My paper presenting exact solutions to classical scalar field theories, with a corresponding quantum formulation, has been accepted for publication in the Journal of Nonlinear Mathematical Physics. The replacement on arxiv will appear tomorrow, the link is here. I would like to thank the Editor, Norbert Euler, and an anonymous referee that pointed out to me the existence of a zero mode in the quantum fluctuations.

## The nature of eta’

30/11/2010is a very peculiar particle. It mixes with that has a lower mass. Recently, in their report on KLOE-2 physics, this group reported here that has a significant glue component besides quarks. This means that understanding its most important decay is not a trivial matter. As my readers may know, I have done a computation in my contribution to proceedings to QCD 10 conference where the decay process is seen to happen through an intermediate step with the resonance followed by the decay of this into two pions. The agreement we get is so good to give a correct estimation of the decay constant of the . This implies that the is a true glue state. Of course our computation is rough enough to exclude mixing with other hadronic states that should exist.

Today, on arxiv, an interesting paper appeared authored by Rafel Escribano, Pere Masjuan, Juan José Sanz-Cillero (see here). These authors give an initial overview of the experimental status of the decay we have considered above. Then, using both the technique of Chiral Perturbation Theory (ChPT) and that of Resonant Chiral Perturbation Theory (RChPT), they try to fit experimental data. I have the luck to hear a talk of Juan José in Montpellier last year about this same matter and I was aware of his struggle to reach an agreement between a successful technique, as ChPT is, and experimental data for this particular process. The leading order of the theory is well below the experimental value and so, already in that first talk, Juan José showed the need for higher order corrections. But he proved that this cannot be enough and said at that time that some other states should be accounted for to reach a satisfactory agreement. This paper goes in this direction showing that if one accounts for the presence of the and a0(980), the latter being dominant, the agreement is reached. These authors were also able to show a consistent relation between ChPT and RChPT that are in some way complimentary.

This paper is relevant as gives a strong support to the idea that I put forward about decay. But these authors go further implying a higher level of understanding accounting for the presence of other hadronic states in a technical affordable way. I expect further improvement by them and it will be interesting to see how these could be obtained.

## Some relevant preprints

29/11/2010My readers know that I keep myself up to date with the daily submissions at arxiv. Of course, I limit myself to the ones very near my research area. Today there have been some quite interesting papers by people working at the question of Green functions in the low-energy limit of QCD. I would like to point out three contributions at the Madrid conference of the last summer: Arlene Aguilar shows how low-energy phenomenology is consistent with our current understanding of the gluon propagator for the decoupling solution (see here). Joannis Papavassiliou shows how the solution for d=2+1 for the gluon propagator, solving numerically Dyson-Schwinger equations, is in agreement with lattice data (see here). Finally, there is the contribution of Eduardo Fraga, Ana Mizher and Maxim Chernodub about the use of a linear sigma model to understand the vacuum of QCD in a strong magnetic field (see here). These people were also present at Gent.

An original research paper appeared from Orlando Oliveira, Pedro Costa and Paulo Silva (see here). Orlando shocked me again with another bright idea. He and his colleagues managed to get an understanding of the behavior of low-energy physics from choosing the propagator through the scaling or the decoupling solution. The smart idea is to do so by deriving a Nambu-Jona-Lasinio model and solving this for known low-energy observables. As you may know, I worked out something like this in this paper that went published in the International Journal of Modern Physics E. I have sent an email to Orlando asking for a contribution by him or his colleagues in this blog. I hope they will take some time to do this as this paper is really striking and the idea really meaningful. Orlando said to me that this paper will appear soon on Physics Letters B.

Finally, I hope you will enjoy reading these papers as I did.

## Some reflections

28/11/2010It is a lot of time that I am thinking about the scenario is emerging from our current understanding of reality through physics. There is a lot to be understood yet, mostly the very nature of space and time and a proof of the real number of dimensions our universe emerged from. Notwithstanding these noteworthy missing questions, we can make an idea of what is going on as the depth of our understanding is already quite sensible.

