## f0(500) and f0(980) are not tetraquarks

27/06/2014

Last week I have been in Giovinazzo, a really beautiful town near Bari in Italy. I participated at the QCD@Work conference. This conference series is now at the 7th edition and, for me, it was my second attendance. The most striking news I heard was put forward in the first day and represents a striking result indeed. The talk was given by Maurizio Martinelli on behalf of LHCb Collaboration. You can find the result on page 19 and on an arxiv paper . The question of the nature of f0(500) is a vexata quaestio since the first possible observation of this resonance. It entered in the Particle Data Group catalog as f0(600) but was eliminated in the following years. Today its existence is no more questioned and this particle is widely accepted. Also its properties as the mass and the width are known with reasonable precision starting from a fundamental work by Irinel Caprini, Gilberto Colangelo and Heinrich Leutwyler (see here). The longstanding question around this particle and its parent f0(980) was about their nature. It is generally difficult to fix the structure of a resonance in QCD and there is no exception here.

The problem arose from famous papers by Jaffe on 1977 (this one and this one) that using a quark-bag model introduced a low-energy nonet of states made of four quarks each. These papers set the stage for what has been the current understanding of the f0(500) and f0(980) resonances. The nonet is completely filled with all the QCD resonances below 1 GeV and so, it seems to fit the bill excellently.

Someone challenged this kind of paradigm and claimed that f0(500) could not be a tetraquark state (e.g. see here and here but also papers by Wolfgang Ochs and Peter Minkowski disagree with the tetraquark model for these resonances). The answer come out straightforwardly from LHCb collaboration: Both f0(500) and f0(980) are not tetraquark and the original view by Jaffe is no more supported. Indeed, people that know the Nambu-Jona-Lasinio model should know quite well where the f0(500) (or $\sigma$ ) comes from and I would also suggest that this model can also accommodate higher states like f0(980).

I should say that this is a further striking result coming from LHCb Collaboration. Hopefully, this should give important hints to a better understanding of low-energy QCD.

$\overline{B}^0\rightarrow J/ψπ^+π^-$ decays arXiv arXiv: 1404.5673v2
Irinel Caprini, Gilberto Colangelo, & Heinrich Leutwyler (2005). Mass and width of the lowest resonance in QCD Phys.Rev.Lett.96:132001,2006 arXiv: hep-ph/0512364v2
Jaffe, R. (1977). Multiquark hadrons. I. Phenomenology of Q^{2}Q[over ¯]^{2} mesons Physical Review D, 15 (1), 267-280 DOI: 10.1103/PhysRevD.15.267
Jaffe, R. (1977). Multiquark hadrons. II. Methods Physical Review D, 15 (1), 281-289 DOI: 10.1103/PhysRevD.15.281
G. Mennessier, S. Narison, & X. -G. Wang (2010). The sigma and f_0(980) from K_e4+pi-pi, gamma-gamma scatterings, J/psi,
phi to gamma sigma_B and D_s to l nu sigma_B Nucl.Phys.Proc.Suppl.207-208:177-180,2010 arXiv: 1009.3590v1

Marco Frasca (2010). Glueball spectrum and hadronic processes in low-energy QCD Nucl.Phys.Proc.Suppl.207-208:196-199,2010 arXiv: 1007.4479v2

## The nature of eta’

30/11/2010

$\eta'$ 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

29/05/2009

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

25/04/2009

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

17/04/2009

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

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.

## Mass of the sigma resonance

09/12/2008

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

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

and

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

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

$m_\sigma=1.198140235\sqrt{\sigma}$

and

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

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.