aainstitutetext: Center for Theoretical Physics and College of Physics, Jilin
University, Changchun 130012, People’s Republic of Chinabbinstitutetext: Max
Planck Institute for Gravitational Physics (Albert Einstein Institute), Am
Mühlenberg 1, 14476 Golm, Germanyccinstitutetext: Department of
Electrophysics, National Chiao Tung University, Hsinchu, ROCddinstitutetext:
Physics Division, National Center for Theoretical Sciences, Hsinchu,
ROCeeinstitutetext: School of physics, University of Chinese Academy of
Sciences, Beijing 100049, Chinaffinstitutetext: Kavli Insititute for
Theoretical Sciences, University of Chinese Academy of Sciences, Beijing
100049, China

# Analytic Study of Magnetic Catalysis in Holographic QCD

Song Hea,b, Yi Yangc,d, Pei-Hung Yuane,f

###### Abstract

We explore the effect of the magnetic field on the QCD phase transition
through AdS/CFT correspondence. By introducing an anisotropic magnetic field
in the Einstein-Maxwell-Scalar system, a family of analytic solutions is
obtained by the potential reconstruction method where the contribution of the
magnetic field in the blackening background can be analytically derived. After
imposing the kinetic gauge function by requesting the linear Regge spectrum of
$J/\psi$ mesons, the contribution of the magnetic field phase diagram can be
demonstrated. The results show that the transition temperature will be raising
as the magnetic field increases, which is the so-call magnetic catalysis
effect. However, if the system is in a strong enough magnetic environment, the
transition temperature will be cool down and display the inverse catalysis
effect.

##  1 Introduction

The recent experiments of heavy ion collision at the Relativistic Heavy Ion
Collider (RHIC) and the Large Hadron Collider (LHC) have produced a strongly
coupled quark-gluon plasma (QGP) and explored the QCD phase structure under
finite baryon density and strong magnetic fields 0907.1396 ; 1103.4239 ;
1111.1949 ; 1201.5108 . In addition, strong magnetic fields are present in
neutron stars and magnetars Astrophys.J.392(1992)L9 ; 0804.0250 , as well as
in the cosmological phase transition during the early stages of the universe
9308270 . Therefore, understanding the effect of magnetic field on QCD phase
transition is an important task in fundamental physics.

The early investigation by lattice QCD showed that the phase transition
temperature increases with the external magnetic field, i.e. magnetic
catalysis (MC) 1005.5365 ; 1203.3360 ; 1209.0374 . The MC phenomenon has been
confirmed by many effective QCD theories 9405262 ; 9412257 ; 9509320 ; 9905116
; 0010211 ; 0205348 ; 0701090 ; 0810.3693 ; 1003.0047 ; 1004.2712 ; 1012.1291
; 1104.5167 ; 1107.4737 ; 1112.5137 ; 1203.4330 ; 1207.5081 ; 1208.0917 ;
1305.4751 ; 1311.3964 ; 1312.6733  as well as holographic QCD (hQCD) models
1110.5902 ; 1511.04042 . See 1207.5081  for a review. However, later lattice
simulations revealed the opposite results by considering the physics quark
mass that the phase transition temperature decreases with the external
magnetic field, i.e. inverse magnetic catalysis (IMC) 1103.2080 ; 1111.4956 ;
1206.4205 ; 1303.1328 ; 1303.3972 ; 1310.7876 ; 1312.5628 ; 1504.08280 ;
1607.08160 ; 1711.02884 ; 1909.09547 . The IMC phenomenon has been
investigated in numerous literature 0803.3156 ; 1201.5881 ; 1207.3190 ;
1207.7094 ; 1208.0536 ; 1208.5025 ; 1209.1319 ; 1211.6245 ; 1305.1100 ;
1404.5577 ; 1404.6969 ; 1406.3885 ; 1406.7408 ; 1407.3503 ; 1410.5247 ;
1412.1647 ; 1502.08011 ; 1509.01181 ; 1602.06448 ; 1603.03847 ; 1712.08384 ;
1907.01852 . Theoretically, we still do not fully understand the response of
the microscopic mechanism for the magnetic field effects. The one
interpretation is that the effect of magnetic field can reduce the number of
the effective dimensions of the quarks, which leads to the chiral condensation
1207.5081 ; 1211.6245 . Comparing with the heavy quark, increasing
temperature, the light sea quarks can be excited easily which induces chiral
condensation of the light quarks. Furthermore, it is believed that the
corresponding effects are due to the competition between the direct valence
effect and the indirect sea effect 1103.2080 ; 1303.3972 ; 1808.07008 ;
1904.10296 . The valence effect induces dynamical mass generation and always
causes MC phenomenon. While including the back-reaction of the sea quarks on
the gauge fields and the screening effect of the gluon interactions will
suppress the phase transition and causes the IMC phenomenon. On the other
hand, the recent lattice simulations have demonstrated that the magnetic
effect depends on quark mass. For light quarks, the sea effect wins the
competition and induces IMC phenomenon. While for heavy quarks, the valence
effect becomes more important and the phase transition behaves as MC
phenomenon 1808.07008 ; 1904.10296 .

Investigating the QCD phase transitions has been a long crusade to understand
the fundamental physics and astrophysics for decades. Since most perturbative
QCD calculations and effective models suggest MC, people believe that IMC is
due to the strongly coupled dynamics near the phase transition. However, the
standard perturbative method can not treat the issue well because of the
strongly coupled region. In addition, the non-perturbative techniques such as
lattice QCD are facing the sign problem for finite baryon densities. In order
to reveal the whole picture in the ”temperature-chemical potential” plane, the
holographic framework 9711200 ; 9802109 ; 9802150  is a good candidate to
fulfill the job. In hQCD, phase transitions in the presence of external
magnetic field has been previously studied for both confinement-deconfinement
phase transition 1501.03262 ; 1505.07894 ; 1511.02721 ; 1604.07197 ;
1709.09258 ; 1710.07310 ; 1811.11724  and chiral condensation 1010.0444 ;
1012.4785 ; 1110.5902 ; 1208.0536 ; 1303.5674 ; 1307.6498 ; 1511.02721 ;
1511.04042 ; 1511.05293 ; 1512.06493 ; 1602.08994 ; 1604.06307 ; 1610.04618 ;
1611.06339 ; 1706.05977 ; 1707.00872 ; 1807.11822 ; 1811.04117 .

One of the ingredients of MC/IMC phenomena is anisotropy. A series study in
1611.06339 ; 1707.00872 ; 1708.05691 ; 1811.11724  reveals that the isotropy
could be broken not only by introducing a magnetic field but also a relevant
or marginal operator. In this work, we study the magnetic effect on phase
transition in QCD for heavy quarks by holographic correspondence. We are going
to explore how the phase transition is affected by the external magnetic
field, which plays the role of anisotropy. We study a hQCD model by
considering an uniform but anisotropic external magnetic field in the
5-dimensional Einstein-Maxwell-Scalar (EMS) system, which is dual to a
4-dimensional QCD theory. This background is a magnetic generalization of the
EMS frameworks considered in 1201.0820 ; 1301.0385 ; 1406.1865 ; 1506.05930 ;
1703.09184 ; 1705.07587 ; 1812.09676 , in which a family of analytic solutions
were obtained, and the phase diagram and the equations of states in QCD have
been extensively studied. To ensure the stability of the gravitational
background, we check the null energy condition (NEC), which induces a
constraint for the profile of scalar field. To study heavy quarks, we fit our
parameters by the Regge linear spectrum of $J/\psi$ mesons. By calculating the
phase transition temperature for different chemical potentials under the
varied magnetic field to obtain the phase structure of QCD. We find the MC/IMC
phenomenon for the small/large external magnetic field. In addition, we locate
the critical end point (CEP) for the QCD phase transition in $(T,\mu,B)$
3-dimensional phase diagram. We find that the CEP moves to the lower chemical
potential with a growing magnetic field. Furthermore, the CEP forms a closed
boundary in the $(\mu,B)$ plane, beyond which the phase transition becomes
crossover.

The organization of the remaining parts of this paper is as follows. The hQCD
model with anisotropic constant magnetic field has been constructed and a
class of analytic gravitational background solutions have been presented in
Sec. 2. The holographic phase transition and CEP with magnetic effects have
been investigated in Sec. 3. Finally, we devote to the conclusions and
discussions.

##  2 Holographic Model

Einstein-Maxwell-Scalar (EMS) system is one of the fundamental frameworks to
build holographic models effectively describing the phenomena of the strongly
coupled field theory. Using EMS system to construct hQCD models was advocated
in 1012.1864 ; 1108.2029 , and has been improved afterward in many literature.

In this work, we will investigate the phase structure in the presence of
external magnetic field in QCD analytically by using a well established
analytical hQCD model in 1301.0385 ; 1406.1865 ; 1506.05930 ; 1703.09184 ;
1705.07587 ; 1812.09676 . Hopefully we can shed a light on this issue and
deliver deeper insights for numerical simulations and experiments.

