Perturbative  QCD fitting of  KEDR and BESIII  ℯ+ℯ- data  for R(s) and αs determination

A. L. Kataev; K. Yu. Todyshev

doi:10.54546/100801

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Прислана: 10.06.2026; Принята: 29.06.2026; Опубликовано 21.07.2026;

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Как цитировать

А. Л. Катаев, К. Ю. Тодышев "Perturbative QCD fitting of KEDR and BESIII ℯ⁺ℯ^- data for R(s) and α_s determination" Natural Sci. Rev. 3 100801 (2026)
https://doi.org/10.54546/100801

А. Л. Катаев^1,2,a, К. Ю. Тодышев^3,4,b

¹Институт ядерных исследований Российской академии наук, 117312, Москва, Россия
²Объединенный институт ядерных исследований, 141980, Дубна, Россия
³Институт ядерной физики им. Г.И. Будкера Российской академии наук, 630090, Новосибирск, Россия
⁴Новосибирский государственный университет, 630090, Новосибирск, Россия

^akataev@ms2.inr.ac.ru
^btodyshev@inp.nsk.su

DOI: 10.54546/100801

Ключевые слова: теория возмущений КХД

Категории: Физика , Физика высоких энергий (эксперимент) , Феноменология в физике высоких энергий , Физика высоких энергий (теория)

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Аннотация

Экспериментальные данные, полученные коллаборациями KEDR и BESIII при энергиях ниже порогов рождения очарованных кварков, сравниваются с выражениями квантовой хромодинамики (КХД) для безмассового случая (отношение \(R\) в процессе \(e^+e^-\)-аннигиляции), обрезанными на различных порядках теории возмущений. Результаты фитирования демонстрируют зависимость определяемых значений \(\alpha_s(M_Z)\) от порядка обрезания соответствующих приближений. Аппроксимация объединенных данных KEDR и BESIII (с учетом обрезания на масштабе массы \(J/\psi\)-мезона) в приближениях следующего за главным (NLO), следующего за следующим за главным (NNLO) и следующего за следующим за следующим за главным (N3LO) порядков дает следующие результаты: \(\alpha_s(M_Z)=0.1151_{-0.0069}^{+0.0052}\), \(\alpha_s(M_Z)=0.1190_{-0.0081}^{+0.0064}\) и \(\alpha_s(M_Z)=0.1283_{-0.0075}^{+0.0028}\). Наблюдаемая тенденция к росту значений \(\alpha_s(M_Z)\) при повышении порядка приближения связана с влиянием кинематических вкладов типа \(\pi^2\) в коэффициенты отношения \(R\) (не полностью контролируемых в рамках асимптотических разложений теории возмущений), которые возникают вследствие аналитического продолжения из области пространственно-подобных энергий в область времени-подобных энергий. Обсуждаются вопросы применения разложений теории возмущений фиксированного порядка и учета эффектов аналитического продолжения.

Поддерживающие организации

The preliminary results of this work were reported at a scientific seminar at the INR RAS Theory Division on 17 November 2025. We would like to express our gratitude to Konstantin Chetyrkin and in particular to Nikolai Krasnikov for supporting conversations. We also wish to thank Andrej Arbuzov, Dmitry Gorbunov, Ivan Logashenko and Sergey Mikhailov for productive discussions. The material of the work was further presented at the Quarks-2026 International Seminar, Perozavodsk, 19 May 2026. We would like to thank Diogo Boito, Tatyana Kharlamova and Oleg Zenin for useful comments.

Библиографические ссылки

[1] J. Gao, Review of different colliders, Int. J. Mod. Phys. A 36 (22) (2021) 2142002. doi:10.1142/S0217751X21420021.

[2] I. B. Logashenko and S. I. Eidelman, Anomalous magnetic moment of the muon, Phys. Usp. 61 (5) (2018) 480–510. doi:10.3367/UFNe.2018.02.038312.

[3] F. Jegerlehner, The Role of σ(e⁺e⁻ → hadrons) in precision tests of the Standard Model, Nucl. Phys. B Proc. Suppl. 131 (2004) 213–222. doi:10.1016/j.nuclphysbps.2004.02.028, arXiv:hep-ph/0312372.

[4] S. Navas et al. (Particle Data Group), Review of particle physics, Phys. Rev. D 110 (3) (2024) 030001. doi:10.1103/PhysRevD.110.030001.

[5] A. Deur, S. J. Brodsky and G. F. de Teramond, The QCD running coupling, Nucl. Phys. 90 (2016) 1. doi:10.1016/j.ppnp.2016.04.003, arXiv:1604.08082.

