The effect of magnetization and electric polarization on the anomalous transport coefficients of a chiral fluid

The effects of finite magnetization and electric polarization on dissipative and non-dissipative (anomalous) transport coefficients of a chiral fluid are studied. First, using the second law of thermodynamics as well as Onsager's time reversal symmetry principle, the complete set of dissipative transport coefficients of this medium is derived. It is shown that the properties of the resulting shear and bulk viscosities are mainly affected by the anisotropy induced by external electric and magnetic fields. Then, using the fact that the anomaly induced currents do not contribute to entropy production, the corresponding algebro-differential equations to non-dissipative anomalous transport coefficients are derived in a certain derivative expansion. The solutions of these equations show that, within this approximation, anomalous transport coefficients are, in particular, given in terms of the electric susceptibility of the medium.

In [14,15], these anomalous transport coefficients are determined within a relativistic hydrodynamical approach in the presence of an external magnetic field. Using the second law of thermodynamics and the fact that the anomaly induced currents do not contribute to entropy production [16], certain algebro-differential equations are derived, whose solutions yield the anomalous transport coefficients corresponding to the aforementioned effects. The main purpose of this paper is to extend the method introduced originally in [14,15] to a medium with finite magnetization and electric polarization. To the best of our knowledge, these in-medium modifications of anomalous transport coefficients, including the linear response of the medium to external electromagnetic fields, are not yet studied in the literature. They seem, however, to be important not only from a theoretical point of view, but also because the experimentally relevant quark-gluon plasma turns out to have finite magnetic [17] and electric susceptibilities [18].
The present paper is organized as follows: In Sec. II, we introduce the ideal electro-magnetohydrodynamical (EMHD) framework with finite magnetization and electric polarization. We essentially follow the method previously used in [19,20]. However, in contrast to these works, where the effect of the external electric field is neglected, we consider the case of non-vanishing magnetic and electric fields, and derive the relevant thermodynamic relations in the presence of finite magnetization and electric polarization. Moreover, an anomalous current will be considered, which includes anomalous transport coefficient as in [14,15]. This brings our derivation in connection with quantum anomalies, where parallel magnetic and electric fields turn out to play a major role. In Sec. III A, we first derive the dissipative transport coefficients of the anomalous EMHD by using the second law of thermodynamics and the Onsager's time reversal symmetry principle. In particular, we show that the dissipative part of the electric current, as well as the viscous stress tensor include a large number of thermal and electric conductivities as well as shear and bulk viscosities. Their properties are mainly affected by the anisotropies induced by external electric and magnetic fields. Our results are therefore a completion of the results presented in [19,20], where the dissipative coefficients arisen from the electric field in the dissipative currents are absent. Moreover, our results include certain, previously discarded, dissipative coefficients which arise from the interplay between external electric and magnetic fields [see (III. 17)]. In Sec. III B, we eventually use the fact that the anomaly induced currents do not contribute to entropy production, and derive the corresponding algebro-differential equations to anomalous transport coefficients of this medium. We show that, within a certain second-order derivative expansion, these equations include, in particular, the electric susceptibility of the medium. We then follow the method introduced in [14], and determine the anomalous transport coefficients by solving the above mentioned equations analytically. The effect of gravitational anomaly [21,22] is not considered in the present work, neither in the case of vanishing nor in the case of non-vanishing susceptibilities. Section IV is devoted to a summary and a number of concluding remarks.

II. IDEAL ANOMALOUS ELECTRO-MAGNETOHYDRODYNAMICS
Electro-magnetohydrodynamics addresses all phenomena related to the interaction of electric and magnetic fields with an electrically conducting magnetized fluid. An ideal and locally equilibrated relativistic fluid is characterized by its long-wavelength degrees of freedom, the four-velocity, the temperature and the chemical potential fields, u µ (x), T (x) and µ(x), respectively. The four-velocity u µ = γ(1, v), with γ ≡ √ 1 − v 2 , is defined by the variation of the four-coordinate x µ with respect to the proper-time τ , and satisfies u µ u µ = 1. 1 In the absence of external electromagnetic fields, the physical observables, the entropy and baryonic currents s µ (0) and n µ b(0) , as well as the fluid energy-momentum tensor T µν F (0) , are expressed in terms of u µ as 2 s µ (0) ≡ su µ , n µ b(0) ≡ n b u µ , T µν F (0) ≡ ǫu µ u ν − p∆ µν , (II.1) with ∆ µν ≡ g µν − u µ u ν , the projector onto the direction perpendicular to u µ . In (II.1), ǫ, p, s, n b are local energy density, thermodynamic pressure, entropy and baryon number densities of the ideal fluid, respectively.