###  2.1 Holographic EMS System with Magnetic Field

We consider a 5-dimensional EMS system as a thermal background for the
corresponding QCD, which is constructed in Einstein frame as

$\displaystyle S_{B}$ $\displaystyle=$ $\displaystyle\frac{1}{16\pi G_{5}}\int
d^{5}x\sqrt{-g}\left[{R-\frac{f\left(\phi\right)}{4}F^{2}}-\frac{1}{2}\left(\partial\phi\right)^{2}-V\left(\phi\right)\right],$
(2.1)

where ${G}_{5}$ labels the 5-dimensional Newtonian constant, $R$ refers to the
Riemann scalar, $\phi$ is a neutral scalar field, and
${{F}}_{\mu\nu}=\partial_{\mu}{A}_{\nu}-\partial_{\nu}{A}_{\mu}$ represents
the field strength originated from the gauge field $A_{\mu}$. $f(\phi)$ is the
gauge kinetic function and $V(\phi)$ is the potential of the $\phi$. The
equations of motion are obtained by varying the Eq. (2.1)

$\displaystyle\nabla^{2}\phi$ $\displaystyle=$ $\displaystyle\frac{\partial
V}{\partial\phi}+\frac{F^{2}}{4}\frac{\partial f}{\partial\phi},$ (2.2)
$\displaystyle\nabla_{\mu}\left[f(\phi)F^{\mu\nu}\right]$ $\displaystyle=$
$\displaystyle 0,$ (2.3) $\displaystyle R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R$
$\displaystyle=$
$\displaystyle\frac{f(\phi)}{2}\left(F_{\mu\rho}F_{\nu}^{~{}\rho}-\frac{1}{4}g_{\mu\nu}F^{2}\right)+\frac{1}{2}\left[\partial_{\mu}\phi\partial_{\nu}\phi-\frac{1}{2}g_{\mu\nu}\left(\partial\phi\right)^{2}-g_{\mu\nu}V(\phi)\right].$
(2.4)

To take account of external magnetic field, we consider the following
anisotropic ansatz of the background blackening metric and ($\phi,~{}A_{\mu}$)
fields in Einstein frame

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
w_{E}(z)^{2}\left[-b(z)dt^{2}+g_{11}(z)dx_{1}^{2}+g_{22}(z)\left(dx_{2}^{2}+dx_{3}^{2}\right)+\frac{dz^{2}}{b(z)}\right],$
(2.5) $\displaystyle\phi$ $\displaystyle=$
$\displaystyle\phi\left(z\right),~{}A_{\mu}=\left(A_{t}(z),0,0,A_{3}(x_{2}),0\right),$
(2.6)

where $w_{E}(z)=\frac{e^{d(z)}}{z}$ is the warped factor with the sub-index
$E$ labeling the Einstein frame, $d(z)$ is the deformed factor which describes
the warping geometry deformed from $AdS$ spacetime, and $b(z)$ stands for the
blackening factor which formulates the black hole background. Conventionally,
$z=0$ corresponds to the conformal boundary of the 5-dimensional space-time,
and we have set the radial of $AdS_{5}$ to be unit by scale invariant.

We consider two components in the gauge field $A_{\mu}$ as Eq. (2.6). The non-
trivial temporal component $A_{t}(z)$ associates with the chemical potential
$\mu$ and the spatial component $A_{3}=Bx_{2}$ introduces an external magnetic
field along the $x_{1}$ direction. The magnetic field in the $x_{1}$ direction
breaks the rotational symmetry $SO(3)$ to the $SO(2)$. Therefore, the metric
at finite temperature becomes magnetic field dependent and anisotropic which
is similar to 1802.05652 . We assume that

$g_{11}=e^{c_{1}C(B)z^{2}},~{}g_{22}=e^{c_{2}C(B)z^{2}},$ (2.7)

where the coefficients $c_{1,2}$ are two constants and $C(B)$ is an arbitrary
function of $B$. Note that, at the boundary $z\to 0$, $b=g_{11}=g_{22}=1$.

The black hole entropy and temperature for this ansatz are well defined as

$\displaystyle s(z_{h})$ $\displaystyle=$
$\displaystyle\frac{w_{E}^{3}g_{22}\sqrt{g_{11}}}{4}\bigg{|}_{z=z_{h}},$ (2.8)
$\displaystyle T(z_{h})$ $\displaystyle=$
$\displaystyle-\frac{\partial_{z}b}{4\pi}\bigg{|}_{z=z_{h}}.$ (2.9)

Before we start to solve the equations of motion, it is worth to verify the
null energy condition (NEC) to guarantee the consistency of the gravitational
model. The NEC can be expressed as

$T_{\mu\nu}N^{\mu}N^{\nu}\geq 0,$ (2.10)

where $T^{\mu\nu}$ is the energy-momentum tensor of the matter fields. The
null vector $N^{\mu}$ satisfies the condition $g_{\mu\nu}N^{\mu}N^{\nu}=0$ and
could be chosen as

$N^{\mu}=\frac{1}{\sqrt{b\left(z\right)}}N^{t}+\frac{\sin\theta\cos\theta}{\sqrt{g_{11}}}N^{x_{1}}+\frac{\sin^{2}\theta}{\sqrt{2g_{22}}}\left(N^{x_{2}}+N^{x_{3}}\right)+\cos\theta\sqrt{b\left(z\right)}N^{z},$
(2.11)

for arbitrary parameter $\theta$. Then the NEC Eq. (2.10) becomes

$\left(A_{t}^{\prime
2}+\frac{B^{2}\sin^{2}\theta}{e^{2c_{2}Cz^{2}}}\right)\frac{f\sin^{2}\theta}{2w_{E}^{4}}+\frac{b\phi^{\prime
2}\cos^{2}\theta}{2w_{E}^{2}}\geq 0,$ (2.12)

which demands that the kinetic gauge function $f$ and $\phi^{\prime 2}$ should
be positive. (Which is the positive energy definite through the Legendre
transformation of Lagrange density.)

Plugging the ansatz Eq. (2.5) and Eq. (2.6) into Eqs. (2.2 \- 2.4), the
equations of motion reduce to

$\displaystyle A_{t}^{\prime\prime}$ $\displaystyle+$
$\displaystyle\left(\frac{f^{\prime}}{f}+\frac{w_{E}^{\prime}}{w_{E}}+\left(c_{1}+2c_{2}\right)Cz\right)A_{t}^{\prime}=0,$
(2.13) $\displaystyle\phi^{\prime 2}$ $\displaystyle+$
$\displaystyle\frac{6w_{E}^{\prime\prime}}{w_{E}}-\frac{12w_{E}^{\prime
2}}{w_{E}^{2}}+2(c_{1}+2c_{2})C+2(c_{1}^{2}+2c_{2}^{2})C^{2}z^{2}=0,$ (2.14)
$\displaystyle b^{\prime\prime}$ $\displaystyle+$
$\displaystyle\left(\frac{3w_{E}^{\prime}}{w_{E}}+c_{1}Cz\right)b^{\prime}-2c_{2}C\left(1+(c_{1}+2c_{2})Cz^{2}+\frac{3zw_{E}^{\prime}}{w_{E}}\right)b-\frac{f}{w_{E}^{2}}\left(A_{t}^{\prime
2}+\frac{B^{2}}{e^{2c_{2}Cz^{2}}}\right)=0,$ (2.15) $\displaystyle V$
$\displaystyle+$
$\displaystyle\frac{3bw_{E}^{\prime\prime}}{w_{E}^{3}}+\frac{6bw_{E}^{\prime
2}}{w_{E}^{4}}+\left(3bC(c_{1}+4c_{2})z+\frac{9b^{\prime}}{2}\right)\frac{w_{E}^{\prime}}{w_{E}^{3}}$
(2.16) $\displaystyle+$
$\displaystyle\frac{fB^{2}}{2w_{E}^{4}e^{2c_{2}Cz^{2}}}+\frac{b^{\prime\prime}+(c_{1}+6c_{2})Czb^{\prime}+4c_{2}C\left(1+(c_{1}+2c_{2})Cz^{2}\right)b}{2w_{E}^{2}}=0,$
$\displaystyle\phi^{\prime\prime}$ $\displaystyle+$
$\displaystyle\left(\frac{3w_{E}^{\prime}}{w_{E}}+\frac{b^{\prime}}{b}+\left(c_{1}+2c_{2}\right)Cz\right)\phi^{\prime}+\left(A_{t}^{\prime
2}-\frac{B^{2}}{e^{2c_{2}Cz^{2}}}\right)\frac{f^{\prime}(\phi)}{2bw_{E}^{2}}-\frac{w_{E}^{2}}{b}V^{\prime}(\phi)=0.$
(2.17)

Since there are four physical quantities $A_{t},\phi,b,V$ so we only need four
equations of motion Eqs. (2.13 \- 2.16). The last Eq. (2.17) is the redundancy
one because of the Bianchi identity.

In order to solve the equations of motion, we need to consider boundary
conditions for the first and second order ODEs. The asymptotic $AdS$ condition
in the UV limit at the conformal boundary $z\to 0$, and the regular condition
in the IR limit at the black hole horizon $z\to z_{h}$ are imposed.

