# Heavy quarkonia in QGP medium in an arbitrary magnetic field

Jobin Sebastian<EMAIL_ADDRESS>School of Physical Sciences,
National Institute of Science Education and Research, An OCC of Homi Bhabha
National Institute, Jatni-752050, India  Lata Thakur<EMAIL_ADDRESS>Asia Pacific Center for Theoretical Physics, Pohang, Gyeongbuk 37673, Republic
of Korea  Department of Physics and Institute of Physics and Applied Physics,
Yonsei University, Seoul 03722, Korea  Hiranmaya Mishra<EMAIL_ADDRESS>School of Physical Sciences, National Institute of Science Education and
Research, An OCC of Homi Bhabha National Institute, Jatni-752050, India
Najmul Haque<EMAIL_ADDRESS>School of Physical Sciences, National
Institute of Science Education and Research, An OCC of Homi Bhabha National
Institute, Jatni-752050, India

###### Abstract

We compute the heavy quarkonium complex potential in an arbitrary magnetic
field strength generated in the relativistic heavy-ion collision. First, the
one-loop gluon polarization tensor is obtained in the presence of an external,
constant, and homogeneous magnetic field using the Schwinger proper time
formalism in Euclidean space. The gluon propagator is computed from the gluon
polarization tensor, and it is used to calculate the dielectric permittivity
in the presence of the magnetic field in the static limit. The modified
dielectric permittivity is then used to compute the heavy quarkonium complex
potential. We find that the heavy quarkonium complex potential is anisotropic
in nature, which depends on the angle between the quark-antiquark ($Q\bar{Q}$)
dipole axis and the direction of the magnetic field. We discuss the effect of
the magnetic field strength and the angular orientation of the dipole on the
heavy quarkonium potential. We discuss how the magnetic field influences the
thermal widths of quarkonium states. Further, we also discuss the limitation
of the strong-field approximation as done in literature in the light of heavy-
ion observables, as the effect of the magnetic field is very nominal to the
quarkonium potential.

Debye mass, heavy quarkonium, retarded potential, QGP, heavy-ion collision,
magnetic field in heavy-ion collision, thermal width.

##  I Introduction

The heavy quarkonia, which are the bound states of heavy quark and its anti-
quark ($c\bar{c},b\bar{b}$), are considered as one of the first proposed
signals to probe the deconfining properties of the strongly interacting matter
known as quark-gluon plasma (QGP) produced in the heavy ion collision. Matsui
and Satz [1] proposed several decades ago that heavy quarkonium production
would be suppressed due to the shrinking of the Debye sphere for color
interactions in the QGP medium. The medium in the presence of various non-
equilibrium effects requires the modification of phenomenological models,
which can be used to study the properties of the QGP medium. The presence of
these non-equilibrium effects such as momentum space anisotropy due to viscous
effects [2, 3, 4, 5, 6, 7, 8, 9, 10, 11], moving medium [12, 13, 14, 15, 16,
17] and magnetic field [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30,
31, 32, 33] can affect the screening phenomena, which results in the
modification of quarkonium properties.

In the past decade, the properties of strongly interacting matter have
attracted much interest in the presence of magnetic field backgrounds. The
non-central heavy ion collision experiments at RHIC and LHC can produce a very
strong magnetic field normal to the reaction plane [34, 35, 36, 37, 38], which
has motivated several interesting phenomenological studies. These studies led
to the determination of novel phenomena such as magnetic catalysis [39, 40,
41] and chiral magnetic effect [42, 43, 44, 45, 46, 47], splitting of open
charm directed flow [48, 49, 51, 52] and modification in properties of heavy
quarkonia and dynamics [53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65,
66, 67]. The potential models have been quite successful in describing the
quarkonium properties both in a vacuum as well as in medium [68, 69, 70, 71].
The quarkonium states are well described by the Cornell potential, which
contains both the perturbative Coulombic and non-perurbative confining terms
[72, 73]. The emergence of the imaginary part of the potential [74, 75, 76,
77, 78, 79, 80] has instigated the study of heavy quarkonium complex potential
[88, 53, 83, 3, 84, 85, 86, 87]. The real part of the quarkonium potential in
the magnetic field background has been studied by lattice QCD in the vacuum
and at finite temperature [89, 90]. To our knowledge, no lattice QCD study has
been conducted on the imaginary part of the heavy quarkonium potential.

In the present work, we aim to study the heavy quarkonium complex potential in
the presence of a general magnetic field without any restriction on its
strength. The heavy quarkonium potential has been studied previously in the
presence of weak and strong magnetic fields based on the assumption that the
potential exhibits isotropic behaviour [53, 55, 56]. Recently, the effect of a
general magnetic field on the modification of the imaginary part of the
potential has been computed in Ref. [54] by considering all Landau level
summations and the general structure of the gluon propagator in the magnetic
field background. In Ref. [54], the authors have shown that the imaginary part
of the potential exhibits anisotropic behavior, but they have not discussed
the real part. In this work, we compute the real and imaginary parts of the
complex heavy quarkonium potential in an arbitrary magnetic field. It would be
essential to study the effect of the magnetic field on the heavy quarkonium
complex potential by assuming the fact that the magnetic field generated in
the heavy ion collisions may not be weak or strong compared to the
temperature. Here, we employ the Schwinger proper time formalism to study the
effect of an arbitrary magnetic field on both the real as well as imaginary
parts of the potential. We also discuss the effect of an arbitrary magnetic
field on the thermal widths of heavy quarkonium states.

