Bottom Quark Mass from $\Upsilon$ Mesons: Charm Mass Effects

The effects of the finite charm quark mass on bottom quark mass determinations from $\Upsilon$ sum rules are examined in detail. The charm quark mass effects are calculated at next-to-next-to-leading order in the non-relativistic power counting for the $\Upsilon$ sum rules and at order $\alpha_s^3$ for the determination of the bottom MSbar mass. For the bottom 1S mass, which is extracted from the $\Upsilon$ sum rules directly, we obtain $M_b^{1S}=4.69\pm 0.03$ GeV with negligible correlation to the value of the strong coupling. For the bottom MSbar mass we obtain $\bar M_b(\bar M_b) = 4.17\pm 0.05$ GeV taking $\alpha_s^{(n_l=5)}(M_Z)=0.118\pm 0.003$ as an input. Compared with an analysis where all quarks lighter than the bottom are treated as massless, we find that the finite charm mass shifts the bottom 1S mass, $M_b^{1S}$, by about -20 MeV and the MSbar mass, $\bar M_b(\bar M_b)$, by -30 to -35 MeV.