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# Is the Chiral Magnetic Effect fast enough?

Jewel K. Ghosh1,2<EMAIL_ADDRESS>Sebastian Grieninger3,4
<EMAIL_ADDRESS>Karl Landsteiner3<EMAIL_ADDRESS>Sergio Morales-Tejera3,4<EMAIL_ADDRESS>1 Independent University
Bangladesh (IUB), Plot 16, Block B, Aftabuddin Ahmed Road, Bashundhara R/A,
Dhaka, Bangladesh
$2$ International Centre for Theoretical Sciences (ICTS-TIFR), Tata Institute
of Fundamental Research, Shivakote, Hesaraghatta, Bangalore 560089, India
3Instituto de Física Teórica UAM/CSIC, c/Nicolás Cabrera 13-15, Universidad
Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain
4Departamento de Física Teórica, Universidad Autónoma de Madrid, Campus de
Cantoblanco, 28049 Madrid, Spain

###### Abstract

It depends: While we find within holography that the lifetime of the magnetic
field for collider energies like the ones achieved at RHIC is long enough to
build up the chiral magnetic current, the lifetime of the magnetic field at
LHC seems to be too short.

We study the real time evolution of the chiral magnetic effect out-of-
equilibrium in strongly coupled holographic gauge theories. We consider the
backreaction of the magnetic field onto the geometry and monitor pressure and
chiral magnetic current. Our findings show that generically at small magnetic
field the pressure builds up faster than the chiral magnetic current whereas
at strong magnetic field the opposite is true. At large charge we also find
that equilibration is delayed significantly due to long lived oscillations. We
also match the parameters of our model to QCD parameters and draw lessons of
possible relevance to the realization of the chiral magnetic effect in heavy
ion collisions. In particular, we find an equilibration time of about $\sim
0.35$ fm/c in presence of the chiral anomaly for plasma temperatures of order
$T\sim 300-400$ M$\mathrm{eV}$.

††preprint: IFT-UAM/CSIC-21-52

###### Contents

  1. I Introduction
  2. II The holographic model
  3. III Results of numerical simulations
    1. III.1 $B$-dependence
    2. III.2 $\alpha$-dependence
    3. III.3 Matching to QCD
  4. IV Conclusions
  5. A Numerical methods
  6. B Simulations for $\mu_{5}=100$MeV.

##  I Introduction

The chiral magnetic effect (CME) is the generation of an electric current in a
chirally imbalanced medium by an applied magnetic field Vilenkin (1980);
Fukushima  _et al._ (2008).111For reviews see Kharzeev (2014); Landsteiner
(2016); Kharzeev and Liao (2021). Formally it is described by

$\vec{J}=8c\mu_{5}\vec{B}\,,$ (I.1)

where $c$ is the coefficient of the axial anomaly and $\mu_{5}$ the axial
chemical potential.

While this formula is rather straightforward to derive its interpretation is
subtle in many respects. The fact that it depends on a chemical potential and
the magnetic field implies that it can be derived in equilibrium quantum field
theory. This is indeed the case but the axial chemical potential is a
thermodynamic variable conjugate to an anomalous charge. The axial current is
not conserved in the quantum theory but obeys the anomalous non-conservation
law Adler (1969); Bell and Jackiw (1969)222In this work, we will not consider
effects of the gravitational contribution to the axial anomaly Kimura (1969).

$\partial_{\mu}J^{\mu}_{5}=c\epsilon^{\mu\nu\rho\lambda}F_{\mu\nu}F_{\rho\lambda}\,.$
(I.2)

Since a chemical potential should in principle be introduced only for
conserved charges the question of the precise meaning of the axial chemical
potential arises. An additional issue is that rather general arguments suggest
that in equilibrium the electric current has to vanish identically Yamamoto
(2015).

This issue is well understood by now. The CME current indeed vanishes in
equilibrium if one introduces the axial chemical potential as a background
value of an axial gauge field $A_{0}^{5}=\mu_{5}$. The axial gauge field gives
rise to an additional contribution to the CME that cancels the CME current
Gynther  _et al._ (2011) (see also Rubakov (2010)). On the other hand, one can
introduce the chemical potential as a property of the initial state and then
consider the time evolution generated by the Hamiltonian $H$ in which no axial
gauge field is present Landsteiner  _et al._ (2013). In that case eq. (I.1)
does in fact apply. However, this already represents a certain amount of non-
equilibrium physics since the axial charge (or chemical potential) has to be
induced by some means in the initial state.

From these considerations it becomes clear that the physical realization of
the CME demands for a certain amount of non-equilibrium physics. Hydrodynamics
only assumes local thermal equilibrium and thus is always intrinsically a non-
equilibrium theory too. In fact, it was shown that the chiral transport can be
derived as a consequence of hydrodynamics with triangle anomalies Son and
Surowka (2009).

If one does not assume local thermal equilibrium but initiates the physical
system in a generic non-equilibrium state the question arises how fast it
evolves towards equilibrium. Since the CME current is formally expressed by a
chemical potential (an equilibrium quantity) the question of how the CME is
realized far from equilibrium is particularly interesting. Answering this
question is particularly important in view of a possible realization of the
CME in heavy ion collisions. The magnetic field in heavy ion collisions is
only present in the initial stages and decays quickly whereas the hot QCD
matter has a short but non-vanishing equilibration time. The question
naturally arising is if the CME builds up fast enough to be measurable before
the magnetic field decays. We want to address this question in this work by
modelling the strongly coupled quark gluon plasma by means of a holographic
model.

The gauge/gravity duality or holographic duality provided many important
insights into this type of questions. Starting with the key insight of the
small value of the shear viscosity to entropy ratio $\eta/s=\frac{1}{4\pi}$
Kovtun  _et al._ (2005), holography has contributed many important results on
hydrodynamics and transport in general. The modern way of understanding
hydrodynamics is strongly influenced by it Baier  _et al._ (2008);
Bhattacharyya  _et al._ (2008). Many insights into anomaly induced transport
phenomena have their origin in the holographic duality Banerjee  _et al._
(2011); Erdmenger  _et al._ (2009); Landsteiner  _et al._ (2011); Ammon  _et
al._ (2021). It also is an ideal tool to study far from equilibrium evolution
of strongly coupled quantum systems Chesler and Yaffe (2009, 2014). In the
present work, we will therefore study the non-equilibrium behavior of the
chiral magnetic effect in a holographic set-up.

Previous studies of out-of-equilibrium chiral transport in holography are Lin
and Yee (2013); Ammon  _et al._ (2016a); Grieninger (2016); Haack  _et al._
(2019); Cartwright (2021). A quantum simulation of the real time evolution of
the CME has been presented in Kharzeev and Kikuchi (2020). In particular, the
question of the timescale on which the CME becomes builds up has been the
subject of Landsteiner  _et al._ (2018); Fernández-Pendás and Landsteiner
(2019); Morales-Tejera and Landsteiner (2020). In these works a holographic
approach was taken based on Vaidya type metrics. The advantage of this
approach is that the background metric can be computed analytically. The time
evolution of chiral transport can then be studied numerically in linear
response for intrinsically small magnetic fields and beyond linear response in
Ammon  _et al._ (2016a); Grieninger (2016) in the probe limit. In contrast,
the purpose of this work will be to study the full back-reaction of the
magnetic field onto the geometry.

The question of how fast the CME current builds up can be studied directly
since it is possible to start with a far from equilibrium state with large
magnetic field in which no CME current is present. Another important
observable that has been in the focus of holographic studies is the pressure
Chesler and Yaffe (2009). The large magnetic field will of course induce a
significant pressure anisotropy. The time evolution of the magnetic field
induced pressure anisotropy in holography has been studied before in Fuini and
Yaffe (2015) albeit without effects of the anomaly. In our case we chose to
start with an initial state in which both, the CME current vanishes and the
pressure anisotropy is not at its equilibrium value. We can then monitor the
equilibration of both of our observables and study their time evolution for
varying parameters. As parameters we chose the total energy $\epsilon$, the
axial charge density $q_{5}$, the magnetic field $B$ and finally also the
strength of the anomaly. The latter is represented in holography by the value
of the Chern-Simons coupling $\alpha$.

