A striking clue and some more



My colleagues participating to “The many faces of QCD” in Ghent last year keep on publishing their contributions to the proceedings. This conference produced several outstanding talks and so, it is worthwhile to tell about that here. I have already said about this here, here and here and I have spent some words about the fine paper of Oliveira, Bicudo and Silva (see here). Today I would like to tell you about an interesting line of research due to Silvio Sorella and colleagues and a striking clue supporting my results on scalar field theory originating by Axel Maas (see his blog).

Silvio is an Italian physicist that lives and works in Brazil, Rio de Janeiro, since a long time. I met him at Ghent mistaking him with Daniele Binosi. Of course, I was aware of him through his works that are an important track followed to understand the situation of low-energy Yang-Mills theory. I have already cited him in my blog both for Ghent and the Gribov obsession. He, together with David Dudal, Marcelo Guimaraes and Nele Vandersickel (our photographer in Ghent), published on arxiv a couple of contributions (see here and here). Let me explain in a few words why I consider the work of these authors really interesting. As I have said in my short history (see here), Daniel Zwanzinger made some fundamental contributions to our understanding of gauge theories. For Yang-Mills, he concluded that the gluon propagator should go to zero at very low energies. This conclusion is at odds with current lattice results. The reason for this, as I have already explained, arises from the way Gribov copies are managed. Silvio and other colleagues have shown in a series of papers how Gribov copies and massive gluons can indeed be reconciled by accounting for condensates. A gluon condensate can explain a massive gluon while retaining  all the ideas about Gribov copies and this means that they have also find a way to refine the ideas of Gribov and Zwanzinger making them agree with lattice computations. This is a relevant achievement and a serious concurrent theory to our understanding of infrared non-Abelian theories. Last but not least, in these papers they are able to show a comparison with experiments obtaining the masses  of the lightest glueballs. This is the proper approach to be followed to whoever is aimed to understand what is going on in quantum field theory for QCD. I will keep on following the works of these authors being surely a relevant way to reach our common goal: to catch the way Yang-Mills theory behaves.

A real brilliant contribution is the one of Axel Maas. Axel has been a former student of Reinhard Alkofer and Attilio Cucchieri & Tereza Mendes. I would like to remember to my readers that Axel have had the brilliant idea to check Yang-Mills theory on a two-dimensional lattice arising a lot of fuss in our community that is yet on. On a similar line, his contribution to Ghent conference is again a striking one. Axel has thought to couple a scalar field to the gluon field and study the corresponding behavior on the lattice. In these first computations, he did not consider too large lattices (I would suggest him to use CUDA…) limiting the analysis to 14^4, 20^3 and 26^2. Anyhow, also for these small volumes, he is able to conclude that the propagator of the scalar field becomes a massive one deviating from the case of the tree-level approximation. The interesting point is that he sees a mass to appear also for the case of the massless scalar field producing a groundbreaking evidence of what I proved in 2006 in my PRD paper! Besides, he shows that the renormalized mass is greater than the bare mass, again an agreement with my work. But, as also stated by the author, these are only clues due to the small volumes he uses. Anyhow, this is a clever track to be pursued and further studies are needed. It would also be interesting to have a clear idea of the fact that this mass arises directly from the dynamics of the scalar field itself rather than from its interaction with the Yang-Mills field. I give below a figure for the four dimensional case in a quenched approximation

I am sure that this image will convey the right impression to my readers as mine. A shocking result that seems to match, at a first sight, the case of the gluon propagator on the lattice (mapping theorem!). At larger volumes it would be interesting to see also the gluon propagator. I expect a lot of interesting results to come out from this approach.



Silvio P. Sorella, David Dudal, Marcelo S. Guimaraes, & Nele Vandersickel (2011). Features of the Refined Gribov-Zwanziger theory: propagators, BRST soft symmetry breaking and glueball masses arxiv arXiv: 1102.0574v1

N. Vandersickel,, D. Dudal,, & S.P. Sorella (2011). More evidence for a refined Gribov-Zwanziger action based on an effective potential approach arxiv arXiv : 1102.0866

Axel Maas (2011). Scalar-matter-gluon interaction arxiv arXiv: 1102.0901v1

Frasca, M. (2006). Strongly coupled quantum field theory Physical Review D, 73 (2) DOI: 10.1103/PhysRevD.73.027701


The Gribov obsession



I have treated the question of Yang-Mills propagators in-depth in my blog being one of my main concerns. There is an important part of the scientific community aimed to understand how these functions behave both at lower energies and overall on the whole energy range. The motivation to write down these few lines today arises from a number of interesting comments that an anonymous reader yielded to this post. If you already read it you know the main history about this matter otherwise you are urged to do so. The competitors in this arena are a pair of different solutions to the question of the propagators: The scaling solution and the decoupling solution. In the former case one expects the gluon propagator to go to zero as momenta lower and the ghost propagator should run to infinity faster than the free case. Similarly, one should have the running coupling to reach a finite value in the same limit. In the other case, the gluon propagator reaches a finite non-zero value toward zero momenta, the ghost propagator behaves as that of a free massless particle and the running coupling seems not to reach any finite value but rather bends significantly toward zero signaling a trivial infrared fixed point for Yang-Mills theory. In this post I would like to analyze the question of the genesis of the scaling solution. It arises from the Gribov obsession.

