Bottom Quark Mass from Upsilon Mesons: Charm Mass Effects

TL;DR

The paper addresses precise bottom-quark mass determinations from Υ sum rules while incorporating finite charm-quark mass effects. It develops NNLO nonrelativistic QCD calculations for the Υ system and computes α_s^3 corrections to the bottom MSbar mass, finding a charm-induced downward shift of about 30 MeV in the MSbar mass and ~20 MeV in the 1S mass. Through a detailed treatment of the static potential, pole–MSbar and pole–1S relations, and the Upsilon expansion, the work demonstrates that charm-mass effects are perturbatively under control and yield refined values: M_b^{1S}=4.69±0.03 GeV and $\bar M_b(\bar M_b)=4.17±0.05$ GeV. These results provide robust bottom-m quark mass inputs for precision flavor physics and CKM related analyses, with controlled theoretical uncertainties and clear guidance on where higher-order effects may appear.

Abstract

The effects of the finite charm quark mass on bottom quark mass determinations from $Υ$ 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 $Υ$ sum rules and at order $α_s^3$ for the determination of the bottom MSbar mass. For the bottom 1S mass, which is extracted from the $Υ$ 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 $α_s^{(n_l=5)}(M_Z)=0.118\pm 0.003$ 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.

Figures (7)

• Figure 1: NLO contribution to the static potential coming from the insertion of a one-loop vacuum polarization of a light quark with finite mass.
• Figure 2: The ratio of the double-insertion contributions to the full result for the NNLO light quark mass corrections in the bottom quark 1S--pole mass relation, $\Delta^{\hbox{\tiny NNLO}}_{\hbox{\tiny massive,d}}/ \Delta^{\hbox{\tiny NNLO}}_{\hbox{\tiny massive}}$ for $M_{\hbox{\tiny Q}}^{\hbox{\tiny pole}}=4.8$ GeV, $\alpha_s^{(5)}(M_Z)=0.118$ and $m=0.1$ GeV (dotted line), $0.5$ GeV (dash-dotted line), $1.0$ GeV (dashed line), $1.5$ GeV (long-dash-dotted line) and $2.0$ GeV (solid line) plotted over the renormalization scale $\mu$.
• Figure 3: Charm mass corrections in the relation between the bottom $\overline{\hbox{MS}}$ mass $\overline M_{\hbox{\tiny b}}(\overline M_{\hbox{\tiny b}})$ and the bottom 1S mass. Figure (a) displays the scale dependence for $M_{\hbox{\tiny b}}^{\hbox{\tiny 1S}}=4.7$ GeV and $\alpha_s^{(5)}(M_Z)=0.118$ and using $\overline m(\overline m)=0.1$ GeV (dotted line), $1.0$ GeV (dash-dotted line), $1.5$ GeV (dashed line) and $2.0$ GeV (solid line) for the $\overline{\hbox{MS}}$ charm quark mass. Figure (b) displays the dependence on $\overline m(\overline m)$ for $\mu=1.5$ GeV (dotted line), $3$ GeV (dashed line) and $5$ GeV (solid line). The other parameters a chosen as in Fig. (a).
• Figure 4: Path of integration to calculate expression (\ref{['Pnnonrelativisticdeformed']}) for the theoretical moments. The dashed line closes the contour at infinity and does not contribute to the integration. The free constant $\gamma$ is chosen large enough to be safely away from the bound state poles, which are indicated by the grey dots on the negative energy axis. The thick grey line on the positive energy axis represents the continuum.
• Figure 5: Results for the allowed range of $M_b^{\hbox{\tiny 1S}}$ for given values of $\alpha_s^{(5)}(M_Z)$ (and the corresponding values of $\alpha_s^{(4)}(2.5\,\hbox{GeV})$) at NLO for different choices of the $\overline{\hbox{MS}}$ charm quark mass. The dots represent points of minimal $\chi^2$ for a large number of random choices within the ranges (\ref{['scaleranges']}) and the sets (\ref{['nsets']}), and randomly chosen values of the strong coupling. Experimental errors at $95\%$ CL are displayed as vertical lines. It is illustrated how the allowed range for $M_b^{\hbox{\tiny 1S}}$ is obtained if $0.115\le\alpha_s^{(5)}(M_Z)\le 0.121$ is taken as an input.