QCD, Weinberg made an educated guess about the convergence properties of the LR two–point function, with the suggestion that the following two sum rules may hold in the underlying theory of the strong interactions [@We67]: $$\label{eq:1stwsr}
\int_{0}^{\infty}dt{\mbox{\rm Im}}\Pi_{LR}(t)\equiv
\int_{0}^{\infty}dt\left[{\mbox{\rm Im}}\Pi_{V}(t)-{\mbox{\rm Im}}\Pi_{A}(t)\right]=0,$$ and $$\label{eq:2ndwsr}
\int_{0}^{\infty}dt\,t{\mbox{\rm Im}}\Pi_{LR}(t)\equiv
\int_{0}^{\infty}dt\,t\left[{\mbox{\rm Im}}\Pi_{V}(t)-{\mbox{\rm Im}}\Pi_{A}(t)\right]=0.$$ They are commonly referred to as the [*1st Weinberg sum rule*]{} and the [*2nd Weinberg sum rule*]{}, and they are examples of the so called [*superconvergent*]{} sum rules for the following reason. Let us assume that the function $\Pi_{LR}(Q^2)$ obeys an unsubtracted dispersion relation: $$\label{eq:udrwsr}
\Pi_{LR}(Q^2)=\int_{0}^{\infty}\frac{dt}{t+Q^2}{\mbox{\rm Im}}\Pi_{LR}(t).$$ The existence of the two sum rules above implies a stronger convergence behaviour for the function $\Pi_{LR}(Q^2)$ in the deep euclidean region than the minimum required for the unsubtracted dispersion relation to hold. Expanding $\frac{1}{t+Q^2}=\frac{1}{Q^2}-
\frac{1}{Q^2}t\frac{1}{Q^2}+\cdots$, we observe that the [*1st Weinberg sum rule*]{} corresponds to the constraint $$\label{eq:ope2}
\lim_{Q^2{\rightarrow}\infty} Q^2\Pi_{LR}(Q^2){\Rightarrow}0,$$ and the [*2nd Weinberg sum rule*]{} to the further constraint $$\label{eq:ope4}
\lim_{Q^2{\rightarrow}\infty} Q^4\Pi_{LR}(Q^2){\Rightarrow}0.$$ It is quite remarkable that, as we shall see in section \[subsec:wsrope\], both constraints are now QCD properties; and in fact, the [*1st Weinberg sum rule*]{} still holds in the presence of finite light quark masses [@FNR79].

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