Higher-order QCD corrections to hadronic $\tau$ decays from Pad\'e approximants

TL;DR

This paper uses Padé approximants to systematically estimate higher-order QCD corrections to hadronic tau decays, improving the precision of strong coupling constant extractions by addressing uncertainties in perturbative series.

Contribution

It introduces a novel application of Padé approximants with convergence strategies to predict unknown higher-order coefficients in QCD perturbation series, validated in the large-beta_0 limit and applied to full QCD.

Findings

Predicted higher-order coefficients: c_{5,1}=277±51, c_{6,1}=3460±690, c_{7,1}=(2.02±0.72)×10^4.

Model-independent results favor fixed-order perturbation theory for tau decay analyses.

Enhanced convergence methods improve the reliability of higher-order QCD correction estimates.

Abstract

Perturbative QCD corrections to hadronic $\tau$ 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, $\alpha_s$, 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 Pad\'e 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-$\beta_0$ 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 Pad\'e 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 $c_{5,1} = 277\pm 51$, $c_{6,1}=3460\pm 690$, and $c_{7,1}=(2.02\pm0.72)\times 10^4$, respectively, with errors to be understood as lower and upper bounds. Our model-independent reconstruction of the perturbative QCD corrections to the $\tau$ hadronic width strongly favours the use of fixed-order perturbation theory (FOPT) for the renormalization-scale setting.