# Chiral Transport Phenomena and Compact Stars

Cristina Manuel  Instituto de Ciencias del Espacio (ICE, CSIC), C. Can Magrans
s.n., 08193 Cerdanyola del Vallès, Catalonia, Spain, and Institut d’Estudis
Espacials de Catalunya (IEEC), C .Gran Capità 2-4, Ed. Nexus, 08034 Barcelona,
Spain<EMAIL_ADDRESS>
###### Abstract

I will review the main chiral transport phemomena arising in systems made up
of (almost) massless fermions associated to the quantum chiral anomaly. These
quantum effects might have relevant implications in compact stars, and I will
review some relevant works that reveal so. I will also show how a conservation
law that has the same form of the chiral anomaly also emerge in perfect
classical fluids, which expresses a conservation law of magnetic, fluid and
mixed helicities for isentropic fluids, and why this should also be relevant
in compact stars.

##  1 Introduction

Understanding the role of symmetries, and also the violation of a symmetry,
has played a pivotal role in different branches of physics. More particularly,
in the development and construction of the Standard Model of particle physics
the study of chiral symmetry was crucial. The weak force was found to be only
interacting with left-handed fermions (or right-handed antifermions),
revealing a breaking of parity. Electromagnetic interactions were known to be
respectful with the chirality of the charged particles. However, at quantum
level the chiral symmetry was found to be not longer preserved. This discovery
had a deep impact on the construction of theoretical particle physics models,
as one had to understand the criteria under which some symmetries could be
broken by quantum effects.

In particle physics the effects of the quantum chiral anomaly come mainly in
the explanation of different anomalous decays, such as that of the neutral
pion into two photons. In many body systems, the quantum chiral anomaly has
also several relevant macroscopic effects, as it has been found out that it
leads to a wide variety of dissipationless transport phenomena that have a
relevant impact in the dynamical evolution of chiral systems. In this talk I
will discuss and review the most relevant aspects of chiral transport
phenomena, and then focus on why this is also relevant in the study of compact
stars. My intention is not to give a complete review on this topic (check the
existing excellent reviews in the literature, see Refs. [1, 2, 3, 4, 5]), so I
apologize if I cannot cover all existing works on this growing topic in this
talk. I will rather pinpoint the most relevant concepts and focus on some
works that reveal its relevance in compact stars.

##  2 Chiral magnetic and vortical effects

Chiral transport phenomena refers to quantum transport effects associated to
the chirality of fermions, and related to the quantum chiral anomaly. One
considers the situation of massless, or quasi-massless fermions, as chirality
is only a well-defined quantum number in that case, as otherwise, the mass
mixes up the dynamical evolution of the different fermion chiralities.

I will concentrate on discussing the chiral magnetic and vortical effects. The
chiral magnetic effect (CME) [6] is a phenomenon in which a magnetic field
generates an electric current in a conducting material, such as a plasma or a
fluid, that contains a population of fermions with a chiral imbalance. One
intuitive way of understanding the CME comes by realizing that for massless
fermions the spin and momentum are either parallel or antiparallel, depending
whether those are right-handed or left-handed (the opposite for the
antifermions). In a magnetic field all the spins are aligned, and this implies
that all particles then move in the direction of the magnetic field. In the
presence of a misbalance among chiralities, these parallel/antiparallel
currents to the magnetic field are not counterbalanced, and this is ultimately
what creates the effect. A similar situation occurs in the presence of
vorticity, as also spins align with the vorticity, then there is a current
parallel to the vorticity vector, and one talks about the chiral vortical
effect (CVE).

