vestifmИзвестия Национальной академии наук Беларуси. Серия физико-математических наукProceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series1561-24302524-2415The Republican Unitary Enterprise Publishing House "Belaruskaya Navuka"10.29235/1561-2430-2023-59-4-338-351vestifm-748Research ArticleФИЗИКАPHYSICSАналитические расчеты электромагнитных поправок пятого порядка к аномальному магнитному моменту лептонов в представлении Меллина – БарнсаAnalytical calculations of fifth-order electromagnetic corrections to the anomalous magnetic moment of leptons within the Mellin-Barnes representationСоловцоваО. П.SolovtsovaO. P.

Соловцова Ольга Павловна – доктор физико-математических наук, главный научный сотрудник научно-исследовательской части, Гомельский государственный технический университет имени П. О. Сухого; ведущий научный сотрудник Лаборатории теоретической физики имени Н. Н. Боголюбова, Объединенный институт ядерных исследований

пр. Октября, 48, 246029, Гомель, Республика Беларусь; ул. Жолио-Кюри, 6, 141980, Дубна, Московская обл., Российская Федерация

Olga P. Solovtsova – Dr. Sc. (Physics and Mathematics), Chief Researcher of the Scientific Research Center, Sukhoi State Technical University of Gomel; Leading Researcher of the Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research

48, Octiabrya Ave., 246029, Gomel, Republic of Belarus; 6, Joliot-Curie Str., 141980, Dubna, Moscow Region, Russian Federation. 

ЛашкевичВ. И.LashkevichV. I.

Лашкевич Василий Иванович – кандидат физико-математических наук, доцент кафедры «Высшая математика»

пр. Октября, 48, 246029, Гомель, Республика Беларусь

Vasil I. Lashkevich – Ph. D. (Physics and Mathematics), Associate Professor of the Higher Mathematics Department 

48, Octiabrya Ave., 246029, Gomel, Republic of Belarus

lashkevich@gstu.gomel.byКаптарьЛ. П.KaptariL. P

Каптарь Леонид Петрович – доктор физико-математических наук, профессор, ведущий научный сотрудник Лаборатории теоретической физики имени Н. Н. Боголюбова

ул. Жолио-Кюри, 6, 141980, Дубна, Московская обл., Российская Федерация

Leonid P. Kaptari – Dr. Sc. (Physics and Mathematics), Professor, Leading Researcher  of  Bogoliubov  Laboratory of Theoretical Physics,

6, Joliot-Curie Str., 141980, Dubna, Moscow Region, Russian Federation

Гомельский государственный технический университет имени П. О. Сухого; Объединенный институт ядерных исследованийSukhoi State Technical University of Gomel; Joint Institute for Nuclear ResearchГомельский государственный технический университет имени П. О. СухогоSukhoi State Technical University of GomelОбъединенный институт ядерных исследованийJoint Institute for Nuclear Research202305012024594338351Copyright © Соловцова О.П., Лашкевич В.И., Каптарь Л.П., 20242024Соловцова О.П., Лашкевич В.И., Каптарь Л.П.Solovtsova O.P., Lashkevich V.I., Kaptari L.P.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://vestifm.belnauka.by/jour/article/view/748

Исследуется явный вид электромагнитных поправок пятого порядка по постоянной тонкой структуре α к аномальному магнитному моменту лептонов aL (L = e, μ, τ) от диаграмм со вставками поляризационного оператора из лептонных петель. Подход основывается на последовательном применении дисперсионных соотношений для поляризационного оператора и преобразования Меллина – Барнса для пропагаторов массивных частиц. Получены явные аналитические выражения для поправок к aL от поляризации вакуума четырьмя одинаковыми лептонными петлями. Найдены асимптотические разложения в пределе как малых, так и больших значений отношения масс лептонов (r = mℓ/mL ), r≪ 1  и r→∞. Полученные разложения сравниваются с соответствующими  выражениями, приведенными в литературе.

We investigate the explicit, analytical expressions for the fifth-order electromagnetic corrections in the fine structure constant α to the anomalous magnetic moment of leptons aL (L = e, μ, τ) from diagrams with insertions of the vacuum polarization operator consisting of pure lepton loops. Our approach is based on the consecutive application of dispersion relations for the polarization operator and the Mellin – Barnes transform for the propagators of massive particles. Exact analytical expressions for the corrections to aL from vacuum polarization by four identical loops are obtained. Asymptotic expansions are found in the limit of both small and large values of the lepton mass ratio  (r = mℓ /mL), r≪ 1 and r→∞ The resulting expansions are compared with the corresponding expressions given in the literature.

