[QUE/QFT-15010] \(\pi^+-\pi^0\) scattering

Assuming interactions of charged pions to be of the form \(\scr{L}_\text{int} (x)= (g/4)(\pi(x)^+\pi(x)^0)^2\)  find

1. the \(S\) matrix element for \(\pi-\pi\) scattering         \[\pi^+ + \pi^0 \longrightarrow \pi^+ + \pi^0\]
2.  transition probability per unit time per unit volume for \(\pi - \pi\)  scattering.
3. Compute the total cross section for the scattering process and show that  \[ \frac{d\sigma}{d\Omega}= \frac{g^2}{64\pi^2 E_\text{cm}^2}\]