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-0 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. Leading contributions of n-photon exchange diagrams produce terms proportional to (\alpha/r)^n. The series of leading contributions are summed. The resulting potentials are functions, in the c.m. frame, of r and of the total energy. Their forms are compatible with Todorov's minimal substitution rules proposed in the quasipotential approach.