# Neutral pion masses within a hot and magnetized medium in a lattice-improved
soft-wall AdS/QCD model

Nanxiang Wen  Department of Physics and Siyuan Laboratory, Jinan University,
Guangzhou 510632, China  Xuanmin Cao  Corresponding author<EMAIL_ADDRESS>Department of Physics and Siyuan Laboratory, Jinan University, Guangzhou
510632, China  Jingyi Chao  Corresponding author<EMAIL_ADDRESS>College of Physics and Communication Electronics, Jiangxi Normal University,
Nanchang, Jiangxi 330022, China  Hui Liu  Department of Physics and Siyuan
Laboratory, Jinan University, Guangzhou 510632, China

###### Abstract

We investigate chiral phase transitions and the screening masses, pole masses,
and thermal widths of neutral pion meson with finite temperature $T$ and
magnetic field $B$ in a lattice-improved AdS/QCD model, which is constructed
by fitting the lattice results of the pseudo-critical temperatures
$T_{\text{pc}}(B)$. Specifically, we have that the chiral condensate $\sigma$
undergoes a crossover phase transition demonstrating distinct magnetic
catalysis and inverse magnetic catalysis effects in very low and high-
temperature regions with fixed finite $B$, respectively. For the screening
masses, we find that the longitudinal component decreases with $B$ at very low
and high temperatures and increases with $B$ near $T_{\text{pc}}$. The
transverse component always increases with $B$ at fixed $T$. However, both the
longitudinal and transverse screening masses increase with $T$ at fixed $B$.
Furthermore, we find that the pole mass decreases with the increasing of $B$
or $T$. Besides, it is interesting to note that the thermal width shows
similar behavior to the longitudinal screening masses in the very high
temperature region.

AdS/QCD, Phase diagram, Neutral pion, Screening mass, Pole mass, Magnetic
field

††preprint: APS

###### Contents

  1. I Introduction
  2. II Gravity background
    1. II.1 EOMs within the EM system
    2. II.2 Complete numerical solutions
  3. III The IMC and MC effects on chiral phase transition
  4. IV Correlation functions and the masses of neutral pion at finite temperature and magnetic field
    1. IV.1 EOMs and its asymptotic expansion of pseudo-scalar channel
    2. IV.2 Numerical results of screening masses of neutral pion
    3. IV.3 Numerical results of pole mass and thermal widths of neutral pion
  5. V Conclusion and discussion
  6. VI Acknowledgments

##  I Introduction

The properties of strong interactions in the presence of a substantial
external magnetic field have been studied extensively in recent years. The
reason for this is that in off-center heavy-ion collisions, strong magnetic
field of about $10^{18}$ to $10^{19}$ Gauss, or $|eB|\sim$ $m_{\pi}^{2}$ to
15$m_{\pi}^{2}$, are produced when two fast-moving nuclei collide [1, 2, 3].
In the presence of such a strong external magnetic field, not only is the
phase diagram of quantum chromodynamics (QCD) matter altered [4, 5, 6], but
also many novel and interesting phenomena will arise due to the interacting
between the magnetic field and the non-perturbative properties of non-Abelian
gauge theory, including the chiral magnetic effect [7, 8, 9, 10], disputable
superconductivity in magnetized vacuum [11, 12], neutral pion condensation
[13], diamagnetism at low temperature and paramagnetism at high temperature
[14].

It has been shown that in a simplified effective theory, certain properties of
the equilibrium plasma can be obtained by a reduction of the dimension from
$D$ to $D-1$ at finite temperature. Furthermore, in the strong $B$ limit the
longitudinal and transverse spaces are decoupled, allowing to reduce the
transverse dimensions. Compared to magnetized matter at zero temperature,
several unexpected phenomena have arisen, showing the interplay between the
anisotropies induced by thermal excitation and the magnetic field. The most
famous effect is that the chiral order parameter, characterized by spontaneous
chiral symmetry breaking (SSB), is catalyzed by the magnetic field in the
vacuum, known as magnetic catalysis (MC) in various models, including two-
flavor QCD system [15, 16, 17]. However, the chiral pseudo-critical
temperature $T_{\text{pc}}$ decreases with increasing $B$ in QCD matter. This
is beyond the prediction of low-energy effective theories or model
calculations, known as inverse magnetic catalysis (IMC) [4, 18]. In order to
solve this puzzle, extensive work has been done by numerous approaches. (see
recent reviews, e.g., Ref. [19]). Among these explanations and calculations,
the proper implementation of thermal modification is particularly important.

Therefore, we pay more attention to the properties of the light mesons at
finite temperature. Since the pattern of SSB is $U(1)_{I_{3}}\otimes
U(1)_{AI_{3}}\to U(1)_{L+R}$, the neutral pion is remained as the only
Goldstone boson in two-flavor QCD. It plays a unique role in low-energy
hadronic physics. [20, 21, 22, 23, 24]. Our motivation for the study of
thermal effects on $\pi^{0}$ is the observation that reliable information on
the temperature modification of magnetized hadronic properties is still
lacking. Since the temperature breaks the Lorentz invariance, it implies that
one needs not only the pole mass $m_{\text{pole}}$, which describes the
positions of the poles in the particle’s propagators, but also the screening
masses $m_{\text{scr}}$, which characterize the exponential decay of static
propagators. The relationship between the screening mass and the pole mass is
determined by the dispersion relation: $m_{\text{scr}}=m_{\text{pole}}/u$. In
addition, there are two types of screening masses because the magnetic field
breaks the $O(3)$ rotation symmetries to $O(2)$: $m_{\text{scr},\parallel}$
and $m_{\text{scr},\perp}$ for the masses along the direction of $B$ and those
perpendicular to it, respectively. Required by the law of causality, as
pointed out by ref. [21], it can be deduced that $u_{\perp,\parallel}<1$.
Also, since motion along the transverse direction is suppressed compared to
motion along the longitudinal direction, a naive estimate is that
$u_{\perp}<u_{\parallel}$. We conclude that $u_{\perp}<u_{\parallel}<1$ and so
then $m_{\text{scr},\perp}>m_{\text{scr},\parallel}>m_{\text{pole}}$ [21].
Over the decades, numerous models have been developed for studying the vacuum
and thermal properties of hadrons with a magnetic field. These frameworks
include Lattice QCD (LQCD) [25, 23, 26, 27, 28, 29], chiral perturbation
theory ($\chi$PT) [30, 31, 32], Nambu–Jona-Lasinio models (NJL) [21, 22, 33,
34, 35], functional renormalization group (FRG) [24, 36, 37].

Besides the traditional methods, the discovery of the famous anti-de
Sitter/conformal field theory (AdS/CFT) correspondence [38, 39, 40] offers a
powerful tool for investigating strongly coupling problems of QCD. In the
framework of the bottom-up approach, several holographic QCD (HQCD) models has
been constructed, e.g. the hard-wall model [41] and soft-wall AdS/QCD
model[42] for chiral dynamics and hadronic physics, the Einstein-Maxwell-
Dilaton system [43, 44, 45, 46] for thermodynamics, the light-front
holographic QCD model [47] for hadronic physics, and so on.

Among those HQCD models, we will take the soft-wall model as the start point
of this work, since it can realize a good description on hadronic physics [48,
49, 50, 51, 52, 53, 54, 55, 56, 57, 58] as well as the spontaneous chiral
symmetry breaking [59, 60]. Furthermore, the mass plane phase diagram from the
soft-wall model [61, 62] are shown to be qualitatively consistent with the so-
called “Columbia plot” [63, 64]. Since the model is constructed based on
symmetry, it is quite direct to introduce the conserve current. Therefore, it
is also convenient to extend the study to many different conditions, such as
finite baryon/isospin densities [65, 66, 67], rotating mediums [68], none-
quilibrium phase transitions [69]. Besides, the thermal properties as well as
the Goldstone nature of pions has been systematically investigated in [70, 71,
72, 73].

Through introducing the magnetic field in the holographic models, it is
possible to study the properties of QCD in the hot magnetized medium [74, 75,
76, 77]. Actually, in ref. [74], the authors obtained the IMC effect on chiral
condensation and its pseudo-critical temperature $T_{\text{pc}}$.
Additionally, result from ref. [77] show that chiral condensation not only
exhibits IMC effect near $T_{\text{pc}}$ but also demonstrates MC effect at
low temperatures. However, those studies are based on perturbative expansion
solutions, and the magnetic field strength can not be extended to large
values. Furthermore, those studies only analyze the order parameter, and the
properties about the Goldstone bosons are absent. Therefore, in this work, we
will try to solve the full magnetized background and study the thermal pions
under this background. Furthermore, the variation of the chiral phase
transition temperature $T_{\text{pc}}$ with $B$ in soft-wall AdS/QCD models
differs from the lattice simulation results [4, 18]. Hence, we posit that
those soft-wall AdS/QCD models should undergo alterations when introducing the
magnetic field. Inspired by the the magnetic field-dependent four-fermion
coupling constant $G(B)$ introduced in ref. [22], we will modify the 5D mass
of the soft-wall AdS/QCD model by introducing a polynomial $g(B)$, which is
considered as an effective coupling between the dilaton and the scalar field,
or in some sense the coupling between the gluonic and chiral sector. We will
fit such an effective coupling by comparing the data of $T_{\text{pc}}(B)$
with the lattice simulation [4]. After fitting the model, we will
systematically study the thermal properties of the neutral pions.

This paper is organized into the following sections. In Section II, after
introducing the Einstein-Maxwell system with back-reaction of magnetic field,
we solve the Einstein-Maxwell equation and obtain the complete numerical
background solutions with different temperatures $T$ and magnetic field
strength $B$. In Section III, we modified the expression of the 5D mass
$m_{5}$ in the AdS/QCD model through fitting the pseudo-critical temperatures
$T_{\text{pc}(B)}$ from the Lattice simulation [4]. Then, we numerically
obtain the chiral condensates with different conditions of $T$ and $B$ and
check out their magnetic catalysis and inverse magnetic catalysis effects. In
Sections IV, we extract and investigate the longitudinal screening masses
$m_{\text{scr},\parallel}$, transverse screening masses
$m_{\text{scr},\perp}$, pole masses $m_{\text{pole}}$, and thermal widths
$\Gamma/2$ of the neutral pion under finite temperature and magnetic field
from the lattice-improved AdS/QCD model. Finally, in Section V, we give a
summary and discussion.

