metric. We shall employ the most convenient choice, $g_{\mu\nu}=\bar g_{\mu\nu}=\delta_{\mu\nu}$. Furthermore, we set $\alpha_\textrm{D}=\alpha_\textrm{YM}=1$ from now on. The remaining calculation is in principle straightforward, but rather lengthy. One has to expand the traces up to terms with two fields $A^a_\mu(x)$ and two derivatives acting on them. Because of the built-in background gauge invariance those terms should combine to $F_{\mu\nu}^a F^{a \mu\nu}$. As a check we verified that this indeed happens.

Let us now discuss the result. Here we specialize for $d=4$ spacetime dimensions; for general $d$ the reader is referred to the Appendix. We present three different formulae for $\eta_F$; they differ with respect to the degree of “RG improvement” they take into account.

To start with, we “switch off” all RG improvements. This means that we discard all terms in $\partial_t\mathcal R_k$ on the RHS of the flow equation where $\partial_t$ hits either a $\mathcal Z_k$-factor or the $\Gamma^{(2)}_k$ in the argument of $\mathcal R_k$. In this way the evaluation of the FRGE amounts to a one-loop calculation, with a non-standard regulator though. We find $$\eta_F=-\frac 6 \pi \,g\, \Phi^1_1(0)-\frac{11}{24\pi^2} N {g_{\textrm{YM}}}^2$$ so that $$\partial_t {g_{\textrm{YM}}}^2 =-\frac 6 \pi \,g\, {g_{\textrm{YM}}}^2\, \Phi^1_1(0)-\frac{11}{24\pi^2} N {g_{\textrm{YM}}}^4\label{4d1loop}$$ Here $\Phi^1_1(0)$ is one of 