  1. (i)

$z\to 0:$

$\displaystyle d(0)=\phi(0)=0,~{}b(0)=1,$ (2.18) $\displaystyle A_{t}(0)=\mu.$
(2.19)

  2. (ii)

$z=z_{h}:$

$A_{t}(z_{h})=b(z_{h})=0,$ (2.20)

where $\mu$ is the chemical potential. According to the holographic dictionary
of the gauge/gravity correspondence,

$A_{t}(z)=\mu-\rho z^{2}+...$ (2.21)

In addition, as the magnetic field shrinks to zero, the spacial symmetry
should restore and return to the asymptotic $AdS$ near the boundary,

$g_{11}=g_{22}=1.$ (2.22)

###  2.2 Analytical Solution

The equations of motion Eqs. (2.13 \- 2.17) are very complicated. However, we
found that the equations system can be analytically solved if we set
$c_{2}=0$. With this choice, the equations reduces to

$\displaystyle
A_{t}^{\prime\prime}+\left(\frac{f^{\prime}}{f}+\frac{w_{E}^{\prime}}{w_{E}}+c_{1}Cz\right)A_{t}^{\prime}=0,$
(2.23) $\displaystyle\phi^{\prime
2}+\frac{6w_{E}^{\prime\prime}}{w_{E}}-\frac{12w_{E}^{\prime
2}}{w_{E}^{2}}+2c_{1}C\left(1+c_{1}Cz^{2}\right)=0,$ (2.24) $\displaystyle
b^{\prime\prime}+\left(\frac{3w_{E}^{\prime}}{w_{E}}+c_{1}Cz\right)b^{\prime}-\frac{f}{w_{E}^{2}}\left(A_{t}^{\prime
2}+B^{2}\right)=0,$ (2.25) $\displaystyle
V+\frac{3bw_{E}^{\prime\prime}}{w_{E}^{3}}+\frac{6bw_{E}^{\prime
2}}{w_{E}^{4}}+\left(3bc_{1}Cz+\frac{9b^{\prime}}{2}\right)\frac{w_{E}^{\prime}}{w_{E}^{3}}+\frac{b^{\prime\prime}+c_{1}Czb^{\prime}}{2w_{E}^{2}}+\frac{fB^{2}}{2w_{E}^{4}}=0.$
(2.26)

The fields $A_{t},\phi^{\prime}$ and $b$ can be analytically integrated from
the Eqs. (2.23 \- 2.25) as

$\displaystyle A_{t}(z)=\mu\left[1-\frac{I_{2}(z)}{I_{2}(z_{h})}\right],$
(2.27)
$\displaystyle\phi^{\prime}=\sqrt{-\frac{6w_{E}^{\prime\prime}}{w_{E}}+\frac{12w_{E}^{\prime
2}}{w_{E}^{2}}-2c_{1}C\left(1+c_{1}Cz^{2}\right)},$ (2.28) $\displaystyle
b(z)=1-\dfrac{I_{1}(z)}{I_{1}(z_{h})}+\frac{\mu^{2}}{I_{2}^{2}(z_{h})I_{1}(z_{h})}\left|\begin{array}[c]{cc}I_{1}(z_{h})&I_{12}(z_{h})\\\
I_{1}(z)&I_{12}(z)\end{array}\right|+\frac{B^{2}}{I_{1}(zh)}\left|\begin{array}[c]{cc}I_{1}(z_{h})&I_{13}(z_{h})\\\
I_{1}(z)&I_{13}(z)\end{array}\right|,$ (2.33)

with the help of the following integrals,

$\displaystyle I_{1}(z)$ $\displaystyle=$
$\displaystyle\int_{0}^{z}\frac{dy}{w_{E}^{3}e^{\frac{1}{2}c_{1}Cy^{2}}},$
(2.34) $\displaystyle I_{2}(z)$ $\displaystyle=$
$\displaystyle\int_{0}^{z}\frac{dy}{w_{E}fe^{\frac{1}{2}c_{1}Cy^{2}}},$ (2.35)
$\displaystyle I_{3}(z)$ $\displaystyle=$
$\displaystyle\int_{0}^{z}w_{E}fe^{\frac{1}{2}c_{1}Cy^{2}}dy,$ (2.36)
$\displaystyle I_{12}(z)$ $\displaystyle=$
$\displaystyle\int_{0}^{z}I^{\prime}_{1}(y)I_{2}(y)dy=I_{1}(z)I_{2}(z)-\tilde{I}_{12}(z),$
(2.37) $\displaystyle I_{13}(z)$ $\displaystyle=$
$\displaystyle\int_{0}^{z}I^{\prime}_{1}(y)I_{3}(y)dy=I_{1}(z)I_{3}(z)-\tilde{I}_{13}(z).$
(2.38)

It is apparent that the blackening factor $b(z)$ receives the contribution
from both chemical potential $\mu$ and magnetic field $B$. In addition, if the
magnetic field is turned off, the family of solutions will reduce to the
original EMS system 1301.0385 .

The potential $V(z)$ can be obtained from Eq. (2.26). It is straightforward to
reconstruct the potential $V(\phi)$ by $z$-expansion of both potential $V(z)$
and scalar field $\phi(z)$,

$\displaystyle V(\phi)$ $\displaystyle=$
$\displaystyle-12+\frac{(-3)}{2}\phi^{2}+...,$ (2.39)

with the coefficient $-3$ being exactly the $m^{2}$ of the scalar field
$\phi$, which satisfies the BF bound implying that the gravitational
background is stable and $-12$ relates to the cosmological constant in
$AdS_{5}$. This process is the so-called potential reconstruction.

To consider meson spectrum, we add a matter action of a probe vector field $V$
into the background,

$S_{m}=-\frac{1}{16\pi G_{5}}\int
d^{5}x\sqrt{-g}{\frac{f\left(\phi\right)}{4}F_{V}^{2}}.$ (2.40)

The equation of motion for the vector field reads

$\nabla_{\mu}[f(\phi)F_{V}^{\mu\nu}]=0.$ (2.41)

We use the gauge invariance to fix the gauge $V_{z}=0$ and write
$V_{i}(\vec{x},z)=\phi(\vec{x})v_{i}(z)$with
$\nabla^{2}\phi(\vec{x})=m^{2}\phi(\vec{x})$. The equation of motion of the
transverse vector field $V_{\mu}$ ($\partial^{\mu}V_{\mu}=0$) in the
background Eq. (2.5) is

$v^{\prime\prime}_{i}+\left(\frac{b^{\prime}}{b}+\frac{w_{E}^{\prime}}{w_{E}}+\frac{f^{\prime}}{f}-c_{1}Cz\right)v^{\prime}_{i}+\frac{m^{2}}{b}v_{i}=0,$
(2.42)

which can be brought to the Schrödinger equation

$\psi_{i}^{\prime\prime}+U(z)\psi+m^{2}\psi_{i}=0,$ (2.43)

where the potential is

$U(z)=\frac{X^{\prime\prime}}{X}-\frac{2X^{\prime 2}}{X^{2}},$ (2.44)

with

$X=\left(\frac{e^{c_{1}Cz^{2}/2}}{bfw_{E}}\right)^{1/2}.$ (2.45)

To realize the linear Regge trajectories for the meson mass spectra, we choose
the gauge kinetic function as

$f(z)=\frac{e^{-\left(R_{gg}+\frac{c_{1}C}{2}\right)z^{2}}}{zw_{E}}.$ (2.46)

At $T=\mu=B=0$, we thus have

$U(z)=\frac{3}{4z^{2}}+R_{gg}^{2}z^{2}.$ (2.47)

which leads to the linear mass spectrum $m_{n}^{2}=4R_{gg}n$, and the
parameter $R_{gg}$ can be fitted by the Regge spectra, for instance the
$J/\psi$ meson.

For the heavy quark sector, we choose the deformed factor
$d(z)=\frac{-R_{gg}}{3}z^{2}-pz^{4}$ so that the warped factor
$w_{E}(z)=\frac{1}{z}\exp\left({\frac{-R_{gg}}{3}z^{2}-pz^{4}}\right)$. The
parameter $p$ recalls the transition point at $\mu=B=0$ which can be fitted
from lattice QCD. The parameter $R_{gg}$ and $p$ have been addressed in
1301.0385 ; 1703.09184  for the vanishing magnetic field with $R_{gg}=1.16$
and $p=0.273$.

With the above choice, the integrals Eqs. (2.34 \- 2.38) are monotonously
growing functions of $z$ from zero, and the integrals Eq. (2.35) and Eq.
(2.36) do not depend on the magnetic field $B$.

To investigate the magnetic effects on the phase diagram, we need to
explicitly fix the function $C(B)$ in the component of the metric
$g_{11}=e^{c_{1}C(B)z^{2}}$. Dimension analysis 111More precisely speaking
that $\frac{T}{\sqrt{B}}$ should be dimensionless. restricts $C(B)\sim B$, we
then take $C(B)=B$ without loss of generality. Subsequently, $c_{1}\leq 0$
ensures the correct monotonous behavior of entropy as the size of BH changes.
For simplicity, we set $c_{1}=-1$ in the later calculations.

##  3 Phase Transition in QCD

In this section, we study the phase structure for the black hole background
which we obtained in the last section. The phase transition between the black
holes correspond to both the confinement-deconfinement phase transition and
the chiral phase transition in the dual holographic QCD theory 1301.0385 .

