An algebraic approach to BCJ numerators

Abstract

One important discovery in recent years is that the total amplitude of gauge theory can be written as BCJ form where kinematic numerators satisfy Jacobi identity. Although the existence of such kinematic numerators is no doubt, the simple and explicit construction is still an important problem. As a small step, in this note we provide an algebraic approach to construct these kinematic numerators. Under our Feynman-diagram-like construction, the Jacobi identity is manifestly satisfied. The corresponding color ordered amplitudes satisfy off-shell KK-relation and off-shell BCJ relation similar to the color ordered scalar theory. Using our construction, the dual DDM form is also established.

Note on symmetric BCJ numerator

“…terms of Kleiss-Kuijf basis [27] color-ordered scalar amplitudes A(1, σ, n) 1 multiplied by BCJ numerators n 1,σ,n [11,23] A tot = σ∈S n−2 n 1,σ,n A(1, σ, n), (1.3) This can be regarded as a result of exchanging the roles of color factors and BCJ numerators in the DDM form [20] A tot = σ∈S n−2 c 1,σ,n A(1, σ, n), (1.4) where c 1,σ,n = f 1σ 1 x 1 f x 2 σ 2 x 3 . .…”

“…An explicit construction was given by Mafra, Schlotterer and Stieberger using the pure spinor language [34]. Alternatively, it was shown that the kinematic factors can be interpreted in terms of diffeomorphism algebra [23,[35][36][37]. In a series of recent papers [38][39][40] another interesting construction was provided by Cachazo, He and Yuan (CHY) using the solutions to the scattering equations.…”

“…The apparently symmetrical structure also suggests mirror versions of the existing color decomposition formulations. In particular that studies of the BCJ duals of the Del Duca-Dixon-Maltoni(DDM) form [20] and the original trace form of color decomposition formulations can be found in [11,[21][22][23][24][25][26]. For example at tree-level, the full Yang-Mills amplitude were shown to be expressible in…”

On three-algebra and bi-fundamental matter amplitudes and integrability of supergravity

“…The BCJ color-kinematics duality for Yang-Mills theory [10], which is known to hold at tree-level [11][12][13][14][15] and conjectured to be valid at loop-level [16], has several interesting consequences. At tree level, it generates non-trivial relations between color-ordered partial amplitudes, so-called BCJ amplitude relations.…”

“…[11,12] (see also refs. [13][14][15]). The conjecture has been extended to loop level [16], where duality satisfying numerators has been found for various amplitudes in different theories [16,18,[46][47][48][49][50][51] and used in gravity constructions [19,20,52,53], though a formal proof is still an open problem.…”