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 $O\left({\alpha }^2\ln \alpha \right)$ corrections known from positronium decays, is demonstrated. It is shown that due to this Coulomb singularity $O\left({\alpha }^2\right)$ 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.