#  Holographic QCD Running Coupling for Light Quarks in Strong Magnetic Field

Irina Ya. Aref’evaa, Ali Hajiloua, Alexander Nikolaeva, Pavel Slepova

###### Abstract

We consider running coupling constant in holographic model with external
magnetic field supported by Einstein-dilaton-three-Maxwell action. We obtain a
significant dependence of the running coupling constant $\alpha$ on chemical
potential, temperature and magnetic field. We use the boundary condition that
ensures the agreement with lattice calculations of string tension between
quarks at zero chemical potential. The location of the 1-st order phase
transitions in $(\mu,T)$-plane does not depend on the dilaton boundary
conditions. We observe that running coupling $\alpha$ decreases with
increasing magnetic field for the fixed values of chemical potential and
temperature. At the 1-st order phase transitions the functions $\alpha$
undergo jumps depending on temperature, chemical potential and magnetic field.

##  1 Introduction

The goal of this paper is to study of the running coupling behavior in strong
external magnetic field. This paper is the generalization of our consideration
of running coupling in the isotropic holographic QCD [1] to anisotropic QCD
[3, 2] originated by external magnetic field. We discuss here only the case of
the light quarks model [2]. The running coupling in QCD can be found
experimentally in wide range of energy. These data mainly concern the case of
small density (small baryon chemical potential) available at LHC and RHIC. The
last experimental data can be found in [4]. One can also compare the
holographic results obtained in [1] with experimental data [5]; see also [6,
7, 8, 9] where holographic data obtained within other holographic models have
been compared with experimental data. To obtain the dependence of the running
coupling on the energy $E$ and the square of the transferred momenta $Q^{2}$,
it is necessary to present a correspondence between them [6, 7, 8, 9]. In this
research we investigated the dependence of the running coupling on the
holographic coordinate $z$ that is related to the energy scale of QCD,
corresponding to the warp factor.

In particular, it would be interesting to consider the holographic isotropic
models of light and heavy quarks to study the running coupling as a function
of the energy scale of QCD and try to fit the results of holographic models
with experiments such as [7, 6]. The dilaton field plays a crucial role in the
holographic approach to study the running coupling and it would be interesting
to find out the effect of imposed proper boundary condition on the running
coupling as a function of the energy scale of QCD for light and heavy quarks.
In addition, in [8] considering holographic approach and using special form of
Fourier transformation from ($z$ coordinate) 5-dimensional space-time in
gravity side to the 4-dimensional one, the behavior of $\alpha$ as a function
of momentum transfer, $Q^{2}$, is studied. We will apply this approach to our
model in future studies. In this paper to obtain the dependence of the running
coupling on the energy $E$ and the square of the transferred momenta $Q^{2}$,
we use a simplest relation between them given by the prefactor in the
5-dimensional metric [6, 7, 5].

The QCD phase transition diagram in the (chemical potential, temperature)
plane is both a challenging and significant topic in high-energy physics. This
area of study is not only of fundamental interest but also has important
implications for the early evolution of the Universe [10] and the interior of
compact stellar objects [11, 12]. One of the primary goals of experiments at
the LHC, RHIC, NICA, and FAIR is to explore the phase structure of strong-
interaction matter under extreme conditions. Traditional theoretical methods
for performing QCD calculations, such as perturbation theory, fail in the
strongly coupled regime. Additionally, lattice theory faces a sign problem for
non-zero chemical potential calculations and, at its current level, cannot
provide adequate information. To describe the physics of the strongly coupled
quark-gluon plasma (QGP) produced in heavy ion collisions (HIC) at RHIC, LHC,
and NICA (see for example [13]), as well as in future experiments, non-
perturbative approaches are required. Over the last three decades, holographic
methods [14] have proven to be a powerful non-perturbative approach suitable
for these purposes [15, 16, 17].

There are several experimental methods to study the QCD phase transition
diagram. One of these methods is the Beam Energy Scan (BES) method [18, 19].
The BES method involves scanning a range of collision energies to search for
signs of a 1-st order phase transition, such as changes in particle yields or
the onset of collective phenomena. Nonmonotonic behavior in both beam energy
and impact parameter dependences, if observed, can be used to identify such a
phase transition.

