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 heavy-ion 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 [7, 6]. 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 μ_B. Currently, there are various QCD inspired effective models, such as the Nambu–Jona–Lasinio (NJL) model [22, 23], Polyakov-loop 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 μ_B. These model calculations have predicted that at high chemical potential, the phase transition is a first-order phase transition, and with decreasing μ_B, 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) [60, 59], and the lattice QCD simulation [61]. The electrical conductivity σ_el, as the response of a medium to an applied electric field, has also attracted attention in high energy physics. The presence of σ_el 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 σ_el 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.

In the initial stages of a HIC, the pressure gradient of the created fireball along the beam direction (denoted as longitudinal z direction) is significantly lower than that along the transverse direction. After the rapid expansion of the medium along the beam direction, the system becomes much colder in the beam direction than the transverse direction, which causes the QGP to possess a local momentum anisotropy, and this anisotropy can survive during the entire evolution of the medium [85]. In addition, the presence of a strong magnetic field can also induce a local anisotropy in the momentum space. Inspired by the presence of momentum-space anisotropy in HICs, the primary objective of the present work is to study phenomenologically its effect on the chiral phase structure, mesonic properties, and transport coefficients in the SU(2) NJL model. To incorporate the momentum anisotropy into numerical calculations, we follow the anisotropic distribution function parametrization method proposed by Romatschke and Strickland [86], $f^{\text{aniso}}(p)\to f^{\text{iso}}(\sqrt{p^2+\xi {(p\cdot n)}^2})$. Here, the isotropic momentum-space distribution function of particle (f^iso) is deformed by rescaling one preferred direction in the momentum space with the unit vector n and introducing the directional dependent parameter ξ, which is used to quantify the degree of momentum-space anisotropy. This method has been extensively employed to study phenomenologically the impacts of momentum anisotropy on various observables, such as photon production [87, 88], parton self-energy [89, 90], heavy-quark potential [91], and various transport coefficients [92-94]. The relativistic anisotropic hydrodynamics (aHydro) models can provide a more accurate description of non-equilibrium dynamics compared to other hydrodynamical models [95]. As in most previous studies, the focus of the present work is on the weakly anisotropic medium (very close to the equilibrium) for which $\left|\xi \right|\ll 1$ and the distribution function can be expanded up to linear order in ξ. We found that even for small values of anisotropy, the effective quark mass and meson masses change significantly compared to the equilibrium result. Unlike most momentum anisotropy studies of transport coefficients in QGP, in which the effect of momentum anisotropy is not considered in the different particle interaction channels, in the present work, we incorporated the momentum anisotropy to the estimation of the relaxation time to better study the impact of ξ on transport properties of quark matter near the phase transition temperature region.

This paper is organized as follows: Sect. 2 provides a brief review of the basic formalism of the 2-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.

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] $\mathcal{L}=\overline{\psi }(i\partial -{\widehat{m}}_0)\psi +G[(\overline{\psi }\psi {)}^2+{(\overline{\psi }i{\gamma }_5\widehat{\tau }\psi )}^2]\mathrm{,}$ (1) where $\psi (\overline{\psi })$ stands for the quark (antiquark) field with two-flavors (u,d) N_f =2 and three colors N_c=3. ${\widehat{m}}_0$ denotes the diagonal matrix of the current quark mass of up and down-quarks, and we take $m_0=m_u^0=m_d^0$ to ensure isospin symmetry of the NJL lagrangian. G is the effective coupling strength of four-point fermion interaction in the scalar and pseudoscalar channels. $\widehat{\tau }$ 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] $m_q=m_0-2Gi\text{Tr}i{S}^{<}(x\mathrm{,}x)\mathrm{,}$ (2) where $S^{<}(x\mathrm{,}y)=i\langle \overline{\psi }(y)\psi (x)\rangle$ with x=(t,x) is the real time Green function in the coordinate space [97, 98], ⟨⋅s⟩ denotes the thermal average, and the trace runs over spin, color, and flavor degrees of freedom. Transforming Eq. (2) to the phase space with the help of Winger transformation and introducing the quasiparticle approximation (see Ref. [99] for details), the gap equation can be further written as [96, 98-100] $m_q=m_0+4N_{\text{f}}{N}_{\text{c}}{\displaystyle \int }\frac{d^{3}p}{{(2\pi )}^3}\frac{m_{\text{q}}}{E_p}(1-f_q(x\mathrm{,}p)-{\overline{f}}_{\overline{q}}(x\mathrm{,}p))\mathrm{,}$ (3) where $E_p=\sqrt{p^2+m_q^2(x)}$ is the quasi-quark energy. As the NJL model is a non-renormalizable model owing to the point-like four fermion interaction in the lagrangian, an ultraviolet cutoff Λ is used to regularize the divergent integral. In the non-equilibrium case, the space–time evolution of the one-particle distribution function f(x,p) in Eq. (3) is described by the Boltzmann–Vlasov transport equation from the NJL model in the Hartree level [100-102]. By solving the Vlasov equation together with the gap equation concurrently, the constituent quark mass affecting the space–time dependence of f(x,p) can be determined self-consistently.

