## Living dangerously

05/11/2013

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

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.

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

## Where does mass come from?

16/12/2012

After CERN’s updates (well recounted here, here and here) producing no real news but just some concern about possible Higgs cloning, I would like to discuss here some mathematical facts about what one should expect about mass generation and why we should not be happy with these results, now coming out on a quarterly basis.

The scenario we are facing so far is one with a boson particle resembling more and more the Higgs particle appearing in the original formulation of the Standard Model. No trace is seen of anything else at higher energies, no evidence of supersymmetry. It appears like no new physics is hiding here rather for it we will have to wait eventually the upgrade of LHC that will start its runs on 2015.

I cannot agree with all of this and this is not the truth at all. The reason to not believe all this is strictly based on theoretical arguments and properties of partial differential equations. We are aware that physicists can be skeptical also about mathematics even if this is unacceptable as mathematics has no other way than being true or false. There is nothing like a half truth but there are a lot of theoretical physicists trusting on it. I have always thought that being skeptical on mathematics is just an excuse to avoid to enter into other work. There could always be the risk that one discovers it is correct and then has to support it.

The point is the scalar field. A strong limitation we have to face when working in quantum field theory is that only small coupling can be managed. No conclusive analysis can be drawn when a coupling is just finite and also lattice computations produce confusion. It seems like small coupling only can exist and all the theory we build are in the hope that nature is benign and yields nothing else than that. For the Higgs field is the same. All our analysis are based on this, the hierarchy problem comes out from this. Just take any of your textbook on which you built your knowledge of this matter and you will promptly realize that nothing else is there. Peschin and Schroeder, in their really excellent book, conclude that strong coupling cannot exist in quantum field theory and the foundation of this argument arises from renormalization group. Nature has only small couplings.

Mathematics, a product of nature, has not just small couplings and nobody can impede a mathematician to take these equations and try to analyze them with a coupling running to infinity. Of course, I did it and somebody else tried to understand this situation and the results make the situation rather embarrassing.

These reflections sprang from a paper appeared yesterday on arxiv (see here). In a de Sitter space there is a natural constant having the dimension of energy and this is the Hubble constant (in natural units). It is an emerging result that a massless scalar field with a quartic interaction in such a space develops a mass. This mass goes like $m^2\propto \sqrt{\lambda}H^2$ being $\lambda$ the coupling coming from the self-interaction and $H$ the Hubble constant. But the authors of this paper are forced to turn to the usual small coupling expansion just singling out the zero mode producing the mass. So, great news but back to the normal.

A self-interacting scalar field has the property to get mass by itself. Generally, such a self-interacting field has a potential in the form $\frac{1}{2}\mu^2\phi^2+\frac{\lambda}{4}\phi^4$ and we can have three cases $\mu^2>0$, $\mu^2=0$ and $\mu^2<0$. In all of them the classical equations of motion have an exact massive free solution (see here and Tao’s Dispersive Wiki) when $\lambda$ is finite. These solutions cannot be recovered by any small coupling expansion unless one is able to resum the infinite terms in the series. The cases with $\mu^2\ne 0$ are interesting in that this term gets a correction depending on $\lambda$ and for the case $\mu^2<0$ one can recover a spectrum with a Goldstone excitation and the exact solution is an oscillating one around a finite value different from zero (it never crosses the zero) as it should be for spontaneous breaking of symmetry. But the mass is going like $\sqrt{\lambda}\Lambda^2$ where now $\Lambda$ is just an integration constant. The same happens in the massless case as one recovers a mass going like $m^2\propto\sqrt{\lambda}\Lambda^2$.  We see the deep analogy with the scalar field in a de Sitter space and these authors are correct in their conclusions.

The point here is that the Higgs mechanism, as has been devised in the sixties, entails all the philosophy of “small coupling and nothing else” and so it incurs in all the possible difficulties, not last the hierarchy problem. A modern view about this matter implies that, also admitting $\mu^2<0$ makes sense, we have to expand around a solution for $\lambda$ finite being this physically meaningful rather than try an expansion for a free field. We are not granted that the latter makes sense at all but is just an educated guess.

