# Chiral magnetic currents with QGP medium response in heavy ion collisions at
RHIC and LHC energies

Duan She  College of Science, China Three Gorges University, Yichang 443002,
China  Sheng-Qin Feng<EMAIL_ADDRESS>College of Science, China
Three Gorges University, Yichang 443002, China  Key Laboratory of Quark and
Lepton Physics (Huazhong Normal Univ.), Ministry of Education, Wuhan 430079,
China  Yang Zhong  College of Science, China Three Gorges University, Yichang
443002, China  Zhong-Bao Yin  Institute of Particle Physics,Central China
Normal University, Wuhan 430079, China

###### Abstract

We calculate the electromagnetic current with a more realistic approach in the
RHIC and LHC energy regions in the article. We take the partons formation time
as the initial time of the magnetic field response of QGP medium. The maximum
electromagnetic current and the time-integrated current are two important
characteristics of the chiral magnetic effect (CME), which can characterize
the intensity and duration of fluctuations of CME. We consider the finite
frequency response of CME to a time-varying magnetic field, find a significant
impact from QGP medium feedback, and estimate the generated electromagnetic
current as a function of time, beam energy and impact parameter.

##  I Introduction

Quantum chromodynamics (QCD) is widely accepted to be the fundamental theory
of the strong interactions. One remarkable feature of QCD is the existence of
configurations of gauge fields characterized by different winding numbers. The
chirality imbalance is induced by the nonzero topological charge through the
axial anomalyBell and Jackiw (1969); Adler (1969); Kharzeev and Warringa
(2009) of QCD as follows:

$\displaystyle\Delta N_{5}=\Delta(N_{R}-N_{L})=-2N_{f}Q,$ (1)

where $N_{f}$ is the number of flavors , and $\Delta N_{5}$ is the change in
chirality ($N_{5}$ ) which is the difference between the number of modes with
right- and left-handed chirality. In the limit of zero quark mass, $N_{5}$ is
equal to the total number of particles plus antiparticles with right-handed
helicity minus the total number of particles plus antiparticles with left-
handed helicity. It has been proposed that if P- and CP-violating processes
are taking place in the quark-gluon plasma (QGP) produced in relativistic
heavy-ion collisions, then positive charges should separate from negative
charges along the direction of angular momentum of the collision. The
underlying mechanism is the so-called chiral magnetic effect (CME)Kharzeev and
Warringa (2009); Fukushima  _et al._ (2008); Kharzeev  _et al._ (2008). The
CME is a particularly interesting phenomenon coming from the interplay of
quantum anomaly with magnetic field. The electric current introduced by the
chirality imbalance Kharzeev and Warringa (2009); Fukushima  _et al._ (2008);
Kharzeev  _et al._ (2008) along an external magnetic field is as follows:

$\displaystyle\vec{J}=\sigma\vec{B}$ (2)

where $\sigma=e^{2}\mu_{5}/(2\pi^{2})$ is the chiral magnetic conductivity
with the chiral chemical potential $\mu_{5}$. One has to sum over the quark
colors and flavors:

$\displaystyle\sigma=N_{c}\Sigma_{f}\frac{Q^{2}_{f}e^{2}\mu_{5}}{2\pi^{2}}$
(3)

where $N_{c}$ is the color number of dynamical quarks. The CME leads to the
event-by-event fluctuations of electric dipole moment of the QGPKharzeev and
Zhitnitsky (2007); Kharzeev (2006); Burnier  _et al._ (2011) in relativistic
heavy ion collisions.

The question is whether CME exists in relativistic heavy ion collisions. Some
analysis shows that the answer seems to be yes. Two necessary conditions for
CME are chirality imbalance and strong magnetic field, which may be met in QGP
generated in relativistic heavy ion collisions. Firstly, enormous magnetic
field can be produced in non-central relativistic heavy-ion collision due to
charged nucleus moving at speed close to the speed of lightMo  _et al._
(2013); Skokov  _et al._ (2009); Voronyuk  _et al._ (2011); Bzdak and Skokov
(2012); Tuchin (2013a); Deng and Huang (2012). Secondly, QCD which describes
the behavior of the QGP allows topological charge changing transition that can
induce chirality imbalance[5]. The magnetic field is the driving force and the
electric charge separation is the manifestation of the CMEWang and Wen (2017).
Therefore, the CME is very likely to exist in relativistic heavy - ion
collisions. Thus, heavy-ion collisions provide a unique terrestrial
environment to investigate QCD matter in strong magnetic fields.

Over the past few years, the CME has been intensively explored by relativistic
heavy ion collisions at the BNL Relativistic Heavy Ion Collider (RHIC) and at
the CERN Large Hadron Collider (LHC), including STARAdamczyk  _et al._ (2014a,
b); Wang (2013); Adamczyk  _et al._ (2013); Voloshin  _et al._ (2011); Abelev
_et al._ (2010, 2009) , ALICEAbelev  _et al._ (2013); Christakoglou (2011) and
for recent reviews see Refs.Kharzeev (2014); Kharzeev  _et al._ (2016).
Observation of the chiral magnetic effect will be direct experimental evidence
for the existence of topologically nontrivial gluon configurations.

