[QUE/QFT-15010] QFT-PROBLEM

$\newcommand{\Lsc}{\mathscr L}$
$\newcommand{\dd}[2][]{\frac{d#1}{d#2}}$
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 \[ \dd[\sigma]{\Omega}= \frac{g^2}{64\pi^2 E_\text{cm}^2}\]