# PNJL model at zero temperature: three-flavor case

O. A. Mattos1, T. Frederico1, C. H. Lenzi1, M. Dutra1, and O. Lourenço1
Departamento de Física, Instituto Tecnológico de Aeronáutica, DCTA, 12228-900,
São José dos Campos, SP, Brazil

###### Abstract

We propose a three-flavor version of the Polyakov-Nambu-Jona-Lasino (PNJL)
model at zero temperature regime, by implementing a traced Polyakov loop
($\Phi$) dependence in the scalar, vector and ’t Hooft channel strengths. We
study the thermodynamics of this model, named as PNJL0, with special attention
for the first order confinement/deconfinement phase transition for which
$\Phi$ is the order parameter. For the symmetric quark matter case, an
interesting feature observed is a strong reduction of the constituent strange
quark mass ($M_{s}$) at the chemical potential related to point where
deconfinement takes place. The emergence of $\Phi$ favors the restoration of
chiral symmetry even for the strange quark. We also investigate the charge
neutral system of quarks and leptons in weak equilibrium. As an application,
we construct a hadron-quark phase transition with a density dependent hadronic
model coupled to the SU(3) PNJL0 model. In this case, the quark side is
composed by deconfined particles. This approach is used to determine mass-
radius profiles compatible with recent data from the Neutron Star Interior
Composition Explorer (NICER) mission.

##  I Introduction

The understanding of the physics related to particles interacting strongly is
a fundamental issue studied for both, theoretical and experimental
researchers. The investigations of collisions at ultrarelativistic energies in
the most sophisticated accelerators of the world, such as the Relativistic
Heavy-Ion Collider (RHIC) rhic  and the Large Hadron Collider (LHC) lhc , for
instance, give the support needed for the study of the strong force probed at
the experimental level. The forthcoming Electron Ion Collider (EIC) eic  at
Brookhaven, the Facility for Antiproton and Ion Research (FAIR) fair , at
Darmstadt, and the Nuclotron-based Ion Collider fAcility (NICA) nica , at the
Joint Institute for Nuclear Research (JINR), in Dubna, are examples of other
facilities with the same purpose. Many aspects of the strongly interacting
matter can be directly or indirectly accessed from experiments, as for
example, the charge-parity violation (CPV) observed in heavy mesons decays.
The reader is addressed to Ref. bediaga  and references therein for an
overview including the role of the strong force in forming the observed CPV
pattern.

From the theoretical point of view, on the other hand, the physics of quarks
and gluons is described by Quantum Chromodynamics (QCD) qcd1 ; qcd2 ; qcd3 .
Its nonperturbative regime, i.e., QCD at large distances, or equivalently, low
energies, can be investigated through the lattice discretization of QCD in the
Euclidean space kogut1 ; kogut2 ; rothe  where the numerical calculations are
based on Monte Carlo simulations landau . However, the so called fermion sign
problem sign  restricts such technique when quarks at finite chemical
potential are included in the system. A second approach applied to study the
infrared region of QCD is the use of continuous methods relying on Dyson-
Schwinger equations (DSE) in Euclidean space dse , used to determine the
equations of motion of the $n$-point functions. Another practical method of
treating quarks and gluons, in systems in which such particles are the
fundamental degrees of freedom, is the use of phenomenological models.
Probably, the simplest one used for this aim is the famous MIT bag model mit .
In its original version, it establishes that quarks are inside a bag that
effectively corresponds to a confining potential. The constant $B$ is added to
the free quark pressure, and all thermodynamics is based on this simple
assumption.

Other widely used effective quark model is the Nambu-Jona-Lasinio (NJL) one
Nambu1 ; Nambu2 ; buballa ; vogl2 ; outros . It was proposed by Y. Nambu and
G. Jona-Lasinio before QCD and originally described nucleons and pions. In the
context of the quark interaction, it presents some similarities with QCD such
as the dynamical breaking of chiral symmetry, and the associated quark mass
generation due to the quark condensates. Similar nonperturbative phenomena
also occur in the Bardeen-Cooper-Schrieffer (BCS) model of superconductivity
bcs  (in which the NJL model was inspired), where interactions give rise to
energy gaps. In the NJL model, the “gap” in the quark mass is produced by the
point-like interactions between Dirac particles.

Although capable of describing some important features of QCD, the NJL model
fails to reproduce a particular phenomenon, namely, the infrared dynamics of
confinement as well as the deconfinement of quarks, related to the so called
“asymptotic freedom”, discovered by Gross, Wilczek and Politzer gross . In the
NJL model there is no information regarding the interaction among the gluons.
Such an issue was corrected by K. Fukushima FUKUSHIMA , who proposed the
inclusion of these mediators through a background field obtained from the
trace of a matrix in color space associated to the gluon gauge field $A_{4}$.
The so-called traced Polyakov loop, $\Phi$ (and its conjugate $\Phi^{*}$),
mimics the gluon dynamics in an effective way. Physically, $\Phi$ can be seen
as the order parameter of the confinement/deconfinement phase transition,
since it is related to $e^{-F/T}$, with $F$ being the free energy of an
external static quark. In the confined phase, the free energy of a single
quark is infinite and, consequently, $\Phi=0$. In the deconfined phase, on the
other hand, $F$ is finite and $\Phi\neq 0$. However, we remind that $\Phi$ is
not enough to have a true confinement in the PNJL model at finite temperature
since it only presents a statistical confinement while asymptotic quark states
still exists. The aforementioned analysis originally does not apply at zero
temperature in the PNJL model and in most of its parametrizations, since in
this case the Polyakov potential vanishes and the modified Fermi-Dirac
distributions reduces to the step functions. Because of that, confinement
effects are lost and it is not possible to investigate high density deconfined
quark systems at $T=0$. In order to circumvent this issue, it was proposed in
Refs. mattos ; epjc  a modification of the strengths of the couplings in the
NJL model by making them dependent on $\Phi$. As a direct consequence, the
Polyakov loop potential became nonvanishing even at $T=0$. In this way, the
new model constructed for flavor SU(2) quarks, named as PNJL0 model, was able
to exhibit the confinement/deconfinement phenomenology at $T=0$. In this work,
we extend these previous studies by generalizing the PNJL0 model to include
flavor SU(3) quarks. We investigate its thermodynamics at $T=0$ and also
perform an application in the context of the hadron-quark phase transition.

The paper is organized as follows. In Sec. II we review the main equations of
the traditional PNJL at finite temperature. In Sec. III, we generalize the
PNJL0 model at $T=0$ by taking into account the dynamics of the strange $s$
quark. The thermodynamics of the SU(3) model is studied for both cases,
namely, symmetric quark matter, and charge neutral system of quarks and
leptons in weak equilibrium. In the former, we verify a strong reduction of
the constituent strange quark mass ($M_{s}$) at the chemical potential where
the finite value of $\Phi$ emerges (deconfined phase). An application to
hybrid stars is also performed. The results indicate that it is possible to
generate hybrid stars compatible with the recent data from the Neutron Star
Interior Composition Explorer (NICER) mission. Finally, summary and concluding
remarks are presented in Sec. IV.

##  II Three-flavor Polyakov-Nambu-Jona-Lasinio model

We start by presenting the Lagrangian density of the three-flavor version of
the PNJL model, that includes a scalar and a vector 4-fermion interaction. It
reads

$\displaystyle\mathcal{L}_{\mbox{\tiny PNJL}}$
$\displaystyle=\bar{q}(i\gamma_{\mu}D^{\mu}-\hat{m})q-\mathcal{U}(\Phi,\Phi^{*},T)$
$\displaystyle+\frac{G_{s}}{2}\sum_{a=0}^{8}\left[(\bar{q}\lambda_{a}q)^{2}-(\bar{q}\gamma_{5}\lambda_{a}q)^{2}\right]$
$\displaystyle-\frac{G_{V}}{2}\sum_{a=0}^{8}\left[(\bar{q}\gamma_{\mu}\lambda_{a}q)^{2}+(\bar{q}\gamma_{\mu}\gamma_{5}\lambda_{a}q)^{2}\right]$
$\displaystyle+K[\mbox{det}_{f}(\bar{q}(1-\gamma_{5})q)+\mbox{det}_{f}(\bar{q}(1+\gamma_{5})q)],$
(1)

where $q$ is a vector of three spinors $q_{f}$ for $f=u,d,s$,
$\hat{m}=\mbox{diag}(m_{u},m_{d},m_{s})$ is a matrix of current quark masses
in flavor space, and $\lambda_{a}$ are the SU(3) Gell-Mann matrices. The last
term, a determinant in flavor space, corresponds to the so called ’t Hooft
interaction thooft  and is responsible for the large value of the
$\eta^{\prime}$ mass due to the anomaly $U_{A}(1)$ vogl . Such a kind of
interactions were firstly considered by Kobayashi and Maskawa km . The
differences between this model and the NJL one Nambu1 ; Nambu2 ; buballa ;
vogl2 ; outros ; ric1 ; ric2 ; ric3  are the replacement of $\partial^{\mu}$
by $D^{\mu}\equiv\partial^{\mu}+iA^{\mu}$, where
$A^{\mu}=\delta^{\mu}_{0}A_{0}$ and $A_{0}=gA_{a}^{0}\lambda_{a}/2$ ($g$ is
the gauge coupling), and the inclusion of the Polyakov potential
$\mathcal{U}(\Phi,\Phi^{*},T)$. It depends on the traced Polyakov loop and its
conjugate, $\Phi$ and $\Phi^{*}$, with $\Phi$ is defined in terms of
$A_{4}=iA_{0}\equiv T\phi$ as

