Meson theory with internal coordinates

This paper is a continuation of the preceding paper in which a generalized Bethe-Salpeter equation in the ladder approximation for a quark-antiquark pair is presented. The generalized equation includes functions of a set of three complex coordinates, called internal coordinates, spanning an abstract three-dimensional complex space. These functions are essential in describing mesons. This generalized equation is treated in this paper. Upon consistent restrictions, it is reduced, in the case of pseudoscalar mesons, to one nine-component tensor equation involving internal coordinates only. Keeping the zero-order S${\mathrm{U}}_3$-symmetry-preserving interaction only, the tensor equation is further reduced to two coupled radial equations and finally to two algebraic recurrence relations for the ${\eta }^{\prime }$ meson. In the $\eta$-meson case, four such relations are obtained. Preliminary treatment of the ${\eta }^{\prime }$ case indicates that the internal interaction is strong. By including the interaction term transforming like the eighth component of an S${\mathrm{U}}_3$ octet vector as a first-order perturbation, the Gell-Mann-Okubo formula is reproduced with the coefficients determined by given relations. Necessary removal of possible degeneracy in the zero-order states leads to mixing of these states.