Classical solutions of Yang-Mills equations

So far, I have posted several posts in this blog about the question of classical solutions to Yang-Mills equations. This has produced some fuzz, mostly arisen from my published papers, as to such solutions may not be correct. Thanks to a wise intervention of Terry Tao, I was able to give a complete understanding of my solutions and a theorem was fully proved in a recent paper of mine to appear in Modern Physics Letters A (see here), agreed with Terry in a private communication. So, I think it is time to give a description of this result here as it appears really interesting showing how, already at a classical level, this theory can display massive solutions and a mass gap is already seen in this case. Then, it takes a really small step to get the corresponding proof in quantum field theory.

To understand how these solutions are obtained, let us consider the following equation for a scalar field

This is a massless self-interacting field. We can select a class of solutions by looking at the case of a rest reference frame. So, we put any dependence on spatial variables to zero and solve the equation

whose solutions are known and given by

being and two integration constants and sn a Jacobi elliptical function. Then, boosting this solution will produce an exact solution of the equation we started from given by

provided the following dispersion relation holds

and we see that, although we started with a massless field, self-interaction provided us massive solutions!

Now, the next question one should ask is if such a mechanism may be at work for classical Yang-Mills equations. These can be written down as

being chosen depending on the gauge choice, the coupling and the structure constants of the gauge group taken to be SU(N). The theorem I proved in my paper above states that the solution given for the scalar field theory is an exact solution of Yang-Mills equations, provided it will not depend on spatial coordinates, for a given choice of Yang-Mills components (Smilga’s choice) and , otherwise the following identity holds

Here is a set of constants arising with the Smilga’s choice. This theorem has the following implications: Firstly, when the coupling become increasingly large, a massless scalar field theory and Yang-Mills theory can be mapped each other. Secondly, already at the classical level, for a coupling large enough, a Yang-Mills theory gets massive solutions. We can see here that a mass gap arises already at a classical level for these theories. Finally, we emphasize that the above mapping appears to hold only in a strong coupling regime while, on the other side, these theories manifest different behaviors. Indeed, we know that Yang-Mills theory has asymptotic freedom while the scalar theory has not. The mapping theorem just mirrors this situation.

We note that these solutions are wave-like ones and describe free massive excitations. This means that these classical theories have to be considered trivial in some sense as these solutions seem to behave in the same way as the plane waves of a free theory.

One can build a quantum field theory on these classical solutions obtaining a theory manifesting a mass gap in some limit. This is has been done in several papers of mine and I will not repeat these arguments here.

A really interesting view about QCD and AdS/CFT

Stan Brodsky is a renowned physicist that has produced a lot of very good works. As I work on QCD, I try to be up-to-date as much as possible and I spend some time to read the most recent literature about. AdS/CFT applied to QCD is a very hot topic these times and I run into a beautiful paper by Stan and Guy de Téramond that was recently published in Physical Review Letters (a preprint is here). Their work is inspired by AdS/CFT in that they are able to map on a five dimensional Anti-de Sitter space a light-front Hamiltonian for QCD, producing a Schrödinger-like equation with a proper potential to get the spectrum of the theory. This equation is depending by a single proper variable and is exactly solvable. Two classes of models can be identified in this way that are those well-known in literature:

• Hard-wall model with a potential described by an infinite potential wall till a given cut-off that fixes the mass scale.
• Soft-wall model with a harmonic potential producing Regge trajectories.

So, these authors are able to give a clever formulation of two known models of QCD obtained from AdS/CFT conjecture and they manage them obtaining the corresponding spectra of mesons and baryons. I would like to emphasize that the hard-wall model was formulated by Joseph Polchinski and Matthew Strassler and was instrumental to show how successful AdS/CFT could be in describing QCD spectrum. This paper appeared in Physical Review Letters and can be found here. Now, leaving aside Regge trajectories, what Stan and Guy show is that the mass spectrum for glueballs in the hard wall model goes like

being an integer and   the angular momentum. This result is interesting by its own. It appears to be in agreement both with my recent preprint and my preceding work and with most of the papers appeared about Yang-Mills theory in 2+1 dimensions. Indeed, they get this spectrum being the zeros of Bessel functions and the cut-off making the scale. Very simple and very nice.

I should say that today common wisdom prefers to consider Regge trajectories being hadron spectroscopy in agreement with them but, as glueballs are not yet identified unequivocally, I am not quite sure that the situation between a soft wall and hard wall models is so fairly well defined. Of course, this is a situation where experiments can decide and surely it is just a matter of a few time.

What is mass?

