of sum over $\Gamma$ and integrations we have implicitly assumed good convergence properties; but in general the product of the distributions $\theta(x)$ and $\int_{0}^{\infty}dt\,\rho(t)\,\Delta^{+}(x;t)$ may not be a well–defined distribution. The ambiguity manifests by the presence of an arbitrary polynomial in $q^2$ in the r.h.s. of the PP–integral $$\label{eq:PPab}
{\mbox{\rm Re}}\Pi(q^2)=\mbox{\rm PP}\!\int_{0}^{\infty}dt\,\frac{1}{t-q^2}\,
\frac{1}{\pi}{\mbox{\rm Im}}\Pi(t) +\mbox{\rm a}+\mbox{\rm b}q^2 +\dots\, .$$ Notice that the coefficients of the arbitrary polynomial have no discontinuities; in other words, the ambiguity of short–distance behaviour reflects only in the evaluation of the real part of the two–point function, not in the imaginary part. The physical meaning of these coefficients depends of course on the choice of the local operator $J(x)$ in the two–point function. In some cases the coefficients in question are fixed by low–energy theorems; e.g. if $\Pi(0)$ is known, we can trade the constant a in eq. (\[eq:PPab\]) for $\Pi(0)$: $${\mbox{\rm Re}}\Pi(q^2)={\mbox{\rm Re}}\Pi(0)+\mbox{\rm
PP}\!\int_{0}^{\infty}\frac{dt}{t}\frac{q^2}{t-q^2}\,
\frac{1}{\pi}{\mbox{\rm Im}}\Pi(t) + \mbox{\rm b}q^2 + \dots\,.$$ In other cases the constants are absorbed by renormalization constants. In general, it is always possible to get rid of the polynomial terms by taking an appropriate number of derivatives with respect to $q^2$. Various examples will be discussed later.

### QCD Moment Sum 