Size shrinking of composite bosons for increasing density in the BCS to Bose-Einstein crossover

We consider a ss stem of fermions in the continuum case at zero temperature , in the strong coupling limit of a short-range attraction when composite b osons form as bound-fermion pairs. We examine the density dependence of the size of the composite bosons at leading order ill the density ("dilute lim it"), and show on general physical grounds that this size should decrease w ith increasing density, both in three and two dimensions. We then compare w ith the analytic zero-temperature mean-field solution, which indeed exhibit s the size shrinking of the composite bosons both in three and two dimensio ns. We arguer nonetheless. that the two-dimensional mean-field solution is not consistent with our general result in the "dilute limit", to the extent that mean field treats the scattering between composite bosons in the Born approximation which is known to break down at low energy in two dimensions .