## And you are calling it a gluon yet…

02/06/2010

One of the more questionable points I have discussed so far is: What are QCD asymptotic states at very low  momenta? This question is not trivial at all. If you will speak with experts in this matter, a common point they will share is that gluons carry color charge and so must form bound states. A claim like this has a strong implication indeed. The implication is that Yang-Mills Hamiltonian must display the same asymptotic states at both ends of the energy range. But the problem is exactly in the self-interaction of the theory that, at very low momenta, becomes increasingly large and gluons, asymptotic states of Yang-Mills theory in the asymptotic freedom regime, are no more good to describe physics. So, what are good states at low energies? I have already answered to this question a lot of times (recently here) and more and more confirmations are around. I would like just to cite a very nice paper I have seen recently on arxiv (see here) by Stanley Brodsky, Guy de Teramond and Alexandre Deur. These authors have nicely exploited AdS/CFT symmetry obatining striking results in the understanding of low-energy QCD. I would like to cite again the work of these authors as their soft-wall model is indeed a strong support to my view. It would be really interesting to get them working out a pure Yang-Mills model obtaining beta function and all that.

What one has at low end of momenta is a new set of states, glue states or glueballs if you prefer, that permits strong interactions. These states have already been seen in most laboratories around the World and belong to the open question of the understanding of the lower part of the hadronic spectrum.

## A really interesting view about QCD and AdS/CFT

07/10/2009

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

$m_n\approx 2n+L$

being $n$ an integer and $L$  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 a glueball?

31/03/2009

Recently I have read a post in Dmitry’s blog by Fabien Buisseret claiming the following conclusion:

“In the present post were summarized various arguments showing that the glueballs and gluelumps currently observed in lattice QCD can be understood in terms of bound states of a few transverse constituent gluons. In this scheme, the lowest-lying glueballs can be identified with two-gluon states, while the lightest negative-C glueballs are compatible with three-gluon states.”

Indeed he considers free gluons interacting each other through a given potential forming bound states. Of course, as all of you may be aware, nobody in the Earth was able to prove that, in the low energy limit, gluons are the right states entering into a quantum Yang-Mills theory. So, this view appears as a well rooted prejudice in the community.

Let me explain what I mean with a classical example. I take the following quartic theory

$\partial^2\phi+\lambda\phi^3=0.$

In the small coupling limit you will get plane waves plus higher order corrections. Assume these plane waves are gluons as we all of us is aware from high-energy QCD. Indeed, these plane waves describe massless excitations. Now I claim that these solutions should hold also when the coupling $\lambda$ becomes increasingly large. But here I have the exact solution

$\phi(x)=\mu\left(\frac{2}{\lambda}\right)^{1\over 4}{\rm sn}(p\cdot x,i)$

being sn a Jacobi snoidal function and $\mu$ an arbitrary constant. But now

$p^2=\mu^2\left(\frac{\lambda}{2}\right)^{1\over 2}$

and I am describing massive excitations that are not resembling at all my plane wave solutions given above. The claim is blatantly wrong already at a classical level with this very simple example.

This proves without any doubt that the view of glueballs as bound states of gluons is plainly wrong as nobody knows the behavior of a Yang-Mills theory in the infrared limit and so, nobody knows what are the right glue excitations for the theory here. As you may have realized, if you would know this you will be just  filed for a Millenium Prize. This means that, unless we learn how to treat the theory at low energies, all this kind of approaches are doomed.

## Gluon condensate

30/03/2009

While I am coping with a revision of a paper of mine asked by a referee, I realized that these solutions of Yang-Mills equations implied by a Smilga’s choice give a proof of existence of a gluon condensate. This in turn means that a lot of phenomenological studies carried out since eighties of the last century are sound as are also their conclusions. E.g. you can check this paper where the authors find a close agreement with my findings about glueball spectrum. The ideas of these authors are founded on the concept of gluon and quark condensates. As they conclusions agree with mine, I have taken some time to think about this. My main conclusion is the following. If you have a gluon condensate, the theory should give $\langle F\cdot F\rangle\ne 0$ being $F_{\mu\nu}^a$ the usual gluon field. So, let us work out this classically. Let us consider a scalar field mapped on the gluon field in such a way to have

