aa r X i v : . [ nu c l - t h ] J u l A field theoretical model for quarkyonic matter
Gaoqing Cao ∗ School of Physics and Astronomy, Sun Yat-sen University, Guangzhou 510275, China.
Jinfeng Liao † Physics Department and Center for Exploration of Energy and Matter,Indiana University, 2401 N Milo B. Sampson Lane, Bloomington, Indiana 47408, USA. (Dated: July 7, 2020)The possibility that nuclear matter at a density relevant to the interior of massive neutron starsmay be a quarkynoic matter has attracted considerable recent interest. In this work, we constructa field theoretical model to describe the quarkyonic matter, that would allow quantitative andsystematic calculations of its various properties. This is implemented by synthesizing the Waleckamodel together with the quark-meson model, where both quark and nucleon degrees of freedomare present based on the quarkyonic scenario. With this model we compute at mean-field level thethermodynamic properties of the symmetric nuclear matter and calibrate model parameters throughwell-known nuclear physics measurements. We find this model gives a very good description of thesymmetric nuclear matter from moderate to high baryon density and demonstrates a continuoustransition from nucleon-dominance to quark-dominance for the system.
PACS numbers: 11.30.Qc, 05.30.Fk, 11.30.Hv, 12.20.Ds
I. INTRODUCTION
To understand the phases and properties of strong in-teraction matter at high baryon density, especially in theregion relevant to the interior of massive neutron stars,is a very active frontier in the research field of nuclearphysics and nuclear astrophysics. The study of highbaryon density region is also very relevant to ongoingexperimental measurements (e.g. STAR at RHIC andHADES at SPS) of heavy ion collisions at low beam en-ergy as well as planned programs at future facilities likethe FAIR, NICA and HIAF. Both neutron star observa-tions and heavy ion experiments will help promote ourunderstanding of the phase diagram over a broad rangeof temperature and baryon density for the strong inter-action matter governed by Quantum Chromodynamics(QCD). For recent reviews, see e.g. [1–5].While a lot has been learned about the QCD mat-ter properties at zero or very small baryon density, thehigh density region remains a significant challenge. Thereare several interesting proposals about possible phases ofhigh density QCD matter, such as (two-flavor) color su-perconductivity [6, 7], color-flavor locking [8], or quarky-onic matter [9]. In the density region comparable withthe neutron star interior, the quarkyonic matter mightbe more directly relevant, thus we shall focus on thatphase in this work. The quarkyonic matter was first pro-posed by following insights from the large N c analysisand emphasizing the coexistence of nucleon/quark de-grees of freedom [9–12]. In the large isospin density (butsmall baryon density) region, an analogous ”quarksonicmatter” was proposed by following similar arguments in ∗ [email protected] † [email protected] Ref. [13]. Recently, there has been increasing interestto explore the possible existence of quarkyonic matterinside compact stars and the consequences for relevantastrophysical observations [14–21].Certain issues require improvements over previousstudies, many of which were based on simple (and oftenoversimplified) picture implementations with crude andad hoc approximations. The chiral symmetry restora-tion with increasing density often lacked a dynamicaltreatment. The important physics constraints from thelower density side, e.g. nuclear matter saturation prop-erties, were not carefully checked. Given these issues, itis therefore important to develop a more sophisticatedfield theoretical model to study the quarkyonic matter –one that would allow systematic calculations of variousproperties and quantitative scrutiny of important physicsconstraints. This is the main goal of our study, with thefirst successful step to be reported in the present paper.The rest of paper is organized as follows. In Sec.II, themodel Lagrangian density is constructed and the mainformalism is developed in great details by focusing onsymmetric nuclear matter. Then in Sec.III, the modelparameters are fixed according to the saturation proper-ties observed from low energy nuclear experiments. InSec.IV, the thermodynamic properties are computed forquarkyonic matter for a wide range of baryon densitieswithin our new model. Finally, we conclude in Sec.V.
