Out-of-Equilibrium Chiral Magnetic Effect at Strong Coupling
aa r X i v : . [ h e p - ph ] A ug RBRC-1018
Out-of-Equilibrium Chiral Magnetic Effectat Strong Coupling
Shu Lin ∗ , Ho-Ung Yee , † RIKEN-BNL Research Center, Brookhaven National Laboratory,Upton, New York 11973-5000 Department of Physics, University of Illinois,Chicago, Illinois 60607
Abstract
We study the charge transports originating from triangle anomaly in out-of-equilibrium conditions in the framework of AdS/CFT correspondence at strongcoupling, to gain useful insights on possible charge separation effects that may hap-pen in the very early stages of heavy-ion collisions. We first construct a gravitybackground of a homogeneous mass shell with a finite (axial) charge density gravi-tationally collapsing to a charged blackhole, which serves as a dual model for out-of-equilibrium charged plasma undergoing thermalization. We find that a finite chargedensity in the plasma slows down the thermalization. We then study the out-of-equilibrium properties of Chiral Magnetic Effect and Chiral Magnetic Wave in thisbackground. As the medium thermalizes, the magnitude of chiral magnetic conduc-tivity and the response time delay grow. We find a dynamical peak in the spectralfunction of retarded current correlator, which we identify as an out-of-equilibriumchiral magnetic wave. The group velocity of the out-of-equilibrium chiral magneticwave is shown to receive a dominant contribution from a non-equilibrium effect,making the wave moving much faster than in the equilibrium, which may enhancethe charge transports via triangle anomaly in the early stage of heavy-ion collisions. ∗ e-mail: [email protected] † e-mail: [email protected] Introduction
Heavy-ion collisions create an interesting new state of matter, quark-gluon plasma of QCD,where confinement is effectively lost due to high temperature above the QCD cross-overline. Although microscopic QCD degrees of freedom of quarks and gluons are expected tobe liberated in this environment, there are many experimental and theoretical indicationsthat the quark-gluon plasma created in the experiments are strongly coupled, which makesthem behaving as nearly perfect liquids with small viscosity [1]. Hydrodynamics has beena powerful tool to describe the long wavelength dynamics of the system without knowingmuch about the microscopic details of the theory except a few transport coefficients.However, going beyond the hydrodynamic regime meets a serious computational challengeof dealing with strongly coupled system of QCD matter. The AdS/CFT correspondencebased on a large N c expansion and strong t’Hooft coupling can be a useful tool to studysuch strongly coupled QCD dynamics.Another approach to circumventing difficulties of strongly coupled dynamics is touse symmetries of the theory and look for interesting observables that are protectedby them. QCD with (approximately) massless quarks has a chiral flavor symmetry SU ( N f ) L × SU ( N f ) R × U (1) V × U (1) A , where the last axial symmetry U (1) A is quantummechanically violated via triangle anomaly, and it is not a true symmetry. The gluoniccontributions in the plasma to the anomalous violation of axial symmetry happen viathermal sphaleron transitions, whose rate in current estimate is about Γ sph ≈ α s T ≈ . α s GeV with T = 250 MeV [2]. This determines the relaxation time scale of axialcharges via fluctuation-dissipation relation as τ R = 2 χT (2 N F ) Γ sph ≈ . × − α s fm , (1.1)where χ ≈ . T ≈ .
06 GeV is the charge susceptibility at T = 250 MeV [3], and N F = 2. This gives τ R ≈
10 fm for α s = 0 .
2, and one could marginally neglect it inheavy-ion experiments with typical lifetime of the plasma being 10 fm ∗ . Another (moreformal) aspect of these gluonic contributions is that they are sub-leading in large N c limit,and would not appear, for example, in the AdS/CFT-based models at leading order. ∗ The above relaxation time formula is highly sensitive to α s : for example, α s = 0 . τ R = 1 . t . ∂ µ J µA = e N c π X F q F ! ~E · ~B , (1.2)where q F is the charge of the quark flavor F . Non-renormalization of this relation underradiative corrections is a rare example where a violation of a symmetry can give us strongconstraints on the predictions of the theory. In the low energy regime of chiral pertur-bation theory, the gauged Wess-Zumino-Witten action accounts for all essential physicsconsequences of the triangle anomaly. However, possible new transport phenomena orig-inating from triangle anomaly in finite temperature or density phases of QCD are lessexplored and have attracted much recent interests from both theorists and experimen-talists. One such phenomenon, the Chiral Magnetic Effect (CME) [6, 7, 8, 9, 10], statesthat in the presence of a magnetic field ~B , a vector (axial) current will be induced by anon-zero axial (vector) chemical potential, ~J V,A = eN c π µ A,V ~B . (1.3)The CME has been confirmed in both weak coupling [11, 12, 13, 14, 15, 16, 17, 18]and strong coupling frameworks [19, 20, 21, 22, 23, 24]. It has also been derived fromthe hydrodynamics [25, 26] and effective action [27, 28, 29, 30, 31, 32]. The off-centralheavy-ion collisions which accompany transient magnetic fields of strength as large as eB ∼ m π are important places to look for possible signals of this effect [7], and there areexperimental indications which favor the existence of the signals that go along with thepredictions from the Chiral Magnetic Effect [33, 34, 35].The two versions of the Chiral Magnetic Effect lead to the existence of a new gaplesssound-like propagating mode of chiral charge densities in the hydrodynamic regime, coinedas Chiral Magnetic Wave (CMW) [36, 37], which has the dispersion relation, ω = ∓ v χ k − iD L k + · · · , (1.4)where the velocity v χ is given by v χ = eN c B π χ , and k is the momentum along the direction ofthe magnetic field. The longitudinal diffusion constant D L depends more on the dynamicsof the theory. The sign in front of the first term that determines the direction of the wavepropagation depends on the chirality of the charge fluctuations, so that a left-handedchiral charge fluctuation moves to the direction opposite to a right-handed chiral charge2uctuation. In off-central heavy-ion collisions, the charge transports via Chiral MagneticWave would induce a net electric quadrupole moment in the fireball [38, 39, 40], whicheventually leads to a charge dependent elliptic flow of pions [38, 39]. Recent analysisfrom STAR seems to support the prediction from the Chiral Magnetic Wave [41, 42].Both Chiral Magnetic Effect and Chiral Magnetic Wave above should be considered aslong wavelength limit of the charge transports originating from triangle anomaly in theequilibrium QCD plasma.In this work, we extend the previous studies in two important aspects: we study Chi-ral Magnetic Effect and Chiral Magnetic Wave in out-of-equilibrium conditions and innon-hydrodynamic regimes. By out-of-equilibrium conditions, we mean that the plasmabackground in question is not thermalized and non-static either. By non-hydrodynamicregimes, we mean the frequency of the probe (in our case, it will be the magnetic field)is comparable or larger than the characteristic time scale of the plasma loosely set bythe late-time temperature or effective collision rate. Our motivation for considering out-of-equilibrium plasma is to study the charge transport originating from triangle anomalyin the early stages of plasma fireball created in heavy-ion collisions where the system isout-of-equilibrium and undergoes thermalization. This is well motivated since the mag-netic field is larger at earlier times and the charge transports via triangle anomaly may besignificant in this out-of-equilibrium stage before local thermalization is achieved. Sincethe thermalization seems to happen relatively fast within 1 fm, how large the net effectscoming from the out-of-equilibrium stage are is an important question to be addressedcarefully. We hope our work lays a useful foundation to answer this question more quan-titatively in the future. The motivation for looking at non-hydrodynamic response to amagnetic field of high frequency, which was first studied in [11] at weak coupling and sub-sequently in [19] at strong coupling, comes from the fact that the magnetic field createdin heavy-ion collisions is highly time-dependent and transient. When the frequency ω ofthe magnetic field is finite, the Chiral Magnetic Effect is generalized to be ~J V ( ω ) = σ χ ( ω ) ~B ( ω ) , (1.5)with the frequency-dependent chiral magnetic conductivity σ χ ( ω ). In the equilibriumQCD plasma, its zero frequency limit is constrained to reproduce the usual Chiral Mag-netic Effect, so that σ χ ( ω →
0) = eN c π µ A , (1.6)whereas the finite frequency behavior depends on the microscopic dynamics of the theory.In our analysis, we look at the same problem in out-of-equilibrium conditions.3e will study these problems in the framework of AdS/CFT correspondence, hopingto gain useful insights on what would be the results at strong coupling † . In AdS/CFT,global symmetries such as vector/axial symmetries appear as 5-dimensional gauge fieldsresiding in the holographic 5 dimensional AdS space. For our purposes, we can focus onsimply U (1) V × U (1) A , and the triangle anomaly manifests itself as a 5 dimensional Chern-Simons term, so that the minimal set-up of our holographic model is the 5 dimensionalEinstein-Maxwell-Chern-Simons theory with U (1) V × U (1) A gauge fields,(16 πG ) L = R + 12 −
12 ( F V ) MN ( F V ) MN −
12 ( F A ) MN ( F A ) MN (1.7)+ κ √− g ǫ MNP QR (cid:16) A A ) M ( F V ) NP ( F V ) QR − ( A A ) M ( F A ) NP ( F A ) QR (cid:17) . The coefficient κ of the Chern-Simons terms should be chosen as κ = − G N c π , (1.8)to reproduce the correct triangle anomaly with a single massless Dirac quark flavor whoseelectromagnetic charge is set to e . Our epsilon symbol is purely numerical with theconvention ǫ zt = 1 ‡ . Note that our vector gauge field A V is defined to be dual to thevector current without e , so that the electromagnetic current is the e times the vectorcurrent obtained from A V . Similarly, an electromagnetic background field will act asa source for the vector current with the coupling e , so that the boundary value of A V will be e times the electromagnetic background field. The generalization to multi-flavorquarks with different electromagnetic charges is straightforward with a few rescalingsof parameters. The 5 dimensional Newton’s constant G in our model can be fixed byconsidering the equation of state of the blackhole solution that describes finite temperatureQCD plasma at high temperatures εT = 3 π G , (1.9)and comparing this with the lattice result for T ≫ T c [57], εT ≈
13 (lattice) , (1.10) † See Refs.[43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56] for previous works on out-of-equilibriumsituations in AdS/CFT correspondence. ‡ Note that our definition of epsilon tensor differs by a sign from that in Ref.[19] because our radialcoordinate z is related to the coordinate r in Ref.[19] by z = r , which is a parity odd transformation.Thus we have an overall plus sign for the Chern-Simons term. G ≈ . § . Independentlyto the Chiral Magnetic Effect, our inclusion of a finite charge is also motivated by the factthat the created fireball in heavy-ion collisions carries a finite vector chemical potentialdue to baryon stopping, and we would like to understand its effect on thermalization ¶ .Our model implicitly assumes the creation of axial charge fluctuation very early in thecollision history, probably by color electric and magnetic fields in the glasma phase [4, 5].The Chern-Simons terms do not play a role in constructing the background solution, andone can patch the known AdS-Reisner-Nordstrom blackhole solution in the UV regionabove the shell with the pure AdS solution in the IR region below the shell. Assumingthe conformal energy-momentum tensor on the shell, the Israel junction conditions [62]result in a simple equation for the time-trajectory of the shell, which we solve numerically.The initial position of the shell at time zero measures the typical virtuality scale of theinitial out-of-equilibrium plasma, and it is natural to set it to be equal to the saturationscale Q s ∼ .
