One of my preferred readings in the blogosphere is Tommaso Dorigo’s blog. I think this is a widely known blog for people interested about physics and got some citation also at New York Times. Quite recently he published a very interesting post (see here) about the fate of our loved Standard Model taking the move from a very nice paper by J.Ellis, J.R.Espinosa, G.F.Giudice, A.Hoecker, and A.Riotto (see here). These authors are well known and really smart at their work and, indeed, I have noticed this paper as it appeared in arxiv. My readers know that I work on a small part (QCD) of the whole picture arisen in sixties and seventies and I have never taken a look from outside. So, while I appreciated this paper I thought it was not the case to comment on it in my blog. But reading Tommaso’s post some thoughts come to my mind and these are really pertinent.
People put out two kind of constraints on the Higgs part of the standard model to have an idea of what to expect. I give you here the Higgs potential for your needs
and one immediately realizes that it introduces two free parameters. The critical one is 
There is no reason whatsoever to believe that we know all one has to know to manage a quantum field theory. Higgs could as well be not that light and strong coupled and there is no reason to think that Nature chose the small coupling case to favor us. Of course, if things will not stay this way I will be happy as a light Higgs is favored by supersymmetry and I like supersymmetry. But I would like also to emphasize that we already have all we need to manage analytically a strong coupled Higgs field. This matter I have discussed widely here and in my published papers.
So, while we all agree that a light Higgs is favored my view is that we should not have any fear of a non-perturbative Higgs field.
The Gauge Connection 
Dear Marco, I think you are actually wrong in your statements that “there is no reason” (for anything). If you don’t see a reason, it doesn’t mean that you have proved that there’s no reason.
Quite generally, a Landau pole is a problem, and almost certainly an inconsistency, and the qualitative behavior is likely to be similar in QED and the Higgs self-coupling. If either of the couplings is large, it goes infinite at some scale, and you would need to be damn lucky for the theory to calm down while remaining a QFT with the same field content.
In fact, I think it is pretty obvious that this can’t happen, either for QED or the Higgs self-coupling, so a large value of the coupling implies an inconsistency.
Hi Luboš,
The question of large coupling and inconsistency in quantum field theory is linked to the existence of a Landau pole as you clearly stated. Let me show you why this idea is plainly wrong being linked again to small perturbation theory that pervades all this field. When I have an equation like
and I decide that I can find a solution series like
and I say that 
is a zero I do a mistake. Exponential function is never zero at any finite value of 
. This is exactly the idea behind Landau pole. One takes few terms in the small perturbation series, compute the beta function and extrapolate the behavior for the full solution.
I would like to remember you (the source is a nice booklet by Warren Siegel but any textbook can help) that the small perturbation series in quantum field theory, after renormalization, when resummed gives infinity again. This is an asymptotic series. So, how do you think to extract any meaningful result at large coupling out from this scenario?
Marco
Dear Marco, I am not sure whether you’re being rational here.
I am in no way saying that whenever first-order perturbative approximations indicate that something diverges, it must diverge.
But you’re simply not right that such a naive, hopelessly approximate argument is the only argument we have to see that the Landau poles exist and they are problems.
It seems that you’re just not quite open-minded about questions that you haven’t been answered (with evidence) yet. In your example with exp(-x), the function never drops to zero, but your example doesn’t describe the Landau pole. Let me be more detailed about this point.
If the prime indicates the derivative with respect to log(mu), the (logged) renormalization scale, and “y” indicates the Higgs self-coupling, then y’+y=0 or y’+power(y)=0 may be a good moral example. But the relevant region of “x” for the discussion of the Landau pole is not the positive values of “x” – small “y” i.e. small “lambda” – but the negative values of “x” where “lambda” i.e. “y” is large!
And the function, y=exp(-x), correctly predicts that the function blows up for “x” going to minus infinity, much like you would think perturbatively. In fact, the speed with which it approaches infinity is speeding up as “lambda” gets bigger: so its worse than what you see perturbatively, not better. This is a conclusion that holds in perturbative expansion but it is one that is unlikely to change – it is more likely to get worse nonperturbatively.
