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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 onprevious theory calculations to achieve accuracies of 1--2\%, giving latticeQCD results that are now more accurate than the experimental values. Inparticular our results transform the theoretical picture for$\eta_c\to\gamma\gamma$ decays. We use gluon field configurations generated bythe MILC collaboration that include $n_f=2+1+1$ flavours of Highly ImprovedStaggered (HISQ) sea quarks at four lattice spacing values from 0.15 fm to 0.06fm and with sea u/d masses down to their physical value. We also implement thevalence $c$ quarks using the HISQ action. We find ${\Gamma (\eta_c \to \gamma\gamma) = 6.788(45)_{\text{fit}}(41)_{\text{syst}} \: \mathrm{keV}}$, in goodagreement with experimental results using $\gamma\gamma \to \eta_c \toK\overline{K}\pi$ but in 4$\sigma$ tension with the Particle Data Group globalfit 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 Dalitzdecay rate $\Gamma (J/\psi \to \eta_c e^+ e^-) =0.01349(21)_{\text{latt}}(13)_{\text{QED}} \; \mathrm{keV}$. We use our resultsto calibrate other theoretical approaches and to test simple relationshipsbetween the form factors and $J/\psi$ decay constant expected in thenonrelativistic limit.

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