Do quarks grant confinement?

In 2010 I went to Ghent in Belgium for a very nice Conference on QCD. My contribution was accepted and I had the chance to describe my view about this matter. The result was this contribution to the proceedings. The content of this paper was really revolutionary at that time as my view about Yang-Mills theory, mass gap and the role of quarks was almost completely out of track with respect to the rest of the community. So, I am deeply grateful to the Organizers for this opportunity. The main ideas I put forward were

• Yang-Mills theory has an infrared trivial fixed point. The theory is trivial exactly as the scalar field theory is.
• Due to this, gluon propagator is well-represented by a sum of weighted Yukawa propagators.
• The theory acquires a mass gap that is just the ground state of a tower of states with the spectrum of a harmonic oscillator.
• The reason why Yang-Mills theory is trivial and QCD is not in the infrared limit is the presence of quarks. Their existence moves the theory from being trivial to asymptotic safety.

These results that I have got published on respectable journals become the reason for rejection of most of my successive papers from several referees notwithstanding there were no serious reasons motivating it. But this is routine in our activity. Indeed, what annoyed me a lot was a refeee’s report claiming that my work was incorrect because the last of my statement was incorrect: Quark existence is not a correct motivation to claim asymptotic safety, and so confinement, for QCD. Another offending point was the strong support my approach was giving to the idea of a decoupling solution as was emerging from lattice computations on extended volumes. There was a widespread idea that the gluon propagator should go to zero in a pure Yang-Mills theory to grant confinement and, if not so, an infrared non-trivial fixed point must exist.

Recently, my last point has been vindicated by a group that was instrumental in the modelling of the history of this corner of research in physics. I have seen a couple of papers on arxiv, this and this, strongly supporting my view. They are Markus Höpfer, Christian Fischer and Reinhard Alkofer. These authors work in the conformal window, this means that, for them, lightest quarks are massless and chiral symmetry is exact. Indeed, in their study quarks not even get mass dynamically. But the question they answer is somewhat different: Acquired the fact that the theory is infrared trivial (they do not state this explicitly as this is not yet recognized even if this is a “duck” indeed), how does the trivial infrared fixed point move increasing the number of quarks? The answer is in the following wonderful graph with the number of quarks (flavours):

From this picture it is evident that there exists a critical number of quarks for which the theory becomes asymptotically safe and confining. So, quarks are critical to grant confinement and Yang-Mills theory can happily be trivial. The authors took great care about all the involved approximations as they solved Dyson-Schwinger equations as usual, this is always been their main tool, with a proper truncation. From the picture it is seen that if the number of flavours is below a threshold the theory is generally trivial, so also for the number of quarks being zero. Otherwise, a non-trivial infrared fixed point is reached granting confinement. Then, the gluon propagator is seen to move from a Yukawa form to a scaling form.

This result is really exciting and moves us a significant step forward toward the understanding of confinement. By my side, I am happy that another one of my ideas gets such a substantial confirmation.

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

Markus Hopfer, Christian S. Fischer, & Reinhard Alkofer (2014). Running coupling in the conformal window of large-Nf QCD arXiv arXiv: 1405.7031v1

Markus Hopfer, Christian S. Fischer, & Reinhard Alkofer (2014). Infrared behaviour of propagators and running coupling in the conformal
window of QCD arXiv arXiv: 1405.7340v1

Advertisement

Nailing down the Yang-Mills problem

Millennium problems represent a major challenge for physicists and mathematicians. So far, the only one that has been solved was the Poincaré conjecture (now a theorem) by Grisha Perelman. For people working in strong interactions and quantum chromodynamics, the most interesting of such problems is the Yang-Mills mass gap and existence problem. The solutions of this problem would imply a lot of consequences in physics and one of the most important of these is a deep understanding of confinement of quarks inside hadrons. So far, there seems to be no solution to it but things do not stay exactly in this way. A significant number of researchers has performed lattice computations to obtain the propagators of the theory in the full range of energy from infrared to ultraviolet providing us a deep understanding of what is going on here (see Yang-Mills article on Wikipedia). The propagators to be considered are those for  the gluon and the ghost. There has been a significant effort from theoretical physicists in the last twenty years to answer this question. It is not so widely known in the community but it should because the work of this people could be the starting point for a great innovation in physics. In these days, on arxiv a paper by Axel Maas gives a great recount of the situation of these lattice computations (see here). Axel has been an important contributor to this research area and the current understanding of the behavior of the Yang-Mills theory in two dimensions owes a lot to him. In this paper, Axel presents his computations on large volumes for Yang-Mills theory on the lattice in 2, 3 and 4 dimensions in the SU(2) case. These computations are generally performed in the Landau gauge (propagators are gauge dependent quantities) being the most favorable for them. In four dimensions the lattice is , not the largest but surely enough for the aims of the paper. Of course, no surprise comes out with respect what people found starting from 2007. The scenario is well settled and is this:

1. The gluon propagator in 3 and 4 dimensions dos not go to zero with momenta but is just finite. In 3 dimensions has a maximum in the infrared reaching its finite value at 0  from below. No such maximum is seen in 4 dimensions. In 2 dimensions the gluon propagator goes to zero with momenta.
2. The ghost propagator behaves like the one of a free massless particle as the momenta are lowered. This is the dominant behavior in 3 and 4 dimensions. In 2 dimensions the ghost propagator is enhanced and goes to infinity faster than in 3 and 4 dimensions.
3. The running coupling in 3 and 4 dimensions is seen to reach zero as the momenta go to zero, reach a maximum at intermediate energies and goes asymptotically to 0 as momenta go to infinity (asymptotic freedom).

