aainstitutetext: Facultad de Ciencias, Universidad Nacional Autónoma de
México, Investigación Científica C.U., 04510 Coyoacan, Ciudad de Mexico,
Mexico bbinstitutetext: Asia Pacific Center for Theoretical Physics, Pohang,
37673, Koreaccinstitutetext: Department of Physics, Pohang University of
Science and Technology, Pohang, 37673, Koreaddinstitutetext: SUBATECH, IMT
Atlantique, 4 Rue Alfred Kastler, La Chantrerie BP 20722, 44307 Nantes,
Franceeeinstitutetext: Nantes Université, 2 Rue de la Houssinière, BP 92208,
44322 Nantes, France

# Chiral Separation Effect from Holographic QCD

Domingo Gallegos b,c Matti Järvinen d,e Eamonn Weitz<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We analyze the chiral separation effect (CSE) in QCD by using the
gauge/gravity duality. In QCD, this effect arises from a combination of chiral
anomalies and the axial $U(1)$ anomaly. Due to the axial gluon anomaly, the
value of the CSE conductivity is not determined by the anomalies of QCD but
receives radiative corrections, which leads to nontrivial dependence on
temperature and density. To analyze this dependence, we use different variants
of V-QCD, a complex holographic model, carefully fitted to QCD data. We find
our results for the anomalous CSE conductivity at small chemical potential and
nonzero temperature to be in good qualitative agreement with recent results
from lattice QCD simulations. We furthermore give predictions for the behavior
of the conductivity at finite (vectorial and axial) chemical potentials.

††preprint: APCTP Pre2024 - 013

##  1 Introduction

Anomalies in field theory leave their imprint on the transport properties of
the matter they describe. They can give rise exotic transport phenomena, which
may lead to observable effects in real-world systems. Examples of such systems
include the Dirac and Weyl semimetals, and perhaps the quark-gluon plasma,
generated in heavy-ion collisions Kharzeev:2024zzm . In this latter context,
the best known exotic phenomenon is the chiral magnetic effect (CME)
Kharzeev:2007jp ; Fukushima:2008xe , where the combination of a magnetic field
and chirality imbalance (i.e., difference of number densities of left and
right handed quarks) leads to a net current in the direction of the magnetic
field. Another related phenomenon is the chiral separation effect (CSE)
Metlitski:2005pr ; Son:2009tf , where magnetic field at finite density leads
to the generation of a chiral current. Interestingly, charge asymmetries
potentially related to these effects have indeed been observed in heavy-ion
collisions Belmont:2014lta ; STAR:2015wza . These effects may also be studied
by using lattice QCD Buividovich:2009wi ; Yamamoto:2011gk ; Yamamoto:2011ks ;
Buividovich:2013hza ; Brandt:2023wgf ; Brandt:2024wlw .

The CME and the CSE are driven by chiral anomalies. That is, the coupling of
the field theory in question to external fields leads to a non-conservation of
the chiral currents. Associated conservation laws are then violated in a
specific way which can be computed from triangle diagrams. This leads to the
anomalous conductivities associated with these effects having specific
“universal” values, which are determined by the anomalies Fukushima:2008xe ;
Son:2009tf ; Neiman:2010zi ; Jensen:2012kj . However, in interesting systems,
the fermions are typically interacting via a strongly coupled gauge theory,
which complicates the physical picture. Namely, the dynamical gauge fields
also give rise to an anomaly. In the case of QCD, the theory describing quark-
gluon plasma, this is the U(1) axial anomaly: the axial current conservation
is violated in QCD per se. As it turns out, the radiative corrections from
dynamical gauge fields affect some of the anomalous conductivities, driving
them away from the universal values Golkar:2012kb ; Jensen:2013vta .

The gauge-gravity duality is a useful tool to study anomalous transport in
strongly coupled systems. From the dual perspective, chiral symmetry is
enhanced to a higher dimensional gauge symmetry, and the corresponding gauge
fields describe the dynamics of the chiral currents. The chiral anomalies are
then implemented through Chern-Simons terms of the gauge fields. In such
setups, universal values for the anomalous conductivities have been
demonstrated to hold quite generally Gynther:2010ed ; Amado:2011zx ;
Landsteiner:2011cp ; Gursoy:2014boa ; Gursoy:2014ela ; Grozdanov:2016ala . Be
that as it may, the anomalies of the dynamical gauge fields arise as some of
the high dimensional gauge fields acquire a mass. In the case of QCD, the
massive field is the U(1) axial gauge field. As usual, gauge symmetry does not
allow one to add a hard mass term, but the mass can be generated by other
means. One possibility, which is the mechanism appearing in gravity duals of
QCD Schwarz:2001sf ; Sakai:2004cn ; Casero:2007ae ; Arean:2013tja ;
Arean:2016hcs is the Stückelberg mechanism where the mass arises due to
coupling of the gauge field to an additional scalar field Jimenez-Alba:2014iia
.111In QCD, this field is dual to the $F\wedge F$ operator and the
corresponding source is the $\theta$-angle. The radiative corrections to
anomalous conductivities in such setups have been studied in Jimenez-
Alba:2014iia ; Jimenez-Alba:2015awa ; Gallegos:2018ozs ; Rai:2023nxe . In
addition, effects of additional mixed gauge-gravitational anomaly have been
studied in Landsteiner:2011iq ; Megias:2013joa . For systematics of
hydrodynamics and transport in the presence of anomalies, see e.g., Son:2009tf
; Neiman:2010zi ; Kalaydzhyan:2011vx ; Megias:2013joa ; Bu:2018psl ;
Bu:2019mow ; Ammon:2020rvg .

We note however that the majority of the analysis of anomalous conductivities
within the gauge/gravity duality has been carried out in generic gravity
backgrounds that are not adjusted to be dual to any specific field theory such
as QCD. That is, while the chiral symmetries of some of the models are the
same as in QCD, the models disagree with lattice or experimental data for QCD.
Even though some work on anomalous transport in specific holographic models
for QCD exists Yee:2009vw ; Rebhan:2009vc ; Gorsky:2010xu ; Ballon-
Bayona:2012qnu ; Fukushima:2021got , radiative corrections due to dynamical
gluons have not been addressed.

An effective approach to holographic QCD uses generic, “bottom-up” models
possessing a large number of free parameters, which are determined through
comparison to QCD data Gursoy:2007cb ; Gursoy:2007er ; Gubser:2008ny ;
DeWolfe:2010he . These models include improved holographic QCD Gursoy:2007cb ;
Gursoy:2007er and its extension to include a fully backreacted quark sector,
the V-QCD model Jarvinen:2011qe . In particular, these models can be fitted
Gursoy:2009jd ; Jokela:2018ers to agree precisely with lattice data for QCD
thermodynamics at finite temperature and low density. Therefore, they have the
potential to produce realistic predictions for various observables in this
region, which are relevant for heavy-ion collisions.

Indeed, our goal here is to carry out an example of such an analysis, using
V-QCD, focusing on anomalous transport in the deconfined quark-gluon plasma
phase.

The complete V-QCD setup Bigazzi:2005md ; Casero:2007ae ; Jarvinen:2011qe is
consistent with global anomalies of QCD, including the U$(1)$ axial anomaly
Arean:2013tja ; Arean:2016hcs . For this model, the fit to lattice data for
QCD thermodynamics has been done in Jokela:2018ers ; Ishii:2019gta , and the
model has been compared to QCD spectrum in Amorim:2021gat ; Jarvinen:2022gcc .
Thus, the remaining task is to carry out a fluctuation analysis in order to
compute the anomalous conductivities, which we will do in this article.

Our computation assumes zero quark masses so that the high-temperature quark-
gluon plasma phase is chirally symmetric. We then compare our results at low
density to the recent lattice analysis Brandt:2023wgf and also give
predictions at higher density. We turn on both vectorial (i.e., baryon number)
and axial chemical potentials. Note that due to the axial anomaly there is no
conserved charge corresponding to the axial chemical potential. Therefore this
chemical potential does not have a clear thermodynamical interpretation, but
should rather be seen as an additional coupling in the QCD action which runs
as a function of the energy scale. Nevertheless, it is possibly that parity-
violating bubbles form in heavy-ion collisions Kharzeev:1998kz ; STAR:2009wot
, so that an effective axial chemical potential is created locally.

The article is organized as follows. In Sec. 2, we give a brief review of the
V-QCD model. In Sec. 3, we show how to compute anomalous conductivities via
fluctuation analysis in V-QCD. In Sec. 4, we present our numerical results for
the chiral separation effect and compare to results from the lattice. We
conclude in Sec. 5. The various Appendices contain additional details on the
definition of the holographic model, its associated equations of motion and
the extraction of observables.

##  2 Holographic Model: V-QCD

The class of bottom-up models we choose to work with is V-QCD Jarvinen:2011qe
; Alho:2012mh , which is based on the combination of two frameworks: Improved
Holographic QCD (IHQCD) Gursoy:2007cb ; Gursoy:2007er , describing the gluonic
sector as well as a setup based on a Dirac-Born-Infeld (DBI) action, which
models the quark sector Bigazzi:2005md ; Casero:2007ae . The two sectors are
fully backreacted in the Veneziano limit where both the number of colors,
$N_{c}$ and the number of flavors, $N_{f}$ are taken to be large:

$\displaystyle N_{c}\rightarrow\infty\ ,\quad N_{f}$
$\displaystyle\rightarrow\infty\ ,\quad\mathrm{with}\ \ \frac{N_{f}}{N_{c}}=x\
\ \mathrm{fixed}\ .$ (1)

The “V” in the name of the models refers to the use of this limit when the
model is defined. That is, in principle, one would like to derive the model
from a concrete string theory background in this limit. However, since finding
such a derivation is extremely challenging, string theory will be used merely
as a guideline on how the Ansatz for the model action is chosen. Consequently,
we switch to a bottom-up approach, where we set $N_{c}=N_{f}=3$ and the
various sectors in the model are directly fitted to available QCD data (mostly
lattice data for the thermodynamics of QCD) so that only the model action is
inspired by string theory. We will now discuss the content of the model in
more detail. See the recent review Jarvinen:2021jbd for a more detailed
discussion.

###  2.1 The action of the model

The fields that we will be concerned with from the IHQCD sector are the
following:

  * •

The dilaton, $\lambda=e^{\Phi}$, which is dual to the scalar gluonic operator,
$\mathrm{Tr}\,G_{\mu\nu}G^{\mu\nu}$ and therefore sources the ’t Hooft
coupling in Yang-Mills theory.

  * •

The metric $g_{MN}$, which is dual to the energy-momentum tensor, $T_{\mu\nu}$
of QCD. In this case, the source is the metric of the field theory, which we
keep fixed as the Minkowski metric.

Our notation is such that capital Latin indices run over all five dimensions
of the bulk theory, whereas the Greek indices denote the Lorentz indices of
the boundary theory. We will also consider an axion field arising from this
sector that we will discuss separately below.

