[QUE/QFT-15005] Charged pion decay \( \pi^- \to e^- \bar{\nu}\)

The interaction Hamiltonian for pion decay \(\pi^-\longrightarrow e^- +
\bar{\nu} \) can be written as \[H_\text{int} = \frac{g}{\sqrt{2}}\bar{\psi}_e(x)\gamma_\mu(1-\gamma_5)\psi(x)_\nu \partial^\mu \phi_\pi^-(x) + h.c. \]

1. Show that the decay rate is given by \[\Gamma = \frac{g^2}{8\pi} \frac{m_e^2(m_\pi^2-m_e^2)^2}{m_\pi^3}.\]
2. Assuming that the coupling constant for \(\pi^-\longrightarrow \mu^- +\bar{\nu} \) is equal to that for the electron decay, calculate the branching ratio \[\frac{\Gamma(\pi^-\longrightarrow \mu^- +\bar{\nu})} {\Gamma((\pi^-\longrightarrow e^- + \bar{\nu})} \]