The nature of eta’


\eta' is a very peculiar particle. It mixes with \eta that has a lower mass. Recently, in their report on KLOE-2 physics, this group reported here that \eta' has a significant glue component besides quarks. This means that understanding its most important decay \eta'\rightarrow\eta\pi^+\pi^- is not a trivial matter. As my readers may know, I have done a computation in my contribution to proceedings to QCD 10 conference where the decay process is seen to happen through an intermediate step with the \sigma resonance followed by the decay of this into two pions. The agreement we get is so good to give a correct estimation of the decay constant of the \eta. This implies that the \sigma is a true glue state. Of course our computation is rough enough to exclude mixing with other hadronic states that should exist.

Today, on arxiv, an interesting paper appeared authored by Rafel Escribano, Pere Masjuan, Juan José Sanz-Cillero (see here). These authors give an initial overview of the experimental status of the decay we have considered above. Then, using both the technique of Chiral Perturbation Theory (ChPT) and that of Resonant Chiral Perturbation Theory (RChPT), they try to fit experimental data. I have the luck to hear a talk of Juan José in Montpellier last year about this same matter and I was aware of his struggle to reach an agreement between a successful technique, as ChPT is, and experimental data for this particular process. The leading order of the theory is well below the experimental value and so, already in that first talk, Juan José showed the need for higher order corrections. But he proved that this cannot be enough and said at that time that some other states should be accounted for to reach a satisfactory agreement. This paper goes in this direction showing that if one accounts for the presence of the \sigma and a0(980), the latter being dominant, the agreement is reached. These authors were also able to show a consistent relation between ChPT and RChPT that are in some way complimentary.

This paper is relevant as gives a strong support to the idea that I put forward about \eta' decay. But these authors go further implying a higher level of understanding accounting for the presence of other hadronic states in a technical affordable way. I expect further improvement by them and it will be interesting to see how these could be obtained.

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.

Yang-Mills theory in D=2+1


There is a lot of work about the pursuing of a deep understanding of Yang-Mills theory in the low energy limit. The interesting case is in four dimensions as our world happens to have such a property. But we also know that a Yang-Mills theory in D=2+1 is not trivial at all and worthwhile to be studied. In this area there has been a lot pioneering work mostly due to V. Parameswaran Nair and Dimitra Karabali . These authors proved that a Hamiltonian formulation may be truly effective to manage this case. Indeed, they obtained a formula for the string tension that works quite well with respect to lattice computations. We would like to remember that, in D=2+1, coupling constant is such that its dimension is [g^2]=[E] while, in D=3+1, is dimensionless.

Quite recently, some authors showed how, from such a formulation, a functional can be given from which one can obtain the spectrum (see here, here and here). These papers went all published on archival journals. Now, these spectra are quite good with respect to lattice computations, after some reinterpretation. We do not know if this is due to some problems in lattice computations or in the theoretical analysis. I leave this to your personal point of view. My idea is that this quenched lattice computations are missing the true ground state of the theory. This happens to be true both for D=3+1 and D=2+1. I do not know why things stay in this way but in this kind of situations are always theoreticians to lose. On the other side, being a physicist means that one should not have a blind faith in anything.

Finally, one may ask how my work performs with respect all this. Yesterday, I spent a few time to try to figure this out. The results I obtain agree fairly well with those of the theoretical papers. I obtain the zero Lionel Brits gets at 0.96m being m a mass proportional to ‘t Hooft coupling. Brits wrote the third of the three papers I cited above. The string tension I get is in agreement with lattice computations. This zero is the problem on lattice computations and the same problem is seen in D=3+1. This fact is at the root of our presenting difficulty to understand what \sigma resonance is. We know that people working on a quenced lattice computation for the propagator do see this resonance. This difference between this two approaches should be understood and an effort in this direction must be made.

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)/\sigma 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 \sigma 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 \pi\pi and KK decays should be almost the same.

