Nailing down the Yang-Mills problem

22/02/2014

ResearchBlogging.org Millennium problems represent a major challenge for physicists and mathematicians. So far, the only one that has been solved was the Poincaré conjecture (now a theorem) by Grisha Perelman. For people working in strong interactions and quantum chromodynamics, the most interesting of such problems is the Yang-Mills mass gap and existence problem. The solutions of this problem would imply a lot of consequences in physics and one of the most important of these is a deep understanding of confinement of quarks inside hadrons. So far, there seems to be no solution to it but things do not stay exactly in this way. A significant number of researchers has performed lattice computations to obtain the propagators of the theory in the full range of energy from infrared to ultraviolet providing us a deep understanding of what is going on here (see Yang-Mills article on Wikipedia). The propagators to be considered are those for  the gluon and the ghost. There has been a significant effort from theoretical physicists in the last twenty years to answer this question. It is not so widely known in the community but it should because the work of this people could be the starting point for a great innovation in physics. In these days, on arxiv a paper by Axel Maas gives a great recount of the situation of these lattice computations (see here). Axel has been an important contributor to this research area and the current understanding of the behavior of the Yang-Mills theory in two dimensions owes a lot to him. In this paper, Axel presents his computations on large volumes for Yang-Mills theory on the lattice in 2, 3 and 4 dimensions in the SU(2) case. These computations are generally performed in the Landau gauge (propagators are gauge dependent quantities) being the most favorable for them. In four dimensions the lattice is (6\ fm)^4, not the largest but surely enough for the aims of the paper. Of course, no surprise comes out with respect what people found starting from 2007. The scenario is well settled and is this:

  1. The gluon propagator in 3 and 4 dimensions dos not go to zero with momenta but is just finite. In 3 dimensions has a maximum in the infrared reaching its finite value at 0  from below. No such maximum is seen in 4 dimensions. In 2 dimensions the gluon propagator goes to zero with momenta.
  2. The ghost propagator behaves like the one of a free massless particle as the momenta are lowered. This is the dominant behavior in 3 and 4 dimensions. In 2 dimensions the ghost propagator is enhanced and goes to infinity faster than in 3 and 4 dimensions.
  3. The running coupling in 3 and 4 dimensions is seen to reach zero as the momenta go to zero, reach a maximum at intermediate energies and goes asymptotically to 0 as momenta go to infinity (asymptotic freedom).

Here follows the figure for the gluon propagator Gluon Propagators

and for the running coupling

RunningCoupling

There is some concern for people about the running coupling. There is a recurring prejudice in Yang-Mills theory, without any support both theoretical or experimental, that the theory should be not trivial in the infrared. So, the running coupling should not go to zero lowering momenta but reach a finite non-zero value. Of course, a pure Yang-Mills theory in nature does not exist and it is very difficult to get an understanding here. But, in 2 and 3 dimensions, the point is that the gluon propagator is very similar to a free one, the ghost propagator is certainly a free one and then, using the duck test: If it looks like a duck, swims like a duck, and quacks like a duck, then it probably is a duck, the theory is really trivial also in the infrared limit. Currently, there are two people in the World that have recognized a duck here:  Axel Weber (see here and here) using renormalization group and me (see here, here and here). Now, claiming to see a duck where all others are pretending to tell a dinosaur does not make you the most popular guy  in the district. But so it goes.

These lattice computations are an important cornerstone in the search for the behavior of a Yang-Mills theory. Whoever aims to present to the World his petty theory for the solution of the Millennium prize must comply with these results showing that his theory is able to reproduce them. Otherwise what he has is just rubbish.

What appears in the sight is also the proof of existence of the theory. Having two trivial fixed points, the theory is Gaussian in these limits exactly as the scalar field theory. A Gaussian theory is the simplest example we know of a quantum field theory that is proven to exist. Could one recover the missing part between the two trivial fixed points as also happens for the scalar theory? In the end, it is possible that a Yang-Mills theory is just the vectorial counterpart of the well-known scalar field, the workhorse of all the scholars in quantum field theory.

Axel Maas (2014). Some more details of minimal-Landau-gauge Yang-Mills propagators arXiv arXiv: 1402.5050v1

Axel Weber (2012). Epsilon expansion for infrared Yang-Mills theory in Landau gauge Phys. Rev. D 85, 125005 arXiv: 1112.1157v2

Axel Weber (2012). The infrared fixed point of Landau gauge Yang-Mills theory arXiv arXiv: 1211.1473v1

Marco Frasca (2007). Infrared Gluon and Ghost Propagators Phys.Lett.B670:73-77,2008 arXiv: 0709.2042v6

Marco Frasca (2009). Mapping a Massless Scalar Field Theory on a Yang-Mills Theory: Classical
Case Mod. Phys. Lett. A 24, 2425-2432 (2009) arXiv: 0903.2357v4

Marco Frasca (2010). Mapping theorem and Green functions in Yang-Mills theory PoS FacesQCD:039,2010 arXiv: 1011.3643v3


