= Solution
The free complex field has the global <unitary group> $U(1)$ symmetry
$$
\Phi(x)\mapsto e^{i\alpha}\Phi(x).
$$
Gauging it permits $\alpha=\alpha(x)$ and introduces a gauge field with
$$
D_\mu=\partial_\mu+ieA_\mu,\qquad
A_\mu\mapsto A_\mu-\frac1e\partial_\mu\alpha.
$$
The <scalar quantum electrodynamics> Lagrangian is
$$
\boxed{\mathcal L
=(D_\mu\Phi)^\dagger D^\mu\Phi-M^2\Phi^\dagger\Phi
-\frac14F_{\mu\nu}F^{\mu\nu}}.
$$
Its interaction terms are
$$
\mathcal L_{\rm int}
=ieA_\mu\{\Phi^\dagger\partial^\mu\Phi
-(\partial^\mu\Phi^\dagger)\Phi\}
+e^2A_\mu A^\mu\Phi^\dagger\Phi.
$$
With momentum flowing along the scalar arrow, the momentum-space <Feynman rules> are:
* an internal scalar line of momentum $p$ contributes $i/(p^2-M^2+i\epsilon)$;
* an internal photon in Feynman gauge contributes $-i\eta_{\mu\nu}/(k^2+i\epsilon)$;
* a scalar-scalar-photon vertex contributes $ie(p+p')^\mu$, where $p$ enters and $p'$ leaves along the scalar line;
* a scalar-scalar-two-photon vertex contributes $2ie^2\eta^{\mu\nu}$;
* every vertex carries momentum conservation, and every undetermined internal momentum is integrated with $d^4q/(2\pi)^4$.
Back to article page