In the current computation, we obtain the in-medium heavy quarkonium complex
potential by modifying the Cornell potential with the dielectric permittivity
which encodes the effects of the magnetized thermal medium [2, 3, 91]. The
dielectric permittivity is computed using the in-medium gluon propagators. The
gluon propagator is obtained from the one-loop polarization tensor in
Euclidean space in an external constant and homogeneous magnetic field. The
computation is done using the Schwinger proper time formalism by considering
the full interaction between the quark and external field [92]. Using the
magnetic field-modified dielectric permittivity, we obtain the heavy
quarkonium complex potential in an arbitrary magnetic field. We find that the
potential obtained exhibits anisotropic behavior because of the quark loop
contribution to the gluon self-energy. We demonstrate the effects of a general
magnetic field on the complex heavy quarkonium potential and decay width
obtained using the imaginary part of the $Q\bar{Q}$ potential and check the
validity of strong field approximation in the potential model of quarkonia.

This paper is structured as follows. In section II, we revisit the derivation
of the gluon polarization tensor in the presence of an arbitrary magnetic
field. We obtain the gluon propagator in the static limit and compute the
dielectric permittivity. In section III, we compute the heavy quarkonium
complex potential and discuss the effect of the magnetic field on it. In
section IV, we estimate the quarkonium state’s thermal width and discuss how
the magnetic field affects them. In section V, the strong field approximation
of polarization tensor is computed, and the corresponding effect on the heavy
quarkonium potential is studied and compared with an arbitrary magnetic field
scenario. We summarize our results in section VI.

##  II Dielectric permittivity

In this section, we derive the dielectric permittivity in the presence of an
arbitrary magnetic field, which we use later to compute the in-medium heavy
quarkonium potential. First, we calculate the gluon self-energies and
propagators in an arbitrary magnetic field.

###  II.1 Gluon self-energy in an arbitrary magnetic field

In the following, we review the computation of the longitudinal component of
gluon self-energy in the one-loop order as followed in Ref. [92] and obtain
the longitudinal component of the gluon self-energy and propagator in the
static limit. Consider a charged particle of charge $q_{f}$ and mass $m$
moving in an external, constant, and homogeneous magnetic field, which is
directed along the z-direction (${\bf B}=B\hat{z}$). Here we choose the
symmetric gauge; therefore, we have

$\displaystyle{A_{0}}(x)=0,\,\,\,\,\,{A_{1}}(x)=-\frac{B}{2}y,$
$\displaystyle{A_{2}}(x)=\frac{B}{2}x,\,\,\,\,\,{A_{3}}(x)=0,$ (1)

In coordinate space, the fermion propagator, as introduced by Schwinger, is
given by [93]

$S(x,x^{\prime})=e^{iex^{\mu}A_{\mu}(x^{\prime})}\widetilde{S}(x-x^{\prime}),$
(2)

where the phase factor $e^{iex^{\mu}A_{\mu}(x^{\prime})}$ does not contribute
to the gluon self-energy with this particular choice of the gauge (1). The
Fourier transform $\widetilde{S}(k)$ of the translational invariant part of
fermion propagator $\widetilde{S}(x-x^{\prime})$ in the proper time formalism
is

$\displaystyle\widetilde{S}(k)$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}ds\,e^{is[k_{0}^{2}-k_{3}^{2}-k^{2}_{\perp}\tan(|q_{f}B|s)/|q_{f}B|s-m^{2}]}$
(3) $\displaystyle\times$
$\displaystyle\\{(k_{0}\gamma_{0}-k_{3}\gamma_{3}+m)[1+\gamma_{1}\gamma_{2}\tan(|q_{f}B|s)]$
$\displaystyle-$ $\displaystyle
k_{\perp}\gamma_{\perp}[1+\tan^{2}(|q_{f}B|s)]\\},$

where $k_{\perp}=(k_{1},k_{2})$ is the transverse momentum. For the finite
temperature case, we note that the bosonic Matsubara modes $\omega_{n}=2n\pi
T$ and fermionic ones $\hat{\omega}_{l}=(2l+1)\pi T$, respectively. The
fermion propagator, in Euclidean space $(k_{0}=i\hat{\omega}_{l})$ along with
$s\rightarrow-is$ as followed in Ref. [92], becomes

$\displaystyle\widetilde{S}_{l}(\bf{k})$ $\displaystyle=$
$\displaystyle-i\int_{0}^{\infty}ds\,e^{-s[\hat{\omega}_{l}^{2}+k_{3}^{2}+k^{2}_{\perp}\tanh(|q_{f}B|s)/|q_{f}B|s+m^{2}]}$
(4) $\displaystyle\times$
$\displaystyle\\{(-\hat{\omega}_{l}\gamma_{4}-k_{3}\gamma_{3}+m)[1-i\gamma_{1}\gamma_{2}\tanh(|q_{f}B|s)]$
$\displaystyle-$ $\displaystyle
k_{\perp}\gamma_{\perp}[1-\tanh^{2}(|q_{f}B|s)]\\},$

where ${\bf{k}}=(k_{\perp},k_{3})$ and $\gamma_{\mu}$ $(\mu=1,2,3,4)$ are the
Euclidean gamma matrices, which satisfy the anticommutation relation
$\\{\gamma_{\mu},\gamma_{\nu}\\}=-2\delta_{\mu\nu}$.

