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\title{
 Superluminal pions in a hadronic fluid
 }
\author{
Neven Bili\'{c}\thanks{E-mail: bilic@thphys.irb.hr}
and
Hrvoje Nikoli\'c  \\
Rudjer Bo\v{s}kovi\'{c} Institute, \\
P.O. Box 180, 10002 Zagreb, Croatia
}
\date{\today}
%-----------------------------------------------------------------------
\begin{document}
%\addtocounter{footnote}{1}
\maketitle
\begin{abstract}
We study the propagation of pions
at finite temperature and finite chemical potential
in the framework of the linear
sigma model with 2 quark flavors and
$N_c$ colors.
The velocity of massless pions in general
differs from that of light.
 One-loop calculations show that
 in the chiral symmetry broken phase pions,
 under certain conditions,
 propagate faster than light.
\end{abstract}

%
%\pacs{11.30.Qc,11.30.Rd}


\section{Introduction}

The linear sigma model,
originally proposed as a  model for
strong nuclear interactions
\cite{gel},
today serves as an effective
model for the low-energy (low-temperature)
phase of quantum chromodynamics.
The model exhibits spontaneous breaking
of chiral symmetry
 and restoration
at finite temperature.
In the chiral symmetry broken phase at zero temperature
pions, being massless,
propagate with the velocity of light.
At finite temperature
one would expect
chiral pions
to propagate slower than light,
owing to medium effects.
Indeed, this expectation has been confirmed by
one-loop
calculations in the linear sigma model with
no fermions \cite{pis2,son1,son2}.

In this paper we discuss
the propagation of pions at nonzero temperature and
nonzero finite baryon density in a model with two quark flavors.
It turns out that  pions in the presence of fermions
become superluminal
in a certain range of temperature and baryon chemical
potential.

A superluminal propagation
has recently been studied in the context of
the Casimir effect in quantum electrodynamics.
Scharnhorst has demonstrated
\cite{sch} that, when vacuum fluctuations
of the electromagnetic field obey periodic boundary conditions in one
spatial dimension (corresponding to parallel Casimir plates),
then the two-loop corrections to the
photon polarization tensor lead to a superluminal propagation of
photons.
A similarity between the effects of Casimir plates
and that of finite temperature on the propagation of photons
has been discussed \cite{lat}, and more general conditions
that lead to a superluminal propagation of photons have
also
been identified.

A similar effect has been found for photons
interacting with fermions
\cite{fer}.
It has been shown that the transverse
photons coupled to fermions obeying periodic boundary
conditions
are  faster than light
when propagating perpendicularly
and are slower than light
when  propagating
parallelly to
the compactified dimenson.
Since
our fermions obey the usual antiperiodic boundary conditions
in  the compact temporal dimension,
one would  expect no superluminal effects.
However, we demonstrate that in a certain region of temperature
and chemical potential
below the symmetry restoration point,
 massles pions propagate faster than light.
Moreover,
if one of the spatial dimensions is compactified,
pions will propagate superluminally or subluminally,
depending on the size of the compact dimension
and on the boundary conditions.

We organize the paper as follows.
In Sec.\ \ref{eff} we describe
the model.
In Sec.\ \ref{velocity}
 we calculate the dependence of the pion velocity on temperature
 and chemical potential.
 We present
the results and discussion in
Sec.\ \ref{results}.
 In the concluding section,
Sec.\ \ref{concl}, we summarize our results.

