## The Saga of Landau-Gauge Propagators: A Short History

I have never discussed too much in-depth the history of the matter of Yang-Mills propagators in Landau gauge even if I often expressed a clearcut position. This is a kind of disclaimer when I say that I would not like to offend the work of anyone but my results agree excellently well with lattice computations that my point of view cannot be much different. But a recent paper on arxiv by Attilio Cucchieri and Tereza Mendes and an email exchange with Attilio motivated the idea to put down these rows to give my audience an idea of the stake we are playing for and why no peace treaty has been signed yet by people working in this area of research.

Firstly, I would like to give an idea of why this part of our scientific community is pursuing such a task to identify the gluon and ghost propagators of a pure Yang-Mills theory. The most obvious reason is to understand confinement. The idea that confinement is coded into these propagators dates back to the works of Vladimir Gribov taken to their natural extension by Daniel Zwanziger. Daniel, that I have had the luck to meet and hear in Ghent last year, did a great job in this direction and proved that, for Yang-Mills theory to be confining, the gluon propagator must go to zero with momenta. This scenario was then named Gribov-Zwanzinger arising from the contributions of these authors. It implies that positivity is maximally violated by the propagator and the real space propagator should be seen to cut the time axis. I would like to emphasize that the propagator is a gauge-dependent quantity, even if the spectrum one could obtain from it is not, and here we aim to talk about Landau-gauge propagators both for gluon and ghost fields that are generally easier to manage both on a lattice and theoretically.

The next and also relevant reason to get such propagators is to understand how Yang-Mills theory behaves at lower energies, as we know quite well its behavior at higher ones, and if a mass gap indeed forms. This could have impact on a lot of activities in high-energy physics and nuclear physics. In accelerator facilities one needs to have an exact idea of what is the background arising from QCD and this is not quite well controlled. We have seen a clear example with the charge asymmetry seen by CDF at Tevatron at 3.4 sigma. We cannot be sure this is new physics yet and if a mass gap exists at lower energies, what happens to such massive particles going to higher energies? So, my conclusion here is that we cannot live forever ignoring low energy behavior of QCD as its complete understanding could have impact at unexpected large scales.

After the work of Zwanzinger, people was motivated to get an explicit form of these propagators. Two approaches were clearly at hand. The first one is the use of large computer facilities to solve the theory on the lattice. The other is to attack the problem theoretically through a non-perturbative hierarchy of equations: Dyson-Schwinger equations. The first technique has the drawback that increasing resources are needed to approach meaningful volumes to get a proper understanding of the the theory. At the start of nineties the computing facilities today available were just a dream. On the side of Dyson-Schwinger equations the problem is mathematically very easy to state but very difficult to solve: How to truncate the hierarchy to get the proper results at lower energies? On 1997 an important paper by Lorenz von Smekal, Andreas Hauck and Reinhard Alkofer made its appearance on Physical Review Letters (see here). The authors of this paper claimed to have found a proper truncation of the Dyson-Schwinger hierarchy of equations showing that the gluon propagator should go to zero at lower momenta while the ghost propagator should go to infinity faster than the free particle case. Also the running coupling should reach a finite non-zero fixed point in the same limit. The so-called “scaling solution” was born. The importance of this paper relies on the fact that it strongly supports the Gribov-Zwanzinger scenario and so the theoretical results of these authors appeared vindicated! The idea of the scaling solution and the school built on it by von Smekal and Alkofer with a lot of students after is one of the main aspects of the history we are telling here.

After this paper, a lot of them followed on high impact journals and all the community working on Landau-gauge propagators was genuinely convinced that this indeed should have been the right behavior of propagators. Also the common wisdom that there was a fixed point at infrared was supported by these results. Indeed, people doing lattice computations seemed to confirm these findings even if the gluon propagator was never seen to converge toward zero. But this was said to depend just on the small volumes used by them with the inherent limitations of the computer facilities at that time. Anyhow, data fits seemed to agree quite well with the scaling solution. At this point, till the begin of the new century, the scaling solution become a paradigm for all the community working on the computation of propagators both theoretically and numerically.

