Is Higgs alone?


I am back after the announcement by CERN of the restart of LHC. On May this year we will have also the first collisions. This is great news and we hope for the best and the best here is just the breaking of the Standard Model.

The Higgs in the title is not Professor Higgs but rather the particle carrying his name. The question is a recurring one since the first hints of existence made their appearance at the LHC. The point I would like to make is that the equations of the theory are always solved perturbatively, even if exact solutions exist that provide a mass also if the theory is massless or has a mass term with a wrong sign (Higgs model). All you need is a finite self-interaction term in the equation. So, you will have bad times to recover such exact solutions with perturbation techniques and one keeps on living in the ignorance. If you would like to see the technicalities involved just take a cursory look at Dispersive Wiki.

What is the point? The matter is rather simple. The classical theory has exact massive solutions for the potential in the form V(\phi)=a\phi^2+b\phi^4 and this is a general result implying that a scalar self-interacting field gets always a mass (see here and here). Are we entitled to ignore this? Of course no. But today exact solutions have lost their charm and we can get along with them.

For the quantum field theory side what could we say? The theory can be quantized starting with these solutions and I have shown that one gets in this way that these massive particles have higher excited states. These are not bound states (maybe could be correctly interpreted in string theory or in a proper technicolor formulation after bosonization) but rather internal degrees of freedom. It is always the same Higgs particle but with the capability to live in higher excited states. These states are very difficult to observe because higher excited states are also highly depressed and even more hard to see. In the first LHC run they could not be seen for sure. In a sense, it is like Higgs is alone but with the capability to get fatter and present himself in an infinite number of different ways. This is exactly the same for the formulation of the scalar field as originally proposed by Higgs, Englert, Brout, Kibble, Guralnik and Hagen. We just note that this formulation has the advantage to be exactly what one knows from second order phase transitions used by Anderson in his non-relativistic proposal of this same mechanism. The existence of these states appears inescapable whatever is your best choice for the quartic potential of the scalar field.

It is interesting to note that this is also true for the Yang-Mills field theory. The classical equations of this theory display similar solutions that are massive (see here) and whatever is the way you develop your quantum filed theory with such solutions the mass gap is there. The theory entails the existence of massive excitations exactly as the scalar field does. This have been seen in lattice computations (see here). Can we ignore them? Of course no but exact solutions are not our best choice as said above even if we will have hard time to recover them with perturbation theory. Better to wait.

Marco Frasca (2009). Exact solutions of classical scalar field equations J.Nonlin.Math.Phys.18:291-297,2011 arXiv: 0907.4053v2

Marco Frasca (2013). Scalar field theory in the strong self-interaction limit Eur. Phys. J. C (2014) 74:2929 arXiv: 1306.6530v5

Marco Frasca (2014). Exact solutions for classical Yang-Mills fields arXiv arXiv: 1409.2351v2

Biagio Lucini, & Marco Panero (2012). SU(N) gauge theories at large N Physics Reports 526 (2013) 93-163 arXiv: 1210.4997v2

Standard Model at the horizon


Hawking radiation is one of the most famous effects where quantum field theory combines successfully with general relativity. Since 1975 when Stephen Hawking uncovered it, this result has obtained a enormousStephen Hawking consideration and has been derived in a lot of different ways. The idea is that, very near the horizon of a black hole, a pair of particles can be produced one of which falls into the hole and the other escapes to infinity and is seen as emitted radiation. The overall effect is to drain energy from the hole, as the pair is formed at its expenses, and its ultimate fate is to evaporate. The distribution of this radiation is practically thermal and a temperature and an entropy can be attached to the black hole. The entropy is proportional to the area of the black hole computed at the horizon, as also postulated by Jacob Bekenstein, and so, it can only increase. Thermodynamics applies to black holes as well. Since then, the quest to understand the microscopic origin of such an entropy has seen a huge literature production with the notable Jacob Bekensteinunderstanding coming from string theory and loop quantum gravity.

