Three-loop free energy for pure gauge QCD

Abstract

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{\ensuremath{\pi}}}^{2}$/9{-1/5+(${\mathit{g}}_{\mathit{A}}$/4\ensuremath{\pi}${)}^{2}$-16/\ensuremath{\surd}3(${\mathit{g}}_{\mathit{A}}$/4\ensuremath{\pi}${)}^{3}$ -48(${\mathit{g}}_{\mathit{A}}$/4\ensuremath{\pi}${)}^{4}$ln(${\mathit{g}}_{\mathit{A}}$/2\ensuremath{\pi}\ensuremath{\surd}3) +(${\mathit{g}}_{\mathit{A}}$/4\ensuremath{\pi}${)}^{4}$[22/3ln\ensuremath{\mu}\ifmmode\bar\else\textasciimacron\fi{}/4\ensuremath{\pi}T+38/3\ensuremath{\zeta}'(-3)/\ensuremath{\zeta}(-3) -148/3\ensuremath{\zeta}'(-1)/\ensuremath{\zeta}(-1)-4${\ensuremath{\gamma}}_{\mathit{E}}$+64/5]+O(${\mathit{g}}_{\mathit{A}}^{5}$)}, where T is the temperature, \ensuremath{\zeta} is the Riemann zeta function, ${\mathit{g}}_{\mathit{A}}$==g(\ensuremath{\mu}\ifmmode\bar\else\textasciimacron\fi{})${\mathit{C}}_{\mathit{A}}^{1/2}$, \ensuremath{\mu}\ifmmode\bar\else\textasciimacron\fi{} is the MS\ifmmode\bar\else\textasciimacron\fi{} renormalization scale, g(\ensuremath{\mu}\ifmmode\bar\else\textasciimacron\fi{}) 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 \ensuremath{\mu}\ifmmode\bar\else\textasciimacron\fi{}. We also give a result for the free energy of scalar ${\mathrm{\ensuremath{\varphi}}}^{4}$ theory, correcting a result previously given in the literature.