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(α²lnα) corrections known from positronium decays, is demonstrated. It is shown that due to this Coulomb singularity O(α²) relativistic corrections to the nonrelativistic cross section of heavy-fermion--antifermion pair production in e⁺e^- annihilation cannot be determined by means of conventional multiloop perturbation theory.