O(5) harmonics and abnormal solutions in the Bethe-Salpeter equation

Abstract

Exact solutions of the covariant Bethe-Salpeter equation in the ladder approximation for two scalar particles bound by a massless particle have been obtained for all energies. By using Fock's stereographic projection, the Bethe-Salpeter equation is transformed on to the surface of a 5-dimensional Euclidean sphere and the solutions are then expressed as a series in O(5) harmonics. The normalization condition has been imposed by requiring that the expectation value of appropriate components of the energy-momentum tensor with respect to the bound states is the total energy of the system; it is found that states corresponding to certain values of the quantum numbers do not satisfy the normalization requirement. These are the so-called abnormal asolutions.