Phenomenological study of the anisotropic quark matter in the 2-flavor Nambu–Jona–Lasinio model

Phenomenological study of the anisotropic quark matter in the 2-flavor Nambu–Jona–Lasinio model 1 2 corresp first-author † zhanghexia@mails.ccnu.edu.cn † zhanghexia@mails.ccnu.edu.cn zhanghexia@mails.ccnu.edu.cn https://orcid.org/0000-0001-6871-1579 1 1 1 2 corresp ‡ bwzhang@mail.ccnu.edu.cn ‡ bwzhang@mail.ccnu.edu.cn bwzhang@mail.ccnu.edu.cn https://orcid.org/0000-0001-6075-6609 Key Laboratory of Quark & Lepton Physics (MOE) and Institute of Particle Physics, Central China Normal University, Wuhan 430079, China Key Laboratory of Quark & Lepton Physics (MOE) and Institute of Particle Physics, Central China Normal University, Wuhan 430079, China Guangdong Provincial Key Laboratory of Nuclear Science, Institute of Quantum Matter, South China Normal University, Guangzhou 510006, China. Guangdong Provincial Key Laboratory of Nuclear Science, Institute of Quantum Matter, South China Normal University, Guangzhou 510006, China. 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/T 3 and scaled electrical conductivity σ el / T exhibited a similar dip structure, with the minima of both η/T 3 and σ el / 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. Heavy-ion collision Momentum anisotropy NJL model Chiral phase transiton Transport coefficient Quark matter introduction 1 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 1 , 2 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 3 - 5 5 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 7 , 6 6 ]. Owing to the so-called fermion sign problem [ 8 8 ], lattice QCD simulation is limited to low finite density [ 9 9 , 10 10 ], even though several calculation techniques, such as the Taylor expansion [ 11 11 , 12 12 ], analytic continuations from imaginary to real chemical potential [ 13 13 , 14 14 ], and multi-parameter reweighting method [ 15 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 16 , 17 17 ]. Alternatively, one also can rely on effective models, the Dyson–Schwinger equation approach [ 18 18 , 19 19 ] and the functional renormalization group approach [ 20 20 , 21 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 22 , 23 23 ], Polyakov-loop enhanced NJL (PNJL) model [ 24 24 , 25 25 ], Quark-Meson (QM) model [ 26 26 , 27 27 ], and Polyakov QM (PQM) model [ 28 28 , 29 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 30 , 31 31 ], magnetic field effects [ 32 32 - 35 35 32-35 ], finite-volume effects [ 36 36 - 40 40 36-40 ], non-extensive effects [ 41 41 , 42 42 ], external electric fields [ 43 43 - 45 45 43-45 ], and chiral chemical potential effects [ 46 46 - 49 49 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 50 - 52 52 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 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 54 - 58 58 54-58 ], the quasiparticle model (QPM) [ 60 60 , 59 59 ], and the lattice QCD simulation [ 61 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 62 ], but also is directly proportional to the emissivity and production of soft photons [ 63 63 , 64 64 ]. The thermal behavior of σ el has been estimated using different approaches, such as the microscopic transport models [ 65 65 - 67 67 65-67 ], lattice gauge theory simulation [ 68 68 , 69 69 ], hadron resonance gas model [ 70 70 , 71 71 ], quasiparticle models [ 72 72 , 73 73 ], effective models [ 54 54 , 74 74 ], string percolation model [ 75 75 ], and holographic method [ 76 76 ]. Recently, studies of electrical conductivity in QGP in the presence of magnetic fields have also been performed [ 77 77 - 79 79 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 80 , 81 81 ] and QGP [ 82 82 , 83 83 ]. In Ref. [ 84 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 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 86 ], <math id="M1"><mrow><msup><mi>f</mi><mrow><mtext>aniso</mtext></mrow></msup><mo stretchy="false">(</mo><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo stretchy="false">)</mo><mo>→</mo><msup><mi>f</mi><mrow><mtext>iso</mtext></mrow></msup><mo stretchy="false">(</mo><msqrt><mrow><msup><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mn>2</mn></msup><mo>+</mo><mi>ξ</mi><msup><mrow><mo stretchy="false">(</mo><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo>⋅</mo><mstyle mathvariant="bold" mathsize="normal"><mi>n</mi></mstyle><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></msqrt><mo stretchy="false">)</mo></mrow></math> . 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 87 , 88 88 ], parton self-energy [ 89 89 , 90 90 ], heavy-quark potential [ 91 91 ], and various transport coefficients [ 92 92 - 94 94 92-94 ]. The relativistic anisotropic hydrodynamics (aHydro) models can provide a more accurate description of non-equilibrium dynamics compared to other hydrodynamical models [ 95 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 <math id="M2"><mrow><mo>|</mo><mi>ξ</mi><mo>|</mo><mo>≪</mo><mn>1</mn></mrow></math> 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 1 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 22 ] <math display="block" id="M3"><mrow><mi mathvariant="script">L</mi><mo>=</mo><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mi>i</mi><mo>∂</mo><mo>−</mo><msub><mover accent="true"><mi>m</mi><mo>^</mo></mover><mn>0</mn></msub><mo stretchy="false">)</mo><mi>ψ</mi><mo>+</mo><mi>G</mi><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><mi>ψ</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><mi>i</mi><msub><mi>γ</mi><mn>5</mn></msub><mover accent="true"><mi>τ</mi><mo>^</mo></mover><mi>ψ</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo stretchy="false">]</mo><mn>,</mn></mrow></math> where <math id="M4"><mrow><mi>ψ</mi><mo stretchy="false">(</mo><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><mo stretchy="false">)</mo></mrow></math> stands for the quark (antiquark) field with two-flavors ( u,d ) N f =2 and three colors N c =3. <math id="M5"><mrow><msub><mover accent="true"><mi>m</mi><mo>^</mo></mover><mn>0</mn></msub></mrow></math> denotes the diagonal matrix of the current quark mass of up and down-quarks, and we take <math id="M6"><mrow><msub><mi>m</mi><mn>0</mn></msub><mo>=</mo><msubsup><mi>m</mi><mi>u</mi><mn>0</mn></msubsup><mo>=</mo><msubsup><mi>m</mi><mi>d</mi><mn>0</mn></msubsup></mrow></math> 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. <math id="M7"><mover accent="true"><mi>τ</mi><mo>^</mo></mover></math> is the vector of the Pauli matrix in the isospin space. In the NJL model, under the mean field (or Hartree) approximation [ 22 22 , 23 23 ], the quark self-energy is momentum-independent and can be identified as the constituent quark mass m q , 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 96 ] <math display="block" id="M8"><mrow><msub><mi>m</mi><mi>q</mi></msub><mo>=</mo><msub><mi>m</mi><mn>0</mn></msub><mo>−</mo><mn>2</mn><mi>G</mi><mi>i</mi><mtext>Tr</mtext><mi>i</mi><msup><mi>S</mi><mo><</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mn>,</mn><mi>x</mi><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where <math id="M9"><mrow><msup><mi>S</mi><mo><</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mn>,</mn><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>i</mi><mrow><mo>〈</mo><mrow><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mo>〉</mo></mrow></mrow></math> with x =(t,x) is the real time Green function in the coordinate space [ 97 97 , 98 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 99 ] for details), the gap equation can be further written as [ 96 96 , 98 98 - 100 100 98-100 ] <math display="block" id="M10"><mrow><msub><mi>m</mi><mi>q</mi></msub><mo>=</mo><msub><mi>m</mi><mn>0</mn></msub><mo>+</mo><mn>4</mn><msub><mi>N</mi><mtext>f</mtext></msub><msub><mi>N</mi><mtext>c</mtext></msub><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mfrac><mrow><msup><mi>d</mi><mn>3</mn></msup><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mo stretchy="false">)</mo></mrow><mn>3</mn></msup></mrow></mfrac><mfrac><mrow><msub><mi>m</mi><mtext>q</mtext></msub></mrow><mrow><msub><mi>E</mi><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle></msub></mrow></mfrac><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msub><mi>f</mi><mi>q</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mn>,</mn><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo stretchy="false">)</mo><mo>−</mo><msub><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mover accent="true"><mi>q</mi><mo>¯</mo></mover></msub><mo stretchy="false">(</mo><mi>x</mi><mn>,</mn><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where <math id="M11"><mrow><msub><mi>E</mi><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle></msub><mo>=</mo><msqrt><mrow><msup><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mn>2</mn></msup><mo>+</mo><msubsup><mi>m</mi><mi>q</mi><mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msqrt></mrow></math> 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 [ 100 100 - 102 102 100-102 ]. 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 23 , 96 96 , 98 98 ]. Following Refs. [ 96 96 , 98 98 ], the explicit form for the pion ( π ) and the sigma meson ( σ ) propagators in the RPA reads as <math display="block" id="M12"><mrow><msub><mi>D</mi><mi>M</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mn>,</mn><msub><mi>k</mi><mn>0</mn></msub><mn>,</mn><mstyle mathvariant="bold" mathsize="normal"><mi>k</mi></mstyle><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn>2</mn><mi>i</mi><mi>G</mi></mrow><mrow><mn>1</mn><mo>−</mo><mn>2</mn><mi>G</mi><msub><mi>Π</mi><mi>M</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mn>,</mn><msub><mi>k</mi><mn>0</mn></msub><mn>,</mn><mstyle mathvariant="bold" mathsize="normal"><mi>k</mi></mstyle><mo stretchy="false">)</mo></mrow></mfrac><mn>.</mn></mrow></math> 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 96 , 98 98 ] <math display="block" id="M13"><mtable><mtr><mtd><msub><mi>Π</mi><mi>M</mi></msub><mo>=</mo><msub><mi>N</mi><mtext>c</mtext></msub><msub><mi>N</mi><mtext>f</mtext></msub><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mfrac><mrow><msup><mi>d</mi><mn>3</mn></msup><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle></mrow><mrow><mn>4</mn><msup><mi>π</mi><mn>3</mn></msup></mrow></mfrac><mfrac><mn>1</mn><mrow><msub><mi>E</mi><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle></msub></mrow></mfrac><mrow><mo>[</mo><mrow><mn>1</mn><mo>−</mo><msub><mi>f</mi><mi>q</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mn>,</mn><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo stretchy="false">)</mo><mo>−</mo><msub><mi>f</mi><mover accent="true"><mi>q</mi><mo>¯</mo></mover></msub><mo stretchy="false">(</mo><mi>x</mi><mn>,</mn><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo stretchy="false">)</mo></mrow><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mo>×</mo><mrow><mo>{</mo><mrow><mn>1</mn><mo>−</mo><mfrac><mrow><mo stretchy="false">(</mo><msubsup><mi>k</mi><mn>0</mn><mn>2</mn></msubsup><mo>−</mo><msup><mstyle mathvariant="bold" mathsize="normal"><mi>k</mi></mstyle><mn>2</mn></msup><mo>−</mo><msubsup><mi>ν</mi><mi>M</mi><mn>2</mn></msubsup><mo stretchy="false">)</mo></mrow><mrow><mn>2</mn><msub><mi>E</mi><mrow><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo>−</mo><mstyle mathvariant="bold" mathsize="normal"><mi>k</mi></mstyle></mrow></msub></mrow></mfrac><mrow><mo>[</mo><mrow><mfrac><mrow><msub><mi>E</mi><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle></msub><mo>+</mo><msub><mi>E</mi><mrow><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo>−</mo><mstyle mathvariant="bold" mathsize="normal"><mi>k</mi></mstyle></mrow></msub></mrow><mrow><msubsup><mi>k</mi><mn>0</mn><mn>2</mn></msubsup><mo>−</mo><msup><mrow><mo stretchy="false">(</mo><msub><mi>E</mi><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle></msub><mo>+</mo><msub><mi>E</mi><mrow><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo>−</mo><mstyle mathvariant="bold" mathsize="normal"><mi>k</mi></mstyle></mrow></msub><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mfrac></mrow></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mrow><mrow><mrow><mo>−</mo><mfrac><mrow><msub><mi>E</mi><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle></msub><mo>−</mo><msub><mi>E</mi><mrow><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo>−</mo><mstyle mathvariant="bold" mathsize="normal"><mi>k</mi></mstyle></mrow></msub></mrow><mrow><msubsup><mi>k</mi><mn>0</mn><mn>2</mn></msubsup><mo>−</mo><msup><mrow><mo stretchy="false">(</mo><msub><mi>E</mi><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle></msub><mo>−</mo><msub><mi>E</mi><mrow><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo>−</mo><mstyle mathvariant="bold" mathsize="normal"><mi>k</mi></mstyle></mrow></msub><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mfrac></mrow><mo>]</mo></mrow></mrow><mo>}</mo></mrow><mn>.</mn></mtd></mtr></mtable></math> Here, ν M denotes the real value of the bound meson energy, and ν M =0 (2 m q ) for the π ( σ ) meson. Constituent quark and meson in an isotropic quark matter 1 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 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, m q , which is temperature- and quark chemical potential-dependent, can be directly calculated from the self-consistent gap equation in momentum space [ 23 23 , 22 22 ]: <math display="block" id="M14"><mrow><msub><mi>m</mi><mi>q</mi></msub><mo>=</mo><msub><mi>m</mi><mn>0</mn></msub><mo>+</mo><mn>4</mn><mi>G</mi><msub><mi>N</mi><mtext>f</mtext></msub><msub><mi>N</mi><mtext>c</mtext></msub><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mfrac><mrow><msup><mi>d</mi><mn>3</mn></msup><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mo stretchy="false">)</mo></mrow><mn>3</mn></msup></mrow></mfrac><mfrac><mrow><msub><mi>m</mi><mi>q</mi></msub></mrow><mrow><msub><mi>E</mi><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle></msub></mrow></mfrac><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msubsup><mi>f</mi><mi>q</mi><mn>0</mn></msubsup><mo stretchy="false">(</mo><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo stretchy="false">)</mo><mo>−</mo><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mover accent="true"><mi>q</mi><mo>¯</mo></mover><mn>0</mn></msubsup><mo stretchy="false">(</mo><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mn>.