On the Integrand-Reduction Method for Two-Loop Scattering Amplitudes

TL;DR

This paper introduces a novel integrand-reduction method for two-loop scattering amplitudes, enabling polynomial fitting of integrands to determine master integrals without prior basis knowledge, advancing automated amplitude calculations.

Contribution

It presents the first implementation of integrand reduction at two loops, extending one-loop techniques to facilitate systematic, semi-analytic amplitude evaluations in gauge theories.

Findings

Successfully applied to planar and non-planar 4- and 5-point MHV amplitudes in N=4 SYM

Demonstrates polynomial residues determine master integral basis

Extends integrand-reduction techniques from one-loop to two-loop amplitudes

Abstract

We propose a first implementation of the integrand-reduction method for two-loop scattering amplitudes. We show that the residues of the amplitudes on multi-particle cuts are polynomials in the irreducible scalar products involving the loop momenta, and that the reduction of the amplitudes in terms of master integrals can be realized through polynomial fitting of the integrand, without any apriori knowledge of the integral basis. We discuss how the polynomial shapes of the residues determine the basis of master integrals appearing in the final result. We present a four-dimensional constructive algorithm that we apply to planar and non-planar contributions to the 4- and 5-point MHV amplitudes in N=4 SYM. The technique hereby discussed extends the well-established analogous method holding for one-loop amplitudes, and can be considered a preliminary study towards the systematic reduction at the integrand-level of two-loop amplitudes in any gauge theory, suitable for their automated semianalytic evaluation.