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 \( \mathcal{O} \)(N2), or one-loop, with exact velocity dependence. By truncating the resulting effective Hamiltonian at \( \mathcal{O} \)(p2) 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.
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