PNJL model at zero temperature: The three-flavor case

O. A. Mattos , T. Frederico , C. H. Lenzi , M. Dutra , and O. LourençoDepartamento de Física, Instituto Tecnológico de Aeronáutica,

DCTA, 12228-900 São Jose dos Campos, SP, Brazil

(Received 31 July 2021; accepted 22 October 2021; published 2 December 2021)

We propose a three-flavor version of the Polyakov-Nambu-Jona-Lasino (PNJL) model at zerotemperature regime, by implementing a traced Polyakov loop (Φ) dependence in the scalar, vector and’t Hooft channel strengths. We study the thermodynamics of this model, named as PNJL0, with specialattention for the first-order confinement/deconfinement phase transition for whichΦ is the order parameter.For the symmetric quark matter case, an interesting feature observed is a strong reduction of the constituentstrange quark mass (Ms) at the chemical potential related to point where deconfinement takes place. Theemergence ofΦ favors the restoration of chiral symmetry even for the strange quark. We also investigate thecharge 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 thiscase, the quark side is composed by deconfined particles. This approach is used to determine mass-radiusprofiles compatible with recent data from the Neutron Star Interior Composition Explorer mission.

DOI: 10.1103/PhysRevD.104.116001

I. INTRODUCTION

The understanding of the physics related to particlesinteracting strongly is a fundamental issue studied forboth theoretical and experimental researchers. The inves-tigations of collisions at ultrarelativistic energies in themost sophisticated accelerators of the world, such as theRelativistic Heavy-Ion Collider [1] and the Large HadronCollider [2], for instance, give the support needed for thestudy of the strong force probed at the experimental level.The forthcoming Electron Ion Collider [3] at Brookhaven,the Facility for Antiproton and Ion Research [4], atDarmstadt, and the Nuclotron-based Ion Collider facility[5], at the Joint Institute for Nuclear Research, in Dubna,are examples of other facilities with the same purpose.Many aspects of the strongly interacting matter can bedirectly or indirectly accessed from experiments, as forexample, the charge-parity violation observed in heavymesons decays. The reader is addressed to Ref. [6] andreferences therein for an overview including the role of thestrong force in forming the observed charge-parity viola-tion pattern.From the theoretical point of view, on the other hand,

the physics of quarks and gluons is described by quantumchromodynamics (QCD) [7–9]. Its nonperturbative

regime, i.e., QCD at large distances, or equivalently,low energies, can be investigated through the latticediscretization of QCD in the Euclidean space [10–12]where the numerical calculations are based on MonteCarlo simulations [13]. However, the so-called fermionsign problem [14] restricts such technique when quarks atfinite chemical potential are included in the system.A second approach applied to study the infrared regionof QCD is the use of continuous methods relying onDyson-Schwinger equations in Euclidean space [15], usedto determine the equations of motion of the n-pointfunctions. Another practical method of treating quarksand gluons, in systems in which such particles are thefundamental degrees of freedom, is the use of pheno-menological models. Probably the simplest one usedfor this aim is the famous MIT bag model [16]. In itsoriginal version, it establishes that quarks are insidea bag that effectively corresponds to a confiningpotential. The constant B is added to the free quarkpressure, and all thermodynamics is based on this simpleassumption.Other widely used effective quark model is the Nambu-

Jona-Lasinio (NJL) one [17–21]. It was proposed byY. Nambu and G. Jona-Lasinio before QCD and originallydescribed nucleons and pions. In the context of the quarkinteraction, it presents some similarities with QCD suchas the dynamical breaking of chiral symmetry, and theassociated quark mass generation due to the quark con-densates. Similar nonperturbative phenomena also occur inthe Bardeen-Cooper-Schrieffer model of superconductivity[22] (in which the NJL model was inspired), where

Published by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license.Further distribution of this work must maintain attribution tothe author(s) and the published article’s title, journal citation,and DOI. Funded by SCOAP3.

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interactions give rise to energy gaps. In the NJL model, the“gap” in the quark mass is produced by the pointlikeinteractions between Dirac particles.Although capable of describing some important features

of QCD, the NJL model fails to reproduce a particularphenomenon, namely, the infrared dynamics of confine-ment as well as the deconfinement of quarks, related tothe so-called asymptotic freedom, discovered by Gross,Wilczek, and Politzer [23]. In the NJL model there is noinformation regarding the interaction among the gluons.Such an issue was corrected by K. Fukushima [24], whoproposed the inclusion of these mediators through abackground field obtained from the trace of a matrix incolor space associated to the gluon gauge field A4. The so-called traced Polyakov loop, Φ (and its conjugate Φ�),mimics the gluon dynamics in an effective way. Physically,Φ 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 isinfinite and, consequently, Φ ¼ 0. In the deconfined phase,on the other hand, F is finite and Φ ≠ 0. However, weremind that Φ is not enough to have a true confinement inthe Polyakov-Nambu-Jona-Lasino (PNJL) model at finitetemperature since it only presents a statistical confinementwhile asymptotic quark states still exists. The aforemen-tioned analysis originally does not apply at zero tempera-ture in the PNJL model and in most of its parametrizations,since in this case the Polyakov potential vanishes and themodified Fermi-Dirac distributions reduces to the stepfunctions. Because of that, confinement effects are lostand it is not possible to investigate high density deconfinedquark systems at T ¼ 0. In order to circumvent this issue,it was proposed in Refs. [25,26] a modification of thestrengths of the couplings in the NJL model by makingthem dependent on Φ. As a direct consequence, thePolyakov loop potential became nonvanishing even atT ¼ 0. In this way, the new model constructed for flavorSU(2) quarks, named as PNJL0 model, was able to exhibitthe confinement/deconfinement phenomenology at T ¼ 0.In this work, we extend these previous studies by general-izing the PNJL0 model to include flavor SU(3) quarks. Weinvestigate its thermodynamics at T ¼ 0 and also performan application in the context of the hadron-quark phasetransition.The paper is organized as follows. In Sec. II we review

the main equations of the traditional the PNJL model atfinite temperature. In Sec. III, we generalize the PNJL0model at T ¼ 0 by taking into account the dynamics of thestrange s quark. The thermodynamics of the SU(3) model isstudied for both cases, namely, symmetric quark matter,and charge neutral system of quarks and leptons in weakequilibrium. In the former, we verify a strong reduction of

the constituent strange quark mass (Ms) at the chemicalpotential where the finite value of Φ emerges (deconfinedphase). An application to hybrid stars is also performed.The results indicate that it is possible to generate hybridstars compatible with the recent data from the NeutronStar Interior Composition Explorer (NICER) mission.Finally, summary and concluding remarks are presentedin Sec. IV.

II. THREE-FLAVOR POLYAKOV-NAMBU-JONA-LASINIO MODEL

We start by presenting the Lagrangian density of thethree-flavor version of the PNJL model, which includes ascalar and a vector four-fermion interaction. It reads

LPNJL ¼ qðiγμDμ − mÞq − UðΦ;Φ�; TÞ

þ Gs

2

X8a¼0

½ðqλaqÞ2 − ðqγ5λaqÞ2�

−GV

2

X8a¼0

½ðqγμλaqÞ2 þ ðqγμγ5λaqÞ2�

þ K½detfðqð1 − γ5ÞqÞ þ detfðqð1þ γ5ÞqÞ�; ð1Þ

where q is a vector of three spinors qf for f ¼ u, d, s,m ¼ diagðmu;md;msÞ is a matrix of current quark massesin flavor space, and λa are the SU(3) Gell-Mann matrices.The last term, a determinant in flavor space, corresponds tothe so-called ’t Hooft interaction [27] and is responsible forthe large value of the η0 mass due to the anomaly UAð1Þ[28]. Such a kind of interactions were firstly consideredby Kobayashi and Maskawa [29]. The differences betweenthis model and the NJL one [17–21,30–32] are thereplacement of ∂μ by Dμ ≡ ∂μ þ iAμ, where Aμ ¼ δμ0A0

and A0 ¼ gA0aλa=2 (g is the gauge coupling), and the

inclusion of the Polyakov potential UðΦ;Φ�; TÞ. It dependson the traced Polyakov loop and its conjugate, Φ and Φ�,with Φ is defined in terms of A4 ¼ iA0 ≡ Tϕ as

Φ ¼ 1

3Tr

�exp

�iZ

1=T

0

dτA4

��;

¼ 1

3½eiðϕ3þϕ8=

ffiffi3

p Þ þ eið−ϕ3þϕ8=ffiffi3

p Þ þ e−2iϕ8=ffiffi3

p�; ð2Þ

written in the Polyakov gauge where ϕ ¼ ϕ3λ3 þ ϕ8λ8. Inthe mean-field approximation, and using that Φ ¼ Φ�[25,26,33–39], Eq. (1) becomes

O. A. MATTOS et al. PHYS. REV. D 104, 116001 (2021)

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LmfaPNJL ¼

Xf

qfðiγμDμ −MfÞqf − Gs

Xf

ρ2sf

þ GV

Xf

ρ2f − 4KYf

ρsf − UðΦ; TÞ: ð3Þ

The grand canonical potential density obtained fromEq. (3) reads

ΩPNJL ¼ Gs

Xf

ρ2sf −GV

Xf

ρ2f þ 4KYf

ρsf

−γ

2π2Xf

ZΛ

0

dk k2ðk2 þM2fÞ1=2

−γ

6π2Xf

Z∞

0

dk k4

ðk2 þM2fÞ1=2

½fðEf; T;ΦÞ

þ fðEf; T;ΦÞ� þ UðΦ; TÞ; ð4Þ

with

ρsf ¼γ

2π2

Z∞

0

dk k2Mf

Ef½fðEf; T;ΦÞ þ fðEf; T;ΦÞ�

−γ

2π2

ZΛ

0

dk k2Mf

Ef;

