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 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: 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 F = − 1 β ∫ ( d k ) ( s log(1 + sf ) − 1 2 ( f − f$_{0}$ 1 + sf$_{0}$ )) f$_{0}$ ( k ) = 1 e β ( ω$_{k}$ − µ ) − s (43) S ( k − k $^{′}$) = k − k ′ − ig/ 4 k k ′ + ig/ 4 (44) G ( | k − k $^{′}$| ) = − 8 πi m | k − k $^{′}$| log ( 1 − img R 16 π | k − k $^{′}$| 1 + img R 16 π k k ′ $_{|}$) = − 16 π m | k − k $^{′}$| arctan ( mg$_{R}$ 16 π | k − k $^{′}$| ) (45) 1 g$_{R}$ = 1 g + m Λ 4 π $_{2}$. (46) G ( k , k $^{′}$) = ∓ 8 π 2 m | k − k $^{′}$| ( d = 3) (47) 17