A beautiful paper on arxiv

Keeping on their way of producing sound work, Bogolubsky, Ilgenfritz, Mueller-Preussker and Sternbeck have got their paper (see here) published on Physics Letters B. This is a collaboration between people working in Russia, Germany and Australia. The main aim of this work is the computation on a lattice of the two-point functions and the running coupling of a pure Yang-Mills theory. They carry on lattice computations from to points entering into a deep enough infrared limit to get a meaningful behavior of the lattice theory in this case. I give below their main results

Gluon propagator

Dressing function of the ghost propagator

Running coupling

These results confirm completely the decoupling solution. The definition of the running coupling is the one proposed by Alkofer and von Smekal and it is my personal conviction that it conveys the right physical behavior of the theory. This is exactly the scenario I have derived in my paper (see here) that has been published on Physics Letters B too and has arisen a lot of rumors around. You will not find this paper cited in this work as these authors have concerns about gauge invariance in my computations. As you may know from my dispute with Terry Tao, gauge invariance is not a problem here. One could ask why a mathematical technique, like a gradient expansion is, should not work for Yang-Mills equations but it does for all other equations of mathematical physics. Anyhow, I am here ready to listen to whoever is able to prove this. With this proof in hand one should also warn all general relativists that use this technique and put it in their handbooks.

The authors conclude their paper by pointing out weaknesses in lattice computations that may bring in discussion their results. Finally, they ask if the other solution, the one with a scaling behavior, can emerge from lattice computations. The understanding of this question is surely of relevant interest. We stay tuned to hear news about.

A sound confirmation of Yang-Mills scenario

Today arxiv contains a very interesting paper by Attilio Cucchieri and Tereza Mendes (see here). They do a significant lattice computation for a SU(2) Yang-Mills theory in 3 and 4 dimensions. Quarks are absent (quenched approximation). Their aim is to verify the current emerging scenario in a particular situation where the coupling on the lattice is taken to go to infinity (). This case, as pointed out by the authors, is unphysical but is a quite interesting test-bed for the behavior of the two-point functions of the theory (gluon and ghost propagators). They do not aim to check the running coupling as this, currently, is matter of hot debate yet.

I would like to remember what is at stake today about this question.  For some years there has been the strong conviction that the gluon propagator should go to zero and ghost propagator should go to infinity faster than the free propagator (conformal solution). This was needed to confirm two confining scenarios that were commonly accepted by the community. Most of these results emerged from an idea due to Gribov that there remains an ambiguity in the gauge also after it is fixed. Taking into account Gribov ambiguity provoked a flourishing of papers all reaching similar conclusions. At that time, due to the small achievable volumes, lattice computations were not able to clarify the situation even if the gluon propagator was never seen to tend to zero in a significant way. As increasing volumes made available a completely new scenario emerged. People obtained that the gluon propagator indeed reaches a finite value at zero momentum while the ghost propagator is seen to behave as that of a free particle (decoupling solution). A commonly accepted definition of running coupling was seen to converge to zero making the theory trivial.

This emerging data prompted several explanations. People started to criticize these lattice computations as maybe there was an accumulation of Gribov copies that modify the right results into the observed ones. Maybe the fixing gauge algorithms should be better analyzed and so on. People from Australia (see here and here) claimed that low energy data should not be trusted. Discarding them one finds again the conformal solution. Cucchieri and Mendes give a sound answer to all these doubts. Indeed, it is not clear why in 2 dimensions one gets the conformal solutions but not in 3 and 4 dimensions, notwithstanding the code used to do these computations is always the same. Further, any reason adduced by Australian group to remove low energy data is proved substantially unfounded and the results obtained by Cucchieri and Mendes represent a correct picture of the case for 3 and 4 dimensions. Indeed, Cucchieri and Mendes show that Gribov copies play no role in the scenario seen at low energies for Yang-Mills theory. This is a crucial point that has been source for misleading research for a lot of years.

So, let us take a look at the scenario found by Cucchieri and Mendes. These authors consider a maximum lattice dimensions of for 3 dimensions and for 4 dimensions. They show without any doubt that one gets the decoupling solution: The gluon propagator reaches a finite value at zero momentum and the ghost propagator is that of a free particle. What is more interesting here are the fits. For the propagator they fit to a sum of Stingl-forms

and for the ghost propagator

being m the gluon mass. The authors tried to avoid to fix the values of their computations with experimental data. As you know, the relevant parameter here is , the string tension. Notwithstanding this operative choice, they get for the gluon mass the following values

using only data with and

using all data. I hope that now some bell is ringing for you as this is the mass of the resonance. This resonance is not seen by people doing quenched computations to obtain the spectrum of a pure Yang-Mills theory. Why? What are they missing with respect to Cucchieri and Mendes? This should not become a longstanding question. We need an answer right now.

