Chiral Magnetic Effect and Three-point Function from AdS/CFT Correspondence
CChiral Magnetic Effect and Three-point Function from AdS/CFT Correspondence
Lei Yin , , De-fu Hou , and Hai-cang Ren , Guangdong Provincial Key Laboratory of Nuclear Science,Institute of Quantum Matter, South China Normal University, Guangzhou 510006, China Guangdong-Hong Kong Joint Laboratory of Quantum Matter,Southern Nuclear Science Computing Center, South China Normal University, Guangzhou 510006, China Institute of Particle Physics and Key Laboratory of Quark and Lepton Physics (MOE),Huazhong Normal University, Wuhan 430079, China and Physics Department, The Rockefeller University, 1230 York Avenue, New York,10021-6399, USA (Dated: February 10, 2021)The chiral magnetic effect for space-time dependent chiral imbalance and magnetic field is exploredfrom AdS/CFT correspondence which relates the AVV three-point function of the N = 4 superYang-Mills at strong coupling on the boundary to the Einstein-Maxwell-Chern-Simons theory inthe bulk. A formulation of the AVV three point function in terms of Heun functions is developedby an iterative solution of the nonlinear equations of motion in Schwarzschild-AdS background,which is free from UV/IR divergence. The low-momentum expansion reveals non-local response ofthe vector current to the chiral imbalance beyond the hydrodynamic approximation and replicatesthe subtlety of the infrared limit discovered in field theoretic approach in weak coupling. Thephenomenological implications to the chiral magnetic effect in the context of relativistic heavy ioncollisions are discussed qualitatively. a [email protected] b Corresponding author: [email protected] c Corresponding author: [email protected] a r X i v : . [ h e p - t h ] F e b I. INTRODUCTION AND SUMMARY
A chiral matter subject to an external magnetic field and/or under rotation exhibits many interesting transportproperties driven by the axial anomaly. Among them are the chiral magnetic effect and chiral vortical effect [1–4].Searching the evidences of these novel anomalous transport phenomena has grown into an active research area for thepast decade with the scope extending from the quark-gluon plasma (QGP) created in relativistic heavy ion collision[5–7] to the Weyl semi-metals [8–11] and involved both theoretician and experimentalist. This theoretical work focuseson the chiral magnetic effect in QGP.The chiral imbalance in QGP is triggered by the topological excitation of QCD and the external magnetic fieldis produced via the off-central collision of heavy ions. The resulting chiral anomaly is manifested by the anomalousWard identity of the axial-vector current J µ A in the presence of vector and axial vector field strengths ( F V ) µν and( F A ) µν ∂ µ J µ A ( x ) = C (cid:15) µνρσ (cid:20) ( F V ) µν ( F V ) ρσ + 13 ( F A ) µν ( F A ) ρσ (cid:21) , (1)and the chiral magnetic current for constant axial chemical potential µ A and magnetic field B takes the simple form J = 8 C µ A B , (2)with the non-renormalization anomaly coefficient C . Here, the constant axial chemical potential serves the Lagrangemultiplier of a grand canonical ensemble of macroscopic chirality. The chiral magnetic effect is thereby a direct probeof the topological structure of QCD, more important than other anomalous transport phenomena in this sense.Theoretically, the chiral magnetic effect has been investigated in different approaches, including the Green functionformalism, [3, 12], kinetic theories [13] and holography [14–17]. Most of these works focus on the situation with a(nearly) constant µ A . Hydrodynamic simulations have also been developed due to RHIC, based on the assumptionthat a net axial charge density is generated in the initial stage of collisions and its characteristic time of variation ismuch longer than the relaxation time to thermal equilibrium [18–20].The chiral magnetic response in the non-equilibrium case, in particular for a spacetime-dependent chiral imbalanceand magnetic field, turns out to be both subtle and more realistic for heavy-ion-collisions. The initial axial charge isgenerally expected to be inhomogeneous across the fireball and furthermore necessarily evolves in time due to randomgluonic topological transitions during the fireball evolution. The magnetic field generated during the collision is alsotransient. The spatial variation length scale and the time evolution scale are not necessarily very large as comparedwith the thermal scale of the medium. Exploring the dynamics of the chiral magnetic effect under inhomogeneousand non-static magnetic field and chiral imbalance is the main target of the present work. To simulate the strongcoupling feature of the QGP created in RHIC, the AdS/CFT duality is employed with the N = 4 super-Yang-Millsat large number of colors and large ‘t Hooft coupling and its global U (1) vector current as the proxies of QCD indeconfinement phase and electric current.In the presence of an axial chemical potential, an external vector potential V µ ( q ) and an axial vector potential A µ ( q ), the response current in 4-momentum representation an be expanded according to the powers of the externalfields, i. e. J µ ( q ) = Π µν ( q ) V ν ( q ) + J µ CME ( q ) + ... (3)where the first term, linear in external fields, represents the ordinary polarization current and the second term,quadratic in external fields, gives rise to the chiral magnetic effect to be discussed in this work. The chiral magneticcurrent J µ CME ( q ) can be divided into two terms, i.e. J µ CME ( q ) = J µ ( q ) + J µ AVV ( q ) (4)The first term consists of only spatial component J ( q ) = µ A K ( q ) B ( q ) (5)that generalizes Eqn.(2) to arbitrary spacetime-dependent magnetic field at a constant axial chemical potential withthe explicit form of the kernel K ( q ) given by (102) in section IV. The second term J µ AVV ( q ) = (cid:90) d q (2 π ) d q (2 π ) (2 π ) δ ( q + q − q )Λ µνρ ( q , q ) A ρ ( q ) V ν ( q ) + ... (6)brings in the spacetime dependence of the chiral imbalance with the integration kernel Λ µνρ ( q , q ) related to the AVVthree point function with q = ( ω , q ) and q = ( ω , q ) the 4-momenta carried by V ν and A ν , respectively. In termsof the standard notation of the AVV three point function∆ ρµν ( k , k ) = (cid:104) J ρA ( − k − k ) J µV ( k ) J νV ( k ) (cid:105) , (7)with J V and J A the vector and axial-vector currents operatorsΛ µνρ ( q , q ) = ∆ ρµν ( − q − q , q ) . (8) J ρA (− k − k ) J μV ( k ) J νV ( k ) FIG. 1. The triangle diagram of AVV three-point function, where the shaded center area implies strong-coupling, contrastingwith one-loop weakly-coupled counterpart.
Because of the anomaly, µ A cannot be not identified with A [21]. Physically, µ A is conjugate to a conserved globalaxial-charge and is thereby a constant. The spacetime variation of the chiral imbalance is attributed to A . Todistinguish their roles mathematically, we impose the condition A ( q ) (cid:12)(cid:12)(cid:12)(cid:12) q =0 = 0 . (9)From holographic perspective, both µ A and A ( q ) pertain to the temporal component of the axial vector potential in the bulk with its value at the horizon equal to µ A and its value on the boundary equal to A ( q ) [14]. Unlike thehydrodynamic approach where the space-time variation of the chiral imbalance is treated as higher orders and thereby | µ A | (cid:29) | A ( q ) | , we consider | µ A | ∼ | A ( q ) | throughout this work in order to investigate the non-equilibrium of chiralimbalance. Taking into account the randomness of the topological transitions, A ( q ) may not be continuous in q , inparticular, A ( q ) may not be small for a small but nonzero q .To calculated the chiral magnetic current in the super-Yang-Mills via AdS/CFT correspondence, we start withEinstein-Maxwell-Chern-Simons action in the AdS bulk and solve the classical equations of motion in the backgroundof a Schwarzschild black hole up to the first order of non-linearity in external gauge potentials. With equal order ofmagnitude of µ A and A µ , the metric fluctuation does not contribute to the terms displayed in the expansion (6), whichimplies that the temperature fluctuation