Gribov ambiguities in gauge theories

[tom.stoer]

Two decades ago, Gribov copies arising in the Lorentz- or Coulomb gauge were considered problematic in non-perturbative calculations, e.g. due to potential failure of cluster decomposition (are there other reasons?)

In the meantime this bug turned into a feature, as IR properties, especially confinement seems to be related to the structure of Gribov horizons.

Can anybody explain briefly the reason behing this new turn?

 

[tom.stoer]

really no idea?

 

[Drakkith]

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[tom.stoer]

what is missing? is the question not clear? silly? ...

 

[daschaich]

Outside my ken... those abstracts I've seen recently seemed less concerned with the residual, discrete gauge ambiguity in the Coulomb or Landau gauge (my understanding of "Gribov copies") than with (what seem to me to be) other aspects of Gribov's work ("Gribov's formula" or "Gribov's confinement scenario" or "the Gribov--Zwanziger framework"). But I haven't actually read beyond any of these abstracts, so I really have no idea what I'm talking about.

http://arxiv.org/abs/0807.3291
http://arxiv.org/abs/0911.0082
http://arxiv.org/abs/1001.3699
http://arxiv.org/abs/1002.2374

 

[tom.stoer]

Let's see if they are telling me something regarding my problem. Thanks anyway

 

[humanino]

It is a difficult question. You may know that Gribov had his personal theory of quark confinement.
http://arxiv.org/abs/hep-ph/9403218
http://arxiv.org/abs/hep-ph/9404332
http://arxiv.org/abs/hep-ph/9905285
http://arxiv.org/abs/hep-ph/9807224
http://arxiv.org/abs/hep-ph/0404216
I was suggested not to "waste my time" with this a couple of years ago by one the major local theoretician, who happened to be a student of Gribov. You may imagine how disappointed I was, especially considering that I had previously followed Dokgarbagezer's lectures who remains enthusiast.

As for the Gribov ambiguity itself, the initial question is hard to answer. I can also provide the following reference
http://arxiv.org/abs/hep-th/0504095v1
So yes, progress was made, but Gribov's own solution remains at large considered at least speculative, and at worse flat wrong.

 

[tom.stoer]

Thanks for the references.

I don't care about Gribov's approach towards confinement, I am only interested in the way the ambiguities are treated today. I know that especially the Coulomb gauge is much studies w.r.t. its IR properties, ghost propagators etc.; and I know that in that gauge Griboc copies arise.

So there must be some way how to deal with them non-pertutbatively. This is what I try to understand.

 

[daschaich]

But do the useful IR properties of the Coulomb gauge have anything to do with the Gribov copies? Or do the Gribov copies simply not interfere with these sorts of calculations? They might arise, but not matter... there may be no need to deal with them, non-perturbatively or otherwise.

This is the sense I get from DeGrand & Detar (page 92): "With either Coulomb or Landau gauge there is a residual, discrete gauge ambiguity discovered by Gribov (1978), which has never caused your authors trouble." They never mention it again.

 

[tom.stoer]

You must do something with these Gribov copies; you must take care not to integrate over unphysical degrees of freedom; they have to be excluded from the PI (or from the state of physical states). Some time ago one was thinking that the 1/det(M) is all you need as it confines the wave functional to regions where this det(M) is non-zero, so it seemed to be sufficient to restrict the range of integration in the PI to the region (containing A=0) where det(M) > 0. But afaik later it became clear that even this restricted region is not free from Gribov copies.

What I saw in some papers (not Gribov's) is that the existence of the Gribov horizon changes the propagators of the gauge and the ghost field in the Coulomb gauge which means that the IR is affected by the existence of a horizon. And these IR properties seem to be related to confinement.

But what about tunneling through the 1/det(M) singularity for vanishing det(M)? How do you treat these processes? They don't violate your gauge condition, but they may violate cluster decomposition or something like that. So I don't see how the PI takes care not to include unphysical effects. It becomes worse if the region containing with det(M) > 0 and A=0 contains ambiguities.

My conclusion was that one must not use gauges suffering trom the Gribov propblem at all; or that one has to restrict their use to A~0 i.e. perturbation theory. I want to understand why one can use these gauges in the IR.

 

[humanino]

I don't care about Gribov's approach towards confinement, I am only interested in the way the ambiguities are treated today.

If you even look at the references provided above, or the rest of the literature, you will quickly realize that people either ignore the problem, or essentially follow Gribov's route. This is the reason I suggested his lectures. They are not merely a model for a specific non-perturbative quantity. They qualify as a theory in the sense that they come with a full interpretation, and a consistent treatment of all observables, together with original predictions. Now, you may not care, but I doubt you will find better out there, unless you decide to publish new work on the subject. In any case, please keep us posted !

 

[Haelfix]

Gribov ambiguities I believe exist in all gauges, provided the theory is nonabelian, since the obstruction to a consistent gauge fixing procedure is topological in origin.

Of course these are irrelevant for perturbation series, so most people ignore them.

However 30 years of work, and people have still not managed to get rid of them nonperturbatively, and the million dollar question in the field is whether they are or are not important there.

