Using only the Britto–Cachazo–Feng–Witten (BCFW) on-shell recursion relation we prove color-order reversed relation, U(1)-decoupling relation, Kleiss–Kuijf (KK) relation and Bern–Carrasco–Johansson (BCJ) relation for color-ordered gauge amplitude in the framework of S-matrix program without relying on Lagrangian description. Our derivation is the first pure field theory proof of the new discovered BCJ identity, which substantially reduces the color-ordered basis from (n−2)! to (n−3)!. Our proof gives also its physical interpretation as the mysterious bonus relation with 1z2 behavior under suitable on-shell deformation for no adjacent pair.
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Feng et al. (2010) studied this question.
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