Another method to identify the 1-st order phase transition is the study of
strange particle production. Anomalies in the production of strange particles,
such as kaons and hyperons, may signal the presence of a 1-st order phase
transition [20]. Analyzing higher-order correlations and cumulants of particle
distributions can provide evidence for critical phenomena associated with 1-st
order phase transitions [21]. Finally, the measurement of elliptic flow as a
function of collision energy can reveal changes in the medium’s properties,
potentially indicating a 1-st order phase transition [22].

Here we would like to discuss another possibility. As shown in [1],
Holographic QCD (HQCD) predicts discontinuities in the dependence of the
running coupling on temperature $T$ and chemical potential $\mu$ akin to a
1-st order phase transition in the $(\mu,T)$-plane. From the positions of
these discontinuities, one can predict the locations of 1-st-order phase
transitions in the $(\mu,T)$-plane. Naturally, this necessitates access to
high-density matter, specifically high chemical potentials, which may be
attainable in future experiments such as NICA and FAIR.

During collisions of heavy ions, strong magnetic fields emerge, which
subsequently influence the behavior of the running coupling constant.
Therefore, it is crucial to assess the impact of strong magnetic fields on the
running coupling in HQCD, or in other words, to determine how the results [1]
are modified in the presence of a magnetic field. This constitutes the primary
objective of this paper. As noted in [1], the structure of the QCD phase
diagram is significantly influenced by the masses of the quarks. Here, for
simplicity, we focus on the scenario involving light quarks.

The typical behavior of the running coupling $\alpha$, specifically the
logarithm of the running coupling $\log\alpha$, as predicted by holographic
models, has been presented in our paper [1]. This behavior is depicted at
fixed energy scales, various temperatures, and chemical potentials, see also
Fig. 4 below. A distinctive feature observed in different parts of this graph
is the appearance of jumps in the coupling constant, indicating 1-st order
phase transitions along the same lines in the ${\mu,T}$ planes, regardless of
different values of energy scales. These jumps cease at the critical endpoint
(CEP), marking the termination of the critical points’ lines. For non-zero
magnetic fields, the 1-st order phase transition persists up to a certain
value of the parameter $c_{B}$, as illustrated in Fig. 1 and Fig. 5.

The paper is organized as follows. In Sect. 2 we present 5-dim holographic
models in the presence of non-zero magnetic field for light quarks. In Sect. 3
we describe the magnetic field influence on running coupling constant for the
light quarks model. In Sect. 4 we summarize obtained results. The paper is
supplemented by Appendix A, in which we briefly recall the basic facts about
our holographic model.

##  2 Holographic model

Studying the effect of external magnetic field on the QCD features via
holography are discussed in [3, 2, 23, 24, 25, 26, 27]. To investigate the
behavior of the running coupling constant we use the holographic model for the
light quarks in a magnetic field with the existence of 1-st order phase
transition [2, 28]. For heavy quarks case the similar model has been proposed
in [3]. These models generalize the corresponding isotropic models without
magnetic fields [30, 29] and anisotropic ones with spacial anisotropy are
related to the non-central HIC [31, 32, 26, 33, 34].

###  2.1 Black hole solution with three Maxwell fields

The 5-dimensional action in the Einstein frame has the following form [2]

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

Here we use the ansatz, where the non-zero components of the electro-magnetic
field and the 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) $R$ is Ricci scalar, $\varphi=\varphi(z)$ is the scalar field,
$f_{1}(\varphi)$, $f_{2}(\varphi)$ and $f_{B}(\varphi)$ 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 constants and
$V(\varphi)$ is the scalar field potential. Thus (2.1) is the extended version
of the action used in [33, 31], where we add an external magnetic field
$F_{\mu\nu}^{(B)}$.

We use the following ansatz for metric [2]:

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

Let us make some notes on the main parameters of the metric (2.3). The
difference between “heavy quarks” and “light quarks” cases lies in the form of
the scale factor ${\mathcal{A}}(z)$ in the warp factor. For the heavy quarks
model see [26] and for simple choice of ${\mathcal{A}}(z)=-\,s\,z^{2}/4$ [33].
To get the “light quarks” case we follow [29] and assume
${\mathcal{A}}(z)=-\,a\ln(bz^{2}+1)$ where the parameters $a=4.046$ and
$b=0.01613$ GeV2 can be obtained by fitting with experimental data [29].

Here parameters $\nu$ and $c_{B}$ have the same meaning as in [3]. The primary
anisotropy parameter $\nu$ describes non-equivalence between longitudinal and
transversal directions. The isotropic case corresponds to $\nu=1$ (that we
used in this paper), while $\nu=4.5$ reproduces the multiplicity of the
charged particles production [32].