At the dawn of the last century, Albert Einstein put forward an important conclusion: mass and energy are the same thing. Indeed, what Einstein had in mind is a deeper understanding of the concept of mass that is for us an important concern. Where does mass come from? What is it made of? At the start of the last century these questions could not have a proper answer. But having reduced the mass to another concept, like energy, was of paramount importance.

On a similar ground it was relevant to understand the role of another, apparently not reducible, concept: Charge. Today we know that the number of charges, the couplings that make all change around us, will be at last reduced to a single number. Maybe. So far we only know for certain that, thanks to Steven Weinberg, Shelly Glashow and Abdus Salam, we have reduced the number of interacting fields. But strong coupling and gravitational constant are still there disconnected if we limit to our current experimental knowledge. Of course, theoretical physics has gone really far in this area and we hope that LHC will help us to give a way to cut out most of what was done here to get the real understanding of the way things work. Somebody will be happy others won’t but this how our World works.

Today, we have a better understanding of the concept of mass and, while waiting for LHC to confirm us our ideas, we can draw some conclusions about all these questions. The fact that mass is energy is an important clue of the idea that this concept is reducible to more fundamentals concepts and that a mechanism for its emergence must exist. Higgs mechanism goes in this direction as also our ideas emerging from QCD about the mass gap that confirm the idea that mass is not a fundamental concept by itself.

So, we can conclude that, so far, our ideas of the World reduce to two fundamental non-reducible ideas: Energy and charge. The former is just a safety lock with respect to the changes provoked by the latter. So, I leave you with a final question: If things stay in this way, does space-time entail a wider concept to embed them or we can reduce also this to them?

## Quote of the day

20/11/2010*“Most days I wish I never met you. Because then I could sleep at night. I wouldn’t have to walk around with the knowledge that someone like you was out there. And I wouldn’t have to watch you throw it all away.”*

Good Will Hunting (1997)

## The many faces of QCD (3)

17/11/2010Today it is appeared my contribution to the proceedings in arxiv (see here). I am happy to hear your comments if any. I prefer to write this kind of papers rather soon, when the memory of the talk is fresh enough to convey the same contents, if possible. As you can see, there has been a lot of maturation of these ideas since their inception and now I am really satisfied on the way the scenario of mass generation and low-energy behavior of Yang-Mills theory is formed. It is also striking to see how all this matter fits well inside the current understanding that is forming in the community as I discussed in the preceding posts. Mass generation is a key matter today, mostly due to the Standard Model and the expectations from LHC (e.g. see here). My view is that our work will be pivotal and it is possible that similar mechanisms are at work here both for Standard Model and QCD. If this is so, Nature will prove to be mind-boggling again.

## The many faces of QCD (2)

10/11/2010Back at home, conference ended. A lot of good impressions both from the physics side and other aspects as the city and the company. On Friday I held my talk. All went fine and I was goodly inspired so to express my ideas at best. You can find all the talks here. The pictures are here. Now it should be easier to identify me.

**Disclaimer**: The talks I will comment on are about results very near my research area. Talks I will not cite are important and interesting as well and the fact that I will not comment about them does not imply merit for good or bad. Anyhow, I will appreciate any comment by any participant to the conference aiming to discuss his/her work.

On Tuesday afternoon started a session about phases in QCD. This field is very active and is a field where some breakthroughs are expected to be seen in the near future. I have had a lot of fun to know Eduardo Fraga that was here with two of his students: Leticia Palhares and Ana Mizher. I invite you to read their talks as this people are doing a real fine work. On the same afternoon I listened to the talk of Pedro Bicudo. Pedro, besides being a nice company for fun, is also a very good physicist performing relevant work in the area of lattice QCD. He is a pioneer in the use of CUDA, parallel computing using graphic processors, and I intend to use his code, produced with his student Nuno Cardoso, on my machine to start doing lattice QCD at very low cost. On his talk you can see a photo of one of my graphic cards. He used lattice computations to understand the phase diagram of QCD. Quite interesting has been the talk of Jan Pawlowski about the phase diagram of two flavor QCD. He belongs to a group of people that produced the so called scaling solution and it is a great moment to see them to recognize the very existence of the decoupling solution, the only one presently seen on lattice computations.

