Multiloop Integrand Reduction for Dimensionally Regulated Amplitudes

TL;DR

This paper introduces a recursive multivariate polynomial division method for integrand reduction in dimensionally regulated Feynman amplitudes, applicable to multi-loop diagrams with arbitrary external legs.

Contribution

It develops a general recursive formula for integrand reduction that works for any number of loops and external legs, streamlining amplitude decomposition.

Findings

Effective reduction of two-loop Feynman diagrams in QED and QCD.

Applicable to integrands with denominators of arbitrary powers.

Can be performed with a finite number of algebraic operations.

Abstract

We present the integrand reduction via multivariate polynomial division as a natural technique to encode the unitarity conditions of Feynman amplitudes. We derive a recursive formula for the integrand reduction, valid for arbitrary dimensionally regulated loop integrals with any number of loops and external legs, which can be used to obtain the decomposition of any integrand analytically with a finite number of algebraic operations. The general results are illustrated by applications to two-loop Feynman diagrams in QED and QCD, showing that the proposed reduction algorithm can also be seamlessly applied to integrands with denominators appearing with arbitrary powers.