= Scalar-QED four-point vertex pole in transverse gauge
{title2=$Z_4=1+3e^2/(8\pi^2\epsilon)+O(e^4)$}
In $D=4-\epsilon$, the two surviving zero-momentum seagull bubbles contribute $-4e_D^4\int_lP^{\mu\nu}(l)/[l^2(l^2-m^2)]+(\mu\leftrightarrow\nu)$. Tensor averaging gives $(D-1)/D$, and the scalar integral has pole $i/(8\pi^2\epsilon)$. Cancellation against $2ie_D^2(Z_4-1)g^{\mu\nu}$ 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.
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