= Scalar QED wave-function pole
{title2=$\delta Z_\phi=e^2(3-\xi)/(16\pi^2\epsilon)$}
In <scalar electrodynamics> with <photon propagator> $-i[g_{\mu\nu}-(1-\xi)l_\mu l_\nu/l^2]/(l^2+i0)$ and <dimensional regularization> $d=4-2\epsilon$, the <minimal subtraction scheme> gives $Z_\phi=1+e^2(3-\xi)/(16\pi^2\epsilon)+O(e^4)$. For an external <four-momentum> $p$, expand the exchange numerator as $4p^2-4\xi p\cdot l+\xi l^2-4(1-\xi)(p\cdot l)^2/l^2$. The logarithmic <ultraviolet poles> of the scalar, vector and tensor integrals give the $p^2$ coefficient $4-2\xi-(1-\xi)=3-\xi$. A massless <photon> <tadpole integral> does not change it. The <scalar QED Ward identity> gives the same vertex <renormalization constant>.
Back to article page