Nature already patched it

09/02/2014

ResearchBlogging.org

Dennis Overbye is one of the best science writer around. Recently, he wrote a beautiful piece on the odd behavior of non-converging series like 1+2+3+4+\ldots and so on to infinity (see here). This article contains a wonderful video, this one

where it shown why 1+2+3+4+\ldots=-1/12 and this happens only when this series is taken going to infinity. You can also see a 21 minutes video on the same argument from these authors

This is really odd as we are summing up all positive terms and in the end one gets a negative result. This was a question that already bothered Euler and is generally fixed with the Riemann zeta function. Now, if you talk with a mathematician, you will be warned that such a series is not converging and indeed intermediate results become even more larger as the sum is performed. So, this series should be generally discarded when you meet it in your computations in physics or engineering. We know that things do not stay this way as nature already patched it. The reason is exactly this: Infinity does not exist in nature and whenever one is met nature already fixed it, whatever a mathematician could say. Of course, smarter mathematicians are well aware of this as you can read from Terry Tao’s blog. Indeed, Terry Tao is one of the smartest living mathematicians. One of his latest successes is to have found a problem in the presumed Otelbaev’s proof of the existence of solutions to Navier-Stokes equations, a well-known millennium problem (see the accepted answer and comments here).

This idea is well-known to physicists and when an infinity is met we have invented a series of techniques to remove it in the way nature has chosen. This can be seen from the striking agreement between computed and measured quantities in some quantum field theories, not last the Standard Model. E.g. the gyromagnetic ratio of the electron agrees to one part on a trillion with the measured quantity (see here). This perfection in the computations was never seen before in physics and belongs to the great revolution that was completed by Feynman, Schwinger, Tomonaga and Dyson that we have inherited in the Standard Model, the latest and greatest revolution seen so far in particle physics. We just hope that LHC will uncover the next one at the restart of operations. It is possible again that nature will have found further ways to patch infinities and one of these could be 1+2+3+4+\ldots=-1/12.

So, we recall one of the greatest principles of physics: Nature patches infinities and use techniques to do it that are generally disgusting mathematicians. I think that diverging series should be taught at undergraduate level courses. Maybe, using the standard textbook by Hardy (see here). These are not just pathologies in an otherwise wonderful world but rather these are the ways nature has chosen to behave!

The reason for me to write about this matter is linked to a beautiful work I did with my colleagues Alfonso Farina and Matteo Sedehi on the way the Tartaglia-Pascal triangle generalizes in quantum mechanics. We arrived at the conclusion that quantum mechanics arises as the square root of a Brownian motion. We have got a paper published on this matter (see here or you can see the Latest Draft). Of course, the idea to extract the square root of a Wiener process is something that was disgusting mathematicians, mostly Didier Piau, that was claiming that an infinity goes around. Of course, if I have a sequence of random numbers, these are finite, I can arbitrarily take their square root. Indeed, this is what one sees working with Matlab that easily recovers our formula for this process. So, what does it happen to the infinity found by Piau? Nothing, but nature already patched it.

So, we learned a beautiful lesson from nature: The only way to know her choices is to ask her.

A. Farina,, M. Frasca,, & M. Sedehi (2014). Solving Schrödinger equation via Tartaglia/Pascal triangle: a possible link between stochastic processing and quantum mechanics Signal, Image and Video Processing, 8 (1), 27-37 DOI: 10.1007/s11760-013-0473-y


Back to work

02/02/2014

ResearchBlogging.org

I would like to have a lot more time to write on my blog. Indeed, time is something I have no often and also the connection is not so good as I would like in the places I spend most of it. So, I take this moment to give an update of what I have seen around in these days.

LHC has found no evidence of dark matter so far (see here). Dark matter appears even more difficult to see and theory is not able to help the search. This is also one of our major venues to go beyond the Standard Model. On the other side, ASACUSA experiment at CERN produced the first beam of antihydpogen atoms (see here, this article is free to read). We expect no relevant news about the very nature of Higgs until, on 2015, LHC will restart. It must be said that the data collected so far are saying to us that this particle is behaving very nearly as that postulated by Weinberg on 1967.

In these days there has been some fuss about the realization in laboratory of a Dirac magnetic monopole (see here).  Notwithstanding this is a really beautiful experiment, nobody has seen a magnetic monopole so far. It is a simulation performed with another physical system: A BEC. This is a successful technology that will permit us an even better understanding of physical systems that are difficult to observe. Studies are ongoing to realize a simulation of  Hawking radiation in such a system.  Even if this is the state of affairs, I have read in social networks and in the news that a magnetic monopole was seen in laboratory. Of course, this is not true.

The question of black holes is always at the top of the list of the main problems in physics. Mostly when a master of physics comes out with a new point of view. So, a lot of  fuss arose from this article in Nature involving a new idea from Stephen Hawking that the author published in a paper on arxiv (see here). Beyond the resounding title, Hawking is just proposing a way to avoid the concept of firewalls that was at the center of a hot debate in the last months. Again we recognize that a journalist is not making a good job but is generating a lot of noise around and noise can hide a signal very well.

