# What is the difference between Chiral anomaly, ABJ anomaly, and Axial anomaly?

+ 3 like - 0 dislike
452 views

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?

This post imported from StackExchange Physics at 2014-11-15 11:17 (UTC), posted by SE-user Eric
Chiral, ABJ, Axial, Gamma5, Triangle anomaly - I think these are the same. Please see "Chiral anomaly" in the wikipedia.

This post imported from StackExchange Physics at 2014-11-15 11:17 (UTC), posted by SE-user noel_lapin

+ 5 like - 0 dislike

The chiral anomaly, axial or the ABJ anomaly (Adler-Bell-Jackiv anomaly) is the type of $U(1)$ global anomalies found in some quantum field theories where some currents are not conserved (they do not follow the typical formula $dJ=0$). This happens when the spinorial representations (fermions) classically exhibit a chiral symmetry (the theory is valid for left- and right-handed representations, i.e. Dirac equation for chiral massless fermions has a chiral symmetry ) nut upon quantization it is lost (because in the path integral, the measure $[D \psi]$ does not have the corresponding symmetry). Usually the anomaly factor is related to some topological invariant and is associated with tunneling and instantons (and the Chern number of the vacuum etc etc...). Another way to express it is to say that the chiral anomaly is a pathology of one-loop fermion triangle diagrams at least at one of the three vertices of the diagram.

The usual example is the chiral anomaly of the Standard Model which results in having only left-handed fermions (right-handed anti-fermions) interacting in the electro-weak processes. The same thing does not happen for QCD though because QCD is invariant under parity. Another example where the chiral anomaly shows up is the neutral pion decay to two photons.

You can find much better information at Peskin & Schroder or similar introductory, and not only, QFT books. If you do not find this answer satisfying enough I could try to throw some maths in but let me know since you can find plenty of information on the subject.

answered Nov 16, 2014 by (3,605 points)
edited Nov 18, 2014

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysics$\varnothing$verflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.