Uniqueness of two-loop master contours

TL;DR

This paper classifies solutions to the maximal cut of two-loop double-box integrals, revealing unique master contours linked to Riemann surfaces, and demonstrates the use of chiral integrals for simplified amplitude calculations.

Contribution

It provides a complete classification of maximal cut solutions for double-box integrals and establishes the uniqueness of master contours, extending the understanding of two-loop amplitude computations.

Findings

Maximal cut solutions are associated with Riemann surfaces determined by vertex states.

Unique master contours exist for four-massless external momenta cases.

Chiral integrals can serve as master integrals, simplifying two-loop amplitude evaluations.

Abstract

Generalized-unitarity calculations of two-loop amplitudes are performed by expanding the amplitude in a basis of master integrals and then determining the coefficients by taking a number of generalized cuts. In this paper, we present a complete classification of the solutions to the maximal cut of integrals with the double-box topology. The ideas presented here are expected to be relevant for all two-loop topologies as well. We find that these maximal-cut solutions are naturally associated with Riemann surfaces whose topology is determined by the number of states at the vertices of the double-box graph. In the case of four massless external momenta we find that, once the geometry of these Riemann surfaces is properly understood, there are uniquely defined master contours producing the coefficients of the double-box integrals in the basis decomposition of the two-loop amplitude. This is in perfect analogy with the situation in one-loop generalized unitarity. In addition, we point out that the chiral integrals recently introduced by Arkani-Hamed et al. can be used as master integrals for the double-box contributions to the two-loop amplitudes in any gauge theory. The infrared finiteness of these integrals allow for their coefficients as well as their integrated expressions to be evaluated in strictly four dimensions, providing significant technical simplification. We evaluate these integrals at four points and obtain remarkably compact results.