To better understand the meson dynamics in HICs, it is useful to study the structure of the meson propagation in the medium. In the framework of the NJL model, mesons are bound states of quarks and antiquarks (collective modes), and the meson propagator can be constructed by calculating the quark–antiquark effective scattering amplitude within the random phase approximation (RPA) [23, 96, 98]. Following Refs. [96, 98], the explicit form for the pion (π) and the sigma meson (σ) propagators in the RPA reads as $D_M(x\mathrm{,}{k}_0\mathrm{,}k)=\frac{2iG}{1-2G{\Pi }_M(x\mathrm{,}{k}_0\mathrm{,}k)}\mathrm{.}$ (4) All information of a meson is contained in the irreducible one-loop pseudoscalar or scalar polarization function ∏M, 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] $\begin{array}{c}{\Pi }_M=N_{\text{c}}{N}_{\text{f}}{\displaystyle \int }\frac{d^{3}p}{4{\pi }^3}\frac{1}{E_p}\left[1-f_q(x\mathrm{,}p)-f_{\overline{q}}(x\mathrm{,}p)\right]\\ \times \{1-\frac{(k_0^2-k^2-{\nu }_M^2)}{2E_{p-k}}[\frac{E_p+E_{p-k}}{k_0^2-{(E_p+E_{p-k})}^2}\\ -\frac{E_p-E_{p-k}}{k_0^2-{(E_p-E_{p-k})}^2}]\}\mathrm{.}\end{array}$ (5) Here, νM denotes the real value of the bound meson energy, and νM=0 (2mq) for the π (σ) meson.

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 space–time 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 potential-dependent, can be directly calculated from the self-consistent gap equation in momentum space [23, 22]: $m_q=m_0+4G{N}_{\text{f}}{N}_{\text{c}}{\displaystyle \int }\frac{d^{3}p}{{(2\pi )}^3}\frac{m_q}{E_p}(1-f_q^0(p)-{\overline{f}}_{\overline{q}}^0(p))\mathrm{.}$ (6) We can see that by taking the thermal equilibrium distribution functions (Fermi–Dirac distributions), Eq. (3) has the same form as Eq. (6). The equilibrium distribution function of (anti-)quark $f_{q(\overline{q})}^0$ is given by $f_{q(\overline{q})}^0(p)=[\exp [(E_p-{\mu }_{q(\overline{q})})\beta ]+{1]}^{-1}\mathrm{,}$ (7) where β=1/T is the inverse temperature of the system, and a uniform quark chemical potential $\mu \equiv {\mu }_{u\mathrm{,}d}\equiv -{\mu }_{\overline{u}\mathrm{,}\overline{d}}$ is assumed. It is noted that the ultraviolet divergence is not presented in the integrand containing Fermi–Dirac distribution functions, so the momentum integral do not need to be regularized for finite temperatures. In the equilibrium, the meson propagator is given as [57] $D_M(k_0\mathrm{,}k)=\frac{2iG}{1-2G{\Pi }_M(k_0\mathrm{,}k)}\mathrm{,}$ (8) where ∏M at an arbitrary temperature and quark chemical potential is given by [103, 104] $\begin{array}{c}{\Pi }_M(k_0\mathrm{,}k)=-\frac{N_{\text{c}}{N}_{\text{f}}}{8{\pi }^2}[((m_q\mp m_q{)}^2-k_0^2+k^2)\\ \times B_0(k_0\mathrm{,}k\mathrm{,}\mu \mathrm{,}T\mathrm{,}{m}_q)+2A(\mu \mathrm{,}T\mathrm{,}{m}_q)]\mathrm{.}\end{array}$ (9) The minus (plus) sign refers to pseudoscalar (scalar) mesons. The function A, which relates to the one-fermion-line integral, in the imaginary time formalism for finite temperatures and quark chemical potentials is given as [103] $A=\frac{16{\pi }^2}{\beta }{\displaystyle \sum _n}\exp (i{\omega }_{n}y){\displaystyle \int }\frac{d^{3}p}{{(2\pi )}^3}\frac{1}{{(i{\omega }_n+\mu )}^2-E_p^2}\mathrm{,}$ (10) 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 $y\to 0$ is to be taken after the Matsubara summation. The function B₀ in Eq. (9) relates to the two-fermion-line integral. At finite μ and T, B₀ is defined as [103] $\begin{array}{l}{B}_0(i{\nu }_l\mathrm{,}k\mathrm{,}\mu \mathrm{,}T\mathrm{,}{m}_q)\\ =\frac{16{\pi }^2}{\beta }{\displaystyle \sum _n}\exp (i{\omega }_{n}y){\displaystyle {\int }_{\left|\overrightarrow{p}\right|<\Lambda }}\frac{d^{3}p}{{(2\pi )}^3}\frac{1}{((i{\omega }_n+\mu {)}^2-E^2)}\times \frac{1}{((i{\omega }_n-i{\nu }_l+\mu {)}^2-E_{}{\text{'}}^2)}\mathrm{,}\end{array}$ (11) where we have abbreviated $E^{\prime }=E_{p-k}=\sqrt{{(p-k)}^2+m_q^2}$ and E=E_p for convenience, and after the Matsubara summation on n is carried out, the complex frequencies iνl are analytically continued to their values on the real plane, i.e., $i{\nu }_l\to k_0-iϵ$ (ϵ>0) with k₀ being the zero component of the associated four-momentum. The full calculations of functions A and B₀ at arbitrary values of temperature and chemical potential can be found in Ref. [105]. After evaluating the Matsubara summation by contour integration in the usual fashion [106], Eq. (10) is given as $A=8{\pi }^2{\displaystyle \int }\frac{d^{3}p}{{(2\pi )}^3{E}_p}(f_q^0+f_{\overline{q}}^0-1)\mathrm{,}$ (12) where the Fermi–Dirac distribution function is introduced. Similar to the treatment of function A, after Matsubara summation over n and the Matsubara frequencies iνl are analytically continued to real values, Eq. (11) can be rewritten in the following form: $\begin{array}{l}{B}_0(k_0\mathrm{,}k\mathrm{,}\mu \mathrm{,}T\mathrm{,}{m}_q)=16{\pi }^2{\displaystyle \int }\frac{d^{3}p}{{(2\pi )}^3}\frac{f_q^0(p)+f_{\overline{q}}^0(p)-1}{2E}\\ \times \left[\frac{1}{{(E+i{\nu }_l)}^2-{(E^{\prime })}^2}+\frac{1}{(E-i{\nu }_l)-{(E^{\prime })}^2}\right]\mathrm{.}\end{array}$ (13) Inserting Eqs. (12) and (13) to Eq. (9), we finally can obtain the expression for ∏M in 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.

