On the Fixed-Point Structure of Scalar Fields

TL;DR

This paper critically examines claims about the fixed-point structure of scalar fields in the Local Potential Approximation, clarifying that previous results do not imply new physics but reveal mathematical properties of the approximation.

Contribution

The authors refute recent claims of a continuum of fixed points in scalar theories, showing that earlier work indicates mathematical features rather than new physical phenomena.

Findings

No evidence for a continuum of fixed points in scalar theories.

Clarification that previous results do not imply new physics.

Highlights the mathematical nature of the Local Potential Approximation.

Abstract

In a recent Letter (K.Halpern and K.Huang, Phys. Rev. Lett. 74 (1995) 3526), certain properties of the Local Potential Approximation (LPA) to the Wilson renormalization group were uncovered, which led the authors to conclude that $D>2$ dimensional scalar field theories endowed with {\sl non-polynomial} interactions allow for a continuum of renormalization group fixed points, and that around the Gaussian fixed point, asymptotically free interactions exist. If true, this could herald very important new physics, particularly for the Higgs sector of the Standard Model. Continuing work in support of these ideas, has motivated us to point out that we previously studied the same properties and showed that they lead to very different conclusions. Indeed, in as much as the statements in hep-th/9406199 are correct, they point to some deep and beautiful facts about the LPA and its generalisations, but however no new physics.