Equating σ with a scattering length a , as appropriate to 2 d , one sees that a diverges in the unitary limit g → ±∞ . Because of the logarithmic dependence on | k$_{1}$ − k$_{2}$ | , the S-matrix does not have a pole. However the denominator is zero when | k$_{1}$ − k$_{2}$ | = 2 k = 2 i Λ$_{∗}$, which is a remanant of the bound state pole in 3 d . The energy of this quasi-bound state is − 2Λ 2 $_{∗}$/m . Note that this quasi-bound state disappears as g$_{0}$ → 0 $^{−}$, since in this limit Λ$_{∗}$ → 0, and this is analagous to the situation in 3 d . We thus expect that the “attractive” case of the unitary limit should be better behaved since this bound state disappears, in contrast to the repulsive case where Λ$_{∗}$ → ∞ as g$_{0}$ → 0 $^{+}$, and this will be born out of our subsequent analysis. At a quantum critical point, the only length scales of the quantum gas are the thermal wavelength λ$_{T}$ = √ 2 π/mT and the length scale n 1 /d set by the density n . Equivalently, one can express physical properties in terms of the only two energy scales, the temperature T and the chemical potential µ , since the density is a function of T, µ . In order to fix normalizations in a meaningful way, it is useful to consider the sim- plest theories with z = 2 scale invariance: free, non-relativistic bosons and fermions. We set k$_{B}$ = ℏ = 1, except in a few select formulas. The free energy density is given by the well-known expression for each particle: where β = 1 /T , ω$_{k}$ = k $^{2}$/ 2 m and s = 1 , − 1 corresponds to bosons, fermions respec- tively. The integrals over wave-vectors can be expressed in terms of polylogarithms 11 F = s β ∫ d $^{d}$k (2 π ) d log ( 1 − se − β ( ω$_{k}$ − µ ) ) (20) III. THERMODYNAMIC SCALING FUNCTIONS AT THE QUANTUM CRITICAL POINT