Anisotropic electrical conductivity of magnetized hot quark matter

We studied the effect of a strong magnetic field ($B$) on the electrical conductivity of hot quark matter. The electrical conductivity is a key transport coefficient determining the time dependence and strength of magnetic fields generated in a relativistic heavy-ion collision. A~magnetic field induces Hall anisotropic conduction, phase-space Landau-level quantization and, if sufficiently strong, interferes with prominent QCD phenomena such as dynamical quark mass generation, likely affecting the quark matter electrical conductivity, which depends strongly on the quark masses. To address these issues, we used a quasi-particle description of quark matter in which the electric charge carriers are constituent quarks with temperature- and magnetic-field-dependent masses predicted by a Nambu--Jona-Lasinio model. The model accurately describes recent lattice QCD results showing magnetic catalysis at low temperatures and inverse magnetic catalysis at temperatures close to the pseudocritical temperature ($T_{\rm pc}$) of the QCD phase transition. We found that the magnetic field increases the conductivity component parallel to it and decreases the transverse component, in qualitative agreement with recent lattice QCD results. In addition, we found that: (1)~for a fixed value of $B$, increasing temperature causes enhancement in all the components of the conductivity, (2)~the anisotropy increases with~$B$, (3)~longitudinal conductivity is increased due to the quantum effects leading to phase-space Landau-level quantization, (4)~lowest Landau level approximation behaves poorly for temperatures close to $T_{\rm pc}$, and (5)~inverse magnetic catalysis leaves a distinctive signal in all components of the conductivity, a kink at $T_{\rm pc}$.


I. INTRODUCTION
Relativistic heavy-ion collisions can produce strong magnetic fields [1,2]. Field strengths comparable to or even larger than the strong-interaction scale Λ QCD ≃ 0.25 GeV can be produced in the collisions. Indeed, field strengths as large as eB ∼ 15m 2 π ≫ Λ 2 QCD have been estimated [3] for Pb-Pb collisions at the Large Hadron Collider. Such strong fields interfere with prominent strong-interaction phenomena such as dynamical chiral symmetry breaking and the chiral anomaly taking place in the hot quark matter-or quark-gluon plasma. Such interferences are predicted to lead to interesting effects, some of which were observed but the interpretation of the observations is ambiguous because alternative explanations are possible-Refs. [4][5][6][7] are recent reviews on this. An adversity here is that the fields are created early in the collision and weaken fast as the system expands [3,8]. On the other hand, the early-produced fields can induce electric currents in the expanding matter which in turn produce magnetic fields that can last while the system exists [9][10][11][12]. A key physical property in the electric-current induction is the electrical conductivity, a quantity poorly constrained by the fundamental theory, quantum chromodynamics (QCD).
The quark matter electrical conductivity is poorly constrained by QCD due to two main reasons. First, the matter produced in a heavy-ion collision is a strongly-interacting manybody system. Second, a strong magnetic field modifies intrinsic properties of the electric charge carriers in the system. A proper treatment of both features requires nonperturbative methods. Lattice QCD, a nonperturbative first-principles numerical method formulated in Euclidean space, can compute the electrical conductivity-and other transport properties such as shear and bulk viscosities-but the computation requires reconstructing spectral functions from Euclidean correlation functions using ill-posed inversion methods from imaginary time to frequency space. For B = 0, the first lattices results for the conductivity in full QCD appeared only recently [13,14], although results from quenched simulations are available for some time [15][16][17]. For B = 0, there are results from quenched simulations of an SU(2) gauge theory of Ref. [18] and from simulations in full QCD with N f = 2 + 1 flavors reported in a very recent preprint [19]. Recent analytical nonperturbative calculations of the conductivity, which do take into account the effects of the magnetic field, have been performed using different phenomenological approaches [20][21][22][23][24][25][26][27][28][29].
Prominent magnetic field effects, however, are neglected in previous studies, namely those that affect intrinsic properties of the electric charge carriers in the medium, which are predominantly the light u and d quarks. This is an important omission: recent lattice QCD results [30][31][32][33] have shown that strong magnetic fields have dramatic effects on the QCD phase diagram, notably in the region close to the pseudocritical temperature T pc ≃ 0.170 GeV, the region associated with the hadron-to-quark transition. The effects are most striking on the u and d quark condensates, namely the condensates increase with the magnetic field for low temperatures and decrease for temperatures close to T pc ; in the first case one refers to magnetic catalysis (MC) and in the latter to inverse magnetic catalysis (IMC)-Refs. [34][35][36] are recent reviews which contain extensive lists of references on this subject. The quark condensates have a direct impact on the effective masses of the u and d quarks, which in turn play an important role in the conductivity.
