Isospin Effects on the Mixed Hadron-Quark Phase at High Baryon Density
M. Di Toro
The phase transition of hadronic to quark matter at high baryon and isospin density is analyzed. Nonlinear relativistic mean field models are used to describe hadronic matter, and the MIT bag model is adopted for quark matter. The boundaries of the mixed phase and the related critical points for symmetric and asymmetric matter are obtained. Isospin effects appear to be rather significant. The binodal transition line of the ($T,\rho_B$) diagram is lowered to a region accessible through heavy ion collisions in the energy range of the new planned facilities, e.g. the $FAIR/NICA$ projects. Some observable effects are suggested, in particular an {Isospin Distillation} mechanism. The dependence of the results on a suitable treatment of isospin contributions in effective QCD Lagrangian approaches, at the level of explicit isovector parts and/or quark condensates, is critically discussed.
mixedphase.tex


\documentclass{elsart}
\usepackage{graphicx}
\usepackage{amssymb} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%
\begin{document}


%\documentclass[aps,preprintnumbers,amsmath,amssymb]
%{revtex4}
%\usepackage{graphicx}
%\textheight 22.5cm \topmargin=1.0cm \textwidth 15.5cm
%\oddsidemargin 1.0cm\evensidemargin 1.0cm \pagestyle{plain}
%\def\baselinestretch{1.3}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frontmatter}

\title{Isospin Effects on the Mixed Hadron-Quark Phase at High
Baryon Density}





\author[catania,ctdip]{M. Di Toro}
\author[ihep,thcas]{B. Liu}
\author[catania,ctdip]{V. Greco}
\author[bucha]{V. Baran}
\author[catania]{M. Colonna}
\author[ctdip,ctsez]{S. Plumari}

\address[catania]{Laboratori Nazionali del Sud INFN, I-95123 Catania, Italy} 
\address[ctdip]{Physics and Astronomy Dept., University of Catania}
\address[ihep]{IHEP, Chinese Academy of Sciences, Beijing, China}
\address[thcas]{Theoretical Physics Center for Scientific Facilities, 
Chinese Academy of Sciences, 100049 Beijing, China}
\address[bucha]{Physics Faculty, Univ. of Bucharest and NIPNE-HH, Romania}
\address[ctsez]{INFN, Sezione di Catania, Italy}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\address{e-mail: ditoro@lns.infn.it}









\begin{abstract}

The phase transition of hadronic to quark matter at high baryon and 
isospin density is analyzed.
Nonlinear relativistic mean field models 
are used to describe hadronic matter, and the MIT bag model is adopted
for quark matter. The boundaries of the mixed phase and the related critical
points for symmetric and asymmetric matter are obtained. Isospin effects 
appear to be rather significant. The binodal transition line of the 
($T,\rho_B$) diagram is lowered to a region accessible through heavy ion 
collisions in the energy range of the new planned facilities, e.g. the
$FAIR/NICA$ projects. Some observable 
effects are suggested, in particular an {\it Isospin Distillation} mechanism.
The dependence of the results on a suitable treatment of 
isospin contributions in effective QCD Lagrangian approaches,
at the level of explicit isovector parts and/or quark condensates,
is critically discussed.

\end{abstract}
%\noindent
\begin{keyword}
Nuclear Matter at High Baryon Density; Symmetry Energy;
Deconfinement Transition; Critical End Point; Effective QCD Lagrangians\\
PACS numbers: {05.70.Fh, 05.70.Ce, 21.65.+f, 12.38Mh}
\end{keyword}
\end{frontmatter}
%\end{quote}}
\date{\today}
%\maketitle
%\normalsize


\section{Introduction}
Several suggestions are already present about the possibility of 
interesting isospin effects on the transition to a mixed hadron-quark phase 
at high baryon density \cite{muller,ditoro,erice08}. This 
seems to be a
very appealing physics program for the new facilities, FAIR at 
GSI-Darmstadt \cite{fair} and NICA at JINR-Dubna \cite{nica}, 
where heavy ion beams (even unstable, with large isospin asymmetry)
will be available with good intensities in the 1-30 AGeV energy region.

The weak point of those predictions is the lack of a reliable equation of 
state that can describe with the same confidence the two phases, hadronic and
deconfined. On the other hand this also represents a strong theory motivation
to work on more refined effective theories for a strong interacting matter.

The main qualitative argument in favor of noticeable isospin effects 
on the hadron-quark 
transition at high density can be derived from
the Fig.\ref{isoparton}, where we compare typical Equations of State (EoS) for 
Hadron (Nucleon) and Quark 
Matter, at zero temperature, for symmetric 
($\alpha \equiv (\rho_n-\rho_p)/\rho_B \equiv -\rho_3/\rho_B=0.0$) 
and neutron matter 
($\alpha=1.0$), where $\rho_{n,p}$ are the neutron/proton densities and
$\rho_B=\rho_n+\rho_p$ the total baryon density.

For the Hadron part we use a Relativistic Mean Field (RMF) EoS
(\cite{SW85,liubo02,baranPR}) 
with non-linear 
terms and effective $\rho-meson$ coupling for the isovector part, largely used 
to study isospin effects in relativistic heavy ion collisions
\cite{erice08,baranPR}. In order to keep a smooth flow of the physics points 
in the discussion, details about the adopted effective nucleon-meson 
Lagrangians
are presented in the Appendix A.

\begin{figure}
\centering
\includegraphics[scale=0.40]{Fig1.eps} 
%\vskip -1.0cm
\caption{Zero temperature $EoS$ of Symmetric/Neutron Matter: 
Hadron ($NL \rho$), solid lines,
vs. Quark (MIT-Bag), dashed lines. $\alpha_{H,Q}$ represent the isospin 
asymmetry parameters respectively of the hadron,quark matter:
$\alpha_{H,Q}=0$, Symmetric Matter; $\alpha_{H,Q}=1$, Neutron Matter.
}
%\vskip -0.3cm
\label{isoparton} 
\end{figure} 

The energy density and the pressure for the quark phase are given by 
the MIT Bag model \cite{MIT} (two-flavor case) and read, respectively:

%\noindent

\begin{eqnarray}\label{edensq}
\epsilon= 3 \times 2 \sum_{q=u,d}\int \frac{{\rm d}^3k}
{(2\pi)^3}\sqrt{k^{2}+m_{q}^{2}}(f_{q}+\bar{f}_{q})
+B~,
\end{eqnarray}


\begin{eqnarray}\label{pressq}
 P =\frac{ 3\times 2}{3}\sum_{q=u,d}\int \frac{{\rm d}^3k}{(2\pi)^3}
\frac{k^2}{\sqrt{k^{2}+m_{q}^{2}}} (f_{q}+\bar{f}_{q})
-B~,
\end{eqnarray}

\noindent
where B denotes the bag constant (the bag pressure), taken as a rather 
standard value from the hadron spectra ($B=85.7~MeV~fm^{-3}$, 
no density dependence), 
%\cite{s10},
$m_{q}$ are the quark masses ($m_u=m_d=5.5~MeV$ choice), 
and $f_q$, $\bar{f}_{q}$ represent
the Fermi distribution functions for quarks and anti-quarks.
The quark number density is given by 

\begin{eqnarray}\label{rhoq}
 \rho_{i}=<q_{i}^{+}q_{i}>={3\times 2} \int \frac{{\rm d}^3k}{(2\pi)^3}
 (f_{i}-\bar{f}_{i})~, ~~~~i=u,d.
\end{eqnarray}


The transition to the more repulsive quark matter will appear around the
crossing points of the two EoS. We see that such crossing for symmetric matter
($\alpha_H=\alpha_Q=0.0$) is located at rather high density, 
$\rho_B \simeq 7\rho_0$, while for pure neutron matter 
($\alpha_H=\alpha_Q=1.0$) it is moving down to about three times
$\rho_0$. Of course the Fig.1 represents just a simple energetic argument to
support the hadron-quark transition to occur at lower baryon densities
for more isospin asymmetric matter. In the rest of the paper we will
rigourously consider the case of a first order phase transition in the Gibbs
frame for a system with two conserved charges (baryon and isospin), in order 
to derive more detailed results. 
Since the first order phase transition presents a jump in the energy, we can 
expect the mixed phase to start at densities even before the crossing points
of the Fig.1. The lower boundary then can be predicted at relatively low 
baryon densities for asymmetric matter, likely
reached in relativistic heavy ion collisions. Moreover this
point is certainly of interest for the structure of the crust and the inner 
core of 
neutron stars, e.g. see refs. \cite{burgio02,nicotra06} and the review 
\cite{page06}. The ref.\cite{burgio02} is particularly interesting since
similar results are obtained
with rather different hadronic approaches, the RMF and the non-relativistic 
Brueckner-Hartree-Fock (BHF) theory. 

We finally note that the above conclusions are rather independent on the 
isoscalar 
part of the used Hadron
EoS at high density, that is chosen to be rather soft in agreement with 
collective flow and
kaon production data \cite{daniel02,fuchs06}.
 
In the used Bag Model no gluon interactions, the $\alpha_s$-strong coupling 
parameter, are 
included. We remark that this in fact would enhance the above effect, since it
represents an attractive correction for a fixed B-constant, 
see \cite{bmuller95}. A reduction of the Bag-constant with increasing 
baryon density, as suggested by various models, see ref.\cite{burgio02}, 
will also go in the direction of an ``earlier'' (lower density) transition,
as already seen in ref.\cite{ditoro}.
At variance, the presence of
explicit isovector contributions in the quark phase could play an 
important role, 
as shown in the following also for other isospin properties inside the mixed 
phase.