The gluonic part of our action is then based on Einstein-dilaton theory,

$S_{g}=M_{p}^{3}N_{c}^{2}\int
d^{5}x\sqrt{-g}\Bigg{(}R-\frac{4}{3}\frac{(\partial_{M}\lambda)^{2}}{\lambda^{2}}+V_{g}(\lambda)\Bigg{)}\
,$ (2)

where $M_{p}$ is the Planck mass. Notice that the non-trivial potential for
the dilaton, $V_{g}(\lambda)$, is introduced to break the conformal symmetry
and to mimic the nontrivial renormalization group flow of the coupling in QCD
Gursoy:2007cb ; Gursoy:2007er . We will discuss explicitly below our choice of
this function, as well as various other functions appearing in the actions for
the other sectors of the model.

Quarks are accounted for in the holographic setup by embedding $N_{f}$ space-
filling $D4$-branes and $N_{f}$ space-filling $\bar{D}4$-branes in the 5D
geometry Bigazzi:2005md ; Casero:2007ae . In this article, we restrict
ourselves to the (exactly) chirally symmetric solutions so that the quark mass
is also taken to be vanishing222Though we do not include it here, chiral
symmetry breaking or quark mass can be introduced in the V-QCD setup through a
tachyon field Bigazzi:2005md ; Casero:2007ae ; Jarvinen:2011qe .. Moreover,
for our purposes, it is enough to consider the Abelian sector of the flavor
fields, i.e., fields proportional to unit matrices in flavor space. In this
case, the low-lying fields on the branes are two gauge fields: $A_{L,\mu}$ on
the flavor branes and $A_{R,\mu}$ on the anti-flavor branes, which are dual to
the currents $\bar{\psi}\gamma_{\mu}(1\pm\gamma_{5})\psi$ in QCD (with a sum
over flavors included implicitly). For our purposes, it will be convenient to
rewrite the gauge fields in the basis of vector and axial vector fields:

$\displaystyle A_{L,M}$ $\displaystyle=V_{M}+A_{M}\ ,$ (3) $\displaystyle
A_{R,M}$ $\displaystyle=V_{M}-A_{M}\ .$ (4)

In this basis, the sources for (the temporal components of) the vectorial and
axial gauge fields are the vectorial and axial quark chemical potentials,
$\mu_{V}$ and $\mu_{A}$, respectively, as we will discuss in more detail
below.

The flavor action is the effective DBI term

$S_{f}=-\frac{xM_{p}^{3}N_{c}^{2}}{2}\int
d^{5}xV_{f}(\lambda)\Big{[}\sqrt{-\det\bm{A}_{+}}+\sqrt{-\det\bm{A}_{-}}\Big{]},$
(5)

where

$\bm{A}_{(\pm)MN}=g_{MN}+w(\lambda)(F^{V}_{MN}\pm F^{A}_{MN}),$ (6)

and

$\displaystyle F^{V}_{MN}$
$\displaystyle=\partial_{M}V_{N}-\partial_{N}V_{M},$ $\displaystyle
F^{A}_{MN}$ $\displaystyle=\partial_{M}A_{N}-\partial_{N}A_{M}.$ (7)

CP-odd effects are then modelled by the addition of the term which combines
contributions both from the gluonic and flavor sectors Casero:2007ae ;
Arean:2013tja ; Arean:2016hcs

$S_{a}=-\frac{M_{p}^{3}N_{c}^{2}}{2}\int
d^{5}x\sqrt{-g}Z(\lambda)(\partial_{M}\mathfrak{a}-2xA_{M})^{2}\ ,$ (8)

where the axion, $\mathfrak{a}$ is dual to the term
$\epsilon^{\mu\nu\rho\sigma}\mathrm{Tr}(G_{\mu\nu}G_{\rho\sigma})$, sourcing
the $\theta$-angle. Here $\epsilon^{\mu\nu\rho\sigma}$ is the Levi-Civita
symbol. This term is essential to our analysis because the mass term for
$A_{M}$ present in the above action is necessary to include gluonic
contributions to the model of anomalous transport Jimenez-Alba:2014iia ;
Gallegos:2018ozs .

The remaining piece, which turns out to be important for the analysis of the
anomalous conductivites, is the Chern-Simons term defined as

$S_{\text{CS}}=-\frac{M_{p}^{3}N_{c}^{2}x}{2}\int
d^{5}x\tilde{\epsilon}^{MNPQR}A_{M}(3\kappa F^{V}_{NP}F^{V}_{QR}+\gamma
F^{A}_{NP}F^{A}_{QR}),$ (9)

where $\tilde{\epsilon}^{MNPQR}$ is the Levi-Civita symbol and $\kappa,\gamma$
are constant coefficients. Actually, the standard expressions for the Chern-
Simons term in the literature Witten:1998qj ; Casero:2007ae have fixed values
for $\kappa$ and $\gamma$:

$\kappa=\gamma=\frac{1}{24\pi^{2}M_{p}^{3}}\ .$ (10)

This is because they are determined by the chiral flavor anomalies of QCD. We
however choose to keep the coefficients unfixed for the moment for generality.
The full Chern-Simons term, including effects from chiral symmetry breaking,
has been analyzed recently in Jarvinen:2022mys .

Putting everything together, the holographic action to be studied is333In
principle one should also add the Gibbons-Hawking and renormalization
counterterms as this action is UV divergent, but these will not play any role
in the analysis of this article.

$S=S_{g}+S_{f}+S_{a}+S_{\text{CS}}.$ (11)

As the full equations of motion are quite cumbersome and provide no physical
insight before an Ansatz (to be presented in Sec. 2.3) is assumed, they are
listed in the App. B.

###  2.2 Choice of the potentials

In order for the model to mimic the physics of QCD precisely, the potentials
$V_{g}(\lambda)$, $V_{f}(\lambda)$, $w(\lambda)$, and $Z(\lambda)$ need to be
chosen carefully. The asymptotics of the three first potentials ($V_{g}$,
$V_{f}$, and $w$) both at weak and strong coupling are constrained by
requiring that the model has fully regular solutions with confinement,
discrete spectrum, correct mass gaps at finite quark mass, reasonable phase
diagram with confined phase extending to finite chemical potential,
asymptotically AdS solutions, and asymptotic freedom Gursoy:2007cb ;
Gursoy:2007er ; Jarvinen:2011qe ; Arean:2012mq ; Arean:2013tja ;
Jarvinen:2015ofa ; Ishii:2019gta .

The details at intermediate coupling can then be fitted to either lattice data
for QCD thermodynamics Gursoy:2009jd ; Panero:2009tv ; Jokela:2018ers ,
experimental meson masses Amorim:2021gat or to both Jarvinen:2022gcc . Since
we are focusing on properties of the model at finite temperature in this
article, we choose potentials which have been carefully fitted to lattice data
for both Yang-Mills theory and full QCD in Jokela:2018ers ; Ishii:2019gta .
This procedure produces a family of models, which depends essentially on a
single parameter, which cannot be determined by fitting. From this family, we
use here the set 7a, which can be seen to be a conservative intermediate
choice. This set has been shown to also produce predictions at high density
that are in good agreement with neutron star observations Ecker:2019xrw ;
Jokela:2020piw ; Jokela:2021vwy ; Demircik:2021zll ; Tootle:2022pvd . See App.
A for explicit expressions for the potentials.

The potential $Z(\lambda)$, which appears in the $S_{a}$ term of the
holographic action controls the physics of the axial anomaly – it is also
constrained by QCD data. Since this potential readily appears in the model for
pure Yang-Mills ($x=0$), it is natural to fit it to the lattice data for
related observables, including the topological susceptibility and the
pseudoscalar glueball masses Gursoy:2009jd . This function also controls the
Chern-Simons diffusion rate Gursoy:2012bt ; Bigazzi:2018ulg , which measures
the rate of change in the topological Chern-Simons number in Yang-Mills
theory.

We expect that the results for the anomalous conductivities in this article
are more sensitive to the choice of $Z(\lambda)$ than the other potentials,
which are quite strictly constrained by the fit to lattice data for the
thermodynamics of QCD. Therefore, we analyze the effect of the choice of
$Z(\lambda)$ on the results of the conductivities, keeping the other
potentials fixed. We consider three choices:

  1. 1.

$Z(\lambda)$ with constant UV asymptotics, i.e.,
$Z(\lambda)\sim\mathrm{const.}$ as $\lambda\to 0$.

  2. 2.

$Z(\lambda)$ with linear UV asymptotics, i.e., $Z(\lambda)\sim\lambda$ as
$\lambda\to 0$.

  3. 3.

$Z(\lambda)$ with quadratic UV asymptotics, i.e., $Z(\lambda)\sim\lambda^{2}$
as $\lambda\to 0$.

In all these scenarios, we choose the IR asymptotics
$Z(\lambda)\sim\lambda^{4}$ as $\lambda\to\infty$. With this choice, the
spectra of the scalar and pseudoscalar glueballs have similar “universal”
asymptotics at high excitation numbers Gursoy:2007er .

These three different choices for $Z(\lambda)$ near the boundary are motivated
as follows. The constant choice was suggested in Gursoy:2007er ; Gursoy:2012bt
as the most general allowed term in analogy with effective theory. However, it
turns out that this choice leads to a peculiar finite anomalous dimension for
the axial chemical potential even in the free field theory limit $\lambda\to
0$ Arean:2016hcs ; Weitz . Moreover the Chern-Simons diffusion rate at high
temperatures is higher with this choice Gursoy:2012bt as compared to the
results from perturbative analysis Arnold:1996dy ; Bodeker:1998hm ;
Bodeker:1999gx ; Moore:2010jd . The linear choice for the asymptotics of
$Z(\lambda)$ leads instead to an anomalous dimension which vanishes linearly
in the coupling in the weak coupling limit. The UV running of the axial
chemical potential is therefore similar to what one obtains for the quark mass
in QCD. Finally, quadratic asymptotics for $Z(\lambda)$ matches with results
from string perturbation theory Gursoy:2007cb and also agrees with
computations in top-down models Son:2002sd ; Craps:2012hd ; Bigazzi:2018ulg ,
even though this comparison is not necessarily applicable at weak coupling.
This choice is also interesting because, unlike the other choices, the running
of the axial chemical potential is suppressed in the weak coupling limit (see
Appendix C) which allows us to define it unambiguously.

We emphasize that matching the weak coupling behavior of the function
$Z(\lambda)$ with QCD results only gives indicative results at best because
the holographic duality (with classical gravity approximation) is not expected
to work at weak coupling. However, since the behavior of $Z(\lambda)$ at
strong coupling is quite strictly fixed by the universality of the glueball
trajectories, the weak coupling behavior is the main source of uncertainty for
our results. Therefore it makes sense to explore different choices. These
choices also drastically change the behavior of the background near the
boundary. Despite this, as we will discuss below, the effects on the results
will be somewhat limited. We fit all functions $Z(\lambda)$ that we use to
lattice data Lucini:2005vg ; Athenodorou:2021qvs ; Bennett:2022gdz for the
topological susceptibility and glueball masses. See App. A for details.

