\documentstyle [12pt] {article}
\def\baselinestretch{1}
\title{Free energy of a four-dimensional\\ chiral bag
\thanks{\it{This work was partially supported by CONICET (Argentina).}}}
\author{M.\ De Francia, H.\ Falomir, \\and E.\ M.\ Santangelo\\ \\
Departamento de F\'{\i}sica\\Fac.\ de
Ciencias Exactas, U.\ N.\ L.\ P.\ \\c.\ c.\ 67, 1900 La Plata, Argentina.}
\date{June 1991}
\begin{document}
\maketitle
\begin{abstract}
Through the application of mathematical methods developed in a recent paper,
we obtain a closed expression for the chiral correction to the free energy
of a bag, due to the presence of the external pionic field in a hedgehog
configuration. Low and high temperature developments are given and, in
particular, the zero temperature limit is shown to be consistent with previous
evaluations of the Casimir energy. The subject of renormalization is also
discussed.
Pacs: 12.40.Aa, 02.90.+p
\end{abstract}
\pagebreak
\section {Introduction}
It is by now a well-established fact that, when field configurations are
subject to boundary conditions, their quantum fluctuations are modified
in a fundamental way. Starting from the well-known work of Casimir
\cite{Casimir}, there has been ever growing evidence of the relevance of
boundary effects on the vacuum structure of confined systems. One of the
best illustrations of this fact is provided by bag models of hadrons, both
in the early MIT version \cite{MIT} and in the more recent chiral version
\cite{chiral}, which provides the original formulation with the ingredient
of axial current conservation.
In a recent paper \cite{2dim}, a simple two-dimensional chiral bag was studied,
as an application of methods developed by R. Forman \cite{Forman} for the
evaluation of Fredholm determinants of quotients of differential operators
in terms of Fredholm determinants of quotients of pseudodifferential operators.
These last (Forman's) operators turn out to be entirely determined by boundary
values of functions in the kernel of the original differential one, which
is independent of boundary conditions. This approach proved comparatively
simpler than a previous one \cite{journal}, based on Seeley's developments
for complex powers of elliptic boundary systems \cite{Seeley}, which requires
the knowledge of the Green's function.
However, the conditions under which Forman's approach can be applied turn
out to be too restrictive when trying to use it for the more realistic case
of a four dimensional chiral bag. For this reason, a generalization of the
results in reference \cite{Forman} has recently been presented \cite{pdet},
which applies to the
case where these Fredholm determinants do not exist, but a regularized
determinant (p-determinant) based on Hilbert's approach does.
It is the aim of this paper to apply the regularization approach provided
by the results in reference \cite{pdet} to evaluate the one-loop contribution
to the free energy of a chiral bag, due to the presence of a hedgehog
configuration of the external pionic field. Even though this approach requires
the introduction of two different boundary conditions, depending on an ad-hoc
parameter $\mu$, an adequate choice of all these elements will allow us
to get complete information about a unique boundary condition for the bag.
In section 2, we give a brief review of the mathematical tool developed
in reference \cite{pdet}.
In section 3, we describe the model to be studied and discuss the definition
of (the chiral correction to) the renormalized free energy in the framework of
this regularization scheme.
Section 4 is devoted to the explicit evaluation of Forman's operator for the
problem at hand. There it is shown that a judicious choice of a basis in
the kernel of the differential operator, suggested by the symmetries of
the problem, allows to reduce Forman's operator to a block-diagonal form,
which greatly simplifies subsequent calculations.
In section 5, we give closed expressions for the chiral correction to the
free energy at any temperature. Making use of asymptotic expansions for
Bessel functions, we also get developments for its behavior at high and
low temperatures. In particular, for the zero-temperature limit, our results
turn out to be consistent with previous evaluations of the Casimir energy
\cite{Zahed}.
Finally, section 6 contains some comments and conclusions.
\section {Brief review of the mathematical tool}
In this section, we will briefly review the main result in a recent paper
\cite{pdet}. There, it was shown that - under certain conditions - a regularized
determinant of the quotient of elliptic operators defined in a region $\Omega
\subset {\bf R}^{N}$ with boundary $\partial \Omega$ is equal to the determinant
of an operator acting on functions defined on $\partial\Omega$. The last
one can be entirely expressed in terms of boundary values of the solutions
of the original elliptic operator, as will become clear in what follows.
This approach allows one to define the (difference of) renormalized free
energies in terms of such boundary values.
We will be interested in elliptic linear differential
operators $L(\mu)$, depending on a parameter $\mu$ and restricted to functions
defined in a region $\Omega \subset {\bf R}^{n}$ and satisfying elliptic
boundary conditions corresponding to the projection $Bf=0$ at $\partial\Omega$.
We will suppose that the homogeneous problem:
\[ L f(x) = 0 , \ x\in \Omega \]
\begin{equation}
B f(x) = 0 , \ x \in \partial\Omega
\label{e1}
\end{equation}
has no nontrivial solution. The Poisson map for the operator $L$ and the
boundary condition $B$,\ \mbox{$P_{B}(\mu)$},
is defined such that the unique solution of
\[ L f(x) = 0 , \ x \in \Omega \]
\begin{equation}
B f(x) = h(x) , \ x \in \partial\Omega
\label{e2}
\end{equation}
is given by
\begin{equation}
f(x)=\mbox{$P_{B}(\mu)$}h(x).
\label{e3}
\end{equation}
We will denote by \mbox{$L_{B}^{-1}(\mu)$},
the Green's function of $L$, satisfying the boundary
condition $B\mbox{$L_{B}^{-1}(\mu)$}=0$ at $\partial\Omega$.
Let us consider a second elliptic boundary condition, defined by the operator
$A$, and its corresponding Green's function \mbox{$L_{A}^{-1}(\mu)$}\
($A\mbox{$L_{A}^{-1}(\mu)$}=0$ at
$\partial\Omega$).
Following reference \cite{Forman}, we will introduce the operator:
\begin{equation}
\mbox{$\Phi_{AB}(\mu)$} = A \mbox{$P_{B}(\mu)$}.
\label{e4}
\end{equation}
Notice that, even though $P_{B}(\mu)$ is difficult to evaluate (it requires
the knowledge of the Green's function $L_{B}^{-1}(\mu))$,
$\mbox{$\Phi_{AB}(\mu)$}$\ is such
that:
\begin{equation}
A \Psi = \mbox{$\Phi_{AB}(\mu)$} B \Psi
\label{e5}
\end{equation}
for any solution of the homogeneous equation:
\begin{equation}
L \Psi = 0 .
\label{e6}
\end{equation}
So, \mbox{$\Phi_{AB}(\mu)$}\ can - in principle - be obtained from the
boundary values of the functions in a basis of the kernel of $L(\mu)$.
In reference \cite{pdet} it has been shown that:
\begin{equation}
\mbox{$det_{p}$} \left( \mbox{$L_{B}^{-1}(\mu)$}\mbox{$L_{A}(\mu)$}
\mbox{$L_{A}^{-1}(0)$}\mbox{$L_{B}(0)$}\right) = \mbox{$det_{p}$}
\left( \mbox{$\Phi_{AB}^{-1}(\mu=0)$}\mbox{$\Phi_{AB}(\mu)$} \right) ,
\label{e7}
\end{equation}
as long these p-determinants, to be defined in the following, exist.
If $M$ is an operator such that $M^{p}$ ($p \geq 1$) is trace class, the
\linebreak p-determinant of $(1-M)$ is defined through \cite{int}:
\begin{equation}
\ln \mbox{$det_{p}$} ( 1 - M ) = - \int_{0}^{1} dz z^{p - 1}
Tr \left\{ M^{p} \left( 1 - z M \right)^{-1} \right\} ,
\label{e8}
\end{equation}
Notice that, if $M$ itself is trace class, this definition reduces to:
\begin{equation}
\ln \mbox{$det_{p}$} ( 1-M ) =
\ln det(1-M) + \sum_{n=1}^{p-1} {{Tr (M^{n})}\over{n}}
\label{e9}
\end{equation}
So, one can see that this regularization amounts to the subtraction of
the first (p-1) (divergent) traces in the development of $\ln det(1-M)$.
