###### Abstract

Chiral anomalous effects in relativistic plasmas are reviewed. The essence of
chiral separation and chiral magnetic effects is explained in simple terms.
Qualitative differences between the two phenomena, both of which are triggered
by background electromagnetic fields, are highlighted. It is shown how an
interplay of the chiral separation and chiral magnetic effects could lead to a
new collective plasma mode, called the chiral magnetic wave. It is argued that
the chiral magnetic wave is overdamped in weakly interacting plasmas. Its fate
in a strongly interacting quark-gluon plasma is less clear, especially in the
presence of a superstrong magnetic field. The observational signatures of the
chiral anomalous effects are discussed briefly.

##  Chapter 0 Anomalous plasma: chiral magnetic effect and all that

Contribution to a Festschrift celebrating the career of Peter Suranyi

###  1 Introduction

A quantum anomaly in quantum field theory describes a situation when the
symmetry of a classical action cannot be extended to the corresponding quantum
theory in a self-consistent way. The first and best-known example of the
quantum anomaly was encountered in gauge theories with a global chiral
symmetry [1, 2]. Mathematically, it showed up in the calculation of the
correlation function of the axial current $j_{5}^{\mu}(x)$ and two
electromagnetic currents $j^{\nu}(x)$ and $j^{\lambda}(x)$. From a physics
viewpoint, such a correlator appears in the theoretical analysis of the
neutral pion decay: $\pi^{0}\to\gamma+\gamma$. Graphically, it is represented
by the well-known triangular Feynman diagrams shown in Fig. 1. The
corresponding formal expression contains several contributions that are ill-
defined due to ultraviolet divergencies. While there are several ways to
regularize the calculation, the final result depends on the regularization
used. Without imposing any additional constraints, some regularizations
produce the result consistent with chiral symmetry (i.e.,
$\partial_{\mu}j_{5}^{\mu}=0$) but violate the conservation of electric charge
(i.e., $\partial_{\nu}j^{\nu}\neq 0$). Others lead to the result compatible
with the electric charge conservation (i.e., $\partial_{\nu}j^{\nu}=0$) but
breaking the chiral symmetry (i.e., $\partial_{\mu}j_{5}^{\mu}\neq 0$). The
practical approach to resolving the dilemma was to employ empirical
observations, namely to require that the regularization be consistent with the
conservation of electric charge.

Most amazingly, the chiral anomaly is not just an obscure mathematical
curiosity in quantum field theory. It has direct observational consequences in
particle physics. Indeed, by treating the chiral anomaly properly, one can
accurately describe the decay rate of a neutral pion into a pair of gammas [3]
and explain the abnormally large mass of the $\eta^{\prime}$ meson [4, 5].
These well-established microscopic phenomena confirm that the chiral anomaly
is a physical reality. They are not the main topic of this presentation,
however.

Figure 1: The triangular diagrams that describe the anomalous correlation
function of the axial current $j_{5}^{\mu}(x)$ (double line) and two
electromagnetic currents $j^{\nu}(x)$ and $j^{\lambda}(x)$ (wavy lines).

This review covers a subset of anomalous phenomena that have macroscopic
consequences in relativistic forms of matter. The corresponding phenomena came
prominently to light only in the last decade in the realm of nuclear
physics.111Some of the anomalous phenomena were proposed by Alexander Vilenkin
in the 1980s [6, 7], but they were not fully understood and did not get much
traction until much later. Among the most interesting of them are the
anomalous phenomena driven by the chiral magnetic [8, 9, 10, 11, 12, 13, 14],
chiral separation [7, 15, 16], and chiral vortical effects [6, 17, 18, 19].
(For some earlier reviews, see also Refs. [20, 21, 22, 23, 24, 25, 26].) These
anomalous effects may lead to observable signatures in heavy-ion collisions
[27, 28, 29, 30, 31, 21, 22, 23, 25, 24], stellar astrophysics [32, 33, 34,
35, 36, 37, 38, 39], cosmology [40, 41, 42, 43, 44, 45, 46, 47], and even in
Dirac and Weyl semimetals [48, 49, 50, 51, 52, 53]. Tentative experimental
verifications of the chiral anomalous effects have been reported in heavy-ion
collisions [54, 55, 56, 57, 58, 59, 60, 61, 62] as well as condensed matter
physics [63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79].
In all fairness, though, the interpretation of the heavy-ion experiments is
not without controversy [80, 81, 82, 83, 84].

To keep the presentation concise and focused, below I will concentrate
primarily on a subset of chiral anomalous phenomena that get triggered by
background (electro-)magnetic fields. These include, in particular, the chiral
magnetic and chiral separation effects. They show the most promise to be
widely realized and verified in experiments. In large part, this is the case
because of the ubiquitous presence of magnetic fields in all types of
relativistic plasmas.

The paradigmatic examples of relativistic plasmas are the quark-gluon plasma
(QGP) created by heavy-ion collisions (or “Little Bangs”) and the primordial
plasma in the Early Universe produced by the “Big Bang”. Many theoretical
studies argue that such primordial plasma was strongly magnetized [85, 86, 87,
88]. Not only is it natural from the theoretical viewpoint but is also
supported by the observational data of present-day astronomy [89, 90, 91, 92].
Very strong magnetic fields are also generated at the early stages of heavy-
ion collisions [93, 94, 95]. In astrophysics, magnetic fields appear and play
an important role in the magnetospheres [96, 97] and the interiors of compact
stars [98, 99, 100].

The presence of magnetic fields modifies many physical properties of
relativistic plasmas. It affects the thermodynamic and transport
characteristics, the emission and absorption of particles, changes the spectra
of collective modes, and more. It is particularly interesting, however, that
magnetic fields can also trigger anomalous phenomena on macroscopic scales. As
expected, their implications are very different from those of the microscopic
anomalous effects in particle physics. Nevertheless, the mere possibility of
such novel phenomena is of great conceptual importance. It may reveal new
aspects of quantum physics that remained in the shadows until now. Some of
them can even find technological applications [101, 102, 103].

