## That Higgs is trivial!

05/04/2015

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“. The 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.

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

## Is Higgs alone?

14/03/2015

I am back after the announcement by CERN of the restart of LHC. On May this year we will have also the first collisions. This is great news and we hope for the best and the best here is just the breaking of the Standard Model.

The Higgs in the title is not Professor Higgs but rather the particle carrying his name. The question is a recurring one since the first hints of existence made their appearance at the LHC. The point I would like to make is that the equations of the theory are always solved perturbatively, even if exact solutions exist that provide a mass also if the theory is massless or has a mass term with a wrong sign (Higgs model). All you need is a finite self-interaction term in the equation. So, you will have bad times to recover such exact solutions with perturbation techniques and one keeps on living in the ignorance. If you would like to see the technicalities involved just take a cursory look at Dispersive Wiki.

What is the point? The matter is rather simple. The classical theory has exact massive solutions for the potential in the form $V(\phi)=a\phi^2+b\phi^4$ and this is a general result implying that a scalar self-interacting field gets always a mass (see here and here). Are we entitled to ignore this? Of course no. But today exact solutions have lost their charm and we can get along with them.

For the quantum field theory side what could we say? The theory can be quantized starting with these solutions and I have shown that one gets in this way that these massive particles have higher excited states. These are not bound states (maybe could be correctly interpreted in string theory or in a proper technicolor formulation after bosonization) but rather internal degrees of freedom. It is always the same Higgs particle but with the capability to live in higher excited states. These states are very difficult to observe because higher excited states are also highly depressed and even more hard to see. In the first LHC run they could not be seen for sure. In a sense, it is like Higgs is alone but with the capability to get fatter and present himself in an infinite number of different ways. This is exactly the same for the formulation of the scalar field as originally proposed by Higgs, Englert, Brout, Kibble, Guralnik and Hagen. We just note that this formulation has the advantage to be exactly what one knows from second order phase transitions used by Anderson in his non-relativistic proposal of this same mechanism. The existence of these states appears inescapable whatever is your best choice for the quartic potential of the scalar field.

It is interesting to note that this is also true for the Yang-Mills field theory. The classical equations of this theory display similar solutions that are massive (see here) and whatever is the way you develop your quantum filed theory with such solutions the mass gap is there. The theory entails the existence of massive excitations exactly as the scalar field does. This have been seen in lattice computations (see here). Can we ignore them? Of course no but exact solutions are not our best choice as said above even if we will have hard time to recover them with perturbation theory. Better to wait.

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

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

Marco Frasca (2014). Exact solutions for classical Yang-Mills fields arXiv arXiv: 1409.2351v2

Biagio Lucini, & Marco Panero (2012). SU(N) gauge theories at large N Physics Reports 526 (2013) 93-163 arXiv: 1210.4997v2

## Waiting for EPS HEP 2013: Some thoughts

13/07/2013

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

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

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.

## Conformal Standard Model is consistent with the observed Higgs particle

12/04/2013

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

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

## Much closer to the Standard Model

18/03/2013

Today, the daily from arxiv yields a contribution from John Ellis and Tevong You analyzing new data presented at Aspen and Moriond the last two weeks by CMS and ATLAS about Higgs particle (see here). Their result can be summarized in the following figure

that is really impressive. This means that the updated data coming out from LHC constraints even more the Higgs particle found so far to be the Standard Model one. Another impressive conclusion they are able to draw is that the couplings appear to be proportional to the masses as it should be expected from a well-behaved Higgs particle. But they emphasize that this is “a” Higgs particle and the scenario is well consistent with supersymmetry. Citing them:

The data now impose severe constraints on composite alternatives to the elementary Higgs boson of the Standard Model. However, they do not yet challenge the predictions of supersymmetric models, which typically make predictions much closer to the Standard Model values. We therefore infer that the Higgs coupling measurements, as well as its mass, provide circumstantial support to supersymmetry as opposed to these minimal composite alternatives, though this inference is not conclusive.

They say that further progress on the understanding of this particle could be granted after the upgraded LHC will run and, indeed, nobody is expecting some dramatic change into this scenario from the data at hand.

John Ellis, & Tevong You (2013). Updated Global Analysis of Higgs Couplings arXiv arXiv: 1303.3879v1

## A Higgs particle but which one?

14/03/2013

After Moriond conference last week, and while Moriond QCD and Aspen conferences are running yet, an important conclusion can be drawn and it is the one given in this CERN press release. The particle announced on 4th July last year is for certain a Higgs particle as it has spin 0, positive parity and couples almost like the Standard Model Higgs particle to all others. The agreement with Standard Model is embarrassingly increasing as cumulated data since last year are analyzed. Today, CMS will also update their results for the decay $H\rightarrow\gamma\gamma$ and we will know if the small deviation observed by ATLAS will be confirmed. It is true that they see such a deviation with a larger dataset but, rather to increase, it has slightly diminished and this is not really encouraging.

So far, no other particle has been seen and no new physics beyond the Standard Model is seen at the horizon. There is some people pushing for a conclusive assignment of the nature of this boson to the vanilla Higgs particle postulated in the sixties. But it is really too early yet to draw such a conclusion and I have explained why in a paper of mine appeared today on arxiv (see here). Indeed, a formulation of the Higgs field is possible such that, at the tree level, coincides with the original Higgs field (a Higgs impostor). This is due to the existence of exact solutions of the equations of motion of such a field (see here). The relevant point to tell which one is realized in nature is through the decay rate in WW and ZZ and, with the current data, there is agreement for both yet. But, being amplitudes exponentially damped, higher excited states of the Higgs boson cannot be easily seen presently and their eventual observation appears as a statistical fluctuation yet. This can be evaluated quantitatively. It is important because the ZZ decay is sensible to higher masses and displays some peaks that reveal themselves as statistical fluctuations. Increasing the number of events could turn these peaks into real observations.

The interesting point here is that we are moving form the discovery moment to the study phase with a lot of room for improving measurements on this Higgs particle. But the analysis for the existence of higher excited states, Higgs’ brothers, is just at its infancy.

Update: This the analogous figure from ATLAS while the figure for $H\rightarrow\gamma\gamma$ from CMS agrees quite well with the Standard Model: $0.8\pm 0.3$.

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

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