$\displaystyle\Phi$
$\displaystyle=\frac{1}{3}\rm{Tr}\left[\,\,\rm{exp}\left(i\int_{0}^{1/T}d\tau\,A_{4}\right)\right]$
$\displaystyle=\frac{1}{3}\left[\rm{e}^{i(\phi_{3}+\phi_{8}/\sqrt{3})}+\rm{e}^{i(-\phi_{3}+\phi_{8}/\sqrt{3})}+\rm{e}^{-2i\phi_{8}/\sqrt{3}}\right],$
(2)

written in the Polyakov gauge where
$\phi=\phi_{3}\lambda_{3}+\phi_{8}\lambda_{8}$. In the mean-field
approximation, and using that $\Phi=\Phi^{*}$ epjc ; mattos ; weise4 ; weise6
; bratovic ; rossner2 ; Schramm ; Dexheimer ; Steinheimer , Eq. (1) becomes

$\displaystyle\mathcal{L}_{\mbox{\tiny PNJL}}^{\mbox{\tiny mfa}}$
$\displaystyle=\sum_{f}\bar{q}_{f}(i\gamma_{\mu}D^{\mu}-M_{f})q_{f}-G_{s}\sum_{f}{\rho_{sf}^{2}}$
$\displaystyle+G_{V}\sum_{f}\rho_{f}^{2}-4K\prod_{f}\rho_{sf}-\mathcal{U}(\Phi,T).$
(3)

The grand canonical potential density obtained from Eq. (3) reads

$\displaystyle\Omega_{\mbox{\tiny
PNJL}}=G_{s}\sum_{f}\rho_{sf}^{2}-G_{V}\sum_{f}\rho_{f}^{2}+4K\prod_{f}\rho_{sf}$
$\displaystyle-\frac{\gamma}{2\pi^{2}}\sum_{f}\int_{0}^{\Lambda}dk\,k^{2}(k^{2}+M_{f}^{2})^{1/2}$
$\displaystyle-\frac{\gamma}{6\pi^{2}}\sum_{f}\int_{0}^{\infty}\frac{dk\,k^{4}}{(k^{2}+M_{f}^{2})^{1/2}}[f(E_{f},T,\Phi)+\bar{f}(E_{f},T,\Phi)]$
$\displaystyle+\mathcal{U}(\Phi,T),$ (4)

with

$\displaystyle\rho_{sf}$ $\displaystyle=$
$\displaystyle\frac{\gamma}{2\pi^{2}}\int_{0}^{\infty}dk\,k^{2}\frac{M_{f}}{E_{f}}\left[f(E_{f},T,\Phi)+\bar{f}(E_{f},T,\Phi)\right]$
(5) $\displaystyle-$
$\displaystyle\frac{\gamma}{2\pi^{2}}\int_{0}^{\Lambda}dk\,k^{2}\frac{M_{f}}{E_{f}}$
$\displaystyle=$ $\displaystyle\left<\bar{q}_{f}q_{f}\right>,$

and

$\displaystyle\rho_{f}$ $\displaystyle=$
$\displaystyle-\frac{\partial\Omega_{\mbox{\tiny PNJL}}}{\partial\mu_{f}}$ (6)
$\displaystyle=$
$\displaystyle\frac{\gamma}{2\pi^{2}}\int_{0}^{\infty}dk\,k^{2}[f(E_{f},T,\Phi)-\bar{f}(E_{f},T,\Phi)]$
$\displaystyle=$ $\displaystyle\left<\bar{q}_{f}\gamma_{0}q_{f}\right>$

Here $\rho_{sf}$ is the quark “scalar” density (quark condensate) related to
flavor $f$, and $\rho_{f}$ is the “vector” quark density. Furthermore,
$E_{f}=(k^{2}+M_{f}^{2})^{1/2}$ and $\gamma=N_{s}\times N_{c}=6$ is the
degeneracy factor given in terms of spin ($N_{s}=2$) and color ($N_{c}=3$)
numbers. The constituent quark masses, $M_{f}$, are given in terms of the
quark condensates as

$\displaystyle M_{f}=m_{f}-2G_{s}\rho_{sf}-2K\prod_{f^{\prime}\neq
f}\rho_{sf^{\prime}}.$ (7)

As in the case of two-flavor PNJL model newratti ; FUKUSHIMA ; weise1 ; weise2
; rossner1 ; rossner2 , the functions $f(E_{f},T,\Phi)$ and
$\bar{f}(E_{f},T,\Phi)$, given by

$\displaystyle f(E_{f},T,\Phi)=$ $\displaystyle\frac{\Phi
e^{2(E_{f}-\tilde{\mu}_{f})/T}+2\Phi e^{(E_{f}-\tilde{\mu}_{f})/T}+1}{3\Phi
e^{2(E_{f}-\tilde{\mu}_{f})/T}+3\Phi
e^{(E_{f}-\tilde{\mu}_{f})/T}+e^{3(E_{f}-\tilde{\mu}_{f})/T}+1}$ (8)

and

$\displaystyle\bar{f}(E_{f},T,\Phi)=$ $\displaystyle\frac{\Phi
e^{2(E_{f}+\tilde{\mu}_{f})/T}+2\Phi e^{(E_{f}+\tilde{\mu}_{f})/T}+1}{3\Phi
e^{2(E_{f}+\tilde{\mu}_{f})/T}+3\Phi
e^{(E_{f}+\tilde{\mu}_{f})/T}+e^{3(E_{f}+\tilde{\mu}_{f})/T}+1},$ (9)

are the generalized Fermi-Dirac distributions for quarks and antiquarks with
$\tilde{\mu}_{f}=\mu_{f}-2G_{V}\rho_{f}$. It is worth to notice that the
vector interaction bratovic ; dynamical ; Vector-Int  produces a shift in the
chemical potentials ($\mu_{f}$), exactly as it occurs in the NJL model. Such
distributions are one of two basic differences between PNJL and NJL models.
The other one is the Polyakov potential $\mathcal{U}(\Phi,T)$, a quantity that
determines the thermodynamics of the gluonic pure gauge sector. It presents
some forms such as the following ones

$\displaystyle\frac{\mathcal{U}_{\mbox{\tiny RTW05}}}{T^{4}}$
$\displaystyle=-\frac{b_{2}(T)}{2}\Phi^{2}-\frac{b_{3}}{3}\Phi^{3}+\frac{b_{4}}{4}\Phi^{4},$
(10) $\displaystyle\frac{\mathcal{U}_{\mbox{\tiny RRW06}}}{T^{4}}$
$\displaystyle=-\frac{b_{2}(T)}{2}\Phi^{2}+b_{4}(T)\mbox{ln}(1-6\Phi^{2}+8\Phi^{3}-3\Phi^{4}),$
(11) $\displaystyle\frac{\mathcal{U}_{\mbox{\tiny FUKU08}}}{b\,T}$
$\displaystyle=-54e^{-a/T}\Phi^{2}-\mbox{ln}(1-6\Phi^{2}+8\Phi^{3}-3\Phi^{4}),$
(12) $\displaystyle\mathcal{U}_{\mbox{\tiny DS10}}$
$\displaystyle=(a_{0}T^{4}+a_{1}\mu^{4}+a_{2}T^{2}\mu^{2})\Phi^{2}+\mathcal{U}_{0}(\Phi),$
(13)

with

$\displaystyle b_{2}(T)$ $\displaystyle=$ $\displaystyle
a_{0}+a_{1}\left(\frac{T_{0}}{T}\right)+a_{2}\left(\frac{T_{0}}{T}\right)^{2}+a_{3}\left(\frac{T_{0}}{T}\right)^{3},$
(14) $\displaystyle b_{4}(T)$ $\displaystyle=$ $\displaystyle
b_{4}\left(\frac{T_{0}}{T}\right)^{3},$ (15)

and

$\displaystyle\mathcal{U}_{0}(\Phi)\equiv
a_{3}T_{0}^{4}\mbox{ln}(1-6\Phi^{2}+8\Phi^{3}-3\Phi^{4}),$ (16)

with $a$, $b$, $a_{0}$, $a_{1}$, $a_{2}$, $a_{3}$, $b_{3}$ and $b_{4}$ being
dimensionless free parameters. $T_{0}$ is defined as the transition
temperature for the pure gauge system. These potentials are named as RTW05
weise1 , RRW06 weise2 ; weise4 , FUKU08 fukushima3  and DS10 Schramm ;
Dexheimer . The traced Polyakov loop is found from
$\partial\Omega_{\mbox{\tiny PNJL}}/\partial\Phi=0$, and the current quark
masses $M_{u,d,s}$ are determined through solutions of Eqs. (7), in which
condensates ${\rho_{s}}_{u,d,s}$ given in Eqs. (5) are also used.

Finally, from Eq. (4) all the other thermodynamical quantities can be
determined, namely, pressure, entropy and energy densities. The expressions
are obtained from $P_{\mbox{\tiny PNJL}}=-\Omega_{\mbox{\tiny PNJL}}$,
$\mathcal{S}_{\mbox{\tiny PNJL}}=-\partial\Omega_{\mbox{\tiny PNJL}}/\partial
T$, and $\mathcal{E}_{\mbox{\tiny PNJL}}=T\mathcal{S}_{\mbox{\tiny
PNJL}}-P_{\mbox{\tiny PNJL}}+\sum_{f}\mu_{f}\rho_{f}$, respectively.

