Scale Invariant Scattering in 2D

For a non-relativistic scale invariant system in two spatial dimensions, the quantum scattering amplitude $f(\theta)$ is given as a dispersion relation, with a simple closed form for ${\rm Im}(f(\theta)$) as well as the integrated cross-section $\sigma \propto {\rm Im}(f(\theta=0))$. For fixed $\theta \neq 0$, the classical limit is straightforward to obtain.