Breaking of a symmetry: A paper


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

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

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

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

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

QCD at finite temperature


The great news for me, in this week, has been the acceptance of my paper of QCD at finite temperature in Physical Review C (see here). This chance materialized after the excellent work of the referee that helped me to improve the paper in a significant way. For a good paper, such a way to review is a fundamental one and should be a rule. I have discussed this paper previously in my blog (see here and here) and I presented its content in a conference in Paris this year (see here). The contribution to proceedings is this.

I think that the main conclusions of this paper that should be emphasized is that the low-energy limit of QCD is a non-local Nambu-Jona-Lasinio model, a critical point exists at finite temperature with zero mass and zero chemical potential and that the instanton liquid picture of the vacuum of QCD is a very good one. These are fundamental questions in QCD that were waiting for an answer for so long.

I would like to thank all the people that, with their efforts and interest about my work, helped me to get these results published, in the end, in such an important journal.

Marco Frasca (2011). Chiral symmetry in the low-energy limit of QCD at finite temperature arXiv arXiv: 1105.5274v4

Marco Frasca (2011). Low-energy limit of QCD at finite temperature arXiv arXiv: 1110.0096v1

Chiral condensates in a magnetic field: A collaboration


So far, it is more than twenty years that I publish in refereed journals and, notwithstanding a lot of exchange with my colleagues, I have never had the chance to work in a collaboration.  The opportunity come thanks to Marco Ruggieri (see here). Me and Marco met in Gent at the Conference “The Many faces of QCD” (see here, here and here). We have had a lot of good time and discussed a lot about physics. I remember a very nice moment discussing with Attilio Cucchieri and Tereza Mendes in a pub, with a good beer, about the history that was going to form on the question of the propagators for Yang-Mills theory in a Landau gauge. About a month ago, Marco wrote to me about his new work in progress. He was managing to analyze the behavior of QCD condensates in a magnetic field through a couple of models: The linear sigma model and the Nambu-Jona-Lasinio model. The formalism for doing this was already known in literature due to the works of Ritus, Leung and Wang (see below) that analyzed the solutions of the Dirac equations in a constant magnetic field giving also the propagator. In our paper we introduce the constant magnetic field into the given phenomenological models through a minimal coupling. It is interesting to note that, while the sigma model is renormalizable, Nambu-Jona-Lasinio model is not and displays explicitly a dependence on a cut-off. This is not a concern here as this cut-off in QCD has a physical meaning as one can already see in asymptotic freedom studies. The motivation for this study was mainly a lattice analysis of this kind of physical situation (see here). The point is that some kind of condensates can form only with the presence of the external magnetic field. We were able to recover the values of magnetic susceptibility and the dependence of the chiral condensate on the magnetic field in the limit of small and large fields. Besides, we obtained an evaluation of the magnetic moment. The agreement with lattice computations is fairly good.

What I have learned from this work is that the use of phenomenological models, particularly their choice, can entail some difficulties with the expected behavior of QCD. First of all, the sigma model and the Nambu-Jona-Lasinio model are not so different: One can be obtained from the other through bosonization techniques. But while the latter cannot be renormalized, implying a contact interaction and a dimensional coupling , the former can. A curious result I obtained working on this paper with Marco is that the Yukawa model, written down as a  non self-interacting scalar field interacting with a massless Dirac field, can be easily transformed into a Nambu-Jona-Lasinio model giving rise to chiral symmetry breaking! If Hideki Yukawa would have had this known, his breakthrough would have been enormous. On the other side, a sigma model is always renormalizable and this implies that any final result of a computation from it is independent on any cut-off used to regularize the theory. This is not what is seen in QCD where a physical scale depending on energy emerges naturally by integrating the equations of motion as already said above. Besides, condensates do depend explicitly on such a cut-off and this means that to regularize a sigma model to describe QCD at very low-energies implies a deviation from physical results. Last but not least, scalar models are trivial at low energies but we know that this is not the case for QCD that has the running coupling reaching a non-trivial fixed point in the infrared limit. For a Nambu-Jona-Lasinio model this is not a concern as it holds when  the infrared limit is already reached with a fixed value of the strong force coupling. My personal view is that one should always use a Nambu-Jona-Lasinio model and reduces to a sigma model after a bosonization procedure so to fix all the parameters of the theory with the physical ones. In this sense, the renormalizability of the sigma model will be helpful to correctly represent mesons and all the low-energy phenomenology. The reason for this is quite simple: Nambu-Jona-Lasinio model is the right low-energy limit of QCD.