We refer to $c_{B}$ as to the magnetic field parameter that describes the non-
centrality of the HIC. Although explicit relation of $c_{B}$ to the magnitude
of the magnetic field is not established, there are considerations suggesting
that $B^{2}=-\,c_{B}$ ${{GeV}^{2}}$ [25, 35, 36]. Therefore, we have
$c_{B}\leq 0$ in all our calculations. Magnetic field parameter $c_{B}$ is an
implicit function of the magnetic charge $q_{B}$ from the ansatz (2.2).
However, we will not need the explicit form of this relation in what follows.
In our model we have two independent parameters, i.e. $c_{B}$ and $q_{B}$ in
the metric (2.4) and action (2.1), respectively. We usually set $q=q_{B}=1$ in
our numerical calculations and vary $c_{B}$. The metric ansatz (2.3) also
contains the blackening function $g(z)$ which has crucial effect to obtain the
temperature and entropy of the black hole.

Applying the stationary action principle, we get the same equations of motion
(EOM) [2, 28, 3]:

$\displaystyle\begin{split}\varphi^{\prime\prime}&+\varphi^{\prime}\left(\cfrac{g^{\prime}}{g}+\cfrac{3\mathfrak{b}^{\prime}}{2\mathfrak{b}}-\cfrac{\nu+2}{\nu
z}+c_{B}z\right)+\left(\cfrac{z}{L}\right)^{2}\cfrac{\partial
f_{1}}{\partial\varphi}\ \cfrac{(A_{t}^{\prime})^{2}}{2\mathfrak{b}g}-\\\
&-\left(\cfrac{L}{z}\right)^{2-\frac{4}{\nu}}\cfrac{\partial
f_{2}}{\partial\varphi}\ \cfrac{q^{2}\ e^{-c_{B}z^{2}}}{2\mathfrak{b}g}\
-\left(\cfrac{z}{L}\right)^{\frac{2}{\nu}}\cfrac{\partial
f_{B}}{\partial\varphi}\ \cfrac{q_{B}^{2}}{2\mathfrak{b}g}\
-\left(\cfrac{L}{z}\right)^{2}\cfrac{\mathfrak{b}}{g}\ \cfrac{\partial
V}{\partial\varphi}=0,\end{split}$ (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)
$\displaystyle\mathfrak{b}^{\prime\prime}-\cfrac{3(\mathfrak{b}^{\prime})^{2}}{2\mathfrak{b}}+\cfrac{2\mathfrak{b}^{\prime}}{z}-\cfrac{4\mathfrak{b}}{3\nu
z^{2}}\left(\cfrac{\nu-1}{\nu}+\left(1-\cfrac{3\nu}{2}\right)c_{B}z^{2}-\cfrac{\nu
c_{B}^{2}z^{4}}{2}\right)+\cfrac{\mathfrak{b}\,(\varphi^{\prime})^{2}}{3}=0,$
(2.8) $\displaystyle 2g^{\prime}\ \cfrac{\nu-1}{\nu}+3g\
\cfrac{\nu-1}{\nu}\left(\cfrac{\mathfrak{b}^{\prime}}{\mathfrak{b}}-\cfrac{4\left(\nu+1\right)}{3\nu
z}+\cfrac{2c_{B}z}{3}\right)+\left(\cfrac{L}{z}\right)^{1-\frac{4}{\nu}}\cfrac{L\,q^{2}\,e^{-c_{B}z^{2}}f_{2}}{\mathfrak{b}}=0,$
(2.9)
$\displaystyle\begin{split}2g^{\prime}\left(1-\cfrac{1}{\nu}+c_{B}z^{2}\right)&+3g\left[\Big{(}1-\cfrac{1}{\nu}+c_{B}z^{2}\Big{)}\left(\cfrac{\mathfrak{b}^{\prime}}{\mathfrak{b}}-\cfrac{4}{3\nu
z}+\cfrac{2c_{B}z}{3}\right)-\cfrac{4\left(\nu-1\right)}{3\nu z}\right]+\\\
&+\left(\cfrac{L}{z}\right)^{1-\frac{4}{\nu}}\cfrac{L\,q^{2}\,e^{-c_{B}z^{2}}\,f_{2}}{\mathfrak{b}}-\left(\cfrac{z}{L}\right)^{1+\frac{2}{\nu}}\cfrac{L\,q_{B}^{2}f_{B}}{\mathfrak{b}}=0,\end{split}$
(2.10)
$\displaystyle\begin{split}\cfrac{\mathfrak{b}^{\prime\prime}}{\mathfrak{b}}&+\cfrac{(\mathfrak{b}^{\prime})^{2}}{2\mathfrak{b}^{2}}+\cfrac{3\mathfrak{b}^{\prime}}{\mathfrak{b}}\left(\cfrac{g^{\prime}}{2g}-\cfrac{\nu+1}{\nu
z}+\cfrac{2c_{B}z}{3}\right)-\cfrac{g^{\prime}}{3zg}\left(5+\cfrac{4}{\nu}-3c_{B}z^{2}\right)+\\\
&+\cfrac{8}{3z^{2}}\left(1+\cfrac{3}{2\nu}+\cfrac{1}{2\nu^{2}}\right)-\cfrac{4c_{B}}{3}\left(1+\cfrac{3}{2\nu}-\cfrac{c_{B}z^{2}}{2}\right)+\cfrac{g^{\prime\prime}}{3g}+\cfrac{2}{3}\left(\cfrac{L}{z}\right)^{2}\cfrac{\mathfrak{b}V}{g}=0.\end{split}$
(2.11)

