# Nonperturbative results for $\phi^3$ theory in dimensions $d>6$?

The theory is nonrenormalizeable in those dimensions, but can you say anything about the theory anyway?

Specifically I am wondering about the status of whether the theory is trivial, i.e. a generalized free field. This is the case for the $$\phi^4$$ theory for dimensions $$d>4$$, however it is clear to me that the analysis for the $$\phi^3$$ theory would be more complicated, since there are nonzero odd correlation functions.

• – Abdelmalek Abdesselam Mar 2 at 16:53
• @DanYand usually one sets the coupling to be imaginary, so that the integral converges. One expects to find a well-defined albeit non-unitary theory. This is when $\phi^3$ is relevant. – Peter Kravchuk Mar 3 at 6:29
• In dimensions where $\phi^3$ is irrelevant, it is indeed unclear what is meant by the $\phi^3$ theory. – Peter Kravchuk Mar 3 at 6:33
• Questions about QFT triviality reminds me of something David Gross said at a round table discussion during the 1994 Paris ICMP (my recollection/paraphrase): I would rather mathematicians proved that $4d$ QCD exists rather than $4d$ $\phi^4$ does not exist. – Abdelmalek Abdesselam Mar 3 at 17:10