Serendipitous Syzygies of Scattering Amplitudes

TL;DR

This paper uncovers a large set of linear relations among one-loop all-plus amplitudes in Yang-Mills theory, revealing only two independent amplitudes for each number of particles and providing explicit relations up to seven particles.

Contribution

It introduces two methods—numerical linear algebra and computational algebraic geometry—to systematically find relations among scattering amplitudes, including explicit forms up to seven particles.

Findings

Number of relations for n≥5 is (n-1)!/2 - 2, leaving only two independent amplitudes.

Explicit relations are provided for all n up to 7 particles.

Includes analysis of tree-level MHV amplitude relations up to n=8, covering known identities.

Abstract

We study linear relations between color-ordered all-plus amplitudes at one loop in Yang--Mills theory. We show that on general grounds, there are $(n-1)!/2-2$ relations for $n\ge 5$, leaving only two independent color-ordered amplitudes. We present two complementary approaches to finding such relations: one using numerical linear algebra and the other using syzygies in computational algebraic geometry. We obtain explicit forms for all relations through $n=7$. We also study relations for the tree-level MHV amplitudes through $n=8$. The latter relations include the well-known color and Bern--Carrasco--Johansson identities.