An article by Steven Weinberg entitled On the Development of Effective Field Theory appeared on the arXiv last night. It’s based on a talk he gave in September, surveys the history of effective field theories and argues for what I’d call the “SM is just a low energy approximation” point of view on fundamental physics. I’ve always found this point of view quite problematic, and think that it’s at the root of the sad state of particle theory these days. That Weinberg gives a clear and detailed version of the argument makes this a good opportunity to look at it carefully.

A lot of Weinberg’s article is devoted to history, especially the history of the late 60s-early 70s current algebra and phenomenological Lagrangian theory of pions. We now understand this subject as a low energy effective theory for the true theory (QCD), in which the basic fields are quarks and gluons, not the pion fields of the effective theory. The effective theory is largely determined by the approximate SU(2) x SU(2) chiral flavor symmetry of QCD. It’s a non-linear sigma model, so non-renormalizable. The non-renormalizability does not make the theory useless, it just means that as you go to higher and higher energies, more possible terms in the effective Lagrangian need to be taken into account, introducing more and more undetermined parameters into the theory. Weinberg interprets this as indicating that the right way to understand the non-renormalizability problem of quantum gravity is that the GR Lagrangian is just an effective theory.

So far I’m with him, but where I part ways is his extrapolation to the idea that all QFTs, in particular the SM, are just effective field theories:

The Standard Model, we now see – we being, let me say, me and a lot of other people – as a low-energy approximation to a fundamental theory about which we know very little. And low energy means energies much less than some extremely high energy scale 10

^{15}−10^{18}GeV.

Weinberg goes on to give an interesting discussion of his general view of QFT, which evolved during the pre-SM period of the 1960s, when the conventional wisdom was that QFTs could not be fundamental theories (since they did not seem capable of describing strong interactions).

I was a student in one of Weinberg’s graduate classes at Harvard on gauge theory (roughly, volume II of his three-volume textbook). For me though, the most formative experience of my student years was working on lattice gauge theory calculations. On the lattice one fixes the theory at the lattice cut-off scale, and what is difficult is extrapolating to large distance behavior. The large distance behavior is completely insensitive to putting in more terms in the cut-off scale Lagrangian. This is the exact opposite of the non-renormalizable theory problem: as you go to short distances you don’t get more terms and more parameters, instead all but one term gets killed off. Because of this, pure QCD actually has no free parameters: there’s only one, and its choice depends on your choice of distance units (Sidney Coleman liked to call this dimensional transvestitism).

The deep lesson I came out of graduate school with is that the asymptotically free part of the SM (yes, the Higgs sector and the U(1) are a different issue) is exactly what you want a fundamental theory to look like at short distances. I’ve thus never been able to understand the argument that Weinberg makes that at short distances a fundamental theory should be something very different. An additional big problem with Weinberg’s argument is its practical implications: with no experiments at these short distances, if you throw away the class of theories that you know work at those distances you have nothing to go on. Now fundamental physics is all just a big unresolvable mystery. The “SM is just a low-energy approximation” point of view fit very well with string theory unification, but we’re now living with how that turned out: a pseudo-scientific ideology that short distance physics is unknowable, random and anthropically determined.

In Weinberg’s article he does give arguments for why the “SM just a low-energy approximation” point of view makes predictions and can be checked. They are:

- There should be baryon number violating terms of order $(E/M)^2$. The problem with this of course is that no has ever observed baryon number violation.
- There should be lepton number violating terms of order $E/M$, “and they apparently have been discovered, in the form of neutrino masses.” The problem with this is that it’s not really true. One can easily get neutrino masses by extending the SM to include right-handed neutrinos and Dirac masses, no lepton number violation. You only get non-renormalizable terms and lepton number violation when you try to get masses using just left-handed neutrinos.

He does acknowledge that there’s a problem with the “SM just a low-energy approximation to a theory with energy scale M=10^{15}−10^{18} GeV” point of view: it implies the well-known “naturalness” or “fine-tuning” problems. The cosmological constant and Higgs mass scale should be up at the energy scale M, not the values we observe. This is why people are upset at the failure of “naturalness”: it indicates the failure not just of specific models, but of the point of view that Weinberg is advocating, which has now dominated the subject for decades.

As a parenthetical remark, I’ve today seen news stories here and here about the failure to find supersymmetry at the LHC. At least one influential theorist still thinks SUSY is our best hope:

Arkani-Hamed views split supersymmetry as the most promising theory given current data.

Most theorists though think split supersymmetry is unpromising since it doesn’t solve the problem created by the point of view Weinberg advocates. For instance:

“My number-one priority is to solve the Higgs problem, and I don’t see that split supersymmetry solves that problem,” Peskin says.