\section{Isospin effects on the Mixed Phase}


We can study in detail the isospin dependence of the transition 
densities
\cite{muller,ditoro,erice08}.
% \cite{MuellerNPA618} in a systematic way,
% exploring also
%the possibility of forming a mixed-phase of quarks and hadrons in
%experiments at energies of the order of a few $GeV$ per nucleon.
%Concerning the hadronic phase, we use the relativistic
%non-linear model of Glendenning-Moszkowski (in particular the ``soft''
%$GM3$ choice) 
%\cite{GlendenningPRL18}, where the isovector part is treated 
%just with $\rho$ meson coupling, and
%the iso-stiffer $NL\rho\delta$ interaction \cite{deconf06}. 
%For the quark phase we consider the $MIT$ bag model 
%\cite{MitbagPRD9}
%with various bag pressure constants.  In particular 
%we are interested in those parameter sets
%which would allow the existence of quark stars
%\cite{DragoPLB511}, i.e. parameters sets for
%which the so-called Witten-Bodmer hypothesis is satisfied
%\cite{WittenPRD30,BodmerPRD4}. 
%One of the
%aim of our work it to show that if quark stars are indeed possible,
%it is then very likely to find signals of the formation of a mixed
%quark-hadron phase in intermediate-energy heavy-ion experiments
%\cite{deconf06}.
The structure of the mixed phase is obtained by
imposing the Gibbs conditions \cite{Landaustat} for
chemical potentials and pressure and by requiring
the conservation of the total baryon and isospin densities:

\noindent
\begin{eqnarray}\label{gibbs}
&&\mu_{B}^{H}(\rho_B^H,\rho_3^H,T)=\mu_{B}^{Q}(\rho_B^Q,\rho_3^Q,T)~, 
\nonumber \\
&&\mu_{3}^{H}(\rho_B^H,\rho_3^H,T)=\mu_{3}^{Q}(\rho_B^Q,\rho_3^Q,T)~, 
\nonumber \\
&&P^{H}(T)(\rho_B^H,\rho_3^H,T)=P^{Q}(T)(\rho_B^Q,\rho_3^Q,T)~, 
\nonumber \\
&&\rho_{B} = (1-\chi)\rho_B^H + \chi \rho_B^Q ~,
\nonumber \\
&&\rho_{3}= (1-\chi)\rho_3^H + \chi \rho_3^Q ~,
\end{eqnarray}

where $\chi$ is the fraction of quark matter in the mixed phase 
and T is the temperature.

The consistent definitions for the densities and chemical potentials
in the two phases are given by :

%\noindent
\begin{eqnarray}\label{hadronrhomu}
&&\rho_{B}^{H}=\rho_{p}+\rho_{n}, ~~~~\rho_3^H=\rho_p-\rho_n~,
\nonumber \\
&&\mu_B^{H} = \frac{\mu_p + \mu_n}{2}, ~~~~~
\mu_3^{H} = \frac{\mu_p - \mu_n}{2}~,
\end{eqnarray}
for the Hadron Phase and

