Construction of an effective Yang-Mills Lagrangian with manifest BCJ duality
Mathias Tolotti, Stefan Weinzierl
TL;DR
This work addresses the problem of realizing color-kinematics (BCJ) duality at the Lagrangian level for tree-level Yang-Mills amplitudes. It introduces an inductive algorithm that augments the standard YM Lagrangian with non-local higher-point interactions so that the resulting Feynman rules reproduce BCJ numerators that satisfy anti-symmetry and Jacobi-like relations. Explicit constructions are provided up to $n=6$, including ${\cal L}^{(2)}$, ${\cal L}^{(3)}$, ${\cal L}^{(4)}$ and corresponding higher-point operators encoded via ${\cal L}^{(n)}=\sum_t\sum_{j=2}^{[n/2]} O^{\mu_1...\mu_n}_{(n,t,j)} \hat D^{-1} \text{Tr}{\bf T}_{\mu_1...\mu_n}^{(n,t)}$, with the difference ${\cal L}_{\text{eff}}-{\\mathcal{L}}_{YM}$ reducing to zero for Born amplitudes. The approach emphasizes the non-uniqueness of BCJ representations and provides a practical path toward exploiting BCJ duality, with potential implications for gravity via the double copy. Overall, the paper offers a constructive framework to study color-kinematics duality directly from a Lagrangian and to extend BCJ ideas to higher-point amplitudes. It thereby paves the way for more systematic explorations of BCJ relations in gauge theory and their gravitational counterparts.
Abstract
The BCJ decomposition is a highly non-trivial property of gauge theories. In this paper we systematically construct an effective Lagrangian, whose Feynman rules automatically produce the BCJ numerators. The effective Lagrangian contains non-local terms. The difference between the standard Yang-Mills Lagrangian and the effective Lagrangian simplifies to zero.