On the issue of quantum gravity, my formative years left me with a different interpretation of the story Weinberg tells about the non-renormalizable effective low-energy theory of pions. This got solved not by giving up on QFT, but by finding a QFT valid at arbitrarily short distances, based on different fundamental variables and different short distance dynamics. By analogy, one needs a standard QFT to quantize gravity, just with different fundamental variables and different short distance dynamics. Yes, I know that no one has yet figured out a convincing way to do this, but that doesn’t imply it can’t be done.

I am just here to give my usual speech: The “naturalness” problems in the standard model are not scientific problems. They are aesthetic problems. They come about because physicists claim an unobservable number that temporarily appears in the math is “unlikely”.

There are two problems with this. First, the debate about the supposed singularity at black hole horizons should have taught physicists that fretting about non-observable issues in mathematical calculations is a waste of time. Second, one can’t speak about probabilities without probability distributions, and we will never be able to obtain an empirically supported probability distribution over unobservable parameters*.

Add to this that “naturalness” arguments haven’t worked with the axion (the original one), haven’t worked with the cosmological constant, haven’t worked with supersymmetry.

(The charm quark prediction btw wasn’t a naturalness argument, it was a good old-fashioned argument from Occam’s razor. It’s just that people at the time used the word “natural” in their argument.)

The bottom line is, naturalness should go out of the window.

For what the SM is concerned, well, it doesn’t contain gravity, so of course the short-distance physics isn’t fundamentally the right one.

—

* Same problem with multiverse arguments.

I’m trying to understand what the central thesis of this post is.

On the one hand, we seem to have the suggestion that the fundamental theory really could be just a QFT which consists of the standard model coupled to a new form of gravity. Well, it’s a striking idea, and one always has the 2009 prediction of the Higgs boson mass by Shaposhnikov and Wetterich, that was premised on exactly such an assumption.

On the other hand, there is the criticism of naturalness, on the grounds that naturalness predicts new weak-scale particles and they haven’t turned up.

I guess I don’t understand whether this constitutes grounds for criticizing the very idea of treating the SM as an effective field theory. Yes, if it’s SM all the way to the Planck scale, then the SM is not just an EFT. But that’s an extremely bold hypothesis that may or may not be true.

Meanwhile, for now, it’s surely legitimate to still be interested in the possibility of new particles or forces. Is it claimed that EFT is a wrong way to model this or a wrong way to be systematic about it?

If naturalness is a bad guide, to me that implies, not that the use of EFT is overall misguided, but that you need to be ready for the coefficients to be larger or smaller than expected.

Lattice QCD is the exact opposite of what you describe: It’s exactly because of the ambiguity in the action that you have “non-universality”. This shows up when trying to take the continuum, or small-coupling, limit. This is a result of the running of the coupling (asymptotic freedom), & shows up in arguments for analyticity in the complex coupling-constant plane (e.g., see ‘t Hooft). It’s also related to the inability of constructive quantum field theorists to prove the existence of theories that aren’t @ least superrenormalizable (besides the instanton problem, also a difficulty on the lattice). These problems also show up in (resummation of) perturbation theory, as renormalons, which require (an infinite number of) new couplings as energies rise (like nonrenormalizable theories), appearing as vacuum values of (color-singlet) composite operators. (Thus, contrary to the belief of loop quantum gravity people, lattices don’t eliminate problems seen in perturbation theory.)

The only known solution is (perturbatively) finite theories, which require something you hate — supersymmetry.

Hi Warren,

If I understand you (and perhaps I don’t) lattice QCD doesn’t suffer from any of the problems you describe (at least according to conventional wisdom. Not theorems). The ambiguities you mention concern semiclassical methods (perturbative, with resummation, summing over saddles points).

It is expected (but not proved) that the only real couplings where physical quantities are not analytic are 1. zero (at $\theta=0$) and 2. some nonzero value (at $\theta=\pi$).

QCD is (presumably) a completely finite theory, with very few parameters (one if there are no quarks and $\theta=0$).

Just to be technically precise, there can be other nonanalyticities, but these are not universal and depend on irrelevant terms in the lattice action.

Sabine, actually this post explains clearly why naturalness is not an esthetic argument but a mathematical one.

QFT is formulated in terms of functional integrals, which are in general divergent and irresolvable. However,

(i) they can be expanded, in various ways: perturbing around the coupling constant (perturbation theory), around scale separations (EFTs), saddle points and so on.

(ii) The divergence can also be regulated in such a way that it does not affect physical quantities, which are generally correlators measured at a certain scale, provided we use some experimental data as input, the bare minimum (which as Peter explains is sufficient for QCD) is the scale.

The most used expansion that includes (i) and (ii) will have a few normalizeable and super-renormalizeable terms (where there are no factors dependent on scale separation) and infinitely many non-renormalizeable and trivial ones that go away if the scale separation is large enough (this last point is where people most often use natuarlness arguments).

