What about ghosts?

As most of my readers know, in order to quantize Yang-Mills field one has to introduce a ghost. This result is due to Fadeev and Popov and since then, technology to work out high-energy behavior of QCD has been widely known. When you have a field you will also have a propagator as also unphysical degrees of freedom, as the ghost is, do propagate. But, in the end of your computations, all these contributions magically disappear giving a meaningful result. When you do your high energy computations you will have as a bare ghost propagator the one of a free particle but this field behaves quite strangely violating spin-statistic theorem. The main question we are concerned here is: How does ghost propagator behave in the infrared (low-energy) limit? Some researchers proposed that this propagator should go to infinity faster than that of a free particle (enhancement): This has been dubbed scaling solution . You  can read a nice paper about by Alkofer and von Smekal describing this kind of solution (see here). Lattice simulations went otherwise. In order to have an idea you can read a paper by Cucchieri and Mendes (see here) that shows, at a leading order of small momenta, that the ghost propagator, in the low-energy limit, is that of a free particle! This kind of solution has been dubbed decoupling solution.

Alike Alkofer and von Smekal, other authors thought to use Dyson-Schwinger equations to get the infrared behavior of such quantities like the ghost propagator. A French group, Boucaud,  Gomez ,  Leroy,  Le Yaouanca,  Micheli, Pene, Rodriguez-Quintero, has produced a lot of important papers showing how decoupling solution indeed comes out. I should say that I am an enthusiastic fan of this group as their results coincide perfectly with my findings. They work mostly numerically on lattice and solving Dyson-Schwinger equations also using interesting theoretical approaches. Quite recently, they put out a beautiful paper (see here) where they solve Dyson-Schwinger equation for the ghost propagator but using a smart trick to make it independent from the one of the gluon propagator. They just take the simplest hypothesis for a gluon propagator

G(p)=\frac{B}{p^2+M^2}

(this is exactly the first term of my propagator!), than they show that the solution for the ghost propagator goes like a free propagator plus a logarithmic correction at higher momenta that they are able to compute.  This solution coincides quite perfectly with lattice computations. Gluon mass is seen to be around 500 MeV as it must be (that is also my case). So, a massive propagator (or a massive gluon) implies necessarily a decoupling solution as is seen on lattice computations. This conclusion is quite striking but is not enough. To have a clear idea of this finding one needs to understand what happens, with such an ansatz, to the scaling solution. This has been obtained in a paper appeared today by Rodriguez-Quintero (see here). The conclusion is again striking: A scaling solution emerges only for a critical coupling when enhancement is asked for in the ghost propagator. This, at best, means that this solution is atypical and this gives also a hint why is not seen on lattice computations for 3d and 4d. I would like to remember that the scaling solution appears in lattice computations in 2d when Yang-Mills theory is trivial and has not dynamics. It would be interesting to add similar terms to their ansatz for the gluon propagator: They should be able to recover my gluon propagator with the right spectrum to be compared with quenched lattice computations for QCD.

These results are really shocking but I should say that most has yet to be done on the way to get a complete understanding of Yang-Mills theory. Papers analyzing both scaling and decoupling solutions are fundamental to learn the relevance of such solutions and how they can come out. Presently, decoupling solution is strongly supported by lattice computations and several theoretical works, not last my papers, and I hope that future analysis could hopefully decide for the right scenario.

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