Bethe-Salpeter Bound-State Solutions: Examining Semirelativistic Approaches

TL;DR

This paper explores simplified semirelativistic approaches to solving the Bethe-Salpeter equation for two-particle bound states, assessing their reliability through rigorous spectral constraints and applications to common potentials.

Contribution

It introduces and evaluates semirelativistic approximations to the Bethe-Salpeter equation, providing a practical framework for estimating meson spectra with simplified models.

Findings

Simplified approaches can approximate bound-state spectra effectively.

Rigorous spectral constraints help validate approximate solutions.

Applications to popular potentials demonstrate the method's utility.

Abstract

Within the formalism of relativistic quantum field theory an adequate framework for the description of two-particle bound states, such as, for instance, all conventional (i.e., non-exotic) mesons, is provided by the Poincar\'e-covariant homogeneous Bethe-Salpeter equation. In applications, however, this approach usually proves to be rather involved, whence it is not always quite easy to extract the predictions sought. In view of this, a coarse idea of the bound-state spectrum to be expected might be gained by adhering to some simplifying approximations - which constitutes an entirely legitimate first step. The reliability of the insights inferred from the arising simpler bound-state equation may be straightforwardly examined by taking into account a couple of rigorous constraints on the obtained discrete spectrum. Application of these tools is illustrated for popular potentials.