Hexadecapolar Kondo effect in URu${}_2$Si${}_2$?

We derive the coupling of a localized hexadecapolar mode to conduction electrons in tetragonal symmetry. The derivation can be easily adapted to arbitrary multipoles in arbitrary environment. We relate our model to the two-channel Kondo (2CK) model and show that for an $f^2$-configuration, a relevant crystal field splitting in addition to the 2CK interaction is intrinsic to tetragonal symmetry. We discuss possible realizations of a hexadecapolar Kondo effect in URu${}_2$Si${}_2$. Solving our model we find good agreement with susceptibility and specific heat measurements in Th${}_{1-x}$U${}_x$Ru${}_2$Si${}_2$ ($x\ll 1$).