</mn></mrow></math> 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 <math id="M15"><mrow><msubsup><mi>f</mi><mrow><mi>q</mi><mo stretchy="false">(</mo><mover accent="true"><mi>q</mi><mo>¯</mo></mover><mo stretchy="false">)</mo></mrow><mn>0</mn></msubsup></mrow></math> is given by <math display="block" id="M16"><mrow><msubsup><mi>f</mi><mrow><mi>q</mi><mo stretchy="false">(</mo><mover accent="true"><mi>q</mi><mo>¯</mo></mover><mo stretchy="false">)</mo></mrow><mn>0</mn></msubsup><mo stretchy="false">(</mo><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">[</mo><mi>exp</mi><mo stretchy="false">[</mo><mo stretchy="false">(</mo><msub><mi>E</mi><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle></msub><mo>−</mo><msub><mi>μ</mi><mrow><mi>q</mi><mo stretchy="false">(</mo><mover accent="true"><mi>q</mi><mo>¯</mo></mover><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mi>β</mi><mo stretchy="false">]</mo><mo>+</mo><msup><mrow><mn>1</mn><mo stretchy="false">]</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mn>,</mn></mrow></math> where β =1/ T is the inverse temperature of the system, and a uniform quark chemical potential <math id="M17"><mrow><mi>μ</mi><mo>≡</mo><msub><mi>μ</mi><mrow><mi>u</mi><mn>,</mn><mi>d</mi></mrow></msub><mo>≡</mo><mo>−</mo><msub><mi>μ</mi><mrow><mover accent="true"><mi>u</mi><mo>¯</mo></mover><mn>,</mn><mover accent="true"><mi>d</mi><mo>¯</mo></mover></mrow></msub></mrow></math> 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 57 ] <math display="block" id="M18"><mrow><msub><mi>D</mi><mi>M</mi></msub><mo stretchy="false">(</mo><msub><mi>k</mi><mn>0</mn></msub><mn>,</mn><mstyle mathvariant="bold" mathsize="normal"><mi>k</mi></mstyle><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn>2</mn><mi>i</mi><mi>G</mi></mrow><mrow><mn>1</mn><mo>−</mo><mn>2</mn><mi>G</mi><msub><mi>Π</mi><mi>M</mi></msub><mo stretchy="false">(</mo><msub><mi>k</mi><mn>0</mn></msub><mn>,</mn><mstyle mathvariant="bold" mathsize="normal"><mi>k</mi></mstyle><mo stretchy="false">)</mo></mrow></mfrac><mn>,</mn></mrow></math> where ∏ M at an arbitrary temperature and quark chemical potential is given by [ 103 103 , 104 104 ] <math display="block" id="M19"><mtable><mtr><mtd><msub><mi>Π</mi><mi>M</mi></msub><mo stretchy="false">(</mo><msub><mi>k</mi><mn>0</mn></msub><mn>,</mn><mstyle mathvariant="bold" mathsize="normal"><mi>k</mi></mstyle><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mfrac><mrow><msub><mi>N</mi><mtext>c</mtext></msub><msub><mi>N</mi><mtext>f</mtext></msub></mrow><mrow><mn>8</mn><msup><mi>π</mi><mn>2</mn></msup></mrow></mfrac><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mi>m</mi><mi>q</mi></msub><mo>∓</mo><msub><mi>m</mi><mi>q</mi></msub><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>−</mo><msubsup><mi>k</mi><mn>0</mn><mn>2</mn></msubsup><mo>+</mo><msup><mstyle mathvariant="bold" mathsize="normal"><mi>k</mi></mstyle><mn>2</mn></msup><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>×</mo><msub><mi>B</mi><mn>0</mn></msub><mo stretchy="false">(</mo><msub><mi>k</mi><mn>0</mn></msub><mn>,</mn><mstyle mathvariant="bold" mathsize="normal"><mi>k</mi></mstyle><mn>,</mn><mi>μ</mi><mn>,</mn><mi>T</mi><mn>,</mn><msub><mi>m</mi><mi>q</mi></msub><mo stretchy="false">)</mo><mo>+</mo><mn>2</mn><mi>A</mi><mo stretchy="false">(</mo><mi>μ</mi><mn>,</mn><mi>T</mi><mn>,</mn><msub><mi>m</mi><mi>q</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mn>.</mn></mtd></mtr></mtable></math> 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 [ 103 103 ] <math display="block" id="M20"><mrow><mi>A</mi><mo>=</mo><mfrac><mrow><mn>16</mn><msup><mi>π</mi><mn>2</mn></msup></mrow><mi>β</mi></mfrac><mstyle displaystyle="true"><munder><mo>∑</mo><mi>n</mi></munder></mstyle><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><msub><mi>ω</mi><mi>n</mi></msub><mi>y</mi><mo stretchy="false">)</mo><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mfrac><mrow><msup><mi>d</mi><mn>3</mn></msup><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mo stretchy="false">)</mo></mrow><mn>3</mn></msup></mrow></mfrac><mfrac><mn>1</mn><mrow><msup><mrow><mo stretchy="false">(</mo><mi>i</mi><msub><mi>ω</mi><mi>n</mi></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>−</mo><msubsup><mi>E</mi><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mn>2</mn></msubsup></mrow></mfrac><mn>,</mn></mrow></math> where ω n =(2 n +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 <math id="M21"><mrow><mi>y</mi><mo>→</mo><mn>0</mn></mrow></math> is to be taken after the Matsubara summation. The function B 0 in Eq. (9) relates to the two-fermion-line integral. At finite μ and T , B 0 is defined as [ 103 103 ] <math display="block" id="M22"><mtable columnalign="left"><mtr><mtd><msub><mi>B</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>i</mi><msub><mi>ν</mi><mi>l</mi></msub><mn>,</mn><mstyle mathvariant="bold" mathsize="normal"><mi>k</mi></mstyle><mn>,</mn><mi>μ</mi><mn>,</mn><mi>T</mi><mn>,</mn><msub><mi>m</mi><mi>q</mi></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mtext>   </mtext><mo>=</mo><mfrac><mrow><mn>16</mn><msup><mi>π</mi><mn>2</mn></msup></mrow><mi>β</mi></mfrac><mstyle displaystyle="true"><munder><mo>∑</mo><mi>n</mi></munder></mstyle><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><msub><mi>ω</mi><mi>n</mi></msub><mi>y</mi><mo stretchy="false">)</mo><mstyle displaystyle="true"><mrow><msub><mo>∫</mo><mrow><mo>|</mo><mover accent="true"><mi>p</mi><mo>→</mo></mover><mo>|</mo><mo><</mo><mi>Λ</mi></mrow></msub></mrow></mstyle><mfrac><mrow><msup><mi>d</mi><mn>3</mn></msup><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mo stretchy="false">)</mo></mrow><mn>3</mn></msup></mrow></mfrac><mfrac><mn>1</mn><mrow><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>i</mi><msub><mi>ω</mi><mi>n</mi></msub><mo>+</mo><mi>μ</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>−</mo><msup><mi>E</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow></mfrac><mo>×</mo><mfrac><mn>1</mn><mrow><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>i</mi><msub><mi>ω</mi><mi>n</mi></msub><mo>−</mo><mi>i</mi><msub><mi>ν</mi><mi>l</mi></msub><mo>+</mo><mi>μ</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>−</mo><msub><mi>E</mi><mrow/></msub><msup><mo>'</mo><mn>2</mn></msup><mo stretchy="false">)</mo></mrow></mfrac><mn>,</mn></mtd></mtr></mtable></math> where we have abbreviated <math id="M23"><mrow><msup><mi>E</mi><mo>′</mo></msup><mo>=</mo><msub><mi>E</mi><mrow><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo>−</mo><mstyle mathvariant="bold" mathsize="normal"><mi>k</mi></mstyle></mrow></msub><mo>=</mo><msqrt><mrow><msup><mrow><mo stretchy="false">(</mo><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo>−</mo><mstyle mathvariant="bold" mathsize="normal"><mi>k</mi></mstyle><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>+</mo><msubsup><mi>m</mi><mi>q</mi><mn>2</mn></msubsup></mrow></msqrt></mrow></math> and E = E p for convenience, and after the Matsubara summation on n is carried out, the complex frequencies iν l are analytically continued to their values on the real plane, i.e., <math id="M24"><mrow><mi>i</mi><msub><mi>ν</mi><mi>l</mi></msub><mo>→</mo><msub><mi>k</mi><mn>0</mn></msub><mo>−</mo><mi>i</mi><mi>ϵ</mi></mrow></math> ( ϵ > 0) with k 0 being the zero component of the associated four-momentum. The full calculations of functions A and B 0 at arbitrary values of temperature and chemical potential can be found in Ref. [ 105 105 ]. After evaluating the Matsubara summation by contour integration in the usual fashion [ 106 106 ], Eq. (10) is given as <math display="block" id="M25"><mrow><mi>A</mi><mo>=</mo><mn>8</mn><msup><mi>π</mi><mn>2</mn></msup><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mfrac><mrow><msup><mi>d</mi><mn>3</mn></msup><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mo stretchy="false">)</mo></mrow><mn>3</mn></msup><msub><mi>E</mi><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle></msub></mrow></mfrac><mo stretchy="false">(</mo><msubsup><mi>f</mi><mi>q</mi><mn>0</mn></msubsup><mo>+</mo><msubsup><mi>f</mi><mover accent="true"><mi>q</mi><mo>¯</mo></mover><mn>0</mn></msubsup><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where the Fermi–Dirac distribution function is introduced. Similar to the treatment of function A , after Matsubara summation over n and the Matsubara frequencies iν l are analytically continued to real values, Eq. (11) can be rewritten in the following form: <math display="block" id="M26"><mtable columnalign="left"><mtr><mtd><msub><mi>B</mi><mn>0</mn></msub><mo stretchy="false">(</mo><msub><mi>k</mi><mn>0</mn></msub><mn>,</mn><mstyle mathvariant="bold" mathsize="normal"><mi>k</mi></mstyle><mn>,</mn><mi>μ</mi><mn>,</mn><mi>T</mi><mn>,</mn><msub><mi>m</mi><mi>q</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn>16</mn><msup><mi>π</mi><mn>2</mn></msup><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mfrac><mrow><msup><mi>d</mi><mn>3</mn></msup><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mo stretchy="false">)</mo></mrow><mn>3</mn></msup></mrow></mfrac><mfrac><mrow><msubsup><mi>f</mi><mi>q</mi><mn>0</mn></msubsup><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mo>+</mo><msubsup><mi>f</mi><mover accent="true"><mi>q</mi><mo>¯</mo></mover><mn>0</mn></msubsup><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn><mi>E</mi></mrow></mfrac></mtd></mtr><mtr><mtd><mtext>    </mtext><mo>×</mo><mrow><mo>[</mo><mrow><mfrac><mn>1</mn><mrow><msup><mrow><mo stretchy="false">(</mo><mi>E</mi><mo>+</mo><mi>i</mi><msub><mi>ν</mi><mi>l</mi></msub><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>−</mo><msup><mrow><mo stretchy="false">(</mo><msup><mi>E</mi><mo>′</mo></msup><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mo stretchy="false">(</mo><mi>E</mi><mo>−</mo><mi>i</mi><msub><mi>ν</mi><mi>l</mi></msub><mo stretchy="false">)</mo><mo>−</mo><msup><mrow><mo stretchy="false">(</mo><msup><mi>E</mi><mo>′</mo></msup><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mfrac></mrow><mo>]</mo></mrow><mn>.</mn></mtd></mtr></mtable></math> 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 1 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 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, <math id="M27"><mrow><msup><mi>f</mi><mrow><mtext>aniso</mtext></mrow></msup><mo stretchy="false">(</mo><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo stretchy="false">)</mo><mo>=</mo><msup><mi>f</mi><mrow><mtext>iso</mtext></mrow></msup><mo stretchy="false">(</mo><msqrt><mrow><msup><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mn>2</mn></msup><mo>+</mo><mi>ξ</mi><msup><mrow><mo stretchy="false">(</mo><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo>⋅</mo><mstyle mathvariant="bold" mathsize="normal"><mi>n</mi></mstyle><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></msqrt><mo stretchy="false">)</mo></mrow></math> , where n and ξ are, respectively, the anisotropy direction and anisotropy parameter. As in previous studies [ 107 107 ], 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 <math display="block" id="M28"><mrow><msup><mi>f</mi><mrow><mtext>aniso</mtext></mrow></msup><mo stretchy="false">(</mo><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mi>exp</mi><mo stretchy="false">[</mo><mo stretchy="false">(</mo><msqrt><mrow><msup><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mn>2</mn></msup><mo>+</mo><mi>ξ</mi><msup><mrow><mo stretchy="false">(</mo><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo>⋅</mo><mstyle mathvariant="bold" mathsize="normal"><mi>n</mi></mstyle><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>+</mo><msubsup><mi>m</mi><mi>q</mi><mn>2</mn></msubsup></mrow></msqrt><mo>−</mo><msub><msup><mi>μ</mi><mo>′</mo></msup><mrow><mi>q</mi><mo stretchy="false">(</mo><mover accent="true"><mi>q</mi><mo>¯</mo></mover><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mo>/</mo><msup><mi>T</mi><mo>′</mo></msup><mo stretchy="false">]</mo><mo>+</mo><mn>1</mn></mrow></mfrac><mn>,</mn></mrow></math> 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 [ 108 108 ]. 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 92 - 94 94 92-94 , 107 107 ]. The anisotropy parameter ξ is defined as <math id="M29"><mrow><mi>ξ</mi><mo>=</mo><mo>〈</mo><msubsup><mi>p</mi><mo>⊥</mo><mn>2</mn></msubsup><mo>〉</mo><mo>/</mo><mo stretchy="false">(</mo><mn>2</mn><mo>〈</mo><msubsup><mi>p</mi><mo>∥</mo><mn>2</mn></msubsup><mo>〉</mo><mo stretchy="false">)</mo><mo>−</mo><mn>1</mn></mrow></math> , where <math id="M30"><mrow><msub><mi>p</mi><mo>⊥</mo></msub><mo>=</mo><mo>|</mo><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo>−</mo><mo stretchy="false">(</mo><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo>⋅</mo><mstyle mathvariant="bold" mathsize="normal"><mi>n</mi></mstyle><mo stretchy="false">)</mo><mo>⋅</mo><mstyle mathvariant="bold" mathsize="normal"><mi>n</mi></mstyle><mo>|</mo></mrow></math> and <math id="M31"><mrow><msub><mi>p</mi><mo>∥</mo></msub><mo>=</mo><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo>⋅</mo><mstyle mathvariant="bold" mathsize="normal"><mi>n</mi></mstyle></mrow></math> 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 <math display="block" id="M32"><mrow><mstyle mathvariant="bold" mathsize="normal"><mi>n</mi></mstyle><mo>=</mo><mo stretchy="false">(</mo><mi>sin</mi><mi>χ</mi><mn>,0,</mn><mi>cos</mi><mi>χ</mi><mo stretchy="false">)</mo><mn>,</mn></mrow></math> <math display="block" id="M33"><mrow><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo>=</mo><mi>p</mi><mo stretchy="false">(</mo><mi>sin</mi><mi>θ</mi><mi>cos</mi><mi>ϕ</mi><mn>,</mn><mi>sin</mi><mi>θ</mi><mi>sin</mi><mi>ϕ</mi><mn>,</mn><mi>cos</mi><mi>θ</mi><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where χ is the angle between p and n, and <math id="M34"><mrow><mi>p</mi><mo>≡</mo><mo>|</mo><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo>|</mo></mrow></math> throughout the computations. With this choice, the spheroidally anisotropic term, ξ (n⋅p) 2 in Eq. (14) can be written as <math id="M35"><mrow><mi>ξ</mi><msup><mrow><mo stretchy="false">(</mo><mstyle mathvariant="bold" mathsize="normal"><mi>n</mi></mstyle><mo>⋅</mo><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>=</mo><mi>ξ</mi><msup><mi>p</mi><mn>2</mn></msup><msup><mrow><mo stretchy="false">(</mo><mo>+</mo><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>=</mo><mi>ξ</mi><mi>c</mi><mo stretchy="false">(</mo><mi>θ</mi><mn>,</mn><mi>ϕ</mi><mn>,</mn><mi>χ</mi><mo stretchy="false">)</mo></mrow></math> . 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 ( <math id="M36"><mrow><mo>|</mo><mi>ξ</mi><mo>|</mo><mo>≪</mo><mn>1</mn></mrow></math> ), 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 [ 107 107 ] <math display="block" id="M37"><mrow><msup><mi>f</mi><mrow><mtext>aniso</mtext></mrow></msup><mo stretchy="false">(</mo><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo stretchy="false">)</mo><mo>=</mo><msup><mi>f</mi><mn>0</mn></msup><mo>−</mo><mfrac><mrow><mi>ξ</mi><msup><mrow><mo stretchy="false">(</mo><mstyle mathvariant="bold" mathsize="normal"><mi>n</mi></mstyle><mo>⋅</mo><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow><mrow><mn>2</mn><msub><mi>E</mi><mrow/></msub><mi>T</mi></mrow></mfrac><msup><mi>f</mi><mn>0</mn></msup><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msup><mi>f</mi><mn>0</mn></msup><mo stretchy="false">)</mo><mn>,</mn></mrow></math> 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 [ 109 109 , 110 110 ] <math display="block" id="M38"><mrow><mi>ξ</mi><mo>=</mo><mfrac><mrow><mn>10</mn></mrow><mrow><mi>T</mi><mi>τ</mi></mrow></mfrac><mfrac><mi>η</mi><mi>s</mi></mfrac><mn>,</mn></mrow></math> 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 <math id="M39"><mrow><msub><mi>T</mi><mtext>c</mtext></msub><mo>≈</mo><mn>160</mn></mrow></math> MeV, <math id="M40"><mrow><mi>τ</mi><mo>≈</mo><mn>6</mn></mrow></math> fm/c and η/s =1/4 π , we can obtain <math id="M41"><mrow><mi>ξ</mi><mo>≈</mo><mn>0.3</mn></mrow></math> . 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 <math display="block" id="M42"><mtable><mtr><mtd><mn>0</mn><mo>=</mo><mrow><mo>[</mo><mrow><mn>2</mn><msub><mi>N</mi><mtext>c</mtext></msub><msub><mi>N</mi><mtext>f</mtext></msub><mi>G</mi><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mi>∞</mi></msubsup></mrow></mstyle><mfrac><mrow><msup><mi>p</mi><mn>2</mn></msup><mi>d</mi><mi>p</mi></mrow><mrow><msup><mi>π</mi><mn>2</mn></msup></mrow></mfrac><mfrac><mrow><msub><mi>m</mi><mi>q</mi></msub></mrow><mi>E</mi></mfrac><mo stretchy="false">(</mo><mo>−</mo><msubsup><mi>f</mi><mi>q</mi><mn>0</mn></msubsup><mo>−</mo><msubsup><mi>f</mi><mover accent="true"><mi>q</mi><mo>¯</mo></mover><mn>0</mn></msubsup><mo>+</mo><mn>1</mn><mo>+</mo><mfrac><mrow><msup><mi>p</mi><mn>2</mn></msup><mi>ξ</mi><msub><mi>F</mi><mtext>p</mtext></msub></mrow><mrow><mn>6</mn><msub><mi>E</mi><mrow/></msub><mi>T</mi></mrow></mfrac><mo stretchy="false">)</mo></mrow><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mo>+</mo><msub><mi>m</mi><mn>0</mn></msub><mo>−</mo><msub><mi>m</mi><mi>q</mi></msub><mn>.</mn></mtd></mtr></mtable></math> Here, we have abbreviated <math id="M43"><mrow><msub><mi>F</mi><mi>p</mi></msub><mo>=</mo><msubsup><mi>f</mi><mi>q</mi><mn>0</mn></msubsup><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msubsup><mi>f</mi><mi>q</mi><mn>0</mn></msubsup><mo stretchy="false">)</mo><mo>+</mo><msubsup><mi>f</mi><mover accent="true"><mi>q</mi><mo>¯</mo></mover><mn>0</mn></msubsup><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msubsup><mi>f</mi><mover accent="true"><mi>q</mi><mo>¯</mo></mover><mn>0</mn></msubsup><mo stretchy="false">)</mo></mrow></math> 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 <math display="block" id="M44"><mrow><mi>A</mi><mo>=</mo><mn>4</mn><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mfrac><mrow><msup><mi>p</mi><mn>2</mn></msup><mi>d</mi><mi>p</mi></mrow><mi>E</mi></mfrac><mo stretchy="false">[</mo><msubsup><mi>f</mi><mi>q</mi><mn>0</mn></msubsup><mo>+</mo><msubsup><mi>f</mi><mover accent="true"><mi>q</mi><mo>¯</mo></mover><mn>0</mn></msubsup><mo>−</mo><mn>1</mn><mo>−</mo><mfrac><mrow><mi>ξ</mi><msup><mi>p</mi><mn>2</mn></msup><msub><mi>F</mi><mi>p</mi></msub></mrow><mrow><mn>6</mn><mi>E</mi><mi>T</mi></mrow></mfrac><mo stretchy="false">]</mo><mn>.