##  III PNJL0 model: three-flavor case

###  III.1 Formulation

At this point, we remind the reader that the the PNJL model equations,
previously presented, are reduced to those related to the NJL one at zero
temperature regime. The origin of this result is twofold. First, one has that
the generalized Fermi-Dirac distributions, given in Eqs. (8) and (9), become
step functions $\theta(k_{Ff}-k)$ at $T=0$ ($k_{Ff}$ is the Fermi momentum of
the quark $f$). Therefore, $\Phi$ disappears in all momentum integrals. The
second reason is due to the gluonic contribution enclosed by the Polyakov
potential. Notice that $\mathcal{U}(\Phi,0)$ vanishes for the most known
versions. In this case, Eq. (4) reads

$\displaystyle\Omega_{\mbox{\tiny
PNJL}}(T=0)=G_{s}\sum_{f}\rho_{sf}^{2}-G_{V}\sum_{f}\rho_{f}^{2}+4K\prod_{f}\rho_{sf}$
$\displaystyle-\frac{\gamma}{2\pi^{2}}\sum_{f}\int_{0}^{\Lambda}dk\,k^{2}(k^{2}+M_{f}^{2})^{1/2}$
$\displaystyle-\frac{\gamma}{6\pi^{2}}\sum_{f}\int_{0}^{k_{Ff}}\frac{dk\,k^{4}}{(k^{2}+M_{f}^{2})^{1/2}}=\Omega_{\mbox{\tiny
NJL}}(T=0).$ (17)

In order to circumvent this problem, we implemented, in Ref. epjc , the
introduction of $\Phi$ in the SU(2) NJL model at $T=0$ by requiring that the
strengths of the scalar and vector channels are vanishing in the deconfined
phase, i.e., at $\Phi=1$. In that approach we incorporate expected effects
from QCD that at low densities, and corresponding large inter-particle
distances, quarks should interact strongly while at short distance the
interaction should be weakened. The former is associated to the
nonperturbative infrared physics from QCD that enhances the interaction
between the effective degrees of freedom, related to the quarks in the model.
The latter should represent the ultraviolet physics of perturbative QCD where
the quarks interact weakly due to the asymptotic freedom phenomenon.
Therefore, this physics is incorporated in a dynamical way such that at low
densities quarks interact strongly and at large densities weakly towards to
the deconfinement phase transition. For this purpose, we use the traced
Polyakov loop as an effective scalar background field at the zero temperature
regime. Although this quantity is not strictly defined at $T=0$, we use it to
bring the complexity of QCD dynamics by incorporating, in a effective way, the
transition from strong (infrared) to weak (ultraviolet) regimes of the quark-
quark interaction. Here, we extended this phenomenology also to the three-
flavor version of the model. The modifications in the couplings for this case
are

$\displaystyle G_{s}\rightarrow\mathcal{G}_{s}(G_{s},\Phi)$ $\displaystyle=$
$\displaystyle G_{s}(1-\Phi^{2}),$ (18) $\displaystyle
G_{V}\rightarrow\mathcal{G}_{V}(G_{V},\Phi)$ $\displaystyle=$ $\displaystyle
G_{V}(1-\Phi^{2}),$ (19)

and

$\displaystyle K\rightarrow\mathcal{K}(K,\Phi)$ $\displaystyle=$
$\displaystyle K(1-\Phi^{2}).$ (20)

Such changes lead to

$\displaystyle\Omega_{\mbox{\tiny PNJL0}}$
$\displaystyle=G_{s}\sum_{f}\rho_{sf}^{2}-G_{V}\sum_{f}\rho_{f}^{2}+4K\prod_{f}\rho_{sf}$
$\displaystyle-\frac{\gamma}{2\pi^{2}}\sum_{f}\int_{0}^{\Lambda}dk\,k^{2}(k^{2}+M_{f}^{2})^{1/2}$
$\displaystyle-\frac{\gamma}{6\pi^{2}}\sum_{f}\int_{0}^{k_{Ff}}\frac{dk\,k^{4}}{(k^{2}+M_{f}^{2})^{1/2}}+\mathcal{U}(\rho_{f},\rho_{sf},\Phi)$
(21)

with

$\displaystyle\mathcal{U}(\rho_{f},\rho_{sf},\Phi)\equiv\mathcal{U}(\rho_{u},\rho_{d},\rho_{s},\rho_{su},\rho_{sd},\rho_{ss},\Phi)=$
$\displaystyle=G_{V}\Phi^{2}\sum_{f}\rho_{f}^{2}-G_{s}\Phi^{2}\sum_{f}\rho_{sf}^{2}-4K\Phi^{2}\prod_{f}\rho_{sf}$
$\displaystyle+\mathcal{U}_{0}(\Phi).$ (22)

By choosing to incorporate the terms $G_{s}\Phi^{2}$, $G_{V}\Phi^{2}$ and
$K\Phi^{2}$ of the grand canonical potential into $\mathcal{U}$ defined in Eq.
(22), it is possible to identify Eq. (21) as the zero temperature version of
Eq. (4). As in Ref. epjc , we refer to this construction as the PNJL0 model,
now written in its SU(3) version. Quark masses and chemical potentials are
given, respectively, by

$\displaystyle
M_{f}=m_{f}-2G_{s}(1-\Phi^{2})\rho_{sf}-2K(1-\Phi^{2})\prod_{f^{\prime}\neq
f}\rho_{sf^{\prime}},\qquad$ (23)

and

$\displaystyle\mu_{f}=(k_{Ff}^{2}+M_{f}^{2})^{1/2}+2G_{V}(1-\Phi^{2})\rho_{f}.$
(24)

where

$\displaystyle\rho_{f}=\frac{\gamma}{6\pi^{2}}k_{Ff}^{3}$ (25)

and

$\displaystyle\rho_{sf}=-\frac{\gamma
M_{f}}{2\pi^{2}}\int_{k_{Ff}}^{\Lambda}\frac{dk\,k^{2}}{(k^{2}+M_{f}^{2})^{1/2}}.$
(26)

Notice that in the three-flavor version of the PNJL0 model, the modification
of the couplings, Eqs. (18)-(20), also induces the definition of a new
Polyakov potential, Eq. (22), in which the backreaction of quarks in the
gluonic sector happens, as well as the inverse one, namely, gluons affecting
quarks. This last effect is intrinsic in the original PNJL model but the
former is absent. In the SU(3) PNJL0 model, the backreaction is complete,
i.e., each sector interacts each other, exactly as in its two-flavor version
epjc . We remark that the modifications pointed out in Eqs. (18)-(20) lead to
the Polyakov potential
$\mathcal{U}=G_{V}\Phi^{2}\sum_{f}\rho_{f}^{2}-G_{s}\Phi^{2}\sum_{f}\rho_{sf}^{2}-4K\Phi^{2}\prod_{f}\rho_{sf}$.
However, this potential is not able to generate nonvanishing solutions for
$\Phi$ when the condition $\partial\Omega_{\mbox{\tiny PNJL0}}/\partial\Phi=0$
is applied. Therefore, we add the term $\mathcal{U}_{0}$ given in Eq. (16) in
the definition of $\mathcal{U}$ and verify that it is responsible to ensure
$\Phi\neq 0$. Moreover, it is also important in order to restrict the traced
Polyakov loop in the range of $0\leqslant\Phi\leqslant 1$ Schramm ; Dexheimer
. Finally, the pressure and energy density of the SU(3) PNJL0 model, obtained
from Eq. (21), are written as

$\displaystyle P_{\mbox{\tiny
PNJL0}}=-G_{s}\sum_{f}\rho_{sf}^{2}+G_{V}\sum_{f}\rho_{f}^{2}-4K\prod_{f}\rho_{sf}$
$\displaystyle+\frac{\gamma}{2\pi^{2}}\sum_{f}\int_{0}^{\Lambda}dk\,k^{2}(k^{2}+M_{f}^{2})^{1/2}$
$\displaystyle+\frac{\gamma}{6\pi^{2}}\sum_{f}\int_{0}^{k_{Ff}}\frac{dk\,k^{4}}{(k^{2}+M_{f}^{2})^{1/2}}$
$\displaystyle-\mathcal{U}(\rho_{f},\rho_{sf},\Phi)+\Omega_{\mbox{\tiny vac}}$
(27)

and

$\displaystyle\mathcal{E}_{\mbox{\tiny
PNJL0}}=G_{s}\sum_{f}\rho_{sf}^{2}+G_{V}\sum_{f}\rho_{f}^{2}+4K\prod_{f}\rho_{sf}$
$\displaystyle-\frac{\gamma}{2\pi^{2}}\sum_{f}\int_{k_{Ff}}^{\Lambda}dk\,k^{2}(k^{2}+M_{f}^{2})^{1/2}-2G_{V}\Phi^{2}\sum_{f}\rho_{f}^{2}$
$\displaystyle+\mathcal{U}(\rho_{f},\rho_{sf},\Phi)-\Omega_{\mbox{\tiny
vac}},$ (28)

respectively. The constant $\Omega_{\mbox{\tiny vac}}=-P_{\mbox{\tiny vac}}$
is added to the pressure and energy density in order to ensure that in vacuum
($\rho_{u}=\rho_{d}=\rho_{s}=0$) one has $\Omega_{\mbox{\tiny
PNJL0}}(\rho_{f}=0)=-P_{\mbox{\tiny PNJL0}}(\rho_{f}=0)=0$ and
$\mathcal{E}_{\mbox{\tiny PNJL0}}(\rho_{f}=0)=0$.

###  III.2 Thermodynamics of the model (symmetric quark matter)

We remark that from now on, all results are obtained at zero temperature
regime.

