pith. sign in

arxiv: 2601.11277 · v2 · submitted 2026-01-16 · ✦ hep-ph · hep-ex

Hadronic tau decays at higher orders in QCD

Gauhar Abbas , Vartika Singh This is my paper

Pith reviewed 2026-05-16 13:34 UTC · model grok-4.3

classification ✦ hep-ph hep-ex
keywords hadronic tau decaysQCD perturbative correctionssequence transformationsShanks transformationWynn epsilon algorithmhigher-order coefficientsfixed-order perturbation theory
0
0 comments X

The pith

Nonlinear sequence transformations applied to the QCD series for hadronic tau decays estimate coefficients through order 12 and predict a fixed-order correction of 0.2119.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper applies Shanks transformations and Wynn's epsilon algorithm to the known low-order terms of the perturbative QCD correction delta^(0) in hadronic tau decays. These nonlinear methods accelerate the series to produce estimates for the unknown coefficients c5,1 through c12,1. The central result is a predicted value for the full fixed-order perturbative correction delta^(0)_FOPT of 0.2119 with combined uncertainties around 0.008. A sympathetic reader cares because hadronic tau decays provide one of the cleanest ways to extract the strong coupling and test QCD at low energies, where higher-order terms have historically limited precision.

Core claim

Applying the Shanks transformation and its generalizations via Wynn's epsilon-algorithm to the fixed-order perturbative expansion of delta^(0) extracts estimates for the higher-order coefficients c5,1 to c12,1, with c5,1 = 298 ± 15, c6,1 = 3431 ± 256, and c7,1 = 2.29 ± 0.29 × 10^4, and yields the prediction delta^(0)_FOPT = 0.2119 ± 0.0040 ± 0.0065_alpha_s.

What carries the argument

Shanks transformation and Wynn's epsilon-algorithm applied to the perturbative series of delta^(0)

If this is right

  • The estimated coefficients c5,1 to c12,1 can be inserted directly into the perturbative expansion to reduce theoretical uncertainty in the tau decay rate.
  • The predicted delta^(0)_FOPT value supplies a concrete input for global fits of the strong coupling from tau data.
  • The same sequence-transformation procedure can be repeated on other slowly convergent QCD series once a few low-order terms are known.
  • Uncertainties quoted from the spread across different transformations provide a practical error estimate when explicit higher-loop results remain unavailable.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the method proves accurate for tau decays, it offers a general shortcut for estimating missing orders in other perturbative QCD observables such as e+e- annihilation or deep-inelastic scattering.
  • A future direct computation of c5,1 or c6,1 would serve as an immediate benchmark to calibrate how much residual bias the transformations carry.
  • The approach could be combined with known Borel-resummation techniques to cross-check the size of non-perturbative contributions in tau decays.

Load-bearing premise

That the chosen sequence transformations applied to the known low-order terms reliably capture the asymptotic behavior of the full perturbative series without introducing uncontrolled bias from the transformation parameters.

What would settle it

A complete five-loop calculation of the coefficient c5,1 that lies well outside the interval 283 to 313 would contradict the central estimate obtained from the transformations.

Figures

Figures reproduced from arXiv: 2601.11277 by Gauhar Abbas, Vartika Singh.

Figure 1
Figure 1. Figure 1: Final prediction of δ (0) in QCD using the higher-order coefficients listed in [PITH_FULL_IMAGE:figures/full_fig_p022_1.png] view at source ↗
read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript applies nonlinear sequence transformations, specifically the Shanks transformation and variants constructed via Wynn's ε-algorithm, to the known low-order terms (up to O(α_s^4)) of the QCD correction δ^{(0)} in hadronic τ decays. It extracts estimates for the higher-order coefficients c_{5,1} through c_{12,1} (e.g., c_{5,1}=298±15, c_{6,1}=3431±256, c_{7,1}=2.29±0.29×10^4) and predicts the fixed-order perturbative value δ^{(0)}_{FOPT}=0.2119±0.0040±0.0065_{α_s}, with uncertainties taken from the spread across the chosen transformations.

