Two-Loop Master Integrals for $γ^* \to 3$ Jets: The non-planar topologies

TL;DR

The paper addresses the challenge of computing two-loop four-point master integrals with one off-shell leg for NNLO predictions in electron-positron annihilation. It extends the differential equation method previously applied to planar topologies to a complete set of non-planar topologies, solving for the master integrals as Laurent expansions in the dimensional regulator and expressing results in terms of 2dHPL. The authors classify the integrals by scale (one-, two-, and three-scale) and provide analytic, all-encompassing results for each topology, with boundary conditions carefully fixed to ensure consistency and correct analytic behavior. These results constitute a crucial ingredient for NNLO virtual corrections to three-jet production and related processes, enabling later combination with real-emission and renormalization effects to yield physical predictions, while also matching existing numerical validations.

Abstract

The calculation of the two-loop corrections to the three-jet production rate and to event shapes in electron--positron annihilation requires the computation of a number of two-loop four-point master integrals with one off-shell and three on-shell legs. Up to now, only those master integrals corresponding to planar topologies were known. In this paper, we compute the yet outstanding non-planar master integrals by solving differential equations in the external invariants which are fulfilled by these master integrals. We obtain the master integrals as expansions in $\e=(4-d)/2$, where $d$ is the space-time dimension. The fully analytic results are expressed in terms of the two-dimensional harmonic polylogarithms already introduced in the evaluation of the planar topologies.