Saying axions, ccs, Higgs mass etc are not natural is a shorthand for saying

that if you take this theory and try to do the functional integral with (i) and (ii) in mind

(i) and (ii) do not quite work for these observables. That is a rigorous mathematical statement about a physical theory, which might indicate problem with the theory, or problems with the approximation (see the discussion after your comment). But it is not just aesthetic. If would be aesthetic if we realized Feynmans dream

————————————————–

https://arxiv.org/abs/2006.08594 “This makes me dream,

or speculate, that maybe there is some way, and we are

just missing it, of evaluating the path integral directly”

——————————————————-

and STILL had to use naturalness. But we did not as yet, so its a legitimate maths question.

Cliff Burgess here

https://twitter.com/CburgesCliff/status/1349579929462198273

characterizes this by

“Very early 70s take on things in that blog post, it seems”

Not quite right, since until 1973 and asymptotic freedom the consensus was that QFT was no good at short distances, or for describing strong interactions. From about 1974 -1984, it’s true that the point of view of this posting may have been the dominant one. Post 1984 things went back to “QFT no good at short distances (need string theory)” and that’s been the prejudice the past 30 years. I think the failure of the field to make any progress during this period argues for going back to before the wrong turn.

One aspect of the history of the subject is that it was only for a short ten year period (1974-1984) that grad students entering the field were being told that maybe QFT works at all distance scales. At every other point during the nearly 100 year history of QFT they’ve been told it fails at high energy.

Michell Porter,

Nothing wrong with EFT. Lots of QFTs are EFTs. All QFTs used in condensed matter are EFTs. I don’t think though that the idea that the SM is not just an EFT is an “extremely bold hypothesis”. All expected failures of the theory based on the idea that it’s just an EFT have not worked out. It agrees with all experimental results and (modulo Higgs +U(1) problems) seems to make perfectly good sense at all distance scales. It should be the baseline conjecture that it can describe all distance scales, until someone comes up with something better. No one has.

Warren Siegel,

I won’t try and argue that issues of the perturbation and semi-classical approximations to a putative rigorous non-perturbative lattice-regularized QCD are well-understood. But all evidence I’m aware of is that (keeping things simple) lattice pure gauge theory is a well-defined non-perturbative theory, with expected infrared and ultraviolet behavior if you take the continuum limit appropriately. And when you do this, the limit is insensitive to the definition at the cutoff scale as I stated.

In any case, even if you can find a problem with this, the larger point is that this is as close as there is to a well-defined 4d theory that makes sense at short distance scales and gives behavior we observe in nature. For no well-defined X is there any evidence that “we should be doing X instead of QFT at short distances”.

@ Peter & Peter:

‘t Hooft’s arguments are based entirely on the running of the coupling, & are not tied to perturbation theory. But the same problems are seen in perturbation theory, & in constructive quantum field theory, which is rigorous & entirely nonperturbative.

Lattice QCD is an alternative approach to perturbation theory that provides results for different observables. But it does not avoid any of the fundamental problems of the theory. The conclusion is that QCD is a low-energy effective theory. Of course, the problem is worse with the Standard Model because of the U(1) gauge group, which is seen to be a problem even @ finite coupling on the lattice. (Problems with asymptotically free couplings show up only near vanishing coupling, as proven by Tomboulis, but in agreement with ‘t Hooft’s argument.)

Hi Warren,

But ‘tHooft’s arguments are based on renormalons/instantons. Although these don’t emerge from perturbation theory per se, they are not pure running-coupling constant arguments.

He also had a program for constructing large-N theories in four dimensions, but I don’t know what came of this (maybe this is what you mean by constructive FT).

By the way, there is a long paper by Magnon, Rivasseu and Seneor, written almost 30 years ago, claiming a construction of SU(2) Yang-Mills in 4 Euclidean dimensions, in finite volume (not the infinite volume limit). If correct, this work goes a long way towards overcoming the problems you raise; in a finite volume, all the same features are present.

… and Peter Woit will tell us soon to shut up and conduct this technical discussion elsewhere.

Peter Orland,

No, technical discussions that are relevant to the topic are encouraged! The question of whether QCD is just an effective theory or not is highly relevant.

Warren Siegel,

My understanding is that there is good evidence (much of it numerical) for the conjecture of the existence (at all distance scales) of a well-defined non-perturbative version of QCD, as specified precisely in the Millenium prize document

https://www.claymath.org/sites/default/files/yangmills.pdf

If Tomboulis or anyone else has a solid argument for non-existence (can you give a reference?), they should be putting in their claim for the \$ 1 million. My suspicion is that what you’re discussing is a different problem involving the subtleties of the perturbation expansion or semi-classical approximation for QCD, for which I’m willing to believe there are all sorts of issues.