One of the main issue to study the CME is the time evolution of the magnetic
field in relativistic heavy ion collisions. This issue has been investigated
by many works in the literatureKharzeev  _et al._ (2008); Mo  _et al._ (2013);
Skokov  _et al._ (2009); Voronyuk  _et al._ (2011); Bzdak and Skokov (2012);
Tuchin (2013a); Deng and Huang (2012). The numerical computations executed by
these works show that an enormous magnetic field ($B\sim 10^{15}T$) can be
generated at the very beginning of the collisions. However, according to these
studies, the strength of the generated magnetic field decreases rapidly with
time. If the lifetime of the magnetic field is so short that we can hardly see
the imprint of the CME, and this is indeed a challenge for the manifestation
of the CME in relativistic heavy-ion collisions. Nevertheless, It is argued
that these studies of the CME are valid only at the early stage of the
collision. It is argued Tuchin (2010); *Tuchin:2010vs2; McLerran and Skokov
(2014); Tuchin (2013b); Zakharov (2014); Tuchin (2015, 2016) that the magnetic
field response from the QGP medium becomes increasingly important at a later
time, and the magnetic field will maintain a much longer time in QGP.

In this paper, we precisely aim to address two pressing issues: (1) the time
evolution of the strong magnetic field (which is the necessary driving force
for CME), (2) the dynamical generation of CME current in response to time-
dependent magnetic field. In order to make progress and to gain valuable
insights on these problems, we chose the simplified model approach and were
able to obtain very interesting results. We consider the finite frequency
response of CME to a time-varying magnetic field, find significant impact from
QGP medium feedback, and estimate the generated electromagnetic current as a
function of time, beam energy and impact parameter, respectively.

This paper is organized as follows: the time evolution of the magnetic field
in relativistic heavy-ion collisions is presented in Sec. II; in Sec. III we
discuss the chiral magnetic conductivity and the electromagnetic current; The
results of the electromagnetic current in the RHIC and LHC energy regions are
exhibited in Sec. IV. The conclusions are summarized in Sec. V.

##  II The magnetic field with the response of QGP medium

In Ref.Kharzeev  _et al._ (2008), Kharzeev, Mclerran, and Warringa (KMW)
published an analytic model to calculate the magnetic field, and assumed a
uniform nucleon density in nucleus rest frame in relativistic heavy-ion
collisions. In Ref. Mo  _et al._ (2013), we improved the calculation of the
magnetic field by using the Woods-Saxon nucleon distribution to replace
uniform nucleon distribution. There also exist many other analyses using
different methodsSkokov  _et al._ (2009); Voronyuk  _et al._ (2011); Bzdak and
Skokov (2012); Tuchin (2013a); Deng and Huang (2012) to calculate the magnetic
field in the vacuum.

However, most of these analyses did not take into account the magnetic field
response of the QGP medium which may obviously influence the time evolution of
the magnetic field. Tuchin first analyzedTuchin (2010); *Tuchin:2010vs2 the
magnetic field feature in QGP medium, and concluded that the magnetic field
was almost constant during the entire plasma lifetime due to high electric
conductivity. Later, it was quantitatively studied in many worksTuchin (2010);
*Tuchin:2010vs2; McLerran and Skokov (2014); Tuchin (2013b); Zakharov (2014);
Tuchin (2015, 2016) . To explore this problem, one needs to consider the
electric conductivity $\sigma$ and chiral magnetic conductivity
$\sigma_{\chi}$ which is induced by the CME. In Ref.McLerran and Skokov
(2014), McLerran and Skokov found that the effects of finite $\sigma_{\chi}$
are not so important for the top RHIC and LHC energies. Therefore, we are not
considering the effects of chiral magnetic conductivity in this paper. For
electric conductivity $\sigma$, there are a lot of theoretical
uncertaintiesFrancis and Kaczmarek (2012); Ding  _et al._ (2011); Arnold  _et
al._ (2003); Gupta (2004).

We adopt the most optimistic assumption proposed in Ref.Deng and Huang (2012),
which assumes the electric conductivity $\sigma$ is large enough that one can
take the QGP as an ideally conducting plasma. We use the following equations
from Maxwell equations:

$\displaystyle\frac{\partial\mathbf{B}}{\partial t}$ $\displaystyle=$
$\displaystyle\nabla\times(\mathbf{v}\times\mathbf{B}),$ (4)
$\displaystyle\mathbf{E}$ $\displaystyle=$
$\displaystyle-\mathbf{v}\times\mathbf{B},$ (5)

where the $\mathbf{v}$ is the flow velocity of QGP.

To study how the magnetic field evolves in the QGP medium, we use the Bjorken
picture for the longitudinal expansionBjorken (1983); Ollitrault (2008) as:

$v_{z}=\frac{z}{t},$ (6)

we adopt a linearized ideal hydrodynamic equation to describe the transverse
velocities as

$\displaystyle v_{x}$ $\displaystyle=$
$\displaystyle\frac{c_{s}^{2}}{a_{x}^{2}}xt,$ (7) $\displaystyle v_{y}$
$\displaystyle=$ $\displaystyle\frac{c_{s}^{2}}{a_{y}^{2}}yt,$ (8)

where $a_{x,y}$ are the root-mean-square widths of the transverse distribution
and $c_{s}$ is the speed of sound. We take $c_{s}^{2}\sim 1/3$ and $a_{x}\sim
a_{y}\sim 3$,respectively.

We can calculate the magnetic field $\mathbf{B}(t)$ at a given initial
condition $\mathbf{B}^{0}(\mathbf{r})=\mathbf{B}(t=t_{0},\mathbf{r})$ by
substituting the velocity into Eqs.(4)-(5), where $t_{0}$ is the formation
time of partons. It is found that only the $y$ component of the magnetic field
of the central point of two nuclei collision remains. Therefore, we only
consider the $y$ component of the magnetic field at the center of the
collision region $\mathbf{r}=\mathbf{0}$, and one gets the following solution

$B_{y}(t\geq
t_{0},\mathbf{0})=\frac{t_{0}}{t}e^{-\frac{c_{s}^{2}}{2a_{x}^{2}}(t^{2}-t_{0}^{2})}B_{y}^{0}(\mathbf{0}).$
(9)

We will calculate the magnetic field $B_{y}(\mathbf{0})$ at $t\leq t_{0}$ at
the central point ($\vec{x}=0$) in which case it is pointing in the $y$
direction, and it is given by contributions of participant and spectator
nucleons in relativistic heavy-ion collisions.

