…would be too massive? Today I was reading the beatiful post of Tommaso Dorigo about Tevatron results on the hunt for Higgs and an old idea come back to my mind. We all know why for the sake of Standard Model one need a small Higgs mass but, so far, such low mass particle has been not seen. We all are aware that the least result of LHC would be the unveiling of the generation mass mechanism in the Standard Model that is a must for the model to survive but we can have surprises. One of these surprises could be a too massive Higgs. This will mean that, at higher energies, our ability to do perturbation theory for electro-weak interactions will blatantly fail and such interactions would become as strong as strong interactions. But there is one more aspect of this situation that is somewhat unexpected. We have recently treated a strongly coupled Goldstone scalar field (here and here) and we have found in the strong coupling limit, besides the ordinary massless excitation, a tower of excited states and this is what one should observe if the Higgs particle is too massive. By “too massive” we mean a mass beyond 1 TeV. I have no checked yet but I believe that also in this case the classical theory admits an exact solution. More to say in the near future.
Update: There has been a post by Lubos Motl (see here) where he argues that Fermilab data favor a light Higgs and supersymmetry. Indeed I hope this will prove to be the right scenario because so we have our cake and we eat it! Finally, D0 Collaboration presented a press release and this seems an important step beyond to Higgs discover.
As I always do to take trace of some computations I am carrying on here and there, I put them on the blog. This time I devised to solve Dirac equation using the massive solution of the massless scalar field given here and here:
is an arbitrary parameter, the coupling and the snoidal Jacobi elliptical function. An arbitrary phase can be added but we take it to be zero in order to keep formulas simpler. Now, we couple this field to a massless fermion field and one has to solve Dirac equation
where we have used the following Yukawa model for the coupling
so that . Dirac equation with such a field can be solved exactly to give
being the mass acquired by the scalar field and and two other Jacobi elliptical functions. This formula says us an interesting thing, that is there is a fermion excitation with zero mass unless a mass is initially given to the fermion. Such a conclusion is reminiscent of the pion status in QCD. So, the computation may seem involved but the conclusion is quite rewarding!
Functional methods are techniques used in these years to manage Yang-Mills theory. This name arose from the various methods people invented to solve Dyson-Schwinger equations. These are a tower of equations, meaning by this that the equation for the two-point function will depend on the three point function and so on. These are exact equations: When you solve them you get all the hierarchy of n-point functions of the theory. So, the only way to manage them to understand the behavior of Yang-Mills theory at lower momenta is by devising a proper truncation of the hierarchy. A similar situation can be found in statistical mechanics with kinetic equations. For a gas we know that collisions with a higher number of particles give smaller and smaller contributions and we are able to provide a meaningful truncation of the hierarchy. For the Dyson-Schwinger equations, generally, we are not that lucky and the choice of a proper truncation can be verified only through lattice computations. This means that the choice of a given truncation scheme may imply an uncontrolled approximation with all the consequences of the case. A beatiful paper about this approach is due to Alkofer and von Smekal (see here). This paper has been published on Physics Report and describes in depth all the elements of functional methods for Yang-Mills theory. Alkofer and von Smekal proposed a truncation scheme for Dyson-Schwinger equations that provided the following scenario:
- Gluon propagator should go to zero at lower momenta.
- Ghost propagator should go to infinity faster than a free particle propagator at lower momenta.
- A proper defined running coupling should reach a fixed point in the infrared.
The reason why this view reached success is due to the fact that gives consistent support to currently accepted confinement scenarios. Today we know as the history has gone. Lattice computations showed instead that
- Gluon propagator reaches a non-zero value at lower momenta.
- Ghost propagator is practically the same of that of a free particle.
- Running coupling as defined by Alkofer and von Smekal goes to zero at lower momenta.
So, after years where people worked to support the scenario coming from functional methods, now the community is trying to understand why the truncation scheme proposed by Alkofer and von Smekal seems to fail. On this line of research, Axel Maas showed recently, with lattice computations, that for D=1+1 the scenario is exactly those Alkofer and von Smekal proposed (see here and here). So, now people is try to understand why for D=1+1 functional method seems to work and for higher dimensions this does not happen.
I think that these are not good news for functional methods. The reason of this is that a pure Yang-Mills theory in D=1+1 is trivial. Trivial here means that this theory has not dynamics at all! This result was obtained some years ago by ‘t Hooft (see here) and published on Nuclear Physics B. He showed this using light cone coordinates. Then, by eliminating gluonic degrees of freedom he obtained a two-dimensional formulation of QCD with non-trivial solutions. In our case this means that the truncation scheme adopted by Alkofer and von Smekal simply does not work because removes all the dynamics of the Yang-Mills field and these are also the implications for the confinement scheme this approach should support. Indeed, a proper numerical solution of Dyson-Schwinger equations proves that the right scenario can also be obtained (see here). These authors met difficulties to get their paper accepted by an archival journal. Today, we should consider this work an important step beyond in our understanding of Yang-Mills theory.