###  2 Chiral separation effect

Let us start the discussion from one of the simplest chiral anomalous effects,
i.e., the chiral separation effect. It comes to light in a magnetized
relativistic (or better yet, ultrarelativistic) plasma when its electric
charge chemical potential $\mu$ is nonzero. (To achieve equilibrium, the
global neutrally of plasma is required. It is assumed that the compensating
background charge is provided, for example, by non-relativistic ions.) In the
simplest consideration, one may assume that charged particles of the
relativistic plasma are massless Dirac fermions ($m=0$). The requirement of
the vanishing mass can be relaxed, of course, when the (ultra-)relativistic
regime is enforced by sufficiently high temperature ($T\gg m$) or density
($\mu\gg m$).

Before proceeding to the discussion of the details of chiral separation
effect, it is instructive to say a few words about the chirality, which is a
characteristics of massless Dirac fermions. The Lagrangian density of the
Dirac field reads

${\cal L}=\bar{\psi}\left(i{\cal D}_{\nu}\gamma^{\nu}+m\right)\psi,$ (1)

where the covariant derivative ${\cal
D}_{\nu}=\partial_{\nu}-ie(A_{\nu}+a_{\nu})$ may include both the background
field $A_{\nu}$ and the quantum gauge field $a_{\nu}$.

In the massless (chiral) limit, $m\to 0$, the Lagrangian density in Eq. (1) is
invariant under the chiral transformation: $\psi\to
e^{i\alpha\gamma_{5}}\psi$. This implies that the classical action has chiral
symmetry. Furthermore, in view of the Noether theorem, the chiral charge
$Q_{5}=\int n_{5}d^{3}\bm{r}$ is expected to be conserved in the classical
field theory. (Here $n_{5}=\bar{\psi}\gamma^{5}\gamma^{0}\psi$ is the chiral
charge density.) However, because of the quantum anomaly, the transformation
$\psi\to e^{i\alpha\gamma_{5}}\psi$ is not a symmetry of the quantum theory.
The latter can be seen most clearly by using the path integral quantization
[104]. The modified continuity equation for the chiral charge takes the
following form:

$\frac{\partial n_{5}}{\partial
t}+\bm{\nabla}\cdot\bm{j}_{5}=-\frac{e^{2}}{2\pi^{2}}\left(\bm{E}\cdot\bm{B}\right),$
(2)

where $\mathbf{j}_{5}\equiv\langle\bar{\psi}\gamma_{5}\bm{\gamma}\psi\rangle$
is the density of chiral current. This is to be compared to the continuity
relation for the exactly conserved electric charge, i.e.,

$\frac{\partial n}{\partial t}+\bm{\nabla}\cdot\bm{j}=0.$ (3)

In the case of a small but nonzero fermion mass $m$, an additional chiral-
charge violating term will appear on the right-hand side of Eq. (2). In the
corresponding quantum operator relation, the extra term reads
$2im\bar{\psi}\gamma^{5}\psi$. When Eq. (2) is interpreted as an equation for
the thermal averages in a relativistic plasma, however, the effect of a
nonzero mass is captured by the term $-\Gamma_{m}n_{5}$, where $\Gamma_{m}$ is
the chirality flip rate. The latter is estimated as follows [105, 106]:

$\Gamma_{m}\simeq\left\\{\begin{array}[]{lll}c_{1}\alpha^{2}\frac{m_{e}^{2}}{T},&\mbox{for}&T\lesssim
m_{e}/\sqrt{\alpha},\\\ c_{2}\alpha\frac{m_{e}^{2}}{T},&\mbox{for}&T\gtrsim
m_{e}/\sqrt{\alpha},\end{array}\right.$ (4)

where $c_{i}$ ($i=1,2$) are constants of order $1$.

Despite the chiral anomaly, the quantum states of massless Dirac fermions can
be classified by a well-defined chirality $\psi_{\chi}$, where $\chi=\pm 1$
labels the right-handed (R) and left-handed (L) chiralities, respectively. By
definition, $\psi_{\chi}$ are the eigenstates of matrix operator
$\gamma_{5}\equiv i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$, i.e.,
$\gamma_{5}\psi_{\chi}=\chi\psi_{\chi}$.

In the absence of background fields, the concept of chirality is closely
related to the particle’s helicity. This can be shown by using the Dirac
equation in momentum space,

$\left(\gamma^{0}p^{0}-\bm{\gamma}\cdot\bm{p}\right)\psi_{p}=0.$ (5)

After multiplying the equation by $\gamma_{5}\gamma^{0}$, using the mass-shell
condition $p^{0}=\mbox{sign}(p^{0})p$, and rearranging the terms, one can
derive the following relation:

$\gamma_{5}\psi_{p}=\mbox{sign}(p^{0})\left(2\bm{S}\cdot\hat{\bm{p}}\right)\psi_{p},$
(6)

where $\hat{\bm{p}}=\bm{p}/p$ and
$S^{j}\equiv\frac{i}{2}\varepsilon^{jkl}\gamma^{k}\gamma^{l}$ is the spin
operator. Note that $\mbox{sign}(p^{0})$ takes values $\pm 1$ for the particle
and antiparticle states, respectively.

Recall that the sign of the fermion’s spin projection on the direction of its
momentum, $\left(2\bm{S}\cdot\hat{\bm{p}}\right)$, is the particle’s helicity
$h$. Thus, from Eq. (6) one concludes that $\chi=\mbox{sign}(p^{0})h$, i.e.,
the chirality of a massless Dirac particle (antiparticle) with a positive
(negative) energy is equivalent (opposite) to its helicity.

Mathematically, the chiral separation effect is described by the following
relation [7, 15]:

$\mathbf{j}_{5}=\frac{e\bm{B}}{2\pi^{2}}\mu,$ (7)

where $\bm{B}$ is the background magnetic field in the plasma.