We will employ a bottom-up approach to construct our holographic model. The
main motivation for choosing a bottom-up approach is that the proper
realization of the CME needs the notion of both an axial ($A)$ and vector
$(V)$ like $U(1)$ symmetry. In holography, we therefore need to introduce two
bulk gauge fields and an appropriate Chern-Simons term representing the mixed
$VAA$ anomaly. It also allows to match the parameters of the model to QCD and
we will do this later on by matching the holographic Chern-Simons coupling to
the axial anomaly of three flavor QCD.

The paper is structured as follows. In section II we present all the details
of our holographic model. We set up the equations of motion and explain some
of the salient features of the numerical methods. Section III contains a scan
through the parameter space. We present the time evolution of the CME current
and the pressure anisotropy for varying magnetic field strength, varying
Chern-Simons coupling and varying axial charge. Of particular interest will be
subsection III.3 in which we match the Chern-Simons coupling to the value of
QCD with $N_{f}=3$. We summarize our findings and conclusions in section IV.
Further details of the numerical methods will be presented in the appendix A.
In appendix 4, we provide results similar to section III.3 for a larger value
of the axial charge density.

##  II The holographic model

We study a holographic quantum field theory with a $U(1)_{A}\times U(1)_{V}$
symmetry. The presence of two gauge fields $A^{\mu}$ and $V^{\mu}$ encodes the
presence of the symmetry in the dual field theory. The field strengths of
these fields are denoted as $F_{5}=dA$ and $F=dV$, respectively. The anomaly
is implemented through a Chern–Simons term which is gauge invariant up to a
total derivative. We work with the consistent form of the anomaly. Combining
these ingredients, the holographic model we consider is the following333In our
notation, Greek letters denote the bulk coordinates and small Latin letters
denote the boundary coordinates.

$\displaystyle S=$
$\displaystyle\frac{1}{2\kappa^{2}}\int_{\mathcal{M}}d^{5}x\sqrt{-g}\left[R+\frac{12}{L^{2}}-\frac{1}{4}F^{2}-\frac{1}{4}F_{(5)}^{2}\right.$
$\displaystyle\left.+\frac{\alpha}{3}\epsilon^{\mu\nu\rho\sigma\tau}A_{\mu}\left(3F_{\nu\rho}F_{\sigma\tau}+F^{(5)}_{\nu\rho}F^{(5)}_{\sigma\tau}\right)\right]$
$\displaystyle\qquad\qquad\qquad\qquad+S_{GHY}+S_{ct}$ (II.1)

where $S_{GHY}$ is the Gibbons-Hawking-York boundary term to make the
variational problem well defined, $L$ is the AdS radius, $\kappa^{2}$ is the
Newton constant and $\alpha$ the Chern-Simons coupling. We also add
appropriate counter-terms $S_{ct}$ to cancel the possible divergences de Haro
_et al._ (2001); Balasubramanian and Kraus (1999); Emparan  _et al._ (1999).
The Levi-Civita tensor is defined as
$\epsilon^{\mu\nu\rho\sigma\tau}=\epsilon(\mu\nu\rho\sigma\tau)/\sqrt{-g}\,$.
The (consistent) anomaly is

$\displaystyle\delta_{5}S=\frac{\alpha}{2\kappa^{2}}\int
d^{4}x\lambda_{5}\epsilon^{abcd}(F_{ab}F_{cd}+\frac{1}{3}F_{5,ab}F_{5,cd})\,.$
(II.2)

The relative factor of $1/3$ reflects the Bose symmetry of the $U_{A}(1)^{3}$
anomaly, $\lambda_{5}$ is an axial gauge parameter.

Varying the fields, we find the equations of motion. They are

$\displaystyle\nabla_{\nu}F^{\nu\mu}+2\alpha\epsilon^{\mu\nu\rho\sigma\tau}F_{\nu\rho}F^{(5)}_{\sigma\tau}=0,$
(II.3)
$\displaystyle\nabla_{\nu}F_{(5)}^{\nu\mu}+\alpha\epsilon^{\mu\nu\rho\sigma\tau}\left(F_{\nu\rho}F_{\sigma\tau}+F_{\nu\rho}^{(5)}F_{\sigma\tau}^{(5)}\right)=0,$
(II.4) $\displaystyle
G_{\mu\nu}-\frac{6}{L^{2}}g_{\mu\nu}-\frac{1}{2}F_{\mu\rho}F_{\nu}^{\
\rho}+\frac{1}{8}F^{2}g_{\mu\nu}-\frac{1}{2}F^{(5)}_{\mu\rho}F_{\nu}^{(5)\rho}$
$\displaystyle+\frac{1}{8}F_{(5)}^{2}g_{\mu\nu}=0.$ (II.5)

We are interested in studying the vector current and pressure living in the
dual field theory. Those can be extracted from the full bulk solution through
the standard holographic prescription, i.e. varying the on-shell action with
respect to the boundary value of the dual field appropriately subtracting the
divergences.

In order to find the bulk solution we choose an ansatz for the metric and the
gauge fields which preserves the symmetries present in the system. We label
the coordinates with $x^{\mu}=(u,v,x,y,z)$, where $u$ is the radial coordinate
and $v$ the time coordinate. In particular, we have translational invariance
of the three spatial directions $(x,y,z)$. Hence, the metric fields can only
depend on the radial $u$ and temporal $v$ coordinates. Besides, the presence
of the magnetic field, which we assume to point in the $z$ direction, breaks
the $SO(3)$ rotational invariance down to $SO(2)$. Consequently, we consider
the metric ansatz to be

$\displaystyle
ds^{2}=-f(v,u)dv^{2}-\frac{2L^{2}}{u^{2}}dvdu+\frac{2}{u^{2}}h(v,u)dvdz$
$\displaystyle+\Sigma(v,u)^{2}\left[e^{\xi(v,u)}(dx^{2}+dy^{2})+e^{-2\xi(v,u)}dz^{2}\right],~{}~{}$
(II.6)

which has been written in infalling Eddington-Finkelstein coordinates, with
the boundary located at $u=0$. Note that $\xi$ parametrizes the anisotropy of
the system. In order to recover asymptotic $AdS_{5}$, we demand

$\begin{split}&\lim_{u\to 0}f(v,u)=\frac{L^{2}}{u^{2}}\,,\hskip
22.76228pt\lim_{u\to 0}h(v,u)=0\,,\\\ &\lim_{u\to
0}\Sigma(v,u)=\frac{L}{u}\,,\hskip 22.76228pt\lim_{u\to
0}\xi(v,u)=0\,.\end{split}$ (II.7)

In this work, we consider a finite axial charge density and a external vector
magnetic field $B$, which are the minimal ingredients required to generate the
CME. We choose to work in the radial gauge $V_{u}=A_{u}=0\,$. Under these
considerations, the only non-trivial component that is turned on for the axial
gauge field is $A_{v}=-Q_{5}(v,u)$, whereas the vector gauge field contains
$B$ and a non-trivial profile in its $z$-component. Therefore, the ansatz for
the gauge fields take the form

$\begin{split}&V_{\mu}=(0,0,-y\,B/2,x\,B/2,V_{z}(v,u))\,,\\\
&A_{\mu}=(-Q_{5}(v,u),0,0,0,0)\,.\end{split}$ (II.8)