can be ignored in evaluating the current to the displayed order. The weakexternal field approximation employed here corresponds to the physical condition T (cid:29) µ A . With this simplifications,we are able to develop an analytic formulation of Λ µj ( q , q ) for arbitrary q and q in terms of two Heun functions,one of which reduces to a hypergeometric function for a homogeneous magnetic field.For low momenta q ( | q | (cid:28) T and | ω | (cid:28) T ), the kernel K ( q ) in (5) approaches to a constant K ( q ) = 8 C (10)recovering the prototype CME formula (2) and the limit q → µνρ ( q , q ) to the leading order in low momenta q and q ( | q , | (cid:28) T and | ω , | (cid:28) T )Λ ij ( q , q ) = 16 C πT ω πT iω − | q | (cid:20) (cid:15) ikj q k + (cid:15) klj q k q l q i πT iω − | q | (cid:21) (11)and Λ j ( q , q ) = − i C π T ω (2 πT iω − | q | )(2 πT iω − | q | ) (cid:15) jkl q k q l , (12)with q = q + q . Eqn. (10). The diffusion denominators in (11) and (12) imply a non-local response of the currentto the chiral imbalance proxied by A . In particular, we find the nontrivial infrared limits:lim ω → lim q → Λ ij ( q , q ) = − i C (cid:15) ikj q k ; (13)lim q → lim ω → Λ ij ( q , q ) = 0 . (14)Consequently, with A ∼ µ A , the CME signal for | q | (cid:28) (cid:112) T | ω | (cid:28) T and | ω | (cid:28) | q | /T (cid:28) T can be quite differentbecause of the AVV contribution. A simultaneous derivative expansion with respect to both V µ and A no longerexists.From field theoretic perspectives[12, 22, 23], the limit (13) follows from the anomalous Ward identity which is robustbecause of the non-renormalization theorem [24, 25] of the chiral anomaly and the limit (14) can be deduced fromColeman-Hill theorem [26]. The expression for arbitrary q and q is subject to higher order corrections, hence theinterpolating formulas (11) and (12) pertains to the strong coupling limit. Moreover, the underlying assumption of theColeman-Hill theorem, the absence of infrared singularity in the zero momentum limit, is verified by our formulation.In addition, we are able to prove UV and IR convergence of the kernel Λ ij ( q , q ) and Λ j ( q , q ). The absence ofUV divergence reflects the finiteness of the underlying dynamics of the N = 4 super-Yang-Mills.This paper is organized as follows. In the next section II, we lay out the Einstein-Maxwell-Chern-Simons action inthe bulk along with the equations of motion and link the solutions of EOM to the vector and axial-vector current on theboundary. The methodology of solving EOM analytically up to the required order is discussed in section III. The mainresults on the chiral magnetic current in the presence of an constant axial chemical potential, a spacetime-dependentmagnetic field and a spacetime-dependent A are presented in sectionIV and the proof of UV/IR convergence ispresented in section V. Section VI conclude the paper. Some technical details behind the solutions of EOM aredeferred to appendices A-D. II. THE EINSTEIN-MAXWELL-CHERN-SIMONS ACTION IN ASYMPTOTIC
AdS BACKGROUND
According to AdS/CFT duality [27, 28], the N = 4 super-Yang-Mills theory at large N c (number of colors)and strong ‘t Hooft in a 3+1 dimensional spacetime corresponds to the classical supergravity limit of the type IIBsuperstring theory in asymptotic AdS with the 3+1 dimensions as its boundary. Consequently, the vector and axial-vector current correlators as well as the chiral anomalies of the super Yang-Mills can be described holographicallywith the following classical Einstein-Maxwell-Chern-Simons action in an asymptotic AdS bulk [14, 16, 29], S = S EH + S MCS + S c.t. . (15)where S EH is the Hilbert-Einstein action S EH = κ EH (cid:90) d x √− g ( R − R and the negative cosmological constant Λ = − L . S MCS is the Maxwell-Chern-Simonsaction and S c.t. is the holographic counter-terms to remove UV divergences caused by various holographic correlationfunctions. The strongly-coupled gauge theory as well as the holographic counter terms reside on the asymptoticAdS-boundary. As will be shown in section V, the AVV three-point function is free from UV divergence and therebydoes not need counter terms. S c.t. is used to cancel the logarithmic divergence in two-point Green’s functions and isnot relevant to us. In terms of the left-hand and right-hand vector potentials A L and A R , the Maxwell-Chern-Simonsaction reads S MCS = κ M (cid:90) d x √− g (cid:20) −
14 (F L ) MN (F L ) MN −
14 (F R ) MN (F R ) MN + κ CS (cid:15) MNOPQ κ M √− g (cid:0) (A L ) M (F L ) NO (F L ) PQ − (A R ) M (F R ) NO (F R ) PQ (cid:1)(cid:21) + S B . (17) The well-known holographic two-point function needs such counter-term to cut-off its UV divergence. In this work V, we will prove thefiniteness of AVV correlation, hence counter-terms have nothing to do for CME three-point function. where
M, N, O, P, Q refers to the indexes of 4+1 dimensional spacetime with the Levi-Civita symbol ε MNOPQ nor-malised according to ε = 1, the field strengths (F L ) MN = ∂ M (A L ) N − ∂ N (A L ) M , (F L ) MN = ∂ M (A L ) N − ∂ N (A L ) M ,and S B is a boundary term to be specified later. The bulk part of this action is invariant under a U L (1) × U R (1) gaugetransformation, (A L ) M → (A L ) M + ∂ M ( φ L ) , (A R ) M → (A R ) M + ∂ M ( φ R ) . (18)Here, the gauge potentials A L and A R stem from the U (1) subgroups of the global U (4) symmetry of the N = 4super-Yang-Mills theory on the boundary. Being tied to a global symmetry on the boundary, the A L and A R do notcontribute the internal lines of the Feynman diagrams of the super Yang-Mills theory and is only employed in itsgravity dual to generate the current correlations and thereby the anomaly-induced transport coefficients of the superYang-Mills plasma.To describe the conserved vector current and anomalous axial vector current on the boundary, it is convenient toexpress the action in terms of the vector and axial vector gauge potentials viaA = 1 √ L − A R ) , V = 1 √ L + A R ) . (19)Integrating by part to remove V outside F V in the Chern-Simon’s term and choose S B = (cid:90) d x κ CS √ ∇ M (cid:2) ε MNOPQ A N V O F V PQ (cid:3) . (20)to cancel the boundary term incurred, we end up with S MCS = κ M (cid:90) d X √− g (cid:20) −
14 F −
14 F A + κ CS √ κ M √− g ε MNOPQ (cid:0) M F V NO F V PQ + A M F ANO F APQ (cid:1)(cid:21) , (21)The Maxwell-Chern-Simons action is invariant under an arbitrary U V (1) gauge transformation V M → V M + ∂ M φ V , butis invariant only under an axial U A (1) transformation A M → A M + ∂ M φ A with φ A = 0 on the boundary. Consequently,the strongly-coupled gauge theory on the boundary maintains only the U V (1) invariance with the parameters κ M and κ CS determined by the coefficient of axial anomaly. S B of (20) plays the role of the Bardeen term. The vector currentassociate to U V (1) is the analog of the electric current underlying CME with the corresponding charge referred to asthe R-charge in the literature of super Yang-Mills.The equation of motion corresponding to (21) can be readily obtained via variational principle, i.e.Vector gauge field : ∇ N (cid:2) (F V ) NM √− g (cid:3) = − √ κ CS κ M · (cid:15) MNOPQ (F A ) NO (F V ) PQ (22)Axial gauge field : ∇ N (cid:2) (F A ) NM √− g (cid:3) = − √ κ CS κ M · (cid:15) MNOPQ (cid:20) (F V ) NO (F V ) PQ + (F A ) NO (F A ) PQ (cid:21) , (23)Metric field : R MN − Rg MN − Λ g MN = − κ M κ EH T MN , (24)with the scaled stress tensor T MN = 2 κ M δS MCS δg MN = (F V ) LM (F V ) NL − g MN (F V ) KL (F V ) KL + (F A ) LM (F A ) NL − g MN (F A ) KL (F A ) KL (25)In the natural units, the mass dimensions of the coupling constants in chiral action (21) are dim κ = 1 and dim κ CS = 0, hence we get the same mass dimension of gauge fields in D = 4 + 1 dimensional spacetime as we did in D = 3 + 1 dimensional QFT: dim A M = dim V M = 1 in coordinate representation.It follows from the dictionary of AdS/CFT that the quantum effective action in the presence of the vector and axialvector gauge potentials V µ , A µ , together with 3+1 dimensional metric g µν corresponds to the classical action (15)evaluated with the solutions of EOM with the AdS-boundary values V µ , A µ and g µν of respective fields. Taking thefunctional derivatives with respect to boundary values V µ and A µ , we derive the holographic formulas of the vectorand axial-vector currents: J µ V ( x ) ≡ δS MCS δ V µ (cid:12)(cid:12)(cid:12)(cid:12) AdS-boundary = (cid:20) − κ M (F V ) µ √− g + 3 κ CS √ (cid:15) µνρσ A ν (F V ) ρσ (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) AdS-boundary (26) J µ A ( x ) ≡ δS MCS δ A µ (cid:12)(cid:12)(cid:12)(cid:12) AdS-boundary = (cid:20) − κ M (F A ) µ √− g + κ CS √ (cid:15) µνρσ A ν (F A ) ρσ (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) AdS-boundary (27)where Greek indexes refer to the 3+1 dimensional spacetime on the boundary. It follows from the EOM (22) and (23)for M = 5 that the vector current is conservative, while the divergence of the axial current acquires an anomaly, i.e., ∂ µ J µ V ( x ) = 0 (28) ∂ µ J µ A ( x ) = C (cid:15) µνρσ (cid:20) F V ) µν ( F V ) ρσ + ( F A ) µν ( F A ) ρσ (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) AdS-boundary , (29)where the anomaly coefficient C is related to the Chern-Simons coupling via C = 3 κ CS √ s = ¯ g MN dx M dx N = ( πLT ) u (cid:32) − f ( u ) d t + (cid:88) i =1 d( x i ) (cid:33) + 14 u f ( u ) d u , f ( u ) = 1 − u , (31)and ¯V M = ¯A M = 0 , (32)where u = 0 is the AdS-boundary and L the AdS radius. he Hawking temperature T of the horizon u = 1 correspondsto the temperature of thermal state of boundary field theory. Notice that for the N = 4 SU ( N c ) super Yang-Mills atlarge N c and strong ‘t Hooft coupling, both κ M and κ M scales with N c as N c → ∞ [29, 31]. The coefficient on RHSof (24), κ M /κ EH = O (1) and is thereby not tunable. Introducing the fluctuations from the background via g µν = ¯ g µν + h µν , A µ = ¯A µ + A µ , V µ = ¯V µ + V µ , (33)under the radial gauge condition V = A = g (cid:63) = 0, EOM (22), (23) and (24) become a set of nonlinear equationsfor the fluctuations. Substituting the solutions into (26) and (27), we obtain the expansion of the vector and axial-vector currents according to the boundary values of the fluctuating fields, which is the holographic version of (6). Itis interesting to notice the following power structure of the respective equations of (22)-(24) in AdS-Schwarzschildgeometry: Vector gauge field : O ( V ) + O ( h V ) = O ( AV ) (34)Axial gauge field : O ( A ) + O ( h A ) = O ( A ) + O ( V ) (35)Metric field : O ( h ) = O ( V ) + O ( A ) . (36)It follows that to the metric fluctuations is quadratic in the fluctuations of gauge fields and therefore do not contributeto the quadratic order of Eqns.(22) and (23) in gauge field fluctuations and thereby do not contribute to the criticalterm of the chiral magnetic current, i.e. O ( AV ) term of the vector current (26). Specifically, all we need to do is tosolve the Maxwell-Chern-Simons Eqns. (22) and (23) with ( V, A ) replaced by the fluctuations and the metric fixed tothe AdS-Schwarzschild background (31), i.e. ∂ N (cid:2) ¯ g NP ¯ g MQ ( F V ) PQ √− ¯ g (cid:3) = − √ κ CS κ M · (cid:15) MNOPQ ( F A ) NO ( F V ) PQ (37) ∂ N (cid:2) ¯ g NP ¯ g MQ ( F A ) PQ √− ¯ g (cid:3) = − √ κ CS κ M · (cid:15) MNOPQ (cid:20) ( F V ) NO ( F V ) PQ + ( F A ) NO ( F A ) PQ (cid:21) , (38)subject to the AdS-boundary conditions V ( x µ ; u ) (cid:12)(cid:12) u → ≡ (cid:126) V ( x µ ; u ) (cid:12)(cid:12) u → (cid:54) = 0 that leads to a spacetime-dependent magnetic field B ( x µ ) ;and A ( x µ ; u ) (cid:12)(cid:12) u → (cid:54) = 0 that is the axialchemical potential A ( x µ ) in the holographic strongly-coupled theory. Substituting the solution to the currents (26)and (27), the term of the spatial component of the vector current is linear in B ( x µ ) and A ( x µ ) respectively. Itscoefficient corresponding to the (0 ij ) component of the AVV triangle diagram in field theory to all orders of the N = 4super-Yang-Mills coupling to be explored analytically in this work. The EMCS equations become a set of couplednonlinear equations with respect to the fluctuating fields and can be solved iteratively. The order of magnitude sortingdescribed in (34), (40) and (41) applies to weak magnetic field, compared to the metric field, and chiral imbalance ina thermal bath of high temperature.In contrast, an alternative background geometric that corresponds to a nonzero temperature and a nonzero chemicalpotential is the AdS-Reissner–Nordstr¨om geometry, which is accompanied by a nonzero background gauge potential, V (cid:54) = 0. The power structure of the EMCS equations becomeVector gauge field : O ( V ) + O ( h ) + O ( h V ) = O ( A ) + O ( AV ) (39)Axial gauge field : O ( A ) + O ( h A ) = O ( h ) + O ( A ) + O ( V ) (40)Metric field : O ( h ) = O ( V ) + O ( V ) + O ( A ) , (41)Consequently, once the background chemical potential is introduced, the metric fluctuations can’t be decoupled evenat linear order of fluctuation in the vector and axial vector gauge fields, which complicates the analytic calculationsfor the three-point functions ∆ ρµν .On the other hand, the expression of the AVV three point function Λ ij ( q , q ) with q = ( ω ,
0) can be deducedfrom the anomalous Ward identity (29) and is thereby robust to all orders of metric fluctuations. The momentumrepresentation of (29) implies that i ( k + k ) µ ∆ µρλ ( k , k ) = 8 C (cid:15) µνρλ k µ k µ (42)with k and k the 4-momenta of the boundary value of the vector gauge potential. It follows from (8) that i q µ Λ µρλ ( q , q ) = 8 C (cid:15) µνρλ q µ q ν (43)For the special q assumed above, it follows readily thatΛ ij ( q , q ) = − i C (cid:15) ikj q k (44)Though the constraint (9) is imposed for the axial vector potential on the boundary, Eq. (44) serves an asymptoticform of J µAV V for a nearly homogeneous A there. III. THE SOLUTION ALGORITHM
For the chiral magnetic effect under an arbitrarily space-time dependent magnetic field and chiral imbalance, weneed the electric current (26) in terms of the boundary values. V µ ( x,
0) = (0 , V ( x )) ; A µ ( x,
0) = ( A ( x ) , ) (45)where ∇ · V = 0 and we have adapted the radial gauge condition V u = A u = 0. The second term on RHS of (26)is already explicit in terms of (45). The explicit expression of the first term on RHS of (26) will be derived in thissection. In what follows, we shall solve the nonlinear Maxwell-Chern-Simons equation (37) and (38) iteratively to findout F µ = ∂ V µ∂u in terms of the boundary values and one iteration serves our purpose.Using Chern-Simons coupling κ CS to track the order of iteration, we have V = V + O ( κ CS ) = V + O ( κ ) (46)and A = A + O ( κ CS ) = A + O ( κ ) (47)where the zeroth order solutions V and A solve the linear equations: ∂ N (cid:2) ¯ g NP ¯ g MQ ( F V ) PQ √− ¯ g (cid:3) = 0 ; (48) ∂ N (cid:2) ¯ g NP ¯ g MQ ( F A ) PQ √− ¯ g (cid:3) = 0 , (49)and the first iteration gives rise to V and A . The differential equations satisfied by V and A are obtained by replacing V and A on RHS of (37) and (38) by the zeroth order solutions V and A , i. e. ∂ N (cid:2) ¯ g NP ¯ g MQ ( F V ) PQ √− ¯ g (cid:3) = − √ κ CS κ M · (cid:15) MNOPQ ( F A ) NO ( F V ) PQ , (50) ∂ N (cid:2) ¯ g NP ¯ g MQ ( F A ) PQ √− ¯ g (cid:3) = − √ κ CS κ M · (cid:15) MNOPQ (cid:20) ( F V ) NO ( F V ) PQ + ( F A ) NO ( F A ) PQ (cid:21) , (51)with F VMN = ∂ M V N − ∂ N V M and F AMN = ∂ M A N − ∂ N A M and are linear partial differential equations with inhomoge-neous terms.Because of the translation invariance with respect to boundary coordinates x µ , it is convenient to introduce themomentum representation via V µ ( x ; u ) = (cid:90) d q (2 π ) e i qx V µ ( q | u ) , A µ ( x ; u ) = (cid:90) d q (2 π ) e i qx A µ ( q | u ) , (52)and the boundary condition (45) becomes V µ ( q |
0) = (0 , V ( q )) A µ ( q |
0) = ( A ( q ) , ) (53)with q · V ( q |
0) = 0 and is satisfied by {V , A} to the leading order and { V , A } to the order O ( κ CS ). The details of theFourier transformation to the momentum representation is described in Appendix A. A. Zeroth Order
