# Quarkonium dissociation properties of hot QCD medium at momentum-anisotropy
in the N-dimensional space using Quasi-particle Debye mass with finite
baryonic chemical potential

Vineet Kumar Agotiyaa<EMAIL_ADDRESS>Siddhartha Solankia
<EMAIL_ADDRESS>Manohar Lala<EMAIL_ADDRESS>aDepartment of Physics, Central University of Jharkhand Ranchi, India, 835 222

###### Abstract

The analytical exact iteration method (AEIM) have been used to calculate the
$N$-dimensional radial schrodinger equation with the real part of complex-
valued potential and it is generalized to the finite value of anisotropy
($\xi$), temperature and baryonic chemical potential. In $N$-dimensional space
the energy eigen values have been calculated for any states (n,l). The present
results have been used to study the properties of quarkonium states (i.e, the
binding energy and mass spectra in the $N$-dimensional space). The influences
of anisotropy ($\xi$) on the dissociation temperature($T_{D}$) has been also
calculated for the ground state of quarkonia at N=$0$ and $\mu=300MeV$. The
anisotropy in oblate case, thus obtained, makes the dissociation
temperature($T_{D}$) lower as compared to the isotropic case . We also seen
that Leading order Debye mass is greater than that of Quasi-particle Debye
mass after introducing the baryonic chemical potential.
PACS: 25.75.-q; 24.85.+p; 12.38.Mh
Keywords : Schrodinger equation, real part of complex-valued potential, Debye
mass, Momentum anisotropy, Heavy quarkonia, dimensionality number and baryonic
chemical potential.

