Strong coupling constant from bottomonium fine structure

Abstract

From a fit to the experimental data on the $b\bar{b}$ fine structure, the two-loop coupling constant is extracted. For the 1P state the fitted value is ${\alpha }_s({\mu }_1)=0.33\pm 0.01(exp)\pm 0.02(th)$ at the scale ${\mu }_1=1.8\pm 0.1$ GeV, which corresponds to the QCD constant ${\Lambda }^{(4)}(2-loop)=338\pm 30$ MeV (n_f = 4) and ${\alpha }_s(M_Z)=0.119\pm \mathrm{0.002.}Forthe2Pstatethevalue$\alpha_s(\mu_2) = 0.40 \pm 0.02(exp)\pm 0.02(th)$atthescale$\mu_2 = 1.02 \pm 0.2$GeVisextracted,whichissignificantlylargerthaninthepreviousanalysisofFulcher(1991)andHalzen(1993),butabout30$\alpha_s(1.0) \approx 0.40$canbeobtainedintheframeworkofthebackgroundperturbationtheory,thusdemonstratingthefreezingof$\alpha_s$.Therelativisticcorrectionsto$\alpha_s$ are found to be about 15%.

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