In the present work we fill this gap in the study of the electrical conductivity of magnetized quark matter by using a quasi-particle model. Much of our understanding of the low-energy regime of QCD, and of the QCD phase diagram in particular, is built on insights gained with quasi-particle models. Among the several existing quasi-particle models, those based on the Nambu-Jona-Lasinio (NJL) model [37,38] have been valuable in this respect-Refs. [39][40][41][42] are extensive reviews on the model in different QCD applications. In these models, the primary electric charge carriers are the u and d quarks and their response to electromagnetic forces are strongly dependent on their effective in-medium masses. The effective quark masses are determined by the in-medium quark condensate. As argued, strong magnetic fields change the condensate and in the present paper take this effect into account in the computation of the electrical conductivity. We use the NJL model of Refs. [43,44], a model that reproduces the lattice QCD data for the quark condensates, showing MC at low temperatures and IMC at temperatures close to T pc . We derive the expressions of the components of the conductivity by solving the Boltzmann equation in the relaxation-time approximation.
For a realistic quantification of magnetic fields effects in a heavy-ion collision, it is imperative to take into account the temperature and magnetic field dependence of the electrical conductivity. Simulations of the field dynamics invariably involve solving relativistic magnetohydrodynamic equations. These equations also require other transport coefficients, as shear and bulk viscosities. The magnetic field dependence of the shear viscosity was computed recently in Refs. [45][46][47][48][49][50][51] and of the bulk viscosity in Refs. [52][53][54][55][56]. Our study of the electrical conductivity adds to this body of work by incorporating important effects neglected in previous studies. Our results should be useful for studies employing magnetohydrodynamics simulations of heavy-ion collisions and for getting insight on lattice QCD results.
The paper is organized as follows. In Sec. II we start reviewing the NJL model quasiparticle description of magnetized quark matter, with particular emphasis on the magnetic catalysis and inverse magnetic catalysis of the chiral condensate. Then we present the derivation of the electrical conductivity by solving the Boltzmann equation in the relaxation time approximation. In Sec. III we present numerical results for the conductivity and discuss their meaning. In particular, we discuss the implications of the changes in the quark masses on two well known magnetic field effects on the conductivity: breaking of the space isotropy of the conductivity, and dimensional reduction of the dynamics along with quantization of phase space (Landau levels). The first happens when mutually perpendicular electric and magnetic fields exist in the conducting medium, field configurations that actually can be generated in a heavy-ion collision [4,9]. The second is a quantum mechanical effect that is particularly important for strong fields. Section IV presents a summary and the perspectives of our study.

II. FORMALISM
We start with a brief review of the NJL model quasi-particle description of quark matter at finite temperature T and in the presence of a magnetic field with strength B. We focus on the implications of the inverse magnetic catalysis (IMC) phenomenon on the quasiparticle effective mass M, the constituent quark mass. Next, we present the derivation of the electrical conductivity within the relaxation time approximation. The main input from the NJL model in these derivations is the T − and B−dependent constituent quark mass M.

A. Quark matter in presence of a magnetic field
The Lagrangian density for the isospin-symmetric two-flavor version of NJL model in presence of an electromagnetic field (A µ ) is given by where ψ is a flavor doublet of u and d quark fields, each being an N c −plet, where N c = 3 is the number of colors, m = diag(u, d) the quark-mass matrix, D µ = i∂ µ − QA µ the covariant derivative, Q = diag(q u = 2e/3, q d = −e/3) the charge matrix, A µ the electromagnetic gauge field, F µν = ∂ µ A ν − ∂ ν A µ , and τ = (τ 1 , τ 2 , τ 3 ) are the isospin Pauli matrices. We work in the approximation of exact isospin symmetry, i.e. m u = m d . The model is solved in the quasi-particle approximation or, equivalently, in the mean-field approximation, which corresponds to the leading-order approximation in the 1/N c expansion. Since the model is unrenormalizable due to the quadratic fermionic interaction, a regularization procedure must be employed. In the present paper we employ a sharp cutoff Λ to regularize ultraviolet divergences.
