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\begin{document}

\begin{titlepage}

\begin{flushright}
\begin{minipage}{3cm}
\begin{flushleft}
SNUTP 02/021\\
DPNU-02-19
\end{flushleft}
\end{minipage}
\end{flushright}

\begin{center}
\LARGE\bf The Vector and Axial-Vector Susceptibilities\\
\vskip -0.5 cm and Effective Degrees of Freedom\\
\vskip -0.5cm in the Vector Manifestation
\end{center}
\vspace{1cm plus 0.5cm minus 0.5cm}
\begin{center}
\large Masayasu Harada$^{(a)}$, Youngman Kim$^{(a)}$~\footnote{and
\it Department of Physics and Astronomy, University of South
Carolina, Columbia, SC 29208},
 Mannque  Rho$^{(b,c)}$~\footnote{and
 {\it School of Physics, Korea Institute for Advanced Study,
Seoul 130-012, Korea.}} \\
and Chihiro Sasaki$^{(d)}$
\end{center}
\vspace{0.5cm plus 0.5cm minus 0.5cm}
\begin{center}
(a)~{\it  School of Physics, Seoul National University,
Seoul 151-742, Korea}\\
(b)~{\it Theory Group, GSI, Planckstr. 1, D-64291 Darmstadt,
Germany}\\
 (c)~{\it Service de Physique Th\'eorique, CEA/DSM/SPhT,
Unit\'e de recherche associ\'ee au CNRS,
CEA/Saclay,  91191 Gif-sur-Yvette c\'edex, France}\\
(d)~{\it Department of Physics, Nagoya University, Nagoya,
464-8602, Japan.}
\end{center}
\vspace{0.6cm plus 0.5cm minus 0.5cm}

\begin{abstract}
The question as to what the relevant degrees of freedom are at the
thermal chiral phase transition is addressed in terms of the
``vector manifestation (VM)" of chiral symmetry \`a la Harada and
Yamawaki. We find that both the vector susceptibility $\chi_V$ and
the axial-vector susceptibility $\chi_A$ at the chiral transition
could receive contributions from other degrees of freedom than
pions in the chiral limit. The possible candidates are zero-mass
vector mesons and quasiquarks.  In the presence of these degrees
of freedom, hidden local symmetry (HLS) theory with vector
manifestation $predicts$ that $\chi_V$ equals $\chi_A$ in
consistency with chiral invariance and that both the time
component $f_\pi^t$ and the space component $f_\pi^s$ of the pion
decay constant vanish at the chiral phase transition. With the VM,
the pion velocity $v_\pi$ proportional to $f_\pi^s/f_\pi^t$ at
$T=T_c$ is predicted to be equal to 1. These results are obtained
in the leading order in power counting but we expect them to hold
more generally in the chiral limit thanks to the VM point.
\end{abstract}

\end{titlepage}

\newpage
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\section{Introduction}
\indent\indent One of the most crucial questions to answer in the
effort to understand chiral restoration in relativistic heavy-ion
collisions is: What are the relevant degrees of freedom just
before and after the phase transition? The standard scenario,
generally accepted in the heavy-ion community, is that the only
relevant thermal excitations in the broken symmetry sector near
the phase transition are the pions, the pseudo-Goldstone modes of
broken chiral symmetry (and possibly a scalar mode that is a
chiral partner to the pions). However there is no a priori reason
to exclude other scenarios. One highly plausible such possibility
is the vector manifestation of Harada and
Yamawaki~\cite{HY:VM,HY:PR}
that requires the presence in the hadronic sector of massless
vector mesons~\cite{HaradaSasaki} and possibly massless
constituent quarks (or quasiquarks)~\cite{HKR}. In fact in a
recent attempt to understand some of the puzzling results coming
out of relativistic heavy ion experiments at RHIC, Brown and
Rho~\cite{QM2002} considered the scenario in which both the
``light" $\rho$ mesons and constituent quarks play a crucial role:
The vector mesons there are ``relayed" via a Higgsing to the
gluons in the QCD sector. The question as to which is the correct
scenario could ultimately be answered by lattice measurements. In
this paper, we study what the implications of the vector
manifestation are on the vector and axial-vector susceptibilities
and on the relevant degrees of freedom for a finite temperature
phase transition. We find that the vector manifestation favors a
scenario which is drastically different from that of the standard
picture.

The issue was brought to highlight by recent beautiful papers by
Son and Stephanov (SS)~\cite{SS:1,SS:2}. As a way of introduction
to the main objective of this paper, we begin by briefly
summarizing their arguments and results.

Consider the vector isospin susceptibility (VSUS in short)
$\chi_V$ (denoted by SS as $\chi_{I}$) and the axial-vector
isospin susceptibility (ASUS) $\chi_A$ (denoted by SS as
$\chi_{I5}$) defined in terms of the vector charge density $V_0^a
(x)$ and the axial-vector charge density $A_0^a (x)$ by the
Euclidean correlators:
 \be
\delta^{ab}\chi_V&=& \int^{1/T}_0 d\tau\int d^3\vec{x}\la V_0^a
(\tau, \vec{x}) V_0^b (0,\vec{0})\ra_\beta,\\
\delta^{ab}\chi_A&=& \int^{1/T}_0 d\tau\int d^3\vec{x}\la A_0^a
(\tau, \vec{x}) A_0^b (0,\vec{0})\ra_\beta
 \ee
where $\la ~\ra_\beta$ denotes thermal average and
 \be
V_0^a\equiv \bar{\psi}\gamma^0\frac{\tau^a}{2}\psi, \ \
A_0^a\equiv \bar{\psi}\gamma^0\gamma^5\frac{\tau^a}{2}\psi
 \ee
with the quark field $\psi$ and the $\tau^a$ Pauli matrix the
generator of the flavor $SU(2)$.

We are interested in these SUS's near the critical temperature
$T=T_c$ at zero baryon density $n=0$. In particular we would like
to compute them ``bottom-up" approaching $T_c$ from below. In
order to do this, we need to resort to effective field theory of
QCD which requires identifying, in the premise of an EFT, $all$
the relevant degrees of freedom.

Let us first assume as done by Son and Stephanov~\cite{SS:1,SS:2}
that the only relevant effective degrees of freedom in heat bath
are the pions (and possibly a light scalar as in linear sigma
model), and that all other degrees of freedom can be integrated
out with their effects incorporated into the coefficients of
higher order terms in the effective Lagrangian. Here the basic
assumption is that near chiral restoration, there is no
instability in the channel of the degrees of freedom that have
been integrated out. In this pion-only case, the appropriate
effective Lagrangian for the axial correlators is the in-medium
chiral Lagrangian dominated by the current algebra terms,
 \be
\L_{eff}=\frac{{f_\pi^t}^2}{4}\left(\Tr\nabla_0 U\nabla_0
U^\dagger - v_\pi^2\Tr\del_i U\del_i U^\dagger\right) -\frac 12
\la\bar{\psi}\psi\ra {\rm Re} M^\dagger
U\label{LA}+\cdots\label{Leff}
 \ee
where $v_\pi$ is the pion velocity, $M$ is the mass matrix
introduced as an external field, $U$ is the chiral field and the
covariant derivative $\nabla_0 U$ is given by $\nabla_0 U=\del_0 U
-\frac i2 \mu_A (\tau_3 U +U\tau_3)$ with $\mu_A$ the axial
isospin chemical potential. The ellipsis stands for higher order
terms in spatial derivatives and covariant
derivatives.~\footnote{The notation here deviates a bit from that
of SS. For example, it will turn out that the pion velocity will
have the form $v_\pi^2=f_\pi^s/f_\pi^t$ (see Eq.(\ref{v2 rel}))
where $f_\pi^t$ ($f_\pi^s$) is the temporal (spatial) component of
the pion decay constant.} Given the effective action described by
(\ref{Leff}) with possible non-local terms ignored, then the ASUS
takes the simple form
 \be
\chi_A=-\frac{\del^2}{\del\mu_A^2}\L_{eff}|_{\mu_A=0}={f_\pi^t}^2.
 \ee
The principal point to note here is that {\it as long as the
effective action is given by local terms (subsumed in the
ellipsis) involving the $U$ field, this is the whole story}: There
is no other contribution to the ASUS than the temporal component
of the pion decay constant.

Next one assumes that at the chiral phase transition point
$T=T_c$, the restoration of chiral symmetry dictates the equality
 \be
\chi_A=\chi_V.
 \ee
While there is no lattice information on $\chi_A$, $\chi_V$ has
been measured as a function of
temperature~\cite{GLTRS,Brown-Rho:96}. In particular, it is
established that
 \be
\chi_V|_{T=T_c}\neq 0,
 \ee
which leads to the conclusion~\cite{SS:1,SS:2} that
 \be
f_\pi^t|_{T=T_c}\neq 0.
 \ee
On the other hand, it is expected and verified by lattice
simulations that the space component of the pion decay constant
$f_\pi^s$ should vanish at $T=T_c$. One therefore arrives at
 \be
v_\pi^2\sim f_\pi^s/f_\pi^t\rightarrow 0, \ \ T\rightarrow T_c.
 \ee
This is the main conclusion of the pion-only theory.

To check whether this prediction is firm, let us see what one
obtains for the VSUS in the same effective field theory approach.
The effective Lagrangian for calculating the vector correlators is
of the same form as the ASUS, Eq.~(\ref{Leff}), except that the
covariant derivative is now defined with the vector isospin
chemical potential $\mu_V$ as $\nabla_0 U=\del_0 U-\frac 12 \mu_V
(\tau_3 U-U\tau_3)$. Now if one assumes as done above for $\chi_A$
that possible non-local terms can be dropped, then the SUS is
given  by
 \be
\chi_V=-\frac{\del^2}{\del\mu_V^2}\L_{eff}|_{\mu_V=0}
 \ee
which can be easily evaluated from the Lagrangian. One finds that
 \be
\chi_V=0
 \ee
{\it for all temperature.} While it is expected to be zero at
$T=0$, the vanishing $\chi_V$ for $T\neq 0$ is at variance with
the lattice data at $T=T_c$. The reason for this defect is
explained in terms of hydrodynamics by Son and
Stephanov~\cite{SS:1,SS:2}. We will return to this issue in
Section 7. Let it suffice to note here that this defect is
identified with the missing ``diffusive modes" in the effective
Lagrangian (\ref{Leff}).

We now turn to the main objective of this paper: the prediction by
the vector manifestation (VM) of
Harada and Yamawaki~\cite{HY:VM}.
Basically
the same scenario was suggested some time ago in conjunction with
Brown-Rho scaling~\cite{Brown-Rho:91,Brown-Rho:01b}. As will be
shown in detail in the following sections, the VM $requires$ that
the vector mesons (and quasiquarks) figure on the same footing
with the pions as $the$ relevant degrees of freedom as the chiral
transition point is approached from below. The key reason for this
conclusion is that the chiral transition coincides with the VM
fixed point at which the vector meson mass (and the quasiquark
mass) must vanish in the chiral limit~\cite{HY:fate}.
This means that the
vector-meson degrees of freedom (as well as the quasiquarks) {\it
cannot} be integrated out near chiral restoration.

Our principal results - which are basically different from the
pion-only scenario - can be summarized as follows. In the presence
of the $\rho$-meson (and isospin-doublet quasiquarks), the only
approach that is consistent with chiral perturbation theory is
Harada-Yamawaki hidden local symmetry (HLS) theory with the VM
fixed point~\footnote{It has been stressed in the literature (see,
e.g., \cite{Georgi,HY:PR}) -- and is stressed again -- that HLS is
a bona-fide effective field theory of QCD {\it only} if the
$\rho$-meson mass is considered as of the same chiral order as the
pion mass. In HLS theory, this condition is naturally met by the
$\rho$-meson mass near the chiral transition point, so chiral
perturbation theory should be more effective in this regime. This
point that underlines our arguments that follow justifies our
one-loop calculation.}. This theory predicts
 \be
f_\pi^t|_{T=T_c}=f_\pi^s|_{T=T_c}=0, \ \
v_\pi|_{T=T_c}=1\label{main1}
  \ee
and
 \be
\chi_A|_{T=T_c}=\chi_V|_{T=T_c}= 2N_f \left[\frac{N_f}{12} T_c^2
+\frac{N_c}{6} T_c^2\right] \ ,\label{main2}
 \ee
where we have included the normalization factor of $2N_f$.
Note that the equality of $\chi_A$ and $\chi_V$ at $T=T_c$ is an
output of the theory. The first term on the RHS of
Eq.~(\ref{main2}) comes from the vector-meson loop and the second
from the quasiquark loop. The latter is equivalent to $\chi_0$ of
Ref.~\cite{Brown-Rho:96}. These results are easy to understand. The
$\rho$ and $\pi$ (and also the quasiquark) enter on the same
footing. At the VM fixed point, the longitudinal components of the
vector mesons and the pions form a degenerate multiplet, the
$\rho$ (and the quasiquark) entering into the hydrodynamic
description as the pion does; the ``diffusive" modes considered by
SS are responsible for the width of the ``collective mode," i.e.,
the vector meson.

The rest of the paper is devoted to the derivation of the main
results (\ref{main1}) and (\ref{main2}). In Section~\ref{sec:HLS},
hidden
local symmetry (HLS) theory with constituent quarks
(``quasiquarks") is briefly introduced. Section 3 describes how
thermal two-point functions are calculated in the HLS theory. In
Section 4, we write down the in-medium vector and axial-vector
current correlators that are needed in what follows. Pion decay
constants and pion velocity are computed in the given framework in
Section 5. The susceptibilities are defined in Section 6 and
computed for temperature $T\sim T_c$. In Section 7, we briefly
comment on the hydrodynamic structure of our theory along the line
developed by Son and Stephanov. The conclusion is given in Section
8. The Appendices contain explicit formulas used in the main text.
A more extensive treatment of the material covered in this paper
together with other issues of finite temperature effective field
theory in the VM is found in Ref.~\cite{HaradaSasaki:prep}.

%YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYyy

\section{Hidden Local Symmetry with Quasiquark}
\label{sec:HLS}
\indent\indent In this Section, we briefly summarize the HLS model
that incorporates fermion degrees of freedom. As mentioned above,
both HLS spin-1 field and constituent quark (referred to in this
paper as quasiquark) field are assumed to be as relevant as the
pion field near chiral restoration. This means that we will have
the quasiquark Lagrangian incorporated into the usual HLS
Lagrangian.
The HLS model~\cite{BKUYY,BKY:88} is based on the $G_{\rm
global} \times H_{\rm local}$ symmetry, where $G =
\mbox{SU($N_f$)}_{\rm L} \times \mbox{SU($N_f$)}_{\rm R}$  is the
global chiral symmetry and $H = \mbox{SU($N_f$)}_{\rm V}$ is the
HLS. The basic quantities are the HLS gauge field $V_\mu$ and two
variables or ``coordinates"
\begin{eqnarray}
&&
\xi_{\rm L,R} = e^{i\sigma/F_\sigma} e^{\mp i\pi/F_\pi}
\ ,
\end{eqnarray}
where $\pi$ denotes the pseudoscalar Nambu-Goldstone (NG) boson
and $\sigma$ the NG boson absorbed into the HLS gauge field
$V_\mu$ (longitudinal $\rho$). $F_\pi$ and $F_\sigma$ are
corresponding decay constants, and the parameter $a$ is defined as
$a \equiv F_\sigma^2/F_\pi^2$. The transformation property of
$\xi_{\rm L,R}$ is given by
\begin{eqnarray}
&&
\xi_{\rm L,R}(x) \rightarrow \xi_{\rm L,R}^{\prime}(x) =
h(x) \xi_{\rm L,R}(x) g^{\dag}_{\rm L,R}
\ ,
\end{eqnarray}
where $h(x) \in H_{\rm local}$ and
$g_{\rm L,R} \in G_{\rm global}$.
The covariant derivatives of $\xi_{\rm L,R}$ are defined by
\begin{eqnarray}
&&
D_\mu \xi_{\rm L} =
\partial_\mu \xi_{\rm L} - i V_\mu \xi_{\rm L}
+ i \xi_{\rm L} {\cal L}_\mu
\ ,
\nonumber\\
&&
D_\mu \xi_{\rm R} =
\partial_\mu \xi_{\rm R} - i V_\mu \xi_{\rm R}
+ i \xi_{\rm R} {\cal R}_\mu
\ ,
\label{covder}
\end{eqnarray}
where
${\cal L}_\mu$ and ${\cal R}_\mu$ denote the external gauge fields
gauging the $G_{\rm global}$ symmetry.
{}From the above covariant derivatives two 1-forms
are constructed as
\begin{eqnarray}
&&
\hat{\alpha}_{\perp}^\mu =
( D_\mu \xi_{\rm R} \cdot \xi_{\rm R}^\dag -
  D_\mu \xi_{\rm L} \cdot \xi_{\rm L}^\dag
) / (2i)
\ ,
\nonumber\\
&&
\hat{\alpha}_{\parallel}^\mu =
( D_\mu \xi_{\rm R} \cdot \xi_{\rm R}^\dag +
  D_\mu \xi_{\rm L} \cdot \xi_{\rm L}^\dag
) / (2i)
\ .
\end{eqnarray}


Now, the HLS Lagrangian to leading order is given
by~\cite{BKUYY,BKY:88}
\begin{equation}
{\cal L} = F_\pi^2 \, \mbox{tr}
\left[ \hat{\alpha}_{\perp\mu} \hat{\alpha}_{\perp}^\mu \right]
+ F_\sigma^2 \, \mbox{tr}
\left[
  \hat{\alpha}_{\parallel\mu} \hat{\alpha}_{\parallel}^\mu
\right]
- \frac{1}{2g^2} \mbox{tr} \left[ V_{\mu\nu} V^{\mu\nu} \right]
\ ,
\label{Lagrangian}
\end{equation}
where $g$ is the HLS gauge coupling and
$V_{\mu\nu} = \partial_\mu V_\nu - \partial_\nu V_\mu
- i [ V_\mu , V_\nu ]$ the gauge field strength.
When the kinetic term of the gauge field
is ignored in the low-energy region,
the second term of Eq.(\ref{Lagrangian}) vanishes
by integrating out $V_\mu$
and only the first term remains.
Then,
the HLS model is reduced to the nonlinear sigma model based on $G/H$.


The terms of the ${\cal O}(p^4)$ Lagrangian relevant to the present
analysis are given by~\cite{Tanabashi,HY:WM}
\begin{equation}
  {\cal{L}}_{(4)} = z_1\mbox{tr}\bigl[ \hat{\cal{V}}_{\mu\nu}
                       \hat{\cal{V}}^{\mu\nu} \bigr] +
                    z_2\mbox{tr}\bigl[ \hat{\cal{A}}_{\mu\nu}
                       \hat{\cal{A}}^{\mu\nu} \bigr] +
                    z_3\mbox{tr}\bigl[ \hat{\cal{V}}_{\mu\nu}
                       V^{\mu\nu} \bigr], \label{eq:L(4)}
\end{equation}
where
 \begin{eqnarray}
  \hat{\cal{A}}_{\mu\nu}=\frac{1}{2}
                         \bigl[ \xi_R{\cal{R}}_{\mu\nu}\xi_R^{\dagger}-
                                \xi_L{\cal{L}}_{\mu\nu}\xi_L^{\dagger}
                         \bigr]\ ,
\nonumber\\
  \hat{\cal{V}}_{\mu\nu}=\frac{1}{2}
                         \bigl[ \xi_R{\cal{R}}_{\mu\nu}\xi_R^{\dagger}+
                                \xi_L{\cal{L}}_{\mu\nu}\xi_L^{\dagger}
                         \bigr]\ ,
  \label{def V mn}
\end{eqnarray}
with ${\cal{R}}_{\mu\nu}\ \mbox{and}\ {\cal{L}}_{\mu\nu}$ being
the field strengths of ${\cal{R}}_{\mu}\ \mbox{and}\ {\cal{L}}_{\mu}$.


The leading order Lagrangian including one quasiquark field and
one anti-quasiquark field is counted as ${\cal O}(p)$ and is given
by
 \begin{eqnarray}
 \delta {\cal L}_{Q(1)} &=& \bar \psi(x)( iD_\mu \gamma^\mu
       + \mu \gamma^0 -m_q )\psi(x)\nonumber\\
 &&+ \bar \psi(x) \left(
  \kappa\gamma^\mu \hat{\alpha}_{\parallel \mu}(x )
+ \lambda\gamma_5\gamma^\mu \hat{\alpha}_{\perp\mu}(x) \right)
        \psi(x) \label{lagbaryon}
 \end{eqnarray}
where $D_\mu\psi=(\partial_\mu -i V_\mu)\psi$ and $\kappa$ and
$\lambda$ are constants to be specified later. At one-loop level
the Lagrangian (\ref{lagbaryon}) generates the ${\cal O}(p^4)$
contributions including hadronic thermal effects as well as
quantum effects. The Lagrangian consisting of (\ref{Lagrangian}),
(\ref{eq:L(4)}) and (\ref{lagbaryon}) defines our theory.


Note that the parameters of the above Lagrangian have the
$intrinsic$ temperature dependence~\cite{HaradaSasaki}. This
effect may break $implicitly$ the Lorentz invariance of the above
Lagrangian. In the present analysis, however, we neglect such
(implicit) Lorentz symmetry violating effects caused by the
intrinsic temperature dependence of the parameters, and use the
Lagrangian with Lorentz invariance even at non-zero temperature.
We believe this approximation to be reliable~\footnote{For
instance, the ``bare" parameters for the pion decay constants we
are concerned with, when the HLS theory is matched to QCD, are
dominated by the $(1+\frac{\alpha_s}{\pi})$ term in the
axial-current correlator with negligible corrections from
Lorentz-violating terms.}. $Explicit$ Lorentz violation in medium
-- which is not negligible -- is of course taken into account (see
Appendices).


\section{Two-Point Functions}
\label{sec:TPF}

 \indent\indent We need to consider two-point
functions involving the isovector vector and axial-vector
currents. Let us calculate them to one-loop order as in
\cite{HY:WM,HY:PR}. The additional ingredient that has not been
considered in previous works of HLS is the presence of
quasiquarks. In calculating the loops, we adopt the background
field gauge (see Refs.~\cite{HY:WM,HY:PR} for details in HLS
theory) and the imaginary time formalism (see, e.g.,
Ref.~\cite{Kap}). For convenience, we introduce the following
Feynman integrals to calculate the one-loop hadronic thermal
corrections and quantum corrections to the two-point functions:
\begin{eqnarray}
A_{0(B,F)}(M;T) &\equiv&
T \sum_{n=-\infty}^{\infty}
\int \frac{d^3k}{(2\pi)^3}
\frac{1}{M^2-k_{(B,F)}^2}
\ ,
\label{def:A0 2}
\\
B_{0(B,F)}(p_0,\bar{p};M_1,M_2;T) &\equiv&
T \sum_{n=-\infty}^{\infty}
\int \frac{d^3k}{(2\pi)^3}
\frac{1}{ [M_1^2-k_{(B,F)}^2] [M_2^2-(k_{(B,F)}-p)^2] }
\ ,
\label{def:B0 2}
\\
B_{(B,F)}^{\mu\nu}(p_0,\vec{p};M_1,M_2;T) &\equiv&
T \sum_{n=-\infty}^{\infty}
\int \frac{d^3k}{(2\pi)^3}
\frac{\left(2k_{(B,F)}-p\right)^\mu \left(2k_{(B,F)}-p\right)^\nu}{%
 [M_1^2-k_{(B,F)}^2] [M_2^2-(k_{(B,F)}-p)^2] }
\ ,
\label{def:Bmunu 2}
\end{eqnarray}
where $\bar{p}\equiv\vert\vec{p}\vert$, the subscript $(B,F)$
refers to the bosonic or fermionic contributions, and
correspondingly the $0$th component of the loop momentum is taken
as $k_{(B)}^0 = i 2 n \pi T$ and $k_{(F)}^0 = i (2n+1) \pi T$,
while that of the external momentum is taken as $p^0 = i 2
n^\prime \pi T$ ($n^\prime$: integer). Using the standard formula
(see, e.g., Ref.~\cite{Kap}), we can convert the Matsubara
frequency sum into an integral over $k_0$ with $k_0$ taken as the
zeroth component of a Minkowski four vector. Accordingly, the
above functions are divided into two parts as
\begin{eqnarray}
A_{0(B,F)}(M;T) &=&
A_{0}^{\rm{(vac)}}(M) + \bar{A}_{0(B,F)}(M;T) \ ,
\nonumber\\
B_{0(B,F)}(p_0,\bar{p};M_1,M_2;T) &=&
B_{0}^{\rm{(vac)}}(p_0,\bar{p};M_1,M_2) +
\bar{B}_{0(B,F)}(p_0,\bar{p};M_1,M_2;T) \ ,
\nonumber\\
B_{(B,F)}^{\mu\nu}(p_0,\vec{p};M_1,M_2;T)
&=&
B^{\rm{(vac)}\mu\nu}(p_0,\vec{p};M_1,M_2)
+
\bar{B}_{(B,F)}^{\mu\nu}(p_0,\vec{p};M_1,M_2;T)
\ ,
\label{B bar defs}
\end{eqnarray}
where $A_{0}^{\rm{(vac)}}$, $B_{0}^{\rm{(vac)}}$ and
$B^{\rm{(vac)}\mu\nu}$ are given by replacing $T
\sum_{n=-\infty}^{\infty}$ with $\int \frac{d k_0}{2\pi i}$ in
Eqs.~(\ref{def:A0 2})--(\ref{def:Bmunu 2}), and
$\bar{A}_{0(B,F)}$, $\bar{B}_{0(B,F)}$ and
$\bar{B}_{(B,F)}^{\mu\nu}$ are defined by Eq.~(\ref{B bar defs}).
The forms of $A_{0}^{\rm{(vac)}}$, $B_{0}^{\rm{(vac)}}$ and
$B^{\rm{(vac)}\mu\nu}$ are equivalent to the zero-temperature
ones. Then, with $p_0$ taken as the 0th component of the Minkowski
four vector, they have no $explicit$ temperature dependence while
the intrinsic dependence mentioned above remains. Therefore, the
functions $A_{0}^{\rm{(vac)}}$, $B_{0}^{\rm{(vac)}}$ and
$B^{\rm{(vac)}\mu\nu}$ represent quantum corrections. In
$\bar{B}_{0(B,F)}$ and $\bar{B}_{(B,F)}^{\mu\nu}$ one can perform
the analytic continuation of $p_0$ to the Minkowski variable after
integrating over $k_0$: Here $p_0$ is understood as $p_0 +
i\epsilon$ ($\epsilon \rightarrow +0$) for the retarded function
and $p_0 - i \epsilon$ for the advanced function. It should be
noticed that the argument $T$ in the above functions refers to
only the temperature dependence arising from the hadronic thermal
effects and $not$ to the intrinsic thermal effects included in the
parameters of the Lagrangian.


