# A systematically study of thermal width of heavy quarkonia in a finite
temperature magnetized background from holography

Sheng-Qin Feng<EMAIL_ADDRESS>College of Science, China Three Gorges
University, Yichang 443002, China  Key Laboratory of Quark and Lepton Physics
(MOE) and Institute of Particle Physics,
Central China Normal University, Wuhan 430079, China  Yan-Qing Zhao  College
of Science, China Three Gorges University, Yichang 443002, China  Xun Chen
Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle
Physics,
Central China Normal University, Wuhan 430079, China

###### Abstract

By simulating the finite temperatures magnetized background in the RHIC and
LHC energies, we systematically study the characteristics of thermal widths
and potentials of heavy quarkonia. It is found that the magnetic field has
less influence on the real potential, but has a significant influence on the
imaginary potential, especially in the low deconfined temperature. Extracted
from the effect of thermal worldsheet fluctuations about the classical
configuration, the thermal width of $\Upsilon(1s)$ in the finite temperature
magnetized background is investigated. It is found that at the low deconfined
temperature the magnetic field can generate a significant thermal fluctuation
of the thermal width of $\Upsilon(1s)$, but with the increase of temperature,
the effect of magnetic field on the thermal width becomes less important,
which means the effect of high temperature completely exceeds that of magnetic
field and magnetic field become less important at high temperature. The
thermal width decreases with the increasing rapidity at the finite temperature
magnetized background. It is also observed that the effect of the magnetic
field on the thermal width when dipole moving parallel to the magnetic field
direction are larger than that moving perpendicular to the magnetic field
direction, which implies that the magnetic field tends to enhance thermal
fluctuation when dipole moving parallel to the direction of magnetic field.
The thermal width of $\Upsilon(1S)$ hardly changes with the increasing
temperature when dipole moving perpendicular to the magnetic field. But when
dipole moving parallel to the magnetic field, the thermal width at low
temperature is obviously larger than that at high temperature.

thermal width, imaginary potential, finite temperature, magnetic field

###### pacs:

11.25.Tq, 25.75.Nq

##  I Introduction

A new state of matter, so-called Quark-Gluon Plasma(QGP), has been generated
in relativistic heavy ion collisions at the Relativistic Heavy Ion Collider
(RHIC) and the Large Hadron Collider (LHC) Adams:2005dq ; Adcox:2004mh ;
Shuryak:2004cy . The heavy quarkonia ($J/\Psi$ and $\Upsilon$ mainly) are
formed in hard processes before the thermalization of the plasma, which are
taken as useful probes to study the formation and evolution of the
QGPShuryak:1980tp . The well-known work of Matsui and Satz Matsui:1986dk
argued that the binding interaction of the heavy quark-antiquark (Q$\bar{Q}$)
pair in a thermal bath is screened by the medium, resulting in the melting of
the heavy quarkonia. In the study of heavy ion collisions, besides high
temperature, another important finding is the generation of a strong magnetic
field of noncentral heavy ion collisions at the RHIC and the LHCSkokov:2009qp
; Voronyuk:2011jd ; Bzdak:2011yy ; Deng:2012pc ; Mo:2013qya ; Zhong:2014cda ;
Zhong:2014sua ; She:2017icp . Since the magnetic field in relativistic heavy-
ion collisions is so great that people believe that the strong magnetic fields
can provide some deep investigations of the dynamics of quantum chromodynamics
(QCD).

The interaction energy $V_{Q\bar{Q}}$ of Q$\bar{Q}$ pair may possess a finite
imaginary part $\textrm{Im}V_{Q\bar{Q}}$, which can be used to calculate a
thermal width of the quarkonium at finite temperatureGubser:1998bc ;
Aharony:1999ti ; Laine:2007gj ; Petreczky:2010tk . The calculations of
$\textrm{Im}V_{Q\bar{Q}}$ related to heavy ion collisions in QCD have been
carried out for static $Q\bar{Q}$ pairs by using lattice QCDBrambilla:2010vq ;
Burnier:2009yu ; Rothkopf:2011db  and perturbative QCDGubser:1998bc . The
dissociation of quarkonia is one of the most important experimental signal for
QGP formation. Some publications arguedLaine:2006ns ; Beraudo:2007ky ;
Brambilla:2008cx ; Escobedo:2014gwa  that the imaginary part of the potential
$\textrm{Im}V_{Q\bar{Q}}$ may be an important reason responsible for this
suppression rather than color screening. The imaginary potential has been
subsequently studied in weakly coupled theories by Refs.Brambilla:2010vq ;
Dumitru:2009fy . However, an available method of the imaginary potential in
recent years Gubser:1998bc ; Margotta:2011ta ; Maldacena:1997re  has be used
in strongly coupled theories with the aid of nonperturbative methods of
AdS/QCD.

