Semiclassics at the cusp

We study cusped Wilson line operators in the Abelian Higgs model in $d=4-\epsilon$ at large external charges. Using a double-scaling limit $Q\to \mathrm{\infty }$, $\epsilon \to 0$ with $Q\epsilon$ fixed, we develop a semiclassical framework that provides analytic control beyond fixed-order perturbation theory. We compute the cusp anomalous dimension for arbitrary charges up to next-to-next-to-leading order in the gauge coupling, while resumming scalar self-interactions to all orders. Our results interpolate between perturbative and large-charge regimes, accessing domains that are invisible in fixed-order perturbation theory. As an application, we provide new predictions for various defect CFT observables, including the Mandelstam-Schwinger-dressed two-point function characterizing the superconducting phase transition.