\section{Effective Lagrangian} \label{eff}


Consider the linear sigma model at finite temperature
and finite baryon density.
The thermal bath provides a medium which may
be assumed to have a homogeneous velocity field.
The dynamics of mesons in such a medium is described by
an effective chirally symmetric Lagrangian of the form
\begin{eqnarray}
\label{eq100}
{\cal{L}}
       \! & \! = \! & \!
 \frac{1}{2}(a\, \eta^{\mu\nu}
 +b\, u^{\mu}u^{\nu})(\partial_{\mu} \sigma
 \partial_{\nu} \sigma
+\partial_{\mu}
\mbox{\boldmath $\pi$}
\partial_{\nu}
\mbox{\boldmath $\pi$})
 - \frac{m_0^{2}}{2} (\sigma^{2} +
\mbox{\boldmath$\pi$}^{2})
-\frac{\lambda}{4} (\sigma^{2} +
\mbox{\boldmath$\pi$}^{2})^{2}
 \nonumber  \\
         &   &
+\bar{\psi} (c\, i \gamma^{\mu} \partial_{\mu} +
d\, i u^{\mu} \partial_{\mu}
+\mu u^{\mu}\gamma_{\mu} +
g (\sigma + i \mbox{\boldmath $\tau \pi$} \gamma_5))\psi
\, ,
\end{eqnarray}
where $u_{\mu}$ is the velocity of the fluid,
and $\eta_{\mu\nu}$ is the flat Lorentzian metric tensor.
The right and left fermions,
$\psi_R=\frac{1}{2} (1+\gamma_5)\psi$,
$\psi_L=\frac{1}{2} (1-\gamma_5)\psi$,
 are assumed to constitute, respectively, the
$ (\frac{1}{2},0)$ and $(0,\frac{1}{2})$
 representation of the chiral SU(2)$\times$SU(2),
 whereas the mesons ($\sigma$,
{\boldmath$\pi$}) belong to
the $(\frac{1}{2},\frac{1}{2})$ representation.
In the original sigma model \cite{gel} the fermion field
was a nucleon.
We consider
 fermions to be
constituent quarks \cite{con,goc,baie,bil1} with
additional $N_c$ degrees of freedom,
``colors'', from the SU($N_{c}$) local gauge group of
an underlying gauge theory (QCD).
The parameters
 $a$, $b$,
 $c $, and $d$ depend on the temperature $T$,
 the chemical potential $\mu$,
 and the parameters of the model $m_0$, $\lambda$,
 and $g$,
 and may be calculated
in perturbation theory.
At $T=\mu=0$ the medium is absent and $
a=c=1$ and $b=d=0$.




