I am assuming that those are bare Green's function in the diagram, and the dotted line is the harmonic oscillator $<a^\dagger a>$ ?

1) The diagram you drew does not "know" about the damping of the harmonic oscillator, (although it is closely related by the optical theorem and such to the diagrams that would calculate the damping, I suppose that is what Isidore is saying). If it is necessary to include the damping then you would have to think of a way to self consistently include it.

2) I would recommend you write everything down in real space coordinates, get the second order term and then Fourier transform it. That should make it clear.

This post imported from StackExchange Physics at 2014-08-22 05:07 (UCT), posted by SE-user BebopButUnsteady