So, what is the Gribov obsession? Let us consider the case of electromagnetism. This does not give full reason to all this matter but just a hint about what is going on. The question bothering people is gauge fixing. To do computations in quantum field theory you need the gauge properly fixed and this is done in different ways. In the Lorenz gauge for example you will be able to do explicitly covariant computations but states have not all positive norm. But if you fix your gauge in the usual way, there is a residual as you can always add a solution of the wave equation for the gauge function and the physics does not change. This residual freedom is just harmless and, indeed, quantum electrodynamics is one of the most successful theories in the history of physics.

In non-Abelian gauge theories, Lorenz gauge is also called Landau gauge and the situation is well richer for residual gauge freedom that gauge fixing does not appear to be enough to grant consistent computations. This question was put forward firstly by Gribov and one has to cope with Gribov copies. Gribov copies should be renamed Gribov obsession as I did. If you want a fine description of the problem you can read this paper by Alfred Actor, appendix H or also the beautiful paper by Silvio Sorella and Rodrigo Sobreiro (see here). Now, we all know that when people is doing perturbation theory in QCD and uncover asymptotic freedom, there is no reason to worry about Gribov copies. They are simply harmless. So, the question is how much are important in the low energy (infrared) case.

This question transformed the original Gribov obsession in the obsession of many. Gribov himself proposed a solution limiting solutions to the so called first Gribov horizon as Gribov pointed out that the set of gauge orbits can be subdivided in regions with the first one having the Fadeed-Popov determinant with all positive eigenvalues and the next ones with eigenvalues becoming zero and then going to negative. In this way he was able to get a confining propagator that unfortunately is not causal. The question is then if limiting in this way the solutions of Yang-Mills theory gives again meaningful physical results. We should consider that this was a conjecture by Gribov and, while surely Gribov copies exist, it could be that imposing such a constraint is simply wrong as could be imposing any other constraint at all. One can also assert with the same right that Gribov copies can be ignored and starting to do physics from this. Now, the point is that the scaling solution arises from the Gribov obsession.

Of course, in my papers I showed (see here and refs therein), through perturbation theory, that in the deep infrared we can completely forget about Gribov copies. This is due to the appearance of an infrared trivial fixed point that makes the theory free in this limit reducing the case to the same of the ultraviolet limit. Starting perturbation theory from this point makes all the matter simply harmless. This scenario has been shown correct by lattice computations that recover the infrared fixed point and so are surely sound. The decoupling solution, now found by many researchers, is there to testify the goodness of the work researchers working with lattices and computers have done so far.

Finally, let me repeat my bet:

I bet 10 euros, or two rounds of beer at the next conference after the result is made manifestly known, that Gribov copies are not important in Yang-Mills theory at very low energies.

Nobody interested?

Actor, A. (1979). Classical solutions of SU(2) Yang—Mills theories Reviews of Modern Physics, 51 (3), 461-525 DOI: 10.1103/RevModPhys.51.461

R. F. Sobreiro, & S. P. Sorella (2005). Introduction to the Gribov Ambiguities In Euclidean Yang-Mills Theories arxiv arXiv: hep-th/0504095v1

Marco Frasca (2010). Mapping theorem and Green functions in Yang-Mills theory arxiv arXiv: 1011.3643v1

SU(2) lattice gauge theory revisited


As my readers know, there are several groups around the World doing groundbreaking work in lattice gauge theories. I would like here to cite names of I. L. BogolubskyE.-M. IlgenfritzM. Müller-Preussker, and  A. Sternbeck jointly working in Russia, Germany and Australia. They have already produced a lot of meaningful papers in this area and today come out with another one worthwhile to be cited (see here). I would like to cite a couple of their results here. Firstly, they show again that the decoupling type solution in the infrared is supported. They get the following figure

The gauge is the Landau gauge. They keep the physical volume constant at 10 fm while varying the linear dimension and the coupling. This picture is really beautiful confirming an emergent understanding of the behavior of Yang-Mills theory in the infrared that we have supported since we opened up this blog. But, I think that a second important conclusion from these authors is that Gribov copies do not seem to matter. Gribov ambiguity has been a fundamental idea in building our understanding of gauge theories and now it just seems it has been a blind alley for a lot of researchers.

All this scenario is fully consistent with our works on pure Yang-Mills theory. As far as I can tell, there is no theoretical attempt to solve these equations than ours being in such agreement with lattice data (running coupling included).

I would finally point out to your attention a very good experimental paper from KLOE collaboration. This is a detector at {\rm DA\Phi NE}  accelerator in Frascati (Rome). They are carrying out a lot of very good work. This time they give the decay constant of the pion on energy ranging from 0.1 to 0.85 {\rm GeV^2} (see here).

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.

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