The Many Colours of Amplitudes

TL;DR

This paper explores how the complexity of scattering amplitudes' colour structures varies with gauge group size and matter representations, revealing new patterns and implications in gauge theory.

Contribution

It provides a detailed analysis of colour-structure counting in Yang-Mills theory, connecting representation theory with amplitude complexity for various gauge groups.

Findings

Colour-structure tensors grow factorially with multiplicity at large gauge rank.

For fixed gauge groups, the growth is at most exponential with multiplicity.

Uncovered surprising structures in colour dependence across different gauge groups.

Abstract

We study the colour-dependence of scattering amplitudes in Yang-Mills theory with arbitrary (but fixed) gauge group and various representations of charged matter. When the rank of the gauge theory is taken arbitrarily large compared to the number of particles involved in an amplitude, it is well known that the number of independent colour-structure tensors grows factorially with multiplicity; however, for any fixed gauge group, this number grows at most exponentially with multiplicity. We review how this counting arises in representation theory and survey its implications for a wide variety of specific gauge groups with various representations of charged matter, uncovering several surprising structures along the way.