Regularisation of non-local actions in two-dimensional field theories

TL;DR

This paper presents a regularisation method for loop calculations in non-local two-dimensional field theories, clarifying gauge invariance issues and confirming the stability of one-loop results at higher orders.

Contribution

It introduces a Pauli-Villars based regularisation approach for non-local 2D field theories and clarifies the relationship between gauge invariance and measure invariance.

Findings

Regularisation method effectively handles one-loop calculations.

Discrepancies in literature regarding invariance are resolved.

Higher-loop results are consistent with one-loop calculations.

Abstract

Taking the induced action for gauge fields coupled to affine currents as an example, we show how loop calculations in non-local two-dimensional field theories can be regulated. Our regularisation method for one loop is based on the method of Pauli and Villars. We use it to calculate the renormalisation factors for the corresponding effective actions, clearing up some discrepancies in the literature. In particular, it will be shown explicitly that vector gauge transformation invariance and Haar invariance of functional integral measures impose different requirements, but they are related by a counterterm (which is local in terms of group variables). For higher loops, we use the method of covariant derivatives combined with Pauli-Villars to argue that the one loop result remains unaltered.