Check of QCD based on theτ-decay data analysis in the complexq2plane

The thorough analysis of the ALEPH data on hadronic $\tau$ decay is performed in the framework of QCD. The perturbative calculations are performed in three- and four-loop approximations. The terms of the operator product expansion (OPE) are accounted for up to the dimension $D=8.\mathrm{}$ The value of the QCD coupling constant ${\alpha }_s{(m}_{\tau }^2)=0.355\mathrm{}\pm 0.025$ is found from the hadronic branching ratio $R_{\tau }.$ The $V+A$ and V spectral functions are analyzed using the analytical properties of the polarization operators in the whole complex $q^2$ plane. Borel sum rules in the complex $q^2$ plane along the rays, starting from the origin, are used. It is demonstrated that QCD with OPE terms is in agreement with the data for a coupling constant close to the lower error edge ${\alpha }_s{(m}_{\tau }^2)=\mathrm{0.330.}\mathrm{}$ The restriction on the value of the gluonic condensate was found to be $〈({\alpha }_s/\pi {)G}^2〉=0.006\mathrm{}\pm 0.012\hspace{0.5em}{\mathrm{GeV}}^2.$ The analytical perturbative QCD is compared with the data. It is demonstrated to be in strong contradiction with experiment. The restrictions on the renormalon contribution are found. The instanton contributions to the polarization operator are analyzed in various sum rules. In the Borel transformation they appear to be small, but not in the spectral moment sum rules.