Suppose the fermion theory with the global group $G$ spontaneously broken to $G' \in G$. Below the SSB scale we integrate the fermions out and are leaved with Goldstone bosons $\varphi_{i}$ parametrizing the coset $H \simeq G/G'$. These bosons are in the non-linear representation of the $G$ group, which is given by $$ U = \text{exp}\left( i\varphi_{i}t_{i}\right) $$

Suppose the group $G$ has non-trivial anomalous structure; i.e., $$ \tag 1 D_{abc} \equiv \text{tr}[t_{a}\{t_{b},t_{c}\}]^{L} - L \leftrightarrow R\neq 0, $$ where $t_{a}$ is the generator of the fermion representation in $G$. Then due to t'Hooft anomaly matching condition the effective field theory with the action $\Gamma[U]$ has to reproduce $(1)$. This is done by the Wess-Zumino term, being $$ \Gamma_{\text{WZ}} = \frac{in}{240\pi^{2}}\int d^{5}x \epsilon^{ijklm}\text{tr}\left[ U\partial_{i}U^{-1}U\partial_{j}U^{-1}U\partial_{k}U^{-1}U\partial_{l}U^{-1}U\partial_{m}U^{-1}\right], $$ where $U$ is extended on the 5-dimensional space with given topology (this is possible if $\pi_{4}(G/G') = 0$).

My question is following. In the case of underlying theory with fermions the internal anomaly $(1)$ doesn't cause observed phenomena. Does it cause some observables for the effective field theory given on the language of $U$? An example of the given theory is the QCD with the chiral global symmetry $SU_{L}(3)\times SU_{R}(3)$ spontaneously broken down to $SU_{V}(3)$, with Goldstones being mesons? I know that it describes processes such as $KK \to 3\pi$, and people call them ''anomalous''. But as far as I know they are anomalous just because they are forbidden because of extra symmetries (which are absent in underlying quark symmetry) of naive chiral perturbation theory.


1 Answer 1


You got it very wrong. Specifically, "In the case of [the] underlying theory with fermions the internal anomaly (1) doesn't cause observed phenomena" flies in the face of the linchpin dramatic demonstration in Witten's 1983 paper that, when coupled to our world's gauge fields, and in particular electromagnetism, the topological action replicates the neutral pion decay, $\pi^0\to \gamma \gamma$, which, hitherto, had only been calculated with fermions in the underlying theory... first protons (Steinberger's triangle diagram) and then quarks.

That is to say, both the fundamental theory and the WZW term actually preserve the axial currents (and of course the vector ones), in the absence of the EM and weak fields, but they can fail to do that consistently (the W-Z consistency conditions), and they are ready for a violation on the r.h.s of (1), which, in the end, coupling to photons achieves.

You would not be asking this question had you done due diligence with the 5-dim action you wrote down: you should expand it to lowest order in pseudoscalars, observe that what you have is a total divergence, so a surface term; so descend to the 4 dims we live in, and inspect the axial symmetry of the 4-d term. (Basically you will have a quintilinear term in the goldstons, with only one of the five not following a gradient operator, $\propto \int d^{4}x \epsilon^{jklm}\text{Tr}\left[ \varphi\partial_{j}\varphi \partial_{k}\varphi\partial_{l}\varphi\partial_{m}\varphi\right]$. But an axial transformation entails shift of $\phi$ by a constant and the action by a total divergence, and thus you may check invariance. You must do this exercise, otherwise all discussions here would be pointless. "Academic"???)

Now, if/when you gauge the electric charge here (Witten does it for you; remember, you are gauging non-anomalous symmetries, not the potentially anomalous axials) you notice the customary $F\tilde{F}$ term in the divergence of the axial current... Consistent, see?

The issue of odd-point pseudoscalar vertices, like the one you are discussing, KKπππ, is a side issue of the types of effective interactions which preserve the axial symmetries but could consistently fail to. Their symmetry structure, dff, involving the symmetric d-coefficients you wrote down (D), is "collateral damage" of the anomaly---you can only have it when you have an anomaly. Note you cannot have a pure SU(2) quintilinear πππππ term for lack of such coefficients for SU(2). I'm not sure it is worthwhile stretching to connect them more directly to a current violation.

  • $\begingroup$ Precisely, the question has arised since, as was shown by Stora and Zumino, the non-abelian anomaly in 4 dimensions is related to the abelian anomaly in 6 dimensions through the Wess-Zumino term $\Gamma[U, A_{L},A_{R}]$. This term isn't zero when $A_{L} = A_{R} =0$, and therefore it seems to me that it is purely anomalous. Therefore I expected that the process $KK \to 3\pi$ is driven by the internal anomaly. Thank You, the last paragraph has helped me. $\endgroup$
    – Name YYY
    Commented Dec 15, 2016 at 9:57
  • $\begingroup$ By the way, I ask You to answer this question, physics.stackexchange.com/questions/298637/…, if You have a wish. $\endgroup$
    – Name YYY
    Commented Dec 15, 2016 at 10:03
  • $\begingroup$ Your question looks too broad and open-ended to have a cogent answer here. Your basic conceit about the interchangeability of chiral fermions with WZW effective terms is standard practice in the industry. We demonstrate chiral delocalization and the bulk-boundary interplay in our Hill & Zachos AnnPhys 323 (2008) 3065 ... $\endgroup$ Commented Dec 15, 2016 at 14:40
  • $\begingroup$ ...and also Hill & Zachos PhysRev D71 (2005) 046002. $\endgroup$ Commented Dec 15, 2016 at 14:40
  • $\begingroup$ Above Ann Phys Link that works. $\endgroup$ Commented Dec 16, 2016 at 18:51

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