  We consider $O(N)$-symmetric bosonic $\phi^4$ field theories above four
dimensions, and propose a new reformulation in terms of an irreducible
tensorial field with a cubic and Yukawa terms. The $\phi^4$ 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.
