Adaptive Integrand Decomposition

TL;DR

This paper introduces a simplified integrand reduction method for multiloop scattering amplitudes that leverages subspace decomposition and Gegenbauer polynomial orthogonality to streamline calculations and eliminate spurious integrals.

Contribution

It presents a novel, lighter polynomial division algorithm and a new approach using Gegenbauer polynomials for efficient integration in multiloop amplitude calculations.

Findings

Effective reduction of multiloop integrals using subspace decomposition.

Elimination of spurious integrals through Gegenbauer polynomial orthogonality.

Applicable to one- and two-loop integrals with arbitrary kinematics.

Abstract

We present a simplified variant of the integrand reduction algorithm for multiloop scattering amplitudes in $d = 4 - 2\epsilon$ dimensions, which exploits the decomposition of the integration momenta in parallel and orthogonal subspaces, $d=d_\parallel+d_\perp$, where $d_\parallel$ is the dimension of the space spanned by the legs of the diagrams. We discuss the advantages of a lighter polynomial division algorithm and how the orthogonality relations for Gegenbauer polynomilas can be suitably used for carrying out the integration of the irreducible monomials, which eliminates spurious integrals. Applications to one- and two-loop integrals, for arbitrary kinematics, are discussed.