# Analytic approach to calculations of mass spectra and decay constants of heavy-light quarkonia in the framework of Bethe-Salpeter equation

## TL;DR

This paper develops an analytic method within the Bethe-Salpeter framework to calculate mass spectra and decay constants of heavy-light quarkonia, improving the treatment of spin structure and including excited states.

## Contribution

It introduces an exact spin structure treatment in the Bethe-Salpeter equation for heavy-light mesons, extending previous approximate methods and analytically solving coupled equations for both ground and excited states.

## Key findings

- Analytic solutions for mass spectra of heavy-light quarkonia.
- Calculated leptonic decay constants for various states.
- Improved accuracy over previous approximate methods.

## Abstract

This work is an extension of the work in \cite{bhatnagar18} to ground and excited states of $0^{++}, 0^{-+}$, and $1^{--}$ of heavy-light ($c\overline{u}, c\overline{s}, b\overline{u}, b\overline{s}$, and $b\overline{c}$) quarkonia in the framework of a QCD motivated Bethe-Salpeter equation (BSE) by making use of the exact treatment of the spin structure $(\gamma_{\mu}\bigotimes\gamma_{\mu})$ in the interaction kernel, in contrast to the approximate treatment of the same in our previous works \cite{hluf16, bhatnagar18}), which is a substantial improvement over our previous works \cite{hluf16,bhatnagar18}. In this $4\times 4$ BSE framework, the coupled Salpeter equations for $Q\overline{q}$ (that are more involved than the equal mass ($Q\overline{Q}$) mesons) are first shown to decouple for the confining part of interaction, under heavy-quark approximation, and analyically solved, and later the one-gluon-exchange interaction is perturbatively incorporated, leading to their mass spectral equations. The analytic forms of wave functions obtained from these equations are then used for calculation of leptonic decay constants of ground and excited states of $0^{-+}$, and $1^{--}$ as a test of these wave functions and the over all framework.