Based on the fermion propagator (4), one can obtain the quark-loop
contribution to the one-loop gluon self-energy in the presence of a magnetic
field as

$\displaystyle\Pi_{n}^{\mu\nu}({\bf{p}},B)$ $\displaystyle=$
$\displaystyle-g^{2}T\sum_{f}\int\frac{d^{3}{\bf
k}}{(2\pi)^{3}}\sum_{l=-\infty}^{\infty}\rm tr\\{\gamma^{\mu}S_{l}({\bf
k})\\}$ (5) $\displaystyle\times$
$\displaystyle\gamma^{\nu}S_{l-n}({\bf{k-p}})\\}+Q^{\mu\nu}(p),$

where $Q^{\mu\nu}(p)$ is the “contact” term, which cancels the ultraviolet
divergences and is independent of both the temperature and magnetic field.
Here, we are interested in the longitudinal component of the gluon
polarization tensor, which we will use further to compute the dielectric
permittivity and hence the in-medium heavy quarkonium potential. The
longitudinal component of the quark contribution to the one-loop gluon self-
energy is obtained after integration over ${\bf k}$ [92].

$\displaystyle\Pi^{44}_{n,q}(\omega_{n},{\bf{p}},B)$ $\displaystyle=$
$\displaystyle-\sum_{f}\frac{g^{2}T}{8\pi^{3/2}}q_{f}B\int\limits_{0}^{\infty}du\
u^{1/2}\int\limits_{-1}^{1}dv\sum_{l=-\infty}^{\infty}\exp\left[\frac{p_{\perp}^{2}}{q_{f}B}\frac{\cosh(q_{f}Buv)-\cosh(q_{f}Bu)}{2\sinh
q_{f}Bu}\right.$ (6)
$\displaystyle\left.-u\left\\{m^{2}+W_{l}^{2}+\frac{1}{4}(1-v^{2})(\omega_{n}^{2}+p_{3}^{2})\right\\}\right]\left[\frac{p_{\perp}^{2}}{2}\frac{\cosh(q_{f}Buv)-v\coth(q_{f}Bu)\,\sinh(q_{f}Buv)}{\sinh(q_{f}Bu)}\right.$
$\displaystyle\left.\hskip
28.45274pt-\frac{1}{u}\coth(q_{f}Bu)\Big{(}1-2\,u\,W_{l}^{2}+\frac{1}{2}u\,v\,\omega_{n}W_{l}-u(1-v^{2})p_{3}^{2}\Big{)}\right]+Q^{44}(p),$

where $W_{l}=\hat{\omega}_{l}-[(1-v)/2]\omega_{n}$. The contact term
$Q^{44}(p)$ is independent of temperature and magnetic field and hence can be
obtained as both $T$ and $B$ approach zero.

After using the Poisson resummation, one can isolate the temperature-
independent and temperature-dependent parts from the longitudinal polarisation
tensor [92]. As we aim to study the effect of the magnetic field on the heavy
quark-antiquark potential in the medium, we only discuss the temperature-
dependent part of the gluon self-energy. The temperature-dependent part of the
longitudinal polarisation tensor in the limit of massless quarks becomes

$\displaystyle\Pi^{44}_{n,q}(\omega_{n},{\bf p},B)$ $\displaystyle=$
$\displaystyle-\sum_{f}\frac{g^{2}}{(4\pi)^{2}}q_{f}B\int_{0}^{\infty}du\int_{-1}^{1}dv\sum_{l\geq
1}(-1)^{l}\exp\Big{[}-\frac{p_{\perp}^{2}}{q_{f}B}\frac{\cosh(q_{f}Bu)-\cosh(q_{f}Buv)}{2\sinh(q_{f}B\,u)}-\frac{1}{4}u\left(1-v^{2}\right)$
(7)
$\displaystyle\times\left(p_{3}^{2}+\omega_{n}^{2}\right)\Big{]}e^{-\frac{l^{2}}{4T^{2}u}}\left[\cos(\pi
ln(1-v))\left\\{p_{\perp}^{2}\frac{\cosh(q_{f}Buv)-v\coth(q_{f}Bu)\,\sinh(q_{f}Buv)}{\sinh(q_{f}Bu)}\right.\right.$
$\displaystyle+\text{p}_{3}^{2}\left(1-v^{2}\right)\coth(q_{f}Bu)\bigg{\\}}-\frac{\coth(q_{f}Bu)}{u}\left\\{\frac{l^{2}}{T^{2}u}\cos\pi
ln(1-v)-2\pi\,l\,n\,v\sin\pi ln(1-v)\right\\}\bigg{]},$

After evaluating the sum over $l$, the above equation (7) in the static limit
becomes

$\displaystyle\Pi_{q}^{44}(0,{\bf p},B)$ $\displaystyle=$
$\displaystyle\Pi_{q}^{L}({\bf p},B)$ (8) $\displaystyle=$
$\displaystyle\sum_{f}\frac{g^{2}}{32\pi^{2}}q_{f}B\int_{0}^{\infty}du\int_{-1}^{1}dv\exp\left[-\frac{1}{4}p_{3}^{2}u(1-v^{2})-\frac{2p_{\perp}^{2}}{q_{f}B}\frac{\cosh(q_{f}Bu)-\cosh(q_{f}Buv)}{\sinh(q_{f}Bu)}\right]$
$\displaystyle\times$ $\displaystyle\bigg{[}\coth q_{f}Bu\
\vartheta_{4}^{(0,1)}(0,e^{-\frac{1}{4T^{2}u}})+\left(1-\vartheta_{4}\left(0,e^{-\frac{1}{4T^{2}u}}\right)\right)$
$\displaystyle\hskip
28.45274pt\times\left\\{p_{\perp}^{2}\frac{\cosh(q_{f}Buv)}{\sinh(q_{f}Bu)}+\coth(q_{f}Bu)\left(p_{3}^{2}-p_{3}^{2}v^{2}-p_{\perp}^{2}v~{}\frac{\sinh(q_{f}Buv)}{\sinh(q_{f}Bu)}\right)\right\\}\bigg{]},$

where $\vartheta_{4}$ is the Jacobi Theta function and obtained as

$\sum_{l=1}^{\infty}(-1)^{l}e^{-al^{2}}=\frac{1}{2}\left[\vartheta_{4}(0,e^{-a})-1\right].$
(9)