The CME or CVE currents depend on the chiral chemical potential, the parameter
that gives account of the chiral misbalance. That the effects are related to
the chiral anomaly can be shown in different ways, but I will highlight that
first used in [7], based on using some effective field theory methods, and
valid at zero temperature. In the Hamiltonian of a system a chemical potential
$\mu$ and chiral chemical potential $\mu_{5}$ enter as new pieces in the
Hamiltonian that go as $\delta H=\mu n+\mu_{5}n_{5}$, where $n/n_{5}$ are the
charge/chiral charges densities. Very often one says that the chemical
potentials act as a ficticious zero component of a vector gauge potential. In
a moving system with velocity $u^{\mu}$ those terms would rather be $\delta
H=\mu u_{\mu}j^{\mu}+\mu_{5}u_{\mu}j^{\mu}_{5}$, where $j^{\mu},j^{\mu}_{5}$
are the (classically conserved) vector and axial vector currents,
respectively. Thus, one sees that $\mu u^{\mu}$ couples to matter as a real
vector gauge field potential $A^{\mu}$ does, while $\mu_{5}u^{\mu}$ couples as
it were an axial vector gauge potential. We can push these analogies, and then
compute the quantum anomaly in the presence of the ficticious vector and axial
vector fields. At a quantum level, we can then use the (covariant) quantum
anomalies [3] for the vector and axial currents, assuming that the vector
gauge field is $eA^{\mu}+\mu u^{\mu}$, while the axial vector field is
$\mu_{5}u^{\mu}$

$\displaystyle\partial_{\mu}j^{\mu}_{5}$ $\displaystyle=$
$\displaystyle-\frac{1}{4\pi^{2}}\epsilon_{\mu\nu\alpha\beta}\left(\partial^{\mu}(eA^{\nu}+\mu
u^{\nu})\partial^{\alpha}(eA^{\beta}+\mu
u^{\beta})+\partial^{\mu}(\mu_{5}u^{\nu})\partial^{\alpha}(\mu_{5}u^{\beta})\right)\
,$ (1) $\displaystyle\partial_{\mu}j^{\mu}$ $\displaystyle=$
$\displaystyle-\frac{1}{2\pi^{2}}\epsilon_{\mu\nu\alpha\beta}\left(\partial^{\mu}(eA^{\nu}+\mu
u^{\nu})\partial^{\alpha}(\mu_{5}u^{\beta})\right)\ .$ (2)

It turns out that all the pieces with a chemical and chiral chemical potential
in the quantum anomaly equations might be re-written down by modifying the
expressions of the vector and axial-vector currents. After integrating by
parts, and discarding surfaces terms, one can re-write the above equations as

$\displaystyle\partial_{\mu}\left(j^{\mu}_{5}+\frac{e\mu}{2\pi^{2}}B^{\mu}+\frac{\mu^{2}+\mu_{5}^{2}}{2\pi^{2}}\omega^{\mu}\right)$
$\displaystyle=$
$\displaystyle-\frac{e^{2}}{16\pi^{2}}\epsilon_{\mu\nu\alpha\beta}F^{\mu\nu}F^{\alpha\beta}\
,$ (3)
$\displaystyle\partial_{\mu}\left(j^{\mu}+\frac{e\mu_{5}}{\pi^{2}}B^{\mu}+\frac{\mu\mu_{5}}{\pi^{2}}\omega^{\mu}\right)$
$\displaystyle=$ $\displaystyle 0\ ,$ (4)

where we have defined

$\omega^{\mu}=\frac{1}{2}\epsilon^{\mu\nu\alpha\beta}u_{\nu}\partial_{\alpha}u_{\beta}\
,\qquad B^{\mu}=\frac{1}{2}\epsilon^{\mu\nu\alpha\beta}u_{\nu}F_{\alpha\beta}\
,$ (5)

which in the rest frame of the system represent the vorticity and magnetic
field vectors, respectively. In a moving frame however, the zero component of
these vectors are associacted to the fluid and mixed helicities (we will later
on discuss on this). We then see that at $T=0$, one could read the
conservation law of the electromagnetic current by multiplying by $e$ the
r.h.s. of Eq. (4), and thus identify the CME and CVE currents in this way.
Please note that induced chiral currents proportional to $B^{\mu}$ and
$\omega^{\mu}$ are also produced, these are called the chiral separation and
chiral vortical effects, respectively.

An interesting remark is that all the chiral transport phenomena that
originate in the chiral quantum anomaly are dissipationless, and thus they do
not imply an increase of entropy. Also that the effects are not corrected
perturbatively, as they rely on the quantum chiral anomaly.

