Sigma resonance again


José Pelaéz and Guillermo Rìos published today a paper on arxiv (see here). The argument is an understanding of the nature of \sigma and \kappa resonances. The technique they use is Chiral Perturbation Theory (ChPT) but the idea is to see the behavior of the amplitudes at increasing number of colors. They get again a confirmation that the very nature of \sigma is not a typical \bar qq state. Rather, a subdominant \bar qq component is seen at larger energies with larger values of the number of colors. This conclusion  agrees with our theorem proved here.

The current situation forces the authors to prudence. They do not draw any conclusion about the real nature of \sigma and \kappa but their results still appear impressive. These authors have a long file of very good works about the quest for an understanding of the lower part of QCD spectrum and they have given the mass and the width of \sigma  with really increased precision. They belong to a group headed by Paco Yndurain. You can find a tribute to Paco by Stephan Narison here.

From my view you can see this as another confirmation to the idea that \sigma is a glueball and the lowest state of a pure Yang-Mills theory. This evidence is becoming overwhelming but other interpretations are not ruled out yet. The fact that \kappa or else f0(980) are glueballs would give further strong support to this as I expect a glueball state at this value of energy.

What is a glueball?


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


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.



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

\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.

Spectrum of two color QCD


In a preceding post (see here) I have shown how SU(2) QCD can be reduced to a theory easily amenable to perturbation treatment. This case is quite easy as, due to the gauge group, algebra is not too much involved. At the leading order one gets the following set of equations

i\gamma^0\partial_\tau q_0+\frac{1}{2\sqrt{2}}\phi_0\Sigma q_0=0


that are very easy to solve and so, we can get an understanding of the mass spectrum both for \phi and the quark field. At a first glance we easily recognize that, at this order, is clear that QCD has a chiral symmetry and this symmetry arises naturally from the strength of the coupling g.

At this stage we want just to write down the mass spectra. For the gluon field we have


exactly as for SU(3). The only change here should be the value of the string tension \sigma.Indeed, the folowing relation should hold \sigma_{SU(2)}=\sqrt{2/3}\sigma_{SU(3)} and so, if \sigma_{SU(3)}=(440MeV)^2 one will have \sigma_{SU(2)}\approx (398MeV)^2. This is in perfect agreement with lattice evidence (see here). This lattice evidence is quite old and should be pursued further. For a quark we have the spectrum


and as for the glueball spectrum we have n=0,1,2,\ldots and K(i) an elliptic integral. From this we say that the lowest state in the spectrum will have zero mass, the pion, and this is just a manifestation of the above approximate chiral symmetry.

The next step will be to go to higher orders and correct these results. But we see that, already at leading order, we conclude that the glueball spectrum must manifest itself at an experimental level exactly as happens to hadronic spectrum. Any other correction to it is just higher order.

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