- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Perturbative QCD below charm threshold: Theory and tensions with e+e− data

Diogo Boito and Marcelle Caram

PDF

Phys. Rev. D 112, 094052 – Published 24 November, 2025Erratum Phys. Rev. D 114, 059902 (2026)

DOI: https://doi.org/10.1103/dtg5-ghdx

Abstract

In this paper we carefully assess the theory prediction for R(s) below charm threshold, Ruds, and address tensions with the existing data, notably with the 2021 BES-III results. We analyze the uncertainty of the perturbative quantum chromodynamics description in the light of renormalons making use of the large-β0 limit and the renormalon-free gluon-condensate scheme. We provide a reliable estimate of the duality violation contributions; we show they are sizable up to 2.5 GeV and improve the agreement between theory and data, but are negligible for higher energies. We then combine the available experimental data for Ruds and find the datasets to be mutually compatible. Finally, we compare theory and data, both locally and in their contributions to the anomalous magnetic moment of the muon. Theory is compatible with the combined data but discrepancies with the BES-III data reach more than 3σ.

View figure in article

Physics Subject Headings (PhySH)

Erratum

Erratum: Perturbative QCD below charm threshold: Theory and tensions with e+e− data [Phys. Rev. D 112, 094052 (2025)]

Diogo Boito and Marcelle Caram
Phys. Rev. D 114, 059902 (2026)

Article Text

References (151)

