That Higgs is trivial!


Notwithstanding LHC has seen the particle, the Higgs sector of the Standard Model has some serious problems. This fact yielded more than one headache to physicists. One of these difficulties is called technically “triviality“. Michael AizenmanThe scalar field theory, that is so well defined classically, does not exist as a quantum field theory unless is non-interacting. There is a wonderful paper by Michael Aizenman that shows that this is true for dimensions 5 and higher. So, one should think that, as we live in four dimensions, there is no reason to worry. The point is that Michael Aizenman left the question in four dimensions open. So, does Higgs particle exist or not and how does it yield mass if it will not interact? CERN said to us that Higgs particle is there and so, in some way, the scalar sector of the Standard Model must properly work. Aizenman’s proof was on 1981 but what is the situation now? An answer is in this article on Scholarpedia. As stated by the author Ulli Wolff

Triviality of lattice phi^4 theory in this sense has been rigorously proven for D>4 while for the most interesting borderline case D=4 we have only partial results but very strong evidence from numerical simulations.

While there is another great expert on quantum field theory, Franco Strocchi, in his really worth to read book saying

The recent proof of triviality of phi^4 in 3 + 1 spacetime dimensions indicates that the situation becomes worse in the real world, and in particular the renormalized perturbative series of the phi^4 model seems to have little to do with the non-perturbative solution.

We see that experts do not completely agree about the fact that a proof exists or not but, for sure, the scalar theory in four dimensions cannot interact and the Standard Model appears in serious troubles.

Before to enter more in details about this matter, let me say that, even if Strocchi makes no citation about where the proof is, he is the one being right. We have proof about this, the matter is now well understood and again we are waiting for the scientific community to wake up. Also, the Standard Model is surely secured and there is no serious risk about the recent discovery by CERN of the Higgs particle.

The proof has been completed recently by Renata Jora with this paper on arxiv. Renata extended the proof an all the energy range. I met her in Montpellier (France) at this workshop organized by Stephan Narison. We have converging interests in research. Renata’s work is based on a preceding proof, due to me and Igor Suslov, showing that, at large coupling, the four dimensional theory is indeed trivial. You can find the main results here and here. Combining these works together, we can conclude that Strocchi’s statement is correct but there is no harm for the Standard Model as we will discuss in a moment. Also the fact that the perturbation solution of the model is not properly describing the situation can be seen from the strictly non-analytical behaviours seen at strong coupling that makes impossible to extend what one gets at small coupling to that regime.

The fact that CERN has indeed seen the Higgs particle and that the Higgs sector of the Standard Model is behaving properly, unless a better understanding will emerge after the restart of the LHC, has been seen with the studies of the propagators of the Yang-Mills theory in the Landau gauge. The key paper is this where the behaviour of the running coupling of the theory was obtained on all the energy range from lattice computations.

Running Coupling

This behaviour shows that, while the theory is trivial at both the extremes of the energy range, there is an intermediate regime where we can trust the theory and treat it as an effective one. There the coupling does not run to zero but moves around some finite non-null value. Of course, all this is just saying that this theory must be superseded by an extended one going to higher energies (supersymmetry? Technicolor?) but it is reasonable to manage the theory as if all this just works at current energies. Indeed, LHC has shown that a Higgs particle is there.

So, triviality is saying that the LHC will find something new for sure. Today, beams moved again inside the accelerator. We are eager to see what will come out form this wonderful enterprise.

Aizenman, M. (1981). Proof of the Triviality of Field Theory and Some Mean-Field Features of Ising Models for Physical Review Letters, 47 (12), 886-886 DOI: 10.1103/PhysRevLett.47.886

Renata Jora (2015). $Φ^4$ theory is trivial arXiv arXiv: 1503.07298v1

Marco Frasca (2006). Proof of triviality of $λφ^4$ theory Int.J.Mod.Phys.A22:2433-2439,2007 arXiv: hep-th/0611276v5

Igor M. Suslov (2010). Asymptotic Behavior of the \Beta Function in the Φ^4 Theory: A Scheme
Without Complex Parameters J.Exp.Theor.Phys.111:450-465,2010 arXiv: 1010.4317v1

