Free energy of hot gauge theories with fermions throughg5

We compute the free energy density F for gauge theories, with fermions, at high temperature and zero chemical potential. In the expansion F=${\mathit{T}}^4$[${\mathit{c}}_0$+${\mathit{c}}_2$ ${\mathit{g}}^2$+${\mathit{c}}_3$ ${\mathit{g}}^3$+(${\mathit{c}}_4^{\prime }$lng+${\mathit{c}}_4$)${\mathit{g}}^4$+(c ${}_5{}^{\prime }{}_{}{}^{}\mathit{g}$+${\mathit{c}}_5$)${\mathit{g}}^5$+O(${\mathit{g}}^6$)] we determine ${\mathit{c}}_5^{\prime }$ and ${\mathit{c}}_5$ analytically by calculating two- and three-loop diagrams. The ${\mathit{g}}^5$ term constitutes the first correction to the ${\mathit{g}}^3$ term and is for the non-Abelian case the last power of g than can be computed within perturbation theory. We find that the ${\mathit{g}}^5$ term receives no contributions from overlapping double-frequency sums and that ${\mathit{c}}_5^{\prime }$ vanishes. © 1995 The American Physical Society.