Phenomenological study of the anisotropic quark matter in the two-flavor Nambu-Jona-Lasinio model

He-Xia Zhang• Yu-Xin Xiao • Jin-Wen Kang • Ben-Wei Zhang

Abstract With the two-flavor Nambu-Jona-Lasinio(NJL)model, we carried out a phenomenological study on the chiral phase structure, mesonic properties, and transport properties of momentum-space anisotropic quark matter.To calculate the transport coefficients we utilized the kinetic theory in the relaxation time approximation, where the momentum anisotropy is embedded in the estimation of both the distribution function and relaxation time.It was shown that an increase in the anisotropy parameter ξ may result in a catalysis of chiral symmetry breaking.The critical endpoint(CEP) is shifted to lower temperatures and larger quark chemical potentials as ξ increases, and the impact of momentum anisotropy on the CEP temperature is almost the same as that on the quark chemical potential of the CEP.The meson masses and the associated decay widths also exhibit a significant ξ dependence.It was observed that the temperature behavior of the scaled shear viscosity η/T3 and scaled electrical conductivity σel/T exhibited a similar dip structure,with the minima of both η/T3 and σel/T shifting toward higher temperatures with increasing ξ.Furthermore,we demonstrated that the Seebeck coefficient S decreases when the temperature rises and its sign is positive,indicating that the dominant carriers for converting the temperature gradient to the electric field are up-quarks. The Seebeck coefficient S is significantly enhanced with a large ξ for a temperature below the critical temperature.

Keywords Heavy-ion collision · Momentum anisotropy ·NJL model · Chiral phase transition · Transport coefficient · Quark matter

1 Introduction

The properties of strongly interacting matter described by the quantum chromodynamics (QCD) in extreme conditions of temperature T and density have aroused a plethora of experimental studies in the last thirty years [1, 2].The experiment studies performed at the Relativistic Heavy Ion Collider (RHIC) in BNL and the Large Hadron Collider(LHC)in CERN have revealed that a new deconfined state of matter, the quark-gluon plasma (QGP), can be created at high temperature.Further,the non-central heavyion collisions produce the strongest magnetic fields and orbital angular momenta, which can induce a number of novel phenomena [3-5]. The lattice QCD calculation,which is a powerful gauge invariant approach to investigate the non-perturbative properties,has also confirmed that the phase transition is a smooth and continuous crossover for vanishing chemical potential[6,7].Owing to the so-called fermion sign problem[8],lattice QCD simulation is limited to low finite density [9, 10], even though several calculation techniques, such as the Taylor expansion[11, 12], analytic continuations from imaginary to real chemical potential [13, 14], and multi-parameter reweighting method [15], have been proposed to address this problem and improve the validity at high chemical potential. More detailed reviews of lattice calculation can be found in Refs. [16,17].Alternatively,one also can rely on effective models, the Dyson-Schwinger equation approach[18,19]and the functional renormalization group approach [20, 21], to study the chiral aspect of QCD for finite baryon chemical potential μB. Currently, there are various QCD inspired effective models, such as the Nambu-Jona-Lasinio (NJL) model [22, 23], Polyakovloop enhanced NJL (PNJL) model [24, 25], Quark-Meson(QM) model [26, 27], and Polyakov QM (PQM) model[28, 29], which not only can successfully describe the spontaneous chiral symmetry breaking and restoration of QCD but also have been applied to explore the QCD phase structure and internal properties of the meson at arbitrary T and μB. These model calculations have predicted that at high chemical potential,the phase transition is a first-order phase transition, and with decreasing μB, the first-order phase transition has to end at a critical end point(CEP)and change into a crossover. At this CEP, the phase transition is of second order. However, owing to various approximations adopted in the model calculations, there is no agreement on the existence and location of the CEP in the phase diagram. Furthermore, the rotation effects [30, 31],magnetic field effects [32-35], finite-volume effects[36-40], non-extensive effects [41, 42], external electric fields [43-45], and chiral chemical potential effects[46-49] have also been considered in the effective models to provide a better insight into the phase transition of the realistic QCD plasma.

