= Massive scalar bubble pole in four dimensions
{title2=$\int\frac{d^{4-\epsilon}k}{(2\pi)^{4-\epsilon}}\frac1{(k^2+m^2)((p-k)^2+m^2)}=\frac1{8\pi^2\epsilon}+O(1)$}
For positive $m^2$, a <Feynman parameter> and a momentum shift turn the integral into $\int_0^1I_{d,2}(\sqrt{m^2+x(1-x)p^2})\,dx$. The <Euclidean massive loop integral> has factor $\Gamma(\epsilon/2)=2/\epsilon+O(1)$, 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>.
Back to article page
