Where does mass come from?



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



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

Back to Earth



Nature publishes, in the last issue, an article about SUSY and LHC (see here).  The question is really simple to state. SUSY (SUperSYmmetry) is a solution to some problems that plagued physics for some time. An important question is the Higgs particle. In order to have the Standard Model properly working, one needs to fine tune the Higgs mass. SUSY, at the price to double all the existing particles, removes this need. But this can be obtained only if a finite parameter space of the theory is considered. This parameter space is what is explored at accelerator facilities like Tevatron and LHC. Tevatron was not able to uncover any SUSY partner for the known particles restricting it. Of course, with LHC opportunities are much larger and, with the recent papers by ATLAS and CMS, the parameter space has become dangerously smaller making somehow more difficult to remove fine tuning for the Higgs mass without fine tuning of the parameters of the SUSY, a paradoxical situation that can be avoided just forgetting about supersymmetry.

But, as often discussed in this blog, there is another way out saving both Higgs and supersymmetry. All the analysis carried out so far about Higgs field are from small perturbation theory and small couplings: This is the only technique known so far to manage a quantum field theory. If the coupling of the Higgs field is large, the way mass generation could happen is different being with a Schwinger-like mechanism. This imposes supersymmetry on all the particles in the model. This was discussed here. But in this way there is no parameter space to be constrainted for fine tuning to be avoided and this is a nice result indeed.

Of course, situation is not so dramatic yet and there is other work to be carried on at CERN, at least till the end of 2012, to say that SUSY is ruled out. Since then, it is already clear to everybody that exciting time are ahead us.

The ATLAS Collaboration (2011). Search for supersymmetry using final states with one lepton, jets, and missing transverse momentum with the ATLAS detector in sqrt{s} = 7 TeV pp
arxiv arXiv: 1102.2357v1

CMS Collaboration (2011). Search for Supersymmetry in pp Collisions at 7 TeV in Events with Jets
and Missing Transverse Energy arxiv arXiv: 1101.1628v1

The ATLAS Collaboration (2011). Search for squarks and gluinos using final states with jets and missing
transverse momentum with the ATLAS detector in sqrt(s) = 7 TeV proton-proton
collisions arxiv arXiv: 1102.5290v1

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

Mass generation: The solution



In my preceding post I have pointed out an interesting mathematicalquestion about the exact solutions of the scalar field theory that I use in this paper


given by

\phi=\mu\left(\frac{2}{\lambda}\right)^\frac{1}{4}{\rm sn}(p\cdot x,i)

that holds for


If you compute the Hamiltonian the energy does not appear to be finite, differently from what the relation dispersion is saying. This is very similar to what happens to plane waves for the wave equation. The way out is to take a finite volume and normalize properly the plane waves. One does this to get the integral of the Hamiltonian finite and all amounts to a proper normalization. In our case where must this normalization enter? The striking answer is: The coupling. This is an arbitrary parameter of the theory and we can properly rescale it to get the right normalization in the Hamiltonian. The final result is a running coupling exactly in the same way as I and others have obtained for the quantum theory. You can see the coupling entering in the right way both in the solution and in the computation of the Hamiltonian.

If you are curious about these computations you can read the revised version of my paper to appear soon on arxiv.

Marco Frasca (2010). Mass generation and supersymmetry arxiv arXiv: 1007.5275v1

Mass generation in the Standard Model


The question of the generation of the mass for the particles in the Standard Model is currently a crucial one in physics and is a matter that could start a revolutionary path in our understanding of the World as it works. This is also an old question that can be rewritten as “What are we made of?” and surely ancient greeks asked for this. Today, with the LHC at work and already producing a wealth of important results, we are on the verge to give a sound answer to it.

The current situation is well-known with a Higgs mechanism (but here there are several fathers) that mimics the second order phase transitions as proposed by Landau long ago. In some way, understanding ferromagnetism taught us a lot and produced a mathematical framework to extract sound results from the Standard Model. Without these ideas the model would have been practically useless since the initial formulation due to Shelly Glashow. The question of mass in the Standard Model is indeed a stumbling block and we need to understand what is hidden behind an otherwise exceptionally successful model.

