Speaker
Summary
Strongly interacting matter at non-zero temperature and chemical
potential is an exciting topic for physicists coming from different
areas, either theoretical or experimental. One of the main goals in
the heavy-ion physics program nowadays is to study the effects of
several macroscopic collective phenomena occurring under extreme
conditions. The discussion about the existence of a tricritical
point (TCP) or a critical end point (CEP) is also a topic of recent
interest. As is well known, a TCP separates the first order
transition at high chemical potential from the second order
transition at high temperatures. If the second order transition is
replaced by a smooth crossover, a CEP which separates the two lines
is found. The existence of the CEP in QCD was suggested at the end
of the eighties \cite{Asakawa:1989NPA}, and its properties in the
context of several models have been studied since then
\cite{Hatta:2003PRD,Schaefer:2006,Costa:2007PLB}.
The possible signatures of the CEP in heavy-ion collisions have been
studied in detail in \cite{Stephanov:1998PRL}. The most recent
lattice results with $N_f=2+1$ staggered quarks of physical masses
indicate the location of the CEP at
$T^{CEP}=162\pm2\mbox{MeV},\,\mu^{CEP}=360\pm40\mbox{MeV}$
\cite{Fodor:2004JHEP}, however its exact location is not yet known.
We perform our calculations in the framework of the three--flavor
NJL model \cite{Buballa:2004PR, Costa:2003PRC}, including the
determinantal 't Hooft interaction that breaks the $U_A(1)$
symmetry. We obtain the baryonic thermodynamic potential
$\Omega (\mu_i ,T)$
from which the relevant equations of state for the entropy $S$, the
pressure $P$ and the particle number $N_i$ can be calculated as
usually \cite{Costa:2003PRC}.
The baryon number susceptibility $\chi_B$ and the specific heat $C$ are
relevant observables concerning the order of the phase transition.
As several thermodynamic quantities diverge at the CEP, we will
focus on the values of a set of indices, the so-called critical
exponents, which describe the behavior near the critical point of
various quantities of interest (in our case $\epsilon$ and $\alpha$
are the critical exponents of $\chi_B$ and $C$, respectively). The
motivation for this study arises from fundamental phase transition
considerations, and thus transcend any particular system. We also
stress that, due to the lack of information from the lattice
simulations, the universality arguments should be confronted with
model calculations.
We verified that the phase diagram in the SU(3) NJL reproduces the
essential features of QCD: a first order phase transition for low
temperatures and the existence of the CEP occur when realistic
values of current quark masses are used. For $m_u=m_d=0$ and
$m_s>m_{s}^{crit}$ ($m_{s}^{crit}=18.3$ MeV) the transition is
second order ending in a first order line at the TCP. As $m_{s}$
increases we have a \textquotedblleft line\textquotedblright of
TCPs. For $m_u=m_d\neq0$ there is a crossover for all the values of
$m_s$ and the \textquotedblleft line\textquotedblright of TCPs
becomes a \textquotedblleft line\textquotedblright of CEPs. The
location of the CEP depends strongly of the strange quark mass.
Around the CEP we have studied the baryon number susceptibility and
the specific heat which are related with event-by-event fluctuations
of $\mu_B$ or $T$ in heavy-ion collisions. In the NJL model, for
$\chi_B$, we conclude that the obtained critical exponents are
consistent with the mean field values $\epsilon=\epsilon'=2/3$. From
our study of the critical exponent for the specific heat, we
conclude that $\alpha$ is different from $\epsilon$. This means
that the specific heat is sensitive to the way we approach the CEP.
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\begin{flushleft} \textbf{Acknowledgments}
\end{flushleft}
Work supported by grant SFRH/BPD/23252/2005 from F.C.T. (P. Costa),
Centro de F\'{\i}sica Te\'orica and FCT under project POCI
2010/FP/63945/2005.
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\end{document}
