values of the quantum fields. With the notation $$\begin{aligned}
\label{gf-vev} A^a_\mu &\equiv& \langle {\cal A}^a_\mu \rangle, \hspace{0.6cm} \bar{a}^a_\mu \equiv \langle a^a_\mu \rangle \\
\label{matric-vev} g_{\mu\nu} &\equiv& \langle \gamma_{\mu\nu} \rangle, \hspace{0.6cm} \bar{h}_{\mu\nu} \equiv \langle h_{\mu\nu} \rangle \\
\label{diff-gh-vev} \xi^\mu &\equiv& \langle {\cal C}^\mu \rangle, \hspace{0.6cm} \bar{\xi}_\mu \equiv \langle \bar{{\cal C}}_\mu \rangle \\
\label{ym-gh-vev} \Upsilon^a &\equiv& \langle \Sigma^a \rangle, \hspace{0.6cm} \bar{\Upsilon}^a \equiv \langle \bar{\Sigma}^a \rangle \end{aligned}$$

the background decomposition (\[decomp\]) now reads $$\begin{aligned}
\label{decomp-vev}
g_{\mu\nu} \equiv \bar{g}_{\mu\nu} + \bar{h}_{\mu\nu} \hspace{0.6cm}\mbox{and}\hspace{0.6cm} A^a_\mu \equiv \bar{A}^a_\mu + \bar{a}^a_\mu\end{aligned}$$

The Legendre transformation of the now $k$-dependent functional ${\rm ln}\:{\cal Z}_k$ with respect to the sources leads to the effective average action then [@avact; @nonabavact; @avactrev]: $$\begin{aligned}
\label{eaa}
\Gamma_k[\bar{h}_{\mu\nu}, \bar{a}^a_\mu, \xi^\mu, \bar{\xi}_\mu, \Upsilon^a, \bar{\Upsilon}^a; \bar{g}_{\mu\nu}, \bar{A}^a_\mu] \equiv \Gamma_k[g_{\mu\nu}, \bar{g}_{\mu\nu}, A^a_\mu, \bar{A}^a_\mu, \xi^\mu, \bar{\xi}_\mu, \Upsilon^a, \bar{\Upsilon}^a]\end{aligned}$$

Its scale dependence is governed by the functional renormalization group equation (FRGE) $$\begin{aligned}
\label{FRGE}
\partial_t \Gamma_k = \frac{1}{2}{\rm STr} \Big[\Big(\Gamma_k^{(2)} + {\cal R}_k\big(\Delta\big)\Big)^{-1}\big(\partial_t{\cal R}_k\big(\Delta\big)\Big)\Big]\end{aligned}$$

where $\Gamma_k^{(2)}$ denotes the Hessian of $\Gamma_k$ with respect to all fluctuation and ghost expectation values, $\Delta$ is some suitably chosen generalized Laplacian, and $t \equiv {\rm ln}\:k$ is the “RG time”.

Since we deal with two gauge invariances, a remark concerning the notation is in order: We 