  We discuss the low-energy physics of the three-orbital Anderson impurity
model with the Coulomb interaction term of the Kanamori form which has orbital
SO(3) and spin SU(2) symmetry and describes systems with partially occupied
$t_{2g}$ shells. We focus on the case with two electrons in the impurity that
is relevant to Hund's metals. Using the Schrieffer-Wolff transformation we
derive an effective Kondo model with couplings between the bulk and impurity
electrons expressed in terms of spin, orbital, and orbital quadrupole
operators. The bare spin-spin Kondo interaction is much smaller than the
orbit-orbit and spin-orbital couplings or is even ferromagnetic. Furthermore,
the perturbative scaling equations indicate faster renormalization of the
couplings related to orbital degrees of freedom compared to spin degrees of
freedom. Both mechanisms lead to a slow screening of the local spin moment. The
model thus behaves similarly to the related quantum impurity problem with a
larger SU(3) orbital symmetry (Dworin-Narath interaction) where this was first
observed. We find that the two problems actually describe the same low-energy
physics since the SU(3) symmetry is dynamically established through the
renormalization of the splittings of coupling constants to zero. The
perturbative renormalization group results are corroborated with the
numerical-renormalization group (NRG) calculations. The dependence of spin
Kondo temperatures and orbital Kondo temperatures as a function of interaction
parameters, the hybridization, and the impurity occupancy is calculated and
discussed.