Significance. If the sequence transformations provide unbiased acceleration of the asymptotic series, the resulting coefficient estimates and δ^{(0)}_{FOPT} prediction would offer a practical tool for improving precision in τ-decay phenomenology and α_s extractions without requiring full multi-loop computations. The systematic nature of the approach could complement existing resummation methods in QCD.

major comments (2)
  1. [Abstract] Abstract: The uncertainties on the extracted coefficients c_{5,1}–c_{12,1} and on δ^{(0)}_{FOPT} are defined exclusively as the numerical spread across the selected Shanks/Wynn transformations. For a divergent asymptotic series with expected factorial growth, this spread alone does not bound systematic bias from transformation parameters or truncation choice; no cross-validation against a known higher-order term, Borel singularity analysis, or independent error model is provided to support the quoted errors.
  2. [Abstract] Abstract: The central numerical claim δ^{(0)}_{FOPT}=0.2119±0.0040±0.0065_{α_s} is obtained by applying the transformations directly to the input low-order coefficients; without an explicit demonstration that the chosen accelerators converge to the true sum (rather than a spurious value) for this particular series, the prediction rests on an untested extrapolation assumption.
minor comments (2)
  1. [Abstract] The abstract would benefit from explicitly listing the input perturbative coefficients (up to O(α_s^4)) used as starting data for the transformations.
  2. [Abstract] Notation for the sequence transformations (e.g., specific variants of the ε-algorithm) should be defined more clearly with references to the original Shanks and Wynn papers for readers unfamiliar with the method.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful and constructive review of our manuscript. We address the major comments point by point below, providing clarifications and indicating where revisions will be made to improve the presentation of our results and their limitations.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The uncertainties on the extracted coefficients c_{5,1}–c_{12,1} and on δ^{(0)}_{FOPT} are defined exclusively as the numerical spread across the selected Shanks/Wynn transformations. For a divergent asymptotic series with expected factorial growth, this spread alone does not bound systematic bias from transformation parameters or truncation choice; no cross-validation against a known higher-order term, Borel singularity analysis, or independent error model is provided to support the quoted errors.

    Authors: The referee is correct that the quoted uncertainties are obtained solely from the numerical spread among the Shanks and Wynn-ε transformations. This dispersion is used as a practical indicator of sensitivity to the choice of accelerator, following standard practice in applications of sequence transformations to asymptotic perturbative series. We acknowledge that this does not constitute a rigorous bound on all possible systematic biases, particularly given the expected factorial growth of the coefficients. In the revised manuscript we will expand the abstract and add a dedicated paragraph in the main text to state explicitly that the errors are indicative rather than exhaustive, to note the absence of an independent error model or Borel analysis, and to reference consistency tests obtained by applying the same transformations to lower-order truncations of the known series. revision: partial

  2. Referee: [Abstract] Abstract: The central numerical claim δ^{(0)}_{FOPT}=0.2119±0.0040±0.0065_{α_s} is obtained by applying the transformations directly to the input low-order coefficients; without an explicit demonstration that the chosen accelerators converge to the true sum (rather than a spurious value) for this particular series, the prediction rests on an untested extrapolation assumption.

    Authors: The prediction follows from applying the selected transformations to the known coefficients up to O(α_s^4). These particular accelerators have documented convergence properties for divergent series with factorial growth in both mathematical literature and prior QCD applications. While we cannot furnish a direct proof of convergence to the unknown exact sum without the higher-order terms themselves, we have performed internal stability checks by varying the order at which the transformation is initiated and by comparing results across the family of Wynn-ε variants. In the revised version we will insert a short subsection that recalls the theoretical justification for these transformations on model series engineered to mimic the QCD asymptotic behavior and that explicitly flags the extrapolation assumption as a limitation of the present approach. revision: partial

standing simulated objections not resolved
  • Direct cross-validation of the extrapolated coefficients c_{5,1}–c_{12,1} or of the predicted δ^{(0)}_{FOPT} against exact higher-order results is impossible without new multi-loop calculations that lie beyond the scope of the present work.

Circularity Check

1 steps flagged

Higher-order coefficients and δ^(0) prediction obtained by construction via sequence transformations on known low-order terms

specific steps
  1. fitted input called prediction [Abstract]
    "we estimate the perturbative coefficients c_{5,1}-c_{12,1}. In particular, we obtain c_{5,1}=298 ± 15, c_{6,1}=3431 ± 256, and c_{7,1}=2.29 ± 0.29×10^4 ... Moreover, we predict the QCD correction δ^{(0)}_{FOPT}=0.2119 ± 0.0040±0.0065_{α_s}."

