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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.

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  • $\begingroup$ try this: journals.aps.org/prd/abstract/10.1103/PhysRevD.95.085001 $\endgroup$ – Abdelmalek Abdesselam Mar 2 at 16:53
  • $\begingroup$ @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. $\endgroup$ – Peter Kravchuk Mar 3 at 6:29
  • $\begingroup$ In dimensions where $\phi^3$ is irrelevant, it is indeed unclear what is meant by the $\phi^3$ theory. $\endgroup$ – Peter Kravchuk Mar 3 at 6:33
  • $\begingroup$ 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. $\endgroup$ – Abdelmalek Abdesselam Mar 3 at 17:10

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