Living dangerously


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


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


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

Some more news on warp drive


Today, New York Times published an article with an interview to Harold “Sonny” White about NASA studies on warp drive (see here). This revamped the interest about what NASA is funding (with a really small budget being  just $50,000) on this that have to be considered forefront research. For the readers that are not aware about what this research is aimed to, I invite them to read the very good article on Wikipedia about Alcubierre drive. As can be easily imagined, this article gets some new adding  each day and moves the curiosity of a myriad of people around the world. So, the activity of this NASA’s group is under a lot of attention by media and, with a lot of skepticism, by the scientific community. Alcubierre itself, the inventor of this idea, does not believe at all that is doable. The main reasons are well explained here (hat tip to Jennifer Ouellette) and one of these, the most important one maybe, is a lot of missing information as studies on this idea showed more its impossibility than else.

Anyhow,  we hope that Harold White will fill all the details at 2013 Starship Congress that he will attend giving a talk (see here). The schedule is here. He will speak on August 17th.

Miguel Alcubierre (2000). The warp drive: hyper-fast travel within general relativity Class.Quant.Grav.11:L73-L77,1994 arXiv: gr-qc/0009013v1

Waiting for EPS HEP 2013: Some thoughts


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


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


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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