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Compact stars with a quark core within NJL model C. H. Lenzi1,2, A. S. Schneider3, C. Providência2, R. M. Marinho Jr.1 1Departamento de F´ısica, Instituto Tecnolo´gico de Aeronáutica, Campo Montenegro, São José dos Campos, SP, 12228-900, Brazil 2Centro de F´ısica Computacional, Department of Physics, University of Coimbra, Rua Larga, Coimbra, 3004-516, Portugal 3Department of Physics, Indiana University, Swain Hall West 117, 727 East Third Street Bloomington, Indiana 47405 (Dated: January 18, 2010) AnultravioletcutoffdependentonthechemicalpotentialasproposedbyCasalbuoniet alisused in the su(3) Nambu-Jona-Lasinio model. The model is applied to the description of stellar quark matterandcompactstars. Itisshownthatwithanewcutoffparametrizationitispossibletoobtain 0 stable hybrid stars with a quark core. A larger cutoff at finite densities leads to a partial chiral 1 symmetryrestorationofquarksatlowerdensities. Adirectconsequenceistheonsetofthesquark 0 in stellar matter at lower densities and a softening of theequation of state. 2 PACSnumbers: n a J I. INTRODUCTION sible patterns for color superconductivity [14] have been 8 conjectured (for a review see e.g. [15, 16] and references 1 therein quoted). Very recently, a new phase of QCD, Compactstarsarecomplexsystemswhichmaycontain named quarkyonic phase, characterized by chiral sym- ] exotic matter such as hyperons, kaon condensation, a h metry and confinement has been predicted [17]. We will non-homogenous mixed quark-hadron phase or, in their t not consider these phases in the present work. - core, a pure quark phase [1, 2]. l c The hadronic phase has been successfully described u within a relativistic mean-field theory with the inclu- n sion of hyperons (for a review see [1]). The quark phase [ hasfrequentlybeen describedbythe schematicMIT bag It was shown in [18] that at very large densities the 1 model [3, 4] orby the Nambu-Jona-Lasinio(NJL) model standardNJL model is notable to reproducethe correct v [5, 6]. The NJL model contains some of the basic sym- QCD behavior of the gap parameter in the quark color 9 metries of QCD, namely chiral symmetry. It has been flavor locked (CFL) phase. In order to solve this prob- 6 very successful in describing the vacuum properties of lem Casalbuoni et al have introduced a ultravioletcutoff 1 3 low lying mesons and predicts at sufficiently high densi- dependent on the baryonic chemical potential [18]. The . ties/temperaturesaphasetransitionto achiralsymmet- dependence of a parameterof the modelon the chemical 1 ric state [7–10]. However, it is just an effective theory potential changes the thermodynamics of the model and 0 0 that does not take into account quark confinement. has to be dealt with care [19]. Within su(2) NJL Baldo 1 Theauthorsof[11]havestudiedthe possibleexistence etalhaveinvestigatedwhetheracutoffdependentonthe : ofdeconfinedquarkmatterintheinteriorofneutronstars chemical potential could solve the problem of star insta- v bility with the onset of the quark phase and concluded i using the NJL model to describe the quark phase and X that this was not a solution [20]. The question that may could show that within this model typical neutron stars r do notpossessanydeconfinedquarkmatter intheir cen- be raisedis whether within the su(3)NJL a differentbe- a havior occurs due to the a different behavior of the s ter. It was shown that as soon as quark matter appears quark constituent mass with density. We will show that thestarbecomesunstableandcollapsesintoablack-hole. a larger cutoff at finite baryonic densities will move the Itwasalsopointedoutthatthelargeconstituentstrange onset of the s quark to smaller densities due to a faster quarkmassobtainedwithNJLoverawide