Integrand Reduction for Two-Loop Scattering Amplitudes through Multivariate Polynomial Division

TL;DR

This paper introduces a new polynomial division-based method for integrand reduction in two-loop scattering amplitudes, enabling complete decomposition for arbitrary amplitudes and applicable to complex multiloop cases.

Contribution

The authors develop a novel integrand reduction technique using multivariate polynomial division and Groebner bases, applicable to any loop order and simplifying the calculation of scattering amplitudes.

Findings

Successfully applied to two-loop five-point amplitudes in super Yang-Mills and supergravity.

Derived polynomial residues and integral bases for complex topologies.

Method is suitable for semi-numerical implementation and generalizes to all orders.

Abstract

We describe the application of a novel approach for the reduction of scattering amplitudes, based on multivariate polynomial division, which we have recently presented. This technique yields the complete integrand decomposition for arbitrary amplitudes, regardless of the number of loops. It allows for the determination of the residue at any multiparticle cut, whose knowledge is a mandatory prerequisite for applying the integrand-reduction procedure. By using the division modulo Groebner basis, we can derive a simple integrand recurrence relation that generates the multiparticle pole decomposition for integrands of arbitrary multiloop amplitudes. We apply the new reduction algorithm to the two-loop planar and nonplanar diagrams contributing to the five-point scattering amplitudes in N=4 super Yang-Mills and N=8 supergravity in four dimensions, whose numerator functions contain up to rank-two terms in the integration momenta. We determine all polynomial residues parametrizing the cuts of the corresponding topologies and subtopologies. We obtain the integral basis for the decomposition of each diagram from the polynomial form of the residues. Our approach is well suited for a seminumerical implementation, and its general mathematical properties provide an effective algorithm for the generalization of the integrand-reduction method to all orders in perturbation theory.