I. L. Bogolubsky, E. -M. Ilgenfritz, M. Müller-Preussker, & A. Sternbeck (2009). Lattice gluodynamics computation of Landau-gauge Green’s functions in the deep infrared Phys.Lett.B676:69-73,2009 arXiv: 0901.0736v3

Gian Giudice and Lisa Randall in Rome


As usual, also for this year there has been the Festival delle Scienze (Festival of the Sciences) in Rome. This lasted for all the last week and ended this sunday. This is the chance to hear from leading scientists the status of forefront research. This year’s theme was “The End of the World – Instructions for the Use”. Two leading theoretical physicists were present in different events: Lisa Randall and Gian Francesco Giudice. I have had not the chance to listen Lisa Randall but something she said come out in Italian newspapers. Lisa declared that KK particles are spies of other dimensions and that these are in the reach of LHC. I think that readers of the blogosphere already know what we are talking about. Indeed, KK stays for Kaluza-Klein and these particles generally arise as an effect of compactification of the other dimensions beyond the four we everyday experience. Lisa has written a paper, in collaboration with Ben Lillie and Lian-Tao Wang, providing an expectation of mass for a KK particle arising as an excitation of gluons at LHC. Some hints in this direction appeared with the measurement of charge asymmetry at Tevatron. I would like to remember that the Randall-Sundrum scenario to explain the hierarchy problem between interactions is one of the most successful ones devised so far due to the real cleverness of the idea. I regret to have missed the opportunity to listen from Lisa due to my very few time, being her present on Friday evening.

Of course, on Saturday I have much more time to spend and so I have had the opportunity to hear from Gian Francesco Giudice from CERN Theoretical Division. His talk was scheduled on Saturday evening at 19 o’clock. The talk was addressed to a non-specialist public so it was also a good opportunity to take my thirteen year’s old boy to listen. The title was “Black holes, accelerators and the end of the World”. I think you have already heard of the fine book Gian Giudice wrote recently in Tommaso Dorigo’s blog. The talk was in-line with the content of the book trying to make common people aware of what are the endeavors we physicists are pursuing with such an enormous enterprise. What makes me hope for the better has been to see a really crowded room such that the saturation point was promptly achieved and the talk started ten minutes in advance with respect to the scheduled time.

Gian started the talk discussing with a lot of irony the question of the LHC and the end of the World. He cited Nostradamus, Apocalypse by S. Giovanni and the date when the construction of the LHC started that sums up to a worrisome 666, the number of the devil. But he pointed out how a fine report to which he collaborated shows that no black hole could possibly form swallowing Earth and its neighborhood. The idea is that cosmic rays already produced even larger energies than LHC and corresponding collisions without ever producing such an effect. In this way, the probability of an unknown event can be evaluated and the event itself ruled out.

He then showed the extraordinary numbers of LHC that prompted a former NASA engineer participating to Apollo project to say that the latter was just a game for children with respect to the machine that was assembling.

Gian clarified that the mass arising from the Higgs field is not the same seen at a macroscopic level. Indeed, this is due to other reasons explained more and more in this blog and gives also another strong motivation to understand the behavior of Yang-Mills theory at low energies. To explain Higgs field, Gian used the example of a fish moving in the water. The fish cannot say there is a medium but if some excitations like waves are perceived these are an evidence in this sense. So, in the same way, we need LHC to get such excitations for the Higgs field and prove its existence. A small boy, claiming to be a physics amateur and well aware of quantum mechanics, asked if such a “Higgs fluid” could slow down particles as happens for normal fluids. Of course, we are talking of different things as relative motion is perceived in a case but not in the other. Higgs vacuum is absolutely indifferent to motion but not in the way it couples to different particles.

A question that naturally arose was if the fact that we have such a fixed space-time stage does not implies a resurrection of Newton’s absolute space. Gian explained with the example of general relativity that this idea is well dead and buried.

An important point Gian made was to note how, starting with a simple field, this field can give the seeds for fluctuations in an otherwise homogeneous space-time producing the galaxies and the large scale structures we observe today in the Universe. So, LHC is an essential starting point to understand our Universe and to answer fundamental questions by observing the particles that are the excitations of this kind of fields. Indeed, he said that are already several years that fields like cosmology, astrophysics and particle physics are going entangled inextricably together. He also pointed out how there are recurring ages in physics when some fields seem to have more results than other but this is just due to the fact that research goes through hits rather than with continuity.