The underlying physics of the chiral separation effect is easy to understand
by considering the well-known Landau-level spectrum of Dirac fermions in a
magnetic field. Simple considerations show that the result in Eq. (7) is
saturated by the contribution of the lowest Landau level (LLL), which is fully
spin polarized. Assuming that the magnetic field points in the $z$ direction
and that the particles are negatively charged (e.g., electrons), all LLL
states have the same spin projection $s_{z}=-1/2$ (opposite to the magnetic
field). It is also useful to remember that the density of states in the LLL is
given by

$\rho(E)=\frac{|eB|}{4\pi^{2}}.$ (8)

At nonzero chemical potential $\mu$, the particle states with the absolute
values of the momenta $p\leq\mu$ are occupied. Because of the spin
polarization, the LLL particles with $p_{z}>0$ (i.e., moving along the field
direction) are left-handed, and the LLL particles with $p_{z}<0$ (i.e., moving
in the direction opposite to the field) are right-handed. This means that both
left-handed particles moving in the $+z$ direction and right-handed particles
moving in the $-z$ direction produce chiral currents going in the same $-z$
direction. Because of the effective one-dimensional nature of the LLL (with
$p_{z}$ as the only momentum component), the currents can be easily computed,
i.e., $j_{5,z}=\sum_{\chi}\chi\,\mbox{sign}(p_{z})\rho(E)\mu=-2\rho(E)\mu$.
Thus, the net chiral current density is indeed given by the expression in Eq.
(7).

The above exercise for the LLL states also provides an intuitive explanation
why the higher Landau levels should not contribute to the chiral separation
effect. Indeed, in the absence of the net spin polarization in the higher
Landau levels, there exist both left- and right-handed states for each value
of the momentum. The contributions of such states to the chiral current cancel
pairwise. As one may anticipate, the result remains the same also in a free
thermal plasma at a nonzero temperature. These findings are consistent with
the conclusion that the chiral anomaly in the presence of an external magnetic
field is generated entirely in the lowest Landau level [107].

Naively, one might claim that the chiral anomaly was not used directly in the
derivation of Eq. (7). The connection to the quantum anomaly can be made more
explicit. Indeed, let us assume that a constant magnetic field points in the
$z$ direction and consider the corresponding relation in the case of a
spatially varying chemical potential $\mu(z)$. This can be modeled, for
example, by a classical electric potential $\Phi(z)=\mu(z)/e$. The
corresponding electric field is $E_{z}=-\partial_{z}\Phi$. In a steady state
with such a field, the anomaly induces a nonzero divergence of the
$z$-component of the axial current density, i.e.,

$\partial_{z}j^{5}_{z}=-\frac{e^{2}}{2\pi^{2}}B_{z}E_{z}=\frac{eB_{z}}{2\pi^{2}}\frac{\partial\mu(z)}{\partial
z}.$ (9)

By integrating over the spatial $z$-coordinate, one reproduces Eq. (7) up to
an integration constant.

It is curious to mention that the chiral separation effect represented by Eq.
(7) appears already in the free theory, where particles do not interact with
one another. The only interaction in the discussion above was the interaction
with the background field. It is natural to ask how interactions will affect
the result. A perturbative investigation of this issue was performed in Ref.
[108]. Naively, the conventional wisdom would suggest that the axial current
should be fixed by the chiral anomaly relation and receive no radiative
corrections. In contrast, the direct calculations performed to the linear
order in the external magnetic field showed that nontrivial corrections do
appear. This was also confirmed independently by using an alternative approach
in Ref. [109]. In general, one finds that the anomalous transport relations
like the one in Eq. (7) are not protected by from radiative corrections when
the theory contains dynamical vector gauge fields.

###  3 Chiral magnetic effect

The chiral magnetic effect is another phenomenon that can be realized in a
chiral magnetized plasma. It is closely related to the chiral separation
effect discussed in the previous section. Conceptually, the chiral magnetic
effect describes a response of a chiral plasma to a nonzero chiral chemical
potential $\mu_{5}\equiv(\mu_{R}-\mu_{L})/2$ [27, 8]. It is given by the
following relation for the electric current density:

$\mathbf{j}=\frac{e\mathbf{B}}{2\pi^{2}}\mu_{5}.$ (10)

In simple terms, it implies that a chiral asymmetry (quantified by $\mu_{5}$)
in a magnetized plasma produces a nonzero charge flow.

By considering the discrete symmetries of the magnetic field (odd under the
time-reversal and event under the parity transformation) and the electric
current (odd under both time-reversal and parity transformations), one finds
that the chiral conductivity, i.e., $\sigma_{\chi}\equiv e\mu_{5}/(2\pi^{2})$,
is odd under the parity transformation and even under the time-reversal. This
implies that the corresponding transport phenomenon is non-dissipative.

One might wonder how a nonzero electric current (10) can exist in a chiral
plasma. This appears to contradict the Bloch theorem [110, 111], stating that
a persistent electric currents must vanish in the ground state. The problem is
resolved by noting that the chiral magnetic effect is a non-equilibrium
phenomenon [112, 113, 114, 115].

To prove the point, let us demonstrate that a plasma with a nonzero chiral
chemical potential $\mu_{5}$ cannot be an equilibrium state. Indeed, by
substituting the electric current (10) into Ampere’s law, one derives

$\bm{\nabla}\times\mathbf{B}=\frac{e\mathbf{B}}{2\pi^{2}}\mu_{5}+\frac{\partial\mathbf{E}}{\partial
t},$ (11)

where $\mathbf{B}$ is the total in-medium magnetic field that includes both
external (background) and dynamical contributions.222The simplified analysis
can be easily generalized to include the Ohm current too [39]. Then, by
calculating the curl on both sides of the equation and using the Faraday law,

$\bm{\nabla}\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t},$ (12)

one arrives at the following equation for the dynamical part of the magnetic
field in the chiral plasma:

$\bm{\nabla}\times(\bm{\nabla}\times\mathbf{B})=\frac{e\mu_{5}}{2\pi^{2}}(\bm{\nabla}\times\mathbf{B})-\frac{\partial^{2}\mathbf{B}}{\partial
t^{2}}.$ (13)