Finally, we hope in a better year in science communication. The start was somewhat disappointing.

Kuroda N, Ulmer S, Murtagh DJ, Van Gorp S, Nagata Y, Diermaier M, Federmann S, Leali M, Malbrunot C, Mascagna V, Massiczek O, Michishio K, Mizutani T, Mohri A, Nagahama H, Ohtsuka M, Radics B, Sakurai S, Sauerzopf C, Suzuki K, Tajima M, Torii HA, Venturelli L, Wu Nschek B, Zmeskal J, Zurlo N, Higaki H, Kanai Y, Lodi Rizzini E, Nagashima Y, Matsuda Y, Widmann E, & Yamazaki Y (2014). A source of antihydrogen for in-flight hyperfine spectroscopy. Nature communications, 5 PMID: 24448273

M. W. Ray,, E. Ruokokoski,, S. Kandel,, M. Möttönen,, & D. S. Hall (2014). Observation of Dirac monopoles in a synthetic magnetic field Nature, 505, 657-660 DOI: 10.1038/nature12954

Zeeya Merali (2014). Stephen Hawking: ‘There are no black holes’ Nature DOI: 10.1038/nature.2014.14583

S. W. Hawking (2014). Information Preservation and Weather Forecasting for Black Holes arXiv arXiv: 1401.5761v1


That strange behavior of supersymmetry…

07/12/2013

ResearchBlogging.org

I am a careful reader of scientific literature and an avid searcher for already published material in peer reviewed journals. Of course, arxiv is essential to accomplish this task and to satisfy my needs for reading. In these days, I am working on Dyson-Schwinger equations. I have written on this a paper (see here) a few years ago but this work is in strong need to be revised. Maybe, some of these days I will take the challenge. Googling around and looking for the Dyson-Schwinger equations applied to the well-known supersymmetric model due to Wess and Zumino, I have uncovered a very exciting track of research that uses Dyson-Schwinger equations to produce exact results in quantum field theory. The paper I have got was authored by Marc Bellon, Gustavo Lozano and Fidel Schaposnik and can be found here. These authors get the Dyson-Schwinger equations for the Wess-Zumino model at one loop and manage to compute the self-energies of the involved fields: A scalar, a fermion and an auxiliary bosonic field. Their equations are yielded for three different self-energies, different for each field. Self-energies are essential in quantum field theory as they introduce corrections to masses in a propagator and so enters into the physical part of an object that is not an observable.

Now, if you are in a symmetric theory like the Wess-Zumino model, such a symmetry, if it is not broken, will yield equal masses to all the components of the multiplet entering into the theory. This means that if you start with the assumption that in this case all the self-energies are equal, you are doing a consistent approximation. This is what Bellon, Lozano and Schaposnik just did. They assumed from the start that all the self-energies are equal for the Dyson Schwinger equations they get and go on with their computations. This choice leaves an open question: What if do I choose different self-energies from the start? Will the Dyson-Schwiner equations drive the solution toward the symmetric one?

This question is really interesting as the model considered is not exactly the one that Witten analysed in his famous paper  on 1982 on breaking of a supersymmetry (you can download his paper here). Supersymmetric model generates non-linear terms and could be amenable to spontaneous symmetry breaking, provided the Witten index has the proper values. The question I asked is strongly related to the idea of a supersymmetry breaking at the bootstrap: Supersymmetry is responsible for its breaking.

So, I managed to numerically solve Dyson-Schwinger equations for the Wess-Zumino model as yielded by Bellon, Lozano and Schaposnik and presented the results in a paper (see here). If you solve them assuming from the start all the self-energies are equal you get the following figure for coupling running from 0.25 to 100 (weak to strong):

All equal self-energies for the Wess-Zumino model

It does not matter the way you modify your parameters in the Dyson-Schwinger equations. Choosing them all equal from the start makes them equal forever. This is a consistent choice and this solution exists. But now, try to choose all different self-energies. You will get the following figure for the same couplings:

Not all equal self-energies for the Wess-Zumino model

This is really nice. You see that exist also solutions with all different self-energies and supersymmetry may be broken in this model. This kind of solutions has been missed by the authors. What one can see here is that supersymmetry is preserved for small couplings, even if we started with all different self-energies, but is broken as the coupling becomes stronger. This result is really striking and unexpected. It is in agreement with the results presented here.

I hope to extend this analysis to more mundane theories to analyse behaviours that are currently discussed in literature but never checked for. For these aims there are very powerful tools developed for Mathematica by Markus Huber, Jens Braun and Mario Mitter to get and numerically solve Dyson-Schwinger equations: DoFun anc CrasyDSE (thanks to Markus Huber for help). I suggest to play with them for numerical explorations.