In the presence of magnetic field, the QCD phase diagram is sensitive to the
background magnetic field. It has been showed that the effect of background
magnetic field on the transition temperature depends on the masses of the
dynamical quarks. For light quarks with physical mass, lattice simulation
reveals the IMC phenomenon for small chemical potential 1111.4956 ; 1206.4205
; 1303.3972 ; 1312.5628 ; 1711.02884 ; 1909.09547 . While for heavy quarks,
the MC phenomenon is observed 1005.5365 ; 1203.3360 ; 1209.0374 . In this
work, we focus on the model of heavy quarks, since we have fixed the model
parameters in terms of linear Regge behavior of $J/\psi$ meson and these
parameters capturing the dynamics of the heavy quarks sector.

###  3.1 Magnetic Effects on Temperature

Using the solutions Eqs. (2.27 \- 2.33), the BH entropy density and the
Hawking temperature are straightforwardly calculated,

$\displaystyle s(z_{h})$ $\displaystyle=$
$\displaystyle\frac{1}{4I^{\prime}_{1}(z_{h})},$ (3.48) $\displaystyle
T(z_{h})$ $\displaystyle=$ $\displaystyle
T_{0}\left(1+\mu^{2}T_{\mu}+B^{2}T_{B}\right),$ (3.49)

where

$T_{0}=\frac{I^{\prime}_{1}(z_{h})}{4\pi
I_{1}(z_{h})},~{}T_{\mu}=-\frac{\tilde{I}_{12}(z_{h})}{I_{2}^{2}(z_{h})},~{}T_{B}=-\tilde{I}_{13}(z_{h}).$
(3.50)

From the explicit expression of the integral $I_{1}(z)$, we can show that

$T_{0}=\frac{I^{\prime}_{1}(z_{h})}{4\pi I_{1}(z_{h})}=\frac{1}{4\pi
w_{E}^{3}(z_{h})\int_{0}^{z_{h}}e^{-\frac{1}{2}B(z_{h}^{2}-y^{2})}\frac{dy}{w_{E}^{3}(y)}}\geq\frac{1}{4\pi
w_{E}^{3}(z_{h})\int_{0}^{z_{h}}\frac{dy}{w_{E}^{3}(y)}}=\left.T_{0}\right|_{B=0},$
(3.51)

which implies that magnetic field will enhance the BH temperature $T(z_{h})$
by $T_{0}$. On the other hand, both $T_{\mu}$ and $T_{B}$ are negative due to
the positive integrals. The effect from $T_{\mu}$ and $T_{B}$ grows for larger
magnetic $B$ and bigger horizon $z_{h}$, so that magnetic field will reduce
the BH temperature eventually for large $B$. The two effects compete each
other as we changing the magnetic field.

Figure 1: Temperature vs horizon. From left to right: $\mu=0$, $\mu=0.5$,
$\mu=\mu_{CEP}(B)$, $\mu=1.0$. In each figure, from top to bottom:
$B=0,0.2,0.4,0.6$. Interestingly, the Hawking-Page transition only happens in
$\mu=B=0$.

The black hole temperature in the presence of the background magnetic field
$B$ for heavy quarks at different chemical potential are plotted in Fig.1. For
small chemical potential $\mu<\mu_{CEP}$, the temperature is a multiple-valued
function of the black hole horizon, which indicates that there would be a
first-order phase transition between the large and the small sizes of black
holes. For $\mu>\mu_{CEP}$, the temperature becomes monotonous and the phase
transition could reduces to a crossover. At the critical point
$\mu=\mu_{CEP}$, we expect a second-order phase transition.

The influence of magnetic field on temperature can be read from Fig.1. The
temperature is enhanced at small horizon $z_{h}$ due to the effect from
$T_{0}$, while is reduced at large $z_{h}$ due to the effect from $T_{\mu}$
and $T_{B}$. To investigate the magnetic effect in detail, we take the
derivative of the temperature with magnetic field,

$\displaystyle\frac{dT}{dB}$ $\displaystyle=$
$\displaystyle\frac{I_{1}^{\prime}}{8\pi
I_{1}}\left(H(z_{h})-\frac{\mu^{2}}{I_{2}^{2}}\int_{0}^{z_{h}}\left[H(z_{h})+\frac{I_{1y}\left(y\right)}{I_{1}\left(y\right)}\right]I_{1}\left(y\right)I_{2}^{\prime}\left(y\right)dy\right)$
(3.52) $\displaystyle-\frac{B^{2}I_{1}^{\prime}}{8\pi
I_{1}}\int_{0}^{z_{h}}\left[H(z_{h})+\frac{I_{1y}\left(y\right)}{I_{1}\left(y\right)}\right]I_{1}\left(y\right)I_{3}^{\prime}\left(y\right)dy-\frac{BI_{1}^{\prime}\tilde{I}_{13}}{2\pi
I_{1}},$

where

$\displaystyle H(z_{h})$ $\displaystyle=$ $\displaystyle
z_{h}^{2}-\frac{I_{1y}\left(z_{h}\right)}{I_{1}\left(z_{h}\right)}\text{ and
}I_{1y}\left(z_{h}\right)=\int_{0}^{z_{h}}y^{2}I_{1}^{\prime}dy.$ (3.53)

It is easy to show that $H(z_{h})\geq 0$ by using the monotonicity of the
integral $I_{1}(z)$.

For small magnetic field, we can neglect the last term in Eq. (3.52),

$\displaystyle\frac{dT}{dB}$ $\displaystyle\simeq$
$\displaystyle\frac{I_{1}^{\prime}}{8\pi
I_{1}}\left(H(z_{h})-\frac{\mu^{2}}{I_{2}^{2}}\int_{0}^{z_{h}}\left[H(z_{h})+\frac{I_{1y}\left(y\right)}{I_{1}\left(y\right)}\right]I_{1}\left(y\right)I_{2}^{\prime}\left(y\right)dy\right),$
(3.54)

which is positive for not too large chemical potential $\mu$ since $I_{1}>0$.
This implies the MC behavior for small chemical potential.

On the other hand, for large magnetic field $B$, only the last term in Eq.
(3.52) is important,

$\displaystyle\frac{dT}{dB}$ $\displaystyle\simeq$
$\displaystyle-\frac{B^{2}I_{1}^{\prime}}{8\pi
I_{1}}\int_{0}^{z_{h}}\left[H(z_{h})+\frac{I_{1y}\left(y\right)}{I_{1}\left(y\right)}\right]I_{1}\left(y\right)I_{3}^{\prime}\left(y\right)dy-\frac{BI_{1}^{\prime}\tilde{I}_{13}}{2\pi
I_{1}}<0,$ (3.55)

which implies the IMC behavior for large magnetic field.

###  3.2 Magnetic Effects on QCD Phase Diagram

As we argued in the previous section, the rough structure of QCD phase
transition can be read from the behavior of black hole temperature. However,
to obtain the exact phase diagram, we need to calculate the free energy of the
thermodynamic system. The free energy in grand canonical ensemble can be
obtained from the first law of thermodynamics,

$F=\epsilon-Ts-\mu\rho-MB,$ (3.56)

where $\rho=\dfrac{\mu}{2I_{2}(z_{h})}$ is the baryon density, $M$ represents
the magnetization which is associated to $B$, and $\epsilon$ labels the
internal energy density. Comparing the free energies of BHs at the same
temperature for certain finite value of chemical potential, we are able to
obtain the phase structure of BHs which gives the phase diagram of the
holographic QCD due to AdS/CFT correspondence. At fixed volume, the
differential of the free energy is defined as 1909.09547

$dF=-sdT-\rho d\mu-MdB.$ (3.57)

For the fixed chemical potential $\mu$ and magnetic field $B$, the free energy
can be evaluated by the following integral,

$F=-\int
sdT=\int_{z_{h}}^{\infty}s(\bar{z_{h}})T^{\prime}(\bar{z_{h}})d\bar{z_{h}}$
(3.58)

where we have normalized the free energy to vanish at $z_{h}\to\infty$.

The thermodynamic properties for heavy quarks without magnetic field has been
studied in 1301.0385 . The temperature vs horizon is plotted in the left
figure of Fig.2. At $\mu=0$, the temperature has a minimum value which implies
the Hawking-Page phase transition. For a finite but small $\mu$, the
temperature drops to zero at certain horizon, but a first order phase
transition still happens at a finite temperature between the local minimum and
maximum temperatures. From the local minimum and maximum for each chemical
potential $\mu<\mu_{CEP}$, we can partition a region where the phase
transition could happen. The CEP of the phase diagram will take place at the
specific temperature $T_{CEP}$ where the local minimum and maximum temperature
are degenerated at characteristic chemical potential $\mu_{CEP}$. When the
chemical potential is beyond a critical value $\mu>\mu_{CEP}$, the temperature
becomes monotonous and the phase transition reduces to a crossover. The exact
phase transition temperature can be obtained from the free energy. The free
energy vs temperature is demonstrated in the middle figure of Fig.2. The
intersection of the swallow-type patterns labels the first order phase
transition temperature at each chemical potential where the free energy and
temperature are equal at different horizons. As the chemical potential
increasing, the swallow-type gradually compresses and eventually becomes a
singular point where character the second order phase transition. When
$\mu>\mu_{CEP}$, the free energy becomes smooth function with respect to
temperature that implies the phase transition is weaken to a crossover. The
phase diagram is delivered in the right figure of Fig.2. The shadow region is
enclosed by the curves of the local minimum and maximum temperatures, which
shrink to the CEP. The phase transition line is within the shadow region. We
should remark that the behavior of the heavy quarks is in contrast to that of
the light quarks, in which crossover occurs at small chemical potential and
becomes phase transition for $\mu>\mu_{CEP}$.

