U(1)-decoupling, KK and BCJ relations in $\mathcal{N}=4$ SYM

TL;DR

The work demonstrates that color-ordered amplitudes in N=4 SYM satisfy color reflection, U(1) decoupling, Kleiss-Kuijf (KK), and Bern-Carrasco-Johansson (BCJ) relations. By leveraging the supersymmetric BCFW recursion and the N=4 on-shell superfield, the authors prove these relations via induction from three-point amplitudes and then show how gluon and matter amplitudes follow by eta-expansion. They reveal that color reflection and U(1) decoupling arise as special KK cases and establish a primary BCJ relation from which all higher-point BCJ relations can be derived using KK relations. The results reinforce the color-kinematics structure in gauge theories and hint at potential extensions to gravity theories through similar recursion-based proofs.

Abstract

We proved the color reflection relation, U(1)-decoupling, Kleiss-Kuijf and Bern-Carrasco-Johansson relation for color-ordered $\mathcal{N}=4$ Super Yang-Mills theory using $\mathcal{N}=4$ SYM version BCFW recursion relation, which depends only on the general properties of super-amplitudes. This verified the conjectured matter fields BCJ relation. We also show that color reflection relation and U(1)-decoupling relation are special cases of KK relation, if we consider the KK relation as a general relation, then the former two relations come out naturally as the special cases.