Abstract
Kinematic numerators of Yang-Mills scattering amplitudes possess a rich Lie algebraic structure that suggest the existence of a hidden infinite-dimensional kinematic algebra. Explicitly realizing such a kinematic algebra is a longstanding open problem that only has had partial success for simple helicity sectors. In past work, we introduced a framework using tensor currents and fusion rules to generate BCJ numerators of a special subsector of NMHV amplitudes in Yang-Mills theory. Here we enlarge the scope and explicitly realize a kinematic algebra for all NMHV amplitudes. Master numerators are obtained directly from the algebraic rules and through commutators and kinematic Jacobi identities other numerators can be generated. Inspecting the output of the algebra, we conjecture a closed-form expression for the master BCJ numerator up to any multiplicity. We also introduce a new method, based on group algebra of the permutation group, to solve for the generalized gauge freedom of BCJ numerators. It uses the recently introduced binary BCJ relations to provide a complete set of NMHV kinematic numerators that consist of pure gauge.
Highlights
From the Lie algebra of the gauge group, the duality implies the existence of a hidden kinematic Lie algebra that builds up the kinematic numerators in a similar fashion
We presented an algebraic framework that generates BCJ numerators of pure YM theory through the NMHV sector; that is, all the terms in the D-dimensional numerators that contain at most two powers of εi·εj
A useful steppingstone is the introduction of pre-numerators, from which one obtains the kinematic numerator of any cubic graph by acting with a nested commutator
Summary
We review and extend some formal notations that were introduced in ref. [122]. We denote a generator of a putative algebra as JU , and consider products of two such generators, JU JV , which we refer to as a fusion product. The t-channel numerator is not a basis element, it is written as Nt = N [1, [2, 3]], 4 , which automatically implies the kinematic Jacobi identity Ns − Nu = Nt. Using the BCJ numerators and color factors in the DDM basis, we can re-write the color-dressed YM amplitude as [1, 128]. Matrix m(σ|ρ) is built out of linear combinations of the scalartype propagators 1/DΓ, as given by the decomposition of the BCJ numerators and color factors into the DDM basis It goes by many names in the literature, it is called the “propagator matrix” [128], the “inverse of the KLT kernel” [69, 129], or the “bi-adjoint scalar amplitude” [129]. E.g., ref. [116] for an all-multiplicity form of BCJ numerators for YM that exhibit Sn−2 crossing symmetry
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