Yang-Mills theory paper gets published!



Exact solutions of quantum field theories are very rare and, normally, refer to toy models and pathological cases. Quite recently, I put on arxiv a pair of papers presenting exact solutions both of the Higgs sector of the Standard Model and the Yang-Mills theory made just of gluons. The former appeared a few month ago (see here) while the latter has been accepted for publication a few days ago (see here). I have updated the latter just today and the accepted version will appear on arxiv on 2 January next year.

What does it mean to solve exactly a quantum field theory? A quantum field theory is exactly solved when we know all its correlation functions. From them, thanks to LSZ reduction formula, we are able to compute whatever observable in principle being these cross sections or decay times. The shortest way to correlation functions are the Dyson-Schwinger equations. These equations form a set with the former equation depending on the higher order correlators and so, they are generally very difficult to solve. They were largely used in studies of Yang-Mills theory provided some truncation scheme is given or by numerical studies. Their exact solutions are generally not known and expected too difficult to find.

The problem can be faced when some solutions to the classical equations of motion of a theory are known. In this way there is a possibility to treat the Dyson-Schwinger set. Anyhow, before to enter into their treatment, it should be emphasized that in literature the Dyson-Schwinger equations where managed just in one way: Carl BenderUsing their integral form and expressing all the correlation functions by momenta. It was an original view by Carl Bender that opened up the way (see here). The idea is to write the Dyson-Schwinger equations into their differential form in the coordinate space. So, when you have exact solutions of the classical theory, a possibility opens up to treat also the quantum case!

This shows unequivocally that a Yang-Mills theory can display a mass gap and an infinite spectrum of excitations. Of course, if nature would have chosen the particular ground state depicted by such classical solutions we would have made bingo. This is a possibility but the proof is strongly related to what is going on for the Higgs sector of the Standard Model that I solved exactly but without other matter interacting. If the decay rates of the Higgs particle should agree with our computations we will be on the right track also for Yang-Mills theory. Nature tends to repeat working mechanisms.

Marco Frasca (2015). A theorem on the Higgs sector of the Standard Model Eur. Phys. J. Plus (2016) 131: 199 arXiv: 1504.02299v3

Marco Frasca (2015). Quantum Yang-Mills field theory arXiv arXiv: 1509.05292v1

Carl M. Bender, Kimball A. Milton, & Van M. Savage (1999). Solution of Schwinger-Dyson Equations for ${\cal PT}$-Symmetric Quantum Field Theory Phys.Rev.D62:085001,2000 arXiv: hep-th/9907045v1

The XV Workshop on Statistical Mechanics and nonperturbative Field Theory



This week I was in Bari as the physics department of that university organized a major event: SM&FT 2011. This is a biennial conference having the aims to discuss recent achievements in fields as statistical mechanics and quantum field theory that have a lot of commonalities. The organizers are well-known physicists and so it was a pleasure for me to see my contribution accepted. Leonardo Cosmai wrote to me confirming my partecipation. Leonardo, together with Paolo Cea, Alessandro Papa and Massimo d’Elia produced a lot of significant works in quantum field theory and a recent paper by Cosmai and Cea arose some fuzz also in the blogosphere (see here). Their forecast for the Higgs boson agrees quite well with my view about this matter. They were also part of the organizing committee. Of course, I was in Bari with my friend Marco Ruggieri that lived there for more than twelve years gaining a PhD in that university.

The scientific content was really interesting an I have had the chance to learn something more about lattice field theory. You can find all the talks here. About this, it should be said that people work with small lattices yet. While this has been a natural way to manage the QCD on the lattice due to missing computational resources, things are rapidly changing due to CUDA as I discussed a lot in my blog and was presented in some talks at this conference. Small groups will be able, with very few bucks of their budgets, to reach a significant ability to analyze increasingly lattice volumes. Besides, also large scale projects in this direction, mostly due to INFN and extending the APE project originated by Nicola Cabibbo and Giorgio Parisi, were presented (see talks by Francesco di Renzo e Piero Vicini). A typical situation in this kind of lattice analysis, improved using CUDA,  was also pointed out by Massimo D’Elia in his talk.  Thanks to this new technology they are increasing significantly the volumes. You can compare the content of his talk with that of his collaborator Francesco Negro, discussing a really interesting problem on the lattice (and a promise for the future with CUDA), with smaller volumes due to reduced computational resources. The interest for the activity of this group and Francesco’s work is strongly linked to a paper that I and Marco Ruggieri wrote together about the QCD vacuum in presence of a magnetic field (see here). The work by Francesco, even if for small volumes, provides interesting conclusions. It should be said that the Nambu-Jona-Lasinio model is there well alive and kicking.

