[This question](https://physics.stackexchange.com/questions/547195/derivation-of-pcac-condition-langle-0j5-mu-ax-pi-bp-rangle-if-pi) reminds you that 
$$\langle 0|J^5_{\mu,0}(x)|\pi^0(p)\rangle=-if_{\pi} e^{-ipx}p_{\mu},$$
the mother of PCAC. That is to say, you *already know* this axial current corresponds to a SSB generator, and so ***is linear in the Goldstone boson corresponding to it***,
$$
J^5_{\mu,0}\propto f_\pi \partial_\mu \pi^0 + ...
$$
where the ellipsis represents terms of higher order in the fields. 

The current is basically the goldston: The corresponding charge pumps such goldstons into and out of the chirally non-invariant vacuum! 

As a consequence, the corresponding term of the effective Lagrangian which gives you the above current divergence is proportional to 
$$
\frac{\pi^0}{f_\pi}F_{\mu\nu}\tilde{F}^{\mu\nu}.
$$
It therefore induces neutral pion decay to two photons, quite observable and physical, really.

This term was an early reassurance of the genius of the WZWN term of flavor-chiral anomalous effective actions.