content_image/328779.jpg content_image/328780.jpg content_image/328781.jpg content_image/328782.jpg content_image/328783.jpg content_image/328784.jpg
function. Finally, the free energy density then takes the simple form: 

where f$_{0}$ is the filling fraction of the free theory: 

Consider first the 1-dimensional bosonic case. The model is integrable, which implies that the N-body S-matrix factorizes into 2-body S-matrices, and the exact free energy is given by the thermodynamic Bethe ansatz[22]. The 2-body S-matrix is ′ 

<img src='content_image/328779.jpg'>

<img src='content_image/328780.jpg'>

<img src='content_image/328781.jpg'>

In the unitary limit g → ±∞ , S = − 1, and the thermodynamic Bethe ansatz reduces to that of a free gas of fermionic particles. 

In 3 spatial dimensions the exact kernel for a single component boson is the following[20]: 

where g$_{R}$ is a renormalized coupling: 

<img src='content_image/328782.jpg'>

<img src='content_image/328783.jpg'>

with Λ an ultra-violet momentum cutoff. As in the 1-dimensional case, as discussed in section II, the unitary limit corresponds to g$_{R}$ → ±∞ where S = − 1. Thus in the unitary limit the kernel becomes 

<img src='content_image/328784.jpg'>