%\noindent
\begin{eqnarray}\label{quarkrhomu}
&&\rho_B^{Q}=\frac{1}{3}(\rho_{u}+\rho_{d})~,
~~~~~~ \rho_3^{Q}=\rho_{u}-\rho_{d}~,
\nonumber \\
&&\mu_B^{Q}=\frac{3}{2}(\mu_u + \mu_d),~~~~
\mu_3^{Q} =\frac{\mu_u - \mu_d}{2}~,
\end{eqnarray}

for the Quark Phase.

%\vskip 0.5cm
The related asymmetry parameters are:
%\noindent
\begin{equation}
\alpha^{H} \equiv -\frac{\rho_3^{H}}{\rho_B^{H}} =
\frac{\rho_n-\rho_p}{\rho_n+\rho_p}~,~~~~~~
\alpha^{Q} \equiv -\frac{\rho_3^{Q}}{\rho_B^{Q}} =
 3 \frac{\rho_d-\rho_u}{\rho_d+\rho_u}~.
\end{equation}
%\vskip 1.0cm

Nucleon and quark chemical potentials, as well as the pressures in the 
two phases, are directly derived from the respective EoS.

In this way we get the $binodal$ surface which gives the phase coexistence 
region
in the $(T,\rho_B,\rho_3)$ space.
%\cite{GlendenningPRD46,MuellerNPA618}. 
For a fixed value of the
total asymmetry  $\alpha_T=-\rho_3/\rho_B$ 
 we will study the boundaries of the mixed phase
region in the $(T,\rho_B)$ plane. 
Since in general the charge chemical potential is related to the symmetry 
term of the EoS, \cite{baranPR},
$
\mu_3 = 2 E_{sym}(\rho_B) \frac{\rho_3}{\rho_B},
$ 
we expect critical and transition densities
rather sensitive to the isovector channel in the two phases.

In the hadron sector we will use the Non-Linear
Relativistic Mean Field models, \cite{liubo02,baranPR,erice08}, with
different structure of the isovector part, already tested to describe
the isospin dependence of
collective flows and meson production for heavy ion collisions 
at intermediate energies, \cite{greco03,gaitanos04,ferini06}.
We will refer to these different Iso-Lagrangians as: i) $NL$, where no 
isovector
meson is included and the symmetry term is only given by the kinetic 
Fermi contribution, ii) $NL\rho$ when the interaction contribution of
an isovector-vector meson is considered and finally iii) $NL\rho\delta$
where also the contribution of an isovector-scalar meson is accounted for.
 See details in Appendix A and refs.\cite{liubo02,baranPR,erice08}.

We will look at the effect on the hadron-quark transition of the different
stiffness of the symmetry term at high baryon densities in the different
parametrizations. As clearly shown in Appendix A, where a rather transparent
form for the density dependence of the symmetry energy
in RMF approaches is discussed, the potential part of the symmetry term
will be proportional to the baryon density in the $NL\rho$ choice
and even stiffer in the $NL\rho\delta$ case.
%At high baryon densities the symmetry energy is increasing in a different
%way with the various
%choices, see \cite{liubo02,baranPR,erice08}, and we will look at the effect 
%on the phase transition, \cite{couplings}.  

As already mentioned, in the quark phase we use the MIT-Bag Model, where 
the symmetry term is only given by the Fermi contribution. The Bag parameter
B is fixed for each baryon density to a constant, rather standard, value
$B^{1/4}=160MeV$, corresponding to a Bag Pressure of $85.7~MeV~fm^{-3}$.


\begin{figure}
\begin{center}
\includegraphics[scale=0.40]{Fig2.eps}
%\vskip -1.0cm
\caption{
Dependence on the Hadron Symmetry Energy of the Lower ($\chi=0.0$) and
Upper ($\chi=1.0$) Boundaries of the Mixed Phase, at zero temperature,
vs. the asymmetry parameter.
Quark $EoS$: $MIT$ bag model with
$B^{1/4}$=160 $MeV$.
}
\vskip 1.0cm
\label{boundaries}
\end{center}
\end{figure}




In general for each effective interactive Lagrangian we can simulate the 
solution of the highly non-linear system of Eqs.(\ref{gibbs}), via an
iterative minimization procedure, in order to determine the binodal 
boundaries and
the Critical End Point (CEP)
($T_c,~\rho^B_c$) of the mixed phase.

A relatively simple calculation can be performed at zero temperature. The 
isospin effect (asymmetry dependence) on the Lower $(\chi=0.0)$ and 
Upper $(\chi=1.0)$ transition densities of the Mixed Phase are shown
in Fig.\ref{boundaries} for various choices of the Hadron EoS. The effect
of a larger repulsion of the symmetry energy in the hadron sector, from
$NL$ to $NL\rho$ and to $NL\rho\delta$,
 is
clearly evident on the lower boundary with a sharp decrease of the transition 
density even at relatively low asymmetries.




\begin{figure}
\begin{center}
\includegraphics[scale=0.40]{Fig3.eps}
%\vskip -1.0cm
\caption{
Binodal surface for symmetric ($\alpha=0.0$) and 
asymmetric ($\alpha=0.2$) matter. 
Hadron EoS from
%Variation of the transition density with proton fraction for various
%hadronic $EoS$ parameterizations. Dotted line: $GM3$ $RMF$-model
%\cite{GlendenningPRL18};
$NL\rho$ interaction. 
Quark $EoS$: $MIT$ bag model with
$B^{1/4}$=160 $MeV$. The grey region corresponds to the conditions reached
in Heavy Ion Collisions simulations at few AGeV beam energies, see
 Section 3.
}
%\vskip -0.7cm
\label{tcrit}
\end{center}
\end{figure}
 
\begin{figure}
\centering
\includegraphics[scale=0.45,angle=-90]{Fig4.ps}
%\vskip -1.0cm
\caption{
Maxwell construction for symmetric ($\alpha=0.0$) matter at temperatures
T=0, 50 and 80 MeV. For Pressure and Chemical Potentials we use: 
Hadron EoS (black curves) from
%Variation of the transition density with proton fraction for various
%hadronic $EoS$ parameterizations. Dotted line: $GM3$ $RMF$-model
%\cite{GlendenningPRL18};
$NL\rho$ interaction;  
 Quark $EoS$ (grey curves) from $MIT$ bag model with
$B^{1/4}$=160 $MeV$. 
}
%\vskip -0.7cm
\label{maxwell}
%\end{center}
\end{figure}


Typical results for isospin effects on the whole binodal ``surface'' are 
presented 
in Fig.\ref{tcrit} for symmetric and asymmetric matter.
For the hadron part we have started from a $NL\rho$ effective 
Lagrangian
very close to other widely used relativistic effective models, e.g. see
the $GM3$ of ref.\cite{GlendenningPRL18} and the $NL3$ interaction of P.Ring
and collaborators \cite{vretenar03}, which has also given good nuclear 
structure 
results, even for exotic nuclei. 

In the symmetric matter case the mixed phase is evaluated from the simpler 
Maxwell conditions. The results are shown in Fig.\ref{maxwell} for the 
same hadron and quark $EoS$'s as in Fig.\ref{tcrit} at temperatures T=0, 
50 and 80 MeV. The equal chemical potential densities (intersection of the 
dotted line in the lower panel) must correspond to the equal 
pressure densities of the upper panels.
We nicely see: i) at T=0 MeV the mixed phase is centered around 
$\rho/\rho_0 \simeq 7.0$ , exactly the $\alpha=0$ crossing point of 
Fig.\ref{isoparton}, confirming our energetic argument about the transition 
location; ii) the size of the mixed phase is shrinking with temperature, it 
is very narrow at T=50 MeV and finally   
at T=80 MeV 
we cannot have anymore a first order transition. In fact the Critical End Point
is found at $T_c \simeq 58 MeV$, $\rho_c/\rho_0 \simeq 3.8$, 
$P_c \simeq 120 MeV/fm^3$ and $\mu_c \simeq 1130 MeV$, 
see also Fig.\ref{tcrit}. The result is dependent on the choice of the Bag 
constant, with increasing of the critical temperature with the Bag value due
to the reduction of the pressure in the quark phase, while the chemical 
potentials are not affected. This point will not be further discussed 
since here the main focus is on the isospin dependence of the mixed phase
at lower temperatures, that can be probed in heavy ion collisions at 
intermediate energies.

As expected, the lower boundary
of the mixed phase is mostly affected by isospin effects. In spite of the 
relatively small
total asymmetry, $\alpha=0.2$, we clearly observe in Fig.\ref{tcrit} a shift 
to the left of the
first transition boundary, in particular at low temperature, and an indication
of a relatively 
``earlier'' Critical End Point,
around ($T_c=50-55 MeV, \rho_c/\rho_0 =2.5-2.8$).

Actually we see from Fig.\ref{tcrit} that
for asymmetric matter, we are not 
able to explicitly reach the $CEP$ of the binodal surface. Since this feature 
appears also in the following figures we must add a general comment.
From the present results we cannot exclude a $CEP$ at higher temperature
and smaller baryon density. However we have also to mention 
some numerical problems.
The solution of the Gibbs conditions, the highly non-linear system
of Eqs.(\ref{gibbs}), is found through an iterative multi-parameter 
minimization procedure (the Newton-Raphson method). When we are close to the 
Critical End Point, for each $\chi$-concentration the baryon and isospin 
densities of the hadron and quark phases become very similar and we 
start to have problems in finding a definite minimum. We are working on 
improving the numerical accuracy but in any case the trends towards 
