Axial charge imbalance is an essential ingredient in novel effects associated with chiral anomaly such as chiral magnetic effects (CME). In a non-Abelian plasma with chiral fermions, local axial charge can be generated a) by topological fluctuations which would create domains with non-zero winding number b) by conventional non-topological thermal fluctuations. We provide a holographic evaluations of medium’s response to dynamically generated axial charge density in hydrodynamic limit and examine if medium’s response depends on the microscopic origins of axial charge imbalance. We show a local domain with non-zero winding number would induce a non-dissipative axial current due to chiral anomaly. We illustrate holographically that a local axial charge imbalance would be damped out with the damping rate related to Chern-Simon diffusive constant. By computing chiral magnetic current in the presence of dynamically generated axial charge density, we found that the ratio of CME current over the axial charge density is independent of the origin of axial charge imbalance in low frequency and momentum limit. Finally, a stochastic hydrodynamic equation of the axial charge is formulated by including both types of fluctuations.
Article funded by SCOAP3
$ in hydrodynamic regime: \be \label{GqqR} G^{qq}_{R}(\o, k) = \frac{1}{2}\[- i \frac{\Gamma_{\CS}}{T} \o -\kappa_{\CS} k^2 \] \, , \ee and $T$ denotes temperature. It is worthy noting that $\kappa_{\CS}$ term in~\eqref{jA_new} is opposite to the direction of diffusive current and non-dissipative. One way to understand current~\eqref{jA_new} is that a topological domain also carries kinetic energy which would be transfered to chiral fermions via anomaly relation~\eqref{anomaly}. In ref.~\cite{Iatrakis-ml-2014dka} by us, we have derived~\eqref{jA_new} based on a generic setting and presented a brief verification of~\eqref{jA_new} in the Sakai-Sugimoto model. We provide more details on this calculation in section~\ref{sec:a2} and elucidate how axial current~\eqref{jA_new} is generated from gravity side of the duality. Our calculation in section~\ref{sec:E0} confirms that a local chiral charge imbalance $n_{A}(t,\vx)$ will induced a non-zero $q(t,\vx)$ and they are related by \be \label{q_nA} q(t,\vx ) = \frac{n_{A}(t,\vx)}{2\tau_{\sph}}\, . \ee $\tau_{\sph}$ here can be interpreted as the axial charge damping time. Indeed, substituting~\eqref{q_nA} into~\eqref{anomaly}, one has: $\pd_{\mu}J^{\mu}_{A} = -n_A/\tau_{\sph}$. Eqs.~\eqref{q_nA} hence implies that a non-zero axial charge density will eventually be damped out by inducing a non-zero $q$. Furthermore, we verify by holographic computations that $\tau_{\sph}$ is related to Chern-Simon diffusive rate $\Gamma_{\CS}$ and susceptibility $\chi$: \be \label{tau_sph} \tau_{\sph} = \frac{\chi T}{2\Gamma_{\CS}}\, . \ee Previously, relation~\eqref{tau_sph} has been derived based on the standard fluctuations-dissipation argument~\cite{Rubakov-ml-1996vz} (see also section~\ref{sec:hydro}). Very recently, $\tau_{\sph}$ has also been computed numerically in a bottom-up holographic model~\cite{Jimenez-Alba-ml-2015awa}. To best of our knowledge, current work is the first direct verification of relation~\eqref{tau_sph} in strong coupling regime. As we find that the axial current in response to axial charge density depend on how such axial charge imbalance is generated, it is natural to ask if chiral magnetic current~\eqref{CME} also depends on the origin of axial charge imbalance. To be quantitative, we consider the ratio between CME current and axial charge density in low frequency, small momentum limit in the presence of constant magnetic field: \be \label{r_CME} \(\chi_{\dyn}\)^{-1} \equiv \lim_{\o, k\to 0} \[ \frac{\mu_{A}(\arguo)} {n_{A}(\arguo)} \] \, , \qquad \mu_{A}(\arguo)\equiv \frac{j^{CME}_{V}(\arguo)}{C_{A}eB}\, . \ee For a system with constant axial charge density $n_{A}$, the ratio $\vj^{CME}_{V}/(CeB)$ equals to axial chemical