Scattering length of composite bosons in the 3D BCS-BEC crossover

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 BEC 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 $a_B = (2/3) a_F$, where $a_F$ is the scattering length of fermions.