Consider a gauge theory of charged scalars and charged fermions interacting with photons, which is a combined form of QED and scalar QED, \[\begin{equation*}\mathcal{L}=\mathcal{L}_{\text {Dirac}}+\mathcal{L}_{\mathrm{KG}}+\mathcal{L}_{\text {Maxwell }}+\mathcal{L}_{\text {int }}^{\mathrm{QED}}+\mathcal{L}_{\text {int }}^{\mathrm{sQED}} \tag{7.6.116}\end{equation*}\]

(a) Calculate the invariant amplitude for the process \(e^{-} e^{+} \rightarrow \phi \phi^{*}\) to lowest order in \(\alpha\), where the \(\phi\) charge is \(q_{c}=+|e|\).

(b) Calculate \(d \sigma / d \Omega\) for this process in the CM frame. Compare with the result for \(e^{+} e^{-} \rightarrow\) \(\mu^{+} \mu^{-}\)in Eq. (7.6.71).