Advances in High Energy Physics

Volume 2017 (2017), Article ID 7306825, 8 pages

https://doi.org/10.1155/2017/7306825

## Radiative Decay Widths of Ground and Excited States of Vector Charmonium and Bottomonium

^{1}Department of Physics, Samara University, P.O. Box 132, Samara, Ethiopia^{2}Department of Physics, University Institute of Sciences, Chandigarh University, Mohali 140413, India

Correspondence should be addressed to Hluf Negash

Received 26 January 2017; Accepted 14 May 2017; Published 20 June 2017

Academic Editor: Thierry Grandou

Copyright © 2017 Hluf Negash and Shashank Bhatnagar. 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

We study the radiative decay widths of vector quarkonia for the process of and (for principal quantum numbers ) in the framework of Bethe-Salpeter equation under the covariant instantaneous ansatz using a form of BSE. The parameters of the framework were determined by a fit to the mass spectrum of ground states of pseudoscalar and vector quarkonia, such as , , , and . These input parameters so fixed were found to give good agreements with data on mass spectra of ground and excited states of pseudoscalar and vector quarkonia, leptonic decay constants of pseudoscalar and vector quarkonia, two-photon decays, and two-gluon decays of pseudoscalar quarkonia in our recent paper. With these input parameters so fixed, the radiative decay widths of ground and excited states of heavy vector quarkonia ( and ) are calculated and found to be in reasonable agreement with data.

#### 1. Introduction

Studies on mass spectra and decays of heavy quarkonia ( and ) have become a hot topic in recent years, due to observation of many new states at various high energy accelerators at BABAR, Belle, CLEO, and BES-III collaborations [1–4]. All this has opened up new challenges in theoretical understanding of heavy hadrons and provide an important tool for exploring the structure of these simplest bound states in QCD and for studying the nonperturbative (long distance) behavior of strong interactions.

As regards the dynamical framework, to investigate these properties is concerned; many nonperturbative approaches, such as lattice QCD [5], chiral perturbation theory [6], QCD sum rules [7], heavy-quark effective theory [8], NRQCD [9], dynamical-equation based approaches like Schwinger-Dyson equation and Bethe-Salpeter equation (BSE) [10–15], and potential models [16] have been proposed to deal with the long distance property of QCD.

Bethe-Salpeter equation (BSE) [11, 15, 17–19] is a conventional approach in dealing with relativistic bound state problems. From the solutions, we can obtain useful information about the inner structure of hadrons, which is also crucial in treating hadronic decays. The BSE framework which is firmly rooted in field theory provides a realistic description for analyzing hadrons as composite objects. Despite its drawback of having to input model-dependent kernel, these studies have become an interesting topic in recent years, since calculations have shown that BSE framework using phenomenological potentials can give satisfactory results as more and more data are being accumulated. The BSE is frequently adopted as starting point of QCD inspired models, due to the fact that this equation has a firm base in quantum field theory.

In a recent work [20, 21], we employed a representation for two-body () BS amplitude for calculating both the mass spectra and the transition amplitudes for various processes. However, the price one pays in this approach is to solve a coupled set of Salpeter equations for both pseudoscalar and vector quarkonia. However, in [21], we explicitly showed that these coupled Salpeter equations can indeed get decoupled in the heavy-quark approximation, leading to mass spectral equations with analytical solutions for both masses, as well as eigenfunctions for all the ground and excited states of pseudoscalar and vector and systems in an approximate harmonic oscillator basis. These analytical forms of eigenfunctions for ground and excited states so obtained were used to evaluate the transition amplitudes for different processes in [21]. Thus, in [21], we had calculated the mass spectrum, weak decay constants, two-photon decay widths and two-gluon decay widths of ground and radially excited () states of pseudoscalar charmonium and bottomonium such as and , as well as the mass spectrum and leptonic decay constants of ground state and excited (, , , , , and ) states of vector charmonium and bottomonium such as and , using this formulation of Bethe-Salpeter equation under covariant instantaneous ansatz (CIA). Our results were in good agreement with data (wherever available) and other models. However, in all the above processes, the quark antiquark loop involved a single hadron-quark vertex, which was simple to handle.

However for the transitions such as , the process requires calculation of triangle quark-loop diagram involving two hadron-quark vertices and is difficult to evaluate in BSE-CIA, which gives rise to complexities in amplitudes. However in [22, 23], they demonstrated an explicit mathematical procedure for handling such problems in representation of BSE. Thus, in the present work, we will precisely apply an instantaneous formalism employing BSE under CIA for transitions involving the process, , where such problems do not enter in the calculations of [21].

This paper is organized as follows. In Section 2, we give the formulation of BSE under CIA. In Section 3, we give the derivation of the hadronic process in the framework of BSE under CIA and calculate its radiative decay widths. The numerical results for radiative decay widths of the processes are worked out. Section 4 is reserved for discussions and conclusion.

#### 2. Formulation of BSE under CIA

We give a short derivation of Salpeter equations in this section, giving only the main steps. The 4D BSE for comprising equal mass fermionic quarks of momenta and masses (=*m*), respectively, is written in representation aswhere the BS wave function is sandwiched between the inverse propagators of the quark and the antiquark, whose individual momenta are related to the internal momentum and total momentum of hadron of mass as . We further decompose the internal momentum, , as the sum of its transverse component, (which is orthogonal to total hadron momentum ), and the longitudinal component, (which is parallel to ). Thus, , where the transverse component, , is an effective 3D vector, while the longitudinal component, , plays the role of the time component. The 4D volume element in this decomposition is . To obtain the 3D BSE and the hadron-quark vertex, use an ansatz on the BS kernel in (1) which is assumed to depend on the 3D variables , asHence, the longitudinal component, of , does not appear in the form of the kernel and we define 3D wave function asSubstituting (3) in (1), with definition of kernel in (2), we get a covariant version of Salpeter equationand the 4D BS wave function can be written aswhereplays the role of hadron-quark vertex function. Following a sequence of steps given in [20, 21], we obtain four Salpeter equations:with the energy projection operators, , , and , where for quarks and antiquarks, respectively. The projected wave functions, , in Salpeter equations are obtained by the operation of the above projection operators on (for details see [20, 21]) as

To obtain the mass spectral equation, we have to start with the above four Salpeter equations and solve the instantaneous Bethe-Salpeter equation. However, the last two equations do not contain eigenvalue and are thus employed to obtain constraint conditions on the Bethe-Salpeter amplitudes associated with various Dirac structures in , as shown in detail in [21]. The framework is quite general so far. In fact the above four equations constitute an eigenvalue problem that should lead to evaluation of mass spectra of pseudoscalar and vector charmonium and bottomonium states such as , , , and (see [21]). The numerical results [21] of mass spectra of* P* and* V* quarkonia participating in the radiative decays studied in this paper are listed in Section 3. We now give details of calculation of decay widths for the process, , in the next section.

#### 3. Electromagnetic Transition of

The lowest order, Feynman diagrams for the process, , are given in Figure 1, where and are vector and pseudoscalar quarkonia, respectively. The second diagram is obtained from the first one by reversing the directions of internal fermionic lines.