Scalar electrodynamics ($\mathcal{L}=-\frac{1}{4}{f}^2+|D\varphi {|}^2-V(|\varphi {|}^2)$)

A charged scalar field coupled to an Abelian gauge field through $D_{\mu }={\partial }_{\mu }-ie{a}_{\mu }$. Under $\varphi \mapsto e^{i\alpha }\varphi$ and $a_{\mu }\mapsto a_{\mu }+e^{-1}{\partial }_{\mu }\alpha$, its gauge covariant derivative transforms with the same phase. The potential determines whether the perturbative spectrum has a massless photon or exhibits the Higgs mechanism.

For unpolarized ultrarelativistic charged spin-half particles annihilating through a photon into minimally coupled unit-charge scalars of mass $m$, set $\beta =\sqrt{1-4m^2/s}$. The scalar vertex contains the difference $k_{-}-k_{+}$ of outgoing four-momenta. Contraction with the spin-averaged initial fermion current gives $\overline{|\mathcal{M}{|}^2}=e^4{\beta }^2{\sin }^2\theta /2$. The two-body final-state flux factor contributes another $\beta$, givingThe ${\sin }^2\theta$ angular dependence and threshold suppression contrast with electron-positron annihilation into a muon pair. These predictions assume the specified scalar electrodynamic coupling, rather than an arbitrary spin-zero interaction.

For $D_{\mu }={\partial }_{\mu }+iq{A}_{\mu }$, the constant phase transformation $\delta \varphi =-iqϵ\varphi$ has the displayed Noether current. Varying the gauge field gives ${\partial }_{\mu }{F}^{\mu \nu }=J^{\nu }$. In this convention $J^{\mu }=iq({\varphi }^{\ast }{\partial }^{\mu }\varphi -\varphi {\partial }^{\mu }{\varphi }^{\ast })-2q^2{A}^{\mu }|\varphi {|}^2$. The matter equations imply ${\partial }_{\mu }{J}^{\mu }=0$; the integrated electric charge additionally requires zero boundary flux. Local phase invariance gives a Noether second theorem identity, rather than independent physical charges for arbitrary functions.^[ref]

With scalar charges $e$ and $2e$, both $\psi$ and ${\varphi }^2$ transform with phase $e^{2i\alpha }$. Therefore $M^2|\psi -b{\varphi }^2{|}^2$ is a gauge-invariant potential term. It contains a nonzero direct cubic interaction $-M^2(b{\psi }^{\ast }{\varphi }^2+h.c.)$. Adding positive quadratic and quartic modulus terms gives a real bounded-below potential. All expanded interactions have field degree at most four.