Gauge Invariance at Large Charge

TL;DR

This paper extends the large-charge expansion method to gauge theories, specifically calculating the scaling dimension of charged operators in the Abelian Higgs model and demonstrating gauge invariance of the results.

Contribution

It develops a gauge-invariant large-charge expansion approach and verifies it through explicit three-loop calculations in the Abelian Higgs model.

Findings

Matching between large-charge and diagrammatic computations.

Gauge independence of the dressed two-point function.

Foundation for applying large-charge methods to gauge theories.

Abstract

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.