In an ideal and locally equilibrated fluid with no electromagnetic fields and sources, the quantities presented in (II.1) are conserved In the presence of electromagnetic fields, however, the total energy-momentum tensor of the ideal fluid is to be modified as [19,20] T µν where the fluid and electromagnetic energy-momentum tensors, T µν F (0) and T µν EM , read as where T µν F (0) , defined in (II.1), is the energy-momentum tensor of an ideal un-magnetized and un-electropolarized fluid. 3 The antisymmetric field strength and polarization tensors, F µν and M µν in (II.4), are expressed in terms of electric and magnetic fields, E and B, as In a frame where the fluid is moving with the velocity u µ , the four-vector of electric and magnetic fields are given by E µ ≡ F µν u ν and B µ ≡ 1 2 ε µναβ F να u β . Here, ε µναβ is the totally antisymmetric Levi-Civita tensor. In the rest frame of the fluid, with u µ = (1, 0), we have therefore E µ = (0, E) and B µ = (0, B). Here, E i = F i0 and B i = −ε ijk F jk /2, as in non-relativistic electrodynamics. The strength of the electric and magnetic fields, E and B, are given by the normalization relations E µ E µ = −E 2 and B µ B µ = −B 2 , which lead immediately to e µ e µ = b µ b µ = −1.
The antisymmetric polarization tensor M µν in (II.5) describes the response of the system to an applied field strength F µν , and leads through the relation F µν − M µν ≡ H µν to the induced field strength tensor H µν , which in terms of e µν and b µν , is given by Here, in analogy to E µ and B µ , we define the four-vectors of induced electric and magnetic fields, D µ ≡ H µν u ν as H µ ≡ 1 2 ε µναβ H να u β . They are in relation to the four-vectors of electric polarization P µ ≡ −M µν u ν and magnetization M µ ≡ 1 2 ε µναβ M να u β . In the rest frame of the fluid, D µ = (0, D) and H µ = (0, H). Similarly, we have P µ = (0, P) and M µ = (0, M). In (II.5) and (II.7), V ≡ |V|, with V = {E, B, P, M, D, H} is used. This fixes the normalization relations The purpose of this paper is to study the effect of electric polarization P and magnetization M on dissipative and non-dissipative transport coefficients of an electromagnetized chiral fluid affected by the quantum anomaly. To this purpose, we introduce the electric and magnetic susceptibilities, χ e and χ m , arising in linearized relations between P and E as well as M and B, P = χ e E and M = χ m B. Moreover, to bring our derivation in connection with quantum anomalies, we will assume, e µ b µν = 0.
(II.8) 3 It is important to notice that the subscript (0) on T µν F (0) in (II.4) refers only to the zeroth-order hydrodynamical derivative expansion of the fluid part of the energy-momentum tensor, T µν F (0) = ǫu µ u ν − p∆ µν from (II.1). Later we will add higher order terms in the derivative expansion to this part of T µν (0) from (II.3), which will contribute to dissipation. We will then determine dissipative coefficients up to first-order derivative expansion.
In the rest frame of the fluid, (II.8) is equivalent to parallel electric and magnetic fields, e b with e ≡ E/E and b ≡ B/B denoting the directions of external electric and magnetic fields. Apart from (II.8), e ν ∂ µ b µν = −b µν ∂ µ e ν = 0 and ∂ µ χ e = ∂ µ χ m = 0 are assumed.
In what follows, the in-medium Maxwell equations will be used to determine the relevant thermodynamic relations for an ideal and locally equilibrated fluid. A number of useful relations will be also derived. We will keep our notations similar to what is presented in [19], where, in contrast to our presentation, the effect of electric field is neglected.