Marc Bellon, Gustavo S. Lozano, & Fidel A. Schaposnik (2007). Higher loop renormalization of a supersymmetric field theory Phys.Lett.B650:293-297,2007 arXiv: hep-th/0703185v1

Edward Witten (1982). Constraints on Supersymmetry Breaking Nuclear Physics B, 202, 253-316 DOI: 10.1016/0550-3213(82)90071-2

Marco Frasca (2013). Numerical study of the Dyson-Schwinger equations for the Wess-Zumino
model arXiv arXiv: 1311.7376v1

Marco Frasca (2012). Chiral Wess-Zumino model and breaking of supersymmetry arXiv arXiv: 1211.1039v1

Markus Q. Huber, & Jens Braun (2011). Algorithmic derivation of functional renormalization group equations and
Dyson-Schwinger equations Computer Physics Communications, 183 (6), 1290-1320 arXiv: 1102.5307v2

Markus Q. Huber, & Mario Mitter (2011). CrasyDSE: A framework for solving Dyson-Schwinger equations arXiv arXiv: 1112.5622v2


Living dangerously

05/11/2013

ResearchBlogging.org

Today, I read an interesting article on New York Times by Dennis Overbye (see here). Of course, for researchers, a discovery that does not open new puzzles is not really a discovery but just the end of the story. But the content of the article is intriguing and is related to the question of the stability of our universe. This matter was already discussed in blogs (e.g. see here) and is linked to a paper by Giuseppe Degrassi, Stefano Di Vita, Joan Elias-Miró, José R. Espinosa, Gian F. Giudice, Gino Isidori, Alessandro Strumia (see here)  with the most famous picture

Stability and Higgs

Our universe, with its habitants, lives in that small square at the border between stability and meta-stability. So, it takes not too much to “live dangerously” as the authors say. Just a better measurement of the mass of the top quark can throw us there and this is in our reach at the restart of LHC. Anyhow, their estimation of the tunnel time is really reassuring as the required time is bigger than any reasonable cosmological age. Our universe, given the data coming from LHC, seems to live in a metastable state. This is further confirmed in a more recent paper by the same authors (see here). This means that the discovery of the Higgs boson with the given mass does not appear satisfactory from a theoretical standpoint and, besides the missing new physics, we are left with open questions that naturalness and supersymmetry would have properly assessed. The light mass of the Higgs boson, 125 GeV, in the framewrok of the Higgs mechanism, recently awarded with a richly deserved Nobel prize to Englert and Higgs, with an extensive use of weak perturbation theory is looking weary.

The question to be answered is: Is there any point in this logical chain where we can intervene to put all this matter on a proper track? Or is this the situation with the Standard Model to hold down to the Planck energy?

In all this matter there is a curious question that arises when you work with a conformal Standard Model. In this case, there is no mass term for the Higgs potential but rather, the potential gets modified by quantum corrections (Coleman-Weinberg mechanism) and a non-null vacuum expectation value comes out. But one has to grant that higher order quantum corrections cannot spoil conformal invariance. This happens if one uses dimensional regularization rather than other renormalization schemes. This grants that no quadratic correction arises and the Higgs boson is “natural”. This is a rather strange situation. Dimensional regularization works. It was invented by ‘t Hooft and Veltman and largely used by Wilson and others in their successful application of the renormalization group to phase transitions. So, why does it seem to behave differently (better!) in this situation? To decide we need a measurement of the Higgs potential that presently is out of discussion.

But there is a fundamental point that is more important than “naturalness” for which a hot debate is going on. With the pioneering work of Nambu and Goldstone we have learned a fundamental lesson: All the laws of physics are highly symmetric but nature enjoys a lot to hide all these symmetries. A lot of effort was required by very smart people to uncover them being very well hidden (do you remember the lesson from Lorentz invariance?). In the Standard Model there is a notable exception: Conformal invariance appears to be broken by hand by the Higgs potential. Why? Conformal invariance is really fundamental as all two-dimensional theories enjoy it. A typical conformal theory is string theory and we can build up all our supersymmetric models with such a property then broken down by whatever mechanism. Any conceivable more fundamental theory has conformal invariance and we would like this to be there also in the low-energy limit with a proper mechanism to break it. But not by hand.

Finally, we observe that all our theories seem to be really lucky: the coupling is always small and we can work out small perturbation theory. Also strong interactions, at high energies, become weakly interacting. In their papers, Gian Giudice et al. are able to show that the self-interaction of the Higgs potential is seen to decrease at higher energies and so, they satisfactorily apply perturbation theory. Indeed, they show that there will be an energy for which this coupling is zero and is due to change sign. As they work at high energies, the form of their potential just contains a quartic term. My question here is rather peculiar: What if exist exact solutions for finite (non-zero) quartic coupling that go like the inverse power of the coupling? We were not able to recover them with perturbation theory  but nature could have sat there. So, we would need to properly do perturbation theory around them to do the right physics. I have given some of there here and here but one cannot exclude that others exist. This also means that the mechanism of symmetry breaking can hide some surprises and the matter could not be completely settled. Never heard of breaking a symmetry by a zero mode?

So, maybe it is not our universe on the verge of showing a dangerous life but rather some of our views need a revision or a better understanding. Only then the next step will be easier to unveil. Let my bet on supersymmetry again.

