Sidney Coleman’s QFT lectures


This post is just to point out to my readers that the lectures of Sidney Coleman on QFT are now available in TeX and pdf format. I have taken this information from Lubos’ site. The link for the full pdf is this. For this excellent work the person to be grateful is Bryan Chen a former Lubos’ student. These lectures give an idea of the greatness of Coleman also as a teacher.

For the readers of my blog pursuing active research in my same areas a relevant paper by him is there are no classical glueballs. Coleman’s conclusion goes like

…there are no finite-energy non-singular solutions of classical Yang-Mills theory in four-dimensional Minkowski space that do not radiate energy out to spatial infinity…

and this agrees quite well with my exact solutions that do not seem to have finite energy (see here) and so, Coleman’s theorem is evaded. Indeed, if you want to have a field to generate a mass, you will need either a finite volume or a running coupling.

Coleman, S. (1977). There are no classical glueballs Communications in Mathematical Physics, 55 (2), 113-116 DOI: 10.1007/BF01626513

Marco Frasca (2009). Exact solutions of classical scalar field equations arxiv arXiv: 0907.4053v2

A wonderful confirmation


Contributions to proceedings to Ghent conference “The many faces of QCD” are starting to appear on arxiv and today appeared one of the most striking one I have heard of at that conference: Orlando Oliveira, Pedro Bicudo and Paulo Silva published their paper (see here). This paper represents a true cornerstone for people doing computations of propagators as the authors for the first time try to connect a gauge-dependent quantity as the gluon propagator is to a gauge-independent one as is the spectrum of Yang-Mills theory, mostly in the way I advocated here and in my papers. The results are given in the following figure

and the data are the following


for one, two, three and four masses respectively. The form of the propagator they consider is the following one


and so the first number above is the maximum momentum considered in the fit, then you have the pairs \{m_n,Z_n\} and the last number is the goodness of the fit as \chi^2/d.o.f.. As you can see from the picture above, the fit goes excellently well on all the range with four masses! The masses they obtain are values that are consistent with hadronic physics and can represent true glueball masses. The series has alternating signs signaling that the match with a true Källén-Lehmann spectral representation is not exact. Finally, the authors show how all the lattice computations performed so far agree well with a value D(0)\approx 8.3-8.5.

Why have I reasons to be really happy? Because all this is my scenario! The paper you should refer to are this and this. The propagator I derive from Yang-Mills theory is exactly the one of the fit of these authors. Besides, this is a confirmation from the lattice that a tower of masses seems to exist for these glue excitations as I showed. The volumes used by these authors are quite large, 80^4, and will be soon accessible also from my CUDA machine (so far I reached 64^4 thanks to a suggestion by Nuno Cardoso), after I will add a third graphics card. Last but not least the value of D(0). I get a value of about 4, just a factor 2 away from the value computed on the lattice, for a string tension of 440 MeV. As my propagator is obtained in the deep infrared, I would expect a better fit in this region.

The other beautiful result these authors put forward is the dependence of the mass on momentum. I have showed that the functional form they obtain is to be seen in the next to leading order of my expansion (see here). Indeed, they show that the fit with a single Yukawa propagator improves neatly with a mass going like m^2=m^2_0-ap^2 and this is what must be in the deep infrared from my computations.

I have already said in my blog about the fine work of these authors. I hope that others will follow these tracks shortly. For all my readers I just suggest to stay tuned as what is coming out from this research field is absolutely exciting.

O. Oliveira, P. J. Silva, & P. Bicudo (2011). What Lattice QCD tell us about the Landau Gauge Infrared Propagators arxiv arXiv: 1101.5983v1

FRASCA, M. (2008). Infrared gluon and ghost propagators Physics Letters B, 670 (1), 73-77 DOI: 10.1016/j.physletb.2008.10.022

FRASCA, M. (2009). MAPPING A MASSLESS SCALAR FIELD THEORY ON A YANG–MILLS THEORY: CLASSICAL CASE Modern Physics Letters A, 24 (30) DOI: 10.1142/S021773230903165X

Marco Frasca (2008). Infrared behavior of the running coupling in scalar field theory arxiv arXiv: 0802.1183v4

The Tevatron affair and the “fat” gluon


Tevatron is again at the forefront of the blogosphere mostly due to Jester and Lubos. Top quark seems the main suspect to put an end to the domain of the Standard Model in particle physics. Indeed, years and years of confirmations cannot last forever and somewhere some odd behavior must appear. But this is again an effect at 3.4 sigma and so all could reveal to be a fluke and the Standard Model will escape again to its end. But in the comment area of the post in the Lubos’ blog there is a person that pointed out my proposal for a “fat” gluon. “Fat” here stays just for massive and now I will explain this idea and its possible problems.

