{Spontaneous Symmetry Breakings in the Linear Sigma Model at finite baryonic density }
{\normalsize Nuclear Theory and Elementary Particle Phenomenology Group,}
{\normalsize Instituto de F\'\i sica, Universidade de S\~ao Paulo }
{\normalsize C.P. 66.318; CEP 05315-970; S\~ao Paulo - SP, Brazil. }
The linear sigma model with pions and sigma coupled to baryons and a (massive) gauge vector meson is considered for the description of a finite baryonic density system. The stability equation for the bound system is studied in the homogeneous limit considering quantum fluctuations for the scalar and pseudo scalar fields within a variational method with truncations. Nearly exact solutions for the stability equation which also solve the field equations are sought semi-analytically. Bosonic fields are found to have non zero expected classical values at finite density corresponding to condensates of dynamical symmetry breakings. The two components of baryon field, neutron and proton for example, acquire different masses due to the isospin symmetry breaking and these states may even oscillate in the medium. There is a non trivial solution for the (massive) vector meson field indicating the occurrence of a gauge dynamical symmetry breaking at finite density typical of a superconductor.
hep-ph0206107.tex
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\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\ba}{\begin{array}}
\newcommand{\ea}{\end{array}}
\newcommand{\cL}{{\cal L}}
\newcommand{\cD}{{\cal D}}
\newcommand{\tC}{{\tilde{C}}}
\newcommand{\tv}{{\tilde{v}}}
\newcommand{\cH}{{\cal H}}
\newcommand{\VM}{{V_{\mu}}}
\newcommand{\VN}{{V_{\nu}}}
\newcommand{\bx}{{\bf x}}
\newcommand{\by}{{\bf y}}
\newcommand{\bp}{{\bf p}}
\newcommand{\bk}{{\bf k}}
\newcommand{\PI}{{\bar{\vec{\pi}}}}
\newcommand{\bpia}{{\bar{\pi}_a}}
\newcommand{\bpi}{{\bar{\vec{\pi}}}}
\newcommand{\bsig}{{\bar{\sigma}}}
\newcommand{\tpi}{{\tilde{\vec{\pi}}}}
\newcommand{\tsig}{{\tilde{\sigma}}}



\begin{document}

                  
\begin{titlepage}


\title{
{\bf Spontaneous Symmetry Breakings in the Linear Sigma 
Model at finite baryonic density } }
\author{  {\bf F\'abio L. Braghin} \thanks{e-mail: 
braghin@if.usp.br} \\
{\normalsize 
 Nuclear Theory and Elementary Particle
 Phenomenology Group,}\\
{\normalsize Instituto de F\'\i sica, Universidade de 
S\~ao Paulo   }\\
{\normalsize C.P. 66.318; CEP 05315-970;  S\~ao Paulo - SP, Brazil. }
}

\date{}
\maketitle
\begin{abstract} 
The 
linear sigma model with pions and sigma  
coupled to baryons and a (massive) gauge vector meson 
is considered for the description of a finite baryonic density system.
The stability equation for the bound system
is studied in the homogeneous limit considering quantum 
fluctuations for the scalar and pseudo scalar fields within a 
variational method with truncations.
Nearly exact solutions for the stability equation which also solve
the field equations are sought semi-analytically.
Bosonic fields are found to have non zero 
expected classical values at finite density
corresponding to condensates of dynamical symmetry breakings.
The two components of baryon field, neutron and proton for example, acquire
different masses due to the isospin symmetry breaking 
and these states may even oscillate in the medium.
There is a non trivial  solution for the (massive) vector meson 
field indicating the occurrence of 
a gauge dynamical symmetry breaking at finite density typical of a 
superconductor. 
\end{abstract}

\vskip 0.4cm

PACS numbers: 11.15.Ex; 11.30.Rd; 12.38.Aw; 12.39.Fe.

\vskip 0.4cm

Key-words: Spontaneous symmetry breaking, 
 finite density, gauge symmetry, chiral 
symmetry, condensates, isospin.

\vskip 0.4cm

IF- USP - 2001



\end{titlepage}

\section{ Introduction}

When a theory, given by a Lagrangian, is invariant under 
a transformation group and the ground state of this theory is not,
this symmetry is spontaneously (or dynamically) broken
(SSB) \cite{SSB}. 

