In fact, the scale Λ$_{∗}$ = Λ e 4 π/mg is an RG invariant: and is the natural coupling constant in this problem that physical quantities are expressed in terms of. The S-matrix can be inferred from the kernel in eq. (48): where | k | = | k$_{1}$ − k$_{2}$ | is the relative momentum of the two incoming particles. Near the fixed point g = 0 $^{−}$, Λ$_{∗}$ = 0 (Λ$_{∗}$ = ∞ ) and the S-matrix S = 1, consistent with the free theory. The same applies to the other side of the fixed point g = 0 + where Λ$_{∗}$ = ∞ . On the other hand, consider the low energy limit of the attractive case where g = −∞ . Here Λ$_{∗}$ = Λ, and this scale is effectively an infra-red cutoff. Thus at low energies | k$_{1}$ − k$_{2}$ | ≈ 2Λ$_{∗}$ and S = − 1. Similar arguments apply to the high energy limit of the repulsive case g = + ∞ . It is clear that, unlike in 3 d , this definition of the unitary limit does not correspond to a renormalization group fixed point in the usual sense of a zero of the beta function and is somewhat engineered; nevertheless it defines a scale-invariant theory. Let us turn now to the scattering length. Using the scattering amplitude M computed in [20], the cross section for relative momenta 2 k is: Near the scale k = Λ$_{∗}$: d Λ$_{∗}$ d log Λ = 0 (14) S ( | k | ) = 4 π/mg + log(2Λ / | k | ) − iπ/ 2 4 π/mg + log(2Λ / k ) + iπ/ 2 (15) = log(2Λ$_{∗}$ / | k | ) − iπ/ 2 log(2Λ$_{∗}$ / k ) + iπ/ 2 (16) σ ( k ) = m $^{2}$g2 8 k 1 (1 + mg 4 π log(Λ /k )) 2 + π 2 (17) = 2 π 2 k 1 log $^{2}$(Λ$_{∗}$ /k ) + 16 π $^{4}$/m$^{2}$g2 (18) σ (Λ$_{∗}$) = 2 Λ$_{∗}$ ( mg 4 π ) 2 (19) 10