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Hadronic tau decays at higher orders in QCD

Gauhar Abbas, Vartika Singh

TL;DR

The paper tackles the slow convergence of the perturbative QCD series in hadronic $\tau$ decays by applying nonlinear sequence-transformations, notably the Shanks transformation and Wynn’s $\varepsilon$-algorithm, to extract higher-order information from fixed-order results of $\delta^{(0)}$. By leveraging several generalizations and regularizations of the algorithm, it estimates the perturbative coefficients $c_{n,1}$ for $n=5$–$12$ and provides a refined prediction for $\delta^{(0)}$ with quantified uncertainties. The approach is validated through cross-checks among multiple methods and input configurations, showing consistency with existing resummation frameworks and offering a systematic, model-independent way to assess missing higher-order effects in the perturbative expansion. The findings have direct implications for precision determinations of $\alpha_s$ from hadronic $\tau$ decays and demonstrate the broader applicability of these sequence-transform techniques to perturbative QCD analyses.

Abstract

We investigate higher-order perturbative corrections to hadronic $τ$ decays by applying nonlinear sequence-transformation techniques to the QCD correction $δ^{(0)}$. 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 $δ^{(0)}$. Within this framework, we estimate the perturbative coefficients $c_{5,1}$-$c_{12,1}$. In particular, we obtain $c_{5,1}=294^{+41}_{-21}$, $c_{6,1}=3415^{+450}_{-467}$, and $c_{7,1}=2.23^{+0.75}_{-0.49}\times 10^4$, where the quoted uncertainties reflect the spread among the different sequence transformations employed. Our analysis demonstrates that Shanks-type sequence transformations based on Wynn's $\varepsilon$-algorithm provide an efficient and systematic tool for probing higher-order perturbative effects in hadronic $τ$ decays in the absence of explicit multi-loop calculations.

Hadronic tau decays at higher orders in QCD

TL;DR

The paper tackles the slow convergence of the perturbative QCD series in hadronic decays by applying nonlinear sequence-transformations, notably the Shanks transformation and Wynn’s -algorithm, to extract higher-order information from fixed-order results of . By leveraging several generalizations and regularizations of the algorithm, it estimates the perturbative coefficients for – and provides a refined prediction for with quantified uncertainties. The approach is validated through cross-checks among multiple methods and input configurations, showing consistency with existing resummation frameworks and offering a systematic, model-independent way to assess missing higher-order effects in the perturbative expansion. The findings have direct implications for precision determinations of from hadronic decays and demonstrate the broader applicability of these sequence-transform techniques to perturbative QCD analyses.

Abstract

We investigate higher-order perturbative corrections to hadronic decays by applying nonlinear sequence-transformation techniques to the QCD correction . In particular, we employ the Shanks transformation and several of its generalisations constructed through Wynn's -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 . Within this framework, we estimate the perturbative coefficients -. In particular, we obtain , , and , where the quoted uncertainties reflect the spread among the different sequence transformations employed. Our analysis demonstrates that Shanks-type sequence transformations based on Wynn's -algorithm provide an efficient and systematic tool for probing higher-order perturbative effects in hadronic decays in the absence of explicit multi-loop calculations.
Paper Structure (17 sections, 56 equations, 1 figure, 15 tables)

This paper contains 17 sections, 56 equations, 1 figure, 15 tables.

Table of Contents

  1. Introduction
  2. The hadronic decay of the $\tau$ lepton
  3. Shanks transformation
  4. Wynn’s $\varepsilon$-algorithm
  5. Generalizations of the $\varepsilon$-algorithm
  6. Vanden Broeck--Schwartz modification of Wynn’s $\varepsilon$-algorithm
  7. Sedogbo--Sablonnière modification of Wynn’s $\varepsilon$-algorithm
  8. A new modification of Wynn’s $\varepsilon$-algorithm
  9. Brezinski’s $\theta$-algorithm
  10. Higher order corrections to hadronic tau decay width
  11. Higher order corrections from the Shanks Transformation
  12. Higher order corrections from the Sedogbo--Sablonnière modification of Wynn’s $\varepsilon$-algorithm
  13. Higher order corrections from the new modification of Wynn’s $\varepsilon$-algorithm
  14. Higher order corrections from the Brezinski’s $\theta$-algorithm
  15. Final predictions of the higher-order coefficients
  16. ...and 2 more sections

Figures (1)

  • Figure 1: Final prediction of $\delta^{(0)}$ in QCD using the higher-order coefficients listed in Table \ref{['tab:final_coeff']}. The shaded regions in the perturbative expansions represent the uncertainties from the coefficients. The solid black curve shown as Shanks sum corresponds to the mean of the values of $\delta^{(0)}$ obtained from different methods. The shaded yellow bands denote spread of the $\delta^{(0)}$ values across the different resummation techniques. The input value $\alpha_s=0.31959$ is used.