  There continues to be enormous interest in the BCS to BEC transition as there
still is no exact theory. We recently reported a revealing reinterpretation of
the condensed phase Boson-Fermion Model (BFM) by comparing it to a cold atom
formulation [1]. While the ground and singly excited states appear to remain
continuous in all models we have examined, the collective modes contain a
singularity due to a Feshbach resonance (tuned by doping) causing a breakdown
of the Migdal theorem. As a result of vertex corrections, there is a
fundamental change in the nature of the superconductivity due to the formation
of preformed pairs as the previously suggested location [1] of a quantum
critical point in the fulleride phase diagram is passed. The result is a
quantum phase transition (QPT) between BCS and BEC-like (or Feshbach resonance)
superconductivity (SC).
  We discuss features of the resonance and the role of the experimentally
observed preformed pair formation in fullerides, essential to the Boson-Fermion
Model (BFM), and often speculated since the work of Nozieres and Schmitt-Rink
[17]. Here, we present arguments to establish a model of the preformed pair
which can be favorably compared to a circular charge density wave (CDW)
isolated on a fulleride molecule. The binding is much larger than a Cooper
pair. The CDW seems to be stabalized by splitting of the Jahn-Teller active
vibrational modes to reduce Coulomb repulsions. Our conclusions are: 1) the
doping of two electrons into triply degenerate orbitals results in the
experimentally observed singlet state (CDW); and 2) this CDW (preformed pair)
has a dual role as doping is varied: suppression of BCS SC and enabling a
Feshbach resonance form of SC.