Since the main equations of the model are defined, we are able to explore some
thermodynamical features of the SU(3) PNJL0 model. Firstly, we need to define
values for its free parameters, namely, $G_{s}$, $G_{V}$, $K$, $m_{u}$,
$m_{d}$, $m_{s}$ and $\Lambda$. Here we use the RKH parametrization buballa ;
rkh  in which $G_{s}=3.67/\Lambda^{2}$, $K=-12.36/\Lambda^{5}$,
$m_{u}=m_{d}=5.5$ MeV, $m_{s}=140.7$ MeV, and $\Lambda=602.3$ MeV. The last
quantity is the cutoff parameter. As in the NJL model, the PNJL0 one is
nonrenormalizable. Therefore, it is needed to adopt a certain regularization
scheme in order to avoid divergent contributions in the momentum integrals.
Here we use a three-momentum cutoff for this purpose buballa ; greiner ;
leupold . This parameters set produces the values of $f_{\pi}=92.4$ MeV,
$m_{\pi}=135$ MeV, $m_{K}=497.7$ MeV, $m_{\eta}=514.8$ MeV, and
$m_{\eta}^{\prime}=957.8$ MeV, respectively, for the pion decay constant and
mass, and for the masses of pseudoscalar mesons $K$, $\eta$, and
$\eta^{\prime}$ buballa ; rkh . For the vacuum, it also leads to
$M_{u}^{\mbox{\tiny(vac)}}=M_{d}^{\mbox{\tiny(vac)}}=367.6$ MeV,
$M_{s}^{\mbox{\tiny(vac)}}=549.5$ MeV,
$\langle\overline{u}u\rangle^{1/3}_{\mbox{\tiny(vac)}}=\langle\overline{d}d\rangle^{1/3}_{\mbox{\tiny(vac)}}=-241.9$
MeV, and $\langle\overline{s}s\rangle^{1/3}_{\mbox{\tiny(vac)}}=-257.7$ MeV.

With regard to the strength of the vector channel, we remind that if a Fierz
transformation of the color-current interaction is performed, the value of
$G_{V}=0.5G_{s}$ is found buballa ; vogl2 . In principle, this “canonical”
value should be used in all calculations involving NJL/PNJL models with the
vector channel included. However, $G_{V}$ is often taken as a free parameter.
Some authors use this freedom to fix the vector meson masses, such as the
$\rho$ meson lutz . Other ones also point out to changes in the magnitude of
$G_{V}$ due to medium effects. For instance, instanton-anti-instanton pairing
effects at high temperature can change its magnitude shuryak . Furthermore, it
is also known that $G_{V}$ plays a significant hole in the phase diagram of
the strongly interacting matter since it shifts the location of the critical
end point (CEP) bratovic ; kashiwa . The increasing of $G_{V}$ induces the
disappearance of the CEP and, consequently, the vanishing of the first order
phase transition line. Therefore, as in Ref. epjc , we adopt here the approach
of setting $G_{V}$ as a free parameter in order to verify its influence in our
calculations.

Finally, the gluonic sector of the model present two more constants, namely,
$T_{0}$ and $a_{3}$, appearing in the last term of the Polyakov potential, Eq.
(22). The former is fixed to $190$ MeV epjc ; ratti1 . Concerning the constant
$a_{3}$, we remind that it appears in the quantity denoted by
$\mathcal{U}_{0}$, Eq. (16), firstly used in Ref. Schramm . In those works,
the authors remark that the free parameters presented in the Polyakov
Potential $\mathcal{U}$, that contains $\mathcal{U}_{0}$, are determined in
order to reproduce lattice data, as well as known information about the phase
diagram. In this case, the value obtained by them is $a_{3}=-0.4$. However, it
is worth to notice that their model is completely different from the PNJL0
model considered here. If we take $a_{3}=-0.4$, no solutions of $\Phi\neq 0$
are found, as we will show in the first figure later on. Therefore, one has
$G_{V}$ and $a_{3}$ as free parameters of the SU(3) PNJL0 model, as well as
they are in its two-flavor version epjc .

At zero temperature, one can also use the thermodynamical relation given by

$\displaystyle E+PV$ $\displaystyle=\mu_{B}N_{B}+\mu_{S}N_{S}+\mu_{I}N_{I}$
$\displaystyle=\mu_{u}N_{u}+\mu_{d}N_{d}+\mu_{s}N_{s},$ (29)

in order to write the quark chemical potentials as

$\displaystyle\mu_{u}=\frac{\mu_{B}}{3}+\frac{\mu_{I}}{2},\,\,\,\mu_{d}=\frac{\mu_{B}}{3}-\frac{\mu_{I}}{2},\,\,\,\mu_{s}=\frac{\mu_{B}}{3}-\mu_{S},$
(30)

i.e., in terms of the baryonic, isospin, and strangeness chemical potentials,
namely, $\mu_{B}$, $\mu_{I}$, and $\mu_{S}$, respectively. Firstly, we work at
the so called symmetric quark matter, the one in which all quark potentials
are the same, namely,

$\displaystyle\mu_{u}=\mu_{d}=\mu_{s}\equiv\mu,$ (31)

situation found by taking $\mu_{I}=\mu_{S}=0$ in Eq. (30). At this point, we
remark to the reader that the concept of symmetric matter in this context is
different from the one used in nuclear matter described by hadronic models
had1 , in which nucleons are in equal number and present identical chemical
potentials, since they have the same rest masses and densities. In the case of
the three-flavor quark system studied here, $m_{u}=m_{d}$ but $m_{s}$ is quite
different. Therefore, densities of the light quarks are equal each other but
different from the strange quark one, despite the validity of Eq. (31).

As a first investigation of the symmetric quark matter described by the SU(3)
PNJL0 model, we show in Fig. 1 $\Omega_{\mbox{\tiny PNJL0}}$ as a function of
the traced Polyakov loop for $G_{V}=0.25G_{s}$ and for the canonical relation
$G_{V}=0.5G_{s}$ found through Fierz transformation of the current-current
interaction buballa ; vogl2 .

Figure 1:  $\Omega_{\mbox{\tiny PNJL0}}$ as a function of $\Phi$ for
$G_{V}=0.25G_{s}$ (a,b,c,d) and $G_{V}=0.5G_{s}$ (e,f,g,h) for different
$a_{3}$. The common quark chemical potential is given by (a and b)
$\mu=M_{u}^{\mbox{\tiny(vac)}}=M_{d}^{\mbox{\tiny(vac)}}=367.6$ MeV, (b and f)
$\mu=387.6$ MeV, (c and g) $\mu=582.3$ MeV, and (d and h) $\mu=\Lambda=602.3$
MeV.

In order to construct these curves, we use Eq. (21) with the constituent quark
masses found by the transcendental equations given by Eqs. (23) and (26).
$\Phi$ is free to run in this case. Our analysis points out that in the range
of chemical potentials from $\mu=M_{u,d}^{\mbox{\tiny(vac)}}$ to
$\mu=\Lambda$, it is possible to find curves in which $\Omega_{\mbox{\tiny
PNJL0}}$ present global minima for nonvanishing values of the traced Polyakov
loop, see Figs. 1c and 1d, for instance. For the curve presenting $a_{3}=-0.2$
and $G_{V}=0.25G_{s}$, we clearly see a single minimum of $\Omega_{\mbox{\tiny
PNJL0}}$ at $\Phi=0$ for $\mu=M_{u,d}^{\mbox{\tiny(vac)}}$ (panel a). However,
$\Phi\sim 0.82$ produces a minimum of $\Omega_{\mbox{\tiny PNJL0}}$ for
$\mu=\Lambda$ (panel d). As in the SU(2) case epjc , we verify a
confinement/deconfinement phase transition in the model, i.e, the existence of
null and nonvanishing values for the order parameter $\Phi$ as a function of
the chemical potential. One can observe in panels e, f, g and h that a similar
behavior is verified for the curves in which $G_{V}=0.5G_{s}$. Another feature
observed here, and also present in the SU(2) version of the model epjc , is
the need of $G_{V}\neq 0$ values in order to become possible the existence of
a global minimum of $\Omega_{\mbox{\tiny PNJL0}}$ at $\Phi\neq 0$. For
$G_{V}=0$, the only minimum is found at $\Phi=0$ for different $a_{3}$ values
(figure not shown). The vector interaction still plays an important role even
in the SU(3) version of the PNJL0 model.

The analysis of the $\Omega_{\mbox{\tiny PNJL0}}\times\Phi$ curves is also
useful in order to determine the chemical potential in which the
confinement/deconfinement phase transition takes place, named here as
$\mu_{\mbox{\tiny conf}}$. For the model with $G_{V}=0.25G_{s}$ and
$a_{3}=-0.1$, for example, we see from Fig. 2 that $\mu_{\mbox{\tiny
conf}}=533.15$ MeV.

Figure 2: $\Omega_{\mbox{\tiny PNJL0}}$ as a function of $\Phi$ for
$G_{V}=0.25G_{s}$ and $a_{3}=-0.1$ for three different values of $\mu$.

This particular chemical potential value is defined as the one that generates
a grand canonical potential presenting two minima with the same value for
$\Omega_{\mbox{\tiny PNJL0}}$, i. e., $\Omega_{\mbox{\tiny
PNJL0}}(\Phi_{1})=\Omega_{\mbox{\tiny PNJL0}}(\Phi_{2})$. In that case,
$\Phi_{1}$ and $\Phi_{2}$ delimit the boundaries of the thermodynamical phases
associated to quark confinement and deconfinement. The traced Polyakov loop is
the order parameter of this transition. The procedure described above can be
used to construct the $\Phi\times\mu$ curve depicted in Fig. 3, in this case
for $G_{V}=0.25G_{s}$ and $G_{V}=0.5G_{s}$.

Figure 3: Traced Polyakov loop as a function of the common quark chemical
potential for $G_{V}=0.25G_{s}$ and $G_{V}=0.5G_{s}$, both for $a_{3}=-0.1$.
These parametrizations lead to $\mu_{\mbox{\tiny conf}}=533.15$ MeV and
$\mu_{\mbox{\tiny conf}}=517.29$ MeV, respectively.