A simple consideration of this equation reveals the existence of unstable
helical modes that grow exponentially with time. Recall that the helical modes
are the eigenstates of the curl operator, i.e.,
$\bm{\nabla}\times\mathbf{B}_{\lambda,k}=\lambda k\mathbf{B}_{\lambda,k}$,
where $\lambda=\pm 1$ represents the helicity of the mode and $k$ is a
dimensionful quantity with the units of a wave vector. An example of such a
solution is given by the following circularly polarized plane wave:

$\mathbf{B}_{\lambda,k}=B_{0}\left(\hat{\bm{x}}+i\lambda\hat{\bm{y}}\right)e^{-i\omega
t+ikz}.$ (14)

Our conclusions will not change if other forms of helical solutions are used.
By substituting the above helical ansatz into Eq. (13) and solving for the
frequency of allowed modes, we obtain333When the electrical conductivity
$\sigma$ is nonzero, the two solutions for the frequency are
$\omega_{1,2}=-\frac{i}{2}\left(\sigma\pm\sqrt{\sigma^{2}+4k(\lambda
k_{\star}-k)}\right)$ [39].

$\omega_{1,2}=\pm\sqrt{k\left(k-\lambda k_{\star}\right)},$ (15)

where $k_{\star}\equiv e\mu_{5}/(2\pi^{2})$. As is easy to check, the
solutions with the helicity $\lambda=\mbox{sign}(e\mu_{5})$ and sufficiently
small wave vectors (i.e., $k<|k_{\star}|$) describe unstable modes that grow
with time as $B_{0}e^{\mbox{\scriptsize Im}(\omega_{2})t+ikz}$, where
$\mbox{Im}(\omega_{2})=\sqrt{k\left(|k_{\star}|-k\right)}$.

The existence of such growing helical solutions is a clear indication of the
plasma being out of equilibrium. A further analysis reveals that the energy
needed for the growing modes comes from the chiral imbalance in the plasma
[33, 46, 47]. It is natural, therefore, that the back reaction of the
dynamical fields leads to the decrease of $\mu_{5}$ and a gradual attenuation
of the instability. In the end, the state with $\mu_{5}=0$ is reached and the
system becomes stable. Since there is no electric current at $\mu_{5}=0$, it
describes the true equilibrium of the plasma.

To sum it up, the relativistic plasma with a nonzero chiral chemical potential
is not an equilibrium state of matter. Thus, the electric current in Eq. (10)
does not contradict the Bloch theorem. In other words, the chiral magnetic
effect is truly a non-equilibrium phenomenon. Curiously, this is is
drastically different from the chiral separation effect discussed in Sec. 2.
The latter can be realized even in an equilibrium state without violating any
fundamental principles. Firstly, the Bloch theorem does not extend to the
chiral current. Secondly, the chiral current can be re-interpreted as a spin
polarization that is well defined in equilibrium.

###  4 Chiral magnetic wave

The interplay of the chiral separation and chiral magnetic effects can be
expected to have interesting consequences. As an example, it was argued to
give rise to a special type of a gapless collective excitation, known as the
chiral magnetic wave (CMW) [116]. The corresponding mode can be thought of as
a self-sustaining propagation of oscillations of the electric and chiral
charge densities along the direction of the background magnetic field. Its
underlying mechanism works as follows. A local perturbation of the electric
charge density $\delta n$ (or $\delta\mu$) induces the chiral separation
effect current $\mathbf{j}^{5}_{\textrm{CSE}}=e\mathbf{B}\delta\mu/(2\pi^{2})$
that perturbs the local axial charge density $\delta n_{5}$ (or
$\delta\mu_{5}$). The latter in turn produces the chiral magnetic effect
current $\mathbf{j}_{\textrm{CME}}=e\mathbf{B}\delta\mu_{5}/(2\pi^{2})$ that
perturbs the local electric charge density, and the cycle repeats [29]. If
realized in heavy-ion collisions, such a wave may lead to interesting
observational signatures, namely the quadrupole correlations of charged
particles [29, 30].

In the simplest background-field approximation, the description of the CMW is
given by the following system of coupled continuity equations in momentum
space [116]:

$\displaystyle\omega\delta n-k\frac{eB}{2\pi^{2}\chi_{5}}\delta n_{5}$
$\displaystyle=$ $\displaystyle 0,$ (16) $\displaystyle\omega\delta
n_{5}-k\frac{eB}{2\pi^{2}\chi}\delta n$ $\displaystyle=$ $\displaystyle 0,$
(17)

where we assumed that the collective mode propagates along the direction as
the magnetic field. (For a general direction of the propagation, the wave
vector $k$ should be replaced by $k\cos\theta$, where $\theta$ is the angle
between vector $\mathbf{k}$ and the magnetic field $\mathbf{B}$.) We also
introduced the number density and chiral charge density susceptibilities,
i.e.,

$\displaystyle\chi$ $\displaystyle=$ $\displaystyle\frac{\partial
n}{\partial\mu}=\frac{\partial}{\partial\mu}\left(\frac{\mu^{3}+3\mu\mu_{5}^{2}+\mu\pi^{2}T^{2}}{3\pi^{2}}\right)=\frac{T^{2}}{3}+\frac{\mu^{2}+\mu_{5}^{2}}{\pi^{2}},$
(18) $\displaystyle\chi_{5}$ $\displaystyle=$ $\displaystyle\frac{\partial
n_{5}}{\partial\mu_{5}}=\frac{\partial}{\partial\mu_{5}}\left(\frac{\mu_{5}^{3}+3\mu_{5}\mu^{2}+\mu_{5}\pi^{2}T^{2}}{3\pi^{2}}\right)=\frac{T^{2}}{3}+\frac{\mu^{2}+\mu_{5}^{2}}{\pi^{2}},$
(19)

respectively. Here, the following convention for the chemical potentials is
used: $\mu_{R}=\mu+\mu_{5}$ and $\mu_{L}=\mu-\mu_{5}$. In other words,
$\mu=(\mu_{R}+\mu_{L})/2$ and $\mu_{5}=(\mu_{R}-\mu_{L})/2$. As we see, the
two susceptibilities are the same. Therefore, without loss of generality, we
will use $\chi_{5}=\chi$ in the following.

