Paper with a proof of confinement has been accepted


Recently, I wrote a paper together with Masud Chaichian (see here) containing a mathematical proof of confinement of a non-Abelian gauge theory based on Kugo-Ojima criterion. This paper underwent an extended review by several colleagues well before its submission. One of them has been Taichiro Kugo, one of the discoverers of the confinement criterion, that helped a lot to improve the paper and clarify some points. Then, after a review round of about two months, the paper has been accepted in Physics Letters B, one of the most important journals in particle physics.

This paper contains the exact beta function of a Yang-Mills theory. This confirms that confinement arises by the combination of the running coupling and the propagator. This idea was around in some papers in these latter years. It emerged as soon as people realized that the propagator by itself was not enough to grant confinement, after extended studies on the lattice.

It is interesting to point out that confinement is rooted in the BRST invariance and asymptotic freedom. The Kugo-Ojima confinement criterion permits to close the argument in a rigorous way yielding the exact beta funtion of the theory.


Dust is finally settling…


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

No scaling solution with massive gluons


Some time ago, while I was just at the beginning of my current understanding of low-energy Yang-Mills theory, I wrote to Christian Fischer to know if from the scaling solution, the one with the gluon propagator going to zero lowering momenta and the ghost propagator running to infinity faster than the free particle in the same limit,  a mass gap could be derived. Christian has always been very kind to answer my requests for clarification and did the same also for this so particular question telling to me that this indeed was not possible. This is a rather disappointing truth as we are accustomed with the idea that short ranged forces need some kind of massive carriers. But physics taught that a first intuition could be wrong and so I decided not to take this as an argument against the scaling solution. Since today.

Looking at arxiv, I follow with a lot of interest the works of the group of people collaborating with Philippe Boucaud.   They are supporting the decoupling solution as this is coming out from their numerical computations through the Dyson-Schwinger equations. A person working with them, Jose Rodríguez-Quintero, is producing several interesting results in this direction and the most recent ones appear really striking (see here and here). The question Jose is asking is when and how does a scaling solution appear in solving the Dyson-Schwinger equations? I would like to remember that this kind of solution was found with a truncation technique from these equations and so it is really important to understand better its emerging. Jose solves the equations with a method recently devised by Joannis Papavassiliou and Daniele Binosi (see here) to get a sensible truncation of the Dyson-Schwinger hierarchy of equations. What is different in Jose’s approach is to try an ansatz with a massive propagator (this just means Yukawa-like) and to see under what conditions a scaling solution can emerge. A quite shocking result is that there exists a critical value of the strong coupling that can produce it but at the price to have the Schwinger-Dyson equations no more converging toward a consistent solution with a massive propagator and the scaling solution representing just an unattainable limiting case. So, scaling solution implies no mass gap as already Christian told me a few years ago.

The point is that now we have a lot of evidence that the massive solution is the right one and there is no physical reason whatsoever to presume that the scaling solution should be the true solution at the critical scaling found by Jose. So, all this mounting evidence is there to say that the old idea of Hideki Yukawa is working yet:  Massive carriers imply limited range forces.

J. Rodríguez-Quintero (2011). The scaling infrared DSE solution as a critical end-point for the family
of decoupling ones arxiv arXiv: 1103.0904v1

J. Rodríguez-Quintero (2010). On the massive gluon propagator, the PT-BFM scheme and the low-momentum
behaviour of decoupling and scaling DSE solutions JHEP 1101:105,2011 arXiv: 1005.4598v2

Daniele Binosi, & Joannis Papavassiliou (2007). Gauge-invariant truncation scheme for the Schwinger-Dyson equations of
QCD Phys.Rev.D77:061702,2008 arXiv: 0712.2707v1

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. 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).

We cannot see the light


An interesting paper appeared today in arxiv by Alkofer,  Huber and Schwenzer (see here). Reinhard Alkofer and Lorenz von Smekal are the proponents of an infrared solution of Yang-Mills theory in D=4 having the following properties

  • Gluon propagator goes to zero at lower momenta
  • Ghost propagator goes to infinity at lower momenta faster than the free propagator
  • Running coupling reaches a fixed point at lower momenta

and this scenario disagrees with lattice evidence in D=4 but agrees with lattice in D=2 when the theory is trivial having no dynamics. After some years that other researchers were claiming that a different solution can be obtained by the same equations, that is Dyson-Schwinger equations, that indeed agrees with lattice computations, Alkofer’s group accepted this fact but with a lot of skepticism pointing out that this solution has several difficulties, last but not least it breaks BRST symmetry. The solution proposed by Alkofer and von Smekal by its side gives no mass gap whatsoever and no low energy spectrum to be compared neither with lattice nor with experiments to understand the current light unflavored meson spectrum. So, whoever is right we are in a damned situation that no meaningful computations can be carried out to get some real physical understanding. The new paper is again on this line with the authors proposing a perturbation approach to evaluate the vertexes of the theory in the infrared and obtaining again comforting agreement with their scenario.