For QED, there are many more things we can say. There exist similar theories with S-duality, like Seiberg duality, where the coupling “g” is equivalent to “1/g” of a different theory. See also cascades etc. These are the smooth theories where the Landau pole can be reached and smoothly transgressed, to get new physics.
But QED with electrons etc. (or one self-coupled Higgs scalar) doesn’t seem to have any simple S-dual description of this kind that could be used around the Landau pole. So it’s a real problem. Even if you wanted to hope that unknown physics fixes the divergence etc., the unknown physics is still unknown! Even in principle, you can’t really say what it is. For example, you can’t put it on the lattice.
Haven’t you thought about it? Do you think that the Landau poles go away if you wisely put a theory on the lattice? They won’t. Why? Well, simply because to make a lattice version of it, you must know how the theory behaves at extremely short distances (you want to send the spacing to zero at the end).
But for theories with Landau poles, you just don’t know this behavior (in fact, because it doesn’t exist, but I don’t want to pressure you to this final conclusion, just feel free to assume that it does exist: but it is unknown). So you don’t have any sensible starting point to create a latticization of the theory.
Even if there were a new UV theory that can be put on a lattice, it’s really new, and probably has completely new UV degrees of freedom. Whether it exists and whatever it is, I don’t see why would you say that it is the same “small” theory (i.e. with one Higgs scalar only) that you saw at weak coupling.
So your statement that there is no problem with the Landau poles is just manifestly composed out of your desire to close your eyes, not to see a problem that you must obviously see if you actually look.
Best wishes
Lubos
Dear Lubos,
Of course having in the hand an exact solution can answer to the question: Is a Landau pole real? Otherwise we should rely on faith matter and calling me not rational is somewhat hazardous.
The question I did about a Landau pole is the following and I think you did not catch it. What you have to solve is an equation like
and you can exploit a theory if the beta function is known. What I am questioning is the fact that you take some expansion of the beta function, that is at best an asymptotic one, and you are claiming to know the behavior of the theory for any
! Indeed, you started with g being small. Here is the point about Landau pole and the reason why it should not be trusted. Of course you can be lucky and can happen that for some theories you hit the jackpot. But we do not know an exact form for beta and we can only hope that our guesses, based on a not converging small perturbation expansion, do work.
For scalar theories the situation is quite different. What really worries me here is not the Landau pole but rather the fact that these theories are all trivial for a coupling going to infinity. This has already been shown by Aizenmann for more than four dimensions. Lattice computations in four dimensions seem to say the same thing.
So, my invitation to you is to look behind the curtains sometime. What you could see there can also shock you.
Best wishes,
Marco
Dear Marco, once again, there are no lattice versions of these theories because these theories don’t have any short-distance physics – or, at least, as everyone should see – no well-known short-distance physics. So all the statements where you use lattices are meaningless.
There are some features the RG flow that are not known but there are also some features of the RG flow that are known, even for every g(mu) that is known for many theories by itself, and the very qualitative point about the existence of a lethal Landau pole for theories with no S-duals, and with large and growing lambda(mu) is one of such conclusions.
It’s just not known that everything is unknown, but even if this were the case, it would still not allow you to make any of your conclusions.
The jackpot analogy is upside down, too. You would have to hit a jackpot to make sense out of a lambda phi^4 theory nonperturbatively, or something like that. Without being extremely lucky, the theory just doesn’t exist at large mu. And no, there’s no asymptotically free UV limit here.
Best wishes
Lubos
[Sorry for any delay in management of comments but I am on holidays and in these days sea and sun have the highest priority. mf]
Hi Lubos,
As always, when the argument becomes consistent you avoid it and turn back to a presumed orthodoxy.
Let me state clearly what a “Landau pole” really means:
A Landau pole means that, at some energy scale, your mathematical technique is failing and this mathematical technique is just small perturbation theory.
You cannot claim anything else and your turning round and round is always relying on the fact the you are convinced that small perturbation theory=quantum field theory and from this comes out all your convictions. This is plainly false.
