#  $J/\Psi$ suppression in a rotating magnetized holographic QGP matter

Yan-Qing<EMAIL_ADDRESS>and Defu Hou1222corresponding
author<EMAIL_ADDRESS>
###### Abstract

We study the dissociation effect of $J/\Psi$ in magnetized, rotating QGP
matter at finite temperature and chemical potential using gauge/gravity
duality. By incorporating angular velocity into the holographic magnetic
catalysis model, we analyze the influence of temperature, chemical potential,
magnetic field, and angular velocity on the properties of $J/\Psi$ meson. The
results reveal that temperature, chemical potential, and rotation enhance the
dissociation effect and increase the effective mass in the QGP phase. However,
the magnetic field suppresses dissociation, and its effect on the effective
mass is non-trivial. Additionally, we explore the interplay between magnetic
field and rotation, identifying a critical angular velocity that determines
the dominant effect. As a parallel study, we also examine the rotation effect
in the holographic inverse magnetic catalysis model, although the magnetic
field exhibits distinctly different behaviors in these two models, the impact
of rotation on the dissociation effect of $J/\Psi$ is similar. Finally, we
investigate the influence of electric field and demonstrate that it also
speeds up the $J/\Psi$ dissociation .

1 Institute of Particle Physics and Key Laboratory of Quark and Lepton Physics
(MOS), Central China Normal University, Wuhan 430079, China

###### Contents

  1. 1 Introduction
  2. 2 Holographic QCD model
  3. 3 The spectral functions
    1. 3.1 Turning on the angular momentum
    2. 3.2 Adding a constant electric field to the background
  4. 4  Numerical Results of spectral functions in a magnetized rotating plasma
    1. 4.1 Turning on a constant magnetic field
    2. 4.2 Turning on angular momentum
      1. 4.2.1  The case of magnetized-rotating QGP with MC
      2. 4.2.2 The case of magnetized-rotating QGP with IMC
    3. 4.3 Turning on electric field
  5. 5 Summary and discussion
  6. A The effect of rotating QGP on $J/\Psi$ dissociation in a holographic inverse magnetic catalysis model

##  1 Introduction

In ultra-relativistic heavy-ion collisions, a new, strongly interacting matter
state known as QGP [1] is created. In order to fully understand the hot and
dense plasma, as one of the best probes, the final yield of dilepton from
heavy quarkonium ($J/\Psi$ and $\Upsilon(1S)$) is used to study the properties
of QGP. The formation time of charm quark is about $\tau_{c}\sim
1/2m_{c}\approx 0.06\,\text{fm/c}$ and about $\tau_{b}\sim 1/2m_{b}\approx
0.02\,\text{fm/c}$ for bottom quark [2]. The original concept of quarkonium
suppression is from Matsui and Satz [3]. The bound state is produced in the
initial stage of heavy-ion collision and then gets dissociated into free
quarks when it passes through the QGP medium, which leads to the decrease of
final output(the dilepton), this phenomenon is called quarkonium suppression.
So far, there are two mechanisms that can be used to explain the reduction of
quarkonium, namely: (1) Hot nuclear matter (HNM) effects—–the reduction is
caused by the existence of QGP medium; The HNM effects include: ① The rotation
of QGP medium, rotating effect; ② The temperature for QGP medium, thermal
effect; ③ The baryon chemical potential, density effect. (2) Cold nuclear
matter (CNM) effects—–the decrease is resulted in by those matters other than
QGP medium. The CNM effects include[4, 5]: ① Parton distribution in the
nuclei, the nPDF effects; ② Inelastic collision between the quarkonium and
nucleons, the nuclear absorption; ③ Interaction of the quarkonium with co-
moving particles, resulting in the melting of bound state, the co-mover
dissociation; ④ The Parton energy loss during multiple scattering process, the
energy loss; ⑤ Electromagnetic field generated by bystanders, the
electromagnetic effect.

Experimentally, the suppression of $J/\Psi$ production has already been
observed in Au+Au collisions at $\sqrt{s_{NN}}=200\,\text{GeV}$ through the
dimuon channel at STAR [6], where the $J/\Psi$ yields are measured in a large
transverse momentum range($0.15\,\text{GeV/c}\leq p_{T}\leq 12\,\text{GeV/c}$)
from central to peripheral collisions. They find the $J/\Psi$ yields are
suppressed by a factor of approximately $3$ for $p_{T}>5\,\text{GeV/c}$ in the
$0-10\%$ most central collisions. Theoretically, heavy quarkonium may survive,
due to the Coulomb attraction between quark and antiquark, as bound states
above the deconfinement temperature. By using the maximum entropy method
(MEM), Ref. [7] studies the correlation functions of $J/\Psi$ at finite
temperature on $32^{3}*(32-96)$ anisotropic lattices, the conclusions indicate
$J/\Psi$ could survive in the plasma up to $T_{d}\sim 1.6T_{c}$ and melt at
$1.6T_{c}\leq T_{d}\leq 1.9T_{c}$. Lattice data [8] find $J/\Psi$ could
survive up to $1.5T_{c}$ and vanish at $3T_{c}$.

Gauge/gravity duality has emerged as a valuable tool for investigating the
properties of heavy quarks. Holographic methods have been extensively employed
to study heavy flavor properties [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37]. In
Ref.[9], we investigate the vector meson spectral function using a dynamical
AdS/QCD model. Our results reveal that the enhancement of heavy quarkonium
dissociation is influenced by the magnetic field, chemical potential, and
temperature. Specifically, we observe non-trivial changes in the peak position
of $J/\Psi$, which we attribute to the interplay between the interaction of
the two heavy quarks and the interaction of the medium with each heavy quark.
Interestingly, by introducing a dilaton in the background action, we
demonstrate that the magnetic field has a more pronounced impact on heavy
meson dissociation when it aligns with the polarization, contrary to the
findings in the EM model discussed in Ref.[32]. In Ref. [33], the binding
energy of heavy quarkonium in the quark-gluon plasma (QGP) and hadronic phase
is examined. The results reveal that the dissociation length of heavy mesons
decreases with increasing temperature or quark chemical potential in the QGP.
However, in the hadronic phase, the dissociation length increases with an
increase in the chemical potential. This phenomenon is attributed to two
distinct dissociation mechanisms: the screening of the interaction between
heavy quarks by light quarks in the deconfinement phase, and the breaking of
the heavy meson into two heavy-light quark bound states in the confinement
phase. Furthermore, Ref.[34] identifies the existence of a second lower
critical temperature for certain magnetic field intensities, below which
stable mesons cease to exist. This is termed Magnetic Meson Melting (MMM),
extending the understanding of meson melting in the presence of varying
magnetic field intensities. Ref.[35] calculates the ratios of dissociation
temperatures for $J/\Psi$ using the U-ansatz potential and finds agreement
with lattice results within a factor of two. Additionally, Ref. [36]
investigates heavy quarkonia spectroscopy at both zero and finite temperature
using a bottom-up AdS/QCD approach, predicting the melting temperature of
$J/\Psi$ to be approximately 415 MeV ($\sim 2.92T_{c}$).