Carrying out the Fourier transformation prescribed in Appendix A for {V , A} , the leading order equation (48) and(49) reduce to w A (cid:48) + f (cid:0) q · A (cid:1) (cid:48) = 0 ; (54) w V (cid:48) + f (cid:0) q · V (cid:1) (cid:48) = 0 ; (55) A (cid:48)(cid:48) − uf (cid:2) q A + w ( q · A ) (cid:3) = 0 ; (56) V (cid:48)(cid:48) − uf (cid:2) q V + w ( q · V ) (cid:3) = 0 ; (57) A (cid:48)(cid:48) k + f (cid:48) f A (cid:48) k + 1 uf (cid:2) w A k + wq k A (cid:3) − uf (cid:2) | q | A k − q k ( q · A ) (cid:3) = 0 ; (58) V (cid:48)(cid:48) k + f (cid:48) f V (cid:48) k + 1 uf (cid:2) w V k + wq k V (cid:3) − uf (cid:2) | q | V k − q k ( q · V ) (cid:3) = 0 , (59)where f is the metric function f ≡ − u in (31) and we define the dimensionless momenta ( w , q ): w = ω πT ; q = q πT . (60)Decomposing V and A into their transverse and longitudinal components with respect to the spatial momentum q , V i ( q | u ) = ( V ⊥ ) i + ( V (cid:107) ) i = ( δ ij − ˆ q i ˆ q j ) V j ( q | u ) + ˆ q i ˆ q j V j ( q | u ) , (61) A i ( p | u ) = ( A ⊥ ) i + ( A (cid:107) ) i = ( δ ij − ˆ q i ˆ q j ) A j ( p | u ) + ˆ q i ˆ q j A j ( p | u ) , (62)with the indices i, j = 1 , , q the unit vector in the direction of q , we find that each component of both V ⊥ and A ⊥ satisfies the following second order linear ordinary differential equation Ψ (cid:48)(cid:48) − u − u Ψ (cid:48) + w − | q | (1 − u ) u (1 − u ) Ψ = 0 , (63)where Ψ ( q | u ) = { V ⊥ , A ⊥ } and we have substituted the explicit form of the background metric (31). The temporalcomponents V (cid:48) , A (cid:48) in Eqns. (54) and (55) can be decoupled from the longitudinal components by eliminating ( q · V )and ( q · A ) from (56) and (57) and we end up with another second order linear ordinary differential equation Φ (cid:48)(cid:48) + 1 − u u (1 − u ) Φ (cid:48) + w − | q | (1 − u ) u (1 − u ) Φ = 0 , (64)for Φ ( q | u ) = {V (cid:48) , A (cid:48) } . The conditions for Eqns. (63) and (64) result from the boundary conditions (53), for thisorder, they read V ⊥ ( q |
0) = V ( q ) ; lim u → u V (cid:48)(cid:48) ( q | u ) = 0 ; A ⊥ ( q |
0) = 0 ; lim u → u A (cid:48)(cid:48) ( q | u ) = | q | A ( q ) , (65)where we have converted the boundary conditions for V and A to that for V (cid:48)(cid:48) and A (cid:48)(cid:48) via Eqns.(56) and (57). Inaddition, the equation for A ⊥ decouples from A and the zero boundary value A ⊥ implies the null solution A ⊥ = 0.Both of Eqn. (63) and (64) are the Fuchs equations with four regular points, and can be transformed into thestandard forms of the Heun equation with details shown in Appendix B, the asymptotic behaviors near the boundaryand near the horizon are exhibited in Tab.I. TABLE I. ∆ Indexes of power series and asymptotic behaviorFluctuation Horizon AdS-boundary u → − u → Φ ( q | u ) ∆ H = ± w i ∆ AdS = 0; 0 (1 − u ) ± i w O (1), O (cid:0) log u (cid:1) V ⊥ ( q | u ) ∆ H = ± w i ∆ AdS = 0; 1 (1 − u ) ± i w O ( u ), O (cid:0) u log u (cid:1) + O (1) For a retarded response in strongly-coupled theory, we choose the in-falling wave solution at horizon [31], whichmeans ∆ H = − w i , and we have: V ⊥ ∝ (1 − u ) − i w ; V (cid:48) ∝ (1 − u ) − i w ; A (cid:48) ∝ (1 − u ) − i w , as u → − . (66)The two linearly independent solutions of Eqn. (63)(64) are denoted as { ψ ( q | u ) , ψ ( q | u ) } and { φ ( q | u ) , φ ( q | u ) } ,respectively. Let ψ ( q | u ) and φ ( q | u ) be the in-falling wave solutions of (63) and (64), normalized at the horizonaccording to , i.e. lim u → − (1 − u ) i w ψ ( q | u ) = 1 ; lim u → − (1 − u ) i w φ ( q | u ) = 1 . (67)As shown in Appendix B, ψ ( q | u ) and φ ( q | u ) can be expressed in terms of two Heun functions. From the boundarycondition (53), the solutions of the zeroth order take the forms V ( q | u ) = V ( q ) ψ ( q | u ) ψ ( q |
0) ; V ( q | u ) = 0 ; A (cid:48) ( q | u ) = | q | A ( q ) φ ( q | u ) D ( q ) ; A ⊥ ( q | u ) = 0 ; (68)with the denominator D ( q ) = lim u → uφ (cid:48) ( q | u ) . (69)The solution of A ( q | u ) at q = 0 is related to the axial chemical potential and requires special consideration inorder to recover Eqn.(2) for a constant magnetic field. It follows from Eqn.(49) for a homogeneous A that ∂ ∂ A = 0 ; ∂ A = 0 , (70)which implies the solution A = au + b with constants a and b , where a is time-independent and b can depend ontime. As only ( F A ) MN contributes to RHS of (50) and (51), the first term of the current (26) does not depend b butthe second term does. Following the gauge invariant definition of the axial chemical potential proposed in [14, 21],we have A ( q | u ) (cid:12)(cid:12)(cid:12)(cid:12) q =0 = µ A u , (71)and thereby A ( q | (cid:12)(cid:12)(cid:12)(cid:12) q =0 ≡ A ( q ) (cid:12)(cid:12)(cid:12)(cid:12) q =0 = 0. The other index ∆ H = + w i assigned to ψ ( q | u ) and φ ( q | u ) leads to the advanced response. B. First Order
To calculate the chiral magnetic current, we need only to carry out the iteration to the first order for the vector gaugepotential V µ . Substituting the zeroth order solution (68) into RHS of Eqn. (50) and making Fourier transformationwith respect to the boundary coordinates x , we obtain that w V (cid:48) + f (cid:0) q · V (cid:1) (cid:48) = κ CS κ M G V ( q | u ) (72) V (cid:48)(cid:48) − uf (cid:2) | q | V + w ( q · V ) (cid:3) = κ CS κ M G V ( q | u ) (73) V (cid:48)(cid:48) k + f (cid:48) f V (cid:48) k + 1 uf (cid:2) w V k + wq k V (cid:3) − uf (cid:2) | q | V k − q k ( q · V ) (cid:3) = κ CS κ M G k V ( q | u ) . (74)Here we need to distinguish A (cid:12)(cid:12)(cid:12)(cid:12) q =0 from A (cid:12)(cid:12)(cid:12)(cid:12) q (cid:54) =0 . In case of the former, it follows from the discussion towards the endof the last sub-section that G V ( q | u ) = − √ µ A ( πT ) Lf B ( q | u ) ; G V ( q | u ) = G V ( q | u ) = 0 . (75)In case of the latter, each component of G M V ( q | u ) is a convolution of the zeroth order solution of V and A , i.e. G M V ( q | u ) = (cid:90) d q (2 π ) d q (2 π ) (2 π ) δ ( q + q − q ) G M V ( q , q | u ) , (76)where the integrands read G ( q , q | u ) = − √ uf ( πT ) L | q | A (cid:48)(cid:48) ( q | u ) (cid:0) q · B ( q | u ) (cid:1) ; (77) G ( q , q | u ) = 3 √ πT ) L f w | q | A (cid:48) ( q | u ) (cid:0) q · B ( q | u ) (cid:1) ; (78) G V ( q , q | u ) = − √ πT ) Lf (cid:20) A (cid:48) ( q | u ) B ( q | u ) − w f | q | A (cid:48) ( q | u ) q × E ( q | u ) − i uf πT | q | A (cid:48)(cid:48) ( q | u ) q × V (cid:48) ( q | u ) (cid:21) . (79)The spatial vectors B ( q | u ) and E ( q | u ) in Eqn. (75)(77)(78)(79) are related to the magnetic field B ( q ) ≡ i q × V ( q )and electric field E ( q | u ) = iω V ( q | u ) on the boundary according to B ( q | u ) = B ( q ) ψ ( q | u ) ψ ( q |
0) ; E ( q | u ) = E ( q ) ψ ( q | u ) ψ ( q | . (80)Taking the transverse component of (79) with respect to q and substituting in the explicit form of the backgroundmetric, we find that V (cid:48)(cid:48)⊥ − u − u V (cid:48)⊥ + w − | q | (1 − u ) u (1 − u ) V ⊥ = κ CS κ M G ⊥ ( q | u ) , (81)where G ⊥ ( q | u ) = (cid:90) d q (2 π ) d q (2 π ) (2 π ) δ ( q + q − q ) G ⊥ ( q , q | u ) (82)with G V ⊥ ( q , q | u ) = G V ( q , q | u ) − q | q | (cid:18) q · G V ( q , q | u ) (cid:19) . (83)1Next, eliminating V (cid:107) from (72) and (73), we end up with V (cid:48)(cid:48)(cid:48) + 1 − u u (1 − u ) V (cid:48)(cid:48) + w − | q | (1 − u ) u (1 − u ) V (cid:48) = κ CS κ M M ( q | u ) , (84)where M ( q | u ) ≡ w uf G V ( q | u ) + 1 uf G V ( q | u ) = (cid:90) d q (2 π ) d q (2 π ) (2 π ) δ ( q + q − q ) M ( q , q | u ) , (85)with M ( q , q | u ) ≡ w uf G V ( q , q | u ) + 1 uf ( uf G V ( q , q | u )) (cid:48) = 3 √ πT ) L uf | q | (cid:18) − w u A (cid:48)(cid:48) ( q | u ) (cid:0) q · B ( q | u ) (cid:1) + w (cid:2) u A (cid:48) ( q | u ) (cid:0) q · B ( q | u ) (cid:1)(cid:3) (cid:48) (cid:19) . (86)The boundary conditions for V ⊥ and V follow from (53) with V and V replaced with V ⊥ and V , i.e. V ⊥ ( q |
