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Question
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
- \(\phi_1 + \phi_2 \longrightarrow \phi_1 +\phi_2\)
- \(\phi_1 + \phi_1 \longrightarrow \phi_1 +\phi_1\)
- \(\phi_1 + \phi_1 \longrightarrow \phi_2 +\phi_2\)
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.
Answer : a. \(4\sigma_0\) b. \(36 \sigma_0\) c. \(4 \sigma_0\)
Hint : Find symmetry appropriate factor.
Remark : Good question
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