Implications of the ALEPH $\tau$-Lepton Decay Data for Perturbative and Non-Perturbative QCD

We use ALEPH data on hadronic $\tau$ decays in order to calculate Euclidean coordinate space correlation functions in the vector and axial-vector channels. The linear combination $V-A$ receives no perturbative contribution and is quantitatively reproduced by the instanton liquid model. In the case of $V+A$ the instanton calculation is in good agreement with the data once perturbative corrections are included. These corrections clearly show the evolution of $\alpha_s$. We also analyze the range of validity of the Operator Product Expansion (OPE). In the $V-A$ channel we find a dimension $d=6$ contribution which is comparable to the original SVZ estimate, but the instanton model provides a different non-singular term of the same magnitude. In the $V+A$ case both the OPE and the instanton model predict the same $d=4$ power correction induced by the gluon condensate, but it is masked by much larger perturbative contributions. We conclude that the range of validity of the OPE is limited to $x\lsim0.3$ fm, whereas the instanton model describes the data over the entire range.