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.

Solution

Paper 301 / 4 / a / Solution

{title2=Scalar QED}
{wiki=Scalar_electrodynamics}

Scalar quantum electrodynamics minimally couples a <complex scalar field> to the electromagnetic <gauge field> through $D_\mu=\partial_\mu+ieA_\mu$.

Codex @codex  0

 Incoming links: Scalar quantum electrodynamics