87 GeV for
RHIC and Q s ∼ .
23 GeV for
LHC k . As for the late-timeequilibrium temperature, we will put T = 300 MeV for RHIC and T = 400 MeV for LHC as exemplary values. With these two scales fixed, the solution is unique giventhe (axial) charge density (or equivalently, the late-time equilibrium value of the (axial)chemical potential, µ eqA ). We will present our results for the values of µ eqA = 50 , , σ χ ( ω ) in an approximation that the mass shellat a given time is nearly static compared to the time scale of the probe (quasi-static ap-proximation) [44]. This brings us some constraints on the validity of our results, and theprecise region of validity will be discussed in detail. The time-trajectory of the mass shellthen allows us to find the time evolution of the chiral magnetic conductivity, σ χ ( ω, t ), inthe quasi-static approximation. Going beyond the quasi-static approximation will be an § See Refs.[58, 59, 60, 61] for works on similar geometries with zero charge density. ¶ The effects of vector and axial charge density will be the same in our model. k Our values are based on the fit formula Q s ≈ . A (cid:0) x . (cid:1) − . in Ref.[63] with x = 0 .
01 for
RHIC and x = 0 .
001 for
LHC . In this section, we will construct a gravitationally collapsing mass shell geometry in asymp-totic
AdS space with a 3-dimensional translational symmetry, generalizing previous worksby including a finite axial charge (below we will simply call charge) density on the shell.This geometry is a toy model for a spatially homogeneous, out-of-equilibrium, chargedplasma which undergoes thermalization. The late-time asymptotic solution which is dualto a thermally equilibrated charged plasma will be the known charged blackhole solutionin AdS . From our action density(16 πG ) L = R + 12 −
12 ( F V ) MN ( F V ) MN −
12 ( F A ) MN ( F A ) MN (2.11)+ κ √− g ǫ MNP QR (cid:16) A A ) M ( F V ) NP ( F V ) QR − ( A A ) M ( F A ) NP ( F A ) QR (cid:17) , with κ = − G N c π , the equations of motion read as R MN + (cid:18) F V ) + 16 ( F A ) (cid:19) g MN − ( F V ) P M ( F V ) PN − ( F A ) P M ( F A ) PN = 0 ,∂ N (cid:16) √− g ( F A ) MN (cid:17) − κ ǫ MNP QR (cid:16) ( F V ) NP ( F V ) QR + ( F A ) NP ( F A ) QR (cid:17) = 0 ,∂ N (cid:16) √− g ( F V ) MN (cid:17) − κ ǫ MNP QR ( F A ) NP ( F V ) QR = 0 . (2.12)6he model has an exact charged black-hole solution which is spatially homogeneous (AdS-Reisner-Nordstrom (AdS-RN) solution), ds = dz f ( z ) z − f ( z ) z dt + ( d~x ) z , A A = − Qz dt , A V = 0 , (2.13)where f ( z ) = 1 − mz + 2 Q z , (2.14)and z H is the location of the blackhole horizon obtained by solving f ( z H ) = 0. Theparameters ( m, Q ) are related to the temperature and (axial) chemical potential ( T, µ A )by T = − f ′ ( z H )4 π , µ A = z H Q . (2.15)The model also has the pure
AdS solution, ds = dz z − dt z + ( d~x ) z , A V = A A = 0 , (2.16)corresponding to the vacuum of the model.We will consider a thin, spatially homogeneous mass shell with a finite charge densitycollapsing from the UV region of small z to the IR region of large z under its own gravity.Following [44], we approximate the thickness of the shell to be infinitesimally small, andthe geometry will be constructed by joining the AdS-RN solution above the shell in theUV region with the pure AdS solution below the shell, across the space-time trajectory ofthe thin mass shell which should be obtained by solving the appropriate Israel junctionconditions [62]. In general, the coordinates ( z, t, ~x ) appearing in the AdS-RN solutionabove the shell should not be identified with the ( z, t, ~x ) in the pure AdS below the shell,and one should specify proper relations between them. One of the junction conditions isthe continuity of the metric across the shell, so that the two metrics evaluated on the 1+3dimensional world volume Σ of the shell should be equal. A part of this condition caneasily be satisfied for the 3-dimensional spatial directions parametrized by ~x , by identifying( z, ~x ) in the AdS-RN and ( z, ~x ) in the pure AdS across the shell, so that the metric part z ( d~x ) in both solutions match across the shell. After this, the time coordinates in theupper region (above the shell) and in the lower region (below the shell) are in generaldifferent, so we call them t U and t L respectively. It is convenient to introduce a 1+3dimensional world-volume coordinate ( τ, ~x ) on the mass shell, and the induced metric onthe shell can always be put into the form ds = − dτ + ( d~x ) ( z ( τ )) , (2.17)7y reparameterizing τ and some function z ( τ ). By identifying ~x on Σ with ~x in thebackground, z ( τ ) is clearly the position of the shell in the z coordinate at time τ . Theremaining relations between t U , t L , and τ , and the mass shell trajectory z ( τ ) (equivalently, z ( t U ) and z ( t L )) should be found by solving the junction conditions.The continuity of the metric across the shell implies that the time component of themetric should match. Writing the trajectory of the shell in the AdS-RN coordinates ( t U , z )parametrized by the world sheet time τ ,( t U , z ) = ( t U ( τ ) , z ( τ )) , (2.18)and comparing the induced metric on the shell from the AdS-RN and (2.17), one obtains f ( z ( τ )) ˙ t U ( τ ) − ˙ z ( τ ) f ( z ( τ )) = 1 , (2.19)where · ≡ ddτ . Similarly, the same trajectory in the pure AdS coordinates( t L , z ) = ( t L ( τ ) , z ( τ )) , (2.20)should satisfy the condition ˙ t L ( τ ) − ˙ z ( τ ) = 1 . (2.21)The (2.19) and (2.21) implicitly give the relation between t U and t L once the trajec-tory z ( τ ) is found. The last ingredient to determine the solution is the Israel junctioncondition ∗∗ [ K ij − γ ij K ] = − πG S ij , (2.22)where [ A ] ≡ A L − A U and S ij is the energy-momentum on the shell, S ij = − √− γ δ ( √− γ L shell ) δγ ij , (2.23)and γ ij is the induced metric on the shell with respect to the shell coordinate ξ i . The K U,Lij are the extrinsic curvatures evaluated on the shell from the upper region (AdS-RNmetric) and the lower region (pure AdS) respectively, K ij = ∂x α ∂ξ i ∂x β ∂ξ j ∇ α n β = − n α (cid:18) ∂ x α ∂ξ i ∂ξ j + Γ αβγ ∂x β ∂ξ i ∂x γ ∂ξ j (cid:19) , (2.24) ∗∗ One can show that the extra terms from the gauge field in the Einstein equation does not modifythe junction condition for the metric, as the field strengths do not contain δ -function singularity. Somederivatives of the field strength such as ∂ z F tz are δ -function singular, and they modify the junctioncondition for the gauge field coming from the Maxwell(-Chern-Simons) equations, which is nothing butthe Gauss’s law across the thin shell. We will not need to consider this in our work. n µ to the surface Σ pointing to the direction of increasing z (thatis, out-going from the upper region of small z to the lower region of large z ). Explicitly, n µ in the upper and lower coordinates are given by n U = (cid:18) z ˙ zf ( z ) (cid:19) ∂∂t + (cid:0) zf ( z ) ˙ t (cid:1) ∂∂z ,n L = ( z ˙ z ) ∂∂t + (cid:0) z ˙ t (cid:1) ∂∂z , (2.25)where all quantities are evaluated on the shell.A straightforward computation gives the non-vanishing components as K Uττ = − ˙ t U z (cid:18) f ( f ′ + 2¨ z )2 ( f + ˙ z ) − fz (cid:19) , K Uij = − ˙ t U fz δ ij , i, j = 1 , , ,K Lττ = − ˙ t L z (cid:18) z z ) − z (cid:19) , K Lij = − ˙ t L z δ ij , i, j = 1 , , . (2.26)where ′ ≡ ddz . To proceed further, we assume that the energy-momentum on the shell hasthe conformal form, S ij = 4 p ( z ) u i u j + γ ij p ( z ) , u i = (cid:18) z , , , (cid:19) , (2.27)with the pressure p ( z ) to be determined, and the junction condition becomes after somemanipulations,˙ t L − f ˙ t U = 8 πG p ( z ) , ˙ t L z ¨ z (1 + ˙ z ) − ˙ t U zf (cid:16) f ′ + ¨ z (cid:17) ( f + ˙ z ) = 4 · πG p ( z ) . (2.28)Removing p ( z ) from the above equations and using˙ t L = √ z , ˙ t U = p f + ˙ z f , (2.29)from (2.19) and (2.21), one finds that the resulting equation for ˙ z is amusingly integrableto give √ z − p f + ˙ z = Cz , (2.30)with a constant of motion C >
0, and hence we obtain˙ z = s(cid:18) Cz m C − Q z C (cid:19) − , (2.31)9 A (MeV) z H (fm) m (fm − ) Q (fm − ) C (fm − )50 0.209 526.8 5.82 264.3100 0.208 535.7 11.7 268.6200 0.206 571.8 23.9 286.0Table 1: The parameters of the numerical solutions for RHIC with the late-time tem-perature T = 300 MeV and several exemplar values of µ A . µ A (MeV) z H (fm) m (fm − ) Q (fm − ) C (fm − )50 0.157 1660.9 10.3 832.7100 0.156 1676.7 20.7 840.4200 0.155 1740.3 42.0 871.2Table 2: The parameters of the numerical solutions for LHC with the late-time temper-ature T = 400 MeV and several exemplar values of µ A .which can be solved numerically given the constant C which should be determined fromthe initial conditions. Once z ( τ ) is found, t U,L ( τ ) and p ( z ) can be found subsequently. p ( z ) turns out to be especially simple p ( z ) = Cz πG . (2.32)We are interested in expressing the falling trajectory in terms of the boundary time t U that can be identified with the time measured in QCD, z ( t U ) = z ( τ ( t U )) , (2.33)so that we can discuss the thermalization history measured in the QCD time. A shortalgebra gives us the equation dzdt U = f ( z ) vuuuut (cid:16) Cz + m C − Q z C (cid:17) − (cid:16) Cz + m C − Q z C (cid:17) − f ( z ) , (2.34)which can be readily solved numerically.Let us discuss the initial conditions in our numerical solutions that are meaningful inheavy-ion experiments at RHIC and
