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

That’s a Higgs but how many?


CMS and ATLAS collaborations are yet up to work producing results from the datasets obtained in the first phase of activity of LHC. The restart is really near the corner and, maybe already the next summer, things can change considerably. Anyway what they get from the old data can be really promising and rather intriguing. This is the case for the recent paper by CMS (see here). The aim of this work is to see if a heavier state of Higgs particle exists and the kind of decay they study is Zh\rightarrow l^+l^-bb. That is, one has a signature with two leptons moving in opposite directions, arising from the dacy of the Z, and two bottom quarks arising from the decay of the Higgs particle. The analysis of this decay aims to get hints of existence of a heavier pseudoscalar Higgs state. This can be greatly important for SUSY extensions of the Standard Model that foresee more than one Higgs particle.

Often CMS presents its results with some intriguing open questions and also this is the case and so, it is worth this blog entry. Here is the main result

CMS study of Zh->llbbThe evidence, as said in the paper, is that there is a 2.6-2.9 sigma evidence at 560 GeV and a smaller one at around 300 GeV. Look elsewhere effect reduces the former at 1.1 sigma and the latter is practically negligible. Overall, this is pretty negligible but, as always, with more data at the restart, could become something real or just fade away. It should be appreciated the fact that a door is left open anyway and a possible effect is pointed out.

My personal interpretation is that such higher excitations do exist but their production rates are heavily suppressed with the respect to the observed ground state at 126 GeV and so, negligible with the present datasets. I am also convinced that the current understanding of the breaking of SUSY, currently adopted in MSSM-like to go beyond the Standard Model, is not the correct one provoking the early death of such models. I have explained this in a coupled of papers of mine (see here and here). It is my firm conviction that the restart will yield exciting results and we should be really happy to have such a powerful machine in our hands to grasp them.

Marco Frasca (2013). Scalar field theory in the strong self-interaction limit Eur. Phys. J. C (2014) 74:2929 arXiv: 1306.6530v5

Marco Frasca (2012). Classical solutions of a massless Wess-Zumino model J.Nonlin.Math.Phys. 20:4, 464-468 (2013) arXiv: 1212.1822v2

Higgs what?


In these days it has been announced the new version of Review of Particle Physics by the Particle Data Group (PDG). This is the bread and butter of any particle physicist and contains all the relevant data about this area of research. It is quite common for us to search the on-line version or using the booklet to know a mass or a decay rate. After the first run of LHC data gathering about Higgs particle, this edition contains a bunch of fundamental informations about it and I post a part of them.

Higgs on PDG

It is Standard Model Higgs! No, not so fast. Take a look at the WW final state. It is somewhat low but yes, it is perfectly consistent with the Standard Model. Also, error bars are somewhat large to conclude something definitive. So, let us take a look nearer at these strengths.

Higgs decay to WWWe discover that the strengths measured by CMS are really low and takes down this value. Indeed, this is consistent with my proposal here. I get 0.68 for both channels WW and ZZ. On the other side, ATLAS moves all upward consistently and there is this strange behaviour compensating each other. So, let us also take a look at the ZZ strength. PDG yields

Higgs decay to ZZagain CMS agrees with my conclusions and ATLAS moves all upward to compensate. But both these results, due to the large error bars, agree rather well with the Standard Model. So, I looked for the publication by CMS  that were produced till today if one or both these analyses were improved. The result was that CMS improved the measure of the strength in the WW channel to leptons (see here). What they measure is


The error is significantly smaller and the result striking. It is bending in the “wrong” turn loosing higgsness. It would be interesting to understand why CMS appear to get results downward for these strengths and ATLAS more upward compensating each other toward the Standard Model. On the other side, I should admire the more aggressive approach by CMS with their results more and more similar to my expectations. I am just curious to see with the restart of LHC what will happen to these data that CMS sharpened to such a point.

Marco Frasca (2013). Scalar field theory in the strong self-interaction limit Eur. Phys. J. C (2014) 74:2929 arXiv: 1306.6530v5

CMS Collaboration (2013). Measurement of Higgs boson production and properties in the WW decay
channel with leptonic final states JHEP 01 (2014) 096 arXiv: 1312.1129v2

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

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.

