I am confused on some identities in Mark Srednicki's QFT book, section76. Let $j^{\mu}=\bar{\psi}\gamma^{\mu}\psi$ and $j^{\nu}_A=\bar{\psi}\gamma^{\nu}\gamma_5\psi$ be $U(1)$ and $U(1)_A$ current, respectively. Then we have three identities (see eqs. (76.17)-(76.19) in the book)
$$ \begin{aligned} &\partial_{\mu} \langle j^{\nu}(x)j^{\nu}(y)j^{\rho}_A(z) \rangle=0 \cr &\partial_{\nu} \langle j^{\nu}(x)j^{\nu}(y)j^{\rho}_A(z) \rangle=0 \cr &\partial_{\rho} \langle j^{\nu}(x)j^{\nu}(y)j^{\rho}_A(z) \rangle=0 \end{aligned} $$
In the next, it says
Note that there are no contact terms in eqs. (76.17)-(76.19), because both $j^{\mu}$ and $j^{\mu}_A$ are invariant under both $U(1)$ transformation.
This is my confusion: why does such invariance give vanishing contact terms?