The axial gauge field still has the gauge freedom to add a general function
$g(v)$ to the temporal component without altering the physics. We fix this
freedom by demanding that $Q_{5}(v,u)$ vanishes at the boundary.
It turns out that such an ansatz allows to set the function $h(v,u)$ in II.6
to zero444The $h$ function would eventually account for the response of the
stress tensor to the CME, yet only with $\mu_{5}$ and $B$ active the stress
tensor receives no contribution., simplifying significantly the system of
equations to solve. Plugging the ansatz II.6 and II.8 into the equations of
motion, setting $L=1$ and manipulating the expressions yields

$\displaystyle
Q_{5}^{\prime}=\frac{q_{5}}{u^{2}\Sigma^{3}}\left(1-8\alpha\frac{BV_{z}}{q_{5}}\right),$
(II.9) $\displaystyle
e^{2\xi}\Sigma(dV_{z})^{\prime}+\dfrac{1}{2}\left(e^{2\xi}\Sigma\right)^{\prime}dV_{z}+\dfrac{1}{2}V_{z}^{\prime}d\left(e^{2\xi}\Sigma\right)$
$\displaystyle-4\alpha BQ_{5}^{\prime}=0\,,$ (II.10)
$\displaystyle\frac{6\Sigma^{\prime}}{u\Sigma}+\frac{3\Sigma^{\prime\prime}}{\Sigma}+\frac{e^{2\xi}}{2\Sigma^{2}}V_{z}^{\prime
2}+\frac{3}{2}\xi^{\prime 2}=0\,,$ (II.11) $\displaystyle
3\frac{d\Sigma^{\prime}}{\Sigma}+\frac{6d\Sigma\Sigma^{\prime}}{\Sigma^{2}}-\frac{1}{4}u^{2}Q_{5}^{\prime
2}+\frac{6}{u^{2}}-\frac{B^{2}e^{-2\xi}}{4u^{2}\Sigma^{4}}=0$ (II.12)
$\displaystyle(d\xi)^{\prime}+\frac{3}{2\Sigma}(\Sigma^{\prime}d\xi+\xi^{\prime}d\Sigma)-\frac{1}{6}\frac{B^{2}e^{-2\xi}}{u^{2}\Sigma^{4}}-\frac{e^{2\xi}}{3\Sigma^{2}}V_{z}^{\prime}dV_{z}=0\,,$
(II.13)
$\displaystyle\dfrac{1}{2}(u^{2}f^{\prime})^{\prime}-\dfrac{3}{2}\xi^{\prime}d\xi+6\dfrac{\Sigma^{\prime}}{\Sigma}\dfrac{d\Sigma}{\Sigma}-\frac{5}{12}\frac{B^{2}e^{-2\xi}}{u^{2}\Sigma^{4}}+\dfrac{2}{u^{2}}$
$\displaystyle-\frac{7}{12}u^{2}Q_{5}^{\prime
2}-\frac{e^{2\xi}}{6\Sigma^{2}}V_{z}^{\prime}dV_{z}=0\,,$ (II.14)
$\displaystyle dd\Sigma+\dfrac{1}{2}\Sigma
d\xi^{2}+\dfrac{1}{2}u^{2}f^{\prime}d\Sigma+\frac{e^{2\xi}}{6\Sigma}dV_{z}^{2}=0\,,$
(II.15)

where II.9 has already been integrated once with integration constant $q_{5}$
which we eventually identify with the axial charge density. More specifically,
setting $\mu=v,u$ in the axial Maxwell equations (II.4) we find

$\displaystyle\partial_{u}\left[u^{2}\Sigma^{3}(v,u)\partial_{u}Q_{5}(v,u)+8\alpha
BV_{z}(v,u)\right]=0,$ (II.16)
$\displaystyle\partial_{v}\left[u^{2}\Sigma^{3}(v,u)\partial_{u}Q_{5}(v,u)+8\alpha
BV_{z}(v,u)\right]=0.$ (II.17)

Hence, the quantity $u^{2}\Sigma^{3}(v,u)\partial_{u}Q_{5}(v,u)+8\alpha
BV_{z}(v,u)$ is independent of $u$ and $v$, and we identify it with the
integration constant $q_{5}$. The prime and the dot denote radial and temporal
derivatives respectively, whereas

$d=\partial_{v}-\dfrac{u^{2}f}{2}\partial_{u}\,$ (II.18)

is the derivative along infalling null geodesics. Its introduction is
customary in this context and decouples some of the differential equations in
a nested structure.

We can use these equations of motion to find the near boundary expansions of
the metric and gauge fields which read

$\displaystyle Q_{5}(v,u)=\frac{u^{2}}{2}q_{5}+\mathcal{O}(u^{3})\,,$ (II.19)
$\displaystyle V_{z}(v,u)=u^{2}\,V_{2}(v)+\mathcal{O}(u^{3})\,,$ (II.20)
$\displaystyle\Sigma(v,u)=\frac{1}{u}+\lambda(v)+\mathcal{O}(u^{5})\,,$
(II.21)
$\displaystyle\xi(v,u)=u^{4}\left(\xi_{4}(v)-\frac{B^{2}}{12}\,\log(u)\right)+\mathcal{O}(u^{5})\,,$
(II.22) $\displaystyle
f(v,u)=\left(\frac{1}{u}+\lambda(v)\right)^{2}+u^{2}\left(f_{2}+\frac{B^{2}}{6}\,\log(u)\right)$
$\displaystyle-2\dot{\lambda}(v)+\mathcal{O}(u^{3})\,.$ (II.23)

The function $\lambda(v)$ is a remnant of diffeomorphism symmetry and thus
arbitrary. We follow Fuini and Yaffe (2015) and use $\lambda$ to keep the
position of the apparent horizon of the black brane at a fixed radial position
$u_{h}=1\,$ throughout the time evolution. The coefficient $f_{2}$ is related
to the energy density of the black brane and the subleading coefficients
$V_{2}(v)$ and $\xi_{4}(v)$ shall give us the vector current and the pressure
anisotropy, respectively. In particular making use of the holographic
prescription described above and substituting the asymptotic solution, we find

$2\kappa^{2}\left<J_{z}\right>=2V_{2}(v)\,,\quad\quad
2\kappa^{2}\left<J_{5}^{0}\right>=q_{5},$ (II.24)

for the currents and

$\begin{split}&\expectationvalue{T^{v}_{\
v}}=\frac{1}{4\kappa^{2}}\left[6f_{2}-B^{2}\textrm{log}(\mu L)\right]\,,\\\
&\expectationvalue{T^{x}_{\ x}}=\expectationvalue{T^{y}_{\
y}}=-\frac{1}{8\kappa^{2}}\left[B^{2}+4f_{2}-16\xi_{4}(v)\right.\\\
&\qquad\qquad\qquad\qquad\qquad\left.-2B^{2}\textrm{log}(\mu L)\right]\,,\\\
&\expectationvalue{T^{z}_{\
z}}=-\frac{1}{4\kappa^{2}}\left[2f_{2}+16\xi_{4}(v)+B^{2}\textrm{log}(\mu
L)\right]\end{split}$ (II.25)

for the stress tensor. We have re-instated the AdS radius $L$ because in the
regularization procedure a renormalization energy scale $\mu$ appears due to
the fact that the magnetic field induces a trace anomaly, breaking conformal
invariance at a microscopic level. All in all, the problem reduces to solving
the full dynamics in the bulk, finding the subleading coefficients of $\xi$
and $V_{z}$ and substituting them into II.24 and II.25. The details about the
numerical implementation of this strategy are summarized in the appendix A.
Our numerical code is implemented in the programming language Julia Bezanson
_et al._ (2015).

The thermodynamic properties of the system are specified by two quantities:
temperature and the axial chemical potential. At late times, the system
equilibrates and we shall label different solutions in terms of their
thermodynamical variables of the final equilibrium state. Those will also be
useful to compare with the QGP produced in heavy-ion collision experiments. In
the dual gravity picture, this equilibration implies that the metric becomes
stationary at late times. The temperature is formally that of the black brane
once equilibrium is reached:

$T=\dfrac{1}{2\pi}\left.\left(-\dfrac{u^{2}}{2}\partial_{u}f(v,u)\right)\right|_{u=u_{h}\,,v\to\infty}\,.$
(II.26)

The chemical potential is computed as the temporal component of the gauge
field at the boundary minus its value at the horizon, i.e.