0) = V ( q ) ; lim u → u V (cid:48)(cid:48) ( q | u ) = 0 . (87)The 2nd equation follows from (73) with V ( q ) = q · V ( q ) = 0 and the limit lim u → u · G ( q , q | u ) = 0, the latter of whichis evident from the asymptotic behavior of V and A as u → { ψ , ψ } of (63) and { φ , φ } of (64) via the method of variation of parameters with detailsshown in Appendix C. The integration constants incurred are fixed by the in-falling wave condition at the horizonand the boundary condition (87). We find that V ⊥ ( q | u ) = C ( q ) ψ ( q | u ) + ψ ( q | u ) κ CS κ M (cid:90) u d ξ G V ⊥ ( q | ξ ) W ⊥ ( ξ ) ψ ( q | ξ ) − ψ ( q | u ) κ CS κ M (cid:90) u d ξ G V ⊥ ( q | ξ ) W ⊥ ( ξ ) ψ ( q | ξ ) , (88)and V (cid:48) ( q | u ) = C ( q ) φ ( q | u ) + φ ( q | u ) κ CS κ M (cid:90) u d ξ M ( q | ξ ) W ( ξ ) φ ( q | ξ ) − φ ( q | u ) κ CS κ M (cid:90) u d ξ M ( q | ξ ) W ( ξ ) φ ( q | ξ ) , (89)where the integral constants C ( q ) , C ( q ) are given by: C ( q ) = V ( q ) ψ ( q | − κ CS κ M (cid:90) d ξ G V ⊥ ( q | ξ ) W ⊥ ( ξ ) ψ ( q | ξ ) + κ CS κ M ψ ( q | ψ ( q | (cid:90) d ξ G V ⊥ ( q | ξ ) W ⊥ ( ξ ) ψ ( q | ξ ) , (90) C ( q ) = κ CS κ M (cid:90) d ξ M ( q | ξ ) W ( ξ ) φ ( q | ξ ) + κ CS κ M D ( q ) D ( q ) (cid:90) d ξ M ( q | ξ ) W ( ξ ) φ ( q | ξ ) , (91)with D ( q ) = lim u → uφ (cid:48) ( q | u ), and W ⊥ ( u ) and W ( u ) standing for Wronskians of { ψ , ψ } and { φ , φ } , respectively. Itfollows from Eqn. (63)(64), we have W ⊥ ( u ) = W [ ψ , ψ ] = const.1 − u ,W ( u ) = W [ φ , φ ] = const. u (1 − u ) , (92)where the overall constants in (92) will be cancelled afterwards.It follows readily from (88), (89) and (92) that V (cid:48)⊥ ( q |
0) = V ( q ) ψ (cid:48) ( q | ψ ( q | − κ CS κ M ψ ( q | (cid:90) d ξ (1 − ξ ) G V ⊥ ( q | ξ ) ψ ( q | ξ ) (93)and V (cid:48) ( q |
0) = κ CS κ M D ( q ) (cid:90) d ξ ξ (1 − ξ ) M ( q | ξ ) φ ( q | ξ ) . (94)2Substituting (93) and (94) into the continuity equation (72), the longitudinal component of V ( q |
0) with its derivativewith respect to u , V (cid:48)(cid:107) ( q | V (cid:48) ( q |
0) = V (cid:48)⊥ ( q | − w q | q | V (cid:48) ( q | − κ CS κ M q | q | G V ( q | . (95)It follows from (26) and F µ = V (cid:48) µ that the vector current of the boundary field theory reads J ( q ) = − π T V (cid:48) ( q | − √ κ CS (cid:90) d q (2 π ) d q (2 π ) (2 π ) δ ( q + q − q ) A ( q ) B ( q ) . (96)The O (1) term of V (cid:48) ( q |
0) above, i.e., the first term of (93), contributes to the polarization current calculated in [31]and the O ( κ CS ) terms give rise to the chiral magnetic current that’s the theme of next section. IV. CHIRAL MAGNETIC CURRENT
With the formulation developed in the preceding section, we are equipped to calculate the chiral magnetic currentfor an arbitrary spacetime dependent chiral imbalance in this section. The chiral imbalance consists of a net axialcharge characterized by a constant axial chemical potential µ A and its spacetime variation proxied by the temporalcomponent of a spacetime dependent axial vector potential A . Adapting the U V gauge invariant definition of µ A inthe holographic environment, we impose the condition that the Fourier component of A with zero spatial momentumvanishes on the AdS-boundary, e.g. (9), which implies that1Ω (cid:90) d r A ( x ) = 0 , (97)with Ω the spatial volume of the system. Correspondingly, the CME current consists of the contribution from theaxial chemical potential and that from the three-point function, i.e. J CME ( q ) = J ( q ) + J AVV ( q ) , (98)with both terms proportional to the anomaly coefficient C .To simplify the notations, we suppress the subscript of the retarded solutions φ and D ( q ), i.e. φ ( q | u ) ≡ φ ( q | u ) ; D ( q ) ≡ D ( q ) = lim u → uφ (cid:48) ( q | u ) (99)and introduce ψ ( q | u ) ≡ ψ ( q | u ) ψ ( q | , (100)for the sake of brevity.The first term of (98) follows readily from (75), we have J ( q ) = 3 √ κ CS µ A (cid:90) d u B ( q | u ) ψ ( q | u ) = µ A K ( q ) B ( q ) (101)with the kernel K ( q ) = 3 √ κ CS (cid:90) d u ψ ( q | u ) . (102)The AVV contribution can be obtained by substituting (76), (82) and (85) into (93), (94) and (95) together withexplicit expressions (77) , (79) , (83) and (86), and then collecting the O ( κ CS ) terms of (96). Finally, we find that J µ AVV ( q ) = (cid:90) d q (2 π ) d q (2 π ) (2 π ) δ ( q + q − q ) J µ ( q , q ) , (103)3with q and q the 4-momenta carried by the magnetic field and the axial-vector potential, respectively. As a result,the spatial component reads J ( q , q ) = 2( πT ) Lκ CS (cid:20)(cid:90) d u (1 − u ) G V ⊥ ( q , q | u ) ψ ( q | u ) + w q | q | D ( q ) (cid:90) d u u (1 − u ) M ( q , q | u ) φ ( q | u ) (cid:21) + 3 √ κ CS q | q | A ( q ) (cid:0) q · B ( q ) (cid:1) − √ κ CS A ( q ) B ( q ) , (104)where the magnetic Gauss law q · B ( q ) = 0 is employed so q · B ( q ) = q · B ( q ). The temporal component of theAVV current is J ( q , q ) = 2( πT ) Lκ CS D ( q ) (cid:90) d u u (1 − u ) M ( q , q | u ) φ ( q | u ) . (105)which represents the charge induced by a spacetime dependent chiral imbalance. As shown in Append. II, ψ ( q | u )reduces to a hypergeometric function for a homogeneous magnetic field, i.e. q = ( ω,
0) and (101) becomes J ( q ) = 3 √ κ CS µ A B ( q ) Γ ( − i w )Γ ( − i w )Γ (1 − i w ) (cid:90) d u (cid:20) (cid:18) − u u (cid:19) − i w F (cid:18) − i w , − i w ; 1 − i w ; 1 − u u (cid:19) (cid:21) . (106)Phenomenologically, a homogeneous magnetic field serves a reasonable approximation for a sufficiently small fireballin RHIC.The low momentum expansion of the solutions ψ ( q | u ) and φ ( q | u ) can be obtained by the transformation ψ ( q | u ) = (1 − u ) − i w G ( q | u ) ; φ ( q | u ) = (1 − u ) − i w F ( q | u ) , (107)and the equations (63) and (64) become G (cid:48)(cid:48) + (cid:18) − u − u + i w − u (cid:19) G (cid:48) + (cid:20) i w − u ) + w (4 + 3 u + u )4 u (1 + u )(1 − u ) − | q | u (1 − u ) (cid:21) G = 0 , (108) F (cid:48)(cid:48) + (cid:20) − u u (1 − u ) + i w − u (cid:21) F (cid:48) + (cid:20) i w (1 + 2 u )2 u (1 − u ) + w (4 + 3 u + u )4 u (1 + u )(1 − u ) − | q | u (1 − u ) (cid:21) F = 0 . (109)Moving the w and q dependent terms in Eqns. (108)(109) to their RHS and solving the equations iteratively startingwith the leading order solutions F (0) = 1 and G (0) = 1, we derive, to the order we need in this section, that G ( q | u ) = 1 + i w u | q | (cid:20) π
12 + Li ( − u ) + ln u ln(1 + u ) + Li (1 − u ) (cid:21) + · · · ; (110) F ( q | u ) = 1 + i w u u + | q | ln 1 + u u + · · · , (111)with Li (u) the Spence function. Combining with the expansion: (1 − u ) − i w = 1 − i w ln(1 − u ) + · · · , we end up with ψ ( q | u ) = 1 + i w u − u + | q | (cid:20) − π ( − u ) + Li (1 − u ) + ln u ln(1 + u ) (cid:21) + · · · , (112) φ ( q | u ) = 1 + i w u − u + | q | ln 1 + u u + · · · , (113)It follows from the definition (69) that D ( q ) = i w − | q | + (cid:20) w i w | q | − ( | q | ) (cid:21) ln 2 + · · · . (114)Eqns. (110)(111) and the first two terms of (114) were derived in Ref.[31] in the context of two-point functions andthe higher order terms of Eqn.(114) is derived in Appendix D. In what follows, we shall apply the low momentumexpansion of (112) and (113) for ψ ( q | u ) and φ ( q | u ) to the AVV current J ( q , q ). Because of the non-locality of theresponse, the orders of the expansion is sorted by introducing a scaling factor λ according to w , → λ w , q , → √ λ q , (115)for arbitrary w , and q , but leaving the boundary values of V ( q ) and A ( q ) unchanged. As λ →
0, the leadingorder and the subleading order expressions described below are accurate to λ and λ respectively. The scale factor λ is set to one in the end for low momenta.4 A. Leading Order
The leading order contribution to the current is given by the O (1) terms of ψ ( q | u ), φ ( q | u ) in (112)(113) and O ( w , q )term of D ( q ) in (114), i. e. ψ ( q | u ) (cid:39) , φ ( q | u ) (cid:39) , (116)and D ( q ) (cid:39) i w − q throughout this subsection. Substituting these approximations to (68), we have V ⊥ ( q | u ) (cid:39) V ( q ) , V (cid:48) ( q | u ) = 0 , A (cid:48) ( q | u ) (cid:39) | q | A ( q ) D ( q ) . (117)It follows from Eqn.(117), (77), (78) and (79) that to the leading order G V ( q , q | u ) (cid:39) − − u ) 3 | q | A ( q ) √ πT ) L D ( q ) B ( q ) , (118) G ⊥ ( q , q | u ) (cid:39) − − u ) 3 | q | A ( q ) √ πT ) L D ( q ) (cid:20) B ( q ) − q | q | (cid:0) q · B ( q ) (cid:1)(cid:21) , (119) M ( q , q | u ) (cid:39) u (1 − u ) 3 w A ( q ) √ πT ) L D ( q ) (cid:0) q · B ( q ) (cid:1) . (120)Substituting (119) and (120) into (104), we obtain the leading order CME current in terms of the axial gauge potentialand magnetic field: J (0) ( q , q ) = − √ κ CS A ( q ) D ( q ) i w (cid:20) B ( q ) + q D ( q ) (cid:0) q · B ( q ) (cid:1)(cid:21) . (121)The corresponding charge density follows from (120) and (105), and reads J ( q , q ) = 3 √ κ CS w D ( q ) D ( q ) A ( q ) (cid:0) q · B ( q ) (cid:1) . (122)The non-local response because of the diffusion denominator D ( q ) is reflected in nontrivial infrared behaviors of J AVV ( q ). For the case | q | (cid:28) w (cid:28)