rangeofden- decrease of the s quark constituent mass with density. sitieswasthecauseofthisbehavior. In[12]itwasshown that for warm neutrino free stellar matter a small quark core could appear at finite temperature. The reason can be traced back to a faster reduction of the s quark con- stituentmassatlowdensitiesand,therefore,theonsetof After a review of the standard su(3) NJL model we the s quark at lower densities. For warm stellar matter will introduce in section II the parametrization of the with an entropy per particle equal or below 2, with or cutoff dependence on the chemical potential, the ther- without trapped neutrinos no quark core was obtained modynamic consistency of the modified model and the [13]. β-equilibrium conditions. In section III we discuss the Over the last decade, it has been realized that strong star stability and the dependence of the maximum mass interacting matter at high density and low temperature configuration on the cutoff. In the last section we draw may possess a large assortment of phases. Different pos- some conclusions. 2 II. THE MODIFIED SU(3) where i=u, j =d and k =s and cyclic permutations. NAMBU-JONA-LASINIO MODEL As shown by authors in [9, 11, 12] we calculate an effective dynamical bag pressure: A. Standard su(3) NJL model B =B −B, (7) eff 0 To describe quark matter phase in neutron star, we where B is given by, use the su(3) NJL model with scalar-pseudoscalar and ’tHooftsix fermioninteraction. TheLagrangiandensity Λ0 p2dp of NJL model is defined by [12]: B =ηN p2+M2− p2+m2 c (2π)2 i 0i 8 Xi Z0 (cid:18)q q (cid:19) L=ψ¯(iγµ∂ +mˆ )ψ+g ψ¯λaψ 2+ ψ¯iγ λaψ 2 −2g hψ¯ψi2−4g hu¯uihd¯dihs¯si, (8) µ 0 s 5 s i t Xa=0h(cid:0) (cid:1) (cid:0) (cid:1) i Xi +g det ψ¯ (1+γ )ψ +det ψ¯ (1−γ )ψ , (1) t i 5 j i 5 j and B =B is a constant. 0 ρu=ρd=ρs=0 where, i(cid:8)n fla(cid:2)vor space, ψ =(cid:3) (u;d;(cid:2)s) denotes the(cid:3)(cid:9)quark Inthisworkweconsiderthefollowingsetofparameters fields and the λa matrices are generators of the u(3) al- [8, 21]: Λ = 631.4 MeV, g Λ2 = 1.829, g Λ5 = −9.4, 0 s t 0 gebra. The term mˆ =diag(m ,m ,m ) is the quark m = m = 5.6 MeV, and m = 135.6 MeV. This 0 0u 0d 0s 0u 0d 0s current mass, which explicitly breaks the chiral symme- set of parameters was chosen in order to fit the vacuum try of the Lagrangian, and g and g are coupling con- values for the pion mass, the pion decay constant, the s t stants of the model and have dimensions of mass−2 and kaon mass, the kaon decay constant and the quark con- mass−5, respectively. densates: m = 139 MeV, f = 93.0 MeV, m = 495.7 π π k The thermodynamic potential density Ω for a given MeV, f = 98.9 MeV, φ = φ = (−246.7MeV)3, and k vd vu baryonic chemical potential µ, at T =0, is given by φ =(−266.9MeV)3. vs Ω=E − µ ρ (2) i i Xi B. Chemical potential-dependent cutoff Λ(µ) where the sum is over the quark flavors (i = u, d and s), µ and ρ are the chemical potential and the density, As proposed by R. Casalbuoni et al in [18] and M. i i respectively, for each quark flavor i and Baldoet alin[20],wewillintroduceachemicalpotential dependency in the NJL model cutoff. This dependence E =−ηN Λ0 d3p p2+m0iMi −2g hψ¯ψi2 implies that the vacuum constituent quark masses Mvi c (2π)3 E s i become chemical potential dependent and the same oc- i Zkfi i i X X curs to the coupling constants, g and g . −2g hu¯uihd¯dihs¯si−E , (3) s t t 0 Inordertoobtaintherenormalizedcouplingconstants we consider that the values of the quark condensates in istheenergydensity. Above,k =θ(µ −M ) µ2−M2 fi i i i i vacuum φ are known properties of the model: φ = is the Fermi momentum of the quark i, the constants vi vd p φ =(−246.7 MeV)3, and φ =(−266.9 MeV)3, η = 2 and N = 3 are the spin and color degeneracies, vu vs c respectively, and the constant E is included to ensure 0 Λ(µ) p2dpM (Λ(µ)) tlahraitzaΩtion=u0ltrianvitohleetvcauctuouffmt.o aTvhoeidΛd0ivteerrgmenicsesainretghue- −ηNcZ0 2π2 Eii(Λ(µ)) =φvi. (9) medium integrals, and it is taken as a parameter of the model. The quarkcondensatesanddensitiesaredefined, The new constituent quark masses Mvi(Λ(µ)) are solu- for each i=u,d,s, respectively, as tionsoftheseequations(Mvu =Mvd becauseφvu =φvd) and the coupling constants are solutions of the two gap Λ0 p2dpM Eqs.