Abstract
In this paper, we derive generalized Bern-Carrasco-Johansson (BCJ) relations for color-ordered Yang-Mills amplitudes by imposing gauge invariance conditions and dimensional reduction appropriately on the new discovered graphic expansion of Einstein-Yang-Mills amplitudes. These relations are also satisfied by color-ordered amplitudes in other theories such as bi-scalar theory and nonlinear sigma model (NLSM). As an application of the gauge invariance induced relations, we further prove that the three types of BCJ numerators in NLSM, which are derived from Feynman rules, Abelian Z-theory and Cachazo-He-Yuan (CHY) formula respectively, produce the same total amplitudes. In other words, the three distinct approaches to NLSM amplitudes are equivalent to each other.
Highlights
Color-kinematic duality (BCJ duality), which was suggested by Bern Carrasco and Johansson [1, 2], provides a deep insight into the study of scattering amplitudes
As an application of the gauge invariance induced relations, we further prove that the three types of BCJ numerators in nonlinear sigma model (NLSM), which are derived from Feynman rules, Abelian Z-theory and Cachazo-He-Yuan (CHY) formula respectively, produce the same total amplitudes
Though BCJ relations are first discovered in Yang-Mills theory, they hold for amplitudes in many other theories including: bi-scalar theory, NLSM [8], which can be uniformly described in the framework of CHY formulation [9,10,11,12]
Summary
Color-kinematic duality (BCJ duality), which was suggested by Bern Carrasco and Johansson [1, 2], provides a deep insight into the study of scattering amplitudes. We derive highly nontrivial generalized BCJ relations (gauge invariance induced relations) by imposing gauge invariance conditions and CHY-inspired dimensional reduction on the recent discovered graphic expansion of color-ordered Einstein-Yang-Mills (EYM) amplitudes [24]. The three distinct approaches: Feyman rules, Abelian Z theory and CHY formula provide different types of half-ladder BCJ numerators, they must produce the same NLSM amplitudes through the dual DDM decomposition. This equivalence condition requires nontrivial relations between color-ordered bi-scalar amplitudes. By using the gauge invariance induced relations and defining partial momentum kernel, we prove that the three distinct constructions of BCJ numerators produce the same NLSM amplitudes precisely.
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