## The question of the mass gap

10/09/2014

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

## Nailing down the Yang-Mills problem

22/02/2014

Millennium problems represent a major challenge for physicists and mathematicians. So far, the only one that has been solved was the Poincaré conjecture (now a theorem) by Grisha Perelman. For people working in strong interactions and quantum chromodynamics, the most interesting of such problems is the Yang-Mills mass gap and existence problem. The solutions of this problem would imply a lot of consequences in physics and one of the most important of these is a deep understanding of confinement of quarks inside hadrons. So far, there seems to be no solution to it but things do not stay exactly in this way. A significant number of researchers has performed lattice computations to obtain the propagators of the theory in the full range of energy from infrared to ultraviolet providing us a deep understanding of what is going on here (see Yang-Mills article on Wikipedia). The propagators to be considered are those for  the gluon and the ghost. There has been a significant effort from theoretical physicists in the last twenty years to answer this question. It is not so widely known in the community but it should because the work of this people could be the starting point for a great innovation in physics. In these days, on arxiv a paper by Axel Maas gives a great recount of the situation of these lattice computations (see here). Axel has been an important contributor to this research area and the current understanding of the behavior of the Yang-Mills theory in two dimensions owes a lot to him. In this paper, Axel presents his computations on large volumes for Yang-Mills theory on the lattice in 2, 3 and 4 dimensions in the SU(2) case. These computations are generally performed in the Landau gauge (propagators are gauge dependent quantities) being the most favorable for them. In four dimensions the lattice is $(6\ fm)^4$, not the largest but surely enough for the aims of the paper. Of course, no surprise comes out with respect what people found starting from 2007. The scenario is well settled and is this:

1. The gluon propagator in 3 and 4 dimensions dos not go to zero with momenta but is just finite. In 3 dimensions has a maximum in the infrared reaching its finite value at 0  from below. No such maximum is seen in 4 dimensions. In 2 dimensions the gluon propagator goes to zero with momenta.
2. The ghost propagator behaves like the one of a free massless particle as the momenta are lowered. This is the dominant behavior in 3 and 4 dimensions. In 2 dimensions the ghost propagator is enhanced and goes to infinity faster than in 3 and 4 dimensions.
3. The running coupling in 3 and 4 dimensions is seen to reach zero as the momenta go to zero, reach a maximum at intermediate energies and goes asymptotically to 0 as momenta go to infinity (asymptotic freedom).

Here follows the figure for the gluon propagator

and for the running coupling

There is some concern for people about the running coupling. There is a recurring prejudice in Yang-Mills theory, without any support both theoretical or experimental, that the theory should be not trivial in the infrared. So, the running coupling should not go to zero lowering momenta but reach a finite non-zero value. Of course, a pure Yang-Mills theory in nature does not exist and it is very difficult to get an understanding here. But, in 2 and 3 dimensions, the point is that the gluon propagator is very similar to a free one, the ghost propagator is certainly a free one and then, using the duck test: If it looks like a duck, swims like a duck, and quacks like a duck, then it probably is a duck, the theory is really trivial also in the infrared limit. Currently, there are two people in the World that have recognized a duck here:  Axel Weber (see here and here) using renormalization group and me (see here, here and here). Now, claiming to see a duck where all others are pretending to tell a dinosaur does not make you the most popular guy  in the district. But so it goes.

These lattice computations are an important cornerstone in the search for the behavior of a Yang-Mills theory. Whoever aims to present to the World his petty theory for the solution of the Millennium prize must comply with these results showing that his theory is able to reproduce them. Otherwise what he has is just rubbish.

What appears in the sight is also the proof of existence of the theory. Having two trivial fixed points, the theory is Gaussian in these limits exactly as the scalar field theory. A Gaussian theory is the simplest example we know of a quantum field theory that is proven to exist. Could one recover the missing part between the two trivial fixed points as also happens for the scalar theory? In the end, it is possible that a Yang-Mills theory is just the vectorial counterpart of the well-known scalar field, the workhorse of all the scholars in quantum field theory.