The imaginary potential of quarkonia for $\mathcal{N}$ = 4 SYM theory was
studied by Noronha and Dumitru in their seminal work Noronha:2009da . This
imaginary contribution originates from thermal fluctuations around the bottom
of the classical sagging string in the bulk that links the heavy quarks
situated at the boundary in the dual gravity picture. The imaginary potential
$\textrm{Im}V_{Q\bar{Q}}$ related to the effect of thermal fluctuations is
due to the interactions between the heavy quarkonia and the medium.
Subsequently, a large number of research work about $\textrm{Im}V_{Q\bar{Q}}$
were carried out with gauge/gravity duality. For instance, the
$\textrm{Im}V_{Q\bar{Q}}$ of static quarkonia was studied in Finazzo:2013aoa
; Fadafan:2013bva . Refs. Finazzo:2014rca ; Ali-Akbari:2014vpa  studied the
effect of moving quarkonia on $\textrm{Im}V_{Q\bar{Q}}$. The influences of
chemical potential and magnetic field on $\textrm{Im}V_{Q\bar{Q}}$ were
investigated in Zhang:2016tem ; Zhang:2018fpe . The studies of
$\textrm{Im}V_{Q\bar{Q}}$ in some AdS/QCD models are provided in
Braga:2016oem ; Sadeghi:2014zya .

In the other hands, strong magnetic field plays an essential roles in non-
central heavy ion collisions at RHIC and LHC Skokov:2009qp ; Voronyuk:2011jd ;
Bzdak:2011yy ; Deng:2012pc ; Mo:2013qya ; Zhong:2014cda ; Zhong:2014sua ;
She:2017icp . Strong magnetic field also provides a good probe of the dynamics
of QCD. To accurately determine the suppression of quarkonia formed in
relativistic heavy ion collisions, it is necessary to evaluate thermal width
of $\Upsilon(1S)$ for moving quarkonia in the finite temperature magnetized
QGP background in the RHIC and LHC energies.

Ref. Faulkner:2008qk  computed the momentum dependence of meson widths within
the gauge/gravity duality. It was proposed that the thermal width becomes very
large for rapidly moving meson, and the imaginary part of rapidly moving
mesons may be already large enough to cause suppression of these states in a
strongly coupled plasma even before complete dissociation. Thus, Refs.
Liu:2006nn ; Liu:2006he ; Ejaz:2007hg  indicated that the dissociation
temperature of meson decreased with the pair’s rapidity. By simulating the
background of finite temperatures magnetized background in the RHIC and LHC
energies, We restrict ourselves to the range of temperature and magnetic field
corresponding to RHIC and LHC energy regions to study the potential and
thermal width for dipole moving parallel and pedicular to the magnetic field.
This paper is organized as follows: in Sec. II, we introduce the setup of the
gravity background with back-reaction of magnetic field through the Einstein-
Maxwell (EM) system. These cases where the $Q\bar{Q}$ dipole moving parallel
and perpendicular to the direction of the magnetic field are discussed in Sec.
III and Sec. IV, respectively. In Sec. V we make a comparison of the results
of dipole moving parallel and perpendicular to the magnetic field direction.
And then we make conclusions in Sec. VI.

##  II The setup

The action of the gravity background with back-reaction of magnetic field
through the Einstein-Maxwell (EM) systemMamo:2015dea ; Dudal:2015wfn ;
Li:2016gfn  is given as:

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

where $R$ is the scalar curvature, $G_{5}$ is $5D$ Newton constant, $g$ is the
determinant of metric $g_{\mu\nu}$, $L$ is the AdS radius and $F_{MN}$ is the
U(1) gauge field Li:2016gfn .

