Classical Dimensional Transmutation and Confinement

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.

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