There are a few different definitions of renormalizability that are standard in quantum field theory textbooks. They're all called the same thing, but I'll make up names to make the distinctions clear.

  1. A theory is counterterm renormalizable if all of its divergences can be absorbed by a finite number of counterterms. Some people call this perturbatively renormalizable.
  2. A theory is superficially renormalizable if its coupling constants all have nonnegative mass dimension. Some people call this perturbatively renormalizable.
  3. A theory is Wilsonian renormalizable if it has a sensible UV limit under RG flow. Some people call this nonperturbatively renormalizable.

My impression is that none of these criteria are equivalent to each other.

  • QED is (1) and (2) but not (3), because of the Landau pole
  • The Standard Model is obviously (2), maybe (3), and when people say the Standard Model was proven renormalizable in the 70's, they mean it was proven to be (1)
  • A non-abelian gauge theory where the gauge symmetry is explicitly broken by a mass term is (2) and (3) but not (1), because the propagator does not fall off fast enough, see here
  • Any theory that is (1) and (3) can be not (2) by rescaling the fields by massive constants, as is done in some conventions for Yang-Mills, see here

A lot of confusion in quantum field theory courses comes from conflating these three notions, and just about every book and set of lecture notes will use all three interchangeably.

I'm including the above just for background. My real question is, why does anybody care about criterion (1) at all? Historically it was a big deal, as the Standard Model was accepted after it was proven to be (1), but I don't see what (1) means physically. In the modern view, there are no divergences because every theory has a cutoff, so it doesn't matter how you would cancel them if they were there.

Despite this, the proof of the (1)-renormalizability of the Standard Model remains the last thing in a lot of textbooks, and the last word in many field theory/particle physics courses. Why is that?

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    $\begingroup$ In the old days whether a theory was renormalizable or not was a useful guide for how seriously we should consider the theory. Would we ever have arrived at QED or the standard model if the modern view had been in vogue? $\endgroup$ – Lewis Miller Mar 23 '18 at 14:21
  • $\begingroup$ @LewisMiller On the contrary, I feel like if people had only used a Wilsonian framework from the start, they would have discovered everything faster. It certainly wouldn't be slower, because you get essentially the same content without having to get bogged down identifying counterterms for $n$-loop divergences. $\endgroup$ – knzhou Mar 23 '18 at 15:39
  • $\begingroup$ I disagree but rather than argue I simply state that my thoughts resonate with the answer by @rparwani $\endgroup$ – Lewis Miller Mar 23 '18 at 19:30

(I answered part of a related question you asked elsewhere. Here is the rest)

Firstly, I unfortunately disagree with many of your comments above and the conclusions you draw from them.

I find your classification of the three types confusing. I propose to relabel the categories as follows for the sake of my comments below:

Category A. Pertubatively renormalisable: All divergences can be absorbed by renormalising a finite number of parameters of the theory while still maintaining all the desired symmetries.

Category B. Power Counting renormalisable: The superficial degree of divergence of graphs is investigated to see if the divergence structure is likely to be manageable. If the couplings are all of positive mass dimension then there is hope that the theory is manageable and also that it might fall under category A (but that requires detailed proof).

Category C. Wilson renormalisation. Here one uses a cutoff, and studies the RG flow of the theory. Effective low energy theories are of interest in condensed matter and also in high energy physics.

Now some comments which address your questions, and also comments on some of you comments that I disagree with:

a. QED has a Landau pole in perturbation theory. No one knows what happens to it non-perturbatively. Indeed long before you reach the Landau pole, the approximation you used to get that pole breaks down.

b. The Standard Model of particle physics, which is still the best experimentally tested theory of the strong, weak and electromagnetic interactions, falls under categories A and B. It can also be studied under category C if you wish, and people have done that to get low energy effective theories for particular applications (see below).

c. I do not know what "nonpertubatively renormalisable" could mean. Any expansion of the quantum theory will be perturbative in some parameter or another. If not the coupling "g", the large N etc. (Unless maybe if you put it on a lattice and study it there...i am not sure even then).

d. The physical meaning of Category A and why its a big deal: If a theory is in category A then it is in some sense self-contained. After fixing a finite number a parameters you can answer many more questions to arbitrary degree of accuracy. Eg the anomalous magnetic moment of the electron agrees with experiment to 10 significant figures. Many many loops calculated by Kinoshita and company over decades. It has been reported as the best verified prediction in the history of physics. So we must be doing something right here. I think we need to pause a few seconds to appreciate this achievement of mankind.

