## 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 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?

### 4 Responses to What is a glueball?

1. dorigo says:

Hey Marco, enlightening post! I hope you go deeper with it. Please explain more about your interpretation of the low-energy limit! I am quite interested in the topic. Add references if you have any that can be digested by experimentalists rusty on QFT!

Cheers,
T.

2. […] link and an invitation for you to join me and ask Marco Frasca to further his already enlightening discussion of glueballs, as I already did in the comments thread of his […]

3. mfrasca says:

Hi Tommaso,

Thank you a lot for your post on your blog http://dorigo.wordpress.com/2009/01/04/what-is-a-glueball/ and the comment about this post. The question runs as follows and is strongly rooted in the work I have done in the latest three years. It is also something that seems quite simple in view of the work condensed matter theorists have been doing since the seminal papers by Landau.

When one considers the Hamiltonian of a solid one general meets difficulties to describe its properties unless the proper states to build perturbation theory are given. These states are known as quasiparticles or dressed particles. After the proper choice is done you can work out perturbation theory and so, a lot of results come down for metals and other materials after this understanding come out. Today, this approach is quite generally accepted in condensed matter physics and has had as a by-product the understanding of superconductivity through BCS theory and Cooper pairs as fundamental states of the Hamiltonian.

For a Yang-Mills theory in the infrared limit you are in a similar situation. At higher energies asymptotic freedom sets in. This mean that you are able to apply small perturbation theory and the states to work with are those of a free theory: Gluons with all their well-known properties as also seen in high energy experiments. But when the coupling in the theory becomes too large the non-linear terms cannot be neglected anymore and you are exactly in the situation of Landau trying to understand why Fermi theory of free electrons appears so good at describing the properties of metals, while these particles should be strongly bounded inside the material.

Indeed, a class of exact classical solutions of Yang-Mills theory exist having a very nice property. They can be represented in quantum field theory with a Gaussian functional and the spectrum of an harmonic oscillator. Ooops! Quasiparticles? Like condensed matter theory?!! This is all the matter since now. The papers you should look at are the following

http://arxiv.org/abs/0807.2179

This gives the classical solutions but the part of QFT is wrong. I have to correct it and I will do it in the due time.

http://arxiv.org/abs/0812.0934

This paper gives a fully QCD computation of the $\sigma$ resonance width and an understanding of observability of pure Yang-Mills spectrum in high-energy experiments.

http://arxiv.org/abs/0709.2042

This has been published (see http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TVN-4TPHRKP-3&_user=10&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=fd5ae79f41aeba1b0ebab2efa65bf73a) and gives a full mathematical account of all this matter.

A full report of all this view is given in

http://arxiv.org/abs/0807.4299

This is the result of my talk at QCD 08 in Montpellier in France. It will appear shortly in the proceedings. Very nice conference. You can look at some of my posts about as https://marcofrasca.wordpress.com/2008/07/14/qcd-08-the-report/

Ciao,

Marco

4. dorigo says:

Thank you for the detailed reply. I fear most of this is over my head, but I will try to absorb something from the references you gave.

Cheers,
T.

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