f^{abc}\lambda^b \big(\bar{A}^c_\mu + a^c_\mu\big)\end{aligned}$$

The crucial idea is to choose a gauge fixing term $S^{\rm gf}$ that breaks only the true gauge invariance but retains background gauge invariance. It gives rise to an associated ghost action $S_{\rm gh}$ in the usual way. Introducing ghost fields ${\cal C}_\mu$ and $\bar{{\cal C}}^\mu$ for the diffeomorphisms and $\Sigma^a$ and $\bar{\Sigma}^a$ for the Yang-Mills transformations, respectively, we arrive at the following path integral which depends parametrically on the background fields: $$\begin{aligned}
\label{PI-more-concrete}
{\cal Z} = \int\:{\cal D}h_{\mu\nu}{\cal D}a^a_\mu{\cal D}{\cal C}_\mu{\cal D}\bar{{\cal C}}^\mu{\cal D}\Sigma^a{\cal D}\bar{\Sigma}^a\:{\rm e}^{-S[\bar{g}+h, \bar{A}+a] - S^{\rm gf} - S_{\rm gh}}\end{aligned}$$

In order to set up a functional RG equation, we follow the well-known construction of the effective average action [@avact; @nonabavact; @avactrev] and add a higher derivative IR cutoff term $\Delta_k S$ that is quadratic in the fluctuations: $$\begin{aligned}
\label{IR-cutoff}
\Delta_k S &=& \frac{1}{2} \kappa^2 \int\:{\rm d}^d x\:\sqrt{\bar{g}}\:h_{\mu\nu}{{\cal R}^{\rm grav}_k [\bar{g}]}^{\mu\nu\rho\sigma}h_{\rho\sigma}\nonumber\\
&& + \frac{1}{2}\int\:{\rm d}^d x\:\sqrt{\bar{g}}\:a^a_\mu{{\cal R}^{\rm YM}_k [\bar{g}, \bar{A}]}^{a\mu b\nu}a^b_\nu\nonumber\\
&& + \sqrt{2} \int\:{\rm d}^d x\:\sqrt{\bar{g}}\:\big(\bar{\cal{C}}, \bar{\Sigma}\big){{\cal R}^{\rm gh}_k [\bar{g}, \bar{A}]}\bigg(\!\!\begin{array}{c} {\cal C} \\[-0.2cm] \Sigma \end{array}\!\! \bigg)\end{aligned}$$

with $\kappa \equiv (32 \pi \hat{G})^{-\frac{1}{2}}$ and $\hat{G}$ denoting Newton’s constant (see below). Furthermore, adding appropriate source terms enables us to easily compute expectation 