Gian presented the contents with beautiful slides and animations keeping always alive the attention of the public. This was confirmed by the large number of questions people asked. What I have found interesting was the numbers Gian declared for “brains at work” for the LHC at CERN. He said that 4500 experimental physicists are involved against a mere 80 theoretical physicists! But the point that appeared to me more exciting was his declaration that the Higgs sector in the Standard Model appears somehow misplaced in an otherwise very beautiful theory and we, the theorists, all suspect that here is hidden the new physics that LHC will uncover for sure. Supersymmetry was in the air more and more, sometime just whispered but it was clear, at least to me, that this is the next actor due to appear on the scene. I strongly agree with this view from my humble side. The fear is that the only finding of LHC will be the Higgs boson and nothing more. This would decree the end of particle physics as devised since now. In any case, due to the needed long times, it is today that we are already doing feasibility studies for the accelerator of the next generation. Gian pointed out that the fact that the LHC will uncover something for sure is inside the Standard Model that seems to fail exactly at the order of energies the LHC works. A fine description of the Higgs particle was also given and this prompted several questions from the public. Indeed, it is easy to think that we are back to ether again but this is easily seen not the case as the Higgs vacuum is invariant by Lorentz transformations. Some people in the public seemed really informed about the experiments at CERN and a question arrived about the heavy ion collisions. Gian was very able to explain what are the aims and the reasons why humankind should keep on pursuing research like this.

The journalist Claudia Di Giorgio of the editorial office of Italian version of Scientific American (“Le Scienze”) was the host. She asked some recurring questions that surely was helpful for the public to be answered. A nice moment was when Gian clarified the question of the name “God particle” given to the Higgs boson by Leon Lederman and that Claudia used frequently asking for a reaction. Indeed, Gian explained that he asked the question to Lederman that claimed that the real title of his book was “The God damn particle” but the editor of the book did not like it and removed the word “damn”.

My son was very enthusiastic about Gian’s presentation and, at the end of the talk, I took him to greet Gian. It was also my chance to shake his hand and to cite him Tommaso Dorigo… Gian Giudice represents a great example of what means following a right track and surely he was one of the right people for my son to be known.

Lillie, B., Randall, L., & Wang, L. (2007). The Bulk RS KK-gluon at the LHC Journal of High Energy Physics, 2007 (09), 74-74 DOI: 10.1088/1126-6708/2007/09/074

The constant fire


As the readers of my blog know I prefer to consider arguments with a transversal view with respect to commonly accepted wisdoms. The reason for this is that behind such ways to reconsider an old problem, most of times, a solution may be hidden and this solution could have breathtaking consequences. It is the case for the book “The Constant Fire”.


The authors is Adam Frank, an astrophysicist and, let me add, a courageous one. The reason is that to face the neverending debate between science and religion is, for a scientist, a true enterprise and a lot of hurdles must be faced before someone can take your arguments as something worthless some attention. But the view presented in this book by Adam is really new and worthwhile to be considered to put all the matter in a fresher view.

As scientists we all know what is a constant fire. It is our inexhaustible research of the way the world works and is, on a different side, the same fire that burns for mystics whose experiences are the grounds of all religions.

Adam puts all matter through the view of experience being this the only thing adding value both to science and religion. The reference author for an understanding of experience in religion is Mircea Eliade. This is a Romanian thinker with a bad side in politics. But we are aware of this kind of misbehavior also for physicists and mathematicians so we care only about ideas. The central idea in Eliade’s thought is the hierophany that is the moment of revelation that is central in all religious thought and is also similar to a scientist’s experience when the discovery moment happens. In both cases the consequences can be of great moment changing forever mankind’s history. So, this is the only valor to be considered and common to both human endeavors. This means that those that appeared as opposite sides have a common central aspect.