$\mu_{5}=\left.Q_{5}(v,u_{h})-Q_{5}(v,0)\right|_{v\to\infty}\,.$ (II.27)

We conclude this section by discussing the initial state of the dual quantum
field theory. The asymptotic form of the metric ansatz II has been chosen so
that it describes an infinite-volume non-expanding plasma. By construction the
plasma has a charged distribution given by $q_{5}\,$, is immersed in a
magnetic field of magnitude $B$ and has some energy density $\epsilon\,$. All
three of them are considered to be homogeneous and constant in time in our
model. Finally, we specify the initial conditions of the evolution by giving a
starting profile to the fields $\xi$ and $V_{z}$. In particular, we choose
them to be zero everywhere, which in turns means that the plasma starts with
vanishing CME current and vanishing dynamical pressure anisotropy, i.e.
vanishing anisotropy generated by $\xi_{4}\,$. Notice that in II.25 the term
proportional to $B$ does include anisotropy in the pressure from the
beginning. That contribution is the same over the evolution and is referred to
as $kinematic$ pressure anisotropy (see Fuini and Yaffe (2015) for further
details). Hence, if there were no magnetic field we could interpret the
initial conditions as being the equilibrium solution; yet since $B$ is present
the system is out-of-equilibrium.

##  III Results of numerical simulations

We scan the parameter space $(\alpha,q_{5},B)$ and study the features
displayed by the chiral magnetic effect through the vector current
$\left<J\right>$555The vector current is parallel to the magnetic field, which
we choose to be along the $z$-direction without loss of generality. and the
dynamical pressure anisotropy, which we denote $\xi_{4}$666Actually $\xi_{4}$
is proportional to the dynamical pressure anisotropy when evaluated at a scale
$\mu=1/L\,$.. The latter simply refers to the subleading coefficient of the
metric field $\xi$, parametrizing the anisotropy. The transverse and
longitudinal pressures can be trivially read off from II.25.

There is a subtlety related to the definition of the energy density due to the
non-trivial renormalization scale dependence in (II.25)

$\epsilon(\mu)=2\kappa^{2}\langle T^{vv}\rangle\,.$ (III.1)

We refer the reader to the discussion in Fuini and Yaffe (2015) for details.
In order to assign a definite value to the energy-momentum tensor, we have to
choose a renormalization scale $\mu$. For computational convenience, we choose
$\mu=1/L$ throughout this section. The scale $1/L$ is, however, not a physical
scale since it can be changed without changing the values of the physical
observables on the boundary due to a scaling symmetry Fuini and Yaffe (2015).
The physically relevant scale is $\mu=\sqrt{B}$ which is also the natural
choice. We will use this scale in section III.3. Both choices are related
through

$\dfrac{\epsilon_{B}}{B^{2}}=\dfrac{\epsilon_{L}}{B^{2}}+\dfrac{1}{4}\textrm{log}(BL^{2})\,,$
(III.2)

where $\epsilon_{L}$ and $\epsilon_{B}$ refer to the energy density at scales
$\mu=1/L$ and $\mu=\sqrt{B}$, respectively.

The parameter scan is performed at fixed energy density
$\epsilon_{L}=12\,$.777Working with a different value for $\epsilon_{L}$ seems
to only modify the final equilibrium state for the pressure anisotropy but
does not alter the relevant qualitative behavior like the build up time and
the presence or absence of oscillations. In section III.3, we match our model
to QCD and give physically relevant values for the parameters.

###  III.1 $B$-dependence

We first study both vector current and dynamical pressure anisotropy as we
vary the vector magnetic field $B$. We keep the anomaly fixed, i.e. fixed
Chern-Simons coupling $\alpha$, and consider two qualitatively different
values of axial charge $q_{5}$. The results are shown in figures 1 and 2. We
choose to work with dimensionless variables: time, pressure and current are
normalized to the energy density $\epsilon_{L}$, whereas we consider the
dimensionless ratio of magnetic field $B$ to temperature squared. All
thermodynamic quantities (chemical potential and temperature) refer to the
final equilibrium state where they are well defined.

As we increase the magnetic field, we observe the appearance of oscillations
in the current $\left<J\right>\,$. This is in agreement with the quasinormal
modes computed in Ammon  _et al._ (2016a); Grieninger (2016), where they found
that the imaginary part of the lowest QNM approaches the real axis for
increasing the magnetic field and hence perturbations become long lived. As a
consequence the equilibration time dramatically increases for increasing $B$.
The oscillatory behavior of the current indicates that the time-evolution is
dominated by the lowest QNM near the real axes. The final equilibrium value
matches the equilibrium value for the CME, i.e.
$2\kappa^{2}\left<J\right>_{eq}=8\alpha\mu_{5}B\,$. Note that oscillatory
behavior in the current indicates that we have not reached the final
equilibrium state yet (and it may take a very long time to get there in the
case of almost undamped oscillations). However, we verified that the axial
chemical potential II.27 read off from these states already closely matches
the values of the would be equilibrium state.

It is also worth noting that the vector current builds up progressively faster
with increasing magnetic field. We expect such a behavior for the following
reason. At large magnetic field all fermions are in the lowest Landau level.
The physics reduces effectively to the motion of the charged particles along
the magnetic field lines and is thus effectively $1+1$ dimensional. In $1+1$
dimensions the following relation between the axial current and the vector
current holds

$J^{5}_{a}=\epsilon_{ab}J^{b}\,,$ (III.3)

(where the $a,b$ indexes are $v,z$). This is an operator relation and hence
valid for matrix elements and expectation values. In contrast, the $3+1$
dimensional chiral magnetic effect depends on the (near-)equilibrium quantity
$\mu_{5}$. Since for larger magnetic field the theory becomes more and more
dominated by the effective $1+1$ dimensional dynamics, we expect the build up
of the vector current to occur already in the non-equilibrium stages in order
to fulfill the $1+1$ dimensional relation between the operators of axial
charge and vector current.

Finally, increasing $q_{5}$ simply increases the absolute value of the final
equilibrium state of the vector current. The effect in the pressure anisotropy
is dramatically different. For large axial charge densities, we clearly
observe oscillatory behavior in $\xi_{4}$. A closer look, however, reveals
that the same type of oscillations are also present for small $q_{5}$ but
their amplitude is significantly smaller and they could not be seen by eye in
figure 1, we have zoomed into one of the curves to clarify this statement.

We conclude this subsection with a discussion of the pressure anisotropy. To
quantify our results we define the time where the first local extremum in the
current and in the pressure anisotropy appears as build up time. The build up
time of the pressure anisotropy decreases slightly with increasing magnetic
field. There then exists a crossover in the system as a whole as we vary $B$:
for small $B$ the pressure anisotropy builds up faster than the current,
whereas for large $B$ such behavior is reversed. As we increase axial charge
the transition is lost and the vector current always builds up faster. We
denote $\Delta_{J}$ and $\Delta_{P}$ for the vector current and pressure,
respectively, and plot the ratio of both quantities in figure 3.

(a)

(b)

Figure 1: Vector current (upper plot) and dynamical pressure anisotropy (lower
plot) for fixed Chern-Simons coupling $\alpha=1.5$ and fixed small axial
charge density $q_{5}=0.2\,$. The magnetic field $B$ is
$\\{0.1\,,0.5\,,1.0\,,2.0\\}\,$. Even though $q_{5}$ is fixed, the
dimensionless ratio of axial chemical potential to the temperature (in the
final state) is different for each simulation. In particular, we find
$\mu_{5}/T=\\{0.11\,,0.06\,,0.04\,,0.02\\}\,$ for the would be final
equilibrium state.