</mn></mrow></math> In the <math id="M45"><mrow><mi>ξ</mi><mo>→</mo><mn>0</mn></mrow></math> limit, the above equation reduces to Eq. (12). Similar to the treatment of function A, the weak momentum anisotropy effects can enter the B 0 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., <math display="block" id="M46"><mrow><mstyle mathvariant="bold" mathsize="normal"><mi>k</mi></mstyle><mo>=</mo><mo stretchy="false">(</mo><mn>0,0,</mn><mi>k</mi><mo stretchy="false">)</mo><mn>,</mn><mtext> </mtext><mo>|</mo><mstyle mathvariant="bold" mathsize="normal"><mi>k</mi></mstyle><mo>|</mo><mo>≡</mo><mi>k</mi><mn>.</mn></mrow></math> We first discuss a simple case, i.e., k=0, k 0 ≠0. The computation of function B 0 in a weakly anisotropic medium is trivial, viz , <math display="block" id="M47"><mtable><mtr><mtd><msub><mi>B</mi><mn>0</mn></msub><mi>s</mi><mo>=</mo><mn>8</mn><msup><mi>π</mi><mn>2</mn></msup><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mfrac><mrow><msup><mi>d</mi><mn>3</mn></msup><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mo stretchy="false">)</mo></mrow><mn>3</mn></msup><mi>E</mi></mrow></mfrac><mo stretchy="false">(</mo><msubsup><mi>f</mi><mi>q</mi><mrow><mtext>aniso</mtext></mrow></msubsup><mo stretchy="false">(</mo><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo stretchy="false">)</mo><mo>+</mo><msubsup><mi>f</mi><mover accent="true"><mi>q</mi><mo>¯</mo></mover><mrow><mtext>aniso</mtext></mrow></msubsup><mo stretchy="false">(</mo><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo stretchy="false">)</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>×</mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mrow><msubsup><mi>k</mi><mn>0</mn><mn>2</mn></msubsup><mo>−</mo><mn>2</mn><mi>E</mi><msub><mi>k</mi><mn>0</mn></msub><mo>+</mo><mi>i</mi><mi>ϵ</mi><mtext>sgn</mtext><mo stretchy="false">(</mo><msub><mi>k</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><msubsup><mi>k</mi><mn>0</mn><mn>2</mn></msubsup><mo>+</mo><mn>2</mn><mi>E</mi><msub><mi>k</mi><mn>0</mn></msub><mo>−</mo><mi>i</mi><mi>ϵ</mi><mtext>sgn</mtext><mo stretchy="false">(</mo><msub><mi>k</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>)</mo></mrow><mn>.</mn></mtd></mtr></mtable></math> In the integrand of the above equation, there are two poles at E = E 0 =± k 0 /2 if <math id="M48"><mrow><mi>m</mi><mo>≤</mo><msub><mi>E</mi><mn>0</mn></msub></mrow></math> . Applying the Cauchy formula <math display="block" id="M49"><mrow><msub><mrow><mtext>lim</mtext></mrow><mrow><mi>ϵ</mi><mo>→</mo><mn>0</mn></mrow></msub><mfrac><mn>1</mn><mrow><mi>x</mi><mo>−</mo><mi>i</mi><mi>ϵ</mi></mrow></mfrac><mo>=</mo><mi mathvariant="script">P</mi><mfrac><mn>1</mn><mi>x</mi></mfrac><mo>+</mo><mi>i</mi><mi>π</mi><mi>δ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where 𝒫 denotes the Cauchy principal value. The function B 0 can finally be rewritten as <math display="block" id="M50"><mtable><mtr><mtd><msub><mi>B</mi><mn>0</mn></msub><mo>=</mo><mn>8</mn><mi mathvariant="script">P</mi><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mfrac><mrow><msup><mi>p</mi><mn>2</mn></msup><mi>d</mi><mi>p</mi></mrow><mrow><mi>E</mi><mo stretchy="false">(</mo><msubsup><mi>k</mi><mn>0</mn><mn>2</mn></msubsup><mo>−</mo><mn>4</mn><msup><mi>E</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow></mfrac><mo stretchy="false">[</mo><msubsup><mi>f</mi><mi>q</mi><mn>0</mn></msubsup><mo stretchy="false">(</mo><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo stretchy="false">)</mo><mo>+</mo><msubsup><mi>f</mi><mover accent="true"><mi>q</mi><mo>¯</mo></mover><mn>0</mn></msubsup><mo stretchy="false">(</mo><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo stretchy="false">)</mo><mo>−</mo><mn>1</mn><mo>−</mo><mfrac><mrow><mi>ξ</mi><msup><mi>p</mi><mn>2</mn></msup></mrow><mrow><mn>6</mn><mi>E</mi><mi>T</mi></mrow></mfrac><msub><mi>F</mi><mi>p</mi></msub><mo stretchy="false">]</mo></mtd></mtr><mtr><mtd><mo>−</mo><mi>i</mi><mi>π</mi><mfrac><mn>2</mn><mrow><msub><mi>k</mi><mn>0</mn></msub></mrow></mfrac><mo stretchy="false">[</mo><mo stretchy="false">(</mo><msubsup><mi>f</mi><mi>q</mi><mn>0</mn></msubsup><mo stretchy="false">(</mo><msub><mi>z</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo>+</mo><msubsup><mi>f</mi><mover accent="true"><mi>q</mi><mo>¯</mo></mover><mn>0</mn></msubsup><mo stretchy="false">(</mo><msub><mi>z</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo>−</mo><mn>1</mn><mo>−</mo><mfrac><mrow><mi>ξ</mi><msubsup><mi>z</mi><mn>0</mn><mn>2</mn></msubsup></mrow><mrow><mn>3</mn><msub><mi>k</mi><mn>0</mn></msub><mi>T</mi></mrow></mfrac><msub><mi>F</mi><mrow><msub><mi>z</mi><mn>0</mn></msub></mrow></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>×</mo><msqrt><mrow><msup><mrow><mo stretchy="false">(</mo><mfrac><mrow><msub><mi>k</mi><mn>0</mn></msub></mrow><mn>2</mn></mfrac><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>−</mo><msubsup><mi>m</mi><mi>q</mi><mn>2</mn></msubsup></mrow></msqrt><mi>Θ</mi><mo stretchy="false">(</mo><mfrac><mrow><msubsup><mi>k</mi><mn>0</mn><mn>2</mn></msubsup></mrow><mn>4</mn></mfrac><mo>−</mo><msubsup><mi>m</mi><mi>q</mi><mn>2</mn></msubsup><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mn>.</mn></mtd></mtr></mtable></math> Here, <math id="M51"><mrow><msub><mi>z</mi><mn>0</mn></msub><mo>=</mo><msqrt><mrow><msup><mrow><mo stretchy="false">(</mo><mfrac><mrow><msub><mi>k</mi><mn>0</mn></msub></mrow><mn>2</mn></mfrac><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>−</mo><msubsup><mi>m</mi><mi>q</mi><mn>2</mn></msubsup></mrow></msqrt></mrow></math> , and Θ is the step function to ensure that the imaginary part appears only for k 0 /2 > m q . Next, in the case of k > 0, k 0 ≠ 0, the expression for function B 0 is slightly complicated and can be written as <math display="block" id="M52"><mtable><mtr><mtd><msub><mi>B</mi><mn>0</mn></msub><mo>=</mo><mfrac><mn>1</mn><mi>k</mi></mfrac><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mi>∞</mi></msubsup></mrow></mstyle><mfrac><mrow><mi>p</mi><mi>d</mi><mi>p</mi></mrow><mi>E</mi></mfrac><mo stretchy="false">(</mo><msubsup><mi>f</mi><mi>q</mi><mrow><mtext>aniso</mtext></mrow></msubsup><mo stretchy="false">(</mo><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo stretchy="false">)</mo><mo>+</mo><msubsup><mi>f</mi><mover accent="true"><mi>q</mi><mo>¯</mo></mover><mrow><mtext>aniso</mtext></mrow></msubsup><mo stretchy="false">(</mo><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo stretchy="false">)</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>×</mo><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mrow><mo>−</mo><mn>1</mn></mrow><mn>1</mn></msubsup></mrow></mstyle><mi>d</mi><mi>x</mi><mo stretchy="false">(</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mo>+</mo><mo stretchy="false">(</mo><msubsup><mi>k</mi><mn>0</mn><mn>2</mn></msubsup><mo>+</mo><mn>2</mn><msub><mi>k</mi><mn>0</mn></msub><mi>E</mi><mo>−</mo><msup><mi>k</mi><mn>2</mn></msup><mo stretchy="false">)</mo><mo>/</mo><mn>2</mn><mi>p</mi><mi>k</mi><mo>−</mo><mi>i</mi><mi>ϵ</mi><mtext>sgn</mtext><mo stretchy="false">(</mo><msub><mi>k</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mfrac></mtd></mtr><mtr><mtd><mo>+</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mo>+</mo><mo stretchy="false">(</mo><msubsup><mi>k</mi><mn>0</mn><mn>2</mn></msubsup><mo>−</mo><mn>2</mn><msub><mi>k</mi><mn>0</mn></msub><mi>E</mi><mo>−</mo><msup><mi>k</mi><mn>2</mn></msup><mo stretchy="false">)</mo><mo>/</mo><mn>2</mn><mi>p</mi><mi>k</mi><mo>+</mo><mi>i</mi><mi>ϵ</mi><mtext>sgn</mtext><mo stretchy="false">(</mo><msub><mi>k</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mfrac><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>=</mo><msubsup><mi>B</mi><mn>0</mn><mrow><mtext>an</mtext></mrow></msubsup><mo>+</mo><msubsup><mi>B</mi><mn>0</mn><mrow><mtext>iso</mtext></mrow></msubsup><mn>,</mn></mtd></mtr></mtable></math> with the abbreviation x =cos θ . The isotropic part <math id="M53"><mrow><msubsup><mi>B</mi><mn>0</mn><mrow><mtext>iso</mtext></mrow></msubsup></mrow></math> reads as <math display="block" id="M54"><mtable columnalign="left"><mtr><mtd><msubsup><mi>B</mi><mn>0</mn><mrow><mtext>iso</mtext></mrow></msubsup><mo>=</mo><mfrac><mn>1</mn><mi>k</mi></mfrac><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mi>∞</mi></msubsup></mrow></mstyle><mfrac><mrow><mi>p</mi><mi>d</mi><mi>p</mi></mrow><mi>E</mi></mfrac><mo stretchy="false">(</mo><msubsup><mi>f</mi><mi>q</mi><mn>0</mn></msubsup><mo>+</mo><msubsup><mi>f</mi><mover accent="true"><mi>q</mi><mo>¯</mo></mover><mn>0</mn></msubsup><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mtext>        </mtext><mo>×</mo><mo stretchy="false">[</mo><mi>log</mi><mo>|</mo><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><msubsup><mi>k</mi><mn>0</mn><mn>2</mn></msubsup><mo>−</mo><msup><mi>k</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>p</mi><mi>k</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>−</mo><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><msub><mi>k</mi><mn>0</mn></msub><mi>E</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><msubsup><mi>k</mi><mn>0</mn><mn>2</mn></msubsup><mo>+</mo><msup><mi>k</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>p</mi><mi>k</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>−</mo><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><msub><mi>k</mi><mn>0</mn></msub><mi>E</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mfrac><mo>|</mo></mtd></mtr><mtr><mtd><mtext>        </mtext><mo>+</mo><mi>i</mi><mi>π</mi><mo stretchy="false">(</mo><mi>Θ</mi><mo stretchy="false">(</mo><mn>2</mn><mi>p</mi><mi>k</mi><mo>−</mo><mo>|</mo><msubsup><mi>k</mi><mn>0</mn><mn>2</mn></msubsup><mo>+</mo><mn>2</mn><msub><mi>k</mi><mn>0</mn></msub><mi>E</mi><mo>−</mo><msup><mi>k</mi><mn>2</mn></msup><mo>|</mo><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mtext>        </mtext><mo>−</mo><mi>Θ</mi><mo stretchy="false">(</mo><mn>2</mn><mi>p</mi><mi>k</mi><mo>−</mo><mo>|</mo><msubsup><mi>k</mi><mn>0</mn><mn>2</mn></msubsup><mo>−</mo><mn>2</mn><msub><mi>k</mi><mn>0</mn></msub><mi>E</mi><mo>−</mo><msup><mi>k</mi><mn>2</mn></msup><mo>|</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mn>.</mn></mtd></mtr></mtable></math> The anisotropic part <math id="M55"><mrow><msubsup><mi>B</mi><mn>0</mn><mrow><mtext>an</mtext></mrow></msubsup></mrow></math> reads as <math display="block" id="M56"><mtable columnalign="left"><mtr><mtd><msubsup><mi>B</mi><mn>0</mn><mrow><mtext>aniso</mtext></mrow></msubsup><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mi>k</mi></mfrac><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mi>∞</mi></msubsup></mrow></mstyle><mfrac><mrow><mi>p</mi><mi>d</mi><mi>p</mi></mrow><mi>E</mi></mfrac><mfrac><mrow><mi>ξ</mi><msup><mi>p</mi><mn>2</mn></msup><msub><mi>F</mi><mi>p</mi></msub></mrow><mrow><mn>2</mn><mi>E</mi><mi>T</mi></mrow></mfrac><mrow><mo>[</mo><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mfrac><mrow><mrow><mo>(</mo><mrow><msubsup><mi>k</mi><mn>0</mn><mn>2</mn></msubsup><mo>−</mo><msup><mi>k</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><msub><mi>k</mi><mn>0</mn></msub><mi>E</mi></mrow><mo>)</mo></mrow></mrow><mrow><mi>p</mi><mi>k</mi></mrow></mfrac></mrow></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mtext>          </mtext><mo>+</mo><msup><mrow><mo>(</mo><mrow><mfrac><mrow><msubsup><mi>k</mi><mn>0</mn><mn>2</mn></msubsup><mo>−</mo><msup><mi>k</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><msub><mi>k</mi><mn>0</mn></msub><mi>E</mi></mrow><mrow><mn>2</mn><mi>p</mi><mi>k</mi></mrow></mfrac></mrow><mo>)</mo></mrow><mn>2</mn></msup><mi>log</mi><mrow><mo>|</mo><mrow><mfrac><mrow><msubsup><mi>k</mi><mn>0</mn><mn>2</mn></msubsup><mo>+</mo><mn>2</mn><msub><mi>k</mi><mn>0</mn></msub><mi>E</mi><mo>−</mo><msup><mi>k</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>p</mi><mi>k</mi></mrow><mrow><msubsup><mi>k</mi><mn>0</mn><mn>2</mn></msubsup><mo>+</mo><mn>2</mn><msub><mi>k</mi><mn>0</mn></msub><mi>E</mi><mo>−</mo><msup><mi>k</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>p</mi><mi>k</mi></mrow></mfrac></mrow><mo>|</mo></mrow></mtd></mtr><mtr><mtd><mtext>          </mtext><mo>−</mo><mfrac><mrow><mrow><mo>(</mo><mrow><msubsup><mi>k</mi><mn>0</mn><mn>2</mn></msubsup><mo>−</mo><msup><mi>k</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><msub><mi>k</mi><mn>0</mn></msub><mi>E</mi></mrow><mo>)</mo></mrow></mrow><mrow><mi>p</mi><mi>k</mi></mrow></mfrac><mo>+</mo><msup><mrow><mo>(</mo><mrow><mfrac><mrow><msubsup><mi>k</mi><mn>0</mn><mn>2</mn></msubsup><mo>−</mo><msup><mi>k</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><msub><mi>k</mi><mn>0</mn></msub><mi>E</mi></mrow><mrow><mn>2</mn><mi>p</mi><mi>k</mi></mrow></mfrac></mrow><mo>)</mo></mrow><mn>2</mn></msup></mtd></mtr><mtr><mtd><mtext>          </mtext><mrow><mrow><mi>log</mi><mrow><mo>|</mo><mrow><mfrac><mrow><msubsup><mi>k</mi><mn>0</mn><mn>2</mn></msubsup><mo>−</mo><mn>2</mn><msub><mi>k</mi><mn>0</mn></msub><mi>E</mi><mo>−</mo><msup><mi>k</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>p</mi><mi>k</mi></mrow><mrow><msubsup><mi>k</mi><mn>0</mn><mn>2</mn></msubsup><mo>−</mo><mn>2</mn><msub><mi>k</mi><mn>0</mn></msub><mi>E</mi><mo>−</mo><msup><mi>k</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>p</mi><mi>k</mi></mrow></mfrac></mrow><mo>|</mo></mrow></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mtext>          </mtext><mo>+</mo><mi>i</mi><mi>π</mi><mrow><mo>(</mo><mrow><mfrac><mrow><msubsup><mi>k</mi><mn>0</mn><mn>2</mn></msubsup><mo>+</mo><mn>2</mn><msub><mi>k</mi><mn>0</mn></msub><mi>E</mi><mo>−</mo><msup><mi>k</mi><mn>2</mn></msup></mrow><mrow><mn>2</mn><mi>p</mi><mi>k</mi></mrow></mfrac><mi>Θ</mi><mrow><mo>(</mo><mrow><mn>2</mn><mi>p</mi><mi>k</mi><mo>−</mo><mrow><mo>|</mo><mrow><msubsup><mi>k</mi><mn>0</mn><mn>2</mn></msubsup><mo>+</mo><mn>2</mn><msub><mi>k</mi><mn>0</mn></msub><mi>E</mi><mo>−</mo><msup><mi>k</mi><mn>2</mn></msup></mrow><mo>|</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mtext>          </mtext><mrow><mrow><mrow><mrow><mo>−</mo><mfrac><mrow><msubsup><mi>k</mi><mn>0</mn><mn>2</mn></msubsup><mo>−</mo><mn>2</mn><msub><mi>k</mi><mn>0</mn></msub><mi>E</mi><mo>−</mo><msup><mi>k</mi><mn>2</mn></msup></mrow><mrow><mn>2</mn><mi>p</mi><mi>k</mi></mrow></mfrac><mi>Θ</mi><mrow><mo>(</mo><mrow><mn>2</mn><mi>p</mi><mi>k</mi><mo>−</mo><mrow><mo>|</mo><mrow><msubsup><mi>k</mi><mn>0</mn><mn>2</mn></msubsup><mo>−</mo><mn>2</mn><msub><mi>k</mi><mn>0</mn></msub><mi>E</mi><mo>−</mo><msup><mi>k</mi><mn>2</mn></msup></mrow><mo>|</mo></mrow></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow><mn>.