= Ward identity for QED renormalization constants
{c}
{title2=$Z_1=Z_2,\quad e_0=\mu_R^\epsilon Z_3^{-1/2}e$}
In a gauge-preserving regularization and consistent <renormalization> convention, the <proper-vertex Ward-Takahashi identity in QED> equates the vertex and electron <wave-function renormalization> constants. With $\psi_0=Z_2^{1/2}\psi$ and $A_0=Z_3^{1/2}A$, the vertex coefficient gives $e_0=\mu_R^\epsilon Z_1e/(Z_2\sqrt{Z_3})$, proving the displayed charge relation in $d=4-2\epsilon$. <Gauge invariance> restricts <counterterms> but does not forbid charge running.
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