##  II Gravity background

For the Gravity background, the authors in Ref. [74] employed an asymptotic
expansion method to obtain an approximate solution of the Einstein-Maxwell
(EM) system. However, the result is only reasonable under the condition $B\ll
T^{2}$. 111In our study, the physical dimension of $B$ is GeV2. To extend to
the large $B$ region, we will introduce the "shooting method" [78, 71] to
solve the EM system and obtain its full solution.

To introduce the magnetic field, we follow the strategy in refs. [79, 80, 74].
The back-reaction of the magnetic field is considered in the EM system. Its
action in five-dimension space-time is

$S_{B}=\frac{1}{16\pi G_{5}}\int
d^{5}x\sqrt{-g}\left(R-F_{MN}F^{{MN}}+\frac{12}{L^{2}}\right),$ (1)

with $R$ the scalar curvature, $L$ the AdS radius and $G_{5}$ the 5D Newton
constant. The notations $M$, $N$ take values of $0,1,\cdots,4$. The $g$ is the
metric determinant of $g_{MN}$. The $F_{MN}$ stands for a $U(1)$ gauge field.
The matter part, described by the soft-wall AdS/QCD model, is considered as a
probe on this background. Thus, the back-reaction from the matter part is
neglected.

###  II.1 EOMs within the EM system

Within the EM system, the equations of motion (EOMs) are

$E_{MN}-\frac{6}{L^{2}}g_{MN}-\left(g^{IJ}F_{IM}F_{JN}-\frac{1}{4}F_{IJ}F^{IJ}g_{MN}\right)=0,$
(2a) $\nabla_{M}F^{MN}=0,$ (2b)

with $E_{MN}$ the Einstein tensor, $R_{MN}$ the $Ricci$ tensor and $R$ the
$Ricci$ scalar, $E_{MN}=R_{MN}-\frac{R}{2}g_{MN}$. For satisfying the field
equation for $F_{MN}$ in eq. (2b), the configuration of a constant magnetic
field could take as

$F=\frac{B}{L}dx_{1}\land dx_{2},$ (3)

where $L$ is set to unity in this work, $L=1$.

In this 5D coordinate system, it takes as $(t,x_{1},x_{2},x_{3},z)$, in which
the coordinate $z$ corresponds to the radial holographic direction. For the
magnetic field align along the $x_{3}$ axis, one has
$B_{x_{3}}=F_{x_{1}x_{2}}=\partial_{x_{1}}A_{x_{2}}-\partial_{x_{2}}A_{x_{1}}=B$,
where the bulk gauge potential is defined as
$A_{\mu}(x_{1},z)=\frac{1}{2}B(x_{1}\delta^{x_{2}}_{\mu}-x_{2}\delta^{x_{1}}_{\mu})$.
As a result, the metric ansatz could be chosen as

$ds^{2}=e^{2A(z)}\left(-fdt^{2}+\frac{1}{f}dz^{2}+h\left(dx_{1}^{2}+dx_{2}^{2}\right)+q\,dx_{3}^{2}\right),$
(4)

with $A(z)=-\text{ln}(z)$. As our study focuses on the thermal properties in
the equilibrium state, the $f$, $h$, and $q$ all are functions only with
respect to the variable $z$.

Under the given metric ansatz in eq. (4), EOM in eq. (2a) reduces to the
following forms:

$f^{\prime\prime}+f^{\prime}\left(\frac{h^{\prime}}{3h}+\frac{q^{\prime}}{6q}-\frac{3}{z}\right)+f\left(-\frac{2h^{\prime}q^{\prime}}{3hq}+\frac{2h^{\prime}}{hz}-\frac{h^{\prime
2}}{3h^{2}}+\frac{q^{\prime}}{qz}\right)-\frac{8B^{2}z^{2}}{3h^{2}}=0,$ (5a)
$q^{\prime\prime}+q^{\prime}\left(\frac{2f^{\prime}}{3f}+\frac{h^{\prime}}{3h}-\frac{2}{z}\right)+q\left(-\frac{8B^{2}z^{2}}{3fh^{2}}-\frac{2f^{\prime}h^{\prime}}{3fh}+\frac{2h^{\prime}}{hz}-\frac{h^{\prime
2}}{3h^{2}}\right)-\frac{q^{\prime 2}}{2q}=0,$ (5b)
$h^{\prime\prime}+h^{\prime}\left(\frac{f^{\prime}}{3f}-\frac{q^{\prime}}{6q}-\frac{1}{z}\right)+h\left(\frac{q^{\prime}}{qz}-\frac{f^{\prime}q^{\prime}}{3fq}\right)-\frac{h^{\prime
2}}{3h}+\frac{4B^{2}z^{2}}{3fh}=0,$ (5c) together with a constrain equation
$\frac{f^{\prime}h^{\prime}}{2fh}+\frac{f^{\prime}q^{\prime}}{4fq}-\frac{3f^{\prime}}{2fz}-\frac{6}{fz^{2}}+\frac{h^{\prime}q^{\prime}}{2hq}-\frac{3h^{\prime}}{hz}+\frac{h^{\prime
2}}{4h^{2}}-\frac{3q^{\prime}}{2qz}+\frac{6}{z^{2}}+\frac{B^{2}z^{2}}{fh^{2}}=0.$
(5d)

###  II.2 Complete numerical solutions

Through a careful analysis of the EOMs in eqs. (5), one can find that there
are singularities at both $z=0$ and $z=z_{h}$, which result in the
difficulties in getting analytical solutions. Nevertheless, we can employ the
numerical algorithm "shooting method", as introduced in refs. [78, 71], to
solve these ODEs.

For eqs. (5), one can obtain the asymptotic expansions around the ultraviolet
(UV) boundary at $z=0$

$\displaystyle f(z)$
$\displaystyle=1+(\frac{2B^{2}}{3h_{0}^{2}})z^{4}\ln(z)+f_{4}z^{4}+\mathcal{O}(z^{5}),\;\;\;\;\;$
(6a) $\displaystyle q(z)$
$\displaystyle=q_{0}+(\frac{2q_{0}B^{2}}{3h_{0}^{2}})z^{4}\ln{(z)}+q_{4}z^{4}+\mathcal{O}(z^{5}),\,$
(6b) $\displaystyle h(z)$
$\displaystyle=h_{0}-(\frac{B^{2}}{3h_{0}})z^{4}\ln{(z)}-\frac{h_{0}q_{4}}{2q_{0}}z^{4}+\mathcal{O}(z^{5}),$
(6c)

with $f_{4}$, $q_{0}$, $q_{4}$ and $h_{0}$ the integration constants.

At the horizon $z=z_{h}$ or the infrared (IR) boundary, in order to get black
hole solution, the condition $f(z=z_{h})=0$ must be satisfied. Thus, for the
asymptotic expansion of $f(z)$ at $z=z_{h}$, there is no constant term. Near
the horizon, one can derive the asymptotic expansions of eqs. (5) as

$\displaystyle f(z)=$ $\displaystyle
f_{h1}(z-z_{h})+\frac{1}{3z_{h}^{2}}\Big{(}\frac{5B^{2}{z_{h}}^{4}}{h_{{h0}}^{2}}+3{z_{h}}f_{{h1}}-6\Big{)}$
(7a) $\displaystyle\times(z-z_{h})^{2}+\mathcal{O}(z-z_{h})^{3},$
$\displaystyle q(z)=$ $\displaystyle
q_{h0}+\frac{2q_{{h0}}}{3f_{{h1}}h_{{h0}}^{2}z_{h}^{2}}$
$\displaystyle\times\left(2B^{2}z_{h}^{4}+3f_{{h1}}h_{{h0}}^{2}z_{h}+12h_{{h0}}^{2}\right)(z-z_{h})$
$\displaystyle+\frac{q_{h0}}{9{z_{h}}^{4}f_{{h1}}^{2}h_{{h0}}^{4}}\Big{[}20B^{4}{z_{h}}^{8}+12B^{2}{z_{h}}^{5}f_{{h1}}h_{{h0}}^{2}$
$\displaystyle+9h_{{h0}}^{4}\left({z_{h}}f_{{h1}}+4\right){}^{2}\Big{]}(z-z_{h})^{2}+\mathcal{O}(z-z_{h})^{3},$
$\displaystyle h(z)=$ $\displaystyle
h_{h0}+\frac{2}{3f_{{h1}}h_{{h0}}z_{h}^{2}}$ (7c)
$\displaystyle\times\left(-4B^{2}z_{h}^{4}+3f_{{h1}}h_{{h0}}^{2}z_{h}+12h_{{h0}}^{2}\right)(z-z_{h})$
$\displaystyle+\frac{1}{9{z_{h}}^{4}f_{{h1}}^{2}h_{{h0}}^{3}}\Big{[}8B^{4}{z_{h}}^{8}-24B^{2}{z_{h}}^{4}h_{{h0}}^{2}$
$\displaystyle\times\left({z_{h}}f_{{h1}}+3\right)+9h_{{h0}}^{4}\left({z_{h}}f_{{h1}}+4\right){}^{2}\Big{]}$
$\displaystyle\times(z-z_{h})^{2}+\mathcal{O}(z-z_{h})^{3},$

where the $f_{h1}$, $q_{h0}$ and $h_{h0}$ are the integration constants.

