The question of the mass gap

Some years ago I proposed a set of solutions to the classical Yang-Mills equations displaying a massive behavior. For a massless theory this is somewhat unexpected. After a criticism by Terry Tao I had to admit that, for a generic gauge, such solutions are just asymptotic ones assuming the coupling runs to infinity (see here and here). Although my arguments on Yang-Mills theory were not changed by this, I have found such a conclusion somewhat unsatisfactory. The reason is that if you have classical solutions to Yang-Mills equations that display a mass gap, their quantization cannot change such a conclusion. Rather, one should eventually expect a superimposed quantum spectrum. But working with asymptotic classical solutions can make things somewhat involved. This forced me to choose the gauge to be always Lorenz because in such a case the solutions were exact. Besides, it is a great success for a physicist to find exact solutions to fundamental equations of physics as these yield an immediate idea of what is going on in a theory. Even in such case we would get a conclusive representation of the way the mass gap can form.

Finally, after some years of struggle, I was able to get such a set of exact solutions to the classical Yang-Mills theory displaying a mass gap (see here). Such solutions confirm both the Tao’s argument that an all equal component solution for Yang-Mills equations cannot hold in any gauge and also my original argument that an all equal component solution holds, in a general case, only asymptotically with the coupling running to infinity. But classically, there exist solutions displaying a mass gap that arises from the nonlinearity of the equations of motion. The mass gap goes to zero as the coupling does. Translating this in the quantum realm is straightforward as I showed for the Lorenz (Landau) gauge. I hope all this will help to better elucidate all the physics around strong interactions. My efforts since 2005 went in that direction and are still going on.

Marco Frasca (2009). Mapping a Massless Scalar Field Theory on a Yang-Mills Theory: Classical Case Mod. Phys. Lett. A 24, 2425-2432 (2009) arXiv: 0903.2357v4

Marco Frasca (2014). Exact solutions for classical Yang-Mills fields arXiv arXiv: 1409.2351v1

Yang-Mills mass gap scenario: Further confirmations

Alexander (Sasha) Migdal was a former professor at Princeton University. But since 1996, he is acting as a CEO of a small company. You can read his story from that link. Instead, Marco Bochicchio was a former colleague student of mine at University of Rome “La Sapienza”. He was a couple of years ahead of me. Now, he is a researcher at Istituto Nazionale di Fisica Nucleare, the same of OPERA and a lot of other striking contributions to physics. With Marco we shared a course on statistical mechanics held by Francesco Guerra at the department of mathematics of our university. Today, Marco posted a paper of him on arXiv (see here). I am following these works by Marco with a lot of interest because they contain results that I am convinced are correct, in the sense that are describing the right scenario for Yang-Mills theory. Marco, in this latter work, is referring to preceding publications from Sasha Migdal about the same matter that go back till ’70s! You can find a recollection of these ideas in a recent paper by Sasha (see here). So, what are these authors saying? Using somewhat different approaches than mine (that you can find well depicted here), they all agree that a Yang-Mills theory has a propagator going like

being some numbers and is given by the zeros of some Bessel functions. This last result seems quite different from mine that I get explicitly but this is not so because, in the asymptotic regime, and zeros for the cosine go like and then, my spectrum is easily recovered in the right limit. The right limit is properly identified by Sasha Migdal from Padè approximants for the propagator that start from the deep Euclidean region , being an arbitrary energy scale entering into the spectrum. So, the agreement between the scenario proposed by these authors and mine is practically perfect, notwithstanding different mathematical approaches are used.

The beauty of these conclusions is that such a scenario for a Yang-Mills theory is completely unexpected but it is what is needed to grant confinement. So, the conclusion about the questions of mass gap and confinement is approaching. As usual, we hope that the community will face these matters as soon as possible making them an important part of our fundamental knowledge.

Marco Bochicchio (2011). Glueballs propagators in large-N YM arXiv arXiv: 1111.6073v1

Alexander Migdal (2011). Meromorphization of Large N QFT arXiv arXiv: 1109.1623v2

Marco Frasca (2010). Mapping theorem and Green functions in Yang-Mills theory PoS FacesQCD:039,2010 arXiv: 1011.3643v3

Current status of Yang-Mills mass gap question

I think that is time to make a point about the question of mass gap existence in the Yang-Mills theory. There are three lines of research in this area: Theoretical, numerical and experimental. I can suppose that the one that mostly interests my readers is the theoretical one. I would like to remember that, in order to get a Millenium Prize, one also needs to prove the existence of the theory. This makes the problem far from being trivial.

