# If a $J/\psi$ decays to an electron-positron pair 5% of the time, how often would a $\phi$ meson decay to a electron-positron pair?

I know the mass of $$J/\psi$$ to be 3097 mev and the mass of phi to be 1018 mev. I know that $$J/\psi$$ decays to electron and positron 5% of the time. I also know the full width of j/psi to be 0.092mev and that the phi meson lives 50 times longer than $$J/\psi$$. My professor claims that if I am given that info and can draw the feynman diagrams for both interactions then it should be possible to make an estimate on how often the phi meson decays to electron-positron pair too. But I dont understand how this can done.

I have drawn both diagrams (c-cbar or s-sbar to electron and positron with photon boson between). I also know the following formulas:

total width =  \hbar / \tau


where tau is the decay time of the particle. I also know that the branching fraction is given as:

BF = partial width / total width


Im not sure how one could estimate the branching fraction of phi to electron-positron pair.

This is the mate of your previous question, with a small difference: the virtual photon producing the $$e^+ e^-$$ pair couples twice as strongly to $$c\bar c$$ as to $$s\bar s$$, so, squaring the diagram, $$\Gamma( \phi\to e^+ e^-) = \Gamma( \psi\to e^+ e^-)/4,$$ while (the BF is closer to 6% than to 5%) $$\Gamma( \psi\to e^+ e^-)\approx 0.06 ~\Gamma_{\psi ~total} \approx 5.6 ~\hbox {keV}.$$
Hence, $$\Gamma( \phi\to e^+ e^-)\approx 1.4 ~\hbox{ keV},$$ compared to the PDG value of 1.3 keV.
The corresponding BF is about 3 $$\cdot 10^{-4}$$. Your professor claims well.