II. AN EFFECTIVE MODEL FORQUARKYONIC MATTERA. Lagrangian and thermodynamic potential
By following the spirit of quark-baryonic (or quarky-onic) matter (QBM) with both quarks and baryons asthe effective degrees of freedom of the strong interac-tion system, we construct a field theoretical model whichcombines the quark-meson (QM) model [22] togetherwith the well-known Walecka model [23]. The quark-meson and Walecka models are common on one aspect: mesons are the “messengers” of the interactions betweenquarks or baryons. The overall Lagrangian density ofour two-flavor model is composed of three parts, that is, L QBM ≡ L q + L N + L M with the following explicit forms: L q =¯ q h i /∂ + (cid:16) µ B /N c + µ I τ (cid:17) γ − g q (cid:0) σ + iγ τ · π (cid:1) i q, L N = ¯ N h i /∂ + (cid:16) µ B + µ I τ (cid:17) γ − g Ns (cid:0) σ + iγ τ · π (cid:1) + g Nv (cid:0) /ρ − γ /A (cid:1) · τ i N, L M =12 ( ∂ µ σ∂ µ σ + D µ π · D µ π ) − λ (cid:0) σ + π · π − υ (cid:1) + c σ + 12 g sv (cid:0) σ + π · π (cid:1) ( ρ µ · ρ µ + A µ · A µ ) −
14 ( D µ ρ ν − D ν ρ µ ) · ( D µ ρ ν − D ν ρ µ ) + 12 m v ρ µ · ρ µ −
14 ( D µ A ν − D ν A µ ) · ( D µ A ν − D ν A µ ) + 12 m a A µ · A µ . (1)Here, the quantum fields are defined as the follow-ing: q ( x ) = ( u ( x ) , d ( x )) T denotes the two-flavor quarkfield with color degrees of freedom N c = 3, N ( x ) =( p ( x ) , n ( x )) T is the two-flavor nucleon field outside theFermi spheres of quarks if exist, σ ( x ) and π ( x ) are thescalar and pseudoscalar mesons, while ρ aµ (with ρ µ the ω meson) and A aµ ( a = 0 , . . . ,
3) are vector and axial vectormesons, respectively. The baryon and isospin chemicalpotentials are given by µ B and µ I , respectively. Theisospin matrices are τ = (cid:18) , τ x − iτ y √ , τ x + iτ y √ , τ z (cid:19) with τ x , τ y and τ z the Pauli matrices in flavor space. Thederivative operators are defined as D = ∂ ∓ iµ I for thecharged π ± , ρ ± µ and A ± µ , and D µ = ∂ µ for the others. Forthe isospin symmetric case with µ I = 0, the Lagrangianhas exact chiral symmetry in the chiral limit c = 0 andwhen chiral anomaly is neglected by choosing m v = m a .In the realistic case, with the linear coefficient c = 0 andthe masses m v < m a , there is only approximate chiralsymmetry in the QBM model.Let us first discuss the vacuum of the above model attemperature T = 0 and chemical potential µ = 0. Inmean field approximation, the thermodynamic potentialis only given by the mesonic part in the vacuum, that is,Ω v = λ (cid:0) h σ i + h π i · h π i − υ (cid:1) − c h σ i , (2)the global minimum of which locates at h π i = 0 and h σ i = X t = ± " c λ + t i r υ − (cid:16) c λ (cid:17) / . (3) It can be checked that we simply have h σ i = υ in thechiral limit c →