As for today, the question of existence of the mass gap both for scalar field theories and Yang-Mills theory should be considered settled. Currently there are two papers of mine, here and here both published in archival journals, proving the existence of the mass gap and give it in a closed analytical form. A proof has been also given by Alexander Dynin at Ohio State University here. Alexander does not give the mass gap in a closed form but gets a lower bound that permits him to conclude that Yang-Mills theory has a discrete spectrum with a mass gap. This is enough to declare this part of the problem solved. It is interesting to note that, differently from Poincaré conjecture, this solution does not require a mathematics that is too much complex. This can be understood from the fact that the corresponding classical equations of the theory already admit  massive solutions of free particle. The quantum theory can be built on these solutions and all this boils down to a trivial fixed point in the infrared for the quantum theory. Such a trivial fixed point, that explains also the lower bound Alexander is able to find, is a good news: We have a set of asymptotic states at diminishing momenta that can be used to do perturbation theory and do computations for physics! The reason why these relevant mathematical results did not get the proper exposition so far escape me and enters into the realm of things that I do not know. It is true that in this area there is a lot of caution and this can be understood as this problem received a lot of attention after Witten and Jaffe proposed it for a big money prize.

But, as I have already said, this problem has two questions to be answered and while computing the mass gap is quite easy, the other question is rather involved. To prove the existence of a quantum field theory is not a trivial matter and, for sure, we know that the Wiener integral exists and the Feynman integral does not (so far and only for mathematicians). What I prove in my papers is that the Euclidean theory exists for the scalar field theory (thanks to Glimm and Jaffe that already proved this) and that this theory matches the Yang-Mills theory in the limit of the gauge coupling going to infinity. It should be an asymptotic existence… Alexander by his side proves existence in a different way but here unfortunately I cannot say too much but I would appreciate that Alexander would write down some lines here about his work.

Other theoretical attempts are based on some educated guess as a starting point as could be the vacuum functional, the beta function or other parts of the theory that, for a full proof, should be derived instead. These attempts give a strong support to my work and that of Alexander. In these papers you will see a discrete spectrum and this is the one of a harmonic oscillator or simply the very existence of the mass gap itself. But, for physicists, the spectrum is the relevant conclusion as from it we can get the masses of physical states to be seen in accelerator facilities. This is the reason why I do not worry too much for mathematicians fussing about my papers.

Finally, I would like to spend a few words about numerical and experimental results. Experiments show clearly always bound states of quarks and gluons that are never seen as free. This is the better proof so far Nature gave us of the existence of the mass gap. Numerically, people computed both Green functions and the spectrum of the theory. I am convinced that these lines should merge. The spectrum on the lattice, both quenched and unquenched, displays the mass gap. Green functions, when one considers just the decoupling solution, are Yukawa-like, both on the lattice and from Dyson-Schwinger equations, and this again is a proof of existence of the mass gap.

I hope I have not forgotten anyone. Please, let me know. If you need explicit references here and there I will be pleased to post here. A lot of people is involved in this kind of research and I am happy to acknowledge the good work.

Finally, I would like to remember that one cannot be skeptical about mathematics as mathematics can only be either right or wrong. No other way.

Sannino and the mass gap in Yang-Mills theory

August is vacation month in Italy and I am not an exception. This is the reason for my silence so far. But, of course, I cannot turn off my brain and physics has always been there. So, reading the daily from arxiv today , I have seen another beautiful paper by Francesco Sannino (you can find his page here) in collaboration with Joseph Schechter that has been his PhD thesis advisor. As you know, Sannino and Ryttov postulated an exact beta function for QCD starting from the exact result in the supersymmetric version of this theory (see here).  The beta function Sannino and Schechter get has a pole. The form is

and, taken as is, this has no fixed point than the trivial one . We know that this seems in agreement with recent lattice computations even if, discussing with Valentin Zakharov at QCD10 (see here), he expressed some skepticism about them.  They point out that the knowledge of this function permits a lot of interesting computations and what they do here is to get the mass gap of the Yang-Mills theory. They also point out as, for all the observables obtainable from such a beta function, the pole is harmless and the results appear really meaningful. Indeed, they get a consistent scenario from that guess.

