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A Note on the Perturbative Expansion of the Schwinger Model on $S^2$

Joseph Smith

Abstract

The Schwinger model is perhaps the simplest non-trivial exactly-solvable QFT. In this note we examine the perturbative structure of the theory on the sphere and show that its quantum corrections match those predicted by the expansion of the exact solution.

A Note on the Perturbative Expansion of the Schwinger Model on $S^2$

Abstract

The Schwinger model is perhaps the simplest non-trivial exactly-solvable QFT. In this note we examine the perturbative structure of the theory on the sphere and show that its quantum corrections match those predicted by the expansion of the exact solution.
Paper Structure (11 sections, 138 equations, 6 figures)

This paper contains 11 sections, 138 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The Schwinger Model on S2: A Brief Review
  3. Perturbation Theory in Stereographic Coordinates
  4. Two-Loop Partition Function and Regularisation
  5. One-Loop Contribution to the Photon Propagator
  6. Perturbation Theory Using an Angular Momentum Expansion
  7. One-Loop Vacuum Polarisation and the Photon Propagator
  8. The Partition Function to All Orders in Perturbation Theory
  9. Conclusion
  10. Conventions
  11. The Gauge Anomaly and the Partition Function

Figures (6)

  • Figure 1: Numerical evaluation of the leading quantum correction to the partition function \ref{['eq: integral']} using the minimum length regularisation \ref{['eq: min length reg']}.
  • Figure 2: Numerical evaluation of the leading quantum correction to the partition function \ref{['eq: integral']} using the Pauli-Villars regularisation \ref{['eq: regulated loop']}.
  • Figure 3: Numerical evaluation of the one-loop correction to the prepotential two-point function \ref{['eq: one-loop prepotential diagram']} using the minimum length regularisation \ref{['eq: min length reg']} and \ref{['eq: min length u reg']} with $\epsilon = 1/5000$.
  • Figure 4: Numerical and proposed leading-order resummed values of $\Tilde{T}_{l L_1}$ and the percentage difference $100\times |\Tilde{T}^{(\text{num.})}_{l L_1} -\Tilde{T}^{(\text{resum.})}_{l L_1} |\,/\,\Tilde{T}^{(\text{num.})}_{l L_1}$ between the two for $l=5$ and $\epsilon=0.005$ up to $k=700$.
  • Figure 5: Numerical evaluation of the one-loop contribution to the divergence of $J^a$ in the minimum length regularization \ref{['eq: min length reg']} with $\epsilon= 0.01$; the exact result for the anomaly coefficient $a=1/2$ is given for comparison.
  • ...and 1 more figures