potential $\mu_{A}$ due to~\eqref{CME}. However, if $n_{A}$ is space-time dependent, the definition of axial chemical potential $\mu_{A}$ is ambiguous. If one takes the ratio $\vj^{CME}_{V}/(CeB)$ as the generalized definition of axial chemical potential, the ratio $n_{A}(\o, k)/\mu_{A}(\o, k)$ can be interpreted as susceptibility. For this reason, we will call $\chi_{\dyn}$ the ``dynamical axial susceptibility''. We would like to emphasis that $\chi_{\dyn}$~\eqref{r_CME} is conceptually different from chiral magnetic conductivity~\cite{Kharzeev-ml-2009pj} which is the proportionality coefficient of CME current to the \emph{time-dependent} magnetic field for a medium with \emph{homogeneous, time-independent} axial chemical potential. In~\eqref{r_CME} however, magnetic field is constant while $n_{A}$ is space-time dependent. It is worthy noting that in realistic situations such as quark-gluon plasma (QGP) created in heavy-ion collisions, $\chi_{\dyn}$ would be a relevant measure of CME as in those situations, the axial charge density is always generated dynamically. Previously, chiral magnetic conductivity has been calculated for plasma in equilibrium at both weak coupling limit~\cite{Kharzeev-ml-2009pj,Hou-ml-2011ze,Yee-ml-2014dxa} and strong coupling~\cite{Yee-ml-2009vw,Gursoy-ml-2014ela,Gursoy-ml-2014boa} and for plasma out-of-equilibrium~\cite{Lin-ml-2013sga}. However, we are not aware any existing literature discussing the ``dynamic axial susceptibility'' $\chi_{\dyn}$ and its universality. We have computed $\chi_{\dyn}$ with $n_{A}(\arguo)$ generated by topological and thermal (non-topological) fluctuations in Sakai-Sugimoto model. We found such ratio $\chi_{\dyn}$~\eqref{r_CME} is \emph{in-dependent of} the origin of axial charge imbalance and equals to static susceptibility $\chi$. Moreover, we derive a simple analytic expression~\eqref{r_CME_holo} relating the (integration of) gravity metric to $\chi_{\dyn}$ which applies to a large class of holographic model. From such expression, we obtain a condition on the universality of $\chi_{\dyn}$. Having established the fact that axial charge generate by both topological fluctuations and thermal fluctuations would contribution to CME current, it is then important to incorporate both fluctuations in the framework of stochastic hydrodynamics. Recently, there are encouraging progress on applying anomalous hydrodynamics to simulate charge seperation effects~\cite{Hirono-ml-2014oda,Yin-ml-2015fca} and chiral magnetic wave effects~\cite{Hongo-ml-2013cqa,Yee-ml-2013cya} in heavy-ion collisions. In those studies, axial charge density enters as the initial conditions while the fluctuations of axial charge density during hydrodynamic evolution have been neglected. Motivated by findings in this paper, we formulate a stochastic hydrodynamic equation of axial charge density in section~\ref{sec:hydro}. Such hydrodynamic equation includes stochastic noise from both topological fluctuations and thermal fluctuations. While it is a direct generalization of the general framework~\cite{Kapusta-ml-2011gt,Kovtun-ml-2012rj,Kapusta-ml-2012zb}, to our knowledge, stochastic equation~\eqref{na_hydro_eq} is new in literature. We hope our theory would be applied to simulate phenomenology of anomalous transport in the future. The paper is organized as follows. We will begin with a brief review of pertinent ingredients of Sakai-Sugimoto model and realization of anomaly relation~\eqref{anomaly} in section~\ref{sec:SS}. Section~\ref{sec:response} is devoted to studying medium's response to axial charge density. The computation of $\chi_{\dyn}$~\eqref{r_CME} is presented in section~\ref{sec:CME}. The stochastic hydrodynamic equation for axial charge is formulated in section~\ref{sec:hydro}. We conclude in section~\ref{sec:sum}. \newpage ]]>
