Solution to the Perturbative Infrared Catastrophe of Hot Gauge Theories

Perturbative calculations of the free energy of a non-Abelian gauge theory at high temperature $T$ break down at order $g^6$. This problem is solved by constructing a sequence of two effective field theories in which the momentum scales $T$ and $\mathrm{gT}$ are integrated out. The free energy is decomposed into contributions from the momentum scales $T$, $\mathrm{gT}$, and $g^{2}T$. The three terms are a power series in $g^2$, a power series in $g$ beginning at order $g^3$, and a power series in $g$ beginning at order $g^6$. In the $g^{2}T$ contribution, the coefficients in the power series can be calculated using lattice simulations of 3-dimensional QCD. Renormalization group equations can be used to sum up the leading logarithms of $T/\left(\mathrm{gT}\right)$ and of $gT/\left(g^{2}T\right)$.