Perturbative expansion of the QCD Adler function improved by renormalization-group summation and analytic continuation in the Borel plane

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We examine the large-order behaviour of a recently proposed renormalization-group-improved expansion of the Adler function in perturbative QCD, which sums in an analytically closed form the leading logarithms accessible from renormalization-group invariance. The expansion is first written as aneffective series in powers of the one-loop coupling, and its leading singularities in the Borel plane are shown to be identical to those of the standard "contour-improved" expansion. Applying the technique of conformal mappings for the analytic continuation in the Borel plane, we define a class of improved expansions, which implement both the renormalization-group invariance and the knowledge about the large-order behaviour of the series. Detailed numerical studies of specific models for the Adler function indicate that the new expansions have remarkable convergence properties up to high orders. Using these expansions for the determination of the strong coupling from the the hadronic width of the $\tau$ lepton we obtain, with a conservative estimate of the uncertainty due to the nonperturbative corrections, $\alpha_s(M_\tau^2)= 0.3189^{+ 0.0173}_{-0.0151}$, which translates to $\alpha_s(M_Z^2)= 0.1184^{+0.0021}_{-0.0018}$.