    The coefficients and the δ^{(0)} value are generated by applying the Shanks and Wynn transformations to the known low-order terms of δ^{(0)}; the quoted numbers and their uncertainties (defined as spread among transformations) are therefore direct outputs of the input series under the chosen accelerators, with no additional QCD input or independent verification.

full rationale

The paper's central results for c_{5,1}–c_{12,1} and the resummed δ^{(0)}_{FOPT} are produced by applying Shanks and Wynn ε-algorithm transformations directly to the input perturbative series truncated at O(α_s^4). The quoted values and uncertainties (spread across transformations) are therefore outputs of the chosen accelerators rather than independent QCD computations. This matches the fitted-input-called-prediction pattern: the 'estimates' and 'prediction' reduce to extrapolations whose validity is assumed from the transformation properties, with no external cross-check or singularity analysis supplied. The derivation chain is self-contained only in the sense that it faithfully implements the transformations; it does not derive new perturbative information from first principles.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that nonlinear sequence transformations known to accelerate certain asymptotic series can be applied directly to the perturbative QCD series for hadronic tau decays without additional QCD-specific justification or validation against known higher-order terms.

axioms (1)
  • domain assumption The perturbative expansion of δ^{(0)} is an asymptotic series whose higher-order behavior can be recovered by Shanks-type and Wynn ε-algorithm transformations applied to the known low-order coefficients.
    Invoked when the authors state that these transformations are used to extract c5,1–c12,1 from the fixed-order expansion.

pith-pipeline@v0.9.0 · 5529 in / 1475 out tokens · 153080 ms · 2026-05-16T13:34:35.307258+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Perturbative QCD fitting of KEDR and BESIII $e^+e^-$ data for R(s) and $\alpha_s$ determination

    hep-ph 2026-03 unverdicted novelty 4.0

    Combined fits of KEDR and BESIII R(s) data yield alpha_s(M_Z) = 0.1179, 0.1221, and 0.1312 at NLO, NNLO, and NNNLO, demonstrating strong dependence on perturbative truncation order.

Reference graph

Works this paper leans on

104 extracted references · 104 canonical work pages · cited by 1 Pith paper · 53 internal anchors

  1. [1]

    The Physics of Hadronic Tau Decays

    M. Davier, A. Hocker and Z. Zhang, Rev. Mod. Phys.78, 1043-1109 (2006) doi:10.1103/RevModPhys.78.1043 [arXiv:hep-ph/0507078 [hep-ph]]. 24

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/revmodphys.78.1043 2006

  2. [2]

    Pich, Prog

    A. Pich, Prog. Part. Nucl. Phys.117, 103846 (2021) doi:10.1016/j.ppnp.2020.103846 [arXiv:2012.04716 [hep-ph]]

    work page doi:10.1016/j.ppnp.2020.103846 2021

  3. [3]

    Navas, et al., Review of particle physics, Phys

    S. Navaset al.[Particle Data Group], Phys. Rev. D110(2024) no.3, 030001 doi:10.1103/PhysRevD.110.030001

    work page doi:10.1103/physrevd.110.030001 2024

  4. [4]

    P. A. Boyle, B. Chakraborty , C. T. H. Davies, T. DeGrand, C. DeTar, L. Del Debbio, A. X. El- Khadra, F. Erben, J. M. Flynn and E. G´amiz,et al.[arXiv:2205.15373 [hep-lat]]

    work page arXiv

  5. [5]

    Davoudi, E

    Z. Davoudi, E. T. Neil, C. W. Bauer, T. Bhattacharya, T. Blum, P. Boyle, R. C. Brower, S. Catterall, N. H. Christ and V. Cirigliano,et al.[arXiv:2209.10758 [hep-lat]]

    work page arXiv

  6. [6]

    FLAG Review 2024

    Y. Aokiet al.[Flavour Lattice Averaging Group (FLAG)], [arXiv:2411.04268 [hep-lat]]

    work page internal anchor Pith review Pith/arXiv arXiv

  7. [7]

    Braaten, S

    E. Braaten, S. Narison and A. Pich, Nucl. Phys. B373(1992), 581-612 doi:10.1016/0550-3213(92)90267-F

    work page doi:10.1016/0550-3213(92)90267-f 1992

  8. [8]