The Einstein equation for the EM system could be derived as follows

$\displaystyle\
E_{MN}-\frac{6}{L^{2}}g_{MN}-2(g^{IJ}F_{MJ}F_{NJ}-\frac{1}{4}F_{IJ}F^{IJ}g_{MN})=0,$
(2)

where $E_{MN}=R_{MN}-\frac{1}{2}Rg_{MN}$, $R_{MN}$ and $R$ are the Einstein
tensor, the Ricci tensor and Ricci scalar, respectively. The ansatz of metric
is taken as Li:2016gfn

$\
ds^{2}=\frac{\textit{R}^{2}}{z^{2}}[-f(z)dt^{2}+h(z)(dx_{1}^{2}+dx_{2}^{2})+q(z)dx_{3}^{2}+\frac{dz^{2}}{f(z)}],$
(3)

where $f(z=z_{h})=0$ locates at horizon $z=z_{h}$, and $q(z)$ together with
$h(z)$ are regular function of $z$ for $0\leq z\leq z_{h}$. The specific
equations of motion derived from the action and the perturbative solution of
$f(z)$, $h(z)$ and $q(z)$ have been discussed by Li:2016gfn . Using the
$r=\frac{\textit{R}^{2}}{z}$($\textit{R}=1$), we can derive a metric with $r$
as follows:

$\
ds^{2}=r^{2}\left(-f(r)dt^{2}+h(r)(dx_{1}^{2}+dx_{2}^{2})+q(r)dx_{3}^{2}\right)+\frac{1}{r^{2}f(r)}dr^{2}.$
(4)

As a first order approximation, one can take the leading expansion in
Refs.Li:2016gfn ; Zhu:2019ujc  as

$f(r)=1-\frac{r_{h}^{4}}{r^{4}}\left(1-\frac{2B^{2}}{3r_{h}^{4}}\log(\frac{r_{h}}{r})\right),$
(5) $q(r)=1+\frac{2}{3}B^{2}\log(\frac{1}{r})\frac{1}{r^{4}},$ (6)
$h(r)=1-\frac{1}{3}B^{2}\log(\frac{1}{r})\frac{1}{r^{4}}.$ (7)

Noticing that $B$ is related to the physical magnetic field $\mathfrak{B}$ at
the boundary by the equation $\mathfrak{B}=\sqrt{3}B$, and $z_{h}$ is the
horizon of the black hole. If we take $B\leq 0.15\textrm{GeV}^{2}$, the
corresponding physical magnetic field is $\mathfrak{B}\leq
0.26\textrm{GeV}^{2}$, which conforms to the magnitude of real magnetic field
generated by in the RHIC and LHC energies. The Hawking temperature with
magnetic field $B$ is computed as

$T(z_{h},B)=\frac{r_{h}}{\pi}-\frac{B^{2}}{6\pi r_{h}^{3}},$ (8)

where $T(z_{h},B)$ is a function of the position of the horizon and the
magnetic field, corresponds to the temperature of the thermal bath in the
gauge theory. At the end of the setup, It’s necessary to check the validity of
the first order pertubative solutions of $f(z)$, $h(z)$ and $q(z)$ in
Eqs.(5-7). It was pointed out Mamo:2015dea  that the perturbative solution can
work well only when $B\ll T^{2}$. After inserting the temperature and magnetic
field into the IR expansion, Refs.Li:2016gfn ; Zhu:2019ujc  made some
comparisons of the leading, next-leading and next-next-leading order of these
perturbative solutions, and found that the approximate of leading order
Eqs.(5-7) is good enough for $T\geq 0.15~{}\textrm{GeV}$ and $B\leq
0.15~{}\textrm{GeV}^{2}$. In the paper, the corresponding temperature range
and magnetic field range are chosen as $0.15~{}\textrm{GeV}\leq T\leq
0.33~{}\textrm{GeV}$ and $0.02~{}\textrm{GeV}^{2}\leq B\leq
0.15~{}\textrm{GeV}^{2}$, which conforms to the range of temperature and
magnetic field in the RHIC and LHC erergies.

##  III Dipole moving parallel to the magnetic feild

In the section, we assume that the initial state of $Q\bar{Q}$ is oriented in
the direction of magnetic field, and the magnetic field direction is along
$x_{3}$ axis. A reference frame is chosen where the plasma is at rest and the
$Q\bar{Q}$ dipole is moving with a constant rapidity, and one can boost to a
reference frame where the dipole is at rest but the plasma is moving past it.
We can utilize this fact to study the effect of the plasma on a $Q\bar{Q}$
pair in the thermal medium. By considering the plasma is at rest, one can
boost our frame in one direction with rapidity $\beta$.

