= Solution
{c}
The <scalar QED Ward identity> equates the <vertex renormalization> and <wave-function renormalization> constants. Consequently, using part (iv),
$$
\boxed{Z_2=Z_1=1+\frac{e^2(3-\xi)}{16\pi^2\epsilon}+O(e^4).}
$$
The <renormalization constants> describe gauge-dependent fields and <off shell> vertices, so their dependence on $\xi$ is permitted. A physical <scattering amplitude> includes all <Feynman diagrams>, the <counterterms>, and the external-state factors from <LSZ reduction>. The <Ward identities> ensure that the dependence on $\xi$ cancels for physical external states. Infrared-sensitive probabilities also require the usual inclusive treatment of unobserved soft <photons>. Thus no dependence on $\xi$ is implied for consistently calculated physical observables.
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