##  I Introduction

The study of the system of heavy quarkonia (such as Bottomonium and
Charmonium) have played an important role for the standard model S.M.Kuchin ,
it also useful for the quantifiable test of Quantum Chromo-Dynamics (QCD),
hence, is a topic of resurgent interest for both the theoretical and
experimental high energy physicists. The radial Schrodinger equation has been
R.Kumar  solved with the real part of complex valued potential, and the
solution can be used to understand many phenomena in the study of atomic and
molecular physics, spectroscopy (Hadronic as well as Molecular), nuclear
physics and also in high energy physics which are not yet understood. Most of
the studies on the study of $N$-dimensional space problem TDas , have been
focused on the lower spatial or dimensional space AAlJamel . The consequences
of $N$-dimensional space have been considered on the energy levels of the
system of quantum mechanics ANIkot . In the $N$-dimensional space, the study
of the harmonic oscillator SMAlJaber  and hydrogen atom AlJaber  have also
been done. Additionally, in $N$-dimensional space, the Schrodinger equation
has also been solved for potentials such as Cornell potential MAbushady ,
fourth order inverse power potential GRKhan , Mie type potential SIkhdair ,
Kratzer potential SMIkhdair , Coulomb potential GChen , Energy dependent
potential HHassanabadi , Global potential GRBoroun , Hua potential
HHassanabadi  and Harmonic potential TDas  etc. In these studies, the
solutions of $N$-dimensional Schrodinger equation for fixed chemical baryonic
potential at finite values of temperature have been used to define the
properties of Charmonium and Bottomonium mesons. The radial schrodinger
equation has been solved by using numerous methods such as, power series
method S.M.Kuchin , Hill determinant method RNChoudhury , numerical methods
LGrIxaru ; PSandin ; ZWang , quasi-linearization method (QLM) EZLiverts ,
point canonical transformation (PCT) RDe  and operator algebric method
JJSakurai  etc. But the N-dimensional radial Schrodinger equation has also ben
solved by other methods like, Nikiforov-Uvarov (UV) method ANIkot ; DAgboola ;
HHassanabadi , power series techniques RKumar12 , Laplace transformation
methods TDas ; GChen , Asymptotic iteration method (AIM) R.Kumar , SUSQM
method, Analytical exact iteration method (AEIM) EMKhokha  and Pekeris type
approximation HHassanabadi ; HRahimov  etc. The ground state of quarkonium
dissociation rates have been studied by direct continuum of thermal
activation, tunnelling, binding energy and phase shift scattering in the hot
quark-gluon plasma (QGP) system for the eigen states of lowest values PSandin
; DKharzeev . Vija and Thoma HVija  extended perturbated gauge theory for the
collisional energy loss in QGP at finite chemical potential and temperature.
The Bottomonium and Charmonium dissociation has been studied through the
correction of cornell potential via hard thermal loop resummed propagator of
the gluon LThakur ; ShiChaoYi .mainly study of the heavy quarkonium binding
energy in details in LThakur16 ; VKAgotiya  and the chemical potential effect
has also been studied by the methods of dissipative hydrodynamic on quark-
gluon plasma, production of photon in QGP and the quark-gluon plasma
thermodynamical properties HGervais ; AMonnai ; SSSingh ; VSFilinov ;
SMSanches . The Alberico et al. WMAlberico , Mocsy et al. AMocsy  and Agotiya
et al. VAgotiya  have solved the Schrodinger equation for the quarkonium
states at finite temperature, using a temperature dependent effective
potential by the linear combination of internal energy and conclude the
spectral function of quarkonium in a quark-gluon plasma. In the present work
we have used the (AEIM) Analytical exact iteration method to solve the
N-dimensional radial Schrodinger equation to investigate the properties of
Quarkonium such as Bottomonium and Charmonium. The main aim of the present
work is to find the solution of $N$-dimensional Schrodinger equation with the
real part of complex valued potential A.Dumitru ; MatthewMargotta  at fixed
value of chemical potential and finite values of temperature with anisotropic
values. An attempt has been made to deduce the exact energy eigen value using
AEIM method for calculating the behaviour of mass spectra, binding energy and
the Dissociation temperature of quarkonium. In this paper we study the
properties of heavy quarkonium (i.e,charmonium and bottomonium) using the
anisotropic plasma in oblate case only. The study of the relativistic heavy
ion collision in the large hadron collider(LHC) at CERN and Brookhaven
National Laboratory(BNL) in USA for understnding the theory of quarkonium
bound states has been looked forward to progress significantly. We have
studied the properties of the quarkonia states in the QGP plasma which
exhibits the anisotropy in momentum space, mainly anisotropy arises because of
the hydrodynamical expansion of the plasma with the non-vanishing shear
viscosity. It leads to an angular dependence of the quarkonia potential
BKPatra . The dissociation of quarkonia in the quasi-particle approach for the
isotropic $(\mbox{\boldmath$\xi$}=0)$ medium has been studied in VKAgotiya ,
the dissociation temperature$(T_{D})$ of quarkonia is estimated by the local
momentum anisotropy. After considering the momentum anisotropy, we consider
the oblate$(\mbox{\boldmath$\xi$}>0)$ case and compared it with the
isotropic$(\mbox{\boldmath$\xi$}=0)$ case. The main motivation to include
anisotropy ($\xi$) for the study of the properties of charmonium and
botommonium comes by the fact that the production of quark gluon-plasma in the
collision of heavy-ions does not possess $\mbox{\boldmath$\xi$}=0$, but in the
momentum-anisotropy it is present in the all dimensions of the direction of
the stages of the off central (heavy-ion) collision. The effect of anisotropy
will significantly revise the values of $T_{D}$, the values of $T_{D}$ has
observed to be lower in the oblate case$(\mbox{\boldmath$\xi$}>0)$ then the
other cases i.e $(\mbox{\boldmath$\xi$}<0)$ and $(\mbox{\boldmath$\xi$}=0)$.
We also know that the ground state is tightly bound and hence higher values of
binding energy(BE/$E_{b}$) and ground state melts later than the other excited
state. The paper is organised as follows. In Section II, we calculate the
Exact Solution of the $N$-dimensional Radial Schrodinger equation with the
real part of complex valued potential. In section III, we consider the debye
mass from a quasiparticle picture of hot QCD and in section IV, we calculate
the binding energy of quarkonium state in $N$-dimensional space. In section V,
we calculate the mass spectra of quarkonium state in $N$-dimensional space and
finally, we conclude the work in Sec. VI.

Figure 1: The variation of potential with distance(r) for the different values
of the temperature (left pannel),for the different values of baryonic chemical
potential (middle pannel) and for the different values of anisotropy but the
value of baryonic chemical potential and temperature is fixed(right pannel).

##  II The Solution of $N$-dimensional Radial Schrodinger equation with the
real part of complex valued potential.

The $N$-dimensional radial Schrödinger equation for two particles interacting
via real part of complex valued potential takes the form A.Dumitru ;
MatthewMargotta

$\frac{\partial^{2}}{\partial
r^{2}}\psi(r)+\frac{N-1}{r}\frac{\partial}{\partial
r}\psi(r)-\left(\frac{l(l+N-2)}{r^{2}}\right)\psi(r)\\\
+2\mu_{Q\bar{Q}}\left[E_{nl}-V(r)\right]\psi(r)=0$ (1)

where $n$,$l$ and $\mu_{Q\bar{Q}}$ are the dimensional number, angular quantum
number and reduced mass of two particles respectively. Now, the value of

$\psi(r)=R(r)/r^{\frac{N-1}{2}}$ (2)

is introduced in the Eq.(1)equation, then we get

$\left[\frac{\partial^{2}}{\partial
r^{2}}-\frac{\lambda^{2}-\frac{1}{4}}{r^{2}}+\left(E_{nl}-V(r)\right)\right]R(r)=0$
(3)

with

$\lambda=l+\left(\frac{N-2}{2}\right)$ (4)

