Hadronic $\tau$ decay, the renormalization group, analiticity of the polarization operators and QCD parameters

The ALEPH data on hadronic τ -decay is throughly analysed in the framework of QCD. The perturbative calculations are performed in 1-4-loop approximation. The analytical properties of the polarization operators are used in the whole complex q 2 plane. It is shown that the QCD prediction for R τ agrees with the measured value R τ not only for conventional Λ conv 3 = (618 ± 29) MeV but as well as for Λ new 3 = (1666 ± 7) MeV . The polarization operator calculated using the renormgroup has nonphysical cut [ − Λ 23 , 0]. If Λ 3 = Λ conv 3 , the contribution of only physical cut is deficient in the explanation of the ALEPH experiment. If Λ 3 = Λ new 3 the contribution of nonphysical cut is very small and only the physical cut explains the ALEPH experiment. The new sum rules which follow only from analytical properties of polarization operators are obtained. Basing on the sum rules obtained, it is shown that there is an essential disagreement between QCD perturbation theory and the τ -lepton hadronic decay experiment at conventional value Λ 3 . In the evolution upwards to larger energies the matching of r ( q 2 ) (Eq.(12)) at the masses J/ψ , Υ and 2 m t was performed. The obtained value α s ( − m 2 z ) = 0 . 141 ± 0 . 004 (at Λ 3 = Λ new 3 ) differs essentially from conventional value, but the calculation of the values R ( s ) = σ ( e + e − → hadrons ) σ ( e + e − → µ + µ − ) , R l