In spherical polar coordinates, equation (8) becomes

$\displaystyle\Pi_{q}^{L}({\bf p},B)$ $\displaystyle=$
$\displaystyle\sum_{f}\frac{g^{2}q_{f}B}{32\pi^{2}}\int_{0}^{\infty}du\int_{-1}^{1}dv\exp\left[-\frac{1}{4}p^{2}\cos^{2}\theta\,u(1-v^{2})-\frac{2p^{2}\sin^{2}\theta}{q_{f}B\sinh(q_{f}Bu)}(\cosh(q_{f}Bu)-\cosh(q_{f}Buv))\right]$
(10) $\displaystyle\times$ $\displaystyle\left[\coth(q_{f}Bu)\
\vartheta_{4}^{(0,1)}(0,e^{-\frac{1}{4T^{2}u}})+\left(1-\vartheta_{4}(0,e^{-\frac{1}{4T^{2}u}})\right)\bigg{\\{}p^{2}\sin^{2}\theta\cosh(q_{f}Buv){\rm
csch}(q_{f}Bu)\right.$
$\displaystyle\left.+\coth(q_{f}Bu)\left(p^{2}\cos^{2}\theta(1-v^{2})-v\,p^{2}\sin^{2}\theta~{}\frac{\sinh(q_{f}Buv)}{\sin(q_{f}Bu)}\right)\bigg{\\}}\right].$

Here the coupling constant $g$ depends upon magnetic field i.e.
$g^{2}(\Lambda^{2},B)=4\pi\alpha_{s}(\Lambda^{2},B)$, where $\alpha_{s}$ is
the one-loop running coupling constant in the magnetic field background as
followed in [94, 31]

$\alpha_{s}(\Lambda^{2},|eB|)=\frac{\alpha_{s}(\Lambda^{2})}{1+b_{1}\alpha_{s}(\Lambda^{2})\ln\left(\frac{\Lambda^{2}}{\Lambda^{2}+|eB|}\right)},$
(11)

and the one-loop strong coupling in the absence of any magnetic field is

$\alpha_{s}(\Lambda^{2})=\frac{1}{b_{1}\ln\left(\frac{\Lambda^{2}}{\Lambda_{\overline{\rm
MS}}^{2}}\right)},$ (12)

where $b_{1}=\frac{(11N_{c}-2N_{f})}{12\pi}$ and $\Lambda_{\overline{\rm
MS}}=176$ MeV for $N_{f}=3$. Here we take $\Lambda$ for quarks as
$\Lambda_{q}=2\pi\sqrt{T^{2}+\mu^{2}/\pi^{2}}$ and for gluons as
$\Lambda_{g}=2\pi T$. We take the zero chemical potential $(\mu)$ here. The
quark loop contribution to the gluon self-energy, $\Pi_{q}^{L}({\bf p})$ for
$B=0$ case can be written as

$\displaystyle\Pi_{q}^{L}({\bf p})$ $\displaystyle=$
$\displaystyle-\frac{3\,g^{2}}{2\,\pi^{2}}\int_{0}^{\infty}\frac{k\,dk}{e^{k/T}+1}$
(13) $\displaystyle\times$
$\displaystyle\left[2+\frac{(p^{2}-4k^{2})}{2\,k\,p}{\rm\log}\left(\frac{p-2\,k}{p+2\,k}\right)\right].$

The magnetic field dependence only comes through the quark loop contribution
to the gluon self-energy, as gluons do not interact with the magnetic field.
Therefore, the gluon contribution to the self-energy remains the same as
without the magnetic field.

$\Pi_{g}^{L}(\omega,{\bf
p})=m_{Dg}^{2}\bigg{[}1-\frac{\omega}{2p}\ln\bigg{(}\frac{\omega+p}{\omega-p}\bigg{)}+i\pi\frac{\omega}{2p}\Theta(p^{2}-\omega^{2})\bigg{]},$
(14)

where $g^{\prime 2}=4\pi\alpha_{s}(\Lambda^{2})$ with
$\alpha_{s}(\Lambda^{2})$ defined in eq. (12).
The above equation (14) can be rewritten in terms of real and imaginary parts
as

$\displaystyle\Re\Pi_{g}^{L}(\omega,{\bf p})$ $\displaystyle=$ $\displaystyle
m_{Dg}^{2}\bigg{[}1-\frac{\omega}{2p}\ln\bigg{(}\frac{\omega+p}{\omega-p}\bigg{)}\bigg{]},$
$\displaystyle\Im\Pi_{g}^{L}(\omega,{\bf p})$ $\displaystyle=$ $\displaystyle
m_{Dg}^{2}\frac{\pi\omega}{2p}\Theta(p^{2}-\omega^{2}).$ (15)

The total longitudinal component of gluon self-energy is the sum of the gluon
and quark contribution

$\Pi^{L}(\omega_{n},{\bf p},B)=\Pi_{g}^{L}(\omega_{n},{\bf
p})+\Pi_{q}^{L}(\omega_{n},{\bf p},B),$ (16)

which can be written in terms of real and imaginary parts. We compute the
gluon self-energy’s real and imaginary parts in the static limit
($\omega\rightarrow 0$). The real part of self-energy reads

$\Re\Pi^{L}(\omega,{\bf p},B)=\Re\Pi_{g}^{L}(\omega,{\bf
p})+\Re\Pi_{q}^{L}(\omega,{\bf p},B).$ (17)

and the imaginary part of the self-energy $\Im\Pi^{L}$ reads

$\displaystyle\Im\Pi^{L}(\omega,{\bf p},B)=\Im\Pi_{g}^{L}(\omega,{\bf
p})+\Im\Pi_{q}^{L}(\omega,{\bf p},B).$ (18)