Marco Frasca, & Marco Ruggieri (2011). Magnetic Susceptibility of the Quark Condensate and Polarization from
Chiral Models arxiv arXiv: 1103.1194v1

RITUS, V. (1972). Radiative corrections in quantum electrodynamics with intense field and their analytical properties Annals of Physics, 69 (2), 555-582 DOI: 10.1016/0003-4916(72)90191-1

C. N. Leung, & S. -Y. Wang (2005). Gauge independent approach to chiral symmetry breaking in a strong
magnetic field Nucl.Phys.B747:266-293,2006 arXiv: hep-ph/0510066v3

P. V. Buividovich, M. N. Chernodub, E. V. Luschevskaya, & M. I. Polikarpov (2009). Chiral magnetization of non-Abelian vacuum: a lattice study Nucl.Phys.B826:313-327,2010 arXiv: 0906.0488v2

A day as a physicist


I speak often in my blog about technical matters as I believe these represent the most important part of my intellectual activity. Indeed, my work is so extended in so much areas of physics due to the fact that I am largely involved about problematics of theoretical physics. So, I would like to tell you about my main activities today as a physicist. In these days I am involved in a collaboration with Marco Ruggieri (see here). Marco is working on the understanding on how QCD vacuum gets modified by strong magnetic fields. This question is relevant as such physical effects could be observed at LHC and so, having a prevision for them is paramount. Hurdles here arise from the fact that we currently miss a low-energy full understanding of QCD and one is forced to use phenomenological models that are more or less useful in this case. Typical choices are Nambu-Jona-Lasinio model and the \sigma model with a pion field coupled. In this way one can arrange some theoretical previsions about magnetic susceptibility and other observables. Strange as may seem, these two models can be made to coincide and so, what one can predict with Nambu-Jona-Lasinio model is there also with the  \sigma model.

The reason why these two largely known models can coincide arise with bosonization techniques (see here). If you start with a linear \sigma model assuming just a mass term and you consider its interaction with a fermion field (Yukawa model), you can integrate away the fermion field. This integration will make the potential of the scalar field absolutely not trivial and the vacuum is no more the standard 0 but you will get a mass gap equation. The old Yukawa model gives a really non trivial physics and chiral symmetry is broken. Now, one can takes the Nambu-jona-Lasinio model and, using a transformation taken from condensed matter physics (Hubbard-Stratonovich transformation), one can change a quartic fermion interaction into a quadratic scalar term with a scalar and a pseudoscalar field (the pion) interacting with the quark fields, the same terms taken into the \sigma model. At this stage one can generate a mass gap equation, the one well-known in literature and the same that can be obtained form the \sigma model. One can make the two models identical at this order. Now, going to higher orders, loop expansion produces kinematic terms in the Nambu-Jona-Lasinio model and higher order corrections to the potential of the  \sigma model. I would like to check these higher order corrections between the two models. But one can see that the Yukawa model can be reduced to a contact interaction between fermions as I have proved quite recently (see here). So, there is a deep relation betwenn the Nambu-Jona-Lasinio model and the Yukawa model. I would like to prove a theorem about but, for the moment, this is just an on-going work with Marco.

Finally, a beautiful way a physicist has to contribute to our community is acting as a referee for journals. I am doing this work since 1996 when American Physical Society hired me through the good help of an associate editor of Physical Review A. This is an important way to help science as progress is achieved through the cooperation of several people in the community and if today we see such a great understanding of the World around us is just because such a cooperation worked satisfactorily well. Indeed, this is a honor for a scientist.

Last but not least, I like to write as I am doing now to let things widely known. This is what I hope I am doing better.

