“The effort to understand the universe is one of the very few things which lifts human life a little above the level of farce and gives it some of the grace of tragedy.”
that holds for
If you compute the Hamiltonian the energy does not appear to be finite, differently from what the relation dispersion is saying. This is very similar to what happens to plane waves for the wave equation. The way out is to take a finite volume and normalize properly the plane waves. One does this to get the integral of the Hamiltonian finite and all amounts to a proper normalization. In our case where must this normalization enter? The striking answer is: The coupling. This is an arbitrary parameter of the theory and we can properly rescale it to get the right normalization in the Hamiltonian. The final result is a running coupling exactly in the same way as I and others have obtained for the quantum theory. You can see the coupling entering in the right way both in the solution and in the computation of the Hamiltonian.
If you are curious about these computations you can read the revised version of my paper to appear soon on arxiv.
Marco Frasca (2010). Mass generation and supersymmetry arxiv arXiv: 1007.5275v1
Being near Christmas I send my wishes to all my friends around the World. Most of them are physicists like Attilio Cucchieri that I cited a lot in this blog with his wife Tereza Mendes for their beautiful works on Yang-Mills propagators. Attilio asked to me again about the behavior of 2d Yang-Mills propagators and my approach to the theory. As you know, lattice computations due to Axel Maas (see here) showed without doubt that here, differently from the higher dimensional case, the scaling solution appears. This is an important matter as ‘t Hooft showed in 1974 (see here) that, in the light cone gauge, Yang-Mills theory has no dynamics. Indeed, the Lagrangian takes a very simple form
and the nonlinear part is zero due to the fact that the field has just two Lorentzian indexes. Please, note that here and . Now, you can choose the time variable as you like and if you take this to be you see that there are no dynamical equations left for the gluon field! You are not able to do this in higher dimensions where the dynamics is not trivial and the field develops a mass gap already classically. Two-dimensional QCD is not trivial as there remains a static (nonlocal) Coulomb interaction between quarks. You can read the details in ‘t Hooft’s paper.
Now, in my two key papers (here and here) I proved that, at the classical level the Yang-Mills field maps on a quartic massless scalar field in the limit of the coupling going to infinity. So, I am able to build a quantum field theory for the Yang-Mills equations in this limit thanks to this theorem. But what happens in two dimensions? The scalar field Lagrangian in the light-cone coordinate becomes
and you can see that, whatever is your choice for the time variable, this field has always a dynamics. In 2d I am not able to map the two fields and, if I choose a different gauge for the Yang-Mills field, I can find propagators that are no more bounded to be massive or Yukawa-like. Indeed, the scaling solution is obtained. This is a bad news for supporters of this solution but this is plain mathematics. It is interesting to note that the scalar field appears to have nontrivial massive solutions also in this case. No mass gap is expected for the Yang-Mills field instead.
Thank you a lot Attilio for pointing me out this question!
Merry Xmas to everybody!
I would like to write down a few lines on a paper published today on arxiv by Axel Maas (see here). This author draws an important conclusion about the propagators in Yang-Mills theories: These functions depend very few on the gauge group, keeping fixed the coupling a la ‘t Hooft as being a Casimir parameter of the group that is N for SU(N). The observed changes are just quantitative rather than qualitative as the author states. Axel does his computations on the lattice in 2 and 3 dimensions and gives an in-depth discussion of the way the scaling solution, the one not seen on the lattice except for the two-dimensional case, is obtained and how the propagators are computed on the lattice. This paper opens up a new avenue in this kind of studies and, as far as I can tell, such an extended analysis with respect to different gauge groups was never performed before. Of course, in d=3 the decoupling solution is obtained instead. Axel also shows the behavior of the running coupling. I would like to remember that a decoupling solution implies a massive gluon propagator and a photon-like ghost propagator while the running coupling is strongly suppressed in the infrared.
The conclusion given in this paper is a strong support to the work all the people is carrying on about the decoupling solution. As you can see from my work (see here), the only dependence seen for my propagators on the gauge group is in the ‘t Hoof t coupling. The same conclusion is true for other authors. It is my conviction that this paper is again an important support to most theoretical work done in these recent years. By his side, Axel confirms again a good nose for the choice of the research avenues to be followed.
