  We present a new many body formalism for BCS-BEC crossover, which represents
a modification of the BCS-Leggett ground state to include 4-fermion, and higher
correlations. In the BEC regime, we show how our approach contains the
\textit{Petrov et al} 4-fermion behavior and associated scattering length
$a_{dd}$ at short distances, and secondly reduces to composite-boson Bogoliubov
physics at long distances. It reproduces the Lee-Yang term, whose numerical
value is also fixed by $a_{dd}$. We have also examined the next term beyond the
Lee-Yang correction in a phenomenological fashion, building on cloud size data
and collective mode experiments. However, one has to view this phenomenological
analysis with some caution since experiments are in a state of flux and are
performed close to unitarity.
