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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 $τ$ 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 $τ$ decay width allows one then to predict the first few moments of the spectral density in $e^+e^-$ annihilations integrated up to $s\sim m_τ^2$ with high accuracy. The proposed tests are in reach of present experimental capabilities.