The Integrand Reduction of One- and Two-Loop Scattering Amplitudes

TL;DR

This work surveys integrand-reduction approaches to scattering amplitudes, highlighting how on-shell singularities organize a decomposition into Master Integrals through residue polynomials and process-dependent coefficients. It introduces Laurent-expansion refinements at one loop and a multivariate polynomial division framework that generalizes reduction to arbitrary loop orders, with a loop-order-agnostic recurrence based on irreducible scalar products. The methods are demonstrated on two-loop five-point amplitudes in N=4 SYM and N=8 SUGRA, using both semi-numerical and analytic Laurent-expansion techniques. Overall, the paper presents a universal, all-orders toolbox for integrand-level reductions with practical implications for precision perturbative calculations in gauge theories and gravity.

Abstract

The integrand-level methods for the reduction of scattering amplitudes are well-established techniques, which have already proven their effectiveness in several applications at one-loop. In addition to the automation and refinement of tools for one-loop calculations, during the past year we observed very interesting progress in developing new techniques for amplitudes at two- and higher-loops, based on similar principles. In this presentation, we review the main features of integrand-level approaches with a particular focus on algebraic techniques, such as Laurent series expansion which we used to improve the one-loop reduction, and multivariate polynomial division which unveils the structure of multi-loop amplitudes.