Massive Nonplanar Two-Loop Maximal Unitarity

TL;DR

This work extends the maximal unitarity program to nonplanar two-loop crossed-box diagrams with up to four external masses by analyzing the seven-cut (hepta-cut) equations that define a nodal genus-3 surface. It reveals two topological pictures (six- or eight-sphere decompositions) and derives a complete set of contour constraints—parity-odd and IBP—governing the extraction of master-integral coefficients across all kinematic configurations. The authors construct explicit master integral projectors for all mass cases, demonstrating that coefficients can be obtained from well-defined contour integrals of tree amplitudes, with a unified topological underpinning. Additionally, they introduce and benchmark a Bezoutian-matrix algorithm for degenerate multivariate residues, showing dramatic efficiency gains for massive two-loop cuts with doubled propagators. These results provide a practical, geometry-grounded framework for nonplanar two-loop amplitude reductions and pave the way for higher-multiplicity extensions and numerical implementations.

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.

Figures (7)

• Figure 1: The two-loop crossed-box integral. All external particles may be massive.
• Figure 2: The first class of two-loop crossed-box integrals includes the four-mass case and related massless limits, i.e. three-mass and short-side two-mass with massless legs in the nonplanar end of the diagram. Massless and massive external legs are denoted by single and doubled lines respectively.
• Figure 3: The three-mass and the two-mass short-side integrals with massless legs in the planar end of the diagram together with diagonal and long-side two-mass integrals, one-mass integrals and finally the zero-mass integral correspond to degenerate massless limits.
• Figure 4: The four possible vertex configurations of the four-mass two-loop crossed box. Black, white and gray blobs denote chiral, antichiral and nonchiral vertices respectively. The opposite-chirality diagrams are not in one-to-one correspondence with the hepta-cut solutions.
• Figure 5: Topological depiction of the genus-3 algebraic curve defined by the hepta-cut of the prime configuration of the two-loop crossed box primitive amplitude. The one-dimensional complex curve should be understood as the filled two-dimensional real surface. Degeneracies appropriate to specific kinematics arise upon contraction of tubes along straight horizontal and vertical lines in the paper plane through the handles of the surface.

Theorems & Definitions (5)

• Theorem 1
• Theorem 2
• Example 1
• Example 2
• Example 3