When the heavy quark is moving parallel to the magnetic field along the
$x_{3}$ direction with rapidity $\beta$, the coordinates are parametrized by

$\differential{t}=\differential{t}^{\prime}\cosh\beta-\differential{x_{3}}^{\prime}\sinh\beta,$
(9)
$\differential{x_{3}}=-\differential{t}^{\prime}\sinh\beta+\differential{x_{3}}^{\prime}\cosh\beta.$
(10)

Substituting (9) and (10) into the metric (4), one can obtain

$\displaystyle\
ds^{2}=\left(-r^{2}f(r)\cosh^{2}\beta+r^{2}q(r)\sinh^{2}\beta\right)dt^{2}-2\sinh\beta\cosh\beta\left(r^{2}q(r)-r^{2}f(r)\right)dtdx_{3}$
$\displaystyle+\left(-r^{2}f(r)\sinh^{2}\beta+r^{2}q(r)\cosh^{2}\beta\right)dx_{3}^{2}+r^{2}h(r)(dx_{1}^{2}+dx_{2}^{2})+\frac{1}{r^{2}f(r)}dr^{2}.$
(11)

By considering the dipole moving parallel to the wind, one can take

$\ t=\tau,x_{1}=x_{2}=0,x_{3}=\sigma,r=r(\sigma),$ (12)

Then the metric is given as

$\displaystyle\
ds^{2}=(-r^{2}f(r)\cosh^{2}\beta+r^{2}q(r)\sinh^{2}\beta)d\tau^{2}$
$\displaystyle+(-r^{2}f(r)\sinh^{2}\beta+r^{2}q(r)\cosh^{2}\beta+\frac{\dot{r}^{2}}{r^{2}f(r)})d\sigma^{2}.$
(13)

Holographically, in the supergravity limit which corresponds to a strongly
coupled plasma, one can evaluate the expectation value of Wilson loop $W(C)$
by the prescription

$\ \langle W(C)\rangle\sim e^{-iS_{NG}},$ (14)

where $S_{NG}$ is the classical Nambu-Goto action of a string in the bulk,
which can be given as:

$\ S_{NG}=-\frac{1}{2\pi\alpha^{\prime}}\int\differential\sigma
d\tau\sqrt{-\det g_{ab}},$ (15)

where the induced metric of the worldsheet $g_{ab}$ is given by

$\ g_{ab}=g_{MN}\partial_{a}X^{M}\partial_{b}X^{N},\quad a,\,b=0,\,1,$ (16)

evaluated at an extremum of the action, $\delta S_{NG}=0$. The string
worldsheet coordinates are given in static gauge,
$X^{\mu}=(t,x_{3},0,0,r(x_{3},t))$, $\tau=t$ and $\sigma=x_{3}$ for the
rectangular Wilson loop.

If the quark and anti-quark are located at $x_{3}=L/2$ and $x_{3}=-L/2$,
respectively, the rectangular Wilson loop of spatial length $L$ and time
extension $\mathcal{T}$ are studied. Plugging back $S_{NG}$ in (14) one can
evaluate the real part of $V_{Q\bar{Q}}$. By considering thermal fluctuations
of the string, one will be able to evaluate the imaginary part of
$V_{Q\bar{Q}}$. With the metric of (III), the Nambu-Goto action (15) can be
calculated as

$\
S_{NG}=\frac{\mathcal{T}}{2\pi\alpha^{\prime}}\int_{-L/2}^{L/2}\differential
x_{3}\sqrt{a(r)+b(r)\dot{r}^{2}}.$ (17)

The Lagrangian density is taken as:

$\mathcal{L}=\sqrt{a(r)+b(r)\dot{r}^{2}},$ (18)

with

$\
a(r)=r^{4}{f(r)q(r)(\cosh^{4}\beta+\sinh^{4}\beta)-\sinh^{2}\beta\cosh^{2}\beta(f^{2}(r)+q^{2}(r))},$
(19)

and

$\ b(r)=\cosh^{2}\beta-\frac{q(r)}{f(r)}\sinh^{2}(\beta).$ (20)

Note that the Lagrangian density does not depend on $x_{3}$ explicitly, then a
conserved quantity can be given as:

$\mathcal{L}-\frac{\partial\mathcal{L}}{\partial\dot{r}}\dot{r}=constant$ (21)

The boundary condition at $x_{3}=0$, when $r=r_{c}$, $\dot{r}_{c}
̇=dr/dx_{3}\mid_{r=r_{c}}=0$. From above, we can derive

$\ \dot{r}=\sqrt{\frac{a(r)^{2}-a(r_{c})a(r)}{a(r_{c})b(r)}}.$ (22)

