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

TL;DR

The study computes two-loop corrections to the electromagnetic vertex for energies near the fermion–antifermion threshold, using NRQCD/NRQED in a $\beta$-expansion. It reveals a new logarithmic Coulomb singularity at NNLO, $O(\alpha^2\ln\beta)$, arising from the fermion–antifermion wave function and requiring all-order resummation for reliable threshold cross sections. The results, obtained via dispersion relations and separated into $2\gamma$ and vacuum-polarization contributions, confirm that fixed-order perturbation theory cannot capture the full NNLO relativistic corrections and provide explicit expressions for the leading singular terms. These insights are crucial for precise threshold predictions in QED and for matching NRQCD descriptions of heavy quark production/decay near threshold, with implications mirroring known positronium corrections.

Abstract

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 $O(α^2\lnα)$ 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.