I will avoid to enter into this neverending controversy about Dyson-Schwinger equations but rather I would ask a more fundamental question: Is it worthwhile an approach that only grants at best saving a phylosophical understanding of confinement without any real understanding of QCD? My view is that one should start from lattice data and try to understand the real mathematical form of the gluon propagator. Why does it resemble the Yukawa form so well? A Yukawa form grants a mass gap and this is elementary quantum field theory. This I would like to see explained. When a method is not satisfactory something must be changed. It is evident that solving Dyson-Schwinger equations requires some new mathematical approach as old views are just confusing this kind of research.

A quite effective QCD theory


As far as my path toward understanding of QCD is concerned, I have found a quite interesting effective theory to work with that is somewhat similar to Yukawa theory. Hideki Yukawa turns out to be more in depth in his hindsight than expected.

yukawa Indeed, I have already showed as the potential in infrared Yang-Mills theory is an infinite sum of weighted Yukawa potentials with the range, at each order, decided through a mass formula for glueballs that can be written down as


being \sigma the string tension, an experimental parameter generally taken to be (440 MeV)^2, and K(i) is an elliptic integral, just a number.

The most intriguing aspect of all this treatment is that an effective infrared QCD can be obtained through a scalar field. I am about to finish a paper with a calculation of the width of the \sigma resonance, a critical parameter for our understanding of low energy QCD. Here I put the generating functional if someone is interested in doing a similar calculation (time is rescaled as t\rightarrow\sqrt{N}gt)

Z[\eta,\bar\eta,j] \approx\exp\left\{i\int d^4x\sum_q \frac{\delta}{i\delta\bar\eta_q(x)}\frac{\lambda^a}{2\sqrt{N}}\gamma_i\eta_i^a\frac{\delta}{i\delta j_\phi(x)}\frac{i\delta}{\delta\eta_q(x)}\right\} \times
\exp\left\{-\frac{i}{Ng^2}\int d^4xd^4y\sum_q\bar\eta_q(x)S_q(x-y)\eta_q(y)\right\}\times
\exp\left\{\frac{i}{2}(N^2-1)\int d^4xd^4y j_\phi(x)\Delta(x-y)j_\phi(y)\right\}.

As always, S_q(x-y) is the free Dirac propagator for the given quark q=u,d,s,\ldots and \Delta(x-y) is the gluon propagator that I have discussed in depth in my preceding posts. People seriously interested about this matter should read my works (here and here).

For a physical understanding of this you have to wait my next posting on arxiv. Anyhow, anybody can spend some time to manage this theory to exploit its working and its fallacies. My hope is that, anytime I post such information on my blog, I help the community to have an anticipation of possible new ways to see an old problem with a lot of prejudices well grounded.

An inspiring paper


These days I am closed at home due to the effects of flu. When such bad symptoms started to relax I was able to think about physics again.  So, reading the daily from arxiv today I have uncovered a truly inspiring paper from Antal Jakovac a and Daniel Nogradi (see here). This paper treats a very interesting problem about quark-gluon plasma. This state was observed at RHIC at Brookhaven. Successful hydrodynamical models permit to obtain values of physical quantities, like shear viscosity, that could be in principle computed from QCD. The importance of shear viscosity relies on the existence of an important prediction from AdS/CFT symmetry claiming that the ratio between this quantity and entropy density can be at least 1/4\pi. If this lower bound would be proved true we will get an important experimental verification for AdS/CFT conjecture.

Jakovac and Nogradi exploit the computation of this ratio for SU(N) Yang-Mills theory. Their approach is quite successful as their able to show that the value they obtain is still consistent with the lower bound as they have serious difficulties to evaluate the error. But what really matters here is the procedure these authors adopt to reach their aim making this a quite simple alley to pursuit when the solution of Yang-Mills theory in infrared is acquired. The central point is again the gluon propagator. These authors assume simply the very existence of a mass gap taking for the propagator something like e^{-\sigma\tau} in Euclidean time. Of course, \sigma is the glueball mass. This is a too simplified assumption as we know that the gluon propagator is somewhat more complicated and a full spectrum of glueballs does exist that can contribute to this computation (see my post and my paper).

So, I spent my day to extend the computations of these authors to a more realistic gluon propagator.  Indeed, with my gluon propagator there is no need of one-loop computations as the identity at 0-loop G_T=G_0 does not hold true anymore for a non-trivial spectrum and one has immediately an expression for the shear viscosity. I hope to give some more results in the near future.

