Adaptive Integrand Decomposition in parallel and orthogonal space

TL;DR

This paper introduces a novel integrand decomposition method for multiloop scattering amplitudes using parallel and orthogonal space-time dimensions, simplifying polynomial division and systematically removing spurious terms.

Contribution

The method extends integrand decomposition to multiloop amplitudes in parallel and orthogonal space, simplifying calculations and applicable to all orders in perturbation theory.

Findings

Simplifies multiloop integrand decomposition using parallel and orthogonal space.

Effectively removes spurious terms via Gegenbauer polynomial orthogonality.

Applicable to planar and non-planar two-loop integrals with up to 8 legs.

Abstract

We present the integrand decomposition of multiloop scattering amplitudes in parallel and orthogonal space-time dimensions, $d=d_\parallel+d_\perp$, being $d_\parallel$ the dimension of the parallel space spanned by the legs of the diagrams. When the number $n$ of external legs is $n\le 4$, the corresponding representation of the multiloop integrals exposes a subset of integration variables which can be easily integrated away by means of Gegenbauer polynomials orthogonality condition. By decomposing the integration momenta along parallel and orthogonal directions, the polynomial division algorithm is drastically simplified. Moreover, the orthogonality conditions of Gegenbauer polynomials can be suitably applied to integrate the decomposed integrand, yielding the systematic annihilation of spurious terms. Consequently, multiloop amplitudes are expressed in terms of integrals corresponding to irreducible scalar products of loop momenta and external momenta. We revisit the one-loop decomposition, which turns out to be controlled by the maximum-cut theorem in different dimensions, and we discuss the integrand reduction of two-loop planar and non-planar integrals up to $n=8$ legs, for arbitrary external and internal kinematics. The proposed algorithm extends to all orders in perturbation theory.