††institutetext: aSteklov Mathematical Institute, Russian Academy of Sciences,
Gubkina str. 8, 119991, Moscow, Russia

# Holographic model for light quarks in anisotropic hot dense QGP with
external magnetic field

Irina Ya. Aref’evaa, Alexey Ermakova, Kristina Rannua and Pavel Slepova
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>slepov@mi-
ras.ru

###### Abstract

We present a five-dimensional twice anisotropic holographic model supported by
Einstein-dilaton-three-Maxwell action describing light quarks. The first of
the Maxwell fields provides finite chemical potential. The second Maxwell
field serves for anisotropy, characterizing spatial anisotropy of the QGP
produced in heavy-ion collisions (HIC). The third Maxwell field is related to
a magnetic field that appears in HIC. The dependence of the 5-dim black hole
solution and confinement/deconfinement phase diagram on this magnetic field is
considered. The effect of the inverse magnetic catalyses for light quarks
phase diagram of is obtained. Positions of critical end points are found. We
also study the behavior of the conductivities for light quarks for both
isotropic and anisotropic cases and show that behaviour of the conductivity
near critical points essentially depend on quark masses, meanwhile at high
temperature they are similar.

###### Keywords:

AdS/QCD, holography, phase transition, electric conductivity, light quarks

##  1 Introduction

Holographic duality provides an effective approach to study nonperturbative
properties of QCD Casalderrey-Solana:2011dxg ; Arefeva:2014kyw ;
DeWolfe:2013cua . One of the achievements of QCD holographic description is
that holographic QCD (HQCD) models can describe QCD at all energy scales
Gursoy:2008za ; Gursoy:2010fj . HQCD reproduces properties of QCD obtained by
others methods, namely perturbative renormalization group and lattice
approach. However neither of these methods can describe QCD with nonzero
chemical potential and therefore predict the phase diagram for quark-gluon
plasma (QGP). Nowadays this goal is achieved by holographic methods only.
There are HQCD models that describe QCD with heavy quarks Yang:2015aia , light
quarks Li:2017tdz  as well as with realistic quarks contents Pirner:2009gr .
These models are based on the so-called “bottom-up” holographic approach.
Holographic method can also describe anisotropic dense matter in external
magnetic field. The goal of this paper is to study the behaviour of
anisotropic QCD with light quarks in strong external magnetic field. This
paper is the generalization of our consideration of anisotropic QCD with heavy
quarks in external magnetic field Arefeva:2020vae  from the one hand and of
our consideration of anisotropic QCD with light quarks at zero magnetic field
Arefeva:2020byn  to the case of non zero external field from other hand.

The model under consideration Arefeva:2022bhx  has been built using a
potential reconstruction method Casalderrey-Solana:2011dxg ; Arefeva:2014kyw ;
DeWolfe:2013cua ; Gursoy:2008za ; Gursoy:2010fj ; He:2013qq ; Yang:2015aia ;
Li:2017tdz ; Arefeva:2018hyo ; Arefeva:2018cli ; Chen:2018msc ;
Arefeva:2020vhf ; Ballon-Bayona:2020xls ; Arefeva:2020byn ; Arefeva:2021kku ;
Slepov:2021gvl ; Rannu:2021pcq ; Li:2020hau ; Arefeva:2021btm ;
Arefeva:2021mag ; Arefeva:2020vae ; Gursoy:2020kjd ; Arefeva:2020bjk ;
Zhou:2021nbp ; Dudal:2021jav ; Arefeva:2021jpa ; Caldeira:2021izy ; Ali-
Akbari:2021ene ; Chen:2021gop . This model is characterized by a 5-dimensional
Einstein-dilaton-Maxwell action and a 5-dimensional metric with a warp factor
that tunes the blackening function and, consequently, the thermodynamics of
the model. The phase transition structure of HQCD strongly depends on the warp
factor choice. The simplest warp factor has an exponential form with a
quadratic polynomial Andreev:2006nw ; Yang:2015aia . It reproduces the first-
order phase transition from lattice calculations (Columbia plots Brown:1990ev
; Philipsen:2016hkv ) for heavy quarks. To reproduce the first-order phase
transition for light quarks one has to use rational functions Gursoy:2010fj ;
Li:2017tdz .

To describe the QGP produced in heavy ion collisions (HIC) by holographic
methods one has to make more modifications of the 5-dimensional metric. These
modifications are related to two types of the anisotropy that take place in
HIC: i) anisotropy between the longitudinal and transversal directions and ii)
anisotropy in transversal plane related with magnetic field produced in non-
central collisions of ions. It was proposed to characterize the first type of
anisotropy by a parameter $\nu$ that is similar to the Lifshitz parameter of
the AdS metric modification. The isotropic model corresponds to $\nu=1$. For
the value of about $\nu=4.5$ the dependence of the entropy produced in the HIC
on energy agrees with the experimental data ALICE:2015juo  for the energy
dependence of the total multiplicity of created particles Arefeva:2014vjl .
The magnetic field created by non-central ion collisions Toneev:2016tgj
produces anisotropy in the plane transverse to the ion collision axis.
Therefore, it is important to consider these types of anisotropy in
holographic models Rougemont:2015oea ; Li:2016gfn ; Gursoy:2016ofp ;
Dudal:2016joz ; Gursoy:2017wzz ; Gursoy:2018ydr ; Bohra:2019ebj ; He:2020fdi ;
Rodrigues:2020ndy ; Arefeva:2021mag ; Arefeva:2020vae . To construct
anisotropic models, one considers an additional Maxwell field to support the
anisotropy in metrics Arefeva:2014vjl ; Arefeva:2018hyo ; Arefeva:2018cli ;
Arefeva:2020vhf ; Arefeva:2020uec ; Arefeva:2020byn ; Arefeva:2021kku ;
Slepov:2021gvl ; Rannu:2021pcq ; Arefeva:2021btm ; Arefeva:2021mag ;
Arefeva:2020vae ; Arefeva:2020bjk ; Arefeva:2021jpa . Note that calculations
in the twice anisotropic model for light quarks are much more complicated than
for heavy quarks fully anisotropic one Arefeva:2021mag ; Arefeva:2020vae ,
since the warp factor and the kinetic gauge function $f{{}_{1}}$ are more
complex. For heavy quark model we have found Arefeva:2020vae  the effect of
the inverse magnetic catalysis (IMC) – larger absolute values of the coupling
coefficient $c_{B}$ lead to decrease of the transition temperature. In the
model for light quarks considered in this paper we also find the IMC. This
work is a generalization of results obtained in Arefeva:2020byn ;
Arefeva:2020vae  to the case of twice anisotropic holographic light quark
model. The results obtained in Arefeva:2020byn  are generalized to the case of
present of an external magnetic field, and the results obtained in
Arefeva:2020vae  to the case of light quarks.

