in terms of $\varphi$ rather than in $A_0$, and $g_k^2=g^2/Z_0$ is nothing but the running coupling at momentum $\vec p^2\sim k_{\rm phys}^2$. Thus we estimate $g_k^2=4\pi \alpha_s(\vec p^2=k_{\rm phys}^2)$. Note that $g_k$ is an RG-invariant. The momentum integration can be performed analytically, and we are led to $$\begin{aligned}
\label{eq:preflowVapp}
  \beta \partial_k \Delta V_k
  = \frac{2}{3 (2 \pi)^2} \frac{(1+\eta_0/5) k^2 }{1+\frac{ g_{k}^2 
      \beta^2}{ k^2 }
    \partial^2_{\varphi} ( V_{\bot,k} + \Delta V_k)}\,, \end{aligned}$$ where $\eta_0$ is given by $$\label{eq:eta0app}
\eta_0=-\partial_t \log \alpha_s\,,$$ as the consistent choice in the given truncation.

Matching scales {#app:match}
===============

The flow of the temporal component of the gauge field, $A_0(\vec x)$, is computed with a three-dimensional regulator, see . In Polyakov gauge $A_0(\vec x)$ only depends on the spatial coordinates, whereas the spatial components $A_\bot(x)$ are four-dimensional fields. For cut-off scales far lower than the temperature, $k/T\ll 1$, also the spatial gauge fields are effectively three-dimensional fields as only the Matsubara zero mode propagates. Hence in this regime we can identify $k=k_\bot$. For large cut-off scales, $k/T\gg 1$, the $A_0$-flow decouples from the theory. A comparison between the two flows can only be done after the summation of the spatial flow over the Matsubara frequencies. In the asymptotic regime $k/T\gg 1$ this leads to the relation $$\label{eq:kTinf}
 \frac{1}{k} \simeq \sum_{n=-\infty}^{\infty} 
  \frac{1}{\omega_n^2 + k_\bot ^2}\to \frac{1}{2 k_\bot}\,,$$ The crossover between these asymptotic regimes happens at about $k/T=
1$. This crossover is implemented with the help of an appropriately chosen interpolating function $f$, $$\begin{aligned}
\label{eq:compare}
 \frac{T}{k^2} f(k/T) &=& T \sum_{n=-\infty}^{\infty} 
  \frac{1}{\omega_n^2 + k_\bot ^2}\,,\end{aligned}$$ A natural choice for $f(k/T)$ is depicted in Fig. \[fig:kbotk\], and has been used in the computation. 