The spin structure of the qq interaction and the mass spectra of bound qq̄ systems: different versions of the 3-dimensional reductions of the Bethe-Salpeter equation

Bound qq̄-systems are considered in the framework of three different versions of the 3-dimensional reduction of the Bethe-Salpeter equation, all having the correct one-body limit when one of the constituent quark masses tends to infinity. The same systems are treated also in the framework of the Salpeter equation. The spin structure of confining qq interaction potential is taken in the form xγ10γ20+(1−x)I1I2, with 0≤x≤1. The problem of existence (nonexistence) of stable solutions of 3-dimensional relativistic equations for bound qq̄ systems is investigated for different values of x from this interval. Some related aspects of this problem are discussed.