Let us consider for a while that the (p-1)-determinants of the operators in
equation (\ref{e7}) also exist. Then, from the following property of
p-determinants\cite{Simon}:
\begin{equation}
\mbox{$det_{p}$} \left( 1 - M \right) =
\exp\left({1 \over p-1} Tr M^{p-1}\right) det_{p-1}( 1 - M ),
\label{e10}
\end{equation}
and equation (\ref{e7}), we conclude that
\[ Tr \left\{ \left( 1 - \mbox{$L_{B}^{-1}(\mu)$}\mbox{$L_{A}(\mu)$}
\mbox{$L_{A}^{-1}(0)$}\mbox{$L_{B}(0)$}\right)^{p-1} \right\} =\]
\begin{equation}
Tr\left\{ \left( 1 - \mbox{$\Phi_{AB}^{-1}(\mu=0)$}\mbox{$\Phi_{AB}(\mu)$}
\right)^{p-1} \right\}
\label{e11}
\end{equation}
When the (p-1)-determinant does not exist, equation (\ref{e11}) can be
considered to be a formal identity, which can be extended to all integer
powers from 1 to (p-1).
In such a way, the (divergent) subtractions in the present
regularization scheme (and, therefore, the finite counterterms required) can be
expressed entirely in terms of the operator \mbox{$\Phi_{AB}(\mu)$}. \bigskip
We will be interested in a situation in which boundary conditions, rather than
operators, depend on $\mu$ through a regular transformation ${\cal U}(\mu)$
such that the relevant boundary values of a solution of $L$, $f$, are given
by $A{\cal U}^{-1}(\mu)f$ ($B{\cal U}^{-1}(\mu)f)$. Again, we will suppose
that the boundary value problem is nonsingular.
We can translate the $\mu$-dependence to the differential operator by noticing
that:
\begin{equation}
\left(
\begin{array}{c}
L \Psi = 0 \\
A \mbox{${\cal U}^{-1}$} \Psi = h
\end{array}
\right)
\Rightarrow
\left(
\begin{array}{c}
\mbox{${\cal U}^{-1}$} L \mbox{${\cal U}$} \chi = 0 \\
A \chi = h
\end{array}
\right), for\ \chi = {\cal U}^{-1} \Psi,
\label{e12}
\end{equation}
and, therefore, apply our previous result to
$\mbox{$L_{A}(\mu)$}\equiv \left( {\cal U}^{-1}L{\cal U} \right)_{A}$.
Since
\begin{equation}
\left( {\cal U}^{-1} L {\cal U}\right)^{-1}_{A} = {\cal U}^{-1}
\mbox{$L_{A{\cal U}^{-1}}^{-1}$} {\cal
U} ,
\label{e13}
\end{equation}
equation (\ref{e7}) implies:
\[ \mbox{$det_{p}$} \left\{ \left( \mbox{${\cal U}^{-1}$}
L \mbox{${\cal U}$}
\right)_{B}^{-1}
\left( \mbox{${\cal U}^{-1}$} L \mbox{${\cal U}$}\right)_{A}
\mbox{$L_{A}^{-1}$}\mbox{$L_{B}$}\right\} = \]
\[ \mbox{$det_{p}$} \left\{ \mbox{${\cal U}^{-1}$}
\mbox{$L_{B{\cal U}^{-1}}^{-1}$}
\mbox{$L_{A{\cal U}^{-1}}$} \mbox{${\cal U}$}
\mbox{$L_{A}^{-1}$}\mbox{$L_{B}$} \right\}
= \]
\begin{equation}
\mbox{$det_{p}$} \left\{ \mbox{$\Phi_{AB}^{-1}(\mu=0)$}
\mbox{$\Phi_{AB}(\mu)$} \right\} ,
\label{e14}
\end{equation}
where
\begin{equation}
\mbox{$\Phi_{AB}(\mu)$} B \mbox{${\cal U}^{-1}$} \Psi = A
\mbox{${\cal U}^{-1}$} \Psi ,
\label{e15}
\end{equation}
for any solution of $L\Psi=0$.
Finally, we wish to stress that, the boundary-value problem of equation
(\ref{e2}) being nonsingular, the projected values of a complete system
of functions in the kernel of $L$ constitute a complete system in the space
of boundary values.
We will use this fact to explicitly evaluate \mbox{$\Phi_{AB}(\mu)$}\ in
following sections.
\section { Definition of the free energy of a chiral bag}
In what follows, we will consider the free energy for a four-dimensional static
chiral bag at temperature $T = 1/\beta$. This amounts to consider a theory of
free massless fermions confined to a spherical cavity of fixed radius $R$ and
interacting at the boundary with a hedgehog configuration of an external pionic
field.
The corresponding partition function is given by:
\begin{equation}
Z= \int {\cal D} \bar{\psi} {\cal D} \psi e^{-S_{E}},
\label{e16}
\end{equation}
where the functional integral is performed over functions satisfying
antiperiodic boundary conditions in the "time" direction, and the Euclidean
action is:
\begin{equation}
S_{E} = \int_{\Omega} d^{4}x \bar{\psi} \mbox{$i\not\!\partial$} \psi -
{{1}\over{2}}
\int_{r = R} d^{3}x \bar{\psi} (e^{-i \theta \vec{\tau}.{\bf n} \gamma_{5}}
+ i \not\!n ) \psi ,
\label{e17}
\end{equation}
where $\Omega$ is the region corresponding to $r \leq R$, and $t \in [0,\beta]$.
Here, ${\bf n}$ is the outward normal to the bag surface and $\tau^{i}\
(i=1,2,3)$ are the Pauli matrices\footnote[1]{Our convention for gamma matrices
is:
\[
\mbox{$\gamma_{0}$} = i \left(
\begin{array}{rr}
1 & 0 \\ 0 & -1
\end{array} \right);
\vec{\gamma} = \left(
\begin{array}{rr}
0 & \vec{\sigma} \\ -\vec{\sigma} & 0
\end{array} \right) ;
\mbox{$\gamma_{5}$} = \left(
\begin{array}{rr}
0 & 1 \\ 1 & 0
\end{array} \right),
\]
where $\sigma^{i}$ are the Pauli matrices.}.
This means that
\begin{equation}
Z= e^{-\beta F} \sim Det (\mbox{$i\not\!\partial$} )_{A}
\label{e18}
\end{equation}
where $(\mbox{$i\not\!\partial$})_{A}$ acts as the differential Dirac
operator \mbox{$i\not\!\partial$}\ on functions
satisfying the chiral boundary condition $A$
\begin{equation}
A \psi = {{1}\over{2}} ( 1 + i\not\!n e^{-i\theta \vec{\tau}.{\bf n}
\gamma_{5}}) \psi = 0 \ at \ r = R,
\label{e19}
\end{equation}
and being antiperiodic in the temperature interval $[0,\beta]$.
Equation (\ref{e18}) is just a formal relation, since the functional determinant
must be defined through a suitable regularization. To do so, our aim is to
employ the scheme developed in reference \cite{pdet} and reviewed in the
previous section.
In order to apply equation (\ref{e14}), we introduce a second boundary operator:
\begin{equation}
B= {{1}\over{2}}(1 + i\not\!n) = A (\theta = 0)
\label{e20}
\end{equation}
and a regular transformation which, at the boundary, takes the value:
\begin{equation}
\mbox{${\cal U}$} (\mu) = e^{-i \mu \vec{\tau}.{\bf n} \gamma_{5}}.
\label{e21}
\end{equation}
Thus,
\begin{equation}
\mbox{$det_{p}$} \left( \mbox{${\cal U}^{-1}$}
(\mbox{$i\not\!\partial$})^{-1}_{B{\cal U}^{-1}}
(\mbox{$i\not\!\partial$})_{A{\cal U}^{-1}}
\mbox{${\cal U}$}
(\mbox{$i\not\!\partial$})^{-1}_{A} (\mbox{$i\not\!\partial$})_{B}\right)
= \mbox{$det_{p}$}
\left(\mbox{$\Phi_{AB}^{-1}(\mu=0)$} \mbox{$\Phi_{AB}(\mu)$} \right),
\label{e22}
\end{equation}
where \mbox{$\Phi_{AB}(\mu)$}\ satisfies equation (\ref{e15}) for any
solution of:
\begin{equation}
(\mbox{$i\not\!\partial$}) \psi = 0.
