{figure}[ht]
\centering
\includegraphics[width=10cm]{images_used/DEM_130698.ps}
\caption{The DEM for off-limb data (where spicules are most easily identified) is calculated
for different abundances using CHIANTI 6.0.1.}
\label{dem}
\end{figure}
A deconvolution method is used to obtain the DEM from measured spectral intensities. In this paper
we use the CHIANTI atomic database version 6.0.1 \citep{Dere2009}. A procedure available along with
the CHIANTI package called chianti\_dem.pro is used to derive the DEM distribution. This program
accepts wavelength, observed intensities and corresponding errors of the spectral lines along with
pressure/density and elemental abundances corresponding to the observed region of the Sun as inputs.
The best fit for the DEM can be controlled by changing the ``mesh points'' (an array that specifies the points for the spline that represent the fitted DEM (see CHIANTI: User Guide for further details)).
\section{Results and Discussion}
We have successfully constructed a DEM distribution of spicules for different solar abundances
that are predefined in the CHIANTI package (see Figure~\ref{dem}). The derived DEM distribution was used to
analyse the thermal response of the AIA 171~\AA\ channel to spicular emission.
\begin{figure}
\centering
\includegraphics[width=10cm]{images_used/171area.ps}
\caption{The AIA 171~\AA\ passband spectral response from the model DEM (solid line). The wavelength
response function of this band is plotted with a dashed line.}
\label{171area}
\end{figure}
In Figure~\ref{171area} we present the spectral response of the AIA 171~{\AA} channel
from our DEM modeling. We found that the radiance in this channel is dominated by emission from
the Fe~{\sc ix}~171.07~\AA\ line and has sparingly little contribution from other lines. However, the product
of the contribution function and DEM, $G(N,T) \times DEM(T)$ provides a better understanding of
the formation temperature of this line for a given phenomenon in the solar atmosphere. This
analysis revealed that, although, the contribution function of the Fe~{\sc ix}~171.07~\AA\ line peaks at
log$T_{max}~(K)~=~5.9$ (the solid line in Figure~\ref{GT171}(a)), the product $G(N,T) \times DEM(T)$
has its maximum at log$T_{max}~(K)~=~5.8$ and, therefore, the emission comes from plasma at transition region temperatures.
This supports the spectral results of \cite{Madjarska2011} that spicules may not attain temperatures greater than 300\,000~K.
\begin{figure}
\centerline{\hspace*{0.015\textwidth}
\includegraphics[width=0.6\textwidth,clip=]{images_used/1contri_fn_limb_171.ps}
\hspace*{-0.03\textwidth}
\includegraphics[width=0.6\textwidth,clip=]{images_used/1contri_fn_prom_171.ps}
}
\vspace{-0.35\textwidth}
\centerline{\bf
\hspace{0.40 \textwidth} \color{black}{(a)}
\hspace{0.60\textwidth} \color{black}{(b)}
\hfill}
\vspace{0.31\textwidth}
\centerline{\hspace*{0.015\textwidth}
\includegraphics[width=0.6\textwidth,clip=]{images_used/1contri_fn_qs_171.ps}
}
\vspace{-0.35\textwidth}
\centerline{\bf
\hspace{0.7\textwidth} \color{black}{(c)}
\hfill}
\vspace{0.31\textwidth}
\caption{The contribution function $G(N,T)$ for Fe~{\sc ix}~171.07~\AA~(black, solid line) is
plotted along with the normalised product $G(N,T)~\times~DEM(T)$ for three different DEMs (a) off-limb, (b) prominence and (c) quiet Sun and for
different solar abundances as indicated in the legend.}
\label{GT171}
\end{figure}
We did a comparative study with prominences since it is known that their plasma properties are similar to that of spicules. As expected
the response of the 171~\AA\ line to the prominence DEM from CHIANTI (see Figure~\ref{GT171}(b)) showed a significant shift in the
emission contribution to log$T_{max}~(K)~=~5.7$ which is even more pronounced than in the case of spicules. Prominences are
visible in the 171~\AA\ passband of AIA but from Figure~\ref{GT171}(b) we see that the contribution from million degree plasma is very insignificant.
We then made a similar plot using the quiet Sun DEM from CHIANTI (see Figure~\ref{GT171}(c)) and found
very contrasting results. The peaks of the line contribution function and the product
$G(N,T) \times DEM(T)$ coincide here, unlike the case with the off-limb DEM and prominence DEM. The differences between
the three DEMs strongly influence the obtained results. We suggest that a quiet Sun DEM could be affected
by emission from plasma at widely differing temperatures as it could originate from numerous
sources like loops, coronal bright points, spicules, filaments etc. in an average quiet Sun region.
From the graphs in Figure~\ref{GT171} we were able to numerically integrate the values to obtain what percent of plasma seen in the
171~\AA\ filter is above million degrees. We find that in the case of spicule or prominence observations the contribution is only 3.6\%
and 0.9\%, respectively, whereas for the quiet Sun it is a little over 15\%.
We then obtained the line emission flux as a function of temperature. Using CHIANTI we calculated
the line emissivity as a function of temperature and then folded it with the QS, prominence and our
off-limb DEM distributions. Here again we find discrepancies for the three DEMs used as shown in
Figure~\ref{emiss171}. While the emission of Fe~{\sc ix} 171.07~\AA\ in response to the off-limb
DEM peaks at log$T_{max}~(K)~=~5.8$ (transition region temperature) and that of the prominence DEM is much lower at log$T_{max}~(K)~=~5.6$,
we find that the maximum flux using the quiet Sun DEM occurs at log$T_{max}~(K)~=~6.0$ (coronal temperature). Since these differences exist in
the use of the Fe~{\sc ix}~171.07~\AA\ line, results pertaining to observations in this line can easily be misrepresented.
The sensitivity of some spectral lines to different DEM has previously been demonstrated by \citet{Brooks2011}. The authors
investigated spectroheliograms of active region fan loops produced from Fe~{\sc viii} and Si~{\sc vii} lines. They found striking similarities
in the appearance of the fan loops despite their different formation temperature, log$T~(K)$ = 5.6 and 5.8 respectively. Note that Fe~{\sc viii}
is the main contributor to the AIA~131~\AA\ channel. It has been found that the Fe~{\sc viii}~185.213~\AA\ line is particularly sensitive to the
slope of the DEM, leading to disproportionate changes in its effective formation temperature. This is similar to the behaviour of the Fe~{\sc ix}
line studied here and its impact to the temperature response of the AIA~171~\AA\ channel.
\begin{figure}
\centering
\includegraphics[width=10cm]{images_used/EmissDEM171.ps}
\caption{A comparison of the Fe~{\sc ix} 171.07~\AA\ line flux as a function of temperature using both off-limb DEM (solid
line), quiet Sun DEM (dashed line) and prominence DEM (dotted line)}.
\label{emiss171}
\end{figure}
\section{Conclusions}
The aim of the present study was to derive a DEM distribution for a region dominated by spicular emission, best described by an
off-limb environment above a coronal hole. Until now spicular studies have always been spoken about in context to the quiet Sun.
However, in this paper using spicular, prominence and quiet-Sun DEM distributions, we demonstrated that there is an obvious
difference in how the Fe~{\sc ix}~171.07~\AA\ emission changes in these regions. From the off-limb DEM we find that during spicule
observations the bulk emission in the AIA 171~\AA\ filter is from cool plasma at log$T~(K)~=~5.8$ to as low as log$T~(K)~=~5.5$ with
only 3.6\% of the plasma being above a million degrees. A similar deduction was made in regard to prominence observations
using this filter where the filter is receptive to emission at log$T~(K)~=~5.7$ to as low as log$T~(K)~=~5.5$ with a meager 0.8\%
being over million degrees. Whereas for a quiet Sun region there is significant emission from plasma at log$T~(K)~=~5.9$ to
log$T~(K)~=~5.55$ with over 15\% being above million degrees. From our study we find that the temperature sensitivity of this filter
depends on the kind of feature that is observed. Although the Fe {\sc ix} line can be formed or has a contribution from plasma
at million degree temperature, our results suggest that spicule observations in this filter cannot be used as a conclusive evidence that these
phenomena are heated to coronal temperatures. We, therefore, emphasise that future studies related to off-limb features, especially
spicules, should adopt the use of the off-limb DEM. We propose that the quiet Sun can be dominated by plasma from neighbouring
structures (e.g. loops \textit{etc}.) and this could give rise to the differences that we point out here.
Our results are in agreement with the conclusions made by \cite{DelZanna2011} where they use data from active region loops to
investigate spectral line contribution to the EUV AIA channels. The outcome of our work shows that mostly transition region emission
is registered through the AIA 171~\AA\ channel during observations of certain phenomena like spicules and prominences. We need
to further explore the issue raised here and have a better understanding of the thermal characteristics of the observed phenomenon
in order to be able to completely and truthfully exploit the state-of-the-art AIA instrument.
\begin{acks}
We would like to thank the anonymous referee for the important suggestions and comments on this manuscript.
K.V, M.M, J.G.D thank ISSI for the support of the team ``Small-scale transient phenomena and
their contribution to coronal heating''. Research at Armagh Observatory is grant-aided by the
N. Ireland Department of Culture, Arts and Leisure. CHIANTI is a collaborative project involving
NRL (USA), RAL (UK), and the Universities: College London (UK), of Cambridge (UK), George Mason
(USA), and of Florence (Italy). The AIA data are courtesy of SDO (NASA) and the AIA consortium.
The SUMER project is financially supported by DLR, CNES, NASA and the ESA
PRODEX programme (Swiss contribution). This work was supported via grant
ST/F001843/1 \& ST/J00135X from the UK Science and Technology Facilities Council.
\end{acks}
\begin{appendix}
As mentioned in Section~2, coronal hole off-limb data are dominated by spicules. We use
EIT images (see Figure~\ref{eit_sumer}) to ensure that no coronal bright points are present in
the SUMER field-of-view since EIT was the only available imager at the time the SUMER reference
spectra were obtained. However, the EIT images do not distinctly reveal spicular structure due
to some instrumental limitations: low spatial resolution and high stray-light contribution. In
order to establish what we see in the EIT image we compare it with co-temporal AIA~304~\AA\
image (see Figure~\ref{aia_eit}) which has a better spatial resolution. While the off-limb EIT
image shows blurred streaks merging together due to its low spatial resolution, the AIA image
reveals numerous narrow spikes, i.e. spicules, confirming the abundance of spicules in off-limb
coronal hole data.
\begin{figure}[ht!]
\centering
\vspace{-1.5cm}
\includegraphics[width=0.99\textwidth]{images_used/aia_eit2.ps}
\vspace{-2cm}
\includegraphics[width=0.99\textwidth]{images_used/aia_eit1.ps}
\vspace{-1cm}
\caption{AIA 304~\AA\ (top) and EIT 304~\AA\ (bottom) images (colour table reversed) of a coronal hole region at the South Pole taken on 14 May 2010.}
\label{aia_eit}
\end{figure}
\end{appendix}
\bibliographystyle{spr-mp-sola-cnd}
|
{
"timestamp": "2012-03-12T01:01:31",
"yymm": "1203",
"arxiv_id": "1203.2073",
"language": "en",
"url": "https://arxiv.org/abs/1203.2073"
}
|
\section{Introduction}
\label{intro}
Recently, optical lattices have acquired primary experimental importance since they allow the study of the main features of systems of well-localized cold atoms: among others, the intriguing interplay of thermal and quantum effects in bosonic gases can thus be investigated rather precisely, with particular attention usually paid to Mott-Hubbard transitions~\cite{BDZ-08,GBMHS-02,HSBBD-06,SPP-07,FWMGB-06}.
A peculiarity of such experiments is the confining of particles within a limited region of the lattice which is normally achieved by introducing a trapping potential. This experimental setup can be mimicked theoretically by the so-called Bose-Hubbard (BH) Hamiltonian~\cite{FWGF-89} reading
\begin{eqnarray}
H_{\rm BH} &=& -{J\over 2}
\sum_{\langle ij\rangle}(b_i^\dagger b_j+b_j^\dagger b_i)
+ {U\over2} \sum_i n_i(n_i-1)+
\nonumber \\&&+
\mu \sum_i n_i + \sum_i V(r_i) n_i ,
\label{bhm}
\end{eqnarray}
where $b^\dagger_i$ and $b_i$ are respectively bosonic creation and destruction operators, $n_i$ is the local density
operator, $\mu$ is the chemical potential, $U$ is the on-site repulsion energy and the sum in the first term
is over the nearest-neighbor sites of a regular $d$-dimensional lattice. As for the trapping potential $V(r_i)$ (being $r_i$ the distance from the center of the trap), a common choice is given by
\begin{equation}
V(r_i)= v^p r_i^p,
\label{vrp}
\end{equation}
being $l \equiv J^{1/p}v^{-1}$ the trap size. The exponent $p$ is clearly even and will be set to $p=2$ in the following. Moreover, the energy unit will be fixed by setting $J=1$ (thus $l=1/v$) while $r$ and $l$ will be measured in units of the lattice spacing $a$~\!\footnote{~\!$a$ will be set to $1$ from now on.} and hence dimensionless.
In the homogenous case (i.e., with vanishing trap) the model undergoes quantum transitions between superfluid and Mott-insulator phases depending on the value of $\mu$
\noindent The introduction of a trapping potential changes the phase diagram~\cite{BRSRMDT-02,WATB-04,GKTWB-06,KPS-02,KSDZ-04}: not only a truly diverging correlation length appears only in the limit $l\rightarrow+\infty$~\cite{BRSRMDT-02,WATB-04} with $\mu$ set to the critical values of the corresponding homogeneous system, but also the scaling properties of any observable generally acquire an extra dependence on the trap size $l$ controlled by the trap exponent $\theta$ given by
\begin{equation}
\theta= {p\over p+2}.
\label{thetaexp}
\end{equation}
A frame to handle this involved scaling is provided by the trap-size scaling (TSS) theory~\cite{CV-10,CV-09}. As a benchmark example, at a quantum critical point TSS prescribes the free-energy density to scale as
\begin{equation}
F(\mu,T,l,r) = l^{-\theta(d+z)}
{\cal F}(\bar{\mu} l^{\theta/\nu},Tl^{\theta z},rl^{-\theta}),
\label{freee}
\end{equation}
with $z$ the dynamical exponent, $\nu$ the critical exponent controlling how the correlation length diverges, $r$ the distance from the middle of the trap,
$\bar{\mu}\equiv \mu-\mu_c$, and $\mu_c$ the critical value of the chemical potential.
TSS has already been applied to the one-dimensional (1D) BH model, both at $T=0$~\cite{CV-10-bh} and at finite temperature~\cite{CTV}: in this paper we extend it to the two-dimensional (2D) BH model at finite temperature. Indeed, 2D systems are relevant not only from a theoretical point of view but have also raised experimental interest~\cite{SPP-07,JCLPPS-10,SPP-08}.
\begin{figure}
\begin{center}
\scalebox{0.9}
{
\begin{tikzpicture}
\draw[->] (0.75,1)--(6.5,1);
\draw[->] (1,0.75)--(1,5.75);
\draw (0.6,5.75) node {$T$};
\draw (6.5,0.75) node {$\mu$};
\draw (5.5,0.75) node {$\mu_c$};
\draw (5.5cm,1cm) arc (10:50:5cm);
\draw[dashed] (5.5,1)--(5.5,5);
\draw[dotted] (3.75cm,4cm) arc (50:70:5cm);
\end{tikzpicture}
}
\end{center}
\caption{A qualitative sketch of the Kosterlitz-Thouless transition (solid line) in the $\mu-T$ plane.}
\label{KT}
\end{figure}
At $T=0$, the 2D BH model (\ref{bhm}) in the hard-core limit (see below) undergoes two phase transitions between superfluid and Mott insulator at $\mu=2$ and $\mu=-2$~\!\footnote{~\!More precisely, the system is in a Mott phase with $\langle n_i\rangle=0$ for $\mu>2$, in a superfluid phase for $|\mu|<2$ and in a Mott phase with $\langle n_i\rangle=1$ for $\mu<-2$. The two transitions share the same critical exponents $\nu=1/2$ and $z=$2 \cite{FWGF-89}.}, while at finite temperature it is well known that the model also develops a Kosterlitz-Thouless (KT) transition~\cite{KT-73,B-72}. In this work we are not going to study the latter but rather perform quantum Monte Carlo (QMC) simulations with chemical potential fixed at $\mu=-2, 0, 2$ only while varying $T$: This is because our aim is to investigate the behavior of the model at the quantum $T=0$ critical points. This choice of the parameters should avoid any crossings of the KT line, as clear from the qualitative diagram in Fig. \ref{KT}.
In this framework, the scaling of the particle density
\begin{equation}
\rho(r_i) \equiv \langle n_{i} \rangle\ ,
\end{equation}
and the density-density correlator
\begin{equation}
G(r_i,r_j) \equiv \langle n_{i} n_{j} \rangle - \langle n_{i} \rangle \langle n_{j} \rangle\ ,
\end{equation}
will be studied at fixed trap size $l$ and compared with TSS predictions. Besides, we extensively study how numerical outcomes for the particle density approach their Local Density Approximation (LDA) predictions at the Mott-to-superfluid transition with non-zero filling and within the superfluid phase.\\
\indent Since scaling properties are expected to be universal with respect to $U$, we will work in the hard-core (HC) limit $U\rightarrow+\infty$ where the particle occupation number at a generic lattice site can be equal to $0$ or $1$ only. This considerably simplifies the simulation algorithm (which is based on the stochastic series expansion~\cite{S-99,SS-02,S-92}).
This paper is organized as follows. In Sec. \!II we provide some details on numerical simulations while in Sec. \!III we start our analysis by studying the $\langle n_i\rangle=0$ Mott transition and compare QMC outcomes with the TSS theory. In Sec. \!IV LDA is numerically estimated for the 2D HC BH model and then applied in Sec. \!V in considering the $\langle n_i\rangle=1$ Mott transition. In Sec. \!VI we analyze the superfluid phase by paying particular attention to the scaling properties close to those lattice sites where the effective chemical potential $\mu_{eff}$, which will be defined later, equals approximately $2$. Finally, we conclude in Sec. \!VII.
\section{Quantum Monte Carlo simulations}
Numerical simulations relied on the directed loop algorithm stemming from the stochastic series expansion method \cite{S-99}: for a generic system with Hamiltonian $H$, its starting point is given by the standard power series expansion of the partition function $Z$, that is
\begin{equation}\label{zed}
Z=Tr\{e^{-\beta H}\}=\sum_{\alpha}\sum_{n=0}^{+\infty}\frac{(-\beta)^n}{n!}\langle\alpha|H^n|\alpha\rangle\ ,
\end{equation}
\noindent being $\{|\alpha\rangle\}$ a basis set. If $H$ can be decomposed as a sum of bond operators $H_{a_i,c_i}
- where $a_i$ labels the bond and $c_i$ refers to whether the operator is diagonal ($c_i=1$) or not ($c_i=2$) with respect to $\{|\alpha\rangle\}$ -, Eq. (\ref{zed}) can be rewritten as
\begin{equation}
Z=\sum_{\alpha}\sum_{n=0}^{+\infty}\sum_{S_n}\frac{(-\beta)^n}{n!}\langle\alpha|\prod_{i=1}^{n}H_{a_i,c_i}|\alpha\rangle\ ,
\end{equation}
\noindent with $S_n$ standing for a sequence $S_n=[a_1,c_1],\ldots,[a_n,c_n]$.
\indent We can easily arrange for this setup with the HC BH model. The basis $\{|\alpha\rangle\}$ is chosen to be the set of eigenvectors of the local density operators $n_i$ and this automatically determines which terms in the Hamiltonian are diagonal: contributions with $b_i^\dagger b_j$ have $c=2$ while those written in terms of the $n_i$'s have $c=1$. Moreover, the former are already bond-like while the latter have to be rewritten: as an example,
\begin{equation}
\mu\sum_in_i \rightarrow \mu\sum_{\langle ij\rangle}\Big(\frac{n_i}{f_i}+\frac{n_j}{f_j}\Big)\ ,
\end{equation}
\noindent where the sum on the right-hand side runs on nearest-neighbor sites and where $f_i$ and $f_j$ are the number of links having respectively site $i$ and site $j$ as one end.\\
\indent Even though the Taylor expansion above converges \cite{S-92}, statistically relevant contributions are basically provided by configurations where the number of bond operators in Eq.(8) is finite and below an opportune value $N_{\rm tr}$; therefore, truncating the series at order $N_{\rm tr}$ for practical purposes should not entail any significant truncation error, as explained in Sec. IIA of \cite{SS-02}. In determining $N_{\rm tr}$, we opted for the standard definition, that is we set $N_{\rm tr}=1.5 \, M_{\rm max} \, N_{\rm{bonds}}/T$, where $M_{\rm max}$ is the highest matrix element of the single-bond Hamiltonians and $N_{\rm{bonds}}$ is the number of interacting site pairs. Besides checking that this cutoff was never crossed during the updating process, fluctuations in the order of the series expansion were monitored to control whether the averaged order was consistently less than $N_{\rm tr}$ (with deviations proportional to the square root of the mean value). As proven in \cite{SS-02}, these criteria ensure that the truncation error is negligible compared to the statistical uncertainty stemming from Monte Carlo fluctuations.\\
\indent Exploiting this truncation, the expression for the partition function can be further simplified, i.e.,
\begin{equation}
\label{genzed}
Z=\sum_{\alpha}\sum_{S_{N_{\rm tr}}}\frac{(-\beta)^n(N_{\rm tr}-n)!}{N_{\rm tr}!}\langle\alpha|\prod_{i=1}^{N_{\rm tr}}H_{a_i,c_i}|\alpha\rangle\ ,
\end{equation}
\noindent where $N_{\rm tr}-n$ identity operators have been inserted in all possible ways in the sequence $S_{N_{\rm tr}}$. It is understood that now the index $c_i$ can assume a third value ($c_i=0$) corresponding to the identity itself.\\
\indent In Eq. (\ref{genzed}) the space of configurations have been generalized to be $\{|\alpha\rangle\}\otimes\{S_{N_{\rm tr}}\}$. This can be sampled by means of two kinds of steps: the first type $M_1$ consists of replacing identity operators in the sequence $S_{N_{\rm tr}}$ with diagonal ones (and vice versa), while the second kind $M_2$ is given by exchanging diagonal operators with non-diagonal ones (and vice versa). An exhaustive description of both steps and of how they are performed can be found in Ref. \cite{SS-02}. Let us just recall that, in implementing kind $M_2$, a set of transition probabilities is needed and must be determined by solving so-called directed loop equations. Depending on the parameters of the Hamiltonian, it is possible to select solutions able to reduce the number of bounces,~\!\footnote{~\!This issue is treated in great detail in Sec. IID of \cite{SS-02} where the XXZ model is studied. Since its matrix elements are in one-to-one correspondence with those of the HC BH model, the discussion can be easily adapted to the present case.} that is to cut the amount of moves where the proposed change is rejected. Such solutions are preferred since they shorten the computer time needed to update the configuration. In our simulations, one bounce was allowed within each group of equations at $\mu=0$ and $\mu=2$ while two bounces entered into play when $\mu=-2$.\\
\indent One MC step is made out of a single step of type $M_1$ followed by a number $N_{\rm{loops}}$ of updates of kind $M_2$. $N_{\rm{loops}}$ is fixed at runtime by imposing that the number of visited vertices is of the order of $N_{\rm tr}$ in a MC step.\\
\indent Runs are performed fixing temperature $T$, chemical potential $\mu$, trap size $l$ and lattice size $L$ with open boundary conditions.
Finite-size effects are avoided by choosing $L$ sufficiently large to obtain $L\to\infty$ data within statistical errors. This condition was fulfilled taking $L/l\approx 3$ when $\mu=2$ and $\mu=0$ and $L/l\approx 5$ when $\mu=-2$.\\
\indent A standard jackknife was employed to assess errorbars, each bin being the mean of $10^4$ MC step measurements. Typical statistics of our QMC simulations range from $2.5\times10^6$ MC steps for simulations at $\mu=2$ to $7.5\times10^6$ MC steps for simulations at $\mu=0$.
\begin{figure}[tbp]
\vspace*{-0.87cm}
\includegraphics*[width=9cm,height=6.5cm]{Figure1.pdf}
\vspace*{-0.8cm}
\caption{(Color online) The particle density at $\mu=2$ with $\tau\equiv Tl=8$ and $\tau=2$ for some values of the trap size $l$.}
\label{densmu2}
\end{figure}
\section{The critical point at\ \ \!$\mu=2$}
We now discuss our QMC results for the density and the density-density correlator
at the Mott-insulator to superfluid transition where $\langle n_i\rangle=0$.
In analogy with the singular part of the free-energy density (\ref{freee}), the scaling ansatz for the two above-mentioned observables~\!\footnote{~\!From the conventions introduced after Eq. (\ref{vrp}), it is clear that both $\rho(r)$ and $G(r,r')$ are dimensionless quantities.} read:
\begin{align}
\label{density}
\rho(r) &= l^{-d\theta} {\cal D} ( \bar\mu l^{2\theta} , Tl^{2\theta} , rl^{-\theta} ) \; ,\\
\label{correlator}
G(r,r') &= l^{-2d\theta} {\cal G} ( \bar\mu l^{2\theta} , Tl^{2\theta} , rl^{-\theta} , r'l^{-\theta} ) \; ,
\end{align}
where the critical exponents for this transition $\nu=1/2$ and $z=2$ have been used. Scaling corrections due to irrelevant perturbations in $l^{-\theta}$ and possible analytic contributions have been neglected.
After setting $d=2$, introducing the scaling coordinates $R=rl^{-\theta}, R'=r'l^{-\theta}$,
and considering the system at criticality (so that $\bar\mu=0$), Eqs. (\ref{density}) and (\ref{correlator}) can be rewritten as
\begin{align}
\label{density2}
l^{2\theta} \rho(r) &\approx \hat{{\cal D}} (\tau,R) \; , \\
\label{correlation2}
l^{4\theta} G(r,r') &\approx \hat{{\cal G}} (\tau,R,R') \; ,
\end{align}
being $\tau \equiv Tl^{2\theta}$ the scaling variable that controls the critical behavior of the system.~\!\footnote{~\!From now on, quantities $R$, $R'$ and $\tau$ will always be defined as in this section unless differently specified.}
The meaning of Eqs. (\ref{density2}) and (\ref{correlation2}) should be pretty clear: A given observable rescaled with the proper power of the trap size $l$ equals a universal function depending on $\tau$, $R$, $R'$, etc. Therefore, data obtained via simulations with values of the parameters tuned in such a way to keep the arguments of the function on the right-hand side of Eqs. (\ref{density2}) and (\ref{correlation2}) constant should collapse on a unique curve once that the proper rescaling has been performed. For the Mott-insulator to superfluid transition in the low-density regime, this condition is fulfilled by performing simulations with fixed $Tl$ since $\theta =1/2$.\\
\begin{figure}[tbp]
\vspace*{-0.87cm}
\includegraphics*[width=9cm,height=6.5cm]{Figure2.pdf}
\vspace*{-0.8cm}
\caption{(Color online) The density-density correlator at $\mu=2$ with fixed $\tau\equiv Tl=8$ for different
values of the trap size $l$.}
\label{corrmu2}
\end{figure}
\indent While in the 1D HC BH model the particle density and the density-density correlator could be treated analytically both at zero and finite temperature (so that numerical outcomes could be compared with their analytical values \cite{CTV}), in the two-dimensional case no exact solution is available. In this study TSS is applied to a 2D-system.\\
\indent Figure \ref{densmu2} shows the rescaled particle density. Data in it are divided into two groups corresponding to simulations performed with fixed $\tau=2$ or $\tau=8$. Since $\tau\equiv Tl$ and since the values of $l$ are essentially the same in both groups, sets with $\tau=2$ are generally related to lower temperatures than those with $\tau=8$. While the latter shows scaling corrections at small $l$, it is evident that the former have a more pronounced tendency to collapse on a universal curve. This comes with no surprise since universality is a feature appearing in proximity of a phase transition, which occurs at $T=0$ when working with chemical potential fixed at $\mu=2$ as in the present case.\\
\indent Figure \ref{corrmu2} contains the rescaled density-density correlator at fixed $\tau=8$ vs.\! $R$. In analogy with the particle density, also for this observable, numerical outcomes after the rescaling prescribed by TSS display a tendency to collapse on a unique curve when increasing $l$, in agreement with the ansatz in Eq. (\ref{correlation2}). Once again, corrections can be noticed only at small values of the trap size.
\section{Local Density Approximation}
In many statistical systems featuring an external potential $V(r)$ varying with the space position, it is common to approximate the ground-state density at point $r$ with the value that the density assumes in the homogeneous system provided with a constant potential fixed everywhere at the value $V(r)$ that the potential takes at point $r$ itself in the inhomogeneous case. This approximation is called Local Density Approximation (LDA).\\
\indent LDA has already been verified to be exact in the 1D HC BH model at zero temperature \cite{CV-10-bh} and it is reasonable to test to which extent it works also in the two-dimensional case at finite $T$. In general, considering a constant potential in Eq. (\ref{bhm}) essentially means to introduce an effective chemical potential $\mu_{\rm eff}(r)$ given by
\begin{equation}\label{effective-potential}
\mu_{\rm eff}(r) \equiv \mu + \frac{r^2}{l^2}\ .
\end{equation}
Therefore, in analogy with the 1D HC BH model, we assume that the LDA of the 2D trapped system equals
\begin{equation}
\rho_{\rm LDA}(r) =
\kern-10pt \quad\left\{
\begin{array}{l@{\ \ }l@{\ \ }l}
0 & {\rm for} & \mu_{\rm eff}(r) > 2\ , \\
\rho_*(\mu_{\rm eff}) &
{\rm for} & -2 \le \mu_{\rm eff}(r) \le 2\ , \\
1 & {\rm for} & \mu_{\rm eff}(r) < -2\ , \\
\end{array} \right.
\end{equation}
where $\rho_*(\mu)$ is the unknown $T=0$ density of the 2D homogeneous system provided with an effective chemical potential given by Eq. (\ref{effective-potential}).\\
\begin{figure}[t]
\vspace*{-0.87cm}
\includegraphics*[width=9cm,height=7cm]{Figure3.pdf}
\caption{(Color online) Numerical outcomes for $\rho_*({\mu})$ vs. $\mu$ for different values of the lattice extent $L$ and temperature $T$. Data were collected in the homogenous system with periodic boundary conditions. The dotted line represents the polynomial fit of the data corresponding to the largest extent.}
\label{LDA_fit}
\end{figure}
\indent In order to obtain an estimate for $\rho_*(\mu)$, we performed simulations of the 2D system without a trap and with periodic boundary conditions~\!\!\footnote{~\!This is expected to reduce finite-size corrections.} in the low-temperature regime for different values of the effective chemical potential. In particular, we employed a set of equally-spaced values covering the range from $-2$ to $+2$.
This setup is easily obtained by setting to zero the trap parameter $v$ in our QMC code and by implementing the specific topology,
all other features of the simulation remaining the same.\\
\indent More specifically, we performed simulations with $L=8$ at $T=1/64$, with $L=16$ at $T=1/128$ and with $L=32$ at $T=1/256$, being $L$ the extent of a square lattice. Following the same criteria of \cite{BBMSTD-02}, we checked that the data were consistent within errorbars so that we could safely assume that the results at $T=1/256$ correspond effectively to the zero-temperature values. Figure \ref{LDA_fit} displays $\rho_*(\mu)$ for the three sets of simulation parameters mentioned above; data basically overlap.\\
\indent Finally we fitted the $T=1/256$ outcomes to a generic polynomial function of degree $n$:
\begin{equation}\label{fit-function}
\rho_*(\mu) = \sum_{i=0}^n c_i \, \mu^i \ ,
\end{equation}
where $n$ was chosen by truncating this Taylor expansion when the $\chi^2$ of the fit stabilized. This was the case with $n=7$ (the reduced $\chi^2$ being approximately $1.5$) though truncations at higher order were also considered without showing meaningful deviations. As expected on theoretical grounds, it turned out that the constant term $c_0$ read $1/2$ (within $10^{-7}$) while even terms were negligible; thus, the only non-trivial contributions are given by the odd powers for which the following estimates were obtained:
\begin{eqnarray}\label{fit-coefficients}
&&c_1=-0.20779(1) \;\;\; ,\ c_3=-0.01323(1) \;\;\;\nonumber\\
&&c_5=+0.00441(1) \;\;\; ,\ c_7=-0.00093(1) \; .
\end{eqnarray}
\indent The function in Eq. (\ref{fit-function}) with $n=7$ and coefficients as given above is plotted in Fig. \ref{LDA_fit} and was used for the data analysis reported in the following sections.
\begin{figure}[tbp]
\vspace*{-0.87cm}
\includegraphics*[width=9cm,height=7cm]{Figure4a.pdf}
\includegraphics*[width=9cm,height=7cm]{Figure4b.pdf}
\includegraphics*[width=9cm,height=7cm]{Figure4c.pdf}
\vspace*{-0.8cm}
\caption{(Color online) The particle density at $\mu=-2$ with fixed $\tau\equiv Tl=2$ (top) and $\tau=8$ (middle) for different
values of the trap size $l$ and the scaling of the subtracted particle density at $\mu=-2$ with $\tau=8$ (bottom). The dotted line in the first two plots represents the numerical estimates of the LDA.}
\label{densmu-2}
\end{figure}
\section{$n=1$\ \ \! Mott transition}
The invariance under the particle-hole exchange entails a similar behavior of the homogeneous HC BH model at the transitions with $\mu=2$ and $\mu=-2$.
However, the trap-size scaling behavior at the $\langle n_i\rangle=1$ transition is expected to be different than in the vacuum-to-superfluid one because the particle-hole symmetry does not hold for a trapped system.\\
\indent Studies of the trapped 1D HC BH model at zero and finite temperature \cite{CV-10-bh,CTV} have revealed how, at the superfluid to Mott transition with non-zero filling, the particle density approaches its Local Density Approximation (LDA) in the large-$l$ limit. In analogy with the one-dimensional case, we thus expect this observable to be given by an expression like
\begin{eqnarray}
\label{LDA}
\rho(r) &=& \rho_{\rm LDA}(rl^{-1}) + l^{-2\theta} \hat{{\cal D}} (Tl^{\theta z},rl^{-\theta})\ =\nonumber\\
&=& \rho_{\rm LDA}(rl^{-1}) + l^{-1} \hat{{\cal D}} (\tau,R) \; ,
\end{eqnarray}
since $\theta=1/2$ and $z=2$ again; irrelevant corrections in $l^{-\theta}$ have been neglected once more. \!Therefore, the scaling quantity should not be the density itself but rather the difference
\begin{equation}
\label{drho}
\Delta\rho(r)\equiv\rho(r)-\rho_{\rm LDA}(rl^{-1})\ .
\end{equation}
\begin{figure}[t]
\vspace*{-0.87cm}
\includegraphics*[width=9cm,height=7cm]{Figure5.pdf}
\vspace*{-0.8cm}
\caption{(Color online) Scaling of the density-density correlator at $\mu=-2$ with fixed $\tau=8$.}
\label{densmu-2s}
\end{figure}
\indent Figure \ref{densmu-2} shows how the particle density converges to the LDA in the two-dimensional model. Since $\tau\equiv Tl$ as in Sec. III, once again data sets with $\tau=2$ correspond to temperatures lower than those of the sets with $\tau=8$, given the common values of the trap size. Besides improving with increasing $l$ in agreement with Eq. (\ref{LDA}), the convergence to the LDA is better at small $T$, as clear from comparing the upper and middle part of Fig. \ref{densmu-2}, since LDA itself is approached at $T\rightarrow0$.\\
\indent The lower part of Fig. \!\ref{densmu-2} illustrates the behavior of $\Delta\rho(r)$ at $\tau=8$ after the rescaling suggested by Eq. (\ref{LDA}) has been performed: a tendency to collapse on a universal curve is evident in a region close to the origin while some transition-like peaks appear at a distance $r\approx2l$ from the center, thus drifting with increasing $l$. Therefore, in the $l\!\!\rightarrow\!\!+\infty$ limit, these peaks disappear and only the region with the universal curve is left, hence confirming TSS predictions in Eq. (\ref{LDA}). The deviations from the expected scaling at finite $l$ will be treated in Sec. VI.\\
\begin{figure}[t]
\vspace*{-0.87cm}
\includegraphics*[width=9cm,height=7cm]{Figure6a.pdf}
\includegraphics*[width=9cm,height=7cm]{Figure6b.pdf}
\vspace*{-0.8cm}
\caption{(Color online) The particle density vs. $r/l$ (top) and the rescaled subtracted particle density (bottom) at $\mu=0$ with fixed $\tau\equiv Tl=2$.}
\label{densmu0noresc}
\end{figure}
\indent As for the density-density correlation function, scaling expectations, again given by Eq. \!\!\!(\ref{correlation2}), are also nicely confirmed, as depicted in Fig. \!\ref{densmu-2s}. At $\tau=8$, besides scaling corrections at small trap size, no strong deviations from a universal curve could be detected within errorbars. In the 1D HC BH a striking result was the universality for the correlator between Mott-to-superfluid transitions at $\langle n_i\rangle=0$ and at $\langle n_i\rangle=1$. A closer inspection at Figs.\! \ref{corrmu2} and \ref{densmu-2s} reveals how this feature seems to hold in the 2D case as well.\\
\indent In the 1D HC BH at zero temperature \cite{CV-10-bh}, a peculiar characteristic of scaling quantities like the subtracted particle density or the density-density correlator was the appearance of modulations depending on the trap size $l$. It was shown that, at fixed $\mu<1$, there are values of the trap size $l$, whose number increases with $l$ itself, for which the energy gap $\Delta E$ between the ground state and the first excited state vanishes. This repeated level crossing leads to modify the scaling ansatz like Eqs. (\ref{density2}) and (\ref{correlation2}) in order to include a further dependence of the universal function on a phase $\phi$ related to the difference between values of $l$ corresponding to zeros of $\Delta E$. At finite temperature, it is expected that this phenomenon plays a lesser and lesser role with increasing $T$ since thermal fluctuations should prevail on quantum effects. This has already been checked in the 1D BH model \cite{CTV} and is confirmed also in two dimensions as Figs. \ref{densmu-2} and \ref{densmu-2s} reveal.~\!\footnote{~\!Recalling the phase diagram, it has to be borne in mind that the condition $\mu<1$ in 1D corresponds to $\mu<2$ in 2D.}
\section{The spatial region where\ \! $\mu_{\rm eff}=2$}
We are now going to study the 2D BH model at $\mu=0$. Since this corresponds to the deep interior of the superfluid phase, no transition is expected to be monitored. However, a closer look at Eq. (\ref{bhm}) suggests to group the last two terms of the BH Hamiltonian, thus giving rise to the effective chemical potential $\mu_{\rm eff}(r)$ already introduced in Eq. (\ref{effective-potential}).\\
\begin{figure}[t]
\vspace*{-0.87cm}
\includegraphics*[width=9cm,height=7cm]{Figure7.pdf}
\vspace*{-0.8cm}
\caption{(Color online) The density-density correlator at $\mu=0$ with fixed $\tau\equiv Tl=2$.}
\label{corrmu0noresc}
\end{figure}
\begin{figure}[t]
\vspace*{-0.87cm}
\includegraphics*[width=9cm,height=7cm]{Figure8a.pdf}
\includegraphics*[width=9cm,height=7cm]{Figure8b.pdf}
\vspace*{-0.8cm}
\caption{(Color online) Scaling of the subtracted particle density (top) and the density-density correlator (bottom) at $\mu=0$ with fixed $Tl^{2/3}=1$ around distance $r_c$ where $\mu_{\rm eff}=2$.}
\label{denscorrmu0}
\end{figure}
\indent After this step, it turns out that, in those regions where $\mu_{\rm eff}(r)\approx 2$, the Hamiltonian is effectively given by that of the homogeneous system at criticality;~\!\footnote{~\!The number of dimensions is understood to be $d=2$, otherwise a critical regime would appear in different sectors of the lattice.} therefore, we might expect some sort of phase transition and universal behavior in these regions even if $\mu$ is not set to a critical value.\\
\indent This seems to be indeed confirmed by the behavior of the particle density portrayed in Fig. \ref{densmu0noresc}. If $\Delta\rho(r)$ defined in Eq. (\ref{drho}) is rescaled with $l$ and plotted versus $r/l$ as suggested by Eq. (\ref{LDA}),~\!\footnote{~\!Recall that $z=1$ in the superfluid phase while $\theta=1$ for smooth modes according to an ansatz verified in Ref. \cite{CV-10-bh}.} no substantial difference with the LDA is observed except for some transition-like peaks at ``critical" distance $r_c=\sqrt{2}l$, corresponding to the distance from the center obtained from Eq. (\ref{effective-potential}) after setting $\mu_{\rm eff}(r)=2$ and $\mu=0$.\\
\indent We have already encountered such a situation in Sec. V. Indeed, by setting $\mu_{\rm eff}(r)=2$ and $\mu=-2$ again in Eq. (\ref{effective-potential}), this results in $r=2l$, corresponding to the distance at which peaks were observed in the $\langle n_i\rangle=1$ Mott-insulator phase (see lower part of Fig. \!\ref{densmu-2}).\\
\indent In order to study the scaling in such spacial sectors of the system, an expansion of $\mu_{\rm eff}(r)$ around $r_c$ is needed, i.e.,
\begin{equation}
\mu_{\rm eff}(r) = 2 + p(2-\mu )^{1-1/p} \, \frac{r-r_c}{l} + O[(r-r_c)^2] \ .
\end{equation}
As pointed out in Ref. \cite{CTV}, the length scale $\xi$ of these critical modes should behave like $\xi\approx l^{\sigma}$, $\sigma$ being the exponent associated to a linear potential. In other words, $\sigma=1/3$ \cite{CV-10-bh} and, again neglecting irrelevant contributions in $l^{-\theta}$, scaling should read
\begin{align}
\label{dr2}
l^{2/3} \Delta\rho(r) &\approx \hat{{\cal D}} (\tau,R) \; , \\
\label{co2}
l^{4/3} G(r,r_c) &\approx \hat{{\cal G}} (\tau,R) \; ,
\end{align}
where $R$ and $\tau$ correspond now to $R=(r-r_c)/l^{1/3}$ and $\tau\equiv Tl^{2/3}$. In deriving the latter exponent the value $z=2$ has been employed since we expect critical modes close to $r_c$ to be controlled by the same dynamical exponent as in the regular Mott-to-superfluid transitions. The exponents in Eqs. (\ref{dr2}) and (\ref{co2}) do not coincide with those in Fig. \ref{densmu0noresc}, where also $T$ and $l$ are not properly tuned, and this explains why data sets do not collapse on a unique curve around $r_c=\sqrt{2}l$. Because of the ``improper" values of the trap size and the temperature, also the particle-particle correlator $G(r,0)$ in Fig. \ref{corrmu0noresc} does not show any particular scaling but simply vanishes after a few lattice spacings.\\
\indent Figure \ref{denscorrmu0} shows the behavior of $\Delta\rho(r)$ and $G(r,r_c)$ vs. $(r-r_c)/l^{1/3}$ at $Tl^{2/3}=1$ after these observables have been rescaled according to Eqs. (\ref{dr2}) and (\ref{co2}). The collapsing of both quantities on a single curve proves the foreseen scaling clearly right.
\section{Conclusions}
We have studied the scaling properties of the two-dimensional Bose-Hubbard model at finite temperature in the presence of a trapping potential at the Mott-insulator to supefluid transitions. The interest in this system and its properties is not only theoretical given that 2D experiments involving cold atoms in optical lattices are currently being carried out~\cite{SPP-07,JCLPPS-10,SPP-08}.\\
\indent Particular attention has been paid to the particle density and the density-density correlator, both computed by means of QMC simulations in the hard-core limit $U\to+\infty$. The latter choice is motivated by the fact that the on-site coupling $U$ should play no role in determining the main properties of the system at criticality. A comparison between numerical outcomes and TSS ansatz has subsequently been performed, revealing that theoretical expectations are well-motivated.\\
\indent An interesting feature arising from our study is how LDA compares to the particle density $\rho(r)$ at finite temperature in two dimensions. As in the one-dimensional case \cite{CTV}, it turns out that also in the 2D HC BH model $\rho(r)$ rapidly converges to the LDA when increasing $l$ even at $T>0$, regardless the value of the chemical potential $\mu$. Even if LDA turned out to be broken in frustrated systems \cite{LM-11} and some shortcomings were proven also in describing the $T=0$ phase diagram of the BH model in two dimensions \cite{MDKKTT-11}, this work shows that LDA is capable of describing at least some properties of the 2D BH model quite nicely, also outside its theoretical range of application.
\acknowledgements \vspace*{-0.08cm} We warmly thank E. Vicari for his suggestions as well as for a careful reading of a first draft of this manuscript. The QMC simulations were performed at the Istituto Nazionale di Fisica Nucleare (INFN) Pisa GRID DATA center, using also the cluster CSN4.
|
{
"timestamp": "2012-06-06T02:03:43",
"yymm": "1203",
"arxiv_id": "1203.2030",
"language": "en",
"url": "https://arxiv.org/abs/1203.2030"
}
|
\section{Introduction\label{sec.introd}}
For numerical analysis of the Solar System dynamics it is chosen as standard of measurement the Astronomical Unit (AU). This standard of measurement
fixes the measurement projection for spatial-lengths and, through Kepler's third law, relates the measurement projection for spatial-lengths with
the measurement projection for temporal-lengths. Specifically the definition of the AU is stated in IS units as~\cite{AU_def,AU_1,AU_2,dG}
\begin{equation}
AU=\left(G\,M_\odot\,\left(\frac{T_{1AU.0}}{2\pi}\right)^2\right)^\frac{1}{3}=149597870700\,m\ ,
\label{AU}
\end{equation}
where $M_{\odot}=1.9891\times 10^{30}\,kg$ is the Sun's mass, $G=6.67\times 10^{-11}\,kg^{-1}\,m^3\,s^{-2}$
is the Gravitational constant, $T_{1AU.0}=31562889.928\ldots\,s$ is the Keplerian orbital period for a point mass in an elliptic orbit with semi-major axis of value $1\,AU$. This definition exactly matches Kepler's third law for a planet orbiting the Sun in an elliptic orbit with semi-major axis of length $r_{1.\mathrm{orb}}=1\,AU$. This definition is based on the existence of a Keplerian constant of motion, specifically the angular momentum $J_0(r_{1.\mathrm{orb}}=1\,AU)=-\sqrt{G\,M\,AU\,(1-e^2)}$, where $e$ is the orbit eccentricity, hence time and space measurements are related by this classical conservation law stated in the definition~(\ref{AU}).
Considering experimental data from optical and range measurements of planetary orbital motion plus range measurement from orbiters and landers it was concluded that a yearly variation of the Astronomical Unit is required to fit the ephemerides in the Solar System. Hence adding
the adicional parameter $\dot{AU}$ to the dynamical model, the best fit to this parameter was originally estimated to be~\cite{AU_1,AU_2}
\begin{equation}
\left<\dot{AU}\right>_{\mathrm Fit}=+(15\pm 4)\,cm\,yr^{-1}\ .
\label{dAU}
\end{equation}
The value for this heuristic fit does not necessarily correspond to a physical radial distance variation in the Solar System, instead it corresponds to a variation of the standard of measurement considered for numerical analysis of the Solar System~(\ref{AU}).
As an alternative to the variation of the AU, it has been suggested that a variation of the product of the Gravitational constant by the Solar mass $(GM_\odot)$ would have a similar effect in the planetary dynamics. Specifically the best fit corresponds to the value $(\dot{GM}_\odot)/(GM_\odot)=-(5\pm 4)\times 10^{-14}\,yr^{-1}$~\cite{dG}. We note that the variation of $(GM_\odot)$ has the effect of increasing both the orbital radius and the orbital period such that angular momentum is approximately conserved. Specifically the period varies as $\dot{T}_0/T_0\approx-2(\dot{GM}_\odot)/(GM_\odot)$ and the orbital radius as $\dot{AU}/AU\approx-(\dot{GM}_\odot)/(GM_\odot)$~\cite{dG1,dG2}. Hence, from Kepler's third law~(\ref{AU}), a variation for $(GM_\odot)$ can be directly mapped to a non-null value for the parameter $\dot{AU}$ while keeping $(\dot{GM}_\odot)$~\cite{dG}
\begin{equation}
\left<\dot{AU}\right>_{\mathrm Fit}=0.75\pm 0.60\,cm\,yr^{-1}\ .
\label{dAU_ephemerids}
\end{equation}
The discrepancy between both fitted values~(\ref{dAU}) and~(\ref{dAU_ephemerids}) is due to the choice of the independent set of parameters considered in the fit. While in~\cite{AU_1} both $AU$ and $\dot{AU}$ are fitted, in~\cite{dG} only $(\dot{GM}_\odot)$ is fitted. This choice is justified in~\cite{dG} by noting that $AU$ and $\dot{AU}$ are correlated by $98.1\%$. We remark that it is also recognized in~\cite{dG} that the choice of whether to fit the parameter $\dot{GM}$ or $\dot{AU}$ is a matter of convenience and that both cannot be fitted simultaneously as the obtained value of $\dot{AU}$ does not exceeds the respective formal error. Actually both these choices are interpreted as a varying scaling of the measurement standard~(\ref{AU}), hence a choice of the measurement projection (see~\cite{Uzan} for further details), being directly mapped into eachother through Kepler's third law.
This discussion is enough to conclude that some sort of unmodelled gravitational interaction seems to be acting in the Solar System which is effectively accounted for by an heuristic rescaling of the relation between spatial lengths and temporal lengths of the dynamical system model. The exact identification of the origin of such interaction is further difficulted due to the only available solution relying on the heuristic fit of one single isotropic parameter $\dot{AU}$ (or equivalently $\dot{GM}$) across the Solar System. In this work our main objective is to employ a background described by an expanding locally anisotropic (ELA) metric~\cite{PLB, eprint} to analytically model the corrections to orbital motion with respect to orbital motion on Schwarzschild background within the Solar System. In particular, based in analytical orbital solutions for the background described by the ELA metric we will explicitly compute the respective corrections to the orbital period due to two distinct contributions, the Sun's mass decrease corrections and the General Relativity corrections with respect to Schwarzschild backgrounds~\cite{review}. It is shown that it is possible to map these corrections to the heuristic variation of the AU, having simultaneously negligible contributions to the remaining orbital parameters. Considering the most reliable estimative for the average value of the parameter $\dot{AU}$~(\ref{dAU_ephemerids}) we will compute bounds for the values of the metric functional parameter which, in turn, will allow to estimate the contributions due to the ELA metric background to the orbital period, orbital precession and orbital radius variation of the several planets within the Solar System.
The ansatz for the ELA metric was originally suggested as a possible description of local matter distributions in the expanding universe~\cite{Hubble,Inflation}, hence interpolating between the Schwarzschild (SC) metric~\cite{Schwarzschild} near the central mass and the Robertson-Walker (RW) metric~\cite{RW} describing the expanding cosmological background far from the central mass. Hence the ELA metric generalizes the isotropic McVittie metric~\cite{McVittie} and the anisotropic metrics considered in~\cite{SCS}, having the novelty of maintaining the SC event horizon and maintain as the only space-time singularity the SC mass pole at the origin such that the value of the Schwarzschild mass pole is maintained~\cite{sing,PLB}. This metric is locally anisotropic consistently with astrophysical observations~\cite{anisotropy,WMAP} converging at large radius to the RW metric such that global isotropy is maintained. Specifically the line-element for the ELA metric is
\be
\ba{rcl}
ds^2&=&\displaystyle\left(1-U_{\mathrm{SC}}\right)c^2\,dt^2-r_1^2\left(d\theta^2+\sin^2\theta d\varphi^2\right)\\[6mm]
& &\displaystyle-\frac{1}{1-U_{\mathrm{SC}}}\left(dr_1-H\frac{\,r_1}{c}\left(1-U_{\mathrm{SC}}\right)^{\frac{\alpha}{2}+\frac{1}{2}}c\,dt\right)^2\ .
\ea
\label{g_generic}
\ee
where $H=\dot{a}/a$ is the time dependent Hubble rate defined by the rate of change of the universe scale factor $a$, $U_{\mathrm{SC}}=2GM/(c^2r_1)$ is the usual Schwarzschild gravitational potential, $G$ is the Gravitational constant, $M$ is the value of the Schwarzschild mass pole for the central mass being considered and $c$ is the speed of light in vacuum. The exponent $\alpha$ is, generally, a function of the radial coordinate, here we will take the simplified ansatz studied in~\cite{PLB}
\be
\alpha(r_1)=(\bar{\alpha}_0-\alpha_1)+\alpha_1\,U_{\mathrm{SC}}(r_1)=(\bar{\alpha}_0-\alpha_1)+\alpha_1\,\frac{2GM}{c^2\,r_1}\ ,
\label{alpha_r_1}
\ee
where $\bar{\alpha}_0$ and $\alpha_1$ are numerical coefficients. The bound $\bar{\alpha}_0\ge 3$ ensures that the SC radius $r_{1.\mathrm{SC}}=2GM/c^2$ is an event horizon and space-time is singularity free at this horizon, while for the bound $\bar{\alpha}_0>5$ space-time is asymptotically
Ricci flat near the event horizon such that the SC metric is a good approximation in a neighbourhood of the point-like mass $M$.
The coefficient $\alpha_1<0$ ensures that the singularity at the origin coincides with the SC mass-pole. In the following analysis
we will consider it arbitrarily close to null, $-1\ll \alpha_1< 0$, such that outside the event horizon its effects are negligible, hence $\alpha\approx \bar{\alpha}_0$ for planetary orbits. For further details in the derivation of the ELA metric ansatz see~\cite{PLB}.
We note that so far no direct physical interpretation for the functional parameter $\alpha$ exists, hence
it is at most considered to be a functional parameterization of corrections to gravitational interactions for intermediate spatial-length scales.
In particular we may assume that it is varying over the radial coordinate allowing to phenomenologically fit existing data hoping that the results obtained may shed some light on its physical interpretation (see for instance~\cite{Pioneer_ELA}).
In order to fit the constant $\bar{\alpha}_0$ to planetary motion within the Solar System maintaining the asymptotic limits of the ELA metric we are considering the simpler possible ansatz for this constant by considering that near the event horizon of the central mass (the Sun) the exponent is some constant $\alpha_{0.0}\geq 3$ and that far from the event horizon (hence for all planetary orbits) the constant $\bar{\alpha}_0$ is approximately given by a constant $\alpha_0$ which can be either positive, either negative. Specifically we are considering the simplified ansatz
\begin{equation}
\bar{\alpha}_0=\left\{\begin{array}{lcl}\displaystyle \alpha_{0.0}\geq 3&,&\displaystyle r_1\sim R_{\odot\mathrm SC}\\[5mm] \displaystyle \alpha_0&,&\displaystyle r_1\gg R_{\odot\mathrm SC}\end{array}\right.
\end{equation}
where $R_{\odot\mathrm SC}=2 G M_\odot/c^2\approx 2952.22\,m$ is the Sun's Schwarzschild radius and $M_\odot$ is the Sun's mass.
Based in this ansatz we will compute the analytical corrections to two body heliocentric orbital motion within the solar system on the ELA metric background~(\ref{g_generic}) with respect to orbital motion on Schwarzschild backgrounds~\cite{Kenyon,Gravitation}. More accurate results including the many body gravitational interactions in the Solar System can only be computed by extensive numerical analysis including 10000 known bodies in the solar system~\cite{Pitjeva}. These analysis are usually carried either in the Post-Newtonian formalism, either in the Post-Post-Newtonian formalism which includes both the General Relativity corrections to the classical Newton law of gravitation on Schwarzschild backgrounds, as well as corrections of extended theories of gravity such as Brans-Dicke gravity~\cite{PPN}.
When required, for numerical evaluation of the Hubble rate $H$ and the deceleration factor $q$ of today's universe, we are considering the values
$H_0=\left.H\right|_{t=t_0}=2.28\times 10^{-18}\,s^{-1}$ and $q_0=\left.-\frac{\ddot{a}}{a}\left(\frac{\dot{a}}{a}\right)^{-2}\right|_{t=t_0}=-0.582$~\cite{WMAP}. As for the planetary orbital parameters considered for numerical evaluations we are considering the data presented in table~\ref{table.planet_data}.
\begin{table}[ht]
\begin{center}
{\small
\begin{tabular}{l|ccccc}
Planet &$r_{1.\mathrm{orb}}\,(\times 10^{11}\,m)$&$e$ &$m\,(\times 10^{24}\,kg)$&$\mathrm{\delta_{orb}\,(deg)}$&$\mathrm{\delta_{per}\,(deg)}$\\\hline
Mercury &$0.579091768$&$0.20563069$&$0.3302$ &$7.00487$ &$77.45645$\\
Venus &$1.08208926$ &$0.00677323$&$4.8685$ &$3.39471$ &$131.53298$\\
Earth &$1.49597887$ &$0.01671022$&$5.9736$ &$0$ &$102.94719$\\
Mars &$2.27936637$ &$0.09341233$&$0.64185$&$1.85061$ &$336.04084$\\
Jupiter &$7.78412027$ &$0.04839266$&$1898.6$ &$1.30530$ &$14.75385$\\
Saturn &$14.26725413$&$0.05415060$&$568.46$ &$2.48446$ &$92.43194$\\
Uranus &$28.70972220$&$0.04716771$&$86.832$ &$0.76986$ &$170.96424$\\
Neptune &$44.98252911$&$0.00858587$&$102.43$ &$1.76917$ &$44.97135$\\
Pluto &$59.06376272$&$0.24880766$&$0.0125$ &$17.14175$&$224.06676$\\\hline
\end{tabular}}
\noindent\hspace{-2cm}\caption{\it \small Planetary orbits parameters considered: the semi-major axis
$r_{1.\mathrm{orb}}$, the eccentricity $e$, the mass, the orbital inclination $\delta_{orb}$ and the longitude of perihelion $\delta_{per}$~\cite{NASA}.\label{table.planet_data}}
\end{center}
\end{table}
This work is organized as follows. In section~\ref{sec.orbits} are computed the analytical solutions for orbital motion on the ELA metric background being derived the General Relativity corrections to orbital precession and orbital period for such backgrounds. This derivation is carried considering a static elliptical orbit approximation. In section~\ref{sec.radius} are analysed circular orbits on the ELA metric background~(\ref{g_generic}) and computed the orbital radius variation within this approximation. In section~\ref{sec.AU} are analysed the corrections to Kepler's third law due to the decrease of the Sun's mass and due to the ELA metric background. In particular are derived estimative for the several contributions that allow to match the heuristic variation of the AU and computed the value for the ELA metric parameter that corresponds to the fitted value of the parameter $\dot{AU}$~(\ref{dAU_ephemerids}), as well as the respective corrections to orbital precession and orbital radius variation. In the conclusions we shortly resume and discuss the results obtained in this work.
\section{Perturbative Static Elliptical Orbit Solutions\label{sec.orbits}}
In this section we derive analytical solutions for elliptical orbits on the background described
by the ELA metric ansatz~(\ref{g_generic}). In particular we will explicitly compute the General
Relativity corrections on such backgrounds to the Keplerian elliptical orbit solution $r_1(\varphi)=1/u_0(\varphi)$ with
\begin{equation}
u_0(\varphi)=\frac{1+e\,\cos(\varphi)}{d}\ \ ,\ \ d=r_{1.\mathrm{orb}}(1-e^2)\ ,
\label{u0.0}
\end{equation}
where $e$ is the orbit eccentricity and $r_{1.\mathrm{orb}}$ is the orbital radius. We note
that such corrections will generally include the General Relativity corrections on Schwarzschild
backgrounds as well as corrections depending on the Hubble rate $H$.
It is hard, if not impossible, to obtain an exact
analytical solution considering the differential equations for a time varying Hubble rate $H$.
The main difficulty is that energy conservation is no-longer given by a constant of motion,
instead we have a non-linear second order differential
equation on the function $t$ coupled to the differential equation
for $r_1$. Hence, for technical simplification purposes,
we are taking the static orbit approximation by considering a fixed Hubble rate $H=H_0$
corresponding to today's measured value for this rate.
From the metric line-element~(\ref{g_generic}) let us consider the Lagrangian definition~\cite{Kenyon,Gravitation} to order $H_0^2$
\be
\ba{rcl}
\displaystyle\frac{{\mathcal{L}}}{m}&=&\displaystyle\left(1-U_{\mathrm{SC}}-\left(H_0\,\frac{r_1}{c}\right)^2\left(1-U_{\mathrm{SC}}\right)^\alpha\right)\,\left(c\,\frac{dt}{d\tau}\right)^2\\[6mm]
&&\displaystyle+2H_0\,\frac{r_1}{c}\,\left(1-U_{\mathrm{SC}}\right)^{\frac{\alpha_0}{2}-\frac{1}{2}}\,c\,\frac{dt}{d\tau}\,\frac{dr_1}{d\tau}-\frac{1}{1-U_{\mathrm{SC}}}\left(\frac{dr_1}{d\tau}\right)^2-r_1^2\left(\frac{d\varphi}{d\tau}\right)^2+O\left(H_0^2t\right)\ .
\ea
\label{A.L_orbit}
\ee
This Lagrangian is a constant ${\mathcal{L}}/m=c$ and it is considered that the orbit of the test body is lying in the plane of constant coordinate $\theta=\pi/2$ such that $d\theta=0$ and $\sin\theta=1$~\cite{Kenyon,Gravitation}. The Lagrangian is independent of the coordinate $\varphi$, hence
a constant of motion corresponding to angular momentum exists being given by the variational
derivation of the Lagrangian with respect to $d\varphi/d\tau$,
\begin{equation}
J=\frac{1}{2m}\frac{\delta{\mathcal{L}}}{\delta\frac{d\varphi}{d\tau}}=-r_1^2\,\frac{d\varphi}{d\tau}\ .
\label{J}
\end{equation}
Also, due to the Lagrangian~(\ref{A.L_orbit}) not depending explicitly on the time coordinate, a conserved constant of motion corresponding
to energy exists being given by the functional variation of the Lagrangian with respect to $c\,dt/d\tau$
\be
\ba{l}
\displaystyle\frac{2E_H}{m\,c}=\displaystyle\frac{1}{m}\,\frac{\delta{\mathcal{L}}}{\delta(c\,dt/d\tau)}\\[6mm]
=\displaystyle 2\left(1-U_{\mathrm{SC}}-\left(H_0\,\frac{r_1}{c}\right)^2\left(1-U_{\mathrm{SC}}\right)^\alpha\right)\,\left(c\,\frac{dt}{d\tau}\right)+2H_0\,\frac{r_1}{c}\,\left(1-U_{\mathrm{SC}}\right)^{\frac{\alpha_0}{2}-\frac{1}{2}}\,\left(\frac{dr_1}{d\tau}\right)\ .
\ea
\ee
This equation can be solved for $c\,dt/d\tau$ such that replacing the obtained solution in the Lagrangian~(\ref{A.L_orbit}), expressing
the derivatives with respect to proper time $dr_1/d\tau$ by the derivatives with respect to $\varphi$,
$dr_1/d\tau=dr_1/d\varphi\times d\varphi/d\tau$ and considering the change of variables $u=1/r$,
further differentiating with respect to $\varphi$ and factoring out an overall factor of $2u'J^2$ (with the primed quantities representing derivation
with respect to $\varphi$), we obtain the approximate differential equation of order $H_0^2$ for the function $u(\varphi)$ describing an orbiting test mass in the gravitational field of a point-like central mass $M$
\be
\ba{rcl}
u''+u&=&\displaystyle\frac{GM}{J^2}+\frac{3GM}{c^2}\,u^2\\[6mm]
&&\displaystyle -\alpha_0\,\frac{GM}{c^2}\left(\frac{H_0}{c}\right)^2\,\left(1-\frac{2GM}{c^2}\,u\right)^{-1+\alpha_0}\\[6mm]
&&\displaystyle -\left(\frac{H_0}{J}\right)^2\,\frac{1}{u^3}\,\left(1-\frac{2GM}{c^2}\,u\right)^{-1+\alpha_0}\,\left(1-\frac{2GM}{c^2}\,u+\frac{\alpha_0\,GM}{c^2}\,u\right)\ .\\[6mm]
\ea
\label{A.Eq_u}
\ee
The terms in the first line match the usual terms obtained for Schwarzschild backgrounds and the terms in the second and third lines are the corrections due to the ELA metric background.
We note that, although maintaining the terms of order $H_0^2$ in the Lagrangian~(\ref{A.L_orbit}) which do not depend explicitly on the time coordinate we have neglected one term containing the factor $H_0^2\,t$. Explicitly it is the term $-2q_0H_0^2\,t\,r_1\,(1-U_{\mathrm{SC}})^{(\alpha_0-1)/2}(dt/d\tau)\,(dr_1/d\tau)$.
Comparing the terms of order $H_0^2$ in the Lagrangian with this term we conclude that this is a valid approximation as long as the value of the time coordinate is below the following bound
\be
t\ll -\frac{1}{q_0}\,\frac{r_1}{dr_1/d\tau}\,(1-U_{\mathrm{SC}})^{\frac{\alpha_0}{2}-\frac{1}{2}}\,\frac{E_H}{mc^2}\sim 10^{13}\ years.
\label{t_bound}
\ee
This bound is well above any astrophysical measurement time span and has been obtained by considering the following simplified assumptions,
within the solar system, from the experimental upper bounds on the orbital radius variations within the Solar System~\cite{Uzan} we cosider
the estimative for the ratio $|r_1/\dot{r}_1|> 10^{20}$, assume weak gravitational field $U_{\mathrm{SC}}\ll 1$ and values of $\alpha_0$ for which the approximation $(1-U_{\mathrm{SC}})^{(\alpha_0-1)/2}\sim 1$ is valid. We note that this approximation will no longer be valid for very large values of the metric exponent, $|\alpha_0|\gg 0$.
Noting that for orbits in the solar system the function $u$ has relatively small values ($0.5\times 10^{-12}<u<0.5\times 10^{-10}\,m^{-1}$, where $r_{1.\mathrm{orb}}$ is the orbit semi-major axis), with the objective of further simplifying the differential equation~(\ref{A.Eq_u}), we consider a series expansion on $u$ of the terms of order $H_0^2$ which is equivalent to an expansion on the weak gravitational field. Specifically, for a generic exponent $p$, the factor $(1-U_{\mathrm{SC}})^p$ has the following series expansion
$\left(1-2GM\,u/c^2\right)^p=1-p\,2GM\,u/c^2+p\left(p-1\right)\left(2GM/c^2\right)^2\,u^2/2-p\,(p-1)(p-2)\left(2GM/c^2\right)^3\,u^3/6+O\left(p^4\left(2GM/c^2\right)^4\,u^4\right)$.
We note that the full series is strictly convergent independently of the value of the exponent $p$ as long as $u<c^2/(2GM)$, however an approximation to first order on $u$ will only be valid as long as $2p\,GM/c^2\,u<1$, otherwise it is required to consider higher order terms to attain a valid approximation. Hence the differential equation~(\ref{A.Eq_u}) is, to order $u^2$, rewritten as
\be
\ba{rcl}
u''(\varphi)+A\,u(\varphi)&\approx&\displaystyle \frac{GM}{J^2}\,B+\frac{3GM}{c^2}\,C\,u^2\\[6mm]
&&\displaystyle-\left(\frac{H_0}{J}\right)^2\,\frac{1}{u^3}+\alpha_0\left(\frac{H_0}{J}\right)^2\,\frac{GM}{c^2}\,\frac{1}{u^2}+O\left(u^3\right)\ .
\ea
\label{orbits_exp}
\ee
We note that with respect to the General Relativity orbit's equation on Schwarzschild backgrounds, there are the extra multiplicative factors $A=1+\delta_A$, $B=1+\delta_B$ and $C=1+\delta_C$. These factors differ from unity by the following additive constants
\be
\ba{rcl}
\delta_A&=&\displaystyle-2(\alpha_0-1)\alpha_0\left(\frac{GM\,H_0}{c^3}\right)^2\left(1+(\alpha_0^2-5\alpha_0+6)\frac{(GM)^2}{3c^2\,J^2}\right)\ ,\\[6mm]
\delta_B&=&\displaystyle-\alpha_0\,\left(\frac{H_0}{c^2}\right)^2\left(J^2+(\alpha_0^2-3\alpha_0+2)\frac{2(GM)^2}{3c^2}\right)\ ,\\[6mm]
\delta_C&=&\displaystyle-\frac{2}{3}\alpha_0\left(\alpha_0^2-3\alpha_0+2\right)\,\left(\frac{GM\,H_0}{c^3}\right)^2\left(1+(\alpha_0^2-7\alpha_0+12)\frac{(GM)^2}{5c^2\,J^2}\right)\ .
\ea
\label{A_B_C}
\ee
So far we have not specify for which values of the coefficient $\alpha_0$ the static approximation considered is valid. By comparing the leading order terms with the next to leading order terms we conclude that this perturbative equation is valid only for absolute values of the parameter $\alpha_0$ up to
\be
|\alpha_0|\,<\,\alpha_{0.\mathrm{max.pert}}\approx \frac{c^2\,r_{1.\mathrm{orb}}}{2GM}\ .
\label{alpha_max_pert}
\ee
Above this value it is either necessary to consider higher order terms of the series expansion or to consider the exact expressions. Nevertheless we remark that, for a fixed positive value of the radial coordinate $r_1$, and larger positive values of the parameter $\alpha_0>\alpha_{0.\mathrm{max.pert}}$ the corrections given by the exact expression due to the ELA metric background will decrease significantly in absolute value becoming, for very large values of the parameter $\alpha_0\gg \alpha_{0.\mathrm{max.pert}}$, negligible, while for larger negative values of the parameter $\alpha_0<-\alpha_{0.\mathrm{max.pert}}$ the corrections become more significant being unbounded from below. Hence, although for positive values of $\alpha_0$ the approximation~(\ref{orbits_exp}) subject to the bound~(\ref{alpha_max_pert}) allows to establish a fairly good estimative for the maximum contribution of the corrections on the ELA metric background, for negative values of $\alpha_0$ no bounds can be set for such contribution. In figure~\ref{fig.orbits_perturbative} are plotted the values of the exact and perturbative correction terms of order $H_0^2$ in the differential equation~(\ref{A.Eq_u}).
\fig{orbits_perturbative.eps}{140mm}{Plot of the exact (dashed line) and perturbative
(continuous line) expressions for the corrections to the orbital differential
equation~(\ref{A.Eq_u}) on the ELA metric background as a function of the
parameter $\alpha_0$ for Earth's orbit. The perturbative regime is valid for $|\alpha_0|<\alpha_{0.\mathrm{max.pert}}\approx 5\times 10^7$~(\ref{alpha_max_pert}):\hfill\break
\ {\bf(a)} plot of the exact expressions for $\alpha_0 > 0$, the corrections asymptotically vanish for large $\alpha_0\gg \alpha_{0.\mathrm{max.pert}}$;\hfill\break
\ {\bf(b)} plot of the exact and perturbative expressions for $\alpha_0\in]-10^8,10^8[$, the perturbative and exact expressions approximately match up to $|\alpha_0|=\alpha_{0.\mathrm{max.pert}}$.}{fig.orbits_perturbative}
In addition, with respect to the bound~(\ref{t_bound}), we note that it is also obeyed as long as the bound~(\ref{alpha_max_pert}) is obeyed, hence for larger values of the coefficient $\alpha_0$ the terms of order $H_0^2$ explicitly depending on the time coordinate become relevant and must be included in the Lagrangian~(\ref{A.L_orbit}). For these cases there is no constant of motion directly associated with energy conservation. Specifically, conserved energy, would be given by the constant $E=\int d\tau(\delta{\mathcal{L}}/\delta \dot{t}-d/d\tau(\delta{\mathcal{L}}/\delta t))$, such that, due to the complexity of the full equations of motion, it would be preferable to consider a numerical analysis to compute orbital motion.
We are proceeding assuming that the upper bound~(\ref{alpha_max_pert}) is obeyed. To solve the differential equation~(\ref{orbits_exp}) we start by solving the differential equation considering only the dominant term in the right-hand side of~(\ref{orbits_exp}), hence obtaining~\cite{Kenyon,Gravitation}
\be
u_{0.H^2}''+A\,u_{0.H^2}=\frac{GM}{J^2}\,B\ \ \Rightarrow\ \ u_{0.H^2}=\frac{1+e\,\cos(\sqrt{A}\,\varphi)}{d}\ ,
\label{u_0_H}
\ee
where $e$ is the elliptical orbit eccentricity and $d$ is defined in terms of the ellipse semi-major axis $r_{1.\mathrm{orb}}$ as
\be
d=r_{1.\mathrm{orb}}(1-e^2)\ .
\label{d}
\ee
The standard General Relativity angular momentum $J_0$ and the angular momentum $J$ are expressed in terms of the parameter $d$ as
\be
\begin{array}{rcl}
J_0&=&\displaystyle-\sqrt{G\,M\,d}\ ,\\[5mm]
J&=&\displaystyle-\sqrt{G\,M\,d\,\frac{B}{A}}\approx \frac{1}{2}\,J_0\left(\delta_B-\delta_A\right)_{J=J_0}\ ,
\end{array}
\label{J_d_H}
\ee
where to evaluate $A$ and $B$, we have approximated the angular momentum by the respective Keplerian quantity, $J\approx J_0$.
Next let us compute the corrections to the solution $u_{0.H^2}$ by considering the remaining terms in the right-hand side of the differential equation~(\ref{orbits_exp}) evaluated for the function $u_{0.H^2}$~(\ref{u_0_H}) such that the full solution is
\be
u=u_{0.H^2}+u_{\mathrm{GR}.H^2}+u_{H^2}\ .
\label{u_H2}
\ee
Here the functions $u_{\mathrm{GR}.H^2}$ and $u_{H^2}$ correspond respectively to the corrections to the Keplerian orbit's solution due to the Schwarzschild background and due to the ELA metric background approximated to order $H_0^2$ being, respectively, the solutions of the following differential equations
\be
\ba{rcl}
\displaystyle
u_{\mathrm{GR}.H^2}''+A\,u_{\mathrm{GR}.H^2}&=&\displaystyle\frac{3GM}{c^2}\,C\,u_{0.H^2}^2=\frac{3GM}{c^2}\,C\,\frac{(1+e\cos(\sqrt{A}\,\varphi))^2}{d^2}\ ,\\[6mm]
\displaystyle u_{H^2}''+A\,u_{H^2}&=&\displaystyle -\left(\frac{H_0}{J}\right)^2\,\frac{1}{u_{0.H^2}^3}+\alpha_0\left(\frac{H_0}{J}\right)^2\,\frac{GM}{c^2}\,\frac{1}{u_{0.H^2}^2}\\[6mm]
&=&\displaystyle -\left(\frac{H_0}{J}\right)^2\,\frac{1}{(1+e\cos(\sqrt{A}\,\varphi))^3}\\[6mm]
&&\displaystyle+\alpha_0\left(\frac{H_0}{J}\right)^2\,\frac{GM}{c^2}\,\frac{1}{(1+e\cos(\sqrt{A}\,\varphi))^2}\ ,
\ea
\ee
such that we obtain
\be
\ba{rcl}
u_{\mathrm{GR}.H^2}&=&\displaystyle\frac{C}{A}\,\frac{\alpha_{\mathrm{GR}}}{d}\left(\left(1+\frac{e^2}{2}\right)-\frac{e^2}{6}\,\cos(2\sqrt{A}\varphi)+\sqrt{A}\,e\,\varphi\sin(\sqrt{A}\,\varphi)\right)\ ,\\[6mm]
u_{H^2}(\varphi)&=&\displaystyle \frac{d^3\,H_0^2}{A\,J^2(1-e^2)}\left(\frac{\alpha_0\,GM}{c^2\,d}+\frac{(-4+e^2)+3e^2\cos(2\sqrt{A}\,\varphi)}{4(1-e^2)(1+e\,\cos(\sqrt{A}\,\varphi))}\right.\\[6mm]
&&\displaystyle\left. -\left(\frac{3}{2(1-e^2)}-\frac{\alpha_0\,GM}{c^2\,d}\right)\frac{2e\,\arctan\left(\sqrt{\frac{1-e}{1+e}}\,\tan\left(\frac{\sqrt{A}\,\varphi}{2}\right)\right)\sin(\sqrt{A}\,\varphi)}{\sqrt{1-e^2}}\right)\ ,
\ea
\label{u_GR_H2}
\ee
where
\begin{equation}
\alpha_{GR}=\frac{3GM}{c^2\,d}\ .
\end{equation}
Both the solutions $u_{\mathrm{GR}.H^2}$ and $u_{H^2}$ have the same structure of the standard solution for Schwarzschild backgrounds~\cite{Kenyon,Gravitation}, the first term is a constant that can be neglected, the second term has a period that is a multiple of the period of solution $u_{0.H^2}$~(\ref{u_0_H}) contributing a small correction to the orbital period and the last term monotonically grows with increasing $\varphi$ contributing to the orbital precession. This last result is justified by noting that the analytic continuation of the inverse of a function corresponds to the argument of the function (in this way $\arctan(\tan \varphi)=\varphi$ increases monotonically with $\varphi$).
Due to the corrections to the Keplerian orbit's solution being small when compared to the dominant term, we can expand the trigonometric functions to lower order~\cite{Kenyon,Gravitation}:
$\cos(\sqrt{A}\varphi)= 1-A^2 \varphi^2/2+O(\varphi^4)$, $\varphi\sin(\sqrt{A}\varphi)=2\sqrt{A}\varphi^2/2+O(\varphi^4)$ and
$\arctan\left(\sqrt{(1-e)/(1+e)}\tan\left(\sqrt{A}\varphi/2\right)\right)=A\sqrt{(1-e)/(1+e)}\varphi^2/2+O(\varphi^4)$. Hence, neglecting the constant terms in the solutions $u_{\mathrm{GR}.H^2}$ and $u_{H^2}$~(\ref{u_GR_H2}) and gathering the several terms and respective coefficients for these lower order expansions, we obtain the full solution $u$~(\ref{u_H2})
\be
\ba{rcl}
u&\approx&\displaystyle\frac{1}{d}\left(1+e\,\cos\left(\left(1-\frac{\Delta\varphi_{\mathrm{GR}}}{2\pi}-\frac{\Delta\varphi_{H^2}}{2\pi}\right)\,\varphi\right)\right)+u_{\mathrm{osc.GR}}+u_{\mathrm{osc}.H^2}\ ,\\[6mm]
\displaystyle\frac{\Delta\varphi_{\mathrm{GR}}}{2\pi}&=&\displaystyle\alpha_{\mathrm{GR}}\\[5mm]
\displaystyle\frac{\Delta\varphi_{H^2}}{2\pi}&=&\displaystyle-\frac{\delta_A}{2}+\alpha_{\mathrm{GR}}\,\delta_C\left(1+\frac{2e^2}{3d}\right)\\[5mm]
&&\hfill\displaystyle+\frac{d^3\,H_0^2}{(1-e)(1+e)^\frac{3}{2}}\,\left(\frac{\alpha_0}{c^2\,d}-\frac{3}{2(1-e^2)\,GM}\right)+O(H_0^4)\ ,\\[6mm]
u_{\mathrm{osc.GR}}&=&\displaystyle -\frac{\alpha_{GR}}{6d}\,e^2\cos(2\varphi)\ ,\\[6mm]
u_{\mathrm{osc}.H^2}&=&\displaystyle \frac{(\delta_C-\delta_A)\alpha_{\mathrm{GR}}}{6}\,e^2\cos(2\varphi)+(H_0\,d)^2\frac{-4+e^2+3e^2\cos(2\varphi)}{4(1-e^2)^2\,GM\,\left(1+e\cos\varphi\right)}+O(H_0^4)
\ea
\label{precession_H2}
\ee
where $\Delta\varphi_{\mathrm{GR}}/(2\pi)$ is the standard precession per turn of the orbit due to General Relativity corrections on Schwarzschild backgrounds and $\Delta\varphi_{H^2}/(2\pi)$ is the precession per turn of the orbit due to the ELA metric background. As for the factor $u_{\mathrm{osc.GR}}$ it is the General Relativity oscillatory factor correction to the orbit solution obtained for Schwarzschild backgrounds and $u_{\mathrm{osc}.H^2}$ is the oscillatory factor correction due to the ELA metric background.
To compute the observable period correction to the orbits due to the ELA metric background it is enough to consider the definition of the constant of motion $J\,d\tau=-d\varphi/u^2$~(\ref{J}) and integrate the infinitesimal proper time displacement $d\tau$ over one turn of the orbit such that we obtain
\begin{equation}
T=-\frac{1}{J}\int_0^{2\pi}\frac{d\varphi}{u^2}\approx -\frac{1}{J}\int_0^{2\pi}d\varphi\,\frac{1}{u_0^2}\,\left(1-\frac{2u_{\mathrm{osc.GR}}}{u_0}-\frac{2u_{\mathrm{osc.H^2}}}{u_0}\right)\ .
\end{equation}
To directly compare the General Relativity corrections to the orbital period on the ELA metric background with the Keplerian orbital period and the General Relativity corrections on Schwarzschild backgrounds this integral can be factorized into the 3 components
\be
\ba{rcl}
T&=&T_0+\Delta T_{\mathrm{GR}}+\Delta T_{H^2}\ ,\\[6mm]
T_0&=&\displaystyle-\frac{1}{J_0}\int_0^{2\pi}d\varphi\frac{1}{u_0^2}\,=\, \frac{2\pi\,r_{1.\mathrm{orb}}^\frac{3}{2}}{\sqrt{GM}}\ ,\\[6mm]
\Delta T_{\mathrm{GR}}&=&\displaystyle+\frac{2}{J_0}\,\int_0^{2\pi}d\varphi\,\frac{u_{\mathrm{osc.GR}}}{u_0^3}\,=\,-\frac{3\pi\,\sqrt{GM}\,r_{1.\mathrm{orb}}^\frac{1}{2}\,e^4}{c^2(1-e^2)^2}\ ,\\[6mm]
\displaystyle\Delta T_{H^2}&=&\displaystyle-\frac{1}{2}\left(T_0+\Delta\,T_{\mathrm{GR}}\right)\left(\delta_A-\delta_B\right)-\frac{2}{|J|}\int_0^{2\pi}d\varphi\,\frac{u_{\mathrm{osc}.H^2}}{u_0^3}\\[4mm]
&\approx&\displaystyle -\left(\delta_A-\delta_B\right)\left(\frac{2\pi\,r_{1.\mathrm{orb}}^\frac{3}{2}}{\sqrt{GM}}-\frac{3\pi\,\sqrt{GM}\,r_{1.\mathrm{orb}}^\frac{1}{2}\,e^4}{2c^2(1-e^2)^2}\right)\\[4mm]
&&\displaystyle+\frac{3\pi\left(\delta_A-\delta_C\right)\,r_{1.\mathrm{orb}}^\frac{3}{2}\,e^4\,\sqrt{GM}}{c^2(1-e^2)}+\frac{\pi\,r_{1.\mathrm{orb}}^\frac{9}{2}\,(4+9e^2)\,H_0^2}{(GM)^\frac{3}{2}}+O(H_0^4)\ ,
\ea
\label{dT_H2}
\ee
where $T_0$ is the classical Keplerian orbit period corresponding to the solution $u_0$~(\ref{u0.0}),
$\Delta T_{\mathrm{GR}}$ is the standard General Relativity period correction on Schwarzschild backgrounds
corresponding to solution $u_{\mathrm{osc.GR}}$~(\ref{precession_H2}) and $\Delta T_{H^2}$ is the General Relativity period correction on the ELA metric background corresponding to solution $u_{\mathrm{osc}.H^2}$~(\ref{precession_H2}).
\begin{table}[ht]
\begin{center}
{\small
\begin{tabular}{l|cc}
Planet& $\displaystyle \Delta T_{\mathrm{GR}}\ (s/yr^{-1})$ & $\displaystyle \frac{\Delta \varphi_{\mathrm{GR}}}{2\pi}\ (arcsec/century^{-1})$\\[2mm]\hline\\[-2mm]
Mercury& $+2.35\times 10^{-3}$ &$10.35$ \\[2mm]
Venus& $+1.36\times 10^{-9}$ &$5.30$ \\[2mm]
Earth& $+3.64\times 10^{-8}$ &$3.84$ \\[2mm]
Mars& $+2.38\times 10^{-5}$ &$2.54$ \\[2mm]
Jupiter& $+4.95\times 10^{-7}$ &$0.74$ \\[2mm]
Saturn& $+4.24\times 10^{-7}$ &$0.40$ \\[2mm]
Uranus& $+1.21\times 10^{-7}$ &$0.20$ \\[2mm]
Neptune& $+8.44\times 10^{-11}$ &$0.13$ \\[2mm]
Pluto & $+5.15\times 10^{-5}$ &$0.10$ \\\hline
\end{tabular}}
\noindent\hspace{-2cm}\caption{\it \small Standard General Relativity corrections to the orbital period $\Delta T_{\mathrm{GR}}$~(\ref{dT_H2})
and orbital precession $\Delta \varphi_{\mathrm{GR}}/2\pi$~(\ref{precession_H2}) for each planet in the Solar System. These corrections are computed
for the Solar Schwarzschild background with respect to the respective quantities for Keplerian orbits.\label{table.GR}}
\end{center}
\end{table}
The values for the standard General Relativity corrections to the orbital period and orbital precession on Schwarzschild backgrounds are listed in table~\ref{table.GR}.
These corrections correspond to orbital motion on Schwarzschild backgrounds being well known and already accounted for in the model employed in the numerical analysis of the Solar System dynamics.
In addition, although on Schwarzschild backgrounds the orbital radius is not varying over time, on expanding backgrounds such as the ones described by the ELA metric, it is expected that the radius does vary as the background expands. The analytical solutions computed so far do not allow to estimate such variation for the orbital radius as we have approximated the ELA metric background by a static background with fixed Hubble rate $H=H_0$~(\ref{A.L_orbit}). In the next section, from conservation of angular momentum, we estimate the orbital radius variation by considering approximately circular orbits.
\section{Circular Orbits Approximation: Time Varying Orbital Radius\label{sec.radius}}
In this section, with the objective of estimating the orbital radius variation on backgrounds described by the ELA metric, we are analysing circular orbits on such backgrounds. In the non-relativistic velocity limit and for relatively small values of the radial coordinate ($r_1\ll l_H=c/H$) the radial acceleration is
\be
\ddot{r}_1\approx\displaystyle -c^2\Gamma^1_{\ 00}\approx -\frac{GM}{r_1^2}+\frac{2(GM)^2}{c^2\,r_1^3}+F_{H^2}+O(r_1^2\,H^4)\ ,
\label{F_Newton_mod}
\ee
where in the right hand side of the equation, the first term is the usual classical Newton gravitational acceleration, the second term is the standard General Relativity correction on Schwarzschild backgrounds and the third term is the General Relativity correction of order $H^2$ for backgrounds described by the ELA metric~(\ref{g_generic})
\be
F_{H^2}=+r_1\left(1-\frac{2GM}{c^2\,r_1}\right)^{\alpha_0}\left(1-\frac{(1-\alpha_0)GM}{c^2\,r_1}-(1+q)\left(1-\frac{2GM}{c^2\,r_1}\right)^{\frac{1}{2}-\frac{\alpha_0}{2}}\right)H^2\ .
\label{F_Newton_mod_I}
\ee
To derive the orbital velocity let us consider the constant of motion corresponding to conservation of angular momentum, $J=-r_1^2\,d\varphi/d\tau$. Particularizing to circular orbits for which the orbital velocity is constant, $v_{\mathrm{orb}}=\sqrt{-r_1\,\ddot{r}_1}$, and considering the usual definition of angular velocity $\dot{\varphi}=\omega=v_{\mathrm{orb}}/r_1$ we obtain the following definition for the angular momentum $J_{\mathrm{circ}}^2=\left.-\gamma^2\,r_1^3\,\ddot{r}_1\right|_{r_1=r_{1.\mathrm{orb}}}$ such that for an orbit of radius $r_{1.\mathrm{orb}}$ we obtain
\be
J_{\mathrm{circ}}^2\approx GM\,r_{1.\mathrm{orb}}\,\left(1+\left(\frac{H\,r_{1.\mathrm{orb}}}{c}\right)^2\left(1-\frac{2GM}{c^2\,r_{1.\mathrm{orb}}}\right)^{\alpha_0}\right)-F_{H^2}\,r_{1.\mathrm{orb}}^3\ .
\label{J_circ}
\ee
Here $\gamma=dt/d\tau$ is the relativistic factor for the ELA metric~(\ref{g_generic}) and dotted quantities represent derivation with respect to the coordinate time $t$. Specifically, in the limit of non-relativistic velocity $\dot{x}^\mu\ll c$, it is
\be
\frac{d^2r_1}{d\tau^2}\approx\gamma^2\,\ddot{r}_1\approx\frac{\ddot{r}_1}{1-\frac{2GM}{c^2r_1}-\left(\frac{H\,r_1}{c}\right)^2\left(1-\frac{2GM}{c^2\,r_1}\right)^{\alpha_0}}\ .
\label{gamma_approx_generic}
\ee
For circular orbits, the main effect obtained due to the corrections on the ELA metric background
correspond to a time varying radius. Such effect can be verified from conservation of angular momentum. To lowest order in time, $H$ is expressed as $H(t)\approx H_0-q_0\,H_0^2\,t$ such that assuming a non-varying Gravitational constant $\dot{G}=0$ and non-varying mass $\dot{M}=0$ we are left with the only possibility of a time-varying orbital radius $\dot{r}_{1.\mathrm{orb}}=0$. Hence differentiating equation~(\ref{J_circ}) and solving the equation $\dot{J}_{\mathrm{circ}}=0$ for $\dot{r}_{1.\mathrm{orb}}$ we obtain, to lowest order in $H_0$, the time dependence of the orbital radius
\be
\ba{rcl}
\displaystyle\frac{\dot{r}_{1.\mathrm{orb}}}{r_{1.\mathrm{orb}}}&\approx&\displaystyle \frac{2q_0\,(H_0\,r_{1.\mathrm{orb}})^3}{GM}\,\left(1-\frac{2GM}{c^2\,r_{1.\mathrm{orb}}}\right)^{\frac{\alpha_0}{2}+\frac{1}{2}}\times\\[6mm]
&&\displaystyle\hfill\times\left(1+q_0-\left(1-\frac{(2-\alpha_0)GM}{c^2\,r_{1.\mathrm{orb}}}\right)\left(1-\frac{2GM}{c^2\,r_{1.\mathrm{orb}}}\right)^{\frac{\alpha_0}{2}-\frac{1}{2}}\right)+O(H_0^5)\ .
\ea
\label{dr_orb_circ}
\ee
As expected from cosmological expansion this expression increases with the orbital radius $r_{1.\mathrm{orb}}$ and decreases with the mass $M$. Consistently at very large radius ($r_1\sim l_H=c/H$) the gravitational potential is negligible ($1/r_1\sim 0$) such that pure cosmological expansion is asymptotically recovered and no stable orbits exist ($\dot{r}_1/r_1\sim 2q_0^2\,H^3r_1^3>0$).
As for the specific dependence of the orbital radius variation on the parameter $\alpha_0$ it
is positive for small values of $\alpha_0\sim 0$ being of the same order of magnitude of the pure expansion effects, for growing positive values of this parameter, the radius variation decreases having a negative minimum value and then asymptotically vanishing in the limit $\alpha_0\to +\infty$ . This is actually expected, we note that in this limit the shift function is null, $\lim_{\alpha_0\to+\infty}(1-U_{\mathrm{SC}})=0$, such that we exactly recover the SC metric, hence a Ricci flat space-time for which $\dot{r}_1/r_1$ is exactly null for all orbits. As for growing negative values of this parameter the radius variation increases up to a maximum positive value and then decreases monotonically. For large negative values of this parameter $\alpha_0\ll 0$ the corrections with respect to Schwarzschild backgrounds become significantly higher with $\dot{r}_1/r_1<0$ being unbounded from below. As an example of the typical values of $\dot{r}_{1.orb}/r_{1.orb}$ as a function of the parameter $\alpha_0$ are plotted in figure~\ref{fig.dr_alpha_circ} the values of $\dot{r}_{1.orb}/r_{1.orb}$ for the Earth-Moon orbit and for Sun-Venus, Sun-Earth and Sun-Mars orbits.
\fig{dr_alpha_circ.eps}{140mm}{Examples of the profiles of the time variation rate of the orbital
radius $\dot{r}_{1.\mathrm{orb}}/r_{1.\mathrm{orb}}$~(\ref{dr_orb_circ}) as a function of the
parameter $\alpha_0$ assuming the circular orbits approximation:\hfill\break
\ {\bf(a)} for the Earth-Moon orbit, the maximum positive variation is
$\dot{r}_{1.\mathrm{orb}}/r_{1.\mathrm{orb}}=1.440\times 10^{-42}\,s^{-1}$
corresponding to $\alpha_0=-3.084\times 10^{10}$ and the minimum negative variation for positive $\alpha_0$ is $\dot{r}_{1.\mathrm{orb}}/r_{1.\mathrm{orb}}=-2.173\times 10^{-44}\,s^{-1}$ corresponding
to $\alpha_0=2.396\times 10^{11}$ ;\hfill\break
\ {\bf(b)} for the Sun-Venus orbit, the maximum positive variation is
$\dot{r}_{1.\mathrm{orb}}/r_{1.\mathrm{orb}}=1.483\times 10^{-41}\,s^{-1}$
corresponding to $\alpha_0=-1.397\times 10^7$ and the minimum negative variation for positive $\alpha_0$ is $\dot{r}_{1.\mathrm{orb}}/r_{1.\mathrm{orb}}=-2.238\times 10^{-43}\,s^{-1}$ corresponding to $\alpha_0=1.085\times 10^8$ ;\hfill\break
\ {\bf(c)} for the Sun-Earth orbit, the maximum positive variation is
$\dot{r}_{1.\mathrm{orb}}/r_{1.\mathrm{orb}}=2.557\times 10^{-40}\,s^{-1}$
corresponding to $\alpha_0=-3.609\times 10^7$
and the minimum negative variation for positive $\alpha_0$ is $\dot{r}_{1.\mathrm{orb}}/r_{1.\mathrm{orb}}=-3.859\times 10^{-42}\,s^{-1}$ corresponding to $\alpha_0=2.803\times 10^8$ ;\hfill\break
\ {\bf(d)} for the Sun-Mars orbit, the maximum positive variation is
$\dot{r}_{1.\mathrm{orb}}/r_{1.\mathrm{orb}}=9.043\times 10^{-40}\,s^{-1}$
corresponding to $\alpha_0=-5.498\times 10^7$
and the minimum negative variation for positive $\alpha_0$ is $\dot{r}_{1.\mathrm{orb}}/r_{1.\mathrm{orb}}=-1.365\times 10^{-41}\,s^{-1}$ corresponding to $\alpha_0=4.271\times 10^8$.}{fig.dr_alpha_circ}
We further note that the estimative for the orbital radius variation just computed is a valid approximation for elliptical orbits of small eccentricity $e\,\ll\,1$, hence a fairly good approximation for all planetary orbits in the Solar system except for Mercury's and Pluto's orbits for which $e\sim 0.2$ such that these estimative correspond at most to a rough approximation to the orbital radius variation for both these planets.
Next, considering the General Relativity corrections on backgrounds described by the ELA metric, we will show that it is possible to map these corrections to the heuristic variation of the standard of measurement discussed in the introduction (either the variation of the AU or the variation of $G$), by matching the corrections to Kepler's third law on such backgrounds to the fitted value of the parameter $\dot{AU}$.
\section{Variation of the Astronomical Unit\label{sec.AU}}
As already discussed in the introduction the heuristic variation of the AU (or equivalently the variation of the Gravitational constant $G$) obtained from numerical analysis of the Solar System dynamics indicates that some unmodelled gravitational interaction is present.
Here we are employing the ELA metric background to model such corrections to gravitational interactions.
From the definition of the AU~(\ref{AU}) and considering a non-varying Gravitational constant $\dot{G}=0$ the effects that can match the heuristic variation of the AU are the decrease of the Sun's mass, the orbital period corrections and the orbital radius variation.
The central mass reduction has the effect of increasing both the orbital radius and the orbital period such that angular momentum is conserved~\cite{dG} and the equality~(\ref{AU}) is maintained. Hence it is mapped to the variation of the AU as a positive contribution approximately given by $\dot{AU}/AU\approx-\dot{M_\odot}/M_\odot$. The estimative for the Sun's mass variation due to radiation and matter emission is (see~\cite{dG,Uzan} and references therein)
\begin{equation}
\frac{\dot{M_\odot}}{M_\odot}=-6.7^{+3.1}_{-3.1}\times 10^{-14}\,(yr^{-1})\ ,
\end{equation}
hence, from this estimative we obtain a equivalent contribution for the variation of the AU of
\begin{equation}
\Delta\dot{AU}_{1AU.\dot{M}}=-\frac{\dot{M_\odot}}{M_\odot}\,AU\,\approx +1.002^{+0.464}_{-0.464}\,(cm\,yr^{-1})\ ,
\label{dAU_dM}
\end{equation}
with $AU$ evaluated in centimetre.
As for the period correction $\Delta T_{H^2}$, due to General Relativity corrections on backgrounds described by the ELA metric, it is mapped to a correction to the variation of the AU given by the following difference
\begin{equation}
\begin{array}{rcl}
\Delta\dot{AU}_{1AU.H^2}&=&\displaystyle\frac{1}{1\,yr}\Bigg(\left(G\,M_\odot\,\left(\frac{T_{1AU.0}+\Delta T_{1AU.\mathrm{GR}}+\Delta T_{1AU.H^2}}{2\pi}\right)^2\right)^\frac{1}{3}\\[3mm]
& &\displaystyle\hfill-\left(G\,M_\odot\,\left(\frac{T_{1AU.0}+\Delta T_{1AU.\mathrm{GR}}}{2\pi}\right)^2\right)^\frac{1}{3}\Bigg)\\[6mm]
&\approx&\displaystyle \frac{2}{3}\,\frac{1}{1\,yr}\,\frac{\Delta T_{1AU.H^2}}{T_{1AU.0}}\,AU\ (cm\,yr^{-1})\ ,
\end{array}
\label{dAU_H2}
\end{equation}
with $AU$ evaluated in centimetre and $T_{1AU.0}$ and $\Delta T_{1AU.H^2}$ corresponding to the period and the period variation over one revolution of the orbit evaluated both in the same units of time (e.g. second). The factor of $yr^{-1}$ is explicitly written converting between the time units for $T_{1AU.0}$ and years such that $1\,yr$ corresponds to one revolution of a Keplerian orbit with semi-major axis of length of $1\,AU$. We remark that one Keplerian Earth year differs from $T_{1AU.0}$ by $5.21\,s$.
In addition, a non-null orbital radius variation, is directly mapped to the variation of the AU such that for an orbital radius of $r_{1.\mathrm{orb}.1AU}=1\,AU$ we obtain
\begin{equation}
\Delta\dot{AU}_{1AU.\dot{r}}=\frac{\dot{r}_{1.\mathrm{orb}.1AU}}{r_{1.\mathrm{orb}.1AU}}\,\frac{T_{1AU.0}}{1\,yr}\,AU\ (cm\,yr^{-1})\ ,
\label{dAU_dr}
\end{equation}
where $AU$ is evaluated in centimetre and $\dot{r}_{1.\mathrm{orb}.1AU}/r_{1.\mathrm{orb}.1AU}$ and $T_{1AU.0}$ are evaluated in the
same units of time (e.g. second).
Both the contributions~(\ref{dAU_H2}) and~(\ref{dAU_dr}) depend on the value of the metric parameter $\alpha_0$. To obtain a precise estimative for the values of these contributions it would be required a numerical analysis of the Solar System dynamics, however to further proceed analytically let us consider the approximate analytical estimative discussed in the previous sections. Specifically, the contribution~(\ref{dAU_H2}) is evaluated by considering the period correction~(\ref{dT_H2}) computed for static analytical orbital solutions and, the contribution~(\ref{dAU_dr}), is evaluated by considering the orbital radius variation~(\ref{dr_orb_circ}) computed for circular orbital solutions. These estimative are a fairly good approximation except for the orbital radius variation for the planet Mercury and the planet Pluto due to the relatively high eccentricity of their orbits. Nevertheless, as we are going to show next, the main contributions that are mapped to the variation of the AU are due to the decrease of the Sun's mass~(\ref{dAU_dM}) and to the General Relativity corrections to the orbital period on the ELA metric background~(\ref{dAU_H2}) such that the contribution due to the orbital radius variation~(\ref{dAU_dr}) is negligible being lower than the remaining contributions by a factor of $10^{-19}$ at Mercury's orbit and by a factor of $10^{-10}$ at Pluto's orbit, hence being negligible for the estimative obtained. For completeness of the analytical expressions we are keeping these contributions in the following derivations.
We are proceeding to evaluate an analytical estimative for the variation of the AU by considering the orbits of the nine planets in the Solar System. We note that for each orbit, the corrections to Kepler's third law are proportional to the orbital radius $r_{1.\mathrm{orb}}$. To relate these corrections to the variation of the AU it is required to rescale the contributions from each planetary orbit to the AU length, hence to multiply each contribution by the factor $AU/r_{1.\mathrm{orb}}$. Specifically, for a given orbit of semi-major axis $r_{1.\mathrm{orb}}$, the respective mapped variation of the AU, for an observer at the Sun, is
\begin{equation}
\dot{AU}_{\mathrm{orbit}}=\left(-\frac{\dot{M}_\odot}{M_\odot}+\frac{\dot{r}_{1.\mathrm{orb}}}{r_{1.\mathrm{orb}}}\frac{T_{1AU.0}}{1\,yr}+\frac{2}{3}\,\frac{1}{yr}\,\frac{\Delta T_{H^2}}{T_{\mathrm{orb}.0}}\frac{T_{1AU.0}}{T_{\mathrm{orb}.0}}\right)\,AU\ ,
\label{dAU_orbit}
\end{equation}
where $T_{\mathrm{orb}.0}$ and $\Delta T_{\mathrm{orb}}$ are, respectively, the Keplerian orbital period and the General Relativity correction to the orbital period of each planetary orbit on the ELA metric background with respect to Schwarzschild backgrounds. $T_{1AU.0}$ is the Keplerian period corresponding to the definition of the AU~(\ref{AU}) and in the last term the factor $T_{1AU.0}/T_{\mathrm{orb}.0}$ scales the temporal variation from each planet year to the AU year. For a given planetary orbit these corrections have the effect of a significant decrease on the variation of the AU for large positive values of $\alpha_0\gg 0$ ($\dot{AU}<0$) and of a significant increase of the AU for large negative values of $\alpha_0\ll 0$ ($\dot{AU}>0$). The dependence of the parameter $\dot{AU}_{\mathrm{orbit}}$ on the values of $\alpha_0$ for the inner planets of the Solar System, Mercury, Venus, Earth and Mars is plotted in figure~\ref{fig.dAU}.
\fig{corrections_planets.eps}{140mm}{Corrections $\dot{AU}_{\mathrm{orbit}}$~(\ref{dAU_orbit}) to the variation of the AU as a function of the constant metric parameter $\alpha_0$ for (a) the orbit of Mercury; (b) the orbit of Venus; (c) the orbit of Earth; (d) the orbit of Mars.}{fig.dAU}
As for the corrections to Kepler's third law as perceived for Earth based range measurements it is computed by evaluating the difference between the corrections corresponding to the geodesic motion of Earth and the corrections corresponding to the geodesic motion of the planet for which the range measurement is being considered. Hence, for range measurements between Earth and any other planet in the Solar System, these corrections are mapped to a variation of the AU by
\begin{equation}
\dot{AU}_{\mathrm{range}}=\left(r_{1.\mathrm{orb.Earth}}\,\dot{AU}_{\mathrm{orbit.Earth}}-r_{1.\mathrm{orb.planet}}\,\dot{AU}_{\mathrm{orbit.planet}}\right)\frac{1}{\left<\Delta r_{1.\mathrm{Earth-planet}}\right>}\ ,
\label{dAU_range}
\end{equation}
where $\dot{AU}_{\mathrm{orbit.Earth}}$ and $\dot{AU}_{\mathrm{orbit.planet}}$ correspond to $\dot{AU}_{\mathrm{orbit}}$~(\ref{dAU_orbit}) evaluated for the orbit of Earth and the orbit of each planet, respectively. The distance $\left<\Delta r_{1.\mathrm{Earth-planet}}\right>$ corresponds to the average distance between Earth and the planet assuming that both have Keplerian orbits. Planetary orbits of distinct bodies in the Solar System are generally not locked to each other, however to compute an estimative to this average value we are parameterizing Earth's orbit by the angular parameter $T_{planet}\,\varphi/(1\,s)$ and the planet orbit, for which the range measurement is being considered, by $T_{Earth}\,\varphi/(1\,s)$ with $\varphi\in[0,2\pi[$ such that the average is performed over ${\mathcal T}=T_{planet}/(1\,s)$ periods of the Earth orbit and ${\mathcal T}_E = T_{Earth}/(1\,s)$ periods of the planet orbit. Further aligning both the Earth and planet perihelion with the $x$ axis, considering a rotation of the planet orbit around the $x$ axis by its orbital inclination angle $\delta_{orb}$ followed by a rotation around the $z$ axis by the angle between the longitudes of the planet's perihelion and Earth's perihelion $\delta_{per}-\delta_{per.E}$ (given in table~\ref{table.planet_data}), we obtain the following Keplerian estimative for the average distance between Earth and any other planet in the Solar System
\begin{equation}
\begin{array}{rcl}
\left<\Delta r_{1.\mathrm{Earth-planet}}\right>&=&\displaystyle\frac{1}{2\pi}\int_0^{2\pi}d\varphi\sqrt{\Delta x^2+\Delta y^2+\Delta z^2}\ ,\\[5mm]
\Delta x&=&\displaystyle
\frac{\cos({\mathcal T}\,\varphi)}{u_{0.E}({\mathcal T}\,\varphi)} -
\frac{1}{u_0({\mathcal T}_E\,\varphi)} \Bigl(\cos({\mathcal T}_E\,\varphi)\cos(\delta_{per}-\delta_{per.E})\ ,\\[5mm]
&&\displaystyle\hfill +
\sin({\mathcal T}_E\,\varphi)\cos(\delta_{orb})\sin(\delta_{per}-\delta_{per.E})\Bigr)\\[5mm]
\Delta y&=&\displaystyle
\frac{\sin({\mathcal T}\,\varphi)}{u_{0.E}({\mathcal T}\,\varphi)} -
\frac{1}{u_0({\mathcal T}_E\,\varphi)} \Bigl(\cos({\mathcal T}_E\,\varphi)\sin(\delta_{per}-\delta_{per.E})\ ,\\[5mm]
&&\displaystyle\hfill -
\sin({\mathcal T}_E\,\varphi)\cos(\delta_{orb})\cos(\delta_{per}-\delta_{per.E})\Bigr)\\[5mm]
\Delta z&=&\displaystyle \frac{\sin({\mathcal T}_E\,\varphi)\sin(\delta_{orb})}{u_0({\mathcal T}_E\,\varphi)}\ ,
\end{array}
\label{dr_E_planet}
\end{equation}
where the functions $u_{0.E}$ and $u_0$ correspond, respectively, to the inverse of the orbital radius for Earth Keplerian orbit and the planet Keplerian orbit~(\ref{u0.0}).
Further noting that, when performing numerical analysis of the Solar System dynamics, the fitted variation of the AU is approximately a linear effect~\cite{dG} being independent of each planet's mass, a estimative for the average value of $\dot{AU}_{\mathrm{orbit}}$~(\ref{dAU_orbit}) can be obtained by a simple average of the contributions due to each planetary orbit
\begin{equation}
\left<\dot{AU}\right>_{\mathrm{orbit}}\,=\,\frac{\displaystyle\sum^{9}_{i=1}\dot{AU}_{\mathrm{orbit}.i}}{\displaystyle 9}\ .
\label{average_AU_orbit}
\end{equation}
As for the average value for the contribution of $\dot{AU}_{\mathrm{range}}$ to the experimental data measurement can be estimated by an average weighted by the number of events $N_{i}$ divided by the respective rms residuals $\sigma_{i}$
\begin{equation}
\left<\dot{AU}\right>_{\mathrm{range}}\,=\,\frac{\displaystyle\sum^{9}_{i=1,i\neq 3}\dot{AU}_{\mathrm{range}.i}\frac{N_{i}}{\sigma_{i}}}{\displaystyle\sum^{9}_{i=1,i\neq 3}\frac{N_{i}}{\sigma_{i}}}\ .
\label{average_AU_range}
\end{equation}
In this expression the index $i$ runs from $1$ to $9$ referring to the planets in the Solar System listed in table~\ref{table.planet_data} and $\dot{AU}_{orbit.i}$ are the variation of the AU for each planetary orbit in the Solar System~(\ref{dAU_orbit}). The values of the weights
$w_{i}=N_{i}/\sigma_{i}$ are computed from table~2 and table~3 of~\cite{dG} being $\omega_1=4.65$, $\omega_2=35692.38$, $\omega_3=0$, $\omega_4=469279.08$, $\omega_5=74.86$, $\omega_6=348.02$, $\omega_7=62.40$, $\omega_8=64.66$ and $\omega_9=38.79$.
Hence, when considering the ELA metric background, there will be two distinct corrections which can be mapped into the heuristic fit of the parameter $\dot{AU}$. Let us recall that the numerical analysis of the Solar System dynamics has, generally, two distinct procedures~\cite{Pitjeva}. First the ephemerides are built considering as the base model only the well established General Relativity gravitation interactions on Schwarzschild backgrounds. Then the ephemerides are numerically integrated by considering a wide number of parameters which generally may include corrections to the gravitational interactions on Schwarzschild backgrounds, for example the PPN parameters $\gamma$, $\beta$ and $\alpha$, as well as the variation of the AU, corresponding to the parameter $\dot{AU}$ or the variation of the Gravitational constant corresponding to the parameter $\dot{G}$. It is from this second numerical analysis that an heuristic fit to the parameter $\left<\dot{AU}\right>_{Fit}$ is obtained~(\ref{dAU_ephemerids}). Therefore when considering the ELA metric background to model gravitational interactions corrections to the gravitational interactions on Schwarzschild backgrounds it is required to consider corrections to both of these procedures. In particular when mapping the modelled correction to an heuristic variation of the AU with respect to some fixed length $AU_0$ we obtain that the average value with respect to the ephemerides data is $\left<AU\right>_{\mathrm{eph}}=AU_0+\left<\dot{AU}\right>_{\mathrm{range}}$ and, when fitting the ephemerides data to a heliocentric model of the Solar System, we obtain $\left<AU\right>_{\mathrm{eph}}=AU_0+\left<\dot{AU}\right>_{Fit}-\left<\dot{AU}\right>_{\mathrm{orbit}}$, where the contribution $\left<\dot{AU}\right>_{Fit}$ corresponds to the heuristic fit~(\ref{dAU_ephemerids}). Hence matching these two expressions we obtain the following map between the heuristic fit to the parameter $\dot{AU}$~(\ref{dAU_ephemerids}) and the modelled average contributions~(\ref{average_AU_orbit}) and~(\ref{average_AU_range})
\begin{equation}
\displaystyle\left<\dot{AU}\right>_{Fit}=\left<\dot{AU}\right>_{\mathrm{orbit}}+\left<\dot{AU}\right>_{\mathrm{range}}\ .
\label{dAU_orbit_estimative}
\end{equation}
Recalling that the ELA metric interpolates between local gravitational backgrounds and the expanding cosmological background we note that the previous discussion is actually consistent with the fact that expansion effects are proportional to the distance between the observer and the observed event, hence for range measurements it is required to consider the difference between geodesical motion of the observer (at Earth) and the geodesical motion of the remaining planets while for heliocentric orbital motion it is considered only the geodesic motion for each planet (the observer is at the Sun).
Generally we could consider a variation of the functional parameter $\alpha$ across the Solar System such that, for each planetary orbit, this metric parameter would be given by an approximate constant value $\alpha_{0.i}$. However an exact fit to ephemerides would
require a full numerical analysis including the gravitational corrections due to the ELA metric background. Nevertheless, for analytical analysis purposes, let us simply consider an approximately constant coefficient $\alpha_0$ across the Solar System. Hence, from the map~(\ref{dAU_orbit_estimative}) and for the value of the heuristic fit~(\ref{dAU_ephemerids}) we obtain
\begin{equation}
\left<\dot{AU}\right>_{\mathrm Fit}=(0.75\pm 0.60)\,cm\,yr^{-1}\ \ \Leftrightarrow\ \ \alpha_0=1.01^{+1.20}_{-1.22}\ .
\label{alpha_fit}
\end{equation}
We note that the dependence of $\dot{AU}$ on $\alpha_0$ has an inflexion point near $\alpha_0=1$ (see figure~\ref{dAU}), the relatively large uncertainty on the value of $\alpha_0$ is mainly due to the proximity to this inflexion point.
\begin{table}[ht]
\begin{center}
{\small
\begin{tabular}{l|cccccc}
Planet& $\displaystyle\left<\Delta r_{1.\mathrm{Earth-planet}}\right>$& $\displaystyle \Delta\dot{AU}_{\mathrm{orb}.\dot{r}}$& $\displaystyle \Delta\dot{AU}_{\mathrm{orb}.H^2}$ & $\displaystyle \dot{AU}_{\mathrm{orbit}}$ & $\dot{AU}_{\mathrm{range}}$\\
& $(\times 10^{11}\,m)$ & $(cm\,yr^{-1})$ & $(cm\,yr^{-1})$ & $(cm\,yr^{-1})$ & $(yr^{-1})$\\[1mm]\hline\\[-2mm]
Mercury& $1.552$ &$5.55\times 10^{-21}$ &$-0.091^{+7.921}_{-7.921}$ &$+1.093^{+7.921}_{-7.921}$ &$0.580^{+1.117}_{-1.117}$\\[2mm]
Venus& $1.700$ &$3.62\times 10^{-20}$ &$-0.036^{+3.101}_{-3.101}$ &$+1.038^{+3.101}_{-3.101}$ &$0.241^{+0.295}_{-0.295}$\\[2mm]
Earth& -- &$9.57\times 10^{-20}$ &$-0.022^{+1.908}_{-1.908}\times 10^{-1}$ &$+1.024^{+1.908}_{-1.908}$ &-- \\[2mm]
Mars& $2.525$ &$3.38\times 10^{-19}$ &$-0.012^{+1.014}_{-1.014}\times 10^{-1}$ &$+1.014^{+1.014}_{-1.014}$ &$-0.309^{+0.215}_{-0.215}$\\[2mm]
Jupiter& $7.842$ &$1.35\times 10^{-17}$ &$-0.018^{+1.607}_{-1.607}\times 10^{-1}$ &$+1.004^{+0.161}_{-0.161}$ &$-0.801^{+0.204}_{-0.204}$\\[2mm]
Saturn& $14.28$ &$8.30\times 10^{-17}$ &$-0.074^{+6.478}_{-6.478}\times 10^{-2}$ &$+1.003^{+0.065}_{-0.065}$ &$-0.895^{+0.135}_{-0.135}$\\[2mm]
Uranus& $28.72$ &$6.76\times 10^{-16}$ &$-0.026^{+2.269}_{-2.269}\times 10^{-2}$ &$+1.005^{+0.023}_{-0.023}$ &$-0.949^{+0.077}_{-0.077}$\\[2mm]
Neptune& $45.00$ &$2.60\times 10^{-15}$ &$-0.013^{+1.157}_{-1.157}\times 10^{-2}$ &$+1.002^{+0.012}_{-0.012}$ &$-0.968^{+0.052}_{-0.052}$\\[2mm]
Pluto & $57.23$ &$5.89\times 10^{-15}$ &$-0.089^{+7.690}_{-7.690}\times 10^{-3}$ &$+1.002^{+0.008}_{-0.008}$ &$-1.008^{+0.042}_{-0.042}$\\\hline
\end{tabular}}
\noindent\hspace{-2cm}\caption{\it \small Corrections to $\dot{AU}$ mapped from the corrections to Kepler's third law on the ELA metric background with respect to Schwarzschild backgrounds. For each planet it is listed, in the first column the average distance between the planet and Earth for Keplerian orbits $\left<\Delta r_{1.Earth-planet}\right>$~(\ref{dr_E_planet}), in the second column the contribution due to the orbital radius variation (the second term in equation~(\ref{dAU_orbit})) for which the uncertainty is at least 8 orders of magnitude below the quoted values, in the third column the contribution due to the period correction on the ELA metric background (the third term in equation~(\ref{dAU_orbit})), in the fourth column the total correction to the variation of the AU for heliocentric orbital motion $\dot{AU}_{\mathrm orbit}$~(\ref{dAU_orbit}) and in the fifth column the total correction to the variation of the AU for Earth based range measurements $\dot{AU}_{range}$~(\ref{dAU_range}). \label{table.planet_dAU_H2}}
\end{center}
\end{table}
In table~\ref{table.planet_dAU_H2} are listed the values of the contributions from each planetary orbit in the Solar System to the estimative~(\ref{alpha_fit}). The correction to the orbital period for the value of the parameter $\alpha_0$ is approximately constant for all planetary orbits
\begin{equation}
\Delta T_{H^2}\,\frac{1}{yr}\frac{T_{1AU.orb.0}}{T_{orb.0}}=-0.069^{+6.038}_{-6.038}\times 10^{-6}\ s\,yr^{-1}\ ,
\end{equation}
where $\Delta T_{H^2}$ corresponds to the period correction per revolution for each planet~(\ref{dT_H2}). Although this correction is enough to map the heuristic fit to the variation of the AU, we note that it is negligible for most of other purposes, even for archaeological fits to the variation of the Solar System parameters we obtain at most a variation of the Earth year by $\pm 1.7\,h$ over a period of $10^9$ years, hence within the uncertainty of such estimative~\cite{Uzan}.
As for the values for the corrections to the orbital precession and orbital radius variation for each planet are listed in table~\ref{table.planet_df_dT}.
\begin{table}[ht]
\begin{center}
{\small
\begin{tabular}{l|cc}
Planet &$\frac{\Delta \varphi_{H^2}}{2\pi}\,(arcsec\,century^{-1})$ & $\frac{\dot{r}_{1.\mathrm{orb}}}{r_{1.\mathrm{orb}}}\,(century^{-1})$ \\[2mm]\hline\\[-2mm]
Mercury &$-2.28\times 10^{-19}$ &$3.71\times 10^{-32}$ \\[2mm]
Venus &$-5.82\times 10^{-19}$ &$2.42\times 10^{-31}$ \\[2mm]
Earth &$-9.46\times 10^{-19}$ &$6.40\times 10^{-31}$ \\[2mm]
Mars &$-1.78\times 10^{-18}$ &$2.26\times 10^{-30}$ \\[2mm]
Jupiter &$-1.12\times 10^{-17}$ &$9.01\times 10^{-29}$ \\[2mm]
Saturn &$-2.79\times 10^{-17}$ &$5.55\times 10^{-28}$ \\[2mm]
Uranus &$-7.96\times 10^{-17}$ &$4.52\times 10^{-27}$ \\[2mm]
Neptune &$-1.56\times 10^{-16}$ &$1.74\times 10^{-26}$ \\[2mm]
Pluto &$-2.35\times 10^{-16}$ &$3.94\times 10^{-26}$ \\\hline
\end{tabular}}
\noindent\hspace{-2cm}\caption{\it \small Corrections to precession and orbital radius variation due to the General Relativity corrections for the background described by the ELA metric. The estimative uncertainty for each of these values is several orders of magnitude below the quoted values.\label{table.planet_df_dT}}
\end{center}
\end{table}
In particular the orbital precession corrections are lower by more than 10 orders of magnitude when compared to the respective values for Schwarzschild backgrounds and the orbital radius variation range from $\dot{r}_{1.\mathrm{orb}}/r_{1.\mathrm{orb}}\sim 10^{-32}\,century^{-1}$ for mercury up to $\dot{r}_{1.\mathrm{orb}}/r_{1.\mathrm{orb}}\sim 10^{-26}\,century^{-1}$, hence being well below any other estimative for these variations~\cite{Uzan}. We recall that a variation of the AU does not imply an orbital radius variation~\cite{dG,dG1,dG2} and further remark that the orbital radial variation obtained from numerical analysis of ephemerides in~\cite{dG} are due to the variation of the Gravitational constant $\dot{G}\neq 0$, hence not directly comparable with the values obtained here, for which the quoted orbital radius variation in table~\ref{table.planet_df_dT} is due to the ELA metric background alone. In the framework discussed here there will be an extra contribution to the orbital radius variation due to the decrease of the Sun's mass of order $10^{-12}\,century^{-1}$ to $10^{-11}\,century^{-1}$ similarly to the fit obtained in~\cite{dG,Oort_cloud}.
With respect to the ELA metric background we note that the value of the metric parameter $\alpha_0$ parameterizes the local anisotropic corrections with respect to the isotropic cosmological background, specifically space-time is locally isotropic for $\alpha_0=0$ which corresponds to the isotropic background described by the McVittie metric~\cite{McVittie}. The value of $\alpha_0\approx 1.01$~(\ref{alpha_fit}) corresponds to a relatively small perturbation to the isotropic background which, as has been shown, corresponds to relatively small corrections to the orbital parameters. Consistently with this discussion we remark that, when compared with a isotropic variation of the $AU$ (or equivalently, to the variation of the Gravitational constant $G$), the corrections due to the ELA metric background to the orbital period are relatively more relevant than the corrections to the orbital precession and orbital radius variation. It is due to the background anisotropy between the radial direction and angular directions that such effect is attainable. In addition we recall that, when considering point-mass objects, as we approach the SC horizon the metric exponent $\alpha$ should be greater or equal to $\alpha(r_{1.\mathrm{SC}})=3$ to ensure that space-time is singularity free at this horizon. This requirement is not absolutely necessary as the real Sun is not a point-like mass being instead an extended spheroid, hence without an event horizon. Nevertheless we further note that the uncertainty on the estimative of the constant $\alpha_0$ is relatively large and we may expect that a functional parameter $\alpha$ varying across the Solar System would possible allow for a better fit to ephemerides, hence we may conjecture that its value should be decreasing with growing heliocentric distances being close to $\alpha=3$ near the Sun. This discussion is not conclusive being required a numerical analysis of the Solar System dynamics including the corrections due to the ELA metric background to actually verify if such a profile for the values of $\alpha$ is the best fit to planetary motion.
\section{Conclusions\label{sec.conclusions}}
In this work we have mapped the General Relativity corrections to Kepler's third law on backgrounds described by the ELA metric~(\ref{g_generic}) to the heuristic variation of the AU estimated from numerical analysis of the Solar System dynamics on Schwarzschild backgrounds. These corrections together with the decrease of the Sun's mass by radiation emission fully account for the fitted value for the variation of the AU~(\ref{dAU_ephemerids}). Reflecting the anisotropic nature of the ELA metric background, the more relevant contribution to such modelling is due to the orbital period corrections, being the contributions from the orbital radius variation negligible.
The constant value for the metric parameter that matches the quoted variation of the AU is $\alpha_0=+1.01^{+1.20}_{-1.22}$~(\ref{alpha_fit}), hence relatively close to the value $\alpha_0=0$ which corresponds to the isotropic background described by the McVittie metric. For completeness let us further note that other effects which may be relevant on backgrounds described by the ELA metric such as the corrections to the Doppler shift for range measurements are, for this range of values of the metric parameter negligible, being of the same order of magnitude of the effects attributed to isotropic cosmological expansion~\cite{McV_light,Pioneer_ELA}. Following the same arguments we further conclude that, within the Solar System the contribution to the cosmological mass-energy density within the Solar System due to the ELA metric background is negligible~\cite{DM_orbits} (for further details see~\cite{Pioneer_ELA}).
Hence we have shown that the heuristic variation of the AU (or equivalently the heuristic variation of the Gravitational constant $G$)~\cite{AU_1,dG} can alternatively be modelled by the ELA metric background parameterizing the corrections to gravitational interactions within the Solar System. For analytical analysis purposes we have considered a constant metric parameter $\alpha_0$ for which we obtain a relatively high uncertainty. More generally a radially symmetric functional parameter with varying value across the Solar System would, in principle, allow for a significant reduction of such uncertainty. The results obtained here are enough to motivate a numerical analysis of the Solar System dynamics including the ELA metric background corrections to gravitational interactions. This framework solves the problem of the unwelcome variation of the measurement standard (whether the AU whether $G$) and constitutes a playground for testing the ELA metric background in the most well known of all the astrophysical systems, the Solar System. We leave such study to another work.
\section*{Acknowledgements}
This work was supported by grant SFRH/BPD/34566/2007 from FCT-MCTES.
Work developed in the scope of the strategical project of GFM-UL PEst-OE/MAT/UI0208/2011.
|
{
"timestamp": "2012-03-09T02:03:42",
"yymm": "1203",
"arxiv_id": "1203.1844",
"language": "en",
"url": "https://arxiv.org/abs/1203.1844"
}
|
\section*{Introduction}
The fractional calculus, \textit{i.e.} the mathematical field dealing with the generalization of the derivative to any real order, plays an increasing role in many varied domains as economy \cite{comt} or probability \cite{levy,stan}. Fractional derivatives also appear in many fields of Physics (see \cite{hilf3}): wave mechanic \cite{alme}, viscoelasticity \cite{bagl}, thermodynamics \cite{hilf2}, fluid mechanic in heterogeneous media \cite{hilf,neel,neel2}, etc. Recently, a subtopic of the fractional calculus gains importance: it concerns the variational principles on functionals involving fractional derivatives. This leads to the statement of fractional Euler-Lagrange equations, see \cite{agra,bale2,riew}.
A direct consequence is the emergence of works concerning a particular class of fractional optimal control problems, see \cite{agra2,agra3,bour7,torr,torr2,jeli} and references therein. Using a Lagrange multiplier technique or not, authors obtain with a calculus of variations a necessary condition for the existence of an optimal control. This condition is commonly given as the existence of a solution of a system of fractional differential equations called \textit{fractional Pontryagin's system}.
Hence, the explicit computation of a potential optimal control, from the above necessary condition, needs the resolution of a fractional Pontryagin's system which is a main drawback. Indeed, solving a fractional differential equation is in general very difficult. Moreover, a fractional Pontryagin's system involves left and right fractional derivatives which is an additional obstruction. \\
In this paper, we then develop a \textit{numerical approach}. Let us remind that there exist many works concerning the statement of discrete operators approaching the fractional derivatives (see \cite{dubo,galu,oldh}) and then concerning numerical schemes for fractional differential equations (see \cite{diet,liu,meer,meer2}). In particular, one can find studies concerning the discretization of fractional Euler-Lagrange equations \cite{agra4,bour} and fractional Pontryagin's systems \cite{agra2,agra3,bale,deft,jeli}.
Nevertheless, a fractional Pontryagin's system admits an intrinsic variational structure: its solutions correspond to the critical points of a cost functional. Moreover, this variational structure induces strong constraints on the qualitative behaviour of the solutions and it seems then important to preserve it at the discrete level. A \textit{variational integrator} is a numerical scheme preserving the variational structure of a system at the discrete level. We refer to Section \ref{section2} for more details concerning the construction of a variational integrator and let us remind that the variational integrators are well-developed in \cite{lubi,mars} for classical Euler-Lagrange equations and in \cite{bour} for fractional ones. In this chapter, we construct a variational integrator for fractional Pontryagin's systems and it is called shifted discrete fractional Pontryagin's system. \\
In \cite{bour7}, we have suggested a deviously way in order to get informations on the solutions of a not solvable fractional Pontryagin's system. Indeed, we have stated a fractional Noether's theorem giving an explicit constant of motion for fractional Pontryagin's systems admitting a symmetry. We refer to \cite{bour7} for more details and we remind that this result is based on a preliminary result proved by Torres and Frederico in \cite{torr,torr2}. In this paper, following the strategy of the continuous case, we introduce the notion of a discrete symmetry for shifted discrete fractional Pontryagin's systems and we finally provide a discrete fractional Noether's theorem giving an explicit computable constant of motion. \\
The paper is organized as follows. Section \ref{section1} is devoted to a reminder on the fractional calculus and on the emergence of fractional Pontryagin's systems in the study of a class of fractional optimal control problems. In Section \ref{section2}, after a reminder concerning discrete fractional derivatives, we focus on the construction of a variational integrator for fractional Pontryagin's systems. We make some numerical tests in Section \ref{section3}. Especially, let us remind that a fractional example is solved in \cite{bour7} in a certain sense. Consequently, we can test the convergence of the variational integrator both in the classical and strict fractional cases. Finally, Section \ref{section4} is devoted to the statement of a discrete fractional Noether's theorem. Technical proofs of Lemmas are provided in Appendix \ref{appA}.
\parindent 0pt
\section{Reminder about fractional Pontryagin's system}\label{section1}
In this section, we first make a reminder about fractional calculus in Section \ref{section11}. Then, in Section \ref{section12}, we remind how fractional Pontryagin's systems emerge from the study of a class of fractional optimal control problems. Let us introduce the following notations available in the whole paper. Let $a < b$ be two reals, let $d$, $m \in \mathbb{N}^*$ denote two dimensions and let $\Vert \cdot \Vert$ be the euclidean norm of $\mathbb{R}^d$ and $\mathbb{R}^m$.
\subsection{Fractional operators of Riemann-Liouville and Caputo}\label{section11}
The fractional calculus concerns the extension of the usual notion of derivative from non-negative integer orders to any real order. Since 1695, numerous notions of fractional derivatives emerge over the year, see \cite{kilb,podl,samk}. In this paper, we only use the notions of fractional integrals and derivatives in the sense of Riemann-Liouville (1847) and Caputo (1967) whose definitions are recalled in this section. We refer to \cite{kilb,podl,samk} for more details. \\
Let $g \in \mathscr{C}^0 ([a,b],\mathbb{R}^d)$ and $\alpha > 0$. The left (resp. right) fractional integral in the sense of Riemann-Liouville with inferior limit $a$ (resp. superior limit $b$) of order $ \alpha $ of $g$ is defined by:
\begin{equation}
\forall t \in ]a,b], \; I^{\alpha}_- g (t) := \dfrac{1}{\Gamma (\alpha)} \displaystyle \int_a^t (t-y)^{\alpha -1} g(y) \; dy
\end{equation}
respectively:
\begin{equation}
\forall t \in [a,b[, \; I^{\alpha}_+ g (t) := \dfrac{1}{\Gamma (\alpha)} \displaystyle \int_t^b (y-t)^{\alpha -1} g(y) \; dy,
\end{equation}
where $\Gamma$ denotes the Euler's Gamma function. For $\alpha =0$, let $I^0_- g = I^0_+ g = g$. \\
Now, let us consider $0 < \alpha \leq 1$. The left (resp. right) fractional derivative in the sense of Riemann-Liouville with inferior limit $a$ (resp. superior limit $b$) of order $\alpha $ of $g$ is then given by:
\begin{equation}
\forall t \in ]a,b], \; D^\alpha_- g(t) := \dfrac{d}{dt} \big( I^{1-\alpha}_- g \big) (t) \quad \Big( \text{resp.} \quad \forall t \in [a,b[, \; D^\alpha_+ g(t) := -\dfrac{d}{dt} \big( I^{1-\alpha}_+ g \big) (t) \Big),
\end{equation}
provided that the right side terms are defined. \\
In the Riemann-Liouville sense, the strict fractional derivative of a constant is not zero. Caputo then suggests the following definition. For $0 < \alpha \leq 1$, the left (resp. right) fractional derivative in the sense of Caputo with inferior limit $a$ (resp. superior limit $b$) of order $\alpha $ of $g$ is given by:
\begin{equation}\label{eq11-1}
\forall t \in ]a,b], \; {}_{{\rm c}} D^\alpha_- g(t) := D^\alpha_- \big( g - g(a) \big) (t) \quad \Big( \text{resp.} \quad \forall t \in [a,b[, \; {}_{{\rm c}} D^\alpha_+ g(t) := D^\alpha_+ \big( g - g(b) \big) (t) \Big),
\end{equation}
provided that the right side terms are defined. Let us note that if $g(a)=0$ (resp. $g(b)=0$), then ${}_{{\rm c}} D^\alpha_- g = D^\alpha_- g$ (resp. ${}_{{\rm c}} D^\alpha_+ g = D^\alpha_+ g$).\\
In the \textit{classical case} $\alpha = 1$, the fractional derivatives of Riemann-Liouville and Caputo both coincide with the classical derivative. Precisely, modulo a $(-1)$ term in the right case, we have $D^1_- = {}_{\text{c}}D^1_- = - D^1_+ = - {}_{\text{c}}D^1_+ = d/dt$.
\subsection{Reminder about a class of fractional optimal control problems}\label{section12}
From now and for all the rest of the paper, we consider $0 < \alpha \leq 1$ and $A \in \mathbb{R}^d$. Let us denote $[\alpha]$ the floor of $\alpha$. \\
In this section, let us remind the following definitions concerning the class of fractional optimal control problems studied in \cite{bour7}:
\begin{itemize}
\item The elements denoted $u \in \mathscr{C}^0 ([a,b],\mathbb{R}^m)$ are called \textit{controls};
\item Let $f$ be a $\mathscr{C}^2$ function of the form:
\begin{equation}
\fonction{f}{\mathbb{R}^d \times \mathbb{R}^m \times [a,b]}{\mathbb{R}^d}{(x,v,t)}{f(x,v,t).}
\end{equation}
It is commonly called the \textit{constraint function}. We assume that $f$ satisfies the following Lipschitz type condition. There exists $M \geq 0$ such that:
\begin{equation}\tag{$f_x$ lip}\label{condf}
\forall (x_1,x_2,v,t) \in (\mathbb{R}^d)^2 \times \mathbb{R}^m \times [a,b], \; \Vert f(x_1,v,t) - f(x_2,v,t) \Vert \leq M \Vert x_1 - x_2 \Vert ;
\end{equation}
\item For any control $u$, let $q^{u,\alpha} \in \mathscr{C}^{[\alpha]} ([a,b],\mathbb{R}^d)$ denote the unique global solution of the following fractional Cauchy problem:
\begin{equation}\tag{CP${}^\alpha_q$}\label{eqcpq}
\left\lbrace \begin{array}{l}
{}_{{\rm c}} D^\alpha_- q =f(q,u,t)\\
q(a) = A.
\end{array}
\right.
\end{equation}
$q^{u,\alpha}$ is commonly called the \textit{state variable} associated to $u$. Its existence and its uniqueness are provided in \cite{bour7} from Condition \eqref{condf};
\item Finally, the fractional optimal control problem studied in \cite{bour7} is the problem of optimization of the following \textit{cost functional}:
\begin{equation}
\fonction{\mathcal{L}^\alpha}{\mathscr{C}^0 ([a,b],\mathbb{R}^m )}{\mathbb{R}}{u}{\displaystyle \int_{a}^{b} L ( q^{u,\alpha},u,t ) \; dt ,}
\end{equation}
where $L$ is a \textit{Lagrangian}, \textit{i.e.} a $\mathscr{C}^2$ application of the form:
\begin{equation}
\fonction{L}{\mathbb{R}^d \times \mathbb{R}^m \times [a,b]}{\mathbb{R}}{(x,v,t)}{L(x,v,t).}
\end{equation}
\end{itemize}
A control optimizing $\mathcal{L}^\alpha$ is called \textit{optimal control}. A necessary condition for a control $u$ to be optimal is to be a \textit{critical point} of $\mathcal{L}^\alpha$, \textit{i.e.} to satisfy:
\begin{equation}
\forall \bar{u} \in \mathscr{C}^0 ([a,b],\mathbb{R}^m ), \; D\mathcal{L}^\alpha(u)(\bar{u}) := \lim\limits_{\varepsilon \rightarrow 0} \dfrac{\mathcal{L}^\alpha(u+\varepsilon \bar{u})-\mathcal{L}^\alpha(u)}{\varepsilon} = 0.
\end{equation}
In \cite{bour7}, we then focused on the characterization of the critical points of $\mathcal{L}^\alpha$. Firstly, we proved with an usual calculus of variations the following Lemma \ref{lem1} giving explicitly the value of the G\^ateaux derivative of $\mathcal{L}^\alpha$:
\begin{lemma}\label{lem1}
Let $u$, $\bar{u} \in \mathscr{C}^0 ([a,b],\mathbb{R}^m )$. Then, the following equality holds:
\begin{equation}
D\mathcal{L}^\alpha(u)(\bar{u}) = \displaystyle \int_a^b \dfrac{\partial L}{\partial x} (q^{u,\alpha},u,t) \cdot \bar{q} + \dfrac{\partial L}{\partial v} (q^{u,\alpha},u,t) \cdot \bar{u} \; dt,
\end{equation}
where $\bar{q} \in \mathscr{C}^{[\alpha]} ( [a,b], \mathbb{R}^d )$ is the unique global solution of the following linearised Cauchy problem:
\begin{equation}\label{eqlcpq}\tag{LCP${}^\alpha_{\bar{q}}$}
\left\lbrace \begin{array}{l}
{}_{{\rm c}} D^\alpha_- \bar{q} = \dfrac{\partial f}{\partial x} (q^{u,\alpha},u,t) \times \bar{q} + \dfrac{\partial f}{\partial v} (q^{u,\alpha},u,t) \times \bar{u} \\[10pt]
\bar{q}(a) = 0.
\end{array}
\right.
\end{equation}
\end{lemma}
This last result not leading to a characterization of the critical points of $\mathcal{L}^\alpha$, we then introduced the following elements stemming from the Lagrange multiplier technique:
\begin{itemize}
\item Let $H$ be the following application
\begin{equation}
\fonction{H}{\mathbb{R}^d \times \mathbb{R}^m \times \mathbb{R}^d \times [a,b]}{\mathbb{R}}{(x,v,w,t)}{L(x,v,t)+w \cdot f(x,v,t).}
\end{equation}
$H$ is commonly called the \textit{Hamiltonian} associated to the Lagrangian $L$ and the constraint function $f$;
\item For any control $u$, let $p^{u,\alpha} \in \mathscr{C}^{[\alpha]} ( [a,b], \mathbb{R}^d )$ denote the unique global solution of the following fractional Cauchy problem:
\begin{equation}\label{eqcpp}\tag{CP${}^\alpha_p$}
\left\lbrace \begin{array}{l}
{}_{{\rm c}} D^\alpha_+ p = \dfrac{\partial H}{\partial x}(q^{u,\alpha},u,p,t) = \dfrac{\partial L}{\partial x}(q^{u,\alpha},u,t) + \left( \dfrac{\partial f}{\partial x}(q^{u,\alpha},u,t)\right)^T \times p \\[10pt]
p(b) = 0.
\end{array}
\right.
\end{equation}
$p^{u,\alpha}$ is commonly called the \textit{adjoint variable} associated to $u$. Its existence and its uniqueness are also provided in \cite{bour7}. Let us note that ${}_{{\rm c}} D^\alpha_+ p^{u,\alpha} = D^\alpha_+ p^{u,\alpha}$ since $p^{u,\alpha} (b) = 0$.
\end{itemize}
Consequently, for any control $u$, the couple $(q^{u,\alpha},p^{u,\alpha})$ is solution of the following \textit{fractional Hamiltonian system}:
\begin{equation}\tag{HS${}^\alpha$}\label{eqhs}
\left\lbrace \begin{array}{l}
{}_{{\rm c}} D^\alpha_- q = \dfrac{\partial H}{\partial w} (q,u,p,t) \\[10pt]
D^\alpha_+ p = \dfrac{\partial H}{\partial x} (q,u,p,t).
\end{array}
\right.
\end{equation}
Finally, the introduction of these last elements allowed us to prove the following theorem:
\begin{theorem}\label{thmfinal}
Let $u \in \mathscr{C}^0 ([a,b],\mathbb{R}^m)$. Then, $u$ is a critical point of $\mathcal{L}^{\alpha}$ if and only if $(q^{u,\alpha},u,p^{u,\alpha})$ is solution of the following \textit{fractional stationary equation}:
\begin{equation}\tag{SE${}^\alpha$}\label{eqse}
\dfrac{\partial H}{\partial v} (q,u,p,t) = 0.
\end{equation}
\end{theorem}
From Theorem \ref{thmfinal}, we retrieved in \cite{bour7} the following result leading to the fractional Pontryagin's system:
\begin{corollary}\label{corfinal}
$\mathcal{L}^\alpha$ has a critical point in $\mathscr{C}^0 ([a,b],\mathbb{R}^m)$ if and only if there exists $(q,u,p) \in \mathscr{C}^{[\alpha]} ([a,b],\mathbb{R}^d) \times \mathscr{C}^0 ([a,b],\mathbb{R}^m) \times \mathscr{C}^{[\alpha]} ([a,b],\mathbb{R}^d)$ solution of the following \textit{fractional Pontryagin's system}:
\begin{equation}\tag{PS${}^\alpha $}\label{eqps}
\left\lbrace \begin{array}{l}
{}_{{\rm c}} D^\alpha_- q = \dfrac{\partial H}{\partial w} (q,u,p,t) \\[10pt]
D^\alpha_+ p = \dfrac{\partial H}{\partial x} (q,u,p,t) \\[10pt]
\dfrac{\partial H}{\partial v} (q,u,p,t) = 0 \\[10pt]
\big( q(a),p(b) \big) = ( A,0 ).
\end{array}
\right.
\end{equation}
In the affirmative case, $u$ is a critical point of $\mathcal{L}^\alpha$ and we have $(q,p) = (q^{u,\alpha},p^{u,\alpha})$.
\end{corollary}
Let us note that the fractional Pontryagin's system \eqref{eqps} is made up of the fractional Hamiltonian system \eqref{eqhs}, the fractional stationary equation \eqref{eqse} and initial and final conditions. \\
In practice, see Examples in \cite{bour7}, we use more Corollary \ref{corfinal} than Theorem \ref{thmfinal}. Let us remind that Corollary \ref{corfinal} was already provided in \cite{agra2,agra3,torr,torr2,jeli} and references therein without Condition \eqref{condf}. However, this result is proved, in each of these papers, using a Lagrange multiplier technique requiring the introduction of an augmented functional. In \cite{bour7}, Condition \eqref{condf} allowed us to give a complete proof of this result using only classical mathematical tools adapted to the fractional case: \textit{calculus of variations}, \textit{Gronwall's Lemma}, \textit{Cauchy-Lipschitz Theorem} and \textit{stability under perturbations of differential equations}. We refer to \cite{bour7} for more details and for a discussion on the subject. \\
As we have seen in this section, fractional Pontryagin's systems emerge from the study of a class of fractional optimal control problems. They have a variational structure in the sense that they are obtained with a calculus of variations on functionals and there resolutions give explicitly the critical points of these functionals. Our aim in this paper is to provide them numerical schemes preserving this strong characteristic at the discrete level. \\
Moreover, let us make the following important remark: since a fractional Pontryagin's system emerges from a fractional optimal control problem, the main unknown is then the control $u$. Consequently, the convergence of the numerical scheme constructed in Section \ref{section2} is going to be considered only with respect to $u$.
\section{Variational integrator for fractional Pontryagin's systems}\label{section2}
In general, fractional differential equations are very difficult to solve. One can find some solved examples in \cite{kilb,podl,samk} using Mittag-Leffler functions, Fourier and Laplace transforms. Additionally, fractional Pontryagin's systems, as fractional Euler-Lagrange equations provided in \cite{agra}, present an asymmetry in the sense that left and right fractional derivatives are involved. It is an additional drawback in order to solve explicitly the most of fractional Pontryagin's systems. In this section, we then develop a numerical approach treating them. \\
Nevertheless, as we have seen in Section \ref{section12}, a fractional Pontryagin's system admits an intrinsic variational structure: its solutions correspond to the critical points of a functional. In this paper, we want to construct a numerical scheme for fractional Pontryagin's systems preserving at the discrete level this strong property. \\
A \textit{variational integrator} is a numerical scheme preserving the variational structure of a system at the discrete level. Precisely, let us consider a differential system coming from a variational principle (\textit{i.e.} \textit{its solutions correspond to the critical points of a functional}). Then, a variational integrator is the numerical scheme constructed as follows:
\begin{itemize}
\item firstly, one have to define a discrete version of the functional;
\item secondly one have to form a discrete variational principle on it.
\end{itemize}
Hence, a numerical scheme is obtained and it is called variational integrator. It preserves the variational structure at the discrete level in the sense that \textit{its discrete solutions correspond to the discrete critical points of the discrete functional}. Let us remind that variational integrators are well-developed for classical Euler-Lagrange equations in \cite{lubi,mars} and let us remind that we have developed a variational integrator for fractional Euler-Lagrange equations in \cite{bour}. In this section, we are going to construct a variational integrator for fractional Pontryagin's systems. \\
Let us introduce the following notations available in the whole paper. Let $N \in \mathbb{N}^*$, $h=(b-a)/N$ denote the step size of discretization and $\boldsymbol{T} = (t_k)_{k=0,\ldots,N} = (a+kh)_{k=0,\ldots,N}$ be the classical partition of the interval $[a,b]$. Let us assume that $N$ is sufficiently large in order to satisfy the following condition:
\begin{equation}\label{condh}\tag{cond $h$}
2h^\alpha M < 1,
\end{equation}
where $M$ is the Lipschitz coefficient of the constraint function $f$, see Condition \eqref{condf}.
\subsection{Reminder about discrete fractional derivatives of Gr\"unwald-Letnikov}\label{section21}
For the sequel, we need the introduction of discrete operators approximating the fractional derivatives of Riemann-Liouville and Caputo. As in \cite{deft,dubo}, let us define $\Delta^\alpha_-$ and $\Delta^\alpha_+$ the following discrete analogous of $D^\alpha_-$ and $D^\alpha_+$ respectively:
\begin{equation}
\fonction{\Delta^\alpha_-}{( \mathbb{R}^d ) ^{N+1}}{( \mathbb{R}^d ) ^{N}}{\boldsymbol{G}}{\left( \dfrac{1}{h^{\alpha}} \displaystyle \sum_{r=0}^{k} \alpha_r G_{k-r} \right) _{k=1,\ldots,N}}
\end{equation}
and
\begin{equation}
\fonction{\Delta^\alpha_+}{( \mathbb{R}^d ) ^{N+1}}{( \mathbb{R}^d ) ^{N}}{\boldsymbol{G}}{\left( \dfrac{1}{h^{\alpha}} \displaystyle \sum_{r=0}^{N-k} \alpha_r G_{k+r} \right) _{k=0,\ldots,N-1},}
\end{equation}
where the elements $(\alpha_r)_{r \in \mathbb{N}}$ are defined by $\alpha_0 := 1$ and
\begin{equation}\label{eqalphar}
\forall r \in \mathbb{N}^*, \; \alpha_r := \dfrac{(-\alpha)(1-\alpha)\ldots(r-1-\alpha)}{r!}.
\end{equation}
These discrete fractional operators are approximations of the continuous ones. Indeed, passing to the limit $h \to 0$, these discrete operators correspond to the definition of the fractional derivatives of Gr\"unwald-Letnikov (1867) coinciding with the Riemann-Liouville's ones. We refer to \cite{podl} for more details. \\
Finally, according to Equation \eqref{eq11-1}, we define ${}_{{\rm c}} \Delta^\alpha_-$ and ${}_{{\rm c}} \Delta^\alpha_+$ the following discrete analogous of ${}_{{\rm c}} D^\alpha_-$ and ${}_{{\rm c}} D^\alpha_+$ respectively:
\begin{equation}
\fonction{{}_{{\rm c}} \Delta^\alpha_-}{( \mathbb{R}^d ) ^{N+1}}{( \mathbb{R}^d ) ^{N}}{\boldsymbol{G}}{\Big( \big( \Delta^\alpha_- (\boldsymbol{G} - G_0) \big)_k \Big) _{k=1,\ldots,N}}
\end{equation}
and
\begin{equation}
\fonction{{}_{{\rm c}} \Delta^\alpha_+}{( \mathbb{R}^d ) ^{N+1}}{( \mathbb{R}^d ) ^{N}}{\boldsymbol{G}}{\Big( \big( \Delta^\alpha_+ (\boldsymbol{G} - G_N) \big)_k \Big) _{k=0,\ldots,N-1}.}
\end{equation}
Let us note that we preserve some continuous properties at the discrete level. In particular, $G_0 = 0$ (resp. $G_N = 0$) implies ${}_{{\rm c}} \Delta^\alpha_- \boldsymbol{G} = \Delta^\alpha_- \boldsymbol{G}$ (resp. ${}_{{\rm c}} \Delta^\alpha_+ \boldsymbol{G} = \Delta^\alpha_+ \boldsymbol{G}$). Additionally, in the classical case $ \alpha = 1$, these discrete fractional derivatives coincide with the usual backward and forward Euler's approximations of $d/dt$ with a $(-1)$ term in the right case:
\begin{equation}
\forall k=1,\ldots,N , \; (\Delta^1_- \boldsymbol{G} )_k= ({}_{{\rm c}} \Delta^1_- \boldsymbol{G})_k = \dfrac{G_k - G_{k-1}}{h}
\end{equation}
and
\begin{equation}
\forall k=0,\ldots,N-1 , \; (\Delta^1_+ \boldsymbol{G} )_k= ({}_{{\rm c}} \Delta^1_+ \boldsymbol{G})_k = \dfrac{G_k - G_{k+1}}{h}.
\end{equation}
\subsection{Results concerning the discrete fractional derivatives}\label{section22}
In this section, we prove two important properties preserved from the continuous level to the discrete one. For the sequel, we first need the introduction of the following \textit{shift operators}:
\begin{equation}
\fonction{\sigma}{(\mathbb{R}^n)^{N+1}}{(\mathbb{R}^n)^{N}}{\boldsymbol{G}}{\big( G_{k+1} \big)_{k=0,\ldots,N-1}} \quad \text{and} \quad \fonction{\sigma^{-1}}{(\mathbb{R}^n)^{N+1}}{(\mathbb{R}^n)^{N}}{\boldsymbol{G}}{ \big( G_{k-1} \big)_{k=1,\ldots,N},}
\end{equation}
where the integer $n$ is $d$ or $m$. \\
The first property is the following: considering the quadrature formula of Gauss as approximation of the integral, we can prove the following discrete fractional integration by parts:
\begin{property}[Discrete fractional integration by parts]\label{lemdfibp}
Let $\boldsymbol{G}^1$, $\boldsymbol{G}^2 \in (\mathbb{R}^d)^{N+1}$ satisfying $G^1_0 = G^2_N = 0$, then we have:
\begin{equation}\tag{DFIBP}\label{eqdfibp}
h \displaystyle \sum_{k=1}^N ({}_{{\rm c}} \Delta^\alpha_- \boldsymbol{G}^1)_k \cdot \sigma^{-1} (\boldsymbol{G}^2)_k = h \displaystyle \sum_{k=0}^{N-1} \sigma (\boldsymbol{G}^1)_k \cdot ({}_{{\rm c}} \Delta^\alpha_+ \boldsymbol{G}^2)_k.
\end{equation}
\end{property}
\begin{proof}
Since $G^1_0 = G^2_N = 0$, we have ${}_{{\rm c}} \Delta^\alpha_- \boldsymbol{G}^1 = \Delta^\alpha_- \boldsymbol{G}^1 $ and ${}_{{\rm c}} \Delta^\alpha_+ \boldsymbol{G}^2 = \Delta^\alpha_+ \boldsymbol{G}^2 $. Then, we have:
\begin{equation}
h \displaystyle \sum_{k=1}^N (\Delta^\alpha_- \boldsymbol{G}^1)_k \cdot \sigma^{-1} (\boldsymbol{G}^2)_k = h \displaystyle \sum_{k=0}^{N-1} (\Delta^\alpha_- \boldsymbol{G}^1)_{k+1} \cdot G^2_k = h^{1-\alpha} \displaystyle \sum_{k=0}^{N-1} \sum_{r=0}^{k+1} \alpha_r G^1_{k+1-r} \cdot G^2_k.
\end{equation}
Finally, since $G^1_0 = G^2_N = 0$, the following equalities hold:
\begin{equation}
\begin{array}{rccl}
& h \displaystyle \sum_{k=1}^N (\Delta^\alpha_- \boldsymbol{G}^1)_k \cdot \sigma^{-1} (\boldsymbol{G}^2)_k & = & h^{1-\alpha} \displaystyle \sum_{k=0}^{N-1} \sum_{r=0}^{k} \alpha_r G^1_{k+1-r} \cdot G^2_k \\
= & h^{1-\alpha} \displaystyle \sum_{r=0}^{N-1} \sum_{k=r}^{N-1} \alpha_r G^1_{k+1-r} \cdot G^2_k & = & h^{1-\alpha} \displaystyle \sum_{r=0}^{N-1} \sum_{k=0}^{N-r-1} \alpha_r G^1_{k+1} \cdot G^2_{k+r} \\
= & h^{1-\alpha} \displaystyle \sum_{k=0}^{N-1} G^1_{k+1} \cdot \left( \sum_{r=0}^{N-k-1} \alpha_r G^2_{k+r} \right) & = & h^{1-\alpha} \displaystyle \sum_{k=0}^{N-1} G^1_{k+1} \cdot \left( \sum_{r=0}^{N-k} \alpha_r G^2_{k+r} \right) ,
\end{array}
\end{equation}
which concludes the proof.
\end{proof}
This last result is very useful for discrete calculus of variations involving discrete fractional derivatives, see proof of Theorem \ref{thmfinald}. Secondly let us prove the following discrete version of the fractional Cauchy-Lipschitz Theorem proved in \cite{bour7}:
\begin{theorem}[Discrete fractional Cauchy-Lipschitz theorem]\label{thmdfcl}
Let $F \in \mathscr{C}^0(\mathbb{R}^d \times [a,b], \mathbb{R}^d)$ satisfying the following Lipschitz type condition:
\begin{equation}\label{eq22}
\exists K \in \mathbb{R}, \; \forall (x_1,x_2,t) \in (\mathbb{R}^d)^2 \times [a,b], \; \Vert F(x_1,t)-F(x_2,t) \Vert \leq K \Vert x_1 - x_2 \Vert,
\end{equation}
with $h^\alpha K < 1$. Then, the following discrete fractional Cauchy problem:
\begin{equation}\label{eq21}
\left\lbrace \begin{array}{l}
{}_{{\rm c}} \Delta^\alpha_- \boldsymbol{Q} = F(\boldsymbol{Q},\boldsymbol{T}) \\
Q_0 = A
\end{array}
\right.
\end{equation}
has an unique solution $\boldsymbol{Q} \in (\mathbb{R}^d)^{N+1}$.
\end{theorem}
\begin{proof}
We are going to construct by induction the solution $\boldsymbol{Q}$ of \eqref{eq21}. Our method uses the classical fix point theorem concerning the contraction mappings. Indeed, let us choose $Q_0 = A$. Then, for any $k=1,\ldots,N$, $Q_k$ has to satisfy:
\begin{equation}
Q_k = h^\alpha F(Q_k,t_k) + Q_0 - \displaystyle \sum_{r=1}^{k-1} \alpha_r (Q_{k-r}-Q_0).
\end{equation}
However, for any $k=1,\ldots,N$, the application $ h^\alpha F(\cdot,t_k) + Q_0 - \sum_{r=1}^{k-1} \alpha_r (Q_{k-r}-Q_0) $ is a contraction and consequently admits an unique fix point. Hence, we first construct $Q_1$, then $Q_2$, etc. By induction, we construct a solution $\boldsymbol{Q}$ of \eqref{eq21} and such a construction assures its uniqueness.
\end{proof}
\subsection{First step of construction}\label{section23}
As said in introduction of this section, in order to complete the first step of construction of a variational integrator, we have to provide a discrete version of $\mathcal{L}^\alpha$. In this way, let us give the following definition:
\begin{itemize}
\item The elements $\boldsymbol{U} \in (\mathbb{R}^m)^{N+1}$ are called the \textit{discrete controls};
\item For any discrete control $\boldsymbol{U}$, let $\boldsymbol{Q}^{\boldsymbol{U},\alpha} \in (\mathbb{R}^d)^{N+1}$ denote the unique solution of the following discrete Cauchy problem:
\begin{equation}\tag{CP${}^\alpha_{\boldsymbol{Q}}$}\label{eqdcpq}
\left\lbrace \begin{array}{l}
{}_{{\rm c}} \Delta^\alpha_- \boldsymbol{Q} =f(\boldsymbol{Q},\boldsymbol{U},\boldsymbol{T})\\
Q_0 = A \in \mathbb{R}^d .
\end{array}
\right.
\end{equation}
$\boldsymbol{Q}^{\boldsymbol{U},\alpha}$ is called the \textit{discrete state variable} associated to $\boldsymbol{U}$. Its existence and its uniqueness are provided by Theorem \ref{thmdfcl} and Conditions \eqref{condf} and \eqref{condh};
\item Finally, we define the following \textit{discrete cost functional}:
\begin{equation}
\fonction{\mathcal{L}^\alpha_h}{(\mathbb{R}^m)^{N+1}}{\mathbb{R}}{\boldsymbol{U}}{h \displaystyle \sum_{k=1}^N L(Q^{\boldsymbol{U},\alpha}_k,U_k,t_k).}
\end{equation}
\end{itemize}
Hence, we have provided a discrete version $\mathcal{L}^\alpha_h$ to the cost functional $\mathcal{L}^\alpha$. Now, the second step of the construction of the variational integrator is to characterize the \textit{discrete critical points} of the discrete cost functional $\mathcal{L}^\alpha_h$ with the help of a discrete calculus of variations. \\
Let us make the following remark: such a characterization implies to be a necessary condition for the existence of an optimizer of the discrete cost functional $\mathcal{L}^\alpha_h$. In fact, in this section, we have defined an actual discrete fractional optimal control problem.
\subsection{Second step of construction}\label{section24}
The second step of construction of a variational integrator consists in forming a discrete variational principle on $\mathcal{L}^\alpha_h$. Precisely, we focus on the characterization of its \textit{discrete critical points}, \textit{i.e.} the elements $\boldsymbol{U} \in (\mathbb{R}^m)^{N+1}$ satisfying:
\begin{equation}
\forall \bar{\boldsymbol{U}} \in (\mathbb{R}^m)^{N+1}, \; D\mathcal{L}^\alpha_h (\boldsymbol{U})(\bar{\boldsymbol{U}}) := \lim\limits_{\varepsilon \to 0} \dfrac{\mathcal{L}^\alpha_h (\boldsymbol{U}+\varepsilon \bar{\boldsymbol{U}})-\mathcal{L}^\alpha_h (\boldsymbol{U})}{\varepsilon} = 0.
\end{equation}
With a discrete calculus of variations, we obtain the following discrete version of Lemma \ref{lem1} giving explicitly the value of the G\^ateaux derivative of $\mathcal{L}^\alpha_h$.
\begin{lemma}\label{lem2}
Let $\boldsymbol{U}$, $\bar{\boldsymbol{U}} \in (\mathbb{R}^m)^{N+1}$. Then, the following equality holds:
\begin{equation}
D\mathcal{L}^\alpha_h (\boldsymbol{U})(\bar{\boldsymbol{U}}) = h \displaystyle \sum_{k=1}^N \left[ \dfrac{\partial L}{\partial x} (Q^{\boldsymbol{U},\alpha}_k, U_k,t_k) \cdot \bar{Q}_k + \dfrac{\partial L}{\partial v} (Q^{\boldsymbol{U},\alpha}_k, U_k,t_k) \cdot \bar{U}_k \right],
\end{equation}
where $\bar{\boldsymbol{Q}} \in (\mathbb{R}^d )^{N+1}$ is the unique solution of the following linearised discrete fractional Cauchy problem:
\begin{equation}\label{eqldcpq}\tag{LCP${}^\alpha_{\bar{\boldsymbol{Q}}}$}
\left\lbrace \begin{array}{l}
{}_{{\rm c}} \Delta^\alpha_- \bar{\boldsymbol{Q}} = \dfrac{\partial f}{\partial x} (\boldsymbol{Q}^{\boldsymbol{U},\alpha},\boldsymbol{U},\boldsymbol{T}) \times \bar{\boldsymbol{Q}} + \dfrac{\partial f}{\partial v} (\boldsymbol{Q}^{\boldsymbol{U},\alpha},\boldsymbol{U},\boldsymbol{T}) \times \bar{\boldsymbol{U}} \\[10pt]
\bar{Q}_0 = 0.
\end{array}
\right.
\end{equation}
\end{lemma}
\begin{proof}
See Appendix \ref{appAc}.
\end{proof}
This last result does not lead to a characterization of the critical points of $\mathcal{L}^\alpha_h$ yet. As in the continuous case, we then introduce the notion of discrete adjoint variable: for any discrete control $\boldsymbol{U}$, let $\boldsymbol{P}^{\boldsymbol{U},\alpha} \in (\mathbb{R}^d)^{N+1}$ denote the unique solution of the following shifted discrete Cauchy problem:
\begin{equation}\tag{$\sigma$CP${}^\alpha_{\boldsymbol{P}}$}
\left\lbrace \begin{array}{rcl}
{}_{{\rm c}} \Delta^\alpha_+ \boldsymbol{P} & = & \dfrac{\partial H}{\partial x} \big( \sigma (\boldsymbol{Q}^{\boldsymbol{U},\alpha}),\sigma (\boldsymbol{U}),\boldsymbol{P},\sigma (\boldsymbol{T}) \big) \\[10pt]
& = & \dfrac{\partial L}{\partial x} \big( \sigma (\boldsymbol{Q}^{\boldsymbol{U},\alpha}),\sigma (\boldsymbol{U}),\sigma (\boldsymbol{T}) \big) + \left( \dfrac{\partial f}{\partial x} \big( \sigma (\boldsymbol{Q}^{\boldsymbol{U},\alpha}),\sigma (\boldsymbol{U}),\sigma (\boldsymbol{T}) \big) \right)^T \times \boldsymbol{P} \\[10pt]
P_N & = & 0.
\end{array}
\right.
\end{equation}
$\boldsymbol{P}^{\boldsymbol{U},\alpha}$ is called the \textit{discrete adjoint variable} associated to $\boldsymbol{U}$. Its existence and its uniqueness are provided by the analogous of Theorem \ref{thmdfcl} for right discrete fractional derivative and by Conditions \eqref{condf} and \eqref{condh}. Let us note that, since $\boldsymbol{P}^{\boldsymbol{U},\alpha}_N = 0$, we can write ${}_{{\rm c}} \Delta^\alpha_+ \boldsymbol{P}^{\boldsymbol{U},\alpha} = \Delta^\alpha_+ \boldsymbol{P}^{\boldsymbol{U},\alpha}$. \\
The presence of shift operators in the definition of the discrete adjoint variable is the consequence of the change of sums in the discrete fractional integration by parts \eqref{eqdfibp} (see Property \ref{lemdfibp}). We refer to the proof of Theorem \ref{thmfinald} for more details. We also refer to Remark \ref{rem0} for a discussion about the presence of the shift operators. \\
Finally, let us note that for any discrete control $\boldsymbol{U}$, the couple $(\boldsymbol{Q}^{\boldsymbol{U},\alpha},\boldsymbol{P}^{\boldsymbol{U},\alpha})$ is solution of the following \textit{shifted discrete fractional Hamiltonian system}:
\begin{equation}\tag{$\sigma$HS${}^\alpha_h$}\label{eqhamsystd}
\left\lbrace \begin{array}{l}
{}_{{\rm c}} \Delta^\alpha_- \boldsymbol{Q} = \dfrac{\partial H}{\partial w} \big( \boldsymbol{Q},\boldsymbol{U},\sigma^{-1}(\boldsymbol{P}),\boldsymbol{T} \big) \\[10pt]
\Delta^\alpha_+ \boldsymbol{P} = \dfrac{\partial H}{\partial x} \big( \sigma (\boldsymbol{Q}),\sigma (\boldsymbol{U}),\boldsymbol{P},\sigma (\boldsymbol{T}) \big).
\end{array}
\right.
\end{equation}
Finally, the introduction of this last discrete element allows us to prove the following theorem:
\begin{theorem}\label{thmfinald}
Let $\boldsymbol{U} \in (\mathbb{R}^m)^{N+1}$. Then, $\boldsymbol{U}$ is a discrete critical point of $\mathcal{L}^{\alpha}_h$ if and only if $(\boldsymbol{Q}^{\boldsymbol{U},\alpha},\boldsymbol{U},\boldsymbol{P}^{\boldsymbol{U},\alpha})$ is solution of the following \textit{shifted discrete fractional stationary equation}:
\begin{equation}\tag{$\sigma$SE${}^\alpha_h$}\label{eqstatfd}
\dfrac{\partial H}{\partial v} \big( \boldsymbol{Q},\boldsymbol{U},\sigma^{-1} (\boldsymbol{P}),\boldsymbol{T} \big) = 0.
\end{equation}
\end{theorem}
\begin{proof}
Let $\boldsymbol{U}$, $\bar{\boldsymbol{U}} \in (\mathbb{R}^m)^{N+1}$. From Lemma \ref{lem2}, we have:
\begin{multline}
h^{-1} D\mathcal{L}^\alpha_h (\boldsymbol{U})(\bar{\boldsymbol{U}}) = \displaystyle \sum_{k=1}^N \left[ \dfrac{\partial L}{\partial x} (Q^{\boldsymbol{U},\alpha}_k, U_k,t_k) + \left( \dfrac{\partial f}{\partial x} (Q^{\boldsymbol{U},\alpha}_k, U_k,t_k) \right)^T \times \sigma^{-1} (\boldsymbol{P}^{\boldsymbol{U},\alpha})_k \right] \cdot \bar{Q}_k \\ - \displaystyle \sum_{k=1}^N \left( \left( \dfrac{\partial f}{\partial x} (Q^{\boldsymbol{U},\alpha}_k, U_k,t_k) \right)^T \times \sigma^{-1} (\boldsymbol{P}^{\boldsymbol{U},\alpha})_k \right)\cdot \bar{Q}_k + \displaystyle \sum_{k=1}^N \dfrac{\partial L}{\partial v} (Q^{\boldsymbol{U},\alpha}_k, U_k,t_k) \cdot \bar{U}_k .
\end{multline}
Then:
\begin{multline}
h^{-1} D\mathcal{L}^\alpha_h (\boldsymbol{U})(\bar{\boldsymbol{U}}) = \displaystyle \sum_{k=0}^{N-1} ({}_{{\rm c}} \Delta^\alpha_+ \boldsymbol{P}^{\boldsymbol{U},\alpha})_k \cdot \sigma (\bar{\boldsymbol{Q}})_k - \displaystyle \sum_{k=1}^N \left( \dfrac{\partial f}{\partial x} (Q^{\boldsymbol{U},\alpha}_k, U_k,t_k) \times \bar{Q}_k \right) \cdot \sigma^{-1} (\boldsymbol{P}^{\boldsymbol{U},\alpha})_k \\ + \displaystyle \sum_{k=1}^N \dfrac{\partial L}{\partial v} (Q^{\boldsymbol{U},\alpha}_k, U_k,t_k) \cdot \bar{U}_k .
\end{multline}
From the discrete fractional integration by parts \eqref{eqdfibp} (see Property \ref{lemdfibp}), we obtain:
\begin{multline}
h^{-1} D\mathcal{L}^\alpha_h (\boldsymbol{U})(\bar{\boldsymbol{U}}) = \displaystyle \sum_{k=1}^{N} \left( ({}_{{\rm c}} \Delta^\alpha_- \bar{\boldsymbol{Q}})_k - \dfrac{\partial f}{\partial x} (Q^{\boldsymbol{U},\alpha}_k, U_k,t_k) \times \bar{Q}_k \right) \cdot \sigma^{-1} (\boldsymbol{P}^{\boldsymbol{U},\alpha})_k \\ + \displaystyle \sum_{k=1}^N \dfrac{\partial L}{\partial v} (Q^{\boldsymbol{U},\alpha}_k, U_k,t_k) \cdot \bar{U}_k .
\end{multline}
Since $\bar{\boldsymbol{Q}}$ is solution of \eqref{eqldcpq}, we have:
\begin{eqnarray*}
h^{-1} D\mathcal{L}^\alpha_h (\boldsymbol{U})(\bar{\boldsymbol{U}}) & = & \displaystyle \sum_{k=1}^{N} \left( \dfrac{\partial f}{\partial v} (Q^{\boldsymbol{U},\alpha}_k, U_k,t_k) \times \bar{U}_k \right) \cdot \sigma^{-1} (\boldsymbol{P}^{\boldsymbol{U},\alpha})_k + \displaystyle \sum_{k=1}^N \dfrac{\partial L}{\partial v} (Q^{\boldsymbol{U},\alpha}_k, U_k,t_k) \cdot \bar{U}_k \\
& = & \displaystyle \sum_{k=1}^{N} \left( \left( \dfrac{\partial f}{\partial v} (Q^{\boldsymbol{U},\alpha}_k, U_k,t_k) \right)^T \times \sigma^{-1} (\boldsymbol{P}^{\boldsymbol{U},\alpha})_k + \dfrac{\partial L}{\partial v} (Q^{\boldsymbol{U},\alpha}_k, U_k,t_k) \right) \cdot \bar{U}_k.
\end{eqnarray*}
Finally:
\begin{equation}
D\mathcal{L}^\alpha_h(\boldsymbol{U})(\bar{\boldsymbol{U}}) = h \displaystyle \sum_{k=1}^{N} \dfrac{\partial H}{\partial v} \big(Q^{\boldsymbol{U},\alpha}_k,U_k,\sigma^{-1}(\boldsymbol{P}^{\boldsymbol{U},\alpha})_k,t_k \big) \cdot \bar{U}_k.
\end{equation}
The proof is completed.
\end{proof}
Finally, from Theorem \ref{thmfinald}, we obtain the following result leading to the variational integrator constructed:
\begin{corollary}\label{corfinald}
$\mathcal{L}^\alpha_h$ has a discrete critical point if and only if there exists $(\boldsymbol{Q},\boldsymbol{U},\boldsymbol{P}) \in (\mathbb{R}^d)^{N+1} \times (\mathbb{R}^m)^{N+1} \times (\mathbb{R}^d)^{N+1}$ solution of the following \textit{shifted discrete fractional Pontryagin's system}:
\begin{equation}\tag{$\sigma$PS${}^\alpha_h$}\label{eqpontsystfd}
\left\lbrace \begin{array}{l}
{}_{{\rm c}} \Delta^\alpha_- \boldsymbol{Q} = \dfrac{\partial H}{\partial w} \big(\boldsymbol{Q},\boldsymbol{U},\sigma^{-1}(\boldsymbol{P}),\boldsymbol{T}\big) \\[10pt]
\Delta^\alpha_+ \boldsymbol{P} = \dfrac{\partial H}{\partial x} \big(\sigma(\boldsymbol{Q}),\sigma(\boldsymbol{U}),\boldsymbol{P},\sigma(\boldsymbol{T})\big) \\[10pt]
\dfrac{\partial H}{\partial v} \big( \boldsymbol{Q},\boldsymbol{U},\sigma^{-1}(\boldsymbol{P}),\boldsymbol{T} \big) = 0 \\[10pt]
( Q_0,P_N ) = ( A,0 ).
\end{array}
\right.
\end{equation}
In this case, $\boldsymbol{U}$ is a discrete critical point of $\mathcal{L}^\alpha_h$ and we have $(\boldsymbol{Q},\boldsymbol{P}) = (\boldsymbol{Q}^{\boldsymbol{U},\alpha},\boldsymbol{P}^{\boldsymbol{U},\alpha})$.
\end{corollary}
Let us note that \eqref{eqpontsystfd} is made up of the shifted discrete Hamiltonian system \eqref{eqhamsystd}, the shifted stationary equation \eqref{eqstatfd} and initial and final conditions. \\
Hence, we have constructed the variational integrator \eqref{eqpontsystfd} for the fractional Pontryagin's system \eqref{eqps}. It is then a numerical scheme for \eqref{eqps} preserving its variational structure in the sense that the discrete solutions $\boldsymbol{U}$ obtained correspond to the discrete critical points of the discrete version $\mathcal{L}^\alpha_h$ of $\mathcal{L}^\alpha$.
\begin{remark}\label{rem0}
Let us note that the variational integrator \eqref{eqpontsystfd} does not correspond with a direct discretization of \eqref{eqps} as it is done in \cite{deft}. There is an emergence of shift operators caused by the conservation at the discrete level of the variational structure. However, it is proved that the use of shifted numerical schemes allows to obtain more stability for some fractional differential equations, see \cite{meer2,meer}.
\end{remark}
\begin{remark}\label{rem1}
Let us remind the following remark: since a fractional Pontryagin's system emerges from a fractional optimal control problem, the main unknown is then the control $u$. Consequently, the convergence of the variational integrator \eqref{eqpontsystfd} is going to be considered only with respect to $u$. Let us note that the value of $U_0$ does not take place in the variational integrator \eqref{eqpontsystfd}: it is a free value. Nevertheless, this is totally coherent with the fact that this value does not take place neither in the definition of $\mathcal{L}^\alpha_h$. Hence, in the following examples in Section \ref{section3}, the error between an exact solution $u$ of \eqref{eqps} and a numerical solution $\boldsymbol{U}$ obtained with \eqref{eqpontsystfd} is going to be evaluated on $\Vert u(t_k) -U_k \Vert$ for $k \in \{ 1,\ldots,N \}$ only.
\end{remark}
\subsection{Link with the discrete fractional Euler-Lagrange equation}\label{section25}
Let us take the constraint function $f(x,v,t) = v$ satisfying \eqref{condf}. In this case, applying Corollary \ref{corfinald}, we know that there exists a critical point of $\mathcal{L}^\alpha_h$ if and only if there exists a solution $(\boldsymbol{Q},\boldsymbol{U},\boldsymbol{P}) \in (\mathbb{R}^d)^{N+1} \times (\mathbb{R}^m)^{N+1} \times (\mathbb{R}^d)^{N+1}$ of the shifted discrete fractional Pontryagin's system \eqref{eqpontsystfd} here given by:
\begin{equation}
\left\lbrace \begin{array}{l}
{}_{{\rm c}} \Delta^\alpha_- \boldsymbol{Q} = \boldsymbol{U} \\[10pt]
\Delta^\alpha_+ \boldsymbol{P} = \dfrac{\partial L}{\partial x} \big(\sigma(\boldsymbol{Q}),\sigma(\boldsymbol{U}),\sigma(\boldsymbol{T})\big) \\[10pt]
\dfrac{\partial L}{\partial v} ( \boldsymbol{Q},\boldsymbol{U},\boldsymbol{T} ) + \sigma^{-1} (\boldsymbol{P}) = 0 \\[10pt]
( Q_0,P_N ) = ( A,0 ).
\end{array}
\right.
\end{equation}
In the affirmative case, it implies that $\boldsymbol{Q}$ is a discrete solution of the following \textit{discrete fractional Euler-Lagrange equation}:
\begin{equation}\tag{EL${}^\alpha_h$}
\dfrac{\partial L}{\partial x} ( \boldsymbol{Q},{}_{{\rm c}} \Delta^\alpha_- \boldsymbol{Q},\boldsymbol{T} ) + \Delta^\alpha_+ \left( \dfrac{\partial L}{\partial v} ( \boldsymbol{Q},{}_{{\rm c}} \Delta^\alpha_- \boldsymbol{Q},\boldsymbol{T} ) \right) = 0.
\end{equation}
Finally, according to our works in \cite{bour}, we then obtain that $\boldsymbol{Q}$ is a critical point of the following \textit{discrete fractional Lagrangian functional}:
\begin{equation}
\boldsymbol{Q} \longrightarrow h \displaystyle \sum_{k=1}^N L \big(Q_k,({}_{{\rm c}} \Delta^\alpha_- \boldsymbol{Q})_k,t_k \big).
\end{equation}
We refer to \cite{bour} for more details concerning discrete fractional Euler-Lagrange equations.
\section{Numerical tests}\label{section3}
In the following numerical tests, according to Remark \ref{rem1}, we are going to give graphic representations only of discrete solutions $\boldsymbol{U}$ and the study of the convergence of the variational integrator \eqref{eqpontsystfd} is only going to be evaluated on the convergence of the discrete control to the continuous one.
\subsection{The linear-quadratic example}\label{section31}
Linear-quadratic examples are often studied in the literature because they are used for tracking problems. The aim of these problems is to determine a control allowing to approach as much as possible reference trajectories, \cite[Part 1.4, p.49]{trel}. In this section, we study such an example, \cite[Part 4.4.3, example 3, p.53]{evan}. More generally, a quadratic Lagrangian is often natural (for example in order to minimize distances) and even if the constraint functions are frequently non linear, we are often leaded to study linearised versions. \\
Let us choose $d = m = A = 1 $ and $[a,b] = [0,1]$. Then, let us take the following quadratic Lagrangian and linear constraint function:
\begin{equation}\label{eq31-1}
\fonction{L}{\mathbb{R}^2 \times [0,1]}{\mathbb{R}}{(x,v,t)}{(x^2+v^2)/2} \quad \text{and} \quad \fonction{f}{\mathbb{R}^2 \times [0,1]}{\mathbb{R}}{(x,v,t)}{x+v.}
\end{equation}
Let us give the graphic representations of the numerical solutions $\boldsymbol{U}$ given by \eqref{eqpontsystfd} for $N=500$ and for $\alpha=1$, $3/4$, $1/2$, $1/4$:
\begin{equation*}
\begin{array}{c}
\includegraphics[width=0.6\textwidth]{discretesolutionUplusieursalphacolor}
\end{array}
\end{equation*}
We have seen in \cite{bour7} that the fractional Pontryagin's system \eqref{eqps} is explicitly solved only in the classical case $\alpha =1$ and we obtained the following unique critical point of $\mathcal{L}^1$:
\begin{equation}
\forall t \in [0,1], \; u(t) = \dfrac{\cosh(\sqrt{2})}{R} \sinh (\sqrt{2}t) - \dfrac{\sinh(\sqrt{2})}{R} \cosh (\sqrt{2}t),
\end{equation}
where $ R = \sqrt{2} \cosh (\sqrt{2}) - \sinh (\sqrt{2}) $. Hence, we can only test the convergence of the variational integrator \eqref{eqpontsystfd} for $\alpha =1$. We give the following graphic representing the logarithm of the error $\max \big( \vert u(t_k)-U_k \vert, k=1,\ldots,N \big)$ versus the logarithm of $h$ and the identity function for comparison:
\begin{equation*}
\begin{array}{c}
\includegraphics[width=0.6\textwidth]{logerroralpha1color}
\end{array}
\end{equation*}
In this example with $\alpha = 1$, the convergence seems then obtained with order $1$. Nevertheless, we do not know the exact solution of \eqref{eqps} in the strict fractional case $0<\alpha <1$. Consequently, we can not study the behaviour of the error in this case.
\subsection{A solved fractional example}\label{section32}
In this section, we are going to compute \eqref{eqpontsystfd} in the framework of an example solved in the strict fractional case in the sense that we know explicitly the unique critical point $u$ of $\mathcal{L}^\alpha_h$ for any $0< \alpha \leq 1$, see \cite{bour7}. Consequently, for this example, we can test the convergence of the variational integrator \eqref{eqpontsystfd} for any $0 < \alpha \leq 1$. \\
Then, let us choose $d=m=A=1$ and $[a,b]=[0,1]$. Then, let us take the following Lagrangian and linear constraint function:
\begin{equation}\label{eq31-1}
\fonction{L}{\mathbb{R}^2 \times [0,1]}{\mathbb{R}}{(x,v,t)}{(1-t)x+(v^2/2)} \quad \text{and} \quad \fonction{f}{\mathbb{R}^2 \times [0,1]}{\mathbb{R}}{(x,v,t)}{x+v.}
\end{equation}
Let us give the graphic representations of the numerical solutions $\boldsymbol{U}$ given by \eqref{eqpontsystfd} for $N=500$ and for $\alpha=1$, $3/4$, $1/2$, $1/4$:
\begin{equation*}
\begin{array}{c}
\includegraphics[width=0.6\textwidth]{discretesolutionUplusieursalphacolor2}
\end{array}
\end{equation*}
As we have seen in \cite{bour7}, the fractional Pontryagin's system \eqref{eqps} is explicitly solved for any $0 < \alpha \leq 1$ and we obtained the following unique critical point of $\mathcal{L}^\alpha$:
\begin{equation}
\forall t \in [0,1], \; u(t) = -(1-t)^{\alpha +1} {\rm E}_{\alpha,\alpha+2} \big( (1-t)^\alpha \big),
\end{equation}
where ${\rm E}_{\alpha,\alpha+2}$ is the Mittag-Leffler function with parameter $(\alpha,\alpha+2)$. Let us the convergence of the variational integrator \eqref{eqpontsystfd} for any $0<\alpha \leq 1$. We give the following graphics representing the logarithm of the error $\max \big( \vert u(t_k)-U_k \vert, k=1,\ldots,N \big)$ versus the logarithm of $h$ and the identity function for comparison for $\alpha=1, \; 3/4, \; 1/2, \; 1/4$:
\begin{equation*}
\begin{array}{cc}
\includegraphics[width=0.5\textwidth]{logerralpha1color2} & \includegraphics[width=0.5\textwidth]{logerralpha2color}
\end{array}
\end{equation*}
\begin{equation*}
\begin{array}{cc}
\includegraphics[width=0.5\textwidth]{logerralpha3color} & \includegraphics[width=0.5\textwidth]{logerralpha4color}
\end{array}
\end{equation*}
For this example, the convergence seems then obtained for any $\alpha=1$, $3/4$, $1/2$, $1/4$ and still with order $1$. Hence, the graphics obtained in these Sections \ref{section31} and \ref{section32} make us confident with respect to the quality of \eqref{eqpontsystfd} both in the classical and strict fractional cases.
\section{A discrete fractional Noether's theorem}\label{section4}
Fractional Pontryagin's systems \eqref{eqps} are very difficult to solve explicitly, see example in Section \ref{section31}. Consequently, a deviously way in order to get informations on the exact solutions is to derive a constant of motion, \textit{i.e.} functions which are constant on each solution. Such conservation laws allow to obtain many informations in the phase space for example or to integrate the equation by quadrature. In \cite{bour7}, we prove a fractional Noether's theorem giving the existence of an explicit conservation law for fractional Pontryagin's systems \eqref{eqps} exhibiting a symmetry. Let us remind that this result is based on a preliminary result proved by Torres and Frederico in \cite{torr,torr2}. \\
In this section, we study the existence of discrete conservation laws for shifted discrete fractional Pontryagin's systems \eqref{eqpontsystfd}. Precisely, following the same strategy, we introduce the notion of \textit{discrete symmetry} for such systems and prove a discrete fractional Noether's theorem providing an \textit{explicit computable} discrete constant of motion. Let us note that this work is strongly inspired from our study in \cite{bour2} where we have provided a discrete fractional Noether's theorem for discrete fractional Euler-Lagrange equations admitting a discrete symmetry. \\
We first review the definition of a one parameter group of diffeomorphisms:
\begin{definition}
Let $n \in \mathbb{N}^*$. For any real $s$, let $\fonctionsansdef{\phi (s,\cdot)}{\mathbb{R} ^n}{\mathbb{R} ^n}$ be a diffeomorphism. Then, $\Phi = \{ \phi (s,\cdot) \}_{s \in \mathbb{R}}$ is a one parameter group of diffeomorphisms of $\mathbb{R}^n$ if it satisfies:
\begin{enumerate}
\item $\phi (0,\cdot) = Id_{\mathbb{R} ^n}$;
\item $\forall s,s' \in \mathbb{R}, \; \phi (s,\cdot) \circ \phi (s',\cdot) = \phi (s+s',\cdot) $;
\item $\phi$ is of class $\mathscr{C}^2$.
\end{enumerate}
\end{definition}
Usual examples of one parameter groups of diffeomorphisms are given by translations and rotations. The action of three one parameter groups of diffeomorphisms on an Hamiltonian allows to define the notion of a discrete symmetry for a shifted discrete fractional Pontryagin's system \eqref{eqpontsystfd}:
\begin{definition}\label{defsymd}
Let $\Phi_i = \{ \phi_i (s,\cdot) \}_{s \in \mathbb{R}}$, for $i=1,2,3$, be three one parameter groups of diffeomorphisms of $\mathbb{R}^d$, $ \mathbb{R}^m$ and $\mathbb{R}^d$ respectively. Let $L$ be a Lagrangian, $f$ be a constraint function and $H$ be the associated Hamiltonian. $H$ is said to be ${}_{{\rm c}} \Delta^\alpha_-$-invariant under the action of $(\Phi_i)_{i=1,2,3}$ if it satisfies: for any $(\boldsymbol{Q},\boldsymbol{U},\boldsymbol{P})$ solution of \eqref{eqpontsystfd} and any $s \in \mathbb{R}$
\begin{multline}\label{eq4-1}
H \Big( \phi_1 \big(s,\boldsymbol{Q}\big), \phi_2 \big(s,\boldsymbol{U}\big) , \phi_3 \big(s,\sigma^{-1}(\boldsymbol{P}) \big) ,\boldsymbol{T} \Big) - \phi_3 \big(s,\sigma^{-1}(\boldsymbol{P}) \big) \cdot {}_{{\rm c}} \Delta^\alpha_- \Big( \phi_1 \big(s,\boldsymbol{Q}\big) \Big) \\ = H\big(\boldsymbol{Q},\boldsymbol{U},\sigma^{-1}(\boldsymbol{P}),\boldsymbol{T} \big) - \sigma^{-1}(\boldsymbol{P}) \cdot {}_{{\rm c}} \Delta^\alpha_- \boldsymbol{Q}.
\end{multline}
\end{definition}
From this notion, we prove the following Lemma:
\begin{lemma}\label{lemtorrd}
Let $L$ be a Lagrangian, $f$ be a constraint function and $H$ be the associated Hamiltonian. Let us assume that $H$ is ${}_{{\rm c}} \Delta^\alpha_-$-invariant under the action of three one parameter groups of diffeomorphisms $(\Phi_i)_{i=1,2,3}$. Then, the following equality holds for any solution $(\boldsymbol{Q},\boldsymbol{U},\boldsymbol{P})$ solution of \eqref{eqpontsystfd}:
\begin{equation}\label{eqlemtorrd}
\dfrac{\partial \phi_1}{\partial s} (0,\boldsymbol{Q}) \cdot \sigma^{-1} ( \Delta^\alpha_+ \boldsymbol{P} ) - {}_{{\rm c}} \Delta^\alpha_- \left( \dfrac{\partial \phi_1}{\partial s} (0,\boldsymbol{Q}) \right) \cdot \sigma^{-1} (\boldsymbol{P} ) = 0.
\end{equation}
\end{lemma}
\begin{proof}
Let us differentiate \eqref{eq4-1} with respect to $s$ and let us invert the operator $ {}_{{\rm c}} \Delta^\alpha_- $ and $\partial / \partial s $. Taking $s=0$, we finally obtain:
\begin{multline}
\dfrac{\partial H}{\partial x} (\star) \cdot \dfrac{\partial \phi_1}{\partial s}(0,\boldsymbol{Q}) + \dfrac{\partial H}{\partial v} (\star) \cdot \dfrac{\partial \phi_2}{\partial s}(0,\boldsymbol{U}) + \dfrac{\partial H}{\partial w} (\star) \cdot \dfrac{\partial \phi_3}{\partial s}\big( 0,\sigma^{-1}(\boldsymbol{P}) \big) \\ - \dfrac{\partial \phi_3}{\partial s}\big( 0,\sigma^{-1}(\boldsymbol{P}) \big) \cdot {}_{{\rm c}} \Delta^\alpha_- \boldsymbol{Q} - \sigma^{-1} (\boldsymbol{P}) \cdot {}_{{\rm c}} \Delta^\alpha_- \left( \dfrac{\partial \phi_1}{\partial s}(0,\boldsymbol{Q}) \right) = 0,
\end{multline}
where $\star = \big( \boldsymbol{Q},\boldsymbol{U},\sigma^{-1}(\boldsymbol{P}),\boldsymbol{T} \big)$. Since $(\boldsymbol{Q},\boldsymbol{U},\boldsymbol{P})$ is solution of \eqref{eqpontsystfd}, we obtain \eqref{eqlemtorrd}.
\end{proof}
Let us note that this last result corresponds to the discrete version of the result proved by Torres and Frederico in \cite{torr,torr2}. Let us remind that our aim is to provide an explicit discrete constant of motion for shifted discrete fractional Pontryagin's systems \eqref{eqpontsystfd} exhibiting a discrete symmetry. Our result is based on Lemma \ref{lemtorrd} and on the following implication:
\begin{equation}\label{eq4-2}
\forall \boldsymbol{G} \in \mathbb{R}^{N+1}, \; \Delta^1_- \boldsymbol{G} = 0 \Longrightarrow \exists c \in \mathbb{R}, \; \forall k=0,\ldots,N, \; G_k = c.
\end{equation}
Namely, if the discrete derivative of $ \boldsymbol{G}$ vanishes, then $ \boldsymbol{G}$ is constant. Consequently, our aim is to write the left term of \eqref{eqlemtorrd} as an explicit discrete derivative (\textit{i.e.} as $\Delta^1_-$ of an explicit quantity). In this way, we are going to use a \textit{discrete transfer formula} as it is done in \cite{bour2} for discrete fractional Euler-Lagrange equations admitting a discrete symmetry. \\
Nevertheless, we have first to introduce some square matrices of length $(N+1)$. First, $B_{1} := \text{Id}_{N+1}$ and then, for any $r \in \{ 2,\ldots,N \}$, the square matrices $B_r \in \mathcal{M}_{N+1}$ defined by:
\begin{equation}
\forall i,j=0,\ldots,N, \; (B_r)_{i,j} := \delta_{\{1 \leq i \leq N-1\}} \delta_{\{1 \leq j \leq N-r\}} \delta_{\{0 \leq i-j \leq r-1\}} - \delta_{\{j=0\}} \delta_{\{r \leq i\}},
\end{equation}
where $\delta$ is the Kronecker symbol. Secondly, we define the square matrices $C_r \in \mathcal{M}_{N+1}$ by:
\begin{equation}
\forall r=1,\ldots,N, \; \forall i,j=0,\ldots,N, \; (C_r)_{i,j} := \delta_{\{r \leq i\}} \delta_{\{j=0\}} .
\end{equation}
Finally, we define the square matrices $A_r \in \mathcal{M}_{N+1}$ by:
\begin{equation}
\forall r=1,\ldots,N, \; A_r := \alpha_r B_r + \beta^\alpha_r C_r ,
\end{equation}
where $\beta^\alpha_r = \sum_{k=0}^{r} \alpha_k $. Examples of matrices $A_r \in \mathcal{M}_{N+1}$ for $N=5$ are given in Appendix \ref{appAd}.
\begin{lemma}[Discrete transfer formula]\label{lemdtransform}
Let $\boldsymbol{G}^1$, $\boldsymbol{G}^2 \in (\mathbb{R}^d)^{N+1}$ satisfying $G^2_N = 0$. Then, the following equality holds:
\begin{equation}\label{eq4-3}
\boldsymbol{G}^1 \cdot \sigma^{-1} ( \Delta^\alpha_+ \boldsymbol{G}^2 ) - ({}_{{\rm c}} \Delta^\alpha_- \boldsymbol{G}^1) \cdot \sigma^{-1} (\boldsymbol{G}^2 ) = h^{1-\alpha} \Delta^1_- \Big[ \displaystyle \sum_{r=1}^N A_r \times \big( \boldsymbol{G}^1 \cdot \sigma^{r-1} (\boldsymbol{G}^2 ) \big) \Big].
\end{equation}
\end{lemma}
\begin{proof} See Appendix \ref{appAd}. \end{proof}
Consequently, combining Lemmas \ref{lemtorrd} and \ref{lemdtransform}, we prove:
\begin{theorem}[Discrete fractional Noether's theorem]\label{thmdfnoether}
Let $L$ be a Lagrangian, $f$ be a constraint function and $H$ be the associated Hamiltonian. Let us assume that $H$ is ${}_{{\rm c}} \Delta^\alpha_-$-invariant under the action of three one parameter groups of diffeomorphisms $(\Phi_i)_{i=1,2,3}$. Then, the following equality holds for any solution $(\boldsymbol{Q},\boldsymbol{U},\boldsymbol{P})$ of \eqref{eqpontsystfd}:
\begin{equation}
\Delta^1_- \left[ \sum_{r=1}^N A_r \times \left( \dfrac{\partial \phi_1}{\partial s} (0,\boldsymbol{Q}) \cdot \sigma^{r-1} (\boldsymbol{P}) \right) \right] = 0.
\end{equation}
\end{theorem}
According to Equation \eqref{eq4-2}, this theorem provides a discrete constant of motion for any shifted discrete fractional Pontryagin's systems \eqref{eqpontsystfd} exhibiting a discrete symmetry. Moreover, this discrete conservation law is not only \textit{explicit} but also \textit{computable} in a finite number of steps. Let us see a concrete example:
\begin{example}\label{ex}
Let us consider $d = m = 2$, the following quadratic Lagrangian and the following linear constraint function:
\begin{equation}
\fonction{L}{\mathbb{R}^2 \times \mathbb{R}^2 \times [a,b]}{\mathbb{R}}{(x,v,t)}{(\Vert x \Vert^2 + \Vert v \Vert^2 )/2} \quad \text{and} \quad \fonction{f}{\mathbb{R}^2 \times \mathbb{R}^2 \times [a,b]}{\mathbb{R}^2}{(x,v,t)}{x+v.}
\end{equation}
Then, we consider the three one parameter groups of diffeomorphisms given by the following rotations:
\begin{equation}
\fonction{\phi_i}{\mathbb{R} \times \mathbb{R}^2}{\mathbb{R}^2}{(s,x_1,x_2)}{\left( \begin{array}{cc} \cos (s \theta_i) & - \sin (s \theta_i) \\ \sin (s \theta_i) & \cos (s \theta_i) \end{array} \right) \left( \begin{array}{c} x_1 \\ x_2 \end{array} \right),}
\end{equation}
for $i=1,2,3$ and where $\theta_1$, $\theta_2 \in \mathbb{R}$ and $\theta_3 = - \theta_1$. With these parameters, one can prove that the Hamiltonian $H$ associated to $L$ and $f$ is ${}_{{\rm c}} \Delta^\alpha_-$-invariant under the action of $(\Phi_i)_{i=1,2,3}$. Consequently, the fractional Pontryagin's system \eqref{eqpontsystfd} admits a symmetry and then admits an explicit discrete conservation law given by the discrete fractional Noether's Theorem \ref{thmdfnoether}. \\
We choose $A=(1,2)$, $N=100$ and $\theta_1 = \theta_2 = - \theta_3 = 1$. Let us compute \eqref{eqpontsystfd} for $\alpha=1$, $3/4$, $1/2$, $1/4$. Then, we denote $\boldsymbol{Q} = (\boldsymbol{Q}^1,\boldsymbol{Q}^2)$ and $\boldsymbol{P} = (\boldsymbol{P}^1,\boldsymbol{P}^2)$ the discrete solutions obtained and we denote $\boldsymbol{G} = \partial \phi_1 / \partial s (0,\boldsymbol{Q}) = (-\boldsymbol{Q}^2,\boldsymbol{Q}^1)$. We are then interested in the value of:
\begin{equation}\label{eq4-4}
\sum_{r=1}^N A_r \times \left( \boldsymbol{G} \cdot \sigma^{r-1} (\boldsymbol{P}) \right).
\end{equation}
Let us see the graphics obtained by the computation of \eqref{eqpontsystfd} and by the computation of the quantity given in Equation \eqref{eq4-4} for $\alpha=1$, $3/4$, $1/2$, $1/4$:
\begin{equation*}
\begin{array}{cc}
\includegraphics[width=0.5\textwidth]{fig1} & \includegraphics[width=0.5\textwidth]{fig5}
\end{array}
\end{equation*}
\begin{equation*}
\begin{array}{cc}
\includegraphics[width=0.5\textwidth]{fig2} & \includegraphics[width=0.5\textwidth]{fig6}
\end{array}
\end{equation*}
\begin{equation*}
\begin{array}{cc}
\includegraphics[width=0.5\textwidth]{fig3} & \includegraphics[width=0.5\textwidth]{fig7}
\end{array}
\end{equation*}
\begin{equation*}
\begin{array}{cc}
\includegraphics[width=0.5\textwidth]{fig4} & \includegraphics[width=0.5\textwidth]{fig8}
\end{array}
\end{equation*}
As expected from Theorem \ref{thmdfnoether}, we obtain discrete constants of motion for this discrete fractional Pontryagin's system \eqref{eqpontsystfd} admitting a discrete symmetry and for any $\alpha=1$, $3/4$, $1/2$, $1/4$. In this specific example, the constant obtained is zero.
\end{example}
|
{
"timestamp": "2012-03-09T02:01:29",
"yymm": "1203",
"arxiv_id": "1203.1707",
"language": "en",
"url": "https://arxiv.org/abs/1203.1707"
}
|
\section{Introduction}
The notion of the dimension of a triangulated category has been introduced implicitly by Bondal and Van den Bergh \cite{BV} and explicitly by Rouquier \cite{R}.
It is defined as the number of extensions necessary to build the category from a single object, up to finite direct sum, direct summand and shift.
Rouquier proved that the bounded derived category of coherent sheaves on a separated scheme of finite type over a perfect field has finite dimension.
Finiteness of the dimension of the bounded derived category of finitely generated modules over a complete local ring with perfect coefficient field was recently proved by Aihara and Takahashi \cite{ddc}.
The concept of a thick subcategory of a triangulated category has been introduced by Verdier \cite{V}, by the name of \'{e}paisse subcategory, to develop the theory of localizations of triangulated categories.
Thick subcategories have been studied widely and deeply so far, mainly from the motivation to classify them; see \cite{BCR,BIK,DHS,FP,Ho,HS,N,stcm,To} for instance.
Since a thick subcategory is a triangulated category, its dimension in the sense of Rouquier can be defined.
It turned out by Oppermann and \v{S}\'{t}ov\'{i}\v{c}ek \cite{OS} that over a noetherian algebra (respectively, a projective scheme) all proper thick subcategories of the bounded derived category of finitely generated modules (respectively, coherent sheaves) containing perfect complexes have infinite dimension.
The concept of a resolving subcategory of an abelian category has been introduced by Auslander and Bridger \cite{AB}.
They proved that in the category of finitely generated modules over a noetherian ring the full subcategory consisting of modules of Gorenstein dimension zero is resolving.
A landmark development concerning resolving subcategories was made by Auslander and Reiten \cite{AR} in connection with tilting theory.
Recently, several studies on resolving subcategories have been done by Dao and Takahashi \cite{radius,resreg,res,stcm,arg,crs}.
In this paper, we introduce an analogue of the notion of the dimension of a triangulated category for full subcategories $\mathcal{X}$ of an abelian category with enough projective objects.
To be precise, we define the dimension of $\mathcal{X}$ as the number of extensions necessary to build $\mathcal{X}$ from a single object in $\mathcal{X}$, up to finite direct sum, direct summand and syzygy.
To state our results, let us fix some notation.
Let $R$ be a Cohen-Macaulay local ring.
Denote by $\operatorname{\mathsf{CM}}(R)$ the category of maximal Cohen-Macaulay $R$-modules, and by $\operatorname{\mathsf{CM}}_0(R)$ the category of maximal Cohen-Macaulay $R$-modules that are locally free on the punctured spectrum.
These two categories are resolving subcategories of the category $\operatorname{\mathsf{mod}} R$ of finitely generated $R$-modules.
The stable categories of $\operatorname{\mathsf{CM}}(R)$ and $\operatorname{\mathsf{CM}}_0(R)$ are denoted by $\operatorname{\underline{\mathsf{CM}}}(R)$ and $\operatorname{\underline{\mathsf{CM}}}_0(R)$, respectively.
When $R$ is Gorenstein, $\operatorname{\underline{\mathsf{CM}}}(R)$ is a triangulated category \cite{B,H}, and $\operatorname{\underline{\mathsf{CM}}}_0(R)$ is a thick subcategory of $\operatorname{\underline{\mathsf{CM}}}(R)$.
The main purpose of this paper is to investigate finiteness of the dimensions of resolving subcategories of $\operatorname{\mathsf{mod}} R$, and the dimensions of thick subcategories of $\operatorname{\underline{\mathsf{CM}}}(R)$ in the case where $R$ is Gorenstein.
Our first main result is a characterization of the isolated singularity of $R$ in terms of the dimensions of $\operatorname{\mathsf{CM}}_0(R)$ and $\operatorname{\underline{\mathsf{CM}}}_0(R)$:
\begin{thm}\label{1.1}
Let $R$ be a Cohen-Macaulay local ring with maximal ideal $\mathfrak{m}$.
\begin{enumerate}[\rm(1)]
\item
Consider the following four conditions.
\begin{enumerate}[\rm(a)]
\item
The dimension of $\operatorname{\mathsf{CM}}_0(R)$ is finite.
\item
The ideal $\bigcap_{i>0,\,M,N\in\operatorname{\mathsf{CM}}_0(R)}\operatorname{\mathsf{Ann}}_R\operatorname{\mathsf{Ext}}_R^i(M,N)$ is $\mathfrak{m}$-primary.
\item
The ideal $\bigcap_{i>0,\,M,N\in\operatorname{\mathsf{CM}}_0(R)}\operatorname{\mathsf{Ann}}_R\operatorname{\mathsf{Tor}}_i^R(M,N)$ is $\mathfrak{m}$-primary.
\item
The ring $R$ has at most an isolated singularity.
\end{enumerate}
Then, the implications ${\rm(a)}\Leftrightarrow{\rm(b)}\Rightarrow{\rm(c)}\Rightarrow{\rm(d)}$ hold.
The implication ${\rm(d)}\Rightarrow{\rm(a)}$ also holds if $R$ is complete, equicharacteristic and with perfect residue field.
\item
Suppose that $R$ is Gorenstein, and consider the following three conditions.
\begin{enumerate}[\rm(a)]
\item
The dimension of the triangulated category $\operatorname{\underline{\mathsf{CM}}}_0(R)$ is finite.
\item
The annihilator of the $R$-linear category $\operatorname{\underline{\mathsf{CM}}}_0(R)$ is $\mathfrak{m}$-primary.
\item
The ring $R$ has at most an isolated singularity.
\end{enumerate}
Then the implications ${\rm(a)}\Leftrightarrow{\rm(b)}\Rightarrow{\rm(c)}$ hold, and so does ${\rm(c)}\Rightarrow{\rm(a)}$ if $R$ is complete, equicharacteristic and with perfect residue field.
\end{enumerate}
\end{thm}
The celebrated Auslander-Huneke-Leuschke-Wiegand theorem states that every Cohen-Macaulay local ring of finite Cohen-Macaulay representation type has at most an isolated singularity.
This was proved by Auslander \cite{A} in the case where the ring is complete, by Leuschke and Wiegand \cite{LW} in the case where the ring is excellent, and by Huneke and Leuschke \cite{HL} in the general case.
Our Theorem \ref{1.1} not only deduces this result but also improves it:
\begin{cor}[Improved Auslander-Huneke-Leuschke-Wiegand Theorem]
Let $R$ be a Cohen-Macaulay local ring.
Suppose that there are only finitely many isomorphism classes of indecomposable maximal Cohen-Macaulay $R$-modules which are locally free on the punctured spectrum.
Then $R$ has at most an isolated singularity.
\end{cor}
This result very easily follows from the first assertion of Theorem \ref{1.1}.
Indeed, the assumption of the corollary implies that the dimension of $\operatorname{\mathsf{CM}}_0(R)$ is zero, hence is finite.
Our Theorem \ref{1.1} also gives rise to finiteness of the dimensions of $\operatorname{\mathsf{CM}}(R)$ and $\operatorname{\underline{\mathsf{CM}}}(R)$ when $R$ has an isolated singularity, the latter of which is a special case of the main result of Aihara and Takahashi \cite{ddc}.
\begin{cor}
Let $R$ be a Cohen-Macaulay excellent local ring with perfect coefficient field.
Suppose that $R$ has at most an isolated singularity.
Then $\operatorname{\mathsf{CM}}(R)$ is of finite dimension.
If $R$ is Gorenstein, then $\operatorname{\underline{\mathsf{CM}}}(R)$ is of finite dimension as a triangulated category.
\end{cor}
Our second main result in this paper concerns finiteness of more general resolving subcategories of $\operatorname{\mathsf{mod}} R$ and thick subcategories of $\operatorname{\underline{\mathsf{CM}}}(R)$.
Denote the $n$-th syzygy of an $R$-module $M$ by $\mathsf{\Omega}^nM$.
\begin{thm}\label{1.2}
Let $R$ be a $d$-dimensional Cohen-Macaulay local ring with residue field $k$.
\begin{enumerate}[\rm(1)]
\item
Let $\mathcal{X}$ be a resolving subcategory of $\operatorname{\mathsf{mod}} R$ containing $\mathsf{\Omega}^dk$ and strictly contained in $\operatorname{\mathsf{CM}}(R)$.
If one of the following three statements holds, then $\mathcal{X}$ has infinite dimension.
\begin{itemize}
\item
$R$ is locally a hypersurface on the punctured spectrum.
\item
$R$ is locally with minimal multiplicity on the punctured spectrum.
\item
$R$ is excellent and locally of finite Cohen-Macaulay representation type on the punctured spectrum, and $\mathcal{X}$ contains a dualizing module.
\end{itemize}
\item
Let $R$ be Gorenstein and locally a hypersurface on the punctured spectrum.
Then every proper thick subcategory of $\operatorname{\underline{\mathsf{CM}}}(R)$ containing $\mathsf{\Omega}^dk$ has infinite dimension.
\item
Let $R$ be a hypersurface.
Then every resolving subcategory of $\operatorname{\mathsf{mod}} R$ containing a nonfree module and strictly contained in $\operatorname{\mathsf{CM}}(R)$ and every nontrivial thick subcategory of $\operatorname{\underline{\mathsf{CM}}}(R)$ have infinite dimension.
\end{enumerate}
\end{thm}
The third assertion of Theorem \ref{1.2} improves for hypersurfaces the main result of Oppermann and \v{S}\'{t}ov\'{i}\v{c}ek \cite{OS}.
Let $\operatorname{\mathsf{D}^b}(\operatorname{\mathsf{mod}} R)$ denote the bounded derived category of $\operatorname{\mathsf{mod}} R$ and $\operatorname{\mathsf{perf}} R$ the full subcategory of perfect complexes.
\begin{cor}
Let $R$ be a local hypersurface.
Let $\mathcal{X}$ be a thick subcategory of $\operatorname{\mathsf{D}^b}(\operatorname{\mathsf{mod}} R)$ with $\operatorname{\mathsf{perf}} R\subsetneq\mathcal{X}\subsetneq\operatorname{\mathsf{D}^b}(\operatorname{\mathsf{mod}} R)$.
Then the Verdier quotient $\mathcal{X}/\operatorname{\mathsf{perf}} R$ has infinite dimension as a triangulated category.
In particular, the dimension of $\mathcal{X}$ is infinite.
\end{cor}
The organization of this paper is as follows.
In Section 2, together with our convention we will recall several basic definitions and fundamental facts for later use.
Section 3 will introduce the notions of the dimensions of subcategories of an abelian category and a triangulated category, and compare them with each other and with the concept of the radius of a subcategory which has been introduced in \cite{radius}. We also compute the dimension of the category of Cohen-Macaulay modules in some small cases, such as rational surface singularities, see Proposition \ref{36}.
In Section 4, we will study the annihilators and supports of $\operatorname{\mathsf{Tor}}$, $\operatorname{\mathsf{Ext}}$ and $\operatorname{\underline{\mathsf{Hom}}}$ as functors on the direct product of given two subcategories $\mathcal{X},\mathcal{Y}$ of $\operatorname{\mathsf{mod}} R$ and $\operatorname{\underline{\mathsf{CM}}}(R)$.
The results stated in this section will become the basis to obtain the main results of this paper.
Section 5 will mainly explore the nonfree loci of subcategories of $\operatorname{\mathsf{CM}}(R)$ and the stable supports of subcategories of $\operatorname{\underline{\mathsf{CM}}}(R)$, using the results obtained in Section 4.
In Section 6, we will consider finiteness of the dimensions of the resolving subcategory $\operatorname{\mathsf{CM}}_0(R)$ of $\operatorname{\mathsf{mod}} R$ and the thick subcategory $\operatorname{\underline{\mathsf{CM}}}_0(R)$ of $\operatorname{\underline{\mathsf{CM}}}(R)$, and give a proof of Theorem \ref{1.1}.
The aim of Section 7 will be to investigate finiteness of the dimensions of more general resolving subcategories of $\operatorname{\mathsf{mod}} R$ and thick subcategories of $\operatorname{\underline{\mathsf{CM}}}(R)$.
We will prove Theorem \ref{1.2} in this section.
\section{Preliminaries}
In this section, we recall basic definitions and fundamental facts for later use.
\begin{conv}
Throughout this paper, unless otherwise specified, we assume:\\
(1) All rings are commutative noetherian rings, and all modules are finitely generated.
All subcategories are nonempty, full and strict (i.e., closed under isomorphism).
Hence, the {\em subcategory} of a category $\mathcal{C}$ consisting of objects $\{M_\lambda\}_{\lambda\in\Lambda}$ always means the smallest strict full subcategory of $\mathcal{C}$ to which $M_\lambda$ belongs for all $\lambda\in\Lambda$.
This coincides with the full subcategory of $\mathcal{C}$ consisting of objects $X\in\mathcal{C}$ such that $X\cong M_\lambda$ for some $\lambda\in\Lambda$.\\
(2) Let $R$ be a ring.
The {\em singular locus} $\operatorname{\mathsf{Sing}} R$ of $R$ is the set of prime ideals $\mathfrak{p}$ of $R$ such that the local ring $R_\mathfrak{p}$ is not regular.
By $\operatorname{\mathsf{Spec}}_0(R)$ we denote the set of nonmaximal prime ideals of $R$.
This is nothing but the {\em punctured spectrum} of $R$ if $R$ is local.\\
(3) The category of $R$-modules is denoted by $\operatorname{\mathsf{mod}} R$, and the subcategory of modules of finite length is denoted by $\operatorname{\mathsf{fl}} R$.
An $R$-module $M$ is called {\em (maximal) Cohen-Macaulay} if $\operatorname{\mathsf{depth}} M_\mathfrak{p}\ge\operatorname{\mathsf{dim}} R_\mathfrak{p}$ for all $\mathfrak{p}\in\operatorname{\mathsf{Spec}} R$.
(Hence the zero module is Cohen-Macaulay.)
The subcategory of $\operatorname{\mathsf{mod}} R$ consisting of Cohen-Macaulay modules is denoted by $\operatorname{\mathsf{CM}}(R)$.\\
(4) For a subcategory $\mathcal{X}$ of an additive category $\mathcal{C}$, we denote by $\operatorname{\mathsf{add}}\mathcal{X}$ (or $\operatorname{\mathsf{add}}_\mathcal{C}\mathcal{X}$, or $\operatorname{\mathsf{add}}_R\mathcal{X}$ when $\mathcal{C}=\operatorname{\mathsf{mod}} R$) the {\em additive closure} of $\mathcal{X}$, namely, the subcategory of $\mathcal{C}$ consisting of direct summands of finite direct sums of objects in $\mathcal{X}$.
When $\mathcal{X}$ consists of a single object $M$, we simply denote it by $\operatorname{\mathsf{add}} M$ (or $\operatorname{\mathsf{add}}_{\mathcal{C}}M$, $\operatorname{\mathsf{add}}_RM$).
For an abelian category $\mathcal{A}$ with enough projective objects, we denote by $\operatorname{\mathsf{proj}}\mathcal{A}$ the subcategory of projective objects.
For $n\ge1$ the $n$-th syzygy of an object $M\in\mathcal{A}$ is denoted by $\mathsf{\Omega}^nM$ (or $\mathsf{\Omega}_\mathcal{A}^nM$, or $\mathsf{\Omega}_R^nM$ when $\mathcal{A}=\operatorname{\mathsf{mod}} R$).
Whenever $R$ is local and $\mathcal{A}=\operatorname{\mathsf{mod}} R$, we use a {\em minimal free resolution} of $M$ to define $\mathsf{\Omega}^nM$, so that it is uniquely determined up to isomorphism.
\end{conv}
\begin{dfn}
Let $\mathcal{A}$ be an abelian category with enough projective objects.
A subcategory $\mathcal{X}$ of $\mathcal{A}$ is called {\it resolving} if $\mathcal{X}$ contains $\operatorname{\mathsf{proj}}\mathcal{A}$ and is closed under direct summands, extensions and kernels of epimorphisms.
The last two closure properties mean that for an exact sequence $0 \to L \to M \to N \to 0$ in $\mathcal{A}$ with $N\in\mathcal{X}$ one has $L\in\mathcal{X}\Leftrightarrow M\in\mathcal{X}$.
\end{dfn}
The notion of a resolving subcategory has been introduced by Auslander and Bridger \cite{AB}.
It is a subcategory such that any two minimal resolutions of a module by modules in it have the same length (cf. \cite[Lemma (3.12)]{AB}).
Every resolving subcategory is closed under finite direct sums.
One can replace closure under kernels of epimorphisms with closure under syzygies (cf. \cite[Lemma 3.2]{Y}).
Clearly, $\operatorname{\mathsf{proj}}\mathcal{A}$ and $\mathcal{A}$ are the smallest and largest resolving subcategories of $\mathcal{A}$, respectively.
A lot of resolving subcategories are known.
For example, $\operatorname{\mathsf{CM}}(R)$ is a resolving subcategory of $\operatorname{\mathsf{mod}} R$ if $R$ is Cohen-Macaulay.
The subcategory of $\operatorname{\mathsf{mod}} R$ consisting of totally reflexive $R$-modules is resolving by \cite[(3.11)]{AB}.
One can construct a resolving subcategory easily by using vanishing of Tor or Ext.
Also, the modules of complexity less than a fixed integer form a resolving subcategory of $\operatorname{\mathsf{mod}} R$.
For the details, we refer to \cite[Example 2.4]{res}.
\begin{dfn}
(1) For $R$-modules $M,N$ we set $\operatorname{\underline{\mathsf{Hom}}}_R(M,N)=\operatorname{\mathsf{Hom}}_R(M,N)/\mathcal{P}_R(M,N)$, where $\mathcal{P}_R(M,N)$ is the set of $R$-homomorphisms $M\to N$ factoring through projective modules, which is an $R$-submodule of $\operatorname{\mathsf{Hom}}_R(M,N)$.\\
(2) The {\em stable category} of $\operatorname{\mathsf{CM}}(R)$, which is denoted by $\operatorname{\underline{\mathsf{CM}}}(R)$, is defined by $\operatorname{\mathsf{Ob}}(\operatorname{\underline{\mathsf{CM}}}(R))=\operatorname{\mathsf{Ob}}(\operatorname{\mathsf{CM}}(R))$ and $\operatorname{\mathsf{Hom}}_{\operatorname{\underline{\mathsf{CM}}}(R)}(M,N)=\operatorname{\underline{\mathsf{Hom}}}_R(M,N)$ for $M,N\in\operatorname{\mathsf{Ob}}(\operatorname{\underline{\mathsf{CM}}}(R))$.
\end{dfn}
Let $R$ be a Gorenstein ring with $\operatorname{\mathsf{dim}} R<\infty$.
Then $R$ is an {\em Iwanaga-Gorenstein} ring.
Taking the syzygy makes an autoequivalence $\mathsf{\Omega}:\operatorname{\underline{\mathsf{CM}}}(R)\to\operatorname{\underline{\mathsf{CM}}}(R)$ of categories, whose quasi-inverse is given by taking the cosyzygy, and $\operatorname{\underline{\mathsf{CM}}}(R)$ is a triangulated category with shift functor $\mathsf{\Sigma}=\mathsf{\Omega}^{-1}$.
For the details, see \cite[Theorem 4.4.1]{B} or \cite[\S2 in Chapter I]{H}.
We can also find in \cite[Remark 1.19]{stcm} how to define an exact triangle.
\begin{dfn}
A {\em thick} subcategory of a triangulated category is defined to be a triangulated subcategory closed under direct summands.
\end{dfn}
The notion of a thick subcategory has been introduced by Verdier \cite{V} by the name of \'{e}paisse subcategory to develop the theory of localizations of triangulated categories.
Every thick subcategory of a triangulated category $\mathcal{T}$ contains the zero object of $\mathcal{T}$, and is closed under shifts, namely, if $M$ is an object in $\mathcal{X}$, then so are $\mathsf{\Sigma} M$ and $\mathsf{\Sigma}^{-1}M$.
Clearly, $\{0\}$ and $\mathcal{T}$ are the smallest and largest thick subcategories of $\mathcal{T}$, respectively.
When $R$ is local, the bounded complexes of $R$-modules having finite complexity form a thick subcategory of the bounded derived category $\operatorname{\mathsf{D}^b}(\operatorname{\mathsf{mod}} R)$ of $\operatorname{\mathsf{mod}} R$.
When $R$ is Gorenstein with $\operatorname{\mathsf{dim}} R<\infty$, for a fixed $R$-module $C$, the Cohen-Macaulay $R$-modules $M$ with $\operatorname{\mathsf{Tor}}_i^R(M,C)=0$ for $i\gg0$ form a thick subcategory of $\operatorname{\underline{\mathsf{CM}}}(R)$.
\begin{dfn}\label{defbar}
(1) For a subcategory $\mathcal{X}$ of $\operatorname{\mathsf{CM}}(R)$, we define the category $\underline\mathcal{X}$ by $\operatorname{\mathsf{Ob}}(\underline\mathcal{X})=\operatorname{\mathsf{Ob}}(\mathcal{X})$ and $\operatorname{\mathsf{Hom}}_{\underline\mathcal{X}}(M,N)=\operatorname{\underline{\mathsf{Hom}}}_R(M,N)$ for $M,N\in\operatorname{\mathsf{Ob}}(\underline\mathcal{X})$.\\
(2) For a subcategory $\mathcal{X}$ of $\operatorname{\underline{\mathsf{CM}}}(R)$, we define the category $\overline\mathcal{X}$ by $\operatorname{\mathsf{Ob}}(\overline\mathcal{X})=\operatorname{\mathsf{Ob}}(\mathcal{X})$ and $\operatorname{\mathsf{Hom}}_{\overline\mathcal{X}}(M,N)=\operatorname{\mathsf{Hom}}_R(M,N)$ for $M,N\in\operatorname{\mathsf{Ob}}(\overline\mathcal{X})$.
\end{dfn}
Let $R$ be a Gorenstein ring of finite Krull dimension.
If $\mathcal{X}$ is a thick subcategory of $\operatorname{\underline{\mathsf{CM}}}(R)$, then $\overline\mathcal{X}$ is a resolving subcategory of $\operatorname{\mathsf{mod}} R$ contained in $\operatorname{\mathsf{CM}}(R)$.
Conversely, if $\mathcal{X}$ is a resolving subcategory of $\operatorname{\mathsf{mod}} R$ contained in $\operatorname{\mathsf{CM}}(R)$, then $\underline\mathcal{X}$ is a thick subcategory of $\operatorname{\underline{\mathsf{CM}}}(R)$ provided that $R$ is a local complete intersection; see \cite[Corollary 4.16]{radius}.
\begin{dfn}
(1) The {\em nonfree locus} $\operatorname{\mathsf{NF}}(M)$ of an $R$-module $M$ is the set of prime ideals $\mathfrak{p}$ of $R$ such that the $R_\mathfrak{p}$-module $M_\mathfrak{p}$ is nonfree.
The {\em nonfree locus} of a subcategory $\mathcal{X}$ of $\operatorname{\mathsf{mod}} R$ is defined by $\operatorname{\mathsf{NF}}(\mathcal{X})=\bigcup_{M\in\mathcal{X}}\operatorname{\mathsf{NF}}(M)$.
For a subset $W$ of $\operatorname{\mathsf{Spec}} R$ we set $\operatorname{\mathsf{NF}}_{\operatorname{\mathsf{CM}}}^{-1}(W)=\{\,M\in\operatorname{\mathsf{CM}}(R)\mid\operatorname{\mathsf{NF}}(M)\subseteq W\,\}$.\\
(2) The {\em stable support} $\operatorname{\underline{\mathsf{Supp}}} M$ of a Cohen-Macaulay $R$-module $M$ is the set of prime ideals $\mathfrak{p}$ of $R$ such that $M_\mathfrak{p}\cong 0$ in $\operatorname{\underline{\mathsf{CM}}}(R_\mathfrak{p})$.
The {\em stable support} of a subcategory $\mathcal{X}$ of $\operatorname{\underline{\mathsf{CM}}}(R)$ is defined by $\operatorname{\underline{\mathsf{Supp}}}\mathcal{X}=\bigcup_{M\in\mathcal{X}}\operatorname{\underline{\mathsf{Supp}}} M$.
For a subset $W$ of $\operatorname{\mathsf{Spec}} R$ we set $\operatorname{\underline{\mathsf{Supp}}}^{-1}W=\{\,M\in\operatorname{\underline{\mathsf{CM}}}(R)\mid\operatorname{\underline{\mathsf{Supp}}} M\subseteq W\,\}$.
\end{dfn}
Recall that a subset $W$ of $\operatorname{\mathsf{Spec}} R$ is called {\em specialization-closed} if $W$ contains $\operatorname{\mathsf{V}}(\mathfrak{p})$ for every $\mathfrak{p}\in W$.
It is equivalent to saying that $W$ is a union of closed subsets of $\operatorname{\mathsf{Spec}} R$.
\begin{rem}
The following hold for $M\in\operatorname{\mathsf{mod}} R$, $N\in\operatorname{\mathsf{CM}}(R)$, $\mathcal{X}\subseteq\operatorname{\mathsf{mod}} R$, $\mathcal{Y}\subseteq\operatorname{\mathsf{CM}}(R)$, $\mathcal{Z}\subseteq\operatorname{\underline{\mathsf{CM}}}(R)$ and $W\subseteq\operatorname{\mathsf{Spec}} R$ (cf. \cite[Propositions 1.14, 1.15, 6.2 and 6.4]{stcm}).\\
(1) $\operatorname{\mathsf{NF}}(M)$ is empty if and only if $M$ is projective.
$\operatorname{\mathsf{NF}}(M)$ contains only maximal ideals if and only if $M$ is locally free on $\operatorname{\mathsf{Spec}}_0(R)$.\\
(2) $\operatorname{\mathsf{NF}}(M)$ is a closed subset of $\operatorname{\mathsf{Spec}} R$ in the Zariski topology.
$\operatorname{\mathsf{NF}}(\mathcal{X})$ is a specialization-closed subset of $\operatorname{\mathsf{Spec}} R$.\\
(3) One has $\operatorname{\mathsf{NF}}(\mathcal{Y})\subseteq\operatorname{\mathsf{Sing}} R$, $\operatorname{\mathsf{NF}}(\operatorname{\mathsf{NF}}_{\operatorname{\mathsf{CM}}}^{-1}(W))\subseteq W$ and $\operatorname{\mathsf{NF}}_{\operatorname{\mathsf{CM}}}^{-1}(W)=\operatorname{\mathsf{NF}}_{\operatorname{\mathsf{CM}}}^{-1}(W\cap\operatorname{\mathsf{Sing}} R)$.\\
(4) One has $\operatorname{\underline{\mathsf{Supp}}} N=\operatorname{\mathsf{NF}}(N)$, $\operatorname{\underline{\mathsf{Supp}}}\underline\mathcal{Y}=\operatorname{\mathsf{NF}}(\mathcal{Y})$, $\operatorname{\underline{\mathsf{Supp}}}\mathcal{Z}=\operatorname{\mathsf{NF}}(\overline\mathcal{Z})$ and $\operatorname{\underline{\mathsf{Supp}}}^{-1}W=\underline{\operatorname{\mathsf{NF}}_{\operatorname{\mathsf{CM}}}^{-1}(W)}$.\\
(5) If $R$ is Cohen-Macaulay, then $\operatorname{\mathsf{NF}}(\operatorname{\mathsf{CM}}(R))=\operatorname{\mathsf{Sing}} R$, and $\operatorname{\mathsf{NF}}_{\operatorname{\mathsf{CM}}}^{-1}(W)$ is a resolving subcategory of $\operatorname{\mathsf{mod}} R$ contained in $\operatorname{\mathsf{CM}}(R)$.\\
(6) If $R$ is Gorenstein with $\operatorname{\mathsf{dim}} R<\infty$, then $\operatorname{\underline{\mathsf{Supp}}}^{-1}W$ is a thick subcategory of $\operatorname{\underline{\mathsf{CM}}}(R)$.
\end{rem}
\begin{dfn}
For an integer $n\ge-1$ we set $\operatorname{\mathsf{CM}}_n(R)=\{\,M\in\operatorname{\mathsf{CM}}(R)\mid\operatorname{\mathsf{dim}}\operatorname{\mathsf{NF}}(M)\le n\,\}$ and $\operatorname{\underline{\mathsf{CM}}}_n(R)=\underline{\operatorname{\mathsf{CM}}_n(R)}=\{\,M\in\operatorname{\underline{\mathsf{CM}}}(R)\mid\operatorname{\mathsf{dim}}\operatorname{\underline{\mathsf{Supp}}} M\le n\,\}$.
\end{dfn}
\begin{rem}
Let $R$ be a $d$-dimensional Cohen-Macaulay local ring with residue field $k$.\\
(1) One has $\operatorname{\mathsf{add}} R=\operatorname{\mathsf{CM}}_{-1}(R)\subseteq\operatorname{\mathsf{CM}}_0(R)\subseteq\operatorname{\mathsf{CM}}_1(R)\subseteq\cdots\subseteq\operatorname{\mathsf{CM}}_d(R)=\operatorname{\mathsf{CM}}(R)$ and $\{0\} =\operatorname{\underline{\mathsf{CM}}}_{-1}(R)\subseteq\operatorname{\underline{\mathsf{CM}}}_0(R)\subseteq\operatorname{\underline{\mathsf{CM}}}_1(R)\subseteq\cdots\subseteq\operatorname{\underline{\mathsf{CM}}}_d(R)=\operatorname{\underline{\mathsf{CM}}}(R)$.\\
(2) One has $\operatorname{\mathsf{CM}}_n(R)=\operatorname{\mathsf{NF}}_{\operatorname{\mathsf{CM}}}^{-1}(\{\,\mathfrak{p}\in\operatorname{\mathsf{Sing}} R\mid\operatorname{\mathsf{dim}} R/\mathfrak{p}\le n\,\})$ and $\operatorname{\underline{\mathsf{CM}}}_n(R)=\operatorname{\underline{\mathsf{Supp}}}^{-1}(\{\,\mathfrak{p}\in\operatorname{\mathsf{Sing}} R\mid\operatorname{\mathsf{dim}} R/\mathfrak{p}\le n\,\})$ for $n\ge-1$.
Hence $\operatorname{\mathsf{CM}}_n(R)$ is a resolving subcategory of $\operatorname{\mathsf{mod}} R$ contained in $\operatorname{\mathsf{CM}}(R)$, and $\operatorname{\underline{\mathsf{CM}}}_n(R)$ is a thick subcategory of $\operatorname{\underline{\mathsf{CM}}}(R)$ if $R$ is Gorenstein.\\
(3) The category $\operatorname{\mathsf{CM}}_0(R)$ consists of the Cohen-Macaulay $R$-modules that are locally free on $\operatorname{\mathsf{Spec}}_0(R)$.
Hence $\operatorname{\mathsf{CM}}_0(R)$ is the smallest subcategory of $\operatorname{\mathsf{mod}} R$ containing $\mathsf{\Omega}^dk$ that is closed under direct summands and extensions; see \cite[Corollary 2.6]{stcm}.
In particular, a resolving subcategory of $\operatorname{\mathsf{mod}} R$ contains $\operatorname{\mathsf{CM}}_0(R)$ if and only if it contains $\mathsf{\Omega}^dk$.
\end{rem}
\section{Definitions of dimensions of subcategories}
This section contains the key definitions and establishes several results.
More precisely, the notions of the dimensions of subcategories of an abelian category and a triangulated category will be introduced.
We will compare them with each other and with the concept of the radius of a subcategory.
Their relationships with representation types will also be explored.
First of all, we recall the definition of a ball given in \cite{BV,radius,R}.
\begin{dfn}
(1) Let $\mathcal{T}$ be a triangulated category.
(a) For a subcategory $\mathcal{X}$ of $\mathcal{T}$ we denote by $\langle\mathcal{X}\rangle$ the smallest subcategory of $\mathcal{T}$ containing $\mathcal{X}$ that is closed under finite direct sums, direct summands and shifts, i.e., $\langle\mathcal{X}\rangle=\operatorname{\mathsf{add}}_\mathcal{T}\{\,\mathsf{\Sigma}^iX\mid i\in\mathbb{Z},\,X\in\mathcal{X}\,\}$.
When $\mathcal{X}$ consists of a single object $X$, we simply denote it by $\langle X\rangle$.
(b) For subcategories $\mathcal{X},\mathcal{Y}$ of $\mathcal{T}$ we denote by $\mathcal{X}\ast\mathcal{Y}$ the subcategory of $\mathcal{T}$ consisting of objects $M$ which fits into an exact triangle $X \to M \to Y \to \mathsf{\Sigma} X$ in $\mathcal{T}$ with $X\in\mathcal{X}$ and $Y\in\mathcal{Y}$.
We set $\mathcal{X}\diamond\mathcal{Y}=\langle\langle\mathcal{X}\rangle\ast\langle\mathcal{Y}\rangle\rangle$.
(c) Let $\mathcal{C}$ be a subcategory of $\mathcal{T}$.
We define the {\it ball of radius $r$ centered at $\mathcal{C}$} as
$$
\langle\mathcal{C}\rangle_r=
\begin{cases}
\langle\mathcal{C}\rangle & (r=1),\\
\langle\mathcal{C}\rangle_{r-1}\diamond\mathcal{C}=\langle\langle\mathcal{C}\rangle_{r-1}\ast\langle\mathcal{C}\rangle\rangle & (r\ge2).
\end{cases}
$$
If $\mathcal{C}$ consists of a single object $C$, then we simply denote it by $\langle C\rangle_r$, and call it the ball of radius $r$ centered at $C$.
We write $\langle\mathcal{C}\rangle_r^{\mathcal{T}}$ when we should specify that $\mathcal{T}$ is the ground category where the ball is defined.\\
(2) Let $\mathcal{A}$ be an abelian category with enough projective objects.
(a) For a subcategory $\mathcal{X}$ of $\mathcal{A}$ we denote by $[\mathcal{X}]$ the smallest subcategory of $\mathcal{A}$ containing $\operatorname{\mathsf{proj}}\mathcal{A}$ and $\mathcal{X}$ that is closed under finite direct sums, direct summands and syzygies, i.e., $[\mathcal{X}]=\operatorname{\mathsf{add}}_\mathcal{A}(\operatorname{\mathsf{proj}}\mathcal{A}\cup\{\,\mathsf{\Omega}^iX\mid i\ge0,\,X\in\mathcal{X}\,\})$.
When $\mathcal{X}$ consists of a single object $X$, we simply denote it by $[X]$.
(b) For subcategories $\mathcal{X},\mathcal{Y}$ of $\mathcal{A}$ we denote by $\mathcal{X}\circ\mathcal{Y}$ the subcategory of $\mathcal{A}$ consisting of objects $M$ which fits into an exact sequence $0 \to X \to M \to Y \to 0$ in $\mathcal{A}$ with $X\in\mathcal{X}$ and $Y\in\mathcal{Y}$.
We set $\mathcal{X}\bullet\mathcal{Y}=[[\mathcal{X}]\circ[\mathcal{Y}]]$.
(c) Let $\mathcal{C}$ be a subcategory of $\mathcal{A}$.
We define the {\it ball of radius $r$ centered at $\mathcal{C}$} as
$$
[\mathcal{C}]_r=
\begin{cases}
[\mathcal{C}] & (r=1),\\
[\mathcal{C}]_{r-1}\bullet\mathcal{C}=[[\mathcal{C}]_{r-1}\circ[\mathcal{C}]] & (r\ge2).
\end{cases}
$$
If $\mathcal{C}$ consists of a single object $C$, then we simply denote it by $[C]_r$, and call it the ball of radius $r$ centered at $C$.
We write $[\mathcal{C}]_r^{\mathcal{A}}$ when we should specify that $\mathcal{A}$ is the ground category where the ball is defined.
\end{dfn}
\begin{rem}
The following statements hold (cf. \cite{radius,R}).\\
(1) Let $\mathcal{T}$ be a triangulated category, and $\mathcal{X},\mathcal{Y},\mathcal{Z},\mathcal{C}$ subcategories.
(a) An object $M\in\mathcal{T}$ belongs to $\mathcal{X}\diamond\mathcal{Y}$ if and only if there is an exact triangle $X \to Z \to Y \to \mathsf{\Sigma} X$ with $X\in\langle\mathcal{X}\rangle$ and $Y\in\langle\mathcal{Y}\rangle$ such that $M$ is a direct summand of $Z$.
(b) One has $(\mathcal{X}\diamond\mathcal{Y})\diamond\mathcal{Z}=\mathcal{X}\diamond(\mathcal{Y}\diamond\mathcal{Z})$ and $\langle\mathcal{C}\rangle_a\diamond\langle\mathcal{C}\rangle_b=\langle\mathcal{C}\rangle_{a+b}$ for all $a,b>0$.\\
(2) Let $\mathcal{A}$ be an abelian category with enough projectives, and $\mathcal{X},\mathcal{Y},\mathcal{Z},\mathcal{C}$ subcategories.
(a) An object $M\in\mathcal{A}$ belongs to $\mathcal{X}\bullet\mathcal{Y}$ if and only if there is an exact sequence $0 \to X \to Z \to Y \to 0$ with $X\in[\mathcal{X}]$ and $Y\in[\mathcal{Y}]$ such that $M$ is a direct summand of $Z$.
(b) One has $(\mathcal{X}\bullet\mathcal{Y})\bullet\mathcal{Z}=\mathcal{X}\bullet(\mathcal{Y}\bullet\mathcal{Z})$ and $[\mathcal{C}]_a\bullet[\mathcal{C}]_b=[\mathcal{C}]_{a+b}$ for all $a,b>0$.
\end{rem}
Now, for a triangulated category and an abelian category with enough projective objects, we can make the definitions of the dimensions of subcategories.
\begin{dfn}
(1) Let $\mathcal{T}$ be a triangulated category.
Let $\mathcal{X}$ be a subcategory of $\mathcal{T}$.
We define the {\it dimension} of $\mathcal{X}$, denoted by $\operatorname{\mathsf{dim}}\mathcal{X}$ (or $\operatorname{\mathsf{dim}}_\mathcal{T}\mathcal{X}$), as the infimum of the integers $n\ge0$ such that $\mathcal{X}=\langle G\rangle_{n+1}^\mathcal{T}$ for some $G\in\mathcal{X}$.\\
(2) Let $\mathcal{A}$ be an abelian category with enough projective objects.
Let $\mathcal{X}$ be a subcategory of $\mathcal{A}$.
We define the {\it dimension} of $\mathcal{X}$, denoted by $\operatorname{\mathsf{dim}}\mathcal{X}$ (or $\operatorname{\mathsf{dim}}_\mathcal{A}\mathcal{X}$), as the infimum of the integers $n\ge0$ such that $\mathcal{X}=[G]_{n+1}^\mathcal{A}$ for some $G\in\mathcal{X}$.
\end{dfn}
The dimension of a subcategory $\mathcal{X}$ of $\mathcal{A}$ is by definition the infimum of $n\ge0$ with $\mathcal{X}\subseteq[G]_{n+1}$ for some $G\in\mathcal{X}$.