Axel Maas (2010). On the gauge-algebra dependence of Landau-gauge Yang-Mills propagators arxiv arXiv: 1012.4284v1
The question of the generation of the mass for the particles in the Standard Model is currently a crucial one in physics and is a matter that could start a revolutionary path in our understanding of the World as it works. This is also an old question that can be rewritten as “What are we made of?” and surely ancient greeks asked for this. Today, with the LHC at work and already producing a wealth of important results, we are on the verge to give a sound answer to it.
The current situation is well-known with a Higgs mechanism (but here there are several fathers) that mimics the second order phase transitions as proposed by Landau long ago. In some way, understanding ferromagnetism taught us a lot and produced a mathematical framework to extract sound results from the Standard Model. Without these ideas the model would have been practically useless since the initial formulation due to Shelly Glashow. The question of mass in the Standard Model is indeed a stumbling block and we need to understand what is hidden behind an otherwise exceptionally successful model.
As many of yours could know, I have written a paper (see here) where I show that if the way a scalar field gets a mass (and so also Yang-Mills field) is identical in the Standard Model, forcefully one has a supersymmetric Higgs sector but without the squared term and with a strong self-coupling. This would imply a not-so-light Higgs and the breaking of the supersymmetry the only way to avoid degeneracy between the masses of all the particles of the Standard Model. By my side I would expect these signatures as evidences that I am right and QCD, a part of the Model, will share the same mechanism to generate masses.
Yet, there is an open question put forward by a smart referee to my paper. I will put this here as this is an interesting question of classical field theory that is worthwhile to be understood. As you know, I have found a set of exact solutions to the classical field equation
from which I built my mass generation mechanism. These solutions can be written down as
being a Jacobi’s elliptic function and provided
From the dispersion relation above we can conclude that these nonlinear waves indeed represent free massive particles of finite energy. But let us take a look to the definition of the energy for this theory, one has
and if you substitute the above exact solutions into this you will get an infinity. It appears like these solutions have infinite energy! This same effect is seen by ordinary plane waves but can be evaded by taking a finite volume, one normalizes the solutions with respect to this volume and so you are done. Of course, you can take finite volume also in the nonlinear case provided you put for the momenta
being as this Jacobi function has period but you should remember that this function is doubly periodic having also a complex period. Now, if you compute for you will get a dispersion relation multiplied by some factors and one of these is the volume. How could one solve this paradox? You can check by yourselves that these solutions indeed exist and must have finite energy.
My work on QCD is not hindered by this question as I work solving the equation and here there are different problems. But, in any case, mathematics claims for existence of these solutions while physics is saying that there is something not so well defined. An interesting problem to work on.
Sometime it is quite interesting to turn back to well-done books to refresh some ideas. This happened to me with Smilga’s book reading again the chapter on classical solutions of Yang-Mills equations. This chapter is greatly important and the reason is quantum field theory. At our undergraduate courses, when we were firstly exposed to quantum field theory we learned that we have to be able to solve the free equations of motion to start quantization of a theory. Indeed, a free theory is generally easy to quantize while some difficulties could appear with gauge theories. But, anyhow, this easiness arises from the Gaussian form the generating functional takes.
When we turn our attention to Yang-Mills theory we have to cope with the nonlinearities appearing in the equations of motion. At first, being not able to solve them exactly, we can consider solutions identical to their Abelian counterpart that is the electromagnetic field. It is easy to verify that both equations of motion can share identical solutions of free plane waves with a dispersion relation of massless particles. To get them you have to properly select a set of components and you are granted that these classical solutions indeed exist. These solutions are well-known and, when we quantize the theory, we recognize them as describing gluons. But when you quantize for this case you immediately recognize that your computations hold when the coupling appearing in the self-interaction terms is going to zero. You are not able to recover any mass gap and this kind of computations does not appear to help to describe low-energy QCD. But you get a formidable agreement with experiments at higher energies and this is where asymptotic freedom sets in. So, quantization of Yang-Mills theory starting with this kind of solutions says us that these are the right ones for the high-energy limit of the theory when the coupling decreases to zero and all our computations are mathematically consistent.
