Scale hierarchy in high-temperature QCD

Because of asymptotic freedom, QCD becomes weakly interacting at high temperature: this is the reason for the transition to a deconfined phase in Yang-Mills theory at temperature $T_c$. At high temperature $T \gg T_c$, the smallness of the running coupling $g$ induces a hierachy betwen the "hard", "soft" and "ultrasoft" energy scales $T$, $g T$ and $g^2 T$. This hierarchy allows for a very successful effective treatment where the "hard" and the "soft" modes are successively integrated out. However, it is not clear how high a temperature is necessary to achieve such a scale hierarchy. By numerical simulations, we show that the required temperatures are extremely high. Thus, the quantitative success of the effective theory down to temperatures of a few $T_c$ appears surprising a posteriori.