Then the only difference between the definitions of $\operatorname{\mathsf{dim}}\mathcal{X}$ and $\operatorname{\mathsf{radius}}\mathcal{X}$ is that we do now require the object $G$ to be in $\mathcal{X}$.
This is subtle but will turn out to be crucial.
For example, let $R=\mathbb{C}[[x,y]]/(x^2y)$.
Then the radius of $\operatorname{\mathsf{CM}}_0(R)$ is $1$ by \cite[Proposition 4.2]{BGS} and \cite[Propositions 2.10]{radius}, in particular, it is finite.
But the dimension of $\operatorname{\mathsf{CM}}_0(R)$ is infinite by Theorem \ref{main}, which will be proved later in this paper.
\begin{rem}
(1) One has $\operatorname{\mathsf{dim}}\mathcal{X}\in\mathbb{N}\cup\{\infty\}$ in both senses.\\
(2)If $\mathcal{X}$ is a triangulated subcategory of $\mathcal{T}$ (respectively, an abelian subcategory of $\mathcal{A}$ containing $\operatorname{\mathsf{proj}}\mathcal{A}$), then $\operatorname{\mathsf{dim}}_\mathcal{T}\mathcal{X}=\operatorname{\mathsf{dim}}_\mathcal{X}\X$ (respectively, $\operatorname{\mathsf{dim}}_\mathcal{A}\mathcal{X}=\operatorname{\mathsf{dim}}_\mathcal{X}\X$).\\
(3) The definition itself works for every subcategory $\mathcal{X}$ of $\mathcal{T}$ (respectively, $\mathcal{A}$).
But the equality $\mathcal{X}=\langle G\rangle_{n+1}^\mathcal{T}$ (respectively, $\mathcal{X}=[G]_{n+1}^\mathcal{A}$) forces $\mathcal{X}$ to be closed under finite direct sums, direct summands and shifts (respectively, to contain the projective objects and be closed under finite direct sums, direct summands and syzygies).
So, basically, a subcategory whose dimension is considered is supposed to be thick (respectively, resolving).\\
(4) The subcategory $\{0\}$ of $\mathcal{T}$ and the subcategory $\operatorname{\mathsf{proj}}\mathcal{A}$ of $\mathcal{A}$ have dimension $0$.
\end{rem}
The dimension of a subcategory of an abelian category is an analogue of the {\em radius} of a subcategory of $\operatorname{\mathsf{mod}} R$ introduced by the authors \cite{radius}.
(In fact it can be defined for a subcategory of an arbitrary abelian category with enough projective objects.)
These have several relationships as follows.
Recall that a Cohen-Macaulay local ring $R$ is said to have {\it finite} (respectively, {\it countable}) {\it Cohen-Macaulay representation type} if there are only finitely (respectively, countably but infinitely) many nonisomorphic indecomposable Cohen-Macaulay $R$-modules.
\begin{prop}\label{35}
\begin{enumerate}[\rm(1)]
\item
One has $\operatorname{\mathsf{radius}}\mathcal{X}\le\operatorname{\mathsf{dim}}\mathcal{X}$ for any subcategory $\mathcal{X}$ of $\operatorname{\mathsf{mod}} R$.
\item
Let $R$ be Gorenstein of finite dimension.
Let $\mathcal{X}$ be a thick subcategory of $\operatorname{\underline{\mathsf{CM}}}(R)$.
Then $\operatorname{\mathsf{dim}}\mathcal{X}\le\operatorname{\mathsf{dim}}\overline\mathcal{X}$ (see Definition \ref{defbar}) holds. We also have $\operatorname{\mathsf{radius}}\operatorname{\mathsf{CM}}(R)=\operatorname{\mathsf{dim}}\operatorname{\mathsf{CM}}(R)$.
\item
If $R$ is a local hypersurface, then one has $\operatorname{\mathsf{dim}}\operatorname{\underline{\mathsf{CM}}}(R)=\operatorname{\mathsf{radius}}\operatorname{\mathsf{CM}}(R)=\operatorname{\mathsf{dim}}\operatorname{\mathsf{CM}}(R)$.
\end{enumerate}
\end{prop}
\begin{proof}
(1) This assertion is by definition.
(2) We claim that for a Cohen-Macaulay $R$-module $G$ and an integer $n\ge1$ every object $M\in[G]_n^{\operatorname{\mathsf{mod}} R}$ belongs to ${\langle G\rangle}_n^{\operatorname{\underline{\mathsf{CM}}}(R)}$.
Indeed, this claim is an easy consequence by induction on $n$.
Now assume $\operatorname{\mathsf{dim}}\overline\mathcal{X}=m<\infty$.
Then there is a Cohen-Macaulay $R$-module $G$ satisfying $\overline\mathcal{X}=[G]_{m+1}^{\operatorname{\mathsf{mod}} R}$.
The claim implies that $\mathcal{X}$ is contained in ${\langle G\rangle}_{m+1}^{\operatorname{\underline{\mathsf{CM}}}(R)}$.
Since $\mathcal{X}$ is thick, it coincides with ${\langle G\rangle}_{m+1}^{\operatorname{\underline{\mathsf{CM}}}(R)}$.
Hence we have $\operatorname{\mathsf{dim}}\mathcal{X}\le m$.
For the second assertion, suppose $\operatorname{\mathsf{radius}}\operatorname{\mathsf{CM}}(R)=n$, so there exists $G \in \operatorname{\mathsf{mod}} R$ such that $\operatorname{\mathsf{CM}}(R) \subseteq [G]_{n+1}$. It follows that for $a$ big enough one has $\operatorname{\mathsf{CM}}(R) = [\mathsf{\Omega}^{-a}\mathsf{\Omega}^aG]_{n+1}$, thus $\operatorname{\mathsf{dim}} \operatorname{\mathsf{CM}}(R) \leq n$ and the equality follows from (1).
(3) The inequalities $\operatorname{\mathsf{dim}}\operatorname{\underline{\mathsf{CM}}}(R)\le\operatorname{\mathsf{radius}}\operatorname{\mathsf{CM}}(R)\le\operatorname{\mathsf{dim}}\operatorname{\mathsf{CM}}(R)$ are obtained by using \cite[Proposition 2.6(1)]{radius} and (1).
The proof of \cite[Proposition 2.6(2)]{radius} actually shows that $\operatorname{\mathsf{dim}}\operatorname{\mathsf{CM}}(R)\le\operatorname{\mathsf{dim}}\operatorname{\underline{\mathsf{CM}}}(R)$.
\end{proof}
Next we calculate some examples of categories with small dimensions.
\begin{prop}\label{36}Let $(R,\mathfrak{m},k)$ be a Cohen-Macaulay local ring.
\begin{enumerate}[\rm(1)]
\item
If $R$ has finite Cohen-Macaulay representation type then $\operatorname{\mathsf{dim}}\operatorname{\mathsf{CM}}(R)=0$. The converse is true of $R$ is hensenlian and Gorenstein.
\item Suppose $\operatorname{\mathsf{dim}} R=2$, $k$ is algebraically closed and $R$ is hensenlian, normal with rational singularity in the sense of \cite{Lip}. Then $\operatorname{\mathsf{dim}}\operatorname{\mathsf{CM}}(R)\leq 1$.
\item
Suppose $R$ is a complete local hypersurface with an algebraically closed coefficient field of characteristic not two.
If $R$ has countable Cohen-Macaulay representation type, then one has $\operatorname{\mathsf{dim}}\operatorname{\mathsf{CM}}(R)=1$.
\end{enumerate}
\end{prop}
\begin{proof}
(1) This follows from (1) of Proposition \ref{35} and \cite[Proposition 2.8]{radius}.
(2) By \cite[Theorem 3.6]{IW1} (which rests on \cite[Theorem 2.1]{Wu}, whose proof goes through in our slightly more general setting), there exists $X \in \operatorname{\mathsf{CM}}(R)$ such that $\mathsf{\Omega}\operatorname{\mathsf{CM}}(R) = \operatorname{\mathsf{add}} X$. Let $M\in \operatorname{\mathsf{CM}}(R)$ be any maximal Cohen-Macaulay module and let $M^{\vee}$ denote $\operatorname{\mathsf{Hom}}_R(M,\omega_R)$. We have an exact sequence:
$$0 \to \mathsf{\Omega} M^{\vee} \to R^n \to M^{\vee} \to 0 $$
Applying $\operatorname{\mathsf{Hom}}_R(-,\omega_R)$ we get
$$0 \to M \to \omega_R^n \to (\mathsf{\Omega} M^{\vee})^{\vee} \to 0 $$
It follows that $\operatorname{\mathsf{CM}}(R) = [X^{\vee} \oplus \omega_R]_2$.
(3) This is shown by (3) of Proposition \ref{35} and \cite[Proposition 2.10]{radius}.
\end{proof}
\section{Annihilators and supports of $\operatorname{\mathsf{Tor}}$, $\operatorname{\mathsf{Ext}}$ and $\operatorname{\underline{\mathsf{Hom}}}$}
In this section, we investigate the annihilators and supports of $\operatorname{\mathsf{Tor}}$, $\operatorname{\mathsf{Ext}}$ and $\operatorname{\underline{\mathsf{Hom}}}$ as functors on the direct product of given two subcategories $\mathcal{X},\mathcal{Y}$ of $\operatorname{\mathsf{mod}} R$ and $\operatorname{\underline{\mathsf{CM}}}(R)$.
Our results stated in this section will be the basis to obtain the main results of this paper.
We start by fixing our notation.
\begin{nota}
\begin{enumerate}[(1)]
\item
For subcategories $\mathcal{X},\mathcal{Y}$ of $\operatorname{\mathsf{mod}} R$, we define:
$$
\operatorname{\mathsf{Tor}}(\mathcal{X},\mathcal{Y})= \textstyle\bigoplus_{i>0,\,X\in\mathcal{X},\,Y\in\mathcal{Y}}\operatorname{\mathsf{Tor}}_i^R(X,Y),\quad
\operatorname{\mathsf{Ext}}(\mathcal{X},\mathcal{Y})= \textstyle\bigoplus_{i>0,\,X\in\mathcal{X},\,Y\in\mathcal{Y}}\operatorname{\mathsf{Ext}}^i_R(X,Y).
$$
If $\mathcal{X}$ (respectively, $\mathcal{Y}$) consists of a single module $M$, we simply write $\operatorname{\mathsf{Tor}}(M,\mathcal{Y})$ and $\operatorname{\mathsf{Ext}}(M,\mathcal{Y})$ (respectively, $\operatorname{\mathsf{Tor}}(\mathcal{X},M)$ and $\operatorname{\mathsf{Ext}}(\mathcal{X},M)$).
\item
For subcategories $\mathcal{X},\mathcal{Y}$ of $\operatorname{\underline{\mathsf{CM}}}(R)$, we define:
$$
\operatorname{\underline{\mathsf{Hom}}}(\mathcal{X},\mathcal{Y}) = \textstyle\bigoplus_{X\in\mathcal{X},\,Y\in\mathcal{Y}}\operatorname{\underline{\mathsf{Hom}}}_R(X,Y).
$$
If $\mathcal{X}$ (respectively, $\mathcal{Y}$) consists of a single module $M$, we simply write $\operatorname{\underline{\mathsf{Hom}}}(M,\mathcal{Y})$ (respectively, $\operatorname{\underline{\mathsf{Hom}}}(\mathcal{X},M)$).
\end{enumerate}
\end{nota}
\begin{rem}
\begin{enumerate}[(1)]
\item
Let $\mathcal{X}\subseteq\mathcal{X}'$ and $\mathcal{Y}\subseteq\mathcal{Y}'$ be subcategories of $\operatorname{\mathsf{mod}} R$.
Then
\begin{align*}
& \operatorname{\mathsf{Supp}}\operatorname{\mathsf{Tor}}(\mathcal{X},\mathcal{Y}) \subseteq \operatorname{\mathsf{Supp}}\operatorname{\mathsf{Tor}}(\mathcal{X}',\mathcal{Y}'),\quad
\operatorname{\mathsf{Supp}}\operatorname{\mathsf{Ext}}(\mathcal{X},\mathcal{Y})\subseteq \operatorname{\mathsf{Supp}}\operatorname{\mathsf{Ext}}(\mathcal{X}',\mathcal{Y}'),\\
& \operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Tor}}(\mathcal{X},\mathcal{Y})) \subseteq \operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Tor}}(\mathcal{X}',\mathcal{Y}')),\quad
\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\mathcal{X},\mathcal{Y}))\subseteq \operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\mathcal{X}',\mathcal{Y}')).
\end{align*}
\item
Let $\mathcal{X},\mathcal{Y}$ be subcategories of $\operatorname{\mathsf{mod}} R$.
Then one has
$$
\operatorname{\mathsf{Supp}}\operatorname{\mathsf{Tor}}(\mathcal{X},\mathcal{Y})\subseteq\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Tor}}(\mathcal{X},\mathcal{Y})),\quad\operatorname{\mathsf{Supp}}\operatorname{\mathsf{Ext}}(\mathcal{X},\mathcal{Y})\subseteq\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\mathcal{X},\mathcal{Y})).
$$
The equalities do not hold in general because $\operatorname{\mathsf{Tor}}(\mathcal{X},\mathcal{Y})$ and $\operatorname{\mathsf{Ext}}(\mathcal{X},\mathcal{Y})$ are usually infinitely generated $R$-modules.
\item
Let $R$ be Gorenstein with $\operatorname{\mathsf{dim}} R<\infty$, and let $\mathcal{X},\mathcal{Y}$ be subcategories of $\operatorname{\underline{\mathsf{CM}}}(R)$.
Suppose that either $\mathcal{X}$ or $\mathcal{Y}$ is closed under shifts.
Then one has the equalities
$$
\operatorname{\mathsf{Ann}}\operatorname{\underline{\mathsf{Hom}}}(\mathcal{X},\mathcal{Y})=\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\overline\mathcal{X},\overline\mathcal{Y}),\quad\operatorname{\mathsf{Supp}}\operatorname{\underline{\mathsf{Hom}}}(\mathcal{X},\mathcal{Y})=\operatorname{\mathsf{Supp}}\operatorname{\mathsf{Ext}}(\overline\mathcal{X},\overline\mathcal{Y}).
$$
Indeed, for all $i>0$, $X\in\mathcal{X}$ and $Y\in\mathcal{Y}$ we have $\operatorname{\mathsf{Ext}}_R^i(X,Y)\cong\operatorname{\underline{\mathsf{Hom}}}_R(\mathsf{\Sigma}^{-i}X,Y)\cong\operatorname{\underline{\mathsf{Hom}}}_R(X,\mathsf{\Sigma}^iY)$ and $\operatorname{\underline{\mathsf{Hom}}}_R(X,Y)\cong\operatorname{\mathsf{Ext}}_R^1(\mathsf{\Sigma} X,Y)\cong\operatorname{\mathsf{Ext}}_R^1(X,\mathsf{\Sigma}^{-1}Y)$.
The assertion is an easy consequence of these isomorphisms.
Using these two equalities, we can translate results on $\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}$ and $\operatorname{\mathsf{Supp}}\operatorname{\mathsf{Ext}}$ into ones on $\operatorname{\mathsf{Ann}}\operatorname{\underline{\mathsf{Hom}}}$ and $\operatorname{\mathsf{Supp}}\operatorname{\underline{\mathsf{Hom}}}$.
\end{enumerate}
\end{rem}
Our first purpose in this section is to analyze the annihilators of $\operatorname{\mathsf{Tor}},\operatorname{\mathsf{Ext}}$ on subcategories of $\operatorname{\mathsf{mod}} R$ by means of the annihilators of $\operatorname{\mathsf{Tor}},\operatorname{\mathsf{Ext}}$ on smaller subcategories:
\begin{prop}\label{a^2}
Let $R$ be local and $M$ be an $R$-module.
Let $a\in R$, $n\in\mathbb{Z}$ and $t\in\mathbb{N}$.
\begin{enumerate}[\rm(1)]
\item
Suppose that $a\operatorname{\mathsf{Tor}}_n^R(M,X)=a\operatorname{\mathsf{Tor}}_{n-1}^R(M,X)=0$ for all $R$-modules $X$ with $\operatorname{\mathsf{dim}} X\le t$.
Then $a^2\operatorname{\mathsf{Tor}}_n^R(M,X)=0$ for all $R$-modules $X$ with $\operatorname{\mathsf{dim}} X\le t+1$.
\item
Suppose that $a\operatorname{\mathsf{Ext}}_R^n(M,X)=a\operatorname{\mathsf{Ext}}_R^{n+1}(M,X)=0$ for all $R$-modules $X$ with $\operatorname{\mathsf{dim}} X\le t$.
Then $a^2\operatorname{\mathsf{Ext}}_R^n(M,X)=0$ for all $R$-modules $X$ with $\operatorname{\mathsf{dim}} X\le t+1$.
\item
Suppose that $a\operatorname{\mathsf{Ext}}_R^n(X,M)=a\operatorname{\mathsf{Ext}}_R^{n+1}(X,M)=0$ for all $R$-modules $X$ with $\operatorname{\mathsf{dim}} X\le t$.
Then $a^2\operatorname{\mathsf{Ext}}_R^n(X,M)=0$ for all $R$-modules $X$ with $\operatorname{\mathsf{dim}} X\le t+1$.
\end{enumerate}
\end{prop}
\begin{proof}
We only prove the first assertion, since the second and third assertions are shown similarly.
Fix an $R$-module $X$ with $\operatorname{\mathsf{dim}} X\le t+1$.
We want to show $a^2\operatorname{\mathsf{Tor}}_n^R(M,X)=0$.
By assumption, we have only to deal with the case $\operatorname{\mathsf{dim}} X=t+1$.
Let $r\in R$ be part of a system of parameters of $X$.
Then we have $\operatorname{\mathsf{dim}} X/rX=t$, and it is easy to see that $\operatorname{\mathsf{dim}}(0:_Xr)\le t$ holds.
Our assumption implies
$$
\text{$a\operatorname{\mathsf{Tor}}_i^R(M,X/rX)=a\operatorname{\mathsf{Tor}}_i^R(M,(0:_Xr))=0$ for $i=n,n-1$.}
$$
There are exact sequences $0 \to (0:_Xr) \to X \to rX \to 0$ and $0 \to rX \to X \to X/rX \to 0$, which give exact sequences
\begin{align}
&\label{colon} \operatorname{\mathsf{Tor}}_n^R(M,X) \xrightarrow{f} \operatorname{\mathsf{Tor}}_n^R(M,rX) \to \operatorname{\mathsf{Tor}}_{n-1}^R(M,(0:_Xr)),\\
&\label{modulo} \operatorname{\mathsf{Tor}}_n^R(M,rX) \xrightarrow{g} \operatorname{\mathsf{Tor}}_n^R(M,X) \to \operatorname{\mathsf{Tor}}_n^R(M,X/rX).
\end{align}
Let $y\in\operatorname{\mathsf{Tor}}_n^R(M,X)$.
By \eqref{modulo} we have $ay=g(z)$ for some $z\in\operatorname{\mathsf{Tor}}_n^R(M,rX)$, and by \eqref{colon} we have $az=f(w)$ for some $w\in\operatorname{\mathsf{Tor}}_n^R(M,X)$.
Hence $a^2y=gf(w)=rw$, and we obtain $a^2\operatorname{\mathsf{Tor}}_n^R(M,X)\subseteq r\operatorname{\mathsf{Tor}}_n^R(M,X)$ for every element $r\in R$ that is part of system of parameters of $M$.
Since the element $r^j$ is also part of system of parameters of $M$ for all $j>0$, the module $a^2\operatorname{\mathsf{Tor}}_n^R(M,X)$ is contained in $\bigcap_{j>0}r^j\operatorname{\mathsf{Tor}}_n^R(M,X)$, which is zero by Krull's intersection theorem.
\end{proof}
Iteration of the above proposition yields the following result; the annihilators of $\operatorname{\mathsf{Tor}},\operatorname{\mathsf{Ext}}$ on $\operatorname{\mathsf{mod}} R$ can be controlled by the annihilators of $\operatorname{\mathsf{Tor}},\operatorname{\mathsf{Ext}}$ on $\operatorname{\mathsf{fl}} R$.
\begin{cor}\label{22d}
Let $R$ be a local ring of dimension $d$.
Let $a\in R$ and $n\in\mathbb{Z}$.
\begin{enumerate}[\rm(1)]
\item
Suppose that $a\operatorname{\mathsf{Tor}}_i^R(M,N)=0$ for all $n-2d\le i\le n$ and $M,N\in\operatorname{\mathsf{fl}} R$.
Then $a^{2^{2d}}\operatorname{\mathsf{Tor}}_n^R(M,N)=0$ for all $M,N\in\operatorname{\mathsf{mod}} R$.
\item
Suppose that $a\operatorname{\mathsf{Ext}}_R^i(M,N)=0$ for all $n\le i\le n+2d$ and $M,N\in\operatorname{\mathsf{fl}} R$.
Then $a^{2^{2d}}\operatorname{\mathsf{Ext}}_R^n(M,N)=0$ for all $M,N\in\operatorname{\mathsf{mod}} R$.
\end{enumerate}
\end{cor}
\begin{proof}
Let us show the first assertion; the second one follows from a similar argument.
First, fix $M\in\operatorname{\mathsf{fl}} R$.
Applying Proposition \ref{a^2}(1) repeatedly, we get:
\begin{align*}
& \text{$a\operatorname{\mathsf{Tor}}_i^R(M,N)=0$ for all $n-2d\le i\le n$ and $N\in\operatorname{\mathsf{fl}} R$},\\
& \text{$a^2\operatorname{\mathsf{Tor}}_i^R(M,N)=0$ for all $n-2d+1\le i\le n$ and $N\in\operatorname{\mathsf{mod}} R$ with $\operatorname{\mathsf{dim}} N\le1$},\\
& \text{$a^4\operatorname{\mathsf{Tor}}_i^R(M,N)=0$ for all $n-2d+2\le i\le n$ and $N\in\operatorname{\mathsf{mod}} R$ with $\operatorname{\mathsf{dim}} N\le2$},\\
& \qquad\cdots
\end{align*}
and we obtain $a^{2^d}\operatorname{\mathsf{Tor}}_i^R(M,N)=0$ for all $n-d\le i\le n$ and $M\in\operatorname{\mathsf{fl}} R$ and $N\in\operatorname{\mathsf{mod}} R$.
Next, fix $N\in\operatorname{\mathsf{mod}} R$.
A similar argument to the above gives:
\begin{align*}
& \text{$a^{2^d}\operatorname{\mathsf{Tor}}_i^R(M,N)=0$ for all $n-d\le i\le n$ and $M\in\operatorname{\mathsf{fl}} R$},\\
& \text{$a^{2^{d+1}}\operatorname{\mathsf{Tor}}_i^R(M,N)=0$ for all $n-d+1\le i\le n$ and $M\in\operatorname{\mathsf{mod}} R$ with $\operatorname{\mathsf{dim}} M\le1$},\\
& \text{$a^{2^{d+2}}\operatorname{\mathsf{Tor}}_i^R(M,N)=0$ for all $n-d+2\le i\le n$ and $M\in\operatorname{\mathsf{mod}} R$ with $\operatorname{\mathsf{dim}} M\le2$},\\
& \qquad\cdots
\end{align*}
and finally we get $a^{2^{2d}}\operatorname{\mathsf{Tor}}_n^R(M,N)=0$ for all $M,N\in\operatorname{\mathsf{mod}} R$.
\end{proof}
In the case where $R$ is Cohen-Macaulay, the annihilators of $\operatorname{\mathsf{Tor}},\operatorname{\mathsf{Ext}}$ on $\operatorname{\mathsf{mod}} R$ can also be controlled by the annihilators of $\operatorname{\mathsf{Tor}},\operatorname{\mathsf{Ext}}$ on $\operatorname{\mathsf{CM}}_0(R)$.
\begin{prop}\label{4d}
Let $R$ be a $d$-dimensional Cohen-Macaulay local ring.
\begin{enumerate}[\rm(1)]
\item
Let $a\in\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Tor}}(\operatorname{\mathsf{CM}}_0(R),\operatorname{\mathsf{CM}}_0(R))$.
Then $a^{2^{2d}}\operatorname{\mathsf{Tor}}_i^R(M,N)=0$ for all $i>4d$ and all $R$-modules $M,N$.
\item
Let $a\in\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\operatorname{\mathsf{CM}}_0(R),\operatorname{\mathsf{CM}}_0(R))$.
Then $a^{2^{2d}(d+1)}\operatorname{\mathsf{Ext}}_R^i(M,N)=0$ for all $i>d$ and all $R$-modules $M,N$.
\end{enumerate}
\end{prop}
\begin{proof}
Let $M,N$ be $R$-modules of finite length.
Note that $\mathsf{\Omega}^dM,\mathsf{\Omega}^dN$ belong to $\operatorname{\mathsf{CM}}_0(R)$.
(1) We have $a\operatorname{\mathsf{Tor}}_i^R(\mathsf{\Omega}^dM,\mathsf{\Omega}^dN)=0$ for every $i>0$, which implies $a\operatorname{\mathsf{Tor}}_i^R(M,N)=0$ for every $i>2d$.
Now let $n>4d$ be an integer.
Then $a\operatorname{\mathsf{Tor}}_i^R(M,N)=0$ for all integers $i$ with $n-2d\le i\le n$.
It follows from Corollary \ref{22d}(1) that $a^{2^{2d}}\operatorname{\mathsf{Tor}}_n^R(X,Y)=0$ for all $R$-modules $X,Y$.
(2) Fix an integer $i>0$.
We have $a\operatorname{\mathsf{Ext}}_R^i(K,L)=0$ for all $K,L\in\operatorname{\mathsf{CM}}_0(R)$.
For each integer $j\ge0$ there is an exact sequence $0 \to \mathsf{\Omega}^{j+1}N \to F_j \to \mathsf{\Omega}^jN \to 0$ such that $F_j$ is free.
Since $F_j$ and $\mathsf{\Omega}^dN$ belong to $\operatorname{\mathsf{CM}}_0(R)$, we have $a\operatorname{\mathsf{Ext}}_R^i(K,F_j)=a\operatorname{\mathsf{Ext}}_R^i(K,\mathsf{\Omega}^dN)=0$.
An exact sequence $\operatorname{\mathsf{Ext}}_R^i(K,F_j) \to \operatorname{\mathsf{Ext}}_R^i(K,\mathsf{\Omega}^jN) \to \operatorname{\mathsf{Ext}}_R^{i+1}(K,\mathsf{\Omega}^{j+1}N)$ is induced, and an inductive argument shows that $a^{d+1}\operatorname{\mathsf{Ext}}_R^i(K,N)=0$.
(Note that in general an exact sequence $A\xrightarrow{\alpha}B\xrightarrow{\beta}C$ yields $\operatorname{\mathsf{Ann}} A\cdot\operatorname{\mathsf{Ann}} C\subseteq\operatorname{\mathsf{Ann}} B$.)
Letting $K:=\mathsf{\Omega}^dM$, we observe that $a^{d+1}\operatorname{\mathsf{Ext}}_R^h(M,N)=0$ for every $h>d$.
Corollary \ref{22d}(2) yields $(a^{d+1})^{2^{2d}}\operatorname{\mathsf{Ext}}_R^h(X,Y)=0$ for all $h>d$ and $X,Y\in\operatorname{\mathsf{mod}} R$.
\end{proof}
Now we can prove that the annihilators of $\operatorname{\mathsf{Tor}},\operatorname{\mathsf{Ext}}$ on $\operatorname{\mathsf{CM}}_0(R)$ are contained in all prime ideals in the singular locus of $R$.
\begin{prop}\label{5.1}
\begin{enumerate}[\rm(1)]
\item
Let $R$ be a Cohen-Macaulay local ring.
Then one has
$$
\operatorname{\mathsf{Sing}} R\subseteq
\begin{cases}
\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Tor}}(\operatorname{\mathsf{CM}}_0(R),\operatorname{\mathsf{CM}}_0(R))),\\
\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\operatorname{\mathsf{CM}}_0(R),\operatorname{\mathsf{CM}}_0(R))).
\end{cases}
$$
\item
Let $R$ be a Gorenstein local ring.
Then
$$
\operatorname{\mathsf{Sing}} R\subseteq\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\underline{\mathsf{Hom}}}(\operatorname{\underline{\mathsf{CM}}}_0(R),\operatorname{\underline{\mathsf{CM}}}_0(R))).
$$
\end{enumerate}
\end{prop}
\begin{proof}
(1) Let $\mathfrak{p}$ be any prime ideal in $\operatorname{\mathsf{Sing}} R$.
Take an element $a\in\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Tor}}(\operatorname{\mathsf{CM}}_0(R),\operatorname{\mathsf{CM}}_0(R))$.
Then, by Proposition \ref{4d}(1) we have $a^{2^{2d}}\operatorname{\mathsf{Tor}}_i^R(R/\mathfrak{p},R/\mathfrak{p})=0$ for $i>4d$.
Localization at $\mathfrak{p}$ shows that $a^{2^{2d}}\operatorname{\mathsf{Tor}}_i^{R_\mathfrak{p}}(\kappa(\mathfrak{p}),\kappa(\mathfrak{p}))=0$ for $i>4d$.
If $a$ is not in $\mathfrak{p}$, then $a^{2^{2d}}$ is a unit in $R_\mathfrak{p}$, and it follows that $\operatorname{\mathsf{Tor}}_i^{R_\mathfrak{p}}(\kappa(\mathfrak{p}),\kappa(\mathfrak{p}))=0$ for $i>4d$.
This is impossible since the local ring $R_\mathfrak{p}$ is nonregular, and thus $a\in\mathfrak{p}$.
The assertion for $\operatorname{\mathsf{Ext}}$ is also proved analogously.
(2) Since $\operatorname{\mathsf{Ann}}\operatorname{\underline{\mathsf{Hom}}}(\operatorname{\underline{\mathsf{CM}}}_0(R),\operatorname{\underline{\mathsf{CM}}}_0(R))$ coincides with $\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\operatorname{\mathsf{CM}}_0(R),\operatorname{\mathsf{CM}}_0(R))$, the assertion follows from (1).
\end{proof}
\begin{rem}
In some results such as Propositions \ref{5.1} and \ref{4.4} the stable category versions are given, because they are used in the proofs of the main results of this paper.
We can also give the stable category versions of other results such as Propositions \ref{4d} and \ref{cep}, but do not, just because they are not necessary to prove our main results.
\end{rem}
Let $R$ be a complete equicharacteristic local ring with residue field $k$.
Let $A$ be a {\em Noether normalization} of $R$, that is, a formal power series subring $k[[x_1,\dots,x_d]]$, where $x_1,\dots,x_d$ is a system of parameters of $R$.
Let $R^e=R\otimes_AR$ be the enveloping algebra of $R$ over $A$.
Define a map $\mu:R^e\to R$ by $\mu(x\otimes y)=xy$ for $x,y\in R$, and put $\mathfrak{N}^R_A=\mu(\operatorname{\mathsf{Ann}}_{R^e}\operatorname{\mathsf{Ker}}\mu)$.
Then $\mathfrak{N}^R_A$ is an ideal of $R$, which is called the {\em Noether different} of $R$ over $A$.
We denote by $\mathfrak{N}^R$ the sum of $\mathfrak{N}^R_A$, where $A$ runs through the Noether normalizations of $R$.
Under a mild assumption, we can substantially refine the statement for $\operatorname{\mathsf{Ext}}$ in the previous proposition, as follows:
\begin{prop}\label{cep}
Let $R$ be a Cohen-Macaulay complete equicharacteristic local ring with perfect residue field.
Then
$$
\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\mathcal{X},\mathcal{Y}))=\operatorname{\mathsf{Sing}} R
$$
for all $\operatorname{\mathsf{CM}}_0(R)\subseteq\mathcal{X}\subseteq\operatorname{\mathsf{CM}}(R)$ and $\operatorname{\mathsf{CM}}_0(R)\subseteq\mathcal{Y}\subseteq\operatorname{\mathsf{mod}} R$.
\end{prop}
\begin{proof}
We have the inclusions below, the first of which follows from Proposition \ref{5.1}(1).
\begin{align*}
\operatorname{\mathsf{Sing}} R
& \subseteq\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\operatorname{\mathsf{CM}}_0(R),\operatorname{\mathsf{CM}}_0(R)))\\
& \subseteq\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\mathcal{X},\mathcal{Y}))\subseteq\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\operatorname{\mathsf{CM}}(R),\operatorname{\mathsf{mod}} R)).
\end{align*}
By virtue of \cite[Corollary 5.13]{W}, the module $\operatorname{\mathsf{Ext}}(\operatorname{\mathsf{CM}}(R),\operatorname{\mathsf{mod}} R)$ is annihilated by the ideal $\mathfrak{N}^R$.
Hence $\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\operatorname{\mathsf{CM}}(R),\operatorname{\mathsf{mod}} R))$ is contained in $\operatorname{\mathsf{V}}(\mathfrak{N}^R)$.
On the other hand, it follows from \cite[Lemma (6.12)]{Y} that $\operatorname{\mathsf{V}}(\mathfrak{N}^R)$ coincides with $\operatorname{\mathsf{Sing}} R$.
\end{proof}
In the rest of this section, we study over an arbitrary local ring how the sets $\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Tor}}(\mathcal{X},\mathcal{Y}))$ and $\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\mathcal{X},\mathcal{Y}))$ vary as $\mathcal{X},\mathcal{Y}$ move subcategories of $\operatorname{\mathsf{mod}} R$.
\begin{prop}\label{doko}
Let $R$ be a local ring of dimension $d$, and let $\mathcal{X}$ be a subcategory of $\operatorname{\mathsf{mod}} R$.
Then one has
\begin{align*}
& \operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Tor}}(\mathcal{X},\mathsf{\Omega}^d\operatorname{\mathsf{mod}} R))\subseteq\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Tor}}(\mathcal{X},\operatorname{\mathsf{fl}} R))\subseteq\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Tor}}(\mathcal{X},\operatorname{\mathsf{mod}} R)),\\
& \operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\mathcal{X},\operatorname{\mathsf{fl}} R))=\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\mathcal{X},\operatorname{\mathsf{mod}} R)),\\
& \operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\operatorname{\mathsf{fl}} R,\mathcal{X}))=\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\operatorname{\mathsf{mod}} R,\mathcal{X})),
\end{align*}
where $\mathsf{\Omega}^d\operatorname{\mathsf{mod}} R$ denotes the subcategory of $\operatorname{\mathsf{mod}} R$ consisting of the $d$-th syzygies of modules in $\mathcal{X}$.
\end{prop}
\begin{proof}
Let $a,b,c\in R$ be elements such that $a\operatorname{\mathsf{Tor}}_i^R(X,M)=b\operatorname{\mathsf{Ext}}_R^i(X,M)=c\operatorname{\mathsf{Ext}}_R^i(Y,M)=0$ for all $i>0$, $X\in\mathcal{X}$ and $M\in\operatorname{\mathsf{fl}} R$.
Then a similar argument to the proof of Corollary \ref{22d} shows that $a^{2^d}\operatorname{\mathsf{Tor}}_i^R(X,M)=b^{2^d}\operatorname{\mathsf{Ext}}_R^j(X,M)=c^{2^d}\operatorname{\mathsf{Ext}}_R^j(M,X)=0$ for all $i>d$, $j>0$, $X\in\mathcal{X}$ and $M\in\operatorname{\mathsf{mod}} R$.
Hence we have
\begin{align*}
\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Tor}}(\mathcal{X},\operatorname{\mathsf{fl}} R) & \subseteq\sqrt{\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Tor}}(\mathcal{X},\mathsf{\Omega}^d\operatorname{\mathsf{mod}} R)},\\
\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\mathcal{X},\operatorname{\mathsf{fl}} R) & \subseteq\sqrt{\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\mathcal{X},\operatorname{\mathsf{mod}} R)},\\
\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\operatorname{\mathsf{fl}} R,\mathcal{X}) & \subseteq\sqrt{\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\operatorname{\mathsf{mod}} R,\mathcal{X})}.
\end{align*}
Therefore $\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Tor}}(\mathcal{X},\mathsf{\Omega}^d\operatorname{\mathsf{mod}} R))$, $\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\mathcal{X},\operatorname{\mathsf{mod}} R))$ and $\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\operatorname{\mathsf{mod}} R,\mathcal{X}))$ are contained in $\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Tor}}(\mathcal{X},\operatorname{\mathsf{fl}} R))$, $\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\mathcal{X},\operatorname{\mathsf{fl}} R))$ and $\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\operatorname{\mathsf{fl}} R,\mathcal{X}))$, respectively.
The other inclusion relations are straightforward.
\end{proof}
\begin{cor}
Let $R$ be a local ring.
The sets $\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\mathcal{X},\mathcal{Y}))$ are constant over the subcategories $\mathcal{X},\mathcal{Y}$ of $\operatorname{\mathsf{mod}} R$ containing $\operatorname{\mathsf{fl}} R$.
\end{cor}
\begin{proof}
First, since $\operatorname{\mathsf{fl}} R\subseteq\mathcal{Y}\subseteq\operatorname{\mathsf{mod}} R$, we have $\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\mathcal{X},\operatorname{\mathsf{fl}} R))\subseteq\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\mathcal{X},\mathcal{Y}))\subseteq\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\mathcal{X},\operatorname{\mathsf{mod}} R))$.
Proposition \ref{doko} implies that the left and right ends are equal.
Next, the inclusions $\operatorname{\mathsf{fl}} R\subseteq\mathcal{X}\subseteq\operatorname{\mathsf{mod}} R$ imply $\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\operatorname{\mathsf{fl}} R,\operatorname{\mathsf{mod}} R))\subseteq\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\mathcal{X},\operatorname{\mathsf{mod}} R))\subseteq\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\operatorname{\mathsf{mod}} R,\operatorname{\mathsf{mod}} R))$, where the left and right ends coincide by Proposition \ref{doko}.
Thus, we obtain $\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\mathcal{X},\mathcal{Y}))=\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\mathcal{X},\operatorname{\mathsf{mod}} R))=\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\operatorname{\mathsf{mod}} R,\operatorname{\mathsf{mod}} R))$.
\end{proof}
\section{Nonfree loci and stable supports of subcategories}
This section mainly investigates the nonfree loci of subcategories of $\operatorname{\mathsf{CM}}(R)$ and the stable supports of subcategories of $\operatorname{\underline{\mathsf{CM}}}(R)$, using the results obtained in the previous section.
First of all, let us study relationships of nonfree loci and stable supports with supports and annhilators of $\operatorname{\mathsf{Tor}}$, $\operatorname{\mathsf{Ext}}$ and $\operatorname{\underline{\mathsf{Hom}}}$.
\begin{prop}\label{4.4}
\begin{enumerate}[\rm(1)]
\item
For an $R$-module $M$ one has equalities
\begin{align*}
\operatorname{\mathsf{NF}}(M)
& = \operatorname{\mathsf{Supp}}\operatorname{\mathsf{Tor}}(M,\operatorname{\mathsf{mod}} R) = \operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Tor}}(M,\operatorname{\mathsf{mod}} R))\\
& = \operatorname{\mathsf{Supp}}\operatorname{\mathsf{Ext}}(M,\operatorname{\mathsf{mod}} R) = \operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(M,\operatorname{\mathsf{mod}} R)).
\end{align*}
\item
Let $R$ be a $d$-dimensional Gorenstein ring.
For an object $M\in\operatorname{\underline{\mathsf{CM}}}(R)$ one has
$$
\operatorname{\underline{\mathsf{Supp}}} M = \operatorname{\mathsf{Supp}}\operatorname{\underline{\mathsf{Hom}}}(M,\operatorname{\underline{\mathsf{CM}}}(R)) = \operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\underline{\mathsf{Hom}}}(M,\operatorname{\underline{\mathsf{CM}}}(R))).
$$
\end{enumerate}
\end{prop}
\begin{proof}
(1) Let us prove the equalities $\operatorname{\mathsf{NF}}(M)=\operatorname{\mathsf{Supp}}\operatorname{\mathsf{Tor}}(M,\operatorname{\mathsf{mod}} R)=\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Tor}}(M,\operatorname{\mathsf{mod}} R))$.
First, let $\mathfrak{p}\in\operatorname{\mathsf{NF}}(M)$.
Then we have $\operatorname{\mathsf{Tor}}_1^R(M,R/\mathfrak{p})_\mathfrak{p}=\operatorname{\mathsf{Tor}}_1^{R_\mathfrak{p}}(M_\mathfrak{p},\kappa(\mathfrak{p}))\ne0$.
Hence $\operatorname{\mathsf{NF}}(M)$ is contained in $\operatorname{\mathsf{Supp}}\operatorname{\mathsf{Tor}}(M,\operatorname{\mathsf{mod}} R)$, which is contained in $\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Tor}}(M,\operatorname{\mathsf{mod}} R))$.
Next, let $\mathfrak{p}\in\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Tor}}(M,\operatorname{\mathsf{mod}} R))$, and suppose $\mathfrak{p}\notin\operatorname{\mathsf{NF}}(M)$.
Then $M_\mathfrak{p}\cong R_\mathfrak{p}^{\oplus n}$ for some $n\ge0$, which gives an exact sequence
$$
0 \to K \to M \xrightarrow{f} R^n \to C \to 0
$$
such that $K_\mathfrak{p}=0=C_\mathfrak{p}$.
We can choose an element $x\in R\setminus\mathfrak{p}$ satisfying $xK=0=xC$.
Taking the image $L$ of $f$ decomposes the above exact sequence into two short exact sequences, which make an exact sequence
\begin{equation}\label{tttt}
\operatorname{\mathsf{Tor}}_i^R(K,N) \to \operatorname{\mathsf{Tor}}_i^R(M,N) \to \operatorname{\mathsf{Tor}}_i^R(L,N)=\operatorname{\mathsf{Tor}}_{i+1}^R(C,N)
\end{equation}
for each $i>0$ and $N\in\operatorname{\mathsf{mod}} R$.
Since $x$ annihilates $\operatorname{\mathsf{Tor}}_i^R(K,N)$ and $\operatorname{\mathsf{Tor}}_{i+1}^R(C,N)$, the element $x^2$ annihilates $\operatorname{\mathsf{Tor}}_i^R(M,N)$.
This means that $x^2$ belongs to $\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Tor}}(M,\operatorname{\mathsf{mod}} R)$, which is contained in $\mathfrak{p}$.
Thus $x$ belongs to $\mathfrak{p}$, which is a contradiction.
Consequently, $\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Tor}}(M,\operatorname{\mathsf{mod}} R))$ is contained in $\operatorname{\mathsf{NF}}(M)$, and now the equalities $\operatorname{\mathsf{NF}}(M)=\operatorname{\mathsf{Supp}}\operatorname{\mathsf{Tor}}(M,\operatorname{\mathsf{mod}} R)=\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Tor}}(M,\operatorname{\mathsf{mod}} R))$ follow.
Along the same lines as in the above, one can prove the equalities $\operatorname{\mathsf{NF}}(M)=\operatorname{\mathsf{Supp}}\operatorname{\mathsf{Ext}}(M,\operatorname{\mathsf{mod}} R)=\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(M,\operatorname{\mathsf{mod}} R))$, using the following instead of \eqref{tttt}:
$$
\operatorname{\mathsf{Ext}}^{i+1}_R(C,N)=\operatorname{\mathsf{Ext}}^i_R(L,N) \to \operatorname{\mathsf{Ext}}^i_R(M,N) \to \operatorname{\mathsf{Ext}}^i_R(K,N).
$$
(2) We have $\operatorname{\underline{\mathsf{Supp}}} M=\operatorname{\mathsf{NF}}(M)=\operatorname{\mathsf{Supp}}\operatorname{\mathsf{Ext}}(M,\operatorname{\mathsf{mod}} R)=\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(M,\operatorname{\mathsf{mod}} R))$ by (1).
Since $R$ is Gorenstein and $M$ is Cohen-Macaulay, the isomorphism $\operatorname{\mathsf{Ext}}_R^i(M,N)\cong\operatorname{\mathsf{Ext}}_R^{i+d}(M,\mathsf{\Omega}^dN)$ holds for all $i>0$ and $N\in\operatorname{\mathsf{mod}} R$.
Note here that $\mathsf{\Omega}^dN$ is Cohen-Macaulay.
Now it is easy to observe that the equalities $\operatorname{\mathsf{Supp}}\operatorname{\mathsf{Ext}}(M,\operatorname{\mathsf{mod}} R)=\operatorname{\mathsf{Supp}}\operatorname{\mathsf{Ext}}(M,\operatorname{\mathsf{CM}}(R))=\operatorname{\mathsf{Supp}}\operatorname{\underline{\mathsf{Hom}}}(M,\operatorname{\underline{\mathsf{CM}}}(R))$ and $\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(M,\operatorname{\mathsf{mod}} R))=\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(M,\operatorname{\mathsf{CM}}(R)))=\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\underline{\mathsf{Hom}}}(M,\operatorname{\underline{\mathsf{CM}}}(R)))$ hold.
\end{proof}
\begin{rem}
\begin{enumerate}[(1)]
\item
Let $M$ be an $R$-module.
Take an ideal $I$ of $R$ with $\operatorname{\mathsf{NF}}(M)=\operatorname{\mathsf{V}}(I)$.
Then Proposition \ref{4.4}(1) implies that there exists an integer $h>0$ such that $I^h$ annihilates $\operatorname{\mathsf{Tor}}_i^R(M,X)$ and $\operatorname{\mathsf{Ext}}_R^i(M,X)$ for all $i>0$ and $X\in\operatorname{\mathsf{mod}} R$.
This is a generalization of \cite[Lemma 4.3]{DV}.
\item
One has $\operatorname{\mathsf{NF}}(\mathcal{X})=\operatorname{\mathsf{Supp}}\operatorname{\mathsf{Ext}}(\mathcal{X},\mathcal{Y})$ for all subcategories $\mathcal{X},\mathcal{Y}$ of $\operatorname{\mathsf{mod}} R$ with $\mathsf{\Omega}\mathcal{X}\subseteq\mathcal{Y}$.
In fact, it is obvious that $\operatorname{\mathsf{NF}}(\mathcal{X})$ contains $\operatorname{\mathsf{Supp}}\operatorname{\mathsf{Ext}}(\mathcal{X},\mathcal{Y})$, and the opposite inclusion relation is obtained by the fact that $\operatorname{\mathsf{NF}}(X)=\operatorname{\mathsf{Supp}}\operatorname{\mathsf{Ext}}_R^1(X,\mathsf{\Omega} X)$ for each $R$-module $X$ (cf. \cite[Proposition 2.10]{res}).
The equality $\operatorname{\mathsf{NF}}(M) = \operatorname{\mathsf{Supp}}\operatorname{\mathsf{Ext}}(M,\operatorname{\mathsf{mod}} R)$ in Proposition \ref{4.4} is also a consequence of this statement.
\end{enumerate}
\end{rem}
Here we need to inspect the annihilators of $\operatorname{\mathsf{Tor}},\operatorname{\mathsf{Ext}},\operatorname{\underline{\mathsf{Hom}}}$ of balls:
\begin{lem}\label{hos}
\begin{enumerate}[\rm(1)]
\item
Let $\mathcal{X},\mathcal{Y}$ be subcategories of $\operatorname{\mathsf{mod}} R$ and $n\ge0$ an integer.
Then:
\begin{align*}
& (\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Tor}}(\mathcal{X},\mathcal{Y}))^n\subseteq
\left\{
\begin{gathered}
\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Tor}}([\mathcal{X}]_n,\mathcal{Y})\\
\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Tor}}(\mathcal{X},[\mathcal{Y}]_n)
\end{gathered}
\right\}
\subseteq\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Tor}}(\mathcal{X},\mathcal{Y}),\\
& (\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\mathcal{X},\mathcal{Y}))^n\subseteq
\left\{
\begin{gathered}
\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}([\mathcal{X}]_n,\mathcal{Y})\\
\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\mathcal{X},[\mathcal{Y}]_n)
\end{gathered}
\right\}
\subseteq\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\mathcal{X},\mathcal{Y}).
\end{align*}
In particular, $\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Tor}}(\mathcal{X},\mathcal{Y}))=\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Tor}}([\mathcal{X}]_n,\mathcal{Y}))=\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Tor}}(\mathcal{X},[\mathcal{Y}]_n))$ and $\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\mathcal{X},\mathcal{Y}))=\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}([\mathcal{X}]_n,\mathcal{Y}))=\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\mathcal{X},[\mathcal{Y}]_n))$ hold.
\item
Suppose that $R$ is Gorenstein of finite Krull dimension.
Let $\mathcal{X},\mathcal{Y}$ be subcategories of $\operatorname{\underline{\mathsf{CM}}}(R)$ and $n\ge0$ an integer.
Then there are inclusions
\begin{align*}
& (\operatorname{\mathsf{Ann}}\operatorname{\underline{\mathsf{Hom}}}(\mathcal{X},\mathcal{Y}))^n\subseteq
\left\{
\begin{gathered}
\operatorname{\mathsf{Ann}}\operatorname{\underline{\mathsf{Hom}}}(\langle\mathcal{X}\rangle_n,\mathcal{Y})\\
\operatorname{\mathsf{Ann}}\operatorname{\underline{\mathsf{Hom}}}(\mathcal{X},\langle\mathcal{Y}\rangle_n)
\end{gathered}
\right\}
\subseteq\operatorname{\mathsf{Ann}}\operatorname{\underline{\mathsf{Hom}}}(\mathcal{X},\mathcal{Y}).
\end{align*}
In particular, one has equalities $\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\underline{\mathsf{Hom}}}(\mathcal{X},\mathcal{Y}))=\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\underline{\mathsf{Hom}}}(\langle\mathcal{X}\rangle_n,\mathcal{Y}))=\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\underline{\mathsf{Hom}}}(\mathcal{X},\langle\mathcal{Y}\rangle_n))$.
\end{enumerate}
\end{lem}
\begin{proof}
Let us only prove the inclusions $(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Tor}}(\mathcal{X},\mathcal{Y}))^n\subseteq\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Tor}}([\mathcal{X}]_n,\mathcal{Y})\subseteq\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Tor}}(\mathcal{X},\mathcal{Y})$; the other inclusions can be shown similarly.
It is clear that the second inclusion holds.
As for the first inclusion, it suffices to show that
$$
\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Tor}}([\mathcal{X}]_n,\mathcal{Y})\supseteq\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Tor}}([\mathcal{X}]_{n-1},\mathcal{Y})\cdot\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Tor}}(\mathcal{X},\mathcal{Y})
$$
holds for any $n\ge1$.
Take elements $a\in\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Tor}}([\mathcal{X}]_{n-1},\mathcal{Y})$ and $b\in\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Tor}}(\mathcal{X},\mathcal{Y})$.
Fix $i>0$, $Z\in[\mathcal{X}]_n$ and $Y\in\mathcal{Y}$.
Then there exists an exact sequence $0 \to L \to M \to N \to 0$ with $L\in[\mathcal{X}]_{n-1}$ and $N\in[\mathcal{X}]$ such that $Z$ is a direct summand of $M$.
This induces an exact sequence $\operatorname{\mathsf{Tor}}_i^R(L,Y) \to \operatorname{\mathsf{Tor}}_i^R(M,Y) \to \operatorname{\mathsf{Tor}}_i^R(N,Y)$.
Since $a$ and $b$ annihilate $\operatorname{\mathsf{Tor}}_i^R(L,Y)$ and $\operatorname{\mathsf{Tor}}_i^R(N,Y)$ respectively, the element $ab$ annihilates $\operatorname{\mathsf{Tor}}_i^R(M,Y)$.
Thus the proof is completed.
\end{proof}
Now we state restriction made by finiteness of dimension.
The following result says that finite-dimensional subcategories of $\operatorname{\mathsf{CM}}(R)$ and $\operatorname{\underline{\mathsf{CM}}}(R)$ containing the Cohen-Macaulay modules that are locally free on $\operatorname{\mathsf{Spec}}_0(R)$ define the biggest nonfree loci and stable supports.
\begin{prop}\label{4.5}
Let $R$ be a Cohen-Macaulay local ring.
\begin{enumerate}[\rm(1)]
\item
Let $\mathcal{X}$ be a subcategory of $\operatorname{\mathsf{CM}}(R)$ containing $\operatorname{\mathsf{CM}}_0(R)$.
If $\mathcal{X}$ has finite dimension, then $\operatorname{\mathsf{NF}}(\mathcal{X})=\operatorname{\mathsf{Sing}} R$.
\item
Suppose that $R$ is Gorenstein.
Let $\mathcal{X}$ be a subcategory of $\operatorname{\underline{\mathsf{CM}}}(R)$ containing $\operatorname{\underline{\mathsf{CM}}}_0(R)$.
If $\mathcal{X}$ has finite dimension, then $\operatorname{\underline{\mathsf{Supp}}}\mathcal{X}=\operatorname{\mathsf{Sing}} R$.
\end{enumerate}
\end{prop}
\begin{proof}
(1) Setting $\operatorname{\mathsf{dim}}\mathcal{X}=n$, we find a module $G\in\mathcal{X}$ with $\mathcal{X}=[G]_{n+1}$.
We have
$$
\operatorname{\mathsf{NF}}(\mathcal{X})\overset{\rm(a)}{=}\operatorname{\mathsf{NF}}(G)\overset{\rm(b)}{=}\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(G,\operatorname{\mathsf{mod}} R))\overset{\rm(c)}{=}\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\mathcal{X},\operatorname{\mathsf{mod}} R))\overset{\rm(d)}{=}\operatorname{\mathsf{Sing}} R,
$$
where each equality follows from the following observation:\\
(a) Let $\mathfrak{p}$ be a prime ideal not in $\operatorname{\mathsf{NF}}(G)$.
Then $G_\mathfrak{p}$ is $R_\mathfrak{p}$-free.
For each $M\in\mathcal{X}$, we have $M_\mathfrak{p}\in[G_\mathfrak{p}]_{n+1}$, which implies that $M_\mathfrak{p}$ is $R_\mathfrak{p}$-free.
Hence $\mathfrak{p}$ is not in $\operatorname{\mathsf{NF}}(\mathcal{X})$.\\
(b) This is obtained by Proposition \ref{4.4}.\\
(c) This follows from Lemma \ref{hos}.\\
(d) We have $\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\mathcal{X},\operatorname{\mathsf{mod}} R))=\operatorname{\mathsf{NF}}(\mathcal{X})\subseteq\operatorname{\mathsf{Sing}} R$ by (a), (b) and (c).
Proposition \ref{5.1} shows that $\operatorname{\mathsf{Sing}} R$ is contained in $\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\operatorname{\mathsf{CM}}_0(R),\operatorname{\mathsf{CM}}_0(R)))$, which is contained in $\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\mathcal{X},\operatorname{\mathsf{mod}} R))$ since $\operatorname{\mathsf{CM}}_0(R)\subseteq\mathcal{X}\subseteq\operatorname{\mathsf{mod}} R$.
(2) This statement is shown by an analogous argument to (1).
\end{proof}
Here, recall that a local ring $(R,\mathfrak{m})$ is called a {\em hypersurface} if the $\mathfrak{m}$-adic completion of $R$ is a residue ring of a complete regular local ring by a principal ideal.
Also, recall that a Cohen-Macaulay local ring $R$ is said to have {\em minimal multiplicity} if the equality $\operatorname{\mathsf{e}}(R)=\operatorname{\mathsf{edim}} R-\operatorname{\mathsf{dim}} R+1$ holds, where $\operatorname{\mathsf{e}}(R)$ denotes the multiplicity of $R$ and $\operatorname{\mathsf{edim}} R$ denotes the embedding dimension of $R$.
The preceding proposition states that finiteness of the dimension of a subcategory $\mathcal{X}$ implies
$$
\operatorname{\mathsf{NF}}(\mathcal{X})=\operatorname{\mathsf{Sing}} R.
$$
Now we are interested in how this condition is close to the condition that $\mathcal{X}=\operatorname{\mathsf{CM}}(R)$.
In fact, it turns out by some results in \cite{stcm,crs} that these two conditions are equivalent under certain assumptions.
We state this here, and by combining it with a result obtained in the previous section we give a criterion for a resolving subcategory to coincide with $\operatorname{\mathsf{CM}}(R)$ in terms of the support and annihilator of $\operatorname{\mathsf{Ext}}$.
\begin{prop}\label{3cond}
Let $(R,\mathfrak{m},k)$ be a Cohen-Macaulay local ring of dimension $d$.
Let $\mathcal{X}$ be a resolving subcategory of $\operatorname{\mathsf{mod}} R$ contained in $\operatorname{\mathsf{CM}}(R)$.
Suppose that one of the following three conditions is satisfied.
\begin{itemize}
\item
$R$ is a hypersurface.
\item
$R$ is locally a hypersurface on $\operatorname{\mathsf{Spec}}_0(R)$, and $\mathcal{X}$ contains $\mathsf{\Omega}^dk$.
\item
$R$ is locally with minimal multiplicity on $\operatorname{\mathsf{Spec}}_0(R)$, and $\mathcal{X}$ contains $\mathsf{\Omega}^dk$.
\item
$R$ is excellent and locally of finite Cohen-Macaulay representation type on $\operatorname{\mathsf{Spec}}_0(R)$, and $\mathcal{X}$ contains $\mathsf{\Omega}^dk$ and a dualizing $R$-module.
\end{itemize}
Then the following statements hold.
\begin{enumerate}[\rm(1)]
\item
One has $\mathcal{X}=\operatorname{\mathsf{NF}}_{\operatorname{\mathsf{CM}}}^{-1}(\operatorname{\mathsf{NF}}(\mathcal{X}))$.
Hence, $\mathcal{X}=\operatorname{\mathsf{CM}}(R)$ if and only if $\operatorname{\mathsf{NF}}(\mathcal{X})=\operatorname{\mathsf{Sing}} R$.
\item
Assume that $R$ is complete and equicharacteristic and that $k$ is perfect.
Then $\mathcal{X}=\operatorname{\mathsf{CM}}(R)$ if and only if $\operatorname{\mathsf{Supp}}\operatorname{\mathsf{Ext}}(\mathcal{X},\mathcal{Y})=\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\mathcal{X},\mathcal{Y}))$ for some subcategory $\mathcal{Y}$ of $\operatorname{\mathsf{mod}} R$ containing $\operatorname{\mathsf{CM}}_0(R)$.
\end{enumerate}
\end{prop}
\begin{proof}
(1) The former assertion follows from \cite[Main Theorem]{stcm} and \cite[Theorem 5.6 and Corollary 6.12]{crs}.
The latter assertion is shown by the former.
(2) We have $\operatorname{\mathsf{Supp}}\operatorname{\mathsf{Ext}}(\operatorname{\mathsf{CM}}(R),\operatorname{\mathsf{mod}} R)\subseteq\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\operatorname{\mathsf{CM}}(R),\operatorname{\mathsf{mod}} R))=\operatorname{\mathsf{Sing}} R$, where the equality follows from Proposition \ref{cep}.
Let $\mathfrak{p}\in\operatorname{\mathsf{Sing}} R$, and set $M=\mathsf{\Omega}^d(R/\mathfrak{p})$.
Then $M$ is Cohen-Macaulay, and $\mathfrak{p}$ belongs to $\operatorname{\mathsf{NF}}(M)$.
We have $\operatorname{\mathsf{NF}}(M)=\operatorname{\mathsf{Supp}}\operatorname{\mathsf{Ext}}_R^1(M,\mathsf{\Omega} M)$ (cf. \cite[Proposition 2.10]{res}), which is contained in $\operatorname{\mathsf{Supp}}\operatorname{\mathsf{Ext}}(\operatorname{\mathsf{CM}}(R),\operatorname{\mathsf{mod}} R)$.
Thus we have $\operatorname{\mathsf{Supp}}\operatorname{\mathsf{Ext}}(\operatorname{\mathsf{CM}}(R),\operatorname{\mathsf{mod}} R)=\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\operatorname{\mathsf{CM}}(R),\operatorname{\mathsf{mod}} R))$, which shows the `only if' part.
Suppose that $\operatorname{\mathsf{Supp}}\operatorname{\mathsf{Ext}}(\mathcal{X},\mathcal{Y})=\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\mathcal{X},\mathcal{Y}))$ for some subcategory $\mathcal{Y}$ of $\operatorname{\mathsf{mod}} R$ containing $\operatorname{\mathsf{CM}}_0(R)$.
It is evident that the inclusions $\operatorname{\mathsf{Supp}}\operatorname{\mathsf{Ext}}(\mathcal{X},\mathcal{Y})\subseteq\operatorname{\mathsf{NF}}(\mathcal{X})\subseteq\operatorname{\mathsf{Sing}} R$ hold.
Proposition \ref{cep} yields the equality $\operatorname{\mathsf{Sing}} R=\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\mathcal{X},\mathcal{Y}))$.
Hence we have $\operatorname{\mathsf{NF}}(\mathcal{X})=\operatorname{\mathsf{Sing}} R$.
It follows from (1) that $\mathcal{X}=\operatorname{\mathsf{CM}}(R)$.
Thus the `if' part is proved.
\end{proof}
\begin{rem}
As to the equivalence in Proposition \ref{3cond}(1), the `only if' part always holds, but `if' part does not hold without the assumption on the punctured spectrum.
Let
$$
R=k[[x,y,z]]/(x^2,y^2z^2),
$$
where $k$ is a field.
This is a $1$-dimensional complete intersection local ring with $\operatorname{\mathsf{Spec}} R=\operatorname{\mathsf{Sing}} R=\{\mathfrak{p},\mathfrak{q},\mathfrak{m}\}$, where $\mathfrak{p}=(x,y),\mathfrak{q}=(x,z),\mathfrak{m}=(x,y,z)$.
Let
$$
\mathcal{X}=\operatorname{\mathsf{res}}_R(\mathfrak{m}\oplus R/(x)).
$$
(For an $R$-module $M$ we denote by $\operatorname{\mathsf{res}}_RM$ the {\em resolving closure} of $M$, i.e., the smallest resolving subcategory of $\operatorname{\mathsf{mod}} R$ containing $M$.)
Then $\mathcal{X}$ is a resolving subcategory of $\operatorname{\mathsf{mod}} R$ contained in $\operatorname{\mathsf{CM}}(R)$ and containing $\mathsf{\Omega}^1k=\mathfrak{m}$.
We have $\operatorname{\mathsf{NF}}(\mathcal{X})=\operatorname{\mathsf{NF}}(\mathfrak{m}\oplus R/(x))=\operatorname{\mathsf{Sing}} R$ by \cite[Corollary 3.6]{res}.
However, $\mathcal{X}$ does not coincide with $\operatorname{\mathsf{CM}}(R)$; the Cohen-Macaulay $R$-module $R/\mathfrak{p}$ does not belong to $\mathcal{X}$.
Indeed, assume $R/\mathfrak{p}\in\mathcal{X}$.
Then $\kappa(\mathfrak{p})$ belongs to $\operatorname{\mathsf{res}}_{R_\mathfrak{p}}((\mathfrak{m}\oplus R/(x))_\mathfrak{p})=\operatorname{\mathsf{res}}_{R_\mathfrak{p}}(R_\mathfrak{p}/xR_\mathfrak{p})$ by \cite[Proposition 3.5]{res}.
Every object of $\operatorname{\mathsf{res}}_{R_\mathfrak{p}}(R_\mathfrak{p}/xR_\mathfrak{p})$ is a Cohen-Macaulay $R_\mathfrak{p}$-module whose complexity is at most that of $R_\mathfrak{p}/xR_\mathfrak{p}$ by \cite[Proposition 4.2.4]{Av}.
Since $R_\mathfrak{p}\cong k[[x,y,z]]_{(x,y)}/(x^2,y^2)$, the complexity of the $R_\mathfrak{p}$-module $R_\mathfrak{p}/xR_\mathfrak{p}$ is $1$.
This shows that $\kappa(\mathfrak{p})$ has complexity at most $1$, which cannot occur because $R_\mathfrak{p}$ is not a hypersurface (cf. \cite[Remark 8.1.1(3)]{Av}).
Thus $R/\mathfrak{p}$ is not in $\mathcal{X}$.
\end{rem}
\section{Dimensions of $\operatorname{\mathsf{CM}}_0(R)$ and $\operatorname{\underline{\mathsf{CM}}}_0(R)$}
In this section, we consider finiteness of the dimensions of $\operatorname{\mathsf{CM}}_0(R)$ and $\operatorname{\underline{\mathsf{CM}}}_0(R)$.
It will turn out that it is closely related to the condition that $R$ is an isolated singularity.
Let us begin with studying over a Cohen-Macaulay local ring $R$ the relationship between finiteness of the dimensions of subcategories of $\operatorname{\mathsf{CM}}(R),\operatorname{\underline{\mathsf{CM}}}(R)$ and the $\mathfrak{m}$-primary property of the annihilator of $\operatorname{\mathsf{Tor}},\operatorname{\mathsf{Ext}},\operatorname{\underline{\mathsf{Hom}}}$ on them, where $\mathfrak{m}$ is the maximal ideal of $R$.
\begin{prop}\label{ky}
Let $(R,\mathfrak{m},k)$ be a Cohen-Macaulay local ring of dimension $d$.
\begin{enumerate}[\rm(1)]
\item
\begin{enumerate}[\rm(a)]
\item
Let $\mathcal{X}$ be a subcategory of $\operatorname{\mathsf{CM}}_0(R)$ with $\operatorname{\mathsf{dim}}\mathcal{X}<\infty$.
Let $\mathcal{Y}$ be any subcategory of $\operatorname{\mathsf{mod}} R$.
Then $\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Tor}}(\mathcal{X},\mathcal{Y})$ and $\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\mathcal{X},\mathcal{Y})$ are $\mathfrak{m}$-primary.
\item
Suppose that $R$ is Gorenstein.
Let $\mathcal{X}$ be a subcategory of $\operatorname{\underline{\mathsf{CM}}}_0(R)$ with $\operatorname{\mathsf{dim}}\mathcal{X}<\infty$.
Let $\mathcal{Y}$ be any subcategory of $\operatorname{\underline{\mathsf{CM}}}(R)$.
Then $\operatorname{\mathsf{Ann}}\operatorname{\underline{\mathsf{Hom}}}(\mathcal{X},\mathcal{Y})$ is $\mathfrak{m}$-primary.
\end{enumerate}
\item
\begin{enumerate}[\rm(a)]
\item
Let $\mathcal{X}$ be a resolving subcategory of $\operatorname{\mathsf{mod}} R$ contained in $\operatorname{\mathsf{CM}}(R)$ and containing $\mathsf{\Omega}^dk$.
Let $\mathcal{Y}$ be a subcategory of $\operatorname{\mathsf{mod}} R$ containing $\mathcal{X}$.
If $\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\mathcal{X},\mathcal{Y})$ is $\mathfrak{m}$-primary, then $\operatorname{\mathsf{dim}}\mathcal{X}<\infty$.
\item
Suppose that $R$ is Gorenstein.
Let $\mathcal{X}$ be a thick subcategory of $\operatorname{\underline{\mathsf{CM}}}(R)$ containing $\mathsf{\Omega}^dk$.
Let $\mathcal{Y}$ be a subcategory of $\operatorname{\underline{\mathsf{CM}}}(R)$ containing $\mathcal{X}$.
If $\operatorname{\mathsf{Ann}}\operatorname{\underline{\mathsf{Hom}}}(\mathcal{X},\mathcal{Y})$ is $\mathfrak{m}$-primary, then $\operatorname{\mathsf{dim}}\mathcal{X}<\infty$.
\end{enumerate}
\end{enumerate}
\end{prop}
\begin{proof}
(1) We prove only the assertion on $\operatorname{\mathsf{Tor}}$ in (a) because the other assertions are similarly shown.
Let $n=\operatorname{\mathsf{dim}}\mathcal{X}$.
Then there exists a module $G\in\mathcal{X}$ such that $\mathcal{X}=[G]_{n+1}$.
Lemma \ref{hos} implies $\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Tor}}(\mathcal{X},\mathcal{Y}))=\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Tor}}(G,\mathcal{Y}))$, which is contained in $\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Tor}}(G,\operatorname{\mathsf{mod}} R))$.
On the other hand, one sees from Proposition \ref{4.4} that $\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Tor}}(G,\operatorname{\mathsf{mod}} R))=\operatorname{\mathsf{NF}}(G)$ holds, and $\operatorname{\mathsf{NF}}(G)$ is contained in $\{\mathfrak{m}\}$ since $G\in\operatorname{\mathsf{CM}}_0(R)$.
Therefore we obtain $\operatorname{\mathsf{V}}(\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Tor}}(\mathcal{X},\mathcal{Y}))\subseteq\{\mathfrak{m}\}$, which shows that $\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Tor}}(\mathcal{X},\mathcal{Y})$ is an $\mathfrak{m}$-primary ideal of $R$.
(2) We prove only the statement (a) because (b) follows from (a) and Proposition \ref{35}(2).
For some integer $h>0$ the ideal $\mathfrak{m}^h$ annihilates $\operatorname{\mathsf{Ext}}(\mathcal{X},\mathcal{Y})$.
Take a parameter ideal $Q$ of $R$ contained in $\mathfrak{m}^h$.
Fix a module $M$ in $\mathcal{X}$.
Then $\mathsf{\Omega}^jM$ is in $\mathcal{X}$ since $\mathcal{X}$ is resolving, and it is also in $\mathcal{Y}$.
Hence we have $Q\operatorname{\mathsf{Ext}}_R^i(M,\mathsf{\Omega}^jM)=0$ for $1\le i,j\le d$.
Since $M$ is Cohen-Macaulay, we can apply \cite[Proposition 2.2]{stcm} to $M$, which shows that $M$ is isomorphic to a direct summand of $\mathsf{\Omega}^d(M/QM)$.
As the ring $R/Q$ is artinian, there exists an integer $r>0$ such that $\mathfrak{m}^r(R/Q)=0$.
We have a filtration
$$
M/QM\supseteq\mathfrak{m}(M/QM)\supseteq\mathfrak{m}^2(M/QM)\cdots\supseteq\mathfrak{m}^r(M/QM)=0
$$
of $R/Q$-submodules of $M/QM$.
Decompose this into exact sequences $0 \to \mathfrak{m}^{i+1}(M/QM) \to \mathfrak{m}^i(M/QM) \to k^{\oplus s_i} \to 0$, where $0\le i\le r-1$.
Taking the $d$-th syzygies, we obtain exact sequences
$$
0 \to \mathsf{\Omega}^d(\mathfrak{m}^{i+1}(M/QM)) \to \mathsf{\Omega}^d(\mathfrak{m}^i(M/QM))\oplus R^{\oplus t_i} \to (\mathsf{\Omega}^dk)^{\oplus s_i} \to 0.
$$
By induction on $i$, we observe that the $R$-module $\mathsf{\Omega}^d(M/QM)$ belongs to $[\mathsf{\Omega}^dk]_r$, and hence $M\in[\mathsf{\Omega}^dk]_r$.
Since $\mathcal{X}$ is resolving and contains $\mathsf{\Omega}^dk$, we have $\mathcal{X}=[\mathsf{\Omega}^dk]_r$.
(Note here that $r$ is independent of the choice of $M$.)
Therefore $\operatorname{\mathsf{dim}}\mathcal{X}\le r-1<\infty$.
\end{proof}
Recall that $R$ is called an {\em isolated singularity} if the local ring $R_\mathfrak{p}$ is regular for all $\mathfrak{p}\in\operatorname{\mathsf{Spec}}_0(R)$.
Recall also that the {\em annihilator} of an $R$-linear additive category $\mathcal{C}$ is defined as:
$$
\bigcap_{M,N\in\mathcal{C}}\operatorname{\mathsf{Ann}}_R\operatorname{\mathsf{Hom}}_\mathcal{C}(M,N).
$$
Combining Propositions \ref{ky}, \ref{5.1} and \ref{cep} yields our first main result of this paper, which is a characterization of the isolated singularity of $R$ in terms of the dimensions of $\operatorname{\mathsf{CM}}_0(R)$ and $\operatorname{\underline{\mathsf{CM}}}_0(R)$:
\begin{thm}\label{main}
Let $R$ be a Cohen-Macaulay local ring with maximal ideal $\mathfrak{m}$.
\begin{enumerate}[\rm(1)]
\item
Set the following four conditions.
\begin{enumerate}[\rm(a)]
\item
The dimension of $\operatorname{\mathsf{CM}}_0(R)$ is finite.
\item
The ideal $\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Ext}}(\operatorname{\mathsf{CM}}_0(R),\operatorname{\mathsf{CM}}_0(R))$ is $\mathfrak{m}$-primary.
\item
The ideal $\operatorname{\mathsf{Ann}}\operatorname{\mathsf{Tor}}(\operatorname{\mathsf{CM}}_0(R),\operatorname{\mathsf{CM}}_0(R))$ is $\mathfrak{m}$-primary.
\item
The ring $R$ is an isolated singularity.
\end{enumerate}
Then, the implications ${\rm(a)}\Leftrightarrow{\rm(b)}\Rightarrow{\rm(c)}\Rightarrow{\rm(d)}$ hold.
The implication ${\rm(d)}\Rightarrow{\rm(a)}$ also holds if $R$ is complete, equicharacteristic and with perfect residue field.
\item
Suppose that $R$ is Gorenstein, and set the following three conditions.
\begin{enumerate}[\rm(a)]
\item
The dimension of the triangulated category $\operatorname{\underline{\mathsf{CM}}}_0(R)$ is finite.
\item
The annihilator of the $R$-linear category $\operatorname{\underline{\mathsf{CM}}}_0(R)$ is $\mathfrak{m}$-primary.
\item
The ring $R$ is an isolated singularity.
\end{enumerate}
Then the implications ${\rm(a)}\Leftrightarrow{\rm(b)}\Rightarrow{\rm(c)}$ hold, and so does ${\rm(c)}\Rightarrow{\rm(a)}$ if $R$ is complete, equicharacteristic and with perfect residue field.
\end{enumerate}
\end{thm}
The celebrated Auslander-Huneke-Leuschke-Wiegand theorem states that every Cohen-Macaulay local ring $R$ of finite Cohen-Macaulay representation type is an isolated singularity.
This was proved by Auslander \cite[Theorem 10]{A} when $R$ is complete, by Leuschke and Wiegand \cite[Corollary 1.9]{LW} when $R$ is excellent, and by Huneke and Leuschke \cite[Corollary 2]{HL} in the general case.
Our Theorem \ref{main} not only deduces this result but also improves it as follows:
\begin{cor}
Let $R$ be a Cohen-Macaulay local ring.
Suppose that there are only finitely many isomorphism classes of indecomposable Cohen-Macaulay $R$-modules which are locally free on $\operatorname{\mathsf{Spec}}_0(R)$.
Then $R$ is an isolated singularity, and hence $R$ has finite Cohen-Macaulay representation type.
\end{cor}
\begin{proof}
Let $M_1,\dots,M_n$ be the nonisomorphic indecomposable Cohen-Macaulay $R$-modules which are locally free on $\operatorname{\mathsf{Spec}}_0(R)$.
Then $\operatorname{\mathsf{CM}}_0(R)$ contains $M:=M_1\oplus\cdots\oplus M_n$.
Since $\operatorname{\mathsf{CM}}_0(R)$ is resolving, it also contains $[M]$.
On the other hand, take $N\in\operatorname{\mathsf{CM}}_0(R)$.
Then each indecomposable summand of $N$ also belongs to $\operatorname{\mathsf{CM}}_0(R)$, so it is isomorphic to one of $M_1,\dots,M_n$.
Hence $N$ is in $\operatorname{\mathsf{add}} M$.
Therefore we have $\operatorname{\mathsf{CM}}_0(R)=\operatorname{\mathsf{add}} M=[M]$.
This implies $\operatorname{\mathsf{dim}}\operatorname{\mathsf{CM}}_0(R)=0<\infty$, and the assertion follows from Theorem \ref{main}(1).
\end{proof}
Our Theorem \ref{main} also gives rise to finiteness of the dimensions of $\operatorname{\mathsf{CM}}(R)$ and $\operatorname{\underline{\mathsf{CM}}}(R)$ as a direct consequence, the latter of which is nothing but \cite[Corollary 5.3]{ddc}.
\begin{cor}\label{comp}
Let $R$ be a Cohen-Macaulay equicharacteristic complete local ring with perfect residue field.
Suppose that $R$ is an isolated singularity.
Then $\operatorname{\mathsf{CM}}(R)$ has finite dimension.
If $R$ is Gorenstein, $\operatorname{\underline{\mathsf{CM}}}(R)$ has finite dimension as a triangulated category.
\end{cor}
\begin{rem}\label{prfhos}
In Corollary \ref{comp}, one can replace the assumption that $R$ is complete with the weaker assumption that $R$ is excellent, using a similar argument to the proof of \cite[Theorem 5.8]{ddc}.
(It is proved in \cite[Theorem 5.8]{ddc} that the latter statement in Corollary \ref{comp} holds true even if $R$ is not complete but excellent.)
Indeed, since the completion $\widehat R$ of $R$ is still an isolated singularity, Corollary \ref{comp} implies that $\operatorname{\mathsf{CM}}(\widehat R)$ has finite dimension.
Putting $n=\operatorname{\mathsf{dim}}\operatorname{\mathsf{CM}}(\widehat R)$, we have $\operatorname{\mathsf{CM}}(\widehat R)=[C]_{n+1}$ for some Cohen-Macaulay $\widehat R$-module $C$.
It follows from \cite[Corollary 3.6]{stcm} that there exists a Cohen-Macaulay $R$-module $G$ such that $C$ is isomorphic to a direct summand of the completion $\widehat G$ of $G$, and hence we have $\operatorname{\mathsf{CM}}(\widehat R)=[\widehat G]_{n+1}$.
Now we claim the following.
\begin{claim*}
Let $m>0$.
For any $N\in[\widehat G]_m$ there exists $M\in[G]_m$ such that $N$ is isomorphic to a direct summand of $\widehat M$.
\end{claim*}
This claim is shown by induction on $m$.
When $m=1$, the module $N$ is isomorphic to a direct summand of a direct sum $\bigoplus_{i=1}^{h}\mathsf{\Omega}^{l_i}\widehat{G}$ for some $l_i\ge0$, and we can take $M=\bigoplus_{i=1}^{h}\mathsf{\Omega}^{l_i}G$.
Assume $m\ge 2$.
There is an exact sequence $\sigma: 0 \to X \to Z \to Y \to 0$ with $X\in[\widehat G]_{m-1}$ and $Y\in[\widehat G]$ such that $N$ is a direct summand of $Z$.
The induction hypothesis implies that there exist $V\in[G]_{m-1}$ and $W\in[G]$ such that $X$ and $Y$ are isomorphic to direct summands of $\widehat{V}$ and $\widehat{W}$, respectively.
We have isomorphisms $\widehat V\cong X\oplus X'$ and $\widehat W\cong Y\oplus Y'$, and get an exact sequence $\sigma':0 \to \widehat V \to Z' \to \widehat W \to 0$, where $Z'=X'\oplus Y'\oplus Z$.
Regard $\sigma'$ as an element of $\operatorname{\mathsf{Ext}}_{\widehat R}^1(\widehat W,\widehat V)$.
We have $\operatorname{\mathsf{Ext}}_{\widehat R}^1(\widehat W,\widehat V)\cong\operatorname{\mathsf{Ext}}_R^1(W,V)^{\widehat{\ }}\cong\operatorname{\mathsf{Ext}}_R^1(W,V)$, where the latter isomorphism follows from the fact that $\operatorname{\mathsf{Ext}}_R^1(W,V)$ has finite length as an $R$-module.
This gives an exact sequence $\tau: 0 \to V \to M \to W \to 0$ such that $\widehat\tau\cong\sigma'$ as $\widehat R$-complexes.
Since $\widehat M\cong Z'=X'\oplus Y'\oplus Z$, the module $N$ is isomorphic to a direct summand of $\widehat M$.
As $M\in [G]_m$, the claim follows.
Let $X\in\operatorname{\mathsf{CM}}(R)$.
Then the completion $\widehat X$ is in $\operatorname{\mathsf{CM}}(\widehat R)=[\widehat G]_{n+1}$.
The claim implies that there exists $M\in[G]_{n+1}$ such that $\widehat{X}$ is isomorphic to a direct summand of $\widehat{M}$.
Hence $\widehat X\in\operatorname{\mathsf{add}}_{\widehat R}(\widehat M)$.
It is seen by \cite[Lemma 5.7]{ddc} that $X$ is in $\operatorname{\mathsf{add}}_RM$, whence $X\in[G]_{n+1}$.
\end{rem}
\section{Dimensions of more general resolving and thick subcategories}
In the preceding section, we studied finiteness of the dimensions of the resolving subcategory $\operatorname{\mathsf{CM}}_0(R)$ of $\operatorname{\mathsf{mod}} R$ and the thick subcategory $\operatorname{\underline{\mathsf{CM}}}_0(R)$ of $\operatorname{\underline{\mathsf{CM}}}(R)$.
The aim of this section is to investigate finiteness of the dimensions of more general resolving subcategories of $\operatorname{\mathsf{mod}} R$ and thick subcategories of $\operatorname{\underline{\mathsf{CM}}}(R)$.
We start by the following theorem.
\begin{thm}\label{mg}
Let $(R,\mathfrak{m},k)$ be a Cohen-Macaulay local ring.
Let $W$ be a specialization-closed subset of $\operatorname{\mathsf{Spec}} R$ contained in $\operatorname{\mathsf{Sing}} R$.
\begin{enumerate}[\rm(1)]
\item
Set the following three conditions.
\begin{enumerate}[\rm(a)]
\item
$\operatorname{\mathsf{NF}}_{\operatorname{\mathsf{CM}}}^{-1}(W)$ has finite dimension.
\item
$W$ is either $\emptyset$ or $\operatorname{\mathsf{Sing}} R$.
\item
$\operatorname{\mathsf{NF}}_{\operatorname{\mathsf{CM}}}^{-1}(W)$ is either $\operatorname{\mathsf{add}} R$ or $\operatorname{\mathsf{CM}}(R)$.
\end{enumerate}
Then ${\rm(a)}\Rightarrow{\rm(b)}\Rightarrow{\rm(c)}$ hold.
\item
Let $R$ be Gorenstein.
Set the following three conditions.
\begin{enumerate}[\rm(a)]
\item
$\operatorname{\underline{\mathsf{Supp}}}^{-1}(W)$ has finite dimension.
\item
$W$ is either $\emptyset$ or $\operatorname{\mathsf{Sing}} R$.
\item
$\operatorname{\underline{\mathsf{Supp}}}^{-1}(W)$ is either $0$ or $\operatorname{\underline{\mathsf{CM}}}(R)$.
\end{enumerate}
Then ${\rm(a)}\Rightarrow{\rm(b)}\Rightarrow{\rm(c)}$ hold.
The implication ${\rm(c)}\Rightarrow{\rm(a)}$ also holds if $R$ is excellent and equicharacteristic and $k$ is perfect.
\end{enumerate}
\end{thm}
\begin{proof}
(1) As to the implication ${\rm(a)}\Rightarrow{\rm(b)}$, we may assume $W\ne\emptyset$.
Then $W$ contains $\mathfrak{m}$, as $W$ is specialization-closed.
Hence we have $\operatorname{\mathsf{CM}}_0(R)=\operatorname{\mathsf{NF}}_{\operatorname{\mathsf{CM}}}^{-1}(\{\mathfrak{m}\})\subseteq\operatorname{\mathsf{NF}}_{\operatorname{\mathsf{CM}}}^{-1}(W)\subseteq\operatorname{\mathsf{CM}}(R)$.
Proposition \ref{4.5} shows $\operatorname{\mathsf{NF}}(\operatorname{\mathsf{NF}}_{\operatorname{\mathsf{CM}}}^{-1}(W))=\operatorname{\mathsf{Sing}} R$.
Since $\operatorname{\mathsf{NF}}(\operatorname{\mathsf{NF}}_{\operatorname{\mathsf{CM}}}^{-1}(W))$ is contained in $W$, we have $W=\operatorname{\mathsf{Sing}} R$.
The implication ${\rm(b)}\Rightarrow{\rm(c)}$ is trivial.
(2) The proof of the implications ${\rm(a)}\Rightarrow{\rm(b)}\Rightarrow{\rm(c)}$ is similar to the corresponding implications in (1).
The implication ${\rm(c)}\Rightarrow{\rm(a)}$ is obtained by \cite[Theorem 5.8]{ddc}.
\end{proof}
We do not know whether the implication $\rm(c)\Rightarrow{\rm(a)}$ in Theorem \ref{mg}(1) holds true:
\begin{ques}
Let $R$ be a Cohen-Macaulay excellent local ring containing a field and with perfect residue field.
Then, is the dimension of $\operatorname{\mathsf{CM}}(R)$ finite?
We do not know the answer even if $R$ is complete or essentially of finite type over a field.
The only answer we know is that this is affirmative when $R$ is an isolated singularity, by Corollary \ref{comp} and Remark \ref{prfhos}.
\end{ques}
In the rest of this section, we give several applications of our Theorem \ref{mg}.
First, we generalize some implications in Theorem \ref{main}.
\begin{cor}
Let $R$ be a Cohen-Macaulay local ring, and let $n$ be a nonnegative integer.
\begin{enumerate}[\rm(1)]
\item
If $\operatorname{\mathsf{dim}}\operatorname{\mathsf{CM}}_n(R)<\infty$, then $\operatorname{\mathsf{dim}}\operatorname{\mathsf{Sing}} R\le n$.
\item
Suppose that $R$ is Gorenstein.
If $\operatorname{\mathsf{dim}}\operatorname{\underline{\mathsf{CM}}}_n(R)<\infty$, then $\operatorname{\mathsf{dim}}\operatorname{\mathsf{Sing}} R\le n$.
The converse also holds if $R$ is excellent, equicharacteristic and with perfect residue field.
\end{enumerate}
\end{cor}
\begin{proof}
Let $W$ be the set of prime ideals $\mathfrak{p}\in\operatorname{\mathsf{Sing}} R$ such that $\operatorname{\mathsf{dim}} R/\mathfrak{p}\le n$.
Then $W$ is a specialization-closed subset of $\operatorname{\mathsf{Spec}} R$ contained in $\operatorname{\mathsf{Sing}} R$.
Since $W$ contains $\mathfrak{m}$, it is nonempty.
(1) Since $\operatorname{\mathsf{CM}}_n(R)=\operatorname{\mathsf{NF}}_{\operatorname{\mathsf{CM}}}^{-1}(W)$, Theorem \ref{mg}(1) shows $W=\operatorname{\mathsf{Sing}} R$, which implies the inequality $\operatorname{\mathsf{dim}}\operatorname{\mathsf{Sing}} R\le n$.
(2) As $\operatorname{\underline{\mathsf{CM}}}_n(R)=\operatorname{\underline{\mathsf{Supp}}}^{-1}(W)$, the assertion follows from Theorem \ref{mg}(2).
\end{proof}
The next application of Theorem \ref{mg} is our second main result of this paper, which provides many sufficient conditions for a subcategory to have {\em infinite} dimension.
\begin{thm}\label{mgc}
Let $(R,\mathfrak{m},k)$ be a $d$-dimensional Cohen-Macaulay local ring.
One has $\operatorname{\mathsf{dim}}\mathcal{X}=\infty$ in each of the following cases:
\begin{enumerate}[\rm(1)]
\item
$R$ is locally a hypersurface on $\operatorname{\mathsf{Spec}}_0(R)$.\\
$\mathcal{X}$ is a resolving subcategory of $\operatorname{\mathsf{mod}} R$ with $\mathsf{\Omega}^dk\in\mathcal{X}\subsetneq\operatorname{\mathsf{CM}}(R)$.
\item
$R$ is Gorenstein and locally a hypersurface on $\operatorname{\mathsf{Spec}}_0(R)$.\\
$\mathcal{X}$ is a thick subcategory of $\operatorname{\underline{\mathsf{CM}}}(R)$ with $\mathsf{\Omega}^dk\in\mathcal{X}\ne\operatorname{\underline{\mathsf{CM}}}(R)$.
\item
$R$ is locally with minimal multiplicity on $\operatorname{\mathsf{Spec}}_0(R)$.\\
$\mathcal{X}$ is a resolving subcategory of $\operatorname{\mathsf{mod}} R$ with $\mathsf{\Omega}^dk\in\mathcal{X}\subsetneq\operatorname{\mathsf{CM}}(R)$.
\item
$R$ is excellent, admits a canonical module $\omega$ and locally has finite Cohen-Macaulay representation type on $\operatorname{\mathsf{Spec}}_0(R)$.\\
$\mathcal{X}$ is a resolving subcategory of $\operatorname{\mathsf{mod}} R$ with $\{\omega,\mathsf{\Omega}^dk\}\subseteq\mathcal{X}\subsetneq\operatorname{\mathsf{CM}}(R)$.
\item
$R$ is a hypersurface.\\
$\mathcal{X}$ is a resolving subcategory of $\operatorname{\mathsf{mod}} R$ with $\operatorname{\mathsf{add}} R\ne\mathcal{X}\subsetneq\operatorname{\mathsf{CM}}(R)$.
\item
$R$ is a hypersurface.\\
$\mathcal{X}$ is a thick subcategory of $\operatorname{\underline{\mathsf{CM}}}(R)$ with $\{0\}\ne\mathcal{X}\ne\operatorname{\underline{\mathsf{CM}}}(R)$.
\end{enumerate}
\end{thm}
\begin{proof}
Note from the assumption on $\mathcal{X}$ that $R$ is nonregular in each of the cases (1)--(4).
By Proposition \ref{3cond}(1), we have $\operatorname{\mathsf{add}} R\ne\mathcal{X}=\operatorname{\mathsf{NF}}_{\operatorname{\mathsf{CM}}}^{-1}(\operatorname{\mathsf{NF}}(\mathcal{X}))$ in the cases (1),(3),(4),(5), and $\{0\}\ne\mathcal{X}=\operatorname{\underline{\mathsf{Supp}}}^{-1}(\operatorname{\underline{\mathsf{Supp}}}\mathcal{X})$ in the cases (2),(6).
Theorem \ref{mg} completes the proof.
\end{proof}
Denote by $\operatorname{\mathsf{D}^b}(\operatorname{\mathsf{mod}} R)$ the bounded derived category of $\operatorname{\mathsf{mod}} R$, and by $\operatorname{\mathsf{perf}} R$ the subcategory of perfect complexes (i.e., bounded complexes of projective modules).
Recently, Oppermann and \v{S}\'{t}ov\'{i}\v{c}ek \cite[Theorem 2]{OS} proved that every proper thick subcategories of $\operatorname{\mathsf{D}^b}(\operatorname{\mathsf{mod}} R)$ containing $\operatorname{\mathsf{perf}} R$ has {\em infinite} dimension.
In the case where $R$ is a hypersurface, we can refine this result as follows:
\begin{cor}
Let $R$ be a local hypersurface.
Let $\mathcal{X}$ be a thick subcategory of $\operatorname{\mathsf{D}^b}(\operatorname{\mathsf{mod}} R)$ with $\operatorname{\mathsf{perf}} R\subsetneq\mathcal{X}\subsetneq\operatorname{\mathsf{D}^b}(\operatorname{\mathsf{mod}} R)$.
Then the Verdier quotient $\mathcal{X}/\operatorname{\mathsf{perf}} R$ has infinite dimension, and in particular so does $\mathcal{X}$.
\end{cor}
\begin{proof}
Note that a thick subcategory of $\operatorname{\mathsf{D}^b}(\operatorname{\mathsf{mod}} R)$ contains $\operatorname{\mathsf{perf}} R$ if and only if it contains $R$.
The equivalence $\operatorname{\underline{\mathsf{CM}}}(R)\cong\operatorname{\mathsf{D}^b}(\operatorname{\mathsf{mod}} R)/\operatorname{\mathsf{perf}} R$ of triangulated categories given by Buchweitz \cite[Theorem 4.4.1]{B} corresponds each thick subcategory of $\operatorname{\underline{\mathsf{CM}}}(R)$ to a subcategory of $\operatorname{\mathsf{D}^b}(\operatorname{\mathsf{mod}} R)/\operatorname{\mathsf{perf}} R$ of the form $\mathcal{X}/\operatorname{\mathsf{perf}} R$, where $\mathcal{X}$ is a thick subcategory of $\operatorname{\mathsf{D}^b}(\operatorname{\mathsf{mod}} R)$ containing $\operatorname{\mathsf{perf}} R$.
Thus Theorem \ref{mgc}(6) implies that $\mathcal{X}/\operatorname{\mathsf{perf}} R$ has infinite dimension.
The last assertion is easy (cf. \cite[Lemma 3.4]{R} or \cite[Lemma 3.5]{ddc}).
\end{proof}
\section*{Acknowledgments}
The authors thank Luchezar Avramov, Craig Huneke, Osamu Iyama and Srikanth Iyengar for their valuable comments and useful suggestions.
This work was done during the visits of the second author to University of Kansas in May, July and August, 2011.
He is grateful for their kind hospitality.
Results in this paper were presented at the seminars in University of Kansas, Nagoya University, University of Nebraska-Lincoln, University of Missouri and University of Isfahan.
The authors thank the organizers of these seminars.
|
{
"timestamp": "2012-05-15T02:00:45",
"yymm": "1203",
"arxiv_id": "1203.1955",
"language": "en",
"url": "https://arxiv.org/abs/1203.1955"
}
|
\section{Introduction}
In this work, we extend the result that perfect isolated-object cloaking is impossible over a nonzero bandwidth~\cite{Pendry06, Miller06} to show that even \emph{imperfect} cloaking of isolated objects necessarily has a bandwidth that decreases with the size of the object. More generally, we show that the cloaking efficiency (the ratio of total cross section to geometric cross section) must worsen proportional to the diameter when averaged over any finite bandwidth for cloaking of isolated objects in air (or any medium of negligible loss and dispersion). Unlike our previous proof that sensitivity to imperfections worsens with diameter at individual frequencies~\cite{Hashemi11}, the result in this paper holds even for perfect fabrication and for materials with negligible absorption over the desired bandwidth. Our proof involves some unusual mathematical techniques. First, we equate the frequency-averaged problem to a single scattering problem at a complex frequency with the help of a version of the optical theorem. Second, we map the complex-frequency problem to an equivalent real-frequency problem with transformed materials. We can then use the fact that $\Im \varepsilon$ and $\Im \mu$ are $> 0$ for any physical causal material at $\Re \omega$ and $\Im \omega > 0$~\cite{Landau84:electro} to analyze an ``effective'' absorption loss in a manner similar to our previous work~\cite{Hashemi11}. Finally, in \secref{num} we numerically verify this scaling in an example of a spherical cloak that is a perfect Pendry cloak at one frequency but has causal dispersion.
The idea of transformation-based invisibility cloaks was proposed in 2006 \cite{Pendry06}, a fascinating idea that was followed by many theoretical works~\cite{Leonhardt06NJP, Schurig06OE, Cummer06, Qiu09, Chen07PRB, Kwon08,Jiang08, Kante08, Ma09, Yan07, Ruan07,Huang07, Zhang08APL, Cai07NP, Zolla07,Tucker05, Nicolet08, Chen08JAP, Leonhardt09, Baile09, Argyropoulos10, Baile10OE, Han10, Han11} and several experimental demonstrations of cloaking of isolated objects at one frequency~\cite{Schurig06, Smolyaninov08, Kante09, Liu09APL}. However, as pointed out by Pendry using a speed-of-light argument~\cite{Pendry06} and by Miller with a more formal approach~\cite{Miller06}, \emph{perfect} cloaking of an isolated object in vacuum over a non-zero bandwidth is impossible due to causality. This severe limitation helped inspire the idea of ground-plane cloaking \cite{Li08,ErginSt10OE, Xu09, Baile10, Landy10} and its experimental demonstrations~\cite{Liu09, Ma09OE, Gabrielli09, Valentine09,Lee09, ErginSt10, Ma10, Zhang11, Chen11, Gharghi11}, which circumvents such causality limitations (although it is still subject to other practical scaling difficulties~\cite{Hashemi10,Hashemi11}). However, the Pendry and Miller proofs say little about imperfect cloaking (a nonzero but small scattering cross-section). By continuity, if near-perfect cloaking is attained at a single frequency, ``good'' cloaking (total cross-section bounded by some given fraction of the geometric cross-section) must persist over some finite bandwidth. We show that causality imposes an even stronger constraint than forbidding perfect cloaking over a finite bandwidth: for a bandwidth-limited cloak, we show that causality constraints imply that the bandwidth of good cloaking scales inversely with the object diameter. (The impossibility of perfect cloaking over a finite bandwidth follows from our results as a special case.) Moreover, our proof holds for isolated objects in any transparent medium (negligible loss and dispersion), not just in vacuum.
The key to our proof in \secref{proof} is the transformation of a frequency-averaged scattering problem (weighted by a Lorentzian window for convenience), which is hard to analyze, to a \emph{single} complex-frequency scattering problem that is easy to analyze by relating the complex frequency to equivalent complex materials. In order to make this transformation, we rely on the optical theorem~\cite{Jackson98}, which relates the total cross section to the imaginary part of a forward scattering amplitude, since the latter is a causal linear response and hence an analytic function in the upper-half complex-frequency plane~\cite{Landau84:electro}. A review of the optical theorem and its history and applications can be found in \cite{Newton76}. The use of the optical theorem, and similar relationships based on conservation of energy, to apply complex analysis to scattering problems is most common in quantum field theory~\cite{Peskin95}. Similar in spirit to this
paper, the optical theorem is also used in the ``ITEP sum rules'' of
quantum chromodynamics to relate integrals of scattering cross
sections multiplied by Lorentzian windows (or powers thereof) to
scattering amplitudes at single points in momentum space via contour
integration~\cite{Peskin95}. On another related note, contour
integration of the imaginary part of Green's functions (and other
scattering amplitudes) is also a key technique for computing Casimir
interactions in quantum field theory~\cite{Johnson11}, where the
relationship between the imaginary part of the Green's function and
the field fluctuation statistics (the fluctuation--dissipation
theorem) is, like the optical theorem, derived from
energy-conservation considerations~\cite{Landau:stat}.
\section{Derivation}
\label{sec:proof}
A transformation-based cloak, as depicted in Fig. 1, works by mapping an object (in ``physical'' space $X$) to a single point (in ``virtual space'' $X'$) and the cloak region (volume $V_c$) to empty space (volume $V_c'$), via a coordinate transformation that is the identity outside the cloak. If the ambient materials are $\varepsilon_a$ and $\mu_a$, then an ideal cloak is obtained by constructing the materials $\varepsilon = \mathcal{J}\varepsilon_a\mathcal{J}'/\det \mathcal{J}$ and $\mu = \mathcal{J}\mu_a\mathcal{J}'/\det \mathcal{J}$ in the cloak, where $\mathcal{J}$ is the Jacobian matrix of the transformation. In our previous work~\cite{Hashemi11} we showed that the cloaking problem at one frequency becomes increasingly difficult as the size of the object being cloaked increases. In particular, we proved that the imperfections due to absorption losses and random fabrication disorder must decrease asymptotically with the object diameter in order to maintain ``good'' cloaking performance: for the total (scattering + absorption) cross-section to be less than a given fraction $f$ of a geometric cross section $s_g$. To prove these results, we assumed bounds on the attainable refractive index contrast in the cloak, $b<n/n_a<B$ ($n=\sqrt{\varepsilon\mu}$), which we showed to be equivalent to bounds on the singular values of $\mathcal{J}$. Using these bounds, we were then able to bound the field amplitudes in the cloak in terms of the field amplitudes in the virtual space (which are a constant for incident planewaves); such bounds, in turn, impose bounds on the allowed absorption losses and random disorder. In particular, we showed in the case of absorption losses that $\Delta \Im \varepsilon$ scales proportionally to $\frac{s_g}{V'_c}$, a measure of the inverse diameter.
We now wish to analyze the effect of other imperfections, especially inevitable material dispersion, on the cloaking performance, even in the idealized case of perfect fabrication and negligible absorption. In particular, we suppose that one is interested in the \emph{average} total cross section $\sigma_\text{tot}$ over a bandwidth $\Delta\omega$ around an operating frequency $\omega_\text{op}$, which (for reasons described below) is convenient to define via a Lorentzian averaging weight as:
\begin{equation}
\langle\sigma_\text{tot} \rangle_\Delta \omega = \int_{-\infty}^{\infty} \sigma_\text{tot}\frac{\Delta \omega/\pi}{(\omega-\omega_\text{op})^2+\Delta \omega^2}d\omega
\label{meanscat}
\end{equation}
As we increase the diameter $d$ of the object and cloak, keeping the materials and transformation mapping fixed and simply rescaling the whole system, we show that the cloaking problem becomes increasingly difficult, in the sense that $\langle \sigma_\text{tot} \rangle_{\Delta \omega}/s_g \sim d$ (where $\sim$ means proportional to), with $s_g$ being a geometric cross sectional area of the object. (This scaling breaks down when $\sigma_\text{tot} / s_g$ is no longer small, i.e. when it is no longer an approximate cloak.) The only assumptions are that the ambient medium $\varepsilon_a$ and $\mu_a$ is approximately lossless and dispersionless over the bandwidth of interest (e.g. for cloaking in air) and that the attainable refractive index contrast (eigenvalues of $\sqrt{\varepsilon \mu / \varepsilon_a \mu_a}$) is $\leq B$ for some finite bound $B$ (similar to our previous work~\cite{Hashemi11}).
Our analysis in the following subsections constitutes the following steps. First, in \secref{fa}, using complex analysis combined with the optical theorem, we relate the frequency average of \eqref{meanscat} to a \emph{single} scattering problem at a \emph{complex} frequency so that we no longer need to consider the cross section at many frequencies at once. Second, in order to understand the precise meaning of this complex-frequency scattering problem, in \secref{optical} we derive a variant of the optical theorem that is particularly easy to analyze. Third, in \secref{freqtomat} we relate this complex-frequency scattering problem to an equivalent scattering problem at a \emph{real} frequency with transformed complex \emph{materials}. Fourth, in \secref{dispersion} we use analysis similar to our proof in~\cite{Hashemi11} to show the ``losses'' introduced by these effective complex materials must scale with diameter, and hence mean $\sigma_\text{tot}$ must scale with diameter. Finally, for the special case of a bandwidth-limited cloak (a cloak that is very good at one frequency but spoiled at other frequencies by material dispersion), we show that the bandwidth must narrow inversely with diameter in \secref{bandwidth-scaling}.
\begin{figure}[t]
\centering
\includegraphics[width=1.0\columnwidth]{cloak-iso2}
\caption{Schematic of transformation-based isolated-object cloak, which works by mapping the object and cloak in physical space $X$ to a virtual space $X'$ in which the object is a single point. The transformation laws of Maxwell's equations~\cite{Ward96, Pendry06} then dictate the required $\varepsilon$ and $\mu$ materials in the cloak volume to produce equivalent solutions in $X$ and $X'$.}
\label{fig:cloak-iso}
\end{figure}
\subsection{Frequency average of the scattering cross-section}
\label{sec:fa}
By the optical theorem~\cite{Jackson98}, $\sigma_\text{tot}(\omega) = \Im f(\omega)$ where $f$ is a forward-scattering amplitude (at least for the case of an incident planewave, but a generalization is given in the next section for any incident field), and so
\begin{equation}
\langle \sigma_\text{tot} \rangle_{\Delta \omega}= \Im \int_{-\infty}^{\infty} f(\omega) \frac{\Delta \omega/\pi}{(\omega-\omega_\text{op})^2+\Delta \omega^2}d\omega.
\end{equation}
Because $f$ is a causal linear response, it is analytic for $\Im \omega>0$, and therefore we can perform a contour integration to obtain $\langle \sigma_\text{tot} \rangle_{\Delta \omega}$ in terms of the residue of the single pole of the Lorentzian in the upper-half complex plane:
\begin{equation}
\langle \sigma_\text{tot} \rangle_{\Delta \omega} = \Im f(\omega_\text{op} + i\Delta\omega) .
\end{equation}
This corresponds to a single scattering problem at a complex frequency, analyzed in more detail below.
\subsection{The optical theorem and analytic continuation to complex $\omega$}
\label{sec:optical}
Consider any finite-volume scatterer (in a linear time-invariant system), described by some change $\Delta\varepsilon$ and $\Delta\mu$ in the permittivity and the permeability compared to the ambient medium. Let the incident field (in the absence of the scatterer) be described by a six-component vector field $\psi_{\text{inc}}= \left( \begin{array}{cr} \vec E_{\text{inc}} \\ \vec H_{\text{inc}} \end{array} \right)$. In the presence of the scatterer, this is modified to a new total field $\psi = \psi_{\text{inc}}+\psi_{\text{scat}}$, the sum of the incident and scattered fields. The charge perturbations in the scatterer are described by bound electric and magnetic polarization currents $\Delta\vec J = -i\omega \Delta\varepsilon \vec E$ and $\Delta\vec K = -i\omega \Delta \mu \vec H$, respectively, which can be combined into a six-component current field $\xi = \left(\begin{array}{c}\Delta\vec J\\ \Delta\vec K\end{array}\right)$. Abstractly, we can write $\xi = A \psi_\text{inc}$ for some linear operator $A$ relating the incident fields to the induced currents, and causality (currents come after fields) implies that $A$ is an analytic function in the upper-half complex-$\omega$ plane~\cite{Landau84:electro}.
Physically, the scattered field is the field produced by these
oscillating induced currents $\xi$ in the scatterer, and the
interactions of the currents and fields provide a simple way to
characterize the absorbed and scattered powers. The total absorbed
power $P_{\text{abs}}$, assuming an ambient medium with negligible
dissipation, equals the total time-average ($\psi$) incoming Poynting
flux:
\begin{equation}
\begin{split}
P_\text{abs} &= \frac{1}{2} \Re \int (\vec E^* \cdot \Delta\vec J + \vec H^* \cdot \Delta\vec K) \\
& = \frac{1}{2} \Re \left<\psi,\xi\right> \\
& =\frac{1}{2}\Re\left<\psi_{\text{inc}},\xi\right>+\frac{1}{2}\Re\left<\psi_{\text{scat}},\xi\right>,
\end{split}
\end{equation}
where we have defined the inner product $\left<\cdots,\cdots\right>$.
Similarly, the scattered power is the work done by the currents on the scattered field: $-\frac{1}{2}\Re \left<\psi_{\text{scat}},\xi\right>$. Therefore, the total ($\mathrm{absorbed} + \mathrm{scattered}$) power is:
\begin{equation}
\begin{split}
P_{\text{tot}} & = P_{\text{abs}}+P_{\text{scat}} \\
& = \left(\frac{1}{2}\Re\left<\psi_{\text{inc}},\xi\right>+\frac{1}{2}\Re\left<\psi_{\text{scat}},\xi\right>\right) -\frac{1}{2}\Re \left<\psi_{\text{scat}},\xi\right> \\
& = \frac{1}{2}\Re \left <\psi_{\text{inc}},\xi\right> \\
& = \frac{1}{2} \Re \left<\psi_{\text{inc}}, A\psi_{\text{inc}}\right> \\
& = \Im f(\omega),
\end{split}
\end{equation}
where we have defined $f(\omega) = \frac{i}{2} \left<\psi_{\text{inc}},A\psi_{\text{inc}}\right>$.
The interpretation of $f$ as a forward-scattering amplitude when $\psi_\text{inc}$ is a planewave is actually irrelevant to our proof, but it follows from the fact that $f$ in that case is simply the forward-planewave Fourier component of the field $\sim i\xi/\omega$ corresponding to the currents $\xi$.
The key question, for our purposes, is to understand what $f$ looks like at a complex frequency. Suppose we have homogeneous, lossless ambient medium with an incident planewave $\psi_\text{inc} = \psi_0 e^{i\omega x/c}$ propagating in the $+x$ direction for a constant amplitude $\psi_0$. Then
\begin{equation}
f(\omega) = \frac{i}{2} \left<\psi_0, e^{-i\omega x/c} A(\omega) e^{+i\omega x /c} \psi_0 \right>,
\end{equation}
moving all the $x$ dependence to the right-hand side of the inner product. At a complex frequency $\omega \to \omega + i\gamma$ in the upper-half plane ($\gamma > 0$), $A(\omega+i\gamma)$ is analytic by causality, while the planewave terms are analytic everywhere (and become exponentially decaying or growing waves in space at complex $\omega$). So, $f$ at a complex frequency represents an overlap with an exponentially growing field of the currents produced (via $A$) by an exponentially decaying source. In the next section, we clarify this picture further by relating the scattering operator $A$ at a complex frequency to scattering at a real frequency with complex materials.
In order to perform contour integrations of $f(\omega)$ multiplied by a Lorentzian, it is not enough for $f$ to be analytic, however: it must also be bounded as $|\omega|\to\infty$ so that we can close the contour above. This fact is used extensively in quantum field theory~\cite{Peskin95}. Intuitively, this occurs because the the exponential growth of $e^{+\gamma x/c}$ cancels the exponential decay of $e^{-\gamma x/c}$. Mathematically, as $|\omega|\to\infty$ the $\Delta\varepsilon$ and $\Delta\mu$ must vanish (since all susceptibilities vanish in the limit of infinite frequency~\cite{Landau84:electro, Jackson98}), and so $A$ simplifies: to lowest order for weak scatterers (i.e., in the first Born approximation), $\xi \approx A\psi_\text{inc} \approx \left( \begin{array}{cr} -i\omega \Delta\varepsilon \vec E_{\text{inc}} \\ -i\omega \Delta\mu \vec H_{\text{inc}} \end{array} \right)$, in which case the exponential factors exactly cancel. We will use a similar procedure, below, to analyze the effects of small imperfections in the cloak due to material dispersion.
\subsection{From complex frequencies to complex materials}
\label{sec:freqtomat}
Combining the previous two sections, we can now relate the frequency-averaged scattering cross-section, for an incident planewave (say in the $x$ direction to the solution of a single scattering problem at a single complex frequency:
\begin{equation}
\langle \sigma_\text{tot} \rangle_{\Delta \omega} = \frac{1}{2} \Re \left<\psi_0, e^{-i\omega x/c} A(\omega) e^{+i\omega x /c} \psi_0 \right>,
\label{transformed-scat}
\end{equation}
evaluated at $\omega = \omega_\text{op} + i\Delta\omega$. The $A(\omega)
e^{+i\omega x /c} \psi_0$ represents the induced currents $\xi$, for
the materials evaluated at the complex $\omega$, in response to an
exponentially decaying incident planewave (multiplied by a constant
amplitude $\psi_0$). The interpretation of this problem is simplified
by the fact that a complex frequency is mathematically equivalent to
using modified \emph{materials} at a real frequency $\omega_\text{op}$. In particular,
consider Maxwell's equations in the frequency domain:
\begin{align}
\nabla \times \vec E & = -i\omega\mu\vec H - \vec K , \\
\nabla \times \vec H & = + i \omega \varepsilon \vec E + \vec J ,
\end{align}
where $\vec J$ and $\vec K$ are (free) electric and magnetic current
densities, respectively. For our complex $\omega$, the $i\omega\varepsilon\vec E$
term becomes $i (\omega_\text{op}+ i\Delta
\omega) \varepsilon(\omega_\text{op}+i\Delta \omega) \vec E = i
\left[\varepsilon(\omega_\text{op}+i\Delta\omega).(1+i\frac{\Delta
\omega}{\omega_\text{op}})\right] \omega_\text{op} \vec E$, which looks exactly like the term for
a real frequency $\omega_\text{op}$ at a modified permittivity $\tilde\varepsilon$
\begin{equation}
\tilde{\varepsilon}(\omega_\text{op}) =
\varepsilon(\omega_\text{op}+i\Delta\omega).\left(1+i\frac{\Delta \omega}{\omega_\text{op}}\right).
\end{equation}
Similarly for $\mu \to \tilde{\mu}$.
Thus, operating at a complex frequency is equivalent to \emph{two}
changes in the materials. First, the multiplication by
$1+i\Delta\omega/\omega_\text{op}$ corresponds to an effective absorption added
throughout all space (including the ambient medium). Second we
evaluate $\varepsilon$ and $\mu$ at $\omega_\text{op}+i\Delta\omega$ rather than at
the real frequency $\omega_\text{op}$, which will change their values in the
presence of material dispersion.
\subsection{Consequences of dispersion on frequency-averaged scattering}
\label{sec:dispersion}
So far, we have shown that the problem of finding the frequency-averaged scattering cross section is equivalent to solving a certain scattering problem at a single complex frequency, which in turn is equivalent to a scattering problem at a single \emph{real} frequency $\omega_\text{op}$ with modified complex materials $\tilde\varepsilon$ and $\tilde\mu$. In this section, we analyze the consequences of those effective material modifications, divided into two separate material changes as discussed above, for cloaking performance.
\subsubsection{Irrelevance of artificial absorption}
\label{sec:false}
The first change in the materials is the multiplication of $\varepsilon$ and $\mu$ by $1+ i\frac{\Delta\omega}{\omega_\text{op}}$. But this change does not
hurt the cloaking performance because it is done uniformly \emph{everywhere} in space. More explicitly, if $\varepsilon = \mathcal{J} \varepsilon_a \mathcal{J}^T/\det
\mathcal{J}$ is a valid transformation-based cloak, so is $\varepsilon.(1+\frac{\Delta
\omega}{\omega_\text{op}}) = \mathcal{J} \varepsilon_a \dot (1+\frac{\Delta\omega}{\omega_\text{op}})\mathcal{J}^T/\det\mathcal{J}$,
except that this is now a transformation-based cloak for a lossy ambient medium
$\varepsilon_a. (1+i\frac{\Delta\omega}{\omega_\text{op}})$, and similarly for $
\mu$. Therefore this change in the materials leaves the scattering cross-section invariant.
\subsubsection{Impact of material dispersion}
\label{sec:real}
The second change in the materials is that we need to evaluate $\varepsilon$
and $\mu$ at the complex frequency $\omega_\text{op}+i\Delta\omega$ instead of $\omega_\text{op}$, and here the presence of material dispersion
(unavoidable for any material other than vacuum) acts to spoil the
cloak. In particular, as reviewed in the Appendix, causality and other fundamental principles imply that $\Im \varepsilon$ and $\Im
\mu$ are both strictly \emph{positive} at $\omega_\text{op} + i\Delta\omega$ for $\Delta\omega>0$~\cite{Landau84:electro}, corresponding to an unavoidable
additional absorption in the complex-frequency scattering problem. Unlike the
artificial absorption in the previous section, this is an absorption
\emph{defect} introduced only in the cloak: $\varepsilon$ (both $\Im$ and $\Re$) differs from the cloaking transformation of the ambient medium by
$$
\Delta\varepsilon = \left[ \varepsilon(\omega_\text{op} + i\Delta\omega) - \varepsilon(\omega_\text{op})\right] \left(1+ i\frac{\Delta\omega}{\omega_\text{op}}\right),
$$
and similarly for $\mu$. The key assumptions here are that the ambient medium has no dispersion over the given bandwidth, so that the ideal cloaking transformation at the complex $\omega$ is given by the second term in $\Delta\varepsilon$, and that the ambient medium is lossless, so that $\varepsilon(\omega_\text{op})$ is real and $\Im\Delta\varepsilon > 0$. More generally, it is sufficient for the dispersion and loss of the ambient medium to be small enough, compared to the dispersion of the cloak materials, such that $\Im\Delta\varepsilon$ is $>0$.
We can analyze the consequences of this effective absorption imperfection similarly to our previous work~\cite{Hashemi11}
It is convenient to first transform the imperfections to virtual space (as shown in \figref{cloak-iso}) where the scattering problem is easier, because in the absence of $\Delta\varepsilon$ and $\Delta\mu$ we have a planewave in a homogeneous medium. In particular, given an upper bound $B$ on the index contrast as discussed above, one obtains $\Delta \varepsilon' \geq \Delta \varepsilon / B$ in virtual space, and similarly for $\mu$~\cite{Hashemi11}. Furthermore we are only interested in the regime in which we have a good cloak---once the imperfections become so large as to make the cloak useless, all of the scaling relations break down and the problem is no longer interesting. This is the regime in which $\Delta\varepsilon$ and $\Delta\mu$ are small, and therefore virtual space at complex $\omega$ is equivalent to a homogeneous medium with small imperfections and perturbative methods are applicable. In particular, the lowest-order scattering current (in virtual space) is simply $\vec J' = -i\omega_\text{op} \Delta \varepsilon' \vec E_\text{inc}$, and similarly for $\vec K'$. In the notation of \secref{optical}, $\xi' = A'\psi_\text{inc} \approx \left( \begin{array}{cr} -i\omega \Delta\varepsilon' \vec E_{\text{inc}} \\ -i\omega \Delta\mu' \vec H_{\text{inc}} \end{array} \right)$. Substituting this into \eqref{transformed-scat}, we find that the exponential factors $e^{\pm \Delta\omega x/c}$ exponential factors exactly cancel, leaving:
\begin{equation}
\langle \sigma_\text{tot} \rangle_{\Delta \omega} \approx \frac{1}{2} \int_{V_c'} \left[ |\vec E_0|^2 \Im \Delta\varepsilon'
+ |\vec H_0|^2 \Im \Delta\mu' \right] .
\end{equation}
This has two main consequences. First, $\operatorname{mean}\sigma_\text{tot} > 0$ for $\Delta\omega>0$ since $\Im \Delta\varepsilon$ and $ \Im \Delta\mu$ are strictly positive as noted above: even if the cloak is a perfect cloak at $\omega_\text{op}$, it is imperfect when averaged over any non-zero bandwidth. This is, therefore, an alternative proof of the results of Pendry~\cite{Pendry06} and Miller~\cite{Miller06} that cloaking of isolated objects over a non-zero
bandwidth is impossible for physical, causal materials. Second, exactly as we showed for other imperfections in previous work~\cite{Hashemi11}, it immediately follows that $\langle \sigma_\text{tot} \rangle_{\Delta \omega}$ grows $\sim V_c' \sim V_c \sim V_o$ and hence the frequency-averaged cloaking efficiency $\langle \sigma_\text{tot} \rangle_{\Delta \omega} / s_g$ scales proportionally to a mean diameter $V_o / s_g$. This is the central result of our proof, but
to better understand its consequences we consider some special cases
in the next section.
\subsection{Scaling of cloaking bandwidth with diameter}
\label{sec:bandwidth-scaling}
As shown in the previous section, the fractional cross-section $\operatorname{mean}
\sigma_\text{tot} / s_g$, averaged over a bandwidth $\Delta\omega$ around $\omega_\text{op}$, must scale
proportional to the diameter $d$, at least as long as $\langle \sigma_\text{tot} \rangle_{\Delta \omega} /
s_g \ll 1$ (i.e. until cloaking breaks down completely). There are
two possible sources of this linear scaling, depending on whether
$\langle \sigma_\text{tot} \rangle_{\Delta \omega}$ is limited by the cross-section at $\omega_\text{op}$ (due to imperfections in the cloak at the design frequency) or by the bandwidth of a dip in the cross-section around
$\omega_\text{op}$ (due to material dispersion degrading a near-perfect single-frequency cloak). In the former case, the physical mechanism is simply that the
losses due to imperfections at $\omega_\text{op}$ scale with diameter, as we already proved in \citeasnoun{Hashemi11}. In the latter case, however, it leads to a new prediction:
in a bandwidth-limited cloak, the bandwidth must narrow as the object
diameter increases. In fact, we show in this section that the bandwidth generically narrows inversely with the diameter in this case.
First, let us consider the bandwidth scaling from generic dimensional
considerations. Because of material dispersion, one expects good
cloaking to only be possible in some limited bandwidth $\sim \Gamma$
around some design frequency $\omega_\text{op}$, in which case a small $\operatorname{mean}
\sigma_\text{tot}/s_g$ requires $\Delta\omega \ll \Gamma$. If we Taylor-expand
$\langle \sigma_\text{tot} \rangle_{\Delta \omega}/s_g$ in $\Delta\omega / \Gamma \ll 1$, we would generically expect an expansion of the form:
$$
\langle \sigma_\text{tot} \rangle_{\Delta \omega}/s_g = \sigma_\text{tot}(\omega_\text{op})/s_g + C \Delta\omega/\Gamma + O[(\Delta\omega/\Gamma)^2]
$$ for some coefficient $C$, where $C$ is generically $>0$ if $\omega_\text{op}$ is chosen to be a minimum of $\sigma_\text{tot}$. For sufficiently small
bandwidths $\Delta\omega \ll \Gamma \sigma_\text{tot}(\omega_\text{op})/s_g$, this is
dominated by the scattering at $\omega_\text{op}$, which scales linearly with
diameter in the presence of imperfections as shown in \citeasnoun{Hashemi11}. On the other hand, for a good single-frequency cloak at
$\omega_\text{op}$, there is a regime $\Gamma \sigma_\text{tot}(\omega_\text{op})/s_g \ll \Delta\omega \ll
\Gamma$ where the second term dominates, i.e. where material
dispersion is the limiting factor. The $C \Delta\omega/\Gamma$ term must therefore also scale linearly
with the diameter in order for $\langle \sigma_\text{tot} \rangle_{\Delta \omega}/s_g$ to scale linearly for such $\Delta\omega$,
and hence we can conclude that the bandwidth $\Gamma$ of the cloak
generally \emph{scales inversely with diameter}.
This is best illustrated by a simple example. Consider the case where
$\sigma_\text{tot}/s_g$ achieves a minimum $f \ll 1$ at $\omega_\text{op}$ with an approximately Lorentzian lineshape of width $\Gamma$, going to~1 at frequencies far from $\omega_\text{op}$:
\begin{equation}
\frac{\sigma_\text{tot}(\omega)}{s_g} = 1 - \frac{\Gamma^2} {(\omega-\omega_\text{op})^2 + \Gamma^2}(1-f).
\end{equation}
The integral of \eqref{meanscat} for this $\sigma_\text{tot}(\omega)$ can be evaluated analytically to obtain
$$
\langle \sigma_\text{tot} \rangle_{\Delta \omega} = 1 - \frac{1-f}{1+\Delta\omega/\Gamma} \approx f + (1-f) \Delta\omega/\Gamma + O[(\Delta\omega/\Gamma)^2].
$$ Since this must scale linearly with diameter for any $f \ll 1$ and
any $\Delta\omega\ll\Gamma$, it follows that both $f$ and $1/\Gamma$
scale linearly with diameter until cloaking breaks down.
\section{Numerical example}
\label{sec:num}
To illustrate and validate our predictions of this scaling of cloaking
bandwidth, we performed explicit numerical calculations of the scaling for an example bandwidth-limited spherical-cloaking problem (near-perfect at one frequency, but degraded by causal dispersion at other frequencies) in vacuum (with nondimensionalized units $\varepsilon_a = \varepsilon_0 = 1$ and $\mu_a = \mu_0 = 1$).
At an operating frequency $\omega_\text{op}$, we use an exact ``Pendry'' cloak~\cite{Pendry06}: a sphere of radius $R_1$ is surrounded by a
cloak of radius $R_2 > R_1$ that is linearly mapped to an empty sphere
($R_1' = 0$), resulting in materials:
$$\varepsilon_r(\omega_\text{op})= \mu_r(\omega_\text{op}) = \frac{R_2-R_1}{R_2}\left(\frac{r-R_1}{r}\right)^2$$
$$\varepsilon_\theta(\omega_\text{op}) = \varepsilon_\phi(\omega_\text{op}) = \mu_\theta(\omega_\text{op}) =
\mu_\phi(\omega_\text{op}) = \frac{R_2}{R_2-R_1}.$$ We fixed $R_2 = 1.5 R_1$, so that the cloaking material
parameters are the same for all $R_1$, merely rescaled in space as the object becomes larger or smaller.
By construction, at $\omega_\text{op}$ we have a \emph{perfect} cloak, and the only limiting factor is the bandwidth: we use idealized lossless materials but with
causal dispersion relations that satisfy the Kramers--Kronig
constraints. In particular, we use a combination of two limiting
cases: a plasma model (a limit of a Drude model as losses go to zero)
and a limit of lossless Lorentzian resonance (corresponding to a
polarization field described by a lossless harmonic oscillator) at a
frequency $\omega_0$. Combined with the prescribed Pendry values at
$\omega_\text{op}$ from above, this results in dispersion relations:
$$\varepsilon_r(r, \omega) = \mu_r(r, \omega) = 1 - \left[1 - \frac{R_2}{R_2-R_1}\left(\frac{r-R_1}{r}\right)^2\right]\frac{\omega_\text{op}^2}{\omega^2} ,$$
$$\varepsilon_\theta=\varepsilon_\phi=
\mu_\theta=\mu_\phi= 1 +
\frac{R_1}{R_2-R_1}\frac{\omega_0^2 - \omega_\text{op}^2}{\omega_0^2-\omega^2} .$$
The Lorentzian resonance frequency $\omega_0$ can be chosen
arbitrarily; we used $\omega_0 = 2\omega_\text{op}$.
This geometry was then simulated using a spectral (spherical-harmonic
expansion) scattering-matrix method as described in
\citeasnoun{QiuHu09}. The continuously varying anisotropic material
parameters are approximated by a large number of piecewise-homogeneous
isotropic layers~\cite{QiuHu09}. The total scattering cross-section
was computed over a range of frequencies for $R_1 = \lambda_\text{op}, 2\lambda_\text{op},
4\lambda_\text{op}$ (where $\lambda_\text{op} = 2\pi c /\omega_\text{op}$), and is plotted in \figref{cs-w}.
The results in \figref{cs-w} are converged with resolution (the number
of spherical layers) to within a few percent accuracy. At $\omega_\text{op}$, the
cloak should theoretically be perfect, but we obtain a small nonzero
$\sigma/s_g$ ($< 10^{-3}$) due to the discretization errors, which
vanishes with increasing resolution.
\begin{figure}[t]
\centering
\includegraphics[width=1.0\columnwidth]{cs-w-db}
\caption{Relative cross-section versus frequency for a spherical cloak
designed to be a perfect Pendry cloak at $\omega_\text{op}$ and showing the
effects of material dispersion at other frequencies, computed by a
spectral scattering-matrix method. As predicted, the cloaking bandwidth
decreases linearly with the object radius, for three object radii relative to $\lambda_\text{op} = 2\pi c/\omega_\text{op}$.}
\label{fig:cs-w}
\end{figure}
As expected, the material dispersion prevents this from being a good
cloak except at frequencies in a narrow bandwidth around $\omega_\text{op}$, and
this bandwidth becomes narrower as the diameter increases.
Quantitatively, if we look at the bandwidth at a fixed $\sigma/s_g$ of
about 1/4 its maximum, we find that the bandwidths for $R_1=2\lambda_\text{op}$ and
$R_1=4\lambda_\text{op}$ are $\approx 1/2.1$ and $1/4.1$ times the bandwidth for $R_1 =
\lambda_\text{op}$, respectively, almost exactly the predicted linear scaling.
[Far from $\omega_\text{op}$, one expects $\sigma_\text{tot}$ to approach twice the total geometric cross section, in the limit of a large scatterer, because in this limit all of the incident light is scattered. The factor of two comes from the definition of scattering cross section~\cite{Jackson98}, in which the scattered power appears twice: once as the scattered waves propagating in other directions, and once as the ``shadow'' canceling the forward-propagating wave. Here, the total geometric cross-section is that of the cloak, $\pi R_2^2$, so we expect $\sigma_\text{tot}/s_g \approx 10 \log \left[2 (\pi R_2^2) / (\pi R_1^2)\right] \approx 6.5$ (dB) away from $\omega_\text{op}$ for $R_1\to\infty$, and this is roughly what is seen in \figref{cs-w}.]
\section{Concluding remarks}
$\langle \sigma_{tot}\rangle_{\Delta\omega}$
In this work, we extended on our previous paper~\cite{Hashemi11} to study the bandwidth limitation of isolated-object cloaking which is a key limiting factor for this type of cloaking~\cite{Pendry06, Miller06}. Although it was known that perfect cloaking was impossible over a nonzero bandwidth, this result did not seem to exclude the possibility of imperfect cloaking over a finite bandwidth. Indeed, imperfect finite-bandwidth cloaking is possible, but we have now shown that it is subject to a severe practical constraint: the bandwidth inevitably narrows proportional to the object diameter, given fixed materials. Although we cannot infer any hard upper bounds on the size or bandwidth of such cloaking without further information about the attainable materials, this result indicates a fundamental challenge in scaling up small experimental demonstrations to larger cloaks.
The use of gain has been proposed to compensate for loss problems in
cloaking~\cite{Han11, Wang10}. Although gain is necessarily nonlinear and can be detected by a sufficiently strong incident field, in the idealization
of linear gain then many of the techniques in this paper are
complicated by the fact that a linear-gain resonance (the
complex-conjugate of an absorption resonance) would be non-analytic
(and have negative imaginary parts) in the upper-half
complex-frequency plane. (The average $\sigma_\text{tot}$ must also be replace by
the average $|\sigma_\text{tot}|$ or similar, since gain can produce a $\sigma_\text{tot} <
0$.) However, the bandwidth limitations described in this paper arise
even for idealized materials with negligible dissipation loss, due to
dispersion in the real part of the $\epsilon$ and $\mu$ alone, in
which case even idealized linear gain is inapplicable. (Furthermore,
as pointed out in our previous work~\cite{Hashemi11}, gain
compensation of absorption must become increasingly perfect as the
object diameter increases, nor does it compensate for scattering from
fabrication disorder.)
Ground-plane cloaking does not suffer from any intrinsic limitation on its bandwidth from causality, nor does our proof apply in that case. The reason our proof does not apply (at least, in the present form) to ground-plane cloaking is the figure of merit is no longer $\sigma_\text{tot}$: a ground-plane is actually \emph{designed} to reflect waves, albeit to reflect them in a way that mimics the ground plane. On the other hand, our previous work~\cite{Hashemi10,Hashemi11} showed that even ground-plane cloaks are increasingly sensitive to imperfections as the size of the object increases. This suggests that ground-plane cloaks should also be increasingly sensitive to material dispersion, for cloaking over a finite bandwidth, as the object size increases, and a challenge for future work is to quantify (or disprove) this relationship.
An alternative direction for cloaking theory (and experiment) is to consider relaxations of the cloaking
problem that might prove more practical. In particular, it would be
valuable to make precise the intuition that the cloaking problem
becomes easier if the incident waves are restricted (e.g. to
plane waves from a certain range of angles) and/or the observer is
limited (e.g. only scattered waves at certain angles are visible, or
only amplitude but not phase can be detected, or sufficiently small time delays are undetectable), since this is arguably
the situation in most experiments. For example, current ``stealth''
aircraft are designed in the radar regime mainly to reduce
back-scattering only~\cite{Nicolai10}. So, it is clear that a sufficiently relaxed cloaking problem is practical even for large objects, and one interesting goal is to find the ``weakest'' relaxation that remains practical at useful scales.
\begin{acknowledgments}
This work was supported in part by the Army Re- search Office through the Institute for Soldier Nanotechnologies (ISN) under contract W911NF-07-D-0004, and by the AFOSR Multidisciplinary Research Program of the University Research Initiative (MURI) for Complex and Robust On-chip Nanophotonics, Grant No. FA9550- 09-1-0704.
\end{acknowledgments}
\section*{Appendix}
Here, we review a known consequence of causality that is the key to
our analysis above: for a passive medium, $\Im \varepsilon (\omega) > 0$ in
the upper half plane $\Im \omega>0$ as proved in~\citeasnoun{Landau84:electro}. A condensed proof of this fact is as follows:
$\Im\varepsilon$ is analytic in $\omega$ in the upper-half plane by causality, and is therefore a harmonic function in the upper-half plane, and so can obtains its minimum
only on the boundary of its domain, except in the trivial case of vacuum where
it is a constant function ($\Im \varepsilon = 0$). In particular, consider the upper-right quadrant of the complex-$\omega$ plane. Along the positive real axis, $\Im\varepsilon \geq 0$ for a passive material (in the absence of gain), even for idealized lossless materials. Along the positive imaginary axis, $\Im \varepsilon
= 0$ since $\varepsilon(-\omega) = \varepsilon(\omega)^*$ for real-valued physical fields~\cite{Jackson98}. As $|\omega|\to\infty$ one must have $\Im\omega \to 0$. Hence the minimum of $\Im \varepsilon$ along the boundary of the upper-right quadrant is zero and it is strictly positive in the interior.
For physical materials, $\Im \omega > 0$ along the positive real-$\omega$ axis except at $\omega=0$ in order to satisfy the second law of thermodynamics, and this is the usual case in which the above statement is proved~\cite{Landau84:electro}. However, it is also interesting to consider the
idealized limit of lossless materials (such as a plasma model or a
lossless resonance), in order to study bandwidth-averaged cloaking with idealized materials. Our proof, above, works even in this case as long as one is a little cautious about the case of poles in $\varepsilon$ lying exactly on the real-$\omega$ axis, in order to exclude the case where $\Im \varepsilon$ diverges to $-\infty$ as the real axis is approached from above. In particular, we must restrict ourselves to idealized materials that are the limit of physical lossy materials as the losses go zero, so that we are taking the limit as poles approach the real axis from below. In this case, $\Im \varepsilon$ in the upper-right quadrants is the limit of a strictly positive quantity and hence cannot go to $-\infty$.
|
{
"timestamp": "2012-03-13T01:00:14",
"yymm": "1203",
"arxiv_id": "1203.2190",
"language": "en",
"url": "https://arxiv.org/abs/1203.2190"
}
|
\section{The Gabriel Quiver of $\mathbf{\B_Q}$}
As before let $k$ be a field and $Q$ a finite, acyclic quiver. Theorem \ref{extkoecher-theo-Extkoecher} is a direct consequence of the next two subsections.
One remark before we start. In \cite{DHST} the authors determined among other things the radical and Gabriel quiver of the monoid algebra of a finite $\mathcal{J}$-trivial monoid. This could be applied here, because $\B_Q$ is -- as the monoid algebra of the Hecke-Kiselman monoid $HK_Q$ -- such a monoid algebra. We repeat the argument mentioned in $\cite{MazorchukKiselman}$: Since being \mbox{$\mathcal{J}$-trivial} is closed under quotients (see \cite{Pin}), it suffices to show, that $HK_Q$ is a quotient of a $\mathcal{J}$-trivial monoid. Now the Hecke-Kiselman monoid $HK_{K_n}$ associated to the quiver $K_n$ with $n$ vertices $\{1,\ldots,n\}$ and arrows $i\rightarrow j$ for each pair $i<j$ is the Kiselman semigroup and $\mathcal{J}$-trivial by \cite{MazorchukKudryavtseva}. Moreover $Q$ can be embedded in the quiver $K_n$ for $n\ldef |Q_0|$ by choosing an enumeration $\{1,\ldots, n\}$ of the vertices $Q_0$ such that $i\rightarrow j$ implies $i<j$. Thus there is the canonical projection introduced in \cite{MazorchukKiselman} from $HK_{K_n}$ onto $HK_{Q}$.
Here, we compute the radical and the Gabriel-quiver of $\B_Q$ directly just using the defining relations.
\subsection{The simples and the radical of $\mathbf{\B_Q}$}\label{extkoecher Abschnitt Einfache und Radikal}
The structure of the simple modules are closely related to those of the $0$-Hecke algebra (see \cite{Norton} and \cite{Hivert}). For every subset $M$ of $Q_0$ we define $\E_M=({k},\delta_M)$ \label{extkoecher-die Einfachen} to be the (one-dimensional) $\B_Q$-module given by the homomorphism $\delta_M\colon \B_Q \longrightarrow {k}\kong{}\End{k}{k}$ of algebras with $X_q\mapsto 1$ if $q\in M$ and $X_q\mapsto 0$ otherwise for all $q\in Q_0$.
In the sequel we compute the radical to show that this family $(\E_M)_{M\subseteq Q_0}$ of $2^{|Q_0|}$ simple modules represents all simple $\B_Q$-modules. For this we construct for each subset $M$ of $Q_0$ a specific monomial \mbox{$X_M \in \B_Q$} to describe a $k$-linear generating set of the radical. To this end we consider the inductively defined sets of sinks $\senke{M}{j}$ at level $j\in \NN$ associated to $M$:
\[
\begin{aligned}
\senke{M}{0} &\ldef \set{q\in M}{ q \text{ is a sink in } Q_M}\\
\vdots&\\
\senke{M}{j+1} &\ldef \set{q\in M}{q \text{ is a sink in } Q_{M\ohne(\senke{M}{0}\cup \senke{M}{1}\cup\ldots \cup \senke{M}{j})}}
\end{aligned}
\]
Since $M$ is finite, there is a uniquely determined index $s(M)\ldef m$ such that \mbox{$\senke{M}{m}\neq \emptyset = \senke{M}{m+1}$} holds. Moreover $Q_{\senke{M}{j}}$ contains no arrows. Hence all $X_p$ and $X_q$ with $p,q \in \senke{M}{j}$ commute and we can thus define:
\[
X_M \ldef \prod_{q\in \senke{M}{m}} \!\!\!X_q \:\:\ldots \prod_{q\in \senke{M}{0}} \!\!\!X_q
\]
Note that by the generalized relations $X_M$ is idempotent, since we have $X_q X_M = X_M = X_M X_q$ for every $q\in M$.
\begin{prop}\label{extkoecher prop Radikal und alle Einfachen}
The radical $\rad{\B_Q}$ of $\B_Q$ is the $k$-linear span of \mbox{$\mathcal{M}\ldef\set{X_{\{w\}}-X_w}{w\in \free{Q}}$}. So $\B_Q$ is a basic algebra and $(\E_M)_{M\subseteq Q_0}$ is a representative system of its simple modules.
\end{prop}
\begin{proof}
Let $\mathcal{V}$ be the $k$-linear span of $\mathcal{M}$ and $\mathcal{I}$ be the intersection of the annihilators \linebreak\mbox{$\Ann{\B_Q}{\E_M}\ldef\set{a\in \B_Q}{\delta_M (a) = 0}$} with $M\subseteq Q_0$. Firstly we observe that $\mathcal{V}$ coincides with the ideal $\mathcal{I}$: by the definition of $\delta_M$ we have for an arbitrary element $b=\sum_{v \in \free{Q}} b_v X_v$ in $\B_Q$:
\[
\delta_{M}(b) = \sum\nolimits_{v \in \free{Q}, \:\:\{v\} \subseteq M} b_v
\]
Thus $b$ lies in $\mathcal{I}$ if and only if $\sum_{v \in \free{Q},\:M=\{v\}}\: b_v \:= 0$ for all $M\subseteq Q_0$, which is in turn equivalent to
\[
b= \sum\nolimits_{M\subseteq Q_0}\sum\nolimits_{v\in \free{Q},M= \{v\}} b_v (X_v-X_M) \in \mathcal{V}
\]
Secondly we show, that $\mathcal{M}$ consists of nilpotent elements. (Hence $\mathcal{V}$ is a nilpotent ideal by a theorem of Wedderburn (see \cite{Pierce}, 4.6)). For this we prove the equality $X_w^{s} = X_{\{w\}}$ for each word $w$ over $Q_0$ and $s\ldef s(\{w\})$ by an induction on $s$: if all the letters occuring in $w$ correspond to sinks, $X_w$ already coincides in $\B_Q$ with $X_{\{w\}}$. So now assume $\senke{\{w\}}{1}\neq \emptyset$. Furthermore let $v$ be the subword of $w$, which arises from $w$ by canceling all sinks $x\in \senke{\{w\}}{0}$ in $w$. Since $s(\{v\})=s-1$, it follows inductively $X_w^{s} = X_v^{s-1}X_w= X_{\{v\}}X_w= X_{\{w\}}$ by the generalized relations and the properties of $X_{\{v\}}$ respective $X_{\{w\}}$. Therefore the element $X_{\{w\}}-X_w\in \B_Q$ is nilpotent:
\[
(X_{\{w\}}- X_w)^{s} = (-X_{\{w\}} +X_w^2)(X_{\{w\}}- X_w)^{s-2}= \ldots = (-1)^{s-1}(X_{\{w\}}- X_w^{s})=0
\]
Now $\mathcal{V}\subseteq \rad{ \B_Q} \subseteq \mathcal{I}$ follows from the different characterizations of the radical of a finite dimensional algebra.
\end{proof}
\subsection{The Gabriel quiver of $\mathbf{\B_Q}$} \label{extkoecher Abschnitt Berechnund des Gabriel Koechers}
We calculate the $k$-dimensions of the extension groups $\Ext{\B_Q}{1}{\E_M,\E_N}$, i.e.\\ the number of arrows from $[\E_M]$ to $[\E_N]$ in the Gabriel quiver $\extko{\B_Q}$ of $\B_Q$. Then the algebra $\B_Q$ is a quotient of the path algebra $k\extko{\B}$ by an ideal $I$ with $\rad{\B_Q}^2\subseteq I \subseteq \rad{\B_Q}^r$ for some $r\in \NN$. We will see that this ideal is zero for the $m$-subspace quiver as well as for some simply shaped quivers. We devote section 5 to the proof that $I$ is generated by the commutativity relations of $\extko{\B_Q}$, if $Q=Q_n$ is the linearly oriented Dynkin quiver of type $A$.
Since the simple modules are one dimensional, it suffices to determine the two dimensional $\B_Q$-modules. The calculations are similar to those for the $0$-Hecke-algebras (type $A$) -- as done for example in \cite{Fayers} with the difference that we have to respect the non-symmetry of the defining relations and the generalisation to finite, acyclic quivers.
Let $M$ and $N$ be subsets of $Q_0$. We consider the characteristic tuples $(m_q\ldef\delta_M(X_q))_{q\in Q_o}$ and $(n_q\ldef \delta_N(X_q))_{q\in Q_0}$ over $0,1 \in k$. Then we call a tuple $a=(a_q)_{q\in Q_0}$ over $k$ (or a function $a\colon Q_0\longrightarrow k, q\mapsto a_q$) admissible or $(M,N)$-admissible, if the assignment
\[
X_{q} \mapsto \begin{pmatrix} n_q & a_q \\ 0 & m_q \end{pmatrix}\rdef A_q \in {k}^{2\times 2}
\]
extends uniquely to an homomorphism from $\B_Q$ to $k^{2\times 2}$. We receive a two-dimensional $\B_Q$-module $W(a,M,N)=W(a)$ and up to equivalence the short exact sequences $\eta$ in $\Ext{A}{1}{\E_M,\E_N}$ are the sequences $\eta_a$:
\begin{center}
\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\matrix(m) [matrix of math nodes, row sep=2em, column sep=2em, text height=1.5ex, text depth=0.25ex]
{\eta_a: & 0 & \E_N & W(a) & \E_M & 0\\};
\path[->,font=\scriptsize]
(m-1-2) edge (m-1-3)
(m-1-3) edge node[above]{$\begin{pmatrix} 1\\ 0 \end{pmatrix}$ } (m-1-4)
(m-1-4) edge node[above]{$\begin{pmatrix} 0 & 1 \end{pmatrix}$} (m-1-5)
(m-1-5) edge (m-1-6);
\end{tikzpicture}
\end{center}
The dimension of $\Ext{\B_Q}{1}{\E_M,\E_N}$ could be described in terms of the following property for the set-theoretic differences $M\ohne N$ and $N\ohne M$: \label{extkoecher-Differenzmenge}
\begin{df}\label{extkoecher-stark verbunden}
Let $P$ and $R$ be two disjoint subsets of $Q_0$. We say that ``$P$ is strongly connected towards $R$`` and write $P\stver R$, if neither $P$ nor $R$ are empty and for every two vertices $p\in P$ and $r \in R$ there exists (at least) one arrow from $p$ to $r$ (in $Q_1$). \footnote{In this case the subquiver of $Q$ with the vertices $P\cup R$ and just those arrows from $Q_1$, which connect vertices from $P$ to vertices from $R$, is a completely bipartite quiver.}
\end{df}
If $P$ is not strongly connected towards $R$ we write $P\nstver R$. Since $Q$ is acyclic, $P\nstver R$ holds if and only if $P=\emptyset$, $R=\emptyset$ or there are vertices $p\in P$ and $r\in R$, such that $p$ is a sink in $Q_{p\konk r}$.
\begin{lem}
(a)\quad We have $\Ext{\B_Q}{1}{\E_M,\E_N} = 0$ if and only if $M\ohne N \nstver N\ohne M$ holds.\newline
(b)\quad We have $\dim_{k}\Ext{\B_Q}{1}{\E_M,\E_N} = 1$ if and only if $M\ohne N \stver N\ohne M$ holds.
\end{lem}
\begin{proof}
Firstly we determine the $(M,N)$-admissible functions. Let $a\colon Q_0\longrightarrow k, q\mapsto a_q$ be a function. The shape of the assigned $A_q$ depends on whether $q\in M\cap N$,\: $q\in Q_0\!\setminus\!(M\cup N) $,\: $q\in M \!\setminus\! N$\: or \: $q\in N\!\setminus\! M$ and is as follows
\[
A_q=\begin{pmatrix} 1 & a_q \\ 0 & 1 \end{pmatrix} \text{,}\quad A_q=\begin{pmatrix} 0 & a_q \\ 0 & 0 \end{pmatrix} \text{,}\quad A_q=\begin{pmatrix} 0 & a_q \\ 0 & 1 \end{pmatrix} \text{,}\quad A_q=\begin{pmatrix} 1 & a_q \\ 0 & 0 \end{pmatrix}
\]
respectively. If $a$ is an admissable function, then all these matrices are idempotent, i.e.\ $a_q=0$ for all $q\in M\cap N$ and for all $q\in Q_0\!\setminus\!(M\cup N)$. Therefore every $(M,N)$-admissible function lies in
\[
\mathcal{F}\ldef \set{a\colon Q_0\rightarrow k}{a|_{M\cap N} = 0\,\text{ and }\quad a|_{Q_0\ohne (M\cup N)} = 0}
\]
and thus it suffices to consider the remaining relations, as for example $A_pA_qA_p=A_qA_pA_q$, just for all $p,q \in M \!\setminus\! N \cup N\!\setminus\! M$.
For example for any two elements $p,q$ in $M \!\setminus\! N$ we conclude $a_p=a_q$ from the conditions, that $A_pA_q$ equals one of the products $A_qA_pA_q$ or $A_qA_p$. Similar considerations show that if $M\ohne N \stver N\ohne M$ holds then the set of all admissible functions is
\[
\set{a \in \mathcal{F}}{\exists\, c_M, c_N\in k: \quad a|_{M\setminus N} = c_M \id_{M\setminus N}\,\text{ and }\quad a|_{N\setminus M} = c_N \id_{N\setminus M}}
\]
Whereas in the case $M\ohne N \nstver N\ohne M$ the existence of $M\!\setminus\! N \ni q \leftarrow p \in N\!\setminus\! M$ yields the equality $A_pA_q = A_pA_qA_p$, i.e. $ a_q+a_p=0 $, hence the set of admissible functions is
\[
\set{a \in \mathcal{F}}{\exists c\in k: \quad a|_{M\setminus N} = c \id_{M\setminus N}\quad \wedge \quad a|_{N\setminus M} = - c \id_{N\setminus M}}
= \langle (m_q-n_q)_{q\in Q_0} \rangle_{k}
\]
So $M\ohne N \nstver N\ohne M$ holds if and only if every $(M,N)$-admissible function lies in \mbox{$\langle (m_q-n_q)_{q\in Q_0} \rangle_{k}$}. Hence it suffices to show that $\Ext{\B_Q}{1}{\E_M,\E_N} = 0$ holds iff every $(M,N)$-admissible function lies in \mbox{$\langle (m_q-n_q)_{q\in Q_0} \rangle_{k}$}. For this let $\eta_0$ denote the trivial short exact sequence and $W(0)$ its middle term; in particular $W(0)\cong \E_M\oplus \E_N$. We show for every admissible function $a$ the equivalence:
\[
(*)\quad \eta_a\sim \eta_0 \Longleftrightarrow a\in \langle (m_q-n_q)_{q\in Q_0} \rangle_{k}
\]
If $c\in k$ and $(a_q = c m_q- cn_q)_{q\in Q_0}$ an admissible function, then a $\B_Q$-homomorphism from $W(a)$ to $W(0)$, which additionally provides $\eta_a\sim \eta_0$, is given by the left multiplication with $\begin{pmatrix} 1 & c \\ 0 & 1 \end{pmatrix}$. On the other hand every homomorphism $\Phi\colon W(a)\longrightarrow W(0)$ providing $\eta_a \sim \eta_0$ is given by a matrix $\begin{pmatrix} 1 & c \\ 0 & 1 \end{pmatrix}$ for a constant $c\in k$ by the commutativity of the corresponding diagram. Since $\Phi$ is in particular a $\B_Q$-homomorphism, it follows $a_p + c m_p = c n_p$ for all $p\in Q_0$. Hence (a) is proved.
Statement (b) follows from (a) if one shows that any two non-trivial short exact sequences are linearly dependent. So let $a$ and $b$ be $(M,N)$-admissible functions such that $[\eta_0]\neq [\eta_a]\neq [\eta_b]\neq [\eta_0]$ holds. By $(*)$ and the previous thoughts there exist $c\neq c'$ resp. $d\neq d'$ in $k$ with
\[
a|_{M\setminus N} = c \id_{M\setminus N}\:\:\text{and }\:\: a|_{N\setminus M} = c' \id_{N\setminus M}\:\:\text{resp. }\:\:b|_{M\setminus N} = d \id_{M\setminus N}\:\:\text{and }\:\:b|_{N\setminus M} = d' \id_{N\setminus M}
\]
Then $e\ldef {d+d'}/{c+c'} \neq 0$ and the left multiplication with $\begin{pmatrix} e & c'e-d' \\ 0 & 1\end{pmatrix}$ is a $\B_Q$-isomorphism from $W(a)$ to $W(b)$ providing
\[
\eta_a\sim e^{-1} \eta_b\colon 0\longrightarrow \E_N \overset{\begin{pmatrix} e & 0 \end{pmatrix}}{\longrightarrow} W(b) \overset{\begin{pmatrix} 0\\ 1 \end{pmatrix}}{\longrightarrow} \E_M \longrightarrow 0
\]
\end{proof}
\subsection{Properties of the Gabriel quiver}\label{subsection properties of Gabriel quiver}
Let $Q$ be a finite acyclic quiver. For this section we abbreviate notation: instead of $[\E_M]$ we just write $M$. Accordingly, the set of vertices of the Gabriel quiver is from now on just the powerset $\pot{Q_0}$ of $Q_0$ . In this notation there is (exactly) one arrow from $M\in \pot{Q_0}$ to $N\in \pot{Q_0}$ in $\extko{\B_Q}$, if $M\ohne N$ is strongly connected towards $N\ohne M$ (w.r.t. $Q$). In particular, the Gabriel quiver has no loops. Obviously, $Q$ can be embedded in $\extko{\B_Q}$. Besides $Q^{op}$ is isomorphic to the full subquiver of $\extko{\B_Q}$ with the vertices $\set{Q_0\ohne \{p\}}{p\in Q_0}$. In general, we have for all $M$ and $N$:
\[
M \rightarrow N \in \extko{\B_Q} \Longleftrightarrow Q_0\ohne N \rightarrow Q_0\ohne M \in \extko{\B_Q}
\]
Therefore the map $\pot{Q_0}\longrightarrow \pot{Q_0},\; M\mapsto Q_0\ohne M$ induces an involution $\iota$ on $\extko{\B_Q}$. In particular we have in the case that $\B_Q$ is already isomorphic to $k\extko{\B_Q}$: $\B_Q\kong{} \B_{Q}^{op} \kong{} \B_{Q^{op}}$.
Now we look at special cases and some examples and end this chapter with general observations.
\begin{itemize}
\item The Gabriel quiver of $\B_{Q_n}$ is described by the following equivalence:
\begin{center}
\textit{There is one arrow $M\rightarrow N$ in $\extko{\B_{Q_n}}$ if and only if there exists exactly one index $i\in \ito{n}$ with $M\ohne N =\{i\}$ and $N\ohne M = \{i+1\}$.}
\end{center}
Therefore only equally large sets are connected in $\extko{\B_{Q_n}}$. Thus the Gabriel quiver of $\B_{Q_n}$ has at least $n+1$ connected components.
Actually $\extko{\B_Q}$ has exactly $n+1$ as we will see.
\item In the $m$-subspace quiver $T_m$ each subset of sources is strongly connected towards the only sink set $\{s\}$. Thus we have:
\begin{center}
\textit{There is one arrow $M\rightarrow N$ in $\extko{\B_Q}$ if and only if there exist two disjoint subsets $M'\neq \emptyset $ and $N'$ of $\ito{m}$ with $M = N'\cup M'$ and $N=N' \cup \{s\}$.}
\end{center}
As in the first special case $\emptyset$ and $(T_m)_0$ are isolated vertices of the Gabriel quiver. Moreover each vertex $\emptyset\neq M\neq (T_m)_0\in\pot{Q_0}$ is either a source (if $M\subseteq \ito{m}$) or a sink (if $s\in M$). Hence there are no paths of length $\geq 2$. Consequently the algebra $k\extko{\B_Q}$ has radical square $0$ and is already isomorphic to $\B_Q$. In particular $\B_Q \cong \B_Q^{op}$.
By the way the only arrow in $\extko{\B_Q}$ which is fixed under the above presented involution $\iota$ is $\ito{n} \rightarrow \{s\}$.
\end{itemize}
We assume $Q$ to be connected. We call $q\in Q_0$ a successor of $p\in Q_0$, if there is a path in $Q$ from $p$ to $q$. By Theorem \ref{extkoecher-theo-Extkoecher} it follows straightforwardly that the sinks (sources) of the Gabriel quiver of $\B_Q$ are precisely the subsets of $Q_0$ which are closed under successors (predecessor) in $Q$. \label{Gabriel quiver remark sources and sinks} Therefore we now know the projective (injectives) amongst the simple $\B_Q$-modules.
We end this section with the proof of Proposition \ref{Gabriel prop connected components}. Successively applying the next lemma shows that the full subquiver $\extko{\B_Q}_j$ of $\extko{\B_Q}$ whose vertices are those subsets of $Q_0$ with exactly $j$ elements is connected for every $j\leq |Q_0|$:
\begin{lem}
For every proper non-empty subset $A$ of $Q_0$, each $a\in A$ and each $b\in Q_0\ohne A$ there exists a walk between $A$ and \mbox{$A\ohne\{a\}\cup \{b\}$} in $\extko{\B_Q}$.
\end{lem}
\begin{proof}
In the sequel we write $x\wa y$, if the vertices $x$ and $y$ are connected by an arrow. Since $Q$ is connected, there is a walk between any two vertices $a$ and $b$ in $Q_0$:
\[
a = x_0 \wa x_1 \wa \ldots \wa x_r \wa x_{r+1} = b
\]
Without loss of generality we can assume, that $x_0, \ldots, x_{r+1}$ are pairwise disjoint. Based on such a walk we construct inductively on the number of the changes from $A$ to $Q_0\ohne A$, i.e.\ on the number $n$ of indices $j\in \oto{r+1}$ with $x_j\in A$ and $x_{j+1}\in Q_0\ohne A$, a walk in $\extko{\B_Q}$ between $A$ and \mbox{$A\ohne\{a\}\cup \{b\}$}.
\newline\case{$n=1$:} Let $a\in A\subseteq Q_0$ and $b\in Q_0\ohne A$, such that there is a walk in $Q$ of the following kind:
\begin{align*}
&\qquad\underbrace{a\wa x_1 \wa \ldots \wa x_{j-1}}_{\in A} \wa \underbrace{x_j \wa \ldots \wa x_{r}\wa b}_{\in Q_0\ohne A}
\shortintertext{Then we obtain the following walk in the Gabriel quiver by Theorem \ref{extkoecher-theo-Extkoecher}:}
A &\wa A\ohne \{x_{j-1}\}\cup \{x_j\} \wa A\ohne \{x_{j-1}\}\cup \{x_{j+1}\}\wa \ldots
\wa A\ohne \{x_{j-1}\}\cup \{x_r\} \wa A\ohne \{x_{j-1}\}\cup \{b\} \\
& \wa A\ohne \{x_{j-2}\}\cup \{b\} \wa \ldots \wa A\ohne \{x_{1}\}\cup \{b\}\wa A\ohne \{a\}\cup \{b\}
\end{align*}
\case{$n\rightarrow n+1$:} Now let $a\in A\subseteq Q_0$ und $b\in Q_0\ohne A$ such that, there is a walk in $Q$ over pairwise disjoint vertices between $a$ and $b$ with more than one change between $A$ and $Q_0\ohne A$. Such a walk is of the kind:
\[
\begin{aligned}
\underbrace{a= x_0\wa\ldots \wa x_{j-1}}_{\in A} &\wa \underbrace{x_j \wa \ldots \wa x_{l-1}}_{\in Q_0\ohne A}
\wa \underbrace{x_l}_{\in A} \wa \ldots \wa \underbrace{x_{r+1} = b }_{\in Q_0\ohne A}
\end{aligned}
\]
with $j-1\geq 0$ and $l-1\geq j$. By the induction hypothesis there exists in $\extko{\B_Q}$ a walk between $A$ and \mbox{$A\ohne \{x_l\} \cup \{b\} \rdef B$}. Since all $x_0,\ldots,x_{r+1}$ are pairwise disjoint the following walk is a walk with just one change (now from $B$ to $Q_0\ohne B$):
\[
\begin{aligned}
\underbrace{a\wa x_1 \wa \ldots \wa x_{j-1}}_{\in B} &\wa \underbrace{x_j \wa \ldots \wa x_{l-1} \wa x_l}_{\in Q_0\ohne B}
\end{aligned}
\]
Again by the induction hypothesis there is a walk in the Gabriel quiver between $B$ and $B\ohne\{a\} \cup \{x_l\} = A\ohne\{a\} \cup \{b\}$ which finishes our walk from $A$ to $A\ohne\{a\} \cup \{b\}$.
\end{proof}
\begin{proof}[Proof of Proposition \ref{Gabriel prop connected components}.]
As we observed above, the Gabriel quiver of $\B_{Q_n}$ has at least $n+1$ connected components. Assertion (a) follows from the last lemma. Now let $Q$ be distinct from any $Q_n$. Therefore $Q$ has a subquiver of the form $x_1\rightarrow s\leftarrow x_2$ or $x_1\leftarrow s \rightarrow x_2$. In each case the subsets $\{x_1,x_2\}$ and $\{s\}$ are strongly connected. Consequently there is an arrow in the Gabriel quiver between $\{x_1,x_2\}$ and $\{s\}$. Thus $\extko{\B_Q}_2$ and $\extko{\B_Q}_1$ are connected. The involution $\iota$ yields an arrow in $\extko{\B_Q}$ between the subsets $Q_0\ohne\{x_1,x_2\}$ and $Q_0\ohne\{s\}$, which connects $\extko{\B_Q}_{n-2}$ and $\extko{\B_Q}_{n-1}$. Now for each subset $D\subseteq Q_0\ohne\{x_1,x_2,s\}$ there is an arrow in $\extko{\B_Q}$ between $D\cup\{x_1,x_2\}$ and $D\cup\{s\}$ linking the subquivers $\extko{\B_Q}_{|D|+1}$ and $\extko{\B_Q}_{|D|+2}$. Hence $\extko{\B_Q}_1, \ldots, \extko{\B_Q}_{n-1}$ are connected. Meanwhile $\extko{\B_Q}_0=\bullet_{\emptyset}$ and $\extko{\B_Q}_{|Q_0|}=\bullet_{Q_0}$.
\end{proof}
\section{Introduction}
The study of $\mathrm{Hom}$-orthogonal subcategories is a classical tool in the representation theory of finite dimensional algebras. The process of passing from the module category to orthogonal subcategories is essential as pointed out for example in \cite{GeigleLenzing}, \cite{Schofield}.
Here the so called projection functors appear naturally. We start by defining them for arbitrary modules $U$ over an associative finite dimensional unital algebra $A$ over a field $k$. We denote by $\mo{A}$ the category of finite dimensional left $A$-modules and by $\gen{U}$ its full subcategory which consists of those modules isomorphic to a quotient of some $U^{\oplus d}$. We first define the endofunctor $\subf{U}$ on $\mo{A}$ by sending a module $M$ to its greatest submodule $\subf{U}(M)$ which lies in $\gen{U}$. This gives a subfunctor of the identity functor $\mathrm{id}_{\mo{A}}$, i.e.\ the embeddings $\iota_M$ of $\subf{U}(M)$ into $M$ for every module $M$ yield a natural transformation $\iota\colon \subf{U} \longrightarrow \mathrm{id}_{\mo{A}}$.
We obtain the so called projection functor $\p{U}\colon\mo{A}\longrightarrow \mo{A}$ by passing to the cokernel $(\pi_M)_{M\in \mo{A}}\colon \mathrm{id}_{\mo{A}}\longrightarrow \p{U}$ of $\iota$.
The question of describing the relations between such functors arises naturally. It fits into the general categorification programmme of realizing Lie-theoretic objects as functors on module categories \cite{Mazorchuk}.
We concretrize this and concentrate just on the multiplicative interplay between certain projection functors. To create the framework we consider the monoid $\monoid{A}$ generated by \mbox{$\set{\p{S}}{S \text{simple}}$} up to natural isomorphism. For this we recall that the composition of endofunctors on $\mo{A}$ induces a multiplication on the isomorphism classes of endofunctors. Hence $\monoid{A}$ is the set of the isomorphism classes $[\p{S_1}\p{S_2}\ldots\p{S_r}]$ with $r\geq 0$ and $S_1,S_2,\ldots, S_r$ simple $A$-modules together with that multiplication. In this sense we speak of monomials over $\set{\p{S}}{S \text{simple }}$, omit the brackets indicating the isomorphism classes and write $1$ for the isomorphism class of the identity functor.
To study this monoid and its monoid algebra a good way to start is finding a set of defining relations. We give some relations:
\begin{prop}\label{relations proposition}
Let $S$ and $T$ be simple $A$-modules without any non-trivial self-extensions. Then the following relations hold:
\begin{itemize}
\item[(a)] $\p{S}\circ\p{S} = \p{S}$
\item[(b)] If $\Ext{A}{1}{T,S}=0$ holds, we have $\p{S}\circ\p{T}\circ \p{S} = \p{T}\circ\p{S}\circ\p{T}= \p{S}\circ\p{T}$.
\end{itemize}
\end{prop}
As a consequence of the second relations we get $\p{S}\circ\p{T}= \p{T}\circ\p{S}$ if $\Ext{A}{1}{T,S}=0=\Ext{A}{1}{S,T}$ holds. Thus certain generators of $\monoid{A}$ satisfy the braid relations of type $A$.
Now if $A$ is the path algebra $kQ$ of a finite acyclic quiver $Q=(Q_0,Q_1)$ over $k$, then each simple $kQ$-module $S_t$ (attached to the vertex $t$) has no non-trivial self-extensions and hence the proposition holds for all generators of $\monoid{Q}\ldef \monoid{kQ}$. We conjecture that these relations are defining for $\monoid{Q}$.
To determine whether the above relations are defining ones we compare $k\monoid{Q}$ with the algebra $B_Q$, which we define by generators $X_t$ with $t\in Q_0$ and the following relations: \begin{itemize}
\item $X_s^2= X_s$ for all $s\in Q_0$
\item $X_sX_tX_s=X_tX_sX_t$ for all $s,t \in Q_0$
\item $X_tX_sX_t= X_sX_t$ for all $s,t \in Q_0$, such that there is no arrow from $t$ to $s$
\end{itemize}
There is an epimorphism $\psi_Q$ of algebras from $\B_Q$ onto $k\monoid{Q}$ with $X_t\mapsto \p{S_t}\rdef \p{t}$ for all $t\in Q_0$ by the above relations of the projection functors since the dimension of $\Ext{kQ}{1}{S_t,S_s}$ coincides with the number of arrows from $t$ to $s$. Note that if this epimorphism $\psi_Q$ is an isomorphism, then $\monoid{Q}$ is isomorphic to the Hecke-Kiselman semigroup associated with $Q$ introduced in \cite{MazorchukKiselman}. We introduce a method to detect when $\psi$ is an isomorphism in section 3. So far we applied it successfully to tree quivers with a specific orientation including bipartite tree quivers, $m$-subspace quivers, star quivers and each Dynkin quiver of type $A$ as well as to a couple of families of symmetrically shaped quivers. In this article we just discuss the former ones (see Prop.\ \ref{Special cases prop normal form of B_n lin.oriented} and Theo.\ \ref{Special cases theo normal form of admissable trees}). As the relations are independent of the number of arrows unless there are none, one expects that multiple arrows have no impact. Moreover the relations are local as they just take direct neighbourhoods into account. We treat these aspects in section 3 for $\B_Q$ where it is obvious and for $k\monoid{Q}$.
In section 4 we study the algebra $\B_Q$ and compute its Gabriel quiver. By definition its underlying monoid is the Hecke-Kiselman semigroup associated with $Q$. These algebras emerge as finite dimensional (see Cor.\ \ref{Special_cases Cor finite}) and basic (see Prop.\ \ref{extkoecher prop Radikal und alle Einfachen}) regardless of the representation type of $Q$. The Gabriel quiver can be described using the combinatorics given by the shape of the original quiver.
\begin{theo}\label{extkoecher-theo-Extkoecher}
The simple modules $E_M$ of $\B_Q$, hence the vertices of the Gabriel quiver $\extko{\B_Q}$, are parametrized by the subsets $M$ of $Q_0$. Moreover there is at most one arrow between two vertices. More precisely we have $[\E_M]\rightarrow [\E_N] \in \extko{\B_Q}$ for two subsets $M,N$ of $Q_0$ if and only if $M\ohne N$ and $N\ohne M$ are non-empty and if for each pair $(m,n)\in M\ohne N\times N\ohne M$ there is an arrow $m\rightarrow n$ in $Q$.
\end{theo}
Since the Hecke-Kiselman semigroup associated with an acyclic finite quiver is $\mathcal{J}$-trivial, these results can be deduced from \cite{DHST}. The author thanks Anne Schilling for pointing this reference out. Let $Q_n$ denote the following linearly oriented Dynkin quiver of type $A$:
\begin{center}
\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}, font=\normalsize]
\matrix(m) [matrix of math nodes, row sep=1em, column sep=2em, text height=1.5ex, text depth=0.25ex]
{1 & 2 & {}\ldots & n\\};
\path[->,font=\scriptsize]
(m-1-1) edge (m-1-2)
(m-1-2) edge (m-1-3)
(m-1-3) edge (m-1-4);
\end{tikzpicture}
\end{center}
This family of quivers has a special role, we show in subsection \ref{subsection properties of Gabriel quiver}:
\begin{prop} \label{Gabriel prop connected components}
Let $Q$ be connected, finite and acyclic.
\begin{itemize}
\item[(a)] If $Q=Q_n$ then the Gabriel quiver of $\B_{Q_n}$ has exactly $n+1$ connected components.
\item[(b)] If $Q$ is distinct from the linearly oriented Dynkin quiver of type $A$, then the Gabriel quiver of $\B_Q$ has exactly $3$ connected components.
\end{itemize}
\end{prop}
Therefore we devote section 5 to the algebra $\B_{Q_n}$, which is isomorphic to $k\monoid{Q_n}$ by section 3. It can be read almost independent. Now the monoid algebra of non-decreasing parking functions $\mathrm{NDPF}_{n+1}$ has the same defining relations as $\B_{Q_n}$ (see \cite{Hivert} or \cite{MazorchukKiselman}) and is thus isomorphic to $\B_{Q_n}$. As shown in \cite{Hivert} it is isomorphic to the incidence algebra $\Inc{\Po_n}$ of the product order $\grn{n}$ on the powerset of $\ito{n}\ldef\{1,2,\ldots, n\}$ if the underlying field $k$ is the field of complex numbers $\CC$. We recall, that for two subsets $K=\{k_1<\ldots < k_r\}$ and $J=\{j_1<\ldots < j_m\}$ of $\ito{n}$ we have $K\grn{n} J$ iff $m=r$ and $k_i\geq j_i$ holds for all $i\in \ito{r}$. They use the representation theory of the symmetric group. In \cite{DHST} this is generalized using the $\mathcal{J}$-triviality of the underlying monoid. Here (see Main Theorem \ref{Dynkin maintheorem}) we inductively construct an isomorphism from $\Inc{\Po_n}$ to $\B_{Q_n}$ independent of the field -- in fact it holds for a commutative ring -- using the structure of the tower of algebras $\B_{Q_1}\subset \B_{Q_2}\subset \ldots$ and the action of $\monoid{Q_n}$ on the (injective indecomposable) $kQ_n$-modules.
We expect that a notion of non-decreasing parking functions for arbitrary quivers determines defining relations for $\monoid{Q}$.
\newline\textbf{Acknowledgments:}
\newline This work is part of my PhD-thesis \cite{Paasch} supervised by Markus Reineke whom I thank for posing this question.
\section{The monoid algebra attached to $\mathbf{Q_n}$}
In this section let $R$ be a field. Recall that $\Po_n$ denotes the poset of the product order $\grn{n}$ defined in the introduction. We first introduce the elements of $\A{n}\ldef R\monoid{Q_n}$ to state the precise isomorphism between the incidence algebra $\Inc{\Po_n}$ and $\A{n}$. Then we prove the stated properties in the subsections 5.1, 5.2 and 5.3.
As is well-known, the incidence algebra $\Inc{\Po_n}$ is the free $R$-space over $X_{(J,I)}$ with $J\grn{n} I$ endowed with the multiplication given by $X_{(K,J)}X_{(J',I)} = X_{(K,I)}$ if $J=J'$ and $X_{(K,J)}X_{(J',I)}=0$ otherwise. It is the path algebra of the Hasse diagramm of the poset $\Po_n$ modulo the ideal which is generated by the commutativity relations (e.g.\ see \cite{Ringel}).
As we have already seen $\A{1}\subseteq \A{2}\subseteq \A{3}\subseteq \ldots$ is a tower of algebras. Thus an inductive description of elements is possible. Local properties such as idempotency or orthogonality are preserved, in contrast to global ones such as centrality or being the unit element $1$ in $\A{n}$, when viewing elements in a greater algebra $\A{n+k}$.
The heart of the definition of the idempotents in $\A{n}$ corresponding to the idempotents $X_{(J,J)}$ in $\Inc{\Po_n}$ are the inductively (on $n$) defined elements $\y{1}{n}, \y{2}{n},\ldots ,\y{n}{n}$ in $\A{n}$. We start with $\y{1}{1}\ldef \p{1}$ and set for all $k\in \ito{n}$:
\begin{align*}
\y{k}{n} \ldef\begin{cases}
\y{1}{n-1}-\p{n}\y{1}{n-1}+ \p{n} \quad &\text{if } k=1 \\[3pt]
\y{k}{n-1}-\p{n}\y{k}{n-1}+ \y{k-1}{n-1}\p{n} \quad &\text{if } 2\leq k \leq n \\[1pt]
0 \quad &\text{if } k>n \\
\end{cases}
\end{align*}
Each of these elements generates an ideal which is closely related to the admissible normal form given in \ref{Special cases prop normal form of B_n lin.oriented} (see Corollary \ref{Dynkin Cor Ideal generators}). Their properties are listed in Lemma \ref{Idempotente Lemma Eigenschaften der y}. From them we conclude that the idempotents in $\A{n}$ corresponding to the connected components of $\Inc{\Po_n}$ are given by:
\[
\z{0}{n} \ldef 1-\y{1}{n}, \quad\ldots,\quad \z{k}{n} \ldef \y{k}{n} - \y{k+1}{n},\quad \ldots, \quad\z{n}{n}\ldef \y{n}{n} - \y{n+1}{n}=\y{n}{n}
\]
Furthermore we consider the inductively defined elements $\g{J}{n}\in \A{n}$ for all $\emptyset\neq J \subseteq \ito{n}$ starting with $\g{\{1\}}{1} \ldef \p{1}$:
\[
\g{J}{n} \ldef \begin{cases}
\g{J}{n-1}-\p{n}\g{J}{n-1} \quad &\text{if } n\notin J\\
\p{n} \quad &\text{if } \{n\}= J\\
\g{J\ohne \{n\}}{n-1}\p{n} \quad &\text{if } n \in J\neq\{n\}\\
\end{cases}
\]
\begin{theo} \label{Dynkin complete pw orthogonal idempotents}
A complete system of pairwise orthogonal idempotents of $\A{n}$ consists of
\[
\f{\emptyset}{n}\ldef\z{0}{n}\quad\text{and}\quad \f{J}{n} \ldef \g{J}{n}\z{|J|}{n}\quad \text{with} \quad \emptyset\neq J\subseteq \ito{n}
\]
\end{theo}
(In \cite{DHST} a complete set of orthogonal idempotents for $\A{n}$ is given explicitly.) Now we turn towards those elements in $\A{n}$ corresponding to $X_{(J,I)}\in \Inc{\Po_n}$ with $J\neq I$. For this we define for all $n\geq j\geq i\geq 1$ the word:
\begin{align*}
j\ru i \ldef j\konk (j-1)\konk\ldots \konk (i+1)\konk i \quad\in \ito{n}^*\\
\shortintertext{and for all subsets $J=j_r>\ldots >j_1$ and $K= k_r>\ldots>k_1$ of $\ito{n}$ with $K\grn{n} J$ the monomial}
\p{(K,J)}\ldef \p{k_1 \ru j_1 \konk \ldots \konk k_r\ru j_r} \quad \in \A{n}\label{definition p(j,K)}
\end{align*}
The already introduced idempotent $\p{J}= \p{j_1\konk j_2 \konk \ldots \konk j_r}$ coincides with $\p{(J,J)}$; meanwhile $\p{(\emptyset,\emptyset)}= 1$ in $\A{n}$. The main difficulty is to show that the elements $\f{K}{n}\p{(K,J)}\f{J}{n}$ are distinct from $0$.
\begin{theo}\label{Dynkin theorem aJI neq 0}
For every pair $K\grn{n} J$ we have: $ \f{K}{n}\p{(K,J)}\f{J}{n} \neq 0$.
\end{theo}
Here we will need two more descriptions of the middle factor, including one inductive one (see subsection 5.2.2) and an inductive description of the elements $\g{J}{n}$ (see Lemma \ref{Dynkin lemma inductive gJn}). Along the way we get all we need to prove the main theorem:
\begin{main}\label{Dynkin maintheorem}
Let $R$ be a field and $n\in \NN$. The $R$-linear map $\Phi\colon \Inc{\Po_n} \longrightarrow \A{n} $ with \mbox{$\Phi(X_{(K,J)})= \f{K}{n}\p{(K,J)}\f{J}{n}$} for all $K\grn{n} J$ is an isomorphism of algebras.
\end{main}
One remark on the field $R$: in our proof we distinguish between two monomials $\p{v}$ and $\p{w}$ by comparing their action on the injective indecomposable $Q_n$ representations $I_0, I_1\ldots I_n$ over $R$. These steps can be replaced by comparing the non-decreasing parking functions $\pi_v$ and $\pi_w$ as functions on $1,\ldots, n+1$ (notation as in \cite{Hivert}). Since we just use the defining relations for $\monoid{Q_n}$ (see Proposition \ref{Special cases prop normal form of B_n lin.oriented}) which are the same as for $\mathrm{NDPF}_{n+1}$ (see \cite{Hivert}), we could replace the field $R$ by an arbitrary commutative ring, as mentioned in \cite{Hivert}) and \cite{DHST}:
\begin{rem}
Let $R$ be a commutative ring. Then the $R$-algebras $R\mathrm{NDPF}_{n+1}$ and $\Inc{\Po_n}$ are isomorphic.
\end{rem}
\subsection{Proof of Theorem \ref{Dynkin complete pw orthogonal idempotents}}
We start with the properties of the elements $\y{1}{n},\ldots,\y{n}{n}$ and the direct conclusions for the elements $\z{0}{n},\ldots,\z{n}{n}$:
\begin{lem}\label{Idempotente Lemma Eigenschaften der y}
The elements $\y{j}{n}$ are central in $\A{n}$ for all $j\in \ito{n}$. Thus the elements $\z{j}{n}$ are central. Meanwhile the idempotency and orthogonality of $\z{0}{n},\ldots,\z{n}{n}$ follows directly from:
\begin{align*}
\y{i}{n}\,\y{j}{n}&=\y{j}{n} \qquad \text{for all} \quad 0<i\leq j\leq n\\
\shortintertext{Each $\y{j}{n}$ is distinct from $0$ since for all subsets $J\subseteq\ito{n}$ with $|J|=j$ the following equation holds:}
\y{j}{n}\p{J} &= \p{J}\\
\end{align*}
\end{lem}
\begin{proof}
Exemplary in more detail, we prove by induction on $n$ that $\y{i}{n}$ is central for each $i\in \ito{n}$.
For $n\in \{1,2\}$ the centrality of $\y{1}{1}=\p{1}$ and of $\y{1}{2}=\p{1}+\p{2}-\p{2\konk 1}$ or $\y{2}{2}=\p{1\konk 2}$ in $\A{1}$ resp. in $\A{2}$ are direct consequences of the defining relations. So let $n>2$. For a generator $\p{j}$ of $\A{n}$ we have:
\[
\p{j}\y{1}{n} -\y{1}{n}\p{j} =(\p{j}\y{1}{n-1}- \y{1}{n-1}\p{j}) + (-\p{j}\p{n}\y{1}{n-1}+\p{n}\y{1}{n-1}\p{j})+ (\p{j}\p{n} - \p{n}\p{j})
\]
For $j<n-1$ this adds up to $0$ by the induction hypothesis and the (commutativity) relations. We consider the cases $j\in \{n-1,n\}$ separately using the induction hypothesis for $j=n-1$:
\begin{align*}
\p{n-1}\y{1}{n} -\y{1}{n}\p{n-1}
&= (-\p{n-1}\p{n} + \p{n}\p{n-1})\y{1}{n-1} + (\p{n-1}\p{n} - \p{n}\p{n-1})\\[2pt]
&= (-\p{n-1}\p{n} + \p{n}\p{n-1})\y{1}{n-2} - (-\p{n-1}\p{n} + \p{n}\p{n-1})\p{n-1}\y{1}{n-2} \\[2pt]
&\qquad + (-\p{n-1}\p{n} + \p{n}\p{n-1})\p{n-1} + (\p{n-1}\p{n} - \p{n}\p{n-1})\\[2pt]
&= (-\p{n-1}\p{n} + \p{n}\p{n-1})\y{1}{n-2} - (-\p{n-1}\p{n} + \p{n}\p{n-1})\y{1}{n-2} \\
&= 0
\shortintertext{and the generalized relations for $j=n$:}
\p{n}\y{1}{n} -\y{1}{n}\p{n}
&= (\p{n}\y{1}{n-1}- \y{1}{n-1}\p{n}) + (-\p{n}\y{1}{n-1}+\y{1}{n-1}\p{n}) =0
\end{align*}
Now we consider the elements $\y{2}{n},\ldots, \y{n}{n}$. For any $k > 1$ and a generator $\p{j}$ of $\A{n}$ we have:
\[
\begin{aligned}
\p{j}\y{k}{n} -\y{k}{n}\p{j}
&=(\p{j}\y{k}{n-1}- \y{k}{n-1}\p{j}) + (-\p{j}\p{n}\y{k}{n-1}+\p{n}\y{k}{n-1}\p{j})+ (\p{j}\y{k-1}{n-1}\p{n} - \y{k-1}{n-1}\p{n}\p{j})
\end{aligned}
\]
Again for $j<n-1$, this adds up to $0$. Meanwhile we use the induction hypothesis and the generalized relations for $j=n-1$:
\begin{align*}
\p{n-1}\y{k}{n} -\y{k}{n}\p{n-1}
&= (-\p{n-1}\p{n}+\p{n}\p{n-1})(\y{k}{n-2}-\p{n-1}\y{k}{n-2}+ \y{k-1}{n-2}\p{n-1}) \\[2pt]
&\qquad + (\y{k-1}{n-2}-\p{n-1}\y{k-1}{n-2}+ \y{k-2}{n-2}\p{n-1})(\p{n-1}\p{n} - \p{n}\p{n-1})\\[2pt]
&= (-\p{n-1}\p{n}+\p{n}\p{n-1})\y{k-1}{n-2}\p{n-1} + (\y{k-1}{n-2}-\p{n-1}\y{k-1}{n-2})(\p{n-1}\p{n} - \p{n}\p{n-1})\\[2pt]
&= -\p{n-1}\p{n}\y{k-1}{n-2}\p{n-1} + \p{n}\y{k-1}{n-2}\p{n-1} + \y{k-1}{n-2}\p{n-1}\p{n} - \p{n}\y{k-1}{n-2}\p{n-1} \\[2pt]
&\qquad -\y{k-1}{n-2}\p{n-1}\p{n} + \p{n-1}\p{n}\y{k-1}{n-2}\p{n-1}\\
&= 0\\
\shortintertext{For $j=n$ the calculation is again simply:}
\p{n}\y{k}{n} -\y{k}{n}\p{n} &=(\p{n}\y{k}{n-1}- \y{k}{n-1}\p{n}) + (-\p{n}\y{k}{n-1}+\y{k}{n-1}\p{n})+ (\y{k-1}{n-1}\p{n} - \y{k-1}{n-1}\p{n}) =0
\end{align*}
Now we prove by induction that $\y{j}{n}\y{k}{n}= \y{k}{n}$ holds for all \mbox{$1\leq j\leq k\leq n$}. The case for $n=1$ is trivial, so we proceed with $n>1$. For this we set $\y{0}{n-1}\ldef 1\in \A{n}$ and rewrite:
\[
\y{j}{n}=\y{j}{n}-\p{n}\y{j}{n-1} + \y{j-1}{n-1}\p{n}=(1-\p{n})\y{j}{n} + \y{j-1}{n-1}\p{n}\in \A{n}
\]
By the generalized relations we get the following equalities in $\A{n}$; the last one by the induction hypothesis:
\[
\begin{aligned}
\y{j}{n}\y{k}{n}
&= (1-\p{n})\y{j}{n-1} (1-\p{n})\y{k}{n-1} + (1-\p{n})\y{j}{n-1} \y{k-1}{n-1}\p{n} \\[2pt]
&\qquad \:+ \y{j-1}{n-1}\p{n}(1-\p{n})\y{k}{n-1} + \y{j-1}{n-1}\p{n}\y{k-1}{n-1}\p{n}\\[3pt]
&= (1-\p{n})\y{j}{n-1} (1-\p{n})\y{k}{n-1} + \y{j-1}{n-1}\y{k-1}{n-1}\p{n}\\[2pt]
&= (1-\p{n})\y{k}{n-1} + \y{k-1}{n-1}\p{n}\\[2pt]
\end{aligned}
\]
For the last statement it is convenient to show by induction on $n$ (simultanously) that for all $\emptyset\neq J\subseteq \ito{n}$ the following two equations hold:
\[
\y{|J|}{n}\p{J} = \p{J} \qquad\text{and }\qquad \y{|J|}{n}\p{n+1}\p{J} = \p{n+1}\p{J}
\]
\end{proof}
\begin{proof}[Proof of Theorem \ref{Dynkin complete pw orthogonal idempotents}.]
By similar computations and case by case analysis we deduce the \linebreak Theorem with the following steps: first show by induction on $n$, that for each $k\in \ito{n}$ the set $\set{\g{J}{n}}{J\subseteq \ito{n},|J|=k}$ is a set of pairwise orthogonal idempotents, whose elements add up to $\y{k}{n}$. Therefore -- by definition of $\f{J}{n}$ and the properties of $\z{k}{n}$ -- the set $\set{\f{J}{n}}{J\subseteq \ito{n},|J|=k}$ consists of pairwise orthogonal idempotents. From $\y{k}{n}\z{k}{n}= \z{k}{n}$ we get $\sum_{J\subseteq \ito{n},|J|=k}\f{J}{n}=\z{k}{n}$. Since $\set{\z{k}{n}}{0\leq k \leq n}$ consists of central pairwise orthogonal elements, which add up to $1\in \A{n}$, Theorem \ref{Dynkin complete pw orthogonal idempotents} follows.
\end{proof}
We finish this subsection with a useful remark on the following chain of ideals in $\A{n}$:
\[
\A{n}= \Jk{0}{n} \supset \Jk{1}{n} \supset \Jk{2}{n} \supset \ldots \supset \Jk{n-1}{n} \supset \Jk{n}{n} = \langle \p{\ito{n}}\rangle_R \supset 0
\]
where $\Jk{k}{n}$ is the ideal generated by the monomials $\p{J}$ with $J\subseteq \ito{n}$ and $|J|=k$. It is directly deduced from the last equation of Lemma \ref{Idempotente Lemma Eigenschaften der y}, that $\Jk{k}{n}$ is contained in the ideal $\y{k}{n}\A{n}$. On the other hand an easy induction shows that $\y{k}{n}$ is contained in $\Jk{k}{n}$. Thus the equalities hold:
\begin{cor}\label{Dynkin Cor Ideal generators}
\[
\Jk{k}{n} = \y{k}{n}\A{n} = \set{a\in \A{n}}{\y{k}{n}a= a}
\]
In particular $\z{k}{n}=\y{k}{n}-\y{k+1}{n}$ annihilates the ideals $\Jk{k+1}{n} \supset \ldots \supset \Jk{n}{n}$.
\end{cor}
\subsection{Proof of Theorem \ref{Dynkin theorem aJI neq 0}}
\subsubsection{Chain description of $\mathbf{\p{K,J}}$}\label{chain description}
For $n\in \NN$ and subsets $J$ and $K$ of $\ito{n}$ we write $K\vgrn{n} J$ if $K$ is a minimal proper successor of $J$ w.r.t. $\grn{n}$ and say that their are neighbours. In general there is more than one $\vgrn{n}$-chain between $K\grn{n} J$. But the monomial $\p{(K,J)}$ is an invariant of such $\vgrn{n}$-chains:
\begin{lem} \label{Idempotente Lemma alternative Beschreibung der mnJK}
Let $K\grn{n} J$. For every $\vgrn{n}$-chain \mbox{$K=H_t \vgrn{n} \ldots \vgrn{n} H_1 = J$} we have:
\[
\p{(K,J)} = \p{{H_t}\konk\ldots\konk{H_1}}
\]
\end{lem}
\begin{proof}
We just show the induction step $ 2\leq n \rightarrow n+1$ for non-empty, proper subsets $n+1 \in K\grn{n+1} J$ of $\ito{n+1}$. Let $1<r\ldef |J| \neq J$ and consider a $\vgrn{n+1}$-chain $K=H_t\vgrn{n+1} \ldots \vgrn{n+1} H_1 = J$.
Let $ h_{r,s}>h_{r-1,s}>\ldots> h_{1,s}$ be the elements of $H_s$ for $s\in \ito{t}$. Since $H_{s+1}\vgrn{n+1} H_{s}$ there is exactly one index $k\in \ito{r}$ with $h_{k,s+1} > h_{k,s}=h_{k,s+1} -1$ and $h_{j,s}=h_{j,s+1}$ for all $j\neq k$.
We denote by $\widetilde{J}$, $\widetilde{K}$ and $\widetilde{H_s}$ the sets $J,K$ and $H_s$ without their maximal elements respectively.
\newline\case{First case: ${n+1 \in J}$}
In particular $k_r \ru j_r = n+1$ holds. Moreover the maximal elements $h_{r,s} = n+1$ for $s\in \ito{t}$ are not involved in the $\vgrn{n+1}$-chain, that is we already have a $\vgrn{n}$-chain:
\begin{align*}
&\hphantom{\p{{H_t}\konk\ldots\konk{H_1}}}\widetilde{K} = K\ohne \{n+1\}= \widetilde{H_t}\vgrn{n} \ldots \vgrn{n} \widetilde{H_1} = J\ohne \{n+1\}=\widetilde{J}\\
\shortintertext{So by the generalized relations and the induction hypothesis we deduce:}
\p{{H_t}\konk\ldots\konk{H_1}} &=\p{\widetilde{H_t}\konk n+1\konk\ldots\konk {\widetilde{H_2}}\konk n+1\konk{\widetilde{H_1}}\konk n+1}= \p{{\widetilde{H_t}}\konk\ldots \konk {\widetilde{H_2}}\konk{\widetilde{H_1}}\konk n+1}
=\p{k_1 \ru j_1 \konk \ldots \konk k_{r-1}\ru j_{r-1}\konk k_r\ru j_r} = \p{(K,J)}
\end{align*}
\case{Second case: ${n+1 \notin J}$}
We divide the chain into two chains, such that one of them contains no $n+1$ but the other does. More precisely, there is an index $s\in \{2,\ldots,t\}$ minimal with $n+1\in H_s$. Then we have: $h_{r,s-1}=n <n+1 = h_{r,s} = \ldots = h_{r,t} = k_r$.
As in the first case we receive:
\[\p{H_t\ldots H_s}= \p{(\widetilde{H_s},\widetilde{H_t})}\p{n+1} \quad\text{and}\quad \p{H_{s-1}\ldots H_1}=\p{(\widetilde{H_{s-1}},\widetilde{H_{1}})}\p{ h_{r,s-1}\ru h_{r,1}}
\]
Now $\p{n+1}$ commutes with all $\p{j}$ for $j\leq h_{r-1,s-1}$, hence with $\p{(\widetilde{H_{s-1}},\widetilde{H_1})}$. Moreover we observe $\widetilde{H}_{s-1} = \widetilde{H}_s$. Therefore we get by applying the induction hypothesis several times:
\[
\begin{aligned}
\p{H_t\ldots H_1} = \p{(\widetilde{H_t},\widetilde{H_s})}\p{(\widetilde{H_{s-1}},\widetilde{H_{1}})}\p{n+1\konk h_{r,s-1}\ru h_{r,1}}=\p{(\widetilde{H_t},\widetilde{H_1})}\p{n+1\ru j_r}=\p{(K,J)}\\
\end{aligned}
\]
\end{proof}
\begin{cor}\label{Idempotente Korollar pJK in den Idealen und invariant unter PJ und PK}
For all $L\grn{n}K\grn{n}J$ the monomial $\p{(K,J)}$ is contained in the ideal $\Jk{|J|}{n}$ and we have $\p{(L,K)}\p{(K,J)}=\p{(L,J)}$, in particular \mbox{$\p{K}\p{(K,J)} = \p{(K,J)} = \p{(K,J)}\p{J}$} holds.
\end{cor}
In fact, the stronger assertion holds:
\begin{lem}\label{Dynkin lemma pKJ not in Jk k+1}
\[
\p{(K,J)} \in \Jk{|J|}{n}\ohne \Jk{|J|+1}{n}
\]
\end{lem}
\begin{proof}
The monomials $\p{(K,J)}$ for maximal $K$ and minimal $J$ (w.r.t. $\kln{n}$) are easier to handle, i.e.\ $K=\{n-r+1, \ldots, n\}$ and $J=\ito{r}$ for some $r\in \oto{n}$. We set $\m{0}{n}\ldef 1\in \A{n}$ and for each $r \in \ito{n}$
\[
\m{r}{n}\ldef \p{(\{n-r+1, \ldots, n\},\ito{r})} =\p{n-r+1\ru 1\konk \ldots \konk n-1\ru r-1\konk n\ru r} =\m{r-1}{n-1}\p{n\ru r}
\]
Now we prove $\m{r}{n} \notin \Jk{r+1}{n}$ by induction on $n$, which is by Corollary \ref{Dynkin Cor Ideal generators} equivalent to:
\[
(1-\y{r+1}{n}) \m{r}{n}\neq 0
\]
In the induction step $n\rightarrow n+1$ the extreme cases $r\in \{0,n+1\}$ are trivial and for the remaining $r$ we get by direct calculations using the generalized relations:
\[
(1-\y{2}{n+1})\m{1}{n+1}= \z{0}{n}\p{n+1\ru 1} \quad\text{and}\quad (1-\y{r+1}{n+1})\m{r}{n+1} = (1-\y{r}{n}) \m{r-1}{n}\p{n+1\ru r}
\]
The induction hypothesis applies to the ladder cases. Thus the linear combinations $\z{0}{n}$ and $(1-\y{r}{n}) \m{r-1}{n}$ of monomials in $\A{n}$ are distinct from $0$. But for any two distinct monomials $\p{v}$ and $\p{w}$ in $\A{n}$ there exists an injective indecomposable $Q_n$-representation $I_y$ with $\p{v}I_y\neq \p{w}I_y$ by the proof of Proposition \ref{Special cases prop normal form of B_n lin.oriented}.
With Lemma \ref{Special_cases lem embedding functor} we conclude that thus $\p{v}\p{n+1\ru r}$ and $\p{w}\p{n+1\ru r}$ act differently on one of the $Q_{n+1}$-representations $I_y$ or $I_{y+1}$. Therefore the monomials $ \m{r}{n+1}$ do not lie in $\Jk{r+1}{n+1}$. Since each monomial $\p{(K,J)}$ is a factor of $\m{|J|}{n}$ by the chain description, $\p{(K,J)}$ does not lie in $\Jk{|J|+1}{n}$.
\end{proof}
\subsubsection{Inductive description of $\mathbf{\p{(K,J)}}$}\label{inductive description}
We denote by $K_{\max}$ the greatest interval (w.r.t. $\groint$, see page \pageref{specialcases_groint}) of a finite subset $K$ of $\NN$, e.g.\ $\{9,8,5,4,2\}_{\max}=\{9,8\}$ and $\{4,3,2\}_{\max}=\{4,3,2\}$ and $\{5,3,2,1\}_{\max}=\{5\}$. As usual we set $K-1\ldef \set{k-1}{k\in K}$ and $\emptyset - 1\ldef \emptyset$.
If $K\grn{n+1} J$ and $n+1$ lies in $J$, then $n+1$ also lies in $K$, moreover $J_{\max}$ is a subset of $K_{\max}$. Therefore the distinction of cases in the next remark is complete.
\begin{rem}
Let $K\grn{n+1} J$ and $K\neq J$. Then we have:
\[
\begin{aligned}
&K\grn{n} J \quad &&\text{if } n+1\notin K\\[2pt]
&K\ohne K_{\max}\cup (K_{\max}-1)\grn{n} J \quad &&\text{if }n+1\in K\ohne J\\[2pt]
& K\ohne K_{\max}\cup ((K_{\max}\ohne J_{\max})-1)\grn{n} J\ohne J_{\max} \quad &&\text{if }n+1\in J
\end{aligned}
\]
\end{rem}
Thus we can define:
\begin{df} \label{Idempotente zu den Wegen korrespond. Elemente in An}
Starting with $\m{\{1\},\{1\}}{1} \ldef \p{1}$ and $\m{\emptyset,\emptyset}{1} = 1\in \A{1}$ we define inductively on $n$ for every pair $K\grn{n+1} J$ the monomial $\m{K,J}{n+1}$ by:
\[
\m{K,J}{n+1}\ldef \begin{cases}
\p{J} \quad &\text{if } J=K\\
\m{K,J}{n}\quad &\text{if } J\neq K , n+1\notin K\\[3pt]
\p{K_{\max}}\:\m{K\ohne K_{\max}\cup (K_{\max}-1)\;,\;J}{n}\quad &\text{if } J\neq K , n+1\in K\ohne J \\[3pt]
\p{K_{\max}}\:\m{K\ohne K_{\max}\cup ((K_{\max}\ohne J_{\max})-1)\;,\;J\ohne J_{\max}}{n} \quad &\text{if } J\neq K, n+1\in J
\end{cases}
\]
\end{df}
An induction on $n$ and a case by case analysis according to the definition of $\m{K,J}{n}$ shows in a straightforward way:
\begin{lem} \label{Idempotente Lemma induktive Beschreibung der mnJK}
Let $K\grn{n} J$. Then we have:
\[
\p{(K,J)} = \m{K,J}{n}
\]
\qed\end{lem}
Before we use the inductive description of $\p{(K,J)}$ we need to introduce two more notation. With them we can formulate an inductive description of $\g{J}{n}$ and hence of $\f{J}{n}$ (see Lemma \ref{Dynkin lemma inductive gJn}).
\begin{df}
For a subset $N$ of $\ito{n}$ we define the element $\yJ{N}$ in $\A{n}$ by
\[
\yJ{N} = \begin{cases}
0 \quad &\text{if } N = \emptyset\\
\yJ{N\ohne\max N} -\p{\max N}\yJ{N\ohne \max N} + \p{\max N}\quad &\text{if } N\neq \emptyset
\end{cases}
\]
\end{df}
So $\yJ{N}$ is similarly defined to $\y{1}{n}$ and has similar properties (w.r.t. to the subalgebra $\A{N}$ of $\A{n}$ generated by $\p{s}$ with $s\in N$), namely: the element $\yJ{N}$ is central in $\A{N}$. Moreover, for all $m\in N$ and each $x>\max N$ we have the identities $\yJ{N}\p{m} = \p{m}$ and $\yJ{N}\p{x}\p{m} = \p{x}\p{m}$ and consequently $\yJ{N}\neq 0$.
We consider these elements for the sets:
\begin{df}
We define for each subset $K$ of $\ito{n}$ the subset $\N{K}{n}$ of $\ito{n}$ by:
\[
\N{K}{n} \ldef \begin{cases}
\emptyset \quad &\text{if } K=\emptyset\\
\set{x\in \ito{n}\ohne K}{x > \min K } \quad&\text{if } K\neq \emptyset
\end{cases}
\]
\end{df}
Some examples are: $\N{\{1,2\}}{5}=\{3,4,5\} = \N{\{2\}}{5}, \N{\{1,3\}}{5}=\{2,4,5\}, \N{1,3,4,5}{5}=\{2\}$ and $\N{\{4,5\}}{5}=\emptyset$.
\begin{lem}\label{Dynkin lemma pJy(NJ)pJ annihilated by z}
For each subset $J$ of $\ito{n}$ the linear combination $\p{J} \yJ{\N{J}{n}} \p{J}$ of monomials in $\A{n}$ lies in the ideal $\Jk{|J|+1}{n}$ and is thus annihilated by $\z{|J|}{n}$ (see Corollary \ref{Dynkin Cor Ideal generators}).
\end{lem}
\begin{proof}
By Corollary \ref{Dynkin Cor Ideal generators} it suffices to show the identity:
\[
\y{|J|+1}{n}\p{J}\,\yJ{\N{J}{n}}\p{J} = \p{J}\,\yJ{\N{J}{n}}\p{J}
\]
This is a straightforward induction on $n$ requiring a case-by-case analysis on the cardinality of $J$ and considering the cases $n+1\in J$ and $n+1\notin J$ separately. One also needs the identity $\y{j}{n}\p{J} = \p{J}$ (see Lemma \ref{Idempotente Lemma Eigenschaften der y}).
\end{proof}
The next rather technical lemma, which we prove in more detail, is the heart of the proof of Theorem \ref{Dynkin theorem aJI neq 0}.
Recall that if $W_n$ as in Proposition \ref{Special cases prop normal form of B_n lin.oriented}, then $\set{\p{w}}{w\in W_n}$ is a basis of $\A{n}$.
\begin{lem}\label{Idempotente Lemma mJK yNK hat nicht mJK}
Let $K\grn{n} J$. Then we have:
\[
\yJ{\N{K}{n}}\,\p{(K,J)}\in \biggl\langle \p{w}\in \A{n} \:\big|\:\: w \in W_n \text{ and } \p{v}\neq \m{K,J}{n}\biggr\rangle_{R}\rdef \U{K,J}{n}
\]
\end{lem}
\begin{proof}
The proof is an induction on $n$.
We start with a remark on the two extreme cases $\N{K}{n}=\emptyset$ and $J=K$ with $\N{K}{n}\neq \emptyset$: in the first case $\yJ{\N{K}{n}}=0$ holds, so the statement is clear. In the second case let $\yJ{\N{K}{n}}=\sum_{w\in W_n} c_w \p{w}$. Then we have $\m{K,K}{n}\,\yJ{\N{K}{n}} = \sum_{w\in W_n} c_w \p{K}\p{w}$. Now for all $w\in W_n$ with $c_w\neq 0$ (i.e.\ $\{w\}\subseteq \N{K}{n} \subset \ito{n}\ohne K$) the functors $\m{K,K}{n}=\p{K}$ and $\p{K}\p{w}$ differ in their action on the injective indecomposable $Q_n$-representations $I_i$ with $i\in \ito{n}\ohne K$. Hence $\m{K,K}{n}\,\yJ{\N{K}{n}}$ lies in $\U{K,K}{n}$.
Since the calculations for $n\in \{1,2,3\}$ are trivial we proceed with the induction step $n\rightarrow n+1$ for $n\geq 3$. Let $K\grn{n+1}J$ such that $J\neq K$ and $\N{K}{n+1}\neq \emptyset$ hold.
\newline\case{$1^{\text{st}}$ case: ${n+1\notin K}$.} Since $\N{K}{n+1}=\N{K}{n}\cup \{n+1\}$ we have
\[
\yJ{\N{K}{n+1}}\m{K,J}{n+1} = \biggl(\yJ{\N{K}{n}}-\p{n+1}\yJ{\N{K}{n}}+\p{n+1}\biggr)\m{K,J}{n} = \yJ{\N{K}{n}}\m{K,J}{n}+\p{n+1}(1-\yJ{\N{K}{n}})\m{K,J}{n}
\]
By the induction hypothesis the first summand $\yJ{\N{K}{n}}\m{K,J}{n}$ lies in $\U{K,J}{n}\subseteq \U{K,J}{n+1}$. Meanwhile each monomial appearing in the second summand starts with $\p{n+1}$, hence does not coincide with $\m{K,J}{n+1} = \m{K,J}{n} \in \A{n}$. (Compare the actions on $I_{n+1}$.)
\newline\case{$2^{\text{nd}}$ case: ${n+1\in K}$}
Then $N\ldef \N{K\ohne\{n+1\}}{n}=\N{K}{n+1} \neq \emptyset$ holds and we have:
\[
K\neq K_{\max}\qquad\text{and}\qquad \max N = \min K_{\max} -1 \neq 0
\]
We denote by $\widetilde{N}$ the set $ N\ohne \{\max N\}$. Let $\widetilde{J}$ and $\widetilde{K}$ be those subsets of $\ito{n}$ given by the definition of $\m{K,J}{n+1}$ (depending on $n+1\in J$ or $n+1\neq J$) such that we have:
\[
\m{K,J}{n+1}=\p{ {K_{\max}}}\m{\widetilde{K},\widetilde{J}}{n}
\]
In the sequel we show:
\begin{align*}
a &\ldef \p{ {K_{\max}}} \yJ{\widetilde{N}} \m{\widetilde{K},\widetilde{J}}{n} \in \U{K,J}{n+1}\\[3pt]
b &\ldef \p{\max N}\p{ {K_{\max}}}\left(1- \yJ{\widetilde{N}}\right) \m{\widetilde{K},\widetilde{J}}{n} \in \U{K,J}{n+1}
\end{align*}
Then the claim follows immediately because we have
\[
\yJ{\N{K}{n+1}}\m{K,J}{n+1}= \left((1-\p{\max N})\yJ{\widetilde{N}}+\p{\max N}\right) \p{ {K_{\max}}}\m{\widetilde{K},\widetilde{J}}{n}=a+b
\]
\newline\case{Proof of $b \in \U{K,J}{n+1}$.} Let $ c_w\p{\max N}\p{ {K_{\max}}}\p{w} \neq 0$ be a summand of $b$. Note that $\{w\}$ is a subset of $\ito{n}$. We look at the action on the injective indecomposable $Q_{n+1}$-representation $I_{n+1}$ to distinguish between $\p{\max N}\p{K_{\max}}\p{w}$ and $\m{K,J}{n+1}$:
\begin{align*}
\p{\max N}\p{ {K_{\max}}}\p{w}(I_{n+1}) &= \p{\max N}\p{ {K_{\max}}}(I_{n+1})
= \p{\max N} (I_{\min K_{\max}-1})
= I_{\min K_{\max}-2 }\\[3pt]
&\neq I_{\min K_{\max}-1 }
= \p{ {K_{\max}}}(I_{n+1})
= \p{ {K_{\max}}}\m{\widetilde{K},\widetilde{J}}{n}(I_{n+1})
=\m{K,J}{n+1}(I_{n+1})
\end{align*}
\case{Proof of $a \in \U{K,J}{n+1}$.} For $\widetilde{N}= \emptyset$ the summand $a$ equals $0$. So we assume $\widetilde{N} \neq \emptyset$.
Now we consider:
\[
s\ldef \begin{cases}
n \quad &\text{if } n+1\notin J\\
j\ldef \min J_{\max} -2 \quad &\text{if } n+1\in J\\
\end{cases}
\]
In each case $\widetilde{J}$ and $\widetilde{K}$ both lie in $\A{s}$ and we have
\begin{align*}
\widetilde{N}= \N{\widetilde{K}}{s} \qquad\text{and}\qquad \m{\widetilde{K},\widetilde{J}}{n} =\m{\widetilde{K},\widetilde{J}}{s}\\
\shortintertext{In particular it follows:}
a = \p{ {K_{\max}}} \yJ{\widetilde{N}}\m{\widetilde{K},\widetilde{J}}{n}=\p{ {K_{\max}}} \yJ{\N{\widetilde{K}}{s}}\m{\widetilde{K},\widetilde{J}}{s}
\end{align*}
Now we consider an arbitrary summand $0\neq c_w\p{w} = c_w\p{K_{\max}}\p{v}$ of $a$. By the induction hypothesis $\p{v}$ and $\m{\widetilde{K},\widetilde{J}}{s}$ are distinct monomials in $\A{s}$. Recall that by the proof of Proposition \ref{Special cases prop normal form of B_n lin.oriented} there thus exist an injective indecomposable $Q_s$-representation $I_j$ with $j\in \ito{s}$ and an index $x \in \oto{\smash{j}}$ with:
\[
I_x=\p{v}(I_j) \neq \m{\widetilde{K},\widetilde{J}}{s}(I_j)
\]
Let $u\in W_s$ with $\m{\widetilde{K},\widetilde{J}}{s} = \p{\widetilde{K}_{\max}}\p{u}$ and let $y\in \oto{\smash{j}}$ be the index such that we have:
\[
I_y=\p{u}(I_j)
\]
In particular we get:
\[
\m{\widetilde{K},\widetilde{J}}{s}(I_j)= \p{\widetilde{K}_{\max}} I_y = \begin{cases}
I_y \quad &\text{if } y\notin \widetilde{K}_{\max}\\
I_{\min \widetilde{K}_{\max} -1} \quad &\text{if } y\in \widetilde{K}_{\max}
\end{cases}
\]
Since $ n+1 > y\notin \widetilde{K}_{\max}$ implies $y\notin K_{\max}$ we conclude $\m{K,J}{n+1}(I_j)= \m{\widetilde{K},\widetilde{J}}{s}(I_j)$ from:
\[
\m{K,J}{n+1}(I_j) = \p{K_{\max}}\p{{\widetilde{K}_{\max}}}\p{u}(I_j) =\begin{cases}
\p{K_{\max}}I_y \quad &\text{if } y\notin \widetilde{K}_{\max}\\
\p{K_{\max}}I_{\min \widetilde{K}_{\max} -1} \quad &\text{if } y\in \widetilde{K}_{\max}
\end{cases}
\]
Meanwhile the action of $\p{w}$ on $I_j$ is:
\[
\p{w}(I_j) = \p{K_{\max}}\p{v}(I_j) = \p{K_{\max}}(I_x) =\begin{cases}
I_x \quad &\text{if } x\notin {K}_{\max}\\
I_{\min K_{\max} -1} \quad &\text{if } x\in K_{\max}
\end{cases}
\]
We finish the proof with a case-by-case-comparison.
\newline\case{$x\notin {K}_{\max}$:} Then $\p{w}(I_j)= \p{v}(I_j) \neq \m{\widetilde{K},\widetilde{J}}{s}(I_j) = \m{K,J}{n+1}(I_j)$.
\newline\case{$x\in {K}_{\max}$:}
If $y\in \widetilde{K}_{\max}$ then $\p{w}$ and $\m{K,J}{n+1}$ act differently on $I_j$ since $\min \widetilde{K}_{\max} < \min K_{\max}$. So we assume $y\notin \widetilde{K}_{\max}$. We have to show, that $y\neq \min K_{\max} -1$. The only case in which $\min K_{\max}-1$ is not contained in $\widetilde{K}_{\max}$ is $n+1\in J$ and $J_{\max} = K_{\max}$. But in that case $y\leq j \leq s= \min J_{\max} -2= \min K_{\max}-2< \min K_{\max}-1 $ holds.
\end{proof}
\subsubsection{An element in $\mathbf{\f{K}{n}\A{n}\f{J}{n}}$, Proof of Theorem \ref{Dynkin theorem aJI neq 0}}
There is also an alternative description of the idempotents $\g{J}{n}$. A straightforward induction on $n$ now shows:
\begin{lem} \label{Dynkin lemma inductive gJn}
For each non-empty subset $J$ of $\ito{n}$ we have:
\[
\g{J}{n} = \p{J} -\yJ{\N{J}{n}}\p{J}
\]
In particular $\g{J}{n}$ is contained in the ideal $\Jk{k}{n}$ with $k=|J|$.
\end{lem}
\begin{proof}[Proof of Theorem \ref{Dynkin theorem aJI neq 0}.]
Let $K\grn{n} J$ and $k\ldef |K|=|J|$. With the previous lemma, Lemma \ref{Dynkin lemma pJy(NJ)pJ annihilated by z}, we gain the first reduction of the term/sum $\f{K}{n}\p{(K,J)}\f{J}{n}$:
\begin{align*}
\f{K}{n}\p{(K,J)}\f{J}{n} &=\quad \z{k}{n}\g{K}{n} \p{(K,J)} \g{J}{n} \:=\quad \z{k}{n}\g{K}{n}\p{(K,J)}\p{J}\biggl(\p{J}-\yJ{\N{J}{n}}\p{J}\biggr)\: = \quad \z{k}{n}\g{K}{n}\p{(K,J)}\\[3pt]
\shortintertext{From the Lemma \ref{Idempotente Lemma Eigenschaften der y} and the chain description of $\p{(K,J)}$ we conclude next:}
\f{K}{n}\p{(K,J)}\f{J}{n} &=\quad\y{k}{n}\g{K}{n}\p{(K,J)} - \y{k+1}{n}\g{K}{n}\p{(K,J)} \:\in\quad \g{K}{n}\p{(K,J)} +\Jk{k+1}{n}
\end{align*}
We finish the proof by looking closer at $\g{K}{n}\p{(K,J)} =\p{(K,J)}- \yJ{\N{K}{n}}\p{(K,J)}$: on the one hand we have $\p{(K,J)} \notin \Jk{k+1}{n}$ as shown in Lemma \ref{Dynkin lemma pKJ not in Jk k+1}. On the other hand
$\p{(K,J)}- \yJ{\N{K}{n}}\p{(K,J)}\neq 0$ holds by Lemma \ref{Idempotente Lemma mJK yNK hat nicht mJK}. Thus it follows $\f{K}{n}\p{(K,J)}\f{J}{n}\neq 0$.
\end{proof}
\subsection{Proof of the Main Theorem \ref{Dynkin maintheorem}}
By Theorem \ref{Dynkin complete pw orthogonal idempotents} and \ref{Dynkin theorem aJI neq 0} $\set{\f{K}{n}\p{(K,J)}\f{J}{n}}{K\grn{n} J}$ is a linearly independent set with exactly $|\Po_n|$ elements. Thus bijectivity follows from Proposition \ref{Special cases prop normal form of B_n lin.oriented}. To see the multiplicity let $M\grn{n} L$ and $K\grn{n} J$. If $K\neq L$ holds, then $\f{L}{n}$ and $\f{K}{n}$ are orthogonal. Hence $ \Phi(X_{(M,L)})\Phi(X_{(K,J)})=0 = \Phi(X_{(M,L)}X_{(K,J)}) = 0$. If $K=L$ we have the following identities, the fourth identity follows from Lemma \ref{Dynkin lemma pJy(NJ)pJ annihilated by z} and the fifth from the chain description):
\[
\begin{aligned}
\Phi(X_{(M,L)})\Phi(X_{(K,J)}) &= \f{M}{n}\p{(M,K)}\f{K}{n}\f{K}{n}\p{(K,J)}\f{J}{n}\\
&= \f{M}{n}\p{(M,K)}\g{K}{n}\p{(K,J)}\f{J}{n}\\
&=\f{M}{n}\p{(M,K)}\biggl(\p{K}-\yJ{\N{K}{n}} \biggr)\p{K}\p{(K,J)}\f{J}{n}\\
&=\f{M}{n}\p{(M,K)}\p{K}\p{K}\p{(K,J)}\f{J}{n}\\
&=\f{M}{n}\p{(M,J)}\f{J}{n}\\
&= \Phi(X_{(M,L)}X_{(K,J)})
\end{aligned}
\]
\qed
\section{Relations}
For every homomorphism $\varphi\colon M\longrightarrow N$ in $\mo{A}$ we get the following commutative diagram defining $\p{U}\varphi$, which we will call the diagram of $\varphi$ induced by $\p{U}$, with exact rows, which we will call the short exact sequence of $M$ resp. $N$ induced by $\p{U}$:
\begin{equation*}
\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\matrix(m) [matrix of math nodes, row sep=3em, column sep=4em, text height=1.5ex, text depth=0.25ex]
{ 0 & \subf{U}M & M & \p{U}M = {M}/{\subf{U}M} & 0 \\
0 & \subf{U}N & N & \p{U}N = {N}/{\subf{U}N} & 0\\ };
\path[->,font=\normalsize]
(m-1-1) edge (m-1-2)
(m-1-2) edge node[above]{$\iota_M$} (m-1-3)
(m-1-2) edge node[auto, swap]{$\subf{U}(\varphi)=\varphi|_{\subf{U}M}$} (m-2-2)
(m-1-3) edge node[auto, swap]{$\varphi$} (m-2-3)
(m-1-3) edge node[above]{$\pi_M$} (m-1-4)
(m-1-4) edge (m-1-5)
(m-1-4) edge [dotted] node[auto, swap]{$\p{U}(\varphi)$} (m-2-4)
(m-2-1) edge (m-2-2)
(m-2-2) edge node[above]{$\iota_N$} (m-2-3)
(m-2-3) edge node[above]{$\pi_N$} (m-2-4)
(m-2-4) edge (m-2-5);
\end{tikzpicture}
\end{equation*}
Other descriptions of the endofunctor $\subf{U}$ are useful. Obviously for every $A$-module $M$, the module $\subf{U}M$ is the sum over all submodules $X$ of $M$ lying in $\gen{U}$. Therefore $\subf{U}M$ is the image of the evaluation map \mbox{$\mathrm{ev}_{U,M}\colon U \otimes_{\End{A}{U}} \Hom{A}{U,M} \longrightarrow M$} with \mbox{$u\otimes \varphi \mapsto \varphi(u)$}.
Now if $U=S$ is simple then the module $\subf{S}M$ is isomorphic to some $S^{\oplus d}$ and \mbox{$\mathrm{ev}\colon S \otimes_{\End{A}{S}} \Hom{A}{S,\_} \longrightarrow \subf{S}$} is thus even a natural isomorphism.
\begin{proof}[Proof of Proposition \ref{relations proposition}(a).]
If $S$ has no non-trivial self-extensions, then $\subf{S}$ is a torsion radical, i.e.\ a subfunctor of the identity functor such that \mbox{$\subf{S}\bigl(M/\subf{S}M\bigr) = 0$} holds for all $M\in \mo{A}$, and $\im\p{S}=\ker\Hom{A}{S,\_}$ holds.
\end{proof}
We will prove (b) of Proposition \ref{relations proposition} with (and after) the following lemma.
\begin{lem}
Let $S$ and $T$ be two non-isomorphic simple modules. Then $\p{T}\circ \p{S}$ and $\p{S\oplus T}$ are naturally isomorphic if and only if $\Ext{A}{1}{T,S}=0$ holds.
\end{lem}
\begin{proof}
If there exists a non-split short exact sequence $0\longrightarrow S \overset{f}{\longrightarrow} X \longrightarrow T \longrightarrow 0$, then
the functors $\p{T}\circ \p{S} $ and $\p{S\oplus T}$ are not naturally isomorphic since we have:
\[
\p{T}\circ \p{S} (X) = \p{T}\bigl(X/f(S)\bigr) = 0 \neq T \cong X/f(S) = \p{S\oplus T} (X)
\]
Now we assume $\Ext{A}{1}{T,S}=0$. Let $M$ be a module. Since $\subf{S\oplus T}M$ is semi-simple it coincides with $\subf{S}M \oplus \subf{T}M$. For that reason the restriction of the canonical projection \mbox{$\pi\colon M \longrightarrow M/\subf{S}M$} to $\subf{S\oplus T} M $ factors through \mbox{$\pi'\colon \subf{S\oplus T} M\longrightarrow \subf{T} (\p{S}M)$}. By passing to the cokernels we get the following commutative diagram whose exact rows are induced by $\p{S\oplus T}$ and $\p{T}$:
\begin{equation*}
\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}, font=\normalsize]
\matrix(m) [matrix of math nodes, row sep=3em, column sep=2em, text height=1.5ex, text depth=0.25ex]
{0 & \subf{S\oplus T} M & M & \p{S\oplus T}M ={M}/\subf{S\oplus T} M & 0\\
0 & \subf{T}(\p{S}M) & \p{S}M & \p{T}\p{S}M = (\p{S}M)/\bigl(\subf{T}(M/\subf{S}M)\bigr) & 0\\};
\path[->,font=\normalsize]
(m-1-1) edge (m-1-2)
(m-1-4) edge (m-1-5)
(m-1-2) edge (m-1-3)
(m-1-2) edge node[auto] {$\pi'$} (m-2-2)
(m-1-3) edge (m-1-4)
(m-1-3) edge node[auto] {$\pi$} (m-2-3)
(m-2-2) edge (m-2-3)
(m-2-3) edge (m-2-4)
(m-2-1) edge (m-2-2)
(m-2-4) edge (m-2-5)
(m-1-4) edge [dotted] node[auto] {$\hat{\pi}=\hat{\pi}_M$} (m-2-4);
\end{tikzpicture}
\end{equation*}
To see that the epimorphism $\hat{\pi}_M$ is an isomorphism, it suffices to show that $\pi'$ is an epimorphism because of $\ker\pi'\cong \ker\pi$. Since $\Hom{A}{T,M}$ and $\Hom{A}{T,\p{S}M}$ are isomorphic by the assumptions, this follows from:
\[
\subf{T}(\p{S}M) \cong T \otimes_{\End{}{T}} \Hom{A}{T,\p{S}M} \cong T \otimes_{\End{}{T}} \Hom{A}{T,M} \cong \subf{T}M\cong \pi'(\subf{S\oplus T}M)
\]
Moreover we get for every morphism $\varphi\colon M\longrightarrow N$ the following cube,in which the sides on the left commute and in which thus the right square commutes. Thus \mbox{$\hat{\pi}$} is a natural isomorphism.
\begin{equation*}
\begin{tikzpicture}
\matrix (m) [matrix of math nodes, row sep=1.5em, column sep=1.5em]
{ & N & &\p{S\oplus T}N &\\
M & & \p{S\oplus T}M & &0\\
& \p{S}N& & \p{T}\p{S}N &\\
\p{S}M & & \p{T}\p{S}M & &\\};
\path[-stealth, font=\scriptsize]
(m-1-2) edge (m-1-4)
edge [densely dotted] (m-3-2)
(m-1-4) edge node[above=15pt, right] {$\hat{\pi}_N$} (m-3-4)
(m-2-1) edge [-,line width=6pt,draw=white] (m-2-3)
(m-2-1) edge (m-2-3) edge (m-4-1)
(m-2-1) edge node[auto, swap] {$\varphi$} (m-1-2)
(m-3-2) edge [densely dotted] (m-3-4)
(m-4-1) edge [densely dotted] node[auto, swap] {$\p{S}\varphi$}(m-3-2)
(m-4-1) edge (m-4-3)
(m-2-3) edge [-,line width=6pt,draw=white] (m-4-3)
edge node[above=15pt, right] {$\hat{\pi}_M$}(m-4-3)
(m-2-3) edge node[auto, swap] {$\p{S\oplus T}\varphi$} (m-1-4)
(m-2-3) edge [-,line width=6pt,draw=white] (m-2-5)
edge (m-2-5)
(m-4-3) edge node[auto, swap] {$\p{T}\p{S}\varphi$} (m-3-4);
\end{tikzpicture}
\end{equation*}
\end{proof}
\begin{proof}[Proof of Proposition \ref{relations proposition}(b).]
By the previous lemma it suffices to show \newline\mbox{\quad(A)\quad $\p{S\oplus T}\circ \p{T} \sim \p{S}\circ\p{T}$}\quad and \mbox{\quad(B)\quad $\p{S}\circ\p{S\oplus T} \sim \p{S}\circ\p{T}$}.
\newline\textit{Proof of (A).} Since $T$ has no non-trivial self-extensions and is as simple as $S$, a natural isomorphism is induced by:
\[
\subf{S\oplus T}(\p{T}M)\cong \subf{S}(\p{T}M)\oplus \subf{T}(\p{T}M) \cong \subf{S}(\p{T}M)
\]
\newline\textit{Proof of (B).} Let $\alpha\colon \p{T}\longrightarrow \p{S\oplus T}$ be the natural transformation given by the following composition $\alpha_M$ of the canonical epimorphism and isomorphism for every module $M$:
\begin{equation*}
\begin{tikzpicture
\matrix(m) [matrix of math nodes, row sep=3em, column sep=3.5em, text height=1.5ex, text depth=0.25ex]
{\p{T}M={M}/{\subf{T}M} &(M/\subf{T}M)\bigg/\biggl((\subf{S}M\oplus \subf{T}M)/\subf{T}M\biggr) & \p{S\oplus T} M \\};
\path[->>, font=\normalsize]
(m-1-1) edge (m-1-2);
\path[->, font=\normalsize]
(m-1-1.north east) edge [bend left = 12] node[auto]{$\alpha_M$} (m-1-3.north west)
(m-1-2) edge node[above]{$\cong$} (m-1-3);
\end{tikzpicture}
\end{equation*}
We claim, that the natural transformation \mbox{$(\p{S}\alpha_M)_{M\in \mo{A}}\colon \p{S}\circ\p{T}\longrightarrow \p{S}\circ \p{S\oplus T}$} is a natural isomorphism. For this we look at the diagram of $\alpha_M$ induced by $\p{S}$ and consider its exact sequence of kernels and cokernels given by the snake lemma:
\[
0\rightarrow \ker{\subf{S}\alpha_M} \overset{\iota}{\longrightarrow} \ker{\alpha_M}\longrightarrow \ker{\p{S}\alpha_M}
\longrightarrow \coker{\subf{S}\alpha_M} \longrightarrow 0 \longrightarrow \coker{\p{S}\alpha_M}\rightarrow 0
\]
Now $\ker \alpha_M\cong \subf{S}M\in \gen{S}$ is a submodule of $\subf{S}\p{T}M$. Hence the monomorphism $\iota$ is an isomorphism. Moreover $\subf{S} \alpha_M$ is surjective. This is seen by using the assumptions on $S$, which yield the surjectivity of $\Hom{A}{S,\alpha_M}$, and the natural isomorphism of $\subf{S}$ and $S\otimes \Hom{A}{S,\_}$.
\end{proof}
\section{The monoid algebras $\mathbf{k\monoid{Q}}$ and $\mathbf{\B_Q}$ for path algebras $\mathbf{kQ}$}
We now turn towards the class of finite dimensional path algebras. So let $Q=(Q_0,Q_1)$ be a finite acyclic quiver, i.e.\ $Q$ is an oriented graph without oriented cycles and with finite sets $Q_0$ and $Q_1$ of vertices and arrows respectively. We will denote an arrow in $Q$ by $\alpha\colon s\rightarrow t \in Q_1$ or $s\overset{\alpha}{\rightarrow} t$, and a path $t_n\overset{\beta_n}{\leftarrow}\ldots\overset{\beta_{2}}{\leftarrow}t_{1}\overset{\beta_1}{\leftarrow}t$ by $\beta_n\ldots\beta_{2}\beta_1$. The path algebra of $Q$ over a field $k$ is denoted by $kQ$. The category $\mo{kQ}$ and the category $\rep{k}{Q}$ of finite dimensional $Q$-representations over $k$ are equivalent and we will not distinguish between them.
Let $(\free{Q},\konk)$ be the free monoid over $Q_0$, i.e.\ the monoid of words over the alphabet $Q_0$ with the concatenation $\konk$ as multiplication. There is the canonical epimorphism $\rho\colon k\free{Q} \longrightarrow \B_Q$ with $q\mapsto X_q$ for all $q\in Q_0$. We will denote the image of a word $w\in \free{Q}$ under $\rho$ by $X_w$. By $\p{w}$ we denote the image of $X_w$ under the canonical epimorphism $\psi_Q\colon \B_Q\longrightarrow k\monoid{Q}$.
\begin{df}
We call a subset $W$ of $\free{Q}$ an admissible normal form associated with $Q$ if the following conditions hold:
\begin{itemize}
\item[(1)] $\{\emptyset\} \cup Q_0 \subseteq W$
\item[(2)] $B_W\ldef \{X_w\mid w\in W\}$ is closed under (right-) multiplication with the generators $X_t$ of $\B_Q$.
\item[(3)] For all words $v\neq w$ in $W$, there is a $Q$-representation $V$ with $\p{v} V \cong\!\!\!\! \left\vert\right.\; \p{w} V$.
\end{itemize}
\end{df}
Obviously the set $\set{X_w}{w\in \free{Q}}$ is a ${k}$-linear generating system of $\B_Q$. This definition extracts suitable conditions on a subset of $\set{X_v}{v\in \free{Q}}$ to be a basis of $\B_Q$ forcing $\psi_Q$ to be an isomorphism: due to the conditions (1) and (2), $B_W$ is a submonoid of $\B_Q$ which contains the generators $X_q$ of $\B_Q$ and the unit $1$. Hence the $k$-linear span of $B_W$ is the monoid algebra $\B_Q$ itself. Furthermore condition (3) ensures that the elements of $B_W$ are indexed by $W$. Thus $B_W$ is a $k$-linear basis of $\B_Q$ with $|W|$ elements. Moreover the canonical epimorphism $\psi_Q$ is an isomorphism because of condition (3).
The existence of an admissible normal form is not obvious. But finding one summarizes our strategy of proving that the relations are defining in several special cases.
\subsection{Tools for (2) and reductions}\label{subsection tools for (2)}
To begin with one needs a better understanding of the multiplication of two arbitrary monomials in $\B_Q$, that is to say of the defining relations. Since $Q$ is acyclic the third relation (under the two first ones) is equivalent to the following two:
\begin{itemize}
\item $X_tX_sX_t= X_sX_t$ for all $s,t \in Q_0$, such that there is an arrow $\alpha\colon s\rightarrow t$
\item $X_sX_t= X_tX_s$ for all vertices $s,t \in Q_0$ which are not connected by an arrow
\end{itemize}
Therefore the underlying monoid of $\B_Q$ is isomorphic to the Hecke-Kiselman semigroup associated with $Q$ introduced in \cite{MazorchukKiselman}. Let us fix some more notation. We will write $\{v\}$ for the set of the letters occuring in the word $v\in \free{Q}$. For example we have $\{v\}= \{1,5,7,15\}$ if $Q_0=\{1,2,\ldots, 15\}$ and $v= 1\konk5\konk15\konk7\konk1\konk5$. For a subset $M$ of $Q_0$ we will denote by $Q_M$ the full subquiver of $Q$ whose underlying set of vertices is exactly $M$ and we will abbreviate $Q_v\ldef Q_{\{v\}}$. A vertex $t \in Q_0$ is called a sink (source), if no arrow has tail (head) $t$. The condition on $s,t\in Q_0$ in the third relation defining $\B_Q$ could be replaced by requiring $t$ to be a sink in $Q_{s\konk t}$. The defining relations of $\B_Q$ generalize to the following identities in $\B_Q$ (and thus in $k\monoid{Q}$):
\begin{lem}\label{Normalform-lem-generalized relations}
For all $t\in Q_0$ and all words $v,w$ over $Q_0$ we have:
\begin{itemize}
\item[] $X_tX_wX_t=X_wX_t$ \qquad\, if $t$ is a sink in the subquiver $Q_{t\konk w}$
\item[] $X_tX_wX_t=X_tX_w$ \qquad\: if $t$ is a source in the subquiver $Q_{t\konk w}$
\item[] $X_vX_w=X_wX_v$ \qquad\: if there is no arrow between the subquivers $Q_{v}$ and $Q_{w}$
\end{itemize}
\end{lem}
\begin{proof}
The second identity follows by duality from the first one, meanwhile the third one results directly from the others. We prove the first identity by an induction on the length of $w$. If $w=s$ the identity is just one of the defining relations of $\B_Q$. So let $w=u\konk s$ for some $s\in Q_0$ and some word $u$. The induction hypothesis applies now to $s$ and to $u$:
\[
X_tX_wX_t= X_tX_uX_sX_t = X_tX_uX_tX_sX_t= X_uX_tX_sX_t= X_uX_sX_t=X_wX_t
\]
\end{proof}
Essentially all calculations are abbreviated using these generalized relations. For example the finiteness of $\monoid{Q}$ can be deduced from them.
\begin{cor}\label{Special_cases Cor finite}
The $k$-algebra $\B_Q$ is finite dimensional. Hence the monoid $\monoid{Q}$ is finite.
\end{cor}
\begin{proof}
If $Q$ just consists of one vertex, $\B_Q$ is two-dimensional. So now we assume $Q$ to have at least two vertices, pick a sink $s\in Q_0$ and consider the quiver $K\ldef Q_{Q_0\setminus \{s\}}$. Inductively $\B_K\subseteq \B_Q$ is finite dimensional, hence has a finite basis $B$ of monomials over $\set{X_t}{t\in Q_0\!\setminus \!\{s\}}$. Due to Lemma \ref{Normalform-lem-generalized relations} every monomial $X_w \in \B_Q$ lies either in $\B_KX_s\B_K$ (if $s\in \{w\}$) or in $\B_K$ (if $s\notin \{w\}$). Thus $B\cup BX_sB$ is a finite $k$-linear generating system of $\B_Q$.
\end{proof}
The next observations enable us to restrict to isomorphism classes of finite, acyclic and connected quivers without multiple arrows. We will call a subquiver $K=(K_0,K_1)$ of $Q$ ``the quiver reduced by multiple arrows of $Q$`` if $K_0=Q_0$ holds and if there is exactly one arrow between two vertices $s$ and $t$ in $K_1$ whenever there exists (at least) one arrow between $s$ and $t$ in $Q_1$.
\begin{prop}\label{specialcases prop BK and BQ}
Let $K, K'$ and $Q$ be finite, acyclic quivers.
\newline (a)\quad If $K$ is a full subquiver of $Q$, then $\B_K$ is a subalgebra of $\B_Q$.
\newline (b)\quad If $Q$ and $K$ are (anti-) isomorphic, then $\B_Q$ and $\B_K$ are (anti-) isomorphic.
\newline (c)\quad If $K$ is the quiver reduced by multiple arrows of $Q$, then $\B_K$ and $\B_Q$ are isomorphic.
\newline (d)\quad If $K$ and $K'$ are the connected components of $Q$, then $\B_{Q}$ and $\B_{K}\otimes_k \B_{K'}$ are isomorphic.
\end{prop}
The analogous statements for $k\monoid{K}$ and $k\monoid{Q}$ are a priori not clear. However, if there is an admissible normal form associated with $K$ most of them hold (see Proposition \ref{specialcases prop monoidK and monoidQ}).
\subsection{Tools for (3) and a criteria for reductions}\label{subsection tools for (3)}
For any vertex $t\in Q_0$ we abbreviate $\p{t}= \p{t}^{(Q)}\ldef \p{S_t}$ and similar for $\subf{S_t}$.
The projection functor $\p{t}$ on $\rep{k}{Q}$ associated with $S_t$ is easily computed for representations $V$ of $Q$, since the functor $\subf{S_t}$ maps $V$ to that submodule $U$ of the socle $\mathrm{Soc}(V)\subseteq V$, which is given by $U_t= (\mathrm{Soc}V)_t$ and $U_s=0$ for all $s\in Q_0\ohne\{t\}$. Therefore $\p{t}V$ is described by $(\p{t}V)_t=V_t/U_t$ and $(\p{t}V)_s=V_s$ for all $s\in Q_0\ohne\{t\}$ and the respectively induced $k$-linear maps. Consequences, in particular for simples and injective indecomposable representations, are summarized in the next remarks. Let $I_x$ be the injective envelope of $S_x$.
\begin{rem}
A $Q$-representation $V$ is fixed under the action of those $\p{t}$ with $t\notin \mathrm{supp} V \linebreak\ldef \set{q\in Q_0}{V(q)\neq 0}$. Thus $\p{w}{V}= V$ holds for all words $w$ over $Q_0\ohne\mathrm{supp}(V)$. In particular we have for all $w\in \free{Q}$:
\[
\p{w}(S_t) = \begin{cases}
S_t \quad &\text{if }\; t\notin \{w\}\\
0 \quad &\text{if }\; t\in \{w\}\\
\end{cases}
\]
\end{rem}
\begin{rem}
The action of $\p{t}$ on the injective indecomposable representation $I_x$ is:
\[
\p{t}(I_x) = \begin{cases}
I_x\quad &\text{if }\; x\neq t\\
I_t/S_t = \bigoplus_{s\rightarrow t} I_s \quad&\text{if }\; x=t
\end{cases}
\]
Hence $I_x$ is fixed under $\p{w}$ for all words $w$ over $Q_0$ not containing $x$.
\end{rem}
Let $K$ be a subquiver of $Q$ and $F\colon\rep{k}{K} \rightarrow \rep{k}{Q}$ the canonical embedding functor. Recall that for every $K$-representation $U$ the $Q$-representation $FU$ is defined by setting for all $t\in Q_0$ and $\alpha\in Q_1$:
\[
(FU)_t \ldef \begin{cases}
U_t \quad &\text{if } t\in K_0\\
0\quad &\text{otherwise}
\end{cases} \qquad \text{and } \quad (FU)_\alpha \ldef \begin{cases}
U_\alpha \quad &\text{if } \alpha\in K_1\\
0\quad &\text{otherwise}
\end{cases}
\]
\begin{lem}\label{Special_cases lem embedding functor}
Let $K$ be a subquiver of $Q$, $x\in K_0$ and $v, w\in\free{K}$.
\newline (a)\quad The functors $F\p{x}^{(K)}$ und $\p{x}^{(Q)}F$ are naturally isomorphic.
\newline (b)\quad If there is a $K$-representation $V$ with \mbox{$\p{v}^{(K)}V\ncong \p{w}^{(K)}V$}, then $\p{v}^{(Q)}$ and $\p{w}^{(Q)}$ are not naturally isomorphic.
\end{lem}
\begin{proof}
We apply $F$ to the short exact sequence of a $K$-representation $U$ induced by $\p{x}^{(K)}$ and get the short exact sequence:
\[
0\longrightarrow F\subf{x}^{(K)} U \longrightarrow FU \longrightarrow F\p{x}^{(K)} U\longrightarrow 0
\]
This is already the short exact sequence of $FU$ induced by $\p{x}^{(Q)}$ because of \mbox{$F\subf{x}^{(K)} U=\subf{x}^{(Q)}FU$} and the uniqueness (up to isomorphism) of the cokernel.
Therefore (a) holds. In particular, we conclude (b) from: $\p{v}^{(Q)}FV \cong F\p{v}^{(K)} V \ncong F\p{w}^{(K)} V \cong \p{w}^{(Q)}F V$.
\end{proof}
\begin{prop} \label{specialcases prop monoidK and monoidQ}
Let $K,K'$ and $Q$ be finite, acyclic quivers. Assume that there are admissible normal forms $W$ and $W'$ associated with $K$ and $K'$ respectively.
\newline (a)\quad If $K$ is a full subquiver of $Q$, then $k\monoid{K}$ is a subalgebra of $k\monoid{Q}$.