\label{e23}
\end{equation}
Notice that
\begin{equation}
A(\theta) \mbox{${\cal U}^{-1}$}
(\mu) \psi = 0 \Rightarrow A(\theta - 2\mu)\psi =0.
\label{e24}
\end{equation}
Then, the spectrum of the Dirac Hamiltonian for this modified boundary condition
will depend on $(\theta - 2\mu)$ and the same will be true for the corresponding
free energy $F=F(\theta-2\mu)$. Consequently, we will adopt the following
definition:
\[
\beta \left\{ F(-2\mu) - F(\theta-2\mu) + F(\theta) - F(0) \right\}
\equiv \ln \mbox{$det_{p}$}
\left(\mbox{$\Phi_{AB}^{-1}(\mu=0)$} \mbox{$\Phi_{AB}(\mu)$}
\right) -
\]
\begin{equation}
\left[ \sum_{q=1}^{p-1} {{1}\over{q}}\ Tr\left\{[1-
\mbox{$\Phi_{AB}^{-1}(\mu=0)$}\mbox{$\Phi_{AB}(\mu)$}]^{q}\right\}
\right]_{renormalized}
\label{e25}
\end{equation}
where the least integer p for which the p-determinant exists will be determined
later in this paper. In this expression we have used the formal identity
established in equation (\ref{e11}).
\bigskip
Now, the parameter $\mu$ is arbitrary and can be chosen so that one gets
the chiral correction to the free energy. Indeed, for $\mu=\theta/2$ and
taking into account that $F(-\theta) = F(\theta)$ (since multiplication
by \mbox{$\gamma_{0}$}\mbox{$\gamma_{5}$}\ changes $\theta$ into
$-\theta$ in the boundary condition and
$E_{n}$ into $-E_{n}$ in the Dirac equation, which does not affect the
fermionic partition function), one obtains:
\[
2\beta [F(\theta)-F(0)] = ln \mbox{$det_{p}$}
\left(\mbox{$\Phi_{AB}^{-1}(\mu=0)$}
\mbox{$\Phi_{AB}(\theta /2)$}\right) - \]
\begin{equation}
\left[ \sum_{q=1}^{p-1} {{1}\over{q}} Tr\left\{\left(
1- \mbox{$\Phi_{AB}^{-1}(\mu=0)$}\mbox{$\Phi_{AB}(\theta /2)$}\right)^{q}
\right\}
\right]_{renormalized}
\label{e26}
\end{equation}
In the next section we will explicitly evaluate the operator
\mbox{$\Phi_{AB}(\mu)$}\ and in section 5 an expression for the chiral free
energy will be given.
\section{ The operator \mbox{$\Phi_{AB}(\mu)$} }
As remarked in Section 2, \mbox{$\Phi_{AB}(\mu)$}\ can be obtained from the
boundary values of a complete system of functions in the kernel of
\mbox{$i\not\!\partial$}. This task is made easier by first studying the
symmetries of the homogeneous Dirac equation and boundary operators
\cite{Mulders,Jezabek}.
While the homogeneous differential equation (\ref{e23}) is invariant under
\linebreak
$SU(2)_{rot.} \otimes SU(2)_{isospin}$, the boundary operator $A(\theta)
\mbox{${\cal U}^{-1}$}
(\mu)$ is invariant only under the diagonal $SU(2)$ subgroup corresponding
to simultaneous rotation in space and isospace. So, following reference
\cite{Jezabek}, we introduce ${\bf K} = {\bf J} + {\bf I}$, where ${\bf J} =
{\bf L} + {\bf S}$ and ${\bf I} = \vec{\tau}/2$.
Moreover, \mbox{$i\not\!\partial$}\ and $A(\theta)\mbox{${\cal U}$}
(\mu)$ also commute with
the parity operator $i\mbox{$\gamma_{0}$} P$ (where $P$ is spatial parity), so
we can classify the solutions of equation (\ref{e23}) through the eigenvalues
of $K^{2}$, $K_{3}$, $J^{2}$ and parity. The corresponding eigenvectors are the
following spin-isospin spherical harmonics:
\begin{equation}
\begin{array}{l}
|1> = | k,j=k+1/2,l=k,m> \\
|2> = | k,j=k-1/2,l=k,m> \\
|3> = | k,j=k+1/2,l=k+1,m> \\
|4> = | k,j=k-1/2,l=k-1,m> ,
\end{array}
\label{e27}
\end{equation}
where the parity of each state is given by $(-)^{l}$. Now, factoring out
the "temporal" dependence and solving the radial equation, it can be easily
seen that the solutions are expressed in terms of modified Bessel functions.
Throughout our calculation, we will only need their boundary values
(conveniently normalized).
For $k > 0$, they can be written as the Dirac spinors \cite{Jezabek}:
\begin{eqnarray}
\Psi^{n,k,m}_{1} (r=R) =
\left(
\begin{array}{c}
|1> \\ \\+ i S_{n} W_{k,n}^{+} |3> \space
\end{array} \right) e^{i \omega_{n} t}, \nonumber \\
\Psi^{n,k,m}_{2} (r=R) =
\left(
\begin{array}{c}
|2> \\ \\-i S_{n} W_{k,n}^{-} |4>
\end{array} \right) e^{i \omega_{n} t}, \nonumber \\
\Psi^{n,k,m}_{3} (r=R) =
\left(
\begin{array}{c}
-i S_{n} W_{k,n}^{+} |3> \\ \\-|1>
\end{array} \right) e^{i \omega_{n} t}, \nonumber \\
\Psi^{n,k,m}_{4} (r=R) =
\left(
\begin{array}{c}
+i S_{n} W_{k,n}^{-} |4>\space \\ \\-|2>
\end{array} \right) e^{i \omega_{n} t},
\label{e28}
\end{eqnarray}
for $n,k \in {\bf Z},\ k \geq 1$ and $ -k \leq m \leq k $.
In equations (\ref{e28}) $\omega_{n} = (2n+1)\pi /\beta$,
$S_{n}= sgn (n+1/2)$, and
\begin{equation}
W_{k,n}^{\pm} = \mbox{${{\mbox{$I_{k \pm 1 +1/2}(|\omega_{n}|R)$}}\over
{\mbox{$I_{k+1/2}(|\omega_{n}|R)$}}}$} \ .
\label{e29}
\end{equation}
In the $k=0$ case, only the solutions $\Psi_{1}^{n,0}$ and $\Psi_{3}^{n,0}$
survive.\bigskip
For this choice of basis in the kernel of \mbox{$i\not\!\partial$}, the
boundary operators $A(\theta)\mbox{${\cal U}^{-1}$}
(\mu)$ and $B \mbox{${\cal U}^{-1}$} (\mu)$ leave the subspaces
characterized by given $k$,$m$,\linebreak $(-)^{l}$ invariant. Thus, the
operator \mbox{$\Phi_{AB}(\mu)$}\ turns out to be block diagonal and it can be
evaluated in each invariant two-dimensional subspace (their directions
corresponding to the values of $J = k \pm 1/2$).
To obtain the projected boundary values, we need the explicit form of the
operator $A(\theta)$, which is given by\footnote[2]{Referred to the basis
of the k,m subspace in equation (\ref{e27}):
\[
(i\vec{\sigma}.{\bf n})=
\left(
\begin{array}{rrrr}
0 & 0 & 1 & 0 \\
0 & 0 & 0 & -1 \\
-1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0
\end{array}
\right) ,\]
\[
(i\vec{\tau}.{\bf n}) = {{1}\over{2k+1}}
\left(
\begin{array}{cccc}
0 & 0 & 1 & -2\sqrt{k(k+1)} \\
0 & 0 & 2\sqrt{k(k+1)} & 1 \\
-1 & -2\sqrt{k(k+1)} & 0 & 0 \\
2\sqrt{k(k+1)} & -1 & 0 & 0
\end{array}
\right)
\label{e30}
\]
(see reference \cite{Jezabek})}:
\begin{equation}
A(\theta) =
\left(
\begin{array}{c}
{\cal A}(\theta) \\
{{i(\vec{\sigma}.{\bf n})}\over{\cos\theta}}
[1+i(\vec{\sigma}.{\bf n})i(\vec{\tau}.{\bf n}) \sin \theta ] {\cal
A}(\theta)
\end{array}
\right),
\label{e31}
\end{equation}
where
\begin{equation}
{\cal A}(\theta) = {{1}\over{2}}
\left[1+i(\vec{\sigma}.{\bf n})i(\vec{\tau}.{\bf n}) \sin \theta ;
i(\vec{\sigma}.{\bf n}) \cos \theta\right] .
