Some Aspects of the Covariant Two-Body Problem. II. The Scattering Problem

By way of extension of a previous study of the bound-state problem within the framework of the covariant Bethe-Salpeter formalism, the scattering of nucleons by nucleons has been investigated in the ladder approximation, assuming scalar nucleons and zero-mass scalar mesons. The solution of the scattering problem can be effected by using an integral-transform method which is similar to that developed for the bound-state problem, subject to the assumption that the nucleon mass is large compared to the kinetic energy in the center-of-mass system. An essential complication which is now encountered is the appearance of an infrared divergence. Two methods of circumventing this difficulty are discussed. The cross section can be calculated from the Green's function by a limiting process; the usual $S$-matrix formalism leads to incorrect results in this case, and an amplitude renormalization is required. In this connection, it is instructive to re-examine, in detail, the Coulomb-scattering problem in momentum space, since this is very closely related to the Bethe-Salpeter scattering problem. The Green's-function method turns out to be unsuitable for the calculation of higher-order corrections to the nucleon-nucleon scattering cross section, so that here a cutoff procedure must be used.