Remark on non-Abelian classical kinetic theory

It is known that non-Abelian classical kinetic theory reproduces the hard thermal loop (HTL) and hard dense loop (HDL) effective actions of QCD, obtained after integrating out the hardest momentum scales from the system, as well as the first higher dimensional operator beyond the HTL and/or 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 color algebra reproduces correctly the prefactor of the 4-point function TrA04 only for matter in asymptotically high dimensional color representations