So, let me point out the main results obtained so far by these people using this approach:

• The beta function for Yang-Mills theory goes to zero with the coupling without displaying non-trivial fixed point but QCD has a non-trivial fixed point (see my paper here).
• Yang-Mills theory has a mass gap.

Numerically their result for the mass gap seems to agree quite well with lattice computations. This should be also the mass for a possible observation of the lightest glueball. My view about is that the lightest glueball is the resonance and recent findings at KLOE-2 seems to point out in this direction. But the exact value of the mass gap is not so relevant. What is relevant is that these researchers have found a quite interesting exact form of the beta function for QCD that describes quite well the current understanding of this theory at lower energies that is slowly emerging.

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 Yang-Mills theory and mass gap

Yesterday I posted a paper on arxiv (see here). In this paper I have given a solution to the classical equations of motion of Yang-Mills theory. Indeed, as already seen for the quartic scalar field (see here), one can show, with undergraduate level arguments, that Yang-Mills theory has a massive exact solution at least at a classical level. This can be seen as a simple proof that a massless theory can indeed produce a mass gap and this mass gap is simply a dynamical effect arising from the self-interaction term of the equations of motion.

As obtained from standard textbooks, the equations of motion for the Yang-Mills potential are

We use latin letters for group indeces and greek letters for space-time indeces. As it is already seen for Maxwell equations, we fix properly the gauge to obtain a solution and we choose the Lorentz gauge following a standard textbook procedure already known for the electromagnetic field. The coupling constant is adimensional and no other constants enter into the theory. are structure constants of the gauge group.

In order to find a solution we make the choice to take all the components of the field equal. So we write with elements for SU(N) and and so we have a total of degrees of freedom. With this substitution the above equation assumes a more familiar form

that is just the equation of the quartic scalar field and we know its solution! Checking back here we can write down the solution

where we have changed the name to the integration constant calling it and introduced another one, a phase, . The incredible interesting new is that the above solution holds only if the following dispersion relation holds

and we see that the classical Yang-Mills field has acquired a mass and this mass goes to zero as the coupling goes to zero. So stronger is the coupling in the self-interacting term and larger is the mass gap. But this says us something more. Indeed, as already seen for the quartic scalar field, we have a mass spectrum that we can write down as

that as we have seen here agrees fairly well with lattice computations but we cannot pursue further this comparison as we are working at a classical level while to compare with lattice and experimental data a quantum field theory is needed. But already at this level the agreement is striking and worthing further analysis.

The lesson to be learnt is that an exact classical solution can give a lot of information about the physics one should expect for a quantum field theory. This is also true for the existence of a mass gap of Yang-Mills theory.

How far are we from a proof of existence of the mass gap?

A well acquired fact is that there are seven problems that will be awarded with a rich prize by the Clay Institute (see here). One of these problems touch us that work on QCD very near. This is the question of the existence of a mass gap for a Yang-Mills theory (see here). I would like to emphasize that Clay Institute is a mathematical institution and, as such, it is acquired that a proof given by a physicist could not be enough to satisfy the criteria of professional mathematicians to be called a proof. Anyhow, one can always suppose that a sound mathematical idea due to some physicists can be made rigorous enough by the proper intervention of a mathematician. But this last passage is generally neither that simple nor obtainable in a short time.

As we have discussed here, present lattice results are already enough to have given to physicists a sound proof of the existence of a mass gap for a Yang-Mills theory in D=3+1. In this post I will avoid to discuss about theoretical work in D=3+1 but rather I would like to point out some relevant work appeared for the case D=2+1. In this case there are two categories of papers to be considered. The first category corresponds to the works of Bruce McKellar and Jesse Carlsson. These works are largely pioneering. In these papers the authors consider the theory on the lattice and try to solve it through analytical means (here, here and here appeared in archival journals). They reached a relevant conclusion:

The spectrum of the Yang-Mills theory in D=2+1 is that of an harmonic oscillator.

This conclusion should be compared with the results in D=3+1 on the lattice. We have seen here that looking in a straightforward way to these computations one arrives easily to the conclusion that the theory is trivial. Yes, it has a mass gap but is trivial. The only missing block is the spectrum. So, Carlsson and McKellar give us the missing step. Also the spectrum is consistent with the view that the theory is trivial.

