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.

A striking confirmation

On arxiv today it is appeared a paper by Stephan Narison,  Gerard Mennessier and Robert Kaminski (see here). Stephan Narison is the organizer of QCD Conferences series and I attended one of this, QCD 08, last year. Narison is located in Montpellier (France) and, together with other researchers, is carrying out research aimed to an understanding of low-energy phenomenology of QCD. So, there is a strong overlapping between their work and mine. Their tools are QCD spectral sum rules and low energy theorems and the results they obtain are quite striking. Narison has written a relevant handbook of QCD (see here) that is a worthwhile tool for people aimed to work with this theory.

The paper gives further support to the idea that the resonance f0(600)/ is indeed a glueball. Currently, researchers have explored another possibility, that this particle is a four quark state. Narison, Mennessier and Kaminski consider that, if this would be true, being this a state with u and d quarks, coupling with K mesons should be suppressed. This would imply that, in a computation for the rates of decays, the contribution coming in the case of K mesons in the final state should be really small. But, for a glueball state, these couplings for and decays should be almost the same.

Indeed, they get the following

that is quite striking indeed. They do the same for f0(980) and, even if they get a similar result, they draw no conclusion about the nature of this resonance.

This, together with the small decay rate in , gives a really strong support to the conclusion that is indeed a glueball. At this stage, we would like to see an improved support from lattice computations. Surely, it is time to revise some theoretical computations of the gluon propagator.

Update: I have received the following correction to above deleted sentence by Stephan Narison. This is the right take:

One should take into account that the sigma to KK is suppressed due to phase space BUT the coupling to KK is very strong. The non-observation of sigma to KK has been the (main) motivation that it can be pi-pi or 4-quark states and nobody has payed attention to this (unobserved) decay.

AdS/CFT and QCD

People really interested about this matter should read the preprint by Elias Kiritsis (see here). This paper gives a full account about this matter and is a recollection of conferences’ contributions yielded by the author.

My point of view about this question, as the readers of the blog may know, is that a general technique to strong coupling problems should be preferred to more aimed approaches. This by no means diminishes the value of these works. Another point I have discussed about the spectrum of AdS/QCD is what happens if one takes the lower state at about 1.19, does one recover the ground state seen in lattice QCD for the glueball spectrum as the next state?

The value of this approach relies on a serious possibility to verify, with a low energy theory, a higher level concept connecting gravity and gauge theories. Both sides have something to be earned.

Narison, Ochs, Mennessier and the width of the sigma

In order to understand what is going on in the lower part of the meson spectrum of QCD that is currently seen in experiments one would like to have an explicit formula for the width of the sigma. The reason is that we would like to have an idea of its broadness. Being this the infrared limit the only known way to get this would be lattice computations but in this case there is no help. Lattice computations see no sigma resonance anywhere. Narison, Ochs and Mennessier were able to obtain an understanding of this quantity by QCD spectral sum rules here and here. They get the following phenomenological equation

being the coupling explaining in this way why this resonance is so broad. Their main conclusion, after computing the width of the reaction , is that this resonance is a glueball.

In our latest paper (see here) we computed the width of the sigma directly from QCD. We obtained the following equation

being , the string tension that we take about 410 MeV, and the pion decay constant. The mass was given by

This permits us to give the coupling in the Narison, Ochs and Mennessier formula as

giving in the end

in very nice agreement with their estimation. We can conclude that their understanding of is quite precise. An interesting conclusion to be drawn here is about how good turn out to be these techniques based on spectral sum rules. The authors call these methods with a single acronym QSSR. They represent surely a valid approach for the understanding of the lower part of QCD spectrum. Indeed, QCD calculations prove that this resonance is a glueball.

What is a glueball?

This question, that seems rather innocuous, is indeed exposed to a lot of prejudices and you will rarely find some expert in the field that will not claim that is just a bound state of gluons that originates from the well known fact that gluons carry color charge. This situation is to be compared to the case of photons that, carrying no electric charge, cannot interact each other (indeed a small effect exists and is called Delbruck scattering and can be obtained from a fully formulation of quantum electrodynamics). Of course one should expect that such states will carry no color charge due to confinement.