\begin{overpic}[scale={0.4}]{fhqwithTTT1.eps} \put(25.0,60.0){\bf{(a)}}
\end{overpic}\begin{overpic}[scale={0.4}]{fhqwithTTT2.eps}
\put(25.0,60.0){\bf{(b)}}
\end{overpic}\begin{overpic}[scale={0.4}]{fhqwithTTT3.eps}
\put(25.0,60.0){\bf{(c)}}
\end{overpic}\begin{overpic}[scale={0.4}]{fhqwithTTTgao1.eps}
\put(25.0,60.0){\bf{(d)}}
\end{overpic}\begin{overpic}[scale={0.4}]{fhqwithTTTgao2.eps}
\put(25.0,60.0){\bf{(e)}}
\end{overpic}\begin{overpic}[scale={0.4}]{fhqwithTTTgao3.eps}
\put(25.0,60.0){\bf{(f)}}
\end{overpic}\begin{overpic}[scale={0.41}]{fhqZeroTTT.eps}
\put(25.0,50.0){\bf{(g)}}
\end{overpic}\begin{overpic}[scale={0.4}]{fhqwithBBB1.eps}
\put(25.0,60.0){\bf{(h)}}
\end{overpic}\begin{overpic}[scale={0.4}]{fhqwithBBB2.eps}
\put(25.0,60.0){\bf{(i)}} \end{overpic}

Figure 1: The $f$, $q$, and $h$ as functions of $z$ are represented by the
lines of red, blue, and cyan, respectively. (a)-(c), magnetic field $B$ is
fixed at 1 GeV2. Temperature $T$ is fixed at 8, 159, 398 MeV, respectively.
(d)-(f), temperature $T$ is fixed at high value, $T\approx 398$MeV. Magnetic
field $B$ is fixed at 0.1, 1, 4 GeV2, respectively. (h)-(i), temperature $T$
is fixed at low value, $T\approx 8$MeV. Magnetic field $B$ is fixed at 0, 0.1,
2 GeV2, respectively.

Physically, when the magnetic field is absent, $B=0$, the metric in eq. (4)
should reduce into

$dS^{2}=\frac{L^{2}}{z^{2}}\left(-fdt^{2}+\frac{1}{f}dz^{2}+dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}\right).$
(8)

This change stems from the restoration of isotropy within the spatial
dimensions $x_{1},x_{2}$ and $x_{3}$. From eq. (8), it becomes evident that
$h=q=1$ at $B=0$. By Considering the zero magnetic field limit in the
asymptotic expansions eq. (6), it is obviously integral constants $q_{0}$ and
$h_{0}$ should satisfy $q_{0}=1$ and $h_{0}=1$. Furthermore, the temperature
is related to the horizon

$T=\Big{|}\frac{f^{\prime}(z)}{4\pi}\Big{|}_{z=z_{h}}.$ (9)

Thus, from eqs. (7) and (9), one can find that $z_{h}$ is a function of $B$
and $T$ (or the temperature-related quantity $f_{h1}$).

Within this system, considering the eq. (5d), the number of constraints is
reduced to five. However, considering the asymptotic expansions of eqs. (5) at
$z=0$ and $z=z_{h}$, the total count of involved unknown integral constants
amounts to 5, which are $f_{4}$ and $q_{4}$ at $z=0$ boundary, $f_{h1}$,
$q_{h0}$ and $h_{h0}$ at horizon.

In mathematics, within this EM system, the integral constants of the EM system
and $z_{h}$ should be clearly determined by the given values of $B$ and
$f_{h1}$. Thus, in our solving process, we treat the $f_{4}$, $q_{4}$,
$q_{h0}$, $h_{h0}$ and $z_{h}$ as unknown quantities for any given $B$ and
temperature-related quantity $f_{h1}$.

\begin{overpic}[scale={1}]{duguiwanquanjie.eps} \put(20.0,1.0){\bf{(a)}}
\end{overpic}\begin{overpic}[scale={0.7}]{zhdexiangduiwucha.eps}
\put(20.0,1.0){\bf{(b)}} \end{overpic}

Figure 2: (a), the horizon boundary $z_{h}$ as a function of temperature $T$
and magnetic field $B$. (b), the distribution of relative error of $z_{h}$,
$\delta=\Delta/z_{h}=(z_{h}-\tilde{z}_{h})/z_{h}$ with $\tilde{z}_{h}$ the
perturbation result, between the results from the perturbative asymptotic
approximation and the full solution across the plane of the temperature $T$
and magnetic field $B$.

To present how the temperature and magnetic field affect the background
metric, we show variation curves of $f(z)$, $q(z)$ and $h(z)$ in Fig. 1. In
Fig. 1(a-c), without loss of generality, we choose the $B$ is fixed at 1 GeV2
as an example to check the temperature effects with $T\approx 0.008$, $0.159$
and $0.398$ GeV2 in (a), (b) and (c), respectively. We can find that, as the
temperature increases, the difference between $q(z)$ and $h(z)$ is vanishing,
and both of them tend to unity. Considering the zero magnetic field limit
metric in eq. (4), this implies that an increase in temperature is beneficial
for restoring the spatial rotation invariance broken by the magnetic field.
222In fact, in our calculations, no matter what value the $B$ is fixed at, the
temperature will always promote spatial isotropy. In physically, in higher
temperature regions, the whole system will appear more chaotic. High
temperatures lead to a large number of inelastic collisions, causing energy
and momentum transfer between various degrees of freedom. Therefore, high
temperatures are beneficial to ”smoothen” the anisotropy brought by the
magnetic field. In Fig. 1(d-f), we show the numerical results at fixed
temperature $T\approx 0.398$ GeV to present the affect of the variation of
magnetic field. we observe that even at very high temperature $T\approx$ 0.398
GeV, the curves of $q(z)$ and $h(z)$, which is almost collapsed at small $B$,
become splitting as the increasing of $B$. This indicates that the increasing
of the magnetic field enhances the spatial anisotropy. Therefore, there exist
competitive between the temperature and magnetic field for the spatial
anisotropy of the background. On the one hand, increasing the magnetic field
enhances the spatial anisotropy. On the other hand, increased temperature
favors the restoration of spatial asymmetry.

Moreover, as shown in Fig. 1(g-i), at very low fixed temperature $T\approx 8$
MeV, as the increasing of $B$, $q(z)$ and $h(z)$ will split, while $f(z)$ and
$q(z)$ will converge. This suggests that at very low temperatures and large
magnetic field, only slight difference exists between the temporal direction
and $x_{3}$-direction. According to the trend, it might be that it will be
identical in the $x_{3}$-direction and the temporal one at $T=0$. The physic
in $x_{3}$-direction and temporal one satisfy Lorentz rotation invariance.
Moreover, in ref. [21], it argued that when $T=0$, $B>0$, there exist
$m_{\text{pole}}=m_{\text{scr},\parallel}$. This is consistent with our
deduction.

In Fig. 2(a), we show the $T$ and $B$ dependence of $z_{h}$ where the magnetic
field $B$ is from 0.01 $\text{GeV}^{2}$ to 4 $\text{GeV}^{2}$ and the
temperature $T$ is from 0.008 GeV to 0.4 GeV. Notably, at lower $T$ and $B$,
the value of $z_{h}$ is significant large. With increasing $T$ and $B$, the
$z_{h}$ undergoes a sharp decrease. Furthermore, we have a compare to the
perturbative asymptotic approximation results given in ref. [74]. In Fig.
2(b), we present the relative error (RE) of $z_{h}$ between full solutions and
the perturbative asymptotic one. The comparison shows that the perturbative
asymptotic approximation agree well with the full solution outcomes in the
region where $B\ll T^{2}$. Nevertheless, the deviations especially emerge at
that region of low values of $T$ and high values of $B$. Hence, when we do
research at the low-temperature and high-magnetic-field regime, the full
solution is necessary to take and the asymptotic approximation is not
suitable.

##  III The IMC and MC effects on chiral phase transition

After obtained the full solution for the EM system, we can explore the chiral
phase transition through the order parameter (the chiral condensation). Among
the various models investigating chiral phase transitions, the soft-wall model
stands out as one of the most effective models. And we investigate the chiral
properties based on the soft-wall model.

In ref. [74], chiral condensation exhibits noticeable IMC behavior and its
$T_{\text{pc}}$ decrease with increasing $eB$. Under this model, we
numerically calculate the full background solution and the chiral
condensation. The results indicate that no MC effect of chiral condensation
exists at low temperatures. However, the LQCD simulations [4, 18, 23] suggest
that the chiral condensation should exhibit distinct MC and IMC behaviors at
low temperatures and near $T_{\text{pc}}$ respectively. The $T_{\text{pc}}(B)$
monotonously decreases with the increasing of $B$. Additionally, in IR-
improved soft-wall AdS/QCD model, the results indicate that the MC effects of
chiral condensation at low temperatures are very weak [77]. Therefore, to more
consistently describe the MC and IMC effects, it might be necessary to
incorporate the magnetic field effects within the soft-wall model. Inspired by
ref. [22], we modify the 5D mass of the soft-wall AdS/QCD model by introducing
a polynomial $g(B)$. The $g(B)$ is constrained by the variation of the
$T_{\text{pc}}$ with $B$ from lattice simulations [4]. In ref. [73], they
attempt to consider coupling between the dilaton $\Phi$ and scalar field $X$.
Furthermore, in this work, we further optimized the AdS/QCD model through
introducing a magnetic field-dependent polynomial $g(B)$ into a effective 5D
mass expression.

