Free energy of QCD at high temperature

Effective-field-theory methods are used to separate the free energy for a non-Abelian gauge theory at high temperature $T$ into the contributions from the momentum scales $T$, $\mathrm{gT}$, and $g^{2}T$, where $g$ is the coupling constant at the scale $2\pi T$. The effects of the scale $T$ enter through the coefficients in the effective Lagrangian for the three-dimensional effective theory obtained by dimensional reduction. These coefficients can be calculated as power series in $g^2$. The contribution to the free energy from the scale $\mathrm{gT}$ can be calculated using perturbative methods in the effective theory. It can be expressed as an expansion in $g$ starting at order $g^3$. The contribution from the scale $g^{2}T$ must be calculated using nonperturbative methods, but nevertheless it can be expanded in powers of $g$ beginning at order $g^6$. We calculate the free energy explicitly to order $g^5$. We also outline the calculations necessary to obtain the free energy to order $g^6$.