= Finite tensor-integral remainder in dimensional regularization
{title2=$\lim_{D\to4}(4/D-1)\int d^Dk\,k^2/(k^2-\Delta+i0)^3=i/(32\pi^2)$}
Include the measure factor $(2\pi)^{-D}$ and a conventional <renormalization scale> factor in the displayed integral. The <Lorentz invariant> replacement $k_\mu k_\nu\mapsto g_{\mu\nu}k^2/D$ must be performed before taking $D\to4$. For $D=4-\varepsilon$, its scalar integral has pole $i/(8\pi^2\varepsilon)$, while $4/D-1=\varepsilon/4+O(\varepsilon^2)$. Their product has the finite value shown. Setting $D=4$ before integration discards this value and can spoil a <Ward identity>, even in a final ultraviolet-finite <scattering amplitude>. This is an elementary example of a vanishing regulator factor multiplying a divergent integral.
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