From a physics viewpoint, Eq. (16) is nothing else but the local continuity
equation for the electric charge (3), while Eq. (17) is its counterpart for
the chiral charge. Note that the anomalous contribution
$\propto(\bm{E}\cdot\bm{B})$ on the right hand side of Eq. (2) was neglected
in Eq. (17). Naively this could be justified when the local electric fields
are screening in a plasma of sufficiently high conductivity.

If the system of linear equations (16) and (17) is taken at face value, the
nontrivial solutions exist when the following characteristic equation is
satisfied:

$\omega^{2}=\left(v^{(0)}_{\rm CMW}\right)^{2}k^{2},$ (20)

where $v^{(0)}_{\rm CMW}=|eB|/(2\pi^{2}\chi)$. The characteristic equation
indicates the existence of a gapless collective mode with a linear dispersion
relation. The speed of the mode is given by $v^{(0)}_{\rm CMW}$ provided the
direction of propagation is along the magnetic field. (For a general
direction, the speed is $v^{(0)}_{\rm CMW}\cos\theta$.)

The existence of the novel gapless collective mode in a chiral plasma is an
interesting prediction. Indeed, the only other known gapless modes in
magnetized charge plasmas are the Alfvén waves (with a linear dispersion) and
the helicons or whistlers (with a quadratic dispersion) in the regimes of high
temperature and high density, respectively. However, both are very different
types of transverse collective modes that appear only in the
magnetohydrodynamic regime. Also, they are sustained by local oscillations of
the flow velocity in the plasma [117]. As is evident from Eq. (16) and (17),
no flow velocity oscillations are needed for the CMW. Instead, its existence
depends on the availability of the approximately conserved chiral charge whose
oscillations are critical for sustaining the propagation of the wave.

Before accepting the possibility of the gapless CMW as a new mode with a
linear dispersion relation determined by Eq. (20), it is instructive to
reexamine critically the main assumptions that lead to Eqs. (16) and (17). To
evaluate the role of high conductivity in the plasma, we will include
explicitly Ohm’s contribution to current $\mathbf{j}_{\rm
Ohm}=\sigma\mathbf{E}$. Since the consistency of such an approximation
requires the inclusion of the local electric field $\mathbf{E}$, it is not
justified to neglect the chiral anomaly term $\propto(\bm{E}\cdot\bm{B})$ in
the continuity equation for the chiral charge. To close the system of
equations, the pair continuity relations for $\delta n$ and $\delta n_{5}$
should be supplemented by the Maxwell equations. In other words, one must take
dynamical electromagnetic fields self-consistently into account [117, 118].

In an improved approximation, the expression for the chiral magnetic effect
current (10) should be replaced by

$\mathbf{j}=\frac{e\mathbf{B}}{2\pi^{2}}\mu_{5}+\sigma\mathbf{E}-\frac{e\tau}{3}\bm{\nabla}n,$
(21)

where the last term is the diffusion current, parametrized by the relaxation
time parameter $\tau$. A similar diffusion contribution should be also added
to the chiral current density $\mathbf{j}_{5}$ in Eq. (7). Then, from the
continuity relations (2) and (3), one derives the following system of
equations for small deviations of densities from their equilibrium values:

$\displaystyle\omega\delta n-k\frac{eB}{2\pi^{2}\chi}\delta
n_{5}+i\frac{\tau}{3}k^{2}\delta n-\frac{1}{e}\sigma k\delta E_{z}$
$\displaystyle=$ $\displaystyle 0,$ (22) $\displaystyle\omega\delta
n_{5}-k\frac{eB}{2\pi^{2}\chi}\delta n+i\frac{\tau}{3}k^{2}\delta
n_{5}-i\frac{e^{2}}{2\pi^{2}}B\delta E_{z}$ $\displaystyle=$ $\displaystyle
0.$ (23)

To close the system of equations, one should also include Gauss’s law, i.e.,

$k\delta E_{z}+ie\delta n=0.$ (24)

It is interesting to note that, for the collective modes propagating along the
magnetic field, the validity of the above system of three coupled equations
also extends to the hydrodynamic regime [117]. It is because the oscillations
of densities $\delta n$ and $\delta n_{5}$ decouple from the fluid velocity
$\delta\mathbf{u}$ for such modes. The same is not true beyond the kinematic
regime with $\mathbf{k}\parallel\mathbf{B}$, however.

After using Gauss’s law (24) to eliminate the electric field $\delta E_{z}$,
we obtain the following system of equations:

$\displaystyle\left(\omega+i\frac{\tau}{3}k^{2}+i\sigma\right)\delta
n-\frac{eBk}{2\pi^{2}\chi}\delta n_{5}$ $\displaystyle=$ $\displaystyle 0,$
(25)
$\displaystyle-\left(\frac{eBk}{2\pi^{2}\chi}+\frac{e^{3}B}{2\pi^{2}k}\right)\delta
n+\left(\omega+i\frac{\tau}{3}k^{2}\right)\delta n_{5}$ $\displaystyle=$
$\displaystyle 0.$ (26)

By solving the corresponding characteristic equation, one obtains the spectrum
of collective modes [117]

$\omega^{(\pm)}=-i\frac{\sigma}{2}\mp
i\frac{\sigma}{2}\sqrt{1-\left(\frac{eB}{\pi^{2}\chi\sigma}\right)^{2}\left(k^{2}+e^{2}\chi\right)}-i\frac{\tau}{3}k^{2}.$
(27)

Let us now consider what happens in the limit of high conductivity. By
expansing the result in Eq. (27) in inverse powers of $\sigma$, we obtain the
following two different dispersion relations:

$\displaystyle\left.\omega^{(+)}\right|_{\sigma\to\infty}$ $\displaystyle=$
$\displaystyle-i\sigma-i\frac{\tau}{3}k^{2}+\ldots,$ (28)
$\displaystyle\left.\omega^{(-)}\right|_{\sigma\to\infty}$
$\displaystyle\simeq$
$\displaystyle-i\frac{e^{2}B^{2}}{4\pi^{4}\chi^{2}\sigma}\left(k^{2}+e^{2}\chi\right)-i\frac{\tau}{3}k^{2}+\ldots,$
(29)

where the ellipses represent subleading terms.

