Well below 1%



When a theory is too hard to solve people try to consider lower dimensional cases. This also happened for Yang-Mills theory. The four dimensional case is notoriously difficult to manage due to the large coupling and the three dimensional case has been treated both theoretically and by lattice computations. In this latter case, the ground state energy of the theory is known very precisely (see here). So, a sound theoretical approach from first principles should be able to get that number at the same level of precision. We know that this is the situation for Standard Model with respect to some experimental results but a pure Yang-Mills theory has not been seen in nature and we have to content ourselves with computer data. The reason is that a Yang-Mills theory is realized in nature just in interaction with other kind of fields being these scalars, fermions or vector-like.

In these days, I have received the news that my paper on three dimensional Yang-Mills theory has been accepted for publication in the European Physical Journal C. Here is tha table for the ground state for SU(N) at different values of N compared to lattice data

N Lattice     Theoretical Error

2 4.7367(55) 4.744262871 0.16%

3 4.3683(73) 4.357883714 0.2%

4 4.242(9)     4.243397712 0.03%

4.116(6)    4.108652166 0.18%

These results are strikingly good and the agreement is well below 1%. This in turn implies that the underlying theoretical derivation is sound. Besides, the approach proves to be successful both also in four dimensions (see here). My hope is that this means the beginning of the era of high precision theoretical computations in strong interactions.

Andreas Athenodorou, & Michael Teper (2017). SU(N) gauge theories in 2+1 dimensions: glueball spectra and k-string tensions J. High Energ. Phys. (2017) 2017: 15 arXiv: 1609.03873v1

Marco Frasca (2016). Confinement in a three-dimensional Yang-Mills theory arXiv arXiv: 1611.08182v2

Marco Frasca (2015). Quantum Yang-Mills field theory Eur. Phys. J. Plus (2017) 132: 38 arXiv: 1509.05292v2

Ground state of charmonium


After the satisfactory derivation of the bottomonium ground state mass (see here) we would like to apply similar concepts to charmonium. Before we go on I would like to mention here the beautiful paper by Nora Brambilla and a lot of other contributors that any serious researcher in the field of heavy quark physics should read (see here). This paper has been published as a yellow report by CERN. What we want to prove here is that the knowledge of the gluon propagator can give a nice understanding of the ground state of quarkonia. Anyhow, for charmonium we could not be that lucky as relativistic effects are more important here than for bottomonium. Besides, if we would like to expand to higher order in r the quark potential we would be no more able to treat the Schroedinger equation unless we treat these terms as a perturbation but this approach is not successful giving at best slowing convergence of the series for bottomonium and an useless result for charmonium.

PDG gives us the data for the ground state of charmonium \eta_c(1S):

m_{\eta_c}=2979.8\pm 1.2 MeV

m_c=1.25\pm 0.09 GeV (\bar{MS} scheme)


and then, our computation gives

m_{\eta_c}=2m_c-\frac{1}{4}\alpha_s^2m_c+0.876\alpha_s\sqrt{\sigma}\approx 2602.8 MeV

that has an error of about 13%. With a quark mass of 1.44 GeV we would get a perfect agreement with \eta_c(1S) mass that makes this computation quite striking together with the analogous computations for the ground state of the bottomonium.

As said at the start, heavy quarkonia are a well studied matter and whoever interested to deepen the argument should read the yellow report by Brambilla and others.

Update:I would like to point out the paper by Stephan Narison (see here and here) that obtains the pole masses of c and b quarks being these the ones I use to obtain the right ground state of charmonium and bottomonium. Striking indeed!

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