Abstract
Colour-kinematics duality suggests that Yang-Mills (YM) theory possesses some hidden Lie algebraic structure. So far this structure has resisted understanding, apart from some progress in the self-dual sector. We show that there is indeed a Lie algebra behind the YM Feynman rules. The Lie algebra we uncover is the Drinfeld double of the Lie algebra of vector fields. More specifically, we show that the kinematic numerators following from the YM Feynman rules satisfy a version of the Jacobi identity, in that the Jacobiator of the bracket defined by the YM cubic vertex is cancelled by the contribution of the YM quartic vertex. We then show that this Jacobi-like identity is in fact the Jacobi identity of the Drinfeld double. All our considerations are off-shell. Our construction explains why numerators computed using the Feynman rules satisfy the colour-kinematics at four but not at higher numbers of points. It also suggests a way of modifying the Feynman rules so that the duality can continue to hold for an arbitrary number of gluons. Our construction stops short of producing explicit higher point numerators because of an absence of a certain property at four points. We comment on possible ways of correcting this, but leave the next word in the story to future work.
Highlights
The Color-Kinematics duality [1] is a statement that the tree-level on-shell scattering amplitudes of Yang-Mills (YM) theory can be written as a sum over cubic graphs, with the contribution of each graph being a product of the group structure constants, kinematic numerator depending on the helicities and momenta of the particles being scattered, as well as the product of propagators, see below for a review
We show that the kinematic numerators following from the YM Feynman rules satisfy a version of the Jacobi identity, in that the Jacobiator of the bracket defined by the YM cubic vertex is cancelled by the contribution of the YM quartic vertex
We have extracted from the YM Feynman rules what we called the YM bracket, which is an anti-symmetric operation that sends a pair of vector fields into a vector field
Summary
The Color-Kinematics duality [1] is a statement that the tree-level on-shell scattering amplitudes of Yang-Mills (YM) theory can be written as a sum over cubic graphs, with the contribution of each graph being a product of the group structure constants (colour), kinematic numerator depending on the helicities and momenta of the particles being scattered (kinematics), as well as the product of propagators, see below for a review. The Jacobi identity satisfied by the kinematic numerators receives the interpretation of that of the Lie algebra of area preserving diffeomorphisms of a certain 2-dimensional space [9] While this result clearly points in the direction of some Lie algebra being behind the general YM amplitude, little is understood in this direction beyond the MHV case. We see that the Jacobi identity established at four points implies a partial cancelation in the sum of kinematic numerators This is incomplete cancelation, and the numerators following from the YM Feynman rules do not satisfy colour-kinematics. We see that certain parts of the Lie algebra Jacobi identity at higher points are missing from what follows from the Feynman rules. The last section is a discussion of how this difficulty may be overcome
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