Precise determination of the bottom-quark on-shell mass using its four-loop relation to the MS̄-scheme running mass

We explore the properties of the bottom-quark on-shell mass ( M b ) by using its relation to the MS ¯ mass ( m ¯ b ). At present, this MS ¯ -on-shell relation has been known up to four-loop QCD corrections, which however still has a ∼ 2% scale uncertainty by taking the renormalization scale as m ¯ b ( m ¯ b ) and varying it within the usual range of [ m ¯ b ( m ¯ b ) / 2 , 2 m ¯ b ( m ¯ b ) ] . The principle of maximum conformality (PMC) is adopted to achieve a more precise 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 MS ¯ mass m ¯ b ( m ¯ b ) = 4.183 ± 0.007 GeV as an input, our predicted value for the bottom-quark on-shell mass becomes M b ≃ 5.372 − 0.075 + 0.091 GeV, where the uncertainty is the squared average of the ones caused by Δ α s ( M Z ), Δ m ¯ b ( m ¯ b ) , and the estimated magnitude of the higher-order terms.