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

TL;DR

The paper presents a new construction of the BCJ dual-trace factor by leveraging the adjoint representation of the kinematic algebra and a singular dual operator M to define a trace-like inner product. This yields a dual-trace factor that naturally encodes cyclic symmetry and mirrors trace-form color decompositions, but it does not generically satisfy KK-relations. The construction reproduces the dual-DDM structure and provides explicit matrix-representation formulas for the trace contractions, offering a trace-based perspective on color–kinematic duality. A discussion clarifies how this approach relates to, yet differs from, the previous Du:2013sha construction, underscoring nontrivial connections and the nuances of KK-relations within this framework.

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.