Now, let us calculate the one-loop corrections to the two-point
function of the axial-vector (background) field $\overline{\cal
A}^\mu$. This is obtained by the sum of one particle irreducible
diagrams with two legs of the axial-vector background field
$\overline{\cal A}^\mu$. In Fig.~\ref{fig:AA} we show the Feynman
diagrams contributing to the $\overline{\cal
A}_\mu$-$\overline{\cal A}_\nu$ two-point function at one-loop
level.
\begin{figure}
\begin{center}
\epsfxsize = 14cm
\ \epsfbox{diagramsAA.eps}
\end{center}
\caption[]{%
Feynman diagrams contributing to the $\overline{\cal
A}_\mu$-$\overline{\cal A}_\nu$ two-point function. Here
$\check{\pi}$ represents the quantum pion field and likewise for
the others.}\label{fig:AA}
\end{figure}
With the help of (\ref{B bar defs}), one can express the one-loop
corrections to the two-point function in a simple form.
The bosonic corrections from $\rho$ and/or $\pi$
shown in Figs.~\ref{fig:AA}(a)--(c)
lead to
the two-point function
\begin{eqnarray}
\left.
  \Pi_\perp^{\mbox{\scriptsize(1-loop)}\mu\nu}(p_0,\vec{p};T)
\right\vert_{(B)}
&=&
- N_f a M_\rho^2 g^{\mu\nu} B_{0(B)}(p_0,\bar{p};M_\rho,0;T)
{}+ N_f \frac{a}{4} B_{(B)}^{\mu\nu}(p_0,\vec{p};M_\rho,0;T)
\nonumber\\
&&
{}+N_f(a-1) g^{\mu\nu} A_{0(B)}(0,T)
\ ,
\label{PiA boson}
\end{eqnarray}
where $B_{0(B)}$-term comes from Fig.~\ref{fig:AA}(a), and
$B_{(B)}^{\mu\nu}$-term and $A_{0(B)}$-term from
Fig.~\ref{fig:AA}(b) and Fig.~\ref{fig:AA}(c), respectively.
The correction from the quasiquark loop shown in
Fig.~\ref{fig:AA}(d) is evaluated as
\begin{eqnarray}
\left.
  \Pi_\perp^{\mbox{\scriptsize(1-loop)}\mu\nu}(p_0,\vec{p};T)
\right\vert_{(F)}
&=&
- \lambda^2 N_c
\Biggl[
  2 g^{\mu\nu} A_{0(F)}(m_q,T)
  {}+ B_{(F)}^{\mu\nu}(p_0,\vec{p};m_q,m_q;T)
\nonumber\\
&&
  {}+ \biggl\{
    - 4 m_q^2 g^{\mu\nu} + (g^{\mu\nu} p^2 - p^\mu p^\nu )
  \biggr\}
  B_{0(F)}(p_0,\bar{p};m_q,m_q;T)
\Biggr]
\ .
\label{PiA fermion}
\end{eqnarray}
Combining the above bosonic and fermionic loop corrections with
the tree contribution given by
\begin{eqnarray}
\Pi_\perp^{{\rm(tree)}\mu\nu}(p_0,\vec{p})
= g^{\mu\nu} F_{\pi,{\rm bare}}^2 +
2 z_{2,{\rm bare}} (g^{\mu\nu} p^2 - p^\mu p^\nu )
\ ,
\label{Pi A tree}
\end{eqnarray}
we have the two-point function of $\overline{\cal
A}_\mu$-$\overline{\cal A}_\nu$ at one-loop level as
\begin{equation}
\Pi_\perp^{\mu\nu}(p_0,\vec{p};T)
= \Pi_\perp^{{\rm(tree)}\mu\nu}(p_0,\vec{p})
+ \Pi_\perp^{\mbox{\scriptsize(1-loop)}\mu\nu}(p_0,\vec{p};T)
\ .
\end{equation}
Analogously to Eq.~(\ref{B bar defs}), we split the two-point
function into two parts as
\begin{equation}
\Pi_\perp^{\mu\nu}(p_0,\bar{p};T)
= \Pi_\perp^{{\rm(vac)}\mu\nu}(p_0,\bar{p})
+ \bar{\Pi}_\perp^{\mu\nu}(p_0,\bar{p};T)
\ ,
\label{Pi A div}
\end{equation}
where $\Pi_\perp^{{\rm(vac)}\mu\nu}$ includes the quantum
correction and the contribution at tree level in Eq.~(\ref{Pi A
tree}), and $\bar{\Pi}_\perp^{\mu\nu}$ represents the hadronic
thermal correction. Since the hadronic thermal correction
$\bar{\Pi}_\perp^{\mu\nu}$ has no divergences, the renormalization
conditions for $F_\pi^2$ and $z_2$ can be determined from
$\Pi_\perp^{{\rm(vac)}\mu\nu}$. For $F_\pi^2$ we adopt the
``on-shell" renormalization condition:
\begin{equation}
\Pi_\perp^{{\rm(vac)}\mu\nu}(p_0=0,\vec{p}=\vec{0})
= g^{\mu\nu} F_\pi^2(0)
\ .
\end{equation}
{}From this renormalization condition, we obtain the
$g^{\mu\nu}$-part of $\Pi_\perp^{{\rm(vac)}\mu\nu}$ in the
form~\cite{HY:PR}
\begin{equation}
p_\mu p_\nu \Pi_\perp^{{\rm(vac)}\mu\nu}(p_0,\vec{p})
= p^2 \left[
  F_\pi^2(0) + \widetilde{\Pi}_\perp^S(p^2)
\right]
\ ,
\end{equation}
where $\widetilde{\Pi}_\perp^S(p^2)$ is the finite renormalization
contribution satisfying
\begin{equation}
\widetilde{\Pi}_\perp^S(p^2=0) = 0 \ .
\end{equation}
For $z_2$ we adopt the renormalization condition that
$\Pi_\perp^{{\rm(vac)}\mu\nu}$ be given by
\begin{eqnarray}
&&
\Pi_\perp^{{\rm(vac)}\mu\nu}(p_0,\vec{p})
=
g^{\mu\nu}
\left[ F_\pi^2(0) + p^2 \widetilde{\Pi}_\perp^S(p^2) \right]
+
(g^{\mu\nu} p^2 - p^\mu p^\nu )
\left[
  2 z_2(M_\rho) + \widetilde{\Pi}_\perp^{LT}(p^2)
\right]
\ ,
\label{PiA T0}
\end{eqnarray}
where $z_2(M_\rho)$ is renormalized at the scale $M_\rho$ and
$\widetilde{\Pi}_\perp^{LT}(p^2)$ is the finite renormalization
subject to the condition
\begin{equation}
\mbox{Re} \,\widetilde{\Pi}_\perp^{LT}(p^2=M_\rho^2) = 0 \ .
\end{equation}


To distinguish the hadronic thermal correction to the pion decay
constant from that to the parameter $z_2$, we decompose the
two-point function of $\overline{\cal A}_\mu$-$\overline{\cal
A}_\nu$ into four components as
\begin{equation}
 \Pi_\perp^{\mu\nu}=u^\mu u^\nu \Pi_\perp^t +
   (g^{\mu\nu}-u^\mu u^\nu)\Pi_\perp^s +
   P_L^{\mu\nu}\Pi_\perp^L + P_T^{\mu\nu}\Pi_\perp^T \ ,
\label{Pi perp decomp}
\end{equation}
where $P_L^{\mu\nu}$ and $P_T^{\mu\nu}$ are the polarization
tensors defined by
\begin{eqnarray}
  P_T^{\mu\nu}
&\equiv&
  g^\mu_i
  \left(
    \delta_{ij} - \frac{\vec{p}_i \vec{p}_j}{ \vert \vec{p} \vert^2}
  \right)
  g_j^\nu
\nonumber\\
&=&
  \left( g^{\mu\alpha} - u^\mu u^\alpha \right)
  \left(
    - g_{\alpha\beta} - \frac{p^\alpha p^\beta}{\bar{p}^2}
  \right)
  \left( g^{\beta\nu} - u^\beta u^\nu \right)
\ ,
\nonumber\\
  P_L^{\mu\nu}
&\equiv&
  - \left( g^{\mu\nu} - \frac{p^\mu p^\nu}{p^2} \right)
  - P_T^{\mu\nu}
\nonumber\\
&=&
  \left( g^{\mu\alpha} - \frac{p^\mu p^\alpha}{p^2} \right)
  u_\alpha
  \frac{p^2}{\vert\vec{p}\vert^2}
  u_\beta
  \left( g^{\beta\nu} - \frac{p^\beta p^\nu}{p^2} \right)
\ .
\label{pols}
\end{eqnarray}
Similarly to the division in Eq.~(\ref{Pi A div}),
it is convenient to divide each component into two parts as
\begin{equation}
\Pi_\perp^t(p_0,\bar{p};T) =
\Pi_\perp^{{\rm(vac)}t}(p_0,\bar{p}) +
\bar{\Pi}_\perp^t(p_0,\bar{p};T)
\ ,
\end{equation}
where $\Pi_\perp^{{\rm(vac)}t}(p_0,\vec{p})$ includes the tree
contribution plus the finite renormalization effect and
$\bar{\Pi}_\perp^t(p_0,\vec{p};T)$ is the hadronic thermal
contribution. {}With Eq.~(\ref{PiA T0}) the functions
$\Pi_\perp^{{\rm(vac)}t,s,L,T}$ can be written as
\begin{eqnarray}
&&
\Pi_\perp^{{\rm(vac)}t}(p_0,\bar{p}) =
\Pi_\perp^{{\rm(vac)}s}(p_0,\bar{p}) =
F_\pi^2(0) + \widetilde{\Pi}_\perp^S(p^2)
\ ,
\nonumber\\
&&
\Pi_\perp^{{\rm(vac)}L}(p_0,\bar{p}) =
\Pi_\perp^{{\rm(vac)}T}(p_0,\bar{p}) =
- p^2 \left[
  2 z_2(M_\rho) + \widetilde{\Pi}_\perp^{LT}(p^2)
\right]
\ .
\end{eqnarray}
The explicit forms of the hadronic thermal corrections
$\bar{\Pi}_\perp^{t,s,L,T}(p_0,\bar{p};T)$
are summarized in Eqs.~(\ref{AA t})--(\ref{AA T})
in Appendix~\ref{app:HTC}.


For the two-point functions of
$\overline{V}_\mu$-$\overline{V}_\nu$,
$\overline{V}_\mu$-$\overline{\cal V}_\nu$ and $\overline{\cal
V}_\mu$-$\overline{\cal V}_\nu$ we adopt similar on-shell
renormalization conditions. The resultant sums of the tree
contributions and quantum corrections take the forms
\begin{eqnarray}
&&
\Pi_V^{{\rm(vac)}t}(p_0,\bar{p}) =
\Pi_V^{{\rm(vac)}s}(p_0,\bar{p})
\nonumber\\
&& \quad
=
\Pi_{V\parallel}^{{\rm(vac)}t}(p_0,\bar{p}) =
\Pi_{V\parallel}^{{\rm(vac)}s}(p_0,\bar{p})
\nonumber\\
&& \quad
=
\Pi_{\parallel}^{{\rm(vac)}t}(p_0,\bar{p}) =
\Pi_{\parallel}^{{\rm(vac)}s}(p_0,\bar{p})
\nonumber\\
&& \quad
=
F_\sigma^2(M_\rho) + \widetilde{\Pi}_V^S(p^2)
\ ,
\nonumber\\
&&
\Pi_V^{{\rm(vac)}L}(p_0,\bar{p}) =
\Pi_V^{{\rm(vac)}T}(p_0,\bar{p})
=
-p^2 \left[
  - \frac{1}{g^2(M_\rho)} + \widetilde{\Pi}_V^{LT}(p^2)
\right]
\ ,
\nonumber\\
&&
\Pi_{V\parallel}^{{\rm(vac)}L}(p_0,\bar{p}) =
\Pi_{V\parallel}^{{\rm(vac)}T}(p_0,\bar{p})
=
-p^2 \left[
  z_3(M_\rho) + \widetilde{\Pi}_{V\parallel}^{LT}(p^2)
\right]
\ ,
\nonumber\\
&&
\Pi_{\parallel}^{{\rm(vac)}L}(p_0,\bar{p}) =
\Pi_{\parallel}^{{\rm(vac)}T}(p_0,\bar{p})
=
-p^2 \left[
  2 z_1(M_\rho) + \widetilde{\Pi}_{\parallel}^{LT}(p^2)
\right]
\ ,
\label{Pis vac}
\end{eqnarray}
where the parameters are renormalized at the scale $M_\rho$ and
the finite renormalization terms satisfy
\begin{eqnarray}
\mbox{Re} \,\widetilde{\Pi}_V^S(p^2=M_\rho^2)
=
\mbox{Re} \,\widetilde{\Pi}_V^{LT}(p^2=M_\rho^2)
=
\mbox{Re} \,\widetilde{\Pi}_{V\parallel}^{LT}(p^2=M_\rho^2)
=
\mbox{Re} \,\widetilde{\Pi}_{\parallel}^{LT}(p^2=M_\rho^2)
= 0
\ .
\label{cond V}
\end{eqnarray}
The hadronic thermal corrections to the above two-point functions
relevant to the present analysis are given in Eqs.~(\ref{rr vv rv
ts}) and (\ref{vv L}) in Appendix~\ref{app:HTC}.


