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 $\beta$ of the fermions in the c.m. frame up to next-to-next-to leading order in $\beta$. The existence of a new Coulomb singularity logarithmic in $\beta$, which is closely related to the $O(\alpha^2\ln\alpha)$ corrections known from positronium decays, is demonstrated. It is shown that due to this Coulomb singularity $O(\alpha^2)$ 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.