Quantum Chromodynamics (QCD) is believed to be the fundamental theory 
for the Strong interactions, 
at least at a certain level.  It has a very complex 
non abelian structure
and strong coupling constants for processes at low energies. 
At not high energies quarks and gluons, whose charge is the SU(3) color, 
are confined and no colored state can be observed.
Due to these features it is very difficult to obtain exact solutions.
Descriptions of low energies processes and systems are
mostly accomplished in finite 
lattices (at zero densities) where space-time is discretized.
One is therefore lead to construct 
effective models which respect the main properties and symmetries of 
the QCD for the range of energy and densities for
the hadronic process of interest.
It is interesting to note that one may have different effective theories 
for certain ranges of these variables which can be different
if one consider different systems such as the vacuum or high energy density 
matter such as formed in relativistic heavy ions collisions in RHIC and CERN.
In the vacuum, the lightest strong interacting particles 
are known to respect, at least approximatedly, 
chiral symmetry whose group is the $SU_L(2) \times
SU_R(2)$ which is spontaneously broken down to $SU(2)$.
This means that the QCD vacuum has a preferencial direction in 
chiral space corresponding to a scalar
state.
Pseudo scalar pions have small masses 
in the hadronic scale and they are viewed as the 
Goldstone bosons of such SSB. 
The vacuum acquires a non trivial structure due 
to the formation of scalar quark - anti-quark condensate $<\bar{q} q>$,
 the order parameter of the Chiral SSB. 
Due to this condensate, vacuum is not invariant under chiral transformations.
These features can be taken into account via sigma models which,
in the linear realization \cite{LSM,SSB} 
with mesons, implement chiral symmetry with two 
fields: the (pseudo-scalars) pions and the (scalar) sigma - 
which is called the chiral partner of the pion.
The expected value of sigma is, in this model, the order parameter of 
chiral SSB, 
$<\sigma> \propto <\bar{q} q>$.
Long time ago S. Weinberg pointed difficulties
of the linear sigma model,
which are being rediscussed, for describing hadronic processes
and proposed the non linear 
realization of chiral symmetry in which the sigma field is 
``eliminated'' \cite{NLSM}. 
There are nowadays strong evidences for the existence of the sigma
\cite{TORNQVIST} although there is no consent \cite{ISGURSPETH}.
Both implementations of chiral 
symmetry have been extensively developped since then.

The linear realization of chiral symmetry exhibits 
advantadges over the non linear realization
 mainly for the description 
of dense systems \cite{BRORHO}.
Pions are expected to have zero classical 
components (condensates) $<\vec{\pi}>_{vac}=0$ in the vacuum.
At finite density, QCD is known and expected to have a very complex phase
diagram with the appearance of other condensates \cite{CONDENSATES} with
the possibility of a restoration of the chiral symmetry when
the chiral condensate $<\bar{q}q>$ is expected to be close to
 zero at high energy densities.
With different approaches, finite density QCD and effective models 
have been intensively 
studied in the last years.  
The behavior of the symmetry order parameter(s) with density 
(and temperature) is a very relevant aspect for the 
understanding of matter and its fundamental interactions as well as
for the description of the evolution of the Universe.


In this work we study
the chiral  Linear Sigma Model (LSM) 
with baryons and a (massive) vector meson.
The mass of the vector meson is considered to be non zero only
when the baryonic density is different from zero.
The exact field equations are truncated to allow for
analytical solutions by considering  particular prescriptions for
the stability condition of the system.
We found that there is a (isospin) spontaneously broken symmetry 
generating non zero expected value for the pion field at finite
densities (not necessarily high).
The  quantum fluctuations of the sigma and pion fields are considered 
in a truncated version of the variational approach with Gaussian
wave functionals.
The solutions for the bosonic fields are discussed
and some other possible consequences  are 
analyzed.



