Two-Loop Four-Fermion Scattering Amplitude in QED

TL;DR

This paper analytically computes the two-loop corrections to four-fermion scattering amplitudes in QED, including both massless and massive leptons, providing essential ingredients for high-precision cross section calculations.

Contribution

It presents the first complete analytic evaluation of two-loop four-fermion scattering amplitudes in QED with mixed massless and massive leptons, using dimensional regularization and advanced polylogarithms.

Findings

The amplitude is expressed as a Laurent series around four dimensions.

Infrared divergences match Soft Collinear Effective Theory predictions.

Results enable next-to-next-to-leading order cross section computations.

Abstract

We present the analytic evaluation of the two-loop corrections to the amplitude for the scattering of four fermions in Quantum Electrodynamics, $f^- + f^+ + F^- + F^+ \to 0$, with $f$ and $F$ representing a massless and a massive lepton, respectively. Dimensional regularization is employed to evaluate the loop integrals. Ultraviolet divergences are removed by renormalizing the coupling constant in the ${\overline{\text{MS}}}$-scheme, and the lepton mass as well as the external fields in the on-shell scheme. The analytic result for the renormalized amplitude is expressed as Laurent series around $d=4$ space-time dimensions, and contains Generalized Polylogarithms with up to weight four. The structure of the residual infrared divergences of the virtual amplitude is in agreement with the prediction of the Soft Collinear Effective Theory. Our analytic results are an essential ingredient for the computation of the scattering cross section for massive fermion-pair production in massless fermion-pair annihilation, i.e. $f^- f^+ \to F^- F^+$, and crossing related processes such as the elastic scattering $f F \to f F$, with up to Next-to-Next to Leading Order accuracy.