= Background gauge invariance of a current generating functional
{title2=$Z[B+\partial\alpha]=Z[B]$}
For a nonanomalous <Dirac current> $j^\mu$, introduce $Z[B]=\int\mathcal D\bar\psi\mathcal D\psi\,e^{iS+i\int B_\mu j^\mu}$. The local phase change of variables has unit <functional Jacobian>, while its kinetic variation cancels the shift of the source. Thus $Z[B+\partial\alpha]=Z[B]$. At first order, <integration by parts> yields $\partial_\mu\delta Z/\delta B_\mu=0$. Charged sources transform too when deriving the full <Ward-Takahashi identity> with <Ward identity contact terms>.
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