(a) Use the definition of the Fermi coupling constant in Eq. (5.2.7) and the low-momentum limit of the tree-level contribution to the scattering \(e^{-} \bar{u}_{e} \rightarrow \mu^{-} \bar{u}_{\mu}\) to show that \(G_{F} / \sqrt{2}=\) \(g^{2} / 8 M_{W}^{2}=1 / 2 v^{2}\). Since we know from many experimental measurements that \(G_{F} \simeq\) \(1.167 \times 10^{-5} \mathrm{GeV}^{-2}\) then \(v=\left(\sqrt{2} G_{F}\right)^{-1 / 2} \simeq 246 \mathrm{GeV}\).

(b) Calculate the total cross-section for \(e^{+} e^{-} \rightarrow Z h\) at tree level in the \(\mathrm{CM}\) frame (the electron-Higgs coupling is very small and can be neglected).

(c) Estimate the total cross-section for \(e^{+} e^{-} \rightarrow\) hadrons at tree level in the CM frame, including both the intermediate \(\gamma\) and \(Z\) contributions to \(\mathcal{M}\) and their interference in the total cross-section, \(\sigma\left(e^{+} e^{-} \rightarrow\right.\) hadrons \()\).