Temperature Dependence of the QCD Coupling

We present a one-loop calculation of a gauge invariant QCD beta function. Using both momentum and temperature renormalization group equations we investigate the running coupling in the magnetic sector as a function of temperature and momentum scale. At fixed momentum scale we find that, in contrast to $\lambda\phi^4$ or QED, high-temperature QCD is strongly coupled, even after renormalization group improvement. However, if the momentum scale is changed simultaneously with temperature in a specified manner, the coupling decreases. We also point out in what regime dimensional reduction occurs. Both the cases $N_f$ smaller and larger than $\frac{11}{2} N_c$ are discussed.