In this work, we investigate the behavior of heavy vector mesons in the
rotating magnetized quark-gluon plasma (QGP) created at RHIC and LHC. As
previously mentioned, although there are numerous non-perturbative methods
available, including the state-of-the-art lattice QCD[38, 39, 40, 41, 42], for
the study of heavy vector meson spectral functions, the emphasis has primarily
been on exploring the thermal and density effects of the quark-gluon plasma
(QGP) and the strong magnetic field effects. However, these properties merely
capture a fraction of the complete QGP environment. Furthermore, the state-of-
the-art lattice data not only support the scenario of inverse magnetic
catalysis(IMC)[43], where the transition temperature decreases as the magnetic
field increases for temperatures slightly above the critical temperature, but
also present compelling evidence for magnetic catalysis(MC) [44] in the
deconfinement phase diagram. In MC, the transition temperature increases as
the magnetic field intensifies for temperatures not significantly higher than
the critical temperature. Therefore, it is of utmost importance to take into
account the MC, which holds an equally prominent position as inverse magnetic
catalysis , along with other factors in order to simulate the influence of the
QGP medium more accurately on the properties of heavy vector mesons. To bridge
the existing gap, we also consider additional factors, such as medium
rotation, electric fields, and magnetic field effects, which pose significant
challenges for the current state-of-the-art lattice techniques. Using
holographic methods, one of the non-perturbative approaches, we explore the
interplay of these factors and their impact on heavy vector meson properties.

The rest structure of this paper is organized as follows. In section 2, we set
up the holographic magnetic catalysis model. In section 3, we display the
specific derivation process for calculating the spectral functions of $J/\Psi$
by introducing different field effects. In section 4, we show and discuss the
meaningful results. Finally, this work is ended with a summary and a
discussion in section 5.

##  2 Holographic QCD model

We consider a 5d EMD gravity system with a Maxwell field and a dilaton field
as a thermal background for the corresponding hot, dense, magnetized QCD. The
action is given as

$\displaystyle\mathcal{S}$ $\displaystyle=$ $\displaystyle\frac{1}{16\pi
G_{5}}\int
d^{5}x\sqrt{-g}\left[\mathcal{R}-\frac{1}{2}\nabla_{\mu}\phi_{0}\nabla^{\mu}\phi_{0}-\frac{f(\phi_{0})}{4}F_{\mu\nu}F^{\mu\nu}-V(\phi_{0})\right].$
(2.1)

where $F_{\mu\nu}$ is the field strength tensor for U(1) gauge field,
$f(\phi_{0})$ is the gauge coupling kinetic function, $\phi_{0}$ is the
dilaton field. $V(\phi_{0})$ is the potential of the $\phi_{0}$. As the dual
system lives in a spatial plane, we choose Poincaré coordinates with
$\tilde{z}$ the radial direction in the bulk. The metric ansatz reads [45]

$\displaystyle\tilde{g}_{\mu\nu}d\tilde{x}^{\mu}d\tilde{x}^{\nu}$
$\displaystyle=w_{E}(z)^{2}\bigg{(}-b(z)d\tilde{t}^{2}+\frac{d\tilde{z}^{2}}{b(z)}+(d\tilde{x}_{2}^{2}+d\tilde{x}_{3}^{2})+e^{-Bz^{2}}d\tilde{x}_{1}^{2}\bigg{)},$
$\displaystyle\phi_{0}$ $\displaystyle=\phi_{0}(\tilde{z}),\quad
A_{\mu}=(A_{t}(\tilde{z}),0,0,A_{3}(\tilde{x}_{2}),0),$ (2.2)

with

$\displaystyle b(\tilde{z})$
$\displaystyle=1-\frac{I_{1}(\tilde{z})}{I_{1}(\tilde{z}_{h})}+\frac{\mu^{2}}{I_{2}^{2}(\tilde{z}_{h})I_{1}(\tilde{z}_{h})}(I_{1}(\tilde{z}_{h})I_{3}(\tilde{z})-I_{1}(\tilde{z})I_{3}(\tilde{z}_{h}))+\frac{B^{2}}{I_{1}(\tilde{z}_{h})}(I_{1}(\tilde{z}_{h})I_{4}(\tilde{z})-I_{1}(\tilde{z})I_{4}(\tilde{z}_{h})),$
$\displaystyle I_{1}(\tilde{z})$
$\displaystyle=\int_{0}^{\tilde{z}}\frac{dy}{w_{E}^{3}e^{-\frac{1}{2}By^{2}}},\quad\quad\quad
I_{2}(\tilde{z})=\int_{0}^{\tilde{z}}\frac{dy}{w_{E}fe^{-\frac{1}{2}By^{2}}},\quad\quad\quad
I_{3}(\tilde{z})=\int_{0}^{\tilde{z}}I_{1}^{\prime}(y)I_{2}(y)dy,$
$\displaystyle I_{4}(\tilde{z})$
$\displaystyle=\int_{0}^{\tilde{z}}I_{1}^{\prime}(y)I_{5}(y)dy,\quad\quad
I_{5}(\tilde{z})=\int_{0}^{\tilde{z}}w_{E}fe^{-\frac{1}{2}By^{2}}dy,$ (2.3)

where
$w_{E}(\tilde{z})=\frac{1}{\tilde{z}}e^{\frac{-c\tilde{z}^{2}}{3}-p\tilde{z}^{4}}$
denotes the warped factor, $c=1.16$, $p=0.273$ determines the transition point
at $\tilde{\mu}=B=0$ fixed by fitting the lattice QCD data [46, 47]. It should
be noted that we use the notation,
$\tilde{x}^{\mu}=({\tilde{t},\tilde{z},\tilde{x}_{1},\tilde{x}_{2},\tilde{x}_{3}})$,
to represent the static frame and $x^{\mu}=({t,z,x_{1},x_{2},x_{3}})$ to
represent the rotating frame. The AdS boundary at $\tilde{z}=0$. Here we have
turned on a constant magnetic field $B$ along the $\tilde{x}_{1}$ direction in
the dual field theory.

The Hawking temperature can be calculated by surface gravity

$T(\tilde{z}_{h},\tilde{\mu},B)=\frac{I_{1}^{\prime}(\tilde{z}_{h})}{4\pi
I_{1}(\tilde{z}_{h})}(1-\tilde{\mu}^{2}\frac{I_{1}(\tilde{z}_{h})I_{2}(\tilde{z}_{h})-I_{3}(\tilde{z}_{h})}{I_{2}^{2}(\tilde{z}_{h})}-B^{2}(I_{1}(\tilde{z}_{h})I_{5}(\tilde{z}_{h})-I_{4}(\tilde{z}_{h}))).$
(2.4)

In this paper, the deconfinement phase transition temperature for zero
chemical potential and magnetic field is at $\tilde{T}_{c}=0.6\text{GeV}$
[45].