Here we see that our current understanding of quantum field theory is just cheating us. We are able to manage quantum field theory for any interaction just with small perturbation theory and almost all our knowledge about comes from such computations. It is not difficult to see that this is a very limited view of the full landscape and we could be easily making mistakes when we try to extend such a small view to the full reality. The question on glueballs is indeed all founded in the infrared limit when small perturbation theory does not apply anymore. In this limit we can rely just on lattice computations and this is already a big limitation notwithstanding the present computational resources.

The right question to be asked here is: Are gluons still the right excitations of the Yang-Mills field in the low energy limit? So far nobody asked this and so nobody has an answer at hand ( I have but this is not the place to discuss it now). So, we are free to call such excitations as glueballs without nobody complaining about.  So, are this bound states of gluons? The answer is no. We are talking about different particles belonging to the spectrum of the same Hamiltonian in a different limit. We can see gluons coming back to reality in the high energy limit due to asymptotic freedom.

Curiously enough, condensed matter theorists seem to be smarter of people like me that worked just on particle physics with the only tool of small perturbation theory. Also condensed matter theorists use such a tool but they, some time ago, asked themselves the right question: What are the right excitations in the given limit? Once you have answered to this question you can safely do ordinary perturbation theory and be happy.

The most important lesson to be learned from all this is that one should not content herself with a theory when it has strong computational limitations. Rather, one should recognize that here there is a serious problem in need of a significant effort to be solved. Of course quantum gravity may be more rewarding but, what if one has a tool to solve any differential equation in physics in a strong coupling regime? Should you call this a scientific revolution?

Nature of f0(980)

As pointed out in my recent post (see here), f0(980) can be a glueball and an excited state of resonance. I have found some theoretical support for this. But it would be enough to have some support from experimental data just for this resonance being a glueball. Such an evidence exists. Firstly I would like to insert here a conclusion from the beautiful paper by Caprini, Colangelo and Leutwyler (see here and here)

“The physics of the σ is governed by
the dynamics of the Goldstone bosons: The properties of
the interaction among two pions are relevant… The properties of the resonance f0(980) are also governed by Goldstone
boson dynamics – two kaons in that case.”

This is just the scenario I depicted in my papers. But I was also able to find a very smart paper by Baru, Haidenbauer, Hanhart, Kalashnikova and Kydryavtsev (see here and here) where a proof is given that this resonance has not a quark structure. This is accomplished through an approach devised by Steven Weinberg that applies to unstable particles.

This is a strong support to our scenario that appears consistently built. In turn, this implies that a clear understanding of the very nature of light unflavored scalar mesons is at hand.

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).

Carriers of the strong force

In my preceding post I discuss the question of the proton spin, experimental measurements and the relevant conclusion that these measurements are consistent with the fact that there is no glue contribution  to spin. Rather, we have a partial contribution by quarks and the remanent should be due to orbital angular momentum. I concluded from this that the infrared carriers of the strong force are not properly gluons but rather spin 0 excitations. In this post I would like to expand on this matter to make clear that all this is plainly obtained from QCD and so, fully consistent with the theory. Anyhow, it should be remembered that the quark sea for nucleons plays a relevant role in this case making quarks dressed.

The question is rather simple. At higher energy QCD tends to become a free theory, that is the coupling becomes increasingly small and the gluon propagator one uses at the tree level is that of a free particle. This in turn means that the non-linear contributions from Yang-Mills theory are small and small perturbation theory applies. In this limit we can identify as the excitations of Yang-Mills theory with ordinary gluons carrying spin one.

In the infrared limit, the case of low energies, the behavior of Yang-Mills theory changes radically. The reason is that in this case the non-linear terms in the equations become so strong that ordinary gluons are no more the fundamental excitations of the theory. In this case one has glueballs and the lower end of the spectrum of the glueballs carries spin zero. This is the reason why COMPASS Collaboration see no spin contributions from glue.

One should see such things in the same way quasi-particles are seen in condensed matter. The free particles that make the theory are not the one of the free theory but will depend upon the way interactions act on them. So, in the ultraviolet limit we can safely call them gluons and in the infrared limit things are quite different. In this light, it would be interesting to try to see similar measurements for charmonium and see in this case what is the contribution of glue. It may happen that this physical situation is radically different from that of the nucleon.