That Higgs is trivial!

Notwithstanding LHC has seen the particle, the Higgs sector of the Standard Model has some serious problems. This fact yielded more than one headache to physicists. One of these difficulties is called technically “triviality“. The scalar field theory, that is so well defined classically, does not exist as a quantum field theory unless is non-interacting. There is a wonderful paper by Michael Aizenman that shows that this is true for dimensions 5 and higher. So, one should think that, as we live in four dimensions, there is no reason to worry. The point is that Michael Aizenman left the question in four dimensions open. So, does Higgs particle exist or not and how does it yield mass if it will not interact? CERN said to us that Higgs particle is there and so, in some way, the scalar sector of the Standard Model must properly work. Aizenman’s proof was on 1981 but what is the situation now? An answer is in this article on Scholarpedia. As stated by the author Ulli Wolff

Triviality of lattice phi^4 theory in this sense has been rigorously proven for D>4 while for the most interesting borderline case D=4 we have only partial results but very strong evidence from numerical simulations.

While there is another great expert on quantum field theory, Franco Strocchi, in his really worth to read book saying

The recent proof of triviality of phi^4 in 3 + 1 spacetime dimensions indicates that the situation becomes worse in the real world, and in particular the renormalized perturbative series of the phi^4 model seems to have little to do with the non-perturbative solution.

We see that experts do not completely agree about the fact that a proof exists or not but, for sure, the scalar theory in four dimensions cannot interact and the Standard Model appears in serious troubles.

Before to enter more in details about this matter, let me say that, even if Strocchi makes no citation about where the proof is, he is the one being right. We have proof about this, the matter is now well understood and again we are waiting for the scientific community to wake up. Also, the Standard Model is surely secured and there is no serious risk about the recent discovery by CERN of the Higgs particle.

The proof has been completed recently by Renata Jora with this paper on arxiv. Renata extended the proof an all the energy range. I met her in Montpellier (France) at this workshop organized by Stephan Narison. We have converging interests in research. Renata’s work is based on a preceding proof, due to me and Igor Suslov, showing that, at large coupling, the four dimensional theory is indeed trivial. You can find the main results here and here. Combining these works together, we can conclude that Strocchi’s statement is correct but there is no harm for the Standard Model as we will discuss in a moment. Also the fact that the perturbation solution of the model is not properly describing the situation can be seen from the strictly non-analytical behaviours seen at strong coupling that makes impossible to extend what one gets at small coupling to that regime.

The fact that CERN has indeed seen the Higgs particle and that the Higgs sector of the Standard Model is behaving properly, unless a better understanding will emerge after the restart of the LHC, has been seen with the studies of the propagators of the Yang-Mills theory in the Landau gauge. The key paper is this where the behaviour of the running coupling of the theory was obtained on all the energy range from lattice computations.

This behaviour shows that, while the theory is trivial at both the extremes of the energy range, there is an intermediate regime where we can trust the theory and treat it as an effective one. There the coupling does not run to zero but moves around some finite non-null value. Of course, all this is just saying that this theory must be superseded by an extended one going to higher energies (supersymmetry? Technicolor?) but it is reasonable to manage the theory as if all this just works at current energies. Indeed, LHC has shown that a Higgs particle is there.

So, triviality is saying that the LHC will find something new for sure. Today, beams moved again inside the accelerator. We are eager to see what will come out form this wonderful enterprise.

Aizenman, M. (1981). Proof of the Triviality of Field Theory and Some Mean-Field Features of Ising Models for Physical Review Letters, 47 (12), 886-886 DOI: 10.1103/PhysRevLett.47.886

Renata Jora (2015). $Φ^4$ theory is trivial arXiv arXiv: 1503.07298v1

Marco Frasca (2006). Proof of triviality of $λφ^4$ theory Int.J.Mod.Phys.A22:2433-2439,2007 arXiv: hep-th/0611276v5

Igor M. Suslov (2010). Asymptotic Behavior of the \Beta Function in the Φ^4 Theory: A Scheme
Without Complex Parameters J.Exp.Theor.Phys.111:450-465,2010 arXiv: 1010.4317v1

I. L. Bogolubsky, E. -M. Ilgenfritz, M. Müller-Preussker, & A. Sternbeck (2009). Lattice gluodynamics computation of Landau-gauge Green’s functions in the deep infrared Phys.Lett.B676:69-73,2009 arXiv: 0901.0736v3

Advertisement

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 enormous 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 understanding 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 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 Hawking 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 term but do perturbation theory (small self-interaction) probably hiding in this way the Unruh-Leahy effect.