We also study electrical transport properties of QGP and its relation to
thermal direct photon production in the line of previous studies of these
problems for other holographic models Wu:2013qja ; CaronHuot:2006te ;
Patino:2012py ; Finazzo:2013efa ; Arciniega:2013dqa ; Iatrakis:2016ugz ;
Arefeva:2016rob ; Avila:2021rcu . Photons do not interact with QGP hadronic
matter and therefore provide information about various characteristics of QGP
including phase structure at different time scales. It is well known that the
direct photons (DP) emission rate is connected to the conductivity of QGP
Erdmenger ; Kapusta:2006pm . In this work we generalize results of
Arefeva:2021jpa  concerning the conductivity and photon production to the case
of light quarks. We show that conductivity both in longitudinal and transverse
directions essentially depends on anisotropy parameter $\nu$. The DP flow also
depends on temperature and other thermodynamic properties of QGP. As in the
case of heavy quarks, for light quarks the higher the magnetic field or/and
chemical potential, the higher the conductivity. We will see that for $\nu=1$
the behavior of the conductivities for heavy and light quarks is different at
temperatures close to critical, but at high temperatures they both saturate to
constant values. For $\nu=4.5$ the behavior of the conductivities for heavy
and light quarks is also different at temperatures close to critical as well
as for high temperatures. Conductivity for light quark model along the
collision direction monotonically decreases with temperature while transversal
components have minimum and increase at large temperatures. Since we expect
isotropisation Giataganas:2012zy ; Strickland:2013uga ; Arefeva:2021kku , i.e.
$\nu\to 1$ about $1-5fm/c\sim 10^{-24}s$, we expect essential change of
conductivity at this time scale.

The paper is organized as follows. In Sect.2 the holographic model is
presented. In Sect.3 the thermodynamics of the model and behaviour of Wilson
loops are described, the confinement/deconfinement phase diagram is obtained.
Sect.4 describes derivation of the direct photons and electrical conductivity
properties. In Sect.5 we summarize our results and discuss future directions
of investigations.

##  2 Model

In this section we present a model suitable for light quarks’ version of the
confinement/deconfinement phase diagram Arefeva:2020byn  in magnetic field
similar to how it was done in our work Arefeva:2020vae .

###  2.1 Metric and EOM

Here we take the same action as in our previous work Arefeva:2020vae . In
Einstein frame it has the following form

$\displaystyle{\mathcal{S}}=\int\cfrac{d^{5}x\,\sqrt{-g}}{16\pi
G_{5}}\left[R-\cfrac{f_{1}(\phi)}{4}\ F_{(1)}^{2}-\cfrac{f_{2}(\phi)}{4}\
F_{(2)}^{2}-\cfrac{f_{B}(\phi)}{4}\
F_{(B)}^{2}-\cfrac{\partial_{\mu}\phi\partial^{\mu}\phi}{2}-V(\phi)\right].$
(2.1)

Here we use the anzats, where the non-zero components of electro-magnetic
field and field strengths are

$\displaystyle A_{\mu}^{(1)}=A_{t}(z)\delta_{\mu}^{0},\quad
F_{y_{1}y_{2}}^{(2)}=q,\quad F_{xy_{1}}^{(B)}=q_{B}.$ (2.2)

In action (2.1) $\phi=\phi(z)$ is the scalar field, $f_{1}(\phi)$,
$f_{2}(\phi)$ and $f_{B}(\phi)$ are the coupling functions associated with the
Maxwell fields $A_{\mu}$, $F_{\mu\nu}^{(2)}$ and $F_{\mu\nu}^{(B)}$
correspondingly, $q$ and $q_{B}$ are the constants and $V(\phi)$ is the scalar
field potential. Thus (2.1) is the extended version of the action used in
Arefeva:2018hyo ; Arefeva:2020byn , where we add an external magnetic field
$F_{\mu\nu}^{(B)}$.

We also use the same metric anzatz Arefeva:2020vae :

$\displaystyle
ds^{2}=\cfrac{L^{2}\,\mathfrak{b}(z)}{z^{2}}\left[-\,g(z)dt^{2}+dx^{2}+\left(\cfrac{z}{L}\right)^{2-\frac{2}{\nu}}\\!\\!dy_{1}^{2}+e^{c_{B}z^{2}}\left(\cfrac{z}{L}\right)^{2-\frac{2}{\nu}}\\!\\!dy_{2}^{2}+\cfrac{dz^{2}}{g(z)}\right],$
(2.3) $\displaystyle\mathfrak{b}(z)=e^{2{\mathcal{A}}(z)}.$ (2.4)

Let us make some notes on the main parameters of metric (2.3). The difference
between “heavy quarks” and “light quarks” cases lies in the form of the warp
factor $\mathfrak{b}(z)$. For heavy quarks we used
$\mathfrak{b}(z)=e^{-\frac{cz^{2}}{2}}$ Arefeva:2018hyo . To get the “light
quarks” version we follow Li:2017tdz  and assume
${\mathcal{A}}(z)=-\,a\ln(bz^{2}+1)$.

Here parameters $\nu$ and $c_{B}$ have the same meaning as in Arefeva:2020vae
. The primary anisotropy parameter $\nu$ describes non-equivalence between
longitudinal and transversal directions. The isotropic case corresponds to
$\nu=1$, while $\nu=4.5$ reproduces the multiplicity of charged particles
production Arefeva:2014vjl .

We refer to $c_{B}$ as to the magnetic field parameter that describes non-
centrality of heavy ion collisions. Although explicit relation of $c_{B}$ to
the magnitude of magnetic field is not established, there are considerations
suggesting that $B^{2}=-\,c_{B}$ Bohra:2019ebj ; DHoker:2009ixq . So we have
$c_{B}<0$ in all our calculations. Magnetic field parameter $c_{B}$ is an
implicit function of the magnetic charge $q_{B}$ from the anzats (2.2).
However, we will not need the explicit form of this relation in what follows.
We usually set $q=q_{B}=1$ in numerical calculations. The metric anzats (2.3)
also contains the blackening function $g(z)$ which will be determined in the
subsequent paragraphs.

Applying the stationary action principle, we get the same equations of motion
(EOM) as in Arefeva:2020vae :

