Quarkonium spectroscopy and perturbative QCD: Massive quark-loop effects

We study the spectra of the bottomonium and $B_c$ states within perturbative QCD up to order ${\alpha }_{\mathrm{s}}^4.$ The $\mathcal{O}\left({\Lambda }_{\mathrm{QCD}}\right)$ renormalon cancellation between the static potential and the pole mass is performed in the ɛ-expansion scheme. We extend our previous analysis by including the (dominant) effects of nonzero charm-quark mass in loops up to the next-to-leading nonvanishing order ${\varepsilon }^3.$ We fix the b-quark $\overline{\mathrm{MS}}$ mass $m_b\equiv m_b^{\overline{\mathrm{MS}}}{(m}_b^{\overline{\mathrm{MS}}})$ on ${\rm Y}\mathrm{}\left(1S\right)\mathrm{}$ and compute the higher levels. The effect of the charm mass decreases $m_b$ by about 11 MeV and increases the $n=2\mathrm{}$ and $n=3\mathrm{}$ levels by about 70–100 MeV and 240–280 MeV, respectively. We provide an extensive quantitative analysis. The size of nonperturbative and higher order contributions is discussed by comparing the obtained predictions with the experimental data. An agreement of the perturbative predictions and the experimental data depends crucially on the precise value (inside the present error) of ${\alpha }_{\mathrm{s}}{(M}_Z).\mathrm{}$ We obtain $m_b^{\overline{\mathrm{MS}}}{(m}_b^{\overline{\mathrm{MS}}})=4190\mathrm{}\pm 20\pm 25\pm 3\mathrm{MeV}.$