 If $m_0^{2} < 0$,  chiral
symmetry will be spontaneously broken.
At the classical level, the $\sigma$ and $\pi$ fields develop
nonvanishing expectation values such that
\begin{equation}\label{eq2}
\langle \sigma \rangle^{2} + \langle \mbox{\boldmath$\pi$} \rangle^{2}=
- {m_0^{2}\over \lambda} \equiv f_{\pi}^{2} \; .
\end{equation}
It is convenient here to choose %(comment on $c \neq 0$ ?)
\begin{equation}\label{eq3}
\langle \pi_{i} \rangle = 0, \;\;\;\;\; \; \langle \sigma \rangle =
f_{\pi} \; .
\end{equation}
At nonzero temperature the expectation value
$\langle \sigma \rangle$  depends on the temperature
and vanishes at the chiral transition point.
 Redefining the fields as
\begin{equation}\label{eq9}
(\sigma,\mbox{\boldmath$\pi$})
\rightarrow
(\sigma,\mbox{\boldmath$\pi$})+
 (\sigma'(x),\mbox{\boldmath$\pi$}'(x)) ,
\end{equation}
where {\boldmath$\pi'$}
and $\sigma'$
are quantum fluctuations around the
constant values {\boldmath$\pi=0$} and $\sigma
=\langle \sigma \rangle$,
respectively,
we obtain
 the effective Lagrangian
 in which
chiral symmetry is explicitly broken:
\begin{eqnarray}\label{eq5}
{\cal{L}'}
       \! & \! = \! & \!
 \frac{1}{2}(a\, g^{\mu\nu}
 +b\, u^{\mu}u^{\nu})
 (\partial_{\mu} \sigma'
 \partial_{\nu} \sigma'
+\partial_{\mu}
\mbox{\boldmath $\pi$}'
\partial_{\nu}
\mbox{\boldmath $\pi$}')
- \frac{m_{\sigma}^{2}}{2} \sigma'^{2}
- \frac{m_{\pi}^{2}}{2}
\mbox{\boldmath$\pi$}'^{2}
-g' \sigma' (\sigma'^2 +
\mbox{\boldmath $\pi$}'^{2})
\nonumber  \\
         &   &
- {\lambda\over 4}
 (\sigma'^{2} +\mbox{\boldmath$\pi$}'^{2})^2
+\bar{\psi} (c \, i \gamma^{\mu} \partial_{\mu} +
d\, i u^{\mu} \partial_{\mu}
+\mu u^{\mu}\gamma_{\mu} +
g (\sigma' + i \mbox{\boldmath $\tau \pi$}' \gamma_5))\psi
\; .
\end{eqnarray}
 The effective masses and
  the trilinear coupling $g'$  are functions
of $\sigma$  defined as
\begin{eqnarray}\label{eq11}
m_{\sigma}^{2} & = &
 m_0^{2} +3 \lambda \sigma^2  \,  ,  \; \;\;\;
  m_{F} =g \sigma \, ,
  \nonumber \\
m_{\pi}^{2} & = & m_0^{2}+\lambda \sigma^{2} \, , \;\;\;\;
g' =  \lambda \sigma  \, .
\end{eqnarray}
At  temperatures below the
chiral transition point the fermion mass remains as in
(\ref{eq11}) and the meson
masses are given by
\begin{equation}\label{eq43}
m_{\pi}^2 =  0\, ; \;\;\;\;\;\;
m_{\sigma}^2 = 2\lambda \sigma^{2} \, ,
\end{equation}
in agreement with the Goldstone theorem.
The temperature dependence of the chiral condensate $\sigma$
is obtained by
minimizing the thermodynamic potential
$\Omega=-(T/V) \ln Z$
with respect to
 $\sigma$
at fixed inverse temperature $\beta$.
At one-loop order,
the extremum condition
reads
\cite{bil1}
\begin{eqnarray}\label{eq032}
\sigma^{2} & = &
f_{\pi}^{2} - {8 g^{2}\over \lambda}     N_{c}
\: \int {d^{3} p\over (2 \pi)^{3}}
\: {1\over 2\omega_F} \;  n_F (\omega_F)
 \nonumber  \\
 &   &
- 3 \: \int  {d^{3} p\over (2 \pi)^{3}}
\: {1\over \omega_{\sigma}} \; n_{B} (\omega_{\sigma})
- 3 \: \int {d^{3} p\over (2 \pi)^{3}}
\: {1\over \omega_{\pi}} \; n_{B} (\omega_{\pi}) \, ,
\end{eqnarray}
where
\begin{equation}\label{eq28}
\omega_{\pi}
=|\mbox{\boldmath $p$}|
\, ; \;\;\;\;\;\;
\omega_{\sigma}
=(\mbox{\boldmath $p$}^2+m_{\sigma}^{2})^{1/2},
\, ; \;\;\;\;\;\;
\omega_F
=(\mbox{\boldmath $p$}^2+m_F^{2})^{1/2},
\end{equation}
\begin{equation}\label{eq29}
n_{F}(\omega) = {1\over e^{\beta(\omega - \mu)} + 1} +
{1\over e^{\beta(\omega + \mu)} + 1}  \, ,
\end{equation}
\begin{equation}\label{eq30}
n_{B}(\omega) = {1\over e^{\beta \omega} - 1}     \, .
\end{equation}
The right-hand side of (\ref{eq032}) depends on
$\sigma$ through the mass
$m_{\sigma}$ given by (\ref{eq43}).
Solutions to  (\ref{eq032})
 are implicit functions of $T$ and $\mu$.
These equations have been derived from the
thermodynamic  potential
in which the loop corrections
have been neglected.
This approximation corresponds to the
leading order in the
$1/N$ expansion, where $N$ is the
number of scalar fields \cite{mey}.
In our case, $N=4$.