Introducing, in analogy to n µ b(0) from (II.1), the four-vector of electric current, n µ e(0) ≡ n e u µ , the inhomogeneous Maxwell equation reads where H µν is defined in (II.7) and n µ e(0) ≡ (n e , n e ). Here, n e is the electric charge density and n e is the corresponding electric current. As expected, (II.9) is consistent with the conservation relation of the electric current, ∂ µ n µ e(0) = 0. Together with the conservation relations ∂ µ s µ (0) = 0 and ∂ µ n µ b(0) = 0 from (II.2), an ideal fluid in the presence of electromagnetic fields is particularly described by the conservation of the full energy-momentum tensor T µν (0) from (II.3), Using T µν EM from (II.4) and the inhomogeneous Maxwell equation (II.9), we arrive at To derive (II.11), we have used the homogeneous Maxwell equation, ε µναβ ∂ β F να = 0. Plugging F µν from (II.5) into this equation, and using the standard relation ε µναβ ε αβρσ = −2(δ µ ρ δ ν σ − δ µ σ δ ν ρ ), the homogeneous Maxwell equation is equivalently given by (II. 13) We use this equation to derive two useful relations, which play important roles in the rest of this work. First, contracting (II.13) with u µ , we obtain where D ≡ u µ ∂ µ and a · b ≡ a µ b µ . The four-vector ω µ , on the right-hand side (r.h.s.) of (II.14), is the vorticity of the fluid, defined by Following the steps described in Appendix A, another useful relation can be derived [see (A.13)-(A.16) for the proof of (A.6) and set ∂ µ χ e = 0 to arrive at (II. 16)]. Contracting further (II.13) with b µ , and using (II.8) as well as ǫ µναβ b µ ∂ β (E ν u α ) = 0, we arrive at another useful relation where θ ≡ ∂ µ u µ . The last useful relation reads To derive (II.18), let us first consider the energy-momentum tensor (II.3). Plugging F µν and M µν from (II.5) into T µν F (0) and T µν EM from (II.4), we arrive after some algebraic manipulations at We further consider ∂ µ T EM µν = n µ tot F µν from (II.11). Contracting this relation with u ν , plugging T µν EM from (II.19) into the left hand side (l.h.s.) of the resulting expression, and using eventually (II.8) as well as b µν ∂ µ e ν = 0 and ∂ µ χ e = 0, which lead to we arrive at The latter yields (II.18) for χ e = −1.
To check the consistency of the thermodynamic relations including electric and magnetic fields, let us contract (II.10) with u ν . Using (II.18) and ∂ µ T µν F (0) = −∂ µ T µν EM , which arises from (II.10) combined with the definition of T µν (0) from (II.4), it is possible to show that u ν ∂ µ T µν F (0) = 0. Plugging then T µν F (0) from (II.19) into this relation, and using u µ b µ = u µ e µ = 0 and u ν ∂ µ u ν = b ν ∂ µ b ν = e ν ∂ µ e ν = 0, we arrive eventually at (II.20) The relevant thermodynamic relation for Dǫ is then derived by using where µ b and µ e are the chemical potentials related to the baryon and electric number densities, n b and n e . Plugging ǫ + p from (II.21) into (II. 20), and using the conservation laws of baryon number density, ∂ µ n µ b(0) = 0, electric number density ∂ µ n µ e(0) = 0 and the entropy density current ∂ µ s µ (0) = 0, we arrive after some algebraic manipulations at which is consistent with the standard thermodynamic relation dǫ = T ds + µ b dn b + µ e dn e − M dB − EdP [23]. Combining at this stage (II.21) with (II.22), the Gibbs-Duhem relation is given by To obtain the standard Gibbs-Duhem relation in the presence of electric and magnetic fields, we definep ≡ p − EP . Using this definition in (II.23), we arrive at Let us reiterate that the main aim of this paper is to study the effect of quantum (axial) anomaly on the EMHD equations once electric polarization P and magnetization M of the underlying fluid are not neglected. To do this, we introduce at this stage a U (1) axial vector current, n µ a(0) ≡ n a u µ , (II. 25) which satisfies the classical conservation law ∂ µ n µ a(0) = 0, (II. 26) in the chiral limit. 4 Denoting the chemical potential associated with n a by µ a , the thermodynamic relations (II.21) and (II.22) turn out to be ǫ + p = T s + µ b n b + µ e n e + µ a n a , (II. 27) and respectively. In the next section, we will consider the axial anomaly of the axial vector current, withF µν ≡ 1 2 ε µνρσ F ρσ and C ≡ e 2 4π 2 , and study its effect on non-dissipative transport coefficients of an electromagnetized relativistic fluid. 5

III. DISSIPATIVE AND ANOMALOUS TRANSPORT COEFFICIENTS OF A CHIRAL FLUID
In the first part of this section, Sec. III A, we will derive the complete set of dissipative transport coefficients in the presence of electric and magnetic fields. To do this, we will follow the formalism of dissipative fluid dynamics from [19]. In Sec. III B, by taking into account the fact that the anomaly induced current is non-dissipative [16], we will then derive the algebro-differential equations leading to anomalous transport coefficients. Similar equations are derived originally in [14,15], where the in-medium effects are neglected. These equations will then be solved in a medium with vanishing (Sec. III B 1) and non-vanishing (Sec. III B 2) electric and magnetic susceptibilities. In what follows, we will first derive the general structure of ∂ µ s µ , with s µ the current of the entropy density of a dissipative anomalous fluid.