Emerging scenario


Reading arxiv dailys today I have found three different papers on the gluon and ghost propagators for Yang-Mills (see here, here and here). These papers prove that this line of research is very strongly alive and that there exist a lot of points to be settled down before to carry on. In this post I would like to point out several evidences that should not be forgotten when one talks about this matter. First of all there are the results of Yang-Mills theory in D=1+1. We know that, for this dimensionality, Yang-Mills theory has no dynamics. Anyhow, several people tried to solve it on the lattice or modified it to try to relate these solutions of the ones of Dyson-Schwinger equations with a given truncation. The bad news is that they find agreement with such solutions of Dyson-Schwinger equations. Why is this bad news? Because this gives, beyond any doubt, a proof that such a truncation of Dyson-Schwinger equations is fault as it removes any dynamics from Yang-Mills theory in higher dimensionality and appears to agree with numerical results just when such a dynamics does not exist. This is already a severe indicator that lattice computations done in higher dimensionality are right. What do they say us about ghost and gluon propagators?

  • Gluon propagator reaches a non-null finite value at zero momenta.
  • Ghost propagator is that of a free particle.
  • Running coupling goes to zero at lower momenta.

This means that the confinement scenarios that are normally considered are faulty and do not work at all. These results demand for a better understanding of the physical situation at hand. It we are not ourselves convinced that they are right, we will keep on fumbling in the dark losing precious resources and time. Evidences are really heavy already at this stage and should be combined with spectra computations carried out so far. Also in this case a lot of work still must be carried out. You can read the beatiful paper of Craig McNeile about (contribution to QCD 08). It is a mistery to me why these ways are seen as different into the understanding of Yang-Mills theory.

A formula I was looking for


As usual I put in this blog some useful formulas to work out computations in quantum field theory. My aim in these days is to compute the width of the \sigma resonance. This is a major aim in QCD as the nature of this particle is hotly debated. Some authors think that it is a tetraquark or molecular state while others as Narison, Ochs, Minkowski and Mennessier point out the gluonic nature of this resonance. We have expressed our view in some posts (see here and here) and our results heavily show that this resonance is a glueball in agreement with the spectrum we have found for a pure Yang-Mills theory.

Our next step is to understand the role of this resonance in QCD. Indeed, we have shown in our recent paper (see here) that, once the gluon propagator is known, it is possible to derive a Nambu-Jona-Lasinio model from QCD with all parameters properly fixed. We have obtained the following:

S_{NJL} \approx \int d^4x\left[\sum_q \bar q(x)(i\gamma\cdot\partial-m_q)q(x)\right.

-\frac{1}{2}g^2\sum_{q,q'}\bar q(x)\frac{\lambda^a}{2}\gamma^\mu q(x)\times

\left.\int d^4yG(x-y)\bar q'(y)\frac{\lambda^a}{2}\gamma_\mu q'(y)\right]

being G(x-y) the gluon propagator with all indexes in color and space-time already saturated. This in turn means that we can use the following formula (see my paper here and here):

e^{\frac{i}{2}\int d^4xd^4yj(x)G(x-y)j(y)}\approx {\cal N}\int [d\sigma]e^{-i\int d^4x\left[\sigma\left(\frac{1}{2}\partial^2\sigma+\frac{Ng^2}{4}\sigma^3\right)-j\sigma\right]}

being again G(x-y) the gluon propagator for SU(N) and {\cal N} a normalization factor. This formula does hold only for infrared limit, that is when the theory is strongly coupled. We plan to extract physical results from this formula and define in this way the role of \sigma resonance.

What makes the proton spin?


There is currently a beautiful puzzle to be answered that relies on sound and beautiful experimental results. The question is how the components of a proton, that is quarks and gluons, concur to determine the value one half for the spin of the particle. During the conference QCD 08 at Montpellier I listened to a beatiful presentation of Joerg Pretz of the COMPASS Collaboration (see here and here). Hearing these results was stunning for me. I explain the reasons in a few words. The spin of the proton should be composed by the spin of the quarks, the contributions of gluons (gluons???) and orbital angular momentum. What happens is that the spin of quarks does not contribute too much. People then thought that the contribution of gluons (gluons again???) should have been decisive. The COMPASS Collaboration realized a beautiful experiment using charmed mesons. This experiment has been described by Pretz at QCD 08. They proved in a striking way that the contribution of the glue to proton spin can be zero and cannot be used to account for the particle spin. Of course, there are beautiful papers around that are able to explain how the proton spin comes out. I have found for example a paper by Thomas and Myhrer at Jefferson Lab (see here and here) that describes quite well an understanding of the puzzle and surely is worthwhile reading. But my question is another: Why the glue  does not contribute?

From our preceding posts one should have reached immediately an answer, the same that come out to my mind when I listened Pretz’s talk. The reason is that, in the infrared, gluons that have spin one are not the true carriers of the strong force. The true carriers have no spin unless higher excited states are considered. This explains why COMPASS experiment did not see any contribution consistently with previous expectations.

This is again a strong support to our description of the gluon propagator (see here). No other theory around shows this.

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