In all the derivations of this effect people generally assumes that the particles are free and there are very good reasons to do so. In this way the theory is easier to manage and quantum field theory on curved spaces yields definite results. The wave equation is separable and exactly solvable (see here and here). For a scalar field, if you had a self-interaction term you are immediately in trouble. Notwithstanding this, in  the ’80 UnruhWilliam Unruh and Leahy, considering the simplified case of two dimensions and Schwarzschild geometry, uncovered a peculiar effect: At the horizon of the black the interaction appears to be switched off (see here). This means that the original derivation by Black-hole-model by Kip ThorneHawking for free particles has indeed a general meaning but, the worst conclusion, all particles become non interacting and massless at the horizon when one considers the Standard Model! Cooper will have very bad times crossing Gargantua’s horizon.

Turning back from science fiction to reality, this problem stood forgotten for all this time and nobody studied this fact too much. The reason is that the vacuum in a curved space-time is not trivial, as firstly noted by Hawking, and mostly so when particles interact. Simply, people has increasing difficulties to manage the theory that is already complicated in its simplest form. Algebraic quantum field theory provides a rigorous approach to this (e.g. see here). These authors consider an interacting theory with a \varphi^3 term but do perturbation theory (small self-interaction) probably hiding in this way the Unruh-Leahy effectValter Moretti.

The situation can change radically if one has exact solutions. A \varphi^4 classical theory can be indeed solved exactly and one can make it manageable (see here). A full quantum field theory can be developed in the strong self-interaction limit (see here) and so, Unruh-Leahy effect can be accounted for. I did so and then, I have got the same conclusion for the Kerr black hole (the one of Interstellar) in four dimensions (see here). This can have devastating implications for the Standard Model of particle physics. The reason is that, if Higgs field is switched off at the horizon, all the particles will lose their masses and electroweak symmetry will be recovered. Besides, further analysis will be necessary also for Yang-Mills fields and I suspect that also in this case the same conclusion has to hold. So, the Unruh-Leahy effect seems to be on the same footing and importance of the Hawking radiation. A deep understanding of it would be needed starting from quantum gravity. It is a holy grail, the switch-off of all couplings, kind of.

Further analysis is needed to get a confirmation of it. But now, I am somewhat more scared to cross a horizon.

V. B. Bezerra, H. S. Vieira, & André A. Costa (2013). The Klein-Gordon equation in the spacetime of a charged and rotating black hole Class. Quantum Grav. 31 (2014) 045003 arXiv: 1312.4823v1

H. S. Vieira, V. B. Bezerra, & C. R. Muniz (2014). Exact solutions of the Klein-Gordon equation in the Kerr-Newman background and Hawking radiation Annals of Physics 350 (2014) 14-28 arXiv: 1401.5397v4

Leahy, D., & Unruh, W. (1983). Effects of a λΦ4 interaction on black-hole evaporation in two dimensions Physical Review D, 28 (4), 694-702 DOI: 10.1103/PhysRevD.28.694

Giovanni Collini, Valter Moretti, & Nicola Pinamonti (2013). Tunnelling black-hole radiation with $φ^3$ self-interaction: one-loop computation for Rindler Killing horizons Lett. Math. Phys. 104 (2014) 217-232 arXiv: 1302.5253v4

Marco Frasca (2009). Exact solutions of classical scalar field equations J.Nonlin.Math.Phys.18:291-297,2011 arXiv: 0907.4053v2

Marco Frasca (2013). Scalar field theory in the strong self-interaction limit Eur. Phys. J. C (2014) 74:2929 arXiv: 1306.6530v5

Marco Frasca (2014). Hawking radiation and interacting fields arXiv arXiv: 1412.1955v1

Living dangerously


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

Where does mass come from?


