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 $\mathrm{I}\mathrm{m}(f(\theta )$) as well as the integrated cross-section $\sigma \propto \mathrm{I}\mathrm{m}(f(\theta =0))$. For fixed $\theta \mathrm{\ne }0$, the classical limit is straightforward to obtain.