the $CEP$ are always quite clearly observed, as a severe reduction
of the mixed phase.

In the following we will concentrate on properties of the mixed phase
mostly located at high density and relatively low temperature, well
described in the calculation and
  within the reach of
heavy ion collisions in the few AGeV range, see Section 3.

\subsection{Inside the Mixed Phase of Asymmetric Matter}

For $\alpha=0.2$ asymmetric matter,
 in the Figs.\ref{NLrmix}, \ref{NLrdmix} we show also the ($T,\rho_B$)
curves inside the Mixed Phase corresponding to a $20\%$ and $50\%$ 
presence of the quark component ($\chi=0.2,0.5$), evaluated respectively
with the
two choices, $NL\rho$ and $NL\rho\delta$, of the symmetry interaction
in the hadron sector. We note, as also expected from Fig.\ref{boundaries},
that in the more repulsive $NL\rho\delta$ case the lower boundary is
much shifted to the left. However this effect is not so evident for the curve
corresponding to a $20\%$ quark concentration, and almost absent for the
$50\%$ case. The conclusion seems to be that for a stiffer symmetry term
in a heavy-ion collision at intermediate energies during the compression stage
we can have more chance
to probe the mixed phase, although in a region with small weight of the
quark component. 

\begin{figure}
\begin{center}
\includegraphics[scale=0.40]{Fig5.eps}
%\vskip -1.0cm
\caption{Asymmetric $\alpha=0.2$ matter.
Binodal surface and ($T,\rho_B$)
curves for various quark concentrations
($\chi=0.2,0.5$) in the mixed phase.
Quark $EoS$: $MIT$ Bag model with
$B^{1/4}$=160 $MeV$. Hadron $EoS$: $NL\rho$ Effective Interaction.
}
%\vskip -0.7cm
\label{NLrmix}
\end{center}
\end{figure}


\begin{figure}
\begin{center}
\includegraphics[scale=0.40]{Fig6.eps}
%\vskip -1.0cm
\caption{As in Fig.\ref{NLrmix}, for the $NL\rho\delta$ Effective Interaction
in the Hadron sector.
}
%\vskip -0.7cm
\label{NLrdmix}
\end{center}
\end{figure}



\begin{figure}
\begin{center}
\includegraphics[scale=0.40]{Fig7.eps}
%\vskip -1.0cm
\caption{Asymmetric $\alpha=0.2$ matter.
Binodal surface and ($T,\rho_B^H,\rho_B^Q$)
curves for various quark concentrations in the mixed phase.
Quark $EoS$: $MIT$ Bag model with
$B^{1/4}$=160 $MeV$. Hadron $EoS$: $NL\rho$ Effective Interaction.
}
%\vskip -0.7cm
\label{NLrmixHQ}
\end{center}
\end{figure}


\begin{figure}
\begin{center}
\includegraphics[scale=0.40]{Fig8.eps}
%\vskip -1.0cm
\caption{As in Fig.\ref{NLrmixHQ}, for the $NL\rho\delta$ Effective 
Interaction in the Hadron sector.
}
%\vskip -0.7cm
\label{NLrdmixHQ}
\end{center}
\end{figure}

In fact from the solution of the system Eq.(\ref{gibbs}) we get 
the baryon densities $\rho_B^H,\rho_B^Q$ in the two phases for any
$\chi$ value. In the Figs. \ref{NLrmixHQ}, \ref{NLrdmixHQ} we present the
results for the same weights $20\%, 50\%$ of the quark phase of the previous 
figures. The quark phase appears always with larger baryon density, even
for the lowest value of the concentration.



Can we expect some signatures related to the subsequent
hadronization in the following expansion?



\begin{figure}
\begin{center}
\includegraphics[scale=0.40]{Fig9.eps}
%\vskip -1.0cm
\caption{
Quark asymmetry in the mixed phase vs. the quark concentration
for asymmetric matter with $T=0$ and $\alpha=0.2$.
$NL\rho$ and $NL\rho\delta$ Effective Hadron Interactions are considered.
Quark $EoS$: $MIT$ bag model with
$B^{1/4}$=160 $MeV$.
}
%\vskip -0.7cm
\label{alphaq}
\end{center}
\end{figure}


An interesting possibility is coming from the study of the asymmetry
$\alpha^Q$ in the quark phase. In fact since the symmetry energy is rather 
different in the two phases we can expect an Isospin Distillation 
(or Fractionation), very similar to the one observed in the Liquid-Gas 
transition in dilute nuclear matter \cite{baran98,chomazPR,baranPR}, this time 
with the larger isospin 
content in the higher density quark phase.

In Fig.\ref{alphaq} we show the asymmetry $\alpha^Q$ in the quark phase
as a function of the quark concentration $\chi$ for the case with global
asymmetry $\alpha=0.2$ (zero temperature). The calculation is performed with 
the two choices of the symmetry term in the hadron sector. We see an impressive
increase of the quark asymmetry when we approach the lower boundary of the
mixed phase, even to values larger than one, likely just for numerical 
accuracy \cite{Qasy}.
Of course the quark asymmetry recovers the global value 0.2 at the upper
boundary $\chi=1$. A simple algebraic calculation allows to evaluate the 
corresponding asymmetries of the hadron phase.
In fact from the charge conservation we have that for any $\chi$-mixture
 the global
asymmetry $\alpha$ is given by:

\begin{equation}\label{alphamix}
\alpha \equiv -\frac{\rho_3}{\rho_B} = 
\frac{(1-\chi)\alpha^H}{(1-\chi)+\chi \frac{\rho_B^Q}{\rho_B^H}} +
\frac{\chi \alpha^Q}{(1-\chi) \frac{\rho_B^H}{\rho_B^Q}+ \chi}
\end{equation}

For any $\chi$, from the calculated $\alpha^Q$ of Fig.\ref{alphaq}
and the $\rho_B^H,\rho_B^Q$ of Figs.\ref{NLrmixHQ}, \ref{NLrdmixHQ},
we can get the correspondent asymmetry of the hadron phase $\alpha^H$.
 For a $20\%$ quark concentration
we have an $\alpha^Q/\alpha^H$ ratio around 5 for $NL\rho$ and around 20
for $NL\rho\delta$, more repulsive in the isovector channel. 
It is also interesting to compare the isospin content $N/Z$ of the high
density region expected from transport simulations without the Hadron-Quark
transition and the effective $N/Z$ of the quark phase in a 
$20\%$ concentration.
In the case of $Au+Au$ (initial $N/Z=1.5$) central collisions at $1AGeV$
in pure hadronic simulations we get in the high density phase a reduced
$N/Z \sim 1.2-1.25$ (respectively with $NL\rho\delta-NL\rho$ interactions)
due to the fast neutron emission \cite{gaitanos04,ferini06}. The corresponding
isospin content of the quark phases is much larger, $N/Z=3$ for $NL\rho$ 
and $N/Z=5.7$ for $NL\rho\delta$. This is the $neutron~trapping$ effect
discussed in the Section 3.
We could expect
a signal of such large asymmetries, coupled to a larger baryon density in 
the quark phase, in the subsequent 
hadronization.

We finally remark that at higher temperature and smaller baryon chemical 
potential (ultrarelativistic collisions) the isospin effects discussed here 
are expected to vanish \cite{sissakian08}, even if other physics can enter 
the game and charge asymmetry effects are predicted also at $\mu_B=0$ and
$T \simeq 170~MeV$ \cite{kogut04,toublan05}.


\subsection*{Isospin in Effective Quark Models}

All the above results will be very sensitive to the explicit
inclusion of isovector interactions in effective 
non-perturbative QCD models in regions at high baryon chemical potentials.
Unfortunately few attempts have been worked out for two main reasons:
i) the difficulties of Lattice-QCD calculations at high baryon densities;
ii) the main interest on the QGP phase transition at high temperature and
small baryon chemical potentials, as probed in the expanding fireball
of ultrarelativistic heavy ion collisions.
A first approach can be supplied by a two-flavor Nambu-Jona Lasinio ($NJL$) 
model \cite{NJL}, which in fact describes the chiral restoration but not
the deconfinement dynamics. 
The isospin asymmetry can be included in a flavor-mixing picture 
\cite{frank03,buballa05}, corresponding to different couplings to
the (u,d) quark-antiquark condensates.
%. These isospin effects are induced by a
%determinant interaction related to the breaking of the axial symmetry.
%The new  Gap Equations are like 
%$M_i=m_i-4G_1 \Phi_i-4G_2 \Phi_j$, $i \not= j,(u,d)$ , where the 
%$\Phi_{u,d}=<\bar u u>,<\bar d d>$ are the two (negative) condensates 
%and $m_{u,d}=m$
%the (equal) current masses. Introducing explicitily a flavor mixing, 
As a consequence we can have now a 
dependence of the constituent mass of a given flavor to both 
quark condensates.
%via $G_1=(1-\alpha) G_0, G_2= \alpha G_0$ we have the coupled equations
%\begin{eqnarray}
%&&M_u=m - 4 G_0 \Phi_u + 4 \alpha G_0 (\Phi_u - \Phi_d), \nonumber \\ 
%&&M_d=m - 4G_0 \Phi_u + 4 (1-\alpha) G_0 (\Phi_u - \Phi_d).
%\label{mix}
%\end{eqnarray}
We devote the Appendix B to a detailed study of this isospin effects in the
$NJL$ chiral dynamics. 

Due to the scalar nature of the interacting part of the
corresponding Lagrangians only the quark effective mass dynamics will 
be affected.
%For $\alpha=1/2$ we have back the usual NJL ($M_u=M_d$), while small/large
%mixing is for $\alpha \Rightarrow 0$/$\alpha \Rightarrow 1$ respectively.
%In neutron rich matter $\mid \Phi_d \mid$ decreases more rapidly due to the 
%larger $\rho_d$ and so $(\Phi_u -\Phi_d)<0$. 
In the ``realistic'' small mixing 
