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Nonlocal PNJL model with wavefunction
renormalization at finite temperature and
chemical potential
We study the chiral phase transition at finite
temperature and chemical potential considering a
non-local Polyakov chiral quark model with
wave-function renormalization (WFR) in the
propagator. In particular we analyze the role
played by the Polyakov loop and compare results
obtained with two different parametrizations and
without the WFR.
For the effective potential we consider (Ratti et
al in Ref.2)
At very low temperatures and densities the two
principal features in the QCD theory are chiral
symmetry breaking and confinement. But when the
temperature and/or chemical potential increase a
chiral symmetry restoration is expected the
interaction among color charges is screened
rather than confined because the short range
interactions are weak due to the asymptotic
freedom. To study the strongly interacting
quark-gluon matter at low energies and try to
interpret lattice QCD results it is necessary
resort to effective models due to the
non-perturbative behavior of the QCD. One of the
most popular is the Nambu-Jona Lasinio (NJL)
model 1, in which the quark fields interact via
local four point vertices, with the quark
condensate and pions as Goldstone bosons emerging
from a chiral invariant if such interaction is
strong enough. The problem with NJL model is that
the confinement is absent. In the context of
NJL-like models the Polyakov loop (PL) has been
considered recently 2 as order parameter to
describe the confinement-deconfinement phase
transition. We study here a non local extension
of the NJL, which includes wave function
renormalization (WFR) and the coupling to the PL,
at finite temperature and chemical potential. Two
parametrizations are used a) an exponential
form, b) a parametrization based on a fit to the
mass and renormalization functions obtained in
lattice calculations was considered.
Parametrization b) has been considered in a
previous work 3 to study the properties of pion
and sigma meson at T?0. Starting with the
Euclidean action for the nonlocal chiral model in
the case of two light flavors 3, and including
the coupling to the PL, we obtain that the mean
field thermodynamical potential ?MFA is
a(T) , b(T) fitted to lattice QCD results.
The presence of the PL implies the existence of
three coupled gap equations,
Depending on the value of temperature one is
expected to find two different types of
transitions i) a crossover at high values of T.
In this region the position of the chiral
transition is determined by the peak of the
chiral susceptibility defined by ii) a first
order transition at low values of T. As usual, in
this case the transition point is determined by
the value of (T,?) at which the two minima of the
thermodynamical potential have the same value.
RESULTS
To calculate the numerical results we use a set
of parameters that allow for a good description
of the pion and sigma vacuum properties 3.
?P
(1)
f (p) NONLOCAL g (p)
REGULATORS
where
and
In eq.(1), represent the MFA values of the
sigma fields, GS, ?P the corresponding coupling
constants (see Ref.3 for details), and is
the MFA Polyakov loop. The latter is, in general
defined as In order to keep ?MFA real valued,
we set ?80 in the present calculations. The
parametrizations of the non-local regulators we
consider are
Fig3 Phase diagram showing the chiral transition
curves in the T-? plane for parametrization a)
with (black line) and without (red line) WRF, in
comparison with parametrization b) (blue line).
Exponential parametrization
a)
Fig4 Phase diagram showing the chiral transition
curves in the T-? plane in the model with PL and
WRF, using the parametrization b) (blue line).
This figure includes the spinodal curves.
Lattice adjusted parametrization
b)
Fig2 Phase diagram showing the chiral transition
curves in the T-? plane for parametrization a)
with (black line) and without (red line) PL.
with
REFERENCES 1 Y. Nambu and G. Jona-Lasinio,
Phys. Rev. 122, 345 (1961) Phys. Rev. 124, 246
(1961). 2 P. N. Meisinger, M. C. Ogilvie, Phys.
Lett. B 379 (1996) 163 K. Fukushima, Phys. Lett.
B 591 (2004) 277 K. Fukushima, Phys. Lett. B
591(2004) 277 E. Megias, E. Ruiz Arriola, L. L.
Salcedo, Phys. Rev. D 74 (2006) 065005 C. Ratti,
M. A. Thaler, W. Weise, Phys. Rev. D 73 (2006)
014019 S. Roessner, C. Ratti, W. Weise, Phys.
Rev. D 75 (2007) 0340007. 3 S. Noguera, N. N.
Scoccola, Phys.Rev. D78 (2008) 114002.