Constituent quark and meson in a weakly anisotropic medium

As mentioned in the introduction, the consideration of momentum anisotropy induced by rapid expansion of the hot QCD medium for existing phenomenological applications is mostly achieved by parameterizing the associated isotropic distribution functions. To proceed with the numerical calculation, a specific form of anisotropic (non-equilibrium) momentum distribution function is required. In this work, we utilized the Romatschke–Strickland (RS) form [86] in which the system exhibits a spheroidal momentum anisotropy, and the anisotropic distribution was obtained from an arbitrary isotropic distribution function by removing the particle with a momentum component along the direction of anisotropy, $f^{\text{aniso}}(p)=f^{\text{iso}}(\sqrt{p^2+\xi {(p\cdot n)}^2})$, where n and ξ are, respectively, the anisotropy direction and anisotropy parameter. As in previous studies [107], we shall restrict ourselves here to a plasma close to local thermal equilibrium, i.e., close to isotropy in the momentum space. Accordingly, the explicit form of anisotropic momentum distribution function in the local rest frame can be written as $f^{\text{aniso}}(p)=\frac{1}{\exp [(\sqrt{p^2+\xi {(p\cdot n)}^2+m_q^2}-{{\mu }^{\prime }}_{q(\overline{q})})/T^{\prime }]+1}\mathrm{,}$ (14) It is worth noting that for the anisotropic (non-equilibrium) matter, the T’ and μ’ appearing in Eq. (14) lose the usual meaning of T and μ in the equilibrium system and may have dimensionful scales related to the mean particle momentum [108]. If we assume the system to be very close to the equilibrium (in the small anisotropy limit), then the parameters T’ and μ’ still could be taken to be the usual T and μ, respectively, as performed in previous studies [92-94, 107]. The anisotropy parameter ξ is defined as $\xi =\langle p_{\perp }^2\rangle /(2\langle p_{\parallel }^2\rangle )-1$, where $p_{\perp }=|p-(p\cdot n)\cdot n|$ and $p_{\parallel }=p\cdot n$ are the momentum components of particles perpendicular and parallel to n, respectively. As the precise time evolution of ξ is still an open question, we assume that the anisotropy parameter ξ in a local anisotropic system is constant and independent of time, where -1<ξ<0 corresponds to a contraction of momentum distribution along the direction of anisotropy and ξ>0 corresponds to a stretching of momentum distribution along the direction of anisotropy. In Eq. (14), the three-velocity of partons and anisotropy unit vector are selected as $n=(\sin \chi \mathrm{,0,}\cos \chi )\mathrm{,}$ (15) $p=p(\sin \theta \cos \varphi \mathrm{,}\sin \theta \sin \varphi \mathrm{,}\cos \theta )\mathrm{,}$ (16) where χ is the angle between p and n, and $p\equiv \left|p\right|$ throughout the computations. With this choice, the spheroidally anisotropic term, ξ(n⋅p)² in Eq. (14) can be written as $\xi {(n\cdot p)}^2=\xi p^2{(+)}^2=\xi c(\theta \mathrm{,}\varphi \mathrm{,}\chi )$. 