## What if the Higgs…

31/07/2008

…would be too massive? Today I was reading the beatiful post of Tommaso Dorigo about Tevatron results on the hunt for Higgs and an old idea come back to my mind. We all know why for the sake of Standard Model one need a small Higgs mass but, so far, such low mass particle has been not seen. We all are aware that the least result of LHC would be the unveiling of the generation mass mechanism in the Standard Model that is a must for the model to survive but we can have surprises. One of these surprises could be a too massive Higgs. This will mean that, at higher energies, our ability to do perturbation theory for electro-weak interactions will blatantly fail and such interactions would become as strong as strong interactions. But there is one more aspect of this situation that is somewhat unexpected. We have recently treated a strongly coupled Goldstone scalar field (here and here) and we have found in the strong coupling limit, besides the ordinary massless excitation, a tower of excited states and this is what one should observe if the Higgs particle is too massive. By “too massive” we mean a mass beyond 1 TeV. I have no checked yet but I believe that also in this case the classical theory admits an exact solution. More to say in the near future.

Update: There has been a post by Lubos Motl (see here) where he argues that Fermilab data favor a light Higgs and supersymmetry. Indeed I hope this will prove to be the right scenario because so we have our cake and we eat it! Finally, D0 Collaboration presented a press release and this seems an important step beyond to Higgs discover.

## Fermions and massless scalar field

30/07/2008

As I always do to take trace of some computations I am carrying on here and there, I put them on the blog. This time I devised to solve Dirac equation using the massive solution of the massless scalar field given here and here:

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

$\mu$ is an arbitrary parameter, $\lambda$ the coupling and ${\rm sn}$ the snoidal Jacobi elliptical function. An arbitrary phase $\varphi$ can be added but we take it to be zero in order to keep formulas simpler. Now, we couple this field to a massless fermion field and one has to solve Dirac equation

$(i\gamma\partial+\beta \phi)\psi=0$

where we have used the following Yukawa model for the coupling

$L_{int}=\Gamma\bar\psi\psi\phi$

so that $\beta=\Gamma\mu\left(2 \over \lambda\right)^{1 \over 4}$. Dirac equation with such a field can be solved exactly to give

$\psi(x)=e^{-iq\cdot x}e^{-i\frac{p\cdot q}{m_0^2}p\cdot x-\beta\frac{\gamma\cdot p}{m_0^2}[\ln({\rm dn}(p\cdot x,i)-i{\rm cn}(p\cdot x,i))-\ln(1-i)]}u_q$

being $m_0=\mu (\lambda /2)^{1\over 4}$ the mass acquired by the scalar field and ${\rm dn}$ and ${\rm cn}$ two other Jacobi elliptical functions. This formula says us an interesting thing, that is there is a fermion excitation with zero mass unless a mass is initially given to the fermion. Such a conclusion is reminiscent of the pion status in QCD. So, the computation may seem involved but the conclusion is quite rewarding!

## Yang-Mills theory in D=1+1

29/07/2008

Functional methods are techniques used in these years to manage Yang-Mills theory. This name arose from the various methods people invented to solve Dyson-Schwinger equations. These are a tower of equations, meaning by this that the equation for the two-point function will depend on the three point function and so on. These are exact equations: When you solve them you get all the hierarchy of n-point functions of the theory. So, the only way to manage them to understand the behavior of Yang-Mills theory at lower momenta is by devising a proper truncation of the hierarchy. A similar situation can be found in statistical mechanics with kinetic equations. For a gas we know that collisions with a higher number of particles give smaller and smaller contributions and we are able to provide a meaningful truncation of the hierarchy. For the Dyson-Schwinger equations, generally, we are not that lucky and the choice of a proper truncation can be verified only through lattice computations. This means that the choice of a given truncation scheme may imply an uncontrolled approximation with all the consequences of the case. A beatiful paper about this approach is due to Alkofer and von Smekal (see here). This paper has been published on Physics Report and describes in depth all the elements of functional methods for Yang-Mills theory. Alkofer and von Smekal proposed a truncation scheme for Dyson-Schwinger equations that provided the following scenario:

• Gluon propagator should go to zero at lower momenta.
• Ghost propagator should go to infinity faster than a free particle propagator at lower momenta.
• A proper defined running coupling should reach a fixed point in the infrared.

