The Colour Dependence of Amplitudes

TL;DR

This paper introduces a systematic method to construct orthogonal colour tensors for scattering amplitudes involving particles in arbitrary gauge theory representations, enabling a universal basis for all perturbative orders.

Contribution

It provides a new, general construction of orthogonal colour tensors from trivalent trees, applicable to any gauge theory and order of perturbation, improving amplitude decomposition methods.

Findings

Constructed a basis of orthogonal colour tensors from trivalent trees.

Systematic decomposition of colour dependence in Feynman diagrams.

Comparison with other tensor bases like multi-traces and f-graphs.

Abstract

We describe how to construct a spanning set of linearly-independent, automatically orthogonal colour tensors for scattering amplitudes involving coloured particles transforming under arbitrary representations of any gauge theory, sufficient to all orders of perturbation theory (or beyond). These tensors are constructed from any choice of a single, trivalent tree graph, with Clebsch-Gordan coefficients at vertices connecting the external particles' representations to internal, irreducible representations via tensor products. We describe how the colour dependence of any Feynman diagram can be systematically decomposed into these bases, and how amplitudes expressed in these bases compare with other choices of tensors such as multi-traces or `f-graphs'.