(6) φ =hψ¯ψi =−ηN i, (4) i i c 2π2 E Zkfi i M (Λ(µ)) =m −4g (Λ(µ))φ −2g (Λ(µ))φ φ vu 0u s vu t vd vs whereM istheconstituentmassofthequarkiandE = i i M (Λ(µ)) =m −4g (Λ(µ))φ −2g (Λ(µ))φ φ . vs 0s s vs t vd vu p2+M2, and i at µ = 0, for the constituent masses M (Λ(µ)) which p kfi p2dp vi ρ =hψ†ψi =ηN . (5) satisfy Eq.(9). i i cZ0 2π2 Inthis paperwemaketwochoicesforthe cutoffchem- icalpotentialdependency. We usethe cutoffproposedin Minimizing the thermodynamic potential with respect [20], to the constituent quark mass M results in three gap i equations, Λ , if µ≤µ 0 0 Λ = (10) M =m −4g φ −2g φ φ , (6) 1 9(µ−µ )2+Λ2, if µ>µ i 0i s i t j k (cid:26) 0 0 0 p 3 stabilizationatsomevalue(onthenextsectionsweshow 850 Λ (µ = 400 MeV) a) 2 b) the effect of the different values of the parameter a in 1 0 ourresults). Fig. 1a)showstheplotsofthetwodifferent Λ (µ = 350 MeV) 800 Λ1(µ0 = 400 MeV) 2Λg.s01 cthuetopffasrafomrettweroµd0iffdeoreenstnvoatlucheasnogfeµt0h.eFroartethofegcruotwoffthΛo2f 2 0 the cutoff, however it changes the region on the chem- Λ (µ = 350 MeV) Mev)750 2 0 0350 40µ0 (Mev4)50 500 iscaamlepfiotgeunrteiawleracnangesewehaelrseotthhee binechraevaisoeroocfctuhres.coIunpltinhge Λ ( constants gs(Λ(µ)) [Fig. 1b)] and gt(Λ(µ)) [Fig. 1c)] −10 c) as a function of the chemical potential. As discussed in 700 [18,20]thecouplingconstantsdecreasewiththeincrease 50 of cutoffs in both cases. Λ−15 g.t 650 3 50 400 450 500 −20350 400 450 500 C. The thermodynamic consistency µ (Mev) µ (Mev) Thechemicaldependentcutoffintroducedinthesu(3) FIG. 1: Dependence on the chemical potential of a) the dif- NJL model gives rise to some modifications in the ther- ferentparametrizationsofthecutoffsdiscussedinthetext;b) modynamics of the system. The baryon thermodynamic thegs and c) gt coupling constants. potential is rewritten as Ω (k ,Λ(µ))=E(k ,Λ(µ))− µ ρ +b(Λ,k ), (13) b f f i i f wherethetermµ isthevalueofchemicalpotentialabove 0 i=u,d,s X which the cutoff becomes a function of the chemical potential. We also propose a new cutoff wherethetermb(Λ,kf)isintroducedinordertomaintain thermodynamical consistency 1 Λ =Λ +aΛ δ − , (11) ∂Ω 2 0 0 0 1+exp µ−µ0 ! ρi =− . (14) b ∂µ i where the constant a determines the(cid:0)maxi(cid:1)mum value of Tocalculatethe the functionbweusethe prescriptionof thecutoffandbhowfastthecutoffincreaseswithdensity. Gorenstein and Yang [19] and we obtain: The constant δ is given by 0 ηN Λ 1 b(Λ,k )= c p2 p2+M2 dp δ0 = 1+exp µc−bµ0 , (12) f 2π2 i=Xu,d,sZΛ0 q i and ensures that Λ2(µc) = (cid:0)Λ0. T(cid:1)he constant µc is +2 Λ φ2∂gs +φ φ φ ∂gt dp, (15) the chemical potential value for which the first drop of  i ∂Λ u d s∂Λ quark matter appears. P. Costa et al have calculated ZΛ0 i=Xu,d,s   this parameter for each of the quarks: µ ≃ 312 MeV u sothatthecondition(∂Ω/∂Λ) =0. InFig. 2weshow T,µ µ = µ ≃ 365 MeV [22, 23] for the parameters chosen d s that the thermodynamic condition defined in Eq.