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We determine the decay rate to leptons of the ground-state $\Upsilon$ mesonand its first radial excitation in lattice QCD for the first time. We useradiatively-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 goodagreement with experiment. The decay constants we obtain are included in asummary plot of meson decay constants from lattice QCD given in theConclusions. We also test time-moments of the vector current-current correlatoragainst values determined from the $b$ quark contribution to $\sigma(e^+e^-\rightarrow \mathrm{hadrons})$ and calculate the $b$-quark piece of thehadronic vacuum polarisation contribution to the anomalous magnetic moment ofthe 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 accurateresult from lattice QCD to date.

The $\Upsilon$ and $\Upsilon^{\prime}$ Leptonic Widths, $a_{\mu}^b$ and $m_b$ from full lattice QCD