\label{e32}
\end{equation}
In order to discard redundant information, we will only retain the upper
spin-isospin component in the projected value. So, we introduce the matrices:
\begin{equation}
H^{I}_{n,k,m} (\mu,\theta) = {\cal A}(\theta) \mbox{${\cal U}^{-1}$} (\mu)
(\Psi_{1},\Psi_{2})^{n,k,m}, \label{e33}
\end{equation}
\begin{equation}
H^{II}_{n,k,m} (\mu,\theta) = {\cal A}(\theta) \mbox{${\cal U}^{-1}$} (\mu)
(\Psi_{3},\Psi_{4})^{n,k,m}, \label{e34}
\end{equation}
where $I$ stands for $(-)^{l} = (-)^{k}$, and $II$ for $(-)^{l} = (-)^{k+1}$.
It is straightforward to obtain their matrix elements from equations
(\ref{e28}), (\ref{e33}) and (\ref{e34}).They are explicitly given in
Appendix 1.
Following equation (\ref{e15}), and taking into account equation (\ref{e20}),
we see the operator \mbox{$\Phi_{AB}(\mu)$}\ is such that:
\begin{equation}
H^{(i)}_{n,k,m} (\mu,\theta) = \mbox{$\Phi_{AB}(\mu)$}
H^{(i)}_{n,k,m} (\mu,\theta =0),\ i=I,II .
\label{e35}
\end{equation}
Therefore, in the $n,k,(-)^{l}$ subspace:
\begin{equation}
[ \mbox{$\Phi_{AB}(\mu)$} ]^{(i)}_{n,k,m} = [ H (\mu,\theta) H^{-1}
(\mu,\theta=0)]^{(i)}_{n,k,m},
\label{e36}
\end{equation}
with $n,k \in {\bf Z},\ k\geq 1, -k\leq m \leq k$, and $i = I,II$.
\bigskip
For $k=0$, the invariant subspaces are one-dimensional and $H^{(i)}_{n,k=0}$
are similarly obtained from $\Psi_{1}^{n,0}$ and $\Psi_{3}^{n,0}$:
\begin{equation}
H^{I}_{n,k=0} (\mu,\theta) = {\cal A} (\theta) \mbox{${\cal U}^{-1}$}
(\mu) \Psi_{1}^{n,0}
\label{e37}
\end{equation}
and
\begin{equation}
H^{II}_{n,k=0} (\mu,\theta) = {\cal A} (\theta) \mbox{${\cal U}^{-1}$}
(\mu) \Psi_{3}^{n,0},
\label{e38}
\end{equation}
for $n \in {\bf Z}$. Their expressions are also given in Appendix 1.
\section { Evaluation of the chiral correction to the free energy}
In section 3, we have proposed a definition for the chiral correction to the
free energy in terms of the p-determinant of [\mbox{$\Phi_{AB}^{-1}(\mu=0)$}
\mbox{$\Phi_{AB}(\theta /2)$}]. Moreover, in the previous section, we have
shown that this operator is block-diagonal. So, its p-determinant reduces to:
\[ \ln \mbox{$det_{p}$}
\left[\mbox{$\Phi_{AB}^{-1}(\mu=0)$}\mbox{$\Phi_{AB}(\theta /2)$}
\right] = \]
\begin{equation}
\sum_{n=-\infty}^{\infty} \sum_{k=0}^{\infty}
\sum_{m=-k}^{m=k} \sum_{i=I,II} \ln \mbox{$det_{p}$}
[\mbox{$\Phi_{AB}^{-1}(\mu=0)$}
\mbox{$\Phi_{AB}(\theta /2)$}]^{(i)}_{n,k,m}\ ,
\label{e39}
\end{equation}
where each term in the series is given by equation (\ref{e9}).
The least p for which the right-hand side in equation(\ref{e39}) exists can be
determined by studying the behavior of
$\left[1-\mbox{$\Phi_{AB}^{-1}(\mu=0)$}\mbox{$\Phi_{AB}(\theta
/2)$}\right]^{(i)}_ {n,k,m}$, for large $n$ and/or $k$.
In what follows, we will - for convenience - study in a separate way the
$k>0$ and $k=0$ cases.
\subsection{ $k \geq 1$ contribution}
In this subspace one obtains, from equation (\ref{e36}) and the results quoted
in Appendix 1:
\[ \left[1-\mbox{$\Phi_{AB}^{-1}(\mu=0)$}\mbox{$\Phi_{AB}(\theta /2)$}
\right]^{(I)}_{n,k,m} = {{1}\over{d}}
\left( \begin{array}{cc} a+b & c-e \\ c+e & a-b \end{array} \right) \]
\begin{equation}
\left[1-\mbox{$\Phi_{AB}^{-1}(\mu=0)$}\mbox{$\Phi_{AB}(\theta /2)$}
\right]^{(II)}_{-n-1,k,m} = {{1}\over{d}} \left(
\begin{array}{cc} a+b & -c+e \\ -c-e & a-b \end{array} \right),
\label{e40} \end{equation}
where:
\[ a = i S_{n} X_{k,n} (1-\cos \theta) + [X_{k,n}^{2} - Y_{k,n}^{2}]
\cos \theta (1-\cos \theta) \]
\[ b= -i S_{n} Y_{k,n} Z_{k,n} (1-\cos \theta)\]
\[ c= -2i \sqrt{k(k+1)} S_{n} Y_{k,n} Z_{k,n} (1-\cos \theta ) \]
\[ d= 1-2i S_{n} X_{k,n} \cos \theta - X_{k,n}^{2} \cos^{2} \theta -
Y_{k,n}^{2} \sin^{2} \theta \]
\begin{equation}
e= -2 \sqrt{k(k+1)} Y_{k,n}^{2} (1-\cos \theta).
\label{e41} \end{equation}
Here we have defined:
\[
X_{k,n} = {{1-W^{+}_{k,n} W^{-}_{k,n}}\over{W_{k,n}^{+} + W_{k,n}^{-}}} =
\]
\begin{equation}
{{1}\over{2}} {{z_{n}}\over{\rho^{2}}} + {{1}\over{2}} {{z_{n}}\over{\rho^{3}}}
\left( {{z_{n}^{2}-2\nu^{2}}\over{2 \rho^{2}}}\right) + {{1}\over{2}}
{{z_{n}}\over{\rho^{4}}} \left( {{(z_{n}^{2}-2\nu^{2})^{2} - 2 \nu^{2}
\rho^{2}}\over{2 \rho^{4}}}\right) + O(1/\rho^{4})
\label{e42}
\end{equation}
\[ Y_{k,n} = {{W_{k,n}^{+} - W_{k,n}^{-}}\over{(2k+1)(W_{k,n}^{+} +
W_{k,n}^{-})}} = \]
\begin{equation}
-{{1}\over{2}} {{1}\over{\rho}} - {{1}\over{4}} {{z_{n}^{2}}\over{\rho^{4}}} +
{{1}\over{4}} {{z_{n}^{2}}\over{\rho^{5}}} \left( -{{1}\over{4}}+ {{5}\over{4}}
{{\nu^{2}}\over{\rho^{2}}}\right) - {{1}\over{8}} {{z_{n}^{4}}\over{\rho^{7}}}
+ O(1/\rho^{4}), \label{e43} \end{equation}
\[
Z_{k,n} = {{1+W^{+}_{k,n} W^{-}_{k,n}}\over{(2k+1)(W_{k,n}^{+} + W_{k,n}^{-})}}
=\]
\begin{equation}
{{z_{n}}\over{\nu \rho}} - {{1}\over{2}}{{z_{n}}\over{\nu \rho^{2}}} +
{{1}\over{2}} {{z_{n}^{3}}\over{\nu \rho^{4}}} + O(1/\rho^{3})
\label{43b}
\end{equation}
where we have used the Debye expansion \cite{Zahed,abramo} of $I_{\nu} (z_{n})
$, for $\nu = k + 1/2$,\linebreak and $z_{n} = |\omega_{n}|R$, and called
$\rho=\sqrt{\nu^{2} + z_{n}^{2}}$ .