At the beginning of the new century things started to change. People started to solve Dyson-Schwinger equations with computers and the results did not appear to agree with the scaling solution. The advantage to solve Dyson-Schwinger equations numerically is that the limitations due to the volume that were plaguing lattice computations, here are absent. It is worthwhile to cite a couple of papers (here and here) that have their culmination in a work by Joannis Papavassiliou, Daniele Binosi and Arlene Aguilar (see here). It should be noticed that the second one of these three papers went unpublished  and the first one met severe difficulties to get published. Indeed, it should be remembered that a paradigm was already formed while these papers contain completely opposite results. What was found was really shocking: the gluon propagator was seen to reach a finite non-zero value and the ghost propagator was going to infinite like that of a free particle! This could be said to be the discover of the “decoupling solution” but it is not completely true. Such a solution was obtained by John Cornwall back in the eighties and since then was waiting for a confirmation (see here). I would like to emphasize that Joannis Papavassiliou worked about this with Cornwall and this work merged in some way with that of solving numerically Dyson-Schwinger equations. The really striking part of these results was that the gluon acquires mass dynamically with a mechanism that is alike the one Schwinger devised long ago and this is an essential starting point to understand confinement.

In parallel to this research line, computers improved and more powerful ones were available during these years to attack the problem on the lattice. Increasing volumes did not seem to change the situation. The scaling solution appeared more and more distant from numerical results. The crushing event happened with the Regensburg conference, Lattice 2007. Large volume computations were finally available. Three contributions appeared by a Brazilian group (Attilio Cucchieri and Tereza Mendes), a Russian-German group (I. Bogolubsky, E.M. Ilgenfritz, M. Muller-Preussker and A. Sternbeck) and a Portuguese-German group (O. Oliveira, P. J. Silva, E.-M.Ilgenfritz, A. Sternbeck). Attilio and Tereza considered huge volumes as 27fm! There was no doubt that the solution with massive gluons, the one contradicting the initial paradigm introduced by Alkofer, von Smekal and Hauck, was the one seen on lattice at large volumes.The propagator was not going to zero but at a finite non-zero value reaching a plateau at lower energies. Also on the lattice the gluon appeared to get a mass at least in four dimensions.

These results were shocking but the question is not settled yet and a lot I would have more to say. People supporting the scaling solution is still there alive and kicking and the propagators war is still on. Nobody wants to cast armies down and surrender. This means that I will have a lot to write yet and the reasons to keep alive this blog are several. For the moment I hope to have kept your attention alive and if you have something more to add or to precise we are open to comments.

von Smekal, L., Hauck, A., & Alkofer, R. (1997). Infrared Behavior of Gluon and Ghost Propagators in Landau Gauge QCD Physical Review Letters, 79 (19), 3591-3594 DOI: 10.1103/PhysRevLett.79.3591

Aguilar, A., & Natale, A. (2004). A dynamical gluon mass solution in a coupled system of the Schwinger-Dyson equations Journal of High Energy Physics, 2004 (08), 57-57 DOI: 10.1088/1126-6708/2004/08/057

Ph. Boucaud, J. P. Leroy, A. Le Yaouanc, A. Y. Lokhov, J. Micheli, O. Pene, J. Rodriguez-Quintero, & C. Roiesnel (2005). The Infrared Behaviour of the Pure Yang-Mills Green Functions arxiv arXiv: hep-ph/0507104v4

Aguilar, A., Binosi, D., & Papavassiliou, J. (2008). Gluon and ghost propagators in the Landau gauge: Deriving lattice results from Schwinger-Dyson equations Physical Review D, 78 (2) DOI: 10.1103/PhysRevD.78.025010

Cornwall, J. (1982). Dynamical mass generation in continuum quantum chromodynamics Physical Review D, 26 (6), 1453-1478 DOI: 10.1103/PhysRevD.26.1453

Attilio Cucchieri, & Tereza Mendes (2007). What’s up with IR gluon and ghost propagators in Landau gauge? A
puzzling answer from huge lattices PoSLAT2007:297,2007 arXiv: 0710.0412v1

I. L. Bogolubsky, E. -M. Ilgenfritz, M. Müller-Preussker, & A. Sternbeck (2007). The Landau gauge gluon and ghost propagators in 4D SU(3) gluodynamics in
large lattice volumes PoSLAT2007:290,2007 arXiv: 0710.1968v2

O. Oliveira, P. J. Silva, E. -M. Ilgenfritz, & A. Sternbeck (2007). The gluon propagator from large asymmetric lattices PoSLAT2007:323,2007 arXiv: 0710.1424v1

### 16 Responses to The Saga of Landau-Gauge Propagators: A Short History

1. abc says:

Hi Marco,
Interesting post!
What is the current argument from the proponents of the scaling solution? I think you had written about it a while ago https://marcofrasca.wordpress.com/2008/12/04/a-shocking-step/. It has something to do with the BRST symmetry (not present on the lattice?). Can you expand it to get it clearer for the beginning students like me?
Cheers,
abc

• mfrasca says:

Dear abc,

Indeed I was somewhat hard in that post with people supporting the scaling solution. The question of broken BRST invariance is one of the arguments these people uses.