Figure 2: (Left) Temperatures vs horizon. From top to bottom:
$\mu=0,0.5,\mu_{CEP},1.0$. (Middle) Free energies vs temperature for the
corresponding $\mu^{\prime}s$. The intersection of the swallow-type patterns
labels the first order phase transition. (Right) QCD phase diagram for heavy
quarks. The phase transition line is within the shadow region which is
enclosed by the curves of the local minimum and maximum temperatures that
shrink to the CEP at $(\mu_{CEP},T_{CEP})=(0.714,0.528).$

In the presence of magnetic field $B$, the behavior of free energy are
qualitatively the same as that with zero magnetic field as showed in the
middle figure of Fig.2. The exact values of the phase transition temperature
changes with both chemical potential $\mu$ and magnetic field $B$ are plotted
in Fig.3.

For fixed $B$ fields, phase transition extends from $\mu=0$ to finite chemical
potentials and terminates at a CEP, the red dots, as showed in the left figure
of Fig.3. Increasing the magnetic field from zero enhances the phase
transition temperature. This confirms the MC behavior as we expected from the
behavior of the temperature. The MC phenomenon we found for the heavy quarks
is consistent with the recent lattice simulations 1808.07008 ; 1904.10296 .
Furthermore, it shows that the CEP of the phase transition moves towards to
the lower chemical potential when the magnetic field increases, that is
consistent with the recent works using PNJL model 1509.01181 ; 1712.08378 ;
1712.08384 .

We plot the phase transition temperature vs magnetic field for different
chemical potentials in the middle figure of Fig.3. For small magnetic field,
it clearly shows that the phase transition temperature increases along the
magnetic field, i.e. MC phenomenon as we have discussed. On the other hand,
when the magnetic field $B$ become large enough, we observe that the phase
transition temperature reduces with the increasing magnetic field. This
justifies our conclusion of IMC behavior at large magnetic field by the
argument of the competition between the contributions from $T_{0}$ and $T_{B}$
to the temperature in Eq. (3.49). In our case, the flipping magnetic field
$B\sim 0.96$ at $\mu=0$, which magnitude is very close to 1111.4956 ;
1501.03262 . Since the sign problem would not encounter at $\mu=0$, we expect
this appearance can be attended in lattice QCD. We also notice that the phase
transition will reach CEP before it turns to IMC from MC if the chemical
potential is too large $\mu\gtrsim 0.6~{}GeV$. Another words, IMC only appears
at relatively low baryon density, which statement has also been declared in
1707.00872 . For each chemical potential, there exists a critical magnetic
field $B_{CEP}(\mu)$ beyond that the phase transition becomes crossover. The
CEPs for different chemical potentials are along the red dashed line with the
shadow region standing for the crossover zone.

The full phase transition structure including both chemical potential and
magnetic field is combined in a 3-dimensional phase diagram as plotted in the
right figure of Fig.3. The left two diagrams are the 3-dimensional phase
diagram projected on the $T-\mu$ and $T-B$ planes, respectively. The four
lines with the fixed magnetic fields in the $T-\mu$ diagram are put with the
corresponding colors in the 3-dimensional phase diagram. In addition, the red
dashed line in the 3-dimensional phase diagram indicates the 3-dimensional CEP
boundary, beyond which the phase transition becomes crossover.

Figure 3: (Left) Phase diagram on $T-\mu$ plane for $B=0,0.2,0.4,0.6$ from
bottom to top. The red dots label the CEPs for different magnetic fields.
(Middle) Phase diagram on $T-B$ plane for $\mu=0,0.1,0.2,0.3,0.4,0.5,0.6$ from
top to bottom. The red dots label the CEPs for different chemical potentials
which are collected in the red dashed line with the shadow region standing for
the crossover zone. (Right) The 3-dimensional phase diagram in $T-\mu-B$ axes.
The curved surface represents the first order phase transition area and
terminates at the red dashed boundary which is the collection of CEPs.

###  3.3 Critical End Point

It is crucial to locate the CEP of the phase transition 0106002 ; 0210284 ;
1712.08378 ; 1908.02000 . It has been showed that the QCD phase transition
temperature gradually cools down as QCD matter being more and more dense, i.e.
increasing chemical potential. The CEP of the first order phase transition can
be evaluated by the free energy Eq. (3.58).

Figure 4: The CEPs on $T-\mu$, $T-B$ and $\mu-B$ planes. We can observe that
the CEP temperature are linear with the magnetic field. On the other hand, as
magnetic field increasing, the chemical potential gradually decreases and
eventually vanishes at $B\sim 1.31~{}GeV^{2}$ which implies that the whole
transition line is weaken to a transition point. We can demonstrate this
tendency in the $T-\mu$ plane. As the imposing magnetic field become stronger,
the CEP temperature become higher but CEP baryon density become looser.
Another words, the first order phase transition line become shorter, the
crossover region become wider, as the magnetic field is stronger.

The authors of 1509.01181 ; 1712.08378 ; 1712.08384  have studied confinement-
deconfinement phase transition by using PNJL model. They showed that the CEP
moves towards lower chemical potentials with increasing magnetic field if
considering a magnetic field dependent coupling $G(B)$, and the CEP could
eventually approaches to the zero chemical potential for large enough magnetic
field. Their result is only for light quarks, here we show that the similar
behavior preserves for heavy quarks.

The tendency of CEP with magnetic field is plotted in the left figure of
Fig.4. When the magnetic field increasing, the CEP moves to the lower chemical
potential and approaches to $T\sim 0.613~{}GeV$ at $\mu=0$. The CEP on $T-B$
plane is plotted in the middle figure of Fig.4. When the chemical potential
increasing, the CEP moves to the smaller magnetic filed and returns to the
original CEP at $B=0$. It is interesting to observe that the CEP temperature
are linear with the magnetic field. Furthermore, the CEP forms a closed
boundary on $\mu-B$ plane as plotted in the right figure of Fig.4. Beyond the
boundary the phase transition becomes crossover.

##  4 Conclusion

In this paper, we have constructed an analytical holographic QCD model for
heavy quarks in the presence of the external magnetic field by using the
Einstein-Maxwell-Scalar system with potential reconstruction approach. We
introduced both electrical and magnetic fields in a single bulk $U(1)$ gauge
field. The electrical component is interpreted as the global baryon number
conservation symmetry on the boundary theory by holography. While the magnetic
component corresponds to the boundary external magnetic field. In this
holographic setup, we have checked the NEC to ensure that the gravitational
background is a stable. In addition, in this background, we have also
considered the linear Regge spectrum of $J/\psi$ mesons. Since the parameters
are fixed by the heavy mesons spectrum, the holographic model captures the
characters of the heavy quarks sector.

We calculated the free energy to obtain the phase transition temperatures for
heavy quarks at different chemical potentials and magnetic fields in the
holographic QCD model in the anisotropic background. For small magnetic field,
we found MC phenomenon that is consistent with the recent lattice results
1808.07008 ; 1904.10296 . While for large enough magnetic field, we found IMC
phenomenon due to the competition between the contributions from $T_{0}$ and
$T_{B}$ to the temperature in Eq. (3.49). Thus the phase transition will
change from MC to IMC as the magnetic field growing. However, we noticed that
the phase transition will reach its critical end point before it turns to IMC
from MC if the chemical potential is large enough $\mu\gtrsim~{}0.6GeV$. See
Fig.3 for the phase diagrams at different chemical potentials and magnetic
fields.

In addition, there exists a extreme point for the phase transition for either
chemical potential or magnetic field, i.e. the CEP. Beyond CEP, the phase
transition becomes to crossover. We discover that, for fixed magnetic fields,
the CEP moves to the lower chemical potential and eventually approaches to
$\mu=0$. While for fixed chemical potentials, the CEP decreases with the
magnetic field growing. It is interesting to observe that the CEP temperature
are linear with the magnetic field. We do not understand the reason of this
linear behavior and it is deserved to study in the future. See Fig.4 for the
CEP at different chemical potentials and magnetic fields.

In this work, we only focused on the thermodynamics of the black hole to
obtain the phase transition between two black holes with different sizes, by
calculating their free energies in the hQCD model. It is also interesting to
investigate the definite order parameters to explore further information
associated with the phase transitions, such as confinement and chiral symmetry
breaking. The two kinds of phase transitions will show different
characteristic behaviors with respect to the magnetic field. We would like to
study these corresponding order parameters to confirm such phenomenon in terms
of holographic approach. We leave this part in the future.

### Acknowledgements

We would like to thank Umut Gürsoy, Danning Li, Xiaoning Wu, Lang Yu for
useful discussion. S.H. also would like to appreciate the financial support
from Jilin University and Max Planck Partner group. This work of Y.Y is
supported by the Ministry of Science and Technology (MOST 106-2112-M-009 -005
-MY3) and National Center for Theoretical Science, Taiwan. The work of P.H.Y.
was supported by the University of Chinese Academy of Sciences.