The distance of the heavy $Q\bar{Q}$ pair can be calculated as

$\
L=2\int^{\infty}_{r_{c}}dr\sqrt{\frac{a(r_{c})b(r)}{a(r)^{2}-a(r_{c})a(r)}}.$
(23)

The real part of the heavy quark potential can be derived as:

$\
\textrm{Re}V_{Q\bar{Q}}=\frac{1}{\pi\alpha^{\prime}}\int^{\infty}_{r_{c}}dr\left(\sqrt{\frac{a(r)b(r)}{a(r)-a(r_{c})}}-\sqrt{b_{0}(r)}\right)-\frac{1}{\pi\alpha^{\prime}}\int^{r_{c}}_{r_{h}}dr\sqrt{b_{0}(r)},$
(24)

where $b_{0}(r)=b(r\rightarrow\infty)$.

The real part of the heavy quark potential as (24) can be computed by using
the classical solution to the Nambu-Goto action (17). To explore
$\textrm{Im}V_{Q\bar{Q}}$ we have to consider the effect of thermal worldsheet
fluctuations of the classical configuration $r=r_{c}(x_{3})$. Although such
fluctuations should be small, but they may turn the integrand of (17) negative
near $x_{3}=0$ and create an imaginary part for the effective string action.
The corresponding physical picture is that some part of the string may reach
the horizon through thermal fluctuations shown in Fig 1. If the bottom of the
classical string solution is close enough to the horizon, thermal worldsheet
fluctuations of very long wavelength may be able to reach the black brane
horizon.

Figure 1:  An illustration of the effect of thermal fluctuations around the
classical string configuration (solid line). The dashed line is for thermal
fluctuations.

It is well known that an imaginary potential $\textrm{Im}V_{Q\bar{Q}}$ is an
imaginary part of the potential, which can be used to define a thermal decay
width. For weak coupling the thermal width is associated with the imaginary
part of the gluon self-energy induced by Landau damping and the $Q\bar{Q}$
color singlet-to-octet thermal break-up. In this approach, the thermal width
of heavy quarkonium states comes from the effect of the thermal fluctuation
due to the interactions between the heavy quarks and the strongly coupled
medium. By considering the thermal worldsheet fluctuations about classical
configuration, one can extract imaginary potential and thermal width, the
detailed analysis can be found in Finazzo:2013aoa ; Fadafan:2013bva . The
imaginary potential is given as:

$\displaystyle\
\textrm{Im}V_{Q\bar{Q}}=-\frac{1}{2\sqrt{2}\alpha^{\prime}}\left(\frac{\sqrt{a^{\prime}(r_{c})}}{2a^{\prime\prime}(r_{c})}-\frac{a(r_{c})}{a^{\prime}(r_{c})}\right)\sqrt{b_{0}(r)},$
(25)

where $a^{\prime}(r_{c})$ and $a^{\prime\prime}(r_{c})$ are the values of the
first and second derivative of $a(r)$ to $r$ at $r_{c}$, respectively.

As follow, we will use a first-order non-relativistic expansion
Finazzo:2013aoa  to estimate the thermal width $\Gamma_{Q\bar{Q}}$ of the
heavy quarkonia

$\
\Gamma_{Q\bar{Q}}=-\langle\psi\mid\textrm{Im}V_{Q\bar{Q}}(L,T)\mid\psi\rangle,$
(26)

where

$\ \langle\vec{r}|\psi\rangle=\frac{1}{\sqrt{\pi}a_{0}^{3/2}}e^{-r/a_{0}}$
(27)

is the ground-state wave function of a particle in a Coulomb-like potential
$V(L)=-K/L$, where the Bohr radius is defined as $a_{0}=2/(m_{Q}K)$, $m_{Q}$
is the mass of the heavy quark $Q$. For the $\Upsilon(1S)$ state, the thermal
width is given as

$\ \frac{\Gamma_{Q\bar{Q}}}{T}=-\frac{4}{Ta_{0}}\int^{\infty}_{0}d\omega
e^{-\frac{2\omega}{Ta_{0}}}\omega^{2}\frac{\textrm{Im}V_{Q\bar{Q}}}{T}(\omega),$
(28)

where $a_{0}\sim 0.6$ GeV-1 and $m_{Q}=4.6\textrm{GeV}$ for the calculation of
$\Upsilon(1S)$ thermal width. Note that the imaginary potential is defined in
the region $(L_{min},L_{max})$ instead of taking the integral from zero to
infinity, which means the imaginary potential starts at a $L_{min}$ which can
be computed by solving $\textrm{Im}V_{Q\bar{Q}}=0$ and ends at a $L_{max}$.
The solution is called as a conservative approach Fadafan:2015kma .