$\displaystyle\begin{split}\phi^{\prime\prime}+\phi^{\prime}\left(\cfrac{g^{\prime}}{g}+\cfrac{3\mathfrak{b}^{\prime}}{2\mathfrak{b}}-\cfrac{\nu+2}{\nu
z}+c_{B}z\right)&+\left(\cfrac{z}{L}\right)^{2}\cfrac{\partial
f_{1}}{\partial\phi}\ \cfrac{(A_{t}^{\prime})^{2}}{2\mathfrak{b}g}\
-\left(\cfrac{L}{z}\right)^{2-\frac{4}{\nu}}\cfrac{\partial
f_{2}}{\partial\phi}\ \cfrac{q^{2}\ e^{-c_{B}z^{2}}}{2\mathfrak{b}g}\ -\\\
&-\left(\cfrac{z}{L}\right)^{\frac{2}{\nu}}\cfrac{\partial
f_{B}}{\partial\phi}\ \cfrac{q_{B}^{2}}{2\mathfrak{b}g}\
-\left(\cfrac{L}{z}\right)^{2}\cfrac{\mathfrak{b}}{g}\ \cfrac{\partial
V}{\partial\phi}=0,\end{split}$ (2.5) $\displaystyle
A_{t}^{\prime\prime}+A_{t}^{\prime}\left(\cfrac{\mathfrak{b}^{\prime}}{2\mathfrak{b}}+\cfrac{f_{1}^{\prime}}{f_{1}}+\cfrac{\nu-2}{\nu
z}+c_{B}z\right)=0,$ (2.6) $\displaystyle
g^{\prime\prime}+g^{\prime}\left(\cfrac{3\mathfrak{b}^{\prime}}{2\mathfrak{b}}-\cfrac{\nu+2}{\nu
z}+c_{B}z\right)-\left(\cfrac{z}{L}\right)^{2}\cfrac{f_{1}(A_{t}^{\prime})^{2}}{\mathfrak{b}}-\left(\cfrac{z}{L}\right)^{\frac{2}{\nu}}\cfrac{q_{B}^{2}\
f_{B}}{\mathfrak{b}}=0,$ (2.7)
$\displaystyle\mathfrak{b}^{\prime\prime}-\cfrac{3(\mathfrak{b}^{\prime})^{2}}{2\mathfrak{b}}+\cfrac{2\mathfrak{b}^{\prime}}{z}-\cfrac{4\mathfrak{b}}{3\nu
z^{2}}\left(1-\cfrac{1}{\nu}+\left(1-\cfrac{3\nu}{2}\right)c_{B}z^{2}-\cfrac{\nu
c_{B}^{2}z^{4}}{2}\right)+\cfrac{\mathfrak{b}\,(\phi^{\prime})^{2}}{3}=0,$
(2.8) $\displaystyle
2g^{\prime}\left(1-\cfrac{1}{\nu}\right)+3g\left(1-\cfrac{1}{\nu}\right)\left(\cfrac{\mathfrak{b}^{\prime}}{\mathfrak{b}}-\cfrac{4\left(\nu+1\right)}{3\nu
z}+\cfrac{2c_{B}z}{3}\right)+\left(\cfrac{L}{z}\right)^{1-\frac{4}{\nu}}\cfrac{L\,q^{2}\,e^{-c_{B}z^{2}}f_{2}}{\mathfrak{b}}=0,$
(2.9)
$\displaystyle\begin{split}\cfrac{\mathfrak{b}^{\prime\prime}}{\mathfrak{b}}&+\cfrac{(\mathfrak{b}^{\prime})^{2}}{2\mathfrak{b}^{2}}+\cfrac{3\mathfrak{b}^{\prime}}{\mathfrak{b}}\left(\cfrac{g^{\prime}}{2g}-\cfrac{\nu+1}{\nu
z}+\cfrac{2c_{B}z}{3}\right)-\cfrac{g^{\prime}}{3zg}\left(5+\cfrac{4}{\nu}-3c_{B}z^{2}\right)+\\\
&+\cfrac{8}{3z^{2}}\left(1+\cfrac{3}{2\nu}+\cfrac{1}{2\nu^{2}}\right)-\cfrac{4c_{B}}{3}\left(1+\cfrac{3}{2\nu}-\cfrac{c_{B}z^{2}}{2}\right)+\cfrac{g^{\prime\prime}}{3g}+\cfrac{2}{3}\left(\cfrac{L}{z}\right)^{2}\cfrac{\mathfrak{b}V}{g}=0,\end{split}$
(2.10)

where ${}^{\prime}=\partial/\partial z$.

Turning off the external magnetic field, i.e. putting $c_{B}=q_{B}=f_{B}=0$
into (2.5)–(2.10), we get the EOM from Arefeva:2020byn . Normalizing to the
AdS-radius, $L=1$, we get the EOM from Arefeva:2018hyo . Excluding anisotropy,
i.e. putting $\nu=1$ and $f_{2}=0$, we get the expressions that fully coincide
with the EOM from Li:2017tdz ; Yang:2015aia . Thus (2.5)–(2.10) are universal
anisotropic EOM, appropriate both for heavy and light quarks description, that
include solution from Li:2017tdz ; Yang:2015aia  as an isotropic special case.
We also consider the general form of the boundary conditions:

$\displaystyle A_{t}(0)=\mu,\quad A_{t}(z_{h})=0,$ (2.11) $\displaystyle
g(0)=1,\quad g(z_{h})=0,$ (2.12) $\displaystyle\phi(z_{0})=0,$ (2.13)

where $z_{0}=0$ corresponds to Li:2017tdz  and $z_{0}=z_{h}$ to
Arefeva:2018hyo . The choice of the boundary condition for the scalar field
was discussed in details in Arefeva:2020byn .

###  2.2 Solution

Just as it was in previous cases, to solve EOM (2.5)–(2.10) we need to
determine the form of the coupling function $f_{1}$. Let us take the same form
as for the light quarks model Arefeva:2020byn . Taking into account the “light
quarks” warp factor, we get

$\displaystyle
f_{1}=e^{-cz^{2}-{\mathcal{A}}(z)}z^{-2+\frac{2}{\nu}}=(1+bz^{2})^{a}\,e^{-cz^{2}}z^{-2+\frac{2}{\nu}}.$
(2.14)

Solving (2.6) with the coupling function (2.14) and boundary conditions (2.11)
gives

$\displaystyle
A_{t}=\mu\cfrac{e^{\left(2c-c_{B}\right)z^{2}/2}-e^{\left(2c-c_{B}\right)z_{h}^{2}/2}}{1-e^{\left(2c-c_{B}\right)z_{h}^{2}/2}}\
\xrightarrow[c_{B}\,\to\,0]{}\ \mu\
\cfrac{e^{cz^{2}}-e^{cz_{h}^{2}}}{1-e^{cz_{h}^{2}}},$ (2.15) $\displaystyle
A_{t}(z)=\mu-\rho\,z^{2}+\dots\ \Longrightarrow\
\rho=-\,\cfrac{\mu\left(2c-c_{B}\right)}{2\left(1-e^{\left(2c-c_{B}\right)z_{h}^{2}/2}\right)}.$
(2.16)

We take $a=4.046$, $b=0.01613$, $c=0.227$ to make our solution agree with
results from Li:2017tdz  in the isotropic case, results from Arefeva:2020byn
in the anisotropic case and results from Arefeva:2020vae  for heavy quarks
version. These values are due to the mass spectrum of $\rho$ meson with its
excitations and lattice results for the phase transition temperature
Li:2017tdz .

A B

Figure 1: Electric potential $A_{t}(z)$ (A) and density $\rho(z_{h})/\mu$ in
logarithmic scale (B) for “light quarks” with (solid lines) and without
(dashed lines) magnetic field, $c_{B}=0,\ -\,1$ (B); $\mu=0.1$, $a=4.046$,
$b=0.01613$, $c=0.227$.

To obtain the blackening function we need to know kinetic function $f_{B}$
describing coupling to the third Maxwell field $F_{\mu\nu}^{(B)}$. System
(2.5)–(2.10) study leads us to the following expression

$\displaystyle
f_{B}=2\left(\cfrac{z}{L}\right)^{-2/\nu}\\!\\!\\!\mathfrak{b}g\,\cfrac{c_{B}z}{q_{B}^{2}}\left(\cfrac{3\mathfrak{b}^{\prime}}{2\mathfrak{b}}-\cfrac{2}{\nu
z}+c_{B}z+\cfrac{g^{\prime}}{g}\right),$ (2.17)

that allows to transform (2.7) into

$\displaystyle
g^{\prime\prime}+g^{\prime}\left(\cfrac{3\mathfrak{b}^{\prime}}{2\mathfrak{b}}-\cfrac{\nu+2}{\nu
z}-c_{B}z\right)-2g\left(\cfrac{3\mathfrak{b}^{\prime}}{2\mathfrak{b}}-\cfrac{2}{\nu
z}+c_{B}z\right)c_{B}z-\left(\cfrac{z}{L}\right)^{2}\cfrac{f_{1}(A_{t}^{\prime})^{2}}{\mathfrak{b}}=0$
(2.18)

