Many body methods and effective field theory

In the framework of pionless nucleon-nucleon effective field theory we study different approximation schemes for the nuclear many body problem. We consider, in particular, ladder diagrams constructed from particle-particle, hole-hole, and particle-hole pairs. We focus on the problem of finding a suitable starting point for perturbative calculations near the unitary limit (kFa)→∞ and (kFr)→0, where kF is the Fermi momentum, a is the scattering length and r is the effective range. We try to clarify the relationship between different classes of diagrams and the large g and large D approximations, where g is the fermion degeneracy and D is the number of space-time dimensions. In the large D limit we find that the energy per particle in the strongly interacting system is 1/2 the result for free fermions.