Sannino and the mass gap in Yang-Mills theory


August is vacation month in Italy and I am not an exception. This is the reason for my silence so far. But, of course, I cannot turn off my brain and physics has always been there. So, reading the daily from arxiv today , I have seen another beautiful paper by Francesco Sannino (you can find his page here) in collaboration with Joseph Schechter that has been his PhD thesis advisor. As you know, Sannino and Ryttov postulated an exact beta function for QCD starting from the exact result in the supersymmetric version of this theory (see here).  The beta function Sannino and Schechter get has a pole. The form is


and, taken as is, this has no fixed point than the trivial one g=0. We know that this seems in agreement with recent lattice computations even if, discussing with Valentin Zakharov at QCD10 (see here), he expressed some skepticism about them.  They point out that the knowledge of this function permits a lot of interesting computations and what they do here is to get the mass gap of the Yang-Mills theory. They also point out as, for all the observables obtainable from such a beta function, the pole is harmless and the results appear really meaningful. Indeed, they get a consistent scenario from that guess.

So, let me point out the main results obtained so far by these people using this approach:

  • The beta function for Yang-Mills theory goes to zero with the coupling without displaying non-trivial fixed point but QCD has a non-trivial fixed point (see my paper here).
  • Yang-Mills theory has a mass gap.

Numerically their result for the mass gap seems to agree quite well with lattice computations. This should be also the mass for a possible observation of the lightest glueball. My view about is that the lightest glueball is the \sigma resonance and recent findings at KLOE-2 seems to point out in this direction. But the exact value of the mass gap is not so relevant. What is relevant is that these researchers have found a quite interesting exact form of the beta function for QCD that describes quite well the current understanding of this theory at lower energies that is slowly emerging.


Mapping is confirmed by lattice computations!


Rafael Frigori is a reader of this blog and I have had a lot of very interesting opinion exchanges with him here.  He belongs to a group of people in Brazil doing groundbreaking work in lattice computations of gauge theories obtaining cornerstone results. Beside him, I would also like to cite Attilio Cucchieri and Tereza Mendes that helped to improve significantly our current understanding on the way Yang-Mills theory behaves at low energies. This time Rafael has done an excellent work to show that Yang-Mills theory in d=2+1 indeed maps on a scalar field theory displaying the same mass spectrum. Actually, this is exactly the content of my mapping theorem that I used to prove that Yang-Mills theory in a strong coupling limit shows a mass gap. You can find Rafael’s paper here. Mapping theorem was firstly proposed by me here and, after Terry Tao pointed out a problem in the proof (see here), the question was finally settled here. Both these papers went published in Physics Letters B and Modern Physics Letters A respectively. The former gives the consequences of this theorem showing how the mass gap can be obtained.

Lattice computations are an essential tool today toward our understanding of quantum field theory in limits where known mathematical techniques fail. So, to see our mathematical result at work in a lattice computation is really striking and open the path toward a new set of mathematical tools to manage these theories in unexpected regimes. This can be beneficial to any area of high-energy physics ranging from string theory to phenomenology. This gives a hint of the importance of Rafael’s paper. It is like a Pandora box is started to be open!

Why is so important to map theories? The main reason to derive mapping is to reduce a complex theory to a simpler one that we are able to manage. In this case, the conclusion is that Yang-Mills theory may belong to the same universality class of the scalar field theory and the Ising model in the infrared limit. This implies that a wealth of results can be immediately taken from a theory to another. What makes the question interesting is the fact that one knows how to manage a scalar field theory in the infrared limit. In a paper I have got published in Physical Review D (see here) I was able to present such techniques deriving the propagator and the spectrum of the theory in this limit.

Having the propagator of the theory gives immediately an effective theory to do computations in the low energy limit. I have had the chance, quite recently, to be in Montpellier thanks to the invitation of Stephan Narison. Stephan organized a very beautiful workshop (see here). You can find all the talks (also mine) here. In this talk I show how computations at low energies for strong interactions can be done. This is a matter I am still working on.

