What about ghosts?


As most of my readers know, in order to quantize Yang-Mills field one has to introduce a ghost. This result is due to Fadeev and Popov and since then, technology to work out high-energy behavior of QCD has been widely known. When you have a field you will also have a propagator as also unphysical degrees of freedom, as the ghost is, do propagate. But, in the end of your computations, all these contributions magically disappear giving a meaningful result. When you do your high energy computations you will have as a bare ghost propagator the one of a free particle but this field behaves quite strangely violating spin-statistic theorem. The main question we are concerned here is: How does ghost propagator behave in the infrared (low-energy) limit? Some researchers proposed that this propagator should go to infinity faster than that of a free particle (enhancement): This has been dubbed scaling solution . You  can read a nice paper about by Alkofer and von Smekal describing this kind of solution (see here). Lattice simulations went otherwise. In order to have an idea you can read a paper by Cucchieri and Mendes (see here) that shows, at a leading order of small momenta, that the ghost propagator, in the low-energy limit, is that of a free particle! This kind of solution has been dubbed decoupling solution.

Alike Alkofer and von Smekal, other authors thought to use Dyson-Schwinger equations to get the infrared behavior of such quantities like the ghost propagator. A French group, Boucaud,  Gomez ,  Leroy,  Le Yaouanca,  Micheli, Pene, Rodriguez-Quintero, has produced a lot of important papers showing how decoupling solution indeed comes out. I should say that I am an enthusiastic fan of this group as their results coincide perfectly with my findings. They work mostly numerically on lattice and solving Dyson-Schwinger equations also using interesting theoretical approaches. Quite recently, they put out a beautiful paper (see here) where they solve Dyson-Schwinger equation for the ghost propagator but using a smart trick to make it independent from the one of the gluon propagator. They just take the simplest hypothesis for a gluon propagator


(this is exactly the first term of my propagator!), than they show that the solution for the ghost propagator goes like a free propagator plus a logarithmic correction at higher momenta that they are able to compute.  This solution coincides quite perfectly with lattice computations. Gluon mass is seen to be around 500 MeV as it must be (that is also my case). So, a massive propagator (or a massive gluon) implies necessarily a decoupling solution as is seen on lattice computations. This conclusion is quite striking but is not enough. To have a clear idea of this finding one needs to understand what happens, with such an ansatz, to the scaling solution. This has been obtained in a paper appeared today by Rodriguez-Quintero (see here). The conclusion is again striking: A scaling solution emerges only for a critical coupling when enhancement is asked for in the ghost propagator. This, at best, means that this solution is atypical and this gives also a hint why is not seen on lattice computations for 3d and 4d. I would like to remember that the scaling solution appears in lattice computations in 2d when Yang-Mills theory is trivial and has not dynamics. It would be interesting to add similar terms to their ansatz for the gluon propagator: They should be able to recover my gluon propagator with the right spectrum to be compared with quenched lattice computations for QCD.

These results are really shocking but I should say that most has yet to be done on the way to get a complete understanding of Yang-Mills theory. Papers analyzing both scaling and decoupling solutions are fundamental to learn the relevance of such solutions and how they can come out. Presently, decoupling solution is strongly supported by lattice computations and several theoretical works, not last my papers, and I hope that future analysis could hopefully decide for the right scenario.

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.

Yang-Mills in D=1+1 strikes back


Today on arxiv I have found a very beatiful paper by Reinhardt and Schleifenbaum (see here). This paper is an important event as the authors present a full account of Yang-Mills theory in D=1+1. As we know, Axel Maas produced a lattice computation of this theory (see here) and found a perfect agreement with truncated Dyson-Schwinger equations. These results disagree completely with those obtained on lattice for D=3+1. From ‘t Hooft’s work we also know that Yang-Mills theory in D=1+1 is completely trivial having no dynamics. This means that the agreement between Maas’ lattice computations and truncated Dyson-Schiwnger equations implies that the truncation eliminates any dynamics from Yang-Mills theory and this explains the disagreement between truncated Dyson-Scwinger equations and lattice Yang-Mills in D=3+1.

