content_image/373525.jpg content_image/373526.jpg content_image/373527.jpg content_image/373528.jpg content_image/373529.jpg content_image/373530.jpg
In fact, the scale Λ$_{∗}$ = Λ e 4 π/mg is an RG invariant: 

<img src='content_image/373525.jpg'>

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): 

<img src='content_image/373526.jpg'>

<img src='content_image/373527.jpg'>

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$_{∗}$: 

<img src='content_image/373528.jpg'>

<img src='content_image/373529.jpg'>

<img src='content_image/373530.jpg'>

