Higgs boson and conformal symmetry



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

Slowness in a quantum world



Some days are gone since my last post but for a very good reason. I was very busy on writing down a new paper of mine. Meanwhile, frantic activity was around in the blogosphere due to EPS Conference in Grenoble. No Higgs yet but we do not despair. Being away from these distractions, I was able to analyze a main problem that was around in quantum mechanics since a nice paper by Karl-Peter Marzlin and Barry Sanders appeared in Physical Review Letters on 2004. My effort is in this paper.

But what is all about? We are aware of the concept of adiabatic variation from thermodynamics and mechanics. We know that there exist physical systems that, if we care on doing a really slow variation in time of their parameters, the state of the system does not change too much. Let me state this with an example taken from mechanics. Consider a table with a hole at the center. Through the hole there is a wire with a little ball suspended. An electric motor is keeping the wire on the table side and the ball is performing small oscillations. If the wire is kept fixed, the period of the small oscillations of this pendulum is given by the square root of the ratio between the length of the wire at the hole position and the gravity acceleration multiplied by 2 times Greek pi. Now, we can turn on the motor and vary the length of the wire with some time varying law. This will imply a variation on the frequency of the oscillations of the ball. Now, if we assume a slow variation of the length it happens a nice thing: The system displays a conserved quantity, a so called adiabatic invariant, that the energy of the system varies proportionally to the frequency. This is just an approximate conserved quantity but it is a characteristic of a slow variation of a parameter of this system. In some way, the initial state of the system, properly evolved, is maintained as time evolves as the phase space occupied by the system keeps its form. This is true provided the rate of change of the length of the wire is much smaller than the frequency of the pendulum. This is a quite general result in classical mechanics.

In 1928, Max Born and Vladimir Fock asked themselves if something similar is also true in a quantum world. In a classical paper, they were able to show that it is indeed so. Given a Schrödinger equation with a time varying Hamiltonian, under a given condition, a system keeps on staying in the same state properly evolved in time, multiplied by some phases. The validity condition is the critical point. This condition is expressed through the ratio between the rate of change of the Hamiltonian itself and the gap between instantaneous eigenvalues, as also eigenvalues as the states evolve in time. The gap condition is a fundamental one and was put forward on1950 by Tosio Kato. This is quite reminiscent to the case we gave about classical mechanics where the rate of change of the length of the wire, entering into the Hamiltonian, should be kept smaller than the frequency of oscillation and so this condition appears surely reasonable. But things are not that simple. You can consider an atom under the effect of a monochromatic radiation inside a cavity. This is generally well-described by a two-level system and people observe this system oscillating between the two states. Well, if the intensity of the field inside the cavity is small enough, one can see these oscillations dubbed Rabi flopping. This phenomenon is ubiquitous wherever a monochromatic field interacts with an atom but, the presence of a continuum of states changes this coherent effect into a decaying one as observed in everyday life. If we apply the condition for adiabatic approximation as devised by Born and Fock we get an inconsistency. The approximation seems to hold but Rabi flopping is there to say us the the state is changing in time. The system does not stay at all in the same state as time evolves notwithstanding our condition seems to say so. This is exactly what Marzlin and Sanders pointed out with an exactly solvable example. The condition found by Born and Fock for quantum slowness does not appear to be sufficient to grant an adiabatic behavior for a quantum system. This is a bad news as, in quantum computation, some promising technological applications are in view with the adiabatic approximation and we must be certain that our system behaves the way we expect. This opened up a hot debate that is yet to be over.

But, what is going on here? The explanation to this inconsistency can be traced back to a couple of papers that I and Ali Mostafazadeh wrote in the nineties (see here and here). What Born and Fock really found is the leading order of a perturbation expansion of the solution of the time dependent Schrödinger equation. This has a deep implication for the validity condition of the adiabatic approximation that represents just the leading order of this series. Indeed, let us consider a system under the effect of a perturbation. There are situations when some corrections grow without bound as time increases. These terms are unphysical as an unbounded solution violates cherished principles of  physics as energy conservation, unitarity  and so on. These are called secularities in literature, due to their timescales, as they were firstly discovered in the perturbation series of computations in astronomy. So, if we stop at the leading order of such a series and blindly apply the condition as devised by Born and Fock and claim for applicability, we can be just wrong. This is exactly what Marzlin, Sanders and others have shown unequivocally.

So, if you want to apply the adiabatic approximation and be sure it works, you have to do a more involved task. You have to compute the next-to-leading order correction at least and, accounting eventually for resonant behavior, identify unbounded terms. Techniques exist to resum them. After you will have this done, you will be able to identify the right condition for the adiabatic approximation to work. So, for the two-level system discussed above you will see that only when a very strong field is applied and Rabi flopping cannot happen you will get a consistent adiabatic behavior.

