## Gluon propagator

Notwithstanding a lot of work on lattice computations, the question of the behavior of the gluon propagator at lower momenta does not seem to be settled yet. The reason for this is that there exists a lot of theoretical work, done by very good physicists, that seems blatantly in contradiction with lattice evidence. One of the pioneers of this work has been Daniel Zwanziger . He is a very smart physicist and he has done a lot of very good work on gauge theories. Just yesterday I was reading a recent paper by him on PRD. This is a beatiful paper and there is proof of the fact the the gluon propagator should have $D(0)=0$ to grant confinement. The argument given by Zwanziger is the following (I copy from the paper):

“We must select the solution to these equations that corresponds
to a probability distribution $Q(A^{tr})$ that vanishes outside
the Gribov horizon. To do so, it is sufficient to impose
any property that holds for this distribution, provided only
that it determines a unique solution of the SD equations.
Besides positivity, which will be discussed in the concluding
section, there are two exact properties that hold for a probability
distribution $P(A^{tr})$ that vanishes outside the Gribov
horizon: (i) the horizon condition and (ii) the vanishing of
the gluon propagator at $k=0$.”

On a similar ground it is obtained that the ghost propagator is infrared singularly enhanced, that is, it goes to infinity faster than the free particle propagator. We see that all the conclusions in this paper rely on Gribov copies and on the fact that fixing the gauge should not be enough for a Yang-Mills field to be completely determined. Gribov’s work has been a reference point for a lot of years working in gauge theories and so it is perfectly acceptable to derive other conclusions from it.

Of course, any acceptable theoretical work must compare with experiment and agree with it. Otherwise is not physics but something else and we, as physicists, can forget it. But in nature a pure Yang-Mills theory does not exists. Gluons interact with quarks and things are not that simple to be understood and compared with theoretical work. So, another approach has been devised using large scale computations on powerful computers. People computed both the spectrum and the propagators in this way. The propagators have been obtained on very large lattices (see here). We have often commented about them and we can give a summary here

• For the gluon propagator $D(0)\neq 0$.
• The ghost propagator is that of a free particle.

We give here the result on the largest lattice $(27fm)^4$ due to Cucchieri and Mendes

A. Cucchieri, T. Mendes - (27fm)^4

where it is seen immediately that the gluon propagator does not go to zero at lower momenta. But one can think that there could be something wrong on these computations even if we know that have been obtained by three different groups independently. There could be something that was not accounted for. But quite recently Axel Maas proved that things went right without really wanting this. How did he do that? He considered Yang-Mills theory in D=1+1 and showed the for this case $D(0)=0$ and the ghost propagator is more singular than the free particle case (see here and here). We know as well from ‘t Hooft’s paper that this case is absolutely trivial (see here). Trivial in this case means that there is no dynamics in D=1+1! So, we recognize that a scenario where the gluon propagator goes to zero only happens when no dynamics exists. We can understand here the reasons of the failure of this scenario: People that derived this case have simply removed any dynamics from Yang-Mills theory.

Now, we can come to the question of Gribov copies. They appear to be essentially irrelevant and useless for the understanding of the behavior of a Yang-Mills theory and have induced a lot of fine people to obtain wrong conclusions. It is the very first time that I see such a situation in physics and I hope it will not end proving to be an example of something bigger going to happen.