Non-Abelian Ward identity for fermion annihilation ($k_{2\nu }{\mathcal{M}}^{ab\mu \nu }\propto k_1^{\mu }$)

For a Dirac field annihilating with its antiparticle into two massless gauge bosons, the two ordered fermion-exchange diagrams leave a colour matrix commutator when one outgoing momentum contracts the scattering amplitude. The three-boson exchange diagram cancels this current term but leaves a term proportional to the other outgoing momentum. With $D=\partial -igA$, $J_c^{\lambda }=\bar{v}{\gamma }^{\lambda }{t}^{c}u$, $q=k_1+k_2$, and the source's positive-sign three-boson vertex, the tree result is $k_{2\nu }{\mathcal{M}}^{ab\mu \nu }=i{g}^2{f}^{abc}(J_c\cdot k_2)k_1^{\mu }/q^2$. It vanishes after contraction with a physical polarization vector on leg 1, although the open-index tensor is generally nonzero.