Precise determination of decay rates for $\eta_c \to \gamma \gamma$, $J/\psi \to \gamma \eta_c$ and $J/\psi \to \eta_c e^+e^-$ from lattice QCD

We calculate the decay rates for $\eta_c \to \gamma \gamma$, $J/\psi \to \gamma \eta_c$ and $J/\psi \to \eta_c e^+e^-$ in lattice QCD with $u$, $d$, $s$ and $c$ quarks in the sea for the first time. We improve significantly on previous theory calculations to achieve accuracies of 1--2\%, giving lattice QCD results that are now more accurate than the experimental values. In particular our results transform the theoretical picture for $\eta_c\to\gamma\gamma$ decays. We use gluon field configurations generated by the MILC collaboration that include $n_f=2+1+1$ flavours of Highly Improved Staggered (HISQ) sea quarks at four lattice spacing values from 0.15 fm to 0.06 fm and with sea u/d masses down to their physical value. We also implement the valence $c$ quarks using the HISQ action. We find ${\Gamma (\eta_c \to \gamma \gamma) = 6.788(45)_{\text{fit}}(41)_{\text{syst}} \: \mathrm{keV}}$, in good agreement with experimental results using $\gamma\gamma \to \eta_c \to K\overline{K}\pi$ but in 4$\sigma$ tension with the global Particle Data Group fit result; we suggest this fit is revisited. We also calculate $\Gamma (J/\psi \to \gamma \eta_c) = 2.219(17)_{\text{fit}}(18)_{\text{syst}}(24)_{\text{expt}}(4)_{\text{QED}} \; \mathrm{keV}$, in good agreement with results from CLEO, and predict the Dalitz decay rate $\Gamma (J/\psi \to \eta_c e^+ e^-) = 0.01349(21)_{\text{latt}}(13)_{\text{QED}} \; \mathrm{keV}$. We use our results to calibrate other theoretical approaches and to test simple relationships between the form factors and $J/\psi$ decay constant expected in the nonrelativistic limit.