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

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.