On the ground state of an impurity in a dilute Fermi gas

The system under consideration is a large collection of identical fermions (B), forming a background, into which is inserted a relatively small number of distinct impurity (I) particles. The background is considered to be dilute in the sense that R ≫ a, where R is the average separation of the B particles, and a is the range of their interaction potential; and the I particles are so dilute with respect to the B particles that I-I interactions can be ignored. The I particles are then all essentially at rest in their ground state. The BB and BI interaction potentials are chosen to be hard cores of the same range a. A series expansion is developed for the ground-state energy of the I particles, and the first four terms are calculated explicitly using two distinct methods, employing Feynman and Goldstone diagrams respectively. It is shown that each method has distinct advantages over the other, and that a judicious combination of both can be used to considerable benefit.