What does all this imply for LHC results? Indeed, if we limit all the analysis to the coupling of the Higgs field with the other fields in the Standard Model, this is not the best way to say we have observed a true Higgs particle as the one postulated in the sixties. It is just curious that no other excitation is seen beyond the (eventually cloned) 126 GeV boson seen so far but we have a big desert to very high energies. Because the very nature of the scalar field is to have massive solutions as soon as the self-interaction is taken to be finite, this also means that other excited states must be seen. This simply cannot be the Higgs particle, mathematics is saying no.

M. Beneke, & P. Moch (2012). On “dynamical mass” generation in Euclidean de Sitter space arXiv arXiv: 1212.3058v1

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

## Breaking of a symmetry: A paper

07/11/2012

I have uploaded a paper on arXiv (see here), following my preceding post,  where I show that supersymmetry has inside itself the seeds for the breaking. I consider a Wess-Zumino model without masses (chiral) and I prove that, at lower momenta, it boils down to a Nambu-Jona-Lasinio model so, breaking supersymmetry through a gap equation that has a solution beyond a critical coupling. An essential assumption is that the coupling in the model is not increasingly smaller but rather increasingly greater. So, bosons and fermions get different masses.

This should open up a new way to see at supersymmetric theories that produce by themselves nonlinearities: It is enough to have such nonlinearities growing bigger. In this way, the large number of parameters that seems a need in the Minimal Supersymmetric Standard Model, arising from the introduction by hand of breaking terms, hopefully should reduce significantly.

Finally, I would like to point out a paper by Jamil Hetzel giving a nice introduction to these problematics (see here). This is a master thesis whose content appeared on JHEP.

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

Jamil Hetzel (2012). Probing the supersymmetry breaking mechanism using renormalisation group
invariants arXiv arXiv: 1211.1157v1

## Breaking of a symmetry

05/11/2012

This week-end has been somewhat longer in Italy due to November 1st holiday and I have had the opportunity to read a very fine book by Ian Aitchison: Supersymmetry in Particle Physics – An Elementary Introduction. This book gives a very clear introduction to SUSY with all the computations clearly stated and going into the details of the Minimal Supersymmetric Standard Model (MSSM). This model was originally proposed by Howard Georgi and Savas Dimopolous (see here) and today does not seem to be in good shape due to recent results from LHC. Authors introduce a concept of a “softly” broken supersymmetry to accomodate the Higgs mechanism in the low-energy phenomenology.  A “soflty” broken supersymmetry is when the symmetry is explicitly broken using mass terms but keeping renormalizability and invariance under the electroweak symmetry group. The idea is that, in this way, the low-energy phenomenology will display a standard Higgs mechanism with a vacuum expectation value different from zero. This fact is really interesting as we know that in a standard electroweak theory the symmetry cannot be explicitly broken as we lose immediately renormalizability but a supersymmetric theory leaves us more freedom. But why do we need to introduce explicit breaking terms into the Lagrangian of the MSSM? The reason is that SUSY is conveying a fundamental message:

There is no such a thing as a Higgs mechanism.

Indeed, one can introduce a massive contribution to a scalar field, the $\mu-$term, but this has just the wrong sign and, indeed, a spontaneously broken supersymmetry is somewhat of a pain. There are some proposed mechanisms, as F or D breaking fields or some dynamical symmetry breaking, but nothing viable for the MSSM. Given the “softly” breaking terms, then the argument runs smoothly and one recovers two doublets and $\tan\beta$ parameter that some authors are fond of.

The question at the root of the matter is that a really working supersymmetry breaking mechanism is yet to be found and should be taken for granted as we do not observe superpartners at accessible energies and LHC has yet to find one if ever. This mechanism also drives the electroweak symmetry breaking. Indeed, supersymmetry properly recovers a quartic self-interaction term but the awkward quadratic term with a wrong sign gives serious difficulties. Of course, the presence of a quartic term into a scalar field interacting with a fermion field, e.g. a Wess-Zumino model, provides the essential element to have a breaking of supersymmetry at lower energies: This model is reducible to a Nambu-Jona-Lasinio model and the gap equation will provide a different mass to the fermion field much in the same way this happens to chiral symmetry in QCD. No explicit mass term is needed but just a chiral model.

