make some comments on these sum rules.

i\) Weinberg in his original paper [@We67] assumed that only the lowest narrow resonant state in the vector and axial–vector spectral functions: $$\label{eq:Vresdom}
\frac{1}{\pi}{\mbox{\rm Im}}\Pi_{V}(t)=f_{V}^2 M_{V}^2\delta(t-M_{V}^2)+\cdots$$ and $$\label{eq:Aresdom}
\frac{1}{\pi}{\mbox{\rm Im}}\Pi_{A}(t)=f_{\pi}^2\delta(t)+ f_{A}^2
M_{A}^2\delta(t-M_{A}^2)+\cdots ,$$ contribute significantly to the sum rules. The first term in the axial–vector spectral function in eq. (\[eq:Aresdom\]) is the contribution from the massless pion. Thus the [*1st and 2nd Weinberg sum rules*]{} in eqs.(\[eq:1stwsr\]) and (\[eq:2ndwsr\]) constrain the couplings and masses of the narrow resonances as follows $$\label{eq:tsc}
\begin{split} f_{V}^2 M_{V}^2 -f_{A}^2 M_{A}^2 &= f_{\pi}^2 \\ f_{V}^2
M_{V}^4 -f_{A}^2 M_{A}^4 &= 0.
\end{split}$$ It is interesting that the original Weinberg’s [*ansatz*]{} in eqs. (\[eq:Vresdom\]) and (\[eq:Aresdom\]) can now be justified within the framework of the large–$N_c$ limit of QCD. Indeed, in this limit, the vector and axial–vector spectral functions are each a sum of an infinite number of narrow sates. The dots in the spectral functions in eqs. (\[eq:Vresdom\]) and (\[eq:Aresdom\]) implicitly assume that the sum of the rest of the narrow states is already globally dual to the onset of the perturbative continuum, $$\label{eq:Vresdomqcd}
\frac{1}{\pi}{\mbox{\rm Im}}\Pi_{V}(t)=f_{V}^2 M_{V}^2\delta(t-M_{V}^2)+
\frac{N_c}{16\pi^2}\frac{2}{3}\theta(t-s_{0})[1+\cdots],$$ and $$\label{eq:Aresdomqcd}
\frac{1}{\pi}{\mbox{\rm Im}}\Pi_{A}(t)=f_{\pi}^2\delta(t)+ f_{A}^2
M_{A}^2\delta(t-M_{A}^2)+\frac{N_c}{16\pi^2}\frac{2}{3}
\theta(t-s_{0})[1+\cdots],$$ where the dots indicate 