1/(2\pi R_4)=M_{KK}/2\pi$~\cite{Aharony-ml-2006da}. Finally, 't Hooft coupling $\lam$ is given by: \be \label{l_coupling} \lam =N_{c}\frac{(2\pi)^2 g_sl_s}{2\pi R_4}=2\pi N_{c}g_s l_s M_{KK}\, . \ee To model gluonic fluctuations, we will consider the dynamics of $C_{7}$ Ramond-Ramond form. The kinetic energy of $C_7$ are given by \begin{align} \label{RR_kinetic} S_{RR}=-\frac{(2\pi l_s)^6}{4\pi}\int_{10}dC_7\wg \(*dC_7\). \end{align} In Sakai-Sugimoto model, right and left handed quarks are introduced by $N_f$ $D8$ branes and $N_f$ ${\bar D}8$ branes~\cite{Sakai-ml-2004cn,Sakai-ml-2005yt}. Right-handed (left-handed) $\U(1)$ gauge filed $A_{R}$ ($A_{L}$) lives on $D8$ ($\bar{D}8$) and is dual to right-handed (left-handed) current $J^{\mu}_{R}$ ($J^{\mu}_{L}$) on the boundary. The $D8$/${\bar D}8$ branes are separated along the $x_4$ direction, with $D8$ branes located at $x_4=0$ and ${\bar D}8$ branes located at $x_4=\pi R_4$. In this work will consider $N_f=1$ though generalization to the case of multi-flavors is straightforward. The action of bulk gauge field $A_{R,L}$ or its field strength $F_{R,L}$ is given by the summation of Dirac-Born-Infeld (DBI) term \be \label{d8_action} S^{R,L}_{DBI}=-\frac{1}{(2\pi)^8l_s^9}\int d^9xe^{-\phi}\sqrt{-det\(g_{MN}+(2\pi\af'\)F^{R,L}_{MN})} \, , \ee and Wess-Zumino (WZ) term which couples $A_{R,L}$ to Ramond-Ramond form: \be \label{WZ_action} S^{R,L}_{WZ}=\pm\int_{\Sig9}\Sig_q C_{q+1}\wg tre^{\frac{F^{R,L}}{2\pi}}\, . \ee We normalize the RR forms $C$ as in~\cite{Sakai-ml-2004cn} and use hermitian worldvolume gauge field $A$. The DBI action is identical for $A_{R}$ and $A_{L}$. In WZ term, plus/minus sign is corresponding to $A_{R}$/$A_{L}$ (i.e.\ $F^{R}_{MN}$/$F^{L}_{MN}$) respectively. Axial gauge field and vector gauge field are related to right-handed and left-handed gauge field by: \begin{align} \label{AV_convention} A= \frac{A^{R}-A^{L}}{2}\, , \qquad V= \frac{A^{R}+A^{L}}{2}\, . \end{align} The total action we will study then becomes: \be \label{action} S = S_{RR} + S_{\rm DBI} + S_{\rm WZ}\, . \ee It is instructive to show how axial anomaly relation~\eqref{anomaly} is realized in the current holographic model. Following holographic dictionary, the axial current $j^{\mu}_{A}$ is given by the variation of holographic on-shell action $S_{\holo}$ with respect the boundary value of $a_{\mu}\equiv A_{\mu}(U\to\infty)$ and $q$ is given by the variation of $S_{\holo}$ with respect to $\theta$: \be \label{Jq_holo} j^{\mu}_{A} = \frac{\delta S_{\holo}}{\delta a_{\mu}}\, , \qquad q=\frac{\dlt S_{\holo}}{\dlt\tht}\, . \ee Here $\theta$ is determined by the holonomy of $C_1$ on the compactified $x_4$ direction~\cite{Witten-ml-1998uka}: \be \label{tht_def} \tht(t,\vx)=\lim_{U\to \infty}\int dx_4(C^{(4)}_1)\, . \ee One may note that the normalization in~\eqref{tht_def} is consistent with the one found by considering action of multiple probe color branes. $C_1$ is related to $C_7$ and $A_{R,L}$ field~\cite{Green-ml-1996dd}: \begin{align}\label{c7c1} dC_1=(2\pi l_s)^6*dC_7 -A^R\wg \(\dlt(x_4-\pi R_4)dx_4\) +A^L\wg \(\dlt(x_4)dx_4\)\, . \end{align} It is clear from~\eqref{c7c1} that $C_{7}$ is invariant under axial gauge transformation: \begin{align} \label{trans} \dlt_{\Lambda} A^R=d\Lambda\, , \qquad \dlt_{\Lambda} A^L=-d\Lambda\, , \qquad \dlt_{\Lambda} C_1=-\Lambda\dlt(x_4)dx_4-\Lambda\dlt(x_4-\pi R_4)dx_4\, , \end{align} where $\Lambda$ is an arbitrary scalar function. At boundary and after integrating out $x_{4}$ dependence, the axial gauge transformation is reduced to \begin{equation} \label{trans_bou} \dlt_{\Lambda} a_{\mu}(t, \vx) = \pd_{\mu}\Lambda (t, \vx)\, , \qquad \dlt_{\Lambda}\theta (t, \vx) = -2\Lambda (t, \vx)\, . \end{equation} Since the action~\eqref{action} expressed in terms of $C_7$ and field strength $F_R/F_L$ are manifestly invariant under axial gauge transformation~\eqref{trans}, $S_{\holo}$ would also be invariant under~\eqref{trans_bou}. We therefore have that for an infinitesimal transformation $\delta \Lambda$: \begin{align} \nn \dlt_{\Lambda} S_{\holo}&=\int d^4x \[\frac{\dlt S_{\holo}}{\dlt