Remarks on the Debye Length and the Topological Susceptibilty in Non-Abelian Gauge Theory

TL;DR

This paper analyzes the uncertainties in calculating the Debye mass and topological susceptibility in high-temperature non-abelian gauge theories, emphasizing the role of infrared sensitivity and effective theories in estimating these quantities.

Contribution

It introduces a Wilsonian effective action approach to estimate the Debye mass and topological susceptibility, quantifying the uncertainties due to infrared sensitivity.

Findings

The uncertainties in $hi$ are of order $g^4$ relative to the semiclassical result.

The effective theory approach provides finite-order estimates for $m_D$ and $hi$.

Infrared sensitivity limits the precision of perturbative calculations in non-abelian gauge theories.

Abstract

We study the Debye mass, $m_D$, and the topological susceptibility, $\chi$, at high temperatures in non-abelian gauge theory. Both exhibit, at some order in the perturbation expansion, infrared sensitivity. As a result, a perturbative analysis can at best provide an estimate of these quantities, subject to some uncertainty. The size of these uncertainties, particularly in the case of $\chi$, has been the subject of some debate. For the perturbative free energy, reframing an analysis of Braaten and Pisarski, the estimate and the associated error, can be understood in terms of a {\it Wilsonian} effective action for the low energy effective three dimensional theory. This action can be obtained completely from a perturbative calculation, which terminates at a finite order. This action provides the desired estimate. The size of the error follows from dimensional analysis in the low energy theory. The Debye length computation and its error can be obtained from a similar study of a non-relativistic effective theory for an adjoint scalar in three dimensions. $\chi$ requires a four dimensional analysis involving finite temperature instantons, but again the dominant sources of uncertainty are three dimensional, and we provide a procedure to estimate and an associated error. This uncertainty is of order $g^4 $ relative to the leading semiclassical result, and in situations of interest is small.