A sound confirmation of Yang-Mills scenario


Today arxiv contains a very interesting paper by Attilio Cucchieri and Tereza Mendes (see here). They do a significant lattice computation for a SU(2) Yang-Mills theory in 3 and 4 dimensions. Quarks are absent (quenched approximation). Their aim is to verify the current emerging scenario in a particular situation where the coupling on the lattice is taken to go to infinity (\beta=0). This case, as pointed out by the authors, is unphysical but is a quite interesting test-bed for the behavior of the two-point functions of the theory (gluon and ghost propagators). They do not aim to check the running coupling as this, currently, is matter of hot debate yet.

I would like to remember what is at stake today about this question.  For some years there has been the strong conviction that the gluon propagator should go to zero and ghost propagator should go to infinity faster than the free propagator (conformal solution). This was needed to confirm two confining scenarios that were commonly accepted by the community. Most of these results emerged from an idea due to Gribov that there remains an ambiguity in the gauge also after it is fixed. Taking into account Gribov ambiguity provoked a flourishing of papers all reaching similar conclusions. At that time, due to the small achievable volumes, lattice computations were not able to clarify the situation even if the gluon propagator was never seen to tend to zero in a significant way. As increasing volumes made available a completely new scenario emerged. People obtained that the gluon propagator indeed reaches a finite value at zero momentum while the ghost propagator is seen to behave as that of a free particle (decoupling solution). A commonly accepted definition of running coupling was seen to converge to zero making the theory trivial.

This emerging data prompted several explanations. People started to criticize these lattice computations as maybe there was an accumulation of Gribov copies that modify the right results into the observed ones. Maybe the fixing gauge algorithms should be better analyzed and so on. People from Australia (see here and here) claimed that low energy data should not be trusted. Discarding them one finds again the conformal solution. Cucchieri and Mendes give a sound answer to all these doubts. Indeed, it is not clear why in 2 dimensions one gets the conformal solutions but not in 3 and 4 dimensions, notwithstanding the code used to do these computations is always the same. Further, any reason adduced by Australian group to remove low energy data is proved substantially unfounded and the results obtained by Cucchieri and Mendes represent a correct picture of the case \beta=0 for 3 and 4 dimensions. Indeed, Cucchieri and Mendes show that Gribov copies play no role in the scenario seen at low energies for Yang-Mills theory. This is a crucial point that has been source for misleading research for a lot of years.

So, let us take a look at the scenario found by Cucchieri and Mendes. These authors consider a maximum lattice dimensions of 100^3 for 3 dimensions and 64^4 for 4 dimensions. They show without any doubt that one gets the decoupling solution: The gluon propagator reaches a finite value at zero momentum and the ghost propagator is that of a free particle. What is more interesting here are the fits. For the propagator they fit to a sum of Stingl-forms

D(x)=\sum_{i=1,2}c_i\cos(b_i+\lambda_i x)e^{-\lambda_i x}

and for the ghost propagator


being m the gluon mass. The authors tried to avoid to fix the values of their computations with experimental data. As you know, the relevant parameter here is \sqrt{\sigma}, the string tension. Notwithstanding this operative choice, they get for the gluon mass the following values

m=0.499 GeV

using only data with p^2<4 GeV and

m=0.466 GeV

using all data. I hope that now some bell is ringing for you as this is the mass of the \sigma resonance. This resonance is not seen by people doing quenched computations to obtain the spectrum of a pure Yang-Mills theory. Why? What are they missing with respect to Cucchieri and Mendes? This should not become a longstanding question. We need an answer right now.

Now, take a look at the fit of the gluon propagator. Try to do a Fourier transform and you will get back something like


and this is shockingly similar to my propagator having the general form


that is a sum of free particle propagators!

I should say that I am somewhat impressed by Cucchieri and Mendes results. They showed that the decoupling scenario is the right one in the physical case of 4 dimensions. My view is that we should move on from the current position and try to find the theoretical framework that better fits the data. It goes without saying what is the one I am supporting.


