Field theory of collective excitations III condensation of four-particle clusters

For many fermions with arbitrary two-body forces we investigate the effective action Г?, α] where ? is the density matrix including pair correlations and α is the four-particle vertex. The physical state including all radiative corrections is obtained by extremizing Г?, α] with respect to ? and α. Just as ? ≠ 0 corresponds to non-zero density and condensation of pairs, α ≠ 0 signals the presence of a condensate of four-particle clusters. A simple loop expansion is given for Г?, α]. To lowest order, the extremum amounts to the Hartree-Fock-Bogoliubov equations. The next two steps bring in the four-particle condensate.