## Paper on critical temperature revised

06/07/2011

Yesterday, I have uploaded a new version of my paper on the critical temperature of chiral symmetry breaking in QCD (see here). The reason for this was that there are some points in need for a better clarification. The main of these is the mapping theorem: I have added a sketch of a proof. The reason for this is that there is a common misunderstanding about it and that some people think  that this theorem is for quantum field theories. Indeed, it just establishes a map between classical solutions of a scalar field and a Yang-Mills field but in the asymptotic limit of a coupling going to infinity. Quantum theory does not enter at all here but these classical asymptotic solutions can be used to build up a perturbation theory for quantum field theory in the infrared, that is for low-energies, that is the range of interest for all the phenomenology we would like to understand.

Another recurring question is if this mapping breaks in some way gauge invariance. The answer is a resounding no as the proof does not select a gauge at the start but anyhow if one wants quantization a gauge must be selected.

Finally, I have better clarified the derivation of the critical temperature and added some more relevant references. I hope in this way that my arguments can be better understood. Indeed, presentation is one of the most difficult aspects of scientific communication and sometime it is a sound explanation of attrition between authors and referees.

## QCD at finite temperature: Does a critical endpoint exist?

05/04/2011

Marco Ruggieri is currently a post-doc fellow at Yukawa Institute for theoretical physics in Kyoto (Japan). Marco has got his PhD at University of Bari in Italy and spent a six months period at CERN. Currently, his main research areas are QCD at finite temperature and high density, QCD behavior in strong magnetic fields and effective models for QCD but you can find a complete CV at his site. So, in view of his expertize I asked him a guest post in  my blog to give an idea of the current situation of these studies. Here it is.

It is well known that Quantum Chromodynamics (QCD) is the most accredited theory describing strong interactions. One of the most important problems of modern QCD is to understand how color confinement and chiral symmetry breaking are affected by a finite temperature and/or a finite baryon density. For what concerns the former, Lattice simulations convince ourselves that both deconfinement and (approximate) chiral symmetry restoration take place in a narrow range of temperatures, see the recent work for a review. On the other hand, it is problematic to perform Lattice simulations at finite quark chemical potential in true QCD, namely with number of color equal to three, because of the so-called sign problem, see here for a recent review on this topic. It is thus very difficult to access the high density region of QCD starting from first principles calculations.

Despite this difficulty, several work has been made to avoid the sign problem, and make quantitative predictions about the shape of the phase diagram of three-color-QCD in the temperature-chemical potential plane, see here again for a review. One of the most important theoretical issues in along this line is the search for the so-called critical endpoint of the QCD phase diagram, namely the point where a crossover and a first order transition line meet. Its existence was suggested by Asakawa and Yazaki (AY) several years ago (see here) using an effective chiral model; in the 2002, Fodor and Katz (FK) performed the first Lattice simulation (see here) in which it was shown that the idea of AY could be realized in QCD with three colors. However, the estimate by FK is affected seriously by the sign problem. Hence, nowadays it is still under debate if the critical endpoint there exists in QCD or not.

After referring to this for a comprehensive review of some of the techniques adopted by the Lattice community to avoid the sign problem and detect the critical endpoint, it is worth to cite an article by Marco Ruggieri, which appeared few days ago on arXiv, in which an exotic possibility to detect the critical endpoint by virtue of Lattice simulations avoiding the sign problem has been detected, see here . We report, after the author permission, the abstract here below:

We suggest the idea, supported by concrete calculations within chiral models, that the critical endpoint of the phase diagram of Quantum Chromodynamics with three colors can be detected, by means of Lattice simulations of grand-canonical ensembles with a chiral chemical potential, $\mu_5$, conjugated to chiral charge density. In fact, we show that a continuation of the critical endpoint of the phase diagram of Quantum Chromodynamics at finite chemical potential, $\mu$, to a critical end point in the temperature-chiral chemical potential plane, is possible. This study paves the way of the mapping of the phases of Quantum Chromodynamics at finite $\mu$, by means of the phases of a fictitious theory in which $\mu$ is replaced by $\mu_5$.

