Spectroscopy of ground and excited states of pseudoscalar and vector charmonium and bottomonium

TL;DR

This paper calculates the mass spectra, decay constants, and decay widths of ground and excited states of pseudoscalar and vector charmonium and bottomonium using the Bethe-Salpeter equation with covariant instantaneous Ansatz, providing results consistent with experimental data.

Contribution

It introduces a comprehensive Bethe-Salpeter framework with a 4x4 representation for two-body amplitudes, enabling analytical solutions for quarkonium spectra and transition amplitudes.

Findings

Results agree well with experimental data and other models.

Analytical solutions for mass spectra and eigenfunctions obtained.

Decoupling of pseudoscalar and vector equations in heavy-quark limit.

Abstract

In this work we calculate the mass spectrum, weak decay constants, two photon decay widths, and two gluon decay widths of ground (1S), and radially excited (2S,3S,...) states of pseudoscalar charmoniuum and bottomonium such as $\eta_{c}$ and $\eta_{b}$, as well as the mass spectrum and leptonic decay constants of ground state (1S), excited (2S,1D,3S,2D,4S,and 3D) states of vector charmonium and bottomonium such as $J/\psi$, and $\Upsilon$, using the formulation of Bethe-Salpeter equation under covariant Instantaneous Ansatz (CIA). Our results are in good agreement with data (where ever available) and other models. In this framework, from the beginning, we employ a $4\times 4$ representation for two-body ($q\overline{q}$) BS amplitude for calculating both the mass spectra as well as the transition amplitudes. However, the price we have to pay is to solve a coupled set of equations for both pseudoscalar and vector quarkonia, which we have explicitly shown get decoupled in the heavy-quark approximation, leading to mass spectral equation with analytical solutions for both masses, as well as eigenfunctions for all the above states, in an approximate harmonic oscillator basis. Further, in the present framework of BSE, the hadron-quark vertex function (and the full 4D BS wave function) for pseudoscalar and vector quarkonia used for calculation of various processes, accommodates all the Dirac structures from their complete set in a natural manner.