There are several different systems where all these ideas might be applied. In
some materials, like the Weyl of Dirac semi-metals, there are quasiparticles
that behave as massless fermions. In these systems the CME has been detected
[9]. In systems at extreme conditions of temperature and/or density, one might
expect that most fermions can be considered as massless whenever their mass
$m$ is such that $m\ll T$ and/or $m\ll\mu$, where $T$ is the temperature.
Thus, one can also expect to find chiral transport phenomena in the quark-
gluon plasma phase studied with heavy-ion collisions. Big efforts by the
different experimental collaborations of both the LHC and RHIC have been
carried out (see for example [11, 10]), but so far it has not been detected,
while there are debates on what the criteria to claim detection should be
[12]. We can also expect that these ideas might be relevant in several
cosmological and astrophysical scenarios, as we will encounter several extreme
conditions where one can take the ultrarelativistic limit to describe the
corresponding quasiparticles.

##  3  Chiral hydrodynamics and chiral kinetic theory

Relativistic hydrodynamics has been naturally applied to a variety of
cosmological, astrophysical and nuclear physics scenarios. The hydrodynamical
equations are the expressions of the conservations laws of a system. In the
presence of chiral fermions, it thus seems natural to incorporate the quantum
chiral anomaly equation in the hydrodynamics [8], the resulting framework is
known as chiral or anomalous hydrodynamics.

An interesting result is that even if the quantum chiral anomaly requires the
computation of one-loop Feynman diagrams, the famous triangle diagrams, for
many-body systems it is possible to give account of it with semi-classical
methods. Chiral or anomalous hydrodynamics might be decuded from the so called
chiral kinetic or transport theory (CKT) [13, 14, 15]. There are several
derivations of CKT, I will focus on those that I am more familiar with. The
physics associated to these chiral imbalanced systems, governed by the chiral
quantum anomaly, might be deduced by incorporating the first quantum
corrections to classical transport equations. One can do that by taking the
Dirac Hamiltonian, in the presence of electromagnetic fields, and diagonalize
it for the particle and antiparticle degrees of freedom in an expansion in
$\hbar$, the Planck constant [16], treating the resulting expression semi-
classically. It is possible also to derive an effective field theory that does
the same at a quantum field theory level [17]. Alternatively, one can derive
transport equations from quantum field theory, and expand the resulting
equations in $\hbar$ [18]. There are some subtleties when one uses one
framework over the other to derive CKT, which has to do with the semi-
classical definition of quasiparticle (and getting rid of the so called
Zitterbewegung effect [19]), but I will not enter in discussing this issue
here.

The transport equation obeyed by the distribution function $f_{p}$ associated
to a fermion of chirality $\chi$ can be written down as (see Refs. [13, 16] )

$\displaystyle\frac{\partial f_{p}}{\partial t}$ $\displaystyle+$
$\displaystyle\left(1+e{\bf
B}\cdot{\bf\Omega}\right)^{-1}\left\\{\left[\tilde{\bf v}+e\ \tilde{\bf
E}\times{\bf\Omega}+e\ {\bf B}(\tilde{\bf
v}\cdot{\bf\Omega})\right]\cdot\frac{\partial f_{p}}{\partial{\bf r}}\right.$
(6) $\displaystyle+$ $\displaystyle\left.e\left[\tilde{\bf E}+\tilde{\bf
v}\times{\bf B}+e{\bf\Omega}\ (\tilde{\bf E}\cdot{\bf
B})\right]\cdot\frac{\partial f_{p}}{\partial{\bf p}}\right\\}=0\ .$

Here ${\bf\Omega}=\chi{\bf p}/p^{3}$ is the so called Berry-curvature, and we
have defined

$\tilde{\bf E}={\bf E}-\frac{1}{e}\frac{\partial\epsilon_{\bf p}}{\partial\bf
r}\ ,\qquad\tilde{\bf v}=\frac{\partial\epsilon_{\bf p}}{\partial{\bf p}}\ ,$
(7)

where $\epsilon_{\bf p}$ is the particle’s energy. Although we work in natural
units, restoring dimensions, it is possible to check that all pieces that
contain the Berry curvature are proportional to $\hbar$, and are pure quantum
effecs that correct the classical terms of the transport equation. The
particle density associated to these chiral fermions reads

