Mapping is confirmed by lattice computations!

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

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. Bogolubsky, E.-M. Ilgenfritz, M. 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  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 (see here).

Great news from CERN

In a press release, CERN informed us that beams have returned to run in the LHC (see here).  These are great news as, from now, we know that they are back on track after the severe setback happened last year.So, great physics awaits us in the next years and I take this opportunity to wish this people the best of lucks for the years to come.

Water on the Moon!

NASA announced today that LCROSS has indeed found water on the moon and in a significant quantity. The announcement is here.

The view of the floor of Cabeus as seen through the LCROSS near-infrared camera. The fresh crater made by the Centaur impact is indicated. Credit: NASA

This is a wonderful result both for the achievement and the technique NASA used. The meaning for the future explorations in space is made brighter by this finding and the reason is that, having significant reserves of water on the moon, building a base for future missions in the solar system is made easier. So, in a situation where funding cuts are the most common actions governments accomplish, such a successful mission proves that smart people always find a way toward the aims. On the other side, we hope that, when economical difficulties in US will be overcome, NASA will be again properly funded to realize the most significant goals in space missions.

Vitaly Ginzburg died

Vitaly Ginzburg,  a Russian physicist that produced a key breakthrough in the understanding of superconductivity together with Lev Landau, died yesterday. He was 93. He left an interview at Physics World quite recently. You can find it here. Ginzburg was awarded a Nobel prize in physics in 2003 along with Abrikosov and Leggett. Ginzburg-Landau equation will stand forever as a key element in condensed matter physics and partial differential equation theory.

Running coupling in QCD

For a quantum field theory it is essential to know the form of the beta function. The reason for this is that this function gives us immediately an understanding on how the theory will behave in different regimes of the energy range. Currently, there is a lot of activity to obtain the full beta function of QCD, the theory of quarks and gluons. The reason for this is that we want to understand the behavior of this theory when the energy becomes lower while we know that at higher energy asymptotic freedom sets in moving the coupling toward zero so that perturbation theory applies. QCD appears as the perfect theory. Indeed, in this case we have no Landau pole or, possibly, no triviality at lower energies and so this appears as the only example in physics of a theory that holds on the full range of energy. I have read three papers about this matter recently (see here, here and here) and all of them give a clear insight about the question of the beta function for Yang-Mills theory and QCD. As my readers know, lattice computations for a pure Yang-Mills theory show clearly that the running coupling goes to zero at lower energies so one may ask if the same can happen when quarks are involved.

Running coupling from lattice computations

From supersymmetric QCD it has been shown that the beta function should have the form

where, apart for some costants here and there, is the number of quark flavors. So, we immediately realize that supersymmetric QCD can develop a fixed point in the infrared limit even if supersymmetric Yang-Mills theory has none! Sannino and Ryttov have been inspired by this beautiful result and proposed a similar beta function for ordinary QCD and the conclusion is the same: Even if Yang-Mills theory has no fixed point in the infrared, QCD has one due to the presence of quarks. So far, lattice computations for fully QCD confirm this scenario and we can be confident that this theory is the most beautiful one being meaningful for all the energy range.

I would like to add a final comment by noting that Ryttov and Sannino give for Yang-Mills theory in the infrared the same functional form I have got in my latest paper (see here)  and that agrees with lattice results. The picture of low-energy QCD is slowly emerging providing to us quite unexpected results and a deeper comprehension of the world as we perceive it.

Higgs mechanism is essential

As the readers of my blog know, I have developed, in a series of papers, the way to manage massive solutions out of massless theories, both in classical and quantum cases. You can check my latest preprints here and here. To have an idea, if we consider an equation

then a solution is

provided

being and two arbitrary constants and a Jacobi elliptical function. We see that a massless theory has massive solutions arising just from a strong nonlinearity into the equation of motion. The question one may ask is: Does this mechanism work to give mass to particles in the Standard Model? The answer is no and this can already be seen at a classical level. To show this, let us consider the following Yukawa model

being a Yukawa coupling. Assuming very large, one is reduced to the solution of the following Dirac equation that holds at the leading order

and this equation is exactly solved in a closed form, provided the fermion has exactly the same mass of the boson, that is . So, we see that the massless fermion acquires mass too but it must be degenerate with respect to the bosonic field. This would imply that one needs a different scalar field for each fermion and such bosons would have the same masses of the fermions. This is exactly what happens in a supersymmetric theory but the theory we are considering is not. So, it would be interesting to reconsider all this with supersymmetry, surely something to do in the near future.

This means that Higgs mechanism is essential yet in the Standard Model to understand how to achieve a finite mass for all particles in the theory. We will see in the future what Nature reserved us about.

Hawking’s successor

Michael Green is the new Lucasian Professor (see here). Best congratulations to a really worthy appointment and wishes for future success to Professor Green. Green is a string theorist famous for being one of the authors of the first string revolution. He was already full professor at Cambridge University and represents a perfect choice as Hawking’s successor.

Remembering Fermi and Feynman

There are some physicists that, besides having produced exceptional results, can also give a track to follow for future generations. It goes without saying that Enrico Fermi and Richard Feynman are two of these. I am remembering them here because of a couple of quotes to be worthwhile to cite, being complementary each other in some way.

“If we are not alone, where are the others?”

“If we are alone why all this room?”

No answer is known yet to these deep questions. Meanwhile, I think it is wise to read classic papers these two authors produced. When you are tired to see how these guys changed our understanding of the World, you can always turn to “Surely you’re joking, Mr. Feynman” to get a moment of humor.

The question of X(3872)

X(3872) is a resonance observed a few years ago at Belle and Tevatron and what hit immediately physicists imagination was that it has roughly two times the mass of meson. This would imply that it could be a neat example of hadron molecule being a combination of two couples of quarks. As you may know, there is a lot of activity in QCD to understand if tetraquarks exist or not and notable physicists are involved in this quest. Several proposals emerged showing how tetraquarks can be the answer to the spectrum of light unflavored mesons. X(3872) could be a particular tetraquark state with diquarks combining with a very low binding energy (about 0.25 MeV) forming a molecule. Whatever its nature, this resonance appears quite exotic indeed. But a recent paper (see here) sheds some light about what this particle cannot be. The authors derive some bounds on the production cross section of it showing that is not plausible to consider this particle as a diquark state. They carry on simulations of production of the resonance proving that is unlikely the formation in S-wave of a molecular state. The paper appeared in this days in Physical Review Letters (see here).

The interest arisen on this particle at the time makes important this article giving a significant clarification about the direction to take to have an understanding of its very nature.