\section{Pion velocity}
\label{velocity}
The propagation of pions is governed by the equation of motion
%
\begin{equation}
\partial_{\mu}
\left[
\,
( a\, \eta^{\mu\nu}+ b\,
u^{\mu}
u^{\nu})\right] \partial_{\nu}\mbox{\boldmath{$\pi$}}
+V(\sigma,
\mbox{\boldmath{$\pi$}},\psi)
\mbox{\boldmath{$\pi$}}=0 \, ,
\label{eq013}
\end{equation}
%
where $V$ is the interaction potential
the form of which is irrelevant  to our
consideration.

In  the simplest case of
   a homogeneous flow,
  Eq.\ (\ref{eq013}) reduces to
  the  wave equation
%
\begin{equation}
(\partial_t^2 -
v^2
\Delta +\frac{v^2}{a}V)
\mbox{\boldmath{$\pi$}}=0,
\label{eq014}
\end{equation}
%
where the quantity $v$ is the pion velocity
given by
%
\begin{equation}
v^2=\left(1+\frac{b}{a}\right)^{-1}  .
\label{eq015}
\end{equation}
As we shall shortly demonstrate, the constants $a$ and $b$
may be derived from the finite-temperature perturbation
expansion of the pion self-energy.

The pion velocity in a sigma model at finite temperature
has been calculated at one-loop level by Pisarski and Tytgat
in the low-temperature approximation
\cite{pis2} and by Son and Stephanov for temperatures
close to the chiral transition point \cite{son1,son2}.
It has been found that the pion velocity vanishes
as one approaches the critical temperature.
Here we analyze the whole range of temperatures in the
chiral symmetry broken phase.