Since both frequencies (28) and (29) are purely imaginary, the corresponding
modes are diffusive. In other words, we find no gapless CMW modes in the case
of a highly conducting plasma. Instead, there are two overdamped (i.e., non-
propagating) modes in the spectrum. This result appears to be puzzling at
first sight. Moreover, it seems unexplainable how this prediction can be so
drastically different from the earlier naive result in Eq. (20), obtained in
the background-field approximation [116].

To clarify the underlying physics behind the overdamped modes in Eqs. (28) and
(29), one can first look at the structure of the corresponding eigenmodes in
the two-dimensional parameter space spanned by $\delta n$ and $\delta n_{5}$.
In the limit $\sigma\to\infty$, they are given by $\left(1,0\right)$ and
$\left(0,1\right)$ for $\omega^{(+)}$ and $\omega^{(-)}$, respectively. In
simple terms, this means that the mode with frequency $\omega^{(+)}$ is driven
almost exclusively by oscillations of the electric charge density $\delta n$.
The other mode (with frequency $\omega^{(-)}$) is supported by oscillations of
the chiral charge density $\delta n_{5}$ only.

The meaning of the results in Eqs. (28) and (29) becomes clear now. Indeed,
the imaginary frequency $\omega^{(+)}\simeq-i\sigma$ describes the damping of
the electric charge oscillations in a conducting plasma due to fast screening.
The corresponding time scale is given by $t_{\rm scr}\simeq 1/\sigma$ as
expected. Similarly, the mode with frequency $\omega^{(-)}$ describes damping
of the chiral charge oscillations. The corresponding rate $\Gamma_{5}\simeq
e^{4}B^{2}/(4\pi^{4}\chi\sigma)$ agrees with the estimates obtained for a
conducting plasma, for example, in Ref. [119].

Now, when the physics behind the overdamped modes in Eqs. (28) and (29) is
understood, it is instructive to reevaluate the validity of the earlier result
in Eq. (20). It was derived from the system of equations (16) and (17), where
the electric field was set to zero. (The charge diffusion currents
$\bm{j}\propto\tau\bm{\nabla}n$ and $\bm{j}_{5}\propto\tau\bm{\nabla}n_{5}$
were neglected as well, but their role is not as critical for the long-
wavelength modes.) It is easy to prove, however, that such an approximation
with $\delta\mathbf{E}=\mathbf{0}$ is inconsistent. This follows immediately
from the Gauss law in Eq. (24). Indeed, the latter reveals that $\delta
E_{z}=0$ is a self-consistent assumption only if one also requires that
$\delta n=0$. From a physics viewpoint, if the electric fields are subject to
fast screening, so are the charge density oscillations. Alternatively, no
oscillations of the local electric charge density are possible without
inducing nonzero electric fields. Without oscillations of $\delta n$, however,
the CMW cannot propagate. These arguments suggests that the CMW cannot be
realized as a propagating mode in a chiral plasma.

Ironically, the corresponding reasoning against the CMW gets relaxed in a
plasma with a relatively low conductivity. By formally considering the extreme
limit of vanishing $\sigma$ in Eq. (27), one derives the following approximate
results for the frequencies:

$\left.\omega^{(\pm)}\right|_{\sigma\to
0}=\mp\frac{eB}{2\pi^{2}\chi}\sqrt{k^{2}+e^{2}\chi}-i\frac{\tau}{3}k^{2}.$
(30)

In the long-wavelength limit (i.e., $k\to 0$), these describe weakly damped
propagating modes. The damping rate is controlled by the charge diffusion
constant $D=\tau/3$.

It should be noted that the collective modes with the dispersions in Eq. (30)
are gapped. The value of the gap given by $M=e^{2}B/(2\pi^{2}\sqrt{\chi})$.
Its origin can be traced back to the last term on the left-hand side of Eq.
(23), which comes from the chiral anomaly. In the long-wavelength limit $k\to
0$, it gives a singular contribution $B\delta E_{z}\simeq-ieB\delta n/k$ that,
in turn, produces a nonzero gap. Since both chiral anomaly and Gauss’s law
play important roles in the derivation, the gap is the result of their
interplay.

While the result in Eq. (30) is encouraging, it cannot be taken seriously.
This is because chiral (i.e., ultra-relativistic) plasmas are realized only at
sufficiently high temperatures or densities. Under such conditions, the
conductivity is usually high. This fact appears to be at odds with the formal
assumption of small $\sigma$ used in the derivation of Eq. (30).

A more refined analysis shows that the conductivity does not have to be
vanishing for the propagating modes to exist. Indeed, as seen from Eq. (27),
the collective modes are strictly diffusive when the following condition is
met:

$\frac{eB}{\pi^{2}\sigma\chi}\sqrt{k^{2}+e^{2}\chi}<1.$ (31)

As we argue below, this inequality is indeed easily satisfied in chiral
plasmas. One may speculate, however, that the inequality can break down in
some special regimes of plasmas with a reduced conductivity and a sufficiently
strong magnetic field. Then, the CMW could become a propagating mode.

To get a better insight into the underlying physics, it is useful to consider
a specific example of a chiral plasma. A hot relativistic QED plasma, made of
electrons and positrons, is the simplest possibility. Note that, its charge
density susceptibility is given by $\chi=T^{2}/3$, provided $\mu=\mu_{5}=0$.
Then, for the long-wavelength modes (i.e., $k\to 0$), the inequality in Eq.
(31) reads

$\frac{\sqrt{3}e^{4}B}{\pi^{2}C_{\sigma}T^{2}}\ln e^{-1}<1,$ (32)

where $C_{\sigma}$ is a constant of order $1$ that determines the electrical
conductivity of the QED plasma, namely $\sigma\simeq C_{\sigma}T/(e^{2}\ln
e^{-1})$ [120]. As expected, the expression for the conductivity is inversely
proportional to the coupling constant. In QED, the strength of the interaction
is determined by the fine structure constant, i.e.,
$\alpha=e^{2}/(4\pi)\approx 1/137$. Since the value of $\alpha$ is so small,
the corresponding plasma has rather high conductivity. This is reasonable
since the mean-free path of weakly interacting particles is large.

