# Impact of Medium Anisotropy on Quarkonium Dissociation and Regeneration

Captain R. Singh<EMAIL_ADDRESS>Department of Physics, Indian
Institute of Technology Indore, Simrol, Indore 453552, India  Mohammad Yousuf
Jamal  Corresponding Author<EMAIL_ADDRESS>School of Physical
Sciences, Indian Institute of Technology Goa, Ponda 403401, Goa, India
Raghunath Sahoo  Corresponding Author<EMAIL_ADDRESS>Department of
Physics, Indian Institute of Technology Indore, Simrol, Indore 453552, India

###### Abstract

Quarkonium production in ultra-relativistic collisions plays a crucial role in
probing the existence of hot QCD matter. This study explores quarkonia states
dissociation and regeneration in the hot QCD medium while considering momentum
anisotropy. The net quarkonia decay width ($\Gamma_{D}$) arises from two
essential processes: collisional damping and gluonic dissociation. The
quarkonia regeneration includes the transition from octet to singlet states
within the anisotropic medium. Our study utilizes a medium-modified potential
that incorporates anisotropy via particle distribution functions. This
modified potential gives rise to collisional damping for quarkonia due to the
surrounding medium, as well as the transition of quarkonia from singlet to
octet states due to interactions with gluons. Furthermore, we employ the
detailed balance approach to investigate the regeneration of quarkonia within
this medium. Our comprehensive analysis spans various temperature settings,
transverse momentum values, and anisotropic strengths. Notably, we find that,
in addition to medium temperatures and heavy quark transverse momentum,
anisotropy significantly influences the dissociation and regeneration of
various quarkonia states.

##  I Introduction

The relativistic heavy-ion collision (HIC) experiments at the Relativistic
Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC) offered an
opportunity to explore the properties of hot dense QCD matter known as the
Quark Gluon Plasma (QGP) expt_rhic1 ; expt_rhic2 ; expt_rhic3 ; expt_rhic4 ;
expt_lhc1 ; expt_lhc2 ; expt_lhc3 . The transient nature of this phase,
resulting from the confinement properties of Quantum Chromodynamics (QCD),
necessitates the study of discernible signatures and probes. One of the most
prominent and analyzed signatures is the suppression of heavy quarkonia
($q-\bar{q}$) due to their interactions with the QGP medium Matsui:1986dk ;
McLerran:1986zb . The pioneering work on the dissolution of various quarkonia
states, attributed to color screening within the finite-temperature QGP
medium, was first undertaken by Matsui and Satz Matsui:1986dk . Since then, a
substantial body of literature has contributed critical refinements to this
research pks1 ; Mocsy:2004bv ; Agotiya:2008ie ; Patra:2009qy ; Datta:2003ww ;
Brambilla:2008cx ; Burnier:2009yu ; Dumitru:2009fy ; Laine:2006ns ; crs1 .
Interestingly, an expected suppression of $J/\psi$ was observed at the Super
Proton Synchrotron (SPS) at CERN NA38:1989yxm . However, this suppression did
not exhibit a corresponding increase with higher beam energies at RHIC
STAR:2009irl . To address this puzzle, the concept of recombination of
$c-\bar{c}$ pairs into $J/\psi$ within the QGP medium was introduced.
Consequently, recombination processes play an important role in understanding
the observed number of $J/\psi$ particles in experiments Silvestre:2008tw ;
Scardina:2013nua .

In HIC, the quarkonia, characterized by their considerable mass, predominantly
emerge during the initial stages directly following the collisions. The
production and evolution of quarkonia can be categorized into two distinct
processes: (i) perturbative production: the generation of quark and anti-quark
($q{\bar{q}}$) bound state through nucleon-nucleon collisions constitutes the
perturbative aspect of quarkonium production bodwin1 . (ii) Non-perturbative
evolution: the subsequent formation and development of bound-state from these
$q{\bar{q}}$ pairs are governed by non-perturbative QCD phenomena. This dual
nature of quarkonia allows us to explore both perturbative and non-
perturbative QCD effects. Theoretical frameworks grounded in QCD, such as
potential non-relativistic QCD (pNRQCD) bodwin2 ; bodwin3  and fragmentation
approaches kang1 ; kang2 , are widely employed to investigate quarkonium
production.

It is reported that the stability of quarkonium states can be compromised by
various QGP effects such as color screening Matsui:1986dk ; crs1 , collisional
damping, and gluonic dissociation nendzig ; wols ; sharma ; crs2 , etc. The
collisional damping arises due to the complex potential between $q-{\bar{q}}$
pairs in the time limit, $t\rightarrow\infty$ represents the thermal decay
width laine . The gluonic dissociation occurs when quarkonia absorb soft
gluons ($E1$ gluons), leading to a transition from a color singlet state to a
color octet state—an unbound quark-antiquark pair brambilla ; peskin ; bhanot
. The mechanism governing quarkonia production in the color singlet state has
been explored in Ref. Adamczyk:2012ey ; Berger:1980ni ; Rapp:2008tf .
Furthermore, the regeneration of quarkonia within the QGP medium is also
possible and offers two primary avenues: (i) Uncorrelated $q-{\bar{q}}$
recombination: this process involves the recombination of uncorrelated
$q-{\bar{q}}$ pairs present within the medium pbm ; andronic ; rapp1 ; rapp2 ;
thews1 ; thews2 ; thews3 . It is particularly significant for charmonium
states like $J/\psi,\chi_{c},\psi^{{}^{\prime}}$, etc. at LHC energies due to
the relatively higher production of $c-\bar{c}$ pairs in HIC compared to
$b-\bar{b}$ pairs. (ii) Correlated $q-{\bar{q}}$ recombination: this process
reverses the gluonic dissociation, with correlated $q-{\bar{q}}$ pairs
transitioning from a color octet state to a color singlet state within the QGP
medium. Notably, regeneration of bottomonium is also possible through the de-
excitation of correlated $b-{\bar{b}}$ pairs crs2 ; crs4 .

