  The equation for the gap parameter represents the main equation of the
pairing theory of superconductivity. Although it is formally defined through a
single-particle property, physically it reflects the pairing correlations
between opposite-spin fermions. Here, we exploit this physical connection and
cast the gap equation in an alternative form which explicitly highlights these
two-particle correlations, by showing that it is equivalent to a
Hugenholtz-Pines condition for fermion pairs. At a formal level, a direct
connection is established in this way between the treatment of the condensate
fraction in condensate systems of fermions and bosons. At a practical level,
the use of this alternative form of the gap equation is expected to make easier
the inclusion of pairing fluctuations beyond mean field. As a proof-of-concept
of the new method, we apply the modified form of the gap equation to the
long-pending problem about the inclusion of the Gorkov-Melik-Barkhudarov
correction across the whole BCS-BEC crossover, from the BCS limit of strongly
overlapping Cooper pairs to the BEC limit of dilute composite bosons, and for
all temperatures in the superfluid phase. Our numerical calculations yield
excellent agreement with the recently determined experimental values of the gap
parameter for an ultra-cold Fermi gas in the intermediate regime between BCS
and BEC, as well as with the available quantum Monte Carlo data in the same
regime.
