Precise determination of the bottom-quark on-shell mass using its four-loop relation to the $\overline{\mathrm{M}\mathrm{S}}$-scheme running mass

In this paper, we explore the properties of the bottom-quark on-shell mass ($M_b$) by using its relation to the $\overline{\mathrm{M}\mathrm{S}}$ mass (${\bar{m}}_b$). At present, this $\overline{\mathrm{M}\mathrm{S}}$-on-shell relation has been known up to four-loop QCD corrections, which however still has a $\sim 2\mathrm{\%}$ scale uncertainty by taking the renormalization scale as ${\bar{m}}_b({\bar{m}}_b)$ and varying it within the usual range of $[{\bar{m}}_b({\bar{m}}_b)\mathrm{/}2,2{\bar{m}}_b({\bar{m}}_b)]$. The principle of maximum conformality (PMC) has been adopted to achieve a more precise $\overline{\mathrm{M}\mathrm{S}}$-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 $\overline{\mathrm{M}\mathrm{S}}$ mass ${\bar{m}}_b({\bar{m}}_b)=4.183\pm 0.007$ GeV as an input, our predicted value for the bottom-quark on-shell mass becomes $M_b\simeq 5.372_{-0.075}^{+0.091}$ GeV, where the uncertainty is the squared average of the ones caused by $\mathrm{\Delta }{\alpha }_s(M_Z)$, $\mathrm{\Delta }{\bar{m}}_b({\bar{m}}_b)$, and the estimated magnitude of the higher-order terms.