content_image/417448.jpg content_image/417447.jpg
The self-energy correction to the phonon propagation line, as given by the diagrams in Fig. 10(b), is 

<img src='content_image/417448.jpg'>

The vacuum bubble (diagram with no external lines, Fig. 10(c)) is 

<img src='content_image/417447.jpg'>

where the 1 / 2 π in the loop energy integral is a Fourier transform normalization factor, and the 2Re comes from including diagrams on both sides of the ρ line. 

Substituting these into Eq. (30) and setting ˙ ρ = 0, we obtain Eq. (A.13) at zeroth order and Eq. (A.33) at second order, i.e. the two methods agree. 

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