Intriguing solutions of the Bethe-Salpeter equation for radially excited pseudoscalar charmonia

Abstract

The purpose of this paper is twofold. The first purpose is to find a fully Poincar\'e invariant solution of the Bethe-Salpeter equation (BSE) for excited quarkonia; however, the second, in fact, major focus is on the relevance of the space-time metric choice and its impact on the correct description of the ground and all excited states. For the first time, we compare the BSE solutions defined independently with the Euclidean and Minkowski metric. For this purpose, the BSE is conventionally defined and solved in Euclidean space with two versions of the propagator: the bare propagator and the confined form of the quark propagator with complex conjugated poles. In both considered cases, there is unexpected doubling of the spectrum when comparing to the experiments as well as to the solutions of the Schr\odinger equation. The quark propagator with complex conjugated singularities allows us to find the BSE solution directly in Minkowski momentum space as well. We find the Minkowski space solution for confining theories is not only numerically accessible but provides a reliable albeit not yet completely satisfactory description of the ground and excited meson states.