Determination of the bottom quark mass from the $Υ(1S)$ system

TL;DR

This paper introduces a renormalon-subtracted (RS) framework to tame the leading infrared renormalon ambiguities in heavy-quark observables, by defining RS and RS' schemes that subtract the closest renormalon from the pole mass and the singlet static potential. Using this approach, the authors determine the bottom and charm quark MS-bar masses from Υ(1S) and B–D mass differences, respectively, providing explicit error budgets that include ultrasoft and higher-order effects. The results, $m_{b,ar{ m MS}}(m_{b,ar{ m MS}})\, o 4.210^{+0.090}_{-0.090}$ MeV and $m_{c,ar{ m MS}}(m_{c,ar{ m MS}})\, o 1.210^{+0.070}_{-0.070}$ GeV (with correlated inputs), illustrate the potential of renormalon-aware threshold masses for precision heavy-quark spectroscopy. The work also discusses nonperturbative uncertainties and outlines paths for further refinement and broader applications in heavy-quark physics.

Abstract

We approximately compute the normalization constant of the first infrared renormalon of the pole mass (and the singlet static potential). Estimates of higher order terms in the perturbative relation between the pole mass and the $\MS$ mass (and in the relation between the singlet static potential and $α_s$) are given. We define a matching scheme (the renormalon subtracted scheme) between QCD and any effective field theory with heavy quarks where, besides the usual perturbative matching, the first renormalon in the Borel plane of the pole mass is subtracted. A determination of the bottom $\MS$ quark mass from the $Υ(1S)$ system is performed with this new scheme and the errors studied. Our result reads $m_{b,\MS}(m_{b,\MS})=4 210^{+90}_{-90}({\rm theory})^{-25}_{+25}(α_s)$ MeV. Using the mass difference between the $B$ and $D$ meson, we also obtain a value for the charm quark mass: $m_{c,\MS}(m_{c,\MS})=1 210^{+70}_{-70}({\rm theory})^{+65}_{-65}(m_{b,\MS})^{-45}_{+45}(λ_1)$ MeV. We finally discuss upon eventual improvements of these determinations.

Figures (7)

• Figure 1: Symbolic relation between observables through the determination of the matching coefficients of the effective field theory.
• Figure 2: $x \equiv {\nu \over m_{\overline{\rm MS}}}$ dependence of $N_m$ for $n_f=4$.
• Figure 3: $x \equiv {\nu r}$ dependence of $N_V$ for $n_f=4$.
• Figure 4: We plot $2m_{b,\rm RS}$ (dashed line), and the LO (dot-dashed line), NLO (dotted line) and NNLO (solid line) predictions for the $\Upsilon(1S)$ mass in terms of $\nu$ in the RS scheme. The value of $m_{b,\rm RS}$ is taken from Eq. (\ref{['MRSdet']}).
• Figure 5: We plot $2m_{b,\rm RS'}$ (dashed line), and the LO (dot-dashed line), NLO (dotted line) and NNLO (solid line) predictions for the $\Upsilon(1S)$ mass in terms of $\nu$ in the RS' scheme. The value of $m_{b,\rm RS'}$ is taken from Eq. (\ref{['MRSprimedet']}).