Media all around the World are spreading the news. A defective apparatus at CERN caused so much ado. Back to Einstein again…

## Dust is finally settling…

10/02/2012The situation about Yang-Mills theory is finally settling down. I do not mean that mathematicians’ community has finally decided the winner of the Millenium prize but rather that people working on the study of two-point functions on a pure Yang-Mills theory have finally a complete scenario for it. These studies have seen very hot debates and breakthrough moments with the use of important computing resources at different facilities. I have tried to sum up this very beautiful piece of history of physical science here. Just today a paper by Attilio Cucchieri, David Dudal and Nele Vandersickel is appeared on arXiv making clear a fundamental aspect of this scenario. Attilio is a principal figure in the Brazilian group that carried out fundamental results in this area of research and was instrumental in the breakthrough at Regensburg 2007. David and Nele were essential into the realization of Ghent conference on 2010 and their work, as we will see in a moment, displays interesting results that could be important for a theoretical understanding of Yang-Mills theory.

The question of the Green functions for Yang-Mills theory can be recounted in two very different views about their behavior at very low energies. Understanding the behavior of these functions in this energy limit could play an essential role to understand confinement, one of the key problems of physics today. Of course, propagators depend on the gauge choice and so, when we talk of them here we just mean in the Landau gauge. But they also code some information that does not depend on the gauge at all as the mass spectrum. So, If one wants to know if the gluon becomes massive and how big is that mass then, she should turn her attention to these functions. But also, if I want to do QCD at very low energies I need these functions to be able to do computations, something that theoretical physicists are not able to perform precisely yet missing this piece of information.

In the ’90, the work performed by several people seemed to convince everyone that the gluon propagator should go to zero lowering momenta and the ghost propagator should run to infinity faster than the case of a free particle. Difficulties with computational resources impeded to achieve the right volume dimensions to draw clearcut conclusions about, working on the lattice. But another solution was emerging, with a lot of difficulties and while a paradigm seemed to be already imposed, proving that the gluon propagator should reach a finite non-null limit at zero momenta and the ghost propagator was behaving like a free particle. A massive gluon propagator was already proposed in the ’80 by John Cornwall and this idea was finally gaining interest. After Regensburg 2007, this latter solution finally come into play as lattice results on huge volumes were showing unequivocally that the massive solution was the right one. The previous solution was then called “scaling solution” while the massive one was dubbed “decoupling solution”.

A striking result obtained by Axel Maas (see here) showed that, in two dimensions, the propagators agree with the scaling solution. This is quite different from the three and four dimensional case where the massive solution is seen instead. This problem was a main concern for people working on the lattice as a theoretical understanding was clearly in need here. Attilio asked to me if I could come out with an explanation with my approach. I have found a possible answer here but this was not the answer Attilio was looking for. With this paper he has found the answer by himself.

The idea is the following. In order to understand the behavior of the propagators in different dimensions one has to solve the set of coupled Dyson-Schwinger equations for the ghost and gluon propagators as one depends on the other. In this paper they concentrate just on the equation for the ghost propagator and try to understand, in agreement with the no-pole idea of Gribov that the ghost propagator must have no poles, when its solution is consistent. This is a generalization of an idea due to Boucaud, Gómez, Leroy, Yaouanc, Micheli, Pène and Rodríguez-Quintero (see here): Consider the equation of the ghost propagator and compute it fixing a form for the gluon propagator, then see when the solution is physically consistent. In their work, Boucaud et al. fix the gluon propagator to be Yukawa-like, a typical massive propagator for a free particle. Here I was already happy because this is fully consistent with my scenario (see here): I have a propagator being the sum of Yukawa-like propagators typical of a trivial infrared fixed point where the theory becomes free. Attilio, David and Nele apply this technique to a propagator devised by Silvio Paolo Sorella, David Dudal, John Gracey, Nele Vandersickel and Henry Verschelde that funded the so-called “Refined Gribov-Zwanziger” scenario (see here). The propagator they get can be simply rewritten as the sum of three Yukawa propagators and so, it is fully consistent with my results. Attilio, David and Nele use it to analyze the behavior of the ghost propagator and to understand its behavior at different dimensions, using Gribov no-pole condition. Their results are indeed striking. They recover a critical coupling at which the scaling solution works in 2 and 3 dimensions: Only when the coupling has this particular value the scaling solution can apply but this is not the real case. Also, as Attilio, David and Nele remeber us, this critical point is unstable as recently showed by Axel Weber (see here). This agrees with the preceding finding by Boucaud et al. but extends the conclusions to different dimensions. In two dimensions a strange thing happen: There is a logarithmic singularity at one-loop for the ghost propagator that can only be removed taking the gluon propagator going to zero and to make the Gribov no-pole condition hold. This is indeed a beautiful physical explanation and gives an idea on what is going on by changing dimensions to these propagators. I would like to emphasize that also the refined Gribov-Zwanziger scenario agrees perfectly well with my idea of a trivial infrared fixed point that is also confirmed by lattice data, having the gluon propagator the sum of Yukawa propagators. I think we can merge our results at some stage fixing the parameters.

Given all this clear view that is finally emerged, maybe it is time to turn to phenomenology. There is a lot of people, for example there at CERN, waiting for fully working models of low-energy QCD. All the people I cited here and a lot more I would like to name have given the answer.

Attilio Cucchieri, David Dudal, & Nele Vandersickel (2012). The No-Pole Condition in Landau gauge: Properties of the Gribov Ghost

Form-Factor and a Constraint on the 2d Gluon Propagator arXiv arXiv: 1202.1912v1

Axel Maas (2007). Two- and three-point Green’s functions in two-dimensional Landau-gauge Yang-Mills theory Phys.Rev.D75:116004,2007 arXiv: 0704.0722v2

Boucaud, P., Gómez, M., Leroy, J., Le Yaouanc, A., Micheli, J., Pène, O., & Rodríguez-Quintero, J. (2010). Low-momentum ghost dressing function and the gluon mass Physical Review D, 82 (5) DOI: 10.1103/PhysRevD.82.054007

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

David Dudal, John Gracey, Silvio Paolo Sorella, Nele Vandersickel, & Henri Verschelde (2008). A refinement of the Gribov-Zwanziger approach in the Landau gauge: infrared propagators in harmony with the lattice results Phys.Rev.D78:065047,2008 arXiv: 0806.4348v2

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

## Numerical evidence for the square root of a Wiener process

02/02/2012Brownian motion is a very kind mathematical object being very keen to numerical simulations. There are a plenty of them for any platform and software so that one is able to check very rapidly the proper working of a given hypothesis. For these aims, I have found very helpful the demonstration site by Wolfram and specifically this program by Andrzej Kozlowski. Andrzej gives the code to simulate Brownian motion and compute Itō integral to verify Itō lemma. This was a very good chance to check my theorems recently given here by some numerical work. So, I have written a simple code on Matlab that I give here (rename from .doc to .m to use with Matlab).

Here is a sample of output:

As you could note, the agreement is almost perfect. I have had to rescale with a multiplicative factor as the square root appears somewhat magnified after the square but the pattern is there. You can do checks by yourselves. So, all my equations are perfectly defined as is a possible square root of a Wiener process.

Of course, improvements, advices or criticisms are very welcome.

**Update**: I have simplified the code and added a fixed scale factor to make identical scale. The code is available at Simulation. Here is an example of output:

Marco Frasca (2012). Quantum mechanics is the square root of a stochastic process arXiv arXiv: 1201.5091v2