and get the solution

$\displaystyle\begin{split}&g=e^{c_{B}z^{2}}\left[1-\cfrac{I_{1}(z)}{I_{1}(z_{h})}+\cfrac{\mu^{2}\,(2c-c_{B})}{L^{2}\left(1-e^{\left(2c-c_{B}\right)z_{h}^{2}/2}\right)^{2}}\right.\times\\\
&\qquad\ \
\left.\times\left\\{\left(e^{\left(2c-c_{B}\right)z^{2}/2}-e^{\left(2c-c_{B}\right)z_{h}^{2}/2}\right)I_{1}(z)-(2c-c_{B})\,I_{2}(z)\left(1-\cfrac{I_{1}(z)}{I_{1}(z_{h})}\,\cfrac{I_{2}(z_{h})}{I_{2}(z)}\right)\right\\}\right],\\\
&I_{1}(z)=\int_{0}^{z}\left(1+b\xi^{2}\right)^{3a}e^{-3c_{B}\xi^{2}/2}\,\xi^{1+\frac{2}{\nu}}\,d\xi,\\\
&I_{2}(z)=\int_{0}^{z}e^{(2c-c_{B})\xi^{2}/2}\int_{0}^{\xi}\left(1+b\chi^{2}\right)^{3a}e^{-3c_{B}\chi^{2}/2}\,\chi^{1+\frac{2}{\nu}}\,d\chi\,d\xi.\end{split}$
(2.19)

However this solution can’t be accepted as the final one and should be
improved. First of all it can’t be shown that our previous “light quarks”
blackening function Arefeva:2020byn  serves as zero magnetic field limit for
(2.19). To overcome this difficulty we simplify the expression via partial
integration:

$\displaystyle
g=e^{c_{B}z^{2}}\left[1-\cfrac{I_{1}(z)}{I_{1}(z_{h})}+\cfrac{\mu^{2}\,(2c-c_{B})\,I_{2}(z)}{L^{2}\left(1-e^{\left(2c-c_{B}\right)z_{h}^{2}/2}\right)^{2}}\left(1-\cfrac{I_{1}(z)}{I_{1}(z_{h})}\,\cfrac{I_{2}(z_{h})}{I_{2}(z)}\right)\right],$
(2.20)

where $I_{1}(z)$ is described by the corresponding expression from (2.19), but
$I_{2}(z)$ was redefined as

$\displaystyle
I_{2}(z)=\int_{0}^{z}\left(1+b\xi^{2}\right)^{3a}e^{\left(c-2c_{B}\right)\xi^{2}}\,\xi^{1+\frac{2}{\nu}}\,d\xi.$
(2.21)

Further solving (2.5)–(2.10) system with coupling functions (2.14) and (2.17)
and boundary condition (2.13) gives

$\displaystyle
f_{2}=2\left(\cfrac{z}{L}\right)^{2-4/\nu}\,\cfrac{\nu-1}{q^{2}\nu^{2}z^{2}}\,\cfrac{e^{c_{B}z^{2}}}{(1+bz^{2})^{2a}}\left\\{\left(2+2\nu-\nu
c_{B}z^{2}+\cfrac{6ab\nu z^{2}}{1+bz^{2}}\right)g-g^{\prime}\nu z\right\\},$
(2.22)
$\displaystyle\phi=\int_{z_{0}}^{z}\sqrt{\cfrac{4(\nu-1)}{\nu^{2}\xi^{2}}-2c_{B}\left(3-\cfrac{2}{\nu}\right)-2c_{B}^{2}\xi^{2}+\cfrac{12ab}{1+b\xi^{2}}\left(1+2\,\cfrac{1+ab\xi^{2}}{1+b\xi^{2}}\right)}\,d\xi,$
(2.23)
$\displaystyle\phi^{\prime}=\sqrt{\cfrac{4(\nu-1)}{\nu^{2}z^{2}}-2c_{B}\left(3-\cfrac{2}{\nu}\right)-2c_{B}^{2}z^{2}+\cfrac{12ab}{1+bz^{2}}\left(1+2\,\cfrac{1+abz^{2}}{1+bz^{2}}\right)},$
(2.24) $\displaystyle\begin{split}&V(z)=-\,\cfrac{(1+bz^{2})^{2a}\ z}{2L^{2}}\
\times\\\
&\times\left[2\left\\{2\,\cfrac{(1+\nu)(1+2\nu)+\bigl{(}2+(2+3a)(3+2\nu)\nu\bigr{)}bz^{2}+(1+\nu+3a\nu)(1+2\nu+6a\nu)b^{2}z^{4}}{(1+bz^{2})^{2}\nu^{2}z^{2}}\right.-\right.\\\
&\left.-\left.c_{B}\left(2+\cfrac{3}{\nu}+\cfrac{12abz^{2}}{1+bz^{2}}\right)+c_{B}^{2}z^{2}\right\\}g\,z-\left(5+3\left(\cfrac{6ab}{1+bz^{2}}-c_{B}\right)z^{2}+\cfrac{4}{\nu}\right)g^{\prime}+g^{\prime\prime}z\right].\end{split}$
(2.25)

Note, that we do not fix coupling function for the 3-rd Maxwell $f_{B}$, but
derive it from the EOM with intent. Fixing, for example, $f_{B}=f_{1}$ makes
system (2.5)–(2.10) not selfconsistent, therefore proper solution with the
$f_{B}=f_{1}$ condition can’t be found.

##  3 Thermodynamics

###  3.1 Temperature and entropy

For metric (2.3) and light quarks warp factor temperature and entropy can be
written as:

$\displaystyle T=\cfrac{|g^{\prime}|}{4\pi}\,\Bigl{|}_{z=z_{h}},\quad
s=\left(\cfrac{L}{z_{h}}\right)^{1+\frac{2}{\nu}}\cfrac{e^{c_{B}z_{h}^{2}/2}}{4\left(1+bz_{h}^{2}\right)^{3a}}.$
(3.1)

A B C

Figure 2: Temperature as function of horizon for different $\mu$ in isotropic
(1-st line, green) and anisotropic cases for $\nu=1.5$ (2-nd line, red),
$\nu=3$ (3-rd line, magenta), $\nu=4.5$ (4-th line, blue) in magnetic field
with $c_{B}=-\,0.001$ (A), $c_{B}=-\,0.01$ (B), $c_{B}=-\,0.1$ (C); $a=4.046$,
$b=0.01613$, $c=0.227$. Lighter curves show chemical potentials for which
second horizon appears and multivalued behavior of $T(z_{h})$-function
actually turns into monotonian.

On Fig.2 temperature as a function of horizon is presented. The multivalued
behavior of $T(z_{h})$ preserves starting from the chemical potential value of
the so called critical end point (CEP) till another specific chemical
potential value for which second horizon appears. For $\nu=1$ $\mu_{CEP}>0$,
while in cases of primary anisotropy ($\nu=1.5,\,3,\,4.5$) $\mu_{CEP}=0$ for
small magnetic fields. Magnetic field’s increasing leads to growth of
$\mu_{CEP}$ values and decrease in characteristic temperatures of the
multivalued behavior, i.e. the temperatures of the background phase transition
(corresponding to black hole – black hole collapse in AdS5). Thus in magnetic
field strong enough background phase transition should occur at high chemical
potentials and negligible temperatures.

###  3.2 Background phase transition

A B

C D

E

Figure 3: Background phase transition diagram $T(\mu)$ in isotropic (A) and
anisotropic cases for $\nu=1.5$ (B), $\nu=3$ (C), $\nu=4.5$ (D) and all
combined (E); $a=4.046$, $b=0.01613$, $c=0.227$.