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 ($\left|\xi \right|\ll 1$), 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] $f^{\text{aniso}}(p)=f^0-\frac{\xi {(n\cdot p)}^2}{2E_{}T}{f}^0(1-f^0)\mathrm{,}$ (17) 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] $\xi =\frac{10}{T\tau }\frac{\eta }{s}\mathrm{,}$ (18) 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 $T_{\text{c}}\approx 160$ MeV, $\tau \approx 6$ fm/c and η/s=1/4π, we can obtain $\xi \approx 0.3$. 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 space–time evolution is not addressed. Therefore, in the non-equilibrium 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., Eq. (2) or Eq. (6), is modified as $\begin{array}{c}0=\left[2N_{\text{c}}{N}_{\text{f}}G{\displaystyle {\int }_0^{\infty }}\frac{p^{2}dp}{{\pi }^2}\frac{m_q}{E}(-f_q^0-f_{\overline{q}}^0+1+\frac{p^2\xi F_{\text{p}}}{6E_{}T})\right]\\ +m_0-m_q\mathrm{.}\end{array}$ (19) Here, we have abbreviated $F_p=f_q^0(1-f_q^0)+f_{\overline{q}}^0(1-f_{\overline{q}}^0)$ for convenience. The momentum anisotropy is also embedded in the study of mesonic properties by substituting the anisotropic momentum distribution in the A function part of Eq. (5), thus obtaining $A=4{\displaystyle \int }\frac{p^{2}dp}{E}[f_q^0+f_{\overline{q}}^0-1-\frac{\xi p^2{F}_p}{6ET}]\mathrm{.}$ (20) In the $\xi \to 0$ limit, the above equation reduces to Eq. (12). Similar to the treatment of function A, the weak momentum anisotropy effects can enter the B₀ function (as given in Eq. (5) or Eq. (13)) by using the anisotropic momentum distribution. Without loss of generality, we selected the coordinate system in such a way that k is parallel to the z-axis, i.e., $k=(\mathrm{0,0,}k)\mathrm{,}\left|k\right|\equiv k\mathrm{.}$ (21) We first discuss a simple case, i.e., k=0, k₀≠0. The computation of function B₀ in a weakly anisotropic medium is trivial, viz, $\begin{array}{c}{B}_{0}s=8{\pi }^2{\displaystyle \int }\frac{d^{3}p}{{(2\pi )}^{3}E}(f_q^{\text{aniso}}(p)+f_{\overline{q}}^{\text{aniso}}(p)-1)\\ \times \left(\frac{1}{k_0^2-2E{k}_0+iϵ\text{sgn}(k_0)}+\frac{1}{k_0^2+2E{k}_0-iϵ\text{sgn}(k_0)}\right)\mathrm{.}\end{array}$ (22) In the integrand of the above equation, there are two poles at E=E₀=± k₀/2 if $m\le E_0$. Applying the Cauchy formula ${\text{lim}}_{ϵ\to 0}\frac{1}{x-iϵ}=\mathcal{P}\frac{1}{x}+i\pi \delta (x)\mathrm{,}$ (23) where 𝒫 denotes the Cauchy principal value. The function B₀ can finally be rewritten as $\begin{array}{c}{B}_0=8\mathcal{P}{\displaystyle \int }\frac{p^{2}dp}{E(k_0^2-4E^2)}[f_q^0(p)+f_{\overline{q}}^0(p)-1-\frac{\xi p^2}{6ET}{F}_p]\\ -i\pi \frac{2}{k_0}[(f_q^0(z_0)+f_{\overline{q}}^0(z_0)-1-\frac{\xi z_0^2}{3k_{0}T}{F}_{z_0})\\ \times \sqrt{{(\frac{k_0}{2})}^2-m_q^2}\Theta (\frac{k_0^2}{4}-m_q^2)]\mathrm{.