The reason why this view reached success is due to the fact that gives consistent support to currently accepted confinement scenarios. Today we know as the history has gone. Lattice computations showed instead that

• Gluon propagator reaches a non-zero value at lower momenta.
• Ghost propagator is practically the same of that of a free particle.
• Running coupling as defined by Alkofer and von Smekal goes to zero at lower momenta.

So, after years where people worked to support the scenario coming from functional methods, now the community is trying to understand why the truncation scheme proposed by Alkofer and von Smekal seems to fail. On this line of research, Axel Maas showed recently, with lattice computations, that for D=1+1 the scenario is exactly those Alkofer and von Smekal proposed (see here and here). So, now people is try to understand why for D=1+1 functional method seems to work and for higher dimensions this does not happen.

I think that these are not good news for functional methods. The reason of this is that a pure Yang-Mills theory in D=1+1 is trivial. Trivial here means that this theory has not dynamics at all! This result was obtained some years ago by ‘t Hooft (see here) and published on Nuclear Physics B. He showed this using light cone coordinates. Then, by eliminating gluonic degrees of freedom he obtained a two-dimensional formulation of QCD with non-trivial solutions. In our case this means that the truncation scheme adopted by Alkofer and von Smekal simply does not work because removes all the dynamics of the Yang-Mills field and these are also the implications for the confinement scheme this approach should support. Indeed, a proper numerical solution of Dyson-Schwinger equations proves that the right scenario can also be obtained (see here). These authors met difficulties to get their paper accepted by an archival journal. Today, we should consider this work an important step beyond in our understanding of Yang-Mills theory.

My view is that we have to improve on the work of Alkofer and von Smekal to make it properly work at higher dimensionalities. This without forgetting all other works that gave the right solution straightforwardly.

## DAMA

28/07/2008

In Italian language DAMA means a lady or draught the well-known game that people commonly plays with a chessboard. But it is also the name of a fundamental experiment carried out at INFN laboratories at Gran Sasso in Italy and headed by physicist Rita Bernabei (you can find here an article by her just in Italian, sorry) . Rita Bernabei was one of my teachers at “La Sapienza” in Rome and I took an examination with her. Recently a preprint by DAMA Collaboration was published on arxiv causing a lot of rumors in the scientific community. The point is that they see a signal that nobody else sees. This evidence cannot be denied and the signal is there.

Personally I am convinced of the goodness of all this matter and sometime happens that something is seen by a group and not by others and things could be generally explained after some time. In my activity field a similar thing happened for the sigma or f0(600) resonance. This particle has a long history as initially was put in the PDG listing and then removed as believed non-existent. After a lot of serious analysis, both theoretical and experimental, its existence has been commonly accepted and now the question is moved to the understanding of its nature. We have discussed this matter here, here and here. This particle is our key to understanding of low energy behavior of QCD and we expect some news about in the near future. Similarly, I think the current position of DAMA collaboration is the same as was for the first observations of the sigma. Being the firsts sometimes requires a lot of patience before general acceptance gets through.

## QCD 08 proceedings

27/07/2008

Today I have posted my contribution to proceedings of QCD 08 at arxiv. It should appear shortly.

I hope to have answered to all open questions about Yang-Mills theory in the infrared. More to say in the following days, having in mind my very near vacations.

Update: The paper appeared today 29th July (see here).

## Rumors on SUSY

25/07/2008

In these days there have been some rumors in the blogosphere about string theorists and SUSY (see Motl, Woit and Dorigo) due to a recent preprint appeared on arxiv. Indeed SUSY is a relevant ingredient of string theory and the latter was the vehicle for the uncovering of this concept that obtained such a fortune in the community. I have listened a talk of Sergio Ferrara at Accademia dei Lincei in Rome a few months ago. Ferrara is one of the discoverers of supergravity and he gave a nice talk on the argument of supersymmetry. He was confident that supersymmetric particles will be seen at LHC. I would like to say that this was also the expectation for LEP and Tevatron but nothing has been seen so far. So the paper above seems like an attempt by a string theorist to be pessimistic and save the day.

