Cooper instability for a given field in a quiescent Fermi gas

The Cooper instability in a Fermi gas is examined using the perturbative diagram approach. A graphical functional derivative technique based on Ward's identity is developed to obtain two-particle interactions and then to calculate the vertex part. The pairing instability for a given interaction, such as a phonon (plasmon, etc.) field, occurs in a quiescent Fermi sea, i.e. without exciting or involving background particles (holes), only if the interaction is attractive, as first proposed by Cooper and adopted in the BCS (Bardeen-Cooper-Schrieffer) theory. The consequence from this technique provides a way to evaluate the effect of the vertex corrections and responses of the Fermi gas in both charge and spin channels incorporating the backward scattering process. The significance of the methodology presented in the present work lies in the fact that it can both reproduce the known results, and, more importantly, be extended to investigate the intermediate or strong coupling case, such as nuclear interactions, where a neglect of vertex corrections may not be a good approximation