Living dangerously

05/11/2013

ResearchBlogging.org

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

Stability and Higgs

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.

Living Dangerously

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


Much closer to the Standard Model

18/03/2013

ResearchBlogging.org

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

Ellis & You: agreement with Standard Modelthat 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


Unbreakable

14/11/2012

Today, at HCP2012, new results on Higgs boson search were made available by CMS and ATLAS. Of course, well aligned with preceding rumors, all in all these appear rather disappointing. Maybe, beyond the increasing agreement with Standard Model expectations, the most delusional result is that the particle announced on July 4th appears to be completely lonely on a desert ranging till almost 1 TeV, at least if one is looking for other Higgs particles behaving Standard Model-like. \tau\tau decay rate is now aligning with expectations even if there is some room for a different outcome. On the other side, both experiments did not update \gamma\gamma findings. The scenario that is emerging from these results is the theorist’s nightmare. Tomorrow, all this will be collected in single talks by CMS and ATLAS speakers.

In a retrospective we could say that people claiming for a prize to discoverers of the Higgs mechanism seem to be vindicated. There appears no sign of supersymmetry that is more and more relinquished in a nowhere land. But, I would like to point out that, if a supersymmetric theory is the right one, there is just one theory to be singled out exploring the parameter space. It is normal in any case to see such a vast epidemic death of theories. It is also possible that theorists should now do a significant effort for new proposals beyond those largely explored in these last thirty years.

Standard Model is even more resembling a perfect theory really unbreakable and mimicking the success of the Maxwell equations put forward 150 years ago.  But we know it must break…

Finally, I would like to conclude this rather fizzling out post by pointing out a rather funny side of this situation. Tommaso Dorigo has a bet on with Gordon Watts and Jacques Distler amounting to $1200 on the non-existence of SUSY partners. This bet has not been payed yet as Distler is claiming there are a lot of “juicy rumors” from CERN and the terms are not fulfilled yet (see comments here).  I do not know what rumors Distler is talking about but, unless CERN is not hiding data (that would appear a rather strange behavior at best), maybe it is time to do a check on who the winner could be.


Mass generation: The solution

26/12/2010

ResearchBlogging.org

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

\Box\phi+\lambda\phi^3=0

given by

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

that holds for

p^2=\mu^2\sqrt{\frac{\lambda}{2}}.

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

20/12/2010

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

\Box\phi+\lambda\phi^3=0

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

p^2=\mu^2\sqrt{\frac{\lambda}{2}}.

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

p_i=\frac{4n_iK(i)}{L_i}

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.


Cabibbo passed away

17/08/2010

This is the kind of news I would not have wanted to write about in my blog. Nicola Cabibbo died yesterday at the Fatebenefratelli Hospital in Rome. He was fighting with a long time illness. Cabibbo has to be considered, for very good reasons, one of the greatest Italian physicists. He is the product of the Roman school of physics that started its glorious traditions with Enrico Fermi. One of his teachers was Edoardo Amaldi, one of the boys of via Panisperna that, together with Fermi, brought Italian physics to a worldwide level. By himself, he contributed to a formidable group of physicists working at the University “La Sapienza” in Rome that gave fundamental contributions to our current understanding of the Standard Model.

Last time I have heard a talk by him was at Accademia dei Lincei last year (see here). It was exciting to hear directly from his voice a layman description of his most fundamental discovery. I did not think that would have been the last time I would have seen him. He has been my teacher at “La Sapienza” when I was studying to get my “laurea” in physics. He firstly exposed me to quantum field theory and so I  learned how to compute the first correction (the Schwinger’s one) of the magnetic moment of the electron. His exposition was always very clear and he made things so easy to understand that this first course is yet a formidable basis to build upon for my career as a physicist. Finally, he was my supervisor for my laurea’s thesis and he was instrumental for me to reach the highest honors in my final examination.

Cabibbo was the president of the Pontificia Accademia delle Scienze as he was a Catholic and he supported his faith in different occasions. One of this was the discussion with Penzias about faith and science organized by Riccardo Chiaberge, an Italian journalist (see here).  The crucial point of his thought was that, as believers, there is no reason to fear science as God may have decided any kind of avenues to pursue His aims.