LHC . One can conveniently measure the time andspace distances in terms of fm (Fermi), and the energy in terms of fm − = 197 MeV. Therelation z H = πT in the neutral blackhole solution ( Q = 0) comes from the Euclidean10 .0 0.2 0.4 0.6 0.8 1.0 1.2 QCD time H fm L H fm L H fm L H fm L Figure 1: The thermalization history of the falling mass shell for
RHIC (left) and
LHC (right). The late-time temperature is T = 300 (400) MeV for RHIC ( LHC ), andthe axial chemical potentials are µ A = 50 MeV (dotted), µ A = 100 MeV (dashed), and µ A = 200 MeV (solid). We observe that the system thermalizes mostly by t . z H = π β where β is the period of the compactified Euclidean time.Since this period (the inverse temperature) is now measured in units of fm, one can alsomeasure the holographic coordinate z in fm. According to the holographic principle, z maps to the inverse energy scale in the QCD which is also measured in fm, but whatis not fixed a priori is a possible numerical rescaling between z measured in fm and theinverse energy scale in QCD also measured in fm. Guided by the relation z H = πT for theneutral blackhole ( Q = 0), we will assume the relation between z and the QCD energyscale E as †† z = 1 πE , (2.35)with both sides being measured in fm. The natural initial condition for the out-of-equilibrium plasma created right after the collision of two heavy-ions is characterizedby the saturation scale Q s , which governs the initial gluon distributions. Roughly speak-ing, gluons with momenta less than Q s are densely saturated in the distribution, whereasthe states with higher momenta than Q s are under-occupied, so that Q s sets a nice bound-ary between the different UV and IR behaviors. Therefore, we naturally set our initialcondition of the falling mass shell to be z ( t U = 0) = z i = 1 πQ s , ˙ z ( t U = 0) = 0 . (2.36) †† This relation in fact depends on what probe we are looking at in the holography. For example,for fundamental quark flavor, the relation between the quark mass and the position z of the probebrane contains an extra factor p g Y M N c . Since the blackhole describes deconfined degrees of freedom ofgluons, and we are mainly interested in thermalization of gluonic degrees of freedom in our description,the mapping (2.35) guided by the blackhole seems appropriate for our purpose. RHIC , we take Q s = 0 .
87 GeV=4.42 fm − , and for LHC we have Q s = 1 . − . To fix m and Q in the solutions, we use the late time temperature T = 300 MeV for RHIC and T = 400 MeV for LHC and several exemplar values for µ A using the relations between them and ( m, Q ) given by (2.15). These data and the aboveinitial conditions are enough to determine the integration constant C and the uniquenumerical solution. See Table 1 and Table 2.In Figure 1 we show the time history of falling mass shell trajectory in QCD time t U for a few exemplar values of µ A = 50 , ,
200 MeV. By the time t . Before going into the detailed computations of chiral magnetic conductivity and chiralmagnetic wave in the solutions obtained in the previous section, it is useful to understandthe global structure of the geometry of the solutions and the quasi-static approximationwe are going to use. This will help us to understand the applicability and the limitationof the quasi-static approximation: the quasi-static approximation will be fine far away12rom equilibrium, but will not be trustable when the mass shell is close enough to theequilibrium horizon.The Penrose diagram of the falling mass shell solution in the previous section is givenin Figure 2(a). The mass shell (the black thick line) falls into a singularity at z = ∞ ,and it crosses an event horizon (denoted as H) in a finite Eddington-Finkelstein time t EFc .Note that the Eddington-Finkelstein time is better suited to correctly capture the causalstructure in the geometry: a light signal sent from the UV boundary z = 0 propagatesinto the bulk geometry whose trajectory is a line of constant Eddington-Finkelstein timeby definition (the dashed line with t EF ). Since any response should remain inside a causallight-cone defined by these light geodesics, these constant Eddington-Finkelstein timelines set a causal structure of the response functions. For example, it is clear that anysignal that is sent after t EFc (the time when the mass shell crosses the event horizon) wouldfeel the full presence of the event horizon, so that the system after t EFc will be a fullythermalized plasma, that is, any response functions after t EFc will precisely be equal to thethermal response functions determined by the event horizon. The signals sent before t EFc may see the presence of the falling mass shell above the horizon (like the one with t EF ), sothe responses from those signals can include non-equilibrium features. This means that t EFc can be interpreted as the thermalization time of the falling mass shell solution.The relation between the time t U in the previous section (QCD time) and the Eddington-Finkelstein time t EF is easily found as t EF = t U − Z z dz ′ f ( z ′ ) , (3.37)so that they agree at the UV boundary z = 0. The AdS-RN metric above the shell looksin terms of t EF as ds = 1 z (cid:0) − f ( z )( dt EF ) − dt EF dz + ( d~x ) (cid:1) . (3.38)The falling trajectory in the previous section is given in terms of t EF as dzdt EF = f ( z ) r(cid:16) Cz + m C − Q z C (cid:17) − r(cid:16) Cz + m C − Q z C (cid:17) − f ( z ) − r(cid:16) Cz + m C − Q z C (cid:17) − . (3.39)Note that in terms of the original time t , it takes an infinite time for the mass shell to crossthe event horizon at z = z H , but it is a coordinate artifact. A finite t EFc is manifested in13 less obvious way in the t coordinate: it is the critical time after which the signal of lightcannot catch the falling shell [58]. We stress, however, that the above argument does nottake into account spatial extension of the background and the probe. Space-like probessuch as strings and Wilson lines can easily give a thermalization time larger than t EFc , seefor example [49].Figure 2(b) shows the Penrose diagram of the quasi-static approximation geometry:the static mass shell (the thick black line) sitting at a constant radius z = z s borders theAdS-RN geometry above the shell ( z < z s ) and the pure AdS below the shell ( z > z s ).We see that for a fixed time t EF , the difference between the full space (Figure 2(a))and the quasi-static geometry may be small around the region of the constant Eddington-Finkelstein time t EF geodesic, if we can safely neglect the velocity of the falling of the massshell at that moment. This is the case where the quasi-static approximation is applicable,and this happens when the mass shell is far away from the horizon describing far out-of-equilibrium situations. However, as the time becomes closer to the critical time t EFc , it isclear that the quasi-static geometry cannot capture the process of thermalization: there isno counterpart of the true event horizon in Figure 2(b). The IR horizon at z = ∞ is non-thermal. The quasi-static geometry at z s = z H is a singular ill-defined geometry wherethe blackhole horizon and the IR horizon overlap with a zero proper length separation. Given the time dependent backgrounds obtained in the previous section as the holographicdescription of out-of-equilibrium plasma with a finite axial charge density, it is interestingto see how the properties of the plasma evolve in time. We are interested in the chargetransports originating from triangle anomaly in the presence of the magnetic field, andin this section we will treat the magnetic field as a probe to the axially charged plasma,and compute the corresponding Chiral Magnetic current, generalizing the results of [19] toout-of-equilibrium case. Although the most precise way of studying the problem would beto solve the time-dependent partial differential equations of the system, we will simplifythe problem by approximating the falling mass shell to be quasi-static compared to thetime-scales of the probes, so that we can solve the time-independent ordinary differentialequations instead. We will discuss the regime of validity of the quasi-static approximationin our results later.Treating the magnetic field as a probe, we will compute the chiral magnetic conduc-14ivity, σ χ , defined by ~J EM = σ χ ( ω ) ~B ( ω ) , (4.40)where the magnetic field (probe) has a definite frequency ω . One naturally expects thatour results for very low ω would not be consistent with the quasi-static approximation,and we will specify precisely where we can trust the results shortly. As the position ofthe quasi-static mass shell changes in time, the chiral magnetic conductivity also evolvesin time. Combining with the time history of the falling mass shell in the previous sectionthen allows us to discuss the time-evolution of the chiral magnetic conductivity in realisticconditions relevant for RHIC and
