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.
An interesting paper appeared today in arxiv by Alkofer, Huber and Schwenzer (see here). Reinhard Alkofer and Lorenz von Smekal are the proponents of an infrared solution of Yang-Mills theory in D=4 having the following properties
- Gluon propagator goes to zero at lower momenta
- Ghost propagator goes to infinity at lower momenta faster than the free propagator
- Running coupling reaches a fixed point at lower momenta
and this scenario disagrees with lattice evidence in D=4 but agrees with lattice in D=2 when the theory is trivial having no dynamics. After some years that other researchers were claiming that a different solution can be obtained by the same equations, that is Dyson-Schwinger equations, that indeed agrees with lattice computations, Alkofer’s group accepted this fact but with a lot of skepticism pointing out that this solution has several difficulties, last but not least it breaks BRST symmetry. The solution proposed by Alkofer and von Smekal by its side gives no mass gap whatsoever and no low energy spectrum to be compared neither with lattice nor with experiments to understand the current light unflavored meson spectrum. So, whoever is right we are in a damned situation that no meaningful computations can be carried out to get some real physical understanding. The new paper is again on this line with the authors proposing a perturbation approach to evaluate the vertexes of the theory in the infrared and obtaining again comforting agreement with their scenario.
I will avoid to enter into this neverending controversy about Dyson-Schwinger equations but rather I would ask a more fundamental question: Is it worthwhile an approach that only grants at best saving a phylosophical understanding of confinement without any real understanding of QCD? My view is that one should start from lattice data and try to understand the real mathematical form of the gluon propagator. Why does it resemble the Yukawa form so well? A Yukawa form grants a mass gap and this is elementary quantum field theory. This I would like to see explained. When a method is not satisfactory something must be changed. It is evident that solving Dyson-Schwinger equations requires some new mathematical approach as old views are just confusing this kind of research.
Today in arxiv appeared a work by Christian Fischer, Axel Maas and Jan Pawlowski (see here
). This work is relevant because, for the first time, three authors that defended functional method so strongly now acknowledge the very existence of another solution in the infrared for the Dyson-Schwinger equations for Yang-Mills theory. Indeed they properly cite the work of Boucaud et al. (see recent preprint
) and Aguilar, Binosi and Papavassiliou (see here
) that found such solution with Dyson-Schwinger equations. I would like to say that these authors find a scenario that agreed with mine and lattice computations (see here
and my contribution here
). This approach has the fault that does not give a closed form to the gluon propagator permitting computations or, at least, to prove the existence of a mass gap. This means that the utility of such understanding, while very important, has some limitations.
Fischer, Maas and Pawlowski paper contains an important new result that none considered before. They prove that the solution that agreed with the scenario seen on the lattice violates BRST invariance. The reason why this is a striking result is that this is what one expects if the particles in the theory acquire a mass. They correctly observe that the gluon acquires a kind of screening mass and cannot be considered a true massive particle. This is truly beautiful because we know that the true carriers of the strong force in the infrared are bounded states of gluons that should be better named glueballs. They also show, but this was an expected result, that the gluon propagator in the infrared has the same behavior independently on the number of colors.
I would like to conclude with praise to this beatiful work that I hope it will appear soon in the published literature. The change of point of view of these researchers has been of great moment.
As always I read the daily arxiv sends to me and I have found a beatiful work due to Alkofer and collaborators. An important reason to mention it too here is that it gives an important tool to work with that can be downloaded. This tool permits to obtain Dyson-Schwinger equations for any field theory. Dyson-Schwinger equations are a tower of equations giving all the correlators of a quantum field theory so, if you know how to truncate this tower you will be able to get a solution to a quantum field theory in some limit.
The paper is here. The link to download the tool for Mathematica version 6.0 and higher is here.
I hope to have some time to study it and try a conversion for Maple. Currently I was not able to test it as on my laptop I have an older version of Mathematica but is just few hours away from testing on my desktop.