## Mapping is confirmed by lattice computations!

16/12/2009

Rafael Frigori is a reader of this blog and I have had a lot of very interesting opinion exchanges with him here.  He belongs to a group of people in Brazil doing groundbreaking work in lattice computations of gauge theories obtaining cornerstone results. Beside him, I would also like to cite Attilio Cucchieri and Tereza Mendes that helped to improve significantly our current understanding on the way Yang-Mills theory behaves at low energies. This time Rafael has done an excellent work to show that Yang-Mills theory in d=2+1 indeed maps on a scalar field theory displaying the same mass spectrum. Actually, this is exactly the content of my mapping theorem that I used to prove that Yang-Mills theory in a strong coupling limit shows a mass gap. You can find Rafael’s paper here. Mapping theorem was firstly proposed by me here and, after Terry Tao pointed out a problem in the proof (see here), the question was finally settled here. Both these papers went published in Physics Letters B and Modern Physics Letters A respectively. The former gives the consequences of this theorem showing how the mass gap can be obtained.

Lattice computations are an essential tool today toward our understanding of quantum field theory in limits where known mathematical techniques fail. So, to see our mathematical result at work in a lattice computation is really striking and open the path toward a new set of mathematical tools to manage these theories in unexpected regimes. This can be beneficial to any area of high-energy physics ranging from string theory to phenomenology. This gives a hint of the importance of Rafael’s paper. It is like a Pandora box is started to be open!

Why is so important to map theories? The main reason to derive mapping is to reduce a complex theory to a simpler one that we are able to manage. In this case, the conclusion is that Yang-Mills theory may belong to the same universality class of the scalar field theory and the Ising model in the infrared limit. This implies that a wealth of results can be immediately taken from a theory to another. What makes the question interesting is the fact that one knows how to manage a scalar field theory in the infrared limit. In a paper I have got published in Physical Review D (see here) I was able to present such techniques deriving the propagator and the spectrum of the theory in this limit.

Having the propagator of the theory gives immediately an effective theory to do computations in the low energy limit. I have had the chance, quite recently, to be in Montpellier thanks to the invitation of Stephan Narison. Stephan organized a very beautiful workshop (see here). You can find all the talks (also mine) here. In this talk I show how computations at low energies for strong interactions can be done. This is a matter I am still working on.

I take this chance to thank Rafael very much for this paper that gives a serious evidence of the correctness of my work and, at the same time, opens up a new significant way toward our understanding of infrared physics.

## SU(2) lattice gauge theory revisited

14/12/2009

As my readers know, there are several groups around the World doing groundbreaking work in lattice gauge theories. I would like here to cite names of I. L. BogolubskyE.-M. IlgenfritzM. Müller-Preussker, and  A. Sternbeck jointly working in Russia, Germany and Australia. They have already produced a lot of meaningful papers in this area and today come out with another one worthwhile to be cited (see here). I would like to cite a couple of their results here. Firstly, they show again that the decoupling type solution in the infrared is supported. They get the following figure

The gauge is the Landau gauge. They keep the physical volume constant at 10 fm while varying the linear dimension and the coupling. This picture is really beautiful confirming an emergent understanding of the behavior of Yang-Mills theory in the infrared that we have supported since we opened up this blog. But, I think that a second important conclusion from these authors is that Gribov copies do not seem to matter. Gribov ambiguity has been a fundamental idea in building our understanding of gauge theories and now it just seems it has been a blind alley for a lot of researchers.

All this scenario is fully consistent with our works on pure Yang-Mills theory. As far as I can tell, there is no theoretical attempt to solve these equations than ours being in such agreement with lattice data (running coupling included).

I would finally point out to your attention a very good experimental paper from KLOE collaboration. This is a detector at ${\rm DA\Phi NE}$  accelerator in Frascati (Rome). They are carrying out a lot of very good work. This time they give the decay constant of the pion on energy ranging from 0.1 to 0.85 ${\rm GeV^2}$ (see here).