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

TL;DR

This paper investigates the properties of a three-dimensional superfluid across the BCS-BEC crossover, focusing on the scattering length of composite bosons and deriving a key relation between bosonic and fermionic scattering lengths.

Contribution

It provides a theoretical derivation of the bosonic scattering length in the BEC regime using Gaussian fluctuation analysis and renormalization techniques.

Findings

The bosonic scattering length is found to be $a_B = (2/3) a_F$.

The analysis includes a low-momentum expansion up to the fourth order.

Renormalization of Gaussian fluctuations yields a reliable equation of state.

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 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.