  It is known that non-Abelian classical kinetic theory reproduces the Hard
Thermal/Dense Loop (HTL/HDL) effective action of QCD, obtained after
integrating out the hardest momentum scales from the system, as well as the
first higher dimensional operator beyond the HTL/HDL level. We discuss here its
applicability at still higher orders, by comparing the exact classical
effective action obtained in the static limit, with the 1-loop quantum
effective potential. We remark that while correct types of operators arise, the
classical colour algebra reproduces correctly the prefactor of the 4-point
function $tr A_0^4$ only for matter in asymptotically high dimensional colour
representations.
