The bottomonium spectrum up to $n = 3$ is studied within Non-Relativistic Quantum Chromodynamics up to N$^3$LO. We consider finite charm quark mass effects both in the QCD potential and the $\overline{ \mathrm{ MS}}$-pole mass relation up to third order in the $\Upsilon$-scheme counting. The $u = 1/2$ renormalon of the static potential is canceled by expressing the bottom quark pole mass in terms of the MSR mass. A careful investigation of scale variation reveals that, while $n = 1, 2$ states are well behaved within perturbation theory, $n = 3$ bound states are no longer reliable. We carry out our analysis in the $n_\ell = 3$ and $n_\ell = 4$ schemes and conclude that, as long as finite $m_c$ effects are smoothly incorporated in the MSR mass definition, the difference between the two schemes is rather small. Performing a fit to $b\bar{ b}$ bound states we find $\overline{ m}_b(\overline{ m}_b) = 4.216\pm 0.039$ GeV. We extend our analysis to the lowest lying charmonium states finding $\overline{ m}_c(\overline{ m}_c)=1.273 \pm 0.054$ GeV. Finally, we perform simultaneous fits for $\overline{ m}_b$ and $\alpha_s$ finding $\alpha_s^{ (n_f=5)}(m_Z)=0.1178\pm 0.0051$. Additionally, using a modified version of the MSR mass with lighter massive quarks we are able to predict the uncalculated $\mathcal{ O}(\alpha_s^4)$ virtual massive quark corrections to the relation between the $\overline{ \mathrm{ MS}}$ and pole masses.
Categories: hep-ph