Here follows the figure for the gluon propagator

and for the running coupling

There is some concern for people about the running coupling. There is a recurring prejudice in Yang-Mills theory, without any support both theoretical or experimental, that the theory should be not trivial in the infrared. So, the running coupling should not go to zero lowering momenta but reach a finite non-zero value. Of course, a pure Yang-Mills theory in nature does not exist and it is very difficult to get an understanding here. But, in 2 and 3 dimensions, the point is that the gluon propagator is very similar to a free one, the ghost propagator is certainly a free one and then, using the duck test: If it looks like a duck, swims like a duck, and quacks like a duck, then it probably is a duck, the theory is really trivial also in the infrared limit. Currently, there are two people in the World that have recognized a duck here:  Axel Weber (see here and here) using renormalization group and me (see here, here and here). Now, claiming to see a duck where all others are pretending to tell a dinosaur does not make you the most popular guy  in the district. But so it goes.

These lattice computations are an important cornerstone in the search for the behavior of a Yang-Mills theory. Whoever aims to present to the World his petty theory for the solution of the Millennium prize must comply with these results showing that his theory is able to reproduce them. Otherwise what he has is just rubbish.

What appears in the sight is also the proof of existence of the theory. Having two trivial fixed points, the theory is Gaussian in these limits exactly as the scalar field theory. A Gaussian theory is the simplest example we know of a quantum field theory that is proven to exist. Could one recover the missing part between the two trivial fixed points as also happens for the scalar theory? In the end, it is possible that a Yang-Mills theory is just the vectorial counterpart of the well-known scalar field, the workhorse of all the scholars in quantum field theory.

Axel Maas (2014). Some more details of minimal-Landau-gauge Yang-Mills propagators arXiv arXiv: 1402.5050v1

Axel Weber (2012). Epsilon expansion for infrared Yang-Mills theory in Landau gauge Phys. Rev. D 85, 125005 arXiv: 1112.1157v2

Axel Weber (2012). The infrared fixed point of Landau gauge Yang-Mills theory arXiv arXiv: 1211.1473v1

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

Marco Frasca (2009). Mapping a Massless Scalar Field Theory on a Yang-Mills Theory: Classical
Case Mod. Phys. Lett. A 24, 2425-2432 (2009) arXiv: 0903.2357v4

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

Ending and consequences of Terry Tao’s criticism

Summer days are gone and I am back to work. I thought that Terry Tao’s criticism to my work was finally settled and his intervention was a good one indeed. Of course, people just remember the criticism but not how the question evolved since then (it was 2009!). Terry’s point was that the mapping given here between the scalar field solutions and the Yang-Mills field in the classical limit cannot be exact as it is not granted that they represent an extreme for the Yang-Mills functional. In this way the conclusions given in the paper are not granted being based on this proof. The problem can be traced back to the gauge invariance of the Yang-Mills theory that is explicitly broken in this case.

Terry Tao, in a private communication, asked me to provide a paper, to be published on a refereed journal, that fixed the problem. In such a case the question would have been settled in a way or another. E.g., also a result disproving completely the mapping would have been good, disproving also my published paper.

This matter is rather curious as, if you fix the gauge to be Lorenz (Landau), the mapping is exact. But the possible gauge choices are infinite and so, there seems to be infinite cases where the mapping theorem appears to fail. The lucky case is that lattice computations are generally performed in Landau gauge and when you do quantum field theory a gauge must be chosen. So, is the mapping theorem really false or one can change it to fix it all?

In order to clarify this situation, I decided to solve the classical equations of the Yang-Mills theory perturbatively in the strong coupling limit. Please, note that today I am the only one in the World able to perform such a computation having completely invented the techniques to do perturbation theory when a perturbation is taken to go to infinity (sorry, no AdS/CFT here but I can surely support it). You will note that this is the opposite limit to standard perturbation theory when one is looking for a parameter that goes to zero. I succeeded in doing so and put a paper on arxiv (see here) that was finally published the same year, 2009.

The theorem changed in this way:

The mapping exists in the asymptotic limit of the coupling running to infinity (leading order), with the notable exception of the Lorenz (Landau) gauge where it is exact.

So, I sighed with relief. The reason was that the conclusions of my paper on propagators were correct. But these hold asymptotically in the limit of a strong coupling. This is just what one needs in the infrared limit where Yang-Mills theory becomes strongly coupled and this is the main reason to solve it on the lattice. I cited my work on Tao’s site, Dispersive Wiki. I am a contributor to this site. Terry Tao declared the question definitively settled with the mapping theorem holding asymptotically (see here).