##  3 The spectral functions

In this section, we will calculate the spectral function for $J/\Psi$ state by
a phenomenological model proposed in Ref. [37]. In order to go smoothly later,
we assume the metric has general forms as follows,

$ds^{2}=-g_{tt}dt^{2}+g_{x_{1}x_{1}}dx_{1}^{2}+g_{tx_{1}}dtdx_{1}+g_{x_{1}t}dx_{1}dt+g_{x_{2,3}x_{2,3}}dx_{2,3}^{2}+g_{zz}dz^{2}.$
(3.1)

The vector field $A_{m}=(A_{\mu},A_{z})(\mu=0,1,2,3)$ is used to represent the
heavy quarkonium, which is dual to the gauge theory current
$J^{\mu}=\overline{\Psi}\gamma^{\mu}\Psi$. The standard Maxwell action takes
the following form

$S=-\int d^{4}xdz\frac{Q}{4}F_{mn}F^{mn},$ (3.2)

where $F_{mn}=\partial_{m}A_{n}-\partial_{n}A_{m}$,
$Q=\frac{\sqrt{-g}}{h(\phi){g_{5}}^{2}}$, $h(\phi)=e^{\phi(z)}$. The function
$\phi(z)$ is used to parameterize vector mesons,

$\phi(z)=\kappa^{2}z^{2}+Mz+\tanh(\frac{1}{Mz}-\frac{\kappa}{\sqrt{\Gamma}}).$
(3.3)

where $\kappa$ labels the quark mass, $\Gamma$ is the string tension of the
quark pair and $M$ denotes a large mass related to the heavy quarkonium non-
hadronic decay. The value of three energy parameters for charmonium in the
scalar field, determined by fitting the spectrum of masses [32], are
respectively:

$\displaystyle\kappa_{c}$
$\displaystyle=1.2\text{GeV},\quad\sqrt{\Gamma_{c}}=0.55\text{GeV},\quad
M_{c}=2.2\text{GeV}\,.$ (3.4)

The spectral functions for $J/\Psi$ state will be calculated with the help of
the membrane paradigm [48]. The equation of motion obtained from Eq. (3.2) are
as follows

$\partial_{m}(QF^{mn})=\partial_{z}(QF^{zn})+\partial_{\mu}(QF^{\mu n}),$
(3.5)

where $F^{mn}=g^{m\alpha}g^{n\beta}F_{\alpha\beta}$, $n=(0,1,2,3,4)$ and
$\mu=(0,1,2,3)$. For the $z$-foliation, the conjugate momentum of the gauge
field $A^{\mu}$ is given by the following formula:

$j^{\mu}=-QF^{z\mu}.$ (3.6)

Supposing plane wave solution for vector field $A^{\mu}$ propagates in the
$x_{1}$ direction. The equation of motion (3.5) can be written as two parts:
longitudinal-the fluctuations along $(t,x_{1})$; transverse-fluctuations along
$(x_{2},x_{3})$. Combined with Eq.(3.6), the dynamical equations for
longitudinal case from components $t,x_{1}$ and $z$ of Eq.(3.5) can be
expressed as

$\displaystyle-\partial_{z}j^{t}-Q(g^{x_{1}x_{1}}g^{tt}+g^{x_{1}t}g^{tx_{1}})\partial_{x_{1}}F_{x_{1}t}$
$\displaystyle=0,$ (3.7)
$\displaystyle-\partial_{z}j^{x_{1}}+Q(g^{tt}g^{x_{1}x_{1}}+g^{tx_{1}}g^{x_{1}t})\partial_{t}F_{x_{1}t}$
$\displaystyle=0,$ (3.8)

The flow $j^{\mu}$ conservation equation and the Bianchi identity can be
written as:

$\displaystyle\partial_{x_{1}}j^{x_{1}}+\partial_{t}j^{t}$ $\displaystyle=0,$
(3.9)
$\displaystyle\partial_{z}F_{x_{1}t}-\frac{g_{zz}}{Q}\partial_{t}[g_{x_{1}x_{1}}j^{x_{1}}+g_{x_{1}t}j^{t}]-\frac{g_{zz}}{Q}\partial_{x_{1}}[g_{tt}j^{t}-g_{tx_{1}}j^{x_{1}}]$
$\displaystyle=0.$ (3.10)

The longitudinal ”conductivity” and its derivative are defined as

$\displaystyle\sigma_{L}(\omega,z)$
$\displaystyle=\frac{j^{x_{1}}(\omega,z)}{F_{x_{1}t}(\omega,z)},$ (3.11)
$\displaystyle\partial_{z}\sigma_{L}(\omega,z)$
$\displaystyle=\frac{\partial_{z}j^{x_{1}}}{F_{x_{1}t}}-\frac{j^{x_{1}}}{F_{x_{1}t}^{2}}\partial_{z}F_{x_{1}t}.$
(3.12)

Kubo’s formula shows that the five-dimensional ”conductivity” at the boundary
is related to the retarded Green’s function:

$\sigma_{L}(\omega)=\frac{-G_{R}^{L}(\omega)}{i\omega}.$ (3.13)

where $\sigma_{L}$ is interpreted as the longitudinal AC conductivity. In
order to obtain flow equation (3.12), we assume $A_{\mu}=A_{n}(p,z)e^{-i\omega
t+ipx_{1}}$, where $A_{n}(p,z)$ is the quasinormal modes. Therefore, we have
$\partial_{t}F_{x_{1}t}=-i\omega F_{x_{1}t}$, $\partial_{t}j^{x_{1}}=-i\omega
j^{x_{1}}$. Finally, by using Eq.(3.8), Eq.(3.9), Eq.(3.10) and taking the
momentum limit $P=(\omega,0,0,0)$, the Eq.(3.12) can be written as

$\partial_{z}\sigma_{L}(\omega,z)=\frac{i\omega
g_{x_{1}x_{1}}g_{zz}}{Q}(\sigma_{L}^{2}-\frac{Q^{2}(g^{tt}g^{x_{1}x_{1}}+g^{tx_{1}}g^{x_{1}t})}{g_{x_{1}x_{1}}g_{zz}}).$
(3.14)

The initial condition for solving the equation can be obtained by requiring
regularity at the horizon $\partial_{z}\sigma_{L}(\omega,z)=0$. The dynamical
equation of transverse channel is as follows:

$\displaystyle\partial_{z}j^{x_{2}}\\!-Q[g^{tx_{1}}g^{x_{2}x_{2}}\partial_{t}F_{x_{1}x_{2}}\\!-g^{tt}g^{x_{2}x_{2}}\partial_{t}F_{tx_{2}}\\!+g^{x_{1}t}g^{x_{2}x_{2}}\partial_{x_{1}}F_{tx_{2}}\\!+g^{x_{1}x_{1}}g^{x_{2}x_{2}}\partial_{x_{1}}F_{x_{1}x_{2}}]$
$\displaystyle=0,$ (3.15)
$\displaystyle\frac{g_{x_{2}x_{2}}g_{zz}}{Q}\partial_{t}j^{x_{2}}+\partial_{z}F_{tx_{2}}$
$\displaystyle=0,$ (3.16)
$\displaystyle\partial_{x_{1}}F_{tx_{2}}+\partial_{t}F_{x_{2}x_{1}}$
$\displaystyle=0.$ (3.17)

The transverse ”conductivity” and its derivative are defined as

$\displaystyle\sigma_{T}(\omega,z)$
$\displaystyle=\frac{j^{x_{2}}(\omega,\overrightarrow{p},z)}{F_{x_{2}t}(\omega,\overrightarrow{p},z)},$
(3.18) $\displaystyle\partial_{z}\sigma_{T}(\omega,z)$
$\displaystyle=\frac{\partial_{z}j^{x_{2}}}{F_{x_{2}t}}-\frac{j^{x_{2}}}{F_{x_{2}t}^{2}}\partial_{z}F_{x_{2}t}.$
(3.19)

