Confinement revisited

27/09/2012

ResearchBlogging.org

Today it is appeared a definitive updated version of my paper on confinement (see here). I wrote this paper last year after a question put out to me by Owe Philipsen at Bari. The point is, given a decoupling solution for the gluon propagator in the Landau gauge, how does confinement come out? I would like to remember that a decoupling solution at small momenta for the gluon propagator is given by a function reaching a finite non-zero value at zero. All the fits carried out so far using lattice data show that a sum of few Yukawa-like propagators gives an accurate representation of these data. To see an example see this paper. Sometime, this kind of propagator formula is dubbed Stingl-Gribov formula and has the property to have a fourth order polynomial in momenta at denominator and a second order one at the numerator. This was firstly postulated by Manfred Stingl on 1995 (see here). It is important to note that, given the presence of a fourth power of momenta, confinement is granted as a linear rising potential can be obtained in agreement with lattice evidence. This is also in agreement with the area law firstly put forward by Kenneth Wilson.

At that time I was convinced that a decoupling solution was enough and so I pursued my analysis arriving at the (wrong) conclusion, in a first version of the paper, that screening could be enough. So, strong force should have to saturate and that, maybe, moving to higher distances such a saturation would have been seen also on the lattice. This is not true as I know today and I learned this from a beautiful paper by Vicente Vento, Pedro González and Vincent Mathieu. They thought to solve Dyson-Schwinger equations in the deep infrared to obtain the interquark potential. The decoupling solution appears at a one-gluon exchange level and, with this approximation, they prove that the potential they get is just a screening one, in close agreement with mine and any other decoupling solution given in a close analytical form. So, the decoupling solution does not seem to agree with lattice evidence that shows a linearly rising potential, perfectly confining and in agreement with what Wilson pointed out in his classical work on 1974. My initial analysis about this problem was incorrect and Owe Philipsen was right to point out this difficulty in my approach.

This question never abandoned my mind and, with the opportunity to go to Montpellier this year to give a talk (see here), I presented for the first time a solution to this problem. The point is that one needs a fourth order term in the denominator of the propagator. This can happen if we would be able to get higher order corrections to the simplest one-gluon exchange approximation (see here). In my approach I can get loop corrections to the gluon propagator. The next-to-leading one is a two-loop term that gives rise to the right term in the denominator of the propagator. Besides, I am able to get the renormalization constant to the field and so, I also get a running mass and coupling. I gave an idea of the way this computation should be performed at Montpellier but in these days I completed it.

The result has been a shocking one. Not only one gets the linear rising potential but the string tension is proportional to the one obtained in d= 2+1 by V. Parameswaran Nair, Dimitra Karabali and Alexandr Yelnikov (see here)! This means that, apart from numerical factors and accounting for physical dimensions, the equation for the string tension in 3 and 4 dimensions is the same. But we would like to note that the result given by Nair, Karabali and Yelnikov is in close agreement with lattice data. In 3 dimensions the string tension is a pure number and can be computed explicitly on the lattice. So, we are supporting each other with our conclusions.

These results are really important as they give a strong support to the ideas emerging in these years about the behavior of the propagators of a Yang-Mills theory at low energies. We are even more near to a clear understanding of confinement and the way mass emerges at macroscopic level. It is important to point out that the string tension in a Yang-Mills theory is one of the parameters that any serious theoretical approach, pretending to go beyond a simple phenomenological one,  should be able to catch. We can say that the challenge is open.

Marco Frasca (2011). Beyond one-gluon exchange in the infrared limit of Yang-Mills theory arXiv arXiv: 1110.2297v4

Kenneth G. Wilson (1974). Confinement of quarks Phys. Rev. D 10, 2445–2459 (1974) DOI: 10.1103/PhysRevD.10.2445

Attilio Cucchieri, David Dudal, Tereza Mendes, & Nele Vandersickel (2011). Modeling the Gluon Propagator in Landau Gauge: Lattice Estimates of Pole Masses and Dimension-Two Condensates arXiv arXiv: 1111.2327v1

M. Stingl (1995). A Systematic Extended Iterative Solution for QCD Z.Phys. A353 (1996) 423-445 arXiv: hep-th/9502157v3

P. Gonzalez, V. Mathieu, & V. Vento (2011). Heavy meson interquark potential Physical Review D, 84, 114008 arXiv: 1108.2347v2

Marco Frasca (2012). Low energy limit of QCD and the emerging of confinement arXiv arXiv: 1208.3756v2

Dimitra Karabali, V. P. Nair, & Alexandr Yelnikov (2009). The Hamiltonian Approach to Yang-Mills (2+1): An Expansion Scheme and Corrections to String Tension Nucl.Phys.B824:387-414,2010 arXiv: 0906.0783v1