The soft-wall AdS/QCD model is constructed within the bottom-up framework [41,
42] through considering the $SU(N_{f})_{L}\times SU(N_{f})_{R}$ gauge
symmetry. The action is expressed as

$\displaystyle S_{M}=$ $\displaystyle\int dz\int
d^{4}x\sqrt{g}e^{-\Phi(z)}\text{Tr}\\{|DX|^{2}-m_{5}^{2}(z)|X|^{2}$ (10)
$\displaystyle-\lambda|X|^{4}-\frac{1}{4g_{5}^{2}}(F_{L}^{2}+F_{R}^{2})\\},$

with $g$ the determinant of the metric $g_{MN}$. And $g_{5}=2\pi$, which is
determined by comparing the large momentum expansion of the correlator of the
vector current $J^{a}_{\mu}=\overline{q}\gamma_{\mu}t^{a}q$ in both the
AdS/QCD and perturbative QCD [41]. The dilaton profile,
$\Phi(z)=\mu_{g}^{2}z^{2}$, with $\mu_{g}$ the constant that is necessary for
the Regge behavior of the meson spectrum [42]. The covariant derivative
$D^{M}$ and the field strength $F^{MN}_{L,R}$ are defined as
$D^{M}X=\partial^{M}X-iA_{L}^{M}X+iXA_{R}^{M}$ and
$F^{MN}_{L,R}=\partial^{M}A^{N}_{L,R}-\partial^{N}A^{M}_{L,R}-i[A^{M}_{L,R},A^{N}_{L,R}]$
with $A^{M}_{L,R}=A^{a,M}_{L,R}t^{a}_{L,R}$. The $t^{a}_{L,R}(a=1,2,3)$ is
defined as the generators of $SU(2)_{L}$ and $SU(2)_{R}$ respectively with
$M,N=0,1,2,3,4$. In this work, we only consider the case of $N_{f}=2$ with
equal $u$ and $d$ quark mass, $m_{q}=m_{u}=m_{d}$.

For the sake of convenience, the gauge field $A_{L/R}$ can generally be
rearranged into the vector field $V^{M}=\frac{1}{2}(A^{M}_{L}+A^{M}_{R})$, and
the axial-vector field $A^{M}=\frac{1}{2}(A^{M}_{L}-A^{M}_{R})$. The vector
field $V^{M}$ correspond as dual entities to the vector current $J_{\mu}^{V}$.
The axial-vector field $A^{M}$ correspond as dual entities to the axial-vector
current $J_{\mu}^{A}$. Accompanied by the transformed chiral gauge field
strengths denote as

$\displaystyle F^{MN}_{A}$ $\displaystyle=\frac{1}{2}(F^{MN}_{L}-F^{MN}_{R})$
$\displaystyle=\partial^{M}A^{N}-\partial^{N}A^{M}-i[V^{M},A^{N}]-i[A^{M},V^{N}],$
$\displaystyle F^{MN}_{V}$ $\displaystyle=\frac{1}{2}(F^{MN}_{L}+F^{MN}_{R})$
$\displaystyle=\partial^{M}V^{N}-\partial^{N}V^{M}-i[V^{M},A^{N}]-i[A^{M},V^{N}].$
Furthermore, the covariant derivative is $\displaystyle
D_{M}X=\partial_{M}X-i[V_{M},X]-i\\{A_{M},X\\}.$ (11c)

Considering the coupling between the scalar field $X$ and dilaton $\Phi$
$m_{5}^{2}(z)$ , it effectively lead to the modification of $m_{5}$,

$m_{5}^{2}(z)=-3\big{[}1+\gamma\tanh(\kappa\Phi)\big{]},$ (12)

with $\gamma,\kappa$ the free parameters. The leading constant term for the 5D
mass, denoted as $m_{5}^{2}=-3$, can be established through the AdS/CFT
dictionary: $m_{5}^{2}=(\Delta-p)(\Delta+p-4)$, with the choice of $p=0$ and
$\Delta=3$, where $\Delta$ represents the dimension of the dual operator
$\overline{q}_{R}\overline{q}_{L}$ [41].

In ref. [73], the chiral condensation has a slight unphysical bumps up at low
temperatures region with the increasing temperature, so that the model
parameters should be adjusted. Our goal is to achieve the pion mass $M_{\pi}$
333The uppercase $M_{\pi}$ in this article specifically denotes the mass of
the $\pi$ meson under conditions of zero temperature, zero density, and zero
magnetic field strength., $M_{\pi}\approx 139.6(\text{MeV})$ [81], and the
saturation chiral condensate $\sigma$, $\sigma\approx 0.0278(\text{GeV}^{3})$
[51] . Details of the new parameters are provided in Table. 2. The selection
and adjustment of parameters involve following steps. Initially, we adjusted
the value of $\mu_{g}$, a parameter associated with the Regge behavior of the
meson spectrum, and fixed it at 0.35 GeV. Subsequently, we adjusted the
magnitude of $\lambda$, which directly influences the value of the chiral
condensation $\sigma$, setting it to 25. Finally, we determined the values of
$\kappa$ and $\gamma$. In our parameter tuning, we found that the values of
$\kappa$ and $\gamma$ not only affect the $T_{\text{pc}}$ but also impact the
magnitude of $\sigma$. In the absence of an external magnetic field, given
that the pseudo-critical temperature $T_{\text{pc}}$ should lie between 150
and 170 MeV, we fit $\kappa=0.85$ and $\gamma=6$. Additionally, as the quark
mass $m_{q}$ is positively correlated with $M_{\pi}$, we fit $m_{q}$ to 3.9
MeV. The computed result with these parameters yields the pseudo-critical
temperature $T_{\text{pc}}(B=0)\approx 161$ MeV.

Furthermore, during the parameter adjustment process, we observed that
changing the coefficient of the dilation function $\Phi(z)$ in eq. (12) alone
could affect both the values of $\sigma$ and the $T_{\text{pc}}$. Therefore,
when introducing the magnetic field-dependent function $g(B)$ into the
effective 5D mass in eq (12), we have $\kappa+g(B)$ as the coefficient of the
dilation $\Phi(z)$. Finally, to make the coefficient dimensionless, we have a
reduced $B$ as $B/\mu_{g}^{2}$. After incorporating the magnetic field-
dependent polynomial $g(B)$, the expression for the effective 5D mass becomes

$\displaystyle
m_{5}^{2}(z)=-3\bigg{\\{}1+\gamma\tanh\Big{[}\Big{(}\kappa+g(B)\Big{)}\Phi\Big{]}\bigg{\\}},$
(13)

where

$\displaystyle
g(B)=\sum_{i}\eta_{i}\bigg{(}\frac{B}{\mu_{g}^{2}}\bigg{)}^{\alpha_{i}},$ (14)

with $\alpha_{i}=2,4,6,8,10$. Since the magnitude of chiral condensation
should not be impacted by the direction of the magnetic field, we set all
$\alpha_{i}$ to be even. In order to ensure that the variation of the
$T_{\text{pc}}(B)$ falls entirely within the error band of LQCD result, we
adjusted $g(B)$ based on the $T_{\text{pc}}$ data points extracted from LQCD
simulations [4]. Furthermore, we observed that it is different in $M_{\pi}$
and $T_{\text{pc}}(B=0)$ for different effective models. Thus, we opted to use
a dimensionless quantity $B/M_{\pi}^{2}$ to measure the strength of the
magnetic field. We also used the normalized pseudo-critical temperature
$T_{\text{pc}}(B)/T_{\text{pc}}(B=0)$ to assess the variation in the pseudo-
critical temperature with $B$. Finally, we get the fitted coefficients as
shown in Table 2.

Table 1:  Parameters of Model.

Parameters | $m_{q}(\text{GeV})$ | $\mu_{g}(\text{GeV})$ | $\kappa$ | $\gamma$ | $\lambda$
---|---|---|---|---|---
Value | $3.90\times 10^{-3}$ | $0.35$ | $0.85$ | $6$ | $25$

Table 2:  Coefficients of $\eta_{i}$.

| $\eta_{1}$ | $\eta_{2}$ | $\eta_{3}$ | $\eta_{4}$ | $\eta_{5}$
---|---|---|---|---|---
| $0.021$ | $-8.65\times 10^{-4}$ | $2.02\times 10^{-5}$ | $-2.37\times 10^{-7}$ | $1.10\times 10^{-9}$

\begin{overpic}[width=208.13574pt,height=130.08731pt]{model1ciralcondensate.eps}
\put(45.0,58.0){\bf{(a)}}
\end{overpic}\begin{overpic}[width=208.13574pt,height=130.08731pt]{model2ciralcondensate.eps}
\put(45.0,58.0){\bf{(b)}} \end{overpic}

Figure 3: (a), the plot illustrates the chiral condensation $\sigma$ as a
function of $T$ under different fixed $B$. The range of $B$ is approximately
from 0 to 51.32 $M_{\pi}^{2}$, which corresponds to 0 to 1 GeV2 in our model.
(b), it depicts $\Delta\sigma=\sigma(B,T)-\sigma(0,T)$ as a function of
$B/M_{\pi}^{2}$ under different fixed $T$, where the temperatures $T$ were
chosen as 70, 88, 130, 140, 162, and 201 MeV, respectively.
\begin{overpic}[scale={0.55}]{NormalizedTcpof2model.eps} \end{overpic} Figure
4: The pseudo-critical temperature scaled by its $B=0$ value as a function of
the $B/M_{\pi}^{2}$. These results compare with those from lattice simulations
[4] and lattice-improved NJL [22]. Given that the value of $M_{\pi}$ in
lattice simulations and the lattice-improved NJL model are 135 and 138 MeV,
respectively, the results have been scaled accordingly for their respective
$M_{\pi}$ in the extracted data.

