\beta_0&=&\frac{1}{16\pi^2}\left(11-\frac{2N_f}{3}\right)\;,\nonumber\\
\beta_1&=&\frac{1}{(16\pi^2)^2}\left(102-\frac{38N_f}{3}\right)\;,\nonumber\end{aligned}$$ assuming vanishing quark masses. In view of the ambiguity in setting the scale in perturbation theory, $\mu T$, we performed a best-fit analysis to fix the scale for the entire temperature range, $1.2\;\lsim\;T/T_c\;\lsim\;2$. We find here $\mu=1.14(2)\pi$ with $T_c/\Lambda_{\overline{MS}}=0.77(21)$ using $T_c\simeq202(4)$ MeV [@Karsch:2000ps] and $\Lambda_{\overline{MS}}\simeq
261(17)$ MeV [@Gockeler:2005rv], which is still in agreement with the lower limit of the range of scales one commonly uses to fix perturbative couplings, $\mu=\pi,...,4\pi$. This is shown by the solid line (fit) in Fig. \[alp\_qqT\] including the error band estimated through $\mu=\pi$ to $\mu=4\pi$ and the error on $T_c/\Lambda_{\overline{MS}}$ (dotted lines). We will turn back to a discussion of the temperature dependence of the coupling above deconfinement in Sec. \[colorscreening\].

At temperatures in the vicinity and below the phase transition temperature, $T\;\lsim\;1.2T_c$, the behavior of $\tilde{\alpha}_{qq}(T)$ is, however, quite different from the perturbative logarithmic change with temperature. The values for $\tilde{\alpha}_{qq}(T)$ rapidly grow here with decreasing temperature and approach non-perturbatively large values. This again shows that $\tilde{\alpha}_{qq}(r,T)$ mimics the confinement part of the zero temperature force still at relatively large distances and that this behavior persists up to temperatures close but above deconfinement. This again demonstrates the persistence of confinement forces at 