The strong coupling from hadronic $τ$-decay data including $τ\toπ^-π^0ν_τ$ from Belle

TL;DR

This work refines the hadronic τ-decay based determination of the strong coupling by constructing a more inclusive non-strange vector spectral function that incorporates Belle’s high-statistics π^-π^0 data alongside ALEPH, OPAL, BaBar electroproduction inputs, with updated HFLAV branching fractions. Using finite-energy sum rules and a DV/OPE framework, the authors extract α_s with a detailed assessment of DV modeling and perturbative uncertainties, obtaining α_s(m_τ^2) ≈ 0.298 with corresponding α_s(m_Z^2) ≈ 0.116, while documenting a downward shift relative to prior results largely due to BF updates. The analysis highlights the impact of data-treatment choices, especially the new combination strategy and the Belle data, and underscores remaining systematic uncertainties tied to duality violations and residual-mode inputs. The results contribute a state-of-the-art τ-based α_s determination and suggest avenues for further improvement through Belle II data and lattice techniques, with implications for hadronic vacuum polarization and the muon g−2 program.

Abstract

In previous work we have combined the $π^-π^0$, $2π^-π^+π^0$ and $π^-3π^0$ spectral data obtained from hadronic $τ$ decays measured by the ALEPH and OPAL experiments, together with electroproduction data for several of the subleading hadronic modes and BaBar data for the $K\bar{K}$ mode to construct an inclusive non-strange vector spectral function entirely based on experimental data, with no Monte-Carlo generated input. In this paper, we include, for the first time, the Belle $τ\toπ^-π^0ν_τ$ high-statistics decay data to construct a new inclusive non-strange vector spectral function that combines more of the world's available data. As no Belle data are at present available for the two $4π$ modes, this requires a revised data analysis in comparison with our previous work. From the resulting new spectral function, we obtain a new determination of the strong coupling, $α_s$, using our previously developed strategy based on finite-energy sum rules. We find, at the $Z$ mass scale, $α_s(m_Z^2)=0.1159(14)$. We discuss the smaller central value and larger error of our new result compared to our previous result, showing the shifts to be due mainly to significant changes in updated HFLAV results for the $π^-3π^0$ decay mode.

Figures (16)

• Figure 1: Results of the fit for the 63-cluster combination of the $\pi^-\pi^0$ exclusive-mode unit-normalized distribution divided by the $s$-dependent kinematic weight factor. The error bars represent the uninflated errors, while inflated errors are represented by the green band.
• Figure 2: Local $p$-value for each cluster, for the 63-cluster $\pi^{-}\pi^{0}$ unit-normalized combined spectral function. The red dots show, up to an arbitrary overall normalization, the corresponding combined $2\pi$ spectral function.
• Figure 3: Ratio between experimental $2\pi$ data and the associated $2\pi$ combination, in the $\rho$ peak region.
• Figure 4: Correlations of the unit-normalized, $4\pi$ exclusive-mode number distribution data sets. (a) ALEPH $2\pi^{-}\pi^{+}\pi^{0}$, (b) ALEPH $\pi^{-}3\pi^{0}$, (c) OPAL $2\pi^{-}\pi^{+}\pi^{0}$ and (d) OPAL $\pi^{-}3\pi^{0}$.
• Figure 5: Results (green points) of the fit for the 46-cluster combination of the ALEPH and OPAL two-mode $4\pi$ spectral function results. The error bars on the green points represent the uninflated errors, while inflated errors are represented by the green band.