After CERN’s updates (well recounted here, here and here) producing no real news but just some concern about possible Higgs cloning, I would like to discuss here some mathematical facts about what one should expect about mass generation and why we should not be happy with these results, now coming out on a quarterly basis.

The scenario we are facing so far is one with a boson particle resembling more and more the Higgs particle appearing in the original formulation of the Standard Model. No trace is seen of anything else at higher energies, no evidence of supersymmetry. It appears like no new physics is hiding here rather for it we will have to wait eventually the upgrade of LHC that will start its runs on 2015.

I cannot agree with all of this and this is not the truth at all. The reason to not believe all this is strictly based on theoretical arguments and properties of partial differential equations. We are aware that physicists can be skeptical also about mathematics even if this is unacceptable as mathematics has no other way than being true or false. There is nothing like a half truth but there are a lot of theoretical physicists trusting on it. I have always thought that being skeptical on mathematics is just an excuse to avoid to enter into other work. There could always be the risk that one discovers it is correct and then has to support it.

The point is the scalar field. A strong limitation we have to face when working in quantum field theory is that only small coupling can be managed. No conclusive analysis can be drawn when a coupling is just finite and also lattice computations produce confusion. It seems like small coupling only can exist and all the theory we build are in the hope that nature is benign and yields nothing else than that. For the Higgs field is the same. All our analysis are based on this, the hierarchy problem comes out from this. Just take any of your textbook on which you built your knowledge of this matter and you will promptly realize that nothing else is there. Peschin and Schroeder, in their really excellent book, conclude that strong coupling cannot exist in quantum field theory and the foundation of this argument arises from renormalization group. Nature has only small couplings.

Mathematics, a product of nature, has not just small couplings and nobody can impede a mathematician to take these equations and try to analyze them with a coupling running to infinity. Of course, I did it and somebody else tried to understand this situation and the results make the situation rather embarrassing.

These reflections sprang from a paper appeared yesterday on arxiv (see here). In a de Sitter space there is a natural constant having the dimension of energy and this is the Hubble constant (in natural units). It is an emerging result that a massless scalar field with a quartic interaction in such a space develops a mass. This mass goes like m^2\propto \sqrt{\lambda}H^2 being \lambda the coupling coming from the self-interaction and H the Hubble constant. But the authors of this paper are forced to turn to the usual small coupling expansion just singling out the zero mode producing the mass. So, great news but back to the normal.

A self-interacting scalar field has the property to get mass by itself. Generally, such a self-interacting field has a potential in the form \frac{1}{2}\mu^2\phi^2+\frac{\lambda}{4}\phi^4 and we can have three cases \mu^2>0, \mu^2=0 and \mu^2<0. In all of them the classical equations of motion have an exact massive free solution (see here and Tao’s Dispersive Wiki) when \lambda is finite. These solutions cannot be recovered by any small coupling expansion unless one is able to resum the infinite terms in the series. The cases with \mu^2\ne 0 are interesting in that this term gets a correction depending on \lambda and for the case \mu^2<0 one can recover a spectrum with a Goldstone excitation and the exact solution is an oscillating one around a finite value different from zero (it never crosses the zero) as it should be for spontaneous breaking of symmetry. But the mass is going like \sqrt{\lambda}\Lambda^2 where now \Lambda is just an integration constant. The same happens in the massless case as one recovers a mass going like m^2\propto\sqrt{\lambda}\Lambda^2.  We see the deep analogy with the scalar field in a de Sitter space and these authors are correct in their conclusions.

The point here is that the Higgs mechanism, as has been devised in the sixties, entails all the philosophy of “small coupling and nothing else” and so it incurs in all the possible difficulties, not last the hierarchy problem. A modern view about this matter implies that, also admitting \mu^2<0 makes sense, we have to expand around a solution for \lambda finite being this physically meaningful rather than try an expansion for a free field. We are not granted that the latter makes sense at all but is just an educated guess.

