Consider a physical system described by the Lagrangian density \[\begin{equation*}\mathcal{L}=\frac{1}{2}\left[\partial_{\mu} \phi \partial^{\mu}\phi+\partial_{\mu} \chi \partial^{\mu} \chi-m_{\phi}^{2} \phi^{2}-m_{\chi}^{2} \chi^{2}\right]-\frac{1}{2} g \chi^{2} \phi-(\lambda / 4 !) \chi^{4} \tag{7.6.115}\end{equation*}

where \(\phi, \chi, m_{\phi}, m_{\chi}, g\) and \(\lambda\) are real.

(a) Calculate \(d \sigma / d \cos \theta\) in the center-of-momentum (CM) frame at tree level for \(\chi \phi \rightarrow \chi \phi\) in terms of CM energy and scattering angle.

(b) Repeat this calculation for \(\chi \chi \rightarrow \chi \chi\) at tree level in the CM frame. Express your answer in terms of the Mandelstam variables \(s, t\) and \(u\).