Scalar-QED four-point vertex pole in transverse gauge ($Z_4=1+3e^2/(8{\pi }^2ϵ)+O(e^4)$)

In $D=4-ϵ$, the two surviving zero-momentum seagull bubbles contribute $-4e_D^4{\int }_l{P}^{\mu \nu }(l)/[l^2(l^2-m^2)]+(\mu \leftrightarrow \nu )$. Tensor averaging gives $(D-1)/D$, and the scalar integral has pole $i/(8{\pi }^2ϵ)$. Cancellation against $2i{e}_D^2(Z_4-1)g^{\mu \nu }$ gives the displayed result. A positive scalar mass separates the ultraviolet pole from the scaleless massless limit. The scalar QED Ward identity gives $Z_4=Z_2$ in this convention.