We determine the decay rate to leptons of the ground-state $Upsilon$ meson and its first radial excitation in lattice QCD for the first time. We use radiatively-improved NRQCD for the $b$ quarks and include $u$, $d$, $s$ and $c$ quarks in the sea with $u/d$ masses down to their physical values. We find $Gamma(Upsilon rightarrow e^+e^-)$ = 1.19(11) keV and $Gamma(Upsilon^{prime} rightarrow e^+e^-)$ = 0.69(9) keV, both in good agreement with experiment. The decay constants we obtain are included in a summary plot of meson decay constants from lattice QCD given in the Conclusions. We also test time-moments of the vector current-current correlator against values determined from the $b$ quark contribution to $sigma(e^+e^- rightarrow mathrm{hadrons})$ and calculate the $b$-quark piece of the hadronic vacuum polarisation contribution to the anomalous magnetic moment of the muon, $a_{mu}^b = 0.271(37) times 10^{-10}$. Finally we determine the $b$-quark mass, obtaining in the $overline{MS}$ scheme, $overline{m}_b(overline{m}_b, n_f=5)$ = 4.196(23) GeV, the most accurate result from lattice QCD to date.
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