Global Residues and Two-Loop Hepta-Cuts

TL;DR

This paper derives compact analytic formulas for two-loop master integral coefficients in massless four-point amplitudes, advancing automated amplitude computations in gauge theories.

Contribution

It introduces a novel approach to compute two-loop nonplanar integral coefficients using residues on higher-dimensional tori, unifying maximal unitarity with integrand reduction methods.

Findings

Derived explicit formulas for two-loop crossed box coefficients

Established equivalence with integrand-level reduction results

Provided several detailed calculations

Abstract

We examine maximal unitarity in the nonplanar case and derive remarkably compact analytic expressions for coefficients of master integrals with two-loop crossed box topology in massless four-point amplitudes in any gauge theory, thereby providing additional steps towards automated computation of the full amplitude. The coefficients are obtained by assembling residues extracted through integration on linear combinations of higher-dimensional tori encircling global poles of the loop integrand. We recover all salient features of two-loop maximal unitarity, such as the existence of unique projectors for each master integral. Several explicit calculations are provided. We also establish exact equivalence of our results and master integral coefficients recently obtained via integrand-level reduction in any renormalizable gauge theory.