increase at smaller temperature and will smoothly match $V(r\equiv\infty)$ [@Digal:2001iu] at zero temperature while it will become negative at high temperature and asymptotically is expected to become proportional to $g^3T$ [@Gava:1981qd; @Kaczmarek:2002mc]. The plateau value of the quark anti-quark free energy at large distances can be used to define non-perturbatively the renormalized Polyakov loop [@Kaczmarek:2002mc], [ *i.e.*]{} $$\begin{aligned}
L^{\text{ren}}(T)&=&\exp\left(-\frac{F_\infty(T)}{2T}\right)\;.\label{renPloop}\end{aligned}$$ As the unrenormalized free energies approach $|\langle L\rangle|^2$ at large distances, this may be reinterpreted in terms of a renormalization constant that has been determined by demanding (\[RGEf\]) to hold at short distances [@Dumitru:2003hp; @Zantow:2003uh], $$\begin{aligned}
L^{\text{ren}}&\equiv&|\langle \left(Z(g,m)\right)^{N_\tau} L \rangle|\;.\end{aligned}$$ The values for $Z(g,m)$ for our simulation parameters are summarized in Table \[tab:ren\]. The normalization constants for the free energies appearing in (\[f1\]-\[fav\]) are then given by $$\begin{aligned}
C=-2 N_\tau Z(g,m).\end{aligned}$$ An analysis of the renormalized Polyakov loop expectation value in high temperature perturbation theory [@Gava:1981qd] suggests at (resummed) leading order[^4], the behavior $$\begin{aligned}
L^{\text{ren}}(T)&\simeq&1+\frac{2}{3}\frac{m_D(T)}{T}\alpha(T)\label{lrenpert}\;\end{aligned}$$ in the fundamental representation. Thus high temperature perturbation theory suggests that the limiting value at infinite temperature, $L^{\text{ren}}(T\to\infty)=1$ is approached from above. An expansion of (\[renPloop\]) then suggests $F_\infty(T)\simeq-\frac{4}{3}m_D(T)\alpha(T)\simeq-{\cal O}(g^3T)$. We thus expect $F_\infty(T)\to-\infty$ in the high temperature limit.

  $\beta$   $Z(g,m)$    $T/T_c$   $L^{\text{ren}}(T)$
  --------- ----------- --------- 