(a)

(b)

Figure 2: Vector current (upper plot) and dynamical pressure anisotropy (lower
plot) for fixed Chern-Simons coupling $\alpha=1.5$ and fixed large axial
charge density $q_{5}=1.5\,$. The magnetic field $B$ is
$\\{0.5\,,1.0\,,1.5\,,2.0\\}\,$. Even though $q_{5}$ is fixed, the
dimensionless ratio $\mu_{5}/T$ (in the final state) is different for each
simulation. In particular, we find
$\mu_{5}/T=\\{0.11\,,0.06\,,0.04\,,0.02\\}\,$ for the would be final
equilibrium state. Figure 3:  Ratio of build up time for current and pressure
anisotropy as a function of the magnetic field for small and high axial charge
for fixed anomaly $\alpha=1.5\,$.

###  III.2 $\alpha$-dependence

In quantum field theory, the anomaly coefficient is fixed by the fermion
spectrum. However, in the context of holography, the anomaly coefficient
appears as a parameter of the holographic model that may be varied at will. We
shall study its effect for two qualitatively different values of axial charge
while keeping the magnetic field fixed at $B=2\,$, or
$B/\sqrt{\epsilon_{L}}=0.58\,$ in dimensionless units. Results for small and
high axial charge are shown in figures 4 and 5, respectively.

The vector current builds up faster and develops oscillatory behavior as we
increase $\alpha$ regardless of the magnitude of $q_{5}\,$. This is the same
effect as we observed in the previous section. Hence, either increasing
$\alpha$ or $B$ results qualitatively in analogous results. At small charge
(see figure 4), the evolution of the pressure seems to be governed solely by
$B$ and is independent of $\alpha$. Zooming in on the tail of the curve shows
that $\xi_{4}$ slightly depends on the anomaly coefficient $\alpha$, however,
due to the small value of the axial charge the effect is negligible. The
situation is clearer for larger axial charges $q_{5}$ (see figure 5):
increasing $\alpha$ yields again undamped oscillations, yet the build up time
remains constant. In figure 6 we show the ratio between the build up time of
the vector current and pressure anisotropies as a function of the Chern-Simons
coupling.

(a)

(b)

Figure 4: Vector current (upper plot) and dynamical pressure anisotropy (lower
plot) for fixed magnetic field $B=2$ and fixed small axial charge
$q_{5}=0.2\,$. In dimensionless units, the simulations are for $B/T^{2}=10.34$
and a final equilibrium value of the axial chemical potential corresponding to
$\mu_{5}/T=\\{0.104\,,0.053\,,0.037\,,0.016\\}$, respectively.

(a)

(b)

Figure 5: Vector current (upper plot) and dynamical pressure anisotropy (lower
plot) for fixed magnetic field $B=2$ and fixed large axial charge
$q_{5}=1.5\,$. In dimensionless units, the simulations are for
$B/T^{2}=\\{10.66\,,10.39\,,10.35\,,10.35\\}\,$ and a final equilibrium value
of the axial chemical potential corresponding to
$\mu_{5}/T=\\{0.802\,,0.396\,,0.278\,,0.119\\}$, respectively. Figure 6: Ratio
of build up time for the current and pressure anisotropy as a function of the
Chern-Simons coupling $\alpha$ for small and high axial charge and fixed
$B=2\,$.

###  III.3 Matching to QCD

### Parameters

In this section, we aim to provide simulations in the parameter range that is
experimentally relevant for the Quark-Gluon plasma (QGP). We obtain estimates
for the parameters in our model by matching to known QCD results, i.e. the
entropy and the anomaly.

Under an axial gauge transformation, our action has the mixed anomaly
$\mathcal{A}_{CS}=\frac{\alpha}{2\kappa^{2}}$. In order to get an estimate for
$\kappa$ we take the entropy of a black brane

$s_{BH}=\dfrac{A}{4G_{N}}=\dfrac{4\pi^{4}T^{3}}{2\kappa^{2}}\,.$ (III.4)

Now we want to match these expressions to the entropy of QCD at finite
temperature and eventually to the anomaly. First, we need to fix how many
flavors we take into account. The up and down quarks are light, whereas the
strange quark has a mass of around $95$ M$\mathrm{e}\mathrm{V}$. The cross
over temperature of QCD is at around $175$ M$\mathrm{e}\mathrm{V}$. Therefore,
we include the strange quark in our counting, i.e. we match to three flavor
QCD. The Stefan-Boltzmann value of the entropy density is

$s_{SB}=4\left(\nu_{b}+\frac{7}{4}\nu_{f}\right)\frac{\pi^{2}T^{3}}{90}\,.$
(III.5)

where $\nu_{b}=2(N_{c}^{2}-1)$ and $\nu_{f}=2N_{c}N_{f}$ with $N_{c}=3$ and
$N_{f}=3$. Note that the Stefan-Boltzmann value of the entropy is reached only
for asymptotically high temperatures. Typically, the entropy at the
temperatures of interest is lower. As a ballpark value, we take a factor of
$3/4$ which is the one that arises in the strongly coupled $\mathcal{N}=4$ SYM
theory Gubser  _et al._ (1996). QCD lattice simulations also indicate a
reduction by a factor of around $0.8$ at moderate temperatures (see e.g.
Borsanyi  _et al._ (2014)). Thus, we match our holographic model to QCD by

$\frac{3s_{SB}}{4}=s_{BH}\,,$ (III.6)

from which we read off $\kappa^{2}=(24\pi^{2})/19\approx 12.5$.

On the other hand, the axial anomaly in three flavor QCD is

$\mathcal{A}_{QCD}=2\,\frac{N_{c}}{32\pi^{2}}\left(\frac{4}{9}+\frac{1}{9}+\frac{1}{9}\right)=\frac{1}{8\pi^{2}}\,,$
(III.7)

where the factor $2$ comes from the sum over right- and left-handed fermions
and we sum over the squares of the electric charges of up, down and strange
quarks in the bracket. We can get the value for the Chern-Simons coupling by
matching the anomaly $\mathcal{A}_{CS}=\mathcal{A}_{QCD}$ and this leads to

$\alpha=\frac{6}{19}\approx 0.316\,.$ (III.8)

Let us finally discuss some physical considerations for the QGP. The strength
of the temperature, magnetic field and chemical potential in typical
nucleations of the QGP at RHIC and LHC are given in table 1.

| RHIC | LHC
---|---|---
$T$ | $300\,\text{M}$\mathrm{eV}$$ | $1000\,\text{M}$\mathrm{eV}$$
$\mu_{5}$ | $10\,\text{M}$\mathrm{eV}$$ | $10\,\text{M}$\mathrm{eV}$$
$B$ | $m_{\pi}^{2}$ | $15\,m_{\pi}^{2}$
Table 1:  Parameters used in our simulations. For the temperature we take a
lower value of roughly twice the critical temperature and a high value of
roughly six times the critical temperature. The values for the magnetic field
are taken from Skokov  _et al._ (2009). Estimates for the axial chemical
potential are very uncertain and we take a small value of 10
M$\mathrm{e}\mathrm{V}$ for both. Due to the considerable uncertainties in the
values of the parameters and also the lifetime of the magnetic field these
should be viewed as ballpark values representative for RHIC and LHC physics.

They provide us with two independent dimensionless quantities, which have to
be adjusted in the numerical simulations with our two free parameters
$(\epsilon_{L},q_{5})$. It turns out that fixing the dimensionless ratio
$\epsilon_{B}/B^{2}$ gives a unique $B/T^{2}$, hence we work with
$\epsilon_{B}$ and then compute the associated $\epsilon_{L}$ for the
simulation through equation (III.2).

