Massive scalar bubble pole in four dimensions ($\int \frac{d^{4-ϵ}k}{(2\pi {)}^{4-ϵ}}\frac{1}{(k^2+m^2)((p-k{)}^2+m^2)}=\frac{1}{8{\pi }^2ϵ}+O(1)$)

For positive $m^2$, a Feynman parameter and a momentum shift turn the integral into ${\int }_0^1{I}_{d,2}(\sqrt{m^2+x(1-x)p^2})dx$. The Euclidean massive loop integral has factor $\Gamma (ϵ/2)=2/ϵ+O(1)$, while its other factors tend to $(4\pi {)}^{-2}$. The pole is independent of the external momentum and the mass, so it is removable by a local counterterm.