</mn></mtd></mtr></mtable></math> When k 0 =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 <math display="block" id="M57"><mtable columnalign="left"><mtr><mtd><mtext>Re</mtext><msub><mi>Π</mi><mi>M</mi></msub><mo stretchy="false">(</mo><msub><mi>k</mi><mn>0</mn></msub><mn>,0</mn><mo stretchy="false">)</mo><mo>=</mo><msub><mi>N</mi><mtext>f</mtext></msub><msub><mi>N</mi><mtext>c</mtext></msub><mi mathvariant="script">P</mi><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mi>∞</mi></msubsup></mrow></mstyle><mfrac><mrow><mi>d</mi><mi>p</mi><msup><mi>p</mi><mn>2</mn></msup></mrow><mrow><msup><mi>π</mi><mn>2</mn></msup><mi>E</mi></mrow></mfrac></mtd></mtr><mtr><mtd><mtext>         </mtext><mo stretchy="false">[</mo><mn>1</mn><mo>−</mo><msubsup><mi>f</mi><mi>q</mi><mn>0</mn></msubsup><mo stretchy="false">(</mo><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo stretchy="false">)</mo><mo>−</mo><msubsup><mi>f</mi><mover accent="true"><mi>q</mi><mo>¯</mo></mover><mn>0</mn></msubsup><mo stretchy="false">(</mo><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo stretchy="false">)</mo><mo>+</mo><mfrac><mrow><mi>ξ</mi><msup><mi>p</mi><mn>2</mn></msup></mrow><mrow><mn>6</mn><mi>E</mi><mi>T</mi></mrow></mfrac><msub><mi>F</mi><mi>p</mi></msub><mo stretchy="false">]</mo><mfrac><mrow><msup><mi>E</mi><mn>2</mn></msup><mo>−</mo><msubsup><mi>ν</mi><mi>M</mi><mn>2</mn></msubsup><mo>/</mo><mn>4</mn></mrow><mrow><msup><mi>E</mi><mn>2</mn></msup><mo>−</mo><msup><mrow><mo stretchy="false">(</mo><msub><mi>k</mi><mn>0</mn></msub><mo>/</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mfrac><mn>,</mn></mtd></mtr></mtable></math> <math display="block" id="M58"><mtable><mtr><mtd><mtext>Im</mtext><msub><mi>Π</mi><mi>M</mi></msub><mo stretchy="false">(</mo><msub><mi>k</mi><mn>0</mn></msub><mn>,0</mn><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msub><mi>N</mi><mtext>c</mtext></msub><msub><mi>N</mi><mtext>f</mtext></msub></mrow><mrow><mn>8</mn><mi>π</mi><msub><mi>k</mi><mn>0</mn></msub></mrow></mfrac><msqrt><mrow><msubsup><mi>k</mi><mn>0</mn><mn>2</mn></msubsup><mo>−</mo><mn>4</mn><msubsup><mi>m</mi><mi>q</mi><mn>2</mn></msubsup></mrow></msqrt><mrow><mo>(</mo><mrow><msubsup><mi>k</mi><mn>0</mn><mn>2</mn></msubsup><mo>−</mo><msubsup><mi>ν</mi><mi>M</mi><mn>2</mn></msubsup></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mrow><mo>[</mo><mrow><mn>1</mn><mo>−</mo><msubsup><mi>f</mi><mi>q</mi><mn>0</mn></msubsup><mo stretchy="false">(</mo><msub><mi>z</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo>−</mo><msubsup><mi>f</mi><mover accent="true"><mi>q</mi><mo>¯</mo></mover><mn>0</mn></msubsup><mo stretchy="false">(</mo><msub><mi>z</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mfrac><mrow><msubsup><mi>z</mi><mn>0</mn><mn>2</mn></msubsup><mi>ξ</mi></mrow><mrow><mn>3</mn><msub><mi>k</mi><mn>0</mn></msub><mi>T</mi></mrow></mfrac><msub><mi>F</mi><mrow><msub><mi>z</mi><mn>0</mn></msub></mrow></msub></mrow><mo>]</mo></mrow><mi>Θ</mi><mrow><mo>(</mo><mrow><msubsup><mi>k</mi><mn>0</mn><mn>2</mn></msubsup><mo>−</mo><mn>4</mn><msubsup><mi>m</mi><mi>q</mi><mn>2</mn></msubsup></mrow><mo>)</mo></mrow><mn>,</mn></mtd></mtr></mtable></math> <math display="block" id="M59"><mtable columnalign="left"><mtr><mtd><mtext>Re</mtext><msub><mi>Π</mi><mi>M</mi></msub><mo stretchy="false">(</mo><mn>0,</mn><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msub><mi>N</mi><mtext>c</mtext></msub><msub><mi>N</mi><mtext>f</mtext></msub></mrow><mrow><msup><mi>π</mi><mn>2</mn></msup></mrow></mfrac><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mi>∞</mi></msubsup></mrow></mstyle><mfrac><mrow><mi>d</mi><mi>p</mi><msup><mi>p</mi><mn>2</mn></msup></mrow><mi>E</mi></mfrac><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mfrac><mrow><msup><mi>k</mi><mn>2</mn></msup><mo>+</mo><msubsup><mi>ν</mi><mi>M</mi><mn>2</mn></msubsup></mrow><mrow><mn>4</mn><mi>p</mi><mi>k</mi></mrow></mfrac><mi>ln</mi><mrow><mo>|</mo><mrow><mfrac><mrow><mi>k</mi><mo>−</mo><mn>2</mn><mi>p</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>2</mn><mi>p</mi></mrow></mfrac></mrow><mo>|</mo></mrow></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mtext>                     </mtext><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><msubsup><mi>f</mi><mi>q</mi><mn>0</mn></msubsup><mo stretchy="false">(</mo><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo stretchy="false">)</mo><mo>−</mo><msubsup><mi>f</mi><mover accent="true"><mi>q</mi><mo>¯</mo></mover><mn>0</mn></msubsup><mo stretchy="false">(</mo><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo stretchy="false">)</mo></mrow><mo>)</mo></mrow><mo>+</mo><mfrac><mrow><msub><mi>N</mi><mi>c</mi></msub><msub><mi>N</mi><mi>f</mi></msub></mrow><mrow><msup><mi>π</mi><mn>2</mn></msup></mrow></mfrac><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mi>∞</mi></msubsup></mrow></mstyle><mfrac><mrow><mi>ξ</mi><msup><mi>p</mi><mn>2</mn></msup><mi>d</mi><mi>p</mi></mrow><mrow><msup><mi>E</mi><mn>2</mn></msup><mi>T</mi></mrow></mfrac><msub><mi>F</mi><mi>p</mi></msub></mtd></mtr><mtr><mtd><mtext>                     </mtext><mrow><mo>[</mo><mrow><mfrac><mrow><msup><mi>p</mi><mn>2</mn></msup></mrow><mn>6</mn></mfrac><mo>+</mo><mfrac><mrow><msup><mi>k</mi><mn>2</mn></msup><mo>+</mo><msubsup><mi>ν</mi><mi>M</mi><mn>2</mn></msubsup></mrow><mn>4</mn></mfrac><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mfrac><mrow><msup><mi>k</mi><mn>2</mn></msup></mrow><mrow><mn>4</mn><mi>p</mi><mi>k</mi></mrow></mfrac><mi>ln</mi><mrow><mo>|</mo><mrow><mfrac><mrow><mi>k</mi><mo>−</mo><mn>2</mn><mi>p</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>2</mn><mi>p</mi></mrow></mfrac></mrow><mo>|</mo></mrow></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow><mn>.</mn></mtd></mtr></mtable></math> 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 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., <math display="block" id="M60"><mrow><mn>1</mn><mo>−</mo><mn>2</mn><mi>G</mi><mtext>Re</mtext><msub><mi>Π</mi><mi>M</mi></msub><mo stretchy="false">(</mo><msub><mi>m</mi><mrow><mi>π</mi><mn>,</mn><mi>σ</mi></mrow></msub><mn>,0</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0.</mn></mrow></math> The solution is a real value for m π,σ < 2 m q , and a meson is stable. However, for m π,σ > 2 m q , 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 <math id="M61"><mrow><msub><mi>Γ</mi><mi>M</mi></msub><mo>=</mo><mtext>Im</mtext><msub><mi>Π</mi><mi>M</mi></msub><mo stretchy="false">(</mo><msub><mi>m</mi><mrow><mi>π</mi><mn>,</mn><mi>σ</mi></mrow></msub><mn>,0</mn><mo stretchy="false">)</mo><mo>/</mo><msub><mi>m</mi><mrow><mi>π</mi><mn>,</mn><mi>σ</mi></mrow></msub></mrow></math> . Transport coefficients in an anisotropic quark matter 1 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. Shear viscosity 2 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 [ 101 101 ] <math display="block" id="M62"><mrow><mo stretchy="false">[</mo><msup><mi>p</mi><mi>μ</mi></msup><msub><mo>∂</mo><mi>μ</mi></msub><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mo>∂</mo><mi>μ</mi></msup><msubsup><mi>m</mi><mi>a</mi><mn>2</mn></msubsup><msubsup><mo>∂</mo><mi>μ</mi><mrow><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">]</mo><msub><mi>f</mi><mi>a</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mn>,</mn><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo stretchy="false">)</mo><mo>=</mo><mi>C</mi><mo stretchy="false">[</mo><msub><mi>f</mi><mi>a</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mn>,</mn><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mn>,</mn></mrow></math> where <math id="M63"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mo>∂</mo><mi>μ</mi></msup><msubsup><mi>m</mi><mi>a</mi><mn>2</mn></msubsup></mrow></math> acts as the force term attributed to the residual mean field interaction. The right-hand side of Eq. (32) is the collision term. Considering that the system has a small departure from the equilibrium due to an external perturbation, the collision term within the RTA can be given as <math display="block" id="M64"><mrow><mi>C</mi><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">]</mo><mo>≃</mo><mo>−</mo><mfrac><mrow><msup><mi>p</mi><mi>μ</mi></msup><msub><mi>u</mi><mi>μ</mi></msub><mo stretchy="false">[</mo><msub><mi>f</mi><mi>a</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mn>,</mn><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo stretchy="false">)</mo><mo>−</mo><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup><mo stretchy="false">(</mo><mi>x</mi><mn>,</mn><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><mrow><msub><mi>τ</mi><mi>a</mi></msub></mrow></mfrac><mo>=</mo><mo>−</mo><mfrac><mrow><msup><mi>p</mi><mi>μ</mi></msup><msub><mi>u</mi><mi>μ</mi></msub><mi>δ</mi><msub><mi>f</mi><mi>a</mi></msub></mrow><mrow><msub><mi>τ</mi><mi>a</mi></msub></mrow></mfrac><mn>,</mn></mrow></math> in which τ a denotes the relaxation time for particle species a and can quantify how fast the system reaches the equilibrium again. Near equilibrium, the distribution function can be expanded about a local equilibrium distribution function for the quarks as <math display="block" id="M65"><mrow><msub><mi>f</mi><mi>a</mi></msub><mo>=</mo><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup><mo stretchy="false">(</mo><mi>x</mi><mn>,</mn><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo stretchy="false">)</mo><mo>+</mo><mi>δ</mi><msub><mi>f</mi><mi>a</mi></msub><mn>,</mn></mrow></math> with <math id="M66"><mrow><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup><mo stretchy="false">(</mo><mi>x</mi><mn>,</mn><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">[</mo><mi>exp</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mi>u</mi><mi>ν</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mi>p</mi><mi>ν</mi></msup><mo>−</mo><msub><mi>μ</mi><mi>a</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>β</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>+</mo><msup><mrow><mn>1</mn><mo stretchy="false">]</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math> , where <math id="M67"><mrow><msup><mi>p</mi><mi>ν</mi></msup><mo>≡</mo><mo stretchy="false">(</mo><msub><mi>E</mi><mi>a</mi></msub><mn>,</mn><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo stretchy="false">)</mo></mrow></math> is the particle four-momentum, and u ν = γ ν (1,u) is the fluid four-velocity with γ ν =(1-u 2 ) 1/2 . δ f a in Eq. (33) is the deviation of the distribution function from the local equilibrium due to the external disturbance, which up to first-order in the gradient expansion can be read as <math display="block" id="M68"><mrow><mi>δ</mi><msub><mi>f</mi><mi>a</mi></msub><mo>=</mo><mo>−</mo><mfrac><mrow><msub><mi>τ</mi><mi>a</mi></msub></mrow><mrow><msup><mi>p</mi><mi>μ</mi></msup><msub><mi>u</mi><mi>μ</mi></msub></mrow></mfrac><mo stretchy="false">[</mo><msup><mi>p</mi><mi>μ</mi></msup><msub><mo>∂</mo><mi>μ</mi></msub><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup><mo>+</mo><msub><mi>m</mi><mi>a</mi></msub><mfrac><mrow><mi>d</mi><msub><mi>m</mi><mi>a</mi></msub></mrow><mrow><mi>d</mi><mi>T</mi></mrow></mfrac><mo stretchy="false">(</mo><msup><mo>∂</mo><mi>μ</mi></msup><mi>T</mi><mo stretchy="false">)</mo><msubsup><mo>∂</mo><mi>μ</mi><mrow><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></mrow></msubsup><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup><mo stretchy="false">]</mo><mn>,</mn></mrow></math> where the four-derivative can be decomposed into <math id="M69"><mrow><msub><mo>∂</mo><mi>μ</mi></msub><mo>≡</mo><mo>∂</mo><mo>/</mo><msup><mo>∂</mo><mi>μ</mi></msup><mo>≡</mo><msub><mi>u</mi><mi>μ</mi></msub><mi>D</mi><mo>+</mo><msub><mo>∇</mo><mi>μ</mi></msub></mrow></math> , and <math id="M70"><mrow><mi>D</mi><mo>≡</mo><msup><mi>u</mi><mi>μ</mi></msup><msub><mo>∂</mo><mi>μ</mi></msub></mrow></math> and <math id="M71"><mrow><msub><mo>∇</mo><mi>ν</mi></msub><mo>≡</mo><msup><mi>Δ</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msup><msub><mo>∂</mo><mi>ν</mi></msub></mrow></math> denote the time derivative and spatial gradient operator in the local rest frame, respectively. g μν = diag (1,-1,-1,-1) is the metric tensor, and Δ μν = g μν -u μ u ν is the projection operator orthogonal to u μ . In the presence of weak momentum anisotropy, the associated covariant version of the weakly anisotropic function <math id="M72"><mrow><msubsup><mi>f</mi><mi>a</mi><mrow><mi>a</mi><mi>n</mi></mrow></msubsup></mrow></math> for particle species a can be written as <math display="block" id="M73"><mrow><msubsup><mi>f</mi><mi>a</mi><mrow><mtext>aniso</mtext></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mn>,</mn><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mi>exp</mi><mo stretchy="false">[</mo><mo stretchy="false">(</mo><msqrt><mrow><msup><mrow><mo stretchy="false">(</mo><msub><mi>p</mi><mi>ν</mi></msub><msup><mi>u</mi><mi>ν</mi></msup><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>+</mo><mi>ξ</mi><msup><mrow><mo stretchy="false">(</mo><msub><mi>p</mi><mi>ν</mi></msub><msup><mi>V</mi><mi>ν</mi></msup><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></msqrt><mo>−</mo><msub><mi>μ</mi><mi>a</mi></msub><mo stretchy="false">)</mo><mi>β</mi><mo stretchy="false">]</mo><mo>+</mo><mn>1</mn></mrow></mfrac><mn>,</mn></mrow></math> <math display="block" id="M74"><mrow><mo>≈</mo><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup><mo>−</mo><mfrac><mrow><mi>ξ</mi><msup><mrow><mo stretchy="false">(</mo><msup><mi>p</mi><mi>ν</mi></msup><msub><mi>V</mi><mi>ν</mi></msub><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow><mrow><mn>2</mn><mi>T</mi><msup><mi>p</mi><mi>ν</mi></msup><msub><mi>u</mi><mi>ν</mi></msub></mrow></mfrac><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where V ν =(0,n) is defined as the anisotropy vector. Employing Eq. (37) in Eq. (35), δ f a can be decomposed into two parts: <math display="block" id="M75"><mrow><mi>δ</mi><msub><mi>f</mi><mi>a</mi></msub><mo>=</mo><mi>δ</mi><msubsup><mi>f</mi><mi>a</mi><mrow><mtext>iso</mtext></mrow></msubsup><mo>+</mo><mi>δ</mi><msubsup><mi>f</mi><mi>a</mi><mrow><mtext>aniso</mtext></mrow></msubsup><mn>.</mn></mrow></math> The first term on the right hand side is <math display="block" id="M76"><mtable><mtr><mtd><mi>δ</mi><msubsup><mi>f</mi><mi>a</mi><mrow><mtext>iso</mtext></mrow></msubsup><mo>=</mo><mfrac><mrow><msub><mi>τ</mi><mi>a</mi></msub></mrow><mrow><msup><mi>p</mi><mi>μ</mi></msup><msub><mi>u</mi><mi>μ</mi></msub></mrow></mfrac><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup><mo stretchy="false">)</mo><mo stretchy="false">[</mo><msup><mi>p</mi><mi>μ</mi></msup><msup><mi>p</mi><mi>ν</mi></msup><mi>β</mi><mo stretchy="false">(</mo><msub><mi>u</mi><mi>μ</mi></msub><mi>D</mi><msub><mi>u</mi><mi>ν</mi></msub><mo>+</mo><msub><mo>∇</mo><mi>μ</mi></msub><msub><mi>u</mi><mi>ν</mi></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>+</mo><msup><mi>p</mi><mi>μ</mi></msup><mo stretchy="false">(</mo><msup><mi>p</mi><mi>μ</mi></msup><msub><mi>u</mi><mi>μ</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>u</mi><mi>μ</mi></msub><mi>D</mi><mi>β</mi><mo>+</mo><msub><mo>∇</mo><mi>μ</mi></msub><mi>β</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mrow><mi>d</mi><msubsup><mi>m</mi><mi>a</mi><mn>2</mn></msubsup></mrow><mrow><mi>d</mi><msup><mi>T</mi><mn>2</mn></msup></mrow></mfrac><mi>D</mi><mi>T</mi><mo stretchy="false">]</mo><mn>.