Functional-integral approach to Gaussian fluctuations in Eliashberg theory

The Eliashberg theory of superconductivity is based on a dynamical electron-phonon interaction as opposed to static present in BCS theory. standard derivation an equation motion approach, which incorporates certain approximations such Migdal's approximation for the pairing vertex. In this paper we provide functional-integral-based and also consider its Gaussian-fluctuation extension. functional approach enables self-consistent method computing mean-field equations, arise saddle-point conditions, here observe that conventional self energy function both appear Hubbard-Stratonovich transformations. An important consequence fact it provides systematic Cooper density-channel interactions Gaussian fluctuation response. We investigate strong-coupling diamagnetic susceptibility near critical temperature.