\section{ Linear sigma model at finite density}

The Lagrangian density of the Linear Sigma Model for
nucleons $N(\bx)$,
sigma and pions $(\sigma,\vec{\pi})$ covariantly
coupled to a massive gauge vector 
meson $\VM$ is given by:
\be \ba{ll} \label{LSM}
\displaystyle{ {\cal L} = \bar{N} (\bx) \left( i \gamma_{\mu}
{\cal D}^{\mu} - g_S (\sigma + i \gamma_5 {\vec{\tau} . \vec{\pi} } )
\right) N (\bx) + \frac{1}{2} \left( \partial_{\mu} \sigma .
\partial^{\mu} \sigma  + \partial_{\mu} \vec{\pi} . \partial^{\mu} {\vec{\pi}} 
\right) + } \\ 
\displaystyle{ - \frac{1}{4} F_{\mu \nu} F^{\mu \nu} + 
\frac{\lambda}{4} \left( (\sigma)^2 + (\vec{\pi})^2 - v^2 \right)^2
+ \frac{1}{2} m^2_V V_{\mu} V^{\mu},}
\ea
\ee
where the covariant derivative is:
$\cD^{\mu} = \partial^{\mu} - i g_V V^{\mu}$, the gauge invariant
tensor is:
$ F^{\mu \nu} = \partial^{\mu} V^{\nu} - \partial^{\nu} V^{\mu}$. 
 $g_V$, $g_S$ and $\lambda$ are the coupling constants
and  $v = f_{\pi}$, the pion decay constant, in the vacuum. 
The gauge invariant coupling of the temporal component of the 
vector meson to the baryons
is equivalent to a redefinition of a chemical potential.
Pion mass is known to break the 
chiral symmetry explicitly, in such case one should
add another (small) term ${\cal L}_{sb} = c \sigma$, 
where c is a constant proportional to the mass.
This would be of no relevance for our main results.
We consider the possible existence of classical expected values 
(condensates) for all the 
mesonic fields and look for solutions for the equations of motion.
In this work the whole baryon masses come from the 
the coupling to the scalar mesonic field  by a Higgs mechanism 
($M^* = g_S \bsig$). 
%In this version the
%$\sigma$ is a Higgs particle.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%The assumption of the 
%existence of an explicit mass term for the baryons in the Lagrangian 
%(due to a quark content)
%does not change the qualitative results of this work. 
%This  would lead
%to an {\it   in medium} mass: $M^* = M+g_S \bsig$ \cite{FLBcont}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%From expression (\ref{LSM}) the 
%Hamiltonian can be calculated.


The vector meson mass $m_V^2$ is considered to be zero at zero baryonic 
density ($\rho_B$) and non zero at $\rho_B \neq 0$.
We will be concerned, in this level of analysis, 
with an homogenous system in which case the other 
(spatial) components of the vector meson $V_i$ would 
%%%%%%%mostly
in part produce small shifts in the numerical results. It would only
appear in the vector meson mass term at finite density and 
in the term of coupling to baryons. A further account of a 
complete coupling will be given elsewhere.


The baryon (nucleon) field is quantized in terms 
of creation and annihilation operators.
Its wave function can be written as superposition of 
spinor ($\chi$), isospinor ($\eta$) and coordinate components 
$u(\bp), \; v(\bp$). 
It generates non zero scalar, baryonic and pseudo-scalar densities
($\rho_s$, $\rho_B$ and $\rho_{PS}$). 
We will not 
explicitly evaluate here all these quantities but only
make use of $\rho_B$
and the quantized fermionic energy density $\rho_f$,
 given respectively by:
\be \ba{ll} \label{RHOF}
\displaystyle{ \rho_B =  4 \int^{k_F} \frac{d^3 k}{(2 \pi)^3},
 }\;\;\;\;\;\;
\displaystyle{
\rho_f = 4\int^{ k_F}
\frac{d^3 k}{(2 \pi)^3}
\sqrt{ \bk^2 + (M^*)^2} .
}
\ea
\ee
In these expressions 
$k_F$ is
the baryon momentum at the Fermi surface.


