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The quadratic Yang-Mills operator has zero directions along gauge orbits, so it has no propagator until one chooses a gauge fixing. The Faddeev-Popov determinant generated by this choice is represented by anticommuting Faddeev-Popov ghost fields, which cancel unphysical gauge-field contributions in loop calculations. A Nakanishi-Lautrup field imposes the gauge condition algebraically and lets the gauge-fixing plus ghost action be written as a BRST-exact term with off-shell nilpotency.

The Yang-Mills action is BRST invariant because its field strength transforms covariantly, and the gauge-fixing contribution is BRST exact. Nilpotence therefore givesThe gauge-fixing functional itself is generally not closed: $sL[A]=L[Dc]$. Applying the graded Leibniz rule givesThe Nakanishi-Lautrup field $B^a$ is auxiliary. Its algebraic equation $B^a=L[A^a]/\xi$ turns the middle terms into $L[A{]}^2/(2\xi )$, while the final term is the Faddeev-Popov ghost field action.

For $L[A^a]={\partial }^{\mu }{A}_{\mu }^a$, eliminating the Nakanishi-Lautrup field gives the covariant gauge LagrangianThe first term supplies the gluon kinetic term, three-gluon vertex, and four-gluon vertex. The second makes the quadratic gauge-field operator invertible. The last supplies the ghost propagator and ghost-antighost-gluon vertex.

The nonzero one-loop contributions to the gluon propagator are a gluon bubble with two three-gluon vertices and a closed ghost bubble with two ghost-antighost-gluon vertices. A four-gluon tadpole is also present with a cutoff regulator; for massless fields it is a scaleless integral and vanishes in dimensional regularization.