Paper with a proof of confinement has been accepted


Recently, I wrote a paper together with Masud Chaichian (see here) containing a mathematical proof of confinement of a non-Abelian gauge theory based on Kugo-Ojima criterion. This paper underwent an extended review by several colleagues well before its submission. One of them has been Taichiro Kugo, one of the discoverers of the confinement criterion, that helped a lot to improve the paper and clarify some points. Then, after a review round of about two months, the paper has been accepted in Physics Letters B, one of the most important journals in particle physics.

This paper contains the exact beta function of a Yang-Mills theory. This confirms that confinement arises by the combination of the running coupling and the propagator. This idea was around in some papers in these latter years. It emerged as soon as people realized that the propagator by itself was not enough to grant confinement, after extended studies on the lattice.

It is interesting to point out that confinement is rooted in the BRST invariance and asymptotic freedom. The Kugo-Ojima confinement criterion permits to close the argument in a rigorous way yielding the exact beta funtion of the theory.


Do quarks grant confinement?


In 2010 I went to Ghent in Belgium for a very nice Conference on QCD. My contribution was accepted and I had the chance to describe my view about this matter. The result was this contribution to the proceedings. The content of this paper was really revolutionary at that time as my view about Yang-Mills theory, mass gap and the role of quarks was almost completely out of track with respect to the rest of the community. So, I am deeply grateful to the Organizers for this opportunity. The main ideas I put forward were

  • Yang-Mills theory has an infrared trivial fixed point. The theory is trivial exactly as the scalar field theory is.
  • Due to this, gluon propagator is well-represented by a sum of weighted Yukawa propagators.
  • The theory acquires a mass gap that is just the ground state of a tower of states with the spectrum of a harmonic oscillator.
  • The reason why Yang-Mills theory is trivial and QCD is not in the infrared limit is the presence of quarks. Their existence moves the theory from being trivial to asymptotic safety.

These results that I have got published on respectable journals become the reason for rejection of most of my successive papers from several referees notwithstanding there were no serious reasons motivating it. But this is routine in our activity. Indeed, what annoyed me a lot was a refeee’s report claiming that my work was incorrect because the last of my statement was incorrect: Quark existence is not a correct motivation to claim asymptotic safety, and so confinement, for QCD. Another offending point was the strong support my approach was giving to the idea of a decoupling solution as was emerging from lattice computations on extended volumes. There was a widespread idea that the gluon propagator should go to zero in a pure Yang-Mills theory to grant confinement and, if not so, an infrared non-trivial fixed point must exist.

Recently, my last point has been vindicated by a group that was instrumental in the modelling of the history of this corner of research in physics. I have seen a couple of papers on arxiv, this and this, strongly supporting my view. They are Markus Höpfer, Christian Fischer and Reinhard Alkofer. These authors work in the conformal window, this means that, for them, lightest quarks are massless and chiral symmetry is exact. Indeed, in their study quarks not even get mass dynamically. But the question they answer is somewhat different: Acquired the fact that the theory is infrared trivial (they do not state this explicitly as this is not yet recognized even if this is a “duck” indeed), how does the trivial infrared fixed point move increasing the number of quarks? The answer is in the following wonderful graph with N_f the number of quarks (flavours):

QCD Running CouplingFrom this picture it is evident that there exists a critical number of quarks for which the theory becomes asymptotically safe and confining. So, quarks are critical to grant confinement and Yang-Mills theory can happily be trivial. The authors took great care about all the involved approximations as they solved Dyson-Schwinger equations as usual, this is always been their main tool, with a proper truncation. From the picture it is seen that if the number of flavours is below a threshold the theory is generally trivial, so also for the number of quarks being zero. Otherwise, a non-trivial infrared fixed point is reached granting confinement. Then, the gluon propagator is seen to move from a Yukawa form to a scaling form.

This result is really exciting and moves us a significant step forward toward the understanding of confinement. By my side, I am happy that another one of my ideas gets such a substantial confirmation.

