Evaluation of a class of diagrams useful in many-body calculations

A recent functional approach to the many-body theory has put in evidence the relevance of the diagrams containing one fermion loop only. We shall prove here a relevant theorem which shows that all (but one) of the θ-functions are irrelevant, the contained information being transferred to analyticity properties. The simplest diagrams are explicitly evaluated and some useful limits are discussed. However, some of these diagrams have been considered also in different approaches to the many-body problem, so their evaluation is of general interest. The low-density behaviour is also considered. (orig.)