Witten-Veneziano Relation for the Schwinger Model

TL;DR

This paper examines the Witten-Veneziano relation in the Schwinger model, comparing different formulations and demonstrating their equivalence in the infinite volume or zero temperature limits using path integral and Hamiltonian methods.

Contribution

It clarifies the relation between topological susceptibility formulas in the Schwinger model, showing their equivalence in specific limits through both path integral and Hamiltonian approaches.

Findings

Both formulas yield the same result in the infinite volume limit.

The contact term and the Seiler-Stamatescu formula are equivalent in the studied limits.

The methods confirm the consistency of topological susceptibility calculations.

Abstract

The Witten-Veneziano relation between the topological susceptibility of pure gauge theories without fermions and the main contribution of the complete theory and the corresponding formula of Seiler and Stamatescu with the so-called contact term are discussed for the Schwinger model on a circle. Using the (Euclidean) path integral and the canonical (Hamiltonian) approaches at finite temperatures we demonstrate that both formulae give the same result in the limit of infinite volume and (or) zero temperature.