Slavnov-Taylor identity

A Slavnov-Taylor identity is a BRST symmetry constraint relating correlation functions or interaction vertices in a gauge-fixed Non-Abelian gauge theory. Unlike a simple statement that every open gauge index is transverse, the general identities include Faddeev-Popov ghost fields. After putting external legs on shell and making all the other external states physical, they imply decoupling when one massless vector polarization vector is replaced by its four-momentum. They also control cancellations of auxiliary covariant components needed for physical unitarity.

For massless $gg\to q\bar{q}$, define $i\mathcal{M}$ by the Feynman-rule product with quark vertex $ig{T}^a{\gamma }^{\mu }$. The ordered quark exchanges contribute $-g^2\bar{u}/k_1[T^a,T^b]v$ after double contraction. Three-gluon exchange contributes $i{g}^2{f}^{abc}\bar{u}/k_1{T}^{c}v$, so the sum vanishes by the Lie algebra. This does not assert off-shell transversality of each open index; the full Slavnov-Taylor identity includes Faddeev-Popov ghost terms.

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.