On stability of the three-dimensional fixed point in a model with three coupling constants from the $\epsilon$ expansion: Three-loop results

TL;DR

This paper investigates the stability of a three-coupling fixed point in a model using three-loop $psilon$-expansion, revealing eigenvalue degeneracy and discussing the method's reliability.

Contribution

It provides a detailed three-loop analysis of the renormalization-group flow in a three-coupling model, highlighting eigenvalue degeneracy and series expansion nuances.

Findings

Degeneracy of eigenvalue exponents at the fixed point.

Potential appearance of $psilon^{1/2}$ terms in series.

Discussion on the reliability of the $psilon$-expansion method.

Abstract

The structure of the renormalization-group flows in a model with three quartic coupling constants is studied within the $\epsilon$-expansion method up to three-loop order. Twofold degeneracy of the eigenvalue exponents for the three-dimensionally stable fixed point is observed and the possibility for powers in $\sqrt{\epsilon}$ to appear in the series is investigated. Reliability and effectiveness of the $\epsilon$-expansion method for the given model is discussed.