The situation can change radically if one has exact solutions. A 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

Igor Suslov and the beta function of the scalar field

I think that blogs are a very good vehicle for a scientist to let his/her work widely known and can be really helpful also for colleagues doing research in the same field. This is the case of Igor Suslov at Kapitza Institute in Moscow. Igor is doing groundbreaking research in quantum field theory and, particularly, his main aim is to obtain the beta function of the scalar field in the limit of a very large coupling. This means that the field of research of Igor largely overlaps mine. Indeed, I have had some e-mail exchange with him and we cited our works each other. Our conclusions agree perfectly and he was able to obtain the general result that, for very large bare coupling one has

where d is the number of dimensions. This means that for d=4 Igor recovers my result. More important is the fact that from this result one can draw the conclusion that the scalar theory is indeed trivial in four dimensions, a long sought result. This should give an idea of the great quality of the work of this author.

On the same track, today  on arxiv Igor posted another important paper (see here). The aim of this paper is to get higher order corrections to the aforementioned result. So, he gives a sound initial explanation on why one could meaningfully take the bare coupling running from 0 to infinity and then, using a lattice formulation of the n components scalar field theory, he performs a high temperature expansion.  He is able to reach the thirteenth order correction! This is an expansion of in powers of and so, for d=4, one gets an expansion in . Again, this Igor’s result is in agreement with mine in a very beautiful manner. As my readers could know, I have been able to go to higher orders with my expansion technique in the large coupling limit (see here and here). This means that my findings and this result of Igor must agree. This is exactly what happens! I was able to get the next to leading order correction for the two-point function and, from this, with the Callan-Symanzik equation, I can derive the next to leading order correction for that goes like with an opposite sign with respect to the previous one. This is Igor’s table with the coefficients of the expansion:

So, from my point of view, Igor’s computations are fundamental for all the understanding of infrared physics that I have developed so far. It would be interesting if he could verify the mapping with Yang-Mills theory obtaining the beta function also for this case. He did some previous attempt on this direction but now, with such important conclusions reached, it would be absolutely interesting to see some deepening. Thank you for this wonderful work, Igor!

I. M. Suslov (2011). Renormalization Group Functions of \phi^4 Theory from High-Temperature
Expansions J.Exp.Theor.Phys., v.112, p.274 (2011); Zh.Eksp.Teor.Fiz., v.139, p.319 (2011) arXiv: 1102.3906v1

Marco Frasca (2008). Infrared behavior of the running coupling in scalar field theory arxiv arXiv: 0802.1183v4

Marco Frasca (2010). Mapping theorem and Green functions in Yang-Mills theory arxiv arXiv: 1011.3643v2

A striking clue and some more

My colleagues participating to “The many faces of QCD” in Ghent last year keep on publishing their contributions to the proceedings. This conference produced several outstanding talks and so, it is worthwhile to tell about that here. I have already said about this here, here and here and I have spent some words about the fine paper of Oliveira, Bicudo and Silva (see here). Today I would like to tell you about an interesting line of research due to Silvio Sorella and colleagues and a striking clue supporting my results on scalar field theory originating by Axel Maas (see his blog).

Silvio is an Italian physicist that lives and works in Brazil, Rio de Janeiro, since a long time. I met him at Ghent mistaking him with Daniele Binosi. Of course, I was aware of him through his works that are an important track followed to understand the situation of low-energy Yang-Mills theory. I have already cited him in my blog both for Ghent and the Gribov obsession. He, together with David Dudal, Marcelo Guimaraes and Nele Vandersickel (our photographer in Ghent), published on arxiv a couple of contributions (see here and here). Let me explain in a few words why I consider the work of these authors really interesting. As I have said in my short history (see here), Daniel Zwanzinger made some fundamental contributions to our understanding of gauge theories. For Yang-Mills, he concluded that the gluon propagator should go to zero at very low energies. This conclusion is at odds with current lattice results. The reason for this, as I have already explained, arises from the way Gribov copies are managed. Silvio and other colleagues have shown in a series of papers how Gribov copies and massive gluons can indeed be reconciled by accounting for condensates. A gluon condensate can explain a massive gluon while retaining  all the ideas about Gribov copies and this means that they have also find a way to refine the ideas of Gribov and Zwanzinger making them agree with lattice computations. This is a relevant achievement and a serious concurrent theory to our understanding of infrared non-Abelian theories. Last but not least, in these papers they are able to show a comparison with experiments obtaining the masses  of the lightest glueballs. This is the proper approach to be followed to whoever is aimed to understand what is going on in quantum field theory for QCD. I will keep on following the works of these authors being surely a relevant way to reach our common goal: to catch the way Yang-Mills theory behaves.

