Crossover from BCS superconductivity to Bose-Einstein condensation: A self-consistent theory

A dilute three-dimensional Fermi liquid is considered with an instantaneous attractive short-range interaction. Two sets of self-consistent equations for the temperature dependent fermion Greens functiong and the four-point vertex function are derived by field theoretic means. The interaction is taken into account using the results of low energys-wave scattering theory. The crossover region between BCS superconductivity and Bose-Einstein condensation of tightly bound pairs is located near the threshold where in the two-particle scattering problem a virtual or quasi-stationary state turns into a bound state. We show how from our self-consistent equations the BCS theory and the theory of a superfluid Bose gas can be recovered in the weak and strong coupling limit, respectively. In the strong coupling limit we find that there is a repulsive interaction between the composite bosons due to the Pauli exclusion principle. It is described by a positive scattering lengtha_B which is twice the scattering length of the fermions,a_B=2a_F. Furthermore we find that this interaction reduces the Bose-Einstein transition temperature to leading order by T_c/T_{c, BE}=–(k_Fa_F)³/3. We show that most of the previous theories of the crossover scenario can be obtained from our theory in mean-field approximation neglecting self consistency.