Apart from the QCD phase structure information, the transport coefficients, characterizing the non-equilibrium dynamical evolution of QCD matter in heavy-ion collisions[50-52],have also attracted significant attention.The shear viscosity η,which quantifies the rate of momentum transfer in a fluid with inhomogeneous flow velocity, has been successfully used in the viscous relativistic hydrodynamic description of the QGP bulk dynamics. The small shear viscosity to entropy density ratio η/s can be extracted from the elliptic flow data [53]. In the literature, there are various frameworks for estimating the η of strongly interacting matter, e.g., the kinetic theory within the relaxation time approximation (RTA), QCD effective models [54-58], the quasiparticle model (QPM) [59, 60], and the lattice QCD simulation [61]. The electrical conductivity σel, as the response of a medium to an applied electric field, has also attracted attention in high energy physics. The presence of σel not only can affect the duration and strength of magnetic fields[62],but also is directly proportional to the emissivity and production of soft photons [63, 64]. The thermal behavior of σel has been estimated using different approaches, such as the microscopic transport models[65-67], lattice gauge theory simulation [68, 69], hadron resonance gas model [70, 71], quasiparticle models[72, 73], effective models [54, 74], string percolation model[75],and holographic method[76].Recently,studies of electrical conductivity in QGP in the presence of magnetic fields have also been performed[77-79].Another less concerned but interesting coefficient is the Seebeck coefficient (also called thermopower). When a spatial gradient of temperature exists in a conducting medium, a corresponding electric field can arise and vice versa, which is the Seebeck effect.When the electric current induced by an electric field can compensate with the current owing to the temperature gradient, the thermal diffusion ends. Accordingly,the efficiency of converting the temperature gradient to an electric field in the open circuit condition is quantified by the Seebeck coefficient S. In past years, the Seebeck effect has been extensively investigated in condensed matter physics. Very recently, the exploration has been extended to the hot QCD matter.For example,the Seebeck coefficient with and without magnetic fields has been studied in both the hadronic matter [80, 81] and QGP[82,83].In Ref.[84],the Seebeck coefficient has also been estimated based on the NJL model, where the spatial gradient of the quark chemical potential is considered in addition to the presence of a temperature gradient.

This paper is organized as follows:Sect. 2 provides a brief review of the basic formalism of the two-flavor NJL model.In Sect. 3 and Sect. 4, we present a brief derivation of the expressions associated with the constituent quark mass and meson mass spectrum in both an isotropic and anisotropic medium. Section 5 includes the detailed procedure for obtaining the formulae of momentum anisotropy-dependent transport coefficients.In Sect. 6,we present the estimation of the relaxation time for(anti-)quarks.The numerical results for various observables are phenomenologically analyzed in Sect. 7.In Sect. 8,the present work is summarized with an outlook. The formulae for the squared matrix elements in different quark-(anti-)quark elastic scattering processes are presented in the Appendix.

2 Theoretical frame

In this work, we start from the standard two-flavor NJL model, which is a purely fermionic theory owing to the absence of all gluonic degrees of freedom. Accordingly,the Lagrangian is given as [22]

where ψ（¯ψ）stands for the quark(antiquark)field with twoflavors (u, d) Nf =2 and three colors Nc =3. ^m0denotes the diagonal matrix of the current quark mass of up and down-quarks,and we take m0=m0u=m0dto ensure isospin symmetry of the NJL Lagrangian. G is the effective coupling strength of four-point fermion interaction in the scalar and pseudoscalar channels. ^τ is the vector of the Pauli matrix in the isospin space.

In the NJL model, under the mean field (or Hartree)approximation [22, 23], the quark self-energy is momentum-independent and can be identified as the constituent quark mass mq, which acts as order parameter for characterizing the chiral phase transition. For an off-equilibrium system, the evolution of the space-time dependence of the constituent quark mass in the real time formalism can be obtained by solving the gap equation [96]

All information of a meson is contained in the irreducible one-loop pseudoscalar or scalar polarization function ΠM,

where the subscript M corresponds to pseudoscalar (π) or scalar (σ) mesons. The space-time dependence of the polarization function in a non-equilibrium system is given as [96, 98]

Here,νMdenotes the real value of the bound meson energy,and νM=0 （2mq） for the π （σ） meson.

