Finite energy chiral sum rules in QCD

A set of well known chiral sum rules, expected to be valid in QCD, is confronted with experimental data on the vector and axial-vector hadronic spectral functions, obtained from tau-lepton decay by the ALEPH collaboration. The Das-Mathur-Okubo sum rule, the first and second Weinberg sum rules, and the electromagnetic pion mass difference sum rule are not well saturated by the data. Instead, a modified set of sum rules having additional weight factors that vanish at the end of the integration range on the real axis, is found to be precociously saturated by the data to a remarkable extent.