It can be easily seen that the matrices in equation (\ref{e40}) are $O(1/\rho)$.
Thus, at least three traces must be subtracted in equation (\ref{e9}). So,
from now on, we will take $p=4$, thus getting for each term in the right
hand side of equation (\ref{e39}):
\[
\ln det_{4} [\mbox{$\Phi_{AB}^{-1}(\mu=0)$}\mbox{$\Phi_{AB}(\theta /2)$}
]^{(i)}_{n,k,m} =
\]
\[
\ln det [\mbox{$\Phi_{AB}^{-1}(\mu=0)$}\mbox{$\Phi_{AB}(\theta /2)$}
]^{(i)}_{n,k,m} + \]
\begin{equation}
\sum_{q=1}^{3} {{1}\over{q}} Tr [(1-\mbox{$\Phi_{AB}^{-1}(\mu=0)$}
\mbox{$\Phi_{AB}(\theta /2)$})^{q}]^{(i)}_{n,k,m},
\label{e45}
\end{equation}
where
\[
\ln det [\mbox{$\Phi_{AB}^{-1}(\mu=0)$}\mbox{$\Phi_{AB}(\theta /2)$}
]^{I}_{n,k,m} = \ln det [\mbox{$\Phi_{AB}^{-1}(\mu=0)$}
\mbox{$\Phi_{AB}(\theta /2)$}]^{II}_{-n-1,k,m} \]
\begin{equation}
= {{(1 - i S_{n} X_{k,n})^{2}}\over{(1-iS_{n} \cos \theta X_{k,n})^{2} -
Y_{k,n} \sin^{2} \theta}} ,
\label{e46}
\end{equation}
and
\[
\sum_{q=1}^{3} {{1}\over{q}}\ Tr \left\{\left(
\left[1-\mbox{$\Phi_{AB}^{-1}(\mu=0)$}\mbox{$\Phi_{AB}(\theta /2)$}
\right]_{n,k,m}^{(I)}\right)^{q}\right\} = \]
\[
\sum_{q=1}^{3} {{1}\over{q}}
Tr\left\{ \left(\left[1-\mbox{$\Phi_{AB}^{-1}(\mu=0)$}
\mbox{$\Phi_{AB}(\theta /2)$}\right]_{-n-1,k,m}^{(II)}\right)^{q}\right\}
= \]
\[
2iS_{n} (X_{k,n} - {{1}\over{3}} X_{k,n}^{3}) (1- \cos \theta) + \]
\[ -2iS_{n} (X_{k,n}Y_{k,n}^{2} + {{1}\over{3}} X_{k,n}^{3}) \cos\theta
\sin^{2}\theta + \]
\begin{equation}
-(X_{k,n}^{2} + Y_{k,n}^{2}) \sin^{2} \theta + O (1/\rho^{4}).
\label{e47}
\end{equation}
Notice that the terms up to order $O(1/\rho^{3})$ in the Debye expansion of the
right hand side of equation (\ref{e47}) exactly cancel the bad behavior of $\ln
det [\mbox{$\Phi_{AB}^{-1}(\mu=0)$}\mbox{$\Phi_{AB}(\theta /2)$}]$ in equation
(\ref{e46}). The rest of the terms contribute to equation (\ref{e39}) with an
absolutely convergent series and will cancell against their contribution to the
renormalized traces in equation (\ref{e26}); so, they can be ignored from now
on\footnote[3]{At this point, we should stress that in the present case - where
we were able to reduce \mbox{$\Phi_{AB}(\mu)$}\ to a block-diagonal form - we
could equally well have studied the pieces leading to divergencies in equation
(\ref{e46}), avoiding the explicit construction of the traces in equation
(\ref{e47}). However, the approach presented in section 2 is valid in the
general case.}.
Consequently, we can write for the $k > 0$ contribution to the chiral
correction of the free energy
\[
\left[F(\theta) - F(0)\right]_{k>0} = \]
\[
= {{1}\over{\beta}} \sum_{n=-\infty}^{\infty}
\sum_{k=1}^{\infty} \nu \left\{\ln \left|
{{(1-iS_{n} X_{k,n})^{2}}\over{(1-iS_{n}
X_{k,n} \cos \theta )^{2} - Y_{k,n}^{2} \sin^{2} \theta }} \right|^{2}
\right. \]
\[ \left.
- {{1}\over{2}}\left[ {{2 z^{2} + \nu^{2}}\over{\rho^{4}}} + {{z^{2}}\over{\rho
^{5}}}
{{2z^{2} - \nu^{2}}\over{\rho^{2}}}\right] \sin^{2} \theta \right\}
+ K \sin^{2} \theta \]
\begin{equation}
+ \sin^{2} \theta\ F.P._{s=0} \left\{{1\over\beta}
\sum_{k=1}^{\infty}
\sum_{n=-\infty}^{\infty}\nu \rho^{-s}{{1}\over{2}} \left[{{2 z^{2} +
\nu^{2}}\over{\rho^{4}}} + {{z^{2}}\over{\rho^{5}}}
{{2z^{2} - \nu^{2}}\over{\rho^{2}}}\right] \right\} ,
\label{e48}
\end{equation}
where we have first summed over the two parities, taken into account
that the terms do not depend on m and introduced a finite counterterm
proportional to $\sin^{2} \theta$, with an undetermined constant
K.\footnote[4]{Expression (\ref{e47}) suggests that, in order to compensate
the arbitrariness in the choice of this regularization scheme, we should also
add finite counterterms proportional to $(1-\cos \theta)$ and $\cos \theta
\sin^{2}\theta$. However, since a discrete chiral \mbox{$\gamma_{5}$}\
transformation \cite{Zahed} leaves the Dirac spectrum invariant, while it turns
$\theta$ into $(\theta + \pi)$ (i.e., $F(\theta + \pi) = F(\theta)$), these
two posibilities are ruled out.}
In order to define the renormalized piece, we have chosen an analytic
regularization in the last term of equation (\ref{e48}) by introducing a factor
$\rho^{-s}$, and taken its finite part at $s=0$. It can be verified that the
residue at $s=0$\ is temperature-independent, so that only temperature
independent divergencies must be removed from the free energy. This can be seen
in the following expansions at low and high temperatures, which are obtained
by means of the Euler-Mac Laurin formula \cite{abramo} \footnote[5]{Notice
that, in the high temperature development, the Euler-Mac Laurin formula was
used to evaluate the sum over $k$, which was the first to performed. The order
of the infinite sums can be freely interchanged for $Re(s)$ large enough, where
the double series is absolutely convergent.}:
\[ C(T)= F.P._{s=0} \left\{
{1\over{2\beta}} \sum_{k=1}^{\infty} \sum_{n=-\infty}^{\infty} \nu \rho^{-s}
\left[ {{2 z^{2} + \nu^{2}}\over{\rho^{4}}} + {{z^{2}}\over{\rho^{5}}} {{2z^{2}
- \nu^{2}}\over{\rho^{2}}} \right] \right\} = \]
\begin{equation}
{{1}\over{4\pi R}}\
F.P._{s=0}\left[ {8\over{15\ s}} - 5.313 + O\left(\left({{2\pi R}\over
\beta}\right)^{5}\right)\right], \ \ for\ {{\beta}\over{2\pi R}} >> 1,
\label{e49}
\end{equation}
\[ = {{1}\over{4\pi R}}\ F.P._{s=0}\left\{ {8\over{15\ s}}+ 1.012 + 2\ln
2\ \left({{2\pi R}\over{\beta}}\right)+ \right.\]
\begin{equation}
\left. {8\over
{15}}\ln\left({\beta \over{2\pi R}}\right)+ O\left({\beta\over{2\pi
R}}\right)\right\}, \ \ for\ {{\beta}\over{2\pi R}} << 1 .