BRST invariance arises when you introduce the ghost field into the Yang-Mills functional to quantize it in a consistent manner. In the infrared all this becomes just trivial when one maps solutions of a massless scalar field to the Yang-Mills field. So, at the leading order one has a trivial fixed point and a Gaussian functional. As the ghost field decouples at this order, BRST invariance does not apply. The next-to-leading order produces the needed corrections to recover BRST perturbatively in a series in the inverse of the gauge coupling. In this sense, if we are able to sum up all this strong coupling series we will recover exact BRST invariance of the full action. So, my view is that this should not be considered a real concern.

Another question is related to Slavnov-Taylor identities. These are the generalization to non-Abelian theories of the Ward identities that are normally used in QED. It is not clear how the massive solution should grant these identities to be satisfied.

Besides, there is no clear understanding of Gribov copies and their fate. The question is that all the matter about the scaling solution arose from taking into account the problem of Gribov copies and Gribov understanding of the way Yang-Mills theory should be managed due to their presence. The decoupling solution is there to show that all the arguments based on that understanding by Gribov seem to fail in a way or another and Gribov copies are not important both for infrared and ultraviolet behavior of the theory. Note that this was difficult to tell through lattice computations as their effect seems to decrease as the volume increases and propagators turn into the decoupling solution ones.

Another question is related to the two-dimensional case. Two-dimensional world is quite strange. E.g. living beings cannot exist because a digestive apparatus is impossible in this case: We will be divided in two parts without any connection! Another strange thing about this case is that gluons do not exist but strong interactions mediated by a two-dimensional Coulomb like potential do. You should also know that gravity too is quite strange here because you need forcefully a cosmological constant to avoid triviality again. But, notwithstanding all these strange behaviors, Axel Mass decided to see what was going on for such a weird space on the lattice (see here). The great surprise was that the scaling solution was seen for the first time in a lattice computations. But this is a bad surprise for the reasons I said before: No dynamics here so the scaling solution at higher dimensions also means no dynamics. No dynamics at higher dimensions implies that, in the truncation scheme used for the Dyson-Schwinger equations, there is something incorrect that removed the interesting parts of the dynamics of the fields. But going beyond this, one would like to understand the appearing of such a scaling in two dimensions. This could help also to understand why this solution is never seen on the lattice at higher dimensions or, else, how to reproduce it in some way. Certainly, from my point of view, I get my mapping theorem not to hold for this case as, in two dimensions, the scalar field has dynamics while the Yang-Mills field has not.

Cheers,

Marco

2. abc says:

Hi Marco,
Thanks for your response! I read your response and then again read the arXiv:0812.0654 which you had mentioned in your blog a while ago. It now made more sense to me. I think the issue that they talk about is somewhat more subtle (at least from what I understood from just reading this paper): in the continuum, the Landau gauge fixing is done by solving $\partial_{\mu}A^{g}_{\mu} = 0$ where subscript g is for the gauge-transforms. Whereas on the lattice, they minimize a function whose first derivatives with respect to the gauge transforms (the lattice version, obviously) are the lattice version of the above Landau gauge equations. In other words, on the lattice they just consider a class of solutions (minima of the function). On the other hand, if they actually solve the lattice version of the $\partial_{\mu}A^{g}_{\mu}=0$ (I don’t know how though! But let’s say it is possible somehow) then the different solutions (Gribov copies) of these equations gives a zero in the end making the observables giving 0 over 0 – basically yielding that there is no BRST symmetry on the lattice! In short, if you do minimization, then you are restricted only in a class of solutions and if you follow the same procedure on the lattice as in the continuum, then you mess up the whole simulations. In other words, what they are saying is that it is not straightforward to compare the lattice and continuum scenarios since you are doing different things. I think in the end the lattice way of gauge-fixing may well be equivalent, but I don’t know if anyone has shown this rigorously. I am not a lattice person and so don’t know the current status on this. But perhaps lattice people may have thought about it already. Do you know about it?
Please let me know if I have understood anything incorrectly.
Cheers,
abc

3. mfrasca says:

Dear abc,

I am not a lattice expert and so it is not easy to arrange an answer to this question. von Smekal by his side has done extensive work on the lattice and so this criticism can be meaningful. Anyhow, there are some considerations that could make such an argument harmless and I take this chance to put forward.

