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