$n=\int\frac{d^{3}p}{(2\pi)^{3}}(1+e\,{\bf B}\cdot{\bf\Omega})f_{p}\ ,$ (8)

while the current reads

${\bf j}=-\int\frac{d^{3}p}{(2\pi)^{3}}\left[\epsilon_{p}\frac{\partial
f_{p}}{\partial{\bf p}}+e{\bf\Omega}\cdot\frac{\partial f_{p}}{\partial{\bf
p}}\epsilon_{p}{\bf B}+\epsilon_{p}{\bf\Omega}\times\frac{\partial
f_{p}}{\partial{\bf r}}-ef_{p}{\bf E}\times{\bf\Omega}\right]\ .$ (9)

After integrating the kinetic equation one then obtains

$\frac{\partial n}{\partial t}+\nabla\cdot{\bf
j}=-e^{2}\int\frac{d^{3}p}{(2\pi)^{3}}\left({\bf\Omega}\cdot\frac{\partial
f_{p}}{\partial{\bf p}}\right){\bf E}\cdot{\bf B}\ .$ (10)

Now considering the two possible chiralities, one can construct both the
vector current and chiral current, combing the contributions of the two
chiralities. Taking an equilibrium form of the distribution function, taking
into account both particle and antiparticle degrees of freedom, one can then
reproduce the quantum chiral anomaly, Eq. (10), while one can obtain the
conservation of the vectorial current. Written in this form, one sees the
clear quantum origin of the non-conservation of the current, as if the quantum
corrections to the classical transport equation are neglected, the chiral
current would be conserved.

It is interesting to study the dynamical evolution of the CME, allowing the
electromagnetic fields to be dynamical. Let us assume that the system is at
rest. After integrating the chiral anomaly equation over space in a closed
volume $V$, it leads to a quantum conservation law that relates the chiral
fermion density, $Q_{5}=\frac{1}{V}\int d^{3}x\,n_{5}(x)$ with the magnetic
helicity of the system

$\frac{dQ_{5}}{dt}=\frac{e^{2}}{2\pi^{2}}\frac{1}{V}\int d^{3}x\,{\bf
E}\cdot{\bf B}=-\frac{e^{2}}{2\pi^{2}}\frac{d{\cal H}}{dt},$ (11)

where

${\cal H}=\frac{1}{V}\int d^{3}x\,{\bf A}\cdot{\bf B}$ (12)

is the magnetic field density. This quantity is gauge invariant provided that
the magnetic field vanishes at, or it is parallel to the boundary of $V$. The
magnetic helicity gives a measure of the linkage and twists of the magnetic
field lines. The above equation tells us that the chirality of the fermions
can be transformed into magnetic helicity, and/or viceversa, and thus
generate/destroy different non-trivial topological field configurations.

As the chiral symmetry is only an approximated symmetry, one should also add
in the chiral anomaly equation a chirality flipping rate $\Gamma_{f}$,
typically giving account of scattering processes that allow for a change of
chirality due to the existence of a small mass. Thus, one should rather write

$\frac{dQ_{5}}{dt}+\frac{e^{2}}{2\pi^{2}}\frac{d{\cal
H}}{dt}=-\Gamma_{f}n_{5}\ .$ (13)

The chiral anomaly coupled to Maxwell’s equations govern the dynamical
evolution of the chiral medium. One can view that the chirality of the
fermions can be transformed into magnetic helicity, and/or viceversa, on time
scales shorter than $t\ll 1/\Gamma_{f}$. An interesting property is that some
electromagnetic modes are unstable, and can grow exponentially. Let us explain
why. Assuming the presence of a CME current along the $z$ direction of an
applied magnetic field, a chiral magnetic instability arises [21, 22].
According to Ampère’s law, a magnetic field is generated in the $\theta$
direction. This induced field, in turn, generates a current in the same
direction through the CME effect, which would then generate a field in the $z$
direction. This process results in an amplified value of the initial magnetic
field in the $z$ direction, leading to a runaway mechanism that causes the
instability. The unstable modes occur for field wavelengths which are less
than $k\sim e^{2}\mu_{5}$. The time scale of growth in an electromagnetic
plasma is of the order $\sim 1/e^{4}\mu_{5}$ [22], while for a conductor, with
electrical conductivity $\sigma$ it is $t_{\rm ins}\sim\sigma/ke^{2}\mu_{5}$
[23]. In the presence of an initial chiral chemical potential there can be a
generation of magnetic helicity with a sort of inverse cascade phenomenon,
when the helicity is transferred from the highest to the lowest modes [23].