$A_\mu^a(t)=\eta_\mu^a \Lambda\left(\frac{2}{3g^2}\right)^\frac{1}{4}{\rm sn}\left[\Lambda\left(\frac{3g^2}{2}\right)^\frac{1}{4}t,i\right]$

being sn a Jacobi snoidal function, and $\eta_\mu^a$ a constant array of elements obtained by a Smilga’s choice. When you work out the product $F\cdot F$ the main contribution will come from the quartic term producing a term $\langle \phi(t)^4 \rangle$. Classically, we substitute the average with $\frac{1}{T}\int_0^T dt$ being the period $T=4K(i)/[\Lambda\left(\frac{3g^2}{2}\right)^\frac{1}{4}]$. The integration is quite straightforward and gives

$\langle \phi(t)^4 \rangle=\frac{\Gamma(1/4)^2}{18K(i)\sqrt{2\pi}}\frac{\Lambda^4}{4\pi\alpha_s}$

I will evaluate this average in order to see if the order of magnitude is the right one with respect to the computations carried out by Kisslinger and Johnson. But the fact that this average is indeed not equal zero is a proof of existence of the gluon condensate directly from Yang-Mills equations.

## Narison, Ochs, Mennessier and the width of the sigma

09/01/2009

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

$\Gamma_\sigma=\frac{|g_{\sigma\pi^+\pi^-}|^2}{16\pi m_\sigma}\sqrt{1-\frac{4m_\pi^2}{m_\sigma^2}}$

being the coupling $|g_{\sigma\pi^+\pi^-}|\approx (4\sim 5)\ GeV$ explaining in this way why this resonance is so broad. Their main conclusion, after computing the width of the reaction $\sigma\rightarrow\gamma\gamma$, 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

$\Gamma_\sigma = \frac{6}{\pi}\frac{G^2_{NJL}}{4\pi\alpha_s} m_\sigma f_\pi^4\sqrt{1-\frac{4m_\pi^2}{m_\sigma^2}}$

being $G_{NJL}\approx 3.76\frac{4\pi\alpha_s}{\sigma}$, $\sigma$ the string tension that we take about 410 MeV, and $f_\pi\approx 93\ MeV$ the pion decay constant. The mass was given by

$m_\sigma\approx 1.198140235\sqrt{\sigma}.$

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

$|g_{\sigma\pi^+\pi^-}|\approx 156.47\sqrt{\frac{\alpha_s}{\sigma}}f^2_\pi$

giving in the end

$|g_{\sigma\pi^+\pi^-}|\approx 3.3\sqrt{\alpha_s}\ GeV$

in very nice agreement with their estimation. We can conclude that their understanding of $\sigma$ 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?

03/01/2009

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 these 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?

## The width of the sigma computed

05/12/2008

One of the most challenging open problems in QCD in the low energy limit is to compute the properties of the $\sigma$ resonance. The very nature of this particle is currently unknown and different views have been proposed (tetraquark or glueball). I have put a paper of mine on arxiv (see here) where I compute the large width of this resonance obtaining agreement with experimental derivation of this quantity. I put here this equation that is

$\Gamma_\sigma = \frac{6}{\pi}\frac{G^2_{NJL}}{4\pi\alpha_s} m_\sigma f_\pi^4\sqrt{1-\frac{4m_\pi^2}{m_\sigma^2}}$

being $G_{NJL}=3.76\frac{4\pi\alpha_s}{\sigma}$ being $\sigma$ the string tension generally taken to be 440 MeV and $f_\pi\approx 93\ MeV$ the pion decay constant. The agreement is obtained with $\alpha_s\approx 2$ giving a consistent result. This is the first time that this rate is obtained from first principles directly from QCD and gives an explanation of the reason why this resonance is so broad. The process considered is $\sigma\to\pi^+\pi^-$ that is dominant. Similarly, the other seen process, $\sigma\to\gamma\gamma$, has been interpreted as due to pion rescattering.

On the basis of these computations, this particle is the lowest glueball state. This is also consistent with a theorem proved in the paper that mixing between glueballs and quarks, in the limit of a very large coupling,  is not seen at the leading order. This implies that the spectrum of pure Yang-Mills theory is seen experimentally almost without interaction with quarks.