Background gauge-field renormalization identity ($Z_g{Z}_A^{1/2}=1$)

For a background gauge field with $A_0=Z_A^{1/2}A$ and $g_0={\mu }^{ϵ/2}{Z}_{g}g$, background gauge covariance preserves the combination $gA$ in the gauge covariant derivative. Its Ward identity therefore gives the displayed relation. This refers to background-field renormalization, not an unrestricted identity between the coupling and the ordinary quantum-gluon wave-function factor. Gauge-invariant one-loop matter determinants can be renormalized with this identity to extract their coupling running from vacuum polarization.