$\Gamma(J/\psi\rightarrow \text{Hadrons})\sim \alpha_s^3$ - why not $\alpha_s^6$? I have seen this in a number of places now (e.g.  Ynduráin (2007), and here) , that the decay of $J/\psi$ goes as $\alpha_s^3$. The decay of $J/\psi$ progresses through three gluon as such each vertex of each of these gluons contributes a factor $\propto \sqrt{4\pi \alpha_s}$ to the matrix element, which thus goes like:
$$M\propto \alpha_s^3$$
to account for the 6 verticies in the diagram. The decay width should goes like $M^2$ and thus we should have:
$$\Gamma(J/\psi\rightarrow \text{Hadrons})\sim \alpha_s^6$$
not
$$\Gamma(J/\psi\rightarrow \text{Hadrons})\sim \alpha_s^3$$
as is commonly stated. My question is why is the latter my commonly written? Does it actually go like $\alpha_s^3$ or is their another reason behind it?
 A: The computation of the decay width for a particular final state is actually complex because this requires a model for the final state non-perturbative hadronisation and non-perturbative QCD is really difficult. But the observable you considered in your question, $\sigma(J/\Psi \to \text{hadrons})$, where "hadrons" stands for any number and flavour of hadrons, is much easier to compute. Indeed we just need to consider all quarks and gluons final state at a given order of perturbation in $\alpha_S$. Since those quark or gluon final state gives a hadron for sure, we only need to compute the square of the matrix elements for $c\bar{c} \to ggg$, etc, and then sum them up. The result is that $\sigma(J/\Psi \to \text{hadrons})/\sigma(J/\Psi \to l^+l^-) \propto \alpha_S^3$ at leading-order, with a next-leading-order correction proportional to $\alpha_S^4$. This is relatively well confirmed experimentally.
Note: what is most reliably computed is this ratio because this cancels out the non-perturbative aspect of the bound-state $J/\Psi$. Note also that when I say "relatively well confirmed", there are actually issues. The value of $\alpha_S$ extracted from the perturbative formula of this ratio does not agree with the value extracted from high energy processes for example. $J/\Psi$ decays are still an ongoing field of research with lingering hypotheses of glue balls in the final state e.g.
