We cannot see the light

An interesting paper appeared today in arxiv by Alkofer,  Huber and Schwenzer (see here). Reinhard Alkofer and Lorenz von Smekal are the proponents of an infrared solution of Yang-Mills theory in D=4 having the following properties

• Gluon propagator goes to zero at lower momenta
• Ghost propagator goes to infinity at lower momenta faster than the free propagator
• Running coupling reaches a fixed point at lower momenta

and this scenario disagrees with lattice evidence in D=4 but agrees with lattice in D=2 when the theory is trivial having no dynamics. After some years that other researchers were claiming that a different solution can be obtained by the same equations, that is Dyson-Schwinger equations, that indeed agrees with lattice computations, Alkofer’s group accepted this fact but with a lot of skepticism pointing out that this solution has several difficulties, last but not least it breaks BRST symmetry. The solution proposed by Alkofer and von Smekal by its side gives no mass gap whatsoever and no low energy spectrum to be compared neither with lattice nor with experiments to understand the current light unflavored meson spectrum. So, whoever is right we are in a damned situation that no meaningful computations can be carried out to get some real physical understanding. The new paper is again on this line with the authors proposing a perturbation approach to evaluate the vertexes of the theory in the infrared and obtaining again comforting agreement with their scenario.

I will avoid to enter into this neverending controversy about Dyson-Schwinger equations but rather I would ask a more fundamental question: Is it worthwhile an approach that only grants at best saving a phylosophical understanding of confinement without any real understanding of QCD? My view is that one should start from lattice data and try to understand the real mathematical form of the gluon propagator. Why does it resemble the Yukawa form so well? A Yukawa form grants a mass gap and this is elementary quantum field theory. This I would like to see explained. When a method is not satisfactory something must be changed. It is evident that solving Dyson-Schwinger equations requires some new mathematical approach as old views are just confusing this kind of research.

A quite effective QCD theory

As far as my path toward understanding of QCD is concerned, I have found a quite interesting effective theory to work with that is somewhat similar to Yukawa theory. Hideki Yukawa turns out to be more in depth in his hindsight than expected.

Indeed, I have already showed as the potential in infrared Yang-Mills theory is an infinite sum of weighted Yukawa potentials with the range, at each order, decided through a mass formula for glueballs that can be written down as

being the string tension, an experimental parameter generally taken to be , and is an elliptic integral, just a number.

The most intriguing aspect of all this treatment is that an effective infrared QCD can be obtained through a scalar field. I am about to finish a paper with a calculation of the width of the resonance, a critical parameter for our understanding of low energy QCD. Here I put the generating functional if someone is interested in doing a similar calculation (time is rescaled as )

As always, is the free Dirac propagator for the given quark and is the gluon propagator that I have discussed in depth in my preceding posts. People seriously interested about this matter should read my works (here and here).

For a physical understanding of this you have to wait my next posting on arxiv. Anyhow, anybody can spend some time to manage this theory to exploit its working and its fallacies. My hope is that, anytime I post such information on my blog, I help the community to have an anticipation of possible new ways to see an old problem with a lot of prejudices well grounded.

An inspiring paper

These days I am closed at home due to the effects of flu. When such bad symptoms started to relax I was able to think about physics again.  So, reading the daily from arxiv today I have uncovered a truly inspiring paper from Antal Jakovac a and Daniel Nogradi (see here). This paper treats a very interesting problem about quark-gluon plasma. This state was observed at RHIC at Brookhaven. Successful hydrodynamical models permit to obtain values of physical quantities, like shear viscosity, that could be in principle computed from QCD. The importance of shear viscosity relies on the existence of an important prediction from AdS/CFT symmetry claiming that the ratio between this quantity and entropy density can be at least . If this lower bound would be proved true we will get an important experimental verification for AdS/CFT conjecture.

Jakovac and Nogradi exploit the computation of this ratio for SU(N) Yang-Mills theory. Their approach is quite successful as their able to show that the value they obtain is still consistent with the lower bound as they have serious difficulties to evaluate the error. But what really matters here is the procedure these authors adopt to reach their aim making this a quite simple alley to pursuit when the solution of Yang-Mills theory in infrared is acquired. The central point is again the gluon propagator. These authors assume simply the very existence of a mass gap taking for the propagator something like in Euclidean time. Of course, is the glueball mass. This is a too simplified assumption as we know that the gluon propagator is somewhat more complicated and a full spectrum of glueballs does exist that can contribute to this computation (see my post and my paper).