</mn></mtd></mtr></mtable></math> Employing the equation of motion in ideal hydrodynamics and ideal thermodynamic relations, <math id="M77"><mrow><mi>δ</mi><msubsup><mi>f</mi><mi>a</mi><mrow><mtext>iso</mtext></mrow></msubsup></mrow></math> can be rewritten as <math display="block" id="M78"><mtable><mtr><mtd><mi>δ</mi><msubsup><mi>f</mi><mi>a</mi><mrow><mtext>iso</mtext></mrow></msubsup><mo>=</mo><mfrac><mrow><msub><mi>τ</mi><mi>a</mi></msub></mrow><mrow><msup><mi>p</mi><mi>μ</mi></msup><msub><mi>u</mi><mi>μ</mi></msub></mrow></mfrac><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup><mo stretchy="false">)</mo><mo>{</mo><mfrac><mrow><msup><mi>p</mi><mi>μ</mi></msup><msup><mi>p</mi><mi>ν</mi></msup></mrow><mi>T</mi></mfrac><msub><mi>σ</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msub></mtd></mtr><mtr><mtd><mo>+</mo><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msup><mi>p</mi><mi>μ</mi></msup><msub><mi>u</mi><mi>μ</mi></msub><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>−</mo><msup><mi>T</mi><mn>2</mn></msup><mfrac><mrow><mi>d</mi><msubsup><mi>m</mi><mi>a</mi><mn>2</mn></msubsup></mrow><mrow><mi>d</mi><msup><mi>T</mi><mn>2</mn></msup></mrow></mfrac><mo stretchy="false">)</mo><msubsup><mi>c</mi><mtext>s</mtext><mn>2</mn></msubsup><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><msub><mi>Δ</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msub><msup><mi>p</mi><mi>μ</mi></msup><msup><mi>p</mi><mi>ν</mi></msup><mo stretchy="false">]</mo><mfrac><mi>θ</mi><mi>T</mi></mfrac><mo>}</mo><mn>,</mn></mtd></mtr></mtable></math> with <math id="M79"><mrow><mi>θ</mi><mo>=</mo><msub><mo>∂</mo><mi>α</mi></msub><msup><mi>u</mi><mi>α</mi></msup></mrow></math> and <math id="M80"><mrow><msubsup><mi>c</mi><mtext>s</mtext><mn>2</mn></msubsup></mrow></math> being the expansion rate of the fluid and squared sound velocity in the medium, respectively. The velocity stress tensor has the usual definition: <math id="M81"><mrow><msup><mi>σ</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msup><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>Δ</mi><mrow><mi>μ</mi><mi>α</mi></mrow></msup><msup><mi>Δ</mi><mrow><mi>ν</mi><mi>β</mi></mrow></msup><mo stretchy="false">(</mo><msub><mo>∇</mo><mi>α</mi></msub><msub><mi>u</mi><mi>β</mi></msub><mo>+</mo><msub><mo>∇</mo><mi>β</mi></msub><msub><mi>u</mi><mi>α</mi></msub><mo>−</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><msup><mi>Δ</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msup><mi>θ</mi><mo stretchy="false">)</mo></mrow></math> . After tedious calculations, one can obtain the second term in Eq. (38): <math display="block" id="M82"><mtable><mtr><mtd><mi>δ</mi><msubsup><mi>f</mi><mi>a</mi><mrow><mtext>aniso</mtext></mrow></msubsup><mo>=</mo><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><msup><mi>p</mi><mi>ν</mi></msup><msub><mi>V</mi><mi>ν</mi></msub><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mi>ξ</mi><mi>β</mi></mrow><mrow><mn>2</mn><msup><mi>p</mi><mi>ν</mi></msup><msub><mi>u</mi><mi>ν</mi></msub></mrow></mfrac><mo stretchy="false">(</mo><mn>2</mn><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mi>δ</mi><msubsup><mi>f</mi><mi>a</mi><mrow><mtext>iso</mtext></mrow></msubsup></mtd></mtr><mtr><mtd><mo>−</mo><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><msup><mi>p</mi><mi>ν</mi></msup><msub><mi>V</mi><mi>ν</mi></msub><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mi>ξ</mi></mrow><mrow><mn>2</mn><mo stretchy="false">(</mo><msup><mi>p</mi><mi>ν</mi></msup><msub><mi>u</mi><mi>ν</mi></msub><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mfrac><mo stretchy="false">[</mo><msup><mi>p</mi><mi>μ</mi></msup><msup><mi>p</mi><mi>ν</mi></msup><mi>β</mi><mo stretchy="false">(</mo><msub><mi>u</mi><mi>μ</mi></msub><mi>D</mi><msub><mi>u</mi><mi>ν</mi></msub><mo>+</mo><msub><mo>∇</mo><mi>μ</mi></msub><msub><mi>u</mi><mi>ν</mi></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>−</mo><msup><mi>p</mi><mi>μ</mi></msup><mo stretchy="false">(</mo><msup><mi>p</mi><mi>μ</mi></msup><msub><mi>u</mi><mi>μ</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>u</mi><mi>μ</mi></msub><mi>D</mi><mi>β</mi><mo>+</mo><msub><mo>∇</mo><mi>μ</mi></msub><mi>β</mi><mo stretchy="false">)</mo><mo>−</mo><mfrac><mrow><mi>d</mi><msubsup><mi>m</mi><mi>a</mi><mn>2</mn></msubsup></mrow><mrow><mi>d</mi><msup><mi>T</mi><mn>2</mn></msup></mrow></mfrac><mi>D</mi><mi>T</mi><mo stretchy="false">]</mo></mtd></mtr><mtr><mtd><mo>×</mo><mfrac><mrow><msub><mi>τ</mi><mi>a</mi></msub></mrow><mrow><msup><mi>p</mi><mi>μ</mi></msup><msub><mi>u</mi><mi>μ</mi></msub></mrow></mfrac><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>=</mo><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><msup><mi>p</mi><mi>ν</mi></msup><msub><mi>V</mi><mi>ν</mi></msub><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mi>ξ</mi><mi>β</mi></mrow><mrow><mn>2</mn><msup><mi>p</mi><mi>ν</mi></msup><msub><mi>u</mi><mi>ν</mi></msub></mrow></mfrac><mo stretchy="false">(</mo><mn>2</mn><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mi>δ</mi><msubsup><mi>f</mi><mi>a</mi><mrow><mtext>iso</mtext></mrow></msubsup></mtd></mtr><mtr><mtd><mo>−</mo><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><msup><mi>p</mi><mi>ν</mi></msup><msub><mi>V</mi><mi>ν</mi></msub><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mi>ξ</mi></mrow><mrow><mn>2</mn><mo stretchy="false">(</mo><msup><mi>p</mi><mi>ν</mi></msup><msub><mi>u</mi><mi>ν</mi></msub><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mfrac><mfrac><mrow><msub><mi>τ</mi><mi>a</mi></msub></mrow><mrow><msup><mi>p</mi><mi>μ</mi></msup><msub><mi>u</mi><mi>μ</mi></msub></mrow></mfrac><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup><mo stretchy="false">)</mo><mo>{</mo><mfrac><mrow><msup><mi>p</mi><mi>μ</mi></msup><msup><mi>p</mi><mi>ν</mi></msup></mrow><mi>T</mi></mfrac><msub><mi>σ</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msub></mtd></mtr><mtr><mtd><mo>−</mo><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msup><mi>p</mi><mi>μ</mi></msup><msub><mi>u</mi><mi>μ</mi></msub><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>−</mo><msup><mi>T</mi><mn>2</mn></msup><mfrac><mrow><mi>d</mi><msubsup><mi>m</mi><mi>a</mi><mn>2</mn></msubsup></mrow><mrow><mi>d</mi><msup><mi>T</mi><mn>2</mn></msup></mrow></mfrac><mo stretchy="false">)</mo><msubsup><mi>c</mi><mi>s</mi><mn>2</mn></msubsup><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><msub><mi>Δ</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msub><msup><mi>p</mi><mi>μ</mi></msup><msup><mi>p</mi><mi>ν</mi></msup><mo stretchy="false">]</mo><mfrac><mi>θ</mi><mi>T</mi></mfrac><mo>}</mo><mn>.</mn></mtd></mtr></mtable></math> Allowing the system to be slightly out of equilibrium, the energy-momentum tensor T μν can be expanded as <math id="M83"><mrow><msup><mi>T</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msup><mo>=</mo><msubsup><mi>T</mi><mn>0</mn><mrow><mi>μ</mi><mi>ν</mi></mrow></msubsup><mo>+</mo><msubsup><mi>T</mi><mrow><mtext>diss</mtext></mrow><mrow><mi>μ</mi><mi>ν</mi></mrow></msubsup></mrow></math> , where <math id="M84"><mrow><msubsup><mi>T</mi><mn>0</mn><mrow><mi>μ</mi><mi>ν</mi></mrow></msubsup></mrow></math> is the ideal perfect fluid form and <math id="M85"><mrow><msubsup><mi>T</mi><mrow><mtext>diss</mtext></mrow><mrow><mi>μ</mi><mi>ν</mi></mrow></msubsup></mrow></math> is the dissipative part of the energy-momentum tensor. In the hydrodynamical description of hot QCD matter, the dissipative part of the energy-momentum tensor up to the first order in the gradient expansion has the following form [ 111 111 ]: <math display="block" id="M86"><mrow><msubsup><mi>T</mi><mrow><mtext>diss</mtext></mrow><mrow><mi>μ</mi><mi>ν</mi></mrow></msubsup><mo>=</mo><msup><mi>π</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msup><mo>−</mo><mi>Π</mi><msup><mi>Δ</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msup><mn>,</mn></mrow></math> where π μν and ∏ are the shear stress tensor and bulk viscous pressure, respectively. In present work, our focus is the shear viscosity component only. In the kinetic theory, the first-order shear stress tensor π μν can be constructed in terms of the distribution functions <math display="block" id="M87"><mrow><msup><mi>π</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msup><mo>=</mo><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mfrac><mrow><msup><mi>d</mi><mn>3</mn></msup><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mo stretchy="false">)</mo></mrow><mn>3</mn></msup></mrow></mfrac><mfrac><mn>1</mn><mrow><mi>u</mi><mo>⋅</mo><mi>p</mi></mrow></mfrac><msubsup><mi>Δ</mi><mrow><mi>ϕ</mi><mi>γ</mi></mrow><mrow><mi>μ</mi><mi>ν</mi></mrow></msubsup><msup><mi>p</mi><mi>ϕ</mi></msup><msup><mi>p</mi><mi>γ</mi></msup><mi>δ</mi><mi>f</mi><mn>.</mn></mrow></math> Here, the double projection operator is defined as <math id="M88"><mrow><msubsup><mi>Δ</mi><mrow><mi>ϕ</mi><mi>γ</mi></mrow><mrow><mi>μ</mi><mi>ν</mi></mrow></msubsup><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><msubsup><mi>Δ</mi><mi>ϕ</mi><mi>μ</mi></msubsup><msubsup><mi>Δ</mi><mi>γ</mi><mi>ν</mi></msubsup><mo>+</mo><msubsup><mi>Δ</mi><mi>γ</mi><mi>μ</mi></msubsup><msubsup><mi>Δ</mi><mi>ϕ</mi><mi>ν</mi></msubsup><mo stretchy="false">)</mo><mo>−</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><msup><mi>Δ</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msup><msup><mi>Δ</mi><mrow><mi>ϕ</mi><mi>γ</mi></mrow></msup></mrow></math> , which can project any rank-2 Lorentz tensor onto its transverse (to u μ ) and traceless part. Inserting Eqs. (38)–(41) to Eq. (42) and comparing with the first-order Navier–Stokes equation π μν =2 ησ μν [ 112 112 ], in the rest frame of the thermal system with u μ ≡(1,0) and <math id="M89"><mrow><msup><mi>p</mi><mi>ν</mi></msup><msub><mi>u</mi><mi>ν</mi></msub><mo>=</mo><msub><mi>E</mi><mi>a</mi></msub></mrow></math> , we finally obtain the expression of the ξ -dependent shear viscosity of particle species a , <math display="block" id="M90"><mtable><mtr><mtd><msub><mi>η</mi><mi>a</mi></msub><mo>=</mo><mo>−</mo><mfrac><mrow><mi>ξ</mi><msub><mi>d</mi><mi>a</mi></msub></mrow><mrow><mn>180</mn><msup><mi>T</mi><mn>2</mn></msup></mrow></mfrac><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mfrac><mrow><mi>d</mi><mi>p</mi></mrow><mrow><msup><mi>π</mi><mn>2</mn></msup></mrow></mfrac><mfrac><mrow><msub><mi>τ</mi><mi>a</mi></msub><msup><mi>p</mi><mn>8</mn></msup></mrow><mrow><msubsup><mi>E</mi><mi>a</mi><mn>3</mn></msubsup></mrow></mfrac><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mn>2</mn><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup><mo>+</mo><mfrac><mi>T</mi><mrow><msub><mi>E</mi><mi>a</mi></msub></mrow></mfrac><mo stretchy="false">)</mo><mn>,</mn></mtd></mtr><mtr><mtd><mo>+</mo><mfrac><mrow><msub><mi>d</mi><mi>a</mi></msub></mrow><mrow><mn>30</mn><mi>T</mi></mrow></mfrac><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mfrac><mrow><mi>d</mi><mi>p</mi></mrow><mrow><msup><mi>π</mi><mn>2</mn></msup></mrow></mfrac><mfrac><mrow><msub><mi>τ</mi><mi>a</mi></msub><msup><mi>p</mi><mn>6</mn></msup></mrow><mrow><msubsup><mi>E</mi><mi>a</mi><mn>2</mn></msubsup></mrow></mfrac><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup><mo stretchy="false">)</mo><mn>,</mn></mtd></mtr></mtable></math> which is consistent with the result from Ref. [ 92 92 ]. For a system consisting of multiple particle species, the total shear viscosity is given as <math id="M91"><mrow><mi>η</mi><mo>=</mo><mstyle displaystyle="true"><msub><mo>∑</mo><mi>a</mi></msub><mrow><msub><mi>η</mi><mi>a</mi></msub></mrow></mstyle></mrow></math> . For SU(2) light quark matter, <math id="M92"><mrow><mi>a</mi><mo>=</mo><mi>u</mi><mn>,</mn><mi>d</mi><mn>,</mn><mover accent="true"><mi>u</mi><mo>¯</mo></mover><mn>,</mn><mover accent="true"><mi>d</mi><mo>¯</mo></mover></mrow></math> and the spin-color degeneracy factor reads explicitly as d a =2 N c . Electrical conductivity and Seebeck coefficient 2 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 e a in the presence of an external electromagnetic field is given by <math display="block" id="M93"><mrow><mo stretchy="false">[</mo><msup><mi>p</mi><mi>μ</mi></msup><msub><mo>∂</mo><mi>μ</mi></msub><mo>+</mo><mo stretchy="false">(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mo>∂</mo><mi>μ</mi></msup><msubsup><mi>m</mi><mi>a</mi><mn>2</mn></msubsup><mo>+</mo><msub><mi>e</mi><mi>a</mi></msub><msup><mi>F</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msup><msub><mi>p</mi><mi>ν</mi></msub><mo stretchy="false">)</mo><msubsup><mo>∂</mo><mi>μ</mi><mrow><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">]</mo><msub><mi>f</mi><mi>a</mi></msub><mo>=</mo><mo>−</mo><mfrac><mrow><msup><mi>p</mi><mi>μ</mi></msup><msub><mi>u</mi><mi>μ</mi></msub><mi>δ</mi><msub><mi>f</mi><mi>a</mi></msub></mrow><mrow><msub><mi>τ</mi><mi>a</mi></msub></mrow></mfrac><mn>,</mn></mrow></math> where F μν is the electromagnetic field strength tensor. We only consider the presence of an external electric field, <math id="M94"><mrow><msup><mi>F</mi><mrow><mi>i</mi><mn>0</mn></mrow></msup><mo>=</mo><mo>−</mo><msup><mi>F</mi><mrow><mn>0</mn><mi>i</mi></mrow></msup><mo>=</mo><mstyle mathvariant="bold" mathsize="normal"><mi>E</mi></mstyle><mo>=</mo><mo stretchy="false">(</mo><mi mathvariant="script">E</mi><mn>,0,0</mn><mo stretchy="false">)</mo></mrow></math> . It is convenient to work in the local rest frame of plasma, and under the steady state assumption ( f a does not depend on time explicitly, <math id="M95"><mrow><mfrac><mrow><mo>∂</mo><msub><mi>f</mi><mi>a</mi></msub></mrow><mrow><mo>∂</mo><mi>t</mi></mrow></mfrac></mrow></math> =0), Eq. (45) can be given by [ 84 84 ] <math display="block" id="M96"><mrow><msub><mstyle mathvariant="bold" mathsize="normal"><mi>v</mi></mstyle><mi>a</mi></msub><mo>⋅</mo><mo>∇</mo><msub><mi>f</mi><mi>a</mi></msub><mo>+</mo><mo stretchy="false">(</mo><msub><mi>e</mi><mi>a</mi></msub><mi mathvariant="script">E</mi><mo>−</mo><msub><mo>∇</mo><mrow/></msub><msub><mi>E</mi><mi>a</mi></msub><mo stretchy="false">)</mo><mo>⋅</mo><mfrac><mrow><mo>∂</mo><msub><mi>f</mi><mi>a</mi></msub></mrow><mrow><mo>∂</mo><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle></mrow></mfrac><mo>=</mo><mo>−</mo><mfrac><mrow><mi>δ</mi><msub><mi>f</mi><mi>a</mi></msub></mrow><mrow><msub><mi>τ</mi><mi>a</mi></msub></mrow></mfrac><mn>,</mn></mrow></math> where we have used of the chain rule <math id="M97"><mrow><mfrac><mrow><mo>∂</mo><msup><mi>p</mi><mn>0</mn></msup></mrow><mrow><mo>∂</mo><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle></mrow></mfrac><mfrac><mo>∂</mo><mrow><mo>∂</mo><msup><mi>p</mi><mn>0</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mo>∂</mo><mrow><mo>∂</mo><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle></mrow></mfrac><mo>→</mo><mfrac><mo>∂</mo><mrow><mo>∂</mo><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle></mrow></mfrac></mrow></math> . e a is the electric charge of particle species a , and <math id="M98"><mrow><msub><mstyle mathvariant="bold" mathsize="normal"><mi>v</mi></mstyle><mi>a</mi></msub><mo>=</mo><mo>∂</mo><msub><mi>E</mi><mi>a</mi></msub><mo>/</mo><mo>∂</mo><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle></mrow></math> is its velocity. To solve Eq. (46), we assume that the deviation of the distribution function in an anisotropic medium satisfies the following linear form: <math display="block" id="M99"><mrow><mi>δ</mi><msub><mi>f</mi><mi>a</mi></msub><mo>=</mo><mo>−</mo><msub><mi>τ</mi><mi>a</mi></msub><mo stretchy="false">(</mo><msub><mi>e</mi><mi>a</mi></msub><mi mathvariant="script">E</mi><mo>−</mo><mfrac><mrow><mo>∂</mo><msub><mi>E</mi><mi>a</mi></msub></mrow><mrow><mo>∂</mo><mstyle mathvariant="bold" mathsize="normal"><mi>x</mi></mstyle></mrow></mfrac><mo stretchy="false">)</mo><mo>⋅</mo><mfrac><mrow><mo>∂</mo><msubsup><mi>f</mi><mi>a</mi><mrow><mtext>aniso</mtext></mrow></msubsup></mrow><mrow><mo>∂</mo><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle></mrow></mfrac><mo>−</mo><msub><mi>τ</mi><mi>a</mi></msub><msub><mstyle mathvariant="bold" mathsize="normal"><mi>v</mi></mstyle><mi>a</mi></msub><mo>⋅</mo><mfrac><mrow><mo>∂</mo><msubsup><mi>f</mi><mi>a</mi><mrow><mtext>aniso</mtext></mrow></msubsup></mrow><mrow><mo>∂</mo><mstyle mathvariant="bold" mathsize="normal"><mi>x</mi></mstyle></mrow></mfrac><mn>,</mn></mrow></math> The spatial gradient of the equilibrium isotropic distribution <math id="M100"><mrow><msub><mo>∂</mo><mstyle mathvariant="bold" mathsize="normal"><mi>x</mi></mstyle></msub><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup></mrow></math> in the presence of a medium-dependent quasiparticle mass can be expressed in the following linear form: <math display="block" id="M101"><mrow><msub><mo>∂</mo><mstyle mathvariant="bold" mathsize="normal"><mi>x</mi></mstyle></msub><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup><mo>=</mo><mo>−</mo><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup><mo stretchy="false">)</mo><mrow><mo>(</mo><mrow><msub><mo>∂</mo><mstyle mathvariant="bold" mathsize="normal"><mi>x</mi></mstyle></msub><mo stretchy="false">(</mo><mfrac><mrow><msub><mi>E</mi><mi>a</mi></msub></mrow><mi>T</mi></mfrac><mo stretchy="false">)</mo><mo>−</mo><msub><mo>∂</mo><mstyle mathvariant="bold" mathsize="normal"><mi>x</mi></mstyle></msub><mo stretchy="false">(</mo><mfrac><mrow><msub><mi>μ</mi><mi>a</mi></msub></mrow><mi>T</mi></mfrac><mo stretchy="false">)</mo></mrow><mo>)</mo></mrow><mn>,</mn></mrow></math> where μ a =t aμ denotes the chemical potential of particle species a and t a =+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. (45) into Eq. (44), the perturbative term δ f a in an anisotropic medium can then be written as <math display="block" id="M102"><mtable><mtr><mtd><mi>δ</mi><msub><mi>f</mi><mi>a</mi></msub><mo>=</mo><msub><mi>H</mi><mi>a</mi></msub><msub><mi>τ</mi><mi>a</mi></msub><mo stretchy="false">(</mo><msub><mi>e</mi><mi>a</mi></msub><mi mathvariant="script">E</mi><msub><mi>v</mi><mi>x</mi></msub><mo stretchy="false">)</mo><mo>−</mo><msub><mi>G</mi><mi>a</mi></msub><msub><mi>τ</mi><mi>a</mi></msub><msub><mo>∂</mo><mi>x</mi></msub><mo stretchy="false">(</mo><mfrac><mrow><msub><mi>μ</mi><mi>a</mi></msub></mrow><mi>T</mi></mfrac><mo stretchy="false">)</mo><msub><mi>v</mi><mi>x</mi></msub></mtd></mtr><mtr><mtd><mo>−</mo><mfrac><mrow><mi>ξ</mi><msup><mrow><mo stretchy="false">(</mo><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo>⋅</mo><mstyle mathvariant="bold" mathsize="normal"><mi>n</mi></mstyle><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow><mrow><msub><mi>E</mi><mi>a</mi></msub><mi>T</mi></mrow></mfrac><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup><mo stretchy="false">)</mo><msub><mi>τ</mi><mi>a</mi></msub><msub><mo>∂</mo><mi>x</mi></msub><msub><mi>E</mi><mi>a</mi></msub><mn>.