Progress on two-loop non-propagator integrals

TL;DR

This work surveys the progress in evaluating two-loop non-propagator integrals relevant for exclusive observables, with emphasis on two-loop four-point functions. It highlights three key technical advances: reduction of vast integral families to a small set of master integrals using IBP and LI identities (Laporta method), computation of these masters via differential equations, and the use of harmonic polylogarithms, extended to two dimensions (2dHPL), to express results. The authors report comprehensive analytic results for massless two-loop four-point functions with all legs on-shell, obtained via Mellin-Barnes techniques and differential equations, expressed in terms of Nielsen polylogarithms, and successful applications to Bhabha scattering and quark–quark scattering. For configurations with one off-shell leg, the differential-equation approach yields master integrals in terms of 2dHPL, with divergent parts linked to Nielsen polylogarithms and finite parts as one-dimensional integrals over generalized polylogarithms; these master integrals are crucial for NNLO corrections to processes like three-jet production and vector-boson–plus–jet production. The outlook emphasizes that full NNLO predictions require combining these virtual corrections with real-emission and PDF effects, with early demonstrations indicating the feasibility of such programs.

Abstract

At variance with fully inclusive quantities, which have been computed already at the two- or three-loop level, most exclusive observables are still known only at one loop, as further progress was hampered up to very recently by the greater computational problems encountered in the study of multi-leg amplitudes beyond one loop. We discuss the progress made lately in the evaluation of two-loop multi-leg integrals, with particular emphasis on two-loop four-point functions.