Rajan Gupta (2011). Equation of State from Lattice QCD Calculations arXiv arXiv: 1104.0267v1

Philippe de Forcrand (2010). Simulating QCD at finite density PoS (LAT2009)010, 2009 arXiv: 1005.0539v2

M. Asakawa, & K. Yazaki (1989). Chiral restoration at finite density and temperature Nuclear Physics A, 504 (4), 668-684 DOI: 10.1016/0375-9474(89)90002-X

Z. Fodor, & S. D. Katz (2001). Lattice determination of the critical point of QCD at finite T and \mu JHEP 0203 (2002) 014 arXiv: hep-lat/0106002v2

Marco Ruggieri (2011). The Critical End Point of Quantum Chromodynamics Detected by Chirally
Imbalanced Quark Matter arXiv arXiv: 1103.6186v1

## QCD at strong magnetic fields

10/02/2011

Today on arxiv it is appeared the contribution to the conference “The many faces of QCD” of my friend Marco Ruggieri. Marco is currently a postdoc student at Yukawa Institute in Tokyo and has been a former student of Raoul Gatto. Gatto is one of the most known Italian physicists that had as students also Gabriele Veneziano and Luciano Maiani. With Marco we have had a lot of fun in Ghent and several interesting discussions about physics. One of the main interests of Marco is to study QCD vacuum under the effect of a strong magnetic field and he pursue this line with Gatto. This is a very rich field of research producing several results that can be compared with lattice computations and LHC findings at last. Marco’s contribution (see here) approaches the question using Nambu-Jona-Lasinio model. Before to enter is some details about Marco’s work, let me explain briefly what is the question here.

As my readers know, there has been so far no widely accepted low-energy limit of QCD rigorously derived from it. Simply, we can do computations of low-energy phenomenology just using some models that we hope, in some approximation, will describe correctly what is going on in this limit. Of course, there have been a number of successful models and Nambu-Jona-Lasinio model is one of this. This model, taken from the original formulation, is not renormalizable and not confining. But it describes fairly well the breaking of chiral symmetry and the way bound states can form from quark fields. Indeed, one is able to get a fine description of the low-energy behavior of QCD notwithstanding the aforementioned shortcomings of this model. In the course of time, this model has been refined and some of its defects have been corrected and today appears a serious way to see the behavior of QCD at very low-energy. But all this success appears somewhat incomprehensible unless someone is able to prove that this model is indeed a low-energy approximation to the QCD quantum  field theory. A couple of proofs are around: One is due to Kei-Ichi Kondo (see here) and the other one is due to your humble writer (see here). Kondo’s work does not reach a value for the NJL coupling while I get one through my gluon propagator that I know in a closed form. Anyhow, I was able to get a fully quantum formulation quite recently and this was published in QCD08 and QCD10 proceedings. But, notwithstanding these achievements, I keep my view that, until the community at large does not recognize these results as acquired, we have to continue to take not proved the fact that NJL is obtainable from QCD.

Given this situation, Marco’s approach is to consider a couple of modified NJL models and applies to them a constant magnetic field. Dirac equation with a constant magnetic field is well-known and exactly solvable producing a set of Landau levels and a closed form fermion propagator. This means that, given the mean field approximation, Marco is able to give well defined conclusions through analytical computations. NJL models Marco is considering have been both tuned to agree with lattice computations. What he finds is that the magnetic field has indeed important effects on the temperature of chiral symmetry restoration and for the deconfining phase. But he claims as a weak point a proper determination of the coupling that appears in the NJL models through the Polyakov loop that enters in the way the NJL models are formulated here. This is work for the future. I would like to emphasize the relevance of this kind of research for our understanding of the low-energy behavior of QCD. I will keep my readers up-to-date about this and I will keep on asking to Marco to clarify what the issues are for his research. What I find really striking here is to see the interplay between a magnetic field and strong force vacuum so entangled to produce really non-trivial results. Other groups around the World are working on this and accelerator facilities as LHC can produce important clues for our understanding of the vacuum of QCD. It will be really interesting to see how the results in this area will reach their maturity.

Marco Ruggieri (2011). Chiral symmetry restoration and deconfinement in strong magnetic fields arxiv arXiv: 1102.1832v1

Kondo, K. (2010). Toward a first-principle derivation of confinement and chiral-symmetry-breaking crossover transitions in QCD Physical Review D, 82 (6) DOI: 10.1103/PhysRevD.82.065024

FRASCA, M. (2009). INFRARED QCD International Journal of Modern Physics E, 18 (03) DOI: 10.1142/S0218301309012781