A Combined Solution of the Schwinger-Dyson and Bethe-Salpeter Equations for Mesons as $q\bar q$ Bound States

TL;DR

This paper combines the Schwinger-Dyson and Bethe-Salpeter equations to accurately compute the mass spectrum of heavy pseudoscalar mesons as quark-antiquark bound states, with implications for future experiments.

Contribution

It presents a novel numerical scheme that effectively solves coupled equations for meson spectra, especially for heavy mesons, improving stability and accuracy.

Findings

Accurately describes ground states of $ ext{-}D_s$ mesons

Extends analysis to excited meson states

Provides results relevant for FAIR physics program

Abstract

The mass spectrum of heavy pseudoscalar mesons, described as quark-antiquark bound systems, is considered within the Bethe-Salpeter formalism with momentum dependent masses of the constituents. This dependence is found by solving the Schwinger-Dyson equation for quark propagators in rainbow-ladder approximation. Such an approximation is known to provide both a fast convergence of numerical methods and accurate results for lightest mesons. However, as the meson mass increases, the method becomes less stable and special attention must be devoted to details of numerical means of solving the corresponding equations. We focus on the pseudoscalar sector and show that our numerical scheme describes fairly accurately the $\pi$, $K$, $D$, $D_s$ and $\eta_c$ ground states. Excited states are considered as well. Our calculations are directly related to the future physics programme at FAIR.