Dynamical gauge fields and anomalous transport at strong coupling

Anomalous transport coefficients are known to be universal in the absence of dynamical gauge fields. We calculate the corrections to these universal values due to dynamical gluon fields at strong coupling, at finite temperature and finite density, using the holographic duality. We show that the consistent chiral magnetic and chiral vortical currents receive no corrections, while we derive a semi-analytic formula for the chiral separation conductivity. We determine these corrections in the large color, large flavor limit, in terms of a series expansion in the anomalous dimension $\Delta$ of the axial current in terms of physical parameters $\Delta$, temperature, electric and chiral chemical potentials and the flavor to color ratio $\frac{N_f}{N_c}$. Our results are applicable to a generic class of chiral gauge theories that allow for a holographic description in the gravity approximation. We also determine the dynamical gluon corrections to the chiral vortical separation current in a particular example in the absence of external axial fields.


I. INTRODUCTION AND SUMMARY
In spite of a long history, dating back to [1], study of unusual transport phenomena induced by chiral anomalies flourished only recently with the hope of discovering such phenomena in real systems such as the quark gluon plasma produced in the heavy ion collisions at RHIC and LHC [2][3][4] 1 . The canonical examples of anomalous transport are the chiral magnetic and vortical effects (CME and CVE), that refer to generation of a macroscopic electric current as a result of an axial anomaly in the presence of magnetic field B or vorticity ω respectively. Whether evidence for the CME and CVE can indeed be found in the heavy ion collisions is still an open issue [6][7][8][9] but there exists strong evidence that anomalous transport finds experimental realization in Dirac and Weyl semimetals [10][11][12][13][14][15]. More theoretically, such anomalous transport phenomena can also be confirmed by lattice QCD studies [16,17].
One should distinguish between two different types of anomalies that contribute to anomalous transport in a chiral gauge theory: i) anomalies caused by external fields, and ii) anomalies caused by dynamical gauge fields. Both appear in the conservation equation for an abelian axial current J A as follows, where F V = dV and F A = dA are the field strengths of external vector and axial gauge fields, G is the field strength of the dynamical gauge fields in the theory, e.g. gluons, and R α βµν is the Riemann tensor of the background geometry. The anomaly coefficients a 1 , a 2 and a 4 are examples of the first type whereas a 3 is of the second type. All are known to be one loop exact [18].
The purpose of our paper is to explore the contribution of the second, dynamical, type of anomalies to anomalous transport in strongly coupled chiral gauge theories. We study the problem using the AdS/CFT correspondence [19][20][21]. In particular we calculate, using AdS/CFT, the transport coefficients that characterize the anomalous transport properties of the system, including the contribution from the dynamical type anomalies. 1 See the review [5] for a recent account.
Anomaly induced axial and vector currents are given in the following Ohm form: where B, B 5 and ω are the external vector and axial magnetic fields and vorticity respectively 2 . Non-vanishing values of the conductivities {σ VV , σ AV , σ VΩ , σ AΩ } lead to the Chiral Magnetic Effect (CME), the Chiral Separation Effect (CSE) , the Chiral Vortical Effect (CVE) and the Chiral Vortical Separation Effect (CVSE) respectively. In equilibrium these conductivities can be calculated from first principles via a Kubo type formula [22] This is the approach we take in this paper.
In the absence of dynamical contribution to the anomaly equation (1), i.e. when a 3 = 0, the anomalous transport coefficients in (4) are universally determined by the values of a 1 , a 2 and a 4 in a given QFT. For example in QCD coupled to external vector and axial gauge fields one finds for the chiral magnetic, chiral separation and chiral vortical conductivities for the covariant current 3 : where µ 5 and µ are the axial and electric chemical potentials and we set the electric charge e = 1. By "universality" we mean a) the same form (with only the coefficients vary depending 2 Even though there are no fundamental axial magnetic fields in Nature, including them as sources is instrumental in calculation of the σ AV conductivity. Furthermore, they appear in the effective description of Weyl semimetals. 3 The story of the chiral vortical separation conductivity is more involved. It involves a term quadratic in temperature, whose coefficient is only very recently understood to be protected by the global gravitational anomaly [23][24][25][26]. on anomaly coeffcients) in any chiral gauge theory and b) non-renormalization of these values by radiative corrections. Universality is suggested by several field theoretic arguments, such as the energy balance [27], Dirac index [28], thermodynamic potential [3] and derivative expansion of the effective action [29] that are all nicely summarized in [3]. Same expressions follow from non-field theoretic approaches, such as the hydrodynamics [30,31], holography [32][33][34][35] and effective field theory [36][37][38]. In particular, Son and Surowka [30] obtained the same results in the hydrodynamic regime with the additional assumption of a local entropy current with non-negative divergence. Eventually the universality of the CME, CVE and CSE coefficients in holography based on two-derivative gravity was established in [39] where they are linked to smooth near-horizon geometry of the corresponding black hole solutions 4 . This holographic demonstration was later extended to higher derivative gravity theories in [40].
