Pressure of hot QCD up to g6ln(1/g)

The free energy density, or pressure, of QCD has at high temperatures an expansion in the coupling constant g, known so far up to order g5. We compute here the last contribution which can be determined perturbatively, g6ln(1/g), by summing together results for the 4-loop vacuum energy densities of two different three-dimensional effective field theories. We also demonstrate that the inclusion of the new perturbative g6ln(1/g) terms, once they are summed together with the so far unknown perturbative and nonperturbative g6 terms, could potentially extend the applicability of the coupling constant series down to surprisingly low temperatures