Symmetric-group decomposition of SU(N) group-theory constraints on four-, five-, and six-point color-ordered amplitudes

TL;DR

This work derives the complete set of SU($N$) group-theory constraints on four-, five-, and six-point color-ordered amplitudes at all loop orders, and shows these constraints decompose naturally into irreducible representations of the symmetric group $S_n$. By constructing projection operators for regular and induced $S_n$ representations, the authors present the null spaces of the trace basis in a compact, basis-independent form, enabling explicit all-loop decompositions for $n=4,5,6$. They provide detailed results for the four-, five-, and six-point cases, including how most constraints mix single-, double-, and triple-trace amplitudes, and they analyze Kleiss-Kuijf relations at higher loops, showing KK relations hold up to certain $n$ and loops but fail for $n\ge 8$ at two loops. The work also discusses a key iterative assumption about building higher-loop color spaces via rung attachments, which, if violated, would affect the number of independent group-theory relations but not their existence. Overall, the paper offers a compact, representation-theory–based framework that clarifies the structure and redundancy of color-ordered amplitudes across loop orders and multiplicities.

Abstract

Color-ordered amplitudes for the scattering of n particles in the adjoint representation of SU(N) gauge theory satisfy constraints that arise from group theory alone. These constraints break into subsets associated with irreducible representations of the symmetric group S_n, which allows them to be presented in a compact and natural way. Using an iterative approach, we derive the constraints for six-point amplitudes at all loop orders, extending earlier results for n=4 and n=5. We then decompose the four-, five-, and six-point group-theory constraints into their irreducible S_n subspaces. We comment briefly on higher-point two-loop amplitudes.