Some time ago, while I was just at the beginning of my current understanding of low-energy Yang-Mills theory, I wrote to Christian Fischer to know if from the scaling solution, the one with the gluon propagator going to zero lowering momenta and the ghost propagator running to infinity faster than the free particle in the same limit, a mass gap could be derived. Christian has always been very kind to answer my requests for clarification and did the same also for this so particular question telling to me that this indeed was not possible. This is a rather disappointing truth as we are accustomed with the idea that short ranged forces need some kind of massive carriers. But physics taught that a first intuition could be wrong and so I decided not to take this as an argument against the scaling solution. Since today.

Looking at arxiv, I follow with a lot of interest the works of the group of people collaborating with Philippe Boucaud. They are supporting the decoupling solution as this is coming out from their numerical computations through the Dyson-Schwinger equations. A person working with them, Jose Rodríguez-Quintero, is producing several interesting results in this direction and the most recent ones appear really striking (see here and here). The question Jose is asking is when and how does a scaling solution appear in solving the Dyson-Schwinger equations? I would like to remember that this kind of solution was found with a truncation technique from these equations and so it is really important to understand better its emerging. Jose solves the equations with a method recently devised by Joannis Papavassiliou and Daniele Binosi (see here) to get a sensible truncation of the Dyson-Schwinger hierarchy of equations. What is different in Jose’s approach is to try an ansatz with a massive propagator (this just means Yukawa-like) and to see under what conditions a scaling solution can emerge. A quite shocking result is that there exists a critical value of the strong coupling that can produce it but at the price to have the Schwinger-Dyson equations no more converging toward a consistent solution with a massive propagator and the scaling solution representing just an unattainable limiting case. So, scaling solution implies no mass gap as already Christian told me a few years ago.

The point is that now we have a lot of evidence that the massive solution is the right one and there is no physical reason whatsoever to presume that the scaling solution should be the true solution at the critical scaling found by Jose. So, all this mounting evidence is there to say that the old idea of Hideki Yukawa is working yet: Massive carriers imply limited range forces.

J. Rodríguez-Quintero (2011). The scaling infrared DSE solution as a critical end-point for the family

of decoupling ones arxiv arXiv: 1103.0904v1

J. Rodríguez-Quintero (2010). On the massive gluon propagator, the PT-BFM scheme and the low-momentum

behaviour of decoupling and scaling DSE solutions JHEP 1101:105,2011 arXiv: 1005.4598v2

Daniele Binosi, & Joannis Papavassiliou (2007). Gauge-invariant truncation scheme for the Schwinger-Dyson equations of

QCD Phys.Rev.D77:061702,2008 arXiv: 0712.2707v1