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

TL;DR

This work delivers the analytic evaluation of the 44 master integrals for the two-loop non-planar topology in μe scattering at NNLO, using a differential equations approach and the Magnus expansion to obtain a canonical basis. The results are expressed as a Taylor series around $d=4$ with coefficients in generalised polylogarithms, under the massless electron and massive muon approximation. The non-planar results, together with prior planar results, furnish the complete set of functions needed for the photonic NNLO corrections and cross-related processes such as $e^+e^-\to\mu^+\mu^-$. The paper also details numerical strategies, including a $d=6$ quasi-finite basis for challenging integrals and cross-checks against SecDec, underpinning the practical computation of the full NNLO amplitude and its extensions to QCD analogues.

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 $μe \to μe$ and $e^+ e^-\toμ^+μ^-$.

Figures (3)

• Figure 1: Two-loop four-point topologies for $\mu e$ scattering.
• Figure 2: The 44 MIs $\mathcal{T}_{1,\ldots,44}$ for the two-loop non-planar topology $T_6$. Thin lines represent massless propagators and thick lines stand for massive ones. Each dot indicates an additional power of the corresponding propagator. Numerator insertions are indicated explicitly on top of each diagram.
• Figure 3: Auxiliary three-point integral family for eq. \ref{['eq:limit_i1112']}. Thick and thin lines represent, respectively, massive and massless propagators. The dashed line corresponds to an external leg with arbitrary squared momentum.