= Conjugate gauge covariant derivative
{title2=$(D_\mu\phi)^*=(\partial_\mu-ieA_\mu)\phi^*$}
If a <complex scalar field> uses $D_\mu\phi=(\partial_\mu+ieA_\mu)\phi$, its conjugate uses $(D_\mu\phi)^*=(\partial_\mu-ieA_\mu)\phi^*$. Under $\phi\mapsto e^{-ie\alpha}\phi$ and $A\mapsto A+d\alpha$, the two derivatives transform with opposite phases. Their product is gauge invariant. Applying the same plus-charge operator to both factors leaves an unwanted term proportional to $\partial_\mu\alpha$ and fails <gauge invariance>.
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