On the Axial Anomaly I know that if we start with a massive theory, the chiral states $L$ and $R$ remain coupled to each other in the massless limit. Because a charged Dirac particle of a given helicity can make a transition into a virtual state of the opposite helicity by emitting a real photon (this is the physical origin of the anomaly). Also, the masslessness of a Dirac field theory is expressed by the invariance under a chiral transformation
$$\psi(x)\rightarrow\mathrm{e}^{-i\omega\gamma_5}\psi(x).$$
From Noether theorem, the chiral invarince gives a conserved axial-vector current 
$$j_{5}^{\mu}=\bar{\psi}\gamma^{\mu}\gamma_{5}\psi,$$ 
and from the E.o.M. for Heisenberg fields we find
$$\partial_{\mu}j_{5}^{\mu}=2mj_{5},$$
where $j_5$ is the chiral density and $m$ is the mass. Here is what I don't understand: I expect that in the limit $m\rightarrow 0$ this should be true $\partial_{\mu}j_{5}^{\mu}\rightarrow 0$. But its not. All the text books give 
$$\partial_{\mu}j_{5}^{\mu}=2mj_{5}+\frac{\alpha_0}{2\pi}\bar{F}^{\mu\nu}F_{\mu\nu}.$$
I can't figure out this result. I don't know how to derive it or what its physical interpretation is.
 A: Let me add a few comments to Michael Brown's answer/comment. As he mentioned, a QFT is well defined with an action $and$ a regulator. We always wish to use regulators that preserve gauge invariance, since that is a redundancy of our description and should not be removed in our quantum theory. However, any regulator that preserves gauge invariance, necessarily violates chiral invariance. P&S mentions the possibility of having gauge non-invariant regulators that preserve chiral invariance but this is not a desirable definition of our theory. 
Another way to see this, is that the usual regulators used to define a theory is dimensional regularization and Pauli-Villars. PV requires introducing a (large) fermion mass and it explicitly breaks chiral symmetry. The problem with Dimreg is more subtle. The chiral symmetry involves the $\gamma^5$ matrix which is only well defined in $d=4$. When one extends dimensions to $d=4-\epsilon$, one has to be careful with the treatment of $\gamma^5$. It turns out that while chiral symmetry is restored in the 4 dimensions it is not in the $-\epsilon$ dimensions (mathematically). This is what gives us the axial anomaly. P&S has a discussion on how to treat the $\gamma^5$ matrix in $d=4-\epsilon$.
All this is discussed in Chapter 19 of P&S
A: You asked the right question, I was thinking of it too when reading about axial anomaly.
Below is how I explained it to myself.
As you mentioned in your question, "... a charged Dirac particle of a given helicity can make a transition into a virtual state of the opposite helicity by emitting a real photon (this is the physical origin of the anomaly)".
Please note that
$\bar{F}^{\mu\nu}F_{\mu\nu}= - 2 \bf{E}\bf{B}$
where $\bf{E}$ and $\bf{B}$ are electric and magnetic field strengths.
In the case of photon $\bf{E} \perp \bf{B}$, and hence $ \bf{E}\bf{B} = 0$.
Consequently, in zero mass limit the chiral symmetry is conserved:
$\partial_{\mu}j_{5}^{\mu}=\frac{\alpha_0}{2\pi}\bar{F}^{\mu\nu}F_{\mu\nu}=0$.
EDIT:
In fact, the spinorial equation for massless fermions (such as Dirac equation where $m=0$) is always equivalent to source-free Maxwell equations. See, e.g., this link.