Gribov and Zwanziger pushed the 'they are important' point of view about as far as you could take it, all the while doing tremendous violence to the Fadeev-Poppov procedure and cutting up the path integral into contours called Gribov horizons. Of course that nowdays its a little more subtle and advanced than simply restricting det(M) > 0, but there are still problems (like still having multiple copies per contour). On the flip side, you can get some very cool nonperturbative physics predictions out of the theory, like the softening of the gluon propagator in qgd in the infrared.

Nowdays this is all very much a subset of lattice QCD work, and I think the story is still very much an open and controversial question mark.

 

[tom.stoer]

If you even look at the references provided above, or the rest of the literature, you will quickly realize that people either ignore the problem, or essentially follow Gribov's route.

...

Now, you may not care

I will definitely study his papers.

And it is not the case that I am not interested in confinement - I am - but first I want to understand the mathematical obstacles.

 

[tom.stoer]

Gribov ambiguities I believe exist in all gauges, provided the theory is nonabelian, since the obstruction to a consistent gauge fixing procedure is topological in origin.

We studied gauge fixing in QCD in the canonical formulation about 15 years ago. Our decision was not to use the Coulomb gauge due to ghosts and gauge-fixing obstacles. The belief was that axial gauge fixing is complete (up to topological zero modes vanishing for large volume) but that no ambiguities arise. Now the message is mixed. Weinberg says that axial gauges are free from Gribov ambiguities whereas there are (afaik) rather general results saying that they must arise in all gauges. Perhaps Weinberg (as we did) confused decoupling of ghosts with absence of gauge ambiguities?

Another related question is whether the axial gauges are ill-defined as there is a different obstacle; instead of cutting fibres multiple times (gauge ambiguities) there are certain fibres which are never cut at all (so the whole sector is unrestricted).
Any ideas regarding these issues?

Another idea: could all this play a role in the electro-weak theory? perhaps this could be related to a new mechanism for Higgsless spontaneous symmetry breaking?

 

[daschaich]

Another idea: could all this play a role in the electro-weak theory? perhaps this could be related to a new mechanism for Higgsless spontaneous symmetry breaking?

How so? The only connection I can imagine off the top of my head would be to "walking" dynamics in the IR. The gross features of dynamical EWSB models should only depend on their chiral symmetry breaking patterns, and not be sensitive to potential ambiguities in the confinement mechanism. But with walking, you are concerned with the IR "phase structure". You want to be "close to" (in some scheme-independent way) an IR fixed point, without actually hitting it.

What do you have in mind?

 

[humanino]

Gribov constructed explicit models of EWSB from modifications of IR QCD dynamics.
http://arxiv.org/abs/hep-ph/9407269
http://arxiv.org/abs/hep-ph/9512352
Unfortunately, it is pretty clear that those are "toy models" and he passed away too early to improve them. They have been pretty much discarded by the community.

 

[tom.stoer]

The gross features of dynamical EWSB models should only depend on their chiral symmetry breaking patterns, and not be sensitive to potential ambiguities in the confinement mechanism.

What do you have in mind?

The Gribov horizon changes the structure of the theory in the IR, especially it changes the gluon propagator. EWSB changes the propagator of the gauge bosons as well, so there could very wellbe a relation. The difference is that IR gluons somehow don't propagate at all, whereas IR EW gauge bosons are only "slowed down".

Anyway, maybe it's nonsense.

 

[tom.stoer]

Does anybody know a good reference regarding axial gauges?

I thought that they are free of ghosts and free of gauge fixing ambiguities. But then I found a remark that "axial gauge fixing is not complete" - w/o any firther explanation.

 

[tom.stoer]

Two recent papers on arxiv discussing gauge fixing and Gribov ambiguities

http://arxiv.org/abs/1010.5718
On gauge fixing
Authors: Axel Maas
(Submitted on 27 Oct 2010)
Abstract: Gauge fixing is a useful tool to simplify calculations. It is also valuable to combine different methods, in particular lattice and continuum methods. However, beyond perturbation theory the Gribov-Singer ambiguity requires further gauge conditions for a well-defined gauge-fixing prescription. Different additional conditions can, in principle, lead to different results for gauge-dependent correlation functions, as will be discussed for the example of Landau gauge. Also the relation of lattice and continuum gauge fixing beyond perturbation theory will be briefly outlined.

http://arxiv.org/abs/1003.5410
Gribov copies and topological charge
Authors: B. Holdom
(Submitted on 29 Mar 2010 (v1), last revised 3 Nov 2010 (this version, v4))
Abstract: The existence of Gribov copies is a central feature of the field configuration space of confining gauge theories. In particular a transition between two Gribov copies with relative winding number implies a space-time configuration with topological charge. We explicitly demonstrate the proliferation of Gribov copies with relative winding number, where our focus is on localized (finite norm) configurations in Coulomb gauge. We then discuss the likelihood that some pairs of such copies are connected by Minkowski space solutions. We also comment on the relative importance of instantons and the connection to confinement.