Following the UMQS model crs1 ; crs2 ; crs3 ; crs4 ; crs5 , centered on the
formulation of gluonic dissociation and collisional damping within an
isotropic medium, the current study extends its scope to accommodate momentum
anisotropy. We aim to assess the decay width and regeneration for $J/\psi$ and
$\Upsilon$(1S) in the QGP medium while considering the presence of momentum
anisotropy. This work holds significant relevance since anisotropy persists
throughout all stages of off-central collisions, making its inclusion
indispensable. The study of anisotropy has been a subject of investigation in
various contexts Romatschke:2006bb ; Schenke:2006yp ; Mauricio:2007vz ;
Dumitru:2007rp ; Baier:2008js ; Dumitru:2007hy ; Carrington:2008sp . In this
article, we adopt an approach inspired by Refs. Romatschke:2003ms ;
Carrington:2014bla ; Jamal:2017dqs , where anisotropy is introduced at the
level of the distribution function by stretching and squeezing it along one
direction. Subsequently, we derive the medium-modified Cornell potential,
considering anisotropic dielectric permittivity. This modification renders the
potential a complex quantity consisting of both real and imaginary parts
Laine:2006ns ; Beraudo:2007ky ; Laine:2007qy ; hatsuda ; Margotta:2011ta ;
Strickland:2011aa ; Thakur:2013nia . Further, we observe the influence of
medium anisotropy and relativistic Doppler shift on the particle’s net decay
width ($\Gamma_{\rm D}$) and reaction reactivity ($\Gamma_{\rm F}$)
responsible for quarkonia regeneration. Transverse momentum ($p_{\rm T}$) and
temperature (T) dependent $\Gamma_{\rm D}$, $\Gamma_{\rm F}$ are obtained for
$J/\psi$, $\Upsilon$(1S) for anisotropic strength, $\zeta=0.2,0.4$ and T =
200, 300, 400 MeV. This study provides a comprehensive method to estimate the
net quarkonia suppression in ultra-relativistic collisions at the RHIC and LHC
energies.

The manuscript is structured as follows: Section II provides the formalism for
the dissociation and recombination of various quarkonia states in the presence
of momentum anisotropy. Section III is dedicated to presenting our results and
engaging in discussions. Finally, in Section IV, we summarize our findings and
contemplate future aspects of the current work. Throughout the text, we employ
natural units with $c=k_{B}=\hbar=1$. Three vectors are in bold typeface,
while four are in regular font. The center dot denotes the four-vector scalar
product, with the metric tensor specified as
$g_{\mu\nu}={\text{diag}}(1,-1,-1,-1)$.

##  II Formalism

The interaction between a ($q-{\bar{q}}$) pair is characterized by the vacuum
Cornell potential, which includes both a Coulombic and a string component
Eichten:1978tg ; Eichten:1979ms . This potential is expressed as:

$\displaystyle{\text{V}(r)}=-\frac{\alpha_{s}}{r}+\sigma r$ (1)

Here, $r$ represents the radius of the quarkonia state being studied,
$\alpha_{s}$ denotes the strong coupling constant, and $\sigma$ represents the
string tension. To account for medium effects, we modify this potential using
the dielectric permittivity, $\epsilon(k)$, which considers the presence of
momentum anisotropy. In Fourier space, this modification is formulated as:

$\displaystyle\grave{V}(k)=\frac{{\bar{\text{V}}}(k)}{\epsilon(k)}$ (2)

where ${\bar{\text{V}}}(k)$ is the Fourier transform of ${\text{V}(r)}$ and is
obtained as:

$\displaystyle{\bar{\text{V}}}(k)=-\sqrt{\frac{2}{\pi}}\bigg{(}\frac{\alpha_{s}}{k^{2}}+2\frac{\sigma}{k^{4}}\bigg{)}$
(3)

Here, the transition from the hadronic phase to the QGP is assumed to be a
crossover Rothkopf:2011db , ensuring that the string tension does not abruptly
disappear at or near the critical temperature $T_{c}$. The derivation of the
medium-modified potential in the presence of anisotropy has been derived by
some of us in Ref. Jamal:2018mog ; we briefly provide the necessary steps for
completeness in the upcoming sections.

###  II.1 Dielectric Permittivity in the Anisotropic QGP Medium

Anisotropy in this formalism is introduced through the medium’s particle
distribution functions, represented as $f_{g}(p)$ and $f_{q/{\bar{q}}}(p)$:

$\displaystyle f_{g}(p)=\frac{1}{e^{\beta
E_{p}}-1},~{}~{}~{}f_{q/{\bar{q}}}(p)=\frac{1}{e^{\beta E_{p}}+1}$ (4)

We adopt the method utilized by Romatschke et al. Romatschke:2003ms ;
Carrington:2014bla ; Jamal:2017dqs , where anisotropic distribution functions
are derived from isotropic ones through rescaling in one direction in momentum
space:

$\displaystyle f({\mathbf{p}})\rightarrow
f_{\zeta}({\mathbf{p}})=C_{\zeta}~{}f(\sqrt{{\bf p}^{2}+\zeta({\bf
p}\cdot{\bf\hat{n}})^{2}})$ (5)

