Fancy and facts in the (d - 2) expansion of non-linear sigma models

TL;DR

This paper reviews the scaling dimensions of higher-derivative operators in non-linear sigma models and challenges previous claims about their relevance and the breakdown of the (d-2) expansion in certain physical models.

Contribution

It provides a critical analysis disputing prior mathematical claims about the significance of these operators and the validity of the (d-2) expansion in specific models.

Findings

Reevaluation of scaling dimensions in non-linear sigma models

Refutation of the supposed breakdown of the (d-2) expansion

Questioning the relevance of higher-derivative operators in certain theories

Abstract

We review the existing results on the scaling dimensions of operators with more than two derivatives in the non-linear sigma models. We argue that the speculations on the relevance of these operators, and correspondingly on the breakdown of the $(d-2)$ expansion for the classical Heisenberg model, or for the one-parameter scaling theory of localization, are based on a dubious mathematical analysis.