(\[Zfactors\]) are correctly chosen. Since we are not going to extract any “extra” background field dependence [@elisa2] we may set $\bar g_{\mu\nu}=g_{\mu\nu}$ and $\bar A^a_\mu=A^a_\mu$ after having found the Hessian.

The truncation contains three running couplings, $\bar{g}_\textrm{YM}(k)$, $G(k)$ and $\bar{\lambda}(k)$. Their beta functions can be found from the FRGE (\[ERGE\]) by “projecting out” the corresponding invariants in the derivative expansion of the traces and equating them to the corresponding field monomials on the LHS of the flow equation. The resulting system of differential equations becomes autonomous if we employ the dimensionless counterparts of $\bar g_\textrm{YM}$, $G$ and $\bar\lambda$ respectively: $${g_{\textrm{YM}}}^2(k)\equiv k^{d-4} Z^{-1}_F(k) \hat{g}_\textrm{YM}^2,\quad g(k)\equiv\frac{k^{d-2}}{32 \pi Z_N(k) \kappa^2}=k^{d-2} G(k),\quad \lambda(k)\equiv k^{-2}\bar\lambda(k)$$ In terms of these variables the three coupled RG equations have the structure $$\begin{aligned}
\partial_t g_\textrm{YM}^2 &=\beta_\textrm{YM}\equiv\left(d-4+\eta_F\right)g_\textrm{YM}^2\\
\partial_t g &=\beta_g \equiv\left(d-2+\eta_N\right)g\\
\partial_t \lambda &=\beta_\lambda
\end{aligned}
\label{SDE}$$ Here we introduced the anomalous dimensions related to the Yang-Mills and the gravitational field, respectively: $$\eta_{F}=-\partial_t \ln Z_F,\qquad \eta_{N}=-\partial_t \ln Z_N$$

In this paper we are only interested in the gravitationally corrected Yang-Mills beta function $\beta_\textrm{YM}$. Therefore it is sufficient to extract the $F_{\mu\nu}^2$-term from the derivative expansion of the traces. For identifying this monomial and reading off its prefactor we may insert any 