Can somebody provide a fairly brief explanation of what the chiral anomaly is? I have been unable to find a concise reference for this. I know QFT at the P&S level so don't be afraid to use math.

# [Physics] Simple explanation of chiral anomaly

chiralityquantum-anomaliesquantum-field-theory

#### Related Solutions

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

**1. How can we show that $\partial\cdot j\equiv 0$ at the quantum level?**

For example, by showing that the Ward Identity holds. It should be more or less clear that the WI holds if and only if $\partial\cdot j=0$. There are multiple proofs of the validity of the WI; some of them assume that $\partial\cdot j=0$, and some of them use a diagrammatic analysis to show that the WI holds perturbatively (and this is in fact how Ward originally derive the identity, cf. 78.182). It is a very complicated combinatiorial problem (you have to show inductively that an arbitrary diagram is zero when you take $\varepsilon^\mu\to k^\mu$), but it can be done. Once you have proven that the WI holds to all orders in perturbation theory, you can logically conclude that $\partial\cdot j\equiv 0$. For a diagrammatic discussion of the WI, see for example Bjorken & Drell, section 17.9. See also Itzykson and Zuber, section 7-1-3. For scalar QED see Schwartz, section 9.4.

Alternatively, you can also show that $\partial\cdot j=0$ by showing that the path integral measure is invariant (à la Fujikawa) under global phase rotations. This implies that the vector current is not anomalous.

**2.a. How does the unphysical photon polarization states appear in the theory through anomaly?**

Take your favourite proof that the WI implies that the unphysical states do *not* contribute to $S$ matrix elements, and reverse it: assume that $\partial \cdot j\neq 0$ to convince yourself that now the unphysical states *do* contribute to $S$ matrix elements. Alternatively, make up your own modified QED theory using a non-conserved current and check for yourself that scattering amplitudes are not $\xi$ independent.

**2.b. And how do their appearance violate the unitarity of the theory?**

Morally speaking, because unphysical polarisations have negative norm. If the physical Hilbert space contains negative-norm states, the whole paradigm of probability amplitudes breaks down.

**3. Why would the vector current anomaly be a problem in QED but not the chiral current anomaly?**

Because in pure QED the axial current is not coupled to a gauge field, and therefore its conservation is not fundamental to the quantum theory. The axial anomaly in pure QED would be nothing but a curiosity of the theory (a nice reminder that classically conserved current need not survive quantisation).

On the other hand, in QED the vector current is coupled to a gauge field, the photon field, and as such its conservation is crucial to the consistency of the theory: without it the WI fails, and therefore we lose unitarity (or covariance, depending on how you formulate the theory).

## Best Answer

There is a very simple and enlightening explanation due to N.V.Gribov given in his following conference article and also beautifully explained by Dmitri Kharzeev in the following arXiv article (section 1). Gribov's argument doesn't involve the heavy machinery of quantum field theory. He actually proves that in the case of colinear electric and magnetic fields acting on gapless fermions, there is a net flow of chiral charge from the Dirac sea as dictated by the anomaly equation.