Following the method presented in [19,20], we start by introducing the first-order dissipative and nondissipative corrections to the conserved quantities of the ideal fluid, T µν (0) , n µ b(0) , n µ e(0) and s µ (0) , Here, the total energy-momentum tensor of the ideal fluid, T µν (0) , is defined in (II.3) and (II.4), and n b , n e and s are the baryonic and electric number densities as well as the entropy density of the ideal fluid, respectively. The coefficients D ω , D B and D E in s µ are associated with the anomaly. The coefficients D ω and D B are originally introduced in [14], where the entropy density current s µ is expanded only in terms of the vorticity ω µ from (II.15) and the external magnetic field B µ . Here, in the presence of magnetic and electric fields, we have also considered the effect of the external electric field E µ , and introduced D E as its coefficient. Later, we will explicitly show that D E = 0. This is also expected from symmetry arguments. To consider the quantum anomaly of the dissipative electromagnetized fluid, we shall also replace the axial vector current n µ a(0) from (II.25) with Whereas in (III.1), τ µν , j µ b and j µ e are dissipative currents, 6 the additional (non-dissipative) current j µ a in (III.2) is introduced in analogy to the considerations in [14,15], and will later be used to determine the algebrodifferential equations leading to anomalous transport coefficients.
Let us notice that j µ i , i = b, e, a are orthogonal to u µ , i.e. u µ j µ i = 0, for i = b, e, a. Moreover, τ µν is a symmetric rank-two tensor, satisfying the orthogonality condition u µ τ µν = 0.
At this stage, we will use the conservation relations associated with the anomalous EMHD together with the second law of thermodynamics, T ∂ µ s µ ≥ 0, in order to arrive at an appropriate relation for T ∂ µ s µ . To do this, we shall first consider the conservation relation (II.11) for the electromagnetic energymomentum tensor T µν EM from (II. 19). Replacing the ideal electric current n µ e(0) on the r.h.s. of (II.11) by n µ e from (III.1), and contracting the resulting expression with u ν , we arrive at This relation replaces (II.18) of the ideal electromagnetized fluid. To derive (III.4), the relations (II.17), together with are used. Then, using the conservation of the total energy-momentum tensor ∂ µ T µν = 0 from (III.3) with T µν from (III.1), and following the same steps leading from u ν ∂ µ T µν (0) = 0 to (II.20), we obtain To arrive at (III.6), the identity (III.4) and are used. Then, expressing j µ s as a linear combination of j µ b , j µ e and j µ a , as in [19], The first four terms on the l.h.s. of (III.9) vanish by making use of (II.28) from ideal EMHD. Replacing n i u µ and su µ on the l.h.s. of (III.9) by their definitions 1), and using the conservation relations for n µ i , i = b, e, a from (III.3), as well as the expansion of j µ s in terms of the other dissipative currents from (III.8), we arrive after some straightforward algebraic manipulations at Here, ∇ µ ≡ ∆ µν ∂ ν and w µν ≡ 1 2 (∇ µ u ν + ∇ ν u µ ) are introduced. To satisfy the positivity condition of T ∂ µ s µ , the expression on the r.h.s. of (III.10) is to be non-negative. This leads immediately to 11) and the general Ansatz where σ µν b , σ µν e and η µναβ include dissipative transport coefficients, and κ i and ξ i , i = B, E, ω are non-dissipative coefficients. The latter can be expressed in terms of anomalous transport coefficients. Let us notice that the dissipative transport coefficients σ µν b , σ µν e and η µνρσ are orthogonal to u µ , and are symmetric under µ ↔ ν. In the next two sections, we will use the Onsager's time-reversal principle to first determine σ µν i , i = b, e and τ µν in terms of thermal conductivity as well as longitudinal and transverse shear and bulk viscosities. This will generalize the standard formulation of magnetohydrodynamics presented in [19] to the case of non-vanishing electric field. We will then consider the anomalous contributions to j µ e and j µ a in (III.10) proportional to κ i and ξ i , i = B, E, ω, and by combining them [15], we will arrive at the algebro-differential equations leading to κ i and ξ i , i = B, E, ω as well as D i , i = B, E, ω in a dense and hot quark matter in the presence of constant E and B fields. This will generalize the results presented in [14,15] to the case of a fluid with finite magnetization M and electric polarization P .

