Color-factor symmetry and BCJ relations for QCD amplitudes

Understand

Tree-level $n$-point gauge-theory amplitudes with $n-2k$ gluons and $k$ pairs of (massless or massive) particles in the fundamental (or other) representation of the gauge group are invariant under a set of symmetries that act as momentum-dependent shifts on the color factors in the cubic decomposition of the amplitude.

• These symmetries lead to gauge-invariant constraints on the kinematic numerators.
• They also directly imply the BCJ relations among the Melia-basis primitive amplitudes previously obtained by Johansson and Ochirov.