This means that the MSSM can be vindicated once one gets rid of an explicit breaking of the supersymmetry and works out in a proper way the infrared limit. There is a fundamental lesson we can learn here: SUSY gives rise to self-interaction and this is all you need to get masses. Higgs mechanism is not a fundamental one.

Dimopoulos, S., & Georgi, H. (1981). Softly broken supersymmetry and SU(5) Nuclear Physics B, 193 (1), 150-162 DOI: 10.1016/0550-3213(81)90522-8

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

## Higgs boson and conformal symmetry

30/08/2012

So far, I believed to be the only man on Earth to trust a complete absence of mass terms in the Standar Model (we call this conformal symmetry). I was wrong.  Krzysztof Meissner and Hermann Nicolai anticipated this idea. Indeed, in a model where mass is generally banned, there is no reason to believe that also the field that is the source of mass should keep a mass term (imaginary or real). We have one more reason to believe in such a scenario and it is the hierarchy problem as the quadratic term in the Higgs field just produces that awkward dependence on the square of the cut-off, the reason why people immediately thought that something else must be in that sector of the model. Meissner and Nicolai obtained their paper published on Physics Letters B and can be found here. As they point out in the article, the problem is to get a meaningful mass for the Higgs field, provided one leaves the self-coupling to be small. I do not agree  at all with the reasons for this, the Landau pole, as I have already widely said in this blog. One cannot built general results starting from perturbation theory. But assuming that this is indeed the case, the only mechanism at our disposal to get a mass is the Coleman-Weinberg mechanism. In this case, radiative corrections produce an effective potential that has a non-trivial minimum. The problem again is that this is obtained using small perturbation theory and so, the mass one gets is too small to be physically meaningful. The authors circumvent the problem adding a further scalar field. In this case the model appears to be consistent and all is properly working. What I would like to emphasize is that, if one assumes conformal symmetry to hold for the Standard Model, a single Higgs is not enough. So, I like this paper a lot and I will explain the reasons in a moment. I am convinced that these authors are on the right track.

Two days ago these authors come out with another paper (see here). They claim that the second Higgs has been already seen at CDF (Tevatron), at about 325 GeV, while we know there is just a hint (possibly a fluke) from CMS and nothing from ATLAS for that mass. Of course, there is always the possibility that this resonance escaped due to its really small width.

My personal view was already presented here. At that time, I was not aware of the work by Meissner and Nicolai otherwise I would have used it as a support. The only point I would like to question is the effective generation of mass. There is no generally accepted quantum field theory for a large coupling, neglecting for the moment attempts arising from string theory. Before to say that string theory grants a general approach for strongly coupled problems I would like to see it to give a solution to the scalar massless quartic field theory in such a case. This is the workhorse for this kind of problems and both the communities of physicists and mathematicians were just convinced that perturbation theory has only one side. As I showed here, this is not true. One can do perturbation theory also when a perturbation is taken to go to infinity. This means that we do not need a Coleman-Weinberg mechanism in a conformal Standard Model but we can do perturbation theory assuming a finite self-interaction: An asymptotic perturbation series can be also obtained in this case. But the fundamental conclusions one can draw from this analysis are the following:

• The theory must be supersymmetric.
• The theory has a harmonic oscillator spectrum for a free particle given by $m_n=(2n+1)(\pi/2K(i))v$, being $K(i)$ an elliptic integral and $v$ an integration constant with the dimension of energy.

Now, let us look at the last point. One can prove that the decays for the higher excited states are increasingly difficult to observe as their decay constants become exponentially smaller with $n$ (see here, eq. 11). But, if the observed Higgs boson has a mass of  about 125 GeV, one has $v=105\ GeV$ and the next excitation is at about 375 GeV, very near the one postulated by Meissner and Nicolai and also near to the bump seen at CDF. This would be an exciting evidence of existence for supersymmetry: The particle seen at CERN would be supersymmetric!

So, what I am saying here is that a conformal Standard Model, not only solves the hierarchy problem, but it is also compelling for the existence of supersymmetry. I think it would be worthy further studies.

