The asymptotic behaviour of the ${\pi }^0$ ${\gamma }^{\ast }$ ${\gamma }^{\ast }$ vertex

The Bjorken-Johnson-Low theorem applied to the $\gamma^\ast \to \gamma^\ast \pi^0$ process provides us with a rather remarkable asymptotic behaviour for the ${\pi }^0{\gamma }^{\ast }{\gamma }^{\ast }$ vertex. We compare our result with previous QCD- inspired estimates and argue that the predicted behaviour is quite consistent with the present data on hadronic $J\mathrm{/}\mathrm{\Psi }$ decays and ensures the matching of long- and short-distance radiative corrections to $\pi^+ \to e^+ \nu_e$.