1, we have J (0) ( q , q ) (cid:39) − √ κ CS A ( q ) B ( q ) , (123)while in the opposite case w (cid:28) | q | (cid:28) J (0) ( q , q ) (cid:39) . (124)As shown at the end Section II, the asymptotic behavior (123) is a direct consequence of the anomalous Ward identity(29). Its validity is not limited to small momenta and can be extended to all orders in metric fluctuations. The asymp-totic behavior (124) is a holographic version of the Coleman-Hill theorem and implies null chiral magnetic currentat µ A = 0, in agreement with the conclusion of [21] for a simplified holographic model. Both asymptotic behaviorsmatch the field theoretic result of the AVV three-point function[12]. Beyond the hydrodynamic approximation, theAVV contribution can significantly impact the chiral magnetic current.Restoring the dimensions of all 4-momenta via (60) and substituting in B i ( q ) = i(cid:15) ikj q k V j ( q ) and the relation be-tween Chern-Simons coupling and axial anomaly coefficient (30), we extract the leading order AVV function exhibitedin (11) and (12). B. Subleading Order
In order to obtain the subleading order of the vector current J µ (1) = {J ( q , q | u ) , J (1) ( q , q | u ) } , we need toinclude the O ( w , | q | ) terms in φ ( q | u ), ψ ( q | u ) and O ( w , w | q | , ( | q | ) ) terms in D ( q ) shown in (112)(113) and (114),5respectively. Substituting (112)(113) into (68) and taking the derivatives with respect to u , we find that A (cid:48) ( q | u ) = A ( q ) D ( q ) | q | (cid:20) i w u − u + | q | ln 1 + u u · · · (cid:21) (125) A (cid:48)(cid:48) ( q | u ) = A ( q ) D ( q ) | q | (cid:20) i w u (1 − u ) − | q | u (1 + u ) + · · · (cid:21) (126) V ( q | u ) = V ( q ) (cid:20) i w u − u + | q | (cid:18) − π ( − u ) + Li (1 − u ) + ln u ln(1 + u ) (cid:19) · · · (cid:21) (127) V (cid:48) ( q | u ) = V ( q )1 − u (cid:20) i w + | q | ln u + · · · (cid:21) , (128)with D ( q ) including all displayed terms of (114).Substituting (125), (126), (127) and (128) into (79), (82) and (86), and then carrying out the integrations in (104),we obtain, to the subleading order, that (cid:90) d u (1 − u ) G ⊥ ( q , q | u ) ψ ( q | u ) = − √ πT ) L A ( q ) D ( q ) (cid:40) | q | B ( q ) + S ( q , q ) − q | q | (cid:20) q · ( | q | B ( q ) + S ( q , q )) (cid:21) · · · (cid:41) (129) (cid:90) d u u (1 − u ) M ( q , q | u ) φ ( q | u ) = 3 √ πT ) L A ( q ) D ( q ) (cid:0) q · B ( q ) (cid:1)(cid:20) w + S ( q , q ) · · · (cid:21) , (130)where for brevity, the two notations S ( q , q ) and S ( q , q ) represent S ( q , q ) = (cid:20) (cid:18) i w + i w + | q | (cid:19) ln 2 − π
12 ( | q | + | q | ) (cid:21) | q | B ( q )+ (cid:18) − π i w + π | q | (cid:19) q × (cid:0) q × B ( q ) (cid:1) + i ln 2 | q | ( q × E ( q )) , (131) S ( q , q ) = − i w ln 2 − i w w ln 2 + ( w | q | + w | q | ) ln 2 − π w | q | . (132)It follows from (104), (105), (130), (131) and (132) that the AVV contribution to the CME that is accurate to thesubleading order reads J ( q , q ) = − √ κ CS A ( q ) D ( q ) i w (cid:20) B ( q ) + q D ( q ) (cid:0) q · B ( q ) (cid:1)(cid:21) + 3 √ κ CS A ( q ) D ( q ) (cid:40)(cid:20) − (cid:18) i w | q | + i w | q | + 12 w (cid:19) ln 2 + π | q | ( | q | + | q | ) (cid:21) B ( q )+ (cid:18) − π i w + π | q | (cid:19) q × (cid:0) q × B ( q ) (cid:1) + i ln 2 | q | ( q × E ( q ))+ (cid:20) π | q | + 1 D ( q ) (cid:18) w w ln 2 + i w | q | ln 2 − w | q | ln 2 + i π w | q | (cid:19) (cid:21)(cid:0) q · B ( q ) (cid:1) q (cid:41) . (133)The first line above takes the same form as the leading order (123) except that the diffusion denominators D ( q ) and D ( q ) maintain the sub-leading term of (114) so that the entire expression is accurate to the required order. Thepresence of | q | in the denominators of the formulas (104) and (83) gives rise to the direction singularity characterisedby ( q i q j ) / | q | . However, like the leading order result (121), the 1 / | q | -term is eventually cancelled in subleadingorder calculation. Though it is not obvious yet, we suspect that this cancellation is generic, not limited to the smallmomenta.The corresponding charge density follows readily from (105), (130) and (132) and we have explicitly that J ( q , q ) = 3 √ κ CS A ( q ) D ( q ) D ( q ) (cid:0) q · B ( q ) (cid:1)(cid:2) w − i w ln 2 − i w w ln 2 + ( w | q | + w | q | ) ln 2 − π w | q | (cid:3) . (134)6Before concluding this section, we would like to comment on the relativistic causality. As discussed above, the 1 / | q | factor is expected to be cancelled so there is no action at a distance. While the diffusion denominator D ( q ) appearssupporting superluminal response to the external sources, this is an artifact of the low momentum expansion. Lowmomentum ( ω, q ) corresponds to large spacetime separation (∆ t, ∆ r ) between a signal origination and its detectionwhere the diffusion profile | ∆ r | ∼ √ ∆ t for ∆ t > | ∆ r | = | ∆ t | and is therebycausal. The bottom lines is that the Maxwell-Chern-Simons equation we are solving is a set of classical field equationsin a curved background without curvature singularity (the outside of the horizon). So there exist local inertial framesattached to each space-time point, where the equations are fully Lorentz covariant and the propagation of signals inthe solution should be superluminal. Let us envisage a solution of the Maxwell-Chern-Simons equation in responseto an external vector or axial current source placed on the boundary. We may construct the solution in two steps:(a) Substituting the bulk solution given the boundary field values to the action (15) and adding the source term onthe boundary to obtain the effective action as a functional of the boundary fields. (b) Solving the boundary fieldsinduced by the boundary sources via variational principle of the effective action. As the two point correlators, AVVcorrelators, etc. extracted from (a) serve the coefficients in the effective action as a power series in boundary fields,the superluminality of the solution should be encoded in the analyticity with respect to complex momenta. On theother hand, a direct proof of the relativistic causality from the analyticity of the correlators appears difficult. See,e.g. [32] for numerical evidences. V. THE UV AND IR CONVERGENCE
Let us recall the AVV three-point function from field theoretic perspectives. The power counting argument leads tothe degree of UV divergence 1, but the vector current conservation factors two powers of external momenta, leaving theeffective degree of divergence −
1. Indeed, the explicit calculation to the one-loop order gives rise to a finite result oncethe U V (1) gauge invariance is maintained through a proper regularization. Taking the Pauli-Villars regularization asan example, the regularized AVV three-point function to one-loop order remains finite in the limit of infinite regulatormass. The above power counting argument applies only to the skeleton diagram. to higher orders in coupling constant,UV divergence emerges via radioactive corrections which includes the self-energy and vertex corrections. Upon thewave function and coupling constant renormalization, the UV divergence is removed leaving the result depending onthe renormalization scale, such as Λ QCD for QCD. In addition to UV divergence, infrared divergence of Yang-Mills atnonzero temperature grows with the order of diagrams and becomes out of control beyond a certain power of couplingconstant. Nonperturbative effect, such as the magnetic mass is expected to eliminate the IR divergence. Being aconformal theory at quantum level, N = 4 super Yang-Mills theory is expected to be UV finite and its gravity dualprovides a nonperturbative approach of calculation. Therefore, both the UV and IR finiteness should be reflected inthe three-point function calculated via AdS/CFT correspondence and we shall prove below that this is the case. Inthis sense, our result also lends a support to the validity of the and the conjectured AdS/CFT duality.It follows from Eqns. (93) and (94) that to prove the UV/IR convergence amounts to prove the convergence of thefollowing integrals I = (cid:90) d u (1 − u ) G ⊥ ( q , q | u ) ψ ( q | u ) (135)and J = (cid:90) d u u (1 − u ) M ( q , q | u ) φ ( q | u ) , (136)with q = q + q , where ψ ( q | u ) and φ ( q | u ) are in-falling solution of (63) and (64) normalized according to (67), and G ⊥ ( q , q | u ) and M ( q , q | u ) are given by (79), (83) and (86) that relate to ψ and φ via (68). The integration limits u = 0 , u = 0 corresponding UV limit and u = 1 toIR limit. Between them (0 < u < G ⊥ ( q , q | u ) and M ( q , q | u ). As long as theintegrand are sufficiently well-behaved near the upper/lower limit, the integrals converge and our theme is proved. Lower Limit (UV)