Similarly, we have $\partial_{t}F_{x_{1}x_{2}}=-i\omega F_{x_{1}x_{2}}$,
$\partial_{t}F_{tx_{2}}=-i\omega F_{tx_{2}}$, $\partial_{t}j^{x_{2}}=-i\omega
j^{x_{2}}$. Then the transverse flow equation (3.19) can be written as

$\displaystyle\begin{split}\partial_{z}\sigma_{T}(\omega,z)&=\frac{i\omega
g_{zz}g_{x_{2}x_{2}}}{Q}(\sigma_{T}^{2}-\frac{Q^{2}g^{zz}g^{tt}}{g_{x_{2}x_{2}}^{2}}).\end{split}$
(3.20)

It is not difficult to find that the metric Eq.(2) restores the SO(3)
invariance and the flow equations Eq.(3.14) and Eq.(3.20) have the same form
when magnetic field $B=0\text{GeV}^{2}$. The spectral function is defined by
the retarded Green’s function

$\rho(\omega)\equiv-ImG_{R}(\omega)=\omega Re\,\sigma(\omega,0)$ (3.21)

###  3.1 Turning on the angular momentum

In the early stage of non-central heavy-ion collisions, produced partons have
a large initial orbital angular momentum $J\propto b\sqrt{s_{NN}}$ where $b$
is the impact parameter and $\sqrt{s_{NN}}$ the nucleon-nucleon center-of-mass
energy. Although, at the stage of initial impact, most of the angular momentum
is carried away by the so-called ”spectators”, there is a considerable part
that remains in the produced QGP [49]. Star collaboration find, by studying
the global $\Lambda$ polarization in nuclear collisions, that the average
vorticity of QGP could reach $\Omega\sim 10^{21}/s$ [50].

Following [51, 52, 53, 54, 29], we extend the holographic magnetic catalysis
model to the situation of rotation with a planar horizon. For a general metric
in the rest frame

$d\tilde{s}^{2}=-\tilde{g}_{tt}d\tilde{t}^{2}+\tilde{g}_{zz}d\tilde{z}^{2}+\tilde{g}_{x_{1}x_{1}}d\tilde{x}_{1}^{2}+\tilde{g}_{x_{2,3}x_{2,3}}d\tilde{x}_{2,3}^{2},$
(3.22)

to introduce the rotation effect, it is convenient to split the 3-dimensional
space into two parts as $\mathcal{M}_{3}=\mathbb{R}\times\Sigma_{2}$. Then we
have

$d\tilde{s}^{2}=-\tilde{g}_{tt}d\tilde{t}^{2}+\tilde{g}_{zz}d\tilde{z}^{2}+\tilde{g}_{x_{1}x_{1}}\l^{2}d\tilde{\theta}^{2}+\tilde{g}_{x_{2,3}x_{2,3}}d\sigma^{2},$
(3.23)

where $l$ denotes the fixed distance to rotating axis and $d\sigma^{2}$
represents the line element of $\Sigma_{2}$. Then the angular momentum will be
turned on in the $l\tilde{\theta}$ direction through the standard Lorentz
transformation,

$\tilde{t}\rightarrow\gamma(t+\Omega
l^{2}\theta),\quad\quad\tilde{\theta}\rightarrow\gamma(\theta+\Omega t),$
(3.24)

where $\gamma=\frac{1}{\sqrt{1-\Omega^{2}l^{2}}}$ is the usual Lorentz factor.
It is estimated that the size of QGP may be around 4-8fm (RHIC) and 6-11fm
(LHC) [55]. Without loss of generality, we use $l=1\,\text{GeV}^{-1}$ in the
subsequent numerical calculation. The metric (3.22) changes to the following
form,

$ds^{2}=\gamma^{2}(\tilde{g}_{x_{1}x_{1}}\Omega^{2}l^{2}-\tilde{g}_{tt})dt^{2}+2\gamma^{2}\Omega
l^{2}(\tilde{g}_{x_{1}x_{1}}-\tilde{g}_{tt})dtd\theta+\gamma^{2}(\tilde{g}_{x_{1}x_{1}}-\Omega^{2}l^{2}\tilde{g}_{tt})l^{2}d\theta^{2}+\tilde{g}_{zz}dz^{2}+\tilde{g}_{x_{2,3}x_{2,3}}d\sigma^{2}.$
(3.25)

Then the Hawking temperature and chemical potential of the rotating black hole
can be calculated by

$\displaystyle T(z_{h},\mu,B,\Omega)$
$\displaystyle=\tilde{T}(\tilde{z}_{h},\tilde{\mu},B)\sqrt{1-\Omega^{2}l^{2}},$
$\displaystyle\mu(\Omega)$
$\displaystyle=\tilde{\mu}\sqrt{1-\Omega^{2}l^{2}}.$ (3.26)

Next, we calculate the spectral function of heavy quarkonium for rotating
case. Suppose that, in the limit of zero momentum, the plane wave solution of
the vector field has the form $A_{\mu}(t,z)=e^{-i\omega t}A_{\mu}(z,\omega)$.
Due to the appearance of rotation and magnetic field destroying the rotational
symmetry of space, the EOM(3.5) can be written in two varying channels:
longitudinal-the direction parallel to the magnetic field; transverse-the
direction perpendicular to the magnetic field. With the help of Eq.(3.14) and
Eq.(3.20), the flow equation can be written as

$\displaystyle\partial_{z}\sigma_{L}(\omega,z)=i\omega\Xi^{//}(\sigma_{L}(\omega,z)^{2}-(\Delta^{//})^{2}),$
$\displaystyle\partial_{z}\sigma_{T}(\omega,z)=i\omega\Xi^{\perp}(\sigma_{T}(\omega,z)^{2}-(\Delta^{\perp})^{2}).$
(3.27)

(1) When the direction of angular velocity is parallel to the direction of the
magnetic field,

$\displaystyle\Xi^{//}$
$\displaystyle=\frac{e^{\frac{-Bz^{2}}{2}+\phi(z)}\left(l^{2}\Omega^{2}e^{Bz^{2}}b(z)-1\right)}{b(z)\left(l^{2}\Omega^{2}-1\right)w_{E}(z)},$
$\displaystyle\Delta^{//}$
$\displaystyle=w_{E}(z)e^{\frac{Bz^{2}}{2}-\phi(z)}\sqrt{\frac{l^{2}\Omega^{2}-1}{l^{2}\Omega^{2}e^{Bz^{2}}b(z)-1}},\quad\quad\quad(\Omega//B//P)$
$\displaystyle\Xi^{\perp}$
$\displaystyle=\frac{e^{\frac{Bz^{2}}{2}+\phi(z)}}{b(z)w_{E}(z)},$
$\displaystyle\Delta^{\perp}$
$\displaystyle=w_{E}(z)e^{-\frac{Bz^{2}}{2}-\phi(z)}\sqrt{\frac{l^{2}\Omega^{2}e^{Bz^{2}}b(z)-1}{l^{2}\Omega^{2}-1}}.\,\,\,\,\quad\quad(\Omega//B\perp
P)$ (3.28)

(2)When the direction of angular velocity is perpendicular to the direction of
the magnetic field,