¼ hqfqfi; ð5Þ

and

ρf ¼ −∂ΩPNJL

∂μf ;

¼ γ

2π2

Z∞

0

dk k2½fðEf; T;ΦÞ − fðEf; T;ΦÞ�;

¼ hqfγ0qfi: ð6Þ

Here ρsf is the quark “scalar” density (quark condensate)related to flavor f, and ρf is the “vector” quark density.Furthermore, Ef ¼ ðk2 þM2

fÞ1=2 and γ ¼ Ns × Nc ¼ 6 isthe degeneracy factor given in terms of spin (Ns ¼ 2) andcolor (Nc ¼ 3) numbers. The constituent quark masses,Mf, are given in terms of the quark condensates as

Mf ¼ mf − 2Gsρsf − 2KYf0≠f

ρsf0 : ð7Þ

As in the case of two-flavor PNJL model [24,36,40–43],the functions fðEf; T;ΦÞ and fðEf; T;ΦÞ, given by

fðEf;T;ΦÞ

¼ Φe2ðEf−μfÞ=Tþ2ΦeðEf−μfÞ=Tþ1

3Φe2ðEf−μfÞ=Tþ3ΦeðEf−μfÞ=Tþe3ðEf−μfÞ=Tþ1ð8Þ

and

fðEf;T;ΦÞ

¼ Φe2ðEfþμfÞ=Tþ2ΦeðEfþμfÞ=Tþ1

3Φe2ðEfþμfÞ=Tþ3ΦeðEfþμfÞ=Tþe3ðEfþμfÞ=Tþ1; ð9Þ

are the generalized Fermi-Dirac distributions for quarks andantiquarks with μf ¼ μf − 2GVρf. It is worth to notice thatthe vector interaction [35,44,45] produces a shift in thechemical potentials (μf), exactly as it occurs in the NJLmodel. Such distributions are one of two basic differencesbetween PNJL and NJL models. The other one is thePolyakov potential UðΦ; TÞ, a quantity that determinesthe thermodynamics of the gluonic pure gauge sector. Itpresents some forms such as the following ones

URTW05

T4¼ −

b2ðTÞ2

Φ2 −b33Φ3 þ b4

4Φ4; ð10Þ

URRW06

T4¼ −

b2ðTÞ2

Φ2 þ b4ðTÞ lnð1 − 6Φ2 þ 8Φ3 − 3Φ4Þ;ð11Þ

UFUKU08

bT¼ −54e−a=TΦ2 − lnð1 − 6Φ2 þ 8Φ3 − 3Φ4Þ;

ð12Þ

UDS10 ¼ ða0T4 þ a1μ4 þ a2T2μ2ÞΦ2 þ U0ðΦÞ; ð13Þ

with

b2ðTÞ ¼ a0 þ a1

�T0

T

�þ a2

�T0

T

�2

þ a3

�T0

T

�3

; ð14Þ

b4ðTÞ ¼ b4

�T0

T

�3

; ð15Þ

and

U0ðΦÞ≡ a3T40 lnð1 − 6Φ2 þ 8Φ3 − 3Φ4Þ; ð16Þ

with a, b, a0, a1, a2, a3, b3, and b4 being dimensionlessfree parameters. T0 is defined as the transition temperaturefor the pure gauge system. These potentials are namedas RTW05 [41], RRW06 [33,42], FUKU08 [46], andDS10 [37,38]. The traced Polyakov loop is found from∂ΩPNJL=∂Φ ¼ 0, and the current quark masses Mu;d;s aredetermined through solutions of Eq. (7), in which con-densates ρsu;d;s given in Eq. (5) are also used.Finally, from Eq. (4) all the other thermodynamical

quantities can be determined: namely, pressure, entropyand energy densities. The expressions are obtained fromPPNJL ¼ −ΩPNJL, SPNJL ¼ −∂ΩPNJL=∂T, and EPNJL ¼TSPNJL − PPNJL þ

Pf μfρf, respectively.

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III. PNJL0 MODEL: THREE-FLAVOR CASE

A. Formulation

At this point, we remind the reader that the PNJL modelequations, previously presented, are reduced to thoserelated to the NJL one at zero temperature regime. Theorigin of this result is twofold. First, one has that thegeneralized Fermi-Dirac distributions, given in Eqs. (8) and(9), become step functions θðkFf − kÞ at T ¼ 0 (kFf is theFermi momentum of the quark f). Therefore, Φ disappearsin all momentum integrals. The second reason is due to thegluonic contribution enclosed by the Polyakov potential.Notice that UðΦ; 0Þ vanishes for the most known versions.In this case, Eq. (4) reads

ΩPNJLðT ¼ 0Þ ¼ Gs

Xf

ρ2sf −GV

Xf

ρ2f þ 4KYf

ρsf

−γ

2π2Xf

ZΛ

0

dk k2ðk2 þM2fÞ1=2

−γ

6π2Xf

ZkFf

0

dk k4

ðk2 þM2fÞ1=2

¼ ΩNJLðT ¼ 0Þ: ð17Þ

In order to circumvent this problem, we implemented, inRef. [26], the introduction of Φ in the SU(2) NJL model atT ¼ 0 by requiring that the strengths of the scalar andvector channels are vanishing in the deconfined phase, i.e.,at Φ ¼ 1. In that approach we incorporate expected effectsfrom QCD that at low densities, and corresponding largeinterparticle distances, quarks should interact stronglywhile at short distance the interaction should be weakened.The former is associated to the nonperturbative infraredphysics from QCD that enhances the interaction betweenthe effective degrees of freedom, related to the quarks in themodel. The latter should represent the ultraviolet physicsof perturbative QCD where the quarks interact weakly dueto the asymptotic freedom phenomenon. Therefore, thisphysics is incorporated in a dynamical way such that at lowdensities quarks interact strongly and at large densitiesweakly towards to the deconfinement phase transition.For this purpose, we use the traced Polyakov loop as aneffective scalar background field at the zero temperatureregime. Although this quantity is not strictly defined atT ¼ 0, we use it to bring the complexity of QCD dynamicsby incorporating, in a effective way, the transition fromstrong (infrared) to weak (ultraviolet) regimes of the quark-quark interaction. Here, we extended this phenomenologyalso to the three-flavor version of the model. The mod-ifications in the couplings for this case are

Gs → GsðGs;ΦÞ ¼ Gsð1 −Φ2Þ; ð18Þ

GV → GVðGV;ΦÞ ¼ GVð1 −Φ2Þ; ð19Þ

and

K → KðK;ΦÞ ¼ Kð1 −Φ2Þ: ð20Þ

Such changes lead to

ΩPNJL0¼Gs

Xf

ρ2sf−GV

Xf

ρ2fþ4KYf

ρsf

−γ

2π2Xf

ZΛ

0

dkk2ðk2þM2fÞ1=2

−γ

6π2Xf

ZkFf

0

dk k4

ðk2þM2fÞ1=2

þUðρf;ρsf;ΦÞ ð21Þ

with

Uðρf; ρsf;ΦÞ≡ Uðρu; ρd; ρs; ρsu; ρsd; ρss;ΦÞ¼ GVΦ2

Xf

ρ2f −GsΦ2Xf

ρ2sf − 4KΦ2Yf

ρsf

þ U0ðΦÞ: ð22Þ

By choosing to incorporate the terms GsΦ2, GVΦ2, andKΦ2 of the grand canonical potential into U defined inEq. (22), it is possible to identify Eq. (21) as the zerotemperature version of Eq. (4). As in Ref. [26], we refer tothis construction as the PNJL0 model, now written in itsSU(3) version. Quark masses and chemical potentials aregiven, respectively, by

Mf ¼ mf − 2Gsð1 −Φ2Þρsf − 2Kð1 −Φ2ÞYf0≠f

ρsf0 ; ð23Þ

and

μf ¼ ðk2Ff þM2fÞ1=2 þ 2GVð1 −Φ2Þρf; ð24Þ

where

ρf ¼ γ

6π2k3Ff ð25Þ

and

ρsf ¼ −γMf

2π2

ZΛ

kFf

dk k2

ðk2 þM2fÞ1=2

: ð26Þ

Notice that in the three-flavor version of the PNJL0 model,the modification of the couplings, Eqs. (18)–(20), alsoinduces the definition of a new Polyakov potential,Eq. (22), in which the backreaction of quarks in thegluonic sector happens, as well as the inverse one, namely,gluons affecting quarks. This last effect is intrinsic in theoriginal PNJL model but the former is absent. In the SU(3)PNJL0 model, the backreaction is complete, i.e., each

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sector interacts each other, exactly as in its two-flavor version [26]. We remark that the modificationspointed out in Eqs. (18)–(20) lead to the Polyakovpotential U¼GVΦ2

Pf ρ

2f−GsΦ2

Pf ρ

2sf−4KΦ2

Qf ρsf.

