Radiativeη′′(2.8)decay and the charmed-quark content ofωandφ

Abstract

We propose a sum rule connecting radiative decay modes of ${\ensuremath{\eta}}^{\ensuremath{'}\ensuremath{'}}$ (the charmed partner of $\ensuremath{\eta}$ and ${\ensuremath{\eta}}^{\ensuremath{'}}$, and presumably the $2\ensuremath{\gamma}$ resonance at 2.8 GeV). The sum rule should allow a determination of the percentage of charmed quarks in the $\ensuremath{\omega}$ and $\ensuremath{\varphi}$ mesons. If the percentage of charmed quarks in the $\ensuremath{\eta}$ and ${\ensuremath{\eta}}^{\ensuremath{'}}$ is large enough, the percentage of charmed quarks in the $\ensuremath{\omega}$ and $\ensuremath{\varphi}$ can also be determined by study of the decay modes $\ensuremath{\psi}\ensuremath{\rightarrow}(\ensuremath{\omega} or \ensuremath{\varphi})+(\ensuremath{\eta} or {\ensuremath{\eta}}^{\ensuremath{'}})$; this possibility is discussed. A phenomenological analysis suggests that the familiar octuplet-singlet pseudoscalar mixing angle ${\ensuremath{\theta}}_{P}$ for $\ensuremath{\eta}$ and ${\ensuremath{\eta}}^{\ensuremath{'}}$ varies from -8\ifmmode^\circ\else\textdegree\fi{} at the $\ensuremath{\eta}$ energy to approximately -38\ifmmode^\circ\else\textdegree\fi{} at the ${\ensuremath{\eta}}^{\ensuremath{'}}$ energy. The latter angle yields predictions $\ensuremath{\Gamma}({\ensuremath{\eta}}^{\ensuremath{'}}\ensuremath{\rightarrow}\mathrm{all})=0.11$ MeV and $\frac{\ensuremath{\Gamma}({A}_{2}\ensuremath{\rightarrow}{\ensuremath{\eta}}^{\ensuremath{'}}\ensuremath{\pi})}{\ensuremath{\Gamma}({A}_{2}\ensuremath{\rightarrow}\mathrm{all})}=0.12%$; both predictions are smaller than those obtained from either the quadratic or linear mass formula values for ${\ensuremath{\theta}}_{P}$. Finally, a relation $\frac{\ensuremath{\Gamma}(\ensuremath{\psi}\ensuremath{\leftarrow}\ensuremath{\eta}\ensuremath{\gamma})}{\ensuremath{\Gamma}(\ensuremath{\psi}\ensuremath{\rightarrow}{\ensuremath{\eta}}^{\ensuremath{'}}\ensuremath{\gamma})}=(\mathrm{p}\mathrm{h}\mathrm{a}\mathrm{s}\mathrm{e}\ensuremath{-}\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{e}\phantom{\rule{0ex}{0ex}}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s})\ifmmode\times\else\texttimes\fi{}\frac{\ensuremath{\Gamma}({\ensuremath{\psi}}^{\ensuremath{'}}\ensuremath{\rightarrow}\ensuremath{\eta}\ensuremath{\gamma})}{\ensuremath{\Gamma}({\ensuremath{\psi}}^{\ensuremath{'}}\ensuremath{\rightarrow}{\ensuremath{\eta}}^{\ensuremath{'}}\ensuremath{\gamma})}$ is shown to follow from the assumption that the $c\overline{c}$ content of $\ensuremath{\psi}$ and ${\ensuremath{\psi}}^{\ensuremath{'}}(3684)$ is negligible compared to the $c\overline{c}$ content of $\ensuremath{\eta}$ and ${\ensuremath{\eta}}^{\ensuremath{'}}$.