In their paper Reinhardt and Schleifenbaum confirm all this but they do a smarter thing. They consider a non trivial Yang-Mills theory in D=1+1 taking a compact manifold {\sl S}^1\times {\mathbb R}. In this case they introduce a length L and this means that the “thermodynamic limit” L\rightarrow\infty should recover the trivial limit of Yang-Mills theory in D=1+1. Of course, due to this deep link between the theory on the compact manifold and the one on the real line, again this case is not representative for Yang-Mills in D=3+1 but, anyhow, can give some hints on how truncated Dyson-Schwinger equations recover these results. However, it should be emphasized that Gribov copies in D=1+1 have a prominent role and this is not generally true in D=3+1. This can yield the false impression to have caught something of the disagreement between functional methods and lattice computations. Of course, this is plainly false. In order to give an idea of what is going on they get a gluon propagator going like D\sim 1/L^2 and this goes to zero in the thermodynamic limit as no dynamics is expected in this case. In D=3+1 there is nothing like this. On a compact manifold for this case, the limit L\rightarrow\infty is absolutely not trivial. Finally, they get an infrared enhanced ghost propagator and the authors claim that the reason why  this is not seen on lattice computations for the D=3+1 case is due to Gribov copies. This conclusion cannot be accepted as the trivial limit of this theory is the D=1+1 case on the real line that has an enhanced ghost propagator too and this must not necessarily be true for D=3+1 where, as said, Gribov copies play no role. This latter fact is the reason of the failure of functional methods and also the reason why dynamics is removed by this approach. Indeed, to account for Gribov copies in D=3+1 one is forced to remove dynamics. This works for D=1+1 where no dynamics exists but fails otherwise.

A note on the running coupling should have been done by the authors. They did not do that but if the gluon propagator goes like \frac{1}{L^2}, whatever else the ghost propagator does, the thermodynamic limit grants that the coupling goes to zero. No dynamics no interaction.

Another interesting result given by the authors is the spectrum for the theory on the compact manifold. They get the spectrum of a rigid free rotator going like j(j+1). This is very nice indeed.

Finally, the conclusion by the authors that functional methods turn out to have got a strong support by their computations cannot be sustained. They just give an understanding, a deep one indeed, of the reason why these methods blatantly fail for the D=3+1 case. This is the role of computations in D=1+1 as already seen with Maas’ work.

The question of the running coupling


Today I was reading a PhD thesis about matters we frequently discuss in this blog (see here). This is a very good work. But when I have come to the question of the running coupling I was somewhat perplexed. Indeed, there is a recurring wishful thinking about running coupling in a Yang-Mills. This prejudice claims that coupling in the low momenta limit should reach a non-trivial fixed point for the theory to be meaningful. Then, if you read the literature since the inception of the success of gauge theories you will read a myriad of papers claiming this “fact” that is not a fact having been never proved.

In this case we have two kind of evidences: lattice and experimental. These evidences show that the coupling at low momenta goes to zero, that is the theory is free also in the infrared! This is a kind of counterintuitive result as are all the results that are coming out from lattice computations. The reason for this relies on the fact that Yang-Mills theory is a scalar theory in disguise and so shares the same fate. But maybe, the most interesting result comes from Giovanni Prosperi and his group at University of Milan. They studied the meson spectrum and showed how the running coupling derived from measurements bends clearly toward zero. Their work has been published on Physical Review Letters (see here and here). They do this studying quarkonium spectra, a matter we discussed extensively in this blog. Their paper has been enlarged and published on Physical Review D (see here and here).

On the lattice the question is linked to the behavior of the gluon and ghost propagators. We have seen that the gluon propagator reach a non-null constant as momentum goes to zero and the ghost propagator behaves as that of a free particle. This means that if we write


for the gluon propagator and


for the ghost propagator, being Z(p^2) and F(p^2) the dressing function, following Alkofer and von Smekal we can define a running coupling as


but the gluon propagator reaches a non-null value for p\rightarrow 0 and so Z(p^2)\sim p^2 and the ghost propagator goes like that of a free particle and so F(p^2)\sim 1. This means nothing else that \alpha(p^2)\rightarrow 0 at low momenta. This is lattice response.

So, why with all this cumulating evidence people does not yet believe it? The reason relies on the fact that is very difficult to remove prejudices and truth takes some time to emerge. We have to live with them for some time to come yet.

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 D(0)=0 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 Q(A^{tr}) 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 P(A^{tr}) that vanishes outside the Gribov
horizon: (i) the horizon condition and (ii) the vanishing of
the gluon propagator at k=0.”

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 D(0)\neq 0.
  • The ghost propagator is that of a free particle.

We give here the result on the largest lattice (27fm)^4 due to Cucchieri and Mendes

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

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 D(0)=0 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.