(14) is forthispaper. Weassumeµ =347MeV,approximately c satisfied when we include the quantity b(Λ,k ) in the f equal to (µ + µ + µ )/3. In order to keep the vac- d u s thermodynamic potential. uum properties the chemical potential dependence is introduced only for µ≥µ . The parameter µ determines c 0 the chemical potential range where the fastest increase D. β-equilibrium condition of the cutoff occurs. Λ is one of the choices considered in [20] which was 1 In the present section we build the equation of state adjustedinordertokeepthe curveproposedin[18]with (EOS) of strange quark stellar matter. We must impose Λ ≃ 580 MeV and µ = 400 MeV on a range of chemi- 0 0 bothβ-equilibriumandelectricchargeneutrality[1]. We cal potential between (400−600) MeV. In present work will consider cold matter, after the neutrinos have dif- we are not concerned with keeping the curve proposed fused out and the neutrino chemical potential is zero. in[18],thereforewetakedifferentvaluesforµ (between 0 Forβ-equilibriummatterweaddthe leptoncontribution 347−400 MeV for both cutoffs) in order to verify the to the thermodynamic potential, effect of this parameter on the results obtained with the model. The cutoff Λ2 of Eq.(11) is a particular choice Ω(kfi,Λ(µ))=Ωb+Ωl, (16) where the numerical coefficients a = 0.17 and b = 0.005 areadjustedinordertoobtainafastincreaseto thecut- where Ω =E (k )− µ ρ is the leptonic contribution l l fl l l l off on a small interval of the chemical potential and a takenas thatofa free Fermigasofelectronsandmuons. P 4 a plateau around B1/4 =161−163 MeV between 3ρ − 0 1.6 5ρ in the standard NJL model. The plateau is due to −∂Ω/∂µ −− Λ 0 1 the partial chiral symmetry restoration of quarks u and 1.4 ρ −− Λ µ0 = 400 MeV d. For the new parametrizations of the cutoff the value 1 1/4 3) −∂Ω/∂µ −− Λ Beff ∼ 162 MeV will occur at lower densities and the −m1.2 2 plateaudisappearsinallcases. Thiseffectoccursbecause ρ) (f ρ −− Λ2 thechemicalpotentialdependenceofthecutoffstartsfor µ , 1 arescthoermatiicoanl fpoortethnteiaqlubarekloswutahnedpda.rtWiael hchavirealmsayrmkemdetthrye ∂ Ω/0.8 onsetofthestrangequarkontheeffectivebagcurvewith ∂ − vertical lines. The effective bag estabilizes much faster ( for Λ = Λ and tends to behave in a similar way to the 0.6 2 MIT bag model with the decrease of the µ . 0 In Fig. 4 we show the quark fractions, Y = ρ /(3ρ ) i i B 0.348 0 400 420 440 460 480 500 [Fig. 4a)], and the constituent masses of the u, d and s µ (MeV) quarksinβ-equilibrium[Fig. 4b)]. Theeffectofdifferent choicesofthecutoffonthequarksisclear: forthefaster FIG. 2: Plot of the thermodynamic condition, Eq. (14), for increaseofΛ(µ)andsmallervalues ofµ the appearance 0 thetwo parametrizations proposed (10,11). of the quark s occurs at lower densities and the its con- stituentmassM approachesthecurrentquarkmassm s 0s at lower densities too. The electron and muon densities are 1 a) ρl = 3π2kf3l. (17) eV)400 Strange M Inβ-equilibriumtheconditionsofchemicalequilibrium M (200 Up and charge neutrality are given by Down 0 2 4 6 8 10 ρ/ρ µs =µd =µu+µe, µe =µµ, (18) 0 ρ +ρ = 1(2ρ −ρ −ρ ). e µ 3 u d s 0.6 Down b) Up 0.4 Yi 0.2 Strange 200 0 2 4 6 8 10 ρ/ρ 0 180 FIG. 4: Quark stellar matter in β-equilibrium with different 1/4eff160 Λ1(µ0 = 400 MeV) cohfotihceesquofartkhseuc,udto,ffs:aas)afrfuacntcitoinosnaonfddebn)sictoyn.stituent masses B Λ (µ = 350 MeV) 1 0 140 Λ (µ = 400 MeV) As can be seen from Figs. 3 and 4 the different slopes 2 0 in the µ dependent cutoff and the different values of the 120 Λ2(µ0 = 350 MeV) chemical potential µ0 change the behavior of the model, namely the constituent quark masses and baryonic den- NJL sity. The EOS becomes softer with these modifications, 100 2 4 6 8 10 as seen in Fig. 5, where the pressure is displayed as a ρ/ρ (Mev) 0 function of the chemical potential for the standard NJL model and the different choices of the cutoff. The cutoff FIG.3: EffectivebagpressuredefinedinEq. (7)fordifferent dependence proposed in Eq. (11) gives the softest EOS. parametrizations of the cutoff and stellar quark matter in β- We also note that decreasing µ favors a deconfinement 0 equilibrium. phase transitionat lowerdensities for both parametrizations of the cutoff, Λ andΛ . In the same figure we 1 2 In Fig. 3 we plot the effective bag pressure, Eq.