A critical point in QCD exists indeed!

After my comeback from the conference in Ghent (see here, here and here), I started a collaboration with Marco Ruggieri. Marco was instrumental in making me aware of that part of the community that does computations in QCD at finite temperature. The aim of these people is to get a full landscape of the ground state of hadronic matter, even when a magnetic field is applied and the vacuum state is expected to change. This is not just an intellectual exercise as recent observations of quark-gluon plasma are there to show and also some important experiments at LHC are now unfolding the complexity of this theory. So, we are saying about the forefront of modern research in the field of nuclear matter that can have significant impact in our understanding of the early universe.

As Marco pointed out in this blog (see here), due to the lack of knowledge of techniques to manage QCD at low-energies, we are not even able to give a definite answer to the question if different phases exist for hadronic matter and if a critical temperature, or a cross-over temperature, can be found from a theoretical standpoint. People use two different approaches to manage this question: lattice computations and phenomenological models like Nambu-Jona-Lasinio or sigma models. Lattice computations displayed a critical temperature at zero quark masses and zero chemical potential  (see here) and a cross-over rather than a phase transition with non-zero quark masses. A critical temperature was found to be about 170 MeV. These studies are yet underway and improve year after year. From a theoretical point of view the situation is less clear even if a Nambu-Jona-Lasinio model can be used to work out a critical temperature. The model should be non-local.

With this scenario in view, it seems not thinkable a proof of existence of a critical point at zero quark masses and zero chemical potential. This is true unless we know how to manage the low-energy behavior of QCD. One should have solved the mass gap problem to say in a few words what is needed here. As my readers know, I am in a position to give a definite answer to such a question and, of course I did. On Friday I have uploaded a new paper of mine on arXiv (see here) and I have obtained an evidence for a critical point in QCD. This is the point in temperature where the chiral symmetry gets restored.

The idea for this paper come after I read a beautiful work by T. Hell, S. Roessner, M. Cristoforetti, W. Weise on the non-local Nambu-Jona-Lasinio model (see here). These authors completely work out the physics of this model. The point is that, as I have shown, this is the right one to describe low-energy physics in QCD. From a comparison with the form factors, mine obtained solving QCD with the mapping theorem and the one of Weise&al. guessed from a model of a liquid of instantons, the agreement is so good that my approach strongly supports the other view.

Finally, I was able to get the long sought equation for the critical temperature at zero quark masses and chemical potential. At this temperature the chiral symmetry appears to be restored. I find really interesting the fact that a similar equation was obtained by Norberto Scoccola and Daniel Gomez Dumm (see here). My equation for the critical temperature is substantially the same  as theirs. Of course, the fundamental difference between my approach and all others relies on the fact that I am able to get the form factor solving QCD. In the preceding works this is just a guess, even if a very good one. Besides, so far, nobody was able to show that a Nambu-Jona-Lasinio model is the right low-energy limit of QCD. This result should be ascribed to me and Ken-Ichi Kondo (see here).

Having proved that a non-local Nambu-Jona-Lasinio model is the right low-energy limit, a prove of existence of a critical point is so obtained. This proof will be presented at the next conference in Paris  on non-perturbative QCD (see here). Me and Marco will be there the next week.

Z. Fodor, & S. D. Katz (2004). Critical point of QCD at finite T and \mu, lattice results for physical
quark masses JHEP 0404 (2004) 050 arXiv: hep-lat/0402006v1

Marco Frasca (2011). Chiral symmetry in the low-energy limit of QCD at finite temperature arXiv arXiv: 1105.5274v2

T. Hell, S. Roessner, M. Cristoforetti, & W. Weise (2008). Dynamics and thermodynamics of a nonlocal Polyakov–Nambu–Jona-Lasinio
model with running coupling Phys.Rev.D79:014022,2009 arXiv: 0810.1099v2

D. Gomez Dumm, & N. N. Scoccola (2004). Characteristics of the chiral phase transition in nonlocal quark models Phys.Rev. C72 (2005) 014909 arXiv: hep-ph/0410262v2

Quote of the day

“Das Wesen der Mathematik liegt in ihrer Freiheit”

Georg Cantor

Tr.: The essence of mathematics lies in its freedom

A briefer history of Stephen Hawking

I am always happy to point out to my readers worthwhile readings from the web and mostly from significant sites. One of my preferred ones is New York Times. This time there is an interview by Claudia Dreifus to the great physicist Stephen Hawking. Hawking is well-known for his fundamental contributions to cosmology and our current understanding of black hole physics positing the foundations to any future theory of quantum gravity. Hawking is also known for his enduring struggle against the motor neuron disease that afflicts him since the times of his youth. Notwithstanding such a hurdle he was able to find his way becoming one of the greatest living theoretical physicists. Hawking has been Lucasian Professor at Cambridge University and left the chair due to the age succeeded by Michael Green, a well-known string theorist.