In the $N_{f}=2$ case, with equal $u$ and $d$ quark mass, $m_{q}=m_{u}=m_{d}$,
the bulk scalar field X can be expressed as

$X=(\frac{\chi}{2}+S)I_{2}e^{2i\pi^{a}t^{a}},$ (15)

where $\chi(z)$ is linked to the vacuum expectation value (VEV) of $X$ and
$I_{2}$ represents the $2\times 2$ identity matrix. Therefore, one can derive
the EOM for $\chi$ as

$\displaystyle\chi^{\prime\prime}+\bigg{(}3A^{\prime}-\Phi^{\prime}+$
$\displaystyle\frac{f^{\prime}}{f}+\frac{h^{\prime}}{h}+\frac{q^{\prime}}{2q}\bigg{)}\chi^{\prime}$
(16)
$\displaystyle-\frac{e^{2A}\chi\left(\lambda\chi^{2}+2m_{5}^{2}\right)}{2f}=0.$

The EOM for $\chi$ in eq. (16) is a nonlinear second order differential
equation with singularities both at $z=0$ and $z=z_{h}$, which makes
analytical solutions difficult to attain. Nevertheless, we can numerically
solve it with a algorithm (“Shooting mehtod”) as utilized in ref. [71]. Within
this algorithm, by considering the eqs. (5), we can derive the asymptotic
expansions of the $\chi$ at both the UV and IR boundaries as

$\displaystyle\chi(z\rightarrow 0)=$ $\displaystyle m_{q}\zeta
z+\frac{\sigma}{\zeta}z+\frac{1}{4}m_{q}\zeta\Big{[}-6\gamma\mu_{g}^{2}g(B)$
$\displaystyle+(4-6\gamma\kappa)\mu_{g}^{2}+\lambda\zeta^{2}m_{q}^{2}\Big{]}z^{3}\ln(z)+\mathcal{O}(z^{4}),$
$\displaystyle\chi(z\rightarrow z_{h})=$
$\displaystyle\chi_{zh0}+\frac{1}{2f_{h1}z_{h}^{2}}\chi_{zh0}\Big{\\{}\lambda\chi_{zh0}^{2}-6-6\gamma$
(17b)
$\displaystyle\times\tanh\Big{[}\big{(}\kappa+g(B)\big{)}\mu_{g}^{2}z_{h}^{2}\Big{]}\Big{\\}}(z-z_{h})$
$\displaystyle+\mathcal{O}(z-z_{h})^{2},$

with integration constants $m_{q}$, $\sigma$ and $\chi_{zh0}$. According to
the holographic dictionary, the two integral constants, $m_{q}$ and $\sigma$,
correspond to the quark mass and the chiral condensate
$\sigma\equiv\left\langle\overline{q}q\right\rangle$, respectively.
Furthermore, the normalization constant $\zeta$ takes the value
$\sqrt{N_{c}}/2\pi$ with $N_{c}=3$, which is determined by matching to 4D QCD
[82].

In Fig. 3, we show the variation of chiral condensation with $T$ (or $B$) when
$B$ (or $T$) is fixed. In Fig. 3(a), under different fixed $B$ and varying
$T$, the chiral phase transition exhibits crossover behavior. The results
suggest that chiral condensation displays evident MC effects at low
temperatures and significant IMC effects near $T_{\text{pc}}(B)$. From Fig.
3(b), on one hand, we observe that chiral condensation exhibits MC effects at
low temperature. As the temperature increases to 140 MeV, chiral condensation
initially shows MC behavior. Then, when the magnetic field strengths $B$ is at
$B\gtrsim$ 34 $M_{\pi}^{2}$, the behavior of chiral condensation transitions
shift from MC to IMC effect. Finally, when the temperature reaches 162 MeV,
chiral condensation completely manifests IMC effects. These findings are
qualitatively consistent with lattice simulation results [4, 18]. On the other
hand, when the temperature is at $T\gtrsim$ 201 MeV, the variation of
$\Delta\sigma$ is almost decoupled from the magnetic field.

As shown in Fig. 4, we have also derived the results for the normalized
pseudo-critical temperature $T_{\text{pc}}(B)/T_{\text{pc}}(0)$. The pseudo-
critical temperature is defined as $(\partial^{2}\sigma/\partial
T^{2})|_{T=T_{\text{pc}}}=0$. We can observe that the normalized
$T_{\text{pc}}(B)$ results fall entirely within the error band of lattice
simulations [4]. Additionally, we extracted results of $T_{\text{pc}}$ from
the lattice-improved NJL model [22] for comparing.

##  IV Correlation functions and the masses of neutral pion at finite
temperature and magnetic field

In the previous section, we construct the lattice-improved soft-wall AdS/QCD
model. We use this model to study the chiral condensate and the chiral phase
transition. In this section, we focus on the calculation of the screening
masses $m_{\text{scr}}$, the pole masses $m_{\text{pole}}$ and the thermal
widths $\Gamma$ of the neutral pion $\pi^{0}$ under different magnetic field
strengths $B$ and temperatures $T$.

The screening mass $m_{\text{scr}}$ describe the exponential decay of the
spatial correlator $G(\vec{x})\sim e^{-m_{\text{scr}}|\vec{x}|}/|\vec{x}|$. Or
it is the pole in the momentum space while the correlation function has the
following form near the pole

$G(0,\textit{{k}})\sim\frac{1}{\textit{{k}}^{2}+m_{\text{scr}}^{2}}\,.$ (18)

Besides the spatial components, the information of the temporal component is
also important for getting a full understanding on the mesonic correlation. It
is described by the pole mass and the thermal width. Both of them appear in
the pole in complex frequency plane, near which the temporal correlation
function takes the form

$G(k_{0},\textbf{0})\sim\frac{1}{k_{0}-(m_{\text{pole}}-i\Gamma/2)}\,.$ (19)

Here we take the four vector $k$ as $k=(k_{0},\textit{{k}})$, and
$k_{0},\textit{{k}}$ represent the frequency and the spatial momentum
respectively.

Thus, to get those quantities, one has to calculate the correlators. Through
the holographic approach, a linkage is forged between the 4D operator
$\hat{O}(x)$ and the 5D field $\phi_{0}(x,z)$ by equating their partition
functions. This methodology stands as a robust means to tackle strong coupling
correlation functions

$\left<e^{i\int
d^{4}x\phi_{0}(x)\hat{O}(x)}\right>=e^{iS_{5D[\phi]}}|_{\phi(x,z=0)=\phi_{0}(x)}.$
(20)

with the field $\phi$. The field $\phi$ corresponds to the classical solution
of the 5D action $S_{5D}$. The boundary value $\phi(x,z=0)$ corresponds to the
4D external source $\phi_{0}(x)$ [38, 39, 40] within 5D space. By taking the
second derivative of the action $S_{5D}$ with respect to the source
$\phi_{0}$, the correlator $\left<\hat{O}(x)\hat{O}(0)\right>$ can be
calculated [83]. In this way, it is possible to get the poles of the
correlation functions and the masses. However, as mentioned in ref. [71], the
poles can be easily obtained by solving the 5D equations of motion under
certain boundary conditions. In this way, one can obtain the screening mass by
solving the equations of motion with a spatial momentum, while replacing the
spatial momentum with frequency for pole mass and thermal width . For details,
please refer to ref. [71].

Actually, those screening and pole masses are not independent and they are
connected by the dispersion relation. Generally, within a thermal medium in an
external magnetic field, the dispersion relation (DR) is represented by

$\displaystyle k_{0}^{2}$
$\displaystyle=\,u_{\perp}^{2}\textbf{{k}}_{\perp}^{2}+u_{\parallel}^{2}\textbf{{k}}_{\parallel}^{2}+m_{\text{pole}}^{2},$
(21)

where we denote that $u_{\perp}=u_{1}=u_{2}$ and $u_{\parallel}=u_{3}$ the
velocities of transverse and longitudinal directions respectively. The
transverse directions are in $x_{1},x_{2}$ directions, and the longitudinal
direction is in $x_{3}$ direction, along which the magnetic field is.
$\textbf{{k}}_{\perp}$ and $\textbf{{k}}_{\parallel}$ are the transverse and
longitudinal momenta respectively. Furthermore, $u_{i}$ represents the pion
sound velocity in the $x_{i}$ direction, respectively. Therefore,
$m_{\text{scr},\perp}$ and $m_{\text{scr},\parallel}$ can be defined in terms
of $m_{\text{pole}}$ and the velocity of sound in that direction, showing as
$m_{\text{scr},\perp}=\frac{m_{\text{pole}}}{u_{\perp}}$ and
$m_{\text{scr},\parallel}=\frac{m_{\text{pole}}}{u_{\parallel}}$. As discussed
in ref. [21] and ref. [33], effective thermal masses will lead to
$u_{\perp}<1$ and $u_{\parallel}<1$. Note also that transversal thermal motion
is not as strong as longitudinal motion due to the dimensional reduction of
magnetic field, giving $u_{\perp}<u_{\parallel}$. Finally, we have
$m_{\text{scr},\perp}>m_{\text{scr},\parallel}>m_{\text{pole}}$.

After the above preparation,in the following subsections we will derive the
EOMs and their asymptotic expansions in the pseudoscalar channel. Then we will
extract the masses and present the numerical results of $m_{\text{scr},\perp}$
, $m_{\text{scr},\parallel}$ and $m_{\text{pole}}$. $u_{\perp}/u_{\parallel}$
will also be examined as it serves to measure the anisotropy induced by the
magnetic field.

###  IV.1 EOMs and its asymptotic expansion of pseudo-scalar channel

As the scalar meson field $S(\chi,z)$ and $\pi(\chi,z)$ is decoupled, the
scalar meson field $S(\chi,z)$ can be to 0. The expression of $X$ in eq. (15)
is reduced to

$X=\frac{1}{2}\chi I_{2}e^{2i\pi^{a}t^{a}}.$ (22)

Moreover, the interaction between the pion field and the longitudinal
component $\varphi^{i}$ of the axial-vector field $a^{i}$ occurs exclusively
in the pseudo-scalar channel. To simplify our analysis, we will adopt the
following decomposition of the gauge field for convenience.

$a_{\mu}^{i}=a_{\mu}^{T,i}+\partial_{\mu}\varphi^{i},$ (23a)
$\partial^{\mu}a_{\mu}^{T,i}=0.$ (23b)

When obtaining the EOMs within our model, we treat the $\pi^{0}$ meson as a
perturbation and therefore neglect terms beyond quadratic order. Additionally,
we adopt the $A_{z}=0$. By combining equations eq. (10), eq. (22) and eq. (23)
and taking the Fourier transformation,

$\pi^{a}(x,z)=\frac{1}{(2\pi)^{4}}\int dk^{4}\,e^{-ikx}\pi^{a}(k,z),$ (24a)
$\varphi^{a}(x,z)=\frac{1}{(2\pi)^{4}}\int dk^{4}\,e^{-ikx}\varphi^{a}(k,z).$
(24b)

we derive the EOMs governing the behavior of the neutral $\pi^{0}$ meson and
$\varphi$

$\displaystyle\partial_{z}(\sqrt{g}e^{-\Phi}g^{zz}\chi^{2}\partial_{z}\pi)-k_{\mu}^{2}\sqrt{g}e^{-\Phi}g^{\mu\mu}\chi^{2}(\pi-\varphi)=0,$
(25a)
$\displaystyle\partial_{z}(\sqrt{g}e^{-\Phi}g^{zz}g^{\mu\mu}\partial_{z}\varphi)-g_{5}^{2}\sqrt{g}e^{-\Phi}g^{\mu\mu}\chi^{2}(\pi-\varphi)=0,$

where $k_{\mu}$ represents the momenta of the neutral pion meson in temporal
and different spatial directions with $\mu$= $t,x_{1},x_{2},x_{3}$.

