It has been proved that the $\phi^4$ theory is trivial in spacetime dimensions $d>4$. By trivial I mean that the field $\phi$ is a generalized free field, or in other words, it's only nonzero connected correlator is the two point correlator. This is a nonperturbative result, which manages to get around the fact that $\phi^4$ is nonrenormalizeable in dimensions $d>4$.
Here is the paper which proves this result: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.47.1
Have there been any results similar to this for the Yang-Mills theory? Yang-Mills is nonrenormalizeable in dimensions $d>4$ as well, so I imagine that if there were a similar result, $d=4$ should also be the critical dimension.
Answer
Good question. I am not aware of similar results for YM. The $\phi^4$ case uses correlation inequalities for ferromagnetic spin systems. Unfortunately, not many of those are known for gauge theories. YM is an example of model with non-Abelian group symmetry like $SU(N)$. Even for much simpler models with $O(N)$ symmetry like $N$-component $\phi^4$ or spherical spins, not much is known as far as correlation inequalities when $N\ge 3$.
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