SU(N) group-theory constraints on color-ordered five-point amplitudes at all loop orders

TL;DR

This work derives SU($N$) group-theory constraints on color-ordered five-point amplitudes at all loop orders by a recursive rung-attachment method seeded with tree-level $U(1)$ decoupling relations. The trace-basis formalism yields a 22-term basis for the five-point color structure, with an all-loop $N$-power expansion, while the color-basis analysis uses null eigenvectors to obtain linear relations among amplitudes. The main results are ten independent constraints at odd loop orders and twelve at even loop orders, which reduce the number of independent amplitudes and relate the most-subleading-color contributions to subleading ones (and, for even $L$, to even more subleading sectors). These group-theory identities pave the way for systematic all-loop control of five-point amplitudes and motivate extensions to higher-point functions and deeper connections to U(1) decoupling and color–trace duality.

Abstract

Color-ordered amplitudes for the scattering of n particles in the adjoint representation of SU(N) gauge theory satisfy constraints arising solely from group theory. We derive these constraints for n=5 at all loop orders using an iterative approach. These constraints generalize well-known tree-level and one-loop group theory relations.