BCS-BEC crossover in a relativistic boson-fermion model beyond mean field approximation

We investigate the fluctuation effect of the di-fermion field in the crossover from Bardeen-Cooper-Schrieffer (BCS) pairing to a Bose-Einstein condensate (BEC) in a relativistic superfluid. We work within the boson-fermion model obeying a global U(1) symmetry. To go beyond the mean field approximation we use Cornwall-Jackiw-Tomboulis formalism to include higher-order contributions. The quantum fluctuations of the pairing condensate is provided by bosons in nonzero modes, whose interaction with fermions gives the two-particle-irreducible effective potential. It changes the crossover property in the BEC regime. With the fluctuations the superfluidity phase transition becomes the first order in a grand canonical ensemble. We calculate the condensate, the critical temperature Tc and particle abundances as functions of a crossover parameter of the boson mass.