case, see also \cite{frank03,shao06}, we get a definite $M_u^*>M_d^*$ 
splitting at high baryon density (before the chiral restoration). 
%This expectation is nicely confirmed by a full calculation \cite{plum_tesi}
%of the coupled gap equations with standard parameters (same as in 
%ref.\cite{frank03}). 

All that can indicate a more fundamental confirmation of the $m^*_p>m^*_n$ 
splitting in the hadron phase,
as suggested by the effective $QHD$ model with the isovector scalar $\delta$ 
coupling, see \cite{liubo02,baranPR}.
However such isospin mixing effect results in a very small variation
of the symmetry energy in the quark phase, related only to the Fermi
kinetic contribution. 
%In fact this represents just a very first step towards a more complete
%treatment of isovector interactions in effective quark models, of large 
%interest for the discussion of the phase transition at high densities.
Moreover we remind that confinement 
is still 
missing in this $NJL$ mean field approach. In any case there are extensive
suggestions about a favored chiral symmetry restoration in systems with
large neutron excess \cite{kaiser09}.

More generally starting from the QCD Lagrangian one can arrive to an 
effective color current-current interaction where an expansion in various 
components can provide isovector contributions, \cite{weise07}.

In this respect we remark another interesting ``indirect'' isospin effect,
i.e. not directly related to isovector terms in the effective Lagrangian,
related to the presence of quark condensates due to the attractive gluon
interaction. We note that
just a color-pairing effect in the two-flavor
system (the $2SC$ phase \cite{alford08}) would imply a stiffer symmetry energy
in the quark EoS since we have a larger attraction when the densities of up 
and down quarks are equal. A first study of the high density hadron-quark 
transition including such gluon correlation in the Bag model has been 
presented very recently in \cite{pagliara10}. 
Now the symmetry energy difference between hadron and quark phases is
partially reduced, at least at low temperatures, and consequently also
the isospin effects discussed in detail in this work will be less robust,
although still present. An interesting point is that in any case the
quark phase is more bound due to the attractive gluon contribution.
Hence the transition to 
the mixed phase will still appear at relatively low baryon densities,
 now for an ``isoscalar'' mechanism,  
within the reach of ``low energy''
heavy ion collisions, i.e. in the range of few AGeV. 
As an intuitive picture we can refer again to the Fig.1. Essentially
the difference between the $\alpha_Q=0.0$ and $\alpha_Q=1$ curves
is increasing but meanwhile both are decreasing.

With increasing temperature the color pairing effect will be in general 
reduced, as confirmed in \cite{huang03} in an extended $NJL$ calculation,
and so isospin effects, as discussed before, will be more relevant. All that, 
as well as the
location of the Critical-End-Point, is naturally related to the used value of 
the superconducting gap,
opening new stimulating perspectives.
In this sense new experiments on mixed phase properties observed with
isospin asymmetric heavy ion collisions, as suggested in the final section,
will be extremely important.





\section{Perspectives and suggested observables}

Based on the qualitative argument of the Introduction and on more detailed
calculations in a first order phase transition scheme, we have predicted
rather ``robust'' isospin effects on the hadron-quark transition at
high baryon densities, not depending on details of the EoS
parametrizations in the hadron and quark phases.

Our results  seem to indicate a specific region where the onset of the
mixed phase should be mainly located: $2<\rho_B/\rho_0<4$, $T \leq 50-60~MeV$,
 for realistic asymmetries $\alpha \sim 0.2-0.3$.
A key question is if such a region of the phase space can be explored by means
of Heavy-Ion-Collisions. In refs.\cite{ditoro,erice08} it is shown that even
collisions of stable nuclei at intermediate energies ($E/A \sim 1-2~GeV$)
make available the pertinent ($T,\rho_B,\alpha$) region where the
phase transition is expected to occur.

%In order to check the possibility of observing some precursor signals
%of a new physics even in collisions of stable nuclei at
%intermediate energies in refs. \cite{ditoro,erice08} some event simulations 
%are shown for the
%collision of very heavy, neutron-rich, elements, using a consistent
%Relativistic Mean Field approach with the same interactions. 

In this respect we can refer to the
reaction $^{238}U+^{238}U$ (average isospin asymmetry $\alpha=0.22$) at
$1~AGeV$ that has been investigated in ref.\cite{ditoro}, using
a consistent Relativistic Mean Field approach with the same interactions,
  for a semicentral 
impact parameter $b=7~fm$, chosen  
just to increase
the neutron excess in the interacting region. 
The evolution of momentum distribution
and baryon/isospin densities in a space cell located in the c.m. of the 
system has been also studied.
After about $10~fm/c$ a local
equilibration is achieved still in the compressed phase, before 
the fast expansion.  We have a unique Fermi distribution and
from a simple fit the ``local'' temperature can be evaluated. 
A rather exotic nuclear matter is formed in a transient
time of the order of $10-20~fm/c$, with baryon density around $3-4\rho_0$,
temperature $50-60~MeV$, and isospin asymmetry
between $0.2$ and $0.3$, likely inside the estimated mixed 
phase region.
We note that high intensity $^{238}U$ beams in this energy range
will be available in the first stage of the FAIR facility
\cite{fair,horst} and also at JINR-Dubna in the Nuclotron
first step of the NICA project \cite{sissakian08}.
%Some signatures
%could come from an expected ``neutron trapping'' effect.
%\begin{figure}
%\begin{center}
%\vskip -1.7cm
%\includegraphics[scale=0.40]{erice9.ps}
%\vskip -0.5cm
%\caption{$^{238}U+^{238}U$, $1~AGeV$, semicentral. Correlation between 
%density, 
%temperature (black values), momentum
%thermalization (3-D plots), inside a cubic cell, 2.5 $fm$ wide, in the center
%of mass of the system.}
%\label{figUU}
%\vskip -0.7cm
%\end{center}
%\end{figure}


Which are the observable effects to look at if we enter and/or cross the 
mixed phase?

As already stressed, a first expectation will be the Isospin Distillation 
effect, a kind of {\it neutron trapping} in the quark phase, supported by 
statistical fluctuations   \cite{ditoro} as well as by a 
symmetry energy difference in the
two phases, as  discussed in Section 2.1.
In fact while in the pure hadron matter (neutron-rich) at high density 
we have a large neutron
potential repulsion (in both $NL\rho$, $NL\rho\delta$ cases), in the
quark phase the $d$-quarks see a much smaller symmetry repulsion only
due to the kinetic contribution 
from the Fermi gas.
As a consequence while in a
pure hadronic phase neutrons are quickly emitted or ``transformed'' in
protons by inelastic collisions \cite{ferini06}, when the mixed phase
starts forming, neutrons are kept in the interacting system,
in the quark phase which is also at larger baryon density, up to the
subsequent hadronization in the expansion stage \cite{erice08}.
Observables related to such neutron ``trapping'' could be an
inversion in the trend of the formation of neutron rich clusters,
an enhancement of the production of isospin-rich nucleon resonances and 
subsequent decays, i.e. an increase of $\pi^-/\pi^+$, $K^0/K^+$ yield ratios 
for reaction
products coming from high density regions, that could be selected looking
at large transverse
momenta, corresponding to a large radial flow. 
%Similar effects can be derived from the inclusion of the strangeness
%in the present model, in both hadron and quark phases, likely with new
%sensitive observables. Some work is going on along this direction.  
 
 If such kinetic selection of particles from the mixed phase can 
really be successful also other potential signatures would become available.
One is related to the general softening of the matter, due to the contribution
of more degrees of freedom, that should show up in the damping of collective 
flows \cite{csernai99}.

The azimuthal distributions (elliptic flows) will be
particularly affected since particles mostly retain their high transverse
momenta escaping along directions orthogonal to the reaction plane without
suffering much rescattering processes. 
Thus a further signature could be the observation, for the selected particles, 
of 
the onset of a quark-number scaling of the elliptic flow: a property of 