In the presence of a small fermion mass, the chiral anomaly equation is also
affected by the presence of quantum coherent mixtures of mixed helicities, as
seen in Ref.[24]. The effect of these genuine quantum states has, however, not
yet been studied.

##  4 Chiral anomaly equation and compact stars

A couple of weeks before giving this talk, a review article on chiral
transport phenomena in astrophysics and cosmology appeared in the arxives [5],
which contains an exhaustive lists of references on this topic. I cannot cover
all works in this talk, but I recommend the previous review for a much more
complete set of references.

Let us start by mentioning that in cosmology the use of the quantum chiral
anomaly has been extensive, as all the baryogenesis models rely on it [25].
The CME and the chiral instabilities have also been used to explain the
generation of primordial magnetic fields, with magnetic helicity (see, for
example [21, 23]). There are also closed related works in cosmology and
axions. Initially axions were proposed to solve the strong CP problem, and
nowadays they are serious dark matter candidates [26]. Axions and axion-like
particles are predicted in many models as pseudo scalar particles that couple
both to fermions and to photons. Most of the experiments to detect axions rely
on how they couple to the electromagnetic fields. Several cosmological models
of axions assume that the axion field $a$ can be treated as a coherent
classical field $a(t)=\sqrt{2\rho_{\rm DM}}{m_{a}}\sin{m_{a}t}$, where
$\rho_{\rm DM}$ is the local dark matter density, and $m_{a}$ is the axion
mass. Then, its time derivative acts as a chiral chemical potential for the
fermions, and would imply that they produce a CME current. A new proposal to
detect axions [28], LACME (low temperature axion chiral magnetic effect), is
based on this fact.

Let me briefly mention how all these ideas are relevant for compact stars.
There has not been the same amount of works on the impact of the chiral
anomaly for compact stars than in cosmology, and most of the discussions are
very qualitative, and simply do some estimates on the scales involved to
assess how relevant chiral transport phenomena could be. Definitively, much
more work is needed in this area of research.

It has been suggested [29] that the CME could explain the magnetic helicity of
neutron stars, whose origin remains unknown. In a proto neutron star one could
create a big chiral misbalance in the population of electrons, which can be
taken as quasi-massless as $\mu_{e}\sim 100$ MeV. The chiral misbalance can be
created in the neutronization process, as only left-handed leptons participate
in this electroweak process. More particularly, when the neutron star is
formed the neutronization or electron capture $p+e_{L}\rightarrow
n+\nu^{e}_{L}$, which is not yet counterbalanced with neutron decay, would
lead to a high population of right-handed electrons. Initial estimates of the
chiral chemical potential assumed these to be close to the QCD scale, creating
a very large magnetic field, of the order of $B\sim 10^{18}$ G. However, it
was later argued [30] that the chirality flipping rate of Rutherford
scattering of electrons off protons would damp the chiral plasma instability.
A more careful analysis [31] however reveals that electron capture rate
depends very much on the temperature, and that one can find for high $T$, but
still for $\mu_{e}\gg T$, that this mechanism is operating, creating magnetic
fields of the order $B\sim 10^{14}$ G.

Another set of ideas come in how the different field configurations of
parallel electric and magnetic fields could create a chiral misbalance. In
particular, these configurations occur in the magnetospheres of supermassive
black holes and of pulsars. In Ref. [32] estimates of the chiral densities
generated in those cases are presented. While for supermassive black holes the
effect seems to be negligible, the authors of Ref. [32] suggest that it could
be substantial for pulsars, estimating the chiral density created with the
large time solution of the chiral anomaly, with a chirality flipping rate due
to electron-electron scattering. Then the effect should be detected by
checking that the electromagnetic radiation in a given specific range of
frequencies should be circular polarized, as it propagates in a chiral medium.

It has also been considered that chiral effects might be relevant for the
explanation of the so called pulsar kicks [33]. Neutron stars typically have a
velocity greater than their progenitors, and many different sort of mechanism
have been considered for the explanation of those kicks. The existence of the
CME and CVE could lead to a possible explanation of the kick [34, 35].

