Proof of the fundamental BCJ relations for QCD amplitudes

Understand

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 ${\mathcal N}=4$ super-Yang-Mills theory.