Comment on ``Hausdorff Dimension of Critical Fluctuations in Abelian Gauge Theories"

TL;DR

This paper critically examines a previously proposed relation between the anomalous scaling dimension and the Hausdorff dimension in U(1) gauge theories, correcting it by introducing a new critical exponent.

Contribution

The authors identify an error in the existing relation and derive a corrected formula that includes an additional critical exponent, advancing the understanding of critical fluctuations in gauge theories.

Findings

The original relation $ ext{eta} + D_H = 2$ is incorrect.

A new relation involving an additional critical exponent is established.

The corrected relation improves the theoretical understanding of critical loop structures.

Abstract

Hove, Mo, and Sudbo [Phys. Rev. Lett. 85, 2368 (2000)] derived a simple connection, $\eta + D_H = 2$, between the anomalous scaling dimension $\eta$ of the U(1) universality class order parameter and the Hausdorff dimension $D_H$ of critical loops in loop representations of U(1) models. We show that the above relation is wrong and establish a correct relation that contains a new critical exponent.