LHC .We turn on a time-dependent magnetic field of frequency ω as a linearized probe, andtry to find the response of the system given by the (quasi-static) mass shell geometry withan axial charge density, ds U = dz f ( z ) z − f ( z ) z dt + ( d~x ) z (upper region : z < z s ) ,ds L = dz z − dt L z + ( d~x ) z (lower region : z > z s ) , (4.41)where z = z s is the (quasi-static) location of the mass shell. Note that we have used thenotation t for the upper part of the metric since it is identified with the QCD time. Thetwo times t and t L are matched at z = z s by p f ( z s ) t = t L , (4.42)in order for the whole metric to be continuous, which is the quasi-static limit of the Israeljunction condition.The matching relation (4.42) in frequency space becomes ω = p f ( z s ) ω L , (4.43)which will be used in solving the equations in the frequency space. Inspecting the lin-earized equations of motion from (2.12), one can easily find that the equation for thevector gauge field A V decouples from those of the metric and the axial gauge field A A ,and since the current and the magnetic field of our interests are all vector quantities, itis enough to consider that equation only, ∂ N (cid:16) √− g ( F V ) MN (cid:17) − κ ǫ MNP QR (cid:16) F (0) A (cid:17) NP ( F V ) QR = 0 , (4.44)15here the vector fields appearing represent linearized fluctuations from our backgroundsolution in the previous section, and F (0) A is the background value of the axial gauge fieldin the solution, given by F (0) A = dA (0) A = − zQdz ∧ dt (upper region) , F (0) A = 0 (lower region) . (4.45)Noting that the shell is vector charge neutral, the natural junction condition for the gaugefield is the continuity of its value and normal derivative. We choose to work in the gauge A z = 0 for both upper and lower regions. From the continuity of the value and the normalderivative, we require [ A M dx M ] = [ F MN n M dx N ] = 0 where n M is the unit normal vectorto the shell. We substitute dx M = ∂x M ∂ξ i dξ i and noting that dξ i can be arbitrary, we endup with [ A M ∂x M ∂ξ i ] = [ F MN n M ∂x N ∂ξ i ] . (4.46)In the quasi-static approximation, we simply set ˙ z = 0 in (2.25), and from the abovejunction conditions, we find A Ut = p f ( z s ) A Lt L , A Ui = A Li ( i = 1 , , , (4.47)whereas the continuity of the normal derivatives gives us ∂ z A Ut = ∂ z A Lt L , p f ( z s ) ∂ z A Ui = ∂ z A Li . (4.48)We have omitted the subscript V without confusion. We solve (4.44) with the abovejunction conditions at z = z s .To introduce a magnetic field along, say, x direction, we consider a fluctuation of A with a momentum along x to have a non-zero F = B , A ( t, ~x, z ) = A ( z ) e − iωt + ikx , (4.49)and the consistency of the equation of motion (4.44) necessitates the introduction of A fluctuation as well, A ( t, ~x, z ) = A ( z ) e − iωt + ikx . (4.50)This coupling between A and A is via the Chern-Simons term, and indeed we willobtain the non-zero chiral magnetic current along x (the direction of the magnetic field)from the induced A fluctuation. Other components of the gauge field can be turned16ff consistently. The equations of motion are explicitly given as (note our convention ǫ zt = 1), ∂ z (cid:18) fz ∂ z A U (cid:19) + 1 z (cid:18) ω f − k (cid:19) A U + 12 iκQzkA U = 0 ,∂ z (cid:18) fz ∂ z A U (cid:19) + 1 z (cid:18) ω f − k (cid:19) A U − iκQzkA U = 0 ,z∂ z A L − ∂ z A L + z (cid:0) ω L − k (cid:1) A L = 0 ,z∂ z A L − ∂ z A L + z (cid:0) ω L − k (cid:1) A L = 0 , (4.51)where the first two equations are in the upper region and the last two in the lower region.In the lower region, the equations are easily solved by Hankel functions, and we requirethe infalling boundary condition for the physical retarded response functions. In theupper region, one has to solve the equation numerically. Since we would like to turn onthe external magnetic field along x direction, the A U field should have a near boundaryexpansion close to z = 0 as A U ( z ) = A (0)2 + A (2)2 z + A h z log z + · · · , (4.52)and the external magnetic field is identified as eB = F (0)12 = ikA (0)2 . (4.53)The A U field should not have any boundary value by the choice of the boundary condition,so its near boundary expansion should be A U ( z ) = A (2)3 z + A h z log z + · · · . (4.54)The infalling boundary condition in the lower region and the above near z = 0 boundarycondition in the upper region uniquely determine the full solution, which is linear in thevalue A (0)2 (and hence the magnetic field) that sets the overall normalization. Note thatwe have to match the solutions in the two regions via the junction conditions (4.47) and(4.48). Once the solution is found given the normalization set by A (0)2 , the current along x direction which is our chiral magnetic current along the direction of the magnetic fieldis obtained as J EM = eJ V = e πG A (2)3 , (4.55)so that the chiral magnetic conductivity is given by σ χ = J B = e πG A (2)3 ikA (0)2 , (4.56)17hich is well-defined independent of the normalization of the solution.The prescription (4.55) needs some explanations. In the careful holographic renor-malization of Einstein-Maxwell-Chern-Simons theory [64], the near boundary expansionof the gauge field is given by A µ = A (0) µ + A (2) µ z + A hµ z log z + · · · . Note that our actiondensity (2.11) has already taken into Bardeen counter term. The current expectationvalue can be obtained from functional derivative of the action as: J µ = 14 πG (cid:0) A (2) µ + A hµ (cid:1) + 3 κ πG ǫ µναβ (cid:16) ( A (0) A ) ν ( F (0) V ) αβ (cid:17) . (4.57)The last contribution from the Chern-Simons term needs a special care. To obtain physicalchiral magnetic effect, one needs to distinguish axial chemical potential µ A and bound-ary value of axial gauge field A (0) A [22, 65]. In Minkowski signature black-hole solution,the time component of A A does not need to vanish at the horizon without causing anysingularity problem [65]. The boundary value of the axial gauge field A (0) A is clearly zeroin real physical configuration created in heavy-ion collisions, while the chemical potentialis simply defined as a work needed to bring a unit charge from infinity to the plasma, sothat the two things are different. For this reason we have chosen the gauge field configura-tion A A to have vanishing boundary value, but correspond to a finite chemical potential.Having zero A (0) A in our plasma in heavy-ion collisions gives no additional contribution to(4.55) from the Chern-Simons term, and it does not affect our formula (4.55) for the J .We are interested in the homogeneous magnetic field with a finite frequency, so wewould like to consider k → B is fixed. This limit can be achieved in thefollowing way [19]. Looking at the equations of motion (4.51), the terms originatingfrom the Chern-Simons term that mix A and A are linear in k , so that one naturallyexpects that the induced A fluctuation from the source of A (the magnetic field) willbe linear in k in k → k → k dependence in A , the other k terms in (4.51) are not relevant, and can be neglected.These considerations lead to expanding the solution in powers of k as A ( z ) = a ( z ) + O ( k ) A ( z ) = ka ( z ) + O ( k ) , (4.58)where a and a satisfy the equations ∂ z (cid:18) fz ∂ z a U (cid:19) + 1 z ω f a U = 0 ,∂ z (cid:18) fz ∂ z a U (cid:19) + 1 z ω f a U − iκQza U = 0 , ∂ z a L − ∂ z a L + zω L a L = 0 ,z∂ z a L − ∂ z a L + zω L a L = 0 , (4.59)with the same junction conditions (4.47) and (4.48). The frequency dependent chiralmagnetic conductivity then becomes σ χ ( ω ) = − i e πG a (2)3 a (0)2 , (4.60)with a similar near boundary expansion as before, a U = a (0)2 + a (2)2 z + · · · , a U = a (2)3 z + · · · . (4.61)The use of (4.56) requires that A U tends to zero as it approaches the boundary. However,fine tuning the boundary value is not numerically convenient. In appendix A, we showhow to calculate numerically both chiral magnetic conductivity and electric conductivityfrom the solutions with non-vanishing boundary value of A U .As remarked previously, the quasi-static approximation has its limitation. It is validwhen the speed of the probe, in this case speed of light for the gauge field, is much greaterthan the falling speed of the shell. As the shell approaches the “horizon”, both the shelland the speed of light are infinitely red-shifted. We expect the quasi-static approximationto break down as z → z H as discussed in section 3. Furthermore, this picture relies on theassumption that we can treat the gauge field as a massless particle. It is justified whenthe wave length of the gauge field is much shorter than curvature of AdS space. Thisis given by ωz &
