A new determination of $α_s (M_τ^2)$ and higher-order QCD corrections to hadronic $τ$ decays

TL;DR

This work introduces Levin-type sequence transformations to accelerate both the fixed-order perturbative expansion of $\delta^{(0)}$ in $\alpha_s$ and the inverted $\alpha_s(\delta^{(0)})$ series, achieving a stable extraction of $\alpha_s(M_\tau^2)$ from hadronic $\tau$ decays. By applying Levin transforms to the divergent inverted series and to the direct $\delta^{(0)}$ expansion, the authors obtain reliable estimates of higher-order coefficients $c_{n,1}$ (for $n=5$–$12$) and demonstrate convergence toward the Levin-summed results from higher orders. The main results include $\alpha_s^{\text{Levin-FOPT}} = 0.3159(\pm 0.0018)(\pm 0.0023)$ and final high-order coefficient predictions that are consistent with existing literature, indicating a robust framework for perturbative QCD analyses of $\tau$ decays. This methodology offers a practical route to constrain $\alpha_s$ at low energies and to quantify uncertainties arising from unknown higher-order contributions, with potential applicability to other asymptotic series in quantum field theory.

Abstract

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 $α_s$, and of the inverted series expressing $α_s$ in powers of $δ^{(0)}$. The method efficiently resums the divergent inverted series, yielding a stable and self-consistent determination of the strong coupling at the $τ$ mass scale. It also provides reliable estimates of higher-order QCD corrections to hadronic $τ$ decays, consistent with existing results. We find $α_s^{\text{Levin-FOPT}} = 0.3159 \pm 0.0018 \, \pm 0.0023 \, $, and predict $c_{5,1} = 269^{+47}_{-45}, \quad c_{6,1} = 3185^{+117}_{-279}, \quad c_{7,1} = (1.9^{+0.9}_{-0.8}) \times 10^4.$ Our results demonstrate that Levin-type transformations provide an efficient framework for analyzing asymptotic perturbative series and improving the extraction of $α_s(M_τ^2)$ from hadronic $τ$ decays.

Figures (3)

• Figure 1: $\alpha_s$ at different orders of $\delta^{(0)}$, using $\delta^{(0)}=0.2027$, in the inverted series Eq. \ref{['Eq:aFOPT']}. The solid black curve shows the mean of Levin-summed values of $\alpha_s$, and the shaded region represents the spread of Levin-sum from the U, T, and D Levin transforms, indicating the uncertainty due to the choice of transformation.
• Figure 2: Perturbative expansions of $\alpha_s$ in FOPT and CIPT using the the higher-order coefficients in Table \ref{['tab:final_coeff_alphas']}. The solid black curve shows the mean of Levin-summed values of $\alpha_s$, and the shaded region represents the spread of Levin-sum from the U, T, and D Levin transforms, indicating the uncertainty due to the choice of transformation. We have used $\delta^{(0)}=0.2027$.
• Figure 3: Final prediction of $\alpha_s$ in QCD using the higher-order coefficients listed in Table \ref{['tab:final_coeff']}. The shaded regions in the perturbative expansions represent the uncertainties in the coefficients. The solid black curves corresponds to the mean of the Levin-summed values of $\alpha_s$. The shaded yellow bands denote the spread of the Levin sums obtained from different Levin-type transformations (U, T, D). The input values $\delta^{(0)}=0.2027$ is used.