Yang-Mills theory with the related question of the mass gap appears today an unsolved problem and, from a mathematical standpoint, the community did not recognized anybody to claim the prize so far. But in physics the answer to this question has made enormous progress mostly by the use of lattice computations and, quite recently, with the support of theoretical analysis. Contrarily to common wisdom, the most fruitful attack to this problem is using Green functions. The reason why this was not a greatly appreciated approach relies on the fact that Green functions are gauge dependent. Anyhow, they contain physical information that is gauge independent and this is exactly what we are looking for: The mass gap.

In order to arrive to such a conclusion a lot of work has been needed since ’80 and the main reason was that at the very start of these studies computational resources were not enough to arrive to a deep infrared region. So, initially, the scenario people supported was not the right one and some conviction arose that the gluon propagator could not say too much about the question of the mass gap. There was no Källen-Lehman representation to help and rather, the propagator seemed to not behave as a massive one but theoretical analysis pointed to a gluon propagator going to zero lowering momenta. This is the now dubbed scaling solution.

Running coupling from the lattice

In the first years of this decade things changed dramatically both due to increase of computational power and by a better theoretical understanding. As pointed out by Axel Weber (see here and here), three papers unveiled what is now called the decoupling solution (see here, here and here). The first two papers were solving Dyson-Schwinger equations by numerical methods while the latter is a theoretical paper solving Yang-Mills equations. Decoupling solution is in agreement with lattice results that in those years started to come out with more powerful computational resources. At larger lattices the gluon propagator reaches a finite non-zero value, the ghost propagator is the one of a free massless particle and the running coupling bends toward zero aiming to a trivial infrared fixed point (see here, here and here). Axel Weber, in his work, shows that the decoupling solution is the only stable one with respect a renormalization group flow.

Gluon propagators for SU(2) from the lattice

These are accepted facts in the physical community so that several papers are now coming out using them. The one I have seen today is from Kenji Fukushima and Kouji Kashiwa (see here). In this case, given the fact that the decoupling solution is the right one, these authors study the data for non-zero temperature and discuss the Polyakov loop for this case. Fukushima is very well-known for his works in QCD at finite temprature.

We can claim, without any possible confutation, that in physics the behavior of a pure Yang-Mills theory is very clear now. Of course, we can miss much of the rigor that is needed in mathematics and this is the reason why no proclamation is heard yet.

Axel Weber (2011). Epsilon expansion for infrared Yang-Mills theory in Landau gauge arXiv arXiv: 1112.1157v2

A. C. Aguilar, & A. A. Natale (2004). A dynamical gluon mass solution in a coupled system of the

Schwinger-Dyson equations JHEP0408:057,2004 arXiv: hep-ph/0408254v2

Ph. Boucaud, Th. Brüntjen, J. P. Leroy, A. Le Yaouanc, A. Y. Lokhov, J. Micheli, O. Pène, & J. Rodríguez-Quintero (2006). Is the QCD ghost dressing function finite at zero momentum ? JHEP 0606:001,2006 arXiv: hep-ph/0604056v1

Marco Frasca (2007). Infrared Gluon and Ghost Propagators Phys.Lett.B670:73-77,2008 arXiv: 0709.2042v6

Attilio Cucchieri, & Tereza Mendes (2007). What’s up with IR gluon and ghost propagators in Landau gauge? A

puzzling answer from huge lattices PoS LAT2007:297,2007 arXiv: 0710.0412v1

I. L. Bogolubsky, E. -M. Ilgenfritz, M. Müller-Preussker, & A. Sternbeck (2007). The Landau gauge gluon and ghost propagators in 4D SU(3) gluodynamics in large lattice volumes PoSLAT2007:290,2007 arXiv: 0710.1968v2

O. Oliveira, P. J. Silva, E. -M. Ilgenfritz, & A. Sternbeck (2007). The gluon propagator from large asymmetric lattices PoSLAT2007:323,2007 arXiv: 0710.1424v1

Kenji Fukushima, & Kouji Kashiwa (2012). Polyakov loop and QCD thermodynamics from the gluon and ghost propagators arXiv arXiv: 1206.0685v1

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