And the real part of complex valued potential $V(r)$ A.Dumitru  is,

$V(r)=-\frac{\alpha}{r}(1+\mu r)exp(-\mu r)+\frac{2\sigma}{\mu}[1-exp(-\mu
r)]\\\ -\sigma rexp[(-\mu r)]-\frac{0.8\sigma}{m_{Q}^{r}}$ (5)

And the value of $\mu$ in the Eq.(5) is,

$\mu=m_{D}\left[1-\xi\left(\frac{3+cos2\Theta}{16}\right)\right]$ (6)

In Eq.(6) we consider the values of theeta is in parallel condition only for
the simplification of the calculation. Using exponential formula $e^{-\mu
r}=\sum_{k=0}^{\infty}\frac{(-\mu r)^{k}}{k!}$ for solving the Eq.(5) and
neglect the higher orders terms such that $\mu r\ll 1$ and then Eq.(5)
becomes,

$V(r,\mu)=-ar^{2}+br-\frac{d}{r}$ (7)

where, the values of $a$,$b$ and $d$ are,
$a=\frac{\alpha\mu^{3}}{2}$,$b=\frac{\alpha\mu^{2}}{2}+\sigma$ an Eq.(3)
$d=\alpha+\frac{0.8\sigma}{m_{Q}^{2}}$ Substituting in Eq.(8), and after
solving we get

${R}^{\prime\prime}(r)=\left[-\varepsilon_{nl}+a_{1}r^{2}+b_{1}r^{2}-\frac{d_{1}}{r}+\frac{\lambda^{2}-\frac{1}{4}}{r^{2}}\right]R(r)$
(8)

where, $\varepsilon_{nl}=2\mu_{Q\bar{Q}}E_{nl}$,
$a_{1}=\left|-2\mu_{Q\bar{Q}}a\right|$, $b_{1}=2\mu_{Q\bar{Q}}b$ and
$d_{1}=2\mu_{Q\bar{Q}}d$ The analytical exact iteration method (AEIM) requires
making the following ansatz for the wave function as in AOBarut ; Sozcelik ;
SMIkhdair .

$R(r)=f_{n}(r)exp[g_{1}(r)]$ (9)

where

$f_{n}(r)=1$ (10)

if n=0 and

$f_{n}(r)=\prod_{i=1}^{n}\left(r-\alpha_{i}^{(n)}\right)$ (11)

for n=1,2,3……..

$g_{1}(r)=-\frac{1}{2}\alpha r^{2}-\beta r+\delta lnr,\alpha>0,\beta>0$ (12)

From Eq.(9), we get

${R}^{\prime\prime}_{nl}(r)=\left({g}^{\prime\prime}_{l}(r)+g_{l}^{{}^{\prime}2}(r)+\frac{{f}^{\prime\prime}_{n}(r)+2g_{l}^{{}^{\prime}}(r)f_{n}^{{}^{\prime}}(r)}{f_{n}(r)}\right)R_{nl}(r)$
(13)

Now compare the Eq.(13) and Eq.(9) yields

$-\varepsilon_{nl}+a_{1}r^{2}+b_{1}r^{2}-\frac{d_{1}}{r}+\frac{\lambda^{2}-\frac{1}{4}}{r^{2}}\\\
={g}^{\prime\prime}_{l}(r)+g_{l}^{{}^{\prime}2}(r)+\frac{{f}^{\prime\prime}_{n}(r)+2g_{l}^{{}^{\prime}}(r)f_{n}^{{}^{\prime}}(r)}{f_{n}(r)}$
(14)

at $n=0$, substitute Eqs.(10),(11),(11) and Eq.(12) into Eq.(14)we get

$a_{1}r^{2}+b_{1}r-\frac{d_{1}}{r}+\frac{\lambda^{2}-\frac{1}{4}}{r^{2}}-\varepsilon_{0l}=\alpha^{2}r^{2}+2\alpha\beta
r-\alpha[1+2(\delta+0)]\\\
+\beta^{2}-\frac{2\beta\delta}{r}+\frac{\delta(\delta-1)}{r^{2}}$ (15)

Now comparing the corresponding power of $r$ on both side of Eq.(15) then
obtains

$\alpha=\sqrt{a_{1}}$ (16) $\beta=\frac{b_{1}}{2\sqrt{a_{1}}}$ (17)
$d_{1}=2\beta(\delta+0)$ (18) $\delta=\frac{1}{2}(1\pm 2\lambda)$ (19)
$\varepsilon_{0l}=\alpha[1+2(\delta+0)]-\beta^{2}$ (20)