In the quasi-particle approximation, the gap equation for the constituent quark mass M at finite temperature T and in the presence of a magnetic field B is given by where ψ f ψ f represents the quark condensate of flavor f , which can be written as a sum of three contributions [43,57,58]: with where q f is the electric charge of the quark with flavor f = (u, d), Γ(x f ) is the Euler gamma function, In addition, in Eq. (6) k indexes Landau levels, with α k = 2−δ k,0 being spin degeneracy, and n F (ω f ) is the Fermi-Dirac distribution function: where Notice that, contrary to the vacuum case, in the presence of a magnetic field the u and d condensates are different due to the different u and d electric charges, but the constituent masses of the u and d quarks are still equal, i.e. M stands for both M u and M d .
As mentioned earlier, the NJL model in the quasi-particle approximation is unable to describe inverse magnetic catalysis (IMC) [30,31], unless one imposes that the coupling constant G of the model is T − and B−dependent [43]. A precise description of the lattice results for the u and d quark condensates is obtained within the NJL model with the parametrization [44]: where c(eB), β(eB), T a (eB) and s(eB) depend only on the magnitude of the external magnetic field. Their values for selected values of B are given in Table 1 of Ref. [44].
All numerical results presented here refer to this parametrization.
The model just described was used to study the effects of a magnetic field on neutral pions in Refs. [59,60], with results that agree with corresponding lattice QCD results [61]. Further evidence for the association of the IMC phenomenon with a magnetic field dependence decreasing the coupling strengths of quark-matter effective degrees of freedom was given in Refs. [62,63].

B. Electrical conductivity in presence of magnetic field: classical and quantum
Here we derive the electrical conductivity of magnetized hot quark matter (composed by the constituent quarks and antiquarks of flavor u and d) employing the relaxation time approximation-closely following the approach of Ref. [23]. Initially we consider the classical description, ignoring the quantization of phase space in terms of Landau levels.
Let us consider a relativistic electrically charged fluid of constituent quarks (and antiquarks) with masses M f and energies ω f (defined in Eq. (8)), characterized by distribution functions f (ω f ) described by the Boltzmann equation. In the electric-charge-transport picture, an external electric field is responsible for driving the system out of equilibrium.
Let f 0 (ω f ) = n F (ω f ) denote the equilibrium distribution, where n F (ω f ) is the Fermi-Dirac distribution given in Eq. (7), and f (ω f ) = f 0 (ω f ) + δf (ω f ) the distribution in the presence the electric force. The electric current J f associated with a given quark flavor f is expressed in terms of the quark distribution function by where q f is the charge of the quark (or antiquark), g a spin-color degeneracy factor, and The Boltzmann equation for the distribution function f (ω) of a quark of a given flavor under the influence of a generic external force F is given by where (∂f /∂t) coll takes into account the collisional effects. For an uniformly distributed system, ∂f /∂x = 0. The force F drives the system out of equilibrium and when the force is removed, scattering events described by (∂f /∂t) coll will restore equilibrium. The relaxation time approximation (RTA) consists in assuming for the collisional term where τ c is the relaxation time, the time required for the system to return to equilibrium after removing F 1 . The determination of τ c is out of the scope of the present paper; when presenting results in the next section we use values in the range 0.2 − 10 fm/c, the latter number being approximately the lifetime of the medium produced at the RHIC and LHC.
In the present context, F is the Lorentz's force, where q is the electric charge (recall that we are suppressing the flavor index f ). It is useful to discuss first the 1 Indeed, for F = 0 the solution of Eq. (12) in the RTA is given by where t 0 is the time at which F is removed.
situation with B = 0; the force is then F = qE. Assuming δf ≪ f 0 , Eq. (12) implies where β = 1/T . Using this into Eq. (11), adding the contributions from quarks and antiquarks of flavor u and d, taking into account that the spin-color degeneracy factor is g = 2×3 and the sum over flavor gives f q 2 f = 5e 2 /9, one obtains for the conductivity with σ = e 2 β 20 9 Next, we proceed with the derivation of σ ij in presence of a magnetic field B of arbitrary strength. In this situation, it is necessary to include the δf contribution in the drift term in Eq. (12); otherwise, the term v × B in the Lorentz gives null contribution, because: Including the δf contribution in the drift term in Eq. (12), one obtains Since any term proportional to p from ∂(δf )/∂p leads to a vanishing contribution, one must have δf ∼ p · F times a function of p 2 , where F is a vector that depends on the E and Specifically: from which one obtains where τ B = ω/qB is the inverse of the cyclotron frequency. Therefore, one obtains for δf : From Ohm's law in Eq. (11) and again adding the contributions from quarks and antiquarks of flavor u and d, and using g = 2 × 3 and f q 2 f = 5e 2 /9, one immediately obtains for σ ij : where σ n , n = 0, 1, 2, are given by σ n = e 2 β 20 9 = e 2 β 20 9 These two ways of expressing the (τ c ,τ B ) dependence will be useful shortly ahead when we discuss the weak-and strong-field limits of the σ n .