Living Dangerously

Giuseppe Degrassi, Stefano Di Vita, Joan Elias-Miró, José R. Espinosa, Gian F. Giudice, Gino Isidori, & Alessandro Strumia (2012). Higgs mass and vacuum stability in the Standard Model at NNLO JHEP August 2012, 2012:98 arXiv: 1205.6497v2

Dario Buttazzo, Giuseppe Degrassi, Pier Paolo Giardino, Gian F. Giudice, Filippo Sala, Alberto Salvio, & Alessandro Strumia (2013). Investigating the near-criticality of the Higgs boson arXiv arXiv: 1307.3536v1

Marco Frasca (2009). Exact solutions of classical scalar field equations J.Nonlin.Math.Phys.18:291-297,2011 arXiv: 0907.4053v2

Marco Frasca (2013). Exact solutions and zero modes in scalar field theory arXiv arXiv: 1310.6630v1


Ending and consequences of Terry Tao’s criticism

21/09/2013

ResearchBlogging.org

Summer days are gone and I am back to work. I thought that Terry Tao’s criticism to my work was finally settled and his intervention was a good one indeed. Of course, people just remember the criticism but not how the question evolved since then (it was 2009!). Terry’s point was that the mapping given here between the scalar field solutions and the Yang-Mills field in the classical limit cannot be exact as it is not granted that they represent an extreme for the Yang-Mills functional. In this way the conclusions given in the paper are not granted being based on this proof. The problem can be traced back to the gauge invariance of the Yang-Mills theory that is explicitly broken in this case.

Terry Tao, in a private communication, asked me to provide a paper, to be published on a refereed journal, that fixed the problem. In such a case the question would have been settled in a way or another. E.g., also a result disproving completely the mapping would have been good, disproving also my published paper.

This matter is rather curious as, if you fix the gauge to be Lorenz (Landau), the mapping is exact. But the possible gauge choices are infinite and so, there seems to be infinite cases where the mapping theorem appears to fail. The lucky case is that lattice computations are generally performed in Landau gauge and when you do quantum field theory a gauge must be chosen. So, is the mapping theorem really false or one can change it to fix it all?

In order to clarify this situation, I decided to solve the classical equations of the Yang-Mills theory perturbatively in the strong coupling limit. Please, note that today I am the only one in the World able to perform such a computation having completely invented the techniques to do perturbation theory when a perturbation is taken to go to infinity (sorry, no AdS/CFT here but I can surely support it). You will note that this is the opposite limit to standard perturbation theory when one is looking for a parameter that goes to zero. I succeeded in doing so and put a paper on arxiv (see here) that was finally published the same year, 2009.

The theorem changed in this way:

The mapping exists in the asymptotic limit of the coupling running to infinity (leading order), with the notable exception of the Lorenz (Landau) gauge where it is exact.

So, I sighed with relief. The reason was that the conclusions of my paper on propagators were correct. But these hold asymptotically in the limit of a strong coupling. This is just what one needs in the infrared limit where Yang-Mills theory becomes strongly coupled and this is the main reason to solve it on the lattice. I cited my work on Tao’s site, Dispersive Wiki. I am a contributor to this site. Terry Tao declared the question definitively settled with the mapping theorem holding asymptotically (see here).

In the end, we were both right. Tao’s criticism was deeply helpful while my conclusions on the propagators were correct. Indeed, my gluon propagator agrees perfectly well, in the infrared limit, with the data from the largest lattice used in computations so far  (see here)

Comparison with lattice dataAs generally happens in these cases, the only fact that remains is the original criticism by a great mathematician (and Terry is) that invalidated my work (see here for a question on Physics Stackexchange). As you can see by the tenths of papers I published since then, my work stands and stands very well. Maybe, it would be time to ask the author.

Marco Frasca (2007). Infrared Gluon and Ghost Propagators Phys.Lett.B670:73-77,2008 arXiv: 0709.2042v6

Marco Frasca (2009). Mapping a Massless Scalar Field Theory on a Yang-Mills Theory: Classical
Case Mod. Phys. Lett. A 24, 2425-2432 (2009) arXiv: 0903.2357v4

Attilio Cucchieri, & Tereza Mendes (2007). What’s up with IR gluon and ghost propagators in Landau gauge? A puzzling answer from huge lattices PoS LAT2007:297,2007 arXiv: 0710.0412v1


