I am considering the calculation of the vertex correction due to electron-phonon interactions. Specifically, I am looking for solutions to the integral (Eqn. 1)
$$\int dq G(q)D(q-p)G(p+k)$$
where $dq=d^4q/(2\pi)^2$. The fermionic Green's function is given by
where $Z$ is the quasiparticle weight, and the Goldstone propagator is given by
where $\omega_s\sim q^2$. When I have seen the calculation done before, the bosonic propagator puts bounds on the integration limits. This is discussed in "Interaction between Electrons and Lattice Vibrations in a Normal Metal", by A.B. Migdal. However, I am interested in explicitly calculating the vertex correction given the explicit form of the phonon propagator, similar to what is done in K. B. Blagoev, J. R. Engelbrecht, and K. S. Bedell Phys. Rev. Lett. 82, 133 (1999).
I would therefore like to request references that explicitly calculate integrals of the form Eqn. 1. Unfortunately, all the major textbooks I have resort to Migdal's theorem, and do not explicitly calculate the integral out. Anything at the level of Mahan would be appreciated, but I am more partial to open source, online references if possible. I am also open to someone answering with a direct solution of the above integral, but I am more interested in the current literature on the topic (as I have been able to find very little). I have my own calculation, but I would like to check my work and see other people's methods.