Scattering length of composite bosons in the three-dimensional BCS-BEC crossover

Abstract

We study the zero-temperature grand potential of a three-dimensional superfluid made of ultracold fermionic alkali-metal atoms in the BCS-BEC crossover. In particular, we analyze the zero-point energy of both fermionic single-particle excitations and bosonic collective excitations. The bosonic elementary excitations, which are crucial to obtain a reliable equation of state in the Bose-Einstein condensate regime, are obtained with a low-momentum expansion up to the forth order of the quadratic (Gaussian) action of the fluctuating pairing field. By performing a cutoff regularization and renormalization of Gaussian fluctuations, we find that the scattering length ${a}_{B}$ of composite bosons, bound states of fermionic pairs, is given by ${a}_{B}=(2/3)\phantom{\rule{4pt}{0ex}}{a}_{F}$, where ${a}_{F}$ is the scattering length of fermions.