Precise Determination of the Bottom-Quark On-Shell Mass Using Its Four-Loop Relation to the \overline\rm MS-Scheme Running Mass

We explore the properties of the bottom-quark on-shell mass (M_b) by using its relation to the \overline\rm MS mass (\overline m_b). At present, this \overline\rm MS-on-shell relation has been known up to four-loop QCD corrections, which however still has a \sim 2\% scale uncertainty by taking the renormalization scale as \overline m_b(\overline m_b) and varying it within the usual range of \overline m_b(\overline m_b)/2, 2 \overline m_b(\overline m_b). The principle of maximum conformality (PMC) is adopted to achieve a more precise \overline\rm MS-on-shell relation by eliminating such scale uncertainty. As a step forward, we also estimate the magnitude of the uncalculated higher-order terms by using the Padé approximation approach. Numerically, by using the \overline\rm MS mass \overline m_b(\overline m_b)=4.183\pm0.007 GeV as an input, our predicted value for the bottom-quark on-shell mass becomes M_b\simeq 5.372^+0.091_-0.075 GeV, where the uncertainty is the squared average of the ones caused by \Delta \alpha_s(M_Z), \Delta \overline m_b(\overline m_b), and the estimated magnitude of the higher-order terms.