Flow equation for Halpern-Huang directions of scalar O(N) models

TL;DR

This paper investigates a class of asymptotically free scalar O(N) models using renormalization group flow equations, providing explicit nonperturbative solutions and insights into symmetry restoration and naturalness in the large-N limit.

Contribution

It derives explicit nonperturbative solutions for the potentials of scalar O(N) models along Halpern-Huang directions in the large-N limit, revealing symmetry restoration and naturalness properties.

Findings

Potentials without symmetry breaking preserve shape in the large-N limit.

Symmetry-breaking potentials become flat and restore symmetry as infrared cutoff vanishes.

Scalar theories studied do not face naturalness problems.

Abstract

A class of asymptotically free scalar theories with O(N) symmetry, defined via the eigenpotentials of the Gaussian fixed point (Halpern-Huang directions), are investigated using renormalization group flow equations. Explicit solutions for the form of the potential in the nonperturbative infrared domain are found in the large-N limit. In this limit, potentials without symmetry breaking essentially preserve their shape and undergo a mass renormalization which is governed only by the renormalization group distance parameter; as a consequence, these scalar theories do not have a problem of naturalness. Symmetry-breaking potentials are found to be ``fine-tuned'' in the large-N limit in the sense that the nontrivial minimum vanishes exactly in the limit of vanishing infrared cutoff: therefore, the O(N) symmetry is restored in the quantum theory and the potential becomes flat near the origin.