I take this chance to thank Rafael very much for this paper that gives a serious evidence of the correctness of my work and, at the same time, opens up a new significant way toward our understanding of infrared physics.

A really interesting view about QCD and AdS/CFT


Stan Brodsky is a renowned physicist that has produced a lot of very good works. As I work on QCD, I try to be up-to-date as much as possible and I spend some time to read the most recent literature about. AdS/CFT applied to QCD is a very hot topic these times and I run into a beautiful paper by Stan and Guy de Téramond that was recently published in Physical Review Letters (a preprint is here). Their work is inspired by AdS/CFT in that they are able to map on a five dimensional Anti-de Sitter space a light-front Hamiltonian for QCD, producing a Schrödinger-like equation with a proper potential to get the spectrum of the theory. This equation is depending by a single proper variable and is exactly solvable. Two classes of models can be identified in this way that are those well-known in literature:

  • Hard-wall model with a potential described by an infinite potential wall till a given cut-off that fixes the mass scale.
  • Soft-wall model with a harmonic potential producing Regge trajectories.

So, these authors are able to give a clever formulation of two known models of QCD obtained from AdS/CFT conjecture and they manage them obtaining the corresponding spectra of mesons and baryons. I would like to emphasize that the hard-wall model was formulated by Joseph Polchinski and Matthew Strassler and was instrumental to show how successful AdS/CFT could be in describing QCD spectrum. This paper appeared in Physical Review Letters and can be found here. Now, leaving aside Regge trajectories, what Stan and Guy show is that the mass spectrum for glueballs in the hard wall model goes like

m_n\approx 2n+L

being n an integer and L  the angular momentum. This result is interesting by its own. It appears to be in agreement both with my recent preprint and my preceding work and with most of the papers appeared about Yang-Mills theory in 2+1 dimensions. Indeed, they get this spectrum being the zeros of Bessel functions and the cut-off making the scale. Very simple and very nice.

I should say that today common wisdom prefers to consider Regge trajectories being hadron spectroscopy in agreement with them but, as glueballs are not yet identified unequivocally, I am not quite sure that the situation between a soft wall and hard wall models is so fairly well defined. Of course, this is a situation where experiments can decide and surely it is just a matter of a few time.

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.

Gluon condensate


While I am coping with a revision of a paper of mine asked by a referee, I realized that these solutions of Yang-Mills equations implied by a Smilga’s choice give a proof of existence of a gluon condensate. This in turn means that a lot of phenomenological studies carried out since eighties of the last century are sound as are also their conclusions. E.g. you can check this paper where the authors find a close agreement with my findings about glueball spectrum. The ideas of these authors are founded on the concept of gluon and quark condensates. As they conclusions agree with mine, I have taken some time to think about this. My main conclusion is the following. If you have a gluon condensate, the theory should give \langle F\cdot F\rangle\ne 0 being F_{\mu\nu}^a the usual gluon field. So, let us work out this classically. Let us consider a scalar field mapped on the gluon field in such a way to have

A_\mu^a(t)=\eta_\mu^a \Lambda\left(\frac{2}{3g^2}\right)^\frac{1}{4}{\rm sn}\left[\Lambda\left(\frac{3g^2}{2}\right)^\frac{1}{4}t,i\right]

being sn a Jacobi snoidal function, and \eta_\mu^a a constant array of elements obtained by a Smilga’s choice. When you work out the product F\cdot F the main contribution will come from the quartic term producing a term \langle \phi(t)^4 \rangle. Classically, we substitute the average with \frac{1}{T}\int_0^T dt being the period T=4K(i)/[\Lambda\left(\frac{3g^2}{2}\right)^\frac{1}{4}]. The integration is quite straightforward and gives

\langle \phi(t)^4 \rangle=\frac{\Gamma(1/4)^2}{18K(i)\sqrt{2\pi}}\frac{\Lambda^4}{4\pi\alpha_s}

I will evaluate this average in order to see if the order of magnitude is the right one with respect to the computations carried out by Kisslinger and Johnson. But the fact that this average is indeed not equal zero is a proof of existence of the gluon condensate directly from Yang-Mills equations.