What is the lesson to be learned here? The simplest one is that looking in some elder literature can often help to solve a problem. Anyhow, deep into an old dusty corner of quantum mechanics is just hidden a fundamental result: Strong perturbations can be managed in quantum mechanics exactly as the weak ones. An entire new world is open from this that our founding fathers cannot be aware of.

Karl-Peter Marzlin, & Barry C. Sanders (2004). Inconsistency in the application of the adiabatic theorem Phys. Rev. Lett. 93, 160408 (2004) arXiv: quant-ph/0404022v6

Marco Frasca (2011). Consistency of the adiabatic theorem and perturbation theory arXiv arXiv: 1107.4971v1

Born, M., & Fock, V. (1928). Beweis des Adiabatensatzes Zeitschrift für Physik, 51 (3-4), 165-180 DOI: 10.1007/BF01343193

Marco Frasca (1998). Duality in Perturbation Theory and the Quantum Adiabatic Approximation Phys.Rev. A58 (1998) 3439 arXiv: hep-th/9801069v3

Ali Mostafazadeh (1996). The Quantum Adiabatic Approximation and the Geometric Phase Phys.Rev. A55 (1997) 1653-1664 arXiv: hep-th/9606053v1

Answer to Terry Tao’s criticism will go published


My paper containing the answer to Terry Tao’s criticism will be published in Modern Physics Letters A. You can get a copy of this preprint from arxiv here.

Thank you very much, folks!

The right mathematical question


After my post on the Higgs field (see here) I would like to explain why there is no reason to be afraid. The point is the right mathematical question to be asked.  So, let me state why, from a strict mathematical standpoint, small perturbation theory is not the whole story. Let us consider the differential equation

\partial_t\psi=(H+\lambda V)\psi.

The exact solution of this equation is the function \psi(t;\lambda). So, one can ask : What happens when \lambda goes to zero? This is a proper mathematical question and the answer, when it exists, is a Taylor series. This is the most celebrated small perturbation theory and the terms of this Taylor series are computed directly from the given differential equation.

Of course, I can also ask what happens to the function \psi(t;\lambda) when \lambda goes to infinity. This is perfectly legal from a mathematical standpoint. This dual limit, when exists, produces an asymptotic series that has a development parameter 1/\lambda. This is a strong perturbation theory when each term is computed directly from the given differential equation.

Indeed, one can build a machinery for this case and prove the very existence of this technique that extends perturbation theory beyond the realm of a research of a small parameter. Rather, one can consider the case of differential equations with a large parameter and solve it producing an analysis of these equations in a range of the parameter space not reachable with small perturbation theory.

Physics is a lucky case for this mathematical question as it is all built on differential equations and having such a technique permits to analyze them in situations never reached before other than with computers.

I would like to emphasize that this is applied mathematics but the solutions one obtains can be interesting for physics. Of course, mathematics cannot be questioned except when is wrong.  But when it is right any discussion  is somewhat grotesque.

Who fears a non-perturbative Higgs field?


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


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.

Quantum field theory and gradient expansion


In a preceding post (see here) I showed as a covariant gradient expansion can be accomplished maintaining Lorentz invariance during computation. Now I discuss here how to manage the corresponding generating functional

Z[j]=\int[d\phi]e^{i\int d^4x\frac{1}{2}[(\partial\phi)^2-m^2\phi^2]+i\int d^4xj\phi}.

This integral can be computed exactly, the theory being free and the integral is a Gaussian one, to give

Z[j]=e^{\frac{i}{2}\int d^4xd^4yj(x)\Delta(x-y)j(y)}

where we have introduced the Feynman propagator \Delta(x-y). This is well-knwon matter. But now we rewrite down the above integral introducing another spatial coordinate and write down

Z[j]=\int[d\phi]e^{i\int d\tau d^4x\frac{1}{2}[(\partial_\tau\phi)^2-(\partial\phi)^2-m^2\phi^2]+i\int d\tau d^4xj\phi}.

Feynman propagator solving this integral is given by


and a gradient expansion just means a series into p^2 of this propagator. From this we learn immeadiately two things:

  • When one takes p=0 we get the right spectrum of the theory: a pole at p_\tau^2=m^2.
  • When one takes p_\tau=0 and Wick-rotates one of the four spatial coordinates we recover the right Feynman propagator.

All works fine and we have kept Lorentz invariance everywhere hidden into the Euclidean part of a five-dimensional theory. Neglecting the Euclidean part gives us back the spectrum of the theory. This is the leading order of a gradient expansion.