A real brilliant contribution is the one of Axel Maas. Axel has been a former student of Reinhard Alkofer and Attilio Cucchieri & Tereza Mendes. I would like to remember to my readers that Axel have had the brilliant idea to check Yang-Mills theory on a two-dimensional lattice arising a lot of fuss in our community that is yet on. On a similar line, his contribution to Ghent conference is again a striking one. Axel has thought to couple a scalar field to the gluon field and study the corresponding behavior on the lattice. In these first computations, he did not consider too large lattices (I would suggest him to use CUDA…) limiting the analysis to , and . Anyhow, also for these small volumes, he is able to conclude that the propagator of the scalar field becomes a massive one deviating from the case of the tree-level approximation. The interesting point is that he sees a mass to appear also for the case of the massless scalar field producing a groundbreaking evidence of what I proved in 2006 in my PRD paper! Besides, he shows that the renormalized mass is greater than the bare mass, again an agreement with my work. But, as also stated by the author, these are only clues due to the small volumes he uses. Anyhow, this is a clever track to be pursued and further studies are needed. It would also be interesting to have a clear idea of the fact that this mass arises directly from the dynamics of the scalar field itself rather than from its interaction with the Yang-Mills field. I give below a figure for the four dimensional case in a quenched approximation

I am sure that this image will convey the right impression to my readers as mine. A shocking result that seems to match, at a first sight, the case of the gluon propagator on the lattice (mapping theorem!). At larger volumes it would be interesting to see also the gluon propagator. I expect a lot of interesting results to come out from this approach.

 

 

Silvio P. Sorella, David Dudal, Marcelo S. Guimaraes, & Nele Vandersickel (2011). Features of the Refined Gribov-Zwanziger theory: propagators, BRST soft symmetry breaking and glueball masses arxiv arXiv: 1102.0574v1

N. Vandersickel,, D. Dudal,, & S.P. Sorella (2011). More evidence for a refined Gribov-Zwanziger action based on an effective potential approach arxiv arXiv : 1102.0866

Axel Maas (2011). Scalar-matter-gluon interaction arxiv arXiv: 1102.0901v1

Frasca, M. (2006). Strongly coupled quantum field theory Physical Review D, 73 (2) DOI: 10.1103/PhysRevD.73.027701

Mass generation: The solution

In my preceding post I have pointed out an interesting mathematicalquestion about the exact solutions of the scalar field theory that I use in this paper

given by

that holds for

If you compute the Hamiltonian the energy does not appear to be finite, differently from what the relation dispersion is saying. This is very similar to what happens to plane waves for the wave equation. The way out is to take a finite volume and normalize properly the plane waves. One does this to get the integral of the Hamiltonian finite and all amounts to a proper normalization. In our case where must this normalization enter? The striking answer is: The coupling. This is an arbitrary parameter of the theory and we can properly rescale it to get the right normalization in the Hamiltonian. The final result is a running coupling exactly in the same way as I and others have obtained for the quantum theory. You can see the coupling entering in the right way both in the solution and in the computation of the Hamiltonian.

If you are curious about these computations you can read the revised version of my paper to appear soon on arxiv.

Marco Frasca (2010). Mass generation and supersymmetry arxiv arXiv: 1007.5275v1

Mass generation in the Standard Model

The question of the generation of the mass for the particles in the Standard Model is currently a crucial one in physics and is a matter that could start a revolutionary path in our understanding of the World as it works. This is also an old question that can be rewritten as “What are we made of?” and surely ancient greeks asked for this. Today, with the LHC at work and already producing a wealth of important results, we are on the verge to give a sound answer to it.