Here, ${\bf{\hat{n}}}$ is a unit vector (${\bf\hat{n}}^{2}=1$) representing
the direction of anisotropy, and $\zeta$ quantifies the anisotropic strength
in the medium. It describes the degree of squeezing ($\zeta>0$, oblate form)
or stretching ($-1<\zeta<0$, prolate form) along the ${\bf\hat{n}}$ direction.
Normalizing the Debye mass $m_{D}$, defined as:

$\displaystyle m^{2}_{D}$ $\displaystyle=$ $\displaystyle
4\pi\alpha_{s}\bigg{(}-2N_{c}\int\frac{d^{3}p}{(2\pi)^{3}}\partial_{p}f_{g}(p)$
(6) $\displaystyle-$ $\displaystyle
N_{f}\int\frac{d^{3}p}{(2\pi)^{3}}\partial_{p}\left(f_{q}(p)+f_{\bar{q}}(p)\right)\bigg{)}$

In both isotropic and anisotropic media, yields the normalization constant
$C_{\zeta}$:

$\displaystyle
C_{\zeta}=\begin{cases}\frac{\sqrt{|\zeta|}}{\tanh^{-1}\sqrt{|\zeta|}}&\text{if
}~{}~{}-1\leq\zeta<0\\\ \frac{\sqrt{\zeta}}{\tan^{-1}\sqrt{\zeta}}&\text{if
}~{}~{}\zeta\geq 0\end{cases}$ (7)

In the limit of small $\zeta$, this expression approximates to:

$\displaystyle
C_{\zeta}=\begin{cases}1-\frac{\zeta}{3}+O\left(\zeta^{\frac{3}{2}}\right)&\text{if
}~{}~{}-1\leq\zeta<0\\\
1+\frac{\zeta}{3}+O\left(\zeta^{\frac{3}{2}}\right)&\text{if }~{}~{}\zeta\geq
0\end{cases}$ (8)

To calculate the dielectric permittivity for the anisotropic QGP medium, it is
essential to consider that perturbative theory at $T>0$ faces infrared
singularities and gauge-dependent results due to the incomplete nature of
perturbative expansion at these temperatures. However, HTL resummation
Braaten:1989mz , semi-classical transport theory, or many-particle kinetic
theory yield consistent results up to one-loop order. These methods lead to
the same expression for the gluon self-energy, $\Pi^{\mu\nu}$, which in turn
affects the medium’s dielectric permittivity Kumar:2017bja . Now, in the
static limit, the dielectric tensor $\epsilon^{-1}({\bf k})$ can be derived
from the temporal component of the propagator Agotiya:2016bqr ; Jamal:2018mog
:

$\displaystyle\epsilon^{-1}({\bf k})=-\lim_{\omega\to
0}k^{2}\Delta^{00}(\omega,{\bf k})$ (9)

The temporal components of the real and imaginary parts of the gluon
propagator are given as follows:

$\displaystyle\Re[\Delta^{00}({\omega=0,\bf k})]$ $\displaystyle=$
$\displaystyle\frac{-1}{k^{2}+m_{D}^{2}}-\zeta\Big{(}\frac{1}{3(k^{2}+m_{D}^{2})}$
(10) $\displaystyle-$
$\displaystyle\frac{m_{D}^{2}(3\cos{2\theta_{n}}-1)}{6(k^{2}+m_{D}^{2})^{2}}\Big{)}$

$\displaystyle\Im[\Delta^{00}({\omega=0,\bf k})]$ $\displaystyle=$
$\displaystyle\pi~{}T~{}m_{D}^{2}\bigg{(}\frac{-1}{k(k^{2}+m_{D}^{2})^{2}}$
(11) $\displaystyle+$
$\displaystyle\zeta\Big{(}\frac{-1}{3k(k^{2}+m_{D}^{2})^{2}}+\frac{3\sin^{2}{\theta_{n}}}{4k(k^{2}+m_{D}^{2})^{2}}$
$\displaystyle-$
$\displaystyle\frac{2m_{D}^{2}\big{(}3\sin^{2}({\theta_{n}})-1\big{)}}{3k(k^{2}+m_{D}^{2})^{3}}\Big{)}\bigg{)}$

These expressions contain direction and strength to account for the anisotropy
in the medium. Now, using Eq.10 and Eq.11 in Eq.9, respectively the real and
imaginary parts of $\epsilon({\bf k})$ can be obtained as,

$\displaystyle\Re[\epsilon^{-1}({\bf k})]$ $\displaystyle=$
$\displaystyle\frac{k^{2}}{k^{2}+m_{D}^{2}}+k^{2}\zeta\Big{(}\frac{1}{3(k^{2}+m_{D}^{2})}$
(12) $\displaystyle-$
$\displaystyle\frac{m_{D}^{2}(3\cos{2\theta_{n}}-1)}{6(k^{2}+m_{D}^{2})^{2}}\Big{)}$

and

$\displaystyle\Im[\epsilon^{-1}({\bf k})]$ $\displaystyle=$ $\displaystyle\pi
Tm_{D}^{2}\bigg{(}\frac{k^{2}}{k(k^{2}+m_{D}^{2})^{2}}-\zeta
k^{2}\Big{(}\frac{-1}{3k(k^{2}+m_{D}^{2})^{2}}$ $\displaystyle+$
$\displaystyle\frac{3\sin^{2}{\theta_{n}}}{4k(k^{2}+m_{D}^{2})^{2}}-\frac{2m_{D}^{2}\big{(}3\sin^{2}({\theta_{n}})-1\big{)}}{3k(k^{2}+m_{D}^{2})^{3}}\Big{)}\bigg{)}$

It is to note that at $T\rightarrow 0$, the real part $\Re[\epsilon(k)]$ goes
to unity, and the imaginary part $\Im[\epsilon(k)]$ vanishes, and we get back
the vacuum Cornell potential.

