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For $\phi^4$ theory, when $d>4$, the theory becomes nonrenormalizable, but when $d>4$, we can use mean field theory to calculate the exact critical exponents. The intuition behind mean field theory is that when d is large, there are more neighbors, so the mean field approximation gets better.

Another example is gravity. We know that gravity is nonrenormalizable theory, but we can use mean field theory to obtain very good results (that is why it took so long to get general relativity).

Is there any relation between nonrenormalizable theory and mean field theory? If so, how can I understand this?

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  • $\begingroup$ There are two problems mixed up here: constructing field theories in the continuum (UV problem), and studying large scale properties of lattice spin systems (IR problem). In high dimension the IR problem simplifies and critical exponents are exactly equal to what one finds in mean field theory. However, the UV problem can't be solved, i.e., the only continuum limits one can hope to get are trivial. $\endgroup$
    – Abdelmalek Abdesselam
    Commented Jun 3, 2020 at 20:39
  • $\begingroup$ @AbdelmalekAbdesselam There is still some indirect relation, no? Take the Ising lattice model. The IR fixed point of the phase transition in $d<4$ is described by a non-trivial Ising CFT. This Ising CFT fixed point can also be reached from the Gaussian fixed point by a relevant perturbation. Now, at $d=4$ these two points merge together, and this is related to the fact that the Gaussian fixed point looses one of the relevant operators ($\phi^4$). After the fps merge, the IR fp of the lattice becomes the gaussian fp. Still, I agree in that I can't see a direct relation between the two problems. $\endgroup$
    – Peter Kravchuk
    Commented Jun 3, 2020 at 20:49
  • $\begingroup$ I didn't say the two problems are not related. What I wanted to make is the OP is aware there are two and not just one problem. Indeed, the relation is the direction of the RG flow going towards instead of away from the Gaussian fp. This direction explains both the mean field behavior in the IR problem and the triviality issues in the UV problem. $\endgroup$
    – Abdelmalek Abdesselam
    Commented Jun 3, 2020 at 20:51

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One can always write down a mean field solution to a field theory. The question is whether this solution is locally/perturbatively stable. Its analogous to local minima of the free energy.

If the upper critical dimension ($d_c$) of a field theory can be correctly identified, then for $d > d_c$, generally, mean-field solutions tend to be stable due to the overall weakness of fluctuations at low energies. For $d< d_c$ if such a locally stable solution is found, then its non-generic, hence highly non-trivial.

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