Marco Frasca (2010). Mapping theorem and Green functions in Yang-Mills theory PoS FacesQCD:039,2010 arXiv: 1011.3643v3

Markus Hopfer, Christian S. Fischer, & Reinhard Alkofer (2014). Running coupling in the conformal window of large-Nf QCD arXiv arXiv: 1405.7031v1

Markus Hopfer, Christian S. Fischer, & Reinhard Alkofer (2014). Infrared behaviour of propagators and running coupling in the conformal
window of QCD arXiv arXiv: 1405.7340v1

QCD 12 and Higgs’ tears


I have spent this week in Montpellier being a participant to QCD 12, a biannual conference organized by Stephan Narison. It is the third time that I go to Montpellier for this conference and there are always very good reasons for being there. Essentially, the quality of physics and beauty of the city are already worthwhile and sound arguments but also the excellent organization  by the host and the attention reserved to the guests are not the least. This year we have had the blessing of a historical event in physics: The discovery at CERN of the Higgs particle. Stephan organized the event with the webcast from CERN the first two hours on Wednesday and so we heard directly from Gianotti and Incandela what they were seeing at LHC.  The conference is a fair interplay between experiment and theory in a field, QCD, that is very active and with several important open problems. Maybe, we would like to emphasize that is QCD that gives mass to everyday things, and not the Higgs boson, and this means that the solution of the mass gap problem and the developing of proper methods to manage non-perturbative regimes are essential to the understanding of our common perception of reality. Indeed, Roberto Mussa of University of Turin remembered us an argument that  makes Higgs boson essential to everyday life: The stability of matter. Without the Higgs boson quarks would have equal masses and so, proton would decay into neutron. The difference in mass between u and d quarks is essential and this originates from Higgs boson.

In this conference several questions emerged that were absolutely exciting. Hadron spectrum is not so well understood both in the low and high part. There is a plenty of experimental results claiming for an explanation. Labs keep on finding resonances that have not an immediate explanation and make hard the life of us theoreticians. One should compare the situation with the case of electromagnetic interactions where a Rydberg formula was promptly found and understanding of bound states is now quite straightforward. For hadrons we have hard times already to catch what the structure of a resonance is. These difficulties arise from the missing of technique to manage non-perturbative problems in a way similar to the weak coupling limit. Indeed, on Wednesday, some approaches were given to manage this kind of situation and, besides my talk, the most common technique is AdS/QCD starting from Maldacena conjecture. This was also the argument of Stefano Nicotri and Floriana Giannuzzi. They are students of Pietro Colangelo and contributed to the organization of Lecce conference. I have spent a lot of good time with them and so we exchanged a lot of opinions about this matter. On this line, Hans Günter Dosch put all us down showing that the situation with this approach is not so fine. Simply, it appears like a proper model for the mapping between gravity and QCD is lacking yet but, of course, people is actively pursuing it.

A talk that gave me some interesting views was the one by Kenichi Konishi. He pointed out how the confinement can emerge looking at the behavior of the supersymmetric counterpart of Yang-Mills theory. He pointed out the problems with the idea of monopoles, already discussed by Kei-Ichi Kondo at Lecce. And you bet, when one looks at SYM one recover the condensation of a scalar field! Konishi works at University of Pisa where teaches quantum mechanics.

On the line of non-perturbative approaches were the talks by Matteo Giordano and Enrico Meggiolaro. They are trying to re-derive from first principles the Froissart bound. This is a bound on hadronic scattering that can be obtained just using unitarity and dispersion relations. This bound depends crucially on the mass gap of the theory and so, again, we are coping with all the problems given above. Meggiolaro showed that, using lattice computations, the limit can be recovered with the proper constant while Matteo is approaching this problem using AdS/QCD. With Matteo we meet again in Montpellier after four years. We remembered each other immediately and drunk a last beer before leaving on Friday night after the social dinner, with Montpellier streets full of people and pleasant noise.

A talk that I followed with a lot of interest was the one given by Pietro Falgari. He is working on the use of perturbation theory at high-energy in QCD to evaluate the production rate of pairs of top quarks. Even if in this limit perturbation theory can be applied in QCD, they have difficulties mostly related to resum a quite singular series with logarithmic contributions. So, also when perturbation theory applies, QCD does not save us from problems. With Pietro I have spent a lot of time in Montpellier and we left the city together on Saturday with the same flight.

