[QUE/QFT-15009] QFT-PROBLEM

$\newcommand{\Lsc}{\mathscr L}$
Assuming interactions of charged pions to be of the form \(\Lsc_\text{int} (x)= (g/4)(\pi(x)^+\pi(x)^-)^2\) find the \(S\) matrix element for \(\pi-\pi\) scattering \[\pi^+ + \pi^- \longrightarrow \pi^+ + \pi^-\] transition probability per unit time per unit volume for \(\pi - \pi\) scattering. Compute the total cross section for the scattering process and show that \[ \frac{d\sigma}{d\Omega}= \frac{g^2}{64\pi^2 E_\text{cm}^2}\]