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 eq(76.17)-(76.19) in the book) \begin{equation} \begin{aligned} &\partial_{\mu} \langle j^{\nu}(x)j^{\nu}(y)j^{\rho}_A(z) \rangle=0\\ &\partial_{\nu} \langle j^{\nu}(x)j^{\nu}(y)j^{\rho}_A(z) \rangle=0\\ &\partial_{\rho} \langle j^{\nu}(x)j^{\nu}(y)j^{\rho}_A(z) \rangle=0\\ \end{aligned} \end{equation} 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?

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?