Ultraviolet pole of the QED vertex (${\Lambda }_{\mathrm{UV}}^{\mu }=e^2{\gamma }^{\mu }/(16{\pi }^2ϵ)$)

Write the QED vertex as $-ie({\gamma }^{\mu }+{\Lambda }^{\mu })$. In Feynman gauge and $d=4-2ϵ$, the one-loop ultraviolet pole has the local form ${\Lambda }_{\mathrm{UV}}^{\mu }=e^2{\gamma }^{\mu }/(16{\pi }^2ϵ)$. Its coefficient follows from the large-momentum gamma matrices contraction ${\gamma }^{\alpha }/k{\gamma }^{\mu }/k{\gamma }_{\alpha }\mapsto (d-2{)}^2{k}^2{\gamma }^{\mu }/d$. It is independent of external kinematics, whereas the full finite vertex requires additional structures off shell. Its cancellation verifies the Ward identity for QED renormalization constants.

For the QED proper Feynman vertex with both external four-momenta zero, a massive internal fermion and a massless internal photon in Feynman gauge, the even numerator reduces under Lorentz covariance to ${\gamma }^{\mu }[\alpha k^2+\beta m^2]$. The gamma matrices identity ${\gamma }^{\nu }{\gamma }^{\mu }{\gamma }_{\nu }=(2-D){\gamma }^{\mu }$ and the integrated replacement $k_{\rho }{k}_{\sigma }\mapsto g_{\rho \sigma }{k}^2/D$ prove the coefficients. The term $m^2/[k^2(k^2-m^2+i0{)}^2]$ has neither an infrared divergence nor an ultraviolet divergence for $m>0$; its integral in four dimensions is $-i/(16{\pi }^2)$. Only the remaining squared massive denominator supplies the ultraviolet pole.