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Physics Letters B 704 (2011) 343346t Scet/loE haGINaArReReAcAvEdKePrTrHaEqMsinint ebe tarkseniffethecooarttopeofofioplan(dtiolitstMmapnivepranticphmisdefr03doSince the matter in the core of compact stars are compresseddensities of several times of saturated nuclear density, it is ex-cted that new degrees of freedom will appear in the interiorthese objects [113]. The hadronquark phase transition is onethe most concerned topics in modern physics related to heavy-n collision experiment and compact star. Because of the com-ication of full calculation of Quantum Chromodynamics (QCD)d the lack of sucient knowledge about the nonpertubtive ande)connement effect, it is dicult to apply full QCD calcula-n to describe the phase transition in astrophysics. Therefore, ineratures the hadronquark phase transition related to compactar are usually described with the simplied, phenomenologicalIT bag model or the effective chiral NambuJona-Lasinio (NJL)odel (e.g., [79,11,14,1619]).The NJL model with chiral dynamics is a prominent one in theplication of astrophysics, but it lacks the connement mecha-sm, one essential characteristic of QCD. Recently, an improvedrsion of the NJL model coupled to Polyakov loop (PNJL) has beenoposed [20]. The PNJL model includes both the chiral dynamicd (de)connement effect, giving a good interpretation of lat-e QCD data [2127]. We have recently studied the hadronquarkase transition relevant to heavy-ion collision in the Hadron-PNJLodel [28]. The calculation shows the color (de)connement effectvery important for the hadronquark phase transition at nitensity and temperature, and improves greatly the results derivedom the Hadron-NJL model [29]. The PNJL model with a chemicalE-mail address: shaogy@pku.edu.cn.potential dependent Poyakov effective potential has also been usedto describe cold neutron star without [30] and with a color su-perconductivity structure [31]. In [32], Fischer et al. discussed theevolution of core collapse supernova with the PNJL model (withdiquark interaction and isoscalar vector interaction) and the pos-sibility of the onset of deconnement in core collapse supernovasimulation. In their study selected proton-to-baryon ratio Yp andthe Maxwell construction are taken for the PNJL model, whichgives a relatively narrow mixed phase than that with Gibbs con-struction. In [33], the authors studied the possibility to probe theQCD critical endpoint during the dynamical black hole formationfrom a gravitational collapse. Several two-avor quark models withdifferent parameters are used in their study, and the calculationshows the Critical Point location of QCD has a strong dependenceon quark models and parameters. A generalization with s quark isneeded to get more reliable results for further investigation.In this Letter we will rstly use the newly improved PNJL modelto describe the evolution of proto-neutron star from its birth withtrapped neutrinos to neutrino-free cold neutron star (NS). TheGibbs criteria will be used to determine the mixed phase un-der the isotropic constraint with and without trapped neutrinos.The emphases are put on the evolution of proto-neutron star witha hadronquark phase transition, the particle distributions alongbaryon density, the EOSs and massradius relations, as well as thestar stability in different snapshots during the evolution.A PNS forms after the gravitational collapse of the core of mas-sive star with the explosion of a supernova. At the beginning ofthe birth of a PNS, the entropy per baryon is about one (S 1)and the number of leptons per baryon with trapped neutrino isContents lists available aPhysics Lwww.elsevier.comvolution of proto-neutron stars with theuo-yun ShaoFN Laboratori Nazionali del Sud, Via S. Soa 62, I-95123 Catania, Italyr t i c l e i n f o a b s t r a c tticle history:ceived 11 July 2011ceived in revised form 11 August 2011cepted 9 September 2011ailable online 16 September 2011itor: W. Haxtonywords:oto-neutron starapped neutrinodronquark phase transitionuation of stateassradius relationThe PoyakovNambuJona-Ladynamics and (de)connemeuse the PNJL model to descristar (PNS) with a hadronqusnapshots of PNS proles, prethe massradius relations at dwhole evolving process, andin the heating stage. In theincreases. In this process a pcollapsing into a black hole.70-2693/$ see front matter 2011 Elsevier B.V. All rights reserved.i:10.1016/j.physletb.2011.09.030iVerse ScienceDirectters Bcate/physletbdronquark phase transitiono (PNJL) model was developed recently, which includes both the chiralffect and gives a good description of lattice QCD data. In this study wehe quark phase, and rst use it to study the evolution of proto-neutronphase transition. Along the line of a PNS evolution, we take severalting the fractions of different