that this truncation is not only sufficient to reproduce one-loop perturbation theory exactly, but even approximates the two-loop result for the beta-function with a small error of a few percent. Therefore we may expect that this truncation, too, is perfectly sufficient as long as $k$ is sufficiently large (well above dynamically generated “confinement” scales, say).

When we insert the truncation ansatz (\[ansatz\]) into the exact FRGE (\[FRGE\]) the supertrace decomposes into a “bosonic” and a ghost contribution: $$\begin{aligned}
\partial_t\Gamma_k &= \frac 1 2 \,\textrm{Tr}\left[\frac{\partial_t \mathcal{R}_k\big(\mathcal{Z}^{-1}_k\breve{\Gamma}^{(2)}_k\big)}{\breve{\Gamma}^{(2)}_k+\mathcal{R}_k\big(\mathcal{Z}^{-1}_k\breve{\Gamma}^{(2)}_k\big)}\right]-\textrm{Tr}\left[\frac{\partial_t \mathcal{R}^{\textrm{gh}}_k\big(\mathcal{Z}^{-1}_{\textrm{gh}} S^{(2)}_{\textrm{gh}}\big)}{S^{(2)}_{\textrm{gh}}+\mathcal{R}^{\textrm{gh}}_k\big(\mathcal{Z}^{-1}_{\textrm{gh}}S^{(2)}_{\textrm{gh}}\big)}\right]\label{ERGE}\end{aligned}$$ Here $\breve\Gamma_k\equiv \Gamma^\textrm{EH}_k+\Gamma^{\textrm{YM}}_k+\Gamma^{\textrm{gf}}_k$ is the bosonic part of the action and $\breve{\Gamma}^{(2)}_k$ is its Hessian. The coarse graining operators in (\[ERGE\]) have the structure $$\mathcal{R}_k(x) =\mathcal{Z}_k k^2 R^{(0)}(x/k^2)\qquad \mathcal{R}^{\textrm{gh}}_k(x) = \mathcal{Z}^{\textrm{gh}}_k k^2 R^{(0)}(x/k^2)$$ where $R^{(0)}(y)$ is a “shape function” continuously interpolating between $R^{(0)}(0)=1$ and $\lim\limits_{y \rightarrow \infty}{ R^{(0)}(y)}=0$. The constants $\mathcal Z_k$ and $\mathcal Z^{\textrm{gh}}_k$ are matrices in field space. They will be adjusted in such a way that if in $\Gamma^{(2)}_k$ a certain mode has the inverse propagator $\zeta_k p^2$ it becomes $\zeta_k\left(p^2 +k^2 R^{(0)}\right)$ when we add $\mathcal R_k$ to $\Gamma^{(2)}_k$. As we shall see, this requirement is met if $\mathcal Z_k$ and $\mathcal Z^{\textrm{gh}}_k$ have the following block structure in 