This is exactly the picture we observe on phase diagrams Fig.3, obtained via
free energy consideration

$\displaystyle F=-\int s\,dT=\int_{z_{h}}^{\infty}s\,T^{\prime}dz,$ (3.2)

where we normalize the free energy to vanish at $z_{h}\to\infty$. The
background phase transition curves become shorter with the magnetic field
increasing (larger $c_{B}$ absolute values). Note, that two opposite
tendencies can be seen: growth of the $\mu_{CEP}$ and decrease in the maximum
chemical potential value. For $c_{B}\approx(-\,0.05;-\,0.03)$ depending on
$\nu$ these tendencies come into conflict, and CEP curve reverses it’s
movement back towards lower values of the chemical potential (Fig.4A).

Primary anisotropy affects the background phase transition for light quarks
QGP in much the same way as it did for heavy quarks Arefeva:2020vae : lowers
transition temperatures and enlarges chemical potentials avaliable. But it
almost doesn’t influence the limit $c_{B}$ values at which phase transition
still exists. Phase transition temperatures reach negligible values at
$c_{B}\approx-\,0.1$ almost regardless of primary anisotropy $\nu$ (Fig.3A-D,
Fig.4B). It’s effect on $c_{B}$ for which $\mu_{CEP}$ becomes nonzero is more
notable: $c_{B}<-\,0.001$ for $\nu=1.5$ and $\nu<-\,0.005$ for $\nu=4.5$
(Fig.4A,C).

We present CEP for $\nu=1,\,1.5,\,3,\,4.5$ in a 3-dimensional space of
parameters ($\mu,c_{B},T$) on Fig.4D. Thus, figures A, B and C are the
projections of a complete picture D on the corresponding planes. Also, one can
see the smooth peach coloured surface of the first order phase transitions
interpolating for intermediate values of anisotropy. The orange curve
indicates boundaries of possible phase transition locations in this space of
parameters.

A B C
D

Figure 4: CEP positions on the background phase transition diagram $T(\mu)$ in
isotropic and anisotropic cases for $\nu=1,\,1.5,\,3,\ 4.5$ on planes $\mu-T$
(A), $c_{B}-T$ (B), $c_{B}-\mu$ (C) and 3D plot $T(c_{B},\mu)$ (D); $a=4.046$,
$b=0.01613$, $c=0.227$.

###  3.3 Temporal Wilson loops

The expectation value of the temporal Wilson loop calculation does not differ
from the heavy quarks case Arefeva:2020vae  untill we substitute the specific
light quarks warp factor into the general expressions for the dynamical wall
(DW) equations:

$\displaystyle\begin{split}DW_{x}:&\quad-\,\cfrac{4abz}{1+bz^{2}}+\sqrt{\cfrac{2}{3}}\
\phi^{\prime}(z)+\cfrac{g^{\prime}}{2g}-\cfrac{2}{z}\
\Big{|}_{z=z_{DWx}}=0,\\\
DW_{y_{1}}:&\quad-\,\cfrac{4abz}{1+bz^{2}}+\sqrt{\cfrac{2}{3}}\
\phi^{\prime}(z)+\cfrac{g^{\prime}}{2g}-\cfrac{\nu+1}{\nu z}\
\Big{|}_{z=z_{DWy_{1}}}=0,\\\
DW_{y_{2}}:&\quad-\,\cfrac{4abz}{1+bz^{2}}+\sqrt{\cfrac{2}{3}}\
\phi^{\prime}(z)+\cfrac{g^{\prime}}{2g}-\cfrac{\nu+1}{\nu z}+c_{B}z\
\Big{|}_{z=z_{DWy_{2}}}=0.\end{split}$ (3.3)

A B

C D

Figure 5: Wilson loop phase transition curves $T(\mu)$ in isotropic (A) and
anisotropic cases for $\nu=1.5$ (B), $\nu=3$ (C) and $\nu=4.5$ (D); $a=4.046$,
$b=0.01613$, $c=0.227$.

Primary anisotropy has significant influence on Wilson loop transformations in
the magnetic field (Fig.5). For $\nu=1$ Wilson loop phase transition curve
becomes less inclined, temperature decreases, while maximum chemical potential
grows. At $c_{B}=-\,0.0000985$ the curve becomes horizontal and for
$c_{B}<-\,0.0000985$ it just disappears (Fig.5A). For $\nu=4.5$ larger
absolute $c_{B}$ also leads to lesser temperatures, but chemical potential
decreases as well, so the Wilson loop curve shrinks clinging to the abscissa
axis $\mu$ at the end. The curve disappears for $c_{B}<-\,0.0858$ (Fig.5D).
For intermediate values of primary anisotropy $\nu=1.5$ (Fig.5B) and $\nu=3$
(Fig.5C) a mixture of both these tendences can be seen. But inverse magnetic
catalysis takes place for any primary anisotropy.

Let us now consider the combination of the 1-st order phase transition and a
crossover for primary anisotropy and magnetic field fixed (while both
transition curves do exist). On Fig.6 regions of their intersection or
approach are shown, and the general view of the phase diagrams can be found on
the inserted mini-plots. Points of intersections, where the
confinement/deconfinement main role passes from the crossover to the 1-st
order phase transitions (or vice versa), are marked as
$(\mu_{by_{2}},T_{by_{2}})$ (Tab.1). Critical end points, where the 1-st order
phase transition begins, are marked as CEPLQ.

For primary isotropic case $\nu=1$ a crossover disappears very soon, so it’s
position relative to the 1-st order phase transition does not have time to
change much. The intersection point just shifts to larger $\mu$ and lesser
$T$.

For $\nu=1.5$ both curves are very close to each other and look cocentric.
They do not intersect, so the crossover defines the phase transition for all
chemical potential values. Magnetic field makes these curves closer to each
other and hardly distinguished, but intersection appears at about
$c_{B}=-\,0.018$ and coincides with the CEPLQ. At this moment phase transition
main role clearly passes from the crossover to the 1-st order phase transition
though for rather a short interval. Before it’s disappearance at
$c_{B}=-\,0.01882$ the crossover line shifts down to lesser temperatures and
becomes almost horizontal like it was for $\nu=1$.

For $\nu=3$ the 1-st order phase transition and a crossover can’t be
distiguished properly. Magnetic field shifts he 1-st order phase transition
curve under the crossover, so that they do not intersect. Under these
circumstances there is no smooth transfer between the crossover and the 1-st
order phase transition, but a jump to lower temperature – temperature of
CEPLQ. Near the limit $c_{B}=-\,0.0631$ the crossover catches up with the 1-st
order phase transition and a smooth transfer appears again just before the
crossover vanishes.

A B C

Figure 6: Phase diagram $(T;\mu)$ as a combination os the 1-st order phase
transition (BB) and a crossover (WL) in isotropic (1-st line, green) with
$c_{B}=0$ (A), $c_{B}=-\,0.00009$ (B), $c_{B}=-\,0.0000985$ (C), and
anisotropic cases for $\nu=1.5$ (2-nd line, red) with $c_{B}=0$ (A),
$c_{B}=-\,0.018$ (B), $c_{B}=-\,0.01882$ (C), $\nu=3$ (3-rd line, magenta)
with $c_{B}=0$ (A), $c_{B}=-\,0.06$ (B), $c_{B}=-\,0.0631$ (C), $\nu=4.5$
(4-th and 5-th line, blue), with $c_{B}=0$ (A), $c_{B}=-\,0.01$ (B),
$c_{B}=-\,0.02$, (C), $c_{B}=-\,0.03$ (A), $c_{B}=-\,0.08$ (B),
$c_{B}=-\,0.0858$ (C); $a=4.046$, $b=0.01613$, $c=0.227$.

