Higher-order analogues of colour-kinematics duality in first-order Yang-Mills Feynman diagrams

Abstract

Colour-kinematics duality is, even at tree level, not manifest from the standard Lagrangian of Yang-Mills theory in that the usual Feynman-diagram expansion does not follow the kinematic Jacobi identities. For a Lagrangian manifesting colour-kinematics duality, the kinematic Jacobi identities of the Feynman-diagram expansion follow from the existence of a second-order differential operator $\mathsf{b}$ acting on the algebra of colour-stripped fields that forms part of the data of a BV$^\square$-algebra. We argue that the existence of differential operators of order greater than two implies weaker but nontrivial fragments of kinematic Jacobi identities. We further show that a superspace formulation of the first-order Yang-Mills action in the Batalin-Vilkovisky formalism admits a differential operator of order six in every spacetime dimension. Therefore, the Feynman-diagram expansion of tree and loop scattering amplitudes of the first-order formulation of Yang-Mills theory automatically enjoy a weak form of colour-kinematics duality.