From this figure, we notice the quark confinement region determined by the
condition given by $\mu<\mu_{\mbox{\tiny conf}}$. On the other hand,
deconfinement is achieved through a first order phase transition, in the
region where $\mu>\mu_{\mbox{\tiny conf}}$, according to the thermodynamics
presented by the model. This feature was also observed in Ref. Schramm  in
which the authors included the traced Polyakov loop in a hadronic SU(3)
nonlinear $\sigma$ model along with quark degrees of freedom. Furthermore, in
Ref. pnjl0outro  another version of a three-flavor PNJL model at zero
temperature was studied. Specifically, a dependence on the quark density was
introduced in the $b_{2}(T)$ function of the Polyakov potential given in Eq.
(11). In that model, a first order phase transition is also verified in the
$\Phi(\mu)$ curve. Another effect verified in Fig. 3 is the decreasing of
$\mu_{\mbox{\tiny conf}}$ as $G_{V}$ increases, exactly as in the SU(2) case
epjc .

Now we evaluate three-flavor PNJL0 model equations by solving Eqs. (23) and
(26) together with condition $\partial\Omega_{\mbox{\tiny
PNJL0}}/\partial\Phi=0$. In the SU(3) case, this procedure consists in
searching for solutions of $M_{u}=M_{d}$, $M_{s}$ and $\Phi$, obtained from
the set of 3 coupled equations aforementioned. By following this method, we
are able to evaluate Eq. (21), with result displayed in Fig. 4.

Figure 4: $\Omega_{\mbox{\tiny PNJL0}}$ as a function of the common quark
chemical potential for $G_{V}=0.25G_{s}$ and $G_{V}=0.5G_{s}$, both for
$a_{3}=-0.1$.

One can notice a typical signal of a system presenting first order phase
transition callen ; yazaki ; theis , since Fig. 4 shows that
$\Omega_{\mbox{\tiny PNJL0}}$ (thermodynamical potential that describes the
system) is multi-valued. A requirement from thermodynamical stability callen
imposes that branches associated to unstable and metastable solutions is
removed from the final curve. The “crossing points” indicate the chemical
potential values related to the first order phase transitions. In the case of
the PNJL0 model, there are two of them, one from confinement/deconfinement and
another one from restored/broken chiral symmetry of the light quarks, that
give rise to $\mu_{\mbox{\tiny conf}}=533.15$ MeV and $\mu_{\mbox{\tiny
chiral}}=372.10$ MeV, respectively, for the parametrization of the model in
which $G_{V}=0.25G_{s}$ and $a_{3}=-0.1$. Notice that at $\mu=\mu_{\mbox{\tiny
conf}}$, we find the same value for $\Omega_{\mbox{\tiny PNJL0}}$ related to
the two global minima of Fig. 2, namely, $\Omega_{\mbox{\tiny
PNJL0}}(\mu_{\mbox{\tiny conf}})\sim-331$ MeV/fm3. This feature confirms the
equivalence of the two methods used to define the first order phase transition
points in such systems. It is also worth to notice the decrease of
$\mu_{\mbox{\tiny conf}}$ as $G_{V}$ increases, exactly as pointed out in Fig.
3.

In the region where $\mu_{\mbox{\tiny chiral}}<\mu<\mu_{\mbox{\tiny conf}}$ we
can recognize, as in the SU(2) version of the model epjc , a particular
thermodynamical phase in which the light quarks present a very low mass in
comparison to their vacuum values (almost fully restored chiral symmetry), and
with $\Phi=0$ (confined quarks), i.e, the so called quarkyonic phase, pointed
out in Refs. fukushima3 ; abuki ; mcnpa09 ; buisseret ; mcnpa07 ; mcnpa08 ,
for example. The quarkyonic phase related to the light quark sector can also
be identified by looking at the constituent quark masses as a function of
$\mu$, as displayed in Fig. 5.

Figure 5: Full and dashed lines: constituent quark masses as a function of the
common quark chemical potential for $G_{V}=0.25G_{s}$ and $G_{V}=0.5G_{s}$,
both for $a_{3}=-0.1$ (PNJL0 model). Dotted lines: SU(3) NJL model.

One can notice two sharp phase transitions occurring at $\mu_{\mbox{\tiny
chiral}}=372.10$ MeV and $\mu_{\mbox{\tiny conf}}=533.15$ MeV, for
$G_{V}=0.25G_{s}$, with the difference defining the “size” of the phase, in
this case, $\Delta\mu=\mu_{\mbox{\tiny conf}}-\mu_{\mbox{\tiny
chiral}}=161.05$ MeV. As we can also see, this difference is decreased for
$G_{V}=0.5G_{s}$. More specifically, for the SU(3) PNJL0 model $\Delta\mu$
depends on both, the strength of the vector channel, $G_{V}$, and the $a_{3}$
parameter related to the Polyakov potential. In Tables LABEL:taba3 and
LABEL:tabgv we present some values of $\Delta\mu$ by running $a_{3}$ and
$G_{V}$ independently.

Table 1: Parameter $a_{3}$ of the SU(3) PNJL0 model along with $\mu_{\mbox{\tiny conf}}$ and $\Delta\mu$. For each line one has $G_{V}=0.25G_{s}$ and $\mu_{\mbox{\tiny chiral}}=372.10$ MeV. $a_{3}$ |  $\mu_{\mbox{\tiny conf}}$ (MeV) |  $\Delta\mu$ (MeV)
---|---|---
$-0.27$ | $602.30$ | $230.20$
$-0.20$ | $576.74$ | $204.64$
$-0.15$ | $555.87$ | $183.77$
$-0.10$ | $533.15$ | $161.05$
$-0.08$ | $523.40$ | $151.30$
$-0.06$ | $513.14$ | $141.04$
$-0.05$ | $507.75$ | $135.65$
Table 2: Parameter $G_{V}$ of the SU(3) PNJL0 model along with $\mu_{\mbox{\tiny conf}}$, $\mu_{\mbox{\tiny chiral}}$ and $\Delta\mu$. For each line we fixed $a_{3}=-0.1$. $G_{V}/G_{s}$ |  $\mu_{\mbox{\tiny conf}}$ (MeV) |  $\mu_{\mbox{\tiny chiral}}$ (MeV) |  $\Delta\mu$ (MeV)
---|---|---|---
$0.50$ | $517.29$ | $382.18$ | $135.11$
$0.45$ | $519.38$ | $380.21$ | $139.17$
$0.40$ | $521.82$ | $378.22$ | $143.60$
$0.35$ | $524.75$ | $376.21$ | $148.54$
$0.30$ | $528.41$ | $374.18$ | $154.23$
$0.25$ | $533.15$ | $372.10$ | $161.05$
$0.20$ | $539.70$ | $370.00$ | $169.70$

From Fig. 5, we can also notice a strong reduction of the constituent strange
quark mass at $\mu=\mu_{\mbox{\tiny conf}}$. At this point, $M_{s}$ undergoes
a reduction of almost $50\%$ for $G_{V}=0.25G_{s}$, while for $G_{V}=0.5G_{s}$
the reduction is somewhat larger. The origin of this feature is the emergence
of $\Phi$ exactly at this chemical potential, see Fig. 3. The decreasing of
the couplings $\mathcal{G}_{s}(\Phi)$, $\mathcal{G}_{V}(\Phi)$ and
$\mathcal{K}(\Phi)$ when $\Phi$ becomes nonvanishing induces a reduction in
the constituent quark masses, including the strange quark one. We reinforce
that such an effect is not present in the original SU(3) NJL model, also show
in Fig. 5 (in that case changes in $M_{s}$ are only gradual, see the dotted
line). For the light sector, on the other hand, it is also verified a
decreasing of $M_{u,d}$ in the PNJL0 model, however, with much smaller
intensity, since at $\mu=\mu_{\mbox{\tiny conf}}$ the light quark masses are
nearly vanishing already.

Figure 6: The same as in Fig. 5, but for $a_{3}=-0.27$.

As already pointed out, the effect of the traced Polyakov loop depends on
$G_{V}$ and $a_{3}$. In Fig. 6 for $a_{3}=-0.27$ and $G_{V}=0.25G_{s}$, we
verify that $\mu_{\mbox{\tiny conf}}$ is equal to the cutoff parameter
$\Lambda$ resulting a smaller reduction of $M_{s}$ with respect to the NJL
model, differently of what is seen in Fig. 5 in the previous case. For the
sake of comparison, we also show the results concerning the parametrization
presenting $G_{V}=0.5G_{s}$ (value obtained from the Fierz transformation of
the current-current interaction). Notice that the decrease of the reduction in
$M_{s}$ is also observed for this case.

###  III.3 Charge neutral system of quarks and leptons in weak equilibrium
within hybrid stars

In this section we proceed to show how the confinement effects described by
the traced Polyakov loop affect the charge neutral system of quarks and
leptons (muons and massless electrons) in weak equilibrium. This study is
relevant in the context of quark stars or hybrid stars leupold ; stars1 ;
stars2 ; stars3 ; stars4 ; stars5 ; stars6 ; stars7 , constructed from a
particular quark model at zero temperature and at high density regime. This is
a system fulfilling the following conditions for the quarks and leptons
chemical potentials: $\mu_{s}=\mu_{d}=\mu_{u}+\mu_{e}$, and
$\mu_{e}=\mu_{\mu}$. It is possible to write $\mu_{u}$, $\mu_{d}$ and
$\mu_{s}$ in terms of a common quark chemical potential ($\mu$), and in terms
of the electron one ($\mu_{e}$) as

$\displaystyle\mu_{u}=\mu-\frac{2}{3}\mu_{e},\quad\mbox{and}\quad\mu_{d}=\mu_{s}=\mu+\frac{1}{3}\mu_{e},$
(32)

in which the relation $\mu_{B}=3\mu$ holds. Furthermore, the requirement of
charge neutral quark matter leads to the additional relation to be satisfied