On Wednesday the morning session continued on the same line of the preceding day. I would like to cite the work of Marco Ruggieri because, besides being a fine drinking companion (see below), he faces an interesting problem: How does the ground state of QCD change in presence of a strong magnetic field? Particularly interesting is to see how the phase diagram gets modified. On the same line were the successive talks of Ana Mizher and Maxim Chernodub. Chernodub presented a claim that in this case vacuum is that of an electromagnetic superconductor due to meson condensation. In this area of research the main approach is to use some phenomenological model. Ana Mizher used a linear sigma model while Marco preferred the Nambu-Jona-Lasinio model. The reason for this is that the low-energy behavior of QCD is not under control and the use of well-supported effective models is the smarter approach we have at our disposal. Of course, this explains why the work of our community is so important: If we are able to model the propagator of the gluon in the infrared, all the parameters of the Nambu-Jona-Lasinio model are properly fixed and we have the true infrared limit of QCD. So, the stake is very high here.

In the afternoon there were some talks that touched very near the question of infrared propagators. Silvio Sorella is an Italian theoretical physicist living in Brazil. He is doing a very good work in this quest for an understanding of the low-energy behavior of QCD. This work is done in collaboration with several other physicists. The idea is to modify the Gribov-Zwanziger scenario, that by itself will produce the scaling solution currently not seen on the lattice, to include the presence of a gluon condensate. This has the effect to produce massive propagators that agree well with lattice computations. In this talk Silvio showed how this approach can give the masses of the lowest states of the glueball spectrum. This has been an important step forward showing how this approach can be used to give experimental forecasts. Daniel Zwanziger then presented a view of the confinement scenario. The conclusion was very frustrating: So far nobody can go to the Clay Institute to claim the prize. More time is needed. Daniel has been the one who proposed the scenario of infrared Yang-Mills theory that produced the scaling solution. The idea is to take into account the problem of Gribov copies and to impose that all the computations must be limited to the first Gribov horizon. If you do this the gluon propagator goes to zero lowering momenta and you get positivity maximally violated obtaining a confining theory. So, this scenario has been called Gribov-Zwanzinger. From lattice computations we learned that the gluon propagator reaches a non zero finite value lowering momenta and this motivated Silvio and others to see if one could maintain the original idea of Gribov horizon and agreement with lattice computations of the Gribov-Zwanzinger scenario. Matthieu Thissier presented a talk with an original view. The idea is to consider QCD with a small perturbation expansion at one loop and a mass term added by hand. He computed the gluon propagator and compared with lattice data till the infrared obtaining a very good agreement. Arlene Aguilar criticized strongly this approach as he worked with a coupling larger than one (a huge one said Arlene) even if he was doing small perturbation theory. I talked about this with Matthieu. My view is that the main thing to learn from this kind of computations is that if you take a Yukawa-like propagator with a mass going at least as (do you remember Orlando Oliveira talk?) the agreement with lattice data is surely fairly good and so, even if you have done something that is mathematically questionable, surely we apprehend an important fact! The afternoon session was concluded by the talk of Daniele Binosi. With Daniele we spent a nice night in Ghent. He is a student of Joannis Papavassiliou and, together with Arlene Aguilar, this group is doing fine work on numerically solving Dyson-Schwinger equations to get the full propagator of Yang-Mills theory. They get a very good agreement with lattice data and support the view that, on the full range of energies, the Cornwall propagator for the gluon with a logarithmic running mass reaching a constant in the infrared is the right description of the theory. Daniele presented a beautiful computation based on Batalin-Vilkoviski framework that supported the conclusions of his group. It should be said that he presented a different definition of the running coupling that grants a non-trivial fixed point at infrared. This is a delicate matter as, already a proper definition of the running coupling for the infrared is not a trivial question. Daniele’s definition is quite different from that given by Andre Sternbeck in his talk as the latter has just the trivial fixed point as is emerging from the lattice computations.