According to Table I, the asymptotic forms of ψ ( q | u ) and φ (cid:48) ( q | u ) as u → ψ ( q | u ) = O (1) + O ( u ln u ) ; φ (cid:48) ( q | u ) = O (ln u ) . (137)7It follows that, V ( q | u ) = O (1) ; V (cid:48) ( q | u ) = O (ln u ) , (138)and A (cid:48) ( q | u ) = O (ln u ) ; A (cid:48)(cid:48) ( q | u ) = O (cid:18) u (cid:19) (139)in accordance with (68). Consequently B ( q | u ) = 2 iπ q × V ( q | u ) = O (1) ; E ( q | u ) = 2 iπ w T V ( q | u ) = O (1) . (140)Substituting (138), (139) and (140) to RHS of (83) and (86), we find that the integrand of (135) and (136)(1 − u ) G ⊥ ( q , q | u ) = O (ln u ) ; (141) u (1 − u ) M ( q , q | u ) φ ( q | u ) = O (ln u ) . (142)So their singularities are not strong enough to give rise to UV divergence.As a side remark, the logarithmic divergence of (137) does show up in the first term of (93). But this divergencepertains to the zeroth power of κ CS and does not contribute to the chiral magnetic current. This UV divergence isthe holographic version of the logarithmic divergence of the self-energy of U V (1) gauge boson in field theory and iscancelled by the holographic counter term S c.t. of Eqn.(15). Upper Limit (IR) As u → − , the in-falling condition (67) implies that the asymptotic forms: V ( q | u ) ∼ (1 − u ) − i w ; (143) V (cid:48) ( q | u ) (cid:39) i w − u ) V ⊥ ( q | u ) ; (144) A (cid:48) ( q | u ) ∼ (1 − u ) − i w ; (145) A (cid:48)(cid:48) ( q | u ) (cid:39) i w − u ) A (cid:48) ( q | u ) ; (146) A (cid:48)(cid:107) ( q | u ) (cid:39) − w | q | (1 − u ) A (cid:48) ( q | u ) . (147)Substituting these asymptotic forms to RHS of (79) and (86), we find that the leading singularity of the order − u )2 get cancelled, leaving G ⊥ ( q , q | u ) ψ ( q | u ) ∼ − u ; (148) M ( q , q | u ) ∼ − u , (149)which make the integral (135) and (136) convergent at the upper limit. The cancellation of the leading singularity in(79) follows from the relations B ( q | u ) = i πT q × V ( q | u ) ; E ( q | u ) = i πT w V ( q | u ) (150)and the cancellation in (86) follows from the observation that (cid:2) u A (cid:48) ( q | u ) (cid:0) q · B ( q | u ) (cid:1)(cid:3) (cid:48) (cid:39) i w − u ) (cid:2) u A (cid:48) ( q | u ) (cid:0) q · B ( q | u ) (cid:1)(cid:3) . (151)Consequently, the chiral magnetic current and its induced charge driven by an external magnetic field and axialvector potential together with the response kernel Λ ij ( q , q ) and Λ j ( q , q ) are free from UV and IR divergenceand our theme is thereby proved.A curious divergence of the AVV three-point function at three loop level was discovered in the axial anomaly inthe context of the massless QED at zero temperature [34], where the two photons emerging from the AVV trianglediagram are re-scattered via a fermion loop. As the vector and axial vector field in the bulk do not contribute tothe internal lines of the boundary field theory. Hence this complication does not arise in the super-Yang-Mills on theboundary.8 VI. CONCLUDING REMARKS
In this work, we developed the holographic formulation of the chiral magnetic current for arbitrary energy-momentaof the external magnetic field B ( q ) and temporal component of the axial-vector potential A ( q ) with the latter proxiesthe space-time variations of the chiral imbalance. The gauge theory on the AdS-boundary is the N = 4, super-Yang-Mills of large N c and strong ‘t Hooft coupling. B ( q ) and A ( q ) come from the boundary values of the bulk vectorand axial-vector potential, which correspond to gauged U (1) subgroups of the global U (4) R-symmetry of the super-Yang-Mills. The kernel relating B ( q ) and A ( q ) to the vector current corresponds to the (0 ij ) component of theAVV three-point function ∆ ρµν ( − q − q , q ). For small but nonzero q , the chiral magnetic response turns out to benon-local because of D ( q ) in the denominator and we replicated the field theoretic result regarding the sensitivityof the three-point function to the order of infrared limit q → q is also small [12, 22]. For arbitrary momenta,the kernel can be expressed in terms of two Heun functions that can’t be turned into simpler expressions because ofthe complexity of the Heun functions. Finally we proved that the AVV contribution does not suffer from the UV andIR divergence, this consequence resonates with the finiteness of the super-Yang-Mills on the boundary.The case of a homogeneous chiral imbalance requires special handling because of the additional integration constantincurred in the 0-th order solution for the bulk axial vector potential component A (0 | u ). To reproduce the classicalCME formula (2), we follow the gauge invariant definition of the chemical potential in [14] by setting A = 0 on theboundary at cost of introducing a nonzero A (0 | u ) at the horizon, a singular field configuration in the local inertialframe there. Though the singularity has no known physical impact, the issue reflects the difficulty of defining theaxial chemical potential associated to a non-conserved axial charge [21]. Nevertheless, this recipe generates the firstterm of the chiral magnetic current (5) which restores (2) as its special case with a constant magnetic field. A by-product of our formulation is an analytic expression for this part of the chiral magnetic current in terms of an ordinaryhypergeometric function for a homogeneous but time-dependent magnetic field and this type of magnetic field wasassumed in some hydrodynamic simulation of CME in RHIC.There is a vast amount of literature on the holographic chiral magnetic effect, most of them works in the probe limitwhich facilitates the expansion to nonlinear orders of external vector or axial vector electromagnetic field withoutconsidering the metric fluctuations [14, 15, 32, 33]. In addition, the condition | µ A | (cid:29) | A ( q ) | (152)was assumed in previous literature, which treats µ A as the gauge field background. What we pointed out here isthat probe limit is not required as far as AVV three-point function is concerned when µ A is treated as a part ofexternal fields, following from the power counting argument in section 2. Regarding µ A as a part of the externalaxial vector potential, the chiral magnetic conductivity evaluated under the condition (152), e. g. References[14, 15]contain all powers of µ A with the leading power corresponds to the AVV three-point function here but with one ofthe independent momentum, q = 0. What we developed is the formulation of the three-point function with both q and q nonzero and thereby exposing the non-local response and non-trivial IR limit reflected in the formulas (11)and (12) for small momenta, which appears more realistic from the dynamics perspective of heavy-ion collisions. Forthe N = 4 super-Yang-Mills on the boundary, the coefficients on RHS of the Einstein equation (24), κ M /κ EH = O (1).To higher powers in external vector and axial vector fields (including µ A beyond the AVV three-point function, themetric fluctuations have to be brought in. Therefore, staying within the probe limit, the coefficient of the currentbeyond bi-linear terms in V ( q ), A ( q ) are no longer pertaining to the N = 4 super-Yang-Mills.The phenomenological implication of the external field approximation is richer than that of the hydrodynamic ap-proximation. In addition to the non-locality introduced by the diffusion denominator, with equal order of magnitudesof µ A and A ( q ), a spacetime dependent A can generate a sizable impact on the chiral magnetic signal through theAVV term of the chiral magnetic current (98), especially for a near homogeneous profile according to the in the limit(123).Our formulation can be readily generalized to explore the chiral separation effect (CSE) under a magnetic field anda space-time dependent chemical potential which is conjugate to the charges associated with the vector potential.As the vector current is conserved, the special treatment, such as (71), of the homogeneous component of the V inthe bulk may be warranted and the ambiguity associated with the IR limit may disappear. We hope to report ourprogress in this direction in near future. ACKNOWLEDGMENTS
