  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.