  1. R. Aliberti et al., The anomalous magnetic moment of the muon in the standard model: An update, Phys. Rep. 1143, 1 (2025).
  2. K. G. Chetyrkin, J. H. Kuhn, A. Maier, P. Maierhofer, P. Marquard, M. Steinhauser et al., Charm and bottom quark masses: An update, Phys. Rev. D 80, 074010 (2009).
  3. K. Chetyrkin, J. H. Kuhn, A. Maier, P. Maierhofer, P. Marquard, M. Steinhauser et al., Precise charm- and bottom-quark masses: Theoretical and experimental uncertainties, Theor. Math. Phys. 170, 217 (2012).
  4. B. Dehnadi, A. H. Hoang, V. Mateu, and S. M. Zebarjad, Charm mass determination from QCD charmonium sum rules at order αs3, J. High Energy Phys. 09 (2013) 103.
  5. B. Dehnadi, A. H. Hoang, and V. Mateu, Bottom and charm mass determinations with a convergence test, J. High Energy Phys. 08 (2015) 155.
  6. J. Erler, P. Masjuan, and H. Spiesberger, Charm quark mass with calibrated uncertainty, Eur. Phys. J. C 77, 99 (2017).
  7. D. Boito, M. Golterman, A. Keshavarzi, K. Maltman, D. Nomura, S. Peris et al., Strong coupling from e+e−→ hadrons below charm, Phys. Rev. D 98, 074030 (2018).
  8. D. Boito and V. Mateu, Precise αs determination from charmonium sum rules, Phys. Lett. B 806, 135482 (2020).
  9. D. Boito and V. Mateu, Precise determination of αs from relativistic quarkonium sum rules, J. High Energy Phys. 03 (2020) 094.
  10. BESIII Collaboration, Measurement of the cross section for e+e−→hadrons at energies from 2.2324 to 3.6710 GeV, Phys. Rev. Lett. 128, 062004 (2022).
  11. KEDR Collaboration, Precise measurement of Ruds and R between 1.84 and 3.72 GeV at the KEDR detector, Phys. Lett. B 788, 42 (2019).
  12. KEDR Collaboration, Measurement of Ruds and R between 3.12 and 3.72 GeV at the KEDR detector, Phys. Lett. B 753, 533 (2016).
  13. KEDR Collaboration, Measurement of R between 1.84 and 3.05 GeV at the KEDR detector, Phys. Lett. B 770, 174 (2017).
  14. Muon g−2 Collaboration, Measurement of the positive muon anomalous magnetic moment to 0.46 ppm, Phys. Rev. Lett. 126, 141801 (2021).
  15. Muon g−2 Collaboration, Measurement of the positive Muon anomalous magnetic moment to 0.20 ppm, Phys. Rev. Lett. 131, 161802 (2023).
  16. Muon g−2 Collaboration, Measurement of the positive Muon anomalous magnetic moment to 127 ppb, Phys. Rev. Lett. 135, 101802 (2025).
  17. Muon g−2 Collaboration, Final report of the muon E821 anomalous magnetic moment measurement at BNL, Phys. Rev. D 73, 072003 (2006).
  18. T. Aoyama, M. Hayakawa, T. Kinoshita, and M. Nio, Complete tenth-order QED contribution to the Muon g−2, Phys. Rev. Lett. 109, 111808 (2012).
  19. S. Volkov, Calculating the five-loop QED contribution to the electron anomalous magnetic moment: Graphs without lepton loops, Phys. Rev. D 100, 096004 (2019).
  20. S. Volkov, Calculation of the total 10th order QED contribution to the electron magnetic moment, Phys. Rev. D 110, 036001 (2024).
  21. T. Aoyama, M. Hayakawa, A. Hirayama, and M. Nio, Verification of the tenth-order QED contribution to the anomalous magnetic moment of the electron from diagrams without fermion loops, Phys. Rev. D 111, L031902 (2025).
  22. R. H. Parker, C. Yu, W. Zhong, B. Estey, and H. Müller, Measurement of the fine-structure constant as a test of the standard model, Science 360, 191 (2018).
  23. L. Morel, Z. Yao, P. Cladé, and S. Guellati-Khélifa, Determination of the fine-structure constant with an accuracy of 81 parts per trillion, Nature (London) 588, 61 (2020).
  24. X. Fan, T. G. Myers, B. A. D. Sukra, and G. Gabrielse, Measurement of the electron magnetic moment, Phys. Rev. Lett. 130, 071801 (2023).