$\displaystyle\Xi^{//}$
$\displaystyle=\frac{e^{Bz^{2}+\phi(z)}\left(1-l^{2}\Omega^{2}b(z)\right)}{b(z)w_{E}(z)\sqrt{e^{Bz^{2}}\left(l^{4}\Omega^{4}-l^{2}\Omega^{2}+1\right)-l^{2}\Omega^{2}+\frac{l^{2}\Omega^{2}\left(1-e^{Bz^{2}}\right)}{b(z)}}},$
$\displaystyle\Delta^{//}$
$\displaystyle=w_{E}(z)e^{-\frac{Bz^{2}}{2}-\phi(z)}\sqrt{\frac{l^{2}\Omega^{2}-1}{l^{2}\Omega^{2}g(z)-1}},\,\quad\quad\quad\quad\quad\quad\quad(\Omega//P\perp
B)$ $\displaystyle\Xi^{\perp}$
$\displaystyle=\frac{\left(1-l^{2}\Omega^{2}\right)e^{\phi(z)}}{w_{E}(z)\sqrt{b(z)\left(b(z)\left(e^{Bz^{2}}\left(l^{4}\Omega^{4}-l^{2}\Omega^{2}+1\right)-l^{2}\Omega^{2}\right)+l^{2}\Omega^{2}\left(1-e^{Bz^{2}}\right)\right)}},$
$\displaystyle\Delta^{\perp}$
$\displaystyle=w_{E}(z)e^{\frac{Bz^{2}}{2}-\phi(z)}\sqrt{\frac{l^{2}\Omega^{2}g(z)-1}{l^{2}\Omega^{2}-1}},\,\,\,\,\quad\quad\quad\quad\quad\quad\quad(\Omega\perp
B//P)$ $\displaystyle\Xi^{\perp}$
$\displaystyle=\frac{\left(1-l^{2}\Omega^{2}\right)e^{Bz^{2}+\phi(z)}}{w_{E}(z)\sqrt{b(z)\left(b(z)\left(e^{Bz^{2}}\left(l^{4}\Omega^{4}-l^{2}\Omega^{2}+1\right)-l^{2}\Omega^{2}\right)+l^{2}\Omega^{2}\left(1-e^{Bz^{2}}\right)\right)}},$
$\displaystyle\Delta^{\perp}$
$\displaystyle=w_{E}(z)e^{-\frac{Bz^{2}}{2}-\phi(z)}\sqrt{\frac{l^{2}\Omega^{2}g(z)-1}{l^{2}\Omega^{2}-1}}.\,\,\quad\quad\quad\quad\quad\quad\quad(\Omega\perp
B\perp P)$ (3.29)

For vanishing angular momentum $\Omega$ and magnetic field $B$, one can easily
check that the Eq.(3.1)-(3.1) have the same forms:

$\displaystyle\Xi^{//}=\Xi^{\perp}=\frac{e^{\phi(z)}}{b(z)w_{E}(z)},\quad\Delta^{//}=\Delta^{\perp}=w_{E}(z)e^{-\phi(z)}.$
(3.30)

###  3.2 Adding a constant electric field to the background

In this subsection, a constant electric field is added on the D-brane, see
Ref.[56] for more information. The field strength tensor can be expressed as
$F=Edt\wedge dx_{1}$ where $E$ is the electric field along the $x_{1}$
direction. Since the equation of motion only depends on the field strength
tensor, this ansatz is still a good solution to supergravity and is the
minimal setup to study the E-field correction for the corresponding field
theory. Then one can write the E-field metric as

$\mathcal{F}_{\mu\nu}=\left(\begin{matrix}0&2\pi\alpha^{\prime}E&0&0\\\
-2\pi\alpha^{\prime}E&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\end{matrix}\right).$ (3.31)

Further, the background metric can be written as
$ds^{2}=G_{\mu\nu}+\mathcal{F}_{\mu\nu}$ where $G_{\mu\nu}$ is from Eq.(3.22).
The flow equation Eq.(3.27) becomes:

(1) When the direction of electric field is parallel to the direction of
magnetic field,

$\displaystyle\Xi^{//}$
$\displaystyle=\frac{e^{\phi(z)-Bz^{2}}}{b(z)w_{E}(z)\sqrt{e^{-Bz^{2}}-\frac{4\pi^{2}\alpha^{2}E^{2}w_{E}(z)^{6}}{b(z)}}},\quad\Delta^{//}=w_{E}(z)e^{\frac{Bz^{2}}{2}-\phi(z)},\quad(E//B//P)$
$\displaystyle\Xi^{\perp}$
$\displaystyle=\frac{w_{E}(z)e^{\phi(z)}}{b(z)\sqrt{e^{-Bz^{2}}w_{E}(z)^{4}-\frac{4\pi^{2}\alpha^{2}E^{2}}{b(z)}}},\quad\Delta^{\perp}=w_{E}(z)e^{-\frac{Bz^{2}}{2}-\phi(z)}.\,\,\,\,\quad\quad(E//B\perp
P)$ (3.32)

(2)When the direction of electric field is perpendicular to the direction of
magnetic field,

$\displaystyle\Xi^{//}$
$\displaystyle=\frac{w_{E}(z)e^{\frac{Bz^{2}}{2}+\phi(z)}}{b(z)\sqrt{\frac{b(z)w_{E}(z)^{4}-4\pi^{2}\alpha^{2}E^{2}}{b(z)}}},\quad\Delta^{//}=w_{E}(z)e^{-\frac{Bz^{2}}{2}-\phi(z)},\,\quad(E//P\perp
B)$ $\displaystyle\Xi^{\perp}$
$\displaystyle=\frac{w_{E}(z)e^{\phi(z)-\frac{Bz^{2}}{2}}}{b(z)\sqrt{\frac{b(z)w_{E}(z)^{4}-4\pi^{2}\alpha^{2}E^{2}}{b(z)}}},\quad\Delta^{\perp}=w_{E}(z)e^{\frac{Bz^{2}}{2}-\phi(z)},\,\,\,\,\,\quad(E\perp
B//P)$ $\displaystyle\Xi^{\perp}$
$\displaystyle=\frac{w_{E}(z)e^{\frac{Bz^{2}}{2}+\phi(z)}}{b(z)\sqrt{\frac{b(z)w_{E}(z)^{4}-4\pi^{2}\alpha^{2}E^{2}}{b(z)}}},\quad\Delta^{\perp}=w_{E}(z)e^{-\frac{Bz^{2}}{2}-\phi(z)}.\,\,\,\quad(E\perp
B\perp P)$ (3.33)

For vanishing E-field and B-field, one can find that the Eq.(3.2)-(3.2) have
the same forms as Eq.(3.30). In addition, one can find when the E-field is
perpendicular to magnetic field, the flow equation has the same form for
$E//P\perp B$ and $E\perp B\perp P$.

##  4  Numerical Results of spectral functions in a magnetized rotating plasma

Figure 1: The spectral functions of the $J/\Psi$ state with different
temperature $T$ (left panel) at $\mu=0\,\text{GeV}$ and $B=0\,\text{GeV}^{2}$
and different chemical potential $\mu$ (right panel) at $B=0\,\text{GeV}^{2}$
and $T=0.6\,\text{GeV}$ for magnetic catalysis model. From top to bottom, the
curves represent $T=0.5,0.6,0.7,0.9\,\text{GeV}$ in the left panel
respectively, and those denote $\mu=0,0.5,0.709,1\,\text{GeV}$ in the right
panel respectively.
Figure 2: The effective mass of $J/\Psi$ state as a function of temperature in
different chemical potentials for magnetic catalysis model.

