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

Hi Marco,

Thanks for putting a whole entry just for the Gribov copies/obsession and thanks for mentioning that our conversation was your motivation for this!

I have been reading about this stuff more and I understand the arguments from the scaling solutions community somewhat better. The argument seems pretty clear now: the BRST construction is quite crucial in the continuum for the Gribov-Zwanziger/Kugo-Ojima scenarios. However, on the lattice, due the Gribov copies cancelling each other exactly, it is NOT possible to restore the BRST symmetry on the lattice. This was first observed by Neuberger already in 80’s. In short, with the current gauge-fixing procedure, there is no BRST on the lattice in all the lattice simulations that have been performed so far. Again, Gribov copies/obsession may or may not be important in general (I don’t want to take your bet on it because I don’t know the evidences or don’t have intuition for that like you do.), but they are ‘important’ in the sense they give rise to the Neuberger problem on the lattice – call them the lattice artifacts if you like.

In other words, it is quite possible that the decoupling solution that people are observing does not need BRST !?! And the scaling solution may still be the BRST-friendly solution and could be observed if there is any prescription for the lattice BRST! You think so? In that case, we still need to give credit to the people who came up with the decoupling solution in that they observed another kind of solution which didn’t need BRST symmetry, and hence was overlooked by the scaling solution community!

Cheers,

abc

Dear abc,

Thank you again for your comment. My site is open to everybody and aims to describe the work our community is doing, independently on people’s view about what should be the truth. I have had fruitful mail exchanges with practically all groups working in this field and whoever is willing to contribute here can do it. I can also accept posts that I will publish without any modification or comment. Just these should be properly signed. I will provide a short bio agreed with the author.

Once I have said this, I would like to turn to the question of BRST. At this point it would be very helpful a view from people working on the lattice as I am convinced that the supporters of the scaling solution already did this, I mean they spoke each other. I guess that an answer could have been provided but I am not aware of this.

The question about the scaling solution is that relies heavily on the idea of Gribov copies that I think has been misleading for a lot of reasons. The decoupling solution, at the leading order, is just saying that BRST is only the identity and is trivial. This is so because the ghost field is decoupled (do you remember the ghost propagator going like that of a free particle?) and the theory appears trivial in the deep infrared (did you recall the running coupling as obtained by Sternbeck and others?). So, this scenario is quite clear and, surely, in the deep infrared the lattice solution is also consistent with BRST but in a trivial way.

Things may be different increasing momenta and we all know that the scaling solution appeared to be correct until the far infrared limit was not reached in 2007 on the lattice. I can think that indeed a scaling could be observed in the intermediate regime but otherwise this solution fails to give a complete overview of all the scenario. But this is the fate of all the theoretical works I have seen so far and mine are not an exception (but I am doing perturbation theory and I can compute corrections). Of course, Cornwall’s case is somewhat different but here there are works in progress yet.

So, I can agree about the missing BRST on the lattice but so why the scaling solution seemed to agree fairly well in the intermediate range of momenta when volumes where so small that Gribov copies where really important?

Cheers,

Marco

Hi Marco,

Thanks for the comments. There can be many reasons to be anonymous and mine is that I am a student soon to be out in the job-market. I don’t want even to sound offending/defending any of the scenario openly, even though I don’t even think of doing that for any of the counter-parts – as community or individual. I hope you would understand this as a valid reason.

Your question about the intermediate momenta is interesting. I don’t know why it was taken as the scaling solution till 2007 (my impression so far was that the diverging propagator – though not properly going to zero – was mistakenly ‘considered’ as diverging, with a blame on the finite volume effects for not really touching the zero. But probably, there must be some more thoughtful reason for that). In any case, I don’t know the answer for this.

Cheers,

abc

[…] 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 […]

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