In contrast to the previous sections, we show the full pressure anisotropy
evaluated at the physical renormalization scale $\mu=\sqrt{B}$ in this
section:

$\delta P_{i}\equiv 2\kappa^{2}\,\frac{\Delta
P_{B}}{B^{2}}=12\,\frac{\xi_{4}(v)}{B^{2}}+\frac{1}{2}\,\log(BL^{2})-\frac{1}{4}.$
(III.9)

Fixing $\epsilon_{B}/B^{2}$ in eq. (III.2) does not fix $f_{2}$ and $B$
uniquely but rather gives us $B(f_{2})$. This means that at fixed
$\epsilon_{B}/B^{2}$ and vanishing initial dynamical anisotropy
$\xi_{4}(0)=0$, we are confronted with a one parameter family of relative
pressures of the initial state (III.9) depending on the value of the magnetic
field $B$ (for $L=1$). We shall exploit this feature to study equilibration of
the pressure and current for several non-equivalent initial states by
considering different values of $\delta P_{i}$.

### Simulation

In figures 7-9, we show the numerical results for the out-of-equilibrium CME
with the physical parameters estimated in the previous sections. We fix our
initial state by setting the dynamical pressure anisotropy to zero, i.e.
$\xi_{4}(0)=0$, fix the ratio $\epsilon_{B}/B^{2}$ and the axial charge
density $q_{5}$ so that we reach the temperature $T$ and the axial chemical
potential $\mu_{5}$ indicated in table 1 as final equilibrium configuration.

In figure 7 and 8, we present the results for the pressure and the current for
the RHIC and LHC parameters, respectively. Neither the vector current nor the
pressure anisotropy shows oscillatory behavior. The former takes slightly more
time to build up than the latter. On one hand, we observe that in the RHIC
simulation in figure 7, the peak in the vector current is reached at
$v_{peak}\sim 0.54\,$fm/c. On the other hand, the pressure anisotropy reaches
the peak at $v_{peak}\sim 0.48\,$fm/c.

We display the equilibration times for the simulation with the RHIC parameters
in table 2. We use the definition of Chesler and Yaffe to label the
equilibration time Chesler and Yaffe (2009), i.e. the time where the pressure
anisotropy and the current, respectively, are within 10% of their final value.
As for the LHC in figure 8, we find the peak in the vector current at
$v_{peak}\sim 0.14\,$fm/c which is also the time where the pressure anisotropy
reaches its peak. We tabulate the equilibration times for the simulation with
the LHC parameters in table 3. Note that the equilibration times for the LHC
parameters are about 1/3 shorter than in the RHIC case.

Another interesting feature is that the equilibration time for the pressure
slightly depends on the initial state in a non-monotonous fashion. The change
in tendency can be understood as a consequence of choosing an inital state
with a pressure anisotropy that is either above or below the final equilibrium
state. Actually, in the regime where the initial and final states do not
differ more than $10\%$ the equilibration time prescribed above is rather ill-
defined, because we could have a curve in whose pressure does not deviate much
from the final value yielding $v_{eq}=0\,$. However, this problem does not
arise for the parameters chosen in our simulations.

An estimate for the lifetime of the magnetic field has recently been given in
Guo  _et al._ (2019) as
$\tau_{B}\sim\frac{115\,\,\textrm{G}$\mathrm{e}\mathrm{V}$\textrm{fm/c}}{\sqrt{s}}$,
where $\sqrt{s}$ is the energy of the collision. At RHIC and LHC the
collisions take place at around $\sqrt{s}\simeq
200\,\textrm{G}$\mathrm{e}\mathrm{V}$$ and $\sqrt{s}\simeq
5000\,\textrm{G}$\mathrm{e}\mathrm{V}$$, respectively, which yield lifetimes
of $\tau_{B}^{\text{RHIC}}\sim 0.6\,\textrm{fm/c}$ and
$\tau_{B}^{\text{LHC}}\sim 0.02\,\textrm{fm/c}\,$. In this context the
equilibration and build up times extracted from our simulations are of high
significance. It is clear from the equilibration times that for the RHIC
parameter choice the current reaches its equilibrium value before the magnetic
field vanishes. On the contrary, for the LHC parameter choice the magnetic
field is short lived and is gone before the current could start to build up.
Hence, we conclude that the chiral magnetic effect should only be observable
at RHIC and not at LHC.

We notice that $\Delta P/B^{2}$ for fixed $\epsilon_{B}/B^{2}$ yields the same
final equilibrium state independent of the value of $B$ as we expect. Even
though the initial state is different for each run all curves cut at the same
point during the evolution. The current is not influenced by the specific
choice of $B$ as long as the dimensionless ratios stay constant.

In heavy-ion collisions, the magnetic field drops almost instantaneously from
its peak value indicated in table 1 where it stays for most of its remaining
lifetime. Since we consider the magnetic field as static and the drop happens
almost instantaneously, we did a second simulation for our parameter estimates
with 10% of the peak magnetic field 1. The corresponding results for the
current are the blue curves in figure 7 and and figure 8 and the results for
the pressure are depicted in figure 9. Even though the smaller magnetic field
influences the overall absolute values of the observables, the equilibration
times remain effectively unchanged which may be seen from the results
tabulated in the the lower columns of table 2 and 3.

(a)

(b)

Figure 7: Vector current (upper plot) and pressure anisotropy (lower plot) as
a function of time for the physical parameter estimates for RHIC in table 1,
i.e. anomaly $\alpha\simeq 0.316$; for $m_{\pi}=140\,\text{M}$\mathrm{eV}$$.
The pressure anisotropy is for $B=m_{\pi}^{2}$, the results for
$B=0.1\,m_{\pi}^{2}$ are shown in figure 9.

(a)

(b)

Figure 8: Vector current (upper plot) and pressure anisotropy (lower plot) for
the physical parameter estimates for LHC in table 1, i.e. anomaly
$\alpha\simeq 0.316\,$. The pressure anisotropy is for $B=15m_{\pi}^{2}$, the
results for $B=1.5m_{\pi}^{2}$ are shown in figure 9.

(a)

(b)

Figure 9: Pressure anisotropy for RHIC (upper plots) and LHC (lower plot) with 10% of the magnetic field compared to the pressures in figure 7 and figure 8. RHIC $B=m_{\pi}^{2}$ |  |  |  |  |
---|---|---|---|---|---
$\delta P_{i}$ | -2.55 | -1.75 | -1.05 | -0.60 | 0.00
$v^{\langle J\rangle}_{\text{eq}}$ in [fm/c] | 0.380 | 0.380 | 0.380 | 0.380 | 0.380
$v^{\langle\Delta P\rangle}_{\text{eq}}$ in [fm/c] | 0.383 | 0.418 | 0.334 | 0.344 | 0.350
RHIC $B=0.1m_{\pi}^{2}$ |  |  |  |  |
$\delta P_{i}$ | -3.70 | -2.90 | -2.55 | -2.21 | -1.75
$v^{\langle J\rangle}_{\text{eq}}$ in [fm/c] | 0.380 | 0.380 | 0.380 | 0.380 | 0.380
$v^{\langle\Delta P\rangle}_{\text{eq}}$ in [fm/c] | 0.383 | 0.418 | 0.310 | 0.334 | 0.344
Table 2: Equilibration times $v_{\text{eq}}$ for the RHIC simulation at $B=m_{\pi}^{2}$ and $B=0.1\,m_{\pi}^{2}$; $\delta P_{i}$ labels the different initial conditions for the pressure anisotropy (III.9). LHC $B=15\,m_{\pi}^{2}$ |  |  |  |  |
---|---|---|---|---|---
$\delta P_{i}$ | -2.55 | -1.75 | -1.40 | -1.05 | -0.60
$v^{\langle J\rangle}_{\text{eq}}$ in [fm/c] | 0.114 | 0.114 | 0.114 | 0.114 | 0.114
$v^{\langle\Delta P\rangle}_{\text{eq}}$ in [fm/c] | 0.114 | 0.187 | 0.085 | 0.098 | 0.103
LHC $B=1.5\,m_{\pi}^{2}$ |  |  |  |  |
$\delta P_{i}$ | -3.70 | -2.90 | -2.55 | -2.21 | -1.75
$v^{\langle J\rangle}_{\text{eq}}$ in [fm/c] | 0.114 | 0.114 | 0.114 | 0.114 | 0.114
$v^{\langle\Delta P\rangle}_{\text{eq}}$ in [fm/c] | 0.114 | 0.187 | 0.085 | 0.098 | 0.103
Table 3: Equilibration times for the LHC simulation at $B=15\,m_{\pi}^{2}$ and
$B=1.5\,m_{\pi}^{2}$; $\delta P_{i}$ labels the different initial conditions
for the pressure anisotropy (III.9).