QCD and lattice computations


QCD in the infrared limit is generally not manageable for computations. We are not able to derive from it masses and other properties of hadrons. So, people thought to use computers to solve it in order to get exact results from it. Since the start, many difficulties were met by people working with this approach pioneered by K. G. Wilson (the Nobel prize winner for the introduction of renormalization group in statistical mechanics). The most serious ones are implied into the limitations of the resources of the computer one uses. On a lattice you have a spacing and you are interested in the continuum limit when the spacing goes to zero. But having the spacing going to zero implies more and more computational resources that are difficult to be found still today. The other question originates on how large is the volume you are using. One should be sure that small volume effects do not enter into our computations so that one is still not into the asymptotic limit is interested on. This latter problem is not so severe even if it has been advocated in computations of gluon and ghost propagators being theoretical expecations seriously at odd with those coming out from lattice.

In a comment about my analysis of quarkonia (see here) it was questioned by James Amundson at Fermilab that my potential does not seem to agree with the one emerging from lattice. People at Fermilab is doing a very good job for lattice QCD and so this comment should be taken rather seriously. Indeed, there is a point I did not emphaise in my answer. I have got an interquark potential


but I do not take \alpha_s to be a constant. Rather, it depends on the energy scale where I am doing computations and this is the key trick that does the job and I get the right answers.

But let me comment about the present situation of lattice QCD. I think that currently the most striking results are given in the following figure

This figure is saying to us that introducing quark sea the precision of computations improves dramatically. This computation was carried on by MILC, HPQCD, UKQCD and Fermilab Lattice Collaborations. In their paper they declare a spacing 1/8 fm and 1/11 fm. Quark masses are generally taken somewhat different from those of PDG as the proper ones require more computational resources. Indeed, the reached volumes are never that large for the reasons seen above. One can look at Gauge Connection to have an idea of the configurations generally involved. So, if volume effects enter in some physical quantity we cannot be aware of them. This is the situation seen on the computations of gluon and ghost propagators (see here). The situation in this case if far more simpler as there are no quarks. This is pure Yang-Mills theory and so people was able to reach volumes till (128)^4 that is a really huge volume.

But for the spectrum of pure Yang-Mills we are not that lucky as computations do not seem to hit the true ground state of the theory. Besides, we have \sigma resonance seen at accelerator facilities but not with lattice computations at any level. Indeed, we know that there is an incongruence between lattice computations of pure Yang-Mills spectra and computations of the gluon propagator (see here and my paper for QCD 08). So, where is the \sigma resonance on the lattice? Full QCD or pure Yang-Mills? In the latter case is a glueball. I think this is one of the main problem to be addressed in the very near future together with the computation of golden-plated quantities. There is too much involved in this to ignore it.

Sorry but your paper is wrong!


In our preceding posts we have largely discussed what are the results emerging from lattice about the gluon and ghost propagators and the running coupling and how functional methods, in the way they are currently adopted, fail to reach agreement with lattice computations at very large volumes. But we want to resume here what are the main conclusions that are obtained from such applications of functional methods. People working in this way fix the gluon propagator as


and for the ghost


then the claim is made that the relation k_D+2k_G=0 does hold while the functions F,H are taken to be regular as momenta go to zero. From the relation between the exponents k_D,k_G we can conclude that, excluding the trivial solution, if the gluon propagator goes to zero at lower momenta, that is k_D>0, than we must have k_G<0 that means that the ghost propagator must go to infinity at lower momenta. What they get is that the ghost propagator should go to infinity faster than a free particle. If this would be true all the confining scenarios (Zwanzinger-Gribov and Kugo-Ojima) hold true. The ghost holds a prominent role and, last but not least, a proper defined running coupling goes to a fixed point to lower momenta.

Lattice computations say that all this is blatantly wrong. Indeed, we have learned from them that

  • Gluon propagator reaches a finite non-null value at lower momenta.
  • Ghost propagator is that of a free particle and so ghosts play no role at lower momenta.
  • Running coupling is seen to approach zero at lower momenta.

From this we can easily derive our exponents as defined by people working with functional methods: k_D=1 and k_G=0 so that k_D+2k_G=1\ne 0 and no relation between exponents is seen to exist. So we have got a clear cut criterion to say when a published paper about infrared behavior of Yang-Mills theory is blatantly wrong independently on the prestige of the journal that publishes it. This happens all the times the relation k_D+2k_G=0 is assumed to hold. I can grant that there are a lot around of these wrong papers published on the highest ranked journals. If you have time and you need fun try to search for them.

I would like to say that this belongs to dynamics of science. We are presently in a transition situation about our matter, a situation similar at that happened after the discovery of the J/\psi resonance that took some time before people agreed on its nature. So, there is nothing to say to editors or referee and also to authors as mistakes are the most common facts in physics and very few people hit the right track after a wide cemetery of mistakes and wrong theories.


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