In passing, we note that the actual values of anomalous conductivities depend on the choice of covariant current (BRST invariant) versus consistent current (satisfying the Wess-Zumino consistency conditions [41]). The two differ by the Chern-Simons current 5 . We calculate the one-point function of the consistent current in (4) in this paper, while most of the early literature on the subject, which we review above, involved the covariant currents.
We present the consistent current version of the universal values 6 for the CME, CSE and CVE conductivities in (5) which follow from (5) upon addition of the Chern-Simons current contribution.
These values are modified when dynamical gauge fields are included, i.e. when a 3 in the anomaly equation (1) is non-vanishing. Indeed it is known that once a 3 = 0 the anomalous conductivities will receive radiative corrections [23,43,44]. This is also clear from the fact that the universal values (5) one finds in the absence of dynamical gluons disagree with lattice QCD calculations, which clearly include such contributions [16,17]. Contribution of dynamical gauge fields to anomaly induced transport recently have been studied in thermal 4 We will make use of this smoothness condition in this paper as well. 5 See [5] for a careful recent presentation in the context of holography. 6 The CME conductivity should vanish in accordance to Bloch's theorem [42] for a system at equilibrium.
The CVE conductivity, in fact, escapes this theorem because a rotating system cannot be in equilibrium in a relativistic theory. The apparent vanishing of the consistent CVE conductivity might just be a choice of fluid frame. To check this one can calculate the energy current T tk . This is beyond the scope of our paper. We thank Karl Landsteiner for pointing this out.
field theory [23,43], effective field theory [44], and holography [45,46]. In [23] a two loop correction to the CVSE conductivity was found while in [44] it was argued that all conductivities should receive corrections once the gluons are turned on. In [45], the holographic dual of dynamical gauge field contribution is shown to be a bulk axion with a Stuckelberg coupling to the bulk gauge field corresponding to the axial current on the boundary. This idea was utilized in [46] to estimate such corrections to CME and CSE conductivities in the case of the conformal plasma.
In general the hydrodynamic approach becomes ambiguous in the presence of dynamical gauge fields since it is based on conservation of charge and the a 3 term in equation (1) violates this inherently 7 . However, there is no obstruction in 't Hooft's large color limit-which is the limit of interest in holography. In this limit, the thermal average of the topological charge G ∧ G in the (deconfined) plasma phase vanish exponentially as exp(−N c × const.) as the only possible contribution from instantons are suppressed as such. This is established both in the lattice studies [47] and in holography [48]. It is important to note that, the exponential suppression of topological charge does not imply exponential suppression of the associated corrections to anomalous transport; akin to the famous result of Witten [49] and However, as noted in [45] that the dynamical corrections should, in fact, be suppressed as N f /N c in the 't Hooft large color limit, just as the mass of the η meson. Thus we are forced to consider the Veneziano limit instead: In the holographic bulk picture, sending N f to infinity necessitates backreaction of the flavor branes to the background geometry, that complicates the gravitational system substantially [51][52][53].
In this paper, assuming such a gravitational background at finite x, we calculate the chiral magnetic, separation and vortical conductivities, including corrections from dynamical gauge fields, following the holographic prescription developed in [45,46]. In particular, we organize the holographic calculation in a series expansion in the anomalous dimension of the axial 7 Unlike the other terms in (1) that can be turned off by turning off external sources. current ∆ and obtain generic formulas-which can be applied to a generic background-for corrections to the consistent currents at first order in ∆. Our main results are that, at this order in ∆ the CME and CVE conductivities remain the same as in (6) and (5) whereas the CSE conductivity is modified, given by equation (87).
The structure of the paper is as follows. In the next two sections we set the stage for the holographic calculation of the anomaly induced conductivities in the presence of dynamical gauge fields. In particular, in section II we explain how the bulk axion is related to the dynamical gluons in the dual field theory, and in section III we determine the ansatz fo the background using fluid-gravity correspondence [54]. In section IV we develop the small ∆ expansion and solve the fluctuation equations to first order in ∆ to determine the anomalous conductivities in this order. Results of this section are applicable to a generic class of holographic theories. In section VI we provide three non-trivial examples and we discuss the restrictions of our results and give an outlook in section VII.

