  Our goal is to understand the phenomena arising in optical lattice fermions
at low temperature in an external magnetic field. Varying the field, the
attraction between any two fermions can be made arbitrarily strong, where
composite bosons form via so-called Feshbach resonances. By setting up
strong-coupling equations for fermions, we find that in spatial dimension $d>2$
they couple to bosons which dress up fermions and lead to new massive composite
fermions. At low enough temperature, we obtain the critical temperature at
which composite bosons undergo the Bose-Einstein condensate (BEC), leading to
BEC-dressing massive fermions. These form tightly bound pair states which are
new bosonic quasi-particles producing a BEC-type condensate. A quantum critical
point is found and the formation of condensates of complex quasi-particles is
speculated over.