So, the next step is to see what happens with an interaction term. I have already solved this problem here and was published by Physical Review D (see here). In this paper I did not care about Lorentz invariance as I expected it would be recovered in the end of computations as indeed happens. But here we can recover the main result of the paper keeping Lorentz invariance. One has

Z[j]=\int[d\phi]e^{i\int d\tau d^4x\frac{1}{2}[(\partial_\tau\phi)^2-(\partial\phi)^2-m^2\phi^2-\frac{\lambda}{2}\phi^4]+i\int d\tau d^4xj\phi}

and if we want something not trivial we have to keep the interaction term into the leading order of our gradient expansion. So we will break the exponent as

Z[j]=\int[d\phi]e^{i\int d\tau d^4x\frac{1}{2}[(\partial_\tau\phi)^2-\frac{\lambda}{2}\phi^4]-i\int d\tau d^4x\frac{1}{2}[(\partial\phi)^2+m^2\phi^2]+i\int d\tau d^4xj\phi}

and our leading order functional is now

Z_0[j]=\int[d\phi]e^{i\int d\tau d^4x\frac{1}{2}[(\partial_\tau\phi)^2-\frac{\lambda}{2}\phi^4]+i\int d\tau d^4xj\phi}.

This can be cast into a Gaussian form as, in the infrared limit, the one of our interest, one can use the following small time approximation

\phi(x,\tau)\approx\int d\tau' d^4y \delta^4(x-y)\Delta(\tau-\tau')j(y,\tau')

being now


that can be exactly solved giving back all the results of my paper. When the Gaussian form of the theory is obtained one can easily show that, in the infrared limit, the quartic scalar field theory is trivial as we obtain again a generating functional in the form

Z[j]=e^{\frac{i}{2}\int d^4xd^4yj(x)\Delta(x-y)j(y)}

being now


after Wick-rotated a spatial variable and having set p_\tau=0. The spectrum is proper to a trivial theory being that of an harmonic oscillator.

I think that all this machinery does work very well and is quite robust opening up a lot of possibilities to have a look at the other side of the world.

Gradient expansions and quantum field theory


It is more than two years that I am working on quantum field theory in the strong coupling limit and I am generally very satisfied with the acceptance by the community about my views. Of course, these are new ideas and may take some time to be accepted. So, I keep on working on them trying to clarify them at best so that people can have a clear understanding of their strengths and weaknesses. One of the ways we researchers have to know how our colleagues consider our views is peer-review. This system is indeed crucial to any serious scientific endeavor and, indeed, I am proud of my achievements only when my peers agree about their value. But peer-review is also useful to my work to know what are the main objections to it. It can happen that sometime these objections are deeply wrong and may be worthwhile to discuss them at length also to have an idea on how such a prejudice arose.

We should know that when a mathematical theory enters into the description of nature, whatever mathematical method one uses to exploit it is always correct. So, natural laws in physics are described by differential equations and  whatever method you know to solve them is good provided is also mathematically legal. You should consider mathematics for physicists as a severe judge that grants no appeal. You are right or wrong depending on the correctness of your computation. But in physics there is something more and these are assumptions we start with. You can do the beautiful mathematics in the world but if you started with a wrong concept about how nature works your computations are simply rubbish.

One of the criticisms I have received on trying to get my papers published is that one cannot do a gradient expansion because this breaks Lorentz/Poincare’ invariance. This is completely wrong from a mathematical standpoint. As an exercise  you can consider the wave equation in two dimensions as


and consider the case where the spatial part is not so important. This can be easily obtained by rescaling time as t\rightarrow\sqrt{\lambda}t and taking the limit \lambda\rightarrow\infty. One gets the solution series


solving the equations




and so on. All this is perfectly legal from a mathematical standpoint and I get a true solution of the wave equation. But, as you can see, I have broken Lorentz invariance, a symmetry of this equation. So, mathematics says yes while physics seems to say no. The answer is quite simple and is known since a long time: The computation is right but Lorentz invariance is no more manifest. This is due to the fact that I have separated time and space. But if I am able to resum all the terms of the expansion series I will get the right answer


that is Lorentz invariant. So, both physics and mathematics give the same answer and is a resounding yes, it works and it works so well that we are left with a kind of strong coupling expansion.

So, what should do a smart referee with such a doubt, admitting that a smart referee does not know such mundane facts of physics and mathematics? It should realize that here one is facing a really interesting problem of physics: Could we formulate a gradient expansion in such a way to have Lorentz invariance manifest? I have not an answer yet to this question but I grant to you that is a matter I would like to publish a paper about  somewhere. This is an interesting mathematical problem as well. We know that people met a similar problem at the start of the deep understanding of QED due to Feynman, Schwinger, Tomonaga and Dyson. I think that an answer to this question would have the same scientific value.

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