A. Dissipative currents of an anomalous chiral fluid
According to the Onsager's principle for transport coefficients [24], the thermal conductivity, σ µν i , i = b, e, corresponding to the diffusive fluxes of the baryonic and electric number density n b and n e shall satisfy Moreover, σ µν b/e have to satisfy the orthogonality condition u µ σ µν i = 0 for i = b, e, and are to be symmetric under µ ↔ ν. The relation e µ b µν = 0 from (II.8) is also to be taken into account.
To build σ µν b/e , we expand it in terms of independent irreducible rank-two tensors, which are built from u µ , g µν , b µ and e µ [19]. The only relevant tensors that are compatible with the above mentioned conditions are thus given by (III.14) Other rank-two tensors like b µν , e [µ, b ν] , · · · are excluded because of the aforementioned conditions. Here, Introducing at this stage three independent thermal conductivity coefficients σ (i) b/e , i = 1, 2, 3, associated with the relevant tensors from (III.14), the dissipative rank-two tensors σ µν b and σ µν e from (III.12) read Then, plugging σ µν i , i = b, e from (III.15) into (III.12), the dissipative part of the baryonic and electric currents reads The coefficients σ Concerning the viscous stress tensor τ µν from (III.12), we apply, as in [19], the Onsager's principle on the rank-four tensor η µνρσ appearing in (III.12). All relevant tensors, compatible with this principle and expressed in terms of u µ , b µ and e µ as well as g µν , b µν and e µν are listed in Table I in four different series.
All these bases fulfill the orthogonality condition, and are symmetric under ρ ↔ σ. The bases appearing in series I and II are previously introduced in [19]. The new bases appearing in series III include only e µ and e µν , and those in series IV include both electric and magnetic fields. Using these bases, and following the method presented in [20] (see Appendix B for more detail), the viscous stress tensor is then given by EB (b µρ e ν e σ + b νρ e µ e σ )w ρσ + 2η As expected, τ µν is symmetric under µ ↔ ν and satisfies u µ τ µν = 0. According to our descriptions in B, shear viscosities η 0 , η B , i = 1, · · · , 4 as well as bulk viscosities ζ ⊥ B and ζ B appear originally in [24] as well as in [19]. Here, we have completed the list of dissipative transport coefficients by considering the additional effect of an external electric field. Let us also notice that all bases including the combination e µ b µν are excluded once the condition (II.8) is taken into account.