###  II.2 Modified Cornell potential in the anisotropic medium

The medium-modified Cornell potential can be obtained using $\epsilon(k)$ in
Eq. (2). The updated potential in the coordinate space can be found by doing
an inverse Fourier transform and is given as,

$\displaystyle
V(r)=\int\frac{d^{3}\mathbf{k}}{(2\pi)^{3/2}}(e^{i\mathbf{k}\cdot\mathbf{r}}-1)\grave{V}(k).$
(14)

Since the medium permittivity is a complex quantity, the updated potential
also contains the real and imaginary parts. Employing Eq. (12) in Eq. (2), the
real part of the potential is obtained as,

$\displaystyle\Re[V]$ $\displaystyle=$
$\displaystyle\alpha_{s}m_{D}\left(-\frac{e^{-s}}{s}-1\right)+\frac{\sigma}{m_{D}}\Big{(}\frac{2e^{-s}}{s}-\frac{2}{s}+2\Big{)}$
(15) $\displaystyle+$
$\displaystyle\alpha_{s}m_{D}\zeta\bigg{[}-\frac{3\cos(2\theta_{r})}{2s^{3}}-\frac{1}{2s^{3}}+\frac{1}{6}+e^{-s}\Big{\\{}\frac{1}{2s^{3}}$
$\displaystyle+$
$\displaystyle\frac{1}{2s^{2}}+\Big{(}\frac{3}{2s^{3}}+\frac{3}{2s^{2}}+\frac{3}{4s}+\frac{1}{4}\Big{)}\cos(2\theta_{r})+\frac{1}{4s}$
$\displaystyle-$
$\displaystyle\frac{1}{12}\Big{\\}}\bigg{]}+\frac{\zeta~{}\sigma}{m_{D}}\bigg{[}\left(\frac{6}{s^{3}}-\frac{1}{2s}\right)\cos(2\theta_{r})+\frac{2}{s^{3}}$
$\displaystyle-$
$\displaystyle\frac{5}{6s}+\frac{1}{3}+e^{-s}\Big{\\{}-\frac{2}{s^{3}}-\frac{2}{s^{2}}+\Big{(}-\frac{6}{s^{3}}-\frac{6}{s^{2}}$
$\displaystyle-$
$\displaystyle\frac{5}{2s}-\frac{1}{2}\Big{)}\cos(2\theta_{r})-\frac{1}{6s}+\frac{1}{6}\Big{\\}}\bigg{]},$

where $s=rm_{D}$. Assuming the limit, $s\ll 1$, Eq.(15) becomes,

$\displaystyle\Re[V]$ $\displaystyle=$
$\displaystyle\frac{s~{}\sigma}{m_{D}}\left(1+\frac{\zeta}{3}\right)-\frac{\alpha_{s}~{}m_{D}}{s}\bigg{(}1+\frac{s^{2}}{2}$
(16) $\displaystyle+$
$\displaystyle\zeta\left(\frac{1}{3}+\frac{s^{2}}{16}\left(\frac{1}{3}+\cos\left(2\theta_{r}\right)\right)\right)\bigg{)}$

Similarly, using Eq.(LABEL:eq:Im_eps), the imaginary part of the potential is
found to be

$\displaystyle\Im[V]$ $\displaystyle=$
$\displaystyle\frac{\alpha_{s}~{}T~{}m^{2}_{D}}{2~{}\pi}\int d^{3}{\bf
k}(e^{i\mathbf{k}\cdot\mathbf{r}}-1)\frac{1}{k}\bigg{[}\frac{-1}{(k^{2}+m_{D}^{2})^{2}}$
$\displaystyle+$
$\displaystyle\zeta[\frac{-1}{3(k^{2}+m_{D}^{2})^{2}}+\frac{3\sin^{2}\theta_{n}}{2(k^{2}+m_{D}^{2})^{2}}-\frac{4m_{D}^{2}(\sin^{2}\theta_{n}-\frac{1}{3})}{(k^{2}+m_{D}^{2})^{3}}]\bigg{]}$
$\displaystyle+$ $\displaystyle\frac{\sigma Tm^{2}_{D}}{\pi}\int
d^{3}\mathbf{k}(e^{i\mathbf{k}\cdot\mathbf{r}}-1)\frac{1}{k^{3}}\bigg{[}\frac{-1}{(k^{2}+m_{D}^{2})^{2}}$
$\displaystyle+$
$\displaystyle\zeta[\frac{-1}{3(k^{2}+m_{D}^{2})^{2}}+\frac{3\sin^{2}\theta_{n}}{2(k^{2}+m_{D}^{2})^{2}}-\frac{4m_{D}^{2}(\sin^{2}\theta_{n}-\frac{1}{3})}{(k^{2}+m_{D}^{2})^{3}}]\bigg{]}$

The analytical solutions of Eq.(LABEL:eq:im_v2) can be obtained in the limit
$s\ll 1$ as,

$\displaystyle\Im[V]$ $\displaystyle=$
$\displaystyle\frac{\alpha_{s}~{}s^{2}~{}T}{3}\Big{\\{}\frac{\zeta}{60}(7-9\cos
2\theta_{r})-1\Big{\\}}\log\left(\frac{1}{s}\right)$ (18) $\displaystyle+$
$\displaystyle\frac{s^{4}~{}\sigma~{}T}{m_{D}^{2}}\Big{\\{}\frac{\zeta}{35}\left(\frac{1}{9}-\frac{1}{4}\cos
2\theta_{r}\right)$ $\displaystyle-$
$\displaystyle\frac{1}{30}\Big{\\}}\log\left(\frac{1}{s}\right)$

Next, using the imaginary potential, we shall study the collisional damping.

