[QUE/QFT-15003] QFT-PROBLEM

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.

• Which of the following interactions will contribute to the pion decay process in the first order?
  • [(i)] \(\phi_{\pi^-}^\dagger(x)\bar{\psi}_e(x) \gamma_\mu(1+\gamma_5) \psi_\nu(x) \)
  • [(ii)] \(\phi_{\pi^-}^\dagger(x)\bar{\psi}_\nu(x) \gamma_\mu(1+\gamma_5) \psi_e(x) \)
  • [(iii)] \(\phi_{\pi^-}(x)\bar{\psi}_e(x) \gamma_\mu(1+\gamma_5) \psi_\nu(x) \)
  • [(iv)] \(\phi_{\pi^-}(x)\bar{\psi}_\nu(x) \gamma_\mu(1+\gamma_5) \psi_e(x) \)
• Is there a term which will contribute to the decay \(\pi^+ \longrightarrow e^+ \nu \)? If yes, which one?
• 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).