  We investigate the normal state of the superconducting compound PuCoGa$_5$
using the combination of density functional theory (DFT) and dynamical mean
field theory (DMFT), with the continuous time quantum Monte Carlo (CTQMC) and
the vertex-corrected one-crossing approximation (OCA) as the impurity solvers.
Our DFT+DMFT(CTQMC) calculations suggest a strong tendency of Pu-5$f$ orbitals
to differentiate at low temperatures. The renormalized 5$f_{5/2}$ states
exhibit a Fermi-liquid behavior whereas one electron in the 5$f_{7/2}$ states
is at the edge of a Mott localization. We find that the orbital differentiation
is manifested as the removing of 5$f_{7/2}$ spectral weight from the Fermi
level relative to DFT. We corroborate these conclusions with DFT+DMFT(OCA)
calculations which demonstrate that 5$f_{5/2}$ electrons have a much larger
Kondo scale than the 5$f_{7/2}$.