\newline (b)\quad If $Q$ and $K$ are isomorphic, then $k\monoid{K}$ and $k\monoid{Q}$ are isomorphic.
\newline (c)\quad If $K$ is the quiver reduced by multiple arrows of $Q$, then $k\monoid{K}$ and $k\monoid{Q}$ are isomorphic.
\newline (d)\quad If $K$ and $K'$ are the connected components of $Q$, then $k\monoid{Q}$ and $k\monoid{K}\otimes_k k\monoid{K'}$ are isomorphic.
\end{prop}
\begin{proof}
By the assumption on $K$ we have $k\monoid{K}\cong \B_K$. In case (a) or (c), the assignment \linebreak\mbox{$\p{x}^{(K)}\mapsto \p{x}^{(Q)}$} for all $x\in K_0$ extends to a homomorphism $k\monoid{K}\longrightarrow k\monoid{Q}$. Its image is \linebreak linearly spanned by the monomials $\p{v}^{(Q)}$ with $v\in W$. But by the previous lemma, Lemma \ref{Special_cases lem embedding functor}, the \linebreak tuple $(\p{v}^{(Q)}\:|\:v\in W)$ is also linearly independent. Thus the assertions (a) - (c) hold for dimensional reasons. Meanwhile an admissible normal form associated with $Q$ in case (d) is given by $W\konk W'$. Therefore $k\monoid{Q}\cong \B_{Q}$ holds and (d) follows with Proposition \ref{specialcases prop BK and BQ}(d).
\end{proof}
\subsection{The linearly oriented Dynkin quiver of type $\mathbf{A}$}
We denote by $\Po_n$ the poset of the product order $\grn{n}$ on the powerset of $\ito{n}$ defined in the introduction. For two intervals $J$ and $I$ in $\ito{n}$ we define $J\groint I$ by requiring $\min J > \min I$ and $\max J > \max I$, in particular $\{\max J, \max I\}\grn{n} \{\min J, \min I\}$. \label{specialcases_groint} The poset $\Po_n$ is in bijection with the set consisting of tuples of intervals $J_r \groint \ldots \groint J_1$ in $\ito{n}$. For an interval $J= \{i,i+1,\ldots,j-1,j\}$ of positive integers let $J$ denote the word $i\konk i+1\konk \ldots\konk j-1\konk j$ as well. The monomial $X_J\in \B_{Q_n}$ is an idempotent. More precisely for all $k\in J$ we have $X_JX_k = X_J$ since $k$ is a source in the subquiver $Q_{k\konk k+1\konk\ldots \konk j-1\konk j}$ (Lemma \ref{Normalform-lem-generalized relations}). We will meet a generalisation of these idempotents to arbitrary finite quivers without oriented cycles to determine the radical of $\B_Q$ in the next section.
\begin{prop} \label{Special cases prop normal form of B_n lin.oriented}
An admissible normal form of $\B_{Q_n}$ is
\[
W_n \ldef \set{J_r\konk \ldots \konk J_1}{r\in \oto{n},\:\: J_r \groint \ldots \groint J_1 \:\text{ intervals in }\: \ito{n}}
\]
\end{prop}
So $\B_{Q_n}$ and $k\monoid{Q_n}$ are isomorphic and have the dimension $C_{n+1}=\frac{1}{n+2}\binom{2(n+1)}{n+1}$ by characterization 6.19.aa. in \cite{Stanley} of the $n+1^{\mathrm{th}}$ Catalan number $C_{n+1}$.
\begin{proof}
Condition (1) is easily verified by looking at $r=0$ and $r=1$. To check condition (2) we just state a multiplication rule:
\bigskip
\newline\textit{Let $n\in \NN$. For all intervals $J$ and $L_1\kleint \ldots \kleint L_s$ in $\ito{n}$ and for $L_0\ldef \emptyset \rdef L_{s+1}$ we define indices $y=y(J,L_1,\ldots,L_s)$ and $ z=z(J,L_1,\ldots,L_s)$ by:
\[
\begin{aligned}
y &\ldef \begin{cases}
\max \set{x\in \ito{s}}{J\groint L_x \text{ and } J\cap(L_x +1) = \emptyset}\quad&\text{ if } L_1\kleint J \text{ and } J\cap(L_1 +1) = \emptyset\\
0\quad&\text{ else}
\end{cases}\\
\text{and} &\\
z &\ldef \begin{cases}
y+1\quad&\text{ if } J\kleint L_{y+1} \\
\min \set{x\in \{y+2,\ldots,s\}}{ L_{y+1} \cup J \kleint L_x}\quad&\text{ if } J \nkleint L_{y+1} \text{ and } J\cup L_{y+1} \kleint L_s\\
s+1\quad&\text{ else, thus if } J \nkleint L_{y+1} \text{ and } J\cup(L_{y+1}) \nkleint L_s\\
\end{cases}\\
\end{aligned}
\]
Then the product of $X_{ {L_1}\ldots {L_s}}$ and $X_{J}$ is given by:
\[
\begin{aligned}
X_{ {L_s}\ldots {L_1}}X_{J}&= X_{ {L_s}\ldots {L_z} {(J\cup \bigcup_{t=y+1}^{z-1} L_t)} {L_y}\ldots {L_1}}
&=\begin{cases}
X_{ {L_s}\ldots L_{y+1} {J} L_{y}\ldots {L_1}}\quad&\text{if } J\kleint L_{y+1}\\
X_{ {L_s}\ldots {L_z} {(J\cup L_{y+1})} {L_y}\ldots {L_1}}\quad&\text{otherwise}
\end{cases}\quad
\end{aligned}
\]
}
The proof is a lengthy but straightforward induction on $s$ requiring case by case analysis.
Now we turn towards condition (3). For this it suffices to consider the injective indecomposable $Q_n$-representations $I_j$ for $j\in \oto{n}$. We set $I_0\ldef 0$.
Because of $\p{j}I_j=I_{j-1}$ we get inductively on $r$ for all intervals $J_r\groint\ldots\groint J_1$ in $\ito{n}$:
\[
\p{J_r\ldots J_1}I_j = \begin{cases}
I_{\min{J_a} -1} \quad &\text{if } j\in \bigcup_{x=1}^r J_x \text{, where } a\in \ito{r} \text{ is min. with } j\in J_a\\
I_j &\text{otherwise}
\end{cases}
\]
Let $J_r\groint \ldots \groint J_1$ and $L_s\groint\ldots \groint L_1$ be two distinct tuples of intervals in $\ito{n}$. Without loss of generality we can pick an index $a\in\ito{r}$ minimal with respect to $J_a$ being distinct from $L_1,\ldots, L_s$. Then we have:
\begin{align*}
\p{J_r\ldots J_1}( I_{\max J_a}) &= I_{\min{J_a} -1}\\
\shortintertext{and}
\p{L_s\ldots L_1}( I_{\max J_a}) &= \begin{cases}
I_{\min{L_b} -1} \: &\text{if } \max J_a \in \bigcup_{x=1}^s L_x \text{, where } b\in \ito{s} \text{ is min. with } \smash{\max J_a \in L_b}\\
I_{\max J_a} &\text{otherwise}
\end{cases}
\end{align*}
We are done if $\max J_a \notin \bigcup_{x=1}^s L_x$. So we assume $\max J_a \in \bigcup_{x=1}^s L_x$. We are done as well if $\min{L_b} \neq \min{J_a}$. Thus let $\min{L_b} = \min{J_a}$. Now we consider the action on $I_{\max L_b}$:
\begin{align*}
\p{L_s\ldots L_1}I_{\max L_b} &= I_{\min L_b -1} = I_{\min J_a -1} \\
\shortintertext{and}
\p{J_1\ldots J_r} I_{\max L_b} &= \begin{cases}
I_{\min J_c-1} \quad &\text{if } \max L_b \in \bigcup_{x=1}^r J_x \text{, where } c \in \ito{r} \text{ is min. with } \smash{\max L_b \in J_c}\\
I_{\max L_b} \quad & \text{otherwise}
\end{cases}\\
\end{align*}
In the first case (for $\max L_b$ ) the inequality holds because of $\max J_c \geq \max L_b > \max J_a$, so $J_c\groint J_a$, hence $\min J_c > \min J_a$. In the second one it holds due to $\max{L_b} > \min L_b-1$.
\end{proof}
The defining relations for $\monoid{Q_n}$ are the same as for the monoid $\mathrm{NDPF}_{n+1}$ of non-decreasing parking functions (see \cite{Hivert} or \cite{MazorchukKiselman}) which is generated by the functions $\pi_j\ldef\binom{1\ldots j-1\, j \, j+1\ldots n+1}{1\ldots j-1\, j \, j\hphantom{+1}\ldots n+1}$. In fact, if $\beta$ is the bijection $\{I_0,I_1,\ldots,I_n\}\longrightarrow \ito{n+1}, I_j\mapsto j+1$ the proof of condition (3) yields the isomorphism $\monoid{Q_n}\longrightarrow \mathrm{NDPF}_{n+1}, \p{v}\mapsto \beta\p{v}\beta^{-1}$, which is the extension of the assignment $\p{j}\mapsto \pi_j$. Thus the full subcategory of the category of covariant functors on $\mo{kQ}$ containing the elements $\p{t_1}\p{t_2}\ldots\p{t_r}$ for $r\geq 0$ and vertices $t_1,t_2,\ldots, t_r\in Q_0$ categorifies the monoid $\mathrm{NDPF}_{n+1}$.
\subsection{Gluing on a sink}
We start with $n$ finite acyclic and pairwise disjoint quivers $Q(1),\ldots, Q(n)$ and pick some vertices \linebreak$p_1^{(1)},\ldots,p_{r_1}^{(1)} \in Q(1),\ldots, p_1^{(n)},\ldots,p_{r_n}^{(n)} \in Q(n)$. Then we consider the quiver $Q$ arising from gluing these quivers together on a new vertex $s$ over new arrows $\alpha_i^{(j)}\colon p_i^{(j)}\rightarrow s$. The shape of $Q$ is sketched below. We will moreover denote by $\Q(j)$ the full subquiver of $Q$ with the vertices $Q(j)_0\cup \{s\}$.
\begin{equation*}
\begin{tikzpicture
\matrix(m) [matrix of math nodes, font=\scriptsize, row sep=0.2cm, column sep=0.4cm, text height=1.5ex, text depth=0.25ex]
{ & & & &s & & & & & &\\
&p_1^{(1)} & & & & & & & &p_{r_n}^{(n)} &\\
& & p_{r_1}^{(1)} &\hphantom{h} &\hphantom{h} & & & p_1^{(n)} & & &Q(n) \\
&Q(1) & & & & & & & & &\\
& & & & & & & & & &\\};
\path[->]
(m-2-2) edge (m-1-5)
(m-3-3) edge (m-1-5)
(m-2-10) edge (m-1-5)
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(m-3-8) edge[loosely dotted] (m-2-10)
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(m-2-2) edge[loosely dotted] (m-3-3);
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\begin{scope}[rotate=40]
\draw[densely dotted,xshift=-2.8cm,yshift=2cm](0,0) ellipse (1.3cm and 1.7cm);
\end{scope};
\begin{scope}[rotate=90]
\draw[densely dotted,xshift=0cm,yshift=-2.8cm](0,0) ellipse (1.2cm and 2.1cm);
\end{scope};
\end{tikzpicture}
\end{equation*}
\begin{lem}\label{Normalformen das Lemma Induktionsargument}
Assume that for every $j\in \ito{n}$ we have admissible normal forms $\W(j)\subseteq \free{Q(j)}$ and $\tW(j)\subseteq \free{\Q(j)}$ associated with $Q(j)$ and $\Q(j)$ respectively, such that $\W(j)\subseteq \tW(j)$ holds and $s$ appears at most once in any word of $\tW(j)$.
Then an admissible normal form $W$ associated with $Q$ consists of the $\prod_{j\in\ito{n}} |\W(j)| + \prod_{j\in\ito{n}} (|\tW(j)|-|\W(j)|)$ words:
\begin{align*}
\text{(a)}\qquad w_1\konk w_2\konk\ldots\konk w_n \qquad &\text{with }\: w_j\in \W(j) \:\text{ for all }\: j\in \ito{n}\\
\shortintertext{and}
\text{(b)}\qquad y_1\konk y_2\konk \ldots\konk y_n \konk s \konk z_1\konk z_2\konk \ldots\konk z_n \qquad & \text{with }\: y_j\konk s \konk z_j \in \tW(j)\setminus\! \W(j)\: \text{ for all }\: j\in \ito{n}
\end{align*}
Thus $\B_Q \cong k\monoid{Q}$ holds. Moreover the dual assertions holds.
\end{lem}
\begin{proof}
Conditions (1) and (2) are verified straightforwardly using the generalized relations. For condition (3) it suffices, by the remarks in subsection \ref{subsection tools for (3)}, to consider two words $v\neq w \in W$ both being either of type (a) or of type (b). In each case there are an index $j\in \ito{n}$ and subwords $v_j, w_j\in \tW(j)$ of $v$ and $w$ respectively such that $v_j\neq w_j$ holds. Now employ the assumption on $\tW(j)$ and Lemma \ref{Special_cases lem embedding functor}.
\end{proof}
A direct application of this lemma gives an admissible normal form associated with the $m$-subspace quiver $T_m$, which is the connected quiver with exactly one sink $s$ and $m$ sources enumerated by $1,\ldots, m$: here $Q(j)$ corresponds just to the vertex $j$ and $\Q(j)$ to $j\rightarrow s$. An admissible normal form associated to the latter is $\{\emptyset, j, s, js, sj\}$ which contains the normal form associated with $j$, i.e.\ $\{\emptyset, j\}$. To fix an order on the vertices we denote for every subset $J= j_1<\ldots <j_k$ of $\ito{m}$ the word $j_1\konk\ldots \konk j_k$ with $w(J)$. Now an admissible normal form associated with $T_m$ has $2^m+3^m$ elements:
\[
\set{w(J)}{J\subseteq\ito{m}}\: \cup\: \set{w(I)\konk s\konk w(J) }{I,J \subseteq \ito{m}, I\cap J = \emptyset}
\]
This can be extended to the star quiver, since we have admissible normal forms associated with its branches, i.e linearly oriented Dynkin quivers of type $A$.
\subsubsection{Tree quivers with a specific orientation}
Furthermore, Lemma \ref{Normalformen das Lemma Induktionsargument} is the induction step for tree quivers with a specific orientation: we will call $Q$ an admissible tree quiver, if each crossing of $Q$, i.e.\ a vertex whose entry degree or exit degree is at least $2$, is either a sink or a source. (So the linearly oriented Dynkin quivers $D_4$ are not subquivers of $Q$.) One can endow every tree with an orientation to obtain an admissible tree quiver. In particular every tree with a bipartite orientation is an admissible tree quiver. Note also that every Dynkin quiver of type $A$ is an admissible tree quiver.
\begin{theo}\label{Special cases theo normal form of admissable trees}
There is an admissible normal form associated with any admissible tree quiver $Q$. In particular the relations are defining for $\monoid{Q}$.
\end{theo}
\begin{proof}\label{Normalformen zulaessige Baumkoecher}
The case $Q$ being $Q_n$ for some $n$ is already done. So assume $Q$ not to be a linear oriented Dynkin quiver of type $A$. In particular we can pick a crossing $s$ of $Q$. To apply Lemma \ref{Normalformen das Lemma Induktionsargument} we just have to check, whether the assumptions hold for the subquivers which are linked to $s$ by one arrow. These subquivers (and their extensions with $s$) are again admissible tree quivers. So it suffices to show:
\newline\textit{Let $K$ be an admissible tree quiver, $y\in K_0$ and $\K$ an extension of $K$ by one (new) vertex $s$ and one (new) arrow $ y\leftarrow s$ or $y\rightarrow s$, so that $\K$ is again an admissible tree quiver.
Then there are admissible normal forms $\W\subseteq \free{K}$ and $\tW\subseteq \free{\K}$ associated with $K$ and $\K$ respectively, such that $\W\subseteq \tW$ and $s$ appears at most once in any word of $\tW$.}
\newline This is proven by induction on the number of vertices of $K$ using Lemma \ref{Normalformen das Lemma Induktionsargument}.
\end{proof}
The induction step, i.e.\ Lemma \ref{Normalformen das Lemma Induktionsargument}, provides a procedure for gaining an admissible normal form associated with $Q$. We illustrate this by the special case of bipartite Dynkin quivers $K_n$ of type $A_n$. Depending on whether $n$ is even or odd $K_n$ has up to anti-isomorphism one of the following shapes:
\begin{equation*}
\begin{tikzpicture
\matrix(m) [matrix of math nodes, row sep=0.3em, column sep=0.5em, font=\scriptsize]
{& 2 & & 4 & & & \vphantom{n}n-1 & & & & & 2 & & 4 & & & n-1 &\\
& & & &\ldots & & & & & \text{or} & & & & &\ldots & & & \\
1& & 3 & & & \vphantom{n-2}n-2 & &n & & &1 & & 3 & & & n-2 & &n\\};
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(m-3-16) edge (m-1-17)
(m-3-18) edge (m-1-17);
\end{tikzpicture}
\end{equation*}
Admissable normal forms $\W_1\subseteq \W_2 \subseteq \W_3$ associated with $K_1\subseteq K_2 \subseteq K_3$ respectively are:
\begin{align*}
\{\emptyset, 1\} &\subseteq \{\emptyset, 1, 2, 1 2 , 2 1\} \subseteq \{{\emptyset}, {1}, {3}, {1 3}, {2}, {1 2}, {2 1}, {3 2}, {2 3}, {1 2 3}, {3 2 1}, {1 3 2}, {2 1 3}\}
\end{align*}
For $ n\geq 4$ the vertex $n-1$ is a crossing and either a sink or a source. Inductively we have admissible normal forms $\W_{n-2} \subseteq \W_{n-1}$ associated with $K_{n-2}\subseteq K_{n-1}$ respectively such that $n-1$ appears at most once in any word of $\W_{n-1}$. On the other hand we have admissible normal forms $\{\emptyset, n\} \subseteq \{\emptyset, n, n-1, n\konk n-1 , n-1\konk n\}$ associated with the quivers $n$ and ${n-1} \rightarrow n$ (or $n-1\leftarrow n$) respectively. Now the Lemma \ref{Normalformen das Lemma Induktionsargument} with $s=n-1$ yields the admissible normal form
\[
\W_n\ldef \W_{n-2} \:\cup\: \W_{n-2}\konk n \bigcup \W_{n-1}\setminus\! \W_{n-2} \bigcup n\konk(\W_{n-1}\setminus\! \W_{n-2}) \bigcup (\W_{n-1}\setminus\! \W_{n-2})\konk n
\]
which contains $\W_{n-1}$ and fulfils the condition on the appearance of $n$. Therefore the dimension $|\W_n|$ of ${k}\monoid{K_n}\cong \B_{K_n}$ can be calculated over the recurrence relation:
\[
|\W_{n}|= 2 |W_{n-2}| + 3(|W_{n-1}| - |W_{n-2}|) = 3|W_{n-1}| -|W_{n-2}|
\]
Hence $(|\W_j|)_{j\in\NN}$ corresponds to the partial sequence $(F_{2j+1})_{j\in\NN}$ of the Fibonacci-sequence \linebreak$(F_n)_{n\in \NN}$ with $F_1=F_2=1$.
|
{
"timestamp": "2012-07-03T02:04:11",
"yymm": "1203",
"arxiv_id": "1203.1943",
"language": "en",
"url": "https://arxiv.org/abs/1203.1943"
}
|
\section{Introduction}
\setcounter{equation}{0}
A great number of physical laws that freshmen are confronted with in the physics lab are linear in nature or can be linearized.
The parameters of these relations are physical constants or characteristics of the system under study.
Most often the aim of the lab is to estimate these constants and if not,
the estimation of these constants is an intermediate step.
Therefore, linear regression is a nearly inevitable statistical tool.
It enables to estimate the constants and to evaluate the validity of the linear relation.
A non-constant linear function has two formulations, it can be inversed and its inverse is linear with as
slope the inverse of the slope of the original function;
if $ y = a x + b$ with $a \ne 0$ then $ x = (1/a) y - b/a$.
It is common in physics labs to use the formulation of a linear relations that emerges naturally as a basis for the regression inconsiderate whether $a$ or $1/a$ is of interest.
Based on theoretical considerations or habit the consequence of an action is place on the left side as a function of the cause(s) on the right side.
If the formulation presented to the students is $ y = a x + b$ and $1/a$ has to be estimated a great number of students first estimates $a$ and subsequently determines the inverse as an estimate for $1/a$.
The assessment of the uncertainty is based on the uncertainty of $a$ and standard error propagation.
We will refer to this approach as the ``inverse slope'' strategy in contrast with the ``direct'' estimation strategy.
Both strategies of estimation are compliant with the international standard ``ISO Guide to the Expression of Uncertainty in Measurement'' (GUM) that provides a basis for the assessment of uncertainty.\cite{GUM1995} The GUM however does not treat regression analysis.
The very accessible introduction to the treatment of experimental data by Young assumes that the measurement uncertainty of the argument in the linear relation can be neglected.\cite{Young1962}
Also more recent treatises of uncertainty in physical measurements attribute the error to one of the variables only -- see e.g.\ \cite{Squires1985}.
Rabinovich explicitly mentions the strategy of inversion in a calibration setting.\cite{Rabinovich2005}
It must be clear that if standard linear regression requires the measurement uncertainty to be insignificant for all but one and only one variable,
the variable to be predicted, the advisable strategy starts with using the variable with the ``least'' measurement uncertainty as argument in the linear relation.
However, measurement uncertainty is dependent on the unit of measurement.
Therefore, if different measurement units are used for the different variables involved, comparison of uncertainties as such is not possible.
The obvious method to eliminate the measurement units is the use of relative errors, yet these errors are not shift invariant.
Moreover, whenever the observed values are near zero the relative error will tend to be high.
Therefore, relative errors are not a good criterion to determine which variable(s) to use as an argument.
When looking at standard multivariate statistical literature the error in the linear regression model is not specifically attributed to one or the other variable -- see e.g.\
\cite{Bhattacharyya1977}.
In Foranasini uncertainty is attributed to both variables involved and shifted to one side of the equation.\cite{Fornasini2008}
Gill pleads to use the regression in a direction opposite to the natural causality when prediction is inverse to the causation.\cite{Gill1987}
In what follows we will follow the strategy of Foranasini and stress the importance of choosing the appropriate formulation when estimating constants describing a system.
In contrast to Gill our paper focuses on the proximity of zero as a disturbing factor in the inversion.
Also using elementary mathematical tools inversion bias is revealed.
The problem will first be approached from a theoretical point of view and illustrated through the simulation of a classical experiment for estimating the
gravitational constant.
\section{Theoretical considerations}
\setcounter{equation}{0}
\subsection{Definition of the problem}
Let $y = a x + b$ with $a \ne 0$, $a$ and $b$ are unknown, and $1/a$ is of interest.
\subsection{Regression model}
Either the values of the variables $y$ and $x$ are measured simultaneously, either one is set and the other measured.
Errors will occur when a variable is measured and also if a variable is set,
the exact effective value will only be known up to an error:
\begin{equation}
\begin{array}{ccc}
X = x + \varepsilon_x & \mbox{ and } & E[\varepsilon_x]=0 \mbox{ , } \\
\end{array}
\label{MF1}
\end{equation}
\begin{equation}
\begin{array}{ccc}
Y = y + \varepsilon_y & \mbox{ and } & E[\varepsilon_y]=0 \mbox{ , } \\
\end{array}
\label{MF2}
\end{equation}
where $\varepsilon_x$ and $\varepsilon_y$ are independent.
After substitution of $x$ and $y$ into the linear relation the following regression models are obtained:
\begin{equation}
\begin{array}{ccc}
Y = a X + b + [\varepsilon_y - a \varepsilon_x ] & \mbox{ let } & \varepsilon_Y=[a \varepsilon_x -\varepsilon_y] \mbox{ (notation Y/X) , } \\
\end{array}
\label{RM1}
\end{equation}
\begin{equation}
\begin{array}{ccc}
X = (1/a) Y - b/a + [ \varepsilon_x -\varepsilon_y/a] & \mbox{ let } & \varepsilon_X=[ \varepsilon_x -\varepsilon_y/a] \mbox{ (notation X/Y) . } \\
\end{array}
\label{RM2}
\end{equation}
Notice that $E[\varepsilon_X] = E[\varepsilon_Y]=0$ and if $\varepsilon_x$ and $\varepsilon_y$ are normally distributed then so are $\varepsilon_X$ and $\varepsilon_Y$.
Standard OLS linear regression requires a linear relation between the variable predicted and the variable measured up to a random variable.
Equations \ref{RM1} and \ref{RM2} however show that in both formulations the variable used as predictor is itself a random variable correlated with the ``error'' terms $\varepsilon_X$ and $\varepsilon_Y$.
The results in a bias towards zero of the coefficient in the linear relation.
This phenomena is well known and generally referred to as ``error in variables'' problem -- see e.g. \cite{Fuller1987}.
In order to use ordinary least squares regression the measurement error of the variable used as argument in the linear relation has to be ``sufficiently small''.
An elaborate treatment and approximation of error made can be found in \cite{Davies1975}.
Most often it is not appropriate to introduce these techniques at a general physics course and repeated measurements are used to reduce the measurement error.
In what follows we will suppose that the error on the arguments are sufficiently reduced through repeated measurement.
If the specific laboratory setting does not allow for precise measurements other more appropriate statistical techniques involving generalized linear models should be used.
\subsection{Bias}
If a random variable $u$ has a continuous density function $f$ which is
strictly positive at zero, it can be shown that $E[1/u]=\int_{-\infty}^{+\infty}(1/u) f(u) du$ does not exist.
Follows that the bias of $1/\hat{a}$ , with $\hat{a}$ the regression estimator of $a$, does not exist.
Although measurement errors are most often assumed to be normally distributed sticking to the former result is not satisfactory.
Errors can not become arbitrarily large in practice and when a value is unrealistic the measurement is discarded and repeated.
Therefore, let us assume that $\hat{a}$ is symmetrically distributed about its expectation $a$ and
has only non zero density in a finite interval $[a-d,a+d]$ that does not contain zero.
Note that as a consequence, the variance $\sigma^2$ exists.
Consider the second order Taylor expansion of $1/u$ at $u=a$:
\begin{equation}
\begin{array}{ccc}
\frac{1}{u} = \frac{1}{a} - \frac{1}{a^2} (u-a) + \frac{1}{\xi^3} (u-a)^{2} & \mbox{ with } & \xi \in ]u,a[ \\
\end{array}
\label{G2}
\end{equation}
\begin{equation}
\begin{array}{ccc}
\frac{1}{u} - \frac{1}{a} + \frac{1}{a^2} (u-a) = \frac{1}{\xi^3} (u-a)^{2} & \mbox{ with } & \xi \in ]u,a[ \\
\end{array}
\label{G3}
\end{equation}
Follows:
\begin{equation}
\begin{array}{ccc}
\frac{1}{u} - \frac{1}{a} + \frac{1}{a^2} (u-a) > \frac{1}{(a+d)^3} (u-a)^{2} & \mbox{ for all } & u \in $]a-d,a+d[$ \\
\end{array}
\label{G4}
\end{equation}
The expectation of the left and the the right side of the inequality results in:
\begin{equation}
\begin{array}{c}
E[\frac{1}{u}] - \frac{1}{a} > \frac{1}{(a+d)^3} \sigma^2 \\
\end{array}
\label{G5}
\end{equation}
A large variance for $u$ will therefore result in a large bias.
Bias does not tell the whole story.
Most often we are interested in a confidence interval.
An interval estimate of $a$ can be transformed into an interval estimate of $1/a$ without loss of accuracy.
An interval that contains $a$ and does not contain zero is mapped to an interval that contains $1/a$ through $1/u$.
A process that generates an interval that will contain $a$ with probability $p=95\%$, generates after transformation $1/u$ an interval about $1/a$ with a $p=95\%$ probability.
Such an interval is not presented as ``estimated value $\pm$ error'', expected from a physics experiment.
When the classical error propagation is used to transform a confidence interval about $\hat{a}$ into a confidence interval about $1/\hat{a}$,
two observations have to be made:
$1/\hat{a}$ is not an unbiased estimator of $1/a$ and for ``small'' values of $u$ the function $1/u$ is not well approximated by its tangent, the approximation being basis of the classical error propagation.
\section{Estimation of the gravitational constant}
\setcounter{equation}{0}
A body in the neighborhood of the earth experiences a nearly constant gravitational force and its acceleration is constant provided that all other forces are neglected.
This constant acceleration parameter is called the gravitational constant $g$.
Consider an object that falls from a height $h$, starting at rest, and the time $t$ it needs to cross this distance.
After a small manipulation of Newton's law,
the following relation between the height $h$ an object starting at rest falls and the time $t$ needed to fall this height is obtained:
\begin{equation}\label{Reg_h_t2}
h = \frac{1}{2} \, g \, t^2 \mbox{ . }
\end{equation}
When the height $h$ and the square of the fall time $t^2$ are obtained for different values for the height it is possible to estimate the gravitational constant through linear regression.
The first approach consists of a linear regression $y = a\,x$ with $h$ used as $y$ variable,
$t^2$ as $x$ variable, and $g=2a$.
Equation \ref{Reg_h_t2} can also be rephrased:
\begin{equation}\label{Reg_t2_h}
t^2 = 2 \, \frac{1}{g} \, h \mbox{ . }
\end{equation}
This leads to a second approach to the estimation of $g$.
First $2/g$ is estimated through linear regression $y=a\,x$ with $t^2$ used as $y$ variable and $h$ as $x$ variable.
The estimate of $g$ is obtained through inversion and doubling of the estimate of $a$: $g = 2/a$.
\subsection{Numerical simulation experiment}
All experiments were conducted using MatLab R2010a (The Mathworks, Inc., Natick, Massachusetts).
A steel bullet is dropped from 10 uniform randomly chosen heights between $0.4m$ en $1m$ in order to estimate the gravitational constant --see Fig.\ \ref{Fig:Drop1}.
\begin{figure}
\begin{center}
\begin{tabular}{c}
\includegraphics*[width=4cm]{./Gravitational_constant_drop_V1.eps}
\end{tabular}
\end{center}
\caption{\small\it A steel bullet is dropped from a height $h$ and the fall time $t$ is measured. \label{Fig:Drop1}}
\end{figure}
To simulate this experiment, 10 random numbers between $0.4$ and $1$ are generated.
For each number a value for the fall time $t$ is calculated using equation \ref{Reg_t2_h} with $g=9.81 m/s^2$.
Height and time are both measured with a measurement error.
In what follows, the measurement error of height and time are assumed to be normally distributed with mean zero and standard deviation $\sigma_h$ and $\sigma_t$ respectively.
The generated values for height and time are disturbed by adding values generated from a normal distribution with zero mean and standard deviation $\sigma_h$ and $\sigma_t$ respectively.
The standard deviation for the time $t$ is chosen to be $0.0001s$.
This relatively small error models a very accurate time measurement.
The former experiment is repeated 1000 times for different values for $\sigma_h$.
For a standard deviation $\sigma_h=0.01 m$, ten repetitions of 1000 experiments were performed.
\noindent {\bf Results.}
The results of the experiments are presented in tables \ref{Tab:1}, \ref{Tab:2} and \ref{Tab:3}.
Although for all criteria the observed standard deviations are comparable for
both estimation methods, the standard deviation of the difference between the gravitational constant and the estimated gravitational constant is systematically smaller when estimated based on the regression $h$ given $t^2$ ($h/t^2$).
The error decreases with decreasing error in height for both methods, but the error obtained through the regression $t^2$ given $h$ ($t^2/h$) is systematically higher than the error obtained through $h/t^2$.
For all chosen values for the standard deviation $\sigma_h$ the mean difference of the gravitational constant and the estimated gravitational constant is positive, although never statistically significant ($p<.05$).
When repeating the series of 1000 experiments 10 times for $\sigma_h = 0.01m$ only one of the mean differences was negative --see table \ref{Tab:2}.
The estimation of the gravitational constant based on ($h/t^2$) is positively biased (Sign test N=10, p=0.21).
\begin{table}
\begin{center}
\begin{tabular}{ccccccc}
height $h$ & \multicolumn{2}{c}{estimate $g$} & \multicolumn{2}{c}{error} & \multicolumn{2}{c}{difference} \\
std & mean & std & mean & std & mean & std \\
($m$) & ($m/s^2$) & & ($m/s^2$) & & ($m/s^2$) & \\
$ 0,1 $ & $ 10,116 $ & $ 0,573 $ & $ 0,640 $ & $ 0,162 $ & $ 0,306 $ & $ 0,573 $ \\
$ 0,05 $ & $ 9,892 $ & $ 0,303 $ & $ 0,333 $ & $ 0,082 $ & $ 0,082 $ & $ 0,303 $ \\
$ 0,03 $ & $ 9,877 $ & $ 0,282 $ & $ 0,323 $ & $ 0,081 $ & $ 0,067 $ & $ 0,282 $ \\
$ 0,01 $ & $ 9,815 $ & $ 0,054 $ & $ 0,058 $ & $ 0,014 $ & $ 0,005 $ & $ 0,054 $ \\
$ 0,005 $ & $ 9,811 $ & $ 0,025 $ & $ 0,028 $ & $ 0,007 $ & $ 0,001 $ & $ 0,025 $ \\
$ 0,001 $ & $ 9,810 $ & $ 0,005 $ & $ 0,005 $ & $ 0,001 $ & $ 0,000 $ & $ 0,005 $ \\
\end{tabular}
\end{center}
\caption{Estimation of $g$ by regressing $t^2$ on $h$\label{Tab:1}}
\end{table}
\begin{table}
\begin{center}
\begin{tabular}{ccccccc}
height $h$ & \multicolumn{2}{c}{estimate $g$} & \multicolumn{2}{c}{error} & \multicolumn{2}{c}{difference} \\
std & mean & std & mean & std & mean & std \\
& ($m/s^2$) & & ($m/s^2$) & & ($m/s^2$) & \\
$ 0,1 $ & $ 9,824 $ & $ 0,571 $ & $ 0,620 $ & $ 0,148 $ & $ 0,014 $ & $ 0,571 $ \\
$ 0,05 $ & $ 9,809 $ & $ 0,301 $ & $ 0,330 $ & $ 0,080 $ & $ -0,001 $ & $ 0,301 $ \\
$ 0,03 $ & $ 9,799 $ & $ 0,280 $ & $ 0,320 $ & $ 0,079 $ & $ -0,011 $ & $ 0,280 $ \\
$ 0,01 $ & $ 9,812 $ & $ 0,054 $ & $ 0,058 $ & $ 0,014 $ & $ 0,002 $ & $ 0,054 $ \\
$ 0,005 $ & $ 9,810 $ & $ 0,025 $ & $ 0,028 $ & $ 0,007 $ & $ 0,000 $ & $ 0,025 $ \\
$ 0,001 $ & $ 9,810 $ & $ 0,005 $ & $ 0,005 $ & $ 0,001 $ & $ 0,000 $ & $ 0,005 $ \\
\end{tabular}
\end{center}
\caption{Estimation of $g$ by regressing $h$ on $t^2$.\label{Tab:2}}
\end{table}
\begin{table}
\begin{center}
\begin{tabular}{cccc}
\multicolumn{2}{c}{estimate $g$} & \multicolumn{2}{c}{error} \\
mean & std & mean & std \\
($m/s^2$) & & ($m/s^2$) & \\
$9,815$ & $0,054$ & $0,005$ & $0,054$ \\
$9,811$ & $0,056$ & $0,001$ & $0,056$ \\
$9,812$ & $0,053$ & $0,002$ & $0,053$ \\
$9,814$ & $0,054$ & $0,004$ & $0,054$ \\
$9,815$ & $0,056$ & $0,005$ & $0,056$ \\
$9,809$ & $0,055$ & $-0,001$ & $0,055$ \\
$9,811$ & $0,049$ & $0,001$ & $0,049$ \\
$9,814$ & $0,058$ & $0,004$ & $0,058$ \\
$9,814$ & $0,048$ & $0,004$ & $0,048$ \\
$9,812$ & $0,064$ & $0,002$ & $0,064$ \\
\end{tabular}
\end{center}
\caption{Estimation of $g$ by regressing $t^2$ on $h$ for 10 repetitions with $\sigma_h = 0.01m$. \label{Tab:3}}
\end{table}
\section{Discussion and conclusions}
\setcounter{equation}{0}
Theory and our numerical experiment show that the choice of formulation can be important when estimating through linear regression.
We proved and demonstrated experimentally that estimation followed by inversion introduces bias and an increased reported error.
The general rule should be that if the value of a variable or constant can be estimated avoiding inversions one should do so since when the original estimate is unbiased the inverse estimate will be biased.
Although differences between methods are not excessive there are no advantages in using a regression followed by an inversion in comparison to the direct estimation.
The combination of regression and inversion scores systematically worse than direct estimation.
The laboratory experiment for deriving the gravitational constant
is an example of a very common exercise for students studying physics.
The most straightforward way of conceiving such experiment consists of
measuring the time of fall of a bullet for a given set of height values.
This suggests time as natural ``dependent variable'' of height and inspires
an approach where $ a = 2/g $ is estimated (according to Eq.\ \ref{Reg_t2_h}).
We have shown that this approach introduces a bias in the estimated
value $ \hat{g} $.
The attentive reader may have noticed that for realistic values of $ \sigma_h $ (e.g.\ 0.001m) this bias can be neglected for the
gravity constant experiment.
However, the point we want to make here is that the students should be warned about the existence of this bias and should be made aware of
the fact that this bias -- which is misleadingly small in an experiment
like the one aiming at estimating the gravity constant -- may be important in
other types of experiments where one equally tends to calculate
$ 1/\hat{a} $ as estimate of a parameter.
\bibliographystyle{plain}
|
{
"timestamp": "2012-11-20T02:00:51",
"yymm": "1203",
"arxiv_id": "1203.2022",
"language": "en",
"url": "https://arxiv.org/abs/1203.2022"
}
|
\section{Introduction}
Structured programming was a great step forward
from the preceding stage of chaotic programming.
The next step, beyond structured programming,
is \emph{verification-driven} programming,
where proof of correctness and code are developed in parallel.
Matrix Code is important for the practicing programmer
because it makes verification-driven programming possible.
But it also helps to solve the problem
faced by an instructor in a first-year programming class:
most of the class understands why a given program works,
but how to help the student,
faced with a blank screen,
\emph{to get started} with the program of the next
assignment?
This is where Matrix Code helps:
there is always something (small) to do,
and when there is nothing more to do,
the code matrix is ready for routine translation to Java
with confidence that the resulting code
has the desired behaviour.
E.W. Dijkstra addressed the same problem when he started
teaching in the 1960's.
His remedy was a detailed step-by-reasoning
and construction resulting in
an Algol-60 program for filling an array with the first
thousand prime numbers.
In this paper we address the same task
to ease comparison with Dijkstra's report \cite{djkddh72}.
\section{Hoare's verification method}
\label{sec:floyd}
As an introduction to the verification method
due to R.W. Floyd and C.A.R. Hoare
we verify a Java version of the prime-number
generating program
developed by Dijkstra in \cite{djkddh72}.
See Figure~\ref{fig:floyd}.
\begin{figure}[htbp]
\begin{center}
\hrule \vspace{0.1in}
\begin{verbatim}
public static void primes(int[] p, int N) {
// S
int j,k,n;
p[0] = 2; p[1] = 3; k = 2;
// A
while (k<N) {
j = p[k-1]+2; n = 0;
// B
while (p[n]*p[n] <= j) {
// C
if (
else {j += 2; n = 0;}
}
p[k++] = j;
}
// H
}
\end{verbatim}
\end{center}
\caption{\label{fig:floyd}
An example of a Java function for filling
{\tt p[0..N-1]} with the first {\tt N} primes.
At the points indicated by the comments S, A, B, C, H
we need assertions to allow verification by Hoare's method.
}
\vspace{0.1in}
\hrule
\end{figure}
The essence of imperative code is that computation
progresses through the code along a well-defined
set of \emph{code locations}.
In Figure~\ref{fig:floyd} some of these locations
are indicated by the comments
\verb"S", \verb"A", \verb"B", \verb"C", and \verb"H".
We think of a computation as a sequence of
\emph{computation states}
each of which consists of a \emph{control state}
(a code location)
and a \emph{data state}
(a vector of values of the variables).
According to the Floyd-Hoare method,
assertions are attached to selected code locations.
The assertions assert that certain relations
between program variables hold at the code locations concerned.
When such an assertion occurs in a loop,
it is the familiar \emph{invariant} of that loop.
In Figure~\ref{fig:floyd} we have indicated by the comments
where these assertions have to be placed.
Figure~\ref{fig:verification} contains the corresponding
assertions and the required Hoare triples
(see following explanation).
\begin{figure}[htbp]
\hrule
\vspace{0.1in}
\begin{verbatim}
Assertions:
S: p[0..N-1] exists and N>1
H: p[0..N-1] are the first N primes
A: S && p[0..k-1] are the first k primes && k <= N
B: A && k<N && relB(p, k, n, j)
C: B && p[n]*p[n] <= j
relB(p,k,n,j)} means that there is no prime
between p[k-1] and j, and that j is not divided
by any prime in p[0..n], and that n<k.
Hoare triples:
{S} p[0]=2; p[1]=3; k=2; {A}
{A && k >= N} {H}
{A && k < N} j=p[k-1]+2; n=0; {B}
{B && p[n]*p[n] <= j} {C}
{B && p[n]*p[n] > j} p[k++] = j {A}
{C &&
{C &&
\end{verbatim}
\caption{\label{fig:verification}
Assertions and Hoare triples for Figure~\ref{fig:floyd}.
The meaning of a Hoare triple {\tt \{A0\} CODE \{A1\}}
is that if assertion {\tt A0} is true
and if {\tt CODE} is executed
with termination, then assertion {\tt A1} is true.
}
\vspace{0.1in}
\hrule
\end{figure}
The verification of the function as a whole
relies on the verification of a number of
implications defined in terms of assertions
and program elements such as tests and statements.
Consider Figure~\ref{fig:floyd}:
because there is an execution path
from \verb+A+ to \verb+B+, one has to show
the truth of\\
\verb" {A && k<N} j=p[k-1]+2; j=0; {B}"\\
It has as meaning:
if \verb"A && k<N" (the \emph{precondition}) is true
and if\\
\verb" j=p[k-1]+2; j=0;"\\
is executed, then \verb"B" (the \emph{postcondition})
is true.
Because of the three elements: precondition,
postcondition, and the item in between,
this is called a \emph{Hoare triple}.
There are many other implementations of the function
in Figure~\ref{fig:floyd} that are verified by the
same set of triples as in Figure~\ref{fig:verification}.
It would be tempting to say that,
once we have a sufficient set of Hoare triples,
we can forget the program in Figure~\ref{fig:floyd}:
all information about it is in the Hoare triples
of Figure~\ref{fig:verification}.
This may seem so because, for example, in\\
\verb" {A && k < N} j=p[k-1]+2; n=0; {B}"\\
{\tt A} stands for the assertion defined earlier in that figure.
What is missing is the fact that assertion \verb"A"
is tied to code location {\tt A}.
But the idea of regarding the set of triples as the
essence is a fruitful one.
It leads one to ask:
what is a format for the information in
Figure~\ref{fig:verification} plus
the fact that the code locations are tied
to the assertions of the same name?
For the answer to this question we propose
Matrix Code.
\section{Matrix Code}
A code matrix can be thought of as
a graph with code locations as nodes
and directed arcs that are labeled with code.
A natural notation for such a graph is a matrix
with columns and rows labeled by nodes
and the arc labels as matrix entries.
When we use this notation for Figure~\ref{fig:verification},
then we get the matrix in
Figure~\ref{fig:primes3}.
This is a \emph{code matrix}.
Although a code matrix arose from a collection of logical
statements, it can also be interpreted as specifying
a set of computations of an abstract machine.
In the explanation below we use a picturesque terminology
that is worth trying on an audience of novices in programming.
A code matrix is executed by a \emph{turtle}
moving over it.
The turtle contains a data state and a knowledge state.
The data state is a vector of the values of the variables
accessible from the code under construction.
The knowledge state is an assertion concerning
the values of these variables.
The turtle has an innate truthfulness
that prevents it from knowing a lie.
In other words,
its knowledge state is always
an assertion that is true of its data state.
The turtle is a logical animal
in the sense that it is endowed with an innate drive
that makes it draw a conclusion from the assertion
that is its knowledge state
and from it data state.
And the turtle only has available for its
conclusions the Hoare triples of the code matrix
and its own data and knowledge states.
The knowledge state specifies a row or column of the code matrix.
In Figure~\ref{fig:primes3} the knowledge state
can have as values \verb"S", \verb"A", \verb"B",
\verb"C", and \verb"H".
The data state is a vector of the values of the variables.
In Figure~\ref{fig:primes3} the data state
has as components the content of array \verb"p"
and the values of the variables \verb"k", \verb"j", and \verb"n".
We call the entries of the code matrix \emph{gates}.
Execution of a code matrix consists of
the turtle performing a sequence of cycles.
The turtle's data and knowledge states are
updated as a consequence of executing the cycle.
At the beginning of the cycle the turtle
enters from the top of the matrix
through the column indicated by
the current knowledge state
until it encounters a gate.
The data state passes the gate or fails to do so.
In the latter case execution terminates with failure.
If the data state passes through the gate,
then the turtle exits the matrix to the right
through the row in which that gate occurs.
The new state has the label of that row
as knowledge state and as data state the one determined by
having had to pass through the gate.
This completes the cycle.
Initially the turtle has knowledge state {\tt S}.
When the knowledge state changes to {\tt H},
execution halts with success.
What determines whether the turtle can pass
through a gate and how does its state change when it does?
If a gate is a boolean expression that evaluates to {\tt false}
in the data state,
then the turtle fails to pass.
If it evaluates to \verb"true",
then the data state passes and remains unchanged.
If a gate is an assignment statement,
then the turtle passes if execution
of the statement is defined and terminates.
When it passes, then the data state is changed as
defined by the semantics of the statement.
In Figure~\ref{fig:primes3} we see
that gates may be composed by means of a semicolon.
In general, if $g$ and $h$ are gates, then
$g;h$ is also a gate.
The data state $s$ passes through
gate $g;h$ yielding state $t$
if it passes through gate $g$ giving $s'$
and if $s'$ passes through gate $h$ giving $t$.
The concept of gate is especially useful
because of the possibility that a gate consisting of
a boolean expression can be composed with an assignment.
Such a gate may block the data state because
the boolean expression evaluates to \verb"false".
When the data state is not thus blocked,
it will be transformed by the assignment statement.
Gates may be composed of boolean expressions
and statements in any order.
For ease of translation to a conventional language like Java,
we ensure that boolean expressions precede statements.
For example, suppose we start execution of Figure~\ref{fig:primes3}
with parameter {\tt N} equal to 1000
and with knowledge state equal to {\tt S}.
As there is only one triple in column {\tt S},
and as this triple occurs in row {\tt A},
the computation continues with column {\tt A}.
Only one of the gates in that column
allows the data state to pass,
so computation continues in the row of that gate,
namely {\tt B}.
Here follows an excerpt of the computation:
\begin{center}
\begin{verbatim}
N = 1000
knowledge | data state
state |
| k j n p
-----------------------------------------
S |
A | 2 {2,3,...}
B | 2 5 0 {2,3,...}
C | 2 5 0 {2,3,...}
B | 2 5 1 {2,3,...}
A | 3 5 1 {2,3,5,...}
... | . . .
H | 1000 7919 23 {2,3,5,...,7919}
\end{verbatim}
\end{center}
\section{Algorithm discovery from first principles}
\label{sec:ratio}
Let us use Matrix Code
to discover an algorithm for filling an array \verb"p[0..N-1]"
with the successive prime numbers
$
p_0 = 2,
p_1 = 3,
p_2 = 5,
\ldots,
p_{N-1}.
$
The specification of the desired function body \verb"G"
can be given as the Hoare triple \verb"{S}G{H}"
with {\tt S} and {\tt H} as in Figure~\ref{fig:primes0}.
This triple becomes part of the final code matrix;
see Figure~\ref{fig:primes0}.
As we don't have an immediate implementation
of gate \verb"G" in Java, we need to expand the code matrix.
One by one we add rows and columns
in such a way that the matrix is expanded
from the top right corner downward and to the left.
\begin{figure}[htbp]
\begin{tabular}{l|l||l}
\lmnt{} & \lmnt{S: p[0..n-1] exists \&\& n>1} & \\
\hline \hline
& \lmnt{/*which G?*/}
& \lmnt{H: p[0..n-1] contains the first n primes} \\
\hline
& & \lmnt{ \\ \\} \\
\end{tabular}
\caption{\label{fig:primes0}
A code matrix solving the problem,
if only we had an easy implementation for gate {\tt G}
such that $\bk{S}{G}{H}$.
We need at least one intermediate assertion;
see Figure~\ref{fig:primes1}.
}
\end{figure}
Assertion {\tt H} is too ambitious to achieve
with a simple gate when the data state is as
described by {\tt S}.
So we need at least one condition,
say, {\tt A}, that is intermediate between {\tt S}
and {\tt H} in the sense
that \bk{S}{G1}{A} and \bk{A}{G2}{H}
for simple {\tt G1} and {\tt G2}.
Less formally: if we can't fill all of a prime-number table
of size $N$ right away,
we can at least fill a small one,
say, of size $k \leq N$.
This suggests as intermediate assertion \verb"A":
the first {\tt k} primes in increasing order are in
{\tt p[0..k-1]} with {\tt 1 < k <= N}.
It is easy to reach
\verb"A"
from
\verb"S":
we put the first two primes in the table
and set \verb"k=2".
This allows us to exit through the new row for {\tt A},
setting the knowledge state to {\tt A}.
In the next step we enter through the column determined by
the knowledge state, hence column {\tt A}.
If the knowledge state is {\tt A}
(and if that assertion holds),
then passing the gate {\tt k >= N}
allows us to halt by exiting through row {\tt H}.
The new row and column update our code matrix to
the one in Figure~\ref{fig:primes1}.
\begin{figure}[htbp]
\begin{center}
\begin{tabular}{l|l|l||l}
\lmnt{} & \lmnt{A:} & \lmnt{S: p[0..N-1] exists \&\& N>1} & \\
\hline \hline
& \lmnt{k >= N} & & \lmnt{H: p[0..N-1] contains the first N primes} \\
\hline
& & \lmnt{p[0] = 2; p[1] = 3; k = 2}
& \lmnt{A: p[0..k-1] contains the first k primes \&\&
k <= N} \\
\hline
& & & \lmnt{ \\ \\} \\
\end{tabular}
\caption{\label{fig:primes1}
Next step after Figure~\ref{fig:primes0}:
in column $A$ the case {\tt k < N} is missing.
This leads to a new row and column labeled {\tt B}
in Figure~\ref{fig:primes2}.
}
\end{center}
\end{figure}
However, when execution enters through column {\tt A}
in Figure~\ref{fig:primes1},
we may have that {\tt k < N},
so that we do not pass the gate {\tt k >= N}.
This points to the need to increase {\tt k},
hence to find the next prime after {\tt p[k-1]}.
Let {\tt j} be the candidate for this next prime.
That suggests including in assertion {\tt B:}
``{\tt A} is true and {\tt k<N} and {\tt j} is such that
there is no prime greater than {\tt p[k-1]}
and less than {\tt j} and {\tt j} is not divisible by any
of {\tt p[0..n]}'',
a statement that we abbreviate to {\tt relB(p,k,n,j)},
as in Figure~\ref{fig:verification}.
Column {\tt A} is now completed with a gate
allowing exit through the new row for {\tt B}.
When we enter through the new column for {\tt B}
we immediately know one of the gates
in column {\tt B} for the easy case
where the candidate {\tt j} for the next prime
actually turns out to be the next prime.
See Figure~\ref{fig:primes2} for
the resulting stage in the development of the code matrix.
\begin{figure}[htbp]
\begin{center}
\begin{tabular}{|l|l|l||l}
\lmnt{B:} & \lmnt{A:} & \lmnt{S: p[0..N-1] exists \&\& N>1} & \\
\hline \hline
& \lmnt{k >= N} & & \lmnt{H: p[0..N-1] contains the first N primes} \\
\hline
\lmnt{p[n]*p[n]>j; p[k++]=j} &
& \lmnt{p[0] = 2; p[1] = 3; k = 2}
& \lmnt{A: p[0..k-1] contains the first k primes \&\&
k <= N} \\
\hline
&\lmnt{k<N; j = p[k-1]+2; n=0} &
& \lmnt{B: A \&\& k<N \&\& relB(p,k,n,j)} \\
\hline
\end{tabular}
\caption{\label{fig:primes2}
Next step after Figure~\ref{fig:primes1}:
in column $A$ we have added a transition in column $A$
for the case that {\tt k < N}.
In that case we can start finding the next prime after
{\tt p[k-1]} because we know that there is enough space
in {\tt p} to store it.
{\tt relB(p,k,n,j)} means that
there is no prime between the last prime found and {\tt j}
and that {\tt n<k}, and that {\tt j} is not divided
by any prime in {\tt p[0..n]}.
The missing entry in column {\tt B} leads to a new row and column
labeled {\tt C} in Figure~\ref{fig:primes3}.
}
\end{center}
\end{figure}
In condition {\tt B}
primeness of {\tt j} can be concluded
for sufficiently large {\tt n}.
Initially this is typically not the case,
hence the need for condition {\tt C}:
{\tt B} is true and the square of {\tt p[n]}
is not greater than {\tt j}.
We need to be assured that {\tt n} does not exceed {\tt k-1},
which is a fact of number theory\footnote{
In \cite{vnm11} we pay due attention to such crucial details.
}.
When entering column {\tt C}
we know that {\tt n} is not large enough to conclude
that {\tt j} is prime.
So we need to test for divisibility of {\tt j} by
{\tt p[n+1]}.
Number theory assures
that {\tt p[n+1]} is a prime that has already been found
\cite{vnm11}.
See Figure~\ref{fig:primes3}.
\begin{figure}[htbp]
\begin{center}
\begin{tabular}{|l|l|l|l||l}
\lmnt{C:}& \lmnt{B:} & \lmnt{A:}
& \lmnt{S: p[0..N-1] exists \&\& N>1} & \\
\hline \hline
& & \lmnt{k >= N} & & \lmnt{H: p[0..N-1] contains the first N primes} \\
\hline
& \lmnt{p[n]*p[n]>j; p[k++]=j} &
& \lmnt{p[0] = 2; p[1] = 3; k = 2}
& \lmnt{A: p[0..k-1] contains the first k primes \&\&
k <= N} \\
\hline
\lmnt{j\%p[n+1]!=0; n++} & &\lmnt{k<N; j = p[k-1]+2; n=0} &
& \lmnt{B: A \&\& k<N \&\& relB(p,k,n,j)} \\
\hline
\lmnt{j\%p[n+1]==0; j += 2; n=0}
& \lmnt{p[n]*p[n] <= j} &
& & \lmnt{C: B \&\& \\ p[n]*p[n] <= j}\\
\hline
\end{tabular}
\caption{\label{fig:primes3}
Next step after Figure~\ref{fig:primes2}:
Change from Figure~\ref{fig:primes2}:
row and column with label $C$ are added.
There are no incomplete columns,
so this is ready for translation to Java;
see Figure~\ref{fig:javaPrimes}.
}
\end{center}
\end{figure}
This code matrix has no incomplete columns,
so all that remains to be done
is a straightforward translation to Java,
which is possible because all boolean expressions
precede assignments.
See Figure~\ref{fig:javaPrimes}.
\section{Conclusions}
We have introduced Matrix Code,
a hybrid between correctness proof
and program in the form of a matrix with rows and columns
labeled by program assertions.
The entries in the matrix consist of tests or statements
or combinations of both.
One can say that Matrix Code is code-independent.
At the same time it \emph{is} code:
one uses it to program an abstract machine.
The existence of a code-independent notation
for the algorithm allows us to \emph{separate concerns}.
The concerns are addressed in two stages:
(1) a code matrix from the specification
and (2) compilable code from the code matrix.
Both stages have been demonstrated in this paper.
The second stage is a routine task
because of the special structure of the code matrix.
We have chosen the generation of the prime number table
because the algorithm is around the upper limit of what
most students in a first-year programming course can grasp.
We believe that with matrix code a larger proportion
of the class will be able to approach their programming
assignments not as a trial-and-error process,
but as a goal-directed activity.
\section{Acknowledgments}
We gratefully acknowledge financial support from
the Canadian Natural Sciences and Engineering Research Council
as well as facilities from the University of Victoria.
|
{
"timestamp": "2012-05-09T02:04:33",
"yymm": "1203",
"arxiv_id": "1203.2296",
"language": "en",
"url": "https://arxiv.org/abs/1203.2296"
}
|
\section{Introduction}
The 5D$_{3/2}$ level in a barium ion has a lifetime of approximately 80 seconds \cite{dehmelt97}, corresponding to a 13 mHz linewidth for the electric quadrupole 6S$_{1/2} \leftrightarrow$ 5D$_{3/2}$ transition. This transition is an important component of a number of proposed experiments. First, it provides an efficient means of populating and detecting 5D$_{3/2}$ sublevels for precision RF spectroscopy of the hyperfine splittings in the 5D$_{3/2}$ manifold in $^{137}$Ba$^+$ (I=3/2), which combined with a similar measurement of the 5D$_{5/2}$ hyperfine splittings will yield the value of the nuclear magnetic octupole moment for this isotope \cite{howell08}. This measurement could provide insight into the unexpectedly large measured value of the nuclear magnetic octupole moment of $^{133}$Cs \cite{tanner03}. Second, the 6S$_{1/2} \leftrightarrow$ 5D$_{3/2}$ transition in $^{137}$Ba$^+$ can be used as a reference for an optical frequency standard. In $^{137}$Ba$^+$ the quadrupole Stark shift due to electric field gradients is equal to zero for both the F=0 and F=2 hyperfine levels 5D$_{3/2}$, making $^{137}$Ba$^+$ an interesting candidate as an ion frequency standard \cite{sherman05}. This shift had been a concern for the clock transition in $^{199}$Hg$^+$ \cite{Itano00}, but was later measured and eliminated at the 10$^{-18}$ level \cite{oskay05}. Finally, this work is an important step towards a measurement of parity non-conservation in a single trapped barium ion \cite{fortson93}.
Driving the 6S$_{1/2}$ to 5D$_{3/2}$ electric quadrupole transition in Ba$^+$ with high fidelity is a necessary step in each of these experiments. We demonstrate the ability to coherently excite a $^{138}$Ba$^+$ ion from the ground state to resolved 5D$_{3/2}$ Zeeman sublevels using a frequency stabilized 2051 nm Tm,Ho:YLF laser.
\section{Apparatus}
To load ions into the trapping apparatus, barium atoms emitted from an oven are singly ionized by a two photon, isotope-selective process\cite{steele07}. First, an external cavity diode laser (ECDL) at 791 nm drives the weak inter-combination transition, 6s$^2 \leftrightarrow$ 6s6p ($J=1$), which has isotope shifts on the order of 1 GHz between the 137 and 138 isotopes of neutral barium \cite{grundevik83}. From this excited state, a pulse from a nitrogen laser at 337 nm excites the electron into the continuum. Since $^{137}$Ba$^+$ represents just 11\% of a natural barium sample, this isotope-selective process allows for faster loading times for $^{137}$Ba$^+$, which will be important for future work.
After photoionization, a single Ba$^+$ ion is then confined by a linear Paul trap in a similar design to the one found in Ref.~\cite{olmschenk07}, driven with a radio frequency (RF) of 22.9 MHz. A pair of Helmholtz coils generate a stable, but adjustable magnetic field of up to 5 Gauss, which provides a quantization axis for the ion. Components of the Earth's magnetic field that are perpendicular to the laboratory field are cancelled by two secondary coils.
\begin{figure}[tbp]
\centering
\includegraphics[]{Figure1.eps}
\caption{Energy levels and important transitions in $^{138}$Ba$^+$. The ion is laser cooled with the 493 nm transition, and is repumped from the long lived 5D$_{3/2}$ state with 650 nm light. The ion can be `shelved' using 455 nm light, via the 6P$_{3/2}$ state, where it will remain in the metastable 5D$_{5/2}$ state until a pulse of 614 nm light returns it to the cooling cycle. The electric quadrupole transition at 2051 nm that is of interest for proposed experiments is driven by a frequency stabilized laser, and detected by means of shelving and deshelving as described in the text.}
\label{fig:ba_138_transitions}
\end{figure}
The energy level diagram along with the relevant transitions for $^{138}$Ba$^+$ is shown in Fig.~\ref{fig:ba_138_transitions}. The ion is cooled to the Doppler limit T$\sim1$mK by addressing the 6S$_{1/2}\leftrightarrow$ 6P$_{1/2}$ transition at 493 nm with a frequency doubled ECDL at 986 nm. Since this state can decay into the long lived 5D$_{3/2}$ state, the ion must be repumped using a second ECDL at 650 nm. Both of these beams are linearly polarized for Doppler cooling. To reduce long term frequency drifts, each laser is frequency stabilized to a Zerodur-spaced optical cavity that is sealed from the atmosphere in a vacuum chamber. Light sent to each of these cavities is frequency shifted using a double passed acousto-optic modulator (AOM) to allow for a tunable offset from the cavity modes. The ion can be optically pumped into either of the 6S$_{1/2}$, m$_J=\pm1/2$ states by switching from a primary 493 nm beam to a second, circularly-polarized beam which is aligned parallel to the quantization axis.
Ion fluorescence from the cooling cycle is collected by a micro-objective lens, then passed through an interference filter centered at 488 nm with a transmission bandwidth of 10 nm to pass photons emitted from the 6S$_{1/2}\leftrightarrow$ 6P$_{1/2}$ transition, and block stray room light as well as scattered 650 nm photons. These 493 nm photons are detected by a photo-multiplier tube (PMT) and imaged by an electron multiplying charged-coupled device (EMCCD) camera.
2051 nm light is generated by a Tm,Ho:YLF diode-pumped solid-state laser manufactured by CLR Photonics (now part of Lockheed Martin) with maximum output power of 40 mW. Changing the temperature of the resonator coarsely tunes the center wavelength of the laser. Fine tuning of the laser frequency is achieved by adjusting the voltage applied to a piezo-electric actuator mounted on the output coupling mirror of the laser cavity, with a small-signal bandwidth of approximately 10 kHz. This light must be frequency stabilized to a reference cavity, as detailed below, to coherently drive the narrow 6S$_{1/2} \leftrightarrow$ 5D$_{3/2}$ electric quadrupole transition. Before interacting with the ion, the 2051 nm beam passes through an AOM driven at 55 MHz (labeled `Shutter AOM' in Figure \ref{fig:schematic}), which acts as a fast shutter.
The 2051 nm beam is linearly polarized and can be aligned parallel or perpendicular to the magnetic field so that $\Delta$m$_J=\pm1,\pm2$ transitions are allowed \cite{roos00}. The $\Delta$m$_J=0$ transitions, which are preferred in an optical frequency standard, are not driven by light parallel or perpendicular to the magnetic field, but are maximized with light at 45 degrees to the magnetic field.
Performing quantum jump laser spectroscopy \cite{dehmelt86} on the 6S$_{1/2}\leftrightarrow$ 5D$_{3/2}$ transition is complicated by the fact that the 2051 nm transition does not remove the ion from the cooling cycle. It is therefore necessary to have some other means of `shelving' the ion. A pulse from a 1W light emitting diode (LED) with a center wavelength of 455 nm and a spectral half-width of 20 nm excites an ion in the 6S$_{1/2}$ level to the 6P$_{3/2}$ level as shown in Figure~\ref{fig:ba_138_transitions}. An ion in the 6P$_{3/2}$ level will spontaneously decay back to the 6S$_{1/2}$ level with a probability of $75.6\%$, to the 5D$_{5/2}$ level with a probability of $21.5\%$, and to the 5D$_{3/2}$ level with a probability of $2.9\%$\cite{kurz08}. A 300 ms pulse of 455 nm light effectively eliminates the possibility of the ion remaining in the 6S$_{1/2}$ level. An ion in the 5D$_{5/2}$ level does not interact with the 493 nm and 650 nm lasers during the cooling cycle and the ion fluorescence signal at 493 nm goes to zero. The ion can be returned to the cooling cycle by applying a 400 ms of 614 nm light that is also generated by a 1W LED with a 617 nm center wavelength, and 18 nm spectral half-width. \footnote{The LEDs in use are Phillips Luxeon III Star LEDs - Royal blue for 455 nm light, and Red-Orange for 617 nm light.}
\section{Tm,Ho:YLF Laser Frequency Stabilization}
The narrow linewidth of the 6S$_{1/2} \leftrightarrow$ 5D$_{3/2}$ transition requires that the frequency of the 2051 nm laser be well stabilized. While reference cavities designed for use at visible and near-infrared wavelengths routinely have finesses greater than $\mathcal{F}\sim10^5$, at the time our system was built, state-of-the-art high reflective coatings at 2051 nm would be limited to $\mathcal{F}\sim20,000$. Additionally, commercial options for electro-optic modulators (EOMs), necessary for a Pound-Drever-Hall (PDH) lock, and broadband AOMs, necessary for a wide tuning range, are not readily available at this wavelength. For these reasons we opted to stabilize the second-harmonic of our laser with a high finesse cavity at 1025 nm.
\subsection{Second Harmonic Generation of 2051 nm Light}
Implementing a robust Pound-Drever Hall lock \cite{hall83} to a 1025 nm reference cavity requires that we generate at least 100 $\mu$W of 1025 nm light so that we can comfortably accommodate power losses from a double passed, frequency shifting AOM. Second harmonic generation (SHG) of light is achieved using a bulk periodically poled lithium niobate (PPLN) crystal that is 4 cm long and has nine parallel tracks with different poling periods. We used a track with a poling period of 30.25 $\mu$m and maintained the crystal temperature at $108\,^{\circ}\mathrm{C}$ for quasi-phase matching. In single-pass configuration, however, the light generated from the PPLN nm was insufficient for a stable PDH lock.
\begin{figure}[htbp]
\includegraphics{Figure2.eps}
\caption{Tm,Ho:YLF laser stabilization schematic. Light from a 2051 nm Tm,Ho:YLF laser passes through an AOM and is frequency doubled with a PPLN crystal in a `bow-tie' cavity. The 1025 nm light is then frequency shifted by a double-passed AOM, and stabilized to a high finesse reference cavity using the PDH method \cite{hall83}. An EOM driven by a 20 MHz source is used to generate the sidebands necessary for frequency stabilization. After passing through a low-pass (LP) filter, high bandwidth proportional feedback (P) is sent to the feedback AOM, while low bandwidth integral feedback (I) is amplified and sent to the tuning piezo inside the laser head.}
\label{fig:schematic}
\end{figure}
1025 nm SHG efficiency is increased substantially by placing the temperature controlled PPLN crystal inside a `bow-tie' enhancement cavity (Fig.~\ref{fig:schematic}) consisting of two flat mirrors and two curved mirrors each with a radius of curvature of 250 mm. The curved mirrors are separated by 325 mm and the PPLN crystal is placed at the midpoint where the beam comes to a 150 $\mu$m waist. The relatively long mirror separation ensures that the beam is not clipped by the 0.5 mm square poling channels. The crystal is anti-reflection (AR) coated for 2051 nm, as well as 1025 nm, and is been polished with a 1 degree wedge to prevent the crystal from acting as an intra-cavity etalon. The length of the enhancement cavity is locked to the wavelength of the 2051 nm laser by sending an error signal derived using the Hansch-Couillaud method\cite{hansch80} to a piezo-electric actuator mounted on one of the flat cavity mirrors. Our coupling efficiency into the TEM00 mode of the enhancement cavity is approximately 75\%. We estimate the cavity build up of 2051 nm light to be approximately 20. Sending the full 40 mW output of the Tm,Ho:YLF laser to the enhancement cavity we were able to generate 2.5 mW of 1025 nm light. When the 55 MHz AOM (labeled `Feedback AOM') in Figure \ref{fig:schematic}, necessary for high bandwidth feedback for the PDH lock to the high finesse cavity, is included in the 2051 nm beam path, approximately 20 mW is sent to the enhancement cavity and we are able to generate approximately 500 $\mu$W of 1025 nm light. This power is sufficient for a stable PDH lock of the laser at 2051 nm.
\subsection{PDH Stabilization with Two-Channel Feedback}
After the enhancement cavity, the 1025 nm beam makes a double pass through a frequency shifting AOM with a center frequency of 200 MHz. This AOM allows for a continuous tuning range of approximately 80 MHz at 1025 nm (equivalent to 40 MHz of tuning at the original 2051 nm wavelength). The beam then passes through a resonant EOM to generate Pound-Drever-Hall sidebands at 20 MHz \cite{hall83} and then through a series of mode matching lenses before coupling into the reference cavity.
Our reference cavity consists of two mirrors held 77.5 mm apart by a spacer made from ultra-low expansion (ULE) glass according to a vibration insensitive design developed at JILA \cite{notcutt05}, resulting in a free spectral range of 1.9 GHz. The mirrors are coated for high reflectivity at the 1025 nm wavelength. The input mirror is flat and the output mirror has a radius of curvature of 500 mm. Cavity ring-down measurements \cite{berden00} indicate the cavity finesse to be greater than 300,000. The cavity is mounted vertically inside a vacuum chamber and maintained at a pressure of less than $10^{-8}$ Torr with an ion pump. The vacuum chamber is enclosed inside a temperature stabilized aluminum box surrounded by insulation to further reduce cavity length variations due to temperature fluctuations.
Light reflected from the cavity is separated from the incident beam using a quarter waveplate and a polarizing beam splitter. The intensity of the rejected beam is detected with an amplified photodetector. The PDH error signal is extracted from the photodetector output using a double balanced mixer, a variable phase shifter, and a 1.9 MHz low-pass filter.
Frequency stabilization of the Tm,Ho:YLF laser is achieved by sending the error signal into high bandwidth and low bandwidth feedback channels. The low bandwidth channel consists of an analog integrator circuit (labeled ``I" in Fig.~\ref{fig:schematic}) followed by a high-voltage amplification stage driving the tuning piezo inside the laser head. The bandwidth of this channel is approximately 10 kHz. Higher bandwidth feedback is necessary to maintain a robust lock to the $\sim$7 kHz wide TEM00 mode of the ULE cavity. To accomplish this, the error signal in the high bandwidth channel (labeled ``P" in Fig.~\ref{fig:schematic}) is sent through an analog amplification stage that provides variable proportional gain, and then to the control input of the voltage controlled oscillator (VCO) driving the feedback AOM shown in Fig.~\ref{fig:schematic}. Both the VCO and the AOM have 2 MHz modulation bandwidths.
\section{2051 nm Spectroscopy Procedure}
The ion is optically pumped into the 6S$_{1/2}$, m$_J=+1/2$ state with circularly polarized, 493 nm light. The 493 nm and 650 nm lasers are then shuttered and the ion is exposed to a pulse of 2051 nm light. The duration of the pulse is optimized to provide the maximum population transfer to the 5D$_{3/2}$ level (i.e. a Rabi $\pi$-pulse). In order to reduce dephasing due to magnetic field fluctuations, the 2051 nm light pulse is triggered by the rising slope of the 60 Hz AC power line. We then apply a 300 ms pulse of 455 nm light. If the 2051 nm transition occurred, then the ion is in the 5D$_{3/2}$ level and the 455 nm light will have no effect. If the 2051 nm transition did not occur, then the ion remains in the 6S$_{1/2}$ level and the 455 nm light will `shelve' the ion into the 5D$_{5/2}$ state, which has a natural lifetime of approximately 32 seconds\cite{kurz08}. The 493 nm and 650 nm lasers are then turned back on and we record whether the ion fluoresces. A failed `shelving' attempt indicates a successful 2051 nm transition.
With appropriate polarization and beam alignment we can drive $\Delta$m = $\pm1,\pm2$ transitions. To identify these transitions between Zeeman levels, we record the frequency offset from the TEM00 mode of the ULE reference cavity for each line for different magnetic field strengths created by the Helmholtz coils. The sensitivity to a known change of the magnitude of the magnetic field can be used to identify the transitions. Driving $\Delta$m $ = \pm2$ transitions will be used to populate the sublevels needed to perform the proposed measurement of the nuclear magnetic octupole moment \cite{howell08}.
\section{Observation of 2051 nm Transitions}
With the 2051 nm beam aligned parallel to the magnetic field, we identified and tuned the 2051 nm laser to the center of the $\Delta$m$=+1$ transition. We then varied the duration of the 2051 nm laser pulse while recording whether the ion is `shelved'. An optical pumping efficiency of approximately 95\% combined with imperfect `shelving' events limits the extrema of the shelving efficiency to a maximum of approximately 80\% and a minimum of 5\%. Fitting a decaying sinusoid to the data indicates a Rabi frequency of 2.0 kHz and a decay time constant for the coherence envelope of 3.2 ms, as shown in Fig.~\ref{fig:rabi_data_fast}. The experimental setup used in this work does not employ magnetic shielding, so the decoherence observed here is most likely caused by short-term fluctuations in the ambient magnetic field, which have been independently observed in measurements of the 6S$_{1/2}$ Zeeman splitting.
\begin{figure}[htbp]
\centering
\includegraphics[]{Figure3.eps}
\caption{Coherent excitation of the 2051 nm transition. The probability of detecting a `shelved' ion is plotted against the time for which the ion is exposed to the 2051 nm laser. The Rabi frequency is calculated using a least-squares fit of an exponentially decaying sinusoid to be 2 kHz with a decay time of 3.2 ms. The decoherence time is consistent with drifts in ambient magnetic fields.}
\label{fig:rabi_data_fast}
\end{figure}
Using an attenuated 2051 nm laser beam and a pulse duration of 1 ms we observed the spectrum shown in Fig.~\ref{fig:2um_peak_broad}. Fitting a sinc function lineshape to this peak, we find that the linewidth is Fourier transform limited by the 1 ms pulse to approximately 700 Hz.
\begin{figure}[htbp]
\centering
\includegraphics[]{Figure4.eps}
\caption{Spectrum of the 6S$_{1/2}$, m = +1/2 $\leftrightarrow$ 5D$_{3/2}$, m = +3/2 transition. The probability of finding the ion in the `shelved' state is plotted against a relative frequency shift to the high finesse optical cavity's TEM00 mode. The FWHM of this spectrum is calculated to be 700 Hz at 2051 nm, with 75\% shelving efficiency, and 95\% optical pumping efficiency. A least-squares fit of a squared sinc function is overlaid, and was used to calculate these parameters.}
\label{fig:2um_peak_broad}
\end{figure}
\begin{figure*}[h!tbp]
\centering
\includegraphics[]{Figure5.eps}
\caption{Observation of the three accessible 6S$_{1/2}$,m=+1/2 $\leftrightarrow$ 5D$_{3/2}$ transitions using the 2051 nm laser. The final Zeeman levels of the 5D$_{3/2}$ manifold are, from left to right, m $=-3/2, -1/2, +3/2$. The m=+1/2 state cannot be addressed from this initial state with our laser alignment. The probably of finding the ion `shelved' after applying a Rabi $\pi$ pulse is plotted against a relative frequency shift to the TEM00 mode frequency of the high finesse optical cavity. A normalized sinc function (solid red line) is overlaid using the known 2051 nm laser pulse duration, as well as the overall shelving efficiency.}
\label{fig:three_plots}
\end{figure*}
With the 2051 nm beam aligned perpendicular to the magnetic field three transitions from the 6S$_{1/2}$, m = +1/2 state can be driven. The spectra showing each of the transitions are shown in Fig.~\ref{fig:three_plots}. The differences in frequency for these Zeeman splittings are 9.387 MHz and 4.693 MHz, so we identify the transition at 200.590 MHz as the $\Delta$m = -2 transition, the one at 205.283 MHz as the $\Delta$m=-1 transition, and the one at 214.670 MHz as the $\Delta$m = +1 transition. Additionally, using the measured Land$\acute{\mathrm{e}}$ g-factors \cite{gfactor}, we can estimate the laboratory magnetic field to be 4.69 Gauss. The linewidths of these spectra are Fourier transform limited by Rabi $\pi$ pulses of 80, 100, and 275 $\mu$s, respectively, which correspond to the maximum Rabi frequencies observed with this configuration.
\section{Conclusions}
Using a diode-pumped solid state Tm,Ho:YLF laser at 2051 nm, we have coherently driven the 6S$_{1/2} \leftrightarrow$ 5D$_{3/2}$ transition in a single trapped $^{138}$Ba$^+$ ion. We are able to address Zeeman levels of the 5D$_{3/2}$ manifold individually and have identified several transitions between Zeeman states. A laser-ion coherence time of 3 ms has been observed in Rabi oscillations on one transition, which is most likely limited by ambient magnetic field noise and can be increased by adding magnetic shielding. Having observed $\Delta$m = 2 transitions, we can proceed with the measurement of the nuclear magnetic octupole moment of $^{137}$Ba$^+$.
\begin{acknowledgments}
The authors wish to thank Amar Andalkar, Warren Nagourney, Will Trimble, and Yonatan Cohen for early work on this project, as well as helpful discussions with other members of the Blinov group, specifically Chen-Kuan Chou, Matt Dietrich, Nathan Kurz, Tom Noel, and Gang Shu. This research was supported by National Science Foundation grant PHY-09-06494.
\end{acknowledgments}
|
{
"timestamp": "2012-03-12T01:00:33",
"yymm": "1203",
"arxiv_id": "1203.1978",
"language": "en",
"url": "https://arxiv.org/abs/1203.1978"
}
|
\section{Introduction}
\label{sec:intro}
Virtually all thermodynamic phase transitions are driven by interactions between particles, which promote symmetry breaking into an ordered state. The phase transition comes about as a result of the competition between the energy, which favours the ordered state, and the entropy, which favours the disordered state.
In contrast, Bose-Einstein condensation (BEC) is a purely statistical phase transition, which at least in principle should not rely on interactions. The transition is instead a direct consequence of the finite-temperature saturation of the number of particles in the excited states of the system \cite{Einstein:1925a, huan87,Pethick:2002,Pitaevskii:2003}.
While this statistical argument does not explicitly invoke interactions between the particles, it does assume that the gas is in thermal equilibrium, which is impossible to attain in a completely noninteracting system\footnote{In the recently observed Bose-Einstein condensation of a photon gas \cite{Klaers:2010}, there is no direct interaction between the light particles. However the interaction with the material environment, which ensures thermalisation, leads to a second-order interaction between the photons.}. This makes it challenging to experimentally observe ideal-gas behaviour and disentangle the role of interactions on the thermodynamics and dynamics of condensation.
In this Chapter we review our recent experiments on this topic \cite{Tammuz:2011,Smith:2011,Smith:2011b}, performed with an ultracold Bose gas of $^{39}$K atoms with tuneable interactions. We were able to identify the interaction regime in which the gas may be considered to be in thermal equilibrium and also to extrapolate our results to the noninteracting limit where direct equilibrium measurements are not possible. This allowed us to verify the statistical-saturation BEC mechanism in the noninteracting limit, and to accurately determine the deviations from ideal-gas behaviour due to interactions; these are seen both in the non-saturation of the excited states and in the shift of the critical point. Before presenting the experimental results we briefly review some background theory that will be useful for our discussion.
\subsection{Noninteracting Bosons}
We start by considering an ideal, noninteracting Bose gas. We first derive the key results for a uniform system, which we then apply to the trapped gas using the local density approximation (LDA).
This ``local" approach will be useful later when we consider the effects of interactions, and in particular for the comparison of a uniform Bose gas with one that is harmonically trapped.
The equilibrium momentum distribution of noninteracting bosons with mass $m$ at a temperature $T$ is given by the Bose distribution function
\begin{equation}\label{eq:bosep}
f_p=\frac{1}{\mathrm{e}^{(p^2/2m-\mu)/k_{\rm B} T}-1} \; ,
\end{equation}
where $p$ is the momentum and $\mu \leq 0$ the chemical potential.
The total particle density $n$ can be found by integrating over all momentum states:
\begin{equation}\label{eq:densityint}
n=\int\frac{\mathrm{d}\mathbf{p}}{(2\pi \hbar)^3}\frac{1}{\mathrm{e}^{(p^2/2m-\mu)/k_{\rm B} T}-1}=\frac{g_{3/2}(\mathrm{e}^{\mu/k_{\rm B} T})}{\lambda^3} \; ,
\end{equation}
where $g_{3/2}(x)=\sum_{k=1}^{\infty}x^{k}/k^{3/2}$ is a polylogarithm function and $\lambda \; = \; [2\pi\hbar^2/(m k_{\rm B} T)]^{1/2}$ is the thermal wavelength. We can re-express this result in terms of the phase space density $D$ as
\begin{equation}\label{eq:psd}
D\equiv n\lambda^3=g_{3/2}(\mathrm{e}^{\mu/k_{\rm B} T}) \; .
\end{equation}
Eq.~(\ref{eq:psd}) shows that there is a maximum value that $D$ can take. This critical value is reached when $\mu=0$ and is given by $D_c=g_{3/2}(1)=\zeta(3/2)\approx2.612$ (where $\zeta$ is the Riemann function). At a given temperature this corresponds to a maximum density.
If this density is reached all the excited states saturate and any additional particles must accumulate in the ground state, forming a Bose-Einstein condensate\footnote{The singular ground-state contribution to the total density is implicitly excluded from the integral in Eq.~(\ref{eq:densityint}). As $\mu$ approaches zero from below the ground state occupation can become arbitrarily large, as can be seen by inspecting Eq.~(\ref{eq:bosep}).}.
At a given density $n$ the BEC transition temperature is given by
\begin{equation}\label{eq:uniformTc}
k_{\rm B} T_c^0=\frac{2\pi \hbar^2}{m}\left(\frac{n}{\zeta(3/2)}\right)^{2/3} \; ,
\end{equation}
where the superscript $^0$ refers to the fact this is an ideal gas result.
For a gas in a potential $V(\mathbf{r})$ we may apply LDA to Eq.~(\ref{eq:densityint}). This amounts to having a local chemical potential
\begin{equation}\label{eq:mulocal}
\mu(\mathbf{r})=\mu-V(\mathbf{r}) \; .
\end{equation}
Specifically for a harmonic trap, $V(\mathbf{r}) = \sum (1/2) m \omega^2_i r_i^2$, where $\omega_i$ (with $i=1,2,3$) are the trapping frequencies along three spatial dimensions.
The local density is then
\begin{equation}\label{eq:nr}
n(\mathbf{r})=\frac{g_{3/2}(\mathrm{e}^{(\mu-V(\mathbf{r}))/k_{\rm B} T})}{\lambda^3} \; ,
\end{equation}
and the local phase space density $D(\mathbf{r})=n(\mathbf{r})\lambda^3$. The total number of particles in the excited states can be found by integrating over all space:
\begin{equation}\label{eq:Ntrap}
N' =\int \frac{g_{3/2}(\mathrm{e}^{(\mu-V(\mathbf{r})/k_{\rm B} T})}{\lambda^3} \, \mathrm{d}\mathbf{r} \; .
\end{equation}
The critical point for a trapped gas is the point at which the maximal local $D$ reaches the critical value of $\zeta(3/2)$. For a fixed $T$ it makes sense to define the critical point in terms of the critical total particle number $N_c$. For a harmonic trap, with the geometric mean of the three trapping frequencies $\bar{\omega}$, the integral in Eq.~(\ref{eq:Ntrap}) for $\mu=0$ gives
\begin{equation}\label{eq:Ncid}
N_{\rm c}^{0}=\zeta(3)\left(\frac{k_{\rm B} T}{\hbar \bar{\omega}}\right)^3 \; ,
\end{equation}
where $\zeta(3)\approx1.202$.
The equivalent expression for the transition temperature at a fixed particle number is given
by\footnote{Finite-size corrections slightly reduce the ideal-gas critical temperature, by $k_{\rm B} \Delta T_c^0 =-\zeta(2)/(2 \zeta(3)) \, \hbar \omega_m\approx - 0.684 \, \hbar \omega_m$, where $\omega_m$ is the algebraic mean of the trapping frequencies \cite{Dalfovo:1999}.}
\begin{equation}\label{eq:Tcid}
k_{\rm B} T_c^0=\hbar \bar{\omega} \left(\frac{N}{\zeta(3)}\right)^{1/3}.
\end{equation}
\begin{figure}[t]
\centering
\includegraphics[width=0.65\columnwidth]{Figure1.pdf}
\caption{Ideal Bose gas condensation. Number of thermal atoms $N'$ (black line) and number of condensed atoms $N_0$ (grey line) are plotted versus the total atom number $N$, at a fixed temperature. As atoms are added to trap $N'=N$ and $N_0=0$ until the critical atom number $N_{\rm c}^{0}$ is reached. At this point the excited states of the system saturate, and for $N> N_{\rm c}^{0}$ we have $N'=N_{\rm c}^{0}$ and $N_0=N-N_{\rm c}^{0}$. }
\label{fig:intro_fig1}
\end{figure}
The ideal-gas picture of Bose-Einstein condensation driven by the purely statistical saturation of the excited states is simply summarised graphically in Fig. \ref{fig:intro_fig1}.
Here we plot the number of atoms in the excited states, $N'$, and in the condensate, $N_0$, as the total atom number $N$ is increased at constant temperature. For $N < N_{\rm c}^{0}$ no condensate is present and $N' = N$. However for $N > N_{\rm c}^{0}$ the thermal component is saturated at $N' = N_{\rm c}^{0}$ and the number of condensed atoms is simply given by $N_0 = N - N_{\rm c}^{0}$.
In sections \ref{sec:saturation} and \ref{sec:Tcshift} we will examine the effects of interactions both on the saturation of the excited states and on the value of the critical atom number.
\subsection{Interacting Bosons}
\label{sec:Interactions}
The dominant effects of interactions on Bose-Einstein condensation are quite different in a uniform system and in the experimentally pertinent case of a harmonically trapped atomic gas. This complex problem has a long history and for reviews we refer the reader to, for example, \cite{Dalfovo:1999,Pethick:2002,Pitaevskii:2003,Andersen:2004RMP}. Here we just briefly introduce some key points.
The simplest theoretical framework to address the effects of interactions in a Bose gas is the Hartree-Fock approximation \cite{Dalfovo:1999}. In this mean-field (MF) model one treats the thermal atoms as a ``noninteracting" gas of density $n'(\bf r)$ that experiences a self-consistent MF interaction potential $g[2n_0({\bf r})+2n'({\bf r})]$, where $g=4 \pi \hbar^2 a /m$, $a$ is the s-wave scattering length, and $n_0(\bf r)$ the condensate density. We can then define an effective total potential
\begin{equation}
V_{\rm eff}({\bf r})=V({\bf r})+2g[n_0({\bf r})+n'({\bf r})] \; ,
\label{eq:Veff}
\end{equation}
and apply the LDA by replacing $V({\mathbf{r}})$ with $V_{\mathrm{eff}}(\mathbf{r})$ in Eq.~(\ref{eq:nr}).
Meanwhile the condensed atoms feel an interaction potential $g[n_0({\bf r})+2n'({\bf r})]$, where the factor of two difference in the condensate self-interaction comes about due to the lack of the exchange interaction term for particles in the same state\footnote{This approach does not take into account the modification of the excitation spectrum due to the presence of the condensate, which is included in more elaborate MF theories such as those of Bogoliubov \cite{Bogoliubov:1947} and Popov \cite{Popov:1987} (see also \cite{Dalfovo:1999}). However, it is often sufficient to give the correct leading order MF results.}.
In a uniform system, the MF potential gives just a spatially uniform energy offset and the most interesting effects arise due to beyond-MF quantum correlations.
On the other hand, in a harmonically trapped gas (with repulsive interactions) the inhomogeneous density results in a mean-field repulsion of atoms from the central high-density region. This geometrical effect often dominates and makes it harder to experimentally observe the more interesting beyond-MF physics.
\subsection{Chapter outline}
In section \ref{sec:experiment} we briefly outline our experimental procedure for performing precision measurements of the effects of interactions on Bose-Einstein condensation of an atomic gas.
In sections \ref{sec:saturation} and \ref{sec:Tcshift} we discuss the effects of interactions on the thermodynamics of a Bose gas with tuneable interactions. Here the range of interaction strengths we explore experimentally is such that the gas can always be assumed to be in thermal equilibrium. In section \ref{sec:saturation} we scrutinise the concept of saturation as the driving mechanism for Bose-Einstein condensation, and in section \ref{sec:Tcshift} we focus on the interaction shift of the critical point for condensation. We compare the experimental results to both MF and beyond-MF theories and discuss how they relate to the case of a uniform Bose gas.
In section \ref{sec:nonequilibrium} we discuss the conditions required for equilibrium measurements, and non-equilibrium effects that are observed when they are violated.
\section{Precision measurements on a Bose gas with tuneable interactions }
\label{sec:experiment}
All the results presented here were obtained by performing conceptually simple experiments which are close in spirit to the ideal-gas theoretical plots of Fig. \ref{fig:intro_fig1}, but are performed for various strengths of repulsive interatomic interactions, characterised by the positive s-wave scattering length $a$.
Our experiments start with a partially condensed gas of $^{39}$K atoms,
produced in an optical dipole trap \cite{Campbell:2010}
with $\bar \omega/2\pi$ varying between 60 and 80\,Hz for data taken at different temperatures.
The strength of interactions in the gas can be tuned by applying a uniform external magnetic field in the vicinity of a Feshbach scattering resonance centred at 402.5 G \cite{Zaccanti:2009}.
In each experimental series we fix $a$ by choosing the value of the Feshbach field, and keep the temperature constant by fixing the depth of the optical trap. The total number of trapped atoms $N$ is then varied by holding the gas in the trap for a variable time $t$, up to tens of seconds. During this time $N$, initially larger than the critical value $N_{\rm c}$, slowly decays and eventually drops below $N_{\rm c}$. Meanwhile elastic collisions between the atoms act to redistribute the particles between the condensed and thermal components of the gas.
In each experimental run within a given series, corresponding to a particular hold-time $t$, the thermal atom number $N'$ and the condensate atom number $N_0$ are extracted from fits to the absorption images of the gas after $18-20\,$ms of free time-of-flight (TOF) expansion from the trap \cite{Ketterle:1999b,Gerbier:2004c}.
The interactions are rapidly turned off at the beginning of TOF, by tuning the Feshbach field to the $a=0$ point. This minimises the condensate expansion and allows us to home in on the critical point by reliably measuring condensed fractions as low as $0.1\,\%$.
For accurate measurements of the small interaction shift of the critical point it is particularly important to minimise various systematic errors, e.g. due to finite-size effects \cite{Dalfovo:1999}, uncertainties in the absolute calibration of $N$ and $\bar{\omega}$, and small anharmonic corrections to the trapping potential \cite{Campbell:2010}. We achieve this by performing ``differential measurements", always concurrently running two experimental series which are identical in every respect except for the choice of the scattering length \cite{Smith:2011}.
\section{Non-saturation of the excited states}
\label{sec:saturation}
\begin{figure}[t]
\centering
\includegraphics[width=0.65\columnwidth]{Figure2.pdf}
\caption{Lack of saturation of the thermal component in a quantum degenerate atomic Bose gas.
$N'$ (black points) and $N_0$ (grey points) are plotted versus the total atom number $N$
at $T=177$\,nK and $a=135\,a_0$. The corresponding predictions for a saturated gas are shown by black and grey solid lines. The critical point $N=N_{\rm c}$ is marked by a vertical dashed line. (Figure adapted from \cite{Tammuz:2011}.)}
\label{fig:figure1}
\end{figure}
In this section, we focus on the concept of the saturation of the excited states as the underlying mechanism driving the BEC transition.
In superfluid $^4$He, which is conceptually associated with BEC, strong interactions preclude direct observation of purely statistical effects postulated by Einstein for an ideal gas. On the other hand it is generally accepted that a close-to-textbook BEC is observed in the weakly interacting atomic gases. Therefore, one might expect that the saturation inequality $N' \leq N_{\rm c}^{0}$ is essentially satisfied in these systems, with just the value of the bound on the right-hand side slightly modified by interactions. However, as shown in Fig. \ref{fig:figure1}, this is far from being the case under typical conditions of an ultracold gas experiment.
Here, in an experimental series taken with $a=135\,a_0$ (where $a_0$ is the Bohr radius) and $T=177$\,nK, the measured critical atom number is
$N_{\rm c} \approx 200\,000$; if the total number of atoms is increased to $450\,000$, only half of the additional atoms accumulate in the condensate.
In order to explore the relationship between the experimentally observed non-saturation of the thermal component and the interatomic interactions, we first identify the relevant interaction energy. As a BEC is formed and then grows, the change in the average density of the condensed atoms is much larger than the change of the thermal density. Therefore one expects the non-saturation of the thermal component to result primarily from its interaction with the condensate (the $2gn_0(\mathbf{r})$ term in Eq.~(\ref{eq:Veff})). The relevant energy scale is then provided by \cite{Dalfovo:1999}
\begin{equation}
\mu_0=gn_0 (\mathbf{r}=0)=\frac{\hbar \bar \omega}{2}
\left(
15 N_0 \frac{a}{a_{\rm ho}}
\right)^{2/5}\ ,
\label{eq:mu}
\end{equation}
where $a_{\rm ho}=(\hbar/m\bar \omega)^{1/2}$ is the spatial extension of the ground state of the harmonic oscillator. The energy $\mu_0$ is the MF result for the chemical potential of a gas with $N_0$ atoms at zero temperature in the Thomas-Fermi limit \cite{Dalfovo:1999}.
Guided by this scaling, for the data shown in Fig.~\ref{fig:figure1} we plot $N'$ as a function of $N_0^{2/5}$ in Fig.~\ref{fig:TwoFifths}. The growth of $N'$ with $N_0^{2/5}$ is not perfectly linear, so we quantify the non-saturation effect with two linear slopes: (1) the initial slope $S_0$ for $N_0 \rightarrow 0$, and (2) the course grained slope $S= \Delta[N']/\Delta [N_0^{2/5}]$ for $0.1<\mu_0\, /\,k_{\rm B} T<0.3$ \cite{Tammuz:2011}.
The data shown in Fig.~\ref{fig:TwoFifths} can also be described by a second-order polynomial fit,
as described later.
\begin{figure}[t]
\centering
\includegraphics[width=0.65\columnwidth]{Figure3.pdf}
\caption{Quantifying the lack of saturation. Here $N'$ is plotted as a function of $N_0^{2/5}$ for the same series as in Fig.~\ref{fig:figure1}. The horizontal dotted line is the saturation prediction $N'=N_{\rm c}$. The two black lines show the initial slope $S_0$ and the slope $S$ for $0.1<\mu_0\, /\,k_{\rm B} T<0.3$. The solid grey line is a guide to the eye based on a second-order polynomial fit. (Figure adapted from \cite{Tammuz:2011}.)}
\label{fig:TwoFifths}
\end{figure}
The initial slope $S_0$ may be compared with the HF model. In order to obtain the non-saturation effect to first order within the HF approach, we only consider the repulsive interaction of the thermal atoms with the condensate and not with other thermal atoms. From Eq.~(\ref{eq:Veff}) this leads to an effective
potential\footnote{Note that $gn_0(\mathbf{r})=\max\{\mu_0-V(r),0\}$.}
\begin{equation}
V_{\rm eff}({\bf r})=V({\bf r})+2gn_0({\bf r})=|V({\bf r})-\mu_0|+\mu_0\ .
\label{eq:Veffsat}
\end{equation}
Note that within this theory $N_c = N_{\rm c}^{0}$ since $V_{\rm eff}({\bf r})=V({\bf r})$ when $N_0=0$. By integrating Eq.~(\ref{eq:Ntrap}) with the effective potential of Eq.~(\ref{eq:Veffsat}) one can predict a linear variation of $N'/N_c^0$ with the small parameter $\mu_0\, /\,k_{\rm B} T$:
\begin{equation}
\frac{N'}{N_c^0}=1 + \alpha \, \frac{\mu_0}{k_{\rm B} T} \; ,
\label{eq:HF}
\end{equation}
with $\alpha = \zeta(2)/\zeta(3)\approx 1.37$. This first order non-saturation result is identical to that obtained in more elaborate MF approximations, which only modify higher order terms.
The origin of the non-saturation effect can be qualitatively understood by noting that interactions with the condensate modify the effective potential seen by the thermal atoms from a parabola into the ``Mexican hat" shape of Eq.~(\ref{eq:Veffsat}); this effectively allows the thermal component to occupy a larger volume, which grows with increasing $N_0$.
From Eqs.~(\ref{eq:mu}) and (\ref{eq:HF}) we define the HF non-saturation slope
\begin{equation} \label{eq:SHF}
S_{\rm HF} = \frac{dN'}{d (N_0^{2/5})} = \frac{\zeta(2)}{\zeta(3)} X
\end{equation}
where $X$ is the dimensionless parameter
\begin{equation}
X=\frac{\zeta(3)}{2}\left(\frac{k_{\rm B} T}{\hbar \bar{\omega}}\right)^2 \left(\frac{15 \, a}{a_{\rm ho}}\right)^{2/5}.
\end{equation}
The measured $S_0$ is found to agree with $S_{\mathrm{HF}}$ for a range of $a$ and $T$ values \cite{Tammuz:2011}.
We now consider the non-saturation at higher $N_0$ values, where the data is not well described by Eq.~(\ref{eq:SHF}). Fig.~\ref{fig:extrapolation} summarises the non-saturation slopes $S(a,T)$ for a wide range of interaction strengths and temperatures. Within experimental error all data points fall onto a straight line with gradient $2.6\pm0.3$ and intercept $S(0) = -20 \pm 100$ when plotted against the dimensionless interaction parameter $X$.
\begin{figure}[t]
\centering
\includegraphics[width=0.65\columnwidth]{Figure4.pdf}
\caption{Deviation from the saturation picture at a range of interaction strengths and temperatures.
The non-saturation slope $S$ is plotted versus the dimensionless interaction parameter $X \propto T^2 a^{2/5}$ (see text). A linear fit (black line) gives $dS/dX = 2.6 \pm 0.3$\ and an intercept $S(0) = -20 \pm 100$, consistent with complete saturation in the ideal-gas limit. The data points are based on measurements with the $^{39}$K gas (closed circles) at a range of scattering lengths ($a = 40 - 356\,a_0$) and temperatures ($T = 115 - 284\,$nK), and two additional experimental series taken with a $^{87}$Rb gas (open circles). (Figure adapted from \cite{Tammuz:2011}.)
}
\label{fig:extrapolation}
\end{figure}
The first and most important thing to notice is that both non-saturation slopes, $S_0$ and $S$, tend to zero for $X \rightarrow 0$.
These experiments thus confirm the concept of a saturated Bose gas, and Bose-Einstein condensation as a purely statistical phase transition in the non-interacting limit.
A question that is still open is the deviation of $S$ from the first order HF result $S_{\mathrm{HF}}$, i.e. $dS/dX \approx 2.6$ versus $\zeta(2)/\zeta(3) \approx 1.37$ predicted from HF theory. This discrepancy can partially be explained by higher order terms in the mean-field theory, either directly from using Eq.~(\ref{eq:Veffsat}) in Eq.~(\ref{eq:Ntrap}), or using more elaborate MF theories such as the Popov approximation \cite{Pethick:2002}. However the effect is far stronger in the experimental results than any of these MF theories predict. To see this we consider the next order term in Eq.~(\ref{eq:HF}), writing $N'/N_{\rm c}^{0}=1+\alpha (\mu_0/k_{\rm B} T)+\alpha_2 (\mu_0/k_{\rm B} T)^2$ with $\alpha=1.37$.
Experimentally, by identifying $S$ with the gradient of this quadratic function evaluated at $\mu_0/k_{\rm B} T= 0.2$, we get $\alpha_2=3\pm0.7$. For comparison the Popov approximation gives $\alpha_2 \approx 0.6$. At present the reason for this discrepancy and the possible role of beyond-MF effects are unclear, and require further investigation.
In summary, we can quantify the non-saturation of the thermal component in a harmonically trapped gas by writing the number of thermal atoms in a partially condensed cloud as
\begin{equation}
N'=N_c+S_0 N_0^{2/5} + S_2 N_0^{4/5} \; ,
\end{equation}
where
\begin{equation}\label{eq:S0}
S_0=\frac{\zeta(2)}{2}\left(\frac{k_{\rm B} T}{\hbar \bar{\omega}}\right)^2 \left(\frac{15a}{a_{\rm ho}}\right)^{2/5}
\end{equation}
and
\begin{equation}\label{eq:S2}
S_2=(3 \pm0.7) \, \frac{\zeta(3)}{4} \, \frac{k_{\rm B} T}{\hbar \bar{\omega}} \, \left(\frac{15a}{a_{\rm ho}}\right)^{4/5}.
\end{equation}
We have seen that in a harmonic trap the dominant non-saturation effect is ``geometric", arising from an interplay of the mean-field repulsion and the inhomogeneity of the condensate density. It is then interesting to consider the case of a uniform system, where this geometric effect is absent. Within MF theory, as the total density of a uniform Bose gas is increased past the critical value the thermal density $n'$ actually decreases. This is due to the fact that the atoms in the condensate have less interaction energy, as discussed in section \ref{sec:Interactions}. In addition, close to the transition, beyond-MF effects are expected to play an important role. This would make measurements in a uniform system (or of the local density in a trapped system) particularly interesting.
\section{Interaction shift of the transition temperature}
\label{sec:Tcshift}
Having considered the effect of interactions on the saturation of the thermal component we now consider the location of the critical point itself.
It is generally accepted that in a uniform system there is no interaction shift of the critical temperature $T_c$ at the level of mean-field theory. However, the beyond-MF correlations between particles which develop near the critical point are expected to shift $T_c$
\cite{Lee:1957TcShift, Lee:1958TcShift2,Bijlsma:1996, Baym:1999, Holzmann:1999, Reppy:2000, Arnold:2001, Kashurnikov:2001, Holzmann:2001, Baym:2001, Kleinert:2003, Andersen:2004RMP, Holzmann:2004}.
For several decades there was no consensus on the functional form, or even on the sign of this $T_c$ shift (for an overview see e.g. \cite{Arnold:2001, Baym:2001, Andersen:2004RMP, Holzmann:2004}). It is now generally believed that the shift is positive and to leading order
given by \cite{Arnold:2001, Kashurnikov:2001}:
\begin{equation}
\frac{\Delta T_c}{T_c^0} \approx 1.3 \, a n^{1/3} \approx 1.8 \, \frac{a}{\lambda_0} \, ,
\label{eq:uniformTcShift}
\end{equation}
where $\Delta T_c = T_c - T_c^0$ and $\lambda_0$ is the thermal wavelength at temperature $T_c^0$. Equivalently, the $n_c$ shift at constant $T$ is $\Delta n_c/n_c^0 \approx - (3/2) \Delta T_c /T_c^0$. The positive $\Delta T_c$ implies that condensation occurs at a phase space density below the ideal-gas critical value of 2.612.
The problem of the $T_c$ shift in a harmonically trapped gas is even more complex. In this case, at least for weak interactions, the shift is dominated by an opposing effect that reduces the critical temperature \cite{Giorgini:1996}.
This negative $T_c$ shift is due to the broadening of the density distribution by repulsive interactions (see Fig.~\ref{fig:TcCartoon}). To leading order it can be calculated analytically using MF theory \cite{Giorgini:1996}, by self-consistently solving Eq.~(\ref{eq:nr}) with $V_{\mathrm{eff}}=V(\mathbf{r})+2gn'(\mathbf{r})$:
\begin{equation}
\frac{\Delta T_c}{T_c^0} \approx - 3.426 \, \frac{a}{\lambda_0} \; .
\label{eq:StringariMF}
\end{equation}
For the experimentally relevant range of interaction strengths, $0<a/\lambda_0<0.05$, we numerically obtain the second-order MF shift, $\approx 11.7 \, (a/\lambda_0)^2$ \cite{Smith:2011b}.
\begin{figure}[t]
\centering
\includegraphics[width=0.65\columnwidth]{Figure5.pdf}
\caption{Opposing effects of interactions on the critical point of a trapped Bose gas. We sketch the density distribution in a harmonic potential $V(r)$ at the condensation point. Compared to an ideal gas (dotted line) at the same temperature, repulsive interactions reduce the critical density, but also broaden the density distribution (solid line). Mean-field theory (dashed line) captures only the latter effect, and predicts an increase of the critical atom number $N_c$ at fixed $T$, equivalent to a decrease of $T_c$ at fixed $N$. (Figure adapted from \cite{Smith:2011}.)}
\label{fig:TcCartoon}
\end{figure}
The two opposing effects of repulsive interactions on the critical point of a trapped gas are visually summarised in Fig.~\ref{fig:TcCartoon}, where we sketch the density distribution at the condensation point for an ideal and an interacting gas at the same temperature\footnote{The shift of the critical point can be equivalently expressed as $\Delta T_c(N)$ or $\Delta N_c(T)$, with $\Delta N_c(T)/N_{\rm c}^{0} \approx -3 \Delta T_c /T_c^0$.}.
In the spirit of LDA, the critical density should be reduced by interactions. However, interactions also broaden the density distribution. For weak interactions the latter effect is dominant, making the overall interaction shift $\Delta N_c(T)$ positive, or equivalently $\Delta T_c(N)$ negative.
The dominance of the negative MF shift of $T_c$ over the positive beyond-MF one goes beyond the difference in the numerical factors in Eqs.~(\ref{eq:uniformTcShift}) and (\ref{eq:StringariMF}). In a harmonic trap, at the condensation point only the central region of the cloud is close to criticality\footnote{The size of the central critical region is $r_c \sim (a/\lambda_0) R_T$, where $R_T = \sqrt {k_{\rm B} T / m\omega^2}$ is the thermal radius of the cloud \cite{Arnold:2001b}.}; this reduces the net effect of critical correlations so that they affect $T_c$ only at a higher order in $a/\lambda_0$. The MF result of Eq.~(\ref{eq:StringariMF}) should therefore be exact at first order in $a/\lambda_0$.
The higher-order beyond-MF shift is still expected to be positive, but the theoretical consensus on its value has not been reached \cite{Houbiers:1997, Holzmann:1999b, Arnold:2001b, Davis:2006, Zobay:2009}.
\subsection{Measurements on a harmonically trapped Bose gas}
Since the early days of atomic BECs there have been several measurements of the interaction $T_c$ shift in a harmonically trapped gas \cite{Ensher:1996, Gerbier:2004,Meppelink:2010}. These experiments,
performed at $a/\lambda_0 $ ranging from $0.007$ \cite{Meppelink:2010} to 0.024 \cite{Gerbier:2004},
nicely confirmed the theoretical prediction for the linear MF shift of Eq.~(\ref{eq:StringariMF}), but could not discern the beyond-MF effects of critical correlations.
The recent measurements \cite{Smith:2011} presented here provided the first clear observation of the beyond-MF $T_c$ shift in a trapped atomic gas. Several improvements contributed to making this possible. First, we explored slightly higher interaction strengths, up to $a/\lambda_0 \approx 0.04$. Second, by performing precision measurements outlined in section \ref{sec:experiment}, and directly accessing the small differential $T_c$ shift due to the variation in $a/\lambda_0$, we significantly reduced the experimental error bars. Third, understanding the non-saturation effects discussed in section \ref{sec:saturation} was also essential for accurately determining the critical point from the measurements performed {\it close} to it (see Fig.~\ref{fig:Nc}).
\begin{figure} [t]
\centering
\includegraphics[width=1.0\columnwidth]{Figure6.pdf}
\caption{Determination of the critical point and the differential interaction shift. (a) Condensed ($N_0$) versus thermal ($N'$) atom number for two concurrently taken data series with $a = 56\,a_0$ (circles) and $a=274\,a_0$ (squares). Note that all points correspond to condensed fractions below $2\,\%$. The data is scaled to the same temperature ($T=240\,$nK) and shows the shift of the critical point in the form $\Delta N_c(T)$. Solid lines show the extrapolation to $N_0=0$, necessary to accurately determine $N_c$. (b) $N'$ is plotted versus $N_0^{2/5}$ for the same data as in (a), showing more clearly the extrapolation procedure. (Figure adapted from \cite{Smith:2011}.)}
\label{fig:Nc}
\end{figure}
Fig.~\ref{fig:Nc} illustrates a differential measurement with $a=56\,a_0$ and $a = 274\,a_0$.
The rise of $N_0$ in Fig.~\ref{fig:Nc}(a) is not simply vertical because the thermal component is not saturated at $N_c$ (see section \ref{sec:saturation}). It is therefore essential to carefully extrapolate $N'$ to the $N_0=0$ limit in order to accurately determine $N_c$. For small $N_0$ the extrapolation is done using $N'=N_c+S_0 N_0^{2/5}$, with the non-saturation slope $S_0(T,\bar{\omega},a)$ given by Eq. (\ref{eq:S0}).
In Fig.~\ref{fig:TcShift} we summarise our measurements of $\Delta T_c / T_c^0$ \cite{Smith:2011}. The data taken with different atom numbers, $N \approx (2-8) \times 10^5$, fall onto the same curve, confirming that the results depend only on the interaction parameter $a/\lambda_0$.
The MF prediction agrees very well with the data for $a/\lambda_0 \lesssim 0.01$, but for larger $a/\lambda_0$ there is a clear deviation from this prediction. All the data are fitted well by a second-order polynomial
\begin{equation}
\frac{\Delta T_c}{T_c^0} \approx b_1 \, \frac{a}{\lambda_0} + b_2 \left( \frac{a}{\lambda_0} \right)^2\, ,
\label{eq:TrapTc}
\end{equation}
with $b_1= - 3.5 \pm 0.3$ and $b_2 = 46 \pm 5$. The value of $b_1$ is in agreement with the MF prediction of $-3.426$. The $b_2$ value strongly excludes the MF result of $b_2^{\rm MF} \approx 11.7$ and its sign is consistent with the expected effect of beyond-MF critical correlations.
Fixing $b_1=-3.426$ (which is expected to be exact even including beyond-MF effects) gives an improved estimate of $b_2=42\pm2$.
\begin{figure} [t]
\centering
\includegraphics[width=0.65\columnwidth]{Figure7.pdf}
\caption{Interaction shift of the critical temperature. Data points were taken with $N \approx 2 \times 10^5$ (open circles), $4 \times 10^5$ (black squares), and $8 \times 10^5$ (open triangles) atoms. The dashed line is the MF prediction. The solid line shows a second-order polynomial fit to the data (see text). Vertical error bars are statistical, while systematic errors in $\Delta T_c / T_c^0$ are assessed to be $<1\,\%$ \cite{Smith:2011}. (Figure adapted from \cite{Smith:2011}.)}
\label{fig:TcShift}
\end{figure}
\subsection{Connection with a uniform Bose gas}
In order to make a connection between the experiments on trapped atomic clouds and the theory of a uniform Bose gas we also need to consider the effect of interactions on the critical chemical potential $\mu_c$.
In a uniform gas the interactions differently affect $T_c$ (or equivalently $n_c$) and $\mu_c$ at both MF and beyond-MF level. The simple MF shift $\beta \mu_c^{\rm MF} = 4 \, \zeta(3/2) \, a/\lambda_0$ (where $\beta = 1/k_{\rm B} T$) has no effect on condensation. To lowest beyond-MF order we have\footnote{Note that $B_2$ is not just a constant but contains logarithmic corrections in $a/\lambda_0$ \cite{Arnold:2001}. We neglect these in our discussion since they are not discernible at the current level of experimental precision.}:
\begin{equation}
\beta \mu_c \approx \beta \mu_c^{\rm MF} + B_2 \left( \frac{a}{\lambda_0} \right)^2 \; .
\label{eq:Muc}
\end{equation}
We see that there is a qualitative difference between Eqs. (\ref{eq:uniformTcShift}) and (\ref{eq:Muc}). Specifically, we have $n_c^{\rm MF} - n_c \propto a/\lambda_0$, but $\mu_c^{\rm MF} - \mu_c \propto (a/\lambda_0)^2$. This difference highlights the fact that the problem of the $T_c$ shift is non-perturbative; near criticality the equation of state does not have a regular expansion\footnote{The non-interacting equation of state (Eq.~(\ref{eq:psd})) cannot be expanded about $D_c$ in $\beta \mu$, but rather in $\sqrt{-\beta \mu}$; up to first order this expansion gives $D=D_c-2\sqrt{\pi}\sqrt{-\beta \mu}$. This scaling goes some way in explaining the qualitative difference between Eqs. (\ref{eq:uniformTcShift}) and (\ref{eq:Muc}) although it cannot be used quantitatively.}
in $\mu$, otherwise one would get $\Delta n_c \propto \mu_c - \mu_c^{\rm MF}$.
For a harmonic trap, within LDA the uniform-system results for $n_c$ and $\mu_c$ apply in the centre of the trap, and elsewhere local $\mu$ is given by Eq.~(\ref{eq:mulocal}). The result for the $T_c$ shift however does not carry over easily to the non-uniform case.
As illustrated in Fig.~\ref{fig:Cartoon}(a), in the centre of the trap we expect $\Delta n_c \propto a/\lambda_0$, but the measured beyond-MF $\Delta N_c \propto (a/\lambda_0)^2$ (see Eq.~(\ref{eq:TrapTc})).
\begin{figure} [t]
\centering
\includegraphics[width=0.65\columnwidth]{Figure8.pdf}
\caption{ Beyond-MF effects near the critical point in a harmonically trapped Bose gas. (a) For a fixed $T$, the density distribution at the critical point $N=N_c < N_c^{\rm MF}$ (solid grey line) is compared with the MF prediction (dashed line). In the trap centre we expect $n_c^{\rm MF} - n_c \propto a/\lambda_0$, characteristic of a uniform system. However the experimentally measured $N_c^{\rm MF} - N_c \propto (a/\lambda_0)^2$ is dominated by the density shift {\it outside} the central critical region,
and is not directly related to the $n_c$ shift.
(b) If $N$ is increased to $N_c^{\rm MF} > N_c$, a condensate induced by critical correlations forms within the critical region of size $\propto a/\lambda_0$. The condensed atom number $N_0 \propto (a/\lambda_0)^4$ directly relates to the critical density shift $\Delta n_c \propto a/\lambda_0$. (Figure adapted from \cite{Smith:2011b}.)}
\label{fig:Cartoon}
\end{figure}
In fact, the experimentally observed $T_c$ shift (Eq.~(\ref{eq:TrapTc})) qualitatively mirrors Eq.~(\ref{eq:Muc}), with the MF term linear in $a/\lambda_0$ and the beyond-MF one quadratic in $a/\lambda_0$ (to leading orders).
This similarity can be understood as follows: The interaction shift of $\mu_c$ affects the density everywhere in the trap; outside the small critical region the equation of state is regular in $\mu$ and the local density shift is simply proportional to the local $\mu$ shift; the contribution to the total $N_c$ shift from the non-critical region greatly outweighs the contribution from within the critical region and therefore one qualitatively expects $N_c - N_c^{\rm MF} \propto \mu_c - \mu_c^{\rm MF}$. More quantitatively, this connection is given by\footnote{Note that this relationship is closely related to Eq.~(\ref{eq:HF}).} \cite{Arnold:2001}
\begin{equation}
\label{eq:B2}
-3(b_2-b_2^{MF})=\frac{\zeta(2)}{\zeta(3)}B_2 \; .
\end{equation}
The key conclusion of this discussion is therefore that the beyond-MF $T_c$ shift observed in a trapped gas is directly related to the beyond-MF $\mu_c$ shift (in either trapped or uniform system). It does not however directly reveal the expected linear $n_c$ shift and the theoretically most intriguing non-perturbative connection between $\mu_c$ and $n_c$ shifts. In fact one could say that the historically puzzling (theoretical) connection between $n_c$ and $\mu_c$ is just replaced by the puzzling connection between (expected) $n_c$ and (measured) $N_c$.
This problem can be partly overcome by studying the condensed fraction $f_0$ in a trapped atomic cloud at the MF-predicted critical point \cite{Smith:2011b}. By definition $f_0$ vanishes within MF theory, and so directly measures the effect of critical correlations which shift $T_c$ above $T_c^{\rm MF}$.
At a fixed $T$ we consider the condensed fraction of a gas, $N_0/N$, at the point where $ N= N_c^{\rm MF} > N_c$, as illustrated in Fig.~\ref{fig:Cartoon}(b).
The analogous quantity for a uniform gas was first theoretically studied by Holzmann and Baym \cite{Holzmann:2003}, who showed
that\footnote{This scaling holds for any distance from the critical point given by $(\mu - \mu_c)(\lambda_0/a)^2 = {\rm const}$. By applying it to the MF critical point we neglect the logarithmic corrections to $\mu_c^{\rm MF} - \mu_c$, which are so far not experimentally observable.}
$n_0/n \propto \Delta n_c \propto a/\lambda_0$.
The condensate density $n_0$ vanishes at the point where the local $\mu = \mu_c$, so from Eqs.~(\ref{eq:mulocal}) and (\ref{eq:Muc}) we get that the spatial extension of the condensate is also $\propto a/\lambda_0$, and hence $f_0 \propto (a/\lambda_0)^4$.
The quartic scaling of $f_0$ with $a/\lambda_0$ is thus directly related to the expected linear scaling of $\Delta n_c$ with $a/\lambda_0$; the two scalings are simply connected by the volume of the critical region, $\propto (a/\lambda_0)^3$. This scaling is indeed confirmed experimentally \cite{Smith:2011b}, as shown in Fig.~\ref{fig:quartic}.
\begin{figure} [t]
\centering
\includegraphics[width=0.65\columnwidth]{Figure9.pdf}
\caption{ Condensed fraction of an atomic gas induced by critical correlations. The condensed fraction $f_0 = N_0/N$ is measured for $N= N_c^{\rm MF} > N_c$. A fit to the data (solid line) with the function $f_0 \propto (a/\lambda_0)^x$ gives an exponent $x=3.9 \pm 0.4$, in agreement with the predicted $x = 4$. (Figure adapted from \cite{Smith:2011b}.)}
\label{fig:quartic}
\end{figure}
One can take the comparison with theory beyond just the scaling of $f_0$ with $a/\lambda_0$, and quantitatively compare the measured $N_0$ at the MF critical point with the Monte-Carlo (MC) calculations \cite{Prokofev:2004} for a uniform gas.
This also works out very well, with the measured and predicted $N_0$ agreeing within a few percent \cite{Smith:2011b}. It is however important to carefully summarise the conceptual steps involved in this comparison:
(i) On the one hand, the measurements of the $T_c$ shift \cite{Smith:2011} experimentally provide (up to logarithmic corrections) the value of $\mu_c^{\rm MF} - \mu_c \propto (a/\lambda_0)^2$, via Eqs.~(\ref{eq:TrapTc}), (\ref{eq:Muc}), and (\ref{eq:B2}) \cite{Smith:2011b}.
(ii) On the other hand, the MC calculations \cite{Kashurnikov:2001, Prokofev:2004} that predict the $n_c$ shift of Eq.~(\ref{eq:uniformTcShift}) also provide tabulated values of the uniform-system condensate density $n_0$ for any $\mu-\mu_c \propto (a/\lambda_0)^2$.
(iii) Combining these two results and the LDA (Eq.~(\ref{eq:mulocal})) we calculate the expected $N_0$ in a trapped gas with $N = N_c^{\rm MF}$ and find excellent agreement with the measurements shown in Fig.~\ref{fig:quartic} \cite{Smith:2011b}.
Overall this provides strong cross-validation of theory and experiment. However it is important to note that the two different measurements of beyond-MF effects (i.e. of the $T_c$ shift and $f_0$) do not provide two \textit{independent} quantitative tests of the uniform-system theory. Instead, what we have shown is that they are consistently connected via the MC calculations for a uniform system. Finally it is also important to stress that we are still lacking a direct measurement of the $n_c$ shift, which would explicitly test the historically most debated theoretical result of Eq.~(\ref{eq:uniformTcShift}).
This goal remains open for future measurements, either of the local density in a harmonically trapped gas or on a uniformly trapped atomic gas.
\section{Equilibrium criteria and non-equilibrium effects}
\label{sec:nonequilibrium}
Finally, we discuss the criteria for the measurements on a trapped Bose gas to faithfully represent its equilibrium properties, and the non-equilibrium effects revealed when they are violated. It is helpful to distinguish two types of non-equilibrium behaviour, transient and intrinsic.
Transient non-equilibrium effects are more familiar, and occur whenever some system parameter, such as the interaction strength, is rapidly changed (quenched). After such a quench the system relaxes towards its new equilibrium.
Classically, one can estimate the relaxation time to be several elastic scattering times $1/\gamma_{\rm el}$, where $\gamma_{\rm el}$ is the elastic scattering rate \cite{Monroe:1993, Arndt:1997, Newbury:1995, Kavoulakis:2000c}.
However a system with continuous dissipation can only be ``close to" thermodynamic equilibrium and is to some extent always intrinsically out of equilibrium. The proximity to equilibrium broadly depends on the competition between relaxation and dissipation. For an atomic gas, this leads to a criterion based on the dimensionless parameter $\gamma_{\rm el} \tau$, where $\tau$ is some characteristic dissipation time, e.g.\ for atom loss. In practice, the relevant $\tau$ and the criteria for equilibrium measurements depend on the required measurement precision.
In the case of the $T_c$ measurements presented in section \ref{sec:Tcshift}, $N_c$ is determined to about $1\,\%$, so we require that the gas continuously (re-)equilibrates on a timescale $\tau$ corresponding to only $1\,\%$ atom-loss. This requires about 100 times higher $\gamma_{\rm el}$ than one would naively conclude by taking the $1/e$ lifetime of the cloud as the relevant timescale.
An interesting question is then what happens if we violate these stringent equilibrium criteria.
In Fig.~\ref{fig:NonEq}(a) we show measurements extending beyond the equilibrium region shown in Fig.~\ref{fig:TcShift}, and in Fig.~\ref{fig:NonEq}(b) we plot the corresponding $\gamma_{\rm el} \tau$.
Individually, $\gamma_{\rm el}$ and $\tau$ vary vastly as a function of $a$ ($\gamma_{\rm el}$ increasing and $\tau$ decreasing) \cite{Smith:2011}, but the breakdown of equilibrium appears to occur at $\gamma_{\rm el} \tau \approx 5$ in both the low- and high-$a$ limit.
\begin{figure}[t]
\centering
\includegraphics[width=0.65\columnwidth]{Figure10.pdf}
\caption{Non-equilibrium effects.
(a) $\Delta T_c / T_c^0$ is determined assuming equilibrium, as in Fig.~\ref{fig:Nc}. At both very low and very high $a$ the apparent $T_c$ deviates from the equilibrium curve. (b)
Equilibrium criteria (see text):
$\gamma_{\rm el} \tau$ (solid squares) is the number of elastic collisions per particle during $1\,\%$ atom-loss; $\gamma_{\rm el}/\bar{\omega} = 1$ (open circles) marks the onset of the hydrodynamic regime. (Figure adapted from \cite{Smith:2011}.)}
\label{fig:NonEq}
\end{figure}
In the small-$a$ limit the apparent $T_c$ is significantly above the equilibrium curve.
We can qualitatively understand this effect within a simple picture. In this regime, losses are dominated by one-body processes which equally affect $N_0$ and $N'$. The net effect of equilibrating elastic collisions would therefore be to transfer atoms from the condensate to the thermal cloud.
However the dissipation rate is too high compared to $\gamma_{\rm el}$, and so
$N_0$ remains non-zero even after the total atom number drops below the equilibrium critical value $N_c$.
In the large-$a$ limit the initial breakdown of equilibrium again appears to result in condensates surviving above the equilibrium $T_c$. However the physics in this regime is much richer, with several potentially competing effects requiring further investigation. For example, three-body decay affects $N_0$ and $N'$ differently, the thermal component is far from saturation, and the thermal component of the gas also enters the hydrodynamic regime, $\gamma_{\rm el}/\bar{\omega} > 1$.
It is interesting that we observe ``superheated" BECs for both very weak and very strong interactions. Further
study of these effects should prove useful for improving the understanding of condensation in intrinsically out-of-equilibrium systems, such as polariton gases.
|
{
"timestamp": "2012-03-12T01:01:27",
"yymm": "1203",
"arxiv_id": "1203.2063",
"language": "en",
"url": "https://arxiv.org/abs/1203.2063"
}
|
\section{Introduction}
{\bf Semantic distance} is a measure of how close or
distant the meanings of two units of language are.
The units of language may be words, phrases, sentences, paragraphs, or
documents.
The nouns {\it dance}
and {\it choreography}, for example, are closer in meaning than the
nouns {\it clown} and {\it bridge},
and so are said to be semantically closer.
The semantic distances between words (or more precisely, between concepts)
can be used as fundamental building blocks for measuring semantic distance between
larger units of language.
The ability to mimic
human judgments of semantic distance is useful in numerous natural language tasks
including machine translation, word sense
disambiguation, thesaurus creation, information retrieval, text
summarization, and identifying discourse structure.
This paper describes the state-of-the-art in corpus-based measures of semantic distance
between these fundamental units of language.
It identifies the significant challenges that existing approaches to semantic
distance face and in the process fleshes out questions that lead to
a better understanding of why two concepts are considered semantically close.
The paper concludes with a discussion of new hybrid approaches, that
show the potential to address these challenges.
Units of language, especially words,
may have more than one possible meaning. However, their context may be used
to determine the intended senses. For example,
{\it star} can mean both {\sc celestial body} and {\sc celebrity}; however, {\it star}
in---{\it Stars are powered by nuclear fusion}---refers only to {\sc celestial body} and is much closer to {\it sun} than to {\it famous}.
Thus, semantic distance between words in context is in fact the distance
between their underlying senses or lexical concepts.
Therefore, in this paper, we take word senses to be a
particular kind of concept. When we refer directly to a concept (written in small capitals),
it is with the understanding that it is the sense of one or more words, as reflected in the name that
we give the concept. We do not, in this paper, consider concepts that are unlexicalized.
Humans consider two concepts to be semantically close if there is a sharing of some meaning.
Specifically, two lexical concepts are semantically close if there is a {\bf lexical semantic relation}
between the concepts. Putting it differently, the reason why two concepts are considered semantically close
can be attributed to a lexical semantic relation that binds them.
According to \namecite{Cruse86}, a lexical semantic relation is a relation between {\bf lexical units}---a surface form along with a sense.
As he points out, the number of semantic relations that bind concepts is innumerable; but certain
relations, such as hyponymy, meronymy, antonymy, and troponymy, are more systematic and have enjoyed more attention in the linguistics
community. However, as \namecite{MorrisH04} point out, these relations are far out-numbered
by others, which they call {\bf non-classical relations}. Here are a few of the kinds of non-classical
relations they observed: positive qualities ({\sc brilliant, kind}), concepts pertaining
to a concept ({\sc kind, chivalrous, formal} pertaining to {\sc gentlemanly}), and
commonly co-occurring words (locations such as {\sc homeless, shelter}; problem--solution
pairs such as {\sc homeless, shelter}).
\subsection{Semantic relatedness and semantic similarity}
Semantic distance is of two kinds: {\bf semantic similarity} and {\bf semantic relatedness}.
The former is a subset of the latter,
but the two terms
may be used interchangeably in certain contexts, making it even more important to be
aware of their distinction.
Two concepts are considered to be semantically similar if there is a synonymy (or near-synonymy),
hyponymy (hypernymy), antonymy, or troponymy relation between them (examples include {\sc apples--bananas},
{\sc doctor--surgeon}, {\sc dark--bright}).
Two word senses are considered to be semantically related if there is any
lexical semantic relation at all between them---classical or non-classical
(examples include {\sc apples--bananas}, {\sc surgeon--scalpel}, {\sc tree--shade}).
\subsection{Human judgments of semantic distance}
Humans are adept at estimating
semantic distance;
but consider the following questions:
How strongly will two people agree/disagree
on distance estimates? Will the agreement vary over different sets of concepts?
Are we equally good at estimating semantic similarity and semantic relatedness?
In our minds, is there a clear distinction between related and unrelated concepts
or are concept-pairs spread across the whole range from synonymous to unrelated?
Some of the earliest work that begins to
address these questions is by \namecite{RubensteinG65}.
They conducted quantitative experiments with human subjects (51 in all)
who were asked to rate 65 English word pairs on a scale from 0.0 to 4.0 as per
their semantic distance.
The word pairs chosen ranged from almost synonymous to unrelated. However, they were all
noun pairs and those that were semantically close were also semantically similar; the dataset did not
contain word pairs that are semantically related but not semantically similar.
The subjects repeated the annotation after two weeks and the new distance values
had a Pearson's correlation $r$ of 0.85 with the old ones.
\namecite{MillerC91} also
conducted a similar study on 30 word pairs taken from the Rubenstein-Goodenough
pairs. These annotations had a high correlation ($r = 0.97$) with the mean annotations
of \namecite{RubensteinG65}.
\namecite{Resnik99} repeated these experiments and found the inter-annotator correlation ($r$)
to be 0.90.
\namecite{ResnikD00} conducted annotations of 48 verb pairs and found
inter-annotator correlation ($r$) to be 0.76 (when the verbs were
presented without context) and 0.79 (when presented in context).
\namecite{Gurevych2005} and \namecite{ZeschGM07}
asked native German speakers to mark two different
sets of German word pairs with distance values. Set 1
was a German translation of the
\namecite{RubensteinG65} dataset. It had 65 noun--noun word
pairs. Set 2 was a larger dataset containing 350 word
pairs made up of nouns, verbs, and adjectives. The semantically close
word pairs in the 65-word set were mostly synonyms or hypernyms (hyponyms) of each
other, whereas those in the 350-word set had both classical and non-classical
relations with each other. Details of these {\bf
semantic distance benchmarks}
are summarized in
Table~\ref{tab:datasets}. Inter-subject correlations (last column in Table \ref{tab:datasets})
are indicative of the degree of ease in annotating the datasets.
\begin{table*}
\begin{center}
\caption{Different datasets that are manually annotated with distance values. Pearson's correlation coefficient ($r$) was used to determine inter-annotator correlation (last column).}
\label{tab:datasets}
\resizebox{\textwidth}{!}{
\begin{tabular}{lcccccc}
\hline
{\bf Dataset} & {\bf Year} & {\bf Language} & {\bf \# pairs} & {\bf PoS} & {\bf \# subjects} & {\bf $r$} \\
\hline
\hline
Rubenstein and Goodenough & 1965 & English & 65 & N & 51 & - \\
Miller and Charles & 1991 & English & 30 & N & 38 & .90 \\
Resnik and Diab & 2000 & English & 27 & V & 10 & .76 and .79 \\
Gurevych & 2005 & German & 65 & N & 24 & .81 \\
Zesch and Gurevych & 2006 & German & 350 & N, V, A & 8 & .69 \\
\hline
\end{tabular}
}
\end{center}
Note: Rubenstein and Goodenough (1965) do not report inter-subject correlation, but determine intra-subject correlation to be 0.85 for 36 (out of the 65) word pairs for which similarity judgments were repeated by 15 (of the 51) subjects.
\normalsize
\end{table*}
It should be noted here that even though the annotators were presented with word-pairs
and not concept-pairs, it is reasonable to assume that they were annotated as per their
closest senses. For example, given the noun pair {\em bank} and {\em interest}, most if not all
will identify it as semantically related even though both words have more than one sense
and many of the sense--sense combinations are unrelated (for example, the {\sc river bank}
sense of {\it bank} and the {\sc special attention} sense of {\it interest}).
The high agreement and correlation values suggest that humans are quite good and consistent at estimating
semantic distance of noun-pairs; however, annotating verbs and adjectives and a
combination of parts of speech is harder. This also means that estimating
semantic relatedness is harder than estimating semantic similarity.
Apart from showing that humans can indeed estimate semantic distance, these
datasets act as ``gold standards" to evaluate automatic
distance measures.
However, lack of large amounts of data from human subject experimentation
limits the reliability of this mode of evaluation. Therefore automatic distance
measures are also evaluated by their usefulness in natural language tasks
such as those mentioned earlier.
\subsection{Automatic measures of semantic distance}
Automatic measures of semantic distance quantify the semantic distance between word pairs.
They give values within a certain range (for example, 0 to 1), such that one end of this range
represent maximal closeness or synonymy, while the other end represents maximal distance.
Depending on which end is which, measures of semantic distance can be classified as
{\bf measures of distance} (larger values indicate greater distance and less closeness) and
{\bf measures of closeness} (larger values indicate shorter distance and more closeness).\footnote{A note
about terminology: In many contexts, the term {\it distance measures} refers to the complete set of measures
(irrespective of what the different ends of the range signify). In certain other contexts (as in this paragraph),
{\it distance measures} refers only to those measures that give larger values to signify greater distance.
The context, usually by its reference to this numeric property or lack thereof will make clear the intended meaning of the term.}
A measure of closeness can be easily converted to a measure of distance by applying a suitable
inverse function,
or vice versa.
Two classes of automatic methods have been traditionally used to determine semantic
distance. {\bf Knowledge-rich measures of concept-distance}, such as those
of \namecite{JiangC97}, \namecite{LeacockC98}, and \namecite{Resnik95},
rely on the structure of a knowledge
source, such as WordNet, to determine the distance between two
concepts defined in it.\footnote{The nodes in WordNet (synsets) represent word senses
and edges between nodes represent semantic relations such as hyponymy and meronymy.}
{\bf Distributional measures of word-distance (knowledge-lean measures)}, such as cosine and
$\alpha$-skew divergence~\cite{Lee01}, rely on the {\bf distributional hypothesis},
which states that two words tend to be semantically close
if they occur in similar contexts \cite{Firth57}.
Distributional measures rely simply on text (and possibly some shallow syntactic processing)
and can give the distance between any two
words that occur at least a few times.
The various WordNet-based measures have been widely studied \cite{BudanitskyH06,PatwardhanBT03}.
The study of distributional measures on the whole has
received much less attention.\footnote{See \namecite{Curran04} and \namecite{WeedsWM04} for other work that compares various distributional
measures.}
Even though, as \namecite{Weeds03} and \namecite{MohammadH06b} show, they perform poorly when compared to WordNet-based
measures, the distributional measures of word-distance have many attractive features, including their
ability to measure both semantic similarity and semantic relatedness. Further,
they are not dependent on costly knowledge sources that do not exist for most languages.
This paper therefore focuses on distributional measures and analyzes
their strengths and limitations. Particular attention is paid to the different
kinds of distributional measures and their components.
The motivation is that a better understanding of distributional measures
will lead to bringing them more in line with human notions of semantic distance,
while still maintaining their applicability to resource-poor languages and their
ability to mimic both semantic similarity and semantic distance.
\section{Knowledge-rich approaches to semantic distance}
Before we begin our examination of distributional measures, we look
briefly at the resource-based measures. In some ways they are
complementary to distributional measures and so the discussion will
set the context for the analysis of distributional measures.
Creation of electronically available ontologies and semantic
networks such as WordNet has allowed their use to help solve numerous
natural language problems including the measurement of semantic
distance.
\namecite{BudanitskyH06}, \namecite{HirstB05}, and \namecite{PatwardhanBT03}
have done an extensive survey of the various WordNet-based measures,
their
comparisons with human judgment on selected word pairs, and
their usefulness in applications such as real-word spelling correction
and word sense disambiguation. Hence, this section provides
only a brief summary of the major knowledge-rich measures
of semantic distance.
\subsection{Measures that exploit WordNet's semantic network}
\label{s:Wnet}
A number of WordNet-based measures consider two concepts to be close if
they are close to each other in WordNet.
One of the earliest and simplest measures is Rada et al.'s \shortcite{RadaMBB89}
{\bf edge-counting} method. The shortest path in the network
between the two target concepts ({\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
hinges on the largely incorrect assumption that all the network edges
correspond to identical semantic distance.
Nodes in a network may be connected by different kinds of
lexical relations such as hyponymy, meronymy, and so on.
Edge counts apart, Hirst and St-Onge's \shortcite{HirstS98} measure takes into account
the fact that if the target path consists of edges that belong
to many different relations, then the target concepts are likely
more distant. The idea is that if we start from a particular node
$c_1$
and take a path via a particular relation (say, hyponymy),
to a certain extent the concepts reached will be semantically related
to $c_1$. 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:
\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,
$d$ is the number of times an edge pertaining to a relation different
from that of the preceding edge is taken, and
$C$ and $k$ are empirically determined constants.
More recently, \namecite{YangP05} proposed a weighted edge-counting method to determine
semantic relatedness using the hypernymy/hyponymy, holonymy/meronymy,
and antonymy links in WordNet.
\namecite{LeacockC98}
used just one relation (hyponymy) and modified the path length
formula to reflect the fact that edges lower down in the hyponymy
hierarchy correspond to smaller semantic distance than the ones
higher up. For example, synsets pertaining to {\it sports car} and {\it car} (low in the hierarchy)
are much
more similar than those pertaining to {\it transport} and {\it instrumentation} (higher
up in the hierarchy) even
though both pairs of nodes are separated by exactly one edge in the
hierarchy. Their formula is:
\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 maximum depth of the taxonomy.
\namecite{Resnik95} suggested a measure that combines corpus statistics
with WordNet. He proposed that since the
{\bf lowest common subsumer} or {\bf lowest superordinate (lso)}
of the target nodes represents what is similar between them,
the semantic similarity between the two concepts
is directly proportional to how specific the lso is.
The more general the lso is, the larger the semantic distance between the target nodes.
This specificity is measured by the formula for information
content (IC)---the negative logarithm of the probability of the lso:
\begin{equation}
{\text{\itshape Res}}(c_1,c_2) = {\text{\itshape IC}}(lso(c_1,c_2)) = -\log {\text{\itshape P}}(lso(c_1,c_2))
\end{equation}
\noindent Observe that using information content has the effect of inherently
scaling the semantic similarity measure by the depth of the taxonomy. Usually, the lower
the lowest superordinate, the lower the probability of occurrence of
the lso and the concepts subsumed by it, and hence, the higher its
information content.
As per Resnik's formula, given a particular lowest superordinate, the exact positions
of the target nodes 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 semantically similar than those that are distant. \namecite{JiangC97} and \namecite{Lin97}
incorporate this notion into their measures which
are arithmetic variations of the same terms.
The \namecite{JiangC97} measure ({\itshape JC}\/) determines how dissimilar each target concept
is from the lso ($IC(c_1) - IC(lso(c_1,c_2))$ and $IC(c_2) - IC(lso(c_1,c_2))$). The final semantic
distance between the two concepts is
then taken to be the sum of these differences.
\namecite{Lin97} (like Resnik) 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. His formula ({\itshape Lin}\/) can be thought of
as taking the Dice coefficient of the information in the two target concepts.
\vspace*{-4mm}
\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}
\vspace*{-4mm}
\namecite{BudanitskyH06} showed that the Jiang-Conrath measure
has the highest correlation (0.850) with the Miller and Charles noun pairs
and performs better than all these measures in a spelling correction task. \namecite{PatwardhanBT03}
achieved similar results using the measure for word sense disambiguation.
All of the approaches described above rely heavily (if not solely) on the
hypernymy/hyponymy network in WordNet; they are designed for, and evaluated on, noun--noun pairs.
However, more recently, \namecite{ResnikD00} and \namecite{YangP06a} developed
measures aimed at verb--verb pairs. \namecite{ResnikD00} ported several measures
which are traditionally applied on the noun hypernymy/hyponymy network
(edge counting, and the measures of \namecite{Resnik95}, and \namecite{Lin97})
to the relatively shallow verb troponymy network. The two information content--based measures
ranked a carefully chosen set of 48 verbs best in order of their semantic distance.\footnote{Only those
verbs were selected which require a theme, and the sub-categorization frames had to match.}
\namecite{YangP06a} ported their earlier work on nouns \cite{YangP05} to verbs.
In order to compensate for the relatively shallow verb troponymy hierarchy
and the lack of a corresponding holonymy/meronymy hierarchy, they proposed
several back-off models---the most useful one being the distance between
a noun pair that has the same lexical form as the verb pair. However, the approach
has too many tuned parameters (9 in all) and performed poorly on a set of 36
TOEFL word-choice questions involving verb targets and alternatives.
\subsection{Measures that rely on dictionaries and thesauri}
\hspace{0.0pt} \namecite{Lesk86} introduced a method to perform word sense disambiguation
using word glosses (definitions). The glosses of the senses of a target word are
compared with those of its context and the number of word overlaps is determined.
The sense with the greatest number of overlaps is chosen as the intended sense of the target.
Inspired by this approach, \namecite{BanerjeeP03} proposed a semantic relatedness measure
that deems two concepts to be more semantically related if there is more overlap in their
glosses. Notably, they overcome the problem of short glosses by considering the glosses
of concepts related to the target concepts through the WordNet lexical semantic relations
such as hyponymy/hypernymy. They also give more weight to larger overlap sequences.
\namecite{PatwardhanP06} proposed another gloss-based semantic relatedness measure
which performed slightly worse than the extended gloss overlap measure in a word
sense disambiguation task, but markedly better at ranking the \namecite{MillerC91} word pairs.
They create
{\bf aggregate co-occurrence vectors} for a WordNet sense by adding
the co-occurrence vectors of the words in its WordNet gloss.
The distance between two senses is then determined by the
cosine of the angle between their aggregate vectors.
Such aggregate co-occurrence
vectors are expected to be noisy because they are created
from data that is not sense-annotated.
\namecite{JarmaszSzpakowicz2003} use the taxonomic structure of {\it Roget's Thesaurus}
to determine semantic similarity. Two words are considered maximally similar if
they occur in the same semicolon group in the thesaurus. Then on, decreasing in similarity
are word pairs in the same paragraph, words pairs in different paragraphs belonging to the same part of speech and within the same
category, word pairs in the category, and so on until word pairs which have nothing in common except
that they are in the thesaurus (maximally distant). They show that this simple
approach performs remarkably well at ranking word pairs and determining the
correct answer in sets of TOEFL, ESL, and {\it Reader's Digest} word choice problems.
\subsection{Challenges}
In this section, we review some of the shortcomings of resource-based
measures in order to motivate and to compare them with distributional measures
that we will introduce in Section \ref{s:dist}.
\subsubsection{Lack of high-quality WordNet-like knowledge sources}
\label{s:Lbottleneck}
Ontologies, WordNets, and semantic networks are available for a few
languages such as English, German, and Hindi.
Creating them requires human experts and it is time intensive.
Thus, for most languages, we cannot use WordNet-based measures simply due to the
lack of a WordNet in that language. Further,
even if 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 only as good as the networks on which
they are based.
On the other hand, distributional measures require only text.
Large corpora, billions 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). This makes distributional
measures very attractive.
\subsubsection{Poor estimation of semantic relatedness}
\label{s:}
As \namecite{MorrisH04} pointed out, a large number
of concept pairs, such as {\sc strawberry--cream} and {\sc doctor--scalpel},
have a non-classical relation between them ({\sc strawberries} are usually eaten with {\sc cream} and
a {\sc doctor} uses a {\sc scalpel} to make an incision).
These words are not semantically similar, but rather semantically related.
An ontology- or WordNet-based measure will correctly identify the amount of
semantic relatedness only if such relations are explicitly coded into the knowledge source.
Further, the most accurate WordNet-based measures rely only on its extensive is-a hierarchy.
This is because networks of other lexical-relations such as meronymy are
much less developed. Further, the networks for different parts of speech
are not well connected.
All this means that, while WordNet-based measures accurately estimate semantic
similarity between nouns, their estimation of semantic relatedness, especially in
pairs other than noun--noun, is at best poor and at worse non-existent.
On the other hand, distributional measures can be used to determine both
semantic relatedness and semantic similarity.
\subsubsection{Inability to cater to specific domains}
\label{s:}
Given a concept pair, measures that rely only on WordNet and no text, such as that of \namecite{RadaMBB89},
give just one distance value.
However, two concepts may be very close in a certain domain but
not so much in another. For example, {\sc space} and {\sc time}
are close 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. Some of the more successful
WordNet-based measures, such as that of \namecite{JiangC97}, that rely also on text,
do indeed capture domain-specificity to some extent, but the distance
values are still largely shaped by the underlying network, which is not domain-specific.
On the other hand, distributional measures rely primarily (if not completely)
on text, and large amounts of
corpora specific to particular domains can easily be collected.
\subsubsection{Computational complexity and storage requirements}
\label{s:Lcomp}
As applications for linguistic distance become more
sophisticated and demanding, it becomes attractive to
pre-compute and store the distance values between all
possible pairs of words or senses.
However both WordNet-based and distributional
measures have large space requirements to do this, requiring
matrices of size $N \times N$, where $N$ is very large.
In case of WordNet-based measures, $N$ is the number of senses (81,000 just for nouns).
In case of distributional measures, $N$ is the size of the
vocabulary (at least 100,000 for most languages).
Given that the above matrices tend to be sparse\footnote{Even though WordNet-based
and distributional measures give non-zero closeness values to a large number of
term pairs, values below a suitable threshold
can be reset to 0.} and that computational
capabilities are continuing to improve, the above limitation may not seem hugely problematic,
but as we see more and more natural language applications in embedded systems and hand-held devices, such as
cell phones, iPods, and medical equipment, memory and computational power become serious constraints.
\subsubsection{Reluctance to cross the language barrier}
\label{s:Lcross}
Both WordNet-based and distributional measures have largely been used in
a monolingual framework. Even though semantic distance
seems to hold promise in tasks such as machine translation and multi-lingual text
summarization that inherently involve two or more languages, automatic measures
of semantic distance have rarely been applied.
With the development of the EuroWordNet, involving interconnected networks
of seven different languages, it is possible that we shall see more cross-lingual
work using WordNet-based measures in the future. However, such an interconnected
network will be very hard to create for more-different
language pairs such as English and Chinese or English and Arabic.
\section{Knowledge-lean, distributional approaches to semantic distance}
\label{s:dist}
\subsection{The distributional hypotheses: the original and the revised}
\label{s:disthypo}
{\bf Distributional measures} are inspired by the maxim ``You
shall know a word by the company it keeps'' \cite{Firth57}.
These measures rely simply on raw text and possibly some shallow
syntactic processing. They are
much less resource-hungry than the semantic measures, but
they measure the distance between words rather than
word-senses or concepts. Two words are
considered close if they occur in similar contexts.
The context of a target word is usually taken to be the set of words within a certain
window around it, for example, $\pm 5$ words or the complete sentence.
The set of contexts of a target word is usually represented by the set of words
in these contexts, their strength of association (SoA) with the target word, and
possibly their syntactic relation with the target, for example verb--object, subject--verb, and so on.
The strength of co-occurrence association between the target and another word quantifies how much more (or less)
than chance the two words occur together in text. Commonly used measures of association
are conditional probability (CP) and pointwise mutual information (PMI).
The distance between the sets of contexts of two target words can be used as a proxy for their semantic distance,
as words found in similar contexts tend to be semantically similar---the
{\bf distributional hypothesis} \cite{Firth57,Harris68}.
The hypothesis makes intuitive sense, as \namecite{BudanitskyH06} point out:
If two words have many co-occurring
words in common, 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.
The distributional hypothesis only mentions semantic similarity and not
semantic relatedness.
This, coupled with the fact that the difference between
semantic relatedness and semantic similarity is somewhat nuanced and can
be missed, meant that almost all work employing the distributional hypothesis
was labeled as estimating semantic similarity. However, it should be noted
that distributional measures can be used to estimate both semantic similarity and
semantic relatedness. Even though \namecite{SchutzeP97} and \namecite{LandauerFL98}, for example,
use the term {\it similarity} and not {\it relatedness}, their
LSA-based distance measures in fact estimate semantic relatedness and not semantic
similarity.
We propose more-specific distributional hypotheses that make clear how
distributional measures can be used to estimate semantic similarity
and how they can be used to measure semantic relatedness:
\begin{quote}
{\bf Hypothesis of the distributionally close and semantically related:} \\
Two target words are distributionally close and semantically related if they have
many common strongly co-occurring words. \\
(For example, {\it doctor}--{\it surgeon} and
{\it doctor}--{\it scalpel}. See example co-occurring words in Table \ref{tab:simrel}.)
\end{quote}
\begin{quote}
{\bf Hypothesis of the distributionally close and semantically similar:} \\
Two target words are distributionally close and semantically similar if they have
many common strongly co-occurring words that each have the same
syntactic relation with the two targets. \\
(For example, {\it doctor}--{\it surgeon},
but not {\it doctor}--{\it scalpel}. See syntactic relations with example co-occurring
words in Table \ref{tab:simrel}.)
\end{quote}
\begin{table}
\caption[Example: Common syntactic relations of target words with co-occurring words.]
{Example: Common syntactic relations of target words with co-occurring words.}
\label{tab:simrel}
\begin{center}
\hspace{-0.05in}
\resizebox{\textwidth}{!}{
\begin{tabular}{lcccccc} \hline
& &\multicolumn{5}{c}{\bf Co-occurring words} \\
& &{\it cut} (v) & &{\it hardworking} (adj) & &{\it patient} (n) \\ \hline
{\bf Semantically similar} & & & & & & \\
{\bf target pair} & & & & & & \\
{\it doctor} (n) & &subject--verb & &noun--qualifier & &subject--object \\
{\it surgeon} (n) & &subject--verb & &noun--qualifier & &subject--object \\
& & & & & & \\
{\bf Semantically related} & & & & & & \\
{\bf target pair} & & & & & & \\
{\it doctor} (n) & &subject--verb & &noun--qualifier & &subject--object \\
{\it scalpel} (n) & &prepositional object--verb & &-- & &prepositional object--object \\
\hline
\end{tabular}
}
\end{center}
\hspace{-0.20in}
\end{table}
\noindent The idea is that both semantically similar and semantically related word pairs
will have many common co-occurring words. However, words that are semantically similar
belong to the same broad part of speech (noun, verb, etc.), but the same need not
be true for words that are semantically related.
Therefore, words that are semantically similar will tend to have
the same syntactic relation, such as verb--object or subject--verb,
with most common co-occurring words.
Thus, the two words are considered semantically related simply if they
have many common co-occurring words. But to be semantically similar as well,
the words must have the same syntactic relation with co-occurring words.
Consider the 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 contribute to a high score of relatedness.
However, they do not have the same syntactic relation with the two targets. (The word
{\em doctor} is almost always used as a noun while {\em operate}
is a verb.) Thus, as per the two revised distributional hypotheses,
{\em doctor} and {\em operate} will correctly be identified as semantically related
but not semantically similar.
The word pair {\em doctor--nurse},
on the other hand, will be identified as both semantically related and semantically similar.
In order to clearly differentiate from the distance as calculated
by a WordNet-based semantic measure (described earlier in Section \ref{s:Wnet}), the distance calculated by a
corpus-based distributional measure will be referred to as
{\bf distributional distance}.
\subsection{Corpus-based measures of distributional distance}
We now describe specific distributional measures that rely on the distributional hypotheses;
depending on which specific hypothesis
they use, they mimic either semantic similarity or semantic relatedness.
\subsubsection{Spatial Metrics: Cos, ${\text L}_1$, ${\text L}_2$}
Consider a multidimensional space in which
the number of dimensions is equal to the size of the vocabulary. A word
$w$ can be represented by a point in this space such that
the component of $\vec w$ in a
dimension (corresponding to word $x$, say) is equal to the strength of association (SoA)
of $w$ with $x$ (${\text \it{SoA}}(w,x)$).
Thus, the vectors corresponding to two words are {\it close} together,
and thereby get a low distributional distance 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.
The distance between two vectors can be calculated in
different ways as described below.
\paragraph{Cosine}
\label{s:cosine}
The {\bf cosine} method (denoted by $\text{\bf Cos}$) is one of the earliest and most widely used 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 $w_2$,
then $\vec w_1 $ and $\vec w_2$ will have a small angle between
them and the cosine will be large; signifying a large
relatedness/similarity between them. The cosine measure gives
scores in the range from $0$ (unrelated) to $1$ (synonymous).
So the higher the value, the less distant the target word-pair is.
\begin{equation}
\label{eq:relCos}
\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 where $C(w)$ is the set of words that co-occur (within a certain window)
with the word $w$ in a corpus.
In this instantiation of the cosine measure, conditional probability
of the co-occurring words given the target words is used as the strength
of association.
The cosine was used, among others, by \namecite{SchutzeP97} and
\namecite{YoshidaYK03}, who suggest methods of automatically generating distributional thesauri from
text corpora. \namecite{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
(singular value decomposition) \cite{SchutzeP97} is 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 target word's vector with each of the other
word vectors is calculated and the
words that give the highest scores comprise the thesaurus entry for the target word.
\namecite{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.
\paragraph{Manhattan and Euclidean Distances}
Distance between two points (words) in vector space can also
be calculated using the formulae for {\bf Manhattan distance} a.k.a.\@ the ${\bf L_1}$ {\bf norm}
(denoted by ${\bf L_1}$)
or {\bf Euclidean distance} a.k.a.\@ the {\bf ${\text L}_2$ norm} (denoted by ${\text L}_2$).
In the Manhattan distance~(\ref{eq:L1}) (\namecite{DaganLP97}, \namecite{DaganLP99},
and \namecite{Lee99}), the difference in strength of association of
$w_1$ and $w_2$ with each word that they co-occur with is summed.
The greater the difference, the greater is the distributional distance between the two words.
Euclidean distance~(\ref{eq:L2}) \cite{Lee99} employs the root mean square of the
difference in association to get the final distributional distance.
Both the ${\text L}_1$ and ${\text L}_2$ norms give scores in the range between 0 (zero
distance or synonymous) 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 a target word as the strength
of association.
\namecite{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: ${\text L}_1$ norm,
cosine, and ${\text L}_2$ norm.
\namecite{Weeds03} determined the correlation of word pair ranking as per
a handful of distributional measures with human rankings (\namecite{MillerC91} word pairs).
She used verb-object pairs from the {\em British
National Corpus (BNC)} and
found the correlation of ${\text L}_1$ norm with human rankings to be 0.39.
\subsubsection{Mutual information--based measures: Hindle, Lin}
\hspace{0.0pt} \namecite{Hindle90} was one of the first to factor the strength
of association of co-occurring words into a distributional similarity measure.\footnote{See \namecite{Grefenstette92}
for an approach that does not incorporate strength of association of co-occurring words. He, like \namecite{Hindle90}, uses syntactic dependencies to
to characterize the set of contexts of a target word. The Jaccard coefficient is used to determine how similar the two sets of contexts are.}
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 the following formula
to determine the distributional similarity of $n_j$ and $n_k$
solely from their occurrences as object of $v_i$:
\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 pointwise mutual information (PMI)
between the noun $n$ and verb $v$
(note that in case of negative PMI values,
the maximum function captures the PMI which is lower in absolute value).
The measure follows from the distributional hypothesis---the more similar the associations of co-occurring words
with the two target words, the more semantically similar they are.
Hindle used PMI\footnote{\namecite{Hindle90} and
\namecite{Lin98C} both refer to pointwise mutual information as mutual
information.} as the strength of association.
Using the minimum of
the two PMIs captures the similarity in the strength of association of $v_i$
with each of the two nouns.
Hindle used an analogous formula to calculate
distributional similarity ($Hin_{subj}$) using the subject--verb relation. The overall
distributional similarity between any two nouns is calculated by the formula:
\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 or synonymous).
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
that mimics semantic similarity and not semantic relatedness. A form of
Hindle's measure where all co-occurring words are used, making it a measure
that mimics semantic 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 where $C(w)$ is the set of words that co-occur with word $w$.
\namecite{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 just subject--verb and
verb--object relations and shows that using multiple relations is beneficial even for
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 $r$.
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 where $T(x)$ 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 \namecite{Hindle90}
where even the cases of negative PMI were considered.
\namecite{ChurchH89} showed that it is hard to accurately predict
negative word association ratios with confidence, and so, co-occurrence pairs with negative
PMI are ignored. The measure gives similarity scores from 0 (maximally dissimilar) to 1 (maximally similar).
Like Hindle's measure, Lin's
is a measure of distributional {\it similarity}. However,
it distinguishes itself from that of Hindle in two respects.
First, Lin normalizes the similarity score between two words (numerator of (\ref{eq:LinCorpus}))
by their cumulative strengths of association with the rest of the co-occurring words (denominator of (\ref{eq:LinCorpus})).
This is a
significant improvement as now high PMI of the target words
with shared co-occurring words alone 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.
Second, Hindle uses the 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.
\namecite{Hindle90} used a portion of the {\em Associated Press} news stories
(6 million words) to classify the nouns into semantically related
classes. \namecite{Lin98C} used his measure to generate a distributional 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 such automatically generated thesauri by comparing
them with WordNet-based and Roget-based thesauri. He shows that the distributional thesaurus
created with his measure is closer to the WordNet and Roget-based thesauri than
that created using Hindle's measure.
\subsubsection{Relative Entropy--Based Measures: KLD, ASD, JSD}
\paragraph{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 $X$, then $D(p\Vert q)$ is the information lost when
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 {\bf KLD}).
\namecite{PereiraTL93} and \namecite{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,
the more these contexts are similar, the more $n_1$ and $n_2$ are 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 {\it 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 {\it Vb}^{\prime} (n_1) \cap \text {\it Vb}^{\prime} (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 $\text{\itshape Vb}^{\prime} (x)$
is the set of verbs that have $x$ as the object.
Note again 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 too is therefore a
measure that mimics semantic similarity and not semantic 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.
One may also estimate semantic relatedness by 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 where $V$ is the vocabulary (all the words found in a corpus).
$C(w)$, as mentioned earlier, is the set of words occurring (within
a certain window) with word $w$.
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
counterintuitive to the general notion of semantic similarity of words, although
\namecite{Weeds03} has argued in favor of asymmetric measures.
Further, it is very likely that there are 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.
\namecite{PereiraTL93} used KLD to create
clusters of nouns from verb-object pairs corresponding
to the thousand most frequent nouns in the {\itshape Grolier's Encyclopedia}, June 1991
version (10 million words).
\namecite{DaganPL94} used KLD 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}.
It should be noted here that 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).
\namecite{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.\\
\paragraph{$\alpha$-skew divergence}
The {\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 where $\alpha$ 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.
\namecite{Lee01} shows that $\alpha$-skew divergence performs better than
Kullback-Leibler divergence in estimating word co-occurrence probabilities. \namecite{Weeds03}
achieves a correlation of $0.48$ 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.\\
\paragraph{Jensen-Shannon divergence}
A relative entropy--based measure that overcomes the problem 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 {\bf 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 the Kullback-Leibler divergence between each of the individual
co-occurrence distributions $d_1$ and $d_2$ of the target words 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. The Jensen-Shannon divergence
is therefore always well defined and, like $\alpha$-skew divergence, obviates the need for smoothed estimates.
The Kullback-Leibler divergence, $\alpha$-skew divergence, and Jensen-Shannon divergence
all give distributional distance scores from 0 (synonymous) to infinity (unrelated).
\subsubsection{Latent Semantic Analysis}
\hspace{0.0pt} \namecite{LandauerFL98} proposed {\bf Latent semantic analysis (LSA)}, which can be used to determine distributional
distance between words or between sets of words.\footnote{\namecite{LandauerFL98}
describe it as a measure of {\em similarity}, but in fact it is a distributional
measure that mimics semantic relatedness.}
Unlike the various approaches described earlier where a word--word co-occurrence
matrix is created, the first step of LSA involves the creation of a word--paragraph,
word--document, or similar such word-passage matrix, where a {\it passage} is some grouping of words.
A cell for word $w$ and passage $p$ is populated with the number of times $w$
occurs in $p$ or, for even better results, a function of this frequency that captures how much information
the occurrence of the word in a text passage carries.
Next, the dimensionality
of this matrix is reduced by applying {\bf singular value decomposition (SVD)}, a
standard matrix decomposition technique. This smaller set of dimensions represents
abstract (unknown) concepts. Then the original word--passage matrix is recreated,
but this time from the reduced dimensions.
\namecite{LandauerFL98} point out that this results in new matrix cell values that
are different from what they were before.
More specifically, words that are expected to occur more often in a passage than
what the original cell values reflect are incremented.
Then a standard vector distance measure, such as cosine, that captures the
distance between distributions of the two target words is applied.
LSA was used by \namecite{SchutzeP97,Turney2001} and \namecite{Rapp03}
to measure distributional distance, with encouraging results.
However, there is no non-heuristic way to determine when the
dimension reduction should stop. Further, the generic concepts represented by the
reduced dimensions are not interpretable; that is, one cannot determine
which concepts they represent in a given sense inventory. This means
that LSA cannot directly be used for tasks such
as unsupervised sense disambiguation or estimating semantic similarity of known concepts.
LSA is computationally expensive as singular value decomposition,
a key component for dimensionality reduction, requires computationally intensive matrix operations.
This makes LSA less scalable to large amounts of text \cite{GormanC06}.
Finally, it too, like other distributional word-distance
measures conflates the many senses of a word
(see Section \ref{s:Lconflation} ahead for more discussion on sense conflation).
\subsubsection{Co-occurrence Retrieval Models}
The distributional measures suggested by \namecite{Weeds03} are based on a 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.
She uses {\bf co-occurrence retrieval} (the retrieval of words that co-occur with a target word
from text) to determine the degree to which one word is substitutable by another.
The degree of substitutability
of $w_2$ with $w_1$ is dependent on how the proportion of co-occurrences of $w_1$
that are also co-occurrences of $w_2$ and the proportion of co-occurrences of $w_2$
that are also co-occurrences of $w_1$.
Thus Weeds's distributional
measures have a precision component and a recall component (which may or may not incorporate
the strength of co-occurrence association). The final
score is a weighted sum of the precision, recall, and standard $F$ measure
(see equation~(\ref{eq:CRMfinal})\footnote{$P$ 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 it results in a symmetric measure which gives
identical scores for the distributional similarity of $w_1$ with $w_2$ and
$w_2$ with $w_1$. Otherwise, we get an asymmetric measure which
assigns different scores to the two cases.
\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.
Both precision and recall can be considered as the product of a
core formula (denoted by $core$) and a penalty function (denoted by $penalty$).
{Weeds03} provides six (three times two) distinct formulae for precision
and recall, depending on the strength of co-occurrence (three alternatives)
and whether or not a penalty is applied for differences in strength of association
of common co-occurring words (two alternatives).
Depending on the strength of association, the CRMs are classified as
{\bf type-based, token-based,} and {\bf mutual information--based}.
The CRMs that use simple counts of the common co-occurrences
and not the strength of associations
as core precision and recall values are called type-based
CRMs (denoted by the superscript {\em type}).
The CRMs that use conditional probabilities of the shared co-occurring words
with the target words are called token-based
CRMs (denoted by the superscript {\em token}).
The CRMs that use pointwise mutual information of the shared co-occurring words with target words
are called 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 where $C(x)$ is the set of words that co-occur with $x$.
Depending on the penalty function, the CRMs are classified as
{\bf additive} and {\bf difference-weighted}.
The CRMs that do not
penalize difference in strength of co-occurrence are called
additive CRMs (denoted by the subscript {\em add});
those that do penalize are called 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 six pairs of precision and recall difference-weighted CRMs are thus as follows:
\begin{eqnarray}
& P_{add}^{type}(w_1,w_2) =& \frac{\mid C(w_1) \cap C(w_2) \mid}{\mid C(w_1) \mid} \\
& R_{add}^{type}(w_1,w_2) =& \frac{\mid C(w_1) \cap C(w_2) \mid}{\mid C(w_2) \mid} \\
& P_{dw}^{type}(w_1,w_2) =& \frac{\sum_{\mid C(w_1) \cap C(w_2) \mid} \frac{\min(P(w|w_1),P(w|w_2))}{P(w|w_1)}}{\mid C(w_1) \mid} \\
& R_{dw}^{type}(w_1,w_2) =& \frac{\sum_{\mid C(w_1) \cap C(w_2) \mid} \frac{\min(P(w|w_1),P(w|w_2))}{P(w|w_2)}}{\mid C(w_2) \mid}
\end{eqnarray}
\begin{eqnarray}
P_{add}^{token}(w_1,w_2)& =& \sum_{w \in C(w_1) \cap C(w_2)} P(w|w_1) \\
R_{add}^{token}(w_1,w_2)& =& \sum_{w \in C(w_1) \cap C(w_2)} P(w|w_2) \\
P_{dw}^{token}(w_1,w_2)& =& \sum_{w \in C(w_1) \cap C(w_2)} \min(P(w|w_1),P(w|w_2)) \\
R_{dw}^{token}(w_1,w_2)& =& \sum_{w \in C(w_1) \cap C(w_2)} \min(P(w|w_2),P(w|w_1))
\end{eqnarray}
\begin{eqnarray}
P_{add}^{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)} \\
R_{add}^{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)} \\
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)} \\
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}
\noindent Note that in case of the difference-weighted token and mutual information--based precision
and recall formulae, there is a cancellation of a pair of terms obtained
from the core formulae and the penalty.
Asymmetry in substitutability is intuitive,
as in many cases
it may be acceptable to substitute a word, say {\em dog}, with another, say {\em animal}, but the
reverse is not acceptable as often. Since Weeds uses
substitutability as 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 asymmetric.
\namecite{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 mimic semantic similarity,
not relatedness.
Correlation with human judgment (Miller and Charles word pairs) showed that
difference-weighted ($r=0.61$) and additive mutual information--based measures
($r=0.62$) performed far better than the other CRMs.
\section{The anatomy of a distributional measure}
\label{s:anatomy}
Even though there are numerous distributional measures, many of which may seem
dramatically different from each other,
all distributional measures
perform two functions: (1) create {\bf distributional profiles (DPs)}, and (2) calculate
the distance between two DPs.
The distributional profile of a word is the strength of association between it and
each of the lexical, syntactic, and/or semantic units that
co-occur with it. Commonly used {\bf measures of strength of
association} are conditional probability (0 to 1) and
pointwise mutual information ($-\infty$ to $\infty$).
Commonly used units of co-occurrence with the target
are other {\it words}, and so we speak of the {\bf lexical
distributional profile of a word (lexical DPW)}.
The
co-occurring words may be all those in a predetermined
window around the target, or may be restricted to those that
have a certain syntactic ({\it e.g.,} verb--object) or
semantic ({\it e.g.,} agent--theme) relation with the target
word. We will refer to the former kind of DPs as {\bf relation-free}.
Usually in the latter
case, separate association values are calculated for each of
the different relations between the target and the
co-occurring units. We will refer to such DPs as {\bf
relation-constrained}.
\begin{table}
\caption[Measures of DP distance, measures of strength of association, and standard combinations.]
{Measures of DP distance, measures of strength of association, and standard combinations.
Measures of strength of association that are traditionally used are marked in bold. The use of
other measures of association remains to be explored.
}
\label{tab:distrib}
\begin{center}
\hspace{-0.05in}
\begin{tabular}{lll} \cline{1-1} \cline{3-3}
{\bf Measures of DP distance} &$\;\;\;\;\;\;\;\;$ &{\bf Measures of strength of association}\\ \cline{1-1} \cline{3-3}
$\alpha$-skew divergence (ASD) & &$\phi$ coefficient (Phi) \\
cosine (Cos) & &{\bf conditional probability (CP)} \\
Dice coefficient (Dice) & &cosine (Cos) \\
Euclidean distance (${\text L}_2$ norm) & &Dice coefficient (Dice) \\
Hindle's measure (Hin) & &odds ratio (Odds) \\
Kullback-Leibler divergence (KLD) & &{\bf pointwise mutual information (PMI)} \\
Manhattan distance (${\text L}_1$ norm) & &Yule's coefficient (Yule) \\
Jensen--Shannon divergence (JSD) & & \\
Lin's measure (Lin) & & \\
\cline{1-1} \cline{3-3}
\end{tabular}
\end{center}
\hspace{-0.20in}
\begin{center}
\begin{tabular}{l} \cline{1-1}
{\bf Standard combinations}\\ \cline{1-1}
$\alpha$-skew divergence with $\phi$ coefficient (ASD--CP)\\
cosine with conditional probability (Cos--CP)\\
Dice coefficient with conditional probability (Dice--CP)\\
Euclidean distance with conditional probability (${\text L}_2$ norm--CP)\\
Hindle's measure with pointwise mutual information (Hin--PMI)\\
Kullback-Leibler divergence with conditional probability (KLD--CP)\\
Manhattan distance with conditional probability (${\text L}_1$ norm--CP)\\
Jensen--Shannon divergence with conditional probability (JSD--CP)\\
Lin's measure with pointwise mutual information (Lin--PMI)\\
\cline{1-1}
\end{tabular}
\end{center}
\hspace{-0.20in}
\end{table}
Typical relation-free DPs are those of \namecite{SchutzeP97} and \namecite{YoshidaYK03}.
Typical relation-constrained DPs are those of
\namecite{Lin98B} and \namecite{Lee01}.
Below are contrived, but plausible, examples of
each for the word {\em pulse};
the numbers are conditional probabilities:
\begin{quote}
{\bf relation-free DP}\\
{\bf \em pulse}: {\em beat} .28, {\em racing} .2, {\em grow} .13,
{\em beans} .09, {\em heart} .04, \ldots
\end{quote}
\begin{quote}
{\bf relation-constrained DP}\\
{\bf \em pulse}: $\langle${\em beat}, subject--verb$\rangle$ .34,
$\langle${\em racing}, noun--qualifying adjective$\rangle$ .22, $\langle${\em
grow}, subject--verb$\rangle$ .14, \ldots
\end{quote}
Since the DPs represent the contexts of the two target words, the
distance between the DPs is the distributional distance and, as per the
distributional hypothesis, a proxy for semantic distance.
A {\bf measure of DP distance}, such as cosine, calculates
the distance between two distributional profiles.
While any of the measures of DP distance may be used with
any of the measures of strength of association,
in practice only certain combinations are used (see Table~\ref{tab:distrib})
and certain other combinations might not be meaningful (for example,
Kullback-Leibler divergence with $\phi$ coefficient).
Observe from Table~\ref{tab:distrib} that all standard-combination distributional measures
(or at least those that are described in this paper) use either conditional
probability or PMI as the measure of association.
\subsection{Simple co-occurrences versus syntactically related words}
\hspace{-1mm}
\namecite{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.
\namecite{DaganLP97}, \namecite{Lee99}, and \namecite{Weeds03}
represent the context of a noun with verbs whose object it is (single
syntactic relation),
\namecite{Hindle90} represents the context of a noun with verbs with which
it shares the verb-object or subject-verb relation, while
\namecite{Lin98C} uses words related to a noun by any of the many pre-decided syntactic relations
to determine distributional similarity.
\namecite{SchutzeP97} and \namecite{YoshidaYK03} use all co-occurring words in a
pre-decided window size.
Although \namecite{Lin98C} shows that the use of multiple syntactic
relations is more beneficial as compared to just one,
\namecite{McCarthyKWC07} show that results obtained using just word co-occurrences
produced almost as good results as those obtained
using syntactically
related words.
Further, use of syntactically related words entails the
requirement of chunking or parsing the data.
\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 of the measures' formula.
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 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.
\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 rather than 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 this theorem:
The product of any two numbers will
always be less than or equal to the square of their average.
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 $1/2$ 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 (formula repeated below for convenience --- equation (\ref{eq:KLDII}))
and Jensen-Shannon divergence do not take absolute
values of the division of co-occurrence probabilities. 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 $w_2$.
Observe however that the weight $P(w|w_1)$ multiplying 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}):\footnote{It should be noted
that any changes to the formula for Kullback-Leibler divergence means that the resulting
measure is no longer Kullback-Leibler divergence; these measures are denoted by KLD (and
a suitable subscript and/or superscript simply to indicate that they are derived from the
Kullback-Leibler divergence.}
\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 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 will be interesting.
\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 a strong indicator of semantic closeness.
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 absolute value of the logarithm is not
taken and because the weight ($P(w|w_1)$) is
dependent on the first word and not the other, Kullback-Leibler divergence is asymmetric.
Below is a symmetric weight function:
\begin{eqnarray}
{\text{\itshape weight}}_{\text{\itshape AvgWt}}(w_1,w_2)& =& \frac{1}{2}\left(P(w|w_1) + P(w|w_2)\right)
\end{eqnarray}
\noindent $L_2$ 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}
A possibly better weight function (which is also symmetric) hinges on
the following hypothesis: The stronger the association of a co-occurring word with a target word,
the better the indicator of semantic properties of the target word it is.
Equation~(\ref{eq:SaifWtdDiv})
shows the corresponding weight function:
\begin{eqnarray}
\label{eq:SaifWtdDiv}
{\text{\itshape weight}}_{\text{\itshape MaxWt}}(w_1,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)}
\end{eqnarray}
\noindent 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 (\namecite{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 \namecite{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.
\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.
~\namecite{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 also use words that co-occur with just one target word.
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.
Consider the classic {\em strong coffee}
vs {\em powerful coffee} example (\namecite{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 distance of $w_1$ with $w_2$ as compared to that of $w_2$
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 (see \cite{MirkinDG07} for work on lexical entailment using
the Kullback-Leibler divergence). \namecite{Weeds03} argues that this behavior is intuitive as
it is more often acceptable 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 counterintuitive, and possibly detrimental. In many natural language tasks,
one needs the distance between two words and there is no order information.
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 (${\text{\em Sim}}_{\text{\em Max}}$),
while still capitalizing
on the asymmetry to generate more suitable distributional distance values for
hypernym-hyponym word pairs. Equation~(\ref{eq:SymMax}) states the formula for the
proposed conversion. A specific implementation of 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}
\noindent Another way to convert an asymmetric measure of distributional
distance 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 equations (\ref{eq:KLDAvg}) through (\ref{eq:KLDAvgEnd}):
\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)\\
\label{eq:KLDAvgEnd}
& & = \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}
\subsection{Summarizing the distributional measures}
Table~\ref{tab:distributional measures of distance}
summarizes the properties of various distributional measures discussed in this paper.
\begin{landscape}
\begin{longtable}{l c c c c c c} \hline \\
\caption[Measures of distributional distance]{Distributional measures and their properties.}
\label{tab:distributional measures of distance}\\
{\bf distributional} &{\bf measure} &{\bf compo-} & & &{\bf symm-} &{\bf strength of}\\
{\bf measure} &{\bf type} &{\bf sitional} &{\bf PCM} &{\bf formula} &{\bf etric} &{\bf association}\\ \hline \hline
\endfirsthead
\multicolumn{7}{l}{Note: For measures that are not compositional, the type of PCM is not applicable.}
\endfoot
\hline\\
\multicolumn{7}{c}{Distributional measures and their properties (continued).}\\
{\bf distributional} &{\bf measure} &{\bf compo-} & & &{\bf symm-} &{\bf strength of}\\
{\bf measure} &{\bf type} &{\bf sitional} &{\bf PCM} &{\bf formula} &{\bf etric} &{\bf association}\\ \hline \hline
\endhead
{\em ASD} &distance &$\checked$ &division &$\sum_{w \in C(w_1) \cup C(w_2)} P(w|w_1) \log \frac{P(w|w_1)}{\alpha P(w|w_2) + (1 - \alpha) P(w|w_1)}$ &X &CP\\
& & & & & & \\
$\cos$ &closeness &X &n.a.
&$\frac{\sum_{w \in C(w_1) \cup C(w_2)} \left( P(w|w_1) \times P(w|w_2) \right) }
{\sqrt{ \sum_{w \in C(w_1)} P(w|w_1)^2 } \times \sqrt{ \sum_{w \in C(w_2)} P(w|w_2)^2 } }$ &$\checked$ &CP\\
& & & & & & \\
{\em CRMs} &closeness &X &n.a. &$\gamma \biggl[ \frac{2 \times P \times R}{P + R} \biggr] + (1 - \gamma) \biggl[ \beta [ P ] + (1 - \beta) [R] \biggr]$ &X &both\\
& & & & & & \\
${\text{\itshape Dice}}^{\text{\itshape CP}}$ &closeness &X &n.a. &$\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)} $ &$\checked$ &CP\\
& & & & & & \\
${\text{\itshape Dif}}\;\; \text{or}\;\; L_1 $ &distance &$\checked$ &difference &$ \sum_{w \in C(w_1) \cup C(w_2)} \mid P(w|w_1) - P(w|w_2) \mid $ &$\checked$ &CP\\
& & & & & & \\
${\text{\itshape Div}}$ &distance &$\checked$ &division &$\sum_{w \in C(w_1) \cup C(w_2)} \left| \log \frac{P(w|w_1)}{P(w|w_2)}\right|$ &$\checked$ &CP\\
& & & & & & \\
${\text{\itshape Hindle}}$ &closeness &X &n.a. &$\sum_{w \in C(w)} \left\{ \begin{array}{l} \min(I(w,w_1), I(w,w_2)),\\ \qquad \quad \text{if}\; I(w,w_1) > 0\; \text{and}\; I(w,w_2) > 0 \\
\mid \max(I(w, w_1), I(w, w_2))\mid,\\ \qquad \quad \text{if}\; I(w, w_1) < 0\; \text{and}\; I(w, w_2) < 0 \\
0,\\ \qquad \quad \text{otherwise} \end{array} \right.$ &$\checked$ &PMI\\
& & & & & & \\
${\text{\itshape Jaccard}}^{\text{\itshape CP}}$ &closeness &X &n.a. &$\frac{\sum_{w \in C(w_1) \cup C(w_2)} \min (P(w|w_1),P(w|w_2))}{ \sum_{w \in C(w_1) \cap C(w_2)} \max (P(w|w_1),P(w|w_2))}$
&$\checked$ &CP\\
& & & & & & \\\hline
{\em JSD} &distance &$\checked$ &division &$\sum_{w \in C(w_1) \cup C(w_2)} \Bigl( P(w|w_1) \log \frac{P(w|w_1)}{\frac{1}{2}\left(P(w|w_1) + P(w|w_2)\right)} +$
&$\checked$ &CP\\
& & & &$\qquad \qquad P(w|w_2) \log \frac{P(w|w_2)}{\frac{1}{2}\left(P(w|w_1) + P(w|w_2)\right)} \Bigr)$ & & \\
& & & & & & \\
$\text{\em KLD}$ &distance &$\checked$ &div. &$\sum_{w \in C(w_1) \cup C(w_2)} P(w|w_1) \log \frac{P(w|w_1)}{P(w|w_2)}$ &X &CP\\
& & & & & & \\
$\text{\em KLD}_{\text{\em Com}}$ &distance &$\checked$ &div. &$\sum_{w \in C(w_1) \cap C(w_2)} P(w|w_1) \log \frac{P(w|w_1)}{P(w|w_2)}$ &X &CP\\
& & & & & & \\
$\text{\em KLD}^{\text{\em Abs}}$ &distance &$\checked$ &div. &$\sum_{w \in C(w_1) \cup C(w_2)} P(w|w_1) \left| \log \frac{P(w|w_1)}{P(w|w_2)} \right|$ &X &CP\\
& & & & & & \\
$\text{\em KLD}_{\text{\em Avg}}$ &distance &$\checked$ &div. &$\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)} $ &$\checked$ &CP\\
& & & & & & \\
$\text{\em KLD}_{\text{\em Max}}$ &distance &$\checked$ &div. &$\max (\text{\em KLD}(w_1,w_2), \text{\em KLD}(w_2,w_1))$ &$\checked$ &CP\\
& & & & & & \\
$L_2$ &distance &$\checked$ &difference &$\sqrt{\sum_{w \in C(w_1) \cup C(w_2)} \left(P\left(w|w_1\right) - P\left(w|w_2\right)\right)^2 }$ &$\checked$ &CP\\
& & & & & & \\
{\em Lin} &closeness &X &n.a. &$\frac{\sum_{(r,w)\, \in\, T(w_{1})\, \cap\, T(w_{2})} (I(w_{1},r,w) + I(w_{2},r,w))}
{{\sum_{(r,w')\, \in\, T(w_1)} I(w_1,r,w') + \sum_{(r,w'')\, \in\, T(w_2)} I(w_2,r,w'')}}$ &$\checked$ &PMI\\
& & & & & & \\
$\text{\itshape{Pdt}}^{\text{\itshape{Avg}}}$ &closeness &$\checked$ &pdt. &$\sum_{w \in C(w_1) \cup C(w_2)} \frac{P(w|w_1) \times P(w|w_2)}{(\frac{1}{2}(P(w|w_1) + P(w|w_2)))^2}$ &$\checked$ &CP\\
& & & & & & \\
${\text{\itshape Pdt}}_{\text{\itshape{AvgWt}}}^{\text{\itshape Avg}}$ &closeness &$\checked$ &pdt. &$\sum_{w \in C(w_1) \cup C(w_2)} \frac{P(w|w_1) \times P(w|w_2)}{\frac{1}{2}(P(w|w_1) + P(w|w_2))}$ &$\checked$ &CP\\
& & & & & & \\
\hline
\end{longtable}
\end{landscape}
\subsection{Challenges}
\subsubsection{Conflation of word senses}
\label{s:Lconflation}
The distributional hypothesis \cite{Firth57} states that
words that occur in similar contexts tend to be semantically
close.
But when words have more
than one sense, it is not at all clear what semantic
distance between them actually means. Further, a word in each of its
senses is likely to co-occur with different sets of words.
For example, {\em bank} in the {\sc financial institution} sense
is likely to co-occur with {\em interest, money, accounts,}
and so on, whereas the {\sc river bank} sense might have words
such as {\em river, erosion,} and {\em silt} around it.
Since words that occur together in text tend to refer
to senses that are closest in meaning to one another, in most natural language applications,
what is needed is the distance between the closest senses of the two target words.
However, because distributional measures calculate distance from
occurrences of the target word in all its occurrences and hence all its senses,
they fail to get the desired result.
Also note that the dimensionality reduction inherent to latent semantic analysis,
a special kind of distributional measure, has the effect of making the
predominant senses of the words more dominant while de-emphasizing the other senses.
Therefore, an LSA-based
approach will also conflate information from the different senses, and even more
emphasis will be placed on the predominant senses.
Given the semantically close target nouns {\em play} and {\em actor}, for example, a distributional
measure will give a score that is some sort of a dominance-based average of the distances
between their senses.
The noun {\em play} has the predominant sense of {\sc children's recreation} (and not {\sc drama}),
so a distributional measure will tend to give the target pair a large (and thus erroneous) distance score.
WordNet-based measures do not suffer from this problem as they give distance between concepts, not words.
\subsubsection{Lack of explicitly-encoded world knowledge and data sparseness}
\label{s:Lsparse}
It is becoming increasingly clear that more-accurate results can be achieved in
a large number of natural language tasks, including the estimation of semantic distance,
by combining corpus statistics with a knowledge source, such as a dictionary, published
thesaurus, or WordNet.
This is because such knowledge sources capture semantic
information about concepts and, to some extent, world knowledge.
For example, WordNet, as discussed earlier, has an extensive is-a hierarchy.
If it lists one concept, say {\sc German Shepherd} as a hyponym of another, say {\sc dog},
then we can be sure that the two are semantically close. On the other hand,
distributional measures do not have access to such explicitly encoded information.
Further, unless the corpus used by a distributional measure has sufficient instances of
{\sc German Shepherd} and {\sc dog}, it will be unable to deem them semantically close.
Since Zipf's law seems to hold even for the largest of corpora, there will always be
words that occur too few times to accurately determine their distributional distance from others.
\subsubsection{Limitations shared with WordNet-based measures}
In addition to the limitations described above, which are unique to the knowledge-lean
distributional measures, like the knowledge-rich measures they also suffer from
problems of requiring the calculation of large distance matrices (as described in Section
\ref{s:Lcomp} earlier) and the reluctance to cross the language barrier (Section
\ref{s:Lcross}).
\section{A hybrid approach: Distributional measures of concept-distance}
So far we have looked at approaches that exploit the
structure of a resource such as WordNet, and corpus-based distributional
approaches that make use of co-occurrence statistics. A {\bf hybrid approach to semantic distance}
is one that reconciling the two, combining the information about concepts, explicitly encoded in a linguistic resource,
with the information about words, implicitly encoded in text by co-occurrence.
\namecite{MohammadH06b} and \namecite{MohammadGHZ07} have proposed one such approach that
combines corpus statistics with a published thesaurus to give the semantic distance between
concepts (rather than words).
\subsection{The distributional hypothesis for concepts}
\label{s:dhc}
As pointed out in Section \ref{s:Lconflation}, words when used in different senses tend to keep different ``company" (co-occurring words).
For example, consider the contrived but plausible distributional profile
of {\it star}:
\begin{quote}
{\bf \em star$\,$}: {\it space} 0.21, {\it movie} 0.16, {\it famous} 0.15, {\it light} 0.12, {\it constellation} 0.11, {\it heat} 0.08, {\it rich} 0.07, {\it hydrogen} 0.07,~\ldots
\end{quote}
Observe that it has words that co-occur both with {\it star}'s
{\sc celestial body} sense and {\it star}'s {\sc celebrity} sense.
Thus, it is clear that different senses of a word may have very different
distributional profiles. Using a
single DP for the word will mean the union of those
profiles. While this might be useful for certain
applications, \namecite{MohammadH06b} argue that in a number of tasks
(including estimating semantic distance), acquiring
different DPs for the different senses is not only more
intuitive, but also, as they show through experiments,
more useful.
They show that the closer the distributional profiles of two concepts, the smaller is their
semantic distance. Below are example distributional profiles of two senses of {\it star}:
\begin{quote}
{\bf {\sc celestial body}}:
{\it space} 0.36, {\it light} 0.27, {\it constellation} 0.11, {\it hydrogen} 0.07,~\ldots\\
{\bf \sc celebrity}:
{\it famous} 0.24, {\it movie} 0.14, {\it rich} 0.14, {\it fan} 0.10,~\ldots
\end{quote}
\noindent
The values are the strength of association (usually pointwise mutual
information or conditional probability) of the target concept
with co-occurring words.
We have seen that creating distributional profiles of words involves
simple word--word co-occurrence counts.
The creation of DPCs, on the other hand, requires: (1) a concept inventory that list all the concepts and words
that refer to them, and (2) counts of how often a concept
co-occurs with a word in text. These two aspects will be discussed in the next two sub-sections; however
once created, any of the many distributional measures can be used to estimate the distance
between the DPs of two target concepts (just as in the case of traditional word-distance measures, where
distributional measures are used to estimate the distance between the DPs of two target words).
For example, here is how Mohammad and Hirst adapt the formula for cosine (described earlier in Section \ref{s:cosine})
to estimate distributional distance between two concepts:
\begin{equation}
\hspace{-8mm} {\textrm {\em Cos}}_{\textrm {\em cp}}(c_1,c_2) =
\frac{\sum_{w \in C(c_1) \cup C(c_2)} \left( P(w|c_1) \times P(w|c_2) \right) }
{\sqrt{ \sum_{w \in C(c_1)} P(w|c_1)^2 } \times \sqrt{ \sum_{w \in C(c_2)} P(w|c_2)^2 } }
\end{equation}
\noindent $C(x)$ is now the set of words that co-occur with {\em concept} $x$
within a pre-determined window. The conditional probabilities in the formula
are taken from the distributional profiles of concepts.
\subsubsection{A suitable knowledge source and concept inventory}
\hspace{0.0pt} \namecite{MohammadH06b} use the categories in the {\it Macquarie Thesaurus}, 812 in all, as very coarse-grained
word senses or concepts,
in contrast to approaches that use WordNet or other similarly fine-grained sense inventories.\footnote{It has been suggested for some time now
that WordNet is much too fine-grained for certain natural language applications (\namecite{AgirreL03} and citations therein).}
Their approach to determining word--concept co-occurrence counts (described in the next sub-section) requires
a set of possibly ambiguous words that together unambiguously represent each concept---for which a thesaurus is a natural choice.
The use of categories in a thesaurus as concepts means that this approach requires a concept--concept
distance matrix of size only $812 \times 812$---much
smaller than (less than 0.01\% of) the matrix required by the WordNet-based and distributional measures.
\subsubsection{Estimating distributional profiles of concepts}
\label{s:DPC}
A {\bf word--category co-occurrence matrix (WCCM)}
is created having word types $w$ as one
dimension and thesaurus categories $c$ as
another.
\begin{displaymath}
\begin{array}{c|c c c c c}
&c_1 &c_2 &\ldots &c_j &\ldots\\
\hline
w_1 &m_{11} &m_{12} &\ldots &m_{1j} &\ldots\\
w_2 &m_{21} &m_{22} &\ldots &m_{2j} &\ldots\\
\vdots &\vdots &\vdots &\ddots &\vdots &\vdots\\
w_i &m_{i1} &m_{i2} &\ldots &m_{ij} &\ldots\\
\vdots &\vdots &\vdots &\ldots &\vdots &\ddots\\
\end{array}
\end{displaymath}
\noindent The matrix is populated with co-occurrence counts from a
large corpus.
A particular cell $m_{ij}$, corresponding to word
$w_i$ and category or concept $c_j$, is populated
with the number of times $w_i$ co-occurs (they use a window of $\pm5$ words) with any word
that has $c_j$ as one of its senses (i.e.,
$w_i$ co-occurs with any word listed under
concept $c_j$ in the thesaurus).
This matrix, created after a first pass of the corpus, is called
the {\bf base word--category co-occurrence matrix (base WCCM)}.
A contingency
table for any particular word $w$ and category $c$ can be easily generated
from the WCCM by collapsing cells for all other words and categories
into one and summing up their frequencies.
\begin{displaymath}
\begin{array}{c|c c}
&c &\neg{c}\\
\hline
w &n_{wc} &n_{w\neg}\\
\neg{w} &n_{\neg{c}} &n_{\neg\neg}
\end{array}
\end{displaymath}
\noindent A suitable statistic, such as pointwise mutual information or conditional probability, will then yield the strength of association between
the word and the category.
As the base WCCM is created from unannotated text, it will be
noisy but nonetheless
capture strong word--category co-occurrence associations reasonably accurately.
This is because the errors in determining the true category that a word co-occurs
with will be distributed thinly across a number of other categories.
(For more discussion of the general principle see \namecite{Resnik98}.)
A second pass of
the corpus is made and the base WCCM is used to disambiguate the words in it.
A new {\bf bootstrapped WCCM} is
created such that each cell $m_{ij}$, corresponding to word
$w_i$ and concept $c_j$, is populated
with the number of times $w_i$ co-occurs with any
word
{\em used in sense} $c_j$.
\namecite{MohammadH06a} showed that this WCCM, created after
simple sense disambiguation, better captures word--concept co-occurrence values,
and hence strengths of association values,
than the base WCCM.
\subsubsection{Mimicking semantic relatedness and semantic similarity}
The distributional profiles created by the above methodology are relation-free.
This is because (1) all co-occurring
words (not just those that are related to the target by certain
syntactic relations) are used, and (2) the WCCM, as described
above, does not maintain separate counts
for the different syntactic relations between the target and
co-occurring words. Thus, distributional measures that use
these WCCMs will estimate semantic {\it relatedness} between concepts. Distributional
measures that mimic semantic {\it similarity}, which require
relation-constrained DPCs, can easily be created from
WCCMs that have rows for each word--syntactic relation pair (rather than just for words).
\subsubsection{Performance}
\hspace{0.0pt} \namecite{MohammadH06b} evaluate this approach on two tasks: ranking word pairs in order of their semantic distance
and correcting real-word spelling errors.
On both tasks, distributional concept-distance measures markedly outperformed distributional
word-distance measures.
The WordNet-based measures performed better in the word-pair ranking task, but in the spelling correction task
three of the four distributional measures outperformed all WordNet-based measures except
the Jiang--Conrath measure. It should be noted, however, that these experiments evaluated only
semantic similarity of noun--noun pairs---for all other part-of-speech combinations
and semantic relatedness estimates, the WordNet-based measures are markedly less accurate.
\subsection{Multilinguality}
\label{s:cross}
Some of the best algorithms for semantic distance cannot be applied to most languages
because of a lack of high-quality linguistic resources.
\namecite{MohammadGHZ07} showed how text in one language $L_1$ can be combined with a knowledge source in another $L_2$ using a bilingual lexicon $L_1$--$L_2$
and a bootstrapping and concept-disambiguation algorithm to create {\bf cross-lingual
distributional profiles of concepts}.
These cross-lingual DPCs model co-occurrence distributions of concepts, as per
a knowledge source in one language, with words from another language,
and obtain semantic distance in a resource-poor language
using a knowledge source from a resource-rich one.
Cross-lingual semantic distance and cross-lingual DPCs are also useful in tasks that inherently involve two or more languages,
such as machine translation, multilingual multidocument tasks of clustering, coreference resolution, and information retrieval.
We summarize their approach here using German as $L_1$ and English as $L_2$; however,
the algorithm is language-pair independent.
\subsubsection{Cross-lingual senses, cross-lingual distributional profiles, and
cross-lingual distributional distance}
\label{s:crossall}
Given a German word $w^{\text {\em de}}$ in context, \namecite{MohammadGHZ07} use a
German--English bilingual lexicon to determine its different possible
English translations. Each English translation $w^{\text {\em en}}$
may have one or more possible coarse senses, as listed in an English
thesaurus. These English thesaurus concepts ($c^{\text {\em en}}$)
will be referred to as the {\bf cross-lingual candidate senses} of the
German word $w^{\text {\em de}}$.
Figure \ref{fig:crossstern} depicts examples.\footnote{They are called ``candidate senses"
because some of the senses of $w^{\text {\em en}}$ might not really be senses
of $w^{\text {\em de}}$. For example, {\sc celestial body} and {\sc celebrity}
are both senses of the English word {\it star}, but
the German word {\it Stern} can only mean {\sc celestial body} and
not {\sc celebrity}.
Similarly, the German {\it Bank} can mean {\sc financial institution}
or {\sc furniture}, but not {\sc river bank} or
{\sc judiciary}.
An automated system has no straightforward method of teasing out the
actual cross-lingual senses of $w^{\text {\em de}}$ from those that are an artifact of the translation step.}
\begin{figure}[t]
\centerline{\scalebox{0.4}{ \input{stern.pstex_t} }}
\caption{The cross-lingual candidate senses of German words {\em Stern} and {\em Bank}.}
\label{fig:crossstern}
\end{figure}
As in the monolingual estimation of distributional concept-distance, the distance between
two concepts is calculated by first determining their DPs.
However, a concept is now glossed by
near-synonymous words in an {\em English} thesaurus, whereas its
profile is made up of the strengths of association with co-occurring
{\em German} words.
Here are constructed example cross-lingual distributional profiles of the two cross-lingual candidate senses of the German word {\em Stern}:\footnote{Vocabulary of German words needed to
understand this discussion:
{\bf \em Bank}: 1.\@ financial institution, 2.\@ bench (furniture);
{\bf \em ber\"{u}hmt}: famous;
{\bf \em Film}: movie (motion picture);
{\bf \em Him\-mels\-k\"{o}rp\-er}: heavenly body;
{\bf \em Konstellation}: constellation;
{\bf \em Licht}: light;
{\bf \em Morgensonne}: morning sun;
{\bf \em Raum}: space;
{\bf \em reich}: rich;
{\bf \em Sonne}: sun;
{\bf \em Star}: star (celebrity);
{\bf \em Stern}: star (celestial body)
}
\begin{quote}
{\bf \sc celestial body} ({\em celestial body, sun, \ldots}): {\em Raum} 0.36, {\em Licht} 0.27, {\it Konstellation} 0.11,~\ldots \\
{\bf \sc celebrity} ({\em celebrity, hero,~\ldots}): {\em ber\"{u}hmt} 0.24, {\em Film} 0.14, {\em reich} 0.14,~\ldots
\end{quote}
\noindent The cross-lingual DPCs are created from a cross-lingual word--category co-occurrence matrix
without the use of any word-aligned parallel corpora or sense-annotated data (as described in the next subsection).
Just as in the case of monolingual distributional concept-distance measures,
distributional measures can be used to estimate the distance between the cross-lingual DPs of two target concepts.
For example, the cosine formula can be adapted to estimate cross-lingual distributional distance between two concepts
as shown below:
\begin{equation}
\hspace{-8mm} {\textrm {\em Cos}}(c_1^{\text {\em en}},c_2^{\text {\em en}}) =
\frac{\sum_{w^{\text {\em de}} \in C(c_1^{\text {\em en}}) \cup C(c_2^{\text {\em en}})} \left( P(w^{\text {\em de}}|c_1^{\text {\em en}}) \times P(w^{\text {\em de}}|c_2^{\text {\em en}}) \right) }
{\sqrt{ \sum_{w^{\text {\em de}} \in C(c_1^{\text {\em en}})} P(w^{\text {\em de}}|c_1^{\text {\em en}})^2 } \times \sqrt{ \sum_{w^{\text {\em de}} \in C(c_2^{\text {\em en}})} P(w^{\text {\em de}}|c_2^{\text {\em en}})^2 } }
\end{equation}
\noindent $C(x)$ is now the set of German words that co-occur with English concept $x$
within a pre-determined window. The conditional probabilities in the formula
are taken from the cross-lingual DPCs.
\subsubsection{Creating cross-lingual word--category co-occurrence matrix}
A German--English cross-lingual word--category co-occurrence matrix has
German word types $w^{\text {\em de}}$ as one dimension and English
thesaurus concepts $c^{\text {\em en}}$ as another.
\begin{displaymath}
\begin{array}{c|c c c c c}
&c_1^{\text{\em en}} &c_2^{\text{\em en}} &\ldots &c_j^{\text{\em en}} &\ldots\\
\hline
w_1^{\text{\em de}} &m_{11} &m_{12} &\ldots &m_{1j} &\ldots\\
w_2^{\text{\em de}} &m_{21} &m_{22} &\ldots &m_{2j} &\ldots\\
\vdots &\vdots &\vdots &\ddots &\vdots &\vdots\\
w_i^{\text{\em de}} &m_{i1} &m_{i2} &\ldots &m_{ij} &\ldots\\
\vdots &\vdots &\vdots &\ldots &\vdots &\ddots\\
\end{array}
\end{displaymath}
\noindent The matrix is populated with co-occurrence counts from a
large German corpus.
A particular cell $m_{ij}$, corresponding to word
$w_i^{\text{\em de}}$ and concept $c_j^{\text{\em en}}$, is populated
with the number of times the German word $w_i^{\text{\em de}}$
co-occurs (say a window of $\pm5$ words) with any German word having $c_j^{\text{\em en}}$ as one
of its {\em cross-lingual candidate senses}.
For example, the {\em Raum}--{\sc celestial body} cell will have the sum
of the number of times {\em Raum} co-occurs with {\em
Himmelsk\"{o}rper, Sonne, Morgensonne, Star, Stern}, and so on (see
Figure \ref{fig:crosscelestial}).
This matrix, created after a first pass of the corpus, is called
the {\bf cross-lingual base WCCM}.
A contingency
table for any particular German word $w^{\text{\em de}}$ and English category $c^{\text{\em en}}$ can be easily generated
from the WCCM by collapsing cells for all other words and categories
into one and summing up their frequencies.
A suitable statistic, such as PMI or conditional probability, will yield the strength of association between
the German word and the English category.
Then a new bootstrapped cross-lingual WCCM is created, just as in the monolingual case.
\begin{figure}[t]
\centerline{\scalebox{0.45}{ \input{celestial.pstex_t} }}
\caption{Words having {\sc celestial body} as one of their cross-lingual
candidate senses.}
\label{fig:crosscelestial}
\end{figure}
\subsubsection{Performance}
\hspace{0.0pt} \namecite{MohammadGHZ07} evaluated the cross-lingual measures of
semantic distance on two tasks: (1) estimating
semantic distance between words and ranking the word pairs according
to semantic distance, and (2) solving {\em Reader's Digest} `Word
Power' problems.
They compared these results with those obtained by conventional
state-of-the-art monolingual approaches with and without a knowledge
source in the target language $L_1$ (GermaNet).
The cross-lingual approach obtained much better results
than monolingual approaches that do not use a knowledge source.
Further, in both tasks, the cross-lingual
measures performed as well if not slightly better than the
GermaNet-based monolingual approaches, as well. This shows that the
cross-lingual approach is able to keep losses due to the translation
step at a minimum, while allowing the use of a superior knowledge
source in another language to get better results.
\subsection{Challenges}
Distributional measures of concept-distance have many desirable features of both
knowledge-rich approaches and strictly corpus-based approaches---they have the high
accuracies of knowledge-rich approaches, they can measure both semantic relatedness and semantic similarity,
and they have a strong corpus-reliance making them domain adaptable. Further, with the cross-lingual approach,
a lack of high-quality knowledge source in the target language is no longer a problem.
However, certain issues remain.
\subsubsection{Integrating domain-specific terminology}
The reliance on a knowledge source means that the approach cannot
measure the distance between words not listed in the thesaurus. This is especially a problem
for domain-specific jargon, which might not find place in a general purpose knowledge source.
Automatic ways of integrating domain-specific terminology into a general purpose knowledge source
will be valuable to this end.
\subsubsection{Choosing the right concept-granularity}
\hspace{0.0pt} \namecite{MohammadH06b} and \namecite{MohammadGHZ07} have reported results
using the categories of the thesaurus as very coarse word senses. This level of granularity
has worked well for the tasks they experimented with; however, a relatively finer
sense inventory may be more suitable for other tasks. Words within a thesaurus category are grouped into paragraphs;
and using them (instead of categories) and determining when this finer-grained sense-inventory
is more suitable for use remains to be explored.
\subsubsection{Identifying lexical semantic relations}
Word pairs can be semantically close because of any of the classical lexical semantic
relations, such as hypernymy, near-synonymy, antonymy, troponymy, and meronymy, or the innumerable
non-classical relations. The various distributional approaches discussed in this paper
determine semantic distance
without explicitly identifying the nature of the relationship.
Already, there is some work on determining lexical entailment \cite{MirkinDG07} and determining near-synonymy \cite{LinZQZ03}.
Identifying antonymy (or more generally,
contrasting word-pairs) is especially useful in many natural language tasks, even if it is simply
to discard them from a list of distributionally close terms.
Also, it will be interesting for measures of semantic distance to
characterize the nature of any non-classical relationship shared by two words---not only
determining if two terms are close but also specifying (in some intuitive way) the aspect of meaning they share.
\section{Conclusion}
A large number of important natural language problems, including machine translation, information retrieval,
and word sense disambiguation, can be viewed in part as semantic distance problems.
Numerous measures of semantic distance exist---those that use a knowledge source and those that rely on corpora. Yet, their use in
real-world applications has been limited. In this paper, we investigated how automatic measures
can be brought more in line with human notions of semantic distance, how they can be made applicable to a large number
of natural language tasks, and how they can be used even for languages deficient in high-quality linguistic resources.
Even though corpus-based distributional measures of distance have traditionally performed poorly when compared to WordNet-based measures,
we have shown that (1) there are a number of reasons that make distributional measures uniquely attractive, and (2) that their potential
is yet to be fully realized.
Distributional measures can be easily applied to most languages (they can make do even with just raw text) and they
can be used to mimic both semantic similarity and semantic relatedness.
With this in mind, the paper presented a detailed study of many important distributional measures, analyzed their limitations, and explained
why their performance has been relatively poor so far. Understanding these limitations is crucial in the development of
new and better approaches, whether they have a distributional base or otherwise.
We concluded the paper with the discussion of a hybrid, yet distinctly distributional approach, that presents
one way to more accurately measure distributional distance without compromising too much on essential
properties such as the applicability to resource-poor languages.
\begin{acknowledgments}
We thank Suzanne Stevenson, Gerald Penn, and the Computational Linguistics group at the University of Toronto for helpful discussions.
This research is financially
supported by the Natural Sciences and
Engineering Research Council of Canada and the University of Toronto.
\end{acknowledgments}
\begin{multicols}{2}
|
{
"timestamp": "2012-03-09T02:04:04",
"yymm": "1203",
"arxiv_id": "1203.1858",
"language": "en",
"url": "https://arxiv.org/abs/1203.1858"
}
|
\section{Introduction}
\vspace{5mm}
It has turned out that supersymmetric Chern-Simons-matter theories play a very important role in the study of M-theory.
The low energy dynamics of M2-branes is described by such a three-dimensional field theory.
Details of the theory, the amount of supercharges and matter contents for
example, are determined by the eleven-dimensional space-time in which the M2-branes live.
M2-branes living in the flat space-time is expected, at least when
the number of the M2-branes are two, to be described by BLG theory \cite{Bagger:2006sk}-\cite{Gustavsson:2007vu}
which can be written as a Chern-Simons-matter theory \cite{VanRaamsdonk:2008ft}.
ABJM theory \cite{Aharony:2008ug} provides a description of an arbitrary number of M2-branes at the tip of an orbifold $\mathbb{C}^4/\mathbb{Z}_k$.
This has ${\cal N}=6$ supersymmetry and ${\rm U}(N)\times{\rm U}(N)$ gauge group with the Chern-Simons levels $(+k,-k)$.
The ${\cal N}=6$ supersymmetry of the ABJM theory is preserved even when the gauge group is replaced with ${\rm U}(N_1)\times {\rm U}(N_2)$
\cite{Hosomichi:2008jb}\cite{Aharony:2008gk}, at least at the classical level.
However, it is argued \cite{Aharony:2008gk} that this theory does not exist as a physically sensible theory when
\begin{equation}
|N_1-N_2|>k
\label{ABJ}
\end{equation}
is satisfied.
This is based on the following observation \cite{Aharony:2008gk}.
When some of the bi-fundamental fields in the theory have non-vanishing vevs, at low energy,
the theory with $N_1\ne N_2$ is reduced to an effective theory including a decoupled
${\cal N}=3$ pure Chern-Simons theory with the gauge group ${\rm SU}(|N_1-N_2|)$ and the level $k$.
The vevs correspond to a point in the moduli space of the ${\cal N}=6$ theory in which no classical vacua are lifted by quantum corrections.
However, it is known \cite{Bergman:1999na}\cite{Ohta:1999iv} that the supersymmetry is spontaneously broken in ${\cal N}=3$ pure Chern-Simons theory
when the parameters satisfy the condition (\ref{ABJ}).
This apparent contradiction would come from the assumption that the ${\cal N}=6$ theory satisfying the condition (\ref{ABJ}) exists.
Supersymmetry breaking in a supersymmetric Chern-Simons theory can be understood most easily when the theory can be realized on some D-branes in
string theory.
For example, ${\cal N}=2$ pure Chern-Simons theory with the gauge group ${\rm SU}(N)$ and the level $k$ is realized on D3-brane segments suspended between
an NS5-brane and a $(1,k)$5-brane \cite{Kitao:1998mf}\cite{Bergman:1999na}, as depicted in Figure \ref{brane1}.
It is known that such a brane system obeys the s-rule \cite{Hanany:1996ie} by which one can check whether the supersymmetry is broken or not.
For the ${\cal N}=2$ pure Chern-Simons theory, it turns out that the supersymmetry is broken completely if and only if
\begin{equation}
N>k
\label{SUSYbreaking}
\end{equation}
is satisfied \cite{Bergman:1999na}\cite{Ohta:1999iv}.
This condition can be phrased as follows:
the supersymmetry is broken when the 't Hooft coupling $\lambda=N/k$ is larger than 1.
It is usually the case that a gravity dual, assuming it exists, will be classical and easy to be handled when the 't Hooft coupling, as well as $N$, is large.
Therefore, the above condition (\ref{SUSYbreaking}) implies that, even if there exists a classical gravity dual for ${\cal N}=2$ pure Chern-Simons theory,
the supersymmetry should be also broken spontaneously in the gravity side.
This kind of argument can apply to other Chern-Simons-matter theories.
If one theory exhibits a pattern of supersymmetry breaking, then the information can be used to discuss the expected gravity dual, whose discussion may include whether
such a dual should exist or not.
If there is no known brane construction for a Chern-Simons-matter theory which one would like to consider, one may try to generalize the arguments in
\cite{Witten:1999ds}\cite{Ohta:1999iv} to the theory.
However, it seems that such generalizations are not always so straightforward technically.
Supersymmetry breaking in Chern-Simons-matter theories was discussed recently in \cite{Niarchos:2009aa}\cite{Morita:2011cs}\cite{Smilga:2012uy}.
A Hamiltonian formulation was recently developed in \cite{Agarwal:2012bn} for supersymmetric Yang-Mills-Chern-Simons theories.
\begin{figure}[tbp]
\begin{center}
\includegraphics{brane1.eps}
\caption{The brane configuration realizing ${\cal N}=2$ pure Chern-Simons theory. }
\label{brane1}
\end{center}
\end{figure}
In this paper, we present a field-theoretic argument to study the supersymmetry breaking in Chern-Simons-matter theories.
It is different from the one given in \cite{Witten:1999ds}\cite{Ohta:1999iv}
for pure (Yang-Mills)-Chern-Simons theories.
In fact, it is not clear in \cite{Witten:1999ds}\cite{Ohta:1999iv} how the supersymmetry is broken.
We will show that, by replacing the Chern-Simons terms with massive fundamental multiplets using the phenomenon found in \cite{Niemi:1983rq}\cite{Redlich:1983dv},
the supersymmetry breaking can be studied
through investigating whether the scalar potential is always positive or not, a quite familiar criterion to show the supersymmetry breaking.
We discuss ${\cal N}=2$ pure Chern-Simons theory and a family of quiver-type theories, but our argument can be generalized to the other theories, as long as
the quantum effects in corresponding theories are well understood.
Our argument can show the occurrence of the supersymmetry breaking in a range of parameters.
For example, for ${\cal N}=2$ pure Chern-Simons theory, we will show that the supersymmetry is broken completely if $N>k+1$, while for the other cases we will
give a plausible argument which is consistent with the pattern derived based on the analysis of the D-brane system \cite{Bergman:1999na}\cite{Ohta:1999iv}.
The basic idea in this paper is in common with the analysis based on brane configurations.
That is, if one would like to know about the low energy physics of a theory, then one may instead investigate any desired theory as long as it shares the same IR
physics with the former.
The worldvolume theory for the brane configuration (Figure \ref{brane1}) is one choice of the UV theory.
An advantage of our choice of the UV theory is that it always exists.
This paper is organized as follows.
Section \ref{review} reviews necessary facts about three-dimensional gauge theories.
${\cal N}=2$ pure Chern-Simons theory is analyzed in section \ref{pureCS}.
Section \ref{quiver type}
analyzes ${\cal N}=2$ Chern-Simons theory with gauge group ${\rm SU}(N_1)\times{\rm SU}(N_2)$ with the level $(+k,-k)$ coupled to bi-fundamental matters.
\vspace{1cm}
\section{Real masses and induced Chern-Simons term} \label{review}
\vspace{5mm}
There is a close similarity between ${\cal N}=2$ supersymmetric gauge theories in three dimensions and
${\cal N}=1$ supersymmetric gauge theories in four dimensions.
Many of the former theories are obtained from the latter ones through the dimensional reduction.
Some properties, for example the existence of the non-renormalization theorem for F-terms, inherit from four dimensions to three dimensions.
Even some non-perturbative phenomena occur in quite similar manner in four and three dimensions.
There are, of course, some properties which are specific to three-dimensional theories.
One example is the presence of the real mass terms, and another is the presence of Chern-Simons terms.
Interestingly, they have a close connection, as will be reviewed in this section.
Some basic facts about three-dimensional supersymmetric theories can be found, for example, in \cite{Aharony:1997bx}.
\vspace{5mm}
The mass terms which are familiar in the four-dimensional theories are the complex mass terms which appear as quadratic terms in the superpotential.
In three dimensions, there is another way to give masses to matter fields while preserving ${\cal N}=2$ supersymmetry.
Such mass terms are known as real mass terms.
The origin of the real mass terms can be easily deduced from the following observation.
The kinetic term of a scalar field $\Phi$ coupled to a gauge field $A_\mu$ in four dimensions provides, through the dimensional reduction,
\begin{equation}
| D_\mu\Phi |^2 \hspace{5mm} \rightarrow \hspace{5mm} | D_{\hat{\mu}}\Phi |^2 + | A_3\Phi |^2,
\end{equation}
where $\hat{\mu}=0,1,2$.
In three dimensions, $A_3$ is a scalar.
If $A_3$ has a non-zero vev, some components of $\Phi$ become massive.
This suggests that a mass term can be introduced as a background vector multiplet $V_m$ in which only the third component of the gauge field, with respect to the
four-dimensional viewpoint, has a non-zero vev,
\begin{equation}
V_m = \theta\bar{\theta}m.
\end{equation}
This indeed gives a mass term as follows,
\begin{equation}
\int d^4\theta\ \Phi^\dag e^{V_m}\Phi = \int d^4\theta\ \Phi^\dag\Phi + m^2|\phi|^2+m\bar{\psi}\psi.
\end{equation}
Since $V_m$ is real, the mass parameter $m$ must be real.
In three dimensions, there is no chiral anomaly.
Therefore, the charge assignment of matter multiplets which couples to $V_m$ is less restrictive.
\vspace{5mm}
Suppose that the mass parameter $m$ is very large.
Naively, one might expect that the chiral multiplet $\Phi$ can be simply integrated out.
However, there must be a remnant of $\Phi$ since the original theory with a non-zero $m$ may break the parity invariance \cite{Deser:1981wh},
and the low energy effective theory must remember it.
The remnant is nothing but a Chern-Simons term \cite{Niemi:1983rq}\cite{Redlich:1983dv}.
It is induced from one-loop diagrams depicted in Figure \ref{1-loop}.
Contributions from the other diagrams are suppressed by $m^{-1}$ which is negligible at low energy.
The Chern-Simons level is determined by the representation of the matter fermions circulating the loop.
If there is a fermion in the representation $R$, the level $k$ is given as
\begin{equation}
k = \frac12\mbox{sgn}(m)C_2(R),
\end{equation}
where $C_2(R)$ is the second Casimir for $R$.
The representation $R$ may be reducible, implying that the level induced by multiple fermions is the sum of contributions from each fermions.
For the gauge group ${\rm SU}(N)$, $C_2(R)$ is normalized such that it is 1 for the fundamental representation.
\begin{figure}[tbp]
\begin{center}
\includegraphics{1-loop.eps}
\caption{The 1-loop diagrams which induce Chern-Simons term. The wavy lines are gauge fields and the solid lines are fermions. The left diagram induces the term with
a derivative, and the right diagram induces the trivalent vertex of the gauge fields. }
\label{1-loop}
\end{center}
\end{figure}
If there is only one massive fundamental chiral multiplet, then it induces a Chern-Simons term with a fractional level, and the gauge symmetry turns out to be
broken.
Suppose that there is a pair $(\Phi, \widetilde{\Phi})$ of chiral multiplets in the representations $(R,\bar{R})$ of a gauge group whose real masses are given as follows,
\begin{equation}
\int d^4\theta \left[ \Phi^\dag e^{V+\theta\bar{\theta}m_1}\Phi + \widetilde{\Phi}e^{-V-\theta\bar{\theta}m_2}\widetilde{\Phi}^\dag \right].
\end{equation}
If both $\Phi$ and $\widetilde{\Phi}$ are integrated out, a Chern-Simons term is induced with the level
\begin{equation}
k = \frac12[ \mbox{sgn}(m_1)-\mbox{sgn}(m_2) ]C_2(R).
\end{equation}
Therefore, if $\mbox{sgn}(m_1m_2)>0$, then no Chern-Simons term is induced, while otherwise $k$ is an integer.
In both cases, the gauge symmetry is preserved.
It would be convenient to define a vector mass $m_v$ and an axial mass $m_a$ as
\begin{equation}
m_v = \frac12(m_1+m_2), \hspace{5mm} m_a = \frac12(m_1-m_2).
\end{equation}
A Chern-Simons term is induced only if $m_a$ is non-zero.
If the gauge group has a ${\rm U}(1)$ factor, then $m_v$ can be absorbed by shifting the scalar field in the ${\rm U}(1)$ vector multiplet.
The effects of the vector mass on the low energy dynamics is studied in \cite{deBoer:1997kr}\cite{Aharony:1997bx}.
\vspace{1cm}
\section{Pure Chern-Simons theory} \label{pureCS}
\vspace{5mm}
In this section, we consider vacuum states of ${\cal N}=2$ pure Chern-Simons theory.
For definiteness, we specify the gauge group to be SU$(N)$ with the Chern-Simons level $k$.
It is known \cite{Bergman:1999na}\cite{Ohta:1999iv} that this theory has supersymmetric vacua when $N\le k$,
which can be shown based on the analysis of the corresponding brane configuration,
while otherwise the supersymmetry is completely broken.
Exactly speaking, \cite{Bergman:1999na}\cite{Ohta:1999iv} discussed the vacuum states of ${\cal N}=2$ Yang-Mills-Chern-Simons theory which appears as the
worldvolume theory on D3-branes in the brane configuration.
However, the properties of the vacuum states of this theory should be the same as those of ${\cal N}=2$
pure Chern-Simons theory since the Yang-Mills term is
irrelevant in three dimensions, and therefore it can be ignored at low energy.
The breakdown of the supersymmetry is rather difficult to show purely from the field theory viewpoint.
For example, it cannot be shown by examining the presence of a non-vanishing vacuum energy.
A rather indirect field theoretic argument was given in \cite{Witten:1999ds} for ${\cal N}=1$ Yang-Mills-Chern-Simons theory, and its extension to the
${\cal N}=2,3$ theories was discussed in \cite{Ohta:1999iv}.
In these analyses, the presence of the Yang-Mills term seems to be crucial.
In the following, we argue the occurrence of supersymmetry breaking in a field theory which flows to ${\cal N}=2$ Yang-Mills-Chern-Simons theory,
and therefore to ${\cal N}=2$ pure Chern-Simons theory, in the IR limit.
As discussed previously, by definition, the UV theory has vacuum states which share the same properties with those
of ${\cal N}=2$ pure Chern-Simons theory.
It will turn out below that the analysis of the former would be more transparent than the analysis of the latter.
It should be noted that our analysis has also a disadvantage: our analysis can show the occurrence of supersymmetry breaking for $N-1>k$ but for the other cases
we cannot draw any definite conclusion.
It is, however, possible to argued that the other cases also seem to be compatible with the known pattern of supersymmetry breaking.
\vspace{5mm}
The theory which will be considered in this section is
${\cal N}=2$ gauge theory in three dimensions with the gauge group SU$(N)$ coupled to $k$ chiral multiplets $Q^I$ $(I=1,2,\cdots,k)$
in the fundamental representation of ${\rm SU}(N)$ and $k$ chiral multiplets $\widetilde{Q}^I$ in the anti-fundamental representation.
All the chiral multiplets have a common axial mass $m$.
The Lagrangian is given as follows,
\begin{equation}
L = \int d^4\theta \left[ Q_Ie^{V+\theta\bar{\theta}m}Q^I+\widetilde{Q}^Ie^{-V+\theta\bar{\theta}m}\widetilde{Q}_I \right]
+ \left[ \int d^2\theta \ \mbox{Tr}\ W^\alpha W_\alpha + \mbox{\rm h.c.} \right],
\label{SQCD}
\end{equation}
where $Q_I=(Q^I)^\dag$ and $\widetilde{Q}_I=(\widetilde{Q}^I)^\dag$.
If $m$ is zero, this theory is nothing but the dimensional reduction of the four-dimensional supersymmetric QCD \cite{Intriligator:1995au} with $N_c=N$ and $N_f=k$.
If $m$ is large, or if one considers this theory at an energy scale much lower than $m$, the chiral multiplets can be integrated out.
As explained in section \ref{review}, this procedure induces a Chern-Simons term with the level $k$.
At low energy, the theory (\ref{SQCD}) becomes equivalent to ${\cal N}=2$ Yang-Mills-Chern-Simons theory without matter.
At a lower energy, the Yang-Mills term becomes irrelevant due to a positive mass dimension of the gauge coupling, and it can be dropped in the IR limit.
Therefore, the theory (\ref{SQCD}) shares the same low energy properties with ${\cal N}=2$ pure Chern-Simons theory.
\vspace{5mm}
\subsection{Classical vacua}
\vspace{5mm}
The supersymmetry is completely broken if and only if the scalar potential ${\cal V}$ of (\ref{SQCD}) is always positive.
At the classical level, since there is no tree-level superpotential, ${\cal V}$ is the sum of the D-term potential $V_D$ and the mass terms including the axial mass,
\begin{equation}
{\cal V} = V_D+\sum_{I=1}^k \Bigl( \bigl| (\phi+m)q^I \bigr|^2+\bigl| (\phi-m)\widetilde{q}_I \bigr|^2 \Bigr),
\label{classicalD}
\end{equation}
where $q^I$ and $\widetilde{q}^I$ are the lowest components of $Q^I$ and $\widetilde{Q}^I$, respectively, and $\phi$ is the adjoint scalar field in the vector multiplet.
In the case $m=0$, it is well-known that the solutions of ${\cal V}=0$ form a continuous family, the moduli space of vacua.
If $N>k$, which will be relevant later, the vevs of $q^I$ and $\widetilde{q}^I$ are
\begin{equation}
q^I = \widetilde{q}_I = \left[
\begin{array}{cccc}
a_1 & & & \\
& a_2 & & \\
& & \ddots & \\
& & & a_k \\
& & &
\end{array}
\right],
\end{equation}
up to a symmetry transformation.
The vev of $\phi$ is then required to satisfy
\begin{equation}
\phi q^I = 0, \hspace{5mm} \phi\widetilde{q}_I = 0.
\label{adjoint mass}
\end{equation}
In general, $\phi$ may be non-zero even when $q^I$ and $\widetilde{q}^I$ are non-zero, and the intersection of the Coulomb branch and the Higgs branch may not be
just a single point.
The presence of a non-zero $m$ changes the classical moduli space of vacua drastically.
The vev of $\phi$ can be always diagonalized by a gauge transformation,
\begin{equation}
\phi = \mbox{diag}(\phi_1, \cdots, \phi_{N}).
\end{equation}
If some of the eigenvalues of $\phi$ is equal to $\pm m$, then $q^I$ and $\widetilde{q}^I$ can have non-zero vevs.
By a gauge transformation, those eigenvalues can be arranged such that
\begin{equation}
\phi_i = \left\{
\begin{array}{cc}
+m, & (1\le i\le l_1) \\ -m, & (l_1<i\le l_2) \\ \mbox{other values} & (l_2<i\le N)
\end{array}
\right.
\end{equation}
For the case $l_1=l_2=0$, the vevs of $q^i$ and $\widetilde{q}^I$ must vanish.
Otherwise, they satisfy
\begin{equation}
q^I_1 = \cdots = q^I_{l_1} = q^I_{l_2+1} = \cdots = q^I_N = 0, \hspace{5mm} \widetilde{q}^I_{l_1+1} = \cdots = \widetilde{q}^I_N = 0.
\end{equation}
The classical D-term potential $V_D$ is
\begin{equation}
V_D = \frac{g_{\rm YM}^2}2\sum_{a=1}^{N^2-1}\left( q_IT^aq^I-\widetilde{q}^IT^a\widetilde{q}_I \right)^2.
\end{equation}
There is a generator $T$ of ${\rm su}(N)$ whose matrix form in the fundamental representation is
\begin{equation}
T = \left[
\begin{array}{ccc}
-I_{l_1} & & \\
& +I_{l_2} & \\
& & \frac{l_1-l_2}{N-l_2}I_{N-l_2}
\end{array}
\right],
\end{equation}
where $I_l$ is the $l\times l$ unit matrix.
The condition $V_D=0$ then implies
\begin{equation}
\left( q_ITq^I-\widetilde{q}^IT\widetilde{q}_I \right)^2 = \left[ \sum_{I=1}^k\left( \sum_{i=l_1+1}^{l_2} |q^I_i|^2 + \sum_{i=1}^{l_1} |\widetilde{q}^I_i|^2 \right) \right]^2 = 0.
\end{equation}
This implies that only $q^I=\widetilde{q}_I=0$ is allowed.
It is well-known that the condition $V_D=0$ can be solved whenever the F-term condition is solved.
See e.g. \cite{Luty:1995sd}.
The above calculation shows that the introduction of the axial mass parameter $m$ lifts all the classical Higgs branch.
\vspace{5mm}
\subsection{Quantum vacua}
\vspace{5mm}
At the quantum level, the scalar potential ${\cal V}$ is modified from the classical one (\ref{classicalD}) by quantum corrections.
We start our discussion of the quantum vacua with the investigation of the quantum F-term potential.
\vspace{5mm}
It was shown in \cite{Aharony:1997bx} that a non-perturbative superpotential
\begin{equation}
W_{\rm np} \propto \left[ Y\det(\widetilde{Q}^IQ^J) \right]^{\frac1{k-N+1}} + \cdots
\label{NPW}
\end{equation}
is induced if $N-1>k$, where $Y$ is a chiral superfield related to the adjoint scalar $\phi$ \cite{deBoer:1997kr}\cite{Aharony:1997bx}.
The dots in (\ref{NPW}) indicate the other terms which do not depend on $Q^I$ nor $\widetilde{Q}^I$.
The F-term potential derived from
(\ref{NPW}) exhibits a runaway behavior along the classical Higgs branch, that is, the potential decreases to zero as $\det(\widetilde{Q}^IQ^J)$ increases.
In the case $m=0$, this theory does not have any stable vacuum state.
This fact alone does not allow one to conclude that the supersymmetry is broken in this theory, as was pointed out in \cite{Intriligator:2005aw}
since there could be a supersymmetric state at infinity.
To check whether the supersymmetry is really broken or not, one has to examine the behavior of the D-term potential at infinity in the classical Higgs branch.
\vspace{5mm}
Since there is no non-renormalization theorem for the D-terms, it is quite difficult to show the behavior of the quantum-corrected D-term potential explicitly.
However, it is still possible to argue that the scalar potential
${\cal V}$ approaches the classical one (\ref{classicalD}) at infinity of the classical Higgs branch at least when $m$ is small.
The argument given below is an analog of the one in \cite{Affleck:1984xz} for four-dimensional theories.
Consider first the case $m=0$.
It was argued in \cite{Witten:1981nf},
whose argument can apply to three dimensions, that the quantum-corrected D-term potential $V_D$ does not lift the classical flat directions.
In fact, this would not be so surprising in three dimensions since the quantum corrections are quite restricted
simply due to the fact that the gauge coupling constant $g_{\rm YM}$ is dimensionful.
The mass terms including $\phi$ should also vanish along the classical moduli space since the vacuum configuration satisfy (\ref{adjoint mass}),
and every $\phi$ in the mass terms must appear in combinations $\phi q^I$ and $\phi\widetilde{q}_I$ due to the gauge invariance.
When the vevs of $q^I$ and $\widetilde{q}^I$ are large and along the classical Higgs branch,
some of the vector multiplets which couple to $q^I$ and $\widetilde{q}^I$ become massive and decouple at a very high energy.
Since the gauge interaction is weak at a high energy scale, the D-term potential has not received large quantum corrections yet.
Below the energy scale set by the vevs, there remain the vector multiplets with the gauge group ${\rm SU}(N-k)$ and the chiral multiplets $M^{IJ}=\widetilde{Q}^IQ^J$ which
are decoupled from the vector multiplets.
Note that a non-generic vevs of $q^I$ and $\widetilde{q}^I$ which may allow a larger gauge group is forbidden by the non-perturbative F-term potential.
Since the low energy effective theory does not provide a non-trivial D-term potential, the total D-term potential is given approximately by the classical one
(\ref{classicalD}).
Now, we introduce a small mass parameter $m$.
As long as the vevs are large enough, the scalar potential along the classical Higgs branch remains the classical one (\ref{classicalD}).
As was shown in the previous subsection, the presence of a non-zero $m$ lifts all the classical Higgs branch, implying that a region of
the classical Higgs branch near infinity is lifted.
Therefore, there must be a point in the middle of the Higgs branch where the quantum-corrected scalar potential has a global minimum.
Since the F-term potential derived from (\ref{NPW}) is exact and turns out to be non-zero at the global minimum,
it is possible to conclude that the supersymmetry is broken completely if
\begin{equation}
N-1>k.
\end{equation}
The remaining task is to take $m$ to be large so that the IR theory is equivalent to ${\cal N}=2$ pure Chern-Simons theory.
We expect that for any finite $m$ there exists a region near infinity of the classical Higgs branch where the D-term potential is approximated by the classical one.
If this is the case, then the supersymmetry is also broken for the theory with finite but large $m$ which then implies the supersymmetry breaking in ${\cal N}=2$ pure
Chern-Simons theory.
It seems natural to expect that the non-zero vacuum energy would increase if $m$ increases, so the supersymmetry breaking seems to be robust against the change of
$m$.
\vspace{5mm}
It should be noted that the field $Y$ should be also fixed to realize a stable vacuum state.
We simply assumed above that those fields would be fixed by the quantum-corrected D-term potential.
This is because a Chern-Simons term, which is induced by integrating out the matters with
a non-zero $m$, provides the vector multiplets with a non-zero mass.
Therefore, the classical flat directions for $\phi$ must be lifted after turning on a non-zero $m$.
Since the superpotential does not depend on the real mass \cite{Aharony:1997bx}, this lift should be due to a non-trivial D-term.
This seems to be reasonable from the fact that the Chern-Simons term appears as a D-term.
For example, the Chern-Simons term for U$(1)$ gauge group is
\begin{equation}
\int d^4\theta\ \Sigma V, \hspace{5mm} \Sigma = \overline{D}DV.
\end{equation}
Since the vacuum state corresponds to a point at the middle of the classical Higgs branch, and
there is no non-renormalization theorem for D-terms for theories with four supercharges, it is quite difficult to explicitly show
how the field $Y$ would be fixed.
There are other fields related to $\phi$ \cite{deBoer:1997kr}\cite{Aharony:1997bx} and contributing to the superpotential (\ref{NPW}).
We also assumed that they are fixed by the D-term potential as for the case of $Y$.
See also the discussions in \cite{Aharony:1997bx} for the fixing of $\phi$ in the presence of an axial mass.
\vspace{5mm}
One might worry about a possibility that there could be a discontinuous change of the non-perturbative superpotential between $m=0$ and $m\ne0$.
If there would be a singularity in the superpotential, it would be due to the appearance of massless particles at $m=0$ which was integrated out in a description for
$m\ne 0$.
In the case considered here, the fields which become massless at $m=0$ are $Q$ and $\widetilde{Q}$.
Since they are retained, there would be no source of singularities in the superpotential in the limit $m\to0$.
Therefore, the superpotential (\ref{NPW}) should be also valid for $m\ne0$.
\vspace{5mm}
It is known that the supersymmetry is also broken when $N-1=k$ \cite{Bergman:1999na}\cite{Ohta:1999iv}.
However, the discussion of this case using the theory (\ref{SQCD}) would not be so simple since there is no superpotential induced in this case \cite{Aharony:1997bx}.
Therefore, the occurrence of supersymmetry breaking would depend on the details of the D-term potential.
The following could be a plausible argument suggesting that the supersymmetry is also broken in this case.
In the case $m=0$, it was shown \cite{Aharony:1997bx} that, although no superpotential is induced, there is a non-trivial constraint on the fields
\begin{equation}
Y\det(\widetilde{Q}^IQ^J) = 1
\label{constraint}
\end{equation}
for a suitable choice of units.
This constraint forbids the origin of the classical Higgs branch to be included in the quantum moduli space of vacua.
In other words, the vevs of $q^I$ and $\widetilde{q}^I$ cannot be zero at any vacuum.
After turning on a non-zero $m$, as explained above, the D-term potential grows toward infinity of the classical Higgs branch.
Therefore, the vacuum would be again at the middle of the classical Higgs branch where, probably, ${\cal V}$ is positive and the supersymmetry is broken.
For the remaining cases $N\le k$, there is again no superpotential, and also no constraints like (\ref{constraint}) which forbids the origin of the classical Higgs branch.
Therefore, the intersection of the Coulomb branch and the Higgs branch would be allowed to be a vacuum state.
As conjectured in the case $m=0$, the intersection point would correspond to a non-trivial IR fixed point.
In the case $m\ne 0$, the fixed point would describe ${\cal N}=2$ pure Chern-Simons theory.
\vspace{5mm}
An interesting point of the analysis shown above is that the supersymmetry breaking occurs in the Higgs branch which is integrated out
when one obtains the low energy description in terms of Yang-Mill-Chern-Simons theory.
From this point of view, it is quite natural that the supersymmetry breaking is rather difficult to see in the latter description
since it occurs in a ``hidden'' part of the theory.
\vspace{5mm}
It should be noted that the behavior of the quantum D-term potential at infinity of the classical Higgs branch, that is,
growing toward infinity for the case $m\ne0$, is crucial for the above argument.
To realize this behavior, the real mass must be the axial mass.
A vector mass, say $\mu$, would give the behavior of the D-term potential
\begin{equation}
{\cal V} \sim V_D+|(\phi+\mu)q^I|^2+|(\phi+\mu)\widetilde{q}_I|^2.
\end{equation}
This may have flat directions near infinity, and therefore, the above argument does not lead us to the conclusion that the supersymmetry must be broken.
Indeed, this seems to be plausible.
The theory with a vector mass is reduced to ${\cal N}=2$ pure Yang-Mills theory, whose superpotential has runaway directions \cite{deBoer:1997kr}\cite{Aharony:1997bx}.
\vspace{1cm}
\section{Chern-Simons-matter theories of quiver type} \label{quiver type}
\vspace{5mm}
In this section, we consider the vacuum states of ${\cal N}=2$ Chern-Simons theory with the gauge group ${\rm SU}(N_1)\times{\rm SU}(N_2)$ and the level $(+k,-k)$
coupled to a number $n_b$ of pairs $(B^n,\widetilde{B}^n)$ $(n=1,\cdots,n_b)$ of bi-fundamental chiral multiplets in the representations $(N_1, \overline{N_2})$ and
$(\overline{N_1}, N_2)$, respectively.
The extension of the following arguments to the case with a general $(k_1,k_2)$ will be straightforward.
Without loss of generality, we assume $N_1\ge N_2$ and $k>0$.
As in the previous section, we add suitable Yang-Mills terms for the gauge fields, and replace the Chern-Simons terms with the matter chiral multiplets in
the representations
\begin{equation}
(N_1,1)^k\oplus (\overline{N_1},1)^k\oplus (1,N_2)^k\oplus (1,\overline{N_2})^k
\end{equation}
of the gauge group with axial masses specified below.
We denote them as $Q_1^I, \widetilde{Q}_1^I, Q_2^I$ and $\widetilde{Q}_2^I$, respectively.
The Lagrangian is
\begin{eqnarray}
L
&=& \int d^4\theta \Bigl[ Q_{1I}e^{V_1+m_1\bar{\theta}\theta}Q_1^I+\widetilde{Q}_1^Ie^{-V_1+m_1\bar{\theta}\theta}\widetilde{Q}_{1I}
+Q_{2I}e^{V_2-m_2\bar{\theta}\theta}Q_2^I+\widetilde{Q}_2^Ie^{-V_2-m_2\bar{\theta}\theta}\widetilde{Q}_{2I} \nonumber \\
& & \hspace*{1cm} +\mbox{tr}_2(B_ne^{V_1}B^ne^{-V_2})+\mbox{tr}_1(\widetilde{B}_ne^{V_2}\widetilde{B}^ne^{-V_1}) \Bigr] \nonumber \\
& & + \int d^2\theta \Bigl[ \mbox{tr}_1(W_1^\alpha W_{1\alpha}) + \mbox{tr}_2(W_2^\alpha W_{2\alpha}) + W_{\rm tree}(B, \widetilde{B}) \Bigr] + \mbox{h.c.}
\label{quiver}
\end{eqnarray}
where tr$_1$ and tr$_2$ are the traces for the fundamental representations of SU$(N_1)$ and SU$(N_2)$, respectively.
\vspace{5mm}
First, let us consider the case $n_b=1$.
The tree level superpotential $W_{\rm tree}$ would be of the form
\begin{equation}
W_{\rm tree} = c\,\mbox{tr}_1(B\widetilde{B}B\widetilde{B}).
\end{equation}
The quadratic term should be absent since we assume the bi-fundamental fields to be massless.
It was argued in \cite{Gaiotto:2007qi} that, in the context of the Chern-Simons-matter theory,
$c=0$ would be unstable and flow to a non-zero but finite value due to $1/k$ corrections.