The spectral function is described by Eq.(3.21). Firstly, we plot our
numerical results of the spectral function for the different temperatures and
chemical potential in Fig.1. One can find the increase of temperature and
chemical potential decrease the height and increase the width of spectral
function peak. The decrease in peak height and the increase in peak width
represent the enhancement of dissociation effect for heavy quarkonium. So we
can get that increasing temperature and chemical potential promote the
dissociation effect of bound state. In Fig.2, we display the effective mass
corresponding to the location of spectral function peak as a function of
temperature in varying chemical potentials. As increasing temperature, the
effective mass remains unchanged in the lower temperature regime, while that
increases in the higher temperature regime. In the lower temperature regime,
chemical potential reduces the effective mass. In the higher temperature
regime, chemical potential increases the effective mass. It is easy to
understand this phenomenon that increasing temperature enlarges the distance
of quark anti-quark pair, which leads to the interaction between quark/anti-
quark and medium becoming stronger. Therefore, the effective mass is larger
with the increase of temperature.

###  4.1 Turning on a constant magnetic field

Figure 3: The spectral functions of the $J/\Psi$ state with different magnetic
field $B$ at $\mu=0\,\text{GeV}$ and $T=0.63\,\text{GeV}$ for magnetic
catalysis model. From bottom to top, the curves represent
$B=0,0.5,0.96,1.3\,\text{GeV}^{2}$ respectively.

Figure 4: The effective mass of $J/\Psi$ state as a function of temperature in
different magnetic fields for magnetic catalysis model. The left picture is
for magnetic field parallel to polarization and the right one is for magnetic
field perpendicular to polarization.

The spectral function with respect to the magnetic field is presented in Fig.3
at $\mu=0\,\text{GeV},T=0.63\,\text{GeV}$. The left panel is for magnetic
field parallel to polarization and the right one is for magnetic field
perpendicular to polarization. A phenomenon completely opposite to the
chemical potential effect and temperature effect is observed. Whether the
magnetic field is parallel or perpendicular to the polarization, the
strengthened magnetic field increases the peak height and reduces the peak
width. That means the presence of magnetic field suppresses the dissociation
of bound state, which is completely opposite to the conclusions, the presence
of magnetic field enhances the dissociation of bound state, obtained from the
inverse magnetic catalysis model in Ref.[9]. While the change of effective
mass is non-trivial as is displayed in Fig.4. When magnetic field is parallel
to the polarization, the effective mass reduces with the increasing magnetic
field in the full temperature regime. When magnetic field is perpendicular to
the polarization, in the lower and higher temperature regimes, the increasing
magnetic field enlarges the effective mass, while that reduces the effective
mass for the middle-temperature regime. In addition, by comparing the left and
right panels, one can easily find that the suppression effect is stronger when
the magnetic field is perpendicular to polarization, but the effective mass is
smaller when the magnetic field is parallel to polarization which is
consistent with the conclusion from inverse magnetic catalysis model in
Ref.[9].

On balance, an interesting conclusion for the $J/\Psi$ state can be obtained:
Increasing temperature and chemical potential all enhance the dissociation
effect, while the magnetic field suppresses the dissociation effect. The
dependence of effective mass on magnetic field and chemical potential is non-
trivial, which is strictly dependent on the temperature.

###  4.2 Turning on angular momentum

####  4.2.1  The case of magnetized-rotating QGP with MC

Figure 5: The spectral functions of the $J/\Psi$ state with different angular
velocity $\Omega l$ at $\mu=0\,\text{GeV}$,$B=0\,\text{GeV}^{2}$ and
$T=0.6\,\text{GeV}$ for magnetic catalysis model. From top to bottom, the
curves represent $\Omega l=0,0.2,0.4,0.6$ respectively.

Figure 6: The effective mass of $J/\Psi$ state as a function of temperature in
different angular velocities for magnetic catalysis model. The left picture is
for angular velocity parallel to polarization and the right one is for angular
velocity perpendicular to polarization.

In Fig.5 we show the behavior of spectral function for different angular
velocities at $\mu=0\,\text{GeV},B=0\,\text{GeV}^{2},T=0.6\,\text{GeV}$. The
left figure is for the rotating direction parallel to polarization and the
right figure is for the rotating direction perpendicular to polarization.
Whether the rotating direction is parallel or perpendicular to the
polarization, increasing angular velocity reduces the height and enlarges the
width, which means angular velocity speed up the dissociation effect. In
addition, by comparing the parallel and the perpendicular cases in Fig.5, it
is found that the dissociation effect is stronger for the perpendicular case.
The angular velocity dependence of effective mass is shown in Fig.6. The
results suggest that angular velocity increases the effective mass. It is
noted that the shadow region denotes the hadronic phase, so the result is
untrustworthy in the regime. Because we introduce the rotation in the black
hole metric on the side of gravity corresponding to the rotation of QGP on the
side of the gauge field. The shadow region should be described by introducing
the rotation in the thermal gas solution rather than the black hole solution.
An interesting behavior can be observed in the perpendicular case when the
angular velocity is larger. The effective mass has a maximum near the phase
transition temperature, which may be regarded as a dissociation signal of the
$J/\Psi$ state. This point can also be checked in Fig.5, the peak is very low
suggesting the quarkonium dissociation. And we find the non-trivial behavior
from effective mass occurs only in the perpendicular case. This conclusion
also well shows that the dissolution effect is stronger in the perpendicular
case, which is consistent with the conclusion obtained by using the height of
the spectral function peak in Fig.5.

Figure 7: The spectral functions of the $J/\Psi$ state, by considering the
superposition effect of magnetic field and rotation, with different angular
velocity $\Omega l$ at $\mu=0\,\text{GeV}$, $B=0.5\,\text{GeV}^{2}$ and
$T=0.63\,\text{GeV}$ for magnetic catalysis model.

Figure 8: The effective mass of the $J/\Psi$ state, by considering the
superposition effect of magnetic field and rotation, with different angular
velocity $\Omega l$ at $\mu=0\,\text{GeV}$, $B=0.5\,\text{GeV}^{2}$ and
$T=0.63\,\text{GeV}$ for magnetic catalysis model.

To sum up, we find that temperature, chemical potential, and angular velocity
promote the dissolution effect, while magnetic field suppresses it. So one can
conclude that there must be a competition effect between magnetic field and
temperature, chemical potential, and angular velocity. As an example, we show
the behavior of spectral function for the superposition effect of rotation and
magnetic field ( here we only show the case of magnetic field parallel to
rotating direction ) in Fig. 7. One can find that there is a critical angular
velocity $\Omega_{crit}(B)l\sim 0.3$ which is the result of the competition
between the magnetic field effect and the rotating effect. Here
$\Omega_{crit}(B)$ denotes that the value of critical angular velocity
$\Omega_{crit}$ depends on the size of magnetic field $B$. When
$\Omega<\Omega_{crit}(B)$, the magnetic field effect plays a leading role and
suppresses the dissociation effect, which leads to the dissociation effect
being stronger when the rotating direction (magnetic field) is parallel to the
polarization. However, when $\Omega>\Omega_{crit}(B)$, the rotating effect is
dominant, which causes the dissolution effect to be more intense when the
rotating direction is perpendicular to the polarization. In Fig.8, we show the
influence of the superposition effect between magnetic field and rotation on
effective mass. One can find, for the parallel case, the magnetic field
suppresses the growth of effective mass caused by rotation, while the magnetic
field speeds up the growth of effective mass for the perpendicular case. But
for this intensity of magnetic field, the behavior of effective mass still is
dominant by rotation in the QGP phase. Although the result from the hadronic
phase is untrustworthy, we can use it as a reference to conclude the effective
mass in hadronic phase is dominant by an intense magnetic field instead of
rotation. We will check this conclusion in the future.