Gluon condensate: The situation


Stephan Narison agreed to contribute to my blog with the following lines describing the current situation about the gluon condensate. It is a pleasure for me to put them here.

The gluon condensate \alpha_s G^2  introduced by Shifman-Vainshtein-Zakharov (SVZ) [see also Zakharov, contribution at Sakurai’s prize 1999: Int. J. Mod. Phys A14 (1999)4865 (here)] within the framework of QCD spectral sum rules  (QSSR) plays also an important role in gluodynamics. The original value of 0.04  GeV^4  obtained by SVZ from charmonium sum rules has been shown by Bell-Bertlmann (BB) to be underestimated by about a factor 2 from their analysis of the non-relativistec version of heavy quark sum rules. The BB result has been confirmed later on from QSSR analyzes of  different channels including  e^+e^-  into hadrons, tau-decay and charmonium by different groups, where the most recent value of  (0.07\pm 0.01) GeV^4 has been obtained [see for a review my 2 books (here and here) and the last paper on tau-decay: PLB673(2009)30 (here) ]. However, in order to extract reliably this (small) quantity one should work with sum rule which can properly disentangle its contribution where some possible competing contributions due to perturbative radiative corrections and to quark mass should not appear. This feature may explain some results in the literature. The non-vanishing of the gluon condensate and its positive sign has been seen in the lattice by  the Pisa group [A. Di Giacomo, G.C. Rossi, PLB100(1981)481 (here);  M. Campostrini, A. Di Giacomo, Y. Gunduc, PLB225(1989)393 (here)] and more recently by P.E. Rakow (here). Its positive sign is expected in a model with a magnetic confinement (H. Nambu), while phenomenologically, its eventual negative value would leave to serious inconsistencies in the QSSR approach. Its negative and non-universal values obtained from some tau-decays analysis can indicate the difficulty to extract its value among the competitive parameters present there where in the tau-decay width the gluon condensate contribution acquires an extra \alpha_s contribution compared to some other non-perturbative contributions. In fact, this peculiar properties have been the most important observation that non-perturbative contributions are small in this observable, then allowing an accurate determination of \alpha_s from tau-decays.

Yang-Mills theory in D=2+1


There is a lot of work about the pursuing of a deep understanding of Yang-Mills theory in the low energy limit. The interesting case is in four dimensions as our world happens to have such a property. But we also know that a Yang-Mills theory in D=2+1 is not trivial at all and worthwhile to be studied. In this area there has been a lot pioneering work mostly due to V. Parameswaran Nair and Dimitra Karabali . These authors proved that a Hamiltonian formulation may be truly effective to manage this case. Indeed, they obtained a formula for the string tension that works quite well with respect to lattice computations. We would like to remember that, in D=2+1, coupling constant is such that its dimension is [g^2]=[E] while, in D=3+1, is dimensionless.

Quite recently, some authors showed how, from such a formulation, a functional can be given from which one can obtain the spectrum (see here, here and here). These papers went all published on archival journals. Now, these spectra are quite good with respect to lattice computations, after some reinterpretation. We do not know if this is due to some problems in lattice computations or in the theoretical analysis. I leave this to your personal point of view. My idea is that this quenched lattice computations are missing the true ground state of the theory. This happens to be true both for D=3+1 and D=2+1. I do not know why things stay in this way but in this kind of situations are always theoreticians to lose. On the other side, being a physicist means that one should not have a blind faith in anything.

Finally, one may ask how my work performs with respect all this. Yesterday, I spent a few time to try to figure this out. The results I obtain agree fairly well with those of the theoretical papers. I obtain the zero Lionel Brits gets at 0.96m being m a mass proportional to ‘t Hooft coupling. Brits wrote the third of the three papers I cited above. The string tension I get is in agreement with lattice computations. This zero is the problem on lattice computations and the same problem is seen in D=3+1. This fact is at the root of our presenting difficulty to understand what \sigma resonance is. We know that people working on a quenced lattice computation for the propagator do see this resonance. This difference between this two approaches should be understood and an effort in this direction must be made.

