On the Convergence of Perturbative QCD at High Temperature

The free energy for QCD at high temperature T is calculated to order $g^5$ using effective-field-theory methods to separate the contributions from the momentum scales $2\pi T$ and $\mathrm{gT}$. The effects of the scale $2\pi T$ enter through the coefficients in the effective Lagrangian for the 3–dimensional effective theory obtained by dimensional reduction. The perturbation series for these coefficients seem to be well behaved if the running coupling constant is sufficiently small: ${\alpha }_s\left(2\pi T\right)\ll 1$. For the contribution to the free energy from the scale $\mathrm{gT}$, the perturbation series is well behaved only if ${\alpha }_s\left(2\pi T\right)$ is an order of magnitude smaller. The implication for applications of perturbative QCD to the quark-gluon plasma is briefly discussed.