Note on Construction of Dual-trace Factor in Yang-Mills Theory

TL;DR

This paper introduces a new method for constructing the BCJ dual-trace factor in Yang-Mills theory using the adjoint representation of kinematic algebra, aligning more closely with color decomposition properties.

Contribution

It presents a novel construction of the dual-trace factor based on the adjoint representation and inner product in the dual space, differing from previous methods.

Findings

Dual-trace factor satisfies cyclic symmetry but not KK-relation.

Construction aligns with color decomposition of Yang-Mills amplitudes.

Method exploits the adjoint representation of kinematic algebra.

Abstract

In this note we provide a new construction of BCJ dual-trace factor using the kinematic algebra proposed in arXiv:1105.2565 and arXiv:1212.6168. Different from the construction given in arXiv:1304.2978 based on the proposal of arXiv:1103.0312, the method used in this note exploits the adjoint representation of kinematic algebra and the use of inner product in dual space. The dual-trace factor defined in this way naturally satisfies cyclic symmetry condition but not KK-relation, just like the trace of U(N) Lie algebra satisfies cyclic symmetry condition, but not KK-relation. In other words the new construction naturally leads to formulation sharing more similarities with the color decomposition of Yang-Mills amplitude.