  We consider the thermodynamics of a homogeneous superfluid dilute Bose gas in
the presence of weak quenched disorder. Following the zero-temperature approach
of Huang and Meng, we diagonalize the Hamiltonian of a dilute Bose gas in an
external random delta-correlated potential by means of a Bogoliubov
transformation. We extend this approach to finite temperature by combining the
Popov and the many-body T-matrix approximations. This approach permits us to
include the quasi-particle interactions within this temperature range. We
derive the disorder-induced shifts of the Bose-Einstein critical temperature
and of the temperature for the onset of superfluidity by approaching the
transition points from below, i.e., from the superfluid phase. Our results lead
to a phase diagram consistent with that of the finite-temperature theory of
Lopatin and Vinokur which was based on the replica method, and in which the
transition points were approached from above.