Turning off the external magnetic field, i.e. putting $c_{B}=q_{B}=f_{B}=0$
into (2.5)–(2.11), we get the EOM from [31]. Normalizing to the AdS-radius,
$L=1$, we get the EOM from [33]. Excluding special anisotropy, i.e. putting
$\nu=1$ and $f_{2}=0$, we get the expressions that fully coincide with the EOM
from [29, 30]. Thus (2.5)–(2.11) are universal anisotropic EOM, appropriate
both for heavy and light quarks description, that include solution from [29,
30] as an isotropic limit. To solve the EOMs, We utilize the general form of
the boundary conditions:

$\displaystyle A_{t}(0)=\mu,\quad A_{t}(z_{h})=0,$ (2.12) $\displaystyle
g(0)=1,\quad g(z_{h})=0,$ (2.13) $\displaystyle\varphi(z_{0})=0\,,$ (2.14)

where $z_{0}=0$ corresponds to [29] and $z_{0}=z_{h}$ to [33]. The choice of
the boundary condition for the scalar field was discussed in details in [31].

###  2.2 Thermodynamics and background phase transition

For the metric (2.3), the temperature and entropy for the light quarks model
can be written as:

$\displaystyle T=\cfrac{|g^{\prime}|}{4\pi}\,\Bigl{|}_{z=z_{h}},\qquad
s=\left(\cfrac{L}{z_{h}}\right)^{1+\frac{2}{\nu}}\cfrac{e^{c_{B}z_{h}^{2}/2}}{4\left(1+bz_{h}^{2}\right)^{3a}}.$
(2.15)

Then, the phase diagrams for light quarks model can be obtained via free
energy consideration

$\displaystyle F=-\int s\,dT=\int_{z_{h}}^{\infty}s\,T^{\prime}dz,$ (2.16)

where we normalized 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). Using free energy (2.16) phase
diagram $(\mu,T)$-plane for light quarks model describing the isotropic case
$c_{B}=0$ and anisotropic cases with different $c_{B}$ is obtained in Fig. 1.
Since in this paper we concentrated on just two different cases, i.e.
$c_{B}=0$ and $c_{B}=-0.05$ ${{GeV}^{2}}$. In the Fig. 1 the associated CEPs
are marked with magenta stars.

Figure 1: Phase diagram $(\mu,T)$-plane for light quarks model describing the
isotropic case $c_{B}=0$ and anisotropic cases with different $c_{B}$. The
magenta star indicates the CEP; $[c_{B}]=$ GeV2

The energy scale $E$ (GeV) as a function of the holographic coordinate $z$
(GeV-1) corresponding to the prefactor of the metric (2.17) is defined as [7]:

$E=\frac{\sqrt{\mathfrak{b}(z)}}{z}=\frac{\left(bz^{2}+1\right)^{-a}}{z}\,,$
(2.17)

where parameters $a$ and $b$ were introduced in Sect. 2.1. The energy scale as
a function of the holographic coordinate $E(z)$ is presented in Fig. 2

Figure 2: Energy scale $E$ (GeV) as a function of the holographic coordinate
$z$ (GeV-1) corresponding to the warp factor $\mathfrak{b}(z)$.