The starting point is the spectrum of Yang-Mills theory that I have obtained recently (see here and here). I have shown that, at very low energies, the gluon field has a propagator proportional to


with the spectrum given by


being \sigma the string tension being about (440\ MeV)^2. If we go beyond the leading order of such a strong coupling expansion one gets that the masses run with momenta. This has been confirmed on the lattice quite recently by Orlando Oliveira and Pedro Bicudo (see here). The interesting point about such a spectrum is that is not bounded from above and, in principle, one could take n large enough to reach TeV energies. These glueballs are very fat indeed and could explain CDF’s results if these should be confirmed by them, their colleagues at D0 and LHC.

It should be emphasized that these excitations of the glue field have spin zero and so will produce t-tbar pairs in a singlet state possibly explaining the charge asymmetry through the production rate of such very massive glueballs.

A problem can be seen immediately from the form of the propagator that has each contribution in the sum exponentially smaller as n increases. Indeed, this has a physical meaning as this is also what appears in the decay constants of such highly massive gluons (see here). Decay constants are fundamental in the computation of cross sections and if they are very near zero so could be the corresponding cross sections. But Oliveira and Bicudo also showed that these terms in the propagator depend on the momenta too, evading the problem at higher energies. Besides, I am working starting from the low energy part of the theory and assuming that such a spectrum will not change too much at such high energies where asymptotic freedom sets in and gluons seem to behave like massless particles. But we know from the classical theory that a small self-interaction in the equations is enough to get masses for the field and massless gluons are due to the very high energies we are working with. For very high massive excitations this cannot possibly apply. The message I would like to convey with this analysis is that if we do not know the right low-energy behavior of QCD we could miss important physics also at high-energies. We cannot live forever assuming we can forget about the behavior of Yang-Mills theory in the infrared mostly if the mass spectrum is not bounded from above.

Finally, my humble personal conviction, also because I like the idea behind Randall-Sundrum scenario, is that KK gluons are a more acceptable explanation if these CDF’s results will prove not to be flukes. The main reason to believe this is that we would obtain for the first time in the history of mankind a proof of existence for other dimensions and it would be an epochal moment indeed. And all this just forgetting what would imply for me to be right…

Frasca, M. (2008). Infrared gluon and ghost propagators Physics Letters B, 670 (1), 73-77 DOI: 10.1016/j.physletb.2008.10.022

Frasca, M. (2009). Mapping a Massless Scalar Field Theory on a Yang–Mills Theory: Classical Case Modern Physics Letters A, 24 (30) DOI: 10.1142/S021773230903165X

P. Bicudo, & O. Oliveira (2010). Gluon Mass in Landau Gauge QCD arxiv arXiv: 1010.1975v1

Frasca, M. (2010). Glueball spectrum and hadronic processes in low-energy QCD Nuclear Physics B – Proceedings Supplements, 207-208, 196-199 DOI: 10.1016/j.nuclphysbps.2010.10.051

A more prosaic explanation


The aftermath of some blogosphere activity about CDF possible finding at Tevatron left no possible satisfactory explanation beyond a massive octet of gluons that was already known in the literature and used by people at Fermilab. In the end we need some exceedingly massive gluons to explain this asymmetry. If you look around in the net, you will find other explanations that go beyond ordinary known physics of QCD. Of course, speaking about known physics of QCD we leave aside what should have been known so far about Yang-Mills theory and mass gap. As far as one can tell, no generally accepted truth is known about otherwise all the trumpets around the World would have already sung.

But let us do some educated guesses using our recent papers (here and here) and a theorem proved by Alexander Dynin (see here). These papers show that the spectrum of a Yang-Mills theory is discrete and the particles have an internal spectrum that is bounded below (the mass gap) but not from above. I can add to this description that there exists a set of spin 0 excitations making the ground state of the theory and ranging to infinite energy. So, if we suppose that the annihilation of a couple of quarks can generate a particle of this with a small chance, having enough energy to decay in a pair t-tbar in a singlet state, we can observe an asymmetry just arising from QCD.

