Classical Dimensional Transmutation and Confinement

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 $λϕ^{4}$ theory and unravel asymptotic freedom and triviality for negative and positives signs of $λ$ respectively. We derive exact classical $β$ 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.