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It has been rigorously proved that the $\phi^4$ theory is trivial, i.e. is a generalized free field, in spacetime dimensions $d>4$. It is also the case that this theory is non-renormalizeable in spacetime dimensions $d>4$. Is this a general feature of renormalization?

I can rationalize this relationship as follows: If an interaction is non-renormalizeable, say the $\lambda\phi^4$ interaction, then the only way to get rid of the infinities which result from this interaction is simply to set $\lambda=0$. Therefore the only way this theory is consistent is if this interaction does not effect any of the dynamics.

Let's say the interaction is easily controlled by some coupling constant (so nothing like a nonlinear sigma model), can we make this assertion reliably that the interaction is trivial when it is non-renormalizeable?

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No, it's not that simple, although the two notions are related.

Renormalizability is a notion that is relative to a renormalization group fixed point. It is possible to have a QFT which flows from a nontrivial UV fixed point to a trivial/free fixed point. In this case it would be non-renormalizable from the point of view of the free theory. A non-unitary example is 2d Gross-Neveu with a slightly higher power of the momentum in the propagator. For a unitary example take 3d Gross-Neveu at large but finite $N$.

Also note that the asymptotic safety program is based on the idea that something similar could happen for gravity.

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