It should be noticed that the renormalized parameters have the
intrinsic temperature dependences in addition to the dependence on the
renormalization point.
Then, the notations used above for the parameters renormalized at
on-shell should be understood as the following abbreviated notations:
\begin{eqnarray}
&& F_\pi(0) \equiv F_\pi(\mu=0;T) \ ,
\nonumber\\
&& F_\sigma(M_\rho) \equiv F_\sigma(\mu=M_\rho(T);T) \ ,
\nonumber\\
&& g(M_\rho) \equiv g(\mu=M_\rho(T);T) \ ,
\nonumber\\
&& z_{1,2,3}(M_\rho) \equiv z_{1,2,3}(\mu=M_\rho(T);T)
\ ,
\end{eqnarray}
where $\mu$ is the renormalization point and the mass parameter
$M_\rho$ is determined from the on-shell condition:
\begin{equation}
M_\rho^2 \equiv M_\rho^2(T)
= g^2(\mu=M_\rho(T);T) F_\sigma^2(\mu=M_\rho(T);T)
\ .
\end{equation}
In addition,
the parameter $a$ appearing in several expressions in Appendices
are defined as
\begin{equation}
a \equiv \frac{F_\sigma^2(\mu=M_\rho(T);T)}{F_\pi^2(\mu=M_\rho(T);T)}
\ .
\end{equation}


\section{Current Correlators}
\indent\indent
We now turn to construct the axial-vector and vector current
correlators from the two-point functions calculated in the previous
section.
The correlators are defined by
\begin{eqnarray}
G_A^{\mu\nu}(p_0=i\omega_n,\vec{p};T) \delta_{ab}
=
\int_0^{1/T} d \tau \int d^3\vec{x}
e^{-i(\vec{p}\cdot\vec{x}+\omega_n\tau)}
\left\langle
  J_{5a}^\mu(\tau,\vec{x}) J_{5b}^\nu(0,\vec{0})
\right\rangle_\beta
\ ,
\nonumber\\
G_V^{\mu\nu}(p_0=i\omega_n,\vec{p};T) \delta_{ab}
=
\int_0^{1/T} d \tau \int d^3\vec{x}
e^{-i(\vec{p}\cdot\vec{x}+\omega_n\tau)}
\left\langle
  J_a^\mu(\tau,\vec{x}) J_b^\nu(0,\vec{0})
\right\rangle_\beta
\ ,
\end{eqnarray}
where $J_{5a}^\mu$ and $J_a^\mu$ are, respectively, the
axial-vector and vector currents, $\omega_n=2n\pi T$ is the
Matsubara frequency, $(a,b)=1,\ldots,N_f^2-1$ denotes the flavor
index and $\langle ~\rangle_\beta$ the thermal average. The
correlators for Minkowski momentum are obtained by the analytic
continuation of $p_0$.


For constructing the axial-vector current correlator
$G_A^{\mu\nu}(p_0,\vec{p};T)$ from the
$\overline{\cal A}_\mu$-$\overline{\cal A}_\nu$ two-point function,
it is convenient to take the unitary gauge of the
background HLS and parameterize the background fields
$\bar{\xi}_{\rm L}$ and $\bar{\xi}_{\rm R}$ as
\begin{equation}
\bar{\xi}_{\rm L} = e^{-\bar{\phi}} \ ,
\quad
\bar{\xi}_{\rm R} = e^{\bar{\phi}} \ ,
\quad
\bar{\phi} = \bar{\phi}_a T_a \ .
\end{equation}
where $\bar{\phi}$ denotes the background field corresponding to
the pion field. In terms of this $\bar{\phi}$, the background
$\overline{\cal A}_\mu$ is expanded as
\begin{equation}
\overline{\cal A}_\mu
= {\cal A}_\mu + \partial_\mu \bar{\phi} + \cdots \ ,
\label{A bar exp}
\end{equation}
where the ellipses stand for the terms that include two or more
fields. Then, the axial-vector current correlator is
\begin{eqnarray}
G_A^{\mu\nu} =
\frac{
  p_\alpha p_\beta \Pi_\perp^{\mu\alpha} \Pi_\perp^{\nu\beta}
}{
  - p_{\bar{\mu}} p_{\bar{\nu}} \Pi_\perp^{\bar{\mu}\bar{\nu}}
}
+ \Pi_\perp^{\mu\nu}
\ ,
\end{eqnarray}
where the first term comes from the $\bar{\phi}$-exchange and the
second term from the direct ${\cal A}_\mu$-${\cal A}_\nu$
interaction. By using the decomposition in Eq.~(\ref{Pi perp
decomp}), this can be rewritten as
\begin{equation}
G_A^{\mu\nu} =
P_L^{\mu\nu} G_A^L + P_T^{\mu\nu} G_A^T
\ ,
\label{GA}
\end{equation}
where
\begin{eqnarray}
G_A^L
&=&
\frac{ p^2 \, \Pi_\perp^t \Pi_\perp^s }{
  - \left[
      p_0^2 \, \Pi_\perp^t - \bar{p}^2 \, \Pi_\perp^s
  \right]
}
+ \Pi_\perp^L
\ ,
\label{GAL}
\\
G_A^T
&=&
- \Pi_\perp^s + \Pi_\perp^T
\ .
\label{GAT}
\end{eqnarray}
One can see from (\ref{GAL}) that the pion exchange contribution
is included only in the longitudinal component $G_A^L$.


To obtain the vector current correlator $G_V$, we first consider
the $\overline{V}$ propagator. By using the fact that the inverse
$\overline{V}$ propagator $i (D^{-1})^{\mu\nu}$ is equal to
$\Pi_V^{\mu\nu}$, the propagator for the field $\overline{V}$ can
be expressed as
\begin{eqnarray}
- i D_V^{\mu\nu}
=
u^\mu u^\nu D_V^t + ( g^{\mu\nu} - u^\mu u^\nu ) D_V^s
+ P_L^{\mu\nu} D_V^L + P_T^{\mu\nu} D_V^T
\ ,
\end{eqnarray}
where
\begin{eqnarray}
&&
D_V^t =
\frac{
  p^2 ( \Pi_V^s - \Pi_V^L )
}{
  p_0^2 \Pi_V^t (\Pi_V^s - \Pi_V^L)
  - \bar{p}^2 \Pi_V^s (\Pi_V^t - \Pi_V^L)
}
\ ,
\label{Dt def}
\\
&&
D_V^s =
\frac{
  p^2 ( \Pi_V^t - \Pi_V^L )
}{
  p_0^2 \Pi_V^t (\Pi_V^s - \Pi_V^L)
  - \bar{p}^2 \Pi_V^s (\Pi_V^t - \Pi_V^L)
}
\ ,
\label{Ds def}
\\
&&
D_V^L =
\frac{
  - p^2 \Pi_V^L
}{
  p_0^2 \Pi_V^t (\Pi_V^s - \Pi_V^L)
  - \bar{p}^2 \Pi_V^s (\Pi_V^t - \Pi_V^L)
}
\ ,
\label{DL def}
\\
&&
D_V^T =
D_V^s - \frac{1}{ \Pi_V^s - \Pi_V^T }
\ .
\label{DT def}
\end{eqnarray}
By using the above propagator $D_V$ and two-point functions of
$\overline{\cal V}_\mu$-$\overline{\cal V}_\mu$ and
$\overline{V}_\mu$-$\overline{\cal V}_\nu$, $G_V$ can be put into
the form
\begin{equation}
G_V^{\mu\nu} = \Pi_{V\parallel}^{\mu\alpha}
i D_{V,\alpha\beta} \Pi_{V\parallel}^{\beta\nu}
+ \Pi_{\parallel}^{\mu\nu}
\ .
\end{equation}
After a lengthy calculation, we obtain
\begin{eqnarray}
G_V^{\mu\nu}
&=&
u^\mu u^\nu
\Biggl[
  \frac{ D_V^L }{ \Pi_V^L }
  \biggl\{
    \frac{\bar{p}^2}{ p^2 } \Pi_{V\parallel}^L
    \left(
      \Pi_V^s \Pi_{V\parallel}^t - \Pi_V^t \Pi_{V\parallel}^s
    \right)
\nonumber\\
&& \qquad\qquad
    {} - \frac{\Pi_{V\parallel}^t}{p^2}
    \biggl(
      -p_0^2 \Pi_{V\parallel}^t (\Pi_V^s - \Pi_V^L)
      + \bar{p}^2 \left(
        \Pi_{V\parallel}^t \Pi_V^s - \Pi_{V\parallel}^s \Pi_V^L
      \right)
    \biggr)
  \biggr\}
  + \Pi_\parallel^t
\Biggr]
\nonumber\\
&&
{} + ( g^{\mu\nu} - u^\mu u^\nu )
\Biggl[
  \frac{ D_V^L }{ \Pi_V^L }
  \biggl\{
    \frac{p_0^2}{ p^2 } \Pi_{V\parallel}^L
    \left(
      \Pi_V^s \Pi_{V\parallel}^t - \Pi_V^t \Pi_{V\parallel}^s
    \right)
\nonumber\\
&& \qquad\qquad
    {} - \frac{\Pi_{V\parallel}^s}{p^2}
    \biggl(
      -p_0^2 \left(
        \Pi_V^t \Pi_{V\parallel}^s - \Pi_{V\parallel}^t\Pi_V^L
      \right)
      + \bar{p}^2 \Pi_{V\parallel}^s \left(
        \Pi_V^t - \Pi_V^L
      \right)
    \biggr)
  \biggr\}
  + \Pi_\parallel^s
\Biggr]
\nonumber\\
&&
{} + P_L^{\mu\nu}
\Biggl[
  \frac{ D_V^L }{ \Pi_V^L }
  \biggl\{
    - \Pi_V^L \Pi_{V\parallel}^t \Pi_{V\parallel}^s
    + \Pi_{V\parallel}^L \left(
      \Pi_{V\parallel}^t \Pi_V^s + \Pi_{V\parallel}^s \Pi_V^t
    \right)
\nonumber\\
&& \qquad\qquad
    {} - \frac{1}{p^2} \left(
      p_0^2 \Pi_V^t - \bar{p}^2 \Pi_V^s
    \right)
    \left( \Pi_{V\parallel}^L \right)^2
  \biggr\}
  + \Pi_\parallel^L
\Biggr]
\nonumber\\
&&
{} + P_T^{\mu\nu}
\Biggl[
  \frac{ D_V^L }{ \Pi_V^L }
  \biggl\{
    - \frac{p_0^2}{p^2} \Pi_{V\parallel}^L
    \left(
      \Pi_V^t \Pi_{V\parallel}^s - \Pi_V^s \Pi_{V\parallel}^t
    \right)
\nonumber\\
&& \qquad\qquad
    {} - \frac{\Pi_{V\parallel}^s}{p^2}
    \left(
      -p_0^2 \left(
        \Pi_V^t \Pi_{V\parallel}^s - \Pi_{V\parallel}^t\Pi_V^L
      \right)
      + \bar{p}^2 \Pi_{V\parallel}^s \left(
        \Pi_V^t - \Pi_V^L
      \right)
    \right)
  \biggr\}
\nonumber\\
&& \qquad\qquad
  {} + \frac{ \left( \Pi_{V\parallel}^s - \Pi_{V\parallel}^T \right)^2
  }{ \Pi_V^s - \Pi_V^T }
  + \Pi_\parallel^T
\Biggr]
\ .
\label{PGP exp 1}
\end{eqnarray}
One might worry that the above form does not satisfy the current
conservation $p_\mu G_V^{\mu\nu} = 0$. However since, as shown in
Eq.~(\ref{rr vv rv ts}), the conditions
\begin{eqnarray}
&& \Pi_V^t = - \Pi_{V\parallel}^t = \Pi_{\parallel}^t \ ,
\nonumber\\
&&
\Pi_V^s = - \Pi_{V\parallel}^s = \Pi_{\parallel}^s
\label{t s equality}
\end{eqnarray}
are satisfied, Eq.~(\ref{PGP exp 1}) can be rewritten as
\begin{eqnarray}
G_V^{\mu\nu}
&=&
P_L^{\mu\nu}
\left[
  \left( \frac{ - D_V^L }{ \Pi_V^L } \right)
  \left\{
    \Pi_V^t \Pi_V^s \left( \Pi_V^L + 2 \Pi_{V\parallel}^L \right)
    + \frac{ p_0^2 \Pi_V^t - \bar{p}^2 \Pi_V^s }{p^2}
    \left( \Pi_{V\parallel}^L \right)^2
  \right\}
  + \Pi_{\parallel}^L
\right]
\nonumber\\
&&
{} +
P_T^{\mu\nu}
\left[
  \frac{
    \Pi_V^s \left( \Pi_V^T + 2 \Pi_{V\parallel}^T \right)
    + \left( \Pi_{V\parallel}^T \right)^2
  }{
    \Pi_V^s - \Pi_V^T
  }
  + \Pi_{\parallel}^T
\right]
\ .
\label{GV mn form}
\end{eqnarray}
Now it is evident that the current is conserved since $p_\mu
P_L^{\mu\nu} = p_\mu P_T^{\mu\nu} = 0$. In the present analysis,
the equality $\bar{\Pi}_V^t = \bar{\Pi}_V^s$ is seen to hold as
shown in Eq.~(\ref{rr vv rv ts}). This implies that $\Pi_V^t =
\Pi_V^s$ is also satisfied since the quantum corrections to
$\Pi_V^t$ and $\Pi_V^s$ are equal to each other due to Lorentz
invariance. Thus, $G_V^{\mu\nu}$ can be written as
\begin{equation}
G_V^{\mu\nu} = P_L^{\mu\nu} G_V^L + P_T^{\mu\nu} G_V^T
\ ,
\label{GV mn}
\end{equation}
where
\begin{eqnarray}
G_V^L
&=&
  \frac{
    \Pi_V^t \left( \Pi_V^L + 2 \Pi_{V\parallel}^L \right)
  }{
    \Pi_V^t - \Pi_V^L
  }
  + \Pi_{\parallel}^L
\label{GVL}
\\
G_V^T
&=&
  \frac{
    \Pi_V^t \left( \Pi_V^T + 2 \Pi_{V\parallel}^T \right)
  }{
    \Pi_V^t - \Pi_V^T
  }
  + \Pi_{\parallel}^T
\ .
\label{GVT}
\end{eqnarray}
Note that, in the above expressions, we have dropped the terms
$\left( \Pi_{V\parallel}^L \right)^2$ and
$\left( \Pi_{V\parallel}^T \right)^2$ since they are of higher order.



