Syzygies Probing Scattering Amplitudes

TL;DR

The paper addresses the challenge of obtaining locally minimal generating sets of syzygies for polynomial ideals and modules, a key step in simplifying scattering-amplitude calculations. It introduces the C2Z algorithm, which builds syzygies by controlled generation and rewriting of $S_{i,j}$-polynomials from critical pairs while enforcing barrier-based irreducibility checks. The method yields an irreducible syzygy basis and an automatically produced Groebner basis, with reported efficiency gains in IBP-related two-loop diagrams compared to established tools. The approach is generalizable to modules and has potential to expose irreducible relations beyond KK and BCJ, offering a broadly applicable framework for algebraic-geometry-inspired physics calculations.

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.

Figures (10)

• Figure 1: Critical pair decomposition.
• Figure 2: Illustrations on the warming-up example.
• Figure 3: Performing syzygies by a series of critical pairs.
• Figure 4: Example for Cell complex.
• Figure 5: Performing a critical pair.

Theorems & Definitions (1)

• proof