Consider the pion
 self-energy  $\Sigma(q,T)$
in the limit when the external momentum
$q$ approaches 0.
The renormalized inverse pion propagator
may, in the limit $q\rightarrow 0$, be expressed in the form
\begin{equation}
 Z_{\pi}\Delta^{-1}=
  q^{\mu}q_{\mu}-
 \frac{1}{2!}
 q^{\mu}q^{\nu}\frac{\partial}{\partial q^{\mu}}
 \frac{\partial}{\partial q^{\nu}}
\left. (\Sigma(q,T)
-  \Sigma(q,0)) \right|_{q=0}
+\dots   \, ,
  \label{eq201}
\end{equation}
where the ellipsis denotes the terms of higher order in
$q^{\mu}$.
The $q^{\mu}$ independent term of the self-energy
absorbs in the renormalized pion mass,
which is equal to zero in the chiral symmetry broken phase.
The subtracted T=0 term  has been
absorbed in the wave function renormalization factor $Z_{\pi}$.
By comparing this equation with
the inverse pion propagator derived directly from
the effective Lagrangian (\ref{eq5})
\begin{equation}
 \Delta^{-1}=(a+b)q_0^2
 -a \mbox{\boldmath $q$}^2,
  \label{eq202}
\end{equation}
we can express
 the parameters $a$ and $b$,
 and hence the pion
velocity, in terms of the second derivatives of
$\Sigma(q,T)$ evaluated at $q^{\mu}=0$.
At one-loop level the diagrams that
give a nontrivial $q$-dependence of $\Sigma$ are the bubble
diagrams. Subtracting the T=0 term
 one finds
\begin{equation}
\Sigma(q)
  \equiv
\Sigma(q,T)
-  \Sigma(q,0)
=\Sigma_B
+  \Sigma_F
  \label{eq203}
\end{equation}
with the contribution of the bosonic and fermionic loops
given by
\begin{eqnarray}
\Sigma_B(q)
  \!&\! = \!&\!
  -4g'^2 \int\!
\frac{d^3p}{(2\pi)^3}
\frac{1}{2\omega_{\pi}
2\omega_{\sigma,q}}
\nonumber\\
  \!&\!\!&\!
  \times\left\{ [n_B(\omega_{\pi})+
n_B(\omega_{\sigma,q})]
\left(\frac{1}{\omega_{\sigma,q}+
\omega_{\pi}-q_0}
  + \frac{1}{\omega_{\sigma,q}+
\omega_{\pi}+q_0}\right)\right. \nonumber\\
  \!&\!\!&\!
   + \left. [n_B(\omega_{\pi})-
n_B(\omega_{\sigma,q})]\left(
  \frac{1}{\omega_{\sigma,q}-
\omega_{\pi}-q_0} +
  \frac{1}{\omega_{\sigma,q}-
\omega_{\pi}+ q_0}\right)\right\},
  \label{eq213}
\end{eqnarray}
\begin{eqnarray}
\Sigma_F(q)
  \!&\! = \!&\!
-8N_cg^2 \int\!\frac{d^3p}{(2\pi)^3} \frac{1}{2\omega_F2\omega_{F,q}}
\nonumber\\
\!&\!\!&\!
\times\left\{ \frac{-q_0\omega_{F,q}+\mbox{\boldmath $q$}
(\mbox{\boldmath $p$}+\mbox{\boldmath $q$})}{e^{\beta(\omega_{F,q}-\mu)
}+1}
\left( \frac{1}{\omega_F+\omega_{F,q}-q_0}
+\frac{1}{\omega_F-\omega_{F,q}+q_0} \right)
\right. \nonumber\\
\!&\!\!&\!
+\frac{-q_0^2-q_0\omega_{F}+\mbox{\boldmath $q$}(\mbox{\boldmath $p$}
+\mbox{\boldmath $q$})}{e^{\beta(\omega_{F}-\mu)}+1}
\left( \frac{1}{\omega_{F,q}+\omega_F+q_0}
+\frac{1}{\omega_{F,q}-\omega_F-q_0} \right)
\nonumber\\
\!&\!\!&\!
+\frac{q_0\omega_{F,q}+\mbox{\boldmath $q$}(\mbox{\boldmath $p$}
+\mbox{\boldmath $q$})}{e^{\beta(\omega_{F,q}+\mu)}+1}
\left( \frac{1}{\omega_F+\omega_{F,q}+q_0}
+\frac{1}{\omega_F-\omega_{F,q}-q_0} \right)
\nonumber\\
\!&\!\!&\! \left.
+\frac{-q_0^2+q_0\omega_{F}+\mbox{\boldmath $q$}(\mbox{\boldmath $p$}
+\mbox{\boldmath $q$})}{e^{\beta(\omega_{F}+\mu)}+1}
\left( \frac{1}{\omega_{F,q}+\omega_F-q_0}
+\frac{1}{\omega_{F,q}-\omega_F+q_0} \right)
\right\}  \, ,
  \label{eq223}
\end{eqnarray}
where
$\omega_{\sigma,q}=
[(\mbox{\boldmath $p$}-
\mbox{\boldmath $q$})^2
+m_\sigma^2]^{1/2}$,
$\omega_{F,q}=[(\mbox{\boldmath $p$}+
 \mbox{\boldmath $q$})^2
+m_F^2]^{1/2}$.
A straightforward evaluation of the second derivatives of
$\Sigma(q)$ at $q_{\mu}=0$ yields
\begin{equation}
a=1+a_B+a_F  \, ,
\label{eq301}
\end{equation}
\begin{equation}
b=b_B+b_F      \, ,
\label{eq302}
\end{equation}
with
\begin{eqnarray}
a_B  =
   \frac{16 g'^2}{m_{\sigma}^4} \int\!
\frac{d^3p}{(2\pi)^3}
  \left[ \frac{n_B(\omega_{\pi})}{4\omega_{\pi}}+
\frac{n_B(\omega_{\sigma})
}{4\omega_{\sigma}}
 - \frac{1}{3}
      \frac{\omega_{\pi}^2}{m_{\sigma}^2}
   \left(
   \frac{n_B(\omega_{\pi})}{\omega_{\pi}} -
\frac{n_B(\omega_{\sigma})
}{\omega_{\sigma}}
\right)\right] \, ,
\label{eq204}
%\end{equation}
\end{eqnarray}
%\begin{equation}
\begin{eqnarray}
%b =
b_B =
   \frac{16g'^2}{m_{\sigma}^4} \int\!
\frac{d^3p}{(2\pi)^3}
  \left[
  \frac{\omega_{\pi} n_B(\omega_{\pi})
  }{m_{\sigma}^2}-
\frac{\omega_{\sigma}
n_B(\omega_{\sigma})
  }{m_{\sigma}^2}
 + \frac{1}{3}
      \frac{\omega_{\pi}^2}{m_{\sigma}^2}
   \left(
   \frac{n_B(\omega_{\pi})}{\omega_{\pi}} -
\frac{n_B(\omega_{\sigma})
}{\omega_{\sigma}}
\right)\right]  \, ,
 \label{eq205}
%\end{equation}
\end{eqnarray}
\begin{eqnarray}
a_F =
N_cg^2\int\!\frac{d^3p}{(2\pi)^3}
\frac{n_F(\omega_F)}{p^2\omega_F}\,,
 \label{eq304}
\end{eqnarray}
\begin{eqnarray}
b_F =
-N_cg^2\int\!\frac{d^3p}{(2\pi)^3}
\frac{m_F^2}{p^2}
\frac{n_F(\omega_F)}{\omega_F^3}\, .
 \label{eq305}
%\end{equation}
\end{eqnarray}
The subscripts $B$ and $F$ denote
the contributions of
the boson and fermion loops,
respectively.
The sign of the fermion contribution in the last equation
is crucial.
The pion velocity $v$ given by Eq.\ (\ref{eq015}) will
become larger than unity when $b<0$, i.e., when the negative fermionic
part exceeds the positive bosonic part.