1. These provide sufficient conditions for quasi-static approximation.In appendix B, we work out more precise conditions to find that a wide region in thefrequency ω space appears to be consistent with the quasi-static approximation. Wesimply quote the results here: ˙ z ≪ H (1)0 ( ωz/ √ f ) H (1)1 ( ωz/ √ f ) p f + ˙ z , p f + ˙ z ≫ H (1)0 ( ωz/ √ f ) H (1)1 ( ωz/ √ f ) ˙ z. (4.62)Figure 3 shows the region of validity for the quasi-static approximation. Generically,the quasi-static approximation corresponds to probing the evolving medium with a planewave, which has infinite resolution ∆ ω = 0 in frequency, but vanishing resolution ∆ t = ∞ in time. We know by uncertainty principle ∆ ω ∆ t ≥ /
2. For a medium evolving19 .2 0.3 0.4 0.5 t (cid:144) fm1234567 Ω (cid:144) GeV 0.2 0.3 0.4 t (cid:144) fm246810 Ω (cid:144) GeV
Figure 3: The region of validity of quasi-static approximation in the frequency space asa function of time for RHIC ( T = 300 MeV, µ A = 50 MeV) and LHC ( T = 400 MeV, µ A = 50 MeV). The shaded region above the curve is consistent with the quasi-staticapproximation.sufficiently slow in time, ∆ t can be made very large, which allows for a small ∆ ω . Thisis the way how the quasi-static approximation works. The breaking down of quasi-staticapproximation at late time seems to suggest that the evolution of medium becomes fasterat late stage of thermalization, while a naive expectation from slow motion of the shellnear horizon that would lead to the opposite conclusion is illusionary, as it is also clearin the Penrose diagrams in section 3.With the falling trajectory obtained in the previous section, z s = z ( t ), one can discusshow σ χ ( ω ) changes in QCD time t in our quasi-static approximation. Note that thechiral magnetic conductivity σ ( ω ) in general has both real and imaginary parts, andwe would like to parametrize it by the magnitude | σ χ ( ω ) | and the response time delay∆ t ( ω ) = arg ( σ χ ( ω )) /ω , defined by J e − iωt = σ χ ( ω ) Be − iωt = | σ χ ( ω ) | B − iω ( t +∆ t ( ω )) . (4.63)As an example, we plot in Figure 4 the time evolution of chiral magnetic conductivitycharacterized by the magnitude and the response time delay for three particular valuesof ω with a fixed µ A . We plot the same quantities in Figure 5 for three values of µ A witha fixed value of ω . We present the results with respect to the equilibrium zero frequencyvalue of chiral magnetic conductivity σ ≡ − κe πG µ A = e N c π µ A , (4.64)where the last equality comes from the relation (1.8): κ = − G N c π . Several conclusionscan be drawn from our results: 20 .1 0.2 0.3 0.4 0.5 t (cid:144) fm0.050.100.150.200.250.300.35Abs @ Σ D Σ (cid:144) fm0.0050.0100.0150.020 D t (cid:144) fm Figure 4: The chiral magnetic conductivity as a function of thermalization history fordifferent frequencies: ω = 200 MeV(blue solid), ω = 300 MeV(red dashed) and ω = 400MeV(green dotted). The thermalization history of the falling mass shell is for RHIC witha final temperature T = 300 MeV and µ A = 50 MeV. The left plot shows the evolutionof the magnitude of chiral magnetic conductivity and the right plot shows the time delayof the response.i) the chiral magnetic conductivity, both its magnitude and the time delay, increasesin general as the medium thermalizes. The increase of the magnitude is consistent withthe naive expectation that as the medium thermalizes, more and more thermalized con-stituents can participate in the formation of chiral magnetic current.ii) From Figure 4 we observe that the magnitude of chiral magnetic conductivitychanges very little as we vary the frequency of the probe, while increase of the latter doesresult in longer delay in the response of the medium. This is in contrast to conventionalelectric property of materials. Simple Drude model of electric conductivity shows thatelectric field of higher frequency results in lower magnitude of electric conductivity andshorter delay in response. The difference should not be surprising as the non-dissipativechiral magnetic conductivity is of different nature from the dissipative electric conductiv-ity. iii) Figure 5 shows that a larger chemical potential gives a smaller ratio of the magni-tude of the conductivity to σ , and a shorter delay in response. However, we should bearin mind that a larger chemical potential also delays the thermalization time. Note thatwe are comparing the conductivity at the same absolute time, which corresponds to lessthermalized medium for larger chemical potential. Therefore the results are consistentwith the observation i). 21 .1 0.2 0.3 0.4 0.5 0.6 0.7 t (cid:144) fm0.050.100.150.200.250.30Abs @ Σ D Σ (cid:144) fm0.0020.0040.0060.0080.0100.012 D t (cid:144) fm Figure 5: The chiral magnetic conductivity as a function of thermalization history at afixed frequency ω = 200 MeV for different axial chemical potentials: µ A = 50 MeV(bluesolid), µ A = 100 MeV(red dashed) and µ A = 200 MeV(green dotted). The thermalizationhistory of the falling mass shell is for RHIC with a final temperature T = 300 MeV. Theleft plot shows the evolution of the magnitude of chiral magnetic conductivity and theright plot shows the time delay of the response. In this section, we study out-of-equilibrium property of charge transports originating fromtriangle anomaly in a different angle: the chiral magnetic wave. The chiral magnetic wavedescribes how (chiral) charge fluctuations behave in the presence of an external magneticfield which we assume to be static. In the equilibrium plasma, the chiral magnetic wavehas a dispersion relation of the form [36] ω = ∓ v χ k − iD L k + · · · , (5.65)with the velocity v χ being proportional to the magnetic field v χ = N c e B π χ , χ ≡ ∂J ∂µ , (5.66)and the sign in the first term (the direction of propagation) depends on the chirality of thefluctuations. Since the chiral magnetic wave is about linearized charge fluctuations, thebackground plasma can be neutral and we consider a neutral (out-of-equilibrium) plasmain this section for simplicity. We are interested in how the dispersion relation of the chiralmagnetic wave changes in time in our out-of-equilibrium conditions represented by fallingmass shell geometries in the previous sections. The charge neutral background can beeasily found by putting Q = 0 in the previous solutions.The dispersion relation of chiral magnetic wave in neutral plasma in equilibrium can beeasily found from poles of retarded current-current correlator in Fourier space. However22he same procedure does not carry over straightforwardly out of equilibrium. We knowthat in equilibrium the poles in the complex ω plane carry the same information asthe full retarded function defined on the real ω axis. This is because the equilibriumretarded correlator has infinite resolution in ω , allowing for an analytic continuation intothe complex plane. For plasma out of equilibrium, the retarded correlator in ω space isonly approximately defined with a finite resolution via Wigner functions, and the analyticcontinuation may not be well justified. Therefore, we stick to work in the real ω domainof the retarded Green’s function, which is more directly relevant to the real-time behaviorof fluctuations. As in the previous section, we will work in the quasi-static approximation.The lowest frequency wave-like excitation, that we call out-of-equilibrium chiral magneticwave, will be identified as a peak in the imaginary part of the correlator (spectral function)below the lightcone ω < k . Higher excitations will in general appear above the lightcone ω > k . We will trace the time evolution of the identified out-of-equilibrium chiral magneticwave peak.To compute the spectral function in the falling mass shell geometry, we turn on astatic, homogeneous magnetic field along x direction ~B = B ˆ x . We consider a weakmagnetic field without backreaction to the shell geometry. A constant magnetic field isa trivial solution of the equations of motion. This constitutes our background solution,from which we consider linearized fluctuations of both axial and vector gauge fields δA A,V that describe chiral charge fluctuations in the QCD side (we omit δ symbol in the belowwithout much confusion). The linearized fluctuations of gauge fields in fact decouple fromthose of the metric in the case of neutral background, and this simplification is one reasonwhy we consider neutral plasma in our study. The linearized equations for the gauge fieldsfrom our main equations (2.12) are diagonalized in the chiral basis defined as A L ≡ A V − A A , A R ≡ A V + A A , (5.67)which represent chiral charge fluctuations. Explicitly, their equations read as ∂ N (cid:0) √− g ( F L,R ) MN (cid:1) ± κeBǫ M QR ( F L,R ) QR = 0 , (5.68)where we have used that the background value of A L,R are given by( F (0) L ) = ( F (0) R ) = eB . (5.69)Since left- and right-handed fluctuations are simply related by B → − B , let us focus onthe right-handed fluctuations only (the lower sign in the above equation) and omit thesubscript R in the following. 