In Fig. 3 2D plots in $(\mu,z_{h})$-plane for $c_{B}=0$ (A) and $c_{B}=-0.05$
(B) are depicted. The magenta curves show coordinates of the 1-st order phase
transition and the black contours correspond to the fixed temperatures and the
brown one corresponds to the $T=0$. The magenta star is CEP. The magenta
curves on each plot consist of two branches: the upper branch and the lower
one are connected at the ”star” (CEP). These two branches are identified
according to the same values of $\mu$ and $T$. The magenta lines indicate the
size of the black holes horizons. Therefore, the regions of the stable black
holes solutions are located above the upper magenta curve and below the lower
branch. The area between the upper and lower branches of the magenta curves
corresponds to the unstable region.

A

B

Figure 3:  2D plots in $(\mu,z_{h})$-plane for $c_{B}=0$ (A) and $c_{B}=-0.05$
GeV2 (B). The magenta curves show coordinates of the 1-st order phase
transition; the black contours correspond to the fixed temperatures and the
brown one corresponds to the $T=0$. The magenta star is CEP; $[T]=$ GeV.

##  3 Running coupling in the strong magnetic field

The characteristic feature of the holographic coupling constant $\alpha$ is
its dependence on the dilaton field and is defined as [37, 38, 6]

$\alpha(z)=e^{\varphi(z)}\,.$ (3.1)

The dilaton field $\varphi$ itself depends on the chosen boundary condition.
We denote the dilaton field with the zero boundary condition at zero
holographic coordinate as $\varphi=\varphi_{0}(z)$, i.e.

$\varphi_{0}(z)\Big{|}_{z=0}=0.$ (3.2)

The dilaton field with the boundary condition at the holographic coordinate
$z=z_{0}$ is denoted by $\varphi_{z_{0}}(z)$. These two solutions are related
by a simple relation

$\varphi_{z_{0}}(z)=\varphi_{0}(z)-\varphi_{0}(z_{0}),$ (3.3)

that gives

$\alpha_{z_{0}}(z)=\alpha_{0}(z)\,\mathfrak{G}(z_{0}),\quad\qquad\alpha_{0}(z)=e^{\varphi_{0}(z)},\quad\mathfrak{G}(z_{0})=e^{-\varphi_{0}(z_{0})}$
(3.4)

Note that $\varphi_{0}(z)$ does not depend on the thermodynamic
characteristics of the model, see [1]. To obtain the dependence of the running
coupling on thermodynamic quantities such as $T$ and $\mu$, one can take
$z_{0}$ related with $z_{h}$, i.e. $z_{0}=\mathfrak{z}(z_{h})$, and get

$\alpha_{\mathfrak{z}}(z;T,\mu)=\alpha_{0}(z)\,\mathfrak{G}(T,\mu),\qquad\quad\mathfrak{G}(T,\mu)=e^{-\varphi_{0}(\mathfrak{z}(z_{h}))}$
(3.5)

One such choice is $\mathfrak{z}(z_{h})=z_{h}$, and the other is given by
exponential function that is named as a second boundary condition at $z_{0}$
[1]:

$z_{0}=\mathfrak{z}(z_{h})=10\,\exp(-\frac{z_{h}}{4})+0.1\,,$ (3.6)

where dilaton field is zero at $z_{0}$, see (2.14). It is important to note
the boundary condition (3.6) that introduced in [31] was obtained by studying
the behavior of QCD string tension as a function of temperature at zero
chemical potential for isotropic case acquired via lattice calculations [39].
This boundary condition (3.6) also is used in this research for anisotropic
model, although there is no lattice results for the anisotropic $c_{B}\neq 0$
case.

To study the behavior of the running coupling constant for magnetic fields
$c_{B}=0$ and $c_{B}=-0.05$ , we utilize the Fig. 3 to respect the physical
regions of the model which is limited by the sizes of black holes, $z_{h}$,
indicate a 1-st order phase transition.