0. Based on the ground state, the sigmaand pion masses can then be derived as [22] m σ = λ (cid:0) h σ i − υ (cid:1) , m π = λ (cid:0) h σ i − υ (cid:1) , (4)which indicate the σ and π mesons as the massive andGoldstone modes, respectively.If we adopt the quark version of Goldberger-Treimanrelation: f π g q = m [24], the expectation value of σ isfound to be h σ i v = f π in vacuum. Then, the parametersin the mesonic sector can be determined by the vacuummasses m σ , m π and pion decay constant f π as λ = m σ − m π f π , υ = m σ − m π m σ − m π f π , c = f π m π . (5)We next discuss the other model parameters in the quarkand baryonic sectors. Firstly, the coupling constants be-tween the scalar sector mesons and quarks or nucleonscan be fixed by their vacuum masses as g q = m v q /f π ≡ m σ / (2 f π ) [22] and g Ns = m v N /f π . The quantities m π , f π and m v N are well determined from the experiments. Theother parameters like m v q (or m σ ), g Nv and g sv will beconstrained later by the empirical saturation propertiesof nuclear matter. Note also that with the additionalscalar-vector interaction, the vector mass is given by m + g sv f π = (785 MeV) in the vacuum.We now turn to compute thermodynamics at finitetemperature and chemical potentials, where quarks andnucleons will also give contributions. In this paper, wewill focus on the (isospin-)symmetric nuclear matter as afirst step, by choosing µ B > µ I = 0. The thermo-dynamic contributions from the quark and baryon sectorsare given below:Ω tq = − N c T X t = ± Z d p (2 π ) ln (cid:16) e − [ E q ( p )+ t µBNc ] /T (cid:17) , (6)Ω tN = − (cid:0) g sv h σ i + m (cid:1) (cid:0) h ω i + ( h ρ i ) (cid:1) − T X t = ± Z d p (2 π ) ln (cid:18) e − [ E N ( p )+ t ( µ B − g Nv h ω i )] /T e − [ E N ( p )+ t ( µ ′ B − g Nv h ω i )] /T (cid:19) , (7)where the dispersion relations are E q ( p ) = (cid:0) p + m (cid:1) / with m q = g q h σ i and E N ( p ) = (cid:0) p + m (cid:1) / with m N = g Ns h σ i . The vector mean-field condensate is subject tothe physical constraint 0 ≤ g Nv h ω i ≤ µ B , that is, thenucleon chemical potential is reduced by h ω i but neverchanges sign.The crucial step here is to implement the quarkyonicpicture in the momentum space, in which the interiorof the Fermi sea is filled up by quarks while the nu-cleons are excluded to reside in an outside shell of theFermi sea [9, 10]. In our model, the boundary for ”Pauli-blocked” nucleon sphere is characterized by an effectivechemical potential µ ′ B . The nucleons in the quarkyonicmatter exist between the Fermi sphere stretched by µ ′ B and µ B . As one can tell in Ω tN : the thermodynamicspotential of the nucleonic part is obtained by subtract-ing the supposed inner contribution (with µ ′ B ) out of thenaive total one (with µ B ). It is important to have an ap-propriate scheme for determining the µ ′ B . One possiblechoice is the µ B -linear form: µ ′ B = µ B − ( N c m q − m N ) , (8)based on comparing kinetic energy of a baryon with that of N c quarks. Another nonlinear choice assumes thatthe momenta of the valence quarks of proton ( uud ) andneutron ( udd ) are the same and nucleons are blocked bythe free quarks from the Fermi sphere [15], that is, µ ′ B = q m + ( N c k F ) , (9)which is smaller than µ B as N c m q > m N . Here, theeffective Fermi momentum of the u and d quarks is k F = h ( µ B /N c ) − m i / . (10)This definition is based on comparing momentum of abaryon with that of N c quarks. We will perform compu-tations with both choices of µ ′ B and compare their resultslater. B. Gap equations and energy density
