Free energy of hot gauge theories with fermions through g5

We compute the free energy density F for gauge theories, with fermions, at high temperature and zero chemical potential. In the expansion F = T4[c0 + c2g2 + c3g3 + (c′4 ln g + c4)g4 +(c′ 5 ln g +c5)g5 +O(g6)], we determine c′5 and c5 analytically by calculating two- and three-loop diagrams. The g5 term constitutes the first correction to the g3 term and is for the non-Abelian case the last power of g that can be computed within perturbation theory. We find that the g5 term receives no contributions from overlapping double-frequency sums and that c′5 vanishes