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