Nailing down the Yang-Mills problem 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 Gluon Propagators

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

9 Responses to Nailing down the Yang-Mills problem

  1. ohwilleke says:

    Off topic: Is there any chance you could comment on ?

    The paper is very understated, but the discrepancies between the pQCD predictions and the experimentally measured hadronic decay cross sections at Belle from high enegy electron-positron collisions is very extreme (in some cases at the ten sigma level), which is not something that I see every day trolling hep at arxiv, and the bare bones four page paper makes almost no effort to analyze why this might be the case. But, on the surface, there doesn’t seem to be any good reason, for example, to think that Belle has grossly understated systemic error in how it conducted these experiments.

    The paper shows a better match of a model with SU(3) with broken flavor symmetry than it does in the case of a model with SU(3) with perfect flavor symmetry, but isn’t even very close to the predictions of either model.

    Do you have any insight into what could be going on in the theoretical calculations that could cause it to be so grossly off at some energies for some decays, while being right on the money for many other decays at certain energies?

    • mfrasca says:

      Dear ohwilleke,

      My take here is that to perform these theoretical computations is not that easy,even if you are in a regime where asymptotic freedom sets in and perturbation theory is reliable. I think that a better insight could come from lattice QCD but I cannot evaluate how much could be helpful at this stage.


  2. Alexander Dynin says:

    Theorem 4.1 of my paper “Quantum Yang-Mills-Weyl dynamics in the Schroedinger paradigm”, Russian Journal of Mathematical Physics,21(2014),No.2,169-188) (arXiv:1005.3779v.3 [math-phys]) claims the relative ellipticity of the infrared Yang-Mills quantum energy-mass operators in von Neumann algebras with regular traces.
    This implies that the spectra of infrared self-adjoint Yang-Mills energy-mass operators in a non-perturbative quantum Yang-Mills theory (with an arbitrary compact simple gauge Lie group) are non-negative sequences of the eigenvalues converging to $+\infty$. The spectra are self-similar in the inverse proportion to the running coupling constant. In particular, they have positive spectral mass gaps. Presumably, this is a solution of the Yang-Mills Millennium problem.

    • mfrasca says:

      Dear Alexander,

      I am pleased that your work finally got published. Just a question as a physicist: Are you able to get an explicit formula for the propagator and the spectrum in the low energy limit? Also the running coupling at lower energies would be interesting to see. The reason for this question is that, if we want to do calculations with QCD, we need these formulas to work with. This was the main motivation behind my post.

      Technically, you seem to have answered the Millenium problem but there is nothing to start to work with. This does not imply that in some limit these expressions could never be obtained by your approach. Just put them out.


      • Alexander Dynin says:

        Dear Marco,

        Actually my mathematical proof is more than just mass gap existence. It involves what I dubbed quantized Galerkin approximations of functional differential operators by partial differential operators with increasing number of variables.
        I would suggest that this might be used for computations.

        On the other hand I am not able to to answer your concerns.

        The bottom line is that my theory is non-perturbative and demonstrates that the YM mass gap is a quantum phenomenon,
        not a (semi)classical one. As I see it, this is the main difference between us.


        • mfrasca says:

          Also for me is a quantum phenomenon but I can also display classical solution in the strong coupling limit with a mass. I have analytical formulas to work with and I can do my work as a physicist. That’s all.


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