An interesting talk was the one given by Eduardo de Rafael about the determination of the g factor of the muon. This is a truly relevant matter as this measurement can give a clue to new physics. But, as de Rafael pointed out, the critical point is the determination of the hadronic contribution. Presently, there is a 3.6 sigmas discrepancy between the theoretical computed value and the measured one. We cannot be confident that the evaluation of the hadronic part is not correctly accomplished.

Last but not least, the current work of Narison on heavy flavors with sum rules. This approach is now fairly well stable and provides results also better than other non-perturbative techniques. This has been shown in the talk by his collaborator Albuquerque from Sao Paolo. Of course, results like these should be a reference for experiments much in the same way are others as lattice computations. Finally, I would like to cite the talk by Robert Kaminski. He presented the fine work done in collaboration with R. Garcia-Martin J. R. Pelaez, J. Ruiz de Elvira aimed to a precise determination of the properties of f0(600) and f0(980). Their results are striking indeed as they fix very precise values to the mass and width of these resonances, in close agreement with preceding works. It is my personal conviction that a serious theoretical approach should be derive both the mass and the width of these resonances deriving at the same time their structure.

Wednesday was the great day. There was a lot of expectation and the great discovery was in the air predated by a lot of rumors here and there. Our organizers did a great work both providing the webcast from CERN and with a pair of talks on Friday from people of CMS and ATLAS. There has been a religious silence during the talks of Incandela and Gianotti just interrupted by applause at the announcements of the 5 sigmas discovery by the two groups. Following this, we discussed a lot about this matter and, besides it is very standard model-like this particle at the moment, we all were very cautious to claim supersymmetry dead. Rather we would like to know more about the rates in the various channels, results to be known in the near future in order to answer the question put forward by CERN director Rolf-Dieter Heuer: Which one? A girl at my conference asked for other four Higgs and we all know why. Talking with a colleague at ATLAS here in Montpellier, he told me a quite interesting figure for the WW channel but I will not disclose it. Work is in progress yet and data are really too fresh to be discussed. It is a matter of few months and we will know better about the nature of this new particle. Meanwhile, I would like to remember Higgs’ tears after the great announcement and the handshaking with Fabiola Gianotti, after the splendid talk by her, confirming the expectation of almost fifty years of waiting with hopes often not coming up. It is an achievement that very few scientists can claim in their lifetime. The same must apply identically to Englert, Brout, Guralnik, Hagen, and Kibble.

On Friday, the program was concluded by the talks of people from CERN, on behalf of ATLAS and CMS Collaborations. Pushpa Bhat from Fermilab talked on behalf of CMS Experiment while Robert Harrington from Particle Physics Experimental Group of University of Edinburgh talked on behalf of ATLAS Experiment. This was a great conclusion for the Conference, hearing directly from people at CERN, about the great achievement announced on Wednesday.

As a final remark, I would like to thank all people with whom I shared beautiful moments at this conference. Besides people I have already mentioned, I would like to thank Stefano Venditti, Antonio Cassese, Andrey Tayduganov, Federico Mescia, Benjamin Obherof. A great thank goes to Stephan Narison for giving me the chance to give a talk here, for giving me the chance to be chairman for the first time, and for the excellent and really enjoying organization in a beautiful city. See you again!

Update: Talks can be downloaded here.

QCD@Work 2012


This week has been of great interest for me being one of the participants to QCD@Work 2012. I have had my contribution accepted by the organizers and so I gave a talk. The conference was held in a really beautiful city, Lecce here in Italy. This conference is organized jointly by University of Bari and University of Salento in Lecce. 

