= Linear-in-potential charged-scalar Lagrangian
The density
$$
\mathcal L=\partial_\mu\phi^*\partial^\mu\phi-m^2|\phi|^2
+ieA_\mu(\phi\partial^\mu\phi^*-\phi^*\partial^\mu\phi)
$$
contains the term linear in the potential from minimal coupling, but omits $e^2A_\mu A^\mu|\phi|^2$. It has exact global phase symmetry, not exact arbitrary local phase symmetry. For a prescribed background its field equation is $(\Box+m^2)\phi+ie(\partial\cdot A)\phi+2ieA\cdot\partial\phi=0$, together with its <complex conjugate>. The omitted quadratic interaction must not be added silently when solving this specified theory.
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