Critical O(2) and O(3) ϕ⁴ theories near six dimensions

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We consider O(N)-symmetric bosonic ϕ⁴ field theories above four dimensions, and propose a new reformulation in terms of an irreducible tensorial field with a cubic and Yukawa terms. The ϕ⁴ 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.