Three-loop free energy for pure gauge QCD

We compute the free energy density for pure non-Abelian gauge theory at high temperature and zero chemical potential. The three-loop result to O(${\mathit{g}}^4$) is F=${\mathit{d}}_{\mathit{A}}$ ${\mathit{T}}^4$ ${\mathrm{\pi }}^2$/9{-1/5+(${\mathit{g}}_{\mathit{A}}$/4π${)}^2$-16/√3(${\mathit{g}}_{\mathit{A}}$/4π${)}^3$ -48(${\mathit{g}}_{\mathit{A}}$/4π${)}^4$ln(${\mathit{g}}_{\mathit{A}}$/2π√3) +(${\mathit{g}}_{\mathit{A}}$/4π${)}^4$[22/3lnμ¯/4πT+38/3ζ’(-3)/ζ(-3) -148/3ζ’(-1)/ζ(-1)-4${\gamma }_{\mathit{E}}$+64/5]+O(${\mathit{g}}_{\mathit{A}}^5$)}, where T is the temperature, ζ is the Riemann zeta function, ${\mathit{g}}_{\mathit{A}}$==g(μ¯)${\mathit{C}}_{\mathit{A}}^{1/2}$, μ¯ is the MS¯ renormalization scale, g(μ¯) is the corresponding coupling constant, and ${\mathit{d}}_{\mathit{A}}$ and ${\mathit{C}}_{\mathit{A}}$ are the dimension and Casimir number of the adjoint representation. We examine the sensitivity of this result to the choice of renormalization scale μ¯. We also give a result for the free energy of scalar ${\mathrm{\phi }}^4$ theory, correcting a result previously given in the literature.