We determine the decay rate to leptons of the ground-state $\mathrm{\Upsilon }$ meson and its first radial excitation in lattice QCD for the first time. We use radiatively improved nonrelativistic QCD 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 $\mathrm{\Gamma }(\mathrm{\Upsilon }\to e^{+}{e}^{-})=1.19(11)\mathrm{keV}$ and $\mathrm{\Gamma }({\mathrm{\Upsilon }}^{\prime }\to e^{+}{e}^{-})=0.69(9)\mathrm{keV}$, both in good agreement with experimental results. 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 $\sigma (e^{+}{e}^{-}\to \text{hadrons})$ and calculate the $b$-quark piece of the hadronic vacuum polarization 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 $\overline{\mathrm{MS}}$ scheme, ${\overline{m}}_b({\overline{m}}_b,n_f=5)=4.196(23)\mathrm{GeV}$, the most accurate result from lattice QCD to date.

• Received 26 August 2014

DOI:https://doi.org/10.1103/PhysRevD.91.074514