Decay of the Pi Meson

It is known experimentally that, relative to the normal pion decay mode $\pi \to \mu +\nu$, the branching ratios $\rho$ and ${\rho }_{\gamma }$ for the alternate decay modes $\pi \to e+\nu$ and $\pi \to e+\nu +\gamma$ are very small ($\rho , {\rho }_{\gamma }\lesssim 5\times {10}^{-5\mathrm{}}$). We investigate the question of whether these limits on the branching ratios are consistent with the idea that pion decay occurs through a universal Fermi interaction via disintegration into a virtual nucleon pair: $\pi \to \mathrm{virtual}\mathrm{nucleon}\mathrm{pair}\to \mu (or e)+\nu$. A value for $\rho$ consistent with the experiments can be obtained if the pseudoscalar coupling constant $g_P$ in the universal Fermi interaction is small compared to the axial vector coupling constant $g_A$. An estimate of ${\rho }_{\gamma }$ is made assuming that the photodecay occurs through the axial vector coupling, and it is found that there is probably no disagreement with experiment on this score. However, the photodecay can also occur through the tensor interaction (forbidden for $\pi \to \mu +\nu$ and $\pi \to e+\nu$). Using the experimental value $\frac{g_A}{g_T}\lesssim 0.02$ obtained from beta-decay experiments, we estimate ${\rho }_{\gamma }\gtrsim 0.025$, a result which is inconsistent with the experimental value. It is shown that the disagreement cannot be removed by using a linear combination of all possible Fermi couplings. An upper limit on $g_P$ for beta decay, valid even in the absence of a universal Fermi interaction, is obtained.