###  2.3 Background solutions

For our analysis, we will solve the gravity backgrounds (charged planar black
holes) for different values of the temperature, quark number chemical
potential, and axial chemical potential. Recall that we are working at exactly
zero quark mass and that we focus on the high-temperature deconfined phase of
QCD so that chiral symmetry is intact. For the metric, we write

$ds^{2}=g_{MN}dx^{M}dx^{N}=\frac{q(a)^{2}}{f(a)}da^{2}+e^{2a}\left(-f(a)dt^{2}+d\mathbf{x}^{2}\right)\
,$ (12)

which is a convenient Ansatz for numerical solutions in this model Alho:2012mh
; Alho:2013hsa . Notice that the holographic coordinate $a$ maps to the
logarithm of the energy scale on the field theory side as one can see from the
warp factor of the spacetime coordinates in Eq. (12). Apart from the metric
and the dilaton $\lambda(a)$, we will turn on the temporal components of both
the vectorial and axial gauge fields, which we denote simply as $V(a)$ and
$A(a)$, respectively. We choose a gauge where the bulk axion as well as the
radial components of the gauge fields ($V_{a}$ and $A_{a}$) vanish444Our
Ansatz for the axial gauge field, including its fluctuations, will be such
that $\partial_{M}A^{M}=0$. This removes the coupling to the axion, which can
be then consistently set to zero.. Therefore the Ansatz is

$V_{M}(x^{N})=\delta_{M}^{t}V(a)\ ,\qquad A_{M}(x^{N})=\delta_{M}^{t}A(a)\ .$
(13)

Note that for much of the rest of this article, we will suppress the
$a$-dependence of $V(a),\,A(a)$.

To summarize, the equations of motion are then given by the Einstein
equations, scalar equation for $\lambda$ and the gauge field equations, which
are given for this setup in Appendix B.

The background solutions can be parameterized in terms of three parameters.
For the numerical solutions, it is convenient to first give these parameters
at the horizon of the black hole. They can be taken to be the horizon value
$\lambda_{h}$ of the dilaton and the derivatives of the gauge fields at the
horizon555Instead of the horizon derivatives, we actually define “charges” at
the horizon that can be mapped to the values of the derivatives. See App. B..
After solving the background numerically, these parameters can be mapped to
the physical parameters, i.e., the temperature and the chemical potentials.

The source of the dilaton field defines an additional parameter, $\Lambda$.
This is the UV scale of the model, which is related to the ’t Hooft coupling
through dimensional transmutation, just as in QCD. See App. B for the precise
holographic definition of $\Lambda$. The overall energy scale is however a
symmetry of the solutions, so that the final three parameters characterizing
the background are the dimensionless ratios $T/\Lambda$, $\mu_{V}/\Lambda$,
and $\mu_{A}/\Lambda$. The physical value of the UV scale, $\Lambda\approx
211$ MeV, is obtained by comparing to lattice data Jokela:2018ers in the same
way as the potential parameters discussed in App. A.

##  3 Anomalous conductivities

The anomalous conductivities for a vectorial current are defined through the
linear response of the vectorial (i.e., quark number) current,

$\delta\langle\mathcal{J}_{V}^{k}\rangle=\sigma_{VV}B^{k}+\sigma_{VA}B_{A}^{k}+\sigma_{V\Omega}\omega^{k}\
,$ (14)

where $k$ indicates spatial indices, $B$ is the (vectorial) magnetic field,
$B_{A}$ is the axial magnetic field and $\omega$ is the vorticity. The
magnetic fields and the vorticity are assumed to be small perturbations
whereas the conductivities $\sigma_{VV}$, $\sigma_{VA}$, and
$\sigma_{V\Omega}$ are associated with the chiral magnetic effect (CME),
chiral separation effect (CSE) and chiral vortical effect (CVE) respectively.

We apply here the definition Eq. (14) for the consistent (rather than
covariant) vectorial current. This is convenient in our model because the
consistent current is divergence free, whereas the covariant current is UV
divergent and needs to be renormalized. In this case, the CME and the CVE are
determined by the anomaly structure Son:2009tf ; Neiman:2010zi ;
Landsteiner:2011cp ; Gursoy:2014boa ; Gursoy:2014ela ; Grozdanov:2016ala but
the CSE does receive radiative corrections666When interpreting these results,
note that most of the literature reports conductivities defined by the
covariant current. Jimenez-Alba:2014iia ; Gallegos:2018ozs . We will therefore
focus here on the CSE, analyzing such effects.

We make use of the holographic dictionary to derive the one-point
function777We do not consider the one point function $\mathcal{J}_{A}$ here as
its computation requires us to solve the equations of motion in full.
Moreover, we only need to know $\mathcal{J}_{V}$ in order to compute the CME,
CSE and CVE.

$\displaystyle\langle\mathcal{J}_{V}^{\mu}\rangle$
$\displaystyle=\frac{1}{V_{4}}\frac{\delta S}{\delta
V_{\mu}^{\mathrm{bdry}}}=\lim\limits_{a\rightarrow\infty}\frac{\partial\mathcal{L}}{\partial\left(\partial_{a}V_{\mu}\right)}=$
(15)
$\displaystyle=\lim\limits_{a\rightarrow\infty}\bigg{\\{}\frac{M_{p}^{3}N_{c}^{2}xV_{f}(\lambda)w(\lambda)}{4}\sum_{s=+,-}\sqrt{-\det\bm{A}_{s}}\Big{[}\left(\bm{A}_{s}^{-1}\right)^{\mu
a}-\left(\bm{A}_{s}^{-1}\right)^{a\mu}\Big{]}+$ $\displaystyle\ \ \ \
+12M_{p}^{3}N_{c}^{2}x\kappa{\epsilon}^{\sigma\nu\rho\mu}A_{\sigma}\partial_{\nu}V_{\rho}\bigg{\\}}\
,$ (16)

where $V_{4}$ is the volume of the space-time, $\bm{A}_{\pm}$ were given in
Eq. (6), the index $a$ denotes the holographic direction and
$V_{\mu}^{\mathrm{bdry}}=\lim_{a\to\infty}V_{\mu}(a)$. Notice that this is the
generic expression which holds for any background assuming that the current is
independent of space-time coordinates. That is, we did not insert the Ansatz
for the background discussed above, but we did assume the standard radial
gauge for the bulk gauge fields. We identify this one-point function as the
consistent vectorial current. Observing the response of this current to a
perturbation will enable us to calculate the anomalous coefficients in Eq.
(14).

Adding small external magnetic fields and vorticity amounts to adding
infinitesimal sources to the field strength tensors of the gauge fields and
the shear components of the metric. Constant magnetic fields are obtained by
turning on perturbations of the gauge fields that are linear in one of the
spatial coordinates, which we choose to be the $z$-coordinate. That is, we
consider the following perturbations888To have a fully consistent system of
fluctuations, one also needs to add terms that are independent of $z$. These
terms however will not contribute to the conductivities.

$\displaystyle\delta V_{k}(x^{M})$ $\displaystyle=z\delta\hat{V}_{k}(a)\
,\qquad\delta A_{k}(x^{M})=z\delta\hat{A}_{k}(a)\ ,$ (17) $\displaystyle\delta
g_{tk}(x^{M})$ $\displaystyle=e^{2a}h_{tk}(x^{M})=e^{2a}z\hat{h}_{k}(a)$ (18)

on top of the background defined in subsection 2.3. Here, only the indices
$k=x,y$ are relevant and we again suppress the explicit a-dependence in much
of what follows.

This perturbation gives rise to the following vorticity and (vectorial)
magnetic field:

$\displaystyle\omega^{i}$
$\displaystyle=\lim_{a\to\infty}\epsilon^{ijk}\partial_{j}h_{tk}=-\epsilon^{zij}\lim_{a\to\infty}\hat{h}_{j}(a)\
,$ (19) $\displaystyle B^{i}-\mu_{V}\omega^{i}$
$\displaystyle=\frac{1}{2}\lim_{a\to\infty}\epsilon^{ijk}\delta
F^{V}_{jk}=-\epsilon^{zij}\lim_{a\to\infty}\delta\hat{V}_{j}(a)\ .$ (20)

$B^{i}$ is the magnetic field in the rest frame of the fluid, corrected by the
field $\mu_{V}\omega^{i}$ arising from the infinitesimal motion of the fluid.
The definition of the axial magnetic field is otherwise similar to the
vectorial field albeit with the additional complication that the field runs
close to the boundary, which needs to be taken into account. See App. C for
details.

The fluctuation equations for the fields $\delta\hat{V}_{k}$,
$\delta\hat{A}_{k}$, and $\hat{h}_{k}$ with $k=x$ and $k=y$ are identical. In
addition, the equations for different values of the index are decoupled.
Therefore we henceforth omit this index and write

$\displaystyle xe^{2a}qZ(\lambda)\delta\hat{A}$
$\displaystyle=\frac{d}{da}\left[\frac{fe^{2a}V_{f}(\lambda)w(\lambda)^{2}}{4qR_{+}}\left(\delta\hat{V}^{\prime}+\delta\hat{A}^{\prime}\right)\right]+\frac{\left(V^{\prime}+A^{\prime}\right)e^{2a}V_{f}(\lambda)w(\lambda)^{2}}{4qR_{+}}\hat{h}^{\prime}$
(21a) $\displaystyle-xe^{2a}qZ(\lambda)\delta\hat{A}$
$\displaystyle=\frac{d}{da}\left[\frac{fe^{2a}V_{f}(\lambda)w(\lambda)^{2}}{4qR_{-}}\left(\delta\hat{V}^{\prime}-\delta\hat{A}^{\prime}\right)\right]+\frac{\left(V^{\prime}-A^{\prime}\right)e^{2a}V_{f}(\lambda)w(\lambda)^{2}}{4qR_{-}}\hat{h}^{\prime}$
(21b) $\displaystyle C_{f}$
$\displaystyle=\frac{\left(V^{\prime}+A^{\prime}\right)xe^{2a}V_{f}(\lambda)w(\lambda)^{2}}{2qR_{+}}\left(\delta\hat{V}+\delta\hat{A}\right)+$
$\displaystyle\ \ \
+\frac{\left(V^{\prime}-A^{\prime}\right)xe^{2a}V_{f}(\lambda)w(\lambda)^{2}}{2qR_{-}}\left(\delta\hat{V}-\delta\hat{A}\right)+\frac{e^{4a}}{q}\hat{h}^{\prime}$
(21c)

where $C_{f}$ is a constant of integration and

$R_{\pm}=\sqrt{1-\frac{e^{-2a}w(\lambda)^{2}\left(A^{\prime}\pm
V^{\prime}\right)^{2}}{q^{2}}}\ .$ (22)

A general solution to this system has five other constants of integration,
which can be associated with the boundary conditions at the horizon. Horizon
regularity however fixes two of these constants, which can be thought of as
derivatives of the gauge fields at the horizon. An additional special
condition for time-independent fluctuations is that the fluctuation of the
metric, $\hat{h}$, must vanish at the horizon for the fluctuated scalar
curvature $R$ to remain finite Gursoy:2014boa , so that the fluctuated action
is normalizable.

The remaining three constants then map to the three observables, i.e., the
conductivities from Eq. (14).