The current situation is well-known with a Higgs mechanism (but here there are several fathers) that mimics the second order phase transitions as proposed by Landau long ago. In some way, understanding ferromagnetism taught us a lot and produced a mathematical framework to extract sound results from the Standard Model. Without these ideas the model would have been practically useless since the initial formulation due to Shelly Glashow. The question of mass in the Standard Model is indeed a stumbling block and we need to understand what is hidden behind an otherwise exceptionally successful model.

As many of yours could know, I have written a paper (see here) where I show that if the way a scalar field gets a mass (and so also Yang-Mills field) is identical in the Standard Model, forcefully one has a supersymmetric Higgs sector but without the squared term and with a strong self-coupling. This would imply a not-so-light Higgs and the breaking of the supersymmetry the only way to avoid degeneracy between the masses of all the particles of the Standard Model. By my side I would expect these signatures as evidences that I am right and QCD, a part of the Model, will share the same mechanism to generate masses.

Yet, there is an open question put forward by a smart referee to my paper. I will put this here as this is an interesting question of classical field theory that is worthwhile to be understood. As you know, I have found a set of exact solutions to the classical field equation

from which I built my mass generation mechanism. These solutions can be written down as

being a Jacobi’s elliptic function and provided

From the dispersion relation above we can conclude that these nonlinear waves indeed represent free massive particles of finite energy. But let us take a look to the definition of the energy for this theory, one has

and if you substitute the above exact solutions into this you will get an infinity. It appears like these solutions have infinite energy! This same effect is seen by ordinary plane waves but can be evaded by taking a finite volume, one normalizes the solutions with respect to this volume and so you are done.  Of course, you can take finite volume also in the nonlinear case provided you put for the momenta

being as this Jacobi function has period but you should remember that this function is doubly periodic having also a complex period. Now, if you compute for you will get a dispersion relation multiplied by some factors and one of these is the volume. How could one solve this paradox? You can check by yourselves that these solutions indeed exist and must have finite energy.

My work on QCD is not hindered by this question as I work solving the equation and here there are different problems. But, in any case, mathematics claims for existence of these solutions while physics is saying that there is something not so well defined. An interesting problem to work on.

What is the right solution?

Sometime it is quite interesting to turn back to well-done books to refresh some ideas. This happened to me with Smilga’s book reading again the chapter on classical solutions of Yang-Mills equations. This chapter is greatly important and the reason is quantum field theory. At our undergraduate courses, when we were firstly exposed to quantum field theory we learned that we have to be able to solve the free equations of motion to start quantization of a theory. Indeed, a free theory is generally easy to quantize while some difficulties could appear with gauge theories. But, anyhow, this easiness arises from the Gaussian form the generating functional takes.

When we turn our attention to Yang-Mills theory we have to cope with the nonlinearities appearing in the equations of motion. At first, being not able to solve them exactly, we can consider solutions identical to their Abelian counterpart that is the electromagnetic field. It is easy to verify that both equations of motion can share identical solutions of free plane waves with a dispersion relation of massless particles. To get them you have to properly select a set of components and you are granted that these classical solutions indeed exist. These solutions are well-known and, when we quantize the theory, we recognize them as describing gluons. But when you quantize for this case you immediately recognize that your computations hold when the coupling appearing in the self-interaction terms is going to zero. You are not able to recover any mass gap and this kind of computations does not appear to help to describe low-energy QCD. But you get a formidable agreement with experiments at higher energies and this is where asymptotic freedom sets in. So, quantization of Yang-Mills theory starting with this kind of solutions says us that these are the right ones for the high-energy limit of the theory when the coupling decreases to zero and all our computations are mathematically consistent.

Now, when we consider the low-energy limit we are in trouble. The reason is that we are not able to solve the equations of motion when the coupling is too large and we are forced to consider them in full with all the nonlinearities in the proper place. But here again Smilga’s book comes to rescue. If you choose judiciously the components of the field and ignore space dependence retaining only time, you will get regular exact solutions that are represented by elliptic Jacobi functions. These are nonlinear standing waves. But looking at them in this way does not help too much. We need also space dependence if we want to extract some physical meaning from these solutions. This is indeed possible looking at a quartic massless scalar field. A quartic massless scalar field with an equation of motion

admits the following exact solution

where and are two integrations constants and is a Jacobi elliptic function. But this holds provided the following dispersion relation does hold

