Massive Nonplanar Two-Loop Maximal Unitarity

TL;DR

This paper advances the maximal unitarity method for nonplanar two-loop integrals with massive external legs, revealing the algebraic geometry of the cuts and providing an algorithm for generalized cuts with higher propagator powers.

Contribution

It introduces a novel geometric analysis of nonplanar two-loop cuts and an algorithm for computing generalized cuts using the Bezoutian matrix method.

Findings

Identifies algebraic curves associated with nonplanar double box cuts.

Classifies topological configurations of external masses into two types.

Provides an algorithm for higher-power propagator cuts.

Abstract

We explore maximal unitarity for nonplanar two-loop integrals with up to four massive external legs. In this framework, the amplitude is reduced to a basis of master integrals whose coefficients are extracted from maximal cuts. The hepta-cut of the nonplanar double box defines a nodal algebraic curve associated with a multiply pinched genus-3 Riemann surface. All possible configurations of external masses are covered by two distinct topological pictures in which the curve decomposes into either six or eight Riemann spheres. The procedure relies on consistency equations based on vanishing of integrals of total derivatives and Levi-Civita contractions. Our analysis indicates that these constraints are governed by the global structure of the maximal cut. Lastly, we present an algorithm for computing generalized cuts of massive integrals with higher powers of propagators based on the Bezoutian matrix method.