From eqs. (25), since the $x_{1}$ and $x_{2}$ directions are perpendicular to
the magnetic field, the $x_{1}$ and $x_{2}$ (transverse) directions possess
isotropic properties. Clearly, for the spatial $x_{1}$ and $x_{2}$ directions,
the system of differential equations are

$\displaystyle\varphi^{\prime\prime}+$
$\displaystyle\left(A^{\prime}+\frac{f^{\prime}}{f}+\frac{q^{\prime}}{2q}-\Phi^{\prime}\right)\varphi^{\prime}$
(26a) $\displaystyle+\frac{e^{2A}g_{5}^{2}\chi^{2}}{f}(\pi-\varphi)=0,$
$\displaystyle\pi^{\prime\prime}+$
$\displaystyle\left(3A^{\prime}+\frac{f^{\prime}}{f}+\frac{q^{\prime}}{2q}+\frac{h^{\prime}}{h}-\Phi^{\prime}+\frac{2\chi^{\prime}}{\chi}\right)\pi^{\prime}$
(26b) $\displaystyle-\frac{k_{1}^{2}}{fh}(\pi-\varphi)=0.$

Similarly, in the $x_{3}$ spatial (longitudinal) direction, the system of
differential equations for $\pi$ and $\varphi$ are

$\displaystyle\varphi^{\prime\prime}+$
$\displaystyle\left(A^{\prime}+\frac{f^{\prime}}{f}-\frac{q^{\prime}}{2q}+\frac{h^{\prime}}{h}-\Phi^{\prime}\right)\varphi^{\prime}$
(27a) $\displaystyle+\frac{e^{2A}g_{5}^{2}\chi^{2}}{f}(\pi-\varphi)=0,$
$\displaystyle\pi^{\prime\prime}+$
$\displaystyle\left(3A^{\prime}+\frac{f^{\prime}}{f}+\frac{q^{\prime}}{2q}+\frac{h^{\prime}}{h}-\Phi^{\prime}+\frac{2\chi^{\prime}}{\chi}\right)\pi^{\prime}$
(27b) $\displaystyle-\frac{k_{3}^{2}}{fq}(\pi-\varphi)=0.$

In the temporal direction, the system of differential equations for $\pi$ and
$\varphi$ are

$\displaystyle\varphi^{\prime\prime}+$
$\displaystyle\left(A^{\prime}+\frac{h^{\prime}}{h}+\frac{q^{\prime}}{2q}-\Phi^{\prime}\right)\varphi^{\prime}$
(28a) $\displaystyle+\frac{e^{2A}g_{5}^{2}\chi^{2}}{f}(\pi-\varphi)=0,$
$\displaystyle\pi^{\prime\prime}+$
$\displaystyle\left(3A^{\prime}+\frac{f^{\prime}}{f}+\frac{q^{\prime}}{2q}+\frac{h^{\prime}}{h}-\Phi^{\prime}+\frac{2\chi^{\prime}}{\chi}\right)\pi^{\prime}$
(28b) $\displaystyle+\frac{k_{0}^{2}}{f^{2}}(\pi-\varphi)=0.$

To solve the eqs. (26), (27), and(28), we can also adopt the “Shooting
method”. This numerical approach enables us to tackle these differential
equations effectively. In our numerical solution process, we initiate by
establishing the asymptotic expansions of the equations at their respective
boundaries $z=0$ and $z=z_{h}$. 444Due to the metric functions and $\chi$
exist in eqs (26)-(28), when seeking their asymptotic expansions at the UV and
IR boundaries, we must also take into account the asymptotic expansions of
eqs. (5) and (16) at the UV and IR regimes, respectively.. At horizon, one
should consider the incoming wave condition [84]. For the spatial direction
along $x_{1}$ and $x_{2}$, the asymptotic expansion of eqs. (26) at both
boundaries can be described as

$\displaystyle\pi(z\rightarrow 0)=$
$\displaystyle\pi_{b0}+\varphi_{0}+\pi_{2}z^{2}+\frac{\pi_{b0}k_{1}^{2}}{2h_{0}}z^{2}\ln(z)$
(29a) $\displaystyle+\mathcal{O}(z^{3}),$ $\displaystyle\varphi(z\rightarrow
0)=$
$\displaystyle\varphi_{0}+\varphi_{2}z^{2}-\frac{1}{2}g_{5}^{2}m_{q}^{2}\zeta^{2}\pi_{b0}z^{2}\ln(z)$
(29b) $\displaystyle+\mathcal{O}(z^{3}),$ $\displaystyle\pi(z\rightarrow
z_{h})=$
$\displaystyle\pi_{h0}+\frac{\pi_{h0}-\varphi_{h0}}{f_{h1}h_{h0}}k_{1}^{2}(z-z_{h})$
(29c) $\displaystyle+\mathcal{O}(z-z_{h})^{2},$
$\displaystyle\varphi(z\rightarrow z_{h})=$
$\displaystyle\varphi_{h0}+\frac{\varphi_{h0}-\pi_{h0}}{f_{h1}z_{h}^{2}}g_{5}^{2}\chi_{zh0}^{2}(z-z_{h})$
(29d) $\displaystyle+\mathcal{O}(z-z_{h})^{2}.$

where $\pi_{b0}$, $\varphi_{0}$, $\pi_{2}$, and $\varphi_{2}$ are the
integration constants for the UV boundary, and $\pi_{h0}$ and $\varphi_{h0}$
are the integration constants for the horizon.

For the EOMs in the longitudinal direction ($x_{3}$ direction), we can obtain
their asymptotic expansions of eqs. (27) at both boundaries as

$\displaystyle\pi(z\rightarrow 0)=$
$\displaystyle\pi_{b0}+\varphi_{0}+\pi_{2}z^{2}+\frac{\pi_{b0}k_{3}^{2}}{2q_{0}}z^{2}\ln(z)$
(30a) $\displaystyle+\mathcal{O}(z^{3}),$ $\displaystyle\varphi(z\rightarrow
0)=$
$\displaystyle\varphi_{0}+\varphi_{2}z^{2}-\frac{1}{2}g_{5}^{2}m_{q}^{2}\zeta^{2}\pi_{b0}z^{2}\ln(z)$
(30b) $\displaystyle+\mathcal{O}(z^{3}),$ $\displaystyle\pi(z\rightarrow
z_{h})=$
$\displaystyle\pi_{h0}+\frac{\pi_{h0}-\varphi_{h0}}{f_{h1}q_{h0}}k_{3}^{2}(z-z_{h})+$
(30c) $\displaystyle+\mathcal{O}(z-z_{h})^{2},$
$\displaystyle\varphi(z\rightarrow z_{h})=$
$\displaystyle\varphi_{h0}+\frac{\varphi_{h0}-\pi_{h0}}{f_{h1}z_{h}^{2}}g_{5}^{2}\chi_{zh0}^{2}(z-z_{h})$
(30d) $\displaystyle+\mathcal{O}(z-z_{h})^{2}.$

where $\pi_{b0}$, $\varphi_{0}$, $\pi_{2}$, and $\varphi_{2}$ are the
integration constants for the UV boundary, and $\pi_{h0}$ and $\varphi_{h0}$
are the integration constants for the horizon.

Similarly, in the temporal direction, we can obtain the asymptotic expansion
of eqs. (28) at both boundaries, given by

$\displaystyle\pi(z\rightarrow 0)=$
$\displaystyle\pi_{b0}+\varphi_{0}+\pi_{2}z^{2}-\frac{\pi_{b0}k_{0}^{2}}{2}z^{2}\ln(z)$
(31a) $\displaystyle+\mathcal{O}(z^{3}),$ $\displaystyle\varphi(z\rightarrow
0)=$
$\displaystyle\varphi_{0}+\varphi_{2}z^{2}-\frac{1}{2}g_{5}^{2}m_{q}^{2}\zeta^{2}\pi_{b0}z^{2}\ln(z)$
(31b) $\displaystyle+\mathcal{O}(z^{5}),$ $\displaystyle\pi(z\rightarrow
z_{h})=$
$\displaystyle\varphi_{h0}+(z-z_{h})^{\frac{ik_{0}}{f_{h1}}}\Big{[}\pi_{h0}$
(31c) $\displaystyle+\mathcal{O}(z-z_{h})^{1}\Big{]},$
$\displaystyle\varphi(z\rightarrow z_{h})=$
$\displaystyle\varphi_{h0}+(z-z_{h})^{\frac{ik_{0}}{f_{h1}}}$ (31d)
$\displaystyle\times\Big{[}\frac{ig_{5}^{2}\pi_{h0}f_{{h1}}\chi_{{zh0}}^{2}}{{z_{h}}^{2}k_{t}\left(f_{{h1}}+ik_{t}\right)}(z-z_{h})$
$\displaystyle+\mathcal{O}(z-z_{h})^{2}\Big{]}.$

The $\pi_{b0}$, $\varphi_{0}$, $\pi_{2}$, and $\varphi_{2}$ are the
integration constants for the UV boundary, and $\varphi_{h0}$ and $\pi_{h0}$
are the integration constants for horizon.