The imaginary contribution from the quark loop can be obtained by using the
identity

$\displaystyle\Im\Pi^{L}_{n,q}(\omega_{n},{\bf p})$ (19) $\displaystyle=$
$\displaystyle\frac{1}{2i}\lim\limits_{\varepsilon\rightarrow
0}\bigg{[}\Pi^{L}_{q}(\omega_{n}+i\varepsilon,{\bf
p})-\Pi^{L}_{q}(\omega_{n}-i\varepsilon,{\bf p})\bigg{]}$

Further, we compute both the real and imaginary part of the longitudinal
component of the gluon propagator using the gluon self-energy. The spectral
function approach, as defined in Ref. [96], is used to obtain the imaginary
part of the gluon propagator as

$\Im{D^{L}}(\omega,{\bf p})=-\pi(1+e^{-\beta\omega})\rm A^{L},$ (20)

where $\rm A^{L}$ is defined as

$\rm A^{L}(\omega,{\bf
p})=\frac{1}{\pi}\frac{e^{\beta\omega}}{e^{\beta\omega}-1}\rho^{L}(\omega,{\bf
p}).$ (21)

The spectral function $\rho^{L}$ can be expressed in the Breit-Wigner form as

$\displaystyle\rho^{L}(\omega_{n},{\bf p},B)$ (22) $\displaystyle=$
$\displaystyle\frac{\Im{\Pi^{L}}(\omega_{n},{\bf
p},B)}{(p^{2}-\Re{\Pi^{L}}(\omega_{n},{\bf
p},B))^{2}+{\Im\Pi^{L}(\omega_{n},{\bf p},B)}^{2}},~{}~{}~{}~{}~{}$

where $p=|{\bm{p}}|$. After substituting eq. (22) in eq. (20), we obtain the
longitudinal component of the gluon propagator, $D^{L}$ in terms of real and
imaginary parts. In the static $(\omega\rightarrow 0)$ and massless light
quark limit, we obtain

$D^{L}({\bf p},B)=\frac{-1}{p^{2}+\Pi^{L}({\bf p},B)}+\frac{i\pi
T\,\Pi^{L}({\bf p},B)}{p(p^{2}+{\Pi^{L}({\bf p},B)})^{2}}.$ (23)

Using the gluon propagator, we obtain the dielectric permittivity as [2, 3,
53]

$\epsilon^{-1}({\bf p},B)=\frac{p^{2}}{p^{2}+\Pi^{L}}-i\pi
T\frac{p\,\Pi^{L}}{(p^{2}+\Pi^{L})^{2}},$ (24)

where $\Pi^{L}\equiv\Pi^{L}({\bf p},B)$.

We use the dielectric permittivity expression (24) to compute the in-medium
heavy quarkonium complex potential in an arbitrary magnetic field.

##  III In-medium heavy quarkonium potential

In this section, we obtain the in-medium heavy quarkonium potential by using
the modified dielectric permittivity computed in the previous section. We
obtain the in-medium heavy quarkonium potential by correcting the Cornell
potential in Fourier space with the dielectric permittivity, which encodes the
effects of the magnetized thermal medium [91, 3, 12]

$\displaystyle
V({\bm{r},T,B})=\int\frac{d^{3}{\bm{p}}}{(2\pi)^{3/2}}~{}(e^{i{\bm{p\cdot
r}}}-1)~{}\frac{V_{\text{Cornell}}(p)}{\epsilon(p,B)},$ (25)

where $V_{\rm Cornell}(p)$ is the Fourier transform of the Cornell potential
$V_{\rm Cornell}(r)=-\alpha/r+\sigma\,r$, which is given by

$V_{\rm
Cornell}(p)=\sqrt{\frac{2}{\pi}}\frac{\alpha}{p^{2}}-2\sqrt{\frac{2}{\pi}}\frac{\sigma}{p^{4}},$
(26)

where $\alpha$ and $\sigma$ are the strong coupling constant and the string
tension, respectively.

Figure 1: The complex potential is plotted as a function of quark-antiquark
separation $r$ for $\Theta=0$ and and $T=170$ MeV. The left panel shows the
variation of the real part of the potential. The right panel shows the
variation of the imaginary part of the potential.

Figure 2: The complex potential is plotted as a function of the strength of
the magnetic field for different values of $\Theta$ when $r=1$ fm and $T=200$
MeV. The left panel shows the variation of the real part of the potential. The
right panel shows the variation of the imaginary part of the potential.

Here we take $\alpha=4/3\,\alpha_{s}(\Lambda^{2},B)$ and $\sigma=0.18$ GeV2
and $\epsilon(p)$ is the dielectric permittivity, which is defined in equation
(24). After substituting equations (24) and (26) in eq. (25), we obtain both
the real as well as imaginary part of the potential, which contains both the
perturbative Coulombic and non-perturbative string terms. The real part of the
potential can be written in terms of Coulombic and string terms as

$\displaystyle\Re V({\bm{r},T,B})=\Re V_{c}({\bm{r},T,B})+\Re
V_{\sigma}({\bm{r},T,B}),$ (27)

where the Coulombic term is

$\displaystyle\Re V_{c}(\bm{r},T,B)$ $\displaystyle=$
$\displaystyle-\frac{\alpha}{2\pi^{2}}\int d^{3}{\bm{p}}$ (28)
$\displaystyle\times$ $\displaystyle\Bigg{[}\frac{e^{i\boldsymbol{p\cdot
r}}}{p^{2}+\Pi^{L}}-\frac{\Pi^{L}}{p^{2}(p^{2}+\Pi^{L})}\Bigg{]},~{}~{}~{}~{}~{}$

and the string term reads

$\displaystyle\Re
V_{\sigma}(\bm{r},T,B)=-\frac{\sigma}{\pi^{2}}\int\\!\\!\frac{d^{3}{\bm{p}}}{(2\pi)^{3}}(e^{i{\bm{p}}\cdot{\bm{r}}}-1)\frac{1}{p^{2}(p^{2}+\Pi^{L})}.\hskip
18.49411pt$ (29)