(8), plot the EOS of the hadronic phase (dotted curve). The for each choice of the cutoff and for quark matter in β- crossing point between the hadronic and the quark EOS equilibrium as a function of the baryonic density ρ = indicatesthephasetransitionfromthehadronicphaseto B (ρ +ρ +ρ )/3. As shownin references [11, 12] there is the quark phase using a Maxwell construction [24]. The u d s 5 EOS of the quark phase constructed with standard NJL whileithasbeenshownin[26]thattheMaxwelldescrip- model does not cross the EOS of the hadronic phase on tion of the mixed phase gives a good description if the the chemical potential range shown. surface tension is very large. The inclusion of a cutoff dependent on the chemical potentialinthe NJL modelinfluences the deconfinement phase transition and, consequently, the stability of the 350 GM1 with hyp. + BPS Q star, as we will show in the next section. 300 Λ (µ = 400 MeV) 1 0 Λ (µ = 350 MeV) 250 1 0 500 NΛ1J(Lµ0 = 400 MeV) −3V.fm)200 ΛΛ2((µµ0 == 430500 MMeeVV)) 400 Λ (µ = 350 MeV) Me150 2 0 −3m)300 Λ12(µ00 = 400 MeV) P (100 M V.f Λ2(µ0 = 350 MeV) 50 e M200 GM1 with hyperons H P ( 0 0 2 4 6 8 10 ρ/ρ 0 100 FIG. 6: EOS of hybrid stellar matter: Maxwell construction for a first order phase transition. Pressure as a function of 0 350 400 450 500 thebaryondensityfordifferentparametrizationsofthecutoff. µ (MeV) The hadronic, mixed and quark phases are identified respectively with a H,M and Q label. FIG. 5: Pressure as a function of the chemical potential for quarkstellarmatterinβ-equilibriumwithdifferentchoicesof For the hadronic sector we use a EOS proposed by thecutoffandstandardNJLmodel. ThehadronicEOS,GM1 Glendenning and Moszkowsky (GM1) [27] with the in- with hyperons[27], is also included (dotted curve). clusionofthe baryonicoctet. Inordertofix the hyperon couplingconstantswe haveusedone ofchoicesdiscussed in literature [1, 27], namely we took for all the hyperons the same coupling constants which are a fraction III. THE NEUTRON STAR STABILITY x of the meson-nucleon coupling constants, x = 0.7, i σ x = 0.783 = x . For low densities (near zero density) ω ρ In this section we investigate the properties of stars we use the Baym, Pethick and Sutherland (BPS) model constructed using the modified su(3) NJL model. The [28]. The standard and the modified su(3) NJL models Maxwell construction [24] is considered for the phase are used to describe quark matter phase. In Fig. 6 we transitionfromthehadronicphasetothequarkphase. In plot the pressure as a function of the baryonic density thiscasethephasetransitionisidentifiedbythecrossing for the complete EOS discussed above. The plateaus, point between the hadronic and the quark EOS in the identified with an M, represent the deconfinement phase pressure versus baryonic chemical potential plane. At transition as a consequence of the first order Maxwell lower densities (below the transition point) an hadronic construction. In the case of cutoff Λ and lower values 1 phase is favoredand at higher densities (above the tran- of µ the transition from hadron to quark phase occurs 0 sition chemical potential) quark matter is favored. atlowervaluesofthepressureandtheplateaudecreases. However, we should point out that the Maxwell con- The same situation occurs with cutoff the Λ2. structionisanapproximationforwhichonlybaryonnum- The presence of strangeness in the core and crust of berconservationisconsideredanddoesnottakecorrectly the star can have an important influence in the stability into account the existence of two charge conserving con- of the star [29, 30]. We