QuAD constrains parity violation on a cosmic scale


QuAD is a collaboration that involves several institutions that aims to measure polarization of photons of the Cosmic Microwave Background radiation that is the relic radiation that started to freely propagating in the universe 400000 years after big bang. QuAD aerialThis radiation is a precious source of information for a deeper understanding of the birth of our universe and a fundamental test-bed for our theories. This collaboration proved that our “standard model” of the universe again agrees with their measurement about polarization of this radiation. But, from these data, physicists are able to get something more. In a paper appeared in the last number of PRL (see here and here), they showed that electrodynamic forces are proved to not violate parity on a cosmic scale and, last but not least, interactions that violate Lorentz invariance are strongly constrained.

I should say that this is an important conclusion to be drawn from an elegant experiment and an unexpected source.

Hawking hospitalized


Today I report bad news. Stephen Hawking, one of the greatest physicists, has been hospitalized and reported being “very ill”. Stephen HawkingYou can find the news here. Hawking is largely known for his works about singularities in general relativity and for the understanding he reached in the behavior of black holes when one takes into account quantum behavior. He become widely known to the public at large writing a book, ” A Brief History of Time”, that is a best-seller.

It is not possible to talk about Hawking without citing the disease he is affected. This is ALS also known as Lou Gehrig’s disease from the name of the baseball player that had this illness.

Stephen Hawking is the holder of the Lucasian chair at University of Cambridge. He is expected to retire later this year.

My wish is that he can turn back to his activities in a very short time.

Update: Expected full recovery for Hawking, Cambridge University has said. Here the news.

A striking confirmation


On arxiv today it is appeared a paper by Stephan Narison,  Gerard Mennessier and Robert Kaminski (see here). Stephan Narison is the organizer of QCD Conferences series and I attended one of this, QCD 08, last year. Narison is located in Montpellier (France) and, together with other researchers, is carrying out research aimed to an understanding of low-energy phenomenology of QCD. So, there is a strong overlapping between their work and mine. Their tools are QCD spectral sum rules and low energy theorems and the results they obtain are quite striking. Narison has written a relevant handbook of QCD (see here) that is a worthwhile tool for people aimed to work with this theory.

The paper gives further support to the idea that the resonance f0(600)/\sigma is indeed a glueball. Currently, researchers have explored another possibility, that this particle is a four quark state. Narison, Mennessier and Kaminski consider that, if this would be true, being this a state with u and d quarks, coupling with K mesons should be suppressed. This would imply that, in a computation for the rates of \sigma decays, the contribution coming in the case of K mesons in the final state should be really small. But, for a glueball state, these couplings for \pi\pi and KK decays should be almost the same.

Indeed, they get the following

|g^{os}_{\sigma\pi +\pi -}|\simeq 6 GeV, r_{\sigma\pi K}\equiv \frac{g^{os}_{\sigma K+K-}}{g^{os}_{\sigma\pi +\pi -}} \simeq 0.8

that is quite striking indeed. They do the same for f0(980) and, even if they get a similar result, they draw no conclusion about the nature of this resonance.

This, together with the small decay rate in \gamma\gamma, gives a really strong support to the conclusion that \sigma is indeed a glueball. At this stage, we would like to see an improved support from lattice computations. Surely, it is time to revise some theoretical computations of the gluon propagator.

Update: I have received the following correction to above deleted sentence by Stephan Narison. This is the right take:

One should take into account that the sigma to KK is suppressed due to phase space BUT the coupling to KK is very strong. The non-observation of sigma to KK has been the (main) motivation that it can be pi-pi or 4-quark states and nobody has payed attention to this (unobserved) decay.