For a generic magnetized plasma, it is natural to expect that $|eB|\lesssim
T^{2}$. Otherwise, the quantizing limit is achieved and the quasiclassical
description fails. While a rigorous treatment of plasma in a quantizing field
is also possible, it requires special considerations. This is clear since its
physical properties (including the conductivity and susceptibility) can be
modified considerably by the field. The corresponding limit of the superstrong
field will not be discussed here, however.

In either case, the inequality in Eq. (32) is easily satisfied even for the
magnetic fields as strong as $|eB|\simeq T^{2}$. In essence, this is because
the coupling constant in QED is very weak. Indeed, the numerator on the left-
hand side of Eq. (32) contains an extra small factor
$e^{3}=(4\pi\alpha)^{3/2}\approx 2.8\times 10^{-2}$. It ensures that the
inequality is satisfied even for rather strong magnetic fields. Therefore, one
must conclude that the CMW is overdamped in a weakly interacting plasma. The
only exception, perhaps, is the regime of the superstrong magnetic field when
$|eB|\gg T^{2}$ [121].

The above example of the QED plasma is very instructive. It demonstrates, in
particular, that the weak interaction is one of the main reasons for the
overdamped nature of the CMW mode. Turning the argument around, one may expect
that the CMW can be revived in a strongly interacting plasma. The deconfined
QGP just above the critical temperature is an example of a system where this
is possible. Indeed, it is a strongly interacting plasma with the coupling
constant $\alpha_{s}=g_{s}^{2}/(4\pi)$ of order $1$.

###  5 CMW in strongly interacting QGP

Let us consider the QGP with two light quark flavors ($f=u,d$). Most of the
qualitative results are not expected to change if the strange quark is also
included. For each quark flavor, one can write down a pair of continuity
relations for the flavor number $\delta n_{f}$ and the flavor-specific chiral
charge $\delta n_{f,5}$. When describing the interaction with the
electromagnetic field, one should take into account that the electric charges
of the up and down quarks are different, i.e., $q_{u}=2/3$ and $q_{d}=-1/3$.

By repeating the same derivations as for the QED plasma in Sec. 4, one arrives
at the following set of linearized equations:

$\displaystyle\omega\delta n_{f}-\frac{eq_{f}Bk}{2\pi^{2}\chi_{f,5}}\delta
n_{f,5}+iD_{f}k^{2}\delta n_{f}-\frac{\sigma_{f}}{eq_{f}}k\delta E_{z}$
$\displaystyle=$ $\displaystyle 0,$ (33) $\displaystyle\omega\delta
n_{f,5}-\frac{eq_{f}Bk}{2\pi^{2}\chi_{f}}\delta n_{f}+iD_{f}k^{2}\delta
n_{f,5}-i\frac{e^{2}q_{f}^{2}}{2\pi^{2}}B\delta E_{z}$ $\displaystyle=$
$\displaystyle 0,$ (34) $\displaystyle k\delta E_{z}+ie\sum_{f}q_{f}\delta
n_{f}$ $\displaystyle=$ $\displaystyle 0,$ (35)

where we used the flavor-dependent fermion number and chiral charge
susceptibilities $\chi_{f}\equiv\partial n_{f}/\partial\mu_{f}$ and
$\chi_{f,5}\equiv\partial n_{f,5}/\partial\mu_{f,5}$. Below, we will assume
that the chiral charge susceptibilities are equal to the fermion number ones,
i.e., $\chi_{f,5}=\chi_{f}$.

The partial flavor contributions to the electrical conductivity are denoted by
$\sigma_{f}=c_{\sigma}e^{2}q_{f}^{2}T$. Note that the total electrical
conductivity $\sigma$ takes the form

$\sigma=\sum_{f}\sigma_{f}=c_{\sigma}C^{\ell}_{\rm em}T,$ (36)

where $C^{\ell}_{\rm em}=e^{2}\sum_{f}q_{f}^{2}=5e^{2}/9\approx 5.1\times
10^{-2}$. For the near-critical deconfined QGP, the electrical conductivity
was obtained numerically by using lattice calculations [122, 123, 124].
According Ref. [124], the value of coefficient $c_{\sigma}$ was found to be
$0.111$, $0.214$, and $0.316$ at the temperatures $200~{}\mbox{MeV}$,
$235~{}\mbox{MeV}$, and $350~{}\mbox{MeV}$, respectively. As expected, the
electrical conductivity decreases with the temperature.

The flavor number susceptibilities $\chi_{f}$ and the diffusion coefficients
$D_{f}$ have the following general structure:

$\displaystyle\chi_{f}$ $\displaystyle=$ $\displaystyle
c_{\chi}\chi^{(SB)}_{f},$ (37) $\displaystyle D_{f}$ $\displaystyle=$
$\displaystyle\frac{c_{D}}{2\pi T}.$ (38)

where $\chi^{(SB)}_{f}\equiv T^{2}/3$ is the Stefan-Boltzmann expression for
the susceptibility. The numerical constants $c_{\chi}$ and $c_{D}$ were
calculated on the lattice in Ref. [124]. In particular, the value of
$c_{\chi}$ was found to be $0.804$, $0.885$, and $0.871$ at the temperatures
$200~{}\mbox{MeV}$, $235~{}\mbox{MeV}$, and $350~{}\mbox{MeV}$, respectively.
Similarly, the value of $c_{D}$ was $0.758$, $1.394$, and $1.826$ for the same
three temperatures. We use below to derive the dispersion relations for the
collective modes in the near-critical QGP.

While the structure of Eqs. (33)–(35) is very similar to Eqs. (22)–(24), the
total number of equations is larger. As a result, the characteristic equation
becomes more complicated and does not have simple analytical solutions.
Nevertheless, as one can verify numerically, the underlying physics remains
essentially the same [118]. Because of the strongly coupled regime, it is
slightly easier to achieve the regime with propagating CMW modes. However, one
still needs very strong background magnetic fields.