Indeed, they get the following

|g^{os}_{\sigma\pi +\pi -}|\simeq 6 GeV, r_{\sigma\pi K}\equiv \frac{g^{os}_{\sigma K+K-}}{g^{os}_{\sigma\pi +\pi -}} \simeq 0.8

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 \gamma\gamma, gives a really strong support to the conclusion that \sigma 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.

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.

Mass of the sigma resonance


One of the most hotly debated properties of the \sigma resonance is the exact determination of its mass. Difficulties arise from its broadness. Indeed, in \pi\pi scattering data this resonance appears with a very large peak that makes difficult a precise determination of the mass and, indeed,  a large body of data is needed to accomplish this. Initially, it was very difficult to accept the existence of this particle and, for some years, disappeared from particle listings of PDG. Recent papers, using Roy equation, proved without doubt the existence of this resonance and gave what appears the most precise determination of the mass and width so far (see here and here). This approach has been recently criticized (see here and, more recently, here) where is claimed that this approach currently underestimates the mass of the particle.

Due to such a situation, we prefer to consider another similar resonance, f0(980), whose mass is better determined giving

m_{f0(980)}=980\pm 10\ MeV.

Theoretically, we have built a full computation, starting with the spectrum of Yang-Mills theory, for the mass of all these resonances (see here). This paper does not use properly the mapping theorem but gives the right results. We have identified two kind of spectra (higher order spectra can also be obtained) giving

m_1(n)=1.198140235\cdot (2n+1)\sqrt{\sigma}


m_2(n,m)=1.198140235\cdot (2n+2m+2)\sqrt{\sigma}

being, as usual, \sigma the string tension, a parameter to be computed experimentally. So, one has the spectrum of the \sigma resonance and its excited states by simply taking m,n=0 giving



m_{\sigma^*}=2\cdot 1.198140235\sqrt{\sigma}.

So, taking \sqrt{\sigma}=410\ MeV we get easily m_{\sigma^*}=982\ MeV in close agreement with experiments, while m_\sigma=491\ MeV showing that, effectively, one has currently an underestimation of this quantity. With these values we will have from the width of the \sigma resonance the QCD constant \Lambda=285\ MeV (see here).

Finally, a derivation of \Lambda and string tension \sigma from other experimental data would be critical to obtain fixed all the constants of QCD, producing immediately a proper understanding of all physics about \sigma resonance.

QCD constants from sigma resonance


In our recent paper we were able to compute a relevant property of the \sigma resonance. We have obtained its decay width as (see here):

\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 is one of the constants to be determined, and f_\pi\approx 93\ MeV the pion decay constant. From asymptotic freedom we know that


being n_f the number of flavors and \Lambda the scale where infrared physics sets in and is another constant to be computed. We have computed the mass of the \sigma resonance from the gluon propagator obtaining

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

and we have all the theoretical data to compute \sqrt{\sigma} and \Lambda from experimental data. This can be accomplished using two main references (here and here) that give:

\sqrt{s_\sigma} = 441^{+16}_{-8}-i279^{+9}_{-14.5}\ MeV


\sqrt{s_\sigma} = 460^{+18}_{-19}-i255^{+17}_{-18}\ MeV

respectively. These produce the following values for QCD constants

\sqrt{\sigma}=368\ MeV\ \Lambda=255\ MeV


\sqrt{\sigma}=384\ MeV\ \Lambda=266\ MeV.

We have not evaluated the errors being this a back of envelope computation. A striking result is that the ratio of these constants is the same in both cases giving the pure number 1.44 that I am not able to explain.

These results, besides being truly consistent, are really striking making possible a deep understanding of QCD. I would appreciate any reference about these values and on their determination, both theoretical and experimental, from other processes in QCD. These values are critical indeed for strong interactions.

The width of the sigma computed


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.

Wonderful QCD!


On Science this week appeared a milestone paper showing two great achievements by lattice QCD:

  • QCD gives a correct description of low energy phenomenology of strong interactions.
  • Most of the ordinary mass (99%) is due to the motion of quarks inside hadrons.

