Determination of $α_s$ and heavy quark masses from recent measurements of $R(s)$

TL;DR

The paper addresses extracting α_s and heavy-quark masses from the hadronic cross section R(s) in e+e− annihilation. It combines continuum data with high-order perturbative QCD predictions to determine α_s and uses low-moment sum rules in the charm-threshold region to obtain m_c(m_c), with a parallel analysis for m_b(m_b). The analysis yields α_s(M_Z)=0.124^{+0.011}_{-0.014}, m_c(m_c)=1.304(27) GeV, and m_b(m_b)=4.191(51) GeV, offering competitive precision and cross-checks with other analyses. The results validate perturbative QCD in the energy ranges used and provide precise inputs for Standard Model phenomenology.

Abstract

In this paper we compare recent experimental data for the total cross section $σ(e^+e^-\to{hadrons})$ with the up-to-date theoretical prediction of perturbative QCD for those energies where perturbation theory is reliable. The excellent agreement suggests the determination of the strong coupling $α_s$ from the measurements in the continuum. The precise data from the charm threshold region, when combined with the recent evaluation of moments with three loop accurracy, lead to a direct determination of the short distance $\bar{\rm MS}$ charm quark mass. Our result for the strong coupling constant $α_s^{(4)}(5 {GeV})=0.235^{+0.047}_{-0.047}$ corresponds to $α_s^{(5)}(M_Z)=0.124^{+0.011}_{-0.014}$, for the charmed quark mass we find $m_c(m_c)=1.304(27)$. Applying the same approach to the bottom quark we obtain $m_b(m_b)=4.191(51)$ GeV. Whereas our result for $α_s(M_Z)$ serves as a useful cross check for other more precise determinations, our values for the charm and bottom quark masses are more accurate than other recent analyses.

Figures (4)

• Figure 1: $R(s)$ in the energy range between $1.8$ GeV and $11.0$ GeV. The solid line corresponds to the theoretical predictions adopting our central values for the input parameters. The theoretical uncertainties are indicated by the dashed curves which are obtained from the variation of the input parameters as described in the text. The two error bars on the data points indicate the statistical (inner) and systematical (outer) uncertainty.
• Figure 2: Feynman diagrams contributing to $R(s)$ at order $\alpha_s^2$. A secondary charm quark pair is produced through gluon splitting.
• Figure 3: $m_c(m_c)$ for $n=1,2,3$ and $4$. For each value of $n$ the results from left to right correspond the inclusion of terms of order $\alpha_s^0$, $\alpha_s^1$ and $\alpha_s^2$ in the coefficients $\bar{C}_n$ (cf. Eq. (\ref{['eq:cn']})). Note, that for $n=3$ and $n=4$ the errors can not be determined with the help of Eq. (\ref{['eq:mc1']}) in those cases where only the two-loop corrections of order $\alpha_s$ are included into the coefficients $\bar{C}_n$ as the equation cannot be solved for $m_c(3~\hbox{GeV})$.
• Figure 4: $m_b(m_b)$ for $n=1,2,3$ and $4$. For each value of $n$ the results from left to right correspond the inclusion of terms of order $\alpha_s^0$, $\alpha_s^1$ and $\alpha_s^2$ in the coefficients $\bar{C}_n$ (cf. Eq. (\ref{['eq:cn']})). Note, that the errors for $n=3$ and both the central value and the errors for $n=4$ can not be determined in those cases where only the two-loop corrections of order $\alpha_s$ are included into the coefficients $\bar{C}_n$ as the corresponding equation cannot be solved for $m_b(10~\hbox{GeV})$.