As many of yours could know, I have written a paper (see here) where I show that if the way a scalar field gets a mass (and so also Yang-Mills field) is identical in the Standard Model, forcefully one has a supersymmetric Higgs sector but without the squared term and with a strong self-coupling. This would imply a not-so-light Higgs and the breaking of the supersymmetry the only way to avoid degeneracy between the masses of all the particles of the Standard Model. By my side I would expect these signatures as evidences that I am right and QCD, a part of the Model, will share the same mechanism to generate masses.

Yet, there is an open question put forward by a smart referee to my paper. I will put this here as this is an interesting question of classical field theory that is worthwhile to be understood. As you know, I have found a set of exact solutions to the classical field equation


from which I built my mass generation mechanism. These solutions can be written down as

\phi(x)=\mu\left(\frac{2}{\lambda}\right)^\frac{1}{4}{\rm sn}(p\cdot x+\theta,i)

being sn a Jacobi’s elliptic function and provided


From the dispersion relation above we can conclude that these nonlinear waves indeed represent free massive particles of finite energy. But let us take a look to the definition of the energy for this theory, one has

H=\int d^3x\left[\frac{1}{2}(\dot\phi)^2+\frac{1}{2}(\nabla\phi)^2+\frac{\lambda}{4}\phi^4\right]

and if you substitute the above exact solutions into this you will get an infinity. It appears like these solutions have infinite energy! This same effect is seen by ordinary plane waves but can be evaded by taking a finite volume, one normalizes the solutions with respect to this volume and so you are done.  Of course, you can take finite volume also in the nonlinear case provided you put for the momenta


being i=x,y,z as this Jacobi function has period 4K(i) but you should remember that this function is doubly periodic having also a complex period. Now, if you compute for H you will get a dispersion relation multiplied by some factors and one of these is the volume. How could one solve this paradox? You can check by yourselves that these solutions indeed exist and must have finite energy.

My work on QCD is not hindered by this question as I work solving the equation \Box\phi+\lambda\phi^3=j and here there are different problems. But, in any case, mathematics claims for existence of these solutions while physics is saying that there is something not so well defined. An interesting problem to work on.

Some reflections


It is a lot of time that I am thinking about the scenario is emerging from our current understanding of reality through physics. There is a lot to be understood yet, mostly the very nature of space and time and a proof of the real number of dimensions our universe emerged from. Notwithstanding these noteworthy missing questions, we can make an idea of what is going on as the depth of our understanding is already quite sensible.

At the dawn of the last century, Albert Einstein put forward an important conclusion: mass and energy are the same thing. Indeed, what Einstein had in mind is a deeper understanding of the concept of mass that is for us an important concern. Where does mass come from? What is it made of? At the start of the last century these questions could not have a proper answer. But having reduced the mass to another concept, like energy, was of paramount importance.

On a similar ground it was relevant to understand the role of another, apparently not reducible, concept: Charge. Today we know that the number of charges, the couplings that make all change around us, will be at last reduced to a single number. Maybe. So far we only know for certain that, thanks to Steven Weinberg, Shelly Glashow and Abdus Salam, we have reduced the number of interacting fields. But strong coupling and gravitational constant are still there disconnected if we limit to our current experimental knowledge. Of course, theoretical physics has gone really far in this area and we hope that LHC will help us to give a way to cut out most of what was done here to get the real understanding of the way things work. Somebody will be happy others won’t but this how our World works.

Today, we have a better understanding of the concept of mass and, while waiting for LHC to confirm us our ideas, we can draw some conclusions about all these questions. The fact that mass is energy is an important clue of the idea that this concept is reducible to more fundamentals concepts and that a mechanism for its emergence must exist. Higgs mechanism goes in this direction as also our ideas emerging from QCD about the mass gap that confirm the idea that mass is not a fundamental concept by itself.

So, we can conclude that, so far, our ideas of the World reduce to two fundamental non-reducible ideas: Energy and charge. The former is just a safety lock with respect to the changes provoked by the latter. So, I leave you with a final question: If things stay in this way, does space-time entail a wider concept to embed them or we can reduce also this to them?

Mass generation and supersymmetry


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!


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