Scale hierarchy in high-temperature QCD

TL;DR

This paper investigates the hierarchy of energy scales in high-temperature QCD and demonstrates through numerical simulations that achieving the ideal scale separation requires extremely high temperatures, questioning the effective theory's applicability at lower temperatures.

Contribution

It provides numerical evidence on the temperature thresholds needed for the scale hierarchy in high-temperature QCD, clarifying the limits of effective theories.

Findings

High temperatures are required for clear scale hierarchy in QCD.

Effective theories work surprisingly well at temperatures only a few times T_c.

Numerical simulations challenge assumptions about the temperature range of effective theory validity.

Abstract

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.