I should confess that one of the reasons why I have chosen to be a physicist is that physics, like no other sciences, is able to give answers to fundamental open questions that until a few years ago were only discussed by philosophers. Most of these questions are ancient as our species and the possibility that we have means to get truth is too strong to lose our time with other activities. So, I managed to learn such means and today I am here writing on this blog trying to explain you what these truths are. Sometime, I am at the forefront of research and so, what can be believed a truth may lose this quality as we deepen our understanding. Indeed, dynamics of science adds one more element of charm to all this matter.

One of such old questions is: “What are we made of?”. This question has been an open question till the dawn of the 20th century with the fundamental experiments carried out by Ernst Rutherford. Till then we have learned so much about matter that this question changed form becoming: “What is mass?”. This question has become compelling with the birth of the Standard Model due to Sheldon Lee Glashow, Steven Weinberg and Abdus Salam. Indeed, in order to maintain symmetry we must ask all particles to be massless and some mechanism must exist giving mass to them. In the sixties and seventies of last century we moved toward a real understanding of this concept. The idea is to rely on the Higgs mechanism and a scalar particle must exist to grant masses to the other particles in the model. As you may know this particle has not yet been seen and it is the only missing element of an otherwise very successful model. We are confident for several reasons that the Higgs mechanism could turn out the right answer to the question on mass but we are no more so confident that should have the simple aspect given originally in the Standard Model. Indeed, this appears as an open door on a Pandora’s vase of new exciting physics.

But whatever will be the mechanism at work for the masses of leptons and quarks, the answer to the main question is not there. For one reason, both electrons and quarks that form protons and neutrons are really light and do not count too much on the determination of our mass. Most of the mass is in the nuclei and we have to understand where such mass comes from. This arises from bound states of quarks glued together in some way as should yield QCD at low energies. This gives you an idea of why is so important to understand QCD at very low energy. In this way we would be able to answer a fundamental question philosophers discussed for so long time.

So, for our everyday life, it is not so relevant to comprehend the real mechanism that gives mass to elementary particles . What we need is to prove the existence of a mass gap in Yang-Mills theory and so the way bound states form in QCD. As you may know, this is not an easy task and involves a lot of talented people around the World that, with a lot of inventive, is trying to do such computations. So far, only computers succeeded in giving an answer and this is so good that we have the most important observed parameters precise to one percent. The hope is to have a technique to work out such computations analytically, as happens for weak coupled physics. I am deeply involved in such enterprise and I think that what will come out will have a large impact on our knowledge. I can only say: Stay tuned!

Answer to Terry Tao’s criticism will go published

My paper containing the answer to Terry Tao’s criticism will be published in Modern Physics Letters A. You can get a copy of this preprint from arxiv here.

Thank you very much, folks!

A confirmation again

One of my main activities in the morning is reading the daily coming from arxiv. Sometime it happens to find significant papers to be put in a post like this. This morning I have found a beautiful paper by a cooperation of people from Germany, Russia and Australia working on lattice QCD (see here). This paper has been written by Igor Bogolubsky, Ernst-Michael Ilgenfritz, André Sternbeck and Michael Mueller-Preussker. I put here the following picture representing one of the main conclusions

This picture gives the gluon propagator with a number of points (96)^4 and shows clearly that it reaches a finite value at smaller momenta implying a massive gluon. Indeed, the authors of the paper extended the lattice computations moving from (80)^4 to (96)^4 points and add some other improvement in the computation itself. The value of beta is quite high being 5.7. The agreement with previous computations of Cucchieri and Mendes is excellent (see here). These latter authors worked with a number of points of (128)^4 while beta was taken to be 2.2.

The other two important conclusions they reach is that the ghost propagator goes like that of a free particle and the running coupling goes to zero at lower momenta. For the running coupling we emphasize that there is no common agreement about its definition in the infrared and the authors properly point out this. But a running coupling that goes to zero does not mean at all that there is no confinement. Quite the contrary as proved by Kazuhiko Nishijima (see here): It gives a proof of confinement.

So, we obtain again a clear proof of the scenario we have already obtained from a theoretical standpoint (see here and here) and we have discussed at length in this blog. I think that evidence of existence of the mass gap both on lattice and from theory are becoming overwhelming. We are just wating the dust to settle down and textbooks reporting these findings.

Update: After an email exchage with Andre Sternbeck he gave further clarifications about his group work correcting something not correct in the post. I post here his corrigenda:

“Our study was for the gauge group SU(3) and not for SU(2). That is
the reason why the Beta-Value is larger than that used for SU(2) by
Cucchieri et al. and by myself et al. in 2007. The lattice spacings are
roughly of the same order, but the numerical effort spent for a 96^4
lattice in SU(3) is much bigger than what had been necessary in SU(2).”

I take this chance to thank him a lot for his comments.

Physics Letters B at last!

