distances analyzed here are not small enough to show the (perturbatively) expected repulsive short distance part. Similar results are obtained at all temperatures analyzed here. In the remainder of this section we study in detail the thermal modifications of these free energies from short to large distances. We begin our analysis of the free energies at small distances in Sec. \[couplatt\] with a discussion of the running coupling which leads to the renormalization of the free energies in Sec. \[renormalization\]. The separation of small and large distances which characterizes sudden qualitative changes in the free energy will be discussed in Sec. \[secshort\]. Large distance modifications of the quark anti-quark free energy will be studied in Sec. \[colorscreening\] at temperatures above and in Sec. \[stringbreaking\] at temperatures below deconfinement.

Our analysis of thermal modifications of the strong interaction will mainly be performed for the color singlet free energy. In this case a rather simple Coulombic $r$-dependence is suggested by perturbation theory at $T=0$ and short distances as well as for large distances at high temperatures. In particular, a proper $r$-dependence of $F_{\bar q q}$ is difficult to establish [@Kaczmarek:2002mc]. This maybe is attributed to contributions from higher excited states [@Jahn:2004qr] or 