have calculated the strangeness ditions,norsurfaceeffectsandtheCoulombfield[25,26]. content of the EOS for the different parametrizations of Instead, we could have considered a Gibbs construction the cutoff. In Fig. 7 we plot the strangeness fraction [1], which takes into account the existence of two charge given by conserving conditions. However, a complete treatment ρ ofthe mixed phase requiresthe knowledgeofthe surface rQS = s, s 3ρ tension between the two phases which is not well estab- lished and may have a value between 10-100 MeV/fm2 for the quark phase and [26]. The Gibbs construction gives results close to the ones obtained with the lower value of the above surface |qB|ρ rQS = B s B, tensionrange,andisrecoveredforazerosurfacetension, s 3ρ P 6 forthehadronicphase. ThetermqB isthestrangecharge baryon B. s 0.3 a) b) The strangeness fraction is strongly modified by the differentchoicesofthe cutoff. AswecanseeinFig. 7,in the case of Λ with µ =400 MeV, the strangeness frac- 1 0 tion decreases in the mixed phase and increases again 0.2 in the pure quark matter. However,with µ =350 MeV 0 thestrangenessfractionincreasesinthemixedphaseand rs continues to increase in the pure quark matter. For the caseofΛ2thestrangenessfractionincreasesinbothcases 0.1 in the mixed phase. These differentbehaviorsare due to GM1 with hyperons GM1 with hyperons the densities at which the mixed phase occurs and the Λ(µ = 400 MeV) Λ(µ = 400 MeV) 1 0 2 0 values of the constituent masses of the strange quark for Λ(µ = 350 MeV) Λ(µ = 350 MeV) these densities. The Table I shows the values of the con- 1 0 2 0 0 stituentmassesofstrangequarkonthemixedphase,the 1 3 5ρ/ρ 7 9 1 3 5ρ/ρ 7 9 value of the densities at the onset of the phase transi- 0 0 tion, the width of the plateau of the mixed phase and the difference of the strangeness fraction between quark FIG. 7: Strangeness fraction rs for different slopes of the and hadronic phase. cutoff for thequark phase: a) Λ1 b) Λ2. TABLEI:Constituentmasses of thestrangequark,densities 1.7 at the onset of the phase transition, width of the plateau of 1.6 a) b) the mixed phases and difference of the strangeness fraction 1.6 between quark and hadronic phase for each parametrization 1.4 of the cutoff. 1.2 Quark phase 1.5 Cutoff (MMesV) (fρmQ−P3) (fm∆−ρ3) (×∆10r−s2) Msun1 Msun1.4 Λ1(µ0 =400MeV) 229.16 1.00 0.32 -1.2 M/0.8 M/ Λ1(µ0 =350MeV) 225.23 0.82 0.23 2.9 1.3 Λ (µ = 400 MeV) Hadron phase 1 0 Λ2(µ0 =400MeV) 197.27 0.69 0.23 8.2 0.6 Λ (µ = 350 MeV) Λ2(µ0 =350MeV) 240.61 0.49 0.14 10.1 1.2 1 0 0.4 Λ (µ = 400 MeV) 2 0 1.1 0.2 Λ (µ = 350 MeV) The values of the density at the onset of the phase 2 0 transitionandthewidthoftheplateaudecreasewhenµ 0 1 decreasesforbothparametrizationsofthecutoff. Onthe0 10 R (km12) 14 1 3 ρc5/ρ0 7 9 other hand, the discontinuity of the strangeness fraction between the two phases becomes positive and increases. FIG. 8: The gravitational mass of the hybrid star is plotted Thevalueoftheconstituentmassofthesquarkgenerally as afunction of a) thestar radiusand b) thecentral density, decreases if µ decreases except for Λ with µ = 350 for the different parametrizations of thecutoff. 0 2 0 MeV that has the biggest constituent mass due to the lowdensityatthephasetransitiondensity. Accordingto references [11, 12] these results are directly relate with plateau in the mass versus central density graph. The the possible existence of deconfined quark matter in the plateauisaconsequenceoftheMaxwellconstructionand interior of neutron star as we will see later corresponds to the mixed phase between a pure hadron We calculate the neutron star configuration for each and a pure quark phase. The cusp occurs at the on- cutoff solving the Tolman-Oppenheimer-Volkoff (TOV) set of the quark phase