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Running coupling and Yang-Mills theory

30/07/2012

ResearchBlogging.org

Forefront research, during its natural evolution, produces some potential cornerstones that, at the end of the game, can prove to be plainly wrong. When one of these cornerstones happens to form, even if no sound confirmation at hand is available, it can make life of researchers really hard. It can be hard time to get papers published when an opposite thesis is supported. All this without any certainty of this cornerstone being a truth. You can ask to all people that at the beginning proposed the now dubbed “decoupling solution” for propagators of Yang-Mills theory in the Landau gauge and all of them will tell you how difficult was to get their papers go through in the peer-review system. The solution that at that moment was generally believed the right one, the now dubbed “scaling solution”, convinced a large part of the community that it was the one of choice. All this without any strong support from experiment, lattice or a rigorous mathematical derivation. This kind of behavior is quite old in a scientific community and never changed since the very beginning of science. Generally, if one is lucky enough things go straight and scientific truth is rapidly acquired otherwise this behavior produces delays and impediments for respectable researchers and a serious difficulty to get an understanding of the solution of  a fundamental question.

Maybe, the most famous case of this kind of behavior was with the discovery by Tsung-Dao Lee and Chen-Ning Yang of parity violation in weak interactions on 1956. At that time, it was generally believed that parity should have been an untouchable principle of physics. Who believed so was proven wrong shortly after Lee and Yang’s paper. For the propagators in the Landau gauge in a Yang-Mills theory, recent lattice computations to huge volumes showed that the scaling solution never appears at dimensions greater than two. Rather, the right scenario seems to be provided by the decoupling solution. In this scenario, the gluon propagator is a Yukawa-like propagator in deep infrared or a sum of them. There is a very compelling reason to have such a kind of propagators in a strongly coupled regime and the reason is that the low energy limit recovers a Nambu-Jona-Lasinio model that provides a very fine description of strong interactions at lower energies.

From a physical standpoint, what does it mean a Yukawa or a sum of Yukawa propagators? This has a dramatic meaning for the running coupling: The theory is just trivial in the infrared limit. The decoupling solution just says this as emerged from lattice computations (see here)

What really matters here is the way one defines the running coupling in the deep infrared. This definition must be consistent. Indeed, one can think of a different definition (see here) working things out using instantons and one see the following

One can see that, independently from the definition, the coupling runs to zero in the deep infrared marking the property of a trivial theory. This idea appears currently difficult to digest by the community as a conventional wisdom formed that Yang-Mills theory should have a non-trivial fixed point in the infrared limit. There is no evidence whatsoever for this and Nature does not provide any example of pure Yang-Mills theory that appears always interacting with Fermions instead. Lattice data say the contrary as we have seen but a general belief  is enough to make hard the life of researchers trying to pursue such a view. It is interesting to note that some theoretical frameworks need a non-trivial infrared fixed point for Yang-Mills theory otherwise they will crumble down.

But from a theoretical standpoint, what is the right approach to derive the behavior of the running coupling for a Yang-Mills theory? The answer is quite straightforward: Any consistent theoretical framework for Yang-Mills theory should be able to get the beta function in the deep infrared. From beta function one has immediately the right behavior of the running coupling. But in order to get it, one should be able to work out the Callan-Symanzik equation for the gluon propagator. So far, this is explicitly given in my papers (see here and refs. therein) as I am able to obtain the behavior of the mass gap as a function of the coupling. The relation between the mass gap and the coupling produces the scaling of the beta function in the Callan-Symanzik equation. Any serious attempt to understand Yang-Mills theory in the low-energy limit should provide this connection. Otherwise it is not mathematics but just heuristic with a lot of parameters to be fixed.

The final consideration after this discussion is that conventional wisdom in science should be always challenged when no sound foundations are given for it to hold. In a review process, as an editorial practice, referees should be asked to check this before to kill good works on shaky grounds.

I. L. Bogolubsky, E. -M. Ilgenfritz, M. Müller-Preussker, & A. Sternbeck (2009). Lattice gluodynamics computation of Landau-gauge Green’s functions in the deep infrared Phys.Lett.B676:69-73,2009 arXiv: 0901.0736v3

Ph. Boucaud, F. De Soto, A. Le Yaouanc, J. P. Leroy, J. Micheli, H. Moutarde, O. Pène, & J. Rodríguez-Quintero (2002). The strong coupling constant at small momentum as an instanton detector JHEP 0304:005,2003 arXiv: hep-ph/0212192v1

Marco Frasca (2010). Mapping theorem and Green functions in Yang-Mills theory PoS FacesQCD:039,2010 arXiv: 1011.3643v3


Millenium prize on Yang-Mills theory: The situation in physics

05/06/2012

ResearchBlogging.org

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


Dust is finally settling…

10/02/2012

ResearchBlogging.org

The 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


A new year full of promises

03/01/2012

ResearchBlogging.org

We have left 2011 with a lot of exciting results from experiments. Neutrinos appear to move a bit faster than expected and Higgs provided some glimpses at CERN. Of course, this kind of Higgs appears somewhat boring at first being in the range of what Standard Model expected. But it is really too early to say something for sure. We expect definite answer for the next summer with a lot more data analyzed by people at CERN.