</mn></mtd></mtr></mtable></math> The expressions of H a and F a in the above equation are as follows, respectively: <math display="block" id="M103"><mtable><mtr><mtd><msub><mi>H</mi><mi>a</mi></msub><mo>=</mo><mfrac><mn>1</mn><mi>T</mi></mfrac><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>ξ</mi><mi>c</mi><mo stretchy="false">(</mo><mi>θ</mi><mn>,</mn><mi>ϕ</mi><mn>,</mn><mi>χ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>−</mo><mfrac><mrow><mi>ξ</mi><msup><mi>p</mi><mn>2</mn></msup><mi>c</mi><mo stretchy="false">(</mo><mi>θ</mi><mn>,</mn><mi>ϕ</mi><mn>,</mn><mi>χ</mi><mo stretchy="false">)</mo></mrow><mrow><mn>2</mn><msub><mi>E</mi><mi>a</mi></msub><msup><mi>T</mi><mn>2</mn></msup></mrow></mfrac><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup><mo stretchy="false">)</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mn>2</mn><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup><mo>+</mo><mfrac><mi>T</mi><mrow><msub><mi>E</mi><mi>a</mi></msub></mrow></mfrac></mrow><mo>)</mo></mrow><mn>,</mn></mtd></mtr></mtable></math> <math display="block" id="M104"><mtable><mtr><mtd><msub><mi>G</mi><mi>a</mi></msub><mo>=</mo><mfrac><mn>1</mn><mrow><msup><mi>T</mi><mn>2</mn></msup></mrow></mfrac><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup><mo stretchy="false">)</mo><mo>−</mo><mfrac><mrow><mi>ξ</mi><msup><mi>p</mi><mn>2</mn></msup><mi>c</mi><mo stretchy="false">(</mo><mi>θ</mi><mn>,</mn><mi>ϕ</mi><mn>,</mn><mi>χ</mi><mo stretchy="false">)</mo></mrow><mrow><mn>2</mn><msub><mi>E</mi><mi>a</mi></msub><msup><mi>T</mi><mn>3</mn></msup></mrow></mfrac><mo stretchy="false">(</mo><msub><mi>E</mi><mi>a</mi></msub><mo>−</mo><msub><mi>μ</mi><mi>a</mi></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>×</mo><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup><mo stretchy="false">)</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mn>2</mn><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup><mo>−</mo><mfrac><mi>T</mi><mrow><msub><mi>E</mi><mi>a</mi></msub><mo>−</mo><msub><mi>μ</mi><mi>a</mi></msub></mrow></mfrac></mrow><mo>)</mo></mrow><mn>.</mn></mtd></mtr></mtable></math> In the linear response theory, the general formula for the electric current density J a due to particle species a in response to an external electric field ( ε ) and temperature gradient ( <math id="M105"><mrow><msub><mo>∇</mo><mi>x</mi></msub><mi>T</mi></mrow></math> ) is given by [ 113 113 ] <math display="block" id="M106"><mrow><msub><mstyle mathvariant="bold" mathsize="normal"><mi>J</mi></mstyle><mi>a</mi></msub><mo>=</mo><msub><mi>σ</mi><mrow><mtext>el</mtext><mn>,</mn><mi>a</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="script">E</mi><mo>−</mo><msub><mi>S</mi><mi>a</mi></msub><msub><mo>∇</mo><mi>x</mi></msub><mi>T</mi><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where σ el, a and S a are the electrical conductivity and Seebeck coefficient of the a -th particle, respectively. In terms of the distribution function, J a within the kinetic theory can be written as <math display="block" id="M107"><mrow><msub><mstyle mathvariant="bold" mathsize="normal"><mi>J</mi></mstyle><mi>a</mi></msub><mo>=</mo><msub><mi>e</mi><mi>a</mi></msub><msub><mi>d</mi><mi>a</mi></msub><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mfrac><mrow><msup><mi>d</mi><mn>3</mn></msup><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mo stretchy="false">)</mo></mrow><mn>3</mn></msup></mrow></mfrac><msub><mstyle mathvariant="bold" mathsize="normal"><mi>v</mi></mstyle><mi>a</mi></msub><mi>δ</mi><msub><mi>f</mi><mi>a</mi></msub><mn>.</mn></mrow></math> Finally, the expressions for σ el, a and S a in the weakly anisotropic medium are respectively obtained as follows: <math display="block" id="M108"><mtable><mtr><mtd><msub><mi>σ</mi><mrow><mtext>el</mtext><mn>,</mn><mi>a</mi></mrow></msub><mo>=</mo><mfrac><mrow><msubsup><mi>e</mi><mi>a</mi><mn>2</mn></msubsup><msub><mi>d</mi><mi>a</mi></msub><msub><mi>τ</mi><mi>a</mi></msub></mrow><mrow><mn>6</mn><mi>T</mi></mrow></mfrac><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mfrac><mrow><mi>d</mi><mi>p</mi></mrow><mrow><msup><mi>π</mi><mn>2</mn></msup></mrow></mfrac><mfrac><mrow><msup><mi>p</mi><mn>4</mn></msup></mrow><mrow><msubsup><mi>E</mi><mi>a</mi><mn>2</mn></msubsup></mrow></mfrac><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup><mo stretchy="false">)</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mfrac><mi>ξ</mi><mn>3</mn></mfrac></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mo>−</mo><mfrac><mrow><mi>ξ</mi><msubsup><mi>e</mi><mi>a</mi><mn>2</mn></msubsup><msub><mi>d</mi><mi>a</mi></msub><msub><mi>τ</mi><mi>a</mi></msub></mrow><mrow><mn>36</mn><msup><mi>T</mi><mn>2</mn></msup></mrow></mfrac><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mfrac><mrow><mi>d</mi><mi>p</mi></mrow><mrow><msup><mi>π</mi><mn>2</mn></msup></mrow></mfrac><mfrac><mrow><msup><mi>p</mi><mn>6</mn></msup></mrow><mrow><msubsup><mi>E</mi><mi>a</mi><mn>3</mn></msubsup></mrow></mfrac><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup><mo stretchy="false">)</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mn>2</mn><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup><mo>+</mo><mfrac><mi>T</mi><mrow><msub><mi>E</mi><mi>a</mi></msub></mrow></mfrac></mrow><mo>)</mo></mrow><mn>,</mn></mtd></mtr></mtable></math> and <math display="block" id="M109"><mtable columnalign="left"><mtr><mtd><msub><mi>S</mi><mi>a</mi></msub><mo>=</mo><mfrac><mn>1</mn><mrow><msub><mi>σ</mi><mrow><mtext>el</mtext><mn>,</mn><mi>a</mi></mrow></msub></mrow></mfrac><mo stretchy="false">[</mo><mfrac><mrow><msub><mi>e</mi><mi>a</mi></msub><msub><mi>d</mi><mi>a</mi></msub><msub><mi>τ</mi><mi>a</mi></msub></mrow><mrow><mn>6</mn><msup><mi>T</mi><mn>2</mn></msup></mrow></mfrac><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mfrac><mrow><mi>d</mi><mi>p</mi></mrow><mrow><msup><mi>π</mi><mn>2</mn></msup></mrow></mfrac><mfrac><mrow><msup><mi>p</mi><mn>4</mn></msup></mrow><mrow><msubsup><mi>E</mi><mi>a</mi><mn>2</mn></msubsup></mrow></mfrac><mo stretchy="false">(</mo><msub><mi>E</mi><mi>a</mi></msub><mo>−</mo><msub><mi>μ</mi><mi>a</mi></msub><mo stretchy="false">)</mo><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mtext>    </mtext><mo>−</mo><mfrac><mrow><msub><mi>e</mi><mi>a</mi></msub><msub><mi>d</mi><mi>a</mi></msub><mi>ξ</mi><msub><mi>τ</mi><mi>a</mi></msub></mrow><mrow><mn>36</mn><msup><mi>T</mi><mn>3</mn></msup></mrow></mfrac><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mfrac><mrow><mi>d</mi><mi>p</mi></mrow><mrow><msup><mi>π</mi><mn>2</mn></msup></mrow></mfrac><mfrac><mrow><msup><mi>p</mi><mn>6</mn></msup></mrow><mrow><msubsup><mi>E</mi><mi>a</mi><mn>3</mn></msubsup></mrow></mfrac><mo stretchy="false">(</mo><msub><mi>E</mi><mi>a</mi></msub><mo>−</mo><msub><mi>μ</mi><mi>a</mi></msub><mo stretchy="false">)</mo><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mtext>    </mtext><mo>×</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mn>2</mn><msubsup><mi>f</mi><mi>a</mi><mn>0</mn></msubsup><mo>−</mo><mfrac><mi>T</mi><mrow><mo stretchy="false">(</mo><msub><mi>E</mi><mi>a</mi></msub><mo>−</mo><msub><mi>μ</mi><mi>a</mi></msub><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>)</mo></mrow><mo stretchy="false">]</mo><mo>=</mo><mfrac><mrow><msub><mi>α</mi><mi>a</mi></msub></mrow><mrow><msub><mi>σ</mi><mrow><mtext>el</mtext><mn>,</mn><mi>a</mi></mrow></msub></mrow></mfrac><mn>,</mn></mtd></mtr></mtable></math> where α a is 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 [ 114 114 , 115 115 ]. Inspired by this, the total Seebeck coefficient in a medium composed of light quarks and antiquarks can be given as <math display="block" id="M110"><mrow><mi>S</mi><mo>=</mo><mfrac><mrow><mstyle displaystyle="true"><msub><mo>∑</mo><mi>a</mi></msub><mrow><msub><mi>S</mi><mi>a</mi></msub><msub><mi>σ</mi><mrow><mtext>el</mtext><mn>,</mn><mi>a</mi></mrow></msub></mrow></mstyle></mrow><mrow><mstyle displaystyle="true"><msub><mo>∑</mo><mi>a</mi></msub><mrow><msub><mi>σ</mi><mrow><mtext>el</mtext><mn>,</mn><mi>a</mi></mrow></msub></mrow></mstyle></mrow></mfrac><mo>=</mo><mfrac><mrow><mstyle displaystyle="true"><msub><mo>∑</mo><mi>a</mi></msub><mrow><msub><mi>α</mi><mi>a</mi></msub></mrow></mstyle></mrow><mrow><mstyle displaystyle="true"><msub><mo>∑</mo><mi>a</mi></msub><mrow><msub><mi>σ</mi><mrow><mtext>el</mtext><mn>,</mn><mi>a</mi></mrow></msub></mrow></mstyle></mrow></mfrac><mo>=</mo><mfrac><mi>α</mi><mrow><msub><mi>σ</mi><mrow><mtext>el</mtext></mrow></msub></mrow></mfrac><mn>,</mn></mrow></math> where the fractional electric charges of up and down (anti-)quarks are given explicitly by <math id="M111"><mrow><msub><mi>e</mi><mi>u</mi></msub><mo>=</mo><mo>−</mo><msub><mi>e</mi><mover accent="true"><mi>u</mi><mo>¯</mo></mover></msub><mo>=</mo><mn>2</mn><mi>e</mi><mo>/</mo><mn>3</mn></mrow></math> and <math id="M112"><mrow><msub><mi>e</mi><mi>d</mi></msub><mo>=</mo><mo>−</mo><msub><mi>e</mi><mover accent="true"><mi>d</mi><mo>¯</mo></mover></msub><mo>=</mo><mo>−</mo><mi>e</mi><mo>/</mo><mn>3</mn></mrow></math> , where the electric charge reads e =(4 πα s ) 1/2 with the fine structure constant <math id="M113"><mrow><msub><mi>α</mi><mtext>s</mtext></msub><mo>≃</mo><mn>1</mn><mo>/</mo><mn>137</mn></mrow></math> . computation of the relaxation time 1 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 57 ] <math display="block" id="M114"><mtable><mtr><mtd><msubsup><mi>τ</mi><mi>l</mi><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>T</mi><mn>,</mn><mi>μ</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>n</mi><mover accent="true"><mi>q</mi><mo>¯</mo></mover></msub><mo stretchy="false">[</mo><msub><mover accent="true"><mi>σ</mi><mo>¯</mo></mover><mrow><mi>u</mi><mover accent="true"><mi>u</mi><mo>¯</mo></mover><mo>→</mo><mi>u</mi><mover accent="true"><mi>u</mi><mo>¯</mo></mover></mrow></msub><mo>+</mo><msub><mover accent="true"><mi>σ</mi><mo>¯</mo></mover><mrow><mi>u</mi><mover accent="true"><mi>u</mi><mo>¯</mo></mover><mo>→</mo><mi>d</mi><mover accent="true"><mi>d</mi><mo>¯</mo></mover></mrow></msub><mo>+</mo><msub><mover accent="true"><mi>σ</mi><mo>¯</mo></mover><mrow><mi>u</mi><mover accent="true"><mi>d</mi><mo>¯</mo></mover><mo>→</mo><mi>u</mi><mover accent="true"><mi>d</mi><mo>¯</mo></mover></mrow></msub><mo stretchy="false">]</mo></mtd></mtr><mtr><mtd><mo>+</mo><msub><mi>n</mi><mi>q</mi></msub><mo stretchy="false">[</mo><msub><mover accent="true"><mi>σ</mi><mo>¯</mo></mover><mrow><mi>u</mi><mi>d</mi><mo>→</mo><mi>u</mi><mi>d</mi></mrow></msub><mo>+</mo><msub><mover accent="true"><mi>σ</mi><mo>¯</mo></mover><mrow><mi>u</mi><mi>u</mi><mo>→</mo><mi>u</mi><mi>u</mi></mrow></msub><mo stretchy="false">]</mo><mn>,</mn></mtd></mtr></mtable></math> where the number density of (anti-)quarks in a weakly anisotropic medium is given as <math id="M115"><mrow><msub><mi>n</mi><mrow><mi>q</mi><mo stretchy="false">(</mo><mover accent="true"><mi>q</mi><mo>¯</mo></mover><mo stretchy="false">)</mo></mrow></msub><mo>=</mo><msub><mi>d</mi><mi>l</mi></msub><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mfrac><mrow><msup><mi>d</mi><mn>3</mn></msup><mi>p</mi></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mo stretchy="false">)</mo></mrow><mn>3</mn></msup></mrow></mfrac><msubsup><mi>f</mi><mrow><mi>q</mi><mo stretchy="false">(</mo><mover accent="true"><mi>q</mi><mo>¯</mo></mover><mo stretchy="false">)</mo></mrow><mrow><mi>a</mi><mi>n</mi></mrow></msubsup></mrow></math> , with <math id="M116"><mrow><msub><mi>d</mi><mi>l</mi></msub><mo>=</mo><msub><mi>d</mi><mover accent="true"><mi>l</mi><mo>¯</mo></mover></msub><mo>=</mo><mn>2</mn><msub><mi>N</mi><mtext>c</mtext></msub></mrow></math> denoting the degeneracy factor. The momenta of the colliding particles for the elastic scattering process <math id="M117"><mrow><mi>a</mi><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mn>1</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mi>b</mi><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mn>2</mn></msub><mo stretchy="false">)</mo><mo>→</mo><mi>c</mi><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mn>3</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mi>d</mi><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mn>4</mn></msub><mo stretchy="false">)</mo></mrow></math> obey the relation <math id="M118"><mrow><msub><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mn>1</mn></msub><mo>+</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mn>2</mn></msub><mo>=</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mn>3</mn></msub><mo>+</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mn>4</mn></msub><mo>=</mo><mstyle mathvariant="bold" mathsize="normal"><mn>0</mn></mstyle></mrow></math> , and we use the notation <math id="M119"><mrow><mo>|</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mn>1</mn></msub><mo>|</mo><mo>=</mo><mo>|</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mn>2</mn></msub><mo>|</mo><mo>=</mo><mi>p</mi></mrow></math> for convenience. In the center-of-mass (cm) frame, the Mandelstam variables s, t, u are given by <math display="block" id="M120"><mtable columnalign="left"><mtr><mtd><mi>s</mi><mo>=</mo><mn>4</mn><msubsup><mi>m</mi><mi>q</mi><mn>2</mn></msubsup><mo>+</mo><mn>4</mn><msup><mi>p</mi><mn>2</mn></msup><mn>,</mn><mi>t</mi><mo>=</mo><mo>−</mo><mn>2</mn><msup><mi>p</mi><mn>2</mn></msup><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>cos</mi><msub><mi>θ</mi><mi>p</mi></msub><mo stretchy="false">)</mo><mn>,</mn></mtd></mtr><mtr><mtd><mi>u</mi><mo>=</mo><mo>−</mo><mn>2</mn><msup><mi>p</mi><mn>2</mn></msup><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>cos</mi><msub><mi>θ</mi><mi>p</mi></msub><mo stretchy="false">)</mo><mn>,</mn></mtd></mtr></mtable></math> where θ p is the scattering angle in the c.m. frame. The Mandelstam variables hold the relation <math id="M121"><mrow><mi>u</mi><mo>+</mo><mi>s</mi><mo>+</mo><mi>t</mi><mo>=</mo><mn>4</mn><msubsup><mi>m</mi><mi>q</mi><mn>2</mn></msubsup></mrow></math> . <math id="M122"><mrow><msub><mover accent="true"><mi>σ</mi><mo>¯</mo></mover><mrow><mi>a</mi><mi>b</mi><mo>→</mo><mi>c</mi><mi>d</mi></mrow></msub></mrow></math> denotes the thermal-averaged elastic scattering cross section in the weakly anisotropic system, which can be written as <math display="block" id="M123"><mtable><mtr><mtd><msub><mover accent="true"><mi>σ</mi><mo>¯</mo></mover><mrow><mi>a</mi><mi>b</mi><mo>→</mo><mi>c</mi><mi>d</mi></mrow></msub><mo>=</mo><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mrow><msub><mi>s</mi><mn>0</mn></msub></mrow><mi>∞</mi></msubsup></mrow></mstyle><mi>d</mi><mi>s</mi><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mrow><msub><mi>t</mi><mrow><mtext>min</mtext></mrow></msub></mrow><mrow><msub><mi>t</mi><mrow><mtext>max</mtext></mrow></msub></mrow></msubsup></mrow></mstyle><mi>d</mi><mi>t</mi><mfrac><mrow><mi>d</mi><msub><mover accent="true"><mi>σ</mi><mo>¯</mo></mover><mrow><mi>a</mi><mi>b</mi><mo>→</mo><mi>c</mi><mi>d</mi></mrow></msub></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><msup><mstyle