B. Anomalous transport coefficients
Let us now consider the remaining terms Series IV ∆ µν e ρ e σ + ∆ ρσ e µ e ν e µ e ν b ρ b σ + b µ b ν e ρ e σ e µ e ν e ρ e σ b ρµ e ν e σ + b ρν e µ e σ + b σµ e ν e ρ + b σν e µ e ρ ∆ µρ e ν e σ + ∆ νρ e µ e σ + ∆ µσ e ν e ρ + ∆ νσ e µ e ρ b (µ, e ν) b (ρ, e σ)  appearing in T ∂ µ s µ from (III.10). In what follows, we will determine the anomalous coefficients D i , κ i and ξ i , i = B, E, ω, introduced in j µ e and j µ a from (III.12). To do this, let us insert j µ a and the anomalous part of j µ e from (III.12) into the above expression and set the resulting expression equal to zero. We arrive first at At this stage, we have to insert the corresponding expressions to ∂ µ ω µ , ∂ µ B µ , and ∂ µ E µ into (III. 19), and after reordering the resulting expression in terms of independent bases 1, B µ , E µ , ω µ , B · ω, E · ω, E · E, B · E, (III. 20) determine their coefficients. In Appendix A, we have determined the exact values of ∂ µ B µ , ∂ µ E µ and ∂ µ ω µ [see (A.11), (A.12) and (A. 22)]. Linearizing these expressions in terms of independent bases (III. 20), and keeping only the terms in the second-order derivative expansion, we arrive at (III.22) Here, we have used the fact that B µ , E µ , ω µ as well as M and P are O(∂) and u µ , ǫ and p are O(1) [see also [14] for the same power counting]. In (III.21), we have kept only terms in O(∂ 2 ), and discard all the remaining terms. This defines our second-order derivative expansion. Let us also note that the only effect of the medium, within this second-order derivative expansion, is reflected in the appearance of non-vanishing electric and magnetic susceptibilities, χ e and χ m in (B.4). Thus, setting χ e = χ m = 0 in (III.21), the results for ∂ µ B µ and ∂ µ ω µ presented in [14] for ∂ µ B µ and ∂ µ ω µ are reproduced. Plugging now (III.21) into (III. 19), and simplifying the resulting expressions, we arrive first at Using then the fact that the bases (III.20) are linear independent, the expressions in front of them can be set independently equal to zero. We arrive immediately at D E = κ E = ξ E = 0, as expected from the symmetry reasons. We conclude that the coefficients proportional to E µ in s µ , j µ e and j µ a do not receive any contribution from anomaly. All the other anomalous transport coefficients satisfy the following algebro-differential equations: Let us reiterate at this stage that to derive the above algebro-differential equations a number of constraints as e µ b µν = 0 from (II.8) as well as e µ ∂ µ b µν = 0, ∂ µ χ e = 0 and χ e = 1 are made. These kinds of constraints, especially those related to (II.8), are used to derive the thermodynamical equations in Sec. II. The latter are then used in Appendix A to derive general expressions for ∂ µ B µ , ∂ µ E µ and ∂ µ ω µ in (A.11), (A.12) and (A.22), respectively. Approximating these relations in an appropriate way (see above) leads to D E = ξ E = κ E = 0 as well as to (III.24), whose solutions yield anomalous transport coefficients D k , ξ k , κ k with k = B, ω. This describes the role played by these constraints to determine these anomalous transport coefficients in this paper. In a medium with vanishing χ e , the above equations (III.24) reduce to the equations appearing in [15], A comparison between (III.24) and (III.25) shows that, within this second-order approximation, only the algebraic equation receives contribution from χ e , the electric susceptibility of the medium. The magnetic susceptibility, χ m , plays thus no role in modifying the anomalous transport coefficients in this approximation. In what follows, we first consider the algebro-differential equations (III. 25), and solve them to determine D k , ξ k , κ k for k = B, ω in terms of thermodynamical quantities ǫ, p, α e and α a . We then present the solutions of (III.24) in an electrically polarized hot and dense medium in the presence of an external magnetic field.

Anomalous transport coefficients in a medium with vanishing χe and χm
To solve (III.25), we use the same method as in [14,15]. Introducing α i = µi T , i = a, e and replacing dµ i = T dα i + α i dT in the Gibbs-Duhem equation dp = sdT + i={e,a} n i dµ i , arising from ǫ + p = T s + i={e,a} n i µ i and We therefore have, dp dT αe,αa = i={e,a} (s + n i α i ) = ǫ + p T , dp dα e αa,T = n e T, dp dα a αe,T = n a T.