To take into account the quantum fluctuations of the sigma 
and pion fields
we consider a truncated version of the 
variational approach using a 
Gaussian trial wave-functional 
in the Schroedinger picture \cite{VARIATIONAL}. 
With the variational principle an estimation for the
energy density of the ground state can be obtained by calculating
the averaged energy with trial wave-functions 
(${\cal H}_{\phi} = <\Phi| \hat{H} |\Phi>$) 
whose (trial) 
parameters are fixed when the energy density is minimum with relation
to its parameters.
%%%%%%%%%%%%%%%%%
Although this statement  is not very precise due to the 
subtraction procedure required by renormalization for the 
ultraviolet divergences one can not 
neglect such idea.
%%%%%%%%%%%%%%%%%%
The averaged value of the Hamiltonian is calculated with
 trial Gaussian wave-functionals for the scalar and pseudo-scalar
fields:
$<\tilde{\Psi} [ \sigma, \vec{\pi}] | H (\sigma,{\vec{\pi}}) | \tilde{\Psi} 
[\sigma, \vec{\pi}] >$.
For the sigma we can write:
\be \label{4} \ba{ll}
\displaystyle{
\Psi\left[\sigma(\bx )\right] = N  exp \left\{ -\frac{1}{4} \int
d \bx d \by \delta\sigma(\bx ) G_S^{-1}(\bx ,\by) 
\delta \sigma(\by) \right\}, }
\ea
\ee
Where $\delta\sigma(\bx) = \sigma(\bx)-\bar{\sigma}$;
 the normalization is  $N$, the variational parameters are   
the condensate $ \bar \sigma = < \Psi | \sigma | \Psi > $, the
quantum fluctuations represented by the width  of the Gaussian  
$ G_{S}(\bx,\by) = <\Psi |\sigma(\bx) \sigma(\by)  | \Psi>$.
An analogous expression for the pion sector is considered with
variational parameters given by:
$\bpi = <\vec \pi>$ and $G_{P}^{a,b}$, which is a  matrix that can 
be considered to be diagonal as a particular case along this work
($G_P^{a,a} = G_P$). This reduces the 
corresponding functional space but it guarantees the explicit 
chiral invariance.
We will assume that these quantum fluctuations only 
intervene for the meson masses
as well as  for a shift of the 
respective fields:
\be \label{SHIFT} \ba{ll}
\displaystyle{ \tpi^2 = \bpi^2 + G_P, \;\;\;\;\;\;\; 
\tsig^2 = \bsig^2 + G_S .}
\ea
\ee
 This 
corresponds to  a truncation on the self consistency
with a particular renormalization energy point.
This approximation corresponds
to neglect terms in the energy density calculated with
the Gaussian functionals. If all terms are considered 
we would only have redefinition of the energy density.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
It does not need self consistency to be fixed.
More exact calculations are being done for this model 
 in \cite{FLBcont}.

The minimizations of the averaged energy
with respect to the Gaussian 
variational parameters  yield the GAP and condensate
equations which define the minimum of the potential for 
these fields.
The following set of equations is obtained for the 
sigma sector:
\be \ba{ll} \label{bsig}
\displaystyle{ \frac{\delta \cH}{\delta \bsig} = 0 \to  \tsig \lambda
\left( \bsig^2 + 3  G_S  + \bpi^2 + G_P - v^2 \right) +
\frac{d \rho_f}{d \tsig} 
=0 ;
}\\
\displaystyle{ \frac{\delta \cH}{\delta G_S} = 0 
\to \frac{d \rho_f}{d G_S} - \frac{G^{-2}_S}{8} 
-\frac{\Delta}{2} + \frac{\lambda}{4} \left( 6 \bsig^2
+ 2 \bpi^2 + 6 G_S + 2 G_P - 2 v^2 \right) 
= 0 . }
\ea
\ee
From this last expression we can write the following expression for 
the sigma mass:
$\mu^2_S =  \lambda \left( 3 \tsig^2 + 
\tpi^2  - v^2 + ...
\right). $
For the sake of clearness we truncate the exact solution
 considering that: 1) fermionic density does not depend 
on the fluctuations $G_i$, 2) fluctuations are small and equal ($G_P=G_S$).
The corresponding equations for the pion field are:
\be \ba{ll} \label{Gmu2b}
\displaystyle{ \lambda  \bpia \left( 
\tpi^2 + 2 G_P+ \tsig^2  -  
v^2 \right) +
 \frac{d \rho_f}{d \bpia} 
=0 ;
}\\
\displaystyle{ 
\mu^2_P =  \lambda \left( 3 \tpi^2 +
\tsig^2 - v^2 + ...
\right) \to 0
}
\ea
\ee
The first of these two expressions contains one of the most relevant results
of the present work. 
As far as the fermionic density depends on $\bsig$, via 
the {\it   in medium} mass $M^*$, and the equation for $\bsig$ 
(expression (\ref{bsig})
does depend on $\bpia$ we find that the pion condensate $\bpia$ must be 
nonzero whenever the fermionic 
density (and therefore baryonic density) is not zero.
We have found therefore that:
\be \label{derivati} \ba{ll}
\displaystyle{ \frac{d \rho_f}{d \bpia} = 
\frac{d \rho_f}{d \bsig} \frac{d \bsig}{d \bpia} \neq 0
.}
\ea
\ee
As discussed above, the expressions 
for the 
meson masses $\mu_S^2 \simeq 500 MeV$ and $\mu_P^2 \simeq 0$, 
and the shift of the fields (\ref{SHIFT}), 
are the only effects of the quantum fluctuations in the 
present work. 
Since we will not explicitly calculate the
two point function $G_i$, there will be no concern with
Ultra Violet divergences here.



