Pairing Correlations and the Pseudo-Gap State: Application of the   Pairing Approximation Theory
  We investigate the pseudogap onset temperature $T^*$, the superconducting transition temperature $T_c$ and the general nature of the pseudogap phase using a diagrammatic BCS-Bose Einstein crossover theory. This decoupling scheme is based on the pairing approximation of Kadanoff and Martin, further extended by Patton (KMP). Our consideration of the KMP pairing approximation is driven by the objective to obtain BCS like behavior at weak coupling, (which does not necessarily follow for other diagrammatic schemes). The breakdown of the Fermi liquid state at $T^*$ is investigated within the lowest order theory and is associated with intermediate values of the coupling. The superconducting instability $T_c$ is evaluated by introducing mode coupling effects, in which the long lived pairs are affected by the single particle pseudogap states and vice versa. Our $T_c$ equations, which turn out to be rather simple as a result of the KMP scheme, reveal a rich structure as a function of $g$ in which the pseudogap is found to compete with superconductivity. Our results are compared with alternate theories in the literature. 
Jiri Maly, Boldizsar Janko, and K. Levin