Abstract
Inspired by recent developments on scattering equations, we present a constructive procedure for computing symmetric, amplitude-encoded, BCJ numerators for n-point gauge-theory amplitudes, thus satisfying the three virtues identified by Broedel and Carrasco. We also develop a constructive procedure for computing symmetric, amplitude-encoded dual-trace functions (tau) for n-point amplitudes. These can be used to obtain symmetric kinematic numerators that automatically satisfy color-kinematic duality. The S_n symmetry of n-point gravity amplitudes formed from these symmetric dual-trace functions is completely manifest. Explicit expressions for four- and five-point amplitudes are presented.
Highlights
Where σj,k = σj −σk and sjk =2 with pj the momenta of the external particles
Inspired by recent developments on scattering equations, we present a constructive procedure for computing symmetric, amplitude-encoded, BCJ numerators for npoint gauge-theory amplitudes, satisfying the three virtues identified by Broedel and Carrasco
In an effort to define an economical and natural representation for the numerators, Broedel and Carrasco [29] enumerated three virtues that kinematic numerators would ideally possess: (1) color-kinematic duality, (2) amplitude-encoding, and (3) symmetry
Summary
We review tree-level amplitudes of gauge, gravity, and double-color scalar theories, and the color-kinematic dualities that relate these various amplitudes. Since the trace basis is independent, the color-ordered amplitudes are well-defined and gauge invariant. Because of the invariance of the full amplitude A(1, 2, · · · , n) under permutations of the arguments and the independence of the trace basis, the color-ordered amplitudes are all related to one another via. Because of the linear dependence of the color factors ci, the kinematic numerators ni(1, 2, · · · , n) in eq (2.1) are not uniquely determined but can undergo what are termed generalized gauge transformations [38] without altering the amplitudes. We will refer to a set of kinematic numerators satisfying eq (2.7) as BCJ numerators Such a choice is not unique: there remain residual generalized gauge transformations that preserve the Jacobi identities (2.7).
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