[6] G. P. Salam, The strong coupling: A theoretical perspective, In: From My Vast Repertoire...: Guido Altarelli’s Legacy, World Scientific, 2019, Eds. S. Forte, A. Levy and G. Ridolfi. doi:10.1142/9789813238053_0007, arXiv:1712.05165.

[7] A. Pich, Precision physics with inclusive QCD processes, Prog. Part. Nucl. Phys. 117 (2021) 103846. doi:10.1016/j.ppnp.2020.103846, arXiv:2012.04716.

[8] D. d’Enterria, S. Kluth, G. Zanderighi, C. Ayala, M. A. Benitez-Rathgeb, J. Bluemlein, D. Boito, N. Brambilla, D. Britzger, S. Camarda, et al. The strong coupling constant: State of the art and the decade ahead, J. Phys. G 51 (9) (2024) 090501. doi:10.1088/1361-6471/ad1a78,

arXiv:2203.08271.

[9] A. De Rujula and H. Georgi, Counting quarks in e+e− annihilation, Phys. Rev. D 13 (1976) 1296–1301. doi:10.1103/PhysRevD.13.1296.

[10] A. Zee, Electron positron annihilation in stagnant field theories, Phys. Rev. D 8 (1973) 4038–4041. doi:10.1103/PhysRevD.8.4038.

[11] T. Appelquist and H. Georgi, e⁺e?− annihilation in gauge theories of strong interactions, Phys. Rev. D 8 (1973) 4000–4002. doi:10.1103/PhysRevD.8.4000.

[12] S. I. Eidelman, L. M. Kurdadze and A. I. Vainshtein, e⁺e⁻ annihilation into hadrons below 2 GeV. Test of QCD predictions, Phys. Lett. B 82 (1979) 278–280. doi:10.1016/0370-2693(79)90755-X.

[13] K. G. Chetyrkin, A. L. Kataev and F. V. Tkachov, Higher order corrections to σ_tot(e⁺e⁻ → hadrons) in quantum chromodynamics, Phys. Lett. B 85 (1979) 277–279. doi:10.1016/0370-2693(79)90596-3.

[14] M. Dine and J. R. Sapirstein, Higher order QCD corrections in e⁺e⁻ annihilation, Phys. Rev. Lett. 43 (1979) 668. doi:10.1103/PhysRevLett.43.668.

[15] W. Celmaster and R. J. Gonsalves, An analytic calculation of higher order quantum chromodynamic corrections in e⁺e⁻ annihilation, Phys. Rev. Lett. 44 (1980) 560. doi:10.1103/PhysRevLett.44.560.

[16] R. M. Barnett, M. Dine and L. D. McLerran, The problem of R in e⁺e⁻ annihilation, Phys. Rev. D 22 (1980) 594. doi:10.1103/PhysRevD.22.594.

[17] H. J. Behrend et al. (CELLO Collab.), Determination of α_s and sin² θ_w from measurements

of the total hadronic cross-section in e⁺e⁻ annihilation, Phys. Lett. B 183 (1987) 400–411. doi:10.1016/0370-2693(87)90986-5.

[18] R. Marshall, A determination of the strong coupling constant α−s from e⁺e− total cross-section data, Z. Phys. C 43 (1989) 595. doi:10.1007/BF01550938.

[19] R. Marshall, e⁺e⁻ annihilation at high energies, Rept. Prog. Phys. 52 (1989) 1329–1420. doi:10.1088/0034-4885/52/11/001.

[20] G. D’Agostini, W. de Boer and G. Grindhammer, Determination of α_s and the Z⁰ mass from

measurements of the total hadronic cross-section in e⁺e⁻ annihilation, Phys. Lett. B 229 (1989) 160–168. doi:10.1016/0370-2693(89)90176-7.

[21] S. G. Gorishnii, A. L. Kataev and S. A. Larin, Next-To-Leading O(α³_s ) QCD correction to

σ_tot(e⁺e⁻ → hadrons): Analytical calculation and estimation of the parameter lambda (MS), Phys. Lett. B 212 (1988) 238–244. doi:10.1016/0370-2693(88)90532-1.

[22] S. G. Gorishnii, A. L. Kataev and S. A. Larin, The O(α³_s )-corrections to σ_tot(e⁺e⁻ →hadrons) and Γ(τ − → ν_τ + hadrons) in QCD, Phys. Lett. B 259 (1991) 144–150. doi:10.1016/0370-2693(91)90149-K.