a_{\mu}(t, \vx)}\pd_{\mu}\(\dlt\Lambda (t, \vx)\) +\frac{\dlt S_{\holo}}{\dlt \tht (t, \vx)}(-2\dlt\Lambda (t, \vx))\]\\ & =\int d^4x\[ j^{\mu}_{A}(t, \vx) \(\pd_{\mu}\delta\Lambda (t, \vx)\) -2 q (t, \vx)\delta\Lambda (t, \vx)\] \nn \\ &= -\int d^4x\[\pd_{\mu}j^{\mu}_{A}(t, \vx)+2q (t, \vx)\]\delta\Lambda (t, \vx) =0\, . \label{S_gauge} \end{align} The anomaly relation~\eqref{anomaly} then follows from the requirement that~\eqref{S_gauge} holds for arbitrary $\delta\Lambda$, This is the holographic realization of axial anomaly in the current model. Realization of axial anomaly in general $Dp/Dq$ brane can be found in~\cite{Casero-ml-2007ae}. ]]>
t'$ when we can pick up the diffusive pole in the lower half plane. We then obtain \be \frac{j_{A}(t,k)}{N_c}= -C_\gm Dk^2I_0(t,k). \ee $I_{0}(k)$ here is defined by: \be I_{0}(k, t)\equiv e^{-i D k^2 t} E_{0}(\o=-iDk^2, k) = \int^{t}_{-\infty}dt' e^{-Dk^2(t-t')} E(t',k)\, . \ee A similar computation gives: \bea \label{nA_tk} \frac{n_{A}(t,k)}{N_c} &=& (-i C_{\gamma}) k I_0(t, k)\, , \\ \label{q_tk} q(t, k)&=& -\(\frac{2K^2}{C_{g}}\)\(-i D k\) I_{0}(t, k)\, . \eea In~\eqref{q_tk}, we have neglected a contribution higher order in $k$. We assume the external field $E_0$ only exists in a finite time window. At sufficiently late time $t$, we can regard $j_A$ and $q$ as responses to $n_A$. Comparing~\eqref{jA_tk} and~\eqref{nA_tk} and return to real space, we obtain Fick's law: \be j_{A}(\argutx)= - D\nabla n_{A}(\argutx)\, . \ee Moreover,~\eqref{nA_tk},~\eqref{q_tk} lead to the relation: \be \label{q_nA_holo} q(\argutx) = \!\(\frac{2K^2}{N_cC_{g}}\)\!\(\frac{D}{C_{\gamma}}\) n_{A}(\argutx) = \(\frac{\Gamma_{\CS}}{T}\)\(\frac{D}{\sigma}\)n_{A}(\argutx) = \(\frac{\Gamma_{\CS}}{\chi T}\) n_{A}(\argutx) = \frac{n_{A}(\argutx)}{2\tau_{\sph}}\, , \ee where we have used~\eqref{G_CS_SS},~\eqref{tau_sph}. \newpage The results of this section can be summarized by the following response matrix, characterizing the response of $q$, $n_A$ and $j_A$ to $\tht$ and $E_A$: \begin{align}\label{response_M} \begin{pmatrix} q\\ n_A\\ j_A \end{pmatrix}= \begin{pmatrix} &\frac{i\omg\Gm_\text{CS}}{2T} &\frac{\Gm_\text{CS}}{T}\frac{i\omg}{k(\omg+iDk^2)}\\ &\frac{\Gm_{\text{CS}}}{T} &\frac{k\sig}{\omg+iDk^2}\\ &ik\(\ka_{\text{CS}}-D\frac{\Gm_{\text{CS}}}{T}\) &\frac{\omg\sig}{\omg+iDk^2} \end{pmatrix} \begin{pmatrix} \tht\\ E_0 \end{pmatrix}. \end{align} We keep only terms lowest order in small $\omg$ and $k$ limit. We further restrict ourselves to the regime $\o\sim k$ for the case with source $\tht$, and to diffusive regime $k^2\sim\o$ for the case with source $E_0$. The transport coefficients we presented are their first nonvanishing order in $N_c$: the responses of $n_A$ and $j_A$ to $E_A$ are at order $O(N_c)$, while the rest responses are at order $O(1)$. Before closing this section, we would like to comment on the $O(1)$ correction to the responses of $n_A$ and $j_A$ to $E_0$ above. This requires us to go beyond leading order in $1/N_{c}$ and compute $E^{(1)}_{A}$ from~\eqref{eom_gio2} with $E_A^{(0)}$ given by~\eqref{E0_sol}. Similar analysis shows that near the boundary, $E_A^{(1)}\sim u^{1/2}+\cdots$ as $u\to\infty$, which would give divergent contributions to $n_A$ and $j_A$. This is because the mixing of the bulk fields changes the dimension of the operator. We note that the change of operator dimension occurs immediately with mixing in bottom-up model in~\cite{Jimenez-Alba-ml-2014iia}, while in our case it occurs from the subleading order in $1/N_c$. As we explained earlier that the solution this order in $1/N_c$ is incomplete without including back-reaction of the flavor branes. It is curious to see if including such backreaction would remove the potential divergence. Although these higher order corrections do not affect the results of our paper, we hope that we could revisit the puzzle in future. ]]>
=0\, ,
\qquad
\<\xi_{i}(t,\vx) \xi_{j}(t,\vx')\>
= 2\sigma T \delta_{ij}\delta(t-t')\delta^3(\vx-\vx')\, .