Cabibbo and quark mixing


Today I have been at Accademia dei Lincei, a renowned institution located here in Rome. There was a talk by Nicola Cabibbo about quark mixing. This conference was addressed also to people not having a proper knowledge of physics, making quite exciting to listen. Cabibbo is member of the Accademia since 1973 and his groundbreaking discovery happened in 1963 (see here). There has been some fuss last year with Nobel prize awarded to Kobayashi and Maskawa but not to Cabibbo (see my post here). So, it was a chance to hear from his voice his personal recollections about all this matter.

He gave a historical overview about quark mixing and how he found himself on the right path. Then, he cited when, on 1963, he met Feynman that said him that things could not work as he got V+A in some cases rather than V-A. Indeed, confirmations arrived later.

Giorgio Salvini, an experimental physicist well-known for his contributions to high-energy physics, introduced Cabibbo recalling as, when in Frascati, after Kobayashi and Maskawa got published their paper, he hit his forehead saying “How did not I think about six quarks?”. Indeed, Cabibbo recalled when, in the sixties, he was in his office at Rome University “La Sapienza” and Shelly Glashow knocked on his door. Glashow asked to Cabibbo if his approach could be used to understand CP violation. At that time, Glashow was frequently in Rome and, some time after, he took Luciano Maiani with himself at Harvard and we all know as history went. To obtain CP violation from Cabibbo mechanism one should introduce a complex phase factor. At that time no more than four quarks were postulated. So, Cabibbo answered that was not possible as, whatever phase one introduces, this can be removed. With six quarks this is no more possible and so, he said, having not thought about six quarks, he missed in this way the possibility to uncover the full CKM matrix.

In the end of the conference, some questions were put forward and Cabibbo promised to put all the talk on paper, after Giorgio Salvini solicited him for doing this. I hope to read his paper. This is a must after such a beautiful talk.

The question of the running coupling


Running coupling is an important quantity in quantum field theory. It identifies the behavior of an interacting theory in the given limit. Indeed, its importance emerged in a full glory after Gross, Wilczek and Politzer showed that QCD becomes a free theory at high energies (asymptotic freedom). For this reason, they were awarded a well deserved Nobel prize in 2004 (see here). Their result implies that the strong interaction coupling goes to zero as the energy involved in the particle interactions becomes larger. Significant experimental confirmations emerged in the course of time but this result already explained the success of Bjorken scaling. The work of Gross, Wilczek and Politzer served also to convince the community that QCD was the right theory to explain strong interactions. This was a long sought result. This matter is so well-acquired today that people is able to work out higher order corrections to this result and found it in close agreement with experimental evidence. So, you can see that this is the story of a great success in physics.

Then, one may ask what happens in the low energy limit. Here we are in troubles as there are no acquired techniques to manage QCD than working with a computer on a lattice. But the situation is not yet at our hands also using computers. The reason relies on the fact that no generally accepted definition of a running coupling constant exists for the low energy limit. But a lot of prejudices exist about. A firm conviction of a large part of the community is that a meaningful running coupling for strong interactions should reach a fixed point as the energy becomes smaller. The reason for this is that, being these interactions so strong, a coupling cannot be zero. Of course, this is wishful thinking and we have no proof whatsoever that things stay that way. So, people work the other way round looking for a definition that grants the existence of a fixed point. An example of this can be found in this paper.

When I read something like this “I reach for my gun” as Hawking would say. The reason is that, in the course of time, we have gathered a lot of expectations and would-be about Yang-Mills theory and QCD in the infrared that we are no more able to distinguish between what is proved and what is not and, mostly, when we are just trying to give a chance to our wishful thinking to be reality. Till now, nobody was able to let us know what are the right excitations of the theory in the low energy limit and so, there is no reason on Earth to believe that the running coupling do not reach zero value lowering the energy. So, your theory could admit bound states with zero charge but surely you do not need a fixed point running coupling here.