For maximal primary anisotropy $\nu=4.5$ the 1-st order phase transition curve
constantly shifts under the crossover curve, so that the intersection point
moves to larger chemical potentials and lower temperatures. On the other hand
CEPLQ also moves, causing a jump transfer from the crossover to the 1-st order
phase transition. The reverse transition is smooth. For $c_{B}=-\,0.03$ the
1-st order phase transition is completely under the crossover so that the
reverse transition does not occur, and further for $c_{B}<-\,0.03$ it occurs
as a jump to higher temperatures. The 1-st order phase transition exists in a
very narrow interval. So the confinement/deconfinement transition should be
defined by a crossover with constantly decreasing temperature except a small
local region of almost zero $T$. Near the limit value $c_{B}=-\,0.0858$ the
crossover catches up with the 1-st order phase transition like it was for
$\nu=3$. Soon after this moment the crossover vanishes, but the 1-st order
phase transition doesn’t survive for long after that.

| $c_{B}$ | $(\mu_{by_{2}},T_{by_{2}})$
---|---|---
| $0$ | $(0.1049,0.1531)$
$\nu=1$ | $-\,0.00009$ | $(0.1405,0.1479)$
| $-\,0.0000985$ | $(0.2308,0.1311)$
| $0$ | no intersection
$\nu=1.5$ | $-\,0.018$ | $(0.4501,0.0699)$
| $-\,0.01882$ | $(0.5622,0.04928)$
| $0$ | $(0.2148,0.1249)$
$\nu=3$ | $-\,0.01$ | $(0.9738,0.03968),\ (1.124,0.01188)$
| $[-\,0.06,-\,0.02]$ | no intersection
| $-\,0.0631$ | $(0.7409,0.006101)$
| $0$ | $(0.03893,0.1222)$
$\nu=4.5$ | $-\,0.01$ | $(0.8903,0.06232)$
| $-\,0.02$ | $(1.123,0.02343)$
| $[-\,0.0858,-\,0.03]$ | no intersection
Table 1: Intersection points from the Wilson loop phase transition curve
(crossover) to the background phase transition curve (1-st order phase
transition) in $(\mu;T)$-coordinates depending on primary anisotropy $\nu$ and
magnetic field’s anisotropy $c_{B}$.

##  4 Electrical conductivity

Electrical conductivity contains information about transport and optical
properties of a physical system. Understanding the behavior of the
conductivity under different conditions is an essential step in studying
properties of direct photons.

###  4.1 Conductivity in anisotropic background

We use the model derived above to investigate properties of direct photons
formed in HIC. The detailed consideration of holographic approach to
electrical conductivity and direct photon emission rate of QGP is presented in
Iatrakis:2016ugz ; Iqbal:2008by ; Arefeva:2021jpa . Below we briefly describe
the framework for holographic investigations of electrical conductivity.

In an anisotropic background 111Note that we get (2.4) from (4.1) after
$x_{1}\to x$, $x_{2}\to y_{1}$ and $x_{3}\to y_{2}$. We call $x_{1}$ the
longitudinal direction, i.e. the direction along the collision axis, $x_{2}$
and $x_{3}$ the transversal directions, and $(x_{1}x_{2})$ are coordinates in
a collision plane. The external magnetic field is aligned along $x_{3}$.

$\displaystyle
ds^{2}=\cfrac{L^{2}\,\mathfrak{b}(z)}{z^{2}}\left[-\,g(z)dt^{2}+\mathfrak{g}_{1}(z)dx_{1}^{2}+\mathfrak{g}_{2}(z)dx_{2}^{2}+\mathfrak{g}_{3}(z)dx_{3}^{2}+\cfrac{dz^{2}}{g(z)}\right]$
(4.1)

we consider a probe Maxwell field

$\displaystyle S_{pert}=-\,\cfrac{1}{4}\int
d^{5}x\sqrt{-g}f_{0}(\phi)F_{MN}F^{MN}.$ (4.2)

We assume this creates no gravitational backreaction. Function $f_{0}(\phi)$
is called a gauge kinetic function and is used to fit the model with lattice
data of isotropic QGP at zero chemical potential and zero magnetic field
Iatrakis:2016ugz .

Transport coefficients may be found within the linear response theory using
the Kubo relations. Namely, the electrical conductivity is related to the
retarded Green’s functions of electric currents. Therefore, one is interested
in a plane wave solution to (4.2), which will be holographically dual to
$U(1)$ electric current $J^{\mu}$ of the boundary theory. We take the ansatz
of a plane wave propagating in $x_{3}$-direction. The wave amplitude is a
function of the “holographic coordinate” $z$ only. The Green’s function in a
low frequency limit may be found either using the prescription by Iqbal and
Liu Iqbal:2008by  or Son and Starinets Son:2002sd . The equivalence of these
approaches is demonstrated in Arefeva:2021jpa . The components of electric
conductivity tensor are then

$\displaystyle\sigma^{11}=\cfrac{2f_{0}(z_{h})}{z_{h}}\,\sqrt{\cfrac{b(z_{h})\mathfrak{g}_{3}(z_{h})\mathfrak{g}_{2}(z_{h})}{\mathfrak{g}_{1}(z_{h})}},$
(4.3)
$\displaystyle\sigma^{22}=\cfrac{2f_{0}(z_{h})}{z_{h}}\,\sqrt{\cfrac{b(z_{h})\mathfrak{g}_{3}(z_{h})\mathfrak{g}_{1}(z_{h})}{\mathfrak{g}_{2}(z_{h})}},$
(4.4)
$\displaystyle\sigma^{33}=\cfrac{2f_{0}(z_{h})}{z_{h}}\,\sqrt{\cfrac{b(z_{h})\mathfrak{g}_{1}(z_{h})\mathfrak{g}_{2}(z_{h})}{\mathfrak{g}_{3}(z_{h})}},$
(4.5)

where $z_{h}$ is position of the event horizon Arefeva:2021jpa .

###  4.2 Model tuning

To tune the model parameters we use the results of lattice calculations from
Aarts:2014nba . These results are known for isotropic QGP ($\nu=1$) with
number of flavours $N_{f}=3$ at zero chemical potential ($\mu=0$) and without
magnetic field ($c_{B}=0$). Thus, we fit the behaviour of electrical
conductivity to the lattice data in this simplest case using the gauge kinetic
function $f_{0}(\phi)$. Then it is possible to calculate the conductivity in
anisotropic case and for nonzero parameters. We take the gauge kinetic
function in the following form

$\displaystyle
f_{0}(\phi)=\cfrac{0.0023}{\text{C}_{em}}\left[0.9\exp\left(-\,\frac{\phi^{2}}{8.8}\right)+0.11\exp(-\,0.05\phi)\right],$
(4.6)

where $\text{C}_{em}=2e^{2}/3$ is electromagnetic constant for $N_{f}=3$ and
$\phi$ is the dilaton field.

Numerical calculation for isotropic conductivity and the gauge function are
shown on Fig.7. Here the critical temperature is $T_{c}=0.155$ GeV. It was
calculated as a temperature of BB phase transition for a given set of
parameters. The conductivity increases with temperature growth and saturates
some value close to the $\mathcal{N}=4$ SYM plasma conductivity.

