Proof of the fundamental BCJ relations for QCD amplitudes

The fundamental BCJ-relation is a linear relation between primitive tree amplitudes with different cyclic orderings. The cyclic orderings differ by the insertion place of one gluon. The coefficients of the fundamental BCJ-relation are linear in the Lorentz invariants 2 p i p j . The BCJ-relations are well established for pure gluonic amplitudes as well as for amplitudes in N = 4 $$ \mathcal{N}=4 $$ super-Yang-Mills theory. Recently, it has been conjectured that the BCJ-relations hold also for QCD amplitudes. In this paper we give a proof of this conjecture. The proof is valid for massless and massive quarks