But if you look back to all that people believing that glueballs are just bound states of gluons, they are in strong need for an infrared fixed point of the theory. Clearly, they hope that describing the theory with two wrong concepts may turn it into a rigth one. Things do not stay that way and evidence is coming out that such bound states that diagonalize the theory are indeed with zero charge. Bound states anyway.

Exact solutions of Yang-Mills theory: The situation


Some time passed by since Terry Tao was so kind to take a look to my work. His concern about a main theorem in my paper, the so called mapping theorem, was motivated by the fact that no proof exists that there are common solutions between Yang-Mills equations and the one of the quartic scalar field. This point is quite crucial as, if such solutions do not exist, I cannot do any claim about Yang-Mills theory.

Some people are in confusion yet about this matter and I find occasionally someone, e.g. the Czech guy, claiming that my paper is false also after I have proved that such solutions exist.

Of course, Terry meant to point out a weakness in the proof given in my paper as I gave no evidence whatsoever of the form of these solutions and so the proof is, at least, incomplete. My next preprint proved that such solutions indeed exist and my argument is true already at level of perturbation theory. The conclusion is straightforward: Smilga’s choice select a class of common solutions between Yang-Mills equations and a quartic scalar field. I have not presented them explicitly in my paper and this is the reason why all this arguing was started. Terry’s suggestion was to complete the proof  and this I have done.

Curiously enough, I was able to see such solutions only in the Smilga’s book. I think this was Smilga’s idea and was also my source of inspiration.  I was in need of these solutions to treat classical Yang-Mills equations with a gradient expansion against a lot of unmanageable chaotic solutions. I would like to remember here that this approach is quite common in physics. For interested readers, I invite them to look at this beautiful Wikipedia entry about BKL solution. This is the way this approach is used in general relativity with a widespread example as the Kasner solution. This is an exact solution of Einstein equations that depends solely on time. Exactly as happens to the solutions obtained by a Smilga’s choice from Yang-Mills equations. Indeed, I suspect that Kasner solution may be helpful to quantize Einstein equations in the infrared limit. Currently I have no time to exploit this but I have given a hint about here.

Dmitry Podolsky (see his blog here) hit correctly the point when asked for the fate of chaotic solutions in the infrared quantum field theory. Presently, the fact that they are not relevant has the status of a conjecture: No quantum field theory can be built out of classical chaotic solutions. I do not even know how to face this kind of question as no closed form chaotic solutions exist to start from.

Finally, this gives the current situation about this matter. My paper that started all this is correct and in agreement with current lattice results. People’s mood about lattice computations range from fully convinced to skeptical.  My view is that they represent correctly the infrared physics at hand but I am a supporter of these people working on lattice computations and so, my judgement should not be counted.

Gluons are not all the story


In a short time, Physical Review Letters will publish a shocking paper by Dan Pirjol and Carlos Schat with a proof of the fact that a simple gluon exchange model for bound states of QCD does not work. The preprint is here. The conclusions drawn by the authors imply that one cannot expect a simple idea of free gluons exchanged by quarks to work. I think the readers of this blog may be aware of the reason why this conclusion is correct and PRL will publish an important paper.

The idea is that, in the low energy limit, the nonlinearities in the Yang-Mills equations modify completely the properties of the glue. We can call these excitations gluons only in the high energy limit were asymptotic freedom grants that nonlinearities can be treated as small perturbations and gluons are what we are acquainted with. But when we have to cope with bound states, we are in a serious trouble as our knowledge of this regime of QCD is really very few helpful for our understanding.

This result of Pirjol and Schat should be taken together with the measurements of the COMPASS Collaboration (see my post) about the spin of the protons. They proved that glue does not contribute to form the spin of the proton. Collecting together all this a conclusion to be drawn is that the high energy excitations of a Yang-Mills theory cannot be the same of the excitations in the low energy limit.

So, let’s move on and take a better look at our equations.

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