When B is in the z direction, which impliesb z = 1 andb x =b y = 0, the conductivity matrix elements are given by: σ zz = σ 0 + σ 2 = σ (longitudinal conductivity), σ xx = σ yy = σ 0 (transverse conductivity), and σ xy = −σ yx = σ 1 (Hall conductivity), with the other components being zero. The Hall components σ xy = −σ xy = σ 1 are absent in the zero magnetic field case, Eq. (17). This reveals that the space-anisotropic nature of the conductivity which, according to Eqs. (26) and (27), is controlled by the ratio between the two time scales τ c and τ B . This anisotropy shows interesting features for weak and strong magnetic fields.
For a weak magnetic field, τ c /τ B is small. Expanding the integrand Eq. (26) up to second order in τ c /τ B one obtains: This shows that a small space-anisotropic conduction is already present in the weak-field limit. The σ xx and σ yy diagonal elements are reduced from the zero-field case, whereas the σ zz = σ 0 + σ 2 = σ component is unchanged. In addition, the Hall conductivity σ xy = σ yx = σ 1 , which vanishes at B = 0, acquires a non-zero value for finite B. For a strong magnetic field, τ c /τ B is large and Eq. (27) up to second order in τ B /τ c can be written as σ n ≃ e 2 β 20 9 Interestingly, in this case the Hall conductivity becomes independent of τ c , indicating a dissipation-free conductivity. On the other hand, as B → ∞ all the components of the conductivity tensor, with the exception of σ zz , vanish .
Up to now, classical physics was used to determine the conductivity. As seen in Sec. (II A), a proper quantum mechanical treatment of electric charge dynamics in a magnetic field leads to one-dimensional dynamics and energy quantization in terms of Landau levels. In practical terms, these effects amount to modifying the results above by the following replacements: where in the last line we used the fact that the lowest Landau level (LLL), k = 0, is spinpolarized-recall that α k = 2−δ k,0 . Therefore, the longitudinal conductivity in the quantum description is given by where the factor 2 comes from summing over the quark and antiquark contributions. The transition between the classical and quantum regimes in σ zz can be obtained by varying τ c for a fixed τ B . This analysis is made in the next section.
In the strong field limit, the dominant contribution to σ zz comes from the lowest Landau level (LLL) because the energy ω k,f , which appears in the denominator in Eq. (32) and in the Fermi-Dirac distributions, increases with B for k = 0. Therefore, the LLL approximation for σ zz leads to where ω 0 = ω k=0,f = p 2 z + M 2 f . Eq. (33) coincides with Eq. (D.5) of Ref. [21]. Since the interest is to take into account the IMC phenomenon in the phenomenology of the electrical conductivity, those parameters were fitted to reproduce the lattice data of Ref. [31]. A precise description of the lattice data for the quark condensate is obtained in [44] when using the parametrization given in Eq. (9).
The values of the parameters c(B), β(B), T a (B), and s(B) defining the fitting formula in Eq. (9) are displayed in Table 1 of Ref. [44]. the value obtained in Eq.(9) that was fitted to the lattice at high temperature region (more details can be found in [44]). For a given momentum, τ B decreases with B and T ; the decrease with B comes mainly from the trivial 1/eB factor, whereas the decrease in T comes from M in ω = (p 2 + M 2 ) 1/2 . But τ B is integrated over a momentum range bounded by the thermal distribution f 0 (ω). To assess the effect of this integration, we have defined the following average: with ω = (p 2 + M 2 ) 1/2 . The temperature dependence of τ B (B, T ) is depicted in Fig. 1(b).
