Bottom and Charm Mass determinations from global fits to $Q\bar{Q}$ bound states at N${}^3$LO

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 $\overline{\mathrm{M}\mathrm{S}}$-pole mass relation up to third order in the $\mathrm{{\rm Y}}$-scheme counting. The $u=1\mathrm{/}2$ 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 $n_{\mathrm{\ell }}=3$ and $n_{\mathrm{\ell }}=4$ 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 ${\bar{m}}_c({\bar{m}}_c)=1.273\pm 0.054$ GeV. Finally, we perform simultaneous fits for ${\bar{m}}_b$ 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 $\overline{\mathrm{M}\mathrm{S}}$ and pole masses.