Non-Trivial Directions for Scalar Fields

TL;DR

This paper investigates the eigenvectors of the renormalization-group matrix for scalar fields at the Gaussian fixed point, identifying relevant directions linked to nontrivial, asymptotically free exponential potentials, and discusses the effects of renormalization away from the fixed point.

Contribution

It reveals the existence of relevant directions corresponding to exponential potentials and characterizes the nature of interactions generated by renormalization.

Findings

Relevant directions correspond to exponential potentials.

Polynomial potentials are irrelevant and lead to trivial theories.

Renormalization generates non-local interactions away from the fixed point.

Abstract

We study the eigenvectors of the renormalization-group matrix for scalar fields at the Gaussian fixed point, and find that that there exist ``relevant'' directions in parameter space. They correspond to theories with exponential potentials that are nontrivial and asymptotically free. All other potentials, including polynomial potentials, are ``irrelevant,'' and lead to trivial theories. Away from the Gaussian fixed point, renormalization does not induce derivative couplings, but it generates non-local interactions.