[23] L. R. Surguladze and M. A. Samuel, Total hadronic cross-section in e⁺e⁻ annihilation at the four loop level of perturbative QCD, Phys. Rev. Lett. 66 (1991) 560–563 [Erratum: Phys. Rev. Lett. 66 (1991) 2416]. doi:10.1103/PhysRevLett.66.560.

[24] K. G. Chetyrkin, Corrections of order α³_s to Rhad in pQCD with light gluinos, Phys. Lett. B 391 (1997) 402–412. doi:10.1016/S0370-2693(96)01478-5. arXiv:hep-ph/9608480.

[25] V. Branchina, M. Consoli, D. Zappala and R. Fiore, Combined fit to R(e⁺e⁻ → hadrons) and data from the CERN e⁺e⁻ collider LEP, Phys. Rev. D 46 (1992) 75–83. doi:10.1103/PhysRevD.46.75.

[26] R. Ammar et al. (CLEO Collab.), A Measurement of the total cross-section for e⁺e⁻ → hadrons at √s = 10.52 GeV, Phys. Rev. D 57 (1998) 1350–1358. doi:10.1103/PhysRevD.57.1350, arXiv:hep-ex/9707018.

[27] J. H. Kuhn and M. Steinhauser, Determination of αs and heavy quark masses from recent measurements of R(s), Nucl. Phys. B 619 (2001), 588–602 [Erratum: Nucl. Phys. B 640 (2002) 415–415]. doi:10.1016/S0550-3213(01)00499-0, arXiv:hep-ph/010908.

[28] D. Besson et al. (CLEO Collab.), Measurement of the total hadronic cross section in e⁺e⁻ annihilations below 10.56 GeV, Phys. Rev. D 76 (2007) 072008. doi:10.1103/PhysRevD.76.072008, arXiv:0706.2813.

[29] J. H. Kuhn, M. Steinhauser and T. Teubner, Determination of the strong coupling constant from the CLEO measurement of the total hadronic cross section in e⁺e⁻ annihilation below 10.56 GeV, Phys. Rev. D 76 (2007) 074003. doi:10.1103/PhysRevD.76.074003, arXiv:0707.2589.

[30] O. V. Zenin, V. V. Ezhela, S. B. Lugovsky, M. R. Whalley, K. Kang and S. K. Kang, A compilation of total cross-section data on e+e− → hadrons and pQCD tests. arXiv:hep-ph/0110176.

[31] D. Boito, M. Golterman, A. Keshavarzi, K. Maltman, D. Nomura, S. Peris and T. Teubner, Strong coupling from e⁺e⁻ → hadrons below charm, Phys. Rev. D 98 (7) (2018) 074030. doi:10.1103/PhysRevD.98.074030, arXiv:1805.08176.

[32] V. V. Anashin et al. Measurement of Ruds and R between 3.12 and 3.72 GeV at the KEDR detector, Phys. Lett. B 753 (2016) 533–541. doi:10.1016/j.physletb.2015.12.059, arXiv:1510.02667.

[33] V. V. Anashin et al. Measurement of R between 1.84 and 3.05 GeV at the KEDR detector, Phys. Lett. B 770 (2017) 174–181. doi:10.1016/j.physletb.2017.04.073, arXiv:1610.02827.

[34] V. V. Anashin et al. (KEDR Collab.), Precise measurement of R_uds and R between 1.84 and 3.72 GeV at the KEDR detector, Phys. Lett. B 788 (2019) 42–51. doi:10.1016/j.physletb.2018.11.012, arXiv:1805.06235.

[35] M. Ablikim et al. (BESIII Collab.), Measurement of the cross section for e⁺e⁻ → hadrons at energies from 2.2324 to 3.6710 GeV, Phys. Rev. Lett. 128 (6) (2022) 062004. doi:10.1103/PhysRevLett.128.062004, arXiv:2112.11728.

[36] D. Boito and M. Caram, Perturbative QCD below charm threshold: Theory and tensions with e⁺e⁻ data, Phys. Rev. D 112 (9) (2025) 9. doi:10.1103/dtg5-ghdx, arXiv:2509.12956.

[37] P. A. Baikov, K. G. Chetyrkin and J. H. Kuhn, Order α⁴_s QCD corrections to Z and τ decays,

Phys. Rev. Lett. 101 (2008) 012002. doi:10.1103/PhysRevLett.101.012002, arXiv:0801.1821.