}\end{array}$ (24) Here, $z_0=\sqrt{{(\frac{k_0}{2})}^2-m_q^2}$, and Θ is the step function to ensure that the imaginary part appears only for k₀/2>mq. Next, in the case of k>0, k₀≠ 0, the expression for function B₀ is slightly complicated and can be written as $\begin{array}{c}{B}_0=\frac{1}{k}{\displaystyle {\int }_0^{\infty }}\frac{pdp}{E}(f_q^{\text{aniso}}(p)+f_{\overline{q}}^{\text{aniso}}(p)-1))\\ \times {\displaystyle {\int }_{-1}^1}dx(\frac{1}{x+(k_0^2+2k_{0}E-k^2)/2pk-iϵ\text{sgn}(k_0)}\\ +\frac{1}{x+(k_0^2-2k_{0}E-k^2)/2pk+iϵ\text{sgn}(k_0)})\\ =B_0^{\text{an}}+B_0^{\text{iso}}\mathrm{,}\end{array}$ (25) with the abbreviation x=cosθ. The isotropic part $B_0^{\text{iso}}$ reads as $\begin{array}{l}{B}_0^{\text{iso}}=\frac{1}{k}{\displaystyle {\int }_0^{\infty }}\frac{pdp}{E}(f_q^0+f_{\overline{q}}^0-1)\\ \times [\log |\frac{{(k_0^2-k^2+2pk)}^2-{(2k_{0}E)}^2}{{(k_0^2+k^2-2pk)}^2-{(2k_{0}E)}^2}|\\ +i\pi (\Theta (2pk-|k_0^2+2k_{0}E-k^2|)\\ -\Theta (2pk-|k_0^2-2k_{0}E-k^2|))]\mathrm{.}\end{array}$ (26) The anisotropic part $B_0^{\text{an}}$ reads as $\begin{array}{l}{B}_0^{\text{aniso}}=-\frac{1}{k}{\displaystyle {\int }_0^{\infty }}\frac{pdp}{E}\frac{\xi p^2{F}_p}{2ET}[(-\frac{\left(k_0^2-k^2+2k_{0}E\right)}{pk}\\ +{\left(\frac{k_0^2-k^2+2k_{0}E}{2pk}\right)}^2\log \left|\frac{k_0^2+2k_{0}E-k^2+2pk}{k_0^2+2k_{0}E-k^2-2pk}\right|\\ -\frac{\left(k_0^2-k^2-2k_{0}E\right)}{pk}+{\left(\frac{k_0^2-k^2-2k_{0}E}{2pk}\right)}^2\\ \log \left|\frac{k_0^2-2k_{0}E-k^2+2pk}{k_0^2-2k_{0}E-k^2-2pk}\right|)\\ +i\pi (\frac{k_0^2+2k_{0}E-k^2}{2pk}\Theta \left(2pk-\left|k_0^2+2k_{0}E-k^2\right|\right)\\ -\frac{k_0^2-2k_{0}E-k^2}{2pk}\Theta \left(2pk-\left|k_0^2-2k_{0}E-k^2\right|\right))]\mathrm{.}\end{array}$ (27) 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 $\begin{array}{l}\text{Re}{\Pi }_M(k_0\mathrm{,0})=N_{\text{f}}{N}_{\text{c}}\mathcal{P}{\displaystyle {\int }_0^{\infty }}\frac{dp{p}^2}{{\pi }^{2}E}\\ [1-f_q^0(p)-f_{\overline{q}}^0(p)+\frac{\xi p^2}{6ET}{F}_p]\frac{E^2-{\nu }_M^2/4}{E^2-{(k_0/2)}^2}\mathrm{,}\end{array}$ (28) $\begin{array}{c}\text{Im}{\Pi }_M(k_0\mathrm{,0})=\frac{N_{\text{c}}{N}_{\text{f}}}{8\pi k_0}\sqrt{k_0^2-4m_q^2}\left(k_0^2-{\nu }_M^2\right)\\ \left[1-f_q^0(z_0)-f_{\overline{q}}^0(z_0)+\frac{z_0^2\xi }{3k_{0}T}{F}_{z_0}\right]\Theta \left(k_0^2-4m_q^2\right)\mathrm{,}\end{array}$ (29) $\begin{array}{l}\text{Re}{\Pi }_M(\mathrm{0,}k)=\frac{N_{\text{c}}{N}_{\text{f}}}{{\pi }^2}{\displaystyle {\int }_0^{\infty }}\frac{dp{p}^2}{E}\left(1+\frac{k^2+{\nu }_M^2}{4pk}\ln \left|\frac{k-2p}{k+2p}\right|\right)\\ \left(1-f_q^0(p)-f_{\overline{q}}^0(p)\right)+\frac{N_c{N}_f}{{\pi }^2}{\displaystyle {\int }_0^{\infty }}\frac{\xi p^{2}dp}{E^{2}T}{F}_p\\ \left[\frac{p^2}{6}+\frac{k^2+{\nu }_M^2}{4}\left(1+\frac{k^2}{4pk}\ln \left|\frac{k-2p}{k+2p}\right|\right)\right]\mathrm{.}\end{array}$ (30) Similar to the treatment in Eq. (6), the vacuum divergence in Eqs. (28)-(30) also need to be regularized. When the effect of momentum anisotropy is turned off (ξ=0), Eqs. (28)–(30) are reduced to the results of Ref. [57] in thermal equilibrium. Once the propagators of mesons are given, their masses can be determined by the pole in Eq. (8) at zero three-momentum, i.e., $1-2G\text{Re}{\Pi }_M(m_{\pi \mathrm{,}\sigma }\mathrm{,0})=0.$ (31) 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 ∏M has an imaginary part that is related to the decay width of the resonance as ${\Gamma }_M=\text{Im}{\Pi }_M(m_{\pi \mathrm{,}\sigma }\mathrm{,0})/m_{\pi \mathrm{,}\sigma }$.