Now, the energy eigen value formula for the ground state is:

$E_{0l}=\sqrt{\frac{a}{2\mu_{Q\bar{Q}}}}(N+2l)-\frac{b^{2}}{4a}$ (21)

Now for the first node (n=$1$), we used the function
$f_{1}(r)=\left(r-\alpha_{1}^{(1)}\right)$ and $g_{1}(r)$ then

$a_{1}r^{2}+b_{1}r-\frac{d_{1}}{r}+\frac{\lambda^{2}-\frac{1}{4}}{r^{2}}-\varepsilon_{1l}=\alpha^{2}r^{2}+2\alpha\beta
r-\alpha[1+2(\delta+1)]\\\
+\beta^{2}-\frac{2[\beta(\delta+1)+\alpha\alpha_{1}^{(1)}]}{r}+\frac{\delta(\delta-1)}{r^{2}}$
(22)

after comparing the coefficients of $r$, the relation between the potential
parameters are:

$\alpha=\sqrt{a_{1}}$ (23) $\beta=\frac{b_{1}}{2\sqrt{a_{1}}}$ (24)
$d_{1}=2\beta(\delta+1)$ (25) $\delta=\frac{1}{2}(1\pm 2\lambda)$ (26)
$\varepsilon_{1l}=\alpha[1+2(\delta+1)]-\beta^{2}$ (27)
$d_{1}-2\beta(\delta+1)=2\alpha\alpha_{1}^{(1)}$ (28)
$(d_{1}-2\beta\delta)\alpha_{1}^{(1)}=2\delta$ (29)

now the energy eigen value formula for first excited state $E_{1l}$ is:

$E_{1l}=\sqrt{\frac{a}{2\mu_{Q\bar{Q}}}}(N+2l+2)-\frac{b^{2}}{4a}$ (30)

Similarly for second node $(n=2)$, we use
$f_{2}(r)=\left(r-\alpha_{1}^{(2)}\right)\left(r-\alpha_{2}^{(2)}\right)$ and
$g_{1}(r)$ and we get,

$a_{1}r^{2}+b_{1}r-\frac{d_{1}}{r}+\frac{\lambda^{2}-\frac{1}{4}}{r^{2}}-\varepsilon_{2l}=\alpha^{2}r^{2}+2\alpha\beta
r-\alpha[1+2(\delta+2)]\\\
+\beta^{2}-\frac{2[\beta(\delta+2)+\alpha(\alpha_{1}^{(2)}+\alpha_{2}^{(2)})]}{r}+\frac{\delta(\delta-1)}{r^{2}}$
(31)

after comparing the coeficients the relation is:

$\alpha=\sqrt{a_{1}}$ (32) $\beta=\frac{b_{1}}{2\sqrt{a_{1}}}$ (33)
$\delta=\frac{1}{2}(1\pm 2\lambda)$ (34)
$\varepsilon_{2l}=\alpha[1+2(\delta+2)]-\beta^{2}$ (35)
$d_{1}-2\beta(\delta+2)=2\alpha(\alpha_{1}^{(2)}+\alpha_{2}^{(2)})$ (36)
$(d_{1}-2\beta\delta)\alpha_{1}^{(2)}\alpha_{2}^{(2)}=2\delta(\alpha_{1}^{(2)}+\alpha_{2}^{(2)})$
(37)
${d_{1}-2\beta(\delta+1)}(\alpha_{1}^{(2)}+\alpha_{2}^{(2)})=4\alpha(\alpha_{1}^{(2)}\alpha_{2}^{(2)})+2(2\delta+1)$
(38)

The energy eigen value formula for $E_{2l}$ is

$E_{2l}=\sqrt{\frac{a}{2\mu_{Q\bar{Q}}}}(N+2l+4)-\frac{b^{2}}{4a}$ (39)

Hence, Iteration method is repeated similarly many times. therefore, exact
energy eigen value for the state of finite temperature and chemical baryonic
potential also with anisotropy in $N$-dimensional space is:

$E_{nl}^{n}=\sqrt{\frac{a}{2\mu_{Q\bar{Q}}}}(N+2l+2n)-\frac{b^{2}}{4a}$ (40)

, $n=0,1,2,3........$.

Figure 2: The variation of quasi-particle debye mass with temperature at
different values of baryonic chemical potential(left pannel) and with baryonic
chemical potential at different values of temperature(right pannel).