However, this potential is not able to generate nonvanishingsolutions for Φ when the condition ∂ΩPNJL0=∂Φ ¼ 0 isapplied. Therefore, we add the term U0 given in Eq. (16) inthe definition of U and verify that it is responsible to ensureΦ ≠ 0. Moreover, it is also important in order to restrictthe traced Polyakov loop in the range of 0 ≤ Φ ≤ 1 [37,38].Finally, the pressure and energy density of the SU(3)PNJL0 model, obtained from Eq. (21), are written as

PPNJL0 ¼ −Gs

Xf

ρ2sf þ GV

Xf

ρ2f − 4KYf

ρsf

þ γ

2π2Xf

ZΛ

0

dk k2ðk2 þM2fÞ1=2

þ γ

6π2Xf

ZkFf

0

dk k4

ðk2 þM2fÞ1=2

− Uðρf; ρsf;ΦÞ þ Ωvac ð27Þ

and

EPNJL0 ¼ Gs

Xf

ρ2sf þ GV

Xf

ρ2f þ 4KYf

ρsf

−γ

2π2Xf

ZΛ

kFf

dk k2ðk2 þM2fÞ1=2 − 2GVΦ2

Xf

ρ2f

þ Uðρf; ρsf;ΦÞ − Ωvac; ð28Þ

respectively. The constant Ωvac ¼ −Pvac is added to thepressure and energy density in order to ensure that invacuum (ρu ¼ ρd ¼ ρs ¼ 0) one has ΩPNJL0ðρf ¼ 0Þ ¼−PPNJL0ðρf ¼ 0Þ ¼ 0 and EPNJL0ðρf ¼ 0Þ ¼ 0.

B. Thermodynamics of the model(symmetric quark matter)

We remark that from now on, all results are obtained atzero temperature regime.Since the main equations of the model are defined, we

are able to explore some thermodynamical features of theSU(3) PNJL0 model. First, we need to define values for itsfree parameters, namely, Gs, GV , K, mu, md, ms, and Λ.Here we use the Rehberg-Klevansky-Hüfner parametriza-tion [19,47] in which Gs ¼ 3.67=Λ2, K ¼ −12.36=Λ5,mu ¼ md ¼ 5.5 MeV, ms ¼ 140.7 MeV, and Λ ¼602.3 MeV. The last quantity is the cutoff parameter. Asin the NJL model, the PNJL0 one is nonrenormalizable.Therefore, it is needed to adopt a certain regularizationscheme in order to avoid divergent contributions in themomentum integrals. Here we use a three-momentumcutoff for this purpose [19,48,49]. This parameters set

produces the values of fπ ¼ 92.4 MeV, mπ ¼ 135 MeV,mK ¼ 497.7 MeV, mη ¼ 514.8 MeV, and m0

η ¼957.8 MeV, respectively, for the pion decay constant andmass, and for the masses of pseudoscalar mesons K, η, and

η0 [19,47]. For the vacuum, it also leads to MðvacÞu ¼

MðvacÞd ¼ 367.6 MeV, MðvacÞ

s ¼ 549.5 MeV, huui1=3ðvacÞ ¼hddi1=3ðvacÞ ¼−241.9MeV, and hssi1=3ðvacÞ ¼ −257.7 MeV.

With regard to the strength of the vector channel, weremind that if a Fierz transformation of the color-currentinteraction is performed, the value of GV ¼ 0.5Gs is found[19,20]. In principle, this “canonical” value should be usedin all calculations involving NJL/PNJL models with thevector channel included. However, GV is often taken as afree parameter. Some authors use this freedom to fix thevector meson masses, such as the ρ meson [50]. Other onesalso point out to changes in the magnitude of GV due tomedium effects. For instance, instanton-anti-instanton pair-ing effects at high temperature can change its magnitude[51]. Furthermore, it is also known that GV plays asignificant hole in the phase diagram of the stronglyinteracting matter since it shifts the location of the criticalend point [35,52]. The increasing of GV induces thedisappearance of the critical end point and, consequently,the vanishing of the first-order phase transition line.Therefore, as in Ref. [26], we adopt here the approachof setting GV as a free parameter in order to verify itsinfluence in our calculations.Finally, the gluonic sector of the model present two more

constants, namely, T0 and a3, appearing in the last term ofthe Polyakov potential, Eq. (22). The former is fixed to190 MeV [26,53]. Concerning the constant a3, we remindthat it appears in the quantity denoted by U0, Eq. (16), firstused in Ref. [37]. In those works, the authors remark thatthe free parameters presented in the Polyakov potential U,which contains U0, are determined in order to reproducelattice data, as well as known information about the phasediagram. In this case, the value obtained by them isa3 ¼ −0.4. However, it is worth to notice that their modelis completely different from the PNJL0 model consideredhere. If we take a3 ¼ −0.4, then no solutions of Φ ≠ 0 arefound, as wewill show in the first figure later on. Therefore,one has GV and a3 as free parameters of the SU(3) PNJL0model, as well as they are in its two-flavor version [26].At zero temperature, one can also use the thermody-

namical relation given by

Eþ PV ¼ μBNB þ μSNS þ μINI;

¼ μuNu þ μdNd þ μsNs; ð29Þ

in order to write the quark chemical potentials as

μu¼μB3þμI

2; μd¼

μB3−μI2; μs¼

μB3−μS; ð30Þ

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i.e., in terms of the baryonic, isospin, and strangenesschemical potentials, namely, μB, μI , and μS, respectively.First, we work at the so called symmetric quark matter, theone in which all quark potentials are the same, namely,

μu ¼ μd ¼ μs ≡ μ; ð31Þ

situation found by taking μI ¼ μS ¼ 0 in Eq. (30). At thispoint, we remark to the reader that the concept ofsymmetric matter in this context is different from theone used in nuclear matter described by hadronic models[54], in which nucleons are in equal number and presentidentical chemical potentials, since they have the same restmasses and densities. In the case of the three-flavor quarksystem studied here, mu ¼ md, but ms is quite different.Therefore, densities of the light quarks are equal each otherbut different from the strange quark one, despite the validityof Eq. (31).As a first investigation of the symmetric quark matter

described by the SU(3) PNJL0 model, we show in Fig. 1ΩPNJL0 as a function of the traced Polyakov loop for GV ¼0.25Gs and for the canonical relation GV ¼ 0.5Gs foundthrough Fierz transformation of the current-current inter-action [19,20]. In order to construct these curves, we useEq. (21) with the constituent quark masses found by thetranscendental equations given by Eqs. (23) and (26). Φ isfree to run in this case. Our analysis points out that in the

range of chemical potentials from μ ¼ MðvacÞu;d to μ ¼ Λ, it is

possible to find curves in which ΩPNJL0 present globalminima for nonvanishing values of the traced Polyakovloop, see Figs. 1(c) and 1(d), for instance. For the curvepresenting a3 ¼ −0.2 and GV ¼ 0.25Gs, we clearly

see a single minimum of ΩPNJL0 at Φ ¼ 0 for μ ¼ MðvacÞu;d

[Fig. 1(a)]. However, Φ ∼ 0.82 produces a minimum ofΩPNJL0 for μ ¼ Λ [Fig. 1(d)]. As in the SU(2) case [26], weverify a confinement/deconfinement phase transition in themodel, i.e., the existence of null and nonvanishing valuesfor the order parameter Φ as a function of the chemicalpotential. One can observe in Figs. 1(e)–1(h) that a similarbehavior is verified for the curves in which GV ¼ 0.5Gs.Another feature observed here, and also present in theSU(2) version of the model [26], is the need of GV ≠ 0values in order to become possible the existence of a globalminimum of ΩPNJL0 at Φ ≠ 0. For GV ¼ 0, the onlyminimum is found at Φ ¼ 0 for different a3 values (figurenot shown). The vector interaction still plays an importantrole even in the SU(3) version of the PNJL0 model.The analysis of the ΩPNJL0 ×Φ curves is also useful in

order to determine the chemical potential in which theconfinement/deconfinement phase transition takes place,named here as μconf . For the model with GV ¼ 0.25Gs anda3 ¼ −0.1, for example, we see from Fig. 2 thatμconf ¼ 533.15 MeV. This particular chemical potentialvalue is defined as the one that generates a grand canonicalpotential presenting two minima with the same value for

ΩPNJL0, i.e.,ΩPNJL0ðΦ1Þ ¼ ΩPNJL0ðΦ2Þ. In that case,Φ1 andΦ2 delimit the boundaries of the thermodynamical phasesassociated to quark confinement and deconfinement. Thetraced Polyakov loop is the order parameter of this tran-sition. The procedure described above can be used toconstruct the Φ × μ curve depicted in Fig. 3, in this casefor GV ¼ 0.25Gs and GV ¼ 0.5Gs. From this figure, wenotice the quark confinement region determined by thecondition given by μ < μconf. On the other hand, deconfine-ment is achieved through a first-order phase transition, in theregion where μ > μconf , according to the thermodynamicspresented by the model. This feature was also observed inRef. [37] in which the authors included the traced Polyakovloop in a hadronic SU(3) nonlinear σmodel alongwith quarkdegrees of freedom. Furthermore, in Ref. [55] anotherversion of a three-flavor PNJL model at zero temperaturewas studied. Specifically, a dependence on the quark densitywas introduced in the b2ðTÞ function of the Polyakovpotential given in Eq. (11). In that model, a first-order phasetransition is also verified in the ΦðμÞ curve. Another effect

-50

0

50

100

150

200

Ω PN

JL0 (

MeV

/fm

3 )

a3 = -0.05

a3 = 0

a3 = 0.02

a3 = 0.1

-0.2 0 0.2 0.4 0.6 0.8 1 Φ

-1200

-1000

-800

-600

-400

a3 = -0.5

a3 = -0.4

a3 = -0.3

a3 = -0.2

a3 = -0.1

-0.2 0 0.2 0.4 0.6 0.8 1

)d()c(

)b()a( μ = 387.6 MeV

μ = 582.3 MeV μ = Λ = 602.3 MeV

μ = M(vac)

u,d

= 367.6 MeV

GV

= 0.25Gs

-50

0

50

100

150

200

Ω PN

JL0 (

MeV

/fm

3 )a

3 = -0.05

a3 = 0

a3 = 0.02

a3 = 0.1

-0.2 0 0.2 0.4 0.6 0.8

Φ

-1200

-1000

-800

-600

-400

a3 = -0.5

a3 = -0.4

a3 = -0.3

a3 = -0.2

a3 = -0.1

-0.2 0 0.2 0.4 0.6 0.8

(g)(h)

