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I get confuse with these three terms: Chiral anomaly, ABJ anomaly, and Axial anomaly. I can not find standard definition of these three. Is there anyone can describe precisely?

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ANS: It is basically the same. All of them are anomalies for chiral fermion in even dimensional spacetime, like 1+1D, 3+1D, etc. Or, in odd dimensional space.

  1. ABJ anomaly is named after the discovers: Adler-Bell-Jackiw.

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  1. Chiral anomaly is named after and implies that the chiral current is not conserved. Chiral, means the left and the right. So the chiral current means the left current $j_L$ and the right current $j_R$. Chiral anomaly is defined to have both

$$\partial_{\mu} j_L^{\mu} \neq 0 \;\;\;\;\text{ and }\;\;\;\; \partial_{\mu} j_R^{\mu} \neq 0, \;\;\;\;\text{ and }\;\;\;\; \partial_{\mu} j_L^{\mu}=-\partial_{\mu} j_R^{\mu} \neq 0$$

  1. Axial anomaly is named after and implies that the axial current is not conserved. Axial, is given that name to compared to the vector. The axial current is $j_A=(j_L-j_R)$. The vector current is the total current summing over left and right $j_V=(j_L+j_R)$. The axial anomalies is defined as $\partial_{\mu}(j_L-j_R)^{\mu} \neq 0$. Since the vector current must be conserved, namely $\partial_{\mu} j_V^{\mu}=0 \Rightarrow \partial_{\mu} (j_L+j_R)^{\mu}=0$, thus what axial anomalies $\partial_{\mu}(j_L-j_R)^{\mu} \neq 0$ implies that

$$\partial_{\mu} j_L^{\mu}=-\partial_{\mu} j_R^{\mu} \neq 0$$

Below are their more detailed mathematical relations. $F$ is the field strength of 1-form $A$ gauge field. In general even $d$-dimensional spacetime, like 1+1D, 3+1D, 5+1D, etc,

$$\partial_{\mu}\, j_{\textrm{L}}=K{F}^{d/2}$$ $$\partial_{\mu}\, j_{\textrm{R}}=-K{F}^{d/2}$$ $$\partial_{\mu}\, j_{\textrm{A}}=\partial_{\mu}\, (j_{\textrm{L}}- j_{\textrm{R}})=2K{F}^{d/2}$$

${F}^{d/2}\equiv F \wedge \dots \wedge F$, of $d/2$ copies of F in wedge product.

In particular, in 1+1D: $$\partial_{\mu}\, j_{\textrm{L}}= \frac{k}{4\pi}\; \varepsilon^{\mu\nu}\,F_{\mu\nu} =\frac{k}{2\pi} \, \varepsilon^{\mu\nu}\,\partial_{\mu} A_{\nu} = \sigma_{xy}\,E_{x}=+J_{y},$$ $$ \partial_{\mu}\, j_{\textrm{R}}=- \frac{k}{4\pi}\; \varepsilon^{\mu\nu}\,F_{\mu\nu} =-\frac{k}{2\pi} \, \varepsilon^{\mu\nu}\,\partial_{\mu} A_{\nu}= -\sigma_{xy}\,E_{x}=-J_{y}. $$

The last part of the formula with $\sigma_{xy}\,E_{x}=J_{y}$, is to compare with the bulk-edge correspondence of Chern-Simons theory in high energy (H.E.) or quantum Hall physics in condensed matter (C.M.). You can skip that part if you are not familiar.

$\bullet$ To introduce the notation and the physics further, you can read this paper from the discussion on chiral fermions in High Energy Physics and C.M.. Also, see its Appendix.

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In 1969 Adler [https://inspirehep.net/record/55000] and, Bell & Jackiw [https://inspirehep.net/record/54998] showed that in the UV divergent triangle Feynman diagram made up of one axial and two vector currents, only the vector current is conserved, whereas the conservation law of the axial current is broken. Hence the names ABJ or chiral anomaly.

$$\partial^\mu j_\mu^5 = \mathcal{A}$$ where $\mathcal{A}$ represents the celebrated Adler-Bell-Jackiw (ABJ) anomaly is given by $$\mathcal{A} = \frac{e^2}{16 \pi^2} \epsilon^{\mu \nu \alpha \beta} F_{\mu \nu} F_{\alpha \beta}.$$

Perhaps the monograph 'Anomalies in Quantum Field Theory' by Bertlmann is the best reference on this subject.

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