\label{e50} \end{equation}
The first term in the right hand side of equation (\ref{e48}) can be
approximated by means of the Debye asymptotic expansion in equations
(\ref{e42}) and (\ref{e43}). Up to the first nonvanishing order in this
expansion, we get for the $k \geq 1$\ contribution:
\[
[ F(\theta) - F(0) ]_{k>0} = A(T) \sin^{2} \theta + B(T) \sin^{4} \theta
+ \]
\begin{equation}
C(T) \sin^{2} \theta + K \sin^{2} \theta ,
\label{51}
\end{equation}
where:
\begin{equation}
A(T) = {{1}\over{\beta}} \sum_{k=1}^{\infty} \nu \sum_{n=-\infty}^{\infty}
{{5}\over{8}} \left[ {{z^{2}}\over{\rho^{6}}} - 5 {{\nu^{2}
z^{4}}\over{\rho^{10}}} \right]
\label{e52}
\end{equation}
\begin{equation}
B(T)= {{1}\over{\beta}} \sum_{k=1}^{\infty} \nu \sum_{n=-\infty}^{\infty}
{{1}\over{16}} \left[ {{8z^{4} + 8 \nu^{2} z^{2} + \nu^{4}}\over {\rho^{8}}}
\right] .
\label{e53}
\end{equation}
As regards the coefficients $A(T)$ and $B(T)$, the sum
over $n$ can be exactly evaluated and, as shown in Appendix 2, they are
given by:
\[
- 4\pi R\ A(T) = {{5}\over{8}} \pi \beta^{3} \left(
{{1}\over{\beta}} {{d}\over{d\beta}}\right) S(\beta ) + {{15}\over{16}} \pi
\beta^{5} \left({{1}\over{\beta}} {{d}\over{d\beta }} \right)^{2} S(\beta) +
\]
\begin{equation}
{{25}\over{96}} \pi \beta^{7} \left({{1}\over{\beta}}
{{d}\over{d\beta}}\right)^{3} S(\beta) + {{25}\over{1536}} \pi \beta^{9} \left(
{{1}\over{\beta}} {{d}\over{d\beta}}\right)^{4} S(\beta)
\label{e54}
\end{equation}
\[
- 4 \pi R\ B(T) = {{\pi}\over{2}} \beta^{3} \left(
{{1}\over{\beta}} {{d}\over{d\beta}}\right) S(\beta) + {{\pi}\over{8}}
\beta^{5} \left({{1}\over{\beta}} {{d}\over{d\beta}}\right)^{2} S(\beta) +
\]
\begin{equation}
{{\pi}\over{384}} \beta^{7} \left( {{1}\over{\beta}}
{{d}\over{d\beta}} \right)^{3} S(\beta) ,
\label{e55}
\end{equation}
with
\begin{equation}
S(\beta) = {{1}\over{\beta}} \sum_{k=1}^{\infty} \nu^{-2} \tanh \left({{\beta
\nu }\over{2 R}}\right).
\label{e56}
\end{equation}
Their high and low temperature limits, also derived in Appendix 2, are:
\[
A(T) =\left[ {{1}\over{4\pi R}} 0.029 + O(e^{-\beta /2 R})\right],
\ \ for\ {{\beta}\over{2\pi R}} >> 1,
\]
\begin{equation}
= \left[ {{1}\over{4\pi R}} 0.514 \left({{\beta }\over{2\pi R}} \right)+
O\left( \left( {\beta \over {2\pi R}} \right)^{3} \right) \right], \ \ for\
{{\beta }\over{2\pi R}} << 1.
\label{e57}
\end{equation}
\[
B(T) =\left[ {{1}\over{4\pi R}} 0.482 + O(e^{-\beta /2 R})\right],
\ \ for\ {{\beta}\over{2\pi R}} >> 1,
\]
\begin{equation}
= \left[ {{1}\over{4\pi R}} 2,673 \left( {{\beta}\over{2\pi R}} \right) +
O\left(\left({\beta\over {2\pi R}}\right)^{3}\right)\right],
\ \ for\ {{\beta}\over{2\pi R}} << 1 .
\label{e58}
\end{equation}
In particular, when $\beta \rightarrow \infty$ we obtain for the chiral
correction to the Casimir energy of the bag, up to this order in the Debye
expansion:
\[
\left[ E(\theta ) - E(0)\right]_{k>0} = \lim_{\beta \rightarrow \infty}
\left[F(\theta) - F(0) \right]_{k>0} = \]
\begin{equation}
= {{1}\over{4\pi R}} (0.029 - 5.313 + 4\pi R K) \sin^{2} \theta +
{{1}\over{4\pi R}} 0.482 \sin^{4} \theta .
\label{e59}
\end{equation}
Notice that the coefficient of $\sin^{4} \theta$, which is renormalization
independent, coincides with the one derived in reference \cite{Zahed} through
the propagator approach. However, our result is finite, the renormalization
prescription has completely removed the divergencies and only an undetermined
constant remains. The corresponding (finite) counterterm can be understood
as a surface integral of a local density in the external pionic field. Indeed,
if we call
\begin{equation}
L_{\alpha} = e^{-i\theta \vec{\tau}.{\bf n}}
\left(i\partial_{\alpha} e^{i\theta \vec{\tau}.{\bf n}}\right),
\end{equation}
we can write:
\begin{equation}
K \sin^{2} \theta = {K\over{16\pi}}\int_{r=R}d^{2}x\ tr\left\{
L_{\alpha}L_{\alpha} - \left(n_{\alpha}L_{\alpha}\right)^{2}\right\} ,
\label{e60}
\end{equation}
showing that $KR$ can be identified with a "surface tension", as discussed
in reference \cite{Zahed}.
\subsection{ $k=0$ contribution}
For $k=0$, from equations (\ref{e37}) and (\ref{e38}) and Appendix 1, we get
\[ [\mbox{$\Phi_{AB}^{-1}(\mu=0)$}\mbox{$\Phi_{AB}(\theta /2)$}
]^{(I)}_{n,k=0} = [\mbox{$\Phi_{AB}^{-1}(\mu=0)$}
\mbox{$\Phi_{AB}(\theta /2)$}]^{(II)}_{-n-1,k=0} = \]
\begin{equation}
= {{(1+iS_{n} W^{+}_{0,n})^{2}}\over{\cos\theta [1-(W_{0,n}^{+})^{2}]+2iS_{n}
W_{0,n}^{+}}} ,
\label{e61}
\end{equation}
(since the corresponding subspaces are one-dimensional).
Taking into account that
\begin{equation}
W_{0,n}^{+} = {{\cosh z_{n}}\over{\sinh z_{n}}} - {{1}\over{z_{n}}},
\label{e62}
\end{equation}
with $z_{n} = |\omega_{n}| R$, we get the asymptotic behavior
\[
\ln [\mbox{$\Phi_{AB}^{-1}(\mu=0)$}\mbox{$\Phi_{AB}(\theta /2)$}
]_{n,k=0}^{(I)} = \ln [\mbox{$\Phi_{AB}^{-1}(\mu=0)$}
\mbox{$\Phi_{AB}(\theta /2)$}]_{-n-1,k=0}^{(II)} = \]
\begin{equation}
= -{{iS_{n}}\over{z_{n}}} (1-\cos \theta) + O(1/z_{n}^{2}),
\label{e63}
\end{equation}
while
\[
[1- \mbox{$\Phi_{AB}^{-1}(\mu=0)$}\mbox{$\Phi_{AB}(\theta /2)$}
]_{n,k=0}^{(I)} = [1- \mbox{$\Phi_{AB}^{-1}(\mu=0)$}
\mbox{$\Phi_{AB}(\theta /2)$}]_{-n-1,k=0}^{(II)}=
\]
\begin{equation}
= {{iS_{n}}\over{z_{n}}} (1-\cos \theta) + O(1/z_{n}^{2}).
\label{e64}
\end{equation}
So, one can see that only the $O(1/z_{n})$ term of the first subtracted trace
in equation (\ref{e45}) (for $k=0$) is needed to cancel the bad behavior
of the right hand side in equation (\ref{e63}). The remaining ones, leading to
convergent series, cancel against their contribution to the renormalized
terms in equation (\ref{e26}) and will be disregarded from now on.
Finally, noticing that terms of $O(1/z_{n})$ cancel when summing over the two
parities, we simply get for the $k=0$ contribution to the chiral correction
of the free energy:
\begin{equation}
\left[F(\theta) - F(0)\right]_{k=0} =
{{1}\over{2\beta}} \sum_{n=-\infty}^{\infty}
\ln \left| {{(1+ iS_{n} W_{0,n}^{+})^{2}}\over{\cos\theta [1-(W_{0,n}^{+})^{2}]
+ 2iS_{n} W_{0,n}^{+}}} \right|^{2} .
