The $Z\to c\bar c\to\gamma\gamma^*$, $Z\to b\bar b\to\gamma\gamma^*$ triangle diagrams and the $Z\to\gamma\psi$, $Z\to\gamma\Upsilon$ decays

We expounded an approach for studying the $Z\to\gamma\psi$ and $Z\to \gamma\Upsilon$ decay based on the sum rules for the $Z\to c\bar c\to \gamma\gamma^*$ and $Z\to b\bar b\to\gamma\gamma^*$ amplitudes and their derivatives. We calculate the branching ratios of the $Z\to\gamma\psi$ and $Z\to\gamma\Upsilon$ decays under different suppositions about the saturation of the sum rules. We find the lower bounds of $\sum_{\psi}BR(Z\to\gamma\psi)= 1.95\cdot10^{-7}$ and $\sum_{\Upsilon}BR(Z\to\gamma\Upsilon)=7.23\cdot10^{-7}$ and discuss deviations from the lower bounds including the possibility of $BR[Z\to\gamma J/\psi(1S)]\sim BR[Z\to\gamma\Upsilon(1S)]\sim10^{-6}$, which is probably measurable at the LHC. Moreover, we calculate the angle distributions in the $Z\to\gamma\psi$ and $Z\to\gamma\Upsilon$ decays.