In the end, we were both right. Tao’s criticism was deeply helpful while my conclusions on the propagators were correct. Indeed, my gluon propagator agrees perfectly well, in the infrared limit, with the data from the largest lattice used in computations so far  (see here)

As generally happens in these cases, the only fact that remains is the original criticism by a great mathematician (and Terry is) that invalidated my work (see here for a question on Physics Stackexchange). As you can see by the tenths of papers I published since then, my work stands and stands very well. Maybe, it would be time to ask the author.

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

Marco Frasca (2009). Mapping a Massless Scalar Field Theory on a Yang-Mills Theory: Classical
Case Mod. Phys. Lett. A 24, 2425-2432 (2009) arXiv: 0903.2357v4

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

Kyoto, arXiv and all that

Today, Kyoto conference HCP2012 has started. There is already an important news from LHCb that proves for the first time the existence of the decay . They find close agreement with the Standard Model (see here). Another point scored by this model and waiting for new physics yet. You can find the program with all the talks to download here. There is a lot of expectations from the update on the Higgs search: The great day is Thursday. Meantime, there is Jester providing some rumors (see here on twitter side) and seem really interesting.

I have a couple of papers to put to the attention of my readers from arXiv. Firstly, Yuan-Sen Ting and Bryan Gin-ge Chen provided a further improved redaction of the Coleman’s lectures (see here). This people is doing a really deserving work and these lectures are a fundamental reading for any serious scholar on quantum field theory.

Axel Weber posted a contribution to a conference (see here) summing up his main conclusions on the infrared behavior of the running coupling and the two-point functions for a Yang-Mills theory. He makes use of renormalization group and the inescapable conclusion is that if one must have a decoupling solution, as lattice computations demand, then the running coupling reaches an infrared trivial fixed point. This is in close agreement with my conclusions on this matter and it is very pleasant to see them emerge from another approach.

Sidney Coleman (2011). Notes from Sidney Coleman’s Physics 253a arXiv arXiv: 1110.5013v4

Axel Weber (2012). The infrared fixed point of Landau gauge Yang-Mills theory arXiv arXiv: 1211.1473v1

Confinement revisited

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

Running coupling and Yang-Mills theory

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

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…

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

Yang-Mills scenario: Yet a confirmation

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 -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

QCD is confining

At Bari Conference , after I gave my talk, Owe Philipsen asked to me about confinement in my approach. The question came out also in the evening, drinking a beer at a pub in the old Bari. Looking at my propagator, it is not so straightforward to see if the theory is confining or not. But we know, from lattice computations, that this must be so. You can realize this from the following figure (see here)

The scale is given by , the so called Sommer’s scale, We note a clear linear rising till about 1.5 fm. A linear rising potential is an evidence of confinement as showed about forty years ago by Kenneth Wilson (see here) with his famous area law. Due to this clear evidence coming from lattice computations, any attempt to explain mass gap must show confinement through a linear rising potential.

Indeed, this is not all the story and going to 1.5 fm cannot be enough to display all the behavior of a Yang-Mills theory. As showed quite recently on the lattice Philippe de Forcrand and Slavo Kratochvila (see here), increasing distance, the potential must saturate. This is an effect of the mass gap that causes screening. This means that, at larger distances, the potential sets on an asymptote becoming horizontal. The linear approximation holds on a finite range.

This is indeed what I observe with my approach. I can prove that the potential has a Yukawa form with a form factor dependent on the distance. The mass scale entering into it is just the mass gap. So, you get a linear fit like the following (see here)

that shows confinement with the area law till 10 fm! If one increases the distance the fit worsens and saturation appears as expected. From this we can easily derive the string tension that is given by . For SU(N), . This is a fine proof of confinement for a Yang-Mills theory and so, for QCD too. This also means that my approach is again consistent with lattice data. Just for completeness, and to give a great thank to Arlene Aguilar and Daniele Binosi, I show the fit of my propagator with the one obtained numerically solving Dyson-Schwinger equations (see here)

The agreement is almost perfect.

Gunnar S. Bali (2000). QCD forces and heavy quark bound states Phys.Rept.343:1-136,2001 arXiv: hep-ph/0001312v2

Wilson, K. (1974). Confinement of quarks Physical Review D, 10 (8), 2445-2459 DOI: 10.1103/PhysRevD.10.2445

Slavo Kratochvila, & Philippe de Forcrand (2003). Observing string breaking with Wilson loops Nucl.Phys. B671 (2003) 103-132 arXiv: hep-lat/0306011v2

Marco Frasca (2011). QCD is confining arXiv arXiv: 1110.2297v1

A. C. Aguilar, D. Binosi, & J. Papavassiliou (2008). Gluon and ghost propagators in the Landau gauge: Deriving lattice
results from Schwinger-Dyson equations Phys.Rev.D78:025010,2008 arXiv: 0802.1870v3