Spin Dependence of Heavy Quarkonium Production and Annihilation Rates: Complete Next-to-Next-to-Leading Logarithmic result
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 $Γ(η_b(1S) \to γγ)=0.659\pm 0.089 ({\rm th.}) {}^{+0.019}_{-0.018} (δα_{\rm s})\pm 0.015 ({\rm exp.}) {\rm keV}$. 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 $α^4\ln^2α$ and $α^5\ln^3α$.