mathsize="140%" displaystyle="true"><mrow><mi>sin</mi></mrow></mstyle><mn>2</mn></msup><msub><mi>θ</mi><mi>p</mi></msub><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mrow><mo>−</mo><mn>1</mn></mrow><mn>1</mn></msubsup></mrow></mstyle><mi>d</mi><msub><mi>x</mi><mn>3</mn></msub></mtd></mtr><mtr><mtd><mo>×</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msubsup><mi>f</mi><mi>c</mi><mrow><mtext>aniso</mtext></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>p</mi><mrow><mtext>cm</mtext></mrow></msub><mn>,</mn><mi>μ</mi><mn>,</mn><msub><mi>x</mi><mn>3</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mrow><mo>−</mo><mn>1</mn></mrow><mn>1</mn></msubsup></mrow></mstyle><mi>d</mi><msub><mi>x</mi><mn>4</mn></msub></mtd></mtr><mtr><mtd><mo>×</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msubsup><mi>f</mi><mi>d</mi><mrow><mtext>aniso</mtext></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>p</mi><mrow><mtext>cm</mtext></mrow></msub><mn>,</mn><mi>μ</mi><mn>,</mn><msub><mi>x</mi><mn>4</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi mathvariant="script">L</mi><mo stretchy="false">(</mo><mi>s</mi><mn>,</mn><mi>μ</mi><mn>,</mn><msub><mi>x</mi><mn>1</mn></msub><mn>,</mn><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mn>,</mn></mtd></mtr></mtable></math> with <math id="M124"><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msubsup><mi>f</mi><mrow><mi>c</mi><mn>,</mn><mi>d</mi></mrow><mrow><mtext>an</mtext></mrow></msubsup><mo stretchy="false">)</mo></mrow></math> 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. <math id="M125"><mrow><mfrac><mrow><mi>d</mi><mover accent="true"><mi>σ</mi><mo>¯</mo></mover></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow><mn>16</mn><mi>π</mi><mi>s</mi><mo stretchy="false">(</mo><mi>s</mi><mo>−</mo><mn>4</mn><msubsup><mi>m</mi><mi>q</mi><mn>2</mn></msubsup><mo stretchy="false">)</mo></mrow></mfrac><mo>|</mo><mover accent="true"><mi>M</mi><mo>¯</mo></mover><msup><mo>|</mo><mn>2</mn></msup></mrow></math> is the differential scattering cross section, with <math id="M126"><mrow><mo>|</mo><mover accent="true"><mi>M</mi><mo>¯</mo></mover><msup><mo>|</mo><mn>2</mn></msup></mrow></math> 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 <math id="M127"><mrow><msub><mi>t</mi><mrow><mtext>min</mtext></mrow></msub><mo>=</mo><mo>−</mo><mn>4</mn><msubsup><mi>p</mi><mrow><mtext>cm</mtext></mrow><mn>2</mn></msubsup><mo>=</mo><mo>−</mo><mo stretchy="false">(</mo><mi>s</mi><mo>−</mo><mn>4</mn><msubsup><mi>m</mi><mi>q</mi><mn>2</mn></msubsup><mo stretchy="false">)</mo></mrow></math> , with <math id="M128"><mrow><msub><mi>p</mi><mrow><mtext>cm</mtext></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><msqrt><mrow><mi>s</mi><mo>−</mo><mn>4</mn><msubsup><mi>m</mi><mi>q</mi><mn>2</mn></msubsup></mrow></msqrt><mo stretchy="false">)</mo><mo>/</mo><mn>2</mn></mrow></math> denoting the momentum in the c.m. frame. The kinematic boundary of s reads <math id="M129"><mrow><msub><mi>s</mi><mn>0</mn></msub><mo>=</mo><mn>4</mn><msubsup><mi>m</mi><mi>q</mi><mn>2</mn></msubsup></mrow></math> . The scattering weighting factor <math id="M130"><mrow><msup><mstyle mathsize="140%" displaystyle="true"><mrow><mi>sin</mi></mrow></mstyle><mn>2</mn></msup><msub><mi>θ</mi><mi>p</mi></msub><mo>=</mo><mfrac><mrow><mo>−</mo><mn>4</mn><mi>t</mi><mo stretchy="false">(</mo><mi>s</mi><mo>+</mo><mi>t</mi><mo>−</mo><mn>4</mn><msubsup><mi>m</mi><mi>q</mi><mn>2</mn></msubsup><mo stretchy="false">)</mo></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mi>s</mi><mo>−</mo><mn>4</mn><msubsup><mi>m</mi><mi>q</mi><mn>2</mn></msubsup><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mfrac></mrow></math> is introduced to exclude the scattering processes with a small initial angle because the large angle scattering dominates in the momentum transport process [ 116 116 ]. In the c.m. frame, the leading-order anisotropic distribution function can be rewritten as <math display="block" id="M131"><mtable><mtr><mtd><msup><mi>f</mi><mrow><mtext>aniso</mtext></mrow></msup><mo stretchy="false">(</mo><msub><mi>p</mi><mrow><mtext>cm</mtext></mrow></msub><mn>,</mn><mi>μ</mi><mn>,</mn><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>f</mi><mn>0</mn></msup><mo stretchy="false">(</mo><msub><mi>p</mi><mrow><mtext>cm</mtext></mrow></msub><mn>,</mn><mi>μ</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>−</mo><mfrac><mrow><msubsup><mi>p</mi><mrow><mtext>cm</mtext></mrow><mn>2</mn></msubsup><mi>ξ</mi><msup><mi>x</mi><mn>2</mn></msup></mrow><mrow><mn>2</mn><msub><mi>E</mi><mrow><mtext>cm</mtext></mrow></msub><mi>T</mi></mrow></mfrac><msup><mi>f</mi><mn>0</mn></msup><mo stretchy="false">(</mo><msub><mi>p</mi><mrow><mtext>cm</mtext></mrow></msub><mn>,</mn><mi>μ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msup><mi>f</mi><mn>0</mn></msup><mo stretchy="false">(</mo><msub><mi>p</mi><mrow><mtext>cm</mtext></mrow></msub><mn>,</mn><mi>μ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mn>,</mn></mtd></mtr></mtable></math> where <math id="M132"><mrow><msub><mi>E</mi><mrow><mtext>cm</mtext></mrow></msub><mo>=</mo><mfrac><mrow><mi>s</mi><mo>−</mo><msubsup><mi>m</mi><mi>q</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>m</mi><mi>q</mi><mn>2</mn></msubsup></mrow><mrow><mn>2</mn><msqrt><mi>s</mi></msqrt></mrow></mfrac><mo>=</mo><msqrt><mi>s</mi></msqrt><mo>/</mo><mn>2</mn></mrow></math> . In Eq. (59), <math id="M133"><mi mathvariant="script">L</mi></math> denotes the probability of finding a quark-(anti)quark pair with the center of mass energy <math id="M134"><mrow><msqrt><mi>s</mi></msqrt></mrow></math> in the anisotropic medium, and it is given by <math display="block" id="M135"><mtable><mtr><mtd><mi mathvariant="script">L</mi><mo>=</mo><mi>C</mi><msqrt><mrow><mi>s</mi><mo stretchy="false">(</mo><mi>s</mi><mo>−</mo><mn>4</mn><msubsup><mi>m</mi><mi>q</mi><mn>2</mn></msubsup><mo stretchy="false">)</mo></mrow></msqrt><msubsup><mi>f</mi><mi>a</mi><mrow><mtext>aniso</mtext></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>p</mi><mrow><mi>c</mi><mi>m</mi></mrow></msub><mn>,</mn><mi>μ</mi><mn>,</mn><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">)</mo><msubsup><mi>f</mi><mi>b</mi><mrow><mtext>aniso</mtext></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>p</mi><mrow><mi>c</mi><mi>m</mi></mrow></msub><mn>,</mn><mi>μ</mi><mn>,</mn><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>×</mo><msub><mi>v</mi><mrow><mtext>rel</mtext></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mn>,</mn></mtd></mtr></mtable></math> where <math id="M136"><mrow><msub><mi>v</mi><mrow><mtext>rel</mtext></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><msqrt><mrow><mfrac><mrow><mi>s</mi><mo>−</mo><mn>4</mn><msubsup><mi>m</mi><mi>q</mi><mn>2</mn></msubsup></mrow><mi>s</mi></mfrac></mrow></msqrt></mrow></math> 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 <math id="M137"><mrow><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mrow><msub><mi>s</mi><mn>0</mn></msub></mrow><mi>∞</mi></msubsup></mrow></mstyle><mi>d</mi><mi>s</mi><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mrow><mo>−</mo><mn>1</mn></mrow><mn>1</mn></msubsup></mrow></mstyle><mi>d</mi><msub><mi>x</mi><mn>1</mn></msub><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mrow><mo>−</mo><mn>1</mn></mrow><mn>1</mn></msubsup></mrow></mstyle><mi>d</mi><msub><mi>x</mi><mn>2</mn></msub><mi mathvariant="script">L</mi><mo>=</mo><mn>1</mn></mrow></math> , where <math id="M138"><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>=</mo><mi>cos</mi><msub><mi>θ</mi><mn>1</mn></msub></mrow></math> and <math id="M139"><mrow><msub><mi>x</mi><mn>2</mn></msub><mo>=</mo><mi>cos</mi><msub><mi>θ</mi><mn>2</mn></msub></mrow></math> , with θ i being the angle between p i and n and <math id="M140"><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>≡</mo><mo>−</mo><msub><mi>x</mi><mn>3</mn></msub></mrow></math> , <math id="M141"><mrow><msub><mi>x</mi><mn>2</mn></msub><mo>≡</mo><mo>−</mo><msub><mi>x</mi><mn>4</mn></msub></mrow></math> . 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 1 Throughout this work, the following parameter set is used: m 0 = m 0, u = m 0, d =5.6 MeV, G Λ 2 =2.44 and Λ=587.9 MeV. These values are taken from Ref. [ 117 117 ], 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 f π =92.4 MeV, and quark condensate <math id="M142"><mrow><mo>−</mo><msup><mrow><mrow><mo>〈</mo><mrow><mi>ψ</mi><mi>ψ</mi></mrow><mo>〉</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>=</mo><mn>241</mn></mrow></math> 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) Fig. 1 (a) , we show the thermal behavior of the light constituent quark mass m q for vanishing quark chemical potential at different ξ . For low temperature, m q remains approximately constant at ( <math id="M143"><mrow><msub><mi>m</mi><mi>q</mi></msub><mo>≈</mo></mrow></math> 400 MeV), and then, with increasing temperature m q , 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. 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 <math id="M146"><mrow><msub><mi>χ</mi><mrow><mtext>ch</mtext></mrow></msub><mo>=</mo><mo>|</mo><mfrac><mrow><mi>d</mi><msub><mi>m</mi><mi>q</mi></msub></mrow><mrow><mi>d</mi><mi>T</mi></mrow></mfrac><mo>|</mo></mrow></math> . 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) 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) 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 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 (2 m q ) is equal to the π mass, so the pion meson is no longer a bound state but only a <math id="M147"><mrow><mi>q</mi><mover accent="true"><mi>q</mi><mo>¯</mo></mover></mrow></math> resonance and obtains a finite decay width. Accordingly, the Mott transition temperature by the definition <math id="M148"><mrow><msub><mi>m</mi><mi>π</mi></msub><mo stretchy="false">(</mo><msub><mi>T</mi><mrow><mtext>Mott</mtext></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><msub><mi>m</mi><mi>q</mi></msub><mo stretchy="false">(</mo><msub><mi>T</mi><mrow><mtext>Mott</mtext></mrow></msub><mo stretchy="false">)</mo></mrow></math> 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) 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. 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 m q for different anisotropy parameters, as shown in Fig. 3 Fig. 3 . We can observe that at a small μ , m q continuously decreases with increasing T , whereas m q 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 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 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 <math id="M149"><mrow><msup><mi>f</mi><mrow><mtext>iso</mtext></mrow></msup><mo stretchy="false">(</mo><msqrt><mrow><msup><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mn>2</mn></msup><mo>+</mo><mi>ξ</mi><msup><mrow><mo stretchy="false">(</mo><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo>⋅</mo><mstyle mathvariant="bold" mathsize="normal"><mi>n</mi></mstyle><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></msqrt><mo stretchy="false">)</mo></mrow></math> 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 ( <math id="M150"><mrow><msub><mi>T</mi><mrow><mtext>CEP</mtext></mrow></msub><mn>,</mn><mi> </mi><msub><mi>μ</mi><mrow><mtext>CEP</mtext></mrow></msub></mrow></math> ) 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 [ 118 118 ] 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 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. 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 Fig 6 , we display the total cross section of quark–quark scattering processes <math id="M151"><mrow><msub><mover accent="true"><mi>σ</mi><mo>¯</mo></mover><mrow><mi>q</mi><mi>q</mi></mrow></msub><mo>=</mo><msub><mover accent="true"><mi>σ</mi><mo>¯</mo></mover><mrow><mi>u</mi><mi>u</mi><mo>→</mo><mi>u</mi><mi>u</mi></mrow></msub><mo>+</mo><msub><mover accent="true"><mi>σ</mi><mo>¯</mo></mover><mrow><mi>u</mi><mi>d</mi><mo>→</mo><mi>u</mi><mi>d</mi></mrow></msub></mrow></math> (plot a ) and the total cross section of quark–antiquark processes <math id="M152"><mrow><msub><mover accent="true"><mi>σ</mi><mo>¯</mo></mover><mrow><mi>q</mi><mover accent="true"><mi>q</mi><mo>¯</mo></mover></mrow></msub><mo>=</mo><msub><mover accent="true"><mi>σ</mi><mo>¯</mo></mover><mrow><mi>u</mi><mover accent="true"><mi>u</mi><mo>¯</mo></mover><mo>→</mo><mi>u</mi><mover accent="true"><mi>u</mi><mo>¯</mo></mover></mrow></msub><mo>+</mo><msub><mover accent="true"><mi>σ</mi><mo>¯</mo></mover><mrow><mi>u</mi><mover accent="true"><mi>d</mi><mo>¯</mo></mover><mo>→</mo><mi>u</mi><mover accent="true"><mi>d</mi><mo>¯</mo></mover></mrow></msub><mo>+</mo><msub><mover accent="true"><mi>σ</mi><mo>¯</mo></mover><mrow><mi>u</mi><mover accent="true"><mi>u</mi><mo>¯</mo></mover><mo>→</mo><mi>d</mi><mover accent="true"><mi>d</mi><mo>¯</mo></mover></mrow></msub></mrow></math> (plot b ) as functions of temperature at different anisotropy parameters for vanishing quark chemical potential. As can be observed, <math id="M153"><mrow><msub><mover accent="true"><mi>σ</mi><mo>¯</mo></mover><mrow><mi>q</mi><mi>q</mi></mrow></msub></mrow></math> and <math id="M154"><mrow><msub><mover accent="true"><mi>σ</mi><mo>¯</mo></mover><mrow><mi>q</mi><mover accent="true"><mi>q</mi><mo>¯</mo></mover></mrow></msub></mrow></math> 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 <math id="M155"><mrow><msub><mover accent="true"><mi>σ</mi><mo>¯</mo></mover><mrow><mi>q</mi><mover accent="true"><mi>q</mi><mo>¯</mo></mover></mrow></msub></mrow></math> is higher than that of <math id="M156"><mrow><msub><mover accent="true"><mi>σ</mi><mo>¯</mo></mover><mrow><mi>q</mi><mi>q</mi></mrow></msub></mrow></math> , 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 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, <math id="M157"><mrow><msub><mover accent="true"><mi>σ</mi><mo>¯</mo></mover><mrow><mi>q</mi><mi>q</mi></mrow></msub></mrow></math> increases in the entire considered temperature domain, whereas <math id="M158"><mrow><msub><mover accent="true"><mi>σ</mi><mo>¯</mo></mover><mrow><mi>q</mi><mover accent="true"><mi>q</mi><mo>¯</mo></mover></mrow></msub></mrow></math> 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 <math id="M159"><mrow><msub><mover accent="true"><mi>σ</mi><mo>¯</mo></mover><mrow><mi>q</mi><mi>q</mi></mrow></msub></mrow></math> at different ξ is nearly in agreement with the respective T c , as the peak positions of <math id="M160"><mrow><msub><mover accent="true"><mi>σ</mi><mo>¯</mo></mover><mrow><mi>q</mi><mover accent="true"><mi>q</mi><mo>¯</mo></mover></mrow></msub></mrow></math> respectively locate at <math id="M161"><mrow><mo>∼</mo><mn>1.07</mn><mi> </mi><msubsup><mi>T</mi><mtext>c</mtext><mrow><mo>−</mo><mn>0.3</mn></mrow></msubsup></mrow></math> , <math id="M162"><mrow><mn>1.07</mn><mi> </mi><msubsup><mi>T</mi><mtext>c</mtext><mn>0</mn></msubsup></mrow></math> , <math id="M163"><mrow><mn>1.10</mn><mi> </mi><msubsup><mi>T</mi><mtext>c</mtext><mrow><mn>0.3</mn></mrow></msubsup></mrow></math> for ξ =-0.3, 0, 0.3, with <math id="M164"><mrow><msubsup><mi>T</mi><mtext>c</mtext><mi>ξ</mi></msubsup></mrow></math> denoting the chiral critical temperature for a fixed ξ . The dependence of total quark relaxation time τ q on temperature for vanishing quark chemical potential at different ξ is displayed in Fig. 7 Fig. 7 . As can be observed, τ q first decreases sharply with increasing temperature, and after an inflection point ( viz , the peak position of <math id="M165"><mrow><msub><mover accent="true"><mi>σ</mi><mo>¯</mo></mover><mrow><mi>q</mi><mover accent="true"><mi>q</mi><mo>¯</mo></mover></mrow></msub></mrow></math> ), τ q changes modestly with temperature. Further, the increase in τ q with ξ is significant at low temperature, whereas at high temperature the reduction in τ q 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 τ q 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. Next, we discuss the results regarding various transport coefficients. In Fig. 8 Fig. 8 , the temperature dependence of scaled shear viscosity η/T 3 in quark matter for different momentum