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.

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] $\begin{array}{c}{\tau }_l^{-1}(T\mathrm{,}\mu )=n_{\overline{q}}[{\overline{\sigma }}_{u\overline{u}\to u\overline{u}}+{\overline{\sigma }}_{u\overline{u}\to d\overline{d}}+{\overline{\sigma }}_{u\overline{d}\to u\overline{d}}]\\ +n_q[{\overline{\sigma }}_{ud\to ud}+{\overline{\sigma }}_{uu\to uu}]\mathrm{,}\end{array}$ (57) where the number density of (anti-)quarks in a weakly anisotropic medium is given as $n_{q(\overline{q})}=d_l{\displaystyle \int }\frac{d^{3}p}{{(2\pi )}^3}{f}_{q(\overline{q})}^{an}$, with $d_l=d_{\overline{l}}=2N_{\text{c}}$ denoting the degeneracy factor. The momenta of the colliding particles for the elastic scattering process $a(p_1)+b(p_2)\to c(p_3)+d(p_4)$ obey the relation $p_1+p_2=p_3+p_4=0$, and we use the notation $\left|p_1\right|=\left|p_2\right|=p$ for convenience. In the center-of-mass (cm) frame, the Mandelstam variables s, t, u are given by $\begin{array}{l}s=4m_q^2+4p^2\mathrm{,}t=-2p^2(1-\cos {\theta }_p)\mathrm{,}\\ u=-2p^2(1+\cos {\theta }_p)\mathrm{,}\end{array}$ (58) where θp is the scattering angle in the c.m. frame. The Mandelstam variables hold the relation $u+s+t=4m_q^2$. ${\overline{\sigma }}_{ab\to cd}$ denotes the thermal-averaged elastic scattering cross section in the weakly anisotropic system, which can be written as $\begin{array}{c}{\overline{\sigma }}_{ab\to cd}={\displaystyle {\int }_{s_0}^{\infty }}ds{\displaystyle {\int }_{t_{\text{min}}}^{t_{\text{max}}}}dt\frac{d{\overline{\sigma }}_{ab\to cd}}{dt}{{\displaystyle \sin }}^2{\theta }_p{\displaystyle {\int }_{-1}^1}d{x}_3\\ \times (1-f_c^{\text{aniso}}(p_{\text{cm}}\mathrm{,}\mu \mathrm{,}{x}_3)){\displaystyle {\int }_{-1}^1}d{x}_4\\ \times (1-f_d^{\text{aniso}}(p_{\text{cm}}\mathrm{,}\mu \mathrm{,}{x}_4))\mathcal{L}(s\mathrm{,}\mu \mathrm{,}{x}_1\mathrm{,}{x}_2)\mathrm{,}\end{array}$ (59) with $(1-f_{c\mathrm{,}d}^{\text{an}})$ denoting the Pauli-blocking factor for the fermions due to the fact that some of the final states are already occupied by other identical (anti-)quarks. $\frac{d\overline{\sigma }}{dt}=\frac{1}{16\pi s(s-4m_q^2)}|\overline{M}{|}^2$ is the differential scattering cross section, with $|\overline{M}{|}^2$ denoting the squared matrix element of a specific scattering process. The formulae of the squared matrix elements for various scattering processes are presented in the Appendix. The integration limits of t are t_max=0 and $t_{\text{min}}=-4p_{\text{cm}}^2=-(s-4m_q^2)$, with $p_{\text{cm}}=(\sqrt{s-4m_q^2})/2$ denoting the momentum in the c.m. frame. The kinematic boundary of s reads $s_0=4m_q^2$. The scattering weighting factor ${{\displaystyle \sin }}^2{\theta }_p=\frac{-4t(s+t-4m_q^2)}{{(s-4m_q^2)}^2}$ is introduced to exclude the scattering processes with a small initial angle because the large angle scattering dominates in the momentum transport process [116]. In the c.m. frame, the leading-order anisotropic distribution function can be rewritten as $\begin{array}{c}{f}^{\text{aniso}}(p_{\text{cm}}\mathrm{,}\mu \mathrm{,}x)=f^0(p_{\text{cm}}\mathrm{,}\mu )\\ -\frac{p_{\text{cm}}^2\xi x^2}{2E_{\text{cm}}T}{f}^0(p_{\text{cm}}\mathrm{,}\mu )(1-f^0(p_{\text{cm}}\mathrm{,}\mu ))\mathrm{,}\end{array}$ (60) where $E_{\text{cm}}=\frac{s-m_q^2+m_q^2}{2\sqrt{s}}=\sqrt{s}/2$. In Eq. (59), $\mathcal{L}$ denotes the probability of finding a quark-(anti)quark pair with the center of mass energy $\sqrt{s}$ in the anisotropic medium, and it is given by $\begin{array}{c}\mathcal{L}=C\sqrt{s(s-4m_q^2)}{f}_a^{\text{aniso}}(p_{cm}\mathrm{,}\mu \mathrm{,}{x}_1)f_b^{\text{aniso}}(p_{cm}\mathrm{,}\mu \mathrm{,}{x}_2)\\ \times v_{\text{rel}}(s)\mathrm{,}\end{array}$ (61) where $v_{\text{rel}}(s)=\sqrt{\frac{s-4m_q^2}{s}}$ is the relative velocity between two incoming particles in the c.m. frame; C is the normalization constant, which is determined from the requirement that ${\displaystyle {\int }_{s_0}^{\infty }}ds{\displaystyle {\int }_{-1}^1}d{x}_1{\displaystyle {\int }_{-1}^1}d{x}_2\mathcal{L}=1$, where $x_1=\cos {\theta }_1$ and $x_2=\cos {\theta }_2$, with θi being the angle between pi and n and $x_1\equiv -x_3$, $x_2\equiv -x_4$. Applying the above formula of relaxation time to Eqs. (44), (54), and (55), we can calculate the transport coefficients in the QCD medium and study their sensitivity to the momentum anisotropy.