##  III  The Debye mass from a quasiparticle picture of hot QCD

The Debye mass in leading order in QCD coupling at high temperature has been
known for a long time and is perturbative in nature. Rebhan ARebhan  has
defined the Debye mass $m_{D}$ by seeing the pole of the static propagator
which is relevant instead of the time-time component of the gluon self energy
and obtained a Debye mass which is gauge independent. This result follows from
the fact that the pole of self energy does not depend on the choice of the
gauge. The Debye mass was calculated for QGP at a high temperature in next to
lead order (NLO) in QCD coupling from correlation of two polyakov loop by
Braaten and Nieta EBraaten , this result agrees with the HTL result ARebhan .
It was pointed out by Arnold and Yaffe PArnold  that the physics of confined
magnetic charge was to be known in order to understand the contribution of
O$(g^{2}T)$ to the Debye mass in QCD, it is also pointed out by them that the
Debye mass as a pole of gluon propagator no longer holds true. Importantly in
lattice QCD, the definition of Debye mass its self encounter difficulty due to
the fact that unlike QED the electric field correlators are not gauge
invariant in QCD. The proposal of this problem is based on effective theories
obtained by dimensional reduction KKajantie , spatial correlation function of
gauge-invariant meson energy and the behavior of color singlet free energies
SNadkarni  has been made Burnier and Rothkopf YBurnier  has attempted to
defined a gauge invariant mass from a complex static in medium heavy-quark
potential obtained from lattice QCD.

Figure 3: Dependence of $J/\psi$ binding energy with temperature (left pannel)
and dependence of $\Upsilon$ binding energy with temperature (right pannel) at
different values of baryonic chemical potential but the value of $N$ and $\xi$
is fixed.

Figure 4: Dependence of $J/\psi$ binding energy with temperature (left pannel)
and dependence of $\Upsilon$ binding energy with temperature (right pannel) at
different values of dimensionality number but the value of baryonic chemical
potential and $\xi$ is fixed.

Several attempts has been made to capture all the interaction effects present
in hot QCD equation of state (EOS) in terms of non-interacting quasi-partons
(quasi-gluons and quasi-quarks). These quasiparton are the excitations of the
interacting quarks and gluons and they can be categorized as, effective mass
model VGoloviznin ; peshier , effective mass model with polykov loop MDElia ,
model based on PNJL and NJL ADumitru  and $(4)$ effective fugacity model
VChandra7 ; VChandra9 . In QCD the quasipartical model is a phenomenological
model which is widely used to describe the nonlinear behaviour of QGP near
phase transition point. In this model a system of interacting massless quarks
and gluon can be described as an ideal gas of ”massive” non interacting
quasiparticle. The mass of the quasiparticle is dependent on the temperature
which arises due to the interaction of gluons and quarks with surrounding
medium. The quasiparticle retain the quantum number of the quarks and gluons
PKSrivastava .

Figure 5: Dependence of $J/\psi$ binding energy with temperature (left pannel)
and dependence of $\Upsilon$ binding energy with temperature (right pannel) at
different values of $\xi$ but the value of $\mu$ and $N$ is fixed.

Figure 6: Dependence of mass of charmonium with temperature (left pannel) and
dependence of mass of bottomonium with temperature (right pannel).

Here, ${m_{D}}$ is the quasi-particle Debye mass. In our calculation, we use
the Debye mass $m_{D}$ in the pure gluonic case as:

$m^{2}_{D}=g^{2}(T)T^{2}\bigg{(}\frac{N_{c}}{3}\times\frac{6PolyLog[2,z_{g}]}{\pi^{2}}\bigg{)}$
(41)

and ${m_{D}}$ for full QCD case is:

$\displaystyle m^{2}_{D}\left(T,\mu_{f}\right)$ $\displaystyle=$
$\displaystyle
g^{2}(T)T^{2}\bigg{[}\bigg{(}\frac{N_{c}}{3}\times\frac{6PolyLog[2,z_{g}]}{\pi^{2}}\bigg{)}$
(42)
$\displaystyle+{\bigg{(}\frac{\hat{N_{f}}}{6}\times\frac{-12PolyLog[2,-z_{q}]}{\pi^{2}}\bigg{]}}$

and

$\displaystyle\hat{N_{f}}$ $\displaystyle=$
$\displaystyle\bigg{(}N_{f}+\frac{3}{\pi^{2}}\sum\frac{\mu^{2}}{T^{2}}\bigg{)}$
(43)

Here, $g(T)$ is the QCD running coupling constant, $N_{c}=3$ ($SU(3)$) and
$N_{f}$ is the number of flavor, the function $PolyLog[2,z]$ having form,
$PolyLog[2,z]=\sum_{k=1}^{\infty}\frac{z^{k}}{k^{2}}$ and $z_{g}$ is the
quasi-gluon effective fugacity and $z_{q}$ is quasi-quark effective fugacity.
These distribution functions are isotropic in nature,

$\displaystyle f_{g,q}=\frac{z_{g,q}exp(-\beta p)}{\left(1\pm
z_{g,q}exp(-\beta p)\right)}$ (44)