To compute the conductivities we therefore need to find three regular
solutions to the fluctuations. Two such solutions can however be found
immediately. For the first solution, $\delta\hat{V}$ is constant:

$\delta\hat{V}=|B|\ ,\qquad\delta\hat{A}=0\ ,\qquad\hat{h}=0\ .$ (23)

To check this solution, note that the coefficient of $\delta\hat{V}$ in Eq.
(21c) is also a constant: it is denoted as $c_{V}$ in Eq. (B.43) in Appendix
B. Therefore, Eq. (23) solves the fluctuation equations for a specific choice
of $C_{f}$. The second solution is

$\delta\hat{V}=-|\omega|V\ ,\qquad\delta\hat{A}=-|\omega|A\
,\qquad\hat{h}=|\omega|f\ .$ (24)

This solution corresponds to an infinitesimal boost of the background. It can
be checked by inserting it in the fluctuation equations and by using the
background equations of motion Eqs. (B.41c), (B.41d) and (B.44). The remaining
solution (for which $\delta\hat{A}$ is non-vanishing at the horizon) can then
be constructed numerically.

Let us then compute the response of the current Eq. (16) to the small
perturbations. First we note that, because the action only depends on
$V_{\mu}$ through the field strength tensor, we can immediately use the
equation of motion for $V_{\mu}$ to write (assuming again radial gauge)

$\delta\langle\mathcal{J}_{V}^{\mu}\rangle=\delta\int_{a_{h}}^{\infty}da\
\partial_{\nu}\frac{\partial\mathcal{L}}{\partial\left(\partial_{\nu}V_{\mu}\right)}\
,$ (25)

where $a_{h}$ denotes the location of the horizon and we assumed that the
perturbation of the IR term (the charge of the black hole) vanishes. For the
perturbation we consider here, only the Chern-Simons action contributes,
leaving us with

$\displaystyle\delta\langle\mathcal{J}_{V}^{i}\rangle$
$\displaystyle=12M_{p}^{3}N_{c}^{2}x\kappa\epsilon^{zij}\int_{a_{h}}^{\infty}da\left(A\
\partial_{z}\partial_{a}\delta V_{j}-\partial_{a}V\ \partial_{z}\delta
A_{j}\right)$ (26)
$\displaystyle=\frac{N_{c}N_{f}}{2\pi^{2}}\,\epsilon^{zij}\int_{a_{h}}^{\infty}da\left(A\
\delta\hat{V}_{j}^{\prime}-V^{\prime}\ \delta\hat{A}_{j}\right)\ ,$ (27)

where we inserted the value of $\kappa$ from Eq. (10).

Notice that the current vanishes for both the solutions Eqs. (23) and (24).
These solutions turn on nonzero vectorial magnetic field and vorticity
respectively. Consequently, the chiral magnetic and vortical effects vanish,

$\sigma_{VV}=0=\sigma_{V\Omega}\ .$ (28)

The third conductivity $\sigma_{VA}$ can be computed numerically. We can
however analyze some of its properties by studying the discrete symmetries of
the action. The fluctuations and equations of motion are unchanged under two
discrete symmetries: the simultaneous change of sign of the vectorial (axial)
gauge field and its fluctuation. The current in Eq. (27) is odd under both
symmetries, whereas the axial magnetic field is even under the vectorial
symmetry but odd under the axial symmetry. Therefore $\sigma_{VA}$ is odd
(even) under the vectorial (axial) symmetry. Moreover, the vectorial (axial)
symmetry changes the sign of the vectorial (axial) chemical potential.
Therefore $\sigma_{VA}$ is an odd function of $\mu_{V}$ and even function of
$\mu_{A}$. In particular, $\sigma_{VA}$ vanishes for $\mu_{V}=0$.

Finally, let us comment on this result in the limit of large temperature. In
this weak coupling limit: the black hole becomes large and the dilaton
$\lambda\sim 1/a$ Gursoy:2007cb ; Gursoy:2007er is small for all values of the
holographic coordinate $a$. If $Z(\lambda)$ vanishes sufficiently fast as
$\lambda\to 0$, it can be set to zero in Eqs. (21a) and (21b). Then the third
solution of the fluctuation equations also becomes simple:

$\delta\hat{V}\approx 0\ ,\qquad\delta\hat{A}\approx|B_{A}|\
,\qquad\hat{h}\approx 0\ .$ (29)

As one can check, this only holds for choice 3 of the function $Z(\lambda)$
with quadratic UV asymptotics; the analysis of Appendix C excludes such a
solution for choices 1 and 2.

For the choice 3 we find

$\delta\langle\mathcal{J}_{V}^{i}\rangle\approx-\frac{N_{c}N_{f}}{2\pi^{2}}\,\epsilon^{zij}\delta\hat{A}_{j}\int_{a_{h}}^{\infty}da\,V^{\prime}=\frac{N_{c}N_{f}}{2\pi^{2}}\mu_{V}B_{A}^{i}\
,$ (30)

so that we recover the universal result Son:2009tf ; Neiman:2010zi ;
Landsteiner:2011cp ; Gursoy:2014boa ; Gursoy:2014ela ; Grozdanov:2016ala

$\sigma^{U}_{VV}=0\ ,\qquad\sigma^{U}_{VA}=\frac{N_{c}N_{f}}{2\pi^{2}}\mu_{V}\
,\qquad\sigma^{U}_{V\Omega}=0\ ,\qquad(T\to\infty)\ .$ (31)

This universal result is also found in the free theory Metlitski:2005pr ;
Brandt:2023wgf so it is indeed the expected high-temperature behavior in QCD,
which becomes weakly coupled as the temperature grows.

##  4 CSE results

Upon solving the fluctuation equations Eq. (21) on top of the background
equations Eq. (B.41), we compute the fluctuation of the vector current, Eq.
(27) from which we can obtain the various anomalous conductivities. While the
conductivity associated with the CME and CVE are zero and therefore do not
deviate from their universal values (see Eqs. (28) and (31)), the CSE does
indeed receive radiative corrections, which we now proceed to discuss.

Throughout our discussion, we generally refrain from commenting on the
absolute size of the radiative corrections that we find, due to the arbitrary
normalization associated with the axial magnetic field, $B_{A}$ for choices
$1$ and $2$ of the $Z(\lambda)$ potential (see Appendix C). For choice $3$,
this arbitrariness is however absent, allowing us to take seriously the
corrections’ size.

The CSE has recently been computed on the lattice Brandt:2023wgf by analyzing
the effect of the magnetic field on the axial current. Said work computes the
CSE as a function of temperature, finding it to be suppressed with respect to
the universal result in the confining regime. A steep rise in the conductivity
around the transition temperature then leads to a marginal enhancement with
respect to the universal result at higher temperatures ($T\gtrsim
400~{}\text{MeV}$).

Figure 1: The ratio $\sigma^{R}_{VA}\equiv\frac{\sigma_{VA}}{\sigma^{U}_{VA}}$
plotted as a function of temperature for different choices of $Z(\lambda)$
potential with $\mu_{V}=1\,\text{keV}$ and zero $\mu_{A}$. Figure 2: The ratio
$\sigma^{R}_{VA}\equiv\frac{\sigma_{VA}}{\sigma^{U}_{VA}}$ plotted as a
function of vector chemical potential for choice 3 of the $Z(\lambda)$
potential. Solid lines correspond to $T=250\,\text{MeV}$, dashed lines to
$T=200\,\text{MeV}$ and dotted lines to $T=150\,\text{MeV}$. The definition of
$\mu_{A}$ varies according to the choice of $Z(\lambda)$ – see App. C.

In Fig. 1, we show our results for the CSE as a function of temperature, for
small vector chemical potential and zero axial chemical potential, so as to
emulate the lattice setup. We observe similar behaviour to that found in
Brandt:2023wgf , at least qualitatively, for all three choices of the
$Z(\lambda)$ potential; a noticeable change in slope is recovered around the
critical temperature, giving rise to a knee-like feature. The results for none
of the choices of potentials seem to agree with the universal or free theory
results at large temperatures. This is not surprising for choices $1$ and $2$
since, as we remarked above, for these choices, the conductivity does not
approach the free theory value in the high-temperature limit, since its
normalization is arbitrary. For choice $3$ however the free theory value will
be eventually obtained as $T\to\infty$, albeit with a very gradual
convergence.

It is perhaps unsurprising that the high-temperature result in V-QCD does not
automatically agree with the free field limit given that V-QCD is by
definition a strongly coupled model, fitted to reproduce low-temperature
features of QCD. Indeed, in general, the gauge/gravity duality is not expected
to work at weak coupling. It would nevertheless be interesting to see how the
addition of quark masses via the tachyon field in our setup would impact the
behaviour of the conductivity around the critical temperature. We will come
back to discuss this in the Conclusion.

In Figs. 2 and 3, we show our results for the CSE computed as a function of
the vectorial chemical potential, for fixed values of the axial chemical
potential. The former shows that for choice $3$ of the $Z(\lambda)$ potential,
the CSE is suppressed with respect to the universal result across a range of
vectorial and axial chemical potentials in the strong-coupling regime. The
ratio to the universal conductivity is an increasing function of all variables
$T$, $\mu_{V}$, and $\mu_{A}$, but the dependence on each of the variables is
somewhat different. The conductivity curves widen non-linearly as a function
of the axial chemical potential at fixed temperature – this is particularly
noticeable for the $T=150~{}\text{MeV}$ curves at low vector chemical
potential. Overall, the corrections clearly exhibit a stronger dependence on
the temperature, which nevertheless begins to disappear for larger values of
$\mu_{V}$, as the various curves appear to asymptotically converge.

Moving on to Fig. 3, we see in general the same behaviour for the CSE for
choices $1$ and $2$ of the $Z(\lambda)$ potential. Even though we cannot
deduce a suppression or enhancement with respect to the universal result, we
note an apparent weakening of the temperature (and axial chemical potential)
dependence for large values of the vector chemical potential. One feature
worth commenting on for choice $1$ of the $Z(\lambda)$ potential is the rather
strong dependence on the axial chemical potential; the nonlinear dependence on
$\mu_{A}$ overtakes the temperature dependence in this case. The large
dissimilarity between choices $1$ and $2$ is apparently due to differences in
the normalization of $\mu_{A}$ between the potentials999A direct way to
estimate the effect of the chemical potential on the background is to evaluate
the charges $n_{A}$ defined in Appendix B. We find that for fixed $\mu_{A}$
the charges for choices $1$ and $3$ are roughly of the same size, while the
charge for the choice $2$ is significantly smaller. This is in rough agreement
with the observed dependence on $\mu_{A}$ in Figs. 2 and 3.. Recall that there
is no obvious way to fix these normalizations (see Appendix C).

Figure 3: The ratio $\sigma^{R}_{VA}\equiv\frac{\sigma_{VA}}{\sigma^{U}_{VA}}$
plotted as a function of vector chemical potential. From top to bottom,
figures are for choices 1 and 2 of the $Z(\lambda)$ potential. Solid lines
correspond to $T=250\,\text{MeV}$, dashed lines to $T=200\,\text{MeV}$ and
dotted lines to $T=150\,\text{MeV}$. The definition of $\mu_{A}$ varies
according to the choice of $Z(\lambda)$ – see App. C.

Notice also that the coefficients $c_{i}$ defined in Appendix A.1 are much
larger for the choice 1 of the $Z(\lambda)$ potential – we remind the reader
that these coefficients are fixed in order to obtain a good agreement with
lattice values of the pseudoscalar glueball spectrum.

