Non-adiabatic corrections to the quasiparticle self-energy

High T-c superconductors and fullerenes seem to be characterized by very small bandwidths of the order of phonon frequencies. This may imply a breakdown of Migdal's theorem for the electron self-energy. There are two different approaches to the problem. The gauge-invariant self-consistent method proposed by Y. Takada includes many vertex corrections using the Ward identity. The other method by C. Grimaldi, L. Pietronero and S. Strassler (GPS) based on Migdal's idea uses the first correction to the unit vertex. These two approaches have been compared and the main results are the following: 1) Takada's method for the self-energy gives incorrect order in the Migdal parameter lambda Omega(ph)/epsilon(F), 2) in GPS's method the momentum cut-off offered by the authors cannot be used as a free parameter, and 3) there is a possible instability which can be ascribed to appearing of polaron states.