The Decay ${\eta }_c\to \gamma \gamma$ : A Test for Potential Models

We use a simple perturbation theory argument and measurements of charmonium leptonic widths $\mathrm{\Gamma }({\psi }_{NS}\to e^{+}{e}^{-})$ to estimate the ratio \mbox{$R_\circ \equiv {\vert \Psi {\eta{c1S}}(0) \vert}^2 /{\vert\Psi_{\psi_{1 S}}(0)\vert}^2$} in the general context of non- relativistic potential models. We obtain $R_{\circ }=1.4\pm 0.1$. We then apply well known potential model formulas, which include lowest order QCD corrections, to find $\Gamma (\eta_c \rightarrow \gamma \gamma )/\Gamma (\psi_{1S} \rightarrow e^+e^-) \approx 2.2\pm 0.2$\mathrm{.}Thecentralvaluefor$\Gamma (\psi_{1S} \rightarrow e^+ e^-)$in the 1992 Particle Data Tables then leads to a (non relativistic) prediction $\Gamma (\eta_c \rightarrow \gamma \gamma )\approx 11.8\pm 0.8 $ keV. This prediction is in good agreement with a recent measurement by the ARGUS collaboration, is consistent with a recent measurement by the L3 collaboration but is significantly higher than several earlier measurements and than previous theoretical estimates, which usually assume $R_{\circ }=1$. The correction to $R_{\circ }=1$ is estimated to be smaller but nonnegligible for the $b\bar{b}$ system. Using the current central measurement for $\mathrm{\Gamma }({\mathrm{{\rm Y}}}_{1S}\to e^{+}{e}^{-})$ we find $\Gamma (\eta_b\rightarrow \gamma \gamma )\approx 0.58\pm 0.03$ keV. A rough estimate of relativistic corrections reduces the expected two photon rates to about 8.8 keV and 0.52 keV for the ${\eta }_c$ and ${\eta }_b$ mesons respectively. Such corrections