##  5 Conclusions

In this article, we have discussed anomalous conductivities in a bottom-up
holographic model, V-QCD. This model has been determined through a careful
comparison to QCD data, in particular lattice data for the thermodynamics of
QCD at nonzero temperatures and small densities.

We considered a simple method, where we turned on constant (vectorial and
axial) magnetic fields as well as vorticity and read off the anomalous
conductivities for the response of the (consistent) vectorial current. Our
results agree with the more general derivations from the literature: in
particular, the CME and CVE vanish in our setup (in the conventions which use
the consistent vectorial current to define the anomalous conductivities).

Therefore, the only nonzero conductivity is that which is related to the CSE.
The value of this conductivity is not fixed by the anomalies but receives
radiative corrections in QCD. We computed these corrections using different
variants of the holographic V-QCD model, and found results in the limit of
small density (see Fig. 1), which qualitatively agree with the recent findings
from lattice QCD simulations Brandt:2023wgf . We gave predictions for the
dependence on vectorial chemical potential in Figs. 2 and 3.

Our analysis could be extended in various ways. The most obvious extension
would be to analyze the axial current in addition to the vectorial current of
Eq. (16). This requires carrying out holographic renormalization, and would
give information about two additional conductivities ($\sigma_{AA}$ and
$\sigma_{A\Omega}$ in notation analogous to Eq. (14)). Another possible avenue
would be to check whether our analysis and results can be connected to the new
method of analyzing holographic transport suggested in Demircik:2023lsn ;
Demircik:2024bxd . It would also be interesting to study dynamical setups,
which mimic more closely the evolution of the plasma after heavy-ion
collisions, for example following recent studies on somewhat simpler
Stückelberg models Grieninger:2023myf ; Grieninger:2023wuq .

Note also that we did not include quark masses – all quark flavors were
assumed be massless and identical. While the light quark masses are expected
to be small perturbations to the current computation, the strange quark mass
might have a significant effect on the results. Checking this properly
requires adding the strange quark in the V-QCD model or in some simpler
holographic model fitted to lattice data. In V-QCD, this extension requires
adding an additional scalar field, the tachyon, in the DBI action Eq. (5) (see
Jarvinen:2015ofa ; CruzRojas:2024etx ). Moreover, as we did not include
electric charges for the quarks, our magnetic fields couple to all quarks in
the same way. Be that as it may, since we are only studying linear
conductivities in these fields, adding charges will not change the results
qualitatively. Related to this, we remark that the lattice study
Brandt:2023wgf did analyze flavor dependence of the CSE and found
qualitatively similar temperature dependence in basically all flavor channels.

It may nevertheless be interesting to try to fit the flavor dependence of the
lattice data more precisely. We expect that doing this accurately requires the
model extension which contains the strange quark mass. It might also be
possible to find a function $Z(\lambda)$ which agrees with the fast approach
to the universal or free quark result with increasing temperature observed on
the lattice. As we argued, this requires that the function vanishes fast
enough as weak coupling, as it controls the renormalization group flow of the
axial chemical potential and magnetic field, which should be suppressed for
the universal result to hold. We speculate that choosing asymptotics that
vanish even faster than the quadratic asymptotic which we studied here might
suppress the flow further and lead to faster convergence towards the universal
result.

###### Acknowledgements.

We thank J. Ghiglieri, S. Grieninger, U. Gürsoy, N. Jokela, N. Poovuttikul for
discussions. M. J. has been supported by an appointment to the JRG Program at
the APCTP through the Science and Technology Promotion Fund and Lottery Fund
of the Korean Government. M. J. has also been supported by the Korean Local
Governments – Gyeongsangbuk-do Province and Pohang City – and by the National
Research Foundation of Korea (NRF) funded by the Korean government (MSIT)
(grant number 2021R1A2C1010834).

##  Appendix A Potential definitions

We pack into this appendix the definitions of the various V-QCD potentials
$V_{g}(\lambda),V_{f}(\lambda)$ and $w(\lambda)$ as well as the potential
$Z(\lambda)$, all of which enter into our holographic model, described in Sec.
2.

We start with the gluonic potential

$V_{g}(\lambda)=12\Bigg{[}1+V_{1}\lambda+\frac{V_{2}\lambda^{2}}{1+\lambda/\lambda_{0}}+V_{\text{IR}}e^{-\lambda_{0}/\lambda}(\lambda/\lambda_{0})^{4/3}\sqrt{\log(1+\lambda/\lambda_{0})}\Bigg{]}.$
(A.32)

The coefficients $V_{1}$ and $V_{2}$ are fixed by requiring the UV RG flow of
the ’t Hooft coupling to be the same as in QCD up to two-loop order. The IR
coefficients are then chosen to reproduce qualitative features of QCD such as
confinement, magnetic charge screening and linear glueball trajectories
through a comparison with lattice data Panero:2009tv . In addition, the
potentials for the flavor sector are

$\displaystyle V_{f}(\lambda)$
$\displaystyle=W_{0}+W_{1}\lambda+\frac{W_{2}\lambda^{2}}{1+\lambda/\tilde{\lambda}_{0}}+12W_{\text{IR}}e^{-\tilde{\lambda}_{0}/\lambda}(\lambda/\tilde{\lambda}_{0})^{2},$
(A.33) $\displaystyle\frac{1}{w(\lambda)}$
$\displaystyle=w_{0}\Bigg{[}1+\bar{w}_{0}e^{\hat{\lambda}_{0}/\lambda}\frac{(\lambda/\hat{\lambda}_{0})^{4/3}}{\log(1+\lambda/\hat{\lambda}_{0})}\Bigg{]}.$
(A.34)

The UV coefficients of $V_{f}$ are determined by comparison with the flavor
dependence of the QCD beta function, while the powers appearing in the IR
dominant terms are chosen to reproduce some of the prominent features of the
QCD phase diagram, such as chiral symmetry breaking and linear meson
trajectories. The coefficient $W_{0}$ cannot however be determined by
comparing to the beta function and remains as a free parameter.

Matching with perturbative QCD gives for the UV coefficients Gursoy:2007cb ;
Jarvinen:2011qe

$\displaystyle V_{1}$ $\displaystyle=\frac{11}{27\pi^{2}}$ , $\displaystyle
V_{2}=\frac{4619}{46656\pi^{4}}$ , $\displaystyle W_{1}$
$\displaystyle=\frac{8+3W_{0}}{9\pi^{2}}$ , $\displaystyle W_{2}$
$\displaystyle=\frac{6488+999W_{0}}{15552\pi^{2}}\ .$ (A.35)

Furthermore, the IR coefficients and $w_{0}$ are fitted to lattice data
Gursoy:2009jd ; Jokela:2018ers ; Ishii:2019gta . We choose the fit 7a in these
references, which means the choice

$\displaystyle V_{\text{IR}}$ $\displaystyle=2.05$ ,
$\displaystyle\lambda_{0}$ $\displaystyle=\frac{8\pi^{2}}{3}$ , $\displaystyle
W_{0}$ $\displaystyle=2.5$ , $\displaystyle W_{\text{IR}}$ $\displaystyle=0.9$
, $\displaystyle\hat{\lambda}_{0}$ $\displaystyle=\frac{8\pi^{2}}{1.18}$ ,
$\displaystyle w_{0}$ $\displaystyle=1.28$ , $\displaystyle\bar{w}_{0}$
$\displaystyle=18\ ,$ (A.36)

for the remaining parameters. Moreover, the lattice fit also fixes

$\Lambda\approx 210.8\ \mathrm{MeV}\ ,\qquad M_{p}\approx 0.1972\ .$ (A.37)

###  A.1 $Z(\lambda)$ Analysis

As has been highlighted in the main text, the inclusion of a mass term for the
$A_{M}$ field (see Eq. (8)) allows for the study of radiative corrections to
the anomalous conductivities, which we present in Sec. 4. We devote the rest
of this appendix to a discussion of the $Z(\lambda)$ potential, considering
various choices and detailing our methods for fixing its coefficients. To
start, let us consider the general Ansatz

$Z(\lambda)=Z_{0}\bigg{(}c_{0}+c_{1}\Big{(}\frac{\lambda}{\tilde{\lambda}_{0}}\Big{)}+c_{2}\Big{(}\frac{\lambda}{\tilde{\lambda}_{0}}\Big{)}^{2}+c_{4}\Big{(}\frac{\lambda}{\tilde{\lambda}_{0}}\Big{)}^{4}\bigg{)},$
(A.38)

where $\tilde{\lambda}_{0}=8\pi^{2}$. Referring to the discussion in Sec. 2.2:

  * •

Choice 1 leads to $c_{0}=1$ and $c_{2}=0$.

  * •

Choice 2 leads to $c_{0}=c_{2}=0$ and $c_{1}=1$.

  * •

Choice 3 leads to $c_{0}=c_{1}=0$ and $c_{2}=1$.

We then follow the procedure described in Gursoy:2007er ; Gursoy:2012bt ,
tweaking the remaining unfixed $c_{i}$ for each choice in such a way that the
pseudo-scalar, lattice-calculated glueball spectra Athenodorou:2021qvs ;
Bennett:2022gdz are best reproduced.

After all of the $c_{i}$ are fixed, the coefficient $Z_{0}$ can then obtained
Gursoy:2007er ; Gursoy:2009jd ; Gursoy:2012bt from the topological
susceptibility (in pure Yang-Mills theory), $\chi_{\text{YM}}$. In $a$
coordinates, it is computed from

$Z_{0}=-\frac{\chi_{\text{YM}}}{M_{p}^{3}}\int_{-\infty}^{\infty}da\frac{e^{-4a}q}{c_{0}+c_{1}\Big{(}\frac{\lambda_{\text{YM}}}{\tilde{\lambda}_{0}}\Big{)}+c_{2}\Big{(}\frac{\lambda_{\text{YM}}}{\tilde{\lambda}_{0}}\Big{)}^{2}+c_{4}\Big{(}\frac{\lambda_{\text{YM}}}{\tilde{\lambda}_{0}}\Big{)}^{4}},$
(A.39)

where $\lambda_{\text{YM}}$ is the dilaton field in the pure glue theory. In
this instance, we use the lattice result
$\chi_{\text{YM}}/\sigma^{2}=0.01937(136)$ Bennett:2022gdz , with $\sigma$ the
string tension. Instead of computing the string tension from the gravity side
of the theory, we choose to work with the dimensionless ratio
$\chi_{\text{YM}}/T_{c}^{4}$; the value of $T_{c}/\sqrt{\sigma}=0.5970(38)$
comes from Lucini:2005vg , so that $\chi_{\text{YM}}/T_{c}^{4}\approx 0.152$.
With this recipe in hand, we can fix all of the coefficients for each choice
of the $Z(\lambda)$ potential. The various inputs are collected in Table. 1.

Choice |  UV behaviour |  $Z_{0}$ |  $c_{0}$ |  $c_{1}$ |  $c_{2}$ |  $c_{4}$
---|---|---|---|---|---|---
1 |  const. |  $0.443387$ |  $1$ |  $21.907$ |  $0$ |  $3$
2 |  $\lambda$ |  $9.04723$ |  $0$ |  $1$ |  $0$ |  $0.2224$
3 |  $\lambda^{2}$ |  $14.0949$ |  $0$ |  $0$ |  $1$ |  $0.68954$
Table 1: Results for different choices of $Z(\lambda)$ potential. Coefficients
are defined according to Eq. (A.39).

