We present a renormalization procedure for Polyakov loops which explicitly implements the fact that the renormalization constant depends only on the ultraviolet cutoff. Using this we study the renormalized Polyakov loops in all representations up to the $\mathbf{27}$ of the gauge group SU(3). We find good evidence for Casimir scaling of the Polyakov loops and for approximate large-$N$ factorization. By studying many loops together, we are able to show that there is a matrix model with a single coupling which can describe the high temperature phase of QCD, although it is hard to construct explicitly. We present the first results for the nonvanishing renormalized octet loop in the thermodynamic limit below the SU(3) phase transition, and estimate the associated string breaking distance and the gluelump binding energy. By studying the connection of the direct renormalization procedure with a generalization of an earlier suggestion which goes by the name of the $Q\overline{Q}$ renormalization procedure, we find that they are functionally equivalent.

• Received 15 November 2007

DOI:https://doi.org/10.1103/PhysRevD.77.034503