D. Ebert (1997). Bosonization in Particle Physics arxiv arXiv: hep-ph/9710511v1

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

QCD at strong magnetic fields


Today on arxiv it is appeared the contribution to the conference “The many faces of QCD” of my friend Marco Ruggieri. Marco is currently a postdoc student at Yukawa Institute in Tokyo and has been a former student of Raoul Gatto. Gatto is one of the most known Italian physicists that had as students also Gabriele Veneziano and Luciano Maiani. With Marco we have had a lot of fun in Ghent and several interesting discussions about physics. One of the main interests of Marco is to study QCD vacuum under the effect of a strong magnetic field and he pursue this line with Gatto. This is a very rich field of research producing several results that can be compared with lattice computations and LHC findings at last. Marco’s contribution (see here) approaches the question using Nambu-Jona-Lasinio model. Before to enter is some details about Marco’s work, let me explain briefly what is the question here.

As my readers know, there has been so far no widely accepted low-energy limit of QCD rigorously derived from it. Simply, we can do computations of low-energy phenomenology just using some models that we hope, in some approximation, will describe correctly what is going on in this limit. Of course, there have been a number of successful models and Nambu-Jona-Lasinio model is one of this. This model, taken from the original formulation, is not renormalizable and not confining. But it describes fairly well the breaking of chiral symmetry and the way bound states can form from quark fields. Indeed, one is able to get a fine description of the low-energy behavior of QCD notwithstanding the aforementioned shortcomings of this model. In the course of time, this model has been refined and some of its defects have been corrected and today appears a serious way to see the behavior of QCD at very low-energy. But all this success appears somewhat incomprehensible unless someone is able to prove that this model is indeed a low-energy approximation to the QCD quantum  field theory. A couple of proofs are around: One is due to Kei-Ichi Kondo (see here) and the other one is due to your humble writer (see here). Kondo’s work does not reach a value for the NJL coupling while I get one through my gluon propagator that I know in a closed form. Anyhow, I was able to get a fully quantum formulation quite recently and this was published in QCD08 and QCD10 proceedings. But, notwithstanding these achievements, I keep my view that, until the community at large does not recognize these results as acquired, we have to continue to take not proved the fact that NJL is obtainable from QCD.

Given this situation, Marco’s approach is to consider a couple of modified NJL models and applies to them a constant magnetic field. Dirac equation with a constant magnetic field is well-known and exactly solvable producing a set of Landau levels and a closed form fermion propagator. This means that, given the mean field approximation, Marco is able to give well defined conclusions through analytical computations. NJL models Marco is considering have been both tuned to agree with lattice computations. What he finds is that the magnetic field has indeed important effects on the temperature of chiral symmetry restoration and for the deconfining phase. But he claims as a weak point a proper determination of the coupling that appears in the NJL models through the Polyakov loop that enters in the way the NJL models are formulated here. This is work for the future. I would like to emphasize the relevance of this kind of research for our understanding of the low-energy behavior of QCD. I will keep my readers up-to-date about this and I will keep on asking to Marco to clarify what the issues are for his research. What I find really striking here is to see the interplay between a magnetic field and strong force vacuum so entangled to produce really non-trivial results. Other groups around the World are working on this and accelerator facilities as LHC can produce important clues for our understanding of the vacuum of QCD. It will be really interesting to see how the results in this area will reach their maturity.

Marco Ruggieri (2011). Chiral symmetry restoration and deconfinement in strong magnetic fields arxiv arXiv: 1102.1832v1

Kondo, K. (2010). Toward a first-principle derivation of confinement and chiral-symmetry-breaking crossover transitions in QCD Physical Review D, 82 (6) DOI: 10.1103/PhysRevD.82.065024

FRASCA, M. (2009). INFRARED QCD International Journal of Modern Physics E, 18 (03) DOI: 10.1142/S0218301309012781

Today on arxiv


As usual I read the daily coming from arxiv for some new papers to talk you about. This morning I have found some interesting ones I would like to say something on. Firstly, I would like to point out to you the paper by Marco Ruggieri and Raoul Gatto (see here). These authors discuss the behavior of QCD in presence of a strong magnetic field. The main tool they consider is the Nambu-Jona-Lasinio model. As you may know, I showed that this is the low-energy limit of QCD (see here and here) but there is also a paper by Kondo (see here) giving the same conclusion even if an expression for the Nambu-Jona-Lasinio constant is not obtained. Gatto and Ruggieri arrive at the important conclusion that a strong magnetic field changes in some way the phase diagram of QCD. I think that this conclusion is strongly supported by the consistency of the model they use. By my side, I think that this area of research is very promising to test my derivation of low-energy QCD.

