All-loop group-theory constraints for color-ordered SU(N) gauge-theory amplitudes

TL;DR

This work derives all-loop color-ordered four-point amplitude constraints in SU($N$) gauge theories that originate solely from group theory. It introduces a recursive rung-attachment method to propagate color-factor relations from $(L-1)$-loop diagrams to $L$-loop diagrams, yielding right-null vectors that enforce constraints among color-ordered amplitudes. The main result is four independent group-theory relations at each loop order (for $L\ge 2$), with special-case counts for $L=0$ and $L=1$, generalizing known tree-level and low-loop identities and aligning with color-kinematic duality ideas. The paper also provides evidence up to four loops and discusses extensions to higher-point functions, while acknowledging the need for a complete all-orders proof that Jacobi-based color-factor generation suffices.

Abstract

We derive constraints on the color-ordered amplitudes of the L-loop four-point function in SU(N) gauge theories that arise solely from the structure of the gauge group. These constraints generalize well-known group theory relations, such as U(1) decoupling identities, to all loop orders.