II. CONSTRUCTION OF THE HOLOGRAPHIC ACTION
To study anomaly induced transport we consider a plasma at finite temperature, electric and axial chemical potentials, and sourced by external axial and chemical magnetic fields.
To include the chiral vortical effect we also need to put the theory on a slightly curved metric that we denote by h µν . We want a plasma as close as possible to the deconfined phase of QCD. This necessitates, in the two-derivative holographic description, to include the following bulk fields corresponding to the marginal or relevant operators: the bulk metric g µν dual to the conserved energy-momentum tensor T µν , a bulk gauge field V dual to the vector current J V , an axial gauge field A the dual to the axial current J A , and a bulk dilaton field φ dual to the scalar glueball operator Tr (G 2 ). The latter breaks the scale invariance as its VeV generates a renormalization group invariant energy scale Λ QCD .
In addition to these fields, as explained in the Introduction, the dynamical gauge field contribution to the anomaly equation, i.e. the a 3 term in equation (1) is realized through a bulk axion field C 0 that couples to the topological charge operator Tr (G ∧ G) [45,46,55,56].
The boundary value of the bulk axion is proportional to the theta parameter: C 0 (x, r) → const. × θ + · · · as r → ∞. The precise form of the axion coupling 8 can be inferred 8 See [45,55,56] for more general possibilities in the brane realizations.
as follows. Consider a space-time dependent chiral transformation Ψ → exp(iα b (x)γ 5 )Ψ, in a gauge theory with massless Dirac fermions Ψ coupled to an axial gauge field A b through a term A b,µ J µ A . Let us, for the moment, ignore the anomalies caused by external sources, i.e. set a 1 = a 2 = a 4 = 0 in (1). Then, the transformation of the QFT lagrangian (that originates from the fermion path integral measure, i.e. the a 3 term) is equivalent to a shift δθ in of the topological charge term θN f 16π 2 Tr (G µν G ρσ ) µνρσ as we would obtain had we also transformed the theta parameter. In the dual gravitational theory, the conservation equation follows from a bulk gauge transformation A → A + dα. As the bulk gauge field and its transformation limit to the boundary values then see that the correct anomaly equation (1) we write the bulk theory in terms of the combination A − dC 0 /Q f and demand invariance under Here Q f is a parameter characterizing the strenght of the CP odd coupling between the axial current and the gluons, and the normalization of the bulk axion C 0 is such that its boundary value is equal to Finally, the external anomaly terms a 1 , a 2 and a 4 in (1) is realized by including a bulk Chern-Simons term of the form A∧ a Having explained the necessary ingredients for a generic holographic theory for a 3+1D chiral gauge theory with anomalies we can now write down the action as where 16πG N = M 3 p N 2 c with M p the five dimension Planck scale and S g , S f , S a , S CS denote the glue, flavor, axion and the Chern-Simons parts respectively 9 , while S GH is the Gibbons-Hawking term and S ct denotes the counterterm action 10 . Labels {V, A} stand for vector and axial bulk fields with their corresponding field strenghts F V /A . We denote the metric of the 5D geometry by G M N which is implicit in the action 11 . The axion enters the action S a in the gauge invariant combinationÃ Thus S a provides both a kinetic term for the axion and a Stückelberg mass term for the axial gauge field.
We allow for arbitrary potentials V (φ), Z V (φ) and Z A (φ) for the dilaton φ and its coupling to the vector and axial gauge fields. The potentials are normalized such that where r denotes the holographic radial coordinate and the AdS like boundary is located at Nc is the Veneziano parameter defined in (7). The scaling with x of each term in the action can be deduced from the original string action as discussed in [57].
The coupling parameter Q f is related to the Veneziano parameter via where m is a constant with mass dimension that in principle can be derived from the original string theory model. In this work we consider Q f as a tunable parameter. We develop a series expansion in Q f , hence assume Q f small. In field theory this corresponds to weak CP odd coupling of the gluons to the fermion. It is important to note that small Q f does not necessarily imply small x and we consider x to be a free parameter.
Finally, in order to fix the Chern-Simons coefficients κ, γ and λ in (13) we consider the variation of the axial gauge field δA = dα under which 9 We present S CS in terms of A instead ofÃ to make the fixing of the coefficients transparent but it has to be noted that once S CT is taken into account the whole action will be written in terms ofÃ [46] 10 See [46] for the explicit form of the counterterms. 11 We use uppercase latin letters for the 5D bulk, greek letters for the 4D boundary and latin indices for the 3D spatial boundary geometries.
Comparing (17) with (1) we find The equations of motion obtained from the variation of the action (9) read as follows.
For the dilaton and the axion fields we obtain For the gauge fields we have, Finally the Einstein's equations are where ω is the spin connetion and Σ L M N is defined as Here M N P QR denotes the 5D Levi-Civita tensor.
As explained in the Introduction our purpose is to calculate the one point function of the vector current and read off the anomalous conductivities. The holographic prescription for this one point function is where˜ µνρσ is the Levi-Civita symbol. This form of the one point function includes holographic renormalization [46].

