Re-running the QCD shear viscosity

The remarkably small shear viscosity to entropy density ratio $\eta/s < 0.5$ of the quark-gluon plasma (QGP) is a key insight from heavy ion experiments. Nonetheless, the basic understanding of this `observable' still seems to be rudimentary, with existing perturbative QCD estimates suggesting $\eta/s > 1$, a view that we scrutinize here. In order to extrapolate the available perturbative approach to phenomenologically relevant temperatures, we consider carefully controllable higher-order corrections: We adapt the leading-order effective kinetic scheme (instead of the catchy next-to-leading log formula $\eta_{_{{\rm NLL}}} \propto [\alpha^2 \log(c/\alpha)]^{-1}$), by using cross sections with a running coupling. This effect should be pertinent (for a QGP) and we argue for a choice of scale-dependence that makes it consistent with thermal screening. We conclude that $\eta/s \approx 0.2$ does not indicate genuine non-perturbative effects, it may rather be understood within resummation-improved perturbative QCD, with some uncertainties from deploying kinetic theory near to its limit.