The Witten’s paradox

17/08/2013

ResearchBlogging.org

Edward Witten is one of the greatest living physicists and also ranks high with mathematicians. He set the agenda for theoretical physics in several areas of research. He is mostly known for championing string theory and being one of few people that revolutionized the field. One of his major contributions to supersymmetry has been a deep understanding of its breaking. In a pair of famous papers (here and here) he put the foundations to our current understanding on the way supersymmetry can break and introduced the well-known Witten index. If a supersymmetric theory breaks supersymmetry then its Witten index is 0. This index is generally very difficult to compute and only perturbative or lattice computations can come to rescue. An important conclusion from Witten’s paper is that the well-known Wess-Zumino model in four dimensions does not break supersymmetry. Witten could rigorously justify this conclusion at small coupling but, at that time, an approach for strong coupling was missing and here Maldacena conjecture cannot help. Anyhow, he concluded that this should be true also for a strongly coupled Wess-Zumino model. Checks to this model in such a regime are rare. After I submitted a paper on arxiv last year (see here) I become aware of an attempt using Dyson-Schwinger equations that confirmed Witten conclusions for small coupling (see here). I have had an interesting mail exchange with one of the authors and this seems a promising approach, given authors’ truncation of Dyson-Schwinger hierarchy. Other approaches consider the Wess-Zumino model in two dimensions on the lattice. So, this appears a rather unexplored area , given the difficulties to cope with a strongly coupled theory, and Witten’s words appear like nails on a coffin to this theory.

I have worked out a lot of techniques to cope with strongly coupled theories and everywhere there is a perturbation going to infinity in a differential equation of any kind and so, I applied these ideas also to this famous model of supersymmetry. The idea is to prove that “supersymmetry has inside itself the seeds of its breaking“. The real issue at stake here is a correct understanding of the way supersymmetry breaks and recover in this way models that now appear to be defeated by data from LHC simply because the idea of symmetry breaking must be applied differently.

Of course, I do not aim to present a claim against the beautiful results given by Witten decades ago but just open up an interesting scientific question. So, considering that the Wess-Zumino model is just a theory of two scalar fields coupled to a Majorana spinor, its equations can be treated classically and so solved both for a strong and a weak coupling limit. I did this in a paper of mine (see here) and this paper has been accepted in these days in the Journal of Nonlinear Mathematical Physics as a letter. The classical solutions contradict the expectations giving a surviving of the supersymmetry at small coupling (as expected from Witten index for the quantum theory) while this does not happen for a strong coupling (formal limit of the coupling going to infinity). This is  a paradox, the Witten paradox, because classical solutions seem to break supersymmetry while the quantum theory does not.  So, we are left with a deep question: How is supersymmetry recovered by quantum corrections?

Marco Frasca (2012). Chiral Wess-Zumino model and breaking of supersymmetry arXiv arXiv: 1211.1039v1

A. Bashir, & J. Lorenzo Diaz-Cruz (1999). A study of Schwinger-Dyson Equations for Yukawa and Wess-Zumino Models J.Phys.G25:1797-1805,1999 arXiv: hep-ph/9906360v1

Marco Frasca (2012). Classical solutions of a massless Wess-Zumino model arXiv arXiv: 1212.1822v2


Waiting for EPS HEP 2013: Some thoughts

13/07/2013

ResearchBlogging.org

On 18th July the first summer HEP Conference will start in Stockholm. We do not expect great announcements from CMS and ATLAS as most of the main results from 2011-2012 data were just unraveled. The conclusions is that the particle announced on 4th July last year is a Higgs boson. It decays in all the modes foreseen by the Standard Model and important hints favor spin 0. No other resonance is seen at higher energies behaving this way. It is a single yet. There are a lot of reasons to be happy: We have likely seen the guilty for the breaking of the symmetry in the Standard Model and, absolutely for the first time, we have a fundamental particle behaving like a scalar. Both of these properties were looked upon for a long time and now this search is finally ended. On the bad side, no hint of new physics is seen anywhere and probably we will have to wait the restart of LHC on 2015. The long sought SUSY is at large yet.

Notwithstanding this hopeless situation for theoretical physics, my personal view is that there is something that gives important clues to great novelties that possibly will transmute into something of concrete at the restart. It is important to note that there seem to exist some differences between CMS and ATLAS  and this small disagreement can hide interesting news for the future. I cannot say if, due to the different conception of this two detectors, something different should be seen but is there. Anyway, they should agree in the end of the story and possibly this will happen in the near future.

The first essential point, that is often overlooked due to the overall figure, is the decay of the Higgs particle in a couple of W or Z. WW decay has a significantly large number of events and what CMS claims is indeed worth some deepening. This number is significantly below one. There is  a strange situation here because CMS gives 0.76\pm 0.21 and in the overall picture just write 0.68\pm 0.20 and so, I cannot say what is the right one. But they are consistent each other so not a real problem here. Similarly, ZZ decay yields 0.91^{+0.30}_{-0.24}. ATLAS, on the other side, yields for WW decay 0.99^{+0.31}_{-0.28} and for ZZ decay 1.43^{+0.40}_{-0.35}. Error bars are large yet and fluctuations can change these values. The interesting point here, but this has the value of a clue as these data agree with Standard Model at 2\sigma, is that the lower values for the WW decay can be an indication that this Higgs particle could be a conformal one. This would mean room for new physics. For ZZ decay apparently ATLAS seems to have a lower number of events as this figure is somewhat larger and the error bar as well. Anyway, a steady decrease has been seen for the WW decay as a larger dataset was considered. This decrease, if confirmed at the restart, would mean a major finding after the discovery of the Higgs particle. It should be said that ATLAS already published updated results with the full dataset (see here). I would like to emphasize that a conformal Standard Model can imply SUSY.

