 Linear-in-potential charged-scalar Lagrangian

The densitycontains the term linear in the potential from minimal coupling, but omits $e^2{A}_{\mu }{A}^{\mu }|\varphi {|}^2$. It has exact global phase symmetry, not exact arbitrary local phase symmetry. For a prescribed background its field equation is $(\square +m^2)\varphi +ie(\partial \cdot A)\varphi +2ieA\cdot \partial \varphi =0$, together with its complex conjugate. The omitted quadratic interaction must not be added silently when solving this specified theory.