species, the equations of state (EOS), andrent stages. The calculation shows the mixed phase may exist during theonset density of quark phase decreases with the radiation of neutrinosling stage, the EOS of the mixed phase softens and the center densityof nuclear matter transforms to quark matter, which may lead to a PNS 2011 Elsevier B.V. All rights reserved.34 B 70apsetrneinconeneevdo1,2,AtimstonbyIntwdethinancaPNLwanartivSUsibecopritcoL(w38UwUbeicth10reququMwtiwAanAThthAlofsusoinKtiustefostwlibtiwbeel4 G.y. Shao / Physics Lettersproximate 0.4 (YLe = Ye + Ye 0.4). In the following 1020conds, neutrinos escape from the star. With the decrease of elec-on neutrino population, the star matter is heated by the diffusingutrinos, and the corresponding entropy density increases, reach-g to S 2 when Y e 0. Following the heating, the star beginsoling by radiating neutrino pairs of all avors, and nally a coldutrons forms [34,35].Along the line of a PNS evolution to the formation of a coldutron star, we take several snapshots to study how the starolves, especially with the appearance of quark degrees of free-m. The snapshots are taken with the following conditions (S =YLe = Ye + Ye = 0.4), (S = 1.5, YLe = Ye + Ye = 0.3), (S =Ye = 0) and (S = 0, Ye = 0), similar with that used in [34,35].each snapshot of a PNS evolution, we take an isentropic approx-ation with which the temperature has a radial gradient in thear. There are also studies related to the properties of PNS basedisothermal approximation (e.g., [15,1719]).For the star matter, the hadronic and quark phase are describedthe non-linear Walecka model and the PNJL model, respectively.the mixed phase between pure hadronic and quark matter, theo phases are connected to each other by the Gibbs conditionsduced from thermal, chemical and mechanical equilibriums. Fore hadron phase we use the Lagrangian given in [36] in which theteractions between nucleons are mediated by , , mesons,d the parameter set GM1 is used in the calculation. The detailsn be found in Refs. [36,5].For the quark phase, we take recently developed three-avorJL model with the Lagrangian densityq = q(i D m0)q + G8k=0[(qkq)2 + (qi5kq)2] K [det f (q(1+ 5)q)+ det f (q(1 5)q)] U([A], [A], T ), (1)here q denotes the quark elds with three avors, u, d, and s,d three colors; m0 = diag(mu,md,ms) in avor space; G and Ke the four-point and six-point interacting constants, respec-ely. The four-point interaction term in the Lagrangian keeps theV (3) SUA(3) UV (1) U A(1) symmetry, while the t Hooftx-point interaction term breaks the U A(1) symmetry.The covariant derivative in the Lagrangian is dened as D = i A . The gluon background eld A = 0A0 is supposed tohomogeneous and static, with A0 = gA0 2 , where2 is SU(3)lor generators. The effective potential U([A], [A], T ) is ex-essed in terms of the traced Polyakov loop = (Trc L)/NC ands conjugate = (Trc L)/NC . The Polyakov loop L is a matrix inlor spacex) = P exp[i0d A4(x, )], (2)here = 1/T is the inverse of temperature and A4 = i A0.Different effective potentials were adopted in literatures [21,37,,30]. The modied chemical dependent one= (a0T 4 + a14 + a2T 22)2 a3T 40 ln(1 62 + 83 34) (3)as used in [30,31] which is a simplication of= (a0T 4 + a14 + a2T 22) a3T 40 ln[1 6 + 4(3 + 3) 3()2] (4) si4 (2011) 343346cause the difference between and is smaller at nite chem-al potential and = at = 0. In the calculation we will usee later one. The related parameters, a0 = 1.85, a1 = 1.44 3, a2 = 0.08, a3 = 0.4, are still taken from [30], which canproduce well the data obtained in lattice QCD calculation.In the mean eld approximation, quarks can be taken as freeasiparticles with constituent masses Mi , and the dynamicalark masses (gap equations) are obtained asi =mi 4Gi + 2K jk (i = j = k), (5)here i stands for quark condensate. The thermodynamic poten-al of the PNJL model in the mean eld level can be derived as= U(,, T ) + 2G(u2 + d2 + s2) 4Ku d s 2d3p(2)33(Eu + Ed + Es) 2Tu,d,sd3p(2)3ln[A(,, Ei i, T )] 2Tu,d,sd3p(2)3ln[A(,, Ei + i, T )], (6)here(,, Ei i, T ) = 1+ 3e(Eii)/T + 3e2(Eii)/T+ e3(Eii)/Td(,, Ei + i, T ) = 1+ 3e(Ei+i)/T + 3e2(Ei+i)/T+ e3(Ei+i)/T .e values of u , d , s , and are determined by minimizinge thermodynamical potentialu= d= s= = = 0. (7)l the thermodynamic quantities relevant to the bulk propertiesquark matter can be obtained from . Particularly, the pres-re and entropy density can be derived with P = ((T ,) (0,0)) and S = /T , respectively.As an effective model, the (P)NJL model is not renormalizable,a cut-off is implemented in 3-momentum space for divergenttegrations. The model parameters: = 603.2 MeV, G2 = 1.835,5 = 12.36, mu,d = 5.5 and ms = 140.7 MeV, determined by t-ng f , M , mK and m to their experimental values [39], areed in the calculation.The Gibbs criteria is usually implemented for a