The second point is a bump found by CMS in the \gamma\gamma channel (see here).  This is what they see

CMS Another Higgs

but ATLAS sees nothing there and this is possibly a fluke. Anyway, this is about 3\sigma and so CMS reported about on a publication of them.

Finally, it is also possible that heavier Higgs particles could have depressed production rates and so are very rare. This also would be consistent with a conformal Standard Model. My personal view is that all hopes to see new physics at LHC are essentially untouched and maybe this delay to unveil it is just due to the unlucky start of the LHC on 2008. Meantime, we have to use the main virtue of a theoretical physicist: keeping calm and being patient.

Update: Here is the press release from CERN.

ATLAS Collaboration (2013). Measurements of Higgs boson production and couplings in diboson final
states with the ATLAS detector at the LHC arXiv arXiv: 1307.1427v1


Return in Paris

15/06/2013

ResearchBlogging.org

After two years since the last edition, I was back in Paris to participate to the Twelfth Workshop on Non-perturbative Quantum Chromodynamics. The conference is organized by high-energy group at Brown University and held at Institut d’Astrophysique de ParisProfessor Chung-I Tan and Professor Berndt Mueller from Duke University are the organizers. As it also happened in the precedent edition, the workshop was really interesting and rich of ideas for research. The first talk was given by Kostantinos Orginos and was about nuclear physics emerging from lattice computations. The Sein (Paris)This is a matter that I am involved into as a “final user” and so, very near my interests. It is noteworthy to point out how current technology permits  to extract such results from lattice QCD making this a useful tool for the understanding of low-energy phenomenology. With Kostantinos,  his wife Vassiliki Panoussi and sons, we have had a nice social dinner on Tuesday night and I have had an interesting discussion about the current situation of lattice computations. The next speaker was Philippe de Forcrand that is well-known for his works on finite temperature QCD on the lattice.   He showed how the effective Yang-Mills theory at high temperature is surprisingly good with respect to lattice results also lowering temperatures at few times the critical temperature. Another interesting talk was the one by Peter Petreczy about the observables of QCD at finite temperature presenting also the most recent value for the critical temperature. As my readers may know, I computed this value in my recent paper on Physical Review C (see here) properly corrected by the mass gap of Yang-Mills theory. Norberto Scoccola and Daniel Gomez-Dumm showed similar results (see here).

On Tuesday it was the ultrarelativistic Heavy-ion collision session. This was particularly interesting and involved the talks of two friends of mine: Marco Ruggieri and Salvatore Plumari. In this area of research there is a really interesting and hot debated situation. On the other side, there is plenty of experimental results from RHIC and LHC. The session chair was Jean-Yves Ollitrault. He put the foundations to the current understanding of the quark-gluon plasma through a hydrodynamic approximation. What is observed in the experiments is the production of a flow of particles in a transverse direction named elliptic flow. Eiffel Tower (Paris)This is a clear evidence of existence for the quark-gluon plasma. Marco and Salvatore work in the group of Vincenzo Greco at University of Catania in Italy. The idea they based their work on is to derive the hydrodynamic equations from a kinetic description as the one provided by the Boltzmann equation. This approach opens up the scene to the possibility to derive such an equation and the full description of the quark-gluon plasma starting directly from QCD and fixing the collisional integral of the kinetic equation. Of course, one should understand the applicability conditions but my take is that, being the running coupling going to zero due to asymptotic freedom, a quark-gluon plasma should have scarce multi-collision effects. On the other side, this is a charged plasma but lives for a very small time. This means that this approach can prove to be really successful. One of the open questions is if, going at higher energies, a state called “color glass condensate” should form and this is a matter of a hot debate in the community. This is creating some tension that is reminiscent of the story I recounted about Landau gauge propagators for pure Yang-Mills theory (see here). A color glass condensate gives an increasing lower bound on the viscosity to entropy ratio by a factor 2 with respect to 1/4\pi, also computed from string theory, and appears less efficient with respect to observed elliptic flow at RHIC (see here). This kind of wars is often unproductive in physics and science at large as it slows down progress and good works could turn out unpublished. In situations like this, researchers should have eyes wide open and open minds granting all the contenders to be fairly listened waiting for experiments or careful lattice computations to say the last word. This should teach the history of Landau gauge propagators and also by looking back to history of physics. Otherwise we will stay on a silly forever war  where we are only able to prove to the rest of mankind that nothing has been learned from the past.

On Wednesday the session was dedicated to AdS/CFT, Holography, and Scattering. There was the talk of Carl Bender that is currently working on PT quantum mechanics. He is the pioneer of strong perturbation for quantum systems and quantum field theory. I often cited his work that has been a source of inspiration. David Dudal also spoke and discussed a holographic model for the analysis of strong ion collisions and the effect of the huge magnetic field generated. He gets results reminiscent of the Nambu-Jona-Lasinio model.  David is one of the proponents of the Refined Gribov-Zwanzinger model (see here). This is a real successful approach to the understanding of Landau gauge propagators and fits quite well with my results in the deep infrared behavior of a Yang-Mills theory as I also pointed out in my talk (see below).