A B

Figure 7: Gauge kinetic function $f_{0}$ as function of the horizon $z_{h}$
(A) and relation of electrical conductivity to temperature $\sigma/T$ as
function of normalised temperature $T/T_{c}$ (B). Blue dots with error bars
are results of lattice calculations from Aarts:2014nba . The red dash-dotted
line on panel (B) is conductivity of $\mathcal{N}=4$ SYM plasma, the blue
curve is result of numerical calculations in our model for $\nu=1$, $c_{B}=0$
and $\mu=0$.

###  4.3 Isotropic case

The conductivity tensor in the isotropic ($\nu=1$) case and zero magnetic
field reduces to the single value $\sigma=\sigma^{ii}$ ($i=1,2,3$), so QGP
provides equal resistance to direct photons in all directions. The dependence
of conductivity to temperature ratio $\sigma^{ii}/T$ on temperature for
different values of chemical potential is shown in Fig.8A. The nonzero
magnetic field kills the degeneracy between two transverse directions
(Fig.8B). The plots presented at Fig.8 demonstrate that near critical
temperature the ratios $\sigma^{ii}/T$ grow with the normalized temperature
till some temperatures around $2T_{c}$ and precise location of maximum depends
on chemical potential and values of magnetic field and then start to decrease
to some constant values near the conductivity of $\mathcal{N}=4$ SYM plasma.
Plots on Fig.8A and 8B demonstrate the growth of conductivity with a growing
magnetic field and/or chemical potential at constant values of temperature.
Also, one can see that conductivity in directions orthogonal to magnetic field
is lower than in a parallel one, since brown curves are lower than the green
ones of the same thickness. Therefore, QGP is more opaque along the external
magnetic fields.

A B

Figure 8: On the left panel the dependence of $\sigma/T$ on normalised
temperature $T/T_{c}$ is presented in isotropic case $\nu=1$ with zero
magnetic field $c_{B}=0$ for different values of chemical potential $\mu$.
Panel (B) shows $\sigma^{33}$ (green curves) and $\sigma^{11}=\sigma^{22}$
(brown curves) for different values of magnetic field parameter $c_{B}$ at
zero chemical potential $\mu=0$. Magnetic field is directed along the $x^{3}$
direction.

###  4.4 Anisotropic case

In this subsection, we study the electrical conductivity of QGP for the value
of anisotropy parameter $\nu=4.5$. The results of numerical calculations of
$\sigma^{33}/T$ (conductivity along external magnetic field) for different
values of magnetic field and chemical potential are presented in Fig.9.

A B

Figure 9: The dependence of $\sigma^{33}/T$ on normalised temperature
$T/T_{c}$ is presented in anisotropic case $\nu=4.5$ for different values of
magnetic field $c_{B}$ for zero chemical potential $\mu=0$ (A) and high value
of chemical potential $\mu=1$ (B). Here critical temperature $T_{c}=0.121$
GeV.

We see that the ratio $\sigma^{33}/T$ first decreases with temperature up to
some value of temperature depending on chemical potential and the value of
external magnetic field, when starts to increase and the growth rate is
greater at zero chemical potential, and the ratio $\sigma^{33}/T$ grows faster
for lower magnetic fields at fixed chemical potential. The effect of the
magnetic field and chemical potential at fixed temperature is to increase the
conductivity. One can see that at high temperatures the curves in Fig.9 are
getting closer to each other and almost coincide. Thus, anisotropic QGP with
$\nu=4.5$ along the direction of the external magnetic field is the most
opaque at critical temperatures, becomes more transparent around $T\approx
2T_{c}$ for $\mu=0$, Fig.9A, and $T\approx 3.5T_{c}$ for $\mu=1$, Fig.9B, and
after these values it again becomes more opaque with increasing temperature.

A B

Figure 10: The dependence of $\sigma^{22}/T$ on normalised temperature
$T/T_{c}$ is presented in anisotropic case $\nu=4.5$ for different values of
magnetic field $c_{B}$ for zero chemical potential $\mu=0$ (panel A) and high
value of chemical potential $\mu=1$ (panel B). Here critical temperature
$T_{c}=0.121$ GeV.

Thermodynamic properties of $\sigma^{22}/T$ are similar to those of
$\sigma^{33}/T$ and are shown in Fig.10. We see that near the phase transition
point $\sigma^{22}/T$ decreases to a minimal value, and then starts to
increase linearly at high temperatures. Plots in Fig.10A,B show that the
higher the chemical potential, the higher the ratio $\sigma^{22}/T$. However,
the growth rate at high temperatures is bigger for $\mu=0$ case. The curves
are almost indistinguishable for high and low magnetic fields, but the
conductivity in higher magnetic fields is greater than in low magnetic fields.
For zero magnetic field there is no difference between $\sigma^{22}$ and
$\sigma^{33}$, it can be seen at non-zero magnetic fields. The value of
$\sigma^{33}/T$ is always larger than $\sigma^{22}/T$.

A B

Figure 11: The dependence of $\sigma^{22}/T$ (red curves) and $\sigma^{33}/T$
(blue curves) on normalised temperature $T/T_{c}$ is presented in anisotropic
case $\nu=4.5$ for the value of magnetic field $c_{B}=-0.07$ for zero chemical
potential $\mu=0$ (panel A) and high value of chemical potential $\mu=1$
(panel B). Here critical temperature $T_{c}=0.121$ GeV.

A B

Figure 12: The dependence of $\sigma^{11}/T$ on normalised temperature
$T/T_{c}$ is presented in anisotropic case $\nu=4.5$ for different values of
magnetic field $c_{B}$ for zero chemical potential $\mu=0$ (panel (A)) and
high value of chemical potential $\mu=1$ (panel (B)). Here critical
temperature $T_{c}=0.121$ GeV.

The results of similar calculations for $\sigma^{11}/T$ are shown in Fig.12.
We see that $\sigma^{11}/T$ monotonically decreases to zero with increasing
temperature, and magnetic field and chemical potential push it down. Or in
other words, QGP along the collision line near critical temperature is almost
opaque and at high temperatures becomes transparent.

##  5 Conclusion

In this paper we have considered a twice anisotropic holographic model for
light quarks Arefeva:2022bhx . It is characterized by Einstein-dilaton-three-
Maxwell action and 5-dimensional metric with the warp factor that has been
considered in isotropic case for light quarks Li:2017tdz . Our model has two
different types of anisotropy mentioned above: the anisotropy related with
parameter $\nu$ (responsible to reproduction of experimental dependence of
multiplicity on energy), and the anisotropy related with external magnetic
field (which will described by the third Maxwell field). The twice anisotropic
model for light quarks is much more complicated than the twice anisotropic
model for heavy quarks Arefeva:2021mag ; Arefeva:2020vae , since the
deformation factor and the kinetic gauge function $f{{}_{1}}$ are more
complicated.

We have investigated the dependence of the phase transition structure on the
anisotropy, characterized by parameter $\nu$ and magnetic fields. The phase
transition structure is set up by the mutual arrangement of two type of phase
transition lines – 1-st order phase transition, originating from the metric
background, and a crossover, originating from temporal Wilson loop behavior.
We have got the following results concerning the phase transition structure.

  * •

For $\nu=1$ the 1-st order phase transition in light quarks model lasts from
$(\mu_{CEP},T_{CEP})$ to $(\mu_{max},0)$. Magnetic field $c_{B}\neq 0$
shortens the 1-st order phase transition, increasing $\mu_{CEP}$ and
decreasing $\mu_{max}$. In heavy quarks model Arefeva:2020vae  1-st order
phase transition lied from $(0,T_{max})$ to $(\mu_{CEP\,HQ},T_{CEP\,HQ})$
almost horizontally. Magnetic field shortened this line just shifting
$\mu_{CEP\,HQ}$ to zero. For light quarks model 1-st order phase transition
line disappears at $c_{B}$ about 7 times larger than it was for heavy quarks
model. Inverse magnetic catalysis found for both models.