##  IV Dipole moving perpendicular to the magnetic feild

In this section, we assume that the initial state of $Q\bar{Q}$ is oriented
transverse to the direction of magnetic field, and the magnetic field
direction is along $x_{3}$ axis. By considering the system moves along the
$x_{1}$ direction with rapidity $\beta$, one can take

$\ t=\tau,x_{2}=x_{3}=0,x_{1}=\sigma,r=r(\sigma).$ (29)

Then the metric of dipole moving perpendicular to the magnetic field becomes

$\displaystyle\
ds^{2}=\left(-r^{2}f(r)\cosh^{2}\beta+r^{2}q(r)\sinh^{2}\beta\right)dt^{2}+\left(r^{2}h(r)+\frac{\dot{r}^{2}}{r^{2}f(r)}\right)dx_{1}^{2}.$
(30)

The Lagrangian density is

$\ \mathcal{L}_{P}=\sqrt{a_{P}(r)+b_{P}(r)\dot{r}^{2}},$ (31)

where

$\ a_{P}(r)=r^{4}f(r)h(r)\cosh^{2}\beta-r^{4}q(r)h(r)\sinh^{2}\beta,$ (32)

and

$\ b_{P}(r)=\cosh^{2}\beta-\frac{q(r)}{f(r)}\sinh^{2}(\beta).$ (33)

The real part of the heavy quark potential can be calculated as

$\displaystyle\
\textrm{Re}V_{Q\bar{Q}}=\frac{1}{\pi\alpha^{\prime}}\int^{\infty}_{r_{c}}dr\left(\sqrt{\frac{a_{P}(r)b_{P}(r)}{a_{P}(r)-a_{P}(r_{c})}}-\sqrt{b_{P0}(r)}\right)-\frac{1}{\pi\alpha^{\prime}}\int^{r_{c}}_{r_{h}}dr\sqrt{b_{P0}(r)}.$
(34)

where $b_{P0}(r)=b_{P}(r\rightarrow\infty)$.

Similarly, one can calculate the imaginary part of the heavy quark potential
when dipole moving perpendicular to the magnetic field

$\
\textrm{Im}V_{Q\bar{Q}}=-\frac{1}{2\sqrt{2}\alpha^{\prime}}\left(\frac{\sqrt{a_{P}^{\prime}(r_{c})}}{2a_{P}^{\prime\prime}(r_{c})}-\frac{a_{P}(r_{c})}{a_{P}^{\prime}(r_{c})}\right)\sqrt{b_{P0}(r)}.$
(35)

##  V Comparison of dipole moving parallel and perpendicular to the magnetic
field

In order to study the effects of the magnetic field on the imaginary potential
and $\Upsilon(1S)$ thermal distributions in finite temperature magnetized
background, we make a comparison of dipole moving parallel and perpendicular
to the magnetic fields cases. As we know, the $Q\bar{Q}$ pair is not generated
in static QGP medium, but in moving QGP medium. Therefore, we should consider
the influence of the moving QGP medium on the $Q\bar{Q}$ pair. In order to
study quarkonia thermal features in RHIC and LHC energies, we choose some
special magnetic fields $B=0.02~{}\textrm{GeV}^{2}$,
$B=0.15~{}\textrm{GeV}^{2}$ and with temperatures as $T=0.15,0.25$ and
$0.33~{}\textrm{GeV}$ in RHIC and LHC energies. Fig. 2 publishes the behavior
of $LT$ as a function of $y_{c}=r_{h}/r_{c}$ for different magnetic fields $B$
with different temperatures. It is found that the magnetic field has a
significant effect on the relationship between $LT$ and $y_{c}$ at the low
deconfined temperature ($T_{c}=0.15\textrm{GeV}$). But as the increase of
temperature $T>T_{c}$, the influence of the magnetic field on the relationship
between $LT$ and $y_{c}$ becomes less and less significant whether dipole
moving parallel or perpendicular to the magnetic field. There is a maximum
value of $LT$ locating at $y_{c,max}$, when $y<y_{c,max}$, $LT$ is an
increasing function of $y_{c}$, but when $y>y_{c,max}$, $LT$ is a decreasing
function of $y_{c}$.