To continue. Historically, the Category A requirement was one of the crucial guiding principle in constructing the Standard Model (see Noble acceptance speech by Weinberg). The theory that was constructed predicted many new particles that had not yet been observed. Many Noble prizes were awarded based on the predictions that came true.

It is always possible to cook up some theory to explain known facts, but to predict something and to find it come true, time and again, is exceptional.

So, a self-contained predictive structure that agrees with Nature. That is what category A is about. That is why it is in the text books.

Physically, it means that Category A theories are not sensitive to the unknown physics at the arbitrarily large cut off you have chosen (taken to infinity eventually) and whose ignorance is partially absorbed in the renormalisation parameters. In this sense category A theories are not different physically from Category C investigations. Its just that in Category A people tried to push the cutoff to infinity and they succeeded.

But the enterprise could have failed badly, then it would have been forgotten and we would have been trying something else (by the way, people were trying something else in the old days, like S-matrix approach and bootstrap when they were groping in the dark...Google it).

e. The fact that it worked tells us that any new physics beyond the Standard Model is at a much higher energy than what we had explored previously. The LHC is trying now to find the potential new physics. Which leads me to the next point.

f. The Standard Model is probably not the end of the story. In fact with the modern perspective that all theories are in some sense effective theories, even the standard model, you can now add non-renormalisable terms to account for potential high energy physics that we have not yet seen. These extra terms you add are constrained by the success of what we have already seen. Weinberg's book and many other places discuss this. So far no new physics beyond that predicted by the Standard model has been seen at LHC (though some people think that the smallness of the likely neutrino masses might be a hint of something beyond the horizon).

g. The Wilsonian perspective, that all theories are effective theories is wonderful. After all, we cant claim to know what is beyond what we have seen so far. His approach was immensely successful in condensed matter physics, and as I mentioned in points above, it is also adopted by many particle physicists. But instead of "integrating our higher degrees of freedom" which is difficult technically (the top down approach) and in practice (what is your top theory? string theory? something else?), most people start at the bottom (renormalisable theory) and add non-renormalisable terms as i mentioned above.

In summary: Category A is the crowning glory of particle physicists. It still provides guidance on the construction of extensions of particle physics theories.

Category C is the modern perspective on what theories are. But as i said above, it doesn't conflict with Category A which was an ambitious programme that somehow succeeded.

There are some language differences between those who are in the Category A camp and those who are in the Category C camp, but I believe its simply a matter of history and convenience.

I recommend the book by Zinn Justin which I believe covers all the Categories: A, B and C. Its a 1000 odd pages, with all details worked out, though the presentation is a bit terse. (I have not read it but flipped through it many years ago). The author is a renowned practitioner in the field of renormalisation with many original contributions.

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    $\begingroup$ See my answer to this question (physics.stackexchange.com/q/284368) for one possible meaning of nonperturbative renormalization. $\endgroup$ – Lewis Miller Mar 24 '18 at 15:49
  • $\begingroup$ @LewisMiller Thanks! Yes, "nonpertubative" has many connotations. In my post above I wanted to highlight that even if its not an expansion in some parameter, it will (in any realistic situation) be some approximation scheme. If I am not wrong, the approxination you used is sometimes called "mean field' or "self-consistent". $\endgroup$ – rparwani Mar 25 '18 at 2:23
  • $\begingroup$ Mean field or self consistent usually represent the result prior to the calculation of the vacuum polarization contribution. What I showed was that the mean field result could also be renormalized with the same counter terms employed in a perturbation expansion. The resulting finite vacuum polarization contribution was the same as that from 1st order (1-loop) perturbation theory except that the sign was reversed. $\endgroup$ – Lewis Miller Mar 26 '18 at 15:56
  • $\begingroup$ staff.science.uu.nl/~hooft101/gthpub/GtH_Yukawa_06.pdf Discusses the difficulties of using cut-offs in continuum non-abelian gauge theories. As far as I know, these have not been resolved. $\endgroup$ – rparwani Mar 27 '18 at 3:59

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