Charmonium: Comparison with experiment

The charmonium model, formulated in detail in an earlier publication, is compared in a comprehensive fashion with the data on the $\psi$ family. The parameters of the "naive" model, in which the system is described as a $c\overline{c}$ pair, are determined from the observed positions of $\psi$, ${\psi }^{\prime }$, and the $P$ states. The model then yields a successful description of the spectrum of spin-triplet states above the charm threshold. It also accounts for the ratio of the leptonic widths of ${\psi }^{\prime }$ and $\psi$. When the $c\overline{c}$ potential is applied to the $\Upsilon$ family, it accounts, without any readjustment of parameters, for the positions of the $2S$ and $3S$ levels and for the leptonic widths of $\Upsilon$ and ${\Upsilon }^{\prime }$ relative to that of $\psi$. The model does not give acceptable values of the absolute leptonic widths, a shortcoming which is ascribed to large quantum-chromodynamic corrections to the van Royen-Weisskopf formula. The calculated $E1$ rates are about twice the values observed in the $\psi$ family. This naive model is also extended with considerable success to mesons composed of one heavy and one light quark. A significant extension of the model is achieved by incorporating coupling to charmed-meson decay channels. This gives a satisfactory understanding of $\psi \mathrm{}\left(3772\right)\mathrm{}$ as the $1{}_{}{}^3{}_{}{}^{}D_1^{}{}_{}{}^{}$ $c\overline{c}$ state, mixed via open and closed decay channels to $2{}_{}{}^3{}_{}{}^{}S$. The model has decay amplitudes that are oscillatory functions of the decay momentum; these oscillations are a direct consequence of the radial nodes in the $c\overline{c}$ parent states. These amplitudes provide a qualitative understanding of the observed peculiar branching ratios into various charmed-meson channels near the resonance at 4.03 GeV, which is assigned to $3{}_{}{}^3{}_{}{}^{}S$. The coupling of the $c\overline{c}$ states below the charm threshold to closed decay channels modifies the bound states and leads to reduction of about 20% in $E1$ rates in comparison to those of the naive model.