Petruzzelli Theater

From a strictly theoretical side, I would like to point out the talks by Giuseppe Mussardo, with which I have had a nice mail exchange and is author of some beautiful books (e.g. see here), and the ones by Adriano Di Giacomo and Valentin Zakharov that seem to have some relevant contact points with my work. There was also a talk by Edward Shuryak, one of the proponents of the instantons liquid for the QCD vacuum that is strongly supported by lattice simulations and theoretical works like mine.

At the end of the social dinner, we have had some interesting discussions with Di Giacomo and Cosmai. There was some excitation about the announced seminar about neutrinos by CERN and INFN. In a pub after the dinner, I have had some interesting discussions about a proposal by Michele Pepe and others (see his talk) that holds the promises to improve significantly lattice computations removing artifacts. It was also the chance to hear the point of view of Owe Philipsen (see his talk) about the current situation on lattice simulations on QCD at finite temperature. As I have discussed in some posts in this blog, this kind of simulations are plagued by the infamous sign problem and most of the work turns back to try to get rid of it. My friend Marco expressed the somewhat pessimistic view that a critical endpoint will never be seen on lattice computations. Indeed, he is the proponent of the use of a chiral chemical potential that does not display this stumbling block on the lattice (see his talk). This approach holds the promises to reach the goal as he showed in a recent paper. His proposal is under scrutiny by the lattice community. The QCD critical endpoint is a Holy Grail for all of us working in this area as QCD displays a quite rich phase diagram and we have also a lot of experimental data in heavy ion collisions to understand. You should take a look both at the talks of Marco and Alessandro Papa.

I would like to have cited all the talks and I apologize for omissions. If my readers have some time to spend usefully just read it all, as the conference was well organized and with very interesting contents in a really nice atmosphere somehow excited by neutrino news in the last two days.

P. Cea, & L. Cosmai (2011). The trivial Higgs boson: first evidences from LHC arXiv arXiv: 1106.4178v1

Marco Frasca, & Marco Ruggieri (2011). Magnetic Susceptibility of the Quark Condensate and Polarization from Chiral Models Phys.Rev.D83:094024,2011 arXiv: 1103.1194v1

Marco Ruggieri (2011). The Critical End Point of Quantum Chromodynamics Detected by Chirally
Imbalanced Quark Matter Phys.Rev.D84:014011,2011 arXiv: 1103.6186v2

Solving Dyson-Schwinger equations


Sunday I posted a paper of mine on arxiv (see here). I was interested on managing a simple interacting theory with the technique of Dyson-Schwinger equations. These are a set of exact equations that permit to compute all the n-point functions of a given theory. The critical point is that a lower order equation depends on higher order n-point functions making the solution of all set quite difficult. The most common approach is to try a truncation at some order relying on some physical insight. Of course, to have a control on such a truncation could be a difficult task and the results of a given computation should be carefully checked. The beauty of these equations relies on their non-perturbative nature to be contrasted with the severe difficulty in solving them.

In my paper I consider a massless \phi^4 theory and I solve exactly all the set of Dyson-Schwinger equations. I am able to do this as I know a set of exact solutions of the classical equation of the theory and I am able to solve an apparently difficult equation for the two point function. At the end of the day,  one gets the exact propagator, the spectrum and the beta function. It is seen that this theory has only trivial fixed points. I was able to get these results on another paper of mine. So, it is surely comforting to get identical results with different approaches.

Finally,  I can apply  the mapping theorem with Yang-Mills theories, recently proved thanks also to Terry Tao intervention, to draw conclusions on them in the limit of a very large coupling. In the paper you can find a formulation of this theorem as agreed with Terry, a direct consequence of my latest accepted paper on this matter (see here).