The Euler Lagrange equation for the  vector meson
was calculated for
a gauge in which only the component $V_0$ 
is non zero and homogeneous. If we consider the other 
components $V_i \neq 0$ the equations are slightly changed, but
the conclusions remain valid with another coupled equation.
The equation is given by:
\be \ba{ll} \label{EQmeson}
\displaystyle{  g_V \left( \rho_B + 
V_0 \frac{d \rho_B}{d V_0} \right) - m_V^2 V_0 
= 0.}
\ea
\ee


The total averaged energy density can be written as:
\be \ba{ll} \label{enden}
\displaystyle{ {\cal H} = \rho_f + g_V V_0 \rho_B - 
\frac{1}{2} m_V^2 V_0^2 
+ \frac{\lambda}{4}(\tsig^2 + \tpi^2 - {v}^2)^2.
}
\ea
\ee



In this work solutions for the above equations 
(\ref{bsig},\ref{Gmu2b},\ref{EQmeson})
are researched  
 such that the main properties of 
a (bound) finite density system are
consistently described. 
The stability condition of the system 
can be written as:
\be \ba{ll} \label{stability}
\displaystyle{ \left. \frac{d {\cal H}}{d\rho_B} 
\right|_{\rho_B = \rho_0}
  = \left.\frac{ \cal H}{\rho_B} \right|_{\rho_B = \rho_0} < 0,
}
\ea
\ee
where $\rho_0$ is the stable density.
To guarantee that this expression is satisfied we
consider some (approximated) prescriptions for the 
dependence of the involved variables in on the baryonic density.
Namely, we divide the expression for the energy density (\ref{enden}) 
for the condition (\ref{stability}) 
into three separated ones:
\be \label{PRESCR} \ba{ll}
\displaystyle{ \frac{d \rho_f}{d \rho_B} = \frac{\rho_f}{\rho_B}, }\\
\displaystyle{ \frac{d (\tsig^2 + \tpi^2 -v^2)}{d \rho_B} = 
\frac{(\tsig^2 + \tpi^2 -v^2)}{ \rho_B} ,}\\
\displaystyle{ \frac{d {\cH}_V}{d \rho_B} = \frac{{\cH}_V}{\rho_B}
.}
\ea
\ee
In this last expression ${\cal H}_V$ is the energy density with contributions
of the vector meson. A detailed comparison of this prescription with
the exact result will be shown elsewhere.
It is certain that the solutions for these (stability) equations do obeys the 
equations of motion of the fields, being this approximation therefore reliable.

From the first of the differential equations (\ref{PRESCR})
 we find a solution for the dependence of $\rho_f$ 
on the baryonic density ($\rho_f = \rho_f(\rho_B)$) which
is in excellent agreement with that resulting from the 
integration of expression (\ref{RHOF}) \cite{FLBcont}.
One more evidence that the prescriptions of expressions (\ref{PRESCR})
are good.


From the second expression in (\ref{PRESCR})
we find a constraint which 
defines a symmetry radius in 
the medium:
\be \label{RAIQUI} \ba{ll}
\displaystyle{ (\tsig^2 + \tpi^2 -v^2 ) = \tC \sqrt{\rho_B}
.}
\ea
\ee
In this expression $\tC$ is  a constant.
Therefore in the vacuum:  $ \tsig^2 \simeq v^2 = f_{\pi}^2$ as discussed
above. We stress that $\bsig = f_{\pi} \neq \tsig$ in the vacuum.
This means that the quantum fluctuations change the condensate value.


