An algebraic approach to BCJ numerators

TL;DR

The paper develops an algebraic, Feynman-diagram–style framework to construct BCJ numerators that satisfy Jacobi identities, using a Fourier-basis Lie algebra with kinematic structure constants. By separating contributions into good (consistent-arrow) and bad parts and employing gauge-averaging techniques, it explicitly builds 4-, 5-, and general n-point numerators that obey KK and BCJ relations and yields dual DDM and related forms. It also demonstrates the consistency of fixed-k amplitudes with off-shell and on-shell BCJ relations and KK-relations, and derives a symmetric KLT/An-DDM-like representation that aligns with double-copy gravity constructions. Collectively, these results provide a systematic path to realizing color-kinematic duality at tree level, with potential implications for loop-level gravity via the double-copy. The work clarifies how gauge freedom and algebraic structure constants underpin the organization of amplitudes into BCJ-compliant forms and dual representations.

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.

Figures (7)

• Figure 1: From left to right, three diagrams represent $f^{12}_{~~~3}, f^{31}_{~~~2}, f^{23}_{~~~1}$ in Eq.(\ref{['eq:3-pt-vertex']}), where we used arrows to distinguish upper from lower indices.
• Figure 2: Graphical representation of contributions from Eq. (\ref{['eq:product-expansion']}) where the first four graphes correspond to contributions from the first line of the equation, and the remaining five graphes, from the second line of the equation.
• Figure 3: A Feynman diagram contributing to $n_{1}^*$
• Figure 4: Five terms with consistent arrow directions contributing to $n_{1}^*$
• Figure 5: Demonstration of color-order reversed relation (part (a)) and the $U(1)$-decoupling relation (part (b))