Multiloop Integrand Reduction for Dimensionally Regulated Amplitudes

TL;DR

This work addresses analytic decomposition of dimensionally regulated multi-loop amplitudes via unitarity cuts. It develops a general integrand reduction using multivariate polynomial division and Gröbner-basis techniques, yielding a recursive formula that expresses any integrand in terms of residues over fewer denominators. The authors illustrate the method with explicit two-loop examples in QED and QCD, including cases with arbitrarily powered propagators, and provide an automated implementation. The approach offers a dimensionally agnostic, top-down framework for systematic, algebraic amplitude decomposition, enabling robust multi-loop computations.

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.

Figures (2)

• Figure 1: Integrand recurrence relation for a generic $\ell$-loop integrand.
• Figure 2: First row: diagrams leading to the two-loop QED corrections to the photon self energy. Second row: two-loop diagrams entering the QCD corrections to $gg \to H$ in the heavy top mass approximation.