[38] F. Herzog, B. Ruijl, T. Ueda, J. A. M. Vermaseren and A. Vogt, On Higgs decays to hadrons and the R-ratio at N⁴LO, JHEP 08 (2017) 113. doi:10.1007/JHEP08(2017)113, arXiv:1707.01044.

[39] A. L. Kataev and V. V. Starshenko, Estimates of the higher order QCD corrections to R(s), R(τ ) and deep inelastic scattering sum rules, Mod. Phys. Lett. A 10 (1995) 235–250. doi:10.1142/S0217732395000272, arXiv:hep-ph/9502348.

[40] D. Boito, P. Masjuan and F. Oliani, Higher-order QCD corrections to hadronic τ decays from Pad´e approximants, JHEP 08 (2018) 075. doi:10.1007/JHEP08(2018)075, arXiv:1807.01567.

[41] G. Abbas and V. Singh, Hadronic tau decays at higher orders in QCD. arXiv:2601.11277.

[42] J. M. Shen, B. H. Qin, J. Yan, S. Q. Wang and X. G. Wu, Novel method to reliably determine the QCD coupling from R_uds measurements and its effects to muon g − 2 and α (M²_Z)within the tau-charm energy region, JHEP 07 (2023) 109. doi:10.1007/JHEP07(2023)109, arXiv:2303.11782.

[43] S. J. Brodsky, M. Mojaza and X. G. Wu, Systematic scale-setting to all orders: The principle of maximum conformality and commensurate scale relations, Phys. Rev. D 89 (2014) 014027. doi:10.1103/PhysRevD.89.014027, arXiv:1304.4631.

[44] A. L. Kataev, Conformal symmetry limit of QED and QCD and identities between perturbative contributions to deep-inelastic scattering sum rules, JHEP 02 (2014) 092. doi:10.1007/JHEP02(2014)092, arXiv:1305.4605.

[45] A. L. Kataev and S. V. Mikhailov, Generalization of the Brodsky–Lepage–Mackenzie optimization within the β-expansion and the principle of maximal conformality, Phys. Rev. D 91 (1) (2015) 014007. doi:10.1103/PhysRevD.91.014007, arXiv:1408.0122.

[46] A. L. Kataev and V. S. Molokoedov, Decomposed photon anomalous dimension in QCD and the β-expanded representations for the Adler function, Phys. Rev. D 108 (9) (2023) 096027. doi:10.1103/PhysRevD.108.096027, arXiv:2309.03994.

[47] S. V. Mikhailov, Adler function, Bjorken polarized sum rule: Confirmation of elements of the β-expansion and the diagrams, JHEP 10 (2024) 166. doi:10.1007/JHEP10(2024)166, arXiv:2406.15014.

[48] D. J. Gross and F. Wilczek, Ultraviolet behavior of non-Abelian gauge theories, Phys. Rev. Lett. 30 (1973) 1343–1346. doi:10.1103/PhysRevLett.30.1343.

[49] H. D. Politzer, Reliable perturbative results for strong interactions?, Phys. Rev. Lett. 30 (1973) 1346–1349. doi:10.1103/PhysRevLett.30.1346.

[50] D. R. T. Jones, Two loop diagrams in Yang–Mills theory, Nucl. Phys. B 75 (1974) 531. doi:10.1016/0550-3213(74)90093-5.

[51] W. E. Caswell, Asymptotic behavior of non-Abelian gauge theories to two loop order, Phys. Rev. Lett. 33 (1974) 244. doi:10.1103/PhysRevLett.33.244.

[52] E. Egorian and O. V. Tarasov, Two loop renormalization of the QCD in an arbitrary gauge, Teor. Mat. Fiz. 41 (1979) 26–32; JINR Preprint E2-11757, Dubna, 1978.

[53] O. V. Tarasov, A. A. Vladimirov and A. Y. Zharkov, The Gell-Mann-Low function of QCD in the three loop approximation, Phys. Lett. B 93 (1980) 429–432. doi:10.1016/0370-2693(80)90358-5.

[54] S. A. Larin and J. A. M. Vermaseren, The three-loop QCD beta function and anomalous dimensions, Phys. Lett. B 303 (1993) 334–336. doi:10.1016/0370-2693(93)91441-O, arXiv:hep-ph/9302208.

[55] T. van Ritbergen, J. A. M. Vermaseren and S. A. Larin, The four-loop beta function in quantum chromodynamics, Phys. Lett. B 400 (1997) 379–384. doi:10.1016/S0370-2693(97)00370-5, arXiv:hep-ph/9701390.

