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I do not know whether I understand the mystery with super-renormalizable terms. Is it that since we can do perturbation theory without paranoia we expect the coupling to be always small enough. But if we consider a higher energy theory the "relevant" operators , get multiplied in the low energy theory by factors of powers of the cutoff which can be made arbitrarily high making us wonder why perturbation theory was successful to begin with? If this is correct does this mean that "relevant" operators must be forbidden by the underlying symmetry of the high energy lagrangian?

This question is inspired by the following paper http://arxiv.org/abs/hep-th/9210046 of Polchinski in particular what is said on page 8 and 9,

"Nonrenormalizable terms are not a problem, but there is a new sort of problem: superrenormalizable terms!: To see why these are bad consider...."

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  • $\begingroup$ As far as I know there is no problem with super-renormalizable theories. You are indeed lucky if the effect you are interested in is described by such a theory, because you are guaranteed that perturbation theory works. The feature you describe about relevant operators not being suppressed by the cutoff, hence making us think twice about the region of applicability of perturbation theory, is a feature of non-renormalizable theories. Could you either edit the question appropriately or, if not convinced, give details of what you claim the problem with super-renormalizable theories to be. $\endgroup$ Commented Oct 25, 2016 at 7:18
  • $\begingroup$ @Heterotic, I made the necessary changes and add a bit more information $\endgroup$ Commented Oct 25, 2016 at 23:32

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Pages 8 and 9 of them mentioned paper discuss naturalness vs renormalizability, so it is important to clarify the difference.

First of all, it is nice to have a renormalizable theory because you can calculate things. However, it is not that "since we can do perturbation theory without paranoia we expect the coupling to be always small enough" as mentioned in the post, but they other way around. Namely, whenever the coupling is small, we can do perturbation theory and we should aways check this and be able to justify why we think the coupling might be small before we start expanding.

That being said, let's assume we have a renormalizable theory, which will of course only involve relevant opearators at low energies. Then (depending on what we are using the theory for) we might also want to check if it is natural or not, in the sense described by Polchinski.

This is better explained by his example: If we take the theory of a free scalar, then renormalizability allows us to add a term $m\phi^2$ to the Lagrangian. We can choose m to have whatever value we like, but no matter what we choose, there are loop corrections to the mass that will contribute to the value we chose. For a free scalar the corrections scale like $\Lambda^2$ which means that at the end of the day the mass of the scalar will be around that scale, unless miraculous cancellations happen in the corrections among terms of different orders. Such cancellations would be un-natural, hence justigying the terminology. Note also that naturalness is intimately related to hierarchies: it is un-natural to have to different scales $m$ and $\Lambda$ that are very different from each other.

So at this point we have two options:

i) we either include a mass term because we might be doing for example an application in condensed matter physics where nothing forces us to demand naturalness or

ii) if we are trying to study a fundamental theory of nature and we want it to be natural then we need to come up with an extra condition that forbids such terms. In QED, we impose the requirement of chiral symmetry, which then forbids these mass terms.

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