A priori, it may not be clear why there are $3$ non-zero $c_{i}$ for the case
of choice $1$, as there are only two non-zero $c_{i}$ for choice $2$ and $3$.
Indeed, for choice $1$, it turns out101010See App. C for more detail. that it
is necessary to have non-zero $c_{1}$ in order to yield the desired UV
asymptotics for the axial gauge field.

##  Appendix B Equations of motion and the background

The equations of motion for the background consist of the Einstein equations,
the equation for the scalar dilaton field and the equations for the vectorial
and axial gauge fields. As usual, this is not a fully independent system of
equations, but for example the dilaton equation can be derived from the rest
(also using the zero energy constraint included in the Einstein equations). We
present here the equations only for the Ansatz we will use in this article,
i.e., taking the metric of Eq. (12), assuming that the dilaton only depends on
$a$ and writing for the gauge fields

$V_{M}(x^{N})=V(a)\delta^{t}_{M}\ ,\qquad A_{M}(x^{N})=A(a)\delta^{t}_{M}\ .$
(B.40)

The equations of motion then reduce to the following system:

$\displaystyle\frac{\left(\lambda^{\prime}\right)^{2}}{\lambda^{2}}$
$\displaystyle=9+\frac{9}{4}\frac{f^{\prime}}{f}+\frac{3}{8}\frac{q^{2}}{f}xV_{f}(\lambda)\left(\frac{1}{R_{+}}+\frac{1}{R_{-}}\right)-\frac{3}{4}\frac{q^{2}}{f}V_{g}(\lambda)-$
$\displaystyle\ \ \ -\frac{3x^{2}Z(\lambda)e^{-2a}q^{2}}{2f^{2}}A^{2}\ ,$
(B.41a) $\displaystyle\frac{q^{\prime}}{q}$
$\displaystyle=4+\frac{f^{\prime}}{f}+\frac{1}{6}\frac{q^{2}}{f}xV_{f}(\lambda)\left(\frac{1}{R_{+}}+\frac{1}{R_{-}}\right)-\frac{1}{3}\frac{q^{2}}{f}V_{g}(\lambda)\
,$ (B.41b) $\displaystyle\frac{e^{2a}qxZ(\lambda)}{f}A$
$\displaystyle=\frac{d}{da}\left[\frac{e^{2a}V_{f}(\lambda)w(\lambda)^{2}\left(V^{\prime}+A^{\prime}\right)}{4qR_{+}}\right]\
,$ (B.41c) $\displaystyle-\frac{e^{2a}qxZ(\lambda)}{f}A$
$\displaystyle=\frac{d}{da}\left[\frac{e^{2a}V_{f}(\lambda)w(\lambda)^{2}\left(V^{\prime}-A^{\prime}\right)}{4qR_{-}}\right]\
,$ (B.41d) $\displaystyle f^{\prime\prime}$
$\displaystyle=\frac{f^{\prime}q^{\prime}}{q}-4f^{\prime}+\frac{e^{-2a}xV_{f}(\lambda)w(\lambda)^{2}}{2}\left[\frac{(V^{\prime}+A^{\prime})^{2}}{R_{+}}+\frac{(V^{\prime}-A^{\prime})^{2}}{R_{-}}\right]+$
$\displaystyle\ \ \ +\frac{4x^{2}Z(\lambda)e^{-2a}q^{2}}{f}A^{2}\ ,$ (B.41e)

where

$R_{\pm}=\sqrt{1-\frac{e^{-2a}w(\lambda)^{2}\left(A^{\prime}\pm
V^{\prime}\right)^{2}}{q^{2}}}\ $ (B.42)

where the primes denote derivatives with respect to $a$. The equations for the
vectorial gauge field (the sum of Eqs. (B.41c) and (B.41d)) and the equation
for the blackening factor $f$ can be integrated once, giving

$\displaystyle\frac{e^{2a}V_{f}(\lambda)w(\lambda)^{2}}{2q}\left[\left(\frac{1}{R_{+}}+\frac{1}{R_{-}}\right)V^{\prime}+\left(\frac{1}{R_{+}}-\frac{1}{R_{-}}\right)A^{\prime}\right]=c_{V}$
(B.43)
$\displaystyle\frac{e^{2a}xV_{f}(\lambda)w(\lambda)^{2}}{2q}\left[\left(\frac{1}{R_{+}}+\frac{1}{R_{-}}\right)\left(VV^{\prime}+AA^{\prime}\right)+\left(\frac{1}{R_{+}}-\frac{1}{R_{-}}\right)\left(AV^{\prime}+VA^{\prime}\right)\right]=$
$\displaystyle=\frac{f^{\prime}e^{4a}}{q}+c_{f},$ (B.44)

where $c_{V}$ and $c_{f}$ are the constants of integration. To obtain the
latter equation here, one also needs to use the equations for the gauge
fields.

The equations of motion are unchanged under the following two transformations:

$\displaystyle a\mapsto a+\log\Lambda_{a}\ ,\qquad V\mapsto\Lambda_{a}V\
,\qquad A\mapsto\Lambda_{a}A\ ;$ (B.45) $\displaystyle f\mapsto\Lambda_{f}f\
,\qquad q\mapsto\sqrt{\Lambda_{f}}q\ ,\qquad V\mapsto\sqrt{\Lambda_{f}}V\
,\qquad A\mapsto\sqrt{\Lambda_{f}}A\ ;$ (B.46)

where $\Lambda_{a}$ and $\Lambda_{f}$ are positive constants. The former
symmetry corresponds to the overall scale transformation discussed also in the
main text. Indeed, the metric Eq. (12) is unchanged under simultaneously
applying Eq. (B.45) and the scaling $x_{\mu}\mapsto x_{\mu}/\Lambda_{a}$ of
all spacetime coordinates. The latter symmetry is similarly identified as the
separate scaling of the time coordinate. This symmetry is fixed by requiring
that the blackening factor $f\to 1$ at the boundary so that the speed of light
equals one (as it should since we are using natural units).

The background solutions are found numerically by shooting towards the
boundary from the horizon $a=a_{h}$, where the blackening factor vanishes,
$f(a_{h})=0$. The gauge fields also vanish at the horizon. We initially fix
the freedom related to the symmetries Eqs. (B.45) and (B.46) by setting
$a_{h}=0$ and $f^{\prime}(0)=1$. The desired values for the energy scale and
asymptotic value of $f$ are obtained thereafter by applying these mappings
with well chosen $\Lambda_{a}$ and $\Lambda_{f}$. Following this, the regular
solutions depend on three parameters; the natural choice for these parameters
appears to be $\lambda_{h}=\lambda(0)$, $V_{h}^{\prime}=V^{\prime}(0)$, and
$A_{h}^{\prime}=A^{\prime}(0)$. This however leads to regularity conditions at
the horizon which cannot be solved analytically. One can try to bypass this
issue by using instead the quantities

$Q_{V}=\frac{V^{\prime}(0)}{q(0)}\ ,\qquad Q_{A}=\frac{A^{\prime}(0)}{q(0)}\
,$ (B.47)

which is actually equivalent to fixing $q$ at the horizon as opposed to
$f^{\prime}$. This choice is however not optimal because it turns out that the
mapping from these variables to the chemical potentials is not monotonic in
the region that we will be interested in. We find that this can be avoided by
rather using the “charges” $n_{V/A}$ defined through

$Q_{V}=\frac{n_{V}}{w(\lambda_{h})\sqrt{V_{f}(\lambda_{h})^{2}w(\lambda_{h})^{2}+n_{V}^{2}}}\
,\qquad
Q_{A}=\frac{n_{A}}{w(\lambda_{h})\sqrt{V_{f}(\lambda_{h})^{2}w(\lambda_{h})^{2}+n_{A}^{2}}}\
.$ (B.48)

The motivation for these definitions is that in the absence of an axial gauge
field, $n_{V}$ is indeed (up to a normalization coefficient) the quark number
density.

Therefore the final input parameters for our numerical solutions are
$\lambda_{h}$, $n_{V}$, and $n_{A}$. We introduce a small cutoff $\epsilon$
and start the numerical evolution of the equations of motion at $a=\epsilon$,
with boundary conditions given by analytic horizon expansions of the regular
solution obtained from the equations of motion111111Starting directly at $a=0$
is not possible due to singular behavior of the system at this point.. Note
that we follow Jarvinen:2011qe ; Alho:2012mh , choosing the sign of $q$ so
that it is always negative. Moreover, the physically relevant branch has
$\lambda^{\prime}(a)<0$.

Finally after the numerical solution has been obtained, we use the symmetry
Eq. (B.46) to set $f=1$ at the boundary, $a\to\infty$. Observables can then be
computed as follows. The UV scale factor $\Lambda$, which appears in the near
boundary expansions of $\lambda$ and the geometry Jarvinen:2011qe , can be
computed by using

$\Lambda=\lim_{a\to\infty}\frac{1}{\ell}\exp\left[a-\frac{8}{9v_{1}\lambda(a)}+\left(\frac{23}{36}-\frac{16v_{2}}{9v_{1}^{2}}\right)\log\left(\frac{9v_{1}\lambda(a)}{8}\right)\right]$
(B.49)

where the coefficients are defined through the weak coupling expansion of the
effective potential

$V_{\mathrm{eff}}(\lambda)=V_{g}(\lambda)-xV_{f}(\lambda)=\frac{12}{\ell^{2}}\left[1+v_{1}\lambda+v_{2}\lambda^{2}+\mathcal{O}\left(\lambda^{3}\right)\right]\
.$ (B.50)

The temperature and the vectorial chemical potential are then computed from

$T=\frac{1}{4\pi}\left|\frac{f^{\prime}(0)}{q(0)}\right|\
,\qquad\mu_{V}=\lim_{a\to\infty}V(a)\ .$ (B.51)

The definition of the axial chemical potential depends on the choice for the
function $Z(\lambda)$ and is discussed in App. C. Finally, the dimensionless
output is then given by the ratios $T/\Lambda$, $\mu_{V}/\Lambda$, and
$\mu_{A}/\Lambda$.