The GAP and condensate equations (\ref{bsig}, \ref{Gmu2b}),
for $\tsig$ 
and for the $\tpi$ can be faced as 
differential equations for $\rho_f$ and they can be written as:
\be \ba{ll} \label{order}
\displaystyle{ (\tsig^2  + \tpi^2 - \tv^2 ) \simeq
\frac{2}{\lambda} \frac{d \rho_f}{d \tsig^2} \simeq
\frac{2}{\lambda} \frac{d \rho_f}{d \tpi^2} 
}
\ea
\ee
Where $\tv^2 = v^2 - G$. 
These equations are isomorfic and show an
equal dependence of $\rho_f$ with each of these
(chiral) fields.
An approximated solution for 
the two condensate equations can be found  as if  
$\rho_f$ were a function of these fields independently.
This yields $\rho_f = \rho_f^{(1)} (\bsig)$ and 
$\rho_f = \rho_f^{(2)} (\bpi)$ with constants fixed for
the boundary $\rho_B = 0$ where $\bpi=0$ and $\bsig=v$.
Equating $\rho_f^{(1)}$ (solution of the first equation) to the 
 second equation we find the following approximated value for the 
pion condensate (if $ |\bpi^2| << v^2$):
\be \label{tpiaprox} \ba{ll}
\displaystyle{ \tpi^2 \simeq \frac{ \tsig^2 ( \tsig^2- \tv^2)
}{ 4 ( -\frac{\tsig^2}{2} \pm \tv^2 )
}
,}
\ea
\ee
In these solutions, as well as in others more exact, 
$\tpi^2$ may be either positive or negative. 
It would correspond respectively to a real or complex condensate
value. This solutions do reproduce values close to 
the exact solutions.
Exact solution for the condensate equations will be given 
elsewhere.


We can also add the two differential equations (\ref{order})
which can be seen  as partial differential equations.
We equate the 
solutions for these equations 
and obtain the following constraint expression for the condensates:
\be \label{contraint2} \ba{ll}
\displaystyle{ C_f \tC \sqrt{\rho_0} = 
\frac{\lambda}{2} \left( (\tsig^2 - \frac{{v}^2}{2})^2
- (\tpi^2 - \frac{{v}^2}{2} )^2 \right).
}
\ea
\ee
Where $C_f$ is a constant from the partial differential equations.
This also expresses the dynamical symmetry breaking which
occurs in the medium mainly because $C_f \tC \sqrt{\rho_B} \neq 0$.
We obtained a consistent basis for the
study of the behavior of the condensates (as order parameters) 
from a Quantum Field Theory 
which has a SSB with  density.


Finally, considering the equation of $V_0$ - expression (\ref{EQmeson}) - 
as a  differential equation
of the baryonic density $\rho_B$ as a function of $V_0$
 we obtain the following
solution:
\be \ba{ll} \label{V0}
\displaystyle{ V_0 (\rho_B) =  \frac{ -g_V \rho_B \pm
\sqrt{ g_V^2 \rho_B^2 - 2 C_V \rho_B m^2_V
 }  }{m_V^2}
,}
\ea
\ee
where $C_V$ is a constant. 
This constant will be the only contribution
of the vector meson sector to the energy density
${\cal H}_V = C_V \rho_B$.
In the limit of zero density we get $V_0 \to 0$. 
%This is consistent with the supposition of equivalence of 
%the redefinition of $V_0$ and the introduction of the chemical potential.
It is seen that the baryonic density generates a non zero value of $V_0$ -
which can be viewd as a condensate. 
This may be another dynamical symmetry breaking.
Requiring the energy to be stable with 
relation to the solution $V_0$ we find that:
$ C_V = 2 g_V \bar{V_0}$.
Where $\bar{V_0}$ is the value of $V_0$ at a stable density $\rho_0$.
In this point we have $m^2_V = - 2 g_V \rho_0/\bar{V_0}$. From this we
see that the mass of the vector meson is proportional to the density, i.e.,
it is an {\it in medium} effect, as required.
The massive vector meson is characteristic from a superconductive state
which occurs at finite density. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%It may be considered to be 
%the photon itself
%which would acquire mass in the presence of the hadronic field.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The question of (non) renormalizability (in a finite density medium) 
of the massive vector field will
not be addressed here.
%The velocity and other properties in the baryonic medium
%of this massive meson will be calculated elsewhere.