Supersymmetry has some problems that still are in need for a satisfactory answer. One is philosophical as one can say that the number of particles simply doubles and so why should we expect such an anti-economical behavior by Nature? Ferrara argued against this question by saying that also with antimatter Nature doubled the number of particles so this would not be the first time that, in order to keep a symmetry, one needs such a doubling. One can say anyhow that for antimatter one has a discrete symmetry on a single field while for supersymmetry is the number of fields that doubles. The other point is about breaking of supersymmetry. There is no satisfactory model so far and such symmetry is not seen at low energies as we know. But this could be just a matter of time before someone finds a way out.

My view is that even if there is no supersymmetry at large, one can save supergravity. Indeed, all one needs is to observe a gravitino, that is a spin 3/2 particle, and we will have a theory of quantum gravity while, at large, no supersymmetry can exist. But this would not be enough for string theory. As a theoretical physicist I would like to see the discover of a gravitino and the failure of supersymmetry at large as this would imply a lot of interesting work to do and an incredible new scenario to understand.

## An useful hint

24/07/2008

Dietmar Ebert is a retired professor of Humboldt University in Berlin. He did relevant work in QCD and particle physics. I have come upon a paper of him at arxiv about the question of bosonization. In a paper of mine I showed how a Nambu-Jona-Lasinio (NJL) model can be derived from QCD using recent results about gluon propagator that is the corner stone of all this construction. In order to make contact with the mesonic spectrum of QCD one needs to manage in some way quark fermionic fields of NJL model to recover bosonic degrees of freedom. In Ebert’s paper this is done through Hubbard-Stratonovich transformation that is a widely known tool to condensed matter theorists. This is a key point to prove that our recent derivation of the width of the sigma resonance given here using a Fermi’s intuition is indeed correct. Ebert obtains by a NJL-model the following bosonic Hamiltonian

$L_{int}=g_{\sigma\pi\pi}\sigma(\sigma^2+\pi^2)+g_{4\pi}(\sigma^2+\pi^2)^2$

being

$g_{\sigma\pi\pi}=\frac{m}{\sqrt{N_c I_2}}$

and

$g_{4\pi}=\frac{1}{8N_c I_2}$

being $N_c$ the number of colors,

$m=m_0+i8mG_{NJL}\int^{\Lambda}\frac{d^4k}{(2\pi)^4}\frac{1}{k^2-m^2}$

quark constituent mass and $m_0$ the quark mass assumed to be equal for u and d, and finally

$I_2=-i\int^{\Lambda}\frac{d^4k}{(2\pi)^4}\frac{1}{(k^2-m^2)^2}.$

In order to make contact with QCD, as we have shown one has

$G_{NJL}=3.761402959\frac{g^2}{\sigma}$

being $g$ the coupling constant and $\sqrt{\sigma}=410\pm 20 \ MeV$ the square root of the string tension.

Ebert’s Lagrangian gives us exactly the term we derived with Fermi’s insight plus other terms implying also the one to compute f0(980) decay rate $2g_{4\pi}\sigma^2\pi^2$. So, as it is well-known, a good idea repeats itself at different levels in the description of Nature. I would call this the “gluonic sector” of QCD. I hope to put down a paper about in the next days.

## Environmental decoherence or not?

23/07/2008

One of the most relevant open questions currently under study in physics is how a classical world emerges from the laws of quantum mechanics. This can be resumed in a standard philosophical question:”How does reality form?”. We have learnt from standard quantum mechanics courses that one just takes the mathematical limit $\hbar\rightarrow 0$ and the classical limit emerges from quantum mechanics. Indeed, as always, things are not that simple as in Nature $\hbar$ is never zero and this means that all objects that are seen should be in a quantum mechanical state. So, why in our everyday life we never observe weird behaviors? What is that cuts out most of the Hilbert space states to maintain a systematic classical behavior in macroscopic objects? More carefully stated the question can be put as:”Where is the classical-quantum border, if any?”. Indeed, if we are able to draw such a border we can buy all the Copenaghen interpretation and be happy.