$B_{y}(t\leq
t_{0},\mathbf{0})=B_{ys}^{+}(t,\mathbf{0})+B_{ys}^{-}(t,\mathbf{0})+B_{yp}^{+}(t,\mathbf{0})+B_{yp}^{-}(t,\mathbf{0}).$
(10)

where $B_{ys}^{\pm}(t,\mathbf{0})$ and $B_{yp}^{\pm}(t,\mathbf{0})$ are the
contributions of the spectators and the participants moving in the positive or
negative $z$ direction, respectively. For spectators, we assume that they do
not scatter at all and that they keep traveling with the beam rapidity
$Y_{0}$. The magnetic field from spectators is given as:

$\displaystyle B_{ys}^{\pm}(t,\mathbf{0})$ $\displaystyle=\pm
Z\alpha_{EM}\sinh{Y_{0}}\int_{V_{s}^{\pm}}{\rm
d}^{3}\vec{x}^{\prime}\rho_{\pm}(\vec{x}^{\prime})$ (11)
$\displaystyle\times\frac{\vec{x}^{\prime}_{\bot}\times\vec{e}_{z^{\prime}}}{[\vec{x}^{\prime
2}_{\bot}+(\pm t\sinh Y_{0}+z^{\prime}\cosh Y_{0})^{2}]^{3/2}},$

where $\rho_{\pm}(\vec{x}^{\prime})$ is the three dimension Wood-Saxon nuclear
density:

$\displaystyle\rho_{\pm}(\vec{x}^{\prime})=\frac{\gamma
n_{0}}{1+\exp\left(\frac{\sqrt{(x^{\prime}\mp{b/2})^{2}+y^{\prime 2}+(\gamma
z^{\prime})^{2}}-{\rm R}}{d}\right)}$ (12)

where $\gamma$ is the Lorentz factor, $n_{0}=0.17\mathrm{fm}^{-3}$,
$d=0.54\mathrm{fm}$, and the radius $R=1.12A^{1/3}\mathrm{fm}$. The
contribution of the participants to the magnetic field can be also given by

$\displaystyle B_{yp}^{\pm}(t,\mathbf{0})$ $\displaystyle=\pm
Z\alpha_{EM}\int_{V_{p}^{\pm}}{\rm
d}^{3}\vec{x}^{\prime}\int_{-Y_{0}}^{Y_{0}}{\rm d}Yf(Y){\sinh
Y}\rho_{\pm}(\vec{x}^{\prime})$ (13)
$\displaystyle\times\frac{\vec{x}^{\prime}_{\bot}\times\vec{e}_{z^{\prime}}}{[\vec{x}^{\prime
2}_{\bot}+(\pm t\sinh Y+z^{\prime}\cosh Y)^{2}]^{3/2}},$

where

$\displaystyle f(Y)=\frac{a}{2\sinh(aY_{0})}{\rm e}^{aY},\hskip 28.45274pt-
Y_{0}\leq{Y}\leq{Y_{0}},$ (14)

and experimental data show that $a\approx 1/2$, consistent with the baryon
junction stopping mechanism.

To study the time evolution of magnetic field from Eq.(9), we must know the
formation time $t_{0}$ of partons and the initial magnetic field
$B_{y}^{0}(\mathbf{0})$ at this time. For the initial magnetic field
$B_{y}^{0}(\mathbf{0})$, we can calculate from Eq.(10) to Eq.(14) which
developed from the method in Ref. Mo  _et al._ (2013). For the formation time
$t_{0}$ of partons Kharzeev and Nardi (2001), the following approximation
formula has been used:

$t_{0}\simeq 1/Q_{s},$ (15)

where $Q_{s}$ is the saturation momentum. According to saturation analysis in
Refs.Kharzeev and Nardi (2001); Kowalski  _et al._ (2008); Lappi (2009),
$Q_{s}$ is related to collision nuclei and collision energy as:

$Q_{s}^{2}\sim A^{1/3}x^{-\lambda},x=Q_{s}/\sqrt{s},$ (16)

where A is the atomic quantity of collision nuclei, and $\lambda$ is between
$0.25$ and $0.3$, we give $\lambda=0.28$ in the paper. Then we get

$Q_{s}^{2}=kA^{(\frac{2}{3(2+\lambda)})}\sqrt{s}^{(\frac{2\lambda}{2+\lambda})},$
(17)

where $k$ is coefficient of proportionality, we take it as a constant. These
$Q_{s}^{2}(\sqrt{s}=130\,\mathrm{GeV},b,A=197)$ were calculated by Tab.2 in
Refs.Kharzeev and Nardi (2001) for different impact parameters at RHIC
$\sqrt{s}=130\,\mathrm{GeV}$ of Au-Au collisions. By comparing with the
results from $\sqrt{s}=130\,\mathrm{GeV}$ of Au-Au collisions, we compute
$Q_{s}^{2}(\sqrt{s},b,A)$ at different $b$, different collision energy
$\sqrt{s}$ and different collision nuclei $A$ as:

