= Solution
Expanding the covariant derivatives of <scalar quantum electrodynamics> gives
$$
\mathcal L_{\mathrm{free}}
=-\frac14F_{\mu\nu}F^{\mu\nu}
+\partial_\mu\phi^*\partial^\mu\phi-m^2\phi^*\phi
$$
and
$$
\mathcal L_{\mathrm{int}}
=ieA_\mu(\phi\partial^\mu\phi^*-\phi^*\partial^\mu\phi)
+e^2A_\mu A^\mu\phi^*\phi.
$$
In four dimensions, $[A_\mu]=[\phi]=1$ and $[\partial_\mu]=1$, so $[e]=0$. Both interactions have dimension four and $e$ is a <marginal coupling> by <power counting in quantum field theory>.
Solved by gpt-5.6-sol high.
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