\ee
Here $\<\ldots\>$ denotes the average over noise and $i,j=1,2,3$ run over
spatial coordinates.
The magnitude of $\vxi(t,\vx)$ is given by the standard fluctuation-dissipation relation.
Furthermore, $q(t,\vx)$ can be related to $n_{A}(t,\vx)$ using~\eqref{q_nA}:
\be
q(t,\vx)= \frac{n_{A}(t,\vx)}{2\tau_{\sph}}+\xi_{q}(t,\vx)\, .
\ee
$\xi_{q}(t,\vx)$ is the noise due to topological fluctuations:
\be
\label{xiq}
\<\xi_{q}(t,\vx)\> =0\, ,
\qquad
\<\xi_{q}(t,\vx) \xi_{q}(t,\vx')\>
= \Gamma_{\CS}\delta(t-t')\delta^3(\vx-\vx')\,,
\ee
and we will assume that there is no cross correlation between two different
types of fluctuations:
\be
\<\xi_{q}(t,\vx) \xi_{i}(t,\vx')\>
= 0 \, .
\ee
This completely specifies our stochastic anomalous hydrodynamic equations.
The noise due to topological fluctuations~\eqref{xiq} have been considered previously in refs.~\cite{Kuzmin-ml-1985mm}.
On the other hand,
for a conserved current (such as vector $\vj_{V}$),
the noise of the form~\eqref{vxi} is standard.
Incorporating both fluctuations in~\eqref{na_hydro_eq} is new to the
extent of our knowledge.
As an application,
we will consider equal time axial charge correlation
function:
\be
\label{Cnn_def}
C_{nn}(t,\vx)
= \<\[n_{A}(t,\vx)-n_{A}(0,\vx)\]\[n_{A}(t,\vx)-n_{A}(0,\vx)\]\>\,
.
\ee
We start with equation for $n_A$ readily followed from
the stochastic hydrodynamic equations~\eqref{hydro_anom}:
\be
\label{na_hydro_eq}
\[ \pd_{t} - D\nabla^2 +\tau^{-1}_{\sph}\]n_{A}(t,\vx)
= - \nabla\vxi(t,\vx)+2\xi_{q}(t,\vx) \, .
\ee
Under the initial condition $n_{A}(0,\vx)=0$,
$C_{nn}(t,\vx)$ characterize the magnitude of axial charge fluctuations at
time $t$ and location $\vx$ due to (both) fluctuations.
To compute~\eqref{Cnn_def},
we first Fourier transform~\eqref{na_hydro_eq} into $\vk$ space (but keep
$t$-dependence):
\be
\label{na_hydro_eqk}
\[ \pd_{t} + D k^2 +\tau^{-1}_{\sph}\]n_{A}(t,\vk)
= \[-i\vk\cdot\vxi(t,\vk)\]+ 2\xi_{q}(t,\vk) \, .
\ee
The solution of~\eqref{na_hydro_eqk} under initial condition
$n_{A}(0,\vx)=0$ reads:
\be
n_{A}(t,\vk)
=
\int^{t}_{0}dt' e^{-\(Dk^2+\tau^{-1}_{\sph}\)(t-t')}\[(-i
\vk)\cdot\vxi(t,\vk)+2\xi_{q}(t,\vk)\] \, .
\ee
We therefore have:
\bea
\nn
\