  Quantum field theories with global symmetries simplify considerably in the
large-charge limit allowing to compute correlators via a semiclassical
expansion in the inverse powers of the conserved charges. A generalization of
the approach to gauge symmetries has faced the problem of defining
gauge-independent observables and, therefore, has not been developed so far. We
employ the large-charge expansion to calculate the scaling dimension of the
lowest-lying operators carrying $U(1)$ charge $Q$ in the critical Abelian Higgs
model in $D=4-\epsilon$ dimensions to leading and next-to-leading orders in the
charge and all orders in the $\epsilon$ expansion. Remarkably, the results
match our independent diagrammatic computation of the three-loop scaling
dimension of the operator $\phi^Q(x)$ in the Landau gauge. We argue that this
matching is a consequence of the equivalence between the gauge-independent
dressed two-point function of Dirac type with the gauge-dependent two-point
function of $\phi^Q(x)$ in the Landau gauge. We, therefore, shed new light on
the problem of defining gauge-independent exponents which has been
controversial in the literature on critical superconductors as well as lay the
foundation for large-charge methods in gauge theories.