  25. A. Czarnecki, W. J. Marciano, and A. Vainshtein, Refinements in electroweak contributions to the muon anomalous magnetic moment, Phys. Rev. D 67, 073006 (2003).
  26. C. Gnendiger, D. Stöckinger, and H. Stöckinger-Kim, The electroweak contributions to (g−2)μ after the Higgs boson mass measurement, Phys. Rev. D 88, 053005 (2013).
  27. J. Lüdtke, M. Procura, and P. Stoffer, Dispersion relations for the hadronic VVA correlator, J. High Energy Phys. 04 (2025) 130.
  28. M. Hoferichter, J. Lüdtke, L. Naterop, M. Procura, and P. Stoffer, Improved evaluation of the electroweak contribution to muon g−2, Phys. Rev. Lett. 134, 201801 (2025).
  29. RBC Collaboration and UKQCD Collaboration, Calculation of the hadronic vacuum polarization contribution to the muon anomalous magnetic moment, Phys. Rev. Lett. 121, 022003 (2018).
  30. ETM Collaboration, Electromagnetic and strong isospin-breaking corrections to the muon g−2 from Lattice QCD+QED, Phys. Rev. D 99, 114502 (2019).
  31. S. Borsányi et al., Leading hadronic contribution to the muon magnetic moment from lattice QCD, Nature (London) 593, 51 (2021).
  32. C. Lehner and A. S. Meyer, Consistency of hadronic vacuum polarization between lattice QCD and the R-ratio, Phys. Rev. D 101, 074515 (2020).
  33. χQCD Collaboration, Muon g−2 with overlap valence fermions, Phys. Rev. D 107, 034513 (2023).
  34. C. Aubin, T. Blum, M. Golterman, and S. Peris, Muon anomalous magnetic moment with staggered fermions: Is the lattice spacing small enough?, Phys. Rev. D 106, 054503 (2022).
  35. M. Cè et al., Window observable for the hadronic vacuum polarization contribution to the muon g−2 from lattice QCD, Phys. Rev. D 106, 114502 (2022).
  36. ETM Collaboration, Lattice calculation of the short and intermediate time-distance hadronic vacuum polarization contributions to the muon magnetic moment using twisted-mass fermions, Phys. Rev. D 107, 074506 (2023).
  37. RBC Collaboration and UKQCD Collaboration, Update of Euclidean windows of the hadronic vacuum polarization, Phys. Rev. D 108, 054507 (2023).
  38. S. Kuberski, M. Cè, G. von Hippel, H. B. Meyer, K. Ottnad, A. Risch et al., Hadronic vacuum polarization in the muon g−2: The short-distance contribution from lattice QCD, J. High Energy Phys. 03 (2024) 172.
  39. A. Boccaletti et al., High precision calculation of the hadronic vacuum polarisation contribution to the muon anomaly, arXiv:2407.10913.
  40. S. Spiegel and C. Lehner, High-precision continuum limit study of the HVP short-distance window, Phys. Rev. D 111, 114517 (2025).
  41. RBC Collaboration and UKQCD Collaboration, The long-distance window of the hadronic vacuum polarization for the muon g−2, Phys. Rev. Lett. 134, 201901 (2025).
  42. D. Djukanovic, G. von Hippel, S. Kuberski, H. B. Meyer, N. Miller, K. Ottnad et al., The hadronic vacuum polarization contribution to the muon g−2 at long distances, J. High Energy Phys. 04 (2025) 098.
  43. ETM Collaboration, Strange and charm quark contributions to the muon anomalous magnetic moment in lattice QCD with twisted-mass fermions, Phys. Rev. D 111, 054502 (2025).
  44. Fermilab Lattice Collaboration, HPQCD Collaboration, and MILC Collaboration, Hadronic vacuum polarization for the muon g−2 from lattice QCD: Complete short and intermediate windows, Phys. Rev. D 111, 094508 (2025).
  45. A. Bazavov et al., Hadronic vacuum polarization for the muon g−2 from lattice QCD: Long-distance and full light-quark connected contribution, Phys. Rev. Lett. 135, 011901 (2025).
  46. A. Keshavarzi, D. Nomura, and T. Teubner, g−2 of charged leptons, α(MZ2), and the hyperfine splitting of muonium, Phys. Rev. D 101, 014029 (2020).