Chiral effects should be relevant to study the dynamics of neutrinos in core-
collapse supernova. Most of the energy released in these supernova explosions
is in the form of neutrinos. The chirality of neutrinos should play a relevant
role, and recently a chiral radiation transport theory has been put forward in
Ref. [36, 37, 38].

##  5 Helicity conservation law

Studies of the relevance of chiral transport phenomena in compact stars are in
their infancy, and typically focus on order of magnitude estimates. The whole
set of hydrodynamical equations, and not only the chiral anomaly, should be
taken into consideration to asses its impact on the physics of compact stars.

To this regard, an interesting claim has been made in the literature
associated to classical hydrodynamics. There is an equation analogous to the
chiral anomaly equation, valid for classical barotropic fluids [39, 40, 41],
which expresses the conservation law of a combination of all the helicities
that can be defined in the fluid. In collaboration with Juan Torres-Rincon, we
have presented our own derivation in Ref. [42], which extends the original
derivation to the isentropic and finite temperature cases. It is easier to
show it for relativistic fluids. It only requires to derive hydrodynamical
equations for the vectors $\omega^{\mu}$ and $B^{\mu}$ from the Euler equation
obeyed by $u^{\mu}$, and use some thermodynamical relations. More
particularly, for isentropic fluids one finds

$\partial_{\mu}(h^{2}\omega^{\mu}+heB^{\mu})=-\frac{e^{2}}{4}\epsilon_{\mu\nu\alpha\beta}F^{\mu\nu}F^{\alpha\beta}\
,$ (14)

where $h$ is the enthalpy density. Thus, for isentropic fluids, there is an
equation very similar to the chiral anomaly equation. This is a classical
conservation law, though. Let us stress that zero component of the
$\omega^{\mu}$ vector gives the fluid helicity, which measures the linkage of
the fluid lines, and the zero component of the $B^{\mu}$ vector gives the
mixed helicity, which measures the linkage of magnetic and fluid lines. This
equation expresses that there is a combination of the fluid, magnetic and
mixed helicities which is conserved for isentropic fluids. As these three
helicities measure linkage, the equation describes a conservation of some
topological properties of the system. It would be interesting to see how
different dissipative effects might affect this conservation law.

We have also formulated how this equation is modified in the presence of a
chiral imbalance, which then reads

$\partial_{\mu}(h^{2}\omega^{\mu}+heB^{\mu})=-\frac{e^{2}}{4}\epsilon_{\mu\nu\alpha\beta}F^{\mu\nu}F^{\alpha\beta}+(2h\omega^{\mu}+eB^{\mu})(T\partial_{\mu}\bar{s}+\mu_{5}\partial_{\mu}x_{5})\
,$ (15)

where $\bar{s}=s/n$ is the entropy per particle, $x_{5}=n_{5}/n$ is the chiral
fraction. Even for isentropic fluids ($\partial_{\mu}\bar{s}=0$), a space-time
dependent chiral misbalance modifies the previous helicity conservation law.
However, in the presence of chiral misbalance, Eq. (15) has to be combined
with the chiral anomaly equation, Eq. (3). The chiral anomaly equation in the
presence of a chiral chemical potential also involves the fluid and mixed
helicities, through the axial vortical and chiral separation effects.
Sometimes the chiral anomaly equation has been taken as a sort of helicity
conservation law [43].

Let us stress that in all estimates of the relevance of the chiral anomaly
presented in the previous section assumed that the fluid is at rest. However,
this is certainly not the case in the astrophysical settings of interest,
where we may expect the presence of vorticity and helicities, so it might be
interesting to revise all those estimates taking into consideration all
hydrodynamical equations. More particularly, it would be interesting to review
the generation of magnetic fields in the proto-nuetron stars, or the magnetic
field dynamics in the magnetospheres of magnetars. We hope to report on this
issue in the near future.

I thank the organizers of this wonderful and unique workshop for the
invitation to give this talk and for financial support. This work was
supported by Ministerio de Ciencia, Investigación y Universidades (Spain)
under the project PID2019-110165GB-I00 (MCI/AEI/FEDER, UE), Generalitat de
Catalunya by the project 2017-SGR-929 (Catalonia). This work was also partly
supported by the Spanish program Unidad de Excelencia Maria de Maeztu
CEX2020-001058-M, financed by MCIN/AEI/10.13039/501100011033

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