A proposal that is going to meet increasing agreement is environmental decoherence. In this case one assumes that an external agent does the job erasing all the quantum behavior of a system and leaving only a superselected set of pointer states that grants a classical behavior. From a mathematical standpoint one can see interference terms disappear but one cannot say what is the exact state the system is left on, leaving in some way the measurement problem unsolved. It should be said that the starting point of this approach has been a pioneering work of Caldeira and Leggett that firstly studied quantum dissipation. In order to have an idea of how environmental decoherence should work, the Einstein’s question: “Do you really believe that the moon is not there when we do not look at it?” is just answered through the external effect of sun radiation and, maybe, the cosmic microwave background radiation that should act as a constant localizing agent even if I have never seen such an interpretation in the current literature. It is clear that to try to explain a physical effect by a third undefined agent is somewhat questionable and sometime happens to read in literature some applications of this idea without a real understanding of what such an agent should be. This happens typically in cosmology where emergence of classicality cannot be easily understood. As unsatisfactory may be such an approach can be seen from the relevant conclusion that it gives strong support to a multiverse interpretation of quantum mechanics. Of course, this can be a welcomed conclusion for string theorists.

Today on arxiv a preprint is appeared by Steven Weinstein of Perimeter Institute that proves a theorem showing that environmental decoherence cannot be effective in producing classical states from generic quantum states. Indeed, all applications of environmental decoherence in literature consider just well built toy models that, due to their construction, produce classicality but this behavior is not generic even if in agreement with the theorem proved by Weinstein. So, the question is still there unanswered:”Where is the classical-quantum border, if any?”. We have already seen here the thermodynamic limit, that is the border at infinity, is an answer, but the question requires a deep experimental investigation. A hint of this was seen in interference experiments with large molecules by Zeilinger’s group. This group is not producing any new result about since 2003 but their latest paper was showing some kind of blurry behavior with larger molecules. This effect has not been confirmed and one cannot say if it was just a problem of the apparatus.

The question is still there and we can state it as:”How does reality form?”.

## Fermi’s insight and the width of the sigma

22/07/2008

Enrico Fermi is one of the greatest Italian physicist . He obtained a wide list of great accomplishments but a reason for him to be remembered in History is that, like the legendary Prometheus did for the fire, he brought mankind into nuclear era building the first nuclear pile in Chicago in 1942. The other reason why I am citing him here is that he uncovered weak interactions by postulating the first beta decay Lagrangian that has been since then the building block for our understanding culminating into the Standard Model of sixties and seventies. Il va sans dire that the paper Fermi wrote about and sent to Nature was rejected. This paper contains a true spectacular insight for a field theory when very few elements are known about a reaction than just the involved fields. Fermi’s insight arose from an analogy with the electromagnetic field. The concept is that when no information is given about the interaction itself the way to get a description of a particle reaction is through a contact interaction. Fermi just takes for the Green function of the interaction the following

$G(x-y)=\frac{G_F}{2}\delta^4(x-y)$

being $G_F$ the Fermi constant and the factor 2 has been introduced just for convenience. Then, if you consider e.g. the reaction $\mu^-\rightarrow e^-+\bar\nu_e+\nu_\mu$ you have just to take the product of the fields to define a Lagrangian as

$L_{int}=\frac{G_F}{2}(\bar\mu\gamma^\mu\nu_\mu)(\bar e\gamma_\mu\nu_e)+h.c.$

and you are done. At this stage you do not have to care about renormalizability (Fermi did not even know the problem at that time (1934)) as you just want a first understanding of the process. You need to fix the Fermi constant by comparison with experimental data and so, all other processes involving muons, electrons and neutrinos are fixed themselves. This is known as the “universality” of Fermi interactions. This model will fail if universality is lost for some reason.

The sigma or f0(600) is a hotly debated resonance seen in strong interactions. It has vacuum quantum numbers and its nature is the most important open question. We know from experimental data that this particle has two decay modes: $\sigma\rightarrow\pi^+\pi^-$ and $\sigma\rightarrow\gamma\gamma$. The first mode is largely dominant. From the paper of Colangelo, Caprini and Leutwyler (see here and here but other determinations exist as here and here) we know the main properties of this particle

$M_\sigma-i\Gamma_\sigma=441^{+16}_{-8}-i272^{+9}_{-12.5} MeV$

and one immediately realize that this resonance is quite broad. Using Fermi’s insight we can put down a contact interaction to compute such a width as

$L_{int}=\frac{G_{NJL}}{2}M_\sigma^3\sigma\pi^\dagger\pi$

being $G_{NJL}$ the Fermi constant for this case and we assume that we can neglect the contribution of the $\gamma\gamma$ decay being this really small. This latter process can also be interpreted as a $\pi\pi$ rescattering as done by Narison, Ochs and Mennessier here explaining in this way the smallness of this contribution. We can determine $G_{NJL}$ from QCD. We have done this here and we have obtained

