Two-Photon Decays ofπ0,η0, andη′0(958)in BrokenSU(3)Symmetry

By using the current algebra and the asymptotic $\mathrm{SU}\left(3\right)\mathrm{}$ symmetry imposed only on the charge operator $V_K$, which is the $\mathrm{SU}\left(3\right)\mathrm{}$ raising or lowering operator in the symmetry limit, we obtain two sum rules for the two-photon decay amplitudes of the ${\pi }^0$, ${\eta }^0$, and ${\eta }^{\prime 0}\mathrm{}\left(958\right)\mathrm{}$. These sum rules, which are consistent with the Gell-Mann-Okubo mass splittings of the $\mathrm{SU}\left(3\right)\mathrm{}$ multiplets, exhibit an interesting interplay between the physical masses, mixing and coupling constants. For example, in the first approximation, we predict $R\equiv \frac{〈\gamma \gamma |{\eta }^0〉}{〈\gamma \gamma |{\pi }^0〉}=\left(\frac{1}{\sqrt{3}}\right)\left(\frac{1}{cos\alpha }\right)\left[\frac{(m_{{\eta }^{\prime }}^{}{}_{}{}^2-m_{\pi }^{}{}_{}{}^2)}{(m_{{\eta }^{\prime }}^{}{}_{}{}^2-m_{\eta }^{}{}_{}{}^2)}\right]\simeq \left(\frac{1}{\sqrt{3}}\right)\times 1.5$ in place of the $\mathrm{SU}\left(3\right)\mathrm{}$ prediction $R=\frac{1}{\sqrt{3}}$. Here $\alpha$ is the $\eta -{\eta }^{\prime }$ mixing angle. In an improved approximation, we obtain $R\simeq \left(\frac{1}{\sqrt{3}}\right)\times 1.7$. The result indicates a significantly larger width for the ${\eta }^0\to 2\gamma$ decay than the $\mathrm{SU}\left(3\right)\mathrm{}$ value—a conclusion that is consistent with the present experimental observation.