  47. L. Di Luzio, A. Keshavarzi, A. Masiero, and P. Paradisi, Model-independent tests of the hadronic vacuum polarization contribution to the Muon g−2, Phys. Rev. Lett. 134, 011902 (2025).
  48. A. Kurz, T. Liu, P. Marquard, and M. Steinhauser, Hadronic contribution to the muon anomalous magnetic moment to next-to-next-to-leading order, Phys. Lett. B 734, 144 (2014).
  49. G. Colangelo, M. Hoferichter, M. Procura, and P. Stoffer, Dispersion relation for hadronic light-by-light scattering: Theoretical foundations, J. High Energy Phys. 09 (2015) 074.
  50. P. Masjuan and P. Sanchez-Puertas, Pseudoscalar-pole contribution to the (gμ−2): A rational approach, Phys. Rev. D 95, 054026 (2017).
  51. G. Colangelo, M. Hoferichter, M. Procura, and P. Stoffer, Dispersion relation for hadronic light-by-light scattering: two-pion contributions, J. High Energy Phys. 04 (2017) 161.
  52. M. Hoferichter, B.-L. Hoid, B. Kubis, S. Leupold, and S. P. Schneider, Dispersion relation for hadronic light-by-light scattering: Pion pole, J. High Energy Phys. 10 (2018) 141.
  53. G. Eichmann, C. S. Fischer, E. Weil, and R. Williams, Single pseudoscalar meson pole and pion box contributions to the anomalous magnetic moment of the muon, Phys. Lett. B 797, 134855 (2019).
  54. J. Bijnens, N. Hermansson-Truedsson, and A. Rodríguez-Sánchez, Short-distance constraints for the HLbL contribution to the muon anomalous magnetic moment, Phys. Lett. B 798, 134994 (2019).
  55. J. Leutgeb and A. Rebhan, Axial vector transition form factors in holographic QCD and their contribution to the anomalous magnetic moment of the muon, Phys. Rev. D 101, 114015 (2020).
  56. L. Cappiello, O. Catà, G. D’Ambrosio, D. Greynat, and A. Iyer, Axial-vector and pseudoscalar mesons in the hadronic light-by-light contribution to the muon (g−2), Phys. Rev. D 102, 016009 (2020).
  57. P. Masjuan, P. Roig, and P. Sánchez-Puertas, The interplay of transverse degrees of freedom and axial-vector mesons with short-distance constraints in g−2, J. Phys. G 49, 015002 (2022).
  58. J. Bijnens, N. Hermansson-Truedsson, L. Laub, and A. Rodríguez-Sánchez, Short-distance HLbL contributions to the muon anomalous magnetic moment beyond perturbation theory, J. High Energy Phys. 10 (2020) 203.
  59. J. Bijnens, N. Hermansson-Truedsson, L. Laub, and A. Rodríguez-Sánchez, The two-loop perturbative correction to the (g−2)μ HLbL at short distances, J. High Energy Phys. 04 (2021) 240.
  60. I. Danilkin, M. Hoferichter, and P. Stoffer, A dispersive estimate of scalar contributions to hadronic light-by-light scattering, Phys. Lett. B 820, 136502 (2021).
  61. D. Stamen, D. Hariharan, M. Hoferichter, B. Kubis, and P. Stoffer, Kaon electromagnetic form factors in dispersion theory, Eur. Phys. J. C 82, 432 (2022).
  62. J. Leutgeb, J. Mager, and A. Rebhan, Hadronic light-by-light contribution to the muon g−2 from holographic QCD with solved U(1)A problem, Phys. Rev. D 107, 054021 (2023).
  63. M. Hoferichter, B. Kubis, and M. Zanke, Axial-vector transition form factors and e+e−→f1π+π−, J. High Energy Phys. 08 (2023) 209.
  64. M. Hoferichter, P. Stoffer, and M. Zillinger, An optimized basis for hadronic light-by-light scattering, J. High Energy Phys. 04 (2024) 092.
  65. E. J. Estrada, S. Gonzàlez-Solís, A. Guevara, and P. Roig, Improved π0,η,η′ transition form factors in resonance chiral theory and their aμHLbL contribution, J. High Energy Phys. 12 (2024) 203.
  66. O. Deineka, I. Danilkin, and M. Vanderhaeghen, Dispersive estimate of the a0(980) contribution to (g−2)μ, Phys. Rev. D 111, 034009 (2025).
