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 :
GeV ( scheme)
and then, our computation gives
that has an error of about 13%. With a quark mass of 1.44 GeV we would get a perfect agreement with 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!