Happy Xmas!


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


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


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)


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


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.

A shocking step


Lorenz von Smekal has been one of the proponents of the functional approach to the understanding of infrared Yang-Mills theory. He is currently working at University of Adelaide where a lot of important work on lattice computations is performed. Today on arxiv appeared a paper by him (see here). I would like to report here his words in the introduction

“Without infrared enhancement of the ghosts in Landau gauge, the global gauge charges of covariant gauge theory are spontaneously broken. Within the language of local quantum field theory the decoupling solution can thus only be realised if and only if it comes along with a Higgs mechanism and massive physical gauge bosons. The Schwinger mechanism can in fact be described in this way, and it can furthermore be shown that a non-vanishing gauge boson mass, by whatever mechanism it is generated, necessarily implies the spontaneous breakdown of global symmetries.”

The “decoupling solution” cited here is the one currently seen on lattice computations having a finite gluon propagator at zero momenta and a ghost propagator behaving like a free particle without any fixed point in the running coupling. The point here is that, in this paragraph, the truth about the real situation of Yang-Mills theory in the infrared is simply exposed. Classical solutions exist that display such dynamical generation of mass for the massless scalar field theory and Yang-Mills theory (see here) and a quantum field theory can be built with them making the above argument truly consistent.

But the point to be emphasized here is the proposal of von Smekal arriving to present a modification of lattice computations. His proposal relies on a recent work done with Andre Sternbeck (see here) where they study the limit \beta\rightarrow\infty of the Yang-Mills theory. Indeed, in this limit they recover the results obtained by functional methods that disagree with lattice computations. Again, they simply freeze the dynamics and get meaningless results as also happens when one compares D=1+1 Yang-Mills theory with no dynamics and the D=4 case. Besides, recent QCD computations on the lattice, relying on known formulations of the Yang-Mills side, give too striking results to go to look for reformulations (see my post).

My view is that functional methods are generally useless. Also when the right solution is hit, of course numerically, one is not able to do any kind of real calculation in QCD. In physics this means that no true understanding is reached. One of the points that should have warned people working with functional methods is that no mass gap is ever obtained and there is no way to recover the low-energy phenomenology of QCD. But having a mass gap produces immediately a Nambu-Jona-Lasinio model from QCD from the ratio g^2 to the square of the mass gap itself and this is a real understanding as Nambu-Jona-Lasinio model gives a lot of comprehension of low energy phenomenology.

I think this paper is worth an in depth reading as it contains several pieces of true awareness. My criticisms should not be of any concern for such a good work.

The whisper of the universe


Current experimental situation about dark matter is becoming increasingly exciting as results are coming out. I would like to emphasize the work of Piergiorgio Picozza that has been my professor teaching me the foundations of nuclear physics. He is currently the leader of PAMELA, an experiment that produced a lot of rumors and striking results as well. The New York Times has, as usual, published a beautiful article on the current situation about dark matter (see here).  It is worthwhile reading and makes quite clear that the situation is becoming stunning day after day but avoids anyway to cite the recent results of CDF Collaboration. Indeed, the link between dark matter and CDF finding is yet in its infancy. We are all awaiting for other confirmations as always happens to experiments. I am confident that these will not miss for too long. Whoever is interested about the relevant discussions about I just point you to the Dorigo’s blog being him one of the authors of the CDF’s paper. This provoked a lot of rumors in the blogosphere that sometime degenerated in hot discussions. We just stay tuned to learn more about.

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