Charm Mass Determination from QCD Charmonium Sum Rules at Order alpha_s^3

TL;DR

The paper refines the determination of the MS-bar charm quark mass by applying charmonium sum rules with perturbative input up to ${\cal O}(\alpha_s^3)$ and by fully exploiting the available $e^+e^-$ hadronic cross-section data up to 10.538 GeV via a data-clustering approach. It systematically analyzes perturbative uncertainties using multiple expansion schemes and independent scale variations, finding a perturbative error around 20 MeV, larger than earlier studies. The study also corroborates the robustness of the result by performing higher-moment analyses and demonstrates convergence of the running coupling and mass down to scales near the charm mass. The final result, $\overline{m}_c(\overline{m}_c) \approx 1.282$ GeV with quantified uncertainties, aligns with other modern determinations but highlights a larger perturbative uncertainty due to methodological improvements and data usage. This work emphasizes the importance of careful uncertainty quantification and data combination for precision quark-mass extraction in QCD.

Abstract

We determine the MS-bar charm quark mass from a charmonium QCD sum rules analysis. On the theoretical side we use input from perturbation theory at O(alpha_s^3). Improvements with respect to previous O(alpha_s^3) analyses include (1) an account of all available e+e- hadronic cross section data and (2) a thorough analysis of perturbative uncertainties. Using a data clustering method to combine hadronic cross section data sets from different measurements we demonstrate that using all available experimental data up to c.m. energies of 10.538 GeV allows for determinations of experimental moments and their correlations with small errors and that there is no need to rely on theoretical input above the charmonium resonances. We also show that good convergence properties of the perturbative series for the theoretical sum rule moments need to be considered with some care when extracting the charm mass and demonstrate how to set up a suitable set of scale variations to obtain a proper estimate of the perturbative uncertainty. As the final outcome of our analysis we obtain m_c(m_c) = 1.282 \pm 0.006_stat \pm 0.009_syst \pm 0.019)_pert \pm 0.010_alpha \pm 0.002_GG GeV. The perturbative error is an order of magnitude larger than the one obtained in previous O(alpha_s^3) sum rule analyses.

Figures (16)

• Figure 1: Path of integration in the complex $\bar{s}$-plane for the computation of the moments.
• Figure 2: Results for $\alpha^{\rm N^3LL}_s(\mu)/\alpha^{\rm N^kLL}_s(\mu)$ (a) and $\alpha^{\rm N^3LL}_s(\mu)/\alpha^{(k)}_s(\mu)$ (b), where $\alpha^{\rm N^kLL}_s$ stands for the $(k+1)$-loop running coupling constant and $\alpha^{(k)}_s$ is the corresponding ${\mathcal{O}}(\alpha_s^{(k+1)})$ fixed-order expression for $\alpha_s$. All orders are run from the common point $\alpha_s(3~{\rm GeV})=0.2535$.
• Figure 3: Results for $\overline{m}^{\rm N^3LL}_c(\mu)/\overline{m}^{\rm N^kLL}_c(\mu)$ (a) and $\overline{m}^{\rm N^3LL}_c(\mu)/\overline{m}^{(k)}_c(\mu)$ (b), where $\overline{m}^{\rm N^kLL}_c$ stands for the $(k+1)$-loop running $\overline{\rm MS}$ charm mass and $\overline{m}^{(k)}_c$ is the ${\mathcal{O}}(\alpha_s^{(k+1)})$ fixed-order expression.
• Figure 4: Results for $\overline{m}_c(\overline{m}_c)$ at various orders, for methods a (graphs 1 and 5), b (2,6), c (3,7), and d (4,8), setting $\mu_\alpha=\mu_m$ (graphs 1-4) and setting $\mu_m=\overline{m}_c(\overline{m}_c)$ (5-8). The shaded regions arise from the variation $2\,{\rm GeV}\le\mu_\alpha\le4\,{\rm GeV}$.
• Figure 5: Estimates of the perturbative error at ${\mathcal{O}}(\alpha_s^3)$. We show the correlated $\mu_m=\mu_\alpha$ variation, (orange), setting $\mu_m=\overline{m}_c(\overline{m}_c)$ and setting $\mu_m=3~$GeV, (blue and red, respectively), and the double scale variation (magenta).