Then we look at the second category of papers. These papers arose from an ingenious idea due to Kim, Karabali and Nair (e.g. see here) that introduced the right variables to manage the theory. In this way one reduces the problem to the one of diagonalizing a Hamiltonian obtaining eigenstates and eigenvalues. Building on this work, Leigh, Minic and Yelnikov were able to postulate a new nontrivial form of the ground state wavefunctional producing the spectrum of the theory in D=2+1 in closed analytical form (see here and here). The spectrum was given as the zeros of Bessel functions that in some approximation can be written as that of an harmonic oscillator. The open problem with this latter approach relies on the proof of existence of the postulated wavefunctional. This may not be easy.

The conclusion to be drawn from this is that we have already sound evidences that a mass gap for Yang-Mills theory exists. These proofs could not be satisfactory for a mathematician but surely for us physicists give a solid ground to work on.

No scaling solution with massive gluons

Some time ago, while I was just at the beginning of my current understanding of low-energy Yang-Mills theory, I wrote to Christian Fischer to know if from the scaling solution, the one with the gluon propagator going to zero lowering momenta and the ghost propagator running to infinity faster than the free particle in the same limit,  a mass gap could be derived. Christian has always been very kind to answer my requests for clarification and did the same also for this so particular question telling to me that this indeed was not possible. This is a rather disappointing truth as we are accustomed with the idea that short ranged forces need some kind of massive carriers. But physics taught that a first intuition could be wrong and so I decided not to take this as an argument against the scaling solution. Since today.

Looking at arxiv, I follow with a lot of interest the works of the group of people collaborating with Philippe Boucaud.   They are supporting the decoupling solution as this is coming out from their numerical computations through the Dyson-Schwinger equations. A person working with them, Jose Rodríguez-Quintero, is producing several interesting results in this direction and the most recent ones appear really striking (see here and here). The question Jose is asking is when and how does a scaling solution appear in solving the Dyson-Schwinger equations? I would like to remember that this kind of solution was found with a truncation technique from these equations and so it is really important to understand better its emerging. Jose solves the equations with a method recently devised by Joannis Papavassiliou and Daniele Binosi (see here) to get a sensible truncation of the Dyson-Schwinger hierarchy of equations. What is different in Jose’s approach is to try an ansatz with a massive propagator (this just means Yukawa-like) and to see under what conditions a scaling solution can emerge. A quite shocking result is that there exists a critical value of the strong coupling that can produce it but at the price to have the Schwinger-Dyson equations no more converging toward a consistent solution with a massive propagator and the scaling solution representing just an unattainable limiting case. So, scaling solution implies no mass gap as already Christian told me a few years ago.

The point is that now we have a lot of evidence that the massive solution is the right one and there is no physical reason whatsoever to presume that the scaling solution should be the true solution at the critical scaling found by Jose. So, all this mounting evidence is there to say that the old idea of Hideki Yukawa is working yet:  Massive carriers imply limited range forces.

J. Rodríguez-Quintero (2011). The scaling infrared DSE solution as a critical end-point for the family
of decoupling ones arxiv arXiv: 1103.0904v1

J. Rodríguez-Quintero (2010). On the massive gluon propagator, the PT-BFM scheme and the low-momentum
behaviour of decoupling and scaling DSE solutions JHEP 1101:105,2011 arXiv: 1005.4598v2

Daniele Binosi, & Joannis Papavassiliou (2007). Gauge-invariant truncation scheme for the Schwinger-Dyson equations of
QCD Phys.Rev.D77:061702,2008 arXiv: 0712.2707v1

What is mass?

I should confess that one of the reasons why I have chosen to be a physicist is that physics, like no other sciences, is able to give answers to fundamental open questions that until a few years ago were only discussed by philosophers. Most of these questions are ancient as our species and the possibility that we have means to get truth is too strong to lose our time with other activities. So, I managed to learn such means and today I am here writing on this blog trying to explain you what these truths are. Sometime, I am at the forefront of research and so, what can be believed a truth may lose this quality as we deepen our understanding. Indeed, dynamics of science adds one more element of charm to all this matter.