\subsection{ Discussion} 

Besides the appearance of another condensate, $\bpi$, at finite baryonic 
density  due to the isospin SSB we find
the possibility, although not very appealing, of further 
chiral symmetry breaking with $\bsig > v$ at finite baryonic density 
as found in \cite{PRD2001}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The numerical studies will be shown in a forthcoming work.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%This also depends on the couplings  $g_S$ and $\lambda$ which 
%are not know experimentally
%\cite{FLBcont}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
With the present results it 
is not yet possible to determine, unambiguously, the symmetry 
behavior with density in the present model 
as precluded in other works \cite{BRORHO}.


We found 
 that, at finite baryonic densities, 
there is a non zero pion classical expected value
(condensate).
Departing from the Lagrangian density or from the
equation of motion of
the baryons  we propose the following definition for their 
effective  masses.
This would also explicit the feature that the nucleon densities depend
on the pion condensate.
From the averaged value of the energy 
we may consider  a matrix for $M^*$
which depends on the isospin (and eventually spin) 
of the  nucleons:
\be \ba{ll} \label{EFMAS} 
\displaystyle{
M^*_{a,b;s} = g_S <\Psi [\sigma, \pi] | .  < N_{a,s} | 
(\sigma + i \gamma_5
{\vec{\tau}. {\PI} }) | N_{b,s} > . | \Psi [\sigma , \pi] > 
= g_S ( \bsig + i \tilde{M}_{(a,b)}^d \pi_d ). }
\ea
\ee
In this expression ${a,b,d}$ 
stands for the isospin number (neutrons/protons), and $\tilde{M}$
is a non diagonal isospin matrix.
This allows for the possibility of different
values of baryonic masses
- i.e. a mass splitting -
but it also means possible oscillations between the isospin states 
(proton and neutron for example) in the medium.
 This matrix
includes a dependence on the baryon spins as well which is not 
discussed further here.
%The nucleonic mass would be therefore obtained 
%the averaged value $\bsig$ plus a contribution
%from the averaged value of the pion field.
However, the quantities calculated with it in the present work 
-mainly the densities ($\rho_f$)- 
were the usual
ones, i.e., with a constant diagonal effective mass due to 
$\bsig$, i.e. 
 $M^*_{a,b}(\bpi) \simeq M^*$, which is the leading term.
This non trivial solution corresponds to a non invariant 
 ground state
under an isospin transformation,
although the Lagrangian is symmetric. 
The massive vector meson is characteristic from a superconductor
- finite density.  It may be  the photon itself
which would acquire mass in the presence of the hadronic field. 
In this case, arguments of the kind of vector meson dominance to 
explain its usual identification with the omega would be appealing.
The velocity and other properties in the baryonic medium
of this massive meson will be calculated elsewhere \cite{FLBcont}.


Phase diagram of QCD would then include this pseudo scalar 
condensates.
Consequences for description of effects in (relativistic) heavy ion collisions
(as well as for astrophysical dense  objects) can be expected:
baryon properties such as masses are modified, with possible splittings and 
baryon states oscillation. 
Pion multiplicities in (relativistic) h.i.c. may increase 
with the presence of $<\vec{\pi}>$ with respect to predictions without
it.
The present model will be considered
for the description of (hadronic) nuclear matter properties
and eventually of atomic nuclei elsewhere as well as 
inhomogeneous situations where topological properties may
arise \cite{FLBcont}. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%We also investigate possible connections between the 
%isospin and gauge spontaneous symmetry breakings.



\vspace{1cm}

\noindent {\Large {\bf Acknowledgement}}

This work was supported by FAPESP, Brazil.