In Fig. 4 density plots with contours for $\log\alpha_{\mathfrak{z}}(E;\mu,T)$
at different energy scales $E=\\{0.5771,0.9373,1.9677\\}$ (GeV) for $c_{B}=0$
are arranged in such a way that they represent a family of sections of three-
dimensional space $(E,\mu,T)$. The selected values of energy scale $E$
correspond to holographic coordinate $z=\\{0.5,\,1,\,1.5$} (GeV-1) according
to the Fig. 2. Each plot has several contours corresponding to fixed values of
$\log\alpha_{\mathfrak{z}}$, these values are indicated in squares. The red
lines in the first two graphs on the left indicate the maximum possible
temperature values at the corresponding energy scale $E$. These restrictions
arise due to the constraint $z\leq z_{h}$. The plot in Fig. 4 actually
reproduces the plot presented earlier in the paper [1]. It can be seen that as
we increase the energy scale, the contours of the fixed values of the running
coupling constant go down to lower temperature. In the paper [1] has been
shown the monotonic behavior of running coupling as a function of temperature.
Thus, we can conclude as the energy scale increases, the running coupling
constant decreases at a fixed temperature.

Figure 4: Density plots with contours for $\log\alpha_{\mathfrak{z}}(E;\mu,T)$
at different energy scales $E=\\{0.5771,0.9373,1.9677\\}$ (GeV) for $c_{B}=0$.
The red lines in the first two graphs on the left indicate the maximum
possible temperature values at the corresponding energy scale $E$.

In Fig. 5 density plots with contours for $\log\alpha_{\mathfrak{z}}(E;\mu,T)$
at different energy scales $E=\\{0.5771,0.9373,1.9677\\}$ (GeV) for
$c_{B}=-0.05$ ${{GeV}^{2}}$ are arranged in such a way that they represent a
family of sections of three-dimensional space $(E,\mu,T)$. In hadronic phase
of Fig. 4 in the isotropic case the running coupling constant almost doesn’t
depend on the chemical potential. For the anisotropic case Fig. 5 this effect
becomes weaker.

Comparing Fig. 4 and Fig. 5 at some fixed parameters $(E,\mu,T)$ shows that
increasing magnetic field decreases value of the running coupling constant.

Figure 5: Density plots with contours for $\log\alpha_{\mathfrak{z}}(E;\mu,T)$
at different energy scales $E=\\{0.5771,0.9373,1.9677\\}$ (GeV) for
$c_{B}=-0.05$ GeV2.

##  4 Conclusion

In this paper we have considered an anisotropic holographic model including a
strong magnetic field for the light quarks [28]. It is characterized by the
Einstein-dilaton-three-Maxwell action and the 5-dimensional metric with the
warp factor that has been considered in isotropic case for the light quarks
[29]. Our model has two different types of anisotropy, i.e. the spatial
anisotropy related to the parameter $\nu$, and the anisotropy related to the
external magnetic field described by the third Maxwell field. Particularly, in
this paper we considered the second type of anisotropy, i.e. non-zero external
magnetic field, and ignored the spatial anisotropy.

It is also very important to note that different boundary conditions lead to
different physical consequences. Only (3.6) boundary condition for light
quarks leads to physical results in accordance with lattice calculations for
non-zero temperature and zero chemical potential [40].

  * •

Magnetic field decreases the value of the running coupling constant $\alpha$
at fixed values of the $T$ and $\mu$.

  * •

The running coupling $\alpha$ decreases with increasing of the energy scale
$E$ at the fixed values of $T$ and $\mu$, and such a behavior also persists in
the presence of the magnetic field.

  * •

In the region of the hadronic phase, the running coupling $\alpha$ changes
slowly with changing in chemical potential, both for an isotropic $c_{B}=0$
and for an anisotropic $c_{B}\neq 0$ cases.

  * •

On the 1-st order transition line, the running coupling has a jump. The
magnitude of the jump decreases along the line of the 1-st order transition to
get zero in the CEP.

Let us remind that the twice anisotropic model for the light quarks is much
more complicated than the twice anisotropic model for heavy quarks [3], since
the deformation factor and the kinetic gauge function $f{{}_{1}}$ have more
compound form. The effect of magnetic field on the running coupling $\alpha$
in heavy quarks model will be studied in future works.

##  5 Acknowledgments

We thank K. Runnu and M. Usova for useful discussions. The work of I.A., P.S.,
A.N is supported by RNF grant 20-12–00200. The work of A.H. was performed at
the Steklov International Mathematical Center and supported by the Ministry of
Science and Higher Education of the Russian Federation (Agreement
No.075-15-2022-265).