in the interior of the star. We equations for a spherically symmetric and static star[31, conclude that for most of the models the deconfinement 32]. Fig. 8showsthegravitationalmassofhybridstarsof phase transition makes the star unstable. However, for themaximummassconfigurationasafunctionof(a)the the cutoff Λ with µ = 350 MeV the maximum mass 2 0 radius and of (b) the central density for each cutoff. As configurationappears after the plateau and the cusp. In wecanseeintheseplotsthemaximummassisinfluenced this case the star configuration with the maximum mass by the cutoff. In the Table II we show the values of the has a quark phase core. We verify that configurations gravitationalmass, central density and the radius of the with a quark phase core are possible only for the chemi- maximum mass star configuration constructed with the calpotentialµ .360MeV.Forvaluesofµ &360MeV 0 0 EOS proposed in the present work as well as with the we have two possibilities: 1) instability of the hybrid EOS proposed by Glendenning and Moszkowsky [1, 27]. starwiththeonsetofaquarkphaseinthestaraswecan The gravitational mass of the hybrid stars is charac- see inthe caseofµ =400MeV; 2)the EOSofhadronic 0 terized by a cusp in the mass versus radius plot and a phase is favored for all densities if µ >430 MeV. 0 7 Theµ rangewithastablequarkcorechangesdepend- 0 ing on the parameters used to Λ2. In Fig. (9) are shown 1.5 a) b) the different configurations for the different values of µ 1.5 0 in the case of Λ with a = 0.17. The value of maxi- 2 Quark phase mum mass decreases if µ0 increases. We have plotted 1.4 the mass of the maximum mass configuration as a func- 1 ttoionthµe0mininiFmigu.m1i0n. thTisheplloimt.itIonfFstiga.bil1it1ytchoerrdeisffpeornednts Msun Msun1.3 star configurations for different values of a and the same M/ Hadron phase M/ µ = 346.5 MeV µ = 347 MeV, are shown. We conclude that increasing 1.2 0 0 0.5 µ = 348 MeV adecreasesthemassofthemaximummassconfiguration 0 andthe correspondingradius,because the EOSbecomes 1.1 µ = 350 MeV 0 softer. µ = 357 MeV 0 It is seen from Table II that a smaller parameter µ 0 0 1 and a harder cutoff Λ gives rise to a smaller maximum 10 R (km12) 14 2 4 ρ/ρ6 8 10 mass. The occurrence of a quark core reduces a lot c the maximum mass but we can still get a reasonable FIG. 9: The gravitational mass of the hybrid star is plotted value, ∼1.45 M , which is consistent with the observed ⊙ as afunction of a) thestar radiusand b) thecentral density, maximum neutron star masses, except for the still not for different values of µ0 for Λ2 with a=0.17. confirmed, highly massive compact stars, the millisec- ond pulsars PSR B1516 + 02B [34], and PSR J1748- 2021B [35] with masses well above 2M . ⊙ 1.49 TABLE II: Maximum gravitational mass and radius of the 1.48 hybridstars(hs)andquarkstars(qs)obtainedwithdifferent parametrizationsofthecutoffΛandtwovaluesofthetransi- 1.47 tion chemical potential µ0. In the last line thevalues for the n maximum mass neutron star (ns) obtained with GM1 with Msu hyperons[27]. M indicates inside the mixed phase. /x1.46 a m Model/Cutoff Mmax R ǫi ǫf ǫc M (M⊙) (km) (fm−4) (fm−4) (fm−4) 1.45 Λ1(µ0 =400MeV) 1.701 11.14 5.97 8.26 M Λ1(µ0 =350MeV) 1.674 11.64 4.69 6.22 M 1.44 Λ2(µ0 =400MeV) 1.621 12.00 3.84 5.29 M Λ2(µ0 =350MeV) 1.456 10.56 2.60 3.42 7.62 1.43 GM1 with hyperons 1.705 11.11 5.99 348 350 352 354 356 358 360 362 µ (MeV) 0 Fig. 12showstheuandsquarkcondensatesasafunc- tionofthechemicalpotentialforbothcutoffparametriza- FIG. 10: Mass of the maximum mass star configuration as a tions. For the cutoff Λ the module of the condensates function of theparameter µ0 for Λ2 with a=0.17. 