Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 304 / 4 / d

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.

For $L[A^a]=n^{\mu }{A}_{\mu }^a$, eliminating $B^a$ gives the axial gauge term $(n\cdot A^a{)}^2/(2\xi )$ and ghost operatorIn the strict $\xi \to 0$ gauge, $n\cdot A=0$. The gauge-field-dependent part of $n\cdot D$ then vanishes, the Faddeev-Popov determinant becomes field independent, and the ghosts decouple. The one-loop gluon propagator therefore receives the gluon bubble but no ghost bubble. As in the covariant gauge, the four-gluon tadpole can occur with a hard cutoff and vanishes as a scaleless integral in dimensional regularization. Gauge-invariant observables agree between the two gauges even though their individual propagators and diagrammatic decompositions differ.