A new determination of higher-order QCD corrections to hadronic $τ$ decays

We employ \textit{Levin-type sequence transformations} to accelerate the convergence of the perturbative fixed-order expansion of the QCD correction $δ^{(0)}$ in terms of the strong coupling $α_s$. The method efficiently resums the series, yielding a stable and self-consistent determination of higher-order QCD corrections to hadronic $τ$ decays, consistent with existing results. We find $δ^{(0)}_{\text{Levin-FOPT}} = 0.2089 \pm 0.0040 \, \pm 0.0060_{α_s} \,$, and predict $c_{5,1} = 278^{+27}_{-19}, \quad c_{6,1} = 3375^{+489}_{-209}, \quad c_{7,1} = (2.03^{+0.41}_{-0.25}) \times 10^4.$ Our results demonstrate that Levin-type transformations provide an efficient framework for analyzing asymptotic perturbative series, and studying the higher order perturbative behaviour of the hadronic $τ$ decays.

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