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