I am looking into the decays of quarkonia, specifically charmonium and bottomonium bound states. The decay widths depend on the quantum numbers $n^{2S+1}L_J$ of the states and the type of force mediating the decay. For example, electromagnetic transitions are related to the fine structure constant $\alpha$, while hadronic transitions to the strong coupling constant $\alpha_s$ from QCD. One can therefore relate the experimentaly found decay widths to the strong coupling constant, which depends on the mass of the state.
The end goal is to produce a rough plot of $\alpha_s$, calculated from the decay widths, versus mass of the quarkonium state. I would like to be able to compare my plot to an analytical expression. As quarkonia are produced in $e^- e^+$ collisions, the mass of the state created depends on the centre of mass energy in the collision, $s$, through (in natural units): $$\sqrt{s} = 2m_q \,.$$ The problem is, I can only find analytical expressions in terms of $Q^2=-q^2$, the negative of 4-momentum transfer (link): $$\alpha_s(Q^2) = \frac{12\pi}{ \left(33-n_f \right)\ln \left(\frac{Q^2}{\Lambda^2}\right)} , $$ where $n_f$ - number of quark flavours and $\Lambda$ - QCD scale (about $250 \, MeV/c$).
Hence, I have a way of relating the quarkonium mass to $s$, but not to $Q^2$. Is there a way to relate $Q^2$ to $\sqrt{s}$ in $e^- e^+$ collisions? Or how else could I check if my values agree with theory?