The relativistic two-body potentials of constraint theory from summation of Feynman diagrams

The relativistic two-body potentials of constraint theory for systems composed of two spin-O or two spin-1/2 particles are calculated, in perturbation theory, by means of the Lippmann-Schwinger type equation that relates them to the scattering amplitude. The cases of scalar and vector interactions with massless photons are considered. The two-photon exchange contributions, calculated with covariant propagators, are globally free of spurious infra-red singularities and produce at leading order O(alpha(4)) effects that can be represented in three-dimensional x-space by local potentials proportional to (alpha/r)(2). An approximation scheme, that adapts the eikonal approximation to the bound state problem, is deviced and applied to the evaluation of leading terms of higher order diagrams. Leading contributions of mn-photon exchange diagrams produce terms proportional to (alpha/r)''. The series of leading contributions are summed. The resulting potentials are functions, in the c.m. frame, of I and of the total energy. Their forms are compatible with Todorov's minimal substitution rules proposed in the quasipotential approach. (C) 1997 Academic Press.