't Hooft anomaly matching condition and the QCD

Question

't Hooft anomaly matching condition states that the (chiral) anomalous structure of the given theory is the same independently on the scale. This means that if we have one particle content $\{\psi\}$ for scales $> \Lambda$ with non-zero anomaly, then another particle content $\{\phi\}$ for scales $<\Lambda$ must reproduce anomaly coefficients $d_{abc}$.

Weinberg (in his QFT Vol. 2 Sec. 22.5) states that in Lorentz invariant theory the states $\{\phi\}$ can be recognized as massless helicity $\frac{1}{2}$ fermions or Goldstone bosons. As an example of the second case it proposes to discuss the QCD, where below the spontaneous symmetry breaking scale the anomalous structure of underlying theory with quarks is reproduced by pseudoscalar mesons.

But I don't understand one thing. These pseudo-scalar mesons actually are pseudo-goldstone bosons with non-zero mass. Therefore it seems that they can't reproduce the anomaly, or some argument referring to the fact that they are true Goldstone bosons in some approximation must be used in order to explain why the anomaly can be reproduced by them.

Do You know this argument?

Comments

The non-zero masses of the pseudo-scalar mesons come from the non-zero masses of the quarks. When the quarks have non-zero masses, there is no chiral symmetry to start with and so there is no discussion of the anomaly.  Only in the zero mass quark case, there is a classical chiral symmetry, which is spontaneously broken, with massless pseudo-scalar mesons (pions...) as Goldstone bosons.

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@40227 : but there anyway is the anomalous part in the conservation law of global currents, as well as internal QCD anomalies, and therefore the Wess-Zumino term in effective field theory for mesons. The latter reproduces all underlying QCD's anomalies and has purely anomalous origin, being the solution of the Wess-Zumino consistency conditions.

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Maybe I am missing something but the discussion of the Wess-Zumino term is also done in the limit of zero quark masses (with massless quarks, we have chiral symmetry, spontaneously broken by a bilinear quark condensate, so at low energy mesons are well-described by a sigma-model of target a quotient space (original symmetry group divided by the unbroken symmetry group) and the Wess-Zumino term is just one term in this sigma-model).

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@40227 : yes, we start from the limit of zero quarks masses. But even in the limit of non-zero masses the Wess-Zumino term remains.