23he chiral magnetic wave is a longitudinal charge-current fluctuation, so we considera longitudinal momentum k along x (the direction of the magnetic field) and turn on A t and A fluctuations in the gauge A z = 0, A t = A t ( z ) e − iωt + ikx , A = A ( z ) e − iωt + ikx , (5.70)where other components of the gauge field can be consistently turned off. The equationsof motion then become ωz ∂ z A t + kfz ∂ z A + 6 κeB ( ωA + kA t ) = 0 , − kzf ( ωA + kA t ) + 6 κeB∂ z A + ∂ z (cid:18) z ∂ z A t (cid:19) = 0 , − ωzf ( ωA + kA t ) − κeB∂ z A t − ∂ z (cid:18) fz ∂ z A (cid:19) = 0 , (5.71)for the upper region z < z s and the equations in the lower region z > z s is the samewith f = 1. We have to match the upper and lower solutions by the previous junctionconditions (4.47) and (4.48). It is more convenient and intuitive to work with a gaugeinvariant variable corresponding to the electric field along x direction defined as E ≡ kA t + ωA , (5.72)for which the equation simply becomes (cid:0) E U (cid:1) ′′ + (cid:18) ω f ′ f ( ω − f k ) − z (cid:19) (cid:0) E U (cid:1) ′ + 1 f (cid:18) ω f − k − (6 κeBz ) + 6 κeBωkzf ′ ω − f k (cid:19) E U = 0 , (5.73)for the upper region z < z s where ′ ≡ ddz , and the equation in the lower region is similarwith replacing f = 1 and ω L = ω/ p f ( z s ). The junction condition in terms of E is E U = p f ( z s ) E L , (cid:0) E U (cid:1) ′ = (cid:0) E L (cid:1) ′ . (5.74)Guided by the equilibrium chiral magnetic wave, we expect to find a chiral magnetic wavepeak in the positive ω axis when k > B > E L at IR infinity. Howeverwe see a subtle problem: due to the (6 κeBz ) term, the solution to E L either divergesor decays exponentially at IR infinity for any frequency momentum. Once we choose theexponentially decaying solution which is naturally real, the full solution will be purely24eal for any ω and k . This means that the imaginary part of the retarded correlator(spectral function) can only have delta-function peaks corresponding to infinitely stablebound states, without any continuum part of our interest that may feature chiral magneticwave as the system thermalizes. This unphysical drawback seems to appear as a resultof our probe limit, where we neglect the backreaction of the B -field to the metric. Inour theory the metric at IR infinity will necessarily be changed in the presence of any B no matter how small B is [66]: the IR geometry should be modified to AdS × R . Oncethis has been taken into account, we checked that the correct geometry does allow the(complex-valued) in-falling IR boundary condition.We defer a full treatment including the back reaction of the B field to the future,and use instead the following approximation that still captures the main physics effect ofthe back reacted geometry: below the shell, we introduce an IR cutoff z c beyond whichwe drop the term (6 κeBz ) such that the solution can be chosen to be in-falling. Thismimics the effect of the AdS × R below the IR cutoff. Above the IR cutoff, we reinstatethe term (6 κeBz ) and find the full solution up to the UV boundary. The cutoff z c isnaturally chosen to be z c = 1 / √ B where the back reaction starts to be important [66].Our treatment is well justified when B ≪ T and the shell is not too close to the horizon.With all these cares, the solution for E U has the following expansion near z = 0, E U ( z ) = E (0) + E (2) z + E h z log z · · · . (5.75)The spectral function χ ( ω, k ) is defined as the imaginary part of the retarded currentcorrelator, χ ( ω, k ) = − Im ( πT ) E (2) πG E (0) . (5.76)In Figure 6 (left figure), we show snapshots of spectral function at different times ofthermalization. Typical spatial momentum relevant to heavy ion collisions is ∼ k ∼
200 MeV. We restrict ourselves to the region below the lightcone,where we expect to find a chiral magnetic wave peak. We also show the equilibriumthermal spectral function as a reference (right figure). We discuss the salient features inFigure 6:i) The snapshots of spectral functions are taken at times before the break-down of thequasi-static approximation. We observe that a sharp peak appears out of the backgroundplateau, which we identify as the out-of-equilibrium chiral magnetic wave.ii) The peak sits close to the left edge of the plateau. The left edge of the plateau isalmost vertical. Further to the left, the spectral function vanishes identically. The location25
92 194 196 198 200 Ω (cid:144) MeV10 - - - Χ
50 100 150 200 Ω (cid:144) MeV0.0010.010.1 Χ Figure 6: The spectral functions in unit of ( πT ) πG as a function of ω at fixed k = 200 MeV.The magnetic field and temperature are chosen for the RHIC: B = m π and T = 300 MeV,which satisfy the condition B ≪ T for the justification of our IR cutoff. The left plotshows the snapshots of the spectral functions at t = 0 .
03 fm(blue solid), t = 0 .
38 fm(reddashed) and t = 0 .
53 fm(green dotted). The right plot shows the equilibrium spectralfunction as a reference.of the left edge can be understood analytically: due to the warping factor f ( z s ), there is amismatch between the frequencies in the upper and lower region ω L = ω/ p f ( z s ). When ω L crosses k from below, the solution in the lower region changes from an exponentiallydecaying real function to a complex-valued in-falling solution (more specifically, it changesfrom a modified Bessel function to a Hankel function). The appearance of in-fallingwave induces a flux toward IR resulting in a non-vanishing imaginary part of the currentcorrelator. Indeed, we have verified numerically that the location of the left edge is givenby ω = p f ( z s ) k with very high accuracy. The right ridge of the plateau in differentsnapshots seem to lie on top of each other.iii) We parametrize the location of our chiral magnetic wave peak by ω = p f ( z s ) k + ∆ ω ( k, B ) , (5.77)where the first piece is the left edge we discussed in (ii) and we have indicated that ∆ ω is afunction of k and B . If we naively extrapolate (5.77) to the equilibrium limit, i.e. z s → ω can reproduce the equilibriumchiral magnetic wave. We have studied the dependence of ∆ ω on k and B , and do find thefeatures characterizing chiral magnetic wave. Figure 7 shows that ∆ ω has an excellentlinear dependence on B and approximate linear dependence on k . These are indeedthe behaviors of chiral magnetic wave in the small magnetic field and long wave lengthlimit. However we mention that the precise connection between out-of-equilibrium chiralmagnetic wave we found and the one in equilibrium is only suggestive, because quasi-static26
00 400 600 800 1000 k (cid:144)
MeV0.20.40.60.8 DΩ (cid:144) MeV 1 2 3 4 B (cid:144) m Π DΩ (cid:144) MeV
Figure 7: Left: ∆ ω as a function of k at fixed B = m π . Right: ∆ ω as a function of B atfixed k = 200 MeV. In both plots, T = 300 MeV and t = 0 .
53 fm.approximation prevents us from going further in time. For the parameters we explored,the group velocity receives most of its contribution from the first term in (5.77), which issignificantly larger than the group velocity of the equilibrium chiral magnetic wave. Thisindicates that the out-of-equilibrium chiral magnetic wave moves the chiral charges muchfaster, potentially enhancing its physical effects in out-of-equilibrium conditions.iv) Note that the chiral magnetic wave for right-handed charges is expected to move tothe same direction as the magnetic field (which means ∆ ω > k >
B >
B <
0) or turned off ( B = 0), whilekeeping the same spatial momentum: see Figure 8 (left figure). We confirm that the peakstructure disappears in the ω > B < B = 0, wesee a structure at ω = p f ( z s ) k , which marks the transition between the in-falling waveand the exponentially decaying real function at the left edge we discussed before. In thecase of B >
0, the structure is shifted away from the left edge to the right, in accordancewith the analysis of the imaginary part.v) We have performed our analysis for different values of IR cutoff: z c = 1 / √ B, . / √ B and 2 / √ B to check how sensitive our results are to the IR cutoff. Different values of IRcutoff change the overall normalization of the spectral function, but do not change ourresults for the peak location which are robust. On the other hand, we have also investi-gated the case with IR cutoff removed, i.e. the equation (5.73) with (6 κeBz ) -term keptall the way through. In this case, the solution in the lower region is always exponentially27
92 194 196 198 200 Ω (cid:144) MeV10 - - - Χ
188 190 192 194 196 198 200 Ω (cid:144) MeV - - - - @ G R D Figure 8: Comparison of spectral functions in unit of ( πT ) πG taken at t = 0 .
