  We consider the problem of the crossover from BCS superconductivity to
Bose-Einstein condensation in three dimensions for a system of fermions with an
attractive interaction, for which we adopt the simplifying assumption of a
suitably regularized point-contact interaction. We examine in a critical way
the fermionic (self-consistent) T-matrix approximation which has been widely
utilized in the literature to describe this crossover above the superconducting
critical temperature, and show that it fails to yield the correct behaviour of
the system in the strong-coupling limit, where composite bosons form as tightly
bound fermion pairs. We then set up the correct approximation for a ``dilute''
system of composite bosons and show that an entire new class of diagrams has to
be considered in the place of the fermionic T-matrix approximation for the
self-energy. This new class of diagrams correctly describes both the weak- and
strong-coupling limits, and consequently results into an improved interpolation
scheme for the intermediate (crossover) region. In this context, we provide
also a systematic mapping between the corresponding diagrammatic theories for
the composite bosons and the constituent fermions. As a preliminary result to
demonstrate the numerical effect of our new class of diagrams on physical
quantities, we calculate the value of the scattering length for composite
bosons in the strong-coupling limit and show that it is considerably modified
with respect to the result obtained within the self-consistent fermionic
T-matrix approximation.