###  II.3 Collisional damping

Collisional damping is an inherent characteristic of the complex potential,
and as mentioned above, quarkonia potential becomes complex within the medium;
the imaginary part of the potential gradually reduces the force fields that
bind $q-\bar{q}$ together. Consequently, quarkonia gets dissociated within the
QGP medium, and the decay width ($\Gamma_{d,nl}$) corresponding to collisional
damping can be obtained by performing a first-order perturbation calculation.
The calculation of $\Gamma_{d,nl}$ involves folding the imaginary part of the
potential (given in Eq. 18) with the radial wave function crs1 ;

$\displaystyle\Gamma_{d,nl}(\tau,p_{T})=\int
g_{nl}(r)^{\dagger}\Im[V]g_{nl}(r)\,dr$ (19)

Here, $g_{nl}(r)$ represents the singlet wave function corresponding to the
specific quarkonia state being investigated, where $n$ and $l$ are the quantum
numbers with their usual meanings. These wave functions are obtained by
solving the Schrödinger equation for various $q-\bar{q}$ states. As the
potential is anisotropic, the respective effect on the particle’s decay width
is observed further.

###  II.4 Gluonic Dissociation

The gluon-induced excitation from color singlet to color octet state is
defined as “Gluonic dissociation.” The cross-section corresponding to the
process is obtained as given by crs3 ;

$\sigma_{gd,nl}(E_{g})=\frac{\pi^{2}\alpha_{s}^{u}E_{g}}{N_{c}^{2}}\sqrt{\frac{{m_{Q}}}{E_{g}+E_{nl}}}\\\
\;\;\;\;\;\times\left(\frac{l|J_{nl}^{q,l-1}|^{2}+(l+1)|J_{nl}^{q,l+1}|^{2}}{2l+1}\right)$
(20)

where, $E_{nl}$ is energy eigenvalues corresponding to $g_{nl}(r)$,
$\alpha_{s}^{u}$ is coupling constant and $m_{Q}$ is the mass of the heavy
quark. The $J_{nl}^{ql^{{}^{\prime}}}$ is the probability density obtained by
using the singlet $g^{*}_{nl}(r)$ and octet $h_{ql^{\prime}}(r)$ wave
functions as follows,

$J_{nl}^{ql^{\prime}}=\int_{0}^{\infty}r\;g^{*}_{nl}(r)\;h_{ql^{\prime}}(r)dr\;$
(21)

Here, $h_{ql^{\prime}}(r)$ is the wave function of the color octet state of
quarkonia. It is obtained by solving the Schrödinger equation with the octet
potential $V_{8}=\alpha_{eff}/8r$. The value of $q$ here, is determined using
conservation of energy, $q=\sqrt{m_{Q}(E_{g}+E_{nl})}$.

Next, we obtained the mean of the gluonic dissociation cross-section to obtain
$\Gamma_{gd,nl}$ of a quarkonia state moving with speed $v$. To do so, we
thermal averaged over the modified Bose-Einstein distribution function for
gluons in the rest frame of quarkonia crs2 . Thus, gluonic dissociation is
found as,

$\Gamma_{gd,nl}(\tau,p_{T},b)=\frac{g_{d}}{4\pi^{2}}\int_{0}^{\infty}\int_{0}^{\pi}\frac{dp_{g}\,d\theta\,\sin\theta\,p_{g}^{2}\sigma_{gd,nl}(E_{g})}{e^{\\{\frac{\gamma
E_{g}}{T_{eff}}(1+v\cos\theta)\\}}-1}$ (22)

where $\gamma$ is a Lorentz factor and $\theta$ is the angle between $v$ and
incoming gluon with energy $E_{g}$. $p_{T}$ is the transverse momentum of the
quarkonia and $g_{d}=16$ is the number of gluonic degrees of freedom.

The quarkonia, being a massive particle, may not experience the same
temperature as the surrounding medium. The relativistic Doppler shift induced
by the relative velocity ($v_{r}$) between the medium and heavy meson causes
an angle-dependent effective temperature ($T_{eff}$), which is expressed as
detailed in Refs. Escobedo:2013tca ; crs2 . As a consequence, the Doppler
effect, a blue-shifted $T_{eff}$ is expected in the forward direction and a
red-shift in the backward direction. In the blue-shifted region $T_{eff}>T$,
however, the blues-shifted is confined to very smaller angles Escobedo:2013tca
. While the red-shifted region dominates over the blue-shifted at all relative
velocities $v_{r}>0$; i.e., $T_{eff}<T$ teff . As the red-shift grows with
increasing velocity, effective temperature decreases further for particles
with high $p_{T}$ values. The $T_{eff}$ is defined as;

$T_{eff}(\theta,|v_{r}|)=\frac{T\;\sqrt{1-|v_{r}|^{2}}}{1-|v_{r}|\;\cos\theta}$
(23)

here $\theta$ is the angle between $v_{r}$ and incoming light partons. To
calculate the relative velocity, $v_{r}$, we have taken medium velocity,
$v_{m}=0.7c$, and quarkonia velocity, $v_{nl}=p_{T}/E_{T}$. Here $p_{T}$ is
transverse momentum of quarkonia and $E_{T}=\sqrt{p_{T}^{2}+M_{nl}^{2}}$ is
its transverse energy, $M_{nl}$ is the mass of corresponding quarkonium state.
We have averaged Eq.(23) over the solid angle and obtained the average
effective temperature given by:

$T_{eff}(\tau,b,p_{T})=T(\tau,b)\;\frac{\sqrt{1-|v_{r}|^{2}}}{2\;|v_{r}|}\;\ln\Bigg{[}\;\frac{1+|v_{r}|}{1-|v_{r}|}\Bigg{]}$
(24)

After incorporating the $T_{eff}$ correction, we have obtained the total
dissociation decay width by taking the sum over the Eqs. (19) and (22)
correspond to the collisional damping and the gluonic dissociation,
respectively.