The fact that you are not aware about Michael Aizenman results is quite strange to me. Then, pretending that scalar field theories cannot be studied on the lattice is somewhat foolish. You should say this to all respectable people that is doing so since eighties showing clear hints of triviality of such theories in the limit of a large coupling. A proof for four dimensions is still lacking.
I would like to emphasize that the only fear that share most of the physicists is to have to face a quantum field theory without a small perturbation theory. It is this fear that I am questioning and I am convinced is overcome.
Best wishes,
Marco
Dear Marco,
It is a nice article!
I just would like to point an up-to-dated lattice computation on electroweak theoretical bounds (I think it will dissipate any doubts about lattice methods and results).
http://xxx.lanl.gov/abs/0902.4135
Best wishes,
Rafael.
Hi Rafael,
Thank you for the link. I am a strong defender of people working on lattice. I see them like Galileo looking through his telescope: Something can be missed but the essence of how it all works is perfectly caught!
Best wishes,
Marco
Dear Marco,
what collapses at the Landau pole, in the case of theories such as a self-coupled scalar and QED, is not just perturbation theory. It’s the whole effective quantum field theory which must therefore be “UV completed”, i.e. understood as a low-energy limit of a broader, consistent, usually asymptotically free theory such as Yang-Mills theory.
The fact that you don’t want to see this full breakdown doesn’t make this breakdown non-existent.
I am not dividing scientific propositions to “orthodoxy” and “not orthodoxy” – which would be very tough (and unscientific) to divide – but to “correct” and “wrong” propositions (which are separated by the scientific method). While it is an ill-defined sociological, pseudoscientific question whether your statements are orthodox or not, it is much more demonstrable that they are incorrect.
There exists no correct paper by Aizenman that would alter the facts about the breakdown of QED or lambda.phi^4 in d=4 at the Landau pole, which is why no sensible person can know such a paper. His statement about d greater than four is just equivalent to the statement that there can’t be any nonvanishing scalar quartic self-interactions that would behave well at all energies.
This statement is also true in d=4 because in d=4, the interaction is marginally irrelevant, which gives the qualitatively same behavior as for d above four (where the interaction is strictly irrelevant).
Best wishes
Lubos
Dear Lubos,
I think that at this point it should be quite clear that your statements are false. Landau pole is an artifact of perturbation theory and this cannot be changed. The fact that you are convinced that this means a breakdown of the whole theory is just unproven as a lot of your statements that till now are already proved wrong.
There is no proof whatsoever that a Landau pole implies what you are saying. It is a matter of controversy for the same identical reasons I have exposed so far. So, what you are doing is claiming something not well acquired as a scientific truth.
The question of triviality is important, even if you are understating it, as it gives a meaning to a scalar theory in the limit of a large coupling where a Landau pole (as you intend it) would imply a not defined theory. As I am giving you mathematical theorems and not widespread legends, I think that, already at this stage, some rethinking about all this matter by your side is a due act.
Best wishes,
Marco
Dear Marco,
your conclusion, so hostile towards the perturbation theory, is wrong for a simple reason.
The amplitudes, dictated by the coupling constant, are not just some abstract numbers. They are actually increasing functions of probabilities that you create new particles in collisions. These probabilities can’t exceed a certain bound – namely 100%, if I talk about the total probability.
So if the coupling constant exceeds a certain bound, it is guaranteed that the theory is inconsistent. In other words, you need to keep “lambda” below a number – usually taken to be either “pi” or “2 pi” – otherwise an inconsistency follows.
But whenever the coupling constant is “small” in this precise sense, the qualitative conclusions of perturbation theory are valid. Perturbation theory actually implies a very large beta function, implying very speedy running that breaks the theory at just a slightly higher scale.
The alternative would be that unknown nonperturbative phenomena stabilize the coupling at a finite value. That would imply the existence of a new conformal theory (fixed point). Such a nontrivial (interacting) fixed point with one scalar in d=4 is not known and most likely doesn’t exist. Even if you thought that we don’t have a full proof that it doesn’t exist, it’s much more important that we don’t know how such a fixed point could be defined.
