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There is one one-particle-irreducible Feynman diagram: an internal photon joins two points of the open fermion line, with the external photon attached to that line between them. It is the upper-left panel of the figure. Route internal photon four-momentum $k$ from the incoming side to the outgoing side; the adjacent internal fermion four-momenta are $p-k$ and $p^{\prime }-k$. The symmetry factor is one.

The vertex renormalization counterterm is an additional local contribution when forming the renormalized Feynman vertex. External-leg self-energy insertions belong to the complete three-point correlation function, rather than this amputated one-particle-irreducible Feynman diagram.

Let the bare fields obey the wave-function renormalization definitions ${\psi }_0=Z_2^{1/2}\psi$, $A_{0\mu }=Z_3^{1/2}{A}_{\mu }$, and put $m_0=Z_{m}m$. Define the vertex renormalization factor by $e_0={\mu }^{ϵ/2}e{Z}_1{Z}_2^{-1}{Z}_3^{-1/2}$. Substitution into the bare QED Lagrangian density givesEquivalently, write $Z_i=1+{\delta }_i$ and ${\delta }_m=Z_2{Z}_m-1$. The counterterm Lagrangian isFor the subsequent Feynman gauge calculation include $-(\partial \cdot A{)}^2/2$ in the gauge-fixed action. The Ward identity for QED renormalization constants implies $Z_1=Z_2$ when the regularization preserves gauge invariance; this is a consequence, rather than an extra independent definition of the renormalization constants.

The scalar QED Ward identity equates the vertex renormalization and wave-function renormalization constants. Consequently, using part (iv),The renormalization constants describe gauge-dependent fields and off shell vertices, so their dependence on $\xi$ is permitted. A physical scattering amplitude includes all Feynman diagrams, the counterterms, and the external-state factors from LSZ reduction. The Ward identities ensure that the dependence on $\xi$ cancels for physical external states. Infrared-sensitive probabilities also require the usual inclusive treatment of unobserved soft photons. Thus no dependence on $\xi$ is implied for consistently calculated physical observables.