On Thursday the first speaker was Attilio Cucchieri. Attilio and his wife, Tereza Mendes, are doing a fine work on lattice computations that reached a breakthrough at Lattice 2007 when they showed, with a volume of , that the gluon propagator in the Landau gauge reaches a finite non-zero value lowering momenta. This was a breakthrough, confirmed at the same conference by two others groups (Orlando Oliveira by one side and I. Bogolubsky, E.M. Ilgenfritz, M. Muller-Preussker and A. Sternbeck by the other side), as for long time it was believed that the only true solution was the scaling one and the gluon propagator should have gone to zero lowering momenta. This became a paradigm so that papers have got rejected on the basis that they were claiming a different scenario. Attilio this time was on a very conservative side presenting an interesting technical problem. Tereza’s talk was more impressive showing that, with higher temperatures and increasing volumes, in the Landau gauge the plateau is still there. With Tereza and Attilio we spent some nice time in a pub discussing together with Marco Ruggeri about the history of their community, how they went to change everything about this matter and their fighting for this. I hope one day this people will write down this history because there is a lot to learn from it. In the afternoon session there was a talk by Reinhard Alkofer. Alkofer has been instrumental in transforming the scaling solution into a paradigm for a lot of years in the community. Unfortunately lattice computations talked against it and, as Bob Dylan one time said, times are changing. He helped the community with discovering a lot of smart students that have given an important contribution to it. In his talk he insisted with his view with a proposal for the functional form for the propagator (this was missing until now for the scaling solution) and a computation of the mass of the . is a very strange particle. From (KLOE-2) we know that this is not just a composite state of quarks but it contains a large part made of glue: It is like to have to cope with an excited hydrogen atom and so, also its decay is to be understood (you can read my paper here). So, maybe a more involved discussion is needed before to have an idea of how to get the mass of this particle. After Alkofer’s talk followed the talks of Aguilar and Papavassiliou. I would like to emphasize the relevance of the work of this group. Aguilar showed how they get an effective quark mass from Schwinger-Dyson equations when there is no enhancement in the ghost propagator. Papavassiliou proposed to extend the background field method to Schwinger-Dyson equations. I invite you to check the agreement they get for the Cornwall propagator of the gluon with lattice data in Arlene’s talk and how this can give the form at lower momenta. My view is that, combining my recent results on strongly coupled expansions for Yang-Mills and scalar field theories and the results of this group, a meaningful scenario is emerging giving a complete comprehension of what is going on for Yang-Mills theory at lower energies. Joannis gave us an appointment for the next year in Trento. I will do everything I can to be there! Finally, the session was completed with Axel Mass’ talk. Axel has been a student of Alkofer and worked with Attilio and Tereza. He put forward a lattice computation of Yang-Mills propagators in two dimensions that, for me, should have completely settled the question but produced a lot of debate instead. He gave in his talk another bright idea: To study on the lattice a scalar theory interacting with gluons. I think that this is a very smart way to understand the mechanism underlying mass generation in these theories. From the works discussed so far it should appear clear that Schwinger mechanism (also at classical level (see my talk)!) is at work here. The talk of Axel manifestly shows this. It would be interesting if he could redo the computations taking a massless scalar field to unveil completely the dynamical generation of masses.

On Friday the morning session started with an interesting talk by Hans Dierckx trying to understand cardiac behavior using string theory. A talk by Oliver Rosten followed. Oliver produced a PhD thesis on the exact renormalization group of about 500 pages (see here). His talk was very beautiful and informative and in some way gave a support to mine. Indeed, he showed, discussing on the renormalization group, how a strong coupling expansion could emerge. In some way we are complimentary. I will not discuss my talk here but you are free to ask questions. The conference was concluded by a talk of Peter van Baal. Peter has a terrible story about him and I will not discuss it here. I can only wish to him the best of the possible lucks.