complicated sys-m with more than one conservation charge. The Gibbs conditionsr the mixed phase of hadronquark phase transition in compactar areH = Q , T H = T Q , P H = P Q , (8)here are usually chosen with n and e . Under the equi-rium with trapped neutrino, the chemical potential of other par-cles including all baryons, quarks, and leptons can be derived byi = bin qie + qie , (9)here bi and qi are the baryon number and electric charge num-r of particle species i, respectively. For the matter with trappedectron neutrinos, Y = (Y + Y ) 0, we do not need to con-L der the contribution from muon and muon neutrino [34,35]. ForB 7Fisnouthinphanwneqtfutio(StrtrwnequthYledetrsiennecoPNhithsptrbycoFimunthnedumevphquofatcithisspdeicplhaattioisEOannmnWpephtophqulathth0)ofergiaPNcrdeG.y. Shao / Physics Lettersg. 1. Relative fractions of different species as functions of baryon density at severalapshots of a PNS evolution. The upper (lower) panels are the results with (with-t) trapped neutrinos.e neutrino-free matter (e = 0), both electrons and muons arecluded in the calculation.The baryon number density and energy density in the mixedase are composed of two parts with the following combinations= (1 )HB + QB , (10)d= (1 )H + Q , (11)here is the volume fraction of quark matter. And the electricutrality is fullled globally withotal = (1 )i=B,lqii + i=q,lqii = 0. (12)In Fig. 1, we display the relative fractions of different species asnctions of baryon density in several snapshots along the evolu-n of a proto-neutron star, from (S = 1, YLe = Ye + Ye = 0.4),= 1.5, YLe = Ye + Ye = 0.3), (S = 2, Ye = 0) to cold neu-on star with (S = 0, Ye = 0). Comparing the upper panels withapped neutrino with the lower panels without trapped neutrino,e nd that the fraction of trapped neutrino affects the protonutron ratio Yp/Yn at lower density before the appearance ofarks. For the hot PNS matter, Yp/Yn with rich e is larger thanat with poor e . According to the Pauli principle, the smallerp/Yn will excite more neutrons to occupy higher energy levels,ading to a stiffer equation of state.Another point we stress is that the onset density of quark phasecreases with the escape of trapped neutrinos, i.e., trapped neu-inos delay the hadronquark phase transition to a higher den-ty. For the neutrino-trapped matter, the fraction of neutrino ishanced with the appearance of quarks, which also affects theutrino opacity. For neutrino-free cases, the lepton population inld neutron star matter (S = 0, Ye = 0) is smaller than that ofS (S = 2, Ye = 0), especially after the appearance of quarks atgh density.The largest center densities of proto-neutron stars at the rstree snapshots taken above are c = 5.23,5.00, and 4.540, re-ectively. The center density decreases with the radiation of neu-inos, because the star expands when the inner matter is heatedthe escaped neutrinos. When neutrinos are free, the PNS beginsoling by radiating neutrino pairs of all avor and the star shrinks th04 (2011) 343346 345g. 2. EOSs of PNS and NS matter at several stages of the star evolution. The dotsark the range of the mixed phase.til the formation of cold neutron star. During the cooling stage,e center density of the star increases. The center density of coldutron star is c = 4.880. The variation of the center densitiesring a PNS evolution is easier to understand by combining theassradius relations that will be given latterly.We present the equations of state of the star matter at differentolving stages in Fig. 2. The dots mark the ranges of the mixedases. For the PNS matter at low density before the onset ofarks, the EOS becomes more and more stiffer with the decreaselepton fraction and the increase of entropy density. This mainlytributes to the decreased Yp/Yn , as shown in Fig. 1, which ex-tes more neutrons to higher energy states. In contrast, the EOS ofe mixed phase with a larger lepton fraction is much stiffer. Thisbecause the pure quark phase has a stiffer EOS, but the corre-onding hadron phase with the same lepton fraction and entropynsity has a softer one. To fulll the Gibbs condition of the chem-al and mechanical equilibrium, the phase transition can only takeace at relatively larger energy density. For the case with a stifferdronic EOS at lower density, the Gibbs condition can be realizedlower energy density to drive the hadronquark phase transi-n. The similar results have been obtained when the NJL modeltaken [34]. The cold neutron star matter at lower density has anS between the initial conditions and the end of heating stage,d a softest one for the mixed phase after quarks appearing.Comparing the results obtained in the NJL model in [34], wed the pressure of the mixed phase of a PNS given by the PNJLodel is larger than that of NJL model. This reects that the con-ement effect (gluon eld) is important at nite