$\displaystyle Q_{s}^{2}(\sqrt{s},b,A)$ $\displaystyle=$
$\displaystyle(\frac{A}{197})^{\frac{2}{3(2+\lambda)}}(\frac{\sqrt{s}}{130})^{\frac{2\lambda}{2+\lambda}}$
$\displaystyle\times$ $\displaystyle Q_{s}^{2}(\sqrt{s}=130GeV,b,A=197).$

From Eq.(15) to Eq.(18), we can calculate the formation time as
$t_{0}=0.16\,\mathrm{fm}$ for $\sqrt{s}=200\,\mathrm{GeV}$ Au-Au collisions
with $b=8\,\mathrm{fm}$, $t_{0}=0.11\,\mathrm{fm}$ for
$\sqrt{s}=2760\,\mathrm{GeV}$ Pb-Pb collisions and $t_{0}=0.10\,\mathrm{fm}$
for $\sqrt{s}=5020\,\mathrm{GeV}$ Pb-Pb collisions with $b=8\,\mathrm{fm}$.
Figure 1(a) provides the comparison of the time evolution of the magnetic
field at the central point with or without considering the QGP response for
Au-Au collisions with $b=8\,\mathrm{fm}$ at $\sqrt{s}=200\,\mathrm{GeV}$. fig.
1(b) and fig. 1(c) are same as the fig.1(a) but for Pb-Pb collisions at
$\sqrt{s}=2760\,\mathrm{GeV}$ and for Pb-Pb collisions at
$\sqrt{s}=5020\,\mathrm{GeV}$, respectively. It is found that the magnetic
field considering the QGP response will maintain a much longer time than that
in the vacuum.

Figure 1: The time evolutions of magnetic field at central point for
$\sqrt{s}=200\,\mathrm{GeV}$ Au-Au collisions with $b=8\,\mathrm{fm}$,
$\sqrt{s}=2760\,\mathrm{GeV}$ Pb-Pb collisions with $b=8\,\mathrm{fm}$ and
$\sqrt{s}=5020\,\mathrm{GeV}$ Pb-Pb collisions with $b=8\,\mathrm{fm}$. The
solid line and dashed line represent with and without considering the response
of QGP medium, respectively. Figure 2: The comparisons of magnetic field time
evolution at central point between for Au-Au collisions with
$b=8\,\mathrm{fm}$ at $\sqrt{s}=200\,\mathrm{GeV}$ and for Pb-Pb collisions
with $b=8\,\mathrm{fm}$ at $\sqrt{s}=2760\,\mathrm{GeV}$ with and without
considering QGP response.

Figure 2 shows a comparison of magnetic field time evolution at central point
between for Au-Au collisions with $b=8\,\mathrm{fm}$ at
$\sqrt{s}=200\,\mathrm{GeV}$ and for Pb-Pb collisions with $b=8\,\mathrm{fm}$
at $\sqrt{s}=2760\,\mathrm{GeV}$ with and without considering QGP response. It
is found that that the magnetic field $B_{t=0}$ at LHC
$\sqrt{s}=2760\,\mathrm{GeV}$ is about 100 times that at RHIC
$\sqrt{s}=200\,\mathrm{GeV}$, but the magnetic field at LHC
$\sqrt{s}=2760\,\mathrm{GeV}$ decreases much faster than that at RHIC
$\sqrt{s}=200\,\mathrm{GeV}$. The formation time $t_{0}$ varies little with
the collision energy $\sqrt{s}$. On the whole, the magnetic field at RHIC is
far larger than the magnetic field at LHC in the case of $t>t_{0}$.

Figure 3: The comparison of various results for the time dependence of the
magnetic field normalized by the peak value of the magnetic field from
different studies: ECHO-QGPInghirami  _et al._ (2016) (dashed line), McLerran-
SkokovMcLerran and Skokov (2014) with different electric conductivity (dotted
line curve for $\sigma=\sigma_{LQCD}$, dash-dotted line for
$\sigma=100\sigma_{LQCD}$).

One potentially very useful application of the results on time-dependent
magnetic fields (in Fig.1) would be serving as an input for recently developed
anomalous hydrodynamic simulations for the CMEJiang  _et al._ (2018); Shi  _et
al._ (2017). As shown in Fig.2 of Ref.Shi  _et al._ (2017), the final CME
signal very sensitively depends on the time-dependence of magnetic field. Many
calculations have been done to study the strength and space-time dependence of
the magnetic field which are among the most crucial factors in quantifying the
CME signal. However, the time evolution of magnetic field (B) remains an open
question. The hot medium created in the collision could delay the decrease of
the magnetic field through the generation of an induction current in response
to the rapidly decaying magnetic field. In fig.3, we show a comparison of
various results for the time dependence of the magnetic field normalized by
the peak value of the magnetic field: the study by McLerran-SkokovMcLerran and
Skokov (2014) with conductivity $\sigma=\sigma_{LQCD}$ and
$\sigma=100\sigma_{LQCD}$, ECHO-QGP simulationInghirami  _et al._ (2016) and
our model. Compared with the other results, the magnetic field given by us is
the fastest to decrease with time.