Here
${\bm{p}}\cdot{\bm{r}}=rp\big{[}\sin\theta\sin\Theta\cos(\phi-\Phi)+\cos\theta\cos\Theta\big{]}$
and the angles $\theta(\Theta)$ and $\phi(\Phi)$ are polar and azimuthal
angles in momentum (coordinate) space, respectively. After integrating over
the azimuthal angle, we obtain

$\displaystyle\Re
V({r,T,B,\Theta})=-\frac{1}{\pi}\int\frac{\sin\theta\,d\theta\,dp}{p^{2}+\Pi^{L}}\Big{[}(\alpha\,\Pi^{L}-2\,\sigma)$
$\displaystyle\times(\alpha\,p^{2}+2\,\sigma)e^{ipr\cos\theta\cos\Theta}J_{0}(p\hskip
0.85358ptr\hskip 0.85358pt\sin\theta\sin\Theta)\Big{]},$ (30)

where $J_{0}$ is the Bessel’s function of the first kind.

Similarly, we can compute the imaginary part of the quarkonium potential. The
imaginary part of the potential is given by

$\displaystyle\Im
V({\bm{r},T,B})=T\int\frac{d^{3}{\bm{p}}}{2\pi}\frac{(e^{i\bm{p\cdot
r}}-1)\Pi^{L}p}{\left(p^{2}+\Pi^{L}\right)^{2}}\left[\frac{\alpha}{p^{2}}+\frac{2\sigma}{p^{4}}\right].~{}~{}~{}~{}$
(31)

After integrating over the azimuthal angle, we obtain

$\displaystyle\Im
V({r,\Theta,T,B})=-T\int\frac{\sin\theta\,d\theta\,dp}{\left(p^{2}+\Pi^{L}\right)^{2}}\Pi^{L}\left[\alpha\hskip
0.85358ptp+\frac{2\sigma}{p}\right]$
$\displaystyle\times\Big{\\{}1-e^{ipr\cos\theta\cos\Theta}J_{0}(pr\sin\theta\sin\Theta)\Big{\\}}.\,\,\,\,\,\,\,\,$
(32)

Figure 3:  The thermal widths of bottomonium and charmonium states as a
function of temperature at $B=0$ and $15m_{\pi}^{2}$ are shown.

We numerically solve the real (30) and imaginary (32) parts of the potential.
In figure 1, we plot the real (left) and imaginary (right) part of the
potential for different strengths of the magnetic field $B$ at $\Theta=0$. We
find that the real part of the potential becomes flattened with the magnetic
field due to an increase in screening with $B$. The imaginary part of the
potential shows different behavior at smaller and larger $r$; it increases
with the magnetic field at smaller $r$ and decreases in magnitude with the
increase in the magnetic field at larger $r$. Note that the magnetic field
dependence is nonsignificant for the potential, especially in the range $B=0$
to $B=15m_{\pi}^{2}$

Figure 2 shows the real and imaginary part of the potential as a function of
magnetic field for different values of $\Theta$ at $r=1$ fm. We observe that
the real part of the potential varies in response to a magnetic field at
different rates according to direction. The magnetic field dependence is found
to be maximum in $\Theta=\pi/2$ direction and minimum along the direction of
the magnetic field, which establishes the anisotropy of the potential in the
magnetic field. The magnitude of the imaginary part of the potential initially
increases when the magnetic field increases, and diminishes as the magnetic
field intensifies. Both components of the potential depend on both the
magnitude of the magnetic field and the angle, however, this dependence is
minimal.

The imaginary part increases in magnitude with the increase in the magnetic
field initially, and the magnitude decreases as the magnetic field increases.
Both parts of the potential depend on the magnetic field and the angle, but
the dependence is negligible.

In the next section, we use the imaginary part of the potential to obtain the
thermal widths of the quarkonium states.

##  IV Thermal Width

Figure 4: Thermal widths of bottomonium (left) and charmonium (right) states
as a function of magnetic field for $T=170$ MeV and $250$ MeV.

In this section, we compute the thermal widths of the quarkonium states. The
imaginary part of the potential estimates the thermal width,
$\Gamma_{Q\bar{Q}}$, when treated as a perturbation of the vacuum potential.
We compute the decay width of the quarkonium states as [3, 88]

$\Gamma_{Q\bar{Q}}(T,B)=-\langle\psi(r)|\Im\,V_{Q\bar{Q}}(r,T,B,\Theta)|\psi(r)\rangle,$
(33)

where $\psi(r)$ is the unperturbed Coulombic wave function. As the leading
contribution to the imaginary potential for a deeply heavy-quark bound state
is Coulombic, which justifies the use of Coulomb wave functions to calculate
the thermal width. The wave function for the ground and excited states is
given by

$\displaystyle\psi_{1s}({r})$ $\displaystyle=$ $\displaystyle\frac{1}{(\pi
a_{0}^{3})^{1/2}}e^{-r/a_{0}},$ $\displaystyle\psi_{2s}({r})$ $\displaystyle=$
$\displaystyle\frac{1}{4(2\pi
a_{0}^{3})^{1/2}}\left(2-\frac{r}{a_{0}}\right)e^{-r/2a_{0}},$ (34)

where $a_{0}=2/C_{F}m_{Q}\alpha_{s}$ is the Bohr radius of the $Q\bar{Q}$
system and $m_{Q}$ is the heavy quark mass. After substituting eqs. (34) and
(32 ) in Eq. (33), we obtain the thermal width of the quarkonium states for
the ground state as