This argument is strongly linked to the idea of Gribov copies. As I said in my precedent comment, these do not appear to be important in the infrared if we consider our current theoretical (not lattice) understanding. But von Smekal is one of the proponents of the scaling solution and so he keeps on taking them seriously. My view is that lattice computations appear to correctly reproduce those theoretical findings that neglect Gribov copies. The existence of a trivial infrared fixed point makes all this matter consistent. You can check my post on the running coupling for this.

Now assuming that one wants to use von Smekal’s argument to claim that the scaling solution is not seen because people doing lattice computations are not seeing the true continuum limit, here we met a couple of problems at least. Firstly, I would like to understand then why the scaling solution is seen in the trivial case of two dimensional lattice computation. Using von Smekal’s point of view this worsens the situation for the reasons I have already expressed above.

Last but not least, I would like to understand why Lattice QCD reproduces at 1% precision the spectrum and other observables on a side and we cannot trust it on the other.

My conservative view is that the way people on the lattice manage the gauge is correct. It is possible that we miss a rigorous proof for the moment about the equivalence but this means nothing. At this stage some expert view would be helpful.

Cheers,

Marco

• abc says:

Hi Marco,
Interesting!
I am not a lattice expert so I shouldn’t comment on it too much either. But just a few other things which I briefly understand from the continuum. Gribov copies are there in the continuum too. But on the lattice, there are additional Gribov copies coming as lattice artifacts. It is these copies, combined with the usual continuum counterparts create the hurdle to construct lattice BRST. Right? So you may be right in that Gribov copies may not be important in the theoretical understanding (I don’t know much about it, but I am taking your words granted for now), but on the lattice the Gribov copies (ok, the lattice artifacts if you like) seem to play crucial role as mentioned above.

You are right. It would be interesting and important to know why the 2D simulations on the lattice did show the scaling solution. But this gets too technical and only the concerned people may be able to answer.

However, on your second reasoning about the spectrum on the lattice: there is no doubt that lattice simulations are giving quite impressively accurate results. But the specific problem we are talking about here is different: the observables for the physical spectrum computed on the lattice are still gauge independent. So you fix gauge or not, they are unaffected – at least they must be unaffected up to some numerics may be. But the gauge propagators are strictly gauge dependent. So only there this gauge fixing issue that the scaling solution people are raising come into the picture. Again, this issue doesn’t make any effect on the general lattice simulations that are impressively precise these days. Or am I misunderstanding your comment?
Note that I am not proponent/opponent of either of the solutions. On the contrary, I am too naive to have any opinion on that ); I already have lots of confusions in the courses I am taking! But I am just objectively trying to understand both sides of the arguments.
Also, please don’t get me wrong here, but I think it is better for us to restraint ourselves to mention specific names in arguing against or even in favour. We can just get away with addressing the scientific paradigms or school of thoughts etc., can’t we? I otherwise think that you are doing a great job in discussing these topics since there is no one else talking about it on the blogosphere.

Cheers,

abs

• mfrasca says:

Dear abc,

The question about Gribov copies on the lattice, as you put it, can be reduced and made even nearer to the continuum case increasing volumes. Do you think that (27fm)^4 is a volume good enough to make the problem negligible? Stated otherwise, what should be the proper critical volume to consider this question harmless or, at least, very similar to the continuum case? I think we have to agree on a point: If I make the lattice spacing going to zero I have to recover the continuum limit otherwise you are right and people doing lattice computations are computing something different. This must be true also for BRST invariance that should be fully recovered in the limit of the spacing going to zero. If we do not agree on this point there is very few progress to do in the discussion.