    P. A. Baikov, K. G. Chetyrkin and J. H. Kuhn, Phys. Rev. Lett.101(2008) 012002 [arXiv:0801.1821 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  9. [10]

    Alpha_s(M_Z) from hadronic tau decays

    K. Maltman and T. Yavin, Phys. Rev. D78, 094020 (2008) doi:10.1103/PhysRevD.78.094020 [arXiv:0807.0650 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.78.094020 2008

  10. [13]

    The 2009 World Average of $\alpha_s$

    S. Bethke, Eur. Phys. J. C64, 689-703 (2009) doi:10.1140/epjc/s10052-009-1173-1 [arXiv:0908.1135 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1140/epjc/s10052-009-1173-1 2009

  11. [18]

    Determination of $\alpha_s(M_\tau^2)$: a conformal mapping approach

    I. Caprini and J. Fischer, Nucl. Phys. B Proc. Suppl.218, 128-133 (2011) doi:10.1016/j.nuclphysbps.2011.06.022 [arXiv:1011.6480 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.nuclphysbps.2011.06.022 2011

  12. [21]

    Recent progress in hadronic tau decays

    M. Jamin, Nucl. Phys. B Proc. Suppl.218, 98-103 (2011) doi:10.1016/j.nuclphysbps.2011.06.017 [arXiv:1101.0681 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.nuclphysbps.2011.06.017 2011

  13. [24]

    Workshop on Precision Measurements of alphas

    M. Beneke and M. Jamin,Fixed-order analysis of the hadronicτdecay width, inWorkshop on Precision Measurements ofα s, ed. S. Bethkeet al, page 25, arXiv:1110.0016

    work page internal anchor Pith review Pith/arXiv arXiv

  14. [25]

    A new determination of \alpha_s from hadronic \tau\ decays

    D. Boito, O. Cata, M. Golterman, M. Jamin, K. Maltman, J. Osborne and S. Peris, Phys. Rev. D84, 113006 (2011) doi:10.1103/PhysRevD.84.113006 [arXiv:1110.1127 [hep- ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.84.113006 2011

  15. [26]

    The strong coupling from the revised ALEPH data for hadronic $\tau$ decays

    D. Boito, M. Golterman, K. Maltman, J. Osborne and S. Peris, Phys. Rev. D91(2015) no.3, 034003 doi:10.1103/PhysRevD.91.034003 [arXiv:1410.3528 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.91.034003 2015

  16. [27]

    Bazavov, N

    A. Bazavov, N. Brambilla, X. G. Tormo, I, P. Petreczky , J. Soto and A. Vairo, Phys. Rev. D90, no.7, 074038 (2014) [erratum: Phys. Rev. D101, no.11, 119902 (2020)] doi:10.1103/PhysRevD.90.074038 [arXiv:1407.8437 [hep-ph]]

    work page doi:10.1103/physrevd.90.074038 2014

  17. [28]

    Boito, A

    D. Boito, A. Eiben, M. Golterman, K. Maltman, L. M. Mansur and S. Peris, Phys. Rev. D111, no.7, 074010 (2025) doi:10.1103/PhysRevD.111.074010 [arXiv:2502.08147 [hep-ph]]

    work page doi:10.1103/physrevd.111.074010 2025

  18. [29]

    alpha_s and the tau hadronic width: fixed-order, contour-improved and higher-order perturbation theory

    M. Beneke and M. Jamin, JHEP09, 044 (2008) doi:10.1088/1126-6708/2008/09/044 [arXiv:0806.3156 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/1126-6708/2008/09/044 2008

  19. [30]

    O. Cata, M. Golterman and S. Peris, Phys. Rev. D77, 093006 (2008) doi:10.1103/PhysRevD.77.093006 [arXiv:0803.0246 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.77.093006 2008

  20. [31]

    Determination of the QCD Coupling from ALEPH $\tau$ Decay Data

    A. Pich and A. Rodr ´ıguez-S´anchez, Phys. Rev. D94, no.3, 034027 (2016) doi:10.1103/PhysRevD.94.034027 [arXiv:1605.06830 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.94.034027 2016

  21. [32]

    Pich and A

    A. Pich and A. Rodr ´ıguez-S´anchez, JHEP07(2022), 145 doi:10.1007/JHEP07(2022)145 [arXiv:2205.07587 [hep-ph]]

    work page doi:10.1007/jhep07(2022)145 2022

  22. [33]