The first day, Monday, at the very beginning, there were both ATLAS and CMS delivering their official results. Of course, gossip arrived also at the conference and people were very aware of it. But both CERN groups presented known results leaving some more room for future developments to be heard very soon. The talks in their pdf format are given here. At this conference I have had the good chance to meet Francesco Sannino. We have had interesting discussions sharing some time outside conference time. His talk was really interesting opening up some new venues to the understanding of what could be going on beyond the Standard Model. I would like to remember the proposal by Francesco and Ryttov on the exact beta function for Yang-Mills theory that appears really insightful (see here and refs therein). The afternoon parallel session concerned more strictly what I am currently doing in QCD. A striking talk was given by Kei-Ichi Kondo. Kondo is currently involved in research about infrared behavior of Yang-Mills theory and his conclusions are very similar to mine: There is an Abelian dominance in the low-energy limit. He was able to get Nambu-Jona-Lasinio model as low-energy limit of QCD much in the same way I did. In his talk he get the interquark potential starting from the idea that confinement arises from non-abelian monopoles in the theory. He also verified his approach through lattice computations. In the successive parallel session I heard the talk from Mirko Serino about a really innovative idea. Mirko is a PhD studend at University of Salento and together with Claudio Coriano, that is his professor, Luigi Delle Rose and Antonio Quintavalle are producing an analytical computation of Standard Model in presence of gravity. This kind of computation is highly non-trivial and quite complicated. The striking result they get is that appears a coupling between a scalar degree of freedom and the gauge field and this appears as a rather interesting new proposal for mass generation. I have talked with the students of Coriano and they were really excited by this result that is indeed really interesting and unexpected. The session ended with my talk.

The next day, Tuesday, it was the day for excursion and social dinner. The program was limited to two sessions in the morning. There were a couple of interesting talks, I mean with respect to my fields of interest, by Huang Mei and Antonio Vairo. The talk by Huang Mei managed to get consistent results about confinement in AdS/QCD. She shows that a way to reconcile confinement and holography is obtained  with the introduction of the condensate of the vector potential. This has been the starting point of a lot of discussions, also fired by an interesting comment by Antonio Vairo.  Vairo’s comment is about the fact that an operator product expansion (OPE) does not produce this condensate at large momenta.  By itself, this condensate is clearly not gauge invariant. We also know that a gluon mass is obtained through this condensate in some scenarios as refined Gribov-Zwanzinger theory that I was able to show is fairly consistent (see here). Of course, I would like to see OPE at small momenta, where the real thing happens, to conclude that Vairo’s comment applies as well. Vairo presented a talk on a non-trivial effect in QCD: Jet quenching. The idea is very similar to the one of a particle going through ordinary matter but in this case the matter is a quark-gluon plasma that has the effect to dump energetically the jet. In the afternoon we headed to Otranto, on the sea, for a guided visit and finally took the social dinner in a very pleasant place.