##  III CHIRAL MAGNETIC CONDUCTIVITY AND CHIRAL MAGNETIC CURRENT

Let us now study the induced current by a magnetic field created in
relativistic heavy ion collisions. As a qualitative research purpose, one
assumes that the magnetic field is homogeneous in space. The induced current
can be given as followsKharzeev and Warringa (2009):

$j(t)=\int_{0}^{\infty}\frac{dw}{\pi}[\sigma^{\prime}_{\chi}(\omega)\cos(\omega
t)+\sigma^{\prime\prime}_{\chi}(\omega)\sin(\omega t)]\tilde{B}(\omega),$ (19)

where

$\displaystyle\tilde{B}(\omega)=\int_{t_{0}}^{\infty}dte^{i\omega t}B(t),$
(20)

and $\sigma^{\prime}_{\chi}(\omega)$, $\sigma^{\prime\prime}_{\chi}(\omega)$
are the real part and imaginary part of the chiral magnetic conductivity. The
so-called Kramers-Kroning relation gives the relationship between the real and
imaginary parts as follows:

$\displaystyle\sigma^{\prime}_{\chi}(\omega)=\frac{1}{\pi}\textbf{P}\int_{-\infty}^{\infty}dq_{0}\frac{\sigma^{\prime\prime}_{\chi}(q_{0})}{q_{0}-\omega},$
(21)
$\displaystyle\sigma^{\prime\prime}_{\chi}(\omega)=-\frac{1}{\pi}\textbf{P}\int_{-\infty}^{\infty}dq_{0}\frac{\sigma^{\prime}_{\chi}(q_{0})}{q_{0}-\omega},$
(22)

where $\sigma_{\chi}(\omega)=\lim_{\vec{p}\rightarrow
0}\sigma_{\chi}(p_{0}=\omega,\vec{p})$ . The symbol P of Eqs.(21) and (22)
denote the integral which avoids the singularity via the upper and the lower
complex plane. The chiral magnetic conductivity will be complex as follows:

$\sigma_{\chi}(p)=\sigma^{\prime}_{\chi}(p)+i\sigma^{\prime\prime}_{\chi}(p)$
(23)

where $\sigma^{\prime}_{\chi}(p)$ and $\sigma^{\prime\prime}_{\chi}(p)$ are
real functions. They can be expressed as:

$\displaystyle\sigma^{\prime}_{\chi}(p)=\frac{1}{p^{i}}\imaginary
G^{i}_{R}(p),$ (24)
$\displaystyle\sigma^{\prime\prime}_{\chi}(\omega)=-\frac{1}{p^{i}}\real
G^{i}_{R}(p),$ (25)

where $G^{i}_{R}(p)=\frac{1}{2}\varepsilon^{ijk}\tilde{\prod}_{R}^{jk}(p)$ is
the retarded correlator.

The retarded correlator $G^{i}_{R}(p)$ is a very important physical quantity
in linear response theory of chiral magnetic conductivity. The detailed
analysis and calculation process of $G^{i}_{R}(p)$ can be found in
Ref.Kharzeev and Warringa (2009). The retarded correlator can be given as
follows:

$\displaystyle G^{i}_{R}(p)$ $\displaystyle=$
$\displaystyle\frac{ie^{2}}{16\pi^{2}}\frac{p^{i}}{p}\frac{p^{2}-p_{0}^{2}}{p^{2}}\int_{0}^{\infty}\differential{q}[f(q)$
$\displaystyle\times$
$\displaystyle\sum_{t=\pm}(2q+tp_{0})\log\frac{(p_{0}+i\varepsilon+tq)^{2}-(q+p)^{2}}{(p_{0}+i\varepsilon+tq)^{2}-(q-p)^{2}}],$

where

$f(q)=\sum_{s=\pm}s[\tilde{n}(q-\mu_{s})-\tilde{n}(q+\mu_{s})],$ (27)

where $\tilde{n}(x)=[1+\exp(\beta x)]^{-1}$ is the Fermi-Dirac distribution
function. To derive the imaginary part of the chiral magnetic conductivity,
one can express the imaginary part of the logarithm in Eq.(21) with $q\geq 0$
and $p=|\vec{p}|\geq 0$ as follows:

$\displaystyle\imaginary$
$\displaystyle\sum_{t=\pm}(2q+tp_{0})\log\frac{(p_{0}+i\varepsilon+tq)^{2}-(q+p)^{2}}{(p_{0}+i\varepsilon+tq)^{2}-(q-p)^{2}}$
$\displaystyle\quad=\pi[2q-|p_{0}|\theta(p_{0}^{2}-p^{2})][\theta(q_{+}-q)-\theta(q_{-}-q)]$
$\displaystyle\quad+\pi
p_{0}\theta(p^{2}-p_{0}^{2})[\theta(q-q_{+})-\theta(q-q_{-})],$

where $q_{\pm}=\frac{1}{2}|p_{0}\pm p|$.

The real and imaginary parts of chiral magnetic conductivity can be computed
by using the Kramers-Kroning relation as shown in Eqs.(21) and (22). Fig.4(a,
b) shows the full frequency and momentum dependence of
$\sigma_{\chi}(\omega,p)$. We can find some features of the magnetic
conductivity as follows:

(1) There is a peak at $p=0$ and $\omega\approx 5.406T$ at high temperature,
which tends to disappear when $p>0$.

(2) The conductivities are not vanishing at $p=0$ and $\omega\neq 0$, and they
still present a discontinuity at $p=0$ and $\omega=0$.

(3) The real part of conductivities can be negative note that the phase angle
is between $0$ and $\pi$. When $p=\omega$, the imaginary part of
conductivities is equal to zero.

(4) The chiral magnetic conductivity is approximately vanishing at high
temperature, in the regime $\omega\gg p$ or $p\gg\omega$.

Figure 4: Frequency and momentum dependence of the chiral magnetic
conductivity normalized to the zero frequency-momentum value, from a numerical
evaluation. (a) shows the real part of chiral magnetic conductivity, and (b)
shows the imaginary part of the conductivity. Where $\mu=10\mathrm{MeV}$,
$\mu_{s}=1\mathrm{MeV}$ , and the temperature $T=200\mathrm{MeV}$.

