= Solution
The <kinetic terms> determine the <engineering dimensions> in $d$ dimensions:
$$
[A_\mu]=\frac{d-2}{2},
\qquad [\psi]=\frac{d-1}{2}.
$$
Requiring $e\bar\psi\gamma^\mu A_\mu\psi$ to have dimension $d$ gives
$$
\boxed{[e]=\frac{4-d}{2}=\frac\epsilon2.}
$$
Thus the electric charge is dimensionless only in four dimensions. In <dimensional regularization> one writes the bare interaction with $e_0=\mu^{\epsilon/2}Z_e e$, where $e$ is dimensionless and the <renormalization scale> $\mu$ supplies the missing dimension.
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