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

TL;DR

This paper computes two-loop master integrals for muon-electron scattering in QED using differential equations, providing results relevant for collider physics and QCD corrections.

Contribution

It introduces a canonical basis of integrals for NNLO virtual corrections in muon-electron scattering using differential equations and Magnus series methods.

Findings

Derived explicit expressions for master integrals as Taylor series with polylogarithm coefficients.

Retained full muon mass dependence while treating the electron as massless.

Results applicable to collider processes like di-muon production and top-quark pair production.

Abstract

We evaluate the master integrals for the two-loop, 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 series to determine a canonical set of integrals, finally expressed as a Taylor series around four space-time dimensions, with coefficients written as 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.