Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 304 / 3 / d / Solution

Taking $q\to 0$ in the Ward-Takahashi identity givesup to the displayed Euclidean $i$ conventions. The ultraviolet divergence of the zero-momentum vertex is therefore exactly the derivative of the fermion self-energy divergence. In renormalization-constant notation this isso the vertex and fermion wave-function renormalization are not independent. The remaining charge renormalization is controlled by photon wave-function renormalization; in a common convention $e_0=Z_3^{-1/2}e$.

The derivation used only an exact change of variables, invariance of the action, and invariance of the measure. It therefore holds nonperturbatively whenever the regulator and definition of the theory preserve the vector symmetry. A symmetry-breaking regulator requires symmetry-restoring counterterms; a genuine quantum anomaly would obstruct the identity, but vector QED has no such anomaly.