In mean field approximation, the total thermodynamicpotential is then Ω = Ω v +Ω tq +Ω tN and the gap equationscan be obtained from the extremal conditions ∂ Ω /∂X =0 ( X = h ω i , h σ i ) as h ω i = − X t = ± Z d p (2 π ) t g Nv m + g sv h σ i e [ E N ( p )+ t ( µ B − g Nv h ω i )] /T + 4 X t = ± Z d p (2 π ) t g Nv m + g sv h σ i e [ E N ( p )+ t ( µ ′ B − g Nv h ω i )] /T , (11) λ (cid:0) h σ i − υ (cid:1) h σ i − c − g sv h σ ih ω i + 4 N c X t = ± Z d p (2 π ) g q m q /E q ( p )1 + e [ E q ( p )+ t µBNc ] /T + 4 X t = ± Z d p (2 π ) g Ns m N /E N ( p )1 + e [ E N ( p )+ t ( µ B − g Nv h ω i )] /T − X t = ± Z d p (2 π ) g Ns m N /E N ( p ) + t ∂µ ′ B /∂ h σ i e [ E N ( p )+ t ( µ ′ B − g Nv h ω i )] /T = 0 , (12)where the derivatives of the effective chemical potential are ∂µ ′ B ∂ h σ i = g Ns − g q N c for the linear choice and ∂µ ′ B ∂ h σ i = µ ′ B (cid:2) g Ns m N − g q N m q (cid:3) for the nonlinear choice, respectively.Furthermore, the baryon number and entropy densities can be derived directly according to the thermodynamicrelationships n B = − ∂ Ω /∂µ B and s = − ∂ Ω /∂T as: n B = − X t = ± Z d p (2 π ) t (cid:18)
11 + e [ E q ( p )+ t µBNc ] /T + 11 + e [ E N ( p )+ t ( µ B − g Nv h ω i )] /T − ∂µ ′ B /∂µ B e [ E N ( p )+ t ( µ ′ B − g Nv h ω i )] /T (cid:19) , (13) s = 4 X t = ± Z d p (2 π ) N c ln (cid:16) e − [ E q ( p )+ t µBNc ] /T (cid:17) + N c E q ( p )+ t µ B T (cid:16) e [ E q ( p )+ t µBNc ] /T (cid:17) + ln (cid:16) e − [ E N ( p )+ t ( µ B − g Nv h ω i )] /T (cid:17) + E N ( p )+ t ( µ B − g Nv h ω i ) T (cid:0) e [ E N ( p )+ t ( µ B − g Nv h ω i )] /T (cid:1) − ln (cid:16) e − [ E N ( p )+ t ( µ ′ B − g Nv h ω i )] /T (cid:17) − E N ( p )+ t ( µ ′ B − g Nv h ω i ) T (cid:0) e [ E N ( p )+ t ( µ ′ B − g Nv h ω i )] /T (cid:1) ! , (14)where the explicit forms of the derivatives of the effective chemical potentials in Eq.(13) are given by ∂µ ′ B ∂µ B = 1 for thelinear choice and ∂µ ′ B ∂µ B = µ B µ ′ B for the nonlinear choice, respectively. Thus, the energy density of the quarkyonic matteris found to be ǫ ≡ Ω + µ B n B + sT − ( T = µ B = 0)= λ (cid:16) h σ i − υ (cid:17) − c h σ i −
12 ( m + g sv h σ i ) h ω i +4 X t = ± Z d p (2 π ) (cid:18) N c E q ( p )1+ e [ E q ( p )+ t µBNc ] /T + E N ( p ) − t g Nv h ω i e [ E N ( p )+ t ( µ B − g Nv h ω i )] /T − E N ( p )+ t [( µ ′ B − µ B ∂µ ′ B /∂µ B ) − g Nv h ω i ]1 + e [ E N ( p )+ t ( µ ′ B − g Nv h ω i )] /T (cid:19) − ( T = µ B = 0) , (15)where we assume m + ( N c k F ) > n B = n qB + n NB − n N ′ B ≡ p π + 2 p π − ∂µ ′ B ∂µ B p ′ F π , (16)where p NF and p N ′ F are the Fermi momenta of the occupied and Pauli-blocked nucleon states, and p qF is the Fermimomentum of the occupied quark states, respectively. The Fermi momenta are related to the chemical potentialsthrough the Fermi energies as E qF ≡ E q ( p qF ) = µ B /N c , E NF ≡ E N ( p NF ) = µ B − g Nv h ω i , E N ′ F ≡ E N ( p N ′ F ) = µ ′ B − g Nv h ω i . (17)In this case, we’re glad that the momentum integrations involved in the gap equations Eqs.