III. BACKGROUND ANSATZ
A. Background at equilibrium We consider a general ansatz for the background obtained by the hydrodynamic setting we want to describe through the fluid-gravity correspondence [54]. First consider an equilibrium configuration characterized by a (boundary) background metric h µν , a four velocity u µ normalized as 12 u ν u µ h µν = −1, a chemical potential µ, an axial chemical potential 13 µ 5 , an equilibrium temperature T , an external vector and axial sourcesṽ andã. At this point all fields are taken to be constant.
To represent magnetic interactions the vector and axial sources are taken transverse to the direction of propagation, i.e. u µṽ µ = 0 and u µã µ = 0. It is possible to use the four velocity to decompose any tensor structure into a projection along the propagation and transverse to it. The transverse projector is which satisfies ∆ µν u µ = 0 and ∆ µρ ∆ ρν = ∆ µ ν . We can then write down the following ansatz for the metric, the gauge fields and the scalars 14 where u = u µ dx µ ,ṽ = v µ dx µ andã = a µ dx µ . The functions f (r) and g(r) are the blackening factors, the functions A t (r) and V t (r) determine the chemical potentials on the boundary 12 Boundary greek indices {µ, ν, ...} are raised and lowered by the metric h µν . 13 As the axial current is non-conserved µ 5 should be thought of as a coupling in the Hamiltonian rather than a true chemical potential [46]. 14 We do not use Eddington-Finkelstein coordinates unlike what is usually done in the fluid gravity correspondence. Our calculations will be at exactly zero frequency and regularity at the horizon is enough to determine the boundary conditions. theory as explained below and V(r), A(r), A (r) and A(r) characterize back reaction of the axial source to the vector and axial bulk gauge fields and the metric. This can be contrasted with the massless case where a constant gauge source does back react onto the rest of the background. Although we will only be interested in the linear response to the sources we will keep the non-linear terms-that naturally appear in the Ansatz-for consistency and generality, until the end of our calculations.
For (27) We also require a non-extremal horizon at r h and regularity at r h for all the other background functions. We read off the temperature from the horizon data as Boundary asymptotics of the gauge fields are (see e.g. [46]) where c i are constant one-forms. The power ∆ corresponds to the anomalous dimension of the axial current on the boundary, given in terms of the parameters in the action as From the powers in the normalizable modes in (34) where prime denotes a radial derivative. Using the regularity of the gauge fields at the equations (36) imply

