[QUE/QFT-15009] Charged pion, \((\pi^+, \pi^-)\), scattering

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

1. the \(S\) matrix element for \(\pi-\pi\) scattering \[\pi^+ + \pi^-\longrightarrow \pi^+ + \pi^-\]
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      \[ \frac{d\sigma}{d\Omega}= \frac{g^2}{64\pi^2 E_\text{cm}^2}\]