    Figueroa

    C. Ayala, G. Cvetic and D. Teca, J. Phys. G50, no.4, 045004 (2023) doi:10.1088/1361- 6471/acbd65 [arXiv:2206.05631 [hep-ph]]

    work page doi:10.1088/1361- 2023

  23. [34]

    A. Deur, S. J. Brodsky and C. D. Roberts, Prog. Part. Nucl. Phys.134(2024), 104081 doi:10.1016/j.ppnp.2023.104081 [arXiv:2303.00723 [hep-ph]]

    work page doi:10.1016/j.ppnp.2023.104081 2024

  24. [35]

    A. Deur, S. J. Brodsky and G. F. de Teramond, Nucl. Phys.90, 1 (2016) doi:10.1016/j.ppnp.2016.04.003 [arXiv:1604.08082 [hep-ph]]. 26

    work page doi:10.1016/j.ppnp.2016.04.003 2016

  25. [36]

    Boito, M

    D. Boito, M. Golterman, K. Maltman and S. Peris, Phys. Rev. D111, no.7, 074019 (2025) doi:10.1103/PhysRevD.111.074019 [arXiv:2402.05588 [hep-ph]]

    work page doi:10.1103/physrevd.111.074019 2025

  26. [37]

    A. A. Pivovarov, Sov. J. Nucl. Phys.54, 676-678 (1991) doi:10.1007/BF01625906 [arXiv:hep-ph/0302003 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/bf01625906 1991

  27. [39]

    R. L. Workmanet al.[Particle Data Group], PTEP2022, 083C01 (2022) doi:10.1093/ptep/ptac097

    work page doi:10.1093/ptep/ptac097 2022

  28. [40]

    A. H. Hoang and C. Regner, Eur. Phys. J. ST230, no.12-13, 2625-2639 (2021) doi:10.1140/epjs/s11734-021-00257-z [arXiv:2105.11222 [hep-ph]]

    work page doi:10.1140/epjs/s11734-021-00257-z 2021

  29. [41]

    Boito, M

    D. Boito, M. Golterman, K. Maltman, S. Peris, M. V. Rodrigues and W. Schaaf, Phys. Rev. D103, no.3, 034028 (2021) doi:10.1103/PhysRevD.103.034028 [arXiv:2012.10440 [hep-ph]]

    work page doi:10.1103/physrevd.103.034028 2021

  30. [43]

    A. H. Hoang and C. Regner, Phys. Rev. D105(2022) 096023 [arXiv:2008.00578 [hep- ph]]

    work page arXiv 2022

  31. [44]

    A. H. Hoang and C. Regner, Eur. Phys. J. ST230(2021) 2625 [arXiv:2105.11222 [hep- ph]]

    work page arXiv 2021

  32. [45]

    Golterman, K

    M. Golterman, K. Maltman and S. Peris, Phys. Rev. D108(2023) no.1, 014007 doi:10.1103/PhysRevD.108.014007 [arXiv:2305.10386 [hep-ph]]

    work page doi:10.1103/physrevd.108.014007 2023

  33. [46]

    N. G. Gracia, A. H. Hoang and V. Mateu, Phys. Rev. D108, no.3, 034013 (2023) doi:10.1103/PhysRevD.108.034013 [arXiv:2305.10288 [hep-ph]]

    work page doi:10.1103/physrevd.108.034013 2023

  34. [47]

    Renormalons

    M. Beneke, Phys. Rept.317, 1-142 (1999) doi:10.1016/S0370-1573(98)00130-6 [arXiv:hep-ph/9807443 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/s0370-1573(98)00130-6 1999

  35. [48]

    M. A. Benitez-Rathgeb, D. Boito, A. H. Hoang and M. Jamin, JHEP07, 016 (2022) doi:10.1007/JHEP07(2022)016 [arXiv:2202.10957 [hep-ph]]

    work page doi:10.1007/jhep07(2022)016 2022

  36. [49]

    M. A. Benitez-Rathgeb, D. Boito, A. H. Hoang and M. Jamin, JHEP09, 223 (2022) doi:10.1007/JHEP09(2022)223 [arXiv:2207.01116 [hep-ph]]

    work page doi:10.1007/jhep09(2022)223 2022

  37. [50]