\section{Pion Decay Constants and Pion Velocity}
\label{sec:PDCV}
 \indent\indent We now proceed to study the
on-shell structure of the pion. For this we look at the pole of
the longitudinal component $G_A^L$ in Eq.~(\ref{GAL}). Since both
$\Pi_\perp^t$ and $\Pi_\perp^s$ have imaginary parts, we choose to
determine the pion energy $E$ from the real part by solving the
dispersion formula
\begin{eqnarray}
  0
&=&
  \left[
    p_0^2 \, \mbox{Re} \Pi^{t}_\perp (p_0,\bar{p};T)
    - \bar{p}^2 \, \mbox{Re} \Pi^{s}_\perp (p_0,\bar{p};T)
  \right]_{p_0=E}
\ ,
\label{pi on shell cond}
\end{eqnarray}
where $\bar{p}\equiv\vert\vec{p}\vert$. As remarked in
Section~\ref{sec:TPF}, in HLS at one-loop level, $\Pi^{t}_\perp
(p_0,\bar{p};T)$ and $\Pi^{s}_\perp (p_0,\bar{p};T)$ are of the
form
\begin{eqnarray}
\Pi^{t}_\perp (p_0,\bar{p};T)
&=&
F_\pi^2(0) + \widetilde{\Pi}_\perp^S(p^2) +
\bar{\Pi}^{t}_\perp (p_0,\bar{p};T)
\ ,
\nonumber\\
\Pi^{s}_\perp (p_0,\bar{p};T)
&=&
F_\pi^2(0) + \widetilde{\Pi}_\perp^S(p^2) +
\bar{\Pi}^{s}_\perp (p_0,\bar{p};T)
\ ,
\label{Pi t s forms}
\end{eqnarray}
where $\widetilde{\Pi}_\perp^S(p^2)$ is the finite renormalization
contribution, and $\bar{\Pi}^{t}_\perp (p_0,\bar{p};T) $ and
$\bar{\Pi}^{s}_\perp (p_0,\bar{p};T) $ are the hadronic thermal
contributions. Substituting Eq.~(\ref{Pi t s forms}) into
Eq.~(\ref{pi on shell cond}), we obtain
\begin{equation}
0 =
\left( E^2 - \bar{p}^2 \right)
\left[
  F_\pi^2(0) +
  \mbox{Re} \,\widetilde{\Pi}_\perp^S( p^2=E^2 - \bar{p}^2 )
\right]
+
  E^2
  \mbox{Re} \,\bar{\Pi}^{t}_\perp (E,\bar{p};T)
  - \bar{p}^2
  \mbox{Re} \, \bar{\Pi}^{s}_\perp (E,\bar{p};T)
\ .
\end{equation}
The pion velocity $v_\pi(\bar{p}) \equiv E / \bar{p}$ is then obtained
by solving
\begin{eqnarray}
v_\pi^2(\bar{p})
&=&
\frac{
  F_\pi^2(0) +
  \mbox{Re} \, \bar{\Pi}^{s}_\perp (\bar{p},\bar{p};T)
}{
  F_\pi^2(0) +
  \mbox{Re} \, \bar{\Pi}^{t}_\perp (\bar{p},\bar{p};T)
}
\ .
\label{v2: form}
\end{eqnarray}
Here we replaced $E$ by $\bar{p}$ in the hadronic thermal terms
$\bar{\Pi}_\perp^t (E,\vec{p})$ and $\bar{\Pi}_\perp^s
(E,\vec{p})$ as well as in the finite renormalization contribution
$\widetilde{\Pi}_\perp^S( p^2 = E^2 - \bar{p}^2)$, since the
difference is of higher order. [Note that $
\widetilde{\Pi}_\perp^S( p^2=0 )= 0$.]


Next we determine the wave function renormalization of the pion
field, which relates the background field $\bar{\phi}$ to the pion
field $\bar{\pi}$ in the momentum space as
 \be
\bar{\phi} = \bar{\pi}/\widetilde{F}(\bar{p};T).
 \ee
We follow the analysis in Ref.~\cite{MOW} to obtain
\begin{equation}
\widetilde{F}^2(\bar{p};T) =
\mbox{Re} \Pi_\perp^t(E,\bar{p};T)
= F_\pi^2(0) +
  \mbox{Re} \, \bar{\Pi}^{t}_\perp (\bar{p},\bar{p};T)
\ .
\label{Ftil def}
\end{equation}
Using this wave function renormalization and the velocity in
Eq.~(\ref{v2: form}), we can rewrite the longitudinal part of the
axial-vector current correlator as
\begin{eqnarray}
G_A^L(p_0,\vec{p})
&=&
\frac{ p^2
  \Pi_\perp^t(p_0,\bar{p};T) \Pi_\perp^s(p_0,\bar{p};T)
  / \widetilde{F}^2(\bar{p};T)
}{
  - \left[
      p_0^2 - v_\pi^2(\bar{p}) \bar{p}^2 + \Pi_\pi(p_0,\bar{p};T)
  \right]
}
+ \Pi_\perp^L(p_0,\bar{p};T)
\ ,
\label{GAL 2}
\end{eqnarray}
where the pion self energy $\Pi_\pi(p_0,\bar{p};T)$ is given by
\begin{eqnarray}
&&
\Pi_\pi(p_0,\bar{p};T)
=
\frac{1}{
  \mbox{Re} \, \Pi^{t}_\perp (E,\bar{p};T)
}
\nonumber\\
&& \quad
\times
\Biggl[
  p_0^2
  \left\{
    \Pi^{t}_\perp (p_0,\bar{p};T)
    -
    \mbox{Re} \, \Pi^{t}_\perp (E,\bar{p};T)
  \right\}
  -
  \bar{p}^2
  \left\{
    \Pi^{s}_\perp (p_0,\bar{p};T)
    -
    \mbox{Re} \, \Pi^{s}_\perp (E,\bar{p};T)
  \right\}
\Biggr]
\ .
\end{eqnarray}

Let us now define the pion decay constant. A natural procedure is
to define the pion decay constant from the pole residue of the
axial-vector current correlator. From Eq.~(\ref{GAL 2}), the pion
decay constant is given by
\begin{eqnarray}
f_\pi^2(\bar{p};T)
&=&
\frac{
 \Pi_\perp^t(E,\bar{p};T)
 \Pi_\perp^s(E,\bar{p};T)
}{ \widetilde{F}^2(\bar{p};T) }
\nonumber\\
&=&
\frac{
 \left[ F_\pi^2(0) + \bar{\Pi}^{t}_\perp (\bar{p},\bar{p};T) \right]
 \left[ F_\pi^2(0) + \bar{\Pi}^{s}_\perp (\bar{p},\bar{p};T) \right]
}{ \widetilde{F}^2(\bar{p};T) }
\ .
\label{fpi2 def}
\end{eqnarray}
We now address how $f_\pi^2(\bar{p};T)$ is related to the temporal
and spatial components of the pion decay constant introduced in
Ref.~\cite{PT:96}. Following their notation, let $f_\pi^t$ denote
the decay constant associated with the temporal component of the
axial-vector current and $f_\pi^s$ the one with the spatial
component. In the present analysis, they can be read off from the
coupling of the $\bar{\pi}$ field to the axial-vector external
field ${\cal A}_\mu$:
\begin{eqnarray}
  f_\pi^t(\bar{p};T)
&\equiv&
  \frac{
    \Pi^{t}_\perp (E,\bar{p};T)
  }{\widetilde{F}(\bar{p};T)}
=
  \frac{ F_\pi^2(0) + \bar{\Pi}^{t}_\perp (\bar{p},\bar{p};T) }
    {\widetilde{F}(\bar{p};T)}
\ ,
\label{fpit def}
\\
  f_\pi^s(\bar{p};T)
&\equiv&
  \frac{
    \Pi^{s}_\perp (\tilde{E},\bar{p};T)
  }{\widetilde{F}(\bar{p};T)}
=
  \frac{ F_\pi^2(0) + \bar{\Pi}^{s}_\perp (\bar{p},\bar{p};T) }
    {\widetilde{F}(\bar{p};T)}
\ .
\label{fpis def}
\end{eqnarray}
Comparing Eqs.~(\ref{fpit def}) and (\ref{fpis def}) with
Eqs.~(\ref{v2: form}), (\ref{Ftil def}) and (\ref{fpi2 def}), we
have~\cite{PT:96,MOW}
\begin{eqnarray}
\widetilde{F}(\bar{p};T)
&=& \mbox{Re} \, f_\pi^t(\bar{p};T)
\ ,
\\
f_\pi^2(\bar{p};T) &=&
  f_\pi^t(\bar{p};T) f_\pi^s(\bar{p};T)
\ ,
\\
v_\pi^2(\bar{p}) &=& \frac{\mbox{Re}
\,f_\pi^s(\bar{p};T)}{\mbox{Re}\,f_\pi^t(\bar{p};T)} \ . \label{v2
rel}
\end{eqnarray}
We should note that there are no thermal quasiquark contributions
to $\bar{\Pi}_\perp^t(\bar{p},\bar{p};T)$ as well as to
$\bar{\Pi}_\perp^s(\bar{p},\bar{p};T)$ to the leading order
because $\bar{B}_{0(F)}(\bar{p},\bar{p};m_q,m_q;T) =
0$.~\footnote{The vanishing of $\bar{B}_{0(F)}$ on-shell follows
from Eqs. (3.2) and (3.4) because in the chiral limit, the pion
mass is zero. This also occurs in dense medium. We should note,
however, that this on-shell condition is not needed for our
argument to be valid. As one can see in Eqs. (A.1) and (A.2), this
quantity appears multiplied by $m_q^2$ which vanishes at the
critical point, so it does not contribute in any case.}


We are now ready to investigate what happens to the above
quantities when the critical temperature $T_c$ is approached. Due
to the VM in hot matter~\cite{HaradaSasaki},  the $parametric$
$\rho$ meson mass goes to zero ($M_\rho\rightarrow0$) and the
parameter $a$ approaches one ($a\rightarrow1$), so we have [see
Eq.~(\ref{PiA ts Tc app})]
\begin{eqnarray}
&&
\bar{\Pi}_\perp^t(\bar{p},\bar{p};T)
\mathop{\longrightarrow}_{T \rightarrow T_c}
- \frac{N_f}{2} \widetilde{J}_{1,(B)}^2(0;T_c)
= - \frac{N_f}{24} T_c^2
\ ,
\nonumber\\
&&
\bar{\Pi}_\perp^s(\bar{p},\bar{p};T)
\mathop{\longrightarrow}_{T \rightarrow T_c}
- \frac{N_f}{2} \widetilde{J}_{1,(B)}^2(0;T_c)
= - \frac{N_f}{24} T_c^2
\ .
\label{PiA ts Tc}
\end{eqnarray}
Substituting these into the expression of the pion velocity
in Eq.~(\ref{v2: form}), we obtain
\begin{equation}
v_\pi^2(\bar{p}) \mathop{\longrightarrow}_{T \rightarrow T_c} 1 \
.
\end{equation}
This is our first main result: in the framework of the VM, the
pion velocity approaches 1 near the critical temperature, not 0 as
in the case of the pion-only situation~\cite{SS:1,SS:2}.

{}From Eq.~(\ref{PiA ts Tc}), we can evaluate the pion decay
constant Eq.~(\ref{fpi2 def}) at the critical temperature which
comes out to be
\begin{equation}
f_\pi^2(\bar{p};T_c) = F_\pi^2(0) - \frac{N_f}{24} T_c^2
\ .
\end{equation}
Since this $f_\pi$ is the order parameter and should vanish at the
critical temperature, the parameter $F_\pi^2(0)$ at $T=T_c$ is
given at $T_c$ as~\cite{HaradaSasaki}
\begin{equation}
F_\pi^2(0)
\ \mathop{\longrightarrow}_{T \rightarrow T_c}\
\frac{N_f}{24} T_c^2
\ .
\label{Fp0 Tc}
\end{equation}
Substituting Eq.~(\ref{PiA ts Tc}) together with Eq.~(\ref{Fp0
Tc}) into Eqs.~(\ref{fpit def}) and (\ref{fpis def}), we conclude
that both temporal and spatial pion decay constants vanish at the
critical temperature~\footnote{These results differ from those
obtained in a framework in which the $only$ relevant degrees of
freedom near chiral restoration are taken to be the
pions~\cite{SS:1,SS:2}. We will explain how this comes about in
the conclusion section.}:
\begin{equation}
f_\pi^t(\bar{p};T_c) = f_\pi^s(\bar{p};T_c) = 0 \ .
\end{equation}
This is our second main result. Note that while $f_\pi^t (T_c)=0$,
$\chi_A (T_c)$ is non-zero in consistency with the lattice result.
Here the hidden gauge boson and quasiquark play an essential role.


