Perturbative QCD corrections to hadronic τ decays and e + e − annihilation into hadrons below charm are obtained from the Adler function, which at present is known in the chiral limit to five-loop accuracy. Extractions of the strong coupling, α s , from these processes suffer from an ambiguity related to the treatment of unknown higher orders in the perturbative series. In this work, we exploit the method of Padé approximants and its convergence theorems to extract information about higher-order corrections to the Adler function in a systematic way. First, the method is tested in the large-β 0 limit of QCD, where the perturbative series is known to all orders. We devise strategies to accelerate the convergence of the method employing renormalization scheme variations and the so-called D-log Padé approximants. The success of these strategies can be understood in terms of the analytic structure of the series in the Borel plane. We then apply the method to full QCD and obtain reliable model-independent predictions for the higher-order coefficients of the Adler function. For the six-, seven-, and eight-loop coefficients we find c 5,1 = 277 ± 51, c 6,1 = 3460±690, and c 7,1 = (2.02±0.72)×10 4 , respectively, with errors to be understood as lower and upper bounds. Our model-independent reconstruction of the perturbative QCD corrections to the τ hadronic width strongly favours the use of fixed-order perturbation theory (FOPT) for the renormalization-scale setting.

Precise extractions of α s from τ → (hadrons) + ν τ and from e + e − → (hadrons) below the charm threshold rely on finite energy sum rules (FESR) where the experimental side is given by integrated spectral function moments. Here we study the renormalons that appear in the Borel transform of polynomial moments in the large-β 0 limit and in full QCD. In large-β 0 , we establish a direct connection between the renormalons and the perturbative behaviour of moments often employed in the literature. The leading IR singularity is particularly prominent and is behind the fate of moments whose perturbative series are unstable, while those with good perturbative behaviour benefit from partial cancellations of renormalon singularities. The conclusions can be extended to QCD through a convenient scheme transformation to the C-scheme together with the use of a modified Borel transform which make the results particularly simple; the leading IR singularity becomes a simple pole, as in large-β 0 . Finally, for the moments that display good perturbative behaviour, we discuss an optimized truncation based on renormalisation scheme (or scale) variation. Our results allow for a deeper understanding of the perturbative behaviour of integrated spectral function moments and provide theoretical support for low-Q 2 α s determinations.

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