What does all this imply for LHC results? Indeed, if we limit all the analysis to the coupling of the Higgs field with the other fields in the Standard Model, this is not the best way to say we have observed a true Higgs particle as the one postulated in the sixties. It is just curious that no other excitation is seen beyond the (eventually cloned) 126 GeV boson seen so far but we have a big desert to very high energies. Because the very nature of the scalar field is to have massive solutions as soon as the self-interaction is taken to be finite, this also means that other excited states must be seen. This simply cannot be the Higgs particle, mathematics is saying no.

M. Beneke, & P. Moch (2012). On “dynamical mass” generation in Euclidean de Sitter space arXiv arXiv: 1212.3058v1

Marco Frasca (2009). Exact solutions of classical scalar field equations J.Nonlin.Math.Phys.18:291-297,2011 arXiv: 0907.4053v2

Breaking of a symmetry: 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

Breaking of a symmetry


This week-end has been somewhat longer in Italy due to November 1st holiday and I have had the opportunity to read a very fine book by Ian Aitchison: Supersymmetry in Particle Physics – An Elementary Introduction. This book gives a very clear introduction to SUSY with all the computations clearly stated and going into the details of the Minimal Supersymmetric Standard Model (MSSM). This model was originally proposed by Howard Georgi and Savas Dimopolous (see here) and today does not seem to be in good shape due to recent results from LHC. Authors introduce a concept of a “softly” broken supersymmetry to accomodate the Higgs mechanism in the low-energy phenomenology.  A “soflty” broken supersymmetry is when the symmetry is explicitly broken using mass terms but keeping renormalizability and invariance under the electroweak symmetry group. The idea is that, in this way, the low-energy phenomenology will display a standard Higgs mechanism with a vacuum expectation value different from zero. This fact is really interesting as we know that in a standard electroweak theory the symmetry cannot be explicitly broken as we lose immediately renormalizability but a supersymmetric theory leaves us more freedom. But why do we need to introduce explicit breaking terms into the Lagrangian of the MSSM? The reason is that SUSY is conveying a fundamental message:

There is no such a thing as a Higgs mechanism.

Indeed, one can introduce a massive contribution to a scalar field, the \mu-term, but this has just the wrong sign and, indeed, a spontaneously broken supersymmetry is somewhat of a pain. There are some proposed mechanisms, as F or D breaking fields or some dynamical symmetry breaking, but nothing viable for the MSSM. Given the “softly” breaking terms, then the argument runs smoothly and one recovers two doublets and \tan\beta parameter that some authors are fond of.

The question at the root of the matter is that a really working supersymmetry breaking mechanism is yet to be found and should be taken for granted as we do not observe superpartners at accessible energies and LHC has yet to find one if ever. This mechanism also drives the electroweak symmetry breaking. Indeed, supersymmetry properly recovers a quartic self-interaction term but the awkward quadratic term with a wrong sign gives serious difficulties. Of course, the presence of a quartic term into a scalar field interacting with a fermion field, e.g. a Wess-Zumino model, provides the essential element to have a breaking of supersymmetry at lower energies: This model is reducible to a Nambu-Jona-Lasinio model and the gap equation will provide a different mass to the fermion field much in the same way this happens to chiral symmetry in QCD. No explicit mass term is needed but just a chiral model.

This means that the MSSM can be vindicated once one gets rid of an explicit breaking of the supersymmetry and works out in a proper way the infrared limit. There is a fundamental lesson we can learn here: SUSY gives rise to self-interaction and this is all you need to get masses. Higgs mechanism is not a fundamental one.

Dimopoulos, S., & Georgi, H. (1981). Softly broken supersymmetry and SU(5) Nuclear Physics B, 193 (1), 150-162 DOI: 10.1016/0550-3213(81)90522-8

Marco Frasca (2011). Chiral symmetry in the low-energy limit of QCD at finite temperature Phys. Rev. C 84, 055208 (2011) arXiv: 1105.5274v4

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


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