  67. G. Eichmann, C. S. Fischer, T. Haeuser, and O. Regenfelder, Axial-vector and scalar contributions to hadronic light-by-light scattering, Eur. Phys. J. C 85, 445 (2025).
  68. J. Bijnens, N. Hermansson-Truedsson, and A. Rodríguez-Sánchez, Constraints on the hadronic light-by-light tensor in corner kinematics for the muon g−2, J. High Energy Phys. 03 (2025) 094.
  69. M. Hoferichter, P. Stoffer, and M. Zillinger, Dispersion relation for hadronic light-by-light scattering: Subleading contributions, J. High Energy Phys. 02 (2025) 121.
  70. S. Holz, M. Hoferichter, B.-L. Hoid, and B. Kubis, Dispersion relation for hadronic light-by-light scattering: η and η′ poles, J. High Energy Phys. 04 (2025) 147.
  71. L. Cappiello, J. Leutgeb, J. Mager, and A. Rebhan, Tensor meson transition form factors in holographic QCD and the muon g−2, J. High Energy Phys. 07 (2025) 033.
  72. G. Colangelo, M. Hoferichter, A. Nyffeler, M. Passera, and P. Stoffer, Remarks on higher-order hadronic corrections to the muon g−2, Phys. Lett. B 735, 90 (2014).
  73. T. Blum, N. Christ, M. Hayakawa, T. Izubuchi, L. Jin, C. Jung et al., Hadronic light-by-light scattering contribution to the Muon anomalous magnetic moment from Lattice QCD, Phys. Rev. Lett. 124, 132002 (2020).
  74. E.-H. Chao, R. J. Hudspith, A. Gérardin, J. R. Green, H. B. Meyer, and K. Ottnad, Hadronic light-by-light contribution to (g−2)μ from lattice QCD: A complete calculation, Eur. Phys. J. C 81, 651 (2021).
  75. E.-H. Chao, R. J. Hudspith, A. Gérardin, J. R. Green, and H. B. Meyer, The charm-quark contribution to light-by-light scattering in the muon (g−2) from lattice QCD, Eur. Phys. J. C 82, 664 (2022).
  76. RBC and UKQCD Collaboration, Hadronic light-by-light contribution to the muon anomaly from lattice QCD with infinite volume QED at physical pion mass, Phys. Rev. D 111, 014501 (2025).
  77. Z. Fodor, A. Gérardin, L. Lellouch, K. K. Szabó, B. C. Toth, and C. Zimmermann, Hadronic light-by-light scattering contribution to the anomalous magnetic moment of the muon at the physical pion mass, Phys. Rev. D 111, 114509 (2025).
  78. T. Aoyama et al., The anomalous magnetic moment of the muon in the standard model, Phys. Rep. 887, 1 (2020).
  79. CMD-3 Collaboration, Measurement of the pion form factor with CMD-3 detector and its implication to the hadronic contribution to Muon (g−2), Phys. Rev. Lett. 132, 231903 (2024).
  80. CMD-3 Collaboration, Measurement of the e+e−→π+π− cross section from threshold to 1.2 GeV with the CMD-3 detector, Phys. Rev. D 109, 112002 (2024).
  81. D. Boito, M. Golterman, K. Maltman, and S. Peris, Evaluation of the three-flavor quark-disconnected contribution to the muon anomalous magnetic moment from experimental data, Phys. Rev. D 105, 093003 (2022).
  82. D. Boito, M. Golterman, K. Maltman, and S. Peris, Data-based determination of the isospin-limit light-quark-connected contribution to the anomalous magnetic moment of the muon, Phys. Rev. D 107, 074001 (2023).
  83. G. Benton, D. Boito, M. Golterman, A. Keshavarzi, K. Maltman, and S. Peris, Data-driven determination of the light-quark connected component of the intermediate-window contribution to the Muon g−2, Phys. Rev. Lett. 131, 251803 (2023).
  84. G. Benton, D. Boito, M. Golterman, A. Keshavarzi, K. Maltman, and S. Peris, Data-driven estimates for light-quark-connected and strange-plus-disconnected hadronic g−2 window quantities, Phys. Rev. D 109, 036010 (2024).
  85. G. Benton, D. Boito, M. Golterman, A. Keshavarzi, K. Maltman, and S. Peris, Data-driven results for light-quark connected and strange-plus-disconnected hadronic g−2 short- and long-distance windows, Phys. Rev. D 111, 034018 (2025).