Axel Maas (2014). Some more details of minimal-Landau-gauge Yang-Mills propagators arXiv arXiv: 1402.5050v1

Axel Weber (2012). Epsilon expansion for infrared Yang-Mills theory in Landau gauge Phys. Rev. D 85, 125005 arXiv: 1112.1157v2

Axel Weber (2012). The infrared fixed point of Landau gauge Yang-Mills theory arXiv arXiv: 1211.1473v1

Marco Frasca (2007). Infrared Gluon and Ghost Propagators Phys.Lett.B670:73-77,2008 arXiv: 0709.2042v6

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

## Ending and consequences of Terry Tao’s criticism

21/09/2013

Summer days are gone and I am back to work. I thought that Terry Tao’s criticism to my work was finally settled and his intervention was a good one indeed. Of course, people just remember the criticism but not how the question evolved since then (it was 2009!). Terry’s point was that the mapping given here between the scalar field solutions and the Yang-Mills field in the classical limit cannot be exact as it is not granted that they represent an extreme for the Yang-Mills functional. In this way the conclusions given in the paper are not granted being based on this proof. The problem can be traced back to the gauge invariance of the Yang-Mills theory that is explicitly broken in this case.

Terry Tao, in a private communication, asked me to provide a paper, to be published on a refereed journal, that fixed the problem. In such a case the question would have been settled in a way or another. E.g., also a result disproving completely the mapping would have been good, disproving also my published paper.

This matter is rather curious as, if you fix the gauge to be Lorenz (Landau), the mapping is exact. But the possible gauge choices are infinite and so, there seems to be infinite cases where the mapping theorem appears to fail. The lucky case is that lattice computations are generally performed in Landau gauge and when you do quantum field theory a gauge must be chosen. So, is the mapping theorem really false or one can change it to fix it all?

In order to clarify this situation, I decided to solve the classical equations of the Yang-Mills theory perturbatively in the strong coupling limit. Please, note that today I am the only one in the World able to perform such a computation having completely invented the techniques to do perturbation theory when a perturbation is taken to go to infinity (sorry, no AdS/CFT here but I can surely support it). You will note that this is the opposite limit to standard perturbation theory when one is looking for a parameter that goes to zero. I succeeded in doing so and put a paper on arxiv (see here) that was finally published the same year, 2009.

The theorem changed in this way:

The mapping exists in the asymptotic limit of the coupling running to infinity (leading order), with the notable exception of the Lorenz (Landau) gauge where it is exact.

So, I sighed with relief. The reason was that the conclusions of my paper on propagators were correct. But these hold asymptotically in the limit of a strong coupling. This is just what one needs in the infrared limit where Yang-Mills theory becomes strongly coupled and this is the main reason to solve it on the lattice. I cited my work on Tao’s site, Dispersive Wiki. I am a contributor to this site. Terry Tao declared the question definitively settled with the mapping theorem holding asymptotically (see here).

In the end, we were both right. Tao’s criticism was deeply helpful while my conclusions on the propagators were correct. Indeed, my gluon propagator agrees perfectly well, in the infrared limit, with the data from the largest lattice used in computations so far  (see here)

As generally happens in these cases, the only fact that remains is the original criticism by a great mathematician (and Terry is) that invalidated my work (see here for a question on Physics Stackexchange). As you can see by the tenths of papers I published since then, my work stands and stands very well. Maybe, it would be time to ask the author.

Marco Frasca (2007). Infrared Gluon and Ghost Propagators Phys.Lett.B670:73-77,2008 arXiv: 0709.2042v6

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

Attilio Cucchieri, & Tereza Mendes (2007). What’s up with IR gluon and ghost propagators in Landau gauge? A puzzling answer from huge lattices PoS LAT2007:297,2007 arXiv: 0710.0412v1

## Large-N gauge theories on the lattice

22/10/2012

Today I have found on arXiv a very nice review about large-N gauge theories on the lattice (see here). The authors, Biagio Lucini and Marco Panero, are well-known experts on lattice gauge theories being this their main area of investigation. This review, to appear on Physics Report, gives a nice introduction to this approach to manage non-perturbative regimes in gauge theories. This is essential to understand the behavior of QCD, both at zero and finite temperatures, to catch the behavior of bound states commonly observed. Besides this, the question of confinement is an open problem yet. Indeed, a theoretical understanding is lacking and lattice computations, especially in the very simplifying limit of large number of colors N as devised in the ’70s by ‘t Hooft, can make the scenario clearer favoring a better analysis.

