The Relevant Scale Parameter in the High Temperature Phase of QCD

Abstract

We introduce the running coupling constant of QCD in the high temperature phase, $\tilde{g}^2(T)$, through a renormalization scheme where the dimensional reduction is optimal at the one-loop level. We then calculate the relevant scale parameter, $Λ_T$, which characterizes the running of $\tilde{g}^2(T)$ with $T$, using the background field method in the static sector. It is found that $Λ_T/Λ_{\overline{\text{MS}}} =e^{(γ_E+1/22)}/(4π)\approx 0.148$. We further verify that the coupling $\tilde{g}^2(T)$ is also optimal for lattice perturbative calculations. Our result naturally explains why the high temperature limit of QCD sets in at temperatures as low as a few times the critical temperature. In addition, our $Λ_T$ agrees remarkably well with the scale parameter determined from the lattice measurement of the spatial string tension of the SU(2) gauge theory at high $T$.