Appendix

##  Appendix A Solution to EOMs

To solve EOM (2.5)–(2.11) we need to determine the form of the coupling
function $f_{1}$. We take them as in the paper [2]. Taking into account the
“light quarks” warp factor, we get

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

where $c=0.227$ GeV2. Solving (2.6) with the coupling function (A.1) and
boundary conditions (2.12) gives

$\displaystyle
A_{t}=\mu\,\cfrac{e^{\left(2c-c_{B}\right)z^{2}/2}-e^{\left(2c-c_{B}\right)z_{h}^{2}/2}}{1-e^{\left(2c-c_{B}\right)z_{h}^{2}/2}}\,.$
(A.2)

We get an expression for the function $f_{B}$

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

and get the solution for blackening function:

$\displaystyle
g=e^{c_{B}z^{2}}\left[1-\cfrac{I_{1}(z)}{I_{1}(z_{h})}+\cfrac{\mu^{2}\,(2c-c_{B})\,I_{2}(z)}{L^{2}\left(1-e^{\left(2c-c_{B}\right)z_{h}^{2}/2}\right)^{2}}\left(1-\cfrac{I_{1}(z)}{I_{1}(z_{h})}\,\cfrac{I_{2}(z_{h})}{I_{2}(z)}\right)\right],$
(A.4)

where

$\displaystyle
I_{1}(z)=\int_{0}^{z}\left(1+b\xi^{2}\right)^{3a}e^{-3c_{B}\xi^{2}/2}\,\xi^{1+\frac{2}{\nu}}\,d\xi,$
(A.5) $\displaystyle
I_{2}(z)=\int_{0}^{z}\left(1+b\xi^{2}\right)^{3a}e^{\left(c-2c_{B}\right)\xi^{2}}\,\xi^{1+\frac{2}{\nu}}\,d\xi.$
(A.6)

Further solving (2.5)–(2.11) system with the coupling functions (A.1) and
(A.3) and the boundary condition (2.14) gives

$\displaystyle
f_{2}=2\left(\cfrac{z}{L}\right)^{2-4/\nu}\,\cfrac{\nu-1}{q^{2}\nu^{2}z^{2}}\,\cfrac{e^{c_{B}z^{2}}}{(1+bz^{2})^{2a}}\left\\{\left(2+2\nu-\nu
c_{B}z^{2}+\cfrac{6ab\nu z^{2}}{1+bz^{2}}\right)g-g^{\prime}\nu z\right\\},$
(A.7)
$\displaystyle\varphi=\int_{z_{0}}^{z}\sqrt{\cfrac{4(\nu-1)}{\nu^{2}\xi^{2}}-2c_{B}\left(3-\cfrac{2}{\nu}\right)-2c_{B}^{2}\xi^{2}+\cfrac{12ab}{1+b\xi^{2}}\left(1+2\,\cfrac{1+ab\xi^{2}}{1+b\xi^{2}}\right)}\,d\xi,$
(A.8)
$\displaystyle\varphi^{\prime}=\sqrt{\cfrac{4(\nu-1)}{\nu^{2}z^{2}}-2c_{B}\left(3-\cfrac{2}{\nu}\right)-2c_{B}^{2}z^{2}+\cfrac{12ab}{1+bz^{2}}\left(1+2\,\cfrac{1+abz^{2}}{1+bz^{2}}\right)},$
(A.9) $\displaystyle\begin{split}&V(z)=-\,\cfrac{(1+bz^{2})^{2a}\ z}{2L^{2}}\
\times\\\
&\left[2\left\\{2\,\cfrac{(1+\nu)(1+2\nu)+\bigl{(}2+(2+3a)(3+2\nu)\nu\bigr{)}bz^{2}+(1+\nu+3a\nu)(1+2\nu+6a\nu)b^{2}z^{4}}{(1+bz^{2})^{2}\nu^{2}z^{2}}\right.\right.\\\
&\left.-\left.c_{B}\left(2+\cfrac{3}{\nu}+\cfrac{12abz^{2}}{1+bz^{2}}\right)+c_{B}^{2}z^{2}\right\\}g\,z-\left(5+3\left(\cfrac{6ab}{1+bz^{2}}-c_{B}\right)z^{2}+\cfrac{4}{\nu}\right)g^{\prime}+g^{\prime\prime}z\right].\end{split}$
(A.10)

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)–(2.11) not selfconsistent, therefore proper solution with the
$f_{B}=f_{1}$ condition can’t be found.

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