Noninvariant renormalization in the background-field method

TL;DR

This paper examines how noninvariant regularization schemes, like cutoff and differential renormalization, lead to incorrect results in the background-field method, especially in theories with nonlinear symmetries.

Contribution

It demonstrates that noninvariant regularizations can cause inconsistencies in the background-field formalism, highlighting the importance of invariant schemes for accurate results.

Findings

Cut-off regularization is incompatible with the background-field method.

Differential renormalization fails in theories with nonlinear symmetries.

Invariant regularizations are necessary for consistent background-field calculations.

Abstract

We investigate the consistency of the background-field formalism when applying various regularizations and renormalization schemes. By an example of a two-dimensional $\sigma$ model it is demonstrated that the background-field method gives incorrect results when the regularization (and/or renormalization) is noninvariant. In particular, it is found that the cut-off regularization and the differential renormalization belong to this class and are incompatible with the background-field method in theories with nonlinear symmetries.