(III. 27) Plugging at this stage, for k = B, ω and ∂ µ p from (III.26) into the first two differential equations in (III.25), we obtain where γ k (α a ), k = B, ω are constants of integration [15]. To arrive at (III.31), the identities (III.27) have been used. Plugging D B and D ω from (III.31) into (III.29) and (III.30), we arrive at which are consistent with the results presented in [14], provided the integration constants γ i , i = 1, 2 are set to be zero (see below). Here, C = e 2 4π 2 is the coefficient of the axial anomaly from (II.29). Let us reiterate that the first term in κ B is the same coefficient of chiral magnetic effect, arising originally in [1,8]. Moreover, κ ω is the coefficient for chiral vortical effect [11][12][13][14], and ξ B as well as ξ ω , the coefficients of chiral vortical as well as chiral vortical separation effects, appeared first in [9,10] as well as in [5]. 7 Let us also note that the contributions from gravitational anomaly to anomalous transport coefficients are not considered in the present work. These kinds of corrections are computed in [21,22] using an appropriate Kubo formalism (see also [25] for a kinetic theory approach). They are shown to appear as additional T 2 dependent terms in κ k , ξ k , k = B, ω. These terms can also be interpreted as contributions from the aforementioned integration constants γ B and γ ω [26]. Their determination turns out to be strongly frame dependent [27,28] (see also [29] for a recent review). In what follows, we will determine D k , ξ k , κ k with k = B, ω for the case of non-vanishing χ e . Requiring that these results lead to the corresponding expressions for χ e = 0, new constants of integrations will be brought in connection with γ B and γ ω . The contributions from gravitational anomaly will not be considered in this framework, as in the case of χ e = 0.

IV. CONCLUDING REMARKS
The anomaly induced effects on magnetized chiral fluids have attracted much attention in recent years. They are all characterized by non-dissipative vector and axial vector currents, which are proportional to either the background magnetic field or the vorticity of the medium. The proportionality factors, whose values are dictated by axial anomaly, represent non-dissipative transport coefficients. They arise naturally within relativistic hydrodynamics, as shown originally by Son and Surowka in [14]. In this paper, we have extended the method previously used in [19,20] to the case of non-vanishing electric field. In addition, an anomalous current has been considered, which includes anomalous transport coefficients as in [14,15]. This brings our derivation in connection with quantum anomalies. In this way, the work of Son and Surowka is generalized to the case of an electromagnetized chiral fluid, which linearly responses to the external electromagnetic field through finite magnetization and electric polarization. We have shown that, within certain approximation, the anomalous transport coefficients are, in particular, given in terms of the electric susceptibility of the medium. Other ingredients are the energy density and thermodynamic pressure of the medium as well as electric and axial charge densities. They are all functions of the temperature T , finite electric and axial chemical potential, µ e and µ a of the fluid, as well as external electric and magnetic fields, E and B acting on the fluid. As a by product, we have also determined the complete set of dissipative transport coefficients arising in the dissipative part of the electric current as well as the viscous stress tensor. This completes the set of coefficients previously obtained in [19,20], where the dissipative coefficients arising from the external electric field were neglected.
This work can be extended in many ways. One possibility is to assume a certain thermodynamic potential for a chiral QCD-like effective model in the presence of parallel electric and magnetic fields. Using standard thermodynamical relations, it is then possible to explicitly determine the energy density, pressure and electric susceptibility of this model in terms of a given set of thermodynamical parameters T, µ e , µ a , E and B. The non-dissipative anomalous transport coefficients, which are presented in this works by certain integration over α e = µe T , can then be determined by numerically performing these integrals for a given set of T, µ e , µ a , E and B. It would be interesting to explore, for instance, the dependence of the anomalous transport coefficients on this set of parameters, especially when the chiral model exhibits a chiral phase transition. The behavior of the anomalous transport coefficients in the vicinity of chiral critical point might be interesting for the physics of quark matter under extreme conditions, and may have phenomenological consequences in HIC experiments. We will postpone these kinds of studies to our future works. Let us consider the fluid energy-momentum tensor from (II.19). Using the definitions of ǫ ′ , p ⊥ and p , it can equivalently be given as where Ξ µν B = ∆ µν + b µ b ν and ∆ µν = g µν − u µ u ν . To determine ∂ µ B µ and ∂ µ E µ , let us consider the combination Using the relations (II.11)-(II.18) as well as the properties (II.8) and u µ e µ = u µ b µ = 0, we arrive first at where ∇ µ = ∆ µν ∂ ν , θ = ∂ µ u µ and E µ = F µν u ν . Combining then these two relations, we get Multiplying at this stage (A.4) with B ρ , and using the definition of F µρ from (II.5), we obtain Plugging B · Du = −u · DB from (II.14) into this relation, and using the following two relations and we arrive after some straightforward computations at Multiplying at this stage (A.4) with E ρ , we arrive first at Then, using (A.6) and (A.7), we obtain .
Appendix B: Relevant bases for the shear and bulk viscosities from (III. 17) In this section we present the bases used to build the rank-two tensors appearing in τ µν from (III.17).