Charmonium properties from lattice QCD + QED: hyperfine splitting, $J\mathrm{/}\psi$ leptonic width, charm quark mass and $a_{\mu }^c$

We have performed the first $n_f=2+1+1$ lattice QCD computations of the properties (masses and decay constants) of ground-state charmonium mesons. Our calculation uses the HISQ action to generate quark-line connected two-point correlation functions on MILC gluon field configurations that include $u\mathrm{/}d$ quark masses going down to the physical point, tuning the $c$ quark mass from $M_{J\mathrm{/}\psi }$ and including the effect of the $c$ quark’s electric charge through quenched QED. We obtain $M_{J\mathrm{/}\psi }-M_{{\eta }_c}$ (connected) = 120.3(1.1) MeV and interpret the difference with experiment as the impact on $M_{{\eta }_c}$ of its decay to gluons, missing from the lattice calculation. This allows us to determine $\mathrm{\Delta }{M}_{{\eta }_c}^{\mathrm{a}\mathrm{n}\mathrm{n}\mathrm{i}\mathrm{h}\mathrm{i}\mathrm{l}\mathrm{n}}$ =+7.3(1.2) MeV, giving its value for the first time. Our result of $f_{J\mathrm{/}\psi }=$ 0.4104(17) GeV, gives $\mathrm{\Gamma }(J\mathrm{/}\psi \to e^{+}{e}^{-})$=5.637(49) keV, in agreement with, but now more accurate than experiment. At the same time we have improved the determination of the $c$ quark mass, including the impact of quenched QED to give ${\bar{m}}_c(3\mathrm{G}\mathrm{e}\mathrm{V})$ = 0.9841(51) GeV. We have also used the time-moments of the vector charmonium current-current correlators to improve the lattice QCD result for the $c$ quark HVP contribution to the anomalous magnetic moment of the muon. We obtain $a_{\mu}^c = 14.638(47) \times 10^{-10}$,whichis2.5$\sigma$ higher than the value derived using moments extracted from some sets of experimental data on $R(e^+e^- \rightarrow $\mathrm{h}\mathrm{a}\mathrm{d}\mathrm{r}\mathrm{o}\mathrm{n}\mathrm{s}$)$\mathrm{.}Thisvaluefor$a_{\mu}^c$ includes our determination of the effect of QED on this quantity, $\delta a_{\mu}^c = 0.0313(28) \times 10^{-10}$.