## References

  * (1) V. Skokov, A. Illarionov, V. Toneev, ”Estimate of the magnetic field strength in heavy-ion collisions”, Int.J.Mod.Phys.A 24 (2009) 5925-5932, arXiv:0907.1396 [nucl-th].
  * (2) V. Voronyuk, V.D Toneev, W. Cassing, E.L. Bratkovskaya, V.P. Konchakovski, S.A. Voloshin, ”Electromagnetic field evolution in relativistic heavy-ion collisions”, Phys.Rev.C 83 (2011) 054911, arXiv:1103.4239 [nucl-th].
  * (3) Adam Bzdak, Vladimir Skokov, ”Event-by-event fluctuations of magnetic and electric fields in heavy ion collisions”, Phys.Lett.B 710 (2012) 171-174, arXiv:1111.1949 [hep-ph].
  * (4) Wei-Tian Deng, Xu-Guang Huang, ”Event-by-event generation of electromagnetic fields in heavy-ion collisions”, Phys.Rev.C 85 (2012) 044907, arXiv:1201.5108 [nucl-th].
  * (5) Robert C. Duncan, Christopher Thompson, ”Formation of very strongly magnetized neutron stars - implications for gamma-ray bursts”, Astrophysical Journal Letters v.392 (1992), p.L9.
  * (6) Sandro Mereghetti, ”The strongest cosmic magnets: Soft Gamma-ray Repeaters and Anomalous X-ray Pulsars”, Astron.Astrophys.Rev. 15 (2008) 225-287, arXiv:0804.0250 [astro-ph].
  * (7) K. Enqvist, P. Olesen, ”On Primordial Magnetic Fields of Electroweak Origin”, Phys.Lett.B 319 (1993) 178-185, arXiv:hep-ph/9308270.
  * (8) M. D’Elia, S. Mukherjee and F. Sanfilippo,, ”QCD Phase Transition in a Strong Magnetic Background”, Phys.Rev.D 82 (2010) 051501, arXiv:1005.5365 [hep-lat].
  * (9) E.-M. Ilgenfritz, M. Kalinowski, M. Muller-Preussker, B. Petersson, A. Schreiber, ”Two-color QCD with staggered fermions at finite temperature under the influence of a magnetic field”, Phys.Rev.D 85 (2012) 114504, arXiv:1203.3360 [hep-lat].
  * (10) M. D’Elia, ”Lattice QCD Simulations in External Background Fields”, Lect.Notes Phys. 871 (2013) 181-208, arXiv:1209.0374 [hep-lat].
  * (11) V.P.Gusynin, V.A.Miransky, I.A.Shovkovy, ”Catalysis of Dynamical Flavor Symmetry Breaking by a Magnetic Field in 2+1 Dimensions”, Phys.Rev.Lett. 73 (1994) 3499-3502, Phys.Rev.Lett. 76 (1996) 1005 (erratum), arXiv:hep-ph/9405262.
  * (12) V.P. Gusynin, V.A. Miransky, I.A. Shovkovy, ”Dimensional Reduction and Dynamical Chiral Symmetry Breaking by a Magnetic Field in 3+1 Dimensions”, Phys.Lett.B 349 (1995) 477-483, arXiv:hep-ph/9412257.
  * (13) V. P. Gusynin, V. A. Miransky, I. A. Shovkovy, ”Dimensional Reduction and Catalysis of Dynamical Symmetry Breaking by a Magnetic Field”, Nucl.Phys.B 462 (1996) 249-290, arXiv:hep-ph/9509320.
  * (14) G. W. Semenoff, I. A. Shovkovy, L. C. R. Wijewardhana, ”Universality and the magnetic catalysis of chiral symmetry breaking”, Phys.Rev.D 60 (1999) 105024, arXiv:hep-th/9905116.
  * (15) J. Alexandre, K. Farakos, G. Koutsoumbas, ”Magnetic catalysis in $QED_{3}$ at finite temperature: beyond the constant mass approximation”, Phys.Rev.D 63 (2001) 065015, arXiv:hep-th/0010211.
  * (16) V. A. Miransky, I. A. Shovkovy, ”Magnetic catalysis and anisotropic confinement in QCD”, Phys.Rev.D 66 (2002) 045006, arXiv:hep-ph/0205348.
  * (17) A. A. Osipov, B. Hiller, A. H. Blin, J. da Providencia, ”Dynamical chiral symmetry breaking by a magnetic field and multi-quark interactions”, Phys.Lett.B 650 (2007) 262-267, arXiv:hep-ph/0701090 .
  * (18) E. S. Fraga and A. J. Mizher, ”Can a strong magnetic background modify the nature of the chiral transition in QCD?”, Nucl.Phys.A 820 (2009) 103C-106C, arXiv:0810.3693 [hep-ph].
  * (19) Kenji Fukushima, Marco Ruggieri, Raoul Gatto, ”Chiral magnetic effect in the PNJL model”, Phys.Rev.D 81 (2010) 114031, arXiv:1003.0047 [hep-ph].
  * (20) A. J. Mizher, M. N. Chernodub, E. S. Fraga, ”Phase diagram of hot QCD in an external magnetic field: possible splitting of deconfinement and chiral transitions”, Phys.Rev.D 82 (2010) 105016, arXiv:1004.2712 [hep-ph].
  * (21) Raoul Gatto, Marco Ruggieri, ”Deconfinement and Chiral Symmetry Restoration in a Strong Magnetic Background”, Phys.Rev.D 83 (2011) 034016, arXiv:1012.1291 [hep-ph].
  * (22) Kouji Kashiwa, ”Entanglement between chiral and deconfinement transitions under strong uniform magnetic background field”, Phys.Rev.D 83 (2011) 117901, arXiv:1104.5167 [hep-ph].
  * (23) Bogdan V. Galilo, Sergei N. Nedelko, ”Impact of the strong electromagnetic field on the QCD effective potential for homogeneous Abelian gluon field configurations”, Phys.Rev.D 84 (2011) 094017, arXiv:1107.4737 [hep-ph].
  * (24) V. Skokov, ”Phase diagram in an external magnetic field beyond a mean-field approximation”, Phys.Rev.D 85 (2012) 034026, arXiv:1112.5137 [hep-ph].
  * (25) Kenji Fukushima, Jan M. Pawlowski, ”Magnetic catalysis in hot and dense quark matter and quantum fluctuations”, Phys.Rev.D 86 (2012) 076013, arXiv:1203.4330 [hep-ph].
  * (26) Igor A. Shovkovy. ”Magnetic Catalysis: A Review”, Lect.Notes Phys. 871 (2013) 13-49, arXiv:1207.5081[hep-ph].
  * (27) Eduardo S. Fraga, ”Thermal chiral and deconfining transitions in the presence of a magnetic background”, Lect.Notes Phys. 871 (2013) 121-141, arXiv:1208.0917 [hep-ph].
  * (28) M. Ferreira, P. Costa, D. P. Menezes, C. Providência and N. Scoccola, ”Deconfinement and chiral restoration within the SU(3) Polyakov–Nambu–Jona-Lasinio and entangled Polyakov–Nambu–Jona-Lasinio models in an external magnetic field”, Phys.Rev.D 89 (2014) 1, 016002, Phys.Rev.D 89 (2014) 1, 019902 (addendum), arXiv:1305.4751 [hep-ph].
  * (29) E. S. Fraga, B. W. Mintz, J. Schaffner-Bielich, ”A search for inverse magnetic catalysis in thermal quark-meson models”, Phys.Lett.B 731 (2014) 154-158, arXiv:1311.3964 [hep-ph].
  * (30) M. Ferreira, P. Costa, C. Providência, ”Deconfinement, chiral symmetry restoration and thermodynamics of (2+1)–flavor hot QCD matter in an external magnetic field”, Phys.Rev.D 89 (2014) 3, 036006,arXiv:1312.6733 [hep-ph].
  * (31) Stefano Bolognesi, David Tong, ”Magnetic Catalysis in AdS4”, Class.Quant.Grav. 29 (2012) 194003, arXiv:1110.5902 [hep-th].
  * (32) D. Dudal, D. R. Granado, T. G. Mertens, ”No inverse magnetic catalysis in the QCD hard and soft wall models”, Phys.Rev.D 93 (2016) 12, 125004, arXiv:1511.04042 [hep-th].
  * (33) M. D’Elia and F. Negro, ”Chiral Properties of Strong Interactions in a Magnetic Background”, Phys.Rev.D 83 (2011) 114028, arXiv:1103.2080 [hep-lat].
  * (34) G. S. Bali, F. Bruckmann, G. Endrodi, Z. Fodor, S. D. Katz, S. Krieg, A. Schafer, K. K. Szabo, ”The QCD phase diagram for external magnetic fields”, JHEP 02 (2012) 044, arXiv:1111.4956 [hep-lat].
  * (35) G. S. Bali, F. Bruckmann, G. Endrodi, Z. Fodor, S. D. Katz, A. Schafer, ”QCD quark condensate in external magnetic fields”, Phys.Rev.D 86 (2012) 071502, arXiv:1206.4205 [hep-lat].