$\Gamma_{1s}(T,B)=-\frac{1}{\pi a_{0}^{3}}\int d^{3}{\bf r}\,e^{-2r/a_{0}}\Im
V\left(r,T,B,\Theta\right),\,$ (35)

$\displaystyle\Gamma_{1s}(T,B)$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\frac{2T}{a_{0}^{3}}\int dr\,d\Theta\,r^{2}\sin\Theta
e^{-2r/a_{0}}\int\frac{\sin\theta\,d\theta\,dp}{\left(p^{2}+\Pi^{L}\right)^{2}}\Pi^{L}\left[\alpha\hskip
0.85358ptp+\frac{2\sigma}{p}\right],\hskip 17.07182pt$ (36)

and for the first excited states ($2S$) as

$\displaystyle\Gamma_{2s}(T,B)=\frac{T}{16a_{0}^{3}}\int
dr\,d\Theta\,r^{2}\sin\Theta\left(2-\frac{r}{a_{0}}\right)^{2}e^{-r/a_{0}}\int\frac{\sin\theta\,d\theta\,dp}{\left(p^{2}+\Pi^{L}\right)^{2}}\Pi^{L}\left[\alpha\hskip
0.85358ptp+\frac{2\sigma}{p}\right].\hskip 17.07182pt$ (37)

We numerically compute the thermal widths of the ground (36) and excited (37)
states of the bottomonium and charmonium states. In Figure 3, we present plots
of the thermal widths of the ground and first excited states of bottomonium
(on the left) and charmonium (on the right) as a function of temperature for
both $B=0$ and $B=15m_{\pi}^{2}$.

It is observed that the thermal width increases with an increase in
temperature, as anticipated. The magnetic field effect is more on the first
excited state than the ground state for both the bottomonium and charmonium
states. At larger $r$, the effect of the magnetic field on the imaginary part
of the potential is more due to the larger size of the excited states. Hence
the excited states are more sensitive to the magnetic field than the ground
state. Figure 4 shows the variation of thermal width with the magnetic field
at different temperatures for bottomonium (left) and charmonium (right)
states. We find that the thermal widths are more sensitive to the magnetic
field at lower temperatures than the higher temperatures. The magnetic field
effects decrease with the increase in heavy quark mass and decrease in the
size of bound states. It can be concluded from the figures that the magnetic
field has only a negligible effect on the thermal width compared to the
temperature.

##  V Strong field approximation

Figure 5: The real part of the potential as a function of $r$ is shown on the
left side for an arbitrary (black solid) and strong magnetic field
approximation (red dashed) at $T=170$ MeV and $\Theta=0$. The right side
illustrates the same potential as a function of $eB$ at these conditions.

In this section, we compute the longitudinal component of the gluon self-
energy in the strong magnetic field approximation ($T\ll\sqrt{B}$). In the
strong magnetic field limit ($|B|\rightarrow\infty$), the fermion propagator
(eq. 4) for massless case becomes

$\displaystyle\widetilde{S}_{l}(\bf{k})$ $\displaystyle=$
$\displaystyle-i\int_{0}^{\infty}ds\,e^{-s[\hat{\omega}_{l}^{2}+k_{3}^{2}+k^{2}_{\perp}/|q_{f}B|s]}$
(38) $\displaystyle\times$
$\displaystyle(-\hat{\omega}_{l}\gamma_{4}-k_{3}\gamma_{3})[1-i\gamma_{1}\gamma_{2}]$
$\displaystyle=$ $\displaystyle
ie^{-k^{2}_{\perp}/|q_{f}B|}\frac{\hat{\omega}_{l}\gamma_{4}+k_{3}\gamma_{3}}{\hat{\omega}_{l}^{2}+k_{3}^{2}}[1-i\gamma_{1}\gamma_{2}],$

which is similar to the expression for the fermion propagator computed for the
lowest Landau level approximation in Ref. [97, 30]. In $|B|\rightarrow\infty$
limit, the temperature dependent part of the longitudinal component of the
gluon self-energy (7) reduces to the dominant term as

$\displaystyle\Pi_{q}^{L}({\omega_{n},\bf p},B)$ $\displaystyle=$
$\displaystyle-\sum_{f}\frac{g^{2}}{(4\pi)^{2}}\frac{(q_{f}B)^{2}}{T^{2}}e^{-p_{\perp}^{2}/2q_{f}B}\int_{0}^{\infty}\frac{du}{u^{2}}$
(39) $\displaystyle\times$ $\displaystyle\int_{-1}^{1}dv\,e^{-cu}\sum_{l\geq
1}(-1)^{(l+1)}l^{2}e^{-d/u}$ $\displaystyle\times$ $\displaystyle\cos\pi
ln(1-v),$

where $c=(1-v^{2})(p_{3}^{2}+\omega_{n}^{2})/(4\,q_{f}B)$ and
$d=l^{2}q_{f}B/4T^{2}$. The integration over $u$ can be done analytically
using the relation

$\int_{0}^{\infty}\frac{du}{u^{2}}e^{-cu-d/u}=2\sqrt{\frac{c}{d}}K_{1}(2\sqrt{c\,d}).$
(40)

Here $K_{n}(z)$ represents the modified Bessel function of the second kind.
Therefore, the longitudinal component of the gluon self-energy (39) for the
strong magnetic field approximation in the static limit
($\omega_{n}\rightarrow 0$) becomes

