Hadronic tau decays at higher orders in QCD

We investigate higher-order perturbative corrections to hadronic $\tau$ decays by applying nonlinear sequence-transformation techniques to the QCD correction $\delta^{(0)}$. In particular, we employ the Shanks transformation and several of its generalisations constructed through Wynn's $\varepsilon$-algorithm, which are known to accelerate the convergence of slowly convergent or divergent series. These methods are used to extract higher-order information from the fixed-order perturbative expansion of $\delta^{(0)}$. Within this framework, we estimate the perturbative coefficients $c_{5,1}$-$c_{12,1}$. In particular, we obtain $c_{5,1}=298 \pm 15$, $c_{6,1}=3431 \pm 256$, and $c_{7,1}=2.29 \pm 0.29\times 10^4$, where the quoted uncertainties reflect the spread among the different sequence transformations employed. Moreover, we predict the QCD correction $\delta^{(0) }_{\text{FOPT}}=0.2119 \pm 0.0040\pm 0.0065_{\alpha_s}$. Our analysis demonstrates that non-linear sequence transformations, such as the Shanks-type, provide an efficient and systematic tool for probing higher-order perturbative effects in hadronic $\tau$ decays in the absence of explicit multi-loop calculations.