Perfect observables for the hierarchical non-linear $O(N)$-invariant $\sigma$-model

TL;DR

This paper analyzes the hierarchical non-linear O(N)-invariant sigma model by computing eigenvalues and eigenvectors of the linear renormalization group transformation using perturbation theory, providing insights into the model's observables.

Contribution

It introduces a method to compute moving eigenvalues and eigenvectors of the RG transformation for the sigma model using perturbation theory.

Findings

Eigenvalues of the RG transformation are explicitly computed.

Eigenvectors satisfy a Callan-Symanzik type equation.

Results enhance understanding of the model's renormalization behavior.

Abstract

We compute moving eigenvalues and the eigenvectors of the linear renormalization group transformation for observables along the renormalized trajectory of the hierarchical non-linear $O(N)$-invariant $\sigma$-model by means of perturbation theory in the running coupling constant. Moving eigenvectors are defined as solutions to a Callan-Symanzik type equation.