Results and discussion

Throughout this work, the following parameter set is used: m₀=m_0,u=m_0,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 m_0,π=135 MeV, pion decay constant fπ=92.4 MeV, and quark condensate $-{\langle \psi \psi \rangle }^{1/3}=241$ 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. 1 (a), we show the thermal behavior of the light constituent quark mass mq for vanishing quark chemical potential at different ξ. For low temperature, mq remains approximately constant at ($m_q\approx$ 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. 1Full-screen View

(Color online) (a) Temperature dependence of the constituent quark mass mq at μ=0 MeV for different fixed anisotropy parameters. (b) Chiral susceptibility χ_ch 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.

In this work, the chiral critical temperature, T_c, was determined by the peak location of the associated chiral susceptibility χ_ch, which is defined as ${\chi }_{\text{ch}}=\left|\frac{d{m}_q}{dT}\right|$. We stress that the criterion of obtaining the chiral critical temperature is different in different studies. Because of some shortcomings of the NJL model, such as parameter ambiguity, nonrenormalization, and the absence of gluonic dynamics, the value of T_c in the present work is not expected to describe the lattice QCD result quantitatively. However, these shortcomings cannot affect our present qualitative results. The temperature dependence of chiral susceptibility χ_ch for different ξ at μ=0 MeV is plotted in Fig. 1 (b). We observe that T_c exhibits a significant ξ dependence. As ξ increases, T_c shifts toward higher temperatures and the height of the peak decreases. The locations of T_c for ξ=-0.3, 0, 0.3 are ~180 MeV, 188 MeV, and 197 MeV, respectively, which means a change of approximately 10% in temperature. Actually, the in-medium meson masses also can be regarded as a signature of chiral phase transition. In Fig. 2(a), we display the variation of π and σ meson masses with temperature for different ξ at μ=0 MeV. As can be observed, the π mass remains approximately constant up to a particular temperature whereas the σ mass first decreases and then increases. As temperature increases further, the difference between the π mass and σ mass decreases and finally vanishes, where σ and π mesons are degenerate and become nonphysical degrees of freedom, which indicates the restoration of chiral symmetry. Further, before π and σ meson masses emerge, the π mass decreases as ξ increase beyond T_c, whereas the σ mass first increases and then decreases with the increase in ξ. Our result slightly differs from that in Ref [40], in which the NJL model is used for a quark matter of finite size. The study shows that below the critical temperature, the π mass enhances as the system size decreases, whereas the σ mass first remains unchanged and then increases. We also see that at a certain temperature, the double constituent quark mass (2mq) is equal to the π mass, so the pion meson is no longer a bound state but only a $q\overline{q}$ resonance and obtains a finite decay width. Accordingly, the Mott transition temperature by the definition $m_{\pi }(T_{\text{Mott}})=2m_q(T_{\text{Mott}})$ can be obtained. The Mott temperatures for ξ=-0.3, 0, and 0.3 turn out to be ~187 MeV, 196 MeV, and 206 MeV, respectively, which are slightly higher than the corresponding T_c. In the vicinity of the Mott temperature, the σ meson features its minimal mass. In Fig. 2(b), we illustrate the variation in the decay widths of both σ and π mesons with temperature for different ξ. As can be observed, the decay width of the σ meson, Γ_σ, is finite in the entire temperature range whereas the decay width of the π meson, Γ_π, starts after the Mott temperature. At high temperature, the merging behaviors of the decay widths for different mesons are also observed. With the increase in ξ, the decay widths of mesons are reduced.

Fig. 2Full-screen View

(Color online) (a) Double mass of the constituent quarks 2m_q (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.

