Advances in High Energy Physics

Volume 2018 (2018), Article ID 7356843, 12 pages

https://doi.org/10.1155/2018/7356843

## Binding Energies and Dissociation Temperatures of Heavy Quarkonia at Finite Temperature and Chemical Potential in the -Dimensional Space

^{1}Department of Applied Mathematics, Faculty of Science, Menoufia University, Shibin El Kom, Egypt^{2}Department of Basic Science, Modern Academy of Engineering and Technology, Cairo, Egypt

Correspondence should be addressed to M. Abu-Shady

Received 23 July 2017; Accepted 29 November 2017; Published 8 January 2018

Academic Editor: Juan José Sanz-Cillero

Copyright © 2018 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

The* N*-dimensional radial Schrödinger equation has been solved using the analytical exact iteration method (AEIM), in which the Cornell potential is generalized to finite temperature and chemical potential. The energy eigenvalues have been calculated in the* N*-dimensional space for any state. The present results have been applied for studying quarkonium properties such as charmonium and bottomonium masses at finite temperature and quark chemical potential. The binding energies and the mass spectra of heavy quarkonia are studied in the* N*-dimensional space. The dissociation temperatures for different states of heavy quarkonia are calculated in the three-dimensional space. The influence of dimensionality number (*N*) has been discussed on the dissociation temperatures. In addition, the energy eigenvalues are only valid for nonzero temperature at any value of quark chemical potential. A comparison is studied with other recent works. We conclude that the AEIM succeeds in predicting the heavy quarkonium at finite temperature and quark chemical potential in comparison with recent works.

#### 1. Introduction

The solution of the radial Schrödinger equation with spherically symmetric potentials has vital applications in different fields of physics such as atoms, molecules, hadronic spectroscopy, and high energy physics. The Schrödinger equation has been solved by operator algebraic method [1], power series method [2, 3], and path integral method [4], in addition to quasi-linearization method (QLM) [5], point canonical transformation (PCT) [6], Hill determinant method [7], and the conventional series solution method [8]. Recently, most of the theoretical studies have been developed to study the solutions of radial Schrödinger equation in the higher dimensions. These studies are general and one can directly obtain the results in the lower dimensions [9–23]. The -dimensional Schrödinger equation has been solved by various methods as the Nikiforov-Uvarov (NU) method [9–12], asymptotic iteration method (AIM) [13], Laplace Transform method [14, 15], supersymmetric quantum mechanics (SUSQM) [16], power series technique [17], Pekeris type approximation [18], and the analytical exact iteration method (AEIM) [19].

The* N*-dimensional radial Schrödinger equation has been solved for different types of spherical symmetric potentials as Coulomb potential [15], pseudo-harmonic potential [20], Mie-type potential [21], energy-dependent potential [11], Kratzer potential [22], and Cornell potential type [13, 23] that consists of the Coulomb term and the linear term, anharmonic potential [14], the Cornell potential with harmonic oscillator potential [12], and the extended Cornell potential [19].

The solution of Schrödinger equation has been used in different studies to describe the properties of heavy-quarkonium systems at finite temperature. Many efforts have been devoted to calculating the mass spectra of charmonium and bottomonium mesons and determining the binding energy and the dissociation temperatures of heavy quarkonia. In [24, 25], the authors have calculated the dissociation rates of quarkonium ground states by tunneling and direct thermal activation to the continuum and the binding energies and scattering phase shifts for the lowest eigenstates in the charmonium and bottomonium systems in hot gluon plasma. In [26, 27], the deconfinement and properties of the resulting quark-gluon plasma (QGP) have been investigated by studying the medium behavior of heavy-quark bound states in statistical quantum chromodynamics and the spectral analysis of quarkonium states in a hot medium of deconfined quarks and gluons and the thermal properties of QGP are discussed. In [28–30], the authors have solved the Schrödinger equation at finite temperature for the charmonium and bottomonium states by employing an effective temperature dependent potential given by a linear combination of the color singlet free and internal energies and discussed the quarkonium spectral functions in a quark-gluon plasma. The dissociation of quarkonia has been studied by correcting the full Cornell potential through the hard-loop resumed gluon propagator and the hard thermal loop (HTL) approximation [31, 32]. Moreover, the binding energies of the heavy quarkonia states are studied in detail in [33, 34].

