Hyperfine splitting of ground-state charmonium

The mass splitting of $\psi \mathrm{}\left(3100\right)\mathrm{}$ and its pseudoscalar partner ${\eta }_c$ is examined within the context of nonrelativistic linear-potential models for bound charmed quarks. If the Coulomb-type short-range gluon interaction is the sole contribution to this hyperfine splitting, then a mass splitting of 15-25 MeV is obtained. However, if the long-range forces that confine quarks also produce shorter-range spin-spin interactions, then the $\psi -{\eta }_c$ mass splitting is predicted to be in the range 40-80 MeV. If one abandons the assignment of ${\psi }^{\prime }\mathrm{}\left(3700\right)\mathrm{}$ as the first radial excitation of $\psi \mathrm{}\left(3100\right)\mathrm{}$, then the ${\eta }_c$ is predicted to lie 150-300 MeV below $\psi \mathrm{}\left(3100\right)\mathrm{}$. This possibility has the consequence that the rate predicted for the yet-unobserved decay $\psi \to {\eta }_c+\gamma$ is uncomfortably large.