We assume in the following that $c$ is non-zero and finite.
A similar theory can be realized as the worldvolume theory on D3-branes suspended between two NS5-branes and a $(1,k)$5-brane.
See Figure \ref{brane2}.
There are $N_1$ D3-brane segments between the left NS5-brane and the $(1,k)$5-brane, and $N_2$ D3-brane segments between the right NS5-brane and the
$(1,k)$5-brane.
The gauge group is ${\rm U}(N_1)\times{\rm U}(N_2)$.
It is always possible for $2N_2$ D3-brane segments in the above two D3-brane stacks to form longer segments suspended between two NS5-branes which can be
detached from the $(1,k)$5-brane.
The remaining $N_1-N_2$ D3-brane segments are still suspended between the left NS5-brane and the $(1,k)$5-brane.
As in the case for pure Chern-Simons theory, the supersymmetry is broken if $N_1-N_2>k$, according to the s-rule \cite{Hanany:1996ie}.
We expect that the presence of ${\rm U}(1)$ factors will not be important in the analysis of supersymmetry breaking in this section.
In terms of the scalar potential, an extra ${\rm U}(1)$ factor provides an additive positive contribution to the D-term potential.
If the supersymmetry is already broken in the theory without the ${\rm U}(1)$ factor, it is also the case in the theory with the ${\rm U}(1)$ factor.
In the following, we analyze the pattern of supersymmetry breaking in the theory (\ref{quiver}) which should be compatible with the pattern expected from the argument
based on the brane configuration in Figure \ref{brane2}.
\begin{figure}[tbp]
\begin{center}
\includegraphics{brane2.eps}
\caption{The brane configuration realizing ${\cal N}=2$ ${\rm U}(N_1)_k\times{\rm U}(N_2)_{-k}$ Chern-Simons theory coupled to 1 pair of bi-fundamental matters. }
\label{brane2}
\end{center}
\end{figure}
\vspace{5mm}
\subsection{Classical vacua}
\vspace{5mm}
Let us start with the analysis of the classical scalar potential ${\cal V}$ of (\ref{quiver}).
The explicit form of ${\cal V}$ is
\begin{eqnarray}
{\cal V}
&=& V_F+V_{D_1}+V_{D_2}+\Bigl| \phi_1b-b\phi_2 \Bigr|^2+\Bigl| \phi_2\widetilde{b}-\widetilde{b}\phi_1 \Bigr|^2 \nonumber \\
& & +\sum_{I=1}^k \Bigl( \bigl| (\phi_1+m_1)q_1^I \bigr|^2+\bigl| (\phi_1-m_1)\widetilde{q}_{1I} \bigr|^2 \Bigr) \nonumber \\
& & +\sum_{I=1}^k \Bigl( \bigl| (\phi_2-m_2)q^I_2 \bigr|^2+\bigl| (\phi_2+m_2)\widetilde{q}_{2I} \bigr|^2 \Bigr).
\end{eqnarray}
If $m_1=m_2=0$, then there exist flat directions consisting of several branches.
Consider the case with non-zero $m_1$ and $m_2$.
The condition $V_F=0$ implies
\begin{equation}
\widetilde{b}b = 0.
\end{equation}
The adjoint scalars $\phi_1$ and $\phi_2$ can be diagonalized by a suitable gauge transformation,
\begin{eqnarray}
\phi_1
&=& \left[
\begin{array}{ccccc}
+m_1\cdot I_{l_1} & & & & \\
& -m_1\cdot I_{l_2} & & & \\
& & +m_2\cdot I_{l_3} & & \\
& & & -m_2\cdot I_{l_4} & \\
& & & & \phi_1'
\end{array}
\right], \\
\phi_2
&=& \left[
\begin{array}{ccccc}
+m_1\cdot I_{l'_1} & & & & \\
& -m_1\cdot I_{l'_2} & & & \\
& & +m_2\cdot I_{l'_3} & & \\
& & & -m_2\cdot I_{l'_4} & \\
& & & & \phi_2'
\end{array}
\right],
\end{eqnarray}
where $\phi'_1$ and $\phi_2'$ are diagonal matrices whose eigenvalues are not $\pm m_1$ nor $\pm m_2$.
Then, the flat directions for the mass terms of the fundamental matters are parametrized as
\begin{equation}
q^I_1 = \left[
\begin{array}{c}
0 \\ q^I_{1,\rm flat} \\ 0 \\ 0 \\ 0
\end{array}
\right], \hspace{5mm} \widetilde{q}^I_{1} = \left[
\begin{array}{c}
\widetilde{q}^I_{1,\rm flat} \\ 0 \\ 0 \\ 0 \\ 0
\end{array}
\right], \hspace{5mm} q^I_2 = \left[
\begin{array}{c}
0 \\ 0 \\ q^I_{2,\rm flat} \\ 0 \\ 0
\end{array}
\right], \hspace{5mm} \widetilde{q}^I_2 = \left[
\begin{array}{c}
0 \\ 0 \\ 00 \\ \widetilde{q}^I_{2,\rm flat} \\ 0
\end{array}
\right],
\end{equation}
and the flat directions of the bi-fundamental matters are
\begin{equation}
b = \left[
\begin{array}{ccccc}
b_1 & & & & \\
& b_2 & & & \\
& & b_3 & & \\
& & & b_4 & \\
& & & & b_5
\end{array}
\right], \hspace{5mm} \widetilde{b} = \left[
\begin{array}{ccccc}
\widetilde{b}_1 & & & & \\
& \widetilde{b}_2 & & & \\
& & \widetilde{b}_3 & & \\
& & & \widetilde{b}_4 & \\
& & & & \widetilde{b}_5
\end{array}
\right]
\end{equation}
The condition $V_{D_2}=0$ then implies
\begin{equation}
\left[ \mbox{tr}_1 \left( b_1T^ab_1^\dag \right)-\mbox{tr}_1 \left( \widetilde{b}_1^\dag T^a\widetilde{b}_1 \right) \right]^2 = 0,
\end{equation}
where $T^a$ are generators of ${\rm su}(l'_1)\subset{\rm su}(N_2)$.
This can be written as
\begin{equation}
\left[ \mbox{tr}_1 \left( b_1T^ab_1^\dag \right) \right]^2+\left[ \mbox{tr}_1 \left( \widetilde{b}_1^\dag T^a\widetilde{b}_1 \right) \right]^2
-2\mbox{tr}_1(\widetilde{b}_1b_1b_1^\dag\widetilde{b}_1^\dag)+\frac 2{l'_1}|b_1|^2|\widetilde{b}_1|^2 = 0.
\end{equation}
Combined with the F-term condition $\widetilde{b}_1b_1=0$, this implies that either $b_1$ or $\widetilde{b}_1$ vanishes.
Similarly, it can be shown that for any $i=1,\cdots,5$ either $b_i$ or $\widetilde{b}_i$ vanishes.
The D-term condition further imposes constraints.
For example, suppose that $b_1$ and $\widetilde{b}_2$ are non-zero.
Let $T$ be a generator of ${\rm su}(l'_1+l'_2)\subset {\rm su}(N_2)$ such that
\begin{equation}
T = \mbox{diag}\left( +l'_2I_{l'_1}, -l'_1I_{l'_2},0,0,0 \right).
\end{equation}
Then the condition $V_{D_2}=0$ implies
\begin{equation}
l'_2|b_1|^2+l'_1|\widetilde{b}_2|^2 = 0,
\end{equation}
concluding that both $b_1$ and $\widetilde{b}_2$ vanish.
This means that if $b_1$ is non-zero, then $\widetilde{b}_2$ must be zero, and the converse is also true.
Similar arguments then imply that either $b$ or $\widetilde{b}$ vanish.
Therefore, the possible flat directions satisfy either $b=0$ or $\widetilde{b}=0$.
This is the enough information of the classical moduli space of vacua of the theory (\ref{quiver}) for the following discussion of the quantum vacua.
\vspace{5mm}
\subsection{Quantum vacua}
\vspace{5mm}
Let us consider the quantum effects.
We first set $m_1=m_2=0$.
Notice that SU$(N_1)$ gauge fields couple to $k+N_2$ pairs of fundamental chiral multiplets ${\cal Q}^A = (Q^I_1, B)$ and anti-fundamental chiral multiplets
$\widetilde{\cal Q}^A = (\widetilde{Q}^I_1, \widetilde{B})$, and SU$(N_2)$ gauge fields couple to $k+N_1$ such pairs.
The dynamics of the former gauge fields will induce a superpotential
\begin{equation}
W_{\rm np} \propto \Bigl[ Y_1\det(\widetilde{\cal Q}^A{\cal Q}^B) \Bigr]^{\frac1{N_2+k-N_1+1}}+\cdots,
\label{quiverNPW}
\end{equation}
if $N_1-1>N_2+k$ is satisfied, while the latter gauge fields will not induce such a superpotential since $N_2-1>N_1+k$ cannot be satisfied for the case $N_1\ge N_2$ which
we assumed.
There are other terms in $W_{\rm np}$ depending only on the adjoint scalars.
The superpotential depends on $Q^I_1$ and $\widetilde{Q}^I_1$ only through the first term in (\ref{quiverNPW}).
The F-term potential vanishes only if
\begin{equation}
\det(\widetilde{\cal Q}^A{\cal Q}^B) \to \infty \hspace{5mm} \mbox{or} \hspace{5mm} \mbox{rank}{\cal Q}^A, \mbox{rank}\widetilde{\cal Q}^A \le N_2.
\end{equation}
The latter possibility is not allowed since in this case $\det(\widetilde{\cal Q}^A{\cal Q}^B)=0$, and therefore, the scalar potential is infinite.
This then implies that the supersymmetry would be preserved only when there is a runaway direction along which $\det(\widetilde{\cal Q}^A{\cal Q}^B)$ diverges.
To find whether there really exists such a runaway direction, it is necessary to examine the D-term potential.
Consider first the region near infinity of the classical flat directions along $q^I_{1,2}$ and $\widetilde{q}^I_{1,2}$ directions.
Since those fields couple to $b$ and $\widetilde{b}$ only through the gauge fields, $b$ and $\widetilde{b}$ will decouple once $q^I_{1,2}$ and $\widetilde{q}^I_{1,2}$
have large vevs so that the corresponding gauge fields become massive and decouple.
Therefore, the D-term potential in these directions approaches the classical one near infinity along the classical flat directions.
This is also the case if the vevs of $b$ and $\widetilde{b}$ become large.
In this case, the gauge group ${\rm SU}(N_1)\times{\rm SU}(N_2)$ would be broken to ${\rm SU}(N_1-N_2-k)$, and all the other light fields are neutral under the remaining
gauge fields.
Therefore, no D-term potential will be induced below the energy scale set by the vevs.
If the vevs are large enough, the D-term potential may be approximated by the classical one.
Now we turn on small axial masses.
As was shown in the previous subsection, the possible flat directions in the presence of the masses satisfy $b=0$ or $\widetilde{b}=0$.
For such a vev, the non-perturbative F-term potential diverges since $\det(\widetilde{\cal Q}^A{\cal Q}^B)$ vanishes.
In other words, all the classical flat directions in the absence of the masses, which might be compatible with the runaway directions of the F-term potential, are lifted
by the masses.
As a result, the vacuum states must correspond to points in the middle of the classical flat directions, at which the F-term potential is non-zero.
This implies that the supersymmetry is completely broken.
This conclusion is valid for
\begin{equation}
N_1-N_2 > k+1,
\end{equation}
for which the non-perturbative superpotential (\ref{quiverNPW}) is induced.
This pattern of the supersymmetry breaking is expected to be the same when $m_1$ and $m_2$ become finite but large, suggesting that the analysis above explains the
supersymmetry breaking of a Chern-Simons-matter theory we would like to discuss.
It would be possible to argue that the supersymmetry is also broken for the case $N_1-N_2=k+1$, while it is preserved otherwise.
Note that this pattern is compatible with the one for the ${\rm U}(N_1)\times{\rm U}(N_2)$ theory discussed previously.
\vspace{5mm}
\subsection{More bi-fundamental matters}
\vspace{5mm}
Next, let us consider the case $n_b\ge 2$.
The theory with $n_b=2$ can be realized, up to ${\rm U}(1)$ factors, on a D3-brane segments in a brane configuration similar to Figure \ref{brane2}.
That is, the brane configuration is obtained from the one in Figure \ref{brane2} by compactifying the horizontal direction and identify two NS5-branes,
as in \cite{Aharony:2008ug}.
The brane configuration suggests that the supersymmetry is broken when $N_1-N_2>k$ based on the similar argument as for $n_b=1$.
For the cases $n_b>2$, there are no known brane configurations realizing such theories.
The argument based on the field theory (\ref{quiver}) is similar to the one in the previous subsection.
A non-perturbative superpotential is generated when
\begin{equation}
N_1-1>n_bN_2+k \hspace{5mm} \mbox{and} \hspace{5mm} N_2-1>n_bN_1+k.
\end{equation}
Under the assumption $N_1\ge N_2$, the second condition is not satisfied, and therefore, a non-perturbative superpotential will be induced only by the dynamics of the
SU$(N_1)$ gauge group.
Then, it is suggested that the supersymmetry would be broken completely if
\begin{equation}
N_1-N_2 > (n_b-1)N_2+k+1.
\end{equation}
For the case $n_b=2$, this condition is consistent with the one expected from the brane configuration, but certainly much weaker.
It should be noted that our argument typically provides a sufficient condition for the supersymmetry breaking.
Even though there is no superpotential, or even though the F-term potential can vanish for some field configurations, there is still a possibility that the D-term potential
would lift such would-be supersymmetric vacua.
To obtain a more stringent condition for the supersymmetry breaking, one has to investigate the detailed form of the quantum corrected D-term potential which is,
however, in general quite difficult for theories with only four supercharges.
A similar complication would appear when one adds to the theory (\ref{quiver}), or even to (\ref{SQCD}), a large number of (anti-)fundamental matters with {\it vector}
masses.
Since such matters would not affect the low energy properties of the theory, the conclusion on the possibility of supersymmetry breaking should not depend on the
presence of those matters.
However, if the number of matters in the (anti-)fundamental representation is too large, the non-perturbative superpotential would not be induced.
In such cases, the only possibility would be that the supersymmetry is broken by the D-term potential, which would be very difficult to show explicitly.
\vspace{2cm}
{\bf \Large Acknowledgements}
\vspace{5mm}
I would like to thank Soo-Jong Rey and Kimyeong Lee for valuable discussions.
This work was supported by the BK21 program of the Ministry of Education, Science and Technology,
National Science Foundation of Korea Grants 0429-20100161, R01-2008-000-10656-0,
2005-084-C00003, 2009-008-0372 and EU-FP Marie Curie Research
\& Training Network HPRN-CT-2006-035863 (2009-06318).
\vspace{1cm}
|
{
"timestamp": "2012-04-11T02:01:42",
"yymm": "1203",
"arxiv_id": "1203.2039",
"language": "en",
"url": "https://arxiv.org/abs/1203.2039"
}
|
\section{Number of defects}
We would like to thank G. Biroli, P. Calabrese, M. Fabrizio, R. Fazio, A. Gambassi, G. Mussardo, C. Kollath, G. Santoro, and in particular M. Marcuzzi for useful and enlightening discussions. A. S. would like to thank the Galileo Galilei Institute in Firenze for hospitality during the completion of this work.
|
{
"timestamp": "2012-08-28T02:01:05",
"yymm": "1203",
"arxiv_id": "1203.2108",
"language": "en",
"url": "https://arxiv.org/abs/1203.2108"
}
|
\section{Introduction}
In magnetic resonance imaging (MRI), the signal of interest $\rho$ represents the magnetization induced by resonance in the imaged tissues. In the standard setting, data acquired in MRI provide complete Fourier, or $\bm{k}$-space, measurements of this signal $\rho$. Accelerating the acquisition process, or equivalently increasing the achievable resolution for a fixed acquisition time, is of major interest for MRI applications. Recent approaches based on compressed sensing seek to reconstruct the signal from incomplete $\bm{k}$-space information, hence defining an ill-posed inverse problem. The ill-posed inverse problem is regularized by the introduction of sparsity priors, acknowledging the fact that many MRI signals are sparse in well-chosen bases; i.e., that their expansion contains only a small number of non-zero coefficients. Such compressed sensing techniques have been developed for static and dynamic imaging \cite{lustig07, jung07, gamper08, jung09, usman11}, parallel MRI \cite{liang09a, lustig10, otazo10}, MR spectroscopic imaging \cite{hu08, hu10, larson10}, and many other applications. Several algorithms have also been proposed to reconstruct MRI signals from under-sampled $\bm{k}$-space measurements (see, e.g., \cite{kim07, trzasko09, kern11}).
In the framework of compressed sensing, signals are usually measured through random matrices to ensure that any sparse signal can be recovered with overwhelming probability. The common approach in MRI consists in also exploiting the fact that the energy of MRI signals is usually concentrated at low spatial frequencies. Therefore a variable density $\bm{k}$-space random sampling where the under-sampling ratio increases at high frequencies is usually used \cite{lustig07}. This method, heuristic in nature, was shown to be very effective in enhancing the signal reconstruction quality when random distributions optimizing the associated point spread function are used. Other approaches that optimize the acquisition procedure have also been proposed: $\bm{k}$-space sampling optimization by Bayesian inference \cite{seeger10}; random $\bm{k}$-space convolution with Toeplitz matrices \cite{sebert08, liang09b, wang09}; encoding by projection onto random waveforms with Gaussian distributions \cite{haldar10}.
In the present work, we propose the use of a spread spectrum technique to accelerate single coil MRI acquisition in the framework of compressed sensing. We study this method, coined spread spectrum MRI or simply s$_2$MRI, theoretically, numerically via simulations, and empirically via real acquisitions. The essence of our strategy consists of pre-modulating the image by a linear chirp, which results from the application of quadratic phase profiles, and then performing random $\bm{k}$-space under-sampling. Images are then reconstructed with non-linear algorithms promoting signal sparsity. The enhancement of the signal reconstruction quality is linked to a decrease of coherence of the measurement system \cite{puy11a, wiaux09b}. In MRI, this type of modulation is known as phase scrambling. It can be obtained by using dedicated coils or by modifying radio frequency (RF) pulses. It has been used for various purposes, such as improving dynamic range \cite{maudsley88, wedeen88}, or reducing aliasing artifacts \cite{pipe95, ito08}, but never in a compressed sensing perspective.
Let us acknowledge that this spread spectrum technique was initially introduced by some of the authors for compressive sampling of pulse trains in \cite{naini09}. Its transfer to a setting encompassing analog signals and modulations was studied in the context of radio interferometry \cite{wiaux09b, wiaux09d, mcewen10}. The effectiveness of the method for MRI was briefly discussed in \cite{puy09, wiaux09c, puy11b}.
The paper is organized as follows. In Section \ref{sec:spread spectrum principle}, we explain the principle of the spread spectrum technique in a simplified analog setting. In Section \ref{sec:MR measurements}, we formulate the inverse problem for image reconstruction from under-sampled $\bm k$-space measurements in the presence of chirp modulation and compare the s$_2$MRI technique to the variable density sampling method on the basis of numerical simulations of MR acquisitions of a brain image. In Section \ref{sec:Real acquisitions}, we describe our implementation of the chirp modulation (on a $7$T scanner) and show the effectiveness of s$_2$MRI using numerical simulations in this precise setting and real acquisitions of phantom and \emph{in vivo} data. Finally, we conclude and discuss potential evolutions of the
technique in Section \ref{sec:Conclusion}.
\section{Spread spectrum principle}
\label{sec:spread spectrum principle}
\subsection{Compressed sensing}
\label{sub:compressed sensing}
The essence of the recent theory of compressed sensing is the merging of data acquisition and compression \cite{candes06a, candes06b, candes06c, donoho06, candes07, baraniuk07a, rauhut10, donoho09}. Beyond MRI, it is well-known that a large variety of natural signals are sparse or compressible in multi-scale bases, such as wavelet bases. In the compressed sensing theory, signals are usually expressed as $N$-dimensional vectors: $\bm{\rho}\in\mathbb{C}^{N}$. By definition, a signal is sparse in some orthonormal basis $\mathsf{\Psi}\in\mathbb{C}^{N\times N}$, called a sparsity basis, if its expansion contains only a small number $K\ll N$ of non-zero coefficients. More generally it is compressible if its expansion contains only a small number of significant coefficients. The decomposition of $\bm{\rho}$ in $\mathsf{\Psi}$ is denoted $\bm{\alpha} \in \mathbb{C}^{N}$, and it satisfies
\begin{equation}
\label{cs1}
\bm{\rho} = \mathsf{\Psi}\bm{\alpha}.
\end{equation}
The theory of compressed sensing demonstrates that a small number $M\ll N$ of linear and non-adaptative measurements $\bm{\nu}\in\mathbb{C}^{M}$ suffices for an accurate and stable reconstruction of the signal $\bm{\rho}$. These measurements may, for example, be obtained by projection onto $M$ randomly selected basis vectors of an orthonormal basis $\mathsf{\Phi}\in\mathbb{C}^{N\times N}$, called a sensing basis. This selection process can be modeled by a multiplication with a rectangular binary matrix $\mathsf{M}\in\mathbb{R}^{M\times N}$ that contains only one non-zero value on each line, at the index of the basis vector to be selected. To model a non-perfect sensing process, these measurements are assumed to be contaminated by independent and identically distributed noise $\bm{n} \in \mathbb{C}^{M}$. The measurement model thus satisfies
\begin{equation}
\label{cs2}
\bm{\nu}=\mathsf{\Theta}\bm{\alpha}+\bm{n}\textnormal{, with }\mathsf{\Theta}=\mathsf{M\Phi^{\star}\Psi}\in\mathbb{R}^{M\times N},
\end{equation}
where the matrix $\mathsf{\Theta}$ identifies the measurement matrix as seen from the sparsity basis ($\cdot^\star$ denotes the conjugate transpose operation).
To reconstruct the signal $\bm{\rho}$ from the measurements $\bm{\nu}$, the compressed sensing framework proposes, among other approaches, to solve the Basis Pursuit (${\rm BP}$) minimization problem \cite{candes06a, candes06b, candes06c, donoho06, candes07, baraniuk07a, donoho09, rauhut10}. This problem regularizes the originally ill-posed inverse problem related to (\ref{cs2}) with an explicit sparsity or compressibility prior on the signal. In the presence of noise, the ${\rm BP}$ problem is the minimization of the $\ell_{1}$ norm\footnote{$\vert\vert \bm{\alpha} \vert\vert_{1}=\sum_{i=1}^{N}\vert \alpha_{i}\vert$ where $\alpha_{i}$ is the $i$th entry of the vector $\bm{\alpha}$.} of $\bm{\alpha}$ under a constraint on the $\ell_{2}$ norm\footnote{$\vert\vert \bm{n} \vert\vert_{2}=(\sum_{i=1}^{M}\vert n_{i}\vert^{2})^{1/2}$ where $n_{i}$ is the $i$th entry of the vector $\bm{n}$.} of the residual noise $\bar{\bm{n}}= \bm{\nu}-\mathsf{\Theta}\bar{\bm{\alpha}}$:
\begin{equation}
\label{cs3}
\bm{\alpha}^* = \arg \min_{\bar{\bm{\alpha}}\in\mathbb{C}^{N}}\vert\vert\bar{\bm{\alpha}}\vert\vert_{1}\textnormal{ subject to }\vert\vert \bm{\nu}-\mathsf{\Theta}\bar{\bm{\alpha}}\vert\vert_{2}\leq\epsilon.
\end{equation}
The corresponding reconstructed signal is $\bm{\rho}^* = \mathsf{\Psi}\bm{\alpha}^*$.
In this setting, the compressed sensing theory shows that if the number of measurements satisfies
\begin{equation}
\label{cs5}
M \geq DNK\mu^{2}\left(\mathsf{\Phi}^{\star},\mathsf{\Psi}\right)\ln^{4}N,
\end{equation}
for a universal constant $D$, then $\bm{\alpha}$ is recovered accurately by solving the $\rm BP$ problem (\ref{cs3}) \cite{rauhut10}. In relation (\ref{cs5}), $\mu\left(\mathsf{\Phi}^{\star},\mathsf{\Psi}\right)$ is the mutual coherence between the sparsity and sensing bases. It is defined as the maximum projection, in absolute value, between the sparsity basis vectors $\bm{\varphi}_{i}$, $1 \leq i \leq N$, and sensing bases vectors $\bm{\psi}_{i'}$, $1 \leq i' \leq N$ \cite{candes07, rauhut10}:
\begin{equation}
\label{cs6}
N^{-1/2} \leq \mu\left(\mathsf{\Phi}^{\star},\mathsf{\Psi}\right) = \max_{1\leq i,i'\leq N}\vert \bm{\varphi}_{i} \bm{\cdot} \bm{\psi}_{i'} \vert \leq 1.
\end{equation}
\begin{figure}
\centering
\includegraphics[width=7cm, height=10cm]{fig1.pdf}
\caption{\label{fig:ssp principle} Spread spectrum principle. A signal $\bm{\rho}$ (top left panel) is modulated by a linear chirp (top right panel). In $\bm{k}$-space, the modulation amounts to the convolution of the Fourier transform of the signal (middle left panel) with that of the chirp (middle right panel). The spectrum of the resulting signal (bottom panel) spreads out in $\bm{k}$-space.}
\vspace{-1mm}
\end{figure}
The mutual coherence $\mu\left(\mathsf{\Phi}^{\star},\mathsf{\Psi}\right)$ plays a crucial role in relation (\ref{cs5}). Indeed, the number of measurements $M^*$ needed to reconstruct $K$-sparse signals increases quadratically with its value: $M^*\propto\mu^{2}(\mathsf{\Phi}^{\star},\mathsf{\Psi})$. In the worst case where $\mu\left(\mathsf{\Phi}^{\star},\mathsf{\Psi}\right) = 1$, $M^*$ is of the order of the signal dimension $N$ and under-sampling is impossible. However, when the mutual coherence is at its minimum, $N^{-1/2}$, $M^*$ is reduced to the order of the sparsity level $K$. This result can intuitively be explained with a simple consideration on the spread of the energy of the sparsity basis vectors in the measurement domain $\mathsf{\Phi}$. In an incoherent orthonormal system, the absolute value of the scalar product between the sensing basis vector $\bm{\varphi}_{i}$ and the sparsity basis vector $\bm{\psi}_{i'}$ is small for all pairs of indices $1\leq i, i' \leq N$. As $\| \mathsf{\Phi}^{\star} \bm{\psi}_{i'} \|_2 = \| \bm{\psi}_{i'} \|_2$, the energy of the sparsity basis vector $\bm{\psi}_{i'}$ spreads equally over the sensing basis vectors $\bm{\varphi}_{i}$. Consequently, whatever index $i$ is selected to perform a measurement, one always gets information concerning all the sparsity basis vectors describing the original signal. The number of measurements needed for accurate recovery thus decreases.
\subsection{Spread spectrum and coherence reduction}
\label{sub:spread spectrum and coherence}
\begin{table}
\caption{\label{tab:table Nmu} Values of $N_{\rm c} \, \mu_{\rm c}^2$ at different chirp rates $\bar{w}$}
\centering
\renewcommand{\arraystretch}{1.05}
\begin{tabular}{c|ccc}
\hline
$N_{\rm c} \, \mu_{\rm c}^2$ & Dirac & Haar & Fourier \\
\hline\hline
$\bar{w} = 0$ & $1.00$ & $256$ & 256 \\
$\bar{w} = 0.1$ & $2.27$ & $43.5$ & 15.5 \\
$\bar{w} = 0.3$ & $2.97$ & $25.9$ & 6.11 \\
$\bar{w} = 0.5$ & $3.29$ & $22.4$ & 4.17 \\
\hline
\end{tabular}
\end{table}
The above considerations suggest that to reduce to the number of measurements needed for accurate recovery of MRI signals, one can try to modify the MRI acquisition procedure in such a way that the energy of the sparsity basis vectors spreads all over the $\bm{k}$-space. Following this idea, the s$_2$MRI technique introduces a linear chirp modulation of the signal of interest before random selection of $\bm{k}$-space coefficients (see Fig. \ref{fig:ssp principle}). Indeed, this modulation conserves the energy of the input signal and corresponds to a convolution that generically spreads the spectrum.
To study theoretically the proposed acquisition scheme, we consider a simplified analog setting where the one-dimensional signal of interest is denoted by a complex-valued function $\rho(x)$ of the position $x \in \mathbb{R}$. This signal is limited on a finite field of view $L$. For the sake of simplicity of the following theoretical result only, we assume that the energy of this signal beyond the spatial frequency $B$ is negligible\footnote{In section \ref{sec:MR measurements}, we will introduce a setting that properly takes into account the fact that, in general, MRI signals are not band limited.}. We thus sample this signal on a discrete uniform grid of $N = 2LB$ points and represent the resulting discrete signal by a vector $\bm{\rho} \in \mathbb{C}^N$. The vector $\bm{\rho}$ is assumed to be $K$-sparse in an orthonormal basis $\mathsf{\Psi} \in \mathbb{C}^{N \times N}$. We also consider a one-dimensional linear chirp, with chirp rate $w \in \mathbb{R}$, that reads as a complex-valued function $c(x) = {\rm e}^{ {\rm i} \pi w x^2 }$. On the field of view $L$, this linear chirp is approximately band limited at its maximum instantaneous frequency $\vert w \vert L/2$. This band limit can be parametrized in terms of a discrete chirp rate $\bar{w} = w L^2/N$ and thus $\vert w \vert L/2 = \vert \bar{w} \vert B$.
In this setting, the s$_2$MRI measurement model is given by equation (\ref{cs2}) with
\begin{eqnarray}
\mathsf{\Phi}^{\star} = \mathsf{F^{\star} CU} \in \mathbb{C}^{N_{\rm c} \times N}.
\end{eqnarray}
In the above equation, the matrix $\mathsf{U}$ represents an up-sampling operator needed to avoid aliasing of the modulated signal due to a lack of sampling resolution in a digital description of the originally analog problem. Indeed, the convolution in Fourier space induced by the analog chirp modulation implies that the band limit of the modulated signal is the sum of the individual band limits of the original signal and of the chirp $c$. Therefore, an up-sampled grid with at least $N_{\rm c}=(1+\vert\bar{w}\vert)N$ points needs to be considered and the modulated signal is correctly obtained by applying the chirp modulation on the signal after up-sampling on the $N_{\rm c}$ points grid. The up-sampling operator $\mathsf{U}$, implemented in Fourier space by zero padding, is thus of size $\mathbb{C}^{N_{c}\times N}$ and satisfies $\mathsf{U}^*\mathsf{U} = \mathsf{I} \in \mathbb{C}^{N \times N}$. The matrix $\mathsf{C} \in \mathbb{C}^{N_{c} \times N_{c}}$ is the diagonal matrix implementing the chirp modulation on the up-sampled grid and the matrix $\mathsf{F} = \left\{\bm{f}_i\right\}_{1 \leq i \leq N_{\rm c}}$ stands for the discrete Fourier basis on the same grid.
The s$_2$MRI sensing matrix $\mathsf{\Phi}$ is not orthogonal because of the presence of the matrix $\mathsf{U}$. Consequently, the recovery condition (\ref{cs5}) does not strictly hold. However, one can obtain a similar recovery condition \cite{puy11a}. In particular, if
\begin{eqnarray}
\label{eq:analog recovery}
M \geq D N_{\rm c} \, \mu_{\rm c}^2(\mathsf{\Phi}^{\star}, \mathsf{\Psi}) K \log^4(N),
\end{eqnarray}
for some universal constant $D$, then the vector $\bm{\rho}$ is accurately recovered with high probability by solving the $\rm BP$ problem (\ref{cs3}). This result shows that the number of measurements $M$ needed to reconstruct $K$-sparse signals is proportional to the product $N_{\rm c} \, \mu_{\rm c}^2(\mathsf{\Phi}^{\star},\mathsf{\Psi})$ rather than to $\mu_{\rm c}^2(\mathsf{\Phi}^{\star},\mathsf{\Psi})$ only. The s$_2$MRI technique is efficient only if this product decreases with the chirp rate $\bar{w}$. In other words, the number of measurements needed for accurate recovery of sparse signals decreases only for sparsity basis $\mathsf{\Psi}$ for which the mutual coherence $\mu_{\rm c}^2(\mathsf{\Phi}^{\star}, \mathsf{\Psi})$ decreases faster than $N_{\rm c}$ increases.
\subsection{Illustration}
\label{sub:illustration}
\begin{figure}
\centering
\includegraphics[width=9.2cm, keepaspectratio]{fig2.pdf}
\caption{\label{fig:theory ssp} Probability of recovery of the signal $\bm{\rho}$ as a function of the number of measurements $M$ for $\bar{w}=0$ (dotted black curve), $\bar{w}=0.1$ (continuous blue curve), $\bar{w}=0.3$ (continuous red curve), $\bar{w}=0.5$ (continuous black curve), and three sparsity bases: the Dirac basis (left); the Haar wavelet basis (middle); the Fourier basis (right).}
\end{figure}
We illustrate here the effect of the chirp modulation on the number of measurements needed for accurate recovery of $K$-sparse signals.
We consider a $1$-dimensional complex signal $\bm{\rho}$ of size $N = 256$ corresponding to one line of an MRI brain image. This signal is decomposed into three different sparsity bases $\mathsf{\Psi}$ and hard-thresholded at a fixed sparsity $K = 25$. The sparsity bases considered are the Dirac basis, the Haar wavelet basis and the Fourier basis. This signal is then probed according to relation (\ref{cs2}) with $\mathsf{\Phi^{\star}}=\mathsf{F^{\star}CU}$ and $\bar{w} \in \{0, 0.1, 0.3, 0.5\}$, and reconstructed from different numbers of measurements $M$ by solving the $\rm{BP}$ problem (\ref{cs3}). Each time, the probability of recovery\footnote{Perfect recovery is considered to occur if the $\ell_2$-norm between the original signal $\bm{\rho}$ and the reconstructed signal $\bm{\rho}^\star$ satisfies: $\|\bm{\rho}-\bm{\rho}^\star\|_2\leq10^{-3}\|\bm{\rho}\|_2$} of the signal is computed over $1000$ simulations. In this experiment, no noise is added to the measurements $\bm{y}$, and the indices $i$ of the selected Fourier basis vectors $\bm{f_i}$ are chosen uniformly at random from $\{0, \ldots, N_{\rm c}-1\}$.
The ${\rm BP}$ reconstructions in the presence of chirp modulation with chirp rate $\bar{w}$ are denoted by ${\rm BP}\bar{w}$. For each sparsity basis and chirp rate considered, the curves of the probability of recovery as a function of the number of measurements $M$ are reported in Fig. \ref{fig:theory ssp}. The corresponding values of the product $N_{\rm c} \, \mu_{\rm c}^2(\mathsf{\Phi}^{\star}, \mathsf{\Psi})$ are reported in Table \ref{tab:table Nmu}.
For the Dirac basis, the number of measurements needed to reach a probability of recovery of $1$ slightly increases with the chirp rate, as suggested by the values of $N_{\rm c} \, \mu_{\rm c}^2(\mathsf{\Phi}^{\star}, \mathsf{\Psi})$ in Table \ref{tab:table Nmu} and relation (\ref{eq:analog recovery}). On the contrary, for the Haar and Fourier bases, the values of $N_{\rm c} \, \mu_{\rm c}^2(\mathsf{\Phi}^{\star}, \mathsf{\Psi})$ in the table predict a drastic improvement of the results with an increase of the chirp rate. This prediction is confirmed by the results presented in Fig. \ref{fig:theory ssp}. Note that for the Haar and Fourier bases, the value of the product $N_{\rm c} \, \mu_{\rm c}^2(\mathsf{\Phi}^{\star}, \mathsf{\Psi})$ decreases much less between $\bar{w} = 0.3$ and $\bar{w} = 0.5$ than between $\bar{w} = 0.1$ and $\bar{w} = 0.3$, suggesting a smaller improvement in the number of measurements needed for accurate recovery. This is also in line with the curves of the probability of recovery in Fig. \ref{fig:theory ssp}. In summary, as predicted by the theory, the effect of the s$_2$MRI technique depends on the sparsity basis. Note that the decrease of the performance for the Dirac basis (optimally incoherent only at $\bar{w} = 0$) is negligible compared to the improvement obtained for the two other bases. Note also that MRI signals are usually sparse in wavelet bases \cite{lustig07}. The results obtained with the Haar wavelet basis therefore suggest strong efficiency of the technique in MRI.
One can wonder if the proposed encoding scheme can result in shift-variant reconstruction quality. Indeed, the phase variation at the center of the chirp is less rapid than at the limit of the field of view. Let us consider the case of a signal sparse in the Haar wavelet basis. This signal can, roughly, be separated in large scale and fine scale sparsity basis vectors. The region where the chirp is oscillating slowly is limited to a small part of the field of view. Consequently, large scale sparsity basis vectors are necessarily affected by the chirp modulation as they have a wide support in signal space. The energy of these basis vectors is thus spread in $\bm{k}$-space thus improving the reconstruction quality of the low scale structures of the signal. On the other hand, the fine scale sparsity basis vectors at the center of the field of view remains not significantly modulated. However, these vectors are already incoherent with the Fourier basis as their energy naturally spreads out in $\bm{k}$-space. The fine scale structure of the signal are well recovered in absence and presence of chirp modulation. In summary, almost no shift-variant reconstruction quality is to be expected.
Let us acknowledge that the idea of convolving the $\bm k$-space to optimize the acquisition procedure in the context of compressed sensing can also be found in \cite{sebert08, liang09b, wang09}. In these works, the $\bm{k}$-space is convolved by a random Toeplitz matrices. We should also note that the spread spectrum technique can be related to the random convolution approach where the signal is convolved by a random sequence and under-sampled in \emph{real space} \cite{romberg09}. In our case, convolution and under-sampling occur in \emph{$\bm k$-space}. Finally, let us mention that in a discrete setting, i.e., in the absence of band limit extension, replacing the linear chirp modulation by a random modulation leads to a universal encoding strategy, i.e., the reconstruction quality does not depend on the sparsity basis \cite{puy11a}. The s$_2$MRI technique tries to emulate this universal encoding strategy. A universal compressed sensing strategy might as well be obtained by projecting the signal $\bm{\rho}$ onto random waveforms with Gaussian distributions \cite{candes06c, donoho06}. In the context of MRI, an encoding strategy based on this sensing scheme was recently proposed in \cite{haldar10}.
\section{Spread spectrum MRI (s$_2$MRI)}
\label{sec:MR measurements}
\begin{figure}
\centering
\includegraphics[width=8.8cm, keepaspectratio]{fig3.pdf}
\caption{\label{fig:original image}Original brain image at $1\,\rm{mm}$ of resolution. The first panel represents its absolute value. The second panel represents its phase mapped between $-\pi$ and $\pi$. The last two panels represent the logarithm of the amplitude of its Fourier transform before (third panel) and after (fourth panel) chirp modulation with chirp rate $\bar{w} = 0.3$. Dark and light regions respectively indicate low and high intensities.}
\end{figure}
\subsection{Quadratic phase profiles}
\label{sub:Quadratic phase profiles}
The spread spectrum principle was explained in the previous section in a simplified analog framework. Here, we move to a setting encompassing realistic analog MR signals which are limited in field of view and are consequently \emph{not} band limited.
Standard MR measurements take the form of $\bm{k}$-space coefficients, with $\bm{k}=(k_x, k_y, k_z)$, of the original three-dimensional image probed representing the tissue magnetization as a function of the position inside a given field of view. This image is complex-valued due to magnetic field inhomogeneities \cite{haake99}. We consequently denote it by a complex-valued function $\rho(\bm{x})$ of the position $\bm{x}\in\mathbb{R}^{3}$, with components $(x,y,z)$ in image space. $(x,y)$ conventionally corresponds to the phase encoding directions, and $z$ corresponds to the readout direction. The field of view is $L=L_x\times L_y\times L_z$. We also consider quadratic phase profiles represented by a linear chirp in the phase encoding directions
\begin{equation}
\label{mri0}
c\left(\bm{x}\right) = {\rm e}^{ {\rm i} \pi \left( w_x x^2 + w_y y^2\right) },
\end{equation}
with chirp rate $(w_x, w_y) \in \mathbb{R}^2$. This linear chirp is characterized by an instantaneous frequency $(w_xx, w_yy)$ at position $\bm{x}$. On the finite field of view $L$, it is therefore approximately a band-limited function with approximate band-limits $\left(\vert w_x\vert L_x/2,\vert w_y\vert L_y/2\right)$ in the phase encoding directions.
In this setting, MR measurements at spatial frequency $\bm{k}\in\mathbb{R}^{3}$ take the general form
\begin{equation}
\label{mri1}
\nu\left(\bm{k}\right) = \int_{\mathbb{R}^{3}} \rho\left(\bm{x}\right) c \left(\bm{x}\right) {\rm e}^{-2{\rm i}\pi \bm{k}\bm{\cdot}\bm{x}} \: {\rm d}^{3}\bm{x}.
\end{equation}
In other words, the measurement $\nu$ corresponds to the coefficient at spatial frequency $\bm{k}$ of a signal obtained as the product of the original image $\rho(\bm{x})$ with the linear chirp modulation $c(\bm{x})$. In the absence of modulation ($w_x=w_y=0$), the measurements simply reduce to standard $\bm{k}$-space measurements. In the presence of quadratic phase profiles, the modulation amounts to the convolution of the Fourier transform of the chirp with that of the original image.
\subsection{Under-sampling in $\bm{k}$-space and the inverse problem}
\begin{figure}
\centering
\includegraphics[width=8.8cm, keepaspectratio]{fig4.pdf}
\caption{\label{fig:simulated reconstruction 2D} Simulated reconstructions for an under-sampling of $20$ percent of $N$ and an input $\rm snr$ of $32$. The top row shows the reconstructions for the variable density sampling (left panel) and the s$_2$MRI (right panel) techniques, respectively. The bottom row shows the error images (difference between the original image and the reconstructions) in the absence (left panel) and presence (right panel) of chirp modulation. For a better visualization, the error images were scaled by a factor of $6$. The colormap for the error images goes from white to black, indicating low and high errors, respectively.}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=16.5cm]{fig5.pdf}
\caption{\label{fig:curves error 2D} Relative reconstruction errors as functions of the input ${\rm snr}$ for the variable density sampling technique (dashed blue curve) and the s$_2$MRI technique with $\bar{w}=0.3$ (dot-dashed red curve) and $\bar{w} = \bar{w}_{\rm opt}$ (continuous black curve). The first to sixth panels show the curves for coverages of $5$, $10$, $15$, $20$, $25$ and $50$ percent of $N$ respectively. All curves represent the mean relative error over $30$ simulations, and the vertical lines represent the error at $1$ standard deviation.}
\vspace{-3mm}
\end{figure*}
Let us assume that we want to probe the signal $\rho$ at a resolution corresponding to a band-limit $B = (B_x, B_y, B_z)$. Note that it does not imply that the signal is band-limited and some energy may remain beyond $B$. At this resolution, the signal of interest is discretized on a discrete uniform grid of $N=N_x\times N_y\times N_z$ spatial frequencies $\bm{k}_{i}\in\mathbb{R}^{3}$, $1\leq i\leq N$, in $\bm{k}$-space, with $N_x=2L_xB_x$, $N_y=2L_yB_y$, and $N_z = 2L_zB_z$. In real space, this discretized signal may equivalently be described by its coefficients on a discrete uniform grid of $N$ points $\bm{x}_{i}$ with $1\leq i\leq N$. On this discrete grid, the linear chirp can be parametrized in terms of discrete chirp rates $(\bar{w}_x, \bar{w}_y)=\left(w_xL_x^2/N_x, w_yL_y^2/N_y\right)$, thus exhibiting approximate band-limits $(\vert \bar{w}_x \vert B_x, \vert \bar{w}_y \vert B_y, 0)$ on the finite field of view $L$.
We assume that the spatial frequencies probed span the $\bm{k}$-space up to the band-limits $B$. We also assume that these frequencies belong to the discrete grid of points $\bm{k}_{i}$, so that we can avoid any re-gridding operation \cite{haake99}. Due to the linear chirp modulation, the measurements can contain significant energy from $\bm{k}$-space coefficients beyond the band-limits $B$. Considering the estimated band-limits of the linear chirp, we choose to reconstruct the original signal $\rho$ on a high resolution grid of $N_{\rm c}=N_x(1+\vert\bar{w}_x\vert)\times N_y(1+\vert\bar{w}_y\vert)\times N_z$ points in order to prevent any aliasing in the reconstruction algorithm from these high spatial frequencies. The sampled signal on this high resolution grid is denoted by a vector $\bm{\rho} \in \mathbb{C}^{N_{\rm c}}$. The reconstructed signal is subsequently down-sampled at the desired resolution by keeping only its spatial frequencies belonging to the previously defined grid of size $N$.
The $\bm{k}$-space coverage provided by the $M$ spatial frequencies probed $\bm{k}_b$, with $1\leq b\leq M$, can be represented by a binary mask in $\bm{k}$-space, equal to $1$ for each spatial frequency probed and $0$ otherwise. The measurements may be denoted by a vector of $M$ $\bm{k}$-space coefficients $\bm{\nu}=\{\nu(\bm{k}_b)\}_{1\leq b\leq M}\in\mathbb{C}^{M}$, possibly affected by noise, which is denoted by the vector $\bm{n}=\{n_b\}_{1\leq b\leq M}\in\mathbb{C}^{M}$. In this setting, we consider an incomplete $\bm{k}$-space coverage ($M<N$), in order to accelerate the acquisition time in comparison with a complete $\bm{k}$-space coverage. In order to satisfy standard MRI constraints, an arbitrary under-sampling can be considered in the phase encoding directions while all spatial frequencies in the readout direction $k_z$ are probed. This form of under-sampling can directly be expressed in terms of an acceleration of the acquisition. Thus, the s$_2$MRI measurement model satisfies
\begin{equation}
\label{mri2}
\bm{\nu}=\mathsf{MF^{\star}CU}\bm{\rho}+\bm{n},
\end{equation}
where the matrix $\mathsf{MF^{\star}CU}\in\mathbb{C}^{M\times N_{\rm c}}$ encodes the complete linear relation between the signal and the measurements. The rectangular matrix $\mathsf{U}\in\mathbb{C}^{N_{\rm u}\times N_{\rm c}}$ represents an up-sampling operation, implemented in $\bm k$-space space by zero padding. This operation is needed to avoid any aliasing of the modulated signal due to a lack of sampling resolution in a discrete description of the originally continuous problem. The modulated signal is correctly obtained by applying the chirp modulation to the signal after up-sampling on the grid of $N_{\rm u} = N_x(1+2\vert\bar{w}_x\vert)\times N_y(1+2\vert\bar{w}_y\vert)\times N_z$ points by zero padding in the $k_x$ and $k_y$ directions. The matrix $\mathsf{C}\in\mathbb{C}^{N_{\rm u}\times N_{\rm u}}$ is the diagonal matrix implementing the chirp modulation on the up-sampled grid. The unitary matrix $\mathsf{F}\in\mathbb{C}^{N_{\rm u}\times N_{\rm u}}$ stands for the discrete Fourier basis on this high resolution grid. The matrix $\mathsf{M}\in\mathbb{R}^{M\times N_{\rm u}}$ is the rectangular binary matrix implementing the mask. It contains only one non-zero value on each line, at the index of the $\bm k$-space coefficient corresponding to each of the spatial frequencies probed.
Regarding the reconstruction of the signal $\bm{\rho}$, (\ref{mri2}) represents the measurement constraint. We take a statistical point of view and consider independent Gaussian noise on each measurement. Considering a candidate reconstruction $\bar{\bm{\rho}}$, the residual noise reads as $\bar{\bm{n}}=\bm{\nu}-\mathsf{MF^{\star}CU}\bar{\bm{\rho}}$. The noise level estimator, defined as twice the negative logarithm of the likelihood associated with $\bar{\bm{\rho}}$, reads as
\begin{equation}
\label{mri3}
\chi^{2}\left(\bar{\bm{\rho}},\bm{\nu}\right)=\sum_{b=1}^{M}\frac{\vert\bar{n}_{b}\vert^2}{[\sigma^{(n_b)}]^2},
\end{equation}
where the symbol $\vert\cdot\vert$ stands for the complex norm and $\sigma^{(n_b)}$ for the standard deviation of each of the real and imaginary parts of the noise component $n_b$. This noise level estimator follows a chi-square distribution with $2M$ degrees of freedom. Typically, this estimator should be minimized by a good candidate for the reconstruction. As explained in \cite{candes06b}, the measurement constraint on the reconstruction may be defined as a bound $\chi^{2}(\bar{\bm{\rho}},\bm{\nu})\leq\epsilon^{2}$, with $\epsilon^{2}$ corresponding to some large percentile of the chi-square distribution. In the remainder of the paper, we choose the 99$\rm th$ percentile.
\subsection{Variable density sampling}
\label{sub:vds}
As discussed in \cite{lustig07}, the method selecting the spatial frequencies $\bm{k}_b$, with $1 \leq b \leq M$, is critical to achieve good reconstruction. Their approach is based on the use of a variable density sampling method. The distributions, which are empirically identified to provide optimized reconstructions, are defined by a power law decaying function $f (\bm{k}) = (1- \vert \bm{k}\vert/\vert\bm{k}_{m}\vert)^p + \beta$ for some power $p > 0$ and real constant $\beta \in [-1, 1]$ ($\vert \bm{k} \vert^2 = k_x^2+k_y^2$ and $\bm{k}_{m}$ is the highest spatial frequency of the $\bm{k}$-space domain to probe). In the present work, the spatial frequencies $\bm{k}_b$ are selected using the method\footnote{Toolbox available at \url{http://www.stanford.edu/~mlustig/SparseMRI.html}} of \cite{lustig07}: the values $f(\bm{k}_i)$ are first thresholded to restrict them to the interval $[0, 1]$; $N$ independent binary random variables $\delta \left(\bm{k}_i\right)$ taking value $1$ with probability $f \left(\bm{k}_i\right)$ are then generated; and finally the mask $\mathsf{M}$ selecting the $\bm{k}$-space coefficients for which $\delta \left(\bm{k}_i\right) = 1$ is created.
Let us highlight the importance of the value $\beta$ on the actual shape of the variable density profile. For a fixed power $p$, this constant is computed beforehand in order to ensure that the number of measurements is, on average, equal to its target value $M$. Given a fixed number $M$ of measurements, we denote $p_M$ the value of $p$ for which $\beta = 0$. For a small value of $p$ ($p<p_M$), one would intuitively predict that the measurements spread all over the $\bm{k}$-space domain with the density of points higher at the center of $\bm{k}$-space. In fact, for $p<p_M$, one has $\beta<0$ and a whole $\bm{k}$-space region at high frequency remains unprobed. On the contrary, we have $\beta>0$ for $p>p_M$ and the $\bm{k}$-space is probed with a non-zero probability at the edges. Consequently, $p=p_M$ is the first power that ensures that the entire $\bm{k}$-space domain is probed with a non-zero probability. Also note that for $p>p_M$, the center of the $\bm{k}$-space is fully sampled, as $\beta>0$, and that the size of the fully sampled region increases when $p$ increases.
In practice, the choice of the power $p$ of the variable density profile is difficult and should be adapted with the number of measurements in order to obtain the best reconstruction qualities. In the absence of chirp modulation, we performed reconstructions for several values of $p$ and noticed that $p=p_M$ leads to the best performance\footnote{In practice, $p_M$ is determined by an iterative process. Starting from $p = 0$, we increase its value by $0.5$ until we obtain $\beta \geq 0$ for the number of measurements considered.}. In the presence of a linear chirp modulation, the power spectrum of the original signal is spread and flattened. However, in the range of the chirp rates studied, the power spectrum of the modulated signal remains peaked at the origin (Fig. \ref{fig:original image}). We therefore also apply a variable density sampling and choose the power $p$ to be $p_M$. Empirically, this value also leads to the best reconstruction qualities. Note that choosing $p = p_M$ seems reasonable, as, given the spread of the information in the phase encoding directions, one wants to distribute the measurements over the entire sampling region of size $B$ and also limit the size of the fully sampled region at the $\bm{k}$-space center.
Let us acknowledge that recent theoretical results obtained in \cite{puy11c} support the choice of such profiles in MRI.
\subsection{Numerical simulations}
\label{sub:simulations}
\subsubsection{Simulation protocol}
To test the proposed technique, a real brain image is acquired on a $7\rm{T}$ short bore, actively shielded, MR scanner (Siemens, Erlangen, Germany). The subjects provided written informed consent prior to the imaging session, according to the guidelines of the local ethics committee. The parameters of the acquisition are as follows: $L_x \times L_y \times L_z = 224 \times 168 \times 208 \, {\rm mm}^3$ with a resolution of $0.5 \times 0.5 \times 4 \, {\rm mm}^3$. The matrix size is thus $N_x \times N_y \times N_z = 448 \times 336 \times 52$. The standard clinical ${\rm MPRAGE}$ sequence is used with echo time $\rm{TE} = 3.53\,\rm{ms}$, inversion time $\rm{TI} = 1.3 \,\rm{s}$, repetition time $\rm{TR} = 3.5\,\rm{s}$, and bandwidth $\rm{BW} = 200\,\rm{Hz}$. Note that, for the sake of simplicity, we restrict our analyses to one two-dimensional $z$-slice of the original three-dimensional acquisition with an under-sampling in both phase encoding directions $k_x$ and $k_y$. Also note that the image used is complex-valued.
In order to model an analog acquisition scheme, the original image at a resolution of $0.5 \, \rm{mm}$ is used to compute the measurements but the reconstruction is performed at a resolution of $1 \, \rm{mm}$. The reconstructed images are compared to the image obtained with a full acquisition at $1 \, \rm{mm}$ of resolution (Fig. \ref{fig:original image}).
The parameters of our analyses are as follows. Firstly, acquisitions are considered for various numbers $M$ of complex measurements corresponding to coverages of $5$, $10$, $15$, $20$, $25$ and $50$ percent of $N$. Secondly, instrumental noise is also added to the measurements as independent and identically distributed zero-mean Gaussian noise. The corresponding standard deviation $\sigma$ is identical for all the frequencies probed and we consider values of input ${\rm snr}$\footnote{The ${\rm snr}$ is defined as the ratio between the mean value of the complex magnitude of the original signal and the standard deviation of the noise $\sigma$.} of $2^j$, with $1\leq j \leq 6$. Thirdly, the chirp modulation studied has the same chirp rate $\bar{w}$ in both phase encoding directions, i.e., $\bar{w}=\bar{w}_x=\bar{w}_y$, with values in the range $[0, 0.5]$. Fourthly, the signals are reconstructed by solving the ${\rm BP}$ problem where the Total Variation (${\rm TV}$) norm of the signal $\bm \rho$ is substituted for the $\ell_1$ norm\footnote{The ${\rm TV}$ norm of a signal is defined as the $\ell_{1}$ norm of the magnitude of its gradient \cite{candes06a, rudin92}. \ Note that the recovery condition (\ref{eq:analog recovery}) does not hold with this norm. However, one can notice that the ${\rm TV}$ norm of a signal $\bm{\rho}$ is very similar to the $\ell_1$ norm of its decomposition in the Haar wavelet basis. In the light of the preliminary results of Section \ref{sub:illustration}, one can thus hope to obtain an improvement of the reconstruction quality in presence of chirp modulation.}. This problem is solved thanks the Douglas Rachford algorithm \cite{combettes11, fadili09}. Note that the ${\rm TV}$ norm in combination with wavelet sparsity basis is, for example, used for reconstructing MRI signals from under-sampled $\bm{k}$-space in \cite{lustig07, liang09a}, or \cite{kern11}. For our problem, we tested several multiscale representations such as Daubechies wavelets, steerable wavelets, or curvelets, but the best reconstructions were obtained with the ${\rm TV}$ norm. Finally, for each value of $M$ and input ${\rm snr}$ considered, $30$ simulations are generated with independent noise and mask realizations, and the relative reconstruction errors $\|\bm{\rho}-\bm{\rho}^\star\|_2/\|\bm{\rho}\|_2$ are computed for the variable density sampling and s$_2$MRI techniques. For each value of $M$ and input ${\rm snr}$, the discrete chirp rate $\bar{w}_{\rm opt}$ that gives the smallest relative error on average over the $30$ simulations is recorded\footnote{s$_2$MRI toolbox available at \url{http://lts2www.epfl.ch/people/gilles}.}.
\subsubsection{Simulation results}
The magnitudes of the reconstructed images obtained with the s$_2$MRI and the variable density sampling techniques for an acceleration factor of $5$ and an input $\rm snr$ of $32$ are presented in Fig. \ref{fig:simulated reconstruction 2D}, along with the corresponding error images (magnitudes of the complex-valued differences between the original image and reconstructed images). The relative errors of the reconstructions as functions of the input ${\rm snr}$ for the six coverages considered and for both methods are reported in Fig. \ref{fig:curves error 2D}.
For acceleration factors larger than $2$, the s$_2$MRI technique with $\bar{w} = 0.3$ provides better reconstruction than the variable density sampling technique with an improvement up to $0.05$ of the relative error. Indeed, the relative error is, on average, lower in the presence of the chirp modulation. The corresponding standard deviations are also much smaller, indicating that the s$_2$MRI technique is more stable\footnote{Some leftover variability for the variable density sampling technique might still be removed by increasing the number of simulations. However, this would not modify the results and comparison with s$_2$MRI.}. At an acceleration factor of $2$, the variable density sampling technique gives slightly better reconstructions than the s$_2$MRI technique with $\bar{w} = 0.3$. However, with $\bar{w} = \bar{w}_{\rm opt}=0.1$, the s$_2$MRI technique provides relative errors similar to those obtained with the variable density sampling technique. These results suggest to reduce the chirp rate as the number of measurements increases. This is coherent with the fact that modulation is not needed in the limit of no under-sampling.
When comparing the magnitudes of reconstructed images in Fig. \ref{fig:simulated reconstruction 2D}, differences between both methods do not appear obvious. However, one can notice that fine details are recovered better with the s$_2$MRI technique: the vessel (white spot) indicated by a blue arrow appears in the s$_2$MRI reconstruction, but not in the variable density sampling reconstruction. The error images bring more information, and one can notice that the errors are smaller in the presence of chirp modulation. In particular, the low scale structures, rendered incoherent with the chirp modulation, are better recovered.
\section{s$_2$MRI implementation and experimental results}
\label{sec:Real acquisitions}
\subsection{Implementation}
The s$_2$MRI technique is tested on the $7\rm{T}$ scanner described in Section \ref{sub:simulations} with $3\rm{D}$ acquisitions of a phantom and a human brain. For the brain experiment, the subjects provided written informed consent prior to the imaging session, according to the guidelines of the local ethics committee.
The chirp modulation is implemented through the use of a second order shim coil $x^2-y^2$. In our implementation, the chirp rate varies linearly with the readout time $t$ (or equivalently $k_z$) and is proportional to the intensity of the quadratic magnetic field $\kappa$: $w(t) = w_x(t) = - w_y(t) = \gamma\kappa t / \pi $, where $\gamma$ is the gyromagnetic factor. The maximum chirp rate $w^{\rm max}$ is reached at $t = {\rm TE} + \Delta t/2$, the mean chirp rate $w^{\rm mean}$ at $t = {\rm TE}$, and the minimum chirp rate $w^{\rm min}$ at $t = {\rm TE} - \Delta t/2$ ($\Delta t$ is the readout duration). This chirp modulation can be introduced in the measurement model by modifying (\ref{mri2}) as follows:
\begin{equation}
\label{eq:real modulation}
\bm{\nu}=\mathsf{MF}^{\star}_{xy}\mathsf{CF}^{\star}_{z}\mathsf{U}\bm{\rho}+\bm{n}.
\end{equation}
In the above equation, $\mathsf{F}^{\star}_{z} \in \mathbb{C}^{N_{\rm u} \times N_{\rm u}}$ and $\mathsf{F}^{\star}_{xy} \in \mathbb{C}^{N_{\rm u} \times N_{\rm u}}$ implement the Fourier transform along the $z$-direction and the $xy$-directions, respectively. The matrix $\mathsf{C} \in \mathbb{C}^{N_{\rm u} \times N_{\rm u}}$ implements the chirp modulation. Signals are thus reconstructed on a grid of $N_{\rm c}=N_x(1+\vert\bar{w}_x^{\rm max}\vert)\times N_y(1+\vert\bar{w}_y^{\rm max}\vert)\times N_z$ points, and $N_{\rm u}=N_x(1+2\vert\bar{w}_x^{\rm max}\vert)\times N_y(1+2\vert\bar{w}_y^{\rm max}\vert)\times N_z$.
This modulation can be decomposed as a quadratic phase modulation in the $(x, y)$ planes with chirp rate $w^{\rm mean}$, combined with a linear phase modulation in $k_z$. This linear phase modulation produces shifts of the original signal by an amount proportional to $\kappa (x^2 - y^2)$ along the $z$ direction. The chirp modulation used is thus not ideal, as it creates distortions of the original object and complicates the measurement matrix. However, the energy of sparsity basis vectors is still spread by the main chirp modulation with chirp rate $w^{\rm mean}$, and the previous conclusions based on theory and simulations should still hold. This will be confirmed by the numerical experiments of Section \ref{sub:numerical experiments}. Note also that the reconstructed images are free of any distortion as the complete effect of the modulation is modeled in (\ref{eq:real modulation}). Let us remark that these distortions might be avoided with the use of RF pulses or dedicated coils applied only during phase encoding.
As in Section \ref{sub:simulations}, the s$_2$MRI technique is compared to the variable density sampling method with $p = p_M$. Full acquisitions ($M/N = 1$) are performed both in the absence and in the presence of the chirp modulation. The number of phase encodings $(k_x, k_y)$ is then reduced retrospectively by applying a mask on the complete data.
\begin{figure}
\centering
\includegraphics[width=8.9cm, keepaspectratio]{fig6.pdf}
\caption{\label{fig:curves error 3D} Relative reconstruction errors as functions of the input ${\rm snr}$ for the variable density sampling technique (dashed blue curve) and the s$_2$MRI technique (dot-dashed red curve) with varying chirp modulation. From left to right and top to bottom, the panels show the curves for coverages of $15$, $25$, $50$ percent of $N$ respectively. All curves represent the mean relative error over $5$ simulations, and the vertical lines represent the error at $1$ standard deviation.}
\vspace{-3mm}
\end{figure}
The noise level is evaluated directly on the under-sampled data available for reconstruction. For each $(k_x, k_y)$ pair measured, all the frequencies $k_z$ are probed so that the signal is available as a function of $z$. The level of the noise is estimated on probed pairs $(k_x, k_y)$ at positions $z$ that do not contain any signal. This noise level is identical in the presence and absence of chirp modulation.
\subsection{Numerical validation}
\label{sub:numerical experiments}
In this section, we perform simulations using the acquisition scheme described above to confirm that, even though the implementation of the chirp modulation is not ideal, the reconstruction quality is still enhanced with the s$_2$MRI technique.
\begin{figure}
\centering
\includegraphics[width=8.8cm, keepaspectratio]{fig7.pdf}
\caption{\label{fig:simulated reconstruction 3D} Simulated reconstructions with varying chirp modulation for an under-sampling of $50$ percent of $N$ and an input $\rm snr$ of $32$. The top row shows the reconstructions of an axial slice for the variable density sampling (left panel) and the s$_2$MRI (right panel) techniques, respectively. The bottom row shows the error images (difference between the original image and the reconstructions) in the absence (left panel) and presence (right panel) of chirp modulation. For a better visualization, the error images were scaled by a factor of $8$. The colormap for the error images goes from white to black, indicating low and high errors, respectively.}
\vspace{-3mm}
\end{figure}
For this numerical experiment, a brain volume was acquired using the standard clinical MPRAGE sequence on a field of view of $L_x = 243\,\rm{mm}$, $L_y = 176\,\rm{mm}$ and $L_z = 256\,\rm{mm}$, with a resolution of $1\,\rm{mm}$ in each direction ($N_x = 243$, $N_y = 176$, $N_z = 256$). The echo time is $\rm{TE} = 4.59\,\rm{ms}$, the inversion time $\rm{TI} = 1.5\,\rm{s}$, the repetition time $\rm{TR} = 3.5\,\rm{s}$, the bandwidth $\rm{BW} = 250\,\rm{Hz}$. As in Section \ref{sub:simulations}, in order to model an analog acquisition scheme, the original image at a resolution of $1 \, \rm{mm}$ is used to compute the measurements, but the reconstruction is performed at a resolution of $2 \, \rm{mm}$. The reconstructed images are compared to the image obtained with a full acquisition at $2 \, \rm{mm}$ of resolution.
The parameters of our experiment are as follows. Firstly, acquisitions are considered for various numbers $M$ of complex measurements corresponding to coverages of $15$, $25$, $50$ percent of $N$. Secondly, instrumental noise is also added to the measurements as independent and identically distributed zero-mean Gaussian noise. The corresponding standard deviation $\sigma$ is identical for all the frequencies probed and we consider values of input ${\rm snr}$ of $2^j$, with $1\leq j \leq 6$. Thirdly, the simulated chirp modulation has a chirp rate varying linearly with $k_z$: $\vert \bar{w}_x \vert \in [0.12, 0.30]$ and $\vert \bar{w}_y \vert \in [0.09, 0.22]$. These values for the discrete chirp rate correspond to those used during the real experiment performed hereafter at $1 \, \rm{mm}$ of resolution. However, relatively to the band-limit at $2 \, \rm{mm}$ of resolution, the spectrum is naturally more spread than relatively to the band-limit at $1 \rm{mm}$ of resolution. Therefore, for a reconstruction at $2 \, \rm{mm}$ of resolution, the spectrum does not need to be spread as much as for a reconstruction at $1 \, \rm{mm}$ of resolution. We thus divided the values of the chirp rate by $2$ in both dimensions. Fourthly, the signals are reconstructed by solving the ${\rm BP}$ problem where the ${\rm TV}$ norm of the signal $\bm \rho$ is substituted for the $\ell_1$ norm. Finally, for each value of $M$ and input ${\rm snr}$ considered, $5$ simulations are generated with independent noise and mask realizations, and the relative reconstruction errors are computed for the variable density sampling and s$_2$MRI techniques.
The relative errors of the reconstructions as functions of the input ${\rm snr}$ for the three coverages considered and for both methods are reported in Fig. \ref{fig:curves error 3D}. The magnitudes of a reconstructed axial slice obtained with the s$_2$MRI and the variable density sampling techniques for an acceleration factor of $5$ and an input $\rm snr$ of $32$ are presented in Fig. \ref{fig:simulated reconstruction 3D}, along with the corresponding error images. Conclusions of Section \ref{sub:simulations} still hold with this acquisition scheme. The relative error of reconstruction is lower in the presence of the chirp for the three coverage considered. When comparing the error images in Fig. \ref{fig:simulated reconstruction 3D}, one can once more notice that the errors are smaller with the s$_2$MRI technique.
\subsection{Experiments}
\begin{figure*}
\centering
\includegraphics[width=18.2cm, keepaspectratio]{fig8.pdf}
\caption{\label{fig:experiments} Phantom and brain reconstructions from real experimental data with the s$_2$MRI technique. The first to fourth columns show the magnitude of the reconstructions from $15$, $25$, $50$, and $100$ percent of phase encodings respectively. The fifth column shows the reference images obtained by inverse Fourier transform (F.T.) of the fully sampled $\bm{k}$-space.}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=18.2cm,keepaspectratio]{fig9.pdf}
\caption{\label{fig:comparisons images} Phantom and brain reconstructions from real experimental data with the s$_2$MRI technique (first and third rows respectively) and the variable density sampling (VDS) technique (second and fourth rows respectively). The first to fourth columns show the magnitude of the reconstructions from $15$, $25$, $50$, and $100$ percent of phase encodings respectively. The fifth column shows the reference images obtained by inverse Fourier transform (F.T.) of the fully sampled $\bm{k}$-space. The white lines on these images indicate the location of the spatial profiles presented in Fig. \ref{fig:comparisons profiles}}
\vspace{-4mm}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=18.5cm, keepaspectratio]{fig10.pdf}\\
\caption{\label{fig:comparisons profiles} Phantom and brain reconstruction profiles from real experimental data with the s$_2$MRI technique (first and third rows respectively) and the variable density sampling (VDS) technique (second and fourth rows respectively). The first to fourth columns show the profile magnitude of the reconstructions (continuous red curve) from $15$, $25$, $50$, and $100$ percent of phase encodings respectively as well as the reference profiles (dotted black curve) obtained by inverse Fourier transform (F.T.) of the fully sampled $\bm{k}$-space.}
\vspace{-4mm}
\end{figure*}
For the phantom experiment, a gradient echo sequence is used on a field of view of $L_x = L_y = L_z = 192\,\rm{mm}$ in the three directions with a resolution of $1\,\rm{mm}$ ($N_x = N_y = N_z = 192$). The echo time is $\rm{TE} = 6\,\rm{ms}$, the repetition time $\rm{TR} = 10\,\rm{ms}$, the bandwidth $\rm{BW} = 400\,\rm{Hz}$, and the quadratic magnetic field intensity $\kappa = 3 000\,\rm{\mu T/m^{2}}$. The discrete chirp rates satisfy $\vert \bar{w}_x \vert = \vert \bar{w}_y \vert \in [0.23, 0.36]$.
For the brain experiment, the standard clinical MPRAGE sequence is used on a field of view of $L_x = 243\,\rm{mm}$, $L_y = 176\,\rm{mm}$ and $L_z = 256\,\rm{mm}$, with a resolution of $1\,\rm{mm}$ in each direction ($N_x = 243$, $N_y = 176$, $N_z = 256$). The echo time is $\rm{TE} = 4.59\,\rm{ms}$, the inversion time $\rm{TI} = 1.5\,\rm{s}$, the repetition time $\rm{TR} = 3.5\,\rm{s}$, the bandwidth $\rm{BW} = 250\,\rm{Hz}$, and the quadratic magnetic field intensity $\kappa = 4 500\,\rm{\mu T/m^{2}}$. The discrete chirp rates satisfy $\vert \bar{w}_x \vert \in [0.25, 0.61]$ and $\vert \bar{w}_y \vert \in [0.18, 0.44]$.
Slices of the $3\rm{D}$ reconstructions obtained with the s$_2$MRI technique for $15$, $25$, $50$ and $100$ percent of phase encodings are presented in Fig. \ref{fig:experiments} for the phantom and brain acquisitions. The images obtained by inverse Fourier transform of the fully sampled $\bm{k}$-space (after correction of the distortions due to the small shifts in the $z$ direction) are also presented. Fig. \ref{fig:comparisons images} provides a comparison of the reconstructed images obtained with the s$_2$MRI and the variable density sampling techniques. In the aim of providing further insight on how each method preserves image resolution or, in other words, captures shape and magnitude of high frequency features, we also provide one-dimensional spatial profiles in Fig. \ref{fig:comparisons profiles}. The white lines in Fig. \ref{fig:comparisons images} indicate the location of these profiles.
Firstly, as one would expect, the visual reconstruction quality improves when the number of phase encodings $M$ increases. In the limit of a coverage of $100$ percent, we cannot identify any loss of details between the reconstructed images and the ones obtained by inverse Fourier transform. Moreover, the reconstructed image contains much less noise than the image obtained by inverse Fourier transform. Indeed, at a coverage of $100$ percent, the problem (\ref{cs3}) is essentially reduced to a denoising problem.
Secondly, as in Section \ref{sub:simulations}, the differences between both methods do not appear at first glance when comparing only the magnitudes of reconstructed images in Fig. \ref{fig:comparisons images}. Unfortunately here, the ground truth image is not accessible, so the corresponding error images cannot be displayed. However, a thorough visual inspection reveals that, for acceleration factors larger than $2$, some fine details are better recovered by our approach. On the phantom reconstructions for acceleration factors of $6.7$ and $4$ (coverages of $15$ and $25$ percent, respectively), the separation between the two biggest circles is more visible. The shapes of the circles are also more curved. On the brain reconstructions for an acceleration factor of $6.7$, the vessels at the center of image are still visible with our method but not with the variable density sampling technique. The cerebral cortex also appears sharper. For an acceleration factor of $2$, the thin layer separating the two hemispheres of the brain remains more visible with the s$_2$MRI technique.
Thirdly, the reconstructed spatial profiles of the phantom presented in Fig. \ref{fig:comparisons profiles} show that the shape and magnitude of high frequency features are, more often, slightly better recovered with the s$_2$MRI technique for acceleration factors larger than $4$ (blue and red arrows indicate features better reconstructed with s$_2$MRI and variable density sampling respectively). This improvement is much more significant on the brain data (see arrows), and holds for acceleration factors larger than $2$.
Finally, the distortions are correctly taken into account by the operator $\mathsf{C}$, as both fully sampled slices of the phantom with and without chirp are identical (see Fig. \ref{fig:comparisons images}). For the brain images, some differences remain due to small movements of the subject between both acquisitions. Let us remark that no fitting of the chirp parameters (center and rates) was performed to improve on the theoretical values. This highlights sufficient stability of the technique relative to parameter approximations.
\section{Conclusion and discussion}
\label{sec:Conclusion}
We presented a spread spectrum technique (s$_2$MRI) designed to accelerate MR acquisitions by compressed sensing. It consists of pre-modulating the original image by a linear chirp, which results from the application of quadratic phase profiles, and then performing random $\bm k$-space under-sampling. Non-linear algorithms promoting signal sparsity are then used for image reconstruction.
In the context of compressed sensing theory, the effectiveness of the technique is supported by a decrease of coherence between the sensing and sparsity bases due to the pre-modulation. Simulations in a simplified analog setting confirm that the enhancement of the image reconstruction quality is linked to the evolution of the mutual coherence. The s$_2$MRI technique was compared with the state-of-the-art variable density sampling using realistic numerical simulations and real acquisitions. Simulation results shows that the s$_2$MRI technique performs slightly better than the variable density sampling technique in terms of relative reconstruction error for acceleration factors larger than $2$. The chirp modulation was also implemented on a $7$T scanner with the use of a second order shim coil. Simulations of this implementation confirms again the slight superiority of s$_2$MRI. Visual inspection of reconstructions obtained from real acquisition of phantom and \emph{in vivo} data also shows that this first (non-ideal) implementation provides slightly better reconstruction qualities than the variable density sampling method. The s$_2$MRI technique thus outperforms the variable density sampling technique in terms of all the criteria used for evaluation.
Regarding future evolutions of the s$_2$MRI technique, an implementation of the linear chirp modulation with the use of RF pulses or dedicated coils could simplify the measurement scheme by applying the chirp modulation only during phase encoding in order to avoid object distortions, and, in turn, further enhance the reconstruction quality. Moreover, fitting the effective chirp center and rates on the basis of the data could improve the measurement model and result in better reconstructions.
Let us also emphasize the potential interest of the s$_2$MRI technique from an enhanced resolution perspective as, in the presence of the chirp modulation, the original image is reconstructed at a high resolution in order to avoid any aliasing problems. One can indeed consider reaching a higher spatial resolution for a fixed acquisition time without probing higher spatial frequencies in practice, which would require stronger gradient coils. In this context, any regularization approach adding a sparsity prior can help to synthesize spatial frequency information higher than that contained in the data. But the chirp modulation implies that high spatial frequency information is actually probed at lower frequencies. For illustration, Fig. \ref{fig:high resolution} shows a reconstructed $(x, y)$ slice for a coverage of $100$ percent on a high resolution grid $N_{\rm c} = 393 \times 255$ as well as the image obtained after downsampling on the grid $N = 243 \times 176$. One can notice that the high resolution image provides sharper details with fewer aliasing artifacts. In particular, the vessels are better resolved.
\section*{Acknowledgments}
\begin{figure}
\centering
\includegraphics[width=8.8cm, keepaspectratio]{fig11.pdf}
\caption{\label{fig:high resolution} Axial $(x, y)$ reconstructed slice for a coverage of $100$ percent in the presence of chirp modulation on a high resolution grid of size $N_{\rm c} = 393 \times 255$ (right panel) and after dowsampling on the original grid of size $N = 243 \times 176$.}
\vspace{-7mm}
\end{figure}
We thank David Shuman for his helpful suggestions to improve the writing quality of the paper, as well as the reviewers for their very useful comments.
|
{
"timestamp": "2012-03-13T01:00:31",
"yymm": "1203",
"arxiv_id": "1203.2205",
"language": "en",
"url": "https://arxiv.org/abs/1203.2205"
}
|
\section{Introduction}
Photoionization of atoms and molecules is one of the most fundamental quantum processes. It played a key role in the early days of quantum mechanics and has ever since been paving the way towards an improved understanding of the structure and dynamics of matter on a microscopic scale. Today, kinematically complete photoionization experiments allow for accurate tests of the most sophisticated ab-initio calculations. Besides, photoionization studies in a new frequency domain are currently becoming feasible by the availability of novel xuv and x-ray radiation sources \cite{FLASH,LCLS,atto}, giving rise to corresponding theoretical developments (see, e.g., \cite{Lambropoulos,Dieter,Santra}).
Various photoionization mechanisms rely crucially on electron-electron correlations. Prominent examples are single-photon double ionization as well as resonant photoionization. The latter proceeds through resonant photoexcitation of an autoionizing state with subsequent Auger decay. In recent years, a similar kind of ionization process has been studied in systems consisting of two (or more) atoms. Here, a resonantly excited atom transfers its excitation energy radiationlessly via interatomic electron-electron correlations to a neighbouring atom leading to its ionization. This Auger-like decay involving two atomic centers is commonly known as interatomic Coulombic decay (ICD) \cite{ICD,ICDrev}. It has been observed, for instance, in noble gas dimers and water molecules \cite{ICDexp}. In metal oxides, the closely related process of multi-atom resonant photoemission (MARPE) was also observed \cite{MARPE}.
We have recently studied resonant two-center photoionization in heteroatomic systems and shown that this ionization channel can be remarkably strong \cite{2CPI,ABV1,ABV2}. In particular, it can dominate over the usual single-center photoionization by orders of magnitude. Besides, characteristic effects resulting from a strong coupling of the ground and autoionizing states by a relatively intense photon field were identified. Also resonant two-photon ionization in a system of two identical atoms was investigated \cite{identical}. We note that photoionization in two-atomic systems was also studied in \cite{Kuhn,Kuhn2} and \cite{Perina,Perina2}. The inverse of two-center photoionization (in weak external fields) is two-center dielectronic recombination \cite{2CDR}.
\begin{figure}[b]
\begin{center}
\includegraphics[width=0.45\textwidth]{Figure1_NJP.eps}
\end{center}
\caption{Schematic illustration of photoionization of an atom $A$ in the presence of an external laser field and two neighbouring atoms $B$ and $B'$. Apart from the direct photoionization of $A$ there are interatomic channels via resonant photoexcitation of the ``molecular'' system $B$-$B'$ and subsequent ICD. }
\label{figure1}
\end{figure}
In the present contribution, we extend our investigations of electron correlation-driven interatomic processes by considering photoionization of an atom $A$ in the presence of \textit{two} neighbouring atoms $B$ (see figure 1). All atoms are assumed to interact with each other and with an external radiation field. We show that the photoionization of atom $A$ via photoexcitation of the system of two neighbouring atoms $B$ and subsequent ICD can be by far the dominant ionization channel. Moreover, we reveal the characteristic properties of the process with regard to its temporal dependence and photoelectron spectra. In particular, by comparing our results with those for photoionization in a system of two atoms $A$ and $B$, we demonstrate the influence which the presence of the second atom $B$ may have.
Atomic units (a.u.) are used throughout unless otherwise stated.
\section{Theoretical Framework}
Let us consider a system consisting
of three atoms, $A$, $B$ and $B'$, where $B$ and $B'$ are
atoms of the same element and $A$ is different.
We shall assume that all these atoms are separated
by sufficiently large distances such that free atomic states
represent a reasonable initial basis set to start with.
Let the ionization potential $ I_A $
of atom $A$ be smaller than
the excitation energy $ \Delta E_B $
of a dipole-allowed transition in atoms $B$ and $B'$.
Under such conditions, if our system is irradiated by
an electromagnetic field with frequency $\omega_0 \approx \Delta E_B$, the
ionization process of this system
(i.e., essentially of the atom $A$) can be qualitatively
different compared to the case when a single, isolated atom
$A$ is ionized. Indeed, in such a case $A$ can be ionized
not only directly but also via resonant photoexcitation
of the subsystem of $B$ and $B'$, with its consequent deexcitation
through energy transfer
to $A$ resulting in ionization of the latter.
In the following, we consider
photoionization in the system of atoms $A$, $B$ and $B'$ in more detail.
For simplicity, we suppose that the nuclei
of all atoms are at rest during photoionization.
Denoting the origin of our coordinate system by $O$,
we assume that the nuclei of the atoms $B$ and $B'$
are located on the $Z$-axis: ${\bf R}_B = (0,0,Z_B)$ and
${\bf R}_{B'} = (0,0,Z_{B'})$. The coordinates of the nucleus of
the atom $A$ are given by ${\bf R}_A = (X_A,Y_A,Z_A)$.
The coordinates of the (active) electron of
atom $\lambda$ with respect to its nucleus are denoted by
${\bf r}_{\lambda}$, where $\lambda \in \{A, B, B'\}$.
The total Hamiltonian describing the
three atoms embedded in an external electromagnetic field reads
\begin{eqnarray}
H = \hat{H}_0 + \hat{V}_{AB} + \hat{V}_{AB'} + \hat{V}_{BB'} + \hat{W}_A
+ \hat{W}_B + \hat{W}_{B'},
\label{hamiltonian}
\end{eqnarray}
where $ \hat{H}_0 $ is the sum of the Hamiltonians
for the noninteracting atoms $A$, $B$ and $B'$.
We shall assume that the (typical) distances
$\Delta R$ between the atoms
are not too large, $ \Delta R \ll c / \Delta E_B $,
where $c$ is the speed of light, such that retardation
effects in the electromagnetic interactions can be ignored.
If transitions of electrons between bound states
in atoms $B$ and $B'$ are of dipole character,
then the interaction between each pair of atoms
$(\lambda,\gamma)$ (with $\lambda, \gamma \in \{A, B, B'\}$)
can be written as
\begin{eqnarray}
\hat{V}_{\lambda,\gamma} &=&
\frac{ \left( {\bf r}_{\lambda} \right)_i \left( {\bf r}_{\gamma}
\right)_j }{R_{\lambda,\gamma}^3}
\left( \delta_{ij} - \frac{ 3 \left( {\bf R}_{\lambda,\gamma} \right)_i \left(
{\bf R}_{\lambda,\gamma} \right)_j }
{R_{\lambda,\gamma}^2} \right),
\label{inter_atomic}
\end{eqnarray}
where $\bf{R}_{\lambda,\gamma} = \bf{R}_{\lambda} - \bf{R}_{\gamma}$
and $\delta_{ij}$ is the Kronecker symbol.
Note that in (\ref{inter_atomic})
a summation over the repeated indices $i$ and $j$ is implied.
In (\ref{inter_atomic}), $\hat{W}_{\lambda}$ denotes
the interaction of the atom $\lambda$ with
the laser electromagnetic field. The latter will be treated
as a classical, linearly polarized field,
described by the vector potential
${\bf A}({\bf r},t)= {\bf A}_0 \cos\left(\omega_0 t \right)$,
where ${\bf A}_0 = c {\bf F}_0/\omega_0$,
$\omega_0 = c k_0 $ is the angular frequency
and ${\bf F}_0$ is the field strength.
The interaction $\hat{W}_{\lambda}$ then reads
\begin{eqnarray}
\hat{W}_{\lambda} = \frac{1}{c} {\bf A}({\bf r}_{\lambda},t) \cdot \hat{\bf
p}_{\lambda},
\label{interaction}
\end{eqnarray}
where $\hat{\bf p}_{\lambda}$ is the momentum operator for
the electron in atom ${\lambda}$.
Our treatment of photoionization will be based on
the following points:
Oscillator strengths for dipole-allowed
bound-bound transitions can be very strong.
This means that, provided that the distances between
all the atoms in our system are
of the same order of magnitude,
the interaction between atoms $B$ and $B'$
is much more effective than the interaction between atoms
$A$ and $B$ (or $A$ and $B'$). Besides, atoms $B$ and $B'$
will, in general, couple much more strongly to
a resonant laser field than atom $A$.
In what follows, we shall assume that
the intensity of the laser field is relatively low such
that the interaction between atoms $B$ and $B'$
changes the states of the system more substantially than
the coupling of these atoms to the laser field.
Therefore, we shall begin with building states of
the $B$-$B'$ subsystem in the absence of the field.
The second step of our treatment will be to
include the interaction of the $B$-$B'$
subsystem with the laser field and,
in the third step, we complete the treatment of ionization
by considering the interaction of atom $A$ with both the laser field
and the field-dressed subsystem of atoms $B$ and $B'$.
\vspace{1cm}
I. We denote the ground and excited states
of the undistorted atoms $B$ and $B'$ by
$\phi_0$, $\phi_e$ and $\phi'_0$, $\phi'_e$, respectively.
Let the corresponding energies of these states be $\varepsilon_0$ and
$\varepsilon_e$.
The state $\psi_{BB'}$ of the $B$-$B'$ subsystem can be expanded
into the ``complete'' set of undistorted atomic states
represented by the configurations
(i) $\phi_0 \phi'_0$, (ii) $\phi_0 \phi'_e$,
(iii) $\phi_e \phi'_0$ and (iv) $\phi_e \phi'_e$.
In the approximation, which neglects the interatomic interaction,
the configurations $\phi_0 \phi'_e$ and $\phi_e \phi'_0$
are characterized by exactly the same value of
the (undistorted) energy $E_{0e} = \varepsilon_0 + \varepsilon_e $.
The latter, in turn, strongly differs from
the energies $E_{00} = 2 \varepsilon_0 $ and
$E_{ee} = 2 \varepsilon_e $ which are
characteristic for the configurations
$\phi_0 \phi'_0$ and $\phi_e \phi'_e$,
respectively. Therefore, provided that the distance
between the atoms is not too small,
the interaction $V_{BB'}$
will strongly mix the configurations (ii) and (iii) only,
while the other configurations (i) and (iv)
will be affected only very weakly.
Taking this into account, it is not difficult
to find the states of the subsystem of
interacting atoms $B$ and $B'$ which read
\begin{eqnarray}
\varphi_{g} &=& \phi_0 \phi'_0
\nonumber \\
\varphi_{+} &=& \frac{1}{\sqrt{2}}
\left( \phi_e \phi'_0 + \phi_0 \phi'_e \right)
\nonumber \\
\varphi_{-} &=& \frac{1}{\sqrt{2}}
\left( \phi_e \phi'_0 - \phi_0 \phi'_e \right)
\nonumber \\
\varphi_{e} &=& \phi_e \phi'_e.
\label{states_of_BB'}
\end{eqnarray}
These two-atomic states are normalized
and mutually orthogonal. They posses energies given by
$E_{g} = 2 \varepsilon_0 $,
$ E_+ = \varepsilon_0 + \varepsilon_e + v_{BB'} $,
$ E_- = \varepsilon_0 + \varepsilon_e - v_{BB'} $
and $E_{e} = 2 \varepsilon_e $, respectively, where
$ v_{BB'} = \left\langle \phi_e \phi'_0 \left| \hat{V}_{BB'} \right|
\phi_0 \phi'_e \right\rangle$. Note that, for definiteness,
$ v_{BB'}$ has been assumed to be real and negative here,
as will always be the case in our examples below (see
section~3).
\vspace{1cm}
II. Let us now consider two interacting atoms $B$ and $B'$
embedded in a resonant laser field. One can look for a state
of such a system by expanding it into the new set of states given by
Eq.~(\ref{states_of_BB'}),
\begin{eqnarray}
\psi(t) &=& g(t)\varphi_{g} + a_{+}(t) \varphi_{+} +
a_{-}(t) \varphi_{-} + b(t) \varphi_{e}.
\label{BB'_in_laser_1}
\end{eqnarray}
Inserting the expansion (\ref{BB'_in_laser_1})
into the corresponding wave equation, we obtain
a set of coupled equations for the unknown time-dependent coefficients
$g(t)$, $a_{+}(t)$, $a_{-}(t)$ and $b(t)$:
\begin{eqnarray}
i \frac{dg}{dt} - E_g g &=&
\left\langle \varphi_g \left|\hat{W}_{B} + \hat{W}_{B'}\right| \varphi_{+}
\right\rangle a_{+} +
\left\langle \varphi_g \left|\hat{W}_{B} + \hat{W}_{B'}\right| \varphi_{-}
\right\rangle a_{-} \nonumber\\
& & +
\left\langle \varphi_g \left|\hat{W}_{B} + \hat{W}_{B'}\right| \varphi_{e}
\right\rangle b
\nonumber \\
i \frac{da_{+}}{dt} - E_{+} a_{+} &=&
\left\langle \varphi_{+} \left|\hat{W}_{B} + \hat{W}_{B'}\right| \varphi_g
\right\rangle g +
\left\langle \varphi_{+} \left|\hat{W}_{B} + \hat{W}_{B'}\right|
\varphi_{-} \right\rangle a_{-} \nonumber\\
& & +
\left\langle \varphi_{+} \left|\hat{W}_{B} + \hat{W}_{B'}\right|
\varphi_{e} \right\rangle b
\nonumber \\
i \frac{da_{-}}{dt} - E_{-}a_{-} &=&
\left\langle \varphi_{-} \left|\hat{W}_{B} + \hat{W}_{B'}\right| \varphi_g
\right\rangle g +
\left\langle \varphi_{-} \left|\hat{W}_{B} + \hat{W}_{B'}\right|
\varphi_{+} \right\rangle a_{+} \nonumber\\
& & +
\left\langle \varphi_{-} \left|\hat{W}_{B} + \hat{W}_{B'}\right|
\varphi_{e} \right\rangle b
\nonumber \\
i \frac{db}{dt} - E_e b &=&
\left\langle \varphi_e \left|\hat{W}_{B} + \hat{W}_{B'}\right| \varphi_g
\right\rangle g +
\left\langle \varphi_e \left|\hat{W}_{B} + \hat{W}_{B'}\right| \varphi_{+}
\right\rangle a_{+} \nonumber\\
& & +
\left\langle \varphi_e \left|\hat{W}_{B} + \hat{W}_{B'}\right| \varphi_{-}
\right\rangle a_{-}.
\label{BB'_in_laser_2}
\end{eqnarray}
The system of equations (\ref{BB'_in_laser_2}) can be greatly simplified
by noting the following. First, all transition matrix
elements of the interaction with the laser field, which involve the
asymmetric
state $\varphi_{-}$, are equal to zero and, thus, only the remaining
three states can be coupled by the field. Second, if we suppose that
the frequency of the laser field is resonant to the transitions
$ \varphi_g \longleftrightarrow \varphi_{+} $ and that the field is
relatively weak
such that the non-resonant transitions $ \varphi_{+} \longleftrightarrow
\varphi_e $
are much less effective than the above resonant ones, the system
(\ref{BB'_in_laser_2}) effectively reduces to
\begin{eqnarray}
i \frac{dg}{dt} - E_g g &=&
\left\langle \varphi_g \left|\hat{W}_{B} + \hat{W}_{B'}\right| \varphi_{+}
\right\rangle a_{+}
\nonumber \\
i \frac{da_{+}}{dt} - E_{+} a_{+} &=&
\left\langle \varphi_{+} \left|\hat{W}_{B} + \hat{W}_{B'}\right| \varphi_g
\right\rangle g,
\label{BB'_in_laser_3}
\end{eqnarray}
which can be readily solved by using the rotating wave approximation.
Assuming that the field is switched on suddenly at $t=0$,
we obtain two solutions
\begin{eqnarray}
\psi_1(t) &=& \frac{1}{z_{2} - z_{1}}
\left[ \left( z_{2} + \omega_0 - E_{+} \right)
{\rm e}^{-i z_{2} t} - \left( z_{1} + \omega_0 - E_{+} \right)
{\rm e}^{-i z_{1} t} \right] \, \varphi_{g}
\nonumber \\
&+& \frac{W_{+,g}}{z_{2} - z_{1}}
\left( {\rm e}^{-i z_{2} t} - {\rm e}^{-i z_{1} t} \right)
{\rm e}^{- i \omega_0 t} \, \varphi_{+}
\label{BB'_in_laser_4}
\end{eqnarray}
and
\begin{eqnarray}
\psi_2(t) &=& \frac{ W_{g,+} }{ z_{2} - z_{1} }
\left( {\rm e}^{-i z_{2} t} - {\rm e}^{-i z_{1} t} \right) \, \varphi_{g}
\nonumber \\
&+& \frac{1}{ z_{2} - z_{1} }
\left[ \left( z_{2} - E_{g} \right) {\rm e}^{-i z_{2} t}
- \left( z_{1} - E_{g} \right)
{\rm e}^{-i z_{1} t} \right] {\rm e}^{- i \omega_0 t} \, \varphi_{+}.
\label{BB'_in_laser_5}
\end{eqnarray}
In the above equations, we have introduced
\begin{eqnarray}
z_{1} &=& \frac{1}{2} %
\left( E_{g} + E_{+} - \omega_0 - \Omega_R \right)
\nonumber \\
z_{2} &=& \frac{1}{2} %
\left( E_{g} + E_{+} - \omega_0 + \Omega_R \right), %
\label{BB'_in_laser_6}
\end{eqnarray}
where $\Omega_R = \sqrt{( E_{+} - E_{g} - \omega_0)^2 + 4 \left| W_{g,+}
\right|^2} $
is the Rabi frequency, $W_{g,+} = \left\langle \varphi_g \left| %
{\bf F}_0 \cdot \left( \hat{\bf p}_B + \hat{\bf p}_{B'} \right)/%
(2 \omega_0) \right| \varphi_{+} \right\rangle $
and $W_{+,g}=(W_{g,+})^*$.
The two solutions in (\ref{BB'_in_laser_5})
correspond to two different initial conditions:
at $t=0$ the system is either in
the state $\varphi_{g}$ or in $\varphi_{+}$.
They are orthogonal to each other and
form a ``complete'' set of field-dressed states
of the subsystem $B$-$B'$.
Note also that we have neglected the spontaneous
radiative decay of the excited state $\varphi_{+}$ which,
in our case, is justified as long as $\left| W_{g,+} \right| \gg \Gamma_r$,
where $\Gamma_r$ is the radiative width of $\varphi_{+}$.
\vspace{1cm}
III. Now, as the last step, we shall add atom $A$ to our consideration.
Let $\chi_0$ and $\chi_{\bf p}$, where ${\bf p}$ is the electron momentum,
be the ground and a continuum state of a single, isolated atom $A$.
The wavefunction of the total system $A$--$B$-$B'$ can be expanded
into the following ``complete'' set of states
\begin{eqnarray}
\Psi(t) = \alpha_0(t) \, \psi_1 \, \chi_0 +
\beta_0(t) \, \psi_2 \, \chi_0 +
\int d^3 {\bf p} \alpha_{\bf p}(t) \, \psi_1 \, \chi_{\bf p} +
\int d^3 {\bf p} \beta_{\bf p}(t) \, \psi_2 \, \chi_{\bf p}.\nonumber\\
\label{ioniz_of_A_1}
\end{eqnarray}
Here, the initial conditions are given
by $\alpha_0(0) = 1$, $ \beta_0(0) = 0 $ and
$ \alpha_{\bf p}(0) = \beta_{\bf p}(0) = 0$.
The coupling of atom $A$ to both
the subsystem $B$-$B'$ and the laser field involves
bound-continuum transitions which are normally much less
effective than the bound-bound ones. For this reason, we
may assume that the interactions of $A$
with the laser field and the $B$-$B'$-subsystem is weak
and consider ionization of atom $A$ in
the lowest order of perturbation theory in these two interactions.
As a result, by inserting the expansion (\ref{ioniz_of_A_1}) into
the corresponding Schr\"odinger equation we obtain
\begin{eqnarray}
i \frac{ d\alpha_{\bf p} }{dt} - \epsilon^A_p \alpha_{\bf p} & = &
\exp(- i \epsilon^A_g t) \left\langle \psi_1 \, \chi_{\bf p}
\left|\hat{W}_{A} +
\hat{V}_{AB} + \hat{V}_{AB'} \right| \psi_1 \, \chi_0 \right\rangle
\nonumber \\
i \frac{ d\beta_{\bf p} }{dt} - \epsilon^A_p \beta_{\bf p} & = &
\exp(- i \epsilon^A_g t) \left\langle \psi_2 \, \chi_{\bf p}
\left|\hat{V}_{AB} +
\hat{V}_{AB'} \right| \psi_1 \, \chi_0 \right\rangle,
\label{ioniz_of_A_2}
\end{eqnarray}
where $\epsilon^A_g$ is the energy of the electron
in the initial state $\chi_0$ of atom $A$
and $\epsilon^A_p$ is the electron energy after the emission.
The probability for ionization of the three-atomic system,
as a function of time, then reads
\begin{eqnarray}
P(t) &=& \int d^3{\bf p}
\left( \mid \alpha_{\bf p}(t) \mid^2 +
\mid \beta_{\bf p}(t) \mid^2 \right).
\label{ioniz_of_A_3}
\end{eqnarray}
Note that equations (\ref{ioniz_of_A_2})
are readily solved analytically. However, the resulting
expressions are somewhat lengthy and will not be given here.
\section{ Results and Discussion }
Based on the results obtained in the previous section,
let us now turn to the discussion of some aspects of
photoionization in a system consisting of one lithium
and two helium atoms. We suppose that in our three-atomic system
the positions of the lithium and helium atoms are given by the vectors
${\bf R}_{\rm Li} = (0,0,0)$, ${\bf R}_{\rm He} = (0,0,Z)$
and ${\bf R}_{{\rm He}'} = (0,0,-Z)$, respectively.
Our system is initially (at time $t=0$) in its ground configuration
and is irradiated by a monochromatic laser field.
The field is linearly polarized along the $Z$-axis and its
frequency is resonant to the $\varphi_g$ - $\varphi_+$ transition
in the He-He subsystem, i.e., $E_{+} - E_{g} - \omega_0
= 0$.
Choosing $Z=14$ a.u. we obtain that the energy spitting
$\Delta E_{\pm} = \left| E_{+} - E_{-} \right|$ between
the states $\varphi_{+}$ and $\varphi_{-}$ of the He-He
subsystem is $5.4 \times 10^{-4}$ eV. Assuming a field strength
of $F_0 = 10^{-5}$ a.u.,
the corresponding Rabi frequency amounts to $\Omega_{R_{\rm He-He}} =
2 \left| W_{g,+} \right| = 1.3 \times 10^{-4}$ eV
which is much less than $\Delta E_{\pm} $.
\begin{figure}[t]
\vspace{-0.25cm}
\begin{center}
\includegraphics[width=0.77\textwidth]{Figure2_NJP.eps}
\end{center}
\vspace{-0.75cm}
\caption{ \footnotesize{
Photoionization probability for Li, Li-He and Li-He-He systems
in an external electromagnetic field, given as a function of time.
The field strength is $F_0=10^{-5}$ a.u.,
the field is linearly polarized and its frequency is resonant to the
corresponding transition in the He or He-He subsystem.
The distances between Li and each of the He atoms is always $14$ a.u.
The atomic positions are aligned along the field polarization with the Li atom
in the middle of the three-atomic system.
The solid, dash and dot curves display results
for Li-He-He, Li-He and Li systems, respectively.
Note that the ionization probability
for an isolated Li atom has been multiplied by a factor of $500$.
For more explanation see the text.} }
\label{time-develop}
\end{figure}
In figure \ref{time-develop}, we present
the probability for ionization
of our system as a function of time.
The probability shows a non-monotonous behaviour
in which time intervals,
when the ionization probability rapidly increases,
are separated by intervals, when the probability
remains practically constant, reflecting oscillations
of the electron populations with the Rabi frequency
$\Omega_{R_{\rm He-He}}$ between
the ground and excited states of the He-He subsystem
in a resonant electromagnetic field.
For comparison, we also show in figure \ref{time-develop} results for
ionization of a single (separated) Li atom and for
ionization in a two-atomic Li-He system.
In the latter case, the lithium atom is located at the origin
(${\bf R}_{\rm Li} = (0,0,0)$) and the coordinates of the helium atom
are ${\bf R}_{\rm He} = (0,0,14\,{\rm a.u.})$. The frequency of the laser field
is assumed to be resonant to the $1s^2\,^1S$--$1s2p\,^1P$ transition frequency of
the corresponding bound states of a single He atom.
In contrast to the single-atom ionization,
in both the two- and three-atomic cases
the ionization probability
demonstrates a step-wise temporal development
in which time intervals of rapid probability growth
are followed by intervals of almost constant probability.
We point out that in
the three-atomic case, however, the size of these
time intervals is shorter by a factor of $\sqrt{2}$.
Compared to ionization of a single Li atom,
ionization in the two-atomic system is very strongly
enhanced \cite{2CPI,ABV1,ABV2}. When the three-atomic system is irradiated, the enhancement increases even further.
In the range of small values of $t$, where
all ionization probabilities still increase monotonously,
this additional enhancement is equal to a factor of $4$.
At larger $t$, however, when the two ionization probabilities
exhibit step-wise behaviours, this additional enhancement
due to the presence of the second He atom
is reduced to a factor close to $2$ on average,
as can also be seen in figure~\ref{time-develop}.
All the above features can be understood by noting the following:
i) For the chosen set of parameters of our two- and three-center systems,
the indirect channels of ionization, which involve two- or three-atomic
correlations, are substantially stronger than the direct one.
Therefore, these correlations have a dominating effect
on the ionization.
ii) At small $t$, ionization in the two- and three-atomic
systems is basically a two-step process: the first step is
photoexcitation in the He or He-He subsystem and the second step is
a consequent energy transfer to Li. In each case,
both these steps are described by basically the same
dipole transition matrix element of the subsystem.
Since, compared to a single He atom, this dipole element
in He-He is by $\sqrt{2}$ larger than in He,
one obtains a factor of 2 for the enhancement in the ionization amplitude,
leading to a factor of $4$ in the ionization probability (see also \cite{collective}).
iii) At larger $t$, when Rabi oscillations show up, the second step
``saturates'' in the sense that the averaged probability to find
the corresponding subsystem in the excited state becomes
equal to 50\%. Therefore, the ionization probability in
the three-atomic system is now larger (on average)
by a factor of $2$ only.
iv) The origin of the step-wise behaviours
of the ionization probabilities
for the two- and three- atomic systems
lies in the oscillations of
the population between the ground and excited states
in the He atom (for the two-atomic case) or in the
He-He subsystem (for the three-atomic case).
The scale of these oscillation is set
by the Rabi frequency and, because in the
He-He subsystem the latter is
larger by a factor of $\sqrt{2}$,
the corresponding time intervals are shorter
by the same factor.
\begin{figure}[t]
\vspace{-0.25cm}
\begin{center}
\includegraphics[width=0.55\textwidth]{Figure3_NJP.eps}
\end{center}
\vspace{-0.75cm}
\caption{ \footnotesize{
Energy spectra of the emitted electrons, as a function
of $\Delta = \epsilon_p^A - \epsilon_0^A - \omega_0$,
for the same parameters as in figure \ref{time-develop}.
The pulse duration is $100$ ps.
a) Solid and dash curves show results for
ionization of Li-He-He and Li-He systems, respectively.
b) Solid and dot curves display results for
ionization of Li-He-He and Li systems, respectively.
The results for the Li system have been multiplied by a factor of $500$. } }
\label{spectra}
\end{figure}
Additional information about the ionization process
can be obtained by considering the energy spectrum
of emitted electrons.
Such a spectrum is shown in figure \ref{spectra} for
the same systems and parameters as in figure \ref{time-develop} and
for a pulse duration of $T=100$ ps.
In panel (a), we compare the energy spectra of
electrons emitted in the process
of photoionization of Li-He-He and Li-He systems.
In both cases, the main feature is the presence
of three pronounced maxima.
The origin of these peaks is similar to the splitting
into three lines of the energy spectrum of photons
emitted during atomic fluorescence in a resonant
electromagnetic field \cite{mollow1969}.
In such a field, the ground and excited levels of the He and
He-He subsystems split into
two sub-levels, which differ by the corresponding Rabi frequency $\Omega_R$.
As a result, the resonant electronic correlations between these subsystems and the Li atom
lead to an energy transfer to the Li which peaks
at $\omega_0$ and $\omega_0 \pm \Omega_R/2$.
Since, as was already mentioned, the Rabi frequencies
of these subsystems differ by a factor of $\sqrt{2}$, the magnitude of
the separation between the corresponding
maxima in panel (a) of figure \ref{spectra} also
differs by this factor.
Note also that the widths of these main maxima as well as
the appearance of additional multiple maxima, seen in the figure,
are related to the finiteness of the pulse duration;
the distance between the latter is roughly given by $2 \pi/T$.
The distinct influence, which the interatomic electron-electron correlations
exert on the shape of the photoelectron spectra, is further highlighted in
panel (b) of figure \ref{spectra}. It compares the energy spectra of photoelectrons
emitted from our Li-He-He system and an isolated Li atom. In the latter case,
there is only one main maximum, while the two main side peaks are missing,
as one would expect (the additional multiple maxima
are related again to the finiteness of the pulse duration).
\section{Conclusion}
We have studied resonant photoionization in a system $A$-$B$-$B'$ consisting of
three atoms, with two atoms $B$ of the same element and one different atom
$A$. We have shown that the mutual correlations among the atoms can largely
enhance the ionization probability and distinctly modify also other
properties of the process in a characteristic manner. In particular, as
compared to the case of resonant photoionization in a two-atom system $A$-$B$,
it has been demonstrated that the presence of a second atom $B$ can (i)
further enhance the photoionization process,
(ii) change the time dependence of the ionization probability and
(iii) move the side peaks in the photoelectron spectrum further apart.
\ack A.B.V. acknowledges the support from the Extreme Matter Institute EMMI.
\section*{References}
|
{
"timestamp": "2012-03-12T01:01:12",
"yymm": "1203",
"arxiv_id": "1203.2038",
"language": "en",
"url": "https://arxiv.org/abs/1203.2038"
}
|
\section{Introduction}
Very many absorption features known as ``diffuse interstellar bands'' (DIBs) appear on the interstellar extinction curve for wavelengths longer than 400 nm. The upper wavelength limit below which DIBs have been observed was recently shifted to 1800 nm (Geballe et al. \cite{geballe11}). The origin of the DIBs remains enigmatic, but numerous atomic, molecular, and solid state carriers have been proposed (for a review, see Sarre \cite{sarre06}). Among these proposals, neutral and ionized polycyclic aromatic hydrocarbons (PAHs) are considered to belong to the most promising candidates (Cox \cite{cox11}).
In addition to electronic absorption bands in the visible and near-IR, which are characteristic for PAH ions and certain sufficiently-sized neutral PAHs, these molecules are characterized by strong absorptions in the UV. Recently, astronomers tried to detect the spectral signatures of individual PAHs in the UV part ($\lambda <$ 400 nm) of the interstellar extinction curve (Clayton et al. \cite{clayton03}, Gredel et al. \cite{gredel11}, Salama et al. \cite{salama11}). While Clayton et al. \cite{clayton03} used low-resolution observations in the 157 $-$ 318 nm range, which were obtained with the Space Telescope Imaging Spectrograph of the Hubble Space Telescope, the surveys reported by Gredel et al. \cite{gredel11} and Salama et al. \cite{salama11} covered high-resolution spectra for wavelengths longer than 305 nm recorded with the UVES spectrograph of the Very Large Telescope. However, besides the well-known UV bump at 217.5 nm, some atomic lines, and bands from small, mostly diatomic molecules, such as CH, CH$^+$, CN, OH$^+$, or NH, no narrow bands related to large gas-phase molecules were found. Based on the signal-to-noise ratio in these observations ($S/N >$ 100), very low upper fractional abundances for \textit{specific} PAHs in the diffuse interstellar medium (ISM) on the order of $N$(PAH)/$N$(H) $\approx$ $10^{-10}$ $-$ $10^{-8}$ (Gredel et al. \cite{gredel11}) were derived. On the other hand, inferred from the aromatic infrared emission bands, PAHs are expected to occur in large quantities in the ISM (estimated \textit{total} PAH abundances on the order of $10^{-7}$; Habart et al. \cite{habart04}; Tielens \cite{tielens08}).
Here, we show that mixtures of isolated and cold PAHs, which were produced in the laboratory under astrophysically relevant conditions, can display absorption curves that are smooth in the UV-visible spectral region if the molecular variety is sufficiently high. Hence, our results demonstrate that the astronomical detection of PAHs as a class of interstellar molecules in the infrared does not necessarily contradict the observed UV properties of interstellar extinction.
\section{Studied PAH mixtures}
The samples studied here were produced by infrared multiphoton dissociation of a gaseous hydrocarbon precursor (ethene, C$_2$H$_4$) followed by the gas phase synthesis of large carbonaceous molecules and grains. The applied CO$_2$ laser pyrolysis method is described elsewhere (J\"ager et al. \cite{jaeger09}; Steglich et al. \cite{steglich10}). The molecular components of the condensate were extracted with the solvents methanol (CH$_3$OH) or dichloromethane (CH$_2$Cl$_2$). The constituents of these extracts, which could unambiguously be identified, are known PAHs. However, PAH-like molecules with multiple hydrogen saturation, i.e., containing CH$_2$ groups and, thus, displaying aliphatic character, must be present as well (see below and J\"ager et al. \cite{jaeger09}). For simplicity, we will refer to all molecular components as ``PAHs''.
The analysis of the soluble condensate was mainly based on high-performance liquid chromatography (HPLC). With this method, different molecular components in a solvent can be separated according to their interaction time with the stationary phase of a column. For this study, we used a column from Jasco, specifically designed for the analytical fractionation and analysis of PAHs. A size separation of a molecular mixture can be realized since larger PAHs have longer retention times on the column than smaller ones. The chromatographic separation was realized with a two-solvent gradient program running at a temperature of 303 K and using methanol and dichloromethane at a flow rate of 1 ml/min. In combination with the retention time, the characteristic electronic absorption spectrum of each species can be used for identification. In our system, the threshold for the detection of a single species is about 5 ng. The HPLC spectrum of the soluble condensate and the absorption spectra of two selected components, which were identified to be the PAHs coronene (C$_{24}$H$_{12}$) and ovalene (C$_{32}$H$_{14}$), are displayed in Fig. \ref{figHPLC}. In the HPLC spectrum, distinct peaks are created only by the species whose product of ``abundance in the solvent'' and ``absorption strength at a given wavelength'' (here 254 nm) is high. Many other PAHs, hidden in the broad underlying continuum, are difficult to identify.
\begin{figure}
\centering
\includegraphics[scale=0.32]{fig1.eps}
\caption{High-performance liquid chromatography spectrum of the soluble components of the laser pyrolysis condensate. The PAHs coronene and ovalene are labeled ``A'' and ``B'', respectively. Inset: Comparison between the UV-visible absorption spectra of coronene and ovalene as measured during the HPLC of the sample (black curves) and their respective reference spectra (light blue curves).}\label{figHPLC}
\end{figure}
Since we are mainly interested in the larger PAHs of the condensate, we chose to divide the soluble extract into three different fractions (see the dashed lines in Fig. \ref{figHPLC}). The fraction with retention times between 0$-$7 min contains small molecules up to the size of molecules containing approximately 22 C atoms such as benzo[ghi]perylene (C$_{22}$H$_{12}$). The fraction ``7$-$20 min'' roughly covers the molecular range from anthanthrene (C$_{22}$H$_{12}$) to ovalene (C$_{32}$H$_{14}$). All larger species should be included in the last fraction ``20$-$35 min''. However, there are a few complications hindering a perfect size separation. First, the retention time is only roughly a monotonic function of the PAH size. The specific shape of each molecule (compact, elongated, or bent structure) does play a significant role, too, and can modify the retention time considerably. Second, larger molecules are more difficult to dissolve. The solubility even changes with the presence or absence of smaller species. This can lead to a ``smearing'' of certain individual components over the whole retention time scale. Especially traces of small PAHs with very high solubilities, such as anthracene (C$_{14}$H$_{10}$), phenanthrene (C$_{14}$H$_{10}$), or pyrene (C$_{16}$H$_{10}$), whose main peaks in the HPLC spectrum appear below 7 min, can also be found in the other two fractions that should normally contain only the larger species (see also the matrix spectra below).
With the HPLC technique, the identification of certain components of the soluble condensate is restricted to well-known molecules whose absorption spectra can be used for comparison. Therefore, only the presence of a few PAHs composed of an even number of C atoms and with ``normal'' H saturation (only aromatic CH groups) could be verified. On the other hand, as revealed by matrix-assisted laser desorption/ionization in combination with time-of-flight mass spectrometry (MALDI-TOF), the condensate is composed of a huge variety of different species. This is demonstrated in Fig. \ref{figMALDI}. In addition to the ``normal'' PAHs with an even number of C atoms, molecules containing an odd number of C atoms are apparent in the mass spectrum. These species, if structurally comparable to the other PAHs, must contain at least one non-aromatic CH$_2$ group to have a closed electronic shell structure. The aliphatic character of some constituents of the laser pyrolysis condensate was also verified via IR absorption spectroscopy, which will be shown in an upcoming publication (for comparison, see also J\"ager et al. \cite{jaeger06}; \cite{jaeger09}). The MALDI-TOF studies have yet another interesting result. Apart from the rather small species that can be analyzed and size-separated by HPLC, very large molecules up to 3000 u, corresponding to PAHs containing more than 200 C atoms, were found (see the inset in Fig. \ref{figMALDI} and J\"ager et al. \cite{jaeger09}). Unfortunately, it is not possible at the moment to enrich mixtures with such large PAHs in sufficient quantities for spectroscopic studies, mainly because of their poor solubility.
\begin{figure}
\centering
\includegraphics[scale=0.32]{fig2.eps}
\caption{Time-of-flight mass spectrum of the laser pyrolysis condensate. Inset: Overview over wider mass range (taken from J\"ager et al. \cite{jaeger09}).}
\label{figMALDI}
\end{figure}
The UV-visible absorption curves of the PAH mixtures presented later were measured by applying the matrix isolation technique. Within this method, the solid extracts have to be transferred to the gas phase before incorporation into the inert gas matrix. For this purpose, we applied thermal evaporation at temperatures $\leq$ 400 $^\circ$C as well as laser desorption with a pulsed Nd:YAG laser, which was operated with 10 Hz at 532 nm ($\leq$ 0.26 mJ/mm$^2$, pulse length ca. 5 ns). To prove how much was actually transferred into the gas phase, the evaporated materials were condensed on a cold shield and subjected to a second HPLC analysis. Compared to the HPLC spectrum of Fig. \ref{figHPLC}, where the HPLC system was optimized for the efficient fractionation of large PAHs, the retention times of the individual components shown in Fig. \ref{figHPLC2} were chosen to be longer, allowing a better tracing of the molecules. Note that the retention times of individual components can vary up to $\approx$ 0.5 min. Comparing the HPLC spectra of the evaporated materials with the spectrum of the original substance, the conclusion can be drawn that the laser desorption method is clearly better suited to transfer the PAH mixtures into the matrix without changing their composition. It follows that matrices doped with laser-evaporated PAHs resemble the original distribution, whereas matrices containing thermally evaporated species are dominated by molecules with high vapor pressures. Therefore, we expect the matrices prepared by laser desorption to reflect the same large variety of different species as encountered in the sample.
\begin{figure}
\centering
\includegraphics[scale=0.33]{fig3.eps}
\caption{Comparison of the HPLC spectrum of the original condensate obtained by laser pyrolysis with the spectra measured after the same material has been evaporated by either a laser or thermally in an oven. The peaks labeled with the letters ``A'' to ``M'' correspond to the displayed identified PAHs.}\label{figHPLC2}
\end{figure}
\section{UV-visible absorption spectra of mixtures of matrix-isolated PAHs}
The absorption spectra of the two condensate fractions ``7$-$20 min'' and ``20$-$35 min'' are presented in Fig. \ref{figMIS}. The molecules of the solid extracts were transferred either via laser desorption or by thermal evaporation into the gas phase and, thereafter, deposited with an excess of Ne atoms onto a cold CaF$_2$ window, which was kept at a temperature of 5.5 K. Based on previous experiments with individual PAHs and PAH mixtures, we can exclude the formation of PAH fragments in the vapor phase and their accumulation in the matrix (see also the discussion along with Fig. \ref{figHPLC2}). For comparison, the spectrum of an undoped Ne matrix of similar thickness is displayed in the top panel of Fig. \ref{figMIS}. Under the present experimental conditions, the Ne matrices are supposed to be mainly crystalline (Cradock \& Hinchcliffe \cite{cradock75}). Site effects causing a molecular band to split up in the matrix spectrum can be excluded, i.e, each band in the matrix spectrum has a one-to-one correspondence to the respective band in the gas phase. The undoped matrix does not exhibit any absorption in the investigated wavelength range. The weak contribution to scattering at short wavelengths can almost be neglected. Basically, we also exclude possible additional light scattering in the PAH-doped matrices as the dimensions of the molecules are much smaller than the wavelength. The extinction of all shown spectra is mainly due to PAH absorption.\footnote{For comparison, see also the spectra of individual PAHs in Ne matrices and as solid films (Steglich et al. \cite{steglich10}, \cite{steglich11}), where the light scattering can be neglected compared to the absorption.} Because of small baseline variations, primarily caused by different sample reflectivities, we shifted all measured spectra at 850 nm to zero absorbance.
As already stated above, the matrices that were doped by laser vaporization contain a larger variety of different species. Obviously, this leads to quite smooth UV-visible absorption curves with almost no fine structure for both fractions. Although this continuous absorption caused by close-lying and overlapping bands is present in all spectra, the matrices prepared by thermal evaporation additionally feature many sharp signatures caused by an overabundance of species with high vapor pressure. A few of these sharp bands originate from known PAHs, such as coronene and ovalene, already previously identified in the HPLC extract ``7$-$20 min''. Unfortunately, many features have to remain unidentified because appropriate spectra of individual PAHs that are larger than, or equal in size to, anthanthrene, isolated in solid Ne are not available for comparison. At least some of the peaks of unknown origin in the matrix spectra probably belong to PAHs that could already be verified via HPLC (see Figs. \ref{figHPLC} and \ref{figHPLC2}).
The matrix spectra of the fraction ``20$-$35 min'', which should contain larger molecules composed of more than 32 C atoms, display only a few weak and narrow bands. Except for the bands at wavelengths longer than 400 nm, they can be considered to be impurities because they belong to the very small, high vapor pressure PAHs anthracene, phenanthrene, and pyrene. As previously mentioned, trace amounts of these small molecules are present in all fractions as a result of imperfect size separation by HPLC. Finally, note that, while the matrices prepared by thermal evaporation exhibit stronger absorbance at short wavelengths compared to the corresponding matrices prepared by laser desorption, the situation is reversed at long wavelengths. The spectra intersect each other somewhere between 300 and 400 nm (see the top panel in Fig. \ref{figMIS}). This can be easily explained by differences in the fractional abundances of larger PAHs with stronger contribution to absorption at longer wavelengths.
\begin{figure*}
\centering
\includegraphics[scale=0.33]{fig4.eps}
\caption{Absorption spectra of PAH mixtures obtained after HPLC fractionization of the soluble laser pyrolysis condensate. The molecules were incorporated into solid Ne matrices kept at 5.5 K. A clean Ne matrix of comparable thickness is shown for comparison. Top panel: Overview over all measurements. The spectra were corrected for weak baseline variations, mainly caused by differences in the sample reflectivity, by shifting them to zero absorbance at 850 nm. Other panels: Numbers given at the right-hand side indicate that the spectra were shifted in $y$-direction for a better comparison. The vertical lines mark the band positions of PAHs that could be identified based on their characteristic peaks in the Ne matrix spectra. The structures of the corresponding molecules are displayed, accordingly.}
\label{figMIS}
\end{figure*}
To summarize, if the original variety of larger PAHs in the size distribution is kept, as is the case when laser desorption is used, the resulting Ne matrix spectrum reveals a fairly smooth decay for wavelengths longer than $\sim$ 250 nm. We do not observe any sharp absorption band that can be assigned to a specific PAH. On the other hand, based on the measured absorbance, we can be sure that the larger PAHs were indeed incorporated into the matrix.
\section{Astrophysical implications -- extension to larger species in the gas phase}
The experiments described above demonstrate that a PAH mixture composed of a sufficient amount of different species exhibits a featureless absorption curve. However, there are a few points that have to be considered if we aim to apply our experimental results to PAH mixtures in the ISM. The first and most obvious point concerns the perturbations induced by the weak interaction forces between the studied molecules and the Ne atoms of the host matrix. Compared to the spectra of completely free gas-phase PAHs, the absorption bands of the individual species in the matrix are redshifted and broadened. Obviously, the broadening promotes the creation of a smooth absorption curve, and some narrow features could perhaps be visible if the PAHs were isolated in the gas phase. However, the matrix-induced broadening usually affects only those bands that involve the first one or two electronic transitions in a closed-shell molecule, i.e., S$_0$$\rightarrow$S$_{1,2}$. The bands of these transitions can be very narrow in the gas phase and would appear substantially broadened in a Ne matrix (see, e.g., Gredel et al. \cite{gredel11}). Nevertheless, as inferred from the aromatic infrared emission bands, the bulk of interstellar PAHs are probably larger than the species we studied here ($\gtrsim$ 40 C atoms; Tielens \cite{tielens08}). The first electronic transitions of these larger molecules are generally located at wavelengths longer than $\approx$ 400 nm (see, e.g., Ruiterkamp et al. \cite{ruiterkamp02}; Malloci et al. \cite{malloci07}). At shorter wavelengths, the bandwidths are usually governed by the short lifetimes of the excited (high-energy) states and by vibrational interactions with energetically lower-lying states (Rouill\'e et al. \cite{rouille09}), i.e., they should be very similar for completely free and matrix-isolated molecules (Steglich et al. \cite{steglich11}). Therefore, a smooth extinction curve at $\lambda \lesssim$ 400 nm can be expected if the molecular variety of the interstellar PAH population is at least comparable to our laboratory-produced mixtures. Probably, the interstellar PAHs show an even higher diversity because the number of isomers raises rapidly with the size of the molecules. Furthermore, free-flying PAHs are exposed in the interstellar space to ionizing irradiation. The formation of ions, which, compared to their neutral precursors, display even broader bands below 400 nm (Steglich et al. \cite{steglich11}), additionally increases the molecular diversity.
A non-negligible part of the interstellar PAHs might also be locked in clusters or on the surface of grains (see, e.g., Fig. 6 in Tielens \cite{tielens08}). The absorption spectrum of larger clusters, if bonded only by van der Waals forces, can be simulated in the laboratory by measuring the spectrum of a thin film of PAHs deposited on a transparent substrate. An example is presented in Fig. \ref{figBUMP}. The film-like deposit of the extract ``7$-$20 min'' was prepared by laser desorption and subsequent deposition onto a 5.5 K cold CaF$_2$ window (without simultaneous Ne flow). The individual molecules of the film were allowed to relax into energetically favored positions by keeping the sample at room temperature and under vacuum for several hours.\footnote{Upon cooling the sample back to cryogenic temperatures afterward, the absorption spectrum does not change anymore. The widths and positions of the spectral features are mainly governed by the structural arrangement of the molecules.} The resulting absorption spectrum does not display any narrow feature. Only a broad bump is apparent at 208 nm. For comparison, the Ne matrix spectrum of the laser-evaporated extract ``7$-$20 min'' is also included in Fig. \ref{figBUMP}. There, the bump appears considerably narrower and further to the blue at 195 nm. It is expected to shift to longer wavelengths for mixtures with a larger mean size of the aromatic planes, approaching the position of the interstellar UV bump at 217.5 nm (Steglich et al. \cite{steglich10}).
\begin{figure}
\centering
\includegraphics[scale=0.31]{fig5.eps}
\caption{Absorption spectrum of a film-like deposit of the HPLC fraction ``7$-$20 min'' compared with the spectrum of the same sample in the Ne matrix. The more noisy gray part of the spectrum was recorded with a vacuum-UV spectrometer.}
\label{figBUMP}
\end{figure}
\section{Conclusions}
We presented experimental UV-visible absorption spectra of mixtures of cold and isolated PAHs, which were produced in the laboratory under astrophysically relevant conditions. We showed that the spectra can be almost completely featureless if the molecular variety is sufficiently high. The present results and their interpretation are also in line with another recent study implying that the DIBs, unlike the interstellar UV bump, very probably do not originate from PAHs (Steglich et al. \cite{steglich11}). This is furthermore supported by an investigation from Xiang et al. (\cite{xiang11}), who were unable to find any correlation between the 2175 $\AA$ feature and nine of the strongest DIBs. Finally, if we assume that the optical emission bands in the Red Rectangle Nebula are caused by some of the DIB carriers (see Sarre \cite{sarre06}), there is yet another argument against a connection between the interstellar PAH population and the DIBs because no correlation was found between these optical bands and the aromatic infrared emission at 3.3 $\mu$m (Sarre \cite{sarre06}).
In summary, we propose that similar to the aromatic emission bands in the infrared, the interstellar PAHs give only rise to collective features also in the UV-visible, i.e., an enhanced but rather smooth extinction culminating in a broad bump at 2175 $\AA$, and not individual fingerprints, which could be used for an identification of specific molecules. The absence of narrow absorption bands related to larger gas-phase polyaromatic molecules on the interstellar extinction curve for 300 nm $< \lambda <$ 400 nm can be explained by a high molecular diversity of the interstellar PAH population. The fractional abundances of individual species are too low to allow their detection on the basis of their characteristic electronic spectra. Nevertheless, high-resolution observations at wavelengths shorter than 300 nm are still lacking. Such measurements could help to verify whether or not one can detect the electronic fingerprints of individual large organic molecules in the ISM.
\begin{acknowledgements}
Part of this work was supported by the \emph{Deut\-sche For\-schungs\-ge\-mein\-schaft, DFG\/} project number Hu 474/21-2. We thank H. Mutschke for providing access to the vacuum-UV spectrometer and G. Born for technical assistance with the chemical extraction and HPLC operation.
\end{acknowledgements}
|
{
"timestamp": "2012-04-09T02:01:19",
"yymm": "1203",
"arxiv_id": "1203.2026",
"language": "en",
"url": "https://arxiv.org/abs/1203.2026"
}
|
\section{Introduction}
\label{sec:introduction}
IC~1396 is a large H\,{\sc ii}\ region in the Cep OB2 association, excited by the O6.5V star HD~206267. Its border includes many bright-rimmed molecular clouds \citep{Weikard1996}, among which IC~1396A is the closest globule, 3.7~pc away from HD~206267 in the plane of the sky assuming a 750~pc distance to IC~1396 \citep{Matthews1979}. IC~1396A has a more negative velocity ($\sim -8$~km\,s$^{-1}$) compared to HD~206267 and several of the other bright-rimmed clouds, indicating that it is located in front of the star and moving toward us \citep{Weikard1996}. IC~1396A thus provides a simple example of photodissociation regions (PDRs) on a globule, illuminated by a single star from one side. Within the globule a cavity associated with the young stellar object (YSO) LkH$\alpha$ 349 is seen in IRAC 8\,$\mu$m, $^{12}$CO and $^{13}$CO(1-0) maps \citep{Nakano1989}, and the extinction map obtained from The Two Micron All-Sky Survey, 2MASS, data \citep{Reach2009}. There are additional cavities associated with YSOs, and \citet{Reach2009} suggested that the globule is being reshaped from the inside, with each protostar residing in a ``compartment'' that it has blown out via its outflow. Therefore, high spatial resolution, as provided by the German REceiver for Astronomy at Terahertz Frequencies \citep[GREAT\footnote{GREAT is a development by the MPI f\"{u}r Radioastronomie and the KOSMA / Universit\"{a}t zu K\"{o}ln, in cooperation with the MPI f\"{u}r Sonnensystemforschung and the DLR Institut f\"{u}r Planetenforschung};][]{Heyminck2012} onboard the Stratospheric Observatory for Infrared Astronomy \citep[SOFIA;][]{Becklin2009,Young2012} at THz-frequencies is essential for understanding the structure and evolution of the IC~1396A globule.
\section{Observation and data reduction}
\subsection{[C\,{\sc ii}]\ observations with SOFIA/GREAT}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{plot_integmap_w_cico_paper.eps}
\caption{Integrated ($-5$ to $-15$~km\,s$^{-1}$) intensity map of the [C\,{\sc ii}]\ emission (color), overlayed with the contours of (a) IRAC 8\,$\mu$m, (b) velocity-integrated [C\,{\sc i}]\ ${}^3P_1 - {}^3P_0$ and (c) CO(4-3). The contour spacing is (a) $10$~MJy\,sr$^{-1}$, (b) 2~K\,km\,s$^{-1}$, and (c) 10~K\,km\,s$^{-1}$. The (0,0)-position is $21^\mathrm{h}36^\mathrm{m}50^\mathrm{s}.7$, $57^\circ31^\prime10^{\prime\prime}$ (J2000) at LkH$\alpha$~349. Five asterisks in (b) mark the positions of the spectra shown in Fig.~\ref{fig:spectra}. HD~206267, the exciting star, is located at ($\Delta$RA, $\Delta$DEC)$=$($17.1$\arcmin, $-1.8$\arcmin) and illuminates the globule from the east.}
\label{fig:cii_integmap}
\end{figure*}
We mapped the [C\,{\sc ii}]\ emission at 1900.5369~GHz (158\,$\mu$m) with GREAT onboard SOFIA during the basic science flights on 22 and 28 July 2011. On the second flight we observed only the [C\,{\sc ii}]\ emission with the L2 channel, whereas on the first flight the L1 channel was operated in parallel tuned to CO(12-11) at 1381.9951050~GHz. The observations were made in total-power on-the-fly (OTF) mode, with 1 second integration time at each dump, and 8\arcsec step size. The OFF position is located east and outside of the globule at $21^\mathrm{h}37^\mathrm{m}27^\mathrm{s}.9$, $57^\circ31^\prime10^{\prime\prime}$ (J2000).
Calibration was made by the standard pipeline \citep{Guan2012}. The forward efficiency is 0.95, and the beam efficiency is 0.54 and 0.51 for L1 and L2, respectively \citep{Heyminck2012}. Because the emission line is sufficiently narrow, only a linear baseline was fitted between $-30$ and $-20$~km\,s$^{-1}$, and between $0$ and $20$~km\,s$^{-1}$. After excluding data suffering from strong standing waves and discarding spectra with excess noise, the data were spectrally resampled to a $0.5$~km\,s$^{-1}$\ resolution and spatially to 25\arcsec\ (0.09~pc at the distance of 750~pc) to obtain a better signal-to-noise ratio (S/N). In the following, we use the data from the XFFTS-backend, because the other backends show fully consistent and hence redundant data \citep[see ][ for details of the spectrometers]{Heyminck2012}.
Fig.~\ref{fig:cii_integmap} shows the integrated line intensity, summed over the velocity range from $-5$ to $-15$~km\,s$^{-1}$. The error in the line-integrated intensity, estimated from the rms noise of the baseline varies between 2--6~K\,km\,s$^{-1}$\ and depends on the number of times that the map point was observed. CO(12-11) is not detected, and the upper limit is 4--13~K\,km\,s$^{-1}$\ (see the appendix for details of the derivation).
\subsection{Complementary observations}
Complementary observations of [C\,{\sc i}]\ ${}^3P_1 - {}^3P_0$-line at 492.1606510~GHz and CO(4-3) at 461.0407682~GHz were performed between October 2000 and April 2001 using the CHAMP array receiver \citep{Guesten1998} at the CSO. The array with $2\times 8$ pixels was operated in frequency-switching mode (throw $\pm 25$~MHz for CO and $\pm 10$~MHz for [C\,{\sc i}], rate $1$~Hz). CO(4-3) data were recorded on-the-fly, the weaker [C\,{\sc i}]\ in raster mode. The auto-correlator back-end array in its high-resolution mode provided 2048 channels with $0.15$ km\,s$^{-1}$\ spectral resolution. Main beams were 15\arcsec\ at 464 GHz and 14.5\arcsec\ at 492 GHz, with respective main beam efficiencies of 0.52 and 0.51 \citep[see ][ for more details of the set-up]{Philipp2006}. Both maps were convolved to 25\arcsec\ angular resolution. The integrated line intensities were obtained in the same manner as for the [C\,{\sc ii}]\ emission (Fig.~\ref{fig:cii_integmap}b, c).
\section{Results and discussion}
\subsection{Dynamics of the [C\,{\sc ii}]\ emitting gas}
Figure~\ref{fig:cii_integmap}a shows an overlay of the [C\,{\sc ii}]\ emission and the IRAC 8\,$\mu$m\ emission, which traces polycyclic aromatic hydrocarbons (PAHs) in PDRs, showing a good spatial match between the two. The [C\,{\sc ii}]\ emission also follows the local cavities originating from the YSO outflows mentioned in Sect.~\ref{sec:introduction}. One exception is the rim at the eastern edge of the globule, where the IRAC 8\,$\mu$m\ has a strong peak while the [C\,{\sc ii}]\ emission is not prominent. The spatial distributions of [C\,{\sc i}]\ and CO(4-3) are different from [C\,{\sc ii}]. CO(4-3) is widely distributed over the globule, having a shallow peak in the middle. The [C\,{\sc i}]\ emission is distributed toward the southeast and farther out compared to CO(4-3), following the ridge-like structure from ($\Delta$RA, $\Delta$DEC)$\sim$($1$\arcmin, $0$\arcmin) to ($0$\arcmin, $-2$\arcmin) in the IRAC 8\,$\mu$m\ emission. The [C\,{\sc ii}]\ emission also follows this rim, but it is much stronger toward the northern part of the globule.
\citet{Wootten1983} showed the intensity distribution and the position-velocity diagram of CO(2-1) and $^{13}$CO(2-1) along the longitude cut that intersects LkH$\alpha$~349 (located at offset (0,0)). The CO(2-1) distribution is similar to that of CO(4-3), and $^{13}$CO(2-1) is similar to [C\,{\sc i}]\ and [C\,{\sc ii}]\ in the sense that the intensity drops around LkH$\alpha$~349. On the other hand, \citet{Nakano1989} showed the cavity near LkH$\alpha$~349 both in CO(1-0) and $^{13}$CO(1-0) although it appears as a much steeper hole in the latter. These differences are compatible with being caused by opacity: $^{12}$CO is more optically thick than $^{13}$CO, and $^{12}$CO(2-1) and (4-3) emissions have a higher optical depth than $^{12}$CO(1-0) in LTE gas with $T\gtrsim 10$~K, although detailed radiative transfer modeling is needed to quantify it.
The channel map of [C\,{\sc ii}]\ (Fig.~\ref{fig:channelmap}) shows that the gas near the rim of the globule is redshifted; going west and toward the center of the globule the velocity becomes more negative. With the globule in front of the exciting star and moving toward us as mentioned in Sect.~\ref{sec:introduction}, this velocity gradient implies that the western part, which is farther away from the illuminated rim, moves faster away from the exciting source. The upper right panel of Fig.~\ref{fig:channelmap} shows two [C\,{\sc ii}]\ components with significantly higher negative velocity (see also the spectra at positions D and E in Fig.~\ref{fig:spectra}). In the southern component (position E) the local outflows from the YSOs may also affect the velocity field because there are several YSOs in the void-like structure from ($\Delta$RA, $\Delta$DEC)$\sim$($1.5$\arcmin, $0$\arcmin) to ($0.5$\arcmin, $-2$\arcmin) \citep{Reach2009}, and the IRAC 8\,$\mu$m\ emission shows a cometary shape toward the southwest.
On the other hand, the [C\,{\sc i}]\ and CO(4-3) emitting gas does not show a significant velocity gradient. Figure~\ref{fig:spectra} shows the line profiles at the five positions marked in Fig.~\ref{fig:cii_integmap}b. CO(4-3) has a slight blueshift at position E, which is also indicated by \citet{Wootten1983} in the CO(2-1) position-velocity diagram along the latitude cut. However, there is a clear difference between the line profiles of [C\,{\sc ii}]\ and CO(4-3). Considering the spatial distribution discussed above, it seems that the gas emitting [C\,{\sc ii}]\ is blown away from the rim. The numerical simulations of the evolution of cometary globules by \citet{Lefloch1994} suggest that such a cometary regime lasts $\sim$ 90\% of the cloud's lifetime, and numerous small clumps that are ejected and accelerated along the globule appear as ``fuzz'' on the blue side of Lefloch \& Lazareff's model position-velocity diagram. The observed structures match this cometary regime.
\begin{figure}
\centering
\includegraphics[width=0.49\textwidth]{plot_channelmap_ic1396a_n01_presen.eps}
\caption{Channel maps of the [C\,{\sc ii}]\ emission overlayed with the contours of IRAC 8\,$\mu$m\ emission.}
\label{fig:channelmap}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.4\textwidth]{pickup_spectra_map_paper.eps}
\caption{Spectra of [C\,{\sc ii}], CO(4-3) and [C\,{\sc i}]\ at the five positions marked in Fig.~\ref{fig:cii_integmap}.}
\label{fig:spectra}
\end{figure}
\subsection{Comparison with PDR models}
\begin{table*}
\caption{PDR model results. Values in parentheses show the ranges defined by the uncertainties.}
\label{table:modelresults}
\centering
\begin{tabular}{|c|ccc|cc|}
\hline
position & \multicolumn{3}{c|}{Model 1 (fixed $\chi$)} & \multicolumn{2}{c|}{Model 2 (fixed $\log(m)=-1$)}\\
\cline{2-6}
& $\log(\chi)$ & $\log(n)$ & $\log(m)$ & $\log(n)$ & $\log(\chi)$\\
\hline
A & $2.0$ & $5.0$ ($4.5$ -- $5.4$) & $-2.3$ ($-3.0$ -- $+ 0.8$) & $5.2$ ($4.2$ -- $6.1$) & $ 2.5$ ($ 1.4$ -- $ 3.1$) \\
B & $1.9$ & \ \ -- \ \ ($4.4$ -- $5.1$) & \ \ -- \ \ ($ 0.0$ -- $+ 3.0$) & $4.1$ ($3.8$ -- $5.0$) & $ 1.0$ ($ 0.6$ -- $ 1.8$) \\
C & $1.9$ & $4.7$ ($4.3$ -- $5.3$) & $-0.8$ ($-3.0$ -- $+ 1.7$) & $4.7$ ($4.0$ -- $5.7$) & $ 1.9$ ($ 1.2$ -- $ 2.7$) \\
D & $1.8$ & $4.9$ ($4.3$ -- $5.1$) & $-2.9$ ($-3.0$ -- $+ 0.2$) & $5.0$ ($4.0$ -- $5.9$) & $ 2.5$ ($ 1.3$ -- $ 3.1$) \\
E & $1.8$ & $4.9$ ($4.4$ -- $5.2$) & $-2.0$ ($-3.0$ -- $+ 0.8$) & $5.0$ ($4.0$ -- $5.9$) & $ 2.2$ ($ 1.2$ -- $ 2.8$) \\
\hline
\end{tabular}
\end{table*}
We compared the observed integrated intensity ratios between [C\,{\sc ii}], [C\,{\sc i}]\ ${}^3P_1 - {}^3P_0$, and CO(4-3) with the KOSMA-$\tau$ PDR model \citep{Roellig2006} at the five positions shown in Fig.~\ref{fig:spectra}. To exclude the contribution of the velocity components in [C\,{\sc ii}]\ that do not match with the CO and [C\,{\sc i}]\ emission, we fitted the [C\,{\sc ii}]\ emission line with the CO(4-3) line profile at each position and adopted the integrated intensity of this fit result. At position E, the intensity obtained this way is 57\% of the total integral of the [C\,{\sc ii}]. We used a single-clump model, which has as free parameters the mean gas density ($n$), the far-ultraviolet (FUV; $h\nu =$ 6--13.6~eV) flux ($\chi$) in units of the Draine field ($2.7\times 10^6$~W\,m$^{-2}$), and the mass of the single clump ($m$) in units of the solar mass ($M_\odot$). We used two simplified model approaches because with three observed lines, there are only two independent ratios but three unknowns. In Model~1, we fixed $\chi$ at the value estimated from the luminosity of the exciting star HD~206267, $10^{5.23}L_\odot$, as an O6.5V star, and the distance to this star from each observed position, assuming that 50\% of the photons have an energy within 6~eV $< h\nu <$ 13.6~eV. The calculated value of $\chi$ is listed in Table~\ref{table:modelresults}. Then we determined $n$ and $m$ by fitting the intensity ratios of [C\,{\sc ii}]/[C\,{\sc i}]\ and [C\,{\sc ii}]/CO(4-3), as listed in Table~\ref{table:modelresults}. To be conservative, we estimated the error by assuming that each line intensity has a systematic uncertainty of 30\%. At position B, no combination of $n$ and $m$ can explain the observed ratio. $m$ is not well constrained because these ratios are insensitive to the mass; the layers that emit [C\,{\sc ii}], [C\,{\sc i}]\ ${}^3P_1 - {}^3P_0$, and CO(4-3) emissions are not the central core but outer layers. With $\chi=10^2$ and $n=10^5$cm$^{-3}$, $A_V$ at the center is $\sim 4$ even for a clump with $m=10^{-3}M_\odot$. Increasing $m$ corresponds to adding layers at deeper $A_v$, which does not significantly change the ratio between the [C\,{\sc ii}], [C\,{\sc i}]\ ${}^3P_1 - {}^3P_0$, and CO(4-3) intensities. Therefore, we fixed $\log(m)$ in Model~2 to a value of $-1$, i.e.\ $0.1M_\odot$, and fitted $n$ and $\chi$. A clump with $m=0.1M_\odot$ and $n=10^5$cm$^{-3}$\ has a radius of $\sim 0.02$~pc, which is $\sim 5\arcsec$ at the distance of 750~pc, small compared to the beam size. The obtained $n$ from both models are consistent within the errors, giving $10^4$--$10^5$cm$^{-3}$\ at position B and $\sim 10^5$cm$^{-3}$\ for the other positions. The fitted $\chi$ in Model~2 is consistent with the values estimated from the luminosity of the star for Model~1 except for position B. At position B, the very low [C\,{\sc ii}]/[C\,{\sc i}]\ ratio compared to the other positions is explained by a low UV field in Model~2. It implies that the UV radiation is shielded at the rim of the globule.
\section{Summary}
We presented mapping observations of the [C\,{\sc ii}]\ emission at 1900.5369~GHz (158\,$\mu$m) with GREAT onboard SOFIA in IC1396A, which is an illuminated globule with internal structures created by embedded YSOs. The [C\,{\sc ii}]\ emission closely follows the IRAC 8\,$\mu$m\ emission, which traces the PAH emission. Together with complementary [C\,{\sc i}]\ ${}^3P_1 - {}^3P_0$ and CO(4-3) observations, we investigated the spatial distributions of the velocity-resolved emission. [C\,{\sc ii}]\ shows significant velocity changes within the globule, while this is not the case for the [C\,{\sc i}]\ and CO(4-3) emission. The spatial distribution is also different: CO(4-3) has a shallow peak at the center of the globule, the [C\,{\sc ii}]\ and [C\,{\sc i}]\ emission is strong at outer region of the globule, but their peaks show different locations. The spatial structure and velocity distribution is consistent with a scenario in which spatially unresolved clumps, which emit mainly [C\,{\sc ii}], are blown away along the globule and are accelerated toward us. A PDR model analysis of the observed intensity ratios indicates densities of $\sim 10^5$ cm$^{-3}$\ and a UV flux consistent with excitation by the external star except for one position, where the UV is indicated to be low because of the shielding at the rim of the globule. IC1396 is selected as one of the key regions to be observed by the GREAT consortium because it provides many bright-rimmed globules with simple exciting sources, and observations in wider regions are expected in future flights.
\begin{acknowledgements}
This work is based in part on observations made with the NASA/DLR Stratospheric Observatory for Infrared Astronomy. SOFIA Science Mission Operations are conducted jointly by the Universities Space Research Association, Inc., under NASA contract NAS2-97001, and the Deutsches SOFIA Institut under DLR contract 50 OK 0901. We thank the SOFIA engineering and operations teams whose support has been essential for the GREAT accomplishments during basic science flights, and the DSI telescope engineering team.
\end{acknowledgements}
\bibliographystyle{aa}
|
{
"timestamp": "2012-03-09T02:02:35",
"yymm": "1203",
"arxiv_id": "1203.1772",
"language": "en",
"url": "https://arxiv.org/abs/1203.1772"
}
|
\section*{Introduction}
The {Friedmann-Lema\^itre-Robertson-Walker} (FLRW) spacetime represents, in conformity with General Relativity, the universe at very large scale, where it is assumed to be homogeneous and isotropic \cite{FRI22de,FRI99en,FRI24,LEM27,ROB35I,ROB35II,ROB35III,WAL37}. At the Big Bang, it exhibits a singularity, considered a breakdown of General Relativity \cite{HP70,Haw76,ASH91,HP96,Ash08,Ash09}. There the tensors involved in the Einstein equations diverge. It was hoped that this problem is a consequence of the high symmetry of the {FLRW} solution, and it disappears in real-life situations, but it turned out that is the norm, rather than the exception, as Hawking's singularity theorem shows \cite{Haw66i,Haw66ii,Haw67iii}.
In this paper we will see that, although Einstein's equation in its standard form breaks down at the Big Bang singularity in the {FLRW} model, it can be rewritten in a form which doesn't break down:
\begin{equation*}
(G\circ g)_{abcd} + \Lambda (g\circ g)_{abcd} = \kappa (T\circ g)_{abcd}.
\end{equation*}
This form is equivalent to the standard form where the metric is not singular, but in addition extends smoothly at the singularities. Like Einstein's equation, this new equation is tensorial, but instead of being expressed in terms of the \textit{Ricci tensor} (of second order), it involves \textit{the Ricci part of the Riemann curvature tensor} (of order $4$). It is obtained by taking the Kulkarni-Nomizu product \eqref{eq_kulkarni_nomizu} between Einstein's equation and the metric tensor.
The proposed new version of Einstein's equation extends naturally and smoothly beyond the Big Bang singularity of the {FLRW} model, and it is based solely on standard General Relativity, without any modifications.
In section \sref{s_flrw} we review briefly the fact that in the {FLRW} model, Einstein's equation blows up. In \sref{s_einstein_eq_expanded} we introduce the expanded version of Einstein's equation, which is equivalent to the latter at the points where the metric is {non{-\penalty0\hskip0pt\relax}singular}, but it is more general. Section \sref{s_beyond_big_bang} contains the central result, a theorem showing that the new equation is smooth everywhere, including at the Big Bang singularity. Section \sref{s_properties} discusses some properties of the proposed equation and solution. We conclude with some observations and implications in \sref{s_conclusions}.
\section{{FLRW} Big-Bang singularity}
\label{s_flrw}
Let $I\subseteq \mathbb{R}$ be an interval representing the time, with the natural metric $-c^2\textnormal{d} t^2$. Let $(\Sigma,g_\Sigma)$ be a three-dimensional Riemannian space, so that at any moment of time $t\in I$ the space is $\Sigma_t=(\Sigma,a^2(t)g_\Sigma)$, where $a: I\to \mathbb{R}$ is a function of time. The {FLRW} spacetime is $I\times\Sigma$, with the metric
\begin{equation}
\label{eq_flrw_metric}
\textnormal{d} s^2 = -c^2\textnormal{d} t^2 + a^2(t)\textnormal{d}\Sigma^2.
\end{equation}
To model the homogeneity and isotropy conditions at large scale, one may take the Riemannian three-manifold $\Sigma$ to be one of the homogeneous spaces $S^3$, $\mathbb{R}^3$, and $H^3$. Then the metric on $\Sigma$ is, in spherical coordinates $(r,\theta,\phi)$,
\begin{equation}
\label{eq_flrw_sigma_metric}
\textnormal{d}\Sigma^2 = \dsfrac{\textnormal{d} r^2}{1-k r^2} + r^2\(\textnormal{d}\theta^2 + \sin^2\theta\textnormal{d}\phi^2\),
\end{equation}
where $k=1,0,-1$, for the $3$-sphere $S^3$, the Euclidean space $\mathbb{R}^3$, or hyperbolic space $H^3$ respectively.
If the universe is filled with a fluid with mass density $\rho(t)$ and pressure density $p(t)$, the stress-energy tensor is
\begin{equation}
\label{eq_friedmann_stress_energy}
T^{ab} = \(\rho + \dsfrac{p}{c^2}\)u^a u^b + p g^{ab},
\end{equation}
where $g(u,u)=-c^2$.
The mass density $\rho(t)$ is determined from $a(t)$ by the \textit{Friedmann equation}
\begin{equation}
\label{eq_friedmann_density}
\rho = \kappa^{-1}\(3\dsfrac{\dot{a}^2 + kc^2}{c^2 a^2} - \Lambda \),
\end{equation}
and pressure density $p(t)$ by the \textit{acceleration equation}
\begin{equation}
\label{eq_acceleration}
\dsfrac{p}{c^2} = \dsfrac{2}{\kappa c^2}\(\dsfrac{\Lambda}{3}-\dsfrac{1}{c^2} \dsfrac{\ddot{a}}{a}\) - \dsfrac \rho 3.
\end{equation}
Both these equations are consequence of the Einstein equation with the stress-energy tensor from equation \eqref{eq_friedmann_stress_energy}.
When $a\to 0$, the metric becomes degenerate, as we can see from equation \eqref{eq_flrw_sigma_metric}. In the same time, equations \eqref{eq_friedmann_density} and \eqref{eq_acceleration} imply that both $\rho$ and $p$ blow up. Consequently, the stress-energy tensor \eqref{eq_friedmann_stress_energy} blows up too. But the expanded stress-energy tensor $(T \circ g)_{abcd}$ is smooth, as it is the expanded Einstein equation which we propose here.
\section{Einstein's equation expanded}
\label{s_einstein_eq_expanded}
The Einstein equation
\begin{equation}
\label{eq_einstein}
G_{ab} + \Lambda g_{ab} = \kappa T_{ab}
\end{equation}
involves the stress-energy tensor $T_{ab}$ of the matter, the \textit{cosmological constant} $\Lambda$, and the constant $\kappa:=\dsfrac{8\pi \mc G}{c^4}$, where $\mc G$ and $c$ are the gravitational constant and the speed of light.
The Einstein tensor
\begin{equation}
\label{eq_einstein_tensor}
G_{ab}:=R_{ab}-\frac 1 2 R g_{ab},
\end{equation}
is obtained from the \textit{Ricci curvature} $R_{ab} := g^{st}R_{asbt}$ and the \textit{scalar curvature} $R := g^{st}R_{st}$.
The \textit{expanded Einstein equation} is
\begin{equation}
\label{eq_einstein_expanded}
(G\circ g)_{abcd} + \Lambda (g\circ g)_{abcd} = \kappa (T\circ g)_{abcd}
\end{equation}
where, for two symmetric bilinear forms $h$ and $k$,
\begin{equation}
\label{eq_kulkarni_nomizu}
(h\circ k)_{abcd} := h_{ac}k_{bd} - h_{ad}k_{bc} + h_{bd}k_{ac} - h_{bc}k_{ad}
\end{equation}
denotes the \textit{Kulkarni-Nomizu product}.
So long as the metric $g$ is {non{-\penalty0\hskip0pt\relax}singular}, the Einstein equation and its expanded version are equivalent. But the expanded version \eqref{eq_einstein_expanded} remains {non{-\penalty0\hskip0pt\relax}singular}, and even smooth, in a wider rage of cases. In the example of the {FLRW} spacetime, the metric becomes degenerate (its determinant cancels), the Einstein tensor $G_{ab}$ becomes singular, but we will see that the Kulkarni-Nomizu product $G\circ g$ tends to $0$ and cancels the blow up of the Einstein tensor.
Explicitly, the expanded Einstein equation \eqref{eq_einstein_expanded} can be rewritten as
\begin{equation}
\label{eq_einstein_expanded_explicit}
2 E_{abcd} - 3 S_{abcd} + \Lambda (g\circ g)_{abcd} = \kappa (T\circ g)_{abcd},
\end{equation}
in terms of the \textit{scalar part}
\begin{equation}
S_{abcd} = \dsfrac{1}{12}R(g\circ g)_{abcd}
\end{equation}
and the \textit{semi-traceless part}
\begin{equation}
E_{abcd} = \dsfrac{1}{2}(S \circ g)_{abcd},
\end{equation}
of the Riemann curvature, where
\begin{equation}
\label{eq_ricci_traceless}
S_{ab} := R_{ab} - \dsfrac{1}{4}Rg_{ab}
\end{equation}
is the traceless part of the Ricci curvature.
These tensors are well-known from the Ricci decomposition of the Riemann curvature tensor:
\begin{equation}
R_{abcd} = S_{abcd} + E_{abcd} + C_{abcd},
\end{equation}
where $C_{abcd}$ is the \textit{Weyl curvature tensor} (see \textit{e.g.} \cite{ST69,BESS87,GHLF04}).
The equation \eqref{eq_einstein_expanded_explicit} is obtained from \eqref{eq_einstein_expanded} and from
\begin{equation}
G_{ab} = S_{ab} - \dsfrac{1}{4}R g_{ab},
\end{equation}
because
\begin{equation}
\label{eq_einstein_tensor_expanded}
\begin{array}{lrl}
(G\circ g)_{abcd} &=& (S \circ g)_{abcd} - \dsfrac{1}{4}R (g\circ g)_{abcd}\\
&=& 2 E_{abcd} - 3 S_{abcd}.
\end{array}
\end{equation}
\section{Beyond the {FLRW} Big-Bang singularity}
\label{s_beyond_big_bang}
\begin{theorem}
For the {FLRW} metric \eqref{eq_flrw_metric},
with $a: I\to \mathbb{R}$ a smooth function of time, the tensors $R_{abcd}$, $S_{abcd}$, and $E_{abcd}$ are smooth, and consequently the expanded Einstein equation is smooth too, even when $a(t)=0$.
\end{theorem}
\begin{proof}
If we denote by $\widetilde T_{ab}:= \kappa T_{ab} - \Lambda g_{ab}$,
\begin{equation}
\label{eq_flrw_curv_ricci}
\begin{array}{lll}
R_{ab} &=& \widetilde T_{ab} - \dsfrac{1}{2}g^{st}\widetilde T_{st} \\
&=& \kappa T_{ab} - \Lambda g_{ab} - \dsfrac{\kappa}{2} g^{st} T_{st} g_{ab} + 2 \Lambda g_{ab} \\
&=& \kappa\(\rho + \dsfrac{p}{c^2}\) u_a u_b + \kappa p g_{ab} - \dsfrac{\kappa}{2} \(-\rho c^2 - p + 4 p\) g_{ab} + \Lambda g_{ab} \\
&=& \kappa\(\rho + \dsfrac{p}{c^2}\) u_a u_b + \dsfrac{\kappa}{2} \(\rho c^2 - p\) g_{ab} + \Lambda g_{ab} \\
\end{array}
\end{equation}
From equations \eqref{eq_friedmann_density}, \eqref{eq_acceleration}, and \eqref{eq_flrw_curv_ricci}, we can see that the Ricci tensor has the form
\begin{equation}
\label{eq_flrw_curv_ricci_a}
R_{ab} = a^{-2}(t)\alpha(t) u_a u_b + a^{-2}(t)\beta(t) g_{ab}.
\end{equation}
where $\alpha(t)$ and $\beta(t)$ are smooth functions.
Similarly,
\begin{equation}
\label{eq_flrw_curv_scalar}
\begin{array}{lll}
R &=& g^{st}R_{st} \\
&=& \kappa\(-\rho c^2 - p + 2 \rho c^2 - 2 p\) + 4\Lambda \\
&=& \kappa\(\rho c^2 - 3p\) + 4\Lambda \\
\end{array}
\end{equation}
and there is a smooth function $\gamma(t)$ so that
\begin{equation}
\label{eq_flrw_curv_scalar_a}
R = a^{-2}(t)\gamma(t).
\end{equation}
We need to check that $a^{-2}(t)$ is compensated in $S_{abcd}$ and $E_{abcd}$, so that $a(t)$ appears to a non-negative power.
Since the {FLRW} metric \eqref{eq_flrw_metric} is diagonal in the standard coordinates, each term in $(g\circ g)_{abcd}$ is of the form $g_{aa}g_{bb}$, with $a\neq b$. This means that at least $a\neq t$ or $b\neq t$ holds, and from \eqref{eq_flrw_metric} we conclude that $g_{aa}g_{bb}$ contains $a(t)$ at least to the power $2$. Therefore, the scalar part of the Riemann curvature, $S_{abcd}$, is smooth.
For the same reason, the Kulkarni-Nomizu product between the metric tensor and the term $a^{-2}(t)\beta(t) g_{ab}$ from the expression of the Ricci curvature \eqref{eq_flrw_curv_ricci_a} is smooth.
The only term from \eqref{eq_flrw_curv_ricci_a} we have to check that is smoothened by the Kulkarni-Nomizu product with $g$ is $a^{-2}(t)u_a u_b$. Since $u_a=g_{as}u^s$ and $g$ is diagonal, it follows that $u_a=g_{aa}u^a$ (without summation). If $b\neq t$ ($a\neq t$ is similar), then $u_b=g_{bb}u^b$ contains the needed $a^2(t)$. In the case when $a=b=t$, $a^{-2}(t) u_t u_t$ is not necessarily smooth, but in the Kulkarni-Nomizu product it will appear only in terms of the form $a^{-2}(t) u_t u_t g_{cc}$, with $c\neq t$. Hence, $\textnormal{Ric}\circ g$ is smooth. From this and from the smoothness of $S_{abcd}$, it follows that $E_{abcd}$ is also smooth.
One of the properties of the {FLRW} metric is that it is conformally flat, that is, $C_{abcd} = 0$. From this it follows that $R_{abcd}=S_{abcd}+E_{abcd}$ is smooth too.
\end{proof}
\section{Properties of the proposed equation}
\label{s_properties}
\subsection{Conservation of energy}
The conservation of energy is usually put in the form
\begin{equation}
-a^3\dot\rho = 3 a^2 \dot a \rho + \dsfrac 3 {c^2} a^2\dot a p,
\end{equation}
which remains valid even when the volume $a^3\to 0$.
\subsection{The metric is parallel}
It is known that if the metric tensor is regular, its covariant derivative vanishes, $g_{ab;c}=0$. For our solution, this is true so long as $a(t)\neq 0$. But if $a=0$, the metric is degenerate, and we have to check that $g_{ab;c}=0$.
The metric being diagonal, its Christoffel symbols of the first kind,
\begin{equation}
\Gamma_{abc}=\dsfrac 1 2 \(g_{bc,a} + g_{ca,b} - g_{ab,c}\)
\end{equation}
which don't vanish are either of the form
\begin{equation}
\Gamma_{aaa}=\dsfrac 1 2 g_{aa,a}
\end{equation}
or, for $a\neq b$,
\begin{equation}
\Gamma_{aab} = -\dsfrac 1 2 g_{aa,b}
\end{equation}
or
\begin{equation}
\Gamma_{aba} = \Gamma_{baa} = \dsfrac 1 2 g_{aa,b}
\end{equation}
Consequently, the Christoffel symbols of the second kind,
\begin{equation}
\Gamma^c_{ab}=g^{cs}\dsfrac 1 2 \(g_{bs,a} + g_{sa,b} - g_{ab,s}\)
\end{equation}
are of the form
\begin{equation}
\Gamma^a_{aa}=\dsfrac 1 2 \dsfrac {g_{aa,a}}{g_{aa}} (!)
\end{equation}
or, for $a\neq b$,
\begin{equation}
\Gamma^b_{aa} = -\dsfrac 1 2 \dsfrac {g_{aa,b}}{g_{bb}} (!)
\end{equation}
or
\begin{equation}
\Gamma^a_{ab} = \Gamma^a_{ba} = \dsfrac 1 2 \dsfrac {g_{aa,b}}{g_{aa}} (!)
\end{equation}
where $(!)$ means ``no summation over the repeated indices''.
It follows that the covariant derivative introduces in the worst case a division by $a^2(t)$.
The covariant derivative of the metric tensor is
\begin{equation}
g_{ab;c} = g_{ab,c} - \Gamma^s_{bc}g_{as} - \Gamma^s_{ac}g_{sb}.
\end{equation}
Obviously $g_{ab,c}$ is smooth, because $g_{ab}$ is smooth. From the other terms, the only ones involving non-vanishing Christoffel symbols are of the form $\Gamma^a_{aa}g_{aa}$, $\Gamma^a_{bb}g_{aa}$, and $\Gamma^a_{ab}g_{aa}$, without summation. Whenever $\Gamma^a_{bc}$ involves $a(t)$ to a negative power, which can only be $1$ or $2$, this is compensated by $g_{aa}$, which contains $a^2(t)$. It follows that the covariant derivative of the metric tensor is smooth, and by continuity is zero even when the metric becomes degenerate (at $a(t)=0$):
\begin{equation}
\label{eq_parallel_metric}
g_{ab;c} = 0.
\end{equation}
\subsection{The Bianchi identity}
We will show that the Riemann curvature tensor satisfies the Bianchi identity
\begin{equation}
R_{(abc)d;e} = 0.
\end{equation}
Given that it holds at all the points for which $a(t)\neq 0$, where the metric is regular, it also holds by continuity at $a(t)= 0$. But we need to check that the covariant derivatives $(\textnormal{Ric}\circ g)_{abcd;e}$ are smooth, because if the Bianchi identity would be between infinte values, there would be no continuity.
Since the Weyl part of the Riemann curvature $C_{abcd}=0$ in the {FLRW} spacetime, and from the equations \eqref{eq_flrw_curv_ricci_a} and \eqref{eq_flrw_curv_scalar_a}, it follows that $R_{abcd}$ has the following form:
\begin{equation}
R_{abcd} = a^{-2}(t)\mu(t) \((u\otimes u)\circ g\)_{abcd} + a^{-2}(t)\nu(t) (g\circ g)_{abcd},
\end{equation}
where the functions $\mu(t)$ and $\nu(t)$ are smooth.
Let's denote by $h_{ij}$, $1\leq i,j\leq 3$, the metric on $\Sigma$. Then $g_{ij}=a^2(t)h_{ij}$. Given that our frame is comoving with the fluid, $u_t=1$, and $u_i=0$ for all $i$. The only terms $a^{-2}(t)\mu(t) \((u\otimes u)\circ g\)_{abcd}$ which don't vanish by containing $u_i=0$ are of the form $a^{-2}(t)\mu(t)u_tu_tg_{ii}$. The covariant derivatives with respect to $t$ cancel one another in the Bianchi identity under the permutation, because the index $t$ is repeated. So we check now those terms of the form $\nabla_j\(a^{-2}(t)\mu(t)u_tu_tg_{ii}\)$, where $i\neq j$. But $\nabla_j\(a^{-2}(t)\mu(t)u_tu_tg_{ii}\)=a^{-2}(t)\mu(t)u_tu_tg_{ii;j}=0$, because only $g_{ii}$ depends on the spacelike direction $x^j$, and because the metric tensor is parallel \eqref{eq_parallel_metric}.
The terms $a^{-2}(t)\nu(t) (g\circ g)_{abcd}$ can only be of the form $a^{-2}(t)\nu(t) g_{aa}g_{bb}$, $a\neq b$. Since $\nabla_i\(a^{-2}(t)\nu(t) g_{aa}g_{bb}\) = a^{-2}(t)\nu(t) \(g_{aa;i}g_{bb} + g_{aa}g_{bb;i}\) = 0$, it follows that the covariant derivatives with respect to $i$ vanish. We check now thosw with respect to $t$. If either the index $a$ or $b$ is equal to $t$, then the cyclic permutation involved in the Bianchi identity vanishes. Then the only remaining possibility is $\nabla_t\(a^{-2}(t)\nu(t) g_{ii}g_{jj}\)$. But $\nabla_t\(a^{-2}(t)\nu(t) g_{ii}g_{jj}\) = -2\dot a(t)a^{-3}(t)\nu(t) g_{ii}g_{jj} + a^{-2}(t)\dot\nu(t) g_{ii}g_{jj}= -2\dot a(t)a(t)\nu(t) h_{ii}h_{jj} + a^{2}(t)\dot\nu(t) h_{ii}h_{jj}$, which is smooth.
Hence, the Bianchi identity makes sense even at the singularity $a(t)=0$.
\subsection{Action principle}
Shortly after Einstein proposed his field equation, Hilbert and Einstein provided a Lagrangian formulation. The Lagrangian density which leads to Einstein's equation with matter given by $\mc L\sqrt{-g}$ and cosmological constant $\Lambda$ is
\begin{equation}
\label{eq_lagrangian}
\dsfrac{1}{2\kappa}\(R\sqrt{-g} - 2\Lambda\sqrt{-g}\) + \mc L\sqrt{-g}.
\end{equation}
In our case, the scalar curvature is singular at $a(t)\to 0$. But this doesn't affect the Lagrangian density, since the density $R\sqrt{-g}$ is smooth \cite{Sto11h}.
Given that our expanded Einstein equation \eqref{eq_einstein_expanded} is equivalent to Einstein's its solutions are extremals of the action given by \eqref{eq_lagrangian}.
\section{Conclusions}
\label{s_conclusions}
The new form of Einstein's equation extends uniquely beyond the Big Bang singularity, as it is represented schematically in Figure \ref{flrw-exp}.
\image{flrw-exp}{0.8}{Schematic representation of a generic {FLRW} spacetime. The solutions of the new equation can be continued naturally before the Big Bang.}
An alternative solution was proposed in \cite{Sto11h}, where the Einstein equation was replaced with a densitized version
\begin{equation}
\label{eq_einstein_idx:densitized}
G_{ab}\sqrt{-g} + \Lambda g_{ab}\sqrt{-g} = \kappa T_{ab}\sqrt{-g}.
\end{equation}
This version is expressed in terms of tensor densities of weight $1$, which appear naturally from the Lagrangean.
The {FLRW} spacetime is an ideal one, based on the assumptions of the \textit{cosmological principle} (that it is homogeneous and isotropic). But the extension proposed here opens new possibilities to explore.
Singularities which are of the type studied here, having the Riemann curvature tensor $R_{abcd}$ smooth, and admitting smooth Ricci decomposition, are in fact more general. In \cite{Sto11e} the Schwarzschild singularity is put in a form in which has these properities, by an appropriate coordinate change. This, and similar results on the Reissner-Nordstr\"om singularity \cite{Sto11f} suggests that we should reconsider the information loss \cite{Sto12e}. More general cosmological models, which are neither homogeneous nor isotropic, are studied in \cite{Sto12c}, and shown to admit a smooth Ricci decomposition, and satisfy the Weyl curvature hypothesis \cite{Pen79}. Implications suggesting to reconsider the the problem of quantization are presented in \cite{Sto12d,Sto12f}.
\textbf{Acknowledgments}
I thank an anonymous referee for the valuable comments and suggestions to improve the clarity and the quality of this paper.
\bibliographystyle{unsrt
|
{
"timestamp": "2012-08-08T02:02:35",
"yymm": "1203",
"arxiv_id": "1203.1819",
"language": "en",
"url": "https://arxiv.org/abs/1203.1819"
}
|
\section{Introduction}
As all good mathematical scientists know, a broad community has contributed to the invention of modern algorithms. Computer scientists, applied mathematicians, statisticians, economists, and physicists, to name just a few, have made lasting contributions. Exposing students to a variety of perspectives outside the realm of their own disciplines sharpens their instincts for modeling and arms them with invaluable tools. In this spirit, the current paper discusses techniques for solving Sudoku puzzles, one of the most popular pastimes in the world. One could make the same points with more serious applications, but it is hard to imagine a more beguiling introduction to the algorithms featured here. Sudoku diagrams are special cases of the Latin squares long familiar in experimental design and, as such, enjoy interesting mathematical and statistical properties \cite{BaiCamCon2008}. The complicated constraints encountered in solving Sudoku puzzles have elicited many clever heuristics that amateurs use to good effect. Here we examine three generic methods with broader scientific and societal applications. The fact that one of these methods outperforms the other two is mostly irrelevant. No two problem categories are completely alike, and it is best to try many techniques before declaring a winner.
The three algorithms tested here are simulated annealing, alternating projections, and backtracking. Simulating annealing is perhaps the most familiar to statisticians. It is the optimization analog of MCMC (Markov chain Monte Carlo) and has been employed to solve a host of combinatorial problems. The method of alternating projections was first proposed by von Neumann \cite{Neu1950} to find a feasible point in the intersection of a family of hyperplanes. Modern versions of alternating projections more generally seek a point in the intersection of a family of closed convex sets. Backtracking is a standard technique taken from the toolkits of applied mathematics and computer science. Backtracking infallibly finds all solutions of a Sudoku puzzle or determines that no solution exists. Its Achilles heel of excessive computational complexity does not come into play with Sudoku puzzles because they are, despite appearances, relatively benign computationally. Sudoku puzzles are instances of the satisfiability problem in computer science. As problem size increases, such problems are combinatorially hard and often defy backtracking. For this reason alone, it is useful to examine alternative strategies.
In a typical Sudoku puzzle, there are 81 cells arranged in a 9-by-9 grid, some of which are occupied by numerical clues. See Figure~\ref{fig:sample_game}. The goal is to fill in the remaining cells subject to the following three rules:
\begin{figure}
\centering
\begin{tikzpicture}[scale=0.75]
\draw[black!50] (0,0) grid (9,9);
\draw[black!100] (0,0) rectangle (3,3);
\draw[black!100] (3,0) rectangle (6,3);
\draw[black!100] (6,0) rectangle (9,3);
\draw[black!100] (0,3) rectangle (3,6);
\draw[black!100] (3,3) rectangle (6,6);
\draw[black!100] (6,3) rectangle (9,6);
\draw[black!100] (0,6) rectangle (3,9);
\draw[black!100] (3,6) rectangle (6,9);
\draw[black!100] (6,6) rectangle (9,9);
\node[regular polygon, regular polygon sides=4] at (3.5,0.5) {$5$};
\node[regular polygon, regular polygon sides=4] at (5.5,0.5) {$6$};
\node[regular polygon, regular polygon sides=4] at (6.5,0.5) {$8$};
\node[regular polygon, regular polygon sides=4] at (7.5,0.5) {$2$};
\node[regular polygon, regular polygon sides=4] at (3.5,1.5) {$3$};
\node[regular polygon, regular polygon sides=4] at (5.5,1.5) {$4$};
\node[regular polygon, regular polygon sides=4] at (7.5,1.5) {$1$};
\node[regular polygon, regular polygon sides=4] at (8.5,1.5) {$9$};
\node[regular polygon, regular polygon sides=4] at (1.5,2.5) {$8$};
\node[regular polygon, regular polygon sides=4] at (4.5,2.5) {$2$};
\node[regular polygon, regular polygon sides=4] at (5.5,2.5) {$1$};
\node[regular polygon, regular polygon sides=4] at (4.5,3.5) {$7$};
\node[regular polygon, regular polygon sides=4] at (8.5,3.5) {$1$};
\node[regular polygon, regular polygon sides=4] at (0.5,4.5) {$2$};
\node[regular polygon, regular polygon sides=4] at (3.5,4.5) {$8$};
\node[regular polygon, regular polygon sides=4] at (4.5,4.5) {$6$};
\node[regular polygon, regular polygon sides=4] at (7.5,4.5) {$7$};
\node[regular polygon, regular polygon sides=4] at (8.5,4.5) {$4$};
\node[regular polygon, regular polygon sides=4] at (0.5,5.5) {$7$};
\node[regular polygon, regular polygon sides=4] at (2.5,5.5) {$4$};
\node[regular polygon, regular polygon sides=4] at (3.5,5.5) {$1$};
\node[regular polygon, regular polygon sides=4] at (5.5,5.5) {$5$};
\node[regular polygon, regular polygon sides=4] at (6.5,5.5) {$2$};
\node[regular polygon, regular polygon sides=4] at (1.5,6.5) {$3$};
\node[regular polygon, regular polygon sides=4] at (2.5,6.5) {$2$};
\node[regular polygon, regular polygon sides=4] at (3.5,6.5) {$9$};
\node[regular polygon, regular polygon sides=4] at (6.5,6.5) {$1$};
\node[regular polygon, regular polygon sides=4] at (7.5,6.5) {$4$};
\node[regular polygon, regular polygon sides=4] at (1.5,7.5) {$4$};
\node[regular polygon, regular polygon sides=4] at (0.5,8.5) {$1$};
\node[regular polygon, regular polygon sides=4] at (1.5,8.5) {$5$};
\node[regular polygon, regular polygon sides=4] at (2.5,8.5) {$7$};
\node[regular polygon, regular polygon sides=4] at (3.5,8.5) {$6$};
\node[regular polygon, regular polygon sides=4] at (4.5,8.5) {$4$};
\node[regular polygon, regular polygon sides=4] at (7.5,8.5) {$8$};
\end{tikzpicture}
\caption{Sample Puzzle\label{fig:sample_game}}
\end{figure}
\begin{itemize}
\item[1.] Each integer between 1 and 9 must appear exactly once in a row,
\item[2.] Each integer between 1 and 9 must appear exactly once in a column,
\item[3.] Each integer between 1 and 9 must appear exactly once in each of the 3-by-3 subgrids.
\end{itemize}
Solving a Sudoku game is a combinatorial task of intermediate complexity. The general problem of filling in an incomplete $n^2 \times n^2$ grid with $n \times n$ subgrids belongs to the class of NP-complete problems \cite{YatSet2003}. These problems are conjectured to increase in computational complexity at an exponential rate in $n$. Nonetheless, a well planned exhaustive search can work quite well for a low value of $n$ such as $9$. For larger values of $n$, brute force, no matter how cleverly executed, is simply not an option. In contrast, simulated annealing and alternating projections may yield good approximate solutions and partially salvage the situation.
In the rest of this paper, we describe the three methods for solving Sudoku puzzles and compare them on a battery of puzzles. The puzzles range in difficulty from pencil and paper exercises to hard benchmark tests that often defeat the two approximate methods. Our discussion reiterates the rationale for equipping students with the best computational tools.
\section{Three methods for solving Sudoku}
\label{sec:methods}
\subsection{Backtracking}
Backtracking systematically grows a partial solution until it becomes a full solution or violates a constraint \cite{Ski2008}. In the latter case it backtracks to the next permissible partial solution and begins the growing process anew. The advantage of backtracking is that a block of potential solutions can be discarded en masse. Backtracking starts by constructing for each empty Sudoku cell
$(i,j)$ a list $L_{ij}$ of compatible digits. This is done by scanning the cell's row, column, and subgrid. The empty cells are then
ordered by the cardinalities of the lists $|L_{ij}|$. For example in Figure~\ref{fig:sample_game}, two cells $(7,4)$ and $(9,5)$ possess lists $L_{74}=\{7\}$ and $L_{95}=\{9\}$ with cardinality 1 and come first. Next come cells such as $(1,6)$ with $L_{16}=\{2,3\}$, $(1,7)$ with $L_{17}=\{3,9\}$, and $(1,9)$ with $L_{19}=\{2,3\}$ whose lists have cardinality 2. Finally come cells such as $(2,9)$ with $L_{29}=\{2,3,5,6,7\}$ whose lists have maximum cardinality 5. Partial solutions are character strings such as $s_{74}s_{95}s_{16}s_{17}s_{19}$ taken in dictionary order with the alphabet at cell $(i,j)$ limited to the list $L_{ij}$. In dictionary order a string such as $7939$ is treated as coming after a string such as $79232$.
Backtracking starts with the string $7$ by taking the only element of $L_{74}$, grows it to $79$ by taking the only element of $L_{95}$, grows it to $792$ by taking the first element of $L_{16}$, grows it to $7923$ by taking the first element of $L_{17}$, and finally grows it to $79232$ by taking the first element of $L_{19}$. At this juncture a row violation occurs, namely a 2 in both cells $(1,6)$ and $(1,9)$. Backtracking discards all strings beginning with $79232$ and moves on to the string $79233$ by replacing the first element of $L_{19}$ by the second element of $L_{19}$. This leads to another row violation with a 3 in both cells $(1,7)$ and $(1,9)$. Backtracking moves back to the string $7929$ by discarding the fifth character of $79239$ and replacing the first element of $L_{17}$ by its second element. This sets the stage for another round of growing.
Backtracking is also known as depth first search. In this setting the strings are viewed as nodes of a tree as depicted in Figure~\ref{fig:backtracking}. Generating strings in dictionary order constitutes a tree traversal that systematically eliminates subtrees and moves down and backs up along branches.
Because pruning large subtrees is more efficient than pruning small subtrees, ordering of cells by cardinality compels the decision tree to have fewer branches at the top. We use the C code from Skiena and Revilla \cite{SkiRev2003} implementing backtracking on Sudoku puzzles. Backtracking has the virtue of finding all solutions when multiple solutions exist. Thus, it provides a mechanism for validating the correctness of puzzles.
\begin{figure}
\centering
\begin{tikzpicture}[level/.style={sibling distance=75mm/#1}, scale=0.75]
\node (z){79}
child {node (a) {792}
child {node (b) {7923}
child {node(c) {79232}
child{node(d) {Infeasible}
child [grow=left] {node (q) {\quad\quad\quad\quad} edge from parent[draw=none]
child [grow=up] {node (r) {$\ldots\VE{s}{19}$ \quad\quad\quad\quad} edge from parent[draw=none]
child [grow=up] {node (s) {$\ldots\VE{s}{17}$ \quad\quad\quad\quad} edge from parent[draw=none]
child [grow=up] {node (t) {$\dots\VE{s}{16}$ \quad\quad\quad\quad} edge from parent[draw=none]
child [grow=up] {node (t) {$\VE{s}{74}\VE{s}{95}$ \quad\quad\quad\quad} edge from parent[draw=none]
}}}}}
}
}
child {node (e) {79233}
child {node (f) {Infeasible}
}}
}
child {node (g) {7929}
child {node (h) {$\vdots$}}}
}
child {node (i) {793}
child {node (j) {$\vdots$}}
}
;
\end{tikzpicture}
\caption{Backtracking on the puzzle shown in Figure~\ref{fig:sample_game}. Starting from $\VE{s}{74}\VE{s}{95} = 79$, the algorithm
attempts and fails to grow the solution beyond $\VE{s}{74}\VE{s}{95}\VE{s}{16}\VE{s}{17}\VE{s}{19} = 79232$. After failing to grow the solution
beyond $\VE{s}{74}\VE{s}{95}\VE{s}{16}\VE{s}{17}\VE{s}{19} = 79233$, all partial solutions beginning with 7923 are eliminated from further consideration.
The algorithm starts anew by attempting to grow $\VE{s}{74}\VE{s}{95}\VE{s}{16}\VE{s}{17} = 7929$. }
\label{fig:backtracking}
\end{figure}
\subsection{Simulated Annealing}
Simulated annealing \cite{Cer1985, KirGelVec1983, PreTeuVet2007} attacks a combinatorial optimization problem by defining a state space of possible solutions, a cost function quantifying departures from the solution ideal, and a positive temperature parameter. For a satisfiability problem, it is sensible to equate cost to the number of constraint violations. Solutions then correspond to states of zero cost. Each step of annealing operates by proposing a move to a new randomly chosen state. Proposals are Markovian in the sense that they depend only on the current state of the process, not on its past history. Proposed steps that decrease cost are always accepted. Proposed steps that increase cost are taken with high probability in the early stages of annealing when temperature is high and with low probability in the late stages of annealing when temperature is low. Inspired by models from statistical physics, simulated annealing is designed to sample the state space broadly before settling down at a local minimum of the cost function.
For the Sudoku problem, a state is a $9 \times 9$ matrix (board) of integers drawn from the set $\{1,\ldots,9\}$. Each integer appears nine times, and all numerical clues are respected. Annealing starts from any feasible board. The proposal stage randomly selects two different cells without clues. The corresponding move swaps the contents of the cells, thus preserving all digit counts. To ensure that the most troublesome cells are more likely to be chosen for swapping, we select cells non-uniformly with probability proportional to
$\exp(i)$ for a cell involved in $i$ constraint violations. Let $\M{b}$ denote a typical board, $c(\M{b})$ its associated cost, and
$n$ the current iteration index. At temperature $\tau$, we decide whether to accept a proposed neighboring board $\M{b}$ by drawing a random deviate $U$ uniformly from $[0,1]$. If $U$ satisfies
\begin{eqnarray*}
U & \le & \min\left\{\exp (\left[c(\M{b}_n)-c(\M{b})\right]/\tau_n),1\right\},
\end{eqnarray*}
then we accept the proposed move and set $\M{b}_{n+1}=\M{b}$. Otherwise, we reject the move and set $\M{b}_{n+1}=\M{b}_n$. Thus, the greater the increase in the number of constraint violations, the less likely the move is made to a proposed state. Also, the higher the temperature, the more likely a move is made to an unfavorable state. The final ingredient of simulated annealing is the cooling schedule. In general, the temperature parameter $\tau$ starts high and slowly declines to 0, where only favorable or cost neutral moves are taken.
Typically temperature is lowered at a slow geometric rate.
\subsection{Alternating Projections}
The method of alternating projections relies on projection operators. In the projection problem, one seeks the closest point
$\V{x}$ in a set $C \subset \mathbb{R}^d$ to a point $\V{y} \in \mathbb{R}^d$. Distance is quantified by the usual Euclidean norm
$\lVert \V{x} - \V{y} \rVert.$ If $\V{y}$ already lies in $C$, then the problem is trivially solved by setting $\V{x} = \V{y}$.
It is well known that a unique minimizer exists whenever the set $C$ is closed and convex \cite{Lan2004}. We will denote the
projection operator taking $\V{y}$ to $\V{x}$ by $P_C(\V{y}) = \V{x}$.
Given a finite collection of closed convex sets with a nonempty intersection, the alternating projection algorithm finds a point in that intersection. Consider the case of two closed convex sets $A$ and $B$. The method recursively generates a sequence $\V{y}_n$
by taking $\V{y}_0 = \V{y}$ and $\V{y}_{n+1} = P_A(\V{y}_{n})$ for $n$ even and $\V{y}_{n+1} = P_B(\V{y}_{n})$ for $n$ odd. Figure~\ref{fig:alternating_projections} illustrates a few iterations of the algorithm. As suggested by the picture, the algorithm does indeed
converge to a point in $A \cap B$ \cite{CheGol1959}. For more than two closed convex sets with nonempty intersection, the method of alternating projections cycles through the projections in some fixed order. Convergence occurs in this more general case as well based
on some simple theory involving paracontractive operators \cite{ElsKolNeu1992}. The limit is not guaranteed to be the closest point in the intersection to the original point $\V{y}$. The related but more complicated procedure known as Dykstra's algorithm \cite{Dyk1983} finds this point.
\begin{figure}[t]
\centering
\includegraphics[scale=0.6]{Alternating_Projection}
\caption{Alternating projections find a point in $A \cap B$, where $A$ and $B$ are closed convex sets. The initial point is $\V{y}$. The sequence of points $\V{y}_n$ is generated by alternating projection onto $A$ with projection onto $B$.}
\label{fig:alternating_projections}
\end{figure}
It is easy to construct some basic projection operators. For instance, projection onto the rectangle
$R = \{\V{x} \in \mathbb{R}^d : a_i \le x_i \le b_i \: \mbox{for all} \: i\}$ is achieved by defining $\V{x}=P_R(\V{y})$ to have components $x_i=\min\{\max\{a_i,y_i\},b_i\}$. This example illustrates a more general rule; namely, if $A$ and $B$ are two closed convex sets, then projection onto the Cartesian product $A \times B$ is effected by the Cartesian product operator
$(\V{x},\V{y}) \mapsto [P_A(\V{x}),P_B(\V{y})]$. When $A$ is an entire Euclidean space, $P_A(\V{x})$ is just the identity map. Projection onto the hyperplane
\begin{eqnarray*}
H & = & \{\V{y} \in \mathbb{R}^d : \V{v}\Tra\V{y} = c\}
\end{eqnarray*}
is implemented by the operator
\begin{eqnarray*}
P_H(\V{x}) & = & \V{x} - \frac{\V{v}\Tra\V{x} - c}{\lVert \V{v} \rVert^2} \V{v} .
\end{eqnarray*}
Projection onto the unit simplex $U = \left\{\V{x} \in \mathbb{R}^d: \sum_{i=1}^d x_i = 1, \; x_i \ge 0 \: \forall i\right\}$ is more subtle. Fortunately there exist fast algorithms for this purpose \cite{DucShaSin2008, Mic1986}.
In either the alternating projection algorithm or Dykstra's algorithm, it is advantageous to reduce the number of participating convex sets to the minimum possible consistent with fast projection. For instance, it is better to take the unit simplex $U$ as a whole rather than as an intersection of the halfspaces $\{\V{x}: x_i \ge 0\}$ and the affine subspace $\{\V{x}: \sum_{i=1}^d x_i =1\}$. Because our alternating projection algorithm for solving Sudoku puzzles relies on projecting onto several simplexes, it is instructive to derive the Duchi et al \cite{DucShaSin2008} projection algorithm. Consider minimization of a convex smooth function $f(\V{x})$
over $U$. The Karush-Kuhn-Tucker stationarity condition involves setting the gradient of the Lagrangian
\begin{eqnarray*}
{\cal L}(\V{x},\lambda,\V{\mu}) & = & f(\V{x})+ \lambda \Big(\sum_{i=1}^d x_i-1\Big) - \sum_{i=1}^d \mu_i x_i
\end{eqnarray*}
equal to $\V{0}$. This is stated in components as the Gibbs criterion
\begin{eqnarray*}
0 & = & \frac{\partial}{\partial x_i} f(\V{x})+ \lambda - \mu_i
\end{eqnarray*}
for multipliers $\mu_i \ge 0$ obeying the complementary slackness conditions $\mu_i x_i = 0$. For the choice
$f(\V{x}) = \frac{1}{2}\|\V{x}-\V{y}\|^2$, the Gibbs condition can be solved in the form
\begin{eqnarray*}
\VE{x}{i} & = & \begin{cases} \VE{y}{i}-\lambda & \VE{x}{i} > 0 \\ \VE{y}{i}-\lambda+\mu_i & \VE{x}{i} = 0. \end{cases}
\end{eqnarray*}
If we let $I_+ = \{i: \VE{x}{i}>0\}$, then the equality constraint
\begin{eqnarray*}
1 & = & \sum_{i \in I_+} \VE{x}{i} \:\;\, = \:\;\, \sum_{i \in I_+} \VE{y}{i} - |I_+| \lambda
\end{eqnarray*}
implies
\begin{eqnarray*}
\lambda & = & \frac{1}{|I_+|} \Big(\sum_{i \in I_+} \VE{y}{i} - 1 \Big) .
\end{eqnarray*}
The catch, of course, is that we do not know $I_+$.
The key to avoid searching over all $2^d$ subsets is the simple observation that the $\VE{x}{i}$ and $\VE{y}{i}$ are consistently ordered.
Suppose on the contrary that $\VE{y}{i}< y_j$ and $x_j < \VE{x}{i}$. For small $s>0$ substitute $\VE{x}{j}+s$ for $x_j$ and $\VE{x}{i}-s$
for $\VE{x}{i}$. The objective function $f(\V{x}) = \frac{1}{2}\lVert \V{x}-\V{y} \rVert^2$ then changes by the amount
\begin{equation*}
\frac{1}{2} \Big[(\VE{x}{i}-s-\VE{y}{i})^2+(x_j+s-y_j)^2-(\VE{x}{i}-\VE{y}{i})^2-(x_j-y_j)^2 \Big] = s(\VE{y}{i}-y_j+x_j-\VE{x}{i})+s^2 ,
\end{equation*}
which is negative for $s$ small. Let $\VE{w}{i}$ denote the $i$th largest entry of $\V{y}$. Then the Gibbs condition implies that
$\VE{w}{1} \geq \VE{w}{2} \geq \ldots \geq \VE{w}{\lvert I_+ \rvert} > \lambda$. Thus, to determine $\lambda$ we seek the largest $k$
such that
\begin{equation*}
\VE{w}{k} > \frac{1}{k} \left ( \sum_{i=1}^k \VE{w}{i} - 1 \right)
\end{equation*}
and set $\lambda$ equal to the right hand side of this inequality. With $\lambda$ in hand, the Gibbs condition implies that $\VE{x}{i} = \max \{\VE{y}{i} - \lambda, 0\}$.
It follows that projection onto $U$ can be accomplished in $O(d \log d)$ operations dominated by sorting. Algorithm~\ref{alg:project_simplex} displays pseudocode for projection onto
$U$.
\begin{algorithm}[t]
\begin{algorithmic}[0]
\State $\V{w} \leftarrow \textsc{sort\_descending}(\V{y}).$
\State $k \leftarrow \max \left \{ j : \VE{w}{j} > \frac{1}{j} \left( \sum_{i=1}^j \VE{w}{i} \right)\right \}$
\State $\lambda \leftarrow \frac{1}{k} \left( \sum_{i=1}^{k} \VE{w}{i} \right)$
\State $\VE{x}{i} \leftarrow \max\{\VE{y}{i} - \lambda, 0\}$.
\end{algorithmic}
\caption{\, \textsc{Projection onto simplex}}
\label{alg:project_simplex}
\end{algorithm}
Armed with these results, we now describe how to solve a continuous relaxation of Sudoku by the method of alternating projections. In the relaxed version of the problem, we imagine generating candidate solutions by random sampling. Each cell $(i,j)$ is assigned a sampling distribution $\TE{p}{ijk}= \Pr(S_{ij}=k)$ for choosing a random deviate $S_{ij} \in \{1,\ldots,9\}$ to populate the cell. If a numerical clue $k$ occupies cell $(i,j)$, then we set $p_{ijl}=1$ for $l=k$ and 0 otherwise. A matrix of sampled deviates $\M{S}$ constitutes a candidate solution. It seems reasonable to demand that the average puzzle obey the constraints. Once we find a feasible 3-dimensional tensor $\T{P} = (\TE{p}{ijk})$ obeying the constraints, a good heuristic for generating an integer solution $\Mhat{S}$ is to put
\begin{eqnarray*}
\hat{s}_{ij} & = & \underset{k \in \{ 1, \ldots, 9 \}}{\max} \TE{P}{ijk}.
\end{eqnarray*}
In other words, we impute the most probable integer to each unknown cell $(i,j)$. It is easy to construct counterexamples where imputation of the most probable integer from a feasible tensor $\T{P}$ of the relaxed problem fails to solve the Sudoku puzzle.
In any case, the remaining agenda is to specify the constraints and the corresponding projection operators. The requirement that each digit appear in each row on average once amounts to the constraint $\sum_{j=1}^9 \TE{p}{ijk} = 1$ for all $i$ and $k$ between 1 and 9. There are 81 such constraints. The requirement that each digit appear in each column on average once amounts to the constraint $\sum_{i=1}^9 \TE{p}{ijk} = 1$ for all $j$ and $k$ between 1 and 9. Again, there are 81 such constraints. The requirement that each digit appear in each subgrid on average once amounts to the constraint $\sum_{j=1}^3 \sum_{j=1}^3 \TE{p}{a+i,b+j,k} = 1$ for all $k$ between 1 and 9 and all $a$ and $b$ chosen from the set $\{0,3,6\}$. This contributes another 81 constraints. Finally, the probability constraints $\sum_{k=1}^9 \TE{p}{ijk} = 1$ for all $i$ and $j$ between 1 and 9 contribute 81 more affine constraints. Hence, there are a total of $324$ affine constraints on the $9^3 = 729$ parameters. In addition there are 729 nonnegativity constraints $\TE{p}{ijk} \ge 0$.
Every numerical clue voids several constraints. For example, if the digit $7$ is mandated for cell (9,2), then we must take
$\TE{p}{927} = 1$, $\TE{p}{92k} = 0$ for $k \ne 7$, $\TE{p}{i27} = 0$ for all $i \not = 9$, $\TE{p}{9j7} = 0$ for all $j \not = 2$, and $\TE{p}{ij7} = 0$ for all other pairs $(i,j)$ in the (3,1) subgrid. In carrying out alternating projection, we eliminate the corresponding variables. With this proviso, we cycle through the simplex projections
summarized in Algorithm~\ref{alg:project_simplex}.
The process is very efficient but slightly tedious to code. For the sake of brevity we omit the remaining details. All code used to generate the subsequent
results are available at \url{https://github.com/echi/Sudoku}, and we direct the interested reader there.
\section{Comparisons}
\label{sec:experiments}
We generated test puzzles from code available online \cite{Wan2007} and discarded puzzles that could be completely solved by filling in entries directly implied by the initial clues. This left 87 easy puzzles, 130 medium puzzles, and 100 hard puzzles. We also downloaded an additional 95 very hard benchmark puzzles \cite{Coc2012,Ste2005}. In simulated annealing, the temperature $\tau$ was initialized to 200 and lowered by a factor of 0.99 after every 50 steps.
We allowed at most $2 \times 10^{5}$ iterations and reset the temperature to 200 if a solution had not been found after $10^{5}$ iterations. For the alternating projection algorithm, we commenced projecting from the origin $\bf 0$.
\begin{table}[th]
\centering
\begin{tabular}{lcccc}
\hline \hline
& Alt. Projection & Sim. Annealing & Backtracking & Number of Puzzles\\ \hline
Easy & 0.85 & 1.00 & 1.00 & 87 \\
Medium & 0.89 & 1.00 & 1.00 & 130 \\
Hard & 0.72 & 0.97 & 1.00 & 100 \\
Top 95 & 0.41 & 0.03 & 1.00 & 95 \\ \hline
\end{tabular}
\caption{Success rates for solving puzzles of varying difficulty.}
\label{tab:comparisons}
\end{table}
\begin{table}[th]
\centering
\begin{tabular}{clccc}
\toprule
\midrule
& & \multicolumn{3}{c}{CPU Time (sec)} \\
\cmidrule(r){3-5}
& & Alt. Projection & Sim. Annealing & Backtracking \\
\midrule
\multicolumn{1}{c}{\multirow{4}{*}{Easy}} &
\multicolumn{1}{l}{Minimum} & 0.032 & 0.006 & 0.007 \\
\multicolumn{1}{l}{} &
\multicolumn{1}{l}{Median} & 0.041 & 0.021 & 0.008 \\
\multicolumn{1}{l}{} &
\multicolumn{1}{l}{Mean} & 0.052 & 0.112 & 0.008 \\
\multicolumn{1}{l}{} &
\multicolumn{1}{l}{Maximum} & 0.237 & 0.970 & 0.009 \\ \\
\multicolumn{1}{c}{\multirow{4}{*}{Medium}} &
\multicolumn{1}{l}{Minimum} & 0.032 & 0.007 & 0.007 \\
\multicolumn{1}{l}{} &
\multicolumn{1}{l}{Median} & 0.051 & 0.037 & 0.008 \\
\multicolumn{1}{l}{} &
\multicolumn{1}{l}{Mean} & 0.062 & 0.231 & 0.008 \\
\multicolumn{1}{l}{} &
\multicolumn{1}{l}{Maximum} & 0.269 & 3.36 & 0.010 \\ \\
\multicolumn{1}{c}{\multirow{4}{*}{Hard}} &
\multicolumn{1}{l}{Minimum} & 0.033 & 0.008 & 0.008 \\
\multicolumn{1}{l}{} &
\multicolumn{1}{l}{Median} & 0.110 & 0.753 & 0.008 \\
\multicolumn{1}{l}{} &
\multicolumn{1}{l}{Mean} & 0.159 & 1.104 & 0.009 \\
\multicolumn{1}{l}{} &
\multicolumn{1}{l}{Maximum} & 0.525 & 7.204 & 0.031 \\
\bottomrule
\end{tabular}
\caption{Summary statistics on the run times for different methods on puzzles of varying difficulty. For the alternating projection and simulated annealing techniques, only successfully solved puzzles are included in the statistics.}
\label{tab:run_times}
\end{table}
Backtracking successfully solved all puzzles. Table~\ref{tab:comparisons} shows the fraction of puzzles the two heuristics were able to successfully complete. Table~\ref{tab:run_times} records summary statistics for the CPU time taken by each method for each puzzle category. All computations were done on an iMac computer with a 3.4 GHz Intel Core i7 processor and 8 GB of RAM. We implemented the alternating projection and simulated annealing algorithms in Fortran 95. For backtracking we relied on the existing implementation in C.
The comparisons show that backtracking performs best, and for the vast majority of $9 \times 9$ Sudoku problems it is probably going to be hard to beat. Simulated annealing
finds the solution except for a handful of the most challenging paper and pencil problems, but its maximum run times are unimpressive. While alternating projection does not perform as well on the pencil and paper problems compared to the other two algorithms, it does not do terribly either. Moreover, we see hints of the tables turning on the hard puzzles.
\begin{figure}[t]
\centering
\begin{tikzpicture}[scale=0.75]
\draw[black!50] (0,0) grid (9,9);
\draw[black!100] (0,0) rectangle (3,3);
\draw[black!100] (3,0) rectangle (6,3);
\draw[black!100] (6,0) rectangle (9,3);
\draw[black!100] (0,3) rectangle (3,6);
\draw[black!100] (3,3) rectangle (6,6);
\draw[black!100] (6,3) rectangle (9,6);
\draw[black!100] (0,6) rectangle (3,9);
\draw[black!100] (3,6) rectangle (6,9);
\draw[black!100] (6,6) rectangle (9,9);
\node[regular polygon, regular polygon sides=4,fill=gray!20,scale=0.75] at (0.5,8.5) {$4$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (1.5,8.5) {$1$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (2.5,8.5) {$7$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (3.5,8.5) {$9$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (4.5,8.5) {$6$};
\node[regular polygon, regular polygon sides=4,fill=gray!70,scale=0.75] at (5.5,8.5) {$2$};
\node[regular polygon, regular polygon sides=4,fill=gray!20,scale=0.75] at (6.5,8.5) {$8$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (7.5,8.5) {$3$};
\node[regular polygon, regular polygon sides=4,fill=gray!20,scale=0.75] at (8.5,8.5) {$5$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (0.5,7.5) {$6$};
\node[regular polygon, regular polygon sides=4,fill=gray!20,scale=0.75] at (1.5,7.5) {$3$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (2.5,7.5) {$2$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (3.5,7.5) {$1$};
\node[regular polygon, regular polygon sides=4,fill=gray!70,scale=0.75] at (4.5,7.5) {$5$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (5.5,7.5) {$8$};
\node[regular polygon, regular polygon sides=4,fill=gray!20,scale=0.75] at (6.5,7.5) {$7$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (7.5,7.5) {$4$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (8.5,7.5) {$9$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (0.5,6.5) {$9$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (1.5,6.5) {$5$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (2.5,6.5) {$8$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (3.5,6.5) {$7$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (4.5,6.5) {$3$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (5.5,6.5) {$4$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (6.5,6.5) {$6$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (7.5,6.5) {$1$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (8.5,6.5) {$2$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (0.5,5.5) {$8$};
\node[regular polygon, regular polygon sides=4,fill=gray!20,scale=0.75] at (1.5,5.5) {$2$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (2.5,5.5) {$5$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (3.5,5.5) {$4$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (4.5,5.5) {$9$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (5.5,5.5) {$7$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (6.5,5.5) {$3$};
\node[regular polygon, regular polygon sides=4,fill=gray!20,scale=0.75] at (7.5,5.5) {$6$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (8.5,5.5) {$1$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (0.5,4.5) {$3$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (1.5,4.5) {$9$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (2.5,4.5) {$1$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (3.5,4.5) {$5$};
\node[regular polygon, regular polygon sides=4,fill=gray!20,scale=0.75] at (4.5,4.5) {$8$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (5.5,4.5) {$6$};
\node[regular polygon, regular polygon sides=4,fill=gray!20,scale=0.75] at (6.5,4.5) {$4$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (7.5,4.5) {$2$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (8.5,4.5) {$7$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (0.5,3.5) {$7$};
\node[regular polygon, regular polygon sides=4,fill=gray!20,scale=0.75] at (1.5,3.5) {$4$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (2.5,3.5) {$6$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (3.5,3.5) {$3$};
\node[regular polygon, regular polygon sides=4,fill=gray!20,scale=0.75] at (4.5,3.5) {$1$};
\node[regular polygon, regular polygon sides=4,fill=gray!70,scale=0.75] at (5.5,3.5) {$2$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (6.5,3.5) {$5$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (7.5,3.5) {$9$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (8.5,3.5) {$8$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (0.5,2.5) {$2$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (1.5,2.5) {$8$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (2.5,2.5) {$9$};
\node[regular polygon, regular polygon sides=4,fill=gray!20,scale=0.75] at (3.5,2.5) {$6$};
\node[regular polygon, regular polygon sides=4,fill=gray!70,scale=0.75] at (4.5,2.5) {$5$};
\node[regular polygon, regular polygon sides=4,fill=gray!20,scale=0.75] at (5.5,2.5) {$3$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (6.5,2.5) {$1$};
\node[regular polygon, regular polygon sides=4,fill=gray!20,scale=0.75] at (7.5,2.5) {$7$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (8.5,2.5) {$4$};
\node[regular polygon, regular polygon sides=4,fill=gray!20,scale=0.75] at (0.5,1.5) {$5$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (1.5,1.5) {$7$};
\node[regular polygon, regular polygon sides=4,fill=gray!20,scale=0.75] at (2.5,1.5) {$3$};
\node[regular polygon, regular polygon sides=4,fill=gray!20,scale=0.75] at (3.5,1.5) {$2$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (4.5,1.5) {$4$};
\node[regular polygon, regular polygon sides=4,fill=gray!20,scale=0.75] at (5.5,1.5) {$1$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (6.5,1.5) {$9$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (7.5,1.5) {$8$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (8.5,1.5) {$6$};
\node[regular polygon, regular polygon sides=4,fill=gray!20,scale=0.75] at (0.5,0.5) {$1$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (1.5,0.5) {$6$};
\node[regular polygon, regular polygon sides=4,fill=gray!20,scale=0.75] at (2.5,0.5) {$4$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (3.5,0.5) {$8$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (4.5,0.5) {$7$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (5.5,0.5) {$9$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (6.5,0.5) {$2$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (7.5,0.5) {$5$};
\node[regular polygon, regular polygon sides=4,scale=0.75] at (8.5,0.5) {$3$};
\end{tikzpicture}
\caption{A typical local minimum that traps simulated annealing in a top 95 puzzle. Clues are shaded light gray. There are two
column constraint violations caused by the cells shaded dark gray. The local minimum is deep in the sense that all one-step swaps
result in further constraint violations. \label{fig:top95_1}}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.75]{Comparison_top95}
\caption{Scatterplot of solution times for the top 95 benchmark problems.}
\label{fig:top}
\end{figure}
Simulated annealing struggles mightily on the 95 benchmark puzzles. Closer inspection of individual puzzles reveals that these very hard puzzles admit many local minima with just a few constraint violations. Figure~\ref{fig:top95_1} shows a typical local minimum that traps the simulated annealing algorithm. Additionally, something curious is happening in Figure~\ref{fig:top}, which plots CPU solution times for alternating projection versus backtracking. Points below the dashed line indicate puzzles that the method of alternating projection solves more efficiently than backtracking. It appears that when the method of alternating projections finds correct solutions to very hard problems, it tends to find them more quickly than backtracking.
\section{Discussion}
\label{sec:discussion}
It goes almost without saying that students of the mathematical sciences should be taught a variety of solution techniques for combinatorial problems. Because Sudoku puzzles are easy to state and culturally neutral, they furnish a good starting point for the educational task ahead. It is important to stress the contrast between exact strategies that scale poorly with problem size and approximate strategies that adapt more gracefully. The performance of the alternating projection algorithm on the benchmark tests suggest it may have a role in solving much harder combinatorial problems. Certainly, the electrical engineering community takes this attitude, given the close kinship of Sudoku puzzles to problems in coding theory \cite{ErlChaGor2009, GunMoo2012, MooGun2006, MooGunKup2009}.
One can argue that algorithm development has assumed a dominant role within the mathematical sciences. Three inter-related trends are feeding this phenomenon. First, computing power continues to grow. Execution times are dropping, and computer memory is getting cheaper. Second, good computing simply tempts scientists to tackle larger data sets. Third, certain fields, notably communications, imaging, genomics, and economics generate enormous amounts of data. All of these fields create problems in combinatorial optimization. For instance, modern DNA sequencing is still challenged by the phase problem of discerning the maternal and paternal origin of genetic variants. Computation is also being adopted more often as a means of proving propositions. The claim that at least 17 numerical clues are needed to ensure uniqueness of a Sudoku solution has apparently been proved using intelligent brute force \cite{McGTugCiv2012}. Mathematical scientists need to be aware of computational developments outside their classical application silos. Importing algorithms from outside fields is one of the quickest means of refreshing an existing field.
\section*{Acknowledgments}
We thank Peter Cock for helpful comments and pointing us to the source and author of the ``Top 95" puzzles.
\bibliographystyle{siammod}
|
{
"timestamp": "2013-05-17T02:02:22",
"yymm": "1203",
"arxiv_id": "1203.2295",
"language": "en",
"url": "https://arxiv.org/abs/1203.2295"
}
|
\section{Introduction}
Recent improvement of the techniques of
ground-based Cherenkov telescopes has increased
the number and variety of TeV gamma-ray objects.
Five gamma-ray binaries have been detected so far, with
all of them known to be high mass X-ray binary systems.
Their common emission mechanism has been vastly investigated
since their discoveries. Each gamma-ray binary is assumed to be composed of
a compact object orbiting around a massive star. Among them,
PSR~B1259-63/LS~2883 is the only system for which the compact
object has been confirmed to be a pulsar.
PSR~B1259-63 is a 48-ms radio pulsar
with a spin down power of $L_{\mathrm{sd}}=8.2\times 10^{35}~\mathrm{erg~s^{-1}}$.
The spectral type of LS~2883 had been known to be B2V
\citep{Johnston1994}.
However,
a recent precise measurement reports
the type to be O9.5V \citep{Negueruela2011}.
This correction implies a change of the assumed distance to the system
from 1.5\,kpc to $2.3\pm 0.4$\,kpc,
as well as a change of the effective temperature of the star.
The orbit has a large eccentricity $e$ of 0.87 and a long period
$P_{\mathrm{orb}}$ of about 3.4~yr.
Non-pulsed and non-thermal emissions from the binary
in the radio \citep{Johnston2005, Moldon2011},
X-ray \citep{Hirayama1996, Uchiyama2009}
and TeV energy ranges \citep{Aharonian2005, Aharonian2009} have been
reported, and flare-like GeV emissions
were detected around the 2010--2011 periastron passage
by the Large Area Telescope (LAT) on board
$Fermi$ \citep{Tam2011, Abdo2011}.
The radio pulse eclipse of about 5 weeks around the periastron
suggests that the pulsar goes to the opposite side
of the Be-disk plane with respect to the observer,
while crossing it plane twice during the course.
The characteristic double-peaked features
observed in the radio and X-ray light curves
\citep{connors02,chernyakova06}
can be mainly attributed to the interactions of the pulsar wind and
the Be-disk during disk crossings by the pulsar.
The peak phases or the peak intensities measured,
extensively in the radio band in particular, vary from orbit to orbit, though.
The observations for 2004 \citep{Aharonian2005}
and 2007 \citep{Aharonian2009} periastron passage indicate that
the TeV light curve also varies from orbit to orbit.
The non-thermal emission mechanisms of the system have been
studied in the framework of leptonic
\citep[e.g.,][]{Tavani1997, Khangulyan2007, Sierpowska2008,
Takata2009, Dubus2010b} and hadronic models (\citealt{Kawachi2004};
see also \citealt{Neronov2007}).
Unfortunately, Be-disk models which have been adopted so far in this field
of research are mostly outdated,
and are significantly different from the model being
widely accepted by the Be star research community currently,
i.e., the viscous decretion disk model
[\citet{Lee1991}; see also \citet{Porter1999} and \citet{Carciofi2006}].
Since the shock distance and its geometry depend on the Be-disk model,
it is important to examine the high-energy emissions with a more
realistic Be-disk model. For instance, disk models with supersonic outflows,
as adopted in most previous studies, have irrelevantly large ram pressure,
and thus give rise to false location of the shock.
Detailed two-dimensional hydrodynamic simulations have also been performed to study the wind-wind
collision interaction in this system \citep[e.g.,][]{Bogovalov2008,Bogovalov2011}, providing important information on the shock structure.
Since the simulations have been limited to 2-D, however,
the orbital motion has not been taken into account, which can have
significant effects near the periastron passage.
Moreover, the presence of the Be-disk can play an important role to the origin
of the high energy emission from this system. In such a case, the behavior of the system
will inevitably manifest in 3-D and be orbital-phase dependent.
In \citet[][hereafter paper I]{Okazaki2011}, we have
studied the interaction between the pulsar and the Be star by,
for the first time, carrying out
3-D hydrodynamic simulations using the viscous decretion disk model.
We have found that
for a Be-disk with typical density, the pulsar wind strips off an
outer part of the Be-disk, truncating the disk at a radius significantly smaller than the pulsar orbit.
This prohibits the pulsar from passing through the disk around periastron passage,
which has been assumed in previous studies.
In other words, a Be-disk with typical density is dynamically unimportant
in this system.
A large H$\alpha$ equivalent width of $-54$\,\AA\
\citep{Negueruela2011}, however, suggests that
the density of the Be-disk of LS~2883 is much higher than typical.
It is, therefore, interesting to see how the interaction
changes if the Be-disk is much denser and dynamically more important
than those studied in paper~I.
Given the double-peaked light curves of PSR B1259-63, a particularly important
issue is whether a reasonably high Be-disk density can lead to a strong pulsar wind confinement, and hence an enhanced emission, during both disk-plane crossings.
In this paper, which is the second of the series, we study high energy emissions from
PSR B1259-63/LS 2883 system, based on the results of numerical simulations
for different values of the Be-disk density.
We review our 3-D hydrodynamic simulations in section~\ref{simulation}
and describe our emission model in section~\ref{model}.
Based on the model light curves, we propose, in section~\ref{x-ray},
a new interpretation of the observed double-peaked feature of the radiation
(in particular in the X-ray band).
The multi-wavelength emission properties are discussed
in section~\ref{gevtev}. Finally, we summarize our results in section~\ref{summary}.
\section{Numerical Model for the Hydrodynamic Interaction between the Pulsar and the Be Star}
\label{simulation}
The simulations presented below are performed with a 3-D SPH code. The code is basically identical to
that used by \citet{Okazaki2002} \citep[see also][]{Bate1995},
except that the current version is adapted to systems with winds and a decretion disk, such as
PSR B1259-63/LS~2883, and takes into account radiative cooling with
the cooling function generated by CLOUDY 90.01 for
an optically thin plasma with solar abundances \citep{Ferland1996}.
Using a variable smoothing length, the SPH equations with a standard
cubic-spline kernel are integrated with an individual time step for
each particle. In our code, the Be-disk, the Be wind, and the pulsar wind are modeled by
ensembles of gas particles of different particle masses with negligible self-gravity,
while the Be star and the pulsar are represented by sink particles
with the appropriate gravitational masses.
Gas particles which fall within a specified accretion radius are accreted by the sink particle.
To reproduce the Be decretion disk, we inject gas particles just outside of the stellar
equatorial surface at a constant rate.
In paper~I, we adopted the injection rate of
$3.5 \times 10^{-9}M_{\odot}\,\mathrm{yr}^{-1}$, which gave rise to a
typical disk base density of $10^{-11}\,{\rm g~cm}^{-3}$
[detailed model fit of the observed H$\alpha$ profiles typically provides a density
between $10^{-12}\,{\rm g~cm}^{-3}$ and several times $10^{-10}\,{\rm g~cm}^{-3}$
\citep[e.g.,][]{Silaj2010}].
In the current study, however, we compare the resulting radiation spectrum/light curve
for three different injection rates, fixing all the other parameters at values adopted
in paper~I. Particularly, for the mass and radius of the Be star, we take values typical for a B2V star to ensure
consistency with the models studied in paper~I. The polar axis of the Be star is tilted from
the binary orbital axis by $45\arcdeg$,
and the azimuth of tilt, i.e. the azimuthal angle of the Be star's polar axis
from the direction of apastron, is $19\arcdeg$.
With this geometry, the pulsar crosses the equatorial plane of the Be star
at $\tau \sim -10\,\mathrm{d}$ and $\tau \sim +25\,\mathrm{d}$,
where $\tau$ is the orbital phase in days relative to the periastron passage.
We choose rates of
$3.5 \times 10^{-9}M_{\odot}\,\mathrm{yr}^{-1}$,
$3.5 \times 10^{-8}M_{\odot}\,\mathrm{yr}^{-1}$, and
$3.5 \times 10^{-7}M_{\odot}\,\mathrm{yr}^{-1}$, corresponding to
disk base densities of
$10^{-11}\,{\rm g~cm}^{-3}$,
$10^{-10}\,{\rm g~cm}^{-3}$, and
$10^{-9}\,{\rm g~cm}^{-3}$, respectively.
Remarkably, we find that the highest injection rate is favored
by best reproducing the observed light curve.
We note that taking the base density of $10^{-9}\,{\rm g~cm}^{-3}$ for this system is not unreasonable,
given that LS~2883 showed the H$\alpha$ equivalent width of $-54$\,\AA\ in quiescence
\citep{Negueruela2011}, which was one of the largest equivalent widths Be stars have ever shown.
The Be wind and pulsar wind
are turned on in the simulation at a certain time after the Be-disk has fully developed
in the tidal simulation (for more details, see paper~I). We started the wind simulation at
$t=11.44\,P_{\rm orb}$, 74 days prior to periastron passage.
For simplicity, we assume that the winds coast without any net external force,
assuming in effect that gravitational forces are
either negligible (i.e. for the pulsar wind) or are cancelled by radiative driving terms (i.e. for the Be wind).
The relativistic pulsar wind is emulated by a non-relativistic flow with
a velocity of $10^{4}\,{\rm km s}^{-1}$ and an adjusted mass-loss rate
so as to provide the same momentum flux as a relativistic flow
with the same assumed energy.
We assume that all the spin down energy
$L_{\rm sd}=8.2 \times 10^{35}\,{\rm erg~s}^{-1}$ goes to
the kinetic energy of a spherically symmetric pulsar wind.
We also assume the Be wind to be spherically symmetric with
a mass loss rate of $10^{-8}\,M_{\odot}\,{\rm yr}^{-1}$.
It is noted that the unshocked pulsar wind
is emulated by a non-relativistic flow with a momentum flux ($L_{\rm sd}/c$),
where $c$ is the speed of light. The global structure of
pulsar wind under the influence of the Be-wind/disk
should be reasonably represented by this treatment.
This is because it depends primarily on the
momentum flux ratio and not on whether the pulsar wind is modeled
as relativistic (see paper I for details). This will
justify the application since we estimate the light curve
and spectrum by integrating all the contributions from each emission region.
The resulting light curve and spectrum should not depend on the detailed local
structure of the shocked pulsar winds. Rather, they should depend mainly on
the global structure. Thus we conclude
the resulting light curve and spectrum
in this study are robust in spite of our non-relativistic treatment.
Of course, as stated in Bogovalov et al. (2008, 2011), the relativistic effects
can affect the detailed structure of unshocked/shocked pulsar wind regions.
Thus it is
our future plan to extend our code to the relativistic regine and examine its
influence. The possibility of relativistic Doppler boost effect is discussed
in section 4.1.
\section{Emission Model for Shocked Pulsar Wind}
\label{model}
We calculate the emission at each orbital phase
using data from the 3-D hydrodynamic simulations.
First, the simulation volume is divided into uniform grids,
and at each grid point the pulsar wind pressure is calculated by counting
only the contribution from the pulsar wind particles.
Then, from the 3-D distribution of the pulsar wind pressure,
synchrotron and inverse-Compton emissions are locally
evaluated by using an assumption on the local magnetic
fields and the calculation scheme
described in the following subsections.
In these calculations, we implicitly assume
that particles are accelerated through the 1st-order Fermi
mechanism at the pulsar wind termination shock, and have a power-law
energy distribution. We anticipate that the motion of each SPH
particle, which represents an ensemble of electrons and positrons, corresponds
to the bulk motion of the pulsar wind particles represented by the
SPH particle.
In the down stream region, we assume that
the motion of the electrons and positrons represented by each SPH particle
is randomized and the energy distribution is described by the power law
function (section~\ref{dis}).
The total emissions are obtained by integrating the local
emissions over the whole simulation volume. We adopt $100^{3}$
uniform grids and have confirmed that
the result does not change even if grids with higher resolution are used.
Although the integration is taken over the whole simulation volume,
virtually all the contribution to
the high energy emission arises from the shocked pulsar wind region,
because the unshocked (upstream) region of the pulsar wind is too
cold to make any significant contribution to the emission.
It has been suggested that the relativistic bulk motion of the upstream
flow may produce a considerable high energy emissions
via the inverse-Compton process.
In this study, however, because
our simulations are done in the non-relativistic limit, we do not
investigate this point.
It has been suggested that the radio emission can come from
a larger volume than emissions in other wavelengths
\citep{Moldon2011}.
This volume is roughly $\sim$ 100\,AU, which is about one order of magnitude larger
than the size of the present simulation volume ($\sim 2a\sim 1.5\times 10^{14}$~cm).
Estimation of emission in the radio band is hence omitted here and postponed to future studies.
\subsection{Magnetic Fields}
Our simulations are performed in the hydrodynamic limit and
there are theoretical uncertainties involved in determining the magnetic field
of the pulsar wind.
The magnetic field in the shock down-stream
is obtained using the Rankine-Hugoniot relations at the shock surface
and the adiabatic expansion-law of ideal MHD flow
as in the cases of steady nebulae
(e.g. Kennel and Coroniti 1984; Tavani and Arons 1997;
Nagataki 2004).
The flow in our simulations, however, is too dynamic for the above
calculation scheme to be applied to. Therefore,
we adopt another approach frequently used in modeling
gamma-ray bursts (e.g. Sari et al. 1998; Xu et al. 2011)
as follows. On the pulsar side, the total pulsar wind pressure $P_{\rm tot}$
at each grid is associated with the local magnetic field $B$ as
\begin{equation}
B=\sqrt{\eta \times 8\pi P_{\rm tot}},
\label{magnetic}
\end{equation}
where $\eta$ is fixed to 0.1, which gives a X-ray flux consistent
with observations of PSR~B1259-63/LS 2883 system.
\subsection{Scheme of Calculation}
\subsubsection{Energy distribution of accelerated particles}
\label{dis}
The synchrotron and inverse-Compton processes are calculated in the
simulation grids where the pulsar wind pressure
$P_\mathrm{tot}$ is non-zero and the contribution of the
wind particles is $P_\mathrm{g}\equiv (1-\eta)P_\mathrm{tot}$.
We assume that the number of shocked
particles per Lorentz factor per volume at each grid point
is given by a single power law function of
\begin{equation}
f(\Gamma)=\frac{K}{4\pi}\Gamma^{-p}~~(\Gamma_\mathrm{min}\le \Gamma\le \Gamma_\mathrm{max}),
\label{func}
\end{equation}
where $\Gamma$ is the Lorentz factor of the particles.
Note that the accelerated particles are strictly relativistic in our model, such that their momenta are simply $m_e c \Gamma$, where $m_e$ is
the electron rest mass.
We will therefore write down their distribution in $\Gamma$ instead of momentum for convenience.
The minimum Lorentz factor $\Gamma_\mathrm{min}$ is similar to
the Lorentz factor of the bulk motion of the
unshocked flow. At the limit of small magnetization parameter
($\sigma\ll 1$), the latter is given by $\sim \sigma_L\Gamma_L$,
where $\sigma_\mathrm{L}$ and $\Gamma_\mathrm{L}$
are the magnetization parameter and the bulk Lorentz factor
at the light cylinder radius of the pulsar.
With $\sigma_\mathrm{L} \sim 10^{3}$ and $\Gamma_\mathrm{L} \sim 10^{2-3}$,
$\Gamma_\mathrm{min}$ of 5$\times 10^{5}$ is obtained.
The maximum Lorentz factor
$\Gamma_\mathrm{max}$ is determined as
$\Gamma_\mathrm{max}=\mathrm{min}(\Gamma_\mathrm{s},\Gamma_\mathrm{c})$;
$\Gamma_s$ is the Lorentz factor
at which the acceleration timescale $t_a\sim \Gamma m_ec/eB$ with $e$ being
electron charge is equal to the synchrotron loss
timescale $t_s\sim 9m_e^3c^5/4e^4B^2\Gamma$,
whereas $\Gamma_c$ is defined as
the Lorentz factor at which
the acceleration timescale $t_a$ is equal to the dynamical timescale
of the shocked pulsar wind $t_c$.
Here, $t_c$ is deduced as $\sim h_p/(c/\sqrt{3})$ with
the scale height $h_p$ of the gas pressure
estimated from the simulation. The typical $\Gamma_\mathrm{max}$
in this work is of order 10$^8$.
The energy density $\epsilon_p$ is related to the
gas pressure $P_g$ as $\epsilon_p=3P_g$. Using the
condition $\epsilon_p=m_ec^2\int\int \Gamma f(\Gamma)d\Gamma d\Omega$, we
calculate the normalization factor $K$ as
\begin{equation}
K=\frac{3P_g}{m_ec^2}\left \{
\begin{array}{@{\,}ll}
\frac{2-p}{\Gamma_{\rm max}^{2-p}-\Gamma_{\rm min}^{2-p}}
& \mathrm{for}~~p\ne 0 \\
\left[\mathrm{ln}(\Gamma_{\rm max}/\Gamma_{\rm min})\right]^{-1}& \mathrm{for}~~p=2
\end{array}
\right .
\label{norm}
\end{equation}
It is well known that the standard 1st-order Fermi mechanism
in the test-particle limit gives $p=(r+2)/(r-1)$,
where $r$ is the compression ratio at the shock wave.
Assuming a strong, non-relativistic shock, the compression ratio
$r=4$ is derived from the Rankine-Hugoniot relation,
so that the index$p=2$. From previous studies, in general we can expect that
$p \geq 2$ for weak, non-relativistic shocks, and
$1.0 \leq p \leq 2.2$ for strong, relativistic shocks,
respectively (see, e.g. Longair 1994 and references therein).
Limited by the spatial resolution, nevertheless,
it is very difficult to determine the shock conditions in the emission region directly from the hydro simulation.
Therefore, unless mentioned otherwise, we will adopt $p=2$ as our canonical value for the results in this paper.
In addition, cases for $p=2 \pm 0.5$ will also be investigated to
illustrate the possible effects of the uncertainty in $p$ on our
results, in particular the multi-wavelength emission spectra.
\subsubsection{Formulae}
The synchrotron power per unit energy emitted by each electron is
calculated according to Rybicki \& Lightman (1979) as
\begin{equation}
P_{\rm syn}(E)=\frac{\sqrt{3}e^2B\sin\theta_p}{hm_ec^2}F_{\rm sy}
\left(\frac{E}{E_{\rm syn}}\right),
\end{equation}
where $E_{\rm syn}=3he\Gamma^2B\sin\theta_p/4\pi m_ec$
is the typical photon energy
and $F_{\rm sy}(x)=x\int_x^{\infty}K_{5/3}(y)dy$, where $K_{5/3}$ is the modified Bessel function of order 5/3. For the pitch angle $\theta_p$, we use the averaged value corresponding to $\sin^2\theta_P=2/3$. The power per unit energy and per unit solid angle of the inverse-Compton process is described by \citep{Bege1987}
\begin{equation}
\frac{dP_{\rm IC}}{d\Omega}={\mathcal D}^2\int_0^{\theta_c}(1-\beta\cos\theta_0)I_b/h\frac{d\sigma'_{KN}}{d\Omega'}d\Omega_0,
\end{equation}
where $d\sigma'/d\Omega'$ is the differential Klein-Nishina cross section,
$\beta=\sqrt{\Gamma^2-1}/\Gamma$,
$\mathcal{D}=\Gamma^{-1}(1-\beta\cos\theta_1)^{-1}$
with $\theta_1$ and $\theta_0$ being the angles between the direction of the particle motion
and the propagating direction of the scattered photon and the background
photons, respectively, $h$ is the Planck constant,
$I_b$ is the background photon field
and $\theta_c$ expresses the angular size of star as seen from the emission
point. For the target stellar photons, a soft photon field
with an effective temperature of 30,000~K is taken,
and the Be-disk emission is omitted for simplicity.
For the Earth viewing angle, we assume that
the inclination angle of the orbital plane with respect to the sky
is $i\sim 23^{\circ}$ and the true anomaly of the direction of Earth is about
$\phi\sim 130^{\circ}$ (Johnston et al. 1996; Negueruela et al. 2011).
The model spectrum of the emission from the shocked wind measured at
Earth is calculated as
\begin{equation}
F_E(E)\sim\frac{e^{-\tau_{\gamma\gamma}}}{D^2}\Sigma_i
\left [ \delta V_i \int_{\Gamma_{\rm min}}^{\Gamma_{\rm max}} f(\Gamma)\left(
P_{\rm syn}+\int\frac{dP_{\rm IC}}{d\Omega}d\Omega \right)d\Gamma\right],
\end{equation}
where $\Sigma_i$ expresses the summation of the each grid and $\delta V_i$ is
the volume of the each grid.
The optical depth, $\tau_{\gamma\gamma}$ for the pair-creation
process between the gamma-rays
emitted by the wind particles and the stellar photons is expressed
by
\begin{equation}
\tau_{\gamma\gamma}=\int_0^{\infty}d\ell\int_{E_{\rm c}}^{\infty}dE_s
\sigma_{\gamma\gamma}dN_s/dE_s,
\end{equation}
where $\ell$ is the propagating distance of the $\gamma$-ray from the
emitted point,
$dN_s/dE_s$ is the distribution of the number density
of the stellar soft photon and
\begin{equation}
\sigma_{\gamma\gamma}(E_{\gamma},E_s)=\frac{3}{16}\sigma_{\rm T}(1-v^2)
\left[(3-v^4)\mathrm{ln}\frac{1+v}{1-v}-2v(2-v^2)\right],
\end{equation}
where $\sigma_T$ is the Thomson cross section,
$v(E_{\gamma},E_s)=\sqrt{1-E_{\rm c}/E_\gamma}$
and $E_{\rm c}=2(m_ec^2)^2/[(1-\cos\theta_{\gamma\gamma})E_s]$
with $\theta_{\gamma\gamma}$ being the collision angle.
We use the distance $D=2.5$~kpc in this work.
\section{Results and Discussion}
\label{result}
In this section, we first summarize the results of the hydrodynamic interaction
between the pulsar wind and the circumstellar material of the Be star.
We then present the multi-wavelength light curves and an time-averaged spectrum
obtained using the simulation data.
We will also briefly discuss the cooling processes and comment on our non-relativistic emulation of the relativistic pulsar wind.
\subsection{Hydrodynamic Interaction between the Pulsar and the Be star}
\label{interaction}
As mentioned in section~\ref{simulation}, we have run 3-D SPH simulations
of hydrodynamic interaction in PSR~B1259-63/LS~2883 with the base density
of the Be decretion disk in the range of $10^{-9}-10^{-11}\,\mathrm{g~cm}^{-3}$.
Figures~\ref{gamma-t-11d} and \ref{gamma-t+33d} show snapshots
of the interaction between the pulsar wind and the circumstellar material of the Be star in the binary orbital plane at two different
epochs, 11 days prior to periastron passage ($\tau = -11~\mathrm{d}$)
and 33 days after it ($\tau = +33~\mathrm{d}$), respectively.
Each figure compares the shock structure in two SPH simulations with
different disk base densities, $\rho_0=10^{-11}\,\mathrm{g~cm}^{-3}$ (upper panels)
and $\rho_0=10^{-9}\,\mathrm{g~cm}^{-3}$ (lower panels): left and right
panels show the distribution of the volume density and
the pulsar wind pressure, respectively. To clarify the location where
the pulsar wind is terminated, the right panel also shows the distribution
of the volume density in contours. These figures highlight the effect of
the Be-disk density on the geometry of
the interaction surface.
For $\rho_0=10^{-11}\,\mathrm{g~cm}^{-3}$, the pulsar wind easily strips off
an outer part of the Be-disk, truncating the disk at a radius significantly
smaller than the pulsar orbit. As a result, the interaction
surface is open and covers only a small solid angle around
the pulsar (Figures~\ref{gamma-t-11d} and \ref{gamma-t+33d}, upper panels),
implying that only a small fraction of the
pulsar wind energy is available for particle acceleration.
In contrast, for $\rho_0=10^{-9}\,\mathrm{g~cm}^{-3}$, the Be-disk
has a large enough inertia not to be pushed away so easily by the pulsar wind.
Thus, as the pulsar approaches the Be-disk before periastron passage,
the distance between the interaction surface and the pulsar rapidly
decreases and finally becomes smaller than the scale-height of the disk.
The pulsar then penetrates the Be-disk, opening a small cavity around it.
The disk gas surrounding the pulsar terminates the pulsar wind over a
large solid angle, converting a large fraction of the bulk pulsar wind
energy into the energy of shocked particles. After periastron,
the pulsar approaches the Be-disk again, now moving away from the Be star.
Since the pulsar wind pushes the disk in the direction along which the disk
density decreases, it can move the disk gas more easily than it could
before periastron. As a result, a slowly expanding shell is formed around
the pulsar, which terminates the pulsar wind again over a large solid angle,
efficiently converting the pulsar wind energy into the energy of shocked
particles (Figure~\ref{gamma-t+33d}, lower panels). The details of these
hydrodynamic simulations will be discussed in a subsequent
paper (Okazaki et al. 2012, in prep)
\subsection{X-ray Light Curves}
\label{x-ray}
Figure~\ref{light} presents the calculated light curves of
the X-ray flux (1-10~keV energy bands) together with the observed fluxes
taken from Neronov\& Chernyakova (2007)
as a function of days relative to the periastron passage, $\tau$.
The solid, dashed and dotted lines are
the results for disk base densities
of $\rho_0=10^{-11}~\mathrm{g~cm^{-3}}$,
$10^{-10}~\mathrm{g~cm^{-3}}$ and $10^{-9}~\mathrm{g~cm^{-3}}$, respectively,
with the typical value of the power index $p=$2 for the accelerated particles.
During the pre-periastron period, up to $\tau=-11\,\mathrm{d}$,
the flux does not depend on the disk base density.
This is because the pulsar wind interacts mainly with the
stellar wind, for which
we assume an identical mass loss rate
of $10^{-8}\,M_{\odot}\,{\rm yr}^{-1}$.
The epochs during which the pulsar crosses the Be-disk plane
are estimated by the simulation to be
about 11 days prior to the periastron passage ($\tau \sim -11\,\mathrm{d}$)
and 25 days after it ($\tau \sim +25\,\mathrm{d}$).
The interaction between the pulsar wind
and Be-disk is expected to be the strongest during the disk crossings.
Remarkably, however,
the X-ray flux for cases of disks with typical base densities $\rho_0 = 10^{-10} - 10^{-11}$ $\mathrm{g~cm^{-3}}$
does not peak at the timing of disk crossings,
but shows a maximum intensity at the periastron. On the other hand,
the X-ray flux for the highest density case
($\rho_0=10^{-9}~\mathrm{g~cm^{-3}}$)
increases distinctively around the phases of the disk crossings,
resulting in a double-peaked structure.
The dependence of the light curves on the Be-disk density
reflects that of the shock geometry and resultant
conversion efficiency from the spin down power to the internal energy of
the shocked pulsar wind (Figure~\ref{ptot}).
For a typical disk density, we see that the pulsar wind can easily
truncate the Be-disk at a radius smaller than the pulsar's orbit.
Therefore, the solid angle as measured from the pulsar,
over which the pulsar wind is stopped by the Be-disk, is small. In this case, the intensity of synchrotron emission is highest
at the periastron because the magnetic fields of the shocked pulsar wind
are highest there (see equation (1)).
The pulsar wind cannot dismiss the disk with
the higher density ($\rho_0=10^{-9}~\mathrm{g~cm^{-3}}$),
i.e., with a large inertia.
The shock is pushed back toward the pulsar
in the first disk plane crossing ($\tau \sim -11\,\mathrm{d}$).
After the periastron, the pulsar approaches the Be-disk again,
but now moving away from the Be star.
Since the pulsar wind pushes the disk in the direction
along which the disk density decreases,
it displaces the disk gas more easily than it did in the
pre-periastron crossing.
As a result, a slowly expanding shell is formed around the pulsar,
which terminates the pulsar wind over a larger solid angle
($\tau \sim +33\,\mathrm{d}$).
The particles at the shock obtain energy from
the bulk kinetic energy of the unshocked pulsar wind, which is in turn provided by
the pulsar spin down power.
At these disk-plane crossing phases,
the conversion efficiency from the pulsar spin down power to
the shock-accelerated particle energy
drastically increases, which leads to the increase in the X-ray flux,
most notably in the post-periastron crossing phase (Figure~\ref{ptot}).
We would like to stress that no fine-tuning has been done in our study
for the sake of reproducing the observed X-ray light curve. Thus it is
remarkable that the calculated flux level and double-peak phases
for the case of $\rho_0=10^{-9}~\mathrm{g~cm^{-3}}$ and $\eta=0.1$ in equation~(\ref{magnetic}) turn out to be very similar to those from observations,
as shown in Figure~\ref{light}. Our model sheds new light on the disk density
as a probe of the high energy emission mechanisms. Interestingly, the favored density in this
study is higher than the typical one for Be-disks.
Our model possesses a few profound features
which distinguish it from previous studies.
First, the present simulations predict that the conversion efficiency
varies with the orbital phase, while the previous studies have
assumed a constant conversion efficiency over the whole orbit
\citep[e.g.,][]{Tavani1997, Takata2009}.
Second, in the present model,
the conversion efficiency and therefore the X-ray flux
acquire the maximum values when the pulsar wind interacts
with the Be-disk. In \citet{Tavani1997}, on the other hand,
the inverse-Compton cooling process which
dominates over other cooling processes
is crucial to reproduce the observed decrease of the flux
near the periastron.
\citet{Takata2009} and \citet{Kong2011} invoked the model that the pulsar
wind parameters at the shock (e.g. the $\sigma$ parameter) vary throughout the
orbital phase.
\subsection{GeV/TeV Light Curves}
\label{gevtev}
Figure~\ref{depen} shows the calculated light curves for
0.1-100~GeV and TeV ($>$300GeV) energy bands.
We find that the gamma-ray light curves show the double-peaked structures
as in their X-ray counterpart.
For 0.1-100~GeV light curves, the peaks
align with those in the X-ray bands, because both X-ray and 100~MeV-1~GeV
emissions are produced by the synchrotron process.
In the recent results from $Fermi$
\citep{Tam2011, Abdo2011}, however, it is likely that
the phases of the peaks in the X-ray and the GeV bands
do not align with each other. Furthermore, $Fermi$ has not detected
a strong and sharp peak before the periastron, which is
seen in the present model. Hence, in the present framework,
there are some discrepancies between the properties of the calculated
and observed GeV light curves.
We would like to note that
the 0.1-1~GeV flux particularly depend on the magnetic field,
because the synchrotron spectrum from the shocked particles
has a cut-off around 1-200~MeV.
In the present model, if the maximum Lorentz factor is determined
by the balance between the
synchrotron cooling timescale and the acceleration timescale,
the maximum Lorentz factor
of the particles is expressed as
$\Gamma_{\mathrm{s}}\sim (9m_e^2c^4/4e^3B)^{1/2}$, which indicates
the synchrotron photon energy of $E_{\mathrm{syn}}=
3\Gamma_{\mathrm{s}}^2eB\sin\theta_p/(4\pi m_ec)
\sim 3^{5/2}m_ec^3h/(2^{7/2}\pi e^2)\sim 200$~MeV.