The results (\ref{eq301}) and (\ref{eq302})
with
(\ref{eq204})-(\ref{eq305})
can easily be extended to include the case of
a hadronic fluid at $T=0$ and $\mu=0$ in
3+1 dimensional space-time with one
spatial dimension compactified
to the size
 $L\equiv\beta$.
In this case, fermions
may be chosen to be periodic or antiperiodic in the
compact dimension.
Clearly, the pion velocity in the subspace orthogonal to the
compact dimension is equal to the velocity of light.
However,  velocity parallel to the compact direction is
 given by
\begin{equation}
v_{||}^2=1+\frac{b'}{a'}\, ,
\label{eq306}
\end{equation}
where $a'=a$ and $b'=b$ for antiperiodic and
\begin{equation}
a'=1+a_B-\bar{a}_F  \, ,
\label{eq401}
\end{equation}
\begin{equation}
b'=b_B-\bar{b}_F      \, ,
\label{eq402}
\end{equation}
for periodic fermions, with
\begin{eqnarray}
\bar{a}_F =
2 N_cg^2\int\!\frac{d^3p}{(2\pi)^3}
\frac{n_B(\omega_F)}{p^2\omega_F}\,,
 \label{eq404}
\end{eqnarray}
\begin{eqnarray}
\bar{b}_F =
-2 N_cg^2\int\!\frac{d^3p}{(2\pi)^3}
\frac{m_F^2}{p^2}
\frac{n_B(\omega_F)}{\omega_F^3}\, .
 \label{eq405}
 \end{eqnarray}
 Hence, for antiperiodic fermions, the velocity $v_{||}$
 given by  (\ref{eq306}) precisely equals the inverse
 $v$ calculated at $T=1/L$, $\mu=0$.
 For periodic fermions, the sign of the fermion loop
 is reversed and the Fermi-Dirac is
 replaced by a Bose-Einstein distribution.

\section{Results and discussion}\label{results}
   In order to proceed with
a quantitative analysis, we have to choose the input parameters
from  particle physics phenomenology.
For the constituent quark mass we take
$m_F=340$ MeV.
The coupling
$g$  is then fixed by (\ref{eq3}) and (\ref{eq11})
$g=m_F/f_{\pi}$.
The  Particle Data Group
gives a rather wide range
400-1200 MeV \cite{part} of the sigma meson masses.
We shall shortly see that the
analysis of the $T=0$, $\mu\neq 0$ case
further restricts this available range.

