Decay of Spin-Zero Mesons into Two Leptons

The following general theorem is formulated: The matrix element for the decay of spin-zero mesons with mass $m$ into two leptons with masses ${\mu }_1$ and ${\mu }_2$ respectively contains only terms proportional to $\frac{{\mu }_1}{m}$ and $\frac{{\mu }_2}{m}$, if in the open polygonal arc of lepton lines the number of matrices ${\gamma }_{\mu }$ plus the number of internal lines ($S_F$-functions) is odd. There can also be terms proportional to $\frac{{\mu }_i}{m}$ if virtual leptons with ${\mu }_i\ne {\mu }_1, {\mu }_2$ appear in the arc. Application of this theorem to the reaction ${\pi }^0\to e^{+}+e^{-}$ leads to a ratio of the one pair to the two pair decay of the order ${\left(\frac{{\mu }_e}{{m_{\pi }}^0}\right)}^2\sim {10}^{-5\mathrm{}}$. A priori one would expect this ratio to be of the order one. Furthermore, the theorem provides a more general basis for the discussion of the relative probability of the reactions $\pi \to \mu +\nu$ and $\pi \to e+\nu$.