$M({\eta }_b)$ and ${\alpha }_s$ from Nonrelativistic Renormalization Group

Abstract

We sum up the next-to-leading logarithmic corrections to the heavy-quarkonium hyperfine splitting using the nonrelativistic renormalization group. On the basis of this result, we predict the mass of the ${\eta }_b$ meson to be $M({\eta }_b)=9419\pm 11(\mathrm{th}){}_{-8}^{+9}(\delta {\alpha }_s)MeV$. The experimental measurement of $M({\eta }_b)$ with a few MeV error would be sufficient to determine ${\alpha }_s(M_Z)$ with an accuracy of $\pm 0.003$. The use of the nonrelativistic renormalization group is mandatory to reproduce the experimental value of the hyperfine splitting in charmonium.

Comments: Eq. (1) corrected, numerical results slightly changed