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Consider pure QED with massless electrons. Due to the axial anomaly the axial current is not conserved:

$$ \tag 1 \partial_{\mu}J^{\mu}_{5} \sim F_{\mu\nu}\tilde{F}^{\mu\nu} $$ On the other hand, it seems that this non-conservation has nothing to do with the particle physics processes, as there is no axial field coupled to $J^{\mu}_{5}$ in the lagrangian.

What is then the physical manifestation of the equation (1)? What are its observable consequences?

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This question 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
$$ \frac{e^2 N_c \pi^0}{48 \pi^2 f_\pi} F_{\mu\nu}\tilde{F}^{\mu\nu}. $$ It therefore induces neutral pion decay to two photons, quite observable and physical, really. enter image description here

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

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  • $\begingroup$ Thanks. I see that in the case of scalars transforming non-trivially under the axial group it is possible to write the non-local anomalous action in the theory with quarks in terms of the local effective lagrangian in a theory with mesons, as the anomaly is scale invariant. Such lagrangian leads to new processes like pion decay, a change of the mass of $\eta'$ ets. But does the axial anomaly manifest itself if there are no such scalars, with fermions only? $\endgroup$
    – Name YYY
    May 23 '20 at 21:48
  • $\begingroup$ Well, to start with, the relevant symmetry (neutral axial generator) is SSBroken, so there Is a Goldstone boson around, whether you can identify it or not. The starting matrix element I write does that: connects a fermion current to the pion, so you can "simulate" the fermion bilinear current as the gradient of the goldston as shown. If the current divergence is now nonzero, as here, you fold this into the picture, as here... $\endgroup$ May 23 '20 at 22:34
  • $\begingroup$ Thanks. By the way, could you please tell me whether the PCAC story has anything to do with the "formula" $\pi^{0} \sim \bar{u}u -\bar{d}d$? I.e., has the mentioned formula any sense (apart from that the pion belongs to the octet representation of the $SU(3)$)? If yes, does it follow from the PCAC? $\endgroup$
    – Name YYY
    May 23 '20 at 22:54
  • $\begingroup$ No, PCAC is not that relevant here. (It just indicates there is yet another piece in the divergence of the above axial current, not due to an anomaly, but due to an explicit breaking of this symmetry by minuscule u and d quark masses... which gives the pion a small mass... skipped here for clarity.) The octet "wave function" for the pion you wrote knows nothing about PCAC... $\endgroup$ May 23 '20 at 23:02

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