Renormalization Group in $2+\epsilon$ Dimensions and $\epsilon\to2$: A simple model analysis

TL;DR

This paper investigates the limitations of dimensional continuation of the etaunction in a simple Higgs--Yukawa model, showing it fails to accurately predict four-dimensional behavior and challenges the concept of asymptotic safety.

Contribution

It demonstrates through an explicit model that the etaunction's dimensional continuation from two to four dimensions is problematic and can lead to spurious fixed points.

Findings

Dimensional continuation of etaunction fails to reproduce four-dimensional behavior.

Mapping between schemes becomes singular as dimension approaches four.

Spurious fixed points suggest asymptotic safety cannot be established in this model.

Abstract

Using a simple solvable model, i.e., Higgs--Yukawa system with an infinite number of flavors, we explicitly demonstrate how a dimensional continuation of the $\beta$ function in two dimensional MS scheme {\it fails\/} to reproduce the correct behavior of the $\beta$ function in four dimensions. The mapping between coupling constants in two dimensional MS scheme and a conventional scheme in the cutoff regularization, in which the dimensional continuation of the $\beta$ function is smooth, becomes singular when the dimension of spacetime approaches to four. The existence of a non-trivial fixed point in $2+\epsilon$ dimensions continued to four dimensions $\epsilon\to2$ in the two dimensional MS scheme is spurious and the asymptotic safety cannot be imposed to this model in four dimensions.