Abelian gauge theory

An Abelian gauge theory has a commutative gauge group. Its gauge field strength has no commutator term, so the pure gauge Lagrangian has no cubic or quartic gauge-boson self-interactions.

For $D_{\mu }={\partial }_{\mu }+ie{A}_{\mu }$, the local U(1) transformation is

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.

A large gauge transformation does not approach the identity at the relevant spatial or asymptotic boundary. Although it leaves the local field strength unchanged, its boundary charge can act nontrivially on physical states and constrain soft limits through a Ward identity.