## Happy Xmas!

23/12/2008

It will be difficult for me to post in the next few days and so here are my warm greetings for these Season’s Holidays to everybody.

## We cannot see the light

23/12/2008

An interesting paper appeared today in arxiv by Alkofer,  Huber and Schwenzer (see here). Reinhard Alkofer and Lorenz von Smekal are the proponents of an infrared solution of Yang-Mills theory in D=4 having the following properties

• Gluon propagator goes to zero at lower momenta
• Ghost propagator goes to infinity at lower momenta faster than the free propagator
• Running coupling reaches a fixed point at lower momenta

and this scenario disagrees with lattice evidence in D=4 but agrees with lattice in D=2 when the theory is trivial having no dynamics. After some years that other researchers were claiming that a different solution can be obtained by the same equations, that is Dyson-Schwinger equations, that indeed agrees with lattice computations, Alkofer’s group accepted this fact but with a lot of skepticism pointing out that this solution has several difficulties, last but not least it breaks BRST symmetry. The solution proposed by Alkofer and von Smekal by its side gives no mass gap whatsoever and no low energy spectrum to be compared neither with lattice nor with experiments to understand the current light unflavored meson spectrum. So, whoever is right we are in a damned situation that no meaningful computations can be carried out to get some real physical understanding. The new paper is again on this line with the authors proposing a perturbation approach to evaluate the vertexes of the theory in the infrared and obtaining again comforting agreement with their scenario.

I will avoid to enter into this neverending controversy about Dyson-Schwinger equations but rather I would ask a more fundamental question: Is it worthwhile an approach that only grants at best saving a phylosophical understanding of confinement without any real understanding of QCD? My view is that one should start from lattice data and try to understand the real mathematical form of the gluon propagator. Why does it resemble the Yukawa form so well? A Yukawa form grants a mass gap and this is elementary quantum field theory. This I would like to see explained. When a method is not satisfactory something must be changed. It is evident that solving Dyson-Schwinger equations requires some new mathematical approach as old views are just confusing this kind of research.

## New evidence for dark energy

18/12/2008

Chandra X-ray observatory has given new evidence of the existence of dark energy. This striking result has been obtained with the observation of Abell 85. This is a galaxy cluster located about 740 million light years from Earth.

I have found another beautiful article on NewYork Times about (see here). Dark energy is currently described by a non-null cosmological constant in the Einstein equations but no explanation is known whatsoever about its origin. Currently, a famous explanation uses the anthropic principle but was criticized by Steven Weinberg (see here). This argument seems to support very strongly the landscape view of string theory but whatever relies on anthropic arguments cannot be said to be really satisfactory for our understanding. So, physicists are looking for something better but nothing is in view. Rather, quantum field theory gives a too large account for this constant. As this is a major breakthrough in our understanding of universe, it is a serious reason to stay tuned waiting for more information.

## Nature of f0(980)

10/12/2008

As pointed out in my recent post (see here), f0(980) can be a glueball and an excited state of $\sigma$ resonance. I have found some theoretical support for this. But it would be enough to have some support from experimental data just for this resonance being a glueball. Such an evidence exists. Firstly I would like to insert here a conclusion from the beautiful paper by Caprini, Colangelo and Leutwyler (see here and here)

“The physics of the σ is governed by
the dynamics of the Goldstone bosons: The properties of
the interaction among two pions are relevant… The properties of the resonance f0(980) are also governed by Goldstone
boson dynamics – two kaons in that case.”

This is just the scenario I depicted in my papers. But I was also able to find a very smart paper by Baru, Haidenbauer, Hanhart, Kalashnikova and Kydryavtsev (see here and here) where a proof is given that this resonance has not a quark structure. This is accomplished through an approach devised by Steven Weinberg that applies to unstable particles.

This is a strong support to our scenario that appears consistently built. In turn, this implies that a clear understanding of the very nature of light unflavored scalar mesons is at hand.

## 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.

## QCD constants from sigma resonance

09/12/2008

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

$\alpha_s(Q^2)=\frac{12\pi}{(33-2n_f)\ln\left(\frac{Q^2}{\Lambda^2}\right)}$

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:

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

and

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

respectively. These produce the following values for QCD constants

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

and

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

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

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.