[QUE/QFT-15003]

Consider the pion decay process \( \pi^- \longrightarrow e^- + \bar{\nu}.\) Let
the fields for \(\pi^-, e^- \text{ and }\nu \) be denoted by \(\phi_{\pi^-}(x),
\psi_{e}(x) \text{ and } \psi_\nu (x)\) respectively.

(a) Which of the following interactions will contribute to the pion decay
process in the first order?

1.   \(\phi_{\pi^-}^\dagger(x)\bar{\psi}_e(x) \gamma_\mu(1+\gamma_5) \psi_\nu(x) \)
2.    \(\phi_{\pi^-}^\dagger(x)\bar{\psi}_\nu(x) \gamma_\mu(1+\gamma_5) \psi_e(x) \)
3.    \(\phi_{\pi^-}(x)\bar{\psi}_e(x) \gamma_\mu(1+\gamma_5) \psi_\nu(x) \)
4.    \(\phi_{\pi^-}(x)\bar{\psi}_\nu(x) \gamma_\mu(1+\gamma_5) \psi_e(x) \)

(b) Is there a term which will contribute to the decay \(\pi^+ \longrightarrow
e^+ \nu \)? If yes,  which one?
(c) Consider processes allowed by different terms (i)-(iv) and check if any of
known selection rules is violated by any of the four expressions (i)-(iv).