One of such old questions is: “What are we made of?”. This question has been an open question till the dawn of the 20th century with the fundamental experiments carried out by Ernst Rutherford. Till then we have learned so much about matter that this question changed form becoming: “What is mass?”. This question has become compelling with the birth of the Standard Model due to Sheldon Lee Glashow, Steven Weinberg and Abdus Salam. Indeed, in order to maintain symmetry we must ask all particles to be massless and some mechanism must exist giving mass to them. In the sixties and seventies of last century we moved toward a real understanding of this concept. The idea is to rely on the Higgs mechanism and a scalar particle must exist to grant masses to the other particles in the model. As you may know this particle has not yet been seen and it is the only missing element of an otherwise very successful model. We are confident for several reasons that the Higgs mechanism could turn out the right answer to the question on mass but we are no more so confident that should have the simple aspect given originally in the Standard Model. Indeed, this appears as an open door on a Pandora’s vase of new exciting physics.

But whatever will be the mechanism at work for the masses of leptons and quarks, the answer to the main question is not there. For one reason, both electrons and quarks that form protons and neutrons are really light and do not count too much on the determination of our mass. Most of the mass is in the nuclei and we have to understand where such mass comes from. This arises from bound states of quarks glued together in some way as should yield QCD at low energies. This gives you an idea of why is so important to understand QCD at very low energy. In this way we would be able to answer a fundamental question philosophers discussed for so long time.

So, for our everyday life, it is not so relevant to comprehend the real mechanism that gives mass to elementary particles . What we need is to prove the existence of a mass gap in Yang-Mills theory and so the way bound states form in QCD. As you may know, this is not an easy task and involves a lot of talented people around the World that, with a lot of inventive, is trying to do such computations. So far, only computers succeeded in giving an answer and this is so good that we have the most important observed parameters precise to one percent. The hope is to have a technique to work out such computations analytically, as happens for weak coupled physics. I am deeply involved in such enterprise and I think that what will come out will have a large impact on our knowledge. I can only say: Stay tuned!

Yang-Mills theory paper gets published!

Exact solutions of quantum field theories are very rare and, normally, refer to toy models and pathological cases. Quite recently, I put on arxiv a pair of papers presenting exact solutions both of the Higgs sector of the Standard Model and the Yang-Mills theory made just of gluons. The former appeared a few month ago (see here) while the latter has been accepted for publication a few days ago (see here). I have updated the latter just today and the accepted version will appear on arxiv on 2 January next year.

What does it mean to solve exactly a quantum field theory? A quantum field theory is exactly solved when we know all its correlation functions. From them, thanks to LSZ reduction formula, we are able to compute whatever observable in principle being these cross sections or decay times. The shortest way to correlation functions are the Dyson-Schwinger equations. These equations form a set with the former equation depending on the higher order correlators and so, they are generally very difficult to solve. They were largely used in studies of Yang-Mills theory provided some truncation scheme is given or by numerical studies. Their exact solutions are generally not known and expected too difficult to find.

The problem can be faced when some solutions to the classical equations of motion of a theory are known. In this way there is a possibility to treat the Dyson-Schwinger set. Anyhow, before to enter into their treatment, it should be emphasized that in literature the Dyson-Schwinger equations where managed just in one way: Using their integral form and expressing all the correlation functions by momenta. It was an original view by Carl Bender that opened up the way (see here). The idea is to write the Dyson-Schwinger equations into their differential form in the coordinate space. So, when you have exact solutions of the classical theory, a possibility opens up to treat also the quantum case!

This shows unequivocally that a Yang-Mills theory can display a mass gap and an infinite spectrum of excitations. Of course, if nature would have chosen the particular ground state depicted by such classical solutions we would have made bingo. This is a possibility but the proof is strongly related to what is going on for the Higgs sector of the Standard Model that I solved exactly but without other matter interacting. If the decay rates of the Higgs particle should agree with our computations we will be on the right track also for Yang-Mills theory. Nature tends to repeat working mechanisms.

Marco Frasca (2015). A theorem on the Higgs sector of the Standard Model Eur. Phys. J. Plus (2016) 131: 199 arXiv: 1504.02299v3

Marco Frasca (2015). Quantum Yang-Mills field theory arXiv arXiv: 1509.05292v1

Carl M. Bender, Kimball A. Milton, & Van M. Savage (1999). Solution of Schwinger-Dyson Equations for ${\cal PT}$-Symmetric Quantum Field Theory Phys.Rev.D62:085001,2000 arXiv: hep-th/9907045v1