[QUE/QFT-15012] QFT-PROBLEM

Consider a system of two real scalar fields \(\phi_1, \phi_2\) described by the Lagrangian density \begin{equation} \Lsc = \frac{1}{2} \partial_\mu \phi_i \partial^\mu \phi_i - \frac{1}{2} m^2 \phi_i\phi_i -\frac{1}{4} \lambda (\phi_i\phi_i)^2. \end{equation} Compute the scattering cross section to the lowest order in \(\lambda\). Find the cross sections for the three processes

1. \(\phi_1 + \phi_2 \longrightarrow \phi_1 +\phi_2\)
2. \(\phi_1 + \phi_1 \longrightarrow \phi_1 +\phi_1\)
3. \(\phi_1 + \phi_1 \longrightarrow \phi_2 +\phi_2\)

Write your answers as a constant times \(\sigma_0\equiv \frac{\lambda^2}{64\pi s}\) where \(s\) is the total energy in the center of mass frame.

• \(4\sigma_0\) &
• \(36 \sigma_0\) &
• \(4 \sigma_0\)