$G_{NJL}=3.761402959\frac{g^2}{\sigma}$

being $g$ the coupling constant and $\sqrt{\sigma}=410\pm 20 MeV$ the square root of the string tension. We will be able to determine $g$ from our Fermi Lagrangian. It should be clear at this point that NJL stays for Nambu-Jona-Lasinio as we have obtained this constant in this context and is equivalent to a Fermi constant for QCD. A standard calculation as given e.g. here yields

$\Gamma_\sigma = \frac{G_{NJL}^2M_\sigma^5}{8\pi}\sqrt{1-\frac{4M_\pi^2}{M_\sigma^2}}$

that can be inverted to give the strong coupling constant

$\alpha_s=\frac{g^2}{4\pi}=\frac{\sigma}{3.761402959 M_\sigma^2}\sqrt{\frac{\Gamma_\sigma}{2\pi \sqrt{M_\sigma^2-4M_\pi^2}}}.$

Using Colangelo, Caprini and Leutwyler data we get the very good result

$\alpha_s\approx 0.16365$

that is physically consistent with expectations for the strong coupling at low energies.

So, we have seen how effective can be Fermi’s insight also for strong interactions. It would be interesting to extend this approach to other kind of resonances to see how far one can go with universality. At this stage we can be really satisfied with our result that justify the broadness of the sigma resonance using just QCD parameters.

## Empire of the Stars

21/07/2008

During my week-end in Soverato I have spent some time reading this book. It was for me a good chance to enter into life of Subramanyan Chandrasekhar. I know him for being a great physicist and I have had the chance to read some of his works. He has been also a collaborator of Enrico Fermi and both produced a pair of relevant papers on magnetohydrodynamics. But my knowledge was rather superficial and I was not aware of the difficulties he met in obtaining his results widely known. He was a pioneer in our current understanding of star evolution and the emerging of singularities in space-time. The difficulties he met were mostly due to Arthur Eddington that employed all his scientific relevance to impede the emerging of this important result for reasons that today have lost any importance and were essentially wrong. This implied that almost forty years were needed before the relevance of Chandra work was generally acknowledged and he was awarded a Nobel prize for this reason in 1983.

We should be aware that this dynamics in the scientific community is still present. Planck used to say that a new idea becomes generally accepted when all the opponents are dead and the new generations are open to it. I think, but this is just my view, that the conservative reasoning is just an heritage of our struggle for surviving. Indeed, entering into a risky situation may happen if your idea is a failure while we know that old methods work so well. And still today communities at large maintain this view. The lucky case in physics is that a new concept cannot be stopped for too long. Scientific method implies that as soon as new experimental results become available, theory advances as well and some apparently questionable concepts may turn out to be the right description of the behavior of Nature. It is just a question of time but we are only humans and can happen that the discoverers of a right concept could be acknowledged late or too late in some cases.

The book has some wrong historical facts. The most blatant I have found was about Lev Landau. Landau was imprisoned by the brutal Stalinist government. Kapitza come in his rescue claiming that only him could solve the “superconductivity” problem he was experimenting on. Everyone well acquainted with history of physics knows that the word “superconductivty” should be changed into “superfluidity”. Explanation of superfluidity earned a Nobel prize to Landau in 1962. Superconductivty has another story. Finally, I have read the italian edition and some imprecisions into the translation can be found here and there. The most entertaining has been the translation of “naked singularities” (“singolarita’ palesi” to be translated as “manifest singularities”) and “duro come la roccia” (“hard like rock”) to indicate how strong is the core of e.g. a neutron star. For this latter expression I hope the author did not use a similar expression in english. I would like to know why italian editors do not ask to competent people for such translations.

Finally, I would not agree fully with the idea the author left to me about Chandra. A kind of angry and always unsatisfied man maybe due to the starting quarrel with Eddington that left an indelible sign for all his lifetime . Indeed, he obtained recognitions of his merit late in his life but indeed his life was great as a physicist both for his accomplishments and for people he happened to meet.

Just a note about LHC in the book. The author claims that one of the aims of this facility is to produce black holes! Maybe an update is needed in view of recent and less recent controversies.