Advances in High Energy Physics

Volume 2019, Article ID 4785615, 10 pages

https://doi.org/10.1155/2019/4785615

## Dissociation of Quarkonium in Hot and Dense Media in an Anisotropic Plasma in the Nonrelativistic Quark Model

^{1}Department of Applied Mathematics, Faculty of Science, Menoufia University, Egypt^{2}Department of Physics, Faculty of Science, Cairo University, Egypt^{3}Institute for Physical Problems, Baku State University, Z. Khalilov St. 23, AZ-1148 Baku, Azerbaijan

Correspondence should be addressed to M. Abu-Shady; moc.liamg@ydahsuba.rd

Received 27 September 2018; Revised 7 December 2018; Accepted 10 January 2019; Published 23 January 2019

Academic Editor: Sally Seidel

Copyright © 2019 M. Abu-Shady et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

In this paper, quarkonium dissociation is investigated in an anisotropic plasma in the hot and dense media. For that purpose, the multidimensional Schrödinger equation is solved analytically by Nikiforov-Uvarov (NU) method for the real part of the potential in an anisotropic medium. The binding energy and dissociation temperature are calculated. In comparison with an isotropic medium, the binding energy of quarkonium is enhanced in the presence of an anisotropic medium. The present results show that the dissociation temperature increases with increasing anisotropic parameter for 1S state of the charmonium and bottomonium. We observe that the lower baryonic chemical potential has small effect in both isotropic and anisotropic media. A comparison is presented with other pervious theoretical works.

#### 1. Introduction

The dissociation of heavy quarkonium has been suggested in the formation of quark-gluon plasma (QGP) in the pioneering work of Matsui and Sats [1]. The production or suppression of quarkonia has been investigated either theoretically or experimentally in Refs. [2–12], and the disassociation temperature has been studied in Refs. [13–17]. In addition, ultrarelativistic heavy-ion collisions (URHIC) are used to explore the quark-gluon plasma (QGP) media as in [18–20]. This shows that the present topic is an interesting research work from both theoretical and experimental views.

One of the most remarkable features of the QGP formation is the color screening of the static chromoelectric fields [21]. The Debye screening in QCD plasma has been studied as a probe of deconfinement in a dense partonic medium which shows a reduction in the interaction between heavy quarks and antiquarks due to color screening leading to a suppression in yields [22, 23]. Hence, the quarkonium in a hot medium is a good tool to examine the confined/deconfined state of matter.

Recently, the quarkonium properties have been studied by modifying both the Coulombic and string terms of the heavy-quark potential using the perturbative hard thermal loop (HTL) dielectric permittivity in both the isotropic and anisotropic media in the static [24–28] and in the moving media [29]. Many attempts have been suggested to calculate the dissociation temperatures of quarkonium states in the deconfined medium by applying the lattice calculations of quarkonium spectral functions [30–33] or nonrelativistic calculations which depend on some effective (screened) potentials [34–39] in an isotropic medium. The nuclei moving towards each other for collision have ultrarelativistic speed and so they are Lorentz contracted. This implies that initially there is a spatial anisotropy in the system. As the system needs to thermalize and isotropize, it generates an anisotropic pressure gradient in all directions. In this way many processes are involved including the formation of unstable modes. These interactions map the spatial anisotropy to momentum anisotropy which remains throughout the evolution of such a system. So, it is important to include momentum anisotropy in the analysis [25]. The effect due to a local anisotropy of the momentum space on the heavy-quark potential is studied in a previous work [40]. The anisotropy is caused by external fields in studying the properties of quarkonium states [41–46]. Besides, the anisotropic quark-gluon plasma concepts are reviewed in [47]. The phenomenological studies that can reproduce the experimentally measured of bottomonia with and without recombination are given in [48, 49], respectively.

So far, most of the previous calculations concentrate on the study of the heavy bound-state properties at vanishing baryonic chemical potential of the thermal medium as in [25, 26, 46]. There are little works which deal with a nonvanishing chemical potential in both perturbative and nonperturbative approaches. The color-screening effect at finite temperature and chemical potential was studied in a thermofield dynamics approach [50, 51] in which the phenomenological potential model [52] and an error-function-type confining force with a color-screened Coulomb-type potential were used. Lattice QCD has also been used to study the color screening in the heavy-quark potential at finite density with Wilson fermions [53]. Kakade and Patra [54] have investigated the quarkonium dissociation at lower temperatures and higher baryonic chemical potential by correcting both the perturbative and nonperturbative approaches of the Cornell potential through the dielectric permittivity in an isotropic medium. The equation of state has successfully been modified to finite baryonic chemical potential using Taylor expansions [55, 56] around a vanishing chemical potential as well as reweighting techniques [57, 58] and using imaginary chemical potentials [59]. The holographic description of the thermal behavior of heavy vector mesons inside a plasma at finite temperature and density is studied in [60].

