Using differential equations to compute two-loop box integrals

TL;DR

The paper addresses the challenge of performing exclusive, multi-leg observables at two loops by reducing the vast space of integrals to a small set of master integrals using integration-by-parts and Lorentz-invariance identities. It then develops a differential-equation framework in external invariants to compute these master integrals without explicit loop integrations, solving the resulting systems order-by-order in the dimensional regularization parameter $ ext{epsilon}$ and expressing results in terms of harmonic polylogarithms. As a concrete result, the authors obtain the ${ m O}( ext{epsilon})$ term for a specific planar massless double-box combination relevant to $2 o 2$ scattering amplitudes, illustrating the method's viability for two-loop four-point functions. The approach offers a practical alternative to direct loop integration and is extendable to more complex multi-leg two-loop calculations, including potential two-loop virtual corrections to jet observables.

Abstract

The calculation of exclusive observables beyond the one-loop level requires elaborate techniques for the computation of multi-leg two-loop integrals. We discuss how the large number of different integrals appearing in actual two-loop calculations can be reduced to a small number of master integrals. An efficient method to compute these master integrals is to derive and solve differential equations in the external invariants for them. As an application of the differential equation method, we compute the ${\cal O}(ε)$-term of a particular combination of on-shell massless planar double box integrals, which appears in the tensor reduction of $2 \to 2$ scattering amplitudes at two loops.