I can understand that this is a really prosaic explanation but it is also true that we cannot live happily forgetting what is going on after a fully understanding of a Yang-Mills theory and that we are not caring too much about. So, before entering into  the framework of very exotic explanations just we have to be sure to have fully understood all the physics of the process and that we have not forgotten anything.

Marco Frasca (2007). Infrared Gluon and Ghost Propagators Phys.Lett.B670:73-77,2008 arXiv: 0709.2042v6

Marco Frasca (2009). Mapping a Massless Scalar Field Theory on a Yang-Mills Theory: Classical
Case Mod. Phys. Lett. A 24, 2425-2432 (2009) arXiv: 0903.2357v4

Alexander Dynin (2009). Energy-mass spectrum of Yang-Mills bosons is infinite and discrete arxiv arXiv: 0903.4727v2

And you are calling it a gluon yet…


One of the more questionable points I have discussed so far is: What are QCD asymptotic states at very low  momenta? This question is not trivial at all. If you will speak with experts in this matter, a common point they will share is that gluons carry color charge and so must form bound states. A claim like this has a strong implication indeed. The implication is that Yang-Mills Hamiltonian must display the same asymptotic states at both ends of the energy range. But the problem is exactly in the self-interaction of the theory that, at very low momenta, becomes increasingly large and gluons, asymptotic states of Yang-Mills theory in the asymptotic freedom regime, are no more good to describe physics. So, what are good states at low energies? I have already answered to this question a lot of times (recently here) and more and more confirmations are around. I would like just to cite a very nice paper I have seen recently on arxiv (see here) by Stanley Brodsky, Guy de Teramond and Alexandre Deur. These authors have nicely exploited AdS/CFT symmetry obatining striking results in the understanding of low-energy QCD. I would like to cite again the work of these authors as their soft-wall model is indeed a strong support to my view. It would be really interesting to get them working out a pure Yang-Mills model obtaining beta function and all that.

What one has at low end of momenta is a new set of states, glue states or glueballs if you prefer, that permits strong interactions. These states have already been seen in most laboratories around the World and belong to the open question of the understanding of the lower part of the hadronic spectrum.

A really interesting view about QCD and AdS/CFT


Stan Brodsky is a renowned physicist that has produced a lot of very good works. As I work on QCD, I try to be up-to-date as much as possible and I spend some time to read the most recent literature about. AdS/CFT applied to QCD is a very hot topic these times and I run into a beautiful paper by Stan and Guy de Téramond that was recently published in Physical Review Letters (a preprint is here). Their work is inspired by AdS/CFT in that they are able to map on a five dimensional Anti-de Sitter space a light-front Hamiltonian for QCD, producing a Schrödinger-like equation with a proper potential to get the spectrum of the theory. This equation is depending by a single proper variable and is exactly solvable. Two classes of models can be identified in this way that are those well-known in literature:

  • Hard-wall model with a potential described by an infinite potential wall till a given cut-off that fixes the mass scale.
  • Soft-wall model with a harmonic potential producing Regge trajectories.

So, these authors are able to give a clever formulation of two known models of QCD obtained from AdS/CFT conjecture and they manage them obtaining the corresponding spectra of mesons and baryons. I would like to emphasize that the hard-wall model was formulated by Joseph Polchinski and Matthew Strassler and was instrumental to show how successful AdS/CFT could be in describing QCD spectrum. This paper appeared in Physical Review Letters and can be found here. Now, leaving aside Regge trajectories, what Stan and Guy show is that the mass spectrum for glueballs in the hard wall model goes like

m_n\approx 2n+L

being n an integer and L  the angular momentum. This result is interesting by its own. It appears to be in agreement both with my recent preprint and my preceding work and with most of the papers appeared about Yang-Mills theory in 2+1 dimensions. Indeed, they get this spectrum being the zeros of Bessel functions and the cut-off making the scale. Very simple and very nice.

I should say that today common wisdom prefers to consider Regge trajectories being hadron spectroscopy in agreement with them but, as glueballs are not yet identified unequivocally, I am not quite sure that the situation between a soft wall and hard wall models is so fairly well defined. Of course, this is a situation where experiments can decide and surely it is just a matter of a few time.

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.


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