An important paper as well is the one posted by BESIII Collaboration (see here). This paper gives the most precise measure of the \eta'\rightarrow\eta\pi\pi decay obtained so far due to their larger statistics. They arrive at the important conclusion that for this decay interactions of the decay products is important. This conclusion is really important as implies a production of intermediate resonances as \sigma and a0(980) as already discussed in my preceding post. The reason why this is so important is that this gives a strong support to the view of the \sigma resonance as a glueball and to our current understanding of QCD given above.

Indeed, today there is again a paper of Juan Sanz-Cillero discussing this matter (see here). A more extended discussion has been given in my post here.

Some relevant preprints


My readers know that I keep myself up to date with the daily submissions at arxiv. Of course, I limit myself to the ones very near my research area. Today there have been some quite interesting papers by people working at the question of Green functions in the low-energy limit of QCD. I would like to point out three contributions at the Madrid conference of the last summer: Arlene Aguilar shows how low-energy phenomenology is consistent with our current understanding of the gluon propagator for the decoupling solution (see here). Joannis Papavassiliou shows how the solution for d=2+1 for the gluon propagator, solving numerically Dyson-Schwinger equations, is in agreement with lattice data (see here). Finally, there is the contribution of Eduardo Fraga, Ana Mizher and Maxim Chernodub about the use of a linear sigma model to understand the vacuum of QCD in a strong magnetic field (see here). These people were also present at Gent.

An original research paper appeared from Orlando Oliveira, Pedro Costa and Paulo Silva (see here). Orlando shocked me again with another bright idea. He and his colleagues managed to get an understanding of the behavior of low-energy physics from choosing the propagator through the scaling or the decoupling solution. The smart idea is to do so by deriving a Nambu-Jona-Lasinio model and solving this for known low-energy observables. As you may know, I worked out something like this in this paper that went published in the International Journal of Modern Physics E. I have sent an email to Orlando asking for a contribution by him or his colleagues in this blog. I hope they will take some time to do this as this paper is really striking and the idea really meaningful. Orlando said to me that this paper will appear soon on Physics Letters B.

Finally, I hope you will enjoy reading these papers as I did.

The question of the conformal solution


There is currently a lot of activity to understand the behavior of the two-point functions in a Yang-Mills theory when quarks are not considered. The relevance of these results relies on the possibility to get working tools to manage low-energy phenomenology of QCD. E.g. we know quite well that the Nambu-Jona-Lasinio model is very successful to describe the behavior of hadronic matter and this model can be easily derived from full QCD if one knows the gluon propagator and the behavior of the ghost field. This idea dates back to a paper by Terry Goldman and Richard Haymaker (see here) on 1981. This can be summed up by saying that: given the gluon propagator one can get back a Nambu-Jona-Lasinio model. One could be able to derive it directly from QCD and this would be a great achievement explaining a lot of work done in several decades by a lot of smart researchers.

In these days it seems that we have to cope with a couple of scenarios and long lasting debate is still alive about them. One of this, the conformal solution, has been supported by a lot of researchers for a long period of time, making sometime very difficult for other scenarios to be commonly accepted and being published on archival journals. The conformal solution is easy to describe with the following sentences:

  • The gluon propagator goes to zero at lower momenta.
  • The ghost propagator goes to infinity at lower momenta faster than the free case.

Supporters of this solution also claimed that the running coupling reaches a fixed point at zero momenta. This fixed point was strongly lowered as lattice computations easily showed that things do not stay this way.

Indeed, lattice computations showed without doubt, increasing the volume, as the gluon propagator reaches a finite value at zero momenta and never converges toward zero. The ghost propagator is seen to behaves exactly as that of a free particle. But notwithstanding such an evidence, much effort is still devoted to understand why the conformal solution is not seen in such computations. A lot of hypotheses were put forward to explain why Yang-Mills propagators do not behave in such a well acquired way.

Axel Maas had a bright idea on this way to understand. He turned his attention to the D=2 model on the lattice (see here). Nobody did this before as this model is known to be trivial as it has no dynamics at all. This was proved by ‘t Hooft long ago. Maas showed that the 2D model on the lattice gives back the conformal solution. This means that a Yang-Mills model without dynamics has two-point functions behaving like those of the conformal solution. Most people saw this as a strong support to the conformal solution but things should be seen the other way round. Removing dynamics makes the conformal solution appear and lattice computations in 3 and 4d, that use the same code, are telling us the right behavior of the two-point functions. This is a serious setback for this approach and my personal view should be that any effort to further support this kind of solution should be  abandoned as is a loss of resources and precious time. But this could be a very difficult choice for a lot of people that for a long time worked on this and contributed to create this scenario.