Now, take a look at the fit of the gluon propagator. Try to do a Fourier transform and you will get back something like

and this is shockingly similar to my propagator having the general form

that is a sum of free particle propagators!

I should say that I am somewhat impressed by Cucchieri and Mendes results. They showed that the decoupling scenario is the right one in the physical case of 4 dimensions. My view is that we should move on from the current position and try to find the theoretical framework that better fits the data. It goes without saying what is the one I am supporting.

Yang-Mills theory in D=2+1

There is a lot of work about the pursuing of a deep understanding of Yang-Mills theory in the low energy limit. The interesting case is in four dimensions as our world happens to have such a property. But we also know that a Yang-Mills theory in D=2+1 is not trivial at all and worthwhile to be studied. In this area there has been a lot pioneering work mostly due to V. Parameswaran Nair and Dimitra Karabali . These authors proved that a Hamiltonian formulation may be truly effective to manage this case. Indeed, they obtained a formula for the string tension that works quite well with respect to lattice computations. We would like to remember that, in D=2+1, coupling constant is such that its dimension is while, in D=3+1, is dimensionless.

Quite recently, some authors showed how, from such a formulation, a functional can be given from which one can obtain the spectrum (see here, here and here). These papers went all published on archival journals. Now, these spectra are quite good with respect to lattice computations, after some reinterpretation. We do not know if this is due to some problems in lattice computations or in the theoretical analysis. I leave this to your personal point of view. My idea is that this quenched lattice computations are missing the true ground state of the theory. This happens to be true both for D=3+1 and D=2+1. I do not know why things stay in this way but in this kind of situations are always theoreticians to lose. On the other side, being a physicist means that one should not have a blind faith in anything.

Finally, one may ask how my work performs with respect all this. Yesterday, I spent a few time to try to figure this out. The results I obtain agree fairly well with those of the theoretical papers. I obtain the zero Lionel Brits gets at being m a mass proportional to ‘t Hooft coupling. Brits wrote the third of the three papers I cited above. The string tension I get is in agreement with lattice computations. This zero is the problem on lattice computations and the same problem is seen in D=3+1. This fact is at the root of our presenting difficulty to understand what resonance is. We know that people working on a quenced lattice computation for the propagator do see this resonance. This difference between this two approaches should be understood and an effort in this direction must be made.

Osaka and Berlin merge their data!

Today in arxiv appeared a relevant paper by Osaka and Berlin groups (see here). This is a really important paper as these two groups merged their data for the lattice computation of the gluon and ghost propagators for SU(3) in the Coulomb gauge. As usual I give here a picture summing up their results about gluon propagator

As you can see one has again the propagator reaching a finite, non-null value in the infrared limit. About the ghost propagator they obtain again a result very near to the case of a free particle. In this case the agreement was perfect for SU(2). For SU(3) there is a tiny disagreement.

I would like to emphasize a couple of points that should be discussed with these results at hand. There is a paper, published on Physical Review Letters, that was claiming that the gluon propagator in the Coulomb gauge should take the Gribov form going to zero at lower momenta. You can find this paper here and here. I think that authors should reconsider their computations as the disagreement with lattice is really serious. All the research lines aimed at a proof of confinement scenarios heavily relying on Gribov ideas seem to have reached a failure point. There could be a lot of reasons for this but it seems to me that, as lattice computations improve, we are left with the only option that the starting points of all these studies are to be reconsidered.

A second point to be made is the completely missing link between people working on the computation of propagators and those working on the spectrum of QCD. I think this is the moment to try to connet these two relevant areas as times are mature to try a consistency check between them. After the failure in view of some functional methods do we have to believe yet that Kaellen-Lehman formula does not apply in the infrared limit?

A confirmation again

One of my main activities in the morning is reading the daily coming from arxiv. Sometime it happens to find significant papers to be put in a post like this. This morning I have found a beautiful paper by a cooperation of people from Germany, Russia and Australia working on lattice QCD (see here). This paper has been written by Igor Bogolubsky, Ernst-Michael Ilgenfritz, André Sternbeck and Michael Mueller-Preussker. I put here the following picture representing one of the main conclusions

This picture gives the gluon propagator with a number of points (96)^4 and shows clearly that it reaches a finite value at smaller momenta implying a massive gluon. Indeed, the authors of the paper extended the lattice computations moving from (80)^4 to (96)^4 points and add some other improvement in the computation itself. The value of beta is quite high being 5.7. The agreement with previous computations of Cucchieri and Mendes is excellent (see here). These latter authors worked with a number of points of (128)^4 while beta was taken to be 2.2.

The other two important conclusions they reach is that the ghost propagator goes like that of a free particle and the running coupling goes to zero at lower momenta. For the running coupling we emphasize that there is no common agreement about its definition in the infrared and the authors properly point out this. But a running coupling that goes to zero does not mean at all that there is no confinement. Quite the contrary as proved by Kazuhiko Nishijima (see here): It gives a proof of confinement.

So, we obtain again a clear proof of the scenario we have already obtained from a theoretical standpoint (see here and here) and we have discussed at length in this blog. I think that evidence of existence of the mass gap both on lattice and from theory are becoming overwhelming. We are just wating the dust to settle down and textbooks reporting these findings.

Update: After an email exchage with Andre Sternbeck he gave further clarifications about his group work correcting something not correct in the post. I post here his corrigenda:

“Our study was for the gauge group SU(3) and not for SU(2). That is
the reason why the Beta-Value is larger than that used for SU(2) by
Cucchieri et al. and by myself et al. in 2007. The lattice spacings are
roughly of the same order, but the numerical effort spent for a 96^4
lattice in SU(3) is much bigger than what had been necessary in SU(2).”

I take this chance to thank him a lot for his comments.

Screening masses in SU(3) Yang-Mills theory

Thanks to a useful comment by Rafael Frigori (see here) I become aware of a series of beautiful papers by an Italian group at Universita’ della Calabria. I was mostly struck by a recent paper written by R. Fiore, R. Falcone, M. Gravina and A. Papa (see here) that appeared in Nuclear Physics B (see here). This paper belongs to a long series of works about the behavior of Yang-Mills theory at non-null temperature and its critical behavior. Indeed, using high-temperature expansion and Polyakov loops one arrives at the main conclusion that the ratio between the lowest and the higher state of the theory must be 3/2. This ratio depends on the universality class the theory belongs to and so, on the kind of effective theory one has in the proper temperature limit (below or above ). It should be said that, in order to get a proper verification of the above prediction, people use lattice computations. Fiore et al. use a lattice of points and, as all this kind of computations are done on lattices having such a dimension, one can cast some doubt about the fact that the true ground state of the theory is really hit. Indeed, this happens in all this kind of computations done to get a glueball spectrum that seem at odd with those giving the gluon propagator producing a lower screening mass at about 500 MeV (see my post here). A state at about 500 MeV is seen at accelerator facilities as resonance or f0(600) but is not predicted by any lattice computation. One of the reasons to reduce lattice volume is that one can reach higher values of granting the reaching of a non-perturbative regime, the one interesting for us.

What can we say about this ratio with our theory? We have put on arxiv a paper that answer this question (see here). These results were also presented at QCD 08 in Montpellier (see here). We assume that the cannot be seen at such small volumes but its excited state can be obtained. This implies that one can exchange the with the lowest state and as the higher one. Then this ratio gives exactly 3/2 as expected. We can conclude on the basis of this analysis that this ratio is the same independently on the temperature but, the one to be properly measured is given in the paper of Craig McNeile (see here) that gives close agreement between lattice and theoretical predictions.

So, we would like to see lattice computations of Yang-Mills spectra at lower lattice spacing and increased volumes granting in this way the proper value of the ground state. This is overwhelming important in view of the fact that no real understanding exists of the existence of the resonance with lattice computations. This will implies, as discussed above, a deeper understanding of the spectrum of the theory also at higher temperatures.

Emerging scenario

Reading arxiv dailys today I have found three different papers on the gluon and ghost propagators for Yang-Mills (see here, here and here). These papers prove that this line of research is very strongly alive and that there exist a lot of points to be settled down before to carry on. In this post I would like to point out several evidences that should not be forgotten when one talks about this matter. First of all there are the results of Yang-Mills theory in D=1+1. We know that, for this dimensionality, Yang-Mills theory has no dynamics. Anyhow, several people tried to solve it on the lattice or modified it to try to relate these solutions of the ones of Dyson-Schwinger equations with a given truncation. The bad news is that they find agreement with such solutions of Dyson-Schwinger equations. Why is this bad news? Because this gives, beyond any doubt, a proof that such a truncation of Dyson-Schwinger equations is fault as it removes any dynamics from Yang-Mills theory in higher dimensionality and appears to agree with numerical results just when such a dynamics does not exist. This is already a severe indicator that lattice computations done in higher dimensionality are right. What do they say us about ghost and gluon propagators?

• Gluon propagator reaches a non-null finite value at zero momenta.
• Ghost propagator is that of a free particle.
• Running coupling goes to zero at lower momenta.

This means that the confinement scenarios that are normally considered are faulty and do not work at all. These results demand for a better understanding of the physical situation at hand. It we are not ourselves convinced that they are right, we will keep on fumbling in the dark losing precious resources and time. Evidences are really heavy already at this stage and should be combined with spectra computations carried out so far. Also in this case a lot of work still must be carried out. You can read the beatiful paper of Craig McNeile about (contribution to QCD 08). It is a mistery to me why these ways are seen as different into the understanding of Yang-Mills theory.

Gluon propagator

Notwithstanding a lot of work on lattice computations, the question of the behavior of the gluon propagator at lower momenta does not seem to be settled yet. The reason for this is that there exists a lot of theoretical work, done by very good physicists, that seems blatantly in contradiction with lattice evidence. One of the pioneers of this work has been Daniel Zwanziger . He is a very smart physicist and he has done a lot of very good work on gauge theories. Just yesterday I was reading a recent paper by him on PRD. This is a beatiful paper and there is proof of the fact the the gluon propagator should have to grant confinement. The argument given by Zwanziger is the following (I copy from the paper):

“We must select the solution to these equations that corresponds
to a probability distribution that vanishes outside
the Gribov horizon. To do so, it is sufficient to impose
any property that holds for this distribution, provided only
that it determines a unique solution of the SD equations.
Besides positivity, which will be discussed in the concluding
section, there are two exact properties that hold for a probability
distribution that vanishes outside the Gribov
horizon: (i) the horizon condition and (ii) the vanishing of
the gluon propagator at .”

On a similar ground it is obtained that the ghost propagator is infrared singularly enhanced, that is, it goes to infinity faster than the free particle propagator. We see that all the conclusions in this paper rely on Gribov copies and on the fact that fixing the gauge should not be enough for a Yang-Mills field to be completely determined. Gribov’s work has been a reference point for a lot of years working in gauge theories and so it is perfectly acceptable to derive other conclusions from it.

Of course, any acceptable theoretical work must compare with experiment and agree with it. Otherwise is not physics but something else and we, as physicists, can forget it. But in nature a pure Yang-Mills theory does not exists. Gluons interact with quarks and things are not that simple to be understood and compared with theoretical work. So, another approach has been devised using large scale computations on powerful computers. People computed both the spectrum and the propagators in this way. The propagators have been obtained on very large lattices (see here). We have often commented about them and we can give a summary here

• For the gluon propagator .
• The ghost propagator is that of a free particle.

We give here the result on the largest lattice due to Cucchieri and Mendes

A. Cucchieri, T. Mendes - (27fm)^4

where it is seen immediately that the gluon propagator does not go to zero at lower momenta. But one can think that there could be something wrong on these computations even if we know that have been obtained by three different groups independently. There could be something that was not accounted for. But quite recently Axel Maas proved that things went right without really wanting this. How did he do that? He considered Yang-Mills theory in D=1+1 and showed the for this case and the ghost propagator is more singular than the free particle case (see here and here). We know as well from ‘t Hooft’s paper that this case is absolutely trivial (see here). Trivial in this case means that there is no dynamics in D=1+1! So, we recognize that a scenario where the gluon propagator goes to zero only happens when no dynamics exists. We can understand here the reasons of the failure of this scenario: People that derived this case have simply removed any dynamics from Yang-Mills theory.

Now, we can come to the question of Gribov copies. They appear to be essentially irrelevant and useless for the understanding of the behavior of a Yang-Mills theory and have induced a lot of fine people to obtain wrong conclusions. It is the very first time that I see such a situation in physics and I hope it will not end proving to be an example of something bigger going to happen.

And the ghosts disappear…

Attilio Cucchieri and Tereza Mendes are two researchers working in Brazil at the forefront of our understanding of the behavior of Yang-Mills theory in the infrared. They do computations on the lattice and presently they have got the record of the largest lattice ever used to compute the behavior of propagators at the low momenta limit. Attilio has also done a lot of theoretical work in the same field. They are married but I think this is not the most relevant information for this post. Today they posted on arxiv the third revision of one of their relevant work about ghost propagator. This is a continuation of another paper of them published on PRL about the gluon propagator (see here and here). In both papers they cited one of my works as also their lattice computations support my theoretical analysis.

From their work we now know that Cucchieri and Mendes are ghostbusters. They proved on the lattice that the ghost behaves like a free particle and we know that “free particle” means no coupling. The ghost has disappeared and the Gribov-Zwanzinger scenario faded away. These authors ask a serious question: What is now the confining mechanism? Indeed, there is another question to be answered: What do Gribov copies serve to? They do not seem to be relevant in any part of QCD and so this also is a question to be answered. We have lived with such ghosts for a lot of time and now time is come to give up to cope with them.

We expect new striking works form Attilio and Tereza about QCD now that they contributed so strongly to set the scenario.