    Beneke and H

    M. Beneke and H. Takaura, PoSRADCOR2023, 062 (2024) doi:10.22323/1.432.0062 [arXiv:2309.10853 [hep-ph]]

    work page doi:10.22323/1.432.0062 2024

  38. [51]

    Beneke and H

    M. Beneke and H. Takaura, [arXiv:2510.12193 [hep-ph]]

    work page arXiv

  39. [52]

    Renormalization Group Summation of Laplace QCD Sum Rules for Scalar Gluon Currents

    F. Chishtie, T. G. Steele and D. G. C. McKeon, Phys. Lett. B754, 43-48 (2016) doi:10.1016/j.physletb.2016.01.008 [arXiv:1506.09018 [hep-ph]]. 27

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.physletb.2016.01.008 2016

  40. [53]

    X. G. Wu, J. M. Shen, B. L. Du, X. D. Huang, S. Q. Wang and S. J. Brodsky , Prog. Part. Nucl. Phys.108, 103706 (2019) doi:10.1016/j.ppnp.2019.05.003 [arXiv:1903.12177 [hep-ph]]

    work page doi:10.1016/j.ppnp.2019.05.003 2019

  41. [55]

    C. N. Lovett-Turner and C. J. Maxwell, Nucl. Phys. B432, 147-162 (1994) doi:10.1016/0550-3213(94)90597-5 [arXiv:hep-ph/9407268 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/0550-3213(94)90597-5 1994

  42. [56]

    P. Ball, M. Beneke and V. M. Braun, Nucl. Phys. B452, 563-625 (1995) doi:10.1016/0550-3213(95)00392-6 [arXiv:hep-ph/9502300 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/0550-3213(95)00392-6 1995

  43. [57]

    A. L. Kataev and V. V. Starshenko, Mod. Phys. Lett. A10, 235-250 (1995) doi:10.1142/S0217732395000272 [arXiv:hep-ph/9502348 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1142/s0217732395000272 1995

  44. [58]

    C. N. Lovett-Turner and C. J. Maxwell, Nucl. Phys. B452, 188-212 (1995) doi:10.1016/0550-3213(95)00383-4 [arXiv:hep-ph/9505224 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/0550-3213(95)00383-4 1995

  45. [59]

    QCD Analysis of Hadronic $\tau$ Decays Revisited

    M. Neubert, Nucl. Phys. B463, 511-546 (1996) doi:10.1016/0550-3213(96)00002-8 [arXiv:hep-ph/9509432 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/0550-3213(96)00002-8 1996

  46. [60]

    RS-invariant all-orders renormalon resummations for some QCD observables

    C. Maxwell, D. Tonge, RS invariant all orders renormalon resummations for some QCD observables, Nucl. Phys. B 481 (1996) 681–703.arXiv:hep-ph/9606392,doi: 10.1016/S0550-3213(96)00532-9

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/s0550-3213(96)00532-9 1996

  47. [62]

    C. J. Maxwell and A. Mirjalili, Nucl. Phys. B611, 423-446 (2001) doi:10.1016/S0550- 3213(01)00327-3 [arXiv:hep-ph/0103164 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/s0550- 2001

  48. [63]

    Jamin, JHEP09, 058 (2005) doi:10.1088/1126-6708/2005/09/058 [arXiv:hep- ph/0509001 [hep-ph]]

    M. Jamin, JHEP09, 058 (2005) doi:10.1088/1126-6708/2005/09/058 [arXiv:hep- ph/0509001 [hep-ph]]

    work page doi:10.1088/1126-6708/2005/09/058 2005

  49. [64]

    The Determination of alpha_s from Tau Decays Revisited

    M. Davier, S. Descotes-Genon, A. Hocker, B. Malaescu and Z. Zhang, Eur. Phys. J. C56, 305-322 (2008) doi:10.1140/epjc/s10052-008-0666-7 [arXiv:0803.0979 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1140/epjc/s10052-008-0666-7 2008

  50. [65]

    Alpha_s Determination from Tau Decays: Theoretical Status

    A. Pich, Acta Phys. Polon. Supp.3, 165-170 (2010) [arXiv:1001.0389 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  51. [66]

    Modified Contour-Improved Perturbation Theory

    G. Cvetic, M. Loewe, C. Martinez and C. Valenzuela, Phys. Rev. D82, 093007 (2010) doi:10.1103/PhysRevD.82.093007 [arXiv:1005.4444 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.82.093007 2010

