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The equation for the gap parameter represents the main equation of thepairing theory of superconductivity. Although it is formally defined through asingle-particle property, physically it reflects the pairing correlationsbetween opposite-spin fermions. Here, we exploit this physical connection andcast the gap equation in an alternative form which explicitly highlights thesetwo-particle correlations, by showing that it is equivalent to aHugenholtz-Pines condition for fermion pairs. At a formal level, a directconnection is established in this way between the treatment of the condensatefraction 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 easierthe inclusion of pairing fluctuations beyond mean field. As a proof-of-conceptof the new method, we apply the modified form of the gap equation to thelong-pending problem about the inclusion of the Gorkov-Melik-Barkhudarovcorrection across the whole BCS-BEC crossover, from the BCS limit of stronglyoverlapping Cooper pairs to the BEC limit of dilute composite bosons, and forall temperatures in the superfluid phase. Our numerical calculations yieldexcellent agreement with the recently determined experimental values of the gapparameter for an ultra-cold Fermi gas in the intermediate regime between BCSand BEC, as well as with the available quantum Monte Carlo data in the sameregime.

Gap equation with pairing correlations beyond mean field and its equivalence to a Hugenholtz-Pines condition for fermion pairs