Singular Bethe-Salpeter Scattering Amplitudes

The Bethe‐Salpeter equation for scattering is investigated in configuration space for the class of singular ``potentials'' (i.e., a λ$\phi$ ⁴ theory in the ladder approximation) which behave as r⁻⁴ near the light cone. The discussion relies on the similarity between solutions of the Bethe‐Salpeter equation and the Schrödinger equation where the corresponding problem is scattering by a r⁻² potential. Through a consideration of the asymptotic properties of the two‐particle, free Green's function, the elastic scattering amplitude is shown to be the coefficient of the outgoing wave part of the wave‐function, just as it is in the nonrelativistic case. At zero total energy, it is just the coefficient of e^−mr/r^½. The differential form of the Bethe‐Salpeter equation is expanded in four‐dimensional spherical harmonics, and the singular part of the potential is incorporated into the differential operator. The resulting equation is formally solved by converting it into what is now a Fredholm integral equation. Care is taken to choose the proper asymptotic behavior for the solutions of the new equation. The discussion of the singular potential is carried out at zero total energy in order to obtain spherical symmetry. The technique for handling the singular potential and extracting the T matrix at zero energy is demonstrated by application to two examples. The exact scattering amplitude is found for exchange of two massless mesons. A first‐order solution is obtained for a phenomenological potential that approximate the exchange of two massive mesons. This solution exhibits many of the features expected from a truly physical potential.