  * •

For $\nu>1$ decreases temperature of the 1-st order phase transition and
increases $\mu_{max}$. It almost doesn’t influence $c_{B}$ value that 1-st
order phase transition line disappears at, like it was for heavy quarks model
Arefeva:2020vae .

  * •

For $\nu=1$ magnetic field decreases Wilson loop curve inclination until it
becomes horizontal. This leads to inverse magnetic catalysis effect for $\mu$
near zero and direct magnetic catalysis for larger chemical potential values.
In heavy quarks model Arefeva:2020vae  Wilson loop curve underwent inverse
magnetic catalysis only and shrink tending to vertical $T$-axis.

  * •

Primary anisotropy $\nu>1$ decreases Wilson loop temperature and region of
direct magnetic catalysis. It also increases $c_{B}$ absolute value at which
Wilson loop survives. For $\nu=4.5$ Wilson loop curve just shrinks tending to
horizontal $\mu$-axis. For heavy quarks model the larger anisotropy was the
weaker it decreased Wilson loop temperature.

  * •

For $\nu=1$ in light quarks model confinement/deconfinement phase transition
is determined by a crossover from $\mu=0$ to the intersection with the 1-st
order phase transition $\mu_{by_{2}}$, where it picks up the main role. On the
opposite, for heavy quarks model Arefeva:2020vae  1-st order phase transition
was decisive from $\mu=0$ to the intersection with the crossover
$\mu_{by_{2}}$.

  * •

Magnetic field $c_{B}\neq 0$ for $\nu=1$ shifts the intersection point
$(\mu_{by_{2}},T_{by_{2}})$ to larger chemical potentials and lower
temperatures, but doesn’t actually change general scheme for light quarks
model. For heavy quarks model Arefeva:2020vae  1-st order phase transition
moved to above the Wilson loop curve and lost it’s influence on the
confinement/deconfinement process.

  * •

Anisotropy $\nu>1$ makes the 1-st order phase transition move under the Wilson
loop curve. Therefore crossover determines the confement/deconfinement process
from $\mu=0$ to $\mu_{CEP}$, where the 1-st order phase transition begins and
takes the main role with a jump. For heavy quarks model Arefeva:2020vae  the
1-st order phase transition also moved under the Wilson loop curve and
determined the confinement/deconfinement process from $\mu=0$ to
$\mu_{CEP\,HQ}$, where the 1-st order phase transition ends and gives the main
role to a crossover with a jump.

We also have studied the DC conductivity of QGP in the deconfined phase under
different conditions. The conductivity has different values and thermodynamic
properties in the HIC direction and in orthogonal plane presented, see
Fig.8-12.

  * •

For $\nu=1$ the conductivity

    * –

along all directions, $\sigma^{ii}/T$ near the critical temperature grows with
the temperature up to $\approx 2T_{c}$ and then starts to decrease to a
constant value near the conductivity of $\mathcal{N}=4$ SYM plasma;

    * –

along the orthogonal to magnetic field directions is lower than in parallel
one;

    * –

the chemical potential as well as value of the external magnetic field
increase conductivity.

  * •

For $\nu=4.5$ the conductivity along

    * –

the magnetic field direction $\sigma^{33}/T$ has a minimum, see Fig.9 and
increases at high temperatures;

    * –

the transversal direction $\sigma^{22}/T$ (Fig.10) also has a minimum followed
by increment at high temperatures. $\sigma^{33}$ and $\sigma^{22}$ may be
distinguished at non-zero magnetic fields, and $\sigma^{33}$ is always greater
than $\sigma^{22}$;

    * –

the longitudinal direction $\sigma^{11}$ monotonically decreases, see Fig.12.

It is instructive to compare conductivity for light and heavy quarks. Let us
remind that for heavy quark model we had Arefeva:2021jpa  the following
picture.

  * •

For $\nu=1$

    * –

for temperatures higher than critical the ratios $\sigma^{ii}/T$ increase to
the constant values depending on $\mu$ and $c_{B}$;

    * –

increasing $\mu$ or magnetic field increases the values of these constants;

    * –

conductivities $\sigma^{22}$ and $\sigma^{33}$ are almost indistinguishable
for small values of an external magnetic field.

  * •

For $\nu=4.5$, the conductivities along

    * –

transversal directions $\sigma^{22}/T$ and $\sigma^{33}/T$ increase with
higher temperature and saturate a certain value
$\sigma^{22}/T\approx\sigma^{33}/T$ depending on the values of $\mu$ and
magnetic field;

    * –

longitudinal direction $\sigma^{11}/T$ approaches zero at high temperatures.

Thus, we see that for $\nu=1$ the behavior of the conductivities for heavy and
light quarks is different at temperatures close to critical, but at high
temperatures they saturate some constant values and these values increase with
increasing chemical potential and magnetic field. The saturation constants are
equivalent at zero $\mu$ and $c_{B}$. Isotropic QGP for light quarks as well
as for heavy quarks is almost transparent in all directions near critical
temperature and become opaque values at high temperatures. The main difference
between them may be seen at intermediate temperatures.

For $\nu=4.5$ the behavior of the conductivities for heavy and light quarks is
different at temperatures close to critical as well as for high temperatures.
In particular, there is no saturation of $\sigma^{33}$ and $\sigma^{22}$ at
high temperatures for light quarks, meanwhile these components of conductivity
for heavy quarks reach a constant value at high temperature. The longitudinal
component of conductivity for heavy quarks decreases to zero as it does for
light quarks.

There are some obvious open questions regarding our model that are worth
exploring. Let us mentioned some of them. Like we perform the study of quark
potential for in heavy quark model Arefeva:2019yzy , it would be interesting
to study it for light quark model considered here.

It would be interesting to study the behavior of the entanglement entropy (EE)
for our model. Note that EE was studied in the Arefeva:2018hyo  model and was
not studied either in the case of the presence of an external magnetic field
for the heavy quark model Arefeva:2020vae , or in the absence of an external
magnetic field for the light quark model Arefeva:2020byn .

In the spirit of studying jet quenching Ageev:2017qpa ; Ageev:2016gtl  for the
simplest Arefeva:2018hyo  model, it would be interesting to study the behavior
of jet quenching for the model considered in this article. Of particular
interest is the study of the influence of an external magnetic field on the
results.

In this paper we determine the MC/IMC as the increase/decrease of the first
order phase transition temperature with increasing the value of magnetic
field. In lattice calculations the MC and IMC are determined as
increase/decrease of the chiral phase transition temperature with increasing
the value of magnetic field Fukushima:2012kc ; Andersen:2014xxa ;
Miransky:2015ava . Note that the chiral phase transition coincides with the
first order phase transition in the holographic model for heavy quarks
Li:2020hau , see also studies of chiral condensate in holography in Li:2016smq
; Fang:2019xbk ; Ballon-Bayona:2020qpq ; Colangelo:2020tpr ; Bohra:2020qom
and refs therein. The study of quark condensate and chiral phase transition is
certainly of interest for the structure of phase transitions for light quarks.

##  6 Acknowledgments

This work is supported by Russian Science Foundation grant 20-12-00200.

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