definition only via its background component $\bar{A}$.

Thus the actual theory space on which we can define an RG flow consists of the functionals $F[h,a,\xi,\bar\xi,\Upsilon,\bar\Upsilon;\bar g,\bar A]\equiv F[g,\bar g,A,\bar A,\xi,\bar \xi,\Upsilon,\bar \Upsilon]$ in $${\cal F}'_{\rm inv} = \{F\:|\:\widetilde{\widetilde{{\cal W^{\rm B}_{\rm D}}}}(v) F = 0 \:\wedge\: {\cal W}^{\rm B}_{\rm YM}(\lambda) F = 0 ~ \forall ~ v^\mu,\:\lambda^a\}$$

Finally, we return to our starting point and compute the $\bar{\Upsilon}$-$\xi$ part of the ghost action, now with the original true diffeomorphism $\delta_{\rm D}^{\rm G}$ replaced by its modified counterpart, $\widetilde{\widetilde{\delta_{\rm D}^{\rm G}}}$: $$\begin{aligned}
\left(S_{\rm gh}\:[\xi, \bar{\Upsilon}; \bar{g}, \bar{A}]\right)_{\bar{\Upsilon}\xi} &= - \int\:{\rm d}^d x\:\sqrt{\bar{g}}\:\big[\bar{\Upsilon}^a \bar{{\cal D}}^{\nu\:ab} \big(\widetilde{\widetilde{\delta_{\rm D}^{\rm G}}}(\xi)a^b_\nu\big)\big]|_{a=0}\\ 
&= - \int\:{\rm d}^d x\:\sqrt{\bar{g}}\:\Big[\bar{\Upsilon}^a \bar{{\cal D}}^{\nu\:ab} \Big(\xi^\rho \partial_\rho \bar{A}^b_\nu + (\partial_\nu \xi^\rho) \bar{A}^b_\rho\\& \hspace{3cm} -\big(\partial_\nu(\bar{A}^b_\rho \xi^\rho) + f^{bcd}\bar{A}^b\bar{A}^d_\rho \xi^\rho\big)\Big)\Big]\\
&= - \int\:{\rm d}^d x\:\sqrt{\bar{g}}\:\Big[ - \bar{\Upsilon}^a \bar{{\cal D}}^{\nu\:ab} \bar{F}^b_{\nu\rho} \xi^\rho\Big]
\end{aligned}\label{ups-xi-geist-neu}$$ This action is obviously invariant under ${\sf SU(N)}_{\rm loc}$ and background diffeomorphisms $\widetilde{\widetilde{\delta_{\rm D}^{\rm B}}}$ for background tensorial ghost expectation values.

The Running Gauge Coupling
==========================

In this section we explicitly evaluate the FRGE on a truncated theory space which is general enough to allow for an approximate determination of the beta function for the scale dependent Yang-Mills coupling $g_\textrm{YM}(k)$. Our truncation is 