(11) and (12) and energydensity Eq.(15) can be carried out explicitely with the help of Fermi momenta as0 = h ω i ( m + g sv h σ i ) − g Nv ∆ 2 p π , (18)0 = λ (cid:0) h σ i − υ (cid:1) h σ i − c − g sv h σ ih ω i + g Ns m N π ∆ (cid:20) E NF p NF − m ln (cid:16) E NF + p NF m N (cid:17)(cid:21) + N c g q m q π (cid:20) E qF p qF − m ln (cid:16) E qF + p qF m q (cid:17)(cid:21) + ∂µ ′ B ∂ h σ i p ′ F π , (19) ǫ = λ (cid:0) h σ i − υ (cid:1) − c h σ i + 12 ( m + g sv h σ i ) h ω i + 14 π ∆ (cid:20) E p NF − m E NF p NF − m ln (cid:16) E NF + p NF m N (cid:17)(cid:21) + N c π (cid:20) E p qF − m E qF p qF − m ln (cid:16) E qF + p qF m q (cid:17)(cid:21) + (cid:18) µ ′ B − µ B ∂µ ′ B ∂µ B (cid:19) p ′ F π − ( µ B = 0) , (20)where the symbol ”∆” means excluding the correspond-ing one with N → N ′ for the energy and momen-tum. Combining Eqs.(16) and (18), we find h ω i ( m + g sv h σ i ) = g Nv ∆ n NB for the linear choice, which actuallyhas a definite physical meaning: the vector condensate isproportional to the nucleon density [23]. III. MODEL PARAMETERS
From the experimental measurements associated withfinite nuclei, some properties of the infinite and isospinsymmetric nuclear matter were extracted: the saturationdensity n ≈ .
16 fm − [25], the energy per nucleon atthis density E/N ≡ ǫ/n B − m v N = −
16 MeV [26, 27] aswell as the compressibility K = 240 ±
20 MeV [28] . Theoretically, they are related with each other as: ∂ ( E/N ) ∂n B (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h σ i n B = n = − ǫn + µ Bc n = P c n = 0 , (21) K = 9 ∂ ( E/N ) ∂ ( n B /n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n B = n , (22)from which it is easy to infer the pressure P c = 0 and thecritical chemical potential µ Bc = E/N + m v N = 923 MeV.Now, we use these saturation properties to fix the re-maining parameters. The dynamical quark mass is variedin the range m v q & m v N / h σ i and h ω i and couplingconstants g Nv and g sv are fixed by solving the gap equa-tions Eqs.(11) and (12), saturation equation Eq.(21) andthe saturation energy E/N = −
16 MeV self-consistently.The extracted results for g sv and the associated nucleonfraction R N ≡ ∆ n NB /n B as functions of m v q are shown to-gether in Fig. 1 for both linear and nonlinear µ ′ B choices.As we can see, the results are quantitatively consistentwith each other for these choices, with only minor differ-ences in the relatively smaller mass region. g sv
320 330 340 350 360 3700.00.20.40.60.81.0 m qv ( MeV ) R N FIG. 1. The extracted values for the coupling g sv and theassociated nucleon fraction R N as functions of the quark vac-uum mass m v q for both linear (red dotted) and nonlinear (bluedashed) µ ′ B choices. In order to further fix the vacuum quark mass in ourmodel, we show our model calculations together with theempirical constraint [28] in Fig.2 for the compressibilityat saturation density n . From the results, we find thebest agreement is achieved for m v q = 370 . ± . m v q = 370 . R N ≈ .