\section{Axial-Vector and Vector Susceptibilities}
\label{sec:SUS}
 \indent\indent In terms of the quantities defined
in the preceding sections, the axial-vector susceptibility
$\chi_A(T)$ and the vector susceptibility $\chi_V(T)$ for
non-singlet currents~\footnote{We will confine ourselves to
non-singlet (that is, isovector) susceptibilities, so we won't
specify the isospin structure from here on.} are given by the
$00$-component of the axial-vector and vector current correlators
in the static--low-momentum limit:
\begin{eqnarray}
&&
\chi_A(T)  = 2 N_f \,
  \lim_{\bar{p}\rightarrow0}
  \lim_{p_0\rightarrow0}
  \left[ G_A^{00}(p_0,\vec{p};T) \right]
\ ,
\nonumber\\
&&
\chi_V(T)
= 2 N_f \, \lim_{\bar{p}\rightarrow0}
  \lim_{p_0\rightarrow0}
  \left[ G_V^{00}(p_0,\vec{p};T) \right]
\ ,
\label{def chiV}
\end{eqnarray}
where we have included the normalization factor of $2 N_f$. Using
the current correlators given in Eqs.~(\ref{GA}) and (\ref{GV mn
form}) and noting that $ \lim_{p_0\rightarrow0} P_L^{00} =
\lim_{p_0\rightarrow0} \bar{p}^2/p^2 = - 1 $, we can express
$\chi_A(T)$ and $\chi_V(T)$ as
\begin{eqnarray}
&&
\chi_A(T)  = - 2 N_f \,\lim_{\bar{p}\rightarrow0}
\lim_{p_0\rightarrow0}
\left[
  \Pi_\perp^L(p_0,\vec{p};T) - \Pi_\perp^t(p_0,\vec{p};T)
\right]
\ ,
\nonumber\\
&&
\chi_V(T)  = - 2 N_f \,\lim_{\bar{p}\rightarrow0}
\lim_{p_0\rightarrow0}
\left[
  \frac{
    \Pi_V^t \left( \Pi_V^L + 2 \Pi_{V\parallel}^L \right)
  }{
    \Pi_V^t - \Pi_V^L
  }
  + \Pi_{\parallel}^L
\right]
\ ,
\end{eqnarray}
where for simplicity of notation, we have suppressed the argument
$(p_0,\vec{p};T)$ in the right-hand-side of the expression for
$\chi_V(T)$. In HLS theory at one-loop level, the susceptibilities
read
\begin{eqnarray}
&&
\chi_A(T)  = 2 N_f \left[
  F_\pi^2(0)
  + \lim_{\bar{p}\rightarrow0}
  \lim_{p_0\rightarrow0}
  \left\{
    \bar{\Pi}_\perp^t(p_0,\vec{p};T) -
    \bar{\Pi}_\perp^L(p_0,\vec{p};T)
  \right\}
\right]
\ ,
\nonumber\\
&&
\chi_V(T)  =
- 2 N_f \,\lim_{\bar{p}\rightarrow0}
\lim_{p_0\rightarrow0}
\left[
  \frac{
    \left( a(0) F_\pi^2(0) + \bar{\Pi}_V^t \right)
    \left( \bar{\Pi}_V^L + 2 \bar{\Pi}_{V\parallel}^L \right)
  }{
    a(0) F_\pi^2(0) + \bar{\Pi}_V^t
    - \bar{\Pi}_V^L
  }
  + \Pi_{\parallel}^L
\right]
\ ,
\label{chiV}
\end{eqnarray}
where the parameter $a(0)$ is defined by
\begin{eqnarray}
a(0) &=&
\frac{ \Pi_V^{{\rm(vac)}t}(p_0=0,\bar{p}=0) }{ F_\pi^2(0) }
= \frac{ \Pi_V^{{\rm(vac)}s}(p_0=0,\bar{p}=0) }{ F_\pi^2(0) }
\ .
\end{eqnarray}
In Ref.~\cite{HY:WM}, $a(0)$ was defined by the ratio
$F_\sigma^2(M_\rho)/F_\pi^2(0)$ without taking into account the
finite renormalization effect which depends on the details of the
renormalization condition. In the present renormalization
condition (\ref{Pis vac}) with Eq.~(\ref{cond V}), the finite
renormalization effect leads to
\begin{equation}
\widetilde{\Pi}_V^S(p^2=0) =
\frac{N_f}{(4\pi)^2} M_\rho^2
\left( 2 - \sqrt{3} \tan^{-1} \sqrt{3} \right)
\ ,
\end{equation}
and then $a(0)$ reads
\begin{equation}
a(0) = \frac{F_\sigma^2(M_\rho)}{F_\pi^2(0)}
+ \frac{N_f}{(4\pi)^2} \frac{ M_\rho^2 }{F_\pi^2(0)}
\left( 2 - \sqrt{3} \tan^{-1} \sqrt{3} \right)
\ .
\label{a0 exp}
\end{equation}
It follows from the static--low-momentum limit of
$(\bar{\Pi}_\perp^t - \bar{\Pi}_\perp^L)$ given in Eq.~(\ref{PiA
tmL SL}) that the axial-vector susceptibility $\chi_A(T)$ takes
the form
\begin{eqnarray}
\chi_A(T) &=& 2N_f
\Biggl[
  F_\pi^2(0)
  - N_f \widetilde{J}_{1,(B)}^2(0;T)
  + N_f a\, \widetilde{J}_{1,(B)}^2(M_\rho;T)
\nonumber\\
&& \qquad
  {}- N_f \frac{a}{M_\rho^2}
  \left\{
    \widetilde{J}_{-1,(B)}^2(M_\rho;T)
    - \widetilde{J}_{-1,(B)}^2(0;T)
  \right\}
  + 4 \lambda^2 N_c \widetilde{J}_{1,(F)}^2(m_q;T)
\Biggr]
\ .
\label{chiA}
\end{eqnarray}
Near the critical temperature ($T\rightarrow T_c$), we have
$M_\rho \rightarrow 0$, $a\rightarrow1$ due to the intrinsic
temperature dependence in the VM in hot
matter~\cite{HaradaSasaki}. As shown in Ref.~\cite{HKR} for the
finite density chiral restoration, $m_q\rightarrow0$ and
$\lambda\rightarrow1$ should be also satisfied at the finite
temperature chiral restoration. Furthermore, from Eq.~(\ref{Fp0
Tc}), we see that the parameter $F_\pi^2(0)$ approaches
$\frac{N_f}{24} T_c^2$ for $T\rightarrow T_c$. Substituting these
conditions into Eq.~(\ref{chiA}) and noting that
\begin{equation}
\lim_{M_\rho\rightarrow0}
\left[
  {}- \frac{1}{M_\rho^2}
  \left\{
    \widetilde{J}_{-1,(B)}^2(M_\rho;T)
    - \widetilde{J}_{-1,(B)}^2(0;T)
  \right\}
\right]
= \frac{1}{2} \widetilde{J}_{1,(B)}^2(0;T) = \frac{1}{24} T^2 \ ,
\end{equation}
we obtain
\begin{equation}
\chi_A(T_c) = 2 N_f \left[
  \frac{N_f}{12} T_c^2 + \frac{N_c}{6} T_c^2
\right]
\ .
\label{axial SUS}
\end{equation}


To obtain the vector susceptibility near the critical temperature,
we first consider $ a(0) F_\pi^2(0) + \bar{\Pi}_V^t$ appearing in the
numerator of the first term in the right-hand-side of
Eq.~(\ref{chiV}). Using Eq.~(\ref{Pir t SL}), we get for the
static--low-momentum limit of $ a(0) F_\pi^2(0) + \bar{\Pi}_V^t$ as
\begin{eqnarray}
&&
\lim_{\bar{p}\rightarrow0}
\lim_{p_0\rightarrow0}
\left[ a(0) F_\pi^2(0) + \bar{\Pi}_V^t(p_0,\bar{p};T) \right]
\nonumber\\
&& \qquad
=
a(0) F_\pi^2(0)
- \frac{N_f}{4} \left[
  2 \widetilde{J}_{-1,(B)}^0(M_\rho;T)
  - \widetilde{J}_{1,(B)}^2(M_\rho;T)
  + a^2 \, \widetilde{J}_{1,(B)}^2(0;T)
\right]
\ .
\label{aF Pit SL}
\end{eqnarray}
{}From Eq.~(\ref{a0 exp}) we can see that $a(0) \rightarrow 1$ as
$T\rightarrow T_c$ since $F_\sigma^2(M_\rho) \rightarrow
F_\pi^2(0)$ and $M_\rho \rightarrow 0$. Furthermore,
$F_\pi^2(0)\rightarrow \frac{N_f}{24}T_c^2$ as we have shown in
Eq.~(\ref{Fp0 Tc}). Then, the first term of Eq.~(\ref{aF Pit SL})
approaches $\frac{N_f}{24}T_c^2$. The second term, on the other
hand, approaches $- \frac{N_f}{24}T_c^2$ as $M_\rho \rightarrow0$
and $a \rightarrow1$ for $T\rightarrow T_c$. Thus, we have
\begin{equation}
\lim_{\bar{p}\rightarrow0}
\lim_{p_0\rightarrow0}
\left[ a(0) F_\pi^2(0) + \bar{\Pi}_V^t(p_0,\bar{p};T) \right]
\mathop{\longrightarrow}_{T \rightarrow T_c}
0 \ .
\end{equation}
This implies that
only the second term $\Pi_\parallel^L$
in the right-hand-side of Eq.~(\ref{chiV}) contributes
to the vector susceptibility
near the critical temperature.
Thus, taking $M_\rho \rightarrow 0$, $a\rightarrow1$,
$m_q\rightarrow0$ and $\kappa\rightarrow1$ in Eq.~(\ref{Piv L SL}),
we obtain
\begin{equation}
\chi_V(T_c) = 2 N_f \left[
  \frac{N_f}{12} T_c^2 + \frac{N_c}{6} T_c^2
\right]
\ ,
\label{vector SUS}
\end{equation}
which agrees with the axial-vector susceptibility in
Eq.~(\ref{axial SUS}). This is a prediction, not an input
condition, of the theory. For $N_f=2$ and $N_c=3$, we have
\begin{equation}
\chi_A(T_c) =
\chi_V(T_c) = \frac{8}{3} T_c^2
\ .
\label{SUS value}
\end{equation}
This is our third main result.

In order to compare our result with the lattice result, we need to
go beyond the one-bubble approximation. We note here that the
value of VSUS (\ref{SUS value}) differs somewhat from the one
bubble result of Nambu-Jona-Lasinio model, $\chi_0
(T_c)=2T_c^2$~\cite{Kunihiro:91}. The difference is easily
understood by the fact that the vector mesons enter in the VM and
contribute the additional $\frac{2}{3}T_c^2$. There is also a
result from a hard thermal loop calculation which gives $\chi_V
(T_c)\approx 1.3 T_c^2$~\cite{CMT}. However this result cannot be
compared to ours for two reasons. First we need to sum higher
loops in our formalism which may be done in random phase
approximation as in \cite{Kunihiro:91}. Second, the perturbative
QCD with a hard thermal loop approximation may not be valid in the
temperature regime we are considering. Even at $T\gg T_c$, the
situation is not clear as pointed out in \cite{blaizot}.


\section{Hydrodynamic Interpretation}
\indent\indent In this section, we make a brief hydrodynamic
analysis~\cite{hh} of how the VM variables (e.g., vector mesons)
modify the structure of the vector correlator. We do this
following closely the arguments of \cite{SS:2}. Our arguments are
heuristic and non-rigorous but we believe the conclusion to be
valid.

Consider first the case where vector mesons and quasiquarks do not
figure as hydrodynamic variables. In this case, apart from
densities of conserved quantities, the pions which are the phases
of the order parameters of broken chiral symmetry are the relevant
hydrodynamic degrees of freedom together with, near chiral
restoration, the order parameters themselves. The dynamics of the vector
isospin charge $V_0^a$ is then dictated by ``diffusive modes" with
the correlator taking the form~\cite{SS:2}
 \be
\int d^4x e^{iq.x}\la V_0^a(x) V_0^b
(0)\ra=\frac{2TD_V\chi_V\vec{q}^2}{q_0^2+D_V^2
\vec{q}^4}\label{Vcorr}
 \ee
where $D_V$ is the temperature-dependent diffusion constant for
the vector charge. The pole $q_0=\pm i D_V \vec{q}^2$ shows that
it is purely diffusive. This diffusive mode is lacking from the
effective Lagrangian (\ref{Leff}) and hence the effective
Lagrangian in the leading order with pions as the $only$ relevant
degrees of freedom cannot describe the vector susceptibility for
$T\neq 0$.

