In 2010 I went to Ghent in Belgium for a very nice Conference on QCD. My contribution was accepted and I had the chance to describe my view about this matter. The result was this contribution to the proceedings. The content of this paper was really revolutionary at that time as my view about Yang-Mills theory, mass gap and the role of quarks was almost completely out of track with respect to the rest of the community. So, I am deeply grateful to the Organizers for this opportunity. The main ideas I put forward were

- Yang-Mills theory has an infrared trivial fixed point. The theory is trivial exactly as the scalar field theory is.
- Due to this, gluon propagator is well-represented by a sum of weighted Yukawa propagators.
- The theory acquires a mass gap that is just the ground state of a tower of states with the spectrum of a harmonic oscillator.
- The reason why Yang-Mills theory is trivial and QCD is not in the infrared limit is the presence of quarks. Their existence moves the theory from being trivial to asymptotic safety.

These results that I have got published on respectable journals become the reason for rejection of most of my successive papers from several referees notwithstanding there were no serious reasons motivating it. But this is routine in our activity. Indeed, what annoyed me a lot was a refeee’s report claiming that my work was incorrect because the last of my statement was incorrect: Quark existence is not a correct motivation to claim asymptotic safety, and so confinement, for QCD. Another offending point was the strong support my approach was giving to the idea of a decoupling solution as was emerging from lattice computations on extended volumes. There was a widespread idea that the gluon propagator should go to zero in a pure Yang-Mills theory to grant confinement and, if not so, an infrared non-trivial fixed point must exist.

Recently, my last point has been vindicated by a group that was instrumental in the modelling of the history of this corner of research in physics. I have seen a couple of papers on arxiv, this and this, strongly supporting my view. They are Markus Höpfer, Christian Fischer and Reinhard Alkofer. These authors work in the conformal window, this means that, for them, lightest quarks are massless and chiral symmetry is exact. Indeed, in their study quarks not even get mass dynamically. But the question they answer is somewhat different: Acquired the fact that the theory is infrared trivial (they do not state this explicitly as this is not yet recognized even if this is a “duck” indeed), how does the trivial infrared fixed point move increasing the number of quarks? The answer is in the following wonderful graph with the number of quarks (flavours):

From this picture it is evident that there exists a critical number of quarks for which the theory becomes asymptotically safe and confining. So, quarks are critical to grant confinement and Yang-Mills theory can happily be trivial. The authors took great care about all the involved approximations as they solved Dyson-Schwinger equations as usual, this is always been their main tool, with a proper truncation. From the picture it is seen that if the number of flavours is below a threshold the theory is generally trivial, so also for the number of quarks being zero. Otherwise, a non-trivial infrared fixed point is reached granting confinement. Then, the gluon propagator is seen to move from a Yukawa form to a scaling form.

This result is really exciting and moves us a significant step forward toward the understanding of confinement. By my side, I am happy that another one of my ideas gets such a substantial confirmation.

Marco Frasca (2010). Mapping theorem and Green functions in Yang-Mills theory PoS FacesQCD:039,2010 arXiv: 1011.3643v3

Markus Hopfer, Christian S. Fischer, & Reinhard Alkofer (2014). Running coupling in the conformal window of large-Nf QCD arXiv arXiv: 1405.7031v1

Markus Hopfer, Christian S. Fischer, & Reinhard Alkofer (2014). Infrared behaviour of propagators and running coupling in the conformal

window of QCD arXiv arXiv: 1405.7340v1

So, does it follow that if there had been only two generations of quarks, instead of three, that there would not be a mass gap?

No, the point in the paper of these authors is that you start with a Yukawa-like propagator and so, you have a mass gap in the Yang-Mills theory as expected and largely proved on lattice computations. But, assuming you are in the limit of exact chiral symmetry, quarks decide for the confinement as a Yukawa-like propagator is just non-confining but only screening. Then, due to the screening effect of quark, also the gluon propagator changes in the way shown in that paper. If confirmed, a propagator with a scaling behaviour is known to be confining and the question of the mass gap does not apply in strict sense for QCD as happens to Yang-Mills theory instead. Indeed, there a deep difference between a pure Yang-Mills theory and QCD in the infrared limit: The former is trivial and the latter is not and one is not expected for the same functional form for the gluon propagator.

Don’t you think these guys were not ethical for not citing you?

Hi Daniel,

This behavior is due to the fact that this people never believed what I have done. This belongs to the dynamics of our community and I don’t care too much about. Indeed, generally it is just a matter of time to see who made the wrong turn and fairness to set in.

Marco