hadronization
by quark coalescence that has been predicted and observed at RHIC energies,
i.e. for the transition at $\mu_B=0$ \cite{fries08}.


We  note that all the above results, on the Binodal Boundaries of the
mixed phase and on the Isospin  Distillation are
sensitive to the symmetry term in the hadron sector, although the main 
isospin effects are present for all the parametrizations of the isovector 
interaction. At variance,
% while they
%are
%rather independent of the used Hadron
%EoS at high density for symmetric matter, that should be relatively soft 
%in agreement with 
%collective flow and
%kaon production data, \cite{daniel02}. 
  for the quark 
sector the lack of explicit isovector
terms could strongly affect the location 
of the phase transition in asymmetric matter and the related 
expected observables.

In conclusion the aim of this work is twofold:
\begin{itemize}
\item
{To stimulate new experiments on isospin effects in heavy ion collisions 
at intermediate energies (in a few $AGeV$ range) with  attention
to the isospin content of produced particles and to elliptic flow properties,
in particular for high-$p_t$ selections.}
\item{To stimulate more refined models of effective Lagrangians for
non-perturbative QCD, where isovector channels are consistently accounted for
 and/or gluon correlations, leading to diquark condensates, can induce 
symmetry 
energy effects.}
\end{itemize}



\subsection*{Acknowledgments}

This project is supported by the National Natural Science Foundation of China
under Grant No. 10875160 and the INFN of Italy. V.B. is grateful for the warm
hospitality at Laboratori Nazionali del Sud, INFN. This collaboration is
supported in part by the Romanian Ministry for Education and Research under 
the CNCSIS contract PNII ID-946/2007.

\vspace{0.5cm}

\appendix

\section{Equation of state for hadronic matter}

\subsection*{Nonlinear (NL) relativistic mean field model with constant 
couplings}
%{\hskip  0.7cm}


A Lagrangian density with ``minimal'' meson channels and non-linear terms
is used. The nuclear interaction is mediated by two isoscalar,
the scalar $\sigma$ and the vector $\omega$, and two isovector, 
 the scalar $\delta$ and the vector $\rho$, mesons. Non linear terms are 
considered only for the $\sigma$ contribution to account for the correct
compressibility around saturation. Constant nucleon-meson couplings are used,
chosen to reproduce the saturation properties and to represent a reasonable
average of the density dependence predicted by Relativistic 
Brueckner-Hartree-Fock calculations 
\cite{hofmann01,goegelein08}, see details in refs.\cite{liubo02,baranPR}.

\noindent
%\begin{widetext}
\begin{eqnarray}\label{lagrdens}
{L} &=& \bar{\psi}[i\gamma_{\mu}\partial^{\mu}-
(M-g_{\sigma}\sigma -g_{\delta}\vec{\tau}\cdot\vec{\delta})
-g{_\omega}\gamma_\mu\omega^{\mu}-g_\rho\gamma^{\mu}\vec\tau\cdot
\vec{b}_{\mu}]\psi \nonumber \\&&
+\frac{1}{2}(\partial_{\mu}\sigma\partial^{\mu}\sigma-m_{\sigma}^2\sigma^2)
-U(\phi)+\frac{1}{2}m^2_{\omega}\omega_{\mu} \omega^{\mu}
+\frac{1}{2}m^2_{\rho}\vec{b}_{\mu}\cdot\vec{b}^{\mu} \nonumber
\\&&
+\frac{1}{2}(\partial_{\mu}\vec{\delta}\cdot\partial^{\mu}\vec{\delta}
-m_{\delta}^2\vec{\delta^2}) -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}
-\frac{1}{4}\vec{G}_{\mu\nu}\vec{G}^{\mu\nu},
\end{eqnarray}
%\end{widetext}

\noindent
where
% $\phi$ is the $\phi$-meson field,
%$\omega_{\mu}$ is the $\omega$-meson field, $\vec{b}_{\mu}$ is
%$\rho$ meson field, $\vec{\delta}$ is the isovector scalar field of the
%$\delta$-meson.
$F_{\mu\nu}\equiv\partial_{\mu}\omega_{\nu}-\partial_{\nu}\omega_{\mu}$,
$\vec{G}_{\mu\nu}\equiv\partial_{\mu}\vec{b}_{\nu}-\partial_{\nu}
\vec{b}_{\mu}$,
and the $U(\sigma)$ is the nonlinear potential of $\sigma$ meson :
$U(\sigma)=\frac{1}{3}a\sigma^{3}+\frac{1}{4}b\sigma^{4}$.


The EoS for nuclear matter
 at finite temperature in the mean-field approximation (MFA) is given by

\noindent
\begin{eqnarray}\label{edens}
&&\epsilon= 2 \sum_{i=n,p}\int \frac{{\rm
d}^3k}{(2\pi)^3}E_{i}^*(k) (f_{i}(k)+\bar{f}_{i}(k))
+\frac{1}{2}m_\sigma^{2}\sigma^2 + U(\sigma) \nonumber \\&&
+\frac{1}{2}\frac{g_\omega^2}{m_\omega^{2}}\rho_B^2
+\frac{1}{2}\frac{g_\rho^2}{m_\rho^{2}}\rho_3^2
+\frac{1}{2}\frac{g_\delta^2}{m_\delta^{2}}\rho_{s3}^{2}~,
%+\frac{1}{2}m_\omega^{2}\omega_0^2
%+\frac{1}{2}m_\rho^{2}b_0^2
%+\frac{1}{2}m_\delta^{2}\delta_{3}^{2}~,
\end{eqnarray}

\noindent
\begin{eqnarray}\label{press}
&& P =\frac{2}{3}\sum_{i=n,p}\int \frac{{\rm d}^3k}{(2\pi)^3}
\frac{k^2}{E_{i}^*(k)} (f_{i}(k)+\bar{f}_{i}(k))
-\frac{1}{2}m_\sigma^2\phi^2
-U(\phi) \nonumber \\&&
+\frac{1}{2}\frac{g_\omega^2}{m_\omega^{2}}\rho_B^2
+\frac{1}{2}\frac{g_\rho^2}{m_\rho^{2}}\rho_3^2
+\frac{1}{2}\frac{g_\delta^2}{m_\delta^{2}}\rho_{s3}^{2}~,
%+\frac{1}{2}m_\omega^2\omega_0^2
%+\frac{1}{2}m_{\rho}^2{b_0^2}
%-\frac{1}{2}m_\delta^{2}\delta_{3}^{2}~,
\end{eqnarray}

where ${E_i}^*=\sqrt{k^2+{M_i^*}^2}$. The nucleon effective masses
are defined as

\noindent
\begin{equation}\label{mstar}
{M_i}^*=M-g_\sigma\phi\mp g_\delta\delta_3~~~ (-~proton, +~neutron).
\end{equation}

The field equations in the relativistic mean field (RMF) approach are


\begin{eqnarray}\label{fields}
&&\sigma=-\frac{a}{m_\sigma^2}\sigma^2-\frac{b}{m_\sigma^2}\sigma^3
+\frac{g_\sigma}{m_\sigma^2}(\rho_{sp}+\rho_{sn})~,\\&&
\omega_0=\frac{g_\omega}{m_\omega^2}\rho~,\\&&
b_0=\frac{g_\rho}{m_\rho^2}\rho_3~,\\&&
\delta_3=\frac{g_\delta}{m_\delta^2}(\rho_{sp}-\rho_{sn})~,
\end{eqnarray}

%\noindent
with the baryon density $\rho\equiv \rho_{B}^{H}=\rho_{p}+\rho_{n}$
and $\rho_3^H=\rho_p-\rho_n$,
$\rho_{sp}$ and $\rho_{sn}$ are the scalar densities for proton and neutron, 
respectively.
The $f_i(k)$ and $\bar{f}_{i}(k)$ in Eqs.(\ref{edens}, (\ref{press}) 
are the fermion and 
antifermion
distribution functions for protons and neutrons ($i=p,n$):

\noindent
\begin{eqnarray}\label{ffbar}
&&f_i(k)=\frac{1}{1+\exp\{({E_i}^*(k)-{\mu_i^*})/T \} }\,,\\&&
\bar{f}_{i}(k)=\frac{1}{1+\exp\{({E_i}^*(k)+{\mu_i^*})/T \} }.
\end{eqnarray}

%\noindent
where the effective chemical potential ${\mu_{i}}^*$
is determined by the nucleon density $\rho_{i}$

%\noindent
\begin{equation}\label{bardens}
\rho_i=2\int\frac{{\rm d}^3k}{(2\pi)^3}(f_{i}(k)-\bar{f}_{i}(k))\,,
\end{equation}

while the scalar density $\rho_{s,i}$, which gives the coupling to the 
scalar fields
is given by
%\noindent
\begin{eqnarray}\label{rhoscal}
\rho_{s,i}=2 \int\frac{{\rm d}^3k}{(2\pi)^3}\frac{M_{i}^*}{E_{i}^*}
(f_{i}(k)+ \bar{f}_{i}(k))~,
\end{eqnarray}
 note the ${M_i^*}/{E_i^*}$ quenching factor at high baryon density.

%\noindent
The $\mu_i^*$ are related to the chemical potentials
$\mu_i=\partial\epsilon/\partial\rho_i$
in terms
of the vector meson mean fields by the equation

%\noindent
\begin{eqnarray}\label{npchempot}
&&\mu_p=\mu_p^\star + \frac{g_\omega^2}{m_\omega^2}\rho_B + 
\frac{g_\rho^2}{m_\rho^2}\rho_3 \nonumber \\&&
\mu_n=\mu_n^\star + \frac{g_\omega^2}{m_\omega^2}\rho_B - 
\frac{g_\rho^2}{m_\rho^2}\rho_3.
\end{eqnarray}

The baryon and isospin chemical potentials in the hadron phase
can be expressed in terms of the (p,n) ones
as 
%\noindent
\begin{eqnarray}\label{hchem}
\mu_B^{H} = \frac{\mu_p + \mu_n}{2}, ~~~~~
\mu_3^{H} = \frac{\mu_p - \mu_n}{2}~.
\end{eqnarray}


In presence of the coupling to the two isovector  $\rho,\delta$-meson fields,
 the expression for the symmetry energy has a simple transparent
form, see \cite{liubo02,grecoPRC03,baranPR}:

%\noindent
\begin{equation}\label{esym}
E_{sym}(\rho)=\frac{1}{6} \frac{k_{F}^{2}}{E_{F}^{\star}}+
\frac{1}{2}[f_{\rho}-f_{\delta}(\frac{M^{\star}}{E_{F}^{\star}})^{2}]\rho~,
\end{equation}

%\noindent
where $M^{\star}=M-g_{\sigma}\sigma$ and
${E_F}^\star=\sqrt{k_{F}^2+{M^\star}^2}$.

Now we easily see that in the $NL\rho\delta$ choice we have a large increase 
of the symmetry energy at high baryon densities. The potential simmetry term
is given by the combination 
$[f_{\rho}-f_{\delta}({M^{\star}}/{E_{F}^{\star}})^{2}]$ of the repulsive 
vector $\rho$ and attractive scalar $\delta$ isovector couplings. Thus, 
when the $\delta$ is included we have to increase the $\rho$-meso coupling 
in order to reproduce the same asymmetry parameter $a_4$ at saturation. 
The net effect will be a stiffer symmetry energy at higher baryon densities 
due to the
$M^{\star}/E_F^{\star}$ quenching of the attractive part.



\subsection*{Parameter determination}

{\hskip  0.7cm}
The coupling constants
are fixed from good 
saturation properties and from Dirac-Brueckner estimations, see the detailed
discussions in refs.\cite{liubo02,grecoPRC03,baranPR}. The isoscalar part of 