We can use an analysis closely paralleling that of \cite{SS:2} to
see how the VM degrees of freedom considered as hydrodynamic
variables modifies the correlator. For this, consider how the
vector meson enters in the discussion. In the VM, the vector meson
becomes massless while the vector coupling goes to zero at
$T=T_c$. Therefore at near $T_c$ we may take the vector meson
field as a hydrodynamic variable~\footnote{Away from $T_c$, the
vector field cannot obviously be taken as a a hydrodynamic. It
will depend on the macroscopic distance and time scale in
question.}, since the vector meson energy can be arbitrarily small
near $T_c$ and relaxation time $\tilde\tau$ becomes arbitrarily
large. Note that in the case of QED, the relaxation time $\tilde
\tau$ for the heavy fermions is given by $1/\tilde\tau\sim
\alpha^2 T\log (1/\alpha)$ \cite{leblac}, where
$\alpha=e^2/(4\pi)$. For simplicity, we shall simply assume that
{\it arbitrarily close to $T_c$}, the time component~\footnote{In
principle we should redefine the $\rho_0$ field such that it
carries no dimension as the pion field $\phi$ in Eq.(\ref{V1})
below. Furthermore we should be using the longitudinal component
rather than the time component. At the VM point, the longitudinal
component gets spewed out of the vector meson as a scalar partner
of the pion.  This field redefinition should be understood in what
follows. A rigorous way of formulating our argument would employ
projection operators which would entail, however, nonlocal
operators away from the VM point. Our argument applies therefore
only very near the critical point.} of the vector meson $\rho_0$
can be taken as a hydrodynamical variable and take the time
derivative of the vector charge density as
 \be
\partial_0 V_0^a=D_V \nabla^2 V_0^a+\partial_i\zeta^{a i} +f_\rho^2\nabla^2
\rho_0^a
 \ee
where $\zeta^{a i}$ is the ``noise" in $V_0^a$. This is identical
in form to the corresponding axial charge density~\cite{SS:2}
 \be
\del_0 A_0^a=D_A \nabla^2 A_0^a+\partial_i\xi^{a i}
+f_\pi^2\nabla^2 \phi^a \label{V1}
 \ee
where $D_A$ is the diffusion coefficient, $\phi$ is the pion field
modulo normalization and $\xi$ is the ``noise."  Now writing
 \be
\partial_0\rho^a_0=\frac{1}{\tilde\chi}
V_0^a+\cdots,\label{ra0}\label{V2}
 \ee
where the ellipsis contains terms that we do not need for our
consideration. Equations (\ref{V1}) and (\ref{V2}) modify the
vector correlator from (\ref{Vcorr}) to
 \be
\int d^4x e^{iq\cdot x}\la V_0^a (x)V_0^b(0)\ra
=\frac{2TD_V\chi_Vq_0^2{\vec q}^2}{ (q_0^2-\tilde\omega^2)^2
+D_V^2q_0^2({\vec q}^2)^2}\label{v0m}
 \ee where
$\tilde\omega^2=f_\rho^2{\bf q}^2/\tilde\chi$ and
$\tilde\chi=\chi_V$. The linear dispersion relation implies that
we are dealing with a Goldstone mode in medium. The integral over
frequencies in the limit $|\vec{q}|\rightarrow 0$ is concentrated
near values $q_0=\pm \tilde\omega$. We see that the $\rho_0$ field
plays the same role as the pion field $\phi$. This is precisely
what we expect from the vector manifestation in which the
longitudinal components of the vector mesons join the pions into a
multiplet of parity-doublet Goldstone excitations.

\section{Summary and Remarks}
\indent\indent The notion of the vector manifestation in chiral
symmetry \`a la Harada and Yamawaki requires that the zero-mass
vector mesons be present at the chiral phase transition. As
discussed by Brown and Rho~\cite{QM2002}, the light vector mesons
near the transition point ``bottom-up" can be considered as
Higgsed gluons in the sense of color-flavor locking in the broken
chiral symmetry sector proposed by Berges and
Wetterich~\cite{wett,berges-wett} and could figure in heavy-ion
processes measured at RHIC energies. In this paper we are finding
that in the VM, the vector mesons (and the quasiquarks) with
vanishing masses at the chiral transition (in the chiral limit)
can figure importantly in the vector and axial-vector
susceptibilities near the chiral transition point. The notable
results are that the VM confirms explicitly the equality
$\chi_V=\chi_A$ at $T_c$ and that both $f_\pi^t$ and $f_\pi^s$
vanish simultaneously with the pion velocity $v_\pi=1$. These
differ from the results expected in a scenario where only the
pions are the relevant effective degrees of freedom.

Understanding the differences between the two scenarios could
provide a valuable insight into some of the basic tenets of
effective field theories.

If one assumes that the only light degrees of freedom near $T_c$
are the pions (and possibly a light scalar, a chiral partner to
the pions), then one can simply take the current algebra terms in
the Lagrangian and the axial-vector susceptibility (ASUS) $\chi_A$
is uniquely given by the temporal component of the pion decay
constant $f_\pi^t$ with the degrees of freedom that are integrated
out renormalizing this constant. Then the unquestionable equality
$\chi_V=\chi_A$ at $T_c$ together with the lattice result
$\chi_V|_{T_C}\neq 0$ leads to the Son-Stephanov result on the
pion velocity $v_\pi=0$. There is however a caveat to this simple
result and it is that the same reasoning fails when one computes
explicitly the vector susceptibility (VSUS) using the same current
algebra Lagrangian.

Positing that the vector mesons enter in the VM near $T_c$
circumvents this caveat and at the same time, makes a concrete
prediction. In this framework, the ASUS is given by a term related
to $f_\pi^t$ plus contributions from the vector-meson (i.e., the
longitudinal component $\sigma$) and quasiquark loops. At $T_c$,
the $f_\pi^t$ vanishes and what remains comes out precisely equal
to the VSUS $\chi_V$ in which the $\sigma$ loop in $\chi_A$ is
replaced by the pion loop. All these are perfectly understandable
in terms of the VM in HGS.

Some remarks on the role of quasiquarks are in order. One might
object that there is a risk of double counting in putting both the
light vector mesons and the quasiquarks. We can answer to this
objection by noting that in our view, the effective fields in the
regime $T\lsim T_c$ are ``relayed" from the QCD fields in the
regime $T\gsim T_c$: The gluons and quarks in $T\gsim T_c$ are
Higgsed \`a la Berges-Wetterich to the vector mesons and the
baryons, respectively. The additional assumption we are making
here is that as in \cite{QM2002}, we can -- and should -- replace
the baryons degrees of freedom by quasiquark degrees of freedom
near the phase transition. This is a picture that resembles the
Georgi-Manohar chiral quark model~\cite{georgi-manohar} except
that the gluons are Higgsed to the vector mesons.

Finally, we should stress that the scenario proposed here can
ultimately be validated or invalidated by lattice calculations. So
far we have not addressed the properties of the quantities we have
studied in this paper away from the critical point $T_c$ (as well
as $n_c$). Confrontation with future lattice data as well as with
RHIC data will require these properties to be worked out.

A more comprehensive discussion of the materials covered in this
paper as well as other issues of HLS-VM in hot bath near chiral
restoration  will be discussed in a future
publication~\cite{HaradaSasaki:prep}.

\subsection*{Acknowledgments}
\indent\indent We acknowledge useful discussions with Gerry Brown
and Bengt Friman. The work of MH was supported in part by the
Brain Pool program (\#012-1-44) provided by the Korean Federation
of Science and Technology Societies and USDOE Grant
\#DE-FG02-88ER40388.  He would like to thank Gerry Brown for
his hospitality during the stay at SUNY at Stony Brook where part of
this work was done.
The work of YK was partially
supported by the Brain Korea 21 project of the Ministry of
Education, by the KOSEF Grant No. R01-1999-000-00017-0,
and by the U.S. NSF Grant No. . He is very
grateful to Kuniharu Kubodera and Fred Myhrer for their
hospitality during his stay at University of South Carolina where
part of this paper was written. The work of MR was supported by
the Humboldt Foundation while he was spending three months at the
Theory Group, GSI (Darmstadt, Germany). He would like to thank GSI
for the hospitality and the Humboldt Foundation for the support.


\newpage

\appendix

\begin{flushleft}
\Large\bf Appendices
\end{flushleft}

\section{Hadronic Thermal Corrections}
\label{app:HTC}
 \indent\indent In this appendix we summarize the
hadronic thermal corrections to the two-point functions of
$\overline{\cal A}_\mu$-$\overline{\cal A}_\nu$,
$\overline{V}_\mu$-$\overline{V}_\nu$, $\overline{\cal
V}_\mu$-$\overline{\cal V}_\nu$ and
$\overline{V}_\mu$-$\overline{\cal V}_\nu$.

The four components of the hadronic thermal corrections to the two
point function of $\overline{\cal A}_\mu$-$\overline{\cal A}_\nu$,
$\Pi_\perp$, are expressed as
\begin{eqnarray}
\bar{\Pi}_{\perp}^t(p_0,\bar{p};T)
&=&
  N_f (a-1) \bar{A}_{0(B)}(0,T)
  - N_f a M_\rho^2 \bar{B}_{0(B)}(p_0,\bar{p};M_\rho,0;T)
\nonumber\\
&&
  {}+ N_f \frac{a}{4} \bar{B}_{(B)}^t(p_0,\bar{p};M_\rho,0;T)
  {} + 4 \lambda^2 N_c m_q^2 \bar{B}_{0(F)}(p_0,\bar{p};m_q,m_q;T)
\ ,
\label{AA t}
\\
\bar{\Pi}_{\perp}^s(p_0,\bar{p};T)
&=&
  N_f (a-1) \bar{A}_{0(B)}(0,T)
  - N_f a M_\rho^2 \bar{B}_{0(B)}(p_0,\bar{p};M_\rho,0;T)
\nonumber\\
&&
  {}+ N_f \frac{a}{4} \bar{B}_{(B)}^s(p_0,\bar{p};M_\rho,0;T)
  {} + 4 \lambda^2 N_c m_q^2 \bar{B}_{0(F)}(p_0,\bar{p};m_q,m_q;T)
\ ,
\label{AA s}
\\
  \bar{\Pi}_{\perp}^L(p_0,\bar{p};T)
&=&
  N_f \frac{a}{4} \bar{B}_{(B)}^L(p_0,\bar{p};M_\rho,0;T)
\nonumber\\
&&
  {}+ \lambda^2 N_c
  \left[
    p^2 \bar{B}_{0(F)}(p_0,\bar{p};m_q,m_q;T)
    - \bar{B}_{(F)}^L(p_0,\vec{p};m_q,m_q;T)
  \right]
\ ,
\label{AA L}
\\
\bar{\Pi}_{\perp}^T(p_0,\bar{p};T)
&=&
  N_f \frac{a}{4} \bar{B}_{(B)}^T(p_0,\bar{p};M_\rho,0;T)
\nonumber\\
&&
  {}+ \lambda^2 N_c
  \left[
    p^2 \bar{B}_{0(F)}(p_0,\bar{p};m_q,m_q;T)
    - \bar{B}_{(F)}^T(p_0,\vec{p};m_q,m_q;T)
  \right]
\ ,
\label{AA T}
\end{eqnarray}
where the functions $\bar{A}_{0(B)}$, $\bar{B}_{0(B,F)}$, and so on
are given in Appendix~\ref{app:Fun}.
To obtain the above results from Eqs.~(\ref{PiA boson}) and
(\ref{PiA fermion}) we used the following relation derived from
Eq.~(\ref{Bts rel}):
\begin{eqnarray}
&&
2 g^{\mu\nu} \bar{A}_{0(F)}(m_q,T)
+ \bar{B}_{(F)}^{\mu\nu}(p_0,\vec{p};m_q,m_q;T)
\nonumber\\
&& \qquad
=
P_L^{\mu\nu} \bar{B}_{(F)}^L(p_0,\bar{p};m_q,m_q;T) +
P_T^{\mu\nu} \bar{B}_{(F)}^T(p_0,\bar{p};m_q,m_q;T)
\ .
\end{eqnarray}



The two components $\bar{\Pi}^t$ and $\bar{\Pi}^s$ of hadronic
thermal corrections to the two-point functions of
$\overline{V}_\mu$-$\overline{V}_\nu$, $\overline{\cal
V}_\mu$-$\overline{\cal V}_\nu$ and
$\overline{V}_\mu$-$\overline{\cal V}_\nu$ are written as
\begin{eqnarray}
&&
\bar{\Pi}_{V}^t(p_0,\bar{p};T)
=
\bar{\Pi}_{V}^s(p_0,\bar{p};T)
\nonumber\\
&& \
=
\bar{\Pi}_{\parallel}^t(p_0,\bar{p};T)
=
\bar{\Pi}_{\parallel}^s(p_0,\bar{p};T)
\nonumber\\
&& \
=
- \bar{\Pi}_{V\parallel}^t(p_0,\bar{p};T)
=
- \bar{\Pi}_{V\parallel}^s(p_0,\bar{p};T)
\nonumber\\
&& \quad
=
- N_f \frac{1}{4}
  \left[ \bar{A}_{0(B)}(M_\rho;T) + a^2 \bar{A}_{0(B)}(0;T) \right]
- N_f M_\rho^2 \bar{B}_{0(B)}(p_0,\bar{p};M_\rho,M_\rho;T)
\ .
\label{rr vv rv ts}
\end{eqnarray}
Among the remaining components only $\bar{\Pi}_\parallel^L$
is relevant to the present analysis.
This is given by~\footnote{%
  The explicit forms of other components will be listed in
  Ref.~\cite{HaradaSasaki:prep}.
}
\begin{eqnarray}
\bar{\Pi}_{\parallel}^L(p_0,\bar{p};T)
&=&
N_f \frac{1}{8} \bar{B}_{(B)}^L(p_0,\bar{p};M_\rho,M_\rho;T)
+ N_f \frac{(2-a)^2}{8} \bar{B}_{(B)}^L(p_0,\bar{p};0,0;T)
\nonumber\\
&&
  {}+ \kappa^2 N_c
  \left[
    p^2 \bar{B}_{0(F)}(p_0,\bar{p};m_q,m_q;T)
    - \bar{B}_{(F)}^L(p_0,\bar{p};m_q,m_q;T)
  \right]
\ .
\label{vv L}
\end{eqnarray}