You’re talking about non-existent beasts and all your reasoning is based on wishful thinking and irrational dissent.
Best wishes
Lubos
Dear Lubos,
After having proved you wrong in several points of your argumentations you turn back here attacking me and claiming that mine is “wishful thinking and irrational dissent”. But as I am talking about mathematics and proved theorems, I fear that this is just your position. Indeed, there is nothing of what you said so far that has a meaningful mathematical proof. Nothing.
Now, you are coming here pretending that unitarity rules out a quantum field theory with a strong coupling, notwithstanding this is proved wrong by lattice computations as also Rafael said before with a link to a paper with a plenty of references about. You said that no lattice computation was possible here and so, either all this people is doing wishful thinking and irrational dissent or you are blatantly wrong. I am sure the latter is the right answer.
Of course, we would like to understand why a classical theory should be defined for any value of the coupling and a quantum field theory should not. Indeed, there is not a single proof here whatsoever but it is enough to have exact solutions of such classical theories to build a meaningful quantum field theory. This shows again that you are wrong.
Interestingly enough, you were not able to counteract any of the arguments that I put out till this point. This is plain mathematics and is very difficult to counteract a mathematical truth.
The conclusion is quite simple: You are disputing here correct mathematical arguments without a shred of a proof supporting your claims. This is worst than irrational. It is just foolish.
Best wishes,
Marco
[…] right mathematical question After my post on the Higgs field (see here) I would like to explain while there is no reason to be afraid. The point is the right mathematical […]
Hi Marco,
I am learning a lot from posts such as this one. Thank you also for the link and the mention of my post.
Best,
T.
Hi Tommaso,
It is always a pleasure to hear from you and to read your beautiful blog. You gave me a wonderful start-up with that post.
Best,
Marco
[…] of the Standard Model,” ArXiv, Last revised 22 Jul 2009. También es interesante leer “Who fears a non-perturbative Higgs field?,” The Gauge Connection, July 28th, 2009, y su secuela “The right mathematical […]
Dear Marco,
today, on arxiv we can see:
Click to access 0908.0780v1.pdf
What do you think about this non-perturbative approach?
Cheers,
Rafael.
Hi Rafael,
I will read that paper but did you see this
http://arxiv.org/abs/0908.1311 ?
If these authors are right I am right too!
Marco
Dear Rafael,
I think I have unleashed my enthusiasm too early. My view is that in Euclidean one should get a solution like
(
is a Bessel)for the propagator while these authors just get
that seems in disagreement with recent computations on large lattices. What is your take about this matter?
About the paper you have pointed out to me, I should say that these authors are quite brave on proposing a Standard Model without Higgs. Do all the formulas they get hold with higher loop corrections? Indeed, they show them to be true at one loop but this is a perturbative result and, as you may know, perturbation theory can hide some traps here and there. My take is that until we are sure Higgs (the particle) does not exist we cannot look further having nothing to rely on. On the other side, Higgs is our passport to new physics and an approach without Higgs should grant a look at it.
Cheers,
Marco
Dear Marco,
As authors have pointed, for “large r” the “massive propagator” transforms (as a limit) in a Yukawa-like one.
They have applied methods from hadron-spectroscopy to extract mass-spectrum for gluons. But, a more complete study, also searching for excited states is needed before they may reach more complete conclusion; since mass-ratios among excited states are universal quantities, while masses may be gauge-dependent.
Cheers,
Rafael.
Dear Rafael,
Indeed, their time-dependent results seem more interesting but they get something
that is not what I would expect from studies of Yang-Mills spectrum on the lattice. But here one should pretend that the two-point function used in this latter case has nothing to do with the propagator. I would not subscribe this.
Cheers,
Marco
[…] mfrasca 2009-07-29 02:33:53 The rest is here: Who fears a non-perturbative Higgs field? […]
[…] I would like to emphasize here is that the possibility of a strongly coupled Higgs is well alive and this can have deep implications for the model and physics at large. There are […]