Are there scaling solutions in the O(N)-models for large N in d>4?

There have been some speculations about the existence of critical unitary O(N)-invariant scalar field theories in dimensions 4<d<6 and for large N. Using the functional renormalization group equation, we show that in the lowest order of the derivative expansion, and assuming that the anomalous dimension vanishes for large $N$, the corresponding critical potentials are either unbounded from below or singular for some finite value of the field.

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