)f()e( μ = 387.6 MeV

μ = 582.3 MeV μ = Λ = 602.3 MeV

μ = M(vac)

u,d

= 367.6 MeV

GV

= 0.5Gs

FIG. 1. ΩPNJL0 as a function of Φ for GV ¼ 0.25Gs (a),(b),(c),(d) and GV ¼ 0.5Gs (e),(f),(g),(h) for different a3. The common

quark chemical potential is given by (a),(e) μ ¼ MðvacÞu ¼

MðvacÞd ¼ 367.6 MeV, (b),(f) μ ¼ 387.6 MeV, (c),(g) μ ¼

582.3 MeV, and (d),(h) μ ¼ Λ ¼ 602.3 MeV.

O. A. MATTOS et al. PHYS. REV. D 104, 116001 (2021)

116001-6

verified in Fig. 3 is the decreasing of μconf as GV increases,exactly as in the SU(2) case [26].Now we evaluate three-flavor PNJL0 model equations

by solving Eqs. (23) and (26) together with condition∂ΩPNJL0=∂Φ ¼ 0. In the SU(3) case, this procedure con-sists in searching for solutions of Mu ¼ Md, Ms, and Φ,obtained from the set of three coupled equations afore-mentioned. By following this method, we are able toevaluate Eq. (21), with result displayed in Fig. 4. Onecan notice a typical signal of a system presenting first-orderphase transition [56–58], since Fig. 4 shows that ΩPNJL0(thermodynamical potential that describes the system) ismultivalued. A requirement from thermodynamical stabil-ity [56] imposes that branches associated to unstable andmetastable solutions is removed from the final curve. The“crossing points” indicate the chemical potential valuesrelated to the first-order phase transitions. In the case of the

PNJL0 model, there are two of them, one from confine-ment/deconfinement and another one from restored/brokenchiral symmetry of the light quarks, that give rise toμconf ¼ 533.15 MeV and μchiral ¼ 372.10 MeV, respec-tively, for the parametrization of the model in which GV ¼0.25Gs and a3 ¼ −0.1. Notice that at μ ¼ μconf , we find thesame value for ΩPNJL0 related to the two global minima ofFig. 2, namely, ΩPNJL0ðμconfÞ ∼ −331 MeV=fm3. Thisfeature confirms the equivalence of the two methods usedto define the first-order phase transition points in suchsystems. It is also worth to notice the decrease of μconf asGV increases, exactly as pointed out in Fig. 3.In the region where μchiral < μ < μconf we can recognize,

as in the SU(2) version of the model [26], a particularthermodynamical phase in which the light quarks present avery low mass in comparison to their vacuum values(almost fully restored chiral symmetry), and with Φ ¼ 0(confined quarks), i.e., the so-called quarkyonic phase,pointed out in Refs. [46,59–63], for example. The quar-kyonic phase related to the light quark sector can also beidentified by looking at the constituent quark masses as afunction of μ, as displayed in Fig. 5. One can notice twosharp phase transitions occurring at μchiral ¼ 372.10 MeVand μconf ¼ 533.15 MeV, for GV ¼ 0.25Gs, with the dif-ference defining the “size” of the phase, in this case,Δμ ¼ μconf − μchiral ¼ 161.05 MeV. As we can also see,this difference is decreased for GV ¼ 0.5Gs. More specifi-cally, for the SU(3) PNJL0 model Δμ depends on both, thestrength of the vector channel, GV , and the a3 parameterrelated to the Polyakov potential. In Tables I and II wepresent some values of Δμ by running a3 and GVindependently.From Fig. 5, we can also notice a strong reduction of the

constituent strange quark mass at μ ¼ μconf . At this point,Ms undergoes a reduction of almost 50% forGV ¼ 0.25Gs,while for GV ¼ 0.5Gs the reduction is somewhat larger.The origin of this feature is the emergence of Φ exactly at

-0,2 0 0,2 0,4 0,6 0,8Φ

-340

-330

-320

ΩPN

JL0 (

MeV

/fm

3 )

p2

GV

= 0.25Gs

a3 = -0.1

μ = μconf

= 533.15 MeV

μ = 531 MeV

μ = 536 MeV

p1

FIG. 2. ΩPNJL0 as a function of Φ for GV ¼ 0.25Gs and a3 ¼−0.1 for three different values of μ.

450 500 550 600 μ (MeV)

0

0.2

0.4

0.6

0.8

1

Φ

GV

= 0.25Gs

μconf

= 533.15 MeV

confinement( μ < μ

conf )

deconfinement( μ > μ

conf )

GV

= 0.5Gs

a3 = -0.1

μconf

= 517.29 MeV

FIG. 3. Traced Polyakov loop as a function of the commonquark chemical potential forGV ¼ 0.25Gs andGV ¼ 0.5Gs, bothfor a3 ¼ −0.1. These parametrizations lead to μconf ¼533.15 MeV and μconf ¼ 517.29 MeV, respectively.

350 400 450 500 550μ (MeV)

-400

-300

-200

-100

0

ΩPN

JL0 (

MeV

/fm

3 )

μconf

= 533.15 MeV

GV

= 0.25Gs

μchiral

= 372.10 MeV

GV

= 0.5Gs

a3 = -0.1

μchiral

= 382.18 MeV

μconf

= 517.29 MeV

FIG. 4. ΩPNJL0 as a function of the common quark chemicalpotential for GV ¼ 0.25Gs and GV ¼ 0.5Gs, both for a3 ¼ −0.1.

PNJL MODEL AT ZERO TEMPERATURE: THE THREE-FLAVOR … PHYS. REV. D 104, 116001 (2021)

116001-7

this chemical potential, see Fig. 3. The decreasing of thecouplings GsðΦÞ, GVðΦÞ, and KðΦÞ when Φ becomesnonvanishing induces a reduction in the constituent quarkmasses, including the strange quark one. We reinforce thatsuch an effect is not present in the original SU(3) NJLmodel, also show in Fig. 5 (in that case changes in Ms areonly gradual, see the dotted line). For the light sector, on theother hand, it is also verified a decreasing of Mu;d in thePNJL0 model, however, with much smaller intensity, sinceat μ ¼ μconf the light quark masses are nearly vanishingalready.As already pointed out, the effect of the traced Polyakov

loop depends on GV and a3. In Fig. 6 for a3 ¼ −0.27 andGV ¼ 0.25Gs, we verify that μconf is equal to the cutoffparameter Λ resulting a smaller reduction of Ms withrespect to the NJL model, differently of what is seen inFig. 5 in the previous case. For the sake of comparison, wealso show the results concerning the parametrizationpresenting GV ¼ 0.5Gs (value obtained from the Fierztransformation of the current-current interaction). Noticethat the decrease of the reduction inMs is also observed forthis case.

C. Charge neutral system of quarks and leptonsin weak equilibrium within hybrid stars

In this section we proceed to show how the confinementeffects described by the traced Polyakov loop affect thecharge neutral system of quarks and leptons (muonsand massless electrons) in weak equilibrium. This studyis relevant in the context of quark stars or hybrid stars[49,64–70], constructed from a particular quark model atzero temperature and at high density regime. This is asystem fulfilling the following conditions for the quarksand leptons chemical potentials: μs ¼ μd ¼ μu þ μe andμe ¼ μμ. It is possible to write μu, μd, and μs in terms of acommon quark chemical potential (μ), and in terms of theelectron one (μe) as

μu ¼ μ −2

3μe; and μd ¼ μs ¼ μþ 1

3μe; ð32Þ

in which the relation μB ¼ 3μ holds. Furthermore, therequirement of charge neutral quark matter leads to theadditional relation to be satisfied

2

3ρu −

1

3ρd −

1

3ρs ¼ ρe þ ρμ; ð33Þ

in which ρe is the electron density. μe and ρe are related toeach other through ρe ¼ μ3e=ð3π2Þ. The muon density is

TABLE I. Parameter a3 of the SU(3) PNJL0 model along withμconf and Δμ. For each line one has GV ¼ 0.25Gs andμchiral ¼ 372.10 MeV.

a3 μconf (MeV) Δμ (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 II. ParameterGV of the SU(3) PNJL0 model along withμconf , μchiral, and Δμ. For each line we fixed a3 ¼ −0.1.

GV=Gs μconf (MeV) μchiral (MeV) Δμ (MeV)

0.50 517.29 382.18 135.110.45 519.38 380.21 139.170.40 521.82 378.22 143.600.35 524.75 376.21 148.540.30 528.41 374.18 154.230.25 533.15 372.10 161.050.20 539.70 370.00 169.70

350 400 450 500 550 600 μ (MeV)

0

100

200

300

400

500M

f (M

eV)

f = sf = u, dNJL

Δμ = μconf

- μchiral

= 161.05 MeV

GV

= 0.25Gs

a3 = -0.1

GV

= 0.5Gs

Δμ = μconf

- μchiral

= 135.11 MeV

GV

= 0.25Gs

GV

= 0.5Gs

FIG. 5. Full and dashed lines: constituent quark masses as afunction of the common quark chemical potential for GV ¼0.25Gs and GV ¼ 0.5Gs, both for a3 ¼ −0.1 (PNJL0 model).Dotted lines: SU(3) NJL model.