  86. M. Davier, A. Hoecker, B. Malaescu, and Z. Zhang, A new evaluation of the hadronic vacuum polarisation contributions to the muon anomalous magnetic moment and to α(mZ2), Eur. Phys. J. C 80, 241 (2020).
  87. P. Masjuan, A. Miranda, and P. Roig, τ data-driven evaluation of Euclidean windows for the hadronic vacuum polarization, Phys. Lett. B 850, 138492 (2024).
  88. J. A. Miranda and P. Roig, New τ-based evaluation of the hadronic contribution to the vacuum polarization piece of the muon anomalous magnetic moment, Phys. Rev. D 102, 114017 (2020).
  89. M. Hoferichter, G. Colangelo, B.-L. Hoid, B. Kubis, J. R. de Elvira, D. Schuh et al., Phenomenological estimate of isospin breaking in hadronic vacuum polarization, Phys. Rev. Lett. 131, 161905 (2023).
  90. G. Colangelo, M. Hoferichter, B. Kubis, and P. Stoffer, Isospin-breaking effects in the two-pion contribution to hadronic vacuum polarization, J. High Energy Phys. 10 (2022) 032.
  91. M. Davier, A. Hoecker, G. Lopez Castro, B. Malaescu, X. H. Mo, G. Toledo Sanchez et al., The discrepancy between τ and e+e− spectral functions revisited and the consequences for the Muon magnetic anomaly, Eur. Phys. J. C 66, 127 (2010).
  92. M. Davier, A. Hoecker, A.-M. Lutz, B. Malaescu, and Z. Zhang, Tensions in e+e−→π+π−(γ) measurements: The new landscape of data-driven hadronic vacuum polarization predictions for the muon g−2, Eur. Phys. J. C 84, 721 (2024).
  93. M. Davier, B. Malaescu, and Z. Zhang, Data-based form factor corrections between the two-pion τ and e+e− spectral functions, arXiv:2504.13789.
  94. G. Colangelo, M. Cottini, M. Hoferichter, and S. Holz, Improved calculation of radiative corrections to τ→ππν decays, arXiv:2510.26871.
  95. A. Keshavarzi, D. Nomura, and T. Teubner, Muon g−2 and α(MZ2): A new data-based analysis, Phys. Rev. D 97, 114025 (2018).
  96. M. Davier, A. Hoecker, B. Malaescu, and Z. Zhang, Reevaluation of the hadronic vacuum polarisation contributions to the standard model predictions of the muon g−2 and α(mZ2) using newest hadronic cross-section data, Eur. Phys. J. C 77, 827 (2017).
  97. P. A. Baikov, K. G. Chetyrkin, and J. H. Kuhn, Order αs4 QCD corrections to Z and τ decays, Phys. Rev. Lett. 101, 012002 (2008).
  98. F. Herzog, B. Ruijl, T. Ueda, J. A. M. Vermaseren, and A. Vogt, On Higgs decays to hadrons and the R-ratio at N4LO, J. High Energy Phys. 08 (2017) 113.
  99. M. Beneke, Renormalons, Phys. Rep. 317, 1 (1999).
  100. M. Beneke and M. Jamin, αs and the τ hadronic width: Fixed-order, contour-improved and higher-order perturbation theory, J. High Energy Phys. 09 (2008) 044.
  101. M. Beneke, D. Boito, and M. Jamin, Perturbative expansion of τ hadronic spectral function moments and αs extractions, J. High Energy Phys. 01 (2013) 125.
  102. D. Boito and F. Oliani, Renormalons in integrated spectral function moments and αs extractions, Phys. Rev. D 101, 074003 (2020).
  103. D. Boito, P. Masjuan, and F. Oliani, Higher-order QCD corrections to hadronic τ decays from Padé approximants, J. High Energy Phys. 08 (2018) 075.
  104. M. A. Benitez-Rathgeb, D. Boito, A. H. Hoang, and M. Jamin, Reconciling the contour-improved and fixed-order approaches for τ hadronic spectral moments. Part I. Renormalon-free gluon condensate scheme, J. High Energy Phys. 07 (2022) 016.
  105. M. A. Benitez-Rathgeb, D. Boito, A. H. Hoang, and M. Jamin, Reconciling the contour-improved and fixed-order approaches for τ hadronic spectral moments. Part II. Renormalon norm and application in αs determinations, J. High Energy Phys. 09 (2022) 223.