53 fm as afunction of ω at fixed k = 200 MeV and T = 300 MeV with different magnetic fields: B = m π (blue solid), B = − m π (red dashed) and B = 0 (green dotted). The right plotshows the real part of retarded correlator in unit of ( πT ) πG with the same parameters andcolor coding. To guide the eyes, we have rescaled the B = m π case by 1 /
200 and the B = 0 case by 10.decaying and is given by E L = e − κBz U ( k − ω L κB , , κBz ) , (5.78)where U is the confluent hypergeometric function. As expected, this case gives vanishingspectral function up to delta-function peaks which are not captured numerically. There-fore, our IR cutoff is crucial for capturing the correct IR physics.Before we close this section, it is interesting to note how the spectral function de-velops its non vanishing smooth part in the ω < k region as the medium thermalizes.Recall that the spectral function is gapped for ω < k in vacuum. In the thermalizingmedium, we observed above that the gap shrinks as the medium thermalizes z s → ω L = ω/ p f ( z s ). As the mediumthermalizes, p f ( z s ) goes to zero, eventually closing the gap. This feature is generic inthe gravitational collapse model of thermalization and insensitive to the presence of themagnetic field. We have studied chiral magnetic conductivity and chiral magnetic wave in out-of-equilibriumconditions that undergo thermalization. Within the quasi-static approximation, we fo-cused on far out-of-equilibrium region and explored the parameters relevant for RHICand LHC. For the chiral magnetic conductivity, we considered both its magnitude and28he time delay in response. We found that the magnitude is insensitive to the frequencyof the magnetic field while the time delay grows with the frequency. This is in contrastto ordinary electric conductivity. As a function of time, both the magnitude and timedelay grows, which can be understood as more and more thermalized constituents becomeavailable as the system thermalizes.For the chiral magnetic wave, far away from equilibrium, we found a sharp peak struc-ture in ω below the lightcone in the spectral function, signaling the out-of-equilibriumchiral magnetic wave. The peak structure is unique to the magnetic field and is a mani-festation of anomaly. The location of the peak in ω relative to a kinematical edge p f ( z s ) k depends linearly on the momentum and the magnetic field, which is a feature same tothose of the chiral magnetic wave in equilibrium. However, the group velocity receivesa sizable contribution from the kinematic p f ( z s ) k piece which makes the wave movingmuch faster than in the equilibrium. The correct physics origin of this behavior is notcompletely clear to us.The results of this work can be generalized in two aspects: the first is to look at chiralmagnetic wave beyond the weak magnetic field limit by including the back reaction of themagnetic field to the metric [66]. This will be more relevant for LHC, which is expected toproduce much stronger magnetic field with only a modest increase of the temperature. Thesecond, perhaps more interestingly, is to go beyond the quasi-static approximation. Thiscan be achieved by raising the temporal resolution and lowering the frequency resolution.This should allow us to extend the coverage of our analysis to near-equilibrium situationsand to address the question on the transition from the out-of-equilibrium chiral magneticwave to the equilibrium chiral magnetic wave. Acknowledgement
We thank Xu-Guang Huang, Dima Kharzeev, Jinfeng Liao, Kiminad Mamo, LarryMcLerran, Todd Springer, Misha Stephanov, Derek Teaney, Raju Venugopalan, Yi Yin foruseful discussions. SL is supported by RIKEN Foreign Postdoctoral Researchers Program.29
Alternative calculation of chiral magnetic conduc-tivity
Let us start by recalling the power expansion in k . A ( z ) = a ( z ) + O ( k ) , A ( z ) = ka ( z ) + O ( k ) , (A.79)where a and a satisfy the equations ∂ z (cid:18) fz ∂ z a U (cid:19) + 1 z ω f a U = 0 ,∂ z (cid:18) fz ∂ z a U (cid:19) + 1 z ω f a U − iκQza U = 0 ,z∂ z a L − ∂ z a L + zω L a L = 0 ,z∂ z a L − ∂ z a L + zω L a L = 0 . (A.80)In the lower region, a L and a L decouple and the solutions are given by Hankel functionswith their ratio unfixed. Requiring the vanishing of a (0)3 would need fine tuning of theratio. This is actually not needed. Suppose we start with a solution in the lower regionwith arbitrary ratio. Matching it to the solution above and integrating to the boundary,we obtain the following electric and magnetic fields to order k : eE = − iωa (0)1 , eE = iωka (0)3 , eB = 0 , eB = ika (0)1 , (A.81)and the currents can be extracted from boundary expansion of a U and a U : J EM = e πG a (2)2 , J EM = e πG ka (2)3 . (A.82)(A.81) and (A.82) are related by electric conductivity σ and chiral magnetic conductivity σ χ : J EM = σE + σ χ B , J EM = σE + σ χ B , (A.83)from which we can solve for σ and σ χ at the same time. It is easy to show that the resultsare independent of the ratio we choose in the lower region. B Region of applicability of quasi-static approxima-tion
In the quasi-static approximation, we neglect terms proportional to ˙ z in the continuitycondition of A µ ∂x µ ∂ξ i , F µν u µ ∂x µ ∂ξ i and F µν n µ ∂x µ ∂ξ i . The conditions from ξ i = x , x (below we30uppress the transverse indices) are given by A Ux = A Lx ,∂ t U A Ux z ˙ zf + ∂ z A Ux ˙ t U zf = ∂ t A Lx z ˙ z + ∂ z A Lx ˙ tz,∂ t U A Ux ˙ t U + ∂ z A Ux ˙ z = ∂ t A Lx ˙ t + ∂ z A Lx ˙ z. (B.84)To neglect the terms proportional to ˙ z on the left hand side (upper region), we need ω ˙ zf ≪ ∂ z A Ux ( ω ) A Ux ( ω ) ˙ t U f ,ω ˙ t U ≫ ∂ z A Ux ( ω ) A Ux ( ω ) ˙ z. (B.85)Using (4.47) and (4.48), we obtain ω ˙ z √ f ≪ ∂ z A Lx ( ω/ √ f ) A Lx ( ω/ √ f ) ˙ t U f,ω ˙ t U f √ f ≫ ∂ z A Lx ( ω/ √ f ) A Lx ( ω/ √ f ) ˙ z. (B.86)Similarly, to neglect the terms proportional to ˙ z on the right hand side (lower region), weobtain ω ˙ z √ f ≪ ∂ z A Lx ( ω/ √ f ) A Lx ( ω/ √ f ) ˙ t L f,ω ˙ t L √ f ≫ ∂ z A Lx ( ω/ √ f ) A Lx ( ω/ √ f ) ˙ z. (B.87)Obviously (B.87) is included in (B.85). Using solution of A Lx in terms of Hankel function,we end up with ˙ z ≪ H (1)0 ( ωz/ √ f ) H (1)1 ( ωz/ √ f ) p f + ˙ z , p f + ˙ z ≫ H (1)0 ( ωz/ √ f ) H (1)1 ( ωz/ √ f ) ˙ z. (B.88) References [1] E. Shuryak, “Physics of Strongly coupled Quark-Gluon Plasma,” Prog. Part. Nucl.Phys. , 48 (2009). 312] G. D. Moore and M. Tassler, “The Sphaleron Rate in SU(N) Gauge Theory,” JHEP , 105 (2011).[3] P. Hegde, F. Karsch and C. Schmidt, “Calculating Quark Number Susceptibilitieswith Domain-Wall Fermions,” PoS LATTICE , 187 (2008).[4] D. Kharzeev, A. Krasnitz and R. Venugopalan, “Anomalous chirality fluctuations inthe initial stage of heavy ion collisions and parity odd bubbles,” Phys. Lett. B ,298 (2002).[5] T. Lappi and L. McLerran, “Some features of the glasma,” Nucl. Phys. A , 200(2006).[6] D. Kharzeev and A. Zhitnitsky, “Charge separation induced by P-odd bubbles inQCD matter,” Nucl. Phys. A , 67 (2007).[7] D. E. Kharzeev, L. D. McLerran and H. J. Warringa, “The Effects of topologicalcharge change in heavy ion collisions: ’Event by event P and CP violation’,” Nucl.Phys. A , 227 (2008).[8] K. Fukushima, D. E. Kharzeev and H. J. Warringa, “The Chiral Magnetic Effect,”Phys. Rev. D , 074033 (2008).[9] D. T. Son and A. R. Zhitnitsky, “Quantum anomalies in dense matter,” Phys. Rev.D , 074018 (2004).