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In the computation of the Adam-Bell-Jackiw anomaly, in Peskin and Schroeder's book they proceed with a regularization of $\partial_\mu j^{\mu 5}$ by evaluating the fields at different spacetime points and inserting a Wilson line to render the whole expression gauge invariant. So far so good, they proceed to calculate the anomaly in 4D and it works just fine (p.659 to 661). Then on p.661 they say that "We can confirm the ABJ anomaly relation by checking that the divergence of the axial vector current has a nonzero matrix element to create two photons. To do this, we must analyze the matrix element

$$\int d^4x \, e^{-iqx} \langle p,k|j^{\mu5}(x)|0\rangle = (2\pi)^4\delta(p+k-q) \epsilon^*_v(p) \epsilon^*_ \lambda(k) \mathcal{M}^{\mu\nu\lambda}(p,k)"$$

and represent such expression with triangle diagrams. My questions are

1) What does that diagram tell you, what actually creates such photons, and how does it relate to the anomaly?

2) Why do we represent such expectation value with triangle diagrams?

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One node of the triangle is given by the current $j^{\mu 5}$ with momentum $q$, the other two nodes represent photons with momenta $p$ and $q$. The current $j^{\mu 5}$ has two fermions and one photon. The fermions give two of the internal lines of the diagram.

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  • $\begingroup$ So the feynman rules for those diagrams are not the same as those for QED? $\endgroup$
    – user35319
    Jan 6 '20 at 13:14
  • $\begingroup$ Yes they are. Why would they be different? $\endgroup$ Jan 6 '20 at 13:20
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    $\begingroup$ Because there's no feynman rule in QED with $j^{\mu5}$ as a node $\endgroup$
    – user35319
    Jan 6 '20 at 13:22

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