What is seen is that confinement is fully preserved, as one gets an exact linear increasing potential in the limit of N going to infinity, and also higher order corrections are obtained diminishing as N increases. They are able to estimate the string tension obtaining (Fig. 7 in their paper):

$\centering{\frac{\Lambda_{\bar{MS}}}{\sigma^\frac{1}{2}}\approx a+\frac{b}{N^2}}.$

This is a reference result for whoever aims to get a solution to the mass gap problem for a Yang-Mills theory as the string tension must be an output of such a result. The interquark potential has the form

$m(L)=\sigma L-\frac{\pi}{3L}+\ldots$

This ansatz agrees with numerical data to distances $3/\sqrt{\sigma}$! Two other fundamental results these authors cite for the four dimensional case is the glueball spectrum:

$\frac{m_{0^{++}}}{\sqrt{\sigma}}=3.28(8)+\frac{2.1(1.1)}{N^2},$
$\frac{m_{0^{++*}}}{\sqrt{\sigma}}=5.93(17)-\frac{2.7(2.0)}{N^2},$
$\frac{m_{2^{++}}}{\sqrt{\sigma}}=4.78(14)+\frac{0.3(1.7)}{N^2}.$

Again, these are reference values for the mass gap problem in a Yang-Mills theory. As my readers know, I was able to get them out from my computations (see here). More recently, I have also obtained higher order corrections and the linear rising potential (see here) with the string tension in a closed form very similar to the three-dimensional case. Finally, they give the critical temperature for the breaking of chiral symmetry. The result is

$\frac{T_c}{\sqrt{\sigma}}=0.5949(17)+\frac{0.458(18)}{N^2}.$

This result is rather interesting because the constant is about $\sqrt{3/\pi^2}$. This result has been obtained initially by Norberto Scoccola and Daniel Gómez Dumm (see here) and confirmed by me (see here). This result pertains a finite temperature theory and a mass gap analysis of Yang-Mills theory should recover it but here the question is somewhat more complex. I would add to these lattice results also the studies of propagators for a pure Yang-Mills theory in the Landau gauge, both at zero and finite temperatures. The scenario has reached a really significant level of maturity and it is time that some of the theoretical proposals put forward so far compare with it. I have just cited some of these works but the literature is now becoming increasingly vast with other really meaningful techniques beside the cited one.

As usual, I conclude this post on such a nice paper with the hope that maybe time is come to increase the level of awareness of the community about the theoretical achievements on the question of the mass gap in quantum field theories.

Biagio Lucini, & Marco Panero (2012). SU(N) gauge theories at large N arXiv arXiv: 1210.4997v1

Marco Frasca (2008). Yang-Mills Propagators and QCD Nuclear Physics B (Proc. Suppl.) 186 (2009) 260-263 arXiv: 0807.4299v2

Marco Frasca (2011). Beyond one-gluon exchange in the infrared limit of Yang-Mills theory arXiv arXiv: 1110.2297v4

D. Gomez Dumm, & N. N. Scoccola (2004). Characteristics of the chiral phase transition in nonlocal quark models Phys.Rev. C72 (2005) 014909 arXiv: hep-ph/0410262v2

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

## Yang-Mills scenario: Yet a confirmation

12/12/2011

While CERN is calming down rumors (see here), research activity on Yang-Mills theories keeps on going on.  A few days ago, a paper by Axel Weber appeared on arxiv  (see here). As my readers know, having discussed this at length, in these last years there has been a hot debate between the proponents of the so called “scaling solution” and the “decoupling solution” for the propagators and the running coupling of a pure Yang-Mills theory in the infrared limit. Scaling solution describes a scenario with the gluon propagator reaching zero with lowering momenta, a ghost propagator enhanced with respect to the tree level one and the running coupling reaching a finite non zero value in the same limit. Decoupling solution instead is given by a gluon propagator reaching a finite non-zero value at lower momenta, a ghost propagator behaving like the one of a free particle (tree level) and the running coupling going to zero in this limit.. It is quite easy to recognize in the decoupling solution all the chrisms of a trivial infrared fixed point for a pure Yang-Mills theory against common wisdom that pervaded the community for a lot of years. So, for some years, having lattice computations unable to tell which solution was the right one, scaling solution seemed the only one to be physically viable and almost accepted by a large part of the community.

Things started to change after the Lattice Conference in Regensburg on 2007 when some groups where able to display lattice computations on very huge volumes. The striking result was that lattice computations confirmed the decoupling solution against common wisdom. What was really shocking here is that the gluon becomes massive at the expenses of the BRST sysmmetry that seems now to acquire an even more relevant role in the understanding of Yang-Mills theory.

