The Relevant Scale Parameter in the High Temperature Phase of QCD

TL;DR

This work defines a temperature-dependent running coupling $\tilde{g}^2(T)$ in high-temperature QCD by enforcing maximal decoupling of non-static modes in the static background-field sector. Using the background-field method, the authors fix a subtraction scale $\mu(T)$ and derive the scale parameter $\Lambda_T$, obtaining $\Lambda_T/\Lambda_{\overline{MS}} = e^{(\gamma_E+c)}/(4\pi)$ (with $c=1/22$ in the quenched case) and $\tilde{g}^2(T)=1/[\beta_0\ln(T^2/\Lambda_T^2)]$. They show the same scheme is optimal for lattice perturbation theory and demonstrate agreement with lattice measurements of the spatial string tension in SU(2) at high $T$, providing a natural explanation for why dimensional reduction sets in at a few times $T_c$. The results also generalize to include light quarks, yielding flavor-dependent shifts in $\Lambda_T$, and the lattice analysis confirms the perturbative picture, reinforcing the utility of the optimal renormalization scheme for studying high-$T$ 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 $T$.