Figure 7: Dependence of $J/\psi$ binding energy with temperature (left pannel)
at different values of $\mu$ and dependence of $\Upsilon$ binding energy with
temperature (right pannel) at different values of $\xi$.

where, $g$ define the study for quasi-gluons and $q$ defines for quasi-quarks.
These fugacities should not be confused with any conservations law (number
conservation) and have merely been introduced to encode all the interaction
effects at high temperature QCD. Both $z_{g}$ and $z_{q}$ have a very
complicated temperature dependence and asymptotically reach to the ideal value
unity VChandra9 . The temperature dependence $z_{g}$ and $z_{q}$ fits well to
the form given below,

$z_{g,q}=a_{q,g}\exp\bigg{(}-\frac{b_{g,q}}{x^{2}}-\frac{c_{g,q}}{x^{4}}-\frac{d_{g,q}}{x^{6}}\bigg{)}.$
(45)

(Here $x=T/T_{c}$ and $a$, $b$ and $c$ and $d$ are fitting parameters), for
both EOS$1$ and EOS$2$. Here, EoS$1$ is the $O(g^{5})$ hot QCD PArnold  and
EoS$2$ is the $O(g^{6}\ln(1/g)$ hot QCD EoS KKajantie  in the quasi-particle
description VChandra7 ; VChandra9  respectively. Now, the expressions for the
Debye mass can be rewritten in terms effective charges for the quasi-gluons
and quarks for pure gauge as:

$m^{2}_{D}\left(T,\mu\right)=Q^{2}_{g}T^{2}\frac{N_{c}}{3}$ (46)

and after introducing Eq.(44) in Eq.(43), then the expression of full QCD
equation becomes:

$m^{2}_{D}\left(T,\mu\right)=T^{2}(\frac{N_{c}}{3}Q^{2}_{g})+\bigg{(}\bigg{(}\frac{N_{f}}{6}+\frac{1}{2\pi^{2}}\bigg{(}\frac{\mu^{2}}{T^{2}}\bigg{)}\bigg{)}Q^{2}_{q}\bigg{)}$
(47)

And in the Eq.(48) the $\mu$ is defined as:

$\displaystyle\mu$ $\displaystyle=$ $\displaystyle\frac{\mu_{b}}{3}$ (48)

where, $(\mu)$ defined the quark-chemical potential and $(\mu_{b})$ is
baryonic chemical potential. where, $Q_{g}$ and $Q_{q}$ are the effective
charges given by the equations:

$\displaystyle Q^{2}_{g}$ $\displaystyle=$ $\displaystyle
g^{2}(T)\frac{6PolyLog[2,z_{g}]}{\pi^{2}}$ $\displaystyle Q^{2}_{q}$
$\displaystyle=$ $\displaystyle g^{2}(T)\frac{-12PolyLog[2,-z_{q}]}{\pi^{2}}.$
(49)

In our present analysis we had used the temperature dependence of the quasi-
particle Debye mass, $m_{D}^{QP}$ in full QCD with $N_{f}=3$ and to determine
the temperature dependence of binding energy and dissociation temperature.

##  IV Binding energy of Quarkonium state in $N-$ dimensional space

The binding energy of quarkonium state such as Bottomonium and Charmonium is
studied in this section. We have calculated by the $N-$dimensional space at
fixed value of $\mu$, at finite temperature and also with the values of
isotropy $({\mbox{\boldmath$\xi$}=0})$ and anisotropy. After using iteration
method, we found the exact energy formula in $N-$dimensional space is:

$E_{nl}^{n}=\sqrt{\frac{a}{2\mu_{Q\bar{Q}}}}(N+2l+2n)-\frac{b^{2}}{4a}$ (50)

Substituting the value of n (i.e, $n=0,1,2,3....$) according to the state of
quarkonia like $1$S and $2$S etc. Now, we substitute the value of $a$ and $b$
in the above Eq.(51) and we get the binding energy for the quarkonium state is

$E_{b}=\sqrt{\frac{\alpha\mu^{3}}{4\mu_{Q\bar{Q}}}}(N+2l+2n)-\frac{(\frac{\alpha\mu^{2}}{2}+\sigma)^{2}}{4\times\frac{\alpha\mu^{3}}{2}}$
(51)

or,

$E_{b}=\sqrt{\frac{\alpha\mu^{3}}{4\mu_{Q\bar{Q}}}}(N+2l+2n)-\left[\frac{\alpha\mu}{8}+\frac{\sigma^{2}}{2\alpha\mu^{3}}+\frac{\sigma}{2\mu}\right]$
(52)

where $\mu_{Q\bar{Q}}$=$\frac{m_{c}}{2}$ and $\frac{m_{b}}{2}$.