The dispersion relations for the CMW-type modes take the following generic
form:

$\omega_{n}^{(\pm)}=\pm E_{n}(k)-i\Gamma_{n}(k),\quad\mbox{with}\quad n=1,2,$
(39)

where $E_{n}(k)$ and $\Gamma_{n}(k)$ are the real and imaginary parts of the
energies (frequencies) of collective modes. Note that the number of modes
doubles because there are two flavors of quarks. In the long-wavelength
regime, one of the modes is the usual CMW, while the other corresponds to a
mode sustained by electrically neutral oscillations with $n_{d}\approx
2n_{u}$.

For the magnetic fields $eB\lesssim m_{\pi}^{2}$, the ratio
$E_{n}(k)/\Gamma_{n}(k)$ is less than $1$ for all CMW-type modes with the wave
vectors $k$ in the range between $50~{}\mbox{MeV}$ and $620~{}\mbox{MeV}$,
which correspond approximately to the wavelengths $\lambda_{k}$ between
$2~{}\mbox{fm}$ and $24~{}\mbox{fm}$. Since $E_{n}(k)<\Gamma_{n}(k)$, the
corresponding modes are strongly overdamped. The dominant damping mechanisms
are the electric conductivity and the charge diffusion at small and large
values of $k$, respectively.

The modes get revived gradually when the magnetic field is about $eB\approx
3m_{\pi}^{2}$. Clearly, this is quite strong even by the standards of heavy-
ion collisions. The corresponding numerical results for the dispersion
relations of collective modes are summarized in Fig. 2. The two panels show
the real parts of the energies $E_{n}(k)$ and the ratios of the real and
imaginary parts $E_{n}(k)/\Gamma_{n}(k)$, respectively. The three colors of
the lines correspond to three fixed temperatures, i.e., $T=200~{}\mbox{MeV}$,
$235~{}\mbox{MeV}$, and $350~{}\mbox{MeV}$. Two shades of the same color
represent two different CMW-type modes. Note that, in the gray shaded regions,
the values of the wavelengths are either too long or too short to be relevant
for the QGP produced in heavy-ion collisions.

Figure 2: The real parts of the energies (left panel) and the ratios of the
real and imaginary parts (right panel) for the CMW-type collective modes in
near-critical regime of QGP.

As is clear from Fig. 2, a substantial damping of collective modes persists
even for the magnetic field as strong as $eB=3m_{\pi}^{2}$. In fact, when
$T=350~{}\mbox{MeV}$, one of the modes is completely diffusive below $k\simeq
44~{}\mbox{MeV}$. Only at relatively low temperature, $T=200~{}\mbox{MeV}$,
the ratio $E_{n}(k)/\Gamma_{n}(k)$ becomes larger than $1$ in a limited window
of wave vectors relevant in heavy-ion collisions. That is where the CMW modes
get partially revived.

The existence (absence) of completely diffusive modes in the spectrum of two-
flavor QCD was investigated in the whole range of relevant model parameters in
Ref. [118]. The corresponding phase diagram in the plane of the wave vector
and the magnetic field was also obtained. It reconfirms that all CMW-type
collective modes are either completely diffusive or strongly overdamped when
$eB\lesssim m_{\pi}^{2}$. This is seen from the ratios of the real and
imaginary parts of their energies, which are less than $1$ for the wave
vectors greater than about $50~{}\mbox{MeV}$.

###  6 Summary

To summarize, the chiral anomaly is predicted to give rise to several unusual
observable phenomena in the bulk of (ultra-)relativistic plasmas. The chiral
separation and the chiral magnetic effects are the simplest examples. In
essence, their signatures are the chiral and electric currents induced by a
background magnetic field in chiral plasmas at nonzero electric and chiral
chemical potentials, respectively.

While the two effects appear to be very similar, they are profoundly
different. The chiral separation effect exists already in an equilibrium
plasma. The associated chiral current can be reinterpreted as a spin
polarization of plasma. As such, it drives no actual transport of conserved
charges or energy.

In contrast, the meaning of the chiral magnetic effect is drastically
different. It is a non-equilibrium phenomenon caused by a chiral imbalance in
the plasma. The chiral imbalance itself, which measures the excess of the
opposite chirality relativistic fermions, is possible only out of equilibrium.
Some background gauge fields can induce the corresponding transient state via
the chiral anomaly. However, it is unstable with respect to a spontaneous
generation of helical electromagnetic waves.

The chiral magnetic effect can be realized presumably in the quark-gluon
plasma produced by ultrarelativistic heavy-ion collisions. While the expected
dipole correlations of charged particles are observed in experiments [54, 55,
56, 57, 58, 59], the identification of the chiral magnetic effect signal from
a large background remains a challenge [80, 81, 82, 83].

There is strong evidence in support of the chiral magnetic effect in
topological semimetals. It causes a negative longitudinal magnetoresistance,
which indeed was observed and reconfirmed in many experiments [63, 64, 65, 66,
67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79]. The chiral magnetic
effect is predicted to cause also the giant planar Hall effect [125]. The
latter was confirmed experimentally as well [126, 127]. A range of other
predictions can be tested in topological semimetals in the future [49].

The status of the chiral magnetic wave, which is a collective mode originating
from an interplay of the chiral magnetic and chiral separation effects, is
less clear. According to the latest theoretical analysis, the corresponding
mode should be diffusive in a weakly interacting relativistic plasma [118,
117]. Because of the strongly interacting regime in the near-critical QGP,
there is a chance that the chiral magnetic wave may survive there. It
requires, however, the presence of a very strong background magnetic field on
the order of $eB\gtrsim 3m_{\pi}^{2}$. One of the proposed signatures of the
chiral magnetic wave is the quadrupole correlation of charged particles in
heavy-ion collisions. Such correlations are observed indeed [60, 61, 62], but
their interpretation is clouded by huge background effects [84].

### Acknowledgements

This work was supported by the U.S. National Science Foundation under Grant
No. PHY-1713950.

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