Advances in High Energy Physics

Volume 2018, Article ID 5961031, 12 pages

https://doi.org/10.1155/2018/5961031

## S-Wave Heavy Quarkonium Spectra: Mass, Decays, and Transitions

Physics Department, Faculty of Arts and Sciences, Ondokuz Mayis University, 55139 Samsun, Turkey

Correspondence should be addressed to Halil Mutuk; moc.liamg@kutumlilah

Received 20 August 2018; Revised 18 October 2018; Accepted 30 October 2018; Published 25 November 2018

Guest Editor: Xian-Wei Kang

Copyright © 2018 Halil Mutuk. 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 we revisited phenomenological potentials. We studied S-wave heavy quarkonium spectra by two potential models. The first one is power potential and the second one is logarithmic potential. We calculated spin averaged masses, hyperfine splittings, Regge trajectories of pseudoscalar and vector mesons, decay constants, leptonic decay widths, two-photon and two-gluon decay widths, and some allowed M1 transitions. We studied ground and 4 radially excited S-wave charmonium and bottomonium states via solving nonrelativistic Schrödinger equation. Although the potentials which were studied in this paper are not directly QCD motivated potential, obtained results agree well with experimental data and other theoretical studies.

#### 1. Introduction

Heavy quarkonium is the bound state of and and one of the most important playgrounds for our understanding of the strong interactions of quarks and gluons. Quantum chromodynamics (QCD) is thought to be the* true* theory of these strong interactions. QCD is a nonabelian local gauge field theory with the symmetry group . In principle, one should be able to calculate hadronic properties such as mass spectrum and transitions by using QCD principles. But QCD does not readily supply us these hadronic properties. This challenge can be attributed to the several features that are not present in other local gauge field theories.

Foremost, being a nonabelian gauge theory, gluons which are gauge bosons, have color charge and interact among themselves. Unlike from quantum electrodynamics (QED), where a photon does not interact with other photon, in QCD one must consider interactions among gluons. This nonabelian nature of the theory makes some calculations complicated, for example, loops in propagators.

There are three other important features of QCD:* asymptotic freedom*,* confinement,* and* dynamical breaking of chiral symmetry*. Asymptotic freedom says that strong interaction coupling constant, , is a function of momentum transfer. When the momentum transfer in a quark-quark collision increases (at short distances), the coupling constant becomes weaker whereas it becomes larger when momentum transfer decreases (at large distances). The idea behind confinement is that, there are no free quarks outside of a hadron; i.e., color charged particles (quarks and gluons) cannot be isolated out of hadrons. Flux tube model gives a reasonable explanation of confinement. When the distance between quark-antiquark (or quarks) pair increases, the gluon field between a pair of color charges forms a flux tube (or string) between them resulting a potential energy which depends linearly on the distance, where is the string constant. As distance increases between quarks, the potential energy can create new quark-antiquark pairs in colorless forms instead of a free quark. Up to now, nobody has been able to prove that confinement from QCD. Lattice QCD calculations simulate this confinement well and give a value for the string tension [1]. The last feature of QCD is the dynamical breaking of chiral symmetry. The QCD Lagrangian with quark flavor has an exact chiral symmetry but breaks down to symmetry because of the nonvanishing expectation value of QCD vacuum [2, 3]. The Goldstone bosons corresponding to this symmetry breaking are the pseudoscalar mesons.

The present aspects of the QCD caused other approaches to deal with these challenges. QCD sum rules, Lattice QCD, and potential models (quark models) are examples of these approaches. These approaches are nonperturbative since the strong interaction coupling constant, which should be the perturbation parameter of QCD is of the order one in low energies, hence the truncation of the perturbative expansion cannot be carried out. Since perturbation theory is not applicable, a nonperturbative approach has to be used to study systems that involve strong interactions. QCD sum rules and lattice QCD are based on QCD itself whereas in potential models, one assumes an interquark potential and solves a Schrödinger-like equation. The advantage of potential model is that, excited states can be studied in the framework of potential models whereas in QCD sum rules and lattice QCD, only the ground state or in some exceptional cases excited states can be studied.

After the discovery of charmonium () states, potential models have played a key role in understanding of heavy quarkonium spectroscopy [4, 5]. These potentials were in type of Coulomb plus linear confining potential with spin dependent interactions. The discovery of bottomonium () states were well described by the potential model picture which was used in the charmonium case. Heavy quarkonium spectroscopy was studied since that era with fruitful works [6–18]. A general review about potential models can be found in [19, 20] and references therein.

