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We calculate the mass difference between the $\Upsilon$ and $\eta_b$ and the$\Upsilon$ leptonic width from lattice QCD using the Highly Improved StaggeredQuark formalism for the $b$ quark and including $u$, $d$, $s$ and $c$ quarks inthe sea. We have results for lattices with lattice spacing as low as 0.03 fmand multiple heavy quark masses, enabling us to map out the heavy quark massdependence and determine values at the $b$ quark mass. Our results are:$M_{\Upsilon} -M_{\eta_b} = 57.5(2.3)(1.0) \,\mathrm{MeV}$ (where the seconduncertainty comes from neglect of quark-line disconnected correlationfunctions) and decay constants, $f_{\eta_b}=724(12)$ MeV and $f_{\Upsilon}=677.2(9.7)$ MeV, giving $\Gamma(\Upsilon \rightarrow e^+e^-) = 1.292(37)(3)\,\mathrm{keV}$. The hyperfine splitting and leptonic width are both in goodagreement with experiment, and provide the most accurate lattice QCD results todate for these quantities by some margin. At the same time results for the timemoments of the vector-vector correlation function can be compared to values forthe $b$ quark contribution to $\sigma(e^+e^- \rightarrow \mathrm{hadrons})$determined from experiment. Moments 4--10 provide a 2\% test of QCD and yield a$b$ quark contribution to the anomalous magnetic moment of the muon of0.300(15)$\times 10^{-10}$. Our results, covering a range of heavy quarkmasses, may also be useful to constrain QCD-like composite theories for beyondthe Standard Model physics.

physical review. d/physical review. d.

Bottomonium precision tests from full lattice QCD: hyperfine splitting,  $\Upsilon$ leptonic width and $b$ quark contribution to $e^+e^- \rightarrow$  hadrons