8% for both choices of µ ′ B at saturation density.We note that at this density there is a nonzero albeit verysmall fraction of quarks that already emerge and coexistwith the nucleons. The corresponding coupling constantsare also fixed to be g Nv ≈ . g sv ≈
81, respectively.At this point, all of our model parameters are fixedand the model satisfactorily catches the nuclear mat-ter properties at saturation density. Lastly we examinethe liquid-gas transition at this density. In Fig. 3, weshow the thermodynamic potential Ω as a function ofthe quark condensate h σ i for both choices of µ ′ B at thecritical chemical potential µ Bc . As one can see, there is atypical first-order transition structure with two degener-ate minima: one at the vacuum value h σ i = f π , and the
320 330 340 350 360 370020040060080010001200 m qv ( MeV ) K ( M e V ) FIG. 2. The compressibility K of quarkyonic matter at sat-uration density n as a function of the quark vacuum mass m v q for both linear (red dotted) and nonlinear (blue dashed) µ ′ B choices. The yellow band is the constraint from experi-ments [28]. other new one at h σ i = 69 . µ Bc , the chiral con-densate jumps from the vacuum value to the smaller one.In the next section we will analyze the matter propertiesat chemical potential beyond this transition point.
40 50 60 70 80 90 100 1100.00.20.40.60.81.01.2 < > (
MeV ) Ω ( M e V ) FIG. 3. The thermodynamic potential Ω as a function of thechiral condensate h σ i at the critical chemical potential µ Bc .The conventions are the same as those in Fig.1. IV. THE QUARKYONIC MATTERPROPERTIES
In this section we present results for quarkyonic mat-ter properties in the region of a few times the saturationdensity. The chiral and vector condensates as well as thecorresponding nucleon ratio are shown in Fig. 4 as func-tions of baryon chemical potential. We find that both h σ i and R N decreas rapidly while h ω i increases with µ B .This implies that with increasing density, the chiral sym-metry gets gradually restored with the quarks becominglighter and more abundant. The increasing of h ω i couldbe understood as due to the enhancement of the nucleondensity with µ B , even though the nucleon fraction R N decreases. < σ > ( M e V ) < ω > ( M e V )
940 960 980 1000 1020 10400.50.60.70.80.91.0 μ B ( MeV ) R N FIG. 4. The chiral condensate h σ i , vector condensate h ω i andnucleon ratio R N as functions of chemical potential µ B in thechiral symmetry partially restored phase. The conventionsare the same as those in Fig.1. We now compute the energy density of the system andpresent the closely related
E/N in the upper panel ofFig. 5. As we can see, the
E/N starts from the mini-mum value of −
16 MeV at the saturation density n andsteadily increases toward higher density. A key quantityrelated to the equation-of-state (EOS) for the quarky-onic matter is the speed of sound C v ≡ q ∂P∂ǫ [29]. Inthe lower panel of Fig. 5, we show C v versus baryondensity for both linear and nonlinear µ ′ B choices, whichshow small deviation from each other. In both cases,the speed of sound increases quickly between 1 ∼ and then approaches the high density asymptotical limitrather smoothly, in consistency with a continuous transi-tion feature [29]. We note that our results are consistentwith those given in Ref. [15, 17] for both small and largedensity, except that the prominent peak structure in theintermediate density is absent in our model. The differ- ence could be due to the hard core feature in Ref. [17]which we do not have. Actually, the monotonous featureof C v was also found in a recent quite convincing studywhen diquark dynamics is ignored [30]. - E / N ( M e V ) n B / n C v FIG. 5. The energy per nucleon
E/N and speed of sound C v as functions of baryon density n B , the range of whichcorresponds to that of µ B in Fig. 4. The conventions are thesame as those in Fig.1. Finally, we proceed to compare our EOS with the ex-perimental extraction as well as other model calcula-tions [31], see Fig. 6. The comparison indicates that ourresults based on quarkyonic matter are reasonably con-sistent with the experimental constraints, especially inthe large density region where quarks become more andmore important.