For a lower magnetic field, the maximum Lorentz factor may be
determined by the balance between the acceleration timescale and
the dynamical timescale of the shocked pulsar wind.
In such a case, the cut-off energy of the synchrotron
radiation is $E<E_{\mathrm{syn}}\sim 200$~MeV, and the 0.1-1~GeV flux
is sensitive to the magnetic field.
Furthermore, contributions by emissions from the high-order generated
pairs \citep{Sierpowska2008} and the
inverse-Compton emission from the unshocked pulsar wind
\citep[e.g.,][]{Khangulyan2011a} to the observed spectrum
below the TeV band have been pointed out.
Therefore, a more detailed modeling of the 0.1-100~GeV emission process is
necessary for a closer comparison with the $Fermi$ results.
For the TeV emissions, which are produced by the inverse-Compton process,
we can see in Figure~\ref{depen} that the second peak of
the light curve aligns with
that of the X-ray band, whereas the first peak comes
around near the periastron. The timing of the first peak
of the TeV lightcurve reflects the fact
that the stellar soft-photon field strength at
the emission region (that is, the shock) reaches its maximum
during periastron passage.
As we described in section~\ref{x-ray},
though the conversion efficiency from the spin down energy
to the particles energy of the shocked wind decreases as the
pulsar moves toward the periastron,
the effect of the increasing strength of the soft photon fields
toward the periastron outweighs the decrease of the conversion efficiency.
The periastron peak does not well reproduce the observations,
at least of the 2004 periastron passage.
It is shown in Figure 5 that the light curve in the $>$ 300 GeV band suffers from photon-photon absorption where soft photons are assumed
to come only from the Be star. In the present calculation, we have taken
a spherically symmetric stellar photon field.
However, \citet{Negueruela2011} estimated a
stellar temperature of $T_{\mathrm{eff}}\sim 3.4\times 10^{4}$~K at the pole
and $T_{\mathrm{eff}}\sim 2.75\times 10^{4}$~K on the equator,
which implies that the stellar photon field does depend on the latitude.
Such a latitudinal dependence of the stellar photon field could affect the
peak phase of $> 300$~GeV emissions.
As mentioned in section~\ref{model}, the current model also omits
the contribution from the disk emission, but it can
become important for the spectrum and the absorption of the GeV-TeV
photons. For example, \citet{vanSoelen2011} studied
the effects of the IR excess on the spectrum and found that
the GeV gamma-ray flux can increase by a factor $\ge 2$.
When we include the contribution from the Be-disk emission,
the absorption effect could be more than doubled, as inferred
from \citet{vanSoelen2011}. Due to this effect, the flux
of $>$ 300 GeV gamma-rays around the periastron phase may be
suppressed. As a result, the peak phase of the
$>$ 300 GeV gamma-ray light curve may be shifted from the periastron
phase to a phase prior to the periastron. We are planning to investigate
this effect using a Monte-Carlo radiative transfer code as our next step.
\subsection{Multi-wavelength Spectra}
In Figure~\ref{spectrum}, we show
the spectra during the periastron passage
in multi-wavelength. On top of them, we plot the observed spectra measured
by various instruments.
To account for the systematic uncertainty in our emission model on
the shock conditions and hence the energy distribution
of the underlying accelerated particles, two model spectra are
calculated using p=1.5 (left panel) and p=2.5
(right panel), respectively. In the figures, the solid, dashed and dotted
lines represent the spectra averaged over three different period
$\tau=-40\;\mathrm{d}--10\;\mathrm{d}$, $\tau=-10\;\mathrm{d} -
+20\;\mathrm{d}$, and $\tau=+20\;\mathrm{d} - +50\;\mathrm{d}$, respectively.
Because we stopped our simulation
at $\tau=+50\;\mathrm{d}$, we cannot calculate the spectrum averaged
over the period covered by $Fermi$ observation.
The model spectra below and above 1~GeV
correspond to the synchrotron emission and the inverse-Compton
process, respectively.
In Figure~\ref{spectrum}, we can see a typical flux and spectral
shape for each of different power indexes.
The spectral slope does not change much from phase
to phase if the power index is constant, although the flux level varies.
Thus having model spectra will help us to infer the actual
power index of the energy distribution of the shocked wind particles from the
photon index of observed spectra.
The present model shows a spectral break of the synchrotron
radiation around 1--10~keV, which corresponds to the minimum
Lorentz factor of the particles, $\Gamma_{\mathrm{min}}=5\times 10^{5}$.
This spectral feature is consistent with the
break around 1~keV measured by the SUZAKU observation
\citep{Uchiyama2009}.
As discussed in section~4.3 and also seen in Figure~\ref{spectrum},
the present model expects that
the synchrotron spectrum extends up to the maximum photon energy
of $E_{\mathrm{syn}}=27m_ec^3h/(16\pi e^2)\sim 200$~MeV,
which corresponds to the Lorentz factor of the particles,
$\Gamma_{\mathrm{max}}\sim
(9m_e^2c^4/4e^3B)^{1/2}=3.6\times 10^{8}(\eta/0.1)^{-1/4}(P_{\mathrm{tot}}/
0.1\mathrm{dyne~cm^{-2}})^{-1/4}$, where we used the equation of
(\ref{magnetic}).
Although the flare-like GeV emissions detected by the $Fermi$ telescope
may not be compared with the result of the
present simple calculation, where we ignore the
effects of the radiative cooling. We would like to remark that
a smaller power index of the particle distribution
is preferred to explain the observed flux of $>$100~MeV emissions.
In fact, Takata \& Taam (2009), who fit the observed X-ray
flux and the photon index for various orbital phases, pointed out
that the expected $>$100~MeV flux for the smaller power index can be
higher than the $Fermi$ sensitivity. A more detailed
modeling for the spectrum and light curve in GeV energy
bands will be done in our subsequent studies.
In the present calculation, the inverse-Compton process between
the accelerated particles with $\Gamma_{min}>5\times 10^{5}$ and the
stellar photons of energy $\sim 1$~eV takes place
in the Klein-Nishina regime.
In such a case, the inverse-Compton spectrum calculated with
the particle power index $p$ is given by $EF_{E}\propto E^{1-p}$ above
the energy $\sim \Gamma_\mathrm{min}m_ec^2\sim 2\times 10^{11}~\mathrm{eV}$.
As we can see in Figure~\ref{spectrum}, the observed spectrum
$EF_E\propto E^{-1}$ by H.E.S.S. (filled circles and triangles)
indicates the power low index $p$ of the distribution of the scattering
particles is $p\sim 2$. The present model also predicts that
there is a change of the spectral slope at
$\sim5\times 10^{11}~\mathrm{eV}$.
\subsection{Effect of Radiative Cooling}
In the shocked regions with strong magnetic fields,
the synchrotron cooling timescale $t_s$ of the accelearated particles may
be less than the crossing time $t_c$ of those regions.
In the present framework, using equations~(\ref{magnetic}),
the total pressure and hence the magnetic field strength
become largest when the pulsar
penetrates the disk, so that the effect of synchrotron cooling
may not be neglected at that phase. In addition,
the inverse-Compton cooling time may be less than $t_c$ at
phases near the periastron. These cooling processes can
affect the resulting spectrum and light curve. For example,
if the inverse-Compton process dominates the other cooling processes
of the TeV particles,
the X-ray emission via synchrotron radiation is weaker in
denser soft photon fields. On the other hand, if the synchrotron
process is the dominant cooling process of the TeV particles,
the TeV emissions via the inverse-Compton process is weaker
in regions with stronger magnetic fields.
For PSR~B1259-63/LS~2883, however,
we expect that the cooling processes have no important effect
on the X-ray light curve.
The synchrotron radiation occurs
in the slow cooling regime
if the particle's Lorentz factor
is smaller than $\Gamma_\mathrm{sy}=(9m_e^3c^6)/(4e^4B^2 \ell)$,
where $\ell$ is the size of the emission cavity. With
$B\sim 0.1~$Gauss and $\ell\sim 10^{13}$~cm as
typical values near the periastron, the critical
energy below which the synchrotron cooling process can be ignored
is $E\sim 7.62\times 10^{7}(B/0.1\mathrm{G})^{-3}(\ell/10^{13}
\mathrm{cm})^{-2})$~eV, which is way above the X-ray energy band.
Moreover, according to \citet{Tavani1997},
the inverse-Compton process between the
accelerated particles and stellar photons enhances the double-peaked structure
in the X-ray light curve. Hence, our model X-ray light curves remain
robust even if the detailed cooling processes will be taken into account.
\section{Summary}
\label{summary}
In our previous study \citep{Okazaki2011},
we have developed 3-D SPH simulations of the interaction
between the pulsar wind and the Be-disk and wind in the gamma-ray binary
PSR~B1259-63/LS~2883. In this paper,
we investigated the high-energy emissions from the
shocked pulsar wind,
calculating the synchrotron radiation and the
inverse-Compton process
on the basis of the simulated shock geometry and pressure distribution
of the pulsar wind. The current study revealed that
the observed double-peaked X-ray light curves are reproduced only if
the Be-disk is denser than typical (with base density
$\sim 10^{-9}\,\mathrm{g~cm^{-3}}$).
The pre- and post-periastron X-ray peaks appear respectively
when the pulsar passes through the disk
prior to the periastron, and when the pulsar wind creates a cavity
in the disk gas after the periastron, in both cases
terminating the pulsar wind over
a large solid angle around the pulsar.
On the other hand, in the model TeV light curve, which also shows a
double peak feature, the first peak appears around the periastron,
which will disagree with the 2004 H.E.S.S. observation showing
the first peak located at a pre-periastron phase.
In a subsequent paper, we will study whether
the effects of the disk emission to the inverse-Compton process and
photon-photon absorption process can shift the first peak to a phase
prior to the periastron passage.
\bigskip
We express our appreciation to an anonymous referee for useful comments.
J.T. thanks K.S. Cheng and R.E. Taam for useful discussions.
S.N. is supported by Grant-in-Aid for Scientific Research on Innovative
Areas No. 23105709 by Ministry of Education, Culture, Sports,
Science and Technology (MEXT), Grant-in-Aid for Scientific Research (S)
No. 19104006 and Scientific Research (B) No. 23340069
by Japan Society for the Promotion of Science (JSPS),
Joint Usage/Research Center for Interdisciplinary Large-scale
Information Infrastructures in Japan
and Grant-in-Aid for the Global COE Program
"The Next Generation of Physics, Spun from Universality and Emergence"
from MEXT of Japan. T.N. is supported by Grant-in-Aid for Scientific
Research (C) No. 23540271 by Japan Society for the Promotion of Science (JSPS).
S.P.O acknowledges partial support from grant \#NNX11AC40G from NASA's
Astrophysics Theory Program. The computation was carried
out on HITACHI SR16000 at Yukawa Institute for Theoretical Physics (YITP),
Kyoto University and on HITACHI SR11000 at the Information
Initiative Center (iiC), Hokkaido university.
In addition to the above grants, this
work was partially supported by
the iiC collaborative research program 2010-2011, the Grant-in-Aid for
Scientific Research (18104003, 19047004, 19740100, 20540236, 21105509,
21540304, 22340045, 22540243, 23105709), and a research grant from
Hokkai-Gakuen Educational Foundation.
|
{
"timestamp": "2012-03-14T01:02:54",
"yymm": "1203",
"arxiv_id": "1203.2179",
"language": "en",
"url": "https://arxiv.org/abs/1203.2179"
}
|
\section{Introduction} It was quite recent that the complete answers were given in \cite{Ueda:AdvMath11,Ueda:MRL} to the questions of factoriality, type classification, fullness and $\mathrm{Sd}$- and $\tau$-invariants for arbitrary free product von Neumann algebras. It is natural as a next project to consider the same questions for more general amalgamated free product von Neumann algebras. Such attempts were already made by us \cite{Ueda:PacificJMath99, Ueda:ASPM04, Ueda:TAMS03} almost 10 years ago for amalgamated free products over Cartan subalgebras. However the results there are far from satisfactory as compared to those on plain free product von Neumann algebras. The aim of this paper is to take a still very first step towards `satisfactory' answers to those questions for amalgamated free product von Neumann algebras. As simple consequences we will give partial answers at least when amalgamated free products are taken over type I von Neumann algebras, which are improvements of our previous works \cite{Ueda:PacificJMath99, Ueda:ASPM04, Ueda:TAMS03, Ueda:JFA05, Ueda:IllinoisJMath08}.
The proofs in \cite{Ueda:AdvMath11,Ueda:MRL} are divided into analytical and combinatorial parts in essence. Combinatorial parts are completed by some `induction arguments', whose essential idea originates in several works due to Dykema, especially \cite{Dykema:DukeMathJ93}. On the other hand, analytical parts are devoted to proving several inequalities involving the Hilbert space norms arising from some states of particular form (instead of so-called free product states themselves), whose essential ideas apparently go back to the ICC argument for factoriality of group von Neumann algebras and the so-called $14\,\varepsilon$-argument both due to Murray and von Neumann. However our problems are of the nature of type III von Neumann algebras, and thus the lack of trace causes main difficulties. Hence the key is to overcome such difficulties. Here we will take up such analytical aspects in the general amalgamated free product setup, and indeed improve the analytical results in \cite{Ueda:AdvMath11,Ueda:MRL} with new techniques from the recent amazing development on type II$_1$ factors opened by several breakthroughs due to Popa. We hope that the technical facts provided in this paper are sufficient as analytical parts in future `best-possible' answers to the questions mentioned above at least in the case where amalgamated free products are taken over type I von Neumann subalgebras.
The organization of this paper is as follows. Section 2 is preliminaries on amalgamated free product von Neumann algebras. In section 3 we provide a non-tracial version of one of the results in Ioana--Peterson--Popa's article \cite[Theorem 1.1]{IoanaPetersonPopa:Acta08}. In relation to it we provide a non-tracial adaptation of the so-called intertwining-by-bimidule criterion due to Popa, which may be of independent interest as future reference. In the same section we also generalize our previous results of controlling central sequences \cite[Proposition 3.5]{Ueda:AdvMath11},\cite[Proposition 3.1]{Ueda:MRL} to the amalgamated free product setting. In section 4, we give several partial answers to the questions mentioned above by utilizing technologies developed in \S3. Those include an answer to the factoriality and non-amenability questions of a given amalgamated free product $(M,E) = (M_1,E_1)\star_N (M_2,E_2)$ when $M_1$ is `diffuse relative to $N$', $M_2$ `non-trivial relative to $N$', and $N$ of type I.
Standard notation rule here follows our previous papers \cite{Ueda:AdvMath11, Ueda:MRL}; for example, the center, the unitary group and the set of projections of a given von Neumann algebra $M$ are denoted by $\mathcal{Z}(M)$, $M^u$ and $M^p$, respectively, and also the central support of $e \in M^p$ in $M$ by $c_e^M$. Notations and facts concerning amalgamated free products of von Neumann algebras will be summarized in next section 2.
\section{Amalgamated Free Product von Neumann Algebras}
Let $M_1 \supseteq N \subseteq M_2$ be $\sigma$-finite von Neumann algebras, and faithful normal conditional expectations $E_1 : M_1 \rightarrow N$, $E_2 : M_2 \rightarrow N$ be given. Their amalgamated free product $(M,E) = (M_1,E_1)\star_N(M_2,E_2)$ is a pair of von Neumann algebra $M$ containing $M_1 \supseteq N \subseteq M_2$ and faithful normal conditional expectation $E : M \rightarrow N$ satisfying (i) $M = M_1\vee M_2$, (ii) $E\!\upharpoonright_{M_k} = E_k$ ($k=1,2$) and (iii) $E\!\upharpoonright_{\Lambda^\circ(M_1^\circ,M_2^\circ)} \equiv 0$, where $\Lambda^\circ(M_1^\circ,M_2^\circ)$ denotes the set of all alternating words in $M_1^\circ := \mathrm{Ker}(E_1)$ and $M_2^\circ := \mathrm{Ker}(E_2)$. The construction of such a pair is a bit complicated, but this simple formulation perfectly serves as a working definition. The construction was introduced in the tracial setting in \cite{Popa:InventMath93} based on the $C^*$-algebraic one \cite{Voiculescu:LNM1132}. Its modular theoretical treatment was given in \cite{Ueda:PacificJMath99}, and will be reviewed below.
Let $\chi$ be a faithful normal semifinite weight on $N$. Then the modular automorphism $\sigma_t^{\chi\circ E}$, $t \in \mathbb{R}$, is simply computed as
\begin{equation}\label{Eq-2.1} \sigma_t^{\chi\circ E}\!\upharpoonright_{M_k} = \sigma_t^{\chi\circ E_k} \quad \quad (k=1,2),\end{equation}
see \cite[Theorem 2.6]{Ueda:PacificJMath99}. This formula together with famous Takesaki's criterion shows that for each $k=1,2$ there is a unique faithful normal conditional expectation $E_{M_k} : M \rightarrow M_k$ characterized by
\begin{equation}\label{Eq-2.2}
E_{M_k}\!\upharpoonright_{\Lambda^\circ(M_1^\circ,M_2^\circ)\setminus M_k^\circ} \equiv 0.
\end{equation}
This fact is easily confirmed in the exactly same way as in \cite[Lemma 2.1]{Ueda:AdvMath11}. It is clear that $E\circ E_{M_k} = E$ holds. Consider the natural inclusion of the so-called continuous cores:
\begin{equation}\label{Eq-2.3}
\widetilde{M} := M\rtimes_{\sigma^{\chi\circ E}}\mathbb{R}\ \supseteq\ \widetilde{M}_k := M_k\rtimes_{\sigma^{\chi\circ E_k}}\mathbb{R}\ (k=1,2)\ \supseteq \widetilde{N} := N\rtimes_{\sigma^\chi}\mathbb{R},
\end{equation}
which is independent of the choice of $\chi$ thanks to Connes's Radon-Nikodym cocycle theorem. The canonical liftings (still being faithful normal conditional expectations) $\widetilde{E} : \widetilde{M} \rightarrow \widetilde{N}$, $\widetilde{E}_k : \widetilde{M}_k \rightarrow \widetilde{N}$ ($k=1,2$) are constructed by
\begin{equation}\label{Eq-2.4}
\widetilde{E} := E\bar{\otimes}\mathrm{Id}_{B(L^2(\mathbb{R}))}\!\upharpoonright_{M\rtimes_{\sigma^{\chi\circ E}}\mathbb{R}}, \quad\quad \widetilde{E}_k := E_k\bar{\otimes}\mathrm{Id}_{B(L^2(\mathbb{R}))}\!\upharpoonright_{M_k\rtimes_{\sigma^{\chi\circ E_k}}\mathbb{R}}.
\end{equation}
Remark that the original $E$ and $E_k$ are recovered as the restrictions of $\widetilde{E}$ and $\widetilde{E}_k$ to $M$ and $M_k$ via the canonical embeddings $M \hookrightarrow \widetilde{M}$ and $M_k \hookrightarrow \widetilde{M}_k$, respectively. Here is a simple but important fact \cite[Theorem 5.1]{Ueda:PacificJMath99} that $\widetilde{M}_1$ and $\widetilde{M}_2$ are freely independent with amalgamation over $\widetilde{N}$ with respect to $\widetilde{E}$, and moreover $\widetilde{M} = \widetilde{M}_1 \vee \widetilde{M}_2$. Consequently the following natural formula holds:
\begin{equation}\label{Eq-2.5}
(\widetilde{M},\widetilde{E}) = (\widetilde{M}_1,\widetilde{E}_1)\star_{\widetilde{N}}(\widetilde{M}_2,\widetilde{E}_2).
\end{equation}
The canonical faithful normal semifinite traces $\mathrm{Tr}_{\widetilde{M}}$, $\mathrm{Tr}_{\widetilde{M}_k}$ ($k=1,2$) and $\mathrm{Tr}_{\widetilde{N}}$ on $\widetilde{M}$, $\widetilde{M}_k$ and $\widetilde{N}$, respectively, (see \cite[Theorem XII.1.1]{Takesaki:Book2}) must satisfy $\mathrm{Tr}_{\widetilde{M}}=\mathrm{Tr}_{\widetilde{N}}\circ\widetilde{E}$ and $\mathrm{Tr}_{\widetilde{M}_k}=\mathrm{Tr}_{\widetilde{N}}\circ\widetilde{E}_k$ (see e.g.~\cite[\S4]{Longo:CMP89}).
Let $M^\omega \supseteq M_k^\omega\ (k=1,2)\ \supseteq N^\omega$ be the ultraproducts of $M \supseteq M_k\ (k=1,2)\ \supseteq N$. Here the inclusion relation is guaranteed by the existence of conditional expectations, and $E$ and $E_k$ ($k=1,2$) can be lifted up to $E^\omega : M^\omega \rightarrow N^\omega$ and $E_k^\omega : M_k^\omega \rightarrow N^\omega$, respectively. All the necessary facts on ultraproducts of von Neumann algebras are summarized in \cite[\S\S2.2]{Ueda:AdvMath11}. Remark that $M_1^\omega$ and $M_2^\omega$ are freely independent with amalgamation over $N^\omega$ with respect to $E^\omega$, see \cite[Proposition 4]{Ueda:TAMS03}. However it is hopeless due to \cite[Lemma 2.2]{Popa:IMRN07} that $M^\omega = M_1^\omega\vee M_2^\omega$ holds.
\section{Technical Results}
\subsection{A non-tracial adaptation of Popa's intertwining-by-bimodule criterion} Let $M$ be an arbitrary $\sigma$-finite (possibly type III) von Neumann algebra, and $A, B$ be its (possibly non-unital) von Neumann subalgebras with units $1_A, 1_B$, respectively. Suppose that $B$ is semifinite with a faithful normal semifinite trace $\mathrm{Tr}_B$ and furthermore that there is a faithful normal conditional expectation $E_B : 1_B M 1_B \rightarrow B$.
\begin{proposition}\label{P-3.1} The following are equivalent{\rm:}
\begin{itemize}
\item[(i)] There is no net $u_\lambda$ of unitaries in $A$ which satisfies $E_B(y^* u_\lambda x) \longrightarrow 0$ $\sigma$-strongly for any $x, y \in \bigcup\big\{ 1_A M p \,|\,p \in B^p; \mathrm{Tr}_N(p)< + \infty\}$.
\item[(ii)] There are a normal {\rm(}possibly non-unital{\rm)} $*$-homomorphism $\rho : A \rightarrow M_n(\mathbb{C})\bar{\otimes}B$ with finite $n \in \mathbb{N}$ and a non-zero partial isometry $w \in M_n(\mathbb{C})\bar{\otimes}M$ such that
\begin{itemize}
\item $(\mathrm{Tr}_n\bar{\otimes}\mathrm{Tr}_B)(\rho(1_A)) < +\infty$,
\item $ww^* \leq e_{11}\otimes 1_A$ and $w^* w \leq \rho(1_A)$, and
\item $(e_{11}\otimes a)w = w\rho(a)$ for all $a \in A$.
\end{itemize}
\item[(iii)] There are non-zero projections $e \in A$, $f \in B$, a normal unital $*$-isomorphism $\theta : eAe \rightarrow fBf$ and a non-zero partial isometry $v \in M$ such that
\begin{itemize}
\item the central support $c_e^A$ is finite in $A$ and $\mathrm{Tr}_B(f) < +\infty$,
\item $vv^* \leq e$ and $v^* v \leq f$, and
\item $xv = v\theta(x)$ for all $x \in eAe$.
\end{itemize}
\end{itemize}
Suppose further that $M$ has an almost periodic weight $\psi$ such that both $A$ and $B$ sit inside the centralizer $M_\psi$, $\psi\!\upharpoonright_B$ is still semifinite, and the $E_B$ is the unique $\psi\!\upharpoonright_{1_B M 1_B}$-preserving one. Then the $w$ in {\rm(ii)} and the $v$ in {\rm(iii)} can be chosen in such a way that there is a common eigenvalue $\lambda$ of $\Delta_\psi$ so that $(\mathrm{id}_n\,\bar{\otimes}\,\sigma_t^\psi)(w) = \lambda^{it}w$ and $\sigma_t^\psi(v) = \lambda^{it}v$ for all $t \in \mathbb{R}$.
\end{proposition}
As usual let us write $A \preceq_M B$ (with $E_B$ and $\mathrm{Tr}_B$) if the above equivalent conditions (i)--(iii) hold. Remark that no assumption on $A$ is necessary. The proof is of course modeled after Popa's original one for finite von Neumann algebras, but some cares are necessary. Indeed we observed this fact with $B$ finite several years ago, through our attempt to get better understanding of the fundamental articles \cite{Popa:AnnMath06, Popa:InventMath06} due to Popa. Houdayer and Vaes informed us that they have also observed it with $B$ finite independently (see \cite[Theorem 2.3]{HoudayerVaes:Preprint12}), and moreover Vaes corrected our misunderstanding on some argument in \cite[\S2]{ChifanHoudayer:DukeMathJ10}. The proof below is just a combination and/or a reformulation of several existing proofs of Popa's criterion \cite[Appendix]{Popa:AnnMath06},\cite[\S2]{Popa:InventMath06} (also see \cite[Appendix F]{BrownOzawa:Book},\cite[Appendix C]{Vaes:Asterisque07} for its exposition) and its variants \cite[\S3]{Asher:PAMS09},\cite[\S2]{ChifanHoudayer:DukeMathJ10},\cite[\S4]{Houdayer:Crelle09}, etc. The same idea as in e.g.~the proof of (1) $\Rightarrow$ (4) in \cite[Proposition C.1]{Vaes:Asterisque07} perfectly works for (ii) $\Rightarrow$ (i). (Note that the proof of (4) $\Rightarrow$ (1) in \cite[Theorem F.12]{BrownOzawa:Book} does not work at this point due to the lack of finite trace. Thus we could not prove (iii) $\Rightarrow$ (i) directly.) Hence the main parts below are (ii) $\Leftrightarrow$ (iii) and (i) $\Rightarrow$ (ii).
\medskip
{\it Proof of {\rm(ii)} $\Rightarrow$ {\rm(i):}} We may assume that $\rho(1_A)=\sum_{k=1}^n e_{kk}\otimes p_k$ with $p_k \in B^p$ thanks to \cite[Corollary 3.20]{Kadison:AmerJMath84}. Since $(\mathrm{Tr}_n\bar{\otimes}\mathrm{Tr}_B)(\rho(1_A)) < +\infty$, one has $w = \sum_{k=1}^n e_{1k}\otimes w_k$ with $w_k = w_k p_k \in \bigcup\big\{ 1_A M p \,|\,p \in B^p; \mathrm{Tr}_N(p)< + \infty\}$. On contrary, suppose that (i) is not true. One can find a net $u_\lambda$ in $A^u$ in such a way that $E_B(w_i^* u_\lambda w_j) \longrightarrow 0$ $\sigma$-strongly for all $i,j$, and hence $\rho(u_\lambda)(\mathrm{id}\bar{\otimes} E_B)(w^* w) = \sum_{i,j=1}^n e_{ij}\otimes E_B(w_i^* u_\lambda w_j) \longrightarrow 0$ $\sigma$-strongly. Therefore, $\Vert(\mathrm{id}_n\bar{\otimes} E_B)(w^* w)\Vert_{\mathrm{Tr}_n\bar{\otimes}\mathrm{Tr}_B} = \Vert\rho(u_\lambda)(\mathrm{id}_n\bar{\otimes} E_B)(w^* w)\rho(1_A)\Vert_{\mathrm{Tr}_n\bar{\otimes}\mathrm{Tr}_B} \longrightarrow 0$, a contradiction to $w\neq 0$. \qed
\medskip
{\it Proof of {\rm(iii)} $\Rightarrow$ {\rm(ii):}} Since $v^* v \in \theta(eAe)'$, one can find a non-zero $z \in \mathcal{Z}(eAe)^p = (\mathcal{Z}(A)e)^p$ in such a way that the normal $*$-homomorphism $x \in zAz = (eAe)z \mapsto \theta(x)v^* v$ is injective. Since $c_e^A$ is finite in $A$, by \cite[Proposition 8.2.1]{KadisonRingrose:Book2} one can find non-zero, mutually orthogonal and equivalent (in $A$) $e_1,\dots,e_n \in A^p$ in such a way that $e_1 \leq z$ and $\sum_{k=1}^n e_k = c_{e_1}^A$. We have $e_1 vv^* \neq 0$, since $\theta(e_1)v^* v = v^* (e_1 vv^*)v$ by the choice of $z$ and $e_1 \leq z$. Then one gets partial isometries $v_1 := e_1, v_2,\dots,v_n \in A$ so that $v^*_k v_k = e_1$ and $v_k v_k^* = e_k$ ($k=2,\dots,n$). Since $e_1 Ae_1 \subseteq eAe$, we can construct a normal $*$-homomorphism $\rho : A \rightarrow M_n(\mathbb{C})\bar{\otimes}B$ by $\rho(a) := \sum_{i,j=1}^n e_{ij}\otimes\theta(v_i^* a v_j)$, $a \in A$. Set $w := \sum_{k=1}^n e_{1k}\otimes v_k v$ with $v$ in (iii), which defines a non-zero partial isometry, since $v^* v_i^* v_j v = \delta_{ij} v^* e_1 v$ and $v^* e_1 v = \theta(e_1)v^* v \neq 0$ as remarked before. Since $\sum_{i=1}^n v_i v_i^* = c_{e_1}^A = c_{e_k}^A$ for all $k=2,\dots,n$, we have $w\rho(a) = \sum_{i,j,k=1}^n e_{1i}e_{jk}\otimes v_i v\theta(v_j^* a v_k) = \sum_{i,k=1}^n e_{1i}e_{ik}\otimes v_i v_i^* a v_k v = \sum_{k=1}^n e_{1k}\otimes c_{e_k}^A a v_k v = (e_{11}\otimes a)w$ for all $a \in A$. Since $\rho(1_A) \leq 1_n\otimes f$, one has $(\mathrm{Tr}_n\bar{\otimes}\mathrm{Tr}_B)(\rho(1_A)) < +\infty$. \qed
\medskip
{\it Proof of {\rm(ii)} $\Rightarrow$ {\rm(iii):}} As in (ii) $\Rightarrow$ (i) we may and do assume that $\rho(1_A) = \sum_{k=1}^n e_{kk}\otimes p_k$ with $\mathrm{Tr}_B$-finite $p_k \in B^p$. Note that any union of finite number of $\mathrm{Tr}_B$-finite projections is again $\mathrm{Tr}_B$-finite thanks to the Kaplansky formula \cite[Theorem 6.1.7]{KadisonRingrose:Book2}. Thus $p=\bigvee_{k=1}^n p_k$ is $\mathrm{Tr}_B$-finite, and replacing $B$ by $pBp$ (if necessary) we may and do assume that $\mathrm{Tr}_B(1_B) < +\infty$. Notice that $A$ must be of the form $A = A_0\oplus\mathrm{Ker}(\rho(-)w^*w)$ with $A_0$ finite, since $\rho(A)$ is finite. Note here that $w^* w \in \rho(A)'$, and thus $\rho(-)w^* w$ is a normal $*$-homomorphism.
Let us first assume that $A_0$ has a type II$_1$ direct summand. By \cite[Lemma 6.5.6]{KadisonRingrose:Book2} one can find nonzero, mutually orthogonal and equivalent (in $A_0$) $e_1,\dots,e_n \in A_0^p$ whose sum is the unit of the type II$_1$ direct summand. With the center-valued trace $\tau : M_n(\mathbb{C})\bar{\otimes}B \rightarrow \mathbb{C}1\bar{\otimes}\mathcal{Z}(B)$ we have $n\tau(\rho(e_1)) \leq \tau(1\otimes1_B) = n\tau(e_{11}\otimes1_B)$, implying that there is a partial isometry $v_1 \in M_n(\mathbb{C})\bar{\otimes}B$ such that $v_1^* v_1 = \rho(e_1)$ and $v_1 v_1^* \leq e_{11}\otimes 1_B$. Since $v_1\rho(e_1)v_1^* = v_1 v_1^* \leq e_{11}\otimes1_B$, we can construct a normal unital $*$-isomorphism $\theta : eAe \rightarrow fBf$ with $e := e_1$, $f := \theta(e)$ in such a way that $e_{11}\otimes\theta(x) = v_1\rho(x)v_1^*$ for $x \in eAe$. Since $w^* w \in \rho(A)'\cap \rho(1_A)\big(M_n(\mathbb{C})\bar{\otimes}M\big)\rho(1_A)$ and $ww^* \in \big(\mathbb{C}e_{11}\bar{\otimes}A\big)'\cap (e_{11}\otimes1_A)\big(M_n(\mathbb{C})\bar{\otimes}M\big)(e_{11}\otimes1_A)$, it is easy to see that $wv_1^*$ is a non-zero partial isometry whose left and right support projections are less than $e_{11}\otimes e$ and $e_{11}\otimes f$, respectively, and hence $wv_1^* = e_{11}\otimes v$ for some non-zero partial isometry $v \in eMf$. Then one has $e_{11}\otimes xv = (e_{11}\otimes x)wv_1^* = w\rho(x)v_1^* = wv_1^* v_1\rho(x)v_1^* = e_{11}\otimes v\theta(x)$ for $x \in eAe$.
We next consider the case that $A_0$ is of type I, that is, there is an abelian (in $A$) $e \in A_0^p$ with $c_e^A = 1_{A_0}$. With a MASA $\mathfrak{A}$ between $\rho(eAe)\oplus\mathbb{C}\rho(e)^\perp \subseteq M_n(\mathbb{C})\bar{\otimes}B$ one can choose, by \cite[Theorem 3.18]{Kadison:AmerJMath84}, mutually orthogonal and equivalent (in $M_n(\mathbb{C})\bar{\otimes}B$) projections $q_1,\dots,q_n$ from $\mathfrak{A}$ with $\sum_{k=1}^n q_k = 1_n\otimes1_B$. Then one immediately observes (by looking at their center-valued traces) that every $q_k$ is equivalent to $e_{11}\otimes1_B$ in $M_n(\mathbb{C})\bar{\otimes}B$. Since $\rho(e)w^*w \neq 0$, some $q:=q_k$ must satisfy $q\rho(e)w^*w \neq 0$. In this way, we can choose a non-zero partial isometry $v_1 \in M_n(\mathbb{C})\bar{\otimes}B$ in such a way $v_1^* v_1 =q\rho(e) (\leq \rho(e))$, $v_1 v_1^* \leq e_{11}\otimes 1_B$, and thus $v_1^* v_1 \in \rho(eAe)'$ and $wv_1^* \neq 0$ (since $q\rho(e)w^* w \neq 0$). Then we can construct a unital normal $*$-homomorphism $\theta : eAe \rightarrow fBf$ with $f := \theta(e)$ by $e_{11}\otimes\theta(x) = v_1\rho(x)v_1^*$ for $x \in eAe$ and a non-zero $y \in eMf$ by $e_{11}\otimes y = wv_1^*$. Moreover we have $e_{11}\otimes xy = (e_{11}\otimes x)wv_1^* = w\rho(x)v_1^* = wv_1^* (v_1\rho(x)v_1^*) = e_{11}\otimes y\theta(x)$ for $x \in eAe$, since $v_1^* v_1 \in \rho(eAe)'$. Hence $xy = y\theta(x)$ for $x \in eAe$. Replacing $e$ by suitable $z \in \mathcal{Z}(eAe)^p$ (if necessary) we can make $\theta$ injective with keeping both $\theta(e) = f$ and $y=eyf$. With the polar decomposition $y=v|y|$ we get $vv^* \leq e$, $v^* v \leq f$ and $xv = v\theta(x)$ for $x \in eAe$. \qed
\medskip
We have two ways for completing the final part of the proof of (i) $\Rightarrow$ (ii) below; one is the use of Haagerup's $L^p$-space technologies and the other that of standard forms due to Araki, Connes and Haagerup. Here we use the latter as easy way. In what follows $(M \curvearrowright \mathcal{H}, J_M, \mathfrak{P}_M^\natural)$ denotes a standard form of $M$, see \cite[Definition IX.1.13]{Takesaki:Book2}.
\medskip
{\it Proof of {\rm(i)} $\Rightarrow$ {\rm(ii):}} Note that $E_B(y^* u_\lambda x) \longrightarrow 0$ $\sigma$-strongly if and only if $\Vert E_B(y^* u_\lambda x)\Vert_{\mathrm{Tr}_B} \longrightarrow 0$ for any $x, y \in \bigcup\{1_A Mp\,|\,p\in B^p; \mathrm{Tr}_B(p) < +\infty\}$. Thus there are $\varepsilon>0$ and $\mathcal{F} \Subset \bigcup\{1_A Mp\,|\,p\in B^p; \mathrm{Tr}_B(p) < +\infty\}$ so that
\begin{equation}\label{Eq-3.1}
\sideset{}{_{x, y \in \mathcal{F}}}\sum \Vert E_B(y^* u x)\Vert_{\mathrm{Tr}_B} \geq \varepsilon \quad \text{for all $u \in A^u$}.
\end{equation}
Each $x \in \mathcal{F}$ has a $\mathrm{Tr}_B$-finite $p_x \in B^p$ with $x = xp_x$, and $p := \bigvee_{x \in \mathcal{F}} p_x$ must be $\mathrm{Tr}_B$-finite as remarked in (ii) $\Rightarrow$ (iii). Thus, replacing $B$ by $pBp$ (if necessary) we may and do assume that $\mathrm{Tr}_B$ is a finite trace, that is, $\mathrm{Tr}_B(1_B) < +\infty$.
Choose a faithful normal state $\varphi_0$ on $1_B^\perp M1_B^\perp$, and set $\hat{B} := B\oplus\mathbb{C}1_B^\perp$ and $E_{\hat{B}} : x \in M \mapsto E_B(1_B x 1_B) + \varphi_0(1_B^\perp x1_B^\perp)1_B^\perp$ giving a faithful normal conditional expectation from the whole $M$ onto $\hat{B}$. Clearly $\hat{B}$ is still finite (since we have assumed that $\mathrm{Tr}_B$ is a finite trace), and the mapping $b+\alpha1_B^\perp \in \hat{B} \mapsto \mathrm{Tr}_B(b)+\alpha \in \mathbb{C}$ defines a faithful normal trace (not weight !) $\mathrm{Tr}_{\hat{B}}$ on $\hat{B}$. Set $\varphi := \mathrm{Tr}_{\hat{B}}\circ E_{\hat{B}}$, a faithful normal positive linear functional on $M$, and let $\xi_0 \in \mathfrak{P}_M^\natural$ be its unique representing vector. It is standard, by a usual exhaustion argument like e.g.~the proof of \cite[Theorem IV.5.5]{Takesaki:Book1}, to see that there is a family of vectors $\{\xi_i\}_{i\in I}$ in $\mathcal{H}$ so that $\xi_0$ is in the family (thus $0$ is regarded as a distinguished element in $I$) and moreover $\mathcal{H} = \sum_{i\in I}^\oplus [J_M\hat{B}J_M\xi_i]$. Therefore, one can construct an isometry $U : \mathcal{H} \rightarrow \ell^2(I)\otimes L^2(\hat{B})$ satisfying $U\xi_0 = \delta_0\otimes\Lambda_{\mathrm{Tr}_{\hat{B}}}(1)$ and $U(J_M x^* J_M) = (1\otimes J_{\hat{B}} x^* J_{\hat{B}})U$ for $x \in \hat{B}$, where $L^2(\hat{B})$ is the usual standard Hilbert space constructed out of $\mathrm{Tr}_{\hat{B}}$, $\Lambda_{\mathrm{Tr}_{\hat{B}}}$ the canonical embedding of $\hat{B}$ to $L^2(\hat{B})$ and $J_{\hat{B}}$ the canonical unitary conjugation on $L^2(\hat{B})$. By the construction we observe that $P:=UU^* \in B(\ell^2(I))\bar{\otimes}\hat{B}$ and moreover that the pair $P\big(B(\ell^2(I))\bar{\otimes}\hat{B}\big)P$ and $P_{\mathbb{C}\delta_0}\otimes1$ with the rank $1$ projection $P_{\mathbb{C}\delta_0}$ onto $\mathbb{C}\delta_0$ is nothing but a concrete realization, modulo the unitary equivalence by $U$, of the basic extension $\langle M, \hat{B}\rangle$ and the Jones projection $e_{\hat{B}}$ associated with $E_{\hat{B}}$. Then
\begin{equation}\label{Eq-3.2} \mathrm{Tr}_{\langle M, \hat{B}\rangle}(-) := (\mathrm{Tr}_{B(\ell^2(I))}\bar{\otimes}\mathrm{Tr}_{\hat{B}})(U(-)U^*)
\end{equation}
with the usual trace $\mathrm{Tr}_{B(\ell^2(I))}$ on $B(\ell^2(I))$ gives a faithful normal semifinite trace on the basic extension $\langle M, \hat{B}\rangle$. For $x \in \hat{B}$ one has $Uxe_{\hat{B}}U^* = P_{\mathbb{C}\delta_0}\otimes x$ and hence $\mathrm{Tr}_{\langle M, \hat{B}\rangle}(x e_{\hat{B}}) =
(\mathrm{Tr}_{B(\ell^2(I))}\bar{\otimes}\mathrm{Tr}_{\hat{B}})(Ux e_{\hat{B}}U^*) =
(\mathrm{Tr}_{B(\ell^2(I))}\bar{\otimes}\mathrm{Tr}_{\hat{B}})(P_{\mathbb{C}\delta_0}\otimes x) = \mathrm{Tr}_{\hat{B}}(x)$.
Therefore, we get
\begin{equation}\label{Eq-3.3}
\mathrm{Tr}_{\langle M, \hat{B}\rangle}(x e_{\hat{B}} y) = \mathrm{Tr}_{\langle M, \hat{B}\rangle}(e_{\hat{B}}yxe_{\hat{B}}) = \mathrm{Tr}_{\langle M, \hat{B}\rangle}(\hat{E}_B(yx)e_{\hat{B}}) =
\varphi(yx), \quad x,y \in M.
\end{equation}
Let $d := \sum_{y \in \mathcal{F}}ye_{\hat{B}}y^* \in \langle M, \hat{B}\rangle^+$, and then $\mathrm{Tr}_{\langle M, \hat{B}\rangle}(d) = \sum_{y \in \mathcal{F}} \varphi(y^* y) < +\infty$ by \eqref{Eq-3.3}. In the exactly same way as in the proof of (1) $\Rightarrow$ (2) of \cite[Theorem F.12]{BrownOzawa:Book} we see, by using \eqref{Eq-3.1}, that the $\sigma$-weakly closed convex hull $\mathfrak{C}$ of $\{ u^* d u\,|\,u \in A^u\}$ does not contain $0$. Moreover, it is plain to see that $J_M 1_B J_M d = d$. Since $1_B \in Z(\hat{B})$ and hence $J_M 1_B J_M \in Z(\langle M, \hat{B} \rangle)$, we conclude that $\mathfrak{C}$ sits in $(1_A J_M 1_B J_M)\langle M, \hat{B} \rangle(1_A J_M 1_B J_M)$. Since $d \geq 0$ and $\mathrm{Tr}_{\langle M, \hat{B} \rangle}(d) < +\infty$, $\mathfrak{C}$ is embedded, as a closed convex set, into $L^2(\langle M, \hat{B} \rangle,\mathrm{Tr}_{\langle M, \hat{B} \rangle})$, the usual GNS Hilbert space associated with $\mathrm{Tr}_{\langle M, \hat{B}\rangle}$. Hence one can choose a unique minimal point $d_0 \in \mathfrak{C}$ with respect to the Hilbert space norm $\Vert-\Vert_{\mathrm{Tr}_{\langle M,\hat{B}\rangle}}$, which in turn falls in $(1_A J_M 1_B J_M)\langle M, \hat{B} \rangle(1_A J_M 1_B J_M)\cap A'$ and satisfies $\mathrm{Tr}_{\langle M,\hat{B}\rangle}(d_0) < +\infty$. Choosing a suitable spectral projection of $d_0$
we get a nonzero projection $e \in \langle M, \hat{B} \rangle\cap A'$ such that $e \leq 1_A J_M 1_B J_M$ and $\mathrm{Tr}_{\langle M, \hat{B}\rangle}(e) < +\infty$.
The projection $e$ apparently gives an $A$--$B$ bimodule $\mathcal{K} := e\mathcal{H}$ with left and right (unital) actions $a\cdot\xi\cdot b := a J_M b^* J_M\xi$ for $a \in A$, $\xi \in \mathcal{K}$, $b \in B$. The GNS representation of $B$ associated with $\mathrm{Tr}_B$ is simply given by the restriction $B \curvearrowright L^2(B) := 1_B L^2(\hat{B})$ with the canonical embedding $\Lambda_{\mathrm{Tr}_B} := \Lambda_{\mathrm{Tr}_{\hat{B}}}\!\upharpoonright_B$, and moreover the canonical unitary conjugation $J_B$ is also just the restriction of $J_{\hat{B}}$ to $L^2(B)$. Thus we get the right $B$-module embedding $U_0 := U\!\upharpoonright_{\mathcal{K}} : \mathcal{K} \hookrightarrow \ell^2(I)\bar{\otimes}L^2(B)_B$ ($\subseteq (1\otimes 1_B)(\ell^2(I)\bar{\otimes}L^2(\hat{B}))$, and $U_0 U_0^* \in B(\ell^2(I))\bar{\otimes} B$ satisfies $(\mathrm{Tr}_{B(\ell^2(I))}\bar{\otimes}\mathrm{Tr}_B)(U_0 U_0^*) = \mathrm{Tr}_{\langle M, \hat{B}\rangle}(e) < +\infty$ by \eqref{Eq-3.2}. By the same reason as in the beginning of the proof of \cite[Proposition F.10]{BrownOzawa:Book} or by \cite[Lemma A.1]{Vaes:Asterisque07} there are $n \in \mathbb{N}$ and a nonzero $z \in \mathcal{Z}(B)^p$ such that $\mathcal{K}_0 := J_M z J_M\mathcal{K}$ is still a non-trivial $A$--$B$ bimodule and $(U_0\!\upharpoonright_{J_M z J_M\mathcal{K}})(U_0\!\upharpoonright_{J_M z J_M\mathcal{K}})^* = (1\otimes J_B z J_B)U_0 U_0^* (1\otimes J_B zJ_B) = (1\otimes z)U_0 U_0^* \precsim P_n\otimes z$ in $B(\ell^2(I))\bar{\otimes}B = (\mathbb{C}1\bar{\otimes}J_B BJ_B)'$, where $P_n$ is a rank $n$ projection in $B(\ell^2(I))$. Choose a partial isometry $v \in (\mathbb{C}1\bar{\otimes}J_B BJ_B)'$ with $v^* v=(U_0\!\upharpoonright_{J_M z J_M\mathcal{K}})(U_0\!\upharpoonright_{J_M z J_M\mathcal{K}})^*$ and $vv^* \leq P_n\otimes z$, and then we can define a right $B$-module embedding $V : \mathcal{K}_0 \hookrightarrow \mathbb{C}^n\bar{\otimes}L^2(B)$ by $V:=v(U_0\!\upharpoonright_{J_M z J_M\mathcal{K}})$ with a fixed identification $P_n\ell^2(I) = \mathbb{C}^n$. The embedding $V$ gives the normal (possibly non-unital) $*$-homomorphism $\rho : a \in A \mapsto VaV^* \in M_n(\mathbb{C})\bar{\otimes}B$.
Let $\delta_i$ ($1\leq i\leq n$) be a standard basis of $\mathbb{C}^n$, and set $\xi_i := V^*(\delta_i\otimes\Lambda_{\mathrm{Tr}_B}(1_B)) \in \mathcal{K}_0$ ($1\leq i \leq n$).
For $a \in A$, write $\rho(a) = \sum_{i,j=1}^n e_{ij}\otimes\rho(a)_{ij}$ with the matrix units $e_{ij}$ associated with the $\delta_i$, and then
\begin{equation}\label{Eq-3.4}
a\xi_j = \sideset{}{_{i=1}^n}\sum J_M\rho(a)_{ij}^* J_M\xi_i, \quad 1\leq j\leq n.
\end{equation}
Consider $\mathbb{M} := M_{n+1}(\mathbb{C})\bar{\otimes}M \curvearrowright L^2(\mathbb{M}) := M_{n+1}(\mathbb{C})\bar{\otimes}\mathcal{H}$ (by left matrix-multiplication) with the canonical unitary conjugation $J_{\mathbb{M}}$ defined by $J_{\mathbb{M}}(e_{ij}\otimes\xi) := e_{ji}\otimes(J_M\xi)$ for $e_{ij}\otimes\xi \in L^2(\mathbb{M})$. The natural cone determined by $(\mathbb{M} \curvearrowright L^2(\mathbb{M}), J_{\mathbb{M}})$ is denoted by $\mathfrak{P}^\natural_{\mathbb{M}}$. Set $\hat{\xi} := \sum_{k=1}^n e_{0k}\otimes \xi_k \in L^2(\mathbb{M})$, and define a normal (possibly non-unital) $*$-homomorphism $\hat{\rho} : A \hookrightarrow \mathbb{M}$ by $\hat{\rho}(a) := e_{00}\otimes a + \sum_{i,j=1}^n e_{ij}\otimes\rho(a)_{ij}$ for $a \in A$. Here a standard matrix unit system $e_{ij}$ in $M_{n+1}(\mathbb{C})$ is indexed by $0,1,\dots,n$. By \eqref{Eq-3.4} one has $\hat{\rho}(a)\hat{\xi} = J_{\mathbb{M}}\hat{\rho}(a)^* J_{\mathbb{M}}\hat{\xi}$ for $a \in A$. A standard fact on polar decomposition in standard forms (c.f.~\cite[Exercise IX.1.2]{Takesaki:Book2},\cite[Lemma 3.1]{Asher:PAMS09}) guarantees the existence of a vector $|\hat{\xi}| \in \mathfrak{P}^\natural_{\mathbb{M}}$ and a partial isometry $\hat{w} \in \mathbb{M}$ satisfying that $\hat{w}|\hat{\xi}| = \hat{\xi}$, $\hat{w}^*\hat{w} = [\mathbb{M}'|\hat{\xi}|]$, $\hat{w}\hat{w}^* = [\mathbb{M}'\hat{\xi}]$ and $\hat{\rho}(a)\hat{w} = \hat{w}\hat{\rho}(a)$ for $a \in A$. Since $(e_{00}\otimes1_A)\hat{\xi} = \hat{\xi}$, one has $(e_{00}\otimes1_A)[\mathbb{M}'\hat{\xi}] = [\mathbb{M}'\hat{\xi}]$, and thus $\hat{w}\hat{w}^* \leq e_{00}\otimes1_A$. Here ($\rho(A) \subseteq$) $M_n(\mathbb{C})\bar{\otimes}M$ is naturally regarded as a corner of $\mathbb{M}$ by the numbering of the matrix units $e_{ij}$'s. Then one has, by \eqref{Eq-3.4} again, $J_{\mathbb{M}}\rho(1_A)J_{\mathbb{M}}\hat{\xi} = \hat{\xi}$, and hence $J_{\mathbb{M}}\rho(1_A)J_{\mathbb{M}}|\hat{\xi}| = |\hat{\xi}|$. By $J_{\mathbb{M}}|\hat{\xi}| = |\hat{\xi}| \in \mathfrak{P}^\natural_{\mathbb{M}}$ we get $\rho(1_A)[\mathbb{M}'|\hat{\xi}|] = [\mathbb{M}'|\hat{\xi}|]$ so that $\hat{w}^*\hat{w} \leq \rho(1_A) \leq \sum_{k=1}^n e_{kk}\otimes1_B$. Therefore, $\hat{w} = \sum_{k=1}^n e_{0k}\otimes w_k$ with $w_k \in 1_A M 1_B$. Letting $w := (e_{10}\otimes 1_A)\hat{w} = \sum_{k=1}^n e_{1k}\otimes w_k \in M_n(\mathbb{C})\bar{\otimes}M$ we have $w^* w \leq \rho(1_A)$, $ww^* \leq e_{11}\otimes1_A$ and $(e_{11}\otimes a)w = (e_{10}\otimes1_A)\hat{\rho}(a)\hat{w}=(e_{10}\otimes1_A)\hat{w}\hat{\rho}(a) = w\rho(a)$ for $a \in A$. We have assumed (by cutting by a projection in $B$) that $\mathrm{Tr}_B(1_B) < +\infty$, and hence $(\mathrm{Tr}_n\bar{\otimes}\mathrm{Tr}_B)(\rho(1_A)) < +\infty$ is now trivial. Hence we are done. \qed
\medskip
{\it Proof of the second part of the assertion{\rm:}} Only the proof of (i) $\Rightarrow$ (ii) needs small modification to prove this. Let us explain this in what follows. The standard form $(M\curvearrowright\mathcal{H},J_M,\mathfrak{P}_M^\natural)$ is constructed from $\psi$ so that $J_M \Delta_\psi J_M = \Delta_\psi^{-1}$. The $\mathrm{Tr}_B$ is given by $\psi\!\upharpoonright_B$. We need an extra argument in relation to the $d_0 \in (1_A J_M 1_B J_M)\langle M, \hat{B} \rangle(1_A J_M 1_B J_M)\cap A'$. By the assumption here the modular operator $\Delta_\psi$ has a diagonalization $\Delta_\psi = \sum_{\lambda>0} \lambda\,e^\psi_\lambda$ and satisfies $\Delta_\psi^{it} \in \langle M, \hat{B} \rangle \cap A'$ for all $t \in \mathbb{R}$. Hence all the $e^\psi_\lambda$'s fall in $\langle M, \hat{B}\rangle \cap A'$. Thus $e^\psi_\lambda\,d_0^{1/2}$ with some $\lambda$ defines a non-zero element in $\langle M, \hat{B} \rangle \cap A'$. Since $e^\psi_\lambda$ commutes with $1_A J_M 1_B J_M$ and since $\mathrm{Tr}_{\langle M, \hat{B}\rangle}(e^\psi_\lambda\,d_0\,e^\psi_\lambda) = \mathrm{Tr}_{\langle M, \hat{B}\rangle}(d_0^{1/2} e^\psi_\lambda d_0^{1/2}) \leq \mathrm{Tr}_{\langle M, \hat{B}\rangle}(d_0) < +\infty$, we may and do assume $d_0 = e^\psi_\lambda\,d_0\,e^\psi_\lambda$. Hence the $A$--$B$ bimodule $\mathcal{K}_0$ can be chosen as a subspace of $e^\psi_\lambda\,\mathcal{H}$. Therefore, the $\hat{\xi} \in L^2(\mathbb{M}) = M_{n+1}(\mathbb{C})\bar{\otimes}\mathcal{H}$ satisfies that $(I_{M_{n+1}(\mathbb{C})}\bar{\otimes}\,\Delta_\psi^{it})\hat{\xi} = \lambda^{it}\hat{\xi}$ for all $t \in \mathbb{R}$. Since $I_{M_{n+1}(\mathbb{C})}\bar{\otimes}\,\Delta_\psi$ is the modular operator of $\mathrm{Tr}_{n+1}\bar{\otimes}\,\psi$ on $\mathbb{M}$, $(I_{M_{n+1}(\mathbb{C})}\bar{\otimes}\,\Delta_\psi^{it})|\hat{\xi}|$ still falls in $\mathfrak{P}_{\mathbb{M}}^\natural$, see \cite[Lemma IX.1.4]{Takesaki:Book2}. Remark here that $J_{\mathbb{M}}$ there is nothing but the one constructed from $\mathrm{Tr}_{n+1}\bar{\otimes}\,\psi$. Hence, by the uniqueness of polar decomposition $(\mathrm{id}\bar{\otimes}\sigma_t^\psi)(\hat{w}) = \lambda^{it}\hat{w}$ and $(I_{M_{n+1}(\mathbb{C})}\bar{\otimes}\,\Delta_\psi^{it})|\hat{\xi}| = |\hat{\xi}|$ hold for every $t \in \mathbb{R}$. These modifications are enough to complete the proof. \qed
\begin{remark}\label{R-3.2} {\rm Let $E_{\hat{B}}$ and $\varphi = \mathrm{Tr}_{\hat{B}}\circ E_{\hat{B}}$ be as in the proof of (i) $\Rightarrow$ (ii) above. Let $\widehat{E_{\hat{B}}} : \langle M, \hat{B}\rangle \rightarrow M$ be the dual operator-valued weight associated with $E_{\hat{B}}$ in the sense of \cite[\S\S1.2]{Kosaki:JFA86}. It is known that the modular operator $\Delta_\varphi$ and Connes's spacial derivative $(d(\varphi\circ\widehat{E_{\hat{B}}}))/(d(\mathrm{Tr}_{\hat{B}}\circ\mathrm{Ad}J_M((-)^*)))$ must coincide, see e.g.~the proof of \cite[Proposition 2.2]{IzumiLongoPopa:JFA98}. Moreover $\Delta_\varphi$ is affiliated with $\langle M, \hat{B} \rangle$, since $\varphi = \mathrm{Tr}_{\hat{B}}\circ E_{\hat{B}}$. With these two facts one can prove that {\it the modular operator $\Delta_\varphi$ is the Radon--Nikodym derivative of $\varphi\circ\widehat{E_{\hat{B}}}$, i.e., $\varphi\circ\widehat{E_{\hat{B}}} = \mathrm{Tr}_{\langle M, \hat{B}\rangle, \Delta_\varphi}$ in the sense of} \cite[Lemma VIII.2.8]{Takesaki:Book2}. This explains, in full generality, the relationship that was pointed out in \cite[Eq.(1.3.1)]{Popa:InventMath06} in the almost periodic case.}
\end{remark}
\subsection{A non-tracial version of Ioana--Peterson--Popa's theorem} Let us investigate an amalgamated free product $(M,E) = (M_1,E_1)\star_N(M_2,E_2)$.
\begin{proposition}\label{P-3.3} Let $A$ be a {\rm(}unital{\rm)} von Neumann subalgebra of the centralizer $(M_1)_\varphi$ of a certain faithful normal state $\varphi$, and\, $\mathfrak{M}_1$ be a {\rm(}possibly non-unital{\rm)} dense {\rm(}in any von Neumann algebra topology{\rm)} $*$-subalgebra of $M_1$ with $E_1(\mathfrak{M}_1) \subseteq \mathfrak{M}_1$. Suppose that there is a net $v_\lambda$ of unitaries in $A$ such that $E_1(y^* v_\lambda x) \longrightarrow 0$ $\sigma$-strongly for all $x,y \in \mathfrak{M}_1$. Then any unitary $u \in M$ with $uAu^* \subseteq M_1$ must fall in $M_1$. In particular, $\mathcal{N}_M(A) = \mathcal{N}_{M_1}(A)$ and $A'\cap M = A'\cap M_1$. Here $\mathcal{N}_P(Q)$ denotes the set of unitaries $u \in P$ with $uQu^* = Q$ for a given unital inclusion $P \supseteq Q$ of von Neumann algebras.
\end{proposition}
This is nothing but a non-tracial version of \cite[Theorem 1.1]{IoanaPetersonPopa:Acta08} due to Ioana, Peterson and Popa. Although the proof below is modeled after their proof, we need to overcome some difficulties due to the lack of trace by utilizing modular theoretic technologies.
\begin{proof} Let $(M \curvearrowright \mathcal{H}, J_M, \mathfrak{P}^\natural_M)$ be a standard form of $M$, and $\xi_0 \in \mathfrak{P}^\natural_M$ be the unique representing vector of $\varphi\circ E_{M_1}$. Let $e_{M_1}$ be the so-called Jones projection associated with $E_{M_1}$, i.e., $e_{M_1}x\xi_0 = E_{M_1}(x)\xi_0$ for $x \in M$, and the basic extension $\langle M, M_1 \rangle$ is defined to be $M \vee \{e_{M_1}\}'' = J_M M_1' J_M \curvearrowright \mathcal{H}$. Consider the projection $p := [A J_M M_1 J_M u^*\xi_0] \in A' \cap(J_M M_1 J_M)' = A' \cap \langle M, M_1\rangle$. Notice that $a J_M x^* J_M u^*\xi_0 = J_M x^* J_M u^*(uau^*)\xi_0$ for $a \in A$ and $x \in M_1$, and moreover that $uau^* \in M_1$ can be approximated in any von Neumann algebra topology, by analytic elements, say $y_\lambda$, in $M_1$ with respect to the modular action $\sigma^\varphi$. Those altogether show that
\begin{align*}
a J_M x^* J_M u^*\xi_0 = \lim_\lambda J_M x^* J_M u^* y_\lambda\xi_0 = \lim_\lambda J_M x^* \sigma_{i/2}^\varphi(y_\lambda)^* J_M u^*\xi_0 \in [J_M M_1 J_M u^*\xi_0]
\end{align*}
thanks to $\sigma_t^{\varphi\circ E_{M_1}}\!\upharpoonright_{M_1} = \sigma_t^\varphi$ ($t \in \mathbb{R}$) and \cite[Lemma VIII.3.18 (ii)]{Takesaki:Book2}. Consequently we get $p \leq [J_M M_1 J_M u^*\xi_0] = u^*e_{M_1}u$, which and $\widehat{E_{M_1}}(e_{M_1}) = 1$ imply $\Vert\widehat{E_{M_1}}(p)\Vert_\infty < +\infty$, where $\widehat{E_{M_1}} : \langle M, M_1 \rangle \rightarrow M$ denotes the dual operator-valued weight of $E_{M_1}$. See \cite[\S\S1.2, Lemma 3.1]{Kosaki:JFA86}. We will prove $(1-e_{M_1})p(1-e_{M_1}) = 0$. In fact, if this is the case, then $p \leq e_{M_1}$ so that $u^*\xi_0 = e_{M_1}u^*\xi_0 = E_{M_1}(u^*)\xi_0$, implying $u = E_{M_1}(u) \in M_1$ since $\xi_0$ is separating for $M \curvearrowright \mathcal{H}$. Since $\Vert \widehat{E_{M_1}}(p)\Vert_\infty < +\infty$ and $\widehat{E_{M_1}}(e_{M_1}) = 1$ as before, any spectral projection $f$ of $(1-e_{M_1})p(1-e_{M_1})$ corresponding to $[\delta,1]$ with arbitrary $\delta > 0$ still satisfies $\Vert\widehat{E_{M_1}}(f)\Vert_\infty < +\infty$. Therefore, it suffices to prove that any projection $f \in A'\cap\langle M, M_1 \rangle$ satisfying both $f \leq 1-e_{M_1}$ and $\Vert\widehat{E_{M_1}}(f)\Vert_\infty < +\infty$ must be $0$.
In what follows we denote by $\mathcal{A}$ the $*$-subalgebra of $M$ consisting of all analytic elements with respect to $\sigma^{\varphi\circ E_{M_1}}$, which is well-known to be dense in any von Neumann algebra topology. Set $\psi := \varphi\circ E_{M_1}\circ \widehat{E_{M_1}}$, a faithful normal semifinite weight on $\langle M, M_1 \rangle$, and let $\langle M, M_1 \rangle \curvearrowright L^2(\langle M, M_1 \rangle, \psi)$ be the GNS representation with canonical embedding $\Lambda_{\psi} : \mathfrak{n}_\psi := \{x \in \langle M, M_1 \rangle\,|\,\psi(x^* x) < +\infty\} \rightarrow L^2(\langle M, M_1 \rangle, \psi)$ and norm $\Vert-\Vert_\psi$ associated with the weight $\psi$. Remark that $E_{M_1}(\mathcal{A}) \subseteq \mathcal{A}$ (thanks to $E_{M_1}\circ\sigma_t^{\varphi\circ E_{M_1}} = \sigma_t^\varphi\circ E_{M_1}$ for all $t \in \mathbb{R}$) and thus $\mathrm{span}(\mathcal{A}e_{M_1}\mathcal{A})$ becomes a dense (in any von Neumann algebra topology) $*$-subalgebra of $\mathfrak{n}_\psi^*\cap\mathfrak{n}_\psi$, and hence $\Lambda_\psi(\mathrm{span}(\mathcal{A}\,e_{M_1}\mathcal{A}))$ is dense in $L^2(\langle M, M_1\rangle,\psi)$ by \cite[Lemma 2.1]{IzumiLongoPopa:JFA98}. Thus one can choose a sequence $T_n \in \mathrm{span}(\mathcal{A}\,e_{M_1}\mathcal{A})$ in such a way that
$\Vert \Lambda_\psi(T_n - f)\Vert_\psi \longrightarrow 0$ as $n \rightarrow \infty$, where note that $f$ clearly falls in $\mathfrak{n}_\psi$. Since $f \leq 1-e_{M_1}$ and $\sigma_t^\psi(e_{M_1}) = e_{M_1}$ ($t \in \mathbb{R}$) \cite[Lemma 5.1]{Kosaki:JFA86}, we also have $\Vert \Lambda_\psi((1-e_{M_1})T_n(1-e_{M_1})-f)\Vert_\psi \longrightarrow 0$ as $n \rightarrow \infty$ so that may and do assume that $T_n = (1-e_{M_1})T_n(1-e_{M_1})$ for all $n$.
On contrary, suppose $f \neq 0$, that is, $\gamma := \Vert\Lambda_\psi(f)\Vert_\psi \gneqq 0$. Then one can choose $T := T_{n_0} \in \mathrm{span}(\mathcal{A}e_{M_1}\mathcal{A})$ with some $n_0$ in such a way that
\begin{equation}\label{Eq-3.5}
\Vert\Lambda_\psi(T)\Vert_\psi \leq 3\gamma/2, \quad
\Vert\Lambda_\psi(T - f)\Vert_\psi \leq \gamma/5.
\end{equation}
For any $v \in A^u$ we compute
\begin{align*}
\gamma^2 - |\psi(T^* vTv^*)|
&\leq
|\psi(fvfv^*) - \psi(T^* vTv^*)| \\
&\leq
|\psi((f-T)^* vfv^*)| + |\psi(T^* v(f-T)v^*)| \\
&\leq
\Vert\Lambda_\psi(f-T)\Vert_\psi \Vert\Lambda_\psi(vfv^*)\Vert_\psi + \Vert\Lambda_\psi(T)\Vert_\psi \Vert\Lambda_\psi(v(f-T)v^*)\Vert_\psi \\
&\leq
\Vert\Lambda_\psi(f-T)\Vert_\psi \Vert\Lambda_\psi(f)\Vert_\psi + \Vert\Lambda_\psi(T)\Vert_\psi \Vert\Lambda_\psi(f-T)\Vert_\psi \\
&\leq \gamma^2/2,
\end{align*}
where the first, the third, the fourth and the fifth inequalities follow from $f \in A'\cap\langle M,M_1\rangle$, the Cauchy--Schwarz inequality, $v \in (M_1)_\varphi \subset \langle M, M_1\rangle_\psi$, and \eqref{Eq-3.5}, respectively. Therefore, $\gamma^2 \leq 2|\psi(T^* v T v^*)|$ holds for all $v \in A^u$. Since $T = (1-e_{M_1})T(1-e_{M_1})$, we can write $T = \sum_{k=1}^m x_k e_{M_1} y_k$ with $x_k, y_k \in \mathcal{A}\cap\mathrm{Ker}(E_{M_1})$. Thus, for every $v \in A^u$ we have
\begin{align*}
\gamma^2
&\leq
2\sideset{}{_{k,l=1}^m}\sum|\psi(y_k^* e_{M_1} x_k^* v x_l e_{M_1} y_l v^*)| \\
&=
2\sideset{}{_{k,l=1}^m}\sum|\psi(y_k^* E_{M_1}(x_k^* v x_l)e_{M_1} y_l v^*)| \\
&=
2\sideset{}{_{k,l=1}^m}\sum|\varphi\circ E_{M_1}(y_k^* E_{M_1}(x_k^* v x_l)y_l v^*)| \\
&=
2\sideset{}{_{k,l=1}^m}\sum|\varphi\circ E_{M_1}(\sigma_i^{\varphi\circ E_{M_1}}(y_l)v^* y_k^* E_{M_1}(x_k^* v x_l))| \\
&\leq
2 \max_{1\leq k \leq m}\Vert y_k\Vert_\infty \max_{1\leq l \leq m} \Vert\sigma_i^{\varphi\circ E_{M_1}}(y_l)\Vert_\infty \sideset{}{_{k,l=1}^m}\sum \Vert E_{M_1}(x_k^* v x_l)\Vert_\varphi.
\end{align*}
Here the third equality is due to $\widehat{E_{M_1}}(e_{M_1}) = 1$, the fourth one follows from $v \in (M_1)_\varphi \subseteq M_{\varphi\circ E_{M_1}}$ and $y_l \in \mathcal{A}$ with the so-called modular condition, and finally the last inequality is due to the Cauchy--Schwarz inequality. Consequently we have chosen $x_1,\dots,x_m \in \mathcal{A}\cap\mathrm{Ker}(E_{M_1})$ and a universal constant $C > 0$ so that
\begin{equation}\label{Eq-3.6}
\gamma^2 \leq C \sideset{}{_{k,l=1}^m}\sum \Vert E_{M_1}(x_k^* v x_l)\Vert_\varphi \quad \text{for all $v \in A^u$}.
\end{equation}
Set $\mathfrak{M}_1^\circ := \mathfrak{M}_1 \cap M_1^\circ$. By the assumption on $\mathfrak{M}_1$ and by the Kaplansky density theorem any element $x \in M_1^\circ$ can be approximated in any von Neumann algebra topology by a bounded net of elements $x_\lambda^\circ = x_\lambda - E_1(x_\lambda) \in \mathfrak{M}_1^\circ$ with $x_\lambda \in \mathfrak{M}_1$, $x_\lambda \longrightarrow x$. Thus $\mathfrak{M}_1 + \mathrm{span}(\Lambda^\circ(\mathfrak{M}_1^\circ,M_2^\circ)\setminus \mathfrak{M}_1^\circ)$ is also dense in $M$ in any von Neumann algebra topology so that the Kaplansky density theorem enables us to approximate each $x_k$ ($=x_k - E_{M_1}(x_k)$) by a net $x_{k,\lambda}$ in $\mathrm{span}(\Lambda^\circ(\mathfrak{M}_1^\circ,M_2^\circ)\setminus\mathfrak{M}_1^\circ)$; namely $\Vert x_{k,\lambda}\Vert_\infty \leq 2\Vert x_k\Vert_\infty$ and $x_{k,\lambda} \longrightarrow x_k$ $\sigma$-$*$-strongly. Then we have, for every $v \in A^u$,
\begin{align*}
&\Vert E_{M_1}(x_k^* v x_l)\Vert_\varphi
\leq
\Vert E_{M_1}(x_{k,\lambda}^* v x_{l,\lambda})\Vert_\varphi
+
\Vert E_{M_1}(x_k^* v x_l - x_{k,\lambda}^* v x_{l,\lambda})\Vert_\varphi \\
&\leq
\Vert E_{M_1}(x_{k,\lambda}^* v x_{l,\lambda})\Vert_\varphi
+ \Vert(x_k - x_{k,\lambda})^* v x_l)\Vert_{\varphi\circ E_{M_1}} + \Vert x_{k,\lambda} v (x_l - x_{l,\lambda})\Vert_{\varphi\circ E_{M_1}} \\
&\leq
\Vert E_{M_1}(x_{k,\lambda}^* v x_{l,\lambda})\Vert_\varphi
+ \Vert \sigma_{i/2}^{\varphi\circ E_{M_1}}(x_l)\Vert_\infty \Vert x_k^* - x_{k,\lambda}^*\Vert_{\varphi\circ E_{M_1}} +
2\Vert x_k\Vert_\infty \Vert x_l - x_{l,\lambda}\Vert_{\varphi\circ E_{M_1}},
\end{align*}
where we used, in the last line, that $x_l \in \mathcal{A}$ with \cite[Lemma VIII.3.18 (ii)]{Takesaki:Book2} and $v \in (M_1)_\varphi$. Let $\varepsilon > 0$ be arbitrary chosen. Then some $\lambda$ (being independent of $v$'s) satisfies that $\gamma^2 \leq \varepsilon + C \sum_{k,l=1}^m \Vert E_{M_1}(x_{k,\lambda}^* v x_{l,\lambda})\Vert_\varphi$ for all $v \in A^u$. Since any element in $\Lambda^\circ(\mathfrak{M}_1^\circ,M_2^\circ)\setminus\mathfrak{M}_1^\circ$ is written as $azb$ with $a,b \in \{1\}\cup \mathfrak{M}_1^\circ$, $z$ an alternating word in $\mathfrak{M}_1^\circ,M_2^\circ$ whose leftmost and rightmost letters are chosen from $M_2^\circ$, there are finitely many such words $a_j^{(i)}z_j^{(i)}b_j^{(i)}$, $i=1,2$, $j = 1,\dots,m'$, and positive constants $C_j > 0$, $j = 1,\dots,m'$, so that
\begin{align*}
\gamma^2 &\leq \varepsilon +
\sideset{}{_{j=1}^{m'}}\sum C_j \Vert E_{M_1}(a_j^{(1)}z_j^{(1)}b_j^{(1)} v a_j^{(2)}z_j^{(2)}b_j^{(2)})\Vert_\varphi \\
&= \varepsilon +
\sideset{}{_{j=1}^{m'}}\sum C_j \Vert a_j^{(1)}E_{M_1}(z_j^{(1)} E_1(b_j^{(1)} v a_j^{(2)})z_j^{(2)})b_j^{(2)}\Vert_\varphi
\end{align*}
for all $v \in A^u$, where the equality comes from the free independence of $M_1, M_2$ and \eqref{Eq-2.2}. Applying the above estimate of $\gamma^2$ to the net $v=v_\lambda$ in our hypothesis we get $\gamma^2 \leq \varepsilon$ (at the limit in $\lambda$), a contradiction to $\gamma \gneqq 0$, since $\varepsilon$ is arbitrary.
\end{proof}
\begin{remark}\label{R-3.4} {\rm It is worth while to note that the inequality \eqref{Eq-3.6} is a general fact. Let $P \supseteq Q$ be $\sigma$-finite von Neumann algebras with a faithful normal conditional expectation $E_Q : P \rightarrow Q$ and $A$ be a von Neumann subalgebra of the centralizer $Q_\varphi$ with some faithful normal state $\varphi$. The middle part of discussion above shows that for each projection $f \in A'\cap\langle P, Q\rangle$ satisfying both $f \leq 1-e_Q$ and $\Vert \widehat{E_Q}(f)\Vert_\infty < +\infty$ there are analytic (with respect to $\sigma^{\varphi\circ E_Q}$) elements $x_1,\dots,x_m \in P$ and a universal constant $C > 0$ such that
\begin{equation*}
\Vert \Lambda_{\varphi\circ E_Q\circ\hat{E}_Q}(f) \Vert_{\varphi\circ E_Q\circ\hat{E}_Q}^2 \leq
C \sideset{}{_{k,l=1}^m}\sum \Vert E_Q(x_k^* v x_l)\Vert_\varphi \quad \text{for all $v \in A^u$}.
\end{equation*}
}
\end{remark}
\subsection{A result for controlling central sequences in amalgamated free products} Let us investigate central sequences in an amalgamated free product $(M,E) = (M_1,E_1)\star_N(M_2,E_2)$. The next result is an adaptation and/or an improvement of the methods of \cite[Proposition 3.5]{Ueda:AdvMath11} and \cite[Proposition 3.1]{Ueda:MRL} to amalgamated free product von Neumann algebras. In this subsection we use the notations and facts summarized in \cite[\S\S2.2]{Ueda:AdvMath11}.
\begin{proposition}\label{P-3.5} Suppose that there is a faithful normal state $\varphi$ on $M_1$ satisfying the following conditions{\rm:}
\begin{itemize}
\item[(a)] $\sigma_t^\varphi(N) = N$ for all $t \in \mathbb{R}$.
\item[(b)] For every $n \in \mathbb{N}$ with $n \geq 2$ there are unitaries $u_k = u_k^{(n)}
,v_k = v_k^{(n)} \in (M_1)_\varphi$, $0 \leq k \leq n-1$, such that $E_1(u_{k_1}^* u_{k_2}) = E_N^\varphi(v_{k_1}^* v_{k_2}) = 0$ for all $0 \leq k_1 \neq k_2 \leq n-1$, where $E_N^\varphi$ denotes the unique $\varphi$-preserving conditional expectation from $M_1$ onto $N$, whose existence follows from {\rm(a)} and Takesaki's criterion.
\end{itemize}
Then, for any $x \in (M_1)_\varphi'\cap M^\omega$, any $y \in M_2^\circ$ and any sequence $(t_m)_m$ of real numbers we have
\begin{align*}
\Vert E_2(y^* y)^{1/2}(x-(E_{M_1})^\omega(x))\Vert_{(\varphi\circ E_{M_1})^\omega}
\leq
\Vert yx - xz \Vert_{(\varphi\circ E_{M_1})^\omega},
\end{align*}
with $z := \big[(\sigma_{t_m}^{\varphi\circ E_{M_1}}(y))_m\big] \in M^\omega$.
\end{proposition}
Remark here that any bounded sequence $(\sigma_{t_m}^{\varphi\circ E_{M_1}}(x(m)))_m$ with arbitrary $(x(m))_m$ giving an element in $M^\omega$ gives again an element in $M^\omega$, as shown in the proof of \cite[Proposition 3.1]{Ueda:MRL}. A key fact behind this is that any modular action $\sigma^\psi$ satisfies $\psi\circ \sigma_{t}^\psi = \psi$ for all $t \in \mathbb{R}$. In particular, the element $z$ in the statement above makes sense.
\begin{proof} Write $M_1^\triangledown := \mathrm{Ker}(E_N^\varphi)$. One can easily see, by using $x \in M_1 \mapsto$ $E_1(x) + (x - E_1(x)) \in N + M_1^\circ$ or $E_N^\varphi(x) + (x - E_N^\varphi(x)) \in N + M_1^\triangledown$, that $\mathrm{span}(\Lambda^\circ(M_1^\circ,M_2^\circ)\setminus M_1^\circ)$ coincides with the linear span of the following sets of words:
\begin{equation*}
\underbrace{M_1^\circ \cdots M_2^\circ}_{\text{alternating}}M_1^\triangledown, \quad \underbrace{M_1^\circ \cdots M_2^\circ}_{\text{alternating}}, \quad
\underbrace{M_2^\circ \cdots M_2^\circ}_{\text{alternating}}M_1^\triangledown, \quad
\underbrace{M_2^\circ \cdots M_2^\circ}_{\text{alternating}}.
\end{equation*}
Define four closed subspaces $\mathcal{X}_1 := \left[\Lambda_{\varphi\circ E_{M_1}}(M_1^\circ \cdots M_2^\circ M_1^\triangledown)\right]$, $\mathcal{X}_2 := \left[\Lambda_{\varphi\circ E_{M_1}}(M_1^\circ \cdots M_2^\circ)\right]$, $\mathcal{X}_3 := \left[\Lambda_{\varphi\circ E_{M_1}}(M_2^\circ \cdots M_2^\circ M_1^\triangledown)\right]$, $\mathcal{X}_4 := \left[\Lambda_{\varphi\circ E_{M_1}}(M_2^\circ \cdots M_2^\circ)\right]$ in $\mathcal{H} := L^2(M,\varphi\circ E_{M_1})$, and clearly
\begin{equation*}
\mathcal{H} = \overline{\Lambda_{\varphi\circ E_{M_1}}(M_1)} \oplus \mathcal{X}_1 \oplus \mathcal{X}_2 \oplus \mathcal{X}_3 \oplus \mathcal{X}_4.
\end{equation*}
Denote by $P_i$, $i=1,2,3,4$, the projection from $\mathcal{H}$ onto $\mathcal{X}_i$. Remark that
\begin{equation}\label{Eq-3.7}
\left(I_{\mathcal{H}} - \sideset{}{_{i=1}^4}\sum P_i\right)\Lambda_{\varphi\circ E_{M_1}}(x) = \Lambda_{\varphi\circ E_{M_1}}(E_{M_1}(x)), \quad x \in M.
\end{equation}
Let $n \in \mathbb{N}$ with $n \geq 2$ be fixed. Define unitary operators $S_k = S_k^{(n)}, T_k = T_k^{(n)}$ ($k = 0,\dots,n-1$) on $\mathcal{H}$ by
\begin{equation*}
S_k\Lambda_{\varphi\circ E_{M_1}}(x) := \Lambda_{\varphi\circ E_{M_1}}(u_k x u_k^*), \quad
T_k\Lambda_{\varphi\circ E_{M_1}}(x) := \Lambda_{\varphi\circ E_{M_1}}(v_k x v_k^*), \quad x \in M,
\end{equation*}
with $u_k = u_k^{(n)}, v_k = v_k^{(n)} \in (M_1)_\varphi \subseteq M_{\varphi\circ E_{M_1}}$ in our hypothesis. Here are simple claims.
\begin{itemize}
\item[(A)] $\{S_k\mathcal{X}_i\}_{k=0}^{n-1}$ is an orthogonal family of closed subspaces, $i=3,4$.
\item[(B)] $\{T_k\mathcal{X}_2\}_{k=0}^{n-1}$ is an orthogonal family of closed subspaces.
\end{itemize}
The proofs of those are essentially same, but (A) is easier than (B). Thus we prove only (B) here and leave (A) to the reader. By using $x \mapsto E_i(x) + (x - E_i(x)) \in N + M_i^\circ$ ($i=1,2$) again and again we have
\begin{align*}
(v_{k_2}&(M_1^\circ\cdots M_2^\circ)v_{k_2}^*)^* (v_{k_1}(M_1^\circ\cdots M_2^\circ)v_{k_1}^*) \\
&= v_{k_2} (M_2^\circ \cdots (M_1^\circ v_{k_2}^* v_{k_1} M_1^\circ)\cdots M_2^\circ) v_{k_1}^* \subseteq v_{k_2} N v_{k_1}^* + v_{k_2}\mathrm{Ker}(E_{M_1})v_{k_1}^*.
\end{align*}
The desired assertion immediately follows from that $v_k \in (M_1)_\varphi$; in fact, if $k_1 \neq k_2$, then
\begin{align*}
&\varphi\circ E_{M_1}(v_{k_2} N v_{k_1}^*) = \varphi(N v_{k_1}^* v_{k_2}) = \varphi(N E_N^\varphi(v_{k_1}^* v_{k_2})) = \{0\}, \\
&\varphi\circ E_{M_1}(v_{k_2}\mathrm{Ker}(E_{M_1})v_{k_1}^*) = \varphi(v_{k_2}E_{M_1}(\mathrm{Ker}(E_{M_1}))v_{k_1}^*) = \{0\}.
\end{align*}
\medskip
Let us choose arbitrary $x \in (M_1)_\varphi'\cap M^\omega$ with representative $(x(m))_m$. For each $\varepsilon > 0$ and each $n \in \mathbb{N}$ with $n \geq 2$ one can choose a neighborhood $W=W_{\varepsilon,n}$ in $\beta(\mathbb{N})$ at $\omega$ so that
\begin{equation*}
\Vert\Lambda_{\varphi\circ E_{M_1}}(x(m)-u_k x(m) u_k^*)\Vert_{\mathcal{H}} < \varepsilon, \quad \Vert\Lambda_{\varphi\circ E_{M_1}}(x(m)-v_k x(m) v_k^*)\Vert_{\mathcal{H}} < \varepsilon
\end{equation*}
for all $0 \leq k \leq n-1$ and $m \in W \cap \mathbb{N}$, where the $u_k = u_k^{(n)}, v_k = v_k^{(n)}$ are as above. For each $i=3,4$ and every $m \in W\cap\mathbb{N}$ we have, with the above $S_k = S_k^{(n)}$,
\begin{align*}
&\Vert P_i\Lambda_{\varphi\circ E_{M_1}}(x(m))\Vert_{\mathcal{H}}^2 \\
&=
\frac{1}{n}\sideset{}{_{k=0}^{n-1}}\sum \Vert S_k P_i\Lambda_{\varphi\circ E_{M_1}}(x(m))\Vert_{\mathcal{H}}^2 \\
&\leq
\frac{2}{n}\sideset{}{_{k=0}^{n-1}}\sum\Big\{
\Vert S_k P_i\Lambda_{\varphi\circ E_{M_1}}(x(m)) -
S_k P_i S_k^*\Lambda_{\varphi\circ E_{M_1}}(x(m))\Vert_{\mathcal{H}}^2 + \Vert S_k P_i S_k^*\Lambda_{\varphi\circ E_{M_1}}(x(m))\Vert_{\mathcal{H}}^2\Big\} \\
&=
\frac{2}{n}\sideset{}{_{k=1}^{n-1}}\sum
\Vert S_k P_i S_k^*\Lambda_{\varphi\circ E_{M_1}}(u_k x(m)u_k^* -x(m))\Vert_{\mathcal{H}}^2 + \frac{2}{n}\sideset{}{_{k=0}^{n-1}}\sum\Vert S_k P_i S_k^*\Lambda_{\varphi\circ E_{M_1}}(x(m))\Vert_{\mathcal{H}}^2 \\
&<
2 \varepsilon^2 + \frac{2}{n}\sideset{}{_{k=0}^{n-1}}\sum\Vert S_k P_i S_{k}^*\Lambda_{\varphi\circ E_{M_1}}(x(m))\Vert_{\mathcal{H}}^2 \\
&\leq
2 \varepsilon^2 + \frac{2}{n}\Vert\Lambda_{\varphi\circ E_{M_1}}(x(m))\Vert_{\mathcal{H}}^2 \quad \text{(by the claim (A))} \\
&\leq
2 \varepsilon^2 + 2\Vert((x(m))_m\Vert_\infty^2/n.
\end{align*}
Similarly, using the claim (B) with $T_k^{(n)}$ instead of $S_k^{(n)}$ we have
\begin{equation*}
\Vert P_2\Lambda_{\varphi\circ E_{M_1}}(x(m))\Vert_{\mathcal{H}}^2 < 2\varepsilon^2 + 2\Vert((x(m))_m\Vert_\infty^2/n
\end{equation*}
for every $m \in W \cap \mathbb{N}$. Since $n$ and $\varepsilon$ are arbitrary, for each $\delta > 0$ one can find a neighborhood $W_\delta$ in $\beta(\mathbb{N})$ at $\omega$ so that
\begin{equation}\label{Eq-3.8}
\Vert (P_2+P_3+P_4)\Lambda_{\varphi\circ E_{M_1}}(x(m))\Vert_{\mathcal{H}} < \delta
\end{equation}
for all $m \in W_\delta \cap \mathbb{N}$.
In the standard embedding $L^2(M^\omega,(\varphi\circ E_{M_1})^\omega) \hookrightarrow \mathcal{H}^\omega$ we have, by \eqref{Eq-3.7} and \eqref{Eq-3.8},
\begin{align*}
&\Big\Vert\Lambda_{(\varphi\circ E_{M_1})^\omega}(y(x - (E_{M_1})^\omega(x))) - \big[(y P_1\Lambda_{\varphi\circ E_{M_1}}(x(m)))_m\big]\Big\Vert_{\mathcal{H}^\omega} \\
&=
\lim_{m\rightarrow\omega}
\big\Vert\Lambda_{\varphi\circ E_{M_1}}(y(x(m) - E_{M_1}(x(m)))) - y P_1\Lambda_{\varphi\circ E_{M_1}}(x(m))\big\Vert_{\mathcal{H}} \\
&=
\lim_{m\rightarrow\omega}
\big\Vert y(P_2 + P_3 + P_4)\Lambda_{\varphi\circ E_{M_1}}(x(m))\big\Vert_{\mathcal{H}} \\
&\leq
\sup_{m \in W_\delta \cap \mathbb{N}} \big\Vert y(P_2 + P_3 + P_4)\Lambda_{\varphi\circ E_{M_1}}(x(m))\big\Vert_{\mathcal{H}} <
\Vert y \Vert_\infty \delta,
\end{align*}
and hence
\begin{equation}\label{Eq-3.9}
\Lambda_{(\varphi\circ E_{M_1})^\omega}(y(x - (E_{M_1})^\omega(x))) = \big[(y P_1\Lambda_{\varphi\circ E_{M_1}}(x(m)))_m\big]
\end{equation}
in $\mathcal{H}^\omega$, since $\delta$ is arbitrary. Trivially, in $\mathcal{H}^\omega$,
\begin{align}\label{Eq-3.10}
\Lambda_{(\varphi\circ E_{M_1})^\omega}(y(E_{M_1})^\omega(x) &- (E_{M_1})^\omega(x)z) \notag\\ &= \big[(\Lambda_{\varphi\circ E_{M_1}}(y E_{M_1}(x(m)) - E_{M_1}(x(m))\sigma_{t_m}^{\varphi\circ E_{M_1}}(y)))_m\big].
\end{align}
Set
\begin{align*}
y_\ell
&:=
\int_{-\infty}^{+\infty} \sigma_t^{\varphi\circ E_{M_1}}(y)\,\frac{e^{-t^2/\ell}\,dt}{\sqrt{\ell\pi}} \\
&=
\int_{-\infty}^{+\infty} [D\varphi\circ E_{M_1}:D\chi\circ E]_t\,\sigma_t^{\chi\circ E}(y)\,[D\varphi\circ E_{M_1}:D\chi\circ E]_t^*\,\frac{e^{-t^2/\ell}\,dt}{\sqrt{\ell\pi}}
\end{align*}
with a fixed faithful normal state $\chi$ on $N$. Clearly $y_\ell$ falls in the $\sigma$-weak (or $\sigma$-strong) closure of $\mathrm{span}(M_1 M_2^\circ M_1)$, since $[D\varphi\circ E_{M_1}:D\chi\circ E]_t = [D\varphi:D\chi\circ E_1]_t \in M_1$ by \cite[Corollary IX.4.22 (ii)]{Takesaki:Book2} and $\sigma_t^{\chi\circ E}(y) \in M_2^\circ$ by \eqref{Eq-2.1}. Set $z_\ell := \big[(\sigma_{t_m}^{\varphi\circ E_{M_1}}(y_\ell))_m\big] \in M^\omega$, which is well-defined as remarked just before the proof. Note that $\sigma_{-i/2}^{\varphi\circ E_{M_1}}(\sigma_{t_m}^{\varphi\circ E_{M_1}}(y_\ell)) = \sigma_{t_m}^{\varphi\circ E_{M_1}}(\sigma_{-i/2}^{\varphi\circ E_{M_1}}(y_\ell))$. For each $\ell$ we have, by \eqref{Eq-3.7}, \eqref{Eq-3.8} as before and by \cite[Lemma VIII.3.18 (ii)]{Takesaki:Book2},
\begin{align*}
&\Big\Vert\Lambda_{(\varphi\circ E_{M_1})^\omega}((x - (E_{M_1})^\omega(x))z_\ell) - \big[(J \sigma_{-i/2}^{\varphi\circ E_{M_1}}(\sigma_{t_m}^{\varphi\circ E_{M_1}}(y_\ell))^* J P_1\Lambda_{\varphi\circ E_{M_1}}(x(m)))_m\big]\Big\Vert_{\mathcal{H}^\omega} \\
&=
\lim_{m\rightarrow\omega}
\big\Vert J \sigma_{-i/2}^{\varphi\circ E_{M_1}}(\sigma_{t_m}^{\varphi\circ E_{M_1}}(y_\ell))^* J \big(\Lambda_{\varphi\circ E_{M_1}}(x(m) - E_{M_1}(x(m))) - P_1\Lambda_{\varphi\circ E_{M_1}}(x(m))\big)\big\Vert_{\mathcal{H}} \\
&\leq
\sup_{m \in W_\delta \cap \mathbb{N}} \big\Vert J \sigma_{-i/2}^{\varphi\circ E_{M_1}}(\sigma_{t_m}^{\varphi\circ E_{M_1}}(y_\ell))^* J(P_2 + P_3 + P_4)\Lambda_{\varphi\circ E_{M_1}}(x(m))\big\Vert_{\mathcal{H}} <
\Vert \sigma_{-i/2}^{\varphi\circ E_{M_1}}(y_\ell) \Vert_\infty \delta
\end{align*}
with the modular conjugation $J$ of $M \curvearrowright \mathcal{H} = L^2(M,\varphi\circ E_{M_1})$. Hence, for each $\ell$,
\begin{align}\label{Eq-3.11}
\Lambda_{(\varphi\circ E_{M_1})^\omega}(&(x - (E_{M_1})^\omega(x))z_\ell) \notag\\
&= \big[(J \sigma_{-i/2}^{\varphi\circ E_{M_1}}(\sigma_{t_m}^{\varphi\circ E_{M_1}}(y_\ell))^* J P_1\Lambda_{\varphi\circ E_{M_1}}(x(m)))_m\big]
\end{align}
in $\mathcal{H}^\omega$, since $\delta$ is arbitrary. Note that
\begin{equation*}
y P_1\Lambda_{\varphi\circ E_{M_1}}(x_m))
\in \overline{\mathrm{span}\Lambda_{\varphi\circ E_{M_1}}(M_2^\circ M_1^\circ \cdots M_2^\circ M_1^\triangledown)}.
\end{equation*}
On the other hand,
\begin{align*}
\Lambda_{\varphi\circ E_{M_1}}(y E_{M_1}(x(m)) &- E_{M_1}(x(m))\sigma_{t_m}^{\varphi\circ E_{M_1}}(y)) \\
&\in \overline{\mathrm{span}\Lambda_{\varphi\circ E_{M_1}}(M_2^\circ)}\oplus \overline{\mathrm{span}\Lambda_{\varphi\circ E_{M_1}}(M_2^\circ M_1^\triangledown)}\\
&\quad\oplus\overline{\mathrm{span}\Lambda_{\varphi\circ E_{M_1}}(M_1^\circ M_2^\circ)}\oplus\overline{\mathrm{span}\Lambda_{\varphi\circ E_{M_1}}(M_1^\circ M_2^\circ M_1^\triangledown)}
\end{align*}
and
\begin{align*}
J \sigma_{-i/2}^{\varphi\circ E_{M_1}}&(\sigma_{t_m}^{\varphi\circ E_{M_1}}(y_\ell))^* J P_1\Lambda_{\varphi\circ E_{M_1}}(x(m)) \\
&\in
\overline{\mathrm{span}\Lambda_{\varphi\circ E_{M_1}}(M_1^\circ\cdots M_2^\circ M_1^\triangledown \sigma_{t_m}^{\varphi\circ E_{M_1}}(y_\ell))} \\
&\subseteq
\overline{\Lambda_{\varphi\circ E_{M_1}}(M_1)} \oplus \overline{\mathrm{span}\Lambda_{\varphi\circ E_{M_1}}(M_1^\circ M_2^\circ \cdots)}.
\end{align*}
Here the last fact follows from \cite[Lemma VIII.3.18 (ii)]{Takesaki:Book2} and that $\sigma_{t_m}^{\varphi\circ E_{M_1}}(y_\ell)$ falls in the $\sigma$-strong closure of $\mathrm{span}(M_1 M_2^\circ M_1)$. Therefore, we see, by \eqref{Eq-3.9}--\eqref{Eq-3.11}, that $\Lambda_{(\varphi\circ E_{M_1})^\omega}(y(x - (E_{M_1})^\omega(x)))$ is orthogonal to both $\Lambda_{(\varphi\circ E_{M_1})^\omega}(y(E_{M_1})^\omega(x) - (E_{M_1})^\omega(x)z)$ and $\Lambda_{(\varphi\circ E_{M_1})^\omega}((x - (E_{M_1})^\omega(x))z_\ell)$. Finally, letting $\hat{x} := \big[(\sigma_{-t_m}^{\varphi\circ E_{M_1}}(x(m)))_m\big]$, $\hat{y} := \big[(\sigma_{-t_m}^{\varphi\circ E_{M_1}}(y))_m\big]$, both of which fall in $M^\omega$ as remarked just before the proof, we have
\begin{align*}
&\big(\Lambda_{(\varphi\circ E_{M_1})^\omega}((x - (E_{M_1})^\omega(x))z)|\Lambda_{(\varphi\circ E_{M_1})^\omega}(y(x - (E_{M_1})^\omega(x)))\big)_{(\varphi\circ E_{M_1})^\omega} \\
&=
(\varphi\circ E_{M_1})^\omega((x-(E_{M_1})^\omega(x))^* y^* (x-(E_{M_1})^\omega(x))z) \\
&=
(\varphi\circ E_{M_1})^\omega((\hat{x}-(E_{M_1})^\omega(\hat{x}))^* \hat{y}^* (\hat{x} - (E_{M_1})^\omega(\hat{x}))y) \\
&=
\lim_{\ell \rightarrow \infty}(\varphi\circ E_{M_1})^\omega((\hat{x}-(E_{M_1})^\omega(\hat{x}))^* \hat{y}^* (\hat{x} - (E_{M_1})^\omega(\hat{x}))y_\ell) \\
&=
\lim_{\ell \rightarrow \infty}(\varphi\circ E_{M_1})^\omega((x-(E_{M_1})^\omega(x))^* y^* (x - (E_{M_1})^\omega(x))z_\ell) \\
&=
\lim_{\ell \rightarrow \infty}
\big(\Lambda_{(\varphi\circ E_{M_1})^\omega}((x - (E_{M_1})^\omega(x))z_\ell|\Lambda_{(\varphi\circ E_{M_1})^\omega}(y(x - (E_{M_1})^\omega(x)))\big)_{(\varphi\circ E_{M_1})^\omega} = 0.
\end{align*}
Consequently we get $\Vert y(x-(E_{M_1})^\omega(x))\Vert_{(\varphi\circ E_{M_1})^\omega} \leq
\Vert yx - xz \Vert_{(\varphi\circ E_{M_1})^\omega}$. We have, by \eqref{Eq-3.9},
\begin{align*}
&\Vert y(x - (E_{M_1})^\omega(x))\Vert_{(\varphi\circ E_{M_1})^\omega}^2 \\
&=
\Big\Vert\big[(y P_1\Lambda_{\varphi\circ E_{M_1}}(x(m)))_m\big]\Big\Vert_{\mathcal{H}^\omega}^2 \\
&=
\lim_{m\rightarrow\omega}
(y P_1 \Lambda_{\varphi\circ E_{M_1}}(x(m))|y P_1 \Lambda_{\varphi\circ E_{M_1}}(x(m)))_{\varphi\circ E_{M_1}} \\
&=
\lim_{m\rightarrow\omega} \Big\{
(E_2(y^*y) P_1 \Lambda_{\varphi\circ E_{M_1}}(x(m))| P_1 \Lambda_{\varphi\circ E_{M_1}}(x(m)))_{\varphi\circ E_{M_1}} \\
&\phantom{aaaaaaaaaa}+ (\underbrace{(y^* y - E_2(y^*y)) P_1 \Lambda_{\varphi\circ E_{M_1}}(x(m))}_{\text{in $\mathcal{X}_3$ orthogonal to $\mathcal{X}_1$}}| \underbrace{P_1 \Lambda_{\varphi\circ E_{M_1}}(x(m))}_{\text{in $\mathcal{X}_1$}})_{\varphi\circ E_{M_1}} \Big\} \\
&=
\lim_{m\rightarrow\omega}
(E_2(y^*y) P_1 \Lambda_{\varphi\circ E_{M_1}}(x(m))| P_1 \Lambda_{\varphi\circ E_{M_1}}(x(m)))_{\varphi\circ E_{M_1}} \\
&=
\Big\Vert\big[(E_2(y^* y)^{1/2} P_1\Lambda_{\varphi\circ E_{M_1}}(x(m)))_m\big]\Big\Vert_{\mathcal{H}^\omega}^2.
\end{align*}
As in showing \eqref{Eq-3.9} one has
\begin{equation*}
\big[(E_2(y^* y)^{1/2} P_1\Lambda_{\varphi\circ E_{M_1}}(x(m)))_m\big] = i
\Lambda_{(\varphi\circ E_{M_1})^\omega}(E_2(y^* y)^{1/2}(x - (E_{M_1})^\omega(x))),
\end{equation*}
and the proof is completed.
\end{proof}
\section{Some Consequences}
We first formulate that $P$ is `non-trivial relative to $Q$' for a given inclusion of von Neumann algebras $P \supseteq Q$, and then provide some technical facts.
\begin{definition}\label{D-4.1} A {\rm(}unital{\rm)} inclusion $P \supseteq Q$ of von Neumann algebras is said to be entirely non-trivial, if no non-zero direct summand of $Q$ is a direct summand of $P$.
\end{definition}
Let $P \supseteq Q$ be an inclusion of von Neumann algebras with a faithful normal conditional expectation $E_Q$. If $zP = Qz$ (as set) for some non-zero $z \in \mathcal{Z}(Q)^p$, then $Pz = Qz$ too by taking adjoints, and thus for $x \in P$ one has $zx = E_Q(zx) = zE_Q(x) = E_Q(x)z = E_Q(xz) = xz$, implying $z \in \mathcal{Z}(P)$. Hence $Qz$ is a direct summand of $P$. Therefore, $P \supseteq Q$ is entirely non-trivial if and only if $Pz \neq Qz$ or equivalently $zP \neq Qz$ for any non-zero projection $z \in \mathcal{Z}(Q)$, where $Pz$ and $zP$ denote the one-sided ideals of all $xz$ and $zx$, respectively, with $x \in P$.
The next simple lemma, especially (3) there, will frequently be used later.
\begin{lemma}\label{L-4.2} Let $P \supseteq Q$ be an inclusion of von Neumann algebras with a faithful normal conditional expectation $E_Q : P \rightarrow Q$.
{\rm (1)} The following are equivalent{\rm:}
\begin{itemize}
\item[(i)] $P \supseteq Q$ is entirely non-trivial.
\item[(ii)] $Pe \neq Qe$ or equivalently $eP \neq eQ$ for any non-zero projection $e \in Q$.
\end{itemize}
{\rm (2)} If $P \supseteq Q$ is entirely non-trivial and $f \in Q$ a projection with $c_f^Q = 1$, then $fPf \supseteq fQf$ is again entirely non-trivial.
{\rm (3)} If $P \supseteq Q$ is entirely non-trivial, then there is a family $\{y_i\}_{i \in I}$ of elements in $\mathrm{Ker}(E_Q)$ so that $\sum_{i \in I} s(E_Q(y_i^* y_i)) = 1$, where $s(x)$ denotes the support projection of $x=x^*$.
\end{lemma}
\begin{proof} (1) By the discussion above (i) is equivalent to $Pz \neq Qz$ or equivalently $zP \neq Qz$ for any non-zero $z \in \mathcal{Z}(Q)^p$. Thus (ii) $\Rightarrow$ (i) is trivial, and it suffices to show (i) $\Rightarrow$ (ii). Suppose that $Pe = Qe$ for some non-zero $e \in Q^p$. By a standard exhaustion argument based on the comparison theorem we can choose an orthogonal family $\{e_i\}_{i \in I}$ of projections in $Q$ such that $e_i \precsim e$ in $Q$ for all $i \in I$ and $c_e^Q = \sum_{i \in I} e_i$. Choose a partial isometry $v_i \in Q$ with $v_i^* v_i = e_i$ and $v_i v_i^* \leq e$, and then $P e_i = P v_i^* v_i \subseteq P e v_i = Q e v_i \subseteq Q e_i$, implying $Pe_i = Q e_i \subseteq Q$. For $x \in P$ one has $xc_e^Q = \sum_{i\in I} xe_i = \sum_{i\in I} E_Q(x)e_i = E_Q(x)c_e^Q$, and therefore $Pc_e^Q = Qc_e^Q$.
(2) By (1) it suffices to prove that $ePf \neq eQf$ for any non-zero $e \in Q^p$ with $e \leq f$. As in (1) one can find an orthogonal family $\{f_i\}_{i\in I}$ of projections in $Q$ such that $f_i \precsim f$ in $Q$ for all $i \in I$ and $\sum_{i\in I} f_i = c_f^Q = 1$. On contrary, suppose that $ePf = eQf$ for some non-zero $e \in Q^p$ with $e \leq f$. Then one has $ePf_i = eQf_i$ in the same way as in (1). Hence, as in the above (1) one can justify, by using $E_Q$, the following computation: $eP = \sum_{i\in I} ePf_i = \sum_{i\in I}eQf_i = eQ$, a contradiction to the entire non-triviality of $P \supseteq Q$ thanks to (1).
(3) Choose a maximal (with respect to set-inclusion) family $\{y_i\}_{i\in I}$ of elements in $\mathrm{Ker}(E_Q)$ so that $\{s(E_Q(y_i^* y_i))\}_{i \in I}$ is an orthogonal family of projections in $Q$. Suppose $\sum_{i\in I}s(E_Q(y_i^* y_i))\neq 1$. Set $e := 1 - \sum_{i\in I}s(E_Q(y_i^* y_i)) \in Q^p\setminus\{0\}$. Since $P \supseteq Q$ is entirely non-trivial, one has $Pe \neq Qe$ by (1), and hence can choose $x \in P$ with $xe \not\in Q$. Hence $xe - E_Q(xe) \neq 0$ and set $y := xe - E_Q(xe) \in \mathrm{Ker}(E_Q)$. Clearly, $ye = y$, and thus $E_Q(y^* y) = e E_Q(y^* y)e$, implying $s(E_Q(y^* y)) \leq e = 1 - \sum_{i\in I} s(E_Q(y_i^* y_i))$, a contradiction to the maximality of $\{y_i\}_{i\in I}$.
\end{proof}
Let $(M,E) = (M_1,E_1)\star_N(M_2,E_2)$ be an amalgamated free product throughout the rest of this section.
\begin{theorem}\label{T-4.3} Assume that there is a faithful normal state $\varphi$ on $M_1$ such that one can find a {\rm(}possibly non-unital{\rm)} dense {\rm(}in any von Neumann algebra topology{\rm)} $*$-subalgebra\, $\mathfrak{M}_1$ of $M_1$ with $E_1(\mathfrak{M}_1) \subseteq \mathfrak{M}_1$ and a net $v_\lambda$ of unitaries in the centralizer $(M_1)_\varphi$ in such a way that $E_1(y^* v_\lambda x) \longrightarrow 0$ $\sigma$-strongly for all $x,y \in \mathfrak{M}_1$. Assume also that $M_2 \supseteq N$ is entirely non-trivial. Then we have{\rm:}
\begin{itemize}
\item[(0)] $((M_1)_\varphi)' \cap M = ((M_1)_\varphi)' \cap M_1$.
\item[(1)] $\mathcal{Z}(M) = \mathcal{Z}(M_1)\cap\mathcal{Z}(M_2)\cap\mathcal{Z}(N)$.
\item[(2)] Let $\chi$ be an arbitrary faithful normal semifinite weight on $N$. Then, if a unitary $u$ in $M$ satisfies $\sigma_t^{\chi\circ E} = \mathrm{Ad}u$ for some $t \in \mathbb{R}$, then $u$ must fall in $N$. In particular, $T(M) = \{ t \in \mathbb{R}\,|\,\sigma_t^{\chi\circ E_1} = \mathrm{Ad}u = \sigma_t^{\chi\circ E_2}\ \text{for some $u \in N^u$}\}$.
\item[(3)] $M$ is semifinite if and only if there is a faithful normal semifinite trace $\mathrm{Tr}_N$ such that both $\mathrm{Tr}_N\circ E_1$ and $\mathrm{Tr}_N\circ E_2$ are traces.
\item[(4)] $\mathcal{Z}(\widetilde{M}) = \mathcal{Z}(\widetilde{M}_1)\cap\mathcal{Z}(\widetilde{M}_2)\cap\mathcal{Z}(\widetilde{N})$.
\end{itemize}
\end{theorem}
\begin{proof} (0) is nothing but what Proposition \ref{P-3.3} says.
(1) Let $x \in \mathcal{Z}(M)$ be arbitrary, and then $x$ must be in $M_1$ by (0). For any $y \in M_2^\circ$ one has $y(x-E_1(x)) + yE_1(x)=yx=xy=E_1(x)y + (x-E_1(x))y$, and thus $\{y E_1(x), E_1(x)y\}$, $y(x-E_1(x))$ and $(x-E_1(x))y$ are orthogonal with respect to $E$ due to the free independence between $M_1$ and $M_2$. Thus $y(x-E_1(x))=0$ so that (by looking at the $E$-value of the product of its adjoint and itself) we get $(x-E_1(x))^* E_2(y^* y) (x-E_1(x)) = 0$. Therefore, $E_2(y^* y)(x-E_1(x))= 0$ for all $y \in M_2^\circ$. By taking its adjoint one can easily see that $(x-E_1(x))^*\!\upharpoonright_{\mathrm{ran}(E_2(y^* y))} \equiv 0$ so that $(x-E_1(x))^* s(E_2(y^* y)) = 0$ for all $y \in M_2^\circ$. By Lemma \ref{L-4.2} (3) one can find a family $\{y_i\}_{i\in I}$ of elements in $M_2^\circ$ so that $\sum_{i\in I}s(E_2(y_i^* y_i)) = 1$, which implies $x = E_1(x) \in N$. The desired assertion is now immediate.
(2) One has $\sigma_t^{\varphi\circ E_{M_1}} = \mathrm{Ad}([D\varphi:D\chi\circ E_1]_t\,u)$ by Connes's Radon--Nikodym cocycle theorem and \cite[Corollary IX.4.20]{Takesaki:Book2}. Since $(M_1)_\varphi \subseteq M_{\varphi\circ E_{M_1}}$, we have $[D\varphi:D\chi\circ E_1]_t\,u \in M_1$ by (0). In particular, $u \in M_1$, since $[D\varphi:D\chi\circ E_1]_t \in M_1^u$. For $y \in M_2^\circ$ we have $\sigma_t^{\chi\circ E}(y)(u-E_1(u)) + \sigma_t^{\chi\circ E}(y)E_1(u) = \sigma_t^{\chi\circ E}(y)u = uy = E_1(u)y + (u-E_1(u))y$, and as in (1) we get $(u-E_1(u))y = 0$, since $\sigma_t^{\chi\circ E}(y) = \sigma_t^{\chi\circ E_2}(y) \in M_2^\circ$ by \eqref{Eq-2.1}. The same argument as in (1) again shows $u = E_1(u) \in N$. The T-set computation is straightforward.
(3) $M$ is semifinite if and only if there is a $1$-parameter unitary group $u(t)$ in $M$ so that $\sigma_t^{\chi\circ E}=\mathrm{Ad}u(t)$, $t \in \mathbb{R}$, for a fixed faithful normal state $\chi$ on $N$. See \cite[Theorem VIII.3.14]{Takesaki:Book2}. Then $u(t) \in N$ by (2). By Stone's theorem $u(t) = H^{it}$ with some positive non-singular, self-adjoint $H$ affiliated with $N$. Since $\sigma_t^\chi(u(t)) = \sigma_t^{\chi\circ E}(u(t))=u(t)$, $H$ must indeed be affiliated with the centralizer $N_\chi$. Hence, by \cite[Lemma VIII.2.8]{Takesaki:Book2} we can construct a faithful normal semifinite weight $\chi_{H^{-1}}$ on $N$, and by the construction we observe that $\chi_{H^{-1}}\circ E = (\chi\circ E)_{H^{-1}}$. Moreover, by \cite[Lemma VIII.2.11]{Takesaki:Book2} we have $\sigma_t^{\chi_{H^{-1}}\circ E} = H^{-it}\sigma_t^{\chi\circ E}(-)H^{it} = \mathrm{id}$. Hence the $\chi_{H^{-1}}$ is a desired faithful normal semifinite trace on $N$.
(4) By (0) together with the same argument as in \cite[Corollary 4]{Ueda:MathScand01} we observe that $((M_1)_\varphi)'\cap(M\rtimes_{\sigma^{\varphi\circ E_{M_1}}}\mathbb{R}) = ((M_1)_\varphi)'\cap(M_1\rtimes_{\sigma^\varphi}\mathbb{R})$, where $(M_1)_\varphi \subset M_1 \subseteq M \hookrightarrow M\rtimes_{\sigma^{\varphi\circ E_{M_1}}}\mathbb{R}$ canonically as in \S2. It follows that $(\widetilde{M}_1)'\cap\widetilde{M} = \mathcal{Z}(\widetilde{M}_1)$, where we need Connes's Radon--Nikodym cocycle theorem together with \cite[Theorem X.1.7]{Takesaki:Book2}. Choose an arbitrary $x \in \mathcal{Z}(\widetilde{M})$. Then $x$ must fall in $\mathcal{Z}(\widetilde{M}_1) \subseteq \widetilde{M}_1$. For $y \in M_2^\circ \subset \widetilde{M}_2^\circ$ one has $y(x-\widetilde{E}(x)) + y\widetilde{E}(x) = yx = xy = \widetilde{E}(x)y + (x-\widetilde{E}(x))y$, and thus $y(x-\widetilde{E}(x)) = 0$ since $\widetilde{M}_1$, $\widetilde{M}_2$ are freely independent with respect to $\widetilde{E}$ as remarked in \S2. In particular, we get $E_2(y^* y) (x -\widetilde{E}(x)) = 0$ for all $y \in M_2^\circ$ as in (1). Therefore, using Lemma \ref{L-4.2} (3) as in (1) once again we can prove $x = \widetilde{E}(x) \in \widetilde{N}$. Hence we are done.
\end{proof}
Let us illustrate how the above theorem is useful by giving next two corollaries. The first corollary shows that Proposition \ref{P-3.1} is useful to confirm the necessary hypothesis of the theorem. The second one does that the theorem is still applicable beyond the case where $N$ is semifinite. Remark that the first one can be viewed as a simultaneous generalization of both \cite[Theorem 3.4]{Ueda:AdvMath11} and \cite[\S4]{Ueda:PacificJMath99}.
\begin{corollary}\label{C-4.4} Assume that $M_1$ is diffuse, $N$ of type I and $M_2 \supseteq N$ entirely non-trivial. Let $z \in \mathcal{Z}(N)$ be the unique projection so that $Nz$ is diffuse and $Nz^\perp$ atomic, and assume further that $M_1 c_{z}^{M_1}$ has no type I direct summand when $z \neq 0$ {\rm(}i.e., this last assumption is fulfilled if $M_1$ has no type I direct summand{\rm)}. Then all the assertions of Theorem \ref{T-4.3} holds with a certain faithful normal state $\varphi$ on $M_1$.
\end{corollary}
\begin{proof} Let us fix a faithful normal semifinite trace $\mathrm{Tr}_N$ on $N$. Write $c := c_z^{M_1}$ for simplicity. Clearly $\sigma_t^{\mathrm{Tr}_N\circ E_1}(c) = c$ for all $t \in \mathbb{R}$, and thus Takesaki's criterion shows that there is a $\mathrm{Tr}_N\circ E_1$-preserving unique conditional expectation $E_L : M_1 \rightarrow L := N \vee \{c\}'' = Nc \oplus Nc^\perp$ ($\supseteq N$). In particular, one observes that $E_1 \circ E_L = E_1$ holds.
As in the proof of \cite[Theorem 3.4]{Ueda:AdvMath11} one can choose a faithful normal state $\varphi$ on $M_1$ such that $(M_1 c)_{\varphi\!\upharpoonright_{M_1 c}}$ has no type I direct summand and $(M_1 c^\perp)_{\varphi\!\upharpoonright_{M_1 c\perp}}$ is just only diffuse. Then it is clear that $(M_1 c)_{\varphi\!\upharpoonright_{M_1 c}} \not\preceq_{M_1 c} Nc$ with $E_L\!\upharpoonright_{M_1 c}$ and $\mathrm{Tr}_N\circ E_1\!\upharpoonright_{Nc}$ and that $(M_1 c^\perp)_{\varphi\!\upharpoonright_{M_1 c^\perp}} \not\preceq_{M_1 c^\perp} Nc^\perp$ with $E_L\!\upharpoonright_{M_1 c}$ and $\mathrm{Tr}_N\circ E_1\!\upharpoonright_{Nc^\perp}$, since $Nc^\perp = (Nz^\perp)c^\perp$ is a reduced von Neumann algebra of the atomic part $Nz^\perp$. Therefore, by the equivalent condition (i) in Proposition \ref{P-3.1} there are two nets $v_\lambda^{(1)}$ and $v_\lambda^{(2)}$ of unitaries in $(M_1 c)_{\varphi\!\upharpoonright_{M_1 c}}$ and $(M_1 c^\perp)_{\varphi\!\upharpoonright_{M_1 c^\perp}}$, respectively, so that $E_L(y_1^* v_\lambda^{(1)} x_1) \longrightarrow 0$ and $E_L(y_2^* v_\lambda^{(2)} x_2) \longrightarrow 0$ $\sigma$-strongly for all $x_1,y_1 \in \bigcup\{M_1 p\,|\, p \in (Nc)^p; \mathrm{Tr}_N\circ E_1(p) < +\infty\}$ and all $x_2,y_2 \in \bigcup\{M_1 p\,|\, p \in (Nc^\perp)^p; \mathrm{Tr}_N\circ E_1(p) < +\infty\}$. Remark that $E_L = (E_L\!\upharpoonright_{M_1 c})\oplus(E_L\!\upharpoonright_{M_1 c^\perp})$ in $M_1 = M_1 c \oplus M_1 c^\perp$ and that $\mathrm{Tr}_N(p) < +\infty$ implies both $\mathrm{Tr}_N\circ E_1(pc)<+\infty$ and $\mathrm{Tr}_N\circ E_1(pc^\perp) < +\infty$ for $p \in N^p$. Thus, letting $v_\lambda := v_\lambda^{(1)}\oplus v_\lambda^{(2)} \in (M_1 c)_{\varphi\!\upharpoonright_{M_1 c}}\oplus(M_1 c^\perp)_{\varphi\!\upharpoonright_{M_1 c^\perp}} = (M_1)_\varphi$ one has, for all $x,y \in \bigcup\{M_1 p\,|\,p \in N^p; \mathrm{Tr}_N(p) < +\infty\}$, $E_L(y^* v_\lambda x) \longrightarrow 0$ $\sigma$-strongly and hence $E_1(y^* v_\lambda x) = E_1(E_L(y^* v_\lambda x)) \longrightarrow 0$ $\sigma$-strongly. Hence we can apply Theorem \ref{T-4.3} with the above $\varphi$ and $\mathfrak{M}_1 := \bigcup\{pM_1 p\,|\,p \in N^p; \mathrm{Tr}_N(p)<+\infty\}$. Note here that $\mathfrak{M}_1$ is indeed a $*$-algebra thanks to the Kaplansky formula \cite[Theorem 6.1.7]{KadisonRingrose:Book2} and dense in any von Neumann algebra topology due to the semifiniteness of $\mathrm{Tr}_N$.
\end{proof}
\begin{corollary}\label{C-4.5} Assume that $(M_1,E_1)$ is one of the following{\rm:} {\rm (i)} $M_1 = N\rtimes_\alpha G$ and $E_1$ is the canonical conditional expectation from $M_1 = N\rtimes_\alpha G$ onto $N$, where $\alpha : G \curvearrowright N$ is an infinite discrete group action preserving a faithful normal state $\psi$ on $N$. {\rm(ii)} $M_1 = Q\bar{\otimes}N$ and $E_1 = \psi\bar{\otimes}\mathrm{id}_N$, where $Q$ is a diffuse von Neumann algebra with a faithful normal state $\psi$. Assume also that $M_2 \supseteq N$ is entirely non-trivial. Then all the assertions of Theorem \ref{T-4.3} holds with $\varphi = \psi\circ E_1$ in {\rm(i)} and with $\varphi = \varphi_0\bar{\otimes}\chi$ in {\rm(ii)}, where $Q_{\varphi_0}$ is diffuse {\rm(}such a state $\varphi_0$ certainly exists{\rm)} and $\chi$ arbitrary.
\end{corollary}
\begin{proof} Case (i): Since $\psi$ is invariant under the action $\alpha$, the restriction $(\psi\bar{\otimes}\mathrm{id}_{B(\ell^2(G))})\!\upharpoonright_{N\rtimes_\alpha G}$ gives a faithful normal conditional expectation from $E_\psi : M_1 = N\rtimes_\alpha G \rightarrow L(G) = \mathbb{C}1\rtimes G$, and it is plain to see that $\psi\circ E_1 := \tau_G\circ E_\psi$ with the canonical tracial state $\tau_G$ on $L(G)$. Clearly $L(G) = \mathbb{C}1\rtimes G$ sits inside $(N\rtimes_\alpha G)_{\psi\circ E_1}$ and is diffuse (see e.g.~\cite[Proposition 5.1]{Dykema:DukeMathJ93}). With $\varphi := \psi\circ E_1 = \tau_G\circ E_\psi$ and $\mathfrak{M}_1 := \mathrm{span}\{ x\lambda_g\,|\, x \in N, g \in G\}$ one can choose a net $v_\lambda$ from $L(G) = \mathbb{C}\rtimes G$ as in Theorem \ref{T-4.3}, since $L(G)$ is diffuse and $E_1\!\upharpoonright_{L(G)=\mathbb{C}1\rtimes G} = \tau_G(-)1$.
Case (ii): As in the proof of \cite[Theorem 2.4]{Ueda:AdvMath11} one can choose a faithful normal state $\varphi_0$ on $Q$ in such a way that the centralizer $Q_{\varphi_0}$ is diffuse. Set $\varphi := \varphi_0\bar{\otimes}\chi$ with a faithful normal state $\chi$ on $N$ and $\mathfrak{M}_1 := Q\odot N = \mathrm{span}\{ x\otimes y\,|\, x \in Q, y \in N\}$. Then one can choose a net $v_\lambda$ from $Q_{\varphi_0}\bar{\otimes}\mathbb{C}1$ as in Theorem \ref{T-4.3}, since $Q_{\varphi_0}$ is diffuse.
\end{proof}
The next lemma seems well-known, but we do give it for the reader's convenience as a reference for the discussions below.
\begin{lemma}\label{L-4.6} Let $(P,F) = (P_1,F_1)\star_Q(P_2,F_2)$ be an amalgamated free product. If a projection $f \in Q$ has $c_f^Q = 1$, then $(fPf,F\!\upharpoonright_{fPf}) = (fP_1 f,F_1\!\upharpoonright_{fP_1 f})\star_{fQf}(fP_2 f,F_2\!\upharpoonright_{fP_2 f})$ holds canonically.
\end{lemma}
\begin{proof} Clearly $fP_1 f$ and $fP_2 f$ are freely independent with respect to $F\!\upharpoonright_{fPf}$, and hence it suffices to see that those generate $fPf$ as von Neumann algebra. As in the proof of Lemma \ref{L-4.2} one can find partial isometries $\{v_i\}_{i\in I}$ in $Q$ such that $\sum_{i\in I}v_i^* v_i = c_f^Q = 1$ and $v_i v_i^* \leq f$ for all $i \in I$. For any alternating word $x = x_1 \cdots x_n \in \Lambda^\circ(P_1^\circ,P_2^\circ)$ one has $fxf = \sum_{i_1,\dots,i_{n-1} \in I} (f x_1 v_{i_1}^*) (v_{i_1} x_2 v_{i_2}^*) \cdots (v_{i_{n-1}} x_n f)$ $\sigma$-strongly, which falls in the $\sigma$-strong closure of the linear span of $\Lambda^\circ((fP_1 f)^\circ, (fP_2 f)^\circ))$. Since $P$ is the $\sigma$-strong closure of $Q + \mathrm{span}\Lambda^\circ(P_1^\circ, P_2^\circ)$, the assertion is immediate.
\end{proof}
\begin{lemma}\label{L-4.7} Let $P \supseteq Q$ be an inclusion of $\sigma$-finite von Neumann algebras with a faithful normal conditional expectation $E_Q : P \rightarrow Q$, and assume that $Q$ is commutative.
{\rm(1)} If $P$ has no type I direct summand and a faithful normal semifinite trace $\mathrm{Tr}_P$ on $P$ with $\mathrm{Tr}_P\circ E_Q = \mathrm{Tr}_P$, then there is a faithful normal state $\chi$ on $Q$ so that for each $n \in \mathbb{N}$ with $n \geq 2$ one can find a unitary $u_n \in P_{\chi\circ E_Q}$ in such a way that $E_Q(u_n^k) = 0$ for all $1\leq k \leq n-1$, i.e., $E_Q(u_n^{k_1}{}^*\,u_n^{k_2}) = 0$ for all $0 \leq k_1 \neq k_2 \leq n-1$.
{\rm(2)} If $P$ is diffuse and $Q$ is atomic, then there is a faithful normal state $\varphi$ on $P$ such that
\begin{itemize}
\item[(a)] the centralizer $P_\varphi$ contains $Q$,
\item[(b)] there are two unitaries $u, v \in P_\varphi$ so that $E_Q(u^k) = E_Q^\varphi(v^k) = 0$ as long as $k\neq 0$, i.e., $E_Q(u^{k_1}{}^*\,u^{k_2}) = E_Q^\varphi(v^{k_1}{}^*\,v^{k_2}) = 0$ for all $k_1 \neq k_2$. Here $E_Q^\varphi$ denotes the unique $\varphi$-preserving conditional expectation from $P$ onto $Q$ whose existence follows from {\rm(a)} and Takesaki's criterion.
\end{itemize}
{\rm(3)} Let $z \in \mathcal{Z}(P)$ be the central support projection of the type I direct summand of $P$. Assume that $P$ is diffuse and $Qz$ atomic. Then there is a faithful normal state $\varphi$ on the continuous core $\widetilde{P}$ of $P$ such that
\begin{itemize}
\item[(a)] the centralizer $(\widetilde{P})_\varphi$ contains $\widetilde{Q}$, where $\widetilde{Q} = Q \rtimes_{\sigma^\chi}\mathbb{R} \hookrightarrow \widetilde{P} = P\rtimes_{\sigma^{\chi\circ E_Q}}\mathbb{R}$ with a faithful normal state or semifinite weight $\chi$ on $Q$,
\item[(b)] for each $n \in \mathbb{N}$ with $n \geq 2$ one can find a unitary $u_n \in (\widetilde{P})_\varphi$ in such a way that $\widetilde{E}_Q(u_n^k) = E_{\widetilde{Q}}^\varphi(u_n^k) = 0$ for all $1 \leq k \leq n-1$, i.e., $\widetilde{E}_Q(u_n^{k_1}{}^*\,u_n^{k_2}) = E_{\widetilde{Q}}^\varphi(v_n^{k_1}{}^*\,v_n^{k_2}) = 0$ for all $0 \leq k_1 \neq k_2 \leq n-1$. Here $\widetilde{E}_Q = (E_Q\bar{\otimes}\mathrm{id}_{B(L^2(\mathbb{R}))})\!\upharpoonright_{\widetilde{P}}$, and $E_{\widetilde{Q}}^\varphi$ denotes the unique $\varphi$-preserving conditional expectation from $\widetilde{P}$ onto $\widetilde{Q}$ as in {\rm(2)}.
\end{itemize}
The same assertion also holds for $P \supseteq Q$ with $E_Q$ themselves, if it is further assumed that $P$ is semifinite and $E_Q$ preserves a faithful normal semifinite trace $\mathrm{Tr}_P$ on $P$.
\end{lemma}
\begin{proof} (1) By assumption $\mathrm{Tr}_P\!\upharpoonright_Q$ is semifinite, and thus one can choose an orthogonal sequence $\{q_m\}_m$ of projections in $Q$ with $\mathrm{Tr}_P(q_m) < +\infty$ and $\sum_{m\in\mathbb{N}} q_m = 1$. Consider the faithful normal state $\chi := \sum_{m\in\mathbb{N}} \frac{1}{2^m \mathrm{Tr}_P(q_m)} \mathrm{Tr}_P\!\upharpoonright_{Qq_m}$ on $Q$. (Remark here that $Q$ is commutative.) Clearly the centralizer $P_{\chi\circ E_Q}$ contains $\sum^\oplus_{m\in\mathbb{N}} q_m P q_m$ $\big(\supseteq \sum^\oplus_{m\in\mathbb{N}} Qq_m = Q\big)$ so that $P_{\chi\circ E_Q}$ must be of type II$_1$. Choose a MASA $\mathfrak{A}$ in $P_{\chi\circ E_Q}$ that contains $Q$. By \cite[Corollary 3.16]{Kadison:AmerJMath84}, for each $n \in \mathbb{N}$ with $n \geq 2$ there are $n$ orthogonal $e_0,\dots,e_{n-1} \in \mathfrak{A}^p$, all of which are equivalent in $P_{\chi\circ E_Q}$, and $\sum_{i=0}^{n-1} e_i = 1$. Then one can construct a unitary $u_n \in P_{\chi\circ E_Q}$ such that $u_n e_0 = e_1 u_n, u_n e_1 = e_2 u_n, \dots, u_n e_{n-1} = e_0 u_n$. Let $E_{\mathfrak{A}} : P \rightarrow \mathfrak{A}$ be the $\chi\circ E_Q$-preserving conditional expectation (whose existence follows from Takesaki's criterion), and clearly $E_Q\circ E_{\mathfrak{A}} = E_Q$. Then, for every $1 \leq k \leq n-1$ one has $E_{\mathfrak{A}}(u_n^k) = 0$ so that $E_Q(u_n^k)= E_Q(E_{\mathfrak{A}}(u_n^k)) = 0$.
(2) Write $Q = \sum_{m \in \mathbb{N}}^\oplus \mathbb{C}q_m$. Clearly $E_Q$ factors as
$P \overset{E_{Q'\cap P}}{\longrightarrow} Q'\cap P \overset{\Psi}{\longrightarrow} Q$, where $Q'\cap P = \sum_{m\in\mathbb{N}}^\oplus q_m P q_m$ and $E_{Q'\cap P}(x) = \sum_{m \in \mathbb{N}} q_m x q_m$ for $x \in P$. Moreover $\Psi$ is of the form $\Psi(\sum_{m\in\mathbb{N}} x_m) = \sum_{m \in \mathbb{N}} \psi_m(x_m)q_m$ for $x_m \in q_m P q_m$ with faithful normal states $\psi_m$ on $q_m P q_m$. Since $P$ is diffuse, so are all $q_m P q_m$; hence by the proof of \cite[Theorem 3.4]{Ueda:AdvMath11} there are faithful normal states $\varphi_m$ on $q_m P q_m$ with $(q_m P q_m)_{\varphi_m}$ diffuse for all $m$. Define $\Phi(\sum_{m\in\mathbb{N}} x_m) = \sum_{m\in\mathbb{N}}\varphi_m(x_m)q_m$ for $x_m \in q_m P q_m$, giving a faithful normal conditional expectation from $Q'\cap P$ onto $Q$. Set $\varphi := \chi\circ\Phi\circ E_{Q'\cap P}$, a faithful normal state on $P$, with a faithful normal state $\chi$ on $Q$. Then $Q' \cap P_{\varphi} = \sum_{m\in\mathbb{N}}^\oplus (q_m P q_m)_{\varphi_m}$, a direct sum of diffuse von Neumann algebras. One can choose, for each $m$, unitaries $u_m, v_m \in (q_m P q_m)_{\varphi_m}$ so that $\varphi_m(u_m^k) = \psi_m(v_m^k) = 0$ as long as $k \neq 0$. (See the proof of \cite[Theorem 3.7]{Ueda:AdvMath11}.) Then $u := \sum_{m\in\mathbb{N}} u_m$, $v := \sum_{m\in\mathbb{N}} v_m$ are unitaries in $Q' \cap P_\varphi$, and moreover $E_Q(u^k) = \Psi(u^k) = 0$ and $E_Q^\varphi(v^k) = \Phi(v^k) = 0$ as long as $k \neq 0$.
(3) Consider $P = Pz \oplus Pz^\perp \supseteq R := Q\vee\{z\}'' = Qz\oplus Qz^\perp \supseteq Q$. Let $\chi$ be an arbitrary faithful normal state on $Q$. As in the proof of Corollary \ref{C-4.4} one can show that there is a unique faithful normal conditional expectation $E_R : P \rightarrow R$ with $E_Q\circ E_R = E_Q$. Then we have
\begin{equation*}
\widetilde{P} = P\rtimes_{\sigma^{\chi\circ E_Q}}\mathbb{R} \overset{\widetilde{E}_R}{\supseteq} \widetilde{R} = R\rtimes_{\sigma^{\chi\circ(E_Q\!\upharpoonright_{R})}}\mathbb{R} \overset{\widetilde{E_Q\!\upharpoonright_R}}{\supseteq} \widetilde{Q} = Q \rtimes_{\sigma^\chi}\mathbb{R},
\end{equation*}
where $\widetilde{E}_R = (E_R\bar{\otimes}\mathrm{id}_{B(L^2(\mathbb{R}))})\!\upharpoonright_{\widetilde{P}}$ and
$\widetilde{{E_Q}\!\upharpoonright_R} =
(({E_Q}\!\upharpoonright_R)\bar{\otimes}\mathrm{id}_{B(L^2(\mathbb{R}))})\!\upharpoonright_{\widetilde{R}} =
{\widetilde{E}_Q}\!\upharpoonright_{\widetilde{R}}$. Since $E_R = ({E_R}\!\upharpoonright_{Pz})\oplus({E_R}\!\upharpoonright_{Pz^\perp})$ in $P = Pz\oplus Pz^\perp$, we have, by \cite[Theorem X.1.7 (ii)]{Takesaki:Book2},
\begin{equation*}
\Big(\widetilde{P} \overset{\widetilde{E}_R}{\supseteq} \widetilde{R}\Big) \cong
\Big(\widetilde{Pz} \overset{\widetilde{{E_R}\!\upharpoonright_{Pz}}}{\supseteq} \widetilde{Qz}\Big)\oplus \Big(\widetilde{Pz^\perp} \overset{\widetilde{{E_R}\!\upharpoonright_{Pz^\perp}}}{\supseteq} \widetilde{Qz^\perp}\Big),
\end{equation*}
where the continuous cores and the conditional expectations in the right-hand side are defined similarly as above. Since $\widetilde{Pz^\perp}$ has no type I direct summand by the assumption here and \cite[Theorem XII.1.1]{Takesaki:Book2} and since $\widetilde{E_R\!\upharpoonright_{Pz^\perp}}$ preserves the canonical trace on $\widetilde{Pz^\perp}$ see e.g.~\cite[\S4]{Longo:CMP89}, we can apply (1) to the second $\Big(\widetilde{Pz^\perp} \supseteq \widetilde{Qz^\perp}\Big)$ with $\widetilde{E_R\!\upharpoonright_{Pz^\perp}}$ directly, and get a faithful normal state $\varphi_{z^\perp}$ on $\widetilde{Pz^\perp}$ with $\varphi_{z^\perp}\circ(\widetilde{E_R\!\upharpoonright_{Pz^\perp}}) = \varphi_{z^\perp}$ such that for each $n \in \mathbb{N}$ with $n \geq 2$ one can find a unitary $u_{z^\perp,n} \in (\widetilde{Pz})_{\varphi_{z^\perp}}$ in such a way that $\widetilde{E}_R(u_{z^\perp,n}^k) = (\widetilde{E_R\!\upharpoonright_{Pz^\perp}})(u_{z^\perp,n}^k) = 0$ for all $1 \leq k \leq n-1$. Write $Qz = \sum_{m\in\mathbb{N}}^\oplus \mathbb{C}e_m$, and $E_R\!\upharpoonright_{Pz}$ factors as $Pz \overset{E_{(Qz)'\cap Pz}}{\longrightarrow} (Qz)'\cap Pz \overset{\Psi}{\longrightarrow} Qz$, where $(Qz)'\cap Pz = \sum_{m\in\mathbb{N}}^\oplus e_m (Pz) e_m$ and $E_{(Qz)'\cap Pz}(x) = \sum_{m \in \mathbb{N}} e_m x e_m$ for $x \in Pz$. Moreover, $\Psi$ is of the form $\Psi(\sum_{m\in\mathbb{N}} x_m) = \sum_{m \in \mathbb{N}} \psi_m(x_m)e_m$ for $x_m \in e_m(Pz)e_m$ with faithful normal states $\psi_m$ on $e_m(Pz)e_m$. By the assumption here $Pz$ is diffuse and of type I, and thus so are the $e_m(Pz)e_m$; hence the centers of those must be diffuse, and so are all the $(e_m(Pz)e_m)_{\psi_m}$. In the same way as in (2), one can find a unitary $u_z \in ((Qz)'\cap Pz)_{\chi_z \circ \Psi}$ with `any' faithful normal state $\chi_z$ on $Qz$ in such a way that $\Psi(u_z^k) = 0$ for all $k \neq 0$. Denote by $\lambda(t)$ the generators of $\mathbb{C}\rtimes\mathbb{R}$ in $\widetilde{Pz} = (Pz)\rtimes_{\sigma^{\chi_z\circ(E_R\!\upharpoonright_{Pz})}}\mathbb{R}$ ($\hookleftarrow (Qz)\rtimes_{\sigma^{\chi_z}}\mathbb{R} = \widetilde{Qz}$ canonically), and set $\varphi_z := \tau\circ(\widetilde{E_R\!\upharpoonright_{Pz}})$, a faithful normal state on $\widetilde{Pz}$, with a fixed faithful normal tracial state $\tau := \chi_z \bar{\otimes}\tau_0$ on $\widetilde{Qz} = Qz\bar{\otimes}\lambda(\mathbb{R})''$. Note that $\lambda(t) u_z = \sigma_t^{\chi_z\circ(E_R\!\upharpoonright_{Pz})}(u_z)\lambda(t) = u_z\lambda(t)$ for all $t \in \mathbb{R}$. Thus, for any finite sum $x = \sum_k x_k \lambda(t_k) \in \widetilde{Pz}$ with $x_k \in Pz$ we have $\varphi_z(u_z x) = \sum_k \tau(\Psi(u_z E_{(Qz)'\cap Pz}(x_k))\lambda(t_k))
= \sum_k \chi_z(\Psi(u_z E_{(Qz)'\cap Pz}(x_k)))\tau_0(\lambda(t_k)) = \sum_k \chi_z(\Psi(E_{(Qz)'\cap Pz}(x_k)u_z))\tau_0(\lambda(t_k)) = \sum_k \varphi_z(x_k u_z \lambda(t_k)) = \varphi_z(x u_z)$. It follows that $u_z$ falls in $(\widetilde{Pz})_{\varphi_z}$. Clearly $\widetilde{E}_R(u_z^k) = (\widetilde{E_R\!\upharpoonright_{Pz}})(u_z^k) = E_R(u_z^k) = \Psi(u_z^k) = 0$ for all $k \neq 0$. Set $\varphi(x) := \frac{1}{2}(\varphi_z(xz) + \varphi_{z^\perp}(xz^\perp))$ for $x \in \widetilde{P}$, and then $\varphi$ becomes a faithful normal state on $\widetilde{P}$ and satisfies $\varphi\circ \widetilde{E}_R = \varphi$, implying the desired condition (a), since $\widetilde{R}$ is commutative. For each $n \in \mathbb{N}$ with $n \geq 2$ we define the unitary $u_n := u_z \oplus u_{z^\perp,n} \in \widetilde{Pz}\oplus\widetilde{Pz^\perp} = \widetilde{P}$, and thus $\widetilde{E}_R(u_n^k) = (\widetilde{E_R\!\upharpoonright_{Pz}})(u_z^k)\oplus(\widetilde{E_R\!\upharpoonright_{Pz^\perp}})(u_{z^\perp,n}^k) = 0$ for all $1 \leq k \leq n-1$. Hence the desired condition (b) is immediate as in (1) from the fact that $\widetilde{E}_Q = \widetilde{E}_Q\circ\widetilde{E}_R$ and $E^\varphi_{\widetilde{Q}} = E^\varphi_{\widetilde{Q}}\circ\widetilde{E}_R$ (the latter follows from $\varphi\circ \widetilde{E}_R = \varphi$). The final assertion is shown in the exactly same way (but easier) as above.
\end{proof}
We will give two applications of Proposition \ref{P-3.5}. The latter is a straightforward generalization of both \cite[Theorem 3.7]{Ueda:AdvMath11} and \cite[Proposition 3.1]{Ueda:MRL}. Remark that the former reproves the assertions (1), (4) in Corollary \ref{C-4.4} without any use of the technologies provided in \S\S3.1--3.2.
\begin{theorem}\label{T-4.8} Assume that $M_1$ diffuse, $N$ of type I and $M_2 \supseteq N$ entirely non-trivial. Let $z \in \mathcal{Z}(N)$ be the unique projection such that $Nz$ is diffuse and $Nz^\perp$ atomic, and assume further that $(M_1)c_z^{M_1}$ has no type I direct summand when $z \neq 0$ {\rm(}i.e., this last assumption is fulfilled if $M_1$ has no type I direct summand{\rm)}. Then $(\widetilde{M})_\omega = \big(\widetilde{M}\big)'\cap \big(\widetilde{M}\big)^\omega = \big(\widetilde{M}\big)'\cap\mathcal{Z}(\widetilde{N})^\omega$. In particular, $\widetilde{M}$ and hence $M$ itself are non-amenable. If $M$ is additionally assumed to be semifinite, then $M_\omega = M' \cap M^\omega = M' \cap \mathcal{Z}(N)^\omega$ also holds.
\end{theorem}
After the completion of the main part of the present work we learned that Houdayer and Vaes have also independently been obtained a similar (but not same) result as above under different assumptions with different (and simpler) methods. See \cite[Theorem 5.8]{HoudayerVaes:Preprint12}. More on this will be discussed at the end of this section.
\begin{proof} Note that $(\widetilde{N} \supseteq N) \cong (N\bar{\otimes}\lambda(\mathbb{R})'' \supseteq N\bar{\otimes}\mathbb{C}1)$. Since $N$ is of type I, one can choose an abelian $f \in N^p$ ($\subset \widetilde{N}^p$) with $c_f^{\widetilde{N}} = 1$. Let us first prove:
\begin{equation}\label{Eq-4.1}
f\big(\widetilde{N}\big)^\omega f = \mathcal{Z}(\widetilde{N})^\omega f.
\end{equation}
For each $x \in \widetilde{N}^\omega$ with representative $(x(m))_m$ one has $fxf = [(fx(m)f)_m]$, and for every $m$ there is a unique $z(m) \in \mathcal{Z}(\widetilde{N})$ with $fx(m)f=z(m)f$. By $c_f^{\widetilde{N}} = 1$ the mapping $x' \in \widetilde{N}' \mapsto x' f \in \widetilde{N}' f$ gives a bijective normal $*$-homomorphism (thus $\Vert-\Vert_\infty$-preserving), and hence $(z(m))_m$ defines $z \in \mathcal{Z}(\widetilde{N})^\omega$. Consequently we get $fxf = zf \in \mathcal{Z}(\widetilde{N})^\omega f$.
By Lemma \ref{L-4.6} together with \eqref{Eq-2.5} we have the identification
\begin{equation}\label{Eq-4.2}
\big(\widetilde{fMf}, \widetilde{E\!\upharpoonright_{fMf}}\big) = \big(\widetilde{fM_1 f}, \widetilde{E_1\!\upharpoonright_{fM_1 f}}\big)\star_{\widetilde{fNf}} \big(\widetilde{fM_2 f}, \widetilde{E_2\!\upharpoonright_{fM_2 f}}\big).
\end{equation}
Let $c \in \mathcal{Z}(M_1)$ be the central support projection of the type I direct summand of $M_1$. Then $e = cf$ is that of $f M_1 f$ too, and $fNf e = \mathcal{Z}(N)fe$ must be atomic (or $0$ if $e = 0$) by the assumption here. In fact, if this was not the case, then $\mathcal{Z}(N) c_e^N \cong \mathcal{Z}(N)e = \mathcal{Z}(N)fe$ is not atomic, and hence $z c_e^N \neq 0$, i.e., $ze \neq 0$, implying $c_z^{M_1} c \geq z c \geq ze \neq 0$, a contradiction to that $M_1 c_z^{M_1}$ has no type I direct summand. Therefore, by Lemma \ref{L-4.7} (3) we can apply Proposition \ref{P-3.5} to \eqref{Eq-4.2} and thus any $x \in \big(\widetilde{fMf}\big)' \cap \big(\widetilde{fMf}\big)^\omega$ and any $y \in \big(\widetilde{fM_2 f}\big)^\circ$ must satisfy that
\begin{equation}\label{Eq-4.3}
(\widetilde{E_2\!\upharpoonright_{fM_2 f}})(y^* y)(x-(\widetilde{E_{fM_1 f}})^\omega(x)) = 0,
\end{equation}
where $E_{fM_1 f}$ is the unique conditional expectation from $fMf$ onto $fM_1 f$ determined as \eqref{Eq-2.2}. Note that $\widetilde{fM_2 f} \supseteq \widetilde{fNf}$ with $\widetilde{E_2\!\upharpoonright_{fM_2 f}}$ contains $fM_2 f \supseteq fNf$ with $E_2\!\upharpoonright_{f M_2 f}$ canonically. Hence, by Lemma \ref{L-4.2} (2), (3) one can find a family $\{y_i\}_{i \in I}$ in $(f M_2 f)^\circ$ in such a way that $\sum_{i\in I} s(E_2(y_i^* y_i)) = f$ ($=1_{fNf}$). Therefore, it follows from \eqref{Eq-4.3} as in the proof of Theorem \ref{T-4.3} that $x = \big(\widetilde{E_{fM_1 f}}\big)^\omega(x) \in \big(\widetilde{fM_1 f}\big)^\omega$. Consequently $\big(\widetilde{fMf}\big)' \cap \big(\widetilde{fMf}\big)^\omega = \big(\widetilde{fMf}\big)' \cap \big(\widetilde{fM_1 f}\big)^\omega$. In the same way as in the proof of Theorem \ref{T-4.3}, we see, by using the above $\{y_i\}_{i\in I}$ again and the free independence between $\big(\widetilde{fM_1 f}\big)^\omega$ and $\big(\widetilde{fM_2 f}\big)^\omega$, that
\begin{equation}\label{Eq-4.4}
\big(\widetilde{fMf}\big)' \cap \big(\widetilde{fMf}\big)^\omega = \big(\widetilde{fMf}\big)' \cap \big(\widetilde{fNf}\big)^\omega.
\end{equation}
Choose a faithful normal semifinite trace $\mathrm{Tr}_N$ on $N$, and $\widetilde{M} \supseteq \widetilde{M}_k\, (k=1,2)\, \supseteq \widetilde{N}$ are realized as
$\widetilde{M} = M\rtimes_{\sigma^{\mathrm{Tr}_N\circ E}}\mathbb{R} \supseteq \widetilde{M}_k = M_k \rtimes_{\sigma^{\mathrm{Tr}_N\circ E_k}}\mathbb{R} \supseteq \widetilde{N} = N\rtimes_{\sigma^{\mathrm{Tr}_N}}\mathbb{R}$. Since $\sigma_t^{\mathrm{Tr}_N\circ E}(f) = f$ for all $t \in \mathbb{R}$, $f\widetilde{M}f \supseteq f\widetilde{M}_k f \supseteq f\widetilde{N}f$ are naturally identified with $\widetilde{fMf} \supseteq \widetilde{fM_k f} \supseteq \widetilde{fNf}$. Hence \eqref{Eq-4.4} and \eqref{Eq-4.1} imply that
\begin{align}\label{Eq-4.5}
\big(\widetilde{M}\big)'f \cap f\big(\widetilde{M}\big)^\omega f
= \big(\widetilde{M}\big)'f \cap f\widetilde{N}^\omega f = \big(\widetilde{M}\big)'f \cap \mathcal{Z}(\widetilde{N})^\omega f.
\end{align}
Let $\pi_f$ be the normal surjective $*$-homomorphism $x \in \big(\widetilde{M}\big)'\cap\big(\widetilde{M}\big)^\omega \mapsto xf \in \big(\widetilde{M}'\cap\big(\widetilde{M}\big)^\omega\big)f = \big(\widetilde{M}\big)'f \cap (f\big(\widetilde{M}\big)^\omega f)$ (c.f.~\cite[Lemma 4.1 (i)]{Voeden:PLMS73}), which is also injective due to $c_f^{\widetilde{N}} = 1$ (and hence $c_f^{\widetilde{M}} = 1$ too). By \eqref{Eq-4.5} we have $\big(\widetilde{M}\big)'\cap\big(\widetilde{M}\big)^\omega = \pi_f^{-1}\big(\big(\widetilde{M}\big)'f \cap \mathcal{Z}(\widetilde{N})^\omega f\big)$. As in the proof of Lemma \ref{L-4.2} one can choose partial isometries $\{v_i\}_{i\in I}$ in $\widetilde{N}$ so that $\sum_{i\in I}v_i^* v_i = c_f^{\widetilde{N}} = 1$ and $v_i v_i^* \leq f$ for all $i \in I$. Then, if $x = zf \in \big(\widetilde{M}\big)'f\cap\mathcal{Z}(\widetilde{N})^\omega f$ with $z \in \mathcal{Z}(\widetilde{N})^\omega$, then we have
$yz = \sum_{i_1,i_2\in I} v_{i_1}^* v_{i_1} y z v_{i_2}^* v_{i_2} = \sum_{i_1,i_2\in I} v_{i_1}^* (v_{i_1} y v_{i_2}^*) x v_{i_2} = \sum_{i_1,i_2\in I} v_{i_1}^* x (v_{i_1} y v_{i_2}^*) v_{i_2} = \sum_{i_1,i_2\in I} v_{i_1}^* v_{i_1} zy v_{i_2}^* v_{i_2} = zy$ for $y \in \widetilde{M}$, implying $z \in \big(\widetilde{M}\big)'\cap\mathcal{Z}(\widetilde{N})^\omega$. Hence $\big(\widetilde{M}\big)'f \cap \mathcal{Z}(\widetilde{N})^\omega f = \big(\big(\widetilde{M}\big)'\cap\mathcal{Z}(\widetilde{N})^\omega\big)f$. Consequently $\big(\widetilde{M}\big)'\cap\big(\widetilde{M}\big)^\omega = \pi_f^{-1}\big(\big(\widetilde{M}\big)'f \cap \mathcal{Z}(\widetilde{N})^\omega f\big) = \pi_f^{-1}\big(\big(\big(\widetilde{M}\big)'\cap\mathcal{Z}(\widetilde{N})^\omega\big)f\big) = \big(\widetilde{M}\big)'\cap\mathcal{Z}(\widetilde{N})^\omega$. Since $\big(\widetilde{M}\big)'\cap\big(\widetilde{M}\big)^\omega = \big(\widetilde{M}\big)'\cap\mathcal{Z}(\widetilde{N})^\omega$ is commutative, it must equal $(\widetilde{M})_\omega$ as observed in \cite[(8) in page 360]{Ueda:TAMS03}.
The final assertion is also shown in the exactly same way as above by using the final assertion in Lemma \ref{L-4.7} (3), since there is a faithful normal semifinite trace $\mathrm{Tr}_N$ on $N$ so that $\mathrm{Tr}_N\circ E_k$ ($k=1,2$) are traces again thanks to Corollary \ref{C-4.4} (3).
\end{proof}
\begin{remark}\label{R-4.9}{\rm
The same type argument as in Theorem \ref{T-4.3} (3) works for constructing a faithful normal state $\chi$ on $N$ with $\sigma_T^{\chi\circ E} = \mathrm{Id}$ with $T = -2\pi/\log\lambda$, $0 < \lambda < 1$, when $M$ is known to be a factor of type III$_\lambda$ under the same set of assumptions as in Theorem \ref{T-4.8}. Hence the discrete core of such $M$ can also be written as an amalgamated free product von Neumann algebra of the same form as the continuous core, and an analogous formula for its asymptotic centralizer holds. In particular, the discrete core of such a factor of type III$_\lambda$ is an $\infty$-amplification of a non-strongly stable type II$_1$ factor. Further and more detailed discussions related to this aspect will be given elsewhere.
}
\end{remark}
\begin{theorem}\label{T-4.10} If $M_1$ is diffuse, $N$ of atomic type I and $M_2 \supseteq N$ entirely non-trivial, then the following hold true{\rm:}
\begin{itemize}
\item[(1)] $M_\omega = M'\cap M^\omega = M' \cap \mathcal{Z}(N)$ {\rm(}$= \mathcal{Z}(M)${\rm)}. Hence $M$ does never have no type III$_0$ direct summand {\rm(}see \cite[Theorem 2.12]{Connes:JFA74}{\rm)}, and becomes full in the sense of Connes {\rm \cite{Connes:JFA74}} under the separability of preduals.
\item[(2)] The Connes $\tau$-invariant $\tau(M)$ {\rm(}see {\rm \cite{Connes:JFA74}}{\rm)} is determined under the separability of preduals as follows. Let $\chi$ be a faithful normal state on $N$. Then $t_m \longrightarrow 0$ in $\tau(M)$ as $m\rightarrow \infty$ if and only if there is a unitary $w \in N$ so that $\sigma_{t_m}^{\chi\circ E} \longrightarrow \mathrm{Ad}w$ in $\mathrm{Aut}(M)$ as $m \rightarrow \infty$.
\end{itemize}
\end{theorem}
\begin{proof} (1) This is proved along the same line as in the proof of Theorem \ref{T-4.8} by using only Lemma \ref{L-4.7} (2) instead together with a well-known fact $\mathcal{Z}(N) = \mathcal{Z}(N)^\omega$ due to the assumption that it is atomic.
(2) We can write $N = \sum_{i \in I}^\oplus B(\mathcal{H}_i)$. Looking at this structure with the given $\chi$ we can choose a collection $\{e_i\}_{i\in I}$ of abelian projections in $N$ with $\sum_{i\in I} e_i = 1$ such that for each $i \in I$ there is a larger abelian $f_i \in N^p$ so that $e_i \leq f_i$, $c_{f_i}^N = 1$ and $\sigma_t^\chi(f_i) = f_i$ ($t \in \mathbb{R}$). Assume that $t_m \longrightarrow 0$ in $\tau(M)$ as $m \rightarrow \infty$. Then there is a sequence $(u_m)_m$ of unitaries in $M$ such that $\mathrm{Ad}u_m\circ\sigma_{t_m}^{\chi\circ E} \longrightarrow \mathrm{id}$ in $\mathrm{Aut}(M)$ as $m\rightarrow \infty$. As observed in the proof \cite[Proposition 3.1]{Ueda:MRL} the $(u_m)_m$ defines a unitary $u \in M^\omega$, and clearly $u f_i = f_i u$ for all $i \in I$. Hence $f_i u$ defines a unitary in $f_i M^\omega f_i = (f_i M f_i)^\omega$, and we denote it by $u_i$ for simplicity. Since $f_i M_1 f_i$ is still diffuse, looking at $f_i M_1 f_i \supseteq f_i N f_i = \mathcal{Z}(N)f_i$ one can choose a faithful normal state $\varphi$ on $f_i M_1 f_i$ as in Lemma \ref{L-4.7} (2). Set $\hat{\varphi}(x) := \varphi(f_i x f_i) + \chi\circ E_1(f_i^\perp x f_i^\perp)$, $x \in M_1$, which becomes a faithful normal positive linear functional on $M_1$. Clearly $f_i \in (M_1)_{\hat{\varphi}}$ and thus $f_i [D\chi\circ E_1:D\hat{\varphi}]_t = [D\chi\circ {E_1}\!\upharpoonright_{f_i M_1 f_i} : D\varphi]_t$ for all $t \in \mathbb{R}$ by the uniqueness part of Connes's Radon-Nikodym cocycle theorem. As observed in the proof of \cite[Proposition 3.1]{Ueda:MRL} again the sequence $v_m := [D\chi\circ E_1 : D\hat{\varphi}]_{t_m}$ defines a unitary $v \in M_1^\omega$ and also the sequence $f_i v_m = v_m f_i$ does a unitary $v_i \in f_i M_1^\omega f_i = (f_i M_1 f_i)^\omega$. Since $\hat{\varphi}\circ {E_{M_1}}\!\upharpoonright_{f_i M f_i} = \varphi\circ({E_{M_1}}\!\upharpoonright_{f_i M f_i})$, we have $yu_i v_i = yuv = \big[(yu_m v_m)_m\big] = \big[ (u_m v_m \sigma_{t_m}^{\varphi\circ({E_{M_1}}\!\upharpoonright_{f_i M f_i})}(y))_m\big] = uvz = u_i v_i z$ for $y \in (f_i M_2 f_i)^\circ$ with $z=\big[(\sigma_{t_m}^{\varphi\circ({E_{M_1}}\!\upharpoonright_{f_i M f_i})}(y))_m\big] \in (f_i M f_i)^\omega = f_i M^\omega f_i$ in the identification $(f_i M f_i, E\!\upharpoonright_{f_i M f_i}) = (f_i M_1 f_i, {E_1}\!\upharpoonright_{f_i M_1 f_i}) \star_{f_i N f_i} (f_i M_2 f_i, {E_2}\!\upharpoonright_{f_i M_2 f_i})$ provided by Lemma \ref{L-4.6}. By Proposition \ref{P-3.5} we get $({E_2}\!\upharpoonright_{f_i M_2 f_i})(y^* y)(u_i v_i - ({E_{M_1}}\!\upharpoonright_{f_i M f_i})^\omega(u_i v_i)) = 0$ for $y \in (f_i M_2 f_i)^\circ$. By using Lemma \ref{L-4.2} (2), (3) twice as in the proof of Theorem \ref{T-4.8} we can prove firstly that $u_i v_i \in (f_i M_1 f_i)^\omega = f_i M_1^\omega f_i$, secondly that $u_i \in f_i M_1^\omega f_i$ (since $v_i \in f_i M_1^\omega f_i$), and finally that $u_i \in f_i N^\omega f_i = \mathcal{Z}(N)^\omega f_i = \mathcal{Z}(N)f_i$. Therefore, $u = \sum_{i \in I} e_i u = \sum_{i \in I} e_i f_i u = \sum_{i\in I} e_i u_i \in N$. Letting $w := u^* \in N^u$ we have $\mathrm{Ad}w^*\circ\sigma_{t_m}^{\chi\circ E} \longrightarrow \mathrm{id}$ in $\mathrm{Aut}(M)$ as $m\rightarrow\infty$.
\end{proof}
The next proposition shows that Proposition \ref{P-3.5} is still useful beyond the case where $N$ is of type I or even semifinite. The proof goes along the same line as that of Theorem \ref{T-4.8} but is easier than it. Hence the proof is left to the reader.
\begin{proposition}\label{P-4.11} Assume that there is a faithful normal state $\varphi$ on $M_1$ satisfying the following conditions{\rm:}
\begin{itemize}
\item[(a)] $\sigma_t^\varphi(N) = N$ for all $t \in \mathbb{R}$.
\item[(b)] For every $n \in \mathbb{N}$ with $n \geq 2$ there are unitaries $u_k = u_k^{(n)}
,v_k = v_k^{(n)} \in (M_1)_\varphi$, $0 \leq k \leq n-1$, such that $E_1(u_{k_1}^* u_{k_2}) = E_N^\varphi(v_{k_1}^* v_{k_2}) = 0$ for all $0 \leq k_1 \neq k_2 \leq n-1$, where $E_N^\varphi$ denotes the unique $\varphi$-preserving conditional expectation from $M_1$ onto $N$, whose existence follows from {\rm(a)} and Takesaki's criterion.
\end{itemize}
Assume also that $M_2 \supseteq N$ is entirely non-trivial. Then $M' \cap M^\omega = M' \cap N^\omega$ holds. Moreover, if it is further assumed that $N$ is finite, then $M_\omega = M'\cap M^\omega = M' \cap N_\omega$.
\end{proposition}
It is easy to confirm that the $(M_1,E_1)$ in Corollary \ref{C-4.5} satisfies the assumption of Proposition \ref{P-4.11}. Thus $M'\cap M^\omega = M'\cap N^\omega$ holds under the set of assumptions in Corollary \ref{C-4.5}.
Assume that $M_1$ is a von Neumann algebra with separable predual and that $N$ is a Cartan subalgebra in $M_1$. It was proved in \cite[Lemma 4.2]{Ueda:PacificJMath99} that if $M_1$ is further assumed to be a non-type I factor, then there are a faithful normal state $\varphi$ on $M_1$ with $\varphi\circ E_1 = \varphi$ and a unitary $u \in (M_1)_\varphi$ such that $E_1(u^k) = 0$ as long as $k \neq 0$. The same assertion can indeed be proved even when $M_1$ is further assumed only to have no type I direct summand (i.e., without being a factor). The proof is similar to \cite[Lemma 4.2]{Ueda:PacificJMath99} but tedious based on disintegration. Hence such $(M_1,E_1)$ satisfies the assumption of Proposition \ref{P-4.11}.
\begin{remark}\label{R-4.12} {\rm
Almost all the results obtained above have appropriate `HNN variants' thanks to tricks given in \cite{Ueda:IllinoisJMath08}. Here it should be emphasized that our results so far essentially need assumptions for only one free component. The notion of HNN extensions of von Neumann algebras as well as their basic properties including their modular theoretic aspects were established in \cite{Ueda:JFA05}.}
\end{remark}
In closing of this section we discuss one of Houdayer and Vaes's results \cite[Theorem 5.8]{HoudayerVaes:Preprint12}. This part of the present paper is added after receiving a draft of \cite{HoudayerVaes:Preprint12} in order to point out only one consequence obtained from this and that papers without any new idea. Therefore, some facts provided in \cite{HoudayerVaes:Preprint12} are necessary below. The original aim of the present work is to provide amalgamated free product counterparts of the results in \cite[\S3]{Ueda:AdvMath11}. One issue to do so is how to formulate a suitable assumption saying that $M_1$ is `diffuse relative to $N$' which corresponds to that $M_1$ is diffuse when $N=\mathbb{C}1$. The requirement for $M_1 \supseteq N$ in Theorem \ref{T-4.3} seems to be one strong form of them without any restriction on $N$, but it seems not so easy to check it in general. Thus we propose the requirement for $M_1 \supseteq N$ in Corollary 4.4 and Theorem 4.8 as such a candidate in the special case when $N$ is of type I. However a more sophisticated one in the special case seems to be that $M_1 \supseteq N$ has no trivial corner, which is proposed in \cite[\S5]{HoudayerVaes:Preprint12} by a different motivation. In fact, Houdayer and Vaes \cite[Theorem 5.8]{HoudayerVaes:Preprint12} give a factoriality and non-amenability result under the set of assumptions that both $M_k \supseteq N$, $k=1,2$, have no trivial corner and that $N$ is of type I, and establish their primeness result under the same set of assumptions. Here an inclusion $P \supseteq Q$ of von Neumann algebras is said to have no trivial corner if $pPp \neq Qp$ for any non-zero projection $p \in Q'\cap P$. Any exact general relationship between theirs and ours is not immediately clear. However the proof of Theorem \ref{T-4.8} and general properties on inclusions without trivial corner provided in \cite[\S\S5.1]{HoudayerVaes:Preprint12} altogether immediately give an improvement of \cite[Theorem 5.8]{HoudayerVaes:Preprint12}, though it is not immediately clear whether the primeness result in \cite[Theorem E]{HoudayerVaes:Preprint12} holds or not under the new set of assumptions.
\begin{theorem}\label{T-4.13} If $M_1 \supseteq N$ has no trivial corner, $N$ is of type I and $M_2 \supseteq N$ entirely non-trivial, then the following hold true{\rm:}
\begin{itemize}
\item[(1)] $\mathcal{Z}(M) = \mathcal{Z}(M_1)\cap\mathcal{Z}(M_2)\cap\mathcal{Z}(N)$.
\item[(2)] $\mathcal{Z}(\widetilde{M}) = \mathcal{Z}(\widetilde{M}_1)\cap\mathcal{Z}(\widetilde{M}_2)\cap\mathcal{Z}(\widetilde{N})$.
\item[(3)] $(\widetilde{M})_\omega = \big(\widetilde{M}\big)'\cap\big(\widetilde{M}\big)^\omega = \big(\widetilde{M}\big)'\cap\mathcal{Z}(\widetilde{N})^\omega$.
\end{itemize}
In particular, {\rm(3)} explains that $M$ does never become amenable.
\end{theorem}
\begin{proof} It is trivial that (3) $\Rightarrow$ (2) $\Rightarrow$ (1), see e.g.~the proof of \cite[Theorem 5.2]{Ueda:JFA05} for (3) $\Rightarrow$ (2) and \cite[Theorem X.II.1.1]{Takesaki:Book2} for (2) $\Rightarrow$ (1). Thus it suffices to prove only (3). The line of the proof below is exactly identical to that of Theorem \ref{T-4.8}, and thus we keep the notations there. In fact, only one modification is sufficient. By \cite[Lemma 5.2, Proposition 5.5]{HoudayerVaes:Preprint12} the inclusion $\widetilde{fM_1 f} \supseteq \widetilde{fNf}$ also has no trivial corner. Then it suffices to prove the exactly same assertion as in Lemma \ref{L-4.7} (1) with replacing the assumption that $P$ has no type I direct summand by that $P \supseteq Q$ has no trivial corner. In fact, by using this new assertion instead of Lemma \ref{L-4.7} (3) one gets the same equation \eqref{Eq-4.3} and the rest of the proof there works well.
Let $P \supseteq Q$ be an inclusion of von Neumann algebras without trivial corner. Assume that $Q$ is commutative, $P$ has a faithful normal semifinite trace $\mathrm{Tr}_P$ and there is a faithful normal conditional expectation $E_Q : P \rightarrow Q$ satisfying $\mathrm{Tr}_P \circ E_Q = \mathrm{Tr}_P$. As in the proof of Lemma \ref{L-4.7} (1) we choose the $q_m$'s and $\chi$. Then we apply \cite[Lemma 5.4 (3)]{HoudayerVaes:Preprint12} (note that it holds without assuming the separability of preduals, see Lemma \ref{L-4.14} below) with $q = p := q_m$ and get a unitary $u_m \in q_m P q_m$ satisfying that $E_Q(u_m^k) = 0$ as long as $k \neq 0$. Letting $u := \sum_{m \in \mathbb{N}} u_m$ we have $u \in P_{\chi\circ E_Q}$ and $E_Q(u^k) = 0$ as long as $k \neq 0$. Hence we are done.
\end{proof}
As remarked in \cite[Lemma 5.3]{HoudayerVaes:Preprint12} the next lemma immediately follows from Rohlin's general theorem on Lebesgue spaces under the separability of preduals. Thus only the advantage of the proof below is no use of disintegration; hence the separability of preduals is not necessary in \cite[Lemma 5.4]{HoudayerVaes:Preprint12}. Although it is a rather minor point, we do give it for the sake of completeness.
\begin{lemma}\label{L-4.14} Let $B \supseteq A$ be {\rm(}unital{\rm)} inclusion of commutative $\sigma$-finite von Neumann algebras with a faithful normal conditional expectation $E_A : B \rightarrow A$. If $Bf \neq Af$ for any nonzero projection $f \in B$, then there is a unitary $u \in B$ such that $E_A(u^k) = 0$ as long as $k \neq 0$.
\end{lemma}
\begin{proof} Choose non-zero $f \in B^p$. Since $Bf \neq Af$, there is $x \in B$ such that $x \not\in Af$ and $0 \leq x \leq f$. Since $E_A(x) \leq E_A(f)$, one can choose a positive contraction $c \in A$ so that $c E_A(f) = E_A(x)$ (since $A$ is commutative). Letting $y := x - cf \in Bf$ we have $y = y^* \neq 0$ (due to $x \not\in Af$) and $E_A(y) = 0$. Therefore, an idea given in the proof of \cite[Lemma 2.1]{CameronFangMukherjee:Preprint11} enables us to construct projections $e_{(\varepsilon_1,\dots,\varepsilon_n)} \in B$, $n \in \mathbb{N}$, $\varepsilon_k \in \{1,2\}$, in such a way that $e_{(\varepsilon_1,\dots,\varepsilon_n)} = e_{(\varepsilon_1,\dots,\varepsilon_n,1)}+e_{(\varepsilon_1,\dots,\varepsilon_n,2)}$ and $E_A(e_{(\varepsilon_1,\dots,\varepsilon_n)}) = \frac{1}{2^n}1$. The proof is done by induction. Assume that we have chosen up to $n$-th stage. Set $\Lambda_e := \{ x = x^* \in Be\,|\,\Vert x \Vert_\infty \leq 1, E_A(x) = 0\}$ with $e := e_{(\varepsilon_1,\dots,\varepsilon_n)}$. It is a $\sigma$-weakly compact convex subset, and thus has sufficiently many extremal points due to the Krein--Milman theorem. Let $a \in \Lambda_e$ be an extremal point. Then it suffices to prove $a = 2e_0 - e$ for some $e_0 \in B^p$ with $e_0 \leq e$, since it clearly implies that $E_A(e_0) = \frac{1}{2}E_A(e)$. On contrary, suppose that it is not the case. By the spectral decomposition of $a$ one can find $\delta>0$ and non-zero $f \in B^p$ in such a way that $f \leq e$ and $-(1-\delta)f \leq af \leq (1-\delta)f$. By what we have shown above, there is a non-zero $y = y^* \in Bf$ such that $-\delta f \leq y \leq \delta f$ and $E_A(y) = 0$, and hence $a+y, a-y \in \Lambda_e$ and $a = \frac{1}{2}(a+y)+\frac{1}{2}(a-y)$, a contradiction. Thus $e_{(\varepsilon_1,\dots,\varepsilon_n,1)} := e_0$ and $e_{(\varepsilon_1,\dots,\varepsilon_n,2)} := e-e_0$ become desired ones in $(n+1)$-th stage. Hence we have proved the claim. Let $(C,\omega)$ be the von Neumann algebraic infinite tensor product of $\mathbb{C}\oplus\mathbb{C}$ with equal weights $\{1/2,1/2\}$. Once passing GNS representations one can construct an injective normal $*$-homomorphism from $C\bar{\otimes}A$ into $B$ which intertwines $\omega\bar{\otimes}\mathrm{id}_A$ and $E_A$. Hence the desired assertion follows, since $(C,\omega) \cong (L(\mathbb{Z}),\tau_{\mathbb{Z}})$ thanks to \cite[Theorem III.1.22]{Takesaki:Book1}.
\end{proof}
The entire non-triviality of an inclusion $P \supseteq Q$ of von Neumann algebras is nothing but just the non-triviality of $P$ when $Q = \mathbb{C}1$, and hence Theorem \ref{T-4.13} is no longer true under assuming only that $M_1 \supseteq N$ is entirely non-trivial instead. In fact, the plain free product of two 2-dimensional algebras with suitable states provides a counter example, see \cite{Ueda:AdvMath11} for suitable references therein. Finally we conjecture that Corollary 4.4, especially a strong kind of irreducibility $((M_1)_\varphi)' \cap M \subseteq M_1$ for some faithful normal state $\varphi$, should also hold under the same set of assumptions of Theorem \ref{T-4.13}. This is rather technical, but such a property may have some potential in further analysis. We will consider it in future work beyond the case where $\mathcal{Z}(M) = \mathcal{Z}(M_1)\cap\mathcal{Z}(M_2)\cap\mathcal{Z}(N)$ need not hold.
\section*{Acknowledgment} I thank Professors Cyril Houdayer and Stefaan Vaes for several fruitful conversations in Dec.~2011 and in Jan.~2012 and also for sending us a draft of \cite{HoudayerVaes:Preprint12} prior to putting it on the ArXiv.
}
|
{
"timestamp": "2012-05-16T02:03:30",
"yymm": "1203",
"arxiv_id": "1203.1806",
"language": "en",
"url": "https://arxiv.org/abs/1203.1806"
}
|
\section{Introduction\label{Introduction} }
\IEEEPARstart{T}{he} calculation of error probability for Linear
Programming (LP) decoding of Binary Phase-Shift Keying (BPSK) modulated
binary codes is often a complex task. This is mainly due to the complexity
of LP Voronoi or decision regions \cite{DecisionRegions} \cite{GraphCover}.
The probability of correct decision in an Additive White Gaussian
Noise (AWGN) channel, can be obtained by integrating a multidimensional
Gaussian distribution over the decision region of the transmitted
codeword (CW).
LP decoding is a relaxed version of the Maximum-Likelihood (ML) decoding.
The \emph{codeword polytope} \cite{UsingLP} of ML is replaced by
a relaxed polytope, called the \emph{fundamental polytope} \cite{UsingLP}.
The fundamental polytope arisen from a given parity check matrix.
Its vertices are every codeword, but it also has some non-codeword.
The vertices of the codeword polytope are the all codewords, and the
vertices of the fundamental polytope are called \emph{pseudocodewords
}(PCWs) \cite{UsingLP}. The additional non-codewords make the decision
region \cite{DecisionRegions} of the LP decoder even more complex
than that of the ML. Therefore, a derivation of analytical bounds
has an important role in evaluating the performance of the LP decoder.
The \emph{fundamental cone} \cite{GraphCover} is the conic hull of
the fundamental polytope. The LP error probability over the fundamental
polytope is equal to that over the fundamental cone \cite{RelaxationBounds}.
Moreover, it is sufficient to consider\emph{ }only the fundamental
cone\emph{ generators }\cite{RelaxationBounds} for evaluating the
performance of the LP decoder.
The well-known upper bound on the error probability of a digital communication
system is the \emph{Union Bound} (UB), which is a first-order \emph{Bonferroni-type}
inequality \cite{Bonferroni} in the probability theory. The UB of
the LP decoder \cite{DecisionRegions} \cite{LP-UB} \cite{LP-UB2}
for High-Density Parity-Check (HDPC) codes presets inaccurate results
due to the high density of fundamental cone generators. In fact, the
union bound sums all of the pairwise error events as if they were
disjoint, but this scenario is far from being the case in LP decoding
of HDPC codes.
Each pseudocodeword in the LP decoder can be located in the BPSK signal
space \cite{GraphCover}. What the LP decoder does, it chooses the
nearest pseudocodeword to the received vector as the most likely transmitted
pseudocodeword. The ML soft decision decoder has such property as
well, but unlike to the LP decoder, its signal space contains only
the set of the all codewords. Thus many of ML upper bounds can be
reused \cite{TSB} \cite{SphereBound} \cite{NewTSB} \cite{ITSB}
in the case of LP decoding.
For a given code, each of its parity-check matrix creates a fundamental
cone with different pseudo-weight spectrum and geometrical structure,
which influences differently on the error probability of the LP decoder.
Therefore, the geometrical properties of the fundamental cone generators
are essential to evaluate with a better accuracy the LP decoding error
probability. Thus ML error probability bounds which use the weight
spectrum of the code or those who sum the error contribution of each
individual codeword become less attractive. In \cite{ITSB} a ML bound
is presented which is based on the second-order upper bound on the
probability of a finite union of events. And indeed, it uses the geometrical
properties of the codewords and considers an intersection of pairwise
error events, but involves relatively high computational complexity.
To explore the density of the fundamental cone generators, we have
defined the \emph{angle graph}: each generator is considered as a
node of a complete undirected graph. The cost of an edge is the angle
between the generators related to the adjacent nodes. The minimum
spanning tree is found and its cost distribution is illustrated. Different
patterns for various parity-check matrices were observed.
In this paper, we propose an upper bound based on the second-order
of Bonferroni-type inequality. The bound needs the fundamental cone
generators rather than their weight spectrum. We call it Improved
Linear Programming Union Bound (ILP-UB). It consists of two parts:
The first term is the LP union bound itself, and the second term is
a second-order correction that can be optimized by a known minimum
spanning tree algorithm. It requires relatively low computational
complexity since it involves only the $Q$-function.
The proposed ILP-UB makes use of an upper bound of the triplet-wise
error probability that has been introduced earlier in the paper. We
derive analytical expression to evaluate the triple-wise error probability
depending on the angle which they create. And for example, the triple-wise
error probability for the minimal-weight generators of the BCH{[}63,57,3{]}
code is calculated. It is compared to the triple-wise error upper
bound and to the UB in different angles and Signal-to-Noise Ratios
(SNRs).
The proposed ILP-UB was tested on three HDPC codes: Golay{[}24,12,8{]},
BCH{[}31,26,3{]}, BCH{[}63 ,57,3{]}, and on the Low-Density Parity-Check
(LDPC) Tanner code {[}155,64,20{]} \cite{Tanner155}. An improvement
of up to 0.37 dB has been demonstrated over the conventional Linear
Programming Union Bound (LP-UB).
This paper is organized as follows. Sec. \ref{Preliminaries} provides
some background on ML and LP decoding. The minimum spanning tree problem
for undirected graph is also reviewed in Sec. \ref{Preliminaries}.
In Sec. \ref{Density} we explore the density of the fundamental cone
generators and we check the effect of that density on the union bound
of the triplet-wise error probability. The problem of finding an LP
dominant error events is discussed in Sec. \ref{DominantGroup}. In
Sec. \ref{ImprovedUB} we propose an improved linear programming error
union bound. Sec. \ref{Results} provides numerical results and discusses
some possible direction for further research on how to improve the
proposed bound. Sec. \ref{Conclusions} concludes the paper.
\renewcommand{\figurename}{\fontsize{9}{10}\selectfont Fig.}
\renewcommand\captionlabeldelim{. }
\section{Preliminaries and Definitions\label{Preliminaries} }
\subsection{ML and LP Decoding}
In this section we briefly review ML and LP decoding \cite{UsingLP}.
Consider a binary linear code $\mathcal{C}$ of length $n$, dimension
$k$ and code rate $R\triangleq k/n$. Let $\mathbb{F}_{2}\triangleq\{0,1\}$
denote the finite field with two elements. The code $\mathcal{C}$
is defined by some $m\times n$ parity-check matrix $H\in\mathbb{F}_{2}^{m\mathrm{x}n}$
with row vectors $\mathrm{\mathbf{h}}_{1},\mathrm{\mathbf{h}}_{2},...,\mathrm{\mathbf{h}}_{m}$,
i.e. $\mathcal{C\triangleq}\{\mathrm{\mathbf{x}}\in\mathbb{F}_{2}^{n}\mid\mathrm{\mathbf{x}}H^{T}=0\}$.
The code will be called an {[}\emph{n,k,d}{]} code, in which \emph{d}
is its minimum \emph{Hamming distance}. The code is used for data
communication over a memoryless binary-input channel with channel
law $P_{Y|X}(y|x)$. We denote the transmitted codeword by $\mathrm{\mathbf{x}}\triangleq(x_{1},...,x_{n})$,
the transmitted signal by $\overline{\mathrm{\mathbf{x}}}\triangleq(\overline{x}_{1},...,\overline{x}_{n})$
and the received signal by $\mathrm{\mathbf{y}}\triangleq(y_{1},...,y_{n})$.
We assume that every codeword $\mathrm{\mathbf{x}}\in\mathcal{C}$
is transmitted with equal probability. Let $\boldsymbol{\lambda}$
denote the Log-Likelihood Ratio (LLR) vector with the LLR components
$\lambda_{i}\triangleq P_{Y|X}(y_{i}|0)/P_{Y|X}(y_{i}|1)$ for $i=1,...,n$.
The block-wise Maximum Likelihood Decoding (MLD) is \vspace{-7mm}
\begin{equation}
\hat{\mathrm{\mathbf{x}}}_{MLD}(\mathrm{\mathbf{y}})\triangleq\underset{\mathrm{\mathbf{x}}\in\mathcal{C}}{\mathrm{arg\: min}}\left\langle \mathrm{\mathbf{x}},\boldsymbol{\lambda}\right\rangle .\label{eq:MLD}
\end{equation}
Where $\left\langle \mathrm{\mathbf{x}},\boldsymbol{\lambda}\right\rangle \triangleq\sum_{i}x_{i}\lambda_{i}$
denote the\emph{ }standard\emph{ inner} \emph{product} of two vectors
of equal length. The ML decoder error probability is independent of
the transmitted CW, therefore, we assume without loss of generality
that the all-zeros codeword $\mathrm{\mathbf{x}}_{0}$ is transmitted.
Then \cite{CommunBook} \vspace{-8mm}
\begin{eqnarray}
P_{r}^{MLD}(error\mid\mathrm{\mathbf{x}}_{0}) & = & P_{r}\left(\hat{\mathrm{\mathbf{x}}}_{MLD}(\mathrm{\mathbf{y}})\neq\mathrm{\mathbf{x}}_{0}\mid\mathrm{\mathbf{x}}_{0}\right)\\
& = & P_{r}\left\{ \bigcup_{\mathrm{x}\in\mathcal{C}\setminus\mathrm{\mathbf{x}}_{0}}||\overline{\mathrm{\mathbf{x}}}-\mathrm{\mathbf{y}}||_{2}\leq|||\mathrm{\overline{\mathbf{x}}}_{0}-\mathrm{\mathrm{\mathbf{y}}}||_{2}\mid\mathrm{\mathbf{x}}_{0}\right\} \label{eq:ML error}\\
& \leq & \sum_{\mathrm{x}\in\mathcal{C}\setminus\mathrm{x}_{0}}P_{r}\left\{ \,||\overline{\mathrm{\mathbf{x}}}-\mathrm{\mathrm{\mathbf{y}}}||_{2}\leq|||\mathrm{\overline{\mathbf{x}}}_{0}-\mathrm{\mathrm{\mathbf{y}}}||_{2}\mid\mathrm{\mathbf{x}}_{0}\,\right\} \\
& = & \sum_{\mathrm{x}\in\mathcal{C}\setminus\mathrm{\mathbf{x}}_{0}}Q\left(\dfrac{d_{\mathrm{\mathbf{x}}}}{2\sigma}\right).\label{eq:ML-UB}
\end{eqnarray}
Where the $Q$-function is defined to be $Q(x)\triangleq\frac{1}{\sqrt{2\pi}}\int_{x}^{\infty}\exp\Bigl(-\frac{t^{2}}{2}\Bigr)dt$
and $||\mathrm{\mathbf{x}}||_{2}\triangleq\sqrt{\sum_{i}x_{i}^{2}}$
denote the $\mathcal{L}_{2}$-norm of a vector \textbf{x}. Eq. \eqref{eq:ML error}
also allows to make a simulation of the error probability contributed
by a subgroup of codewords. Eq. \eqref{eq:ML-UB} is the ML union
bound, where $d_{\mathrm{\mathbf{x}}}\triangleq||\overline{\mathrm{\mathbf{x}}}-\mathrm{\overline{\mathbf{x}}}_{0}||_{2}=2\sqrt{RE_{b}w_{H}(\mathrm{\mathbf{x}})}$
is the Euclidean distance from $\overline{\mathrm{\mathbf{x}}}$ to
the transmitted signal $\mathrm{\overline{\mathbf{x}}}_{0}$.
The MLD \eqref{eq:MLD} can be formulated \cite{UsingLP} as the following
equivalent optimization problem:\vspace{-6mm}
\begin{equation}
\hat{\mathrm{\mathbf{x}}}_{MLD}(\mathrm{\mathbf{y}})\triangleq\underset{\mathrm{\mathbf{x}}\in conv(\mathcal{C})}{\mathrm{arg}\:\mathrm{min}}\left\langle \mathrm{\mathbf{x}},\boldsymbol{\lambda}\right\rangle .\label{eq:OriginalLP}
\end{equation}
$conv(\mathcal{C})$ is called the \emph{codeword polytope} \cite{UsingLP},
which is the convex hull of all possible codewords. The vertices of
the codeword polytope are the all codewords. The number of inequalities
needed to describe it grows exponentially in the code length. Therefore,
solving this linear programming problem is not practical for codes
with reasonable block length. To make this problem more feasible it
was suggested \cite{UsingLP} to replace $conv(\mathcal{C})$ by a
relaxed polytope $\mathcal{P}\triangleq\mathcal{P}(H)$, called the
\emph{fundamental polytope. }\vspace{-4mm}\emph{ }
\begin{eqnarray*}
\mathrm{\mathcal{P}\triangleq\overset{m}{\underset{\mathit{j}=1}{\bigcap}}\, conv}(\mathcal{C}_{\text{\ensuremath{j}}})\qquad\mathrm{with}\quad\mathcal{C}_{\text{\ensuremath{j}}} & \triangleq & \left\{ \mathrm{\mathbf{x}}\in\mathbb{F}_{2}^{n}\mid\mathrm{\mathbf{x}}\mathrm{\mathbf{h}}_{j}^{T}=0\right\} .
\end{eqnarray*}
Where $conv(\mathcal{C})\subseteq conv(\mathcal{C}_{\text{\ensuremath{j}}})$
for $\text{\ensuremath{j}}=1,...,m$ and hence $conv(\mathcal{C})\subseteq\mathcal{P}(H)\subset[0,1]^{n}$.
The number of inequalities that describe $\mathcal{P}(H)$ is typically
much smaller than those of $conv(\mathcal{C})$. The Linear Programming
Decoding (LPD) is then \vspace{-6mm}
\begin{equation}
\hat{\boldsymbol{\omega}}_{LPD}(\mathrm{\mathbf{y}})\triangleq\underset{\boldsymbol{\omega}\in\mathcal{P}}{\mathrm{arg\:}\mathrm{min}}\left\langle \boldsymbol{\omega},\boldsymbol{\lambda}\right\rangle .
\end{equation}
In the case of $conv(\mathcal{C})=\mathcal{P}(H)$ the relaxed LP
solution equals to that of ML. In the case of $conv(\mathcal{C})\subset\mathcal{P}(H)$
the relaxed LP problem represents a suboptimal decoder which has vertices
in $\mathcal{P}(H)$ which are not in $conv(\mathcal{C})$. The vertices
of $\mathcal{P}(H)$, denoted by $\mathcal{V}(\mathcal{P}(H))$, are
called LP pseudocodewords.
The \emph{fundamental cone} \cite{GraphCover} $\mathcal{K}(H)\triangleq\mathcal{K}$
is defined to be the conic hull of the fundamental polytope i.e. the
set that consists of all possible conic combinations of all the points
in $\mathcal{P}(H)$ and hence $\mathcal{P}(H)\subset\mathcal{K}(H)$.
The LP decoding error probability over the fundamental polytope is
equal to that over the fundamental cone \cite{RelaxationBounds}.
We let $\mathcal{\mathbb{R}}$ and $\mathbb{R_{\mathrm{+}}}$ be the
set of real numbers and the set of non-negative real numbers, respectively.
\medskip{}
\begin{defn}
\label{def:generators}( \cite{RelaxationBounds}, \cite{LiftingFund})
A set $\mathcal{G}(\mathcal{\mathcal{K}})\triangleq\{\mathrm{\mathbf{g}}_{1},\mathrm{\mathbf{g}}_{2},...,\mathrm{\mathbf{g}}_{M}\mid\mathbf{g}_{i}\in\mathbb{R}_{+}^{n},\; i=1,...,M\}$
of \emph{M} linearly independent vectors where $\mathcal{\mathcal{K}}=\left\{ \overset{M}{\underset{i=1}{\sum}}\alpha_{i}\mathrm{\mathbf{g}}_{i}\mid\alpha_{i}\in\mathbb{R}\right\} $
are called the \emph{generators} of the cone $\mathcal{\mathcal{K}}$.
\begin{comment}
how to find generator of code: relaxation...
\end{comment}
\qed
\end{defn}
\bigskip{}
It follows from Def. \ref{def:generators} that a vector $\mathrm{\mathbf{x}}$
is in $\mathcal{K}$ if and only if $\mathrm{\mathbf{x}}$ can be
written as a nonnegative linear combination of the generators, i.e.
$\mathrm{\mathbf{x}}=\overset{M}{\underset{i=1}{\sum}}\alpha_{i}\mathrm{\mathrm{\mathbf{g}}}_{i}$
where $\alpha_{i}\in\mathbb{R}$. Note that a set of generators is
not unique, and that the all-zeros codeword $\mathrm{\mathrm{\mathbf{x}}}_{0}\notin\mathcal{G}(\mathcal{\mathcal{K}}).$
We assume an AWGN channel, where each \emph{i}-th transmitted bit
perturbed by a white Gaussian noise $z_{i}$ with a zero mean and
noise power $\sigma^{2}\triangleq N_{0}/2$. The received signal is
$\mathbf{\mathrm{\mathbf{y}}}=\overline{\mathrm{\mathbf{x}}}+\mathrm{\mathbf{z}}$,
where $\mathrm{\mathbf{z}}$ designates an\emph{ n}-dimensional Gaussian
noise vector with independent components $z_{1},z_{2},...,z_{n}$.
We consider a BPSK modulation: the transmitted signal is $\overline{\mathrm{\mathbf{x}}}=\gamma\left(1-2\mathrm{\mathbf{x}}\right)$,
where $\gamma\triangleq\sqrt{RE_{b}}$ in which $E_{b}$ is the information
bit energy. The signal-to-noise ratio is defined to be $\mathrm{SNR}\triangleq E_{b}/N_{0}$.
Following from the above, the LLR vector is $\boldsymbol{\lambda}=4\frac{\sqrt{RE_{b}}}{N_{0}}\mathrm{\mathbf{y}}$
\cite{GraphCover}, and therefore, the LPD will be considered henceforth
\vspace{-10mm}
\begin{equation}
\hat{\boldsymbol{\omega}}_{LPD}=\underset{\boldsymbol{\omega}\in\mathcal{P}}{\mathrm{arg\:}\mathrm{min}}\left\langle \boldsymbol{\omega},\mathrm{\mathbf{y}}\right\rangle .
\end{equation}
\begin{defn}
(\cite{GraphCover}, \cite{PhDWiberg}, \cite{EffectiveWeights})
Let $\boldsymbol{\omega}\in\mathbb{R}_{+}^{n}.$ The AWGN channel
pseudo-weight $w_{p}^{AWGNC}(\boldsymbol{\omega})$ of $\boldsymbol{\omega}$
is given by \vspace{-10mm}
\end{defn}
\begin{equation}
w_{p}^{AWGNC}(\boldsymbol{\omega})\triangleq\frac{||\boldsymbol{\omega}||_{1}^{2}}{||\boldsymbol{\omega}||_{2}^{2}}\text{,}
\end{equation}
where $||\mathrm{\mathbf{x}}||_{1}\triangleq\sum_{i}|x_{i}|$ denote
the $\mathcal{L}_{1}$-norm of a vector x. If $\boldsymbol{\omega}=0$
we define $w_{p}^{AWGNC}(\boldsymbol{\omega})\triangleq0$, and in
the case of $\boldsymbol{\omega}\in\{0,1\}^{n}$ we have $w_{p}^{AWGNC}(\boldsymbol{\omega})$
= $w_{H}(\boldsymbol{\omega})$\emph{.}\qed
\bigskip{}
\noindent For an easier notation, as we discuss in this paper only
AWGN channel, we will use the shorter notation $w_{p}(\boldsymbol{\omega})$
instead of $w_{p}^{AWGNC}(\boldsymbol{\omega})$.
\medskip{}
Due to the symmetry property of the fundamental polytope the probability
that the LP decoder fails is independent of the codeword that was
transmitted \cite{UsingLP}. Therefore, we henceforth assume without
loss of generality when analyzing LPD error probability, that the
all-zeros codeword $\mathrm{\mathbf{x}}_{0}$ is transmitted.
The set of optimal solutions of a closed convex LP problem always
includes at least one vertex of the polytope. Therefore, the LPD error
probability is \vspace{-5mm}
\begin{equation}
P_{r}^{LPD}(error\mid\mathrm{\mathbf{x}}_{0})=P_{r}\left\{ \bigcup_{\boldsymbol{\omega}\in\mathcal{V}(\mathcal{\mathcal{P}}(H))\setminus\mathrm{\mathbf{x}}_{0}}\left\langle \boldsymbol{\omega},\mathrm{\mathbf{y}}\right\rangle \leq0\mid\mathrm{\mathbf{x}}_{0}\right\} .\label{eq:LP-errorP}
\end{equation}
A pseudocodeword $\mathrm{\mathbf{p}}\in\mathcal{V}(\mathcal{P})$
also belongs to the fundamental cone. Thus it can be written as a
non-negative linear combination of the generators, i.e. $\mathrm{\mathbf{p}}=\overset{M}{\underset{i=1}{\sum}}\alpha_{i}\mathrm{\mathrm{\mathbf{g}}}_{i}$
with $\alpha_{i}\geq0$. Therefore, if there is $\mathrm{\mathbf{p}}\in\mathcal{V}(\mathcal{P})$
such that $\left\langle \mathrm{\mathbf{p}},\mathrm{\mathbf{y}}\right\rangle =\overset{M}{\underset{i=1}{\sum}}\alpha_{i}\left\langle \mathrm{\mathrm{\mathbf{g}}}_{i},\mathrm{\mathbf{y}}\right\rangle <0$,
then there must be at least one generator $\mathrm{\mathbf{g}}_{i}\mathcal{\in G}(\mathcal{\mathcal{K}})$
such that $\left\langle \mathrm{\mathrm{\mathbf{g}}}_{i},\mathrm{\mathbf{y}}\right\rangle <0.$
Therefore, the union of the pseudocodewords' error events in \eqref{eq:LP-errorP}
can be replaced by the union of the generators' error events.
A vector $\boldsymbol{\omega}\in\mathbb{R}_{+}^{n}$ which is not
codeword can be located into the signal space in the same way as a
codeword, i.e $\mathrm{\boldsymbol{\overline{\omega}}}=\gamma\left(1-2\mathrm{\mathbf{\boldsymbol{\omega}}}\right)$.
The vector $\mathrm{\boldsymbol{\omega}}_{virt}\triangleq\tfrac{||\mathrm{\boldsymbol{\omega}}||_{1}}{||\mathrm{\boldsymbol{\omega}}||_{2}^{2}}\boldsymbol{\omega}$
was introduced by Vontobel and Koetter \cite{GraphCover}. They showed
that the decision hyperplane of $\boldsymbol{\omega}$ in the signal
space, is at the same Euclidean distance from $\mathrm{\overline{\mathbf{x}}}_{0}$
and from $\boldsymbol{\overline{\omega}}_{virt}$. Note that if $\boldsymbol{\omega}\in\mathcal{C}\subseteq\{0,1\}^{n}$,
then $\boldsymbol{\omega}_{virt}=\boldsymbol{\omega}$. From the above,
the LP error probability is then expressed in the signal space as
follows.\vspace{-4mm}
\begin{equation}
P_{r}^{LPD}(error\mid\mathrm{\mathbf{x}}_{0})=P_{r}\left\{ \bigcup_{\boldsymbol{\omega}\in\mathcal{G}(\mathcal{\mathcal{K}}(H))}||\overline{\boldsymbol{\omega}}_{virt}-\mathrm{\mathbf{y}}||_{2}\leq||\overline{\mathrm{\mathbf{x}}}_{0}-\mathrm{\mathbf{y}}||_{2}\mid\mathrm{\mathbf{x}}_{0}\right\} .\label{eq:LP error}
\end{equation}
Evaluating the LP error probability by simulating Eq. \eqref{eq:LP error}
is not practical, since it involves enormous number of generators.
However, it allows to make a simulation of the error probability contributed
by a subgroup of generators.
Let $E_{\mathrm{\mathbf{x}}_{0}\rightarrow\boldsymbol{\omega}}=\left\{ \,||\overline{\boldsymbol{\omega}}_{virt}-\mathrm{\mathbf{y}}||_{2}\leq||\overline{\mathrm{\mathbf{x}}}_{0}-\mathrm{\mathbf{y}}||_{2}\mid\mathrm{\mathbf{x}}_{0}\,\right\} $
denote the LP pairwise error event where the received vector $\mathrm{\mathbf{y}}$
is closer to $\overline{\boldsymbol{\omega}}_{virt}$ than to the
transmitted signal $\overline{\mathrm{\mathbf{x}}}_{0}$. Thus the
LP error probability \eqref{eq:LP error} can be written: \vspace{-7mm}
\begin{equation}
P_{r}^{LPD}(error\mid\mathrm{\mathbf{x}}_{0})=P_{r}\left\{ \bigcup_{\boldsymbol{\omega}\in\mathcal{G}(\mathcal{\mathcal{K}}(H))}E_{\mathrm{\mathbf{x}}_{0}\rightarrow\boldsymbol{\omega}}\right\} ,\label{eq:LP errorUn}
\end{equation}
and the LP union bound is \vspace{-8mm}
\begin{equation}
P_{r}^{LPD}(error\mid\mathrm{\mathbf{x}}_{0})\leq\sum_{\boldsymbol{\omega}\in\mathcal{G}(\mathcal{\mathcal{K}}(H))}P_{r}\{E_{\mathrm{\mathbf{x}}_{0}\rightarrow\boldsymbol{\omega}}\}.\label{eq:UB-PW}
\end{equation}
Let $r_{\boldsymbol{\omega}}\triangleq\tfrac{||\overline{\boldsymbol{\omega}}_{virt}-\overline{\mathrm{\mathbf{x}}}_{0}||_{2}}{2}=\gamma\sqrt{w_{p}(\boldsymbol{\omega})}$
denote the Euclidean distance from $\overline{\mathrm{\mathbf{x}}}_{0}$
or from $\overline{\boldsymbol{\omega}}_{virt}$ to the decision boundary
line. Thus the LP pairwise error probability \cite{GraphCover} \vspace{-7mm}
\begin{equation}
P_{r}(E_{\mathrm{\mathbf{x}}_{0}\rightarrow\boldsymbol{\omega}})=Q\left(\dfrac{r_{\boldsymbol{\omega}}}{\sigma}\right),\label{eq:LPairWise}
\end{equation}
and the LP-UB in Eq. \eqref{eq:UB-PW} can be written as follows \cite{DecisionRegions}
\cite{LP-UB2}. \vspace{-7mm}
\begin{equation}
P_{r}^{LPD}(error\mid\mathrm{\mathbf{x}}_{0})\leq\sum_{\boldsymbol{\omega}\in\mathcal{G}(\mathcal{\mathcal{K}}(H))}Q\left(\dfrac{r_{\boldsymbol{\omega}}}{\sigma}\right).\label{eq:LP-UB}
\end{equation}
\subsection{Undirected Graphs }
In this section, we give a brief overview of some terms from graph
theory. By a graph we will always mean an undirected graph without
loops and multiple edges. We let $|V|$ denote the size of a set $V$.
\begin{defn}
\label{def:UndirectedGraph} (\cite{DataStruct&Alg}) An \emph{undirected
graph} \emph{$G(V,\mathcal{E})$} consists of a set of nodes $V$
and a set of edges $\mathcal{E}$. An edge is an unordered pair of
nodes $(v_{i},v_{j})$. Associated with each edge $(v_{i},v_{j})\in\mathcal{E}$
is a cost $c(v_{i},v_{j})$.
\end{defn}
\vspace{-10mm}\qed
\begin{defn}
(\cite{DataStruct&Alg}) A \emph{spanning tree} of an undirected graph
\emph{$G(V,\mathcal{E}$)}, is a subgraph \emph{$T(V,\mathcal{E}'$)}
that is a tree and connects all the nodes in \emph{$V$}. It has $|V|$
nodes and $|\mathcal{E}'|=|V|-1$ edges, in which \emph{$\mathcal{E}'$}
is a subset of \emph{$\mathcal{E}$. }The cost of a spanning tree
T, denoted by $cost(T)$, is the sum of the costs of all the edges
in the tree. i.e. $cost(T)=\underset{(v_{i},v_{j})\in T}{\sum}c(v_{i},v_{j}).$\qed
\end{defn}
\begin{defn}
(\cite{DataStruct&Alg}) A spanning tree of a graph \emph{$G(V,\mathcal{E})$}
is called a\emph{ Minimum Spanning Tree }(MST), if its cost is less
than or equal to the cost of every other spanning tree \emph{$T(V,\mathcal{E}')$}
of \emph{$G(V,\mathcal{E})$.} \qed
\end{defn}
Two popular algorithms for finding an MST in undirected graph are
Prim's \cite{Prim} and Kruskal's \cite{Kruskal}. A simple implementation
of Prim's algorithm can shows $O(|V|^{2})$ running time, and both
can be implemented with complexity of $O(|\mathcal{E}|\, log|V|)$.
\section{Generator Density Characterization\label{Density} }
In this section, we explore the density of the fundamental cone generators
and we compare it to that of ML codewords. As a result, we will later
examine how the union bound is affected by that density. Let $0\leq\theta_{ij}\leq\pi$
denote the positive angle formed by the vectors $\boldsymbol{\omega}_{i}$
and $\boldsymbol{\omega}_{j}$, which is equal to the angle formed
by the vectors $\,\overrightarrow{\overline{\mathrm{\mathbf{x}}}_{0}\boldsymbol{\overline{\omega}}_{i,}}{}_{virt}$
and $\overrightarrow{\overline{\mathrm{\mathbf{x}}}_{0}\boldsymbol{\overline{\omega}}_{j,}}{}_{virt}$
in a BPSK signal space.
\bigskip{}
\begin{defn}
\label{Def:AngleGraph} Let $\boldsymbol{\omega}_{1},\boldsymbol{\omega}_{2},...,\boldsymbol{\omega}_{M}\in\mathbb{R}_{+}^{n}$
be a set of vectors. Consider each vector as a node of an undirected
graph $G(V,\mathcal{E})$, with an undirected edge joining each pair
of nodes $\boldsymbol{\omega}_{i}$ and $\boldsymbol{\omega}_{j}$,
denoted by $(\boldsymbol{\omega}_{i},\boldsymbol{\omega}_{j})$. An
edge $(\boldsymbol{\omega}_{i},\boldsymbol{\omega}_{j})\in\mathcal{E}$
has a cost that equal to the angle between the vectors related to
the adjacent nodes, i.e, $c(\boldsymbol{\omega}_{i},\boldsymbol{\omega}_{j})=\theta_{ij}$.
The graph $G(V,\mathcal{E})$\emph{ }will be called the\emph{ angle
graph}. Note that the angle graph is a \emph{complete} graph; it has
$|V|$ nodes and $|V|(|V|-1)/2$ edges. \qed
\end{defn}
\begin{defn}
\label{def:angle dist.}Let $T(V,\mathcal{E}')$ be an MST of the
angle graph $G(V,\mathcal{E})$ in Def. \ref{Def:AngleGraph}. The\emph{
MST angle distribution} is defined to be the cost distribution of
the all edges $(\boldsymbol{\omega}_{i},\boldsymbol{\omega}_{j})$
in the graph $T(V,\mathcal{E}')$. For easier notation, we will use
the shorter term\emph{ angle distribution} instead. \qed\end{defn}
\begin{example}
\label{Ex:GolayAngDis} Let $H_{G'}$ \cite{DecisionRegions} and
$H_{G''}$ \eqref{eq:HG''} be parity-check matrices for the extended
Golay{[}24,12,8{]} code. The former matrix was introduced by Halford
and Chugg \cite{Golay-H}, the latter is a systematic parity-check
matrix. Fig. \ref{fig:AngleDistribution} presents the angle distributions
of the first 759 minimal-weight generators of $H_{G'}$ and $H_{G''}$
(generators with equal pseudo-weight were ordered randomly.). For
a comparison, the angle distribution of the 759 minimal-weigh ML codewords
is presented as well. The average angle of $H_{G'}$, $H_{G''}$ generators
and of ML codewords are : $1.43^{\circ},$ $10.69^{\circ}$ and $60^{\circ}$,
respectively; and their Standard Deviations (STDs) are: $3.38^{\circ},$
$8.72^{\circ}$ and $0^{\circ}$, respectively. Note that $H_{G'}$
and $H_{G''}$ have two different \emph{generator matrices}, however,
both have the same angle distribution for their $759$ minimal-weight
CWs. It is clear from Fig. \ref{fig:AngleDistribution}, that $H_{G'}$
generators are much crowded than those of $H_{G''}$, and between
these three distributions the ML codewords are spread most widely
and evenly in the Euclidean space.
\begin{spacing}{0.9}
\setlength\arraycolsep{0.2em}
\begin{equation}
H_{G''}=\left(\begin{array}{cccccccccccccccccccccccc}
0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
1 & 1 & 1 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
1 & 1 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
1 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
1 & 1 & 1 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
1 & 1 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\
1 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\
1 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\
1 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\
1 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\
1 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\
1 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1
\end{array}\right)\;\label{eq:HG''}
\end{equation}
\end{spacing}
\end{example}
\begin{figure}[H]
\centering{}\includegraphics[scale=0.14]{pic66} {\footnotesize \caption{{\footnotesize \label{fig:AngleDistribution}}\fontsize{9}{10}\selectfont
Angle distributions for the extended Golay{[}24, 12, 8{]} code of
the first 759 minimal-weight generators of the parity-check matrices
$H_{G'}$ and $H_{G''}$, compared to the angle distribution of the
759 minimal-weight ML codewords.}
}
\end{figure}
\qed
\begin{example}
\smallskip{}
The error probability contributed by two vectors depends on the angle
between them. Let $\boldsymbol{\omega}_{i},\boldsymbol{\omega}_{j}\in\mathbb{R}_{+}^{n}$
be vectors with an equal pseudo-weight, and let $\xi_{1}$ and $\xi_{2}$
be the two independent Gaussian random variables obtained by projecting
the noise vector $\mathbf{\mathrm{\mathbf{z}}}$ onto the plan determined
by the vectors $\overrightarrow{\overline{\mathrm{\mathbf{x}}}_{0}\boldsymbol{\overline{\omega}}_{i,}}{}_{virt}$
and $\overrightarrow{\overline{\mathrm{\mathbf{x}}}_{0}\boldsymbol{\overline{\omega}}_{j,}}{}_{virt}$.
We refer to the probability $P_{\text{\ensuremath{r}}}\left\{ E_{\mathrm{\mathbf{x}}_{0}\rightarrow\boldsymbol{\omega}_{i}}\bigcup E_{\mathrm{\mathbf{x}}_{0}\rightarrow\boldsymbol{\omega}_{j}}\right\} $
as the triplet-wise error probability, that is $\boldsymbol{\omega}_{i}$
or $\boldsymbol{\omega}_{j}$ was decoded when the all-zeros codeword
was transmitted. The triplet-wise error probability depends on the
angle $\theta_{ij}$ and it can be obtained by integrating a two dimensional
Gaussian distribution over the darkened regions $\mathcal{R}_{1}$
and $\mathcal{R}_{2}$ in Fig. \ref{fig:RealcorrectRegion} \cite{NewTechniques}.
Without loss of generality, we assume that $\boldsymbol{\omega}_{j}$
is placed on $\xi_{1}$ axis. $r_{\boldsymbol{\omega}_{i}}$ and $r_{\boldsymbol{\omega}_{j}}$
denote the Euclidean distances from the decision boundaries lines
of $\boldsymbol{\omega}_{i}$ and $\boldsymbol{\omega}_{j}$, respectively,
to the all-zeros codeword. In the case of vectors of equal pseudo-weight,
$r_{\boldsymbol{\omega}_{i}}=r_{\boldsymbol{\omega}_{j}}$. The decision
region boundary lines of $\boldsymbol{\omega}_{i}$ and $\boldsymbol{\omega}_{j}$
are $\xi_{2}=-a\xi_{1}+b$ and $\xi_{1}=r_{\boldsymbol{\omega}_{j}}$,
respectively. The $\boldsymbol{\omega}_{i}$ boundary line crosses
$\xi_{2}$ axis at point $b=r_{\boldsymbol{\omega}_{i}}/sin\theta_{ij}$
and its slope is $a=\tan(90-\theta_{ij})$. The intersection between
the two boundary lines occurs at point $(\xi'_{1},\xi'_{2})=(r_{\boldsymbol{\omega}_{j}},\:-ar_{\boldsymbol{\omega}_{j}}+b)$.
There are various numerical integration ways \cite{Numerical} to
evaluate the triplet-wise error probability. Another possibility,
is to approximate it by sum of $Q$-functions as follows.
\end{example}
\smallskip{}
$P_{\text{\ensuremath{r}}}\left\{ E_{\mathrm{\mathbf{x}}_{0}\rightarrow\omega_{i}}\bigcup E_{\mathrm{\mathbf{x}}_{0}\rightarrow\omega_{j}}\right\} =P_{r}\{\mathcal{R}_{1}\}+P_{r}\{\mathcal{R}_{2}\}\approx Q\left(\dfrac{r_{\boldsymbol{\omega}_{i}}}{\sigma}\right)+$
\vspace{-6mm}
\begin{equation}
\overset{\left\lfloor \frac{\xi_{1,max}}{\bigtriangleup\xi_{1}}\right\rfloor }{\underset{k=0}{\sum}}\left[1-Q\left(\frac{-a(\xi'_{1}+k\bigtriangleup\xi_{1})+b}{\sigma}\right)\right]\left[Q\left(\frac{\xi'_{1}+k\bigtriangleup\xi_{1}}{\sigma}\right)-Q\left(\frac{\xi'_{1}+(k+1)\bigtriangleup\xi_{1}}{\sigma}\right)\right].\label{eq:RealCalc}
\end{equation}
$P_{r}\{\mathcal{R}_{1}\}$ is equal to an LP pairwise error probability
\eqref{eq:LPairWise}. $P_{r}\{\mathcal{R}_{2}\}$ is calculated as
follows. The region $\mathcal{R}_{2}$ is divided into rectangles
of a width $\bigtriangleup\xi_{1}$ which are parallel to the $\xi_{2}$
axis, as shown in Fig. \ref{fig:RealcorrectRegion}. Each rectangle
starts from a point on the decision boundary line of $\boldsymbol{\omega}_{i}$
and goes to infinity in the opposite direction of $\xi_{2}$ axis.
The multiplication inside the sum of Eq. \eqref{eq:RealCalc} is the
probability that the noise components $\xi_{1}$ and $\xi_{2}$ are
within the \emph{k}-th rectangle. Since a two dimensional Gaussian
distribution converges to zero as $\xi_{1}$ goes to infinity, it
will be sufficient to sum from $k=0$ to a large $k$ such as $\left\lfloor \frac{\xi_{1,max}}{\bigtriangleup\xi_{1}}\right\rfloor $,
where all the rectangles are located on the left side of the line
$\xi_{2}=\xi_{1,max}$.
\bigskip{}
\begin{figure}[H]
\centering{}\includegraphics[scale=0.1]{pic69} {\scriptsize \caption{{\scriptsize \label{fig:RealcorrectRegion}}{\footnotesize{} }\fontsize{9}{10}\selectfont
The LP triplet-wise error region in the signal space.}
}
\end{figure}
\begin{example}
\label{exa:RealErrorBCH} Consider the BCH{[}63,57,3{]} code. The
fundamental cone of the systematic parity-check matrix created by
the \emph{generator polynomial} $x^{6}+x+1$ has $11,551$ minimal-weight
generators of pseudo-weight three. The angles between them varied
from $5.85^{\circ}$ to $90^{\circ}$. The triplet-wise error probability
of its two minimal-weight generators depends on $\theta_{ij}$ is
presented in Fig. \ref{fig:UpBVsReal}. It was calculated by Eq. \eqref{eq:RealCalc}
for $0$ and $8$ dB SNR in different angles. The triplet-wise union
bound which is $2Q\left(\tfrac{r_{\boldsymbol{\omega}}}{\sigma}\right)$
is presented as well. $\xi_{1,max}$ and $\bigtriangleup\xi_{1}$
was chosen to be 2000 and 1/2000, respectively. From Fig. \ref{fig:UpBVsReal}
one can observe that the lower the SNR and the smaller the angle are,
the worse is the UB. The figure also presents a triplet-wise error
probability upper bound which is tighter than the UB and it will be
introduced in Sec. \ref{ImprovedUB}
\begin{figure}[H]
\begin{raggedright}
\begin{minipage}[c][1\totalheight][t]{0.45\textwidth}%
\selectlanguage{british}%
\begin{flushleft}
\subfloat[\selectlanguage{english}%
\fontsize{9}{10}\selectfont SNR = 0 dB\selectlanguage{british}%
]{\selectlanguage{english}%
\includegraphics[scale=0.1]{pic49}
\selectlanguage{british}%
}
\par\end{flushleft}\selectlanguage{english}%
\end{minipage}\hfill{}%
\begin{minipage}[c][1\totalheight][t]{0.45\textwidth}%
\medskip{}
\selectlanguage{british}%
\begin{flushleft}
\subfloat[\selectlanguage{english}%
\fontsize{9}{10}\selectfont SNR = 8 dB\selectlanguage{british}%
]{\selectlanguage{english}%
\begin{raggedright}
\includegraphics[scale=0.1]{pic46}
\par\end{raggedright}
\selectlanguage{british}%
}
\par\end{flushleft}\selectlanguage{english}%
\end{minipage}\hfill{}
\par\end{raggedright}
\caption{\label{fig:UpBVsReal}\fontsize{9}{10}\selectfont Comparison between
the triplet-wise error probability, its union bound and the upper
bound in different angles of two minimal-weight generators of BCH{[}63,57,3{]},
when the all-zeros word was transmitted. }
\end{figure}
\qed
\end{example}
\section{The Problem of Locating Dominant Error Events of LPD \label{DominantGroup} }
Consider a ML decoding of a binary-linear code BPSK-modulated over
an AWGN channel. The decoder performance can be evaluated by considering
the contributions of the most dominant error events to the probability
of error. That dominant error events, especially in the higher SNR
region, are the minimal weight codewords.
In this section, we will examine whether the minimal-weight generators
of LP decoding have such a property as well. We let $w_{H}(\mathrm{\mathbf{x}})$\emph{
}denote\emph{ }the\emph{ Hamming weight} of $\mathrm{\mathbf{x}}$,
which is the number of non-zero positions of $\mathrm{\mathbf{x}}$.
Let $w_{H}^{min}(\mathcal{C})$ denote the minimum Hamming weight
of a linear code $\mathcal{C}$, and let $w_{p}^{min}(H)$ denote
the minimum AWGN channel pseudo-weight of a linear code defined by
the parity-check matrix \emph{H}. We will use the shorter notations
$w_{H}^{min}$ and $w_{p}^{min}$ in case where the discussed code
and matrix are mentioned explicitly. We let $\mathrm{\mathcal{K}_{sub}\mathcal{\subset K}}$
denote a sub-cone of the fundamental cone which created by a chosen
subgroup of generators. The LPD$(\mathrm{\mathcal{K}_{sub}})$ Frame
Error Rate (FER) can be obtained by simulating Eq. \eqref{eq:LP error}.
In the next example, we will study the error probability contributed
by a subgroup of codewords and generators for the extended Golay{[}24,12,8{]}
code.
\begin{example}
The extended Golay{[}24,12,8{]} code has a total $4\text{,}096$ codewords
of which 759 have minimal Hamming weight of $w_{H}^{min}=8$. The
fundamental cone of the parity check-matrix $H_{G'}$ has a total
of $231,146,333$ generators of which two have minimal-weight of $w_{p}^{min}=3.6$
\cite{DecisionRegions}. Simulating the error probability by Eq. \eqref{eq:ML error}
shows that the minimal-weight CWs describe well the MLD performance
at the whole range of SNR, which is not the case for the first 759
minimal-weight generators for LPD. For instance, consider the error
rate of $10^{\lyxmathsym{\textminus}2}$, it was found that the difference
between LPD$(\mathrm{\mathcal{K}_{sub}})$ and LPD$(\mathrm{\mathcal{K}})$
is about 2.5 dB. The angle distributions which were presented in Fig.
\ref{fig:AngleDistribution} support this result: the average angle
of that group of generators is as small as $1.43^{\circ}$, and the
average angle of the ML minimal-weight CWs is $60^{\circ}$.\qed
\end{example}
There are number of reasons why the minimal-weight generators are
often not a dominant subgroup of LPD: (a) There is no guarantee for
significant number of generators with minimal pseudo-weight. The fundamental
cone of $H_{G'}$ for example, has only two. (b) A subgroup of generators
can be very crowded, which significantly reduces their contribution
to the error probability. (c) Unlike MLD which has distinct subgroup
of minimal-weight codewords, LPD often has a continuous-like weight
distribution. For example, the BCH{[}31,21,5{]} code of parity-check
matrix $H_{BCH_{[31,21]}}$ \eqref{eq:HBCH31-21 matrix} has 627,052,479
generators. The pseudo-weight distribution of these generators is
presented in Fig. \ref{fig:BCH[31,21] weight}. Its smooth distribution
makes it difficult to locate a minimal-weight dominant subgroup.
In LPD, a potential subgroup to be a dominant is taking all generators
of weight $w_{p}\leq w_{H}^{min}$. This group is not empty since
$w_{p}^{min}\leq w_{H}^{min}$ \cite{PCWs}, however, it may contains
enormous number of generators. For example, Golay{[}24,12,8{]} has
only $759$ minimal-weight CWs of $w_{H}^{min}=8$, but the fundamental
cone of parity-check matrix $H_{G''}$ has $143,757,418$ generators
of weight $w_{p}\leq w_{H}^{min}=8$.
\bigskip{}
\begin{spacing}{1}
\setlength\arraycolsep{0.1em}
\begin{equation}
H_{BCH_{[31,21]}}=\left(\begin{array}{ccccccccccccccccccccccccccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0\\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0\\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1\\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0\\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 1\\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1\\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 1\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 1
\end{array}\right)\qquad\;\label{eq:HBCH31-21 matrix}
\end{equation}
\end{spacing}\vspace{2mm}
\begin{figure}[H]
\begin{centering}
\includegraphics[scale=0.15]{pic33}
\par\end{centering}
\caption{{\scriptsize \label{fig:BCH[31,21] weight}}\fontsize{9}{10}\selectfont
A complete generators' pseudo-weight distribution for the BCH{[}31,21,5{]}
code of $H_{BCH_{[31,21]}}$ with 627,052,479 generators.}
\end{figure}
\section{Improved LP Union Bound\label{ImprovedUB} }
In this section, we propose an improved union bound for LP decoding
of a binary linear code transmitted over a binary-input AWGN channel.
This bound is based on the second-order of Bonferroni-type inequality
in probability theory \cite{Bonferroni}, also referred to as Hunter
bound \cite{HunterBound}. For any set of events $E_{1},E_{2},...,E_{M}$
and their \emph{complementary }events, denoted by $E_{1}^{c},E_{2}^{c},...,E_{M}^{c}$,
\begin{equation}
P_{r}\left(\overset{M}{\underset{i=1}{\bigcup}}E_{i}\right)=\overset{M}{\underset{i=1}{\sum}}P_{r}\left(E_{i}\bigcap\left[\,\overset{i-1}{\underset{j=1}{\bigcup}}E_{j}^{c}\right]\right).\label{eq:unionevent}
\end{equation}
\vspace{-1mm} \smallskip{}
Let denote the $M!$ possible permutations of the indices of the error
events $E_{1},E_{2},...,E_{M}$ by $\Pi$(1,2,...,\emph{M}) = $\{\pi_{1},\pi_{2}$,...,$\pi_{M}\}$.
For a given $\Pi$, let $\Lambda=\{\hat{\pi}_{2},\hat{\pi}_{3},...,\hat{\pi}_{M}\}$
denote the $(M^{2}-M)/2$ possible sets of indices in which $\hat{\pi}_{i}\in\{\pi_{1},\pi_{2},...,\pi_{i-1}\}$
for $i=2,3,...,M$. Hunter \cite{HunterBound} presented the second-order
bound of Eq. \eqref{eq:unionevent} as follows.\vspace{-3mm}
\begin{equation}
P_{r}\left(\overset{M}{\underset{i=1}{\bigcup}}E_{i}\right)\leq\overset{M}{\underset{i=1}{\sum}}P_{r}(E_{\pi_{i}})-\overset{M}{\underset{i=2}{\sum}}P_{r}(E_{\pi_{i}}\cap E_{\hat{\pi}_{i}}).\label{eq:hunter3}
\end{equation}
\vspace{-4mm}
Minimization of the Right-Hand Side (RHS) of Eq. \eqref{eq:hunter3}
is required to achieve the tightest second-order bound. Using the
sets of the indices $\Lambda$ and $\Pi$, the minimization problem
can be written as follows \cite{NewTSB} \cite{HunterBound}.\vspace{4mm}
\noindent
\begin{equation}
P_{r}\left(\overset{M}{\underset{i=1}{\bigcup}}E_{i}\right)\leq\overset{M}{\underset{i=1}{\sum}}P_{r}(E_{i})+\underset{\Pi,\Lambda}{min}\left\{ -\overset{M}{\underset{i=2}{\sum}}P_{r}(E_{\pi_{i}}\cap E_{\hat{\pi}_{i}})\right\} .\label{eq:hunter4}
\end{equation}
\noindent The first sum goes through over all the indices 1 to \emph{M}
of the error events, thus $E_{\pi_{i}}$ could be changed to $E_{i}$.
Consider each of the random events $E_{i}$ as a node of an undirected
graph $G$ and the intersection $(E_{i}\cap E_{j})$ as an undirected
edge joining the nodes $E_{i}$ and $E_{j}$, denoted by $(i,j)$,
with a cost $c(i,j)=P_{r}(E_{i}\cap E_{j})$. Hunter \cite{HunterBound}
showed that a set of $(M-1)$ intersections may be used in the second
term of Eq. \eqref{eq:hunter4} if and only if it forms a spanning
tree of the nodes $\left\{ E_{i}\right\} _{i=1}^{M}$. Thus the minimization
problem of Eq. \eqref{eq:hunter4} can be written equivalently \cite{HunterBound},
\cite{NewTSB}, \vspace{-2mm}
\begin{equation}
P_{r}\left(\overset{M}{\underset{i=1}{\bigcup}}E_{i}\right)\leq\overset{M}{\underset{i=1}{\sum}}P_{r}(E_{i})+\underset{\tau}{min}\left\{ -\overset{}{\underset{(i,j)\in\tau}{\sum}}P_{r}(E_{i}\cap E_{j})\right\} .\label{eq:hunter5}
\end{equation}
\noindent Where $\tau$ is a spanning tree of the graph $G$. The
problem is to find a graph $\tau$ which minimizes Eq. \eqref{eq:hunter5}
over all possible spanning trees. The solution for that is known as
the solution of the minimum spanning tree problem and has been proposed
by Prim \cite{Prim} and Kruskal \cite{Kruskal}.
\bigskip{}
Consider the event $E_{i}$ as the pairwise error event $E_{\mathrm{\mathbf{x}}_{0}\rightarrow\boldsymbol{\omega}_{i}}$.
In order to upper bound the LP decoding error probability in Eq. \eqref{eq:LP errorUn}
by the second-order upper bound \eqref{eq:hunter5}, the probability
$P_{\text{\ensuremath{r}}}\left\{ E_{\mathrm{\mathbf{x}}_{0}\rightarrow\boldsymbol{\omega}_{i}}\bigcap E_{\mathrm{\mathbf{x}}_{0}\rightarrow\boldsymbol{\omega}_{j}}\right\} $
is required, or instead, its lower bound. The probability of intersection
of two events can be expressed using the \emph{inclusion-exclusion}
principle in probability theory, \vspace{-7mm}
\begin{equation}
P_{\text{\ensuremath{r}}}\left\{ E_{\mathrm{\mathbf{x}}_{0}\rightarrow\boldsymbol{\omega}_{i}}\bigcap E_{\mathrm{\mathbf{x}}_{0}\rightarrow\boldsymbol{\omega}_{j}}\right\} =P_{\text{\ensuremath{r}}}\left\{ E_{\mathrm{\mathbf{x}}_{0}\rightarrow\boldsymbol{\omega}_{i}}\right\} +P_{\text{\ensuremath{r}}}\left\{ E_{\mathrm{\mathbf{x}}_{0}\rightarrow\boldsymbol{\omega}_{j}}\right\} -P_{\text{\ensuremath{r}}}\left\{ E_{\mathrm{\mathbf{x}_{0}}\rightarrow\boldsymbol{\omega}_{i}}\bigcup E_{\mathrm{\mathbf{x}}_{0}\rightarrow\boldsymbol{\omega}_{j}}\right\} .\label{eq:Intersect}
\end{equation}
The first and the second terms in the RHS of Eq. \eqref{eq:Intersect}
are the LP pairwise error probability \eqref{eq:LPairWise}, the third
term can be upper bounded by the following theorem.
\bigskip{}
\begin{thm}
\label{thm:UB TWE} Let $\boldsymbol{\omega}_{i},\boldsymbol{\omega}_{j}\in\mathbb{R}_{+}^{n}$
be vectors of a pseudo-weight $w_{p}(\boldsymbol{\omega}_{i})\neq w_{p}(\boldsymbol{\omega}_{j})$.
The LP triplet-wise error probability \vspace{-8mm}
\end{thm}
\begin{equation}
P_{\text{\ensuremath{r}}}\left\{ E_{\mathrm{\mathbf{x}_{0}}\rightarrow\boldsymbol{\omega}_{i}}\bigcup E_{\mathrm{\mathbf{x}}_{0}\rightarrow\boldsymbol{\omega}_{j}}\right\} \leq min\left\{ \begin{array}{c}
Q\left(\dfrac{min(r_{\boldsymbol{\omega}_{i}},r_{\boldsymbol{\omega}_{j}})}{\sigma}\right)+\dfrac{\theta_{ij}}{2\pi}e^{-\frac{max(r_{\boldsymbol{\omega}_{i}}^{2},r_{\boldsymbol{\omega}_{j}}^{2})}{2\sigma^{2}}},\\
Q\left(\dfrac{r_{\boldsymbol{\omega}_{i}}}{\sigma}\right)+Q\left(\dfrac{r_{\boldsymbol{\omega}_{j}}}{\sigma}\right)-Q\left(\dfrac{r_{\boldsymbol{\omega}_{i}}}{\sigma}\right)Q\left(\dfrac{r_{\boldsymbol{\omega}_{j}}}{\sigma}\right)
\end{array}\right\} .\label{eq:TriUpperBd}
\end{equation}
\bigskip{}
\begin{IEEEproof}
Let $\tilde{\xi}\triangleq\xi_{1}^{2}+\xi_{2}^{2}$ be a random variable
with Chi-square distribution \cite{Chai} with two degrees of freedom,
i.e. \vspace{-8mm}
\begin{equation}
f(\tilde{\xi})=\frac{1}{2\sigma^{2}}e^{-\frac{\tilde{\xi}}{2\sigma^{2}}}U(\tilde{\xi}),\label{eq:ChiDis}
\end{equation}
in which $U(\cdot)$ is the unit step function. Without loss of generality
we assume that $w_{p}(\boldsymbol{\omega}_{i})<w_{p}(\boldsymbol{\omega}_{j})$.
With the help of Fig. \ref{fig:Error-probability-area diff} the triplet-wise
error probability,\vspace{-7mm}
\begin{eqnarray}
P_{\text{\ensuremath{r}}}\left\{ E_{\mathrm{\mathbf{x}_{0}}\rightarrow\boldsymbol{\omega}_{i}}\bigcup E_{\mathrm{\mathbf{x}}_{0}\rightarrow\boldsymbol{\omega}_{j}}\right\} & \leq & P_{r}\left(\overset{4}{\underset{i=1}{\bigcup}}\mathcal{R}_{i}\right)\leq\overset{4}{\underset{i=1}{\sum}}P_{r}\{\mathcal{R}_{i}\}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\\
& = & \underset{P_{r}(\mathcal{R}_{1})+P_{r}(\mathcal{R}_{2})}{\underbrace{Q\left(\dfrac{r_{\boldsymbol{\omega}_{j}}}{\sigma}\right)}}+\underset{P_{r}(\mathcal{R}_{3})}{\underbrace{\frac{\theta_{ij}}{2\pi}P_{r}\left(\tilde{\xi}>r_{\boldsymbol{\omega}_{j}}^{2}\right)}}+\underset{P_{r}(\mathcal{R}_{4})}{\underbrace{Q\left(\dfrac{r_{\boldsymbol{\omega}_{i}}}{\sigma}\right)-Q\left(\dfrac{r_{\boldsymbol{\omega}_{j}}}{\sigma}\right)}}\label{eq:UsingChi}\\
& = & Q\left(\dfrac{r_{\boldsymbol{\omega}_{i}}}{\sigma}\right)+\dfrac{\theta_{ij}}{2\pi}e^{-\frac{r_{\boldsymbol{\omega}_{j}}^{2}}{2\sigma^{2}}}.
\end{eqnarray}
\noindent \medskip{}
From the noise symmetry, each of the probabilities $P_{r}(\mathcal{R}_{1})$
or $P_{r}(\mathcal{R}_{2})$ equal to $\tfrac{1}{2}Q\left(\tfrac{r_{\boldsymbol{\omega}_{j}}}{\sigma}\right)$.
$P_{r}(\mathcal{R}_{3})$ is the probability that of $\xi_{1}^{2}+\xi_{2}^{2}$
lies in the region outside a circle of a radios $r_{\omega_{j}}$
created by the central angle $\theta_{ij}$. $P_{r}\left(\tilde{\xi}>r_{\boldsymbol{\omega}_{j}}^{2}\right)$
was calculated in Eq. \eqref{eq:UsingChi} by integrating the Chi-square
distribution \eqref{eq:ChiDis} from $r_{\boldsymbol{\omega}_{j}}^{2}$
to $\infty$. Thus for two vectors of pseudo-weight $w_{p}(\boldsymbol{\omega}_{i})\neq w_{p}(\boldsymbol{\omega}_{j})$
\vspace{-4mm}
\begin{equation}
P_{\text{\ensuremath{r}}}\left\{ E_{\mathrm{\mathbf{x}_{0}}\rightarrow\boldsymbol{\omega}_{i}}\bigcup E_{\mathrm{\mathbf{x}}_{0}\rightarrow\boldsymbol{\omega}_{j}}\right\} \leq Q\left(\dfrac{min(r_{\boldsymbol{\omega}_{i}},r_{\boldsymbol{\omega}_{j}})}{\sigma}\right)+\dfrac{\theta_{ij}}{2\pi}e^{-\frac{max(r_{\boldsymbol{\omega}_{i}}^{2},r_{\boldsymbol{\omega}_{j}}^{2})}{2\sigma^{2}}}.\label{eq:BoundUniondiffEvent}
\end{equation}
\begin{flushleft}
\smallskip{}
The triplet-wise error probability can also be bounded using the
inclusion\textendash{}exclusion principle as follows. \vspace{-6mm}
\par\end{flushleft}
\begin{spacing}{1}
\begin{eqnarray}
P_{\text{\ensuremath{r}}}\left\{ E_{\mathrm{\mathbf{x}_{0}}\rightarrow\boldsymbol{\omega}_{i}}\bigcup E_{\mathrm{\mathbf{x}}_{0}\rightarrow\boldsymbol{\omega}_{j}}\right\} & = & P_{\text{\ensuremath{r}}}\left\{ E_{\mathrm{\mathbf{x}}_{0}\rightarrow\boldsymbol{\omega}_{i}}\right\} +P_{\text{\ensuremath{r}}}\left\{ E_{\mathrm{\mathbf{x}}_{0}\rightarrow\boldsymbol{\omega}_{j}}\right\} -P_{\text{\ensuremath{r}}}\left\{ E_{\mathrm{\mathbf{x}}_{0}\rightarrow\boldsymbol{\omega}_{i}}\bigcap E_{\mathrm{\mathbf{x}}_{0}\rightarrow\boldsymbol{\omega}_{j}}\right\} \label{eq:exl-incl}\\
\nonumber \\
& \leq & Q\left(\dfrac{r_{\boldsymbol{\omega}_{i}}}{\sigma}\right)+Q\left(\dfrac{r_{\boldsymbol{\omega}_{j}}}{\sigma}\right)-Q\left(\dfrac{r_{\boldsymbol{\omega}_{i}}}{\sigma}\right)Q\left(\dfrac{r_{\boldsymbol{\omega}_{j}}}{\sigma}\right).\label{eq:90d bound-d}
\end{eqnarray}
\end{spacing}
\vspace{5mm}
\noindent The transition from Eq. \eqref{eq:exl-incl} to Eq. \eqref{eq:90d bound-d}
was done by lower bounding $P_{\text{\ensuremath{r}}}\left\{ E_{\mathrm{\mathbf{x}}_{0}\rightarrow\boldsymbol{\omega}_{i}}\bigcap E_{\mathrm{\mathbf{x}}_{0}\rightarrow\boldsymbol{\omega}_{j}}\right\} $
at its lowest value $Q\left(\tfrac{r_{\boldsymbol{\omega}_{i_{\,}}}}{\sigma}\right)Q\left(\tfrac{r_{\boldsymbol{\omega}_{j}}}{\sigma}\right)$
accepted in $\theta_{ij}=90^{0}$. Finally, selecting the minimum
between Eq. \eqref{eq:BoundUniondiffEvent} and Eq. \eqref{eq:90d bound-d}
completes the proof.
\end{IEEEproof}
\begin{figure}[H]
\centering{}\includegraphics[scale=0.1]{pic58} {\footnotesize \caption{\label{fig:Error-probability-area diff}\fontsize{9}{10}\selectfont
The region in the signal space used to bound the LP triplet-wise error
probability ($w_{p}(\boldsymbol{\omega}_{i})\neq w_{p}(\boldsymbol{\omega}_{j})).$}
}
\end{figure}
\begin{example}
We continue Ex. \ref{exa:RealErrorBCH}. The triplet-wise error probability
upper bound of Theorem \ref{thm:UB TWE} was calculated for two minimal-weight
generators of the BCH{[}63,57,3{]} code. It is presented in Fig. \ref{fig:UpBVsReal}
together with the previous results of Ex. \ref{exa:RealErrorBCH}.
We can see that the smaller the angle and lower the SNR, the more
improvement the triplet-wise error upper bound has over the union
bound. Note that because $\tfrac{r_{\boldsymbol{\omega}}}{\sigma}\propto\sqrt{\mathrm{SNR\cdot}w_{p}(\boldsymbol{\omega})}$,
changing the pseudo-weight of the generators will have the same effect
as changing the SNR. Thus this bound is expected to have more improvement
on low pseudo-weight generators. \qed
\end{example}
\noindent In the next theorem, we propose an improved UB for the LP
decoding.
\begin{thm}
\label{thm:ILP-UB} Let $\mathcal{G}(\mathcal{K}(H))$ be a set of
cone generators of a parity-check matrix H. For each \textup{$\boldsymbol{\omega}_{i}\in\mathcal{G}$}
the pairwise error event $E_{\mathrm{\mathbf{x}}_{0}\rightarrow\boldsymbol{\omega}_{i}}$
is considered as a node of a complete graph $G(V,\mathcal{E})$. Let
$(\boldsymbol{\omega}_{i},\boldsymbol{\omega}_{j})$ denote an undirected
edge joining the nodes related to the events $E_{\mathrm{\mathbf{x}}_{0}\rightarrow\boldsymbol{\omega}_{i}}$
and $E_{\mathrm{\mathbf{x}}_{0}\rightarrow\boldsymbol{\omega}_{j}}$.
$\tau(V,\mathcal{E}')$ is denoted for a spanning tree of $G(V,\mathcal{E})$.
The LP decoding error probability can be upper-bounded by
\end{thm}
\setlength\arraycolsep{0.05em} \vspace{-12mm}
\begin{eqnarray}
P_{r}^{LPD}(error\mid\mathrm{\mathbf{x}}_{0}) & \leq & \overset{}{\underset{\boldsymbol{\omega}\in\mathcal{G}(\mathcal{\mathcal{K}}(H))}{\sum}}Q\left(\dfrac{r_{\boldsymbol{\omega}}}{\sigma}\right)\nonumber \\
& + & \underset{\tau}{min}\left\{ \overset{}{\underset{(\boldsymbol{\omega}_{i},\boldsymbol{\omega}_{j})\in\tau}{\sum}}min\left\{ \begin{array}{c}
-Q\left(\dfrac{max(r_{\boldsymbol{\omega}_{i}},r_{\boldsymbol{\omega}_{j}})}{\sigma}\right)+\dfrac{\theta_{ij}}{2\pi}e^{-\frac{max(r_{\boldsymbol{\omega}_{i}}^{2},r_{\boldsymbol{\omega}_{j}}^{2})}{2\sigma^{2}}},\\
-Q\left(\dfrac{r_{\boldsymbol{\omega}_{i}}}{\sigma}\right)Q\left(\dfrac{r_{\boldsymbol{\omega}_{j}}}{\sigma}\right)
\end{array}\right\} \right\} \:\label{eq:ILP-UB}
\end{eqnarray}
\\
We call this bound the \emph{Improved LP Union Bound} (ILP-UB). The
first term is the LP union bound itself \eqref{eq:LP-UB}, the second
term is a second-order correction.
\begin{IEEEproof}
To prove this, we will apply Hunter bound for the LP error probability.
