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 $\Gamma(\eta_b(1S) \rightarrow \gamma\gamma)=0.659\pm 0.089
({\rm th.}) {}^{+0.019}_{-0.018} (\delta \alpha_{\rm s})\pm 0.015 ({\rm
exp.})\; {\rm keV}$. 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
$\alpha^4\ln^2\alpha$ and $\alpha^5\ln^3\alpha$.