My view is that we have to improve on the work of Alkofer and von Smekal to make it properly work at higher dimensionalities. This without forgetting all other works that gave the right solution straightforwardly.
In Italian language DAMA means a lady or draught the well-known game that people commonly plays with a chessboard. But it is also the name of a fundamental experiment carried out at INFN laboratories at Gran Sasso in Italy and headed by physicist Rita Bernabei (you can find here an article by her just in Italian, sorry) . Rita Bernabei was one of my teachers at “La Sapienza” in Rome and I took an examination with her. Recently a preprint by DAMA Collaboration was published on arxiv causing a lot of rumors in the scientific community. The point is that they see a signal that nobody else sees. This evidence cannot be denied and the signal is there.
Personally I am convinced of the goodness of all this matter and sometime happens that something is seen by a group and not by others and things could be generally explained after some time. In my activity field a similar thing happened for the sigma or f0(600) resonance. This particle has a long history as initially was put in the PDG listing and then removed as believed non-existent. After a lot of serious analysis, both theoretical and experimental, its existence has been commonly accepted and now the question is moved to the understanding of its nature. We have discussed this matter here, here and here. This particle is our key to understanding of low energy behavior of QCD and we expect some news about in the near future. Similarly, I think the current position of DAMA collaboration is the same as was for the first observations of the sigma. Being the firsts sometimes requires a lot of patience before general acceptance gets through.
Today I have posted my contribution to proceedings of QCD 08 at arxiv. It should appear shortly.
I hope to have answered to all open questions about Yang-Mills theory in the infrared. More to say in the following days, having in mind my very near vacations.
Update: The paper appeared today 29th July (see here).
In these days there have been some rumors in the blogosphere about string theorists and SUSY (see Motl, Woit and Dorigo) due to a recent preprint appeared on arxiv. Indeed SUSY is a relevant ingredient of string theory and the latter was the vehicle for the uncovering of this concept that obtained such a fortune in the community. I have listened a talk of Sergio Ferrara at Accademia dei Lincei in Rome a few months ago. Ferrara is one of the discoverers of supergravity and he gave a nice talk on the argument of supersymmetry. He was confident that supersymmetric particles will be seen at LHC. I would like to say that this was also the expectation for LEP and Tevatron but nothing has been seen so far. So the paper above seems like an attempt by a string theorist to be pessimistic and save the day.
Supersymmetry has some problems that still are in need for a satisfactory answer. One is philosophical as one can say that the number of particles simply doubles and so why should we expect such an anti-economical behavior by Nature? Ferrara argued against this question by saying that also with antimatter Nature doubled the number of particles so this would not be the first time that, in order to keep a symmetry, one needs such a doubling. One can say anyhow that for antimatter one has a discrete symmetry on a single field while for supersymmetry is the number of fields that doubles. The other point is about breaking of supersymmetry. There is no satisfactory model so far and such symmetry is not seen at low energies as we know. But this could be just a matter of time before someone finds a way out.
My view is that even if there is no supersymmetry at large, one can save supergravity. Indeed, all one needs is to observe a gravitino, that is a spin 3/2 particle, and we will have a theory of quantum gravity while, at large, no supersymmetry can exist. But this would not be enough for string theory. As a theoretical physicist I would like to see the discover of a gravitino and the failure of supersymmetry at large as this would imply a lot of interesting work to do and an incredible new scenario to understand.
Dietmar Ebert is a retired professor of Humboldt University in Berlin. He did relevant work in QCD and particle physics. I have come upon a paper of him at arxiv about the question of bosonization. In a paper of mine I showed how a Nambu-Jona-Lasinio (NJL) model can be derived from QCD using recent results about gluon propagator that is the corner stone of all this construction. In order to make contact with the mesonic spectrum of QCD one needs to manage in some way quark fermionic fields of NJL model to recover bosonic degrees of freedom. In Ebert’s paper this is done through Hubbard-Stratonovich transformation that is a widely known tool to condensed matter theorists. This is a key point to prove that our recent derivation of the width of the sigma resonance given here using a Fermi’s intuition is indeed correct. Ebert obtains by a NJL-model the following bosonic Hamiltonian
being the number of colors,
quark constituent mass and the quark mass assumed to be equal for u and d, and finally
In order to make contact with QCD, as we have shown one has
being the coupling constant and the square root of the string tension.
Ebert’s Lagrangian gives us exactly the term we derived with Fermi’s insight plus other terms implying also the one to compute f0(980) decay rate . So, as it is well-known, a good idea repeats itself at different levels in the description of Nature. I would call this the “gluonic sector” of QCD. I hope to put down a paper about in the next days.