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In the Peskin & Schröder (chapter 16.6) they use the background field method and spilt the gauge field into an background field $A$ and a fluctuation field $\mathcal{A}$. Next they claim that the Lagrangian (16.98) is invariant under the background gauge symmetry $$A^a_\mu\rightarrow A^a_\mu+D_\mu\beta^a\tag{16.99a}$$ $$ \mathcal{A}^a_\mu\rightarrow \mathcal{A}^a_\mu-f^{abc}\beta^b\mathcal{A}_\mu^a\tag{16.99b} $$

(+ transformations of ghosts and fermions) where $D$ is the covariant derivative constructed from the background field. However I don’t see why the gauge fixing term $D^\mu\mathcal{A_\mu^a}$ should be invariant under this transformation.

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Hints:

  1. One trick is to rewrite the (infinitesimal) background gauge transformation (16.99) as $$\begin{align} \delta_B A_{\mu} ~=~& [D_{\mu},\beta],\cr \delta_B D_{\mu} ~=~& -i[D_{\mu},\beta],\cr \delta_B {\cal A}_{\mu} ~=~& -i[{\cal A}_{\mu},\beta], \end{align} \tag{16.99} $$ using the conventions (15.44) & (16.91). Here $[\cdot,\cdot]$ denotes a commutator or a Lie bracket depending on context.

  2. The Lorenz gauge function $[D^{\mu},{\cal A}_{\mu}]$ therefore transforms covariantly $$ \delta_B [D^{\mu},{\cal A}_{\mu}] ~\stackrel{(16.99) + \text{Jac.id.}}{=}~ -i[[D^{\mu},{\cal A}_{\mu}],\beta]. $$

  3. The gauge-fixing term $$\kappa\left([D^{\mu},{\cal A}_{\mu}],~[D^{\nu},{\cal A}_{\nu}]\right)$$ is invariant under the background gauge transformation (16.99) because the Killing form $\kappa(\cdot,\cdot)$ is associative/invariant.

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