V. CONCLUSIONS
In this work, we propose a field theoretical model forquarkyonic matter by combining the Walecka model to-gether with the quark-meson model. We have systemat-ically calibrated the model parameters based on varioushadron properties in the vacuum as well as nuclear mat-ter properties a the saturation density. Based on that, wethen extend our calculations to the large baryon densityregion and find a number of interesting results. Firstly,the chiral symmetry is partially and smoothly restoredwith increasing baryon chemical potential µ B (see theupper panel of Fig.4), contrary to the first-order transi-tion and nearly full restoration in Nambu–Jona-Lasiniomodel [24]. Secondly, the vector condensate increaseswith µ B (see the middle panel of Fig.4) as the nucleondensity increases, which can be easily expected from the FIG. 6. The comparison between our quarkyonic mat-ter model results with those from experimental constraints(shadow region) and other model predictions (colored lines)for the pressure P of symmetric nuclear matter as a functionof baryon density. Note that this plot is made via adaptingan original figure extracted from Re. [31], for which we keepthe original notations of the various physical quantities. Inparticular, the baryon density ρ ( ρ ) in this plot correspondsto n B ( n ) we use in other places of the paper. proportionality shown in Eq.(18). Thirdly, the nucleonratio reduces (see the lower panel of Fig.4) as the quark density enhances more quickly than the nucleon density,which indicates gradual dominance of the quark degreesof freedom at larger chemical potential. Finally we havecalculated the equation of state and especially the speedof sound for quarkyonic matter in this model. The resultsare found to be consistent with predictions of variousother models as well as with experimental constraints forsymmetric nuclear matter at a few times the saturationdensity. Our overall conclusion is that, quantitative re-sults from our field theoretical model of quarkyonic mat-ter provide a satisfactory description about the propertiesof vacuum as well as nuclear matter up to several timesthe saturation density. Apart for the first-order liquid-gas transition at µ Bc = 923 MeV, the results feature acontinuous transition from nucleon-dominated regime toquark-dominated one along with gradual restoration ofthe chiral symmetry. It will be a natural step to furtherexplore the implications of this quarkyonic matter modelfor the interiors of neutron stars, such as had been donein Ref. [15]. The results shall be reported in a futurepublication. Acknowledgments — The authors are grateful toCharles Horowitz and Larry McLerran for very helpfuldiscussions. G.C. is supported by the National NaturalScience Foundation of China with Grant No. 11805290and Young Teachers Training Program of Sun Yat-senUniversity with Grant No. 19lgpy282. J.L. is supportedin part by the U.S. NSF Grant No. PHY-1913729 and bythe U.S. DOE Office of Science, Office of Nuclear Physics,within the framework of the Beam Energy Scan Theory(BEST) Topical Collaboration. [1] D. Page and S. Reddy, Ann. Rev. Nucl. Part. Sci. , 327-374 (2006) doi:10.1146/annurev.nucl.56.080805.140600[arXiv:astro-ph/0608360 [astro-ph]].