  106. M. A. Benitez-Rathgeb, D. Boito, A. H. Hoang, and M. Jamin, Reconciling the FOPT and CIPT predictions for the hadronic Tau decay rate, SciPost Phys. Proc. 16, 007 (2025).
  107. O. Catà, M. Golterman, and S. Peris, Duality violations and spectral sum rules, J. High Energy Phys. 08 (2005) 076.
  108. O. Catà, M. Golterman, and S. Peris, Unraveling duality violations in hadronic tau decays, Phys. Rev. D 77, 093006 (2008).
  109. D. Boito, I. Caprini, M. Golterman, K. Maltman, and S. Peris, Hyperasymptotics and quark-hadron duality violations in QCD, Phys. Rev. D 97, 054007 (2018).
  110. D. Boito, A. Eiben, M. Golterman, K. Maltman, L. M. Mansur, and S. Peris, Strong coupling from hadronic τ-decay data including τ→π−π0ντ from Belle, Phys. Rev. D 111, 074010 (2025).
  111. D. Boito, M. Golterman, K. Maltman, S. Peris, M. V. Rodrigues, and W. Schaaf, Strong coupling from an improved τ vector isovector spectral function, Phys. Rev. D 103, 034028 (2021).
  112. I. Caprini, Higher-order perturbative coefficients in QCD from series acceleration by conformal mappings, Phys. Rev. D 100, 056019 (2019).
  113. G. Abbas, B. Ananthanarayan, I. Caprini, and J. Fischer, Expansions of τ hadronic spectral function moments in a nonpower QCD perturbation theory with tamed large order behavior, Phys. Rev. D 88, 034026 (2013).
  114. I. Caprini, J. Fischer, G. Abbas, and B. Ananthanarayan, Perturbative expansions in QCD improved by conformal mappings of the borel plane, arXiv:1711.04445.
  115. I. Caprini, Renormalization-scheme variation of a QCD perturbation expansion with tamed large-order behavior, Phys. Rev. D 98, 056016 (2018).
  116. A. Rodríguez-Sánchez, Hadronic tau decays, arXiv:2504.21732.
  117. F. Le Diberder and A. Pich, The perturbative QCD prediction to Rτ revisited, Phys. Lett. B 286, 147 (1992).
  118. A. A. Pivovarov, Renormalization group analysis of the τ-lepton decay within QCD, Sov. J. Nucl. Phys. 54, 676 (1991).
  119. A. H. Hoang and C. Regner, Borel representation of τ hadronic spectral function moments in contour-improved perturbation theory, Phys. Rev. D 105, 096023 (2022).
  120. A. H. Hoang and C. Regner, On the difference between FOPT and CIPT for hadronic tau decays, Eur. Phys. J. Special Topics 230, 2625 (2021).
  121. M. Golterman, K. Maltman, and S. Peris, Difference between fixed-order and contour-improved perturbation theory, Phys. Rev. D 108, 014007 (2023).
  122. N. G. Gracia, A. H. Hoang, and V. Mateu, Mathematical aspects of the asymptotic expansion in contour improved perturbation theory for hadronic tau decays, Phys. Rev. D 108, 034013 (2023).
  123. K. G. Chetyrkin and A. Kwiatkowski, Mass corrections to the tau decay rate, Z. Phys. C 59, 525 (1993).
  124. K. G. Chetyrkin and J. H. Kuhn, Mass corrections to the Z decay rate, Phys. Lett. B 248, 359 (1990).
  125. R. Brüser, A. H. Hoang, and M. Stahlhofen, Three-loop OPE Wilson coefficients of dimension-four operators for (axial-)vector and (pseudo-)scalar current correlators, J. High Energy Phys. 12 (2024) 103.
  126. D. Boito, O. Catà, M. Golterman, M. Jamin, K. Maltman, J. Osborne et al., A new determination of αs from hadronic τ decays, Phys. Rev. D 84, 113006 (2011).
  127. D. Boito, M. Golterman, K. Maltman, and S. Peris, Quark-hadron duality and the determination of αs from hadronic τ decay: Facts vs myths, Phys. Rev. D 111, 074019 (2025).
  128. M. Gonzalez-Alonso, A. Pich, and J. Prades, Violation of quark-hadron duality and spectral chiral moments in QCD, Phys. Rev. D 81, 074007 (2010).