In the parameter estimates in table 1, the estimate for the axial chemical
potential is the most uncertain one in the literature. To prove that our
estimations for the build up and equilibration times of the current and the
pressure are not influenced by choosing the particular value of
$\mu_{5}=10\textrm{M}$\mathrm{e}\mathrm{V}$$, we provide analogous simulations
at a ten times larger axial chemical potential of
$\mu_{5}=100\textrm{M}$\mathrm{e}\mathrm{V}$$ in appendix B. The time
dependent current and the pressure anisotropy are depicted in figure 11 and
figure 12, respectively. Furthermore, we tabulated the equilibration and build
up times in table 4. The bottom line is that our results for the build up
times and thus the presence of the chiral magnetic effect in heavy ion
collisions at RHIC and LHC remain qualitatively unchanged at the larger axial
chemical potential.

##  IV Conclusions

In this work, we investigated the out-of-equilibrium behavior of the chiral
magnetic effect in the presence of strong external magnetic fields. We
characterize how the chiral anomaly, the magnetic field and the axial charge
density influence the non-equilibrium response of the chiral magnetic vector
current and the pressure anisotropy and how they affect their equilibration
and build up times.

To quantify the real time response of the vector current and the anisotropy,
we performed a parameter scan. For a fixed strength of the chiral anomaly, we
investigated the dependence of the response on the magnetic field $B$ for a
small and large value of the axial charge density $q_{5}$. Increasing the
magnetic field at fixed strength of $q_{5}$ eventually leads to long lived
oscillations in the vector current which send the equilibration time to
infinity. This is in agreement with the QNM results for our system obtained in
Ammon  _et al._ (2016a); Grieninger (2016). Furthermore, the build up time of
the vector current gets progressively smaller for increasing the magnetic
field. Both effects might be rooted in the presence of Landau levels in our
system. For large magnetic fields, the system is effectively 1+1 dimensional
and the physics is totally dictated by the lowest Landau level. Keeping the
magnetic field constant while increasing the axial charge density simply
increases the final value of the current. The build up time for the anisotropy
also decreases for increasing the magnetic field even though the effect is
small. However, increasing the axial charge density dramatically affects the
pressure anisotropy since it induces long lived oscillations which appear to
be absent in the setup without chiral anomalies Fuini and Yaffe (2015).
Indeed, we show explicitly that the anomaly coefficient has to be sufficiently
large in order to observe these long lived oscillations.

Interestingly, we observe a crossover in the build up times of the vector
current and the anisotropy at small axial charge. For small magnetic fields,
the pressure anisotropy builds up faster while at large magnetic field the
roles are reversed. For large axial charges, the vector current always builds
up faster than the anisotropy independent of the magnetic field.

Finally, we aim to provide insights on the build up time of the chiral
magnetic current in heavy ion collision experiments at RHIC and LHC. Within
our setup, the build up time of the chiral magnetic effect is smaller than the
lifetime of the magnetic field and should thus be an observable in heavy ion
collisions at RHIC Shi  _et al._ (2020). However, the lifetime of the magnetic
field at LHC seems to be so short that the magnetic field already drops to
zero before the chiral magnetic current can build up in a meaningful way.
Furthermore, we find that in both cases, the build up time of the chiral-
magnetic current is approximately as fast as the build up time of the pressure
anisotropy. Interestingly, in the RHIC case we find in presence of the chiral
anomaly a shorter equilibration time of $\sim 0.35$ fm/c (for an initial state
with $\delta P_{i}(0)=0$) compared to the result of Chesler and Yaffe which
estimates the equilibration time as $\sim 0.5$ fm/c Chesler and Yaffe (2009).
This is in agreement with the equilibration time estimate of $\sim 0.3$ fm/c
for plasma temperatures of $T\sim 300-400$ MeV Heinz (2004). Note that the
build up time of the chiral magnetic current is with $\sim 0.38$ fm/c in the
same parameter range. The parameter estimate for the axial chemical potential
seems to be the most uncertain one in the literature. To prove that our
results do not rely on the given parameter estimate of
$\mu_{5}=10\textrm{M}$\mathrm{e}\mathrm{V}$$, we verified that our results for
the build up and equilibration times remain qualitatively unchanged for a ten
times larger axial chemical potential.

In the future, it would be interesting to generalize our setup to more
realistic case of asymmetric shockwave collisions as initiated in Müller  _et
al._ (2020); Waeber  _et al._ (2019). It would also be very interesting to
consider time dependent, dynamical magnetic fields as they are present in
heavy-ion collisions. Finally, it would be interesting to understand how the
quantum critical point investigated in D’Hoker and Kraus (2010, 2013); Ammon
_et al._ (2016b) influences the non-equilibrium dynamics of the system. We
leave these questions open for future investigations.

###### Acknowledgements.

SG is supported by the ‘Atracción de Talento’ program (2017-T1/TIC-5258,
Comunidad de Madrid) and through the grants SEV-2016-0597 and
PGC2018-095976-B-C21. SMT is supported by an FPI-UAM predoctoral fellowship.

##  Appendix A Numerical methods

In this appendix, we explain the numerical methods used throughout this work.
We use the so-called characteristic formulation of Bondi and Sachs888Note that
there is a second approach from numerical relativity established in Heller
_et al._ (2012). established in holographic setups by Chesler and Yaffe (2009,
2010, 2011, 2014); Baggioli  _et al._ (2020a). The big advantage of this
approach is that a set of coupled partial differential equations decouple in a
nested structure of ordinary differential equations in which the equations may
be solved successively. In terms of the characteristic formulation, we solve
the ordinary differential equations on a given time-slice by means of pseudo-
spectral methods. The main idea of pseudo-spectral methods (for an
introduction see Boyd (2003); here we follow Grieninger (2020); Baggioli and
Grieninger (2019); Baggioli  _et al._ (2020b)) is to expand the solution
$u(x)=\sum_{n=0}^{\infty}\,c_{n}\phi_{n}(x)$ to the differential equation in a
basis $\\{\phi_{n}(x)\\}$ and approximate the exact solution $u(x)$ by a
finite number $N$ of basis polynomials $\phi_{n}(x)$

$u(x)\approx u_{N}(x)=\sum\limits_{n=0}^{N}\,c_{n}\,\phi_{n}(x).$ (A.1)

As basis functions we choose Chebychev functions

$T_{k}(x)=\cos(k\,\arccos(x)).$ (A.2)

We can re-write first and second derivative by using the derivatives of the
basis functions i.e.
$\phi^{\prime}_{m}(x)=\sum_{n=0}^{N}\hat{D}_{mn}\phi_{n}(x),\
\phi^{\prime\prime}_{m}(x)=\sum_{n,l=0}^{N}\hat{D}_{mn}\hat{D}_{nl}\,\phi_{l}(x)$.
With the differentiation matrices, we can rewrite derivatives so that they act
on the coefficients, for example

$\displaystyle
u^{\prime}(x)\approx\\!\sum\limits_{n=0}^{N}c_{n}\,\phi_{j}^{\prime}(x)=\\!\sum\limits_{n,m=0}^{N}c_{n}\,\hat{D}_{nm}\phi_{m}(x)\\!=\\!\sum\limits_{n=0}^{N}c_{n}^{\prime}\,\phi_{n}(x).$
(A.3)