[56] M. Czakon, The four-loop QCD beta function and anomalous dimensions, Nucl. Phys. B 710 (2005) 485–498. doi:10.1016/j.nuclphysb.2005.01.012, arXiv:hep-ph/0411261.

[57] P. A. Baikov, K. G. Chetyrkin and J. H. K¨uhn, Five-loop running of the QCD coupling constant, Phys. Rev. Lett. 118 (8) (2017) 082002. doi:10.1103/PhysRevLett.118.082002, arXiv:1606.08659.

[58] F. Herzog, B. Ruijl, T. Ueda, J. A. M. Vermaseren and A. Vogt, The five-loop beta function of Yang–Mills theory with fermions, JHEP 02 (2017) 090. doi:10.1007/JHEP02(2017)090, arXiv:1701.01404.

[59] T. Luthe, A. Maier, P. Marquard and Y. Schr¨oder, The five-loop beta function for a general gauge group and anomalous dimensions beyond Feynman gauge, JHEP 10 (2017) 166. doi:10.1007/JHEP10(2017)166, arXiv:1709.07718.

[60] N. V. Krasnikov and A. A. Pivovarov, The influence of the analytical continuation effects on the value of the QCD scale parameter Λ extracted from the data on charmonium and upsilon hadon decays, Phys. Lett. B 116 (1982) 168–170. doi:10.1016/0370-2693(82)91001-2.

[61] A. V. Radyushkin, Optimized Λ-parametrization for the QCD running coupling constant in space-like and time-like regions, JINR Preprint E2-82-159, Dubna, 1982; JINR Rapid Commun. 78 (1996) 96–99. arXiv:hep-ph/9907228

[62] M. R. Pennington, R. G. Roberts and G. G. Ross, How to continue the predictions of perturbative QCD from the space-like region where they are derived to the time-like regime where experiments are performed, Nucl. Phys. B 242 (1984) 69–80. doi:10.1016/0550-3213(84)90134-2.

[63] N. V. Krasnikov, On the ultraviolet behavior of the invariant charge in quantum electrodynamics. arXiv:2603.24092.

[64] J. D. Bjorken, Two topics in Quantum Chromodynamics, Preprint SLAC-PUB-5103, 1989.

[65] A. P. Bakulev, S. V. Mikhailov and N. G. Stefanis, Higher-order QCD perturbation theory in different schemes: From FOPT to CIPT to FAPT, JHEP 06 (2010) 085. doi:10.1007/JHEP06(2010)085, arXiv:1004.4125.

[66] A. V. Nesterenko, Electron–positron annihilation into hadrons at the higher-loop levels, Eur. Phys. J. C 77 (12) (2017) 844. doi:10.1140/epjc/s10052-017-5405-5, arXiv:1707.00668.

[67] M. S. A. Alam Khan, Renormalization group summation and analytic continuation from spacelike to timelike regions, Phys. Rev. D 108 (1) (2023) 014028. doi:10.1103/PhysRevD.108.014028, arXiv:2306.10262.

[68] 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 (3) (2023) 034013. doi:10.1103/PhysRevD.108.034013, arXiv:2305.10288.

[69] A. A. Pivovarov, Renormalization group analysis of the tau lepton decay within QCD, Z. Phys. C 53 (1992) 461–464. doi:10.1007/BF01625906, arXiv:hep-ph/0302003.

[70] F. Le Diberder and A. Pich, The perturbative QCD prediction to Rτ revisited, Phys. Lett. B 286 (1992) 147–152. doi:10.1016/0370-2693(92)90172-Z.

[71] D. V. Shirkov and I. L. Solovtsov, Analytic model for the QCD running coupling with universal ¯αs(0) value, Phys. Rev. Lett. 79 (1997) 1209–1212. doi:10.1103/PhysRevLett.79.1209, arXiv:hep-ph/9704333.

[72] D. V. Shirkov and I. L. Solovtsov, Ten years of the Analytic Perturbation Theory in QCD, Theor. Math. Phys. 150 (2007) 132–152. doi:10.1007/s11232-007-0010-7, arXiv:hep-ph/0611229.

[73] A. L. Kataev, G. Parente and A. V. Sidorov, Improved fits to the xF3 CCFR data at the next-to-next-to-leading order and beyond, Phys. Part. Nucl. 34 (2003) 20–46; Erratum: Phys. Part. Nucl. 38 (6) (2007) 827. doi:10.1134/S1063779607060068, arXiv:hep-ph/0106221.

