Maximal Unitarity for the Four-Mass Double Box

TL;DR

This paper extends the maximal unitarity method to two-loop double-box integrals with four massive external legs, providing formulas for amplitude coefficients as products of tree amplitudes over specific contours, relevant for complex scattering processes.

Contribution

It introduces a novel extension of the maximal unitarity formalism to four-mass double-box integrals, detailing contour conditions and algebraic structures, with no IBP constraints on contours in this case.

Findings

Contours are unconstrained by IBP identities in four-mass case

Formulas express amplitude coefficients as products of tree amplitudes

Algebraic varieties and symmetries largely determine contour constraints

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.