Background gauge invariance of a current generating functional ($Z[B+\partial \alpha ]=Z[B]$)

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.