Massless scalar bubble pole
= Massless scalar bubble pole
{title2=$f_d(P^2)\sim\frac{2}{16\pi^2(4-d)}$}
With a mostly-plus <metric signature>, define $f_d=(1/i)\int d^dk/[(2\pi)^d(k^2-i\epsilon)((P-k)^2-i\epsilon)]$. <Wick rotation> and a <Feynman parameter> give $f_d=\Gamma(2-d/2)(4\pi)^{-d/2}\int_0^1[x(1-x)P^2]^{d/2-2}dx$. At nonzero Euclidean $P^2$, the <Gamma function> supplies the pole. Exceptional massless <momentum> can introduce infrared ambiguity.