With the new year, I would like to point out to my readers a couple of nice papers that are really worthwhile reading. About CUDA and lattice QCD, my Portuguese friends, Pedro Bicudo and Nuno Cardoso,  made a relevant step beyond and made available their code for working for a generic SU(N) gauge group (see here, their code is here). As I have some time I will try their code. The work of these people is excellent and making their code worldwide available is really helpful for all our community.

Finally, Axel Maas put forward a revision of his very good review paper (see here). Axel gave important contributions to the current understanding of Yang-Mills theory and his paper yields a lucid description of these ideas that rely on a large effort on lattice computations and functional methods. Often, I complain about the fact that the community at large seems to not consider these lines of research reliable yet to work with. This is not true as the results they were able to get give since now sound results to work with and the most important of these are that Yang-Mill theory has indeed a mass gap and that this theory appears to display a running coupling reaching zero lowering momenta, a completely unexpected result that goes against common wisdom but this is just what lattice put out.

So, let me wish to you a great 2012 and I hope to share with you the excitement physics research is promising.

Nuno Cardoso, & Pedro Bicudo (2011). Generating SU(Nc) pure gauge lattice QCD configurations on GPUs with
CUDA and OpenMP arXiv arXiv: 1112.4533v1

Axel Maas (2011). Describing gauge bosons at zero and finite temperature arXiv arXiv: 1106.3942v2


Yang-Mills scenario: Yet a confirmation

12/12/2011

ResearchBlogging.org

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 \epsilon-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


Yang-Mills mass gap scenario: Further confirmations

28/11/2011

ResearchBlogging.org

Alexander (Sasha) Migdal was a former professor at Princeton University. But since 1996, he is acting as a CEO of a small company. You can read his story from that link. Instead, Marco Bochicchio was a former colleague student of mine at University of Rome “La Sapienza”. He was a couple of years ahead of me. Now, he is a researcher at Istituto Nazionale di Fisica Nucleare, the same of OPERA and a lot of other striking contributions to physics. With Marco we shared a course on statistical mechanics held by Francesco Guerra at the department of mathematics of our university. Today, Marco posted a paper of him on arXiv (see here). I am following these works by Marco with a lot of interest because they contain results that I am convinced are correct, in the sense that are describing the right scenario for Yang-Mills theory. Marco, in this latter work, is referring to preceding publications from Sasha Migdal about the same matter that go back till ’70s! You can find a recollection of these ideas in a recent paper by Sasha (see here). So, what are these authors saying? Using somewhat different approaches than mine (that you can find well depicted here), they all agree that a Yang-Mills theory has a propagator going like

G(p)=\sum_{n=0}^\infty\frac{Z_n}{p^2-m_n^2+i\epsilon}

being Z_n some numbers and m_n is given by the zeros of some Bessel functions. This last result seems quite different from mine that I get explicitly m_n=(n+1/2)m_0 but this is not so because, in the asymptotic regime, J_k(x)\propto \cos(x-k\pi/2-\pi/4)/\sqrt{x} and zeros for the cosine go like (n+1/2)\pi and then, my spectrum is easily recovered in the right limit. The right limit is properly identified by Sasha Migdal from Padè approximants for the propagator that start from the deep Euclidean region \Lambda\rightarrow\infty, being \Lambda an arbitrary energy scale entering into the spectrum. So, the agreement between the scenario proposed by these authors and mine is practically perfect, notwithstanding different mathematical approaches are used.

The beauty of these conclusions is that such a scenario for a Yang-Mills theory is completely unexpected but it is what is needed to grant confinement. So, the conclusion about the questions of mass gap and confinement is approaching. As usual, we hope that the community will face these matters as soon as possible making them an important part of our fundamental knowledge.

Marco Bochicchio (2011). Glueballs propagators in large-N YM arXiv arXiv: 1111.6073v1

Alexander Migdal (2011). Meromorphization of Large N QFT arXiv arXiv: 1109.1623v2

Marco Frasca (2010). Mapping theorem and Green functions in Yang-Mills theory PoS FacesQCD:039,2010 arXiv: 1011.3643v3


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