Two - Loop Corrections to the Electromagnetic Vertex for Energies close to Threshold

Two-loop contributions to the electromagnetic form factors are calculated in the kinematic regime close to the fermion-antifermion threshold. The results are presented in an expansion in the velocity β of the fermions in the c.m. frame up to next-to-next-to leading order in β. The existence of a new Coulomb singularity logarithmic in β, which is closely related to the O(α 2 ln α) corrections known from positronium decays, is demonstrated. It is shown that due to this Coulomb singularity O(α 2) relativistic corrections to the non-relativistic cross section of heavy fermion-antifermion pair production in e + e − annihilation cannot be determined by means of conventional multi-loop perturbation theory.