Advances in High Energy Physics

Volume 2018, Article ID 4549705, 7 pages

https://doi.org/10.1155/2018/4549705

## Nonrelativistic Arbitrary -States of Quarkonium through Asymptotic Iteration Method

^{1}Gazi Üniversitesi, Fen-Edebiyat Fakültesi, Fizik Bölümü, 06500 Teknikokullar-Ankara, Turkey^{2}Department of Basic Sciences, Faculty of Maritime, Mersin University, Mersin, Turkey

Correspondence should be addressed to Hakan Ciftci; rt.ude.izag@ictfich and Hasan Fatih Kisoglu; rt.ude.nisrem@khitafnasah

Received 15 February 2018; Accepted 27 March 2018; Published 29 May 2018

Academic Editor: Chun-Sheng Jia

Copyright © 2018 Hakan Ciftci and Hasan Fatih Kisoglu. 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 energy eigenvalues with any states and mass of heavy quark-antiquark system (quarkonium) are obtained by using Asymptotic Iteration Method in the view of nonrelativistic quantum chromodynamics, in which the quarks are considered as spinless for easiness and are bounded by Cornell potential. A semianalytical formula for energy eigenvalues and mass is achieved via the method in scope of the perturbation theory. The accuracy of this formula is checked by comparing the eigenvalues with the ones numerically obtained in this study and with exact ones in literature. Furthermore, semianalytical formula is applied to c, b, and c meson systems for comparing the masses with the experimental data.

#### 1. Introduction

Investigation of an atomic or subatomic system is done by achieving an energy spectrum of the system. This is carried out, generally, for the events in which the system is bounded by a potential function. Besides, the scattering states or resonance states can also be observed in the investigation of the system. The eigenvalues (or eigenenergies) of Hamiltonian of this system is obtained for a given potential function. In order to do this, various mathematical methods are used in quantum mechanics. One of these, named Asymptotic Iteration Method (AIM), has been commonly used since 2003 [1]. AIM can be used for analytically as well as numerically (or approximately) solvable problems [2–4]. Moreover, it can be used for obtaining the perturbative energy eigenvalues of the system without any need of the unperturbative eigenstate [5, 6].

As a subatomic system, a quarkonium that is composed of a heavy quark-antiquark (q) pair has attracted attention of particle physicists since the first half of 1970, and [7–11] are just a few studies of them. In most of these studies, for easiness, the system is examined via Schrödinger equation in nonrelativistic quantum chromodynamics (NRQCD), assuming that the quarks are spinless [12–15]. Cornell potential is one of the potential functions that represent interactions between the quarks in such a q system. It is used for obtaining the mass and energy spectrum of the quarkonium and obtaining the hadron decay widths [7–9, 16]. Cornell potential is given aswhere and are positive constants ( is in energy dimension). As is seen in (1), Cornell potential has two parts: one is the Coulombic term and the other is the linear part. For obtaining the energy levels and mass of the quarkonium, and may be fitted to the first-few states. Therefore, the full spectrum of the quarkonium can be constructed through these potential parameters.

In literature, it is possible to find many studies in which the solutions of Schrödinger equation for Cornell potential have been obtained. For example, in [17], Hall has found an approximate energy formula to construct an energy spectrum of Schrödinger equation for Cornell potential, under some conditions. Jacobs et al. [13] have compared the eigenvalues of Schrödinger and spinless Salpeter equations in the cases of Cornell potential and Wisconsin potential [18]. Vega and friends have obtained, for* l*=0 states, the energy spectrum, mass, and wavefunctions at the origin for c, b, and b mesons by using the usual variation method in the scope of supersymmetric quantum mechanics (SUSYQM) [19, 20], in [12]. They have also compared their results with the exact ones in literature and with the experimental data.

In this study, we attempted to get the energy eigenvalues (for any states) and masses of heavy mesons by using Asymptotic Iteration Method in the view of NRQCD, in which the quarks are considered as spinless for easiness and are bounded by Cornell potential. We achieved a semianalytical formula for constructing the energy spectrum and obtaining the masses of the mesons, using the method in scope of the perturbation theory. The accuracy of this formula was cross-checked by comparing the eigenvalues with the ones numerically obtained in this study and with the exact ones in literature. Furthermore, semianalytical formula was applied to c, b, and c heavy mesons for comparing the masses with the experimental data.

AIM has been firstly applied to Schrödinger equation for Cornell potential by Hall and Saad in [21]. They have used Airy function as an asymptotic form of the wavefunction and have got highly-accurate numerical results in their study. Alternatively, we obtained a semianalytical mass-energy formula for quarkonium by having differential equation which gives polynomial solutions for asymptotic forms of the wavefunction of the system.

This paper is organized as follows: we give a short summary of AIM in Section 2, while Section 3 includes the main problem. In Section 4, we give numerical results for the eigenenergies and obtain semianalytical energy formula by applying perturbation theory to our problem in the view of AIM. Furthermore, in Section 4, we compare our energy spectrum and masses with the exact ones in literature and with the experimental data. Finally, Section 5 includes some comments about our results.

#### 2. The Asymptotic Iteration Method (AIM)

According to the organization of the paper, we summed up AIM in this section, while it is comprehensively introduced in [1]. The AIM is used to solve second-order homogeneous linear differential equations in the following form:where and have continuous derivatives in the defined interval of the independent variable. If there is an asymptotic condition such asfor , where is large enough, the general solution of (2) is obtained aswith the functions

As a field of application, AIM can be used to deal with Schrödinger equation (or energy eigenvalue problem) in mathematical physics. The eigenvalues can be obtained through the following quantization condition:

If the energy eigenvalues () can be obtained from (6), independently from the variable, the problem is exactly solvable. In this case, the eigenvalue and eigenfunction of th energy level can be derived in explicit algebraic form via iterations. However, there are limited numbers of suitable potentials for this case.

As for the approximately (or numerically) solvable problems, depends on both and . In this case, an appropriate value, , should be determined to solve with respect to [2, 6]. The energy eigenvalue of an th level is obtained through iterations where .

#### 3. Formulation of the Problem

Consider the following Cornell potential:where , are real and positive constants ( is in energy dimension) and . If we substitute into Schrödinger equation in three dimensions, we havewhere , , and . and are energy eigenvalue of th level and reduced mass of the q system, respectively ( and are quark masses). Besides, we study in natural units (i.e., , ) for the system. After changing the variable, in (8), as , then substituting , we get

If one puts into (9), in accordance with the domain of the problem, we havewhere , , , and . The final equation is suitable for applying AIM. After this point, we can apply AIM to the problem in two different ways: one is direct application (i.e., approximate solution) to get the numerical results and the other is usage of the method in scope of perturbation theory to obtain perturbative energies through a perturbation expansion as follows: where , , ,… are perturbation expansion coefficients. These can be obtained independently from the potential parameters. Thus, we can get a semianalytical formula for the energy eigenvalues. One can also achieve the mass-energy of the system by using this formula, as given in Section 4.

##### 3.1. Numerical Results

In this section, we directly apply AIM to (10) to get the energy eigenvalues for different potential parameters, and we compare our results with the perturbative energies, for which (28) in the next section has been used.

From (12), it is easily seen that and according to (2). We tabulate the results of direct application of AIM in Tables 1, 2, and 3. For simplicity, in the calculations, the reduced mass has been considered GeV. In Table 1 the potential parameters have been chosen as and GeV while , GeV in Table 2, and , GeV in Table 3. , seen in the tables, is for the comparison and has been obtained by using (28).