  129. S. J. Brodsky and E. De Rafael, Suggested boson-lepton pair couplings and the anomalous magnetic moment of the muon, Phys. Rev. 168, 1620 (1968).
  130. B. E. Lautrup and E. De Rafael, Calculation of the sixth-order contribution from the fourth-order vacuum polarization to the difference of the anomalous magnetic moments of muon and electron, Phys. Rev. 174, 1835 (1968).
  131. P. A. Baikov, K. G. Chetyrkin, and J. H. Kühn, Five-loop running of the QCD coupling constant, Phys. Rev. Lett. 118, 082002 (2017).
  132. Particle Data Group, Review of particle physics, Phys. Rev. D 110, 030001 (2024).
  133. D. J. Broadhurst and A. G. Grozin, Matching QCD and heavy-quark effective theory heavy-light currents at two loops and beyond, Phys. Rev. D 52, 4082 (1995).
  134. M. Beneke and V. M. Braun, Naive nonAbelianization and resummation of fermion bubble chains, Phys. Lett. B 348, 513 (1995).
  135. D. J. Broadhurst, Large N expansion of QED: Asymptotic photon propagator and contributions to the muon anomaly, for any number of loops, Z. Phys. C 58, 339 (1993).
  136. M. Beneke, Large order perturbation theory for a physical quantity, Nucl. Phys. B405, 424 (1993).
  137. D. Boito, M. Jamin, and R. Miravitllas, Scheme variations of the QCD coupling and hadronic τ decays, Phys. Rev. Lett. 117, 152001 (2016).
  138. L. S. Brown, L. G. Yaffe, and C.-X. Zhai, Large order perturbation theory for the electromagnetic current current correlation function, Phys. Rev. D 46, 4712 (1992).
  139. S. G. Gorishnii, A. L. Kataev, and S. A. Larin, Three loop corrections of order O(m2) to the correlator of electromagnetic quark currents, Nuovo Cimento Soc. Ital. Fis. 92A, 119 (1986).
  140. F. J. Yndurain, Electromagnetic corrections to asymptotic freedom predictions in e+e− annihilation and deep inelastic scattering, Nucl. Phys. B136, 533 (1978).
  141. A. L. Kataev, Higher order O(α2) and O(ααs) corrections to σtot(e+e−→hadrons) and Z boson decay rate, Phys. Lett. B 287, 209 (1992).
  142. L. R. Surguladze, O(αnαsm) corrections in e+e− annihilation and τ decay, arXiv:hep-ph/9803211.
  143. BES Collaboration, Measurement of the total cross-section for hadronic production by e+e− annihilation at energies between 2.6–5 GeV, Phys. Rev. Lett. 84, 594 (2000).
  144. BES Collaboration, Measurements of the cross-section for e+e−→hadrons at center-of-mass energies from 2 to 5 GeV, Phys. Rev. Lett. 88, 101802 (2002).
  145. BES Collaboration, Measurements of the cross-sections for e+e−→hadrons at 3.650, 3.6648, 3.773 GeV and the branching fraction for ψ(3770)→ non-DD¯, Phys. Lett. B 641, 145 (2006).
  146. BES Collaboration, Measurements of the continuum Ruds and R values in e+e− annihilation in the energy region between 3.650 and 3.872 GeV, Phys. Rev. Lett. 97, 262001 (2006).
  147. BES Collaboration, R value measurements for e+e− annihilation at 2.60, 3.07 and 3.65 GeV, Phys. Lett. B 677, 239 (2009).
  148. L. Criegee and G. Knies, e+e− physics with the PLUTO detector, Phys. Rep. 83, 151 (1982).
  149. A. Pineda, Determination of the bottom quark mass from the ϒ(1S) system, J. High Energy Phys. 06 (2001) 022.
  150. A. H. Hoang, A. Jain, I. Scimemi, and I. W. Stewart, R-evolution: Improving perturbative QCD, Phys. Rev. D 82, 011501 (2010).
  151. A. H. Hoang, A. Jain, C. Lepenik, V. Mateu, M. Preisser, I. Scimemi et al., The MSR mass and the O(ΛQCD) renormalon sum rule, J. High Energy Phys. 04 (2018) 003.

Outline

Information

Phys. Rev. D 112, 094052– Published 24 November, 2025
  • Received 19 September 2025
  • Accepted 12 November 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation