Spectra of light and heavy mesons with \(J \le 5\) in a relativistic Bethe–Salpeter approach

Abstract

We extend the range of application of the relativistic Dyson–Schwinger/Bethe–Salpeter approach from previously discussed mesons with total angular momentum \(J \le 3\) to the ones with \(J=4,5\). On a technical level, the new element is the general Dirac tensor representations for the latter which, to our knowledge, are presented here for the first time. As a first application, we provide an exploratory spectrum for these mesons in a rainbow-ladder truncation of Dyson–Schwinger and Bethe–Salpeter equations. We discuss the merits and limitations of this truncation and explore the shape of the heavy-quark potential corresponding to the underlying effective running coupling. With our predictions for the masses of ground state mesons with quantum numbers \(J^{\text {PC}}=3^{--}, 4^{++}, 5^{--}\) we identify Regge trajectories in channels where the interaction model can be trusted on a semi-quantitative level. In other channels, discrepancies with experiments confirm the well-known need to go beyond rainbow-ladder in the Dyson–Schwinger/Bethe–Salpeter approach by using more sophisticated interactions.