where − , + corresponds to g being just below, above g$_{∗}$ , where the scattering length a = + ∞ (BEC side) and a = −∞ (BCS side) respectively. It should be kept in mind that the underlying interactions are attractive in both cases since the fixed point occurs at negative g . In two spatial dimensions the single boson kernel obtained in [20] is G ( | k | ) = − 4 i m log   1 + mg 4 π ( log ( 2Λ | k | ) − iπ/ 2 ) 1 + mg 4 π log 2Λ | k | + iπ/ 2   = − 8 m arctan   mg/ 8 1 + mg 4 π log ( 2Λ | k | )   = − 8 m arctan ( 2 π log(2Λ$_{∗}$ / | k | ) ) (48) where Λ$_{∗}$ is defined in eq. (13), and | k | = | k$_{1}$ − k$_{2}$ | is the relative momentum. In the unitary limit g → ±∞ , the theory is at the scale Λ$_{∗}$ and one should consider | k$_{1}$ − k$_{2}$ | ≈ 2Λ$_{∗}$. The result is that G becomes a constant in this unitary limit: In the attractive case, | k − k $^{′}$| approaches 2Λ$_{∗}$ from above as g → −∞ , and thus corresponds to the + sign above. The − sign then corresponds to the repulsive case where 2Λ$_{∗}$ is approached from below. For two-component fermions, the phase space factors I in [20] are doubled, and since G ∝ 1 / I , the kernels have an extra 1 / 2 in the fermionic case: The above unitary limit of the kernels leads to a scale-invariant integral equation for the pseudo-energy, which in turn leads to the scaling forms of the previous section. This will be described in detail for the d = 2 case in subsequent sections. Note 18 G ( | k | ) = ∓ 4 π m ( d = 2) (49) G$_{fermi}$ = 1 2 G$_{bose}$ (50)