To discretize the differential equations in the radial direction, we use a
Chebychev-Lobatto grid with $N$ gridpoints given by

$x_{i}=\cos\varphi_{i}=\cos\frac{\pi\,i}{N}.$ (A.4)

We may solve the equations of motion for the axial gauge field by introducing

$q_{5}\equiv 8\alpha\,B\,V(v,u)+u^{2}Q_{5}^{\prime}(v,u)\Sigma(v,u)^{3}$ (A.5)

and simplify it further by introducing
$\tilde{q}_{5}(v,u,\alpha,B)=q_{5}-8\alpha\,B\,V(v,u)$. As explained in
Chesler and Yaffe (2014), the condition for fixing the apparent horizon to
$u_{h}=1$ on the initial time slice reads $\mathrm{d}\Sigma(u_{h})=0.$ We can
keep the apparent horizon fixed at $u_{h}=1$ by imposing that the time
derivative of the aforementioned equation vanishes. By using the equations of
motion we find that we can implement this as a boundary condition on the
blackening factor $f$ at the horizon, i.e.

$\displaystyle
2\,f(v,1)\left(B^{2}\Sigma(v,1)^{2}e^{-2\,\xi(v,1)}+\tilde{q}_{5}^{2}-24\,\Sigma(v,1)^{6}\right)$
(A.6)
$\displaystyle\\!\\!\\!-2\,\Sigma(v,1)^{4}\\!\left(\mathrm{d}V(v,1)^{2}\,e^{2\,\xi(v,1)}\\!+3\,\mathrm{d}\xi(v,1)^{2}\,\Sigma(v,1)^{2}\right)\\!\\!=0$

We start with an initial profile for $V$ and $\xi$. The initial data also
contains the energy density $\epsilon=-3\,f_{2}$, the axial charge density
$q_{5}$, the Chern-Simons coupling $\alpha$ and the magnetic field $B$. On a
given time slice we solve the set of equations in the following order: eq.
(II.11) for $\Sigma$, eq. (II.12) for $\mathrm{d}\Sigma$, (II.10) and eq.
(II.13) for $\mathrm{d}V$ and $\mathrm{d}\xi$, eq. (II.14) for the blackening
factor $f$. Eq. (II.15) functions as a constraint. Every time step, we can
extract $\dot{\lambda}$ by reading of $\dot{\lambda}=-f_{s}(u=0)/2$. We impose
the constraint equation (II.15) in terms of the horizon boundary condition for
$f$ (A.6) and at the conformal boundary in terms of
$\mathrm{d}\Sigma_{s}(u=0)=0$. Finally, to evolve in time we use a forth order
Runge-Kutta with an appropriately small timestep.

To improve the convergence we subtract the logarithmic terms up to
appropriated order and work with redefined functions which are given by

$\displaystyle\Sigma(v,u)=$
$\displaystyle\frac{1}{u}+\lambda(v)+u^{5}\Sigma_{s}(v,u)$
$\displaystyle\mathrm{d}\Sigma(v,u)=$
$\displaystyle\frac{1}{2}\,\Sigma(v,u)^{2}+\mathrm{d}\Sigma_{s}(v,u)+\epsilon\,\left(\frac{1}{u}+\lambda(v)\right)^{-2}$
$\displaystyle-\frac{B^{2}}{12}\log\\!\left(\frac{1}{u}+\lambda(v)\right)\,\left(\frac{1}{u}+\lambda(v)\right)^{-2}$
$\displaystyle\xi(v,u)=$
$\displaystyle\left(\frac{1}{u}+\lambda(v)\right)^{-3}\,\xi_{s}(v,u)$
$\displaystyle-\frac{B^{2}}{12}\log\\!\left(\frac{1}{u}+\lambda(v)\right)\,\left(\frac{1}{u}+\lambda(v)\right)^{-4}$
$\displaystyle\mathrm{d}\xi(v,u)=$
$\displaystyle\left(\frac{1}{u}+\lambda(v)\right)^{-2}\,\mathrm{d}\xi_{s}(v,u)$
$\displaystyle-\frac{B^{2}}{6}\log\left(\frac{1}{u}+\lambda(v)\right)\,\left(\frac{1}{u}+\lambda(v)\right)^{-3}$
$\displaystyle f(v,u)=$
$\displaystyle\left(\frac{1}{u}+\lambda(v)\right)^{2}\\!\\!+f_{s}(v,u)+2\epsilon\,\left(\frac{1}{u}+\lambda(v)\right)^{-2}$
$\displaystyle-\frac{B^{2}}{6}\log\\!\left(\frac{1}{u}+\lambda(v)\right)\,\left(\frac{1}{u}+\lambda(v)\right)^{-2}$
$\displaystyle V(v,u)$
$\displaystyle=\left(\frac{1}{u}+\lambda(v)\right)^{-1}\,V_{s}(v,u)$
$\displaystyle\mathrm{d}V(v,u)$ $\displaystyle=\mathrm{d}V_{s}(v,u).$

We monitor the accuracy of our numerical algorithm by different methods.
Firstly, we check the constraint equation throughout the time evolution and
monitor how accurately the apparent horizon stays at one. Secondly, we
compared our solution to a solution with a larger number of gridpoints and
checked that it does not change significantly. Lastly, we checked the
Chebychev coefficients of our numerical solution, as presented in figure 10
for a given time and ensured that the coefficients drop to the required
precision.

(a)  Figure 10: Chebychev coefficients for a large magnetic field
$B/\sqrt{\epsilon}=2.31,\,q_{5}/\epsilon^{3/4}=0.31,\,\alpha=6/19$.

##  Appendix B Simulations for $\mu_{5}=100$MeV.

In this appendix, we provide simulations with a ten times larger chemical
potential compared to the parameter estimates given in table 1. We keep all
the other parameters the same and figure 11 has to be compared with figure 7
(for RHIC at $B=m_{\pi}^{2}$) and figure 12 with figure 8 (for LHC at
$B=15\,m_{\pi}^{2}$). We tabulate the equilibration and build up times for
these simulations in table 4.

(a)

(b)

Figure 11: Vector current (upper plot) and pressure anisotropy (lower plot)
for the physical parameter estimates for RHIC at chemical potential
$\mu_{5}=100\,\textrm{M}$\mathrm{e}\mathrm{V}$$ (otherwise with the values
from table 1) and anomaly $\alpha\simeq 0.316\,$.

(a)

(b)

Figure 12: Vector current (upper plot) and pressure anisotropy (lower plot) for the physical parameter estimates for LHC at chemical potential $\mu_{5}=100\,\textrm{M}$\mathrm{e}\mathrm{V}$$ (otherwise with the values from table 1) and anomaly $\alpha\simeq 0.316\,$. $\mu_{5}=100\textrm{M}$\mathrm{e}\mathrm{V}$$ | RHIC ($B=m_{\pi}^{2}$) | LHC ($B=15\,m_{\pi}^{2}$)
---|---|---
$\delta P_{i}$ | -2.21 | -1.05 | -2.55 | -0.60
$v^{\langle J\rangle}_{\text{eq}}$ in [fm/c] | 0.380 | 0.380 | 0.114 | 0.114
$v^{\langle\Delta P\rangle}_{\text{eq}}$ in [fm/c] | 0.393 | 0.336 | 0.113 | 0.102
$v_{\text{peak}}^{\langle J\rangle}$ in [fm/c] | 0.537 | 0.537 | 0.161 | 0.161
$v_{\text{peak}}^{\langle\Delta P\rangle}$ in [fm/c] | 0.477 | 0.437 | 0.142 | 0.133
Table 4: Equilibration and build up times for the RHIC and LHC simulation at
ten times the axial charge density.

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