Gap energy and long-range order in the boson-fermion model of superconductivity

From the observations of very small coherent lengths for all high-temperature superconductors, we conclude that the pair state is reasonably well localized in the coordinate space and therefore can be represented phenomenologically by a local boson field $\ensuremath{\varphi}$. The underlying mechanism for superconductivity is assumed to be through the "$s$-channel" reaction $2e\ensuremath{\rightarrow}\ensuremath{\varphi}\ensuremath{\rightarrow}2e$. This leads to a mixed boson-fermion model. We examine the long-range order, gap energy, and Meissner effect in such a theory.