[2] J. M. Lattimer and M. Prakash, Phys. Rept. , 127-164 (2016) doi:10.1016/j.physrep.2015.12.005[arXiv:1512.07820 [astro-ph.SR]].[3] P. Braun-Munzinger, V. Koch, T. Schfer andJ. Stachel, Phys. Rept. , 76-126 (2016)doi:10.1016/j.physrep.2015.12.003 [arXiv:1510.00442[nucl-th]].[4] A. Bzdak, S. Esumi, V. Koch, J. Liao, M. Stephanovand N. Xu, Phys. Rept. , 1-87 (2020)doi:10.1016/j.physrep.2020.01.005 [arXiv:1906.00936[nucl-th]].[5] X. Luo and N. Xu, Nucl. Sci. Tech. , no. 8, 112 (2017)doi:10.1007/s41365-017-0257-0 [arXiv:1701.02105 [nucl-ex]].[6] M. G. Alford, K. Rajagopal and F. Wilczek, Phys. Lett.B , 247 (1998) doi:10.1016/S0370-2693(98)00051-3[hep-ph/9711395].[7] R. Rapp, T. Schfer, E. V. Shuryak andM. Velkovsky, Phys. Rev. Lett. , 53 (1998)doi:10.1103/PhysRevLett.81.53 [hep-ph/9711396].[8] M. G. Alford, K. Rajagopal and F. Wilczek, Nucl. Phys.B , 443 (1999) doi:10.1016/S0550-3213(98)00668-3[hep-ph/9804403]. [9] L. McLerran and R. D. Pisarski, Nucl. Phys.A , 83 (2007) doi:10.1016/j.nuclphysa.2007.08.013[arXiv:0706.2191 [hep-ph]].[10] L. McLerran, K. Redlich and C. Sasaki, Nucl. Phys. A , 86-100 (2009) doi:10.1016/j.nuclphysa.2009.04.001[arXiv:0812.3585 [hep-ph]].[11] A. Andronic, D. Blaschke, P. Braun-Munzinger, J. Cley-mans, K. Fukushima, L. McLerran, H. Oeschler,R. Pisarski, K. Redlich, C. Sasaki, H. Satz andJ. Stachel, Nucl. Phys. A , 65-86 (2010)doi:10.1016/j.nuclphysa.2010.02.005 [arXiv:0911.4806[hep-ph]].[12] T. Kojo, Y. Hidaka, L. McLerran and R. D. Pis-arski, Nucl. Phys. A , 37-58 (2010)doi:10.1016/j.nuclphysa.2010.05.053 [arXiv:0912.3800[hep-ph]].[13] G. Cao, L. He and X. G. Huang, Chin. Phys. C ,no. 5, 051001 (2017) doi:10.1088/1674-1137/41/5/051001[arXiv:1610.06438 [nucl-th]].[14] J. Steinheimer, S. Schramm and H. Stocker, Phys. Rev.C , 045208 (2011) doi:10.1103/PhysRevC.84.045208[arXiv:1108.2596 [hep-ph]].[15] L. McLerran and S. Reddy, Phys. Rev. Lett. , no.12, 122701 (2019) doi:10.1103/PhysRevLett.122.122701[arXiv:1811.12503 [nucl-th]].[16] K. Fukushima and T. Kojo, Astrophys. J. , no.2, 180 (2016) doi:10.3847/0004-637X/817/2/180[arXiv:1509.00356 [nucl-th]].[17] K. S. Jeong, L. McLerran and S. Sen, arXiv:1908.04799[nucl-th].[18] S. Sen and N. C. Warrington, [arXiv:2002.11133 [nucl-th]].[19] D. C. Duarte, S. Hernandez-Ortiz and K. S. Jeong,[arXiv:2003.02362 [nucl-th]].[20] T. Zhao and J. M. Lattimer, [arXiv:2004.08293 [astro-ph.HE]].[21] C. J. Xia, S. S. Xue and S. G. Zhou, JPS Conf. Proc. ,011010 (2018) doi:10.7566/JPSCP.20.011010[22] B. J. Schaefer and J. Wambach, Phys. Rev. D ,085015 (2007) doi:10.1103/PhysRevD.75.085015 [hep-ph/0603256].[23] J.D. Walecka, Ann. of Phys. 83, 491 (1974).[24] S. P. Klevansky, ”The Nambu-Jona-Lasinio model of quantum chromodynamics,” Rev. Mod. Phys. , 649(1992).[25] R. Hofstadter, Rev. Mod. Phys., 28:214 (1956).[26] A. E. S. Green and D. F. Edwards, Phys. Rev., 91:46(1953).[27] A. E. S. Green, Phys. Rev., 95:1006 (1954).[28] S. Shlomo, V. M. Kolomietz, G. Col‘o, Eur. Phys. J. A30, 23-30 (2006).[29] G. Baym, T. Hatsuda, T. Kojo, P. D. Powell, Y. Songand T. Takatsuka, Rept. Prog. Phys.81