Integrand-Level Reduction of Loop Amplitudes by Computational Algebraic Geometry Methods

TL;DR

This paper introduces an algebraic geometry-based algorithm for reducing multi-loop amplitudes at the integrand level, enabling efficient reconstruction of integrands in quantum field theory calculations.

Contribution

It presents a novel computational algebraic geometry approach, including a Mathematica package, for integrand reduction applicable to multi-loop amplitudes in various dimensions.

Findings

Successfully applied to two and three-loop examples

Works in both four and D=4-2ε dimensions

Provides a systematic algebraic method for amplitude reduction

Abstract

We present an algorithm for the integrand-level reduction of multi-loop amplitudes of renormalizable field theories, based on computational algebraic geometry. This algorithm uses (1) the Gr\"obner basis method to determine the basis for integrand-level reduction, (2) the primary decomposition of an ideal to classify all inequivalent solutions of unitarity cuts. The resulting basis and cut solutions can be used to reconstruct the integrand from unitarity cuts, via polynomial fitting techniques. The basis determination part of the algorithm has been implemented in the Mathematica package, BasisDet. The primary decomposition part can be readily carried out by algebraic geometry softwares, with the output of the package BasisDet. The algorithm works in both D=4 and $D=4-2\epsilon$ dimensions, and we present some two and three-loop examples of applications of this algorithm.