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Color kinematic symmetric (BCJ) numerators in a light-like gauge

Color-ordered tree level scattering amplitudes in Yang-Mills theories can be written as a sum over terms which display the various propagator poles of Feynman diagrams. The numerators in these expressions which are obtained by straightforward application of Feynman rules are not satisfying any particular relations, typically. However, by reshuffling terms, it is known that one can arrive at a set of numerators which satisfy the same Jacobi identity as the corresponding color factors. By extending previous work by us we show how this can be systematically accomplished within a Lagrangian framework. We construct an effective Lagrangian which yields tree-level color-kinematic symmetric numerators in Yang-Mills theories in a light-like gauge at five-points. The five-point effective Lagrangian is non-local and it is zero by Jacobi identity. The numerators obtained from it respect the original pole structure of the color-ordered amplitude. We discuss how this procedure can be systematically extended to higher order.