For obtaining the pion decay constants and velocity in
Section~\ref{sec:PDCV} we need the limit of $p_0 = \bar{p}$ of
$\bar{\Pi}_\perp^t$ and $\bar{\Pi}_\perp^s$ in Eqs.~(\ref{AA t})
and (\ref{AA s}). {}From Eq.~(\ref{B0F 0}) we see that the
fermionic contribution vanishes, and only $\rho$ and/or $\pi$
loops contribute. Then, with Eq.~(\ref{B0 Bts VM limits}),
$\bar{\Pi}_\perp^t$ and $\bar{\Pi}_\perp^s$ reduce to the
following forms in the limit $M_\rho \rightarrow0$ and
$a\rightarrow1$:
\begin{eqnarray}
&&
\bar{\Pi}_\perp^t(p_0=\bar{p}+i\epsilon,\bar{p};T)
\ \mathop{\longrightarrow}_{M_\rho \rightarrow 0,\, a\rightarrow1} \
- \frac{N_f}{2} \widetilde{J}_{1,(B)}^2(0;T)
= - \frac{N_f}{24} T^2
\ ,
\nonumber\\
&&
\bar{\Pi}_\perp^s(p_0=\bar{p}+i\epsilon,\bar{p};T)
\ \mathop{\longrightarrow}_{M_\rho \rightarrow 0,\, a\rightarrow1} \
- \frac{N_f}{2} \widetilde{J}_{1,(B)}^2(0;T)
= - \frac{N_f}{24} T^2
\ .
\label{PiA ts Tc app}
\end{eqnarray}


{}In the static--low-momentum limits of the functions listed in
Eqs.~(\ref{JB SL}) and (\ref{JF SL}), the $(\bar{\Pi}_\perp^t -
\bar{\Pi}_\perp^L)$ appearing in the axial-vector susceptibility
becomes
\begin{eqnarray}
&&
\lim_{\bar{p}\rightarrow0}
\lim_{p_0\rightarrow0}
\left[
  \bar{\Pi}_\perp^t(p_0,\bar{p};T) - \bar{\Pi}_\perp^L(p_0,\bar{p};T)
\right]
\nonumber\\
&& \quad
=
- N_f \widetilde{J}_{1,(B)}^2(0;T)
+ N_f a\, \widetilde{J}_{1,(B)}^2(M_\rho;T)
- N_f \frac{a}{M_\rho^2}
\left[
  \widetilde{J}_{-1,(B)}^2(M_\rho;T)
  - \widetilde{J}_{-1,(B)}^2(0;T)
\right]
\nonumber\\
&& \qquad
{}+ 4 \lambda^2 N_c \widetilde{J}_{1,(F)}^2(m_q;T)
\ .
\label{PiA tmL SL}
\end{eqnarray}
For the functions appearing in the vector susceptibility
relevant to the present analysis
we have
\begin{eqnarray}
&&
\lim_{\bar{p}\rightarrow0}
\lim_{p_0\rightarrow0}
\left[ \bar{\Pi}_V^t(p_0,\bar{p};T) \right]
=
- \frac{N_f}{4} \left[
  2 \widetilde{J}_{-1,(B)}^0(M_\rho;T)
  - \widetilde{J}_{1,(B)}^2(M_\rho;T)
  + a^2 \, \widetilde{J}_{1,(B)}^2(0;T)
\right]
\ ,
\label{Pir t SL}
\\
&&
\lim_{\bar{p}\rightarrow0}
\lim_{p_0\rightarrow0}
\left[ \bar{\Pi}_{\parallel}^L(p_0,\bar{p};T) \right]
=
- N_f \frac{1}{4} \left[
  M_\rho^2 \widetilde{J}_{1,(B)}^0(M_\rho;T)
  + 2 \widetilde{J}_{1,(B)}^2(M_\rho;T)
\right]
\nonumber\\
&& \qquad
{}- N_f \frac{(2-a)^2}{2} \widetilde{J}_{1,(B)}^2(0;T)
- 2 \kappa^2 N_c \left[
  m_q^2  \widetilde{J}_{1,(F)}^0(m_q;T)
  + 2\widetilde{J}_{1,(F)}^2(m_q;T)
\right]
\ .
\label{Piv L SL}
\end{eqnarray}




\section{Functions}
\label{app:Fun}
 \indent\indent In this appendix we list the explicit forms of the
functions that figure in the hadronic thermal corrections,
$\bar{A}_{0(B,F)}$, $\bar{B}_{0(B,F)}$ and
$\bar{B}_{(B,F)}^{\mu\nu}$ in various limits relevant to the
present analysis.

The functions
$\bar{A}_{0(B)}(M;T)$ and $\bar{A}_{0(F)}(m_q;T)$ are
expressed as
\begin{eqnarray}
&&
\bar{A}_{0(B)}(M;T) = \widetilde{J}_{1,(B)}^2(M;T)
\ ,
\nonumber\\
&&
\bar{A}_{0(F)}(m_q;T) = - \widetilde{J}_{1,(F)}^2(m_q;T)
\ ,
\label{A0 J}
\end{eqnarray}
where $\tilde{J}_{1,(B)}^2(M;T)$ and
$\tilde{J}_{1,(F)}^2(m_q;T)$ are defined by
\begin{eqnarray}
&&
\widetilde{J}_{l,(B)}^n(M;T)
= \int \frac{d^3\vec{k}}{(2\pi)^3}
\frac{1}{ e^{\omega(\vec{k};M)/T} - 1 }
\frac{ \vert \vec{k} \vert^{n-2} }{[ \omega(\vec{k};M) ]^l }
\ ,
\nonumber\\
&&
\widetilde{J}_{l,(F)}^n(m_q;T)
= \int \frac{d^3\vec{k}}{(2\pi)^3}
\frac{1}{ e^{\omega(\vec{k};m_q)/T} + 1 }
\frac{ \vert \vec{k} \vert^{n-2} }{[ \omega(\vec{k};m_q) ]^l }
\ ,
\end{eqnarray}
with $l$ and $n$ being integers and $\omega(\vec{k};M) \equiv
\sqrt{ M^2 + \vert\vec{k}\vert^2 }$. In the massless limit $M=0$
or $m_q=0$, the above integrations can be performed analytically.
Here we list those results relevant to the present analysis:
\begin{eqnarray}
&&
\widetilde{J}_{1,(B)}^2(0;T) =
\widetilde{J}_{-1,(B)}^0(0;T) = \frac{1}{12} T^2 \ ,
\nonumber\\
&&
\widetilde{J}_{1,(F)}^2(0;T) = \frac{1}{24} T^2 \ .
\end{eqnarray}




It is convenient to decompose $\bar{B}_{(B,F)}^{\mu\nu}$
into four components as done for $\Pi_\perp^{\mu\nu}$
in Eq.~(\ref{Pi perp decomp}):
\begin{equation}
\bar{B}_{(B,F)}^{\mu\nu}
 =u^\mu u^\nu \bar{B}_{(B,F)}^t +
   (g^{\mu\nu}-u^\mu u^\nu) \bar{B}_{(B,F)}^s +
   P_L^{\mu\nu} \bar{B}_{(B,F)}^L + P_T^{\mu\nu}\bar{B}_{(B,F)}^T
\ .
\label{Bmn decomp}
\end{equation}
We note here that, by explicit computations, the following
relations are satisfied:
\begin{eqnarray}
&&
\bar{B}_{(B)}^t(p_0,\bar{p};M,M;T) =
\bar{B}_{(B)}^s(p_0,\bar{p};M,M;T) =
- 2 \bar{A}_{0(B)}(M;T) = -2 \widetilde{J}_{1,(B)}^2(M;T) \ ,
\nonumber\\
&&
\bar{B}_{(F)}^t(p_0,\bar{p};m_q,m_q;T) =
\bar{B}_{(F)}^s(p_0,\bar{p};m_q,m_q;T) =
- 2 \bar{A}_{0(F)}(m_q;T) = 2 \widetilde{J}_{1,(F)}^2(m_q;T) \ .
\label{Bts rel}
\end{eqnarray}





To obtain the pion decay constants and velocity in
Section~\ref{sec:PDCV} we need the limit of $p_0 = \bar{p}$ of the
functions in Eqs.~(\ref{AA t}) and (\ref{AA s}). The fermionic
contribution in $\bar{\Pi}_\perp^t$ and $\bar{\Pi}_\perp^s$ [see
Eqs.~(\ref{AA t}) and (\ref{AA s})] vanishes for $p_0 = \bar{p}$:
\begin{equation}
\bar{B}_{0(F)}(p_0=\bar{p}+i\epsilon,\bar{p};m_q,m_q;T)
=
0 \ ,
\label{B0F 0}
\end{equation}
where we put $\epsilon \rightarrow +0$ to make the analytic
continuation of the frequency $p_0=i2\pi n T$ to the Minkowski
variable.
As for the functions $M_\rho^2\bar{B}_{0(B)}$,
$\bar{B}_{(B)}^t$ and $\bar{B}_{(B)}^s$ appearing
in Eqs.~(\ref{AA t}) and (\ref{AA s}),
we find
that, in the limit of $M_\rho$ going to zero,
they reduce to
\begin{eqnarray}
&&
M_\rho^2 \bar{B}_0(p_0=\bar{p} + i \epsilon,\bar{p};M_\rho,0;T)
\ \mathop{\longrightarrow}_{M_\rho \rightarrow 0} \
0 \ ,
\nonumber\\
&&
\bar{B}^t(p_0=\bar{p}+ i\epsilon,\bar{p};M_\rho,0;T)
\ \mathop{\longrightarrow}_{M_\rho \rightarrow 0} \
- 2 \widetilde{J}_{1,(B)}^2(0;T) = - \frac{1}{6} T^2 \ ,
\nonumber\\
&&
\bar{B}^s(p_0=\bar{p}+ i\epsilon,\bar{p};M_\rho,0;T)
\ \mathop{\longrightarrow}_{M_\rho \rightarrow 0} \
- 2 \widetilde{J}_{1,(B)}^2(0;T) = - \frac{1}{6} T^2 \ .
\label{B0 Bts VM limits}
\end{eqnarray}


The static--low-momentum limits of the functions appearing in the
bosonic corrections to the axial-vector and vector susceptibility
are summarized as
\begin{eqnarray}
&&
\lim_{\bar{p}\rightarrow0}
\lim_{p_0\rightarrow0}
\left[ M_\rho^2 \bar{B}_{0(B)}(p_0,\bar{p};M_\rho,0;T) \right]
=
- \widetilde{J}_{1,(B)}^2(M_\rho;T) + \widetilde{J}_{1,(B)}^2(0;T)
\ ,
\nonumber\\
&&
\lim_{\bar{p}\rightarrow0}
\lim_{p_0\rightarrow0}
\left[
  \bar{B}_{(B)}^t(p_0,\bar{p};M_\rho,0;T)
  - \bar{B}_{(B)}^L(p_0,\bar{p};M_\rho,0;T)
\right]
\nonumber\\
&& \qquad
=
\frac{-4}{M_\rho^2} \left[
  - \widetilde{J}_{-1,(B)}^2(M_\rho;T)
  + \widetilde{J}_{-1,(B)}^2(0;T)
\right]
\ ,
\nonumber\\
&&
\lim_{\bar{p}\rightarrow0}
\lim_{p_0\rightarrow0}
\left[ M_\rho^2 \bar{B}_{0(B)}(p_0,\bar{p};M_\rho,M_\rho;T) \right]
=
\frac{1}{2}
\left[
  \widetilde{J}_{-1,(B)}^0(M_\rho;T)
  - \widetilde{J}_{1,(B)}^2(M_\rho;T)
\right]
\ ,
\nonumber\\
&&
\lim_{\bar{p}\rightarrow0}
\lim_{p_0\rightarrow0}
\left[ \bar{B}_{(B)}^L(p_0,\bar{p};M_\rho,M_\rho;T) \right]
=
- 2 M_\rho^2 \widetilde{J}_{1,(B)}^0(M_\rho;T)
- 4 \widetilde{J}_{1,(B)}^2(M_\rho;T)
\ ,
\nonumber\\
&&
\lim_{\bar{p}\rightarrow0}
\lim_{p_0\rightarrow0}
\left[ \bar{B}_{(B)}^L(p_0,\bar{p};0,0;T) \right]
=
- 4 \widetilde{J}_{1,(B)}^2(0;T)
\ ,
\label{JB SL}
\end{eqnarray}
and the functions that appear in the fermionic corrections are
given in the limit by
\begin{eqnarray}
&&
\lim_{\bar{p}\rightarrow0}
\lim_{p_0\rightarrow0}
\left[ m_q^2 \bar{B}_{0(F)}(p_0,\bar{p};m_q,m_q;T) \right]
=
- \frac{1}{2}
\left[
  \widetilde{J}_{-1,(F)}^0(m_q;T) - \widetilde{J}_{1,(F)}^2(m_q;T)
\right]
\ ,
\nonumber\\
&&
\lim_{\bar{p}\rightarrow0}
\lim_{p_0\rightarrow0}
\left[ \bar{B}_{(F)}^L(p_0,\bar{p};m_q,m_q;T) \right]
=
2m_q^2  \widetilde{J}_{1,(F)}^0(m_q;T)
+ 4 \widetilde{J}_{1,(F)}^2(m_q;T)
\ .
\label{JF SL}
\end{eqnarray}


%XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
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\end{thebibliography}


\end{document}


