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.