Claudia in this interview gives relevance to Hawking’s disease and tries to give a picture on how Stephen was able to reach such high goals despite of this. It is also interesting to point out a couple of questions about LHC and the recent finding at Fermilab of a claim for a new particle discover (see here). All this makes the interview a worthwhile reading.

Today in arXiv (2)

Today I have found some papers in the arXiv daily that makes worthwhile to talk about. The contribution by Attilio Cucchieri and Tereza Mendes at Ghent Conference “The many faces of QCD” is out (see here). They study the gluon propagator in the Landau gauge at finite temperature at a significantly large lattice. The theory is SU(2) pure Yang-Mills. As you know, the gluon propagator in the Landau gauge at finite temperature is assumed to get two contributions: a longitudinal and a transverse one. This situation is quite different form the zero temperature case where such a distinction does not exist. But, of course, such a conclusion could only be drawn if the propagator is not the one of massive excitations and we already know from lattice computations that massive solutions are those supported. In this case we should expect that, at finite temperature, one of the components of the propagator must be suppressed and a massive gluon is seen again. Tereza and Attilio see exactly this behavior. I show you a picture extracted from their paper here

The effect is markedly seen as the temperature is increased. The transverse propagator is even more suppressed while the longitudinal propagator reaches a plateau, as for the zero temperature case, but with the position of the plateau depending on the temperature making it increase. Besides, Attilio and Tereza show how the computation of the longitudinal component is really sensible to the lattice dimensions and they increase them until the behavior settles to a stable one.  In order to perform this computation they used their new CUDA machine (see here). This result is really beautiful and I can anticipate that agrees quite well with computations that I and Marco Ruggieri are performing  but yet to be published. Besides, they get a massive gluon of the right value but with a mass decreasing with temperature as can be deduced from the moving of the plateau of the longitudinal propagator that indeed is the one of the decoupling solution at zero temperature.

As an aside, I would like to point out to you a couple of works for QCD at finite temperature on the lattice from the Portuguese group headed by Pedro Bicudo and participated by Nuno Cardoso and Marco Cardoso. I have already pointed out their fine work on the lattice that was very helpful for  my studies that I am still carrying on (you can find some links at their page). But now they moved to the case of finite temperature (here and here). These papers are worthwhile to read.

Finally, I would like to point out a really innovative paper by Arata Yamamoto (see here). This is again a lattice computation performed at finite temperature with an important modification: The chiral chemical potential. This is an important concept introduced, e.g. here and here, by Kenji Fukushima, Marco Ruggieri and Raoul Gatto. There is a fundamental reason to introduce a chiral chemical potential and this is the sign problem seen in lattice QCD at finite temperature. This problem makes meaningless lattice computations unless some turn-around is adopted and the chiral chemical potential is one of these. Of course, this implies some relevant physical expectations that a lattice computation should confirm (see here). In this vein, this paper by Yamamoto is a really innovative one facing such kind of computations on the lattice using for the first time a chiral chemical potential. Being a pioneering paper, it appears at first a shortcoming the choice of too small volumes. As we already have discussed above for the gluon propagator in a pure Yang-Mills theory, the relevance to have larger volumes to recover the right physics cannot be underestimated. As a consequence the lattice spacing is 0.13 fm corresponding to a physical energy of 1.5 GeV that is high enough to miss the infrared region and so the range of validity of a possible Polyakov-Nambu-Jona-Lasinio model as currently used in literature. So, while the track is open by this paper, it appears demanding to expand the lattice at least to recover the range of validity of infrared models and grant in this way a proper comparison with results in the known literature. Notwithstanding these comments, the methods and the approach used by the author are a fundamental starting point for any future development.

Attilio Cucchieri, & Tereza Mendes (2011). Electric and magnetic Landau-gauge gluon propagators in
finite-temperature SU(2) gauge theory arXiv arXiv: 1105.0176v1

Nuno Cardoso, Marco Cardoso, & Pedro Bicudo (2011). Finite temperature lattice QCD with GPUs arXiv arXiv: 1104.5432v1

Pedro Bicudo, Nuno Cardoso, & Marco Cardoso (2011). The chiral crossover, static-light and light-light meson spectra, and
the deconfinement crossover arXiv arXiv: 1105.0063v1

Arata Yamamoto (2011). Chiral magnetic effect in lattice QCD with chiral chemical potential arXiv arXiv: 1105.0385v1

Fukushima, K., Ruggieri, M., & Gatto, R. (2010). Chiral magnetic effect in the Polyakov–Nambu–Jona-Lasinio model Physical Review D, 81 (11) DOI: 10.1103/PhysRevD.81.114031

Fukushima, K., & Ruggieri, M. (2010). Dielectric correction to the chiral magnetic effect Physical Review D, 82 (5) DOI: 10.1103/PhysRevD.82.054001