temperature.ith the PNJL model, the pressure of quark matter at nite tem-rature are much smaller than that of NJL model, therefore thease transition can only take place at relatively larger densityassure the quark-phase pressure can match that of the hadronase. The details can be found in Refs. [28,29] where the hadronark phase transition for symmetric and asymmetric matter re-ted to heavy-ion collision experiment has been investigated ine Hadron-(P)NJL model, and the same results were obtained.The above illustration also explains the inverse of pressure ofe mixed phase in the cases of (S = 2, Ye = 0) and (S = 0, Ye =as given by the NJL model in [34]. All the features of equationsstate will be reected in the massradius relations.Finally, in Fig. 3 we plot the massradius relations of sev-al snapshots taken above along the evolution a PNS. This gureves us a more intuitive picture about the PNS evolution withhadronquark phase transition. Firstly, in the heating stage theS expands with the decrease of lepton fraction (YLe) and the in-ease of entropy density (S). Simultaneously, the center densitycreases with the expansion of the star. When neutrinos are free,e star begins cooling, and then the star shrinks, leading to the34 B 70FilureceabearethJ1nadistmsldehaIfstaincacerainofdeanclinphexorQThtiinnoObeefoftewindrrachimthtrqudevateinRe[[[[[[[[[[1[1[1[1[1[1[1[1[1[1[2[2[2[2[2[2[2[2[2[2[3[3[3[3[3[3[3[3[3[3[46 G.y. Shao / Physics Lettersg. 3. Massradius relations of PNS and NS at several snapshots along a PNS evo-tion. The solid curves are the results without quarks and the dash curves are thesults with hadronquark phase transition.nter density increasing again. In this stage for a star with quarks,part of nuclear matter transforms to quark matter and the EOScomes softer in the mixed phase. The star may collapse intoblack hole in the cooling process if the EOS in the core cannotsist the gravity. This point is different from the result derived ine NJL model.The radio timing observations of the binary millisecond pulsar614-2230 with a strong general relativistic Shapiro delay sig-ture, implies that the pulsar mass is 1.97 0.04M [40]. Thescovery of this massive pulsar rules out many soft equations ofate. With the improved quark model, our calculation shows theaximum mass of cold neutron star with deconned quarks isightly larger than 2M . Because the quark model parameters aretermined by experiments and lattice QCD simulation, a stifferdronic EOS is needed to fulll this constraint. On the other hand,a vector interaction is included for quark matter, the EOS will beiffer and the maximum mass of hybrid star will be improved.In summary, we have studied the evolution of PNS withhadronquark phase transition with a more reliable quark modelcluding both chiral dynamics and (de)connement effect. Thelculation shows the quark phase may exist in the whole pro-ss of a PNS evolution. The trapped neutrinos affect greatly thetion of Yp/Yn and the EOS at low density before quarks appear-g. In the heating stage, with the deleptonization and the increaseentropy density, the PNS expands and the corresponding centernsity decreases. In contrast, in the cooling stage, the PNS shrinksd the center density increases. During this process a part of nu-ear matter transforms to quark matter, and the PNS may collapseto a black hole if the EOS in the core is not stiff enough.In this study, only the qq interaction is considered for quarkase because the relevant model parameters can be xed withperiments and lattice QCD simulation. In analogy to BCS the-y, color superconductivity in low-temperature and high-densityCD matter may appear and there may exist rich phase diagram.e coupling constant of this interaction channel affects the equa-on of state of quark matter and the onset density of quark phasecompact star. One question is that the coupling parameter can-t be xed from heavy-ion experiment or lattice QCD simulation.n the other hand, the (isoscalar) vector interaction channel hasen included in some studies. Such an interaction reduces thefective quark chemical potential but contributes to the pressurequark matter. This interaction stiffens the EOS of quark mat-r and increases the maximum mass of a hybrid star. Comparedith the hadron Walecka model, the (isoscalar) vector interactionquark matter plays the role corresponding to the meson. Oneawback is that the coupling constant is usual taken as a free pa-4 (2011) 343346meter and its strength affects greatly the Critical End Point ofiral symmetry restoration relevant to heavy-ion collision exper-ent. Besides, the isovector vector interaction (corresponding toe role of meson in hadron phase and inuencing the symme-y energy of quark matter) can be also introduced for asymmetricark matter. 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