that is the one of  a free massive particle! So, a massless classical theory produced massive solutions due to the nonlinear term. That the origin of the mass can be this can be easily understood when you take the limit . The theory becomes massless in this limit and if you do standard perturbation theory the mass term will be hidden in the series and you are not granted you will recover the mass. This result is really beautiful and we see that these solutions are very similar, from a mathematical standpoint, to the ones Smilga considered for classical Yang-Mills equations. So, it is very tempting to try to match these theories. Indeed, this is a truth coded into a (mapping) theorem that I have given in two papers here and here published in archival journals. This mapping holds perturbatively for the coupling going to infinity and this is what we need for studying the opposite limit with respect to gluons. So, I have gone further: These are the right solutions to build a low-energy limit quantum field theory for Yang-Mills equations. This implies that

Mass gap question is settled for Yang-Mills theory.

This is the main conclusion to be drawn: When you build your quantum field theory be careful in the choice of the right classical solutions!

Triviality

One of the questions that is not that easy to answer is: When is a quantum field theory exactly solved? Of course, we have the example of a free theory. When one is able to put the generating functional into a Gaussian form, the spectrum of the theory is that of a harmonic oscillator and when the coupling is zero, one is left with a possibly solved theory. But this case is trivial and does not say anything about the case of an exactly solved but interacting quantum field theory. An immediate answer to this question is: When one is able to get all the n-point functions. This implies that, if you are able to solve all the hierarchy of Dyson-Schwinger equations, you are done. Solving this set of equations is practically impossible in almost all the interesting case. But there is an exception and a notable one. So, consider the case of a massless quartic scalar field theory. Lattice computations in d=3+1 strongly hint toward triviality in the low-energy limit. Better, for d>3+1 there is a beautiful proof by Michael Aizenman that went published here. In this case the hierarchy is exactly solved (see here) and this is true also for d=3+1. So far, triviality and exact solution indeed are the same thing. But why does an interacting theory become trivial? The reason is in the behavior of the running coupling as the energy varies. We have learned from quantum field theory that couplings have not always the same value. Rather, their value is varying depending on the energy scale they are measured. In a trivial theory, couplings happen to go to zero in the given limit and an interacting theory becomes free!

For the scalar theory in the low-energy limit (infrared) in d=3+1, evidence is becoming wider that the beta function, the function that determines the behavior of the running coupling, goes like

being the coupling and space-time dimension. I have proved this firstly here for d=3+1 but other authors arrived to an identical conclusion by different means (see here and here). But there is a surprise here: Some authors, a few years ago, proved an identical result for Yang-Mills theory (see here) with lattice computations. So, this is again a striking proof of the correctness of my mapping theorem but an indirect one. Then, we can conclude this post by stating a shocking result: Yang-Mills theory is trivial in the infrared even if QCD is not. But this result is enough to make QCD manageable at very low-energies.

Classical Yang-Mills theory and mass gap

One of the key ingredients to build up a quantum field theory is to have a set of solutions of classical equations of motion to start with. Then, given such solutions, we are able to perform perturbation theory and to extract results from the theory to be compared with experiment. I think that my readers are familiar with standard approach having free equations of the theory solved. When path integrals are used, we solve for the Green function of the free theory but we are talking about the same thing: we know how to solve our theory in some limit and then we build on it. So, to give an answer to the question of the mass gap for Yang-Mills theory, we have to know how to solve the theory in a limit we are not so familiar: strong coupling limit. So far, very few was known about this limit except knowledge acquired through lattice computations. Also in this latter case, for several years a lot of confusion pervaded the field: Does gluon propagator go to zero or not? Enlarging volumes produced an answer that is a reason for hot debate yet: Gluon propagator does not go to zero at very low energy but reaches a finite value. In literature this is known as the decoupling solution to be contrasted with the scaling solution having a propagator going to zero at very small momenta. If we know gluon propagator, we are able to compute the behavior of QCD at very low energies (see here) and this is a well-known fact since eighties.

The question of existence of a class of solutions for Yang-Mills theory to work with at low energies has been successfully answered quite recently. I have written a nice pair of papers that went published in respectful journals and permitted to solve all this matter (see here and here). Two papers were needed because Terry Tao showed that a proof in a key theorem (mapping theorem) was not correct. After this, I was able to give  an answer that both agreed. My aim in this post is to explain, with some simple mathematics, what is the content of this theorem that produces a set of classical solutions to build up a quantum field theory in the low-energy limit for Yang-Mills theory and so QCD.