B. Fluctuations
To study fluctuations around the equilibrium configuration (27) contributions will be disregarded.
The full hydrodynamic Ansatz is then given by 16 where the index I runs over all the magnetic sources and we fix the thermodynamic sources to constant values. The metric and gauge field solutions are ordered such that the boundary coordinates only appear through the sources while all the dependence on the radial coordinate 17 is in the functions f, g,Ã t , µ t , V t ,V, A, A, A , α, β, κ and γ. To ensure regularity of the Ricci scalar at the horizon the functions γ I and A should satisfy [39] γ I (r h ) = A(r h ) = 0 .
Finally, the magnetic field forms B V /A = B V /A µ dx µ and the vorticity form B ω = ω = ω µ dx µ are defined by In above we set the axial gauge V r = 0 and A r = 0 18 . 16 A more general Ansatz reads [32,58] where Π µ , L µ , V ⊥ µ and A ⊥ µ are all transverse to the fluid velocity. When these terms are evaluated on an equilibrium configuration only the terms shown in (39) remain. 17 Due to the mass term the blackening functions are corrected byã 2 . This plays no role in the linear response regime considered in this paper but might become relevant in another holographic context. 18 We also moved the functional dependence of A r is to a.

IV. SOLUTION TO FLUCTUATIONS AT SMALL ∆
In this section we solve the background equations of motion that we derived in section II on the Ansatz of section III B perturbatively in the parameter Q f . In particular we will be interested in the solution up to O(Q 2 f ). Assumption of small Q f corresponds to small anomalous dimension ∆, c.f. equation (35), hence, a weak contribution of the mixed gaugeglobal axial anomaly in internal Feynman diagrams. Recalling the derivative expansion in the hydrodynamic picture, that we denoted by in section III B, any field Φ formally admits a double expansion of the form We only consider contributions up to n = 1 in the hydrodynamic expansion, as this is sufficient for our purpose to compute the conductivities. Similarly, we keep only terms up to m = 2 in the Q f expansion. We note that, thi series actually start at m = 2 as the mass term in (12) first appears at this order. Furthermore, the two expansions are of different nature therefore they do not mix.
The axion a is assumed to be at least of O (1) while the functions {V, A, A, A , κ} are Scaling properties of these functions then allow the O (∂ 0 ) metric components G µν to split into a part g µν with the lowest term O(Q 0 f ) and a part G with the lowest term O Q 2 As we keep terms only up to Q 2 f , we can write the bulk metric and its inverse as with g µν and G µν given by Two comments are in order. The form of the inverse metric is exact at all orders in Q f in the linear response regime. Moreover, the Q f expansion of the anomalous dimension ∆ is given by ∆ =

A. The axion
The equation of motion (20) for the axion can be expanded as which, in turn can easily be integrated as, Here the non-normalizable mode a 0 (x) is independent of the radial coordinate r, and it corresponds to a space-time dependent θ-term in QCD. More precisely, from equation (8), (14) and the discussion in between, we have the constant (x-independent) part of a 0 equals N f θ/4π 2 a 3 . To derive (51) from (50) we used g(r h ) = 0, and The first one is just incompressibility of the fluid at linear order and the second one is best understood in the rest frame where it means that there are no electric components in the external gauge fieldã.
The normalizable mode given by the integral expression in (51) is identified with the expectation value Tr (G ∧ G) . This implies that Tr (G ∧ G) ∼ O (∂). However, as discussed in [59], validity of the hydrodynamic expansion requires that the O(∂) contribution to Tr (G ∧ G) vanish, to prevent any contribution of gluons to entropy production. This means that we need to require While the first condition is again natural since we require the sourceã be magnetic-like hence divergenceless, the second one imposes a non-trivial restriction on the external sources we consider in this paper 19 . This means, That is, consistency with positive local entropy production requires axion to be constant up to second order in derivatives.

B. The vector field
The vector field equation of motion (21) at the zeroth order in the derivative expansion which results in translating in an expression for the conserved charge Q, and a relationship between the background functions A and V To study the vector equation at O (∂) we need the CS form F V ∧ F A and the membrane currentJ ν = √ −GxZ V F V,rν up to this order in the derivative expansion. We find The membrane current can schematically be expressed as where we defined the functionsJ 1I ,J 2I andJ 3Ī What is relevant for our calculation below is thatJ ν (r h ) = xQu ν which follows from the boundary conditions γ I (r h ) = A(r h ) = g(r h ) = f (r h ) = 0. Using (57), (58) and (59) where˜ µναβ denotes the Levi-Civita symbol. The bulk version of the consistent vector current (25) is obtained by shiftingJ with the Chern-Simons current as 20 20J ν itself gives the covariant current. and extracting its boundary value From (63) the holographic vector current is found to be given bỹ The holographic current (66) contrary to the membrane current in (63) has a well defined r → ∞ limit. Equation (66) together with (56) constitute the main results of this section.

