Momentum Space Regularizations and the Indeterminacy in the Schwinger Model

TL;DR

This paper examines the indeterminacies in the Schwinger model's calculations across different regularization methods, revealing how certain schemes influence gauge invariance and the mathematical properties of key tensors.

Contribution

It demonstrates how introducing parameters captures the indeterminacies and shows that Pauli-Villars regularization removes these ambiguities, ensuring gauge invariance.

Findings

Indeterminacies depend on regularization scheme used.

Pauli-Villars regularization eliminates indeterminacy.

Mathematical properties of the vacuum polarization tensor are affected.

Abstract

We revisited the problem of the presence of finite indeterminacies that appear in the calculations of a Quantum Field Theory. We investigate the occurrence of undetermined mathematical quantities in the evaluation of the Schwinger model in several regularization scenarios. We show that the undetermined character of the divergent part of the vacuum polarization tensor of the model, introduced as an {\it ansatz} in previous works, can be obtained mathematically if one introduces a set of two parameters in the evaluation of these quantities. The formal mathematical properties of this tensor and their violations are discussed. The analysis is carried out in both analytical and sharp cutoff regularization procedures. We also show how the Pauli Villars regularization scheme eliminates the indeterminacy, giving a gauge invariant result in the vector Schwinger model.