First, we find a lower bound for $P_{\text{\ensuremath{r}}}\left\{ E_{\mathrm{\mathbf{x}}_{0}\rightarrow\boldsymbol{\omega}_{i}}\bigcap E_{\mathrm{\mathbf{x}}_{0}\rightarrow\boldsymbol{\omega}_{j}}\right\} $:
by substituting the upper bound of $P_{\text{\ensuremath{r}}}\left\{ E_{\mathrm{\mathbf{x}_{0}}\rightarrow\boldsymbol{\omega}_{i}}\bigcup E_{\mathrm{\mathbf{x}}_{0}\rightarrow\boldsymbol{\omega}_{j}}\right\} $
\eqref{eq:TriUpperBd} into the inclusion\textendash{}exclusion principal
\eqref{eq:Intersect}, we will have
\begin{flushleft}
\smallskip{}
$P_{\text{\ensuremath{r}}}\left\{ E_{\mathrm{\mathbf{x}}_{0}\rightarrow\boldsymbol{\omega}_{i}}\bigcap E_{\mathrm{\mathbf{x}}_{0}\rightarrow\boldsymbol{\omega}_{j}}\right\} \geq$\vspace{-10mm}
\par\end{flushleft}
\begin{eqnarray}
& \geq & Q\left(\dfrac{r_{\boldsymbol{\omega}_{i}}}{\sigma}\right)+Q\left(\dfrac{r_{\boldsymbol{\omega}_{j}}}{\sigma}\right)-min\left\{ \begin{array}{c}
Q\left(\dfrac{min(r_{\boldsymbol{\omega}_{i}},r_{\boldsymbol{\omega}_{j}})}{\sigma}\right)+\dfrac{\theta_{ij}}{2\pi}e^{-\frac{max(r_{\boldsymbol{\omega}_{i}}^{2},r_{\boldsymbol{\omega}_{j}}^{2})}{2\sigma^{2}}},\\
Q\left(\dfrac{r_{\boldsymbol{\omega}_{i}}}{\sigma}\right)+Q\left(\dfrac{r_{\boldsymbol{\omega}_{j}}}{\sigma}\right)-Q\left(\dfrac{r_{\boldsymbol{\omega}_{i}}}{\sigma}\right)Q\left(\dfrac{r_{\boldsymbol{\omega}_{j}}}{\sigma}\right)
\end{array}\right\} \\
& & \tfrac{}{}\nonumber \\
& = & max\left\{ \begin{array}{c}
Q\left(\dfrac{max(r_{\boldsymbol{\omega}_{i}},r_{\boldsymbol{\omega}_{j}})}{\sigma}\right)-\dfrac{\theta_{ij}}{2\pi}e^{-\frac{max(r_{\boldsymbol{\omega}_{i}}^{2},r_{\boldsymbol{\omega}_{j}}^{2})}{2\sigma^{2}}},\\
Q\left(\dfrac{r_{\boldsymbol{\omega}_{i}}}{\sigma}\right)Q\left(\dfrac{r_{\boldsymbol{\omega}_{j}}}{\sigma}\right)
\end{array}\right\} .\label{eq:InterLower}
\end{eqnarray}
\noindent \smallskip{}
\noindent Applying Hunter bound \eqref{eq:hunter5} for LP decoding
error probability \eqref{eq:LP errorUn} and substituting into it
the expression in \eqref{eq:InterLower} together with the LP pairwise
error probability \eqref{eq:LPairWise}, will give the desired result.
\end{IEEEproof}
Given a set of generators $\mathcal{G}$, the running time of ILP-UB
is equal to that of finding an MST on a complete graph $G(V,\mathcal{E})$.
It can be obtained by Prim's algorithm with a complexity of $O(|\mathcal{G}|^{2})$.
The LP-UB for a comparison, for a given set of generators has running
time of $O(|\mathcal{G}|)$.
\section{Results and Discussion\label{Results} }
In this section, we provide results to show the improvement of ILP-UB
over LP-UB. For this purpose, we examine four codes, three HDPC codes:
extended Golay{[}24,12,8{]}, BCH{[}31,26,3{]}, BCH{[}63,57,3{]}; and
one LDPC Tanner code {[}155,64,20{]} \cite{Tanner155}. The parity-check
matrices we use for Golay{[}24,12,8{]} and BCH{[}31,26,3{]} are $H_{G}''$
\eqref{eq:HG''} and $H_{BCH_{[31,26]}}$ \eqref{eq:HBCH31-26 matrix},
respectively; and for the BCH{[}63,57,3{]} we use a systematic parity-check
matrix created by the generator polynomial $x^{6}+x+1$. The minimal
pseudo-weight of the extended Golay{[}24,12,8{]} is $w_{p}^{min}=3.2$.
BCH{[}31,26,3{]} and BCH{[}63,57,3{]} have the same minimal pseudo-weight:
$w_{p}^{min}=3$; and the Tanner code {[}155,64,20{]} has $w_{p}^{min}\approx16.403$
\cite{DecisionRegions}.
Because of the enormous number of cone generators, we chose representative
subgroups: for the BCH{[}31,26,3{]}, BCH{[}63,57,3{]} and Tanner code
{[}155,64,20{]} we chose all the minimal-weight generators that are
1,185 , 11,551 and 465 generators, respectively. Because the extended
Golay{[}24,12,8{]} code has only 165 minimal-weight generators we
chose for it the first 231 generators of a weight equal or less than
$w_{p}=3.25$.
\bigskip{}
\begin{spacing}{1}
\setlength\arraycolsep{0.1em}
\begin{equation}
H_{BCH_{[31,26]}}=\left(\begin{array}{ccccccccccccccccccccccccccccccc}
1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 0 & 1 & 0\\
0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 0 & 1\\
0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 0 & 1 & 0 & 1 & 0 & 0\\
0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 0 & 1 & 0 & 1 & 0\\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 0 & 1 & 0 & 1
\end{array}\right)\qquad\;\;\;\label{eq:HBCH31-26 matrix}
\end{equation}
\end{spacing}
\bigskip{}
\bigskip{}
Fig. \ref{fig:ResultesAngledistribution} presents the angle distributions
according to Def. \ref{def:angle dist.} for the aforementioned codes:
extended Golay{[}24,12,8{]}, BCH{[}31,26,3{]} and BCH{[}63,57,3{]}.
Their average angles are $19.85^{\circ}$, $29.58^{\circ}$, $21.87^{\circ}$,
respectively; and their STDs are $13.44^{\circ}$, $13.94^{\circ}$,
$13.84^{\circ}$, respectively.
\begin{flushleft}
\begin{figure}[H]
\begin{raggedleft}
\begin{minipage}[c][1\totalheight][t]{0.45\textwidth}%
\selectlanguage{british}%
\begin{flushleft}
\subfloat[{\selectlanguage{english}%
{\scriptsize \label{fig:HG'' AnglesDis}}\fontsize{9}{10}\selectfont
Golay{[}24,12,8{]} code: angle distribution for all the $231$ generators
with $w_{p}\leq3.25$.\selectlanguage{british}%
}]{\selectlanguage{english}%
\begin{centering}
\includegraphics[scale=0.1]{pic76}
\par\end{centering}
\selectlanguage{british}%
}
\par\end{flushleft}\selectlanguage{english}%
\end{minipage}\hfill{}%
\begin{minipage}[c][1\totalheight][t]{0.45\textwidth}%
\medskip{}
\selectlanguage{british}%
\begin{flushright}
\subfloat[{\selectlanguage{english}%
{\scriptsize \label{fig:BCH(31,26) AnglesDis}}\fontsize{9}{10}\selectfont
BCH{[}31,26,3{]} code: angle distribution of all the 1,185 minimal-weight
generators.\selectlanguage{british}%
}]{\selectlanguage{english}%
\begin{centering}
\includegraphics[scale=0.1]{pic72}
\par\end{centering}
\selectlanguage{british}%
}
\par\end{flushright}\selectlanguage{english}%
\end{minipage}\hfill{}
\par\end{raggedleft}
\begin{centering}
\begin{minipage}[c][1\totalheight][t]{0.45\textwidth}%
\bigskip{}
\selectlanguage{british}%
\begin{flushleft}
\subfloat[{\selectlanguage{english}%
{\scriptsize \label{fig:BCH(63,57) anglesDis}}\fontsize{9}{10}\selectfont
BCH{[}63,57,3{]} code: angle distribution of all the 11,551 minimal-weight
generators.\selectlanguage{british}%
}]{\selectlanguage{english}%
\begin{centering}
\includegraphics[scale=0.1]{pic70}
\par\end{centering}
\selectlanguage{british}%
}
\par\end{flushleft}\selectlanguage{english}%
\end{minipage}
\par\end{centering}
\caption{\label{fig:ResultesAngledistribution}\fontsize{9}{10}\selectfont
Angle distributions.}
\begin{comment}
it will be nice to see CWs distribution for each code
\end{comment}
\end{figure}
\par\end{flushleft}
Fig. \ref{fig::ResultesILP-UB} presents results of the: ILP-UB$(\mathrm{\mathcal{K}_{sub}})$,
LP-UB$(\mathrm{\mathcal{K}_{sub}})$ and LPD$(\mathrm{\mathcal{K}_{sub}})$
for the chosen subgroups of generators. It presents the LPD FER as
well. The ILP-UB optimized by Prim's algorithm. The ILP-UB presents
an improvement over the LP-UB. For instance, we consider the error
rate of $10^{\lyxmathsym{\textminus}2}$. For the extended Golay{[}24,12,8{]},
the difference between LP-UB$(\mathrm{\mathcal{K}_{sub}})$ and LPD$(\mathrm{\mathcal{K}_{sub}})$
is about 0.9 dB while ILP-UB$(\mathrm{\mathcal{K}_{sub}})$ shows
an improvement of 0.37 dB over LP-UB$(\mathrm{\mathcal{K}_{sub}})$.
For BCH{[}31,26,3{]}, the difference between LP-UB$(\mathrm{\mathcal{K}_{sub}})$
and LPD$(\mathrm{\mathcal{K}_{sub}})$ is about 0.47 dB while ILP-UB$(\mathrm{\mathcal{K}_{sub}})$
shows an improvement of 0.13 dB. And for BCH{[}63,57,3{]}, the difference
between LP-UB$(\mathrm{\mathcal{K}_{sub}})$ and LPD$(\mathrm{\mathcal{K}_{sub}})$
is about 0.62 dB while ILP-UB$(\mathrm{\mathcal{K}_{sub}})$ shows
an improvement of 0.16 dB.
The results of the LDPC Tanner code were omitted, since the improvement
of the ILP-UB$(\mathrm{\mathcal{K}_{sub}})$ over the LP-UB$(\mathrm{\mathcal{K}_{sub}})$
at error rate of $10^{\lyxmathsym{\textminus}3}$ is dropped to about
0.05 dB. The reason for that is twofold. First, the Tanner code has
a large average angle: $35.16^{\circ}$. Second, the generators have
an high pseudo-weight: $w_{p}^{min}\approx16.403$. These two values
are high as compared to the other tested codes.
\begin{flushleft}
\begin{figure}[H]
\begin{raggedleft}
\begin{minipage}[c][1\totalheight][t]{0.45\textwidth}%
\selectlanguage{british}%
\begin{flushleft}
\subfloat[{\selectlanguage{english}%
{\scriptsize \label{fig:ILP-UB HG''} }\fontsize{9}{10}\selectfont
Golay{[}24,12,8{]} code: results for $231$ generators with $w_{p}\leq3.25$
($w_{p}^{min}=3.2$).\selectlanguage{british}%
}]{\selectlanguage{english}%
\begin{centering}
\includegraphics[scale=0.1]{pic97}
\par\end{centering}
\selectlanguage{british}%
}
\par\end{flushleft}\selectlanguage{english}%
\end{minipage}\hfill{}%
\begin{minipage}[c][1\totalheight][t]{0.45\textwidth}%
\begin{flushright}
\smallskip{}
\foreignlanguage{british}{}\subfloat[{{\scriptsize \label{fig:BCH(31,26) ILP-UB} }\fontsize{9}{10}\selectfont
BCH{[}31,26,3{]} code: results of all the $1,185$ minimal-weight
generators ($w_{p}^{min}=w_{H}^{min}=3$).}]{\includegraphics[scale=0.1]{pic98}
\selectlanguage{british}%
\raggedright{}\selectlanguage{english}%
}
\par\end{flushright}%
\end{minipage}\hfill{}
\par\end{raggedleft}
\centering{}%
\begin{minipage}[c][1\totalheight][t]{0.45\textwidth}%
\selectlanguage{british}%
\begin{flushleft}
\subfloat[{\selectlanguage{english}%
{\footnotesize \label{fig:BCH(63,57) ILP-UB}}\fontsize{9}{10}\selectfont
BCH{[}63,57,3{]} code: results of all the $11,551$ minimal-weight
generators ($w_{p}^{min}=w_{H}^{min}=3$).\selectlanguage{british}%
}]{\selectlanguage{english}%
\begin{centering}
\includegraphics[scale=0.1]{pic96}
\par\end{centering}
\selectlanguage{british}%
\raggedleft{}}
\par\end{flushleft}\selectlanguage{english}%
\end{minipage}\caption{\label{fig::ResultesILP-UB}\fontsize{9}{10}\selectfont A comparison
between ILP-UB, LP-UB, LPD and LPD FER for HDPC codes.}
\end{figure}
\par\end{flushleft}
Fig . \ref{fig::ResultesILP-UB} together with Fig. \ref{fig:ResultesAngledistribution}
show that the lower the average angle is, the more improvement the
ILP-UB has. A small average angle is typical for HDPC codes, therefore,
the advantage of ILP-UB over the LP-UB will be reflected better on
such type of codes. But on the other hand, as the larger the average
angle is, the better the LP-UB will be. Fig. \ref{fig:ILP-UB HG''}
presents the highest improvement of the ILP-UB$(\mathrm{\mathcal{K}_{sub}})$
among the other codes. This result correlates to Golay's smallest
average angle: $19.85^{\circ}$. However, it presents the largest
gap to its $\mathrm{LPD}(\mathrm{\mathcal{K}_{sub}})$. This apparently
happens because there are a significant probabilities of intersections
between three error events or more.
Buksz\'ar and Pr\'ekopa have suggested \cite{CherryTrees} a third
order upper bound on the probability of a finite union of events.
Their bound considers intersections of two and three events. They
proved that this third order bound, which is obtained by the use of
a type of graph called cherry tree, is at least as strong as the second-order
bound. Therefore, implementing such a bound will improve (or at least
will be equal to) the proposed ILP-UB.
\section{Conclusions\label{Conclusions} }
In this paper, we have presented an improved union bound on the error
probability of LP decoding of binary linear HDPC codes transmitted
over a binary-input AWGN channel. It is based on the second-order
upper bound on the probability of a finite union of events. It has
low computational complexity since it only involves the Q-function.
It can be implemented with running time of $O(|\mathcal{G}|^{2})$,
where $\mathcal{G}$ is a set of generators of the fundamental cone
arisen from a given parity check matrix. We examined the proposed
bound for several HDPC codes: Golay{[}24,12,8{]}, BCH{[}31,26,3{]},
BCH{[}63,57,3{]}, and for the LDPC Tanner {[}155,64,20{]} code. The
improvement of the proposed bound over the union bound presents dependency
on the pseudo-weight of the generators and their density. We studied
and compared the generator density through the angle distribution
of various codes and parity-check matrices. Finally, a third order
upper bound was proposed, it is based on a type of graph called cherry
tree, and is left open for further research.
|
{
"timestamp": "2012-03-09T02:03:49",
"yymm": "1203",
"arxiv_id": "1203.1850",
"language": "en",
"url": "https://arxiv.org/abs/1203.1850"
}
|
\section{Introduction}\label{sec:Introduction}
The study of face numbers of polytopes is a classical problem.
For a simplicial $d$-polytope $P$ let $f_i(P)$ denote the number of its $i$-dimensional faces, where $-1\leq i\leq d-1$ ($f_{-1}(P)=1$ for the emptyset).
The numbers $f_i(P)$ are conveniently described by the \emph{$h$-numbers}, defined by $h_i(P)=\sum_{j=0}^i (-1)^{j-i} {d-j \choose i-j} f_{j-1}$ for $0\leq i\leq d$. The Dehn-Sommerville relations assert that $h_i(P)=h_{d-i}(P)$ for all $0\leq i\leq \lfloor \frac{d}{2}\rfloor$, generalizing the Euler-Poincar\'{e} formula.
In 1971, McMullen and Walkup \cite{McMullenWalkup:GLBC-71} posed the following \emph{generalized lower bound conjecture} (GLBC), generalizing Barnette's \emph{lower bound theorem} (LBT) \cite{Barnette-LBTfacets-71,Barnette:LBT-73}.
\begin{conjecture}\label{conj:GLBTpolytopes}(McMullen--Walkup)
Let $P$ be a simplicial $d$-polytope. Then
\begin{itemize}
\item[(a)] $1=h_0(P)\leq h_1(P)\leq \dots \leq h_{\lfloor \frac{d}{2} \rfloor}(P)$.
\item[(b)] for an integer $1\leq r\leq \frac{d}{2}$, the following are equivalent:
\begin{itemize}
\item[(i)] $h_{r-1}(P)=h_{r}(P)$.
\item[(ii)] $P$ is $(r-1)$-stacked, namely, there is a triangulation $K$ of $P$ all of whose faces of dimension at most $d-r$ are faces of $P$.
\end{itemize}
\end{itemize}
\end{conjecture}
Around 1980 the \emph{$g$-theorem} was proved, giving a complete characterization of the face numbers of simplicial polytopes. It was conjectured by
McMullen \cite{McMullen-g-conj},
sufficiency of the conditions was proved by Billera--Lee \cite{Billera-Lee} and necessity by Stanley \cite{Stanley:NumberFacesSimplicialPolytope-80}.
Stanley's result establishes part (a) of the GLBC.
As for part (b), the implication $(\mathrm{ii})\Rightarrow (\mathrm{i})$ was shown in \cite{McMullenWalkup:GLBC-71}. The implication $(\mathrm{i})\Rightarrow (\mathrm{ii})$ is
easy for $r=1$, and was proved for $r=2$ as part of the LBT \cite{Barnette-LBTfacets-71}.
The main goal of this paper is to prove the remaining open part of the GLBC.
In particular, it follows that $(r-1)$-stackedness of
a simplicial $d$-polytope where $r \leq \frac d 2 $ only depends on its face numbers.
McMullen \cite{McMullen:Triangulations} proved that,
to study Conjecture \ref{conj:GLBTpolytopes}(b),
it is enough to consider combinatorial triangulations.
Thus we write a statement in terms of simplicial complexes.
For a simplicial complex $\Delta$ on the vertex set $V$ and a positive integer $i$, let
$$\Delta(i):=\{F\subseteq V: \skel_{i}(2^F)\subseteq \Delta\},$$
where $\skel_{i}(2^F)$ is the $i$-skeleton of the simplex defined by $F$, namely the collection of all subsets of $F$ of size at most $i+1$.
For a simplicial $d$-polytope $P$
with boundary complex $\Delta$,
we say that a simplicial complex $K$ is a \emph{triangulation} of $P$ if its geometric realization is homeomorphic to a $d$-ball and its boundary is $\Delta$.
A triangulation $K$ of $P$ is \emph{geometric} if in addition
there is a geometric realization of $K$ whose underlying space is $P$.
\begin{theorem}\label{thmIntro:GLBCpolytopes}
Let $P$ be a simplicial $d$-polytope with the $h$-vector $(h_0,h_1,\dots,h_d)$,
$\Delta$ its boundary complex, and $1\leq r\leq \frac{d}{2}$ an integer.
If $h_{r-1}=h_{r}$ then $\Delta(d-r)$ is the unique geometric triangulation of $P$
all of whose faces of dimension at most $d-r$ are faces of $P$.
\end{theorem}
Note that the uniqueness of such a triangulation was proved by McMullen \cite{McMullen:Triangulations}.
Moreover, it was shown by Bagchi and Datta \cite{Bagchi-Datta:StellatedSpheres} that if Conjecture \ref{conj:GLBTpolytopes}(b) is true then the triangulation must be $\Delta(d-r)$.
Since the above theorem is described in terms of simplicial complexes,
it would be natural to ask if a similar statement holds for triangulations of spheres, or more generally homology spheres.
Indeed, we also prove an analogous result for homology spheres satisfying a certain algebraic property
called the \emph{weak Lefschetz property} (WLP, to be defined later).
\begin{theorem}\label{thmIntro:GLBCspheres}
Let $\Delta$ be a homology $(d-1)$-sphere having the WLP over a field of characteristic $0$, $(h_0,h_1,\dots,h_d)$ the $h$-vector of $\Delta$, and $1\leq r\leq \frac{d}{2}$ an integer.
If $h_{r-1}=h_{r}$ then $\Delta(d-r)$ is the unique homology $d$-ball with no interior faces of dimension at most $d-r$
and with boundary $\Delta$.
\end{theorem}
Note that an algebraic formulation of the $g$-conjecture (for homology spheres) asserts that any homology sphere has the WLP, see e.g. \cite[Conjecture 4.22]{Swartz-SpheresToManifolds} for a stronger variation.
If this conjecture holds, then Theorem \ref{thmIntro:GLBCspheres} will extend to all homology spheres.
Indeed, the case $r=2$ in Theorem \ref{thmIntro:GLBCspheres} was proved by Kalai \cite{Kalai-LBT}, without the WLP assumption, as part of his generalization of the LBT to homology manifolds and beyond.
Further, note that for $r\leq d/2$, if a homology $(d-1)$-sphere $\Delta$ satisfies that $\Delta(d-r)$ is a homology $d$-ball with boundary $\Delta$, then $\Delta$ satisfies all the numerical conditions in the $g$-conjecture (including the nonlinear Macaulay inequalities), as was shown by Stanley \cite{Stanley:CMcomplexes77}.
This paper is organized as follows:
In Section \ref{sec:Triangulations} we give preliminaries on triangulations and prove the uniqueness claim in the above two theorems.
In Section \ref{sec:CM} we prove that $\Delta(d-r)$ satisfies a nice algebraic property called the Cohen--Macaulay property.
In Section \ref{sec:polytopes}, by using this result together with a geometric and topological argument, we show that $\Delta(d-r)$ triangulates $P$ in Theorem \ref{thmIntro:GLBCpolytopes}.
In Section \ref{sec:LefschetzSpheres} we prove Theorem \ref{thmIntro:GLBCspheres} based on the theory of canonical modules in commutative algebra.
Lastly, in Section \ref{sec:conclude} we give some concluding remarks and open questions.
\section{Triangulations}\label{sec:Triangulations}
In this section, we provide some preliminaries and notation on triangulations, and prove the uniqueness statements in Theorems \ref{thmIntro:GLBCpolytopes} and \ref{thmIntro:GLBCspheres}.
Let $\Delta$ be an (abstract) \emph{simplicial complex} on vertex set $V$, namely, a collection of subsets of $V$ such that, for any $F \in \Delta$ and $G \subset F$,
one has $G \in \Delta$.
An element $F \in \Delta$ is called a {\em face} of $\Delta$ and a maximal face (under inclusion) is called a {\em facet} of $\Delta$.
A face $F \in \Delta$ is called an {\em $i$-face} if $\#F=i+1$,
where $\# X$ denotes the cardinality of a finite set $X$.
The {\em dimension} of $\Delta$ is $\dim \Delta=\max\{\# F -1: F \in \Delta \}$.
For $0 \leq k \leq \dim \Delta$,
we write $\mathrm{skel}_k(\Delta)=\{F \in \Delta: \#F \leq k+1\}$ for the {\em $k$-skeleton} of $\Delta$.
Let $f_i=f_i(\Delta)$ be the number of $i$-faces of $\Delta$.
The {\em $h$-vector} $h(\Delta)=(h_0(\Delta),h_1(\Delta),\dots,h_d(\Delta))$ of $\Delta$
is a sequence of integers defined by
$$
h_i(\Delta)=\sum_{j=0}^i (-1)^{j-i} {d-j \choose i-j} f_{j-1}
$$
for $i=0,1,2,\dots,d$, where $d=\dim \Delta+1$ and where $f_{-1}=1$.
If $\Delta$ is the boundary complex of a simplicial polytope $P$,
we also call $h(\Delta)$ the $h$-vector of $P$.
Let $\Delta$ be a simplicial complex on vertex set $V$.
A subset $F \subset V$ is called a \emph{missing face} of $\Delta$ if $F \notin \Delta$ and all proper subsets of $F$ are faces of $\Delta$.
Note that the set of the missing faces of $\Delta$ determines $\Delta$ itself
since it determines all subsets of $V$ which are not in $\Delta$.
It is not hard to see that, by definition, the simplicial complex $\Delta(i)$, defined in the Introduction, is the simplicial complex whose missing faces are the missing faces $F$
of $\Delta$ with $\#F \leq i+1$.
In particular, for $j\leq i$, one has $\Delta(j)=\Delta(i)$ if and only if
$\Delta$ has no missing $k$-faces for $j+1\leq k\leq i$.
The following relation between face numbers and missing faces will be used in the sequel. It was first proved by Kalai \cite[Proposition 3.6]{Kalai:Aspects-94} when $d > 2r+1$,
and was later generalized by Nagel \cite[Corollary 4.8]{Nagel:Empty}.
\begin{lemma}
\label{lem:NoMissingFaces}
Let $\Delta$ be the boundary complex of a simplicial $d$-polytope.
If $h_{r-1}(\Delta)=h_r(\Delta)$ then $\Delta(r-1)=\Delta(d-r)$.
\end{lemma}
\begin{remark}
\label{rem:NoMissingFaces}
Nagel \cite{Nagel:Empty} writes a statement only for simplicial polytopes,
but his proof works for homology spheres admitting the WLP which we study in Section 5.
\end{remark}
Next, we prove the uniqueness statements in Theorems \ref{thmIntro:GLBCpolytopes} and \ref{thmIntro:GLBCspheres}.
We start with some notations and definitions.
Let $\kk$ be a field.
For a simplicial complex $\Delta$,
let $\widetilde H_i(\Delta;\kk)$ be the $i$th reduced homology group of $\Delta$ with coefficients in $\kk$,
and let
$$\lk_\Delta(F)=\{ G \in \Delta: F \cup G \in \Delta,\ F \cap G = \emptyset\}$$
be the {\em link} of $F$ in $\Delta$.
A $d$-dimensional simplicial complex $\Delta$
is said to be a {\em homology $d$-sphere} (over $\kk$)
if the homology groups
$\widetilde H_{d-\#F-i} (\lk_\Delta(F);\kk)$ are isomorphic to $\kk$ for $i=0$ and vanish for all $i>0$, for all $F \in \Delta$ (including the empty face $\emptyset$).
Also, a {\em homology $d$-ball} (over $\kk$) is
a $d$-dimensional simplicial complex $\Delta$ such that the homology groups $\widetilde H_{d-\#F-i} (\lk_\Delta(F);\kk)$ are either $\kk$ or $0$ for $i=0$ and vanish for $i>0$, for all $F \in \Delta$,
and moreover, its boundary complex
$$\partial \Delta =\{F \in \Delta: \widetilde H_{d-\#F} (\lk_\Delta(F);\kk) =0\}$$
is a homology $(d-1)$-sphere.
We say that a simplicial complex $\Delta$ is a \emph{triangulation} of a topological space $X$ if its geometric realization is homeomorphic to $X$.
Note that a triangulation of a $d$-sphere (resp.\ $d$-ball) is a homology $d$-sphere (resp.\ $d$-ball) over any field.
Let $\Delta$ be a homology $d$-ball.
The faces in $\Delta-\partial \Delta$ are called the \emph{interior faces} of $\Delta$.
If $\Delta$ has no interior $k$-faces for $k \leq d-r$ then $\Delta$ is said to be \emph{$(r-1)$-stacked}.
An {\em $(r-1)$-stacked} sphere (resp.\ homology sphere) is the boundary complex
of an $(r-1)$-stacked triangulation of a ball (resp.\ homology ball).
Recall that a triangulation of a simplicial $d$-polytope $P$ with boundary complex $\Delta$
is a triangulation $K$ of a $d$-ball such that $\partial K=\Delta$.
McMullen \cite[Theorem 3.3]{McMullen:Triangulations} proved that,
for $r \leq \frac d 2$, an $(r-1)$-stacked triangulation $K$ of a simplicial $d$-polytope $P$ is unique.
Moreover, Bagchi and Datta \cite[Corollary 3.6]{Bagchi-Datta:StellatedSpheres}
proved that such a triangulation must be equal to $\Delta(d-r)$
(they actually proved a more general statement for PL-spheres).
We generalize these statements for homology spheres
based on an idea of Dancis \cite{Dancis} who proved that a homology $d$-sphere is determined by its $\lceil \frac{d}{2}\rceil$-skeleton (generalizing a previous work of Perles who showed it for polytopes).
In particular, our result answers \cite[Question 6.4]{Bagchi-Datta:StellatedSpheres}.
\begin{theorem}\label{thm:Uniqueness}
Let $\Delta$ be a homology $(d-1)$-sphere and $1\leq r\leq \frac{d+1}{2}$ an integer.
\begin{itemize}
\item[(i)]
If $\Delta(d-r)$ is a homology $d$-ball with $\partial \Delta(d-r)=\Delta$
then it is $(r-1)$-stacked.
\item[(ii)]
If $\Delta'$ is an $(r-1)$-stacked homology $d$-ball with $\partial \Delta'=\Delta$
then $\Delta'=\Delta(d-r)$.
\end{itemize}
\end{theorem}
\begin{proof}
The statement (i) is obvious since $\Delta(d-r)$ and $\partial \Delta(d-r)=\Delta$ have the same $(d-r)$-skeleton.
We prove (ii).
Since $\Delta'$ is $(r-1)$-stacked, $\Delta'$ has the same $(d-r)$-skeleton as $\Delta$,
and therefore has the same $(d-r)$-skeleton as $\Delta(d-r)$ by definition.
Thus what we must prove is that $\Delta'$ has no missing faces of cardinality $> d-r+1$.
Let $F$ be a $(k+1)$-subset of $[n]$ with $k > d-r$
such that all its proper subsets are in $\Delta'$.
We claim $F \in \Delta'$.
Consider the homology $d$-sphere $S=\Delta'\cup (\{v\}*\Delta)$, where $v$ is a new vertex
and where $\{v\}*\Delta= \Delta \cup \{ \{v\} \cup F: F \in \Delta\}$ is the cone of $\Delta$ with the vertex $v$.
For a subset $W \subset V$, where $V$ is the vertex set of $S$,
let $S|_W=\{G \in S: G \subset W\}$ be the induced subcomplex of $S$ on $W$.
Since all proper subsets of $F$ are in $\Delta'$ and $\Delta'$ is an induced subcomplex of $S$,
to prove $F \in \Delta'$,
it is enough to show that $S|_F$ is not a $(k-1)$-sphere, equivalently that $\widetilde H_{k-1} (S|_F;\kk)=0$.
Since $S-S|_F$ is homotopy equivalent to $S|_{V-F}$ (e.g.\ \cite[Lemma 70.1]{Munkres}),
by Alexander duality (e.g.\ \cite[Theorem 71.1]{Munkres}) and the universal coefficient theorem with field coefficients
we have
$$
\widetilde H_{k-1} (S|_F;\kk) \cong \widetilde H_{d-k}(S-S|_F;\kk) \cong \widetilde H_{d-k}(S|_{V-F};\kk),
$$
so we need to show $\widetilde H_{d-k}(S|_{V-F};\kk)=0$.
Since $d-k \leq r-1 \leq d-r$,
we have $\mathrm{skel}_{d-k}(S|_{V-F})=\mathrm{skel}_{d-k}((\{v\} * \Delta)|_{V-F})$ and $S|_{V-F} \supset (\{v\} * \Delta)|_{V-F}$.
Then, by the definition of the simplicial homology,
we have
$$\dim_\kk \widetilde H_{d-k}(S|_{V-F};\kk) \leq \dim_\kk \widetilde H_{d-k}((\{v\} * \Delta)|_{V-F};\kk).$$
Recall that $v \not \in F$.
The right-hand side of the above inequality is equal to zero
since $(\{v\}*\Delta)|_{V-F}=\{v\}*(\Delta|_{V-F-\{v\}})$ is a cone.
Hence $\widetilde H_{d-k}(S|_{V-F}:\kk)=0$.
\end{proof}
Unlike $(r-1)$-stacked polytopes with $ r \leq \frac d 2$,
$(\frac {d-1} 2)$-stacked simplicial $d$-polytopes cannot be characterized by their $h$-vectors
since $h_{ ({d-1})/ 2}=h_{({d+1})/ 2}$ holds for all simplicial $d$-polytopes when $d$ is odd.
On the other hand, Theorem \ref{thm:Uniqueness} says that
$(\frac {d-1} 2)$-stacked simplicial $d$-polytopes still have a nice combinatorial property.
It would be of interest to have a nice combinatorial characterization of these polytopes.
\section{Cohen--Macaulayness}\label{sec:CM}
In this section, we prove that the simplicial complexes $\Delta(d-r)=\Delta(r-1)$ in Theorems \ref{thmIntro:GLBCpolytopes} and \ref{thmIntro:GLBCspheres} (the equality holds by Lemma \ref{lem:NoMissingFaces} and Remark \ref{rem:NoMissingFaces} respectively)
satisfy a nice algebraic condition, called the Cohen--Macaulay property.
We first introduce some basic tools in commutative algebra.
\subsection*{Stanley--Reisner rings}
Let $S=\kk[x_1,\dots,x_n]$
be a polynomial ring over an infinite field $\kk$.
For a subset $F \subset [n]=\{1,2,\dots,n\}$,
we write $x_F=\prod _{k \in F} x_k$.
For a simplicial complex $\Delta$ on $[n]$,
the ring
$$\kk[\Delta]=S/I_\Delta$$
where $I_\Delta=(x_F: F \subset [n],\ F \not \in \Delta)$,
is called the {\em Stanley--Reisner ring} of $\Delta$.
The simplicial complex $\Delta(i)$ has a simple expression in terms of Stanley--Reisner rings.
For a homogeneous ideal $I \subset S$, let $I_{\leq k}$ be the ideal generated by all elements in $I$ of degree $ \leq k$.
Since the missing faces of $\Delta$ correspond to the minimal generators of $I_\Delta$
and since $\Delta(i)$ is the simplicial complex whose missing faces are the missing faces $F$ of $\Delta$
with $\#F \leq i+1$,
one has
$$I_{\Delta(i)}=(I_\Delta)_{\leq i+1}.$$
\subsection*{Cohen--Macaulay property}
Let $I \subset S$ be a homogeneous ideal and $R=S/I$.
The Krull dimension $\dim R$ of $R$ is the minimal number $k$
such that there is a sequence of linear forms $\theta_1,\dots,\theta_k \in S$
such that $\dim_\kk S/(I+(\theta_1,\dots,\theta_k))< \infty$.
If $d=\dim R$, then a sequence $\Theta=\theta_1,\dots,\theta_d$ of linear forms such that
$\dim_\kk S/(I+(\Theta))< \infty$ is called a {\em linear system of parameters} of $R$ (l.s.o.p.\ for short).
A sequence of homogeneous polynomials $f_1,\dots,f_r$ of positive degrees is called
a {\em regular sequence} of $R$ if $f_i$ is a non-zero divisor of $S/(I+(f_1,\dots,f_{i-1}))$
for all $i=1,2,\dots,r$.
We say that $R$ is {\em Cohen--Macaulay} if every l.s.o.p.\ of $R$ is a regular sequence of $R$.
A simplicial complex $\Delta$ is said to be {\em Cohen--Macaulay} (over $\kk$)
if $\kk[\Delta]$ is a Cohen--Macaulay ring.
The following topological criterion for the Cohen--Macaulay property was proved by Reisner \cite{Reisner}.
\begin{lemma}[Reisner's criterion]\label{lem:Reisner}
A simplicial complex $\Delta$ is Cohen--Macaulay (over $\kk$) if and only if,
for any face $F \in \Delta$,
$\widetilde H_i(\lk_\Delta(F); \kk)=0$ for all $i \ne \dim \lk_\Delta(F)$.
\end{lemma}
\subsection*{The weak Lefschetz property}
Let $I \subset S$ be a homogeneous ideal such that $R=S/I$ has dimension $0$.
We write $R=\bigoplus_{i=0}^s R_i$, where $R_i$ is the homogeneous component of $R$ of degree $i$ and where $R_s \ne 0$.
We say that $R$ has the {\em weak Lefschetz property} (WLP for short) if there is a linear form $w \in R_1$,
called a {\em Lefschetz element} of $R$,
such that the multiplication $\times w :R_k \to R_{k+1}$
is either injective or surjective for all $k$.
We say that a ring $R=S/I$ of dimension $d>0$,
where $I$ is a homogeneous ideal, has the WLP if it is Cohen--Macaulay
and there is an l.s.o.p.\ $\Theta$ of $R$ such that $S/(I+(\Theta))$
has the WLP.
Also, a simplicial complex $\Delta$ is said to have the WLP (over $\kk$)
if $\kk[\Delta]$ has the WLP.
It is known that the boundary complex of a simplicial polytope has the WLP
over $\QQ$.
See \cite[Section 5.2]{Fulton}.
It is known that, for any homogeneous ideal $I \subset S$,
the Hilbert series $H(S/I,t)=\sum_{i=0}^\infty (\dim_\kk (S/I)_i) t^i$ of the ring $S/I$ can be written in the form
$$H(S/I,t)= \frac {h_0+h_1t + \dots + h_s t^s} {(1-t)^d}$$
where $d=\dim S/I$ and where $h_s \ne 0$.
See \cite[Corollay 4.1.8]{Bruns-Herzog}.
The vector $h(S/I)=(h_0,h_1,\dots,h_s)$ is called the $h$-vector of $S/I$.
If $S/I$ has the WLP then its $h$-vector is unimodal,
namely it satisfies $h_0 \leq \dots \leq h_p \geq h_{p+1} \geq \dots \geq h_s$
for some $p$.
Indeed, let $R=S/(I+(\Theta))$, where $\Theta$ is an l.s.o.p.\ of $S/I$.
Then $h_k=\dim_\kk R_k$ for all $k$.
If $R$ has the WLP and $h_p \geq h_{p+1}$ for some $p$,
then the multiplication $w : R_{k} \to R_{k+1}$ is surjective
for $k=p$. The multiplication map is also surjective for all $k \geq p$ as $S$ is generated by elements of degree $1$ and $p+1\geq 1$,
and we have $h_p \geq \dots \geq h_s$.
\subsection*{Generic initial ideals}
Here we briefly recall generic initial ideals.
We do not give details on this subject.
\cite{Gr} and \cite[Section 4]{HibiHerzog} are good surveys on generic initial ideals.
Let $>_\rev$ be the degree reverse lexicographic order induced by the ordering $x_1 >_\rev \dots >_\rev x_n$.
For a homogeneous ideal $I \subset S$,
let $\init_{>_\rev}(I)$ be the initial ideal of $I$ w.r.t.\ $>_\rev$.
Let $\mathrm{GL}_n(\kk)$ be the general linear group with coefficients in $\kk$.
Any $\varphi=(a_{ij}) \in \mathrm{GL}_n(\kk)$ induces an automorphism of $S$,
again denoted by $\varphi$,
$$\varphi\big(f(x_1,\dots,x_n)\big)=
f\left(\sum_{k=1}^n a_{k1} x_k,\dots,\sum_{k=1}^n a_{kn} x_k\right)$$
for any $f \in S$.
It was proved by Galligo that $\init_{>_\rev}(\varphi(I))$ is constant for a generic choice of $\varphi \in \mathrm{GL}_n(\kk)$.
See \cite[Theorem 1.27]{Gr}.
This monomial ideal $\init_{>_\rev}(\varphi(I))$ is called the {\em generic initial ideal} of $I$ w.r.t.\ $>_{\rev}$,
and denoted $\gin(I)$.
We need the following well-known property on the WLP.
\begin{lemma}
\label{gin}
Let $I \subset S$ be a homogeneous ideal and $d=\dim S/I$.
\begin{enumerate}
\item[(i)] $S/I$ is Cohen--Macaulay if and only if $S/\gin(I)$ is Cohen--Macaulay.
\item[(ii)] $S/I$ has the WLP if and only if $S/\gin(I)$ has the WLP.
Moreover, if $S/I$ has the WLP, then
$x_n,\dots,x_{n-d+1}$ is an l.s.o.p.\ of $S/\gin(I)$
and
$x_{n-d}$ is a Lefschetz element of $S/(\gin(I)+(x_n,\dots,x_{n-d+1}))$.
\end{enumerate}
\end{lemma}
See \cite[Corollary 4.3.18]{HibiHerzog} for the first statement.
The second statement follows from \cite[Lemmas 4.3.7]{HibiHerzog}
together with the facts that, for $\theta_1,\dots,\theta_{d+1} \in S$ generic linear forms,
$\theta_1,\dots,\theta_d$ is an l.s.o.p.\ of $S/I$ and $\theta_{d+1}$ is a Lefschetz element of $S/(I+(\theta_1,\dots,\theta_d))$,
and that for a generic choice of $\varphi \in \mathrm{GL}_n(K)$ the linear forms
$x_n,\dots,x_{n-d}$ are generic for $S/\varphi(I)$.
The following result due to Mark Green \cite[Proposition 2.28]{Gr} is crucial to prove the Cohen--Macaulay property of $\Delta(r-1)$.
\begin{lemma}[Crystallization Principle]
\label{crysterization}
Suppose $\mathrm{char}(\kk)=0$.
Let $I \subset S$ be a homogeneous ideal generated by elements of degree $\leq m$.
If $\gin(I)$ has no minimal generators of degree $m+1$
then $\gin(I)$ is generated by elements of degree $\leq m$.
\end{lemma}
\begin{theorem}\label{thm:CM}
Suppose $\mathrm{char}(\kk)=0$.
Let $I \subset S$ be a homogeneous ideal such that $S/I$ has the WLP,
and let $h(S/I)=(h_0,h_1,\dots,h_s)$.
If $h_{r-1} = h_r = h_{r+1}$ for some $ 1 \leq r \leq s-1$,
then $S/I_{\leq r}$ is Cohen--Macaulay of dimension $\dim S/I +1$.
\end{theorem}
\begin{proof}
Let $J= \gin(I)$ and $d=\dim S/I$.
We first claim that $S/J_{\leq r}$ is Cohen--Macaulay.
Observe that $J$ is a monomial ideal.
By Lemma \ref{gin}, $S/J$ is Cohen--Macaulay of dimension $d$,
and $J$ has no minimal generators which are divisible by one of $x_n,\dots,x_{n-d+1}$.
Also, since $ h_{r-1}=h_r=h_{r+1}$, the WLP shows that the multiplication
\begin{align}
\label{wlp}
\times x_{n-d} : S/\big(J+(x_n,\dots,x_{n-d+1})\big)_j \to
S/\big(J+(x_n,\dots,x_{n-d+1})\big)_{j+1}
\end{align}
is injective for $ j \leq r$,
which implies that $J$ has no minimal generators of degree $\leq r+1$ which are divisible by $x_{n-d}$.
Indeed, if there is a minimal generator of the form $ux_{n-d}$, then $u$ is in the kernel of the map \eqref{wlp}.
Thus $J_{\leq r}$ has no minimal generators which are divisible by
one of $x_n,\dots,x_{n-d}$.
Thus $x_n,\dots,x_{n-d}$ is a regular sequence of $S/J_{\leq r}$.
In particular, we have $\dim S/J_{\leq r} \geq d+1$
since the length of a regular sequence is bounded by the dimension (\cite[Proposition 1.2.12]{Bruns-Herzog}).
It is left to show that the quotient by this regular sequence is a finite dimensional vector space over $\kk$.
Since
the multiplication map \eqref{wlp}
is surjective when $j=r-1$, $(S/J+(x_n,\dots,x_{n-d}))_{r}=0$
and $J$ contains all monomials in $\kk[x_1,\dots,x_{n-d-1}]$ of degree $r$.
Thus $\dim_\kk S/(J_{\leq r}+(x_n,\dots,x_{n-d})) < \infty$, and $S/J_{\leq r}$ is Cohen--Macaulay of dimension $d+1$ with an l.s.o.p.\ $x_n,\dots, x_{n-d}$.
Next, we prove $\gin(I_{\leq r})=\gin(I)_{\leq r}$.
By the Crystallization principle, what we must prove is that $\gin(I_{\leq r})$ has no minimal generators of degree $r+1$.
Since $I_{\leq r} \subset I$ and $(I_{\leq r})_{r}=I_{r}$,
it is enough to prove that $\gin(I)$ has no minimal generators of degree $r+1$.
Indeed, we already showed that $J=\gin(I)$ has no minimal generator of degree $r+1$ which
is divisible by one of $x_n,\dots,x_{n-d+1},x_{n-d}$.
We also showed that $J$ contains all monomials in $\kk[x_1,\dots,x_{n-d-1}]$
of degree $r$.
These facts guarantee that $J=\gin(I)$ has no minimal generators of degree $r+1$, as desired.
We proved that $S/\gin(I_{\leq r})=S/\gin(I)_{\leq r}$ is Cohen--Macaulay of dimension $d+1$.
Then the desired statement follows from Lemma \ref{gin}(i).
\end{proof}
\begin{corollary}\label{cor:CM}
Suppose $\mathrm{char}(\kk)=0$.
Let $\Delta$ be a homology $(d-1)$-sphere having the WLP over $\kk$.
If $h_{r-1}(\Delta)=h_r(\Delta)$ for some $r \leq \frac d 2$,
then $\Delta(r-1)$ is Cohen--Macaulay over $\kk$ and has dimension $d$.
\end{corollary}
\begin{proof}
Recall that
the $h$-vector of $\Delta$ coincides with the $h$-vector
of its Stanley--Reisner ring $\kk[\Delta]$.
Since the $h$-vector of $\Delta$ is symmetric,
the WLP shows $h_{r-1}(\Delta)=h_r(\Delta)=\dots=h_{d-r+1}(\Delta)$.
Since $I_{\Delta(r-1)}=(I_\Delta) _{\leq r}$,
Theorem \ref{thm:CM}
says that $\kk[\Delta(r-1)]$ is Cohen-Macaulay of dimension $d+1$.
Thus $\Delta(r-1)$ is Cohen--Macaulay of dimension $d$.
\end{proof}
\begin{remark}\label{rem:vanKampen}
The weaker assertion that $\dim(\Delta(d-r))\leq d$ for $r\leq \frac{d}{2}$ is true for any simplicial $(d-1)$-sphere $\Delta$, and more generally for any simplicial complex $\Delta$ which embeds in the $(d-1)$-sphere.
This can be shown using van-Kampen obstruction to embedability, see \cite{VanKampen, Shapiro, Wu}, for cones over Flores complexes \cite{Flores}.
If we assume $\dim(\Delta(d-r))>d$ then, for $d$ even, $\Delta$ contains
$\skel_{d/2}(2^{[d+2]})$, hence it contains the cone over Flores complex $L=\skel_{\frac{d}{2}-1}(2^{[d+1]})$. (Here $[i]:=\{1,2,\dots,i\}$.)
By the non-vanishing on $L$ of the van-Kampen obstruction to embedability in the $(d-2)$-sphere, we conclude that the cone over $L$ does not embed in the $(d-1)$-sphere, a contradiction.
The argument for $d$ odd is similar.
\end{remark}
\section{GLBC for polytopes}\label{sec:polytopes}
In this Section we prove the existence part of Theorem \ref{thmIntro:GLBCpolytopes}.
\begin{theorem}\label{thm:GLBCpolytopes}
Let $P$ be a simplicial $d$-polytope with the $h$-vector $(h_0,h_1,\dots,h_d)$,
$\Delta$ its boundary complex, and $1\leq r\leq \frac{d}{2}$ an integer.
If $h_{r-1}=h_{r}$ then $\Delta(d-r)$ is a geometric triangulation of $P$.
\end{theorem}
In the rest of this section,
we fix a simplicial $d$-polytope $P$ satisfying the assumption of Theorem \ref{thm:GLBCpolytopes},
and prove the theorem for $P$.
We may assume $P \subset \RR^d$.
Let $V=\{v_1,v_2,\dots,v_n\} \subset \RR^d$ be the vertex set of $P$
and let $\Delta$ be the boundary complex of $P$.
For a subset $T =\{v_{i_1},\dots,v_{i_k}\} \subset V$,
we write $[T]=\mathrm{conv}(v_{i_1},\dots,v_{i_k})$ for the convex hull of the vertices in $T$.
Let $\Delta'=\Delta(r-1)$.
\begin{lemma}\label{lem:D'embedded}
The set $\{[F]: F \in \Delta'\}$ is a geometric realization of $\Delta'$,
namely,
\begin{itemize}
\item[(i)] $[F_1]\cap[F_2]=[F_1\cap F_2]$ for all $F_1,F_2 \in \Delta'$, and
\item[(ii)] $\dim [F]=\# F -1$ for all $F \in \Delta'$.
\end{itemize}
\end{lemma}
\begin{proof}
The proof is similar to that of \cite[Proposition 3.4]{Bagchi-Datta:StellatedSpheres}.
(i)
Assume by contradiction that $F_1,F_2 \in \Delta'$ form a counterexample to (i) with the size $\# F_1 + \# F_2$ minimal.
Note that the convex set $[F_1]\cap [F_2]$ is not contained in the boundary of $P$, as otherwise it would equal a single face $[F]$ with $F\in \Delta$ and thus $F_1\cap F_2=F$, which says that (i) holds for $F_1$ and $F_2$.
In particular,
we have $F_1 \not \in \Delta$ and $F_2 \not \in \Delta$.
We prove the following properties for $F_1$ and $F_2$.
\begin{itemize}
\item[(a)] Any $p \in [F_1] \cap [F_2] \setminus [F_1 \cap F_2]$ is in the relative interior of both $[F_1]$ and $[F_2]$.
\item[(b)] $F_1\cap F_2=\emptyset.$
\item[(c)] $[F_1]$ and $[F_2]$ intersect in a single point.
\end{itemize}
We first prove (a).
Suppose to the contrary that $p$ is in the boundary of $[F_1]$.
Then there is an $u \in F_1$ such that $p \in [F_1 -\{u\}]$.
Since $p \not \in [F_1 \cap F_2]$,
we have $p \in [F_1 -\{u\}] \cap [F_2] \setminus [(F_1- \{u\}) \cap F_2]$,
contradicting the minimality of $F_1$ and $F_2$.
Hence (a) holds.
Next we show (b).
Let $p \in [F_1] \cap [F_2] \setminus [F_1 \cap F_2]$.
By (a), there are convex combinations with positive coefficients
$\sum_{v\in F_1}a_v v=p=\sum_{v\in F_2}b_v v$ with $\# F_1 \geq 2$ and $\#F_2\geq 2$.
If there is $u\in F_1\cap F_2$, say with $a_u\leq b_u$, then by subtracting $a_u u$ from both sides and by normalizing them
we get a point $q$ which is contained in $[F_1-\{u\}] \cap [F_2]$.
Since $q$ is in the relative interior of $[F_1-\{u\}]$ by the construction,
we have $q \notin [(F_1- \{u\}) \cap F_2]$, contradicting the minimality.
Hence (b) holds.
We finally prove (c).
Suppose to the contrary that $[F_1] \cap [F_2]$ contains two different points $p$ and $q$.
Let $\ell$ be the line through them.
Then the endpoints of the line segment $\ell \cap [F_1]\cap[F_2]$ must be on the boundary of either $[F_1]$ or $[F_2]$,
contradicting (a) as $[F_1 \cap F_2]$ is empty by (b).
Hence (c) holds.
We now complete the proof of (i).
By (a) and (c), the intersection of $[F_1]$ and $[F_2]$ equals the intersection of their affine hulls,
as otherwise the neighborhood of $p$ in $[F_1]\cap[F_2]$ is not a single point.
This fact and (b) say $\# F_1+ \#F_2 \leq d+2$.
However,
since $F_1$ and $F_2$ are not in $\Delta$
and since $\Delta'=\Delta(d-r)$ and $\Delta$
have the same $(d-r)$-skeleton,
we have $\#F_1 \geq d-r+2$ and $\#F_2 \geq d-r+2$, a contradiction.
Hence we conclude that (i) holds.
(ii)
Lemma \ref{lem:NoMissingFaces} and Theorem \ref{thm:CM} show that $\Delta'$ is $d$-dimensional and pure, namely all of its facets have cardinality $d+1$.
Thus it is enough to show that if $F=\{v_{i_1},\dots,v_{i_{d+1}}\}$ is a facet of $\Delta'$ then $\dim [F]=d$. Suppose to the contrary that $\dim [F]<d$.
Then $v_{i_1},\dots,v_{i_{d+1}}$ are in the same hyperplane in $\RR^d$.
Thus by Radon's theorem there is a partition $F=F'\cup F''$ such that $[F']\cap [F'']\neq \emptyset$. This contradicts (i).
\end{proof}
Let $[\Delta']=\cup_{F \in \Delta'} [F]$ be the underlying space of the geometric simplicial complex $\{[F]:F \in \Delta'\}$.
To complete the proof of Theorem \ref{thm:GLBCpolytopes}, it is left to show
\begin{lemma}\label{lem:D'equalsP}
$[\Delta']=P$.
\end{lemma}
\begin{proof}
Observe that $[\Delta']\subseteq P$.
Assume by contradiction that there is a point $p\in P-[\Delta']$.
We assume that $[\Delta']$ and $P$ are embedded in $S^d$
via the natural homeomorphism $\RR^d \cong S^d - v \subset S^d$,
where $v$ is a point in $S^d$.
Let $q \in \RR^d - P$.
Since $[\Delta']$ contains the boundary of $P$,
$p$ and $q$ are in different connected components in $S^d-[\Delta']$.
Thus $S^d-[\Delta']$ is not connected.
By Alexander duality,
we have $\widetilde H_{d-1}([\Delta'];\QQ) \cong \widetilde H_0(S^d-[\Delta']; \QQ) \ne 0$.
Recall that $\Delta$ has the WLP over $\QQ$.
Thus $\Delta'$ is Cohen--Macaulay over $\QQ$ of dimension $d$ by Corollary \ref{cor:CM}. By Lemma \ref{lem:D'embedded} $[\Delta']$ is the underlying space of a geometric realization of $\Delta'$.
Thus Reisner's criterion (Lemma \ref{lem:Reisner}) says $\widetilde H_{d-1}([\Delta'];\QQ)=0$, a contradiction.
\end{proof}
\section{GLBC for Lefschetz spheres}\label{sec:LefschetzSpheres}
In this section we prove the existence part in Theorem \ref{thmIntro:GLBCspheres}.
The proof is algebraic
and we assume familiarity with $\ZZ^n$-graded commutative algebra theory.
See e.g.\ \cite{MillerSturmfels} for the basics of this theory.
First, we set some notation.
Let $\ee_i \in \ZZ^n$ be the $i$th unit vector of $\ZZ^n$.
We consider the $\ZZ^n$-grading of $S=\kk[x_1,...,x_n]$ defined by $\deg x_i = \ee_i$.
For a $\ZZ^n$-graded $S$-module $M$ and for $\aaa=(a_1,\dots,a_n) \in \ZZ^n$,
we denote by $M_\aaa$ the graded component of $M$ of degree $\aaa \in \ZZ^n$.
Let $\mideal=(x_1,\dots,x_n)$ be the graded maximal ideal of $S$.
We regard $\kk$ as a graded $S$-module by identification $\kk=S/\mideal$.
We recall a few known properties on $\Tor_i^S(\kk,-)$.
\begin{lemma}
\label{koszul}
Let $C$ be a graded $S$-module.
If $C_k =0$ for all $k \leq r$ then one has
$\Tor_{i}(\kk,C)_{i+j}= 0$ for all $i$ and $j \leq r$.
\end{lemma}
\begin{proof}
Let $\mathcal K_\bullet= \mathcal K_\bullet (x_1,\dots,x_n)$ be the Koszul complex of $x_1,\dots,x_n$ (see, e.g., \cite[\S 1.6]{Bruns-Herzog}).
Since $\mathcal K_\bullet$ is the minimal free resolution of $\kk$,
$$\Tor_i(\kk,C)_{i+j} \cong H_i(\mathcal K_\bullet \otimes C)_{i+j}.$$
On the other hand, all the elements in $\mathcal K_i$ have degree $\geq i$
and all the elements in $C$ have degree $\geq r+1$ by the assumption.
These facts imply that
$(\mathcal K_i \otimes C)_{i+j}=0$ for $j \leq r$.
Hence $H_i(\mathcal K_\bullet \otimes C)_{i+j}=0$ for all $j \leq r$.
\end{proof}
The following fact on generic initial ideals is well-known.
See \cite[Theorem 2.27]{Gr}.
\begin{lemma}[Bayer--Stillman]
\label{reg}
Suppose $\mathrm{char}(\kk)=0$.
Let $I \subset S$ be a homogeneous ideal.
If $\gin(I)$ is generated by monomials of degree $\leq m$
then
$\Tor_i^S(\kk,S/I)_{i+j}=0$ for all $j \geq m$.
\end{lemma}
We also recall some basic facts on canonical modules.
For a subset $F \subset [n]$,
let $\ee_F= \sum_{i \in F} \ee_i$.
For a Cohen--Macaulay $\ZZ^n$-graded ring $R=S/I$ of dimension $d$,
the module $\omega_R=\Ext_S^{n-d}(R,S(-\ee_{[n]}))$ is called the {\em canonical module} of $R$.
An important property of a canonical module is that
it is isomorphic to the Matlis dual of the local cohomology module $H_\mideal^d(R)$
by the local duality
(see \cite[Corollay 3.5.9]{Bruns-Herzog}).
Now suppose that $R=\kk[\Delta]$.
Then the local duality and the Hochster's formula for local cohomology \cite[Theorem 5.3.8]{Bruns-Herzog} imply that, for any $F \in \Delta$, one has
\begin{align}
\label{3}
\dim_\kk (\omega_{\kk[\Delta]})_{\ee_F}
= \dim_\kk (H_\mideal^d(\kk[\Delta]))_{-\ee_F}
=\dim_\kk \widetilde H_{d-1-\# F}(\mathrm{lk}_\Delta(F)).
\end{align}
Recall that by Reisner's criterion homology balls and spheres are Cohen--Macaulay.
The next result and Theorem \ref{thm:Uniqueness} prove Theorem \ref{thmIntro:GLBCspheres}.
\begin{theorem}\label{thm:GLBCLefschetzSpheres}
Suppose $\mathrm{char}(\kk)=0$.
Let $\Delta$ be a homology $(d-1)$-sphere having the WLP.
If $h_{r-1}(\Delta)=h_r(\Delta)$ for some $r \leq \frac d 2$
then $\Delta(r-1)$ is a homology $d$-ball whose boundary complex is $\Delta$.
\end{theorem}
\begin{proof}
\textit{Step 1:}
Let $\Delta'=\Delta(r-1)$ and $C=I_{\Delta}/I_{\Delta'}$.
For a graded $S$-module $M$, let $\mathrm{ann}_S(M)=\{g \in S: gf = 0 \mbox{ for all }f \in M\}.$
We first show that $C$ satisfies the following conditions:
\begin{enumerate}
\item[(i)] $\mathrm{ann}_S(C)=I_{\Delta'}$.
\item[(ii)] $C$ is Cohen--Macaulay of dimension $d+1$.
\item[(iii)] $\Tor_{n-d-1}^S (\kk,C)\cong \kk(-\ee_{[n]})$.
\end{enumerate}
(i)
$\mathrm{ann}_S(C) \supset I_{\Delta'}$ is clear.
It is enough to show that there is an element $f \in I_{\Delta}$
such that $ g f \not \in I_{\Delta'}$ for all $g \in S$ with $g \not \in I_{\Delta'}$.
Let $F_1,\dots,F_s$ be the facets of $\Delta'$.
By Corollary \ref{cor:CM} each $F_i$ is of size $d+1$.
We claim that the polynomial $f= \sum_{i=1}^s x_{F_i} \in I_\Delta$ satisfies the desired property.
To prove this,
since $C$ contains $\bigoplus_{i=1}^s x_{F_i} \cdot(S/(x_k: k \not \in F_i))$ as a submodule,
it is enough to show that, for any $g \not \in I_{\Delta'}$, $g x_{F_i} \ne 0$ in $x_{F_i} \cdot (S/(x_k: k \not \in F_i))$ for some $i$.
Moreover,
since $x_{F_i} \cdot (S/(x_k: k \not \in F_i))$ is $\ZZ^n$-graded,
we may assume that $g=x_{i_1}^{a_1} \cdots x_{i_t}^{a_t}$, where $a_1,\dots,a_t$ are not zero.
Since $x_{i_1}^{a_1} \cdots x_{i_t}^{a_t} \not \in I_{\Delta'}$, we have $\{i_1,\dots,i_t\} \in \Delta'$.
Then since $\Delta'$ is Cohen--Macaulay, $\Delta'$ is pure and there is a facet $F_i$ which contains $\{i_1,\dots,i_t\}$.
Then we have $x_{i_1}^{a_1} \cdots x_{i_t}^{a_t}x_{F_i} \ne 0$ in $x_{F_i}\cdot (S/(x_k: k \not \in F_i))$ as desired.
(ii)
Consider the
short exact sequence
\begin{align}
\label{2}
0 \longrightarrow
C\longrightarrow
S/I_{\Delta'}
\longrightarrow
S/I_{\Delta}
\longrightarrow
0.
\end{align}
Since $S/I_{\Delta'}$ is Cohen--Macaulay of dimension $d+1$
and since $S/I_{\Delta}$ is Cohen--Macaulay of dimension $d$,
we conclude that $C$ is Cohen--Macaulay of dimension $d+1$ (e.g.\ use the depth lemma \cite[Proposition 1.2.9]{Bruns-Herzog}).
(iii)
It remains to prove
$\Tor_{n-d-1}^S(\kk,C) \cong \kk(-\ee_{[n]})$.
Note that $\Tor_{n-d}^S(\kk,S/I_{\Delta'})=0$
since $S/I_{\Delta'}$ is Cohen--Macaulay of dimension $d+1$.
Then the short exact sequence \eqref{2} induces the exact sequence
$$
0
\longrightarrow
\Tor_{n-d}^S(\kk,S/I_{\Delta})_j
\longrightarrow
\Tor_{n-d-1}^S(\kk,C)_j
\longrightarrow
\Tor_{n-d-1}^S(\kk,S/I_{\Delta'})_j
\longrightarrow \cdots
$$
for all $j \geq 0$.
Since $\Delta$ is a homology $(d-1)$-sphere,
$\Tor_{n-d}^S(\kk,S/I_{\Delta})\cong \kk(-\ee_{[n]})$.
On the other hand,
since $\gin(I_{\Delta'})$ has no generators of degrees $\geq r+1$
as we showed in the proof of Theorem \ref{thm:CM},
we have
$\Tor_{n-d-1}^S(\kk,S/I_{\Delta'})_j=0$ for $j \geq n-d-1+r$
by Lemma \ref{reg}.
These facts and the exact sequence imply
$$\bigoplus_{j \geq n-d-1+r} \Tor_{n-d-1}^S(\kk,C)_j\cong \kk(-\ee_{[n]}).$$
On the other hand, since $I_{\Delta'}=(I_{\Delta})_{\leq r}$, we have $C_k = 0$ for $k \leq r$.
This implies $\Tor_{n-d-1}^S(\kk,C)_j=0$ for $j < n-d-1+r$ by Lemma \ref{koszul}, and (iii) follows.
\textit{Step 2:}
We show that $C \cong \Ext_S^{n-d-1}(\kk[\Delta'],S(-\ee_{[n]}))=\omega_{\kk[\Delta']}$.
It is standard in commutative algebra that conditions (i), (ii) and (iii) imply this isomorphism, but we include its proof.
Since $C$ is Cohen--Macaulay of dimension $d+1$,
it follows from \cite[Theorem 3.3.10]{Bruns-Herzog}
that
\begin{align}
\label{2-1}
\Ext_S^{n-d-1}\big(\Ext_S^{n-d-1}\big(C,S(-\ee_{[n]})\big),S(-\ee_{[n]})\big)=C.
\end{align}
On the other hand, by the duality on resolutions of $C$ and $\Ext_S^{n-d-1}(C,S(-\ee_{[n]}))$,
we have
$$\Tor_0^{S}(\kk,\Ext_S^{n-d-1}(C,S(-\ee_{[n]})))_{\aaa} \cong \Tor_{n-d-1}^S(\kk,C)_{\ee_{[n]}-\aaa}$$
for all $\aaa \in \ZZ^n$
(see \cite[Corollary 3.3.9]{Bruns-Herzog}).
Then the condition (iii) of Step 1 implies that $\Ext_S^{n-d-1}(C,S(-\ee_{[n]}))$ has a single generator in degree $0$,
so $\Ext_S^{n-d-1}(C,S(-\ee_{[n]}))\cong S/J$ for some ideal $J$.
We claim that $J=I_{\Delta'}$, equivalently $\mathrm{ann}_S(\Ext_S^{n-d-1}(C,S(-\ee_{[n]})))=I_{\Delta'}$.
Since $\mathrm{ann}_S(M) \subset \mathrm{ann}_S(\Hom_S(M,N))$ for all $S$-modules $M$ and $N$,
\eqref{2-1} says
$$
\mathrm{ann}_S(C) \subset \mathrm{ann}_S(\Ext_S^{n-d-1}(C,S(-\ee_{[n]})))
\subset \mathrm{ann}_S(C),
$$
which implies $\mathrm{ann}_S(\Ext_S^{n-d-1}(C,S(-\ee_{[n]})))=\mathrm{ann}_S(C)=I_{\Delta'}$ by (i) of Step 1.
Now $C \cong \Ext_S^{n-d-1}(\kk[\Delta'],S(-\ee_{[n]}))$ follows from \eqref{2-1}
since $\Ext_S^{n-d-1}(C,S(-\ee_{[n]})) \cong S/I_{\Delta'}=\kk[\Delta']$.
\textit{Step 3:}
We now prove the theorem.
By the Hochster's formula \eqref{3},
for any $F \in \Delta'$
we have
\begin{align*}
\dim_\kk \widetilde H_{d -\#F} \big(\mathrm{lk}_{\Delta'}(F)\big)=
\dim_\kk (\omega_{\kk[\Delta']})_{\ee_F}=
\dim_\kk (I_{\Delta}/I_{\Delta'})_{\ee_F}
=
\begin{cases}
1, & \mbox{ if } F \not \in \Delta, \\
0, & \mbox{ otherwise.}
\end{cases}
\end{align*}
Clearly the above equation
together with $\Delta'$ being Cohen--Macaulay
imply that $\Delta'$ is a homology ball whose boundary complex is equal to $\Delta$.
\end{proof}
The proof given in this section is quite algebraic.
It would be of interest to have a combinatorial or a topological proof of Theorem \ref{thm:GLBCLefschetzSpheres}.
\section{Concluding Remarks}\label{sec:conclude}
It is easy to see that ($1$-)stacked spheres are boundaries of stacked polytopes,
and that their stacked triangulations are shellable.
Then it is natural to ask
\begin{question}
\label{6-1}
Let $\Delta$ be an $(r-1)$-stacked $d$-ball with $r \leq \frac{d+1}{2}$.
Then
\begin{itemize}
\item[(i)] is it true that $\partial \Delta$ is polytopal?
\item[(ii)] is it true that $\Delta$ is shellable?
\end{itemize}
\end{question}
The next examples show that the answers to the above questions
are negative.
\begin{example}
\label{e1}
Let $B$ be Rudin's non-shellable triangulation of a $3$-ball \cite{Rudin}. Its $f$-vector is $(1,14,66,94,41)$ and its $h$-vector is $(1,10,30,0,0)$.
Let $K$ be the join of $B$ and a simplex $\sigma$ of dimension $k \geq 2$.
Then $K$ is a $(k+4)$-ball.
Also, the interior faces of $K$ are exactly those containing both $\sigma$ and an interior face of $B$.
Then, since $B$ contains no interior vertices,
$K$ is $2$-stacked.
On the other hand, $K$ is not shellable since $B$ is not shellable. Indeed, a shelling order on $K$ would induce a shelling order on $B$ by deleting $\sigma$ from all facets in the shelling order of $K$.
Also, $\partial K$ is non-polytopal. Indeed, assume the contrary, then for $v$ a vertex of $\sigma$, $\lk_{\partial K}(\sigma -\{v\})=B \cup (\{v\} * \partial B)$ is the boundary complex of a polytope. Thus, there is a Bruggesser--Mani \emph{line shelling} of $\lk_{\partial K}(\sigma -\{v\})$ which adds the facets with $v$ last (see \cite[Section 8.2]{Ziegler} for details),
so first it shells $B$, a contradiction.
\end{example}
\begin{example}
\label{e2}
There exists a large number of shellable $(r-1)$-stacked $d$-balls with $r \leq \frac d 2$
whose boundary is non-polytopal. Indeed, fixing $d$,
Goodman and Pollack \cite{GoodmanPollack:fewPolytopes-86} showed that the log of the number of combinatorial types of boundaries of simplicial $d$-polytopes on $n$ vertices is at most $O(n \log(n))$.
On the other hand,
the log of the number of Kalai's squeezed $(d-1)$-spheres satisfying $h_{r-1}=h_r$, where $r \leq \frac d 2$,
is at least $\Omega(n^{r-2})$
(see \cite{Kalai-manyspheres} for the details).
Since Kalai's squeezed spheres satisfying $h_{r-1}=h_r$ are known to be the boundaries of $(r-1)$-stacked shellable balls (see \cite{Kalai-manyspheres} and \cite{Kleinschmidt-Lee84} for details),
they give a large number of $(r-1)$-stacked triangulations of a $d$-ball whose boundary is non-polytopal when $r \geq 4$.
\end{example}
Although the answers to Question \ref{6-1} are negative in general,
it would be of interest to study these problems for special cases.
Below, we write a few open questions on stacked balls and spheres.
\begin{conjecture}\label{Q:shellablePolytopes}
Let $P$ be an $(r-1)$-stacked $d$-polytope with $r \leq \frac{d+1}{2}$.
\begin{itemize}
\item[(i)] (McMullen \cite{McMullen:Triangulations})
The $(r-1)$-stacked triangulation of $P$ is {\em regular}.
\item[(ii)] (Bagchi--Datta \cite{Bagchi-Datta:StellatedSpheres})
The $(r-1)$-stacked triangulation of $P$ is shellable.
\end{itemize}
\end{conjecture}
Note that Conjecture \ref{Q:shellablePolytopes}(i)
implies Conjecture \ref{Q:shellablePolytopes}(ii).
McMullen's original conjecture considered the case $r \leq \frac d 2$,
but we want to include the case $r= \frac {d+1} 2$ in view of Theorem \ref{thm:Uniqueness}.
Also, it would be of interest to study the geometric meaning of the triangulation given in Theorem \ref{thmIntro:GLBCpolytopes}.
We see that there exists a non-shellable $2$-stacked ball whose boundary is non-polytopal in Example \ref{e1},
and that there even exists a shellable $3$-stacked balls whose boundary is non-polytopal in Example \ref{e2}.
But the following question is open.
\begin{question}\label{Q:polytopal}
Let $\Delta$ be a $2$-stacked triangulation of a $d$-ball which is shellable.
Is $\partial \Delta$ polytopal?
\end{question}
Finally, we raise the following question concerning Theorem \ref{thmIntro:GLBCspheres}.
\begin{question}
With the same notation as in Theorem \ref{thmIntro:GLBCspheres},
is it true that if $\Delta$ is a triangulation of a sphere then $\Delta(d-r)$ is a triangulation of a ball?
\end{question}
It seems to be plausible that if $\Delta$ is a PL-sphere then $\Delta(d-r)$ is a PL-ball.
But we do not have an answer even for this case.
\section*{Acknowledgments}
We would like to thank Gil Kalai and Isabella Novik for helpful comments on an earlier version of this paper.
|
{
"timestamp": "2012-04-06T02:03:00",
"yymm": "1203",
"arxiv_id": "1203.1720",
"language": "en",
"url": "https://arxiv.org/abs/1203.1720"
}
|
\section{Introduction}
The symmetric derivative of function $f$ at point $x$
is defined as $\lim_{h \rightarrow 0} (f(x+h)-f(x-h))/(2h)$.
The notion of symmetrically differentiable is interesting
because if a function is differentiable at a point then
it is also symmetrically differentiable, but the converse
is not true. The best known example of this fact
is the absolute value function: $f(x) = |x|$ is not
differentiable at $x = 0$ but is symmetrically differentiable
at $x = 0$ with symmetric derivative zero \cite{Thomsom}.
Quantum calculus is, roughly speaking, the equivalent to traditional
infinitesimal calculus but without limits \cite{Kac}. Therefore,
one can introduce the symmetric quantum derivative of $f$ at $x$
by $(f(x+h)-f(x-h))/(2h)$. As in any calculus, it is then natural
to develop a corresponding integration theory, looking to such integral
as the inverse operator of the derivative.
The main goal of this paper is to study
the properties of a general symmetric quantum integral
that we call, due to the so-called N\"{o}rlund sum \cite{Kac},
the $\alpha,\beta$-symmetric N\"{o}rlund sum.
The paper is organized as follows. In Section~\ref{sec:2}
we define the forward and backward N\"{o}rlund sums.
Then, in Section~\ref{sec:sns}, we introduce the
$\alpha,\beta$-symmetric N\"{o}rlund sum and give some
of its properties. We end with Section~\ref{sec:ineq},
proving $\alpha,\beta$-symmetric versions of H\"{o}lder's,
Cauchy--Schwarz's and Minkowski's inequalities.
\section{Forward and backward N\"{o}rlund sums}
\label{sec:2}
This section is dedicated to the inverse operators of
the $\alpha$-forward and $\beta$-backward differences,
$\alpha > 0$, $\beta > 0$, defined respectively by
\[
\Delta_{\alpha}\left[ f\right] \left( t\right) :=\frac{f\left(
t+\alpha\right) -f\left( t\right) }{\alpha}\, , \quad
\nabla_{\beta}\left[ f\right] \left( t\right) :=\frac{f\left( t\right)
-f\left(t-\beta\right)}{\beta}.
\]
\begin{definition}
\label{def:o1}
Let $I \subseteq \mathbb{R}$ be such that
$a,b\in I$ with $a<b$ and $\sup I=+\infty$.
For $f:I\rightarrow\mathbb{R}$ and $\alpha >0$ we define the N\"{o}rlund sum
(the $\alpha$-forward integral) of $f$ from $a$ to $b$ by
\[
\int_{a}^{b}f\left( t\right) \Delta_{\alpha}t=\int_{a}^{+\infty}f\left(
t\right) \Delta_{\alpha}t-\int_{b}^{+\infty}f\left( t\right)
\Delta_{\alpha}t,
\]
where
$\displaystyle \int_{x}^{+\infty}f\left( t\right) \Delta_{\alpha}t=\alpha\sum
_{k=0}^{+\infty}f\left( x+k\alpha\right)$,
provided the series converges at $x=a$ and $x=b$. In that case, $f$ is
said to be $\alpha$-forward integrable on $\left[ a,b\right]$. We say that $f$
is $\alpha$-forward integrable over $I$ if it is $\alpha$-forward integrable
for all $a,b\in I$.
\end{definition}
Until Definition~\ref{beta} (the backward/nabla case),
we assume that $I$ is an interval of $\mathbb{R}$
such that $\sup I=+\infty$. Note that if
$f:I\rightarrow\mathbb{R}$ is a function
such that $\sup I<+\infty$, then we can extend function
$f$ to $\tilde{f}:\tilde{I}\rightarrow\mathbb{R}$,
where $\tilde{I}$ is an interval with $\sup\tilde{I}=+\infty$,
in the following way: $\tilde{f}|_{I}=f$ and
$\tilde{f}|_{\tilde{I}\backslash I}=0$.
Using the techniques of Aldwoah in his Ph.D. thesis \cite{Aldwoah},
it can be proved that the $\alpha$-forward integral has the following properties:
\begin{theorem}
If $f,g: I \rightarrow\mathbb{R}$ are
$\alpha$-forward integrable on $[a,b]$,
$c\in\left[a,b\right]$, $k\in\mathbb{R}$, then
\begin{enumerate}
\item $\displaystyle\int_{a}^{a}f\left( t\right)\Delta_{\alpha}t=0$;
\item $\displaystyle\int_{a}^{b}f\left( t\right)\Delta_{\alpha}t
=\int_{a}^{c}f\left( t\right) \Delta_{\alpha}t+\int_{c}^{b}f\left( t\right)
\Delta_{\alpha}t$, when the integrals exist;
\item $\displaystyle\int_{a}^{b}f\left( t\right)\Delta_{\alpha}t
=-\int_{b}^{a}f\left( t\right) \Delta_{\alpha}t$;
\item $kf$ is $\alpha$-forward integrable on $\left[a,b\right]$ and
$\displaystyle \int_{a}^{b}kf\left(t\right)\Delta_{\alpha}t
=k\int_{a}^{b}f\left(t\right) \Delta_{\alpha}t$;
\item $f+g$ is $\alpha$-forward integrable on $\left[a,b\right]$ and
\[
\int_{a}^{b}\left( f+g\right) \left( t\right) \Delta_{\alpha}t=\int
_{a}^{b}f\left( t\right) \Delta_{\alpha}t+\int_{a}^{b}g\left( t\right)
\Delta_{\alpha}t\text{;}
\]
\item if $f\equiv0$, then $\displaystyle\int_{a}^{b}f\left( t\right)
\Delta_{\alpha}t=0$.
\end{enumerate}
\end{theorem}
\begin{theorem}
Let $f: I \rightarrow\mathbb{R}$ be $\alpha$-forward integrable
on $\left[a,b\right]$.
If $g:I\rightarrow\mathbb{R}$ is a nonnegative
$\alpha$-forward integrable function on $\left[a,b\right]$,
then $fg$ is $\alpha$-forward integrable on $\left[a,b\right]$.
\end{theorem}
\begin{proof}
Since $g$ is $\alpha$-forward integrable, then both series
$\alpha\sum_{k=0}^{+\infty}g\left( a+k\alpha\right)$
and $\alpha\sum_{k=0}^{+\infty}g\left( b+k\alpha\right)$
converge. We want to study the nature of series
$\alpha\sum_{k=0}^{+\infty}fg\left( a+k\alpha\right)$
and $\alpha\sum_{k=0}^{+\infty}fg\left( b+k\alpha\right)$.
Since there exists an order $N\in\mathbb{N}$ such that
$\left\vert fg\left( b+k\alpha\right) \right\vert \leqslant g\left(
b+k\alpha\right)$ and $\left\vert fg\left( a+k\alpha\right)
\right\vert \leqslant g\left( a+k\alpha\right)$
for all $k>N$, then both
$\alpha\sum_{k=0}^{+\infty}fg\left(a+k\alpha\right)$
and $\alpha\sum_{k=0}^{+\infty}fg\left( b+k\alpha\right)$
converge absolutely. The intended conclusion follows.
\end{proof}
\begin{theorem}
\label{p}
Let $f:I\rightarrow\mathbb{R}$ and $p>1$.
If $\left\vert f\right\vert $ is $\alpha$-forward integrable
on $\left[ a,b\right]$, then $\left\vert f\right\vert ^{p}$
is also $\alpha$-forward integrable on $\left[a,b\right]$.
\end{theorem}
\begin{proof}
There exists $N\in\mathbb{N}$ such that
$\left\vert f\left(b+k\alpha\right)\right\vert^{p}
\leqslant
\left\vert f\left(b+k\alpha\right) \right\vert$
and
$\left\vert f\left(a+k\alpha\right)\right\vert^{p}
\leqslant
\left\vert f\left(a+k\alpha\right) \right\vert$
for all $k>N$. Therefore, $\left\vert f\right\vert^{p}$
is $\alpha$-forward integrable on $\left[a,b\right]$.
\end{proof}
\begin{theorem}
\label{desigualdade}
Let $f,g:I\rightarrow\mathbb{R}$ be $\alpha$-forward integrable on $\left[ a,b\right]$.
If $\left\vert f\left( t\right) \right\vert \leqslant g\left( t\right)$
for all $t\in\left\{ a+k\alpha:k\in\mathbb{N}_{0}\right\}$,
then for $b\in\left\{ a+k\alpha:k\in\mathbb{N}_{0}\right\}$ one has
\[
\left\vert \int_{a}^{b}f\left( t\right) \Delta_{\alpha}t\right\vert
\leqslant\int_{a}^{b}g\left( t\right) \Delta_{\alpha}t.
\]
\end{theorem}
\begin{proof}
Since $b\in\left\{ a+k\alpha:k\in\mathbb{N}_{0}\right\}$,
there exists $k_{1}$ such that $b=a+k_{1}\alpha$. Thus,
\begin{align*}
\left\vert \int_{a}^{b}f\left( t\right) \Delta_{\alpha}t\right\vert &
=\left\vert \alpha\sum_{k=0}^{+\infty}f\left( a+k\alpha\right) -\alpha
\sum_{k=0}^{+\infty}f\left( a+\left( k_{1}+k\right) \alpha\right)
\right\vert \\
&=\left\vert \alpha\sum_{k=0}^{+\infty}f\left( a+k\alpha\right) -\alpha
\sum_{k=k_{1}}^{+\infty}f\left( a+k\alpha\right) \right\vert
=\left\vert \alpha\sum_{k=0}^{k_{1}-1}f\left( a+k\alpha\right) \right\vert\\
&\leqslant \alpha\sum_{k=0}^{k_{1}-1}\left\vert f\left( a+k\alpha\right)
\right\vert
\leqslant\alpha\sum_{k=0}^{k_{1}-1}g\left( a+k\alpha\right) \\
& =\alpha\sum_{k=0}^{+\infty}g\left( a+k\alpha\right) -\alpha\sum_{k=k_{1}
}^{+\infty}g\left( a+k\alpha\right)
=\int_{a}^{b}g\left( t\right) \Delta_{\alpha}t.
\end{align*}
\end{proof}
\begin{corollary}
\label{desigualdade2}
Let $f,g:I\rightarrow\mathbb{R}$ be
$\alpha$-forward integrable on $\left[a,b\right]$
with $b = a+k\alpha$ for some $k\in\mathbb{N}_{0}$.
\begin{enumerate}
\item If $f\left( t\right) \geqslant 0$
for all $t\in\left\{ a+k\alpha:k\in\mathbb{N}_{0}\right\}$,
then $\int_{a}^{b}f\left( t\right) \Delta_{\alpha}t \geqslant 0$.
\item If $g\left( t\right) \geqslant f$ $\left( t\right)$ for all
$t\in\left\{ a+k\alpha:k\in\mathbb{N}_{0}\right\}$, then
$\int_{a}^{b}g\left( t\right) \Delta_{\alpha}t\geqslant\int_{a}^{b}f\left(
t\right) \Delta_{\alpha}t$.
\end{enumerate}
\end{corollary}
We can now prove the following fundamental
theorem of the $\alpha$-forward calculus.
\begin{theorem}[Fundamental theorem of N\"{o}rlund calculus]
Let $f:I\rightarrow\mathbb{R}$ be $\alpha$-forward integrable
over $I$. Let $x\in I$ and define
$F\left( x\right) :=\int_{a}^{x}f\left( t\right) \Delta_{\alpha}t$.
Then, $\Delta_{\alpha}\left[ F\right] \left( x\right) =f\left( x\right)$.
Conversely,
$\int_{a}^{b}\Delta_{\alpha}\left[ f\right] \left( t\right) \Delta_{\alpha}t
=f\left( b\right) -f\left( a\right)$.
\end{theorem}
\begin{proof}
If $G\left( x\right)=
-\int_{x}^{+\infty}f\left(t\right)\Delta_{\alpha}t$, then
\begin{align*}
\Delta_{\alpha}\left[ G\right] \left( x\right)
&=\frac{G\left(x+\alpha\right) -G\left( x\right) }{\alpha}
=\frac{-\alpha\sum_{k=0}^{+\infty}f\left( x+\alpha+k\alpha\right)
+\alpha\sum_{k=0}^{+\infty}f\left( x+k\alpha\right) }{\alpha}\\
& =\sum_{k=0}^{+\infty}f\left( x+k\alpha\right)
-\sum_{k=0}^{+\infty}f\left( x+\left( k+1\right) \alpha\right)
=f\left( x\right).
\end{align*}
Therefore,
$\Delta_{\alpha}\left[ F\right] \left( x\right)
=\Delta_{\alpha}\left(\int_{a}^{+\infty}f\left( t\right)
\Delta_{\alpha}t-\int_{x}^{+\infty}
f\left( t\right) \Delta_{\alpha}t\right)
=f\left( x\right)$.
Using the definition of $\alpha$-forward difference operator, the second part
of the theorem is also a consequence of the properties of Mengoli's series.
Since
\begin{align*}
\int_{a}^{+\infty}\Delta_{\alpha}\left[ f\right] \left( t\right)
\Delta_{\alpha}t & =\alpha\sum_{k=0}^{+\infty}\Delta_{\alpha}\left[
f\right] \left( a+k\alpha\right)
=\alpha\sum_{k=0}^{+\infty}\frac{f\left( a+k\alpha+\alpha\right) -f\left(
a+k\alpha\right) }{\alpha}\\
& =\sum_{k=0}^{+\infty}\bigg(f\left( a+\left( k+1\right) \alpha\right)
-f\left( a+k\alpha\right) \bigg) =-f\left( a\right)
\end{align*}
and
$\int_{b}^{+\infty}\Delta_{\alpha}\left[ f\right] \left( t\right)
\Delta_{\alpha}t=-f\left( b\right)$,
it follows that
\begin{equation*}
\int_{a}^{b}\Delta_{\alpha}\left[ f\right] \left( t\right) \Delta_{\alpha}t
=\int_{a}^{+\infty}f\left( t\right) \Delta_{\alpha}t-\int_{b}^{+\infty}
f\left( t\right) \Delta_{\alpha}t
=f\left( b\right) -f\left( a\right).
\end{equation*}
\end{proof}
\begin{corollary}[$\alpha$-forward integration by parts]
\label{partes}
Let $f,g:I\rightarrow\mathbb{R}$.
If $f g$ and $f\Delta_{\alpha}\left[ g\right] $ are $\alpha
$-forward integrable on $\left[ a,b\right] $, then
\[
\int_{a}^{b}f\left( t\right) \Delta_{\alpha}\left[ g\right] \left(
t\right) \Delta_{\alpha}t=f\left( t\right) g\left( t\right)
\bigg|_{a}^{b}-\int_{a}^{b}\Delta_{\alpha}\left[ f\right] \left( t\right)
g\left( t+\alpha\right) \Delta_{\alpha}t
\]
\end{corollary}
\begin{proof}
Since
$\Delta_{\alpha}\left[ fg\right] \left( t\right) =\Delta_{\alpha}\left[
f\right] \left( t\right) g\left( t+\alpha\right) +f\left( t\right)
\Delta_{\alpha}\left[ g\right] \left( t\right)$, then
\begin{align*}
\int_{a}^{b}f\left( t\right) \Delta_{\alpha}\left[ g\right] \left(
t\right) \Delta_{\alpha}t & =\int_{a}^{b}\bigg(\Delta_{\alpha}\left[
fg\right] \left( t\right) -\Delta_{\alpha}\left[ f\right] \left(
t\right) g\left( t+\alpha\right) \bigg)\Delta_{\alpha}t\\
& =\int_{a}^{b}\Delta_{\alpha}\left[ fg\right] \left( t\right)
\Delta_{\alpha}t-\int_{a}^{b}\Delta_{\alpha}\left[ f\right] \left(
t\right) g\left( t+\alpha\right) \Delta_{\alpha}t\\
& =f\left( t\right) g\left( t\right) \bigg|_{a}^{b}-\int_{a}^{b}
\Delta_{\alpha}\left[ f\right] \left( t\right) g\left( t+\alpha\right)
\Delta_{\alpha}t\text{.}
\end{align*}
\end{proof}
\begin{remark}
Our study of the N\"{o}rlund sum is in agreement with
the Hahn quantum calculus \cite{Aldwoah,withMiguel01,MalinowskaTorres}.
In \cite{Kac}
$\int_{a}^{b}f\left( t\right) \Delta_{\alpha}t=\alpha\left[ f\left(
a\right) +f\left( a+\alpha\right)
+ \cdots +f\left( b-\alpha\right) \right]$
for $a<b$ such that $b-a\in\alpha\mathbb{Z}$,
$\alpha\in\mathbb{R}^{+}$. In contrast with \cite{Kac}, our definition is valid
for any two real points $a,b$ and not only for those points belonging to the time
scale $\alpha\mathbb{Z}$. The definitions (only) coincide
if function $f$ is $\alpha$-forward
integrable on $\left[a,b\right]$.
\end{remark}
Similarly, we introduce the $\beta$-backward integral.
\begin{definition}
\label{beta}
Let $I$ be an interval of $\mathbb{R}$ such that $a,b\in I$
with $a<b$ and $\inf I=-\infty$. For $f:I\rightarrow\mathbb{R}$
and $\beta >0$ we define the $\beta$-backward integral of $f$ from $a$ to $b$ by
\[
\int_{a}^{b}f\left( t\right) \nabla_{\beta}t=\int_{-\infty}^{b}f\left(
t\right) \nabla_{\beta}t-\int_{-\infty}^{a}f\left( t\right) \nabla_{\beta}t,
\]
where $\displaystyle \int_{-\infty}^{x}f\left( t\right)\nabla_{\beta}t
=\beta\sum_{k=0}^{+\infty}f\left( x-k\beta\right)$,
provided the series converges at $x=a$ and $x=b$. In that case, $f$ is
called $\beta$-backward integrable on $\left[a,b\right]$. We say that $f$
is $\beta$-backward integrable over $I$ if it is $\beta$-backward integrable
for all $a,b\in I$.
\end{definition}
The $\beta$-backward N\"{o}rlund sum has similar results and properties as the
$\alpha$-forward N\"{o}rlund sum. In particular, the $\beta$-backward integral
is the inverse operator of $\nabla_\beta$.
\section{The $\alpha,\beta$-symmetric N\"{o}rlund sum}
\label{sec:sns}
We define the $\alpha,\beta$-symmetric integral
as a linear combination of the $\alpha$-forward
and the $\beta$-backward integrals.
\begin{definition}
\label{def:3}
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ and $a,b\in\mathbb{R}$, $a<b$.
If $f$ is $\alpha$-forward and $\beta$-backward integrable
on $\left[ a,b\right]$, $\alpha, \beta \ge 0$ with $\alpha + \beta > 0$,
then we define the $\alpha,\beta$-symmetric integral of $f$ from $a$ to $b$ by
\[
\int_{a}^{b}f\left( t\right) d_{\alpha,\beta}t=\frac{\alpha}{\alpha+\beta
}\int_{a}^{b}f\left( t\right) \Delta_{\alpha}t+\frac{\beta}{\alpha+\beta
}\int_{a}^{b}f\left( t\right) \nabla_{\beta}t\text{.}
\]
Function $f$ is $\alpha,\beta$-symmetric integrable if it is
$\alpha,\beta$-symmetric integrable for all $a,b\in\mathbb{R}$.
\end{definition}
\begin{remark}
Note that if $ \alpha\in\mathbb{R}^{+}$ and $\beta=0$,
then $\displaystyle\int_{a}^{b}f\left( t\right)
d_{\alpha,\beta}t=\int_{a}^{b}f\left( t\right) \Delta_{\alpha}t$
and we do not need to assume in Definition~\ref{def:3} that
$f$ is $\beta$-backward integrable;
if $\alpha=0$ and $\beta\in\mathbb{R}^{+}$, then
$\displaystyle\int_{a}^{b}f\left( t\right) d_{\alpha,\beta}t
=\int_{a}^{b}f\left( t\right) \nabla_{\beta}t$
and we do not need to assume that
$f$ is $\alpha$-forward integrable.
\end{remark}
\begin{example}
Let $f\left(t\right)= 1/t^{2}$. Then
$\displaystyle \int_{1}^{3}\frac{1}{t^{2}}d_{2,2}t = \frac{10}{9}$.
\end{example}
The $\alpha,\beta$-symmetric integral has the following properties:
\begin{theorem}
\label{propriedades}
Let $f,g:\mathbb{R}\rightarrow\mathbb{R}$
be $\alpha,\beta$-symmetric integrable on $\left[a,b\right]$.
Let $c\in\left[ a,b\right]$ and $k\in\mathbb{R}$. Then,
\begin{enumerate}
\item $\displaystyle\int_{a}^{a}f\left( t\right) d_{\alpha,\beta}t=0$;
\item $\displaystyle\int_{a}^{b}f\left( t\right) d_{\alpha,\beta}t=\int
_{a}^{c}f\left( t\right) d_{\alpha,\beta}t+\int_{c}^{b}f\left( t\right)
d_{\alpha,\beta}t$, when the integrals exist;
\item $\displaystyle\int_{a}^{b}f\left( t\right) d_{\alpha,\beta}t
=-\int_{b}^{a}f\left( t\right) d_{\alpha,\beta}t$;
\item $kf$ is $\alpha,\beta$-symmetric integrable on $\left[a,b\right]$
and
$\displaystyle \int_{a}^{b}kf\left(t\right) d_{\alpha,\beta}t
=k\int_{a}^{b}f\left(t\right) d_{\alpha,\beta}t$;
\item $f+g$ is $\alpha,\beta$-symmetric integrable
on $\left[a,b\right]$ and
\[
\int_{a}^{b}\left( f+g\right) \left( t\right) d_{\alpha,\beta}t=\int
_{a}^{b}f\left( t\right) d_{\alpha,\beta}t+\int_{a}^{b}g\left( t\right)
d_{\alpha,\beta}t\text{;}
\]
\item $fg$ is $\alpha,\beta$-symmetric integrable on $\left[ a,b\right]$
provided $g$ is a nonnegative function.
\end{enumerate}
\end{theorem}
\begin{proof}
These results are easy consequences of the $\alpha$-forward
and $\beta$-backward integral properties.
\end{proof}
The next result follows immediately from Theorem~\ref{p} and the
corresponding $\beta$-backward version.
\begin{theorem}
\label{modulo}
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ and $p>1$.
If $\left\vert f\right\vert $ is symmetric $\alpha,\beta$-integrable
on $\left[ a,b\right]$, then $\left\vert f\right\vert ^{p}$
is also $\alpha,\beta$-symmetric integrable on $\left[a,b\right]$.
\end{theorem}
\begin{theorem}
\label{des}
Let $f,g:\mathbb{R}\rightarrow\mathbb{R}$
be $\alpha,\beta$-symmetric integrable functions on $\left[a,b\right]$,
$\mathcal{A} := \left\{ a+k\alpha:k\in\mathbb{N}_{0}\right\}$
and $\mathcal{B} := \left\{ b-k\beta:k\in\mathbb{N}_{0}\right\}$.
For $b\in\mathcal{A}$ and $a\in\mathcal{B}$ one has:
\begin{enumerate}
\item if $\left\vert f\left( t\right) \right\vert
\leqslant g\left(t\right)$ for all $t\in\mathcal{A}\cup\mathcal{B}$, then
$\displaystyle \left\vert \int_{a}^{b}f\left( t\right) d_{\alpha,\beta}t\right\vert
\leqslant\int_{a}^{b}g\left( t\right) d_{\alpha,\beta}t$;
\item if $f\left( t\right) \geqslant0$ for all
$t\in\mathcal{A}\cup\mathcal{B}$, then
$\displaystyle \int_{a}^{b}f\left( t\right) d_{\alpha,\beta}t\geqslant0$;
\item if $g\left( t\right) \geqslant f\left( t\right)$
for all $t\in\mathcal{A}\cup\mathcal{B}$, then
$\displaystyle \int_{a}^{b}g\left( t\right)
d_{\alpha,\beta}t\geqslant\int_{a}^{b}f\left(t\right)d_{\alpha,\beta}t$.
\end{enumerate}
\end{theorem}
\begin{proof}
It follows from Theorem~\ref{desigualdade} and Corollary~\ref{desigualdade2}
and the corresponding $\beta$-backward versions.
\end{proof}
In Theorem~\ref{thm:mvt} we assume that $a,b\in\mathbb{R}$ with
$b\in\mathcal{A} := \left\{ a+k\alpha:k\in\mathbb{N}_{0}\right\}$ and
$a\in\mathcal{B} := \left\{ b-k\beta:k\in\mathbb{N}_{0}\right\}$, where
$\alpha,\beta\in\mathbb{R}_{0}^{+}$, $\alpha + \beta \ne 0$.
\begin{theorem}[Mean value theorem]
\label{thm:mvt}
Let $f,g:\mathbb{R}\rightarrow\mathbb{R}$
be bounded and $\alpha,\beta$-symmetric integrable on $[a,b]$
with $g$ nonnegative. Let $m$ and $M$ be
the infimum and the supremum, respectively, of function $f$.
Then, there exists a real number $K$ satisfying the inequalities
$m\leqslant K\leqslant M$ such that
$\displaystyle \int_{a}^{b}f\left( t\right) g\left( t\right) d_{\alpha,\beta}t
=K\int_{a}^{b}g\left( t\right) d_{\alpha,\beta}t$.
\end{theorem}
\begin{proof}
Since $m\leqslant f\left( t\right) \leqslant M\text{ for all }t\in\mathbb{R}$
and $g\left( t\right) \geqslant 0$, then
$mg\left( t\right) \leqslant f\left( t\right) g\left( t\right) \leqslant
Mg\left( t\right)$ for all $t\in\mathcal{A} \cup \mathcal{B}$.
All functions $mg$, $fg$ and $Mg$ are
$\alpha,\beta$-symmetric integrable on $[a,b]$.
By Theorems~\ref{propriedades} and \ref{des},
$m\int_{a}^{b}g\left( t\right) d_{\alpha,\beta}t
\leqslant\int_{a}^{b}f\left(t\right) g\left( t\right) d_{\alpha,\beta}t$
$\leqslant M\int_{a}^{b}g\left(
t\right) d_{\alpha,\beta}t$.
If $\int_{a}^{b}g\left( t\right) d_{\alpha,\beta}t=0$, then
$\int_{a}^{b}f\left( t\right) g\left( t\right)
d_{\alpha,\beta}t=0$; if $\int_{a}^{b}g\left( t\right)
d_{\alpha,\beta}t>0$, then
$m\leqslant\frac{\int_{a}^{b}f\left( t\right) g\left( t\right)
d_{\alpha,\beta}t}{\int_{a}^{b}g\left( t\right)
d_{\alpha,\beta}t} \leqslant M$.
Therefore, the middle term of these inequalities
is equal to a number $K$, which yields the intended result.
\end{proof}
\section{$\alpha,\beta$-Symmetric Integral Inequalities}
\label{sec:ineq}
Inspired in the work by Agarwal et al. \cite{Agarval},
we now present $\alpha,\beta$-symmetric versions of H\"{o}lder,
Cauchy--Schwarz and Minkowski inequalities. As before,
we assume that $a,b\in\mathbb{R}$ with
$b\in\mathcal{A} := \left\{ a+k\alpha:k\in\mathbb{N}_{0}\right\}$ and
$a\in\mathcal{B} := \left\{ b-k\beta:k\in\mathbb{N}_{0}\right\}$, where
$\alpha,\beta\in\mathbb{R}_{0}^{+}$, $\alpha + \beta \ne 0$.
\begin{theorem}[H\"{o}lder's inequality]
\label{Holders Inequality}
Let $f,g:\mathbb{R}\rightarrow\mathbb{R}$
and $a,b\in\mathbb{R}$ with $a<b$.
If $\left\vert f\right\vert $ and $\left\vert g\right\vert$ are
$\alpha,\beta$-symmetric integrable on $\left[a,b\right]$, then
\begin{equation}
\label{eq:hi}
\int_{a}^{b}\left\vert f\left( t\right) g\left( t\right) \right\vert
d_{\alpha,\beta}t\leqslant\left( \int_{a}^{b}\left\vert f\left( t\right)
\right\vert ^{p}d_{\alpha,\beta}t\right) ^{\frac{1}{p}}\left(
\int_{a}^{b}\left\vert g\left( t\right)\right\vert^{q}
d_{\alpha,\beta}t\right)^{\frac{1}{q}},
\end{equation}
where $p>1$ and $q=p/(p-1)$.
\end{theorem}
\begin{proof}
For $\alpha,\beta\in\mathbb{R}_{0}^{+}$, $\alpha + \beta \ne 0$,
the following inequality holds:
$\alpha^{\frac{1}{p}}\beta^{\frac{1}{q}}\leqslant\frac{\alpha}{p}+\frac{\beta}{q}$.
Without loss of generality, suppose that
$\displaystyle \left( \int_{a}^{b}\left\vert f\left( t\right) \right\vert ^{p}
d_{\alpha,\beta}t\right) \left( \int_{a}^{b}\left\vert g\left( t\right)
\right\vert ^{q}d_{\alpha,\beta}t\right) \neq 0$
(note that both integrals exist by Theorem~\ref{modulo}). Set
$\xi\left( t\right)
=\left\vert f\left( t\right) \right\vert ^{p}/
\int_{a}^{b}\left\vert f\left( \tau\right) \right\vert ^{p}d_{\alpha,\beta}\tau$
and $\gamma\left( t\right)
=\left\vert g\left(t\right)\right\vert ^{q}/\int_{a}^{b}\left\vert g\left(\tau\right)
\right\vert ^{q}d_{\alpha,\beta}\tau$.
Since both functions $\alpha$ and $\beta$ are symmetric
$\alpha,\beta$-integrable on $\left[a,b\right]$, then \eqref{eq:hi} holds:
\begin{align*}
\int_{a}^{b}&\frac{\left\vert f\left( t\right) \right\vert }{\left(
\int_{a}^{b}\left\vert f\left( \tau\right) \right\vert ^{p}d_{\alpha,\beta
}\tau\right) ^{\frac{1}{p}}}\frac{\left\vert g\left( t\right) \right\vert
}{\left( \int_{a}^{b}\left\vert g\left( \tau\right) \right\vert
^{q}d_{\alpha,\beta}\tau\right) ^{\frac{1}{q}}}d_{\alpha,\beta}t
=\int_{a}^{b}\xi\left( t\right) ^{\frac{1}{p}}\gamma\left( t\right)
^{\frac{1}{q}}d_{\alpha,\beta}t\\
& \leqslant\int_{a}^{b}\left( \frac{\xi\left( t\right) }{p}+\frac
{\gamma\left( t\right) }{q}\right) d_{\alpha,\beta}t\\
& =\frac{1}{p}\int_{a}^{b}\left( \frac{\left\vert f\left( t\right)
\right\vert ^{p}}{\int_{a}^{b}\left\vert f\left( \tau\right) \right\vert
^{p}d_{\alpha,\beta}\tau}\right) d_{\alpha,\beta}t
+\frac{1}{q}\int_{a}^{b}\left( \frac{\left\vert g\left( t\right)
\right\vert ^{q}}{\int_{a}^{b}\left\vert g\left( \tau\right)
\right\vert ^{q}d_{\alpha,\beta}\tau}\right) d_{\alpha,\beta}t = 1.
\end{align*}
\end{proof}
The particular case $p=q=2$ of \eqref{eq:hi}
gives the Cauchy--Schwarz inequality.
\begin{corollary}[Cauchy--Schwarz's inequality]
Let $f,g:\mathbb{R}\rightarrow\mathbb{R}$
and $a,b\in\mathbb{R}$ with $a<b$. If $f$ and $g$
are $\alpha,\beta$-symmetric integrable on $\left[ a,b\right]$, then
\[
\int_{a}^{b}\left\vert f\left( t\right) g\left( t\right) \right\vert
d_{\alpha,\beta}t\leqslant\sqrt{\left( \int_{a}^{b}\left\vert f\left(
t\right) \right\vert ^{2}d_{\alpha,\beta}t\right) \left(
\int_{a}^{b}\left\vert g\left( t\right)
\right\vert ^{2}d_{\alpha,\beta}t\right)}\text{.}
\]
\end{corollary}
We prove the Minkowski inequality using H\"{o}lder's inequality.
\begin{theorem}[Minkowski's inequality]
Let $f,g:\mathbb{R}\rightarrow\mathbb{R}$
and $a,b,p\in\mathbb{R}$ with $a<b$ and $p>1$.
If $f$ and $g$ are $\alpha,\beta$-symmetric integrable
on $\left[a,b\right]$, then
\[
\left( \int_{a}^{b}\left\vert f\left( t\right) +g\left( t\right)
\right\vert ^{p}d_{\alpha,\beta}t\right) ^{\frac{1}{p}}\leqslant\left(
\int_{a}^{b}\left\vert f\left( t\right) \right\vert ^{p}d_{\alpha,\beta
}t\right) ^{\frac{1}{p}}+\left( \int_{a}^{b}\left\vert g\left( t\right)
\right\vert ^{p}d_{\alpha,\beta}t\right) ^{\frac{1}{p}}.
\]
\end{theorem}
\begin{proof}
One has
\begin{multline*}
\int_{a}^{b}\left\vert f\left( t\right)
+g\left( t\right) \right\vert^{p}d_{\alpha,\beta}t
=\int_{a}^{b}\left\vert f\left( t\right) +g\left(
t\right) \right\vert ^{p-1}\left\vert f\left( t\right) +g\left( t\right)
\right\vert d_{\alpha,\beta}t\\
\leqslant\int_{a}^{b}\left\vert f\left( t\right) \right\vert \left\vert
f\left( t\right) +g\left( t\right) \right\vert ^{p-1}d_{\alpha,\beta
}t+\int_{a}^{b}\left\vert g\left( t\right) \right\vert \left\vert f\left(
t\right) +g\left( t\right) \right\vert ^{p-1}d_{\alpha,\beta}t.
\end{multline*}
Applying H\"{o}lder's inequality (Theorem~\ref{Holders Inequality}) with
$q=p/(p-1)$, we obtain
\begin{align*}
& \int_{a}^{b}\left\vert f\left( t\right) +g\left( t\right) \right\vert
^{p}d_{\alpha,\beta}t
\leqslant \left( \int_{a}^{b}\left\vert f\left(
t\right) \right\vert ^{p}d_{\alpha,\beta}t\right) ^{\frac{1}{p}}\left(
\int_{a}^{b}\left\vert f\left( t\right) +g\left( t\right) \right\vert
^{\left( p-1\right) q}d_{\alpha,\beta}t\right) ^{\frac{1}{q}}\\
& +\left( \int_{a}^{b}\left\vert g\left( t\right) \right\vert ^{p}
d_{\alpha,\beta}t\right) ^{\frac{1}{p}}\left( \int_{a}^{b}\left\vert
f\left( t\right) +g\left( t\right) \right\vert ^{\left( p-1\right)
q}d_{\alpha,\beta}t\right) ^{\frac{1}{q}}\\
=& \left[ \left( \int_{a}^{b}\left\vert f\left( t\right) \right\vert
^{p}d_{\alpha,\beta}t\right) ^{\frac{1}{p}}+\left( \int_{a}^{b}\left\vert
g\left( t\right) \right\vert ^{p}d_{\alpha,\beta}t\right) ^{\frac{1}{p}
}\right]
\left( \int_{a}^{b}\left\vert f\left( t\right) +g\left(
t\right) \right\vert ^{\left( p-1\right) q}d_{\alpha,\beta}t\right)
^{\frac{1}{q}}.
\end{align*}
Therefore,
\begin{equation*}
\frac{\int_{a}^{b}\left\vert f\left( t\right)
+g\left( t\right) \right\vert^{p}d_{\alpha,\beta}t}{\left(
\int_{a}^{b}\left\vert f\left(
t\right) +g\left( t\right) \right\vert ^{\left( p-1\right) q}
d_{\alpha,\beta}t\right)^{\frac{1}{q}}}
\leqslant
\left( \int_{a}^{b}\left\vert f\left(
t\right) \right\vert ^{p}d_{\alpha,\beta}t\right)^{\frac{1}{p}}
+\left(\int_{a}^{b}\left\vert g\left( t\right) \right\vert ^{p}d_{\alpha,\beta
}t\right) ^{\frac{1}{p}}.
\end{equation*}
\end{proof}
Our $\alpha,\beta$-symmetric calculus is more general
than the standard $h$-calculus. In particular, all our results give,
as corollaries, results in the classical quantum $h$-calculus
by choosing $\alpha=h>0$ and $\beta=0$.
\begin{acknowledgement}
Work supported by FEDER and Portuguese funds,
COMPETE reference FCOMP-01-0124-FEDER-022690,
and CIDMA and FCT, project PEst-C/MAT/UI4106/2011.
Brito da Cruz is also supported by FCT through
the Ph.D. fellowship SFRH/BD/33634/2009.
\end{acknowledgement}
|
{
"timestamp": "2012-03-13T01:00:35",
"yymm": "1203",
"arxiv_id": "1203.2212",
"language": "en",
"url": "https://arxiv.org/abs/1203.2212"
}
|
\section{Introduction}
Nowadays the equation of state (EOS) of isospin symmetric nuclear
matter is now relatively well determined mainly by studying
collective flows in heavy-ion collisions and nuclear giant
monopole resonances \cite{pd02,youngblood99}. The major remaining
uncertainty about the EOS of symmetric nuclear matter is due to
our poor knowledge about the density dependence of the nuclear
symmetry energy \cite{LCK08,Bar05,pd02,pie04,colo04}, which is
crucial for understanding many interesting issues in both nuclear
physics and astrophysics \cite{Bro00,Sum94,Lat04,Ste05a}. And it
is also crucial in connection with the structure of neutron stars
and the dynamical evolution of proto-neutron stars \cite{mk94}.
Nowadays considerable progress has been made recently in
determining the density dependence of the nuclear symmetry energy
around the normal nuclear matter density. However, much more work
is still needed to probe the high-density behavior of the nuclear
symmetry energy. Currently, to pin down the symmetry energy, the
National Superconducting Cyclotron Laboratory (NSCL) at Michigan
State University, the Gesellschaft fuer Schwerionenforschung (GSI)
at Darmstadt, the Rikagaku Kenkyusho (RIKEN, the Institute of
Physical and Chemical Research) of Japan, and the Cooler Storage
Ring (CSR) in Lanzhou are planning to do related experiments to
probe the symmetry energy.
The neutron-proton differential transverse flow and the difference
of neutron and proton flows are both sensitive to the symmetry
energy \cite{LCK08,Bar05}, but the used transport models always
adopt different momentum dependent interactions among nucleons.
The importance of the momentum dependence of nuclear symmetry
potential on the two kinds of nuclear flows was seldom mentioned.
Considering the momentum dependence of nuclear symmetry potential
is quite controversial \cite{chen11}, in the framework of the
isospin-dependent Boltzmann-Uehling- Uhlenbeck transport model, we
find that, besides the ratio of $\pi^{-}/\pi^{+}$ \cite{gao11},
the momentum dependence of nuclear symmetry potential affects the
neutron-proton differential transverse flow more evidently than
the difference of neutron and proton transverse flows as well as
the difference of proton and neutron elliptic flows. It is thus
better to probe the symmetry energy by using the difference of
nucleonic transverse (or elliptic) flows.
\section{THE IBUU04 TRANSPORT MODEL}
The present study is based on the IBUU04 transport model
\cite{LCK08}. The initial neutron and proton density distributions
of the projectile and target are obtained by using the
relativistic mean field theory. Although nuclear flow may be
affected by the initializations of colliding nuclei \cite{init11},
our main results in the present studies are not sensitive to the
initializations of target and projectile nuclei. The experimental
free-space nucleon-nucleon (NN) scattering cross sections and the
in-medium NN cross sections can be used optionally. In the present
work, we did not use the isospin-dependent in-medium NN elastic
cross sections from the scaling model according to nucleon
effective masses \cite{li05,epj11}, although it is similar to the
Brueckner approach calculations \cite{ligq93,fuchs01,zhf0710}.
This is because the momentum-independent symmetry potential (MID)
can not use the isospin-dependent in-medium NN elastic cross
sections from the scaling model according to nucleon effective
masses. Thus all the studies here optionally use the experimental
free-space nucleon-nucleon (NN) scattering cross sections for
consistence. For the inelastic cross sections people use the
experimental data from free space NN collisions since the
in-medium inelastic NN cross sections are still very much
controversial. The total and differential cross sections for all
other particles are taken either from experimental data or
obtained by using the detailed balance formula. In the model,
besides nucleons, $\Delta $ and $N^{\ast }$ resonances as well as
pions and their isospin-dependent dynamics are included. The
isospin dependent phase-space distribution functions of the
particles involved are solved by using the test-particle method
numerically. The isospin-dependence of Pauli blockings for
fermions is also considered. The momentum-dependent single nucleon
potential (MDI) adopted here is \cite{das03
\begin{eqnarray}
U(\rho ,\delta ,\mathbf{p},\tau ) &=&A_{u}(x)\frac{\rho _{\tau ^{\prime }}}
\rho _{0}}+A_{l}(x)\frac{\rho _{\tau }}{\rho _{0}} \nonumber \\
&&+B(\frac{\rho }{\rho _{0}})^{\sigma }(1-x\delta ^{2})-8x\tau \frac{B}
\sigma +1}\frac{\rho ^{\sigma -1}}{\rho _{0}^{\sigma }}\delta \rho
_{\tau
^{\prime }} \nonumber \\
&&+\frac{2C_{\tau ,\tau }}{\rho _{0}}\int d^{3}\mathbf{p}^{\prime }\frac
f_{\tau }(\mathbf{r},\mathbf{p}^{\prime
})}{1+(\mathbf{p}-\mathbf{p}^{\prime
})^{2}/\Lambda ^{2}} \nonumber \\
&&+\frac{2C_{\tau ,\tau ^{\prime }}}{\rho _{0}}\int d^{3}\mathbf{p}^{\prime
\frac{f_{\tau ^{\prime }}(\mathbf{r},\mathbf{p}^{\prime })}{1+(\mathbf{p}
\mathbf{p}^{\prime })^{2}/\Lambda ^{2}}. \label{potential}
\end{eqnarray
In the above equation, $\delta =(\rho _{n}-\rho _{p})/(\rho
_{n}+\rho _{p})$ is the isospin asymmetry parameter, $\rho =\rho
_{n}+\rho _{p}$ is the baryon density and $\rho _{n},\rho _{p}$
are the neutron and proton densities, respectively. $\tau
=1/2(-1/2)$ for neutron (proton) and $\tau \neq \tau ^{\prime }$,
$\sigma =4/3$, $f_{\tau }(\mathbf{r},\mathbf{p})$ is the
phase-space distribution function at coordinate $\mathbf{r}$ and
momentum $\mathbf{p}$. The parameters $A_{u}(x),A_{l}(x),B,C_{\tau
,\tau }$, $C_{\tau ,\tau ^{\prime }}$ and $\Lambda $ were set by
reproducing the momentum-dependent potential $U(\rho ,\delta
,\mathbf{p},\tau )$ predicted by the Gogny Hartree-Fock and/or the
Brueckner-Hartree-Fock calculations, the saturation properties of
symmetric nuclear matter and the symmetry
energy of about $32$ MeV at normal nuclear matter density $\rho _{0}=0.16$ f
$^{-3}$. The propagations of nucleon are according to Hamilton's
equations
\begin{eqnarray}\label{heq}
dp_{i}/dt&=&-\nabla_{r} U(r_{i})+q_{i}\vec{E},\nonumber\\
dr_{i}/dt&=&p_{i}/\sqrt{m^{2}+p_{i}^{2}}+\nabla_{p}
U(r_{i},p_{i}),
\end{eqnarray}
where $q_{i}$ is the charge of particle, $\vec{E}$ is the Coulomb
field of particle felt. The incompressibility of symmetric nuclear
matter at normal density is set to be $211$ MeV. According to
essentially all microscopic model calculations, the EOS for
isospin asymmetric nuclear matter can be expressed as
\begin{equation}
E(\rho ,\delta )=E(\rho ,0)+E_{\text{sym}}(\rho )\delta ^{2}+\mathcal{O
(\delta ^{4}),
\end{equation
where $E(\rho ,0)$ is the energy per nucleon of symmetric nuclear
matter, and $E_{\text{sym}}(\rho )$ is the nuclear symmetry
energy. With the single particle potential $U(\rho ,\delta
,\mathbf{p},\tau )$, for a given
value $x$, one can readily calculate the symmetry energy $E_{\text{sym
}(\rho )$ as a function of density. Because the purpose of present
studies is just to see how large the effect of momentum dependence
of nuclear symmetry potential on the transverse and elliptic
flows, we let the variable $x$ be $1$, since the IBUU04 model
gives a super-soft symmetry energy at higher densities
\cite{xiao09}. In fact, behavior of nuclear symmetry energy at
supra-densities is still in controversy. The main characteristic
of the present single particle is the momentum dependence of
nuclear symmetry potential, which has evident effect on energetic
free $n/p$ ratio in heavy-ion collisions \cite{IBUU04}. In the
present studies, to show the effects of the momentum dependence of
symmtry potential, we kept the isoscalar potential fixed while
changing the symmetry potential from the momentum dependent
symmetry potential to the momentum independent symmetry potential
and keep the symmetry energy fixed \cite{IBUU04}.
\begin{figure}[th]
\begin{center}
\includegraphics[width=0.5\textwidth]{fig1.eps}
\end{center}
\caption{(Color online) The momentum dependent symmetry potential
(MD) as a function of momentum and the momentum independent
symmetry potential (MID) as a function of density (upper
x-coordinate) from the Gogny interaction \cite{IBUU04}.}
\label{symp}
\end{figure}
Fig.~\ref{symp} shows the momentum dependent symmetry potential
(MD) and the momentum independent symmetry potential (MID) from
the Gogny interaction. We can see that at higher densities
strength of the momentum independent symmetry potential (MID) is
always larger than that of the momentum dependent symmetry
potential. In the following, we give our results of the momentum
dependence of nuclear symmetry potential on nuclear transverse and
elliptic flows at higher densities (The density reached in the
$^{132}$Sn+$^{124}$Sn at a beam energy of 400 MeV/nucleon is about
2 times saturation density \cite{yong06}).
\section{Results and discussions}
\begin{figure}[th]
\begin{center}
\includegraphics[width=0.5\textwidth]{fig2.eps}
\end{center}
\caption{(Color online) Rapidity distribution of free neutron to
proton ratio n/p in the reaction $^{132}$Sn+$^{124}$Sn at a beam
energy of 400 MeV/nucleon and an impact parameter of 5 fm with and
without momentum dependence of nuclear symmetry potential, signed
with MD and MID, respectively.} \label{yrnp}
\end{figure}
We first study the neutron to proton ratio n/p of free nucleons as
a function of rapidity as shown in Fig.~\ref{yrnp}. It is seen the
case with momentum dependence of nuclear symmetry potential causes
lower neutron to proton ratio, whereas the case without momentum
dependence of nuclear symmetry potential causes higher neutron to
proton ratio, especially for nucleons at large rapidities. Because
the momentum dependence of nuclear symmetry potential decreases
the strength of nuclear symmetry potential at high densities or
high nucleonic momenta as shown in Fig.~\ref{symp}. The small
symmetry potential decreases the free neutron to proton ratio
\cite{IBUU04}. From this plot, we can see that while using the
neutron to proton ratio n/p of free nucleons to probe the symmetry
energy, one should keep in mind that this observable is sensitive
to the the momentum dependence of nuclear symmetry potential used,
and thus may cause uncertainties while compared with the
experimental data.
\begin{figure}[th]
\begin{center}
\includegraphics[width=0.5\textwidth]{fig3.eps}
\end{center}
\caption{(Color online) Neutron and proton transverse flows
analysis in the reaction $^{132}$Sn+$^{124}$Sn at a beam energy of
400 MeV/nucleon and an impact parameter of 5 fm with and without
momentum dependence of nuclear symmetry potential, signed with MD
and MID, respectively.} \label{sflow}
\end{figure}
Fig.~\ref{sflow} shows neutron and proton transverse flows
analysis in the reaction $^{132}$Sn+$^{124}$Sn at a beam energy of
400 MeV/nucleon and an impact parameter of 5 fm with and without
momentum dependence of nuclear symmetry potential. We can see that
without (with) momentum dependence of nuclear symmetry potential,
the strength of the nucleon flow (especially neutron flow)
increases (decreases). This is understandable since the momentum
independence of nuclear symmetry potential overall increases the
strength of nuclear symmetry potential as shown in
Fig.~\ref{symp}, thus neutrons are repelled more strongly than the
case with the momentum dependent symmetry potential. From
Fig.~\ref{sflow} we can also see that effect of the momentum
dependence of nuclear symmetry potential on the proton flow is
less evident owing to the Coulomb potential added on protons. The
Coulomb potential (always repulsive for protons) decreases the
strength of symmetry potential (attractive for protons at high
densities) added on protons.
\begin{figure}[th]
\begin{center}
\includegraphics[width=0.5\textwidth]{fig4.eps}
\end{center}
\caption{(Color online) Difference of proton and neutron
transverse flows analysis in the reaction $^{132}$Sn+$^{124}$Sn at
a beam energy of 400 MeV/nucleon and an impact parameter of 5 fm
with and without momentum dependence of nuclear symmetry
potential, signed with MD and MID, respectively.} \label{dflow}
\end{figure}
The difference of proton and neutron transverse flows has been
studied previously and was shown to be sensitive to the symmetry
energy \cite{Bar05}. In order to show if the difference of proton
and neutron transverse flows is sensitive to the momentum
dependence of nuclear symmetry potential, we plot
Fig.~\ref{dflow}, the difference of proton and neutron transverse
flows analysis in the reaction $^{132}$Sn+$^{124}$Sn at a beam
energy of 400 MeV/nucleon and an impact parameter of 5 fm with and
without momentum dependence of nuclear symmetry potential. From
this plot, we see that the difference of proton and neutron
transverse flows is insensitive to the momentum dependence of
nuclear symmetry potential. It is thus better to be used to probe
the symmetry energy since the the momentum dependence of nuclear
symmetry potential is still an open question \cite{chen11}.
\begin{figure}[th]
\begin{center}
\includegraphics[width=0.5\textwidth]{fig5.eps}
\end{center}
\caption{(Color online) Neutron proton differential transverse
flow analysis in the reaction $^{132}$Sn+$^{124}$Sn at a beam
energy of 400 MeV/nucleon and an impact parameter of 5 fm with and
without momentum dependence of nuclear symmetry potential, signed
with MD and MID, respectively.} \label{cflow}
\end{figure}
The neutron proton differential transverse flow also has been
studied extensively and was shown to be sensitive to the symmetry
energy \cite{LCK08,IBUU04}. To see if neutron proton differential
transverse flow is sensitive to the momentum dependence of nuclear
symmetry potential, we give Fig.~\ref{cflow}, the neutron proton
differential transverse flow analysis in the reaction
$^{132}$Sn+$^{124}$Sn at a beam energy of 400 MeV/nucleon and an
impact parameter of 5 fm with and without momentum dependence of
nuclear symmetry potential. From this plot, we see that the
neutron proton differential transverse flow is sensitive to the
momentum dependence of nuclear symmetry potential as shown in
\cite{IBUU04}.
The neutron-proton differential transverse flow is defined as
\cite{yong06,yong09}
\begin{eqnarray}
F_{n-p}^{x}(y) &\equiv
&\frac{1}{N(y)}\sum_{i=1}^{N(y)}p_{i}^{x}(y)w_{i}
\nonumber \\
&=&\frac{N_{n}(y)}{N(y)}\langle p_{n}^{x}(y)\rangle -\frac{N_{p}(y)}{N(y)
\langle p_{p}^{x}(y)\rangle \label{npflow}
\end{eqnarray
where $N(y)$, $N_{n}(y)$ and $N_{p}(y)$ are the number of free
nucleons, neutrons and protons, respectively, at rapidity $y$;
$p_{i}^{x}(y)$ is the transverse momentum of the free nucleon at
rapidity $y$; $w_{i}=1$ $(-1)$ for neutrons (protons); and
$\langle p_{n}^{x}(y)\rangle $ and $\langle p_{p}^{x}(y)\rangle $
are respectively the average transverse momenta of neutrons and
protons at rapidity $y$. One can see from Eq. (\ref{npflow}) that
the constructed neutron-proton differential transverse flow
depends not only on the proton flow and neutron flow but also on
their relative multiplicities. Therefore the neutron-proton
differential transverse flow is not simply the difference of the
neutron and proton transverse flows, it in fact depends also on
the isospin fractionation at the rapidity $y$. If neutrons and
protons have the same average transverse momentum in the reaction
plane but different multiplicities in each rapidity bin, i.e.,
$\langle p_{n}^{x}(y)\rangle =\langle p_{p}^{x}(y)\rangle =\langle
p^{x}(y)\rangle $, and $N_{n}(y)\neq N_{p}(y)$, then Eq.
(\ref{npflow}) is reduced to
\begin{equation}
F_{n-p}^{x}(y)=\frac{N_{n}(y)-N_{p}(y)}{N(y)}\langle
p^{x}(y)\rangle =\delta (y)\cdot \langle p^{x}(y)\rangle,
\end{equation
reflecting effects of the isospin fractionation. On the other
hand, if neutrons and protons have the same multiplicity but
different average transverse momenta, i.e., $N_{n}(y)=N_{p}(y)$
but $\langle
p_{n}^{x}(y)\rangle \neq \langle p_{p}^{x}(y)\rangle $, then Eq. (\re
{npflow}) is reduced t
\begin{equation}
F_{n-p}^{x}(y)=\frac{1}{2}(\langle p_{n}^{x}(y)\rangle -\langle
p_{p}^{x}(y)\rangle ).
\end{equation
In this case it reflects directly the difference of the neutron
and proton transverse flows. Because the effect of the momentum
dependence of nuclear symmetry potential for neutron to proton
ratio is very large (shown in Fig.~\ref{yrnp}) and the difference
of the proton and neutron transverse flows has no such
combination, we see larger effect of the momentum dependence of
nuclear symmetry potential on the neutron proton differential
transverse flow than the difference of proton and neutron
transverse flows. It is thus better to probe the symmetry energy
by using the difference of neutron and proton flows since the
momentum dependence of nuclear symmetry potential is still an open
question \cite{chen11}.
\begin{figure}[th]
\begin{center}
\includegraphics[width=0.5\textwidth]{fig6.eps}
\end{center}
\caption{(Color online) Difference of proton elliptic flow and
neutron elliptic flow analysis in the reaction
$^{132}$Sn+$^{124}$Sn at a beam energy of 400 MeV/nucleon and an
impact parameter of 5 fm with and without momentum dependence of
nuclear symmetry potential, signed with MD and MID, respectively.}
\label{eflow}
\end{figure}
The elliptic flow $v_{2}(y,p_t)$, which
is derived as the second coefficient from a Fourier expansion of
the azimuthal distribution
$N(\phi,y,p_{t})=v_{0}(1+v_{1}cos(\phi)+2v_{2}cos(2\phi))$, can be
expressed as
\begin{equation}
v_2=<\frac{p^2_x-p^2_y}{p^2_t}>,
\end{equation}
where $p_t=\sqrt{p^2_x+p^2_y}$ is the transverse momentum
\cite{ditorof,ditorof2}. The difference of proton and neutron
elliptic flows is also shown to be sensitive to the symmetry
energy \cite{ditorof,ditorof2}. It is thus also interesting to see
if the difference of proton elliptic flow and neutron elliptic
flow is sensitive to the momentum dependence of nuclear symmetry
potential. Fig.~\ref{eflow} shows the difference of proton
elliptic flow and neutron elliptic flow analysis in the reaction
$^{132}$Sn+$^{124}$Sn at a beam energy of 400 MeV/nucleon and an
impact parameter of 5 fm with and without momentum dependence of
nuclear symmetry potential. Again, we see that the difference of
proton elliptic flow and neutron elliptic flow is not sensitive to
the symmetry energy.
\section{Conclusions}
Based on the IBUU04 transport model, effect of the momentum
dependence of nuclear symmetry potential on nuclear transverse and
elliptic flows in the neutron-rich reaction $^{132}$Sn+$^{124}$Sn
at a beam energy of $400$ MeV/nucleon is studied. It is found that
the momentum dependence of nuclear symmetry potential affects the
rapidity distribution of the free neutron to proton ratio, neutron
flow and proton flow as a function of rapidity. The momentum
dependence of nuclear symmetry potential affects neutron-proton
differential transverse flow more evidently than the difference of
neutron and proton transverse flows as well as the difference of
proton elliptic flow and neutron elliptic flow. Therefore it is
better to probe the symmetry energy by using the difference of
neutron flow and proton flow since the momentum dependence of
nuclear symmetry potential is still an open question. And it is
better to probe the momentum dependence of nuclear symmetry
potential by using the neutron-proton differential transverse flow
and the rapidity distribution of the free neutron to proton ratio.
\section*{Acknowledgments}
The author Y. Gao thanks Prof. Bao-An Li for providing the code
and useful guidance while he stayed at Institute of Modern
Physics, Chinese Academy of Sciences and thanks Prof. Wei Zuo for
helpful discussions. The work is supported by the National Natural
Science Foundation of China (10975064, 11175074, 11175219 and
11105035) and the Zhejiang Provincial Natural Science Foundation
(Y6110644).
|
{
"timestamp": "2012-03-09T02:01:47",
"yymm": "1203",
"arxiv_id": "1203.1724",
"language": "en",
"url": "https://arxiv.org/abs/1203.1724"
}
|
\section{Introduction}
The interaction between light and matter are interaction between two groups of physical quantities i.e. quantities derived from electromagnetic waves, electric fields, $\vec{E}(\vec{r},t)$, and magnetic fields, $\vec{H}(\vec{r},t)$, with quantities derived from the material i.e. electric susceptibility, $\chi_{e}$, magnetic susceptibility, $\chi_{m}$, and electrical conductivity, $\sigma$.
In a non-conducting materials, electrical conductivity is zero, so that the material response only expressed by $\chi_{e}$ and $\chi_{m}$. The values $\chi_{e}$ and $\chi_{m}$ in the material are influenced by various factors such as frequency of the incident wave \cite{azad,kaya,thamizhmani}, temperature \cite{baeraky,krupka}, pressure and porosity \cite{bourlange,donnelly,saar}, inhomogeneous properties \cite{agron,huttunen,longhi,saville}.
Reflection and refraction of electromagnetic waves in inhomogeneous material are an interesting phenomenon and have several applications, such as: developing radar signal absorbing material \cite{saville}, developing photonic crystals \cite{longhi}, investigations of biological materials \cite{huttunen}, and investigations of stratified media \cite{agron}.
Inhomogeneous properties which are used in \cite{agron,huttunen,longhi,saville} derived from the values of materials susceptibility are only changing in spatial coordinates. Research on inhomogeneous materials which are involving changes in susceptibility values by time has not found widely. By using the concept of inhomogeneous medium in space and time, we expect to find more complete picture of reflection and refraction of electromagnetic wave at the surface of anisotropic, inhomogeneous and linear medium.
\section{Medium Properties}
In this study we use medium which is anisotropic, linear, without charge density and inhomogeneous. All types of materials generally have an anisotropic and linear properties except in certain materials such as FeF$_{2}$ \cite{roniyus1,roniyus2,roniyus3,roniyus4,roniyus5} dan GeSe$_{4}$ \cite{patrick}. Both of these anisotropic materials, beside having linear properties, also have nonlinear properties. Material without free charge density is an insulating material. Example of an insulating materials which are commonly used in optical experiments are glass based materials: fluoride glass \cite{lucas}, silica \cite{tan}, soda glass and copper glass \cite{Smith}, selenium glass \cite{patrick} and non-oxide glass \cite{adam}. Some of ceramic based materials are Nd YAG \cite{qu} and NAT \cite{wang}. Some examples of polymer-based materials are polycarbonate (PC) and polyethylene (PET) \cite{yong}.
Mathematically, anisotropic medium is a medium with the values of linear electric susceptibility, $\chi_{e}$, or linear magnetic susceptibility, $\chi_{m}$, has the form of rank-2 tensor (matrix of ordo 3$\times$3). Inhomogeneous medium in this study is a model of electric and magnetic susceptibility tensor that all components are functions of frequency, position and time respectively, have the form $\overleftrightarrow{\chi}_{e}(\omega,r,t)$ and $\overleftrightarrow{\chi}_{m}(\omega,r,t)$ with $ij$ subscript in form tensor components to $ij$. Theoretical and computational studies in this research is done by involving all components (18 entries) susceptibility tensor of the medium to obtain optimal use.
All components of the tensor model above are included in the calculation and will be used for testing a real medium. Considering FeF$_{2}$ magnet materials have been widely used in optics research \cite{roniyus1,roniyus2,roniyus3,roniyus4,roniyus5} and has been qualified as a medium with properties: linear, ansitropic and no charge density, so that the calculation for a real medium will use this material. The tensor form of FeF$_{2}$ are as follow:
\begin{equation}\label{1}
\overleftrightarrow{\chi}_{e}(\omega,r,t)=
\begin{bmatrix}
4.5 & 0 & 0 \\
0 & 4.5 & 0 \\
0 & 0 & 4.5 \\
\end{bmatrix}
\end{equation}
\begin{equation}\label{2}
\overleftrightarrow{\chi}_{m}(\omega,r,t)=
\begin{bmatrix}
\chi_{11}(\omega,r,t) & i\chi_{12}(\omega,r,t) & 0 \\
-i\chi_{21}(\omega,r,t) & \chi_{22}(\omega,r,t) & 0 \\
0 & 0 & \chi_{33}(\omega,r,t) \\
\end{bmatrix}.
\end{equation}
\section{Basic Formulation}
Electromagnetic wave propagation in the medium is shown by Maxwell equations. Maxwell's equations in international units (SI) for no free charge volume density have form as follows \cite{khan}:
\begin{equation}\label{3}
\nabla.\vec{D}(\vec{r},t)=0
\end{equation}
\begin{equation}\label{4}
\nabla.\vec{B}(\vec{r},t)=0
\end{equation}
\begin{equation}\label{5}
\nabla\times\vec{E}(\vec{r},t)+\frac{\partial\vec{B}(\vec{r},t)}{\partial t}=0
\end{equation}
\begin{equation}\label{6}
\nabla\times\vec{H}(\vec{r},t)-\frac{\partial\vec{D}(\vec{r},t)}{\partial t}=0
\end{equation}
where
\begin{equation}\label{7}
\vec{D}(\vec{r},t)=\epsilon \vec{E}(\vec{r},t)
\end{equation}
\begin{equation}\label{8}
\vec{B}(\vec{r},t)=\mu \vec{H}(\vec{r},t)
\end{equation}
with $\epsilon$ and $\mu$ respectively permittivity and permeability of the medium, i.e. medium response when subjected to electric and magnetic fields. Vectors $\vec{E}(\vec{r},t)$ and $\vec{H}(\vec{r},t)$ are respectively the vector amplitude of electric and magnetic waves.
When the medium of propagation of electromagnetic waves is vacuum, then the value of medium permittivity, $\epsilon$, equal to the vacuum permittivity, $\epsilon_{0}$, and medium permeability, $\mu$, equal to vacuum permeability, $\mu_{0}$. The value of vacuum permittivity, $\epsilon_{0}$, is {8.85$\times$10}$^{-12}$ $C^{2}$/$N.m^{2}$ and vacuum permeability, $\mu_{0}$, is {4$\pi$$\times$10}$^{-7}$ $T.m/A$.
When the electromagnetic wave propagates in non vacuum medium, then the value of $\epsilon$ is equal to $\epsilon$ = $\epsilon_{0}$$\epsilon_{r}$, where $\epsilon_{r}$ is the relative permittivity of the medium and medium permeability value, $\mu$, equal to $\mu_{0}$$\mu_{r}$, where $\mu_{r}$ is relative permeability of the medium. This study has used an electric susceptibility, $\chi_{e}$, and magnetic susceptibility, $\chi_{m}$, of material as response to the presence of electric and magnetic fields.
Linear anisotropic materials have a relative permittivity value of the medium as follows,
\begin{equation}\label{9}
\epsilon^{(r)}_{ii}=1+{\chi}^{(e)}_{ii},~~\epsilon^{(r)}_{ij} = \chi^{(e)}_{ij}
\end{equation}
and relative permeability value of the medium as follows,
\begin{equation}\label{10}
\mu^{(r)}_{ii}=1+{\chi}^{(m)}_{ii},~\mu^{(r)}_{ij}={\chi}^{(m)}_{ij}
\end{equation}
with superscript $r$, $e$ and $m$ respectively have the meanings $relative$, $electric$ and $magnetic$, while the subscript $i$ and $j$ in form of matrix components at $ii$ and $ij$.
Propagation of electromagnetic waves in the medium are expressed by the governing equation of electromagnetic waves in the medium. The formula can be found by put on $curl$ to eq.(\ref{4}) then use eq.(\ref{6}), eq.(\ref{7}) and eq.(\ref{8}) until it is found governing equation of electromagnetic waves in the medium, as follows :
\begin{equation}\label{11}
\nabla^2\vec{E}(\vec{r},t)-\mu_{0}\frac{\partial^2\vec{D}(\vec{r},t)}{\partial t^2}+\frac{1}{\epsilon_{0}}\nabla[\nabla.\vec{P}(\vec{r},t)]-\mu_{0}\frac{\partial}{\partial t}\left[\nabla\times\vec{M}(\vec{r},t)\right]=0.
\end{equation}
Eq.(\ref{11}) will be required to determine value of the vector wave refraction, $k^{(t)}$, in the medium.
\section{Theoretical and Computational Study of Reflection and Refraction of Electromagnetic Waves}
Illustration of propagation of electromagnetic waves can be seen in Fig.1 below.
\begin{center}
\includegraphics[scale=0.7]{Figure1}\\
Figure 1. The events of reflection and refraction of electromagnetic waves with $p$-polarized of electric waves incident (located in the y-z plane) at the left boundary surface of the material.
\end{center}
\subsection{The intensity of the incident wave}
Based on Fig.1, it informs that the incident wave vector in vacuum space is
\begin{equation}\label{12}
\vec{k}^{(i)}= k^{(i)}\hat{y}.
\end{equation}
Consider the incident wave comes from vacuum space, so that the incident wave vector is \cite{wangsness}
\begin{equation}\label{13}
k^{(i)}=\omega\sqrt{\epsilon_{0}\mu_{0}}.
\end{equation}
Vector amplitude of the $p$-polarized incident electric wave is defined as follows
\begin{equation}\label{14}
\vec{E}^{(i)}={E}_{0}\hat{z}.
\end{equation}
Based on eq.(\ref{5}), the vector amplitude of incident magnetic wave is
\begin{equation}\label{15}
\vec{H}^{(i)}=\frac{k^{(i)}}{\mu_{0}\omega}E_{0}\hat{x}.
\end{equation}
The intensity of the incident wave can be found by using the Poynting vector \cite{wangsness}, i.e.
\begin{equation}\label{16}
\langle\vec{S}\rangle=\frac{1}{2}\Re \{\vec{E}(\vec{r},t)\times{\vec{H}}^{*}(\vec{r},t)\},
\end{equation}
then, eq.(\ref{14}) and eq.(\ref{15}) are substituted into eq.(\ref{16}) to obtain the intensity of the incident wave equation
\begin{equation}\label{17}
\langle\vec{S}^{(i)}\rangle=\frac{k^{(i)}{E_{0}}^{2}}{2\mu_{0}\omega}\hat{y}.
\end{equation}
\subsection{The intensity of the reflected wave}
There are two types of reflections that can occur when the electromagnetic waves penetrate the material surface i.e., $ps$ and $pp$-reflection \cite{roniyus3,roniyus4,roniyus5}. Reflection of $ps$, is a reflection of $p$-polarized incident electric waves which generate $s$-polarized reflected electric waves (perpendicular to the plane of incident, i.e. plane of $\textit{x-y}$) with the vector amplitude of the reflected electric wave on $\textit{x}$-axis. Reflection of $pp$, is a reflection of $p$-polarized incident electric waves that generate $p$-polarized of reflected electric waves (parallel to the incident plane, i.e. $\textit{y-z}$ plane) with the vector amplitude of the reflected electric wave on $\textit{z}$-axis.
The results of theoretical studies using susceptibility tensor of vacuum and FeF$_{2}$ show that, in propagation of light parallel to the $\textit{y}$-axis with $p$-polarized of incident electric waves, the reflectance coefficient of $ps$ has been found equal to zero. Percentage of reflectance totally in the $pp$ reflectance, therefore the $ps$ reflection will not be discussed in further. Reflection of $pp$ produces amplitude vector of reflected electric waves on the $\textit{z}$-axis, so that the vector amplitude of the electric waves is
\begin{equation}\label{18}
\vec{E}^{(r_{pp})}=-r_{pp}E_{0}(\hat{z}).
\end{equation}
where $r_{pp}$ is a reflection coefficient of $p$-polarized incident electric wave amplitude.
Based on Snell's Law, the boundary conditions for reflected wave between two different media expressed for the reflected wave vector is
\begin{equation}\label{19}
\vec{k}^{(r_{pp})}=-\vec{k}^{(i)}.
\end{equation}
Furthermore, by using eq.(\ref{5}), eq.(\ref{18}) and eq.(\ref{19}), it can be found amplitude vector for magnetic waves, i.e.
\begin{equation}\label{20}
\vec{H}^{(r_{pp})}=\frac{k^{(i)}}{\mu_{0}\omega}r_{pp}E_{0}(\hat{x}).
\end{equation}
Eq.(\ref{18}) and eq.(\ref{20}) are used to calculate the intensity of reflected waves. The equation of $pp$ reflection intensity for reflected wave is obtained as
\begin{equation}\label{21}
\langle\vec{S}^{(r_{pp})}\rangle=\frac{k^{(i)}{E_{0}}^{2}}{2\mu_{0}\omega}{\mid{r_{pp}}\mid}^{2}\hat{y}.
\end{equation}
\subsection{The intensity of the refraction wave}
The presence of electromagnetic waves on the material surface can raises refraction events \cite{wangsness}. Discussion of wave refraction is started from the relationship between electric field and magnetic field in the material. Eq.(\ref{5}) and eq.(\ref{8}) in forms of the value of electric field related to the magnetic field are
\begin{equation}\label{22} E^{(t)}_{x}=-\frac{\mu_{0}}{k^{(t)}}\bigl(\omega(1+\chi_{33})+i\frac{\partial\chi_{33}}{\partial t}\bigr)H^{(t)}_{z},
\end{equation}
\begin{equation}\label{23} E^{(t)}_{z}=\frac{\mu_{0}}{k^{(t)}}\bigl(\omega(1+\chi_{11})+i\frac{\partial\chi_{11}}{\partial t}\bigr)H^{(t)}_{x}.
\end{equation}
Eq.(\ref{6}) and eq.(\ref{7}) in forms of the wave propagation in the $\textit{y}$-axis direction, so that it can be found the relationship between magnetic field and electric field of the form
\begin{equation}\label{24}
H^{(t)}_{x}=\frac{\epsilon_{0}\omega}{k^{(t)}}\bigl(1+\chi_{e}\bigr)E^{(t)}_{z},
\end{equation}
\begin{equation}\label{25}
H^{(t)}_{z}=-\frac{\epsilon_{0}\omega}{k^{(t)}}\bigl(1+\chi_{e}\bigr)E^{(t)}_{x}.
\end{equation}
If it is assumed that the solution of eq.(\ref{11}) is a harmonic wave function, then by substituting eq.(\ref{22}), eq.(\ref{23})), eq.(\ref{24}) and eq.(\ref{25}) into eq.(\ref{11}), it can be found characteristics matrix of wave refraction, in the form
\begin{equation}\label{26}
\begin{bmatrix}
\frac{\mu_{0}\epsilon_{0}}{k^{(t)}}\omega(1+\chi_{e})P+\mu_{0}\epsilon_{0}\omega(1+\chi_{e})R-{k^{(t)}}^{2} & 0 \\
0 & \frac{\mu_{0}\epsilon_{0}}{k^{(t)}}\omega(1+\chi_{e}) M+\mu_{0}\epsilon_{0}\omega(1+\chi_{e})O-{k^{(t)}}^{2}\\
\end{bmatrix}=0,
\end{equation}
where
\begin{equation}\label{27}
P=\frac{\partial}{\partial t}\frac{\partial\chi_{33}}{\partial y}-i\omega\frac{\partial\chi_{33}}{\partial y},
\end{equation}
\begin{equation}\label{28}
R=\omega(1+\chi_{33})+i\frac{\partial\chi_{33}}{\partial t},
\end{equation}
\begin{equation}\label{29}
M=\frac{\partial}{\partial t}\frac{\chi_{11}}{\partial y}-i\omega\frac{\partial\chi_{11}}{\partial y},
\end{equation}
\begin{equation}\label{30}
O=\omega(1+\chi_{11})+i\frac{\partial\chi_{11}}{\partial t}.
\end{equation}
The values of refraction wave vector can be obtained by solving the determinant of characteristics matrix in eq.(\ref{26}) using MATLAB software. The intensity of the electromagnetic wave refraction has form
\begin{equation}\label{31} \langle\vec{S}^{(t)}\rangle=\mu_{0}\epsilon_{e}{\omega}^2(1+\chi_{e})\left|\Re\Bigl\{\frac{t.t^{*}}{k^{(i)}{k^{(t)}}^{*}}\Bigr\}\right|\hat{y},
\end{equation}
\begin{equation}\label{32} \langle\vec{S}^{(r_{pp})}\rangle=\left|\Re\bigl\{-r_{pp}.r_{pp}^{*}\bigl\}\right|\hat{y}.
\end{equation}
Mark of $(^{*})$ in eq.(\ref{31}) and eq.(\ref{32}) means conjugate vector. Refraction coefficient, $t$, and reflection coefficients, $r_{pp}$, are obtained by equating the tangential components of the amplitude vector at the boundary between two media. The value of $t$ and $r_{pp}$ are as follows:
\begin{equation}\label{33}
t=\frac{2k^{(i)}k^{(t)}}{\mu_{0}\epsilon_{0}{\omega}^2(1+\chi_{e})+k^{(i)}k^{(t)}},
\end{equation}
\begin{equation}\label{34}
r_{pp}=1-t.
\end{equation}
Reflectance and transmittance are calculated as follows :
\begin{equation}\label{35} T=\left|\frac{\langle\vec{S}^{(t)}\rangle.\hat{y}}{\langle\vec{S}^{(i)}\rangle.\hat{y}}\right|,
\end{equation}
\begin{equation}\label{36} R_{pp}=\left|\frac{\langle\vec{S}^{(r_{pp})}\rangle.\hat{y}}{\langle\vec{S}^{(i)}\rangle.\hat{y}}\right|.
\end{equation}
The values of $T$ and $R_ {pp}$ will be visualized with MATLAB software.
\subsection{Transmittance and reflectance in the vacuum and FeF$_{2}$}
To find out the truth of $T$ and $R_ {pp}$ formulation above, it will be tested using vacuum as a first test and then FeF$_2$ magnetic material as a second test material. Based on the law of conservation of energy, in a vacuum, all the energy which arrive should in refraction, while in the non-conducting material, energy which arrive can be reflected and refractived but there is no absorption of energy by the material.
Given the percentage of the total energy which arrive is 1 (100$\%$), so that the total of false intensity is defined not equal to 1. The results of calculations using MATLAB software for transmittance, $T$, and reflectance, $R_{pp}$, values from vacuum and FeF$_2$ in the influence of external magnetic field 3 Tesla can be seen in Fig.2 below
\begin{center}
\includegraphics[scale=0.27]{Figure2}\\
Figure 2. The graph of $T$ and $R_{pp}$ calculation result for (a) vacuum and (b) FeF$_2$ magnetic material with $Total$ as additional graph.
\end{center}
Fig.2 (a) and (b) respectively are result graphs of calculations $T$ and $R_{pp}$ from vacuum and $T$, $R_{pp}$ and $Total$ from FeF$_2$ magnetic materials.
It is shown in Fig.2 (a) that the value of transmittance curve, $T$, is 1 and $R$ is 0. These results are consistent with the theory of refraction in vacuum. The frequency written in $cm^{-1}$ unit is standard which is used by researchers in the field of optical physics \cite{ritchmyer}.
Fig.2 (b) shows the graph of $T$, $R_{pp}$ and $Total$ calculated from FeF$_2$ magnetic materials. Theoretical study with FeF$_2$ materials generally uses far infrared electromagnetic waves with high intensity in the frequency range 48 cm$^{-1}$ to 58 cm$^{-1}$. In this study, we use the frequency range from 51 cm$^{-1}$ to 53.8 cm$^{-1}$ to shorten the running time because of limitations of computer's processor.
Selection of frequency range from 51 cm$^{-1}$ to 53.8 cm$^{-1}$ is related with material resonance frequency i.e. at 52.45 cm$^{-1}$ \cite{roniyus3}. In Fig.2 (b), it can be seen the value of $T$ and $R_{pp}$ fluctuated around the resonance frequency. Furthermore, it appears the value of $Total$ coincides with value of 1. The $Total$ values which coincide with value of 1 can be used as an indicator that the calculations which are performed using the material FeF$_2$ are correct \cite{roniyus3}.
\subsection{Transmittance and reflectance in an anisotropic, inhomogeneous and linear medium}
Medium with properties of anisotropic, inhomogeneous and linear used in this study is a model of susceptibility tensor as shown in eq.(\ref{1}) and eq.(\ref{2}) with shape of susceptibility curve similar to the curve of FeF$_2$ magnetic susceptibility at far infrared frequency.
Consider that the FeF$_2$ susceptibility curve has similar to the fluctuations tangent function. After the modification, it can be obtained tangent function approaches the curve of FeF$_2$ magnetic materials susceptibility under the influence of external magnetic field of 1 Tesla, which is
\begin{equation}\label{37}
\chi_{ij}(\omega,r,t)=\rho(r,t)\beta_{ij}\tan\left(\frac{\zeta(r,t)-82.26}{\omega}\right),
\end{equation}
with $\beta_{11}=0.7/8.55\times10^{\pi+3}$, $\beta_{12}=\beta_{21}=1/8.55\times10^{\pi+1.46}$, $\beta_{22}=\beta_{33}=1/8.55\times10^{\pi}$.
Factor of $\rho(r,t)$ and $\zeta(r,t)$ in eq.(\ref{37}) is the homogeneity parameter of materials. Homogeneity parameter of materials is defined with:\\
$$\vbox{\offinterlineskip
\halign{\strut \quad $#$\quad & \hfil \quad #\quad \hfil & \quad $#$\quad & \hfil \quad #\quad \hfil \cr
\noalign{}
$Homogeneous$ & when\;$\rho(r,t)=c$, $c\in\Re$ & $when$\;\zeta(r,t)=c, \;c\in\Re,\cr
\noalign{}
$Inhomogeneous$ & when\;$\rho(r,t)=\rho^{L}(r)\pm\rho^{L}(t)$ &$when$\;\zeta(r,t)=\zeta^{L}(r)\pm\zeta^{L}(t).\cr
\noalign{}
}}$$
The sign of ($..^{L}$) means linear. The figures below are visualization of eq.(\ref{37}) when the medium is homogeneous $\rho(r,t)=1$, $\zeta(r,t)=0$ and magnetic susceptibility FeF$ _2$ in the influence of external magnetic field of 1 Tesla \cite{roniyus3} :
\begin{center}
\includegraphics[scale=0.27]{Figure3ab}\\
\includegraphics[scale=0.27]{Figure3c}\\
Figure 3. Magnetic susceptibility curves of models and FeF$_2$ appears to fluctuate around the frequency 52.45 cm$^{-1}$, i.e. the resonance frequency of FeF$_2$. (a) $\chi_{11}$, model (red) and $\chi_{11}$, FeF$_2$ (blue), (b) $\chi_{12}$, $\chi_{21}$, models (red) and $\chi_{12}$, $\chi_{21}$, FeF$_2$ (blue and black), (c) $\chi_{22}$, $\chi_{33}$, model (red) and FeF$_2$ (blue and black).
\end{center}
Based on eq.(\ref{37}) note that the value of the magnetic susceptibility is influenced by two homogeneity parameters of the medium i.e., $\rho(r,t)$ (rho) and $\zeta(r,t)$ (zeta). Formulation the value of $\rho(r,t)$ and $\zeta(r,t)$, to simplify the problem, can use Agron and Gogineni approach on stratified materials \cite {agron}.
In stratified materials, review is performed in the direction of light propagation. Given the direction of light propagation in this study is $y$-axis, then it can be selected $\rho(r,t)=1-y$, $\zeta(r,t)=0$ and $\rho(r,t)=1$, $\zeta(r,t)=y$ for an example of calculation. The calculation results of $T$ and $R_{pp}$ values using these parameters at the point (0, 0, 0), are shown in Fig.4 as follows
\begin{center}
\includegraphics[scale=0.27]{Figure4}\\
Figure 4. (a) The comparison values of $T$ and $R_{pp}$ of the model depend on the position and (b) depend on the position of the magnetic susceptibility at the origin (0,0,0).
\end{center}
Fig.4 shows the effect of inhomogeneous magnetic susceptibility values in the direction of light propagation. The values of transmittance is shown in Fig.4 (a), whereas the reflectance values is shown in Fig.4 (b). Both figures show that there are no change in the pattern of transmittance and reflectance from initially state.
The parameters of inhomogeneous by time can be expressed by functions $\rho(r,t)=1-t$, $\zeta(r,t)=0$ and $\rho(r,t)=1$, $\zeta(r,t)=t$. The results of calculation of $T$ and $R_{pp}$ using linear functions at the point (0, 0, 0) and $t$ = 0, 0.5 $s$ are shown in Fig.5 as follow
\begin{center}
\includegraphics[scale=0.27]{Figure5}\\
Figure 5. The comparison values of $T$ and $R_{pp}$ from the magnetic susceptibility model of time dependent and time independent at the origin (0, 0, 0) when $t$ = 0, 0.5 $s$. (a) The values of $T$ on $t$ = 0 $s$ (blue rectangle and black circle), $t$ = 0.5 $s$ (green dot and pink dot), (b) The value of $R_{pp}$ on $t$ = 0 $s$ (blue rectangle and black circle), $t$ = 0.5 $s$ (green dot and pink dot).
\end{center}
Fig.5 shows the effect of inhomogeneous magnetic susceptibility values by time. Transmittance value, $T$, is shown in Fig.5 (a), whereas the reflectance value $R_{pp}$ is shown in Fig.5 (b). Both curves, $T$ and $R_{pp}$, (blue rectangle and black circle) in the beginning ($t$ = 0 $s$), coincide with transmittance and reflectance curves of the medium which are independent of time. But in further time ($t$ = 0.5 $s$), the fluctuations of $T$ and $R_{pp}$ (green dot and pink dot) are decreased and shifted relative to its original state. Changes in $T$ and $R_{pp}$ at inhomogeneous medium thus can be generated by the parameters of time.
\section{Conclusions}
Computational results using vacuum susceptibility shows that the value of transmittance, $T$, is 1 ($ 100 \% $), the reflectance, $R_ {pp}$, is 0 ($ 0 \% $) and the total intensity is 1 (100 $ \% $), according to the theory of reflection and refraction of electromagnetic waves that propagate in vacuum. Computational results of FeF$ _2$ magnetic material shows that the total value of intensity is 1 (100 $ \% $), according to the results of previous studies.
The results of theoretical and computational studies on the surface of an anisotropic, inhomogeneous and linear medium shows that the values of transmittance, $T$, and reflectance, $R_ {pp}$, are not affected by variations of susceptibility values in the direction of light propagation. The changes in values of transmittance, $T$, and reflectance, $R_ {pp}$, on the medium surface can be caused by time.
In other words, in the direction of light propagation ($y$-axis), the values of transmittance, $T$, and reflectance, $R_{pp}$, in the surface of medium are not affected by the changes of magnetic susceptibility values �as shown in Fig.4 (a) and (b). Variations in medium susceptibility values by time can change the fluctuation of transmittance, $T$, and reflectance, $R_{pp}$, at the surface of medium by two mechanisms as shown in Fig.5 (a) and (b).
\section{Acknowledgments}
ASH thank to Computational Laboratory, University of Lampung for supporting software and hardware. Thank to all of Physics Department Staffs, colleagues, friends whom I am not able to mention one by one. Thank for everything. In depth, ASH thank to beloved Mother for sincere and limitless love. Aliya Syauqina Hadi for her cuteness and cheerfulness.
\section{Introduction}
The interaction between light and matter are interaction between two groups of physical quantities i.e. quantities derived from electromagnetic waves, electric fields, $\vec{E}(\vec{r},t)$, and magnetic fields, $\vec{H}(\vec{r},t)$, with quantities derived from the material i.e. electric susceptibility, $\chi_{e}$, magnetic susceptibility, $\chi_{m}$, and electrical conductivity, $\sigma$.
In a non-conducting materials, electrical conductivity is zero, so that the material response only expressed by $\chi_{e}$ and $\chi_{m}$. The values $\chi_{e}$ and $\chi_{m}$ in the material are influenced by various factors such as frequency of the incident wave \cite{azad,kaya,thamizhmani}, temperature \cite{baeraky,krupka}, pressure and porosity \cite{bourlange,donnelly,saar}, inhomogeneous properties \cite{agron,huttunen,longhi,saville}.
Reflection and refraction of electromagnetic waves in inhomogeneous material are an interesting phenomenon and have several applications, such as: developing radar signal absorbing material \cite{saville}, developing photonic crystals \cite{longhi}, investigations of biological materials \cite{huttunen}, and investigations of stratified media \cite{agron}.
Inhomogeneous properties which are used in \cite{agron,huttunen,longhi,saville} derived from the values of materials susceptibility are only changing in spatial coordinates. Research on inhomogeneous materials which are involving changes in susceptibility values by time has not found widely. By using the concept of inhomogeneous medium in space and time, we expect to find more complete picture of reflection and refraction of electromagnetic wave at the surface of anisotropic, inhomogeneous and linear medium.
\section{Medium Properties}
In this study we use medium which is anisotropic, linear, without charge density and inhomogeneous. All types of materials generally have an anisotropic and linear properties except in certain materials such as FeF$_{2}$ \cite{roniyus1,roniyus2,roniyus3,roniyus4,roniyus5} dan GeSe$_{4}$ \cite{patrick}. Both of these anisotropic materials, beside having linear properties, also have nonlinear properties. Material without free charge density is an insulating material. Example of an insulating materials which are commonly used in optical experiments are glass based materials: fluoride glass \cite{lucas}, silica \cite{tan}, soda glass and copper glass \cite{Smith}, selenium glass \cite{patrick} and non-oxide glass \cite{adam}. Some of ceramic based materials are Nd YAG \cite{qu} and NAT \cite{wang}. Some examples of polymer-based materials are polycarbonate (PC) and polyethylene (PET) \cite{yong}.
Mathematically, anisotropic medium is a medium with the values of linear electric susceptibility, $\chi_{e}$, or linear magnetic susceptibility, $\chi_{m}$, has the form of rank-2 tensor (matrix of ordo 3$\times$3). Inhomogeneous medium in this study is a model of electric and magnetic susceptibility tensor that all components are functions of frequency, position and time respectively, have the form $\overleftrightarrow{\chi}_{e}(\omega,r,t)$ and $\overleftrightarrow{\chi}_{m}(\omega,r,t)$ with $ij$ subscript in form tensor components to $ij$. Theoretical and computational studies in this research is done by involving all components (18 entries) susceptibility tensor of the medium to obtain optimal use.
All components of the tensor model above are included in the calculation and will be used for testing a real medium. Considering FeF$_{2}$ magnet materials have been widely used in optics research \cite{roniyus1,roniyus2,roniyus3,roniyus4,roniyus5} and has been qualified as a medium with properties: linear, ansitropic and no charge density, so that the calculation for a real medium will use this material. The tensor form of FeF$_{2}$ are as follow:
\begin{equation}\label{1}
\overleftrightarrow{\chi}_{e}(\omega,r,t)=
\begin{bmatrix}
4.5 & 0 & 0 \\
0 & 4.5 & 0 \\
0 & 0 & 4.5 \\
\end{bmatrix}
\end{equation}
\begin{equation}\label{2}
\overleftrightarrow{\chi}_{m}(\omega,r,t)=
\begin{bmatrix}
\chi_{11}(\omega,r,t) & i\chi_{12}(\omega,r,t) & 0 \\
-i\chi_{21}(\omega,r,t) & \chi_{22}(\omega,r,t) & 0 \\
0 & 0 & \chi_{33}(\omega,r,t) \\
\end{bmatrix}.
\end{equation}
\section{Basic Formulation}
Electromagnetic wave propagation in the medium is shown by Maxwell equations. Maxwell's equations in international units (SI) for no free charge volume density have form as follows \cite{khan}:
\begin{equation}\label{3}
\nabla.\vec{D}(\vec{r},t)=0
\end{equation}
\begin{equation}\label{4}
\nabla.\vec{B}(\vec{r},t)=0
\end{equation}
\begin{equation}\label{5}
\nabla\times\vec{E}(\vec{r},t)+\frac{\partial\vec{B}(\vec{r},t)}{\partial t}=0
\end{equation}
\begin{equation}\label{6}
\nabla\times\vec{H}(\vec{r},t)-\frac{\partial\vec{D}(\vec{r},t)}{\partial t}=0
\end{equation}
where
\begin{equation}\label{7}
\vec{D}(\vec{r},t)=\epsilon \vec{E}(\vec{r},t)
\end{equation}
\begin{equation}\label{8}
\vec{B}(\vec{r},t)=\mu \vec{H}(\vec{r},t)
\end{equation}
with $\epsilon$ and $\mu$ respectively permittivity and permeability of the medium, i.e. medium response when subjected to electric and magnetic fields. Vectors $\vec{E}(\vec{r},t)$ and $\vec{H}(\vec{r},t)$ are respectively the vector amplitude of electric and magnetic waves.
When the medium of propagation of electromagnetic waves is vacuum, then the value of medium permittivity, $\epsilon$, equal to the vacuum permittivity, $\epsilon_{0}$, and medium permeability, $\mu$, equal to vacuum permeability, $\mu_{0}$. The value of vacuum permittivity, $\epsilon_{0}$, is {8.85$\times$10}$^{-12}$ $C^{2}$/$N.m^{2}$ and vacuum permeability, $\mu_{0}$, is {4$\pi$$\times$10}$^{-7}$ $T.m/A$.
When the electromagnetic wave propagates in non vacuum medium, then the value of $\epsilon$ is equal to $\epsilon$ = $\epsilon_{0}$$\epsilon_{r}$, where $\epsilon_{r}$ is the relative permittivity of the medium and medium permeability value, $\mu$, equal to $\mu_{0}$$\mu_{r}$, where $\mu_{r}$ is relative permeability of the medium. This study has used an electric susceptibility, $\chi_{e}$, and magnetic susceptibility, $\chi_{m}$, of material as response to the presence of electric and magnetic fields.
Linear anisotropic materials have a relative permittivity value of the medium as follows,
\begin{equation}\label{9}
\epsilon^{(r)}_{ii}=1+{\chi}^{(e)}_{ii},~~\epsilon^{(r)}_{ij} = \chi^{(e)}_{ij}
\end{equation}
and relative permeability value of the medium as follows,
\begin{equation}\label{10}
\mu^{(r)}_{ii}=1+{\chi}^{(m)}_{ii},~\mu^{(r)}_{ij}={\chi}^{(m)}_{ij}
\end{equation}
with superscript $r$, $e$ and $m$ respectively have the meanings $relative$, $electric$ and $magnetic$, while the subscript $i$ and $j$ in form of matrix components at $ii$ and $ij$.
Propagation of electromagnetic waves in the medium are expressed by the governing equation of electromagnetic waves in the medium. The formula can be found by put on $curl$ to eq.(\ref{4}) then use eq.(\ref{6}), eq.(\ref{7}) and eq.(\ref{8}) until it is found governing equation of electromagnetic waves in the medium, as follows :
\begin{equation}\label{11}
\nabla^2\vec{E}(\vec{r},t)-\mu_{0}\frac{\partial^2\vec{D}(\vec{r},t)}{\partial t^2}+\frac{1}{\epsilon_{0}}\nabla[\nabla.\vec{P}(\vec{r},t)]-\mu_{0}\frac{\partial}{\partial t}\left[\nabla\times\vec{M}(\vec{r},t)\right]=0.
\end{equation}
Eq.(\ref{11}) will be required to determine value of the vector wave refraction, $k^{(t)}$, in the medium.
\section{Theoretical and Computational Study of Reflection and Refraction of Electromagnetic Waves}
Illustration of propagation of electromagnetic waves can be seen in Fig.1 below.
\begin{center}
\includegraphics[scale=0.7]{Figure1}\\
Figure 1. The events of reflection and refraction of electromagnetic waves with $p$-polarized of electric waves incident (located in the y-z plane) at the left boundary surface of the material.
\end{center}
\subsection{The intensity of the incident wave}
Based on Fig.1, it informs that the incident wave vector in vacuum space is
\begin{equation}\label{12}
\vec{k}^{(i)}= k^{(i)}\hat{y}.
\end{equation}
Consider the incident wave comes from vacuum space, so that the incident wave vector is \cite{wangsness}
\begin{equation}\label{13}
k^{(i)}=\omega\sqrt{\epsilon_{0}\mu_{0}}.
\end{equation}
Vector amplitude of the $p$-polarized incident electric wave is defined as follows
\begin{equation}\label{14}
\vec{E}^{(i)}={E}_{0}\hat{z}.
\end{equation}
Based on eq.(\ref{5}), the vector amplitude of incident magnetic wave is
\begin{equation}\label{15}
\vec{H}^{(i)}=\frac{k^{(i)}}{\mu_{0}\omega}E_{0}\hat{x}.
\end{equation}
The intensity of the incident wave can be found by using the Poynting vector \cite{wangsness}, i.e.
\begin{equation}\label{16}
\langle\vec{S}\rangle=\frac{1}{2}\Re \{\vec{E}(\vec{r},t)\times{\vec{H}}^{*}(\vec{r},t)\},
\end{equation}
then, eq.(\ref{14}) and eq.(\ref{15}) are substituted into eq.(\ref{16}) to obtain the intensity of the incident wave equation
\begin{equation}\label{17}
\langle\vec{S}^{(i)}\rangle=\frac{k^{(i)}{E_{0}}^{2}}{2\mu_{0}\omega}\hat{y}.
\end{equation}
\subsection{The intensity of the reflected wave}
There are two types of reflections that can occur when the electromagnetic waves penetrate the material surface i.e., $ps$ and $pp$-reflection \cite{roniyus3,roniyus4,roniyus5}. Reflection of $ps$, is a reflection of $p$-polarized incident electric waves which generate $s$-polarized reflected electric waves (perpendicular to the plane of incident, i.e. plane of $\textit{x-y}$) with the vector amplitude of the reflected electric wave on $\textit{x}$-axis. Reflection of $pp$, is a reflection of $p$-polarized incident electric waves that generate $p$-polarized of reflected electric waves (parallel to the incident plane, i.e. $\textit{y-z}$ plane) with the vector amplitude of the reflected electric wave on $\textit{z}$-axis.
The results of theoretical studies using susceptibility tensor of vacuum and FeF$_{2}$ show that, in propagation of light parallel to the $\textit{y}$-axis with $p$-polarized of incident electric waves, the reflectance coefficient of $ps$ has been found equal to zero. Percentage of reflectance totally in the $pp$ reflectance, therefore the $ps$ reflection will not be discussed in further. Reflection of $pp$ produces amplitude vector of reflected electric waves on the $\textit{z}$-axis, so that the vector amplitude of the electric waves is
\begin{equation}\label{18}
\vec{E}^{(r_{pp})}=-r_{pp}E_{0}(\hat{z}).
\end{equation}
where $r_{pp}$ is a reflection coefficient of $p$-polarized incident electric wave amplitude.
Based on Snell's Law, the boundary conditions for reflected wave between two different media expressed for the reflected wave vector is
\begin{equation}\label{19}
\vec{k}^{(r_{pp})}=-\vec{k}^{(i)}.
\end{equation}
Furthermore, by using eq.(\ref{5}), eq.(\ref{18}) and eq.(\ref{19}), it can be found amplitude vector for magnetic waves, i.e.
\begin{equation}\label{20}
\vec{H}^{(r_{pp})}=\frac{k^{(i)}}{\mu_{0}\omega}r_{pp}E_{0}(\hat{x}).
\end{equation}
Eq.(\ref{18}) and eq.(\ref{20}) are used to calculate the intensity of reflected waves. The equation of $pp$ reflection intensity for reflected wave is obtained as
\begin{equation}\label{21}
\langle\vec{S}^{(r_{pp})}\rangle=\frac{k^{(i)}{E_{0}}^{2}}{2\mu_{0}\omega}{\mid{r_{pp}}\mid}^{2}\hat{y}.
\end{equation}
\subsection{The intensity of the refraction wave}
The presence of electromagnetic waves on the material surface can raises refraction events \cite{wangsness}. Discussion of wave refraction is started from the relationship between electric field and magnetic field in the material. Eq.(\ref{5}) and eq.(\ref{8}) in forms of the value of electric field related to the magnetic field are
\begin{equation}\label{22} E^{(t)}_{x}=-\frac{\mu_{0}}{k^{(t)}}\bigl(\omega(1+\chi_{33})+i\frac{\partial\chi_{33}}{\partial t}\bigr)H^{(t)}_{z},
\end{equation}
\begin{equation}\label{23} E^{(t)}_{z}=\frac{\mu_{0}}{k^{(t)}}\bigl(\omega(1+\chi_{11})+i\frac{\partial\chi_{11}}{\partial t}\bigr)H^{(t)}_{x}.
\end{equation}
Eq.(\ref{6}) and eq.(\ref{7}) in forms of the wave propagation in the $\textit{y}$-axis direction, so that it can be found the relationship between magnetic field and electric field of the form
\begin{equation}\label{24}
H^{(t)}_{x}=\frac{\epsilon_{0}\omega}{k^{(t)}}\bigl(1+\chi_{e}\bigr)E^{(t)}_{z},
\end{equation}
\begin{equation}\label{25}
H^{(t)}_{z}=-\frac{\epsilon_{0}\omega}{k^{(t)}}\bigl(1+\chi_{e}\bigr)E^{(t)}_{x}.
\end{equation}
If it is assumed that the solution of eq.(\ref{11}) is a harmonic wave function, then by substituting eq.(\ref{22}), eq.(\ref{23})), eq.(\ref{24}) and eq.(\ref{25}) into eq.(\ref{11}), it can be found characteristics matrix of wave refraction, in the form
\begin{equation}\label{26}
\begin{bmatrix}
\frac{\mu_{0}\epsilon_{0}}{k^{(t)}}\omega(1+\chi_{e})P+\mu_{0}\epsilon_{0}\omega(1+\chi_{e})R-{k^{(t)}}^{2} & 0 \\
0 & \frac{\mu_{0}\epsilon_{0}}{k^{(t)}}\omega(1+\chi_{e}) M+\mu_{0}\epsilon_{0}\omega(1+\chi_{e})O-{k^{(t)}}^{2}\\
\end{bmatrix}=0,
\end{equation}
where
\begin{equation}\label{27}
P=\frac{\partial}{\partial t}\frac{\partial\chi_{33}}{\partial y}-i\omega\frac{\partial\chi_{33}}{\partial y},
\end{equation}
\begin{equation}\label{28}
R=\omega(1+\chi_{33})+i\frac{\partial\chi_{33}}{\partial t},
\end{equation}
\begin{equation}\label{29}
M=\frac{\partial}{\partial t}\frac{\chi_{11}}{\partial y}-i\omega\frac{\partial\chi_{11}}{\partial y},
\end{equation}
\begin{equation}\label{30}
O=\omega(1+\chi_{11})+i\frac{\partial\chi_{11}}{\partial t}.
\end{equation}
The values of refraction wave vector can be obtained by solving the determinant of characteristics matrix in eq.(\ref{26}) using MATLAB software. The intensity of the electromagnetic wave refraction has form
\begin{equation}\label{31} \langle\vec{S}^{(t)}\rangle=\mu_{0}\epsilon_{e}{\omega}^2(1+\chi_{e})\left|\Re\Bigl\{\frac{t.t^{*}}{k^{(i)}{k^{(t)}}^{*}}\Bigr\}\right|\hat{y},
\end{equation}
\begin{equation}\label{32} \langle\vec{S}^{(r_{pp})}\rangle=\left|\Re\bigl\{-r_{pp}.r_{pp}^{*}\bigl\}\right|\hat{y}.
\end{equation}
Mark of $(^{*})$ in eq.(\ref{31}) and eq.(\ref{32}) means conjugate vector. Refraction coefficient, $t$, and reflection coefficients, $r_{pp}$, are obtained by equating the tangential components of the amplitude vector at the boundary between two media. The value of $t$ and $r_{pp}$ are as follows:
\begin{equation}\label{33}
t=\frac{2k^{(i)}k^{(t)}}{\mu_{0}\epsilon_{0}{\omega}^2(1+\chi_{e})+k^{(i)}k^{(t)}},
\end{equation}
\begin{equation}\label{34}
r_{pp}=1-t.
\end{equation}
Reflectance and transmittance are calculated as follows :
\begin{equation}\label{35} T=\left|\frac{\langle\vec{S}^{(t)}\rangle.\hat{y}}{\langle\vec{S}^{(i)}\rangle.\hat{y}}\right|,
\end{equation}
\begin{equation}\label{36} R_{pp}=\left|\frac{\langle\vec{S}^{(r_{pp})}\rangle.\hat{y}}{\langle\vec{S}^{(i)}\rangle.\hat{y}}\right|.
\end{equation}
The values of $T$ and $R_ {pp}$ will be visualized with MATLAB software.
\subsection{Transmittance and reflectance in the vacuum and FeF$_{2}$}
To find out the truth of $T$ and $R_ {pp}$ formulation above, it will be tested using vacuum as a first test and then FeF$_2$ magnetic material as a second test material. Based on the law of conservation of energy, in a vacuum, all the energy which arrive should in refraction, while in the non-conducting material, energy which arrive can be reflected and refractived but there is no absorption of energy by the material.
Given the percentage of the total energy which arrive is 1 (100$\%$), so that the total of false intensity is defined not equal to 1. The results of calculations using MATLAB software for transmittance, $T$, and reflectance, $R_{pp}$, values from vacuum and FeF$_2$ in the influence of external magnetic field 3 Tesla can be seen in Fig.2 below
\begin{center}
\includegraphics[scale=0.27]{Figure2}\\
Figure 2. The graph of $T$ and $R_{pp}$ calculation result for (a) vacuum and (b) FeF$_2$ magnetic material with $Total$ as additional graph.
\end{center}
Fig.2 (a) and (b) respectively are result graphs of calculations $T$ and $R_{pp}$ from vacuum and $T$, $R_{pp}$ and $Total$ from FeF$_2$ magnetic materials.
It is shown in Fig.2 (a) that the value of transmittance curve, $T$, is 1 and $R$ is 0. These results are consistent with the theory of refraction in vacuum. The frequency written in $cm^{-1}$ unit is standard which is used by researchers in the field of optical physics \cite{ritchmyer}.
Fig.2 (b) shows the graph of $T$, $R_{pp}$ and $Total$ calculated from FeF$_2$ magnetic materials. Theoretical study with FeF$_2$ materials generally uses far infrared electromagnetic waves with high intensity in the frequency range 48 cm$^{-1}$ to 58 cm$^{-1}$. In this study, we use the frequency range from 51 cm$^{-1}$ to 53.8 cm$^{-1}$ to shorten the running time because of limitations of computer's processor.
Selection of frequency range from 51 cm$^{-1}$ to 53.8 cm$^{-1}$ is related with material resonance frequency i.e. at 52.45 cm$^{-1}$ \cite{roniyus3}. In Fig.2 (b), it can be seen the value of $T$ and $R_{pp}$ fluctuated around the resonance frequency. Furthermore, it appears the value of $Total$ coincides with value of 1. The $Total$ values which coincide with value of 1 can be used as an indicator that the calculations which are performed using the material FeF$_2$ are correct \cite{roniyus3}.
\subsection{Transmittance and reflectance in an anisotropic, inhomogeneous and linear medium}
Medium with properties of anisotropic, inhomogeneous and linear used in this study is a model of susceptibility tensor as shown in eq.(\ref{1}) and eq.(\ref{2}) with shape of susceptibility curve similar to the curve of FeF$_2$ magnetic susceptibility at far infrared frequency.
Consider that the FeF$_2$ susceptibility curve has similar to the fluctuations tangent function. After the modification, it can be obtained tangent function approaches the curve of FeF$_2$ magnetic materials susceptibility under the influence of external magnetic field of 1 Tesla, which is
\begin{equation}\label{37}
\chi_{ij}(\omega,r,t)=\rho(r,t)\beta_{ij}\tan\left(\frac{\zeta(r,t)-82.26}{\omega}\right),
\end{equation}
with $\beta_{11}=0.7/8.55\times10^{\pi+3}$, $\beta_{12}=\beta_{21}=1/8.55\times10^{\pi+1.46}$, $\beta_{22}=\beta_{33}=1/8.55\times10^{\pi}$.
Factor of $\rho(r,t)$ and $\zeta(r,t)$ in eq.(\ref{37}) is the homogeneity parameter of materials. Homogeneity parameter of materials is defined with:\\
$$\vbox{\offinterlineskip
\halign{\strut \quad $#$\quad & \hfil \quad #\quad \hfil & \quad $#$\quad & \hfil \quad #\quad \hfil \cr
\noalign{}
$Homogeneous$ & when\;$\rho(r,t)=c$, $c\in\Re$ & $when$\;\zeta(r,t)=c, \;c\in\Re,\cr
\noalign{}
$Inhomogeneous$ & when\;$\rho(r,t)=\rho^{L}(r)\pm\rho^{L}(t)$ &$when$\;\zeta(r,t)=\zeta^{L}(r)\pm\zeta^{L}(t).\cr
\noalign{}
}}$$
The sign of ($..^{L}$) means linear. The figures below are visualization of eq.(\ref{37}) when the medium is homogeneous $\rho(r,t)=1$, $\zeta(r,t)=0$ and magnetic susceptibility FeF$ _2$ in the influence of external magnetic field of 1 Tesla \cite{roniyus3} :
\begin{center}
\includegraphics[scale=0.27]{Figure3ab}\\
\includegraphics[scale=0.27]{Figure3c}\\
Figure 3. Magnetic susceptibility curves of models and FeF$_2$ appears to fluctuate around the frequency 52.45 cm$^{-1}$, i.e. the resonance frequency of FeF$_2$. (a) $\chi_{11}$, model (red) and $\chi_{11}$, FeF$_2$ (blue), (b) $\chi_{12}$, $\chi_{21}$, models (red) and $\chi_{12}$, $\chi_{21}$, FeF$_2$ (blue and black), (c) $\chi_{22}$, $\chi_{33}$, model (red) and FeF$_2$ (blue and black).
\end{center}
Based on eq.(\ref{37}) note that the value of the magnetic susceptibility is influenced by two homogeneity parameters of the medium i.e., $\rho(r,t)$ (rho) and $\zeta(r,t)$ (zeta). Formulation the value of $\rho(r,t)$ and $\zeta(r,t)$, to simplify the problem, can use Agron and Gogineni approach on stratified materials \cite {agron}.
In stratified materials, review is performed in the direction of light propagation. Given the direction of light propagation in this study is $y$-axis, then it can be selected $\rho(r,t)=1-y$, $\zeta(r,t)=0$ and $\rho(r,t)=1$, $\zeta(r,t)=y$ for an example of calculation. The calculation results of $T$ and $R_{pp}$ values using these parameters at the point (0, 0, 0), are shown in Fig.4 as follows
\begin{center}
\includegraphics[scale=0.27]{Figure4}\\
Figure 4. (a) The comparison values of $T$ and $R_{pp}$ of the model depend on the position and (b) depend on the position of the magnetic susceptibility at the origin (0,0,0).
\end{center}
Fig.4 shows the effect of inhomogeneous magnetic susceptibility values in the direction of light propagation. The values of transmittance is shown in Fig.4 (a), whereas the reflectance values is shown in Fig.4 (b). Both figures show that there are no change in the pattern of transmittance and reflectance from initially state.
The parameters of inhomogeneous by time can be expressed by functions $\rho(r,t)=1-t$, $\zeta(r,t)=0$ and $\rho(r,t)=1$, $\zeta(r,t)=t$. The results of calculation of $T$ and $R_{pp}$ using linear functions at the point (0, 0, 0) and $t$ = 0, 0.5 $s$ are shown in Fig.5 as follow
\begin{center}
\includegraphics[scale=0.27]{Figure5}\\
Figure 5. The comparison values of $T$ and $R_{pp}$ from the magnetic susceptibility model of time dependent and time independent at the origin (0, 0, 0) when $t$ = 0, 0.5 $s$. (a) The values of $T$ on $t$ = 0 $s$ (blue rectangle and black circle), $t$ = 0.5 $s$ (green dot and pink dot), (b) The value of $R_{pp}$ on $t$ = 0 $s$ (blue rectangle and black circle), $t$ = 0.5 $s$ (green dot and pink dot).
\end{center}
Fig.5 shows the effect of inhomogeneous magnetic susceptibility values by time. Transmittance value, $T$, is shown in Fig.5 (a), whereas the reflectance value $R_{pp}$ is shown in Fig.5 (b). Both curves, $T$ and $R_{pp}$, (blue rectangle and black circle) in the beginning ($t$ = 0 $s$), coincide with transmittance and reflectance curves of the medium which are independent of time. But in further time ($t$ = 0.5 $s$), the fluctuations of $T$ and $R_{pp}$ (green dot and pink dot) are decreased and shifted relative to its original state. Changes in $T$ and $R_{pp}$ at inhomogeneous medium thus can be generated by the parameters of time.
\section{Conclusions}
Computational results using vacuum susceptibility shows that the value of transmittance, $T$, is 1 ($ 100 \% $), the reflectance, $R_ {pp}$, is 0 ($ 0 \% $) and the total intensity is 1 (100 $ \% $), according to the theory of reflection and refraction of electromagnetic waves that propagate in vacuum. Computational results of FeF$ _2$ magnetic material shows that the total value of intensity is 1 (100 $ \% $), according to the results of previous studies.
The results of theoretical and computational studies on the surface of an anisotropic, inhomogeneous and linear medium shows that the values of transmittance, $T$, and reflectance, $R_ {pp}$, are not affected by variations of susceptibility values in the direction of light propagation. The changes in values of transmittance, $T$, and reflectance, $R_ {pp}$, on the medium surface can be caused by time.
In other words, in the direction of light propagation ($y$-axis), the values of transmittance, $T$, and reflectance, $R_{pp}$, in the surface of medium are not affected by the changes of magnetic susceptibility values �as shown in Fig.4 (a) and (b). Variations in medium susceptibility values by time can change the fluctuation of transmittance, $T$, and reflectance, $R_{pp}$, at the surface of medium by two mechanisms as shown in Fig.5 (a) and (b).
\section{Acknowledgments}
ASH thank to Computational Laboratory, University of Lampung for supporting software and hardware. Thank to all of Physics Department Staffs, colleagues, friends whom I am not able to mention one by one. Thank for everything. In depth, ASH thank to beloved Mother for sincere and limitless love. Aliya Syauqina Hadi for her cuteness and cheerfulness.
|
{
"timestamp": "2012-03-09T02:01:48",
"yymm": "1203",
"arxiv_id": "1203.1725",
"language": "en",
"url": "https://arxiv.org/abs/1203.1725"
}
|
\section{Introduction}
In this paper we consider optimization problems where the objective is a sum of
two terms: The first term is separable in the variable blocks, and the second
term is separable in the difference between consecutive variable blocks. One
example is the Fused Lasso method in statistical learning,
\cite{Tibshirani-Saunders-Rosset-Zhu-Knight-05}, where the objective includes
an $\ell_1$-norm penalty on the parameters, as well as an $\ell_1$-norm
penalty on the difference between consecutive parameters. The first penalty
encourages a sparse solution, \emph{i.e.~}, one with few nonzero entries, while the
second penalty enhances block partitions in the parameter space.
The same ideas have been applied in many other areas, such as Total Variation
(TV) denoising, \cite{Rudin:1992:NTV:142273.142312}, and segmentation of
ARX models, \cite{OhlssonLB:10}
(where it is called sum-of-norms regularization).
Another example is multi-period portfolio optimization, where the
variable blocks give the portfolio in different time periods, the first
term is the portfolio objective (such as risk-adjusted return), and
the second term accounts for transaction costs.
In many applications, the optimization problem involves a large number of
variables, and cannot be efficiently handled by generic optimization solvers.
In this paper, our main contribution is to derive an efficient and scalable
optimization algorithm, by exploiting the structure of the optimization
problem. To do this, we use a distributed optimization method called
Alternating Direction Method of Multipliers (ADMM). ADMM was developed in the
1970s, and is closely related to many other optimization algorithms
including Bregman iterative algorithms for $\ell_1$ problems, Douglas-Rachford
splitting, and proximal point methods; see \cite{Eckstein92onthe, 4407760}.
ADMM has been applied in many areas, including image and signal processing,
\cite{DBLP:journals/ijcv/Setzer11}, as well as large-scale problems in
statistics and machine learning, \cite{DBLP:journals/ftml/BoydPCPE11}.
We will apply ADMM to $\ell_1$ mean filtering and $\ell_1$ variance filtering
(\cite{BW_Asilomar}), which are important problems in signal processing with
many applications, for example in financial or biological data analysis. In
some applications, mean and variance filtering are used to pre-process data
before fitting a parametric model. For non-stationary data it is also important
for segmenting the data into stationary subsets. The approach we present is
inspired by the $\ell_1$ trend filtering method described in
\cite{Kim-Koh-Boyd-Gorinevsky-09}, which tracks changes in the mean value of
the data. (An example in this paper also tracks changes in the variance of the
underlying stochastic process.) These problems are closely related to the
covariance selection problem, \cite{Dempster-72}, which is a convex
optimization problem when the inverse covariance is used as the optimization
variable, \cite{Banerjee-ElGhaoui-dAspremont-08}. The same ideas can also be
found in \cite{Kim-Koh-Boyd-Gorinevsky-09} and
\cite{Friedman-Hastie-Tibshirani-08}.
This paper is organized as follows. In Section \ref{sec:admm} we review the
ADMM method. In Section \ref{sec:sep}, we apply ADMM to our optimization
problem to derive an efficient optimization algorithm. In Section
\ref{sec:l1mean} we apply our method to $\ell_1$ mean filtering, while in
Section \ref{sec:l1var} we consider $\ell_1$ variance filtering.
Section \ref{sec:num}
contains some numerical examples, and Section \ref{sec:con} concludes the
paper.
\section{Alternating Direction Method of Multipliers (ADMM) }
\label{sec:admm}
In this section we give an overview of ADMM. We follow closely the development in Section 5 of
\cite{DBLP:journals/ftml/BoydPCPE11}.
Consider the following optimization problem
\begin{equation}\label{e-constrained-problem}
\begin{array}{ll}
\mbox{minimize} & f(x)\\
\mbox{subject to} & x \in {\mathcal C}
\end{array}
\end{equation}
with variable $x\in \mathbb{R}^n$, and where $f$ and $\mathcal{C}$ are convex.
We let $p^\star$ denote the optimal value of (\ref{e-constrained-problem}).
We first re-write the problem as
\begin{equation}\label{e-admm-problem}
\begin{array}{ll}
\mbox{minimize} & f(x) + I_\mathcal{C}(z)\\
\mbox{subject to} & x = z,
\end{array}
\end{equation}
where $I_\mathcal{C}(z)$ is the indicator function on $\mathcal{C}$ (\emph{i.e.~},
$I_\mathcal{C}(z) = 0$ for $z\in\mathcal{C}$, and $I_\mathcal{C}(z) = \infty$
for $z\notin\mathcal{C}$). The augmented Lagrangian for this problem is
\[
L_\rho (x, z, u) = f(x) + I_\mathcal{C}(z) + (\rho/2)\|x-z+u\|_2^2,
\]
where $u$ is a scaled dual variable associated with the constraint $x = z$,
\emph{i.e.~}, $u = (1/\rho)y$, where $y$ is the dual variable for $x = z$.
Here, $\rho > 0$ is a penalty parameter.
In each iteration of ADMM, we perform alternating minimization of the augmented
Lagrangian over $x$ and $z$. At iteration $k$ we carry out the following steps
\begin{align}
x^{k+1} &:= \mathop{\rm argmin}_x\{f(x) +(\rho/2)\|x - z^k +
u^k\|_2^2\} \label{eq:admm1}\\
z^{k+1} &:= \Pi_{\mathcal{C}}(x^{k+1} + u^k) \label{eq:admm2}\\
u^{k+1} &:= u^k + (x^{k+1} - z^{k+1}) \label{eq:admm3},
\end{align}
where $\Pi_{\mathcal C}$ denotes Euclidean projection onto $\mathcal{C}$.
In the first step of ADMM, we fix $z$
and $u$ and minimize the augmented Lagrangian over $x$; next, we fix $x$ and
$u$ and minimize over $z$; finally, we update the dual variable $u$.
\subsection{Convergence}
Under mild assumptions on $f$ and $\mathcal{C}$, we can show that the
iterates of ADMM converge to a solution; specifically, we have
\[
f(x^k) \rightarrow p^\star, \quad x^k-z^k\rightarrow 0,
\]
as $k\rightarrow\infty$. The rate of convergence, and hence the number of
iterations required to achieve a specified accuracy, can depend strongly on the
choice of the parameter $\rho$. When $\rho$ is well chosen, this method can
converge to a fairly accurate solution (good enough for many applications),
within a few tens of iterations. However, if the choice of $\rho$ is poor,
many iterations can be needed for convergence. These
issues, including heuristics for choosing $\rho$, are discussed in more detail
in \cite{DBLP:journals/ftml/BoydPCPE11}.
\subsection{Stopping criterion}
The primal and dual residuals at iteration $k$ are given by
\[
e_p^k = (x^k-z^k), \quad
e_d^k = -\rho (z^k-z^{k-1}).
\]
We terminate the algorithm when the primal and dual residuals satisfy a
stopping criterion (which can vary depending on the requirements of the
application). A typical criterion is to stop when
\[
\|e_p^k\|_2 \leq \epsilon^\mathrm{pri},\quad
\|e_d^k\|_2 \leq \epsilon^\mathrm{dual}.
\]
Here, the tolerances
$\epsilon^\mathrm{pri} > 0$ and $\epsilon^\mathrm{dual} > 0$ can
be set via an absolute plus relative criterion,
\begin{align*}
&\epsilon^\mathrm{pri} =
\sqrt{n} \epsilon^\mathrm{abs} + \epsilon^\mathrm{rel}
\max\{\|x^k\|_2, \|z^k\|_2\}, \\
&\epsilon^\mathrm{dual} =
\sqrt{n} \epsilon^\mathrm{abs} + \epsilon^\mathrm{rel}
\rho \|u^k\|_2,
\end{align*}
where $\epsilon^\mathrm{abs} > 0$ and $\epsilon^\mathrm{rel} > 0$ are absolute
and relative tolerances (see \cite{DBLP:journals/ftml/BoydPCPE11} for details).
\section{Problem formulation and method}
\label{sec:sep}
In this section we formulate our problem and derive an efficient distributed
optimization algorithm via ADMM.
\subsection{Optimization problem}
We consider the problem
\begin{equation}\label{e-our-problem}
\begin{array}{ll}
\mbox{minimize} & \sum_{i=1}^N \Phi_i(x_i)+\sum_{i=1}^{N-1} \Psi_i(r_i)\\
\mbox{subject to} & r_i=x_{i+1}-x_i,\quad i = 1,\ldots,N-1
\end{array}
\end{equation}
with variables $x_1, \ldots, x_N,r_1, \ldots, r_{N-1}\in\mathbf R^n$,
and where $\Phi_i:\mathbf R^n\rightarrow\mathbf R\cup\{\infty\}$ and
$\Psi_i:\mathbf R^n\rightarrow\mathbf R\cup\{\infty\}$ are convex functions.
This problem has the form (\ref{e-constrained-problem}), with variables
$x = (x_1,\ldots,x_N)$, $r = (r_1,\ldots,r_{N-1})$, objective function
\[
f(x,r) = \sum_{i=1}^N \Phi_i(x_i)+\sum_{i=1}^{N-1} \Psi_i(r_i)
\]
and constraint set
\begin{equation}\label{e-constraint-set}
\mathcal{C} = \{ (x, r) \mid r_i = x_{i+1}-x_i, \;i=1,\ldots,N-1\}.
\end{equation}
The ADMM form for problem (\ref{e-our-problem}) is
\begin{equation}\label{e-our-problem-admm}
\begin{array}{ll}
\mbox{minimize} & \sum_{i=1}^N \Phi_i(x_i)+\sum_{i=1}^{N-1} \Psi_i(r_i) +
I_\mathcal{C}(z,s) \\
\mbox{subject to}
& r_i = s_i, \quad i = 1,\ldots,N-1 \\
& x_i = z_i, \quad i = 1,\ldots,N,
\end{array}
\end{equation}
with variables $x = (x_1,\ldots,x_N)$, $r = (r_1,\ldots,r_{N-1})$, $z
= (z_1,\ldots,z_N)$, and $s = (s_1,\ldots,s_{N-1})$. Furthermore, we let $u =
(u_1,\ldots,u_N)$ and $t = (t_1,\ldots,t_{N-1})$ be vectors of scaled dual
variables associated with the constraints $x_i = z_i$, $i = 1,\ldots,N$, and
$r_i = s_i$, $i = 1,\ldots,N-1$ (\emph{i.e.~}, $u_i = (1/\rho)y_i$, where $y_i$ is the
dual variable associated with $x_i = z_i$).
\subsection{Distributed optimization method}
Applying ADMM to problem (\ref{e-our-problem-admm}), we carry out the following
steps in each iteration.
\paragraph*{Step 1.}
Since the objective function $f$ is separable in $x_i$ and $r_i$, the
first step (\ref{eq:admm1}) of the ADMM algorithm consists of $2N-1$
separate minimizations
\begin{equation}\label{e-admm-11}
x_i^{k+1} :=
\mathop{\rm argmin}_{x_i} \{\Phi_i(x_i) +(\rho/2)\|x_i - z_i^k + u_i^k\|_2^2\},
\end{equation}
$i = 1,\ldots,N$, and
\begin{equation}\label{e-admm-12}
r_i^{k+1} :=
\mathop{\rm argmin}_{r_i} \{\Psi_i(r_i) +(\rho/2)\|r_i - s_i^k + t_i^k\|_2^2\},
\end{equation}
$i = 1,\ldots,N-1$. These updates can all be carried out in parallel. For
many applications, we will see that we can often solve (\ref{e-admm-11})
and (\ref{e-admm-12}) analytically.
\paragraph*{Step 2.}
In the second step of ADMM, we project $(x^{k+1} + u^k, r^{k+1} + t^k)$ onto the constraint
set $\mathcal{C}$, \emph{i.e.~},
\[
(z^{k+1}, s^{k+1}) := \Pi_\mathcal{C}((x^{k+1}, r^{k+1}) + (u^k, t^k)).
\]
For the particular constraint set (\ref{e-constraint-set}), we will show in
Section \ref{s-projection} that the projection can be performed extremely
efficiently.
\paragraph*{Step 3.}
Finally, we update the dual variables:
\[
u_i^{k+1} := u_i^k + (x_i^{k+1}-z_i^{k+1}), \quad i = 1,\ldots,N
\]
and
\[
t_i^{k+1} := t_i^k + (r_i^{k+1}-s_i^{k+1}), \quad i = 1,\ldots,N-1.
\]
These updates can also be carried out independently in parallel, for each
variable block.
\subsection{Projection}\label{s-projection}
In this section we work out an efficient formula for projection onto the
constraint set $\mathcal{C}$ (\ref{e-constraint-set}). To perform the
projection
\[
(z, s) = \Pi_\mathcal{C}((w, v)),
\]
we solve the optimization problem
\[
\begin{array}{ll}
\mbox{minimize} & \|z - w\|_2^2 + \|s - v\|_2^2 \\
\mbox{subject to} & s = Dz,
\end{array}
\]
with variables $z = (z_1,\ldots,z_N)$ and $s = (s_1,\ldots,s_{N-1})$, and where
$D\in\mathbf R^{(N-1)n\times Nn}$ is the forward difference operator, \emph{i.e.~},
\[
D = \left[
\begin{array}{lllll}
-I & I & & & \\
& -I & I & & \\
& & \ddots & \ddots & \\
& & & -I & I \\
\end{array}
\right].
\]
This problem is equivalent to
\[
\begin{array}{ll}
\mbox{minimize} & \|z - w\|_2^2 + \|Dz - v\|_2^2.
\end{array}
\]
with variable $z = (z_1,\ldots,z_N)$. Thus to perform the projection we first
solve the optimality condition
\begin{equation}\label{e-opt-cond}
(I + D^TD)z = w + D^Tv,
\end{equation}
for $z$, then we let $s = Dz$.
The matrix $I + D^TD$ is block tridiagonal, with diagonal blocks equal to
multiples of $I$, and sub/super-diagonal blocks equal to $-I$. Let $LL^T$ be
the Cholesky factorization of $I + D^TD$. It is easy to show that $L$ is block
banded with the form
\[
L = \left[
\begin{array}{lllll}
l_{1,1} & & & & \\
l_{2,1} & l_{2,2} & & & \\
& l_{3,2} & l_{3,3} & & \\
& & \ddots & \ddots & \\
& & & l_{N,N-1} & l_{N,N}
\end{array}
\right] \otimes I,
\]
where $\otimes$ denotes the Kronecker product. The coefficients $l_{i,j}$ can
be explicitly computed via the recursion
\[
\begin{array}{l}
l_{1,1} = \sqrt{2}, \\
l_{i+1,i} = -1/l_{i,i}, \;\; l_{i+1,i+1} = \sqrt{3-l_{i+1,i}^2},
\;\; i = 1,\ldots,N-2, \\
l_{N,N-1} = -1/l_{N-1,N-1}$, \quad $l_{N,N} = \sqrt{2-l_{N,N-1}^2}.
\end{array}
\]
The coefficients only need to be computed once, before the projection
operator is applied.
The projection therefore consists of the following steps
\begin{enumerate}
\item Form $b := w + D^Tv$:
\[
\begin{array}{l}
b_1 := w_1 - v_1, \quad b_N := w_N + v_{N-1}, \\
b_i := w_i + (v_{i-1} - v_i),\quad i = 2,\ldots,N-1.
\end{array}
\]
\item Solve $Ly = b$:
\begin{align*}
y_1 &:= (1/l_{1,1})b_1, \\
y_i &:= (1/l_{i,i})(b_i - l_{i,i-1}y_{i-1}),\quad i = 2,\ldots,N.
\end{align*}
\item Solve $L^Tz = y$:
\begin{align*}
z_N &:= (1/l_{N,N})y_N, \\
z_i &:= (1/l_{i,i})(y_i-l_{i+1,i}z_{i+1}),\quad i = N-1,\ldots,1.
\end{align*}
\item Set $s = Dz$:
\[
s_i := z_{i+1}-z_i,\quad i = 1,\ldots,N-1.
\]
\end{enumerate}
Thus, we see that we can perform the projection very efficiently,
in $\mathcal{O}(Nn)$ flops (floating-point operations). In fact,
if we pre-compute the inverses $1/l_{i,i}$, $i = 1,\ldots,N$, the only
operations that are required are multiplication, addition, and
subtraction. We do not need to perform division, which can be expensive
on some hardware platforms.
\section{Examples}
\subsection{$\ell_1$ Mean filtering}
\label{sec:l1mean}
Consider a sequence of vector random variables
\[
Y_i\sim {\mathcal{N}}(\bar y_i, \Sigma),
\quad i = 1,\ldots,N,
\]
where $\bar y_i\in\mathbf R^n$ is the mean, and $\Sigma\in{\mbox{\bf S}}^n_+$ is the
covariance matrix. We assume that the covariance matrix is known, but the mean
of the process is unknown. Given a sequence of observations $y_1,\ldots,y_N$,
our goal is to estimate the mean under the assumption that it is piecewise
constant, \emph{i.e.~}, $\bar y_{i+1} = \bar y_i$ for many values of $i$.
In the Fused Group Lasso method, we obtain our estimates by solving
\[
\begin{array}{ll}
\mbox{minimize} & \sum_{i=1}^N\frac 1 2
(y_i-x_i)^T\Sigma^{-1}(y_i-x_i)+\lambda \sum_{i=1}^{N-1}
\|r_i\|_2\\
\mbox{subject to} & r_i=x_{i+1}-x_i, \quad i = 1,\ldots,N-1,
\end{array}
\]
with variables $x_1,\ldots,x_N$, $r_1,\ldots,r_{N-1}$. Let
$x_1^\star,\ldots,x_N^\star$, $r_1^\star,\ldots,r_{N-1}^\star$
denote an optimal point, our estimates
of $\bar y_1,\ldots,\bar y_N$ are $x_1^\star,\ldots,x_N^\star$.
This problem is clearly in
the form (\ref{e-our-problem}), with
\[
\Phi_i(x_i) = \frac{1}{2} (y_i-x_i)^T\Sigma^{-1}(y_i-x_i),\quad
\Psi_i(r_i) = \lambda \|r_i\|_2.
\]
\newpage
\paragraph*{ADMM steps.}
For this problem, steps (\ref{e-admm-11}) and (\ref{e-admm-12}) of ADMM
can be further simplified. Step (\ref{e-admm-11}) involves minimizing an
unconstrained quadratic function in the variable $x_i$, and can be
written as
\[
x_i^{k+1} = (\Sigma^{-1}+\rho I)^{-1}
(\Sigma^{-1} y_i + \rho(z_i^k-u_i^k)).
\]
Step (\ref{e-admm-12}) is
\[
r_i^{k+1} := \mathop{\rm argmin}_{r_i} \{\lambda \|r_i\|_2+(\rho/2)\|r_i
- s_i^k + t_i^k\|_2^2\},
\]
which simplifies to
\begin{equation}\label{eq:thresh}
r_i^{k+1} ={\mathcal{S}}_{\lambda/\rho}(s_i^k -t_i^k),
\end{equation}
where $\mathcal{S}_\kappa$ is the vector soft thresholding operator, defined as
\[
{\mathcal{S}}_\kappa({a})=(1-\kappa/\|a\|_2)_+ {a},\quad {\mathcal{S}}_\kappa({0})=0.
\]
Here the notation $(v)_+ = \max\{0, v\}$ denotes the positive part of the vector $v$.
(For details see \cite{DBLP:journals/ftml/BoydPCPE11}.)
\paragraph*{Variations.}
In some problems, we might expect that individual components of $x_t$ will be
piecewise constant, in which case we can instead use the standard Fused Lasso
method. In the standard Fused Lasso method we solve
\[
\begin{array}{ll}
\mbox{minimize} & \sum_{i=1}^N\frac 1 2
(y_i-x_i)^T\Sigma^{-1}(y_i-x_i)+\lambda \sum_{i=1}^{N-1}
\|r_i\|_1 \\
\mbox{subject to} & r_i=x_{i+1}-x_i,\quad i = 1,\ldots,N,
\end{array}
\]
with variables $x_1,\ldots,x_N$, $r_1,\ldots,r_{N-1}$. The ADMM updates are the
same, except that instead of doing vector soft thresholding for step
(\ref{e-admm-12}), we perform scalar componentwise soft thresholding, \emph{i.e.~},
\[
(r_i^{k+1})_j ={\mathcal{S}}_{\lambda/\rho}((s_i^k -t_i^k)_j),\quad j=1,\ldots,n.
\]
\subsection{$\ell_1$ Variance filtering}
\label{sec:l1var}
Consider a sequence of vector random variables (of dimension $n$)
\[
Y_i\sim {\mathcal{N}}(0,\Sigma_i), \quad i = 1,\ldots,N,
\]
where $\Sigma_i\in{\mbox{\bf S}}^n_+$ is the covariance matrix for $Y_i$ (which
we assume is fixed but unknown). Given observations of $y_1,\ldots,y_N$, our
goal is to estimate the sequence of covariance matrices
$\Sigma_1,\ldots,\Sigma_N$, under the assumption that it is
piecewise constant, \emph{i.e.~}, it is often the case that $\Sigma_{i+1} =
\Sigma_i$. In order to obtain a convex problem, we use the
inverse covariances $X_i = \Sigma_i^{-1}$ as our variables.
The Fused Group Lasso method for this problem involves solving
\[
\begin{array}{ll}
\mbox{minimize} & \sum_{i=1}^N \mathop{\bf Tr}(X_iy_iy_i^T)-\log\det X_i
+\lambda \sum_{i=1}^{N-1}\|R_i\|_F \\
\mbox{subject to} & R_i = X_{i+1}-X_i, \quad i = 1,\ldots,N-1,
\end{array}
\]
where our variables are $R_i\in{\mbox{\bf S}}^n$, $i = 1,\ldots,N-1$, and
$X_i\in{\mbox{\bf S}}^n_+$, $i = 1,\ldots,N$. Here,
\[
\|R_i\|_F = \sqrt{\mathop{\bf Tr}(R_i^TR_i)}
\]
is the Frobenius norm of $R_i$. Let $X_1^\star, \ldots,X_N^\star$,
$R_1^\star,\ldots,R_{N-1}^\star$ denote an optimal point, our estimates
of $\Sigma_1,\ldots,\Sigma_N$ are $(X_1^\star)^{-1},\ldots,(X_N^\star)^{-1}$.
\paragraph*{ADMM steps.}
It is easy to see that steps (\ref{e-admm-11}) and (\ref{e-admm-12})
simplify for this problem. Step (\ref{e-admm-11}) requires solving
\[
X_i^{k+1} := \mathop{\rm argmin}_{X_i\succ 0} \{
\Phi_i(X_i)
+(\rho/2)\|X_i - Z_i^k + U_i^k\|_2^2\},
\]
where
\[
\Phi_i(X_i) = \mathop{\bf Tr}(X_iy_iy_i^T)-\log\det X_i.
\]
This update can be solved analytically, as follows.
\begin{enumerate}
\item Compute the eigenvalue decomposition of
\[
\rho\left(Z_i^k
-U_i^k\right)-y_iy_i^T=Q\Lambda Q^T
\]
where $\Lambda={\bf diag}(\lambda_1,\ldots , \lambda_n)$.
\item Now let
\[
\mu_j := \frac{\lambda_j+\sqrt{\lambda_j^2+4\rho}}{2\rho},\quad j = 1,\ldots,n.
\]
\item Finally, we set
\[
X_i^{k+1} = Q \mathop{\bf diag}(\mu_1,\ldots,\mu_n) Q^T.
\]
\end{enumerate}
For details of this derivation, see Section 6.5 in
\cite{DBLP:journals/ftml/BoydPCPE11}.
Step (\ref{e-admm-12}) is
\[
R_i^{k+1} := \mathop{\rm argmin}_{R_i}
\{\lambda \|R_i\|_F+(\rho/2)\|R_i - S_i^k + T_i^k\|_2^2\},
\]
which simplifies to
\[
R_i^{k+1} ={\mathcal{S}}_{\lambda/\rho}(S_i^k - T_i^k),
\]
where $\mathcal{S}_\kappa$ is a matrix soft threshold operator, defined as
\[
{\mathcal{S}}_\kappa(A)=(1-\kappa/\|A\|_F)_+ A,\quad
{\mathcal{S}}_\kappa({0})=0.
\]
\paragraph*{Variations.}
As with $\ell_1$ mean filtering, we can replace the Frobenius norm penalty with
a componentwise vector $\ell_1$-norm penalty on $R_i$ to get the problem
\[
\begin{array}{ll}
\mbox{minimize} & \sum_{i=1}^N
\mathop{\bf Tr}(X_iy_iy_i^T)-\log\det X_i
+ \lambda \sum_{i=1}^{N-1}\|R_i\|_1 \\
\mbox{subject to} & R_i=X_{i+1}-X_i,
\quad i = 1,\ldots,N-1,
\end{array}
\]
with variables $R_1,\ldots,R_{N-1}\in{\mbox{\bf S}}^n$, and
$X_1,\ldots,X_N\in{\mbox{\bf S}}^n_+$, and where
\[
\|R\|_1 = \sum_{j,k} |R_{jk}|.
\]
Again, the ADMM updates are the same, the only difference is that in step
(\ref{e-admm-12}) we replace matrix soft thresholding with a componentwise
soft threshold, \emph{i.e.~},
\[
(R_i^{k+1})_{l,m} = \mathcal{S}_{\lambda/\rho}((S_i^k
-T_i^k)_{l,m}),
\]
for $l = 1,\ldots,n$, $m = 1,\ldots,n$.
\subsection{$\ell_1$ Mean and variance filtering}
\label{sec:l1mean_var}
Consider a sequence of vector random variables
\[
Y_i\sim {\mathcal{N}}(\bar y_i, \Sigma_i),
\quad i = 1,\ldots,N,
\]
where $\bar y_i\in\mathbf R^n$ is the mean, and $\Sigma_i\in{\mbox{\bf S}}^n_+$ is the
covariance matrix for $Y_i$. We assume that the mean and covariance matrix of
the process is unknown. Given observations $y_1,\ldots,y_N$, our goal is to
estimate the mean and the sequence of covariance matrices
$\Sigma_1,\ldots,\Sigma_N$, under the assumption that they are piecewise
constant, \emph{i.e.~}, it is often the case that $\bar y_{i+1} = \bar y_i$ and
$\Sigma_{i+1} = \Sigma_i$. To obtain a convex optimization problem, we use
\[
X_i=-\frac{1}{2}\Sigma_t^{-1},\quad m_i=\Sigma_t^{-1}x_i,
\]
as our variables. In the Fused Group Lasso method, we obtain our estimates by solving
\[
\begin{array}{ll}
\mbox{minimize} & \sum_{i=1}^N -(1/2)\log\det(-X_i)-\mathop{\bf Tr}(X_iy_iy_i^T)\\
& \quad\qquad - m_i^T y_i -(1/4)\mathop{\bf Tr}(X^{-1}_im_im_i^T)\\
& \quad\qquad + \lambda_1 \sum_{i=1}^{N-1} \|r_i\|_2+\lambda_2 \sum_{i=1}^{N-1} \|R_i\|_F\\
\mbox{subject to} & r_i=m_{i+1}-m_i, \quad i = 1,\ldots,N-1, \\
& R_i=X_{i+1}-X_i, \quad i = 1,\ldots,N-1,
\end{array}
\]
with variables $r_1,\ldots,r_{N-1} \in \mathbf R^n$, $m_1,\ldots,m_{N}\in \mathbf R^n$, $R_1,\ldots,R_{N-1}\in{\mbox{\bf S}}^n$, and
$X_1,\ldots,X_N\in{\mbox{\bf S}}^n_+$.
\paragraph*{ADMM steps.}
This problem is also in the form (\ref{e-our-problem}), however, as far as we
are aware, there is no analytical formula for steps (\ref{e-admm-11}) and
(\ref{e-admm-12}). To carry out these updates, we must solve semidefinite
programs (SDPs), for which there are a number of efficient and reliable
software packages (\cite{TTT:99,Stu:99}).
\section{Numerical Example}
\label{sec:num}
In this section we solve an instance of $\ell_1$ mean filtering with $n = 1$,
$\Sigma = 1$, and $N = 400$, using the standard Fused Lasso method. To improve
convergence of the ADMM algorithm, we use over-relaxation with $\alpha=1.8$,
see \cite{DBLP:journals/ftml/BoydPCPE11}. The parameter $\lambda$ is chosen as
approximately 10\% of $\lambda_\mathrm{max}$, where $\lambda_\mathrm{max}$ is
the largest value that results in a non-constant mean estimate. Here,
$\lambda_\mathrm{max} \approx 108$ and so $\lambda=10$. We use an absolute
plus relative error stopping criterion, with $\epsilon^\mathrm{abs} = 10^{-4}$
and $\epsilon^\mathrm{rel} = 10^{-3}$. Figure \ref{ex1} shows convergence of the
primal and dual residuals. The resulting estimates of the means
are shown in Figure~\ref{ex2}.
\begin{figure}[ht]
\begin{center}
\includegraphics[width = \columnwidth]{residuals.pdf}
\caption{Residual convergence: Primal residual $e_p$ (solid line), and dual residual $e_d$
(dashed line).}\label{ex1}\end{center}
\end{figure}
\begin{figure}[ht]
\begin{center}
\includegraphics[width = \columnwidth]{estimates.pdf}
\caption{Estimated means (solid line), true means (dashed line) and
measurements (crosses).}\label{ex2}\end{center}
\end{figure}
We solved the same $\ell_1$ mean filtering problem using CVX, a package for
specifying and solving convex optimization problems (\cite{cvxProg}). CVX
calls generic SDP solvers SeDuMi (\cite{TTT:99}) or SDPT3 (\cite{Stu:99}) to
solve the problem. While these solvers are reliable for wide classes of
optimization problems, and exploit sparsity in the problem formulation,
they are not customized for particular problem families, such as ours.
The computation time for CVX is approximately 20 seconds. Our
ADMM algorithm (implemented in C), took $2.2$ \textit{milliseconds} to produce
the same estimates. Thus, our algorithm is approximately 10000 times faster
compared with generic optimization packages. Indeed, our implementation
does \textit{not} exploit the fact that steps 1 and 3 of ADMM can be
implemented independently in parallel for each measurement. Parallelizing
steps 1 and 3 of the computation can lead to further speedups. For example,
simple multi-threading on a quad-core CPU would result in a
further $4 \times$ speed-up.
\section{Conclusions}
\label{sec:con}
In this paper we derived an efficient and scalable method for an optimization
problem (\ref{e-our-problem}) that has a variety of applications in control and
estimation. Our custom method exploits the structure of the problem via a
distributed optimization framework. In many applications, each step of the
method is a simple update that typically involves solving a set of linear
equations, matrix multiplication, or thresholding, for which there are
exceedingly efficient libraries. In numerical examples we have shown that we
can solve problems such as $\ell_1$ mean and variance filtering many orders of
magnitude faster than generic optimization solvers such as SeDuMi or SDPT3.
The only tuning parameter for our method is the regularization parameter
$\rho$. Finding an optimal $\rho$ is not a straightforward problem, but
\cite{DBLP:journals/ftml/BoydPCPE11} contains many heuristics that work
well in practice. For the $\ell_1$ mean filtering example, we find that
setting $\rho \approx \lambda$ works well, but we do not have a formal
justification.
|
{
"timestamp": "2012-03-09T02:03:21",
"yymm": "1203",
"arxiv_id": "1203.1828",
"language": "en",
"url": "https://arxiv.org/abs/1203.1828"
}
|
\section{Introduction}
The wealth of interesting physical phenomena that can be found for equiatomic intermetallic compounds, like superconductivity, heavy-fermion or Kondo behavior, magnetic ordering, or thermopower, is not only due
to the vast amount of possibilities of combining two or more different elements from the periodic table. The interplay between electronic structure, crystal structure and chemical bonding leads to an additional degree of variability. The ternary equiatomic compound TiGePt is such an example.\cite{Aackerbauer10} This intermetallic adopts two different crystal structures (Fig.~\ref{structureX}). The low temperature (LT) modification of TiGePt forms in the MgAgAs-type structure\cite{Nowotny41} ("half-Heusler"). Here, Ti and Ge atoms form a sodium chloride type lattice, in which Pt atoms are inserted in half of the tetrahedral voids. By heating above 885 $^{\mathrm{o}}$C, TiGePt transforms via a reconstructive transition into an orthorhombic TiNiSi-type structure\cite{Shoemaker65} with a considerably lowered symmetry. The high temperature (HT) modification of TiGePt can be quenched down to low temperatures.
Its crystal structure can be regarded as a three-dimensional network formed by edge-sharing six-membered puckered rings of Pt and Ge atoms, interlinked along the [100] direction through short Pt-Ge contacts.
In the [010] direction, large eight-membered Pt-Ge rings form channels in which Ti atoms are embedded.
It is remarkable that the volume is reduced by over 10\% in going from the LT to the HT phase.\cite{Aackerbauer10} Furthermore, the occurrence of an insulator-metal transition was suggested based on electrical resistivity measurements.\cite{Aackerbauer10} In the LT phase, TiGePt revealed a semiconducting behavior, while in the HT modification it showed a more metallic temperature dependence with three to four orders of magnitude smaller resistivity values. The structural changes are caused by differences in chemical bonding. Analysis of the atomic interactions within the electron density/electron localizability approach revealed strong differences in atomic interactions between the LT and HT modifications.\cite{Aackerbauer10}
A similar polymorphism has been reported for YbPdSb \cite{Mishra02}, YbAuBi \cite{Merlo90}, GdNiSb \cite{Skolozdra97} and VFeSb \cite{VFeSbT}. All these compounds crystallize in the cubic structure isotypic to MgAgAs at low temperatures. At elevated temperatures, they adopt an AlB$_2$-related crystal structure - being the aristotype of the structure family to which TiNiSi belongs to - with lower symmetry and larger crystal density.
In the Yb-based systems, the structural transitions are accompanied by changes in valence state of Yb.\cite{Mishra02, Merlo90}
For GdNiSb, an insulator-metal transition has been predicted based on $ab$ $initio$ electronic structure calculations, but not been confirmed experimentally yet.\cite{GdNiSb}
Electrical resistivity measurements for VFeSb suggest a transition from a highly doped semiconductor to a metallic-like conductor at a temperature of 1042 K.\cite{VFeSbT} So far, the change in its electrical properties has not been inspected in detail by means of an electronic structure study.
\begin{figure}
\includegraphics[width=0.4\textwidth]{figure_1.eps}\\
\caption{(Color online) Crystal structures of the LT phase (cubic MgAgAs-type, left) and the HT phase (orthorhombic TiNiSi-type, right) of TiGePt.\cite{Aackerbauer10} Ti, Ge, Pt atoms are shown as white, blue and grey spheres, respectively.}
\label{structureX}
\end{figure}
Here, we report on the electronic structure of TiGePt in both the LT and HT phases. The objective of our study is to establish the relationship between the crystal structure and the electronic properties of TiGePt. To this end we will employ X-ray photoelectron and absorption spectroscopies in combination with full potential electronic band structure calculations.
\section{Methods}
\label{sec:methods}
The samples were
prepared and characterized as described in Ref.~\onlinecite{Aackerbauer10}.
All spectroscopic measurements were carried out at room temperature.
The soft X-ray photoelectron spectroscopy (PES) and absorption
spectroscopy (XAS) experiments were performed at the Dragon
beamline of the NSRRC in Taiwan, using an ultra-high vacuum system
with a pressure in the low 10$^{-10}$ mbar range. For the PES, a
Scienta SES-100 electron energy analyzer was used and the overall
energy resolution was set to 150 meV FWHM at 190 eV photon energy, and
to 350 meV FWHM at 700 eV photon energy. The energy calibration has been done by
using the Fermi cut-off of a polycrystalline Pt metal reference
which was also taken as the zero of the binding energy scale. The
4$f_{7/2}$ core level of the Pt metal was used as an energy
reference.
The XAS spectra at the Ti $L_{2,3}$-edges were
taken in the total electron yield mode with energy resolution of
the photons of 150 meV. A SrTiO$_3$ single crystal was measured
simultaneously as an energy reference for the XAS. Before the
measurements, the polycrystalline TiGePt samples were fractured
\textit{in-situ} to obtain clean surfaces.
The XAS spectra at the Ge $K$--edge were obtained in a transmission
arrangement at the EXAFS beamline C of HASYLAB at DESY, equipped with a Si (111) double
crystal monochromator which yielded an experimental resolution (FWHM) of approximately 3 eV
at the Ge $K$ threshold of about 11100 eV. Powdered materials
were mixed with small amounts of B$_{4}$C and mounted on a sample holder (1 cm$^{2}$ window) using
paraffin wax. The data were recorded together with powdered Ge as an external reference.
The electronic structure of the two modifications of TiGePt was computed using lattice parameters and atomic positions obtained experimentally at room temperature.\cite{Aackerbauer10}
First--principles band structure calculations were performed using the full--potential local-orbital code FPLO (version 9.01-35)\cite{FPLOKoepernik99} in the fully relativistic mode.
In this method, the four-component Kohn-Sham-Dirac equation containing spin-orbit (SO) coupling to all orders is solved self-consistently.
The Perdew-Wang parametrization\cite{PerdewWang92} of the exchange-correlation potential within the local density approximation (LDA) was employed.
The Brillouin zone was sampled by a well--converged mesh of 27000 $k$-points (30$\times$30$\times$30 mesh, 1368 points in the irreducible wedge of the Brillouin zone) for the cubic LT phase and 10260 $k$-points (20$\times$27$\times$19 mesh, 1540 points in the irreducible wedge of the Brillouin zone) for the orthorhombic HT phase.
To explain the near-edge structures of the Ge $K$ XAS spectra, we carried out band structure calculations by the full potential linearized augmented plane wave (FP--LAPW) method\cite{LAPW} as implemented in the Wien2k\_07 code\cite{Wien2k}. In the scalar--relativistic calculations, exchange-correlation effects were treated within the LDA approximation in the form proposed by Perdew and Wang\cite{PerdewWang92}. Spin-orbit coupling was included in the second variational method using the scalar-relativistic eigenfunctions as basis.\cite{SO}
By comparing the resulting total densities of states (DOS) and band structures with those derived from the fully relativistic calculations using the FPLO code we verified the sufficient accuracy of our FP-LAPW computational results.
The near-edge spectra were calculated according to the formalism described in Refs \onlinecite{XANESt, XANESt1, XANESt2}.
For dipole-allowed transitions, energy dependent matrix elements containing radial transition probabilities were multiplied with the partial DOS. The results were convoluted by the pseudo--Voigt function with a FWHM of 1.5 eV for the Lorentzian and of 2.5 eV for the Gaussian components, respectively, to mimic the instrumental resolution and the lifetime broadening effects. Finally, the calculated curves were shifted by 11101.2 eV in order to match the experimental energy scales.
\section{results and discussion}
Fully relativistic electronic structure calculations support the experimental observation that the LT phase is the more stable modification of TiGePt. The calculated difference in the total energy between the two phases is about 0.19 eV per formula unit, which is of the same order as the energy scale of the observed transition temperature of about 1160 K.
\begin{figure}
\includegraphics[width=0.4\textwidth]{figure_2.eps}\\
\caption{(Color online) Total and partial density of states (DOS) from fully relativistic electronic structure calculations of TiGePt: the results for the low-temperature (LT) phase are shown in the bottom panel and the high-temperature (HT) phase in the top panel. The common vertical dashed line indicates the position of the Fermi level.}
\label{LDA}
\end{figure}
Fig.~\ref{LDA} shows the calculated total electronic DOS of the LT phase (bottom panel) and HT phase (top panel) of TiGePt together with the partial DOS of the Pt 5$d$, Ge 4$p$ and 4$s$, and the Ti 3$d$, which are the relevant states composing the valence band. The obtained electron counts for the valence states of the two modifications of TiGePt having the same nominal composition are very similar and amount to about 8.6, 2.7, 1.5, and 2.5 for the Pt 5$d$, Ge 4$p$, Ge 4$s$, and Ti 3$d$ orbital, respectively. Nevertheless, the essential differences in crystal structure and chemical bonding properties between the two phases lead to substantial differences in the DOS as explained below.
For LT-TiGePt there is a band gap of about 0.8~eV, consistent with the semiconducting behavior in the resistivity measurements.\cite{Aackerbauer10} The HT phase, on the other hand, is a metal with a rather low value of the DOS at the Fermi level, $i.e.$ about 0.3 states per eV and formula unit, in line with the results of our thermodynamic and transport study.\cite{Aackerbauer10, Gamta11} For this phase, the DOS above the Fermi level exhibits a pseudogap with a width of about 0.3 eV.
In comparing the valence band of HT-TiGePt with that of the LT phase, one can see immediately that the former has a noticeably larger band width than the latter: 7.2 eV vs 6.0 eV.
The observed band broadening originates from the altered chemical bonding situation related to the change in local atomic environments, followed by the larger orbital overlap caused by the volume reduction, as argued in Ref. \onlinecite{Aackerbauer10}.
Interestingly, the shallow core Ge 4$s$--like band also broadens accordingly, although it is located far below the Fermi level and is well--separated from the rest of the valence band.
Moreover, this shallow core band is positioned between 9.0 eV and 11 eV binding energy in the LT phase, whereas in HT-TiGePt it is appreciably further away from the Fermi level, namely between 9.7 eV and 12.0 eV binding energy.
The two modifications of TiGePt differ also in the overall shape of the valence band. For the LT phase, the valence band can even be divided into two parts: a~Ti~$3d$--Ge~$4p$ derived band with a sizable admixture of Pt states in the binding energy range from 2 eV to $E_{\mathrm{F}}$ and a broader Pt $5d$ dominated part between 2.5 eV and 6 eV. These features, by contrast, are washed out in the HT phase: one can only recognize that the Pt $5d$ states are more pronounced in the energy region above 2~eV while the Ti $3d$ states contribute more at the lower binding energy part. The Ge $4p$ states are even almost equally distributed over the entire valence band.
One should note that for HT-TiGePt the presented results of the fully relativistic electronic structure calculations are very similar to those obtained recently using the scalar relativistic approach.\cite{Aackerbauer10} In case of LT-TiGePt, however, the inclusion of the SO coupling has a significant impact on the calculated DOS. It affects the $d$ states of Pt and Ti, the latter ones due to their strong hybridization with the Pt 5$d$ states. Consequently, the width of the valence band is larger than that previously reported and the calculated band gap is smaller by about 0.15 eV.
\begin{figure}
\includegraphics[width=0.4\textwidth]{figure_3.eps}\\
\caption{(Color online) Valence band spectra of the low-temperature (LT) phase (bottom panel) and the high-temperature (HT) phase of TiGePt (top panel) in comparison with the broadened and photoionization-cross-section weighted partial DOS. The spectra were taken with 700 eV photon energy. }
\label{PESLDA700}
\end{figure}
The PES results taken at 700 eV photon energy are shown in Fig.~\ref{PESLDA700}. To facilitate the comparison with the band structure results, the experimental spectra of the LT phase (bottom panel) and the HT phase (top panel) are plotted together with their respective calculated DOS. The partial DOS are multiplied with the Fermi-Dirac distribution function, weighted by their respective tabulated photoionisation cross-sections, \cite{Yeh85} and broadened to account for the experimental resolution and lifetime effects. Finally, the commonly used integral-type of background - as indicated by the dotted lines in Fig.~\ref{PESLDA700} - is added to account for the presence of secondary electrons during the photoemission process. The cross-sections per electron at 700 eV photons are 7.4, 1.9, 3.0, and 1.7 kb/e for the Pt $4d$, Ge $4p$, Ge $4s$, and Ti $3d$, \cite{Yeh85} respectively. The Pt $5d$ and - to a lesser extent - the Ge $4s$, thus dominate at this photon energy.
A very good correspondence between the computational and the experimental results can clearly be seen in Fig.~\ref{PESLDA700}. The essential features in the experimental data are all well reproduced, including the energy gap between the Ge $4s$-like shallow core states and the remainder of the valence band. The experiment confirms that most of the Pt $5d$ spectral weight is concentrated at the high-binding-energy side of the valence band, and that the Ti $3d$ states contribute significantly to the features near the Fermi level. Most importantly, the broadening of the Pt $5d$ and Ti $3d$ derived bands in the HT phase as compared to the LT phase is also clearly revealed by the experiment.
\begin{figure}
\includegraphics[width=0.40\textwidth]{figure_4.eps}
\vspace{-2mm} \caption{(Color online) Pt 4$f$ core level photoemission spectra of TiGePt in the low-temperature (LT, top, blue solid line) and high-temperature (HT, top, red dashed line) phase, and of elemental Pt metal (bottom, black solid line). The spectra were taken with 700 eV photon energy. Solid vertical lines represent the peak positions of the $4f_{7/2}$ levels, dashed vertical lines the center of gravity positions (see text). }
\label{CoreLevels4f}
\end{figure}
As a further check we also study the Pt 4$f$ core levels of TiGePt and compare them to those of elemental Pt. The experimental
spectra are displayed in Fig.~\ref{CoreLevels4f} and exhibit the characteristic spin-orbit splitting giving the 4$f_{5/2}$ and
4$f_{7/2}$ peaks. For the LT phase of TiGePt, the peak positions are 75.7 eV and 72.4 eV, respectively. The HT phase has peaks at 75.5 eV and 72.2 eV, while elemental Pt shows peaks at 74.4 eV and 71.1 eV, respectively. The spin-orbit splitting is thus 3.3 eV for all the three materials. This compares well with the calculated spin-orbit splitting of about 3.45--3.46 eV for TiGePt in both
modifications and elemental Pt.
In TiGePt the Pt 4$f$ peaks are shifted by 1.1--1.3 eV to higher binding energies in comparison to those of Pt metal. Similar shifts have also been observed in other noble-metal intermetallic compounds,\cite{Franco03,Gegner06,Gegner08,RosnerT} indicating a lowered averaged electron density around the noble-metal sites. To compare this chemical shift to the results of LDA calculations, one has to take into account that LDA does not incorporate many-body effects of the final state, as manifested in the asymmetric line shape in the spectra of the elemental Pt, as we will discuss below in more detail. Yet, it can be shown that final-state effects do not alter the average energy of the spectrum.\cite{Lundqvist68} If we determine the center of gravity of the 4$f_{7/2}$, we find a binding energy of 72.4 eV for the LT phase of TiGePt, 72.2 eV for the HT phase, and 71.9$\pm$0.2\,eV for Pt metal. These centers of gravity are indicated by dashed lines in Fig.~\ref{CoreLevels4f}. Thus, the experimental chemical shift between the LT and the HT phase TiGePt, and Pt metal is about 0.1--0.5 and 0.3--0.7 eV, respectively. This is in reasonable agreement with the shift obtained from our band structure calculations which is about 0.71/0.74 eV.
We note that the line-shape of the core levels in TiGePt is not as asymmetric as for Pt metal. An asymmetry in the line-shape is caused by the presence of electron-hole pair excitations upon the creation of the core hole, i.e. screening of the core
hole by conduction-band electrons, and can be well understood in terms of the Doniac-Sunjic theory.\cite{Doniach70} The strong
asymmetry of the 4$f$ of Pt metal can therefore be taken as an indication for the high DOS with Pt character at the
$E_F$.\cite{Huefner75} The rather symmetric line shape of the 4$f$ of TiGePt, on the other hand, indicates a rather small DOS at the $E_{\mathrm{F}}$. Indeed, all this confirms the results of the valence band measurements: the main intensity of the Pt 5$d$ band is between 2 and 6 eV binding energies, with little weight at $E_{\mathrm{F}}$.
\begin{figure}
\includegraphics[width=0.5\textwidth]{figure_5.eps}\\
\caption{X-ray absorption spectra of the Ti $L_{2,3}$-edge of TiGePt in the low-temperature (LT) and high-temperature (HT) phase in comparison with Ti$_2$O$_3$ and SrTi$_2$O$_3$ as references for Ti $d^1$ and Ti $d^0$, respectively.}
\label{XAS}
\end{figure}
We now focus our attention to the contribution of Ti to the
electronic structure of the material. Fig.~\ref{XAS} shows the Ti
$L_{2,3}$ ($2p$$\rightarrow$$3d$) XAS spectra for the two phases
of TiGePt. It is important to note that XAS spectra are highly
sensitive to the valence state: an increase of the valence of a
transition metal ion by one causes a shift of the $L_{2,3}$ XAS
spectra by one eV or more towards higher
energies.\cite{Chen90,Hu98,Hu00,Burnus2008a} Therefore, as a
reference we include also the spectra of Ti$_2$O$_3$, a nominally
Ti$^{3+}$ ($3d^{1}$) compound, and SrTiO$_3$, a nominally
Ti$^{4+}$ ($3d^{0}$) system.
From the experimental spectra we can estimate their center of gravity, and after correcting for the background, we obtain energy
positions of roughly 460.7, 461.0, 461.5, and 462.9 eV for HT-TiGePt, LT-TiGePt, Ti$_2$O$_3$, and SrTiO$_3$, respectively.
This suggests that the valence of the Ti ions in the two modifications of TiGePt is rather similar, but appreciably
smaller than in Ti$_2$O$_3$ and SrTiO$_3$. This finding is in agreement with the effective atomic charges of titanium (+1.4 in LT and +1.3 in HT phase, respectively) obtained from the bonding analysis by means of electron density.\cite{Aackerbauer10} Further, the FPLO calculations result in the Ti $3d$ occupation of about 2.54~e and 2.53~e for HT- and LT-TiGePt, respectively, and 2.24~e and 2.14~e for Ti$_2$O$_3$ and SrTiO$_3$, respectively. These numbers follow the trend of the XAS energy positions, confirming again the consistency of the calculations.
Apart from this, one can clearly see that the spectra of Ti$_2$O$_3$ and SrTiO$_3$ show distinct multiplet structures whereas the features observed in the TiGePt spectra are much broader. This is fully consistent with the more ionic nature of the oxides as compared to the TiGePt, where covalent interactions play a significant role. In addition, the structures for HT-TiGePt are broader than for the LT phase, which is in line with our band structure calculations predicting broader bands for the metallic phase than for the semiconductor.
\begin{figure}
\includegraphics[width=0.4\textwidth,angle=0]{figure_6.eps}
\caption{\small (color online) X-ray absorption (XAS) spectra at the Ge $K$-edge for TiGePt in the low-temperature (LT) and high-temperature (HT) phase (black solid lines with experimental points), together with the data for the reference system Ge (blue dashed lines with experimental points) and with the calculated XAS spectra (red solid lined).
The experimental spectra were normalized using a standard method as implemented in the Athena program.\cite{Athena} The position of the absorption edges determined by taking the maximum in the first derivative of the normalized spectra is indicated by vertical dashed lines.
The theoretical curves were scaled to match the maximum in the near edge XAS region of the experimental spectra.
}
\label{XAS2p}
\end{figure}
To study in more detail the conduction band of TiGePt, we also have performed Ge $K$ near edge structure measurements. The results are displayed in Fig.~\ref{XAS2p} together with that of the elemental Ge as a reference compound.
To interpret the TiGePt spectra, we compare them with the calculated unoccupied 4$p$ partial DOS, weighted with the energy dependent transition probabilities calculated as described in the Section \ref{sec:methods}.
We can observe that most of the experimental features can be satisfactorily reproduced. The intensities are, however, not correct, but this to be expected since our calculations do not take into account the core hole effect.
It is important to note that the energy position of the Ge $K$-edge in both phases of TiGePt is the same as for elemental Ge, within the experimental error bars. This finding is in line with the basically neutral charge state of germanium in both phases of TiGePt obtained from the analysis of the electron density based on the quantum theory of atoms in molecules (QTAIM).\cite{Aackerbauer10}
\begin{figure*}
\includegraphics[width=0.8\textwidth]{figure_7.eps}\\
\caption{(Color online) Density of states and photoemission spectra near the Fermi level of the low-temperature (LT, left panel) and high-temperature (HT, right panel) phase of TiGePt. Top curves: density of states; middle curves: density of states multiplied by the Fermi distribution function at 300 K; bottom curves: photoemission spectra taken using 190 eV photons (dots) and density of states multiplied by the Fermi function and broadened by the experimental resolution.}
\label{PES190}
\end{figure*}
Next, we focus on the states near the Fermi level. Fig. \ref{PES190} shows a close-up of the valence band
photoemission spectrum (dots) and the calculated DOS (solid lines) in the vicinity of the Fermi level. The photoemission spectra were taken using 190 eV photons, with an overall energy resolution of about 150 meV. To facilitate the comparison, we have multiplied the DOS with the Fermi distribution function at 300 K (dashed lines) and broadened with the experimental energy resolution.