As explained in ref. [72], due to the presence of the term $(\pi-\varphi)$ in
eqs. (25), if one makes a transformation of assuming $\pi_{c}=\pi+c$ and
$\varphi_{c}=\varphi+c$ with $c$ a nonzero constant, one can find that the
solution is still available. Consequently, we can take integration constant
$\varphi_{0}$ as an extraneous free parameter. For convenience, we let
$\varphi_{0}=0$. Additionally, due to the linear nature of eqs. (25), for the
integral constants $\pi_{h0}$ at the IR boundary, we can set it to unity.

With the asymptotic expansion solutions, we can use the "shooting method" to
numerically solve the differential equation system (25) for $\pi$ and
$\varphi$. Since the second order differential equations, one should matching
the function values and its first order derivatives. Therefore, there are four
constraints. However, as per the analysis presented above, the integration
constants that we need to solve are five: $\pi_{b0}$ $\pi_{2}$, $\varphi_{2}$,
$\varphi_{h0}$ and the $k_{i}$ for each direction. However, a physically
analyzing of the solutions of differential equations and two-point Retarded
correlator allow us to determine another integration constant, $\pi_{b0}$, for
each direction, which will be explored and discussed in the subsequent
subsections.

\begin{overpic}[scale={0.61}]{twomodelsx3screenwithTT.eps}
\put(80.0,58.0){\bf{(a)}}
\end{overpic}\begin{overpic}[scale={0.61}]{twomodelsx1x2screenwithTT.eps}
\put(80.0,58.0){\bf{(b)}} \end{overpic}

Figure 5: The plots illustrate the $m_{\text{scr},\parallel}$ and
$m_{\text{scr},\perp}$ as a function of $T$ under different fixed $B$
respectively. The fixed magnetic field $B$ is about 0, 5.56, 21.04, 36.44 and
51.32 $M_{\pi}^{2}$, which corresponds to 0, 0.11, 0.41, 0.71, and 1 GeV2,
respectively in our model. The inset in (a) and (b) is a zoom for the range
$T\in[0.05,0.14]$ GeV. For the $B=$ 0, 5.56, 21.04, 36.44, 51.32
$M_{\pi}^{2}$, their corresponding $T_{\text{pc}}$ of chiral phase transition
are 161.0, 160.2, 156.0, 146.5, 138.8 MeV, respectively.

\begin{overpic}[scale={0.61}]{twomodelsx3NormalizedscreenwithBB.eps}
\put(59.0,63.0){\bf{(a)}}
\end{overpic}\begin{overpic}[scale={0.61}]{twomodelsx1x2NormalizedscreenwithBB.eps}
\put(60.0,63.0){\bf{(b)}} \end{overpic}

Figure 6: The plots depict normalized screening mass,
$m_{\text{scr}}(B,T)/m_{\text{scr}}(0,T)$, as a function of $B/M_{\pi}^{2}$
under different fixed $T$, where the temperatures $T$ are chosen as 70, 158,
212 and 398 MeV, respectively. The figures (a) and (b) are described for
normalized screening masses in longitudinal and transverse screening masses,
respectively.

###  IV.2 Numerical results of screening masses of neutral pion

In this subsection we will proceed to numerically solve the EOMs for $\pi$ and
$\varphi$, eqs. (26) and (27). Then we will show the transverse and
longitudinal screening masses of the neutral pion.

Following the holographic dictionary, $\pi_{b0}$, as shown in the asymptotic
expansions in eqs. (29) and (30), is interpreted as the external sources
$J_{\pi}$. Actually, considering the spatial correlator in eq. (18), the
desired values of $m_{\text{scr},\perp}$ or $m_{\text{scr},\parallel}$ for the
neutral pion meson is situated at the pole of the Green’s function,
representing its singularities. Consequently, at the UV boundary, the integral
constants $\pi_{b0}$ in the asymptotic expansion can be set to zero [72, 73].
Thus, the $m_{\text{scr},\perp}$ and $m_{\text{scr},\parallel}$ are the
imaginary part of $k_{1}$ and $k_{3}$.

After numerically solving the EOMs of $\pi$ and $\varphi$ in different
directions by combining eqs.(5) and (16), we show the solution of the
screening mass of the neutral pion in Figs. 5 and 6.

The behavior of the screening masses with respect to $T$ at different fixed
$B$ is shown in Fig. 5. At low temperature, $m_{\text{scr},\parallel}$ and
$m_{\text{scr},\perp}$ are both slightly affected by $T$. As shown in the
inset, $m_{\text{scr},\parallel}$ and $m_{\text{scr},\perp}$ show decreasing
and increasing behaviors with $B$ at low temperature region. In the critical
region, close to $T_{\text{pc}}(B)$, both screening masses suddenly increase.
As the temperature increases, the two screen masses are characterized by the
linear thermal mass and tend to be the same for different given $B$ at
extremely high temperatures.

\begin{overpic}[scale={0.615}]{twomodelssoundspeedwithTTT.eps}
\put(50.0,20.0){\bf{(a)}}
\end{overpic}\begin{overpic}[scale={0.604}]{twomodelssoundspeedwithBBB.eps}
\put(60.0,20.0){\bf{(b)}} \end{overpic}

Figure 7: (a), the plot illustrates the ratio of sound velocity
$u_{\perp}/u_{\parallel}$ as a function of $T$ under different fixed $B$. The
range of $B$ is approximately from 0 to 51.32 $M_{\pi}^{2}$, which corresponds
to 0 to 1 GeV2 in our model. (b), it depicts the ratio of sound velocity
$u_{\perp}/u_{\parallel}$ as a function of $B/M_{\pi}^{2}$ under different
fixed $T$, where the temperatures $T$ were chosen as 50, 150, 201, 251 and 398
MeV, respectively. For the $B=$ 0, 10.78, 21.04, 31.31, 51.32 $M_{\pi}^{2}$,
their corresponding $T_{\text{pc}}$ of chiral phase transition are 161.0,
158.7, 156.1, 150.2, 138.8 MeV respectively. Furthermore, we have extracted
results from the lattice-improved NJL model [22], represented by dots in the
graph. The displayed results from lattice-improved NJL model have been scaled
according to $M_{\pi}$ of its model $M_{\pi}=138$ MeV.

In Fig. 6, we plot the normalized screening mass of neutral pion mesons with
$B$ at different fixed $T$. As shown, the normalized screening masses
($m_{\text{scr},\parallel}$ and $m_{\text{scr},\perp}$), either at low
($T=70$) MeV or at higher temperatures ($T$= 212 and 398 MeV), depend slightly
on $B$, which corresponds to the main properties of a neutral particle. More
precisely, at $T=70$ MeV the $m_{\text{scr},\parallel}$ decrease with
increasing $B$, but the $m_{\text{scr},\perp}$ increase with increasing $B$.
At $T=212$ MeV, $m_{\text{scr},\parallel}$ first decrease and then start to
increase as $B$ grows. And they always increase with $B$ when the temperature
is $T<398$ MeV. At a higher given temperature, $T=398$ MeV,
$m_{\text{scr},\parallel}$ show the behavior of decreasing with $B$ again, and
the decrease is very small. This is in agreement with the lattice result [23].
However, at a certain temperature where $T=158$ MeV near $T_{\text{pc}}(B)$,
the normalized screening masses increase significantly with $B$, which can be
understood as the magnetic dependence being enhanced by the critical
fluctuation.

From the energy dispersion relation in eq. (21), It is known that the ratio
$u_{\perp}/u_{\parallel}=m_{\text{scr},\parallel}/m_{\text{scr},\perp}$. In
Fig. 7(a), we present the $T$ dependence of the ratio
$u_{\perp}/u_{\parallel}$ at different $eB$. For finite $eB$,
$u_{\perp}/u_{\parallel}$ is almost independent of temperature at low $T$. As
the temperature increases to near $T_{\text{pc}}(B)$,
$u_{\perp}/u_{\parallel}$ show a non-monotonic bump, and the bump is more
obvious as $eB$ increases. When the temperature is $T>T_{\text{pc}}(B)$, all
curves increase with increasing $T$. When $T\gg T_{\text{pc}}(B)$, all curves
approach 1. Mathematically, increasing $T$ will cause both $q(z)$ and $h(z)$
to approach 1, as shown in the subsection II.2, and then it induces that eq.
(26) is almost the same as eq. (27). This indicates that in all three spatial
dimensions, the thermal fluctuations dominate over the anisotropy induced by
the magnetic field. Furthermore, in Fig. 7(b), we study the $B$ dependence of
the ratio $u_{\perp}/u_{\parallel}$. The ratio $u_{\perp}/u_{\parallel}$
decreases with increasing $B$, which is the consequence of the decoupling of
the transverse dimension in the strong limit of the magnetic field. Again, in
the region of the critical temperature, the curve decays faster with $B$, as
expected, as shown by the orange line.

###  IV.3 Numerical results of pole mass and thermal widths of neutral pion

\begin{overpic}[scale={0.615}]{twomodelspolewithTTT.eps}
\put(85.0,55.0){\bf{(a)}}
\end{overpic}\begin{overpic}[scale={0.61}]{twomodelsthermalwidthwithTTT.eps}
\put(82.0,55.0){\bf{(b)}} \end{overpic}

Figure 8: The plots illustrate the $m_{\text{pole}}$ and $\Gamma/2$ as a
function of $T$ under different fixed $B$ respectively. The selection of $B$
is about 0, 15.91, 31.31, 41.57 and 51.32 $M_{\pi}^{2}$, which corresponds to
0, 0.51, 0.81 and 1 GeV2 respectively in our model. The inset in (b) is a zoom
for the range $T\in[0.05,0.14]$ GeV.

\begin{overpic}[scale={0.625}]{twomodelsNormalizedpolewithBBB.eps}
\put(20.0,55.0){\bf{(a)}}
\end{overpic}\begin{overpic}[scale={0.605}]{twomodelsNormalizedthermalwidthwithBBB.eps}
\put(15.0,60.0){\bf{(b)}} \end{overpic}

Figure 9: The plots depict normalized $m_{\text{pole}}$ and normalized
$\Gamma/2$ as a function of $B/M_{\pi}^{2}$ under different fixed $T$, where
the temperatures $T$ are chosen as 50, 80, 105, 125 MeV, and 120, 140, 169,
185, 205, 299 MeV respectively.