1 starts to increase for µ & 475 MeV. This is not the case Λ : the module of the condensates decreases with den- 2 sity leading the system to chiral symmetry restauration. of the consequences of increasing the cutoff is a faster Therefore, the cutoff proposed in this work is physically decrease the constituent s quark mass with density and, favored to cutoff Λ . We believe the behavior of Λ at therefore, the onset of s quark at lower densities, giving 1 1 highdensities is due to the highvalue ofthe cutoffΛ at rise to a larger pressure for the same chemical potential. 1 these densities. The phase transitionto a deconfinedquark phase occurs at smaller densities and pressures and the density discontinuity at the phase transition is smaller. For cutoff IV. SUMMARY Λ (a = 0.17), stars with a quark core are obtained for 2 a choice of the parameterµ <360MeV. The maximum 0 We have studied the possibility of formation of sta- mass of these stars is between (1.46-1.51) M⊙, and is ble compact stars with a quark core within the su(3) compatible with most of the compact star observations. NJL model with a chemical potential dependent ultra- However,thehighlymassivestarsPSRB1516+02B[34], violet cutoff. We use a su(3) NJL model parametriza- and PSR J1748-2021B [35], in case they are confirmed, tion which describes the vacuum properties of low mass would not reproduced. mesons (pions and kaons) and choose the parametriza- According to Baldo et al the instability of compact tion of the cutoff so that it increases with density. One stars within the NJL model is probably due to the lack 8 of confinement in this model [20], since the authors of 1.5 a) b) [36] were able to obtain stable stars with a quark core 1.5 introducingaconfiningpotentialinthe NJLmodel. The confiningpotentialinthe approachof[36]is switchedoff Quark phase 1.4 at the chiral phase transition. In [20] the authors have 1 triedtogetstable compactstarswitha quarkcoreusing Msun Msun1.3 spuo(t2e)ntNiaJlLamndodweelrweintohtasucuccteosffsfduelp.eUnsdinengttohnetshamecehdeempiecna-l Hadron phase M/ M/ dence of the chemicalpotential, but introducing also the 1.2 a=0.17 strangeflavorwewerealsonotabletoobtainstablecom- 0.5 a=0.21 pact stars with a quark core. However, when we use the 1.1 a=0.27 new cutoff proposed in this paper it is possible to get a=0.33 stable compactstarswith a quarkcoreifthe strangefla- vor is included. We believe the stability of core quark 0 1 10 12 14 2 4 6 8 10 occurs due to the fast increase of the cutoff Λ allowing R (km) ρ/ρ 2 0 forachiralsymmetryrestorationforthes-quarkatmuch lowerdensitiesthantheonespredictedbyaconstantcut- FIG.11: Thegravitational mass ofthehybridstarisplotted off. The result is an EOS soft enough to give rise to a as a function of a) thestar radiusand b) thecentral density, quark core stable in a hybrid star. The stabilization of for different valuesof a for Λ2 with µ0 =347 MeV. thecutoffathighdensitiesisanimportantcharacteristic because it repares divergence problems due to the fast increase of the cutoff. −20 The effect of color superconductivity was not considered in the present work and will be the subject of a −40 Down future work. NJL −60 Λ (µ = 350 MeV) V) 2 0 3 (Me −80 Λ1 (µ0 = 350 MeV) Acknowledgments 1/ φ −100 We would like to thank the fruitful discussions Strange with João da Provid¨ı¿1ncia, Pedro Costa and To- −120 2 bias Frederico. This work was partially supported by FEDER and Projects PTDC/FP/64707/2006 and −1403 50 375 400 425 450 475 500 525 CERN/FP/83505/2008, and by COMPSTAR, an ESF µ (MeV) Research Networking Programme. CHL thanks to CAPES by the fellowship 2071/07-0 and the interna- FIG. 12: Quark u and s condensates versus density for the tional cooperation program Capes-FCT between Brazil- different cutoff parametrizations. Portugal. [1] N. K. 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