The key element is a mapping theorem. We map two classical theories, one of this we are able to solve exactly. So, consider a massless scalar field theory

Contrarily to common wisdom, we are able to solve this exactly. Our solution can be written down as

provided that

Hera is an integration constant with the dimension of energy, is another integration constant and is the snoidal Jacobi’s elliptic function. Why is this solution so interesting? The reason is that we started with a massless equation and the solution describes a wave with a massive dispersion solution of a free particle! This is the famous mass gap when we translate this result to quantum field theory. I have done this here. So, the classical theory already has the feature of a mass gap. Scalar theory proves to be trivial for the simple reason that we produce, in the low-energy limit, free massive excitations. This is a long awaited result that is going to get increasingly confirmed from other theoretical studies. I will discuss this issue in another post.

What is the relation, if any, between a massless scalar field theory and Yang-Mills theory? Indeed, there exists a deep relation in the low-energy limit, when the coupling becomes increasingly large, as the solutions of the two theories can be mapped. So, for SU(3), mapping theorem shows that

being our solution above provided the substitution . This is a very beautiful result as this gives at once the following conclusions:

• Strong coupling solutions of classical Yang-Mills theory are free massive waves.
• Yang-Mills theory displays massive solutions already at classical level.
• Quantum theory maintains such conclusions as I showed in my papers.

Lattice computations beautifully confirmed this mapping theorem in d=2+1 as showed by Rafael Frigori in a very nice paper (see here). Strong hints are also seen in d=3+1 by other authors and it would be very nice to see an extended computation in this case as the one Frigori did in d=2+1. For yourselves, you can check with Mathematica or Maple the equations given above. You will also see that gauge invariance is not hindered.

Classical solutions of Yang-Mills equations

So far, I have posted several posts in this blog about the question of classical solutions to Yang-Mills equations. This has produced some fuzz, mostly arisen from my published papers, as to such solutions may not be correct. Thanks to a wise intervention of Terry Tao, I was able to give a complete understanding of my solutions and a theorem was fully proved in a recent paper of mine to appear in Modern Physics Letters A (see here), agreed with Terry in a private communication. So, I think it is time to give a description of this result here as it appears really interesting showing how, already at a classical level, this theory can display massive solutions and a mass gap is already seen in this case. Then, it takes a really small step to get the corresponding proof in quantum field theory.

To understand how these solutions are obtained, let us consider the following equation for a scalar field

This is a massless self-interacting field. We can select a class of solutions by looking at the case of a rest reference frame. So, we put any dependence on spatial variables to zero and solve the equation

whose solutions are known and given by

being and two integration constants and sn a Jacobi elliptical function. Then, boosting this solution will produce an exact solution of the equation we started from given by

provided the following dispersion relation holds

and we see that, although we started with a massless field, self-interaction provided us massive solutions!

Now, the next question one should ask is if such a mechanism may be at work for classical Yang-Mills equations. These can be written down as

being chosen depending on the gauge choice, the coupling and the structure constants of the gauge group taken to be SU(N). The theorem I proved in my paper above states that the solution given for the scalar field theory is an exact solution of Yang-Mills equations, provided it will not depend on spatial coordinates, for a given choice of Yang-Mills components (Smilga’s choice) and , otherwise the following identity holds

Here is a set of constants arising with the Smilga’s choice. This theorem has the following implications: Firstly, when the coupling become increasingly large, a massless scalar field theory and Yang-Mills theory can be mapped each other. Secondly, already at the classical level, for a coupling large enough, a Yang-Mills theory gets massive solutions. We can see here that a mass gap arises already at a classical level for these theories. Finally, we emphasize that the above mapping appears to hold only in a strong coupling regime while, on the other side, these theories manifest different behaviors. Indeed, we know that Yang-Mills theory has asymptotic freedom while the scalar theory has not. The mapping theorem just mirrors this situation.

We note that these solutions are wave-like ones and describe free massive excitations. This means that these classical theories have to be considered trivial in some sense as these solutions seem to behave in the same way as the plane waves of a free theory.

One can build a quantum field theory on these classical solutions obtaining a theory manifesting a mass gap in some limit. This is has been done in several papers of mine and I will not repeat these arguments here.