$\displaystyle\frac{2}{3}\rho_{u}-\frac{1}{3}\rho_{d}-\frac{1}{3}\rho_{s}=\rho_{e}+\rho_{\mu},$
(33)

in which $\rho_{e}$ is the electron density. $\mu_{e}$ and $\rho_{e}$ are
related to each other through $\rho_{e}=\mu_{e}^{3}/(3\pi^{2})$. The muon
density is $\rho_{\mu}=[(\mu_{\mu}^{2}-m_{\mu}^{2})^{3/2}]/(3\pi^{2})$, and
$m_{\mu}=105.7$ MeV. In this case, the total energy density and total pressure
are given, respectively, by

$\displaystyle\mathcal{E}_{tq}=\mathcal{E}_{\mbox{\tiny
PNJL0}}+\frac{\mu_{e}^{4}}{4\pi^{2}}+\frac{1}{\pi^{2}}\int_{0}^{\sqrt{\mu_{\mu}^{2}-m^{2}_{\mu}}}dk\,k^{2}(k^{2}+m_{\mu}^{2})^{1/2},\qquad$
(34)

and

$\displaystyle P_{tq}$ $\displaystyle=P_{\mbox{\tiny
PNJL0}}+\frac{\mu_{e}^{4}}{12\pi^{2}}+\frac{1}{3\pi^{2}}\int_{0}^{\sqrt{\mu_{\mu}^{2}-m^{2}_{\mu}}}\frac{dk\,k^{4}}{(k^{2}+m_{\mu}^{2})^{1/2}}.$
(35)

We start by showing in Fig. 7 the $\mu_{B}$ dependence of the quark fractions
defined as $Y_{i}=\rho_{i}/(\rho_{u}+\rho_{d}+\rho_{s})$, for the SU(3) PNJL0
model with $G_{V}=0.25G_{s}$ and different values of $a_{3}$.

Figure 7: Quark fractions as a function of $\mu_{B}$ for the charge neutral
system of quarks and leptons in weak equilibrium. PNJL0 parametrization for
$G_{V}=0.25G_{s}$ and different values of $a_{3}$.

We clearly verify that $d$ quark fraction gradually reduces to a value close
to the $u$ quark fraction. Furthermore, at the chemical potential related to
the deconfinement transition, point in which $\Phi$ becomes nonvanishing, one
observes that this reduction is anticipated by a sharp variation. For the $s$
quark fraction, the opposite is found, namely, $Y_{s}$ approaches to $Y_{u}$
with sharp variations produced by the emergence of $\Phi$. In the case of
$Y_{u}$, we find no significant variations along the $\mu_{B}$ dependence in
comparison with both previous cases. From the figure, it is also clear that
the critical baryonic chemical potential moves to the direction of increasing
$\mu_{B}$ as $|a_{3}|$ increases, feature also present in symmetric quark
matter, as shown in Table LABEL:taba3.

As mentioned before, an important application of a quark model available at
zero temperature is the description of hybrid stars, in which a hadron-quark
phase transition takes place at high density regime. A recent study based on a
model-independent analysis, performed in Ref. nat , suggests evidences for
massive stars composed by cores of quark matter. Here we take the PNJL0 model
to take into account the quark side of the equations of state used to describe
stellar matter. For the hadronic side, we use the relativistic mean-field
model DDH$\delta$ given by the following Lagrangian density

$\displaystyle\mathcal{L}_{\mbox{\tiny
HAD}}=\overline{\psi}(i\gamma^{\mu}\partial_{\mu}-M)\psi+\Gamma_{\sigma}(\rho)\sigma\overline{\psi}\psi-\Gamma_{\omega}(\rho)\overline{\psi}\gamma^{\mu}\omega_{\mu}\psi$
$\displaystyle-\frac{\Gamma_{\rho}(\rho)}{2}\overline{\psi}\gamma^{\mu}\vec{\rho}_{\mu}\vec{\tau}\psi+\Gamma_{\delta}(\rho)\overline{\psi}\vec{\delta}\vec{\tau}\psi+\frac{1}{2}(\partial^{\mu}\sigma\partial_{\mu}\sigma-m^{2}_{\sigma}\sigma^{2})$
$\displaystyle-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}+\frac{1}{2}m^{2}_{\omega}\omega_{\mu}\omega^{\mu}-\frac{1}{4}\vec{B}^{\mu\nu}\vec{B}_{\mu\nu}+\frac{1}{2}m^{2}_{\rho}\vec{\rho}_{\mu}\vec{\rho}^{\mu}$
$\displaystyle+\frac{1}{2}(\partial^{\mu}\vec{\delta}\partial_{\mu}\vec{\delta}-m^{2}_{\delta}\vec{\delta}^{2}),$
(36)

where

$\displaystyle\Gamma_{i}(\rho)=\Gamma_{i}(\rho_{0})a_{i}\frac{1+b_{i}(\rho/\rho_{0}+d_{i})^{2}}{1+c_{i}(\rho/\rho_{0}+d_{i})^{2}},$
(37)

for $i=\sigma,\omega$, and

$\displaystyle\Gamma_{i}(\rho)=\Gamma_{i}(\rho_{0})[a_{i}e^{-b_{i}(\rho/\rho_{0}-1)}-c_{i}(\rho/\rho_{0}-d_{i})].$
(38)

for $i=\rho,\delta$. In Eq. (36) $\psi$ is the nucleon field whereas $\sigma$,
$\omega^{\mu}$, $\vec{\rho}_{\mu}$, and $\vec{\delta}$ are the scalar, vector,
isovector-vector, and isovector-scalar fields, that represents mesons
$\sigma$, $\omega$, $\rho$, and $\delta$, respectively. The antisymmetric
tensors $F_{\mu\nu}$ and $\vec{B}_{\mu\nu}$ are written as
$F_{\mu\nu}=\partial_{\nu}\omega_{\mu}-\partial_{\mu}\omega_{\nu}$ and
$\vec{B}_{\mu\nu}=\partial_{\nu}\vec{\rho}_{\mu}-\partial_{\mu}\vec{\rho}_{\nu}$.
The nucleon rest mass is $M_{\mbox{\tiny nuc}}$, and the mesons masses are
$m_{\sigma}$, $m_{\omega}$, $m_{\rho}$, and $m_{\delta}$. A more simplified
version of this model was firstly proposed by J. D. Walecka in Ref. walecka
in which the couplings between mesons and nucleons are constants. Here we
choose to use a more sophisticated structure for the hadronic model in which
such couplings are density dependent, namely, $\Gamma_{i}(\rho)$, with $\rho$
being the sum of the densities of proton ($\rho_{p}$) and neutrons
($\rho_{n}$). In Eqs. (37) and (38) $\rho_{0}$ is the nuclear matter
saturation density and $\Gamma_{i}(\rho_{0})$, $a_{i}$, $b_{i}$, $c_{i}$, and
$d_{i}$ are free parameters of the model, that along with the nucleon and
meson masses, define a specific hadronic parametrization. In particular, we
use here the DDH$\delta$ ddhd  one. It is one of the 263 parametrizations
tested against a set of constraints related to symmetric nuclear matter, pure
neutron matter, symmetry energy and its slope (evaluated at $\rho=\rho_{0}$)
PRC90-055203 . It was also shown that this particular model is capable to
generate massive neutron stars with mass around two solar masses PRC93-025806
.

Concerning the inclusion of more baryons in the hadronic sector, we remind the
reader that some studies point out to a possible suppression of the hyperons
population in hadron-quark phase transitions stars6 ; sup  due to the onset of
hyperons occurring above the onset of quark matter. Furthermore, the inclusion
of hyperons soften the hadronic equation of state with a direct consequence of
producing not so massive stars in comparison with the case in which only
nucleons are considered. Because of that, and since the main focus of this
work is to analyze the thermodynamical structure of a quark model presenting
deconfinement effects at zero temperature regime (PNJL0 model), we decide to
use a RMF model only with nucleons, avoiding a proliferation of more coupling
constants to be fixed in the hadronic sector (hyperons interactions are not
completely determined and some assumptions are required).

The Euler-Lagrange equations are used in order to obtain the field equations
of the model. In addition, the implementation of the Hartree approximation (no
Fock term) serot ; walecka  in these equations, leading to
$\sigma\rightarrow\left<\sigma\right>\equiv\sigma$,
$\omega_{\mu}\rightarrow\left<\omega_{\mu}\right>\equiv\omega_{0}$,
$\vec{\rho}_{\mu}\rightarrow\left<\vec{\rho}_{\mu}\right>\equiv\bar{\rho}_{0(3)}$,
and $\vec{\delta}\rightarrow\,\,<\vec{\delta}>\equiv\delta_{(3)}$, allows the
determination of the energy-momentum tensor, quantity used to calculate the
energy density and pressure of the hadronic model. These last two
thermodynamical equations of state are given by