C. The axial field
The Maxwell equation for the axial gauge field at O (∂ 0 ) reads This equation decouples into the equations for the background and a coupled equation for A and A that can, alternatively, be written in terms of V as Equations (68) and (69) are the main results that is used below.

D. Einstein equations
At zeroth order the radial and spatial parts of Einstein's equations do no mix, giving rise to four sets of independent equations. As detailed in Appendix A, rhe radial projection yields where R rr is shown in (A9) and ρ is defined as On the other hand, the projection along ∆ µν reads From this equation it follows that whereρ is some function. Similarly, the projection along u µ u ν is given by Equations (72) and (74) can be combined as whereρ is some function. Equation (75) implies that up to O (Q f ), or alternatively in the linear response regime, there exists a conserved gravitational charge. We note that for the set of equations (72)-(74) to be consistent, both f and g should receive an O ã 2 Q 2 f correction. This will have no consequence in the analysis of this paper as we are only interested in linear response. A similar analysis for the projection along u µãν also yields a total derivative It is possible to integrate equation (76) twice by making use of equations (75) and (69) together with their universal near horizon behavior, all in all, resulting in the following simpler formula which is the main result of this analysis. This equation, together with (69) yields a solution to A and V up to O Q 2 f . We emphasize that both (77) and (69) become exact in the linear response regime.

E. The A − V subsystem
The back reaction functions A and V, or equivalently A, satisfy a closed system of second order differential equations given by (69) and (77). This system can be solved by noting that the right hand side of (69) has no dependence on the back reaction functions 21 . Combining (69) and (77) we find the solutions as where we used the boundary conditions (31) and introduced the following functions To evaluate A(r h ) we use the boundary condition and obtain Note that the limit is finite as the apparent log divergence is exactly cancelled by the divergent part of L(r).

A. General results
To find the linear response of the one point function it will be convenient to rewrite (66) in the reference frame of the solution at equilibrium: We are interested in calculating the vector current Using (84) We now read off the anomalous conductivities from (85) as using where we used (18). Equations (86) and (87) are the main results of our paper. As a consistency check, we observe that our expressions yield the known results (6) in the limit taking Q f → 0. It is important to emphasize that to arrive to this solution only the universality of the horizon was used and as such these results are valid in a generic class of gravitational theories described by the generic action (9). Finally, we note that higher order corrections in Q f can easily be computed by solving (77) and (69) in a series expansion.

VI. EXAMPLES: CHIRAL CHARGE SEPARATION
In this section we provide three examples, two of which concern the chiral separation effect using our master formula (87) and one calculating the chiral vortical separation effect-which we did not discuss in detail above-in a specific simple setting.

A. Reissner-Nordstrom Blackhole Background
The double charged Reissner-Nordstrom blackhole solution, with vanishing dilaton φ = 0hence dilaton potential equals the cosmological constant V = −12 (with AdS lenght = 1) -can easily be found by using the constants of motion (56), (68) and (75) with Q f = 0: where the mass M , the effective charge squareQ 2 , and the horizon radius r ± are given by The temperature of this solution is For this background the functions D(r), L(r) and H(r) defined in section IV E read We express (99) in terms of physical parameters as We observe that the CSE conductivity can be schematically rewritten as where σ U CSE is the universal value of the chiral separation conductivity in the absence of dynamical gauge fields (6) and we replaced the gluon-axial current coupling Q f by the anomalous dimension ∆. We plot the correctionσ in figure 1 as a function ofμ for fixedμ 5 and as a function ofμ 5 for fixedμ. We observe in figure 1 thatσ is symmetric, positive definite, and bounded from both above and below. It is also a decreasingly monotonic function ofμ 2 , i.e. for fixedμ 5 it converges to a global minimum atμ → ±∞ and a global maximum atμ = 0. As a function ofμ 5 for fixedμ it attains a global maximum at some value±µ 5c , a global minimum at µ 5 = 0 and a local minimum at µ 5 → ∞. All in all we find thatσ is bounded aŝ As the upper bound is substantial, in order to obey our assumption of a perturbative expansion in ∆ we see that ∆ may have to be very small. Otherwise the calculation is invalid for certain values of µ and µ 5 that yield largeσ. For larger O(1) values of ∆, a nonperturbative solution of the A − V system of equations in section IV E will be needed, which can be obtained numerically for a given background.

