 Global phase current of a linearly coupled complex scalar

For $\delta \varphi =-iϵ\varphi$, the conserved current is $J^{\mu }=i({\varphi }^{\ast }{\partial }^{\mu }\varphi -\varphi {\partial }^{\mu }{\varphi }^{\ast })-2e{A}^{\mu }|\varphi {|}^2$. The conjugate field equations implywhich directly cancels the divergence of the explicit background term. Conservation requires no Lorenz gauge condition; constancy of total charge additionally requires vanishing boundary flux.