  52. [67]

    QCD Description of Hadronic Tau Decays

    A. Pich, Nucl. Phys. B Proc. Suppl.218, 89-97 (2011) doi:10.1016/j.nuclphysbps.2011.06.016 [arXiv:1101.2107 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.nuclphysbps.2011.06.016 2011

  53. [68]

    Understanding perturbative results for decays of \tau leptons into hadrons

    S. Groote, J. G. Korner and A. A. Pivovarov, Phys. Part. Nucl.44, 285-298 (2013) doi:10.1134/S1063779613020147 [arXiv:1212.5346 [hep-ph]]. 28

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1134/s1063779613020147 2013

  54. [69]

    Scheme variations of the QCD coupling and hadronic $\tau$ decays

    D. Boito, M. Jamin and R. Miravitllas, Phys. Rev. Lett.117, no.15, 152001 (2016) doi:10.1103/PhysRevLett.117.152001 [arXiv:1606.06175 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.117.152001 2016

  55. [70]

    Higher-order QCD corrections to hadronic $\tau$ decays from Pad\'e approximants

    D. Boito, P. Masjuan and F. Oliani, JHEP08, 075 (2018) doi:10.1007/JHEP08(2018)075 [arXiv:1807.01567 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep08(2018)075 2018

  56. [71]

    A Study of Ultraviolet Renormalon Ambiguities in the Determination of $\as$ from $\tau$ Decay

    G. Altarelli, P. Nason and G. Ridolfi, Z. Phys. C68, 257-268 (1995) doi:10.1007/BF01566673 [arXiv:hep-ph/9501240 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/bf01566673 1995

  57. [72]

    Resummation of the hadronic tau decay width with the modified Borel transform method

    G. Cvetic, C. Dib, T. Lee and I. Schmidt, Phys. Rev. D64, 093016 (2001) doi:10.1103/PhysRevD.64.093016 [arXiv:hep-ph/0106024 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.64.093016 2001

  58. [73]

    On the determination of alpha_s from hadronic tau decays with contour-improved, fixed order and renormalon-chain perturbation theory

    S. Menke, [arXiv:0904.1796 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv

  59. [74]

    $\alpha_s$ from $\tau$ decays: contour-improved versus fixed-order summation in a new QCD perturbation expansion

    I. Caprini and J. Fischer, Eur. Phys. J. C64, 35-45 (2009) doi:10.1140/epjc/s10052- 009-1142-8 [arXiv:0906.5211 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1140/epjc/s10052- 2009

  60. [75]

    A note on renormalon models for the determination of alpha_s(M_tau)

    S. Descotes-Genon and B. Malaescu, [arXiv:1002.2968 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv

  61. [76]

    Expansion functions in perturbative QCD and the determination of $\alpha_s(M_\tau^2)$

    I. Caprini and J. Fischer, Phys. Rev. D84, 054019 (2011) doi:10.1103/PhysRevD.84.054019 [arXiv:1106.5336 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.84.054019 2011

  62. [77]

    Determination of $\alpha_s(M_{\tau}^2)$ from Improved Fixed Order Perturbation Theory

    G. Abbas, B. Ananthanarayan and I. Caprini, Phys. Rev. D85, 094018 (2012) doi:10.1103/PhysRevD.85.094018 [arXiv:1202.2672 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.85.094018 2012

  63. [78]

    Perturbative expansion of tau hadronic spectral function moments and alpha_s extractions

    M. Beneke, D. Boito and M. Jamin, JHEP01, 125 (2013) doi:10.1007/JHEP01(2013)125 [arXiv:1210.8038 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep01(2013)125 2013

  64. [79]

    Perturbative expansion of the QCD Adler function improved by renormalization-group summation and analytic continuation in the Borel plane

    G. Abbas, B. Ananthanarayan, I. Caprini and J. Fischer, Phys. Rev. D87, no.1, 014008 (2013) doi:10.1103/PhysRevD.87.014008 [arXiv:1211.4316 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.87.014008 2013

  65. [80]

    Expansions of $\tau$ hadronic spectral function moments in a nonpower QCD perturbation theory with tamed large order behavior

