## 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

## 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