350 400 450 500 550 600 μ (MeV)

0

100

200

300

400

500

Mf (

MeV

)

f = sf = u, dNJL

GV

= 0.25Gs

μconf

= Λ = 602.3 MeV

μconf

= 580.63 MeV

GV

= 0.25Gsμ

chiral = 382.18 MeV

a3 = -0.27

GV

= 0.5Gs

GV

= 0.5Gs

FIG. 6. The same as in Fig. 5, but for a3 ¼ −0.27.

O. A. MATTOS et al. PHYS. REV. D 104, 116001 (2021)

116001-8

ρμ ¼ ½ðμ2μ −m2μÞ3=2�=ð3π2Þ, and mμ ¼ 105.7 MeV. In this

case, the total energy density and total pressure are given,respectively, by

Etq¼EPNJL0þμ4e4π2

þ 1

π2

Z ffiffiffiffiffiffiffiffiffiffiμ2μ−m2

μ

p

0

dkk2ðk2þm2μÞ1=2; ð34Þ

and

Ptq¼PPNJL0þμ4e

12π2þ 1

3π2

Z ffiffiffiffiffiffiffiffiffiffiμ2μ−m2

μ

p

0

dkk4

ðk2þm2μÞ1=2

: ð35Þ

We start by showing in Fig. 7 the μB dependence of thequark fractions defined as Yi ¼ ρi=ðρu þ ρd þ ρsÞ, for theSU(3) PNJL0 model with GV ¼ 0.25Gs and differentvalues of a3. We clearly verify that d quark fractiongradually reduces to a value close to the u quark fraction.Furthermore, at the chemical potential related to thedeconfinement transition, point in which Φ becomes non-vanishing, one observes that this reduction is anticipated bya sharp variation. For the s quark fraction, the opposite isfound, namely, Ys approaches to Yu with sharp variationsproduced by the emergence of Φ. In the case of Yu, we findno significant variations along the μB dependence incomparison with both previous cases. From the figure, itis also clear that the critical baryonic chemical potentialmoves to the direction of increasing μB as ja3j increases,feature also present in symmetric quark matter, as shown inTable I.As mentioned before, an important application of a quark

model available at zero temperature is the description ofhybrid stars, in which a hadron-quark phase transition takesplace at high density regime. A recent study based on amodel-independent analysis, performed in Ref. [71], sug-gests evidences for massive stars composed by cores ofquark matter. Here we take the PNJL0 model to take into

account the quark side of the equations of state used todescribe stellar matter. For the hadronic side, we use therelativistic mean-field model DDHδ given by the followingLagrangian density

LHAD¼ ψðiγμ∂μ−MÞψþΓσðρÞσψψ−ΓωðρÞψγμωμψ

−ΓρðρÞ2

ψγμρμτψþΓδðρÞψ δ τψþ1

2ð∂μσ∂μσ−m2

σσ2Þ

−1

4FμνFμνþ

1

2m2

ωωμωμ−

1

4BμνBμνþ

1

2m2

ρρμρμ

þ1

2ð∂μδ∂μδ−m2

δδ2Þ; ð36Þ

where

ΓiðρÞ ¼ Γiðρ0Þai1þ biðρ=ρ0 þ diÞ21þ ciðρ=ρ0 þ diÞ2

; ð37Þ

for i ¼ σ, ω, and

ΓiðρÞ ¼ Γiðρ0Þ½aie−biðρ=ρ0−1Þ − ciðρ=ρ0 − diÞ�; ð38Þ

for i ¼ ρ, δ. In Eq. (36) ψ is the nucleon field, whereas σ,ωμ, ρμ, and δ are the scalar, vector, isovector-vector, andisovector-scalar fields that represent mesons σ, ω, ρ, and δ,respectively. The antisymmetric tensors Fμν and Bμν are

written as Fμν ¼ ∂νωμ − ∂μων and Bμν ¼ ∂νρμ − ∂μρν. Thenucleon rest mass is Mnuc, and the mesons masses are mσ,mω, mρ, and mδ. A more simplified version of this modelwas firstly proposed by J. D. Walecka in Ref. [72] in whichthe couplings between mesons and nucleons are constants.Here we choose to use a more sophisticated structure forthe hadronic model in which such couplings are densitydependent, namely, ΓiðρÞ, with ρ being the sum of thedensities of proton (ρp) and neutrons (ρn). In Eqs. (37) and(38) ρ0 is the nuclear matter saturation density and Γiðρ0Þ,ai, bi, ci, and di are free parameters of the model, that alongwith the nucleon and meson masses, define a specifichadronic parametrization. In particular, we use here theDDHδ [73] one. It is one of the 263 parametrizations testedagainst a set of constraints related to symmetric nuclearmatter, pure neutron matter, symmetry energy and its slope(evaluated at ρ ¼ ρ0) [74]. It was also shown that thisparticular model is capable to generate massive neutronstars with mass around two solar masses [75].Concerning the inclusion of more baryons in the had-

ronic sector, we remind the reader that some studies pointout to a possible suppression of the hyperons population inhadron-quark phase transitions [69,76] due to the onset ofhyperons occurring above the onset of quark matter.Furthermore, the inclusion of hyperons soften the hadronicequation of state with a direct consequence of producingnot so massive stars in comparison with the case in whichonly nucleons are considered. Because of that, and since the

1200 1400 1600 1800μ

B (MeV)

0

0.1

0.2

0.3

0.4

0.5

0.6

Yi

a3 = -0.20

a3 = -0.15

a3 = -0.10

a3 = -0.05

i = d

GV

= 0.25Gs

i = u

i = s

FIG. 7. Quark fractions as a function of μB for the chargeneutral system of quarks and leptons in weak equilibrium. PNJL0parametrization for GV ¼ 0.25Gs and different values of a3.

PNJL MODEL AT ZERO TEMPERATURE: THE THREE-FLAVOR … PHYS. REV. D 104, 116001 (2021)

116001-9

main focus of this work is to analyze the thermodynamicalstructure of a quark model presenting deconfinementeffects at zero temperature regime (PNJL0 model), wedecide to use a relativistic mean-field model only withnucleons, avoiding a proliferation of more coupling con-stants to be fixed in the hadronic sector (hyperons inter-actions are not completely determined and someassumptions are required).The Euler-Lagrange equations are used in order to obtain

the field equations of the model. In addition, the imple-mentation of the Hartree approximation (no Fock term)[72,77] in these equations, leading to σ → hσi≡ σ,ωμ → hωμi≡ ω0, ρμ → hρμi≡ ρ0ð3Þ, and δ → hδi≡ δð3Þ,allows for the determination of the energy-momentumtensor quantity used to calculate the energy density andpressure of the hadronic model. These last two thermody-namical equations of state are given by

EHAD ¼ 1

2m2

σσ2 −

1

2m2

ωω20 −

1

2m2

ρρ20ð3Þ þ

1

2m2

δδ2ð3Þ

þ ΓρðρÞ2

ρ0ð3Þρ3 þ1

π2

ZkFp

0

dkk2ðk2 þM�2p Þ1=2

þ ΓωðρÞω0ρþ1

π2

ZkFn

0

dkk2ðk2 þM�2n Þ1=2 ð39Þ

and

PHAD ¼ ρΣRðρÞ −1

2m2

σσ2 þ 1

2m2

ωω20 þ

1

2m2

ρρ20ð3Þ

−1

2m2

δδ2ð3Þ þ

1

3π2

ZkFp

0

dkk4

ðk2 þM�2p Þ1=2

þ 1

3π2

ZkFn

0

dkk4

ðk2 þM�2n Þ1=2 ; ð40Þ

with

ΣRðρÞ ¼∂Γω

∂ρ ω0ρþ1

2

∂Γρ

∂ρ ρ0ð3Þρ3 −∂Γσ

∂ρ σρs

−∂Γδ

∂ρ δð3Þρs3; ð41Þ

ρs ¼ ρsp þ ρsn;

¼ M�

π2

�ZkFp

0

dkk2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 þM�2

p

q þZ

kFn

0

dkk2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 þM�2

n

p�; ð42Þ

ρ3 ¼ ρp − ρn, and ρp;n ¼ kF3p;n=3π2. Furthermore, thefields are determined as σ ¼ ΓσðρÞρs=m2

σ ,ω0 ¼ ΓωðρÞρ=m2

ω, ρ0ð3Þ ¼ ΓρðρÞρ3=2m2ρ, and δð3Þ ¼

ΓδðρÞρs3=m2δ with ρs3 ¼ ρsp − ρsn. The nucleon effective

masses read

M�p;n ¼ Mnuc − ΓσðρÞσ � ΓδðρÞδð3Þ; ð43Þ

with (−) for protons and (þ) for neutrons. Finally, theproton and neutron chemical potentials, obtained throughμp;n ¼ ∂EHAD=∂ρp;n, are given by