The precision reached has no precedent. The authors are able to get a control of involved  errors in such a way to reach an agreement of about 1% into the computation of nucleon masses. Frank Wilczek gives here a very readable account of these accomplishments and is worthwhile reading. These results open a new era into this kind of method to extract results to be compared with experiments for QCD and give an important confirmation to our understanding of strong interactions. But I would like to point out Wilczek’s concern: Until we will not have a theoretical way to obtain results from QCD in the low energy limit, we will miss a great piece of understanding of physics. This is a point that I discussed largely with my posts in this blog but it is worthwhile repeating here coming from such an authoritative voice.

An interesting point about these lattice computations can be made by observing that again no \sigma resonance is seen. I would like to remember that in these computations entered just u, d and s quarks as the authors’ aims were computations of bound states of such quarks. Some authoritative theoretical physicists are claiming that this resonance should be a tetraquark, that is a combination of u and d quarks and their antiparticles. What we can say about from our point of view? As I have written here some time ago, lattice computations of the gluon propagator in a pure Yang-Mills theory prove that this can be fitted with a Yukawa form


being m\approx 500 MeV. This is given in Euclidean form. This kind of propagators says to us that the potential should be Yukawa-like, that is


if this is true no tetraquark state can exist for lighter quarks. The reason is that a Yukawa-like potential heavily damps any van der Waals kind of residual potential. But, due to asymptotic freedom, this is no more true for heavier quarks c and b  as in this case the potential is Coulomb-like and, indeed, such kind of states could have been seen at Tevatron.

We expect that the glueball spectrum should display itself in the observed hadronic spectrum. This means that a major effort in lattice QCD computations should be aimed in this direction now that such a deep understanding of known hadronic states has been reached.

Screening masses in SU(3) Yang-Mills theory


Thanks to a useful comment by Rafael Frigori (see here) I become aware of a series of beautiful papers by an Italian group at Universita’ della Calabria. I was mostly struck by a recent paper written by R. Fiore, R. Falcone, M. Gravina and A. Papa (see here) that appeared in Nuclear Physics B (see here). This paper belongs to a long series of works about the behavior of Yang-Mills theory at non-null temperature and its critical behavior. Indeed, using high-temperature expansion and Polyakov loops one arrives at the main conclusion that the ratio between the lowest and the higher state of the theory must be 3/2. This ratio depends on the universality class the theory belongs to and so, on the kind of effective theory one has in the proper temperature limit (below or above T_c). It should be said that, in order to get a proper verification of the above prediction, people use lattice computations. Fiore et al. use a lattice of 16^3 \times 4 points and, as all this kind of computations are done on lattices having such a dimension, one can cast some doubt about the fact that the true ground state of the theory is really hit. Indeed, this happens in all this kind of computations done to get a glueball spectrum that seem at odd with those giving the gluon propagator producing a lower screening mass at about 500 MeV (see my post here). A state at about 500 MeV is seen at accelerator facilities as \sigma resonance or f0(600) but is not predicted by any lattice computation. One of the reasons to reduce lattice volume is that one can reach higher values of \beta granting the reaching of a non-perturbative regime, the one interesting for us.

What can we say about this ratio with our theory? We have put on arxiv a paper that answer this question (see here). These results were also presented at QCD 08 in Montpellier (see here). We assume that the \sigma cannot be seen at such small volumes but its excited state \sigma^* can be obtained. This implies that one can exchange the \sigma^* with the lowest state and 0^+ as the higher one. Then this ratio gives exactly 3/2 as expected. We can conclude on the basis of this analysis that this ratio is the same independently on the temperature but, the one to be properly measured is given in the paper of Craig McNeile (see here) that gives close agreement between lattice and theoretical predictions.

So, we would like to see lattice computations of Yang-Mills spectra at lower lattice spacing and increased volumes granting in this way the proper value of the ground state. This is overwhelming important in view of the fact that no real understanding exists of the existence of the \sigma resonance with lattice computations. This will implies, as discussed above, a deeper understanding of the spectrum of the theory also at higher temperatures.


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