[10] M. A. Metlitski and A. R. Zhitnitsky, “Anomalous axion interactions and topologicalcurrents in dense matter,” Phys. Rev. D , 045011 (2005).[11] D. E. Kharzeev and H. J. Warringa, “Chiral Magnetic conductivity,” Phys. Rev. D , 034028 (2009).[12] D. K. Hong, “Anomalous currents in dense matter under a magnetic field,” Phys.Lett. B , 305 (2011).[13] D. Hou, H. Liu and H. -c. Ren, “Some Field Theoretic Issues Regarding the ChiralMagnetic Effect,” JHEP , 046 (2011).[14] R. Loganayagam and P. Surowka, “Anomaly/Transport in an Ideal Weyl gas,” JHEP , 097 (2012). 3215] D. T. Son and N. Yamamoto, “Berry Curvature, Triangle Anomalies, and the ChiralMagnetic Effect in Fermi Liquids,” Phys. Rev. Lett. , 181602 (2012).[16] M. A. Stephanov and Y. Yin, “Chiral Kinetic Theory,” Phys. Rev. Lett. , 162001(2012).[17] I. Zahed, “Anomalous Chiral Fermi Surface,” Phys. Rev. Lett. , 091603 (2012).[18] J. -W. Chen, S. Pu, Q. Wang and X. -N. Wang, “Berry curvature and 4-dimensionalmonopole in relativistic chiral kinetic equation,” [ arXiv:1210.8312 [hep-th]].[19] H. -U. Yee, “Holographic Chiral Magnetic Conductivity,” JHEP , 085 (2009).[20] A. Rebhan, A. Schmitt and S. A. Stricker, “Anomalies and the chiral magnetic effectin the Sakai-Sugimoto model,” JHEP , 026 (2010).[21] A. Gorsky, P. N. Kopnin and A. V. Zayakin, “On the Chiral Magnetic Effect inSoft-Wall AdS/QCD,” Phys. Rev. D , 014023 (2011).[22] A. Gynther, K. Landsteiner, F. Pena-Benitez and A. Rebhan, “Holographic Anoma-lous Conductivities and the Chiral Magnetic Effect,” JHEP , 110 (2011).[23] T. Kalaydzhyan and I. Kirsch, “Fluid/gravity model for the chiral magnetic effect,”Phys. Rev. Lett. , 211601 (2011).[24] C. Hoyos, T. Nishioka and A. O’Bannon, “A Chiral Magnetic Effect from AdS/CFTwith Flavor,” JHEP , 084 (2011).[25] D. T. Son and P. Surowka, “Hydrodynamics with Triangle Anomalies,” Phys. Rev.Lett. , 191601 (2009).[26] Y. Neiman and Y. Oz, “Relativistic Hydrodynamics with General AnomalousCharges,” JHEP , 023 (2011).[27] M. Lublinsky and I. Zahed, “Anomalous Chiral Superfluidity,” Phys. Lett. B ,119 (2010).[28] A. V. Sadofyev, V. I. Shevchenko and V. I. Zakharov, “Notes on chiral hydrodynamicswithin effective theory approach,” Phys. Rev. D , 105025 (2011).[29] S. Lin, “An anomalous hydrodynamics for chiral superfluid,” Phys. Rev. D , 045015(2012). 3330] V. P. Nair, R. Ray and S. Roy, “Fluids, Anomalies and the Chiral Magnetic Effect:A Group-Theoretic Formulation,” Phys. Rev. D , 025012 (2012).[31] J. Bhattacharya, S. Bhattacharyya and M. Rangamani, “Non-dissipative hydrody-namics: Effective actions versus entropy current,” JHEP , 153 (2013).[32] T. Kalaydzhyan, “Chiral superfluidity of the quark-gluon plasma,” [ arXiv:1208.0012[hep-ph]].[33] S. A. Voloshin, “Parity violation in hot QCD: How to detect it,” Phys. Rev. C ,057901 (2004).[34] B. I. Abelev et al. [STAR Collaboration], “Azimuthal Charged-Particle Correlationsand Possible Local Strong Parity Violation,” Phys. Rev. Lett. , 251601 (2009).[35] I. Selyuzhenkov [ALICE Collaboration], “Anisotropic flow and other collective phe-nomena measured in Pb-Pb collisions with ALICE at the LHC,” Prog. Theor. Phys.Suppl. , 153 (2012).[36] D. E. Kharzeev and H. -U. Yee, “Chiral Magnetic Wave,” Phys. Rev. D , 085007(2011).[37] G. M. Newman, “Anomalous hydrodynamics,” JHEP , 158 (2006).[38] Y. Burnier, D. E. Kharzeev, J. Liao and H. -U. Yee, “Chiral magnetic wave at finitebaryon density and the electric quadrupole moment of quark-gluon plasma in heavyion collisions,” Phys. Rev. Lett. , 052303 (2011).[39] Y. Burnier, D. E. Kharzeev, J. Liao and H. -U. Yee, “From the chiral magnetic waveto the charge dependence of elliptic flow,” [arXiv:1208.2537 [hep-ph]].[40] E. V. Gorbar, V. A. Miransky and I. A. Shovkovy, “Normal ground state of denserelativistic matter in a magnetic field,” Phys. Rev. D , 085003 (2011).[41] G. Wang [STAR Collaboration], “Search for Chiral Magnetic Effects in High-EnergyNuclear Collisions,” [arXiv:1210.5498 [nucl-ex]].[42] H. Ke [STAR Collaboration], “Charge asymmetry dependency of π + /π − elliptic flowin Au + Au collisions at √ s NN = 200 GeV,” J. Phys. Conf. Ser. , 012035 (2012).3443] S. Lin and E. Shuryak, “Toward the AdS/CFT gravity dual for High Energy Colli-sions: I.Falling into the AdS,” Phys. Rev. D , 085013 (2008).[44] S. Lin and E. Shuryak, “Toward the AdS/CFT Gravity Dual for High Energy Colli-sions. 3. Gravitationally Collapsing Shell and Quasiequilibrium,” Phys. Rev. D ,125018 (2008).[45] S. Bhattacharyya and S. Minwalla, “Weak Field Black Hole Formation in Asymptot-ically AdS Spacetimes,” JHEP , 034 (2009).[46] G. Beuf, M. P. Heller, R. A. Janik and R. Peschanski, “Boost-invariant early timedynamics from AdS/CFT,” JHEP , 043 (2009).[47] P. M. Chesler and L. G. Yaffe, “Boost invariant flow, black hole formation, and far-from-equilibrium dynamics in N = 4 supersymmetric Yang-Mills theory,” Phys. Rev.D , 026006 (2010).[48] H. Bantilan, F. Pretorius and S. S. Gubser, “Simulation of Asymptotically AdS5Spacetimes with a Generalized Harmonic Evolution Scheme,” Phys. Rev. D ,084038 (2012).[49] V. Balasubramanian, et al. , “Thermalization of Strongly Coupled Field Theories,”Phys. Rev. Lett. , 191601 (2011).[50] S. Caron-Huot, P. M. Chesler and D. Teaney, “Fluctuation, dissipation, and ther-malization in non-equilibrium AdS black hole geometries,” Phys. Rev. D , 026012(2011).[51] D. Garfinkle and L. A. Pando Zayas, “Rapid Thermalization in Field Theory fromGravitational Collapse,” Phys. Rev. D , 066006 (2011).[52] B. Wu and P. Romatschke, “Shock wave collisions in AdS5: approximate numericalsolutions,” Int. J. Mod. Phys. C , 1317 (2011).[53] D. Galante and M. Schvellinger, “Thermalization with a chemical potential from AdSspaces,” JHEP , 096 (2012).[54] E. Caceres and A. Kundu, “Holographic Thermalization with Chemical Potential,”JHEP , 055 (2012). 3555] W. Baron, D. Galante and M. Schvellinger, “Dynamics of holographic thermaliza-tion,” JHEP , 070 (2013).[56] P. M. Chesler and D. Teaney, “Dilaton emission and absorption from far-from-equilibrium non-abelian plasma,” arXiv:1211.0343 [hep-th].[57] O. Philipsen, “The QCD equation of state from the lattice,” Prog. Part. Nucl. Phys. , 55 (2013).[58] J. Erdmenger and S. Lin, “Thermalization from gauge/gravity duality: Evolution ofsingularities in unequal time correlators,” JHEP , 028 (2012).[59] R. Baier, S. A. Stricker, O. Taanila and A. Vuorinen, “Holographic Dilepton Produc-tion in a Thermalizing Plasma,” JHEP , 094 (2012).[60] D. Steineder, S. A. Stricker and A. Vuorinen, “Holographic Thermalization at Inter-mediate Coupling,” Phys. Rev. Lett. , 101601 (2013).[61] D. Steineder, S. A. Stricker and A. Vuorinen, “Probing the pattern of holographicthermalization with photons,” [arXiv:1304.3404 [hep-ph]].[62] W. Israel, “Singular hypersurfaces and thin shells in general relativity,” Nuovo Cim.B , 1 (1966) [Erratum-ibid. B , 463 (1967)] [Nuovo Cim. B , 1 (1966)].[63] H. Kowalski, T. Lappi and R. Venugopalan, “Nuclear enhancement of universal dy-namics of high parton densities,” Phys. Rev. Lett. , 022303 (2008).[64] B. Sahoo and H. -U. Yee, “Electrified plasma in AdS/CFT correspondence,” JHEP , 095 (2010).[65] D. E. Kharzeev and H. -U. Yee, “Chiral helix in AdS/CFT with flavor,” Phys. Rev.D , 125011 (2011).[66] E. D’Hoker and P. Kraus, “Magnetic Brane Solutions in AdS,” JHEP0910