 At $T=0$ the extremum condition (\ref{eq032}) for
nonnegative $\mu$ and $N_c=3$ reads
\begin{equation}
\sigma^2 =f_{\pi}^2 -\frac{3g^2}{\lambda\pi^2}\:\vartheta (\mu -
 g\sigma)\left[ \mu\sqrt{\mu^2 -g^2\sigma^2}+g^2\sigma^2\ln
 \frac{g\sigma}{\mu +\sqrt{\mu^2 -g^2\sigma^2}} \right] \, ,
  \label{eq206}
\end{equation}
which determines the extremum of the thermodynamic potential
\begin{eqnarray}
\Omega(\sigma,\mu) \!&\! = \!&\!
\frac{\lambda}{4}\sigma^4
-\frac{\lambda}{2}f_{\pi}^2\sigma^2
\nonumber\\
\!&\!\!&\!
-\frac{\vartheta (\mu -g\sigma)}{(2\pi)^2}
\left[ \mu\sqrt{\mu^2-g^2\sigma^2}(2\mu^2-5g^2\sigma^2)
+3g^4\sigma^4\ln\frac{\mu +\sqrt{\mu^2 -g^2\sigma^2}}{g\sigma}\right].
\label{eq206'}
\end{eqnarray}
The chemical potential $\mu_c$ at which chiral symmetry
gets restored is given by the condition
$\Omega(0,\mu_c)=\Omega(\sigma,\mu_c)$.
For $\mu_c<gf_{\pi}$, the solution of (\ref{eq206}) is
$\sigma=f_{\pi}$, so (\ref{eq206}) leads to
\begin{equation}
 \mu_c=\left( \frac{\lambda\pi^2}{2} \right)^{1/4} f_{\pi}.
 \label{eq207}
\end{equation}
Since $\mu_c$ is the threshold where extended nuclear
matter forms,
a reasonable  assumption,
confirmed
by the strong-coupling QCD analysis \cite{bil2,bil3},
is
\begin{equation}
\mu_c \leq m_F        \, .
  \label{eq208}
\end{equation}
It may be easily verified that with this assumption
the pion velocity is constant and
equal to light velocity up to the transition point
$\mu_c$, where it drops to zero.

It is remarkable that Eq. (\ref{eq208}) yields
an upper bound on the sigma mass
\begin{equation}
 m_{\sigma} \leq \frac{2m_F^2}{\pi f_{\pi}}.
\label{eq209}
\end{equation}
If we take $f_\pi=92.4$ MeV, $m_F=340$ MeV, and saturate the bound,
we find  $m_{\sigma}= 796.5$ MeV.
In the following numerical analysis we take this
value as an input parameter.
Any other choice below this value would not alter our qualitative
picture.

\begin{figure}
\label{fig1}
\begin{center}
\includegraphics[width=.6\textwidth,trim= 0 2cm 0 2cm]{fig1.ps}
\caption{
Pion velocity squared as a function of temperature and
chemical potential.
The dashed line corresponds to a thermodynamically unstable
solution. $T$ and $\mu$ are in MeV.
}
\end{center}
\end{figure}



Next we analyze the case of nonzero temperature.
In Fig.\ 1 we plot
the pion velocity $v$ as a function of temperature
for various fixed $\mu$ (upper plots) and as a function
of chemical potential for various fixed $T$ (lower plots).
For each $T$ and $\mu$
the chiral condensate
 $\sigma$
is calculated by
solving  Eq.\ (\ref{eq032})
numerically.
The dashed line represents the velocity
obtained with thermodynamically unstable
solutions for $\sigma$.
The actual phase transition takes place at the
point $T_c$ (for a given $\mu=\mu_c$) or at $\mu_c$
(for a given $T=T_c$) where the two minima of the
thermodynamic potential at $\sigma=0$
and $\sigma(T_c,\mu_c)$ are leveled \cite{bil1}.
At the critical point, the pion velocity drops abruptly to zero.
We note that there is always a region of temperatures
or chemical potentials below the critical point where
the pion velocity becomes superluminal.