μp ¼ ðk2Fp þM�p2Þ1=2 þ Γωω0 þ

Γρ

2ρ0ð3Þ þ ΣR ð44Þ

and

μn ¼ ðk2Fn þM�n2Þ1=2 þ Γωω0 −

Γρ

2ρ0ð3Þ þ ΣR; ð45Þ

respectively. We also implement charge neutrality andchemical equilibrium in the hadronic side by includingmuons and massless electrons in the system, in the sameway it was done in the quark sector. This leads to thefollowing relations: ρp − ρe ¼ ρμ and μn − μp ¼ μe,with μμ ¼ μe. In this case, total energy density and totalpressure are constructed as EtH ¼ EHAD þ Ee þ Eμ andPtH ¼ PHAD þ Pe þ Pμ. Moreover, Ee;μ and Pe;μ can beextracted from Eqs. (34) and (35).In Fig. 8(a) we display total pressure as a function of μB

for the hadronic model and for three different parametriza-tions of the PNJL0 quark model. In the hadronic side, onehas μB ¼ μn. For the quark sector we construct the PNJL0parametrizations as follows: first we choose some valuesfor the ratio GV=Gs, namely, 0.15, 0.25, 0.35, and 0.5. Foreach of these values we impose that the point related to theconfinement/deconfinement phase transition takes placeexactly at the match with the hadronic equation of state.The value of a3 is then determined for this purpose. Thisprocedure, also adopted in Ref. [55], generates the linkedmodels presented in Fig. 8(b), in which the entire curvesstart with the hadronic model until the crossing point, andchange to the deconfined quark model thereafter. This is thebasis of the Maxwell construction, that imposes equalpressures and chemical potentials at both phases, for a fixedtemperature, with a sharp first-order phase transition. ThePNJL0 parametrizations constructed from this methodpresent (i) set 1: GV=Gs ¼ 0.15, a3 ¼ −0.052; (ii) set 2:GV=Gs ¼ 0.25, a3 ¼ −0.135; (iii) set 3: GV=Gs ¼ 0.35,a3 ¼ −0.241; and (iv) set 4: GV=Gs ¼ 0.5, a3 ¼ −0.458.Notice that, in set 4, where GV=Gs ¼ 0.5, we found avalue very close (a3 ¼ −0.458) to one used inRef. [37] (a3 ¼ −0.4).In Fig. 9 it is depicted how Pt depends on Et for the

linked models. It is clear that the effect of the first-orderphase transition is to establish a plateau in the pressure witha gap in the energy density. The values of the transitionpressure (Ptrans) and the energy density gap at the transitionpoint (ΔEtrans) for each PNJL0 parametrization are alsoshown. Another interesting feature present in this analysisis that the two quantities, Ptrans and ΔEtrans, increase with

O. A. MATTOS et al. PHYS. REV. D 104, 116001 (2021)

116001-10

GV , or equivalently, with ja3j, both free parameters relatedto the PNJL0 quark model. The increasing of Ptrans meansthat the hadronic side of the linked model persists at higher

values of μB. This leads to a decreasing of the quark side,consequently, whilst the increasing of ΔEtrans indicates anincreasing 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-Volkoffequations [78–80] given by (G ¼ c ¼ 1)

dpðrÞdr

¼ −½ϵðrÞ þ pðrÞ�½mðrÞ þ 4πr3pðrÞ�

r2½1 − 2mðrÞ=r� ; ð46Þ

dmðrÞdr

¼ 4πr2ϵðrÞ; ð47Þ

whose solution is constrained to pð0Þ ¼ pc (central pres-sure) and mð0Þ ¼ 0. At the star surface, one has pðRÞ ¼ 0andmðRÞ≡M, with R defining its radius. The equations ofstate used as input to solve Eqs. (46)–(47) are given by totalpressure and total energy density of the linked modelspresented before. Furthermore, at the hadronic side wedescribe the crust of the star by the model developed byBaym, Pethick, and Sutherland [81] in a density regionfrom ρ ¼ 0.16 × 10−10 fm−3 to ρ ¼ 0.89 × 10−2 fm−3.The mass-radius profiles of the hybrid stars obtained fromthe models studied here are shown in Fig. 10. In this figurewe compare our results with some important observationaldata. Two of them are related to the mass values of theobjects PSR J1614-2230 and PSR J0348þ 0432 withM ¼ ð1.97� 0.04Þ M⊙ [82] and M ¼ ð2.01� 0.04Þ M⊙[83], respectively, and the recent one related to the MSPJ0740þ 6620 pulsar at 68.3% credible level, namely,M ¼2.14 þ0.10

−0.09 M⊙ [84]. As one can see, the hybrid model isable to produce stars with masses inside these boundaries.For the sake of comparison, we also depict in Fig. 10 therecent data extracted from the NICER mission concerning

5 10 15 20

εt (fm

-4)

0

1

2

3

4

Pt (

fm-4

)

DDHδ-PNJL0 (set 1)DDHδ-PNJL0 (set 2)DDHδ-PNJL0 (set 3)DDHδ-PNJL0 (set 4)

Ptrans

= 2.4 fm-4

Δεtrans

= 8.1 fm-4

Ptrans

= 1.6 fm-4

Δεtrans

= 6.3 fm-4

Ptrans

= 1.9 fm-4

Δεtrans

= 7.0 fm-4

Ptrans

= 3.3 fm-4

Δεtrans

= 11 fm-4

FIG. 9. Total pressure as a function of total energy density forthe linked models.

1500 1600 1700 1800 1900 2000μ

B (MeV)

1

1.5

2

2.5

3

3.5

4

Pt (

fm-4

)

DDHδPNJL0 (set 1)PNJL0 (set 2)PNJL0 (set 3)PNJL0 (set 4)

set 1:G

V = 0.15G

sa

3 = -0.052

set 2:G

V = 0.25G

sa

3 = -0.135

set 3:G

V = 0.35G

sa

3 = -0.241

(a) two models

set 4:G

V = 0.5G

sa

3 = -0.458

1500 1600 1700 1800 1900 2000μ

B (MeV)

1

1.5

2

2.5

3

3.5

4

Pt (

fm-4

)

DDHδ-PNJL0 (set 1) DDHδ-PNJL0 (set 2)DDHδ-PNJL0 (set 3)DDHδ-PNJL0 (set 4)

set 1:G

V = 0.15G

sa

3 = -0.052

set 2:G

V = 0.25G

sa

3 = -0.135

set 4:G

V = 0.5G

sa

3 = -0.458

(b) linked models

set 3:G

V = 0.35G

sa

3 = -0.241

FIG. 8. Total pressure as a function of μB for (a) two models:DDHδ and different PNJ0 parametrizations, and (b) linkedmodels built through Maxwell construction.

9 10 11 12 13 14 15R (km)

0

0.5

1

1.5

2

2.5

M /

MO.

DDHδ-PNJL0 (set 1)DDHδ-PNJL0 (set 2)DDHδ-PNJL0 (set 3)DDHδ-PNJL0 (set 4)Riley, et al. [86] (NICER)Miller, et al. [85] (NICER)Raaijmakers, et al. [87] (NICER)

PSR J0704+6620

PSR J1614-2230PSR J0348+0432

FIG. 10. Hybrid star mass overM⊙ (solar mass) as a function ofits radius for the DDHδ-PNJL0 models. Bands extracted fromRefs. [82–84]. Circle, squares, and diamonds with error bars arerelated to the NICER data [85–87].

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116001-11

the radius related to the stars of masses equal to 2.08 M⊙and 1.4 M⊙. Such points are given by M ¼ 1.4 M⊙with R ¼ ð12.45� 0.65Þ km [85], M ¼ ð2.08�0.07Þ M⊙ with R ¼ ð12.35� 0.75Þ km [85] (squares);M ¼ 2.072þ0.067

−0.066 M⊙ with R ¼ 12.39þ1.30−0.98 km [86]

(circle); M ¼ 1.4 M⊙ with R ¼ 12.33þ0.76−0.81 km [87], M ¼

1.4 M⊙ with R ¼ 12.18þ0.56−0.79 km [87] (diamonds). We also

observe that hybrid star mass-radius curve is essentiallydominated by the hadronic phase, leaving little room to thedetailed dynamics from the PNJL0 model, unless for starswith the smallest radius, where the mass is somewhatdecreasing by shrinking its size. That effect comes fromthe softening of the repulsion in the PNJL0 model when thedeconfinement phase transition is approached, withoutthe 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 starconstructed from the four sets of the DDHδ-PNJL0 model.In this case we chose the M ¼ 2.098 M⊙ star. Notice thatthis particular star is predicted by all the sets consideredhere. In this figure, the kinks identify the quark cores ofthe star with radius Rq ∼ 1.0 km for set 1. For sets 2 and 3we found practically the same result of Rq ∼ 1.5 km andRq ∼ 1.6 km, respectively. Finally, set 4 predicts a quarkcore of Rq ∼ 1.3 km. In our approach, the core is composedby deconfined quarks. On the other hand, hadrons are thedegrees of freedom of the star from r ¼ Rq to r ¼ R.As a last remark of our results concerning the mass-

radius diagrams, we observe that for each parametrizationconstructed, namely, sets 1 to 3 of the DDHδ-PNJL0hadron-quark model, one identifies linear branches inwhich M decreases as R decreases. From the investigationof the static condition for the star’s stability, this kind ofbranch would indicate unstable configurations [88,89].

However, a dynamical analysis of the star stability cangive further insights by verifying its response to smallradial perturbations, see for instance Refs. [90–96] fordetails. By using this criterion, stars are said to be stable ifsuch perturbations produce well-defined oscillations. Onthe other hand, an indefinite increasing of the perturbationsamplitude characterizes unstable stars. The set of differ-ential equation that need to be solved, coupled to theTolman-Oppenheimer-Volkof ones, are given by [90–96]

dξdr

¼ −1

r

�3ξþ Δp

Γp

�−dpdr

ξ

ðpþ ϵÞ ð48Þ

and

dΔpdr

¼ ξ

�ω2eλ−νðpþ ϵÞr − 4

dpdr

�

þ ξ

��dpdr

�2 rpþ ϵ

− 8πeλðpþ ϵÞpr�

þ Δp�dpdr

1

pþ ϵ− 4πðpþ ϵÞreλ

�; ð49Þ

with eλ ¼ 1–2mðrÞ=r, dν=dr ¼ −2ðdp=drÞðpþ ϵÞ−1, andΓ ¼ ðρ=pÞðdp=dρÞ. The relative radial displacement andthe perturbation of the pressure, ξ ¼ Δr=r and Δp,respectively, are assumed to have a time dependence ofeiωt with ω being the eigenfrequency.Furthermore, other important aspect that must be men-

tioned is the kind of phase transition chosen. In stars withtwo phases, transitions are classified as “fast” or “slow”depending on the timescale of the transition compared tothe timescale of the fundamental oscillation [97]. As we cansee in Refs. [91–93], slow transitions have importantconsequences in the microscopic structure of compact starswith two different phases. It is well known that in stablestars the fundamental frequency has to satisfy the conditionof ω2 > 0, and the last stable star occurs at ω ¼ 0.In general, this property coincides with the maximummass in the mass-radius diagram. However, when a starpresents two phases and the transition is slow, the last stablestar (ω ¼ 0) can occur after the point of maximum mass, aswe can verify in the results shown in Ref. [91], for instance.By following this approach, we verify that all branchesanalyzed in the parametrizations shown in Fig. 10 arestable; i.e., we found no points in which ω ¼ 0.

