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Anomalous dimensions of monopole operators in three-dimensional quantum electrodynamics

Silviu S. Pufu

TL;DR

The paper computes the scaling dimensions of monopole operators in three-dimensional QED with $N_f$ fermion flavors by mapping local monopole insertions to states on $S^2 \times \mathbb{R}$ and performing a $1/N_f$ expansion via the state-operator correspondence.Leading order comes from integrating out fermions in a fixed monopole background, while next-to-leading order arises from Gaussian fluctuations of the U(1) gauge field; the calculation hinges on the fermion Green's function and the gauge-field kernel in the monopole background.For the unit monopole ($n=1$), the dimension at the infrared fixed point is found to be $\text{dim} = 0.265\,N_f - 0.0383 + O(1/N_f)$, reflecting a small but nonzero $O(1)$ correction to the leading result.The analysis reveals exact UV divergence cancellations between electric and magnetic-type gauge-field fluctuations and provides a framework to assess confinement/deconfinement by comparing monopole dimensions to the spacetime dimension.The results are consistent with lattice and theoretical expectations about the critical flavor number for confinement and highlight the potential for applying similar methods to non-supersymmetric gauge theories.

Abstract

The space of local operators in three-dimensional quantum electrodynamics contains monopole operators that create $n$ units of gauge flux emanating from the insertion point. This paper uses the state-operator correspondence to calculate the anomalous dimensions of these monopole operators perturbatively to next-to-leading order in the $1/N_f$ expansion, thus improving on the existing leading order results in the literature. Here, $N_f$ is the number of two-component complex fermion flavors. The scaling dimension of the $n=1$ monopole operator is $0.265 N_f - 0.0383 + O(1/N_f)$ at the infrared conformal fixed point.

Anomalous dimensions of monopole operators in three-dimensional quantum electrodynamics

TL;DR

The paper computes the scaling dimensions of monopole operators in three-dimensional QED with $N_f$ fermion flavors by mapping local monopole insertions to states on $S^2 \times \mathbb{R}$ and performing a $1/N_f$ expansion via the state-operator correspondence.Leading order comes from integrating out fermions in a fixed monopole background, while next-to-leading order arises from Gaussian fluctuations of the U(1) gauge field; the calculation hinges on the fermion Green's function and the gauge-field kernel in the monopole background.For the unit monopole ($n=1$), the dimension at the infrared fixed point is found to be $\text{dim} = 0.265\,N_f - 0.0383 + O(1/N_f)$, reflecting a small but nonzero $O(1)$ correction to the leading result.The analysis reveals exact UV divergence cancellations between electric and magnetic-type gauge-field fluctuations and provides a framework to assess confinement/deconfinement by comparing monopole dimensions to the spacetime dimension.The results are consistent with lattice and theoretical expectations about the critical flavor number for confinement and highlight the potential for applying similar methods to non-supersymmetric gauge theories.

Abstract

The space of local operators in three-dimensional quantum electrodynamics contains monopole operators that create units of gauge flux emanating from the insertion point. This paper uses the state-operator correspondence to calculate the anomalous dimensions of these monopole operators perturbatively to next-to-leading order in the expansion, thus improving on the existing leading order results in the literature. Here, is the number of two-component complex fermion flavors. The scaling dimension of the monopole operator is at the infrared conformal fixed point.

Paper Structure

This paper contains 16 sections, 99 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Setup and conventions
  3. Leading order free energy
  4. Beyond leading order: The strategy
  5. Fermion Green's function
  6. Evaluating the gauge field kernel
  7. General formulas
  8. Particular cases
  9. $n=0$
  10. $n=1$
  11. Numerics for $n = 1$
  12. Discussion
  13. Monopole harmonics via conformal transformation
  14. Scalar harmonics
  15. Spinor and vector harmonics
  16. ...and 1 more sections

Figures (2)

  • Figure 1: The quantities $F^{(1) E}_{1}$ (brown) and $F^{(1) B}_{1}$ (red) evaluated with the relativistic cutoff \ref{['IsotropicCutoff']}. They each diverge logarithmically as $L \to \infty$.
  • Figure 2: The correction $F^{(1)}_1 = F^{(1)E}_{1} + F^{(1)B}_{1}$ to the free energy evaluated by summing up the expressions in \ref{['deltaFEB']} with the relativistic cutoff \ref{['IsotropicCutoff']}. The orange points are obtained by evaluating \ref{['deltaFEB']} numerically, and the solid line is a cubic fit.