to the repulsive contributions from states with static charges fixed in an octet configuration.

The running coupling at $T\neq0$ {#couplatt}
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We extend here our studies of the coupling at zero temperature to finite temperatures below and above deconfinement following the conceptual approach given in [@Kaczmarek:2004gv]. In this case the appropriate observable is the color singlet quark anti-quark free energy and its derivative. We use the perturbative short and large distance relation from one gluon exchange [@Nadkarni:1986as; @Nadkarni:1986cz; @McLerran:1981pb], [*i.e.*]{} in the limit $r\Lambda_{\text{QCD}}\ll1$ zero temperature perturbation theory suggests $$\begin{aligned}
F_1(r,T)\;\equiv\;V(r)&\simeq&-\frac{4}{3}\frac{\alpha(r)}{r}\;,\label{alp_rT1}\end{aligned}$$ while high temperature perturbation theory, [*i.e.*]{} $rT\gg1$ and $T$ well above $T_c$, yields $$\begin{aligned}
F_1(r,T)&\simeq&-\frac{4}{3}\frac{\alpha(T)}{r}e^{-m_D(T)r}\;.\label{alp_rT2}\end{aligned}$$ In both relations we have neglected any constant contributions to the free energies which, in particular, at high temperatures will dominate the large distance behavior of the free energies. Moreover, we already anticipated here the running of the couplings with the expected dominant scales $r$ and $T$ in both limits. At finite temperature we define the running coupling in analogy to $T=0$ as (see [@Kaczmarek:2002mc; @Kaczmarek:2004gv]),$$\begin{aligned}
  \alpha_{qq}(r,T)&\equiv&\frac{3}{4}r^2 \frac{dF_1(r,T)}{dr}\;.\label{alp_rT}\end{aligned}$$ With this definition any undetermined constant contributions to the free energies are eliminated and the coupling defined here at finite temperature will recover the coupling at 