####  4.2.2 The case of magnetized-rotating QGP with IMC

Figure 9: The spectral functions of the $J/\Psi$ state with different angular
velocity $\Omega l$ at $\mu=0\,\text{GeV}$,$B=0\,\text{GeV}$ and
$T=0.6\,\text{GeV}$ for inverse magnetic catalysis model. From top to bottom,
the curves represent $\Omega l=0,0.2,0.4,0.6$ respectively.
Figure 10: The spectral functions of the $J/\Psi$ state, by considering the
superposition effect of magnetic field and rotation, with different angular
velocity $\Omega l$ at $\mu=0\,\text{GeV}$,$B=0.5\,\text{GeV}$ and
$T=0.63\,\text{GeV}$ for inverse magnetic catalysis model.

In addition, as a parallel study, we introduce the rotation in a holographic
inverse magnetic catalysis model(refer to appendixA for more details). As we
focus primarily on the impact of rotation on the dissociation effect of
$J/\Psi$ in the two different magnetic field models, we present only the
behavior of the spectral function in Fig.9 and Fig.10. From Fig.9, one can
find that growing angular velocity promotes the dissociation effect and the
melting effect is stronger for the direction of angular velocity perpendicular
to the direction of polarization. In addition, by considering the
superposition effect of magnetic field and rotation, for a smaller angular
velocity, Fig.10 straightforwardly illustrates that the dissociation effect is
stronger for the direction of angular velocity parallel to the direction of
polarization, which indicates the dissociation effect is dominant by magnetic
field. As pointed out in Ref.[9], the melting effect produced by the magnetic
field is stronger in the parallel case. For a larger angular velocity, an
opposing conclusion is obtained that suggests the dissociation effect is
controlled by angular velocity. Although the magnetic field exhibits
distinctly different behaviors in these two models, the effect of rotation on
the dissociation effect of $J/\Psi$ is similar.

In a word, the influences of rotation on the dissociation effect of the bound
state can be summarized as follows: (1) As the increase of angular velocity,
the dissociation effect becomes stronger and the effective mass becomes larger
and the dissociation effect is stronger for the perpendicular case. (2)The
superposition effect of magnetic field and rotation,

Figure 11: Left: The spectral functions of the $J/\Psi$ state with different
electric field $E$ at $\mu=0\,\text{GeV}$,$B=0\,\text{GeV}^{2}$ and
$T=0.6\,\text{GeV}$ for magnetic catalysis model. Right: The effective mass of
$J/\Psi$ state as a function of temperature in different electric fields for
magnetic catalysis model.

the strength of the dissolution effect depends entirely on the interplay
between the magnetic field and the rotation effect and the behavior of
effective mass is non-trivial. (3)Whether the magnetic field behaves as
magnetic catalysis or inverse magnetic catalysis, the conclusion from the
rotating effect is similar.

###  4.3 Turning on electric field

Figure 12: The spectral functions of the $J/\Psi$ state with different
electric field $E$ at $\mu=0\,\text{GeV}$, $B=0.5\,\text{GeV}^{2}$ and
$T=0.63\,\text{GeV}$ for magnetic catalysis model.

Figure 13: The effective mass of the $J/\Psi$ state, by considering the
superposition effect of magnetic field and electric field, with different
electric field $E$ at $\mu=0\,\text{GeV}$, $B=0.5\,\text{GeV}^{2}$ and
$T=0.63\,\text{GeV}$ for magnetic catalysis model.

In Fig.11, we draw the spectral function and effective mass of $J/\Psi$ state
for varying electric field at
$B=0\,\text{GeV}^{2},\mu=0\,\text{GeV},T=0.6\,\text{GeV}$. The result shows
that the dissociation effect and the effective mass are enhanced by the
increasing electric field. But one can find that, whether E-field is parallel
or perpendicular to polarization, the dissociation effect is the same.

Since the electric field and magnetic field have opposite interactions on the
dissociation effect and effective mass. Next, we take into account the
superposition of the magnetic field and electric field. As an example, we plot
the spectral function and effective mass with respect to the electric field
for the electric field parallel to the magnetic field in Fig.12 and Fig.13,
respectively. One can find that the difference from the dissociation effect
between the magnetic field parallel and perpendicular to polarization reduces
with the increase of electric field and the difference will vanish when the
electric field is large enough. The effective mass in the lower temperature
regime entirely depends on the electric field, however, in the higher
temperature regime, that is determined by the interplay between the magnetic
field and electric field.

To make a long story short, the influence of electric field on spectral
function can be summarized as follows: (1) The electric field enhances the
dissociation effect and enlarges the effective mass of the bound state
$J/\Psi$. (2) The dissociation effect is the same for the electric field
parallel and perpendicular to polarization. (3) Increasing electric field
reduces the difference from the dissociation effect between the magnetic field
parallel and perpendicular to polarization and the difference will vanish for
a sufficiently large electric field.

##  5 Summary and discussion

In this paper, by calculating the spectral function described by Eq.(3.21), we
investigate the dissociation effect of bound state $J/\Psi$ in a holographic
magnetic catalysis model. In order to more truly simulate the non-central
collision environment of extremely relativistic heavy ions, we consider a
total of five effects, thermal effect, density effect, magnetic effect,
rotation effect, and electric effect.

The results show that both temperature and chemical potential promote the
dissociation effect and enlarge the effective mass of heavy quarkonium
$J/\Psi$ in the QGP phase, while the magnetic field suppresses the
dissociation effect and the behavior of effective mass is non-trivial.
Interestingly, increasing magnetic field reduces the effective mass for the
parallel case but enlarges the effective mass in the lower and higher
temperature regimes for the perpendicular case. In addition, by considering
the rotating QGP background, we obtain that the dissociation effect becomes
stronger and the effective mass becomes larger with the increase of angular
velocity.

In view of the completely opposite behavior of the spectral functions
generated by magnetic field and rotation, we consider the superposition effect
of magnetic field and angular velocity. As an example, we show the behavior of
spectral function in the case of a magnetic field parallel to the rotating
direction. It is found that there exists a critical angular velocity
$\Omega_{crit}(B)$ depending on the magnetic field. For
$\Omega<\Omega_{crit}(B)$, the magnetic field plays a leading role and
suppresses the dissociation effect, which leads to the dissociation effect
being stronger and the effective mass being smaller when the rotating
direction (magnetic field) is parallel to the polarization. For
$\Omega>\Omega_{crit}(B)$, the rotating effect is dominant, which causes the
dissolution effect to be more intense and the effective mass is larger when
the rotating direction is perpendicular to the polarization. As a parallel
study, we also examine the rotation effect in the holographic inverse magnetic
catalysis model, although the magnetic field exhibits distinctly different
behaviors in these two models, the impact of rotation on the dissociation
effect of $J/\Psi$ is similar.