IV. SUMMARY AND CONCLUDING REMARKS

In this work we generalize the study performed inRef. [26], where we have proposed a version of thePolyakov-Nambu-Jona-Lasino model that exhibits effectsof the confinement/deconfinement phase transition at zerotemperature regime for the two-flavor case. We improveour analysis in order to take into account the strangeness inthe dense system by introducing the dynamics of

0 2 4 6 8 10 12r (km)

0

1

2

3

4

p (f

m-4

)

DDHδ-PNJL0 (set 1)DDHδ-PNJL0 (set 2)DDHδ-PNJL0 (set 3)DDHδ-PNJL0 (set 4)

FIG. 11. Pressure as a function of the radial coordinate for theM ¼ 2.098 M⊙ star obtained from the DDHδ-PNJL0 model (sets1, 2, 3, and 4). The vertical lines mark the kinks related to thequark cores of each parametrization.

O. A. MATTOS et al. PHYS. REV. D 104, 116001 (2021)

116001-12

the strange quark s, whose current mass is considered asms ¼ 140.7 MeV. We start with the Nambu-Jona-LasinioLagrangian density model with the ’t Hooft interactionincluded. We recall the NJL grand canonical potentialwritten in terms of the strengths of the scalar, vector, andquark mixture channels, namely, Gs, GV , and K, respec-tively. The implementation of the infrared quark-gluondynamics is performed by making these couplings depen-dent on the traced Polyakov loop as follows, Gs→GsðGs;ΦÞ¼Gsð1−Φ2Þ, GV →GVðGV;ΦÞ¼GVð1−Φ2Þ,and K → KðK;ΦÞ ¼ Kð1 −Φ2Þ. The motivation forthis procedure is to ensure vanishing quark interactionsat Φ ¼ 1 (deconfined phase). By using these new functionsGsðK;ΦÞ, GVðK;ΦÞ, andKðK;ΦÞ, it is possible to define anew Polyakov potential now with strangeness included inthe 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 ofΦ at a fixed (common) quark chemical potential. In thisinvestigation, we focus on the case of symmetric quarkmatter, namely, the system presenting μu ¼ μd ¼ μs. Thesame study applied to other values of μ generates a typicalfirst-order phase transition for the order parameter Φ as afunction of μ. Therefore, the system presents two phasetransitions, namely, the one defined by Φ, and the otherrelated to the broken/restored chiral symmetry phasetransition, in which the quark condensates are the orderparameters. This last one is already presented in the originalNJL model. The region between these two transitions canbe identified, as in the SU(2) case, as the quarkyonic phase,defined here as the phase in which the light quarks presentrestored chiral symmetry but are still confined (Φ ¼ 0).Another interesting feature of the SU(3) version of the

PNJL0 model is the strong reduction of the constituentstrange quark mass (Ms) at the deconfinement phasetransition. We verify that Ms undergoes a reduction of

almost 50% for the parametrization in which GV ¼ 0.25Gsand a3 ¼ −0.1. The dynamics of the traced Polyakov loopat T ¼ 0 favors the restoration of chiral symmetry evenfor the strange quark. This effect is not present in theoriginal SU(3) NJL model since Φ is always vanishing inthis case.We also implement the chemical equilibrium and charge

neutrality in the SU(3) PNJL0 model in order to apply it inthe description of hybrid stars. For the hadronic side we usea density dependent model and match both equations ofstate exactly at the point where quarks are deconfined.Different PNJL0 parametrizations were constructed bychanging the values of GV and a3. We verify that theincreasing of both, GV and ja3j, increases the transitionpressure and the gap in the energy density of the hadron-quark phase transition. Finally, we generate the mass-radiusprofiles determined by the parametrizations of the PNJL0model (different GV and ja3j values) coupled to thehadronic model. Our results indicate that it is possible toconstruct hybrid stars compatible with some observationaldata. In particular, the recent ones proposed by the NICERmission [85–87].

ACKNOWLEDGMENTS

This work is a part of the project INCT-FNA proc.No. 464898/2014-5. It is also supported by ConselhoNacional de Desenvolvimento Cientfico e Tecnolgico(CNPq) under Grants No. 406958/2018-1, No. 312410/2020-4 (O. L.), No. 433369/2018-3 (M. D.), andNo. 308486/2015-3 (T. F.). We also acknowledgeFundação de Amparo a Pesquisa do Estado de SãoPaulo (FAPESP) under Thematic Project No. 2017/05660-0 (O. L., M. D., C. H. L, T. F) and GrantNo. 2020/05238-9 (O. L., M. D., C. H. L). O. A. M. thanksfor the fellowship provided by CNPq—Brazil.

[1] www.bnl.gov/rhic/.[2] https://home.cern/science/accelerators/large-hadron-

collider.[3] www.bnl.gov/eic/.[4] www.gsi.de/fair.[5] http://nica.jinr.ru/.[6] I. Bediaga and C. Göbel, Prog. Part. Nucl. Phys. 114,

103808 (2020).[7] S. Weinberg, Phys. Rev. Lett. 31, 494 (1973).[8] H. Fritzsch, M. Gell-Mann, and H. Leutwyler, Phys. Lett.

47B, 365 (1973).[9] K. Huang, Quarks, Leptons and Gauge Fields (World

Scientific, Singapore, 1992).

[10] J. B. Kogut, Rev. Mod. Phys. 51, 659 (1979).[11] J. B. Kogut, Rev. Mod. Phys. 55, 775 (1983).[12] H. J. Rothe, Lattice Gauge Theories: An Introduction,

3rd ed. (World Scientific, Singapore, 2005).[13] D. P. Landau and K. Binder, A Guide to Monte Carlo

Simulations in Statistical Physics (Cambridge UniversityPress, Cambridge, England, 2000).

[14] K. Splittorff and J. J. M. Verbaarschot, Phys. Rev. Lett. 98,031601 (2007); S. Muroya, A. Nakamura, C. Nonaka, andT. Takaishi, Prog. Theo. Phys. 110, 615 (2003).

[15] C. D. Roberts and A. G. Williams, Prog. Part. Nucl. Phys.33, 477 (1994); R. Alkofer and L. von Smekal, Phys. Rep.353, 281 (2001).

PNJL MODEL AT ZERO TEMPERATURE: THE THREE-FLAVOR … PHYS. REV. D 104, 116001 (2021)

116001-13

[16] A. Chodos, R. L. Jaffe, K. Johnson, C. B. Thorn, and V. F.Weisskopf, Phys. Rev. D 9, 3471 (1974); A. Chodos, R. L.Jaffe, K. Johnson, and C. B. Thorn, Phys. Rev. D 10, 2599(1974); T. DeGrand, R. L. Jaffe, K. Johnson, and J. Kiskis,Phys. Rev. D 12, 2060 (1975).

[17] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345 (1961).[18] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 124, 246 (1961).[19] M. Buballa, Phys. Rep. 407, 205 (2005).[20] U. Vogl and W. Weise, Prog. Part. Nucl. Phys. 27, 195

(1991).[21] S. P. Klevansky, Rev. Mod. Phys. 64, 649 (1992); T. Hatsuda

and T. Kunihiro, Phys. Rep. 247, 221 (1994).[22] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev.

106, 162 (1957).[23] D. J. Gross and F. Wilczek, Phys. Rev. Lett. 30, 1343

(1973); H. D. Politzer, Phys. Rev. Lett. 30, 1346 (1973).[24] K. Fukushima, Phys. Lett. B 591, 277 (2004).[25] O. A. Mattos, O. Lourenço, and T. Frederico, J. Phys. Conf.

Ser. 1291, 012031 (2019).[26] O. A. Mattos, T. Frederico, and O. Lourenço, Eur. Phys. J. C

81, 24 (2021).[27] G. ’t Hooft, Nucl. Phys. B72, 461 (1974); Phys. Rev. D 14,

3432 (1976); Phys. Rep. 142, 357 (1986); Phys. Rev. Lett.37, 8 (1976).

[28] S. Klimt, M. Lutz, U. Vogl, and W. Weise, Nucl. Phys.A516, 429 (1990).

[29] M. Kobayashi, H. Kondo, and T. Maskawa, Prog. Theor.Phys. 49, 634 (1973); M. Kobayashi and T. Maskawa, Prog.Theor. Phys. 44, 1422 (1970).

[30] S. S. Avancini, R. L. S. Farias, M. B. Pinto, W. R. Tavares,and V. S. Timóteo, Phys. Lett. B 767, 247 (2017).

[31] R. L. S. Farias, V. S. Timóteo, S. S. Avancini, M. B. Pinto,and G. Krein, Eur. Phys. J. A 53, 101 (2017).

[32] R. L. S. Farias, K. P. Gomes, G. Krein, and M. B. Pinto,Phys. Rev. C 90, 025203 (2014).