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

Krzysztof A. Meissner, & Hermann Nicolai (2012). A 325 GeV scalar resonance seen at CDF? arXiv arXiv: 1208.5653v1

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

Marco Frasca (2010). Glueball spectrum and hadronic processes in low-energy QCD Nucl.Phys.Proc.Suppl.207-208:196-199,2010 arXiv: 1007.4479v2

## Today in arXiv

08/04/2011

After the excitation for the findings at Tevatron, we turn back to routine. Of course, I have never forgotten to cast a glance at arXiv where it is crystal clear the vitality of the physics community. I want to put down these few lines to point to your attention a couple of papers appeared today on the preprint archive. Today, Nele Vandersickel uploaded her PhD Thesis (see here). She has got her PhD on March this year. Nele was one of the organizers of the beautiful and successful conference in Ghent (Belgium) where I was present last year (see here, here and here). But most important is her research work with the group of Silvio Sorella and David Dudal that is the central theme of her thesis. Nele does an excellent job in presenting a lot of introductory material, difficult to find in the current literature, beside her original research. Sorella and Dudal have accomplished an interesting research endeavor by supporting the Gribov-Zwanziger scenario, at odds in the initial formulation with lattice data, with their view that condensates must be accounted for. In this way, Gribov-Zwanziger scenario can be taken to agree with lattice computations.  These theoretical studies describe a consistent approach and these authors were able to obtain the masses of the first glueball states. I would like to conclude with my compliments for the PhD reached by Nele and for the excellent wotk her and the other people in the group were able to realize.

The other fine paper I have found is a report by a group of authors, “Discoverig Technicolor”, giving a full account of the current situation for this theoretical approach to the way particles acquire their masses. As you know, the original formulation of the Higgs particle that entered into the Standard Model contains some drawbacks that motivated several people to find better solutions. Technicolor is one of these. One assumes the existence of a set of Fermions with a self-interaction. We know that this kind of models, as Nambu-Jona-Lasinio is, are able to break symmetries and generate masses to massless particles. Indeed, one can formulate a consistent theory with respect to all the precision tests of the Standard Model as also discussed in this report. This means in turn that in accelerator facilities one should look for some other Fermions and their bound states that can also mimic a standard Higgs scalar boson. It is important to note that in this way some drawbacks of the original Higgs mechanism are overcome. Of course, the relevance of this report cannot be underestimated in view of the results coming out from LHC and we could know very soon if an idea like Technicolor is the right  one or not. For sure, this is time for answers in the end.

Nele Vandersickel (2011). A study of the Gribov-Zwanziger action: from propagators to glueballs arXiv arXiv: 1104.1315v1

J. R. Andersen, O. Antipin, G. Azuelos, L. Del Debbio, E. Del Nobile, S. Di Chiara, T. Hapola, M. Jarvinen, P. J. Lowdon, Y. Maravin, I. Masina, M. Nardecchia, C. Pica, & F. Sannino (2011). Discovering Technicolor arXiv arXiv: 1104.1255v1

## Mass generation and supersymmetry

30/07/2010

I have uploaded a paper on arxiv with a new theorem of mine. I have already exposed the idea in this blog but, so far, I have had no much time to make it mathematically sound.  The point is that the mechanism I have found that gives mass to Yang-Mills and scalar fields implies supersymmetry. That is, if I try to apply it to the simplest gauge theory, in a limit of a strong self-interaction of a massless Higgs field, all the fields entering into the theory acquire identical masses  and the couplings settle down to the proper values for a supersymmetric model. Being this result so striking, I was forced to produce a theorem at the classical level, as generally done with the standard Higgs mechanism, and let it widely known. My next step is to improve the presentation and extend this result after a fully quantum treatment. This is possible as I have already shown in the case of a Yang-Mills theory.

My view is that just a mechanism could be seen in Nature to produce masses and I expect that this is the same already seen for QCD. So, supersymmetry is mandatory. This will imply a further effort for people at work to uncover Higgs particle as they should also say to us what kind of self-interaction is in action here and if it is a supersymmetric particle, as it should.

The interesting point is that all the burden of the spectrum of the standard model will rely, not on the mechanism that generates masses but on the part of the model that breaks supersymmetry.

Interesting developments are expected in the future. Higgs is always Higgs but a rather symmetric one. So, stay tuned!