##  V Mass Spectra of Quarkonium state in $N$-dimensional space

For calculating the mass spectra of heavy quarkonia the relation is:

$M=2m_{Q}+E_{nl}^{n}$ (53)

Here, mass spectra is equal to the sum of the exact formula of energy and
twice the quark-mass. Now we substitute the values of $E_{nl}^{n}$ from
Eq.(51) in the Eq.(54) as,

$M=2m_{Q}+\sqrt{\frac{a}{2\mu_{Q\bar{Q}}}}(N+2l+2n)-\frac{b^{2}}{4a}$ (54)

Now, substitute the values of $a$ and $b$ in the Eq.(55) we get:

$M=2m_{Q}+\sqrt{\frac{\alpha\mu^{3}}{4\mu_{Q\bar{Q}}}}(N+2l+2n)-\frac{(\frac{\alpha\mu^{2}}{2}+\sigma)^{2}}{4\times\frac{\alpha\mu^{3}}{2}}$
(55)

and

$M=2m_{Q}+\sqrt{\frac{\alpha\mu^{3}}{4\mu_{Q\bar{Q}}}}(N+2l+2n)-\left[\frac{\alpha\mu}{8}+\frac{\sigma^{2}}{2\alpha\mu^{3}}+\frac{\sigma}{2\mu}\right]$
(56)

where $m_{Q}$ is used for the mass of quarkonium state such as charmonium and
bottomonium mass.

##  VI Results and Conclusion

In the present analysis, the value of critical temperature ($T_{c}=197MeV$)
have been fixed, and find the various quantities after considering the
anisotropy values in oblate case only in the hot quantum-chromodynamics (QCD)
plasma. The value of ($\xi$) have been increased in the right pannel of
figure1, noticed that the separation of potential near the axis of origin is
slightly increased, and same behaviour is also shown if the value of chemical
potential have been increased in the figure 1 (middle pannel). But in figure 1
(left pannel) the separation of potential near the axis of origin is slightly
decreased after increased the value of temperature. from this we conclude that
there is little effect of anisotropy on the real part of the potential. Figure
2, shows the variation of quasiparticle debye mass with temperature at
different values of baryonic chemical potential(left pannel) and with baryonic
chemical potential at different values of temperature (right pannel). Where
the baryonic chemical potential$(\mu_{b})$ is related to the three times of
the quark-chemical potential$(\mu)$. The screening mass at baryon density and
temperature has studied by the lattice taylor’s expansion method MDoring . In
the figure 2 it has been observed that with increase in the values of chemical
potential, the debye mass increases (left pannel), same behaviour for the
debye mass has been seen for the temperature (right pannel). from the energy
eigen values obtained through N-dimensional Schrodinger equation using AEIM
method, we calculate the binding energy, mass spectra and dissociation
temperature. In figures 3, 4 and 5 shows, the variation of binding energy of
$J/\psi$ and $\Upsilon$ with temperature. After increased the value of
anisotropy ($\xi$) in figure 5, the variation of binding energy is decreased.
Because there are mainy two cases: first one is that, the value of anisotropy
have been increased, the binding energy is stronger in the comparison of
isotropic and prolate case. Since the potential is more deep with the
increases of ($\xi$) because of weaker screening. The screening of the string
part and coloumb part are less attenuated in that case the quarkonium binding
energy becomes larger then the case of isotropic medium, and also there is a
strong decreasing trend with the temperature. It has also been noticed that,
the variation of binding energy is depend upon the mass of the state, if the
value of the quark-mass of the state is increased, the binding energy
decreased. The effect of dimensionality number$(N)$ on the binding energy for
the quarkonium state $J/\psi$ and $\Upsilon$ with temperature have been shown
in figure 4. The value of dimensionality number have been increased then the
binding energy of the state of quarkonia is also increased at the fixed value
of baryonic chemical potential($\mu$) and ($\xi$). Similarly in figure 3, we
also noticed that the binding energy is increased with increasing the values
of chemical potential at constant value of anisotropy and dimensionality
number. But in figure 6, the variation of mass spectra of heavy quarkonia with
temperature for $1S$ state of charmonium (left pannel) and $1S$ of bottomonium
(right pannel) is observed using two different value of masses of charmonium
and bottomonium. If we increased the value of quark mass, the mass spectra of
heavy quarkonia is also increased WMAlberico . And the same variation of mass
spectra is also observed for the excited states of quarkonia. In figure 7, we
have compared the binding energy of Leading order(LO) and Quasi-Particle(QP)
Debye mass with chemical potential (in left pannel) and with anisotropy (in
right pannel), and it has been noticed that the value of binding energy in
terms of LO debye mass is greater than that of QP debye mass. The LO Debye
mass at high temperature range is known from long time and is perturbative
E.Shuryak  in nature. But in the present analysis, we consider the LO and QP
debye mass in terms of chemical potential. However, we also know that about
the lattice parameterized form($m_{D}^{L}$) is $1.4$ times the LO debye mass,
in that case, we observed that ($m_{D}^{L}$) the variation of binding energy
of lattice parameterized form of Debye mass is greater than that of LO debye
mass. Thus the relation between binding energy and temperature is poised to
give the information about the pattern of dissociation temperature($T_{D}$) in
thermal medium, which will now be used to calculate the $T_{D}$ for the
different quarkonium states. Recent studies shows various methods for
obtaining the dissociation temperature for different states of quarkonia.
PSandin , authors have investigate the $T_{D}$ of heavy quarkonia by the
thermal width. AMocsy , authors have apply a condition for calculating the
$T_{D}$ i.e, thermal width is greater than equal to the twice of the real part
of binding energy. VAgotiya , authors have calculated the lower and upper
bound state of the $T_{D}$ by applying the condition for the dissociation i.e,
$E_{bin}=3T_{D}$ and $E_{bin}=T_{D}$ respectively. UK , authors obtained the
$T_{D}$ of the states of quarkonia when the binding energy is of the order of
their chemical baryonic potential.