Finally, I would like to thank the organizers for the beautiful conference they gave me the chance to join. The place was very nice (thanks Nele!) and city has an incredible beauty. I think these few lines do not do justice to them and all the participants for what they have given. See you again folks!

## The many faces of QCD

02/11/2010After a long silence, due to technical impediments as many of you know, I turn back to you from Ghent (Belgium). I am participating at the conference “The many faces of QCD”. You can find the program here. The place is really beautiful as the town that I had the chance to look out yesterday evening. Organizers programmed a visit downtown tomorrow and I hope to see this nice town also at the sun light. The reason why this conference is so relevant is that it gathers almost all the people working on this matter of Green functions of Yang-Mills theory and QCD whose works I cited widely in my blog and in my papers. Now, I have the chance to meet them and speak to them. I am writing after the second day ended. The atmosphere is really exciting and discussion is always alive and it happens quite often that speakers are interrupted during their presentations. The situation this field is living is simply unique in the scientific community. They are at the very start of a possible scientific revolution as they are finally obtaining results of non-perturbative physics in a crucial field as that of QCD.

**Disclaimer**: The talks I will comment on are about results very near my research area. Talks I will not cite are important and interesting as well and the fact that I will not comment about them does not imply merit for good or bad. Anyhow, I will appreciate any comment by any participant to the conference aiming to discuss his/her work.

I would like to cite some names here but I fear to forget somebody surely worthwhile to be named. From my point of view, there have been a couple of talks that caught my attention more strongly than others, concerning computations on the lattice. This happened with the talk of Tereza Mendes yesterday and the one of Orlando Oliveira today. Tereza just studied the gluon propagator at higher temperatures obtaining again striking and unexpected results. There is this plateau in the gluon propagator appearing again and again when lattice volume is increased. It would have been interesting to have also a look to the ghost and the running coupling. Orlando, by his side, showed for the first time an attempt to fit with the function that you can recognize as the one I proposed since my first analysis to explain the infrared behavior of Yang-Mills theory. But Orlando went further and found the next to leading order correction to the mass appearing in a Yukawa-like propagator. The idea is to see if the original hypothesis of Cornwall can agree with the current lattice computations. So, he shows that for the sum of propagators one can get even better agreement in the fitting increasing the number of masses (at least 4) and for the Cornwall propagator you will need a mass corrected as . Shocking as may seem, I computed this term this summer and you can find it in this paper of mine. Indeed, this is a guess I put forward after a referee asked to me an understanding of the next-to-leading corrections to my propagator and, as you can read from my paper, I guessed it would have produced a Cornwall-like propagator. Indeed, this is just a first infrared correction that can arise by expanding the logarithm in the Cornwall’s formula.

The question of the gluon condensate, that I treated in my blog extensively thanks to the help of Stephan Narison, has been presented today by Olivier Péne through a lattice computation. Olivier works in the group of Philippe Boucaud and contributed to the emerging of the now called decoupling solution for the gluon propagator. The importance of this work relies on the fact that a precise determination of the gluon condensate from lattice is fundamental for our understanding of low-energy behavior of QCD. For this analysis is important to have a precise determination of the constant . Boucaud’s group produced an approach to this aim. Similarly, Andre Sternbeck showed how this important constant could be obtained by a proper definition of the running coupling and he showed a very fine agreement with the result of Boucaud’s group.

Finally, I would like to remember the talk of Valentine Zakharov. I talked extensively about Valentine in my previous blog’s entries. His discoveries in this area of physics are really fundamental and so it is important to have a particular attention to his talks. Substantially, he mapped scalar fields and Yang-Mills fields to get an understanding of confinement! As I am a strong supporter of this view, as my readers may know from my preceding posts, I was quite excited to see such a an idea puts forward by Valentine.

As conference’s program unfolds I will take you updated with an eyes toward the aspects that are relevant to my work. Meantime, I hope to have given to you the taste of the excitement this area of research conveys to us that pursue it.