Rigorous QCD evaluation of spectrum and ground state properties of heavy $q\overline{q}$ systems, with a precision determination of $m_b$, $M\left({\eta }_b\right)$

We present an evaluation of heavy quarkonium states $b\overline{b}$, $c\overline{c}$ from first principles. We use tree-level QCD (including relativistic corrections) and the full one-loop potential; nonperturbative effects are taken into account at the leading order through the contribution of the gluon condensate $〈{\alpha }_s{G}^2〉$. We use the values $\Lambda (2\mathrm{}\mathrm{loops},4\mathrm{}\mathrm{flavors})={200}_{-60\mathrm{}}^{+80\mathrm{}}$ MeV, $〈{\alpha }_s{G}^2〉=0.042\mathrm{}\pm 0.020{\mathrm{GeV}}^4$, but we trade the value of the quark mass with the masses of $\frac{J}{\psi }$, $\Upsilon$ as input. We get good agreement in what is essentially a zero parameter evaluation for the masses of the $1S$, $2S$, $2P$ states of $b\overline{b}$, the $1S$ state for $c\overline{c}$, and the decay $\Upsilon \to e^{+}{e}^{-}$. As outstanding results we obtain the precise determination of $m_b$ as well as an estimate of the hyperfine splitting $M\left(\Upsilon \right)-M\left({\eta }_b\right):{\overline{m}}_b\left({\overline{m}}_b^2\right)={4397}_{-2\mathrm{}}^{+7\mathrm{}}{\left(\Lambda \right)}_{+4\mathrm{}}^{-3\mathrm{}}{\left(〈{\alpha }_s{G}^2〉\right)}_{-32\mathrm{}}^{+16\mathrm{}}\left(\mathrm{syst}\right)$ MeV, $M\left(\Upsilon \right)-M\left({\eta }_b\right)={36}_{-7\mathrm{}}^{+13\mathrm{}}{\left(\Lambda \right)}_{-6\mathrm{}}^{+3\mathrm{}}{\left(〈{\alpha }_s{G}^2〉\right)}_{-5\mathrm{}}^{+11\mathrm{}}\left(\mathrm{syst}\right)$ MeV, the first error due to that in $\Lambda$, the second to that in $〈{\alpha }_s{G}^2〉$ (varied independently). For the $c$ quark, we find ${\overline{m}}_c\left({\overline{m}}_c^2\right)={1306}_{-34\mathrm{}}^{+21\mathrm{}}{\left(\Lambda \right)}_{+6\mathrm{}}^{-6\mathrm{}}\left(〈{\alpha }_s{G}^2〉\right)$ MeV (up to systematic errors). The $\overline{\mathrm{MS}} b$, $c$ masses correspond to pole mass values of $m_b\left(\mathrm{pole}\right)={4906}_{-51\mathrm{}}^{+69\mathrm{}}{\left(\Lambda \right)}_{+4\mathrm{}}^{-4\mathrm{}}{\left(〈{\alpha }_s{G}^2〉\right)}_{-40\mathrm{}}^{+11\mathrm{}}\left(\mathrm{syst}\right)$ MeV and $m_c\left(\mathrm{pole}\right)={1570}_{-19\mathrm{}}^{+19\mathrm{}}{\left(\Lambda \right)}_{+7\mathrm{}}^{-7\mathrm{}}\left(〈{\alpha }_s{G}^2〉\right)$ MeV.