Strong coupling constant from bottomonium fine structure

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 $\alpha_s(M_Z) = 0.119 \pm 0.002. For the 2P state the value$\alpha_s(\mu_2) = 0.40 \pm 0.02(exp)\pm 0.02(th)$at the scale$\mu_2 = 1.02 \pm 0.2$GeV is extracted, which is significantly larger than in the previous analysis of Fulcher (1991) and Halzen (1993), but about 30$\alpha_s(1.0) \approx 0.40$can be obtained in the framework of the background perturbation theory, thus demonstrating the freezing of$\alpha_s$. The relativistic corrections to$\alpha_s$ are found to be about 15%.