The width of the sigma computed


One of the most challenging open problems in QCD in the low energy limit is to compute the properties of the \sigma resonance. The very nature of this particle is currently unknown and different views have been proposed (tetraquark or glueball). I have put a paper of mine on arxiv (see here) where I compute the large width of this resonance obtaining agreement with experimental derivation of this quantity. I put here this equation that is

\Gamma_\sigma = \frac{6}{\pi}\frac{G^2_{NJL}}{4\pi\alpha_s} m_\sigma f_\pi^4\sqrt{1-\frac{4m_\pi^2}{m_\sigma^2}}

being G_{NJL}=3.76\frac{4\pi\alpha_s}{\sigma} being \sigma the string tension generally taken to be 440 MeV and f_\pi\approx 93\ MeV the pion decay constant. The agreement is obtained with \alpha_s\approx 2 giving a consistent result. This is the first time that this rate is obtained from first principles directly from QCD and gives an explanation of the reason why this resonance is so broad. The process considered is \sigma\to\pi^+\pi^- that is dominant. Similarly, the other seen process, \sigma\to\gamma\gamma, has been interpreted as due to pion rescattering.

On the basis of these computations, this particle is the lowest glueball state. This is also consistent with a theorem proved in the paper that mixing between glueballs and quarks, in the limit of a very large coupling,  is not seen at the leading order. This implies that the spectrum of pure Yang-Mills theory is seen experimentally almost without interaction with quarks.

Screening masses in SU(3) Yang-Mills theory


Thanks to a useful comment by Rafael Frigori (see here) I become aware of a series of beautiful papers by an Italian group at Universita’ della Calabria. I was mostly struck by a recent paper written by R. Fiore, R. Falcone, M. Gravina and A. Papa (see here) that appeared in Nuclear Physics B (see here). This paper belongs to a long series of works about the behavior of Yang-Mills theory at non-null temperature and its critical behavior. Indeed, using high-temperature expansion and Polyakov loops one arrives at the main conclusion that the ratio between the lowest and the higher state of the theory must be 3/2. This ratio depends on the universality class the theory belongs to and so, on the kind of effective theory one has in the proper temperature limit (below or above T_c). It should be said that, in order to get a proper verification of the above prediction, people use lattice computations. Fiore et al. use a lattice of 16^3 \times 4 points and, as all this kind of computations are done on lattices having such a dimension, one can cast some doubt about the fact that the true ground state of the theory is really hit. Indeed, this happens in all this kind of computations done to get a glueball spectrum that seem at odd with those giving the gluon propagator producing a lower screening mass at about 500 MeV (see my post here). A state at about 500 MeV is seen at accelerator facilities as \sigma resonance or f0(600) but is not predicted by any lattice computation. One of the reasons to reduce lattice volume is that one can reach higher values of \beta granting the reaching of a non-perturbative regime, the one interesting for us.

What can we say about this ratio with our theory? We have put on arxiv a paper that answer this question (see here). These results were also presented at QCD 08 in Montpellier (see here). We assume that the \sigma cannot be seen at such small volumes but its excited state \sigma^* can be obtained. This implies that one can exchange the \sigma^* with the lowest state and 0^+ as the higher one. Then this ratio gives exactly 3/2 as expected. We can conclude on the basis of this analysis that this ratio is the same independently on the temperature but, the one to be properly measured is given in the paper of Craig McNeile (see here) that gives close agreement between lattice and theoretical predictions.

So, we would like to see lattice computations of Yang-Mills spectra at lower lattice spacing and increased volumes granting in this way the proper value of the ground state. This is overwhelming important in view of the fact that no real understanding exists of the existence of the \sigma resonance with lattice computations. This will implies, as discussed above, a deeper understanding of the spectrum of the theory also at higher temperatures.

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