Spectral moments of two-point correlators in perturbation theory and beyond

We discuss the choice of weight functions for the moments of the spectral density for two-point correlators of hadronic currents over a finite energy interval. Of phenomenological relevance is an analysis of the spectra of $\tau$ lepton decay on the energy interval $\left[{0,M}_{\tau }\right]$ and the low-energy hadron production in $e^{+}{e}^{-}$ annihilation. General arguments are given for the calculability of such moments in perturbation theory using both a finite-order analysis and infinite resummation on the contour. Nonperturbative contributions emerging from the operator product expansion for two-point correlators are discussed within explicit models for the physical spectra. The quantitative analysis strongly disfavors weight functions that suppress the high-energy contribution to theoretical moments. This is in agreement with expectations from qualitative considerations in perturbative QCD with asymptotic freedom. We discuss the implication of our results for the ultimate accuracy that can be reached in $\tau$ decays and low-energy $e^{+}{e}^{-}$ annihilation into hadrons with present experimental data.