Classical Dimensional Transmutation and Confinement

TL;DR

This paper reveals classical renormalization group structures, including dimensional transmutation, in a $bb\u00b4$ theory, showing how external sources expose asymptotic freedom, triviality, and the energy behavior of sources and dipoles.

Contribution

It demonstrates classical renormalization group phenomena and derives an exact classical $eta$ function in a $bb\u00b4$ theory, linking classical and quantum concepts.

Findings

Isolated source has infinite energy and cannot exist as an asymptotic state.

Dipoles have finite positive energy and their interaction potential grows as the third root of distance.

The classical $eta$ function reveals asymptotic freedom and triviality depending on the sign of $b$.

Abstract

We observe that probing certain classical field theories by external sources uncovers the underlying renormalization group structure, including the phenomenon of dimensional transmutation, at purely-classical level. We perform this study on an example of $\lambda\phi^{4}$ theory and unravel asymptotic freedom and triviality for negative and positives signs of $\lambda$ respectively. We derive exact classical $\beta$ function equation. Solving this equation we find that an isolated source has an infinite energy and therefore cannot exist as an asymptotic state. On the other hand a dipole, built out of two opposite charges, has finite positive energy. At large separation the interaction potential between these two charges grows indefinitely as a distance in power one third.