His dead has been a bad news for me and I think that, behind the sorrow, we will greatly miss him for the clearity of his thought and the excitement he always transmitted about science, this kind of excitement that he also gave to me, one of his students.

Note: Cabibbo was awarded this year with the Dirac medal by ICTP (see here and here).


Rumors on Higgs at Tevatron

10/07/2010

It is not my habit to put rumors about as my readers know, but the news is really sensational. Tommaso Dorigo in his blog told  that rumors are leaking about a light Higgs seen in one of the two collaborations at Tevatron. Tommaso is working there too.

Rumors say of a Higgs particle having a mass of 115 GeV, very near the limit identified at LEP and so a reason to regret for CERN. This will support the view of a supersymmetric particle. Supersymmetry, for consistency reasons, requires Higgs to be light. On the other side, we know that the Standard Model cannot hold with a superheavy Higgs particle. This implies that a similar identification at LHC should be near and this machine should work out supersymmetry in all its glory.

Finally, let me point out a similar post by the Czech guy.

Update: Fermilab denied rumors about Higgs finding at Tevatron (see here).


Higgs mechanism is essential

30/10/2009

As the readers of my blog know, I have developed, in a series of papers, the way to manage massive solutions out of massless theories, both in classical and quantum cases. You can check my latest preprints here and here. To have an idea, if we consider an equation

\Box\phi+\lambda\phi^3=0

then a solution is

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

provided

p^2=\mu^2\left(\frac{\lambda}{2}\right)^\frac{1}{2}

being \mu and \theta two arbitrary constants and \rm sn a Jacobi elliptical function. We see that a massless theory has massive solutions arising just from a strong nonlinearity into the equation of motion. The question one may ask is: Does this mechanism work to give mass to particles in the Standard Model? The answer is no and this can already be seen at a classical level. To show this, let us consider the following Yukawa model

L=\bar{\psi}(i\gamma\cdot\partial-g\phi)\psi+\frac{1}{2}(\partial\phi)^2-\frac{\lambda}{4}\phi^4

being g a Yukawa coupling. Assuming \lambda very large, one is reduced to the solution of the following Dirac equation that holds at the leading order

\left(i\gamma\cdot\partial-g\mu\left(\frac{2}{\lambda}\right)^\frac{1}{4}{\rm sn}(p\cdot x+\theta,i)\right)\psi=0

and this equation is exactly solved in a closed form, provided the fermion has exactly the same mass of the boson, that is g=\sqrt{\lambda/2}. So, we see that the massless fermion acquires mass too but it must be degenerate with respect to the bosonic field. This would imply that one needs a different scalar field for each fermion and such bosons would have the same masses of the fermions. This is exactly what happens in a supersymmetric theory but the theory we are considering is not. So, it would be interesting to reconsider all this with supersymmetry, surely something to do in the near future.

This means that Higgs mechanism is essential yet in the Standard Model to understand how to achieve a finite mass for all particles in the theory. We will see in the future what Nature reserved us about.


What is mass?

30/08/2009

I should confess that one of the reasons why I have chosen to be a physicist is that physics, like no other sciences, is able to give answers to fundamental open questions that until a few years ago were only discussed by philosophers. Most of these questions are ancient as our species and the possibility that we have means to get truth is too strong to lose our time with other activities. So, I managed to learn such means and today I am here writing on this blog trying to explain you what these truths are. Sometime, I am at the forefront of research and so, what can be believed a truth may lose this quality as we deepen our understanding. Indeed, dynamics of science adds one more element of charm to all this matter.

One of such old questions is: “What are we made of?”. This question has been an open question till the dawn of the 20th century with the fundamental experiments carried out by Ernst Rutherford. Till then we have learned so much about matter that this question changed form becoming: “What is mass?”. This question has become compelling with the birth of the Standard Model due to Sheldon Lee Glashow, Steven Weinberg and Abdus Salam. Indeed, in order to maintain symmetry we must ask all particles to be massless and some mechanism must exist giving mass to them. In the sixties and seventies of last century we moved toward a real understanding of this concept. The idea is to rely on the Higgs mechanism and a scalar particle must exist to grant masses to the other particles in the model. As you may know this particle has not yet been seen and it is the only missing element of an otherwise very successful model. We are confident for several reasons that the Higgs mechanism could turn out the right answer to the question on mass but we are no more so confident that should have the simple aspect given originally in the Standard Model. Indeed, this appears as an open door on a Pandora’s vase of new exciting physics.