anisotropy parameters at a vanishing chemical potential is displayed. We observe that with increasing temperature, η/T 3 first decreases, reaches a minimum around the critical temperature, and increases afterwards. The temperature position for the minimum η/T 3 is consistent with the temperature for the peak <math id="M166"><mrow><msub><mover accent="true"><mi>σ</mi><mo>¯</mo></mover><mrow><mi>q</mi><mover accent="true"><mi>q</mi><mo>¯</mo></mover></mrow></msub></mrow></math> . This dip structure of η/T 3 mainly depends on the competition between the quark distribution function <math id="M167"><mrow><msubsup><mi>f</mi><mi>q</mi><mn>0</mn></msubsup></mrow></math> and the quark relaxation time τ q in the integrand of Eq. (44). The decreasing feature of η/T 3 in the low temperature domain is governed by τ q , whereas in the high temperature domain, the increasing behavior of <math id="M168"><mrow><msubsup><mi>f</mi><mi>q</mi><mn>0</mn></msubsup></mrow></math> overwhelms the decreasing behavior of τ q , resulting in η/T 3 as an increasing function of temperature. In addition, we observe that with an increase in ξ , η/T 3 has an overall enhancement and the minimum of η/T 3 shifts to higher temperatures. This behavior of η/T 3 can 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/T 3 for the inclusion of positive ξ . Meanwhile, at high T , the qualitative and quantitative behavior of η/T 3 with ξ is dominated by the second term. The location of the minimum for η/T 3 at different ξ is consistent with the peak position of <math id="M169"><mrow><msub><mover accent="true"><mi>σ</mi><mo>¯</mo></mover><mrow><mi>q</mi><mover accent="true"><mi>q</mi><mo>¯</mo></mover></mrow></msub></mrow></math> . We further observe that η/T 3 decreases 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/T 3 in the hadron resonance gas (HRG) model [ 70 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 57 ] in the N f =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 59 ] is a logarithmically increasing function of temperature beyond the critical temperature, and it is quantitively 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 58 ] for the N f =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. In Fig. 9 Fig. 9 , we plot the thermal behavior of scaled electrical conductivity σ el / T at μ =0 MeV for different ξ . Similar to the temperature dependence of η/T 3 , σ el / 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 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 σ el / T using the microscopic simulation code SMASH [ 65 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 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 71 ] (cyan dash-dotted line) and the result obtained from the partonic cascade BAMPS [ 67 67 ] (gray thick dash-dotted line) are qualitatively and quantitively 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 N f =3 NJL model [ 54 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 σ el / T . At high T , the decreasing feature of relaxation time with ξ can weaken the increasing behavior of σ el / T with ξ , and the values of σ el / T for different ξ gradually approach and eventually overlap. Our qualitative result of σ el / T is different from the result in Ref. [ 73 73 ], where the σ el / 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 σ el / T is only determined by the anisotropic distribution function. We also observe that with the increase in ξ , the minimum of σ el / T shifts to higher temperatures, which is similar to η/T 3 . However, the height of the minimum increases, which is opposite to η/T 3 . Finally, we study the Seebeck coefficient S in quark–antiquark matter. Owing to the sensitivity of S to the charge of particle species, at a vanishing chemical potential, quark number density n q is equal to antiquark number density <math id="M170"><mrow><msub><mi>n</mi><mover accent="true"><mi>q</mi><mo>¯</mo></mover></msub></mrow></math> , and the contribution of quarks to S is exactly cancelled 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 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 , n q is larger than <math id="M171"><mrow><msub><mi>n</mi><mover accent="true"><mi>q</mi><mo>¯</mo></mover></msub></mrow></math> , and the contribution of quarks to total S in magnitude is always prominent. As shown in Fig. 10 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 ( E q -μ q ) in the integrand of Eq. (55). In Ref. [ 83 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 84 ] at μ =100 MeV in the N f =2 NJL model (the green dots) is very different from ours. In Ref. [ 84 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 84 ] was estimated using the averaged transition rate <math id="M172"><mrow><msub><mover accent="true"><mi>w</mi><mo>¯</mo></mover><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></math> , 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 74 ]); (2) in Ref. [ 84 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 ( E q -ω/n q ) with ω denoting the enthalpy density in the associated formalism. Given that the single-particle energy E q remains smaller than ( ω/n q ), S in Ref. [ 84 84 ] is negative. We also observed that with increasing temperature, S sharply decreased below T c , whereas the decreasing feature of S was inconspicuous above T c . 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 84 ], where the absolute value of S in quark matter increased with increasing temperature because of the increasing behaviors of both the factor <math id="M173"><mrow><mrow><mo>|</mo><mrow><mo>−</mo><mi>ω</mi><mo>/</mo><msub><mi>n</mi><mi>q</mi></msub></mrow><mo>|</mo></mrow></mrow></math> and the equilibrium distribution function. In addition, the Seebeck coefficient in the HRG model [ 80 80 , 81 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 <math id="M174"><mrow><mn>1</mn><mo>/</mo><msub><mi>σ</mi><mrow><mtext>el</mtext></mrow></msub></mrow></math> has a cancellation effect on the increase in S . At sufficiently high temperature, the rise in <math id="M175"><mrow><mn>1</mn><mo>/</mo><msub><mi>σ</mi><mrow><mi>e</mi><mi>l</mi></mrow></msub></mrow></math> can 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. Summary 1 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 2-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/T 3 , scaled electrical conductivity σ el / 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/T 3 and σ el / T have a dip structure around the critical temperature. Within the consideration of momentum anisotropy, η/T 3 decreases as ξ increases and the minimum shifts to higher temperatures. With the increase in ξ , σ el / 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/T 3 with 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. 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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. 10.1007/s41365-022-01129-4 This work was supported by the Guangdong Major Project of Basic and Applied Basic Research (No. 2020B0301030008), the Natural Science Foundation of China (No. 11935007), and the Science and Technology Program of Guangzhou (No. 2019050001). 2022-07-31 2022-09-27 2022-09-28 2022-11-26 2022-12-16 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2022 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-022-01129-4 http://dx.doi.org/10.1007/s41365-022-01129-4 . Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. 2022 Appendix 2 In the N f =2 NJL model, there are 12 different elastic scattering processes: <math display="block" id="M176"><mtable columnalign="left"><mtr><mtd><mi>u</mi><mover accent="true"><mi>u</mi><mo>¯</mo></mover><mo>→</mo><mi>u</mi><mover accent="true"><mi>u</mi><mo>¯</mo></mover><mn>,</mn><mi> </mi><mi>u</mi><mover accent="true"><mi>d</mi><mo>¯</mo></mover><mo>→</mo><mi>u</mi><mover accent="true"><mi>d</mi><mo>¯</mo></mover><mn>,</mn><mi> </mi><mi>u</mi><mover accent="true"><mi>u</mi><mo>¯</mo></mover><mo>→</mo><mi>d</mi><mover accent="true"><mi>d</mi><mo>¯</mo></mover><mn>,</mn></mtd></mtr><mtr><mtd><mi>u</mi><mi>u</mi><mo>→</mo><mi>u</mi><mi>u</mi><mn>,</mn><mi> </mi><mi>u</mi><mi>d</mi><mo>→</mo><mi>u</mi><mi>d</mi><mn>,</mn><mi> </mi><mover accent="true"><mi>u</mi><mo>¯</mo></mover><mover accent="true"><mi>u</mi><mo>¯</mo></mover><mo>→</mo><mover accent="true"><mi>u</mi><mo>¯</mo></mover><mover accent="true"><mi>u</mi><mo>¯</mo></mover><mn>,</mn></mtd></mtr><mtr><mtd><mover accent="true"><mi>u</mi><mo>¯</mo></mover><mover accent="true"><mi>d</mi><mo>¯</mo></mover><mo>→</mo><mover accent="true"><mi>u</mi><mo>¯</mo></mover><mover accent="true"><mi>d</mi><mo>¯</mo></mover><mn>,</mn><mi> </mi><mi>d</mi><mover accent="true"><mi>d</mi><mo>¯</mo></mover><mo>→</mo><mi>d</mi><mover accent="true"><mi>d</mi><mo>¯</mo></mover><mn>,</mn><mi> </mi><mi>d</mi><mover accent="true"><mi>d</mi><mo>¯</mo></mover><mo>→</mo><mi>u</mi><mover accent="true"><mi>u</mi><mo>¯</mo></mover><mn>,</mn></mtd></mtr><mtr><mtd><mi>d</mi><mover accent="true"><mi>u</mi><mo>¯</mo></mover><mo>→</mo><mi>d</mi><mover accent="true"><mi>u</mi><mo>¯</mo></mover><mn>,</mn><mi> </mi><mi>d</mi><mi>d</mi><mo>→</mo><mi>d</mi><mi>d</mi><mn>,</mn><mi> </mi><mover accent="true"><mi>d</mi><mo>¯</mo></mover><mover accent="true"><mi>d</mi><mo>¯</mo></mover><mo>→</mo><mover accent="true"><mi>d</mi><mo>¯</mo></mover><mover accent="true"><mi>d</mi><mo>¯</mo></mover><mn>.</mn></mtd></mtr></mtable></math> The explicit expressions of the squared matrix elements for the <math id="M177"><mrow><mi>u</mi><mover accent="true"><mi>u</mi><mo>¯</mo></mover><mo>→</mo><mi>u</mi><mover accent="true"><mi>u</mi><mo>¯</mo></mover></mrow></math> , <math id="M178"><mrow><mi>u</mi><mover accent="true"><mi>d</mi><mo>¯</mo></mover><mo>→</mo><mi>u</mi><mover accent="true"><mi>d</mi><mo>¯</mo></mover></mrow></math> , and <math id="M179"><mrow><mi>u</mi><mi>d</mi><mo>→</mo><mi>u</mi><mi>d</mi></mrow></math> processes via exchange of scalar and/or pseudoscalar mesons to 1/ N c order are given as <math display="block" id="M180"><mtable columnalign="left"><mtr><mtd><mo>|</mo><msub><mover accent="true"><mi>M</mi><mo>¯</mo></mover><mrow><mi>u</mi><mover accent="true"><mi>u</mi><mo>¯</mo></mover><mo>→</mo><mi>u</mi><mover accent="true"><mi>u</mi><mo>¯</mo></mover></mrow></msub><msup><mo>|</mo><mn>2</mn></msup><mo stretchy="false">(</mo><mi>s</mi><mn>,</mn><mi>t</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mtext>    </mtext><mo>=</mo><msup><mi>s</mi><mn>2</mn></msup><msup><mrow><mo>|</mo><mrow><msubsup><mi>D</mi><mi>s</mi><mi>π</mi></msubsup></mrow><mo>|</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mi>t</mi><mn>2</mn></msup><msup><mrow><mo>|</mo><mrow><msubsup><mi>D</mi><mi>t</mi><mi>π</mi></msubsup></mrow><mo>|</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mo>(</mo><mrow><mi>s</mi><mo>−</mo><mn>4</mn><msup><mi>m</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><mn>2</mn></msup><msup><mrow><mo>|</mo><mrow><msubsup><mi>D</mi><mi>s</mi><mi>σ</mi></msubsup></mrow><mo>|</mo></mrow><mn>2</mn></msup></mtd></mtr><mtr><mtd><mtext>    </mtext><mo>+</mo><msup><mrow><mo>(</mo><mrow><mi>t</mi><mo>−</mo><mn>4</mn><msup><mi>m</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><mn>2</mn></msup><msup><mrow><mo>|</mo><mrow><msubsup><mi>D</mi><mi>t</mi><mi>σ</mi></msubsup></mrow><mo>|</mo></mrow><mn>2</mn></msup></mtd></mtr><mtr><mtd><mtext>    </mtext><mo>+</mo><mfrac><mn>1</mn><mrow><msub><mi>N</mi><mi>c</mi></msub></mrow></mfrac><mi>R</mi><mi>e</mi><mrow><mo>[</mo><mrow><mi>s</mi><mi>t</mi><msubsup><mi>D</mi><mi>s</mi><mrow><msup><mi>π</mi><mo>*</mo></msup></mrow></msubsup><msubsup><mi>D</mi><mi>t</mi><mi>π</mi></msubsup><mo>+</mo><mi>s</mi><mrow><mo>(</mo><mrow><mn>4</mn><msup><mi>m</mi><mn>2</mn></msup><mo>−</mo><mi>t</mi></mrow><mo>)</mo></mrow><msubsup><mi>D</mi><mi>s</mi><mrow><msup><mi>π</mi><mo>*</mo></msup></mrow></msubsup><msubsup><mi>D</mi><mi>t</mi><mi>σ</mi></msubsup></mrow></mrow></mtd></mtr><mtr><mtd><mtext>    </mtext><mo>+</mo><mi>t</mi><mrow><mo>(</mo><mrow><mn>4</mn><msup><mi>m</mi><mn>2</mn></msup><mo>−</mo><mi>s</mi></mrow><mo>)</mo></mrow><msubsup><mi>D</mi><mi>t</mi><mi>π</mi></msubsup><msubsup><mi>D</mi><mi>s</mi><mrow><msup><mi>σ</mi><mo>*</mo></msup></mrow></msubsup></mtd></mtr><mtr><mtd><mtext>    </mtext><mrow><mrow><mo>+</mo><mrow><mo>(</mo><mrow><mi>s</mi><mi>t</mi><mo>+</mo><mn>4</mn><msup><mi>m</mi><mn>2</mn></msup><mrow><mo>(</mo><mrow><mi>s</mi><mo>+</mo><mi>t</mi></mrow><mo>)</mo></mrow><mo>−</mo><mn>16</mn><msup><mi>m</mi><mn>4</mn></msup></mrow><mo>)</mo></mrow><msubsup><mi>D</mi><mi>t</mi><mi>σ</mi></msubsup><msubsup><mi>D</mi><mi>s</mi><mrow><msup><mi>σ</mi><mo>*</mo></msup></mrow></msubsup></mrow><mo>]</mo></mrow><mn>,</mn></mtd></mtr></mtable></math> <math display="block" id="M181"><mtable columnalign="left"><mtr><mtd><msup><mrow><mo>|</mo><mrow><msub><mover accent="true"><mi>M</mi><mo>¯</mo></mover><mrow><mi>u</mi><mover accent="true"><mi>d</mi><mo>¯</mo></mover><mo>→</mo><mi>u</mi><mover accent="true"><mi>d</mi><mo>¯</mo></mover></mrow></msub></mrow><mo>|</mo></mrow><mn>2</mn></msup><mrow><mo>(</mo><mrow><mi>s</mi><mn>,</mn><mi>t</mi></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mo>=</mo><mn>4</mn><msup><mi>s</mi><mn>2</mn></msup><msup><mrow><mo>|</mo><mrow><msubsup><mi>D</mi><mi>s</mi><mi>π</mi></msubsup></mrow><mo>|</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mi>t</mi><mn>2</mn></msup><msup><mrow><mo>|</mo><mrow><msubsup><mi>D</mi><mi>t</mi><mi>π</mi></msubsup></mrow><mo>|</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mo>(</mo><mrow><mi>t</mi><mo>−</mo><mn>4</mn><msup><mi>m</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><mn>2</mn></msup><msup><mrow><mo>|</mo><mrow><msubsup><mi>D</mi><mi>t</mi><mi>σ</mi></msubsup></mrow><mo>|</mo></mrow><mn>2</mn></msup><mo>−</mo></mtd></mtr><mtr><mtd><mfrac><mn>1</mn><mrow><msub><mi>N</mi><mi>c</mi></msub></mrow></mfrac><mi>R</mi><mi>e</mi><mrow><mo>[</mo><mrow><mn>2</mn><mi>s</mi><mi>t</mi><msubsup><mi>D</mi><mi>s</mi><mrow><msup><mi>π</mi><mo>*</mo></msup></mrow></msubsup><msubsup><mi>D</mi><mi>t</mi><mi>π</mi></msubsup><mo>+</mo><mn>2</mn><mi>s</mi><mrow><mo>(</mo><mrow><mn>4</mn><msup><mi>m</mi><mn>2</mn></msup><mo>−</mo><mi>t</mi></mrow><mo>)</mo></mrow><msubsup><mi>D</mi><mi>s</mi><mrow><msup><mi>π</mi><mo>*</mo></msup></mrow></msubsup><msubsup><mi>D</mi><mi>t</mi><mi>σ</mi></msubsup></mrow><mo>]</mo></mrow><mn>,</mn></mtd></mtr></mtable></math> <math display="block" id="M182"><mtable columnalign="left"><mtr><mtd><msup><mrow><mo>|</mo><mrow><msub><mover accent="true"><mi>M</mi><mo>¯</mo></mover><mrow><mi>u</mi><mi>d</mi><mo>→</mo><mi>u</mi><mi>d</mi></mrow></msub></mrow><mo>|</mo></mrow><mn>2</mn></msup><mrow><mo>(</mo><mrow><mi>t</mi><mn>,</mn><mi>u</mi></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mo>=</mo><mn>4</mn><msup><mi>u</mi><mn>2</mn></msup><msup><mrow><mo>|</mo><mrow><msubsup><mi>D</mi><mi>u</mi><mi>π</mi></msubsup></mrow><mo>|</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mi>t</mi><mn>2</mn></msup><msup><mrow><mo>|</mo><mrow><msubsup><mi>D</mi><mi>t</mi><mi>π</mi></msubsup></mrow><mo>|</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mo>(</mo><mrow><mi>t</mi><mo>−</mo><mn>4</mn><msup><mi>m</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><mn>2</mn></msup><msup><mrow><mo>|</mo><mrow><msubsup><mi>D</mi><mi>t</mi><mi>σ</mi></msubsup></mrow><mo>|</mo></mrow><mn>2</mn></msup></mtd></mtr><mtr><mtd><mo>−</mo><mfrac><mn>1</mn><mrow><msub><mi>N</mi><mi>c</mi></msub></mrow></mfrac><mi>R</mi><mi>e</mi><mrow><mo>[</mo><mrow><mn>2</mn><mi>u</mi><mi>t</mi><msubsup><mi>D</mi><mi>u</mi><mrow><msup><mi>π</mi><mo>*</mo></msup></mrow></msubsup><msubsup><mi>D</mi><mi>t</mi><mi>π</mi></msubsup><mo>+</mo><mn>2</mn><mi>u</mi><mrow><mo>(</mo><mrow><mn>4</mn><msup><mi>m</mi><mn>2</mn></msup><mo>−</mo><mi>t</mi></mrow><mo>)</mo></mrow><msubsup><mi>D</mi><mi>u</mi><mrow><msup><mi>π</mi><mo>*</mo></msup></mrow></msubsup><msubsup><mi>D</mi><mi>t</mi><mi>σ</mi></msubsup></mrow><mo>]</mo></mrow><mn>.</mn></mtd></mtr></mtable></math> 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 [ 120 120 , 57 57 ]. Nuclear Science and Techniques 11 33