 Pole residue

The coefficient of a simple one-particle pole in a propagator is its pole residue. If $G(p)=i/[p^2-m_0^2-\Sigma (p^2)]$ and $m$ is the pole mass, then $Z_{\mathrm{pole}}=[1-{\Sigma }^{\prime }(m^2){]}^{-1}$. An on-shell renormalization scheme may impose unit residue by a finite or divergent wave-function renormalization; mere ultraviolet finiteness does not force the residue to be one.