In this subsection, we present the behavior of the pole mass $m_{\text{pole}}$
and thermal width $\Gamma/2$ of the neutral pion meson at given $T$ and $B$.

For asymptotic expansion in eq. (31), according to the holographic dictionary,
the integration constant $\pi_{b0}$ is understood to correspond to the
external sources denoted as $J_{\pi}$. From the temporal correlator eq. (19),
we know that the complex frequencies $k_{0}$ precisely lie in the location of
the poles in the Green’s function. Evidently, this corresponds exactly to the
position of singularities. Thus, for the integration constant $\pi_{b0}$, its
value must be 0 [72, 73]. Hence, taking into account the analysis from the
Section (IV.1), we have already determined that $\varphi_{0}=0$, $\pi_{b0}=0$,
and $\pi_{h0}=1$. Similarly, we employ the "shooting method" to numerically
solve the differential eqs. (28). Thus, the $m_{\text{pole}}$ and thermal
width are the real and imaginary part of $k_{0}$, respectively.

In Fig. 8, we study the temperature dependence of $m_{\text{pole}}$ and
thermal widths $\Gamma/2$ of the neutral pion meson under different fixed $B$.
From Fig. 8(a), $m_{\text{pole}}$ decrease with temperature and go to zero
near $T_{\text{pc}}$. On the contrary, in Fig. 8(b), for given $eB$, the
thermal widths ($\Gamma/2$) are small at low temperatures. And then, in the
region above the critical temperature, they increase rapidly with the increase
of $T$. Since the thermal width is related to the dissociation level, such
curves naively indicate that $\pi^{0}$ is a well-behaved bound state at low
$T$ and becomes loose at high $T$. Indeed, without $B$, the corresponding
behavior of the real and imaginary parts of the neutral pion propagator is
consistent with results from LQCD simulations [85], $\chi$PT [86] as well as
holography results [71, 72].

In Fig. 9, we have extracted the normalized $m_{\text{pole}}$ and $\Gamma/2$
as functions of $B/M_{\pi}^{2}$ at different fixed $T$. Clearly, in Fig. (9a),
the normalized $m_{\text{pole}}$ decreases with increasing $B$ at finite
temperatures. In vacuum, such a similar behavior has already been found in
lattice simulations [87] and studied by model calculations [88, 89, 90].
Meanwhile, in Fig. 9(b), we observe that when $T\lesssim T_{\text{pc}}(B)$,
$\Gamma/2$ exhibits strong dependence on the magnetic field and increases with
$B$. When $T\gtrsim T_{\text{pc}}(B)$, $\Gamma/2$ is slightly affected by the
magnetic field and the thermal width is almost controlled by the magnitude of
the temperatures. We conclude that the magnetic field is more involved in the
thermal effects below than above the critical temperature, i.e. temperature
and magnetic field are mostly entangled at moderate $T$. Moreover, at high
temperature, $T\gg T_{\text{pc}}(B)$, $\Gamma/2$ decrease with increasing $B$.
It is interesting to note that, at high temperature, the behavior of
$\Gamma/2$ is similar with $m_{\text{scr},\parallel}$.

##  V Conclusion and discussion

In this work, we mainly investigate the thermal properties of neutral pions
within a hot and magnetized medium, including their screening masses, pole
masses, and thermal widths, by using a lattice-improved soft-wall AdS/QCD
model.

We introduce the magnetic field within the Einstein-Maxwell holographic model
following refs. [79, 80, 74]. By numerically solving the equations of motion,
we obtain the full solutions with constant magnetic field. Compared with the
expanding solutions in refs. [79, 80, 74], the full solutions turn out to be
valid in a wider range of temperature and magnetic field. Based on those
numerical solutions, we consider the chiral dynamics in the soft-wall AdS/QCD
model. By considering an effective coupling between the dilaton field and the
flavor sector and introducing a magnetic-field-dependent coupling function
$g(B)$, which is fitted by results for $T_{\text{pc}}(B)$ from lattice
simulation [4], we successfully capture the MC effect of chiral condensation
at low temperatures and the IMC effect near the chiral transition temperature
$T_{\text{pc}}(B)$. It is worth of mentioning that a better fitting of the
chiral transition temperature $T_{\text{pc}}(B)$ has been obtained, and all
the other qualitative behaviors are in good agreement with the lattice
simulations results [18], which shows a good description of chiral phase
transition from such an effective holographic QCD model.

Then we study the thermal properties of the neutral pion, which is still the
Goldstone boson of the chiral symmetry breaking even within a magnetic field.
The temperature and magnetic field dependence of the screening masses
$m_{\text{scr},\parallel}$ and $m_{\text{scr},\perp}$, characterizing the
mesonic correlations in perpendicular and vertical directions respectively,
are extracted. If one fixes the magnetic field, both
$m_{\text{scr},\parallel}$ and $m_{\text{scr},\perp}$ are slightly affected by
$T$ at low temperature. In the temperature region close to transition
temperatures $T_{\text{pc}}(B)$, the screening masses significantly increase.
Such enhanced behavior shows the connection between the screening masses and
the chiral phase transition. After the phase transition, the mesons are not
tightly bounded and the hot and magnetized medium start imposing strongly
impact on their properties. Then, when the temperature increases further, the
two screening masses are shown to increase linearly with $T$ and tend to be
the same for different given $B$ at extremely high temperatures. Those results
are consistent with the results from effective models like NJL, confirming the
effectiveness of the soft-wall holographic description on chiral dynamics.

Furthermore, we fixed temperature and study the $B$ dependence of screening
masses. Roughly speaking, the screening masses do not change much below
$T_{\text{pc}}$. This is reasonable, since the neutral pions from our
holographic model are still tightly bound states and as neutral particles
their properties would not be changed significantly by the magnetic field. But
if we look carefully into the results, we can find that at low temperature
$m_{\text{scr},\parallel}$ decreases as $B$ grows, while
$m_{\text{scr},\perp}$ increases, showing an anisotropic effect of the
magnetic field at low temperature. Such a qualitative behavior can also be
found from NJL model calculation [22]. Then, when the temperature is taken to
be near $T_{\text{pc}}$, both $m_{\text{scr},\parallel}$ and
$m_{\text{scr},\perp}$ increase with $B$. At high temperature region,
$m_{\text{scr},\perp}$ still increase as $B$ grow but
$m_{\text{scr},\parallel}$ show behavior of decreasing with increasing $B$
again. It is worth mentioning that a similar behavior can be seen in the
lattice study [23]. Moreover, we also study the ratio
$u_{\perp}/u_{\parallel}$. When $T\gg T_{\text{pc}}(B)$, all the curves
approach to 1. This indicates that the difference between $u_{\perp}$ and
$u_{\parallel}$ is vanishing. For the $B$ dependence of the ratio
$u_{\perp}/u_{\parallel}$. The ratio $u_{\perp}/u_{\parallel}$ decrease with
the increase of $B$. This indicates the destruction of spatial symmetry by the
magnetic field. We would like to emphasize that by simply fitting data of
$T_{\text{pc}}(B)$ from lattice simulation, all the qualitative behaviors for
the pion properties can be consistent with lattice simulation simultaneously.
In some sense, it confirms the effectiveness of holographic QCD method.

The screening mass depicts the spatial effect, and we have seen that it can be
well described by the holographic method. As for the temporal effect on the
pionic sector, there are still not many data from lattice simulations. But the
extension in hologarphic method is quite directly. We can obtain the poles in
complex frequency plane and get the pole mass $m_{\text{pole}}$ and the
thermal widths $\Gamma/2$ from their real and imaginary part. If we fix $B$,
we find that $m_{\text{pole}}$ decreases with temperature and goes to zero
near $T_{\text{pc}}$, while the thermal widths $\Gamma/2$ increase
monotonically with increasing $T$. It is interesting to see the pole mass
decrease below $T_{\text{pc}}$, which are consistent with the analysis from
scaling law in the finite temperature chiral perturbation theory in [91].
Also, it is consistent with the lattice study in [85], in which the mass of
pions are taken slightly higher than their physical values. Finally, at
different fixed temperatures, $m_{\text{pole}}$ monotonically decrease with
$B$ upto $1\rm{GeV}^{2}$. This behavior is qualitatively consistent with the
lattice-improved NJL study in [22] while it differs from the NJL calculation
in ref. [35], in which $m_{\text{pole}}$ decreases when $B<0.8\rm{GeV}^{2}$
and increases when $B>0.8\rm{GeV}^{2}$. Also, being similar with the behavior
of $m_{\text{scr},\parallel}$, we can see that the thermal widths of the
$\pi^{0}$ slightly decrease as $B$ grows above the critical temperature. In
this work, we found that the thermal properties of magnetized pions are mainly
determined by the temperature itself at high $T$. These are preliminary
results of the dependence on the magnetic field, since such a weak dependence
would be easily washed out by other effects, for example the backaction from
the metric, and a full calculation will be required in further work.

In the present study, we see that by fitting our holographic model using the
condensation data from lattice, the model can capture most of the qualitative
behaviors, especially for the thermal properties of pions. Due to the
limitation from our current numerical techniques, we can not compare the zero
temperature behavior of the pole masses under magnetic field and check the
main difference between our results and the NJL studies ref. [35]. We will
leave it to the future. Furthermore, the recent LQCD simulation results
indicate that within the $B$ range of approximately 4 to 9 $\text{GeV}^{2}$,
chiral condensation switches from a crossover to first order at a critical
endpoint located in this range [92]. It is also interesting to extend our
study to stronger magnetic field case.

##  VI Acknowledgments

NW thanks Wei-Jiang Liang for valuable discussions. This work is supported by
the National Natural Science Foundation of China (NSFC) under Grant Nos.
12275108, 12305142, Science and Technology Planning Project of Guangzhou,
China under Grant No. 2024A04J3243 and the Fundamental Research Funds for the
Central Universities under Grant No. 21622324. JC is supported by the National
Natural Science Foundation of China (NSFC) under Grant No. 12365020.

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