$\sqrt{\frac{\alpha\mu^{3}}{4\mu_{Q\bar{Q}}}}(N+2l+2n)-\left[\frac{\alpha\mu}{8}+\frac{\sigma^{2}}{2\alpha\mu^{3}}+\frac{\sigma}{2\mu}\right]=3(T_{D})$
(57)

where $\mu_{Q\bar{Q}}$=$\frac{m_{c}}{2}$ and $\frac{m_{b}}{2}$.

Table 1: The dissociation temperature(is in unit of $T_{c}$) of QP Debye mass for lower bound state $(T_{D})$ with $T_{c}=197MeV$ for the different ground states of quarkonia i.e, ${J/\psi}$ and ${\Upsilon}$ at $\mu=300MeV$ and for $N=0$. $State$ | $\mbox{\boldmath$\xi$}=0$ | $\mbox{\boldmath$\xi$}=0.3$ | $\mbox{\boldmath$\xi$}=0.7$
---|---|---|---
$J/\psi$ | 3.54060 | 3.14720 | 2.65228
$\Upsilon$ | 2.05583 | 1.82741 | 1.53553
Table 2: The dissociation temperature(is in unit of $T_{c}$) of LO Debye mass for lower bound state $(T_{D})$ with $T_{c}=197MeV$ for the different ground states of quarkonia i.e, ${J/\psi}$ and ${\Upsilon}$ at $\mu=300MeV$ and for $N=0$. $State$ | $\mbox{\boldmath$\xi$}=0$ | $\mbox{\boldmath$\xi$}=0.3$ | $\mbox{\boldmath$\xi$}=0.7$
---|---|---|---
$J/\psi$ | 4.03553 | 3.57868 | 3.03299
$\Upsilon$ | 2.38578 | 2.15736 | 1.85279

But in the present study, the dissociation temperature is calculate only for
the ground states of quarkonia, ${J/\psi}$ and ${\Upsilon}$ at $\mu=300MeV$
and $N=0$ for the lower bound state of the $T_{D}$ by applying the condition
i.e, $E_{bin}=3(T_{D})$ (is in unit of $T_{c}$). If the dimensionality number
is increased then the values of dissociation temperature and binding energy is
also increased. The value of anisotropy ($\xi$) have been increased in oblate
case only then the values of $T_{D}$ is decreased. The values of dissociation
temperature given in table-$1$ for QP debye mass and in table-$2$ for LO debye
mass is expected for the ground states of quarkonia. Hence we conclude that
after noticed the variation of binding energy in figure 7, we have been
conclude that the value of QP debye mass is smaller than that of LO and
lattice parameterized form of Debye mass. These observations can be understood
from the hierarchy in their numerical values of these debye masses,
$m_{D}^{QP}$ $<$ $m_{D}^{LO}$ $<$ $m_{D}^{L}$, and $(B.E)^{QP}$ $<$
$(B.E)^{LO}$ $<$ $(B.E)^{L}$. And this behaviour is also satisfied by the
values of table-$1$ and table-$2$ because the dissociation of LO Debye mass is
greater then the QP Debye mass. Hence we also conclude that $T_{D}^{QP}$ $<$
$T_{D}^{LO}$ $<$ $T_{D}^{L}$.

##  VII Acknowledgments

One of the authors, VKA acknowledges the Science and Engineering Research
Board (SERB) Project No. EEQ/2018/000181 New Delhi for the research support in
basic sciences.

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