= Ultraviolet pole of QED vacuum polarization
{title2=$\Pi_{\rm UV}=-e^2/(12\pi^2\epsilon)$}
For one unit-charge <Dirac field> in $d=4-2\epsilon$, define the insertion as $i\Pi_{\mu\nu}=i(q^2g_{\mu\nu}-q_\mu q_\nu)\Pi$. The one-loop <ultraviolet pole> is $\Pi_{\rm UV}=-e^2/(12\pi^2\epsilon)$. The <Ward identity> cancels any apparent divergent vector <mass> term. With photon <counterterm> insertion $-i(q^2g_{\mu\nu}-q_\mu q_\nu)\delta_3$, this gives $\delta_3=-e^2/(12\pi^2\epsilon)$ in the <minimal subtraction scheme>.
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