While CERN is calming down rumors (see here), research activity on Yang-Mills theories keeps on going on. A few days ago, a paper by Axel Weber appeared on arxiv (see here). As my readers know, having discussed this at length, in these last years there has been a hot debate between the proponents of the so called “scaling solution” and the “decoupling solution” for the propagators and the running coupling of a pure Yang-Mills theory in the infrared limit. Scaling solution describes a scenario with the gluon propagator reaching zero with lowering momenta, a ghost propagator enhanced with respect to the tree level one and the running coupling reaching a finite non zero value in the same limit. Decoupling solution instead is given by a gluon propagator reaching a finite non-zero value at lower momenta, a ghost propagator behaving like the one of a free particle (tree level) and the running coupling going to zero in this limit.. It is quite easy to recognize in the decoupling solution all the chrisms of a trivial infrared fixed point for a pure Yang-Mills theory against common wisdom that pervaded the community for a lot of years. So, for some years, having lattice computations unable to tell which solution was the right one, scaling solution seemed the only one to be physically viable and almost accepted by a large part of the community.

Things started to change after the Lattice Conference in Regensburg on 2007 when some groups where able to display lattice computations on very huge volumes. The striking result was that lattice computations confirmed the decoupling solution against common wisdom. What was really shocking here is that the gluon becomes massive at the expenses of the BRST sysmmetry that seems now to acquire an even more relevant role in the understanding of Yang-Mills theory.

The idea of Axel Weber is to perform an -expansion for the Yang-Mills Lagrangian with a massive term to fix the scale. The striking result he gets is that both the scaling and the decoupling solutions are there but the former is unstable with respect to the renormalization group flow in dimensions greater than 2. So, this computation confirms again the scenario that I and other authors were able to devise.

Today, we have reached a deep understanding of the infrared physics of a Yang-Mills field theory. Scientific community is urged to take a look to the work of these people that could accelerate progress in a large body of physics research.

Axel Weber (2011). Epsilon expansion for infrared Yang-Mills theory in Landau gauge arXiv arXiv: 1112.1157v1

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

Dear Marco,

I just want to bring your attention on two papers that precede Weber work (and were not cited by him !), where the infrared propagators of the gluons and ghosts are in very nice agreement with the lattice simulation.

http://arxiv.org/abs/1004.1607

http://arxiv.org/abs/1105.2475

Cheers,

Adam

Dear Adam,

Thank you for pointing out these papers. I have known Matthieu at Ghent Conference where we had a very nice exchange of our respective points of view. This approach was heavily criticized at that conference by Arlene Aguilar. She said that hey were using a weak perturbation technique where is known that the coupling is strong, i.e. in the infrared limit.

My personal view is somewhat different from that of Arlene. Studies of her group often claim that there exists an infrared non-trivial fixed point for a pure Yang-Mills theory. This point view can be dated back to the work of Cornwall in the ’80 and, more recently, Cornwall and Papavassiliou that is the main inspiration for their work. My take is that the running coupling of a pure Yang-Mills theory reaches a trivial infrared fixed point going to zero lowering momenta, in agreement with lattice computations, and so, the ideas of Matthieu Tissier and Nicolas Wschebor can be well founded.

Cheers,

Marco

Recently, it became known that a complete solution beyond all perturbaion orders of PDEs can be found via Homotopic Analysis Method.

http://en.wikipedia.org/wiki/Homotopy_analysis_method

http://www.springerlink.com/content/k7556511j8556730/

How is it that it did not appear yet in QM / QFT calculations?

Dear Theophanes,

I am not able to see a single paper of this author so it is difficult for me to evaluate such an approach. But if he is able to produce exact classical solutions to a quartic scalar field theory and Yang-Mills theory I would be happy to compare them with mine.

Marco

That’s strange! I checked my self and prof Liao’s pages are all down. I ‘va had a long correspondence with him just before new year;s eve and he was keen to cooperate to figure out the significance of his methods for QFT of which he was unaware. He is basically a hydrodynamicist in Sgangai Naval Eng. Dept. I have kept copies of his pages, MAPLE codes and papers. I will send a link at your mail. It might be a temp. server dysfunction but the thing is that I am still waiting for his answer at my mail. You see, Shangai Univ. is also under the party’s influence and the times are “tricky” if you know what I mean so I am a little worried to see if the server will come up again

Feel free to send copies of papers of this author at my mail address that you can find on arxiv. From my experience, it is generally difficult, if not impossible, to reach colleagues in China. I have always got expired waiting time and the mail back.

Marco