Consider the spontaneously broken theory of a complex scalar field in Section 3.3.3, which is described after symmetry breaking by the Lagrangian density in (3.155) for a rectangular basis, or by (3.163) in a polar basis.
(a) Show that the on-shell amplitudes for ϕ1′(p1)ϕ2′(p2)→ϕ1′(p3)ϕ2′(p4) and η(p1)ξ(p2)→η(p3)ξ(p4) are the same at tree-level.
(b) Now consider pseudoscalar-fermion scattering ϕ2′(p1)ψ(p2)→ϕ2′(p3)ψ(p4). The Yukawa interactions are given for three parametrizations in (3.159), (3.164), and (3.166), and the corresponding scalar interactions in (3.155) and (3.163). Show that the tree-level amplitudes are the same in all three parametrizations.