$\Gamma_{D,nl}=\Gamma_{d,nl}+\Gamma_{gd,nl}$ (25)

Next, we shall discuss the regeneration of the bound state in the presence of
the anisotropic QGP medium.

###  II.5 Regeneration of quarkonia in the anisotropic QGP medium

The regeneration via correlated $q-\bar{q}$ pairs is considered through the
de-excitation of the octet state to a singlet state via emitting a gluon. The
recombination cross-section $\sigma_{f,nl}$ is obtained from gluonic
dissociation cross-section $\sigma_{gd,nl}$ using the detailed balance
approach crs1 ,

$\sigma_{f,nl}=\frac{48}{36}\sigma_{gd,nl}\frac{(s-M_{nl}^{2})^{2}}{s(s-4\;m_{Q}^{2})}$
(26)

Here, $s$ is the Mandelstam variable, $s=({\bf p_{q}+p_{\bar{q}}})^{2}$, where
$\bf p_{q}$ and $\bf p_{\bar{q}}$ are four momentum of $q$ and $\bar{q}$,
respectively. The recombination factor, $\Gamma_{F,nl}$ is obtained by taking
the thermal average of the product of recombination cross-section and relative
velocity $v_{rel}$ between $q$ and $\bar{q}$ as thews3 :

$\Gamma_{F,nl}=<\sigma_{f,nl}\;v_{rel}>_{p_{q}}$ (27)

rewriting Eq.(27) as,

$\Gamma_{F,nl}=\frac{\int_{p_{q,min}}^{p_{q,max}}\int_{p_{\bar{q},min}}^{p_{\bar{q},max}}dp_{q}\;dp_{\bar{q}}\;p_{q}^{2}\;p_{\bar{q}}^{2}\;f_{q}\;f_{\bar{q}}\;\sigma_{f,nl}\;v_{rel}}{\int_{p_{q,min}}^{p_{q,max}}\int_{p_{\bar{q},min}}^{p_{\bar{q},max}}dp_{q}\;dp_{\bar{q}}\;p_{q}^{2}\;p_{\bar{q}}^{2}\;f_{q}\;f_{\bar{q}}}$
(28)

where, $p_{q}$ and $p_{\bar{q}}$ are respectively, $3$-momentum of heavy quark
and heavy anti-quark under investigation. The relative velocity of $q-\bar{q}$
pair in the medium is given as,

$v_{rel}=\sqrt{\frac{({\bf
p_{q}^{\mu}\;p_{\bar{q}\mu}})^{2}-m_{Q}^{4}}{p_{q}^{2}\;p_{\bar{q}}^{2}+m_{Q}^{2}(p_{q}^{2}+p_{\bar{q}}^{2}+m_{Q}^{2})}}$
(29)

The $f_{q,\bar{q}}$ is the modified Fermi-Dirac distribution function of heavy
quark and heavy anti-quark given as,

$\displaystyle
f_{q,\bar{q}}=\lambda_{q,\bar{q}}/(e^{E_{q,\bar{q}}/T_{eff}}+1)$ (30)

Here $E_{q,\bar{q}}=\sqrt{p_{q,\bar{q}}^{2}+m_{Q}^{2}}$ is the energy of heavy
quark and heavy anti-quark, in medium and $\lambda_{q,\bar{q}}$ is their
respective fugacity terms. Next, we shall discuss the obtained results.

##  III Results and Discussion

Figure 1: (Color Online) The net decay width, $\Gamma_{D}$ for $J/\psi$ as a
function of transverse momentum, $p_{T}$ and effective temperature, $T_{eff}$
is demonstrated for the anisotropic parameter, $\zeta$ = 0, 0.2, 0.4 and
medium temperature, T = 200, 300, 400 MeV.

Figure 2: (Color Online) The net decay width, $\Gamma_{D}$ for $\Upsilon$(1S)
as a function of transverse momentum, $p_{T}$ and effective temperature,
$T_{eff}$ is demonstrated for the anisotropic parameter, $\zeta$ = 0, 0.2, 0.4
and medium temperature, T = 200, 300, 400 MeV.

