Counting number of gluon fields in ABJ anomaly of QCD

Question

In QCD, the axial singlet quark current $j_5$ is anomalous, which was found by Adler, Bell, and Jackiw (here is a review):

\begin{equation} \partial_\mu j_5^\mu=G\tilde{G}, \end{equation}

where $G$ is the gluon field strength and $\tilde{G}$ its dual. As QCD is a non-Abelian gauge theory, $G\tilde{G}$ contains terms with up to four gluon fields $A$.

However, the anomaly arises from the computation of triangle diagrams with only two external gluon lines.

Now my question is: How can we obtain the above anomaly from such a diagram, even though the term $G\tilde{G}$ involves terms with more than two gluons fields? Do the terms with higher order in $A$ somehow cancel out in the computation?

Answer 1

The ABJ anomaly is determined by the triangle diagram, in the sense that once we've calculated the triangle diagrams we can immediately infer the whole result, but the box diagram also contributes if the gauged group is nonabelian. This is stated explicitly in Bertlmann (1996), Anomalies in Quantum Field Theory, section 4.7:

...we get two contributions for the singlet anomaly, the triangle and the quadrangle diagrams. But these are not independent of each other. If, for example, the triangle result vanishes, the quadrangle result is zero too. Hence it is enough to consider just the simple triangle diagram in search of an anomaly.

The last sentence explains why some less-explicit texts might leave the impression that the triangle diagram is the only one that contributes.

For comments about the role of counterterms (in the context of the nonabelian anomaly instead of the singlet anomaly), see The non-abelian chiral anomaly and one-loop diagrams higher than the triangle one