Now, when we consider the low-energy limit we are in trouble. The reason is that we are not able to solve the equations of motion when the coupling is too large and we are forced to consider them in full with all the nonlinearities in the proper place. But here again Smilga’s book comes to rescue. If you choose judiciously the components of the field and ignore space dependence retaining only time, you will get regular exact solutions that are represented by elliptic Jacobi functions. These are nonlinear standing waves. But looking at them in this way does not help too much. We need also space dependence if we want to extract some physical meaning from these solutions. This is indeed possible looking at a quartic massless scalar field. A quartic massless scalar field with an equation of motion
admits the following exact solution
where and are two integrations constants and is a Jacobi elliptic function. But this holds provided the following dispersion relation does hold
that is the one of a free massive particle! So, a massless classical theory produced massive solutions due to the nonlinear term. That the origin of the mass can be this can be easily understood when you take the limit . The theory becomes massless in this limit and if you do standard perturbation theory the mass term will be hidden in the series and you are not granted you will recover the mass. This result is really beautiful and we see that these solutions are very similar, from a mathematical standpoint, to the ones Smilga considered for classical Yang-Mills equations. So, it is very tempting to try to match these theories. Indeed, this is a truth coded into a (mapping) theorem that I have given in two papers here and here published in archival journals. This mapping holds perturbatively for the coupling going to infinity and this is what we need for studying the opposite limit with respect to gluons. So, I have gone further: These are the right solutions to build a low-energy limit quantum field theory for Yang-Mills equations. This implies that
Mass gap question is settled for Yang-Mills theory.
This is the main conclusion to be drawn: When you build your quantum field theory be careful in the choice of the right classical solutions!
As I pointed out in a recent post, the question of the mass gap for Yang-Mills theory should be considered settled. This implies an understanding of the way mass arises in our world. It is seen that mass is a derived concept and not a fundamental one. I have given an explanation of this here. In a Yang-Mills theory, massive excitations appear due to the presence of a finite nonlinearity. The same effect is seen for a massless quartic scalar field and, indeed, these fields map each other at a classical level. It is interesting to note that a perturbation series with a coupling going to zero can miss this conclusion and we need a dual perturbation with the coupling going to infinity to uncover it. The question we would like to ask here is: What does all this mean for string theory?
As we know, string theory has been claimed not to have any single proposal for an experimental verification. But, of course, without entering into a neverending discussion, there are some important points that could give strong support to the view string theory entails. Indeed, so far there are two essential points that research on string theory produced and that should be confirmed as soon as possible: AdS/CFT correspondence and supersymmetry. Both these theoretical results are strongly supported by the research pursued by our community. For the first point, understanding of QCD spectrum, with or without quarks, through the use of AdS/CFT correspondence is a very active field of research with satisfactory results. I have discussed here this matter several times and I have pointed out the very good work of Stan Brodsky and Guy de Teramond as an example for this kind of research (e.g. see this). Soft-wall model discussed by these authors seems in a very good agreement with the current scenario that is arisen in our understanding of Yang-Mills theory that I emphasized several times in this blog.
About supersymmetry I should say that I am at the forefront since I have presented this paper. The mass gap obtained in Yang-Mills theory arising from nonlinearities has an interesting effect when considered for the quartic scalar field interecting with a gauge field and spinor fields. Taking a coupling for the self-interaction of the scalar field being very large, all the conditions for supersymmetry are fulfilled and all the interacting fields get identical masses and coupling. This implies that, if the mechanism that produces mass in QCD and Standard Model is the same, the Higgs field must be supersymmetric. I call this field Higgs, notwithstanding it has lost some important characteristics of a Higgs field, because is again a scalar field inducing masses to the other fields interacting with it. So, if current experiments should confirm this scenario this would be a big hit for physics ending with a complete understanding of the way mass arises in our world both for the macroscopic and the microscopic world.
So, we can conclude that our research area is producing some relevant conclusions that could address research in more fundamental areas as quantum gravity in a well-defined direction. I think we will get some great news in the near future. As for the present, I am happy to have given an important contribution to this research line.