In my opinion, the most severe drawback of this solution is that it goes against the original idea by Goldman and Haymaker that the gluon propagator should give back the Nambu-Jona-Lasinio model. Indeed, the conformal solution does not even give back a mass gap and it is practically useless to answer to a lot of open questions in the low-energy phenomenology. In some way it remembers to me the position  of the bootstrap model against the gauge paradigm that in the end proved to be the more fruitful approach. The last paper by Cucchieri and Mendes (see here) proves without doubt that the right form is a sum of Yukawa propagators and this gives back immediately a Nambu-Jona-Lasinio model.

As I cannot see reasons for people supporting the conformal solution to surrender, I think a lot of time will pass yet before truth will be acquired. Meantime, we stay here looking at the fight between such  strong contenders.

A formula I was looking for


As usual I put in this blog some useful formulas to work out computations in quantum field theory. My aim in these days is to compute the width of the \sigma resonance. This is a major aim in QCD as the nature of this particle is hotly debated. Some authors think that it is a tetraquark or molecular state while others as Narison, Ochs, Minkowski and Mennessier point out the gluonic nature of this resonance. We have expressed our view in some posts (see here and here) and our results heavily show that this resonance is a glueball in agreement with the spectrum we have found for a pure Yang-Mills theory.

Our next step is to understand the role of this resonance in QCD. Indeed, we have shown in our recent paper (see here) that, once the gluon propagator is known, it is possible to derive a Nambu-Jona-Lasinio model from QCD with all parameters properly fixed. We have obtained the following:

S_{NJL} \approx \int d^4x\left[\sum_q \bar q(x)(i\gamma\cdot\partial-m_q)q(x)\right.

-\frac{1}{2}g^2\sum_{q,q'}\bar q(x)\frac{\lambda^a}{2}\gamma^\mu q(x)\times

\left.\int d^4yG(x-y)\bar q'(y)\frac{\lambda^a}{2}\gamma_\mu q'(y)\right]

being G(x-y) the gluon propagator with all indexes in color and space-time already saturated. This in turn means that we can use the following formula (see my paper here and here):

e^{\frac{i}{2}\int d^4xd^4yj(x)G(x-y)j(y)}\approx {\cal N}\int [d\sigma]e^{-i\int d^4x\left[\sigma\left(\frac{1}{2}\partial^2\sigma+\frac{Ng^2}{4}\sigma^3\right)-j\sigma\right]}

being again G(x-y) the gluon propagator for SU(N) and {\cal N} a normalization factor. This formula does hold only for infrared limit, that is when the theory is strongly coupled. We plan to extract physical results from this formula and define in this way the role of \sigma resonance.

QCD and gluon propagator


As I have pointed out in a preceding post, I have got a paper of mine accepted for publication (see here) where I derive a Nambu-Jona-Lasinio model from QCD. I am able to fix the following parameters

G_{NJL}\approx 3.76 \frac{g^2}{\sigma}



being as always \sigma=(440MeV)^2 the string tension and g the gauge coupling. Indeed, this model is very similar to the one presented in a paper by Kurt Langfeld, Christiane Kettner, and  Hugo Reinhardt (see here and here). They solved the model through the numerical solution of the Dyson-Schwinger equations. One can verify that in this case a mean field approximation does not work. In our case, with the above parameters, one gets

\frac{G_{NJL}\Lambda^2_{NJL}}{16\pi^2}\approx 1.117

when \alpha_s\approx 2.6 and we see that we have built a consistent model. Indeed, the authors of the aforementioned paper get the following table

These results are really striking. For \alpha_s=5.3 our model obtain a substantial agreement with experiments. In the near future we hope to find some analytical treatment to work out these values. At this stage we can be really satisfied of this achievement.

Indeed, this comparison is somewhat rough. We have to obtain the Dyson-Schwinger equation for our model and carry out the above computations by ourselves. This is a program to work out in the future. But already at this stage, the consistency with values of \alpha_s we obtain is really satisfactory.

%d bloggers like this: