difficult to establish at temperatures in the close vicinity of the confinement deconfinement crossover.

We also analyzed the size of the maximum that the running coupling $\alpha_{qq}(r,T)$ at fixed temperature exhibits at a certain distance, $r_{max}$, [*i.e.*]{} we identify a temperature dependent coupling, $\tilde{\alpha}_{qq}(T)$, defined as $$\begin{aligned}
\tilde{\alpha}_{qq}(T)&\equiv&\alpha_{qq}(r_{max},T)\;.\label{alp_Tdef} \end{aligned}$$ The values for $r_{max}$ will be discussed in Sec \[secshort\] (see Fig. \[onset\]). Values for $\tilde{\alpha}_{qq}(T)$ are also available in pure gauge theory [@Kaczmarek:2004gv] at temperatures above deconfinement[^2]. Our results for $\tilde{\alpha}_{qq}(T)$ in $2$-flavor QCD and pure gauge theory are shown in Fig. \[alp\_qqT\] as function of temperature, $T/T_c$. At temperatures above deconfinement we cannot identify significant differences between the data from pure gauge and 2-flavor QCD[^3]. Only at temperatures quite close but above the phase transition small differences between full and quenched QCD become visible in $\tilde{\alpha}_{qq}(T)$. Nonetheless, the value of $\tilde{\alpha}_{qq}(T)$ drops from about $0.5$ at temperatures only moderately larger than the transition temperature, $T\;\gsim\;1.2T_c$, to a value of about $0.3$ at $2T_c$. This change in $\tilde{\alpha}_{qq}(T)$ with temperature calculated in $2$-flavor QCD does not appear to be too dramatic and might indeed be described by the $2$-loop perturbative coupling, $$\begin{aligned}
g_{\text{2-loop}}^{-2}(T)=2\beta_0\ln\left(\frac{\mu T}
{\Lambda_{\overline{MS}}}\right)+\frac{\beta_1}{\beta_0}
\ln\left(2\ln\left(\frac{\mu T}{\Lambda_{\overline{MS}}}\right)\right),\nonumber\\
\label{2loop}\end{aligned}$$ with $$\begin{aligned}