On Wednesday there was a number of talks very near my interests. The first one was delivered by Luigi Capozza on behalf of COMPASS collaboration. I always find the results of this group really striking. Their aim is to take a measurement of the components of the proton spin. These are usually divided in three parts: Quark contribution, gluon contribution and orbital contribution. The striking part of the measurement is that the gluon contribution is compatible with zero! From what we know in the high-energy limit, where the gluon concept is well-defined, these are spin-1 objects and so, it is not so straightforward to have zero contribution from them to the proton spin and indeed this is an important open problem in theoretical physics. The following talk was delivered by Mark Alford. With Mark we spent a lot of time going around Lecce and taking meats looking for very good restaurants.  I was very impressed both by his humor, sometime really sharp, and by his thorough knowledge of a lot of arguments and physics was surely not the last of them. The talk he delivered left me somewhat impressed and the reason is that he and some other few people is managing a really pioneering question: What is going on to nuclear matter inside collapsed stars? He modeled the matter inside a neutron star as a liquid. This liquid has essentially two components and one of this is a superfluid. Sometime, the fluid motion happens in such a way to produce a quadrupole configuration emitting gravitational waves and so the star loses energy. This mechanism is characterized by large amplitude waves. Mark said to me that they are in difficulty due to the strongly coupled situation of such a plasma that has the properties of a superfluid. He called them speculations but much of us know that also special relativity in 1905 was just speculation and there was no hope to see an experimental test in a short time. Next talk was delivered by Maxim Chernodub. Maxim was one of the companions, together with Mark, spending around time in Lecce. We have had several discussions on a lot of arguments. His talk was about his very important work on superconductor properties of the vacuum of QCD. He proved that \rho mesons undergo condensation. This effect appears when a strong magnetic field is applied and this is a typical situation in collisions at RHIC or LHC with heavy ions even if this effect could be very well-hidden. On the other side, surely on lattice computations this could be seen quite straightforwardly. I have asked to Maxim if current approximations in lattice computations can give off the mark results but he said that the choice of quark masses, even if nonphysical,  are not a concern as it should also be for the choice of the lattice spacing. From a theoretical standpoint, he was able to show this effect with some smart computations recently appeared (see here). The final talk of this session was delivered by Marco Ruggieri. Marco is one of my best friends and he gave me a lot of really helpful comments as you can see from the acknowledgment in my talk. The argument was Yang-Mills thermodynamics well above the critical temperature that is 270 MeV. This work, done at University of Catania, is providing really important results. It should be stated that we are again in a strongly coupled regime. But, if you remember condensed matter questions, generally the effect of strongly coupled degrees of freedom goes to dressing particles, called quasi-particles, leaving you with a manageable description of the physical picture. What Marco and the group he works with have proven is that it appears that a quasi-particle description does work in this limit of very high-temperature. The order parameter they identify for the phases is the Polyakov loop and they provide support to the existence of a condensate of Z(3) lines. Z(3) symmetry arises from the fact that only non-colored objects can propagate in Yang-Mills theory and so, only objects with this symmetry are physical degrees of freedom (see here). These results provide an innovative view of what is found on lattice computations and are really challenging for theoreticians. In the afternoon there has been another session with some other interesting talks. The first of this was delivered by Hiroaki Abuki. Abuki’s approach follows an interesting research path to study QCD near the critical point. The idea is to do a Ginzburg-Landau ansatz expanding the Gibbs free energy in terms of chiral fields till fourth power. I think it would be interesting to see a complete justification of this approach starting from QCD as also happens for a superconductor and BCS theory. Hiroaki and his students were very nice companions during launches and dinners at Lecce. Following talk was delivered by Andreas Schmitt. Andreas is doing an important theoretical work in trying to justify phenomenological behavior of nuclear matter in extreme physical conditions starting from models of QCD. This work is essential to give an in-depth comprehension of the work of Mark Alford and collaborators. This is a very challenging activity working with a strongly coupled theory. The starting point for Andreas is Nambu-Jona-Lasinio model and I should say that this is very sound. He is able to show that chiral symmetry is recovered under large magnetic fields. Finally, I have listened the talk of Motoi Tachibana. Spending time with Motoi in Lecce was really enjoying. His talk was about a collaboration with Marco Ruggieri and people at INFN Gran Sasso. This work is aimed to a deeper understanding of the ultradense nuclear matter on the same line of the preceding ones. This new path of research I was not aware of entails the same difficulties of other standard approaches to QCD: One has to cope with a strongly coupled theory. Of course, recurring to Nambu-Jona-Lasinio model is the right way but all the parameters should be properly fixed using QCD and here lies the main question. As my readers probably know, I have approached this problem in one of my most recent papers (see here) and I have presented part of it to this conference. It would be interesting to extend possible applications of it further in such very extreme conditions.

I have to say that this proved to be a very interesting conference, well-organized and I have had the chance to say this personally to Pietro Colangelo, one of the organizers. Knowing how Italian universities are managed by our political class should make clearer yet how great the job has been done by the organizers to get all this machine properly work. Finally, Lecce is a very beautiful city and is worthwhile a staying there. It is another part of this conference that I enjoyed a lot.