This commonality can be extended to permit to acquire a unique knowledge element to improve the future of mankind and avoiding further useless fighting between the two sides. Indeed, we are fully aware, being on the privileged side of the scientist, as people needs something that can match their archetype structure and this cannot be simply science with its mathematics. The only element able to produce this food are myths and these supply the same elements mathematics yields in science for the sacred aspect people needs. Science produces new myths, a common behavior with respect to religion.

Scientists have hierophanies as happens to mystical people. Some notable examples are reported in the book. The most relevant one is surely Wolfgang Pauli and his longstanding relation with Carl Gustav Jung. Pauli is surely the scientist that mostly combined the research in the sensible world and his inside world producing a lot of great material about. Indeed, this part of the book was for me a real discover about this great scientist and his ability to produce imaginative ideas both in science and other human endeavor.

The conclusion to be expected is that, in the view of their common experience, science and religion must merge in an unique ethical view of the World forcing mankind into a maturation stage where we will be able to manage our knowledge to improve our pursuing of life on Earth avoiding any risk of self-destruction.

I should say as an essential point that skeptical environmentalists should have some problems here and there in the book. The author has had an experience working in the climate science when he was 24 and since then he got imprinted by this matter. So, someone could not share some conclusions about.

The book contains a lot of anecdotes both for scientists and not scientists and there is a lot to learn in several unexpected fields. It shows quite clearly that the author underwent significant pursuits for his accomplishment. It is surely a worthwhile reading and can be a significant improvement in fields of endeavor where people working in science can be not well accustomed.

A blog of the author can be found here.

Empire of the Stars


During my week-end in Soverato I have spent some time reading this book. It was for me a good chance to enter into life of Subramanyan Chandrasekhar. I know him for being a great physicist and I have had the chance to read some of his works. He has been also a collaborator of Enrico Fermi and both produced a pair of relevant papers on magnetohydrodynamics. But my knowledge was rather superficial and I was not aware of the difficulties he met in obtaining his results widely known. He was a pioneer in our current understanding of star evolution and the emerging of singularities in space-time. The difficulties he met were mostly due to Arthur Eddington that employed all his scientific relevance to impede the emerging of this important result for reasons that today have lost any importance and were essentially wrong. This implied that almost forty years were needed before the relevance of Chandra work was generally acknowledged and he was awarded a Nobel prize for this reason in 1983.

We should be aware that this dynamics in the scientific community is still present. Planck used to say that a new idea becomes generally accepted when all the opponents are dead and the new generations are open to it. I think, but this is just my view, that the conservative reasoning is just an heritage of our struggle for surviving. Indeed, entering into a risky situation may happen if your idea is a failure while we know that old methods work so well. And still today communities at large maintain this view. The lucky case in physics is that a new concept cannot be stopped for too long. Scientific method implies that as soon as new experimental results become available, theory advances as well and some apparently questionable concepts may turn out to be the right description of the behavior of Nature. It is just a question of time but we are only humans and can happen that the discoverers of a right concept could be acknowledged late or too late in some cases.

The book has some wrong historical facts. The most blatant I have found was about Lev Landau. Landau was imprisoned by the brutal Stalinist government. Kapitza come in his rescue claiming that only him could solve the “superconductivity” problem he was experimenting on. Everyone well acquainted with history of physics knows that the word “superconductivty” should be changed into “superfluidity”. Explanation of superfluidity earned a Nobel prize to Landau in 1962. Superconductivty has another story. Finally, I have read the italian edition and some imprecisions into the translation can be found here and there. The most entertaining has been the translation of “naked singularities” (“singolarita’ palesi” to be translated as “manifest singularities”) and “duro come la roccia” (“hard like rock”) to indicate how strong is the core of e.g. a neutron star. For this latter expression I hope the author did not use a similar expression in english. I would like to know why italian editors do not ask to competent people for such translations.

Finally, I would not agree fully with the idea the author left to me about Chandra. A kind of angry and always unsatisfied man maybe due to the starting quarrel with Eddington that left an indelible sign for all his lifetime . Indeed, he obtained recognitions of his merit late in his life but indeed his life was great as a physicist both for his accomplishments and for people he happened to meet.

Just a note about LHC in the book. The author claims that one of the aims of this facility is to produce black holes! Maybe an update is needed in view of recent and less recent controversies.

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