I think the misunderstanding in Warren Siegel’s comment is this:

“Lattice QCD is an alternative approach to perturbation theory that provides results for different observables.”

The lattice can of course be used as a regulator in perturbation theory, and this theory has the sames issues as weak coupling perturbation theory using other regulators. (The lattice has a convergent strong coupling expansion, albeit with a finite radius of convergence). However, the main point is that the lattice provides a fully non-perturbative definition of the theory. There is indeed no proof, but we have strong physical arguments, and plenty of numerical evidence, that the continuum and infinite volume limits exist, and that they define the theory we observe in nature.

This means that QCD is a perfect theory, one that can be extended to arbitrarily short distances. But in practice this does not help, because QCD is embedded into electroweak theory, and we have an equally strong expectation (and, again, numerical evidence) that the $U(1)$ and scalar sectors cannot be extended to arbitrarily short distances. This means that the SM, even without gravity, must be viewed as an EFT.

What is maybe somewhat unusual is that the only estimate of the breakdown scale that we have right now is from RG running of the scalar sector. It would be nice to directly observe a higher dimension operator. It is possible that neutrino mass comes from a higher dimension operator, but until we observe a Majorana mass we don’t know (which is why double beta decay experiments are so important).

Thomas,

To be clear, since the SU(3) and SU(2) gauge theories are asymptotically free, the remaining problem is the U(1) and Higgs. The U(1) “Landau pole” problem is at scales way above the Planck scale. The Higgs problem is intriguing, indicating borderline instability up near the Planck scale.

I agree that the SM cannot just stand on its own, ultimately one wants to unify it with a quantum theory of gravity. To me, the simplest scenario would be a unification with a gravity QFT, that would resolve the issues of the high energy limit of the U(1) and the Higgs. Then, yes, the SM decoupled from gravity would not be fully consistent, but on the other hand characterizing it as just a low energy effective approximation would be misleading (since to a large degree the theory would work at all distance scales).

Lun,

Although it is true that renormalizable “unnatural” theories are not fully consistent at high energies, those energies are EXTREMELY high.

For example, QED is unnatural, but its cut-off cannot be predicted from the theory itself. If we didn’t know about SU(2)$\times$U(1), we would not know where (below $10^19\;eV$) QED breaks down.

In this sense, Sabine is completely correct. There is no ability to predict where the theory breaks down, and naturalness is an attempt to do the impossible (to predict what cannot be predicted). We only know that such theories must break down, below some very large momentum.

That a QFT has a Landau pole is a problem only for approaches that regulate the theory with a noncovariant cutoff. This includes lattice approximation.

However, a Landau pole is not a problem for covariant regularizations.

In particular, in causal perturbation theory, the renormalization is done covariantly at an arbitrarily chosen renormalization energy parameter. The theory is well-defined for any choice far away from the Landau pole and, by the Peterman-Stückelberg renormalization group, is independent of this choice — only the quality of approximation depends on the choice.

The Landau pole only says that one cannot choose the renormalization energy parameter close to the pole without getting meaninglessly inaccurate results.

(This is unlike renormalization through cutoffs, where the covariant theory is only obtained in a limit, a process that suffers from a UV induced Landau pole.)

Relevant in this context is also the not widely known fact that QCD has like QED a Landau pole. But while for QED the UV induced Landau pole is at physically inaccessible energies (far larger than the Planck energy), the IR induced QCD Landau pole is at physically realized energies! Nevertheless the pole does not invalidate predictions at these energies. Thus arguing for inconsistency based on Landau poles is a relic from ancient times where QFT was not yet well enough understood.

For further details see my article on Causal perturbation theory at https://www.physicsforums.com/insights/causal-perturbation-theory/

and the discussion at https://www.physicsoverflow.org/32752/ and

https://www.physicsoverflow.org/21328/

The Landau pole is an artifact of the one-loop (or finite loop) approximation. The pole is present in a region of momentum space where this approximation can’t be used.

I don’t think it is helpful to think of triviality/unnaturalness in connection with Landau poles. In Wilson’s discussion (see Sections 12 and 13 of Wilson and Kogut’s Physics Reports article on the renormalization group) concerning (non)triviality, the Landau pole is not mentioned at all. Corrections to Landau’s mean field theory (an entirely different development) and Ginzburg’s criterion are, however, relevant.

Peter Orland, for sure “naturalness” is not a very useful tool to understand what exactly is wrong (the last decades of theoretical physics have conclusively shown this). Rather, lack of naturalness is a symptom of a problem, and its wrong to say the problem is aesthetic, it is mathematical consistency rather then aesthetics.