[74] W. Bernreuther and W. Wetzel, Decoupling of heavy quarks in the minimal subtraction scheme, Nucl. Phys. B 197 (1982) 228–236 [Erratum: Nucl. Phys. B 513 (1998) 758–758]. doi:10.1016/0550-3213(82)90288-7.

[75] S. A. Larin, T. van Ritbergen and J. A. M. Vermaseren, The large quark mass expansion of Γ(Z⁰ → hadrons) and Γ(τ − → ν_τ + hadrons) in the order α³_s , Nucl. Phys. B 438 (1995) 278–306. doi:10.1016/0550-3213(94)00574-X, arXiv:hep-ph/9411260.

[76] K. G. Chetyrkin, B. A. Kniehl and M. Steinhauser, Decoupling relations to O(α³_s ) and their connection to low-energy theorems, Nucl. Phys. B 510 (1998) 61–87. doi:10.1016/S0550-3213(98)81004-3, arXiv:hep-ph/9708255.

[77] K. G. Chetyrkin, J. H. Kuhn, C. Sturm, QCD decoupling at four loops, Nucl. Phys. B 744 (2006) 121–135. doi:10.1016/j.nuclphysb.2006.03.020, arXiv:hep-ph/0512060.

[78] B. A. Kniehl, A. V. Kotikov, A. I. Onishchenko and O. L. Veretin, Strong-coupling constant with flavor thresholds at five loops in the anti-MS scheme, Phys. Rev. Lett. 97 (2006) 042001. doi:10.1103/PhysRevLett.97.042001, arXiv:hep-ph/0607202.

[79] K. G. Chetyrkin, J. H. Kuhn and M. Steinhauser, RunDec: A Mathematica package for running and decoupling of the strong coupling and quark masses, Comput. Phys. Commun. 133 (2000) 43–65. doi:10.1016/S0010-4655(00)00155-7, arXiv:hep-ph/0004189.

[80] F. Herren and M. Steinhauser, Version 3 of RunDec and CRunDec, Comput. Phys. Commun. 224 (2018) 333–345. doi:10.1016/j.cpc.2017.11.014, arXiv:1703.03751.

[81] A. Pich and A. Rodr´ıguez-S´anchez, Violations of quark-hadron duality in low-energy determinations of αs, JHEP 07 (2022) 145. doi:10.1007/JHEP07(2022)145, arXiv:2205.07587.

[82] 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 (7) (2025) 074019. doi:10.1103/PhysRevD.111.074019, arXiv:2402.05588.

[83] M. Beneke and H. Takaura, Gradient-flowed operator product expansion without IR renormalons, JHEP 03 (2026) 033. doi:10.1007/JHEP03(2026)033, arXiv:2510.12193.

[84] M. A. Shifman, Quark hadron duality, in: Proc. 8th Int. Symp. on Heavy Flavor Physics (Heavy Flavor 8), World Scientific, Singapore, 2000. V. 3(7), pp. 1447–1494; Int. Workshop “Gribov-70” (What High-Energy Physicists Would Have Liked to Discuss with Vladimir Gribov on his 70th Birthday?), Orsay, France, 2000. doi:10.1142/9789812810458_0032 arXiv:hep-ph/0009131.

[85] D. I. Kazakov and D. V. Shirkov, Scheme invariant perturbation theory in massive QCD, Yad. Fiz. 42 (1985) 768–776 (in Russian).

[86] J. Chyla, A. Kataev and S. Larin, Renormalization scheme dependence and infrared behavior in e⁺e⁻ annihilations and τ lepton decay at the next-to-next-to-leading order of perturbative QCD, Phys. Lett. B 267 (1991) 269–276. doi:10.1016/0370-2693(91)91259-X.

[87] P. M. Stevenson, Optimization of QCD Perturbation Theory: Results for R_e⁺e⁻ at fourth order, Nucl. Phys. B 868 (2013) 38–64. doi:10.1016/j.nuclphysb.2012.11.005, arXiv:1210.7001.

[88] 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 Cim. A 92 (1986) 119–131. doi:10.1007/BF02727185.

[89] K. G. Chetyrkin, R. Harlander, J. H. Kuhn and M. Steinhauser, Mass corrections to the vector current correlator, Nucl. Phys. B 503 (1997) 339–353. doi:10.1016/S0550-3213(97)00383-0, arXiv:hep-ph/9704222.

[90] A. I. Onishchenko, Three-loop moments and spectral density of photon polarization function in QED, Natural Sci. Rev. 1 (2024) 2. doi:10.54546/NaturalSciRev.100102.