  * (36) G. S. Bali, F. Bruckmann, G. Endrodi, F. Gruber, A. Schaefer, ”Magnetic field-induced gluonic (inverse) catalysis and pressure (an)isotropy in QCD”, JHEP 04 (2013) 130, arXiv:1303.1328 [hep-lat].
  * (37) F. Bruckmann, G. Endrodi, T. G. Kovacs, ”Inverse magnetic catalysis and the Polyakov loop”, JHEP 04 (2013) 112, arXiv:1303.3972 [hep-lat].
  * (38) E.-M. Ilgenfritz, M. Muller-Preussker, B. Petersson, A. Schreiber, ”Magnetic catalysis (and inverse catalysis) at finite temperature in two-color lattice QCD”, Phys.Rev.D 89 (2014) 5, 054512, arXiv:1310.7876 [hep-lat].
  * (39) V. G. Bornyakov, P. V. Buividovich, N. Cundy, O. A. Kochetkov and A. Schäfer, ”Deconfinement transition in two-flavour lattice QCD with dynamical overlap fermions in an external magnetic field”, Phys.Rev.D 90 (2014) 3, 034501, arXiv:1312.5628 [hep-lat].
  * (40) Gergely Endrodi, ”Critical point in the QCD phase diagram for extremely strong background magnetic fields”, JHEP 07 (2015) 173, arXiv:1504.08280 [hep-lat].
  * (41) C. Bonati, M. D’Elia, M. Mariti, M. Mesiti, F. Negro, A. Rucci and F. Sanfilippo, ”Magnetic field effects on the static quark potential at zero and finite temperature”, Phys.Rev.D 94 (2016) 9, 094007, arXiv:1607.08160 [hep-lat].
  * (42) Akio Tomiya, Heng-Tong Ding, Swagato Mukherjee, Christian Schmidt, Xiao-Dan Wang, ”Chiral phase transition of three flavor QCD with nonzero magnetic field using standard staggered fermions”, EPJ Web Conf. 175 (2018) 07041, arXiv:1711.02884 [hep-lat].
  * (43) V. V. Braguta, M. N. Chernodub, A. Yu. Kotov, A. V. Molochkov, A. A. Nikolaev, ”Finite-density QCD transition in magnetic field background”, Phys.Rev.D 100 (2019) 11, 114503, arXiv:1909.09547 [hep-lat].
  * (44) N. O. Agasian, S. M. Fedorov, ”Quark-hadron phase transition in a magnetic field”, Phys.Lett.B 663 (2008) 445-449, arXiv:0803.3156 [hep-ph].
  * (45) Eduardo S. Fraga, Leticia F. Palhares, ”Deconfinement in the presence of a strong magnetic background: an exercise within the MIT bag model”, Phys.Rev.D 86 (2012) 016008, arXiv:1201.5881 [hep-ph].
  * (46) Raoul Gatto, Marco Ruggieri, ”Quark Matter in a Strong Magnetic Background”, Lect.Notes Phys. 871 (2013) 87-119, arXiv:1207.3190 [hep-ph].
  * (47) Eduardo S. Fraga, Jorge Noronha, Leticia F. Palhares, ”Large Nc Deconfinement Transition in the Presence of a Magnetic Field”, Phys.Rev.D 87 (2013) 11, 114014, arXiv:1207.7094 [hep-ph].
  * (48) Florian Preis, Anton Rebhan, Andreas Schmitt, ”Inverse magnetic catalysis in field theory and gauge-gravity duality”, Lect.Notes Phys. 871 (2013) 51-86, arXiv:1208.0536 [hep-ph].
  * (49) M. N. Chernodub, ”Electromagnetic superconductivity of vacuum induced by strong magnetic field”, Lect.Notes Phys. 871 (2013) 143-180, arXiv:1208.5025 [hep-ph].
  * (50) Kenji Fukushima, Yoshimasa Hidaka, ”Magnetic Catalysis vs Magnetic Inhibition”, Phys.Rev.Lett. 110 (2013) 3, 031601, arXiv:1209.1319 [hep-ph].
  * (51) Dmitri E. Kharzeev, Karl Landsteiner, Andreas Schmitt, Ho-Ung Yee, ”Strongly interacting matter in magnetic fields”: an overview”, Lect.Notes Phys. 871 (2013) 1-11, arXiv:1211.6245 [hep-ph].
  * (52) Jingyi Chao, Pengcheng Chu, Mei Huang, ”Inverse magnetic catalysis induced by sphalerons”, Phys.Rev.D 88 (2013) 054009, arXiv:1305.1100 [hep-ph].
  * (53) M. Ferreira, P. Costa, O. Lourenço, T. Frederico, C. Providência, ”Inverse magnetic catalysis in the (2+1)-flavor Nambu–Jona-Lasinio and Polyakov–Nambu–Jona-Lasinio models”, Phys.Rev.D 89 (2014) 11, 116011, arXiv:1404.5577 [hep-ph].
  * (54) Lang Yu, Hao Liu, Mei Huang, ”Spontaneous generation of local CP violation and inverse magnetic catalysis”, Phys.Rev.D 90 (2014) 7, 074009, arXiv:1404.6969 [hep-ph].
  * (55) Alejandro Ayala, M. Loewe, Ana Julia Mizher, R. Zamora, ”Inverse magnetic catalysis for the chiral transition induced by thermo-magnetic effects on the coupling constant”, Phys.Rev.D 90 (2014) 3, 036001, arXiv:1406.3885 [hep-ph].
  * (56) Alejandro Ayala, M. Loewe, R. Zamora, ”Inverse magnetic catalysis in the linear sigma model with quarks”, Phys.Rev.D 91 (2015) 1, 016002, arXiv:1406.7408 [hep-ph].
  * (57) E. J. Ferrer, V. de la Incera, X. J. Wen, ”Quark Antiscreening at Strong Magnetic Field and Inverse Magnetic Catalysis”, Phys.Rev.D 91 (2015) 5, 054006, arXiv:1407.3503 [nucl-th].
  * (58) Jens O. Andersen, William R. Naylor, Anders Tranberg, ”Inverse magnetic catalysis and regularization in the quark-meson model”, JHEP 02 (2015) 042, arXiv:1410.5247 [hep-ph].
  * (59) Bo Feng, Defu Hou, Hai-cang Ren, ”(Inverse) Magnetic Catalysis in Bose-Einstein Condensation of Neutral Bound Pairs”, Phys.Rev.D 92 (2015) 6, 065011, arXiv:1412.1647 [cond-mat.quant-gas].
  * (60) Niklas Mueller, Jan M. Pawlowski, ”Magnetic catalysis and inverse magnetic catalysis in QCD”, Phys.Rev.D 91 (2015) 11, 116010, arXiv:1502.08011 [hep-ph].
  * (61) MM. Ferreira, P. Costa and C. Providência, ”The QCD phase diagram in the presence of an external magnetic field: the role of the inverse magnetic catalysis”, arXiv:1509.01181 [hep-ph].
  * (62) Aftab Ahmad, Alfredo Raya, ”Inverse magnetic catalysis and confinement within a contact interaction model for quarks”, J.Phys.G 43 (2016) 6, 065002, arXiv:1602.06448 [hep-ph].
  * (63) R. L. S. Farias, V. S. Timoteo, S. S. Avancini, M. B. Pinto, G. Krein, ”Thermo-magnetic effects in quark matter: Nambu–Jona-Lasinio model constrained by lattice QCD”, Eur.Phys.J.A 53 (2017) 5, 101, arXiv:1603.03847 [hep-ph].
  * (64) M. Ferreira, P. Costa and C. Providência,, ”Magnetized QCD phase diagram”, Acta Phys.Polon.Supp. 10 (2017) 4, 1197-1202, arXiv:1712.08384 [hep-ph].
  * (65) Hardik Bohra, David Dudal, Ali Hajilou, Subhash Mahapatra, ”Anisotropic string tensions and inversely magnetic catalyzed deconfinement from a dynamical AdS/QCD model”, Phys.Lett.B 801 (2020) 135184, arXiv:1907.01852 [hep-th].
  * (66) M. D’Elia, F. Manigrasso, F. Negro and F. Sanfilippo, ”QCD phase diagram in a magnetic background for different values of the pion mass”, Phys.Rev.D 98 (2018) 5, 054509, arXiv:1808.07008 [hep-lat].
  * (67) Gergely Endrodi, Matteo Giordano, Sandor D. Katz, Tamas G. Kovacs, Ferenc Pittler, ”Magnetic catalysis and inverse catalysis for heavy pions”, JHEP 07 (2019) 007, arXiv:1904.10296 [hep-lat].
  * (68) Juan M. Maldacena, ”The Large N Limit of Superconformal Field Theories and Supergravity”, Int.J.Theor.Phys. 38 (1999) 1113-1133, Adv.Theor.Math.Phys. 2 (1998) 231-252, arXiv:hep-th/9711200.
  * (69) S.S. Gubser, I.R. Klebanov, A.M. Polyakov, ”Gauge Theory Correlators from Non-Critical String Theory”, Phys.Lett.B 428 (1998) 105-114, arXiv:hep-th/9802109.
