= Solution
At order $g_0^2$, use the two-point <sunset diagram>: two quartic vertices are joined by three internal propagators, with one external line attached to each vertex. Unlike the one-vertex <tadpole diagram>, its self-energy $\Sigma(p)$ depends nontrivially on the external momentum $p$. The coefficient of $p^2$ in the expansion of $\Sigma(p)$ changes the kinetic term, so normalizing that term requires <wave-function renormalization> and gives a nonzero field <anomalous dimension>.
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