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.

Advertisement

Good news from Moriond

Some days ago, Rencontres of Moriond 2018 ended with the CERN presenting a wealth of results also about the Higgs particle. The direction that the two great experiments, ATLAS and CMS, took is that of improving the measurements on the Standard Model as no evidence has been seen so far of possible new particles. Also, the studies of the properties of the Higgs particle have been refined as promised and the news are really striking.

In a communicate to the public (see here), CERN finally acknowledge, for the first time, a significant discrepancy between data from CMS and Standard Model for the signal strengths in the Higgs decay channels. They claim a 17% difference. This is what I advocated for some years and I have published in reputable journals. I will discuss this below. I would like only to show you the CMS results in the figure below.

ATLAS, by its side, is seeing significant discrepancy in the ZZ channel () and a compatibility for the WW channel. Here are their results.

On the left the WW channel is shown and on the right there are the combined and ZZ channels.

The reason of the discrepancy is due, as I have shown in some papers (see here, here and here), to the improper use of perturbation theory to evaluate the Higgs sector. The true propagator of the theory is a sum of Yukawa-like propagators with a harmonic oscillator spectrum. I solved exactly this sector of the Standard Model. So, when the full propagator is taken into account, the discrepancy is toward an increase of the signal strength. Is it worth a try?

This means that this is not physics beyond the Standard Model but, rather, the Standard Model in its full glory that is teaching something new to us about quantum field theory. Now, we are eager to see the improvements in the data to come with the new run of LHC starting now. In the summer conferences we will have reasons to be excited.

It was twenty years ago today . . .

With these beautiful words starts a recollection paper by the founder of arXiv, Paul Ginsparg. This is worth the reading as this history spans a number of years exactly overlapping the computer revolution that definitely changed our lives. What Paul also changed through these new information tools was the way researchers should approach scientific communication. It is a revolution that is not stopped yet and all the journals I submit my papers have a link to arXiv for direct uploading of the preprint. This change has had also a great impact on the way these same journals should present to authors, readers and referees as well at their website.

For my readers I would like just to point out how relevant was all this for our community with the Grisha Perelman’s case. I think all of you are well aware that Perelman never published his papers on a journal: You can find both of them on arXiv. Those preprints paid as much as a Fields medal and a Millenium prize. Not bad I should say for a couple of unpublished papers. Indeed, it is common matter to have a paper largely discussed well before its publication and often a preprint becomes a case in the community without not even seeing the light of a publication. It is quite common for us doing research to console colleagues complaining about the harsh peer-review procedure by saying that today exists arXiv and that is enough to make your work widely known.

I was a submitter since 1994, almost at the very start, and I wish that the line of successes of this idea will never end.

Finally, to prove how useful is arXiv for our community, I would like to point out to you, for your summer readings a couple of papers. The first one is this from R. Aouane, V. Bornyakov, E.-M. Ilgenfritz, V. Mitrjushkin, M. Müller-Preussker, A. Sternbeck. My readers should know that these researchers always do a fine work and get important results on their lattice computations. The same happens here where they study the gluon and ghost propagators at finite temperature in the Landau gauge. Their conclusion about Gribov copies is really striking, comforting my general view on this matter (see here), that Gribov copies are not essential not even when one rises the temperature. Besides, they discuss the question of a proper order parameter to identify the phase transition that we know exists in this case.

The next paper is authored by Tereza Mendes, Axel Maas and Stefan Olejnik (see here). The idea in this work is to consider a gauge, the -gauge, with a free parameter interpolating between different gauges to see the smoothness of the transition and the way of change of the propagators. They reach a volume of 70^4 but Tereza told me that the errors are too large yet for a neat comparison with smaller volumes. In any case, this is a route to be pursued and I am curious about the way the interpolated propagator behaves at the deep infrared with larger lattices.

Discussions on Higgs identification are well alive yet ( you can see here). take a look and enjoy!

Paul Ginsparg (2011). It was twenty years ago today … arXiv arXiv: 1108.2700v1

R. Aouane, V. Bornyakov, E. -M. Ilgenfritz, V. Mitrjushkin, M. Müller-Preussker, & A. Sternbeck (2011). Landau gauge gluon and ghost propagators at finite temperature from
quenched lattice QCD arXiv arXiv: 1108.1735v1

Axel Maas, Tereza Mendes, & Stefan Olejnik (2011). Yang-Mills Theory in lambda-Gauges arXiv arXiv: 1108.2621v1

What’s going on with Higgs particle?