The idea of Axel Weber is to perform an $\epsilon$-expansion for the Yang-Mills Lagrangian with a massive term to fix the scale. The striking result he gets is that both the scaling and the decoupling solutions are there but the former is unstable with respect to the renormalization group flow in dimensions greater than 2. So, this computation confirms again the scenario that I and other authors were able to devise.

Today, we have reached a deep understanding of the infrared physics of a Yang-Mills field theory. Scientific community is urged to take a look to the work of these people that could accelerate progress in a large body of physics research.

Axel Weber (2011). Epsilon expansion for infrared Yang-Mills theory in Landau gauge arXiv arXiv: 1112.1157v1

Marco Frasca (2007). Infrared Gluon and Ghost Propagators Phys.Lett.B670:73-77,2008 arXiv: 0709.2042v6

## Yang-Mills mass gap scenario: Further confirmations

28/11/2011

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

$G(p)=\sum_{n=0}^\infty\frac{Z_n}{p^2-m_n^2+i\epsilon}$

being $Z_n$ some numbers and $m_n$ is given by the zeros of some Bessel functions. This last result seems quite different from mine that I get explicitly $m_n=(n+1/2)m_0$ but this is not so because, in the asymptotic regime, $J_k(x)\propto \cos(x-k\pi/2-\pi/4)/\sqrt{x}$ and zeros for the cosine go like $(n+1/2)\pi$ 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 $\Lambda\rightarrow\infty$, being $\Lambda$ 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

## A Millenium Problem issue

07/11/2011

As my readers know, a recurring question in this blog is the solution to the Millenium Problem on Yang-Mills theory. So far, we have heard no fuzz about this matter and the page at the Clay Institute is no more updated since 2004. But in these years, activity on this problem has been significant and my aim here is to take this to your attention. In these days, a revision to a paper by Alexander Dynin is appeared (see here). The main conclusion in this paper is theorem 3.1 on page 17 that makes a clear statement about the spectrum of Yang-Mills theory: This must go like the one of a harmonic oscillator at least. This agrees perfectly well with the conclusions in my paper published in Physics Letters B (see here). As I prove that the theory is trivial in the infrared limit, a result that could be inferred but it is not stated in the Dynin’s paper, this limit gives a theory that exists, being it free. Anyhow, Dynin claims a complete proof of existence and this must be true all the way down the infrared limit starting from the ultraviolet one. Having the theory two trivial fixed points, at high and low energies, in these limits we are certain that the theory must exist. This is so because the theory becomes free. This should make easier a proof of existence of the theory for all the energy range. In any case, it would appear rather strange if the theory would exist just in its limit cases and not otherwise. So, if Dynin’s proof is correct, this should be checked promptly as all this would represent a significant breakthrough in our current understanding of quantum field theory. Dynin’s paper does not provide neither an explicit mass spectrum nor a techniques to do computations in quantum field theory. These are given in my papers and, all in all, we have here a complete new mathematical setup to manage also strongly coupled quantum field theories. It is interesting to show here a clear evidence of this situation for Yang-Mills theory through  the following picture obtained from lattice computations for the running coupling (see here and Physics Letters B)

This picture gives a blatant evidence of the scenario I was able to obtain mathematically and that is consistent with the mathematical proof given in Dynin’s paper.

On the other side, the spectrum is the one of a harmonic oscillator at lower energies. This means that, at lower energies. a Yukawa propagator should fit the bill rather well and a plateau must be observed from lattice computations. This is indeed the real situation (see here for a recent review).  To support further this scenario, the ghost propagator is the one of a free particle in the same limit. This shows again that the theory is infrared free. So, there is a situation, both from a mathematical side and a physical one clearly showing an explicit solution to the Yang-Mills question and that should be addressed rapidly.

Of course, behind all this, there is a lot of work of very good people that moved our knowledge to the present point and that is cited in the papers I presented here. It is my view that, whatever would be any other contribution to this research area, the acquired scenario is the one I described above as strongly emerged from lattice computations. My hope is that this will become part of our knowledge in a reasonable time.

Alexander Dynin (2009). Energy-mass spectrum of Yang-Mills bosons is infinite and discrete arXiv arXiv: 0903.4727v4

Marco Frasca (2007). Infrared Gluon and Ghost Propagators Phys.Lett.B670:73-77,2008 arXiv: 0709.2042v6

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

Ph. Boucaud, J. P. Leroy, A. Le Yaouanc, J. Micheli, O. Péne, & J. Rodríguez-Quintero (2011). The Infrared Behaviour of the Pure Yang-Mills Green Functions arXiv arXiv: 1109.1936v1