    G. Abbas, B. Ananthanarayan, I. Caprini and J. Fischer, Phys. Rev. D88, no.3, 034026 (2013) doi:10.1103/PhysRevD.88.034026 [arXiv:1307.6323 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.88.034026 2013

  66. [81]

    Caprini, Phys

    I. Caprini, Phys. Rev. D100, no.5, 056019 (2019) doi:10.1103/PhysRevD.100.056019 [arXiv:1908.06632 [hep-ph]]

    work page doi:10.1103/physrevd.100.056019 2019

  67. [82]

    Caprini, Phys

    I. Caprini, Phys. Rev. D102, no.5, 054017 (2020) doi:10.1103/PhysRevD.102.054017 [arXiv:2006.16605 [hep-ph]]

    work page doi:10.1103/physrevd.102.054017 2020

  68. [83]

    Caprini, Phys

    I. Caprini, Phys. Rev. D108, no.11, 114031 (2023) doi:10.1103/PhysRevD.108.114031 [arXiv:2312.07143 [hep-ph]]

    work page doi:10.1103/physrevd.108.114031 2023

  69. [84]

    Caprini, Phys

    I. Caprini, Phys. Rev. D109, no.7, 074002 (2024) doi:10.1103/PhysRevD.109.074002 [arXiv:2403.10844 [hep-ph]]

    work page doi:10.1103/physrevd.109.074002 2024

  70. [85]

    Schilcher, C

    K. Schilcher, C. Sebu and H. Spiesberger, Phys. Lett. B852, 138590 (2024) doi:10.1016/j.physletb.2024.138590 [arXiv:2401.17131 [hep-ph]]. 29

    work page doi:10.1016/j.physletb.2024.138590 2024

  71. [86]

    Strange Quark Mass Determination from Cabibbo-Suppressed Tau Decays

    A. Pich and J. Prades, JHEP10(1999), 004 doi:10.1088/1126-6708/1999/10/004 [arXiv:hep-ph/9909244 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/1126-6708/1999/10/004 1999

  72. [87]

    Laporta and E

    S. Narison and A. Pich, Phys. Lett. B211(1988), 183-188 doi:10.1016/0370- 2693(88)90830-1

    work page doi:10.1016/0370- 1988

  73. [88]

    X. G. Wu, J. M. Shen, B. L. Du and S. J. Brodsky , Phys. Rev. D97(2018) no.9, 094030 doi:10.1103/PhysRevD.97.094030 [arXiv:1802.09154 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.97.094030 2018

  74. [89]

    Renormalization-scheme variation of a QCD perturbation expansion with tamed large-order behavior

    I. Caprini, Phys. Rev. D98(2018) no.5, 056016 doi:10.1103/PhysRevD.98.056016 [arXiv:1806.10325 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.98.056016 2018

  75. [90]

    K. G. Wilson, Phys. Rev.179(1969), 1499-1512 doi:10.1103/PhysRev.179.1499

    work page doi:10.1103/physrev.179.1499 1969

  76. [91]

    The strong coupling from $e^+e^-\to$ hadrons below charm

    D. Boito, M. Golterman, A. Keshavarzi, K. Maltman, D. Nomura, S. Peris and T. Teubner, Phys. Rev. D98, no.7, 074030 (2018) doi:10.1103/PhysRevD.98.074030 [arXiv:1805.08176 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.98.074030 2018

  77. [92]

    Nearly perturbative lattice-motivated QCD coupling with zero IR limit

    C. Ayala, G. Cvetic, R. Kogerler and I. Kondrashuk, J. Phys. G45, no.3, 035001 (2018) doi:10.1088/1361-6471/aa9ecc [arXiv:1703.01321 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/1361-6471/aa9ecc 2018

  78. [93]

    J. L. Kneur and A. Neveu, Phys. Rev. D88, no.7, 074025 (2013) doi:10.1103/PhysRevD.88.074025 [arXiv:1305.6910 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.88.074025 2013

  79. [94]

    Levin,Development of non-linear transformations for improving convergence of se- quences, Int

    D. Levin,Development of non-linear transformations for improving convergence of se- quences, Int. J. Comput. Math. B3(1973), 371 - 388

    work page 1973

  80. [95]

    Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series,

    E. J. Weniger, “Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series,”Comput. Phys. Rep.10(1989) 189–371

    work page 1989

Showing first 80 references.