3 Constituent quark and meson in an isotropic quark matter

In an expanding system (e.g., the dynamical process of heavy-ion collisions), the space-time dependence in the phase-space distribution function is hidden in the spacetime dependence of temperature and chemical potential.However, for a uniform temperature and chemical potential,i.e.,for a system in global equilibrium,the distribution function is well defined and independent of space-time.Therefore, in the equilibrium (isotropic) state, to investigate the chiral phase transition and mesonic properties within the NJL model,one can employ the imaginary-time formalism.Actually,the results in Ref.[96]have indicated that the real-time calculation of the closed time-path Green’s function reproduces exactly the finite temperature result of the NJL model obtained from the Matsubara’s temperature Green’s function in the thermodynamical equilibrium limit. In the following, we will briefly present the procedure for the derivation of the polarization function in the imaginary time formalism.In an equilibrium system,mq, which is temperature- and quark chemical potentialdependent, can be directly calculated from the self-consistent gap equation in momentum space [22, 23]:

where ωn=（2n+1）π/β are the fermionic Matsubara frequencies and the sum of n runs over all positive and negative integer values.It is to be understood that the limit y →0 is to be taken after the Matsubara summation. The function B0in Eq. (9) relates to the two-fermion-line integral. At finite μ and T, B0is defined as [103]

where the Fermi-Dirac distribution function is introduced.Similar to the treatment of function A, after Matsubara summation over n and the Matsubara frequencies iνlare analytically continued to real values, Eq. (11) can be rewritten in the following form:Inserting Eqs. (12) and (13) to Eq. (9), we finally can obtain the expression for ΠMin the equilibrium state,which is formally the same as Eq. (5), except that the distribution functions are ideal Fermi-Dirac distribution functions rather than space-time-dependent distribution functions.

4 Constituent quark and meson in a weakly anisotropic medium

where χ is the angle between p and n, and p ≡|p|throughout the computations. With this choice, the spheroidally anisotropic term ξ（n·p）2in Eq. (14) can be written as ξ（n·p）2=ξp2（sin χ cos φ sin θ+cos χ cos θ）2=ξc（θ，φ，χ）.We further assume that n points along the beam(z)axis,i.e.,n=（0，0，1）.It is essential to note that we shall restrict ourselves here to a plasma close to the equilibrium state and have small anisotropy around the equilibrium state.Therefore, in the weak anisotropy limit (|ξ|≪1), one can expand Eq. (14)around the isotropic limit and retain only the leading order in ξ. Accordingly, the anisotropic momentum distribution function in the local rest frame can be further written as[107]

where the second term is the anisotropic correction to the equilibrium distribution, which is also related to the leading-order viscous correction to the equilibrium distribution in viscous hydrodynamics. For a fluid expanding one-dimensionally along the direction n in the Navier-Stokes limit, the explicit relation is given as [109, 110]

which indicates that the non-zero shear viscosity (finite momentum relaxation rate) in an expanding system can also explicitly lead to the presence of momentum-space anisotropy. At the RHIC energy with the critical temperature Tc ≈160 MeV, τ ≈6 fm/c and η/s=1/4π, we can obtain ξ ≈0.3. In principle, for the non-equilibrium dynamics of the chiral phase transition, a self-consistent numerical study must be performed by solving the Boltzmann-Vlasov transport equation together with gap equation in terms of the space-time-dependent quark distribution as mentioned in Sect. 2.However,because the non-equilibrium distribution function in the local rest frame of weakly anisotropic systems has a specific form and the temperature and chemical potential appearing in Eq. (17) are still considered as free parameters, the spacetime evolution is not addressed. Therefore, in the nonequilibrium states possessing small momentum space anisotropy, just by solving the gap equation with anisotropic momentum distribution, i.e., Eq. (17), it is possible for us to investigate phenomenologically the impact of momentum anisotropy on the temperature and quark potential dependence of the constituent quark mass. Accordingly,the gap equation, i.e., Eqs. (2) or (6), is modified as

where P denotes the Cauchy principal value. The function B0can finally be rewritten as

When k0=0, the imaginary part of Eq. (25) vanishes.Therefore, the real and imaginary parts of meson polarization functions for different cases in a weakly anisotropic medium are given by

The solution is a real value for mπ，σ＜2mq, and a meson is stable.However,for mπ，σ＞2mq,a meson dissociates to its constituents and becomes a resonant state.Accordingly,the polarization function is a complex function and ΠMhas an imaginary part that is related to the decay width of the resonance as ΓM=ImΠM（mπ，σ，0）/mπ，σ.

