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    $\begingroup$ I am a little confused by the statement "...a spontaneous symmetry breaking caused by quantum anomaly effect". I am not sure I understand why the quantum anomaly can be seen as 'spontaneous symmetry breaking' of symmetry...? Let me make a few naive comments on this, which might very well be wrong. Usually anomalies are described as the impossibility to find a regularization scheme in which the anomalous symmetry is explicitly preserved, and thus it implies that the anomaly is a problem of the UV completion. $\endgroup$
    – Heidar
    Commented Aug 21, 2013 at 18:37
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    $\begingroup$ However, as you very likely know well, the 't Hooft anomaly matching condition says that the anomaly is scale independent, the UV and IR anomalies must match. For me, this seems to imply that the anomalous symmetry is not a symmetry of the theory at all in any limit. It just appears to be a symmetry in the naive classical limit, but that's just an illusion. Thus its not a spontaneous nor explicit breaking of symmetry, but absence of symmetry. This line of reasoning might very likely be wrong, though. $\endgroup$
    – Heidar
    Commented Aug 21, 2013 at 18:37
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    $\begingroup$ TO Heidar: I do agree with what Heidar said. My original understanding is that the quantum anomaly for a current non-conservation respect to G is a fact that there is no that G symmetry at all for that system from the beginning. So that is why we do not have $\eta'$ meson, as I gave the example. However, I read this statement from Fuijikawa's book "Path Integral and Quantum Anomaly": Sec 5.6.2 "Nambu-Goldstone bosons do not generally appear for a spontaneously broken symmetry if the relevant global symmetry is broken by the effects of the anomaly and and instantons" $\endgroup$
    – wonderich
    Commented Aug 22, 2013 at 7:47
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    $\begingroup$ continue: and Weinberg's book QFT II, Sec. 22.7 Anomalies and Goldstone bosons. " the gauged effective field theory of Goldstone bosons must have an anomaly for the fictions symmetry which is equal to what has been produced from the underlying (fermion) theory." I thought there may be some room to think a bit more for this question? $\endgroup$
    – wonderich
    Commented Aug 22, 2013 at 7:52
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    $\begingroup$ Continue: I mean, for example,can one do an explicit analytic exercise using Fujikawa's path integral approach, where a global symmetry transformation G (of underlying fermionic) is broken by a quantum anomaly, but this symmetry G is explicitly the same symmetry of a bosonic fields potential(say G broken down to N, such as O(N) broken down to O(N-1), or $U(N)$ broken down to $SU(N)$)? I wonder whether it makes sense to do this analysis to see there are/aren't |G/N| number of Goldstone bosons? $\endgroup$
    – wonderich
    Commented Aug 22, 2013 at 8:05