##  Appendix C Axial gauge field asymptotics

In this Appendix, we describe the asymptotic behaviour of the background axial
gauge field, $A$, along with its associated fluctuation, $\delta A$, which are
needed in order to extract the associated axial chemical potential, $\mu_{A}$
and axial magnetic field, $B_{A}$ respectively. We begin by assuming the
Ansatz

$\lim_{a\to\infty}A^{i}(a)\sim\mu^{i}_{A}e^{-\Delta_{i}a}a^{p_{i}}\bigg{[}1+\frac{1}{a}\Big{(}k^{i}_{1}+k^{i}_{11}\log{a}\Big{)}+\frac{1}{a^{2}}\Big{(}k^{i}_{2}+k^{i}_{22}\log{a}\Big{)}\bigg{]},$
(C.52)

where the $i$ superscript corresponds to the choice of the potential
$Z(\lambda)$ and $\mu_{A}$ is the axial chemical potential. Then, by expanding
either Eq. (B.41c) or Eq. (B.41d) in $1/a$ at the boundary, one can calculate
$\Delta^{i},p^{i}$ and the various $k^{i}$. With these coefficients in hand,
we can read off the value of the axial chemical potential for each case

$\displaystyle\mu^{1}_{A}$
$\displaystyle=\mathcal{C}_{1}\lim_{a\to\infty}\frac{A^{1}(a)}{e^{-\Delta_{1}a}a^{p_{1}}\Big{[}1+\frac{1}{a}\Big{(}k_{1}^{1}+k^{1}_{11}\log{a}\Big{)}\Big{]}}\,,$
(C.53) $\displaystyle\mu^{2}_{A}$
$\displaystyle=\mathcal{C}_{2}\lim_{a\to\infty}\frac{A^{2}(a)}{a^{p_{2}}\Big{[}1+\frac{1}{a}\Big{(}k_{1}^{2}+k^{2}_{11}\log{a}\Big{)}+\frac{1}{a^{2}}\Big{(}k_{2}^{2}+k^{2}_{22}\log{a}\Big{)}\Big{]}}\,,$
(C.54) $\displaystyle\mu^{3}_{A}$
$\displaystyle=\lim_{a\to\infty}\frac{A^{3}(a)}{1+\frac{1}{a}\Big{(}k_{1}^{3}+k^{3}_{11}\log{a}\Big{)}+\frac{1}{a^{2}}\Big{(}k_{2}^{3}+k^{3}_{22}\log{a}\Big{)}}\,,$
(C.55)

where the various coefficients are listed in Table. 2. Notice that for choice
3 in Eq. (C.55) the chemical potential is directly related to the boundary
value of the gauge field and the correction term in the denominator only
improves convergence as $a\to\infty$. For the other choices, the correction
term is required to obtain a finite value. Notice also that the normalization
of the correction cannot be determined unambiguously for the choices 1 and 2,
which is reflected in a similar normalization ambiguity of the chemical
potentials. This is why we have included the coefficients $\mathcal{C}_{i}$ in
the formulas. For the numerical results presented in the text, we used
$\mathcal{C}_{i}=10^{p_{i}}$. This was motivated by the fact that the value of
$a$ at which the near boundary expansions start to converge is around $a=10$,
so the coefficients cancel the large numerical factors arising from the power
corrections $a^{p_{i}}$ at this point.

$i$ |  $\Delta_{i}$ |  $p_{i}$ |  $k_{1}^{i}$ |  $k_{11}^{i}$ |  $k_{2}^{i}$ |  $k_{22}^{i}$
---|---|---|---|---|---|---
1 |  $-0.47048$ |  $2.13839$ |  $9.23171\newline -2.13839\mathcal{B}$ |  $0.831595$ |  neg. |  neg.
2 |  $0$ |  $4.933$ |  $25.5208\newline -4.993\mathcal{B}$ |  $1.94172$ |  $263.614\newline -103.847\mathcal{B}\newline +12.465\mathcal{B}^{2}$ |  $38.4996\newline -9.69504\mathcal{B}$
3 |  $0$ |  $0$ |  $-2.59291$ |  $0$ |  $6.10334\newline -2.59291\mathcal{B}$ |  $1.008351$
Table 2: Coefficients defining the asymptotic behaviour of the axial gauge
field, $A(a)$. The quantity $\mathcal{B}\equiv\log\frac{l}{\Lambda_{a}}$ is
background-dependent. For choice 1, we neglect the coefficients
$k^{2}_{2},\,k^{2}_{22}$.

We note that $\Delta_{1}\equiv\Delta$ is indeed the anomalous dimension
associated with the axial chemical and must therefore obey an additional
constraint; as was pointed out in Jimenez-Alba:2014iia ; Weitz , for
$\Delta<-1$, the axial counterpart of Eq. (16) would become an irrelevant
operator, thereby destroying the desired $\text{AdS}_{5}$ asymptotics.
Consequently, for choice 1 of potential, we must require $Z_{0}<0.906$.
Referring to the discussion in App. A.1, for choice 1 with $c_{1}=c_{2}=0$,
one obtains a value of $Z_{0}>0.906$, thus necessitating the inclusion of a
non-zero $c_{1}$.

Following an asymptotic analysis of Eqs. (21a) and (21b), we see that the
fluctuation of the axial gauge field possesses the same behaviour in the UV as
its background counterpart. We may thus define the axial magnetic fields

$\displaystyle B^{1\,j}_{A}-\mu^{1}_{A}\omega^{j}$
$\displaystyle=-\epsilon^{zjk}\mathcal{C}_{1}\lim_{a\to\infty}\frac{\delta\hat{A}_{k}^{1}(a)}{e^{-\Delta_{1}a}a^{p_{1}}\Big{[}1+\frac{1}{a}\Big{(}k_{1}^{1}+k^{1}_{11}\log{a}\Big{)}\Big{]}}\,,$
(C.56) $\displaystyle B^{2\,j}_{A}-\mu^{2}_{A}\omega^{j}$
$\displaystyle=-\epsilon^{zjk}\mathcal{C}_{2}\lim_{a\to\infty}\frac{\delta\hat{A}_{k}^{2}(a)}{a^{p_{2}}\Big{[}1+\frac{1}{a}\Big{(}k_{1}^{2}+k^{2}_{11}\log{a}\Big{)}+\frac{1}{a^{2}}\Big{(}k_{2}^{2}+k^{2}_{22}\log{a}\Big{)}\Big{]}}\,,$
(C.57) $\displaystyle B^{3\,j}_{A}-\mu^{3}_{A}\omega^{j}$
$\displaystyle=-\epsilon^{zjk}\lim_{a\to\infty}\frac{\delta\hat{A}^{3}_{k}(a)}{1+\frac{1}{a}\Big{(}k_{1}^{3}+k^{3}_{11}\log{a}\Big{)}+\frac{1}{a^{2}}\Big{(}k_{2}^{3}+k^{3}_{22}\log{a}\Big{)}}\,,$
(C.58)

where the $j$ index refers to the spatial index of the axial magnetic field.

