Global Residues and Two-Loop Hepta-Cuts

TL;DR

This work extends maximal unitarity to the nonplanar two-loop four-point amplitude, deriving compact analytic expressions for the coefficients of two master integrals in the massless crossed-box topology. By formulating maximal cuts as contour integrals around global poles in higher dimensions and enforcing reduction identities, the authors construct unique master integral projectors and express the master-coefficient results in terms of residues of products of tree amplitudes. They demonstrate exact equivalence with integrand-level reduction results across renormalizable gauge theories and provide explicit helicity configurations to validate the formalism. The approach paves the way for automated two-loop amplitude computations and offers a framework that can be extended to massive legs, $D$-dimensional unitarity, and higher-loop generalizations.

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.

Figures (9)

• Figure 1: The pentabox-triangle and planar double box topologies appearing in the color-decomposition of the two-loop four-point amplitude.
• Figure 2: The four-point massless one-loop box diagram.
• Figure 3: The massless four-point planar double box diagram. External momenta are by convention taken as outgoing.
• Figure 4: Momentum flow for the twoloop crossed box.
• Figure 5: A view of the global structure of the eight on-shell solutions for the massless twoloop crossed box. The set of solutions has ten intersections and each branch is topologically equivalent to a Riemann sphere. Our convention is to denote holomorphic and antiholomorphic vertices is by $\oplus$ and $\ominus$ respectively.