5 Transport coefficients in an anisotropic quark matter

In this section,we study the effects due to the local anisotropy of the plasma in the momentum space on the transport coefficients(shear viscosity,electrical conductivity,and Seebeck coefficient). The calculation is performed in the kinetic theory that is widely used to describe the evolution of the non-equilibrium many-body system in the dilute limit.Assuming that the system has a slight deviation from the equilibrium, the relaxation time approximation (RTA) can be reasonably employed. The momentum anisotropy is encoded in the phase-space distribution function, which evolves according to the relativistic Boltzmann equation.We provide the following procedures for deriving the ξ-dependent transport coefficients.

5.1 Shear viscosity

The propagation of a single quasiparticle with temperature- and chemical potential-dependent mass in an anisotropic medium is described by the relativistic Boltzmann-Vlasov equation [101]

which is consistent with the result from Ref. [92]. For a system consisting of multiple particle species, the total shear viscosity is given as η=∑a ηa. For SU(2) light quark matter, a=u，d， ¯u， ¯d and the spin-color degeneracy factor reads explicitly as da=2Nc.

5.2 Electrical conductivity and Seebeck coefficient

We also investigate the effect of momentum anisotropy on the electrical conductivity and thermoelectric coefficient. Under the RTA, the relativistic Boltzmann-Vlasov equation for the distribution function of single-quasiparticle of charge eain the presence of an external electromagnetic field is given by

where μa=taμ denotes the chemical potential of particle species a and ta=+1（-1） for the quark (antiquark).Considering μ to be homogeneous in space and a temperature gradient only existing along the x-axis, and inserting Eq. (47) into Eq. (46), the perturbative term δfain an anisotropic medium can then be written as

where σel，aand Saare the electrical conductivity and Seebeck coefficient of the a-th particle, respectively. In terms of the distribution function,Jawithin the kinetic theory can be written as

where αais the thermoelectric conductivity due to particle species a. In the isotropic limit ξ →0, Eqs. (54)-(55)reduce to the formulae in the equilibrium. In condensed physics, a semiconductor can exhibit either electron conduction (negative thermopower) or hole conduction (positive thermopower). The total thermopower in a material with different carrier types is given by the sum of these two contributions weighted by their respective electrical conductivity values [114, 115]. Inspired by this, the total Seebeck coefficient in a medium composed of light quarks and antiquarks can be given as

6 Computation of the relaxation time

To quantify the transport coefficients, one needs to specify the relaxation time. In present work, the scattering processes of (anti)quarks through the exchange of mesons are encoded into the estimation of the relaxation time.

The relaxation times of(anti)quarks are microscopically determined by the thermal-averaged elastic scattering cross section and particle density.For light quarks,the relaxation time in the RTA can be written as [57]

7 Results and discussion

Throughout this work, the following parameter set is used: m0=m0，u=m0，d=5.6 MeV, GΛ2=2.44 and Λ=587.9 MeV. These values are taken from Ref. [117],where these parameters are determined by fitting quantities in the vacuum (T =μ=0 MeV). At T =0, the chiral symmetry is spontaneously broken and one obtains the current pion mass m0，π=135 MeV, pion decay constant fπ=92.4 MeV, and quark condensate-〈ψψ〉1/3=241 MeV.

In the NJL model, the constituent quark mass is a good indicator and an order parameter for analyzing the dynamical feature of chiral phase transition. In the asymptotic expansion-driven momentum anisotropic system,the anisotropy parameter ξ is always positive owing to the rapid expansion along the beam direction.However,in the presence of a strong magnetic ξ, it becomes negative because of the reduction in transverse momentum due to Landau quantization. As we restrict the analysis to only a weakly anisotropic medium, the anisotropy parameter we address here is artificially taken as ξ=-0.3， 0.0， 0.3 to investigate phenomenologically the effect of ξ on various quantities.In Fig. 1a,we show the thermal behavior of the light constituent quark mass mqfor vanishing quark chemical potential at different ξ. For low temperature, mqremains approximately constant at (mq≈400 MeV), and then,with increasing temperature mq,it continuously drops to near zero. The transition to small mass occurs at higher temperature for a higher value of ξ. These phenomena imply that at zero chemical potential,the restoration of the chiral symmetry (the chiral symmetry is not strictly restored because the current quark mass is nonzero) in an(an-)isotropic quark matter takes place as crossover phase transition, and an increase in ξ can lead to a catalysis of chiral symmetry breaking.

