Charmonium and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>ϒ</mml:mi></mml:math>systems with small hyperfine splittings

Abstract

The parameters of the simple Coulomb plus linear effective potential with lowest-order relativistic corrections are determined by the charmonium states other than the $^{1}S_{0}$ states and by the leptonic decay widths of $\frac{J}{\ensuremath{\psi}}$ and ${\ensuremath{\psi}}^{\ensuremath{'}}$. The effective coupling constant ${\ensuremath{\alpha}}_{s}$ is found to be 0.21. Within this framework the hyperfine splittings are small and so are the consequent $M1$ transitions (well below the experimental upper limit of 1.2 keV). The $^{1}S_{0}$ partners of $\ensuremath{\psi}$ and ${\ensuremath{\psi}}^{\ensuremath{'}}$ are predicted to be ${\ensuremath{\eta}}_{c}(3020)$ and ${{\ensuremath{\eta}}_{c}}^{\ensuremath{'}}(3638)$ and are not to be identified with the experimentally uncertain $X(2830)$ and $\ensuremath{\chi}(3450)$. The same effective potential, chainging only the constituent quark mass, gives a good description of the $\ensuremath{\Upsilon}$ system. The spin-independent relativistic corrections play an important role in obtaining the equality ${m}_{{\ensuremath{\Upsilon}}^{\ensuremath{'}}}\ensuremath{-}{m}_{\ensuremath{\Upsilon}}\ensuremath{\approx}{m}_{{\ensuremath{\psi}}^{\ensuremath{'}}}\ensuremath{-}{m}_{\ensuremath{\psi}}$.