the EoS is chosen to be rather soft at high densities, see \cite{liubo05},
in order to satisfy the experimental constraints from collective flows and
kaon production in intermediate energy heavy ion collisions
\cite{daniel02,fuchs06}.

The coupling constants, $f_{i}\equiv g_{i}^{2}/m_{i}^{2}$,
$i=\sigma, \omega, \rho, \delta$, and the two parameters of the $\sigma$
self-interacting terms : $A\equiv a/g_{\sigma}^{3}$ and $B\equiv
b/g_{\sigma}^{4}$ are reported in Table 1. The $\sigma $ mass is fixed at
$550MeV$.
The corresponding properties of nuclear matter are listed in Table 2.
Here the binding energy is defined $E/A=\epsilon/\rho-M$.

\par
\vspace{0.3cm}
\noindent

\begin{center}
{{\large \bf Table 1.}~Parameter set.}
\par
\vspace{0.5cm}
\noindent

\begin{tabular}{c|c|c} \hline
$Parameter~~Set$  &NL$\rho$  &NL$\rho\delta$ \\ \hline
$f_\sigma~(fm^2)$  &10.32924  &10.32924    \\ \hline
$f_\omega~(fm^2)$  &5.42341   &5.42341     \\ \hline
$f_\rho~(fm^2)$    &0.94999   &3.1500      \\ \hline
$f_\delta~(fm^2)$  &0.000     &2.500       \\ \hline
$A~(fm^{-1})$      &0.03302   &0.03302     \\ \hline
$B$                &-0.00483  &-0.00483    \\ \hline
\end{tabular}
\end{center}
\vspace{0.5cm}
\noindent

%\vspace{0.5cm}
%\noindent
\begin{center}

{{\large \bf Table 2.}~Saturation properties of nuclear matter.}

\par
\vspace{0.5cm}

\noindent
\begin{tabular}{ c c c } \hline
$\rho_{0}~(fm^{-3})$ &0.16   \\ \hline
$E/A ~(MeV)$         &-16.0  \\ \hline
$K~(MeV)$            &240.0   \\ \hline
$E_{sym}~(MeV)$      &31.3    \\ \hline
$M^{*}/M $           &0.75    \\ \hline
\end{tabular}
\end{center}

We finally note that these Lagrangians have been already used for flow
\cite{greco03}, pion production \cite{gaitanos04}, isospin tracer
\cite{gaitanosplb04} and kaon production \cite{ferini06} calculations 
for relativistic heavy ion collisions with
an overall good agreement to data.



\section{Nambu-Iona Lasinio model for asymmetric matter} 


From the above discussion it appears (extremely) important to include the 
Isospin degree of freedom in any effective QCD dynamics. 
A first approach can be supplied by a two-flavor Nambu-Jona Lasinio 
model where the isospin asymmetry can be included in a flavor-mixing picture 
\cite{frank03,buballa05}. 
%These isospin effects are induced by a
%determinant interaction related to the breaking of the axial symmetry.
The lagrangian is given by
\begin{equation}
{ L} = {L}_0 + {L}_1 + {L}_2 , 
\end{equation}
with ${ L}_0$ the free part 
$$ 
{ L}_0 = {\bar \psi} ( i\not \!\partial - m ) \psi, 
$$
and the two different interaction part given by
\begin{eqnarray}
&&{L}_1 = G_1 \Big\{ ({\bar \psi}
\psi)^2 + ({\bar \psi}\,\vec\tau \psi)^2 + ({\bar \psi}\,i\gamma_5 \psi)^2 + 
({\bar
\psi}\,i\gamma_5\vec\tau \psi)^2 \Big\} \nonumber \\&&
{L}_2 = 
%&=& G_2 \left\lbrace {\rm det}_f [\bar{\psi} (1+\gamma_5) \psi] 
%+ {\rm det}_f
%[\bar{\psi} (1-\gamma_5) \psi] \right\rbrace= \nonumber \\
 G_2 \Big\{ ({\bar \psi} \psi)^2 - ({\bar \psi}\,\vec\tau \psi)^2 - ({\bar
\psi}\,i\gamma_5 \psi)^2 + ({\bar \psi}\,i\gamma_5\vec\tau \psi)^2 \Big\}.
\end{eqnarray}
In the mean field approximation the new Gap Equations are 
$M_i=m_i-4G_1 \Phi_i-4G_2 \Phi_j$, $i \not= j,(u,d)$ , where the 
$\Phi_{u,d}=<\bar u u>,<\bar d d>$ are the two (negative) condensates 
which are given by
\begin{equation}
\Phi_f = - 2N_c \int \frac{d^3p}{(2 \pi)^3} \,
\frac{M_f}{E_{p,f}}\Big\{ 1 - f^-(T,\mu_f) 
- f^+(T,\mu_f) \Big\}.
\end{equation}
and $m_{u,d}=m$
the (equal) current masses. 

Introducing explicitily a flavor mixing, i.e.
the dependence of the constituent mass of a given flavor to both condensate,
via $G_1=(1-\beta) G_0, G_2= \beta G_0$ we have the coupled equations
\begin{eqnarray}
&&M_u=m - 4 G_0 \Phi_u + 4 \beta G_0 (\Phi_u - \Phi_d), \nonumber \\ 
&&M_d=m - 4G_0 \Phi_u + 4 (1-\beta) G_0 (\Phi_u - \Phi_d).
\label{mixing}
\end{eqnarray}
For $\beta=1/2$ we have back the usual NJL ($M_u=M_d$), while small/large
mixing is for $\beta \Rightarrow 0$/$\beta \Rightarrow 1$ respectively.
The value of $\beta$ has a consequence on the structure 
of the phase diagram in the region of low temperatures 
and high chemical potential. In fact as shown in \cite{frank03,buballa05} 
for $\beta = 0$ there are two distinct phase transitions 
for the up quarks and for the down quarks, but for this 
value the interaction is symmetric under $U_A(1)$ 
transformations and it is unrealistic. While for 
$\beta \geq 0.1$ the $U_A(1)$ symmetry becomes explicitly
broken and there is only a single first order phase transition.
Realistic estimations of $\beta$ fitting the physical $\eta$-meson  mass
give a value of $\beta \approx 0.11$  \cite{frank03,shao06}.


%\begin{figure}
%\centering
%\includegraphics[angle=-90,width=9.0cm]{restorudmass.ps} 
%\vskip 1.0cm
%\caption{$NJL$ results with flavor mixing $\alpha=0.2$ and isospin 
%asymmetry $(N-Z)/A=0.4$: costituent u- d- masses vs.baryon density.
%}
%\vskip -0.5cm
%\label{MuMd} 
%\end{figure} 


In neutron rich matter $\mid \Phi_d \mid$ decreases more rapidly due to the 
larger $\rho_d$ and so $(\Phi_u -\Phi_d)<0$. In the ``realistic'' small mixing 
case we will get a definite $M_u>M_d$ 
splitting at high baryon density (before the chiral restoration). 
This expectation is confirmed by a full calculation of the coupled gap 
equations 
with standard parameters  \cite{erice08,plum_tesi}. 
All that can indicate a more fundamental confirmation of the $m^*_p>m^*_n$ 
splitting in the hadron phase,
as suggested by the effective $QHD$ model with the isovector scalar $\delta$ 
coupling, see \cite{liubo02,baranPR}.

However such isospin mixing effect results in a very small variation
of the symmetry energy in the quark phase, still related only to the Fermi
kinetic contribution. 
 In fact this represents just a very first step towards a more complete
treatment of isovector contributions in effective quark models, of large 
interest for the discussion of the phase transition at high densities. 
%We remind that confinement is still missing in these mean field approaches. 





%\setcounter{section}{1}


%\newpage
%\section*{References}
\begin{thebibliography}{99}

\bibitem{muller}H.  M$\ddot{u}$ller,  {\it Nucl.\ Phys.}  {\bf A618} 
(1997) 349.

\bibitem{ditoro} M.Di Toro, A. Drago, T. Gaitanos, V. Greco, A. Lavagno,
      {\it Nucl.\ Phys.} {\bf A775} (2006) 102.

\bibitem{erice08} 
M.Di Toro et al. {\it Progr.\  Part.\  Nucl.\  Phys.} 
{\bf 62} (2009) 389-401.

\bibitem{fair}
P. Senger  et al., {J.\ Phys. \rm G \ Nucl.\ Phys.} {\bf 36} 
(2009) 064037.

\bibitem{nica}
See the Website $<http://nica.jinr.ru/>$.

\bibitem{SW85} B.D. Serot and J.D. Walecka, {\it Adv.\  Nucl.\ Phys.} 
{\bf 16} (1985) 1.

\bibitem{liubo02} B. Liu, V. Greco, V. Baran, M. Colonna, M. Di Toro,
      {\it Phys.\  Rev.}{\bf C65} (2002) 045201.

\bibitem{baranPR} V. Baran, M. Colonna, V. Greco, M. Di Toro,
            {\it Phys.\  Rep.} {\bf 410} (2005) 335-466.

\bibitem{MIT}  A. Chodos et al., {\it Phys.\  Rev.} {\bf D9} (1974) 3471.

\bibitem{burgio02}
G.F. Burgio, M. Baldo, P.K. Sahu, H.-J. Schulze, 
 {\it Phys.\  Rev.} {\bf C66} (2002) 025802.

\bibitem{nicotra06}
O.E. Nicotra, M. Baldo, G.F. Burgio, H.-J. Schulze,
{\it Phys.\  Rev.} {\bf D74} (2006) 123001.

\bibitem{page06}
D. Page, S. Reddy, 
{\it Ann.\  Rev.\  Nucl.\  Part.\  Sci.} {\bf 56} (2006) 327. 

\bibitem{daniel02}
P. Danielewicz, R. Lacey, W.G. Lynch, 
{\it Science} {\bf 298} (2002) 1592.

\bibitem{fuchs06}
C. Fuchs,
{\it Prog. \ Part.\ Nucl.\ Phys.} {\bf 56} (2006) 1-103.

\bibitem{bmuller95}
B. M$\ddot{u}$ller,
{\it Rep.\  Prog.\ Phys.} {\bf 58} (1995) 611.

\bibitem{Landaustat} 
 L.D. Landau and L. Lifshitz,
{\em Statistical Physics}
 Pergamon Press 1969, Oxford.

\bibitem{greco03}
V. Greco et al., {\it Phys.\ Lett.} {\bf B562} (2003) 215.

\bibitem{gaitanos04}
T. Gaitanos et al., {\it Nucl.\ Phys.} {\bf A732} (2004) 24.

\bibitem{ferini06}
G. Ferini, T. Gaitanos, M. Colonna, M. Di Toro. H.H. Wolter,
 {\it Phys.\ Rev.\ Lett.} {\bf 97}  (2006) 202301.

\bibitem{GlendenningPRL18} 
N.K. Glendenning, S.A. Moszkowski,  
{\it Phys.\ Rev.\ Lett.} {\bf 67} (1991) 2414.

\bibitem{vretenar03}
D. Vretenar, T. Niksic, P. Ring,
 {\it Phys.\ Rev.} {\bf C68} (2003) 024310, and refs. therein.


%\bibitem{couplings}
%The meson-nucleon couplings are considered not density dependent and
%chosen in order to reproduce good saturation properties and average
%Dirac-Brueckner-Hartree-Fock predictions at high density 
%\cite{hofmann01,goegelein08}, see details in refs.\cite{liubo02,baranPR}.


\bibitem{baran98}
V. Baran, M. Colonna, M. Di Toro, A. Larionov,