\label{e65}
\end{equation}
In this expression we have again discarded a finite counterterm proportional
to $(1-\cos \theta)$ on physical grounds (see footnote 4).
A straightforward algebra allows one to write equation (\ref{e65}) in the
form:
\begin{equation}
\left[ F(\theta) - F(0) \right]_{k=0} = A' + B' \ ,
\label{e66}
\end{equation}
with:
\[ A' = {{1}\over{2\beta}} \sum_{n=-\infty}^{\infty} \ln {{1+ 2e^{-4z_{n}} +
e^{-8z_{n}}}\over{1 + 2e^{-4z_{n}} \cos (2\theta ) + e^{-8z_{n}}}} =
{{1}\over{\beta}} \ln {{\theta_{3} (0)}\over{\theta_{3}(\theta )}} \]
\[
B' = {{1}\over{2\beta }} \sum_{n=-\infty}^{\infty} \left\{\ln \left[
1+ {{ - {{2 a_{n}}\over{z_{n}}} + {{1}\over{z_{n}^{2}}}}\over{1 + a_{n}^{2}}}
\right]^{2}- \right. \]
\begin{equation} \left.
\ln \left[ 1+ {{ 4\left( -{{2a_{n}}\over{z_{n}}} + {{1}\over{z_{n}^{2}}}
\right) + \left[ 2(1-a_{n}^{2}) + \left( {{2a_{n}}\over{z_{n}}} -
{{1}\over{z_{n}^{2}}} \right) \right] \left( {{2a_{n}}\over{z_{n}}} -
{{1}\over{z_{n}^{2}}} \right) \cos^{2} \theta }\over{ 4 a_{n}^{2} +
(1-a_{n}^{2})^{2} \cos^{2} \theta}}\right] \right\},
\label{e67}
\end{equation}
where $\theta_{3}$ is the Jacobi's theta function \cite{tabla}, and $a_{n}$
stands for $\coth z_{n}$. These expressions can be studied in the low
temperature limit by making use of the Euler-Mac Laurin summation formula
\cite{abramo}. In particular, $B'$ can be estimated in this limit from the
asymptotic behavior of the integrand. As concerns the high temperature limit,
it can be obtain from suitable expansions of the general terms in equations
(\ref{e67}). We thus get:
\[
A' = {{1}\over{4\pi R}} \theta^{2} + O\left({{2\pi R}\over\beta}\right) \ ,\
for\ {{\beta}\over{2\pi R}} >> 1 \]
\begin{equation}
\ = {{1}\over{4\pi R}} 8 \left( {{2\pi R}\over{\beta}}\right) e^{-{{4\pi
R}/{\beta}} } + O(e^{-{{8 \pi R}/{\beta}}}), \ \ for\ {{\beta}\over
{2\pi R}} << 1
\label{e68}
\end{equation}
and
\[
B' \simeq {{1}\over{4\pi R}} \left(2.587 \sin^{2} \theta
+ 0.922 \sin^{4} \theta \right)+ O\left(\left({{2\pi R}\over \beta}\right)^{3}
\right) ,\ for\ {{\beta}\over{2\pi R}} >> 1 \]
\begin{equation}
\ = {{1}\over{4\pi R}} \pi^{2} \left( {{\beta}\over{2\pi R}}\right) \sin^{2}
\theta + O\left( \left({\beta \over{2\pi R}}\right)^{2} \right) ,\ \ for\
{{\beta}\over{2\pi R}} << 1
\label{e69}
\end{equation}
In particular, when $\beta \rightarrow \infty $, we obtain for the $k =
0$ contribution to the chiral
correction to the Casimir energy:
\begin{equation}
\left[ E(\theta) - E(0) \right]_{k=0} \simeq {{1}\over{4\pi R}} \theta^{2} +
{{1}\over{4\pi R}} \left( 2.587 \sin^{2} \theta + 0.922 \sin^{4} \theta\right),
\label{e70}
\end{equation}
which is to be compared with the corresponding result in reference
\cite{Zahed}, where the $\theta^{2}$\ term is obtained from the asymptotic
behavior of the Dirac spectrum.
\section {Comments and conclusions}
In this paper, we have applied the results in reference \cite{pdet} to evaluate
the one-loop contribution to the free energy of a four-dimensional chiral bag
due to the presence of an external pionic field in a hedgehog configuration. In
that reference, it has been established that the p-determinant of a quotient of
elliptic differential operators, defined on a space of functions living in a
region $\Omega$ and subject to given elliptic boundary conditions at
$\partial\Omega$, equals the p-determinant of a quotient of pseudodifferential
(Forman's) operators, acting on functions defined just on $\partial\Omega$, the
later being entirely expressible in terms of boundary values of functions in
the kernel of the original differential operator.
In order to define renormalized free energies on the basis of this regularized
functional determinants, one must readd the subtracted traces in the
p-determinant, duely renormalized. As discussed in reference \cite{pdet},
these can be also be expressed in terms of Forman's operators.
Even though this approach relies on the simultaneous consideration of two
distinct boundary conditions (each depending on an ad-hoc parameter $\mu$),
a choice of these conditions and of $\mu$ adequate to the symmetries of the
problem, allowed us to get the difference between the (one-loop) renormalized
free energies of the bag with and without the external hedgehog pionic field,
$\Delta F(\theta) = F(\theta) - F(0)$.
A convenient selection of a basis in the kernel of the differential operator
allowed us to show that the relevant Forman's operator is block-diagonal.
In this way, the evaluation of its p-determinant was performed in each
finite-dimensional invariant subspace, thus greatly simplifying the task.
Through an analytic regularization of the subtracted traces, together with
a finite part prescription, we got a closed expression, in the form of a
double series, for $\Delta F(\theta)$ at any finite temperature. We have
verified that the residue at the discarded pole is temperature-independent,
so that only temperature-independent divergencies were removed.
Notice that the only term affected by the renormalization procedure, i.e., that
proportional to $\sin^{2} \theta$, can be expressed as a boundary integral of a
local density in the pionic field. Thus, its undetermined coefficient can be
understood as a phenomenological "surface tension" (as discussed in reference
\cite{Zahed}).
In the $k>0$ sector, a Debye expansion of Bessel functions was employed;
up to the first nonvanishing order, the sum over $n$ was explicitly performed
and high and low temperature expansions of the remaining sum were given.
In particular, in the zero-temperature limit, our results are consistent
with the Casimir energy in reference \cite{Zahed}. In fact, the coefficient
of $\sin^{4} \theta$ (which is renormalization independent) exactly coincides
with the one obtained in that reference. But our result is free of divergencies
and $\sin^{2} \theta$ is affected by an undetermined factor as a consequence
of renormalization.
For the $k=0$ sector, the asymptotic expansion of Bessel functions for large
arguments was used to again obtain high and low temperature developments.
In particular, for $T=0$, we obtained a contribution proportional to
$\theta^{2}$, which coincides with the one derived in reference \cite{Zahed}
from the asymptotic behaviour of the Dirac spectrum.
Finally, we wish to stress that those quantities which can be derived from
the renormalized energy turn out to be automatically finite in this approach.
For example, the flux of the vacuum axial current, which in reference
\cite{Zahed} is related to the spectral asymmetry of the energy, can be
defined from equations (\ref{e16}) and (\ref{e17}) as
\[
\Phi (\theta) = {{d}\over{d\theta}} [ E(\theta) - E(0) ] .
\]
So, from equations (\ref{e59}) and (\ref{e70}), we get:
\[
\Phi (\theta) = {\theta\over{2\pi R}} + {constant\over{4\pi R}} \sin 2\theta
+ {{(0.482 + 0.922)}\over{2\pi R}} \sin^{2} \theta \sin 2\theta + ... ,
\]
which depends on the undetermined constant $K$ only through the coefficient
of $\sin 2\theta$\ in the second term on the right hand side of this
expression.