Besides, we consider the influence of electric field on spectral functions,
the calculation indicates the increasing electric field enhances the
dissociation effect and enlarges the effective mass. It is noted that whether
the electric field is parallel or perpendicular to polarization, the
dissociation effect is the same. In addition, we find that increasing electric
field could reduce the difference in spectral functions between magnetic field
parallel and perpendicular to polarization until the difference vanishes.

According to the interesting conclusion of this paper, in the future, it will
be desirable to study the magneto-rotational and electric-rotational
dissociation effects of heavy mesons and analyze the competition between the
two effects. Of course, one can also study other heavy vector mesons, such as
$\Upsilon(1S)$, or other physical quantities, such as configuration entropy
and QNMs [57], the spin density matrix element $\rho_{00}$ [58, 59]. As
paper[9] points out, the properties of $J/\Psi$ and $\Upsilon(1S)$ are very
different. In addition, one can also consider the influence of differential
rotation on the dissociation effect. Because the rotation generated in the
process of heavy ion collision depends on the distance to the rotating axis,
which implies that angular velocity has an attenuation along the rotating
radius, it is very necessary to consider a differential rotation to simulate
the RHIC and LHC experiments. To establish a connection between holographic
theory and experimental observations, one can consult Ref.[60] for the
calculation of collision energy $\sqrt{S_{NN}}$.

## Acknowledgements

We would like to thank Song He and Hai-cang Ren for the useful discussions.
This work is supported in part by the National Key Research and Development
Program of China under Contract No. 2022YFA1604900. This work is also partly
supported by the National Natural Science Foundation of China (NSFC) under
Grants No. 12275104, No. 11890711, No. 11890710, and No. 11735007.

##  Appendix A The effect of rotating QGP on $J/\Psi$ dissociation in a
holographic inverse magnetic catalysis model

Here, we consider a 5d EMD gravity system whose Lagrangian is given as

$\mathcal{L}=\sqrt{-g}(R-\frac{g_{1}(\phi_{0})}{4}F_{(1)\mu\nu}F^{\mu\nu}-\frac{g_{2}(\phi_{0})}{4}F_{(2)\mu\nu}F^{\mu\nu}-\frac{1}{2}\partial_{\mu}\phi_{0}\partial^{\mu}\phi_{0}-V(\phi_{0})).$
(A.1)

where $F_{(i)\mu\nu}$ ($i=1,2$) is the field strength tensor for U(1) gauge
field, $g_{i}(\phi_{0})$ ($i=1,2$) denotes the gauge coupling kinetic
function, $\phi_{0}$ represents the dilaton field, $V(\phi_{0})$ is the
potential of the $\phi_{0}$ (see[61] for exact expression). By introducing an
external magnetic field in $x_{1}$ direction, the metric ansatz in Einstein
frame can be written as [61]

$\displaystyle ds^{2}$
$\displaystyle=\frac{R^{2}S(z)}{z^{2}}(-f(z)dt^{2}+dx_{1}^{2}+e^{B^{2}z^{2}}(dx_{2}^{2}+dx_{3}^{2})+\frac{dz^{2}}{f(z)}),$
$\displaystyle\phi_{0}$ $\displaystyle=\phi_{0}(z),\quad
A_{(1)\mu}=A_{t}(z)\delta_{\mu}^{t},\quad F_{(2)}=Bdx_{2}\wedge dx_{3},$ (A.2)

with

$\displaystyle f(z)$
$\displaystyle=1+\int_{0}^{z}d\xi\xi^{3}e^{-B^{2}\xi^{2}-3A(\xi)}[K+\frac{\widetilde{\mu}^{2}}{2R_{gg}R^{2}}e^{R_{gg}\xi^{2}}],$
$\displaystyle K$
$\displaystyle=-\frac{1+\frac{\widetilde{\mu}^{2}}{2R_{gg}R^{2}}\int_{0}^{z_{h}}d\xi\xi^{3}e^{-B^{2}\xi^{2}-3A(\xi)+R_{gg}\xi^{2}}}{\int_{0}^{z_{h}}d\xi\xi^{3}e^{-B^{2}\xi^{2}-3A(\xi)}},$
$\displaystyle\widetilde{\mu}$
$\displaystyle=\frac{\mu}{\int_{0}^{z_{h}}d\xi\frac{\xi
e^{-B^{2}\xi^{2}}}{g_{1}(\xi)\sqrt{S(\xi)}}},$ (A.3)

where $R$ is the AdS radius, $S(z)$ labels the scale factor, $f(z)$ is the
blackening function and $\mu$ denotes the chemical potential. The asymptotic
boundary is at $z=0$ and $z=z_{h}$ denotes the location of the horizon where
$f(z_{h})=0$. The concrete form of gauge coupling function $g_{1}$ can be
determined by fitting the vector meson mass spectrum. The linear Regge
trajectories for $B=0$ can be restored when

$g_{1}(z)=\frac{e^{-R_{gg}z^{2}-B^{2}z^{2}}}{\sqrt{S(z)}}.$ (A.4)

Here, this $B$, in units GeV, is the 5d magnetic field. The 4d physical
magnetic field is $e\mathcal{B}\sim\frac{const}{R}\times B$ where $const=1.6$
(See details in Ref.[62]). By taking $S(z)=e^{2A(z)}$, one can obtain
$R_{gg}=1.16\,\text{GeV}^{2}$ for heavy meson state $J/\psi$. In the following
calculation, we take $A(z)=-az^{2}$ where $a=0.15\,\text{GeV}^{2}$ matching
with the lattice QCD deconfinement temperature at $B=0\,\text{GeV}$ [63].

The Hawking temperature has the following form,

$T(z_{h},\mu,B)=\frac{-z_{h}^{3}e^{-3A(z_{h})-B^{2}z_{h}^{2}}}{4\pi}(K+\frac{\widetilde{\mu}^{2}}{2R_{gg}R^{2}}e^{R_{gg}z_{h}^{2}}).$
(A.5)

The paper [61] assumes that the dilaton field $\phi$ remains real everywhere
in the bulk, which leads to magnetic field $B\leq B_{c}\simeq
0.61\,\text{GeV}$. In this holographic inverse magnetic catalysis model, the
deconfinement temperature is $T_{c}=0.268\,\text{GeV}$ at zero chemical
potential and magnetic field.

In the rotating background, the metric uses Eq.(3.25), and the temperature and
chemical potential takes Eq.(3.1). The flow equation takes Eq.(3.14) and
Eq.(3.20). In this holographic inverse magnetic catalysis model, we have
$\tilde{g}_{tt}=\frac{f(z)S(z)}{z^{2}},\tilde{g}_{x_{1}x_{1}}=\frac{S(z)}{z^{2}},\tilde{g}_{x_{2}x_{2}}=g_{x_{3}x_{3}}=\frac{e^{B^{2}z^{2}}S(z)}{z^{2}},\tilde{g}_{zz}=\frac{S(z)}{z^{2}f(z)}$.
Finally, the spectral function can be obtained. We show the behavior of
spectral functions for different angular velocities in Fig.9. In Fig.10, the
influence of the superposition of angular velocity and magnetic field on the
spectral function is studied. One can find the conclusion is similar to the
holographic magnetic catalysis model.

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