##  IV THE RESULTS OF THE ELECTROMAGNETIC CURRENT

In the above chapter, we have computed the real and imaginary parts of the
magnetic conductivity, which are shown in fig.4. And then we will use the
Eq.(19) to calculate the electromagnetic current. It can be seen from the
Eqs.(19-20) that in order to calculate the electromagnetic current we must
study the dependence of the magnetic field on time after the formation of the
parton. The detailed evolution formula of the magnetic field with time after
$t\geq t_{0}$ is given by Eq.(9). The dependence of the magnetic field on time
before the formation of the part ($t\leq t_{0}$)is given by Eqs. (10-14).

The dependencies of the electromagnetic current on time with or without the
QGP response in the RHIC and LHC are shown in Fig.5. The time dependencies of
chiral magnetic currents and magnetic field are calculated by using
Eqs.(19-23) and Eq.(9). It is found that the strength of the electromagnetic
current considering the QGP response is larger than that of the
electromagnetic current in the vacuum. In order to make a related study of the
experimental results of CMS collaboration in Refs.Sirunyan  _et al._ (2017),
we also compute the electromagnetic current at LHC $\sqrt{s}=5020\mathrm{GeV}$
for Pb - Pb collisions with $b=8\mathrm{fm}$.

Figure 5: The dependencies of the electromagnetic current on the time with the
QGP response or in the vacuum in the RHIC $\sqrt{s}=200\mathrm{GeV}$ and in
the LHC $\sqrt{s}=2760\mathrm{GeV}$ and $\sqrt{s}=5020\mathrm{GeV}$, the solid
line represents the result of considering the QGP response, and the dashed
line represents the result in the vacuum.

The comparisons of the dependencies of the electromagnetic current on the time
in the RHIC $\sqrt{s}=200\mathrm{GeV}$, and in the LHC
$\sqrt{s}=2760\mathrm{GeV}$ and $\sqrt{s}=5020\mathrm{GeV}$ are given in
Fig.6(a,b). It is found that the strength of the electromagnetic current at
RHIC $\sqrt{s}=200\mathrm{GeV}$ for Au-Au collisions with $b=8\mathrm{fm}$ far
outweigh that of the electromagnetic current at LHC
$\sqrt{s}=2760\mathrm{GeV}$ and $\sqrt{s}=5020\mathrm{GeV}$ for Pb - Pb
collisions with $b=8\mathrm{fm}$. The maximum electromagnetic current can
reach $800\mathrm{MeV}^{2}$ at RHIC $\sqrt{s}=200\mathrm{GeV}$, but the
maximum electromagnetic current is only $11\mathrm{MeV}^{2}$ at LHC
$\sqrt{s}=2760\mathrm{GeV}$ and then decreases to $4\mathrm{MeV}^{2}$ at LHC
$\sqrt{s}=5020\mathrm{GeV}$. From Fig.(5) and Fig.6, it is found that the
electromagnetic current in the LHC energy region is so small that it is
difficult to study CME. The main reason for such a small electromagnetic
current in the LHC energy region is due to the sharply decrease of the
magnetic field with time in the LHC energy region. Such observation is
important for understanding why the recent CMS measurements Sirunyan  _et al._
(2017) at LHC $\sqrt{s}=5020\mathrm{GeV}$ see no CME signal.

The study of the CME should be mainly concentrated in the RHIC energy region,
so we have carried out the study of the dependencies of electromagnetic
current on collision energy, impact parameter and temperature in the RHIC
energy region.

Figure 6: The comparisons of the dependencies of the electromagnetic current
on the time with the QGP response (Fig.7(a)) or in the vacuum (Fig.7(b))in the
RHIC $\sqrt{s}=200\mathrm{GeV}$, LHC $\sqrt{s}=2760\mathrm{GeV}$ and
$\sqrt{s}=5020\mathrm{GeV}$. The solid line represents the result at RHIC
$\sqrt{s}=200\mathrm{GeV}$ for Au-Au collisions, the dashed line represents
the result at LHC $\sqrt{s}=2760\mathrm{GeV}$ and the dotted line represents
the result at LHC $\sqrt{s}=5020\mathrm{GeV}$.

It is an interesting subject to study the relationship between electromagnetic
current and magnetic field. It is well known that the magnetic field decreases
monotonically with time in relativistic heavy ion collisions. A magnetic field
in relativistic heavy ion collisions is a uniform function of time. As can be
seen from Fig.7, the electromagnetic current increases with time at first, and
then decreases with the increase of time. By comparing the dependence of the
electromagnetic current and the magnetic field on the time variation,
respectively, we can establish the relationship between the electromagnetic
current and the magnetic field.

Figure 7: The dependence of the electromagnetic current on the magnetic field
at RHIC $\sqrt{s}=200\mathrm{GeV}$ for Au-Au collisions with $b=8\mathrm{fm}$.
The solid line represents the result by considering the QGP response, and the
dashed line represents the result in the vacuum. The formation time is given
as $t_{0}=0.16\mathrm{fm}$.

Fig.7 shows the dependence of the electromagnetic current on the magnetic
field by considering the QGP response and in the vacuum at RHIC
$\sqrt{s}=200\mathrm{GeV}$ for Au-Au collisions with $b=8\mathrm{fm}$. It is
found that the fast decaying magnetic field can give rise to a non-negligible
current which firstly increases to the maximum at $eB\sim 1.25\times
10^{3}\mathrm{MeV^{2}}$($eB\sim 2.51\times 10^{2}\mathrm{MeV^{2}}$ )
considering QGP response (in the vacuum), then diminishes. One also finds that
the electromagnetic current considering the QGP response is larger than the
electromagnetic current in the vacuum in most of the magnetic field regions.

