Maximal Unitarity for the Four-Mass Double Box

TL;DR

The paper extends maximal unitarity to two-loop planar double-box integrals with four external masses, formulating master-integral coefficients as generalized discontinuity operator contour integrals over products of tree-level amplitudes. Coefficients are extracted by sevenfold localization to the heptacut surface and a final contour on a Riemann surface of global poles, with parity-odd and IBP constraints shaping the allowed contours. A key finding is that IBP identities do not constrain the four-mass contours, in contrast to fewer-mass cases, while equal-mass kinematics introduce a specific reduction of master integrals. The work ties the contour structure to the underlying algebraic geometry of the Feynman graphs and outlines a path toward an IBP-informed basis for higher-point, multi-loop amplitudes.

Abstract

We extend the maximal-unitarity formalism at two loops to double-box integrals with four massive external legs. These are relevant for higher-point processes, as well as for heavy vector rescattering, VV -> VV. In this formalism, the two-loop amplitude is expanded over a basis of integrals. We obtain formulas for the coefficients of the double-box integrals, expressing them as products of tree-level amplitudes integrated over specific complex multidimensional contours. The contours are subject to the consistency condition that integrals over them annihilate any integrand whose integral over real Minkowski space vanishes. These include integrals over parity-odd integrands and total derivatives arising from integration-by-parts (IBP) identities. We find that, unlike the zero- through three-mass cases, the IBP identities impose no constraints on the contours in the four-mass case. We also discuss the algebraic varieties connected with various double-box integrals, and show how discrete symmetries of these varieties largely determine the constraints.

Figures (3)

• Figure 1: The double-box integral.
• Figure 2: A representation of the pinched torus solution space for the class (a) heptacut kinematics, showing the two independent solutions ${\cal S}_i$, and the locations of the eight global poles ${\cal G}_i$. The small white, black and gray blobs indicate the pattern of chiral, antichiral and nonchiral kinematics, respectively, at the vertices of a double-box integral. Complex-conjugate pairs of poles are identified by reflection through the center of the torus.
• Figure 3: Representations of the solution space for the class (a), (b) and (c) heptacut equations, showing the independent solutions ${\cal S}_i$, and the locations of the global poles ${\cal G}_j$.