The aftermath of the EPS Conference is quite exciting on a side. Higgs hunting points to an unexpected direction even if some residuals of an old expectation are still there. I just want to show you the graphs of this conference from Tevatron and LHC

From these it is very clear that the excluded range of mass is become significantly large restricting the possibilities to the intervals of a mass around 140 GeV or to a massive Higgs implying a strongly coupled theory. The evidence for a 140 GeV Higgs particle is yet small, about two sigmas, and we cannot exclude that this is a fluke but, to support this clue, it appears both at Tevatron and LHC. A small peak at around 250 GeV is seen only by ATLAS and could disappear in the future.

What I would like to emphasize here is that the possibility of a strongly coupled Higgs is well alive and this can have deep implications for the model and physics at large. There are several reasons for this. First of all, a strongly coupled Higgs field implies supersymmetry (see here). This result is inescapable and some breaking pattern of supersymmetry must be devised to get the right mass spectrum of the Standard Model. But this is already old and well-acquired matter. The most important point is that there will be a completely new way to approach quantum field theory. So far, quantum field theory has been managed just using weak perturbation theory but a strongly coupled Higgs would mean that we will also have to devise a perturbative technique the other way round, i.e. with a coupling increasingly large.

So, I will keep on support this view of a heavy Higgs as, being a theoretical physicist, consequences will be devastating and largely more exciting of any other possibility. We will be eager to see the improvement in the next months from the measurement datasets. Certainly, on 2012 all the curtains will be definitely down.

Marco Frasca (2010). Mass generation and supersymmetry arXiv arXiv: 1007.5275v2

Paper on critical temperature revised

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.

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

Back from Paris

It is several days that I have no more posted on the blog but for a very good reason: I was in Paris for the Eleventh Workshop on Non-Perturbative Quantum Chromodynamics (see here). It has been a beautiful chance to see Paris with the eyes of a tourist and being immersed in a lot of physics in the area I am currently contributing. The conference was held at the Institut d’Astrophyisique de Paris. This week was indeed plenty of information for people in high-energy physics due to the release by D0 of their measurements on the Wjj data, showing that the almost 5 sigma bump of CDF was not there (see here, here and here). In the conference there has been room for talks by experimentalists too and it was the most shocking part as I will explain below.

The talks were somehow interesting with a couple of days mostly dedicated to AdS/CFT approach for QCD. So, string theory got a lot of space even if I should say that more promising approaches seem to exist. The first day there have been a couple of talks that were very near my interest by Dario Zappalà and Marco Ruggieri. They were reporting on their very recent papers (here and here). With Marco, I spent all the week together while with Dario we have had a nice dinner near Latin Quartier. The question Dario presented was about the existence of massive excitations (let me say “persistence”) also beyond the critical temperature for Yang-Mills theory. We discussed together with Marco this result and Marco claimed that massive excitations should have melted beyond the critical temperature while my view is that the residual of mass should be due to temperature corrections to the mass spectrum of the theory. Marco in his talk presented the idea of measuring the chiral chemical potential on the lattice as this could give plain evidence of existence for the critical endpoint without the annoying sign problem. A proof of existence of the critical endpoint is somehow the Holy Grail of finite temperature QCD and something under a lot of studies both theoretically and on the lattice. So, Marco’s proposal can turn out a significant shortcut toward the reaching of this goal.

The second day Carl Bender gave a very beautiful talk telling us about PT invariant quantum mechanics. PT stays for Parity and Time reversal. The point to start from is the Dirac postulate about the Hamiltonian being Hermitian self-adjoint. Differently from the other postualates of quantum mechanics, this one is too much a mathematical requirement and one could ask if can be made somewhat looser. The paradigm Hamiltonian has the from . The answer is yes of course and we were left with the doubt that maybe this is the proper formulation of quantum mechanics rather the standard one. I suspect that this could represent a possible technique useful in quantum gravity studies.

I have already said of the two days on string theory. I have just noticed the talk by Luca Mazzucato showing how, with his approach, my scaling with for the energy spectrum could be recovered in a strong coupling expansion being the ‘t Hooft coupling. Unfortunately, Gabriele Veneziano could not partecipate.

On Wednesday there was the most shocking declaration from an experimentalist: “We do not understand the proton”. The reason for this arises from the results presented by people from CERN working at LHC. They showed a systematic deviation of their Montecarlo simulations from experimental data. This means for us, working in this area, that their modeling of low-energy QCD is bad and their possible estimation of the background unsure. There is no way currently to get an exact evaluation of the proton scattering section. I am somewhat surprised by this as so far, as I have always pointed out in this blog, at least the structure of the gluon propagator at low energies should be known exactly from the lattice. So, modeling the proton in such Montecarlo models should be a mitigated issue. This does not seem to be so and these different communities do not seem to talk each other at all. After these shocking news, the evening we took an excellent social dinner and I have had some fine discussions with foreigners colleagues that were well aware of the books from Umberto Eco. One of these, Karl Landsteiner, suggested us to visit the Pantheon to look at the Foucault pendulum. I, Marco Ruggieri and Orlando Oliveira took this initiative the next day and it was a very nice place to visit. If you are a physicist you can understand the emotion of being there seeing that sphere moving like Newton’s equations demand and inexorably proving the rotation of the Earth. Karl gave an interesting talk that day where AdS/CFT is used to obtain transport coefficients in heavy ion collisions.

In the same day, Orlando Oliveira gave his talk. Orlando is a friend of mine and gave relevant contribution to our understanding of the behavior of low-energy gluon propagator. He has been the author of one of the papers that, at Regensburg on 2007, started the end of the so called “scaling solution” for the gluon propagator (see here). Orlando is going ahead, starting from the acquired form of the gluon propagator, to understand low-energy phenomenology of nuclear forces. In this work, he and his colleagues introduce an octect of scalar fields having the aim to produce the gluon mass through a non-zero vacuum expectation value (see here) producing chiral symmetry breaking. My work and that of Orlando are somewhat overlapped in the initial part where we have an identical understanding of the low-energy behavior of  Yang-Mills theory.

On Friday, there have been a couple of significant events. The first one was my talk. This is a report on my recent paper. I will not discuss this point further leaving this material to your judgement. The second relevant event was given in the talks by Thierry Grandou and our Chairman and Organizer Herbert Fried. The relevant paper is here. While Grandou made a more mathematical introduction with a true important result: the resummation of all gluon exchange diagrams realizing some dream of having completely solved QCD, Fried provided a more concrete result giving the binding potential between quarks analytically obtained from the preceding theorem. We were somehow astonished by this that seems just a small step away from the Millenium prize. Berndt Mueller, one of the Organizers, suggested to Fried to determine the mass gap and wait a couple of years to get the prize. Indeed, this appears a true striking exact result in the realm of QCD.

All in all, an interesting conference in a special place: Paris. For me, it has been a very nice period of full immersion in physics with the company of very nice friends.

Update: Mary Ann Rotondo put online the slides of the talks (see here).

P. Castorina, V. Greco, D. Jaccarino, & D. Zappalà (2011). A reanalysis of Finite Temperature SU(N) Gauge Theory arXiv arXiv: 1105.5902v1

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

Irene Amado, Karl Landsteiner, & Francisco Pena-Benitez (2011). Anomalous transport coefficients from Kubo formulas in Holography JHEP 05 (2011) 081 arXiv: 1102.4577v3

O. Oliveira, W. de Paula, & T. Frederico (2011). Linking Dynamical Gluon Mass to Chiral Symmetry Breaking via a QCD Low
Energy Effective Field Theory arXiv arXiv: 1105.4899v1

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

H. M. Fried, Y. Gabellini, T. Grandou, & Y. -M. Sheu (2009). Gauge Invariant Summation of All QCD Virtual Gluon Exchanges Eur.Phys.J.C65:395-411,2010 arXiv: 0903.2644v2

CDF bump at 4.8 sigma!

In these days I am exceeding with exclamation marks but let me say that there are sound reasons for this. I will keep on staying on a prudence line as my more renowned colleagues are doing but there is a talk by Giovanni Punzi, the spokesman of CDF Collaboration at Tevatron, (see here), presenting the following picture

They increased the number of events and the bump is still there. They have consistently found for the mass of this presumed particle. Of course, we have to wait for D0 and LHC to confirm this finding but this is becoming more and more a real discovery. But since the first inception, theoretical physicists have come out with possible explanations, the most reasonable of these seems a new U(1) interaction with Z’ leptophobic boson, where leptophobic just means that this particle has essentially strong decay modes. People at CDF has discarded several possible mundane explanations that emerged at the dawn of the first announcement but, as my readers know, it should be stated that our current understanding of low-energy QCD is somewhat in development and some modeling used by people at CERN or Tevatron can be not so accurate. My personal view is that is time to wait yet notwithstanding the good news.

For more relevant posts see here, here and here.

Kingman Cheung, & Jeonghyeon Song (2011). Baryonic Z’ Explanation for the CDF Wjj Excess arXiv arXiv: 1104.1375v3

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