At finite temperature and chemical potential, Vija and Thoma [35] have extended the effective perturbation theory for gauge theories at finite temperature and chemical potential for studying the collisional energy loss of heavy quarks in QGP. In [36, 37], the authors have generalized a thermodynamic quasi-particle description of deconfined matter to finite chemical potential and analyzed the response of color singlet and color averaged heavy-quark free energies to a nonvanishing baryon chemical potential. On the same hand, the effect of chemical potential is studied on the photon production of quantum chromodynamics (QCD) plasma, dissipative hydrodynamic effects on QGP, and thermodynamic properties of the QGP [38–42] by using different methods. At finite chemical potential and small temperature region, the dissociation of quarkonia states has been studied in a deconfined medium of quarks and gluons in [43].

The aim of this work is to find the analytic solution of the* N*-dimensional radial Schrödinger equation with generalized Cornell potential at finite temperature and chemical potential using the analytical exact iteration method (AEIM) to obtain the energy eigenvalues, where the energy eigenvalues are only valid for nonzero temperature for any value of quark chemical potential. So far no attempt has been made to solve the* N*-dimensional radial Schrödinger when finite temperature and chemical potential are included by using AEIM. In addition, the application of present results on quarkonium properties has been investigated such as the mass spectra of heavy quarkonium and the dissociation temperature for different states of heavy quarkonia. The influence of the dimensionality number, which is not considered in many recent works, has been investigated on the binding energy and the dissociation temperature at finite temperature and chemical potential.

The paper is organized as follows: the background of the study of previous efforts is introduced in Section 1. In Section 2, the analytic solution of the -dimensional radial Schrödinger equation is derived. In Section 3, the results are discussed. In Section 4, summary and conclusion are presented.

#### 2. Analytic Solution of the -Dimensional Radial Schrödinger Equation with the Cornell Potential at Finite Temperature and Chemical Potential

The -dimensional radial Schrödinger equation for two particles interacting via a spherically symmetric potential takes the following form [14, 44]:where , , and are the angular quantum number, the dimensional number, and reduced mass of the two particles , respectively.

Inserting in (1), we obtainwith , where is the Cornell potential that takes the following form [45]:with GeV^{2} and The potential is modified in QGP to study the binding energy and dissociation temperature by including Debye screening mass as follows [45, 46]:where is the Debye screening mass at finite temperature and quark chemical potential [37]:where is the number of quark flavors, is the number of colors, and is the QCD coupling constant at finite temperature [47]:Using in (4) with neglecting the higher orders at , thus (4) takes the following form:where Substituting (7) into (2), we obtain where The analytical exact iteration method (AEIM) requires the following ansatz for the wave function as in [48–50]:whereFrom (11), we obtainBy comparing (9) and (14), we obtain At , substituting (12) and (13) into (15) givesBy comparing the corresponding powers of* r *on both sides of (16), one obtainsFrom (17a)–(17e) and (10), by taking the positive sign in (17d), then the ground state energy isFor the first node , we use the functions and from (13). Equation (15) takes the following form:Then, the relations between the parameters of the potential and the coefficients , and are given byUsing (20a)–(20g) and (10), we obtain the formula asFor the second node , we use and from (13) to solve (15) which givesThus, the relations between the coefficients , and are given byHence, the formula is given byThen, the iteration method is repeated many times. Therefore, the exact energy formula for any state in the* N*-dimensional space is written asAccording to (8), we note that (25) is only valid for finite temperature, since, at , the parameter equals zero. Therefore, the energy eigenvalues diverge at this case.

#### 3. Discussion of Results

In the first part of this section, we compare between the exact potential in (4) and the approximate potential in (7) for different values of chemical potential and temperature.

In Figure 1, the exact and the approximate potential are plotted for different values of chemical potentials. We note that there is a good qualitative agreement between exact potential and approximate potential. In Figure 2, we note a good qualitative agreement between two potentials. By increasing temperature, the positive part of two potentials is reduced. Thus, the present potential gives a good accuracy in comparison with original potential.