On your second point, I strongly disagree on your view. The reason is that I believe that in a gauge dependent function as is the gluon propagator is coded a fundamental gauge-independent observable: The spectrum of the theory. I have asked, also in Ghent, to my friends doing lattice computations to check if the glueball spectrum can emerge out from their numbers. The surprise was that Orlando Oliveira already tried it! You can read his talk at Ghent here. I always said here and to them that this is a sanity check for their computations and, indeed, just look at the numbers Orlando gets out from the gluon propagator and then take a look to the current literature about. I think that further pursuing this road will close any criticism one can think of about what this people is doing.

Finally, a line for disclaimer. I have great estimation and respect for the people pursuing this research, independently from their views about what should be the truth. Of course, I have my personal point view arising from my researches in this field but I hope that this will not be a reason to be offended as I never had any intention to attack on a personal ground anybody.

Cheers,

Marco

• abc says:

Hi Marco,
Yes, I agree that 27fm^4 is not a small size and we should at least hope to get the continuum limit. However, shall we also agree that this is the continuum limit of the gauge-fixing procedure that is done on the lattice: at this volume, we approach the continuum version of the gauge-fixing function (whose first derivatives with respect to gauge transforms are the Landau gauge fixing equations) and that for each gauge-orbit there is precisely one minimum is taken (it may not be the global minimum since it is ‘almost’ impossible to get the global minimum of this kind of nonlinear function)?
If you agree to that, then it means that we should compare the continuum version of the prapagators that are computed with this ‘minimization gauge fixing’ in the continuum (I don’t know what to call it!). And we couldn’t, at least as they are given, compare the continuum propagators computed with solving $\partial_{\mu} A_{\mu} = 0$. Am I right?
If you are ok with this so far, then to summarize it, what we could compare the lattice data with in the continuum is: the continuum propagators computed with the ‘minimization gauge fixing’ where one (or probably more, but not all) copies per gauge-orbit obtained from the first Gribov region (and usually, NOT from the fundamental modular region).

On the second point, I don’t have much knowledge about the glueball spectrum. I thought you were talking about the particle spectrum on the lattice. And so now I don’t want to make any stupid comments here on my own);

Cheers,
abc

4. mfrasca says:

Dear abc,

Let me state your point, please correct if I am wrong, about lattice computations of the propagators for Yang-Mills theory just for my readers that could not be too much aware about. Now, I choose just pure gauge computations to make things more dramatic.

The question is to get the propagators of a pure Yang-Mills theory in the Landau gauge. The point you raise is that the gauge fixing condition is introduced on the lattice through a minimization condition but this condition is not granted to get a global minimum, being this generally too involved to compute, and, as for each gauge orbit one has a different minimum, what we have at the end of the computation is not just the true propagators but the propagators proper to the particular result of this minimization condition. Of course, this could be a problem for a gauge dependent quantity as a propagator is as we are not able in this way to take properly into account Gribov copies. The fundamental modular region that could not be hit in this way is essential to recover the scaling solution that would have been proved to be the right one.

Of course, assuming this is true, you should admit that the decoupling solution happens more often than the scaling one that has not been seen so far in the lattice except for the trivial two-dimensional case (and let me emphasize this latter point). This makes me suspect that all this matter is just grasping at straws. But I want to do one step more:

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.

About the question of the spectrum, talking about glueballs or quark bound states does not change things too much. But surely the spectrum of the theory is gauge-invariant and independent from the gauge-fixing procedure. So, if people doing computation of the propagators on the lattice will be able to get back the right spectrum they will have also proved the soundness of all their approach. So, there is no need to find a global minimum at all.

Cheers,

Marco

• abc says:

Hi Marco,
Thanks for putting the things in a better way. Although these are not my arguments, but the arguments from the scaling solution community); The decoupling solution seems indeed a right solution. Btw, do you know if there is any restriction that only one type of the solutions can exist? Can’t both be there? (Of course, the scaling one is not found yet in a generic case, I know I know!) But may be everybody is right in the end making it a win-win situation for all);
Cheers,
abc

• mfrasca says:

Dear abc,

Your part as devil’s advocate was so perfect that I felt like the Exorcist. 🙂

Jokes aside, I do not know how the scaling solution could enter at the end of the game. Surely, one would like to know the reasons why it comes out from Dyson-Schwinger equations and some researchers are working on this (e.g. Boucaud and his group in France). Time will say.

Thank you for these interesting comments that help my readers to better understand the situation.

Cheers,

Marco

• abc says:

Hi Marco,
Thanks for your kind comments. However, trust me, it was me who learned the most out of the arguments, than anyone else!
Cheers,
abc

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