I think this paper adds an important contribution to our understanding of Dyson-Schwinger equations presenting an exact non-trivial solution of them.

Who fears a non-perturbative Higgs field?


One of my preferred readings in the blogosphere is Tommaso Dorigo’s blog. I think this is a widely known blog for people interested about physics and got some citation also at New York Times. Quite recently he published a very interesting post (see here) about the fate of our loved Standard Model taking the move from a very nice paper by J.Ellis, J.R.Espinosa, G.F.Giudice, A.Hoecker, and A.Riotto (see here). These authors are well known and really smart at their work and, indeed, I have noticed this paper as it appeared in arxiv. My readers know that I work on a small part (QCD) of the whole picture arisen in sixties and seventies and I have never taken a look from outside. So, while I appreciated this paper I thought it was not the case to comment on it  in my blog. But reading Tommaso’s post some thoughts come to my mind and these are really pertinent.

People put out two kind of constraints on the Higgs part of the standard model to have an idea of what to expect. I give you here the Higgs potential for your needs


and one immediately realizes that it introduces two free parameters. The critical one is \lambda and let me explain why. When one does quantum field theory, the only real tool that she has to do any meaningful computation is small perturbation theory. The word “small” is never said but it should be said in any circumstance as this technique only works if you have a small parameter in your theory (a coupling) to use as a development parameter. Otherwise we are lost and all starts to become foggy and not so well-defined. Today, nobody knows how to manage a theory with a strong coupling. Parameter \lambda is exactly such a coupling and we are able to manage a Higgs field when this parameter is small. But when you do small perturbation theory in quantum field theory you realize immediately that infinities come out and you are not able to obtain meaningful results going beyond the first order. For the most interesting theories around we are lucky:  Schwinger, Tomonaga, Feynman and Dyson invented renormalization and this works to remove infinities at each order of perturbation theory in the Standard Model and also for the Higgs, if the coupling is small. We are so accustomed to such a situation that we think that this is all one needs to know to understand quantum field theory: Perturbation theory and renormalization. We think that small perturbation theory is the perturbation theory and nothing else. So, we hope also the Higgs field should fulfill such requirements. Indeed, we are already in trouble in QCD for these same reasons but I have discussed at lengthy such a situation before here and I do not want to repeat myself.

There is no reason whatsoever to believe that we know all one has to know to manage a quantum field theory. Higgs could as well be not that light and strong coupled and there is no reason to think that Nature chose the small coupling case to favor us. Of course, if things will not stay this way I will be happy as a light Higgs is favored by supersymmetry and I like supersymmetry. But I would like also to emphasize that we already have all we need to manage analytically a strong coupled Higgs field. This matter I have discussed widely here and in my published papers.

So, while we all agree that a light Higgs is favored my view is that we should not have any fear of a non-perturbative Higgs field.

Exact solutions on arxiv


As promised in my preceding post (see here), I have posted yesterday a preprint on this matter on arxiv (see here). I have presented all the exact solutions I was able to obtain at a classical level and I have given a formulation of the quantum field theory for a massless quartic theory. The key point in this case is the solution of the equation for the propagator


being \phi_c the given exact classical solution. As usual, I have used a gradient approximation and the solution of the equation


that I know when the phase in \phi_c(t,0) is quantized as (4n+1)K(i), being n an integer and K(i) an elliptic integral. This gives back a consistent result in the strong coupling limit, \lambda\rightarrow\infty, with my preceding paper on Physical Review D (see here).

The conclusion is rather interesting as quantum field theory, given from such subset of classical solutions, is trivial when the coupling becomes increasingly large as one has a Gaussian generating functional and the spectrum of a harmonic oscillator. This is in perfect agreement with common wisdom about this scalar theory. So, in some way, Jacobi elliptical functions that describe nonlinear waves behave as plane waves for a quantum field theory in a regime of a strong coupling.

