Master integrals for the NNLO virtual corrections to $\mu e$ scattering in QED: the non-planar graphs

TL;DR

This paper computes the complex two-loop non-planar master integrals for muon-electron scattering in QED, providing essential functions for precise NNLO virtual correction calculations, with broader applications in collider physics.

Contribution

It introduces a canonical basis of master integrals for non-planar diagrams using differential equations and Magnus exponential, including full muon mass dependence and massless electron approximation.

Findings

Derived master integrals as Taylor series with polylogarithm coefficients.

Complete set of integrals needed for NNLO QED corrections to mu-e scattering.

Results applicable to related collider processes like di-muon production.

Abstract

We evaluate the master integrals for the two-loop non-planar box-diagrams contributing to the elastic scattering of muons and electrons at next-to-next-to-leading order in QED. We adopt the method of differential equations and the Magnus exponential to determine a canonical set of integrals, finally expressed as a Taylor series around four space-time dimensions, with coefficients written as a combination of generalised polylogarithms. The electron is treated as massless, while we retain full dependence on the muon mass. The considered integrals are also relevant for crossing-related processes, such as di-muon production at $e^+e^-$ colliders, as well as for the QCD corrections to top-pair production at hadron colliders. In particular, our results, together with the planar master integrals recently computed, represent the complete set of functions needed for the evaluation of the photonic two-loop virtual next-to-next-to-leading order QED corrections to $\mu e \to \mu e$ and $e^+ e^-\to\mu^+\mu^-$.