  We generalize the Beliaev-Popov diagrammatic technique for the problem of
interacting dilute Bose gas with weak disorder. Averaging over disorder is
implemented by the replica method. Low energy asymptotic form of the Green
function confirms that the low energy excitations of the superfluid dirty Boson
system are sound waves with velocity renormalized by the disorder and
additional dissipation due to the impurity scattering. We find the
thermodynamic potential and the superfluid density at any temperature below the
superfluid transition temperature and derive the phase diagram in temperature
vs. disorder plane.
