= Solution
The operator $A^\mu\bar\psi\gamma_\mu\psi$ has <mass dimension> $1+3=4$, so $[e]=0$. It is the \b[renormalizable electromagnetic current coupling] when completed as <quantum electrodynamics>: a massless Abelian <gauge field> couples minimally to the conserved Dirac current, with consistent <gauge fixing> and <Ward identities>. Indeed, replacing $\partial_\mu$ by a <gauge covariant derivative> in the <Dirac equation> produces this interaction. The gauge structure, rather than dimension counting alone, ensures the necessary cancellations.<Quantum Field Theory by David Tong>[§6.3.1, coupling the electromagnetic field to fermions]{ref}
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