In the potential models, many features such as mass spectra and decay properties of heavy quarkonium could be described by an interquark potential in two-body Schrödinger equation. Interquark potentials are obtained both from phenomenology and theory. In the phenomenological method, it is assumed that a potential exist with some parameters to be determined by fits to the data. In the theory side, one can use perturbative QCD to determine the potential form at short distances and use lattice QCD at long distances [19]. These potentials can be classified as QCD motivated potentials [21–25] and phenomenological potentials [26–31]. The most commonly used phenomenological potentials are power-law potentials, for example [26] and logarithmic potentials, for example [30]. The detailed properties of these type potentials are studied extensively in [29]. All the potentials which are mentioned here have almost similar behaviour in the range of which is characteristic region of charmonium and bottomonium systems [32, 33]. Outside the range, the behaviour of potentials differ. Up to now, no one was able to obtain a potential which is compatible at the whole range of distances by using QCD principles.

The potential model calculations have been quite successful in describing the hadron spectrum. Most of the phenomenological potentials must satisfy the following conditions:It means that static potential is a monotone nondecreasing and concave function of which is a general property of gauge theories [34].

The great success of quarkonium phenomenology was somehow cracked at 2003 after the observation of [35]. The properties of this exotic particle are not compatible with the conventional quark model, the reason why it is named* exotic*. For example in [36], the authors studied near threshold zero in the S-wave. There are other exotic states, , and the exotic particle zoo is growing. In this paper we will present some exotic states in the framework of quark model.

Energy spectra of heavy quarkonium are a rich source of the information on the nature of interquark forces and decay mechanisms. The prediction of mass spectrum in accordance with the experimental data does not verify the validity of a model for explaining hadronic interactions. Different potentials can produce reliable spectra with the experimental data. Thus other physical properties such as decay constants, leptonic decay widths, radiative decay widths, etc. need to be calculated.

A specific form of the QCD potential in the whole range of distances is not known. Therefore one needs to use potential models. In this work we revisited a power-law potential [26] and a logarithmic potential [30] to study S-wave heavy quarkonium. These potentials satisfy Eqn. (1), i.e. having nonsingular behaviour for . For our purposes, it must be mentioned that power-law and logarithmic potentials have nice scaling properties when used with a nonrelativistic Schrödinger equation [19]. We generated S-wave charmonium and bottomonium mass spectrum with the decays and M1 transitions. At Section 2 we give out theoretical model. In Sections 3 and 4, we generate S-wave heavy quarkonium spectrum, decays and transitions. In Section 5 we discuss our results and in Section 6 we conclude our results.

#### 2. Formulation of the Model

When quark model was proposed, many authors treated baryons in detail with the harmonic oscillator quark model by using harmonic oscillator wave functions [37–39]. Mesons comparing to baryons are simpler objects since they are composites of two quarks. The reason for using harmonic oscillator wave function is that they form a complete set for a confining potential [40].

In order to obtain mass spectra, we solved Schrödinger equation by variational method. The variational method by using harmonic oscillator wave function gave successful results for heavy and light meson spectrum [15, 41, 42]. The procedure for this method is calculating expectation value of the Hamiltonian via the trial wave function:The mass of the meson is found by adding two times the mass of quark to the eigenenergyThe Hamiltonian we consider iswhere , is the relative momentum, is the reduced mass, and is the potential between quarks. The spectrum can be obtained via solving Schrödinger equationwith the harmonic oscillator wave function defined asHere is the radial wave function given aswith the associated Laguerre polynomials and the normalization constant is the well-known spherical harmonics.

Armed with these, the expectation value of the given Hamiltonian can be calculated. In the variational method, one chooses a trial wave function depending on one or more parameters and then finds the values of these parameters by minimizing the expectation value of the Hamiltonian. It is a good tool for finding ground state energies but as well as energies of excited states. The condition for obtaining excited states energies is that the trial wave function should be orthogonal to all the energy eigenfunctions corresponding to states having a lower energy than the energy level considered. In (7), is treated as a variational parameter and it is determined for each state by minimizing the expectation value of the Hamiltonian.

In the following sections we study power-law and logarithmic potentials in order to obtain full spectrum.

#### 3. Mass Spectra of Power-Law and Logarithmic Potentials

Power-law potential is given by [26]They showed that upsilon and charmonium spectra can be fitted with that potential. The small power of refers to a situation in which the spacing of energy levels is independent of the quark masses. This situation is also valid for the purely logarithmic potential [30]

At first step we obtained spin averaged mass spectrum for and systems, respectively. The constituent quark masses are and for power-law potential and and for logarithmic potential. Table 1 shows the charmonium spectrum and Table 2 shows the bottomonium spectrum.