Figure 2:  LT versus $y_{c}$ when dipole moving parallel and perpendicular to
the magnetic fields for different temperatures and magnetic fields at fixed
moving rapidity($\beta=0.5$). Fig.2(a)and Fig.2(b) are for the case of dipole
moving parallel and perpendicular to the magnetic field, respectively. Solid
line is for $B=0.02\textrm{GeV}^{2}$ and dashed line is for
$B=0.15\textrm{GeV}^{2}$.

The maximum values of $LT$ ($LT_{\textrm{max}}$), which defines a dissociation
length for $Q\bar{Q}$Liu:2006nn ; Liu:2006he , as a function of magnetic field
$B$ has been studied in Fig. 3. Fig. 3 illustrates that increasing $B$ reduces
$LT_{\textrm{max}}$ for the moving $Q\bar{Q}$. It is found that the magnetic
field has a significant effect on the dissociation length for $Q\bar{Q}$ at
the low deconfined temperature $T_{c}$, however, with the increase of
temperature, the effect of magnetic field on dissociation length
$LT_{\textrm{max}}$ become less and less significant. When the temperature
reaches $0.33\textrm{GeV}$, $LT_{\textrm{max}}$ remains unchanged with the
increase of magnetic field. In this case, the dominant configuration for
$S_{NG}$ should be two straight strings running from the boundary to the
horizon. Moreover, the dissociation properties of heavy quarkonia should be
sensitive to the imaginary part of the potential.

Figure 3:  The dependencies of maximum value of $LT$ ($LT_{\textrm{max}}$) on
magnetic field $B$ with different temperatures for dipole moving parallel(a)
and perpendicular(b) to the magnetic field direction, respectively

Fig. 4(a, b) include comparisons of the real potential of $Q\bar{Q}$ pair
versus $LT$ for a dipole moving parallel and perpendicular to the magnetic
field, respectively. Generally speaking, the different of the effects of
dipole moving parallel, and perpendicular to magnetic fields on the
relationship between real potential and $LT$ is not very obvious. From
Fig.4(a, b), we find out that no matter how the magnetic field or temperature
changes, the real potential remains unchanged in the RHIC and LHC energy
regions.

Figure 4:  The real potential of quark-antiquark pair versus $LT$ with
different magnetic fields and temperatures for dipole moving parallel(a), and
perpendicular (b) to magnetic field, respectively

As shown in Fig. 5(a, b), The imaginary potential starts at a $L_{min}$ which
can be computed by solving $\textrm{Im}V_{Q\bar{Q}}=0$ and ends at a
$L_{max}$. We also find that at the low deconfined temperature
($T=T_{c}=0.15~{}\textrm{GeV}$) increasing the magnetic field leads to an
increase of the absolute value of the imaginary potential, and makes the
imaginary potential $\textrm{Im}V_{Q\bar{Q}}$ occurring at a smaller inter-
quark distance. When the temperature increases to $T=0.33~{}\textrm{GeV}$, the
effect of the magnetic field on $\textrm{Im}V_{Q\bar{Q}}$ becomes less and
less important. Comparing Fig. 5 (a) with Fig. 5 (b), we find that the
absolute value of the imaginary potential in the case of dipole moving
parallel to the magnetic field is apparently bigger than the case of dipole
moving perpendicular to the magnetic field.

Figure 5:  The imaginary potential $\textrm{Im}V_{Q\bar{Q}}$ of quark-
antiquark pair versus $LT$ with different magnetic field and temperature for
dipole moving parallel (a), and perpendicular (b) to the magnetic
field,respectively. Solid line is $B=0.02$ GeV 2, and dashed line is
$B=0.15$GeV2 .

Fig.6 shows the thermal width of the $\Upsilon(1S)$ with non-zero magnetic
fields at different finite temperatures for dipole moving parallel and
perpendicular to the magnetic field case. We find that with increasing
magnetic field leads to an significant increase of thermal width at the low
deconfined temperature ($T=0.15\textrm{GeV}$). But with the increase of
temperature, the effect of the magnetic field on thermal width becomes less
and less significant Comparing Fig. 6 (a) with Fig. 6 (b), we find that the
thermal width in the case of dipole moving parallel to the magnetic field is
apparently bigger than the case of dipole moving perpendicular to the magnetic
field. It is well known that the imaginary potential can be utilized to assess
the thermal width of heavy quarkonium. It is argued that a large thermal width
corresponds to a large dissociation length. Thus, the magnetic field has the
effect of increasing the dissociation length or increasing the thermal width
at the low deconfined temperature ($T=T_{c}=0.15~{}\textrm{GeV}$). However,
with the increase of temperature, the effect of magnetic field on thermal
width and the dissociation length will gradually weaken.

