An algebraic approach to BCJ numerators

TL;DR

This paper introduces an algebraic method for constructing BCJ numerators in gauge theory amplitudes, ensuring Jacobi identities are satisfied explicitly and connecting to known amplitude relations.

Contribution

It provides a novel algebraic approach to explicitly construct BCJ numerators that satisfy Jacobi identities and related amplitude relations.

Findings

Jacobi identities are manifestly satisfied in the construction

Color ordered amplitudes obey off-shell KK and BCJ relations

Dual DDM form is established using the new construction

Abstract

One important discovery in recent years is that the total amplitude of gauge theory can be written as BCJ form where kinematic numerators satisfy Jacobi identity. Although the existence of such kinematic numerators is no doubt, the simple and explicit construction is still an important problem. As a small step, in this note we provide an algebraic approach to construct these kinematic numerators. Under our Feynman-diagram-like construction, the Jacobi identity is manifestly satisfied. The corresponding color ordered amplitudes satisfy off-shell KK-relation and off-shell BCJ relation similar to the color ordered scalar theory. Using our construction, the dual DDM form is also established.