Charmonium andϒsystems with small hyperfine splittings

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 $\frac{J}{\psi }$ and ${\psi }^{\prime }$. The effective coupling constant ${\alpha }_s$ 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 $\psi$ and ${\psi }^{\prime }$ are predicted to be ${\eta }_c\mathrm{}\left(3020\right)\mathrm{}$ and ${{\eta }_c}^{\prime }\mathrm{}\left(3638\right)\mathrm{}$ and are not to be identified with the experimentally uncertain $X\left(2830\right)\mathrm{}$ and $\chi \mathrm{}\left(3450\right)\mathrm{}$. The same effective potential, chainging only the constituent quark mass, gives a good description of the $\Upsilon$ system. The spin-independent relativistic corrections play an important role in obtaining the equality $m_{{\Upsilon }^{\prime }}-m_{\Upsilon }\approx m_{{\psi }^{\prime }}-m_{\psi }$.