Excited s-wave 1−− vector mesons, their leptonic decays and (in-)complete absence of abnormal states as seen from the constituent quark BSE

Within a lattice inspired interaction between quark and antiquark, we obtain a hierarchy of solutions to the Bethe-Salpeter equation (BSE) for vector quarkonia excited states in the constituent quark mass approximation. As a toy model, we apply the similar to calculate ground and excited states of ϕ and $\omega /\rho$ meson. Through detailed numerical searches, our study provides evidence that the single-valued constant mass quark propagator does not yield known meson spectra without the simultaneous presence of abnormal (unphysical) solutions. We classify normal and abnormal solutions and discuss necessary changes in the calculation scheme to avoid the spectrum of inconsistent solutions. While all experimental narrow vector mesons are identified with a normal state of the BSE, the occurrence of mutually cancelled normal-abnormal states are reported. The results are slightly different for the heavy and light mesons, but in both cases they seem to result from the inconsistent use of the homogeneous BSE to describe broad resonances.