The Decay $η_c \rightarrow γγ$ : A Test for Potential Models

Abstract

We use a simple perturbation theory argument and measurements of charmonium leptonic widths $Γ(ψ_{NS} \rightarrow e^+e^-)$ to estimate the ratio \mbox{$R_\circ \equiv {\vert Ψ_{η_{c1S}}(0) \vert}^2 /{\vertΨ_{ψ_{1 S}}(0)\vert}^2$} 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 $Γ(η_c \rightarrow γγ)/Γ(ψ_{1S} \rightarrow e^+e^-) \approx 2.2\pm 0.2$. The central value for $Γ(ψ_{1S} \rightarrow e^+ e^-)$in the 1992 Particle Data Tables then leads to a (non relativistic) prediction $Γ(η_c \rightarrow γγ)\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 $Γ(Υ_{1S}\rightarrow e^+e^-)$ we find $Γ(η_b\rightarrow γγ)\approx 0.58\pm 0.03$ keV. A rough estimate of relativistic corrections reduces the expected two photon rates to about 8.8 keV and 0.52 keV for the $η_c$ and $η_b$ mesons respectively. Such corrections