Figure 6:  The thermal width $\Gamma_{Q\bar{Q}}$ of $\Upsilon(1S)$ state
divided by the temperature $T$ versus $Ta_{0}$ for different magnetic fields
and temperatures for dipole moving parallel(a) and perpendicular(b) to the
magnetic field, respectively. The fixed rapidity is given as $\beta=$ 0.5.

Fig.7 shows the thermal width of the $\Upsilon(1S)$ with different rapidity
and temperatures for dipole moving parallel and perpendicular to the magnetic
field case, respectively. It is found that the thermal width decreases as the
increasing rapidity at a fixed temperature, which is in agreement with that
computed by Refs.Escobedo:2011ie ; Ali-Akbari:2014vpa . The thermal width of
$\Upsilon(1S)$ hardly changes with temperature at a fixed rapidity and
magnetic field in the case of dipole moving perpendicular to the magnetic
field as shown in Fig.7(b). But in the case of dipole moving parallel to the
magnetic field as shown in Fig.7(a), the thermal width at low deconfined
temperature ($T=0.15~{}\textrm{GeV}$) is obviously larger than that at high
temperature.

Figure 7:  The thermal width $\Gamma_{Q\bar{Q}}$ of $\Upsilon(1S)$ state
divided by the temperature $T$ versus $Ta_{0}$ for different rapidity and
temperature for dipole moving parallel(a) and perpendicular(b) to the magnetic
field, respectively. The fixed magnetic field is given as
$B=0.15\textrm{GeV}^{2}$.

##  VI SUMMARY AND CONCLUSION

The heavy-quark potential, whether it is a real potential or a imaginary
potential, both are very important quantity in gauge theories at finite
temperature. It also has great relevance in connection with experimental
programs in heavy ion collisions in the RHIC and LHC energies. The melting of
heavy quarkonia in a medium is considered to be one of the main experimental
signatures for QGP formation. Current analyses of available researches
indicate that the matter formed in such collisions is strongly coupled. Thus,
the study of the heavy quark potential requires strong-coupling techniques,
such as the AdS/CFT correspondence. There has been a lot of interests on the
heavy quarkonium suppression which has been observed in the RHICAdams:2005dq
and LHC Chatrchyan:2012lxa ; Abelev:2012rv . The suppression is a signal of
deconfinement and it was suggested that the bound states dissociate in the hot
thermal bath. It was proposed that the imaginary part of the potential
$\textrm{Im}V_{Q\bar{Q}}$ may be an important reason responsible for this
suppression rather than color screening.

By simulating the finite temperature and magnetic field in the RHIC and LHC
energy regions of relativistic heavy ion collisions, we restrict ourselves to
the range of temperature and magnetic field corresponding to RHIC and LHC
energy regions to study the potential and thermal width for dipole moving
parallel and pedicular to the magnetic field. It is found that the magnetic
field has less influence on the real potential, but has greater influence on
the imaginary potential. Extracting from the effect of thermal worldsheet
fluctuations about the classical configuration, we investigate the thermal
width of $\Upsilon(1s)$ in the finite temperature magnetized background. The
thermal width of $\Upsilon(1s)$ increases with the increasing magnetic field
at the low deconfined temperature ($T_{c}=0.15\textrm{GeV}$), but with the
increase of temperature ($T>T_{c}$), the thermal width hardly changes with the
increase of magnetic field, which means the effect of high temperature
completely exceeds that of magnetic field. The thermal width decreases with
the increasing rapidity at the finite temperature magnetized background.

It is also found that the effects of a magnetic field on the thermal width
when dipole moving parallel to the magnetic field direction are larger than
dipole moving perpendicular to the magnetic field direction, which implies
that the magnetic field tends to enhance thermal fluctuation when dipole
moving parallel to the magnetic field. The thermal width of $\Upsilon(1S)$
hardly changes with temperature at fixed rapidity and magnetic field in the
case of dipole moving perpendicular to the magnetic field. But in the case of
dipole moving parallel to the magnetic field, the thermal width at low
temperature ($T=0.15~{}\textrm{GeV}$) is obviously larger than that at high
temperature.

## Acknowledgments

This work was supported by National Natural Science Foundation of China
(Grants No. 11875178, No. 11475068, and No. 11747115) and the CCNU-QLPL
Innovation Fund (Grant No. QLPL2016P01).

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