$\displaystyle\Pi_{q}^{L}({\bf p},B)$ $\displaystyle=$
$\displaystyle-\sum_{f}\frac{g^{2}q_{f}B}{8\pi^{2}T}e^{-p_{\perp}^{2}/2q_{f}B}\,\sum_{l\geq
1}(-1)^{(l+1)}$ (41) $\displaystyle\times\int\limits_{-1}^{1}dv\,l\
p_{3}\sqrt{1-v^{2}}K_{1}\left(\frac{l\,p_{3}}{2T}\sqrt{1-v^{2}}\right).$

Further, we compute the Debye screening mass from eq. (7) in the limit
$p\rightarrow 0$, for the regime where $T\ll\sqrt{eB}$

$\displaystyle m_{D}^{2}(T,B)$ $\displaystyle=$ $\displaystyle-\lim_{{\bf
p}\to 0}\Pi^{L}_{q}({\bf p},B)$ (42) $\displaystyle=$
$\displaystyle\sum_{f}\frac{g^{2}q_{f}B}{8\pi^{2}T^{2}}\int_{0}^{\infty}\frac{du}{u^{2}}\coth
q_{f}B\,u\sum_{l\geq 1}(-1)^{l+1}l^{2}$ $\displaystyle\vspace{0.5cm}\ \times\
e^{-l^{2}/4uT^{2}}$ $\displaystyle=$
$\displaystyle\sum_{f}\frac{g^{2}q_{f}B}{4\pi^{2}}.$

In the absence of a magnetic field, the Debye screening mass becomes

$\displaystyle m_{D}^{2}(T)$ $\displaystyle=$
$\displaystyle\frac{1}{3}C_{A}g^{2}T^{2}+\sum_{f}\frac{g^{2}}{8\pi^{2}T^{2}}\int_{0}^{\infty}\frac{du}{u^{3}}$
(43) $\displaystyle\times\sum_{l\geq 1}(-1)^{l+1}l^{2}e^{-l^{2}/4uT^{2}}$
$\displaystyle=$
$\displaystyle\frac{1}{3}C_{A}g^{2}T^{2}+\frac{1}{6}N_{f}g^{2}T^{2}.$

The longitudinal component of gluon self-energy in strong field approximation
obtained in eq. (41) can be substituted in eq. (25) to study the behavior of
quarkonium potential in the strong field approximation. In figure 5, we show
the effect of arbitrary magnetic fields and the strong field approximation on
the real part of the potential. We find that the real part of the potential is
more suppressed for the case of an arbitrary magnetic field as compared to
strong field approximation due to larger screening in an arbitrary $B$. We can
see that the potential with approximation differs in large values from the
exact potential for any realistic magnetic field magnitude, and the difference
gradually reduces as the magnetic field increases. Hence we can say that the
strong field approximation is invalid, and one should consider the general
case while studying the properties of quarkonium states.

##  VI Summary

In this current work, we have evaluated the influence of a magnetic field on
the heavy quarkonium complex potential. We initially computed the dielectric
permittivity from the static limit of the gluon propagator. This propagator
was derived from the one-loop gluon self-energy in the presence of an external
magnetic field, which was evaluated using Schwinger’s proper time formalism in
Euclidean space. The effect of the magnetic field enters through the quark-
loop contribution to the gluon self-energy and coupling constant. Then, we
computed the in-medium heavy quarkonium complex potential using the modified
dielectric permittivity. Results showed that this potential is anisotropic and
varies with magnetic field strength and angle $\Theta$ between quark-antiquark
axis and direction of magnetic field.

At very high magnetic field strengths, the real part of the potential is
flattened due to an increase in screening with $B$. Conversely, the imaginary
part of the quarkonium potential experiences a rise in magnitude at short
distances, followed by a decrease at long distances. Furthermore, the
inclusion of a magnetic field introduces an angular dependence into the
potential. Finally, we observed that its overall effect of the magnetic field
on the complex potential is minimal.

We computed the decay widths of the ground and first excited states of
bottomonium ($\Upsilon,\Upsilon^{\prime}$) and charmonium
($J/\psi,\psi^{\prime}$) using the imaginary part of the potential. We found
that the excited states $(\Upsilon^{\prime},\psi^{\prime})$ are more sensitive
to the magnetic field than the ground states $(\Upsilon,J/\psi)$. The effect
of magnetic fields decreases with increasing heavy quark mass and decreasing
size, making charmonium states more sensitive to $B$ than bottomonium states.
As temperature increases, sensitivity to magnetic fields decreases, eventually
disappearing at high temperatures.

We have compared our results with the strong-field approximated potential. We
found that the potential doesn’t even come close to any realistic magnetic
field value generated in the heavy ion collisions. The approximation makes the
screening much weaker but gradually increases as the magnetic field increases.
As done in previous works, we invalidate the strong magnetic field
approximation in studying quarkonium complex potential; one needs to consider
the general magnetic field. Moreover, the weak-field expansion introduces new
divergences in the gluon propagators, and one needs a way to regulate it.
Hence, it is essential to study the effect of arbitrary magnetic fields on the
heavy quarkonium complex potential and the properties of quarkonium states.

In the future, we would like to extend our computation for the moving medium
in the presence of an arbitrary magnetic field.

##  VII Acknowledgment

J. S. would like to thank R. Ghosh for the useful discussions. L. T. is
supported by the Korean Ministry of Education, Science and Technology,
Gyeongsangbuk-do and Pohang City at the APCTP and National Research Foundation
(NRF) funded by the Ministry of Science of Korea (Grant No. 2021R1F1A1061387).
N.H. is supported in part by the SERB-MATRICS under Grant No. MTR/2021/000939.

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