The Free Energy of Hot Gauge Theories with Fermions Through g^5

We compute the free energy density $F$ for gauge theories, with fermions, at high temperature and zero chemical potential. In the expansion $F=T^4 [c_0+c_2 g^2+c_3 g^3+(c'_4\ln g+c_4)g^4+ (c'_5\ln g+c_5)g^5+O(g^6)]$, we determine $c'_5$ and $c_5$ analytically by calculating two- and three-loop diagrams. The $g^5$ term constitutes the first correction to the $g^3$ term and is for the non-Abelian case the last power of $g$ that can be computed within perturbation theory. We find that the $g^5$ term receives no contributions from overlapping double-frequency sums and that $c'_5$ vanishes.