We continue the analysis in the finite quark chemical potential case to investigate the effect of momentum anisotropy on the phase boundary and CEP position. First, we display the temperature- and quark chemical potential-dependence of constituent quark mass mq for different anisotropy parameters, as shown in Fig. 3. We can observe that at a small μ, mq continuously decreases with increasing T, whereas mq has a significant discontinuity or a sharp drop along the T-axis at sufficiently high μ, which is usually considered as the appearance of a first-order phase transition. To visualize the phase diagram, we use the significant divergency of χ_ch at sufficiently high chemical potential as the criterion for a first-order phase transition, as shown in Fig. 4. With the decrease in μ, the first-order phase transition terminates at a CEP, where the phase transition is expected to be of second order. As μ decreases further, the maximum of the chiral susceptibility (χ_ch) as the crossover criterion. The full chiral boundary lines in the (μ-T) plane for three different values of ξ are displayed in Fig. 5. We observe that as ξ increases, the phase boundary shifts toward higher quark chemical potentials and higher temperatures. We observe that the CEP appears in the low temperature and high chemical potential regions. Once the effect of momentum anisotropy (ξ>0) is taken into account, the rapid expansion and fast cooling of the created fireball along the beam direction make the temperature of the anisotropic system lower than that of the isotropic system under the same conditions. By modifying the distribution function in the gap equation, we can study the influence of momentum anisotropy on the position of the CEP. It is noted that the criteria for determining the CEP position remain the same for both isotopic and anisotropic systems. Therefore, as ξ increases, the momentum component (temperature) in the anisotropic distribution function $f^{\text{iso}}(\sqrt{p^2+\xi {(p\cdot n)}^2})$ increases (decreases). Accordingly, the chemical potential in the anisotropic function for determining the CEP position is greater than that in the isotropic distribution function. The CEP locations ($T_{\text{CEP}}\mathrm{,}{\mu }_{\text{CEP}}$) in this work are presented at (298.70 MeV, 88.2 MeV), (321.8 MeV, 82.4 MeV), (348.4 MeV, and 74.2 MeV) for ξ=-0.3, 0, 0.3, respectively. The position of the CEP for ξ=0 in this work is almost consistent with the existing result [118] for the same parameter set. The value of μ_CEP (T_CEP) from ξ=-0.3 to ξ=0.3 increases (decreases) by approximately 16% (17%), indicating that the influence degree of momentum anisotropy on the temperature of the CEP is almost the same as that on the quark chemical potential of the CEP. This is different from the effect due to the finite volume in Ref. [38], which has indicated that in the PNJL model, the finite volume affects the CEP shift along the temperature stronger than along the quark chemical potential shift. When the system size is reduced to 2 fm, the CEP in the PNJL model vanishes and the whole chiral phase boundary becomes a crossover curve. Based on this result, there also exists a possibility that if ξ further increases, the CEP may disappear from the phase diagram.

Fig. 3Full-screen View

(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. 4Full-screen View

(Color online) Three-dimensional plot of chiral susceptibility χ_ch for ξ=-0.3 in the entire μ and T ranges of interest. The gray area means that χ_ch 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. 5Full-screen View

(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 towards larger values of the quark chemical potential but smaller values of the temperature for higher anisotropy parameters.

To better understand the qualitative behavior of the transport coefficients in the quark matter, we first discuss the results of the scattering cross sections and the relaxation time. In Fig 6, we display the total cross section of quark–quark scattering processes ${\overline{\sigma }}_{qq}={\overline{\sigma }}_{uu\to uu}+{\overline{\sigma }}_{ud\to ud}$ (plot a) and the total cross section of quark–antiquark processes ${\overline{\sigma }}_{q\overline{q}}={\overline{\sigma }}_{u\overline{u}\to u\overline{u}}+{\overline{\sigma }}_{u\overline{d}\to u\overline{d}}+{\overline{\sigma }}_{u\overline{u}\to d\overline{d}}$ (plot b) as functions of temperature at different anisotropy parameters for vanishing quark chemical potential. As can be observed, ${\overline{\sigma }}_{qq}$ and ${\overline{\sigma }}_{q\overline{q}}$ have similar peak structures in their temperature dependence, i.e., the scattering cross sections first increase, reach a peak, and decrease with increasing temperature afterwards. In addition, the magnitude of ${\overline{\sigma }}_{q\overline{q}}$ is higher than that of ${\overline{\sigma }}_{qq}$, and this is mainly due to the additional s-channel contribution to a resonance of the exchanged meson with the incoming quark and antiquark, which leads to a large peak in the cross section [74]. We further observe that the scattering cross sections in the weakly anisotropic medium have the same behaviors as those in the isotropic medium. As ξ increases, ${\overline{\sigma }}_{qq}$ increases in the entire considered temperature domain, whereas ${\overline{\sigma }}_{q\overline{q}}$ first decreases as ξ increases, and then increases as ξ increases. With an increase in ξ, the maximum of the scattering cross section shifts toward higher temperatures. The location of the maximum for ${\overline{\sigma }}_{qq}$ at different ξ is nearly in agreement with the respective T_c, as the peak positions of ${\overline{\sigma }}_{q\overline{q}}$ respectively locate at $\sim 1.07T_{\text{c}}^{-0.3}$, $1.07T_{\text{c}}^0$, $1.10T_{\text{c}}^{0.3}$ for ξ=-0.3, 0, 0.3, with $T_{\text{c}}^{\xi }$ denoting the chiral critical temperature for a fixed ξ.

Fig. 6Full-screen View

(Color online) (a) Total cross section of the quark–quark scattering processes ${\overline{\sigma }}_{qq}$ as a function of temperature at μ=0 MeV for different anisotropy parameters. (b) The total cross section of quark-antiquark scattering processes ${\overline{\sigma }}_{q\overline{q}}$ 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 T_c= 180 MeV, 188 MeV, and 197 MeV for ξ=-0.3, 0, 0.3, respectively.

The dependence of total quark relaxation time τq on temperature for vanishing quark chemical potential at different ξ is displayed in Fig. 7. As can be observed, τq first decreases sharply with increasing temperature, and after an inflection point (viz, the peak position of ${\overline{\sigma }}_{q\overline{q}}$), τq changes modestly with temperature. Further, the increase in τq with ξ is significant at low temperature, whereas at high temperature the reduction in τq with 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 τq is mainly determined by the inverse quark number density, whereas at high temperature it is primarily governed by the inverse total cross section, even though this effect is largely cancelled out by the inverse quark density effect.