Gauge Invariance at Large Charge

Quantum field theories with global symmetries simplify considerably in the large-charge limit allowing us 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-ε dimensions to leading and next-to-leading orders in the charge and all orders in the ε expansion. Remarkably, the results match our independent diagrammatic computation of the three-loop scaling dimension of the operator ϕ^{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 ϕ^{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.