\vskip 0.2cm




\begin{thebibliography}{ll}
\bibitem{SSB} Y. Nambu, Phys. Rev. Lett. {\bf 4}, 380 (1960), Phys. Rev. 
          {\bf 117}, 648 (1960). Y. Nambu and G. Jona-Lasinio, Phys. Rev.
          {\bf 122} 354 (1961), {\bf 124} 246 (1961).
             J. Goldstone, Nuovo Cimento {\bf 19} 154 (1961).

\bibitem{LSM} M. Gell Mann and M. L\'evy, Nuovo Cimento {\bf 16} 705 (1960).

\bibitem{TORNQVIST} N.A. Tornqvist and M. Roos, Phys. Rev. Lett. {\bf 76},
      1575 (1996).
 I. Bediaga, Nuc. Phys. {\bf B} Proc. Suppl. {\bf 96}, 225 (2001).

\bibitem{ISGURSPETH} N. Isgur and J. Speth, Phys. Rev. Lett. {\bf 77}, 2332 (1996).

\bibitem{NLSM} S. Weinberg, Phys. Rev. Lett. {\bf 18} 154 (1961).

\bibitem{BRORHO} G.E. Brown and M. Rho, Phys. Rep. {\bf 269}, 333 (1996).

\bibitem{CONDENSATES} R. Rapp, T. Sch\"afer, E. Shuryak, M. Velkovsky, 
      Phys. Rev. Lett., 53 (1998); 
M. Alford, K. Rajagopal, F. Wilczek, Phys. Lett. {\bf B 422},
247 (1998).

\bibitem{FLBcont} F.L. Braghin, {\it in preparation}; 

\bibitem{VARIATIONAL}  P.M. Stevenson, Phys. Rev. {\bf D 32}, 1389 (1985);
	            W.A. Bardeen, M. Moshe, Phys. Rev. {\bf D 28}, 1372 (1983);
                     A.K. Kerman, C. Martin, D. Vautherin, Phys. Rev. 
                       {\bf D 47}, 632 (1993) and references therein.

\bibitem{PRD2001}   F.L. Braghin, Phys. Rev. {\bf D 64}, 125001 (2001).
% - hep-ph/0106338.




\end{thebibliography}


\end{document}

\newpage


hep-ph0206107.extracted_bib
\begin{thebibliography}{ll}
\bibitem{SSB} Y. Nambu, Phys. Rev. Lett. {\bf 4}, 380 (1960), Phys. Rev. 
          {\bf 117}, 648 (1960). Y. Nambu and G. Jona-Lasinio, Phys. Rev.
          {\bf 122} 354 (1961), {\bf 124} 246 (1961).
             J. Goldstone, Nuovo Cimento {\bf 19} 154 (1961).

\bibitem{LSM} M. Gell Mann and M. L\'evy, Nuovo Cimento {\bf 16} 705 (1960).

\bibitem{TORNQVIST} N.A. Tornqvist and M. Roos, Phys. Rev. Lett. {\bf 76},
      1575 (1996).
 I. Bediaga, Nuc. Phys. {\bf B} Proc. Suppl. {\bf 96}, 225 (2001).

\bibitem{ISGURSPETH} N. Isgur and J. Speth, Phys. Rev. Lett. {\bf 77}, 2332 (1996).

\bibitem{NLSM} S. Weinberg, Phys. Rev. Lett. {\bf 18} 154 (1961).

\bibitem{BRORHO} G.E. Brown and M. Rho, Phys. Rep. {\bf 269}, 333 (1996).

\bibitem{CONDENSATES} R. Rapp, T. Sch\"afer, E. Shuryak, M. Velkovsky, 
      Phys. Rev. Lett., 53 (1998); 
M. Alford, K. Rajagopal, F. Wilczek, Phys. Lett. {\bf B 422},
247 (1998).

\bibitem{FLBcont} F.L. Braghin, {\it in preparation}; 

\bibitem{VARIATIONAL}  P.M. Stevenson, Phys. Rev. {\bf D 32}, 1389 (1985);
	            W.A. Bardeen, M. Moshe, Phys. Rev. {\bf D 28}, 1372 (1983);
                     A.K. Kerman, C. Martin, D. Vautherin, Phys. Rev. 
                       {\bf D 47}, 632 (1993) and references therein.

\bibitem{PRD2001}   F.L. Braghin, Phys. Rev. {\bf D 64}, 125001 (2001).
% - hep-ph/0106338.




\end{thebibliography}