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This is actually about two related doubts and I hope is appropriate for a single question (if not, I will happily divide it). So, my problems are related to the analysis and calculation of chiral anomalies in QFT (and are independent of a specific scenario).

(i) Reading a paper of Gross and Jackiw on the topic, it is said that was the chiral current $J_{5\mu}$ to be conserved also in the quantum case, then the ward identity wouldn't be spoiled (which I understand), but then proceed to write this statement explicitly as

$$ p^\mu\int d^4x \;d^4y \; e^{ipx}e^{iqy} \langle0|T{J_{5\mu}(x)J_{5\nu}(y)J_{5\alpha}(0)}|0\rangle=0. $$

What I don't understand is why the currents' Green function (and not the fields') also obey the same kind of ward identity and have never seen it be derived. Why is that? Is it actually a consequence of the 'rightful' Green function Ward identity?

(ii) He then goes on to show that this is actually false, implying that the chiral current is not conserved. He does it in the normal way, that is, setting the above fourier transform of a correlator equal to the triangle diagram (and permutation). This is a basic thing I have never understood: how can we write a diagram for a current expected value? Is just brute force, that is, writing e.g. $J_{5\mu}(x)=e\bar{\psi}(x)\gamma_\mu \gamma_5 \psi(x)$ and evaluating the expected value by expanding $J_{5\mu}(x)J_{5\nu}(y)J_{5\alpha}(0)e^{-iS_{\text{int}}}$ and applying Wick's theorem? By the way that (literally) every author I've read so far made it look as if it was obvious I feel like there some intuition I'm missing.

I suspect a lot of this is related to the Schwinger-Dyson equation and probably points to my poor understanding of the Ward identities, but I cannot pin it down. Any related insights are also greatly appreciated.

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