This average increases with T despite the decrease of M with T . This is because f 0 (ω) dominates the temperature dependence of the integrand; although the numerator and denominator in Eq. (34) both grow with T , the numerator grows faster. This feature has important consequences for the temperature dependence of the anisotropy of the conduc-tivity. The vertical arrow in Fig. 1(b) indicates the direction of growth of τ B with B and identifies regions of weak (τ c < τ B ) and strong (τ c > τ B ) magnetic field. Next, we present our results for the electrical conductivity-we recall the B is taken in the z direction. We start with the classical results. We recall that for zero magnetic field, the electrical conductivity tensor is diagonal, with all the diagonal elements being equal to σ, given by Eq. (17). When B = 0, there is anisotropy in the conductivity, as indicated by Eqs. (26) and (27). To quantify the anisotropy, we define the differences ∆σ n /σ = (σ n −σ)/σ. τ c = 5 fm/c characterizes a strong magnetic field whereas τ c = 0.2 fm/c characterizes a weak magnetic field. It can be seen from Fig. 2 that σ n increase with the temperature having a kink at T ≃ T pc , after which the rate of increase is substantially reduced. We conclude that the electrical conductivity becomes almost independent of T for T > T pc , when the constituent quarks stops changing with T , since then M ≈ m. In addition, σ xx = σ yy < σ zz for τ c = 5 fm/c (strong field), whereas σ xx = σ yy ≈ σ zz for τ c = 0.2 fm/c (weak field), as expected.
Interesting insight is gained by examining the conductivity as a function of the magnetic field at the pseudocritical temperature. Figure 3 displays the B dependence of ∆σ n /σ and σ n /T at T = T pc = 0.170 GeV, for τ c = 1 fm/c. The anisotropy is enhanced as B increases.
The Hall conductivity, σ xy = −σ xy = σ 1 , presents a peak at eB ≃ 10m 2 π . A peak in the Hall conductivity at some value of B is easily understood by using the simplified Drude relation [25,64,65] for the conductivity: where q f and n f are respectively the quark electric charge and density which, together with the quark mass M are assumed to be independent of B. One obtains dσ 1 /dB = 0 for  Figure 6 displays the comparison: the kink is due to a rapid inversion of the temperature dependence of M when G = G(eB, T ). In addition, the effect of IMC on the conductivities is more clear in the Hall conductivity σ xy = σ 1 : IMC makes the peak more prominent at eB ≃ 10m 2 π . This is a clear-cut prediction of the model that can be tested by lattice QCD simulations.

IV. SUMMARY AND PERSPECTIVES
Relativistic heavy collisions can produce strong magnetic fields. Strong magnetic fields have striking effects on properties of the hot quark matter created in such collisions. Measurability of many of the effects depend on the duration of the field produced in a collision.
A key physical property determining the duration of the field is the electrical conductivity of the medium: the conductivity is responsible for the induction of electric currents which in turn can produce magnetic fields that can last while the systems exists. Simulations of the field dynamics involves solving relativistic magnetohydrodynamics equations which, for a realistic quantification require the temperature and magnetic field dependence of the electrical conductivity (and, of course, of other transport coefficients). We focused on the implications for the conductivity of changes induced by a magnetic field on the effective quark masses, with special interest on the importance of magnetic catalysis (MC) and inverse magnetic catalysis (IMC)-while MC occurs at low temperatures, IMC occurs at temperatures close to the pseudocritical temperature (T pc ) of the QCD phase transition. We employed a quasi-particle model for the medium, implemented by a Nambu-Jona-Lasinio model with a temperature-and magnetic-field-dependent coupling constant adjusted to lattice QCD data on MC and IMC. One of our main finding was that while the longitudinal conductivity (σ zz ) increases with B, the transverse component (σ xx = σ yy ) decreases with B, in qualitative agreement with the very recent lattice QCD results of Ref. [19]. Moreover, our study revealed that IMC leaves a distinctive signal in all components of conductivity, a kink at T pc .
In addition, IMC makes the peak more prominent in the Hall conductivity (σ xy = −σ yx ) at eB ≃ 10m 2 π . Such a peak appears when the two time scales controlling the conductivity, i.e. the relaxation time τ c and the inverse of the cyclotron frequency τ B are comparable.
This feature of IMC on the Hall conductivity is a prediction of the model and is testable with lattice QCD simulations. Additional findings in our study were: (1) for a fixed value of B, all components of the conductivity increase with the temperature, (2) the anisotropy in the conductivity increases with B, (3) quantum effects leading to phase-space Landau-level