Figure 8: Time dependence of the induced electromagnetic current normalized to
zero frequency chiral magnetic conductivity. The results are plotted with
different values of the temperature above the QCD phase transition. Top (a):
Au-Au collisions with $\sqrt{s}=200\mathrm{GeV}$ and $b=8\mathrm{fm}$ in RHIC
energy region. Bottom (b): Pb-Pb collisions with $\sqrt{s}=2760\mathrm{GeV}$
and $b=8\mathrm{fm}$ in LHC energy region.

In fig.8 it can be seen that even in the non-interacting case there still is
sizable response at high temperature. Hence even for such fast changing fields
there will really be an induced current. One finds that thermal fluctuations
will increase in magnitude when the temperature is increased. These
fluctuations can cause the spins of the particles to align along the fast
decaying magnetic field, it clearly takes some time for the current to
respond. By keeping $\sqrt{s},b$ fixed and increasing the temperature, the
system will respond faster to the changing magnetic field and the maximal
current will be larger.

We denote the $t_{\textrm{response}}$ as the response time of the
electromagnetic current from the beginning of the current production to the
maximum value $j_{\textrm{max}}$ , and as the maximum electromagnetic current
of the system. The maximum electromagnetic current and the response time are
important physical quantities, and they represent the intensity of the
electromagnetic signal and the speed of the response of the CME of the system.

As mentioned above, CME is mainly in the RHIC energy region, so below we will
make a detailed analysis of the characteristics of the electromagnetic current
in the RHIC energy region. We will carry out the study of the dependencies of
electromagnetic current on collision energy, impact parameter and temperature
in the RHIC energy region. In order to study the dependencies of
electromagnetic current on collision energy ($\sqrt{s}$), we calculate the
energy dependence of formation time $t_{0}$ and magnetic field $eB^{0}_{y}(0)$
at $t=t_{0}$ at central point $\mathbf{r}=\mathbf{0}$ for Au-Au collisions as
shown in table 1.

Table 1: Energy dependence of formation time $t_{0}$ and initial magnetic field $eB^{0}_{y}(0)$ at $t=t_{0}$ for $b=8\mathrm{fm}$ Au-Au collisions  $\sqrt{s}(\mathrm{GeV})$ | $t_{0}(\mathrm{fm})$ | $eB_{y}^{0}(\mathrm{MeV}^{2})$
---|---|---
7.7 | 0.231 | $1.03\times 10^{4}$
10 | 0.224 | $1.30\times 10^{4}$
20 | 0.206 | $2.01\times 10^{4}$
30 | 0.196 | $2.19\times 10^{4}$
40 | 0.189 | $2.09\times 10^{4}$
50 | 0.184 | $1.89\times 10^{4}$
60 | 0.180 | $1.67\times 10^{4}$
70 | 0.176 | $1.47\times 10^{4}$
80 | 0.174 | $1.29\times 10^{4}$
90 | 0.171 | $1.13\times 10^{4}$
100 | 0.169 | $1.00\times 10^{4}$
110 | 0.167 | $8.65\times 10^{3}$
120 | 0.165 | $7.98\times 10^{3}$
130 | 0.164 | $7.18\times 10^{3}$
140 | 0.162 | $6.50\times 10^{3}$
150 | 0.161 | $5.90\times 10^{3}$
160 | 0.160 | $5.39\times 10^{3}$
170 | 0.158 | $4.94\times 10^{3}$
180 | 0.157 | $4.55\times 10^{3}$
190 | 0.156 | $4.20\times 10^{3}$
200 | 0.155 | $3.89\times 10^{3}$

The dependence of the maximum electromagnetic current on the center-of-mass
energy is presented in fig. 9. Figure 9 shows that it is not a monotonic
change of the dependence of the maximum electromagnetic current on the center-
of-mass energy. Closely related to the beam energy dependence, the results of
fig.9 also demonstrate a strong decrease with increasing collision energy as
$\sqrt{s}>30\mathrm{GeV}$. Figure 9 also suggests that the CME signal could
vanish at LHC energies.

The dependence of the time-integrated current signal ($Q=\int j(t)dt$) on the
center-of-mass energy is shown in fig.10. We find that the time-integrated
current signal reaches the maximum at $\sqrt{s}\approx 30\mathrm{GeV}$, then
decreases with the increase of $\sqrt{s}$. The relation of the time-integrated
current with the energy change is consistent with the maximum electromagnetic
current with the collision energy.

From fig. 10, we can find that the peak value of time-integrated current
appears near $\sqrt{s}\approx 30\mathrm{GeV}$, and then decreases with the
increase of energy, which indicates that the CME at this collision energy
$\sqrt{s}\approx 30\mathrm{GeV}$ is the most obvious. The qualitative trend of
fig.9 and fig.10 is in agreement with STAR BES analysis results for a wide
range of beam energyAdamczyk  _et al._ (2014b), which are based on the two-
component decomposition method given by Refs.Bzdak  _et al._ (2013); Liao
(2015).

Figure 9: The dependence of the maximum electromagnetic current on the
central-of-mass energy for Au-Au collisions with $T=200\mathrm{MeV}$ and
$b=8\mathrm{fm}$, the solid line represents the result of considering the QGP
response, and the dashed line represents the result in the vacuum. The
relation between the response time and the collision energy is shown in the
subgraph.  Figure 10: The dependence of the time-integrated current signal
($Q=\int j(t)dt$) on the central-of-mass energy for Au-Au collisions with
$T=200\mathrm{MeV}$ and $b=8\mathrm{fm}$, the solid line represents the result
of considering the QGP response, and the dashed line represents the result in
the vacuum.

