New High Order Relations between Physical Observables in Perturbative QCD

We exploit the fact that within massless perturbative QCD the same Green's function determines the hadronic contribution to the $\tau$ decay width and the moments of the $e^{+}{e}^{-}$ cross section. This allows one to obtain relations between physical observables in the two processes up to an unprecedented high order of perturbative QCD. A precision measurement of the $\tau$ decay width allows one then to predict the first few moments of the spectral density in $e^{+}{e}^{-}$ annihilations integrated up to $s\sim m_{\tau }^2$ with high accuracy. The proposed tests are in reach of present experimental capabilities.