The Construction of Dual-trace Factor in Yang-Mills Theory

TL;DR

This work addresses the systematic construction of dual-trace factors $\tau$ for Yang–Mills tree amplitudes from BCJ numerators within the dual-DDM framework. It establishes a concrete procedure that links $n_{1\sigma n}$ to $\tau_{1\sigma n}$ via $n_{1\sigma_2...\sigma_{n-1}n}=\tau_{1[\sigma_2,[...,[\sigma_{n-1},n]...]]}$, and solves for $\tau$ by inverting a Gram-like matrix $G_{1,n}$ to yield $(n-2)!$ independent traces, with all others generated by KK-relations and cyclic symmetry. The authors prove that the resulting $\tau$'s satisfy a natural relabeling property, and they demonstrate this through explicit 3–7-point examples, confirming consistency with permutations and the underlying color-kinematic duality. They also discuss applications of relabeling to reduce unknowns and to fix $\tau$ from a single solution, highlighting two practical approaches. Overall, the paper strengthens the practical toolkit for dual-trace representations and clarifies how relabeling symmetry intertwines with BCJ numerators in constructing complete dual-trace decompositions.

Abstract

Recently, a BCJ dual of the color-ordered formula for Yang-Mills amplitude was proposed, where the dual-trace factor satisfies cyclic symmetry and KK-relation. In this paper, we present a systematic construction of the dual-trace factor based on its proposed relations to kinematic numerators in dual-DDM form. We show that the construction presented respects relabeling symmetry. In addition, we show that using relabeling symmetry as conditions, the same construction can be solved independently.