L.Yin is supported by Guangdong Major Project of Basic and Applied Basic Research No. 2020B0301030008. D-f.Hou and H-c. Ren are supported in part by the NSFC Grant Nos. 11735007, 11890711. L.Yin is also supported byScience and Technology Program of Guangzhou No. 2019050001.9
Appendix A: Inhomogeneous Maxwell Equation in Coordinate and Fourier Space
Substituting the AdS-Schwarzschild metric (31) and the gauge condition V u = A = 0 = V u = A = 0 into (50) and(51), we obtain explicit coordinate representation of the field equations up to the linear order in κ CS , i.e. ∂ ∂ V − f ∂ k ∂ V k = − √ πT ) L κ CS κ M ε kij (cid:2) F A k F Vij + F V k F Aij (cid:3) (A1) ∂ ∂ A − f ∂ k ∂ A k = − √ πT ) L κ CS κ M ε kij (cid:2) F V k F Vij + F A k F Aij (cid:3) (A2) ∂ V − πT ) uf ∂ k (cid:2) ∂ k V − ∂ V k (cid:3) = − √ πT ) L κ CS κ M ε kij (cid:2) ∂ A k F Vij + ∂ V k F Aij (cid:3) (A3) ∂ A − πT ) uf ∂ k (cid:2) ∂ k A − ∂ A k (cid:3) = − √ πT ) L κ CS κ M ε kij (cid:2) ∂ V k F Vij + ∂ A k F Aij (cid:3) (A4) ∂ (cid:2) f ∂ V k (cid:3) − ∂ (cid:2) ∂ V k − ∂ k V (cid:3) (2 πT ) uf + ∂ l [ ∂ l V k − ∂ k V l ](2 πT ) u = − √ πT ) L κ CS κ M ε kij (cid:2)(cid:0) ∂ V F Aij + ∂ A F Vij (cid:1) − (cid:0) ∂ V i F A j + ∂ A i F V j (cid:1)(cid:3) (A5) ∂ (cid:2) f ∂ A k (cid:3) − ∂ (cid:2) ∂ A k − ∂ k A (cid:3) (2 πT ) uf + ∂ l [ ∂ l A k − ∂ k A l ](2 πT ) u = − √ πT ) L κ CS κ M ε kij (cid:2) ( ∂ A F Aij + F Vij ∂ V ) − ∂ A i F A j + F V j ∂ V i ) (cid:3) (A6)where ∂ V µ ≡ ∂ V µ∂u and we have separated the time index “0” and the AdS radial index “5” from the spatial indiceson the boundary.Making Fourier transformation with respect to the boundary coordinate x µ on both sides with ∂ µ → iq µ = i ( − ω, q ) (A7)we find that w A (cid:48) + f (cid:0) q · A (cid:1) (cid:48) = κ CS κ M G A ( q | u ) (A8) w V (cid:48) + f (cid:0) q · V (cid:1) (cid:48) = κ CS κ M G V ( q | u ) (A9) A (cid:48)(cid:48) − uf (cid:2) | q | A + w ( q · A ) (cid:3) = κ CS κ M G A ( q | u ) (A10) V (cid:48)(cid:48) − uf (cid:2) | q | V + w ( q · V ) (cid:3) = κ CS κ M G A ( q | u ) (A11) A (cid:48)(cid:48) k + f (cid:48) f A (cid:48) k + 1 uf (cid:2) w A k + wq k A (cid:3) − uf (cid:2) | q | A k − q k ( q · A ) (cid:3) = κ CS κ M G kA ( q | u ) (A12) V (cid:48)(cid:48) k + f (cid:48) f V (cid:48) k + 1 uf (cid:2) w V k + wq k V (cid:3) − uf (cid:2) | q | V k − q k ( q · V ) (cid:3) = κ CS κ M G kV ( q | u ) (A13)where we have introduced dimensionless 4-momenta via (60) and denoted derivative with respect to u by a primefor brevity. The explicit form of the inhomogeneous terms G V ( q | u ), G V ( q | u ) and G kV ( q | u ) are displayed in (76) as aconvolution with the explicit form of the integrand given by (77), (78) and (79). The parallel expressions of G V ( q | u ), G V ( q | u ) and G kV ( q | u ), which are not used in this work, are summarized in the following convolution form. G M A ( q | u ) = (cid:90) d q (2 π ) d q (2 π ) (2 π ) δ ( q + q − q ) G M A ( q , q | u ) ≡ (cid:90) q ,q G µ V ( q , q | u ) , (A14)0where the integrand G ( p, l | u ) = − √ p ( πT ) L V ⊥ ( p | u ) · B ( p | u ) (A15) G ( p, l | u ) = − √ πT ) L V ⊥ ( p | u ) · B ( p | u ) (A16) G Ak ( p, l | u ) = − √ πT ) Lf (cid:2) V (cid:48) ( p | u ) B k ( p | u ) − (cid:0) E × V (cid:48)⊥ (cid:1) k ( p | u ) (cid:3) , (A17)where, for u →
0, the classical fields B ( p | u ) and E ( p | u ) become the magnetic field and electric field on quantum field,respectively (80). Appendix B: Zeroth Order Solution in Terms of Heun Functions
Each of the differential equations (63) and (64) is a Fuchs equation with four regular points at u = 0 , , − , ∞ .Making the transformation Ψ = (1 − u ) − i w (cid:18) u (cid:19) w f I Φ = (1 − u ) − i w (cid:18) u (cid:19) w f II (B1)and z = − u , f s with s = I , II satisfy the standard Heun equation z ( z − z − a s ) d f s dz + [( α s + β s + 1) z − [ α s + β s + 1 − δ s + ( γ s + δ s ) a s ] z + a s γ s ] df s dz + α s β s ( z − b s ) f s = 0 (B2)where the parameters a I = a II = 12 α I = α II = 12 (1 − i ) w γ I = γ II = 1 − i w δ I = δ II = 1 + w β I = 12 (1 − i ) w + 1 β II = 12 (1 − i ) w + 2 b I = w + (cid:0) − + i (cid:1) w − | q | i w − (1 − i ) w b II = (cid:0) − i (cid:1) w + (cid:0) − i (cid:1) w − | q | i w − − i ) w (B3)The indices at the regular points z = 0 , , a s , ∞ are (0 , − γ s ), (0 , − δ s ), (0 , − (cid:15) s ) and ( α s , β s ) with (cid:15) s = α s + β s − γ s − δ s + 1. In terms of the standard notation of the Heun function in [30], the in-falling solutions normalized by theconditions (67) are given by (67): ψ ( q | u ) = (1 − u ) − i w (cid:18) u (cid:19) w F (cid:18) a I , b I ; α I , β I , γ I , δ I ; 1 − u (cid:19) φ ( q | u ) = (1 − u ) − i w (cid:18) u (cid:19) w F (cid:18) a I I , b II ; α II , β II , γ II , δ II ; 1 − u (cid:19) (B4)Interestingly, the infinity of (63) becomes an ordinary point in the homogeneous limit, q →
0, and the Heun equationis reduced to a hypergeometric equation with the in-falling solution ψ ( q | u ) = (cid:18) − u u (cid:19) − i w F (cid:18) − i w , − i w ; 1 − i w ; 1 − u u (cid:19) (B5)1and ψ ( q |
0) = Γ (1 − i w )Γ (cid:0) − i w (cid:1) Γ (cid:0) − i w (cid:1) . (B6) Appendix C: Special Solutions by Variation of Parameter
It follows from the method of variation of parameters that the general solution of an inhomogeneous 2nd-orderdifferential equation L u Ψ( u ) = g ( u ) with differential operator L u := d d u + p ( u ) dd u + q ( u ) (C1)is given by Ψ( u ) = c ψ ( u ) + c ψ ( u ) + ψ ( u ) (cid:90) u d ξ g ( ξ ) W ( ξ ) ψ ( ξ ) − ψ ( u ) (cid:90) u d ξ g ( ξ ) W ( ξ ) ψ ( ξ ) . (C2)where ψ ( u ) and ψ ( u ) are the two linearly independent solution of the homogeneous equation L u ψ ( u ) = 0 and W ( u )is their Wronskian. The constants c and c are determined by appropriate boundary conditions.For the inhomogeneous equations (81) and (84) for V ⊥ and V (cid:48) with in-falling and outgoing solutions of the homo-geneous equation, ψ ( u ) and ψ ( u ), at u = 1, the in-falling condition of Ψ( u ) there set c = 0. The Dirichlet likeboundary condition at u = 0 for V ⊥ gives rise to the constant c = Ψ(0) ψ (0) − (cid:90) d ξ g ( ξ ) W ( ξ ) ψ ( ξ ) + ψ (0) ψ (0) (cid:90) d ξ g ( ξ ) W ( ξ ) ψ ( ξ ) , (C3)and the solution (88) together with (90) follow then. The Newman like boundary condition for V (cid:48) , uV (cid:48)(cid:48) ( u ) → u → c = − (cid:90) d ξ g ( ξ ) W ( ξ ) ψ ( ξ ) + ψ (cid:48) (0) ψ (cid:48) (0) (cid:90) d ξ g ( ξ ) W ( ξ ) ψ ( ξ ) , (C4)and gives rise to the solution (89) together with (91). Appendix D: Low Momentum Expansion of the Diffusion Denominator
To derive the low momentum expansion of the diffusion denominator D ( q ), eq. (114), we convert the differentialequation (109) into an integral equation via the method of variation of parameters, subject to the boundary conditionof F = 1 at u = 1. We have F = 1 + 12 (cid:90) u dξξ (1 − ξ ) E ln ξ − ξ −
12 ln u − u (cid:90) u dξξ (1 − ξ ) E (D1)where E = − i w − u F (cid:48) − i w (1 + 2 u )2 u (1 − u ) F − w (4 + 3 u + u )4 u (1 + u )(1 − u ) F + | q | u (1 − u ) F (D2)As F (cid:39) D ( q ) ln u as u →
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