Critical $O(2)$ and $O(3)$ $ϕ^4$ theories near six dimensions

We consider $O(N)$-symmetric bosonic $ϕ^4$ field theories above four dimensions, and propose a new reformulation in terms of an irreducible tensorial field with a cubic and Yukawa terms. The $ϕ^4$ field theory so rewritten exhibits real and nontrivial IR-stable fixed points near and below six dimension, for low values of $N$ such as $N=2$ and $N=3$. The so-defined UV completions of the $O(2)$ and $O(3)$ models hence constitute precious examples of asymptotically safe quantum field theories. The possibility of an extension of our results to five dimensions is discussed.

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