In order to study the dependencies of electromagnetic current on the impact
parameter ($b$), we calculate the energy dependence of formation time $t_{0}$
and magnetic field $eB^{0}_{y}(0)$ at $t=t_{0}$ at central point
$\mathbf{r}=\mathbf{0}$ for Au-Au collisions as shown in table 2.

Table 2: Impact parameter dependence of formation time $t_{0}$ and magnetic field $eB^{0}_{y}(0)$ at $t=t_{0}$ for $\sqrt{s}=200\mathrm{GeV}$ Au-Au collisions. $b(\mathrm{fm})$ | $t_{0}(\mathrm{fm})$ | $eB_{y}^{0}(\mathrm{MeV}^{2})$
---|---|---
1 | 0.131 | $8.74\times 10^{2}$
2 | 0.132 | $1.71\times 10^{3}$
3 | 0.133 | $2.47\times 10^{3}$
4 | 0.135 | $3.11\times 10^{3}$
5 | 0.138 | $3.59\times 10^{3}$
6 | 0.142 | $3.91\times 10^{3}$
7 | 0.147 | $4.00\times 10^{3}$
8 | 0.155 | $3.89\times 10^{3}$
9 | 0.167 | $3.49\times 10^{3}$
10 | 0.184 | $2.89\times 10^{3}$
11 | 0.214 | $2.05\times 10^{3}$

Figure 11 shows the dependence of the maximum electromagnetic current on the
impact parameter. When considering the QGP response, one finds that the
response time is larger than that in the vacuum. The maximum electromagnetic
current shows no monotonic dependence on the impact parameter. It is found
that the maximum electromagnetic current $j_{max}$ increases first and reaches
its maximum at $b\approx 8\mathrm{fm}$, then decreases with the increase of
the impact parameter when considering the QGP response. For the centrality
dependence shown in fig.11, what is the reason behind such dependence? An
important point made in Ref.Bloczynski  _et al._ (2013) and the azimuthal
fluctuations bring important change for the centrality dependence. our
discussion is consistent with that of Ref.Bloczynski  _et al._ (2013).

Figure 11: The dependence of the maximum electromagnetic current on the impact
parameter $b$ for RHIC $\sqrt{s}=200\mathrm{GeV}$ Au-Au collisions with
$T=200\mathrm{MeV}$ , the solid line represents the result of considering the
QGP response, and the dashed line represents the result in the vacuum. The
relation between the response time and impact parameter $b$ is shown in the
subgraph.

Figure 12 shows the dependence of the maximum electromagnetic current on the
temperature. Figure 12 indicates that the maximum electromagnetic current
increases monotonically with the increase of the temperature, but the response
time decreases monotonically with the increase of the temperature. The
relationships among the maximum current and response time with the temperature
in the vacuum are also shown in fig.12.

Figure 12: The dependence of the maximum electromagnetic current on the
temperature for RHIC $\sqrt{s}=200\mathrm{GeV}$ Au-Au collisions with
$b=8\mathrm{fm}$ , the solid line represents the result of considering the QGP
response, and the dashed line represents the result in the vacuum. The
relation between the response time and the temperature is shown in the
subgraph.

##  V Summary

It is argued that nonzero chirality by gluon configurations with nonzero
topological charge can be generated in a quark gluon plasma. In the presence
of a strong magnetic field, an electromagnetic current could be produced along
the field by nonzero chirality. This is the chiral magnetic effect which can
potentially lead to observable effects in relativistic heavy ion collisions.
Since the QGP medium is an electrically conducting medium, it is desirable to
study the chiral magnetic effect in a time-dependent magnetic field by
considering the response of the QGP medium.In this paper, we precisely aim to
address the following two pressing issues: (1) the time evolution of the
strong magnetic field which is the necessary driving force for CME, (2) the
dynamical generation of CME current in response to time-dependent magnetic
field. We chose the simplified model approach to gain valuable insights on
these problems.

By considering the QGP response, we systematically study the dependencies of
the electromagnetic current on temperature, collision energy and impact
parameter in the RHIC and LHC energy regions. It is found that the
electromagnetic current in the LHC energy region is so small that it is
difficult to produce CME. The main reason for such a small electromagnetic
current in the LHC energy region is due to the sharp decrease of the magnetic
field with time in the LHC energy region. Such observation is important for
understanding why the recent CMS measurements Sirunyan  _et al._ (2017) at LHC
$\sqrt{s}=5020\mathrm{GeV}$ see no CME signal. On the other hand, the initial
magnetic field ($B(t=0)$)in the RHIC region is smaller than LHC, but the
magnetic field time decreases slower with the time evolution, leading to an
electromagnetic current which is much larger than that of LHC.

As we further our study of the CME research in the RHIC energy region, we find
that the strongest electromagnetic current signal is not at the top RHIC
energy $\sqrt{s}=200\mathrm{GeV}$ but at collision energy $\sqrt{s}\sim
30\mathrm{GeV}$ for Au-Au collisions at $b=8\mathrm{fm}$. We find that with
the increase of collision energy(as $\sqrt{s}>30\mathrm{GeV}$ ), the
corresponding electromagnetic currents become smaller and smaller, which is
consistent with some experimental studies.

## Acknowledgments

This work was supported by National Natural Science Foundation of China (Grant
Nos. 11747115, 11475068), the CCNU-QLPL Innovation Fund (Grant No.
QLPL2016P01), the Excellent Youth Foundation of Hubei Scientific Committee
(Grant No. 2006ABB036).

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