We have shown the results of the lattice computations of the gluon propagator here. It would be important to get an understanding of what is going on in the low momentum limit. Indeed, this is not impossible but rather very easy. The reason for this relies on the fact that we assume that a mass gap forms in the limit of low momenta explaining the short range proper to nuclear forces. But there is more to say. This mass gap must represent an observable resonance in the spectrum of the light scalar mesons. We expect this as Yang-Mills theory is the one describing nuclear forces and if this theory is the true one, apart mixing with quarks states, its spectrum must be observed in nature.
In order to pursue our aims we consider the data coming from Attilio Cucchieri and Teresa Mendes being those for the largest volume here. If there is a mass gap the simplest form of propagator to use (in Euclidean metric) is the Yukawa one
being a constant and the mass gap entering into the fit. We obtain the following figure
for and . This result is really shocking. The reason is that a resonance with this mass has been indeed observed. This is f0(600) or and recent analysis by S. Narison, W. Ochs and others ( see here and here) and chiral perturbation theory as well (see here or here) have shown that this resonance has indeed a large gluonic content. But this is not enough.
If we assume that the computation by Cucchieri and Mendes used a QCD constant of about 440 MeV we can take the ratio with the mass gap to obtain an adimensional number of about 1.25. But for the QCD constant, that should be fixed in any computation on the lattice, there is no general agreement about and this constant should be obtained from experimental data. When one is able to solve Yang-Mills theory by analytical means this constant is an integration constant arising from the conformal invariance of the theory. The other value that is generally chosen by lattice groups is 410 MeV. This would give a mass gap of about 512 Mev that is very near the value obtained from experimental data ( e.g. see here).
So, the conclusions to be drawn from lattice computations are really striking. We have seen that the ghost behaves like a free particle, i.e. decouples from the gluon field. On the same ground it is also seen that lowering momenta makes the running coupling going to zero. Together with a fit to a Yukawa propagator we recognize here all the chrisms of a trivial theory! The same that is believed to happen to scalar field theory in the same limit in four dimensions.
Indeed, this is not all the story. The spectrum of a Yang-Mills theory is more complex and higher excitations than are expected. We will discuss this in the future and we will see how to recover the spectrum of light scalar mesons, at least for the gluonic part, without losing the property of triviality proper to the theory.