Fig. 1 (Color online) (a) Temperature dependence of the constituent quark mass mq at μ=0 MeV for different fixed anisotropy parameters. (b) Chiral susceptibility χch at μ=0 MeV for different fixed anisotropy parameters.The broad dashed lines,dashed lines,and solid lines correspond to the results for ξ=-0.3, ξ=0, and 0.3,respectively

Fig. 2 (Color online) a Double mass of the constituent quarks 2mq(gray lines),π meson mass(blue lines),and σ meson mass(red lines)as a function of temperature at μ=0 GeV for different anisotropy parameters ξ.b Temperature dependences of π(blue lines)and σ(red lines) meson decay widths for different ξ. The long dashed lines,dashed lines,and solid lines represent the results for ξ=-0.3,ξ=0,and 0.3, respectively, with the corresponding Mott temperatures approximately given by 187 MeV, 196 MeV, and 206 MeV

Fig. 3 (Color online) Three-dimensional plot of the constituent quark mass mq with respect to temperature and quark chemical potential for different anisotropy parameters (ξ=-0.3， 0， 0.3)

Fig. 4 (Color online) Three-dimensional plot of chiral susceptibility χch for ξ=-0.3 in the entire μ and T ranges of interest. The gray area means that χch is divergent. The values remain finite due to numerical problems(differential quotient).The peak height at high μ is two orders of magnitude higher than that in cases with low μ and can be considered ‘‘divergent’’

Fig. 5 (Color online) Chiral phase diagram for different anisotropy parameters in the Nambu-Jona-Lasinio (NJL) model. The inside curve is for ξ=-0.3, the next curve is for ξ=0, and the outermost curve is for ξ=0.3. The solid lines denote the first-order phase transition curves, the dashed lines denote the crossover transition curves,and the solid dots represent the critical endpoints(CEPs).We observe that the CEP is shifted toward larger values of the quark chemical potential but smaller values of the temperature for higher anisotropy parameters

The dependence of total quark relaxation time τqon temperature for vanishing quark chemical potential at different ξ is displayed in Fig. 7. As can be observed, τqfirst decreases sharply with increasing temperature,and after an inflection point (viz, the peak position of ¯σq¯q), τqchanges modestly with temperature.Further,the increase in τqwith ξ is significant at low temperature, whereas at hightemperature, the reduction in τqwith ξ is imperceptible.This is the result of the competition between the quark number density and the total scattering cross section in Eq. (57). At low temperature, the ξ dependence of τqis mainly determined by the inverse quark number density,

Fig. 6 (Color online) (a) Total cross section of the quark-quark scattering processes ¯σqq as a function of temperature at μ=0 MeV for different anisotropy parameters. (b) The total cross section of quark-antiquark scattering processes ¯σq¯q as a function of temperature at μ=0 MeV for different anisotropy parameters, i.e., ξ=-0.3(orange long dashed line),ξ=0(blue dashed line),and ξ=0.3(red solid line). The gray vertical lines (from left to right) represent the critical temperatures Tc = 180 MeV, 188 MeV, and 197 MeV for ξ=-0.3， 0， 0.3, respectively

Fig. 7 (Color online) Relaxation time of quarks at μ=0 MeV as a function of temperature for different anisotropy parameters, i.e., ξ=-0.3 (orange long dashed line), ξ=0 (blue dashed line), and ξ=0.3 (red solid line)

Fig. 8 (Color online) Temperature dependence of scaled shear viscosity η/T3 in quark matter at vanishing chemical potential for different anisotropy parameters, i.e., ξ=-0.3 (orange long dashed line),ξ=0(blue dashed line), and ξ=0.3(red solid line).The thick cyan dotted line represents the result in the Nf =3 quasiparticle model(QPM)[59],which is an effective model for the description of non-perturbative QCD. The purple dash-dotted line shows the result obtained in the Nf =2 NJL model by Zhuang et al. [57]. The brown dots show the result from hadron resonance gas (HRG) model [70].The green dots correspond to the result of Rehberg et al.in the Nf =3 NJL model [58] using the averaged transition rate method for the estimation of relaxation time

whereas at high temperature, it is primarily governed by the inverse total cross section, even though this effect is largely canceled out by the inverse quark density effect.