In this study, we rigorously investigate how the anisotropic properties of the
medium impact the decay width ($\Gamma_{D}$) and reaction reactivity
($\Gamma_{F}$) of $J/\Psi$ and $\Upsilon$(1S). We determine $\Gamma_{D}$ and
$\Gamma_{F}$ using the expressions presented in Eqs. (25) and (28),
respectively. We thoroughly analyze how these quantities depend on the degree
of medium anisotropy, $\zeta$. Additionally, we examine the relativistic
Doppler effect, which arises from particle motion and leads to changes in the
effective temperature of the particles in relation to the medium. We quantify
this effect using the parameter $T_{eff}$. Our results demonstrate the
dependence of $\Gamma_{D}$ on both $T_{eff}$ and transverse momentum
($p_{T}$). We omit the detailed discussion of the $T_{eff}$ dependence for
$\Gamma_{F}$, as it is difficult to show effective temperature for the quark-
antiquark pair as depending upon their velocity $q-\bar{q}$ may have different
values of $T_{eff}$. However, the effective temperature for $q-\bar{q}$ is
considered while calculating the $\Gamma_{F}$. Our analysis considers three
distinct temperature values, T = 200, 300, and 400 MeV, and two values for
anisotropic parameter $\zeta$ = 0.2 and 0.4. These specific temperature and
anisotropy choices are made to elucidate the qualitative impact of medium
anisotropy on the dissociation of quarkonia within the QGP medium.

Fig. 1 shows the change in $\Gamma_{D}$ for $J/\psi$ with respect to $p_{T}$
and $T_{eff}$, at different values of $T$ and $\zeta$. The $\Gamma_{D}$
increases with increasing $\zeta$ from 0 to 0.4 and further grows with respect
to the change in the medium temperature from 200 MeV to 400 MeV. At
low-$p_{T}$ and large $T_{eff}$, the impact of anisotropy on $\Gamma_{D}$ is
distinguishable, while with increasing $p_{T}$, this difference diminishes
($T_{eff}$ also decreases). Initially, at $p_{T}<3$ GeV, there is a slight
increase in the $\Gamma_{D}$ for all $\zeta$ values. This rising trend is a
consequence of the $T_{eff}$, as $T_{eff}$ increases up to $p_{T}<3$ GeV, and
after that, it decreases with increasing $p_{T}$, and as a result,
$\Gamma_{D}$ also decreases for $p_{T}>3$ GeV. It suggests that the particle
dissociation rate will be high if the particle temperature is large and
comparable to medium temperature. Quarkonia with high-$p_{T}$ quickly
traverses through the medium; therefore, high-$p_{T}$ $J/\psi$ feels less
heat, and survival of such particle becomes more probable.

The decay width of $\Upsilon$(1S) is shown in Fig. 2; it follows the same
reasoning as Fig. 1 except the difference in $\Gamma_{D}$ corresponding to
$\zeta$ values remains consistent throughout $p_{T}=30$ GeV at T = 200 and 300
MeV. While at T = 400 MeV, $\Gamma_{D}$ is almost constant up to
$T_{eff}\approx$360 MeV, and then a sudden change is observed, which further
decreases the $\Gamma_{D}$. This anomaly in decay width for $\Upsilon$(1S) at
T = 400 MeV comes because the change in the $T_{eff}$ is almost negligible up
to $p_{T}\sim 25$ GeV, and as a consequence, change in $\Gamma_{D}$ becomes
stagnant at $p_{T}<25$ GeV. It is to be noted that the difference in the
analyses of $J/\psi$ and $\Upsilon$(1S) relies on their masses. The
$\Upsilon$(1S), being a heavy particle, dissociates at higher temperatures
than $J/\psi$. Various articles have studied collisional damping from
different perspectives, including the anisotropic medium. However, in most of
these analyses, the aspect of gluonic dissociation is missing. Fig. 1 & 2
contain both the collisional damping as well as gluonic dissociation
considering the anisotropy in the QGP medium.

Figure 3: (Color Online) The recombination reactivity, $\Gamma_{F}$ for
$J/\psi$ is demonstrated as a function of transverse momentum, $p_{T}$ with
medium anisotropy parameter, $\zeta$ = 0, 0.2, 0.4 for T = 200, 300, 400 MeV.

Figure 4: (Color Online) The recombination reactivity, $\Gamma_{F}$ for
$\Upsilon$(1S) is demonstrated as a function of transverse momentum, $p_{T}$
with medium anisotropy parameter, $\zeta$ = 0, 0.2, 0.4 for T = 200, 300, 400
MeV.

In Fig. 3, the variation of recombination reactivity, $\Gamma_{F}$ of
$J/\Psi$, is shown with respect to $p_{T}$ at different anisotropic strength
and temperature values. It is found that $\Gamma_{F}$ has a non-trivial
dependence on the three i.e., $p_{T}$, $T$ and $\zeta$. Regeneration of
$J/\psi$ is almost independent of medium anisotropy at T = 200 MeV. Small
dependence on anisotropy can be observed in the intermediate $p_{T}$
($5<p_{T}<17$ GeV) at T = 300 MeV. It depicts that an anisotropic medium is
less favorable to $J/\psi$ regeneration than an isotropic medium. As
anisotropy induces or supports $J/\psi$ dissociation, it is obvious that it
would restrict the particle regeneration, though its impact is negligible.
Overall, regeneration increases with increasing $p_{T}$, except at mid-$p_{T}$
where particle interactions and large medium temperature constrain the
$J/\psi$ to form. While at $p_{T}>17$ GeV $c-\bar{c}$ pair feels a lower
temperature than the medium, and as they are moving at a high speed, they
easily traverse through the medium. Consequently, we get a linear increase in
regeneration reactivity with $p_{T}$. As we move to a very high temperature,
T= 400 MeV, the probability of recombination suppresses to a very narrow range
of $p_{T}$. Here, $\Gamma_{F}$ showed a non-monotonic behavior, as medium
temperature T = 400 MeV is larger than the dissociation temperature of
$J/\psi$ ($T_{D}\sim$350 MeV); therefore a large number of $c-\bar{c}$ pair
are expected, and as the interaction probability is large at mid-$p_{T}$, most
of the $c-\bar{c}$ recombine to form $J/\psi$. Hence, a maximum recombination
probability is attained at $p_{T}\sim 8-10$ GeV for the isotropic case. At
$p_{T}>15$ GeV for T = 400 MeV, the order of recombination becomes almost
equivalent to the T = 200 and 300 MeV, following a similar explanation as the
previous one corresponding to high-$p_{T}$. Moreover, regeneration due to un-
correlated $q-\bar{q}$ pair dominates at low-$p_{T}$ and decreases rapidly at
high-$p_{T}$. Thus, charmonia regeneration due to correlated $c-\bar{c}$ pair
is contrary to the regeneration due to un-correlated $q-\bar{q}$ pair.

