now gluonic corrections. The vector and axial–vector spectral functions of the pQCD continuum are the same in the chiral limit.

ii\) In order to make a “prediction” for the axial mass $M_{A}$, Weinberg in his original paper also makes the assumption that $$f_{V}^2=2\frac{f_{\pi}^2}{M_{V}^2},$$ which was called at the time the KSFR relation [@KS66; @FR66]. This, together with eqs. (\[eq:tsc\]) leads to the prediction $M_{A}=\sqrt{2}M_{V}$ which is satisfied by the central values of the presently known masses within an accuracy of less than $\sim 12\%$. It can be shown [@EGLPR89] that the KSFR–relation follows from the assumption that both the electromagnetic pion form factor and the axial form factor in $\pi{\rightarrow}e\nu_{e}\gamma$ obey unsubtracted dispersion relations, plus the dynamical input of resonance dominance.

iii\) Inverse moments of the difference of vector and axial–vector spectral functions with the pion pole removed, i.e. integrals like $$\int_{0}^{\infty}dt\frac{1}{t^p}\left(\frac{1}{\pi}{\mbox{\rm Im}}\Pi_{LR}(t)
+f_{\pi}^2\delta(t)\right),$$ with $p=1,2,3,\cdots$, correspond to non–local order parameters which govern the couplings of local operators of higher and higher dimensions in the low energy effective chiral Lagrangian \[[See Pich’s lectures [@PiclesH] in this school.]{}\]. For example, the first inverse moment is related to the ${{\cal O}}(p^4)$ coupling constant $L_{10}$ as follows [@GL84] $$-4L_{10}=\int_{0}^{\infty}\frac{dt}{t}\,
\left(\frac{1}{\pi}{\mbox{\rm Im}}\Pi_{V}(t)-
\frac{1}{\pi}
{\mbox{\rm 