The exploration of high baryon densities and the moderate-temperature region of the QCD phase diagram is possible with the upcoming compressed baryonic matter (CBM) experiment at the Facility for Antiproton and Ion Research (FAIR) [54]. Moreover, model calculations based on transport and hydrodynamical equations show that the highest net baryon densities () carried out in the center of collision are ~6 to 12 times the density of normal nuclear matter for the most central collision () [61].

The aim of this work is to study quarkonium dissociation of the quark matter at finite temperature and baryonic chemical potential in an anisotropic medium in comparison with an isotropic medium, in which the N-dimensional Schrödinger equation is analytically solved using the Nikiforov-Uvarov (NU) method. In Section 2, the multidimensional Schrödinger equation is introduced with heavy-quark potential in an anisotropic medium. Moreover, the solution of the N-dimensional Schrödinger equation is given by using the NU method. In Section 3, the binding energy and dissociation temperature are studied with different parameters in the three-dimensional space. In Section 4, the summary and conclusion are presented.

#### 2. The N-Dimensional Schrödinger Equation at Finite Temperature in an Anisotropic Medium and Baryonic Chemical Potential

The Schrödinger equation for two particles interacting via an effective potential in the -dimensional space is given [14, 62–64]:where , and are the reduced mass for the quarkonium particle, the angular momentum quantum number, and the dimensionality number, respectively. Setting the wave function , the following radial Schrödinger equation is obtained

##### 2.1. Real Part of the Potential in a Anisotropic Medium

Here, we aim to find the potential due to the presence of a dissipative anisotropic hot QCD medium. The in-medium modification can be obtained in the Fourier space by dividing the heavy-quark potential by the medium dielectric permittivity, , as follows:and by taking the inverse Fourier transform, the modified potential is obtained as follows:where is the Fourier transform of Cornell potential that gives may be calculated found from the self-energy using finite temperature QCD. By applying hard thermal loop resummation technique as in [25, 46], the static gluon propagator which represents the inelastic scattering of an off-shell gluon to a thermal gluon is defined as follows:The dielectric tensor can be obtained in the static limit in Fourier space, from the temporal component of the propagator asTo calculate the real part of the interquark potential in the static limit, one can obtain first the temporal component of real part of the retarded propagator in Fourier space at finite temperature and chemical potential as given in [54] as follows:The medium dielectric permittivity is then givenSubstituting (5) and (9) into (3) and then taking its inverse Fourier transform, we can write the real part of the potential for as follows:where is the anisotropic parameter. and are the temperature and the baryonic chemical potential, respectively. In (10), the potential depends on which is the angle between the particle momentum and the direction of anisotropy. We note that the potential in (10) reduces to the Cornell potential for and details, see [46]. In the present work, the Debye mass is given as in [65, 66] bywhere is the coupling constant as defined in [51], is the quark chemical potential , is number of flavours, and is number of colors. The NU method [67] is briefly given here to solve the form of the following:where and are polynomials of maximum second degree and is a polynomial of maximum first degree with an appropriate coordinate transformation. We try to find a particular solution by separation of variables, if one deals with the transformationEquation (13) is written aswhere is a polynomial of degree which satisfies the hypergeometric equation, taking the formwhere is a normalization constant and is a weight function which satisfies the following: is a polynomial of the first degree. The values of in the square root of (21) are possible to calculate if the function under the square is a square of a function. This is possible if its discriminant is zero. For parallel to the direction of of anisotropy at , the potential is given bywhereBy applying the above method to the potential given in (22), we obtain the energy eigenvalues as follows:Similarly, for perpendicular to the direction of of anisotropy at , the potential is given bywhereand the energy eigenvalues are given as follows:where is a parameter determined as in [68].

#### 3. Discussion of the Results

In the present analysis, the various quantities are computed, using the weakly anisotropy parameter () at finite temperature and finite baryonic chemical potential in a QCD plasma. First, we discuss the Debye mass which plays an important role in the present study, since the Debye mass is inserted through the potential and also in the binding energy. We assume the Debye mass intact from the effects of anisotropy present in the media so that it remains the same in both media (an isotropic and an anisotropic) as done in [43, 46]. In Figure 1, the Debye mass is plotted as a function of the baryonic chemical potential, for three values of temperatures (T = 0.23 GeV), (T = 0.25 GeV), and (T = 0.27 GeV). The Debye mass is an increasing function of the chemical potential. By increasing the temperature, we note that the Debye mass shifts to higher values. Hence, the Debye mass is affected in an isotropic medium when the finite baryonic chemical potential is included. In [54], the authors studied the quarkonium mass at the lower temperatures and higher chemical potentials. They found that the Debye mass increases with increasing the chemical potential which is compatible with Figure 1. In addition, the Debye mass increases with increasing temperatures, where they studied the Debye mass below the range of temperatures 0.02 to and up 0.12 GeV. This conclusion is observed in [50], in which the color-screening effects in the QGP are noticeable at higher temperatures and lower baryon density in the anisotropic medium.