Scalar electrodynamics is the gauge invariant minimal coupling of a charged scalar field and the electromagnetic field. It has the Lagrangian density \(\mathcal{L}=-\frac{1}{4} F_{\mu u} F^{\mu u}+\left(D_{\mu} \phi\right)^{*}\left(D^{\mu} \phi\right)-\) \(m^{2} \phi \phi^{*}\), where the covariant derivative is defined as \(D_{\mu} \equiv \partial_{\mu}+i q A_{\mu}\) and where \(q\) is the electric charge of the scalar particle.