Suppose a theory with a fermion sector $\psi$ having some global chiral symmetry group $G$ without internal anomalies (i.e., a group whose algebra generators $t_{a}$ give zero coefficients $D_{abc}\equiv \text{tr}[t_{a},\{t_{b},t_{c}\}]$). Such summetry may be $U_{L}(1)\times U_{R}(1)$, $SU_{L}(2)\times SU_{R}(2)$, and so on.

Suppose now this symmetry is broken spontaneously. Fermions form condensate, and Goldstone bosons $\varphi_{i}$ appear. If we add the gauge gauge group $C$, then non-zero $D_{\tilde{a}\tilde{b}\tilde{c}}$ may appear, and therefore anomalies exist. Therefore there must be terms in corresponding effective action $\Gamma[A_{L},A_{R},U]$ (with $U = \text{exp}\big(it_{a}\varphi_{a}(x) \big)$), whose gauge variation $\delta_{\epsilon}\Gamma_{\text{WZ}}$ reproduces all anomalies $\text{A}_{a}$ of the underlying theory. They are called the Wess-Zumino terms.

The general essence of Wess-Zumino terms in a case of absence of internal anomalies is that they can be given as a 4-dimensional integral of a local lagrangian density (see, for example, this question): $$ \delta_{\epsilon}\Gamma_{\text{WZ}}[U,A_{L},A_{R}] \textbf{ (local) } = \int d^{4}x \epsilon_{a}\text{A}_{a}(x) \text{ (local)} $$ In contrast with mentioned above, the chiral anomaly in underlying theory of fermions is a non-local phenomena (in a sense that it is produced by a massless pole in the triangle vertex function, and therefore can't be generated by a polynomial of fields and their derivatives in the action): $$ \delta_{\epsilon} \Gamma [\psi, A_{L},A_{R}] \textbf{ (non-local) } = \int d^{4}x \epsilon_{a}\text{A}_{a}(x) \text{ (local)} $$ Therefore, it is surprisingly for me that it is possible to obtain the local expression in the case of spontaneously symmetry breaking generating the anomaly.

My question: is the locality of the Wess-Zumino terms connected to masslessness of Goldstone bosons $\varphi_{a}$?


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