But whatever will be the mechanism at work for the masses of leptons and quarks, the answer to the main question is not there. For one reason, both electrons and quarks that form protons and neutrons are really light and do not count too much on the determination of our mass. Most of the mass is in the nuclei and we have to understand where such mass comes from. This arises from bound states of quarks glued together in some way as should yield QCD at low energies. This gives you an idea of why is so important to understand QCD at very low energy. In this way we would be able to answer a fundamental question philosophers discussed for so long time.

So, for our everyday life, it is not so relevant to comprehend the real mechanism that gives mass to elementary particles . What we need is to prove the existence of a mass gap in Yang-Mills theory and so the way bound states form in QCD. As you may know, this is not an easy task and involves a lot of talented people around the World that, with a lot of inventive, is trying to do such computations. So far, only computers succeeded in giving an answer and this is so good that we have the most important observed parameters precise to one percent. The hope is to have a technique to work out such computations analytically, as happens for weak coupled physics. I am deeply involved in such enterprise and I think that what will come out will have a large impact on our knowledge. I can only say: Stay tuned!


Who fears a non-perturbative Higgs field?

28/07/2009

One of my preferred readings in the blogosphere is Tommaso Dorigo’s blog. I think this is a widely known blog for people interested about physics and got some citation also at New York Times. Quite recently he published a very interesting post (see here) about the fate of our loved Standard Model taking the move from a very nice paper by J.Ellis, J.R.Espinosa, G.F.Giudice, A.Hoecker, and A.Riotto (see here). These authors are well known and really smart at their work and, indeed, I have noticed this paper as it appeared in arxiv. My readers know that I work on a small part (QCD) of the whole picture arisen in sixties and seventies and I have never taken a look from outside. So, while I appreciated this paper I thought it was not the case to comment on it  in my blog. But reading Tommaso’s post some thoughts come to my mind and these are really pertinent.

People put out two kind of constraints on the Higgs part of the standard model to have an idea of what to expect. I give you here the Higgs potential for your needs

V_H=\frac{1}{4}\lambda(\phi^\dagger\phi-v^2)^2

and one immediately realizes that it introduces two free parameters. The critical one is \lambda and let me explain why. When one does quantum field theory, the only real tool that she has to do any meaningful computation is small perturbation theory. The word “small” is never said but it should be said in any circumstance as this technique only works if you have a small parameter in your theory (a coupling) to use as a development parameter. Otherwise we are lost and all starts to become foggy and not so well-defined. Today, nobody knows how to manage a theory with a strong coupling. Parameter \lambda is exactly such a coupling and we are able to manage a Higgs field when this parameter is small. But when you do small perturbation theory in quantum field theory you realize immediately that infinities come out and you are not able to obtain meaningful results going beyond the first order. For the most interesting theories around we are lucky:  Schwinger, Tomonaga, Feynman and Dyson invented renormalization and this works to remove infinities at each order of perturbation theory in the Standard Model and also for the Higgs, if the coupling is small. We are so accustomed to such a situation that we think that this is all one needs to know to understand quantum field theory: Perturbation theory and renormalization. We think that small perturbation theory is the perturbation theory and nothing else. So, we hope also the Higgs field should fulfill such requirements. Indeed, we are already in trouble in QCD for these same reasons but I have discussed at lengthy such a situation before here and I do not want to repeat myself.

There is no reason whatsoever to believe that we know all one has to know to manage a quantum field theory. Higgs could as well be not that light and strong coupled and there is no reason to think that Nature chose the small coupling case to favor us. Of course, if things will not stay this way I will be happy as a light Higgs is favored by supersymmetry and I like supersymmetry. But I would like also to emphasize that we already have all we need to manage analytically a strong coupled Higgs field. This matter I have discussed widely here and in my published papers.

So, while we all agree that a light Higgs is favored my view is that we should not have any fear of a non-perturbative Higgs field.


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