The regeneration of $\Upsilon$(1S) is depicted in Fig. 4, there is a slight
increase in $\Gamma_{F}$ magnitude at T = 300 MeV than 200 MeV temperature.
However, it follows the same trend at both temperature values. The effect of
anisotropy on $\Upsilon$(1S) regeneration is visible, though it decreases with
increasing temperature. As shown in the figure, at T = 200 and 300 MeV,
$\Upsilon$(1S) regeneration is dominated for the isotropic medium, which is
equivalent to $J/\psi$ as shown in Fig. 3. Initially, $\Gamma_{F}$ increases
because of the abundance of $b-\bar{b}$ octet states at mid-$p_{T}$ range but
at high-$p_{T}$, this abundance decreases for low temperatures, and so does
the $\Gamma_{F}$. The physics of $\Upsilon$(1S) regeneration probability at T
= 400 MeV is as follows: since gluonic dissociation increases with the
increase in temperature, it produces a significant number of $b-\bar{b}$ octet
states for temperature more than $300$ MeV. Such that the de-excitation of
$b-\bar{b}$ octet states to $\Upsilon(1S)$ enhances the regeneration of
$\Upsilon(1S)$ at T = 400 MeV, which can be seen in Fig 4, where the value of
$\Gamma_{F}$ is higher at T = 400 MeV as compared with T = 300 MeV. It also
shows that high temperature produces enough $b-\bar{b}$ pair even at
high-$p_{T}$ and therefore, we get an increasing pattern in $\Upsilon$(1S)
regeneration with $p_{T}$. As gluonic excitation decreases at high-$p_{T}$, so
the de-excitation of $b-\bar{b}$ octet state into $\Upsilon(1S)$ becomes more
probable with increasing $p_{T}$ at T = 400. For isotropic case ($\zeta=0$),
$\Gamma_{F}$ decreases between $p_{T}$ 1 to 10 GeV, because as it is shown in
Fig. 2 the dissociation $\Upsilon$(1S) is more favorable at this $p_{T}$
range. Meanwhile, $\Gamma_{F}$ for $\Upsilon$(1S) always has a rising trend
for the anisotropic case at T = 400 MeV. It seems that at high temperature
where anisotropy increases the decay width but also favor the $\Upsilon$(1S)
to regenerate. Due to high dissociation temperature, $\Upsilon$(1S) can
regenerate even at T = 400 MeV or more, and therefore, $\Gamma_{F}$ for
$\Upsilon$(1S) is almost two times larger than $J/\psi$.

##  IV Summary

This research delves into the modification in quarkonia potential induced by
momentum anisotropy within the anisotropic hot QCD medium. This anisotropic
medium intricately influences the gluon distribution, leading to significant
changes in quarkonia dissociation and regeneration phenomena. We
comprehensively examine the interplay of dissociation mechanisms, including
gluonic dissociation and collisional damping, considering various strengths of
the medium anisotropy through the parameter, $\zeta$. Moreover, we
meticulously explore the role of the relativistic Doppler effect (RDE) on
quarkonia dissociation by considering the effective temperature, $T_{eff}$.

Our findings reveal distinct effects of medium anisotropy and RDE on the
dissociation and regeneration of $J/\psi$ and $\Upsilon$(1S). Notably,
$\Upsilon$(1S) dissociation is enhanced in the presence of anisotropy at all
temperature levels, while $J/\psi$ is more profoundly affected at higher
temperatures and lower transverse momentum ($p_{T}$) values. The influence of
anisotropy on the regeneration of $J/\psi$ is marginal, whereas $\Upsilon$(1S)
regeneration is particularly dependent on the degree of medium anisotropy. We
have observed that medium anisotropy increases and supports particle
dissociation by reducing its binding strength within the QGP medium as it
alters the particle’s potential. Therefore, if the medium is anisotropic, then
quarkonia states become more vulnerable and can have a higher decay width than
an isotropic medium.

The primary objective of this study was to elucidate the effects of medium
anisotropy at various temperatures and on various dissociation mechanisms.
These insights can serve as corrections for future investigations into the
suppression of bottomonium and charmonium in heavy-ion collisions at
facilities such as RHIC at Brookhaven National Laboratory (BNL) and LHC at the
European Organization for Nuclear Research (CERN). As the future upgrades at
the LHC focus on the measurement of heavy flavors (along with some others)
with increased low-$p_{T}$ reach and vertexing efficiencies close to the
interaction point; this study will help the theoretical model tuning to have a
better understanding of quarkonia production dynamics at the RHIC and LHC
energies.

##  V Acknowledgement

Captain R. Singh and Raghunath Sahoo gratefully acknowledge the DAE-DST,
Government of India funding under the mega-science project “Indian
Participation in the ALICE experiment at CERN” bearing Project No.
SR/MF/PS-02/2021-IITI (E-37123). MY Jamal would like to acknowledge the SERB-
NPDF (National postdoctoral fellow) File No. PDF/2022/001551.

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