B. Full analytic solution in the probe limit
A non perturbative solution of the A − V system can be obtained in the probe limit, that is, ignoring the metric fluctuations. This amounts to considering (69) as the only relevant equation in the system, i.e. setting V = 0 and disregarding Einstein's equations, namely ignoring equation (77). We consider the AdS-RN blackhole as the fixed background: In this limit the CSE conductivity becomes where the position of the horizon can be set to r h = 1. A satisfies Regularity at the horizon together with the asymptotic behavior lim r→∞ A = r r h ∆ fixes the solution as where functions F 1 and F 2 are defined as and the functions D(∆) and D 1 (∆) read The CSE conductivity is then given by The integrals in (112) We define a covariant bulk axial currentJ ν A as from which we obtain the renormalized axial one point function [46] 16πG The solution to (113) in a Q f expansion is given by with Using the solution (116) together with (115) we finally obtain the CVSE conductivity as We observe that the T 2 contribution to the CVSE conductivity also receives correction from dynamical gluons in strongly coupled N = 4 sYM. Just like the CSE, this correction is positive.

VII. DISCUSSION
We developed a semi-analytic procedure to calculate the chiral magnetic, chiral separation, and chiral vortical conductivities in the presence of dynamical gauge fields in strongly interacting chiral gauge theories in the Veneziano large color, large flavor limit using the holographic correspondence. We find that, while the CME and CVE conductivities do not receive corrections, the CSE conductivity does, and it is given by two background functions {A, V} that satisfy a closed and coupled system of differential equations. We solved this system of equations in a perturbative expansion in the anomalous dimension of the axial current ∆ and obtained an analytic expression at first order.
Quite generally -for theories that can be described by two-derivative gravity -we find that the correction to the CSE conductivity due to the dynamical gauge fields are positive definite. This is to be contrasted with negative corrections obtained in some lattice calculations [16,17]. There is no clash in these results however, since our calculation is strictly valid in the Veneziano limit, at large 't Hooft coupling λ and in a class of gauge theories that are related to but not the same as QCD. It will be interesting to extend our results beyond these limitations, in particular to determine the 1/λ corrections. Whereas existing holographic studies [39,40] indicate no deviation from the universal values, i.e. no 1/λ corrections in the absence of dynamical gauge fields, there is no analogous result when such dynamical gauge fields are taken into account. This is an open problem.
The example we provide in section VI A show that the validity of the small ∆ expansion should be checked carefully. There is a range of parameters µ and µ 5 where the perturbative expansion makes sense only for very small choices of ∆. This prompts us to look for alternatives to the small ∆, or equivalently small Q f expansion. One such powerful approach would solving the A − V system of differential equations. This provides a straightforward way to obtain, at least numerically, the non perturbative solutions. We provided an example of such a calculation in the case of AdS blackhole with no backreaction in section VI B. This particular example was first discussed in [46] our analytic results are consistent with the numerical results in this paper. We should stress that the method we use is very different than the method of [46]. Whereas [46] solves the second order fluctuation equations numerically and obtain the conductivity from the Kubo formula, we solve the second order A − V ODE system, which seems to provide a simpler way to obtain the conductivity with full back reaction either numerically or analytically.
Finally, we should remark that we treated the anomalous dimension of the axial current ∆ as a (small) tuneable parameter in our model. In reality, ∆ itself should be determined in terms of the anomaly coefficients [60]. Whereas, this can be achieved in perturbative QFT, it is not at all clear how to proceed at strong coupling. We suspect however a holographic relation between ∆ and the anomaly coefficients might exist. We plan to return this problem in the future.