$\displaystyle\mathcal{E}_{\mbox{\tiny HAD}}$
$\displaystyle=\frac{1}{2}m^{2}_{\sigma}\sigma^{2}-\frac{1}{2}m^{2}_{\omega}\omega_{0}^{2}-\frac{1}{2}m^{2}_{\rho}\bar{\rho}_{0(3)}^{2}+\frac{1}{2}m^{2}_{\delta}\delta^{2}_{(3)}$
$\displaystyle+\frac{\Gamma_{\rho}(\rho)}{2}\bar{\rho}_{0(3)}\rho_{3}+\frac{1}{\pi^{2}}\int_{0}^{{k_{F}}_{p}}dk\,k^{2}(k^{2}+M_{p}^{*2})^{1/2}$
$\displaystyle+\Gamma_{\omega}(\rho)\omega_{0}\rho+\frac{1}{\pi^{2}}\int_{0}^{{k_{F}}_{n}}dk\,k^{2}(k^{2}+M_{n}^{*2})^{1/2}$
(39)

and

$\displaystyle P_{\mbox{\tiny HAD}}$
$\displaystyle=\rho\Sigma_{R}(\rho)-\frac{1}{2}m^{2}_{\sigma}\sigma^{2}+\frac{1}{2}m^{2}_{\omega}\omega_{0}^{2}+\frac{1}{2}m^{2}_{\rho}\bar{\rho}_{0(3)}^{2}$
$\displaystyle-\frac{1}{2}m^{2}_{\delta}\delta^{2}_{(3)}+\frac{1}{3\pi^{2}}\int_{0}^{{k_{F}}_{p}}\frac{dk\,k^{4}}{(k^{2}+M_{p}^{*2})^{1/2}}$
$\displaystyle+\frac{1}{3\pi^{2}}\int_{0}^{{k_{F}}_{n}}\frac{dk\,k^{4}}{(k^{2}+M_{n}^{*2})^{1/2}}$
(40)

with

$\displaystyle\Sigma_{R}(\rho)$
$\displaystyle=\frac{\partial\Gamma_{\omega}}{\partial\rho}\omega_{0}\rho+\frac{1}{2}\frac{\partial\Gamma_{\rho}}{\partial\rho}\bar{\rho}_{0(3)}\rho_{3}-\frac{\partial\Gamma_{\sigma}}{\partial\rho}\sigma\rho_{s}$
$\displaystyle-\frac{\partial\Gamma_{\delta}}{\partial\rho}\delta_{(3)}\rho_{s3},$
(41) $\displaystyle\rho_{s}$ $\displaystyle=\rho_{sp}+\rho_{sn}$
$\displaystyle=\frac{M^{*}}{\pi^{2}}\left[\int_{0}^{{k_{F}}_{p}}\frac{dk\,k^{2}}{\sqrt{k^{2}+M_{p}^{*2}}}+\int_{0}^{{k_{F}}_{n}}\frac{dk\,k^{2}}{\sqrt{k^{2}+M_{n}^{*2}}}\right],$
(42)

$\rho_{3}=\rho_{p}-\rho_{n}$, and $\rho_{p,n}={k_{F}}_{p,n}^{3}/3\pi^{2}$.
Furthermore, the fields are determined as
$\sigma=\Gamma_{\sigma}(\rho)\rho_{s}/m^{2}_{\sigma}$,
$\omega_{0}=\Gamma_{\omega}(\rho)\rho/m_{\omega}^{2}$,
$\bar{\rho}_{0(3)}=\Gamma_{\rho}(\rho)\rho_{3}/2m_{\rho}^{2}$, and
$\delta_{(3)}=\Gamma_{\delta}(\rho)\rho_{s3}/m_{\delta}^{2}$ with
$\rho_{s3}=\rho_{sp}-\rho_{sn}$. The nucleon effective masses read

$\displaystyle M^{*}_{p,n}=M_{\mbox{\tiny
nuc}}-\Gamma_{\sigma}(\rho)\sigma\pm\Gamma_{\delta}(\rho)\delta_{(3)}$ (43)

with $(-)$ for protons and $(+)$ for neutrons. Finally, the proton and neutron
chemical potentials, obtained through
$\mu_{p,n}=\partial\mathcal{E_{\mbox{\tiny HAD}}}/\partial\rho_{p,n}$, are
given by

$\displaystyle\mu_{p}$ $\displaystyle=$
$\displaystyle(k_{Fp}^{2}+{M_{p}^{*}}^{2})^{1/2}+\Gamma_{\omega}\omega_{0}+\frac{\Gamma_{\rho}}{2}\bar{\rho}_{0(3)}+\Sigma_{R}$
(44)

and

$\displaystyle\mu_{n}$ $\displaystyle=$
$\displaystyle(k_{Fn}^{2}+{M_{n}^{*}}^{2})^{1/2}+\Gamma_{\omega}\omega_{0}-\frac{\Gamma_{\rho}}{2}\bar{\rho}_{0(3)}+\Sigma_{R},$
(45)

respectively. We also implement charge neutrality and chemical equilibrium in
the hadronic side by including muons and massless electrons in the system, in
the same way it was done in the quark sector. This leads to the following
relations: $\rho_{p}-\rho_{e}=\rho_{\mu}$ and $\mu_{n}-\mu_{p}=\mu_{e}$, with
$\mu_{\mu}=\mu_{e}$. In this case, total energy density and total pressure are
constructed as $\mathcal{E}_{tH}=\mathcal{E}_{\mbox{\tiny
HAD}}+\mathcal{E}_{e}+\mathcal{E}_{\mu}$ and $P_{tH}=P_{\mbox{\tiny
HAD}}+P_{e}+P_{\mu}$. Moreover, $\mathcal{E}_{e,\mu}$ and $P_{e,\mu}$ can be
extracted from Eqs. (34) and (35).

In Fig. 8a we display total pressure as a function of $\mu_{B}$ for the
hadronic model and for three different parametrizations of the PNJL0 quark
model.

Figure 8: Total pressure as a function of $\mu_{B}$ for (a) two models:
DDH$\delta$ and different PNJ0 parametrizations, and (b) linked models built
through Maxwell construction.

In the hadronic side, one has $\mu_{B}=\mu_{n}$. For the quark sector we
construct the PNJL0 parametrizations as follows: first we choose some values
for the ratio $G_{V}/G_{s}$, namely, $0.15$, $0.25$, $0.35$ and $0.5$. For
each of these values we impose that the point related to the
confinement/deconfinement phase transition takes place exactly at the match
with the hadronic equation of state. The value of $a_{3}$ is then determined
for this purpose. This procedure, also adopted in Ref. pnjl0outro , generates
the linked models presented in Fig. 8b, in which the entire curves start with
the hadronic model until the crossing point, and change to the deconfined
quark model thereafter. This is the basis of the Maxwell construction, that
imposes equal pressures and chemical potentials at both phases, for a fixed
temperature, with a sharp first-order phase transition. The PNJL0
parametrizations constructed from this method present (i) set 1:
$G_{V}/G_{s}=0.15$, $a_{3}=-0.052$; (ii) set 2: $G_{V}/G_{s}=0.25$,
$a_{3}=-0.135$; (iii) set 3: $G_{V}/G_{s}=0.35$, $a_{3}=-0.241$; and (iv) set
4: $G_{V}/G_{s}=0.5$, $a_{3}=-0.458$. Notice that in set 4, where
$G_{V}/G_{s}=0.5$, we found a value very close ($a_{3}=-0.458$) to one used in
Ref. Schramm  ($a_{3}=-0.4$).

In Fig. 9 it is depicted how $P_{t}$ depends on $\mathcal{E}_{t}$ for the
linked models. It is clear that the effect of the first-order phase transition
is to establish a plateau in the pressure with a gap in the energy density.
The values of the transition pressure ($P_{\mbox{\tiny trans}}$) and the
energy density gap at the transition point ($\Delta\mathcal{E}_{\mbox{\tiny
trans}}$) for each PNJL0 parametrization are also shown.

Figure 9: Total pressure as a function of total energy density for the linked
models.

Another interesting feature present in this analysis is that the two
quantities, $P_{\mbox{\tiny trans}}$ and $\Delta\mathcal{E}_{\mbox{\tiny
trans}}$, increase with $G_{V}$, or equivalently, with $|a_{3}|$, both free
parameters related to the PNJL0 quark model. The increasing of $P_{\mbox{\tiny
trans}}$ means that the hadronic side of the linked model persists at higher
values of $\mu_{B}$. This leads to a decreasing of the quark side,
consequently, whilst the increasing of $\Delta\mathcal{E}_{\mbox{\tiny
trans}}$ indicates an increasing of the coexistence region of hadrons and
(deconfined) quarks.

In order to describe a spherically symmetric hybrid star of mass $M$, we solve
the Tolman-Oppenheimer-Volkoff (TOV) equations tov39 ; tov39a ; glen  given by
($G=c=1$)

$\displaystyle\frac{dp(r)}{dr}$
$\displaystyle=-\frac{[\epsilon(r)+p(r)][m(r)+4\pi
r^{3}p(r)]}{r^{2}[1-2m(r)/r]},$ (46) $\displaystyle\frac{dm(r)}{dr}$
$\displaystyle=4\pi r^{2}\epsilon(r),$ (47)

whose solution is constrained to $p(0)=p_{c}$ (central pressure) and $m(0)=0$.
At the star surface, one has $p(R)=0$ and $m(R)\equiv M$, with $R$ defining
its radius. The equations of state used as input to solve Eqs.(46)-(47) are
given by total pressure and total energy density of the linked models
presented before. Furthermore, at the hadronic side we describe the crust of
the star by the model developed by Baym, Pethick and Sutherland (BPS) bps  in
a density region from $\rho=0.16\times 10^{-10}\,\mbox{fm}^{-3}$ to
$\rho=0.89\times 10^{-2}\,\mbox{fm}^{-3}$. The mass-radius profiles of the
hybrid stars obtained from the models studied here are shown in Fig. 10.

Figure 10: Hybrid star mass over $M_{\odot}$ (solar mass) as a function of its
radius for the DDH$\delta$-PNJL0 models. Bands extracted from Refs. Demorest ;
Antoniadis ; Cromartie . Circle, squares and diamonds with error bars are
related to the NICER data a79 ; a80 ; a81 .