On Thursday there was my talk in Paris. I will not comment about. On the morning, I heard the talk by Chung-I Tan, one of the organizers. He uses holographic techniques and the running coupling he obtains is similar to mine into an expansion in the inverse of the square root of the coupling. This is a nice result and it would be interesting to compare both of them numerically. One of the most interesting talks I heard was the one by Guy de Teramond. I have had reason to appreciate his work with Stanley Brodsky about holographic QCD and reduction to a Schrödinger-like equation to identify hadronic states (see here). With Guy I exchanged some interesting words and he was so kind to make compliments to my blog. A couple of talks were presented by cosmologists. The one by Patrick Peter about decoherence and cosmology struck me once again. I heard before about this matter and what makes me surprise is that the question of decoherence for a closed quantum system is stopped yet at the old Bohm pilot wave or a multiverse. Eiffel Tower SeesightThis should not be considered serious ways anymore because there is a theorem due to Barry Simon and Elliott Lieb, two giants of mathematical physics, that states that the limit of a large number of particles, in a reasonable many-body quantum system, reaches a Thomas-Fermi limit (see here where you can download a pdf). It is known that the Thomas-Fermi limit is just a semiclassical limit and the behavior of the matter is essentially classical. This means that one has not to recur to exotic hypothesis to understand what went on in the primordial universe and its fluctuations. I have recounted all this matter here. The final talk was given by Herbert Fried and was about a new understanding of dark matter and the universe using a new view of quantum electrodynamics.

It was a great workshop and I have been very happy to be there also this year. I hope people at Brown University will repeat this again. Thanks a lot!

Marco Frasca (2011). Chiral symmetry in the low-energy limit of QCD at finite temperature Phys. Rev. C 84, 055208 (2011) arXiv: 1105.5274v4

D. Gomez Dumm, & N. N. Scoccola (2004). Characteristics of the chiral phase transition in nonlocal quark models Phys.Rev. C72 (2005) 014909 arXiv: hep-ph/0410262v2

Ollitrault, J. (1992). Anisotropy as a signature of transverse collective flow Physical Review D, 46 (1), 229-245 DOI: 10.1103/PhysRevD.46.229

M. Ruggieri, F. Scardina, S. Plumari, & V. Greco (2013). Elliptic Flow from Nonequilibrium Color Glass Condensate Initial
Conditions arXiv arXiv: 1303.3178v1

David Dudal, John Gracey, Silvio Paolo Sorella, Nele Vandersickel, & Henri Verschelde (2008). A refinement of the Gribov-Zwanziger approach in the Landau gauge:
infrared propagators in harmony with the lattice results Phys.Rev.D78:065047,2008 arXiv: 0806.4348v2

Lieb, E., & Simon, B. (1973). Thomas-Fermi Theory Revisited Physical Review Letters, 31 (11), 681-683 DOI: 10.1103/PhysRevLett.31.681

Lieb, E., & Simon, B. (1977). The Thomas-Fermi theory of atoms, molecules and solids Advances in Mathematics, 23 (1), 22-116 DOI: 10.1016/0001-8708(77)90108-6

Marco Frasca (2006). Thermodynamic Limit and Decoherence: Rigorous Results Journal of Physics: Conference Series 67 (2007) 012026 arXiv: quant-ph/0611024v1


Higgs and beyond

06/06/2013

I am writing these few lines while the conference “Higgs and beyond” is still going on at Tohoku University (Sendai) in Japan. Talks can be found here. Both ATLAS and CMS presented a lot of results about Higgs particle and the most relevant of them is the combination of the data from the two experiments (see here). I am following the excellent recount by Richard Ruiz on twitter (@bravelittlemuon) that also takes care of CERN’s blog. Some interesting point is that there seems to be a bump in Z\gamma channel that is persistent also in other channels. About decay rates, improvements confirm yet nearly Standard Model behavior of the Higgs particle but with the rates of WW and ZZ going down with a too large error bars yet (see my preceding post).  Hopes are that CMS and ATLAS could combine also these data reducing error bars. No other Standard Model heavy Higgs particle is seen. Both CMS and ATLAS are looking for evidence of more Higgs particles to no avail yet. Of course, my view is that these excitations should be searched with somewhat different rates from Standard Model expectations. In any case, Standard Model confirms itself as one of the most successful theories in the history of physics. As said by one of ATLAS speakers: “There is overwhelming evidence for a new boson; there is overwhelming evidence for nothing else.” Both experiments plan to complete the analysis of data at 8 TeV for the summer conferences. My personal expectations are that just improvements in the precision of the measurements of the decay rates could eventually give hints of new physics. To fulfill other hopes, we need LHC upgrade that will restart operations on the spring of 2015, hopefully.


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