Next, we discuss the results regarding various transport coefficients. In Fig. 8, the temperature dependence of scaled shear viscosity η/T3in quark matter for different momentum anisotropy parameters at a vanishing chemical potential is displayed. We observe that with increasing temperature, η/T3first decreases, reaches a minimum around the critical temperature, and increases afterward.The temperature position for the minimum η/T3is consistent with the temperature for the peak ¯σq¯q. This dip structure of η/T3mainly depends on the competition between the quark distribution function f0qand the quark relaxation time τqin the integrand of Eq. (44). The decreasing feature of η/T3in the low temperature domain is governed by τq,whereas in the high temperature domain,the increasing behavior of f0qoverwhelms the decreasing behavior of τq, resulting in η/T3as an increasing function of temperature. In addition, we observe that with an increase in ξ, η/T3has an overall enhancement and the minimum of η/T3shifts to higher temperatures. This behavior of η/T3can be understood from the related expression in Eq. (44), where apart from the ξ-dependent relaxation time, the first term in the integrand of Eq. (44)has an additional ξ factor, which leads to an overall enhancement of the absolute first term at low T. The variation of the first term at low T is larger than the counterpart of the second term, which results in a suppression of η/T3for the inclusion of positive ξ. Meanwhile, at high T, the qualitative and quantitative behavior of η/T3with ξ is dominated by the second term. The location of the minimum for η/T3at different ξ is consistent with the peak position of ¯σq¯q. We further observe that η/T3decreases as ξ increases in the entire temperature region. In addition, we also compare our result for ξ=0 with the results reported in the literature.The calculation of η/T3in the hadron resonance gas (HRG) model [70](brown dots) using the RTA is a decreasing function with temperature, which is qualitatively similar to ours below the critical temperature. The quantitative difference between the HRG model result and ours can be attributed to the use of different degrees of freedom and scattering cross sections. The result of Zhuang et al[57]in the Nf =2 NJL model(purple dash-dotted line)is of the same order of magnitude as ours, whereas at high temperature, their result still has a decreasing feature because an ultraviolet cutoff is used in all momentum integrals whether the temperature is finite or zero. The result estimated in the quasiparticle (QPM) [59] is a logarithmically increasing function of temperature beyond the critical temperature,and it is quantitatively larger than ours beyond the critical temperature owing to the differences in both the effective quark mass and relaxation time.The result of Rehberg et al[58] for the Nf =3 NJL model in the temperature regime close to the critical temperature is smaller than ours, and the obvious dip structure is not observed because the momentum cutoff is also used at finite temperature.

Fig. 9 (Color online) Temperature dependence of scaled electrical conductivity σel/T in quark matter at vanishing chemical potential for different anisotropy parameters, i.e., ξ= -0.3 (orange long dashed line), ξ=0 (blue dashed line), and ξ=0.3 (red solid line).The green dotted line shows the result of Marty et al. in the Nf =3 NJL model[54].The thick gray dash-dotted line represents the result from the pQCD-based microscopic Boltzmann approach to multiparton scatterings (BAMPS) transport model [67] with running coupling constant. The brown stars present the result in the partonhadron-string dynamics (PHSD) transport approach [66]. The cyan dot dashed line shows the result within the excluded volume hadron resonance gas (EVHRG) model with the RTA [71]. The darkyellow dots are the lattice date obtained from Ref.[69].The red open circles are the calculation for hadronic gas in the transport approach simulating many accelerated strongly interacting hadrons (SMASH)[65] based on the Green-Kubo formalism