{\it Nucl. \ Phys.} {\bf A632} (1998) 287-303.

\bibitem{chomazPR}
P. Chomaz, M. Colonna, J. Randrup.
{\it Phys.\ Rep.} {\bf 389} (2004) 263-440.

\bibitem{Qasy}
 In principle there is nothing wrong with $\alpha^Q>1$ evaluations
(although we note that for pure "neutron" matter the quark phase 
asymmetry should be 1, since $\rho_d=2\rho_u$). Indeed for low $\chi$-values,
$\chi<0.1$, very small quark concentrations, we can get $\alpha^Q$ values 
slightly 
larger than 1. However we are cautious about these results since we can 
expect also some numerical problems. In fact for very small $\chi$ values
the weight of the $\alpha^Q$ contribution in the minimization procedure 
is expected not too relevant, as we can clearly see from the 
Eq.(\ref{alphamix}). 
In any case the important point is that this is not affecting the 
general discussion about the isospin distillation.


\bibitem{sissakian08}
A.N. Sissakian, A.S. Sorin, V.D. Toneev,
{\it Phys.\ Part.\ Nucl.} {\bf 39} (2008) 1062.

\bibitem{kogut04}
J.B. Kogut, D.K. Sinclair,
 {\it Phys.\ Rev.} {\bf D70} (2004) 095401.

\bibitem{toublan05}
D. Toublan, J.B. Kogut, 
 {\it Phys.\ Lett.\rm B} {\bf 605} (2005) 129.

\bibitem{NJL}
Y. Nambu and G. Jona-Lasinio, 
 {\it Phys.\ Rev.} {\bf 122} (1961) 345; {\bf 124} (1961) 246.

\bibitem{frank03}
M. Frank, M. Buballa, M. Oertel, 
 {\it Phys.\ Lett.} {\bf B562} (2003) 221.

\bibitem{buballa05}
M. Buballa, 
 {\it Phys.\ Rep.} {\bf 407} (2006) 205-376.

\bibitem{shao06}
Guo-yun Shao et al., 
 {\it Phys.\ Rev.} {\bf D73} (2006) 076003

\bibitem{plum_tesi}
S. Plumari, Ph.D.Thesis, 2009  Univ.Catania.

\bibitem{kaiser09}
N. Kaiser, W. Weise, 
{\it Phys.\ Lett.} {\bf B671} (2009) 25.

\bibitem{weise07}
W. Weise,
{\it Prog.\ Theor.\ Phys.\ Suppl.} {\bf 170} (2007) 161.

\bibitem{alford08}
M.G. Alford, A. Schmitt, K. Rajagopal, T. Schafer,
{\it Rev.\ Mod..\ Phys.} {\bf 80} (2008) 1455.

\bibitem{pagliara10}
G. Pagliara, J. Schaffner-Bielich,
{\it Phase transition from nuclear matter to color superconducting
 quark matter: the effects of isospin}, arXiv:1003.1017[nucl-th].

\bibitem{huang03}
Mei Huang, Pengfei Zhuang, Weiqin Chao,
 {\it Phys.\ Rev.} {\bf D67} (2003) 065015

\bibitem{horst}
H. St$\ddot{o}$ecker, 2008 private communication.

\bibitem{csernai99}
L. Csernai, D. Rohrich, 
{\it Phys.\ Lett.} {\bf B458} (1999) 454.

\bibitem{fries08}
R.J. Fries, V. Greco V, P. S$\ddot{o}$rensen, 
 {\it Ann.\ Rev.\ Nucl.\ Part.\ Sci.} {\bf 58} (2008) 177.

\bibitem{hofmann01}
F. Hofmann, C.M. Keil, H. Lenske, 
{\it Phys.\ Rev.} {\bf C64} (2001) 034314.

\bibitem{goegelein08}
P. Goegelein, E.N.E. van Dalen, C.  Fuchs, H. M$\ddot{u}$ther,  
{\it Phys.\ Rev.} {\bf C77} (2008) 025802.

\bibitem{grecoPRC03} 
V. Greco, M. Colonna, M. Di Toro, F. Matera,
 {\it Phys.\ Rev.} {\bf C67} (2003) 015203.

\bibitem{liubo05}
B. Liu, H. Guo, M. Di Toro, V. Greco, 
{\it Eur.\ Phys.\ J.\ \rm A} {\bf 25} (2005) 293.

\bibitem{gaitanosplb04}
T. Gaitanos, M. Colonna, M. Di Toro, H.H. Wolter,
 {\it Phys.\ Lett.} {\bf B595} (2004) 209
.
\end{thebibliography}
\end{document}













0909.3247.extracted_bib
\begin{thebibliography}{99}

\bibitem{muller}H.  M$\ddot{u}$ller,  {\it Nucl.\ Phys.}  {\bf A618} 
(1997) 349.

\bibitem{ditoro} M.Di Toro, A. Drago, T. Gaitanos, V. Greco, A. Lavagno,
      {\it Nucl.\ Phys.} {\bf A775} (2006) 102.

\bibitem{erice08} 
M.Di Toro et al. {\it Progr.\  Part.\  Nucl.\  Phys.} 
{\bf 62} (2009) 389-401.

\bibitem{fair}
P. Senger  et al., {J.\ Phys. \rm G \ Nucl.\ Phys.} {\bf 36} 
(2009) 064037.

\bibitem{nica}
See the Website $<http://nica.jinr.ru/>$.

\bibitem{SW85} B.D. Serot and J.D. Walecka, {\it Adv.\  Nucl.\ Phys.} 
{\bf 16} (1985) 1.

\bibitem{liubo02} B. Liu, V. Greco, V. Baran, M. Colonna, M. Di Toro,
      {\it Phys.\  Rev.}{\bf C65} (2002) 045201.

\bibitem{baranPR} V. Baran, M. Colonna, V. Greco, M. Di Toro,
            {\it Phys.\  Rep.} {\bf 410} (2005) 335-466.

\bibitem{MIT}  A. Chodos et al., {\it Phys.\  Rev.} {\bf D9} (1974) 3471.

\bibitem{burgio02}
G.F. Burgio, M. Baldo, P.K. Sahu, H.-J. Schulze, 
 {\it Phys.\  Rev.} {\bf C66} (2002) 025802.

\bibitem{nicotra06}
O.E. Nicotra, M. Baldo, G.F. Burgio, H.-J. Schulze,
{\it Phys.\  Rev.} {\bf D74} (2006) 123001.

\bibitem{page06}
D. Page, S. Reddy, 
{\it Ann.\  Rev.\  Nucl.\  Part.\  Sci.} {\bf 56} (2006) 327. 

\bibitem{daniel02}
P. Danielewicz, R. Lacey, W.G. Lynch, 
{\it Science} {\bf 298} (2002) 1592.

\bibitem{fuchs06}
C. Fuchs,
{\it Prog. \ Part.\ Nucl.\ Phys.} {\bf 56} (2006) 1-103.

\bibitem{bmuller95}
B. M$\ddot{u}$ller,
{\it Rep.\  Prog.\ Phys.} {\bf 58} (1995) 611.

\bibitem{Landaustat} 
 L.D. Landau and L. Lifshitz,
{\em Statistical Physics}
 Pergamon Press 1969, Oxford.

\bibitem{greco03}
V. Greco et al., {\it Phys.\ Lett.} {\bf B562} (2003) 215.

\bibitem{gaitanos04}
T. Gaitanos et al., {\it Nucl.\ Phys.} {\bf A732} (2004) 24.

\bibitem{ferini06}
G. Ferini, T. Gaitanos, M. Colonna, M. Di Toro. H.H. Wolter,
 {\it Phys.\ Rev.\ Lett.} {\bf 97}  (2006) 202301.

\bibitem{GlendenningPRL18} 
N.K. Glendenning, S.A. Moszkowski,  
{\it Phys.\ Rev.\ Lett.} {\bf 67} (1991) 2414.

\bibitem{vretenar03}
D. Vretenar, T. Niksic, P. Ring,
 {\it Phys.\ Rev.} {\bf C68} (2003) 024310, and refs. therein.


%\bibitem{couplings}
%The meson-nucleon couplings are considered not density dependent and
%chosen in order to reproduce good saturation properties and average
%Dirac-Brueckner-Hartree-Fock predictions at high density 
%\cite{hofmann01,goegelein08}, see details in refs.\cite{liubo02,baranPR}.


\bibitem{baran98}
V. Baran, M. Colonna, M. Di Toro, A. Larionov,
{\it Nucl. \ Phys.} {\bf A632} (1998) 287-303.

\bibitem{chomazPR}
P. Chomaz, M. Colonna, J. Randrup.
{\it Phys.\ Rep.} {\bf 389} (2004) 263-440.

\bibitem{Qasy}
 In principle there is nothing wrong with $\alpha^Q>1$ evaluations
(although we note that for pure "neutron" matter the quark phase 
asymmetry should be 1, since $\rho_d=2\rho_u$). Indeed for low $\chi$-values,
$\chi<0.1$, very small quark concentrations, we can get $\alpha^Q$ values 
slightly 
larger than 1. However we are cautious about these results since we can 
expect also some numerical problems. In fact for very small $\chi$ values
the weight of the $\alpha^Q$ contribution in the minimization procedure 
is expected not too relevant, as we can clearly see from the 
Eq.(\ref{alphamix}). 
In any case the important point is that this is not affecting the 
general discussion about the isospin distillation.


\bibitem{sissakian08}
A.N. Sissakian, A.S. Sorin, V.D. Toneev,
{\it Phys.\ Part.\ Nucl.} {\bf 39} (2008) 1062.

\bibitem{kogut04}
J.B. Kogut, D.K. Sinclair,
 {\it Phys.\ Rev.} {\bf D70} (2004) 095401.

\bibitem{toublan05}
D. Toublan, J.B. Kogut, 
 {\it Phys.\ Lett.\rm B} {\bf 605} (2005) 129.

\bibitem{NJL}
Y. Nambu and G. Jona-Lasinio, 
 {\it Phys.\ Rev.} {\bf 122} (1961) 345; {\bf 124} (1961) 246.

\bibitem{frank03}
M. Frank, M. Buballa, M. Oertel, 
 {\it Phys.\ Lett.} {\bf B562} (2003) 221.

\bibitem{buballa05}
M. Buballa, 
 {\it Phys.\ Rep.} {\bf 407} (2006) 205-376.

\bibitem{shao06}
Guo-yun Shao et al., 
 {\it Phys.\ Rev.} {\bf D73} (2006) 076003

\bibitem{plum_tesi}
S. Plumari, Ph.D.Thesis, 2009  Univ.Catania.

\bibitem{kaiser09}
N. Kaiser, W. Weise, 
{\it Phys.\ Lett.} {\bf B671} (2009) 25.

\bibitem{weise07}
W. Weise,
{\it Prog.\ Theor.\ Phys.\ Suppl.} {\bf 170} (2007) 161.

\bibitem{alford08}
M.G. Alford, A. Schmitt, K. Rajagopal, T. Schafer,
{\it Rev.\ Mod..\ Phys.} {\bf 80} (2008) 1455.

\bibitem{pagliara10}
G. Pagliara, J. Schaffner-Bielich,
{\it Phase transition from nuclear matter to color superconducting
 quark matter: the effects of isospin}, arXiv:1003.1017[nucl-th].

\bibitem{huang03}
Mei Huang, Pengfei Zhuang, Weiqin Chao,
 {\it Phys.\ Rev.} {\bf D67} (2003) 065015

\bibitem{horst}
H. St$\ddot{o}$ecker, 2008 private communication.

\bibitem{csernai99}
L. Csernai, D. Rohrich, 
{\it Phys.\ Lett.} {\bf B458} (1999) 454.

\bibitem{fries08}
R.J. Fries, V. Greco V, P. S$\ddot{o}$rensen, 
 {\it Ann.\ Rev.\ Nucl.\ Part.\ Sci.} {\bf 58} (2008) 177.

\bibitem{hofmann01}
F. Hofmann, C.M. Keil, H. Lenske, 
{\it Phys.\ Rev.} {\bf C64} (2001) 034314.

\bibitem{goegelein08}
P. Goegelein, E.N.E. van Dalen, C.  Fuchs, H. M$\ddot{u}$ther,  
{\it Phys.\ Rev.} {\bf C77} (2008) 025802.

\bibitem{grecoPRC03} 
V. Greco, M. Colonna, M. Di Toro, F. Matera,
 {\it Phys.\ Rev.} {\bf C67} (2003) 015203.

\bibitem{liubo05}
B. Liu, H. Guo, M. Di Toro, V. Greco, 
{\it Eur.\ Phys.\ J.\ \rm A} {\bf 25} (2005) 293.

\bibitem{gaitanosplb04}
T. Gaitanos, M. Colonna, M. Di Toro, H.H. Wolter,
 {\it Phys.\ Lett.} {\bf B595} (2004) 209
.
\end{thebibliography}