'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?