Rules {#subsec:msrs}

In QCD the number of derivatives which is required to obtain a well defined two–point function is fixed by the asymptotic freedom property of the theory. For a gauge invariant local operator $J(x)$, the asymptotic behaviour of the associated two–point function is of the type $$\label{eq:asymsp}
\lim_{t{\rightarrow}\infty}\frac{1}{\pi}{\mbox{\rm Im}}\Pi(t){\Rightarrow}{\mbox{\rm
A}}t^{d}\left\{1+a_{1}\frac{{\alpha_{\mbox{\rm {\scriptsize s}}}}(t)}{\pi}+\dots\,\right\},$$ with A and $a_{1}$ calculable coefficients, and $d$ a known integer $d=0,1,2,\dots\,$, depending on the dimensions of the operator $J(x)$. It is then sufficient to take $d+1$ derivatives with respect to $q^2$ to get rid of the arbitrary polynomial and obtain a convergent integral. The functions defined by the [*moment integrals*]{} ($Q^2\equiv -q^2$): $$\begin{split}
\Pi^{(m)}(Q^2) &= \frac{(-1)^m}{(m-d-1)!}\,(Q^2)^{m-d}\,
\frac{\partial^m}{(\partial Q^2)^m}\,\Pi(q^2) \\
&=\int_{0}^{\infty}dt\,\frac{m(m-1)\cdots
(m-d)}{(t+Q^2)^{d+1}}\,\left(\frac{Q^2}{t+Q^2}\right)^{m-d}
\frac{1}{\pi}{\mbox{\rm Im}}\Pi(t),
\end{split}$$ for $m\ge d+1$ are then well defined functions calculable in pQCD at sufficiently large $Q^2$–values.

### The Adler Function {#subsec:adlerfunction}

An example of a QCD moment sum rule is the Adler function corresponding to the two–point function associated with the vector–isovector quark current $$V^{\mu}_{\mbox{\scriptsize\rm I}=1}(x)=\frac{1}{2}\left[:\!{\bar
u}(x)\gamma^{\mu}u(x)\!:- :\!{\bar d}(x)\gamma^{\mu}d(x)\!:\right].$$ Since the current is conserved, the two–point function $$\label{eq:adlsef}
\begin{split}
\Pi_{\mbox{\scriptsize\rm I}=1}^{\mu\nu}(q) &= i\int d^{4}x
e^{iq\cdot x}
\langle 0\vert\mbox{\rm T}
\left\{V_{\mbox{\scriptsize\rm I}=1}^{\mu}(x) V_{\mbox{\scriptsize\rm
I}=1}^{\nu}(0)\right\}\vert 0\rangle \\
 &= -(g^{\mu\nu}q^2 -q^{\mu}q^{\nu})\Pi_{\mbox{\scriptsize\rm
I}=1}(q^2),
\end{split}$$ has one invariant 