Anomalous K-K*-γ interaction and similar D-D*-γ and B-B*-γ?

In trying to better understand the chiral-anomaly WZW-term interaction of the heavy vector mesons, I have trouble finding much relevant literature, but maybe I am looking in the wrong place.

One useful article I found is this one:

In equation (3) they state a WZW summand proportional to this term:

$$A \wedge d K^+ \wedge d K^- \wedge d \pi^0 \phantom{AAAAAAAA} (3)$$

$$K^*$$ could be regarded as a pair of $$K$$ and $$\pi$$

and then they list electromagnetic interactions, of which their equation (9) is the term proportional to this term:

$$d A \wedge \big( d K^+ \wedge {K^*}^- - d K^- \wedge {K^*}^+ \big) \phantom{AAAAAAA} (9)$$

Question 1: Is (9) supposed to be the result of (3) after "regarding $$K^*$$ as a pair of $$K$$ and $$\pi$$" (and then doing an integration by parts)?
• Shouldn't $\pi^0$ come with a $d$ since in their paper it comes with a derivative? Also, how should we interpret the pair here? Are we supposed to add $K$ and $\pi$ or wedge them? Apr 11 '20 at 16:19
• Regarding the missing $d$: yes thanks, fixed now. Regarding the pairing: that's part of my question, so I don't know. Apr 11 '20 at 16:35
• Indeed, the bulk of the decays of K* go through $K\partial_m \pi$. Have you looked at similar anomalous vertices involving a ρ and a π ? May 1 '21 at 17:00