Spin Dependence of Heavy Quarkonium Production and Annihilation Rates: Complete Next-to-Next-to-Leading Logarithmic result

Abstract

The ratio of the photon mediated production or annihilation rates of spin triplet and spin singlet heavy quarkonium states is computed to the next-to-next-to-leading logarithmic accuracy within the nonrelativistic renormalization group approach. The result is presented in analytical form and applied to the phenomenology of $t\bar{t}$, $b\bar{b}$ and $c\bar{c}$ systems. The use of the nonrelativistic renormalization group considerably improves the behaviour of the perturbative expansion and is crucial for accurate theoretical analysis. For bottomonium decays we predict $\Gamma ({\eta }_b(1S)\to \gamma \gamma )=0.659\pm 0.089(th.){}_{-0.018}^{+0.019}(\delta {\alpha }_{\mathrm{s}})\pm 0.015(exp.)\mathrm{keV}$. Our results question the accuracy of the existing extractions of the strong coupling constant from the bottomonium annihilation. As a by-product we obtain novel corrections to the ratio of the ortho- and parapositronium decay rates: the corrections of order ${\alpha }^4{\ln }^2\alpha$ and ${\alpha }^5{\ln }^3\alpha$.

Comments: Appendices A.4, A.5 and B corrected