## References

  * (1) D. E. Kharzeev, J. Liao, and P. Tribedy, Chiral Magnetic Effect in Heavy Ion Collisions: The Present and Future, [2405.05427].
  * (2) D. E. Kharzeev, L. D. McLerran, and H. J. Warringa, The Effects of topological charge change in heavy ion collisions: ’Event by event P and CP violation’, Nucl. Phys. A 803 (2008) 227 [0711.0950].
  * (3) K. Fukushima, D. E. Kharzeev, and H. J. Warringa, The Chiral Magnetic Effect, Phys. Rev. D 78 (2008) 074033 [0808.3382].
  * (4) M. A. Metlitski and A. R. Zhitnitsky, Anomalous axion interactions and topological currents in dense matter, Phys. Rev. D 72 (2005) 045011 [hep-ph/0505072].
  * (5) D. T. Son and P. Surowka, Hydrodynamics with Triangle Anomalies, Phys. Rev. Lett. 103 (2009) 191601 [0906.5044].
  * (6) ALICE Collaboration, R. Belmont, Charge-dependent anisotropic flow studies and the search for the Chiral Magnetic Wave in ALICE, Nucl. Phys. A 931 (2014) 981 [1408.1043].
  * (7) STAR Collaboration, L. Adamczyk et al., Observation of charge asymmetry dependence of pion elliptic flow and the possible chiral magnetic wave in heavy-ion collisions, Phys. Rev. Lett. 114 (2015) 252302 [1504.02175].
  * (8) P. V. Buividovich, M. N. Chernodub, E. V. Luschevskaya, and M. I. Polikarpov, Numerical evidence of chiral magnetic effect in lattice gauge theory, Phys. Rev. D 80 (2009) 054503 [0907.0494].
  * (9) A. Yamamoto, Chiral magnetic effect in lattice QCD with a chiral chemical potential, Phys. Rev. Lett. 107 (2011) 031601 [1105.0385].
  * (10) A. Yamamoto, Lattice study of the chiral magnetic effect in a chirally imbalanced matter, Phys. Rev. D 84 (2011) 114504 [1111.4681].
  * (11) P. V. Buividovich, Anomalous transport with overlap fermions, Nucl. Phys. A 925 (2014) 218 [1312.1843].
  * (12) B. B. Brandt, G. Endrődi, E. Garnacho-Velasco, and G. Markó, The Chiral Separation Effect from lattice QCD at the physical point, [2312.02945].
  * (13) B. B. Brandt, G. Endrődi, E. Garnacho-Velasco, and G. Markó, On the absence of the Chiral Magnetic Effect in equilibrium QCD, [2405.09484].
  * (14) Y. Neiman and Y. Oz, Relativistic Hydrodynamics with General Anomalous Charges, JHEP 03 (2011) 023 [1011.5107].
  * (15) K. Jensen, R. Loganayagam, and A. Yarom, Thermodynamics, gravitational anomalies and cones, JHEP 02 (2013) 088 [1207.5824].
  * (16) S. Golkar and D. T. Son, (Non)-renormalization of the chiral vortical effect coefficient, JHEP 02 (2015) 169 [1207.5806].
  * (17) K. Jensen, P. Kovtun, and A. Ritz, Chiral conductivities and effective field theory, JHEP 10 (2013) 186 [1307.3234].
  * (18) A. Gynther, K. Landsteiner, F. Pena-Benitez, and A. Rebhan, Holographic Anomalous Conductivities and the Chiral Magnetic Effect, JHEP 02 (2011) 110 [1005.2587].
  * (19) I. Amado, K. Landsteiner, and F. Pena-Benitez, Anomalous transport coefficients from Kubo formulas in Holography, JHEP 05 (2011) 081 [1102.4577].
  * (20) K. Landsteiner, E. Megias, and F. Pena-Benitez, Gravitational Anomaly and Transport, Phys. Rev. Lett. 107 (2011) 021601 [1103.5006].
  * (21) U. Gursoy and J. Tarrio, Horizon universality and anomalous conductivities, JHEP 10 (2015) 058 [1410.1306].
  * (22) U. Gürsoy and A. Jansen, (Non)renormalization of Anomalous Conductivities and Holography, JHEP 10 (2014) 092 [1407.3282].
  * (23) S. Grozdanov and N. Poovuttikul, Universality of anomalous conductivities in theories with higher-derivative holographic duals, JHEP 09 (2016) 046 [1603.08770].
  * (24) J. H. Schwarz and E. Witten, Anomaly analysis of brane - anti-brane systems, JHEP 03 (2001) 032 [hep-th/0103099].
  * (25) T. Sakai and S. Sugimoto, Low energy hadron physics in holographic QCD, Prog. Theor. Phys. 113 (2005) 843 [hep-th/0412141].
  * (26) R. Casero, E. Kiritsis, and A. Paredes, Chiral symmetry breaking as open string tachyon condensation, Nucl. Phys. B 787 (2007) 98 [hep-th/0702155].
  * (27) D. Areán, I. Iatrakis, M. Järvinen, and E. Kiritsis, The discontinuities of conformal transitions and mass spectra of V-QCD, JHEP 11 (2013) 068 [1309.2286].
  * (28) D. Arean, I. Iatrakis, M. Järvinen, and E. Kiritsis, CP-odd sector and $\theta$ dynamics in holographic QCD, Phys. Rev. D 96 (2017) 026001 [1609.08922].
  * (29) A. Jimenez-Alba, K. Landsteiner, and L. Melgar, Anomalous magnetoresponse and the Stückelberg axion in holography, Phys. Rev. D 90 (2014) 126004 [1407.8162].
  * (30) A. Jimenez-Alba, K. Landsteiner, Y. Liu, and Y.-W. Sun, Anomalous magnetoconductivity and relaxation times in holography, JHEP 07 (2015) 117 [1504.06566].
  * (31) A. D. Gallegos and U. Gürsoy, Dynamical gauge fields and anomalous transport at strong coupling, JHEP 05 (2019) 001 [1806.07138].
  * (32) N. Rai and E. Megias, Anomalous conductivities in the holographic Stückelberg model, JHEP 06 (2023) 215 [2301.00361].
  * (33) K. Landsteiner, E. Megias, L. Melgar, and F. Pena-Benitez, Holographic Gravitational Anomaly and Chiral Vortical Effect, JHEP 09 (2011) 121 [1107.0368].
  * (34) E. Megias and F. Pena-Benitez, Holographic Gravitational Anomaly in First and Second Order Hydrodynamics, JHEP 05 (2013) 115 [1304.5529].
  * (35) T. Kalaydzhyan and I. Kirsch, Fluid/gravity model for the chiral magnetic effect, Phys. Rev. Lett. 106 (2011) 211601 [1102.4334].
  * (36) Y. Bu, T. Demircik, and M. Lublinsky, Nonlinear chiral transport from holography, JHEP 01 (2019) 078 [1807.08467].
  * (37) Y. Bu, T. Demircik, and M. Lublinsky, Chiral transport in strong fields from holography, JHEP 05 (2019) 071 [1903.00896].
  * (38) M. Ammon, S. Grieninger, J. Hernandez, et al., Chiral hydrodynamics in strong external magnetic fields, JHEP 04 (2021) 078 [2012.09183].
  * (39) H.-U. Yee, Holographic Chiral Magnetic Conductivity, JHEP 11 (2009) 085 [0908.4189].
  * (40) A. Rebhan, A. Schmitt, and S. A. Stricker, Anomalies and the chiral magnetic effect in the Sakai-Sugimoto model, JHEP 01 (2010) 026 [0909.4782].
  * (41) A. Gorsky, P. N. Kopnin, and A. V. Zayakin, On the Chiral Magnetic Effect in Soft-Wall AdS/QCD, Phys. Rev. D 83 (2011) 014023 [1003.2293].
  * (42) A. Ballon-Bayona, K. Peeters, and M. Zamaklar, A chiral magnetic spiral in the holographic Sakai-Sugimoto model, JHEP 11 (2012) 164 [1209.1953].
  * (43) K. Fukushima and A. Okutsu, Electric conductivity with the magnetic field and the chiral anomaly in a holographic QCD model, Phys. Rev. D 105 (2022) 054016 [2106.07968].
  * (44) U. Gursoy and E. Kiritsis, Exploring improved holographic theories for QCD: Part I, JHEP 02 (2008) 032 [0707.1324].
  * (45) U. Gursoy, E. Kiritsis, and F. Nitti, Exploring improved holographic theories for QCD: Part II, JHEP 02 (2008) 019 [0707.1349].
  * (46) S. S. Gubser and A. Nellore, Mimicking the QCD equation of state with a dual black hole, Phys. Rev. D 78 (2008) 086007 [0804.0434].
  * (47) O. DeWolfe, S. S. Gubser, and C. Rosen, A holographic critical point, Phys. Rev. D 83 (2011) 086005 [1012.1864].
  * (48) M. Jarvinen and E. Kiritsis, Holographic Models for QCD in the Veneziano Limit, JHEP 03 (2012) 002 [1112.1261].
  * (49) U. Gursoy, E. Kiritsis, L. Mazzanti, and F. Nitti, Improved Holographic Yang-Mills at Finite Temperature: Comparison with Data, Nucl. Phys. B 820 (2009) 148 [0903.2859].
  * (50) N. Jokela, M. Järvinen, and J. Remes, Holographic QCD in the Veneziano limit and neutron stars, JHEP 03 (2019) 041 [1809.07770].
  * (51) F. Bigazzi, R. Casero, A. L. Cotrone, E. Kiritsis, and A. Paredes, Non-critical holography and four-dimensional CFT’s with fundamentals, JHEP 10 (2005) 012 [hep-th/0505140].
  * (52) T. Ishii, M. Järvinen, and G. Nijs, Cool baryon and quark matter in holographic QCD, JHEP 07 (2019) 003 [1903.06169].
  * (53) A. Amorim, M. S. Costa, and M. Järvinen, Regge theory in a holographic dual of QCD in the Veneziano limit, JHEP 07 (2021) 065 [2102.11296].
  * (54) M. Järvinen, E. Kiritsis, F. Nitti, and E. Préau, The V-QCD baryon : numerical solution and baryon spectrum, [2212.06747].
  * (55) D. Kharzeev, R. D. Pisarski, and M. H. G. Tytgat, Possibility of spontaneous parity violation in hot QCD, Phys. Rev. Lett. 81 (1998) 512 [hep-ph/9804221].
  * (56) STAR Collaboration, B. I. Abelev et al., Azimuthal Charged-Particle Correlations and Possible Local Strong Parity Violation, Phys. Rev. Lett. 103 (2009) 251601 [0909.1739].
  * (57) T. Alho, M. Järvinen, K. Kajantie, E. Kiritsis, and K. Tuominen, On finite-temperature holographic QCD in the Veneziano limit, JHEP 01 (2013) 093 [1210.4516].
  * (58) M. Järvinen, Holographic modeling of nuclear matter and neutron stars, Eur. Phys. J. C 82 (2022) 282 [2110.08281].
  * (59) E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150].
  * (60) M. Järvinen, E. Kiritsis, F. Nitti, and E. Préau, Tachyon-dependent Chern-Simons terms and the V-QCD Baryon, [2209.05868].
  * (61) D. Arean, I. Iatrakis, M. Järvinen, and E. Kiritsis, V-QCD: Spectra, the dilaton and the S-parameter, Phys. Lett. B 720 (2013) 219 [1211.6125].
  * (62) M. Järvinen, Massive holographic QCD in the Veneziano limit, JHEP 07 (2015) 033 [1501.07272].
  * (63) M. Panero, Thermodynamics of the QCD plasma and the large-N limit, Phys. Rev. Lett. 103 (2009) 232001 [0907.3719].
  * (64) C. Ecker, M. Järvinen, G. Nijs, and W. van der Schee, Gravitational waves from holographic neutron star mergers, Phys. Rev. D 101 (2020) 103006 [1908.03213].
  * (65) N. Jokela, M. Järvinen, G. Nijs, and J. Remes, Unified weak and strong coupling framework for nuclear matter and neutron stars, Phys. Rev. D 103 (2021) 086004 [2006.01141].
  * (66) N. Jokela, M. Järvinen, and J. Remes, Holographic QCD in the NICER era, Phys. Rev. D 105 (2022) 086005 [2111.12101].
  * (67) T. Demircik, C. Ecker, and M. Järvinen, Dense and Hot QCD at Strong Coupling, Phys. Rev. X 12 (2022) 041012 [2112.12157].
  * (68) S. Tootle, C. Ecker, K. Topolski, T. Demircik, M. Järvinen, and L. Rezzolla, Quark formation and phenomenology in binary neutron-star mergers using V-QCD, SciPost Phys. 13 (2022) 109 [2205.05691].
  * (69) U. Gürsoy, I. Iatrakis, E. Kiritsis, F. Nitti, and A. O’Bannon, The Chern-Simons Diffusion Rate in Improved Holographic QCD, JHEP 02 (2013) 119 [1212.3894].
  * (70) F. Bigazzi, A. L. Cotrone, and F. Porri, Universality of the Chern-Simons diffusion rate, Phys. Rev. D 98 (2018) 106023 [1804.09942].
  * (71) E. Weitz, Anomalous Transport from Dynamical Gauge Fields and Holography, Master’s thesis, Utrecht U., 2020.
  * (72) P. B. Arnold, D. Son, and L. G. Yaffe, The Hot baryon violation rate is O (alpha-w**5 T**4), Phys. Rev. D 55 (1997) 6264 [hep-ph/9609481].
  * (73) D. Bodeker, On the effective dynamics of soft nonAbelian gauge fields at finite temperature, Phys. Lett. B 426 (1998) 351 [hep-ph/9801430].
  * (74) D. Bodeker, G. D. Moore, and K. Rummukainen, Chern-Simons number diffusion and hard thermal loops on the lattice, Phys. Rev. D 61 (2000) 056003 [hep-ph/9907545].
  * (75) G. D. Moore and M. Tassler, The Sphaleron Rate in SU(N) Gauge Theory, JHEP 02 (2011) 105 [1011.1167].
  * (76) D. T. Son and A. O. Starinets, Minkowski space correlators in AdS / CFT correspondence: Recipe and applications, JHEP 09 (2002) 042 [hep-th/0205051].
  * (77) B. Craps, C. Hoyos, P. Surowka, and P. Taels, Chern-Simons diffusion rate in a holographic Yang-Mills theory, JHEP 11 (2012) 109 [1209.2532].
  * (78) B. Lucini, M. Teper, and U. Wenger, Properties of the deconfining phase transition in SU(N) gauge theories, JHEP 02 (2005) 033 [hep-lat/0502003].
  * (79) A. Athenodorou and M. Teper, SU(N) gauge theories in 3+1 dimensions: glueball spectrum, string tensions and topology, JHEP 12 (2021) 082 [2106.00364].
  * (80) E. Bennett, D. K. Hong, J.-W. Lee, et al., Color dependence of the topological susceptibility in Yang-Mills theories, Phys. Lett. B 835 (2022) 137504 [2205.09254].
  * (81) T. Alho, M. Järvinen, K. Kajantie, E. Kiritsis, C. Rosen, and K. Tuominen, A holographic model for QCD in the Veneziano limit at finite temperature and density, JHEP 04 (2014) 124 [1312.5199].
  * (82) T. Demircik, D. Gallegos, U. Gürsoy, M. Järvinen, and R. Lier, A Novel Method for Holographic Transport, [2311.00042].
  * (83) T. Demircik, D. Gallegos, U. Gürsoy, M. Järvinen, and R. Lier, Holographic transport in anisotropic plasmas, [2402.12224].
  * (84) S. Grieninger and S. Morales-Tejera, Real-time dynamics of axial charge and chiral magnetic current in a non-Abelian expanding plasma, Phys. Rev. D 108 (2023) 126010 [2308.14829].
  * (85) S. Grieninger and D. E. Kharzeev, Spacetime dynamics of chiral magnetic currents in a hot non-Abelian plasma, Phys. Rev. D 108 (2023) 126004 [2308.14825].
  * (86) J. Cruz Rojas, T. Gorda, C. Hoyos, et al., Estimate for the bulk viscosity of strongly coupled quark matter, [2402.00621].