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

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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 tt, bb and cc 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(ηb(1S) → γγ)=0.659± 0.089 (th.) ⁺⁰.⁰¹⁹₋₀.₀₁₈ (δα s)± 0.015 (exp.) 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 α⁴²α and α⁵³α.