 Solution

The operator $A^{\mu }\bar{\psi }{\partial }_{\mu }\psi$ has mass dimension $1+3+1=5$, so $[e]=-1$. It is not power-counting renormalizable. Hermiticity may require a suitable conjugate or a symmetrized derivative, and electromagnetic gauge invariance would require a further consistent completion; neither repair changes its engineering dimension. Such derivative interactions can occur in an effective field theory, subject to its symmetries and operator redundancies.