In Fig. 9, we plot the thermal behavior of scaled electrical conductivity σel/T at μ=0 MeV for different ξ.Similar to the temperature dependence of η/T3,σel/T also exhibits a dip structure in the entire temperature region of interest. We also present the comparison with other previous results.The result obtained from the PHSD approach[66](brown stars),where the plasma evolution is solved by a Kadanoff-Baym type equation, also has a valley structure, even though the location of the minimum is different from ours. We also observe that in the temperature region dominated by the hadronic phase, the thermal behavior of σel/T using the microscopic simulation code SMASH[65](pink open circles) is similar to ours. Furthermore, our result is much larger than the lattice QCD data(dark yellow dots) taken from Ref. [69] owing to the uncertainty in the parameter set and absence of gluonic dynamics. The result within the excluded volume hadron resonance gas(EVHRG)model[71](cyan dash-dotted line)and the result obtained from the partonic cascade BAMPS [67] (gray thick dash-dotted line) are qualitatively and quantitatively similar to our calculations below the critical temperature and beyond the critical temperature, respectively. Our result is similar to that of Marty et al. obtained within the Nf =3 NJL model [54] (green dotted line), with the numerical discrepancy mainly coming from the differences in the values of the model parameter set and scattering cross sections. At low T, the absolute values of both the first and second terms in Eq. (54) increase as ξ increases.However,the variation of the first term is larger than that of the second term,which results in an enhancement of σel/T.At high T, the decreasing feature of relaxation time with ξ can weaken the increasing behavior of σel/T with ξ, and the values of σel/T for different ξ gradually approach and eventually overlap. Our qualitative result of σel/T is different from the result in Ref. [73], where the σel/T of the QGP is a monotonic decreasing function of ξ. This occurs because the effect of momentum anisotropy is not incorporated in the calculation of the relaxation time and the effective mass of quasiparticles, as the ξ dependence of σel/T is only determined by the anisotropic distribution function. We also observe that with the increase in ξ, the minimum of σel/T shifts to higher temperatures, which is similar to η/T3. However, the height of the minimum increases, which is opposite to η/T3.

Fig. 10 (Color online) Temperature dependence of the Seebeck coefficient in quark matter at μ=100 MeV for different anisotropy parameters, i.e., ξ=-0.3 (orange broad dashed line), ξ=0 (blue dashed line), and ξ=0.3 (red solid line). The brown dotted line corresponds to the result for the QGP in the quasiparticle model [83]at μq =50 MeV. The cyan thick-dotted line represents the result in the hadron resonance gas model for μB =0.1 GeV [81]. The mauve dash-dotted line and green dots, respectively, represent the results in the HRG model for μB =50 MeV [80] and the Nf =2 NJL model for μ=100 MeV [84], where the gradient of the quark chemical potential apart from a spatial gradient in temperature is also included

Finally, we study the Seebeck coefficient S in quarkantiquark matter.Owing to the sensitivity of S to the charge of particle species,at a vanishing chemical potential,quark number density nqis equal to antiquark number density n¯q,and the contribution of quarks to S is exactly canceled by that of antiquarks.Thus,a finite quark chemical potential is required to obtain a non-zero thermoelectric current in the medium.In Fig. 10,we plot the variation of S with respect to temperature for different ξ at μ=100 MeV. The comparison with other previous calculations, which were all performed in the kinetic theory under the RTA, is also presented.We remind the reader that at finite μ,nqis larger than n¯q, and the contribution of quarks to total S in magnitude is always prominent. As shown in Fig. 10, the sign of S in our investigation is positive,which indicates that the dominant carriers converting the heat gradient to the electric field are positively charged quarks, i.e., up quarks.Actually, the positive or negative sign of S is mainly determined by the factor （Eq-μq） in the integrand of Eq. (55). In Ref. [83], the Seebeck coefficient studied in the QPM (brown dotted line) at μ =50 MeV also exhibits a decreasing feature with increasing temperature. The result of Abhishek et al [84] at μ=100 MeV in the Nf =2 NJL model(the green dots)is very different from ours.In Ref. [84], S is negative and its absolute value is an increasing function with temperature. The reasons behind this quantitative and qualitative discrepancy are twofold:(1)the relaxation time in Ref.[84]was estimated using the averaged transition rate ¯wij, whereas our relaxation time was obtained from the thermally averaged cross section of elastic scattering (a detailed comparison of the two methods can be found in Ref. [74]); (2) in Ref. [84],the spatial gradient of chemical potential was also included apart from the temperature gradient and accordingly,the sign of S was primarily determined by the factor （Eq-ω/nq） with ω denoting the enthalpy density in the associated formalism.Given that the single-particle energy Eqremains smaller than （ω/nq）, S in Ref. [84] is negative. We also observed that with increasing temperature, S sharply decreased below Tc, whereas the decreasing feature of S was inconspicuous above Tc. Further, the value of S at low T was much larger than that at high T.This is also different from the result in Ref. [84], where the absolute value of S in quark matter increased with increasing temperature because of the increasing behaviors of both the factor |-ω/nq| and the equilibrium distribution function. In addition, the Seebeck coefficient in the HRG model [80, 81] is also positive (negative) without (with) the spatial gradient of chemical potential (cyan thick dotted line and mauve dash-dotted line). Nevertheless, the absolute value of S in hadronic matter is still an increasing function of temperature regardless of the spatial gradient of μ. We also observed that as ξ increases, S has a quantitative enhancement,which is primarily due to a significant rise in the thermoelectric conductivity α,even though 1/σel has a cancelation effect on the increase in S.At sufficiently high temperature, the rise in 1/σelcan almost compensate the reduction in α,and as a result,S varies insignificantly with the ξ of interest, compared to the value of S itself.

