= Solution
<Gauge invariance> and <power counting> permit the same local operators as in the original <Renormalizability>[renormalizable action]. With renormalized fields, a convenient <counterterm> parametrization is
$$
\boxed{\mathcal L_{\mathrm{ct}}=-\delta Z_\phi(D_\mu\phi)^\dagger D^\mu\phi-\delta M^2\phi^\dagger\phi-\frac{\mu^\epsilon\delta\lambda}{4}(\phi^\dagger\phi)^2-\frac{\delta Z_A}{4}F_{\mu\nu}F^{\mu\nu}.}
$$
Here $D_\mu=\partial_\mu-i\mu^{\epsilon/2}eA_\mu$. Expanding the first term supplies kinetic, scalar-scalar-photon, and <Seagull vertex>[seagull] <counterterms> with the relative coefficients required by the <Ward identity>. Equivalently introduce separate <vertex> constants first and then impose that identity. For $\phi_0=Z_\phi^{1/2}\phi$, $A_0=Z_A^{1/2}A$, and $e_0=\mu^{\epsilon/2}Z_e e$, it gives $Z_eZ_A^{1/2}=1$. Thus <charge renormalization> can be represented through the <photon> field factor, rather than adding an independent gauge-noninvariant <Interaction term>[interaction].
The <mass> coefficient defined here is $\delta M^2=Z_\phi m_0^2-m^2$; it must be distinguished from $m_0^2-m^2$. <Gauge fixing> may require the corresponding <Gauge-fixing parameter>[gauge-parameter] redefinition, and <vacuum diagrams> can require an additive <Vacuum energy>[vacuum-energy] <counterterm>. No <photon> <mass> <counterterm> is allowed by the <Gauge symmetry>[unbroken gauge symmetry]. The <Quartic interaction>[quartic] <Scalar field>[scalar] <counterterm> remains necessary for the full theory even at a scale where the renormalized $\lambda$ is zero, because gauge <Interaction term>[interactions] generate <Scalar field>[scalar] <Four-point function>[four-point] divergences.
Back to article page