Yesterday, the Editor of Physics Letters B communicated to me that my paper (see here) was accepted for publication. This was great for at least one reason: Physics Letters B is one of the most important journals in our area of activity and the paper that was accepted gives a sensible mathematical proof of the form of the gluon and ghost propagators in the infrared and relative mass spectrum that implies the very existence of a mass gap for the Yang-Mills theory. The key theorem is what I called the “mapping theorem” where a SU(N) Yang-Mills theory is mapped on a theory whose solution in the low energy limit I presented here and Physical Review D (see here). This analysis is in perfect agreement with the scenario emerging from lattice computations but we have the nice situation of explicit formulas for the gluon propagator and the spectrum permitting explicit computations wherever needed.

Also in this case the peer-review system worked at best. Both Editor and referees permitted to correct what appeared a serious difficulty in the proof of the mapping theorem. Once I solved this the paper was straightforwardly approved for publication. I take this chance to thank them all publicly.

I give here the formulas for the gluon propagator in the Landau gauge (the ghost propagator is that of a free particle) and the spectrum:

being

and

being an integration constant of Yang-Mills theory, arising from conformal invariance, to be fixed experimentally and an elliptic integral that is about 1.3111028777. From the mass spectrum is clearly seen the mass gap when is taken. Nature decides what is but an higher order theory should be able to derive it. We see that the spectrum of the theory is made of massive excitations that should not be called gluons. I think that glueballs is more appropriate.

So, this is a key moment for Yang-Mills theory. It implies a great understanding of the behavior of the theory in a regime not accessible before. Knowing the gluon propagator means that a Nambu-Jona-Lasinio model describes correctly the phenomenology at low energies. This we proved quite recently (see this post).

Lower part of QCD spectrum

Today in arxiv appeared the contribution of Heinrich Leutwyler at QCD 08 (see here). According to Britannica online Leutwyler is one of the inventors of the concept of color (see here) together with Harald Fritzsch and Murray Gell-Mann. I have the luck to listen to his talk at Montpellier for QCD 08 and I was aware of his work due to a very beatiful paper he published recently with Irinel Caprini and Gilberto Colangelo (see here and here) where they get the mass and the width of the resonance and her sibling f0(980). This paper is absolutely relevant for an understanding of the nature of this resonance that is is the vault key to connect theoretical analysis and experimental data for a Yang-Mills theory. A different approach by Yndurain, Garcia-Martin and Pelaez (see here and here) gave slightly different values for the .

This Leutwyler’s paper gives a deep insight of our current understanding of the lower part of the QCD spectrum. What really striked me is the deep relation between and f0(980) as these resonances seem to emerge from such analysis always together. This is in deep agreement both with my analysis (see here) and that of Narison, Mennessier and Ochs (see here). We note as the QCD mass gap is appreciably lower (pion mass) than that of a pure Yang-Mills when the nature of the is elucitated being a glueball. We discussed this in a post. This means that a strict connection can exist between the phenomenological analysis of Leutwyler and our current understanding of a Yang-Mills theory in the infrared. What we miss now is a substantial support from lattice QCD (see McNeile’s paper).

Corrigenda: I have to correct the fact that Leutwyler is a proponent of color degree of freedom. This is not true. He is one of the proponents of QCD using color as charge. This is the same information correctly conveyed by Britannica online.

Sergei Matinyan and George Savvidy

Some months ago a paper of mine was rejected by an editor of JHEP in a few minutes as this person pretended that my choice of the initial solution of Yang-Mills theory to build a strong coupled QFT was “ad hoc”. The point is that I assumed that the work of George Savvidy and Sergei Matinyan was universally known. These authors proved without any doubt that Yang-Mills mechanics is generally chaotic and so, if I would like to use a gradient expansion to build a QFT I am in a serious difficulty as a QFT built on chaotic solutions cannot exist.

Matinyan summed up most of these results here and was kind enough to cite a paper of mine. So, we all known that for small perturbations one can use free particle solutions but what one can do for the strong coupling case, what are the solutions to start from?

In this post I showed that a non chaotic classical solution exists that has the property to manifest a massive dispersion relation. This classical solution is obtained when one takes all the components of the Yang-Mills to be equal (see here). This is a kind of “replica trick”, that is, we replicate a massless scalar field a number of times enough to solve exactly Yang-Mills equations of motion. So, if we want the quantum theory to produce a mass gap, this is the only choice we have. A QFT can be straightforwardly built and we can manage strong interactions in the strong coupling limit.

So, aside from rejection records, there are serious reasons for satisfaction as a meaningful theory exists that is general enough to treat a quantum field in the strong coupling limit due also to the relevant contribution of Matinyan and Savvidy. The starting point is anyhow a gradient expansion. We will have more to say about in future posts.