negligible and the running coupling is controlled by distance only. Thus we can fix the free energies at small distances to the heavy quark potential, $F_1(r\ll1/T,T)\simeq V(r)$, and the renormalization group equation (RGE) will lead to $$\begin{aligned}
\lim_{r\to0}T\frac{dF_1(r,T)}{dT}&=&0\;,\label{RGEf}\end{aligned}$$ where we already have assumed that the continuum limit, $a\to0$, has been taken. On basis of the analysis of the coupling in Sec. \[couplatt\] and experiences with the quark anti-quark free energy in pure gauge theory [@Kaczmarek:2002mc; @Kaczmarek:2004gv] we assume here that the color singlet free energies in 2-flavor QCD calculated on finite lattices with temporal extent $N_\tau=4$ already have approached appropriate small distances, $r\ll1/T$, allowing for renormalization.

The (renormalized) color singlet quark anti-quark free energies, $F_1(r,T)$, and the heavy quark potential, $V(r)$ (line), were already shown in Fig. \[fes\](a) as function of distance at several temperatures close to the phase transition. From that figure it can be seen that the quark anti-quark free energy fixed at small distances approaches finite, temperature dependent plateau values at large distances signaling color screening ($T\;\gsim\;T_c$) and string breaking ($T<T_c$). These plateau values, $F_\infty(T)\equiv
F_1(r\to\infty,T)$, are decreasing with increasing temperature in the temperature range analyzed here. In general it is expected that $F_\infty(T)$ will continue to 