Boundqq¯systems in the framework of the different versions of the three-dimensional reductions of the Bethe-Salpeter equation

Abstract

Bound $q\overline{q}$ systems are studied in the framework of different three-dimensional relativistic equations derived from the Bethe-Salpeter equation with the instantaneous kernel in the momentum space. Except the Salpeter equation, all these equations have a correct one-body limit when one of the constituent quark masses tends to infinity. The spin structure of the confining $\mathrm{qq}$ interaction potential is taken in the form $x{\ensuremath{\gamma}}_{1}^{0}{\ensuremath{\gamma}}_{2}^{0}+(1\ensuremath{-}{x)I}_{1}{I}_{2},$ with $0<~x<~1.$ At first stage, the one-gluon-exchange potential is neglected and the confining potential is taken in the oscillator form. For the systems $(u\overline{s}),(c\ifmmode \bar{u}\else \={u}\fi{}),(c\overline{s})$ and $(u\ifmmode \bar{u}\else \={u}\fi{}),(s\overline{s})$ a comparative qualitative analysis of these equations is carried out for different values of the mixing parameter x and the confining potential strength parameter. We investigate (1) the existence/nonexistence of stable solutions of these equations, (2) the parameter dependence of the general structure of the meson mass spectum and leptonic decay constants of pseudoscalar and vector mesons. It is demonstrated that none of the three-dimensional equations considered in the present paper simultaneously describes even general qualitative features of the whole mass spectrum of $q\overline{q}$ systems. At the same time, these versions give an acceptable description of the meson leptonic decay characteristics.