# Relation between relevant/irrelevant/marginal and renormalizability

I have few more conceptual questions on the renormalization group (I began here). Let me briefly summarize what I've learned so far.

There are few types of interactions in QFT:

1. Relevant

2. Irrelevant

3. Marginal

All those properties can be understood at the classical level by studying the dimensions of the corresponding coupling constants. The third type can be further classified due to the possible presence of the quantum corrections as:

3 a) Exactly marginal ($\beta$-function vanishes at all orders)

3 b) Marginally relevant

3 c) Marginally irrelevant

Now, let me ask few closely related questions about these five cases.

1. Is it possible to unambiguously put items {1, 2, 3a), 3b), 3c)} into boxes {non-renormalizable, renormalizable, super-renormalizable}? Or do we still need to perform some further research in certain cases?

To ask the subsequent questions, I will need to clearly define what I mean by 'renormalization'. I think of it not just as of 'getting rid of unphysical cutoff' (which can clearly happen only in theories, where we have to introduce this cutoff), but as of the presence of the running coupling constant and non-trivial RG flow (which, to my understanding, can happen even in theories with no divergences).

In other words, the $\beta$-function should be generally defined via differentiating with respect to the physical energy scale $\mu$, and not with respect to the unphysical cutoff $\Lambda$ (even though, in theories with such a cutoff formally it's the same thing).

2. Is everything OK with what I said about the renormalization?

3. Of those five cases, in which ones do we have the phenomenon of renormalization? Speaking of the second classification, my guess is that we cannot talk about renormalization for non-renormalizable theories. Do we always have non-trivial RG flow in renormalizable theories? What about super-renormalizable?

I apologise in advance for so many text-book questions. But I feel like it will not take long for a specialist to answer. While in textbooks, all this knowledge is always hiding behind some monstrous calculations.

• Please read our FAQ on writing question titles. – DanielSank Nov 25 '16 at 6:13
• Do you mean renormalizable or perturbatively renormalizable? These two are quite different. For example, quantum General Relativity in 3 spacetime dimensions is perturbatively nonrenormalizable (its gravitational coupling is irrelevant), and yet it was proven by Witten that there is (at least one) consistent nonperturbative quantization of GR in 3d. – Prof. Legolasov Nov 26 '16 at 1:09
• In general, the relation between Wilsonian classification of couplings and perturbative renormalizability is subtle because renormalizability of models is proven in the context of perturbative theory, while Wilsonian classification is (formally, of course) nonperturbative. Another example would be QED in 4d spacetime. It is perturbatively renormalizable (there's a proof of this claim), but its coupling is known to be marginally irrelevant (the Landau pole problem). – Prof. Legolasov Nov 26 '16 at 1:14
• Thanks for comments, they are helpful. I will need to think for some time, since I'm not quite familiar with these topics, and trying so far to set up the general picture. – mavzolej Nov 27 '16 at 3:01