\pagebreak
\appendix
\section*{Appendix 1: Matrix elements of $H_{n,k,m}^{(i)}(\mu,\theta)$}
For $k \geq 1$, the elements of the boundary values matrices deffined in
equations (\ref{e33}) and (\ref{e34}) are given by:
\[ \left(H^{I}_{n,k,m} (\mu,\theta)\right)_{1,1} =\]\[ \left[\cos (\mu) + i
S_{n} \cos (\theta - 2\mu) W_{k,n}^{+}\right]- {{1}\over{2k+1}}\left[\sin
(\theta - 2\mu) - iS_{n} \sin (\mu) {W_{k,n}^{+}}\right], \]
\[ \left(H^{I}_{n,k,m} (\mu,\theta)\right)_{1,2} =
-{{2\sqrt{k(k+1)}}\over{2k+1}} \left[\sin (\theta - 2\mu) - i S_{n} \sin (\mu)
W_{k,n}^{-}\right], \]
\[ \left(H^{I}_{n,k,m} (\mu,\theta)\right)_{2,1} =
-{{2\sqrt{k(k+1)}}\over{2k+1}} \left[\sin (\theta - 2\mu) - i S_{n} \sin (\mu)
W_{k,n}^{+}\right], \]
\begin{equation}
\left(H^{I}_{n,k,m} (\mu,\theta)\right)_{2,2} = \]\[ \left[\cos (\mu) + i
S_{n} \cos (\theta - 2\mu) W_{k,n}^{-}\right]+ {{1}\over{2k+1}} \left[\sin
(\theta - 2\mu) - iS_{n} \sin (\mu) W_{k,n}^{-}\right], \end{equation}
\[ \left(H^{II}_{n,k,m} (\mu,\theta)\right)_{1,1} =\]\[ \left[\cos (\theta -
2\mu) - i S_{n} \cos (\mu) W_{k,n}^{+}\right] +{{1}\over{2k+1}} \left[\sin
(\mu) + iS_{n} \sin (\theta - 2\mu) W_{k,n}^{+}\right], \]
\[ \left(H^{II}_{n,k,m} (\mu,\theta)\right)_{1,2} =
-{{2\sqrt{k(k+1)}}\over{2k+1}} \left[\sin (\mu) + i S_{n} \sin (\theta - 2\mu)
W_{k,n}^{-}\right], \]
\[ \left(H^{II}_{n,k,m} (\mu,\theta)\right)_{2,1} =
-{{2\sqrt{k(k+1)}}\over{2k+1}} \left[\sin (\mu) + i S_{n} \sin (\theta - 2\mu)
W_{k,n}^{+}\right], \]
\[ \left(H^{II}_{n,k,m} (\mu,\theta)\right)_{2,2} = \]\[ \left[\cos (\theta -
2\mu) - i S_{n} \cos (\mu) W_{k,n}^{-}\right])- {{1}\over{2k+1}}\left[\sin
(\mu) + iS_{n} \sin (\theta - 2\mu) W_{k,n}^{-}\right]. \]
\bigskip
For $k = 0$, equations (\ref{e37}) and (\ref{e38}) give:
\[
H^{I}_{n,k=0} (\mu,\theta) = \]
\[
= \left[\cos(\mu) + i S_{n} \cos (\theta - 2\mu) W_{0,n}^{+}\right] -
\left[\sin (\theta - 2\mu)- i S_{n} \sin (\mu) W_{0,n}^{+}\right]
\label{e37p}
\]
and
\[
H^{II}_{n,k=0} (\mu,\theta) = \]
\[
= \left[\cos (\theta - 2\mu) - i S_{n} \cos(\mu) W_{0,n}^{+}\right]
+ \left[\sin (\mu) + i S_{n} \sin (\theta - 2\mu) W_{0,n}^{+}\right].
\label{e38p}
\]
\section*{Appendix 2: Evaluation of $A(T)$ and $B(T)$}
When evaluating $A(T)$ and $B(T)$ one must consider sums of the form:
\begin{equation}
F(\beta' \nu;n) = \sum_{n'=-\infty}^{\infty} {{1}\over{\left[
(\beta'\nu)^{2}+(n'+1/2)\right]^{n}}}
\label{ea1}
\end{equation}
with $n\in {\bf N}$. This can be generated from $F(\beta'\nu;1)$ as:
\begin{equation}
F(\beta'\nu;n+1) = {{1}\over{n!}} (-1)^{n} {{1}\over{(2\nu^{2})^{n}}} \left[
{{1}\over{\beta'}} {{d}\over{d\beta'}} \right]^{n} F(\beta'\nu;1).
\label{ea2}
\end{equation}
The generating series can be exactly evaluated through the Poisson summation
formula \cite{abramo} and it is easily seen to be given by:
\begin{equation}
F(\beta'\nu ;1) = {{\pi}\over{\beta'\nu}} \tanh (\pi \beta' \nu)
\label{ea3}
\end{equation}
A straightforward algebra leads from equations (\ref{e52}) and (\ref{e53})
in \linebreak Section 5 to
\[
- 4\pi R\ A(T) = {{5}\over{8}} \pi \beta^{3} \left(
{{1}\over{\beta}} {{d}\over{d\beta}}\right) S(\beta ) + {{15}\over{16}} \pi
\beta^{5} \left({{1}\over{\beta}} {{d}\over{d\beta }} \right)^{2} S(\beta) +
\]
\begin{equation}
{{25}\over{96}} \pi \beta^{7} \left({{1}\over{\beta}}
{{d}\over{d\beta}}\right)^{3} S(\beta) + {{25}\over{1536}} \pi \beta^{9} \left(
{{1}\over{\beta}} {{d}\over{d\beta}}\right)^{4} S(\beta)
\label{ea4}
\end{equation}
\[
- 4 \pi R\ B(T) = {{\pi}\over{2}} \beta^{3} \left(
{{1}\over{\beta}} {{d}\over{d\beta}}\right) S(\beta) + {{\pi}\over{8}}
\beta^{5} \left({{1}\over{\beta}} {{d}\over{d\beta}}\right)^{2} S(\beta) +
\]
\begin{equation}
{{\pi}\over{384}} \beta^{7} \left( {{1}\over{\beta}}
{{d}\over{d\beta}} \right)^{3} S(\beta) ,
\label{ea5}
\end{equation}
with
\begin{equation}
S(\beta) = {{1}\over{\beta}} \sum_{k=1}^{\infty} \nu^{-2} \tanh \left({{\beta
\nu }\over{2 R}}\right).
\label{ea6}
\end{equation}
In the low temperature limit ($\beta /2\pi R >>1 $) one has:
\begin{equation}
\tanh \left({{\beta\nu }\over{2 R}}\right) = 1 + O(e^{-\beta\nu/R}),
\label{ea7}
\end{equation}
which gives for $S(\beta )$:
\begin{equation}
S(\beta )= {4\over\beta} \left[ {{\pi^{2}}\over 8} - 1\right] + O\left(
{{e^{-\beta/2 R}}\over \beta } \right).
\label{ea8}
\end{equation}
When replaced in equations (\ref{ea4}) and (\ref{ea5}) this gives:
\begin{equation}
A(T) =\left[ {{1}\over{4\pi R}} 0.029 + O(e^{-\beta /2 R})\right],
\label{ea9}
\end{equation}
and
\begin{equation}
B(T) =\left[ {{1}\over{4\pi R}} 0.482 + O(e^{-\beta /2 R})\right].
\label{ea10}
\end{equation}
For the high temperature limit ($\beta/2\pi R <<1$), $S(\beta )$ can be
evaluated through the Euler-Mac Laurin formula, which gives:
\begin{equation}
S(\beta ) = -{1\over{2R}} \log \left({\beta \over{4R}}\right) + constant
+ O \left( \left({\beta\over{2\pi R}}\right)^{2} \right).
\label{ea11}
\end{equation}
So, in this limit, we obtain:
\begin{equation}
A(T) = \left[ {{1}\over{4\pi R}} 0.514 \left({{\beta }\over{2\pi R}} \right)+
O\left( \left( {\beta \over {2\pi R}} \right)^{3} \right) \right],
\label{ea12}
\end{equation}
and
\begin{equation}
B(T) = \left[ {{1}\over{4\pi R}} 2,673 \left( {{\beta}\over{2\pi R}} \right)
+ O\left(\left({\beta\over {2\pi R}}\right)^{3}\right)\right].
\label{ea13}
\end{equation}
\pagebreak
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