Marco Frasca (2012). Condensates in the refined Gribov-Zwanziger scenario arXiv arXiv: 1202.4105v2

M. N. Chernodub (2010). Superconductivity of QCD vacuum in strong magnetic field Phys.Rev.D82:085011,2010 arXiv: 1008.1055v2

M. Ruggieri, P. Alba, P. Castorina, S. Plumari, C. Ratti, & V. Greco (2012). Polyakov Loop and Gluon Quasiparticles in Yang-Mills Thermodynamics arXiv arXiv: 1204.5995v1

Marco Frasca (2011). Chiral symmetry in the low-energy limit of QCD at finite temperature Phys. Rev. C 84, 055208 (2011) arXiv: 1105.5274v4

Evidence of a QCD critical endpoint at RHIC


A critical endpoint in QCD is a kind of holy grail in nuclear physics. It has been theorized as a point where deconfinement occurs and hadronic matter leaves place to some kind of plasma of quarks and gluons. We know that the breaking of chiral symmetry is something that people has proposed several years ago and we recently gave a proof of existence of such a transition (see here). But here the situation is more complex: We have essentially two physical variables to describe the phase diagram and these are temperature and chemical potential. This makes lattice computations a kind of nightmare. The reason is the sign problem. Some years ago Zoltan Fodor and Sandor Katz come out with a pioneering paper (see here) doing lattice computation and seeing the chemical potential taking an imaginary factor: The infamous sign problem. Discretization implies it but a theoretical physicist can happily lives just ignoring it. Fodor and Katz evaded the problem just taking an absolute value but this approach was criticized casting doubt on their results at chemical potential different from zero. It should be said that they gave evidence of existence for the critical point and surely their results are unquestionably correct with zero chemical potential in close agreement with my and others findings. A lucid statement of the problems of lattice computations for finite temperatures and densities was recently given by Philippe de Forcrand (see here).

So far, people has produced several results just working around with phenomenological model like a Nambu-Jona-Lasinio or sigma model. This way of work arises from our current impossibility to manage QCD at very low energies but, on the other side, we are well aware that these models seem to represent reality quite well. The reason is that a Nambu-Jona-Lasinio is really the low-energy limit for QCD but I will not discuss this matter here having done this before (see here). Besides, the sigma model arises naturally in the low-energy limit interacting with quarks. The sigma field is a true physical field that drives the phase transitions in low-energy QCD.

While the hunt for the critical point in the lattice realm is already open since the paper by Fodor and Katz, the experimental side is somewhat more difficult to exploit. The only facility we have at our disposal is RHIC and no much proposals are known to identify the critical point from the experimental data were available since a fine proposal by Misha Stephanov a few years ago (see here and here). The idea runs as follows.  At the critical point, fluctuations are no more expected to be Gaussian and all the correlations are extended to all the hadronic matter  as the correlation length is diverging. Non-Gaussianity implies that if we compute cumulants, linked to higher order moments of the probability distribution, these will depend on the correlation length with some power and, particularly, moments like skewness and kurtosis, that are a measure of deviation from Gaussianity, start to change. Particularly, kurtosis is expected to change sign. So, if we are able to measure such a deviation in a laboratory facility we are done and we get evidence for a critical point and critical behavior of hadronic matter. We just note that Stephanov accomplishes his computations using a sigma model and this is a really brilliant hindsight.

At RHIC a first evidence of this has been obtained by STAR Collaboration (see here). These are preliminary results but further data are expected this year. The main result is given in the following figureWe see comparison with data from lattice as red balls for Au+Au collisions and the kurtosis goes down to negative values! The agreement with lattice data is striking and this is already evidence for a critical endpoint. But this is not enough as can be seen from the large error bar. Indeed further data are needed to draw a definitive conclusion and, as said, these are expected for this year. Anyhow, this is already a shocking result. Surely, we stay tuned for this mounting evidence of a critical endpoint. This will represent a major discovery for nuclear physics and, in some way, it will make easier lattice computations with a proper understanding of the way the sign problem should be settled.

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

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

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

M. A. Stephanov (2008). Non-Gaussian fluctuations near the QCD critical point Phys.Rev.Lett.102:032301,2009 arXiv: 0809.3450v1

Christiana Athanasiou, Krishna Rajagopal, & Misha Stephanov (2010). Using Higher Moments of Fluctuations and their Ratios in the Search for
the QCD Critical Point Physical review D arXiv: 1006.4636v2

Xiaofeng Luo (2011). Probing the QCD Critical Point with Higher Moments of Net-proton
Multiplicity Distributions arXiv arXiv: 1106.2926v1

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

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

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