Edward Witten


Today I was at the Festival of Mathematics 2009 in Rome to listen a talk by Edward Witten. festivaldellamatematica2009Witten is one of the greatest living physicists and his contributions to mathematics were so relevant that he was awarded a Fields medal. This was for me a great chance to see him personally and hear at his way of doing physics for everybody. This is a challenging task for anyone and mostly for the most relevant personalities of our community. I was there with my eleven years old son and two of his friends. Before the start of the talk, John Nash come out near our row of seats and my son and his friends suggested to go to him asking for an autograph. Indeed, he seemed in real difficulty as some people was around him asking for an handshaking or something else. Somebody took him away and this was a significant help.

I showed Witten immediately at my company. He was there speaking and greeting people around. He appeared a tall and a very cordial man.

Marco Cattaneo, deputy director of the Italian edition of Scientific American (Le Scienze), introduced Witten with a very beautiful and well deserved presentation. Witten of course speaks Italian being his wife Chiara Nappi, an Italian physicist. Witten started to talk in Italian saying that he was very happy to be in Italy to meet his wife parents but his Italian was not enough to sustain a talk like the one he was giving.

The talk was addressed to a generic public. It was very well presented and my company found it very interesting. Witten did not use any formulas rather than Einstein’s E=mc^2 and the parabola y=x^2 and this is enough to keep up the attention of the public for all the time.

Witten pointed out that quantum field theory represents the greatest achievement ever for physicists. This theory is so deep and complex that mathematicians still fail to go through it fully and most of these results, widely used by physicists, are presently out of reach for mathematical proofs. He also said clearly, showing it explicitly, that the problem implied in the vertexes of ordinary Feynman diagrams are removed by string theory making all the machinery less singular.

He did a historical excursus starting from Einstein and arriving to string theory. He showed the famous Anderson’s photograph blatantly proving the very existence of antimatter. A great success of the wedding between special relativity and quantum mechanics. This wedding produced such a great triumph as quantum field theory. Witten showed this with the muon magnetic moment, emphasizing the precise agreement between theory and experiment, but saying that the small discrepancy may be or not real new physics being just at 1\sigma.

He emphasized the long path it takes to physicists to achieve our present understanding of quantum field theory and cited several Nobel prize winners that gave key contributions for this goal. He pointed out the relevance of the seventies of the last century that become a cornerstone moment for our current view.

Starting from Gabriele Veneziano‘s insight, Witten arrived to our current view about string theory. He said that this theory has had some frailty aspects that put it, sometime, on the border of a gulch. But, as we know, recoveries happened. He said that strings set the rules and not the other way round as happens with the Standard Model. He gave the example of the Veneziano’s model for strong interactions that was there pretending spin two excitations. This made the model better suited for other aims as indeed happened.

Witten hopes that LHC will unveil supersymmetry. He showed a detector of this great accelerator that we will see at work at the end of this year. Discovery of supersymmetry will be a great achievement for humankind as it will be the first evidence for a world with more than four dimensions. Anyhow, Witten said, string theory put out several elements, quantum gravity and supersymmetry are two of them, that make this theory compelling.

After the talk, some questions by the public were about ten or eleven dimensions in string theory. Witten avoided to be too technical. But the most interesting question was the one by Marco Cattaneo. He asked about critics of string theory and its present inability to do predictions. Witten’s answer was quite unexpected. He said that it is a good fact that a theory has critics. It is some kind of praise for it. But he also said, and his answer was quite similar to the one of Nicola Cabibbo, that there are a lot of things to be understood yet but such a richness physicists run into cannot be just a matter of chance with no significance.

Surely, this has been a very well paid waiting!

Updated paper


After a very interesting analysis about classical solutions of Yang-Mills equations, in this blog and elsewhere in the web, and having recognized that a paper of mine was in great need for corrections (see here) I have finally done it.

I have replaced the paper on arxiv a few moments ago (see here). I do not know if it is immediately available or you have to wait for tomorrow morning. In any case, the only new result added, with respect to material already discussed in this blog, is the first order correction to the propagator of the massless scalar theory. This goes like 1/\lambda making all the argument consistent. This asymptotic series should be modified as the limit \lambda\rightarrow\infty becomes more and more difficult to be applied and this should be in a kind of intermediate region that, presently, I have no technique to manage. This is matter for future work. The perspective is the ability to recover the solution of a scalar field theory for all energy range.

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