Nonrenormalizability and the Short-Range Force in Some Field-Theoretic Models

Abstract

Bethe-Salpeter equations for the scattering Green's functions are discussed in some nonrenormalizable models. The models involve the multiple exchanges of pairs of Dirac particles, coupled to the scattering particles by a four-Fermi interaction. The Green's function is constructed from a Bethe-Salpeter scattering wave function. For certain forms of coupling the forces are repulsive at short distances and analogous to potentials in a nonrelativistic scattering problem which behave as $r^{-6\mathrm{}}$ at the origin. In these cases there is a unique, well defined, and Fourier transformable solution for the Green's function in space-time; and the scattering amplitude exists. A class of terms corresponding to delta-function potentials may be included in the interaction kernel without changing the solution. In a case with zero-mass particles and zero total energy an exact solution for the scattering amplitude is obtained.