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The holomorphic coupling is $\tau =\vartheta /(2\pi )+4\pi i/g^2$. Perturbative coefficients are independent of the topological angle $\vartheta$. A holomorphic function invariant under continuous shifts of $\mathrm{Re}\tau$ is constant by the Cauchy-Riemann equations, hence a perturbative local superpotential coefficient cannot depend on $\tau$. Terms in $e^{2\pi i\tau }$ can evade this argument nonperturbatively. This does not forbid renormalization of the separate gauge kinetic F-term, or wave-function renormalization from the Kähler potential.

For a cubic graph with $E$ external legs, $3V=2I+E$ and $L=I-V+1$. Its superficial degree of divergence is therefore $6L-2I=6-2E$. Divergences in vacuum, one-, two-, and three-point functions can be absorbed into vacuum energy, a linear term, mass, wave-function renormalization, and cubic-coupling counterterms. Higher-point proper diagrams are superficially convergent after subtraction of divergent subgraphs. This establishes perturbative renormalizability, without asserting a nonperturbative positive measure for the unstable real cubic potential.

A sunset diagram is a two-point diagram with two interaction vertices joined by three internal propagators. Its dependence on the external momentum produces wave-function renormalization in scalar quartic theory.

In a gauge-preserving regularization and consistent renormalization convention, the proper-vertex Ward-Takahashi identity in QED equates the vertex and electron wave-function renormalization constants. With ${\psi }_0=Z_2^{1/2}\psi$ and $A_0=Z_3^{1/2}A$, the vertex coefficient gives $e_0={\mu }_R^{ϵ}{Z}_{1}e/(Z_2\sqrt{Z_3})$, proving the displayed charge relation in $d=4-2ϵ$. Gauge invariance restricts counterterms but does not forbid charge running.

With interaction $-{\lambda }_0{\varphi }^4/4!$, the tadpole diagram has symmetry factor one half. After Wick rotation and a rotationally invariant Euclidean cutoff, its self-energy is $({\lambda }_0/2){\int }_{|k|<\Lambda }{d}^{4}k(2\pi {)}^{-4}(k^2+m_0^2{)}^{-1}$. The sphere area $2{\pi }^2$ and radial integral ${\int }_0^{\Lambda }{k}^{3}dk/(k^2+m_0^2)=\frac{1}{2}[{\Lambda }^2-m_0^2\log (1+{\Lambda }^2/m_0^2)]$ give the displayed shift. It is independent of external momentum, so no one-loop momentum-dependent wave-function renormalization arises from this graph. Different regulator schemes can treat power divergences differently while defining the same chosen pole mass.