  * (70) Edward Witten, ”Anti De Sitter Space And Holography”, Adv.Theor.Math.Phys. 2 (1998) 253-291, arXiv:hep-th/9802150.
  * (71) Kiminad A. Mamo, ”Inverse magnetic catalysis in holographic models of QCD”, JHEP 05 (2015) 121, arXiv:1501.03262 [hep-th].
  * (72) Romulo Rougemont, Renato Critelli, Jorge Noronha, ”Holographic calculation of the QCD crossover temperature in a magnetic field”, Phys.Rev.D 93 (2016) 4, 045013, arXiv:1505.07894 [hep-th].
  * (73) Kaddour Chelabi, Zhen Fang, Mei Huang, Danning Li, Yue-Liang Wu, ”Realization of chiral symmetry breaking and restoration in holographic QCD”, Phys.Rev.D 93 (2016) 10, 101901, arXiv:1511.02721 [hep-ph].
  * (74) Si-wen Li, Tuo Jia, ”Dynamically flavored description of holographic QCD in the presence of a magnetic field”, Phys.Rev.D 96 (2017) 6, 066032, arXiv:1604.07197 [hep-th].
  * (75) Diego M. Rodrigues, Eduardo Folco Capossoli, Henrique Boschi-Filho, ”Deconfinement phase transition in a magnetic field in 2+1 dimensions from holographic models”, Phys.Lett.B 780 (2018) 37-40, arXiv:1709.09258 [hep-th].
  * (76) Diego M. Rodrigues, Eduardo Folco Capossoli, Henrique Boschi-Filho, ”Magnetic catalysis and inverse magnetic catalysis in (2+1)-dimensional gauge theories from holographic models”, Phys.Rev.D 97 (2018) 12, 126001, arXiv:1710.07310 [hep-th].
  * (77) Umut Gursoy, Matti Jarvinen, Govert Nijs, Juan F. Pedraza, ”Inverse Anisotropic Catalysis in Holographic QCD”, JHEP 04 (2019) 071, arXiv:1811.11724 [hep-th].
  * (78) Veselin G. Filev, Radoslav C. Rashkov, ”Magnetic Catalysis of Chiral Symmetry Breaking. A Holographic Prospective”, Adv.High Energy Phys. 2010 (2010) 473206, arXiv:1010.0444 [hep-th].
  * (79) Florian Preis, Anton Rebhan, Andreas Schmitt, ”Inverse magnetic catalysis in dense holographic matter”, JHEP 03 (2011) 033, arXiv:1012.4785 [hep-th].
  * (80) N. Callebaut, D. Dudal, ”On the transition temperature(s) of magnetized two-flavour holographic QCD”, Phys.Rev.D 87 (2013) 10, 106002, arXiv:1303.5674 [hep-th].
  * (81) Alfonso Ballon-Bayona, ”Holographic deconfinement transition in the presence of a magnetic field”, JHEP 11 (2013) 168, arXiv:1307.6498 [hep-th].
  * (82) Brett McInnes, ”Inverse Magnetic/Shear Catalysis”, Nucl.Phys.B 906 (2016) 40-59, arXiv:1511.05293 [hep-th].
  * (83) Kaddour Chelabi, Zhen Fang, Mei Huang, Danning Li, Yue-Liang Wu, ”Chiral Phase Transition in the Soft-Wall Model of AdS/QCD”, JHEP 04 (2016) 036, arXiv:1512.06493 [hep-ph].
  * (84) M. Ruggieri, G. X. Peng, ”Quark Matter in a Parallel Electric and Magnetic Field Background: Chiral Phase Transition and Equilibration of Chiral Density”, Phys.Rev.D 93 (2016) 9, 094021, arXiv:1602.08994 [hep-ph].
  * (85) Nick Evans, Carlisson Miller, Marc Scott, ”Inverse Magnetic Catalysis in Bottom-Up Holographic QCD”, Phys.Rev.D 94 (2016) 7, 074034, arXiv:1604.06307 [hep-ph].
  * (86) Danning Li, Mei Huang, Yi Yang, Pei-Hung Yuan, ”Inverse Magnetic Catalysis in the Soft-Wall Model of AdS/QCD”, JHEP 02 (2017) 030, arXiv:1610.04618 [hep-th].
  * (87) U. Gürsoy, I. Iatrakis, M. Järvinen and G. Nijs, ”Inverse Magnetic Catalysis from improved Holographic QCD in the Veneziano limit”, JHEP 03 (2017) 053, arXiv:1611.06339 [hep-th].
  * (88) Alfonso Ballon-Bayona, Matthias Ihl, Jonathan P. Shock, Dimitrios Zoakos, ”A universal order parameter for Inverse Magnetic Catalysis”, JHEP 10 (2017) 038, arXiv:1706.05977 [hep-th].
  * (89) Umut Gursoy, Matti Jarvinen, Govert Nijs, ”Holographic QCD in the Veneziano limit at finite Magnetic Field and Chemical Potential”, Phys.Rev.Lett. 120 (2018) 24, 242002, arXiv:1707.00872 [hep-th].
  * (90) Diego M. Rodrigues, Danning Li, Eduardo Folco Capossoli, Henrique Boschi-Filho, ”Chiral Symmetry Breaking and Restoration in (2+1) Dimensions from Holography: Magnetic and Inverse Magnetic Catalysis”, Phys.Rev.D 98 (2018) 10, 106007, arXiv:1807.11822 [hep-th].
  * (91) Diego M. Rodrigues, Danning Li, Eduardo Folco Capossoli, Henrique Boschi-Filho, ”Holographic Description of Chiral Symmetry Breaking in a Magnetic Field in 2+1 Dimensions with an Improved Dilaton”, EPL 128 (2019) 6, 61001, arXiv:1811.04117 [hep-ph].
  * (92) Dimitrios Giataganas, Umut Gürsoy, Juan F. Pedraza, ”Strongly-coupled anisotropic gauge theories and holography”, Phys.Rev.Lett. 121 (2018) 12, 121601, arXiv:1708.05691 [hep-th].
  * (93) R. G. Cai, S. He and D. Li, ”A hQCD model and its phase diagram in Einstein-Maxwell-Dilaton system”, JHEP 1203 , 033 (2012), arXiv:1201.0820 [hep-th].
  * (94) Song He, Shang-Yu Wu, Yi Yang, Pei-Hung Yuan, ”Phase Structure in a Dynamical Soft-Wall Holographic QCD Model”, JHEP 04 (2013) 093, arXiv:1301.0385 [hep-th].
  * (95) Yi Yang, Pei-Hung Yuan, ”A Refined Holographic QCD Model and QCD Phase Structure”, JHEP 11 (2014) 149, arXiv:1406.1865 [hep-th].
  * (96) Yi Yang, Pei-Hung Yuan, ”Confinement-Deconfinment Phase Transition for Heavy Quarks”, JHEP 12 (2015) 161, arXiv:1506.05930 [hep-th]
  * (97) Meng-Wei Li, Yi Yang, Pei-Hung Yuan, ”Approaching Confinement Structure for Light Quarks in a Holographic Soft Wall QCD Model”, Phys.Rev.D 96 (2017) 6, 066013, arXiv:1703.09184 [hep-th].
  * (98) Yi Yang, Pei-Hung Yuan, ”Universal Behaviors of Speed of Sound from Holography”, Phys.Rev.D 97 (2018) 12, 126009, arXiv:1705.07587 [hep-th].
  * (99) Meng-Wei Li, Yi Yang, Pei-Hung Yuan, ”Imprints of Early Universe on Gravitational Waves from First-Order Phase Transition in QCD”, arXiv:1812.09676 [hep-th].
  * (100) Oliver DeWolfe, Steven S. Gubser, Christopher Rosen, ”A holographic critical point”, Phys.Rev.D 83 (2011) 086005, arXiv:1012.1864 [hep-th].
  * (101) Oliver DeWolfe, Steven S. Gubser, Christopher Rosen, ”Dynamic critical phenomena at a holographic critical point”, Phys.Rev.D 84 (2011) 126014, 1108.2029 [hep-th].
  * (102) Irina Aref’eva, Kristina Rannu, ”Holographic Anisotropic Background with Confinement-Deconfinement Phase Transition”, JHEP 1805 (2018) 206, arXiv:1802.05652 [hep-th].
  * (103) Z. Fodor, S.D. Katz, ”Lattice determination of the critical point of QCD at finite T and $\mu$”, JHEP 03 (2002) 014, arXiv:hep-lat/0106002.
  * (104) Yoshitaka Hatta, Takashi Ikeda, ”Universality, the QCD critical/tricritical point and the quark number susceptibility”, Phys.Rev.D 67 (2003) 014028, arXiv:hep-ph/0210284.
  * (105) M. Ferreira, P. Costa and C. Providência, ”Multiple critical end points in magnetized three flavor quark matter”, Phys.Rev.D 97 (2018) 1, 014014, arXiv:1712.08378 [hep-ph].
  * (106) Xun Chen, Danning Li, Defu Hou, Mei Huang, ”Quarkyonic phase from quenched dynamical holographic QCD model”, JHEP 03 (2020) 073, arXiv:1908.02000 [hep-ph].