For the LT phase, one can clearly observe a very good agreement between experiment and theory. The gentle slope and vanishing weight at the top of the valence band of this semiconductor is well reproduced. For the HT phase, the high spectral weight in the 0.2-0.8 eV region is also well explained by the theory. Yet, the observed Fermi cut-off is not in agreement with the calculated DOS. The calculations show a more reduced DOS close to the Fermi level. We currently have no explanation for this discrepancy and would like to remark that the slope of the measured spectrum in the Fermi level region matches very well the slope in the top of the calculated occupied DOS. This may suggest that the DOS of the measured material has somehow been shifted rigidly towards the Fermi level by about 80~meV. It could be that the measured material has some surface defects or imperfections which cause such a shift of the chemical potential.
Finally, we discuss the nature of the bandgap changes in going from the LT to the HT phase.
The formation of a band gap in MgAgAs-type compounds with a valence electron count of 18 per formula unit is a well--studied issue which has been the subject of many reports within the last decade.\cite{Ogut94, Tobola96, Pierre97, Jung99, Kandpal06, Offernes06, Koehler07, Gegner08}
To get insight into the cause of the gap closure in the HT modification of TiGePt, we analyse the effect
of volume reduction first. In contrast to naive expectations, that the band broadening should decrease the gap, we find that the calculated gap size increases slightly with decreasing unit cell volume of the LT phase (10\% volume contraction leads to the increase of gap by $\sim$10\%).
Thus, the closing of the band gap in HT-TiGePt can not be understood by solely considering the volume change.
The absence of the gap results rather from a change in Ti local environment. In LT-TiGePt, Ti atoms are tetrahedrally coordinated by Pt atoms with a short distance of 2.57 {\AA}, suggesting strong Ti-Pt interactions. Such interactions
were found to be crucial for the formation of a band gap in "half-Heusler"-type compounds.\cite{Ogut94, Tobola96, Pierre97, Jung99, Kandpal06, Offernes06, Koehler07, Gegner08}
The transition from LT-TiGePt to the HT phase requires a breaking of the Ti-Pt bonds.\cite{Aackerbauer10}
In HT-TiGePt, the nearest neighbors of Ti are five Ge atoms with an average distance of 2.71~{\AA}, followed by six Pt atoms with a much longer average distance of 2.98 {\AA}.
The drastic change in the local coordination of Ti is reflected in the partial DOS. The sizable admixture of the Ti 3$d$ states visible for the LT phase in the binding energy region above $\sim$4 eV, resulting from the hybridization with Pt 5$d$ orbitals, is clearly reduced in the HT modification.
The essential weakening of the Ti-Pt interaction in HT-TiGePt and a corresponding increase in bonding interaction between Ti and Ge atoms have been confirmed by the combined topological analysis of the electron localizability indicator and the electron density.\cite{Aackerbauer10}
\section{summary}
We have determined the electronic structure of the low-temperature (LT) and high-temperature (HT) phases of TiGePt by means of
photoelectron spectroscopy, X-ray absorption spectroscopy and band structure calculations.
The combined theoretical and experimental study revealed substantial differences in the electronic structure for the two TiGePt modifications, although they have the same nominal composition and show similar electron counts for particular valence band states.
Most importantly, we have confirmed that the structural change in TiGePt is accompanied by an insulator-to-metal transition with an appreciable band broadening and a closing of the band gap.
The good correspondence between the computational results and the spectroscopic data for both the occupied and the unoccupied states
indicates that our calculations based on the LDA approximation provide a reasonable description of the electronic structure of the two modifications of TiGePt at ambient conditions.
Thus, the LDA level of theory can be regarded as a good starting point for a future theoretical study aiming to identify the mechanism of the structural and electronic transition in TiGePt and its driving force.
\section{Acknowledgments}
We gratefully acknowledge the NSRRC staff for providing us with
beamtime. The research in Cologne is supported by the Deutsche
Forschungsgemeinschaft through SFB 608. The authors are grateful
to Dr Ulrich Burkhardt from the MPI CPfS and Dr D. Zajac from
Hasylab for their helpful assistance during the Ge-K XAS
experiment. M.~Gam\.za is grateful for the financial support from
the DAAD foundation.
|
{
"timestamp": "2012-03-12T01:01:44",
"yymm": "1203",
"arxiv_id": "1203.2095",
"language": "en",
"url": "https://arxiv.org/abs/1203.2095"
}
|
\section{Introduction}
In quantum information theory, quantum systems can be used to transmit classical information. But quantum systems can be neither unambiguously distinguished~\cite{holevo1,pang}, nor perfectly cloned~\cite{noclone} in general. Usually the information encoded in quantum systems cannot be transmitted without any distortion. Even if the transmitted quantum states are not disturbed during the transmission, there is a Holevo bound that limits the accessible information of the receivers~\cite{holevo,nielsen}.
The information transmission process can be described as follows: there is a classical information source which produces symbols $i= 1,...,n$ according to a probability distribution $p_1,...,p_n$. The classical information is quantified by the Shannon entropy $H(p_i)=-\sum_i p_i\mathrm{log}p_i$, where the
base of the logarithm function is 2 in this paper. The message sender Alice encodes the information into the quantum state $\rho_i$ with the probability $p_i$ where $i=1,...,n$. The receiver Bob performs a measurement described by the positive operator valued measure (POVM), $\{E_j\}=\{E_1,...,E_m\}$, to gain the information~\cite{nielsen}. If the measured state is $\rho_i$, the probability of obtaining output $j$ is $p_{j|i}=\mathrm{Tr}(E_j\rho_i)$, and $p_{ij}=p_i\mathrm{Tr}(E_j\rho_i)$. The accessible information on Bob is $I_{acc}=H(p_i)+H(p_j)-H(p_{ij})$. The Holevo bound is $\chi=S(\rho)-\sum_i p_i S(\rho_i)$~\cite{holevo,nielsen}, we have $I_{acc} \leq S(\rho)-\sum_i p_i S(\rho_i)$, where $\rho=\sum_i p_i \rho_i$, and $S(\rho)$ is the von Neumann entropy of the state $\rho$. From the properties of the von Neumann entropy~\cite{nielsen,wehrl}, we get that $I_{acc}\leq S(\rho)-\sum_i p_i S(\rho_i)\leq H(p_i)$ which means that the accessible information on the receiver is less than the original information.
From another perspective, the Holevo bound $\chi$ is equal to the mutual information $S(A:B)$ of the bipartite state $\rho_{AB}=\sum_i p_i |i\rangle_{A}\langle i| \otimes \rho_{iB}$, where $\{|i\rangle_{A}\}$ is an orthonormal basis, and $\rho_{iB}=\rho_i$ is the transmitted state. From the aspect of the quantum correlation theory~\cite{discord1, discord2, luo, luo1, wu}, the mutual information $S(A:B)$ is considered to be the total correlation of subsystems $A$ and $B$. The maximal classical information that Bob can gain from states $\{\rho_{iB}\}$ is the classical correlation $I_{\max}(A:B)$ of state $\rho_{A,B}$ which is defined as $I_{\max}(A:B)=\max_{\{E_j\}}\{H(p_i)+H(p_j)-H(p_{ij})\}$. It has been proven that $I_{\max}(A:B) \leq S(A:B)$~\cite{wu,barnett}, so we have $I_{\max}(A:B) \leq \chi$, which provides an alternative proof of the Holevo bound.
In actual experiments, for the technical limits or some special purposes, the interactions between quantum systems and measuring devices may not be very strong. So, it is interesting to study the relationship between the interaction strength and the information gain of the measuring device. In this paper, we calculate the value of the information gain as a function of the coupling strength. From our intuition, the information gain of a measuring device increases with the coupling strength between the device and the quantum system. For qubit systems, we prove that our intuition is actually true. We also prove that the information gain of the projective measurement along the $x$ direction decreases with an increase in the measurement strength along the $z$ direction. Based on the monotonicity, we obtain a complementarity of the information gain in the measurements along two perpendicular directions.
\section{The information gain}
The quantum states sent by Alice constitute an ensemble $\{p_i, \rho_{iB}\}$, specified by Alice sending state $\rho_i$ with probability $p_i$, where
$i = 1, ..., n$. The ensemble can be described by the density matrix $\rho_{B}=\sum_{i}p_i \rho_{iB}$. The state of the measuring device is $|\Phi\rangle_{D}$; the interaction between the quantum systems and the measuring device is assumed to be impulsive which can be described as~\cite{w1,w2,w3,w4}
\begin{equation}\label{eq01}
H_{int}=g\delta(t-t_0)B\otimes D,
\end{equation}
where $B$ is an observable operator of the quantum systems, $D$ is an operator of the measuring device, and $g$ is the coupling strength with the assumption that $g\geq 0$. We introduce a fictitious auxiliary system $A$ which can be thought of as the "preparation" system. The auxiliary system has an orthonomal basis $\{|i\rangle_{A}\}$ whose elements correspond to the labels $1,2,...,n$ on the possible preparations for the transmitted system, $B$. The states of $A$ can be considered as the memory of the original information source. Before the interaction, the overall state of $A$, $B$, and the measuring device $D$ is
\begin{equation}\label{eq02}
\rho_{ABD}=\sum_i p_i |i\rangle_{A}\langle i|\otimes \rho_{iB}\otimes |\Phi\rangle_D\langle\Phi|.
\end{equation}
After the interaction the overall state evolves into
\begin{equation}\label{eq03}
\rho_{ABD}'=\sum_i p_i |i\rangle_{A}\langle i|\otimes U \rho_{iB} \otimes |\Phi\rangle_D\langle\Phi|U^{\dagger},
\end{equation}
where $U=e^{-i\int H_{int} \mathrm{d}t}=e^{-igB\otimes D}$, with $\hbar=1$ throughout this paper. It is assumed that the complete orthonormal eigenstates of the observable $B$ are $\{|b_m\rangle\}$, and the corresponding eigenvalues are $\{b_m\}$. States $\rho_{iB}$ and $\rho_{B}$ can be written as
\begin{equation}\label{eq04}\begin{split}
\rho_{iB}=\sum_{mn}\rho_{mn}^{i}|b_m\rangle\langle b_n|,\\
\rho_{B}=\sum_{mn}\rho_{mn}|b_m\rangle\langle b_n|.
\end{split}\end{equation}
After the interaction, state $\rho_{iBD}=\rho_{iB}\otimes|\Phi\rangle_D\langle\Phi|$ evolves into
\begin{equation}\label{eq05}
\rho_{iBD}'=\sum_{mn}\rho_{mn}^{i}e^{-igb_m D}|b_m\rangle\langle b_n|\otimes|\Phi\rangle_D\langle\Phi|e^{igb_n D}.
\end{equation}
We get the measuring device's state
\begin{equation}\label{eq06}
\rho_{iD}'=\mathrm{tr}_{B}(\rho_{iBD}')=\sum_{m}\rho_{mm}^{i}e^{-igb_m D}|\Phi\rangle_D\langle\Phi|e^{igb_m D}.
\end{equation}
Similarly, we can obtain the final overall state of the measuring device and the information source,
\begin{equation}\label{eq07}
\rho_{AD}'=\mathrm{tr}_{B}(\rho_{ABD}')=\sum_i p_i |i\rangle_{A}\langle i|\otimes\rho_{iD}'.
\end{equation}
The mutual information $S(A:D)$ of the measuring device and system $A$ represents the correlation of the measuring device and the information source~\cite{luo2}, so we define the information gain of the measuring device,
\begin{equation}\label{eq08}\begin{split}
I_a=S(A:D)&=S(\rho_A')+S(\rho_D')-S(\rho_{AD}')\\
&=S(\rho_D')-\sum_i p_i S(\rho_{iD}'),
\end{split}\end{equation}
where $\rho_A'=\rho_A=\sum_i p_i |i\rangle_{A}\langle i|$ is the total density matrix of the information source, and $\rho_D'=\sum_i p_i \rho_{iD}'$ is the total density matrix of the measuring device. From another perspective, the information gain $I_a$ is the Holevo bound for the case that the classical information is encoded in the measuring device's states $\{\rho_{iD}'\}$ with the probabilities $\{p_i\}$.
Now, we prove that the information gain $I_a$ is less than the Holevo bound $\chi=S(\rho_B)-\sum_i p_i S(\rho_{iB})$. From the theory of relative entropy~\cite{nielsen,vedral}, we have
\begin{equation}\label{eq09}\begin{split}
\chi &=S(\rho_{AB}\otimes |\Phi\rangle_D\langle\Phi| \parallel \rho_{A}\otimes \rho_{B}\otimes|\Phi\rangle_D\langle\Phi|)\\
&=S(\rho_{ABD}' \parallel U_{ABD} \rho_{A}\otimes \rho_{B}\otimes|\Phi\rangle_D\langle\Phi|U_{ABD}^{\dagger})
\end{split}\end{equation}
where $U_{ABD}=I_{A}\otimes e^{-i\int H_{int} \mathrm{d}t}$ is a unitary operator and $\rho_{ABD}'=U_{ABD}\rho_{AB}\otimes |\Phi\rangle_D\langle\Phi |U_{ABD}^{\dagger}$. Based on the monotonicity of relative entropy~\cite{nielsen,vedral}, we obtain
\begin{equation}\label{eq10}\begin{split}
\chi&\geq S(\mathrm{tr}_{B}(\rho_{ABD}') \parallel\mathrm{tr}_{B}( U_{ABD}\rho_{A}\otimes \rho_{B}\otimes|\Phi\rangle_D\langle\Phi| U^{\dagger}_{ABD}))\\
&=S(\rho_{AD}' \parallel \rho_{A}'\otimes \rho_{D}') \\
&=S(\rho_D')-\sum_i p_i S(\rho_{iD}')=I_a.
\end{split}\end{equation}
Thus we obtain that the information gain $I_{a}$ is less than the Holevo bound $\chi$.
Without loss of generality, the initial state of the measuring device is assumed to be a Gaussian wave function centered on $q=0$
\begin{equation}\label{eq11}
\Phi(q)=\frac{1}{ (2 \pi\Delta^{2})^{\frac{1}{4}}}\exp({-\frac{q^2}{4\Delta^2}}),
\end{equation}
where the standard deviation $\Delta q= \Delta$. The original density matrix of the measuring device is
\begin{equation}\label{eq12}
\rho_{D}=\frac{1}{ (2 \pi\Delta^{2})^{\frac{1}{2}}}\int\int e^{-\frac{q^2}{4\Delta^2}}e^{-\frac{q'^2}{4\Delta^2}}|q\rangle\langle q'|\mathrm{d}q\mathrm{d}q'.
\end{equation}
The interaction Hamiltonian considered is $H_{int}=g\delta(t-t_0)B\otimes p$. From Eq. (\ref{eq06}), we obtain the density matrix $\rho_{iD}'$ is
\begin{equation}\label{eq13}\begin{split}
\rho_{iD}'&=\sum_{m}\rho_{mm}^{i}e^{-igb_m p}\rho_{D}e^{igb_m p}\\
&=\sum_{m}\frac{\rho_{mm}^{i}}{ (2 \pi\Delta^{2})^{\frac{1}{2}}}\int\int e^{-\frac{(q-gb_m)^2}{4\Delta^2}}e^{-\frac{(q'-gb_m)^2}{4\Delta^2}} |q\rangle\langle q'| \mathrm{d}q\mathrm{d}q'.
\end{split}\end{equation}
Since the $\rho_{iD}'$ is a continuum variable density matrix, it is not easy to calculate its von Neumann entropy directly. We can introduce an auxiliary system $R$ to purify the state of the measuring device, and the state of the combined system is
\begin{equation}\label{eq14}
|\Psi_i\rangle_{DR}=\sum_m\frac{\sqrt{\rho_{mm}^{i}}}{ (2 \pi\Delta^{2})^{\frac{1}{4}}}\int e^{-\frac{(q-gb_m)^2}{4\Delta^2}}|m\rangle_R |q\rangle \mathrm{d}q,
\end{equation}
where $\{|m\rangle_{R}\}$ is an orthonormal basis of the auxiliary system, and $\rho_{iD}'=\mathrm{tr}_R(|\Psi_i\rangle_{DR}\langle\Psi_i|)$. As $|\Psi_i\rangle_{DR}$ is a pure state, we have
\begin{equation}\label{eq15}
S(\rho_{iD}')=S(\rho_{iR}),
\end{equation}
and the density matrix $\rho_{iR}$ is
\begin{equation}\label{eq16}\begin{split}
\rho_{iR}=\mathrm{tr}_{D}(|\Psi_i\rangle_{DR}\langle\Psi_i|)=\sum_{mn} \rho_{mn}^{iR}|m\rangle\langle n|.
\end{split}\end{equation}
We can obtain the matrix elements of $\rho_{iR}$
\begin{equation}\label{eq17}\begin{split}
\rho_{mn}^{iR}&=\frac{\sqrt{\rho_{mm}^{i}\rho_{nn}^{i}} }{ (2 \pi\Delta^{2})^{\frac{1}{2}}}\int\int e^{-\frac{(q-gb_m)^2}{4\Delta^2}}e^{-\frac{(q'-gb_n)^2}{4\Delta^2}}\langle q'| q\rangle\mathrm{d}q\mathrm{d}q'\\
&=\sqrt{\rho_{mm}^{i}\rho_{nn}^{i}} e^{-\frac{g^2(b_m-b_n)^2}{8\Delta^2}}.
\end{split}\end{equation}
It can be seen that the dimension of the matrix $\rho_{iR}$ is the same with the observable $B$. By a similar derivation, we obtain
\begin{equation}\label{eq18}
S(\rho_D')=S(\rho_{R})=S(\sum_{mn} \rho_{mn}^{R}|m\rangle\langle n|),
\end{equation}
and the matrix element $\rho_{mn}^{R}=\sqrt{\rho_{mm}\rho_{nn}} e^{-\frac{g^2(b_m-b_n)^2}{8\Delta^2}}$. So we can get the von Neumann entropy of the measuring device by calculating the entropy of the auxiliary system $R$. From Eqs. (\ref{eq08}), (\ref{eq15}), and (\ref{eq18}), the information gain of the measuring device is
\begin{equation}\label{eq19}
I_a=S(\rho_{R})-\sum_{i}p_iS(\rho_{iR}).
\end{equation}
When the coupling strength is strong (i.e., $g\gg \Delta$), and the eigenvalues of $B$ are nondegenerate, we will prove that the information gain is equal to the information $I_p$ extracted by the projective measurement along the orthonormal eigenstates $\{|b_{m}\rangle \}$ of $B$. The information $I_p$
obtained in the projective measurement along the basis $\{|b_{m}\rangle]\}$ is
\begin{equation}\label{new1}
I_p=H(p_m)+H(p_i)-H(p_{im}).
\end{equation}
where $H(p_m)=-\sum_m p_m\mathrm{log}p_m$ is the Shannon entropy, $p_m=\mathrm{tr}(|b_m\rangle\langle b_m|\rho_{B})$, and the joint probability $p_{im}=p_i\mathrm{tr}(|b_m\rangle\langle b_m|\rho_{iB})$.
When $g\gg \Delta$, and $m\neq n$, we have $e^{-{g^2(b_m-b_n)^2}/{8\Delta^2}} \to 0 $, the nondiagonal elements of matrices $\rho_{iR}$ and $\rho_{R}$ are approximatively equal to 0, and we have
\begin{equation}\label{eq20}
\rho_{iR}\approx \sum_{m} \rho_{mm}^{i}|m\rangle\langle m|, \rho_{R}\approx \sum_{m} \rho_{mm}|m\rangle\langle m|.
\end{equation}
From Eq. (\ref{eq04}), we have $\rho_{iR}=\rho_{iB}'=\sum_m \langle b_m|\rho_{iB}|b_m\rangle |b_m\rangle\langle b_m|$, and $\rho_{R}=\rho_{B}'=\sum_m \langle b_m|\rho_{B}|b_m\rangle |b_m\rangle\langle b_m|$ which are the states after the projective measurements on states $\rho_{iB}$ and $\rho_{B}$ along the basis $\{|b_m\rangle\}$, respectively. From Eq. (\ref{eq19}), the information gain is
\begin{equation}\label{eq21}\begin{split}
I_a&=S(\rho_{B}')-\sum_i p_i S(\rho_{iB}')\\
&=H(p_m)-\sum_i p_i H(p_{im|i})\\
&=H(p_m)+H(p_i)-H(p_{im})=I_p.
\end{split}\end{equation}
Thus we have proved that the information gain $I_a$ equals the information obtained by measuring the states transmitted $\{\rho_{iB}\}$ along the basis $\{|b_m\rangle\}$. It can be seen that when the coupling strength is large, the information gain of the measuring device is equal to the information obtained in the ideal projective measurement, which is consistent with our expectation.
\section{The monotonicity of the information gain}
Now we study the monotonicity of the information gain $I_a$ and the coupling strength $g$ when the transmitted quantum systems are qubits. For two-dimensional systems, the orthonomal eigenstates of observable $B$ can be denoted as $\{|0\rangle, |1\rangle\}$, and without loss of generality, the corresponding eigenvalues are assumed to be $\{1,-1\}$. The general state of a qubit can be represented as a point in the Bloch sphere~\cite{nielsen}. We can use three parameters, $r$ (radius), $\theta$ (polar angle), and $\phi$ (phase angle), to define a qubit state, where $0 \leq r \leq 1$, $0\leq \theta < \pi$, and $0 \leq \phi< 2\pi$. In the representation $\{|0\rangle, |1\rangle\}$, the transmitted state $\rho_{iB}$ can be written as
\begin{equation}\label{eq22}
\rho_{iB}=\left(
\begin{array}{ccc}
\frac{1+r_i\cos{\theta_i}}{2} & \frac{r_i\sin{\theta_i}e^{-i\phi_i}}{2} \\
\frac{r_i\sin{\theta_i}e^{i\phi_i}}{2} & \frac{1-r_i\cos{\theta_i}}{2} \\
\end{array}
\right).
\end{equation}
Then $\rho_{11}^{i}= \frac{1+r_i\cos{\theta_i}}{2}$ and $\rho_{22}^{i}= \frac{1-r_i\cos{\theta_i}}{2}$, and from Eqs. (\ref{eq16}) and (\ref{eq17}), we have
\begin{equation}\label{eq23}
\rho_{iR}=\left(
\begin{array}{ccc}
\frac{1+r_i\cos{\theta_i}}{2} & \sqrt{\frac{1-r_i^2\cos^2{\theta_i}}{4}}e^{-\frac{g^2}{2\Delta^2}} \\
\sqrt{\frac{1-r_i^2\cos^2{\theta_i}}{4}}e^{-\frac{g^2}{2\Delta^2}} & \frac{1-r_i\cos{\theta_i}}{2} \\
\end{array}
\right).
\end{equation}
Then we obtain the entropy of the state $\rho_{iR}$ as
\begin{equation}\label{eq24}
S(\rho_{iR})=H_B\left(\lambda_i\right),
\end{equation}
where $\lambda_i=({1+({r_i^2\cos^2{\theta_i}+(1-r_i^2\cos^2{\theta_i})e^{-\frac{g^2}{\Delta^2}}})^{1/2}})/{2}$, and $H_B(\lambda_i)=-\lambda_i\mathrm{log}\lambda_i-(1-\lambda_i)\mathrm{log}(1-\lambda_i)$ is the binary Shannon entropy. By similar calculations, we have
\begin{equation}\label{eq25}
S(\rho_{R})=H_B\left(\frac{1+s^{1/2}}{2}\right),
\end{equation}
where $s=(\sum_i p_i r_i\cos{\theta_i})^2+(1-(\sum_i p_i r_i\cos{\theta_i})^2)e^{-\frac{g^2}{\Delta^2}}$. From Eqs. (\ref{eq19}), (\ref{eq24}), and (\ref{eq25}), the information gain of the measuring device is
\begin{equation}\label{eq26}
I_{a}=H_B\left(\frac{1+s^{1/2}}{2}\right)-\sum_i p_i H_B\left(\lambda_i\right).
\end{equation}
In the following theorem, we present that the information gain $I_a$ increases with the coupling strength.
\begin{theorem} \label{thm01}
When the transmitted systems are qubits, the information gain $I_a$ monotonically increases with the coupling strength $g$.
\end{theorem}
The proof of this theorem is given in the Appendix.
Now we consider the case when information eavesdroppers are in. In this case, an eavesdropper named Eve intercepts the qubits which are transmitted from Alice to Bob, performs a measurement on the qubits for extracting the information sent to Bob, and resends the states to Bob. The interaction Hamiltonian between the quantum systems and Eve's measuring device is
\begin{equation}\label{eq27}
H_{int}=g\delta(t-t_0)\sigma_z\otimes p.
\end{equation}
Without loss of generality, we have assumed that this measurement is along the $z$ direction. The information gain $I_{a,z}$ of Eve is given by Eq. (\ref{eq26}). After the measurement performed by Eve, the state $\rho_{iB}$ given in Eq. (\ref{eq22}) is changed into
\begin{equation}\label{eq28}\begin{split}
\rho_{iB}'&=\mathrm{tr}_D(\rho_{iBD}')\\
&=\left(
\begin{array}{ccc}
\frac{1+r_i\cos{\theta_i}}{2} &\frac{ r_i\sin{\theta_i}e^{-i\phi_i}e^{-\frac{g^2}{2\Delta^2}}}{2} \\
\frac{r_i\sin{\theta_i}e^{i\phi_i}e^{-\frac{g^2}{2\Delta^2}}}{2} & \frac{1-r_i\cos{\theta_i}}{2} \\
\end{array}
\right),
\end{split}\end{equation}
and the total density matrix of the ensemble evolves into
\begin{equation}\label{eq29}\begin{split}
\rho_{B}'&=\sum_i p_i\rho_{iB}\\
&=\left(
\begin{array}{ccc}
\frac{1+\sum_i p_i r_i\cos{\theta_i}}{2} & \frac{\sum_i p_ir_i\sin{\theta_i}e^{-i\phi_i}e^{-\frac{g^2}{2\Delta^2}}}{2} \\
\frac{\sum_i p_i r_i\sin{\theta_i}e^{i\phi_i}e^{-\frac{g^2}{2\Delta^2}}}{2} & \frac{1-\sum_i p_i r_i\cos{\theta_i}}{2} \\
\end{array}
\right).
\end{split}\end{equation}
Finally, the legitimate receiver Bob performs a projective measurement on his received quantum system along the $x$ direction to gain the information from Alice. The projective measurement operators are $\{|+\rangle\langle+|, |-\rangle\langle-|\}$, where $|+\rangle=\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$ and $|-\rangle=\frac{1}{\sqrt{2}}(|0\rangle-|1\rangle)$. The information gain $I_{a,x}$ of Bob is
\begin{equation}\label{eq30}
I_{a,x}=H(p_i)+H(p_j)-H(p_{ij}),
\end{equation}
where $j=+,-$, $p_+=\mathrm{tr}(|+\rangle\langle+|\rho_{B}')$, $p_-=\mathrm{tr}(|-\rangle\langle-|\rho_{B}')$, $p_{i+}=p_i\mathrm{tr}(|+\rangle\langle+|\rho_{iB}')$, and $p_{i-}=p_i\mathrm{tr}(|-\rangle\langle-|\rho_{iB}')$.
From Eqs. (\ref{eq28}), (\ref{eq29}), and (\ref{eq30}), we have
\begin{equation}\label{eq31}\begin{split}
I_{a,x}=&H_B\left( \frac{1+\sum_i p_i r_i \sin{\theta_i}\cos{\phi_i}e^{-\frac{g^2}{2\Delta^2}}}{2} \right)\\
&-\sum_i p_i H_B\left( \frac{1+ r_i \sin{\theta_i}\cos{\phi_i}e^{-\frac{g^2}{2\Delta^2}}}{2} \right).
\end{split}\end{equation}
Now, we give a theorem to show that the information gain $I_{a,x}$ decreases with an increase in the coupling strength $g$.
\begin{theorem} \label{thm02}
For qubit systems, the information gain $I_{a,x}$ of the projective measurement along the $x$ direction monotonically decreases with the measurement coupling strength $g$ along the $z$ direction.
\end{theorem}
This monotonicity is consistent with the widely studied information-disturbance trade-off relation~\cite{fuchs, maccone, barbieri, luo3}. The proof of this theorem is given in the Appendix.
\section{Complementarity of the information gain}
From Theorem 1, we know that the information gain $I_{a,z}$ of Eve increases with $g$. For $g\to +\infty$, from Eq. (\ref{eq21}), the information gain is equal to the information gain of the projective measurement along the basis $\{|0\rangle, |1\rangle\}$, which is
\begin{equation}\label{eq32}\begin{split}
I_{z}&=H(p_i)+H(p_{j,z})-H(p_{ij,z})\\
&=H(p_{j,z})-\sum_i p_i H(p_{j,z|i}),
\end{split}\end{equation}
where $p_{1,z}=\mathrm{tr}(|0\rangle\langle0|\rho_B )$, $p_{2,z}=\mathrm{tr}(|1\rangle\langle1|\rho_B )$, $p_{i1,z}=p_i\mathrm{tr}(|0\rangle\langle0|\rho_{iB})$, $p_{i2,z}=p_i\mathrm{tr}(|1\rangle\langle1|\rho_{iB})$, and we have $I_{a,z}\leq I_{z}$. In Theorem 2, it is shown that the information gain $I_{a,x}$ decreases with the coupling strength $g$, when $g=0$, we have
\begin{equation}\label{eq33}\begin{split}
I_{x}&=H(p_i)+H(p_{j,x})-H(p_{ij,x})\\
&=H(p_{j,x})-\sum_i p_i H(p_{j,x|i}),
\end{split}\end{equation}
where $p_{1,x}=\mathrm{tr}(|+\rangle\langle+|\rho_B )$, $p_{2,x}=\mathrm{tr}(|-\rangle\langle-|\rho_B )$, $p_{i1,x}=p_i\mathrm{tr}(|+\rangle\langle+|\rho_{iB})$, $p_{i2,x}=p_i\mathrm{tr}(|-\rangle\langle-|\rho_{iB})$, and we have $I_{a,x}\leq I_x$. Then we obtain
\begin{equation}\label{eq34}\begin{split}
I_{a,z}+I_{a,x} & \leq I_{z} +I_{x}\\
&=H(p_{j,z})+H(p_{j,x})-\sum_i p_i(H(p_{j,z|i})+H(p_{j,x|i})),
\end{split}\end{equation}
where the conditional probabilities $p_{1,z|i}=\mathrm{tr}(|0\rangle\langle0|\rho_{iB})$, $p_{2,z|i}=\mathrm{tr}(|1\rangle\langle1|\rho_{iB})$, $p_{1,x|i}=\mathrm{tr}(|+\rangle\langle+|\rho_{iB})$, and $p_{2,x|i}=\mathrm{tr}(|-\rangle\langle-|\rho_{iB})$. From the entropic uncertainty
relation given in~\cite{berta} and \cite{guo}, we have
\begin{equation}\label{eq35}
H(p_{j,z|i})+H(p_{j,x|i})\geq 1+S(\rho_{iB}).
\end{equation}
As $\{p_{j,z}\}$ and $\{p_{j,x}\}$ are the two-outcome probability distributions, we have $H(p_{j,z})+H(p_{j,x})\leq 2$, and from Eqs. (\ref{eq34}) and (\ref{eq35}), we obtain
\begin{equation}\label{eq36}
I_{a,z}+I_{a,x} \leq 1- \sum_i p_i S(\rho_{iB}),
\end{equation}
which is a complementarity of the information gain of the measurements along two mutually unbiased bases~\cite{ivonovic,wu1,wu2}. By complementarity, we mean that the more information eavesdropper Eve extracted from the measurement along the $z$ direction, the less information Bob could gain by the projective measurement along the $x$ direction. Numerical calculations indicate that there is the bound $I_{a,z}+I_{a,x}\leq \chi =S(\rho_B)-\sum_i p_i S(\rho_{iB})$, which is much tighter than the one given in Eq. (\ref{eq36}). Unfortunately, we do not know how to prove this inequality.
Now, we give a simple example to show the complementarity of $I_{a,z}$ and $I_{a,x}$. In the BB84 quantum key distribution protocol~\cite{bb84}, Alice sends the states $\{|0\rangle, |1\rangle, |+\rangle, |-\rangle\}$ with equal probability, and the Holevo bound is $\chi=1$. By simple calculation, we obtain the information gain,
\begin{equation}\label{eq37}\begin{split}
I_{a,z}=\frac{1}{2}H_B \left( \frac{1+e^{-g^2/2\Delta^2}}{2} \right), \\
I_{a,x}=\frac{1}{2}-\frac{1}{2}H_B \left( \frac{1+e^{-g^2/2\Delta^2}}{2} \right),
\end{split}\end{equation}
and $I_{a,z}+I_{a,x}=\frac{1}{2} < \chi$. In Fig. 1, the relationship between the information gain and the coupling strength $g$ is depicted. We can see that the information gain $I_{a,z}$ increases with $g$, while $I_{a,x}$ decreases with the value of $g$. This means that the measurement performed by Eve along the $z$ direction destroys the information that Bob could gain in the projective measurement along the $x$ direction.
\begin{figure}[t]
\centering \includegraphics[scale=0.5]{1} \caption{(Color online)Information gain $I_{a,z}(g)$ and $I_{a,x}$ for the BB84 protocol. }
\label{fig:01}
\end{figure}
\section{conclusions}
In conclusion, we have studied the relationship between information gain and measurement coupling strength. For a finite interaction, the information gain of the measuring device is calculated when the measuring device's states are of the Gaussian type. When the coupling strength is high, we have shown that the information gain of the measuring device is equal to the information obtained in the projective measurement. It has been proved that the information gain increases with the coupling strength $g$ monotonously for qubit systems. Complementarity of the information obtained in the measurements along two different mutually unbiased bases is given. The research in this paper is useful for evaluating the information gain in finite-interaction measurements.
\section*{Acknowledgments}
This work was financially supported by the National Natural Science Foundation of China (Grants No. 11075148, and No. 11175063).
\section{Appendix}
\subsection{Proof of Theorem 1}
\begin{proof}
Here, we show that the information gain $I_a$ is a monotonic function of $g$. As $g\geq 0$, let $t=\frac{g^2}{\Delta^2}$, $s=(\sum_i p_i r_i\cos{\theta_i})^2+(1-(\sum_i p_i r_i\cos{\theta_i})^2)e^{-t}$, and $s_i=(r_i\cos{\theta_i})^2+(1-( r_i\cos{\theta_i})^2)e^{-t}$, we have
\begin{equation}\label{a01}\begin{split}
F&=\frac{\mathrm{d}I_a}{\mathrm{d}t} \\
&=\frac{(1-(\sum_i p_i r_i\cos{\theta_i})^2)e^{-t}}{4s^{1/2}}\mathrm{log}\frac{1+s^{1/2}}{1-s^{1/2}}\\
&-\sum_i p_i \frac{(1-(r_i\cos{\theta_i})^2)e^{-t}}{4s_i^{1/2}}\mathrm{log}\frac{1+s_i^{1/2}}{1-s_i^{1/2}}
\end{split}\end{equation}
To show the monotonicity of $I_a$, we only need to prove that $F\geq 0$. Let $a=e^{-t}$, and we define a function
\begin{equation}\label{a02}
h(x)=\frac{(1-x^2)a}{4(x^2+(1-x^2)a)^{1/2}}\mathrm{log}\frac{1+(x^2+(1-x^2)a)^{1/2}}{1-(x^2+(1-x^2)a)^{1/2}},
\end{equation}
where $x\in[-1,1]$, we get
\begin{equation}\label{a03}
F=h(\sum_i p_i r_i\cos{\theta_i})-\sum_i p_i h(r_i \cos{\theta_i}).
\end{equation}
Since $\sum_i p_i =1$, if we could prove that $h(x)$ is a concave function, we will get $F\geq0$. The second derivative of $h(x)$ is
\begin{equation}\label{a04}\begin{split}
\frac{\mathrm{d}^2h(x)}{\mathrm{d}x^2}= \frac{(1-a)^2aC(x,a)}{4(x^2-1)(a+x^2-ax^2)^{1/2}D(a,x)\ln 2},
\end{split}\end{equation}
where $C(x,a)=2(a+x^2-ax^2)^{1/2}(2x^2+a(x^2-1))+(x^2-1)(2x^2+a^2(x^2-1)-a(1+3x^2))\ln \frac{1+(x^2+(1-x^2)a)^{1/2}}{1-(x^2+(1-x^2)a)^{1/2}}$ and $ D(a,x)=(a + x^2 - 2 a x^2 + a^2 (-1 + x^2))^2$; Let $w=(a+x^2-ax^2)^{1/2}$, we have $0\leq w\leq 1$ and $x^2=\frac{w^2-a}{1-a}$. Let
\begin{equation}\label{a05}\begin{split}
G(w,a)&=\frac{C(x,a)}{x^2-1}\\
&=\frac{2(2+a)w^3-6aw}{w^2-1}+((2-a)w^2-3a)\ln\frac{1+w}{1-w}.
\end{split}\end{equation}
For a fixed value of $w$, we search the extreme value of $G(w,a)$, from
\begin{equation}\label{a06}
\frac{\partial G(w,a)}{\partial a}=\frac{2w^3-6w}{w^2-1}-(3+w^2)\ln \frac{1+w}{1-w}=0,
\end{equation}
we obtain
\begin{equation}\label{a07}
\mathrm\ln\frac{1+w}{1-w}=\frac{2w^3-6w}{(w^2-1)(3+w^2)}.
\end{equation}
Substituting this solution into Eq. (\ref{a05}), the extreme value of this function is
\begin{equation}\label{a08}
G(w,a)_{\mathrm{ext}}=\frac{8w^5}{(w^2-1)(3+w^2)}.
\end{equation}
Since $0\leq w\leq 1$, we have $G(w,a)_{\mathrm{ext}} \leq 0$. As $0<a=e^{-t}\leq 1$, for $a=0$, we have
\begin{equation}\label{a09}
G(w,0)=\frac{2w^2}{w^2-1}(2w-(1-w^2)\ln\frac{1+w}{1-w} ).
\end{equation}
Let $K(w)=2w-(1-w^2)\ln\frac{1+w}{1-w}$, the first derivative of $K$ is
\begin{equation}\label{a10}
\frac{\mathrm{d}K}{\mathrm{d}w}=2w\ln \frac{1+w}{1-w}.
\end{equation}
Since $0\leq w\leq 1$, we have $\frac{\mathrm{d}K}{\mathrm{d}w}\geq 0$, as $K(0)=0$, so $K(w)\geq 0$. From Eq. (\ref{a09}), we have $G(w,0)\leq 0$. When $a=1$, we obtain
\begin{equation}\label{a11}
G(w,1)=6w+(w^2-3)\ln \frac{1+w}{1-w}.
\end{equation}
The partial derivative of $G(w,1)$ is
\begin{equation}\label{a12}\begin{split}
\frac{\partial G(w,1)}{\partial w} &=\frac{2w}{w^2-1}(2 w-(1-w^2)\ln \frac{1+w}{1-w})\\
&=\frac{2w}{w^2-1}K.
\end{split}\end{equation}
Since $K \geq 0$ and $0\leq w\leq 1$, so $\frac{\partial G(w,1)}{\partial w}\leq 0$, and as $G(0,1)=0$, we have $G(w,1)\leq 0$. Now we have proved that $G(w,0)\leq 0$, $G(w,1)\leq 0$, and the extreme value $G(w,a)_{\mathrm{ext}}\leq 0$, and as the function $G(w,a)$ is a continuum function of $a$, $0\leq w\leq 1$, and $0 <a \leq 1$, we have $G(w,a) \leq 0$. From Eqs. (\ref{a04}) and (\ref{a05}), we have
\begin{equation}\label{a13}
\frac{\mathrm{d}^2h(x)}{\mathrm{d}x^2}\leq 0,
\end{equation}
and $h(x)$ is a concave function. From Eq. (\ref{a03}), we have $F \leq 0$. Thus we have proved Theorem 1.
\end{proof}
\subsection{Proof of Theorem 2}
\begin{proof}
As $g\geq 0$, let $t=\frac{g^2}{\Delta^2}$, $s=\sum_i p_i r_i \sin{\theta_i}\cos{\phi_i}$, and $s_i=r_i \sin{\theta_i}\cos{\phi_i}$, we have
\begin{equation}\label{a14}\begin{split}
W=\frac{\mathrm{d}{I_{a,x}}}{\mathrm{d}t}=&\frac{s e^{-t/2}}{4}\mathrm{log}\frac{1+s e^{-t/2}}{1-s e^{-t/2}}\\
&-\sum_i p_i\frac{ s_i e^{-t/2}}{4}\mathrm{log}\frac{1+s_i e^{-t/2}}{1- s_i e^{-t/2}}.
\end{split}\end{equation}
To show the monotonicity of $I_{a,x}$, we only need to prove that $W \leq 0$. We define a function $f(x)=\frac{xe^{-t/2}}{4}\mathrm{log}\frac{1+xe^{-t/2}}{1-xe^{-t/2}}$, where $x\in[-1,1]$. We have
\begin{equation}\label{a15}
W=f(\sum_i p_i r_i \sin{\theta_i}\cos{\phi_i})-\sum_i p_i f(r_i \sin{\theta_i}\cos{\phi_i}).
\end{equation}
The second derivative of $f(x)$ is
\begin{equation}\label{a16}
\frac{\mathrm{d}^2f(x)}{dx^2}=\frac{e^{-t}}{(1 - e^{-t} x^2)^2 \ln 2}
\end{equation}
Since $0<e^{-t}\leq 1$ and $-1\leq x\leq 1$, we have $\frac{\mathrm{d}^2f(x)}{\mathrm{d}x^2}>0$, so $f(x)$ is a convex function. Since $\sum_i p_i =1$, from Eq. (\ref{a15}), we obtain that $W \leq 0$. Thus, it has been proven that the information gain $I_{a,x}$ monotonically decreases with the coupling strength $g$.
\end{proof}
|
{
"timestamp": "2012-05-08T02:01:27",
"yymm": "1203",
"arxiv_id": "1203.2251",
"language": "en",
"url": "https://arxiv.org/abs/1203.2251"
}
|
\section{Introduction}
The standard vision on Hawking radiation (HR) of massive particles dates back to~\cite{Page:1976df}. It states essentially that there is a cut-off in the emission of massive particles because the Hawking temperature $T_H$ of the black hole must deliver $E \geq m$. Therefore, as anticipated in~\cite{Hawking:1974sw}, ``there will not be much emission of particles of rest mass $m$ unless the temperature $\kappa/2\pi$ is greater than $m$''. The example of an electron gives $m_e \approx 10^{-30}kg$, hence $T \sim 10^9 K$ and so the black hole mass must decrease to $M\sim 10^{-16} m_{\astrosun}\sim 10^{21} m_{Pl}$ ($m_{\astrosun}$ and $m_{Pl}$ are the solar and Planck mass, respectively) before electrons can be emitted in any significant amount. Through the emission of massless particles, the black hole will evaporate and this threshold eventually be reached. However, during most of the lifetime of an astrophysical black hole, there will be no significant emission of massive particles.
This standard vision sees HR as a ``black box'' whose outcome is measured at asymptotic infinity. It must be amended if one views HR as a near-horizon process. This allows, e.g., to reconcile the two heuristic interpretations of HR as mode conversion between positive/negative-norm partners, related to pair creation (from vacuum fluctuations) just \emph{inside} the horizon, or just \emph{outside} the horizon. Obviously, any calculation should give the same result in both interpretations. A thermal spectrum is indeed detected at asymptotic infinity in both cases. But the mass decrease of the black hole is due to the emission of the positive partner in the first case, and to the absorption of the negative partner (in the second). There might be good reasons for the emission of positive partners to be subject to the threshold $E\geq m$, but there is no reason why the absorption of negative partners should obey a similar condition. How can these two heuristic visions then be equivalent?
\section{Dispersion relation}
The simplest dispersion relation for massive modes/particles is (in units $\hbar=1$)
\begin{equation}\label{dispersion-relation}
(\omega-Uk)^2=m^2+c^2k^2,
\end{equation}
where $U(r)<0$ is the free-fall velocity of an observer starting at rest at infinity. \mbox{$U(r)=-c\sqrt{r_h/r}$} for a Schwarzschild black hole (with $r_h$ the horizon or Schwarzschild radius, i.e. $U(r_h)=-c$), and we have defined the ``mass'' $m=\bar m c^2$. Such a Klein-Gordon dispersion relation can be derived for example from the Painlev\'e-Gullstrand-Lema\^itre form of the Schwarzschild metric
\[
ds^2=[c^2-U(r)^2]dt^2 - 2U(r) dt\, dr -dr^2 ,
\]
which provides an intuitive analogy with sound or surface waves counter-propagating with local velocity $c$ against a background fluid flowing at a speed $U$ in the laboratory frame. $\omega_0=\sqrt{m^2+k^2}$ is then the frequency in the co-moving reference frame, and $\omega=\omega_0+\bf k\cdot \bf U$ is the Doppler-shifted frequency (corresponding to the laboratory or ``black hole rest frame''). Note that $\omega$ is a conserved quantity (for stationary spacetimes), and therefore provides a good basis for a semiclassical treatment (see below).
A graphical analysis of the mode conversion characteristic of Hawking radiation for this dispersion relation is given in Fig.~\ref{Fig:dispersion-relation}~\cite{Jannes:2011vb}. The key point here is that the appearance of the negative mode solution (negative co-moving frequency, i.e.: bottom part of the figure) is identical for the various upper parts of the figure, i.e.: the positive-negative mode conversion around the horizon is totally independent of the ratio $E/m$ (there is no threshold $E \geq m$) and in fact identical to the case of a massless mode~\cite{Jannes:2011vb}.
It is also easy to realize that there exists a critical value $U^*$ for the counterflow in the case $E<m$, such that there are two positive mode solutions when $|U|>|U^*|$, and there is none left when $|U|<|U^*|$. $U^*$ constitutes a turning point or saddle-node bifurcation and its value is $U^*=\pm c\sqrt{1-\left(\frac{\omega}{m}\right)^2}$~\cite{Jannes:2011vb}. This shows that the $E<m$ modes cannot escape to infinity but eventually turn around like a boomerang at a location $r^*=r_h(c/U^*)^2$ (which depends on their ratio $E/m$ through $U^*$) and fall back into the black hole (see~\cite{Jannes:2011vb} for a wave packet simulation). Therefore, they do not contribute to the evaporation of the black hole. Nevertheless, they can (at least in principle) be detected at any finite distance from the horizon. Putting in numbers gives $r^*\approx 3r_h$ for $E=0.8m$, or $r^*\approx 50r_h$ for $E=0.99m$. Taking into account that, for a solar-mass black hole, $r_h\sim 3km$, and moreover that the resolution of any detection will be limited by the time until re-absorption in the black hole, it seems that this boomerang effect only has a purely theoretical interest for astrophysical black holes (except perhaps for supermassive ones).
\begin{figure}
$\begin{array}{cccc}
\includegraphics[height=.18\textheight]{dispersionKGUneq0min}&
\includegraphics[height=.18\textheight]{dispersionKGUneq0}&
\\
\includegraphics[height=.18\textheight]{dpn}&
\includegraphics[height=.18\textheight]{dpn}
\end{array}$
\caption{Positive and negative mode solutions for the dispersion relation~\eqref{dispersion-relation} for different values of the ``counter-current velocity'' $U$ and (a) $E<m$; (b) $E>m$. The mechanism of appearance of negative-norm solutions (bottom) is independent of the ratio $E/m$, and actually equal to the massless case~\cite{Jannes:2011vb}.}\label{Fig:dispersion-relation}
\end{figure}
\section{Tunneling formalism}
The tunneling probability $W$ in a semiclassical formalism is $W\propto \exp[-2\text{Im}S]$, where $\text{Im}S=\text{Im}\int p_r(r)dr $ is the imaginary part of the action along the semiclassical trajectory, and $p_r(r)$ is obtained from the dispersion relation~\eqref{dispersion-relation}, which we now write \mbox{$(E-{\bf p}\cdot {\bf U})^2=m^2+p^2$}. The expected final result is of the form $W(E)\propto \exp[-2\text{Im}S_1]\exp[-2\text{Im}S_2] $, where $S_1$ corresponds to tunneling through the horizon, and $S_2$ stems from the second classically prohibited region beyond the horizon in the case $E<m$. Recall that for $U(r \to \infty)=0$, $E^2=m^2+p^2$ (i.e., the standard Minkowski dispersion relation), and hence particles with $E<m$ are certainly classically forbidden at flat asymptotic infinity. From the dispersion relation, $p$ can be written
\[
p
=-\frac{EU}{1-U^2} + \frac{1}{1-U^2}\sqrt{m^2(U^2-1)+E^2}\\
=p_1+p_2.
\]
$S_1$ is obtained from $p_1$ in the usual way~\cite{Volovik:1999fc}. We shift the contour of integration to the complex plane, apply the residue theorem $\int_C f(z)=2\pi i\text{Res} f$, find a pole along the radial path: $U=-1$ at $r=r_h$, and obtain $\text{Res} p_1=\text{Res}\frac{-EU}{(1+U)(1-U)}=\frac{E}{2U'(r_h)}$. This leads to the standard Hawking result: $\tilde{W_1}(E)\propto \exp[-E/T_H]$, with $T_H=|U'(r_h)|/2\pi$, where the prime denotes $d/dr$. This confirms that the presence of a mass has absolutely no influence on the probability of tunneling across the horizon.
We now consider a detector at a distance $R$ and examine the second contribution $S_2$. There will be an imaginary contribution if $E^2<m^2(1-U^2)$ (i.e., $E<E_c(R)=m\left(1-\frac{r_h}{R} \right)^{1/2}$ for given $R$, or $R>r_c(E)=\frac{1}{1-\frac{E^2}{m^2}}r_h$ for given $E$). Ffor $R\to \infty$, we recover the condition $E<m$.
For a Schwarzschild profile, we obtain $\text{Im}S_2 =\int_{r_c}^R dr \frac{1}{1-r_h/r}\sqrt{m^2(1-r_h/r)-E^2}$, see Fig.~\ref{Fig:S2}. In the limit $R\gg r_h$, the integral is dominated by the contributions $U\to 0$ and one can write
$\text{Im}S_2=\sqrt{m^2-E^2}\int_{r_c}^R dr\approx R \sqrt{m^2-E^2}$ (assuming $R\gg r_c$). The overall tunneling rate then becomes
\[
W(E)\propto \exp\left[\frac{-E}{T_H}\right]\exp\left[-2R\sqrt{m^2-E^2}\right],
\]
Other simple analytical results can be obtained in the near-horizon limit $R \to r_h$~\cite{Jannes:2011qp}.
\begin{figure}
\includegraphics[height=.16\textheight]{im-p2}
\caption{Semiclassical barrier Im$p_2$ (thick lines) and tunneling action $\text{Im}S_2$ (dashed lines) beyond the horizon $r_h$ for various values of $E/m<1$~\cite{Jannes:2011qp}.}\label{Fig:S2}
\end{figure}
\section{Analogue gravity}
As mentioned above, the emission of $E<m$ particles has little practical impact for astrophysical black holes. However, in the case of analogue-gravity systems with a massive dispersion relation, they become more interesting. Indeed, in such systems, particles can be detected inside the black hole also, so there is no limit on the spectral resolution. Moreover, the region where $U \neq 0$ can be extended far away from the horizon. Candidate systems include acoustic waves in ion rings, massive phonons from 2-component BECs, Langmuir waves in a moving plasma, barotropic waves (inertia-gravity or Poincar\'{e} waves), and spin waves in magnetic media~\cite{Jannes:2011vb}. Note that the existence of two imaginary contributions for $E<m$ indicates the presence of a double barrier, and hence the possibility of creating resonant states according to the Bohr-Sommerfeld quantization condition. These resonant states could be interesting candidates for detection and hence for a confirmation of some of the curious features of Hawking radiation described here.
\bibliographystyle{aipproc}
|
{
"timestamp": "2012-03-09T02:01:53",
"yymm": "1203",
"arxiv_id": "1203.1729",
"language": "en",
"url": "https://arxiv.org/abs/1203.1729"
}
|
\section{Introduction}
Recall that a metric space $(X,\rho)$ is called an
\emph{ultrametric space} if for every $x,y,z\in X$ we have $\rho(x,y)
\leq \max\{ \rho (x,z), \rho(z,y) \}$. Such spaces naturally appear and
have applications in
various areas such as number theory, $p$-adic analysis, and computer
science (see \cite{Lem1}, \cite{Lemin}, \cite{Mo1, Mo2}).
Let us briefly review several results with respect to isometric embedding
of ultrametric spaces. Timan
and Vestfrid \cite{VeTi, VeTi2} proved that any separable ultrametric space embed
isometrically into $\ell_2$. Vestfrid \cite{Ves2} later proved that the result is also true if one
replace $\ell_2$ by $\ell_1$ and $c_0$ by constructing a universal
ultrametric space for the class of separable ultrametric space and
using its property. Vestfrid \cite{Ves} also proved that a certain
class of countable ultrametric spaces embed isometrically into $\ell_p$
for $p\geq 1$.
Lemin \cite{Lemin} proved that any separable ultrametric space embed isometrically
into the Lebesgue space. He also raised a problem whether any separable
ultrametric space embed isometrically into any infinite dimensional
Banach space. Motivated by Lemin's problem, Shkarin \cite{Shk} proved that every finite ultrametric space embeds into every
infinite dimensional Banach space. From these results ultrametric spaces
have attracted much attention in embedding theory.
In this paper we tackle Lemin's problem in the case where the target
Banach space is $\ell_p$. It is already well-known that every separable ultrametric space embeds
isometrically into the function space $L_p$ for any $p\geq 1$. In fact,
it follows from Timan and Vestfrid's result mentioned above and the
fact that $\ell_2$ embeds isometrically into $L_p$. Since $\ell_2$ does
not embed bi-Lipschitzly into $\ell_p$ for any $p\neq 2$
(\cite[Corollary 2.1.6]{alkal}), embedding separable ultrametric spaces
into $\ell_p$ is left as a problem. Our main theorem is the following: Recall that a metric space is \emph{proper} if every
closed ball in $X$ is compact.
\begin{thm}\label{MT}Every proper ultrametric space isometrically embeds into
$\ell_p$ for any $p\geq 1$.
\end{thm}
The case of general separable ultrametric spaces remains open. A similar method of the proof of Theorem \ref{MT} also implies an
isometric embedding into $c_0$ (see Remark \ref{rem1}).
Our construction of isometric embeddings into $\ell_1$, $\ell_2$, and
$c_0$ is different from the one by \cite{VeTi, VeTi2}, \cite{Ves, Ves2} in the
case of proper ultrametric spaces.
As an application of Theorem \ref{MT} we obtain an $\ell_p$-version of
nonlinear Dvoretzky's theorem, see Section $3$.
\section{Proof of the main theorem}
We use some basic
facts of compact ultrametric spaces (see \cite{hugh}, \cite[Section 2]{MN112}). Let $(X,\rho)$ be a compact ultrametric space and put $r_0:=\diam X$.
Consider the relation $\sim_0$ on $X$ given by $x \sim_0 y$
$\Longleftrightarrow $ $\rho(x,y)< r_0$. Since $\rho$ is ultrametric
$\sim_0$ is an equivalence relation on $X$. The compactness of $X$
implies that each equivalence
class is a closed ball of radius strictly less than $r_0$ (see
\cite[Section $2$]{MN112}). Since the distance between two distinct equivalence classes
is exactly $r_0$ and $X$ is totally bounded, there are only finitely
many equivalence
classes, say, $\{B_1, \cdots, B_{k_1}\}$, where each $B_i$ is a closed
ball of radius $r_i=\diam B_i<r_0$. Note that for any $x\in B_i$ and $y\in
B_j$ $(i\neq j)$ we have $\rho(x,y)=r_0$. For each $i$ we choose
$x_i\in B_i$ and fix it. As above for each $i_1=1,\cdots,k_1$ we
consider the equivalence relation $\sim_{i_1}$ on $B_{i_1}$ given by $x \sim_{i_1} y$
$\Longleftrightarrow $ $\rho(x,y)< r_{i_1}$. Then we
can divide $X_{i_1}$ into finitely many equivalence classes, i.e., $B_{i_1}=
\amalg_{i_2=1}^{k(i_1)} B_{i_1 i_2}$, where $B_{i_1 i_2}$ is a closed ball of
radius $r_{i_1 i_2}=\diam B_{i_1 i_2}<r_{i_1}$. We may assume that
$x_{i_1}\in B_{i_1 1}$. For each $i_1,i_2$, we choose a point
$x_{i_1i_2}\in B_{i_1 i_2}$ so that $x_{i_1 1}= x_{i_1}$ and we fix
$x_{i_1 i_2}$. Repeatedly we get a sequence $\mathcal{P}_k= \{ B_{i_1 \cdots i_k}
\}_{i_1,\cdots, i_k}$ of partitions of $X$ satisfying
the following:
\begin{enumerate}
\item Each $B_{i_1 \cdots i_k}$ is a closed ball of radius $r_{i_1
\cdots i_k}=\diam B_{i_1 \cdots i_k}$.
\item If $r_{i_1 \cdots i_k}\neq 0$, then $r_{i_1 \cdots i_k}> r_{i_1
\cdots i_{k+1}}$.
\item $B_{i_1 \cdots i_{k-1}}= \amalg_{i_k}B_{i_1 \cdots i_{k-1}i_k}$.
\end{enumerate}For each $i_1, \cdots, i_k$ we choose $x_{i_1\cdots i_k}\in B_{i_1\cdots i_k}$ so that
$x_{i_1 \cdots i_k 1 \cdots 1}=x_{i_1 \cdots i_k}$. The
compactness of $X$ yields the following:
\begin{lem}[{cf.~\cite[Section 2]{MN112}}]\label{fact1}$\lim_{k\to \infty} \max_{i_1,\cdots, i_k} r_{i_1 \cdots i_k}=0 $.
\end{lem}In particular, $\bigcup_{k=1}^{\infty}\{x_{i_1 \cdots i_k}
\}_{i_1,\cdots,i_k} $ is a countable dense subset of $X$.
\begin{lem}[{cf.~\cite[Section 2]{MN112}}]\label{fact2} For every closed
ball $B$ in $X$, there exist $k$ and $B_{i_1 \cdots i_k}\in
\mathcal{P}_k$ such that $B=B_{i_1\cdots i_k}$.
\end{lem}
\begin{proof}[Proof of Theorem \ref{MT}]
We first prove the theorem for compact ultrametric
spaces. Let $(X,\rho)$ be a compact ultrametric space and let
$\mathcal{P}_k = \{ B_{i_1 \cdots i_k} \}_{i_1,\cdots, i_k}$, $r_{i_1 \cdots i_k}$, and
$x_{i_1 \cdots i_k}$ as above. Put $N_k:= \# \mathcal{P}_k$. We consider each coordinate of an element of
$\ell_p^{N_k}$ is indexed by $(i_1, \cdots, i_k)$. We define a map
$f_k: \{ x_{i_1 \cdots i_k} \}_{i_1 , \cdots , i_k}\to \ell_p^{N_k}$
as follows: $(f_k(x_{i_1 \cdots i_k}))_{(j_1, \cdots, j_{k})}: = 0$
if $(j_1,\cdots, j_k) \neq (i_1,\cdots, i_k)$ and
\begin{align*}(f_1(x_{i_1}))_{i_1}:=\frac{(r_0^p-r_{i_1}^p)^{\frac{1}{p}}}{2^{\frac{1}{p}}}
\text{ and }
(f_k(x_{i_1 \cdots i_k}))_{(i_1, \cdots, i_k)}:=
\frac{(r_{i_1 \cdots i_{k-1}}^p-r_{i_1 \cdots
i_k}^p)^{\frac{1}{p}}}{2^{\frac{1}{p}}} \text{ if }k\geq 2.\end{align*}
Note that $f_k(x_{i_1\cdots i_k})\perp f_k(x_{j_1 \cdots j_k})$ for two distinct $(i_1,\cdots , i_k)$, $(j_1,\cdots, j_k)$.
We define a map $f:\bigcup_{k=1}^{\infty} \{
x_{i_1 \cdots i_k}\}_{i_1, \cdots, i_k} \to \ell_p $ as
follows. For each $x_{i_1 \cdots i_k}$, putting
$i_m:=1$ for $m>k$, we define
\begin{align*}
f(x_{i_1 i_2 \cdots i_k}) :=
(f_1(x_{i_1}), f_2(x_{i_1 i_2}), \cdots , f_m(x_{i_1\cdots i_m}),
\cdots ).
\end{align*}The right-hand side in the above definition is
actually the element of $\ell_p$ since
\begin{align*}
\sum_{m=1}^{\infty} \| f_m(x_{i_1 \cdots i_m})\|_p^p=
\sum_{m=1}^{\infty} \frac{r_{i_1 \cdots i_{m-1}}^p - r_{i_1 \cdots
i_m}^p}{2} = \frac{r_{0}^p}{2} <+\infty
\end{align*}by Lemma \ref{fact1}. Note that $f$ is well-defined in the
sense that $f(x_{i_1 \cdots
i_k 1 \cdots 1})=f(x_{i_1 \cdots i_k})$.
We shall prove that $f$ is an isometric embedding. Since $\bigcup_{k=1}^{\infty} \{
x_{i_1 \cdots i_k}\}_{i_1, \cdots, i_k}$ is dense in $X$ this implies
the theorem. Taking two distinct elements $x_{i_1 \cdots i_k}$ and $x_{j_1 \cdots
j_l}$ we may assume that $k\leq l$. Put $i_m:=1 $ for $m>k$. Then we have $(i_1,\cdots, i_l)\neq (j_1,\cdots,j_l)$. Letting
\begin{align*}
n:=\min \{ m \leq l \mid i_m \neq j_m\}
\end{align*}we get
$\rho(x_{i_1 \cdots i_k}, x_{j_1 \cdots j_l})= \diam B_{i_1 \cdots
i_{n-1}}=r_{i_1 \cdots i_{n-1}}$ if $n\geq 2$ and $\rho(x_{i_1 \cdots
i_k}, x_{j_1 \cdots j_l})=r_0$ if
$n=1$. Since $f_m(x_{i_1 \cdots i_m})=f_m(x_{j_1 \cdots j_m})$ for $m<n$ and
$f_m(x_{i_1 \cdots i_m}) \perp f_m(x_{j_1 \cdots j_m})$ for $m\geq n$,
\begin{align*}
\| f(x_{i_1 \cdots i_k})-f(x_{j_1 \cdots j_l})\|_p^p=\ &
\sum_{m=n}^{\infty}\|f(x_{i_1 \cdots i_m}) \|_p^p +
\sum_{m=n}^{\infty} \| f(x_{j_1 \cdots j_m}) \|_p^p\\
=\ & r_{i_1
\cdots i_{n-1}}^p\\ =\ & \rho (x_{i_1 \cdots i_k},x_{j_1 \cdots j_l})^p.
\end{align*}
This completes the proof of the theorem for compact ultrametric spaces.
Let $(X,\rho)$ be a proper ultrametric space and fix a point $x_0\in
X$. For any $r>0$ we denote by $B(x_0,r)$ the closed ball of radius
$r$ centered at $x_0$. For any $R>0$ let
$f_1:B(x_0,R)\to \ell_p$ be an isometric embedding constructed as in
the above way. It suffices to prove that for any $R'>R$ we can
construct an isometric embedding $f_2:B(x_0,R')\to \ell_p$ as in the above way, which
extends $f_1$ in the following sense: There exists an isometry
$T:\ell_p \to \ell_p$ such that $T \circ f_2|_{B(x_0,R)}=f_1$.
This is possible by the above construction. In fact, keep dividing
$B(x_0,R')$ as in the above way. Then at finite steps we reach at $B(x_0,R)$ by
Lemma \ref{fact2} since $B(x_0,R')$ is compact. From the above
construction we easily see the existence of $f_2$ and $T$ . This
completes the proof of the theorem.
\end{proof}
\begin{rem}\label{rem1}\upshape A similar method of the above proof implies new
isometric embeddings of proper ultrametric spaces into $c_0$. In fact,
let us consider first the case of compact ultrametric spaces. Using the same notation as above, for each $k$ we define $g_k: \{ x_{i_1 \cdots i_k}\}_{i_1, \cdots, i_k}\to
\ell_{\infty}^{N_k}$ as follows: $(g_k(x_{i_1 \cdots i_k}))_{(j_1, \cdots, j_{k})}: = 0$
if $(j_1,\cdots, j_k) \neq (i_1,\cdots, i_k)$ and
\begin{align*}(g_1(x_{i_1}))_{i_1}:=r_0
\text{ and }
(g_k(x_{i_1 \cdots i_k}))_{(i_1, \cdots, i_k)}:=
r_{i_1 \cdots i_{k-1}} \text{ if }k\geq 2.\end{align*}Then we define a
map $g:\bigcup_{k=1}^{\infty} \{ x_{i_1 \cdots i_k}\}_{i_1,\cdots,
i_k}\to c_0$ by
\begin{align*}
g(x_{i_1 i_2 \cdots i_k}):= (g_{1}(x_{i_1}), g_2(x_{i_1 i_2} ),
\cdots, g_m(x_{i_1 \cdots i_m}),\cdots),
\end{align*}where as in the above proof we put $i_m:=1$ for
$m>k$. Note that the right-hand side of the above definition is in
$c_0$ by Lemma \ref{fact1}. We can easily check that the map
$g:\bigcup_{k=1}^{\infty} \{ x_{i_1 \cdots i_k} \}_{i_1, \cdots, i_k}
\to c_0$ is an isometric embedding. As in the proof of Theorem \ref{MT} this construction also implies an
isometric embedding from every proper ultrametric space into $c_0$.
\end{rem}
\section{$\ell_p$-version of nonlinear Dvoretzky's theorem}
In this section we apply Theorem \ref{MT} to obtain an
$\ell_p$-version of nonlinear Dvoretzky's theorem. Refer to
\cite{blmn2}, \cite{ChKa} for the case of finite metric spaces.
We say that a metric space $X$ is \emph{embedded with distortion} $D\geq 1$
in a metric space $Y$ if there exist a map $f:X\to Y$ and a constant $r>0$ such that
\begin{align*}
r \dist_X(x,y) \leq \dist_Y(f(x),f(y))\leq D r \dist_X(x,y) \text{ for all
}x,y\in X.
\end{align*}
Dvoretzky's theorem states that for every $\e>0$, every
$n$-dimensional normed space contains a $k(n,\e)$-dimensional subspace
that embeds into a Hilbert space with distortion $1+\e$
(\cite{Dvo60}). This theorem was conjectured by Grothendieck
(\cite{groth}). See \cite{Mil71} and \cite{MilSch96}, \cite{Sch} for
the estimate of $k(n,\e)$. Bourgain, Figiel, and Milman \cite{BFM86} first studied Dvoretzky's
theorem in the nonlinear setting. They obtained that for every $\e>0$, every finite metric
space $X$ contains a subset $S$ of sufficiently large size which embeds into a Hilbert space with
distortion $1+\e$. See \cite{blmn}, \cite{MN07}, \cite{NT} for further
investigation. Recently Mendel and Naor \cite{MN112, MN11} studied an another variant of nonlinear
Dvoretzky's theorem, answering a question by T.~Tao. For example they
obtained the following: For a metric space $X$ we denote by $\dim_H(X)$
the Hausdorff dimension of $X$.
\begin{thm}[{cf.~\cite[Theorem 1.7]{MN11}}]\label{MNDV}There exists a universal constant $c\in
(0,\infty)$ such that for every $\e \in (0,\infty)$, every compact
metric space $X$ contains a closed subset $S\subseteq X$ that embeds with distortion
$2+\e$ in an ultrametric space, and
\begin{align*}
\dim_H(S)\geq \frac{c\e}{\log (1/\e)}\dim_H(X).
\end{align*}
\end{thm}
Note that since every separable ultrametric space isometrically embed
into $\ell_1$, $\ell_2$, and $c_0$ (\cite{Ves2}), the above $S$ embeds into these spaces.
Applying Theorem \ref{MT} to Theorem \ref{MNDV} we obtain the following
$\ell_p$-version of nonlinear Dvoretzky's theorem:
\begin{cor}There exists a universal constant $c\in
(0,\infty)$ such that for every $\e \in (0,\infty)$, every compact
metric space $X$ contains a closed subset $S\subseteq X$ that embeds with distortion
$2+\e$ in $\ell_p$, and
\begin{align*}
\dim_H(S)\geq \frac{c\e}{\log (1/\e)}\dim_H(X).
\end{align*}
\end{cor}
Mendel and Naor also obtained the following impossibility result for
distortion less than $2$:
\begin{thm}[{cf.~\cite[Theorem 1.8]{MN11}}]\label{hanrei}For every $\alpha>0$ there exists a compact metric space $(X,\dist)$ of Hausdorff
dimension $\alpha$, such that if $S\subseteq X$ embeds into a Hilbert space with distortion strictly smaller
than $2$ then $\dim_H(S) = 0$.
\end{thm}
We shall consider an impossibility problem for the $\ell_p$-version of nonlinear
Dvoretzky's theorem.
In the proof of Theorem \ref{hanrei} Mendel and Naor used the following
result: Let $G$ be the random graph on $n$-vertices of the
Erd\"os-Reyni model $G(n,1/2)$, i.e., every edge is present
independently with probability $1/2$. From $G$ we construct a metric
space $W_n$ by assigning the distance between each two vertices of $G$ by
$1$ if they are joined by an edge, and $2$ if they are not joined by an edge. Then the obtained metric space $W_n$ satisfies the following
property (\cite{blmn}). There exists $K\in (0,\infty)$ such that for any $n\in \mathbb{N}$
there exists an $n$-point metric space $W_n$ such that for every $\delta
\in (0,1)$ any subset
of $W_n$ of size larger than $2\log_2 n + K (\delta^{-2}\log(2/\delta))^2$
must incur distortion at least $2-\delta$ when embedded into $\ell_2$.
Bartal, Linial, Mendel, and Naor obtained a similar result for the
same $W_n$ when considering $\ell_p$ instead of $\ell_2$
(\cite{blmn2}). Then Charikar and Karagiozova \cite[Theorem 1.3]{ChKa} improved the result in
\cite{blmn2}: For any $\delta\in (0,1)$ and
$p\geq 1$, there is a constant $c(p,\delta)$ depending only on $p$
and $\delta $ such that any subset of $W_n$ of size larger than
$c(p,\delta) \log n$ must incur distortion at least
$2-\delta$ when embedded into $\ell_p$.
Then applying this result to the proof in \cite[Section
7.3]{MN11} implies the following:
\begin{prop}For every $p\geq 1$ and $\alpha>0$, there exists a compact metric space
$(X,\dist)$ with $\dim_H(X,\dist)=\alpha$, such that if $S\subseteq
X$ embeds into $\ell_p$ with distortion strictly smaller than $2$
then $\dim_H(S)=0$.
\end{prop}
The case of the distortion $2$ remains open for any $p\geq 1$.
\begin{ack}\upshape
The author wish to express his gratitude to Mr. Ryokichi
Tanaka for discussion. The author also would like to express his thanks
to Professor Assaf Naor for valuable comments and suggestions which improved the preliminary version of this paper.
\end{ack}
|
{
"timestamp": "2012-03-09T02:02:26",
"yymm": "1203",
"arxiv_id": "1203.1761",
"language": "en",
"url": "https://arxiv.org/abs/1203.1761"
}
|
\section{Conclusion}
\label{sec:Conclusion}
In this paper, we study the mixing time of Markov Chain Monte Carlo (MCMC) for the integer least-square optimization problem. It is found that the mixing time of MCMC for the integer least-square problem depends on the structure of the underlying lattice. More specifically, the mixing time of MCMC is found to be closely related to whether there is a local minimum in the lattice structure of the integer least-square problem. For some lattices, the mixing time of the Markov chain is independent of the signal-to-noise ratio; while for some lattices, the mixing time is correlated with the signal-to-noise ratio. We also derive the probability that there exist local minima in an integer least-square problem, which can be as high as $\frac{1}{3}-\frac{1}{\sqrt{5}}+\frac{2\arctan(\sqrt{\frac{5}{3}})}{\sqrt{5}\pi}$. Both theoretical and empirical results suggest that to ensure fast mixing for the MCMC for the integer least-square problem, the temperature for MCMC should often grow as the signal-noise-ratio increases.
\section{Gibbs Sampling and Mixing Time}
\label{sec:Gibbs_sampling}
In this paper, we investigate one kind of MCMC detector called Gibbs sampler which follows a reversible Markov chain and asymptotically converges to the stationary distribution \cite{Mackay_03}. Under the stationary distribution, the Gibbs sampler has a certain probability of visiting the optimal solution. So if run for sufficiently long time, the Gibbs sampler will be able to find the optimal solution to \eqref{EQ:Original_minimization_problem}.
More specifically, the Gibbs sampler starts with a certain $N$-dimensional feasible vector $\hat{{\bf x}}^{(0)}$ among the set $\{-1,+1\}^{N}$ of cardinality $2^{N}$. Then the Gibbs sampler performs a random walk over $\{-1,+1\}^{N}$ based on the following reversible Markov chain. Assume that we are at time index $l$ and the current state of the Markov chain is $\hat{{\bf x}}^{(l)} \in \{-1,+1\}^{N}$. In the next step, the Markov chain uniform randomly picks one position index $j$ out of $\{1,2, ..., N\}$ and keeps the symbols of $\hat{{\bf x}}^{(l)}$ at other positions fixed. Then the Gibbs sampler computes the conditional probability of transferring to each constellation point at the $j$-th index. With the symbols at the $(N-1)$ other positions fixed, the probability that the $j$-th symbol adopts the value $\omega$, is given by
\begin{equation}
\label{Eq:Prob_of_symbol_MCMC}
p\left( {\hat{{\bf x}}_j^{(l+1)} = \omega \left|{ \theta }\right.} \right) = \frac{e^{-\frac{1}{2\alpha^2} \left\| {\bf y} - \sqrt{\frac{\mbox{SNR}}{N}} {\bf H} \hat{{\bf x}}_{j \left|{\omega}\right.} \right\|^2 }}{ \sum\limits_{\hat{{\bf x}}_{j \left|{\tilde{\omega}}\right.} \in \Omega}{e^{-\frac{1}{2\alpha^2} \left\| {\bf y} - \sqrt{\frac{\mbox{SNR}}{N}} {\bf H} \hat{{\bf x}}_{j \left|{ \tilde{\omega} }\right.} \right\|^2 } }} \ ,
\end{equation}
where $\tilde{{\bf x}}_{j \left|{\omega}\right. }^T \triangleq \left[\hat{{\bf x}}_{1:j-1}^{(l)}, \omega, \hat{{\bf x}}_{j+1:N}^{(l)} \right]^T$ and $\theta = \left\{ \hat{{\bf x}}^{(l)}, j, {\bf y}, {\bf H} \right\}$. So conditioned on the $j$-th position is chosen, the Gibbs sampler will with probability $p\left( {\hat{{\bf x}}_j^{(l+1)} = \omega \left|{\theta}\right.} \right)$ keep $\omega$ at the $j$'th index in estimated symbol vector. The initialization of the symbol vector $\hat{{\bf x}}^{(0)}$ can either be chosen randomly or other heuristic solutions. $\alpha$ represents a tunable positive parameter which controls the mixing time of the Markov chain, this parameter is also sometimes called the ``temperature". The smaller $\alpha$ is, the larger the stationary probability for the optimal solution will be, and the easier for the Gibbs sampler to find the optimal solution in the stationary distribution. But as we will show in the paper, there is often a lower bound on $\alpha$, in order to ensure the fast mixing of the Markov chain to the stationary distribution.
It is not hard to see that the Markov chain of Gibbs sampler is reversible and has $2^{N}$ states with the stationary distribution $e^{-\frac{1}{2\alpha^2} \left\| {\bf y} - \sqrt{\frac{\mbox{SNR}}{N}} {\bf H} \hat{{\bf x}} \right\|^2 }$ for an state $\hat{{\bf x}}$. The $2^N \times 2^N$ transition matrix is denoted by $P$, and the element $P_{i,j}$ in the $i$-th ( $1\leq i \leq N$) row and $j$-th ( $1\leq j \leq N$) column is the probability of transferring to state $j$ conditioned on the previous state is $i$. So each row of $P$ sums up to $1$ and the transition matrix after $t$ iterations is $P^{t}$. Denoting the vector for the stationary distribution as $\mathbf{\pi}$, then for an $\epsilon>0$, the mixing time $t(\epsilon)$ is a parameter describing how long it takes for the Markov chain to get close to the stationary distribution, namely,
\begin{equation*}
t_{mix}(\epsilon):=\min\{t: \max_{\tilde{{\bf x}}} \|P^{t}(\tilde{{\bf x}},\cdot)-\mathbf{\pi}\|_{TV}\},
\end{equation*}
where $\|\mu-\nu\|_{TV}$ is the usual total variation distance between two distributions $\mu$ and $\nu$ over the state space $\{+1,-1\}^{N}$.
\begin{equation*}
\|\mu-\nu\|_{TV}=\frac{1}{2}\sum_{{\bf y} \in \{+1,-1\}^{N}} |\mu({\bf y})-\nu({\bf y})|.
\end{equation*}
The mixing time is closely related to the spectrum of the transition matrix $P$. More precisely, for a reversible Markov chain, its mixing time is generally small when the gap between the largest and the second largest eigenvalue of $P$, namely $1-\lambda_2$, is large. The inverse of this gap $\frac{1}{1-\lambda_2}$ is called the relaxation time for this Markov chain. In the next few sections, we will discuss how the mixing time is related to specific system structures.
\section{Introduction}
\label{sec:Introduction}
The integer least-square problem is an NP-hard optimization problem which has received attention in many research areas, for example, communications, global navigation satellite systems, radar imaging, Monte Carlo second-moment estimation, bioinformatics and lattice design \cite{Agrell_et_al_02, Borno}. A computationally efficient way of exactly solving the integer LS problem is the sphere decoder (SD) \cite{Damen_et_al, Hochwald_Ten-Brink_03, Hassibi_1, Agrell_et_al_02}. It is known that for a moderate problem size and a suitable range of Signal-to-Noise Ratios ($SNR$), SD has low computational complexity, which can be significantly smaller than an exhaustive search solver. But for a large problem size and fixed $SNR$, the average computational complexity of SD is still exponential in the problem dimension\cite{Ottersten_05}. So for large problem sizes, (for example large-scale Multiple-Input Multiple-Output (MIMO) systems with many transmit and receive antennas), SD still has high computational complexity and is thus computationally infeasible.
Unlike SD, MCMC algorithms perform a random walk over the signal space in the hope of finding the optimal solution. Gibbs sampling (or Glauber dynamics) is a popular MCMC method which performs the random walk according to the transition probability determined by the stationary distribution of a reversible Markov chain \cite{Levin} \cite{Haggstrom_02}. The Gibbs sampler has been proposed for detection purposes in wireless communication \cite{Zhu_Farhang_Boroujeny_05, Wang_Poor_03} (see also the references therein). These MCMC methods are able to provide the optimal solution if they are run for a sufficiently long time; and empirically MCMC methods are observed to provide near-optimal solutions in a reasonable amount of computational time even for large problem dimensions \cite{Zhu_Farhang_Boroujeny_05, Wang_Poor_03, Hassibi_Globecom}. \cite{Hassibi_Globecom} gave a characterization of the MCMC temperature parameter such that the optimal solution can be found in polynomial time assuming stationary distribution has been reached. However, the understanding of the mixing time (or the convergence rate, namely how fast a Markov chain converges to the stationary distribution) of these MCMC methods is still limited\cite{Hassibi_Globecom, Farhang_Boroujeny_06, ChenRong}.
In this paper, we are interested in deriving the mixing time of the Gibbs sampler for integer LS problems. We derive upper and lower bounds on the mixing time and show how the mixing time is related to the structures of integer LS problems. Our work furthers the understanding of the mixing time in MCMC for integer LS problems, and is helpful in optimizing the MCMC parameter for better computational performance.
Our paper is organized as follows. In Section \ref{sec:System_model} we present the system model. The MCMC method and related background knowledge are introduced in Section \ref{sec:Gibbs_sampling}. Section \ref{sec:mixing_time_ortho}, \ref{sec:mixing_time_local},\ref{sec:local_minimum} and \ref{sec:choice_alpha} derive the bounds on the mixing time and discuss how to optimize MCMC parameters to ensure fast mixing. Simulation results are given in Section \ref{sec:sim_results}. Section \ref{sec:Conclusion} concludes this paper.
\section{Simulation Results}
\label{sec:sim_results}
In this section we present simulation results for an $N \times N$ system with a full square channel matrix containing i.i.d. Gaussian entries.
In Figure \ref{fig:N_expected}, we plot the expected number of local minima in a system as the problem dimension $N$ grows. For each $N$, we generate $100$ random channel matrices and for each matrix, we examine the number of local minima by exhaustive search. As the problem dimension $N$ grows, the number of local minima grows rapidly.
In Figure \ref{fig:N_frequency}, we plot the probability of there existing a local minimum as the problem dimension $N$ grows. For each $N$, we generate $100$ random channel matrices and for each matrix, we examine whether there exists local minimum by exhaustive search. As $N$ grows, the empirical probability of there existing at least one local minimum approaches $1$. It is interesting to see that for $N=2$, our theoretical result $\frac{1}{3}-\frac{1}{\sqrt{5}}+\frac{2\arctan(\sqrt{\frac{5}{3}})}{\sqrt{5}\pi}\approx0.15$ matches well with the simulations.
We also examine how the spectral gap for MCMC is related to the existence of local minima. For $N=5$ and $SNR=10$, we randomly generated $10$ problem instances and keep the temperature $\alpha^2=1$ the same as the noise variance. Out of the $10$ trials, the number of local minima are $2, 1, 0, 0, 0, 2, 0,0, 2$ and $0$. The corresponding spectral gaps are respectively $0.0037, 0.0008, 0.1244, 0.1957, 0.1989$, $0.0011$, $0.1698$, $0.1764$, $5 \times 10^{-10}$, and $0.1266$. It can be seen that when there exist local minima, the spectral gap is significantly smaller than the cases without local minima. This implies a slower mixing for the systems with local minima, which is consistent with our theoretical results.
\begin{figure}[tb]
\centering
\includegraphics[width=3.5in, height=2.5in]{Figures/N_expected.eps}
\caption{Average Number of Local Minima}
\label{fig:N_expected}
\end{figure}
\begin{figure}[tb]
\centering
\includegraphics[width=3.5in,height=2.5in]{Figures/N_frequency.eps}
\vspace{-2mm}
\caption{The Probability of Having Local Minima}
\label{fig:N_frequency}
\end{figure}
\section{System Model}
\label{sec:System_model}
In this paper, we consider a real-valued integer least-square problem with $N$ transmit and $N$ receive dimensions, targeting applications in block-fading MIMO antenna systems with known channel coefficients. The received signal ${\bf y} \in \mathbb{R}^N$ can be expressed as
\begin{equation}
\label{EQ:sig_model_matrix}
{\bf y} = \sqrt{\frac{\mbox{SNR}}{N}}{{\bf H}{{\bf x}}} + {\bf v} \ ,
\end{equation}
where ${{\bf x}} \in \Omega^{N}$ is the transmitted signal, and $\Omega$ denotes the constellation set. To simplify the derivations in the paper we will assume that $\Omega = \left\{\pm 1\right\}$. ${\bf v} \in \mathbb{R}^{N}$ is the noise vector where each entry is Gaussian $\mathcal{N} \left(0,1\right)$ and independent identically distributed (i.i.d.), and ${\bf H} \in \mathbb{R}^{N \times N}$ denotes the channel matrix with i.i.d. $\mathcal{N} \left(0,1\right)$ entries. The signal-to-noise ratio is defined as
\begin{equation}
\label{EQ:SNR}
\begin{split}
\mbox{SNR} &= \frac{\mathcal{E} \left\|\sqrt{\frac{\mbox{SNR}}{N}}{\bf H}{{\bf x}}\right\|^2}{\mathcal{E} \|{\bf v}\|^2} \ ,
\end{split}
\end{equation}
which is done in order to take into account the total transmit energy
Without loss of generality, we assume that the all minus one vector was transmitted, ${{\bf x}} = -\text{\boldmath $1$}$. Therefore
\begin{equation}
{\bf y} = {\bf v}-\sqrt{\frac{\mbox{SNR}}{N}}{\bf H}\text{\boldmath $1$} \ .
\end{equation}
To minimize the average error probability, we need to perform Maximum Likelihood Sequence Detection (here simply referred to as ML detection) given by
\begin{equation}
\label{EQ:Original_minimization_problem}
{{\bf x}}^* = \arg \mathop {\text{min} }\limits_{{{\bf x}} \in \Omega^{N}} \ \ \left\| {\bf y} - \sqrt{\frac{\mbox{SNR}}{N}}{\bf H} {{\bf x}} \right\|^2,
\end{equation}
which is exactly an integer LS problem.
\section{Choice of Temperature $\alpha$ in High $SNR$}
\label{sec:choice_alpha}
In previous sections, we have looked at the mixing time of MCMC for an integer LS problem. Now we use the results we have accumulated so far to help choose the appropriate temperature of $\alpha$ to ensure that the MCMC mixes fast and that the optimal solution also comes up fast when the system is in a stationary distribution.
When $SNR \rightarrow \infty$, the integer LS problem will have the same local minima as the case ${\bf v}=0$. From the derivations and simulations, it is suggested that with high probability there will be at least one local minimum in the integer LS problem, especially for large problem dimension $N$.
So following from Lemma \ref{thm:mixing_scale_alpha} and the reasoning therein, to ensure there is an upper bound on the mixing time as $SNR\rightarrow \infty$, the temperature $\alpha$ should at least grow at a rate such that
\begin{equation*}
\max_{\tilde{{\bf x}}} \min_{\tilde{{\bf x}}'}\frac{\frac{SNR}{N} \left(\|-{\bf H}\mathbf{1}-\tilde{{\bf x}}'\|^2-\|-{\bf H}\mathbf{1}-\tilde{{\bf x}}\|^2\right)}{2\alpha^2} \leq C,
\end{equation*}
where $\tilde{{\bf x}}$ is a local minimum and $\tilde{{\bf x}}'$ is a neighbor of $\tilde{{\bf x}}$, and $C$ is a constant.
This will require that $\alpha^2$ grow as fast as $\Omega(SNR)$ to ensure fast mixing with the existence of local minima. This explains that if we keep the temperature at the noise level, it will lead to slow convergence in the high SNR regime \cite{Farhang_Boroujeny_06}.
\section{The Presence of Local Minima}
\label{sec:local_minimum}
In this section, we look at the problem of how many local minima there are in an integer least-square problem, especially when the $SNR$ is high.
\begin{theorem}
There can be exponentially many local minima in an integer least-quare problem.
\end{theorem}
\begin{proof}
Let $N$ be an even integer. Consider a matrix whose first $\frac{N}{2}$ columns ${\bf h}_{i}$, $1\leq i \leq \frac{N}{2}$ have unit norms and are orthogonal to each other. For the other $\frac{N}{2}$ columns ${\bf h}_{i}$, $\frac{N}{2} +1\leq i \leq N$, ${\bf h}_{i}=-(1+\epsilon){\bf h}_{i-\frac{n}{2}}$, where $\epsilon$ is a sufficiently small positive number ($\epsilon<1$). We also let ${\bf y}={\bf H} (-\mathbf{{1}})$, where $\mathbf{{1}}$ is an all-$1$ vector. So $-\mathbf{{1}}$ is a globally minimum point for this integer LS problem.
Consider all those vectors ${\tilde{{\bf x}}}'$ which, for any $1\leq i\leq \frac{N}{2}$, its $i$-th element and $i+\frac{N}{2}$-th element are either simultaneously $+1$ or simultaneously $-1$. When $\epsilon$ is smaller than $1$, we claim that any such a vector except the all $-1$ vector ${\tilde{{\bf x}}}$, is a local minimum, which shows that there are at least $2^{\frac{N}{2}}-1$ local minima.
Assume that for a certain $1\leq i\leq \frac{N}{2}$, the $i$-th element and $(i+\frac{N}{2})$-th element of ${\tilde{{\bf x}}}'$ are simultaneously $-1$. Then if we change the $i$-th element to $+1$, $\|{\bf y}-{\bf H}{\tilde{{\bf x}}}'\|^2$ increases by $4$; and if we change the $(i+\frac{N}{2})$-th element to $+1$, $\|{\bf y}-{\bf H}{\tilde{{\bf x}}}'\|^2$ increases by $4(1+\epsilon)^2$. This is true because the $i$-th and $(i+\frac{N}{2})$-th columns are orthogonal to other $(N-2)$ columns.
Similarly, assume that for a certain $1\leq i\leq \frac{N}{2}$, the $i$-th element and $(i+\frac{N}{2})$-th element of ${\tilde{{\bf x}}}'$ are simultaneously $+1$. Then if we change the $i$-th element to $-1$, $\|{\bf y}-{\bf H}{\tilde{{\bf x}}}'\|^2$ increases by $4(1+\epsilon)^2-4\epsilon^2$; and if we change the $(i+\frac{N}{2})$-th element to $-1$, $\|{\bf y}-{\bf H}{\tilde{{\bf x}}}'\|^2$ increases by $4-4\epsilon^2$.
\end{proof}
Now we study how often we encounter a local minimum in the specific inter least-square problem model. Without loss of generality, we assume that the transmitted sequence is an all $-1$ sequence. We first give the condition for ${\tilde{{\bf x}}}$ to be a local minimum. We assume that ${\tilde{{\bf x}}}$ is a vector which has $k$ `$+1$' over an index set $K$ with $|K|=k$ and $(N-k)$ `$-1$' over the set $\overline{K}=\{1,2,...,N\}\setminus {K}$.
\begin{lemma}
${\tilde{{\bf x}}}$ is a local minimum if and only if ${\tilde{{\bf x}}}$ is not a global minimum; and
\begin{itemize}
\item $\forall i \in K$,
\begin{eqnarray}
{\bf h}_{i}^{T}(\sum_{j \in K}{\bf h}_{j}-\frac{{\bf v}}{2})<\frac{\|{\bf h}_{i}\|^2}{2}
\end{eqnarray}
\item $\forall i \in \overline{K}$,
\begin{eqnarray}
{\bf h}_{i}^{T}(\sum_{j \in K}{\bf h}_{j}-\frac{{\bf v}}{2})>-\frac{\|{\bf h}_{i}\|^2}{2}.
\end{eqnarray}
\end{itemize}
\label{lemma:localcondition}
\end{lemma}
\begin{proof}
For a position $i \in K$, when we flip ${\tilde{{\bf x}}}_{i}$ to $1$, $\|{\bf y}-{\bf H}{\tilde{{\bf x}}}'\|^2$ is increased, namely,
\begin{eqnarray}
&&\|{\bf y}-{\bf H}{\tilde{{\bf x}}}\|^2-\|{\bf y}-{\bf H}{\tilde{{\bf x}}}_{\sim i}\|^2 \nonumber \\
&=& \|-2\sum_{j \in K}{\bf h}_{j}+{\bf v}\|^2-\|-2\sum_{j \in K, j \neq i}{\bf h}_{j}+{\bf v}\|^2 \nonumber \\
&=& 4\|{\bf h}_{i}\|^2+4{\bf h}_{i}^{T}(2\sum_{j \in K, j \neq i}{\bf h}_{j}-{\bf v}) \nonumber\\
&<&0,
\end{eqnarray}
where $\tilde{{\bf x}}_{\sim i}$ is a neighbor of $\tilde{{\bf x}}$ by changing index $i$.
This means
\begin{eqnarray}
{\bf h}_{i}^{T}(\sum_{j \in K}{\bf h}_{j}-\frac{{\bf v}}{2})<\frac{\|{\bf h}_{i}\|^2}{2}.
\end{eqnarray}
For a position $i \in \overline{K}$, when we flip ${\tilde{{\bf x}}}_{i}$ to $-1$, $\|{\bf y}-{\bf H}{\tilde{{\bf x}}}'\|^2$ is also increased, namely,
\begin{eqnarray}
&&\|{\bf y}-{\bf H}{\tilde{{\bf x}}}\|^2-\|{\bf y}-{\bf H}{\tilde{{\bf x}}}_{\sim i}\|^2 \nonumber \\
&=& \|-2\sum_{j \in K}{\bf h}_{j}+{\bf v}\|^2-\|-2\sum_{j \in K}{\bf h}_{j}-2{\bf h}_{i}+{\bf v}\|^2 \nonumber \\
&=& -4\|{\bf h}_{i}\|^2+4{\bf h}_{i}^{T}(-2\sum_{j \in K}{\bf h}_{j}+{\bf v}) \nonumber\\
&<&0.
\end{eqnarray}
This means
\begin{eqnarray}
({\bf h}_{i})^{T}(\sum_{j \in K}{\bf h}_{j}-\frac{{\bf v}}{2})>-\frac{\|{\bf h}_{i}\|^2}{2}.
\end{eqnarray}
\end{proof}
It is not hard to see that when $SNR \rightarrow \infty$, ${\bf v}$ is comparatively small with high probability, so we have the following lemma.
\begin{lemma}
When $SNR \rightarrow \infty$, with high probability, ${\tilde{{\bf x}}}$ is a local minimum if and only if ${\tilde{{\bf x}}} \neq -\mathbf{1}$; and
\begin{itemize}
\item $\forall i \in K$,
\begin{eqnarray}
{\bf h}_{i}^{T}(\sum_{j \in K}{\bf h}_{j})<\frac{\|{\bf h}_{i}\|^2}{2}
\end{eqnarray}
\item $\forall i \in \overline{K}$,
\begin{eqnarray}
{\bf h}_{i}^{T}(\sum_{j \in K}{\bf h}_{j})>-\frac{\|{\bf h}_{i}\|^2}{2}.
\end{eqnarray}
\end{itemize}
\end{lemma}
\begin{theorem}
Consider a $2 \times 2$ matrix ${\bf H}$ whose two columns are uniform randomly sampled from the unit-normed $2$-dimensional vector.
When ${\bf v}=0$, the probability of there existing a local minimum for such an ${\bf H}$ is $\frac{1}{3}$.
\end{theorem}
\begin{proof}
When ${\bf v}=0$, clearly ${\tilde{{\bf x}}}=(-1,-1)$ is a global minimum point, not a local minimum point. It is also clear that ${\tilde{{\bf x}}}=(-1,1)$ or ${\tilde{{\bf x}}}=(1,-1)$ can not be a local minimum point since they are neighbors to the global minimum solution. So the only possible local minimum point is ${\tilde{{\bf x}}}=(1,1)$.
From Lemma \ref{lemma:localcondition}, the corresponding necessary and sufficient condition is
\begin{equation*}
{\bf h}_{1}^{T}{\bf h}_{2} < -\frac{\|{\bf h}_{1}\|^2}{2}=-\frac{\|{\bf h}_{2}\|^2}{2}=-\frac{1}{2}.
\end{equation*}
This means the angle $\theta$ between the two 2-dimensional vectors ${\bf h}_{1}$ and ${\bf h}_{2}$ satisfy $\cos(\theta) <-\frac{1}{2}$. Since ${\bf h}_{1}$ and ${\bf h}_{2}$ are two independent uniform randomly sampled vector, the chance for that to happen is $\frac{\pi-\arccos{(-\frac{1}{2})}}{\pi}=\frac{1}{3}$.
\end{proof}
\begin{theorem}
Consider a $2 \times 2$ matrix ${\bf H}$ whose elements are independent $\mathcal{N} (0,1)$ Gaussian random variables.
When ${\bf v}=0$, the probability of there existing a local minimum for such an ${\bf H}$ is $\frac{1}{3}-\frac{1}{\sqrt{5}}+\frac{2\arctan(\sqrt{\frac{5}{3}})}{\sqrt{5}\pi}$.
\label{thm:22Gaussian}
\end{theorem}
\begin{proof}
When ${\bf v}=0$, clearly ${\tilde{{\bf x}}}=(-1,-1)$ is a global minimum point, not a local minimum point. It is also clear that ${\tilde{{\bf x}}}=(-1,1)$ or ${\tilde{{\bf x}}}=(1,-1)$ can not be a local minimum point since they are neighbors to the global minimum solution. So the only possible local minimum point is ${\tilde{{\bf x}}}=(1,1)$.
From Lemma \ref{lemma:localcondition}, the corresponding necessary and sufficient condition is
\begin{equation*}
{\bf h}_{1}^{T}{\bf h}_{2} < -\max\left\{ \frac{\|{\bf h}_{1}\|^2}{2}, \frac{\|{\bf h}_{2}\|^2}{2} \right\}.
\end{equation*}
This means the angle $\theta$ between the two 2-dimensional vectors ${\bf h}_{1}$ and ${\bf h}_{2}$ satisfy
\begin{equation*}
r_1 r_2 \cos(\theta) < -\frac{\max\left\{r_1^2, r_2^2\right\}}{2},
\end{equation*}
where $r_1$ and $r_2$ are respectively the $\ell_2$ norm of ${\bf h}_{1}$ and ${\bf h}_{2}$.
Because the elements of ${\bf H}$ are independent Gaussian random variables, $r_1$ and $r_2$ are thus independent random variables following the Rayleigh distribution
\begin{eqnarray*}
p(r_1)&=&r_1 e^{-\frac{r_1^2}{2}}\nonumber\\
p(r_2)&=&r_2 e^{-\frac{r_2^2}{2}};
\end{eqnarray*}
while $\theta$ follows a uniform distribution over $[0, 2\pi)$
By symmetry, for $t\geq 1$,
\begin{eqnarray*}
&&P(\frac{\max\left\{r_1^2, r_2^2\right\}}{r_1 r_2}>t) \\
&=&2\int_{0}^{\infty} r_1 e^{-\frac{r_1^2}{2}} \times {\int_{0}^{\frac{r_1}{t}} r_2 e^{-\frac{r_2^2}{2}} \,dr_2} \,dr_1\\
&=&2\int_{0}^{\infty} r_1 e^{-\frac{r_1^2}{2}} \times {(1-e^{-\frac{r_1^2}{2}})} \,dr_1\\
&=&2(1-\int_{0}^{\infty} r_1 e^{-(\frac{1}{2}+\frac{1}{2t^2})r_1^2} \,dr_1)\\
&=&\frac{2}{t^2+1}.
\end{eqnarray*}
Since $\theta$ is an independent random variable satisfying $\cos(\theta) < -\frac{\max\left\{r_1^2, r_2^2\right\}}{2r_1 r_2}$ and $\cos(\theta) \geq -1$, the probability that ${\tilde{{\bf x}}}=(+1,+1)$ is a local minimum is given by
\begin{eqnarray*}
&P=& \int_{1}^{2} (1-\frac{2}{t^2+1})' (1-\frac{\arccos(-\frac{t}{2})}{\pi}) \,dt\\
&=& \int_{1}^{2} \frac{4t}{(t^2+1)^2} (1-\frac{\arccos(-\frac{t}{2})}{\pi}) \,dt. \\
&=&\frac{1}{3}-\frac{1}{\sqrt{5}}+\frac{2\arctan(\sqrt{\frac{5}{3}})}{\sqrt{5}\pi},
\end{eqnarray*}
which is approximately $0.145696$.
\end{proof}
For higher dimension $N$, it is hard to directly estimate the probability of a vector being a local minimum based on the conditions in Lemma \ref{lemma:localcondition}. Simulation results instead suggest that for large $N$, with high probability, there exists at least one local minimum. The following lemma gives us a sufficient condition. For example, if the sum of $k$ columns has a very small $\ell_2$ norm, that will very likely lead to a local minimum.
\begin{lemma}
\begin{eqnarray}
\|\sum_{j \in K}{\bf h}_{j} -\frac{{\bf v}}{2}\| < \min_{i} \frac{\|{\bf h}_{i}\|}{2}.
\end{eqnarray}
\label{lemma:localsufficient}
\end{lemma}
\begin{proof}
This follows from $|{\bf h}_{i}^T (\sum_{j \in K}{\bf h}_{j} -\frac{{\bf v}}{2})| < \frac{\|{\bf h}_{i}\|^2}{2}$.
\end{proof}
\begin{theorem}
Consider an $N \times N$ matrix ${\bf H}$ whose $N$ columns are uniform randomly sampled from the unit-normed $N$-dimensional vector.
When ${\bf v}=0$, then the expected number of local minima for such an ${\bf H}$ is $\mathcal{E}(N_{local}) \geq \sum_{k=2}^{N}{\binom{N}{k}}P_{k}$, where $P_{k}$ is the probability that the magnitude of the sum of $k$ uniform randomly sampled vectors is less than $\frac{1}{2}$.
\end{theorem}
\begin{proof}
This follows from Lemma \ref{lemma:localcondition} and the fact that there are $\binom{N}{k}$ vectors for ${\tilde{{\bf x}}}$ which have exactly $k$ +1 in it.
\end{proof}
\section{Mixing Time with local Minima}
\label{sec:mixing_time_local}
In this section, we consider the mixing time for integer LS problems which have local minima besides the global minimum point.
\begin{definition}
A local minimum $\tilde{{\bf x}}$ is a state such that $\tilde{{\bf x}}$ is not a global minimizer for $\min_{{\bf s} \in \{-1,+1\}^N}\|{\bf y}-{\bf H}{\bf s}\|^2$; and any of its neighbors which differ from $\tilde{{\bf x}}$ in only one position index, denoted by $\tilde{{\bf x}}'$, satisfies $\|{\bf y}-{\bf H}\tilde{{\bf x}}'\|^2>\|{\bf y}-{\bf H}\tilde{{\bf x}}\|^2$.
\end{definition}
We will use the following theorem about the spectral gap of Markov chain to evaluate the mixing time.
\begin{theorem}[Jerrum and Sinclair 1989 \cite{JerrumSinclair}, Lawler and Sokal (1988) \cite{Lawler}, \cite{Levin}] Let $\lambda_2$ be the second largest eigenvalue of a reversible transition matrix $P$, and let $\gamma=1-\lambda_2$. Then
\begin{equation*}
\frac{\Phi_{*}^2}{2}\leq \gamma \leq 2\Phi_{*},
\end{equation*}
where $\Phi_{*}$ is the bottleneck ratio defined as
\begin{equation*}
\Phi_{*}=\min_{\pi(S) \leq \frac{1}{2}} \frac{Q(S,S^{c})}{\pi(S)}.
\end{equation*}
Here $S$ is any subset of the state spaces with stationary measure no bigger than $\frac{1}{2}$, $S^{c}$ is its complement set, and $Q(S,S^c)$ is the probability of moving from $S$ to $S^c$ in one step when starting with the stationary distribution.
\label{thm:gap_bottle}
\end{theorem}
\begin{theorem}
If there is a local minimum ${\tilde{{\bf x}}}$ in an integer least-square problem and we denote its neighbor differing only at the $k$-th ($1\leq k \leq N$) location as ${\tilde{{\bf x}}}_k$, then the mixing time of the Gibbs sampler is at least
\begin{equation}
t_{mix}(\epsilon) \geq \log(\frac{1}{2\epsilon})(\frac{1}{\gamma}-1),
\end{equation}
where
\begin{equation}
\gamma=\sum_{k=1}^{N} \frac{2}{N} {\frac{e^{-\frac{\|{\bf y}-{\bf H}{\tilde{{\bf x}}}_{k}\|^2}{2\alpha^2}}}{e^{-\frac{\|{\bf y}-{\bf H}{\tilde{{\bf x}}}_{k}\|^2}{2\alpha^2}}+e^{-\frac{\|{\bf y}-{\bf H}{\tilde{{\bf x}}}\|^2}{{2\alpha^2}}}} }
\end{equation}
The parameter $\gamma$ is upper bounded by
\begin{equation}
\frac{2}{1+e^{\frac{\min_{k}{\|{\bf y}-{\bf H}{\tilde{{\bf x}}}_{k}\|^2}-\|{\bf y}-{\bf H}{\tilde{{\bf x}}}\|^2}{2\alpha^2}}}
\end{equation}
\label{thm:gap_local}
\end{theorem}
\begin{proof}
We apply Theorem \ref{thm:gap_bottle} to prove this result. We take a local minimum point ${\tilde{{\bf x}}}$ as the single element in the bottle-neck set $S$. Since ${\tilde{{\bf x}}}$ is a local minimum, $\pi(S) \leq \frac{1}{2}$.
\begin{equation}
Q(S,S^{c})=\frac{\pi(S)}{N}\sum_{k=1}^{N} {\frac{e^{-\frac{\|{\bf y}-{\bf H}{\tilde{{\bf x}}}_{k}\|^2}{2\alpha^2}}}{e^{-\frac{\|{\bf y}-{\bf H}{\tilde{{\bf x}}}_{k}\|^2}{2\alpha^2}}+e^{-\frac{\|{\bf y}-{\bf H}{\tilde{{\bf x}}}\|^2}{{2\alpha^2}}}} }
\end{equation}
Dividing by $\pi(S)$, by the definition of $\Phi_{*}$
\begin{equation}
\Phi_{*}\leq \frac{Q(S,S^{c})}{\pi(S)}=\frac{1}{N}\sum_{k=1}^{N} {\frac{e^{-\frac{\|{\bf y}-{\bf H}{\tilde{{\bf x}}}_{k}\|^2}{2\alpha^2}}}{e^{-\frac{\|{\bf y}-{\bf H}{\tilde{{\bf x}}}_{k}\|^2}{2\alpha^2}}+e^{-\frac{\|{\bf y}-{\bf H}{\tilde{{\bf x}}}\|^2}{{2\alpha^2}}}} }
\end{equation}
So we know $\gamma \leq 2 \frac{1}{N}\sum_{k=1}^{N} {\frac{e^{-\frac{\|{\bf y}-{\bf H}{\tilde{{\bf x}}}_{k}\|^2}{2\alpha^2}}}{e^{-\frac{\|{\bf y}-{\bf H}{\tilde{{\bf x}}}_{k}\|^2}{2\alpha^2}}+e^{-\frac{\|{\bf y}-{\bf H}{\tilde{{\bf x}}}\|^2}{{2\alpha^2}}}} }$.
From a well-known theorem for the relationship between $t_{mix}(\epsilon)$ and $\gamma$:
$t_{mix}(\epsilon) \geq (\frac{1}{\gamma}-1) \log(\frac{1}{2\epsilon})$ \cite{Levin},
our conclusion follows.
\end{proof}
\begin{theorem}
For an integer least-square problem where no two vectors give the same objective distance, the relaxation time (the inverse of the spectral gap) of MCMC is upper bounded by a constant as the temperature $\alpha \rightarrow 0$ if and only if there is no local minimum. Moreover, when there is a local minimum, as $\alpha \rightarrow 0$, the mixing time of Markov chain $t_{mix}(\epsilon) =e^{\Omega(\frac{1}{2\alpha^2})}$. \footnote{In this paper, $\Omega(\cdot)$, $\Theta(\cdot)$, and $O(\cdot)$ are the usual scaling notations as in computer science}
\label{thm:mixing_scale_alpha}
\end{theorem}
\begin{proof}
First, when there is a local minimum, from Theorem \ref{thm:gap_local} and Theorem \ref{thm:gap_bottle}, the spectral gap $\gamma$ is lower bounded by
\begin{equation}
\gamma=\frac{2}{N}\sum_{k=1}^{N} {\frac{e^{-\frac{\|{\bf y}-{\bf H}{\tilde{{\bf x}}}_{k}\|^2}{2\alpha^2}}}{e^{-\frac{\|{\bf y}-{\bf H}{\tilde{{\bf x}}}_{k}\|^2}{2\alpha^2}}+e^{-\frac{\|{\bf y}-{\bf H}{\tilde{{\bf x}}}\|^2}{{2\alpha^2}}}} }
\end{equation}
As the temperature $\alpha \rightarrow 0$, the spectral gap upper bound
\begin{equation}
\frac{2}{1+e^{\frac{\min_{k}{\|{\bf y}-{\bf H}{\tilde{{\bf x}}}_{k}\|^2}-\|{\bf y}-{\bf H}{\tilde{{\bf x}}}\|^2}{2\alpha^2}}}
\end{equation}
decreases at the speed of $\Theta(e^{-\frac{\min_{k}{\|{\bf y}-{\bf H}{\tilde{{\bf x}}}_{k}\|^2}-\|{\bf y}-{\bf H}{\tilde{{\bf x}}}\|^2}{2\alpha^2}})$. So the relaxation time of the MCMC is lower bounded by $t_{mix}(\epsilon) =e^{\Omega(\frac{1}{2\alpha^2})}$, which grows unbounded as $\alpha \rightarrow 0$.
Suppose instead that there is no local minimum. We argue that as $\alpha \rightarrow 0$, the spectral gap of this MCMC is lower bounded by some constant independent of $\alpha$. Again, we look at the bottle neck ratio and use Theorem \ref{thm:gap_bottle} to bound the spectral gap.
Consider any set $S$ of sequences which do not include the global minimum point ${{\bf x}^*}$. As $\alpha \rightarrow 0$, the measure of this set of sequences $\pi(S)\leq \frac{1}{2}$. Moreover, as $\alpha \rightarrow 0$, any set $S$ with $\pi(S)\leq \frac{1}{2}$ can not contain the global minimum point ${{\bf x}^*}$. Now we look at the sequence ${\tilde{{\bf x}}}'$ which has the smallest distance $\|{\bf y}-{\bf H}{\tilde{{\bf x}}}'\|$ among the set $S$. Since there is no local minimum, ${\tilde{{\bf x}}}'$ must have at least one neighbor ${\tilde{{\bf x}}}''$ in $S^{c}$ which has smaller distance than ${\tilde{{\bf x}}}'$. Otherwise, this would imply ${\tilde{{\bf x}}}'$ is a local minimum. So
\begin{equation}
Q(S,S^{c}) \geq \pi({\tilde{{\bf x}}}') \times \frac{1}{N} {\frac{e^{-\frac{\|{\bf y}-{\bf H}{\tilde{{\bf x}}}''\|^2}{2\alpha^2}}}{e^{-\frac{\|{\bf y}-{\bf H}{\tilde{{\bf x}}}''\|^2}{2\alpha^2}}+e^{-\frac{\|{\bf y}-{\bf H}{\tilde{{\bf x}}}'\|^2}{{2\alpha^2}}}} }
\end{equation}
As $\alpha \rightarrow 0$, $\frac{\pi({\tilde{{\bf x}}}')}{\pi(S)} \rightarrow 1$. So for a given $\epsilon>0$, as $\alpha \rightarrow 0$
\begin{equation}
\frac{Q(S,S^{c})}{\pi(S)} \geq \frac{1-\epsilon}{N} {\frac{e^{-\frac{\|{\bf y}-{\bf H}{\tilde{{\bf x}}}''\|^2}{2\alpha^2}}}{e^{-\frac{\|{\bf y}-{\bf H}{\tilde{{\bf x}}}''\|^2}{2\alpha^2}}+e^{-\frac{\|{\bf y}-{\bf H}{\tilde{{\bf x}}}'\|^2}{{2\alpha^2}}}} },
\end{equation}
which approaches $\frac{(1-\epsilon)}{N}$ as $\alpha \rightarrow 0$ because $\|{\bf y}-{\bf H}{\tilde{{\bf x}}}''\|^2 < \|{\bf y}-{\bf H}{\tilde{{\bf x}}}'\|^2$.
From Theorem \ref{thm:gap_bottle}, the spectral gap $\gamma$ is at least $\frac{(\frac{Q(S,S^{c})}{\pi(S)})^2}{2}$, which is lower bounded by a constant as $\alpha \rightarrow 0$.
\end{proof}
So from the analysis above, the mixing time is closely related to whether there are local minima in the problem. In the next section, we will see there often exist local minima, which implies very slow convergence rate for MCMC when the temperature is kept at the noise level in the high SNR regime.
\section{Mixing Time without Local Minima}
\label{sec:mixing_time_ortho}
In this section, we consider the mixing time for MCMC for integer LS problems and study how the mixing time for integer LS problem depends on the linear matrix structure and $SNR$. As a first step, we consider a linear matrix ${\bf H}$ with orthogonal columns. As shown later, the mixing time for this matrix has an upper bound independent of $SNR$. In fact, this is a general phenomenon for integer LS problems without local minima.
For simplicity, we incorporate the $SNR$ term into ${\bf H}$, and the model we are currently considering is
\begin{equation}
\label{eq:orthogonal}
{\bf y}={\bf H} {\bf x} +{\bf v},
\end{equation}
where the columns of ${\bf H}$ are orthogonal to each other. We will also incorporate the $SNR$ term into ${\bf H}$ this way in the following sections unless stated otherwise.
\begin{theorem}
Independent of the temperature $\alpha$ and $SNR$, the mixing time of the Gibbs sampler for orthogonal-column integer least-square problems is upper bounded by $N \log(N)+\log(1/\epsilon)N$.
\end{theorem}
This theorem is an extension of the mixing time for regular random walks on an $N$-dimensional hypercube \cite{Levin}. The only difference here is that the transition probability follows (\ref{Eq:Prob_of_symbol_MCMC}) and that the transition probability depends on $SNR$.
\begin{proof}
When the $k$-th index was selected to update in the Gibbs sampler, since the columns of ${\bf H}$ are orthogonal to each other, the probability of updating ${\bf x}_{k}$ to $-1$ is $\frac{1}{1+e^{\frac{2{\bf y}^{T} {\bf{h}}_{k}}{\alpha^2}}}$. We note that this probability is independent of the current state of Markov chain $\hat{{\bf x}}$. So we can use the classical coupling idea to get an upper bound on the mixing time of this Markov Chain.
Consider two separate Markov chains starting at two different states ${\bf x}_{1}$ and ${\bf x}_{2}$. These two chains follow the same update rule according to (\ref{Eq:Prob_of_symbol_MCMC}) and, by using the random source, each step they select the same position index to update and they update that position to the same symbol. Let $\tau_{couple}$ be the first time the two chains come to the same state. Then by a classical result, the total variation distance
\begin{equation}
\label{Eq:TV_coupling}
d(t)=\max_{\tilde{{\bf x}}} \|P^{t}(\tilde{{\bf x}},\cdot)-\mathbf{\pi}\|_{TV} \leq \max_{{\bf x}_{1},{\bf x}_{2}} p_{{\bf x}_1,\xb2} \{\tau_{couple}>t\}.
\end{equation}
Note that the coupling time is just time for collecting all of the positions where ${\bf x}_1$ and ${\bf x}_2$ differ, as in the coupon collector problem. From the coupon collector results, for any ${\bf x}_1$ and ${\bf x}_2$,
\begin{equation}
\label{Eq:TV_coupling2}
d(N \log(N)+cN)\leq p_{{\bf x}_1,{\bf x}_2} \{\tau_{couple}>N \log(N)+cN \} \leq e^{-c}.
\end{equation}
So the conclusion follows.
\end{proof}
|
{
"timestamp": "2012-03-13T01:00:36",
"yymm": "1203",
"arxiv_id": "1203.2213",
"language": "en",
"url": "https://arxiv.org/abs/1203.2213"
}
|
\section{Introduction/Context}
All stars with masses initially between $\sim$8 and
$\sim$150\,M$_\odot$\ eventually form a degenerate core that inevitably
collapses to form a proto-neutron star. Much less certain is its subsequent
evolution, the potential formation of a black hole, and the powering of a
supernova (SN) explosion, sometimes associated with a long-duration
$\gamma$-ray\ burst (LGRB). The situation is deceptively simple and the
outcome rests fundamentally on the solution to an energy problem. An
explosion or a fizzle depends on the efficiency with which the system
can extract the prodigious gravitational energy released during
collapse. There are two principal forms of energy at disposal. The
first one is the gravitational-binding energy liberated by the
collapsing star and in particular its degenerate core. The second is
its rotational energy (actually drawn from gravitational energy),
which is a function of the angular-momentum distribution and budget in
the progenitor star. Understanding how these two energy sources can
be channeled to power relativistic and non-relativistic ejecta in
core-collapse SNe and leave behind a neutron star, a fast-rotating
pulsar, a magnetar, or a black hole, has been the subject of numerous
studies and the source of much debate \citep{bethe:90, woosley:93,
herant:94, bhf:95, jankamueller:96, wheeler:00, kitaura:06,
buras:06a, burrows:06, murphy:08, nordhaus:10, pejcha:11,hanke:11,
takiwaki:11}.
Thermal MeV neutrinos are abundantly radiated from the
optically-thick, dense, and hot proto-neutron star, allowing its internal energy to
be released on a diffusion timescale. In the neutrino mechanism for
core-collapse SN explosions \citep{bethewilson:85}, the absorption in
the infalling mantle of $\sim$10\% of this neutrino flux may alone
lead to the revival of the stalled shock and the ejection of the
progenitor envelope with an asymptotic kinetic energy of up to 1\,B
($10^{51}\,\mathrm{erg}$) \citep{kitaura:06,buras:06b,buras:06a}. But
this generic mechanism should not, however, be the origin of the
larger explosion energies of $\sim$\,10\,B inferred for a small
fraction of core-collapse SNe. Instead, their scarcity calls for
exceptional circumstances, which seem intricately related to fast
progenitor-core rotation \citep{burrows:07b,takiwaki:11}. It is
probable that most stars contain some angular momentum at the time of
death, either because they did not lose it completely through the
combined effects of magnetic torques and stellar-wind mass loss
\citep{maeder:00, meynet:00, heger:00, hirschi:04, meynet:05,
hirschi:05, heger:05}, or because they gained it from a companion
star in a binary system \citep{wellstein:99,petrovic:05,cantiello:07}.
As the envelope collapses, the rotational energy increases.
During this process, the inner core
($\lesssim\,0.5\,M_\odot$) spins up by about three orders of magnitude and
remains in solid body rotation, while the outer core develops a
differentially-rotating profile \citep{ott:06spin}. The energy
associated with rotation can be large ($\mathcal{O}(10\,\text{B})$)
and tapped by instabilities developing at the surface of the proto-neutron star, in
particular the magneto-rotational instability (MRI;
\citealt{balbus:91,akiyama:03,obergaulinger:09}). Numerical
simulations for fast-rotating progenitor stars suggest that the
magneto-rotational mechanism of explosion is promising and offers a
very attractive explanation for the existence of highly-energetic SNe
\citep{leblanc:70, bisno:76, wheeler:00, yamada:04, moiseenko:06,
burrows:07b, dessart:08a, takiwaki:11}. However, this mechanism
relies fundamentally on the assumption that the MRI can increase the
magnetic field to the required values. An assumption that has not yet
been shown numerically in the full core-collapse context, although
preliminary investigations in this sector are promising
\citep{obergaulinger:09}. Hence, combined with the diversity of
progenitor-core properties, these two mechanisms alone, the neutrino
and the magneto-rotational mechanism, may explain the diversity of
core-collapse SNe, potentially encompassing two orders of magnitude in
explosion energy, from the low-luminosity SNe II-Plateau
\citep{pastorello:04} to highly energetic SNe Ic \citep{mazzali:02}.
A great puzzle is then to understand the necessary departures from
this general core collapse scenario to produce an LGRB in addition to
a SN explosion, as spectroscopically confirmed in, to date, six
LGRB/SNe pairs, (for a recent compilation, see \citealt{berger:11}).
This requires that $\sim$\,0.1-1\,B be injected into a low-mass,
baryon-deficient collimated region (at the origin of $\gamma$-rays) and that
at the same time $\sim$10\,B be injected quasi-isotropically to eject
the progenitor envelope (at the origin of the SN thermal emission
observed in the optical). The very low occurrence rate of LGRB/SN per
core-collapse SN of $\lesssim$\,1\% \citep{guetta:07} calls for
progenitor properties that are rarely encountered in star
formation/evolution. Interestingly, an unambiguous diversity emerges
among LGRB/SN observations, necessarily translating into a significant
range for the inferred properties of the SN ejecta,
with proposed masses and kinetic energies possibly varying by a factor
of 5--10 for both \citep{berger:11}. Unfortunately, a significant
uncertainty is associated with such inferences. For example,
\citet{iwamoto:98} propose an ejecta mass of 11\,M$_\odot$\ with a total
energy of 20--50\,B for GRB980425/SN1998bw, but \citet{woosley:99}
reproduce the light curve with an ejecta of $\sim$5\,M$_\odot$\ and a
total energy of 22\,B. Such differences are not surprising since
both spectra and light curves must be modeled simultaneously and with
allowance for the complicated non-LTE non-thermal and time-dependent
effects controlling the radiative transfer. The exceptionally fast
ejecta expansion of hypernovae is expected to strengthen the
time-dependent effects seen in ``standard" core-collapse SNe
\citep{dessart:08} while the large production of $^{56}{\rm Ni}$\ and
significant mixing may sizably affect line-profile shapes from which
the expansion rate is inferred \citep{dessart:12}.
Two LGRB central-engine models are currently favored. They suggest
the key components for a successful LGRB/SN are a compact progenitor
with a short light-crossing time of $\sim$\,1\,s and fast rotation at
the time of collapse. One is the collapsar model \citep{woosley:93,
macfadyen:99, macfadyen:01}: A fast-rotating progenitor fails to
explode in its early post-bounce phase and instead forms a black hole,
while the in-falling envelope eventually forms a Keplerian disk
feeding the hole on an accretion/viscous timescale comparable to that
of the LGRB. It is within this disk that the SN explosion is triggered
and the $^{56}{\rm Ni}$\ synthesized. The other model involves a
proto-magnetar \citep{wheeler:00, bucciantini:08, metzger:10c,
metzger:11} in which the LGRB is born after a successful SN
explosion (either by the neutrino or the magneto-rotational mechanism,
although the latter seems more likely given the rapid
rotation required for the magnetar) and the ejection of the overlying
envelope (or at least the onset of the ejection of the inner envelope
layers that clear the proto-neutron star\ surface). Fast rotation in the
proto-neutron star\ permits the huge enhancement of the magnetic-field energy and
stresses, which strengthen as the proto-neutron star\ cools and contracts,
eventually giving rise to relativistic ejecta ($\sim$\,10\,s after the
onset of collapse, once the neutrino driven wind decays away). In
both models, rotation is key to control the dynamics in a unique
way. It is also key to allow the simultaneous ejection of
baryon-deficient material at relativistic speeds over a small solid
angle and the quasi-spherical ejection of the progenitor envelope.
A critical difference between the collapsar model and the
proto-magnetar model is that the collapsar has to form a black hole.
Being so central to the model, it is legitimate to investigate what
conditions black-hole formation in this context
\citep{woosley:11a}. Surprisingly, little has been done on this
problem. Numerous simulations so far have focused on the early
collapse phase and the revival of the SN shock, stopping too early to
make any statement concerning black-hole formation. In 1D, several
studies have investigated the neutrino signal and progenitor
dependence of non-rotating failed SNe \citep{liebendoerfer:04,
sumiyoshi:07, fischer:09a, oconnor:11}. \citet{sekiguchi:11a} have
performed 2D simulations of black-hole formation and the subsequent
formation of an accretion disk, although with initial conditions that
are incommensurate with currently suggested LGRB progenitors. In 3D,
\cite{ott:11a} performed fully general-relativistic simulations of
black-hole formation, however using a simplified soft equation of
state that favored it. Finally, other simulations have started from a
pre-existing black hole\ and investigated the powering of the jet at the
origin of the LGRB \citep{aloy:00,proga:03,zhang:04,lindner:10}, or
the longer-term synthesis of $^{56}{\rm Ni}$\ in this unusual context
\citep{milosavljevic:12}. No simulation has ever demonstrated from
first principles, and thus convincingly, the validity of the collapsar
model, i.e., that the progenitors proposed for this model would indeed
proceed through each and every necessary step: Collapse, formation of
a proto-neutron star, failure of the shock revival, formation of a black hole\ followed by
that of a Keplerian disk, and finally the powering of both the LGRB
and the SN, including the synthesis of a generally large amount of
$^{56}{\rm Ni}$\ by core-collapse SN standards. This is an obvious shortcoming
of all theoretical investigations on the collapsar model and its
proposed progenitors. Similar gaps in the modeling of proto-magnetar
driven LGRBs exist: The early magneto-rotational core-collapse SN
evolution has been modeled in 2D \citep{burrows:07b,takiwaki:11} and
so has the phase in which relativistic outflows are driven
\citep{bucciantini:07,bucciantini:08,bucciantini:09,komissarov:07},
but the evolution connecting the two phases has not been modeled. The
robustness of the magneto-rotational explosion mechanism largely rests
on the efficiency of angular-momentum transport, and in particular the
extraction of the free-energy stored (and replenished through
accretion) in differential rotation at the surface of the
proto-neutron star. The failure to extract this energy on short time
scales may, however, facilitate black-hole formation, although it may
also compromise energy extraction in the collapsar model. These
complicated issues require detailed modeling to build upon the
promising results of \citet{thompson:05} and \citet{obergaulinger:09}.
To this day, the collapsar model has been studied more extensively
than the proto-magnetar model for LGRBs, so the later may look more
promising in some ways in part because of the lesser scrutiny it has
received.
In this paper, we focus on one important aspect of the collapsar model
to validate, or invalidate, the assumption, often made but so far
never checked, that the LGRB progenitor models available in the
literature indeed collapse to form a black hole. We do this by performing
hydrodynamical simulations of the LGRB progenitor models of
\citet{woosley:06} using the code \code{GR1D} \citep{oconnor:10}.
This issue is critical for testing the potential of progenitor stars
for producing LGRBs via the collapsar mechanism, but may also serve to
diagnose an attractive channel for the formation of
proto-magnetars. Such ``failed'' collapsars (because they explode
before forming a black hole) represent a serious alternative for the
production of LGRBs, although they have their own caveats
\citep{metzger:11}. Admittedly, the phenomenon of core collapse,
bounce, and the events that follow are fundamentally multi-D. We
believe, however, that much can be learned from 1D simulations of the
kind presented here. For example, the mass-accretion rate onto the
proto-neutron star\ is largely determined by the angular-averaged density profile of
the progenitor star, which we capture accurately. Our 1D exploration
reveals the landscape of core properties at bounce and quantifies
fundamental differences between progenitors. In the next section, we
start by reviewing results from stellar-evolution models for LGRB
progenitors. We then describe our methodology for the \code{GR1D}
simulations of LGRB progenitors available in the literature. We make a
short digression in Section~\ref{sect:rot} to discuss the rotational
properties of the collapsed cores of massive stars. In
Section~\ref{sect_results}, we present our results before concluding
in Section~\ref{sect_conc}.
\section{Stellar-evolution models of LGRB progenitors}
\label{sect_presn}
Stellar-evolution calculations have been performed to investigate the
mass, rotation, and metallicity requirements for producing
fast-rotating pre-collapse stars. Both single-
\citep{hirschi:05,yoon:05,woosley:06,georgy:09} and binary-star
\citep{petrovic:05,cantiello:07} evolutionary scenarios have been
investigated. Fast rotation of the proto-stellar core is clearly
critical to procure a large angular momentum to the star initially. If
the rotation rate attained is sufficiently large, the star may even
evolve chemically homogeneously and avoid a supergiant phase, which is
known to sap the core of its rotation through the effects of magnetic
torques. Such fast-rotating chemically-homogeneous stars also
naturally die as H-deficient He-poor Wolf-Rayet (WR) stars. Low
metallicity quenches the stellar-wind mass loss rate, a condition that
may be more important for a single-star scenario than for the
binary-star scenario \citep{yoon:06,cantiello:07}. While the general
outcome of these simulations is that it is possible to produce massive
stars with a rapidly spinning core/envelope at death, it is difficult
to compare the final properties of published models. Indeed, models
are rarely evolved all the way to an iron core. The treatments of
mixing, mass loss, and angular-momentum loss/transport
differ. Magnetic-fields may or may not be included and when they are
the prescription may differ \citep{spruit:02,zahn:07}.
Furthermore, all these studies remain speculative about the outcome of
collapse for such progenitors. They argue for black hole formation
and the formation of a disk based on order of magnitude estimates,
rather than detailed numerical simulations. For a start, of all the
LGRB progenitor simulations, only those of \citet{woosley:06} are
evolved all the way to the onset of collapse. In simulations halted
well before, the iron-core mass is estimated from the CO-core mass
\citep{hirschi:05} or is simply not considered in the discussion
\citep{yoon:05}. Most studies consider a model viable for producing a
collapsar based exclusively on the angular-momentum budget of the
inner 3\,M$_\odot$, and perform no checks on the likelihood of forming a
black hole: its formation is deemed so obvious that the discussion of any
alternate scenario is generally omitted.
Differing in their criteria and approaches for selecting collapsar
candidates, some studies may yield progenitor-mass ranges that do not
even overlap: Using an angular-momentum criterion,
\citet{yoon:05,woosley:06,yoon:06} favor progenitor stars with a
main-sequence mass below $\sim$30\,M$_\odot$. In contrast,
\citet{hirschi:05}, arguing for the need for both a large angular
momentum, a large iron core, and a WO stellar type at death, favor
progenitors with a main-sequence masses above $\sim$35\,M$_\odot$\
(magnetic fields are not treated in this work, though).
Recent studies suggest that selecting collapsar progenitors based
exclusively on a large angular-momentum budget may be too simplistic.
For example, the magneto-rotational explosion invoked to explain
hypernovae derives its energy from this same large core angular
momentum (via the MRI and the strongly differentially-rotating layers
in the post-shock region). This mechanism does not obviously
accommodate the formation of a black hole, as demonstrated by
\citet{dessart:08a}. They simulated the collapse of the core and the
development of a magneto-rotational explosion in model 35OC of
\citet{woosley:06} and found that rotational energy of order 10\,B is
readily available to launch a SN ejecta on a timescale of a few
100\,ms. They furthermore found that accretion is easily shut off by
the developing explosion and that the proto-neutron star\ mass fails to grow to the
instability threshold for black-hole formation. However, the
simulations of \citet{dessart:08a} did not resolve the MRI but instead
used an equipartition ansatz to estimate the magnitude of the
MRI-amplified magnetic fields. In reality, magnetic-field reconnection
may, for example, compromise the dynamical potential of magnetic
stresses, channeling magnetic energy into heat to be radiated away by
neutrinos. This and other alternatives have been studied by
\citet{thompson:05} under the general form of viscous
dissipation. They find that the extra energy deposition can, in some
cases, considerably alter the post-bounce dynamics and generate a
vigorous explosion.
\citet{oconnor:11} studied black-hole formation based on a variety of
progenitor models characterized by different main-sequence mass,
metallicity, and rotation rate. They find that the outcome of collapse
can be anticipated from the bounce compactness of the progenitor core
structure and in particular that of the region inside 2.5\,M$_\odot$,
which corresponds approximately to the maximum mass that a proto-neutron star\ can
have and remain in hydrostatic equilibrium. They find that higher
mass progenitors published in the literature do not always have larger
iron cores and therefore that they are not necessarily more prone to
black-hole formation. They also reveal considerable diversity in
progenitor core structure, even for the same main-sequence mass. Some
stellar evolution studies obtain a monotonic increase of the iron-core
mass (or bounce compactness; see Fig.~9 of \citealt{oconnor:11}) versus
main-sequence mass (\citealt{limongi:06}; see also
\citealt{hirschi:04,hirschi:05}), while some show an anti-correlation
beyond $\sim$40\,M$_\odot$\ \citep{woosley:07}. The primary
reason for this are differing prescriptions for rate and time of
mass loss, one of the major uncertainties in massive star evolution
(see also the discussions in \citealt{hirschi:05} and
\citealt{oconnor:11}).
\begin{deluxetable*}{l@{\hspace{-11mm}}cc|ccc|cccc|ccc|c}
\tabletypesize{\scriptsize}
\tablecolumns{14}
\tablewidth{0pc}
\tablecaption{Progenitor Model Properties}
\tablehead{Model\tablenotemark{a} & $\xi_{2.5}$ &
$\Omega_c$\tablenotemark{b} & $t_\mathrm{BH}$\tablenotemark{c} & $M_\mathrm{b, max}$\tablenotemark{d}
& $M_\mathrm{g, max}$\tablenotemark{e} &
$M_\mathrm{b,BH}^\mathrm{DF}$\tablenotemark{f} &
$a^\star_\mathrm{BH,DF}$\tablenotemark{g} &
$t_\mathrm{DF}$\tablenotemark{h} & $M_\mathrm{preSN} -
M_\mathrm{b,BH}^\mathrm{DF}$\tablenotemark{i} &
$\bar{\eta}_\mathrm{heat}^\mathrm{crit}$ & $P_\text{ref}$\tablenotemark{j} &
$I$\tablenotemark{k} &$F_\mathrm{rot}^\mathrm{100ms}$~~\tablenotemark{l} \\
&&(s$^{-1}$) & (s)&($M_\odot$) &($M_\odot$) & ($M_\odot$) & & (s) &
($M_\odot$) & & (ms) & ($10^{45}$\,g\,cm$^2$) & (B)}
\startdata
12SA & 0.003 & 0.000 & $\cdots$ & (1.56) & (1.47) & (10.9)\phm{0} & $\cdots$ & $\cdots$ & $\cdots$ & 0.167 & $\cdots$ & $\cdots$ & $\cdots$ \\
12SG & 0.239 & 0.198 & 2.728 & 2.33 & 2.12 & \phm{0}(7.57) & $\cdots$ & $\cdots$ & $\cdots$ & 0.124 & 18.1\phm{0} & 3.37 & 0.034 \\
12SH & 0.141 & 0.144 & $\cdots$ & (2.08) & (1.91) & \phm{0}(5.43) & $\cdots$ & $\cdots$ & $\cdots$ & 0.148 & 25.5\phm{0} & 3.35 & 0.008 \\
12SI & 0.075 & 0.208 & $\cdots$ & (1.75) & (1.64) & \phm{0}(6.95) & $\cdots$ & $\cdots$ & $\cdots$ & 0.153 & 24.7\phm{0} & 3.17 & 0.010 \\
12SJ & 0.121 & 0.751 & $\cdots$ & (2.05) & (1.90) & \phm{0}6.77 & 0.470 & \phm{0}91.6\phm{0} & \phm{0}2.27 & 0.166 & \phm{0}4.10 & 3.53 & 0.305 \\
\hline
12OA & 0.011 & 0.000 & $\cdots$ & (1.52) & (1.44) & (11.9)\phm{0} & $\cdots$ & $\cdots$ & $\cdots$ & 0.104 & $\cdots$ & $\cdots$ & $\cdots$ \\
12OG & 0.029 & 0.149 & $\cdots$ & (1.88) & (1.75) & \phm{0}4.50 & 0.217 & 4.42$\times10^6$ & \phm{0}7.32 & 0.179 & 22.5\phm{0} & 3.17 & 0.006 \\
12OH & 0.090 & 0.285 & $\cdots$ & (1.84) & (1.71) & \phm{0}7.62 & 0.210 & 710.\phm{00} & \phm{0}0.07 & 0.135 & 17.1\phm{0} & 3.30 & 0.020 \\
12OI & 0.095 & 1.061 & $\cdots$ & (1.86) & (1.73) & \phm{0}5.91 & 0.535 & \phm{0}68.8\phm{0} & \phm{0}3.81 & 0.181 & \phm{0}4.49 & 3.27 & 0.270 \\
12OL & 0.076 & 0.299 & $\cdots$ & (1.75) & (1.64) & \phm{0}6.99 & 0.259 & 554.\phm{00} & \phm{0}0.36 & 0.136 & 19.7\phm{0} & 3.61 & 0.017 \\
12ON & 0.170 & 1.709 & $\cdots$ & (2.21) & (2.02) & \phm{0}2.67 & 0.496 & \phm{00}8.51 & \phm{0}8.26 & 0.145 & \phm{0}2.38 & 3.27 & 1.013 \\
\hline
12TA & 0.008 & 0.000 & $\cdots$ & (1.59) & (1.49) & (12.0)\phm{0} & $\cdots$ & $\cdots$ & $\cdots$ & 0.117 & $\cdots$ & $\cdots$ & $\cdots$ \\
12TG & 0.034 & 0.148 & $\cdots$ & (1.91) & (1.77) & \phm{0}4.59 & 0.228 & 3.73$\times10^6$ & \phm{0}7.35 & 0.182 & 24.9\phm{0} & 3.43 & 0.007 \\
12TH & 0.107 & 1.042 & $\cdots$ & (1.93) & (1.79) & \phm{0}6.67 & 0.495 & \phm{0}85.2\phm{0} & \phm{0}2.56 & 0.138 & \phm{0}5.06 & 3.59 & 0.313 \\
12TI & 0.145 & 1.323 & $\cdots$ & (2.02) & (1.86) & \phm{0}3.33 & 0.507 & \phm{0}15.3\phm{0} & \phm{0}7.46 & 0.144 & \phm{0}3.26 & 3.48 & 0.610 \\
12TJ & 0.517 & 1.281 & 0.853 & 2.51 & 2.37 & \phm{0}2.97 & 0.640 & \phm{00}2.64 & \phm{0}8.58 & 0.191 & \phm{0}1.10 & 4.31 & 3.286 \\
\hline
16SA & 0.101 & 0.000 & $\cdots$ & (1.88) & (1.74) & (14.6)\phm{0} & $\cdots$ & $\cdots$ & $\cdots$ & 0.138 & $\cdots$ & $\cdots$ & $\cdots$ \\
16SG & 0.109 & 0.203 & $\cdots$ & (1.91) & (1.77) & \phm{0}9.79 & 0.404 & 2.71$\times10^7$ & \phm{0}2.16 & 0.141 & 20.8\phm{0} & 3.27 & 0.010 \\
16SH & 0.081 & 0.341 & $\cdots$ & (1.76) & (1.64) & \phm{0}(7.70) & $\cdots$ & $\cdots$ & $\cdots$ & 0.182 & 16.8\phm{0} & 2.68 & 0.019 \\
16SI & 0.380 & 0.189 & 1.132 & 2.38 & 2.20 & \phm{0}(9.85) & $\cdots$ & $\cdots$ & $\cdots$ & 0.158 & 11.9\phm{0} & 3.82 & 0.062 \\
16SL & 0.075 & 0.207 & $\cdots$ & (1.73) & (1.62) & \phm{0}(6.30) & $\cdots$ & $\cdots$ & $\cdots$ & 0.130 & 28.4\phm{0} & 3.31 & 0.007 \\
16SM & 0.121 & 0.229 & $\cdots$ & (2.02) & (1.85) & \phm{0}(8.31) & $\cdots$ & $\cdots$ & $\cdots$ & 0.145 & 20.0\phm{0} & 3.51 & 0.016 \\
16SN & 0.496 & 0.455 & 0.777 & 2.42 & 2.27 & \phm{0}9.45 & 0.508 & \phm{0}40.6\phm{0} & \phm{0}1.77 & 0.187 & \phm{0}3.25 & 3.69 & 0.451 \\
\hline
16OA & 0.144 & 0.000 & $\cdots$ & (2.15) & (1.96) & (15.8)\phm{0} & $\cdots$ & $\cdots$ & $\cdots$ & 0.133 & $\cdots$ & $\cdots$ & $\cdots$ \\
16OG & 0.193 & 0.176 & 3.437 & 2.33 & 2.11 & \phm{0}7.16 & 0.230 & 4.37$\times10^6$ & \phm{0}8.49 & 0.168 & 17.6\phm{0} & 3.82 & 0.018 \\
16OH & 0.185 & 0.248 & $\cdots$ & (2.21) & (2.00) & \phm{0}9.18 & 0.133 & 810.\phm{00} & 7.9$\times10^{-5}$ & 0.150 & 20.5\phm{0} & 3.59 & 0.023 \\
16OI & 0.344 & 0.733 & 1.449 & 2.37 & 2.19 & \phm{0}7.10 & 0.553 & \phm{0}18.2\phm{0} & \phm{0}5.11 & 0.152 & \phm{0}3.21 & 4.06 & 0.700 \\
16OL & 0.124 & 0.316 & $\cdots$ & (2.02) & (1.86) & \phm{0}8.63 & 0.200 & 593.\phm{00} & \phm{0}0.05 & 0.138 & 14.1\phm{0} & 3.41 & 0.030 \\
16OM & 0.172 & 1.059 & $\cdots$ & (2.17) & (1.98) & \phm{0}5.64 & 0.590 & \phm{0}24.5\phm{0} & \phm{0}6.31 & 0.177 & \phm{0}2.60 & 3.02 & 0.480 \\
16ON & 0.357 & 1.382 & 1.458 & 2.40 & 2.22 & \phm{0}3.36 & 0.582 & \phm{00}4.70 & 10.8\phm{0} & 0.162 & \phm{0}1.40 & 3.59 & 2.082 \\
\hline
16TA & 0.070 & 0.000 & $\cdots$ & (1.76) & (1.64) & (16.0)\phm{0} & $\cdots$ & $\cdots$ & $\cdots$ & 0.148 & $\cdots$ & $\cdots$ & $\cdots$ \\
16TG & 0.288 & 0.242 & 1.738 & 2.35 & 2.16 & 13.3\phm{0} & 0.366 & 3.44$\times10^4$ & \phm{0}2.41 & 0.174 & 10.4\phm{0} & 4.02 & 0.083 \\
16TH & 0.434 & 0.598 & 0.958 & 2.41 & 2.25 & \phm{0}8.01 & 0.511 & \phm{0}23.7\phm{0} & \phm{0}3.44 & 0.151 & \phm{0}2.57 & 3.80 & 0.599 \\
16TI & 0.242 & 1.367 & 2.791 & 2.41 & 2.21 & \phm{0}3.51 & 0.554 & \phm{0}10.6\phm{0} & 10.4\phm{0} & 0.150 & \phm{0}2.17 & 3.77 & 1.341 \\
\hline
35OA & 0.178 & 0.289 & $\cdots$ & (2.26) & (2.05) & (12.9)\phm{0} & $\cdots$ & $\cdots$ & $\cdots$ & 0.153 & 14.6\phm{0} & 3.53 & 0.044 \\
35OB & 0.537 & 1.545 & 0.776 & 2.42 & 2.25 & 16.4\phm{0} & 0.545 & \phm{0}31.5\phm{0} & \phm{0}4.80 & 0.198 & \phm{0}1.44 & 3.48 & 3.617 \\
35OC & 0.458 & 1.980 & 0.972 & 2.43 & 2.29 & \phm{0}4.44 & 0.622 & \phm{00}4.84 & 23.6\phm{0} & 0.162 & \phm{0}1.10 & 4.49 & 7.521 \\
\hline
HE16C & 0.137 & 0.133 & $\cdots$ & (2.06) & (1.90) & \phm{0}(5.15) & $\cdots$ & $\cdots$ & $\cdots$ & 0.134 & 25.5\phm{0} & 3.11 & 0.007 \\
HE16D & 0.283 & 0.440 & 1.706 & 2.35 & 2.16 & \phm{0}8.65 & 0.367 & 117.\phm{00} & \phm{0}0.88 & 0.171 & \phm{0}4.74 & 3.65 & 0.206 \\
HE16E & 0.129 & 1.428 & $\cdots$ & (2.00) & (1.85) & \phm{0}6.82 & 0.594 & \phm{0}43.1\phm{0} & \phm{0}6.05 & 0.153 & \phm{0}3.77 & 3.27 & 0.447 \\
HE16F & 0.496 & 1.096 & 0.837 & 2.48 & 2.33 & \phm{0}4.29 & 0.567 & \phm{00}8.97 & 10.5\phm{0} & 0.202 & \phm{0}1.46 & 4.28 & 2.323 \\
HE16H & 0.610 & 1.196 & 0.641 & 2.52 & 2.38 & \phm{0}4.12 & 0.597 & \phm{00}5.95 & 11.6\phm{0} & 0.204 & \phm{0}1.26 & 4.23 & 3.335 \\
HE16K & 0.132 & 0.134 & $\cdots$ & (2.04) & (1.88) & \phm{0}(5.16) & $\cdots$ & $\cdots$ & $\cdots$ & 0.140 & 26.6\phm{0} & 3.11 & 0.007 \\
HE16L & 0.316 & 0.315 & 1.497 & 2.36 & 2.17 & \phm{0}9.34 & 0.286 & 195.\phm{00} & \phm{0}0.24 & 0.165 & \phm{0}6.73 & 3.73 & 0.116 \\
HE16M & 0.111 & 1.206 & $\cdots$ & (1.91) & (1.77) & 10.4\phm{0} & 0.532 & \phm{0}94.0\phm{0} & \phm{0}2.61 & 0.134 & \phm{0}4.42 & 3.23 & 0.348 \\
HE16N & 0.198 & 1.203 & 3.424 & 2.36 & 2.15 & \phm{0}7.81 & 0.604 & \phm{0}35.9\phm{0} & \phm{0}7.15 & 0.154 & \phm{0}2.59 & 3.84 & 0.970 \\
HE16O & 0.298 & 1.209 & 1.891 & 2.39 & 2.20 & \phm{0}6.77 & 0.594 & \phm{0}25.0\phm{0} & \phm{0}8.86 & 0.127 & \phm{0}1.98 & 3.61 & 1.414 \\
HE16P & 0.573 & 1.038 & 0.672 & 2.49 & 2.35 & \phm{0}6.42 & 0.584 & \phm{0}14.2\phm{0} & \phm{0}9.46 & 0.235 & \phm{0}1.58 & 4.16 & 2.523
\enddata
\tablenotetext{a}{Model designation from \citet{woosley:06}. See text
for details.}
\tablenotetext{b}{Initial central angular velocity.}
\tablenotetext{c}{Time elapsed between bounce and black hole formation, $\cdots$ indicates that no black hole formed within 3.5\,s of bounce.}
\tablenotetext{d}{Baryonic mass of the proto-neutron star\ at the time of black hole formation. If no black hole forms in 3.5\,s, we give the proto-neutron star\ baryonic mass at 3.5\,s.}
\tablenotetext{e}{Gravitational mass of the proto-neutron star\ at the time of black hole formation. If no black hole forms in 3.5\,s, we give the proto-neutron star\ gravitational mass at 3.5\,s.}
\tablenotetext{f}{Baryonic mass interior to the innermost stable circular orbit at the time of disk formation. If the angular momentum is too low to foster disk formation, we give the progenitor mass in parentheses instead.}
\tablenotetext{g}{Dimensionless spin of the black hole when the disk forms. $\cdots$ indicates no disk forms.}
\tablenotetext{h}{Twice the free-fall time of the mass element at the innermost stable circular orbit. $\cdots$ indicates no disk forms.}
\tablenotetext{i}{Baryonic mass outside of the black hole at disk formation. $\cdots$ indicates no disk forms.}
\tablenotetext{j}{Rotational period $P_\text{ref}=2\pi/\bar{\omega}$ computed by appoximating $\bar{\omega}$ as
the ratio of the total angular momentum to the moment of inertia of the proto-neutron star, Eq.~\ref{eq:omegabar}. $\cdots$ indicates a non-rotating model.}
\tablenotetext{k}{Proto-neutron star moment of inertia, note this can be up to two times the value for a non-rotating cold neutron star.
This is the origin of the discrepancy with the values presented by \citet{woosley:06},
who consider a cold non-rotating 1.4\,M$_\odot$\ neutron star with a moment of inertia $I = 1.4\times10^{45}\,\mathrm{g}\,\mathrm{cm}^2$. $\cdots$ indicates a non-rotating model.}
\tablenotetext{l}{Free energy stored in differential rotation. This amounts
to the energy difference between the proto-neutron star\ we obtain with \code{GR1D} and the corresponding proto-neutron star\ with the same total angular
momentum and moment of inertia but assuming solid-body rotation. $\cdots$ indicates a non-rotating model.}
\label{tab:results}
\end{deluxetable*}
\section{Methods \& Initial Model Set}
\label{sect_method}
In this work, we use the open-source, spherically symmetric, general
relativistic, Eulerian hydrodynamics code \code{GR1D}
\citep{oconnor:10}. Rotation is included through a
centrifugal-acceleration term in the momentum equation --- this is the
most important dynamical feature of rotation relevant to core
collapse. However, \code{GR1D} cannot account for the associated
deviations from spherical symmetry nor any angular-momentum
redistribution. We select the equation of state (EOS) from
\citet{lseos:91} characterized by a nuclear incompressibility of
$220\,$MeV (hereafter referred to as the LS220 EOS). This EOS
provides the best match to both mass and mass-radius constraints from
observations and nuclear theory
\citep{demorest:10,oezel:10b,steiner:10,hebeler:10}. \code{GR1D} uses
an efficient neutrino leakage/heating scheme that qualitatively
reproduces the salient features of neutrino transport. We refer the
reader to \citet{oconnor:10,oconnor:11} for additional details on
\code{GR1D} and our methodology.
As described above, the only stellar-evolutionary models for LGRB
progenitors that are evolved until the onset of collapse are those
proposed by \citet{woosley:06}. We thus focus on their model dataset
for our investigation on the dynamics of the core-collapse SN
engine and the potential formation of a black hole in the collapsar
context. Using \code{KEPLER}, \citet{woosley:06} investigated a
rather narrow range of progenitor masses, but varied the initial
rotation rate (solid-body rotation is assumed initially) and
environmental metallicity from solar to 1\% solar (with an additional
tunable factor as low as 0.1 for the metallicity-dependent mass loss
rate, equivalent to a reduction in metallicity by a factor of
100 in their mass-loss prescription). Arguing that the inferred
mass of LGRB/SN ejecta known in 2006 is on the order of 10\,M$_\odot$, and since
higher-mass stars may lose too much angular momentum through stellar
winds (even at low metallicity), they focused primarily on lower-mass
progenitors, with main-sequence masses of 12 and 16\,M$_\odot$,\footnote{They
also perform simulations for 16-M$_\odot$\ helium cores and find
comparable outcomes.} with the exception of one 35\,M$_\odot$\ model
set.
We adopt the same nomenclature as for their 12-, 16- and 35-$M_\odot$
models. It comprises the model's main-sequence mass, followed by a
letter denoting the environmental metallicity (`S' for solar, `O' for
10\% solar, and `T' for 1\% solar). An additional letter is appended
to individualize the models done with different WR mass-loss rate
prescriptions, allowing or not for magnetic effects, and the total
angular momentum of the star. 16-$M_\odot$ helium models are denoted
by `HE16' followed by an individualizing capital letter.
In this work, we simulate the collapse and post-bounce evolution with
\code{GR1D} for all these progenitor models, with a primary focus on
determining their ability to produce the key features of the collapsar
model: A black hole\ together with a Keplerian disk. As we
discuss in the following section, black-hole formation is not obviously
guaranteed in any of these dying stars.
\section{Notes on Rotating Core Collapse}
\label{sect:rot}
Since LGRBs seem fundamentally related to rapid rotation, it is useful
to summarize a few facts and concepts related to the gravitational
collapse of rotating iron cores in massive stars. First, it is
reasonable to assume (which is borne out by simulations, e.g.,
\citealt{heger:05}) that the iron core, in its pre-collapse state,
will be approximately uniformly rotating. Such a solid-body rotation
corresponds to the lowest energy state at fixed total angular momentum
and will be assumed on a secular timescale by any rotating fluid that
has some means to redistribute angular momentum.
\begin{figure}[t!]
\includegraphics[width=0.97\columnwidth]{fig1.pdf}
\caption{Angular velocity $\Omega(r)$ versus radius $r$ at both the
pre-SN stage (dashed lines) and at core bounce (solid lines)
for selected models of \cite{woosley:06}. The inner homologously
collapsing core maintains its initial uniform rotation throughout
collapse. \label{fig:omega} }
\end{figure}
Rotating core collapse, even for the high pre-collapse rotation rates
of some of the potential LGRB progenitors that we consider in this
study, proceeds qualitatively in a very similar fashion to
non-rotating collapse as long as the ratio of the centrifugal
acceleration $a_\text{cent}$ to the gravitational acceleration
$a_\text{grav}$, is small,
\begin{equation}
\frac{a_\mathrm{cent}}{a_\mathrm{grav}} = \frac{\Omega^2(r) r}{G M(r) r^{-2}} = \frac{\Omega^2(r) r^3}{G M(r)} \ll 1\,\,.
\end{equation}
Due to angular-momentum conservation, the angular velocity behaves as
$\Omega(r) \propto r^{-2}$. $M(r)$ stays constant for a collapsing
mass shell, and, thus, the above ratio increases during collapse as
$r^{-1}$ and may potentially become large for small radii.
In the case of $a_\mathrm{cent}/ a_\mathrm{grav} \ll 1$, the
collapsing rotating iron core will behave like a non-rotating core and
separate into a subsonically collapsing inner core ($|v_r(r)| <
c_s(r)$) and a supersonically collapsing outer core ($|v_r(r)| >
c_s(r)$). The inner core exhibits a self-similar (homologous) velocity
profile, $v(r) \propto r$, until core bounce and shock formation
\citep{goldreich:80}. After core bounce, the inner core material forms
the core of the proto-neutron star\ and outer core material accumulates at its edge.
The mass of the inner core at bounce is typically $\sim$$0.5\,M_\odot$
for non-rotating cores, set by the EOS of relativistic electrons, the
trapped lepton number, core entropy, and gravity \citep{burrows:83}.
It increases monotonically, though slowly, with increasing pre-collapse
rotation rate. For cores that reach $a_\mathrm{cent} / a_\mathrm{grav}
\approx 1$, the mass of the inner core will be increased to $\gtrsim
0.7\,M_\odot$ \citep{dimmelmeier:08}.
Since the inner core is collapsing homologously, we can introduce
a homology parameter $\alpha(t)$, so that
\begin{equation}
r(t) = \alpha(t)\, r_0\,\,,\label{eq:selfsimilar}
\end{equation}
where $r(t)$ is the radius of a collapsing fluid element at time $t$
and $r_0$ is its initial radius. This must hold for any mass shell
within the inner core. Two collapsing fluid elements
located initially at $r_1$ and $r_2$, and rotating with a frequency
$\Omega_1 = j_1/r_1^2$ and $\Omega_2 = j_2/r_2^2$ conserve angular
momentum, $\Omega'_1 {r'_1}^2 = j'_1 = j_1 = \Omega_1r_1^2$.
Homology implies $r'_1 = \alpha(t) r_1$, and $r'_2 = \alpha(t) r_2$, therefore
\begin{equation}
\frac{\Omega_1'(r_1')}{\Omega'_2(r'_2)} = \frac{j_1 / {r'_1}^2}{j_2/{r'_2}^2} =
\frac{j_1}{j_2}\frac{\alpha(t)^2 r_2^2}{\alpha(t)^2 r_1^2} = \frac{\Omega_1}{\Omega_2} \,\,.
\end{equation}
Since this property holds for any mass shell within the inner core,
the rotational profile must be preserved under homologous collapse. In
practice, because the pre-collapse cores of massive stars are always
in solid-body rotation, so are the inner cores of the proto-neutron star\ at bounce.
Outside of the homologously collapsing core the self similar relation
of Eq.~\ref{eq:selfsimilar} does not hold. Most generally, gradients
in the rotation rate develop due to the underlying density gradients
in the hot postshock region and in the supersonically infalling region
ahead of the accretion shock.
Based on these arguments, we expect an early post-bounce rotational
profile that is approximately uniform within the inner
$0.5-0.7\,M_\odot$ (out to $\sim$$10-15\,\mathrm{km}$ in radius) and
strongly differential at larger radii. This is confirmed by
Fig.~\ref{fig:omega} which shows $\Omega(r)$ at bounce as obtained
with \code{GR1D} for a variety of models considered in this study. We
also show the rotational profile at the onset of collapse. This result
is not entirely new, but has previously been pointed out by
\cite{ott:06spin} in the context of 1D and 2D rotating core collapse
simulations.
Since uniform rotation is the lowest energy state, the \emph{shear
energy} of differential rotation is to be interpreted as a free
energy that will be tapped by any process (e. g., nonaxisymmetric
rotational shear instabilities, viscosity, or the MRI) capable of
redistributing angular momentum. Viscosity would lead to additional
heating in the postshock region to enhance the neutrino mechanism
\citep{thompson:05} while the MRI action could strengthen the magnetic
fields, driving bipolar outflows in the magneto-rotational mechanism
\citep{burrows:07b}.
In our simulations, we estimate the available free energy of
differential rotation by computing the difference in rotational energy
of the proto-neutron star\ model in \code{GR1D} and the rotational energy of a
uniformly spinning proto-neutron star\ of the same angular momentum and moment of
inertia,
\begin{equation}
F_\text{rot} = T - \frac{I\bar{\omega}^2}{2}\,,
\label{eq:freeenergy}
\end{equation}
\noindent
where from \cite{oconnor:10}, for \code{GR1D},
\begin{eqnarray}
T &=& {4 \pi \over 3} \int_0^{R_\text{PNS}} \rho h X W^2 v^2_\varphi r^2 dr\,\,,\\
I &=& {8 \pi \over 3} \int_0^{R_\text{PNS}} \rho h X W^2 r^4 dr\,\,,\\
\bar{\omega} &= & \int_0^{R_\text{PNS}} \rho h X W^2 rv_\varphi r^2 dr \bigg/
\int_0^{R_\text{PNS}} \rho h X W^2 r^4 dr\label{eq:omegabar}
\end{eqnarray}
\noindent
and $T$ is the rotational energy, $I$ is the moment of inertia, and
$\bar{\omega}$ is the uniform rotation frequency, $h$ is the specific
enthalpy, $X^2$ is the $g_{rr}$ component of the metric, $W$ is the
Lorentz factor, and $v_\varphi$ is the angular velocity. We take
$R_\text{PNS}$ to be the radius where the matter density, $\rho =
10^{10}\,$g\,cm$^{-3}$.
\section{Results}
\label{sect_results}
We have performed simulations for the entire set of \code{KEPLER}
models published in \citet{woosley:06}\footnote{\url{Models are
available from http://homepages.spa.umn.edu/\textasciitilde
alex/GRB2/}}. We first consider the rapidly spinning progenitors
evolved without magnetic fields. All these models have a
dimensionless Kerr spin ($a^\star = Jc/GM^2$) at 3\,M$_\odot$\ greater
than unity (with the exception of model HE16J, which has $a^\star =
0.91$) and are thus considered as promising collapsar candidates by
\citet{woosley:06}. Unfortunately, when evolved with \code{GR1D}, the
collapsing iron core of all such models halts its collapse and expands
--- these models do not experience core bounce within a few seconds of
evolution in \code{GR1D}. We associate this problem with the neglect
of the centrifugal acceleration in the momentum equation in
\code{KEPLER}, an approximation that fails in the fastest rotating
models. This term is included in \code{GR1D}. The mismatch suggests
that their fastest models may be significantly affected by the
addition of this term. Even if they did collapse, it is not clear that
such extremely fast rotating cores would avoid a centrifugal bounce.
In the remainder of this paper, we thus limit our discussion to models
evolved with magnetic fields and therefore subject to magnetic torques
during their evolution. Of the 46 models that fulfill this criterion,
we identify 4 additional models (12OM, 16TJ, 35OD, and HE16G) that do
not collapse but instead expand when restarted with \code{GR1D}. We
exclude these as well from our study. Finally, for reference and
completeness, we include the non-rotating models associated with each
series (12SA, 12OA, 12TA, 16SA, 16OA, 16TA), making a total of 48
models. Each simulation is continued after core bounce until a
black hole\ forms or until a time of $3.5\,$s has passed, whichever
comes first. We present the results for these 48 models evolved with
magnetic fields in Table~\ref{tab:results} (for the table layout, we
group models in bundles first of increasing mass, then of decreasing
metallicity, and finally in alphabetical order which generally
corresponds to an increased initial rotation rate).
As advocated by \citet{oconnor:11}, the bounce compactness is a robust
quantity for diagnosing the propensity to black-hole formation, which
is largely determined by the spatial extent encompassed by the
2.5\,M$_\odot$\ Lagrangian mass coordinate at core bounce in the
progenitor core. To avoid introducing biases associated with the
non-uniform conditions in the progenitor simulations (\code{KEPLER}
models are not all evolved to the same central density on their
collapse trajectory) this compactness is unambiguously evaluated at
the time of bounce. The formal definition of this core compactness is
\begin{equation}
\xi_{M} = {M / \,M_\odot \over R(M_\mathrm{bary} =
M) / 1000\,\mathrm{km}}\Big|_{t =t_{\mathrm{bounce}}}\,,\label{eq:bouncecompactness}
\end{equation}
where we take $M=2.5\,M_\odot$.
$R(M_\mathrm{bary}=2.5\,M_\odot)$ is the radial coordinate that
encloses 2.5\,$M_\odot$ of baryonic material at the time of core bounce
\citep{oconnor:11}.
Our simulations first demonstrate that most of the models have a small
core compactness $\xi_{2.5}$. \citet{oconnor:11} argues that a
compactness of 0.45 represents a threshold value, for the neutrino
mechanism, since above it an unrealistic neutrino-heating
efficiency is required to prevent black-hole formation. We further
confirm this by determining the critical heating efficiency for the
models in Table~\ref{tab:results} via the same procedure as in
\citet{oconnor:11}, to which we refer the reader for full details. We
note that this criterion is for explosions via the neutrino mechanism
and therefore neglects any magneto-rotational contribution to the
powering of an explosion \citep{dessart:08a}, so this threshold value
is probably a lower limit. We also stress that this criterion is
based on spherically symmetric simulations with an efficient, but
crude, approximation to neutrino transport. Therefore we
suggest caution when attempting to interpret the outcome of a model
based solely on the bounce compactness, since the adopted threshold
value for BH formation is a semi-quantitative estimate based on
where the slope of the required heating efficiency begins to clearly
increase (see the discussion in \citealt{oconnor:11}). When
plotting the critical heating efficiency of the non-rotating solar
metallicity stars from \citet{woosley:07} determined in
\citet{oconnor:11}, together with the generally fast-rotating
progenitors of \citet{woosley:06}, we find that both datasets in fact
overlap for the most part (Fig.~\ref{fig:eta_xi}). In other words, in
terms of compactness, most of these progenitors are similar to garden
variety, low-mass, non-rotating, progenitors and do not seem to have
any more reason to form a black hole\ than, e.g., the RSG progenitors
expected to produce SNe II-Plateau. As shown in
Table~\ref{tab:results}, provided no explosion is launched, half of
these models have not formed a black hole\ after 3.5\,s, and only $\sim$15\%
do within $\sim$1\,s. We note that this result is not so surprising
given the small iron-core mass ($\sim 1.4\,M_\odot$) of most
\citet{woosley:06} models (see their Tables~1 and 2). The first
conclusion from our work is therefore that most of the models
presented here are rather unlikely to form a black hole and thus may
fail in a very fundamental way to produce a collapsar, irrespective of
their angular-momentum budget.
\begin{figure}[htbp]
\includegraphics[width=0.97\columnwidth]{fig2.pdf}
\caption{ Illustration of the critical heating efficiency
$\bar{\eta}_\mathrm{heat}^\mathrm{crit}$ versus bounce compactness
$\xi_{2.5}$ for our \code{GR1D} simulations of the
\citet{woosley:06} models, whose properties are summarized in
Table~1 (blue diamonds). For comparison, we overplot the same
quantity for the standard non-rotating core-collapse SN progenitor
models of \citet{woosley:07} evolved at solar metallicity (red
squares). For the most part, the two distributions overlap,
suggesting that the propensity to black-hole formation and explosion
is comparable for both. Only models with the fastest rotation rates
achieve a larger compactness in excess of 0.4--0.5, but these may
then be diverted from black-hole formation through an early
magneto-rotational explosion.
\label{fig:eta_xi}
}
\end{figure}
Within each sequence presented in Table~\ref{tab:results}, the models
that form a black hole\ within 3.5\,s of core bounce, and thus at least in
principle susceptible to form a collapsar, are the faster rotating
ones characterized by very weak mass-loss rates. These properties
conspire to produce larger CO cores, more typical of more massive
stars that do not rotate. In the following discussion, we group these
models into several categories.
The first category are models which obviously do not give rise to a
LGRB, either by the standard Type-I collapsar or the proto-magnetar
mechanism, because they contain too little angular momentum.
Optimistically assuming a failed core-collapse SN, which is
unlikely given the modest values of $\xi_{2.5}$, models 12SG
($\xi_{2.5} = 0.239$), 16OG ($\xi_{2.5} = 0.193$), 16SI ($\xi_{2.5} =
0.380$), and 16TG ($\xi_{2.5} = 0.288$) possess too
little angular momentum in the remainder of the star to form a disk
about the central black hole within $10^6$\,s of collapse. This
behavior is reflected by the stellar type at the time of death, i.e. a
BSG star for model 12SG and a RSG star for models 16OG and 16TG, only
16SI is a WR star at the time of death. Quantitatively, this can be
further inspected in Table~\ref{tab:results} where we include the disk
formation time, the black hole mass and spin at that time, and the mass
exterior to the disk. We define disk formation
to be when the accreting material will first be supported at the
innermost stable circular orbit\ about a black hole with the enclosed mass and angular momentum
using the formulae of \citet{bardeen:72}. We estimate the disk
formation time as twice the free fall time of the innermost mass element
that reaches a Keplerian velocity \citep{oconnor:11, burrows:86bh}.
\begin{equation}
t_\text{DF} = 2\times \pi \sqrt{\frac{[r_\text{pre-SN}(M_\text{disk})]^3}{8GM_\text{disk}}}
\end{equation}
\noindent
where $r_\text{pre-SN}(M_\text{disk})$ is the radius of the
disk-forming Lagrangian mass element in the pre-SN model. If no such
mass element exists, no disk will form. In this case we include,
instead of the enclosed black-hole mass, the total pre-SN stellar mass
in parentheses. In the four models mentioned above, either no disk
forms or the disk formation time is $\gtrsim 10^6\,$s.
Additionally, we can discuss the potential for these models to form a
LGRB via the proto-magnetar model. Using
Eqs.~\ref{eq:freeenergy}~-~\ref{eq:omegabar}, we calculate the free
energy available in differential rotation at 100\,ms after bounce. We
also calculate a reference spin period ($P_\text{ref}
=2\pi/\bar{\omega}$), measured at the onset of the neutrino driven
explosion, by assuming solid-body rotation for the entire proto-neutron star\ with
the same total angular momentum and moment of inertia. The
corresponding values are given in Table~\ref{tab:results}. In
Fig.~\ref{fig:freeenergy_period}, we show the free energy available in
differential rotation at 100\,ms and the spin period of the proto-neutron star\ at
the onset of explosion. The total rotational energy of the proto-neutron star,
estimated as $I\bar{\omega}^2/2$, will increase as the proto-neutron star\ cools and
contracts. In models 12SG, 16OG, 16SI, and 16TG, which
are contained within the green (lightest shade) box of
Fig.~\ref{fig:freeenergy_period}, $\lesssim\,$0.1B of free energy
could be extracted from differential rotation via the MRI and
converted to explosion energy, much less than is needed for a
magneto-rotational explosion. Also, the proto-neutron star\ spin periods are
$\gtrsim$\,10\,ms,\footnote{Even taking into account the spin up due
to the PNS cooling and contraction, which will decrease the moment
of inertia from the value in Table~\ref{tab:results} to $\sim 0.4
M_\text{PNS} R_\text{PNS}^2 \sim
1.6\times10^{45}(M/1.4\,M_\odot)(R/12\,\text{km})^2$
\citep{metzger:11}, or roughly a factor of 2, the spin periods are
$\gtrsim 5\,$ms.} significantly larger than the $\lesssim 2\,$ms
periods required for the proto-magnetar model to reproduce classical
LGRB energies \citep{metzger:11}.
\begin{figure}[htbp]
\includegraphics[width=0.97\columnwidth]{fig3.pdf}
\caption{Reference proto-neutron star\ spin period, $P_{\rm ref}$, taken at the
onset of explosion (left axis, blue dots; Eq.~\ref{eq:omegabar}) and
the free energy stored in differential rotation 100\,ms after bounce
$F_\mathrm{rot}^\mathrm{100ms}$ (right axis, red stars;
Eq.~\ref{eq:freeenergy}) versus bounce compactness $\xi_{2.5}$ for
all rotating models in Table~\ref{tab:results}. While models with a
low bounce compactness show a diversity in core-rotation properties,
those with a high bounce compactness systematically have short spin
periods and a large budget of free energy stored in the differential
rotation. Shaded boxes refer to specific groupings of
models discussed in the text. Using $\xi_{2.5} > 0.45$ as a
black-hole formation criterion for non-rotating progenitors, we can
qualitatively compare the reference spin periods of this figure to
\citet{metzger:11}, who sketches the outcome of collapse as a
function of progenitor spin and mass. From this, one would predict
that none of the LGRB progenitor models studied here formed black
holes.}
\label{fig:freeenergy_period}
\end{figure}
The second category are models with a larger angular-momentum budget
but unfavorable bounce compactness. Although compact enough to lead to
black-hole formation within 3.5\,s of core bounce, we find that the
predicted critical heating efficiencies are similar to that expected
for a standard 15\,M$_\odot$\ non-rotating RSG progenitor star
\citep{oconnor:11}. These properties make them unlikely collapsar
progenitors, but in contrast, make them ideal candidates for
proto-magnetar formation, and perhaps LGRBs through that channel.
These models include 16OI ($\xi_{2.5} = 0.344$), 16ON ($\xi_{2.5} =
0.357$), 16TH ($\xi_{2.5} = 0.434$), 16TI ($\xi_{2.5} = 0.242$), HE16D
($\xi_{2.5} = 0.283$), HE16L ($\xi_{2.5} = 0.316$), HE16N ($\xi_{2.5}
= 0.198$), and HE16O ($\xi_{2.5} = 0.298$) and are
contained in the orange (medium shade) box of
Fig.~\ref{fig:freeenergy_period}. In addition to having critical
heating efficiencies similar to what is needed to explode typical
low-mass massive stars, the free energy available in rotation is
$\mathcal{O}$(1\,B). This energy may be converted to explosion energy
via the magneto-rotational mechanism. The spin period of these proto-neutron star s
is in the range 1-6\,ms, thus on the order of what is needed for the
proto-magnetar model of LGRBs \citep{metzger:11}.
Eventually, the fastest rotating progenitor models evolved with a
strongly inhibited stellar-wind mass loss represent more suitable
collapsar candidates, although each model has caveats. This set is
contained in the purple (darkest shade) box of
Fig.~\ref{fig:freeenergy_period} and includes models 12TJ ($\xi_{2.5}
= 0.517$), 16SN ($\xi_{2.5} = 0.496$), 35OB ($\xi_{2.5} = 0.537$), and
35OC ($\xi_{2.5} = 0.458$). Model 12TJ will form a
2.37\,M$_\odot$\ (gravitational mass) black hole\ 0.85\,s after core bounce,
followed by a Keplerian disk after 2.64\,s, with a potential ejecta
mass of 8.57\,M$_\odot$. However, much like the models in the previous
category, model 12TJ has $\sim$3\,B of free energy available in
rotation that may lead to a magneto-rotational explosion early-on,
preventing collapsar formation. This model is evolved at 1\% solar
metallicity, with an additional mass-loss rate scaling of 0.1,
equivalent to an overall evolution at $10^{-4}$ solar metallicity,
much below that observed for LGRB/SN sites. We find that models
HE16F, HE16H, and HE16P have similar characteristics to model 12TJ.
Model 16SN forms a 2.27\,M$_\odot$\ black hole\ 0.78\,s after bounce. Being
evolved at an effective metallicity of 0.01 solar, it has a lower
angular-momentum budget at death and is thus more likely to avoid a
magneto-rotational explosion. However, it forms a Keplerian disk only
40.6\,s after core bounce, with only 1.77\,M$_\odot$\ left over for the SN
ejecta. Such characteristics might in fact be more amenable to
reproduce recent observations of LGRB/SNe characterized by a very
early and narrow light-curve peak, as witnessed for example for
GRB100316D/SN 2010bh \citep{chornock:10}. They may even explain why
no SN is found in association with some nearby LGRBs \citep{fynbo:06}.
Finally, models 35OB and 35OC form a black hole within 0.78 and
0.97\,s of bounce, respectively. Model 35OB will accrete
$\sim$\,16.4\,M$_\odot$\ before a Keplerian disk forms $\sim$31.5\,s after
the onset of collapse, 4.8\,M$_\odot$\ is then available for the SN
ejecta. With the 35OC model, a disk forms very quickly after
collapse, in 4.8\,s, and a significant amount of mass is exterior to
the disk, $23.6M_\odot$, and thus much too large to accommodate
inferred LGRB/SN ejecta masses. However, the propensity to collapsar
formation of the 35OB and 35OC model may be ill-founded if the MRI is
successful at powering a magneto-rotational explosion. The free energy
available in rotation is huge, i.e.\ on the order of 4-7.5\,B. In
fact, in the 2D magneto-hydrodynamic simulations of
\citet{dessart:08a} based on the 35OC model, it was found that,
despite the large progenitor compactness, a magneto-rotational
explosion was initiated $\sim$200\,ms\ after core bounce and that the
proto-neutron star\ mass decreased thereafter, never reaching the mass threshold for
black-hole formation. In our models, the protoneutron stars in models
35OB and 35OC have $\sim 30-70\,$B of total rotational energy at the
onset of explosion, amply matching the inferred energies of observed
hypernovae. These inferences are based on the assumption that energy
extraction from the differentially-rotating layers at the
proto-neutron star surface is efficient and can power an
explosion. Failing to do so, black-hole formation would result,
although the question of energy extraction from the disk for the
powering of a GRB would then arise. Detailed multi-dimensional
core-collapse simulations need to be carried out to investigate the
efficiencies of magnetic/rotational/hydrodynamical instabilities for
the transport of angular momentum and the extraction of rotational
energy.
\section{Discussion}
\label{sect_conc}
In this paper, we have performed 1D general-relativistic
hydrodynamical simulations with \code{GR1D} of the collapse, bounce,
and post-bounce phases of the LGRB candidates of \citet{woosley:06} to
investigate their propensity to black-hole and disk formation. We find that
these progenitors are at odds with the proposed criteria for a
collapsar progenitor or with the inferred properties of observed
LGRB/SNe, namely a H-deficient He-poor WR star with a massive iron
core (equivalent to a large compactness), a large angular momentum to
form a Keplerian disk soon after black-hole formation, an ejecta of
$\sim$10\,M$_\odot$, and an evolution at about 0.1 solar metallicity. A
critical aspect that we focus on in this study is the compactness of
the progenitor cores at bounce, a quantity that helps diagnose the
likelihood of black-hole formation.
We group the \citet{woosley:06} models in different categories
according to their suitability for producing collapsars:
\begin{enumerate}
\item Models with a dimensionless Kerr spin parameter greater
than unity at an enclosed mass of 3\,M$_\odot$, i.e. the models
identified by \citet{woosley:06} as having the best potential for
collapsar formation, fail to collapse when evolved with \code{GR1D}.
Their cores are so fast spinning that the associated centrifugal
acceleration leads them into expansion. All these models are evolved
until death in \code{KEPLER} without magnetic fields and centrifugal
forces, which seems questionable given the unrealistically short
spin periods at collapse.
\item Models evolved with magnetic fields produce much lower
rotation rates and most collapse with \code{GR1D}.
\item Of those evolved with magnetic fields, models with moderate
rotation produce progenitors with a small compactness comparable to
that characterizing the low-mass massive-star models proposed as
progenitors of garden-variety core-collapse SNe. A small fraction of
these is endowed with sufficient angular momentum to make a
proto-magnetar, and thus a potential channel for producing LGRBs.
\item A few models (12TJ, 16SN) with the fastest rotation possess a
large compactness favorable for black-hole formation and sufficient
angular momentum for the formation of a Keplerian disk, but they
require evolution at metallicities in the range 0.0001-0.01\,Z$_{\rm
sol}$, significantly lower than the metallicity of a few tenths
solar or even higher at which these LGRB/SNe are found
\citep{modjaz:08,levesque:10a,levesque:10b}. Although in many
respects very attractive for forming a collapsar (if we ignore its
huge core angular momentum), model 35OC is characterized by a large
envelope mass of $\sim$23\,M$_\odot$, which is a factor 2-10 times
larger than the inferred ejecta mass of LGRB/SNe discovered so far
(for a summary, see \citealt{berger:11}). We note that these models
have a large angular momentum in the core, as models in the previous
category, and may thus experience a magneto-rotational explosion
preventing collapsar formation.
\end{enumerate}
Our quantitative study thus spells out the various
shortcomings of these progenitor stars for producing collapsars. Even
in those models that have the right compactness for black-hole formation and
sufficient angular momentum for disk formation, it is still unresolved
today how they would avoid the magneto-rotational mechanism of
explosion that is used to explain hypernovae \citep{leblanc:70, bisno:76,
wheeler:00, yamada:04, moiseenko:06,burrows:07b, dessart:08a, takiwaki:11}
The difficulty of forming a black hole\ and avoiding a magneto-rotational explosion
in fast-rotating cores, at least in the models of \cite{woosley:06}, lends credence
to the proto-magnetar model of LGRB/SNe.
Uncertainties in mass loss at low metallicity, and in particular
during transient phases of dynamical mass loss as observed in some
Luminous Blue Variable stars, is an issue, since it may completely
dominate the mass lost in the form of a weaker, but secular,
steady-state wind \citep{owocki:04}. This uncertain mass-loss rate
plagues more severely the evolution of higher-mass stars, since
15-20\,M$_\odot$\ stars stay further away from the Eddington limit, and
overall lose little mass, even at solar metallicity. By what
mechanism, at what rate, and during what phases a
100\,M$_\odot$\ star loses mass (and angular momentum) is much less well
known and this directly conditions the final mass and iron core mass
at collapse.
Overall, this suggests that studies of collapsar progenitors
would benefit from a second look. Angular momentum is key in the
current collapsar and proto-magnetar models, but there is a stiff
requirement on the progenitor compactness to speculate on its
propensity for forming a black hole, and thus for producing a LGRB through
one or the other channel. A major step forward in resolving those
issues would be to conduct massive-star evolution with rotation,
centrifugal force, and magnetic fields always all the way to the
formation of a degenerate neutronized core on the verge of
collapse. This would allow a straightforward comparison of results
between groups, and an easy determination of the compactness using
\code{GR1D} to test the suitability of the core for black-hole
formation.
The ultimate check on the collapsar model requires multi-dimensional
simulations covering the whole evolution from progenitor collapse,
bounce, failed explosion during the proto-neutron star\ phase, formation of a
black hole\ followed by the formation of a Keplerian disk, and the powering
of a $\sim$\,10\,B SN ejecta. As we emphasize, black-hole
formation is perhaps one of the most difficult steps in this
sequence of events, and in that respect, renders the proto-magnetar
channel quite attractive for the production of hypernovae and LGRBs.
The diversity of LGRB/SNe, the existence of SN-less LGRBs and of
LGRB-less hypernovae, may in fact call for a variety of formation
channels for these rare events, including both collapsars and
proto-magnetars.
\section*{Acknowledgements}
We acknowledge fruitful discussions with R.~Hirschi, A.~Beloborodov,
and T.~Piro. We also thank Stan Woosley for his comments on a draft
version of this paper. This research is supported in part by the
National Science Foundation under grand Nos.\ AST-0855535 and
OCI-0905046 and by the Sherman Fairchild Foundation. E.O. is
supported in part by a post-graduate fellowship from the Natural
Sciences and Engineering Research Council of Canada (NSERC). The
computations were performed at Caltech's Center for Advanced Computing
Research on the cluster ``Zwicky'' funded through NSF grant
no.\ PHY-0960291 and the Sherman Fairchild Foundation.
|
{
"timestamp": "2013-01-31T02:01:03",
"yymm": "1203",
"arxiv_id": "1203.1926",
"language": "en",
"url": "https://arxiv.org/abs/1203.1926"
}
|
\section{Introduction}
Efficient data gathering in sensor networks has been the topic of many research projects where different applications have been considered.
One of the concerns in data gathering is to take care of \textit{inter-node redundancy} during the transmission.
When the knowledge of inter-node dependency is known at the encoders (\textit{i.e.} sensor nodes) and the decoder node(s), distributed source coding \cite{xiong2004distributed} and optimal packet forwarding is the best transmission method, in terms of achieved information rates \cite{NCCorr_NIFwithCOrrSo}.
However, flexibility and robustness to network changes, and
no need (et the encoders) to the knowledge of inter-node dependency has drawn attention to random linear network coding \cite{NC_RLNCtoMulticast} as an alternative transmission method \cite{Ho04networkcoding}.
Recently, the concepts of \textit{compressed sensing} \cite{1614066} have been used to perform an embedded distributed source coding in linear network coding of correlated or sparse messages
\cite{rabbat,CdataGathering,sFeiz,feizi2011power}.
Joint source, channel, and network coding is studied in \cite{sFeiz,feizi2011power}, where analogue network coding \cite{katti2007embracing} is used as a linear mapping to decrease temporal and spatial redundancy of sensor data.
In \cite{naba}, we proposed Quantized Network Coding (QNC) with $\ell_1$-min decoding , where the sparse messages can be recovered from smaller number of packets compared to the conventional linear network coding \cite{NC_RLNCtoMulticast}.
To guarantee robust $\ell_1$-min recovery of messages from an under-determined set of linear measurements, the total measurement matrix has to be appropriate (or in other words satisfy some special properties).
For instance, if it satisfies Restricted Isometry Property (RIP) of appropriate order, then $\ell_1$-min recovery is feasible \cite{candes,LinProg}.
However, the literature of compressed sensing-based network coding does not include any result discussing theoretical (or even practical) requirements for robust $\ell_1$-min recovery of linear network coded messages.
In this paper, we discuss theoretical guarantees for $\ell_1$-min decoding of quantized network coded messages, based on RIP.
Specifically, we discuss the satisfaction of RIP and its implications for the measurement matrix, resulting from the design of local network coding coefficients, proposed in \cite{naba}.
The description of data gathering scenario and formulation of our proposed quantized network coding \cite{naba} is presented in section~\ref{sec:QNC}.
This is followed by a discussion on choosing appropriate local network coding coefficients, which result in zero mean Gaussian entries for the measurement matrix, in section~\ref{sec:DesignNCodes}.
In section~\ref{sec:TailProbRIP}, we derive the relation between the tail probability of $\ell_2$-norms and satisfaction of RIP, and discuss satisfaction of RIP for our designed measurement matrices.
In section~\ref{sec:Numerical}, a numerical example is presented, which compares the measurement matrix, resulting from our QNC scenario with the case of perfect Gaussian measurement matrix.
Finally, in section~\ref{sec:Conclusions}, we discuss our concluding remarks on satisfaction of RIP in our QNC scenario.
\section{Quantized Network Coding with $\ell_1$-min Decoding in Lossless Networks}\label{sec:QNC}
In this paper, we consider a lossless sensor network, represented by a directed graph, $\mathcal{G}=(\mathcal{V},\mathcal{E})$, where $\mathcal{V}=\{1,\ldots,n\}$ is the set of its nodes.
$\mathcal{E}=\{1,\ldots,|\mathcal{E}|\}$ is also the set of edges (links), where each edge, $e \in \mathcal{E}$, maintains a lossless communication from $tail(e)$ node to $head(e)$ node, at a maximum rate of $C_e$ bits per link use.
As a result, the input content of edge $e$ at time $t$, represented by $y_e(t)$ (since the links are lossless, input and output contents of each edge are the same), is from a discrete finite alphabet of size $2^{L C_e}$.
Time index, $t$, is integer and a time unit represents the time in which blocks of length $L$ are transmitted over all edges.
The sets of incoming and outgoing edges of node $v$, are defined respectively:
\begin{eqnarray}
\textit{In}(v)&=&\{e: head(e)=v, e \in \mathcal{E}\}, \nonumber \\
\textit{Out}(v)&=&\{e: tail(e)=v, e \in \mathcal{E}\}. \nonumber
\end{eqnarray}
We assume that each node $v$ has a random information source, $X_v$, which generates (random) message, called $x_v$, where $x_v \in \mathbb{R}$.
Furthermore, we consider the case where the messages, $\underline{x}=[x_v:v \in \mathcal{V}] \in \mathbb{R}^n$, are such that there is a linear transform matrix, $\phi_{n \times n}$, for which $\underline{x}=\phi \cdot \underline{s}$, and $\underline{s}$ is $k$-sparse (has at most $k$ non-zero elements).
In (single session) data gathering, all the messages, $x_v$'s, are to be transmitted to a single gateway (or decoder) node, represented by $v_0$, where $v_0 \in \mathcal{V}$.
QNC at each node, $v \in \mathcal{V}$, was defined in \cite{naba}, as follows:
\begin{equation}\label{Eq:QNC1}
y_e(t)= \textbf{Q}_e \Big [\sum_{e' \in \textit{In}(v)} \beta_{e,e'}(t)\cdot y_e(t-1)+\alpha_{e,v}(t)\cdot x_v \Big ],
\end{equation}
where $\textbf{Q}_e[ \centerdot]$ is the quantizer (designed based on the value of $C_e$ and the distribution of incoming contents and messages), associated with the outgoing edge $e \in \textit{Out}(v)$, and $\beta_{e,e'}(t)$ and $\alpha_{e,v}(t)$ are the corresponding network coding coefficients, picked from real numbers.
Initial rest condition is also assumed to be satisfied in our QNC scenario: $y_e(1)=0,~\forall~e \in \mathcal{E}.$
We represent the quantization error of $\textbf{Q}_e[ \centerdot]$ by $n_e(t)$, which implies:
\begin{equation}\label{Eq:QNC2}
y_e(t)= \sum_{e' \in \textit{In}(v)} \beta_{e,e'}(t)\cdot y_e(t-1)+\alpha_{e,v}(t)\cdot x_v + n_e(t).
\end{equation}
Equivalently, we have \cite{naba}:
\begin{equation}\label{Eq:matrixForm}
\underline{y}(t)=F(t) \cdot \underline{y}(t-1)+A(t) \cdot \underline{x}+\underline{n}(t),
\end{equation}
where $\underline{y}(t)=[y_e(t):e \in \mathcal{E}]$, $\underline{n}(t)=[n_e(t):e \in \mathcal{E}]$, and,
\begin{equation}
F(t)_{|\mathcal{E}|\times |\mathcal{E}|}: \{F(t)\}_{e,e'}=\left\{
\begin{array}{l l}
\beta_{e,e'}(t) & ,~\scriptsize tail(e)=head(e') \\
0 & ,~\mbox{otherwise} \\ \end{array} \right. \nonumber
\end{equation}
\begin{equation}\label{Eq:defineAt}
A(t)_{|\mathcal{E}|\times |\mathcal{V}|}: \{A(t)\}_{e,v}=\left\{
\begin{array}{l l}
\alpha_{e,v}(t) & ,~tail(e)=v \\
0 & ,~\mbox{otherwise} \\ \end{array} \right. . \nonumber
\end{equation}
By using linearity in the QNC scenario, the \textit{marginal measurements} at time $t$, represented by $\{z(t)\}_i$'s, where $\underline{z}(t)=[y_e(t):e \in \textit{In}(v_0)]$, are calculated as:
\begin{equation}\label{Eq:measForm1}
\underline{z}(t)=B(t) \cdot \underline{y}(t)= \Psi(t) \cdot \underline{x}+\underline{n}_{eff}(t).
\end{equation}
In Eq.~\ref{Eq:measForm1}, $\Psi(t)$ and $\underline{n}_{eff}(t)$ are defined as:
\begin{eqnarray}
\Psi(t)&=&B(t) \cdot \sum_{t'=2}^{t} \prod_{t''=t}^{t'+1} F(t'') \cdot A(t'), \label{Eq:DefPsi} \\
\underline{n}_{eff}(t)&=&B(t) \cdot \sum_{t'=2}^{t} \prod_{t''=t}^{t'+1} F(t'') \cdot \underline{n}(t'), \label{Eq:DefNeff}
\end{eqnarray}
and $B(t)$ is defined such that:
\begin{equation}
\{B(t)\}_{i,e}=\left\{
\begin{array}{l l}
b_{i,e}(t) & ,~i~\mbox{corresponds to}~e,~e \in \textit{In}(v_0) \\
0 & ,~\mbox{otherwise} \\ \end{array} \right. . \nonumber
\end{equation}
We store marginal measurements, at the decoder, and build up \textit{total measurements vector}, called $\underline{z}_{tot}(t)$, as follows:
\begin{equation}\label{Eq:totMeasEq}
\underline{z}_{tot}(t)=\left[ {\begin{array}{*{20}c}
\underline{z}(2) \\
\vdots \\
\underline{z}(t) \\
\end{array} } \right]_{m \times 1},
\end{equation}
where
\begin{equation}
m=(t-1) |\textit{In}(v_0)| ,
\end{equation}
and for which we have \cite{naba}:
\begin{equation}\label{Eq:measEq}
\underline{z}_{tot}(t)= \Psi_{tot}(t) \cdot \underline{x} + \underline{n}_{eff,tot}(t),
\end{equation}
where the \textit{total measurement matrix}, $\Psi_{tot}(t)$, and \textit{total effective noise vector}, $\underline{n}_{eff,tot}(t)$, are calculated as follows:
\begin{equation}
\Psi_{tot}(t)=\left[ {\begin{array}{*{20}c}
\Psi(2) \\
\vdots \\
\Psi(t) \\
\end{array} } \right],~~
\underline{n}_{eff,tot}(t)=\left[ {\begin{array}{*{20}c}
\underline{n}_{eff}(2) \\
\vdots \\
\underline{n}_{eff}(t) \\
\end{array} } \right].
\end{equation}
Since (\ref{Eq:measEq}) is in the form of a noisy linear measurement equation, compressed sensing decoding (\textit{i.e.} $\ell_1$-min recovery of Eq.~14 in \cite{naba}) can be applied, even if $m$ is smaller than $n$.
However, robust $\ell_1$-min decoding requires the total measurement matrix, $\Psi_{tot}(t)$, to satisfy some conditions \cite{LinProg,candes}.
Specifically, to ensure that the upper bound of Eq.~15 in \cite{naba} holds, we have to investigate the satisfaction of RIP for $\Psi_{tot}(t)$, in our QNC scenario.
In \cite{naba}, we proposed an appropriate design for network coding coefficients, which resulted in improved delay-quality performance for our QNC, compared to conventional packet forwarding.
In this paper, we analyze the satisfaction of RIP for $\Psi_{tot}(t)$, resulting from the proposed design of local network coding coefficients in \cite{naba} (also described in section~\ref{sec:DesignNCodes}).
\section{Design of Network Coding Coefficients}\label{sec:DesignNCodes}
Matrices with good norm conservation property are shown to be good choices for measurement in compressed sensing \cite{candes}.
RIP characterizes the norm conservation such that an $m \times n$ matrix, $\Theta$, is said to satisfy \textit{RIP of order $k$ with constant $\delta_k$}, if:
\begin{equation}\label{Eq:RIPexact}
1-\delta_k \leq \frac{ \vectornorm{\Theta \cdot \underline{s} }^2}{\vectornorm{\underline{s}}^2} \leq 1+ \delta_k,~\forall~\underline{s} \in \mathbb{R}^n,~\vectornormZero{\underline{s}} \leq k.
\end{equation}
Random matrices with Independently and Identically Distributed (i.i.d.) zero mean Gaussian entries are proved to satisfy RIP with an overwhelming probability \cite{simpleProof}.
Explicitly, for an $m \times n$ matrix with i.i.d. zero mean Gaussian entries of variance $\frac{1}{m}$, called $G_{m \times n}$, the probability of satisfying RIP of order $k$ and constant $\delta_k$ is exceeding
\begin{equation}\label{Eq:GoverWprob}
1-e^{-\kappa_2 m},
\end{equation}
(also referred as \textit{overwhelming}) where
\begin{equation}\label{Eq:GnumMeas}
m > \kappa_1 k \log(\frac{n}{k}),
\end{equation}
and $\kappa_1, \kappa_2$ only depend on $\delta_k$ (theorem~5.2 in \cite{simpleProof}).
In the following, we mention a design (originally proposed in \cite{naba}) for the local network coding coefficients, $\beta_{e,e'}(t)$ and $\alpha_{e,v}(t)$, which results in zero mean Gaussian entries for $\Psi_{tot}(t)$.
Then in section~\ref{sec:TailProbRIP}, we derive an upper probability bound for satisfying RIP in our QNC scenario with the proposed coefficients.
\begin{theorem}\label{th:Gaussian1}
Consider a quantized network coding scenario, in which the network coding coefficients, $\alpha_{e,v}(t)$ and $\beta_{e,e'}(t)$, are such that:
\begin{itemize}
\item $\alpha_{e,v}(t)=0,~\forall t>2,$
\item $\alpha_{e,v}(2)$'s are independent zero mean Gaussian random variables,
\item $\beta_{e,e'}(t)$'s are deterministic.
\end{itemize}
For such a scenario, the entries of the resulting $\Psi_{tot}(t)$ are zero mean Gaussian random variables, and the entries of different columns of $\Psi_{tot}(t)$, \textit{i.e.} $\{\Psi_{tot}(t)\}_{iv}$ and $\{\Psi_{tot}(t)\}_{i'v'}$, where $v,v' \in \mathcal{V},~v \neq v',$ are independent.
\end{theorem}
\begin{proof}
By choosing $\alpha_{e,v}(t)=0,~\forall~t>2,$ we have:
\begin{equation}
\Psi(t)=B(t) \cdot F(t) \cdots F(3) \cdot A(2), \label{Eq:Psit}
\end{equation}
which implies that each entry of $\Psi(t)$'s and also $\Psi_{tot}(t)$ is a linear combination of entries of $A(2)$.
Moreover, since entries of $A(2)$ are zero mean Gaussian random variables, then the entries of $\Psi(t)$'s and also $\Psi_{tot}(t)$ are zero mean Gaussian random variables.
Since entries in different columns of $\Psi_{tot}(t)$, are linear combinations of two independent sets of random variables, \textit{i.e.} entries of $A(2)$, then they are also independent.
However, such conclusion can not be made for entries of the same column of $\Psi_{tot}(t)$.
\end{proof}
\section{RIP Analysis and Tail Probability of $\ell_2$-norms}\label{sec:TailProbRIP}
Satisfaction of RIP for random matrices is usually characterized by its probability (or its lower probability bounds) \cite{simpleProof}.
Moreover, to approach the probabilistic satisfaction of RIP, we first need to derive an expression for the tail probability of $\ell_2$ norms \cite{simpleProof,CSbook}.
Specifically, a well behaved $\Psi_{tot}(t)$ (\textit{i.e.} $\Psi_{tot}(t)$ with high RIP probability) should be such that
\begin{equation}\label{Eq:defineTailProb1}
\textbf{P}(\Big |\vectornorm{{\Psi_{tot}(t) \cdot \underline{x}}}^2-1 \Big| \geq \epsilon),
\end{equation}
is very small, for all $\underline{x}$, with $\vectornorm{\underline{x}}=1$.
In the following, we calculate this tail probability for our QNC scenario with the proposed network coding coefficients, and then present a theorem which explicitly describes the relation between the satisfaction of RIP and the tail probability of Eq.~\ref{Eq:defineTailProb1}.
In the rest of this section, we assume that the conditions of Theorem~\ref{th:Gaussian1} hold.
Consider
\begin{equation}
\underline{z}'=\Psi_{tot}(t) \cdot \underline{x},
\end{equation}
where $\underline{x} \in \mathbb{R}^n$, and $\vectornorm{\underline{x}}=1$.
Since the conditions of Theorem~\ref{th:Gaussian1} are satisfied, Eq.~\ref{Eq:Psit} holds, and therefore:
\begin{equation}
\Psi_{tot}(t) = \left[ {\begin{array}{*{20}c}
\Psi(2) \\
\vdots \\
\Psi(t) \\
\end{array} } \right]
= \Omega(t) \cdot A(2),
\end{equation}
where
\begin{equation}
\Omega(t)=\left[ {\begin{array}{*{20}c}
B(2) \\
B(3) F(3) \\
\vdots \\
B(t) F(t) \cdots F(3) \\
\end{array} } \right].
\end{equation}
This implies:
\begin{equation}
\underline{z}'=\Omega(t) ~ A(2) \cdot \underline{x},
\end{equation}
or equivalently:
\begin{eqnarray}
z'_i&=&\sum_{v=1}^n \{\Psi_{tot}(t)\}_{iv}~ x_v \nonumber \\
&=& \sum_{v=1}^n \sum_{e=1}^{|\mathcal{E}|} \{\Omega(t)\}_{ie} ~\{A(2)\}_{ev}~ x_v.
\end{eqnarray}
By expanding $z'^2_i$, and using the fact that $\{A(2)\}_{ev}$ is non-zero only when $tail(e)=v$, we have:
\begin{eqnarray}
\vectornorm{\underline{z}'}^2 &=& \sum_{i=1}^{m} z'^2_i \label{Eq:equation1} \\
& = &\sum_{e=1}^{|\mathcal{E}|} \sum_{e'=1}^{|\mathcal{E}|} \gamma_{e,e'}(\underline{x}) \{A(2)\}_{e,tail(e)} \{A(2)\}_{e',tail(e')}, \nonumber
\end{eqnarray}
where:
\begin{equation}
\gamma_{e,e'}(\underline{x}) = \sum_{i=1}^{m} \{\Omega(t)\}_{ie}~\{\Omega(t)\}_{ie'} \cdot x_{tail(e)}~x_{tail(e')}
\end{equation}
Using eigen-decomposition, (\ref{Eq:equation1}) simplifies to:
\begin{eqnarray}
\vectornorm{\underline{z}'}^2= \sum_{e=1}^{|\mathcal{E}|} \lambda_{e}(\underline{x}) \cdot \chi^2_{e},
\end{eqnarray}
where $\lambda_e(\underline{x})$'s are eigen-values of the symmetric matrix
\begin{equation}
\Gamma(\underline{x})=[\gamma_{e,e'}(\underline{x})]_{|\mathcal{E}| \times |\mathcal{E}|},
\end{equation}
and $\chi^2_e$'s are independent Chi-Square random variables of first order.
Moreover, for the characteristic function of $\vectornorm{\underline{z}'}^2$, we have:
\begin{eqnarray}
\textbf{E}[e^{j\omega \vectornorm{\underline{z}'}^2 }] &=&
\textbf{E}[e^{j\omega \sum_{e=1}^{|\mathcal{E}|} \lambda_{e}(\underline{x}) \chi^2_{e} }] \\
&=& \prod_{e=1}^{|\mathcal{E}|} \textbf{E}[e^{j\omega \lambda_{e}(\underline{x}) \chi^2_{e} }] \label{Eq:prove111} \\
&=& \prod_{e=1}^{|\mathcal{E}|} \frac{1}{\sqrt{1-j2\omega \lambda_e(\underline{x}) }},
\end{eqnarray}
where (\ref{Eq:prove111}) is derived from independence of $\chi^2_e$'s.
By using the inverse formula of characteristic function, Eqs.~\ref{Eq:provepart1}-\ref{Eq:tailProb} can be obtained, where $\textbf{p}_{\vectornorm{\underline{z}'}^2} (\centerdot)$ is the probability density function of $\vectornorm{\underline{z}'}^2$, and (\ref{Eq:prove1}) is resulted from the integral property of the Fourier transform.
\begin{figure*}[t]
\begin{eqnarray}
\textbf{P}\Big(\Big | \vectornorm{\underline{z}'}^2 -1 \Big |> \epsilon \Big)
&=& 1+\int_{-\infty}^{1-\epsilon} \textbf{p}_{\vectornorm{\underline{z}'}^2} (\nu) d\nu
- \int_{-\infty}^{1+\epsilon} \textbf{p}_{\vectornorm{\underline{z}'}^2} (\nu) d\nu \label{Eq:provepart1} \\
&=& 1+\frac{1}{2\pi} \int_{-\infty}^{+\infty} \frac{\textbf{E}[e^{j\omega \vectornorm{\underline{z}'}^2 }]}{-j\omega}e^{-j\omega(1-\epsilon)} d\omega
- \frac{1}{2\pi} \int_{-\infty}^{+\infty} \frac{\textbf{E}[e^{j\omega \vectornorm{\underline{z}'}^2 }]}{-j\omega}e^{-j\omega(1+\epsilon)} d\omega \label{Eq:prove1} \\
&=&1-\frac{1}{\pi} \int_{-\infty}^{+\infty} \frac{ e^{-j\omega} \sin(\epsilon \omega)}{\omega \prod_{e=1}^{|\mathcal{E}|} \sqrt{1-j2\omega \lambda_e(\underline{x})} } d\omega, \label{Eq:tailProb}
\end{eqnarray}
\end{figure*}
The right hand side of (\ref{Eq:tailProb}) is the expression for the tail probability of $\ell_2$-norms, for a specific $\underline{x}$, resulting from our proposed network coding coefficients.
In the following, we present Theorem~\ref{theorem:RIP1}, which clarifies the relation between the tail probability of (\ref{Eq:defineTailProb1}) and the probability of satisfying RIP \textit{for a general case}.
\begin{theorem}\label{theorem:RIP1}
Consider $\Phi$ for which we have:
\begin{eqnarray}\label{Eq:RIP1condition}
\textbf{p}_{tail}({\Phi},\epsilon)&=&\max_{\underline{x}} \textbf{P}\Big(\Big |\vectornorm{{\Phi \cdot \underline{x}}}^2-1 \Big| \geq \epsilon \Big), \nonumber \\
&& s.t.~\vectornorm{\underline{x}}=1
\end{eqnarray}
In such case, for every orthonormal $\phi$, $\Theta=\Phi \cdot \phi$ satisfies RIP of order $k$ and constant $\delta_{k}$, with a probability exceeding,
\begin{equation}\label{Eq:lowerRIPbound1}
\textbf{p}_{RIP}\Big(\Phi,k,\delta_k \Big)=1- \left(
\begin{array}{c}
n\\
k
\end{array}
\right) (\frac{42}{\delta_k})^k ~ \textbf{p}_{tail}({\Phi},\epsilon=\frac{\delta_k}{\sqrt{2}}).
\end{equation}
\end{theorem}
\begin{proof}\footnote{Most of the proof is similar to the proof of Theorem~7.3 in \cite{CSbook}.}
To prove that RIP holds, we should show that inequality of (\ref{Eq:RIPexact}) is satisfied, for all $k$-sparse vectors, $\underline{s}$.
We only need to show it is satisfied, for vectors, with $\vectornorm{\underline{s}}=1$, since $\vectornorm{\Phi \phi \cdot \underline{s}}$ is proportional with $\vectornorm{\underline{s}}$.
Now, fix a set $T \subset \{1,2,\ldots,n\}$, with $|T|=k$, and let $\Gamma_T$ be the subspace of $k$-dimensional vectors, $\underline{s}_T$, spanned by columns of $\Phi$, with indexes in $T$.
According to lemma 7.5 in \cite{CSbook}, we can choose a finite set of vectors, $\underline{w}_T \in \mathcal{W}_T$, where $\mathcal{W}_T \subset \Gamma_T$ and $\vectornorm{\underline{w}} \leq 1$, such that for all $\underline{s}_T \in \Gamma_T$, with $\vectornorm{\underline{s}_T} \leq 1$, we have:
\begin{equation}
\vectornorm{\underline{s}_T-\underline{w}_T} \leq \frac{\delta_k}{14},
\end{equation}
conditioned on:
\begin{equation}
|\Gamma_T| \leq (\frac{42}{\delta_k})^k.
\end{equation}
There are $\left(
\begin{array}{c}
n\\
k
\end{array}
\right)$ different $T$'s, for which we repeat the above procedure and obtain: $$\mathcal{W}= \bigcup_T \mathcal{W}_T.$$
By using the union bound and the fact that for every $\underline{x}=\phi \cdot \underline{w}$, where $\underline{w} \in \mathcal{W}$, Eq.~\ref{Eq:RIP1condition} implies:
\begin{equation}
\textbf{P}\Big(\Big |\vectornorm{{\Phi \cdot \underline{x}}}^2-1 \Big| \geq \epsilon \Big) \leq
\textbf{p}_{tail}({\Phi},\epsilon).
\end{equation}
Therefore, for every $\underline{w} \in \mathcal{W}$, the inequality
\begin{equation}\label{Eq:inequality110}
(1-\frac{\delta_k}{\sqrt{2}}) \vectornorm{\underline{w}}^2 \leq \vectornorm{\Phi \phi \cdot \underline{w}}^2 \leq (1+\frac{\delta_k}{\sqrt{2}}) \vectornorm{\underline{w}}^2,
\end{equation}
holds with a probability exceeding
\begin{equation}
1-\left(
\begin{array}{c}
n\\
k
\end{array}
\right) (\frac{42}{\delta_k})^k ~\textbf{p}_{tail}({\Phi},\frac{\delta_k}{\sqrt{2}}).
\end{equation}
The rest of the proof uses the same reasoning procedure, as in the proof of Theorem~7.3 in \cite{CSbook}.
\end{proof}
It can be concluded from Theorem~\ref{theorem:RIP1} that in order to have a good RIP satisfaction (\textit{i.e.} high upper probability bound for satisfaction of RIP), a small worst case tail probability, $\textbf{p}_{tail}(\centerdot,\frac{\delta_k}{\sqrt{2}})$, is required.
In section~\ref{sec:Numerical}, we compare $\textbf{p}_{tail}(\centerdot,\frac{\delta_k}{\sqrt{2}})$'s, corresponding to our designed $\Psi_{tot}(t)$ and i.i.d. Gaussian matrix, to numerically evaluate their RIP behaviour.
\begin{figure*}[!t]
\centering
\subfigure[$1100$ edges]{
\resizebox{!}{.33\textheight}{
\includegraphics{tailProb1100edges2.pdf}}
\label{fig:subfig1100}
}
\subfigure[$1400$ edges]{
\resizebox{!}{.33\textheight}{
\includegraphics{tailProb1400edges2.pdf}}
\label{fig:subfig1400}
}
\subfigure[$1800$ edges]{
\resizebox{!}{.33\textheight}{
\includegraphics{tailProb1800edges2.pdf}}
\label{fig:subfig1800}
} \\
\caption{Logarithmic tail probability versus logarithmic ratio of minimum required number of measurements in
our QNC scenario and i.i.d. Gaussian measurement matrices, for $n=100$, different RIP constants, and different number of edges
\label{fig:subfigureExample}.
}
\end{figure*}
By using the derived tail probability of (\ref{Eq:tailProb}), and applying Theorem~\ref{theorem:RIP1}, the following theorem can be obtained, which suggests an upper probability bound on the satisfaction of RIP, in our QNC scenario.
\begin{theorem}\label{th:RIPqnc}
For a quantized network coding scenario, in which the network coding coefficients hold the conditions of Theorem~\ref{th:Gaussian1}, for every orthonormal $\phi$, the resulting $\Theta=\Psi_{tot}(t) \cdot \phi$ satisfies RIP of order $k$, and constant $\delta_k$, with a probability exceeding $$\textbf{p}_{RIP}\Big(\Psi_{tot}(t),k,\delta_k \Big),$$ defined in Eq.~\ref{Eq:definePrip}.
\begin{figure*}[]
\begin{equation}\label{Eq:definePrip}
\textbf{p}_{RIP}\Big(\Psi_{tot}(t),k,\delta_k \Big)=
1-\left(
\begin{array}{c}
n\\
k
\end{array}
\right) (\frac{42}{\delta_k})^k ~ \Big ( 1-\frac{1}{\pi} \min_{~\underline{x},~\vectornorm{\underline{x}}=1~} \int_{-\infty}^{+\infty} \frac{ e^{-j\omega} \sin(\frac{\delta_k}{\sqrt{2}} \omega)}{\omega \prod_{e=1}^{|\mathcal{E}|} \sqrt{1-j2\omega \lambda_e(\underline{x})} } d\omega \Big)
\end{equation}
\end{figure*}
\end{theorem}
It is however difficult to derive the number of required measurements, $m$, from the expression of Eq.~\ref{Eq:definePrip}; we use numerical evaluations, in section~\ref{sec:Numerical}, to explore the properties of our QNC design.
\section{Numerical Evaluations and Discussion}
\label{sec:Numerical}
In order to evaluate the RIP satisfaction of $\Psi_{tot}(t)$, resulting from the proposed network coding coefficients, we use the worst case tail probability, $\textbf{p}_{tail}(\centerdot,\frac{\delta_k}{\sqrt{2}})$.
This is because of the deterministic (linear) relation between $\textbf{p}_{tail}(\centerdot,\frac{\delta_k}{\sqrt{2}})$ and the proposed upper probability bound in Theorem~\ref{theorem:RIP1}.
Moreover, we calculate the worst case tail probability, corresponding to an i.i.d. Gaussian matrix, called $G_{m \times n}$, and compare it with that of our $\Psi_{tot}(t)$.
For an $m \times n$ i.i.d. Gaussian matrix, $G_{m \times n}$, the worst case tail probability, $\textbf{p}_{tail}(G_{m \times n},\frac{\delta_k}{\sqrt{2}})$, can be calculated as:
\footnote{This can be obtained similar to the reasoning procedure for Eq.~\ref{Eq:tailProb}.}
\begin{equation}\label{Eq:GaussianTail}
\textbf{p}_{tail}(G_{m \times n},\frac{\delta_k}{\sqrt{2}})=
1-\frac{1}{\pi} \int_{-\infty}^{+\infty} \frac{ e^{-j\omega} \sin( \omega\frac{ \delta_k }{\sqrt{2}})}{\omega~ (1-2j\frac{\omega }{m} )^{m/2} } d\omega.
\end{equation}
To present our numerical evaluations, for each value of tail probability, represented by $\textbf{p}_{tail}$, the minimum number of required measurements in $\Psi_{tot}(t)$, resulting from our QNC scenario (with the designed network coding coefficients, as in Theorem~\ref{th:Gaussian1}) and $G_{m \times n}$, are calculated.
This is done by generating random deployments of networks and calculating the worst case tail probability of (\ref{Eq:tailProb}) and (\ref{Eq:GaussianTail}) in each generated deployment.
The resulting tail probabilities, and corresponding number of measurements are then averaged over different realizations of network deployments.
In Fig.~\ref{fig:subfigureExample}, $\textbf{p}_{tail}$ is drawn versus $m$ in logarithmic scale, for $\Psi_{tot}(t)$ (QNC) and $G_{m \times n}$ (Gaussian), and different values of RIP constant, $\delta_k$ ($\delta_k=0.41421 \simeq \sqrt(2)-1$ is the largest RIP constant for which Theorem~4.1 of \cite{naba} can be applied).
The statistical characteristics of the resulting $\Psi_{tot}(t)$ and its worst case tail probability vary by changing the network deployment parameters, like the distribution of edges in the network.
In Figs.~\ref{fig:subfig1100} to \ref{fig:subfig1800}, the curves correspond to different deployments with $n=100$ nodes, and $|\mathcal{E}|=1100,1400,1800$ uniformly distributed edges, respectively.
To generate the network coding coefficients, $\alpha_{e,v}(t)$'s and $\beta_{e,e'}(t)$'s, we make sure that the conditions of Theorem~\ref{th:Gaussian1} are satisfied.
Moreover, for $\beta_{e,e'}(t)$'s, it was experimentally understood that the resulting $\Psi_{tot}(t)$ has a better behavior in terms of RIP satisfaction (and also $\ell_1$-min recovery) if in any two outgoing edges, $\beta_{e,e'}$'s are orthogonal.
By studying the curves in Fig.~\ref{fig:subfigureExample}, the following arguments can be made:
\begin{itemize}
\item
The minimum number of required measurements for $\Psi_{tot}(t)$ to achieve a worst case tail probability as a perfect i.i.d. Gaussian measurement matrix is in the same order as that of i.i.d. Gaussian (the logarithmic difference between the number of measurements for QNC and Gaussian cases is less than $1$).
Therefore, the number of required measurements in our QNC, for an {overwhelming} probability of RIP satisfaction (Eq.~\ref{Eq:GoverWprob}) is in the same order as that of an i.i.d. Gaussian matrix.
Furthermore, this behavior is improved when the number of edges in the network increases, or the corresponding RIP constant is increased.
\item
By applying Theorem~\ref{theorem:RIP1}, on the resulting $\textbf{p}_{tail}$, the lower bound on the RIP satisfaction for each sparsity, $k$, can be obtained.
Therefore, \emph{{as an implication of RIP}} (Theorem~4.1 in \cite{naba}), we can make the following probabilistic statement about $\ell_1$-min recovery error, in QNC scenario:
\textit{Consider the QNC scenario, described in Theorem~4.1 of \cite{naba}, in which we transmit $k$-sparse messages.
In such a scenario, if the resulting $\Psi_{tot}(t)$ corresponds to a point with $\textbf{p}_{tail}(\Psi_{tot}(t),\frac{\delta_{2k}}{\sqrt{2}})$ on one of the evaluated curves of Fig.~\ref{fig:subfigureExample}, then the $\ell_2$-norm of recovery error, using the $\ell_1$-min decoder of Eq.~14 in \cite{naba}, is upper bounded according to (15) in \cite{naba}, with a probability exceeding $\textbf{p}_{RIP}\Big(\Psi_{tot}(t),2k,\delta_{2k} \Big)$.}
\item
By calculating the lower bound for RIP satisfaction (using Eq.~\ref{Eq:lowerRIPbound1}), corresponding to one of the points on the curves of Fig.~\ref{fig:subfigureExample}, it would be clear that the possible sparsity, $k$, for which the resulting $\textbf{p}_{RIP}(\Psi_{tot}(t),2k,\delta_{2k})$ approaches $1$, is very small.
In other words, QNC requires a lot of measurements to guarantee the upper bound of (15) in \cite{naba}, with an overwhelming probability.
However, this is also the case for i.i.d. Gaussian matrices, as it has been previously pointed out by the authors of \cite{phaseTransitions,sharpRIP,bah2010improved}, that the RIP analysis for i.i.d Gaussian matrices proposes an exaggerated minimum number of measurements, required for robust $\ell_1$-min recovery.
In conclusion, the minimum number of measurements, required for guaranteeing robust $\ell_1$-min decoding, using our proposed $\Psi_{tot}(t)$, is in the same order as that of i.i.d. Gaussian matrix.
The aforementioned fact (on exaggerated required number of measurements) can be considered as a weakness of RIP analysis, used in the compressed sensing literature.
\end{itemize}
\section{Conclusions}
\label{sec:Conclusions}
Joint distributed source coding and network coding of sparse messages with compressed sensing perspective was discussed in this paper.
We investigated the satisfaction of RIP, in a modified random linear network coding scenario, called quantized network coding.
This was explicitly done by using mathematical derivation for the tail probability of the resulting measurement matrix in our QNC scenario, and that of i.i.d. Gaussian matrix.
It was numerically shown that our linear measurements have the same RIP behavior (in terms of order of minimum number of required measurements) as i.i.d. Gaussian measurements.
Our RIP analysis provided us with the preliminaries for guaranteeing robust $\ell_1$-min decoding, in QNC scenario.
\section*{Acknowledgement}
This work was supported by Hydro-Québec, the Natural Sciences and Engineering Research Council of Canada and McGill University in the framework of the NSERC/Hydro-Québec/McGill Industrial Research Chair in Interactive Information Infrastructure for the Power Grid.
\bibliographystyle{ieeetr}
|
{
"timestamp": "2012-03-16T01:01:08",
"yymm": "1203",
"arxiv_id": "1203.1892",
"language": "en",
"url": "https://arxiv.org/abs/1203.1892"
}
|
\section{Introduction}
The goal of this paper is to give a unified theory for {\it integrable
surfaces} using the real forms of complex extended framings of complex
{\sc CMC}-immersions and the generalized Weierstra{\ss} type
representation for complex {\sc CMC}-immersions.
It is classically known that {\sc CMC} surfaces with nonzero mean
curvature, or equivalently constant positive Gau{\ss}ian curvature
({\sc CPC} for short) surfaces as parallel surfaces,
and constant negative Gau{\ss}ian curvature
({\sc CNC} for short) surfaces in $\mathbb R^3$ are characterized by
the transformations of real (or complex) tangential line congruences between
surfaces with special properties, which are commonly called ``(Bianchi)
B\"{a}cklund transformations''. In modern terminology, such
classes of surfaces are characterized by the (Lorentz) harmonicities of
their Gau{\ss} maps, and they are equivalent to the
existence of families of flat connections on $\mathcal M \times SO(3)$, where
$\mathcal M$ is $\mathbb C$ for {\sc CMC}-immersions or $\mathbb R^{1,1}$
for {\sc CNC}-immersions, see \cite{DPW} and \cite{MS:PSsurfaces}.
Spacelike or timelike constant positive or negative Gau{\ss}ian
curvature surfaces in $\mathbb R^{1,2}$ are less known, however,
they are also characterized by the (Lorentz) harmonicities of their Gau{\ss}
maps, or equivalently, the existence of
families of flat connections on $\mathcal M \times
SO(2,1)$, where $\mathcal M$ is $\mathbb C$ for spacelike {\sc
CNC}-immersions, or equivalently spacelike {\sc CMC}-immersions,
and timelike {\sc CNC}-immersions, or $\mathbb R^{1, 1}$ for spacelike {\sc
CPC}-immersions and timelike {\sc CPC}-immersions, or equivalently
timelike {\sc CMC}-immersions,
see \cite{DIT:Timelike} and \cite{Klotz:Harm-Mink}.
On the one hand, to classify all {\sc CMC}-cylinders in $\mathbb R^3$,
in \cite{DK:cyl} we gave a natural complexification of the extended
framing, a moving frame with spectral parameter and an element in the
$SU(2)$ loop group, of a {\sc CMC}-immersion, which is called the {\it
complex extended framing}.
Moreover in \cite{DKP:Complex}, we introduced holomorphic
immersions in $\mathbb C^3$ associated with the complex extended
framings and a natural definition of the complex mean curvature for a
holomorphic immersion. Then a holomorphic immersion with
complex constant mean curvature $H \in \mathbb C$ is naturally called
the {\it complex {\sc CMC}-immersion}.
Similar to the real case, a holomorphic immersion with complex constant Gau{\ss}
curvature $K \in \mathbb C^*$ ({\sc CGC} for short) is obtained as the parallel
immersion of a complex {\sc CMC}-immersion with nonzero complex
constant mean curvature $H \in \mathbb C^*$.
In this paper, we shall interpret those complex {\sc CGC}-immersions, or
equivalently {\sc CMC}-immersions by the parallel immersions,
as {\it complexifications} for the surfaces discussed above. These
real surfaces are then obtained by the real form surfaces of a complex {\sc
CGC}-immersion, which are defined from the real forms of
the Maurer-Cartan form of the complex extended framing of a complex {\sc
CMC}-immersion. It is known that the twice central extensions of a loop algebra $\Lambda
\mathfrak{sl}(2, \mathbb C)_{\sigma}$ is a twisted affine Kac-Moody Lie
algebra of $A_1^{(1)}$ type. The classification of the real forms of affine
Kac-Moody Lie algebras was given in \cite{BBBR:almostsplit} and
\cite{BR:almostcompact}. It follows that the classification of real forms of $\Lambda
\mathfrak{sl}(2, \mathbb C)_\sigma$ is also given. In particular, the real forms of
$\Lambda \mathfrak{sl}(2, \mathbb C)_{\sigma}$ consist of seven classes:
three classes are called the {\it almost split} and the other four
classes are called the {\it almost compact}, according to the types of
the semi-linear involutions of the real forms.
Thus there are seven classes of surfaces, which are called {\it integrable
surfaces}, according to the classification of the real forms of
$\Lambda \mathfrak{sl} (2, \mathbb C)_\sigma$. Spacelike or timelike {\sc CMC} or {\sc CGC}
surfaces in $\mathbb R^3$ or $\mathbb R^{1,2}$ form the six classes of
integrable surfaces, and {\sc CMC} surfaces with mean curvature $|H| <1$
in $H^3$ form the last class of integrable surfaces (Theorem
\ref{thm:compactsplit} and Corollary \ref{coro:compactsplit}).
Moreover, all integrable surfaces are characterized by (Lorentz) harmonicities of
their Gau{\ss} maps, which are maps into symmetric spaces $S^2$, $H^2$,
$S^{1,1}$ and the $4$-symmetric space $SL(2, \mathbb C)/U(1)$
respectively, Theorem \ref{thm:Gaussmap}.
The generalized Weierstra{\ss} type representation for complex {\sc
CMC}-immersions is a procedure to construct complex {\sc
CMC}-immersions in $\mathbb C^3$, see Section \ref{subsc:DPW} for more
details: {\bf 1.} Define pairs of holomorphic
potentials, which are pairs of holomorphic 1-forms $\check \eta =
(\eta, \tau)$ with $\eta = \sum_{j \geq -1}^{\infty} \eta_j \lambda^j$
and $\tau = \sum_{-\infty}^{j \leq 1} \tau_j \lambda^j$. Here $\lambda$
is the complex parameter, the so-called ``spectral parameter'', $\eta_j$ and $\tau_j$ are diagonal
(resp. off-diagonal) holomorphic 1-forms
depending only on one complex variable if $j$ is even (resp. $j$ is
odd). {\bf 2.} Solve the pair of ODE's $d (C, L) = (C, L) \check \eta$
with some initial condition $(C(z_*), L(w_*))$, and perform the
generalized Iwasawa decomposition, Theorem
\ref{doublesplitting},
for $(C, L)$, giving $ (C, L)= (F, F)(V_+, V_{-})$.
It is known that $F \cdot l$ is the complex extended framing of some complex
{\sc CMC}-immersion Theorem \ref{thm:DKP-Extendedframings},
where $l$ is some $\lambda$-independent
diagonal matrix. {\bf 3.} Form a complex {\sc CMC}-immersion
by the Sym formula $\varPsi$ via the complex extended framing
$F \cdot l$, Theorem \ref{thm:Sym-Bob}.
Since each class of integrable surfaces is defined
by a real form of $\Lambda \mathfrak{sl}(2, \mathbb C)_\sigma$, there exists a
unique semi-linear involution $\rho$ corresponding to each class of
integrable surfaces. Then these semi-linear involutions
naturally define the pairs of semi-linear involutions on pairs of
holomorphic potentials $\check \eta = (\eta, \tau)$.
It follows that the generalized Weierstra{\ss} type representation for
each class of integrable surfaces can be formulated by the above
construction with a pair of holomorphic potentials which is invariant
under a pair of semi-linear involutions, Theorem
\ref{thm:DPWforint}.
In this way we give a unified theory for all integrable surfaces.
More precisely, in Section \ref{sc:Pre}, we give a brief review of the
basic results for complex {\sc CMC} and {\sc CGC}-immersions. In
Section \ref{subsc:holonull}, holomorphic null
immersions and the basic facts for holomorphic null
immersions are considered. In Section \ref{subsc:complexCMC}, the basic
facts and results for complex {\sc CMC}-immersions are
given. Analogously to the complex {CMC}-immersions,
the definition and the basic facts for complex {\sc CGC}-immersions
are given, Theorem \ref{thm:Sym-Bob}.
In Section \ref{subsc:realform}, the classification of real forms of $\Lambda
\mathfrak{sl}(2, \mathbb C)_{\sigma}$ is given, Theorem
\ref{thm:almostcompact} and Theorem \ref{thm:almostsplit}.
In Section \ref{sc:Realforms}, we give a classification of integrable
surfaces. In Section \ref{subsc:Integrablesurf}, it is shown that all
integrable surfaces are obtained from real forms of
$\Lambda \mathfrak{sl}(2, \mathbb C)_\sigma$,
Theorem \ref{thm:compactsplit}. In Section
\ref{subsc:gaussmap}, the Gau{\ss} maps of integrable surfaces are
characterized using the Gau{\ss} map of a complex {\sc CMC}-immersion and
the real forms of the complex extended framing, Theorem \ref{thm:Gaussmap}.
In Section \ref{sc:DPW}, we give a construction of all integrable surfaces
via the generalized Weierstra{\ss} type representation.
In Section \ref{subsc:DPW}, the pairs of semi-linear
involutions, which are determined from classes of integrable
surfaces, are considered. Then a construction of all
integrable surfaces is discussed via the pairs of holomorphic
potentials which are invariant under the pairs of
semi-linear involutions, Theorem \ref{thm:DPWforint}.
In Appendix \ref{BasicResult}, we give the basic results for affine
Kac-Moody Lie algebras and the loop algebras. In Section
\ref{subsc:Affine}, Kac-Moody Lie algebras are considered.
In Section \ref{loopgroups}, loop groups and a realization of affine
Kac-Moody Lie algebras via the twice central extensions of loop
algebras are discussed, Theorem \ref{thm:Kac}.
In Section \ref{nsc:doubleIwasawa}, double
loop groups are defined and the Iwasawa decomposition theorem
for the double loop groups are given, Theorem \ref{doublesplitting}.
\section{Preliminaries}\label{sc:Pre}
In this preliminary section, we give a brief review of the basic
results for holomorphic null immersions, complex {\sc
CMC}-immersions and complex {\sc CGC}-immersions.
We also give a brief review of the basic facts about loop
algebras and their real forms.
Throughout this paper, $\mathbb C^3$ is identified with
$\mathfrak{sl}(2, \mathbb C)$ as follows:
\begin{equation}\label{eq:ident1}
(a, b, c)^t \in \mathbb C^3 \leftrightarrow -\frac{i a}{2} \sigma_1
-\frac{i b}{2} \sigma_2 - \frac{i c}{2} \sigma_3 \in \mathfrak{sl} (2,
\mathbb C)\;\;,
\end{equation}
where $\sigma_j\;(j=1, 2, 3)$ are Pauli matrices as follows:
\begin{equation}\label{eq:ident2}
\sigma_1 = \begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix}, \;\;\sigma_2 =
\begin{pmatrix} 0 & -i \\ i & 0
\end{pmatrix}\;\; \mbox{and}\;\;\sigma_3 = \begin{pmatrix} 1 & 0 \\ 0 & -1\end{pmatrix}\;.
\end{equation}
\subsection{Holomorphic null immersions in $\mathbb
C^3$}\label{subsc:holonull}
In this subsection, we show the basic results for holomorphic
immersions $\varPsi$ from $\mathfrak D^2 \subset \mathbb C^2$ into
$\mathbb C^3$. We give natural definitions of complex mean curvature (Definition
\ref{def:Mean}) and complex Gau{\ss} curvature (Definition
\ref{def:Gauss}) for a holomorphic immersion analogous to the mean
curvature and the Gau{\ss} curvature of a surface in $\mathbb R^3$.
We refer to \cite{DKP:Complex} for more details.
Let $\mathcal M$ be a simply connected $2$-dimensional Stein manifold, and let $\varPsi:
\mathcal M \to \mathfrak{sl}(2, \mathbb C)$ be a holomorphic immersion,
i.e., the complex rank of $d\varPsi$ is two. We consider the following bilinear
form on $\mathfrak{sl}(2, \mathbb C) \cong \mathbb C^3$:
\begin{equation}\label{eq:bilinear}
\langle a, b \rangle = -2 {\rm Tr}\; a b\;,
\end{equation}
where $a, b \in \mathfrak{sl}(2, \mathbb C)$. We note that the bilinear
form \eqref{eq:bilinear} is a $\mathbb C$-bilinear form on $\mathbb
C^3$ by the identification \eqref{eq:ident1}. Then it is known that,
for a neighborhood $\widetilde {\mathcal M}_p \subset \mathcal M$ around
each point $p \in \mathcal {M}$, the bilinear form \eqref{eq:bilinear}
induces a holomorphic Riemannian metric on $\widetilde{\mathcal M}_p$,
i.e., a holomorphic covariant symmetric 2-tensor
$g$, see \cite{LeBrun:Complex-Riemannian} and \cite{DKP:Complex}. From
\cite{DKP:Complex}, it is also known that there exist
special coordinates $(z, w) \in \mathfrak D^2 \subset \mathbb C^2$ such
that a holomorphic Riemannian metric $g$ can be
written as follows:
\begin{equation}\label{eq:nullmetric}
g = e^{u(z, w)}dz dw\;,
\end{equation}
where $u(z, w) : \mathfrak D^2 \to \mathbb C$ is some holomorphic
function. The special coordinates defined above are called {\it
null coordinates}. From now on, we always assume a holomorphic immersion
$\varPsi : \mathcal M \to \mathfrak{sl}(2, \mathbb C)$ has null
coordinates. A holomorphic immersion with null
coordinates is also called the {\it holomorphic null immersion}.
\begin{Remark}
The assumption ``Stein'' is used for the existence of the form of a
holomorphic Riemannian metric $g =e^{u(z, w)} dz dw$ defined in
\eqref{eq:nullmetric} and the existence of a well-defined pair of
holomorphic potentials on $\mathcal M$ for the generalized Weierstra{\ss}
type representation in Section \ref{sc:DPW}, see \cite{DKP:Complex} for more
details.
\end{Remark}
We now define a vector $N \in \mathfrak{sl}(2, \mathbb C)$ as follows:
\begin{equation}
\label{eq:GaussMap}
N := 2 i e^{-u} [\varPsi_{w},\; \varPsi_z ]\;,
\end{equation}
where the subscripts $z$ and $w$ denote the partial derivatives with
respect to $z$ and $w$, respectively.
It is easy to verify that $\langle \varPsi_z, N\rangle =\langle
\varPsi_w, N\rangle = 0$ and the $\langle N, N\rangle =1$. Thus $N$ is
a transversal vector to $ d \varPsi$. Therefore it is natural to call $N$
the {\it complex Gau{\ss} map} of $\varPsi$.
From \cite{DKP:Complex}, we quote the following theorem:
\begin{Theorem}[\cite{DKP:Complex}]\label{thm:MovingFrame}
Let $\varPsi : \mathcal M \to \mathbb C^3 (\cong \mathfrak{sl}(2, \mathbb C))$ be a holomorphic null
immersion. Then there exists a $SL(2, \mathbb C)$ matrix $F$ such that
the following equations hold:
\begin{equation}
\label{eq:Lax-complex}
\begin{array}{lcr}
F_{z} = F U, \\
F_{w} = F V,
\end{array}
\end{equation}
where
\begin{equation}\label{eq:U-V}
\left\{
\begin{array}{lcr}
U =
\begin{pmatrix}
\frac{1}{4} u_z & -\frac{1}{2} H e^{u/2} \\
Q e^{-u/2} & -\frac{1}{4} u_{z}
\end{pmatrix},
\\ \\
V =
\begin{pmatrix}
-\frac{1}{4} u_w & - R e^{-u/2} \\
\frac{1}{2} H e^{u/2} & \frac{1}{4} u_w
\end{pmatrix},
\end{array}
\right.
\end{equation}
with $ Q := \langle \varPsi_{zz}, N \rangle $, $R:=\langle
\varPsi_{ww}, N\rangle $ and $H := 2 e^{-u} \langle \varPsi_{z w}, N
\rangle $.
\end{Theorem}
We call $F : \mathcal M \to SL(2, \mathbb C)$ the {\it moving
frame} of $\varPsi$. Then the compatibility condition for the equations
in \eqref{eq:Lax-complex} is
\begin{equation}\label{eq:compatibilit}
U_w - V_z + [V, U] = 0 .
\end{equation}
A direct computation shows that the equation \eqref{eq:compatibilit} can be rephrased as follows:
\begin{equation}\label{eq:GC}
\left\{
\begin{array}{lcr}
u_{zw} - 2 R Q e^{-u} + \frac{1}{2}H^2 e^u =0,\\[0.2cm]
Q_w - \frac{1}{2} H_z e^u= 0, \\[0.2cm]
R_z - \frac{1}{2} H_w e^u = 0.
\end{array}
\right.
\end{equation}
The first equation in \eqref{eq:GC} will be called the {\it
complex Gau{\ss} equation}, and the second and third equations in
\eqref{eq:GC} will be called the {\it complex Codazzi
equations}.
From the discussion above we know that all holomorphic null immersions $\varPsi :
\mathcal M \rightarrow \mathbb C^3$ satisfy the complex Gau{\ss}-Codazzi
equations \eqref{eq:GC}. We note that, setting $\alpha = F^{-1} d
F = U dz + V dw$, the equations in \eqref{eq:GC} are equivalent to
$$d \alpha + \frac{1}{2} [\alpha \wedge \alpha] = 0.$$
Using the functions $u$, $Q$, $R$ and $H$ defined in
\eqref{eq:nullmetric} and \eqref{eq:U-V} respectively, the symmetric quadratic form
$I\!I : = - \langle d \varPsi , d N \rangle$ can be represented as follows:
\begin{equation}\label{eq:second-fund}
I\!I :=- \langle d \varPsi , d N\rangle = Q dz^2 + e^u H dz dw + R dw^2\;.
\end{equation}
The symmetric quadratic form $I \! I$ is called the {\it second
fundamental form} for a holomorphic null immersion $\varPsi$.
Then the complex mean curvature and the complex Gau{\ss} curvature for a
holomorphic null immersion $\varPsi$ are defined as follows.
\begin{Definition}\label{def:Mean}
Let $\varPsi : \mathcal M \to \mathbb C^3$ be a holomorphic null
immersion. Then the function $H = 2 e^{-u}\langle \varPsi_{z
w}, N \rangle$ will be called the {\rm complex mean curvature} of $\varPsi$.
\end{Definition}
\begin{Remark}\label{Rem:Tr}
From the forms of the holomorphic metric $g$ defined in \eqref{eq:nullmetric}
and the second fundamental form $I\!I$ defined in
\eqref{eq:second-fund}, the equality $H = \tfrac{1}{2}{\rm Tr} (\tilde
I^{-1} \cdot \widetilde {I\!I})$ holds, where
$\tilde I$ (resp. $\widetilde {I\!I}$) is the coefficient matrix of $g$ (resp. $I\!I$).
\end{Remark}
\begin{Definition}\label{def:Gauss}
We retain the notation in Remark \ref{Rem:Tr}.
Then the function $K = {\rm det} (\tilde I^{-1} \cdot \widetilde
{I\!I}) = H^2 - 4 e^{-2u} Q R$ will be called the {\rm complex Gau{\ss}
curvature} of $\varPsi$.
\end{Definition}
\subsection{Complex CMC and CGC immersions in $\mathbb
C^3$}\label{subsc:complexCMC}
In this subsection, we give characterizations of complex constant
mean curvature immersions via loop groups, see Appendix \ref{loopgroups}
for the definitions of loop groups.
There is a useful formula representing complex {\sc CMC}-immersions,
which is a generalization of the Sym formula
for {\sc CMC}-immersions in $\mathbb R^3$, see also
\cite{DK:cyl}. There is also a formula for complex {\sc
CGC}-immersions given by the parallel holomorphic immersions of complex
{\sc CMC}-immersions with $H \in \mathbb C^*$.
The notions of a complex {\sc CMC}-immersion and a {\sc CGC}-immersion are defined
analogous to the notions of a {\sc CMC}-immersion and a {\sc
CGC}-immersion in $\mathbb R^3$, see also \cite{DKP:Complex}.
\begin{Definition}
Let $\varPsi: \mathcal M \to \mathbb C^3$ be a holomorphic null
immersion, and let $H$ (resp. K) be its complex mean curvature (resp. Gau{\ss} curvature).
Then $\varPsi$ is called a {\it complex constant mean curvature ({\sc CMC} for short)
immersion} (resp. a {\it complex constant Gau{\ss} curvature ({\sc
CGC} for short) immersion}) if $H$ (resp. $K$) is a complex constant.
\end{Definition}
\begin{Remark}
Since we are interested in complexifications of {\sc CMC}
(resp. {\sc CGC}) surfaces with nonzero mean curvature $H \in \mathbb R^*$
(resp. Gau{\ss} curvature $K \in \mathbb R^*$), from now on, we always assume that the
complex mean curvature $H$ (resp. the complex Gau{\ss} curvature $K$) is
a nonzero constant.
\end{Remark}
From \cite{DKP:Complex}, we quote the following characterizations
of a complex {\sc CMC}-immersion:
\begin{Lemma}
\label{lem:complex-CMC}
Let $\mathcal M$ be a connected 2-dimensional Stein manifold,
and let $\varPsi :\mathcal M \to \mathbb C^3 \cong \mathfrak{sl}(2, \mathbb C)$
be a holomorphic null immersion. Further, let $Q$, $R$, $H$ and $N$ be the complex
functions defined in \eqref{eq:U-V} and the {Gau\ss}
map defined in \eqref{eq:GaussMap}, respectively.
Then the following statements are equivalent:
\begin{enumerate}
\item $H$ is a nonzero constant;
\item $Q$ depends only on $z$ and $R$ depends only on
$w$;
\item $N_{z w} = \rho N$, for some holomorphic function $\rho :
\mathcal M\rightarrow \mathbb C$.
