Syzygies Probing Scattering Amplitudes

TL;DR

This paper introduces an efficient algorithm for computing minimal syzygy generators, which are crucial in analyzing scattering amplitudes, especially for deriving IBP relations in two-loop Yang-Mills diagrams.

Contribution

The paper presents a novel algorithm that effectively computes minimal syzygy sets, improving the analysis of scattering amplitudes and their related algebraic structures.

Findings

Algorithm efficiently computes minimal syzygy generators.

Applicable to modules and complex scattering amplitude problems.

Illustrated with two-loop Yang-Mills IBP relations.

Abstract

We propose a new efficient algorithm to obtain the locally minimal generating set of the syzygies for an ideal, i.e. a generating set whose proper subsets cannot be generating sets. Syzygy is a concept widely used in the current study of scattering amplitudes. This new algorithm can deal with more syzygies effectively because a new generation of syzygies is obtained in each step and the irreducibility of this generation is also verified in the process. This efficient algorithm can also be applied in getting the syzygies for the modules. We also show a typical example to illustrate the potential application of this method in scattering amplitudes, especially the Integral-By-Part(IBP) relations of the characteristic two-loop diagrams in the Yang-Mills theory.