  We introduce a novel point-particle effective description of ANO vortex
solitons in the critical Abelian Higgs Model (AHM) in $d=2+1$ based on the
small winding expansion. Identifying the effective vortices with the elementary
quanta of a complex scalar field, relativistic vortex-vortex scattering
amplitudes are calculated as a diagrammatic, perturbative expansion in the
winding number $N$. Making use of powerful techniques recently developed for
analyzing the post-Minkowskian two-body problem in general relativity, we
efficiently extract the contribution to the loop integrals from the classical
potential region, with the resulting velocity expansion subsequently resummed
to all orders. The main result of this paper is an analytic expression for the
classical, vortex-vortex potential at $\mathcal{O}\left(N^2\right)$, or
one-loop, with exact velocity dependence. By truncating the resulting effective
Hamiltonian at $\mathcal{O}\left(p^2\right)$ we derive an analytic,
perturbative expression for the metric on the 2-vortex moduli space. Finally,
the emergence of the critical AHM from the classical limit of the
$\mathcal{N}=2$ supersymmetric AHM, and the resulting constraints on the
point-particle EFT is described in detail using an on-shell superspace
construction for BPS states in $d=2+1$.
