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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.

The proper two-point correction to the gluon propagator in QCD. At one loop it receives quark, gluon, Faddeev-Popov ghost field and four-gluon tadpole diagram contributions. The Slavnov-Taylor identity makes the total tensor transverse, $k_{\mu }{\Pi }^{\mu \nu ,ab}=0$, although the individual gluon and ghost loop contributions need not be transverse. The sign of the symbol must be tied to whether the amputated insertion is defined as $i\Pi$ or $-i\Pi$.

The double-longitudinal Ward identity for gluon fusion uses the defining Lie algebra identity $[T^a,T^b]=i{f}^{abc}{T}^c$ makes the boxed amplitude vanish:In QED, both external gauge bosons are photons, the charge matrices commute, and there is no three-photon vertex. The two fermion-exchange graphs then cancel without a third graph. In QCD, omitting the three-gluon interaction would leave a generally nonzero colour commutator: gauge-boson self-interaction is essential to the cancellation.

This is a tree-level manifestation of a Ward identity and, more generally in a gauge-fixed non-Abelian theory, the Slavnov-Taylor identities. They connect the matter representation, the structure constants and the gauge interactions, ensuring that auxiliary longitudinal polarizations do not enter physical probabilities. With all other external legs physical and on shell, replacing one external massless gauge-boson polarization by its momentum leaves zero. The double-contraction result proved here is one consequence, not by itself a proof that every open-index amplitude tensor is transverse off shell. At loop order the corresponding non-Abelian identities also involve Faddeev-Popov ghosts and a consistent symmetry-preserving renormalization.

The Slavnov-Taylor identity implies transversality of the full perturbative gluon self-energy:The quark loop is separately transverse. This follows without evaluating its Feynman integral: contract its vector Feynman vertex with $k_{\mu }$ and use $/k=(/l+/k-m)-(/l-m)$ between its two Dirac propagators. One obtains a difference of two loop integrals that cancel after a shift of $l$ in dimensional regularization. The colour factor is proportional to $\mathrm{tr}(T^a{T}^b)$ and does not obstruct this argument.

The four-gluon tadpole diagram vanishes after removal of $m_g$. Thus the nontransverse parts of the two-three-gluon-vertex loop and the ghost loop must cancel. Contracting the preceding ghost-loop pole of the gluon self-energy givesFor QCD, set $C_A=3$. No evaluation of the remaining three-gluon Feynman diagram is required. Its uncontracted transverse part is not fixed by this identity.

The non-Abelian counterpart of the photon Ward identity is a Slavnov-Taylor identity, derived from BRST symmetry. A scattering amplitude with all other external states physical vanishes when one external massless gauge boson polarization vector is replaced by its four-momentum. An amputated tensor with the remaining indices open need not be transverse: part d gives its explicit nonzero longitudinal remainder.

Consequently, amplitudes involving several unphysical covariant polarization vectors need not vanish. For example, contracting the result of part d with a timelike vector $n_{\mu }$ gives $i{g}^2{ϵ}^{abc}(J_c\cdot k_2)(n\cdot k_1)/s$, nonzero in the explicit configuration there for $n=(1,0,0,0)$. Even with other states physical, the pure-gauge vector $k$ is the relevant null combination; separate timelike and spatial longitudinal basis vectors are not themselves pure gauge, and their separate amplitudes need not be zero.

These auxiliary components do not define separately observable positive probabilities. Physical states are classes in BRST cohomology, and the S-matrix preserves that physical state space in this anomaly-free vectorlike gauge spectrum. There is no physical transition from a physical state to a nonphysical state. The incorrect stronger statement would be that every covariant component called “unphysical” has a zero matrix element.

For loop Feynman diagrams or unitarity cuts, summing covariant gauge indices with $-g_{\mu \nu }$ propagates those auxiliary components. In a Non-Abelian gauge theory, their cancellation requires the interacting Faddeev-Popov ghost fields as well as the gauge boson diagrams. Ghost contributions carry their Grassmann field loop signs and restore the Slavnov-Taylor identities and physical unitarity. One may instead use a formulation with only physical propagating polarization vectors, but one cannot simply discard the ghost diagrams while retaining the Feynman gauge rules from part a.

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.