So, I spent my day to extend the computations of these authors to a more realistic gluon propagator.  Indeed, with my gluon propagator there is no need of one-loop computations as the identity at 0-loop does not hold true anymore for a non-trivial spectrum and one has immediately an expression for the shear viscosity. I hope to give some more results in the near future.

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.

A formula I was looking for

As usual I put in this blog some useful formulas to work out computations in quantum field theory. My aim in these days is to compute the width of the resonance. This is a major aim in QCD as the nature of this particle is hotly debated. Some authors think that it is a tetraquark or molecular state while others as Narison, Ochs, Minkowski and Mennessier point out the gluonic nature of this resonance. We have expressed our view in some posts (see here and here) and our results heavily show that this resonance is a glueball in agreement with the spectrum we have found for a pure Yang-Mills theory.

Our next step is to understand the role of this resonance in QCD. Indeed, we have shown in our recent paper (see here) that, once the gluon propagator is known, it is possible to derive a Nambu-Jona-Lasinio model from QCD with all parameters properly fixed. We have obtained the following:

being the gluon propagator with all indexes in color and space-time already saturated. This in turn means that we can use the following formula (see my paper here and here):

being again the gluon propagator for SU(N) and a normalization factor. This formula does hold only for infrared limit, that is when the theory is strongly coupled. We plan to extract physical results from this formula and define in this way the role of resonance.

What makes the proton spin?

There is currently a beautiful puzzle to be answered that relies on sound and beautiful experimental results. The question is how the components of a proton, that is quarks and gluons, concur to determine the value one half for the spin of the particle. During the conference QCD 08 at Montpellier I listened to a beatiful presentation of Joerg Pretz of the COMPASS Collaboration (see here and here). Hearing these results was stunning for me. I explain the reasons in a few words. The spin of the proton should be composed by the spin of the quarks, the contributions of gluons (gluons???) and orbital angular momentum. What happens is that the spin of quarks does not contribute too much. People then thought that the contribution of gluons (gluons again???) should have been decisive. The COMPASS Collaboration realized a beautiful experiment using charmed mesons. This experiment has been described by Pretz at QCD 08. They proved in a striking way that the contribution of the glue to proton spin can be zero and cannot be used to account for the particle spin. Of course, there are beautiful papers around that are able to explain how the proton spin comes out. I have found for example a paper by Thomas and Myhrer at Jefferson Lab (see here and here) that describes quite well an understanding of the puzzle and surely is worthwhile reading. But my question is another: Why the glue  does not contribute?

From our preceding posts one should have reached immediately an answer, the same that come out to my mind when I listened Pretz’s talk. The reason is that, in the infrared, gluons that have spin one are not the true carriers of the strong force. The true carriers have no spin unless higher excited states are considered. This explains why COMPASS experiment did not see any contribution consistently with previous expectations.

This is again a strong support to our description of the gluon propagator (see here). No other theory around shows this.

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 . In this case they introduce a length and this means that the “thermodynamic limit” 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 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 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 , 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 . 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 and 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 and so and the ghost propagator goes like that of a free particle and so . This means nothing else that 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 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.

Pion mass

We have seen in preceding posts how good was the computation of ground states of quarkonia obtaining the interquark potential from the gluon propagator and solving the Dirac equation (see here). Here we try a more ambitious aim: We compute the pion mass from the interquark potential in the limit of very light quarks but assuming them to be not relativistic that is a drastic assumption. So, the interquark potential is given by (see here)

but in case of light quarks we can take just the first term, that is

and so our problem reduces to the one of solving the Schroedinger equation with the Yukawa potential. This is a well-known problem. To get the ground state we have used this paper by A.E.S. Green. Then, our final formula is

being MeV the constituent mass quark, the strong coupling constant, with the string tension, , and two constants of the energy level computation from Green’s paper. So, finally we get the satisfactory value MeV in good agreement with experimental value taking into account of how rough was our computation.

We cannot claim this as a full success but rather as a simple exercise showing how knowing the proper gluon propagator can give a serious hint on computation of all the relevant quantities in QCD and this has been the main aim of such analysis.