8 Summary

We phenomenologically investigated the impact of weak momentum-space anisotropy on the chiral phase structure, mesonic properties, and transport properties of quark matter in the two-flavor NJL model.The momentum anisotropy, which is induced by the initial preferential expansion of the created fireball in heavy-ion collisions along the beam direction, can be incorporated in the calculation through the parameterization of the anisotropic distribution function. Our result has shown that the chiral phase transition is a smooth crossover for vanishing quark chemical potential, independent of the anisotropy parameter ξ,and an increase in ξ can even hinder the restoration of the chiral symmetry. We found that the CEP is highly sensitive to the change in ξ.With the increase in ξ,the CEP shifts to higher μ and smaller T, and the momentum anisotropy affects the CEP temperature to almost the same degree as it affects the CEP chemical potential. Before the merge of π and σ meson masses,the ξ dependence of the π meson mass is opposite to that of the σ meson mass.

We also studied the thermal behavior of various transport coefficients, such as the scaled shear viscosity η/T3,scaled electrical conductivity σel/T, and Seebeck coefficient S at different ξ. The associated ξ-dependent expressions are derived by solving the relativistic Boltzmann-Vlasov transport equation in the relaxation time approximation, and the momentum anisotropy effect is also embedded in the estimate of relaxation time.We found that η/T3and σel/T have a dip structure around the critical temperature. Within the consideration of momentum anisotropy, η/T3decreases as ξ increases and the minimum shifts to higher temperatures.With the increase in ξ,σel/T significantly increases at low temperature, whereas its sensitivity to ξ at high temperature is significantly reduced,which is different from the behavior of η/T3with respect to ξ.We also found that the sign of S at μ=100 MeV was positive, indicating that the dominant carriers for converting the thermal gradient to the electric field are up quarks.With increasing temperature, S first decreases sharply and then almost flattens out.At low temperature,S significantly increases with the increase in ξ, whereas at high temperature,the rise is marginal compared to the value of S itself.

We note that it is of considerable interest to include the Polyakov-loop potential in the present model to study both chiral and confining dynamics in a weakly anisotropic quark matter. A more general ellipsoidal momentum anisotropy characterized by two independent anisotropy parameters is then needed to gain a deeper understanding of the QGP properties.In the present work,no proper time dependence was given to the anisotropy parameter. However, in a realistic case, ξ varies with the proper time starting from the initial proper time up to a time when the system becomes isotropic. Thus, a proper time-dependent anisotropy parameter[119]needs to be introduced to better explore the effect of time-dependent momentum anisotropy on chiral phase transition. For the strongly longitudinal expanding QCD matter, the investigation of chiral phase transition needs to be performed by numerically solving both the Vlasov equation and gap equation concurrently and continuously. In this case, the phase diagram of a strongly expanding system is a map in the space-time plane rather than in the T -μ plane.In addition,the investigation of the thermoelectric coefficients, especially the magneto-Seebeck coefficient and Nernst coefficient in magnetized quark matter, based on the PNJL model would be an attractive direction, and we plan to work on it in the near future.

Appendix

The meson propagators in the above processes are ξ-dependent. Based on the above formulae of three scattering processes, the squared matrix element for the remaining scattering processes can be obtained through charge conjugation and crossing symmetry [57, 120].

Author ContributionsAll authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by He-Xia Zhang, Yu-Xin Xiao and Jin-Wen Kang.The first draft of the manuscript was written by He-Xia Zhang,and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.