Next we discuss the case
$T=0$ and $\mu=0$ with one
spatial dimension compactified
to the size
 $L\equiv\beta$.
The pion velocity $v_{||}$ parallel to the compact direction is
plotted in Fig.\ 2 for both periodic
and antiperiodic fermions.
In the case of antiperiodic fermions,
the compactification size plays
the role of inverse temperature and chiral
symmetry gets restored at the critical size
$L_c=1/T_c$.
In this case, the velocity plot
is just the  inverse of the first plot in Fig. 1.

For periodic fermions, we find no restoration
of chiral symmetry, the minimum of the thermodinamical potential
corresponding to  a nonzero $\sigma$ is always below
the minimum corresponding to $\sigma=0$.
Owing to the opposite sign of the fermion contribution
the velocity $v_{||}$ is always superluminal, monotonously
approaching the velocity of light as the size of the
compactification $L$ approaches infinity.

\begin{figure}
\label{fig2}
\begin{center}
\includegraphics[width=0.6\textwidth,trim= 0 2cm 0 2cm]{fig2.ps}
\caption{
Pion velocity squared as a function of inverse
compactification length with
antiperiodic (solid line)
and periodic (dot-dashed line) fermions.
The dashed line corresponds to a physically
unstable solution.
}
\end{center}
\end{figure}



\section{Summary and conclusion}
\label{concl}
We have analyzed the
 propagation of massless pions at nonzero temperature and
nonzero finite baryon density in a sigma model with two quark flavors.
By calculating the pion dispersion relation at
one-loop order we have shown
that pions in the presence of fermions
become superluminal
in a certain range of temperatures and baryon chemical
potential.

Furthermore, we have studied the case
when one of the spatial dimensions is compactified
with  fermions obeying periodic or antiperiodic boundary
conditions.
Restricting attention to $T=\mu=0$, we have calculated
the pion velocity $v_{||}$ along the compact direction.
We have found that for antiperiodic fermions,
pions will propagate superluminally or subluminally,
depending on the size of the compact dimension.
With periodic fermions, the velocity $v_{||}$
is always larger than the velocity of light.


A superluminal propagation of massless particles
may naively seem to contradict
special relativity and Lorentz invariance.
The most disturbing consequence of a superluminal propagation
would be an apparent violation of causality.
However, it has been convincingly argued
that superluminal effects realized in quantum field
theories with a nontrivial vacuum
do not lead to causal paradoxes \cite{lib}.
The basic argument goes as follows:
 once the conditions
that determine the vacuum fluctuations are fixed, the propagation
velocity in a given reference frame is {\em unique}.
This implies
that it is not possible to send signals both forwards and backwards
in time within one reference frame.
In other words, the causal loops are forbidden.

It is important to bear in mind that we have considered an idealized
situation when chiral symmetry of the original Lagrangian
is exact and pions are exactly massless in the broken
symmetry phase.
%In reality, pions are massive particles
%owing to an explicit symmetry breaking term in the Lagrangian
%and hence will always propagate slower than light.
In reality, chiral symmetry is explicitly broken owing to
nonvanishing current quark masses.
As a consequence, pions become massive
even in the spontaneously broken symmetry phase and always
propagate slower than light.

\section*{Acknowledgments}
We thank I. Dadi\'c for valuable discussions.
This  work is supported  by
the Ministry of Science and Technology of the Republic of Croatia
under Contract No.\ 0098002.

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\newpage

\begin{figure}[p]
\label{fig7}
\end{figure}
\end{document}
\begin{center}
\begin{large}
{\bf Figure Captions}
\end{large}
\end{center}
%Uvodni komentari:
%2. Pri crtanju svih slika uzete su vrijednosti (na T=0)
%    m_{F} = 340 MeV
%    m_{\sigma} = 1000 MeV
%    f_{\pi} = 92.4 MeV





%\begin{figure}
%\caption{Quark condensate
%as a function of temperature,
%calculated at $\mu =0$.}
%% $\langle \bar{\Psi} \Psi \rangle =0$ at $T=T_{c}$.
%\label{fig10}
%\end{figure}

\end{document}