CMS harbors new physics beyond the Standard Model


In these days is ongoing LHCP 2013 (First Large Hadron Collider Physics Conference) and CMS data seem to point significantly toward new physics. Their measurements on the production modes for WW and ZZ are agreeing with my recent computations (see here) and overall are deviating slightly from Standard Model expectations giving

\frac{\sigma}{\sigma_SM}=0.80\pm 0.14

Note that Standard Model is alive and kicking yet but looking at the production mode of WW you will read

\frac{\sigma_{WW}}{\sigma_{WW\ SM}}=0.68\pm 0.20

in close agreement with results given in my paper and improved respect to Moriond that was 0.71\pm 0.21. The reason could be that: Higgs model is a conformal one. Data from ZZ yield

\frac{\sigma_{ZZ}}{\sigma_{ZZ\ SM}}=0.92\pm 0.28

that is consistent with the result for WW mode, though. I give here the full table from the talk

CMS at LHCP2013

For the sake of completeness I give here also the same results from ATLAS at the same conference that, instead, seems to go the other way round obtaining overall 1.30\pm 0.20 and this is already an interesting matter.


At CMS, new physics beyond the Standard Model is peeping out and, more inteestingly, the Higgs model tends to be a conformal one. If this is true, supersymmetry is an inescapable consequence (see here). I would like to conclude citing the papers of other people working on this model and that will be largely cited in the foreseeable future (see here and here).

Marco Frasca (2013). Revisiting the Higgs sector of the Standard Model arXiv arXiv: 1303.3158v1

Marco Frasca (2010). Mass generation and supersymmetry arXiv arXiv: 1007.5275v2

T. G. Steele, & Zhi-Wei Wang (2013). Is Radiative Electroweak Symmetry Breaking Consistent with a 125 GeV
Higgs Mass? Physical Review Letters 110, 151601 arXiv: 1209.5416v3

Krzysztof A. Meissner, & Hermann Nicolai (2006). Conformal Symmetry and the Standard Model Phys.Lett.B648:312-317,2007 arXiv: hep-th/0612165v4

Conformal Standard Model is consistent with the observed Higgs particle


Robert Garisto is an Editor of Physical Review Letters, the flagship journal of American Physical Society and the one with the highest impact factor in physics. I follow him on twitter (@RobertGaristo) and he points out interesting papers that appear in the journal he works in. This time I read the following

Tweets from Garisto

and turned immediately my attention to the linked paper: This one (if you have not a subscription you can find it at arxiv) by Tom Steele and Zhi-Wei Wang showing, with the technique of Padè approximants and an average method how to compute the exact mass of Higgs particle from Coleman-Weinberg mechanism arriving to estimate the ninth order contribution. This is so beacuse they need a stronger coupling with respect to the original Higgs mechanism. They reach an upper bound of 141 GeV for the mass and 0.352 for the self-coupling while they get the mass of 124 GeV for a self-coupling of 0.23. This shows unequivocally that the quadratic term, the one generating the hierarchy problem, is absolutely not needed and the Standard Model, in its conformal formulation, is able to predict the mass of the Higgs particle. Besides, the production rates are identical to the original model but differ for the production of Higgs pairs and this is where one could tell which way nature has chosen. This implies that, at the moment, one has no way to be sure this is the right solution but we have to wait till 2015 after LHC upgrade. So, once again, the precise measurements of these decay rates are essential to tell if we are coping with the original Higgs mechanism or something different or if we need two more years to answer this question. In any case, it is possible that Nobel committee has to wait yet before to take a decision. However, in the sixties that formulation was the only possible and any other solution would have been impossible to discover for the lack of knowledge. They did a great job even if we will prove a different mechanism at work as they provided credibility to the Standard Model and people could trust it.

Finally, I would like to note how the value of the coupling is consistent with my recent estimation where I get 0.36 for the self-interaction. I get different production rates and I would be just curious to see how pictures from ATLAS and CMS would change comparing differently from the Standard Model in order to claim no other Higgs-like particle is seen.

What we can conclude is that the conformal Standard Model is in even more better shape than before and just a single Higgs particle would be needed. An astonishing result.

Steele, T., & Wang, Z. (2013). Is Radiative Electroweak Symmetry Breaking Consistent with a 125 GeV Higgs Mass? Physical Review Letters, 110 (15) DOI: 10.1103/PhysRevLett.110.151601

Marco Frasca (2013). Revisiting the Higgs sector of the Standard Model arXiv arXiv: 1303.3158v1


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