\item\label{itm:extend} There exists $\tilde F(z, w, \lambda) \in
\Lambda SL(2, \mathbb C)_{\sigma}$ such that
\begin{equation*}
\begin{array}{l}
\tilde F(z, w, \lambda)^{-1} d\tilde F(z, w, \lambda)
=\tilde U dz + \tilde V dw, \\
\end{array}
\end{equation*}
where
\begin{equation*}
\left\{
\begin{array}{lcr}
\tilde U =
\begin{pmatrix}
\frac{1}{4} u_z & -\frac{1}{2}\lambda^{-1} H e^{u/2} \\
\lambda^{-1} Q e^{-u/2} & -\frac{1}{4} u_{z}
\end{pmatrix},
\\ \\
\tilde V =
\begin{pmatrix}
-\frac{1}{4} u_w & - \lambda R e^{-u/2} \\
\frac{1}{2}\lambda H e^{u/2} & \frac{1}{4} u_w
\end{pmatrix},
\end{array}
\right.
\end{equation*}
\end{enumerate}
and $\tilde F(z, w, \lambda =1)=F(z, w)$ is the moving frame of $\varPsi$ in \eqref{eq:Lax-complex}.
\end{Lemma}
The $\tilde F(z, w, \lambda)$ defined in \eqref{itm:extend} of
Lemma \ref{lem:complex-CMC} is called the {\it complex extended framing} of a complex
{\sc CMC}-immersion $\varPsi$.
From now on, for simplicity, the symbol $F(z, w, \lambda)$
(resp. $U(z, w, \lambda)$ or $V(z, w, \lambda)$) is used instead of $\tilde
F(z, w, \lambda)$ (resp. $\tilde U(z, w, \lambda)$ or $\tilde V(z, w,
\lambda)$).
There is an immersion formula for a complex {\sc CMC}-immersion using
the complex extended framing $F(z, w, \lambda)$ for a complex {\sc
CMC}-immersion $\varPsi$, the so-called ``Sym formula'', see
\cite{DKP:Complex}. We show a similar immersion formula for a
complex {\sc CGC}-immersion using the same complex extended framing
$F(z, w, \lambda)$ of a complex {\sc CMC}-immersion $\varPsi$.
\begin{Theorem}
\label{thm:Sym-Bob}
Let $F(z, w, \lambda)$ be the complex extended framing of some complex
CMC-immersion defined as in
Lemma \ref{lem:complex-CMC}, and let $H$ be its nonzero complex constant mean curvature. We set
\begin{equation}\label{eq:Sym-Bobenko}
\left\{
\begin{array}{l}
\displaystyle \varPsi = -\frac{1}{2H}\left( i \lambda \partial_{\lambda} F (z, w, \lambda)
\cdot F(z, w, \lambda)^{-1} + \frac{i}{2}F(z, w, \lambda) \sigma_3 F(z, w, \lambda)^{-1}\right), \\[0.1cm]
\displaystyle \varPhi = -\frac{1}{2H}\left( i \lambda \partial_{\lambda} F (z, w, \lambda)
\cdot F(z, w, \lambda)^{-1} \right),
\end{array}
\right.
\end{equation}
where $\sigma_3$ has been defined in \eqref{eq:ident2}.
Then $\varPsi$ (resp. $\varPhi$) is, for
every $\lambda \in \mathbb C^\ast$, a complex constant mean curvature
immersion (resp. complex constant Gau{\ss}ian curvature
immersion, possibly degenerate) in
$\mathbb C^3$ with complex mean curvature $H \in \mathbb C^*$
(resp. complex Gau{\ss}
curvature $K = 4 H^2 \in \mathbb C^*$),
and the Gau{\ss} map of $\varPsi$ (resp. $\varPhi$) can be described
by $\tfrac{i}{2}F(z, w, \lambda)\sigma_3 F(z, w, \lambda)^{-1}$.
\end{Theorem}
\begin{proof}
The proof for complex {\sc CMC}-immersions follows from
\cite{DKP:Complex}. We show that the second formula $\varPhi$ in
\eqref{eq:Sym-Bobenko} defines a complex {\sc CGC}-immersion.
Let $\varPhi$ be a map in the second formula in \eqref{eq:Sym-Bobenko}.
Let $N=\tfrac{i}{2}F(z, w, \lambda)\sigma_3 F(z, w, \lambda)^{-1}$ be
the complex Gau{\ss} map for $\varPsi$. Since $\langle N, N \rangle =1$
and the relation $\varPhi = \varPsi + \tfrac{1}{2 H} N$
holds for the formulas in \eqref{eq:Sym-Bobenko}, $N$ is also the Gau{\ss} map for
$\varPhi$, i.e., $\langle \varPhi_{z}, N\rangle$ =
$\langle\varPhi_{w}, N \rangle = 0$. We also denote the Gau{\ss} map for
$\varPhi$ by $N$.
We then compute the holomorphic metric $g$ and the second fundamental
form $I\!I$ for the holomorphic map $\varPhi$. Using the bilinear form
in \eqref{eq:bilinear}, we have
\begin{equation*}
\left\{
\begin{array}{l}
\displaystyle \langle \varPhi_{z}, \varPhi_{z}\rangle = -2 {\rm Tr} \left(\varPhi_{z}
\cdot \varPhi_{z} \right) = \frac{\lambda^2}{2 H^2} {\rm Tr}\left({\rm
Ad} (F)U_{\lambda}^2\right), \\
\displaystyle \langle \varPhi_{w}, \varPhi_{w} \rangle = -2 {\rm Tr} \left(\varPhi_{w}
\cdot \varPhi_{w} \right) =\frac{\lambda^2}{2 H^2} {\rm Tr}
\left({\rm Ad} (F)V_{\lambda}^2\right),
\end{array}
\right.
\end{equation*}
where $U$ and $V$ are defined in Lemma \ref{lem:complex-CMC}, and the
subscript $z$ (resp. $w$ or $\lambda$) denotes the partial derivative
with respect to $z$ (resp. $w$ or $\lambda$). Since the trace of a matrix
is invariant under the map ${\rm Ad} (F)$, and using the form of $U$ in
Lemma \ref{lem:complex-CMC}, we have
\begin{equation*}
\left\{
\begin{array}{l}
\displaystyle \langle \varPhi_{z}, \varPhi_{z}\rangle = \frac{\lambda^2}{2 H^2} {\rm Tr}
(U_{\lambda}^2) = -\frac{1}{2 H^2}
\lambda^{-2} HQ, \\[0.3cm]
\displaystyle \langle \varPhi_{w}, \varPhi_{w}\rangle = \frac{\lambda^2}{2 H^2} {\rm Tr}
(V_{\lambda}^2) =-\frac{1}{ 2H^2}
\lambda^2 H R.
\end{array}
\right.
\end{equation*}
Using again the invariace of trace of a matrix under the map ${\rm Ad (F)}$ and
the forms of $U$ and $V$ in Lemma \ref{lem:complex-CMC}, we have
\begin{flalign*}
\langle \varPhi_{z}, \varPhi_{w}\rangle &= -2 {\rm Tr}\left( -\frac{i \lambda}{2 H} {\rm
Ad} (F)U_{\lambda} \times-\frac{i\lambda}{2 H} {\rm
Ad} (F)V_{\lambda} \right)\\ &= \frac{1}{2 H^2} \left(\frac{1}{4} H^2 e^u +
Q R e^{-u}\right).
\end{flalign*}
Therefore, we have the following first fundamental form for the
holomorphic map $\varPhi$:
\begin{equation}\label{eq:firstfork}
g = \begin{pmatrix}dz & dw\end{pmatrix}
\begin{pmatrix}
-\frac{1}{2 H^2} \lambda^{-2} H Q & \frac{1}{2 H^2}\left( \frac{1}{4} H^2 e^u + Q R e^{-u}\right) \\[0.5cm]
\frac{1}{2 H^2}\left( \frac{1}{4} H^2 e^u + Q R e^{-u}\right) & -\frac{1}{2 H^2} \lambda^{2} H R
\end{pmatrix}
\begin{pmatrix} dz \\ dw\end{pmatrix}
\;.
\end{equation}
Let $\tilde I$ denote the coefficient matrix for $g$ in
\eqref{eq:firstfork}. Then it is easy to verify that $\det \tilde I =
-(\frac{1}{4} H^2 e^u - Q R e^{-u})^2/(4 H^4)$.
Thus the holomorphic map $\varPhi$ actually defines a holomorphic immersion
under the condition $e^{2 u} \neq 4 H^{-2} Q R$.
Next, the second fundamental form $I\!I$ for $\varPhi$ is computed
as follows. Using again the invariace of the trace of a matrix under the map ${\rm Ad (F)}$,
$\langle \varPhi_{z}, N_{z}\rangle$ is computed as follows:
\begin{flalign*}
\langle \varPhi_{z}, N_{z}\rangle & = -2 {\rm Tr}\left( -\frac{i
\lambda}{2 H} {\rm Ad} (F)U_{\lambda}
\times \frac{i}{2} {\rm Ad} (F) [U,\sigma_3]\right)
\\ & = -\frac{1}{2 H} {\rm Tr}
(U_{\lambda}\cdot [U, \sigma_3])\;.
\end{flalign*}
From the form of $U$ in Lemma
\ref{lem:complex-CMC}, we have $\langle \varPhi_{z}, N_{z}\rangle = 0$.
A similar argument holds for $\langle \varPhi_{w}, N_{w} \rangle$, where $U$ is replaced by $V$.
Thus we have $\langle \varPhi_{w}, N_{w} \rangle = 0$.
Using again the invariance of the trace of a matrix under the map ${\rm Ad} (F)$
and the form of $U$ in Lemma
\ref{lem:complex-CMC}, we obtain
\begin{flalign*}
\langle \varPhi_{z}, N_{w}\rangle &= -2 {\rm Tr}\left( -\frac{i
\lambda}{2 H} {\rm Ad} (F)U_{\lambda}
\times \frac{i}{2} {\rm Ad} (F) [V,\sigma_3]\right)
\\& = -\frac{1}{2 H} \left(\frac{1}{2}
H^2 e^u - 2 QR e^{-u}\right).
\end{flalign*}
Since $N$ is the Gau{\ss} map of $\varPhi$, i.e., $\langle \varPhi_{z}, N\rangle$ =
$\langle\varPhi_{w}, N \rangle = 0$ and $\langle N,
N\rangle=1$, we obtain $\langle \varPhi_{z},
N_{w}\rangle =- \langle \varPhi_{wz}, N\rangle = \langle
\varPhi_{w}, N_{z}\rangle$.
Finally, the second fundamental form $I\!I$ for $\varPhi$ has
the following form:
\begin{equation}\label{eq:secondfork}
I\!I = \begin{pmatrix}dz & dw\end{pmatrix}
\begin{pmatrix} 0 & \frac{1}{2 H}\left( \frac{1}{2} H^2 e^u -2 Q R e^{-u}\right) \\[0.5cm]
\frac{1}{2 H}\left( \frac{1}{2} H^2 e^u -2 Q R e^{-u}\right)& 0
\end{pmatrix}
\begin{pmatrix} dz \\ dw\end{pmatrix}
\;.
\end{equation}
Let us denote the coefficient matrix of $I\!I$ by $\widetilde {I\!I}$.
Then, using \eqref{eq:firstfork} and \eqref{eq:secondfork}, the
complex Gau{\ss} curvature $K$ for $\varPhi$ is computed as
\begin{equation*}
K = {\rm det }( \tilde I^{-1} \cdot \widetilde {I\!I} ) = 4 H^2 \in
\mathbb C^*\;.
\end{equation*}
This completes the proof.
\end{proof}
Since the Gau{\ss} maps of a complex {\sc CMC}-immersion $\varPsi$
and the corresponding complex {\sc CGC}-immersion $\varPhi$ are the same, which is
$N = \tfrac{i}{2}F(z, w, \lambda)\sigma_3 F(z, w, \lambda)^{-1}$,
we have the following corollary:
\begin{Corollary}
Let $\varPsi$ be a complex {\sc CMC}-immersion, and let $H$ (resp. $u$,
$Q$ and $R$) be its nonzero constant mean curvature (resp. the
functions defined in \eqref{eq:nullmetric} and Theorem \ref{thm:MovingFrame}).
Moreover, let us assume $e^{2 u} \neq 4 H^{-2} Q R$. Then there exists the parallel complex
{\sc CGC}-immersion with Gau{\ss} curvature $K = 4 H^2 \in \mathbb C^*$.
\end{Corollary}
\subsection{Real forms of $\Lambda \mathfrak{sl}(2, \mathbb
C)_{\sigma}$}\label{subsc:realform}
In this subsection, we give the classification of real forms
for the twisted $\mathfrak{sl}(2, \mathbb C)$ loop algebra $\Lambda
\mathfrak{sl}(2, \mathbb C)_{\sigma}$, see Appendix \ref{BasicResult}
for the notation and the definitions of loop algebras.
First we recall the basic facts about real forms for complex Kac-Moody
Lie algebras.
Let $\mathfrak g$ be a Kac-Moody Lie algebra over $\mathbb C$. Then a Lie
subalgebra $\mathfrak g_{\mathbb R} \subset \mathfrak g$ over
$\mathbb R$ will be called the {\it real form} of $\mathfrak g$ if there is an
isomorphism between $\mathfrak g$ and the complexification $\mathfrak
g_{\mathbb R} \otimes \mathbb C$.
We note that all real forms of $\mathfrak g$ correspond to semi-linear
involutions of $\mathfrak g$, see, for example
\cite{BR:Kac-Moody}, i.e., each real form is
defined by an automorphism $\rho$ of $\mathfrak g$ such that
\begin{equation}\label{eq:semi-involution}
\left\{
\begin{array}{l}
\rho^2 = {\rm id}, \\
\rho(\ell x) = \bar \ell \rho (x)\;\;\; \mbox{for}\;\; \ell \in
\mathbb C.
\end{array}
\right.
\end{equation}
Let $\mathfrak h$ denote the {\it standard Cartan subalgebra} of $\mathfrak
g$, which is a maximal $\mbox{ad}(\mathfrak g)$--diagonalizable
subalgebra of $\mathfrak g$. Let $\Delta$ be the corresponding root
system, and let $\mathfrak g_{\alpha}$ denote the root space corresponding
to $\alpha$ in $\Delta$. Then the root space
decomposition for $\mathfrak g$ is as follows:
\begin{equation*}
\mathfrak g = \mathfrak h \oplus \left(\bigoplus_{\alpha \in \Delta}
\mathfrak g_{\alpha}\right)
\;.
\end{equation*}
It is known that $\Delta$ can be decomposed as $\Delta = \Delta^{+}
\cup \Delta^{-}$, where $\Delta^{+}$ (resp. $\Delta^{-}$) is the set of
positive (resp. negative) roots.
Then a subalgebra of $\mathfrak g$ is said to be a {\it Borel subalgebra} if it is a maximal
completely solvable subalgebra. And the {\it standard positive (resp. negative) Borel
subalgebra} $\mathfrak b^{+}$ (resp. $\mathfrak b^{-}$) of $\mathfrak
g$ is defined as follows:
\begin{equation*}
\mathfrak b^{\pm} = \mathfrak h \oplus \left(\bigoplus_{\beta \in \Delta^{\pm}}
\mathfrak g_{\beta}\right)
\;.
\end{equation*}
If a linear or semi-linear automorphism $\rho$ for $\mathfrak g$ transforms a
Borel subalgebra into a Borel subalgebra of the same (resp. opposite)
sign, then $\rho$ is said to be the {\it first kind} (resp. {\it second kind}).
We now give definitions of the almost split real forms and the almost
compact real forms of $\mathfrak g$.
\begin{Definition}
Let $\mathfrak g$ be a Kac-Moody Lie algebra over $\mathbb C$, and let
$\mathfrak g_{\mathbb R}$ be a real form of $\mathfrak g$. Moreover,
let $\rho$ be the semi-linear involution corresponding to $\mathfrak
g_{\mathbb R}$. Then the real
form $\mathfrak g_{\mathbb R}$ is called {\rm almost split}
(resp. {\rm almost compact}) if the corresponding semi-linear
involution $\rho$ is of the first kind (resp. second kind).
\end{Definition}
It is clear that the real subalgebra of $\mathfrak g$ generated by
$\{\mathfrak h, e_j, f_j \;;\;j = 1, 2, \dots, n \}$, the Cartan
subalgebra and the Chevalley generators of
the Kac-Moody Lie algebra $\mathfrak g$, see Appendix
\ref{subsc:Affine}, is
an almost split real form, which is called the {\it standard split
form}. The corresponding semi-linear involution of
the first kind $\sigma_n^{\prime}$ is called the {\it standard normal
semi-involution} of $\mathfrak g$. The map $e_j \mapsto -f_j$, $f_j
\mapsto -e_j$ and $h \mapsto -h$, $h \in \mathfrak h$ for $\{\mathfrak h, e_j,
f_j \;;\;j = 1, 2, \dots, n \}$ can be extended to an involution
$\omega$ of $\mathfrak g$. The $\omega$ is called the {\it Cartan
involution} of $\mathfrak g$. It is known that the standard normal
semi-involution and the Cartan involution commute.
\begin{Definition}
Let $\omega$ and $\sigma_n^{\prime}$ be the Cartan involution and the
standard normal semi-involution respectively, and let
$\omega^{\prime}$ be $\omega^{\prime} = \sigma_n^{\prime} \omega =
\omega \sigma_n^{\prime}$. Then $\omega^{\prime}$ is called {\rm the
standard Cartan semi-involution},
and the corresponding almost compact real form is called {\rm the
standard compact form}. Moreover, a conjugation of $\omega^{\prime}$ is
called a {\rm Cartan semi-involution}.
\end{Definition}
We quote the following theorem about the real forms of
Kac-Moody Lie algebras \cite{BBBR:almostsplit},
\cite{BR:almostcompact}.
\begin{Theorem}[Theorem 4.4 in \cite{BBBR:almostsplit}, Proposition 2.9
in \cite{BR:almostcompact}]\label{thm:Ross}
Let us consider the following:
\begin{enumerate}
\item[(1)] The semi-linear involutions $\rho$ of $\mathfrak{g}$ of the
second kind (resp. the first kind).
\item[(2)] The involutions $\theta$ of $\mathfrak{g}$ of the first
kind (resp. the second kind).
\item[(3)] The relation $\rho \thickapprox \theta$ if and only if
\begin{itemize}
\item[(a)] $\omega^{\prime} = \theta \rho = \rho \theta$ is a Cartan semi-involution.
\item[(b)] $\theta$ and $\rho$ stabilize the same Cartan subalgebra $\mathfrak{h}$.
\item[(c)] $\mathfrak{h}$ is contained in a minimal $\rho$-stable
positive parabolic subalgebra.
\end{itemize}
\end{enumerate}
Then the relation induces a bijection between the conjugacy classes
under $Aut(\mathfrak{g})$ of semi-linear involutions of the second kind
(resp. the first kind) and conjugacy classes of involutions of
the first kind (resp. the second kind).
\end{Theorem}
We also quote the following theorem about the classification of the
involutions of the affine Kac-Moody Lie algebra of $A_1^{(1)}$ type
\cite{Kob:AutoKac}.
\begin{Theorem}[Theorem 3 in \cite{Kob:AutoKac}]\label{thm:ZKob}
All involutions on the affine Kac-Moody Lie algebra of type $A_1^{(1)}$
are given as follows:
\begin{itemize}
\item[(a)] $e_1 \longmapsto - e_1, \;\;\;\; f_1 \longmapsto -f_1, \;\;\;\; e_2 \longmapsto
-e_2, \;\;\;\; f_2 \longmapsto - f_2$,\vspace{0.2cm}
\item[(a$^{\prime}$)]$e_1 \longmapsto e_2, \;\;\;\; f_1 \longmapsto f_2, \;\;\;\; e_2 \longmapsto
e_1, \;\;\;\; f_2 \longmapsto f_1$,\vspace{0.2cm}
\item[(b)]$e_1 \longmapsto e_1, \;\;\;\; f_1 \longmapsto f_1, \;\;\;\; e_2 \longmapsto
-\frac{1}{2}[[f_2, f_1],f_1], \;\;\;\; f_2 \longmapsto -
\frac{1}{2}[[e_2, e_1],e_1]$,\vspace{0.2cm}
\item[(b$^{\prime}$)]$e_1 \longmapsto -e_1, \;\;\;\; f_1 \longmapsto -f_1, \;\;\;\; e_2 \longmapsto
\frac{1}{2}[[f_2, f_1],f_1], \;\;\;\; f_2 \longmapsto
\frac{1}{2}[[e_2, e_1],e_1]$,\vspace{0.2cm}
\item[(b$^{\prime \prime}$)]$e_1 \longmapsto f_2, \;\;\;\; f_1 \longmapsto e_2, \;\;\;\; e_2 \longmapsto
f_1, \;\;\;\; f_2 \longmapsto e_1$,\vspace{0.2cm}
\item[(c)]$e_1 \longmapsto e_1, \;\;\;\; f_1 \longmapsto f_1, \;\;\;\; e_2 \longmapsto
-e_2, \;\;\;\; f_2 \longmapsto - f_2$,
\end{itemize}
where $e_j, f_j$ for $j \in \{1, 2\}$ are the Chevalley generators of
the affine Kac-Moody Lie algebra of $A_1^{(1)}$ type. Moreover the cases
{\rm (a), (a$^{\prime}$)} and {\rm (c)} (resp. {\rm
(b)}, {\rm (b$^{\prime}$)} and {\rm (b$^{\prime \prime}$)})
are involutions of the first kind (resp. the second kind).
\end{Theorem}
It is well known that the twice central
extensions of the untwisted $\mathfrak{sl}(2, \mathbb C)$ loop algebra
$\Lambda \mathfrak{sl}(2, \mathbb C)$ is an affine
Kac-Moody Lie algebra of $A_{1}^{(1)}$ type, see Theorem
\ref{thm:Kac}. It is also known that the
twisted $\mathfrak{sl}(2, \mathbb C)$ loop algebra $\Lambda
\mathfrak{sl}(2, \mathbb C)_{\sigma}$ and the
untwisted $\mathfrak{sl}(2, \mathbb C)$ loop algebra $\Lambda \mathfrak{sl}(2, \mathbb C)$
are isomorphic by the following map from $\Lambda \mathfrak{sl} (2,
\mathbb C)$ to $\Lambda \mathfrak{sl} (2, \mathbb C)_\sigma$, see also \cite{Kac:Kac-Moody}:
\begin{equation}\label{eq:isountwist}
g(\lambda) \in \Lambda \mathfrak{sl} (2, \mathbb C) \mapsto {\rm Ad}
\left(\begin{smallmatrix} \sqrt{\lambda} & 0
\\0& \sqrt{\lambda}^{-1}\end{smallmatrix}\right) g(\lambda^2) \in \Lambda \mathfrak{sl} (2, \mathbb C)_{\sigma}\;.
\end{equation}
Therefore we have the following
classification of all real forms for $\Lambda \mathfrak{sl}(2, \mathbb C)_\sigma$.
\begin{Theorem}\label{thm:almostcompact}
Let $\mathfrak{c}_j$ for $j \in \{1, 2, 3, 4 \}$ be the following
involutions on $\Lambda \mathfrak{sl} (2, \mathbb C)_\sigma$:
\begin{equation}\label{eq:classcom}
\begin{array}{ll}
\mathfrak{c}_1 : g(\lambda) \mapsto -\overline{g(-1/\bar
\lambda)}^{t},\;\; & \mathfrak{c}_2: g(\lambda) \mapsto \overline{g \left(- 1/\bar \lambda\right)},\\
\mathfrak{c}_3: g(\lambda) \mapsto - \overline{g \left(1/\bar \lambda\right)}^t\;,\;\; &
\mathfrak{c}_4: g(\lambda) \mapsto
-{\rm Ad} \left(\begin{smallmatrix} 1/\sqrt{i} & 0 \\ 0 & \sqrt{i}
\end{smallmatrix}\right) \overline{g(i/ \bar \lambda)}^{t},
\end{array}
\end{equation}
where $g(\lambda) \in \Lambda \mathfrak{sl}(2, \mathbb C)_\sigma$.
Then, the almost compact real forms of
$\Lambda \mathfrak{sl}(2, \mathbb C)_{\sigma}$
are the following real Lie subalgebras of $\Lambda \mathfrak{sl}(2, \mathbb C)_{\sigma}$:
\begin{equation}
\Lambda \mathfrak{sl}(2, \mathbb C)_{\sigma}^{(\mathfrak{c}, j)} = \left\{ g(\lambda ) \in \Lambda
\mathfrak{sl}(2, \mathbb C)_{\sigma} \;\left|\; \mathfrak{c}_j \circ g (\lambda) = g(\lambda) \right. \right\}
\;\;\mbox{for}\;\;j\in \{1, 2, 3, 4\}.
\end{equation}
\end{Theorem}
\begin{Theorem}\label{thm:almostsplit}
Let $\mathfrak{s}_j$ for $j \in \{1, 2, 3\}$ be the following
involutions on $\Lambda \mathfrak{sl} (2, \mathbb C)_\sigma$:
\begin{equation}\label{eq:classsplit}
\begin{array}{ll}
\mathfrak{s}_1 : g(\lambda) \mapsto -\overline{g(-\bar
\lambda)}^{t},\;\; &
\mathfrak{s}_2 : g(\lambda) \mapsto \overline{g \left(- \bar \lambda\right)},\\
\mathfrak{s}_3: g(\lambda) \mapsto - {\rm Ad}
\left(\begin{smallmatrix} \lambda & 0 \\ 0 & \lambda^{-1}
\end{smallmatrix}\right) \overline{g \left( \bar
\lambda\right)}^t\;,\;\; &
\end{array}
\end{equation}
where $g(\lambda) \in \Lambda \mathfrak{sl}(2, \mathbb C)_\sigma$.
Then, the almost split real forms of $\Lambda
\mathfrak{sl}(2, \mathbb C)_{\sigma}$ are the following real Lie
subalgebras of $\Lambda \mathfrak{sl}(2, \mathbb C)_{\sigma}$:
\begin{equation}
\Lambda \mathfrak{sl}(2, \mathbb C)_{\sigma}^{(\mathfrak{s}, j)} = \left\{ g(\lambda ) \in \Lambda
\mathfrak{sl}(2, \mathbb C)_{\sigma} \;\left|\; \mathfrak{s}_j \circ g (\lambda) = g(\lambda) \right. \right\}
\;\;\mbox{for}\;\;j\in \{1, 2, 3\}.
\end{equation}
\end{Theorem}
\begin{proof}
Since all linear involutions of the first kind and the
second kind for the affine Kac-Moody Lie algebra of $A_1^{(1)}$ type
are classified in Theorem \ref{thm:ZKob}, and using Theorem \ref{thm:Ross},
all semi-linear involutions $\rho$ of the first kind (resp. the second
kind) can be represented as follows:
$$
\rho = \omega^{\prime} \theta\;,
$$
where $\omega^{\prime}$ is the Cartan semi-involution and
$\theta$ is the linear involution of the second kind (resp. the
first kind). Noting that the identity map is also the trivial involution of
the first kind, we have the four classes of semi-linear involutions of
the second kind and the three classes of semi-linear involutions of the
first kind. Since the affine Kac-Moody Lie algebra of $A_1^{(1)}$ type can be
realized by the twice central extensions of the loop algebra $\Lambda
\mathfrak{sl}(2, \mathbb C)$ (see Theorem \ref{thm:Kac}), the real
forms of the loop algebra $\Lambda \mathfrak{sl}(2, \mathbb C)$ are
derived. Finally, using the isomorphism
\eqref{eq:isountwist}, the real forms of $\Lambda
\mathfrak{sl}(2, \mathbb C)$ are transformed into the real forms of
$\Lambda \mathfrak{sl} (2, \mathbb C)_{\sigma}$, which are obtained in
\eqref{eq:classcom} and \eqref{eq:classsplit} by a direct calculation.
This completes the proof.
\end{proof}
\begin{Remark}
The identification \eqref{eq:isountwist} implies, in the untwisted
setting, the involution $\mathfrak{s}_3$ can be rephrased as follows:
\begin{equation}\label{eq:invs3}
\mathfrak{s}_3: g(\lambda) \mapsto -\overline{g \left( \bar
\lambda\right)}^t
\;\; \mbox{for}\;\;g(\lambda) \in \Lambda \mathfrak{sl}(2, \mathbb C).
\end{equation}
From now on
we use the involution \eqref{eq:invs3} instead of the original
involution $\mathfrak{s}_3$ in \eqref{eq:classsplit}.
\end{Remark}
\section{Real forms of complex CGC-immersions}\label{sc:Realforms}
In this section, we show one of the main theorems in this paper (Theorem
\ref{thm:compactsplit}), which is the classification of ``integrable
surfaces'' obtained from all the real forms of the twisted $\mathfrak{sl}(2,
\mathbb C)$ loop algebra $\Lambda \mathfrak{sl}(2, \mathbb
C)_{\sigma}$.
\subsection{Integrable surfaces as real forms of complex
CGC-immersions}\label{subsc:Integrablesurf}
Let $F(z, w, \lambda) \in \Lambda SL(2, \mathbb C)_{\sigma}$ be the
complex extended framing of some complex {\sc CGC}-immersion
$\varPhi$. And let $\alpha(z, w, \lambda) = F(z, w, \lambda)^{-1} d F(z, w,
\lambda)$ be the Maurer-Cartan form of $F(z, w, \lambda)$.
From the forms of $U$ and $V$ defined as in Lemma \ref{lem:complex-CMC}, we
set $\alpha_i \; (i \in \{-1, 0, 1\})$ as follows:
\begin{equation}\label{eq:alpha}
\alpha(z, w, \lambda) = F^{-1} d F= U dz + V dw = \lambda^{-1} \alpha_{-1} + \alpha_0 + \lambda \alpha_1 \;,
\end{equation}
where
\begin{equation}\label{eq:alpha2}
\left\{
\begin{array}{l}
\alpha_{-1} = \begin{pmatrix}0 & -\frac{1}{2} H e^{u/2}dz, \\ Q
e^{-u/2}dz & 0\end{pmatrix},\\[0.5cm]
\alpha_{0} =\begin{pmatrix} \frac{1}{4} u_z dz - \frac{1}{4} u_w dw & 0
\\ 0 & -\frac{1}{4} u_z dz + \frac{1}{4} u_w dw\end{pmatrix},\\[0.5cm]
\alpha_{1} =\begin{pmatrix} 0 & -R e^{-u/2}dw \\ \frac{1}{2} H e^{u/2}dw
& 0 \end{pmatrix}.
\end{array}
\right.
\end{equation}
We denote the space of $\Lambda \mathfrak{sl}(2, \mathbb C)_{\sigma}$ valued
$1$-forms by $\Omega (\Lambda \mathfrak{sl}(2, \mathbb C)_\sigma)$.
Similar to the involutions in Theorem \ref{thm:almostcompact}
(resp. Theorem \ref{thm:almostsplit}), we define the involutions
$\tilde {\mathfrak{c}}_j$ (resp. $\tilde{\mathfrak{s}}_j$) for $
g(\lambda) \in \Omega
(\Lambda \mathfrak{sl}(2, \mathbb C)_\sigma)$ as follows:
{\small
\begin{equation}\label{eq:inv-1-forms}
\begin{array}{ll}
\left\{
\begin{array}{l}
\tilde{\mathfrak{c}}_1 : g(\lambda) \mapsto -\overline{g(-1/\bar
\lambda)}^{t},\;\; \\[0.2cm]
\tilde{\mathfrak{c}}_2: g(\lambda) \mapsto \overline{g \left(- 1/\bar \lambda\right)},\\[0.2cm]
\tilde{\mathfrak{c}}_3: g(\lambda) \mapsto - \overline{g \left(1/\bar \lambda\right)}^t\;,\;\; \\[0.2cm]
\tilde{\mathfrak{c}}_4: g(\lambda) \mapsto -{\rm Ad} \left(\begin{smallmatrix} 1/\sqrt{i} & 0 \\ 0 & \sqrt{i}
\end{smallmatrix}\right) \overline{g(i/ \bar \lambda)}^{t},
\end{array}
\right.
\hspace{1cm}
\left\{
\begin{array}{l}
\tilde{ \mathfrak{s}}_1 : g(\lambda) \mapsto -\overline{g(-\bar
\lambda)}^{t},\;\; \\[0.2cm]
\tilde{ \mathfrak{s}}_2 : g(\lambda) \mapsto \overline{g \left(- \bar
\lambda\right)}, \\[0.2cm]
\tilde{ \mathfrak{s}}_3: g(\lambda) \mapsto - \overline{g \left( \bar \lambda\right)}^t.
\end{array}
\right.
\end{array}
\end{equation}
}
Then the real forms of $\Omega(\Lambda \mathfrak{sl} (2, \mathbb
C)_{\sigma}^{(\mathfrak{c}, j )})$ are defined as follows:
\begin{equation}\label{eq:involutionscj}
\begin{array}{l}
\Omega(\Lambda \mathfrak{sl} (2, \mathbb C)_{\sigma}^{(\mathfrak{c}, j )}) =
\left\{ g \in \Omega(\Lambda
\mathfrak{sl}(2, \mathbb C)_{\sigma})
\;\left|\; \tilde {\mathfrak{c}}_j \circ g(\lambda ) =
g(\lambda) \right. \right\}\;, \\
\Omega(\Lambda \mathfrak{sl} (2, \mathbb C)_{\sigma}^{(\mathfrak{s}, j )}) =
\left\{ g \in \Omega(\Lambda
\mathfrak{sl}(2, \mathbb C)_{\sigma})
\;\left|\; \tilde{\mathfrak{s}}_j \circ g(\lambda ) =
g(\lambda) \right. \right\}\;.
\end{array}
\end{equation}
From now on, for simplicity, we use the symbols $\mathfrak{c}_j$ and
$\mathfrak{s}_j$ instead of $\tilde{\mathfrak{c}}_j$ and $\tilde{\mathfrak{s}}_j$
respectively.
We now consider the following conditions on $\alpha(z, w, \lambda)$:
\begin{itemize}
\item\label{itm:Almostcompact} {\bf Almost compact cases $(C, j)$:} $\alpha(z,
w, \lambda)$ is an element in the real form $\Omega(\Lambda
\mathfrak{sl} (2, \mathbb C)_{\sigma}^{(\mathfrak{c}, j )})$. \\[0.05cm]
\item {\bf Almost split cases $(S, j)$:} $\alpha(z, w, \lambda)$
is an element in the real form $\Omega(\Lambda
\mathfrak{sl} (2, \mathbb C)_{\sigma}^{(\mathfrak{s}, j )})$.
\end{itemize}
A straightforward computation shows that the conditions above,
which are the almost compact cases $(C, j)$ and the
almost split cases $(S, j)$, are equivalent to the
following equations for $\alpha_i \;(i \in\{ -1, 0, 1\})$:
\begin{equation}\label{eq:conditionA}
\left\{
\begin{array}{ll}
\alpha_0 = - \overline{ \alpha_0}\;\;\mbox{and} \;\;\alpha_{\pm j} = \overline
{\alpha_{\pm 1}}^t\; & \mbox{for the $(C, 1)$ or $(S, 1)$ case,} \\
\alpha_0 = \overline{ \alpha_0}\;\;\mbox{and}\;\;\alpha_{\pm j} = - \overline
{\alpha_{\pm 1}}\; & \mbox{for the $(C, 2)$ or $(S, 2)$ case,} \\
\alpha_0 = - \overline{ \alpha_0}\;\;\mbox{and}\;\;\alpha_{\pm j} = - \overline
{\alpha_{\pm 1}}^t\; & \mbox{for the $(C, 3)$ or $(S, 3)$ case,} \\[0.1cm]
\alpha_0 = -\overline{ \alpha_0}\;\;\mbox{and}\;\;\alpha_{-1} = i {\rm
Ad } \left(\begin{smallmatrix} 1/\sqrt{i} & 0 \\ 0 & \sqrt{i}\end{smallmatrix}\right)\overline
{\alpha_{1}}^t\; & \mbox{for the $(C, 4)$ case,}
\end{array}
\right.
\end{equation}
where $j = -1$ (resp. $j = 1$) if $\alpha$ satisfies one of the
conditions for almost compact cases (resp. almost split cases).
From the symmetry between $\alpha_1$ and $\alpha_{-1}$ for the almost compact
cases and the symmetries on each of $\alpha_1$ and $\alpha_{-1}$ for the almost
split cases, we obtain
\begin{equation}\label{eq:coordinates}
\left\{
\begin{array}{l}
w = \bar z\;\; \mbox{for the almost compact cases $(C, j)$,} \\
z = \bar z\;\;\mbox{and}\;\; w = \bar w\;\;\mbox{for the almost split
cases $(S, j)$.}
\end{array}
\right.
\end{equation}
Moreover the following choices, which are unique up to
constants, of $u$, $Q$, $R$ and $H$ for $\alpha(z, w,
\lambda)$ in \eqref{eq:alpha2} give solutions for
\eqref{eq:conditionA}:
\begin{equation}\label{eq:solforMaurer}
\left\{
\begin{array}{cr}
u \in \mathbb R\;, R = - \bar Q\;, H \in i
\mathbb R^*\;\; &\mbox{for the $(C, 1)$ case,} \\[0.05cm]
u \in i \mathbb R\;, R = Q = - \frac{1}{2}
\bar H\;, H \in \mathbb C^*\;\; &\mbox{for the $(C, 2)$ case,} \\[0.05cm]
u \in \mathbb R\;, R = \bar Q\;, H \in\mathbb R^*\;\; &\mbox{for
the $(C, 3)$ case,}\\[0.05cm]
u \in \mathbb R\;, R = \bar Q\;, H \in i \mathbb R^*\;\; &\mbox{for the $(C, 4)$ case,}
\\[0.1cm]
u \in i \mathbb R\;, Q = R = - \frac{1}{2}\bar H, H \in\mathbb C^* \;\;
&\mbox{for the $(S, 1)$ case,} \\[0.05cm]
u \in \mathbb R\;, Q, R \in i \mathbb R, H \in i \mathbb R^*\;\; &\mbox{for the $(S, 2)$ case,}\\[0.05cm]
u \in i \mathbb R\;, Q = R = \frac{1}{2} \bar H, H \in \mathbb C^* \;\;
&\mbox{for the $(S, 3)$ case.}
\end{array}
\right.
\end{equation}
We denote loop groups whose loop algebras are $\Lambda \mathfrak{sl}(2,
\mathbb C)_{\sigma}^{(\mathfrak{c}, j)}$ and $\Lambda \mathfrak{sl}(2, \mathbb
C)_{\sigma}^{(\mathfrak{s}, j)}$ by
\begin{equation}\label{eq:RealformGroup}
\Lambda SL(2, \mathbb C)_{\sigma}^{(\mathfrak{c}, j)}\;\mbox{for}\;\; j \in \{1, 2, 3, 4\} \;\;\mbox{and}\;\;
\Lambda SL(2, \mathbb C)_{\sigma}^{(\mathfrak{s}, j)}\;\mbox{for}\;\; j
\in \{1, 2, 3\}.
\end{equation}
If the Maurer-Cartan form $\alpha =F^{-1}dF$ is in
$\Omega(\Lambda\mathfrak{sl}(2, \mathbb C)_{\sigma}^{(\mathfrak{c},
j)})$ for $j \in \{1, 2, 3, 4\}$ (resp. $\Omega(\Lambda \mathfrak{sl}(2, \mathbb
C)_{\sigma}^{(\mathfrak{s}, j)})$ for $j \in \{1, 2, 3\}$), the corresponding complex
extended framing $F$ is in $\Lambda SL(2, \mathbb
C)_{\sigma}^{(\mathfrak{c}, j)}$ (resp. $\Lambda SL(2, \mathbb
C)_{\sigma}^{(\mathfrak{s}, j)}$) under the initial condition $F(z_*, w_*,
\lambda) ={\rm id}$ with $(z_*, w_*) = (z_*, \bar z_*) \in \mathfrak
D^2$ (resp. $(z_*, w_*) = (\bar z_*, \bar w_*) \in \mathfrak D^2$). We
denote the complex extended framing $F$ which is a loop in $\Lambda
SL(2, \mathbb C)_{\sigma}^{(\mathfrak{c}, j)}$ (resp. $\Lambda SL(2, \mathbb
C)_{\sigma}^{(\mathfrak{s}, j)}$) by $F^{(\mathfrak{c}, j)}$
(resp. $F^{(\mathfrak{s}, j)}$).
We now set the following formulas $\varPhi^{(\mathfrak{c},j)}$ for
$j \in \{1, 2, 3, 4\}$ (resp. $\varPhi^{(\mathfrak{s},j)}$ for $j
\in \{1, 2, 3\}$) analogous to the second formula in \eqref{eq:Sym-Bobenko}:
\begin{align}
\varPhi^{(\mathfrak{c},j)} &= \displaystyle
\left.-\frac{1}{2|H|}\left( i \lambda \partial_{\lambda}
F^{(\mathfrak{c}, j)} (z, \bar z, \lambda)
\cdot F^{(\mathfrak{c}, j)} (z, \bar
z, \lambda)^{-1}
\right)\right|_{\lambda \in S^1} \mbox{for $j
\in \{1, 2, 3\},$} \label{eq:Sym-Bobenko-2}\\
\varPhi^{(\mathfrak{c},4)} &= \frac{1}{2}\left.\left(F^{(\mathfrak{c}, 4)} (z, \bar z,
\lambda) \left( \begin{smallmatrix} e^{q/2} & 0 \\ 0 &
e^{-q/2}\end{smallmatrix}\right) (F^{(\mathfrak{c},
4)} (z, \bar z, \lambda))^*\right)\right|_{\lambda \in S^r}\;, \label{eq:Sym-Bobenko-3}\\
\varPhi^{(\mathfrak{s},j)} &= \displaystyle
\left.-\frac{1}{2|H|}\left( \lambda \partial_{\lambda} F
^{(\mathfrak{s}, j)}(x, y, \lambda) \cdot F^{(\mathfrak{s}, j)}(x, y,
\lambda)^{-1} \right)\right|_{\lambda \in \mathbb R^*} \mbox{for $j
\in \{1, 2, 3\}$}, \label{eq:Sym-Bobenko-4}
\end{align}
where $\lambda = \exp (i t) \in S^1$ or $\lambda = \exp (
q/2+i t) \in S^r$ for \eqref{eq:Sym-Bobenko-2} or
\eqref{eq:Sym-Bobenko-3} (resp. $\lambda = \pm \exp (t) \in \mathbb R^*$
for \eqref{eq:Sym-Bobenko-4}) with $t, q \in \mathbb R$, and where $*$
denotes $X^*= \bar X^t$ for $X \in
M_{2 \times 2}(\mathbb C)$.
We note that $w = \bar z$ (resp. $z = \bar z = x \in \mathbb R$ and $w =
\bar w = y \in \mathbb R$) for $\varPhi^{(\mathfrak{c},j)}$
(resp. $\varPhi^{(\mathfrak{s},j)}$), from \eqref{eq:coordinates}.
Then, for each $\lambda \in S^1$ or $\lambda \in S^r$
(resp. $\lambda \in \mathbb R^*$), the formula
$\varPhi^{(\mathfrak{c},j)}$ (resp.
$\varPhi^{(\mathfrak{s},j)}$) defines a map into
one of the following spaces:
\begin{equation*}
\left\{
\begin{array}{cl}
\mathfrak{su}(1, 1) \cong \mathbb R^{1,2} &\mbox{for the $(C, 1)$ and $(S, 1)$ cases,}\\
\mathfrak{sl}_*(2, \mathbb R)\cong \mathbb R^{1,2} &\mbox{for the $(C, 2)$ and $(S, 2)$
cases,}\\
\mathfrak{su}(2) \cong \mathbb R^3 &\mbox{for the $(C, 3)$ and $(S, 3)$
cases,}\\
SL(2, \mathbb C)/SU(2) \cong H^3 &\mbox{for the $(C, 4)$ case,}
\end{array}
\right.
\end{equation*}
where $\mathfrak{sl}_*(2, \mathbb R) = \{ g \in \mathfrak{sl} (2,
\mathbb C) \;| \;g = \left(\begin{smallmatrix}a & b\\c & -a
\end{smallmatrix}\right), a \in \mathbb
R,\;b, c \in i \mathbb R\}$, which is isomorphic to $\mathfrak{sl} (2,
\mathbb R)$. Here $\mathbb R^{1,2}$ and $\mathbb
R^3$ can be identified with $\mathfrak{su}(1, 1)$, $ \mathfrak{sl}_*(2,
\mathbb R)$ and $\mathfrak{su}(2)$ analogous to the identification
\eqref{eq:ident1}. Minkowski $4$-space $\mathbb R^{3,1}$ can be identified with ${\rm
Herm}(2):=\left\{ X \in M_{2 \times 2}(\mathbb C) \;|\; \bar X^t = X
\right\}$ via the map $$(x_1, x_2, x_3, x_0) \mapsto
\frac{1}{2}\begin{pmatrix}x_0+x_3 &
x_1+i x_2 \\ x_1-i x_2 & x_0-x_3 \end{pmatrix},
$$
then $H^3 \subset \mathbb R^{3,1}$ can be identified with ${\rm Herm}(2)$
with the determinant $1/4$.
Then the inner product for $\mathfrak{su}(1, 1) \cong \mathbb R^{1,2}$
(resp. $\mathfrak{sl}_*(2, \mathbb R) \cong \mathbb R^{1,2}$ or
$\mathfrak{su}(2) \cong \mathbb R^{3}$) can be defined by $\langle a,
b\rangle = -2 {\rm Tr} \;(a b)$ for $a, b \in \mathfrak{su}(1, 1)$
(resp. $a, b \in \mathfrak{sl}_*(2, \mathbb R)$ or $a, b \in
\mathfrak{su}(2)$). The inner product for ${\rm Herm}(2) \cong \mathbb R^{3,1}$ can
be defined by $\langle a, b\rangle = -2 {\rm Tr}\; (a \sigma_2 b^t \sigma_2)$
for $a, b \in {\rm Herm}(2)$, where $\sigma_2$ is defined in \eqref{eq:ident2}.
From now on, we always assume that the
spectral parameter $\lambda$ is in $S^1$ or $S^r$ for the almost
compact cases and $\lambda$ is in $\mathbb R^*$ for the almost split
cases, respectively.
\begin{Remark}
For the $(C, 4)$ case, the complex Gau{\ss} equation in \eqref{eq:GC} can be reduced to
the elliptic cosh-Gordon type equation by the choices of functions in
\eqref{eq:solforMaurer}. It is known that the Gau{\ss} equation for
{\sc CMC} surfaces with mean curvature $|H| <1$ in
$H^3$ is the elliptic cosh-Gordon type equation, see
\cite{BB:MiniH3}. Therefore it is
natural to use the Sym formula defined in \eqref{eq:Sym-Bobenko-3} for
the $(C, 4)$ case.
\end{Remark}
We denote the metrics for $\varPhi^{(\mathfrak{c}, j)}$ by
$g^{(\mathfrak{c}, j)}$ (resp. $\varPhi^{(\mathfrak{s}, j)}$ by $g^{(\mathfrak{s}, j)}$), and
also denote the coefficient matrices for the metrics $g^{(\mathfrak{c}, j)}$
by $\tilde I^{(\mathfrak{c}, j)}$ for
$j \in \{1, 2, 3, 4\}$ (resp. $g^{(\mathfrak{s}, j)}$ by $\tilde I^{(\mathfrak{s}, j)}$ for $j \in \{1, 2, 3\}$).
Since $\lambda \in S^1$ or $S^r$ for the almost compact cases and $\lambda
\in \mathbb R^*$ for the almost split cases, $\tilde
I^{(\mathfrak{c}, j)}$ and $\tilde I^{(\mathfrak{s}, j)}$ are given as follows:
\begin{equation}\label{eq:firstforreal}
\left\{
\begin{array}{ll}
\tilde I^{(\mathfrak{c}, j)} =
{\displaystyle \frac{1}{2 |H|^2}} \begin{pmatrix}
\mathfrak{a} + \mathfrak{b} + \mathfrak{c} & i( \mathfrak{b} -\mathfrak{c}) \\
i( \mathfrak{b} -\mathfrak{c}) & \mathfrak{a} -\mathfrak{b}-\mathfrak{c}
\end{pmatrix}\;\;\mbox{for $j \in \{1, 2, 3\}$},
\\[0.5cm]
\tilde I^{(\mathfrak{c}, 4)} =
\displaystyle - H^2 e^{u} \cosh^2 (q) \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix},
\\[0.5cm]
\tilde I^{(\mathfrak{s}, j)} =
{\displaystyle \frac{1}{2 |H|^2}}
\begin{pmatrix}
-\mathfrak{b}&-\frac{1}{2} \mathfrak{a} \\
-\frac{1}{2} \mathfrak{a} & -\mathfrak{c}
\end{pmatrix}\;\;\mbox{for $j \in \{1, 2, 3\}$} ,
\end{array}
\right.
\end{equation}
where $\mathfrak{a} = H^2 e^u/2 + 2 Q R e^{-u}$,
$\mathfrak{b} = - \lambda^{-2} H Q$, $\mathfrak{c} = - \lambda^2 H
R$ and $u$, $Q$, $R$ and $H$ are solutions defined in
\eqref{eq:solforMaurer}. Therefore it is easy to verify that the
determinants of $ \tilde I^{(\mathfrak{c}, j)}$ and $ \tilde
I^{(\mathfrak{s}, j)}$ are as follows:
\begin{equation}\label{eq:firstforreal2}
\left\{
\begin{array}{l}
\det \widetilde{I}^{(\mathfrak{c}, j)} =\displaystyle \frac{1}{|H|^4} \left(\frac{1}{4}H^2
e^u - Q R e^{-u}\right)^2 \;\;\mbox{for $j \in \{1, 2, 3 \}$, }
\\[0.3cm]
\det \widetilde{I}^{(\mathfrak{c}, 4)} =\displaystyle H^4 e^{2 u} \cosh^4 (q),\\[0.1cm]
\det \widetilde{I}^{(\mathfrak{s}, j)} =\displaystyle -\frac{1}{4|H|^4} \left(\frac{1}{4}H^2
e^u - Q R e^{-u}\right)^2 \;\;\mbox{for $j \in \{1, 2, 3 \}$}.
\end{array}
\right.
\end{equation}
From \eqref{eq:firstforreal2} one can verify that
$\varPhi^{(\mathfrak{c},j)}$ and $\varPhi^{(\mathfrak{s},j)}$
actually define immersions if and only if $e^{u} \neq 4 H^{-2} Q R$ for
$j \in \{1, 2, 3\}$, and for the $(C, 4)$ case,
$\varPhi^{(\mathfrak{c},4)}$ always defines an immersion.
From \eqref{eq:solforMaurer}, the immersions
$\varPhi^{(\mathfrak{c}, j)}$ for $j \in \{1, 3, 4\}$ and
$\varPhi^{(\mathfrak{s}, j)}$ for $j \in \{1, 3\}$ are spacelike, and
the immersions $\varPhi^{(\mathfrak{c}, 2)}$ and
$\varPhi^{(\mathfrak{s}, 2)}$ are timelike.
\begin{Remark}
Since we consider Minkowski space $\mathbb R^{1,2}$ as the three
dimensional vector space $\{ (x_1, x_2, x_3)
\;|\; x_j \in \mathbb R\}$ endowed with the metric $g = dx_1^2 - dx_2^2
-dx_3^2$, the spacelike, timelike and lightlike vectors are $\langle a, a
\rangle < 0$, $\langle b, b \rangle > 0$ and $\langle c, c \rangle = 0$ for $a, b, c \in \mathbb
R^{1,2}$, respectively.
\end{Remark}
Let $N^{(\mathfrak{c}, j)}$ and $N^{(\mathfrak{s}, j)}$ be the following maps:
\begin{equation}\label{eq:Gaussmap}
\left\{
\begin{array}{l}
N^{(\mathfrak{c}, j)} := \frac{\ell}{2}{\rm Ad} (F^{(\mathfrak{c}, j)})\sigma_3,
\;\mbox{for}\;\; j \in \{1, 2, 3\}\;,\\[0.25cm]
N^{(\mathfrak{c}, 4)} := \frac{1}{2}F^{(\mathfrak{c},
4)}\left(\begin{smallmatrix} e^{q/2} & 0 \\ 0 & -e^{-q/2}
\end{smallmatrix} \right)
(F^{(\mathfrak{c},
4)})^*
\;\;,\\[0.25cm]
N^{(\mathfrak{s}, j)} := \frac{\ell}{2}{\rm Ad} (F^{(\mathfrak{s}, j)})
\sigma_3\;\;\mbox{for}\;\; j \in \{1, 2, 3\}\;,
\end{array}
\right.
\end{equation}
where $\ell$ is $i$ (resp. 1) for $j \in \{1, 3\}$ (resp. $j = 2$).
It is clear that $N^{(\mathfrak{c}, j)}$ and $
N^{(\mathfrak{s}, j)} $ are the Gau{\ss} maps of the immersions
$\varPhi^{(\mathfrak{c}, j)}$ and $\varPhi^{(\mathfrak{s}, j)}$
respectively.
The second fundamental forms
$I\!I^{(\mathfrak{c}, j)}$ and $I\!I^{(\mathfrak{s}, j)}$
for the immersions $\varPhi^{(\mathfrak{c}, j)}$ and
$\varPhi^{(\mathfrak{s}, j)}$ are defined by (see \cite[page
107]{Oneill:Semi-Riemannian})
\begin{equation*}
\left\{
\begin{array}{l}
I\!I^{(\mathfrak{c}, j)} = - \langle d
\varPhi^{(\mathfrak{c}, j)}, d N^{(\mathfrak{c}, j)}\rangle
\;\;\mbox{for}\;j\in \{1, 2, 3, 4\},\\[0.2cm]
I\!I^{(\mathfrak{s}, j)} = - \langle d
\varPhi^{(\mathfrak{s}, j)}, d N^{(\mathfrak{s},
j)}\rangle\;\;\mbox{for}\; j \in \{1, 2, 3\}.
\end{array}
\right.
\end{equation*}
We denote the coefficient matrices of $I\!I^{(\mathfrak{c}, j)}$
by $\widetilde{I\!I}{}^{(\mathfrak{c}, j)}$
(resp. $I\!I^{(\mathfrak{s}, j)}$ by
$\widetilde{I\!I}{}^{(\mathfrak{s}, j)}$). A straightforward
computation (see also the proof of Theorem \ref{thm:Sym-Bob}) shows
that the $\widetilde {I\!I}{}^{(\mathfrak{c}, j)}$ for $j \in \{1, 2,
3, 4\}$ and $\widetilde {I\!I}{}^{(\mathfrak{s}, j)}$ for $j \in \{1,
2, 3\}$ are as follows:
\begin{equation}\label{eq:secondforint}
\left\{
\begin{array}{l}
\widetilde{I\!I}{}^{(\mathfrak{c}, j)} =\displaystyle -\frac{2i \ell}{|H|}\left(\frac{1}{4} H^2 e^u - Q R e^{-u}\right)
\begin{pmatrix} 1 & 0 \\ 0& 1 \end{pmatrix}
\; \mbox{for $j \in \{1, 2, 3\}$, } \\[0.5cm]
\widetilde{I\!I}{}^{(\mathfrak{c}, 4)} = \begin{pmatrix} \mathfrak{d} +2 {\rm Re}\; \mathfrak{e} &
-2 {\rm Im}\; \mathfrak{e} \\ -2 {\rm Im}\; \mathfrak{e}
& \mathfrak{d}
-2 {\rm Re}\; \mathfrak{e}
\end{pmatrix}, \\[0.5cm]
\widetilde{I\!I}{}^{(\mathfrak{s}, j)} = \displaystyle
- \frac{\ell}{|H|}\left(\frac{1}{4} H^2 e^u -Q R e^{-u}\right)
\begin{pmatrix} 0 & 1 \\ 1& 0 \end{pmatrix}
\; \mbox{for $j \in \{1, 2, 3\}$, }
\end{array}
\right.
\end{equation}
where $\mathfrak{d} = H^2 e^u \cosh (q) \sinh (q)$ and $\mathfrak{e} =
H Q \cosh (q)e^{- 2i t}$.
We recall that the Gau{\ss} curvatures $K^{(\mathfrak{c}, j)}$ and
$K^{(\mathfrak{s}, j)}$ (resp. the mean curvature $H^{(\mathfrak{c},
4)}$) of the immersions $\varPhi^{(\mathfrak{c},
j)}$ and $\varPhi^{(\mathfrak{s}, j)}$ for $j \in \{ 1, 2, 3\}$
(resp. $\varPhi^{(\mathfrak{c}, 4)}$) are defined as follows (see
also \cite[page 157, (93)]{Weinstein:Lorentz}):
\begin{equation*}
\left\{
\begin{array}{l}
K^{(\mathfrak{c}, j)}:= \pm \det \left(\tilde{I}^{(\mathfrak{c},
j)-1}\widetilde{I\!I}{}^{(\mathfrak{c}, j)}\right)
\;\mbox{and}\; K^{(\mathfrak{s}, j)}:= \pm \det \left(\tilde I^{(\mathfrak{s},
j)-1}\widetilde{I\!I}{}^{(\mathfrak{s}, j)}\right) \; \mbox{for} \; j \in \{1, 2, 3\}, \\[0.3cm]
H^{(\mathfrak{c}, 4)} := \displaystyle \frac{1}{2} {\rm Tr} \left(\tilde
I^{(\mathfrak{c}, 4) -1}\widetilde{I\!I}{}^{(\mathfrak{c}, 4)}\right),
\end{array}
\right.
\end{equation*}
where the plus sign (resp. the minus sign) has been chosen if the
surface is in $\mathbb R^3$ or timelike in $\mathbb R^{1,2}$,
i.e., $\varPhi^{(\mathfrak{c}, j)}$ and $\varPhi^{(\mathfrak{s}, j)}$
for $j \in \{2, 3\}$ (resp. spacelike in
$\mathbb R^{1, 2}$, i.e., $\varPhi^{(\mathfrak{c}, 1)}$ and
$\varPhi^{(\mathfrak{s}, 1)}$), see \cite{Weinstein:Lorentz}.
Combining \eqref{eq:firstforreal2} with \eqref{eq:secondforint}, we
finally obtain
\begin{equation*}
\left\{
\begin{array}{l}
K^{(\mathfrak{s}, 1)} = K^{(\mathfrak{s}, 2)} = K^{(\mathfrak{c}, 3)} = 4 |H|^2 >0,\\[0.1cm]
K^{(\mathfrak{c}, 1)}= K^{(\mathfrak{c}, 2)}=K^{(\mathfrak{s}, 3)} = -
4|H|^2 < 0 , \\[0.1cm]
H^{(\mathfrak{c}, 4)} = \tanh (-q).
\end{array}
\right.
\end{equation*}
The above discussion is summarized in the following theorem:
\begin{Theorem}\label{thm:compactsplit}
Let $F(z, w, \lambda)$ be the complex extended framing of some complex
{\sc CGC}-immersion $\varPhi$. Then the following statements hold:
\begin{enumerate}
\item[$(C, 1)$] If $F^{-1} d F$ is in $\Omega(\Lambda \mathfrak{sl}(2,
\mathbb C)_{\sigma}^{(\mathfrak{c}, 1)})$, then for each $\lambda \in S^1$ the Sym formula in
\eqref{eq:Sym-Bobenko-2} defines a {\rm spacelike
constant negative {Gau\ss ian} curvature surface} in
$\mathbb R^{1,2}$.
\item[$(C, 2)$] If $F^{-1} d F$ is in $\Omega(\Lambda \mathfrak{sl}(2,
\mathbb C)_{\sigma}^{(\mathfrak{c}, 2)})$, then for each $\lambda \in S^1$ the Sym formula in
\eqref{eq:Sym-Bobenko-2} defines
a {\rm timelike constant negative Gau{\ss}ian curvature
surface} in $\mathbb R^{1,2}$.
\item[$(C, 3)$] If $F^{-1} d F$ is in $\Omega(\Lambda \mathfrak{sl}(2,
\mathbb C)_{\sigma}^{(\mathfrak{c}, 3)})$, then for each $\lambda \in
S^1$ the Sym formula in
\eqref{eq:Sym-Bobenko-2} defines a
{\rm constant positive {Gau\ss ian} curvature surface} in
$\mathbb R^3$.
\item[$(C, 4)$] If $F^{-1} d F$ is in $\Omega(\Lambda \mathfrak{sl}(2,
\mathbb C)_{\sigma}^{(\mathfrak{c}, 4)})$, then for each
$\lambda \in S^r$ the Sym formula in
\eqref{eq:Sym-Bobenko-3} defines
a {\rm constant mean curvature surface} with mean
curvature $|H^{(\mathfrak{c}, 4)}| < 1$ in $H^3$.
\item[$(S, 1)$] If $F^{-1} d F$ is in $\Omega(\Lambda \mathfrak{sl}(2,
\mathbb C)_{\sigma}^{(\mathfrak{s}, 1)})$, then for each $\lambda \in \mathbb R^*$ the Sym formula in
\eqref{eq:Sym-Bobenko-4} defines a {\rm spacelike
constant positive Gau{\ss}ian curvature surface} in
$\mathbb R^{1,2}$.
\item[$(S, 2)$] If $F^{-1} d F$ is in $\Omega(\Lambda \mathfrak{sl}(2,
\mathbb C)_{\sigma}^{(\mathfrak{s}, 2)})$, then for each $\lambda \in \mathbb R^*$ the Sym formula in
\eqref{eq:Sym-Bobenko-4} defines a
{\rm timelike constant positive Gau{\ss}ian curvature surface} in
$\mathbb R^{1,2}$.
\item[$(S, 3)$] If $F^{-1} d F$ is in $\Omega(\Lambda \mathfrak{sl}(2,
\mathbb C)_{\sigma}^{(\mathfrak{s}, 3)})$, then for each $\lambda \in \mathbb R^*$ the Sym formula in
\eqref{eq:Sym-Bobenko-4} defines a {\rm constant negative
Gau{\ss}ian curvature surface} in $\mathbb R^3$.
\end{enumerate}
\end{Theorem}
\begin{Definition}
Let $F^{(\mathfrak c, j)}(z, \bar z, \lambda)$ for $j \in \{1, 2, 3, 4\}$
(resp. $F^{(\mathfrak s, j)}(x, y, \lambda)$ for $j \in \{1, 2, 3\}$) be
the complex extended framings, which are elements in ${\Lambda SL(2,
\mathbb C)_\sigma}^{(\mathfrak c, j)}$ (resp. ${\Lambda SL(2, \mathbb
C)_\sigma}^{(\mathfrak s, j)}$). Then $F^{(\mathfrak c, j)}(z, w,
\lambda)$ (resp. $F^{(\mathfrak s, j)}(x, y, \lambda)$) is called the
{\rm extended framing for the immersion}
$\varPhi^{(\mathfrak{c}, j)}$ (resp. $\varPhi^{(\mathfrak{s}, j)}$).
\end{Definition}
It is known that for three classes of surfaces in the above seven
classes, there exist parallel constant mean curvature
surfaces in $\mathbb R^3$ or $\mathbb R^{1,2}$, see also
\cite{Inoguchi:timelike} and \cite{Inoguchi:Minkowski}.
\begin{Corollary}\label{coro:compactsplit} We retain the assumptions in Theorem
\ref{thm:compactsplit}. Then we have the following:
\begin{enumerate}
\item[$(C, 1M)$] For the $(C, 1)$ case in Theorem \ref{thm:compactsplit},
there exists a parallel spacelike constant mean
curvature surface with mean curvature $H^{(\mathfrak{c},
1)} = |H|>0$ in $\mathbb R^{1,2}$.
\item[$(C, 3M)$] For the $(C, 3)$ case in Theorem \ref{thm:compactsplit},
there exists a parallel constant mean curvature surface
with mean curvature $H^{(\mathfrak{c}, 3)} =
|H|>0$ in $\mathbb R^3$.
\item[$(S, 2M)$] For the $(S, 2)$ case in Theorem \ref{thm:compactsplit},
there exists a parallel timelike constant mean curvature
surface with mean curvature $H^{(\mathfrak{s}, 2)} = |H|>0$ in $\mathbb R^{1,2}$.
\end{enumerate}
\end{Corollary}
\begin{proof}
Let $\varPhi^{(\mathfrak{c}, 1)}$, $\varPhi^{(\mathfrak{c},
3)}$ and $\varPhi^{(\mathfrak{s}, 2)}$ be a spacelike constant
negative Gau{\ss}ian curvature surface
in $\mathbb R^{1,2}$, a constant positive Gau{\ss}ian curvature surface
in $\mathbb R^3$ and a timelike constant positive Gau{\ss}ian curvature
surface in $\mathbb R^{1,2}$, as defined in Theorem \ref{thm:compactsplit},
respectively. Let $N^{(\mathfrak{c}, 1)}$, $N^{(\mathfrak{c},
3)}$ and $N^{(\mathfrak{s},
2)}$ be the Gau{\ss} maps for $\varPhi^{(\mathfrak{c}, 1)}$, $\varPhi^{(\mathfrak{c},
3)}$ and $\varPhi^{(\mathfrak{s}, 2)}$ defined in \eqref{eq:Gaussmap}, respectively.
Then the parallel surfaces for $\varPhi^{(\mathfrak{c},
1)}$, $\varPhi^{(\mathfrak{c}, 3)}$ and
$\varPhi^{(\mathfrak{s}, 2)}$ are defined by
\begin{equation*}
\left\{
\begin{array}{l}
\varPsi^{(\mathfrak{c}, j)} := \varPhi^{(\mathfrak{c}, j)} +
\frac{1}{2 |H|} N^{(\mathfrak{c}, j)} \;\;\;\mbox{for $j \in \{1, 3\}$,}\\[0.2cm]
\varPsi^{(\mathfrak{s}, 2)} := \varPhi^{(\mathfrak{s}, 2)} +
\frac{1}{2 |H|} N^{(\mathfrak{s}, 2)}.
\end{array}
\right.
\end{equation*}
Then the first fundamental forms and the second
fundamental forms for these immersions can be computed explicitly, and
we can easily show that these immersions $\varPsi^{(\mathfrak{c}, 1)}$,
$\varPsi^{(\mathfrak{c}, 3)}$ and $\varPsi^{(\mathfrak{s}, 2)}$
define a spacelike constant mean curvature surface with mean curvature
$H^{(\mathfrak c, 1)} =|H|$ in $\mathbb R^{1,2}$, a constant mean
curvature surface with mean curvature $H^{(\mathfrak c, 3)} =|H|$ in
$\mathbb R^3$ and a timelike constant mean curvature surface with mean
curvature $H^{(\mathfrak s, 2)} =|H|$ in $\mathbb R^{1,2}$,
respectively. This completes the proof.
\end{proof}
\begin{Definition}
The surfaces defined in Theorem \ref{thm:compactsplit} and
Corollary \ref{coro:compactsplit} are called the {\rm integrable surfaces}.
\end{Definition}
\begin{Remark}
For the three classes of surfaces in Theorem \ref{thm:compactsplit}, which
are spacelike constant positive Gau{\ss}ian curvature surfaces
in $\mathbb R^{1, 2}$, constant negative Gau{\ss}ian curvature surfaces in $\mathbb R^3$ and
timelike constant negative Gau{\ss}ian curvature surfaces in $\mathbb
R^{1,2}$, there never exist parallel constant mean curvature
surfaces.
\end{Remark}
\subsection{Gau{\ss} maps of integrable surfaces}\label{subsc:gaussmap}
In this subsection, we consider the Gau{\ss} maps of integrable
surfaces defined in the previous section.
From \cite{DKP:Complex}, it is known that the complex Gau{\ss} map $N$
of a complex {\sc CMC}-immersion $\varPsi$ with null coordinates
($N$ is also the complex Gau{\ss} map of the parallel complex {\sc
CGC}-immersion) satisfies the following equation:
\begin{equation}\label{eq:normal}
N_{z w} = \rho N \;,
\end{equation}
where $(z, w) \in \mathfrak D^2 \subset \mathbb C^2$
and the function $\rho : \mathfrak D^2 \to \mathbb C$ is defined by
$\rho \cdot i\sigma_3 = [\alpha_{-1}, [\alpha_{1}, i \sigma_3]]$ with
$\alpha_{j}$ as defined in \eqref{eq:alpha2}.
From Theorem \ref{thm:Sym-Bob}, we note that the complex Gau{\ss}
map $N$ is represented by $ N = \frac{i}{2}{\rm Ad} (F)\; \sigma_3,$
where $F$ is the complex extended framing of the complex {\sc
CMC}-immersion $\varPsi$.
Let $F^{(\mathfrak{c}, j)}$ (resp. $F^{(\mathfrak{s}, j)}$) be the
extended framing of $\varPhi^{(\mathfrak{c}, j)}$ for $j \in \{1, 2, 3, 4\}$
(resp. $\varPhi^{(\mathfrak{s}, j)}$ for $j \in \{1, 2, 3 \}$).
Using \eqref{eq:Gaussmap}, we can easily verify that $N^{(\mathfrak{c}, j)}$
and $N^{(\mathfrak{s}, j)}$ are maps into the following spaces:
\begin{equation*}
\left\{
\begin{array}{l}
H^2 = SU(1, 1)/U(1) \;\;\mbox{for}\;\; j =1, \\[0.1cm]
S^{1, 1} = SL_*(2, \mathbb R)/K\;\;\mbox{for}\;\; j =2, \\[0.1cm]
S^2 = SU(2)/U(1)\;\;\mbox{for}\;\; j =3, \\[0.1cm]
SL(2, \mathbb C)/U(1) \;\;\mbox{for}\;\; j =4,
\end{array}
\right.
\end{equation*}
where $K = \left\{{\rm diag}[a, a^{-1}] \;|\; a \in \mathbb R^* \right\}$,
which is isomorphic to $\mathbb R^*$, and $SL_*(2, \mathbb R) =\{ g \in
SL(2, \mathbb C)\;|\;g =\left(\begin{smallmatrix} a & b\\ c&
d\end{smallmatrix}\right), a, d \in
\mathbb R, b, c \in i \mathbb R \}$, which is isomorphic to $SL(2,
\mathbb R)$. It is known that the space $SL(2, \mathbb C)/U(1)$ is a $4$-symmetric
space via the fourth order automorphism
$$
X \mapsto {\rm Ad}\begin{pmatrix}1/\sqrt{i} & 0 \\ 0& \sqrt{i}
\end{pmatrix} \left(\bar X^{t}\right)^{-1}\;\; \mbox{for}\;\; X \in SL(2, \mathbb C).
$$ The choices of coordinates in
\eqref{eq:coordinates}, the functions in \eqref{eq:solforMaurer} and
the relation $\rho i \sigma_3 =[\alpha_{-1}[\alpha_1, i \sigma_3]]$ imply
\begin{equation}\label{eq:Gaussforint}
\left(N^{(\mathfrak{c}, j)}\right)_{z \bar z} = \rho^{(\mathfrak{c}, j)} N^{(\mathfrak{c}, j)}
\;\;\mbox{and}\;\; \left(N^{(\mathfrak{s},j)}\right)_{x y} =
\rho^{(\mathfrak{s}, j)}N^{(\mathfrak{s},j)}\;\;,
\end{equation}
where $\rho^{(\mathfrak{c}, j)} : \mathfrak D \subset \mathbb C \to \mathbb R$
and $\rho^{(\mathfrak{s}, j)} : \mathfrak D \subset \mathbb R^2 \to
\mathbb R$.
It is well known that the equations in \eqref{eq:Gaussforint} for $j
\in \{1, 2, 3 \}$ are equivalent to the harmonicities (resp. Lorentz
harmonicities) of Gau{\ss} maps $N^{(\mathfrak{c},
j)}$ (resp. $ N^{(\mathfrak{s}, j)}$) with respect to the second fundamental
forms defined in the first equations of \eqref{eq:secondforint}
(resp. third equations of \eqref{eq:secondforint}), see Theorem 13 in
\cite{Klotz:Harm-Mink}.
We then have the following theorem:
\begin{Theorem}\label{thm:Gaussmap}
Let $\varPhi^{(\mathfrak{c}, j)}$ for $j \in \{1, 2, 3, 4\}$
and $\varPhi^{(\mathfrak{s}, j)}$ for $j \in \{1, 2, 3\}$ be
the integrable surfaces defined in Theorem \ref{thm:compactsplit}
respectively. Moreover, let $N^{(\mathfrak{c}, j)}$ and
$N^{(\mathfrak{s}, j)}$ be their Gau{\ss} maps respectively.
Then the Gau{\ss} maps $N^{(\mathfrak{c}, j)}$ and
$N^{(\mathfrak{s}, j)}$ are characterized as follows:
\begin{enumerate}
\item[$(C, S, 1)$] The Gau{\ss} map $N^{(\mathfrak{c}, 1)}$
(resp. $N^{(\mathfrak{s}, 1)}$) is a harmonic
(resp. Lorentz harmonic) map into $H^2$.
\item[$(C, S, 2)$] The Gau{\ss} map $N^{(\mathfrak{c}, 2)}$
(resp. $N^{(\mathfrak{s}, 2)}$) is a harmonic
(resp. Lorentz harmonic) map into $S^{1, 1}$.
\item[$(C, S, 3)$] The Gau{\ss} map $N^{(\mathfrak{c}, 3)}$
(resp. $N^{(\mathfrak{s}, 3)}$) is a harmonic
(resp. Lorentz harmonic) map into $S^{2}$.
\item[$(C, 4)$\;\;\;\;] The Gau{\ss} map $N^{(\mathfrak{c}, 4)}$ is a
harmonic map into $SL(2, \mathbb C)/U(1)$.
\end{enumerate}
\end{Theorem}
\begin{proof}
Let us show the $(C, 4)$ case.
We recall that a map from a Riemann surface into a $k$-symmetric space
$N = G/K$ is harmonic (see, for example, \cite[page
242]{BP:Adler}) if
\begin{equation*}
\begin{array}{l}
[\alpha_{\mathfrak m}^{\prime} \wedge \alpha_{\mathfrak m}^{\prime \prime}]_{\mathfrak m} =0,\\[0.1cm]
d \alpha_{\lambda} +\frac{1}{2}[\alpha_{\lambda} \wedge \alpha_{\lambda}] =0,
\end{array}
\end{equation*}
where $\alpha_{\lambda} = \lambda^{-1} \alpha_{\mathfrak m}^{\prime} +
\alpha_{\mathfrak k} + \lambda \alpha_{\mathfrak m}^{\prime \prime}$,
$\mathfrak g = \mathfrak{k} \oplus \mathfrak{m}$ is the reductive
decomposition and ${}^{\prime}$
(resp. ${}^{\prime \prime}$) denotes the $(1,0)$-part (resp. $(0,1)$-part).
Since the map $N^{(\mathfrak c, 4)}$ has the lift $F^{(\mathfrak c, 4)}
: \mathfrak D \to \Lambda SL(2, \mathbb C)_\sigma^{(\mathfrak c, j)}$ which is
defined from the complex extended framing $F$ with the conditions in
\eqref{eq:solforMaurer}, the Maurer-Cartan
form
$\alpha_{\lambda} = F^{(\mathfrak c,
4)-1} d F^{(\mathfrak c, 4)}$ has the form $\alpha_{\lambda} =
\lambda^{-1} \alpha_{-1}+ \alpha_0+\lambda \alpha_1$ and satisfies the
Maurer-Cartan equation $d \alpha_{\lambda} +
\frac{1}{2}[\alpha_{\lambda} \wedge \alpha_{\lambda}] =0$,
see \eqref{eq:alpha} and Lemma \ref{lem:complex-CMC}.
The conditions in \eqref{eq:solforMaurer} imply $\alpha_0 = \alpha_{\mathfrak k}$,
$\alpha_{-1} = \alpha_{\mathfrak m}^{\prime}$ and $\alpha_{1} =
\alpha_{\mathfrak m}^{\prime \prime}$, where $\mathfrak{sl}(2, \mathbb
C) = \mathfrak{k} \oplus \mathfrak{m}$ is the reductive decomposition
associated to $SL(2, \mathbb C)/U(1)$. Moreover, since $\alpha_{-1}$
and $\alpha_{1}$ have off-diagonal forms, it follows that
$[\alpha_{\mathfrak m}^{\prime} \wedge \alpha_{\mathfrak m}^{\prime
\prime}]_{\mathfrak m} =0$. Therefore $N^{(\mathfrak c, 4)}$ is a
harmonic map into the $4$-symmetric space $SL(2, \mathbb C)/U(1)$. For
other cases, since the target spaces are symmetric spaces,
the condition $[\alpha_{\mathfrak m}^{\prime} \wedge \alpha_{\mathfrak m}^{\prime
\prime}]_{\mathfrak m} =0$ is vacuous. Thus the Maurer-Cartan equation
$d \alpha_{\lambda} + \frac{1}{2}[\alpha_{\lambda} \wedge
\alpha_{\lambda}] =0$ with $\alpha_{\lambda} = \lambda^{-1}
\alpha_{-1}+ \alpha_0+\lambda \alpha_1$ is equivalent to the map
being harmonic or Lorentz harmonic. This completes the proof.
\end{proof}
\begin{Remark}
In fact, in the $(C, 4)$ case, the harmonic map $N^{(\mathfrak{c}, 4)}$
into the $4$-symmetric space $SL(2, \mathbb C)/U(1)$ is known as the
so-called {\rm Legendre harmonic map} \cite{Ishihara:GG}. We will
discuss this topic in a separate publication \cite{DIK:H3}.
\end{Remark}
\begin{table}
\extrarowheight=1mm
\begin{tabular}{|c|c|c|c|}
\hline
{\small Surfaces class} & {\small Gau{\ss} curvature} & {\small Gau{\ss}
curvature} &{\small Parallel {\sc CMC}}\\[1mm]\hline
{\small Surfaces in $\mathbb R^3$ }& {\small $K^{(\mathfrak{s}, 3)} = -4 |H|^2$}& {\small $K^{(\mathfrak{c}, 3)}= 4|H|^2$}& $H^{(\mathfrak c,
3)}= |H|$ \\[1mm] \hline
{\small Spacelike surfaces in $\mathbb R^{1,2}$} & {\small $K^{(\mathfrak{s}, 1)} = 4
|H|^2$}& {\small $K^{(\mathfrak{c}, 1)}=-4 |H|^2\;$}&$H^{(\mathfrak c, 1)} = |H|$ \\[1mm] \hline
{\small Timelike surfaces in $\mathbb R^{1,2}$} & {\small $K^{(\mathfrak{c}, 2)} = -4 |H|^2$} &
{\small $K^{(\mathfrak{s}, 2)}=4 |H|^2$} & $H^{(\mathfrak s,
2)} = |H|$\\ \hline
{\small Surfaces in $H^3$} & & & $H^{(\mathfrak c, 4)} = \tanh
(-q)$ \\[1mm] \hline
\end{tabular}
\caption{Integrable surfaces defined by the real forms of $\Lambda \mathfrak{sl}(2, \mathbb C)_{\sigma}$}
\end{table}
\section{The generalized Weierstra{\ss} type representation for integrable
surfaces}\label{sc:DPW} The generalized Weierstra{\ss} type
representation for complex {\sc CMC}-immersions, or equivalently {\sc
CGC}-immersions as the parallel immersions,
is the procedure of a
construction of complex {\sc CMC}-immersions from
a pair of holomorphic potentials, see \cite{DKP:Complex}. In the
previous section, we classified all integrable surfaces according to the
classification of real forms of $\Lambda \mathfrak{sl}(2, \mathbb
C)_{\sigma}$. In this section, we show how all integrable surfaces
are obtained from the pairs of holomorphic potentials in the generalized
Weierstra{\ss} type representation.
\subsection{Integrable surfaces via the generalized Weierstra{\ss} type
representation}\label{subsc:DPW}
The generalized Weierstra{\ss} type representation for complex {\sc
CMC}-immersions, or equivalently {\sc CGC}-immersions as the parallel
immersions,
is divided into the following 4 steps, see also
\cite{DKP:Complex} for more details:
\begin{description}
\item[Step 1] Let $\Check{\eta} =
(\eta (z, \lambda), \tau(w, \lambda))$ be a pair of holomorphic
potentials of the following forms:
\begin{equation}
\label{eq:eqforcheketa}
\Check{\eta} = (\eta (z, \lambda),\;\; \tau (w, \lambda)) =
\left(\sum_{k=-1}^{\infty} \eta_k (z) \lambda^{k}, \;\; \sum_{m=-\infty}^{1} \tau_m (w) \lambda^m \right)\;\;,
\end{equation}
where $(z, w) \in \mathfrak D^2$ and where $\mathfrak D^2$ is some
holomorphically convex domain in $\mathbb C^2$, $\lambda \in
\mathbb C^\ast$, $|\lambda| = r$ $(0 < r < 1)$, and $\eta_k$ and
$\tau_m$ are $\mathfrak{sl}(2, \mathbb C)$-valued
holomorphic differential 1-forms. Moreover $\eta_k (z)$ and
$\tau_k (w)$ are diagonal (resp. off-diagonal) matrices if $k$ is even
(resp. odd). We also assume that the upper right entry
of $\eta_{-1} (z)$ and the lower left entry $\tau_{1} (w)$
do not vanish for all $(z, w) \in \mathfrak D^2$.
\item[Step 2] Let $C$ and $L$ denote
the solutions to the following linear ordinary differential equations
\begin{equation}\label{eq:eqforC^3}
d C = C \eta \;\;\mbox{and}\;\;d L = L \tau\;\;\mbox{with}\;\;C(z_*, \lambda) = L(w_*, \lambda) = {\rm id},
\end{equation}
where $(z_*, w_*) \in \mathfrak D^2$ is a fixed base point.
\item[Step 3] We factorize the pair of matrices $(C, L)$ via the
generalized Iwasawa decomposition of Theorem
\ref{doublesplitting} as follows:
\begin{equation}
\label{eq:splittingCR}
(C,\;\; L) = (F,\;\; F) ({\rm id} ,\;\;W) (V_+,\;\; V_-)\;\;,
\end{equation}
where $V_{\pm} \in \Lambda^{\pm} SL(2, \mathbb C)_{\sigma}$.
\end{description}
\begin{Theorem}[\cite{DKP:Complex}]\label{thm:DKP-Extendedframings}
Let $F$ be a $\Lambda SL(2, \mathbb C)_\sigma$-loop defined by the
generalized Iwasawa decomposition in \eqref{eq:splittingCR}. Then there
exists a $\lambda$-independent diagonal matrix $l(z, w) \in SL(2,
\mathbb C)$ such that $F\cdot l$ is a complex extended framing of some
complex {\sc CMC}-immersion, or equivalently the complex {\sc
CGC}-immersion as the parallel immersion.
\end{Theorem}
\begin{description}
\item[Step 4] The Sym formula defined in \eqref{eq:Sym-Bobenko} via
$F(z, w, \lambda)l (z, w)$ represents
a complex {\sc CMC}-immersion and a complex {\sc CGC}-immersion
in $\mathfrak{sl}(2, \mathbb C) \cong \mathbb C^3$.
\end{description}
Let $\mathfrak{c}_j $ for $j \in \{1, 2, 3, 4\}$ and $\mathfrak{s}_j$
for $j \in \{1, 2, 3\}$ be the involutions defined in
\eqref{eq:inv-1-forms}, respectively. Then we define the following pairs of involutions on
$\check\eta = (\eta, \tau) \in \Omega(\Lambda \mathfrak{sl}(2, \mathbb
C)_\sigma) \times \Omega(\Lambda \mathfrak{sl}(2, \mathbb C)_\sigma)$:
\begin{equation}\label{eq:involutions}
\mathfrak{r}_j : (\eta, \tau) \longmapsto (\mathfrak{c}_j \tau, \;\mathfrak{c}_j \eta)
\;\;\mbox{and}\;\;\mathfrak{d}_j : (\eta, \tau) \longmapsto (\mathfrak{s}_j \eta, \;\mathfrak{s}_j \tau).
\end{equation}
We now prove the following theorem.
\begin{Theorem}\label{thm:DPWforint}
Let $\check \eta = (\eta(z, \lambda), \tau (w, \lambda))$ be a pair of
holomorphic potentials defined as in \eqref{eq:eqforcheketa}, and let
$\mathfrak{r}_j$ for $ \;j \in \{1, 2, 3, 4\}$ and $\mathfrak{d}_j$
for $j \in \{1, 2, 3\}$ be the pairs of involutions defined in
\eqref{eq:involutions}, respectively. Then the following statements hold:
\begin{enumerate}
\item[$(C, 1)$] If $ \mathfrak{r}_1 (\check \eta) = \check \eta$, then the
resulting immersions given by the generalized
Weierstra{\ss} type representation are
spacelike constant negative Gau{\ss}ian curvature surfaces
in $\mathbb R^{1,2}$.
\item[$(C, 2)$] If $\mathfrak{r}_2 (\check \eta) = \check \eta$, then the
resulting immersions given by the generalized
Weierstra{\ss} type representation are timelike constant
negative Gau{\ss}ian curvature surfaces
in $\mathbb R^{1,2}$.
\item[$(C, 3)$] If $\mathfrak{r}_3 (\check \eta) = \check \eta$, then the
resulting immersions given by the generalized
Weierstra{\ss} type representation are
constant positive Gau{\ss}ian curvature surfaces
in $\mathbb R^{3}$.
\item[$(C, 4)$] If $\mathfrak{r}_4 (\check \eta) = \check \eta$, then the
resulting immersions given by the generalized
Weierstra{\ss} type representation are
constant mean curvature surfaces with mean curvature
$|H^{(\mathfrak{c}, 4)}| < 1$ in $H^3$.
\item[$(S, 1)$] If $\mathfrak{d}_1 (\check \eta) = \check \eta$, then the
resulting immersions given by the generalized
Weierstra{\ss} type representation are
spacelike constant positive Gau{\ss}ian curvature surfaces
in $\mathbb R^{1,2}$.
\item[$(S, 2)$] If $\mathfrak{d}_2 (\check \eta) = \check \eta$, then the
resulting immersions given by the generalized
Weierstra{\ss} type representation are
timelike constant positive Gau{\ss}ian curvature surfaces
in $\mathbb R^{1,2}$.
\item[$(S, 3)$] If $\mathfrak{d}_3 (\check \eta) = \check \eta$, then the
resulting immersions given by the generalized
Weierstra{\ss} type representation are
constant negative Gau{\ss}ian curvature surfaces in
$\mathbb R^{3}$.
\end{enumerate}
\end{Theorem}
\begin{proof}
Since the pairs of holomorphic potentials are invariant under the
involutions $\mathfrak{r}_j$ or $\mathfrak{d}_j$, the coordinates $(z,
w) \in \mathfrak D^2$ satisfy the following relations:
\begin{equation}\label{eq:conjasymp}
\left\{
\begin{array}{l}
w = \bar z \;\;\mbox{if} \;\; \mathfrak{r}_j (\check \eta) = \check \eta, \\
z = \bar z \;\mbox{and}\; w = \bar w\;\;\mbox{if} \;\; \mathfrak{d}_j (\check \eta) = \check \eta.
\end{array}
\right.
\end{equation}
Let $(C, L)$ be the pair of solutions of the differential equations in
\eqref{eq:eqforC^3} with the initial conditions $C(z_*) = L(w_*) = {\rm id}$,
where $(z_*, w_*) \in \mathfrak D^2$ satisfies one of the conditions in
\eqref{eq:conjasymp}.
Let $\mathcal R_j$ for $j \in \{1, 2, 3, 4\}$ (resp. $\mathcal D_j$ for
$j \in \{1, 2, 3\}$) be the following pair of involutions on $\Lambda
SL(2, \mathbb C)_{\sigma} \times \Lambda SL(2, \mathbb C)_\sigma$:
\begin{equation}\label{eq:mathcalG}
\mathcal{R}_j (C, L):= (\mathcal C_j (L), \mathcal C_j (C))
\;\;\mbox{and} \;\; \mathcal D_j (C, L):= (\mathcal S_j (C), \mathcal
S_j (L)) \;\;,
\end{equation}
where $\mathcal C_j$ (resp. $\mathcal S_j$) are the involutions on
$\Lambda SL(2, \mathbb C)_{\sigma}$ corresponding to the
involutions $\mathfrak{c}_j$ (resp. $\mathfrak{s}_j$) as in
Theorem \ref{thm:almostcompact} (resp. Theorem \ref{thm:almostsplit}), e.g.,
\begin{equation*}
\mathcal C_1 : C(\lambda) \to \overline{C(-1/\bar
\lambda)}^{t -1}\;\;\mbox{and}\;\;\mathcal S_1 : C(\lambda) \to
\overline{C(- \bar \lambda)}^{t -1} \;\;\mbox{for}\;\;
C(\lambda) \in \Lambda SL(2, \mathbb C)_\sigma.
\end{equation*}
Noting the conditions in \eqref{eq:conjasymp}, the involutions
$\mathfrak{r}_j$ for $j \in \{1, 2, 3, 4\}$ and $\mathfrak{d}_j$ for $j
\in \{1, 2, 3\}$ in \eqref{eq:involutions} define symmetries on the
pair of solutions $(C, L)$ as follows:
\begin{equation}\label{eq:symmholoext}
\left\{
\begin{array}{l}
\mathcal R_j (C(z, \lambda), L(\bar z, \lambda)) = (C(z, \lambda), L(\bar z,
\lambda)) \;\;\mbox{if $\mathfrak{r}_j (\check \eta) = \check \eta$ for} \;\; j \in \{1, 2, 3, 4\}, \\[0.1cm]
\mathcal D_j (C(x, \lambda), L(y, \lambda)) = (C(x, \lambda), L(y,
\lambda)) \;\;\mbox{if $\mathfrak{d}_j (\check \eta) = \check \eta$ for}\;\; j \in \{1, 2, 3\},
\end{array}
\right.
\end{equation}
where $x = z = \bar z \in \mathbb R$ and $y = w = \bar w \in \mathbb
R$. Applying the generalized Iwasawa decomposition of Theorem
\ref{doublesplitting} for $(C, L) \in \Lambda SL(2, \mathbb C)_{\sigma}
\times \Lambda SL(2, \mathbb C)_{\sigma}$, we have
\begin{equation}\label{eq:Iwasawadecom}
(C, L) = (F, F) ({\rm id}, W) (V_{+}, V_-)\;,
\end{equation}
where $V_{\pm} \in \Lambda^{\pm} SL(2, \mathbb C)_{\sigma}$. If $(z, w)
\in \mathfrak{D}^2$ is sufficiently close to $(z_*, w_*) \in \mathfrak
D^2$, then the middle term $W$ of the generalized
Iwasawa decomposition is identity.
Since the left component $F$ of the generalized Iwasawa
decomposition in \eqref{eq:Iwasawadecom} can be rephrased as $ F = C
V_+^{-1} = L V_{-}^{-1}$, we have
\begin{equation*}
C^{-1} L = V_{+}^{-1} V_{-}\;.
\end{equation*}
From the symmetries on $(C, L)$ in \eqref{eq:symmholoext}, $V_{-}$ and
$V_{+}$ have the following relations:
\begin{equation*}
\left\{
\begin{array}{l}
\mathcal
C_j (V_{\pm}(z, \bar z, \lambda)) = k^{(\mathfrak{c}, j)}(z,\bar
z)^{-1}V_{\mp}(z, \bar z, \lambda)\;\;\mbox{if $\mathfrak{r}_j (\check \eta) = \check \eta$ for} \;\; j \in \{1, 2, 3, 4\}, \\[0.2cm]
\mathcal S_j (V_{\pm}(x, y, \lambda) ) = k^{(\mathfrak{s}, j)}(x,y)^{-1}
V_{\pm}(x, y, \lambda) \;\;\mbox{if $\mathfrak{d}_j (\check \eta) = \check \eta$ for}\;\; j \in \{1, 2, 3\},
\end{array}
\right.
\end{equation*}
where $k^{(\mathfrak{c}, j)}(z,\bar z)$ and $k^{(\mathfrak{s}, j)}(x,
y)$ are $\lambda$-independent diagonal matrices satisfying the
symmetries $ \mathcal C_j
(k^{(\mathfrak{c}, j)}(z, \bar z)) = k^{(\mathfrak{c}, j)}(z, \bar
z)^{-1}$ and $ \mathcal S_j(k^{(\mathfrak{s}, j)}(x, y)) = k^{(\mathfrak{s}, j)}(x,
y)^{-1}$ respectively. From the discussion above $F$ has the symmetry as follows:
\begin{equation}\label{eq:invarianceofF}
\left\{
\begin{array}{l}
\mathcal C_j (F(z, \bar z, \lambda)) = F(z, \bar z, \lambda)k^{(\mathfrak{c},j)} (z,
\bar z) \;\;\mbox{if} \;\; \mathfrak{r}_j(\check \eta) = \check
\eta\;\;\mbox{for}\;j \in \{1, 2, 3, 4\}, \\[0.1cm]
\mathcal S_j (F(x, y, \lambda)) =
F(x, y, \lambda)k^{(\mathfrak{s}, j)}(x, y)
\;\;\mbox{if} \;\;\mathfrak{d}_j(\check \eta) = \check \eta \;\;\mbox{for}\;j \in \{1, 2, 3\}.
\end{array}
\right.
\end{equation}
Let $F^{(\mathfrak{c}, j)}$
(resp. $F^{(\mathfrak{s}, j)}$) denote the left components $F$ of the
generalized Iwasawa decomposition in \eqref{eq:Iwasawadecom} which have
the symmetries in \eqref{eq:invarianceofF} by $\mathcal C_j$ (resp. $\mathcal S_j$).
Let $\tilde k^{(\mathfrak{c}, j)} (z, \bar z)$ and $\tilde
k^{(\mathfrak{s}, j)}(x, y)$ be the $\lambda$-independent diagonal
matrices such that $\tilde k^{(\mathfrak{c}, j)} (z, \bar z)^2 =
k^{(\mathfrak{c}, j)}(z, \bar z)$ and $\tilde k^{(\mathfrak{s}, j)} (x,
y)^2 = k^{(\mathfrak{s}, j)}(x, y)$, respectively.
Setting $\tilde F^{(\mathfrak{c}, j)}(z, \bar z, \lambda) =
F^{(\mathfrak{c}, j)}(z, \bar z, \lambda) \tilde k^{(\mathfrak{c}, j)}
(z, \bar z)$ and $\tilde F^{(\mathfrak{s}, j)}(x, y, \lambda) =
F^{(\mathfrak{s}, j)}(x, y, \lambda) \tilde k^{(\mathfrak{s}, j)}(x,
y)$, we have
\begin{equation*}
\left\{
\begin{array}{l}
\mathcal C_j (\tilde F^{(\mathfrak{c}, j)}(z, \bar z, \lambda)) =
\tilde F^{(\mathfrak{c}, j)}(z, \bar z, \lambda) \;\;\mbox{if } \;\;
\mathfrak{r}_j (\check \eta) =\check \eta \;\;\mbox{for}\;j \in \{1, 2, 3, 4\}, \\[0.1cm]
\mathcal S_j(\tilde F^{(\mathfrak{s}, j)} (x, y, \lambda)) = \tilde
F^{(\mathfrak{s}, j)} (x, y, \lambda) \;\;\mbox{if} \;\; \mathfrak{d}_j
(\check \eta) = \check \eta \;\;\mbox{for}\;j \in \{1, 2, 3\}.
\end{array}
\right.
\end{equation*}
Moreover a straightforward calculation shows that $\alpha^{(\mathfrak{c}, j)} :=\tilde
F^{(\mathfrak{c}, j)-1} d \tilde F^{(\mathfrak{c}, j)}$ and $\alpha^{(\mathfrak{s}, j)} :=\tilde
F^{(\mathfrak{s}, j)-1} d \tilde F^{(\mathfrak{s}, j)}$ have the forms in
\eqref{eq:alpha} with the properties in \eqref{eq:solforMaurer}, i.e., $\tilde
F^{(\mathfrak{c}, j)} \in \Lambda SL(2, \mathbb
C)_\sigma^{(\mathfrak{c}, j)}$ and $\tilde F^{(\mathfrak{s}, j)} \in
\Lambda SL(2, \mathbb C)_\sigma^{(\mathfrak{s}, j)}$ are the
extended framings.
From the argument in Theorem \ref{thm:compactsplit}, the Sym formulas
$\varPhi^{(\mathfrak{c}, j)}$ for $j \in \{1, 2,
3\}$ in \eqref{eq:Sym-Bobenko-2} via $\tilde F^{(\mathfrak{c}, j)}$,
$\varPhi^{(\mathfrak{c}, 4)}$ in \eqref{eq:Sym-Bobenko-3} via
$\tilde F^{(\mathfrak{c}, 4)}$ and $\varPhi^{(\mathfrak{s}, j)}$ for
$j \in \{1, 2, 3\}$ in \eqref{eq:Sym-Bobenko-4} via $\tilde
F^{(\mathfrak{s}, j)}$ define immersions
which have the properties as desired. This completes the proof.
\end{proof}
\begin{Remark}
From the forms of pairs of involutions $\mathfrak{r}_j$ for $j \in \{1, 2, 3,
4\}$ defined in \eqref{eq:involutions}, the pairs of holomorphic
potentials $\check \eta$ for $(C, j)$ cases in Theorem
\ref{thm:DPWforint} are generated by a single potential,
i.e., $\check \eta = (\eta, \tau ) = (\eta, \mathfrak{c}_j ( \eta ))$,
where $\mathfrak{c}_j$ for $j \in \{1, 2, 3, 4\}$ are involutions defined
in \eqref{eq:inv-1-forms}.
\end{Remark}
\begin{Remark}
In the proof of Theorem \ref{thm:DPWforint}, we assume our domain
$\mathfrak D^2 \subset \mathbb C^2$ is sufficiently small around the
initial point $(z_*, w_*) \in \mathfrak D^2$ so that the middle term
$w_n$ of the generalized
Iwasawa decomposition of Theorem \ref{doublesplitting} is in the
identity component. In
general, if we consider the larger domain $\widetilde{\mathfrak
D}^2$ such that $\mathfrak D^2 \subset \widetilde{\mathfrak D}^2$, then
the middle terms $w_n$ have many components. Therefore the extended
framing $F$ could have singularities on $\widetilde{\mathfrak D}^2$.
\end{Remark}
|
{
"timestamp": "2012-03-09T02:01:42",
"yymm": "1203",
"arxiv_id": "1203.1718",
"language": "en",
"url": "https://arxiv.org/abs/1203.1718"
}
|
\subsection*{Einleitung}
Außer den verschiedenen Rechenarten, die im allgemeinen in der Zahlentheorie zu behandelt werden pflegen und quasi den praktischen Teil dieser Disziplin festsetzen, ist der theoretische Teil derselben, der sich mit dem Untersuchen der Natur der Zahlen beschäftigt, schon einst begonnen worden nicht weniger behandelt zu werden, wie sich aus Euklid und Diophant einsehen lässt, wo man außerordentliche Eigenschaften der Zahlen entdeckt und bewiesen vorfindet. Je mehr aber darauf die Mathematiker die Gestalt und die Beschaffenheit der Zahlen untersucht haben, haben sie um Vieles mehr Eigenschaften derer beobachtet, woher sie die schönsten Theoreme, die die Natur der Zahlen aufzeigen, berechnet haben, die teils durch Beweise untermauert worden sind, teils diese immer noch brauchen, weil sie entweder von den Autoren nicht gefunden worden sind oder mit der Zeit verloren gegangen sind; auf diese Art tauchen verstreut viele zahlentheoretische Theoreme dieser Art auf, deren Beweise man noch heute ersehnt, auch wenn sich deren Gültigkeit nicht in Zweifel ziehen lässt. Und hier müssen wir einen riesigen Unterschied, welcher zwischen zahlentheoretischen und geometrischen Theoremen einhergeht, nicht wenig bewundern, weil kaum irgendeine Proposition hervorgebracht werden kann, die nicht klar ist, entweder Wahres oder Falsches zu zeigen, während dagegen viele Propositionen über die Natur der Zahlentheorie bekannt sind, deren Gültigkeit sich von uns erkennen lässt, aber sich in kleinster Weise beweisen lässt. Man hat eine große Menge von Theoremen dieser Art, die von Fermat hinterlassen wurde, deren Beweise er versicherte zum größten Teil gefunden zu haben, welche zum außerordentlichen Verlust für diese Wissenschaft zu bedauern sind, mit seinen verloren gegangen zu sein. Wie viele Beweise solcher Theoreme aber entweder bekannt sind oder wiederentdeckt sind, wird sich bei diesen gewiss um vieles größere Kraft des Geistes zeigen, die wir kaum bei irgendeiner anderen Art von Beweisen entdecken; daher ist bei dieser Aufgabe nicht so sehr die Nützlichkeit, durch die Wissenschaft der Zahlen illustriert wird, zu schätzen, wie die größte Freiheit, durch die die Beweise dieser Art sich in Bezug auf andere unterscheiden. Und deswegen, weil ich schon öfter, als es den Meisten als angemessen erscheinen kann, in diesem Gebiet gearbeitet habe, glaube ich für meine Person, die Arbeit nicht verschwendet zu haben und glaube auch jetzt nicht, dass die Theoreme, die ich hier vorlege, frei von Nutzen sind. Besonders bemerkenswert schien jenes Theorem von Fermat, in welchem er versichert, dass in dieser Formel $a^{p-1} - 1$ enthaltenen Zahlen immer durch die Zahl $p$ teilbar sind, wenn sie natürlich prim war und auch $a$ trotzdem durch sie keine Teilung zulässt, von welchem Theorem ich schon zwei Beweise gegeben habe. Nun betrachte ich selbiges, aber in weiterem Sinne und untersuche es in der Art, wenn der Teiler keine Primzahl war, sondern irgendeine Zahl $N$, ein Exponent von welcher Art der Potenz zugeteilt werden muss, dass der Ausdruck $a^n - 1$ immer durch die Zahl $N$ teilbar ist, während die Zahl $a$ mit ihr keinen gemeinsamen Teiler hat. Ich habe aber entdeckt, dass dies immer passiert, sooft der Exponent $n$ gleich der Menge der Teiler der Zahlen kleiner als $N$ war, die zu $N$ prim sind. Um das also zu beweisen, braucht man vor allem Theoreme solcher Art, aus denen für irgendeine vorgelegte Zahl $N$ erkannt werden kann, wie viele unter den Zahlen kleiner als sie selbst zu ihr prim sein werden, oder die mit ihr keinen gemeinsamen Teiler haben; diese Theoreme scheinen nun selbst einen weiteren Nutzen zu haben und zu anderen verborgeneren Eigenschaften der Zahlen einen Zugang zu verschaffen. Nachdem diese Dinge vorausgeschickt worden sind, ist der Beweis der vorgelegten Wahrheit deart beschaffen, dass er größerer Aufmerksamkeit nicht unwürdig sein wird.
\section*{Theorem 1}
\paragraph{§1}
Wenn die Terme einer arithmetischen Progression, deren Differenz eine zu $n$ prime Zahl sei, durch irgendeine Zahl $n$ geteilt wurden, werden unter den Resten alle Zahlen kleiner als die Zahl $n$ auftauchen.
\subsection*{Beweis}
Es sei der erste Term der arithmetischen Progression gleich $a$ und die Differenz gleich $d$, die eine zu $n$ prime Zahl sei oder die mit der Zahl $n$ keinen gemeinsamen Teiler außer der Einheit habe, sodass die arithmetische Progression
\[
a,\quad a+d,\quad a+2d,\quad a+3d,\quad a+4d,\quad a+5d,\quad \text{etc}
\]
sein wird, und ich sage, wenn die einzelnen Terme durch die Zahl $n$ geteilt werden, dass unter den Resten alle Zahlen kleiner als $n$ auftauchten. Um das zu zeigen, wird es genügen, nur die $n$ Terme dieser Progression betrachtet zu haben, die
\[
a,\quad a+d,\quad a+2d,\quad a+3d,\quad \dots \quad a+(n-1)d
\]
sind. Wenn daher also diese einzelnen Terme durch $n$ geteilt werden, müssen alle Reste zueinander verschieden sein. Wenn nämlich zwei Terme, z.\,B. $a + \mu d$ und $a + \nu d$, während $\mu$ und $\nu$ Zahlen kleiner als $n$ selbst sind, diese durch $n$ geteilt gleiche Reste liefern würden, würde deren Differenz $(\nu - \mu )d$ jedenfalls durch $n$ teilbar sein. Weil aber die Zahlen $d$ und $n$ keinen gemeinsamen Teiler haben, ist es notwendig, dass $(\nu - \mu )$ eine Teilung durch $n$ zuließen. Weil daher all jene Reste verschieden sind und ja die Anzahl der Terme von der Zahl her gleich $n$ ist, werden bei diesen natürlich alle Zahlen kleiner als $n$ auftauchten, natürlich
\[
0,\quad 1,\quad 2,\quad 3,\quad 4,\quad 5,\quad \dots\quad n-1,
\]
wenn freilich die Differenz der Progression $d$ eine zum vorgelegten Teiler $n$ prime Zahl ist.
\hfill\textsc{Q.E.D.}
\subsection*{Korollar 1}
\paragraph{§2}
Unter diesen Termen dieser arithmetischen Progression, deren Anzahl $n$ ist, solange seine Differenz eine zu $n$ prime Zahl ist, wird man also gewiss eine finden, die durch $n$ teilbar ist; dann aber wird auch eine da sein, die durch $n$ geteilt einen gegebenen Rest $r$ zurücklassen wird.
\subsection*{Korollar 2}
\paragraph*{§3}
Wenn also die Zahl $d$ zu $n$ prim war, kann immer eine Zahl dieser Zahl dieser Form $a + \nu d$ beschafft werden, während $a$ irgendeine Zahl ist und $\nu$ kleiner als $n$ ist, die durch die Zahl $n$ teilbar ist, und es wird auch unter denselben Bedingungen immer eine solche Zahl $a+rd$ gegeben sein, die durch $n$ geteilt einen gegebenen Rest $r$ lassen wird.
\subsection*{Korollar 3}
\paragraph{§4}
Nachdem also Zahlen $a$ und $d$ gegeben worden sind, von denen diese $d$ zu $n$ prim sei, lassen sich immer Zahlen $\mu$ und $\nu$ finden, dass dieser Gleichung
\[
a + \nu d = \mu n
\]
oder auch dieser
\[
a + rd = \mu n + r
\]
genügt wird, was für eine Zahl kleiner als $n$ für $r$ auch immer angenommen wird.
\subsection*{Bemerkung}
\paragraph{§5}
Was wir über die Anzahl $n$ der Terme der arithmetischen Progression bewiesen haben, gilt auch über die ganze Progression ins Unendliche fortgesetzt; die Terme, die nach jenen $n$ Termen folgen, erzeugen nämlich in derselben Reihenfolge die Reste, wenn sie durch $n$ geteilt werden. So stimmen die Reste der Terme, die nach $a + (n-1)d$ folgen und $a+nd$, $a+(n+1)d$, $a+(n+2)d$, etc sind, durch $n$ geteilt mit den Resten überein, die aus den anfänglichen Termen $a$, $a+d$, $a+2d$, etc entstehen. Und wenn die ganze Reihe in unendlich viele Perioden aufgeteilt werden, indem man irgendeiner $n$ Terme auf diese Weise zuteilt
\[
a, \quad a+d, \dots, a+(n-1)d | a+nd, \dots, a+(2n-1)d | a+2nd, \dots, a+(3n-1)d | \dots
\]
werden die Terme einer beliebigen Periode dieselben Reste, in derselben Reihenfolge geordnet, liefern; die ersten und zweiten und dritten Terme aller Perioden werden nämlich gleich bleiben die gleichen Reste geben. Wenn wir daher die Art der Reste erkennen wollen, genügt es, eine einzige Periode untersucht zu haben.
\section*{Theorem 2}
\paragraph{§6}
In einer arithmetischen Progression, deren Anzahl an Termen gleich $n$ ist, werden so viele Terme zur Zahl $n$ prim sein, wie unter den Zahlen kleiner als $n$ selbst prime zu $n$ gegeben sind, solange die Differenz der Progression zu $n$ prim war.
\subsection*{Beweis}
Es sei nämlich $a$ der erste Term und $d$ die Differenz der Progression, die zu $n$ prim sei, und daher ist die Progression, die $n$ Terme enthält,
\[
a,\quad a+d,\quad a+2d,\quad a+3d,\quad \dots \quad a +(n-1)d.
\]
Weil ja also, wenn diese Terme durch $n$ geteilt werden, natürlich alle Reste der Zahl kleiner als $n$ selbst auftauchen, wollen wir setzen, dass aus irgendeinem Term $a+\nu d$ der Rest $r$ resultiert und es ist klar, wenn $r$ eine zu $n$ prime Zahl war, dass auch jener Term $a + \nu d$ zu $n$ prim sein wird; wenn aber $r$ mit $n$ einen gemeinsamen Teiler hat, wird selbiger auch gemeinsamer Teiler der Zahlen $n$ und $a+\nu d$ sein. Wie viele Zahlen daher unter den Zahlen kleiner als $n$ prim zu $n$ waren, man wird genauso viele zu $n$ prime Zahlen unter den Termen der vorgelegten arithmetischen Progression haben.
\hfill\textsc{Q.E.D.}
\subsection*{Korollar 1}
\paragraph{§7}
Wenn $n$ eine Primzahl war, weil alle Zahlen kleiner al selbige zu dieser auch prim sind, deren Anzahl also gleich $n-1$ ist, werden auch in jener arithmetischen Progression alle Terme aus einem einzigen zu $n$ prim sein, weil ja ein einziger durch $n$ teilbar ist.
\subsection*{Korollar 2}
\paragraph{§8}
Wenn aber $n$ eine zusammengesetzte Zahl war, sind zwischen den Zahlen kleiner als selbige solche gegeben, die mit ihr einen gemeinsamen Teiler haben, und man wird in der Tat auch so viele in der arithmetischen Progression finden, bei denen dieselben gemeinsamen Teiler mit $n$ übereinstimmen.
\subsection*{Korollar 3}
\paragraph{§9}
Wenn $n=6$ war, werden, weil unter den Zahlen kleiner als $6$ zwei zu $6$ prim aind, natürlich $1$ und $5$, in der ganzen arithmetischen Progression der $6$ Terme
\[
a,\quad a+d,\quad a+2d,\quad a+3d,\quad a+4d,\quad a+5d
\]
nur zwei zu $6$ prim sein, solange die Differenz $d$ eine zu $6$ prime Zahl ist. Wenn man so $a=4$, $d=5$ nimmt, sind zwei dieser sechs Zahlen $4$, $9$, $14$, $19$, $24$, $29$, natürlich $19$ und $29$, zu $6$ prim, eine, $24$, ist durch $6$ teilbar, die übrigen $4$, $9$, $14$ sind in der Tat genauso wie $2$, $3$, $4$ zu $6$ nicht prim.
\subsection*{Bemerkung}
\paragraph{§10}
Diese Theoreme haben in der Lehre und der Betrachtung der Natur der Zahlen einen riesigen Nutzen, hier scheint es aber nur ratsam diese zu verwenden, um diese Frage zu erörtern, nachdem irgendeine Zahl $n$ vorgelegt worden ist, wie viele unter den Zahlen kleiner als $n$ zu derselben Zahl $n$ prim sind. Es ist freilich sofort klar, wenn $n$ eine Primzahl ist, dass alle Zahlen kleiner als selbige zugleich zu ihr prim sein werden und deren Anzahl deshalb gleich $n-1$ sein wird. Wenn aber $n$ eine zusammengesetzte Zahl ist, ist die Menge der zu ihr primen Zahlen kleiner als selbige; wie groß sie aber in jedem Fall ist, kann nicht so leicht angegeben. Wenn $n=12$ ist, findet man unter den Zahlen kleiner als diese nur vier zu $12$ prime, natürlich $1$, $5$, $7$, $11$, und wenn $n=60$ ist, sind die kleineren zu dieser Zahl primen
\[
1,~~ 7,~~ 11,~~ 13,~~ 17,~~ 19,~~ 23,~~ 29,~~ 31,~~ 37,~~ 41,~~ 43,~~ 47,~~ 49,~~ 53,~~ 59
\]
deren Anzahl $16$ ist, woher die übrigen $43$ alle mit $60$ einen gemeinsamen Teiler haben. Hier sollte man sich aber daran erinnern, dass die Einheit eine zu allen Zahlen prime Zahl ist, auch wenn sie ein Teiler aller ist; das ist aus der Definition klar, nach welcher Zahlen bezeichnet werden, zueinander prim zu sein, die außer der Einheit keinen anderen Teiler haben.
\section*{Theorem 3}
\paragraph{§11}
Wenn $n$ irgendeine Potenz der Primzahl $p$ oder $n=p^m$ ist, werden zwischen den Zahlen größer als selbige so viele zu ihr prim sein, wie Einheiten in
\[
p^m - p^{m-1} = p^{m-1}(p-1)
\]
enthalten sind.
\subsection*{Beweis}
Die Menge aller Zahlen kleiner als die Potenz $n=p^m$ ist $p^m - 1$, unter diesen findet man aber einige, die zu $n$ nicht prim sind, natürlich alle Vielfachen von $p$ kleiner als $n$ und zusätzlich keine anderen; daraus werden die folgenden Zahlen zu $n$ nicht prim sein
\[
p,\quad 2p,\quad 3p,\quad 4p, \quad \dots \quad p^m - p
\]
deren Anzahl $p^{m-1}-1$ ist; nachdem diese von der Zahl aller $n=p^m$ abgezogen wurde, findet man die Menge der kleineren, die zu $p^m$ prim sind, deren Anzahl deshalb gleich $p^m - p^{m-1} = p^{m-1}(p-1)$ ist.
\hfill\textsc{Q.E.D}.
\subsection*{Korollar 1}
\paragraph{§12}
Daher folgt also, was an sich klar ist, wenn $n=p$ war, während $p$ eine Primzahl ist, dass die Anzahl aller zu ihr primen Zahlen kleiner als selbige gleich $p-1$ ist, weil ja alle Zahlen kleiner als dieselbe zugleich zu ihr prim sind.
\subsection*{Korollar 2}
\paragraph{§13}
Aber wenn $n=p^2$ ist, es die Menge unter den Zahlen kleiner als selbige von diesen, die zu ihr prim sind, gleich $pp-p = p(p-1)$; die übrigen, deren Anzahl $p-1$ ist, werden zu $n=p^2$ nicht prim sein oder durch $p$ teilbar.
\subsection*{Korollar 3}
\paragraph{§14}
Nachdem aber irgendeine Potenz $n=p^m$ der Primzahl vorgelegt wurde, findet man unter den Zahlen kleiner als selbige, deren Menge gleich $p^m - 1$ ist, $p^{m-1}-1$, die durch $p$ teilbar sind und daher zu $p^m$ nicht prim sind; aber alle übrigen, deren Anzahl gleich $p^m - p^{m-1} = p^{m-1}(p-1)$ ist, sind zu $p^m$ prim.
\subsection*{Bemerkung}
\paragraph{§15}
Wenn also die vorgelegte Zahl $n$ eine Potenz einer Primzahl $n$ war, werden wir mithilfe dieser Regel angeben können, wie viele unter allen Zahlen kleiner als selbige zu dieser prim sein werden. Wann immer aber die Zahl $n$ aus zweien oder mehr Primzahlen zusammengesetzt war, kann daher diese Frage nicht erledigt werden; durch Anwenden der vorhergehenden Theoreme können wir diese sich weiter erstreckende Frage lösen.
\section*{Theorem 4}
\paragraph{§16}
Wenn die Zahl $n$ das Produkt zweier Primzahlen $p$ und $q$ war oder $n=pq$, ist die Menge aller zu ihr primen Zahlen kleiner als selbige gleich $(p-1)(q-1)$.
\subsection*{Beweis}
Weil die Anzahl aller Zahlen kleiner als $n=pq$ gleich $pq-1$ ist, müssen daher die zuerst ausgeschlossen werden, die durch $p$ teilbar sind, darauf aber auch die, die durch $q$ teilbar sind; und nach dem Streichen dieser wird man die gesuchte Menge finden. Man notiere also von der Einheit bis hin zu $pq$ die Zahlen, die zu $p$ prim sind, auf diese Weise
\[
\begin{array}{cccccc}
1 & 2 & 3 & 4 & \dots & p-1 \\
p+1 & p+2 & p+3 & p+4 & \dots & 2p-1 \\
2p+1 & 2p+2 & 2p+3 & 2p+4 & \dots & 3p-1 \\
3p+1 & 3p+2 & 3p+3 & 3p+4 & \dots & 4p-1 \\
\vdots & \vdots & \vdots & \vdots & & \vdots \\
(q-1)p+1 & (q-1)p+2 & (q-1)p+3 & (q-1)p+4 & \dots & pq-1
\end{array}
\]
und nun müssen aus diesen nur die ausgewählt werden, die zugleich auch zu $q$ prim sind. Man betrachte also die vertikalen Reihen, deren Anzahl $p-1$ ist; eine beliebige aber enthält $q$ Terme, die in einer arithmetischen Progression wachsen, während die Differenz $p$ ist, die eine zu $q$ prime Zahl ist. In einer beliebigen vertikalen Reihe werden also alle Terme außer einem zu $q$ prim sein (nach §$7$); daher enthält jede vertikale Reihe $q-1$ zu $q$ prime Zahlen. Weil daher die Anzahl der vertikalen Reihen $p-1$ ist, sind in allen gleichzeitig $(p-1)(q-1)$ zur Zahl $q$ prime Zahlen enthalten und dieselben werden also auch zum Produkt $pq$ prim sein; als logische Konsequenz wird man unter allen Zahlen kleiner als $pq$ selbst $(p-1)(q-1)$ zu $pq$ prime Zahlen finden.
\hfill\textsc{Q.E.D.}
\subsection*{Korollar 1}
\paragraph{§17}
Weil die Menge aller Zahlen kleiner als das Produkt $pq$ gleich $pq-1$ ist, sind unterdessen immer $(p-1)(q-1) = pq-p-q+1$ zu $pq$ prim, die übrigen aber, deren Anzahl $p+q-2$ ist, sind zu ihr nicht prim oder haben mit ihr als gemeinsamen Teiler entweder $p$ oder $q$.
\subsection*{Korollar 2}
\paragraph{§18}
Das ist auch daher klar, weil unter den Zahlen kleiner als das Produkt $pq$ $q-1$ durch $p$ teilbare Zahlen sind, natürlich
\[
p,\quad 2p,\quad 3p,\quad 4p,\quad \dots \quad (q-1)p
\]
darauf unter denselben $p-1$ Zahlen durch $q$ teilbar sind, nämlich
\[
q,\quad 2q,\quad 3q,\quad 4q,\quad \dots \quad (p-1)q;
\]
weil ja von jenen alle verschieden sind, wird man im Ganzen
\[
(q-1) + (p-1) = p+q-2
\]
Zahlen haben, die zu $pq$ nicht prim sind.
\subsection*{Korollar 3}
\paragraph{§19}
Wenn man also sucht, wie viele von den Zahlen $1$ bis hin zu $15$ zu $15$ prime Zahlen sind, lehrt wegen $p=3$ und $q=5$ die Regel, dass die Anzahl derer $2\cdots 4 = 8$ ist, die ja sind
\[
1, \quad 2,\quad 4,\quad 7,\quad 8,\quad 11,\quad 13,\quad 14
\]
sind. Auf die gleiche Weise ist die Menge der zu $35$ primen Zahlen von $1$ bis $35$ wegen $p=5$ und $q=7$ $4\cdot 6 = 24$ und diese Zahlen sind
\[
1,\,2,\,3,\,4,\,5,\,6,\,8,\,9,\,11,\,12,\,13,\,16,\,17,\,18,\,19,\,22,\,23,\,24,\,26,\,27,\,29,\,31,\,32,\,33,\,34
\]
\subsection*{Bemerkung}
\paragraph{§20}
Weil ja hier die Frage über Zahlen geht, die zu einer Zahl prim sind und kleiner als sie, werden diese sich angenehm als zu dieser Zahl prime Teile bezeichnen lassen. Wenn so die Primzahl $p$ vorgelegt war, wird die Anzahl der zu ihr primen Teile gleich $p-1$ sein; wenn die vorgelegte Zahl eine Potenz der Primzahl, $p^n$, ist, wird die Anzahl der zu ihr primen Zahlen gleich $p^n - p^{n-1}(p-1)$ sein; aber wenn die vorgelegte Zahl das Produkt zweier verschiedenen Primzahlen gleich $pq$ ist, ist die Anzahl der zur ihr primen Teile gleich $(p-1)(q-1)$; und auf diese Weise können wir die Umschweife beim Sprechen verkürzen; auf die gleiche Weise können wir zeigen, wenn die vorgelegte Zahl das Produkt aus drei verschiedenen Primzahlen gleich $pqr$ ist, dass die Anzahl der zu ihr primen Teile gleich $(p-1)(q-1)(r-1)$ sein wird; und das ließe sich auch auf Produkte mehrerer ausdehnen. Aber die folgende Proposition umfasst all diese Fälle in sich.
\section*{Theorem 5}
\paragraph*{§21}
Wenn $A$ und $B$ zwei zueinander prime Zahlen sind und die Anzahl der zu $A$ primen Teile gleich $a$ ist, die Anzahl der zu $B$ primen Teile gleich $b$ ist, dann wird die Anzahl der zum Produkt $AB$ primen Teile gleich $ab$ sein.
\subsection*{Beweis}
Es seien $1$, $\alpha$, $\beta$, $\gamma$, $\dots$, $\omega$ zu $A$ prime Zahlen und kleiner als selbige oder zu $A$ prime Teile, die Anzahl welcher Teile also per Annahme gleich $a$ ist. Es werden also so viele Zahlen zu $A$, wie von $A$ zu $2A$ prim sein, und ebenso von $2A$ zu $3A$, usw. Auf diese Weise können alle zu $A$ prime Zahlen von der Einheit bis hin zur vorgelegten Zahl $AB$ beschafft werden, welche das folgende Schema beschaffen wird:
\[
\begin{array}{ccccc}
1 & \alpha & \beta & \dots & \omega \\
A+1 & A+\alpha & A+\beta & \dots & A+\omega \\
2A + 1 & 2A+\alpha &2A+\beta & \dots & 2A + \omega \\
3A+1 & 3A+\alpha & 3A+\beta & \dots & 3A + \omega \\
\vdots & \vdots & \vdots & & \vdots \\
(B-1)A+1 & (B-1)A+\alpha & (B-1)A+\beta & \dots & (B-1)A+\omega
\end{array}
\]
Hier enthalten die einzelnen horizontalen Reihen $a$ Terme und die Anzahl aller horizontalen Reihen ist gleich $B$, woher alle Reihen zusammen $aB$ Terme ergeben, die schon alle zu $A$ prim sein werden. Daher müssen also noch die ausgeschlossen werden, die zu $B$ nicht prim sind, damit auf diese Weise die zurückgelassen werden, die nicht nur zu $A$, sondern auch zu $B$ und daher zum Produkt $AB$ prim sind; oder es müssen aus diesen Reihen nur die Terme gezählt werden, die auch zu $B$ prim sind. Für dieses Ziel wollen wir die Reihen vertikal betrachten; und weil die Anzahl der vertikalen Reihen gleich $a$ ist, wird eine beliebige vertikale Reihe $B$ Terme enthalten, die in einer aritmetischen Progression vermehrt worden sind; weil deren Differenz gleich $A$ ist und eine zu $B$ prime Zahl ist, wird durch Theorem $2$ eine beliebige vertikale Reihe so viele zu $B$ prime Terme enthalten, wie zur Zahl $B$ prime Teile gegeben sind. Die Anzahl derer ist also per Annahme gleich $b$. Weil also die einzelnen vertikalen Reihen $b$ zu $B$ prime Terme enthalten, die deshalb auch zum Produkt $B$ prim sein werden, wird die Anzahl aller zu $AB$ primen Terme, das heißt zu dieser Zahl $AB$ primen Teile, gleich $ab$ sein.
\hfill\textsc{Q.E.D.}
\subsection*{Korollar 1}
\paragraph*{§22}
Wenn darüber heraus eine dritte Zahl $C$ hinzugefügt wird, die zu jeder der beiden vorhergehenden $A$ und $B$ oder zum Produkt der $AB$ prim ist, und die Anzahl der zu $C$ primen Teile gleich $c$ ist, dann wird die Anzahl der zum Produkt $ABC$ primen Teile gleich $abc$ sein. Es kann nämlich das Produkt $AB$ als eine Zahl betrachtet werden, die Menge der zu ihr primen Teile gleich $ab$ ist; und weil $c$ zu $AB$ prim ist, hat das Theorem hier Geltung.
\subsection*{Korollar 2}
\paragraph*{§23}
Weil also jede einzelne Zahl in $N$ zueinander prime Faktoren aufgelöst werden kann, welche einzelnen entweder selbst Primzahlen sind oder Potenzen von Primzahlen, wird mit Hilfe dieser Regel die Menge der zu irgendeiner Zahl $N$ primen Fälle angegeben werden können.
\subsection*{Korollar 3}
\paragraph*{§24}
Während natürlich $p$, $a$, $r$, $s$ etc Primzahlen sind, wird jeder Zahl $N$ in einer Form dieser Art $N = p^{\lambda} q^{\mu}r^{\nu}s^{\xi}$ erfasst werden, woher die Anzahl der zu $N$ primen Teile
\[
p^{\lambda-1}(p-1)\cdot p^{\mu - 1}(q-1)\cdot r^{\nu - 1}(\nu - 1)\cdot s^{\xi - 1}(s-1)
\]
sein wird.
\subsection*{Korollar 4}
\paragraph*{§25}
Für einfachere Formen von Zahlen wird sich die Menge der zu diesen primen Teile so verhalten:
\begin{tabular}{p{1.7cm} p{4cm} >{\centering\arraybackslash}p{1.7cm} >{\centering\arraybackslash}p{3cm}}
\toprule
\small\centering Vorgelegte Zahl & \small \centering Menge der zu ihr primen Teile & \small\centering Vorgelegte Zahl & \small \centering Menge der zu ihr primen Teile
\tabularnewline
\midrule
\small
$p$ & $p-1$ & $2$ & $1$ \\
\midrule
$pp$ & $p(p-1)$ & $3$ & $2$ \\
$pq$ & $(p-1)(q-1)$ & $4$ & $2$ \\
\midrule
$p^3$ & $pp(q-1)$ & $5$ & $4$ \\
$p^2q$ & $p(p-1)(q-1)$ & $6$ & $2$ \\
$pqr$ & $(p-1)(q-1)(r-1)$ & $7$ & $6$ \\
\midrule
$p^4$ & $p^3(p-1)$ & $8$ & $4$ \\
$p^3 q$ & $p^2(p-1)(q-1)$ & $9$ & $6$ \\
$p^2q^2$ & $p(p-1)q(q-1)$ & $10$ & $4$ \\
$p^2qr$ & $p(p-1)(q-1)(r-1)$ & $11$ & $10$ \\
$pqrs$ & $(p-1)(q-1)(r-1)(s-1)$ & $12$ & $4$ \\
\midrule
$p^5$ & $p^4(p-1)$ & $13$ & $12$ \\
$p^4q$ & $p^3(p-1)(q-1)$ & $14$ & $6$ \\
$p^3q^2$ & $p^2(p-1)q(q-1)$ & $15$ & $8$ \\
$p^2qr$ & $p^2(p-1)(q-1)(r-1)$ & $16$ & $8$ \\
$p^2q^2r$ & $p(p-1)q(q-1)(r-1)$ & $17$ & $16$ \\
$p^2qrs$ & $p(p-1)(q-1)(r-1)(s-1)$ & $18$ & $6$ \\
$pqrst$ & $(p-1)(q-1)(r-1)(s-1)(t-1)$ & $19$ & $18$ \\
\bottomrule
\end{tabular}
\subsection*{Korollar 5}
\paragraph*{§26}
Nachdem daher also irgendeine Zahl vorgelegt wurde, wird die Menge der zur ihr primen Zahlen angenehm bestimmt werden. Wenn z.\,B. 360 vorgelegt wird, wird, weil $360 = 2^3 \cdot 3^2 \cdot 5$ ist, die Menge der zu 360 primen Teile gleich $4\cdot 9\cdot 4 = 96$ sein.
\subsection*{Bemerkung}
\paragraph*{§27}
Dies kann über die Menge der zu einer Zahl primen Teile für unser Unternehmen ausreichen. Dennoch wird es förderlich sein, über die zu einer Zahlen primen Teile selbst dies bemerkt zu haben: Wenn die vorgelegte Zahl $N$ war und unter den zu ihr primen Teilen die Zahl $\alpha$ auftaucht, wird da selbst auch die Zahl $N-\alpha$ auftauchen, weil ja, während $\alpha$ zu $N$ prim ist, auch $N-\alpha$ zu $N$ prim sein wird. Daher wird es also genügen für eine beliebige Zahl nur die Hälfte ihrer kleineren Teile gefunden zu haben, weil die übrigen deren Komplemente zur Zahl $N$ selbst sind. Auf die gleiche Weise wird, wenn $N$ eine gerade Zahl ist, unter den zu $N$ primen Zahlen auch $\frac{1}{2}N-\alpha$ auftauchen, dann aber auch $\frac{1}{2}N+\alpha$. Wenn so $N$ durch irgendeine Zahl $n$ teilbar ist, werden unter den zu ihr primen Teilen auch diese Zahlen auftauchen
\[
\frac{1}{n}A\pm\alpha, \quad \frac{2}{n}N\pm\alpha,\quad \dots \quad \frac{n-1}{n}N\pm \alpha \quad \text{und}\quad N-\alpha
\]
und daher werden um Vieles leichter die Teile selbst tatsächlich beschafft werden können.
\section*{Theorem 6}
\paragraph*{§28}
Wenn eine Zahl $x$ prim zu $N$ war, dann werden alle Potenzen von $x$ durch $N$ geteilt Reste lassen, die zur Zahl $N$ prim sein werden.
\subsection*{Beweis}
Weil nämlich $x$ eine zu $N$ prime Zahl ist, werden alle Potenzen von ihr auch zu $N$ prim sein, und wenn sie daher durch $N$ geteilt werden, werden auch die Reste zu $N$ prime Zahlen sein.
\hfill\textsc{Q.E.D.}
\subsection*{Korollar 1}
\paragraph*{§29}
Unter den Resten der Potenzen von $x$, die durch $N$ geteilt werden, tauchen also keine anderen Zahlen auf, wenn diese nicht die zu $N$ prime Teile sind; weil die Anzahl deren für die Gestalt der Zahl $N$ bestimmt ist, existieren unzählige Potenzen von $x$, die durch $N$ geteilt gleiche Reste zurücklassen.
\subsection*{Korollar 2}
\paragraph*{§30}
Unter den Resten, die aus der Teilung der Potenzen von $x$ durch die Zahl $N$ entstehen, wird man aber immer die Einheit finden, deshalb weil zwischen den Potenzen von $x$ auch $x^0 = 1$ gezählt werden muss. Ob aber außer der Einheit auch alle übrigen zu $N$ primen Teile unter den Resten auftauchen oder nicht, werden wir bald sehen.
\subsection*{Korollar 3}
\paragraph*{§31}
Wenn man für $x$ die Einheit nimmt, werden alle Reste die Einheit sein, welche Zahl für $N$ auch immer angenommen worden war. Wenn man darauf $x = N-1$ nimmt, welche Zahl zu $N$ auch prim ist, wird man bei den Resten, die aus der Teilung der Potenzen
\[
(N-1)^0, \quad (N-1)^1, \quad (N-1)^2, \quad (N-1)^3, etc
\]
entstehen, nur zwei verschiedene finden, natürlich 1 und $N-1$, die ununterbrochen abwechselnd auftauchen.
\subsection*{Korollar 4}
\paragraph*{§32}
Je nachdem wie also die Zahl $x$ von der Art zu $N$ beschaffen war, kann es natürlich geschehen, dass unter den Resten aller Potenzen von $x$ nicht alle zum Teiler $N$ primen Teile auftauchen.
\subsection*{Korollar 5}
\paragraph*{§33}
Wenn also alle zur Zahl $N$ primen Teile $1$, $a$, $b$, $c$, $d$, $e$, \dots sind, deren Anzahl gleich $n$ ist, werden unter den erwähnten Resten entweder all diese Teile auftauchen oder nur gewisse, unter denen man aber immer die Einheit finden wird.
\subsection*{Korollar 6}
\paragraph*{§34}
Wenn daher nicht all jene Teile in den Resten, die aus der Teilung der Potenzen von $x$ durch die Zahl $N$ zurückgelassen werden, auftauchen, werden jene Teile in zwei Klassen aufgeteilt werden, deren eine die Teile enthalten wird, die in den Resten auftauchen, die andere aber die Teile, die nicht in den Resten auftauchen.
\section*{Theorem 7}
\paragraph*{§35}
Wenn die Reihe der Potenzen $x^0$, $x^1$, $x^2$, $x^3$, $x^4$, $x^5$, etc durch die Zahl $N$, die zu $x$ prim sei, geteilt wird, werden bis dorthin verschiedene Reste hervorgehen, bis man zu einer Potenz gelangt, die wiederum die Einheit für den Rest liefert.
\subsection*{Beweis}
Weil ja in der Reihe der Potenzen $1, x, x^2, x^3, x^4$, etc ins Unendliche fortgesetzt die Reste nicht alle verschieden sein können, ist nötig, dass schließlich ein bestimmter von den vorhergehenden wieder auftaucht; und ich sage, dass die Einheit dieser Rest ist, weil er als erster von allen auftaucht. Wenn irgendeiner das verneinen sollte, sei $x^{\mu}$ jene Potenz, deren Rest als erstes in den folgenden aus der Potenz $x^{\mu + \nu}$ wieder auftaucht; weil also die Potenzen $x^{\mu}$ und $x^{\mu + \nu}$ die gleichen Reste liefern werden, wird deren Differenz $x^{\mu + \nu} - x^{\mu} = x^{\mu}(x^{\nu}-1)$ durch die Zahl $N$ teilbar sein. Aber der erste Faktor des Produktes $x^{\mu}(x^{\nu}-1)$ ist eine zu $N$ prime Zahl, also ist notwendig, dass der andere $x^{\nu}-1$ durch $N$ teilbar ist. Daher würde aber die Potenz $x^{\nu}$ durch $N$ geteilt den Rest 1 geben und so wird die Einheit unter den folgenden Resten schneller auftauchen als der Rest einer Potenz $x^{\mu}$, welcher ja per Annahme erst in der höheren Potenz $x^{\mu + \nu}$ wiederkehrt. Daraus ist klar, dass kein Rest nochmal auftauchen kann, wenn nicht zuvor die Einheit dazwischen aufgetaucht sein wird.
\hfill\textsc{Q.E.D.}
\subsection*{Korollar 1}
\paragraph*{§36}
Nachdem die Teilung der Terme der Reihe $1$, $x$, $x^2$, $x^3$, $x^4$, etc durch die $N$, die zu $x$ prim ist, von Anfang verschiedene Reste gegeben hatte, z.\,B. $1$, $\alpha$, $\beta$, $\gamma$, etc, wird schließlich wieder der erste Rest $1$ auftauchen; wenn er daher also aus der Potenz $x^{\nu}$ entsteht, wird die Anzahl der vorhergehenden verschiedenen Reste gleich $\nu$ sein.
\subsection*{Korollar 2}
\paragraph*{§37}
Wann immer aber die Potenz $x^{\nu}$ den Rest $1$ gibt, ist es derselbe, den der erste Term $x^0$ gibt, wird die folgende Potenz $x^{\nu + 1}$ denselben Rest geben, den $x^n$ gibt. Und irgendeine der folgenden gibt denselben, welchen die Potenz $x^{\mu}$ gibt. Weil nämlich die Differenz $x^{\nu + \mu}-x^{\mu} = x^{\mu}(x^{\nu}-1)$ durch $N$ teilbar ist, ist notwendig, dass beide Terme $x^{\nu + \mu}$ und $x^{\mu}$ durch $N$ geteilt denselben Rest geben.
\subsection*{Korollar 3}
\paragraph*{§38}
Weil nach der Potenz $x^{\nu}$ dieselben Reste $1$, $\alpha$, $\beta$, $\gamma$, etc der Reihe nach auftauchen, werden die Potenzen $x^{3\nu}$, $x^{4\nu}$, $x^{5\nu}$, etc alle durch $N$ geteilt denselben Rest übrig lassen. Ja es werden sogar alle Potenzen $x^{\mu}$, $x^{\mu + \nu}$, $x^{\mu + 2\nu}$, $x^{\mu + 3\nu}$, $x^{{\mu+4\nu}}$, etc die gleichen Reste liefern.
\subsection*{Korollar 4}
\paragraph*{§39}
Wenn also $x^{\nu}$ die unterste Potenz war, die nach $x^0 - 1$ wiederum die Einheit für den Rest liefert, wird die Anzahl der verschiedenen Reste $\nu$ sein. Weil also die Anzahl der zur Zahl $N$ primen Teile gleich $n$ ist, kann es gewiss nicht geschehen, dass $\nu > n$ ist; es wird also $\nu = n$ oder $\nu < n$ sein.
\subsection*{Korollar 5}
\paragraph*{§40}
Wenn also die Reihe der Potenzen $1$, $x$, $x^2$, $x^3$, etc bis hin zu $x^n$ fortgesetzt wird, wird man unter diesen gewiss zumindest eine einzige außer dem ersten Term $1$ finden, die durch $N$ geteilt die Einheit zurücklässt. Es werden vielleicht irgendwann viele Potenzen dieser Art, aber niemals weniger als eine existieren.
\subsection*{Bemerkung}
\paragraph*{§41}
Die Reste werden ausschließlich immer Zahl kleiner als der Teiler $N$ sein, aber nichts hindert daran, dass wir auch größere Zahlen als Reste betrachten, von welcher Art sie zurückgelassen werden, wenn der Quotient zu klein angenommen wird. Wenn so bei der Teilung der Zahl durch $N$ $N+\alpha$ zurückgelassen wird, muss dieser Rest $\alpha$ selbst äquivalent angesehen werden; und daher, wenn von Resten die Rede ist, sind all diese Zahlen $\alpha$, $N+\alpha$, $2N+\alpha$, $3N+\alpha$ etc gleich einem einzigen Rest $\alpha$ anzusehen. Natürlich verändern irgendwelche Vielfachen des Teilers $N$ entweder hinzugefügt oder von einem Rest $\alpha$ abgezogen seine Natur nicht und auf diese Weise werden auch die negativen Zahlen angenehm unter die Reste gezählt, wie z.\,B. $\alpha - N$ für denselben Rest zu halten ist wie $\alpha$ und der Rest $-1$ dem Rest $N-1$ äquivalent ist. Aus diesen erreicht man, dass alle Zahlen, die durch $N$ geteilt denselben Rest $\alpha$ beschaffen, für denselben Rest gehalten werden können; daraus ensteht nämlich aus einer Zahl durch Teilung ein zu kleiner Quotient, indem man entweder $N+\alpha$ oder $2N+\alpha$ oder $3N+\alpha$ etc, nimmt, aus derselben entsteht, indem man den vollen Quotienten nimmt, der Rest $\alpha$; dann aber wird man eben daher, wenn der Quotient zu groß genommen wird, negative Reste $\alpha - N$ oder $\alpha -2N$ oder $\alpha - 3N$ etc erhalten, die also auch so zu verstehen sind, sich nicht von $\alpha$ zu unterscheiden.
\section*{Theorem 8}
\paragraph*{§42}
Wenn, während die Terme der Progression $1$, $x$, $x^2$, $x^3$, $x^4$, etc durch die zu $x$ prime Zahl $N$ geteilt werden, die Reste $1$, $a$, $b$, $c$ etc waren, werden irgendwelche Produkte entweder zweier oder dreier oder beliebig vieler miteinander multiplizierten auftauchen.
\subsection*{Beweis}
Es mögen also die Reste $a$, $b$, $c$, etc aus den Potenzen $x^{\alpha}$, $x^{\beta}$, $x^{\gamma}$, etc entstehen und, indem man auch größere Zahlen als $N$ bei den Resten zulässt, werden aus den Potenzen $x^{2\alpha}$, $x^{3\alpha}$, $x^{4\alpha}$, etc die Reste $a^2$, $a^3$, $a^4$, etc entspringen, die also auch in der Reihe der Reste $1$, $a$, $b$, $c$, etc enthalten sein werden. Dann aber werden die Potenzen $x^{\alpha + \beta}$, $x^{\alpha + \gamma}$, $x^{\alpha + \beta + \gamma}$, etc die Reste $ab$, $ac$, $abc$, etc zurücklassen, die also auch in der Reihe der Reste der gefunden werden müssen werden. Die Produkte werden wie auch immer aus den Resten $1$, $a$, $b$, $c$, etc durch Multiplikation gebildet alle in derselben Reihe der Reste auftauchen, wenn natürlich die einzelnen durch Wegschaffen des Teilers $N$, sooft das gemacht werden kann, auf die kleinste Form gebracht werden.
\hfill\textsc{Q.E.D.}
\subsection*{Korollar 1}
\paragraph*{§43}
Diese Formen der Reste würden sich umso deutlicher zeigen, wenn anstelle derer jene Potenzen von $x$, woher sie entstanden sind, eingesetzt werden; dann tauchen nämlich natürlich nicht nur alle Potenzen dieser Potenzen, sondern auch irgendwelche Produkte derer in den Resten auf.
\subsection*{Korollar 2}
\paragraph*{§44}
Und dennoch wird daher die Anzahl der Reste nicht unbestimmt; so wie wir nämlich schon gesehen haben, dass aus unzähligen Potenzen die gleichen Reste hervorgehen, so werden, wenn all diese Reste, die aus gegenseitiger Multiplikation entstehen, auf die kleinste Form gebracht werden, auf eine gemäßigte Menge zurückgeführt.
\subsection*{Korollar 3}
\paragraph*{§45}
Wenn so die kleinste Potenz, die durch $N$ geteilt wiederum die Einheit zurücklässt, $x^{\nu}$ war, sodass die Anzahl der Reste $1$, $a$, $b$, $c$, etc gleich $\nu$ ist, dann werden in derselben Zahl alle Produkte, die aus der Multiplikation der Zahlen $a$, $b$, $c$, etc entstehen, enthalten sein, wenn natürlich von diesen der Teiler $N$ sooft, wie es gemacht werden kann, weggeschafft wird.
\subsection*{Bemerkung}
\paragraph*{§46}
Es wird ein einziges Beispiel genügen uns alle Zweifel, die vielleicht über diese sich zeigende Menge an Resten entstehen kann, aufzulösen. Es sei also $x=2$ und für den Teiler nehme man $N=15$, welcher natürlich zu $2$ prim ist; nun werden die einzelnen Potenzen von $2$ durch $15$ geteilt die folgende Reste zurücklassen:
\begin{tabular}{lllllllllllll}
Potenzen & $1$, & $2$, & $2^2$, & $2^3$, & $2^4$, & $2^5$, & $2^6$, & $2^7$, & $2^8$, & $2^9$, & $2^{10}$, & etc \\
Reste & $1$, & $2$, & $4$, & $8$, & $1$, & $1$, & $2$, & $4$, & $8$, & $1$, & $2$, & etc \\
\end{tabular}
Die Potenz also, die als erste wieder die Einheit ergibt, ist $2^4$, von welcher die Reste immer in derselben Reihenfolge $1$, $2$, $4$, $8$ wiederholt werden, sodass nur vier verschiedene Reste auftauchen. Hier ist nun klar, wie auch immer die Reste miteinander multipliziert werden, dass aber niemals Zahlen erzeugt werden, die nicht im selben Quadrupel eingeschlossen sind, nachdem sie natürlich durch Wegschaffen des Teilers $15$ auf die kleinste Zahl zurückgeführt worden sind. In diesem Beispiel tauchen auch unter den Resten nicht alle zu $15$ primen Teile auf, sondern es werden diese Teile $7$, $11$, $13$, $14$ ausgeschlossen, die in gleicher Weise zu $15$ prim sind; daher wird die oben [§34] gemachte Verteilung unter den zum Teiler primen Teile, die in den Resten auftauchten und die nicht auftauchten, illustriert, auf die man im folgenden besonders achte.
\section*{Theorem 9}
\paragraph*{§47}
Bei den Resten, die aus der Teilung der Potenzen einer Zahl durch einen zu ihr primen Teiler zurückbleiben, tauchen entweder alle zum Teiler primen Teile auf oder die Anzahl der nicht auftauchenden Teile wird gleich sein oder wird in einem vielfachen Verhältnis zur Zahl der Teile stehen, die die Reste festsetzen.
\subsection*{Beweis}
Es sei die Reihe der Potenzen $1$, $x$, $x^2$, $x^3$, $x^4$, $x^5$, etc und der zu $x$ prime Teiler sei $N$, dessen Anzahl der zu selbiger primen Teile gleich $n$ sei. Es sei weiter $x^{\nu}$ die kleinste Potenz, die durch $N$ geteilt wieder die Einheit zurücklässt, sodass die Anzahl aller verschiedenen Reste gleich $\nu$ ist; weil diese alle zu $N$ prime Zahlen sind, wird deren Anzahl entweder gleich $n$ oder kleiner sein, und im ersten Fall werden unter den Resten jedenfalls alle zu $N$ primen Teile auftauchen. Wir wollen also den Fall betrachten, in dem $\nu < n$ ist, und es seien $1$, $a$, $b$, $c$, $d$, etc alle Reste, die aus der Teilung der Potenzen $1$, $x$, $x^2$, $x^3$, $x^4$, \dots $x^{n-1}$ durch den Teiler $N$ zurückgelassen werden; weil deren Anzahl gleich $\nu$ ist, werden dort nicht alle zu $N$ primen Teile auftauchen. Es sei also $\alpha$ der Teil dieser Art, der in den Resten nicht auftaucht und es kann bewiesen werden, dass auch keine dieser Zahlen $\alpha a$, $\alpha b$, $\alpha c$, $\alpha d$, etc in den Resten auftaucht. Denn wenn $\alpha a$ ein Rest wäre, der der Potenz $x^{\lambda}$ entspricht, würde, weil $a$ auch ein Rest ist, der aus einer Potenz, z.\,B. $x^{\xi}$, entsteht, $x^{\lambda} = an + \alpha a$ und $x^{\xi} = BN + a$ und daher $x^{\lambda} - \alpha x^{\xi} = (A-\alpha B)N$ durch $N$ teilbar sein. Weil aber $x^{\xi}$ eine zu $N$ prime Zahl ist und $x^{\lambda} - \alpha x^{\xi} = x^{\xi}(x^{\lambda - \xi} - \alpha)$ ist, würde die Zahl $x^{\lambda - \xi} - \alpha$ durch $N$ teilbar und so würde die Potenz $x^{\lambda - \xi}$ durch $N$ geteilt entgegen der Annahme den Rest $\alpha$ übrig lassen. Weil also $\alpha$, $\alpha a$, $\alpha b$, $\alpha c$, etc, deren Anzahl gleich $\nu$ ist, zu $N$ prime Zahlen sind und durch Teilung durch $N$ auf zu $N$ prime Teile zurückgeführt werden können, wird man sofort auch einen einzigen Teil, der zu $N$ prim ist, in den Resten nicht finden, zugleich werden auch $\nu$ Teile solcher Art angegeben werden können, die in den Resten nicht auftauchen. Die Anzahl der nicht auftauchenden Teile ist also, wenn sie nicht $0$ ist, mindestens gleich $\nu$ und wenn außerdem ein zu $N$ primer Teiler $\beta$ in diesen Nicht-Resten nicht enthalten war, wird man erneut $\nu$ neue Teile haben, die in den Resten nicht auftauchen, und so weiter. Wenn daher nicht alle zum Teiler $N$ primen Teile in den Resten auftauchen, ist die Anzahl der Teile, die nicht auftauchen, notwendigerweise entweder gleich $\nu$ oder gleich $2\nu$ oder gleich $3\nu$ oder irgendein anderes Vielfaches von $\nu$, das heißt der Anzahl der verschieden Reste.
\hfill\textsc{Q.E.D.}
\subsection*{Korollar 1}
\paragraph*{§48}
Nachdem also der Unterschied zwischen den zum Teiler $N$ primen Teilen, die Reste sind, und denen, die keine Reste sind, festgesetzt worden ist, ist aus dem Beweis klar, dass das Produkt aus dem Rest und Nicht-Rest immer in der Klasse der Nicht-Reste enthalten ist. Wenn so $a$ ein Rest ist, $\alpha$ ein Nicht-Rest, wird also gewiss das Produkt $\alpha a$ kein Rest sein.
\subsection*{Korollar 2}
\paragraph*{§49}
Hingegen haben wir schon oben gesehen, dass das Produkt aus $2$ oder mehreren Resten in der Klasse der Reste gefunden wird. Daher folgt, dass das Produkt aus einem Nicht Rest und irgendwelchen Resten in der Klasse der Nicht-Reste auftauchen muss.
\subsection*{Bemerkung}
\paragraph*{§50}
Die Art dieses Beweis ist also auf dieses Fundament gestützt, wenn also unter den Resten diese zum Teiler primen Teile $1$, $a$, $b$, $c$, $d$, etc auftauchen und $\alpha$ auch ein zum Teiler primer Teil war, der in diesen Resten nicht enthalten ist, dass dann alle Produkte $\alpha a$, $\alpha b$, $\alpha c$, $\alpha d$, etc nicht nur in den Resten auftauchen, was freilich perfekt bewiesen worden ist, sondern dass sie auch zum Teiler $N$ prime Teile sind und alle untereinander verschieden sind oder, wenn sie tatsächlich durch $N$ geteilt werden, die verschiedenen Reste zurückgelassen werden. Jenes ist freilich per se klar; weil nämlich $\alpha$ wie $a$, $b$, $c$, $d$, etc zu $N$ prime Zahlen sind, ist notwendig, dass auch deren Produkte zu $N$ prim sind. Dass aber die Produkte $\alpha a$, $\alpha b$, $\alpha c$, $\alpha d$, etc, die auf $N$ bezogen, alle zueinander verschieden sind, sieht man ein, weil, wenn z.\,B. $\alpha a$ und $\alpha b$ durch $N$ geteilt die gleichen Reste gäben, deren Differenz $\alpha b - \alpha a = \alpha (b-a)$ durch $N$ teilbar wäre und daher auch $b-a$; das widerspricht aber der Annahme, dass $a$ und $b$ zu $N$ prime verschiedene Teile sind.
\section*{Theorem 10}
\paragraph*{§51}
Der Exponent der kleinsten Potenz $x^{\nu}$, die durch eine zu $x$ prime Zahl $N$ geteilt die Einheit übrig lassen, ist entweder gleich der Anzahl der zu $N$ primen Teile oder der Hälfte dieser Anzahl oder einem anderen echten Teil.
\subsection*{Beweis}
Es sei $n$ die Anzahl der zu $N$ primen Teile; weil $\nu$ von diesen die Reste festsetzen, wird die Anzahl der Nicht-Reste $n-\nu$ sein. Wir haben aber gesehen, dass diese Zahl entweder gleich $0$ oder gleich $\nu$ oder $2\nu$ oder ein anderes Vielfaches vom Exponent $\nu$ ist. Es sei also $n-\nu = (m-1)\nu$, sodass $m$ entweder die Einheit oder eine andere ganze Zahl bezeichnet, und daher werden wir $n=m\nu$ und $\nu = \frac{n}{m}$ erhalten; daher ist klar, dass der Exponent der kleinsten Potenz von $x$, die durch $N$ geteilt die Einheit zurücklässt, entweder gleich $\nu$ oder $m=1$ ist, oder gleich $\frac{n}{2}$, wenn $m=2$ ist, oder dass sie im Allgemeinen ein echter Teil der Zahl $n$ ist, die die Menge der zum Teiler $N$ primen Teile ausdrückt.
\hfill\textsc{Q.E.D.}
\subsection*{Korollar 1}
Wenn $x^{\nu}$ die kleinste Potenz war, die durch eine zu $x$ prime Zahl $N$ geteilt die Einheit zurücklässt, sind die folgenden Potenzen, die die selben Reste zurücklassen, $x^{2\nu}$, $x^{3\nu}$, $x^{4\nu}$, $x^{5\nu}$, etc und es sind zusätzlich keine anderen gegeben, die durch $N$ geteilt die Einheit zurücklassen.
\subsection*{Korollar 2}
\paragraph*{§53}
Der Exponent dieser kleinsten Potenz ist also immer mit der Anzahl der zu $N$ primen Teile so verknüpft, dass sie entweder jener selbst oder irgendeinem echten Teil von ihr gleich ist.
\subsection*{Bemerkung}
\paragraph*{§54}
Um dieses Verhältnis besser zu erkennen, wird es förderlich sein, einige einfache Fälle betrachtet zu haben. Es sei also $x=2$ und für $N$ wollen wir nacheinander ungerade zu $x=2$ prime Zahlen nehmen und wollen die kleinste Potenz von $2$ beschaffen, die durch eine ungerade Zahl geteilt die Einheit übrig lässt.
\begin{tabular}{>{\centering}p{1.7cm} >{\centering}p{4cm} >{\centering\arraybackslash}p{5cm}}
\toprule
\small\centering Teiler & \small \centering Anzahl der zu ihm primen Teile $n$ & \small\centering Die kleinste Potenz $2^{\nu}$, die durch $N$ geteilt die Einheit zurücklässt
\tabularnewline
\midrule
\small
$3$ & $2$ & $2^2$, also $\nu = n$ \\
$5$ & $4$ & $2^{4}$, also $\nu = n$ \\
$7$ & $6$ & $2^{3}$, also $\nu = \tfrac{1}{2}n$ \\
$9$ & $6$ & $2^{6}$, also $\nu = n$ \\
$11$ & $10$ & $2^{10}$, also $\nu = n$ \\
$13$ & $12$ & $2^{12}$, also $\nu = n$ \\
$15$ & $8$ & $2^{4}$, also $\nu = \tfrac{1}{2}n$ \\
$17$ & $16$ & $2^{8}$, also $\nu = \tfrac{1}{2}n$ \\
$19$ & $18$ & $2^{18}$, also $\nu = n$ \\
$21$ & $12$ & $2^{6}$, also $\nu = \tfrac{1}{2}n$ \\
$23$ & $22$ & $2^{11}$, also $\nu = \tfrac{1}{2}n$ \\
$25$ & $20$ & $2^{20}$, also $\nu = n$ \\
$27$ & $18$ & $2^{18}$, also $\nu = n$ \\
$29$ & $28$ & $2^{28}$, also $\nu = n$ \\
$31$ & $30$ & $2^{5}$, also $\nu = \tfrac{1}{6}n$ \\
\bottomrule
\end{tabular}
\section*{Theorem 11}
\paragraph*{§55}
Wenn $N$ eine zu $x$ prime Zahl war und $n$ die Anzahl der zu $N$ primen Teiler, dann wird die Potenz $x^n$ um die Einheit vermindert immer durch die Zahl $N$ teilbar sein.
\subsection*{Beweis}
Es sei $x^{\nu}$ die kleinste Potenz, die durch $N$ geteilt die Einheit zurücklässt, und es wird $\nu$ entweder der Zahl $n$ gleich sein oder einem echten Teil $\frac{n}{m}$ von ihr. Weil also $x^{\nu}-1$ durch $N$ teilbar ist, wird, weil die Form $x^{\nu m}-1$ den Faktor $x^{\nu}-1$ hat, auch diese Form oder $x^n - 1$ durch $N$ teilbar sein.
\hfill\textsc{Q.E.D.}
\subsection*{Korollar 1}
\paragraph*{§56}
Wenn also der Teiler $N$ eine Primzahl ist und $x$ durch $p$ nicht teilbar ist, dann wird $x^{p-1}-1$ immer durch die Primzahl $p$ teilbar sein, wie ich schon längst bewiesen habe.
\subsection*{Korollar 2}
\paragraph*{§57}
Wenn außerdem $p$, $q$, $r$, etc Primzahlen sind und $x$ keine derer teilt, folgt aus diesem Theorem,
\begin{center}
\begin{tabular}{lc}
dass diese Formen &\quad durch \dots teilbar sein werden
\tabularnewline
\small
$x^{p-1}-1$ & $p$ \\
$x^{(p-1)(q-1)}-1$ & $pq$ \\
$x^{pp(p-1)}-1$ & $p^3$ \\
$x^{p(p-1)(q-1)}-1$ & $ppq$ \\
$x^{(p-1)(q-1)(r-1)}-1$ & $pqr$ \\
\end{tabular}
\end{center}
\subsection*{Korollar 3}
\paragraph*{§58}
Wenn $x$ und $y$ zum Teiler $N$ prim sind, deren Anzahl der zu ihr primen Teile gleich $n$ sei, wird, weil $x^{n-1}$ und $y^{n}-1$ durch $N$ teilbar ist, auch $x^n - y^n$ immer durch die Zahl $N$ teilbar sein, welches Theorem allgemeiner ist.
\subsection*{Korollar 4}
\paragraph*{§59}
Nachdem also irgendeine Zahl $N$ vorgelegt wurde, deren Anzahl zu selbiger primen Teile gleich $n$ sei, wird, welche zu $N$ prime Zahl auch immer für $x$ genommen wird, die Formel $x^n = 1$ immer durch die Zahl $N$ teilbar sein.
\subsection*{Korollar 5}
\paragraph*{§60}
Oftmals kann es aber auch passieren, dass eine einfachere Form dieser Art, wie z.\,B. $x^{\frac{1}{2}n - 1}$ oder $x^{\frac{1}{3}n} - 1$ oder $x^{\frac{1}{4}}-1$, durch die Zahl $N$ teilbar ist, welcher Umstand von der Gestalt der Zahl $x$ abhängt.
\subsection*{Bemerkung}
\paragraph*{§61}
Sieh also diesen neuen Beweis des Fermat'schen Satzes, dass, wenn $p$ eine Primzahl war, alle Zahlen, die in dieser Form $a^{p-1}-1$ enthalten sind, durch $p$ teilbar sind, solange $a$ nicht durch $p$ teilbar ist. Ich hatte aber schon längst zwei Beweise dieses Theorems gegeben, aber der, den ich hier beschafft habe, scheint jenen vorzustehen, weil er nicht nur auf $p$ als Primzahl beschränkt ist. Denn welche Zahl $N$ auch immer für den Teiler angenommen wird, wird, solange $a$ zu ihr prim ist, diese Zahl $a^n - 1$ immer durch $N$ teilbar sein, wenn natürlich $n$ die Anzahl der zu $N$ primen Teile bezeichnet, welche Proposition sich um Vieles weiter erstreckt als die Fermat'sche. Daraus zeigt sich umso mehr die Nützlichkeit der ersten Theoreme, durch die ich die Anzahl der zu irgendeiner Zahl primen Teile bestimmt habe, was ohne diese Anwendung als zu unfruchtbar hätte erscheinen können.
\end{document}
|
{
"timestamp": "2012-03-12T01:00:41",
"yymm": "1203",
"arxiv_id": "1203.1993",
"language": "de",
"url": "https://arxiv.org/abs/1203.1993"
}
|
\section*{\hfil #1\hfil}}
\bibliographystyle{abbrv}
\newenvironment{proof}{\paragraph{\bf Proof:}}{\hspace*{\fill}\(\Box\)}
\title{Matrix Representation of Iterative Approximate Byzantine Consensus in Directed Graphs\thanks{This research is supported
in part by
National
Science Foundation award CNS 1059540 and Army Research Office grant W-911-NF-0710287. Any opinions, findings, and conclusions or recommendations expressed here are those of the authors and do not
necessarily reflect the views of the funding agencies or the U.S. government.}
}
\author{Nitin Vaidya\\ Department of Electrical and Computer Engineering\\ University of Illinois at Urbana-Champaign\\ nhv@illinois.edu}
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\begin{document}
\date{~}
\maketitle
\centerline{March 8, 2012}
~
\begin{abstract}
This paper presents a proof of correctness of an iterative
approximate Byzantine
consensus (IABC) algorithm for directed graphs. The iterative algorithm allows
fault-free nodes to reach approximate conensus despite the presence of up
to $f$ Byzantine faults. Necessary conditions on the underlying network
graph for the existence of a correct IABC algorithm were shown in our
recent work \cite{IBA_sync,us}. \cite{IBA_sync} also analyzed
a specific IABC algorithm and showed that it performs correctly in any
network graph that satisfies the necessary condition, proving that the
necessary condition is also sufficient. In this paper, we present an
alternate proof of correctness of the IABC algorithm, using a familiar
technique based on transition matrices \cite{jadbabaie_consensus,Benezit,vaidyaII,Zhang}.\\
The key contribution of this paper is to exploit the
following observation: for a {\em given} evolution of the state vector
corresponding to the state of the fault-free nodes,
many alternate state transition matrices may be chosen to model
that evolution correctly. For a given state evolution, we identify one approach
to suitably ``design'' the transition matrices so that the standard tools
for proving convergence can be applied to the Byzantine fault-tolerant
algorithm as well.
In particular, the transition matrix for each iteration
is designed such that each row of the matrix contains a large enough number
of elements that are bounded away from 0. \\
\end{abstract}
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\section{Introduction}
\label{sec:intro}
Dolev et al. \cite{AA_Dolev_1986} introduced the notion of
{\em approximate Byzantine consensus} by relaxing the requirement
of {\em exact} consensus \cite{AA_nancy}.
The goal in approximate consensus is to allow the fault-free nodes to agree on values that are approximately equal to each other (and {\em not necessarily}
exactly identical).
In presence of Byzantine faults, while {\em exact} consensus
is impossible in {\em asynchronous} systems \cite{FLP_one_crash}, approximate
consensus is achievable \cite{AA_Dolev_1986}.
The notion of approximate consensus is of interest in {\em synchronous}
systems as well, since approximate consensus can be achieved using
simple distributed algorithms that do {\em not} require complete knowledge of
the network topology \cite{AA_convergence_markov}.
In this paper, we are interested in iterative algorithms
for achieving approximate Byzantine consensus in synchronous point-to-point
networks that are modeled by arbitrary {\em directed}\, graphs.
The {\em iterative
approximate Byzantine consensus} (IABC) algorithms of interest have
the following properties, which we will soon state more formally:
\begin{itemize}
\item {\em Initial state} of each node is equal to a real-valued
{\em input} provided to that node.
\item {\em Validity} condition: After each iteration of an IABC algorithm, the state of each fault-free node
must remain in the {\em convex hull} of the states of the fault-free nodes
at the end of the {\em previous} iteration.
\item {\em Convergence} condition:
For any $\epsilon>0$, after a sufficiently large number of iterations,
the states of the fault-free nodes are guaranteed to be within $\epsilon$
of each other.
\end{itemize}
Certain IABC algorithms have been shown to satisfy the above properties
in {\em fully
connected} graphs \cite{AA_Dolev_1986,AA_nancy}, and in {\em arbitrary
directed} graphs satisfying a tight necessary condition
\cite{IBA_sync,us}.
Please refer to \cite{IBA_sync,us} for a summary of the related work.
The main contribution of this paper is to develop an alternate proof of
correctness for a IABC algorithm, which was proved correct
in arbitrary graphs that satisfy a necessary condition developed
in our prior work \cite{IBA_sync}.
The alternate proof is based on
transition matrices that capture the behavior of the IABC algorithm
executed by the fault-free nodes. This work is inspired
by, and borrows some matrix analysis tools from, other work that also uses
transition matrices in
related contexts \cite{jadbabaie_consensus,Benezit,vaidyaII,Zhang}.
This paper exploits the
following observation: for a {\em given} evolution of the state vector
corresponding to the state of the fault-free nodes,
many alternate state transition matrices may potentially be chosen to emulate
that evolution correctly. For a given state evolution, we identify one approach
to suitably ``design'' the transition matrices so that the standard tools
can be applied to prove convergence of the Byzantine fault-tolerant
algorithm in {\em all networks} that satisfy a necessary condition
(proved in \cite{us}) on the network communication graph.
In particular, the transition matrix for each iteration
is designed such that each row of the matrix contains a large enough number
of elements that are bounded away from 0.
\section{Network and Failure Models}
\paragraph{Network Model:}
The system is assumed to be {\em synchronous}.
The communication network is modeled as a simple {\em directed} graph $G(\scriptv,\scripte)$, where $\scriptv=\{1,\dots,n\}$ is the set of $n$ nodes, and $\scripte$ is the set of directed edges between the nodes in $\scriptv$.
Node $i$ can reliably transmit messages to node $j$ if and only if
the directed edge $(i,j)$ is in $\scripte$.
Each node can send messages to itself as well, however,
for convenience, we {\em exclude self-loops} from set $\scripte$.
That is, $(i,i)\not\in\scripte$ for $i\in\scriptv$.
With a slight abuse of terminology, we will use the terms {\em edge}
and {\em link} interchangeably in our presentation.
For each node $i$, let $N_i^-$ be the set of nodes from which $i$ has incoming
edges.
That is, $N_i^- = \{\, j ~|~ (j,i)\in \scripte\, \}$.
Similarly, define $N_i^+$ as the set of nodes to which node $i$
has outgoing edges. That is, $N_i^+ = \{\, j ~|~ (i,j)\in \scripte\, \}$.
Since we exclude self-loops from $\scripte$,
$i\not\in N_i^-$ and $i\not\in N_i^+$.
However, we note again that each node can indeed send messages to itself.
A necessary condition for correctness of an IABC algorithm for $f>0$ is that
$|N_i^-|>2f$ \cite{IBA_sync}.
Node $j$ is said to be an {\em incoming neighbor} of node $i$,
if $j\in N_i^-$. Similarly, $j$ is said to be an {\em outgoing neighbor}
of node $i$, if $j\in N_i^+$.
\paragraph{Failure Model:}
We consider the Byzantine failure model, with up to $f$ nodes becoming faulty. A faulty node may {\em misbehave} arbitrarily. Possible misbehavior includes sending incorrect and mismatching (or inconsistent) messages to different neighbors. The faulty nodes may potentially collaborate with each other. Moreover, the faulty nodes are assumed to have a complete knowledge of the execution of
the algorithm, including the states of all the nodes,
contents of messages the other nodes send to each other,
the algorithm specification, and the network topology.
\section{Iterative Approximate Byzantine Consensus (IABC)}
\label{sec:iabc}
Each node $i$ maintains state $v_i$, with $v_i[t]$ denoting the state
of node $i$ at the {\em end}\, of the $t$-th iteration of the algorithm.
Initial state of node $i$,
$v_i[0]$, is equal to the initial {\em input}\, provided to node $i$.
At the {\em start} of the $t$-th iteration ($t>0$), the state of
node $i$ is $v_i[t-1]$.
Let $\scriptf$ denote the set of faulty nodes.
Thus, the nodes
in $\scriptv-\scriptf$ are non-faulty.\footnote{\normalsize For sets $X$ and $Y$, $X-Y$ contains elements that are in $X$ but not in $Y$. That is, $X-Y=\{i~|~ i\in X,~i\not\in Y\}$.}
\begin{itemize}
\item $U[t] = \max_{i\in\scriptv-\scriptf}\,v_i[t]$. $U[t]$ is the largest state among the fault-free nodes at the end of the $t$-th iteration.
Since the initial state of each node is equal to its input,
$U[0]$ is equal to the maximum value of the initial input at the fault-free nodes.
\item $\mu[t] = \min_{i\in\scriptv-\scriptf}\,v_i[t]$. $\mu[t]$ is the smallest state among the fault-free nodes at the end of the $t$-th iteration.
$\mu[0]$ is equal to the minimum value of the initial input at the
fault-free nodes.
\end{itemize}
The following conditions must be satisfied by an IABC algorithm
in presence of up to $f$ Byzantine faulty nodes:
\begin{itemize}
\item {\em Validity:} $\forall t>0,
~~\mu[t]\ge \mu[t-1]
~\mbox{~~and~~}~
~U[t]\le U[t-1]$
\item {\em Convergence:} $\lim_{\,t\rightarrow\infty} ~ U[t]-\mu[t] = 0$.
Equivalently, $\lim_{\,t\rightarrow\infty} ~ v_i[t]-v_j[t] = 0$, for
$i,j\in \scriptv-\scriptf$.
\end{itemize}
An iterative algorithm is said to be {\em correct} if it satisfies
the {\em validity} and {\em convergence} conditions.
We will prove the correctness of Algorithm 1 below
in all graphs that satisfy the
necessary condition in Theorem 2 of \cite{us}. The algorithm
should be performed by each node $i$ in the
$t$-th iteration, $t\geq 1$. The faulty nodes may deviate from the
algorithm specification. If a fault-free node does not receive an
expected message from an incoming neighbor (in the {\em Receive step}
below), then that message is assumed to have some default value.
\vspace*{8pt}\hrule
{\bf Algorithm 1}
\vspace*{4pt}\hrule
~
Steps to be performed by node $i$ in the $t$-th iteration:
\begin{enumerate}
\item {\em Transmit step:} Transmit current state $v_i[t-1]$ on all outgoing edges.
\item {\em Receive step:} Receive values on all incoming edges. These values form
vector $r_i[t]$ of size $|N_i^-|$.
\item {\em Update step:}
Sort the values in $r_i[t]$ in an increasing order, and eliminate
the smallest $f$ values, and the largest $f$ values (breaking ties
arbitrarily).
Let $N_i^*[t]$ denote the identifiers of nodes from
whom the remaining $|N_i^-| - 2f$ values were received, and let
$w_j$ denote the value received from node $j\in N_i^*[t]$.
For convenience, define $w_i=v_i[t-1]$.
Observe that
if $j\in \{i\}\cup N_i^*[t]$ is fault-free, then $w_j=v_j[t-1]$.
Define
\begin{eqnarray}
v_i[t] ~ = ~\sum_{j\in \{i\}\cup N_i^*[t]} a_i \, w_j
\label{e_Z}
\end{eqnarray}
where
\[ a_i ~=~ \frac{1}{|N_i^-|-2f+1} ~=~
\frac{1}{|N_i^*[t]|+1}
\]
Recall that
$i\not\in N_i^*[t]$
because $(i,i)\not\in\scripte$.
The ``weight'' of each term on the right-hand side of
(\ref{e_Z}) is $a_i$, and these weights add to 1.
Observe that $0<a_i\leq 1$.
For future reference, let us define $\alpha$ as:
\begin{eqnarray}
\alpha = \min_{i\in \scriptv}~a_i
\label{e_alpha}
\end{eqnarray}
Note that $0<\alpha\leq 1$.
Specifically, $\alpha$ is a positive constant
that is dependent only on $f$ and the graph $G(\scriptv,\scripte)$.
\end{enumerate}
~
\hrule
~
~
Similar algorithms have been proven to work correctly in
{\em fully connected} graphs \cite{AA_Dolev_1986,IBA_sync}
and {\em arbitrary directed} graphs
satisfying the necessary condition stated in
\cite{IBA_sync}.
In this paper, we provide an alternate proof of correctness
in such arbitrary graphs, using an alternate form of the
necessary condition \cite{us}.
\comment{
\section{Related Work}
\label{sec:related}
Some of the related work has already been discussed earlier in the paper.
In this section, we discuss other related work.
There have been previous attempts at achieving approximate
consensus iteratively in {\em partially} connected graphs. Kieckhafer and Azadmanesh
examined the necessary conditions in order to achieve ``local'' convergence in
synchronous \cite{AA_PCN_Local} and asynchronous \cite{AA_async_PCN} systems.
\cite{AA_PFCN} presents a specific class of networks in which convergence
condition can be satisfied using iterative algorithms.
Zhang and Sundaram \cite{Zhang}
consider a
{\em restricted} fault model in which the faulty nodes are restricted
to sending identical messages to their neighbors. In contrast, our
Byzantine fault model allows a faulty node to send different messages to different
neighbors.
In particular,
under the {\em restricted} model,
Zhang and Sundaram \cite{Zhang} develop {\em sufficient}\,
conditions for iterative consensus algorithm assuming a ``local" fault
model (in their ``local'' model, a bounded number of each node's neighbors
may be faulty). LeBlanc and Koutsoukos \cite{Leblanc_HSCC_1} address a continuous
time version of the Byzantine consensus problem in {\em complete} graphs.
Under the above {\em restricted} fault model, as well as our fault model, LeBlanc and Koutsoukos \cite{Leblanc_HSCC_2}
have identified some sufficient conditions under which iterative consensus can be
achieved; however, these sufficient conditions are {\em not} tight.
For the {\em restricted} model, recently
LeBlanc et al. \cite{diss_Sundaram} have obtained tight necessary
and sufficient conditions; but these conditions are not adequate
for our Byzantine fault model.
Iterative approximate consensus algorithms that {\em do not} tolerate faulty behavior
have also been studied extensively (e.g., \cite{jadbabaie_consensus, AA_convergence_markov}).
}
\section{Matrix Preliminaries}
We use boldface upper case letters to denote matrices,
rows of matrices, and their elements. For instance,
$\bfH$ denotes a matrix, $\bfH_i$ denotes the $i$-th row of
matrix $\bfH$, and $\bfH_{ij}$ denotes the element at the
intersection of the $i$-th row and the $j$-th column
of matrix $\bfH$.
\begin{definition}
\label{d_stochastic}
A vector is said to be {\em stochastic} if all the elements
of the vector are {\em non-negative}, and the elements add up to 1.
A matrix is said to be row stochastic if each row of the matrix is a
stochastic vector.
\end{definition}
For a row stochastic matrix $\bfA$,
coefficients of ergodicity $\delta(\bfA)$ and $\lambda(\bfA)$ are defined as
\cite{Wolfowitz}:
\begin{align}
\delta(\bfA) & := \max_j ~ \max_{i_1,i_2}~ | \bfA_{i_1\,j}-\bfA_{i_2\,j} |, \label{e_delta} \\
\lambda(\bfA) & := 1 - \min_{i_1,i_2} \sum_j \min(\bfA_{i_1\,j} ~, \bfA_{i_2\,j}). \label{e_lambda}
\end{align}
It is easy to see that $0\leq \delta(\bfA) \leq 1$ and $0\leq \lambda(\bfA) \leq 1$, and that the rows are all identical if and only if $\delta(\bfA)=0$. Additionally, $\lambda(\bfA) = 0$ if and only if $\delta(\bfA) = 0$.
The next result from \cite{Hajnal58} establishes a relation between the coefficient of ergodicity $\delta(\cdot)$ of a product of row stochastic matrices, and the coefficients of ergodicity $\lambda(\cdot)$ of the individual matrices defining the product.
\begin{claim}
\label{claim_delta}
For any $p$ square row stochastic matrices $\bfQ(1),\bfQ(2),\dots \bfQ(p)$,
\begin{align}
\delta(\bfQ(1)\bfQ(2)\cdots \bfQ(p)) ~\leq ~
\Pi_{i=1}^p ~ \lambda(\bfQ(i)).
\end{align}
\end{claim}
Claim \ref{claim_delta} is proved in \cite{Hajnal58}. It implies that
if, for all $i$, $\lambda(\bfQ(i))\leq 1-\gamma$ for some $\gamma>0$, then $\delta(\bfQ(1),\bfQ(2)\cdots \bfQ(p))$ will approach zero as $p$ approaches $\infty$.
\begin{definition}
A row stochastic
matrix $\bfH$ is said to be a {\em scrambling}\, matrix, if $\lambda(\bfH)<1$
{\normalfont \cite{Hajnal58,Wolfowitz}}.
\end{definition}
In a scrambling matrix $\bfH$, since $\lambda(\bfH)<1$, for each pair of
rows $i_1$ and $i_2$, there exists a column $j$ (which may depend on
$i_1$ and $i_2$) such that
$\bfH_{i_1\,j}>0$ and $\bfH_{i_2\,j}>0$, and vice-versa \cite{Hajnal58,Wolfowitz}.
As a special case, if any one column of a row stochastic matrix $\bfH$
contains only non-zero elements that are lower bounded by some
constant $\gamma>0$, then $\bfH$ must be scrambling, and $\lambda(\bfH)\leq 1-\gamma$.
\comment{======================
\begin{definition}
\label{d_type}
Two matrices of identical size are said to be of the same ``type'' if
they contain non-zero elements in identical positions.
\end{definition}
Let us denote by $\T{\bfH}$ the {\em type} of matrix $\bfH$.
A partial order can be
defined on the matrix types. Specifically, for matrices $\bfH$ and $\bfK$,
$\T{\bfH}\leq \T{\bfK}$ provided that matrix $\bfK$ is non-zero in each position
where $\bfH$ is non-zero.
\begin{lemma}
\label{l_scambling_1}
For any two row stochastic matrices $\bfH,~\bfK$ of the same size,
if $\T{\bfH}\leq \T{\bfK}$ and $\bfH$ is a scrambling matrix,
then $\bfK$ is a scrambling matrix.
\end{lemma}
\begin{proof}
Follows immediately from the definition of matrix {\em type}
and {\em scrambling} matrices.
\end{proof}
~
\begin{lemma}
\label{l_scrambling_2}
Consider a sequence $\bfH(1),\bfH(2),\cdots,\bfH(t)$ of square row stochastic matrices
with non-zero diagonals. For any subset $N$ of $\{1,2,\cdots,t\}$,
\[
\T{\Pi_{i\in N} \bfH(i)} ~ \leq ~ \T{\Pi_{1\leq i\leq t} \bfH(i)}
\]
\end{lemma}
\begin{proof}
The proof follows from the definition of matrix {\em type}, and the fact
the row stochastic matrices above have non-zero diagonals.
\end{proof}
~
=================================================}
\section{Matrix Representation of Algorithm 1}
\label{s_claim}
Recall that $\scriptf$ is the set of faulty nodes.
Let $|\scriptf|=\phi$.
Without loss of generality, suppose that nodes 1 through $(n-\phi)$ are
fault-free, and if $\phi>0$, nodes $(n-\phi+1)$ through $n$ are faulty.
Denote by $\bfv[0]$ the column vector consisting of the initial states of
all the {\em fault-free} nodes.
Denote by $\bfv[t]$, where $t\geq 1$, the column vector consistsing of
the states of all the {\em fault-free} nodes
at the end of the $t$-th iteration, $t\geq 1$.
The $i$-th element
of vector $\bfv[t]$ is state $v_i[t]$. The size of the column
vector $\bfv[t]$ is
$(n-\phi)$.
~
\begin{claim}
\label{claim_1}
{
We can express the iterative update of the state
of a fault-free node $i$ $(1\leq i\leq n-\phi)$
performed in (\ref{e_Z}) using the matrix form in (\ref{e_matrix_i})
below,
where $\bfM_i[t]$ satisfies the following four conditions.
\begin{eqnarray}
v_i[t] & = & \bfM_i[t] ~ {\bfv}[t-1]
\label{e_matrix_i}
\end{eqnarray}
}
In addition to $t$, the row vector $\bfM_i[t]$
may depend on the state vector $\bfv[t-1]$ as well as the
behavior of the faulty
nodes in $\scriptf$. For simplicity, the notation $\bfM_i[t]$ does not
explicitly represent this dependence.
\begin{enumerate}
\item $\bfM_i[t]$ is a {\em stochastic} row vector of size $(n-\phi)$.
Thus,
$\bfM_{ij}[t]\geq 0$, for $1\leq j\leq n-\phi$, and
\[
\sum_{1\leq j\leq n-\phi}~\bfM_{ij}[t] ~ = ~ 1
\]
\item $\bfM_{ii}[t]$ equals $a_i$ defined in Algorithm 1.
Recall that $a_i\geq \alpha$.
\item $\bfM_{ij}[t]$ is non-zero
{\bf only if} $(j,i)\in\scripte$ or $j=i$.
\item At least $|N_i^-\cap\,(\scriptv-\scriptf)| - f+1$ elements in $\bfM_i[t]$
are lower bounded by some constant $\beta>0$, to be defined later
($\beta$ is independent of $i$).
Note that $N_i^-\cap\,(\scriptv-\scriptf)$ is the set of fault-free
incoming neighbors of node $i$.
\end{enumerate}
\end{claim}
\begin{proof}
The proof of this claim is presented in Section \ref{ss_claim_1} below.
The last condition above plays an important role in the proof, and the
main contribution of this paper is to ``design'' $\bfM_i[t]$ to make this
condition true.
\end{proof}
~
By ``stacking'' (\ref{e_matrix_i}) for different
$i$, $1\leq i\leq n-\phi$, we can
represent the state update for all the fault-free nodes together
using (\ref{e_matrix})
below, where $\bfM[t]$ is a $(n-\phi)\times (n-\phi)$ matrix, with its $i$-th row
being equal to $\bfM_i[t]$ in (\ref{e_matrix_i}).
\begin{eqnarray}
{\bfv}[t] & = & \bfM[t] ~ {\bfv}[t-1]
\label{e_matrix}
\end{eqnarray}
The four properties of $\bfM_i[t]$ imply that $\bfM[t]$ is a
row stochastic matrix with a non-zero diagonal.
Also, the $i$-th row of $\bfM[t]$ contains $|N_i^-\cap\,(\scriptv-\scriptf)| - f+1$
elements lower bounded by $\beta$ ($\beta$ will be defined later).
This property of $\bfM[t]$ turns out to be important in proving
convergence of Algorithm 1.
$\bfM[t]$ is said to be a {\em transition matrix}.
By repeated application of (\ref{e_matrix}), we obtain:
\begin{eqnarray*}
\bfv[t] & = & \left(\,\Pi_{i=1}^t \bfM[i]\,\right)\, \bfv[0]
\end{eqnarray*}
\subsection{Correctness of Claim \ref{claim_1}}
\label{ss_claim_1}
Figure \ref{f_sets} illustrates the various sets used here.
Some of the sets in this figure are not yet defined, and will be defined
later in the paper.
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{sets.eps}
\caption{Illustration of sets $\scriptv$, $\scriptf$, $N_i^-$,
$N_i^*[t]$, $L^*$ and $S^*$}
\label{f_sets}
\end{figure}
We prove the correctness of Claim \ref{claim_1} by constructing $\bfM_i[t]$
for $1\leq i\leq n-\phi$ that satisfies the conditions in Claim \ref{claim_1}.
Recall that nodes 1 through $n-\phi$ are fault-free, and the remaining
$\phi$ nodes ($\phi\leq f$) are faulty.
Consider a fault-free node $i$ performing the {\em Update step}
in Algorithm 1.
Recall that the largest $f$ and the smallest $f$ values are eliminated
from $r_i[t]$. Let us denote by $L$ and $S$, respectively, the set
of nodes\footnote{Although $L$ and $S$ may be different for each $t$,
for simplicity, we do not explicitly represent this dependence
on $t$ in the notations $L$ and $S$.}
from whom the largest $f$ values and the smallest $f$ values
were received by node $i$ in iteration $t$.
Thus,
$|L|=|S|=f$, $N_i^*[t] = N_i^- - (L\cup S)$,
and $|N_i^*[t]|=|N_i^--(L\cup S)| = |N_i^-|-2f$.
For any set of nodes $X$ here, let $\delta_X$ and $g_X$ respectively
denote the number of faulty nodes, and the number of fault-free nodes, in set $X$.
For instance, $\delta_L$ and $g_L$ denote, respectively, the number
of faulty and fault-free nodes in set $L$.
Thus,
\[ \delta_L+g_L~=~\delta_S+g_S~=~f\]
Let
\[
\delta = |N_i^-\cap \scriptf|
\]
That is, the number of faulty incoming neighbors of node $i$ is denoted
as $\delta$.
Therefore, $\delta\leq\phi\leq f$,
and
\[
\delta = \delta_L+\delta_S+\deltaC
\]
Then, it follows that
\begin{eqnarray}
g_L & = & f-\delta_L ~= ~ \delta_S+\deltaC+(f-\delta), \mbox{~and} \label{g_L}\\
g_S & = & f-\delta_S~=~\delta_L+\deltaC+(f-\delta) \label{g_S}
\end{eqnarray}
For fault-free node $i$,
we now define the elements of row $\bfM_i[t]$.
We consider two cases separately: (i) $f-\delta+\deltaC=0$,
and (ii) $f-\delta+\deltaC>0$.
\subsubsection{$f-\delta+\deltaC=0$}
\label{ss_1}
We know that $(f-\delta) \geq 0$ and $\deltaC\geq 0$.
Therefore, $f-\delta+\deltaC=0$ implies that $f=\delta$
and $\deltaC=0$. Thus, in this case, all the nodes in $N_i^*[t]$
are fault-free.
\begin{itemize}
\item
For each
$j \in \{i\}\cup N_i^*[t]$, define $\bfM_{ij}[t] = a_i$.
Element $\bfM_{ij}[t]$ corresponds to the term $a_iw_j$ in (\ref{e_Z}).
Recall that $a_i\geq \alpha$, and that each node in $\{i\}\cup N_i^*[t]$
in this case is fault-free.
\item
For each $j$ such that $j\in\scriptv-\scriptf$
and $j\not\in \{i\}\cup N_i^*[t]$, define $\bfM_{ij}[t] = 0$.
\end{itemize}
Observe that with the above definition of elements of $\bfM_i[t]$,
\[ \bfM_i[t] \bfv[t-1] = \sum_{k\in \{i\}\cup N_i^*[t]} a_iw_k
\]
In the above procedure, we have set $|N_i^*[t]|+1$ elements of $\bfM_i[t]$
equal to $a_i$ (recall that $a_i\geq \alpha$).
Now, because $\delta=f$ and $|N_i^*[t]|=|N_i^-|-2f$, we have
$|N_i^-\cap(\scriptv-\scriptf)|-f+1=|N_i^-|-\delta-f+1
= |N_i^-|-2f+1 = |N_i^*[t]|+1$.
Also, in this case $a_i=1/(|N_i^*[t]|+1)$. Thus, it should
be easy to see that the conditions in Claim \ref{claim_1} are satisfied
by defining $\beta=\alpha$.
~
\subsubsection{$f-\delta+\deltaC>0$}
Since $0\leq \deltaC\leq\delta\leq f$,
$f-\delta+\deltaC>0$ implies that $f>0$.
When $f>0$, the necessary condition in \cite{IBA_sync} implies that
$|N_i^-|\geq 2f+1$. Therefore, the set $N_i^*[t]$ is non-empty.
As per (\ref{e_Z}), each node $k\in N_i^*[t]$ contributes $a_i\, w_k$
to the new state $v_i[t]$ of node $i$. We will define elements
of $\bfM_i[t]$ to account for the contribution of each node $k\in N_i^*[t]$.
Define subsets $L^*$ and $S^*$ such that
$L^*\subseteq L$, $S^*\subseteq S$, $L^*\cap\scriptf=S^*\cap\scriptf=\Phi$,
and $|L^*|=|S^*|=f-\delta+\deltaC$. That is, sets $L^*$ and $S^*$
are subsets of $L$ and $S$, respectively, each of size $f-\delta+\deltaC$,
and containing only fault-free nodes.
Expressions (\ref{g_L}) and (\ref{g_S}) for $g_L$ and $g_S$
imply that such subsets exist.
Let
\[ L^*=\{l_j~ | ~ 1\leq j\leq f-\delta+\deltaC\}\]
and \[ S^*=\{s_j~ | ~ 1\leq j\leq f-\delta+\deltaC\}.\]
Consider any node $k\in N_i^*[t]$.
For each $j$, $1\leq j\leq f-\delta+\deltaC$,
\[ v_{s_j}[t-1] \leq w_k \leq v_{l_j}[t-1] \]
Therefore,
we can find weights $\lambda_{k,j}\geq 0$ and $\psi_{k,j}\geq 0$
such that
\[
\lambda_{k,j} + \psi_{k,j}~=~1\] and
\[
w_k~=~ ~\lambda_{k,j}\, v_{l_j}[t-1] ~ + ~
\psi_{k,j} \, v_{s_j}[t-1]
\]
Clearly, at least one of the weights $\lambda_{k,j}$ and $\psi_{k,j}$
must be $\geq 1/2$.
Now, observe that
\begin{eqnarray}
a_i\, w_k & = &
\frac{a_i}{f-\delta+\deltaC} ~
\sum_{1\leq j\leq f-\delta+\deltaC}
\left(
\lambda_{k,j}\, v_{l_j}[t-1] +
\psi_{k,j} \, v_{s_j}[t-1]
\right)
\label{e_faulty}
\end{eqnarray}
The above equality is true independent of whether $k$ is fault-free or faulty.
We will later use the above equality for the case when $k$ is a faulty node.
When $k$ is fault-free, \[w_k=v_k[t-1],\] and we can similarly obtain the equality below.
\begin{eqnarray}
a_iw_k & = & \frac{a_i}{2}v_k[t-1]
~+~
\frac{a_i}{2(f-\delta+\deltaC)} ~
\sum_{1\leq j\leq f-\delta+\deltaC}
\left(
\lambda_{k,j}\, v_{l_j}[t-1] +
\psi_{k,j} \, v_{s_j}[t-1]
\right) \nonumber\\
\label{e_faultfree}
\end{eqnarray}
We now use (\ref{e_Z}), (\ref{e_faulty}) and (\ref{e_faultfree}) to define elements
of $\bfM_i[t]$ in the following four cases:
\begin{itemize}
\item {\bf Case 1: Node $i$}\\ Define $\bfM_{ii}[t]=a_i$. This is obtained by observing in (\ref{e_Z}) that
the contribution of node $i$ to the new state $v_i[t]$ is $a_iw_i=a_iv_i[t-1]$.
\item {\bf Case 2: Fault-free nodes in $N_i^*[t]$}\\
For each $k\in N_i^*[t]\cap (\scriptv-\scriptf)$,
define $\bfM_{ik}[t] = \frac{a_i}{2}$. This choice is motivated by (\ref{e_faultfree})
wherein the contribution of node $k$ to $a_iw_k$ is $\frac{a_i}{2}v_k[t-1]$.
In Case 2, $|N_i^*[t]\cap(\scriptv-\scriptf)|=|N_i^-|-\delta$ elements
of $\bfM_i[t]$ are defined.
\item {\bf Case 3: Nodes in $L^*$ and $S^*$}\\ For $1\leq j\leq f-\delta+\deltaC$, consider $l_j\in L^*$. In this
case,
\begin{eqnarray*}
\bfM_{il_j}[t] &=&
\sum_{k\in N_i^*[t] \cap \scriptf} \frac{a_i}{f-\delta+\deltaC} \lambda_{k,j}
~+~
\sum_{k\in N_i^*[t]\cap (\scriptv-\scriptf)}
\frac{a_i}{2(f-\delta+\deltaC)} \lambda_{k,j}
\end{eqnarray*}
Similarly, for $1\leq j\leq f-\delta+\deltaC$, consider $s_j\in S^*$. In this
case,
\begin{eqnarray*}
\bfM_{is_j}[t] &=&
\sum_{k\in N_i^*[t] \cap \scriptf} \frac{a_i}{f-\delta+\deltaC} \psi_{k,j}
~+~
\sum_{k\in N_i^*[t]\cap (\scriptv-\scriptf)}
\frac{a_i}{2(f-\delta+\deltaC)} \psi_{k,j}
\end{eqnarray*}
These expressions are obtained by summing (\ref{e_faulty})
and (\ref{e_faultfree}), respectively, over the faulty and fault-free nodes
in $N_i^*[t]$, and then identifying the contribution
of each node in $L^*$ and $S^*$ to this sum.
Recall the earlier observation that at least one of $\lambda_{k,j}$
and $\psi_{k,j}$ must be $\geq 1/2$ for each pair $k,j$ where $k\in N_i^*[t]$
and $1 \leq j\leq f-\delta+\deltaC$.
Therefore, it follows that
at least $f-\delta+\deltaC$ elements of $\bfM_i[t]$ defined
in Case 3 must be $\geq \frac{a_i}{4(f-\delta+\deltaC)}$.
\item {\bf Case 4: Nodes in $(\scriptv-\scriptf)-(\{i\}\cup N_i^*[t]\cup L^*\cup S^*)$}\\
These fault-free nodes have not yet been considered in Cases 1, 2 and 3.
For each node $k\in (\scriptv-\scriptf)-(\{i\}\cup N_i^*[t]\cup L^*\cup S^*)$,
we assign $\bfM_{ik}[t]=0$.
\end{itemize}
Observe that above the definition of the elements of $\bfM_i[t]$
ensures that
\[ \sum_{j\in\{i\}\cup N_i^*[t]}a_iw_j~=~\bfM_i[t]\bfv[t-1]\]
However, the contribution by the
faulty nodes in $N_i^*[t]$ in (\ref{e_Z}) is
now replaced by an
equivalent contribution by the nodes in $L^*$ and $S^*$.
Now let us verify that the four conditions in Claim \ref{claim_1} hold
for the above assignments to the elements of $\bfM_i[t]$.
\begin{enumerate}
\item
Observe that all the elements of $\bfM_i[t]$ are non-negative.
Case 1 specifies just $\bfM_{ii}[t]=a_i$.
The elements of $\bfM_i[t]$ specified in Case 2
add up to
\[
\frac{a_i}{2} ~ |N_i^*[t]\cap (\scriptv-\scriptf)|
\]
Recall that for each $j$, $1\leq j\leq (f-\delta+\deltaC)$,
$\lambda_{k,j}+\psi_{k,j}=1$ for $k\in N_i^*[t]$.
Therefore, when added over all $k\in N_i^*[t]$
and $1\leq j\leq (f-\delta+\deltaC)$, the elements of $\bfM_i[t]$ specified in Case 3
add up to
\[
a_i ~ |N_i^*[t]\cap \scriptf| ~+~
\frac{a_i}{2} ~ |N_i^*[t]\cap (\scriptv-\scriptf)|
\]
Therefore, when all the elements of $\bfM_i[t]$ defined in Cases 1,
2 and 3 are added together, we get
\begin{eqnarray*}
a_i ~+~ a_i\, |N_i^*[t]\cap\scriptf|
~+~
a_i |N_i^*[t]\cap (\scriptv-\scriptf)| ~=~
a_i (|N_i^*[t]|+1) ~=~1
&&
\end{eqnarray*}
because
$a_i=1/(|N_i^*[t]|+1)$.
Now observe that the elements specified in Cases 1, 2 and 3 are clearly
$\leq 1$.
In the expression for $\bfM_{il_j}[t]$ in Case 3, observe that the two
summations on the right side together contain $|N_i^*[t]|$ terms,
and in these terms, observe that $\lambda_{k,j}\leq 1$,
$f-\delta+\deltaC\geq 1$ and $a_i=\frac{1}{|N_i^*[t]|+1}$. Therefore,
$\bfM_{il_j}[t]<1$.
Similarly, we can show that $\bfM_{is_j}[t]<1$ as well.
Thus, we have shown that $\bfM_i[t]$ is a stochastic vector.
\item $\bfM_{ii}[t]=a_i$ as specified in Case 1.
\item Since $\bfM_{ij}[t]$ is defined to be non-zero only in Cases 1, 2 and 3,
which consider the nodes only in $\{i\}\cup N_i^-$, it follows
that $\bfM_{ij}[t]$ is non-zero {\em only if} $(j,i)\in\scripte$
or $j=i$.
\item
Cases 1 and 2 together set $1+|N_i^*[t]\cap(\scriptv-\scriptf)|=
1+|N_i^*[t]|-\deltaC$
elements of $\bfM_i[t]$ to be $\geq a_i/2$. We observed
earlier that Case 3 results in at least
$f-\delta+\deltaC$ elements of $\bfM_i[t]$
being $\geq \frac{a_i}{4(f-\delta+\deltaC)}$.
Also, observe that the elements of $\bfM_i[t]$ specified
in Cases 1 and 2 are distinct from those specified in Case 3,
and that $\frac{a_i}{2} \geq \frac{a_i}{4(f-\delta+\deltaC)}$. Thus, overall, at
least
\begin{eqnarray*}
(1+|N_i^*[t]|-\deltaC) ~+~
f-\delta+\deltaC ~=~ |N_i^*[t]|+f-\delta+1
~=~ |N_i^-|-f-\delta+1 && \\ ~=~ |N_i^-\cap(\scriptv-\scriptf)|-f-1
&& \end{eqnarray*}
elements of $\bfM_i[t]$ are set
$\geq \frac{a_i}{4(f-\delta+\deltaC)}$.
Derivation of the above equation uses the facts that
$|N_i^*[t]|=|N_i^-|-2f$ and
$|N_i^-\cap(\scriptv-\scriptf)|=|N_i^-|-\delta$. Then
by defining $\beta$ as below, condition 4 in Claim \ref{claim_1}
holds true.
\[
\beta = \frac{\alpha}{4(f-\delta+\deltaC)}
\]
\end{enumerate}
Therefore, Claim \ref{claim_1} is proved correct.
~
\subsection{Correspondence Between Sufficiency Condition and $\bf M[t]$}
Let us define set $R_\scriptf$ of subgraphs of $G(\scriptv,\scripte)$ as follows.
\begin{eqnarray}
R_\scriptf & = & \{ H ~ | ~ H \mbox{~ is obtained by removing all the faulty}
\nonumber \\
&& \mbox{ nodes
from $\scriptv$ along with their edges, and then} \nonumber
\\ && \mbox{ removing any additional $f$ incoming edges}\nonumber \\
&&
\mbox{ at each fault-free node} \}
\end{eqnarray}
Thus, $\scriptv-\scriptf$ is the set of nodes in each graph in $R_\scriptf$.
Let $\tau$ denote $|R_\scriptf|$.
$\tau$ depends on $\scriptf$ and the underlying network, and it is finite.
\begin{claim}
\label{claim_suff}
Suppose that graph $G(\scriptv,\scripte)$ satisfies the necessary condition in
Theorem 2 in \cite{us}. Then it follows that in each $H\in R_\scriptf$,
there exists at least one node that has directed paths to all the nodes in $H$
(consisting of the edges in $H$).
\end{claim}
\begin{proof}
The proof follows from Theorem 2 of \cite{us}.
\end{proof}
In this discussion, let us denote a graph by an italic upper case letter,
and the corresponding {\em connectivity matrix} using the same letter
in boldface upper case. Thus,
$\bfH$ will denote the connectivity matrix for graph $H\in R_\scriptf$;
$\bfH$ is defined as follows:
(i) for $1\leq i,j\leq n-\phi$,
if there is a directed link from node $j$ to node $i$ in
graph $H$ then $\bfH_{ij}=1$,
and (ii) $\bfH_{ii}=1$ for $1\leq i\leq n-\phi$.
Note that in our notation, the $i$-th row of $\bfH$ (that is, $\bfH_i$)
corresponds to the incoming links at node $i$, and the self-loop
at node $i$.
The connectivity matrix $\bfH$ for any $H\in R_\scriptf$
has a non-zero diagonal.
\begin{lemma}
\label{l_one_column}
For any $H\in R_\scriptf$, $\bfH^{n-\phi}$ has at least one non-zero column.
\end{lemma}
\begin{proof}
By Claim \ref{claim_suff}, in graph $H$ there exists at least one node, say
node $k$, that has a directed path in $H$ to all the remaining nodes in $H$.
Since the length of the path from $k$ to any other node in $H$ can contain
at most $n-\phi-1$ directed edges,
the $k$-th column of matrix ${\bfH^{n-\phi}}$ will
be non-zero.\footnote{That is, all the elements of the column will be
non-zero (more precisely, positive, since the elements of matrix $\bfH$
are non-negative).
Also, such a non-zero column will exist in $\bfH^{n-\phi-1}$ too.
We use the loose bound of $n-\phi$ to simplify the presentation. }
\end{proof}
~
\begin{definition}
We will say that an element of a matrix is ``non-trivial'' if it is lower
bounded by $\beta$.
\end{definition}
\begin{definition}
For matrices $\bfA$ and $\bfB$ of identical size, and
a scalar $\gamma$, $\bfA\leq \gamma \, \bfB$ provided
that $\bfA_{ij}\leq \gamma\, \bfB_{ij}$ for all $i,j$.
\end{definition}
\begin{lemma}
\label{l_H}
For any $t\geq 1$, there exists a graph $H[t]\in R_\scriptf$ such that
$\beta \, \bfH[t] ~ \leq ~ {\bfM[t]}$.
\end{lemma}
\begin{proof}
Observe that the $i$-th row of the transition matrix $\bfM[t]$ corresponds to the state update
performed at fault-free node $i$. Recall from Claim \ref{claim_1} that
the $\bfM_{ij}$ is non-zero {\bf only if} link $(j,i)\in\scripte$.
Also, by Claim \ref{claim_1},
$\bfM_i[t]$ (i.e., the $i$-th row of $\bfM[t]$)
contains at least $|N_i^-\cap\,(\scriptv-\scriptf)| - f+1$
{\em non-trivial} elements corresponding
to {\bf fault-free} incoming neighbors of node $i$ and itself (i.e., the
diagonal element).
Now observe that, for any subgraph $H\in R_\scriptf$, $i$-th row of
$\bfH$ contains exactly $|N_i^-\cap\,(\scriptv-\scriptf)| - f+1$ non-zero elements,
including the diagonal element.
Considering the above two observations, and the definition of set $R_\scriptf$,
the lemma follows.
\end{proof}
~
\section{Correctness of Algorithm 1}
The proof below uses techniques also applied in prior work
(e.g., \cite{jadbabaie_consensus,Benezit,vaidyaII,Zhang}),
with some similarities to the arguments used in \cite{vaidyaII,Zhang}.
\begin{lemma}
\label{l_product_H}
In the product below of $\bfH[t]$ matrices for consecutive
$\tau(n-\phi)$ iterations, at least one column is non-zero.
\[
\Pi_{t=z}^{z+\tau(n-\phi)-1} \, \bfH[t]
\]
\end{lemma}
\begin{proof}
Since the above product consists of $\tau(n-\phi)$ matrices
in $R_\scriptf$,
at least one of the $\tau$ distinct connectivity matrices
in $R_\scriptf$, say matrix $\bfH_*$, will appear in the above
product at least $n-\phi$ times.
Now observe that: (i)
By Lemma \ref{l_one_column}, $\bfH_*^{n-\phi}$ contains a non-zero
column, say the $k$-th column is non-zero,
and (ii) all the $\bfH[t]$ matrices in the product contain a non-zero diagonal.
These two observations together imply that the $k$-th column in the above product
is non-zero.
\end{proof}
Let us now define a sequence of matrices $\bfQ(i)$ such that
each of these matrices is a product of $\tau(n-\phi)$ of the
$\bfM[t]$ matrices. Specifically,
\[
\bfQ(i) ~=~ \Pi_{t=(i-1)\tau(n-\phi)+1}^{i\tau(n-\phi)} ~ \bfM[t]
\]
Observe that
\begin{eqnarray}
\bfv[k\tau(n-\phi)] & = & \left(\, \Pi_{i=1}^k ~ \bfQ(i) \,\right)~\bfv[0]
\end{eqnarray}
\begin{lemma}
\label{l_Q}
For $i\geq 1$, $\bfQ(i)$ is a scrambling row stochastic matrix,
and $\lambda(\bfQ(i))$ is bounded from above by a constant
smaller than 1.
\end{lemma}
\begin{proof}
$\bfQ(i)$ is a product of row stochastic matrices ($\bfM[t]$), therefore,
$\bfQ(i)$ is row stochastic.
From Lemma \ref{l_H}, for each $t$,
\[
\beta \, \bfH[t] ~ \leq ~ \bfM[t]
\]
Therefore,
\[
\beta^{\tau(n-\phi)} ~ \Pi_{t=(i-1)\tau(n-\phi)+1}^{i\tau(n-\phi)} ~ \bfH[t] ~ \leq
~ \bfQ(i)
\]
By using $z=(i-1)(n-\phi)+1$ in Lemma \ref{l_product_H},
we conclude that the matrix product on the left side
of the above inequality contains a non-zero column. Therefore, $\bfQ(i)$ contains
a non-zero column as well. Therefore, $\bfQ(i)$ is a scrambling matrix.
Observe that $\tau(n-\phi)$ is finite, therefore, $\beta^{\tau(n-\phi)}$
is non-zero. Since the non-zero terms in $\bfH[t]$ matrices are all 1,
the non-zero elements in $\Pi_{t=(i-1)\tau(n-\phi)+1}^{i\tau(n-\phi)} \bfH[t]$
must each be $\geq$ 1. Therefore, there exists a non-zero column in $\bfQ(i)$
with all the elements in the column being $\geq \beta^{\tau(n-\phi)}$.
Therefore $\lambda(\bfQ(i))\leq 1-\beta^{\tau(n-\phi)}$.
\end{proof}
\begin{theorem}
\label{t}
Algorithm 1 satisfies the validity and the convergence conditions.
\end{theorem}
\begin{proof}
Since $\bfv[t]=\bfM[t]\,v[t-1]$, and $\bfM[t]$ is a row stochastic matrix, it
follows that
Algorithm 1 satisfies the validity condition.
By Claim \ref{claim_delta},
\begin{eqnarray}
\lim_{t\rightarrow \infty} \delta(\Pi_{i=1}^t \bfM[t])
& \leq & \lim_{t\rightarrow\infty} \Pi_{i=1}^t \lambda(\bfM[t]) \\
& \leq & \lim_{i\rightarrow\infty} \Pi_{i=1}^{\lfloor\frac{t}{\tau(n-\phi)}\rfloor} \lambda(\bfQ(i)) \\
& = & 0
\end{eqnarray}
The above argument makes use of the facts that
$\lambda(\bfM[t])\leq 1$ and $\lambda(\bfQ(i))\leq (1-\beta^{\tau(n-\phi)})<1$.
Thus, the rows of $\Pi_{i=1}^t \bfM[t]$ become identical in the limit.
This observation, and the fact that $\bfv[t]=(\Pi_{i=1}^t \bfM[i])\bfv[t-1]$ together imply that
the state of the fault-free nodes satisfies the
convergence condition.
Now, the validity and convergence conditions
together imply that
there exists a positive scalar $c$ such that
\[
\lim_{t\rightarrow\infty}
\bfv[t] ~ = ~ \lim_{t\rightarrow\infty} \left( \Pi_{i=1}^t \bfM[i]) \right)\,
\bfv[0] ~ = ~ c\,{\bf 1}
\]
where {\bf 1} denotes a column with all its elements being 1.
\end{proof}
\section{Extension of Above Results}
\label{s_extend}
In this paper, we
analyzed IABC Algorithm 1 designed for synchronous systems. Similar
analysis also applies for IABC Algorithm 2 presented
in \cite{us} for asynchronous systems.
The analysis will also naturally extend to an IABC algorithm for the
{\em partially synchronous algorithmic} model
presented in \cite{AA_convergence_markov}, which assumes a bounded
delay in propagation of state between neighbors, and a bounded delay between
consecutive state updates at each node in the network.
The generalization of Algorithm 1 to the
{\em partially synchronous algorithmic} model will allow a node $i$,
if performing state update in iteration $t$, to form vector $r_i[t]$
using the most recent known states of its incoming neighbors; these states
of the neighbors may correspond to any of the prior $B$ iterations,
for some bounded $B$.
A similar IABC algorithm can also be used
in time-varying network topologies (i.e., networks wherein the set
of links available in iteration $t$ varies with $t$);
the above analysis will
then extend to time-varying topologies as well, with the
algorithm performing correctly so long as
the connectivity matrices for the graphs at different $t$
jointly satisfy some reasonable properties, as in \cite{jadbabaie_consensus,Benezit,vaidyaII}.
\section{Summary}
We presented a proof of validity and convergence of Algorithm 1 by
expressing the algorithm in the matrix form. The main contribution of
the paper is to express the algorithm in matrix form that allows us to
prove its convergence under certain necessary conditions on the underlying
communication graph. Thus, the proof implies that the necessary conditions
are also sufficient.
The key to the proof is to ``design'' the transition matrix for each
iteration such that each row of the matrix contains a large enough number
of elements that are bounded away from 0.
|
{
"timestamp": "2012-03-09T02:04:25",
"yymm": "1203",
"arxiv_id": "1203.1888",
"language": "en",
"url": "https://arxiv.org/abs/1203.1888"
}
|
\section{Introduction}
Inflation \cite{Guth:1980zm,Linde:1981mu, Albrecht:1982wi} is remarkably successful at accounting for the primordial perturbations. While it is a relatively simple matter to arrange for an inflationary phase in the early universe, the challenge is to make it end, and end in such a way that the resulting universe resembles our observed universe. Single field inflation achieves this by utilizing a scalar field slowly rolling on a flat potential. Inflation ends when the potential becomes too steep relative to its height and the slow roll conditions are violated. This occurs at a particular point on the potential, and thus one may think of the position of the field as a clock which tells the time before inflation ends. In order to get sufficient inflation and solve the flatness and horizon problems, the field must roll slowly. In canonical inflation, this translates into a requirement that the potential be nearly flat relative to its height. Maintaining slow roll requires that the curvature of the potential also be small. While it is a simple matter to write down suitable potentials which have the desired properties, these potentials require a high degree of fine tuning in order that they remain stable against radiative corrections. This is known as the `eta' problem. Natural Inflation solves this problem by using an axion \cite{Freese:1990rb, Adams:1992bn} whose potential is protected from such corrections by a shift symmetry. Unfortunately for Natural Inflation, matching cosmic microwave background observations requires the model to have a Planck-scale axion decay constant, $f$ \cite{Freese:2004un}. Such a setup seems to be difficult, if not impossible, to realize in string theory \cite{Banks:2003sx}.
In our recently proposed theory of Chromo-Natural Inflation \cite{Adshead:2012kp}, we demonstrated that it may be possible to alleviate this problem by coupling the axion to non-Abelian gauge fields in a classical, rotationally invariant configuration. The interactions between the axion and
the gauge fields generate a slowly rolling inflationary solution for a wide range of parameters, including
both large and small values for the axion decay constant. Importantly, this range includes values $f \ll M_{\rm pl}$.
Our scenario differs in important ways from the many others that involve either axions, vector-like fields, or both. The seminal example of an axionic inflationary theory,
Natural Inflation, requires $f \sim M_{\rm pl}$ as we noted before \cite{Freese:1990rb}. Within axionic theories, a variety of methods have been attempted to cure the need for Planckian decay constants, e.g. \cite{Kim:2004rp, Dimopoulos:2005ac, Easther:2005zr, Silverstein:2008sg, Germani:2010hd}. In a closer approach to our model, Ref.\ \cite{Anber:2009ua} uses the copious emission of Abelian gauge quanta to permit inflation with $f\ll M_{\rm pl}$. The authors of \cite{Gumrukcuoglu:2010yc,Kanno:2010nr,Watanabe:2009ct,Watanabe:2010fh, Yamamoto:2012sq} study models of inflation with a uniform gauge-kinetic coupling of the inflaton to multiple vector fields, while models of inflation where the curvature perturbations are partially produced by the vacuum fluctuations of a vector multiplet were considered by \cite{Dimopoulos:2008yv,Bartolo:2009pa,Bartolo:2009kg, Dimastrogiovanni:2010sm, Bartolo:2011ee}. A number of models have been proposed where gauge or vector-like fields have classical background values that play a central role in the inflationary mechanism (e.g. \cite{Ford:1989me,ArmendarizPicon:2004pm, Koivisto:2008xf, Golovnev:2008cf,Golovnev:2009ks, Alexander:2011hz}), but wherein the inherent anisotropy of the vector fields is cured by invoking many such fields, rather than through the rotationally invariant background configuration we consider. We note that the models of \cite{Ford:1989me,ArmendarizPicon:2004pm, Golovnev:2008cf,Golovnev:2009ks}, but not \cite{Alexander:2011hz}, were shown to be unstable in \cite{Himmetoglu:2008zp,Himmetoglu:2008hx,Golovnev:2009rm,Himmetoglu:2009qi}. More generally, other researchers have found that non-minimally coupled p-form fields can also generate inflationary backgrounds without fundamental scalar fields \cite{Koivisto:2009sd, Germani:2009iq, Germani:2009gg}; however, the stability of these models still appears to be an open question \cite{Golovnev:2009rm}. Finally, other work has demonstrated that self interacting 3-form fields can give rise to accelerating cosmologies \cite{Koivisto:2009ew, Koivisto:2009fb}.
Classical, cosmological solutions for gauge fields have a history going back to the late 1970's, when there was a search for solutions to the Einstein-Yang-Mills field equations \cite{Cervero:1978db, Henneaux:1982vs, Hosotani:1984wj, Moniz:1990hf}. More recently, these configurations have been studied in the context of dark energy \cite{ Gal'tsov:2010dd, Galtsov:2011aa,Elizalde:2012yk} and it was noticed that such a configuration allows for inflation -- Gauge-flation -- without the presence of a scalar field \cite{Maleknejad:2011jw,Maleknejad:2011sq} (see also \cite{Galtsov:2011aa}). While such gauge field configurations have been studied in the context of inflating backgrounds before \cite{Moniz:1991kx}, in that earlier work the authors only studied situations in which the inflaton was charged under the gauge group but otherwise the gauge fields played no role in generating the inflationary epoch.
This paper is organized as follows. In Sec.\ \ref{sec:scni} we provide a detailed description of the trajectories in Chromo-Natural Inflation. In Sec.\ \ref{sec:params}, we describe the space of parameters which generate sufficient inflation. In Sec.\ \ref{sec:GFvsCNI} we elucidate the relationship between the model of Chromo-Natural Inflation \cite{Adshead:2012kp} and the model of Gauge-flation \cite{Maleknejad:2011jw,Maleknejad:2011sq}. We conclude in Sec.\ \ref{sec:concl}. Throughout this work, we use natural units where the reduced Planck mass $M_{\rm pl}= c = \hbar = 1$.
\section{The Space of Chromo-Natural Inflation}\label{sec:scni}
In our previous work \cite{Adshead:2012kp}, we proposed a model of inflation with an axion, $\mathcal{X}$, and three non-Abelian SU(2) gauge-fields:
\begin{align}\label{eqn:action}\nonumber
\mathcal{L} = \sqrt{-g} & \bigg[-\frac{R}{2}-\frac{1}{4}F_{\mu\nu}^{a}F_{a}^{\mu\nu} - \frac{1}{2}(\partial \mathcal{X} )^2
\\ & - \mu^4\left(1+\cos\left(\frac{\mathcal{X} }{f}\right)\right)-\frac{\lambda}{8f} \mathcal{X} F^{a}_{\mu\nu}\tilde F_{a}^{\mu\nu}\bigg],
\end{align}
where $F^{a}_{\mu\nu}$ is the usual non-Abelian gauge field strength tensor
\begin{align}
F^{a}_{\mu\nu} = \partial_{\mu}A^a_{\nu} - \partial_{\nu}A^a_{\mu} - \tilde g f^{a}_{bc}A^{b}_{\mu}A^c_{\nu},
\end{align}
$\tilde g$ is the gauge field coupling, $f^{a}_{bc}$ are the structure constants of SU(2), and $\tilde F_{a}^{\mu\nu} = \epsilon^{\mu\nu\alpha\beta}F_{a\alpha\beta}$. Following usual practice, Greek letters indicate spacetime indices while Roman letters represent gauge indices.
We take the gauge fields to be in a classical configuration given by the ansatz,
\begin{align}
A^{a}_{0} = 0,\quad A^{a}_i =\psi(t) \,a(t)\delta^{a}_i.
\end{align}
That is, our gauge sector has a vacuum expectation value (VEV).
This configuration of the gauge fields is notable because it leads to an isotropic and spatially homogeneous cosmological solution where isotropy is protected by the non-Abelian fields' gauge invariance -- rotations in real space are `undone' by rotations in gauge space, leaving the fields invariant. The SU(2) gauge symmetry is crucial here, as its global part can be associated with rotations in 3-space leading to the invariance.
{ In the slow roll approximation, we can diagonalize the resulting equations of motion for the velocities, which gives
\begin{align}
\label{Xdiag}
\Bigl(3H+\frac{g^2\lambda^2}{H f^2}\psi^4\Bigr)\dot\mathcal{X} &= \frac{\mu^4}{f}\sin(\mathcal{X}/f) - \frac{g\lambda}{f} H\psi^3 + \frac{2g^3\lambda}{ f H} \psi^5 \\
\label{psidiag}
\Bigl(3H+\frac{g^2\lambda^2}{H f^2}\psi^4\Bigr)\dot\psi &= -2H^2\psi - 2g^2\psi^3 - \frac{g^2\lambda^2}{f^2}\psi^5 \nonumber \\
& + \frac{g\lambda}{3H f^2}\psi^2\mu^4\sin({\cal X}/f) \ .
\end{align}
}
As was demonstrated in \cite{Adshead:2012kp}, the interacting system of a scalar field and non-Abelian gauge fields described by the action in Eqn.\ (\ref{eqn:action}) and the equations of motion, Eqns. (\ref{Xdiag}) and (\ref{psidiag}), is extremely effective at generating slow roll inflation in the presence of steep bare scalar potentials. This is because the gauge field VEV is dynamically forced to a trajectory where it generates a very flat potential for the scalar {since the scalar feels an extra damping from the gauge field.} In our case, where the potential is assumed to be a cosine, this trajectory is very accurately given by (see Fig.\ \ref{fig:Psi})
\begin{align}\label{eqn:psimin}
\psi_{\rm min} = \left(\frac{\mu^4\sin\(\frac{\mathcal{X}}{f}\)}{3\tilde g \lambda H}\right)^{1/3}.
\end{align}
On this trajectory, the change in potential energy as the axion rolls is almost entirely converted into gauge field energy rather than the axion's kinetic energy. This means that the axion rolls only slowly on its otherwise steep potential. {The gauge field's VEV, in turn, is being classically sourced by the axion's roll, which is why the VEV remains approximately constant despite
the presence of an exponentially expanding space \footnote{In our previous work, \cite{Adshead:2012kp}, we considered the question of whether the emission of gauge quanta, which happens in addition to the classical generation of the gauge field, will spoil our set-up. A rough calculation, found in a footnote there, suggests that the quantum emission is far smaller than
the classical generation. Additionally, we note that the gauge field gets a dynamical mass around the minimum given in the text, Eqn. \ref{eqn:psimin}, of ${\cal O}(H)$. This effective mass will
further suppress the emission of gauge quanta beyond the calculation that we mention in the previous work.}.}
This solution makes no assumptions about where the axion is on its potential. The only assumption needed is that we have chosen parameters such that
\begin{equation}
3f^{2}H^2 \ll \tilde g^2\psi^4 \lambda^2,
\end{equation} which amounts to choosing $\lambda \gg 1$. For our purposes, $\lambda \sim \mathcal{O}(100)$ is sufficient. This permits us to achieve observationally
viable inflation with $f \ll M_{\rm pl}$. A typical inflationary path for $\mathcal{X}$ is illustrated in Fig. \ref{fig:axion}. Note that inflation proceeds over nearly the entire range of the axion's potential.
\begin{figure*}[t]
\centerline{\psfig{file=GaugePhase.pdf, width=3.5in} \psfig{file=Psi.pdf, width=3.5in}}
\caption{The behaviour of the gauge field during inflation for the choice of parameters $\{\mu,f,\tilde g, \lambda\} = \{3.16 \times 10^{-4}, 0.01, 2.0 \times10^{-6}, 200\}$. In the left panel we show the phase portrait of the gauge field. The solid black curve corresponds to the period of exponential expansion, inflation. The red curve is the $\sim 8$ observable efoldings, 50 efoldings before the end of inflation, where the cosmic microwave background fluctuations are produced. Inflation ends where the dashed black line begins, and the gauge field decays. In the right hand panel we show the behavior of the gauge field as inflation proceeds, the observable window 50 efoldings before the end of inflation is shown in red. Plotted in blue is the value of the gauge field that minimizes its effective potential, Eqn.\ (\ref{eqn:psimin}).}
\label{fig:Psi}
\end{figure*}
\begin{figure}[t*]
\centering
\includegraphics[width=0.45 \textwidth]{Axion.pdf}
\caption{The part of the bare axion potential traversed 50 efoldings before inflation ends (upper panel) is shown in red over the top of the parts of the potential that are probed by the axion during the entire period of inflation for the choice of parameters $\{\mu,f,\tilde g, \lambda\} = \{3.16 \times 10^{-4}, 0.01, 2.0 \times10^{-6}, 200\}$. In the lower panel we show the full range of values the axion takes during inflation as a function of the efolding number. The axion's position in between 50 - 42 efoldings before the end of inflation is shown in red.}
\label{fig:axion}
\end{figure}
\begin{figure*}[t]
\centerline{\psfig{file=JustmuandConstNlog.pdf, width=3.5in} \psfig{file=JustgandLambdalog.pdf, width=3.5in}}
\caption{We show the number of efoldings of inflation produced as the various parameters of Chromo-Natural Inflation are varied. In the upper left panel, we show how the total amount of inflation varies as we vary the energy scale of the axion's potential, $\mu$. In the upper right panel we show how the total amount of inflation varies as we vary the gauge field coupling strength $\tilde g$. In the lower left panel, we show that the number of efoldings of inflation is kept constant if the parameters gauge field strength and the axion energy scale are covaried while keeping the ratio $2\tilde g/\mu^2$ constant. We also show the effect of these variations at various values of the coupling strength between the gauge and axion sectors, $\lambda$. In the lower right panel, we show the effect of varying $\lambda$ while keeping the remaining parameters fixed. Unless otherwise noted, all other parameters are fixed at the values $\{\mu,f,\tilde g, \lambda\} = \{3.16 \times 10^{-4}, 0.01, 2.0 \times10^{-6}, 200\}$. {This parameter set is one that we estimate will give the appropriate level of cosmological perturbations. In this figure, however, deviations from this set are not guaranteed to generate the appropriate level of perturbations, and generically will not.}}
\label{fig:params}
\end{figure*}
It is instructive to ask what brings about the end of inflation. With this goal in mind, we can calculate the slow roll parameter $\epsilon_H =-\dot{H}/H^2$, where here and throughout, and overdot denotes derivatives with respect to cosmic time; inflation ceases when $\epsilon_H = 1$, and \cite{Adshead:2012kp}
\begin{align}\label{eqn:epsilon}
\epsilon_{H} \approx \frac{3\tilde g^2 \psi^4}{\mu^4\(1+\cos\(\frac{\mathcal{X}}{f}\)\)}+\psi^2.
\end{align}
This equation has a simple interpretation. The second term ($\psi^2$) is unimportant at the end of inflation and can be neglected. The numerator of the first term is related to the gauge field energy density, while the denominator is the axion's potential energy. Hence, inflation ends when the gauge field energy density becomes comparable to the axion's potential energy; in other words, when the Universe becomes radiation dominated rather than vacuum energy dominated. Since the gauge field energy is small and nearly constant throughout inflation, this happens when the axion reaches the bottom of its potential.
In Chromo-Natural Inflation we are not required to make any assumptions about region of the potential where the axion is located -- we can simply evolve the system over the entire range,
$0 < \mathcal{X}/f < \pi.$ Precisely how close the system can get to $\mathcal{X}/f = \pi$ is determined by the first term in Eqn.\ (\ref{eqn:epsilon}).
The system of equations corresponding to Chromo-Natural Inflation (see \cite{Adshead:2012kp})
is straightforward to solve numerically. For the plots in Figs.\ \ref{fig:axion} and \ref{fig:Psi}, we work with the parameter set from \cite{Adshead:2012kp}
\begin{equation}
\label{eqn:params}
\{\mu,f,\tilde g, \lambda\} = \{3.16 \times 10^{-4}, 0.01, 2.0 \times10^{-6}, 200\},
\end{equation}
and take the following initial conditions. The axion begins at $\mathcal{X}(t_0) = 5\times10^{-4}$, with a velocity given by
\begin{align}
\left.\frac{d \mathcal{X}}{dt}\right|_{t_{0}} = -\frac{\lambda}{f}\tilde{g}\dot{\psi}\psi^2.
\end{align}
For convenience, we initialize the gauge field at its attractor value,
\begin{align}
\psi(t_0) = \left(\frac{\mu^4\sin\(\frac{\mathcal{X}(t_0)}{f}\)}{3\tilde g \lambda H}\right)^{1/3}.
\end{align}
However, let us emphasize that this is not necessary. If the gauge field starts away from this value, it relaxes to it within a few efolds.
For similar reasons, we choose for the gauge field to have a small initial velocity,
\begin{align}
\frac{\dot\psi}{H} = -1\times10^{-6}.
\end{align}
In Figure \ref{fig:axion} we show the region of field space over which the axion ranges during all of the inflationary period. The upper panel illustrates the potential, the lower panel the position of the axion as a function of the time before inflation ends. In both Figures \ref{fig:axion} and \ref{fig:Psi}, areas where the line is colored red indicate the region where the observable fluctuations are produced.
In the left panel of Figure \ref{fig:Psi}, we show the phase space of the gauge field. We show in solid lines the region that corresponds to the inflationary epoch, in red is the region where the fluctuations are produced, 50 efoldings before inflation ends. We also show, in dashed lines, the post inflationary evolution of the gauge field. Notice that the end of inflation and the post inflationary epoch exhibit similar behavior to that noted by the authors of \cite{Maleknejad:2011jw,Maleknejad:2011sq}. This period corresponds to the decay of the gauge field VEV and would likely provide a mechanism for reheating.
In the right hand panel of Figure \ref{fig:Psi}, we show the evolution of the gauge field as a function of the number of efoldings before inflation ends. As with the left hand panel, the observable range is shown in red. We also plot the curve corresponding to the condition in Eqn.\ (\ref{eqn:psimin}), noting the excellent agreement throughout the inflating epoch.
\begin{figure*}[t]
\centerline{\psfig{file=CNIvsGF.pdf, width=3.5in} \psfig{file=CNIvsGFaxion.pdf, width=3.5in}}
\caption{The behaviour of the gauge field during inflation for the choice of parameters that corresponds to the Gauge-flation model of \cite{Maleknejad:2011jw} $\{\mu,f,\tilde g, \lambda\} = \{4 \times 10^{-2}, 0.01, 2.5 \times10^{-3}, 12158\}$. In the left panel we show a phase portrait of the gauge field. The solid black curve corresponds to the period of exponential expansion, inflation. The red curve shows the final $\sim 60$ efoldings of inflation and Chromo-Natural inflation ends at the end of the red curve here where the gauge field decays. The dashed blue line here (which begins 60 efoldings before the end of inflation, and thus includes the red curve) shows the result of evolving the equations that follow from the action after the axion has been integrated out, Eqn.\ (\ref{eqn:actionGF}), where $\{\tilde g, \kappa\} = \{2.5\times10^{-3}, 1.73\times 10^{14}\}$. In the inset panel we show in more detailed the post-inflationary region.
In the right hand panel we show the full range of the axion as Chromo-Natural inflation proceeds. In blue we show the region that corresponds to the gaugeflation regime, which for these parameters is occurring for 60 efoldings. In the inset panels we show the region of the evolution of the axion where the two theories overlap.}
\label{fig:CNIvsGF}
\end{figure*}
\section{The parameter space of Chromo-Natural Inflation}\label{sec:params}
We have so far focussed on the parameter set in Eqn. (\ref{eqn:params}) because we were able to show that it gives
an observationally viable amplitude and running of the spectrum of adiabatic density perturbations, at least
at the level of approximation that was used in \cite{Adshead:2012kp}. Our estimate was based on the fact that the axion's position on its potential sets the `clock' for our model, as is the case in most single field inflationary models.
Curvature fluctuations in inflation are due to spatial fluctuations in the time at which inflation ends. Our estimates were made assuming that the curvature fluctuations are generated by quantum fluctuations of the axion along its effectively flat direction. The spatial variation of the fluctuations translate into spatial fluctuations of the clock and hence produce a shift in the time inflation ends from place to place - a curvature fluctuation. We stress that this kind of calculation can only give rough estimates of the amplitude and tilt of the fluctuations. {From \cite{Adshead:2012kp}, this calculation gives
\begin{equation}
{\cal R} \simeq \frac{1}{2 \pi} \frac{H}{\mathcal{X}'} \sim \frac{1}{10} \frac{\lambda \mu^2}{f} \sim 10^{-4}
\end{equation}
where $\cal R$ is the curvature perturbation, $\mathcal{X}'$ is the axion's velocity in e-folding time, and in the final $\sim$ equivalences we approximate $\mathcal{X}' \sim f/\lambda$, $H \sim \mu^2$. Of course, a full analysis of all of the degrees of freedom of the theory is necessary to go beyond this estimate. Preliminary results suggest that the full story may be somewhat more complicated \cite{precite}.}
We do not expect this particular choice to exhaust the possible parameter space of viable models. Hence, in this section we explore the parameter space of models which generate sufficient inflation, and therefore solve the horizon and flatness problems. We have verified directly that it is very easy to achieve sufficient inflation ($N_{\rm efolds}>60$) in our model; all that is required is that our parameter $\lambda \gtrsim 100$, and of course one needs to start with the axion far enough up its potential. For our fiducial parameter set, $\mathcal{X}_0 /(f\pi) < 0.4$ is sufficient. Any larger value of $\lambda$ will easily generate enough inflation, with mild assumptions about the other parameters (see Fig \ref{fig:params}).
The behavior of the axion on its effective potential so well describes the trajectory of the axion that we can simply integrate our approximations to find the total number of efoldings that our model produces. The number of efoldings is found by integrating the expression \cite{Adshead:2012kp},
\begin{equation}\label{eqn:efoldings}
N(\mathcal{X}_0) = \int_{\frac{\mathcal{X}_0}{f}}^\pi \hspace{-4 pt} \frac{{1\over2} \(3 \tilde g^{2} \lambda ^{4} \tilde \mu^{4} \left( 1+\cos x \right)^{2} \sin x\)^{1/3}}{ \( \lambda ^{2} \tilde \mu^{8} \left( 1+\cos x \right)^{4}\)^{1/3}+ \(3 \tilde g^{2} \sin x\)^{2/3} } dx,
\end{equation}
over the entire range of the axions motion, $\mathcal{X}/f \in (0, \pi)$. Unfortunately, this expression does not seem to admit any useful approximations which are valid over a large range of parameters, or at all points in the axion's range, and thus must be evaluated numerically. We note that this expression accurately matches the results from integrating
the full set of equations of motion across the full range of cases we consider.
In Fig.\ \ref{fig:params}, we illustrate the parameter space of Chromo-Natural Inflation. We show how the total amount of inflation varies as we move through the ranges of the parameters.
Notice that, if $\tilde g$ and $\mu$ are varied while keeping the ratio $\tilde g/\mu^2$ constant, the the total number of efoldings depends only on the value of $\lambda$. In the plot, we have chosen to hold $\tilde g/ \mu^2 = 1/2$ as an illustration of this effect.
It appears that the axion decay constant plays little to no role in this story.
\section{Relationship of Chromo-Natural Inflation and Gauge-flation}\label{sec:GFvsCNI}
References \cite{Maleknejad:2011jw,Maleknejad:2011sq} propose the model of Gauge-flation,
which is closely related to the one we have written down in Eqn. (\ref{eqn:action}).
However, the authors of \cite{Maleknejad:2011jw,Maleknejad:2011sq} did not include the axion explicitly, but instead included a higher-order interaction
for the gauge fields:
\begin{align}\label{eqn:actionGF}
\mathcal{L_{\rm GF}} = \sqrt{-g} & \bigg[-\frac{R}{2}-\frac{1}{4}F_{\mu\nu}^{a}F_{a}^{\mu\nu} + \frac{\kappa}{384} (F^{a}_{\mu\nu}\tilde F_{a}^{\mu\nu})^2\bigg].
\end{align}
Comparison between the models will immediately reveal that this $(F^{a}_{\mu\nu}\tilde F_{a}^{\mu\nu})^2$ interaction term is precisely what one
obtains upon integrating out the axion in our model. Integrating out a field can only be done when that field can be
assumed to be in the minimum of its effective potential. Indeed, in \cite{SheikhJabbari:2012qf} it is explicitly shown that the Gauge-flation model corresponds to the Chromo-Natural Inflation model with the axion very near the bottom of its potential ($\mathcal{X} \simeq \pi f$). In terms of the parameters of Chromo-Natural Inflation,
\begin{align}
\kappa = 3 \frac{\lambda^2}{\mu^4},
\end{align}
and thus it follows that in the large axion regime ($\mathcal{X} \simeq \pi f$), the theory of Chromo-Natural Inflation reduces to that of Gauge-flation. In order that such a model generates sufficient inflation, the axion-gauge field coupling $\lambda$ must be at least an order of magnitude larger than the minimum values needed in \cite{Adshead:2012kp}. This is hardly surprising, since one now needs to generate 60 efoldings of inflation while the axion is very near its minimum. For the choice of parameters from \cite{Adshead:2012kp} (Eqn.\ (\ref{eqn:params})), only 110 efoldings are generated in total and, as discussed above and illustrated in Fig. \ref{fig:axion}, during the required 60 efoldings the axion rolls more than half of the total distance in field space.
At the classical level, there is no impediment that we see to using this Gauge-flation parameter space to generate an inflating background solution.
However, it is also clear that this is a special case of the general model, Eqn. (\ref{eqn:action}). When the axion is included explicitly, one can describe cases far away from both the minimum, $\mathcal{X}/f \simeq \pi$, and maximum, $\mathcal{X}/f \simeq 0$,
of the axion's potential. In fact, given that one can always integrate out the axion near the bottom of its potential, Gauge-flation type trajectories
can always be matched to solutions of the full Chromo-Natural inflation model.
In the right panel of Fig.\ \ref{fig:CNIvsGF}, we demonstrate that the choice of parameters corresponding to the gaugeflation model of \cite{Maleknejad:2011jw} leads to more than 2500 efoldings of inflation in total in Chromo-Natural Inflation. The last 60 efoldings (shown in red) here lie well within the range where one can safely integrate out the axion to obtain the Gauge-flation model. The insets provide a more detailed view of the region where the theories overlap.
In the left hand panel of Fig.\ \ref{fig:CNIvsGF}, we show the full phase space of the Chromo-Natural Inflation trajectory. We also overplot the Gauge-flation result, which overlaps with the Chromo-Natural inflation result in the final 60 efoldings (shown in red here). We have inset a zoomed in plot of the `reheating' phase to demonstrate the agreement in this region. As one would expect, the curves are virtually indistinguishable.
\section{Conclusions}\label{sec:concl}
In this work, we have described the space of trajectories in Chromo-Natural inflation in detail and demonstrated that sufficient inflation can be generated for a wide range of parameter values. We have shown that the Gauge-flation model of \cite{Maleknejad:2011jw} is subsumed by the more general model of Chromo-Natural inflation; Chromo-Natural inflation reduces to Gauge-flation when the axion is close to the minimum of its potential and can thus be integrated out. Integrating out the axion is an entirely valid way of simplifying the theory when the axion is near an extremum, and is a good way of describing the theory when the axion is near the minimum of its potential. In particular, one only recovers precisely the results of Gauge-flation once the axion nears the minima of its bare potential.
This is not a surprising result. In the Gauge-flation model, the axion is essentially non-dynamical. It simply supplies the vacuum energy on which the universe inflates. While these trajectories are also described by Chromo-Natural Inflation, a much wider range of trajectories are also available where the dynamical behaviour of the axion over a large field range is important for the evolution of the system. Thus Chromo-Natural Inflation has a much larger model space. In particular, Chromo-Natural Inflation has two additional parameters compared to Gauge-flation, indicative of the greater freedom. A complete analysis of the perturbations in the various inflationary regimes \cite{precite} is required to discover if any are observationally viable,
and which observables might be able to discriminate among them.
\acknowledgements
We thank Richard Easther for comments and M.M. Sheikh-Jabbari for very helpful correspondence and for a sharing an early draft of his upcoming article. This work was supported in part by the Kavli Institute for Cosmological Physics at the University of Chicago through grants NSF PHY-0114422 and NSF PHY-0551142 and an endowment from the Kavli Foundation and its founder Fred Kavli. MW was supported by U.S. Dept. of Energy contract DE-FG02-90ER-40560.
|
{
"timestamp": "2012-08-21T02:12:24",
"yymm": "1203",
"arxiv_id": "1203.2264",
"language": "en",
"url": "https://arxiv.org/abs/1203.2264"
}
|
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