[33] S. Rößner, C. Ratti, and W. Weise, Phys. Rev. D 75, 034007(2007).

[34] S. Rößner, T. Hell, C. Ratti, and W. Weise, Nucl. Phys.A814, 118 (2008).

[35] N. Bratovic, T. Hatsuda, and W. Weise, Phys. Lett. B 719,131 (2013).

[36] S. Rößner, Phases of QCD, Ph.D thesis, Technische Uni-versität München, 2009.

[37] V. A. Dexheimer and S. Schramm, Phys. Rev. C 81, 045201(2010); Nucl. Phys. A827, 579c (2009).

[38] V. Dexheimer, R. O. Gomes, T. Klähn, S. Han, and M.Salinas, Phys. Rev. C 103, 025808 (2021).

[39] J. Steinheimer, V. Dexheimer, M. Bleicher, H. Petersen, S.Schramm, and H. Stöcker, Phys. Rev. C 81, 044913 (2010).

[40] H. Hansen, W.M. Alberico, A. Beraudo, A. Molinari, M.Nardi, and C. Ratti, Phys. Rev. D 75, 065004 (2007).

[41] C. Ratti, M. A. Thaler, and W. Weise, Phys. Rev. D 73,014019 (2006).

[42] C. Ratti, S. Rößner, M. A. Thaler, and W. Weise, Eur. Phys.J. C 49, 213 (2007).

[43] S. Rößner, Field theoretical modelling of the QCD phasediagram, Diploma thesis, Technische Universität München,2006.

[44] T. E. Restrepo, J. C. Macias, M. B. Pinto, and G. N. Ferrari,Phys. Rev. D 91, 065017 (2015).

[45] M. Kitazawa, T. Koide, T, Kunihiro, and Y. Nemoto, Prog.Theor. Phys. 108, 929 (2002).

[46] K. Fukushima, Phys. Rev. D 77, 114028 (2008).[47] P. Rehberg, S. P. Klevansky, and J. Hüfner, Phys. Rev. C 53,

410 (1996).[48] I. N. Mishustin, L. M. Satarov, H. Stöcker, and W. Greiner,

Phys. Rev. C 62, 034901 (2000).[49] K. Schertler, S. Leupold, and J. Schaffner-Bielich, Phys.

Rev. C 60, 025801 (1999).[50] M. Lutz, S. Klimt, and W. Weise, Nucl. Phys. A542, 521

(1992).[51] T. Schafer, E. V. Shuryak, and J. J. M. Verbaarschot, Phys.

Rev. D 51, 1267 (1995); T. Schafer and E. V. Shuryak, Rev.Mod. Phys. 70, 323 (1998).

[52] T. Hell, K. Kashiwa, and W. Weise, J. Mod. Phys. 4, 31915(2013).

[53] C. Ratti, M. A. Thaler, and W. Weise, Phys. Rev. D 73,014019 (2006).

[54] O. Lourenço, M. Dutra, and D. P. Menezes, Phys. Rev. C 95,065212 (2017); O. Lourenço, B. M. Santos, M. Dutra, andA. Delfino, Phys. Rev. C 94, 045207 (2016); O. Lourenço,M. Dutra, C. H. Lenzi, S. K. Biswal, M. Bhuyan, and D. P.Menezes, Eur. Phys. J. A 56, 32 (2020).

[55] O. Ivanytskyi, M. A. Perez-García, V. Sagun, and C.Albertus, Phys. Rev. D 100, 103020 (2019).

[56] H. B. Callen, Thermodynamics and an Introduction toThermostatistics, 2nd ed. (John Wiley & Sons Inc., NewYork, 1985).

[57] M. Asakawa and K. Yazaki, Nucl. Phys. A504, 668 (1989).[58] J. Theis, G. Graebner, G. Buchwald, J. Maruhn, W. Greiner,

H. Stöcker, and J. Polonyi, Phys. Rev. D 28, 2286 (1983).[59] H. Abuki, R. Anglani, R.Gatto, G.Nardulli, and M.

Ruggieri, Phys. Rev. D 78, 034034 (2008).[60] L. McLerran, K. Redlich, and C. Sasaki, Nucl. Phys. A824,

86 (2009).[61] F. Buisseret and G. Lacroix, Phys. Rev. D 85, 016009

(2012).[62] L. McLerran and R. D. Pisarski, Nucl. Phys. A796, 83

(2007).[63] Y. Hidaka, L. McLerran, and R. D. Pisarski, Nucl. Phys.

A808, 117 (2008).[64] I. F. Ranea-Sandoval, M. G. Orsaria, D. Curin, M. Mariani,

G. A.Contrera, andO. M.Guilera, Symmetry11, 425 (2019).[65] M. Hanauske, L. M. Satarov, I. N. Mishustin, H. Stöcker,

and W. Greiner, Phys. Rev. D 64, 043005 (2001).[66] D. P. Menezes, C. Providência, and D. B. Melrose, J. Phys.

G 32, 1081 (2006).[67] C. H. Lenzi, A. S. Schneider, C. Providência, and R. M.

Marinho, Jr., Phys. Rev. C 82, 015809 (2010).[68] A. Drago and G. Pagliara, Phys. Rev. D 102, 063003 (2020).[69] R. C. Pereira, P. Costa, and C. Providência, Phys. Rev. D 94,

094001 (2016).[70] M. Ferreira, R. C. Pereira, and C. Providência, Phys. Rev. D

102, 083030 (2020).[71] E. Annala, T. Gorda, A. Kurkela, J. Nättilä, and A.

Vuorinen, Nat. Phys. 16, 907 (2020).[72] J. D. Walecka, Ann. Phys. (N.Y.) 83, 491 (1974).[73] T. Gaitanos, M. Di Toro, S. Typel, V. Baran, C. Fuchs, V.

Greco, and H. H. Wolter, Nucl. Phys. A732, 24 (2004); S. S.Avancini, L. Brito, J. R. Marinelli, D. P. Menezes, M. M.W.

O. A. MATTOS et al. PHYS. REV. D 104, 116001 (2021)

116001-14

de Moraes, C. Providência, and A. M. Santos, Phys. Rev. C79, 035804 (2009); M. Fortin, C. Providência, Ad. R.Raduta, F. Gulminelli, J. L. Zdunik, P. Haensel, and M.Bejger, Phys. Rev. C 94, 035804 (2016).

[74] M. Dutra, O. Lourenço, S. S. Avancini, B. V. Carlson, A.Delfino, D. P. Menezes, C. Providência, S. Typel, and J. R.Stone, Phys. Rev. C 90, 055203 (2014).

[75] M. Dutra, O. Lourenço, and D. P. Menezes, Phys. Rev. C 93,025806 (2016); 94, 049901(E) (2016).

[76] N. Yasutake, G. F. Burgio, and H. J. Schulze, Phys. At.Nucl. 74, 1502 (2011).

[77] B. D. Serot and J. D. Walecka, Adv. Nucl. Phys. 16, 1 (1986).[78] R. C. Tolman, Phys. Rev. 55, 364 (1939).[79] J. R. Oppenheimer and G. M. Volkoff, Phys. Rev. 55, 374

(1939).[80] N. K. Glendenning, Compact Stars, 2nd ed. (Springer, New

York, 2000).[81] G. Baym, C. Pethick, and P. Sutherland, Astrophys. J. 170,

299 (1971).[82] P. B. Demorest, T. Pennucci, S.M. Ransom,M. S. E. Roberts,

and J.W. T. Hessels, Nature (London) 467, 1081 (2010).[83] J. Antoniadis, P. C. C. Freire, N. Wex, T. Tauris et al.,

Science 340, 6131 (2013).[84] H. T. Cromartie, E. Fonseca, S. M. Ransom, P. Demorest

et al., Nat. Astron. 4, 72 (2020).

[85] M. C. Miller et al., Astrophys. J. Lett. 918, L28 (2021).[86] T. E. Riley et al., Astrophys. J. Lett. 918, L27 (2021).[87] G. Raaijmakers et al., Astrophys. J. Lett. 918, L29 (2021).[88] B. K. Harrison, K. S. Thorne, M. Wakano, and J. A.

Wheeler, Gravitation Theory and Gravitational Collapse(Chicago Press University, Chicago, IL, 1965).

[89] D.W. Meltzer and K. S. Thorne, Astrophys. J. 145, 514(1966).

[90] D. Gondek, P. Haensel, and J. L. Zdunik, Astron. Astrophys.325, 217 (1997).

[91] A. Parisi, C. V. Flores, C. H. Lenzi, Chian-Shu Chena, andG. Lugones, J. Cosmol. Astropart. Phys. 06 (2021) 042.

[92] J. P. Pereira, C. V. Flores, and G. Lugones, Astrophys. J.860, 12 (2018).

[93] M. Mariani, M. G. Orsaria, I. F. Ranea-Sandoval,and G. Lugones, Mon. Not. R. Astron. Soc. 489, 4261(2019).

[94] J. D. V. Arbañil and M. Malheiro, Phys. Rev. D 92, 084009(2015).

[95] Ting-Ting Sun, Zi-Yue Zheng, Huan Chen, G. F. Burgio,and H.-J. Schulze, Phys. Rev. D 103, 103003 (2021).

[96] J. C. Jimenez and E. S. Fraga, Phys. Rev. D 100, 114041(2019).

[97] P. Haensel, J. L. Zdunik, and R. Schaeffer, Astron.Astrophys. 217, 137 (1989).

PNJL MODEL AT ZERO TEMPERATURE: THE THREE-FLAVOR … PHYS. REV. D 104, 116001 (2021)

116001-15