In this figure we compare our results with some important observational data.
Two of them are related to the mass values of the objects PSR J1614-2230 and
PSR J0348+0432 with $M=(1.97\pm 0.04)M_{\odot}$ Demorest  and $M=(2.01\pm
0.04)M_{\odot}$ Antoniadis , respectively, and the recent one related to the
MSP J0740+6620 pulsar at $68.3\%$ credible level, namely,
$M=2.14{{+0.10}\atop{-0.09}}M_{\odot}$ Cromartie . As one can see, the hybrid
model is able to produce stars with masses inside these boundaries. For the
sake of comparison, we also depict in Fig. 10 the recent data extracted from
the Neutron Star Interior Composition Explorer (NICER) mission concerning the
radius related to the stars of masses equal to $2.08M_{\odot}$ and
$1.4M_{\odot}$. Such points are given by $M=1.4M_{\odot}$ with $R=(12.45\pm
0.65)$ km a79 , $M=(2.08\pm 0.07)M_{\odot}$ with $R=(12.35\pm 0.75)$ km a79
(squares); $M=2.072_{-0.066}^{+0.067}M_{\odot}$ with $R=12.39_{-0.98}^{+1.30}$
km a80  (circle); $M=1.4M_{\odot}$ with $R=12.33_{-0.81}^{+0.76}$ km a81 ,
$M=1.4M_{\odot}$ with $R=12.18_{-0.79}^{+0.56}$ km a81  (diamonds). We also
observe that hybrid star mass-radius curve is essentially dominated by the
hadronic phase, leaving little room to the detailed dynamics from the PNJL0
model, unless for stars with the smallest radius, where the mass is somewhat
decreasing by shrinking its size. That effect comes from the softening of the
repulsion in the PNJL0 model when the deconfinement phase transition is
approached, without the star loosing the stability, as we discuss below.

As an illustration, we plot in Fig. 11 how the pressure depends on the radial
coordinate for a particular hybrid star constructed from the four sets of the
DDH$\delta$-PNJL0 model. In this case we chose the $M=2.098M_{\odot}$ star.

Figure 11: Pressure as a function of the radial coordinate for the
$M=2.098M_{\odot}$ star obtained from the DDH$\delta$-PNJL0 model (sets 1, 2,
3 and 4). The vertical lines mark the kinks related to the quark cores of each
parametrization.

Notice that this particular star is predicted by all the sets considered here.
In this figure, the kinks identify the quark cores of the star with radius
$R_{q}\sim 1.0$ km for set 1. For sets 2 and 3 we found practically the same
result of $R_{q}\sim 1.5$ km and $R_{q}\sim 1.6$ km, respectively. Finally,
set 4 predicts a quark core of $R_{q}\sim 1.3$ km. In our approach, the core
is composed by deconfined quarks. On the other hand, hadrons are the degrees
of freedom of the star from $r=R_{q}$ to $r=R$.

As a last remark of our results concerning the mass-radius diagrams, we
observe that for each parametrization constructed, namely, sets 1 to 3 of the
DDH$\delta$-PNJL0 hadron-quark model, one identifies linear branches in which
$M$ decreases as $R$ decreases. From the investigation of the static condition
for the star’s stability, this kind of branch would indicate unstable
configurations harrison ; meltzer . However, a dynamical analysis of the star
stability can give further insights by verifying its response to small radial
perturbations, see for instance Refs. rad1 ; rad2 ; rad3 ; rad4 ; rad5 ; rad6
; rad7  for details. By using this criterion, stars are said stable if such
perturbations produce well defined oscillations. On the other hand, an
indefinite increasing of the perturbations amplitude characterizes unstable
stars. The set of differential equation that need to be solved, coupled to the
TOV ones, are given by rad1 ; rad2 ; rad3 ; rad4 ; rad5 ; rad6 ; rad7

$\displaystyle\frac{d\xi}{dr}=-\frac{1}{r}\left(3\xi+\frac{\Delta p}{\Gamma
p}\right)-\frac{dp}{dr}\frac{\xi}{(p+\epsilon)}$ (48)

and

$\displaystyle\frac{d\Delta p}{dr}$
$\displaystyle=\xi\left[\omega^{2}e^{\lambda-\nu}(p+\epsilon)r-4\frac{dp}{dr}\right]$
$\displaystyle+\xi\left[\left(\frac{dp}{dr}\right)^{2}\frac{r}{p+\epsilon}-8\pi
e^{\lambda}(p+\epsilon)pr\right]$ $\displaystyle+\Delta
p\left[\frac{dp}{dr}\frac{1}{p+\epsilon}-4\pi(p+\epsilon)re^{\lambda}\right],$
(49)

with $e^{\lambda}=1-2m(r)/r$, $d\nu/dr=-2(dp/dr)(p+\epsilon)^{-1}$, and
$\Gamma=(\rho/p)(dp/d\rho)$. The relative radial displacement and the
perturbation of the pressure, $\xi=\Delta r/r$ and $\Delta p$, respectively,
are assumed to have a time dependence of $e^{i\omega t}$ with $\omega$ being
the eigenfrequency.

Furthermore, other important aspect that must be mentioned is the kind of
phase transition chosen. In stars with two phases, transitions are classified
as “fast” or “slow” depending on the timescale of the transition compared to
the timescale of the fundamental oscillation rad0 . As we can see in Refs.
rad3 ; rad4 ; rad2 , slow transitions have important consequences in the
microscopic structure of compact stars with two different phases. It is well
known that in stable stars the fundamental frequency has to satisfy the
condition of $\omega^{2}>0$, and the last stable star occurs at $\omega=0$. In
general, this property coincides with the maximum mass in the mass-radius
diagram. However, when a star presents two phases and the transition is slow,
the last stable star ($\omega=0$) can occur after the point of maximum mass,
as we can verify in the results shown in Ref rad2 , for instance. By following
this approach, we verify that all branches analyzed in the parametrizations
shown in Fig. 10 are stable, i.e., we found no points in which $\omega=0$.

##  IV Summary and concluding remarks

In this work we generalize the study performed in Ref. epjc , where we have
proposed a version of the Polyakov-Nambu-Jona-Lasino that exhibits effects of
the confinement/deconfinement phase transition at zero temperature regime for
the two-flavor case. We improve our analysis in order to take into account the
strangeness in the dense system by introducing the dynamics of the strange
quark $s$, whose current mass is considered as $m_{s}=140.7$ MeV. We start
with the Nambu-Jona-Lasinio Lagrangian density model with the ’t Hooft
interaction included. We recall the NJL grand canonical potential written in
terms of the strengths of the scalar, vector and quark mixture channels,
namely, $G_{s}$, $G_{V}$ and $K$, respectively. The implementation of the
infrared quark-gluon dynamics is performed by making these couplings dependent
on the traced Polyakov loop as follows,
$G_{s}\rightarrow\mathcal{G}_{s}(G_{s},\Phi)=G_{s}(1-\Phi^{2})$,
$G_{V}\rightarrow\mathcal{G}_{V}(G_{V},\Phi)=G_{V}(1-\Phi^{2})$, and
$K\rightarrow\mathcal{K}(K,\Phi)=K(1-\Phi^{2})$. The motivation for this
procedure is to ensure vanishing quark interactions at $\Phi=1$ (deconfined
phase). By using these new functions $\mathcal{G}_{s}(K,\Phi)$,
$\mathcal{G}_{V}(K,\Phi)$ and $\mathcal{K}(K,\Phi)$, it is possible to define
a new Polyakov potential now with strangeness included in the back reaction
(quarks and gluons affecting each other).

We also analyze how the confinement/deconfinement phase transition takes place
from the study of the minima of $\Phi$ at a fixed (common) quark chemical
potential. In this investigation, we focus on the case of symmetric quark
matter, namely, the system presenting $\mu_{u}=\mu_{d}=\mu_{s}$. The same
study applied to other values of $\mu$ generates a typical first order phase
transition for the order parameter $\Phi$ as a function of $\mu$. Therefore,
the system presents two phase transitions, namely, the one defined by $\Phi$,
and the other related to the broken/restored chiral symmetry phase transition,
in which the quark condensates are the order parameters. This last one is
already presented in the original NJL model. The region between these two
transitions can be identified, as in the SU(2) case, as the quarkyonic phase,
defined here as the phase in which the light quarks present restored chiral
symmetry but are still confined ($\Phi=0$).

Another interesting feature of the SU(3) version of the PNJL0 model is the
strong reduction of the constituent strange quark mass ($M_{s}$) at the
deconfinement phase transition. We verify that $M_{s}$ undergoes a reduction
of almost $50\%$ for the parametrization in which $G_{V}=0.25G_{s}$ and
$a_{3}=-0.1$. The dynamics of the traced Polyakov loop at $T=0$ favors the
restoration of chiral symmetry even for the strange quark. This effect is not
present in the original SU(3) NJL model since $\Phi$ is always vanishing in
this case.

We also implement the chemical equilibrium and charge neutrality in the SU(3)
PNJL0 model in order to apply it in the description of hybrid stars. For the
hadronic side we use a density dependent model and match both equations of
state exactly at the point where quarks are deconfined. Different PNJL0
parametrizations were constructed by changing the values of $G_{V}$ and
$a_{3}$. We verify that the increasing of both, $G_{V}$ and $|a_{3}|$,
increases the transition pressure and the gap in the energy density of the
hadron-quark phase transition. Finally, we generate the mass-radius profiles
determined by the parametrizations of the PNJL0 model (different $G_{V}$ and
$|a_{3}|$ values) coupled to the hadronic model. Our results indicate that it
is possible to construct hybrid stars compatible with some observational data.
In particular, the recent ones proposed by the NICER mission a79 ; a80 ; a81 .

## ACKNOWLEDGMENTS

This work is a part of the project INCT-FNA proc. No. 464898/2014-5. It is
also supported by Conselho Nacional de Desenvolvimento Científico e
Tecnológico (CNPq) under Grants No. 406958/2018-1, 312410/2020-4 (O.L.), No.
433369/2018-3 (M.D.), and 308486/2015-3 (T.F.). We also acknowledge Fundação
de Amparo à Pesquisa do Estado de São Paulo (FAPESP) under Thematic Project
2017/05660-0 (O.L., M.D., C.H.L, T.F) and Grant No. 2020/05238-9 (O.L., M.D.,
C.H.L). O.A.M. thanks for the fellowship provided by CNPq - Brazil.

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