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 $T$ and $gT$. The effects of the scale $T$ 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: $\alpha_s(2 \pi T) \ll 1$. For the contribution to the free energy from the scale $gT$, the perturbation series is well-behaved only if $\alpha_s(2 \pi T)$ is an order of magnitude smaller. The implication for applications of perturbative QCD to the quark-gluon plasma are briefly discussed.