аномальные магнитные моменты лептоновквантово-электродинамические поправки высших порядковпреобразование Меллина – Барнсаanomalous magnetic moment of the leptonshigher-order quantum electromagnetic correctionsMellin – Barnes transformРабота выполнена при поддержке Международной программы сотрудничества между Республикой Беларусь и ОИЯИ.This work was supported by the International Program of Cooperation between the Republic of Belarus and JINR.ReferencesDirac, P. A. M. The quantum theory of the electron / P. A. M. Dirac // Proc. R. Soc. London, Ser. A. – 1928. – Vol. 117, № 778. – P. 610–624. https://doi.org/10.1098/rspa.1928.0023Dirac P. A. M. The quantum theory of the electron. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 1972, vol. 117, no. 778, pp. 610–624. https://doi.org/10.1098/rspa.1928.0023Jegerlehner, F. The Anomalous Magnetic Moment of the Muon / F. Jegerlehner. – Springer Cham, 2017. – 693 p. – (Springer Tracts in Modern Physics). https://doi.org/10.1007/978-3-319-63577-4Jegerlehner F. The Anomalous Magnetic Moment of the Muon. Springer Tracts in Modern Physics. Springer Cham, 2017. 693 p. https://doi.org/10.1007/978-3-319-63577-4The anomalous magnetic moment of the muon in the Standard Model / T. Aoyama [et al.] // Phys. Rep. – 2020. – Vol. 887. – P. 1–166. https://doi.org/10.1016/j.physrep.2020.07.006Aoyama T., Asmussen N., Benayoun M., Bijnens J., Blum T., Bruno M. [et al.]. The anomalous magnetic moment of the muon in the Standard Model. Physics Reports, 2020, vol. 887, pp. 1–166. https://doi.org/10.1016/j.physrep.2020.07.006Measurement of the fine-structure constant as a test of the Standard Model / R. H. Parker [et al.] // Science. – 2018. – Vol. 360, № 6385. – P. 191–195. https://doi.org/10.1126/science.aap7706Parker R. H., Chenghui Y., Zhong W., Estey B., Müller H. Measurement of the fine-structure constant as a test of the Standard Model. Science, 2018, vol. 360, no. 6385, pp. 191–195. https://doi.org/10.1126/science.aap7706Determination of the fine-structure constant with an accuracy of 81 parts per trillion / L. Morel [et al.] // Nature. – 2020. – Vol. 588. – P. 61–65. https://doi.org/10.1038/s41586-020-2964-7Morel L., Yao Z., Cladé P., Guellati-Khélifa S. Determination of the fine-structure constant with an accuracy of 81 parts per trillion. Nature, 2020, vol. 588, pp. 61–65. https://doi.org/10.1038/s41586-020-2964-7Measurement of the Positive Muon Anomalous Magnetic Moment to 0.46 ppm. / B. Abi [et al.]; (Muon g – 2 Collaboration) // Phys. Rev. Lett. – 2021. – Vol. 126. – ID 141801. – 11 p. https://doi.org/10.1103/PhysRevLett.126.141801Abi B., Albahri T., Al-Kilani S., Allspach D., Alonzi L. P., Anastasi A., Anisenkov A. [et al.]; (Muon g – 2 Collaboration). Measurement of the Positive Muon Anomalous Magnetic Moment to 0.46 ppm. Physical Review Letters, 2021, vol. 126, art. ID 141801. https://doi.org/10.1103/PhysRevLett.126.141801Schwinger, J. S. Quantum electrodynamics. III: The electromagnetic properties of the electron: radiative corrections to scattering / J. S. Schwinger // Phys. Rev. – 1949. – Vol. 76. – P. 790–817. https://doi.org/10.1103/PhysRev.76.790Schwinger J. S. Quantum electrodynamics. III: The electromagnetic properties of the electron: radiative corrections to scattering. Physical Review, 1949, vol. 76, pp. 790–817. https://doi.org/10.1103/PhysRev.76.790Petermann, A. Fourth order magnetic moment of the electron / A. Petermann // Nucl. Phys. – 1958. – Vol. 5. – P. 667– 683. https://doi.org/10.1016/0029-5582(58)90065-8Petermann A. Fourth order magnetic moment of the electron. Nuclear Physics, 1958, vol. 5, pp. 667–683. https://doi. org/10.1016/0029-5582(58)90065-8Sommerfield, C. M. Magnetic dipole moment of the electron / C. M. Sommerfield // Phys. Rev. – 1957. – Vol. 107, № 1. – P. 328–329. https://doi.org/10.1103/PhysRev.107.328Sommerfield C. M. Magnetic dipole moment of the electron. Physical Review, 1957, vol. 107, pp. 328–329. https://doi. org/10.1103/PhysRev.107.328Laporta, S. High-precision calculation of the 4-loop contribution to the electron g − 2 in QED / S. Laporta // Phys. Lett. – 2017. – Vol. 772. – P. 232–238. https://doi.org/10.1016/j.physletb.2017.06.056Laporta S. High-precision calculation of the 4-loop contribution to the electron g − 2 in QED. Physics Letters B, 2017, vol. 772, no. 1, pp. 232–238. https://doi.org/10.1016/j.physletb.2017.06.056Laporta, S. High-precision calculation of the 4-loop QED contribution to the slope of the Dirac form factor / S. Laporta // Phys. Lett. B. – 2020. – Vol. 800. – Art. ID 135137. https://doi.org/10.1016/j.physletb.2019.135137Laporta S. High-precision calculation of the 4-loop QED contribution to the slope of the Dirac form factor. Physics Letters B, 2020, vol. 800, art. ID 135137. https://doi.org/10.1016/j.physletb.2019.135137Laporta, S. The analytical contribution of the sixth order graphs with vacuum polarization insertions to the muon (g–2) in QED / S. Laporta // Il Nuovo Cim. A. – 1993. – Vol. 106. – P. 675–683. https://doi.org/10.1007/bf02787236Laporta S. The analytical contribution of the sixth order graphs with vacuum polarization insertions to the muon (g–2) in QED. Il Nuovo Cimento A, 1993, vol. 106, pp. 675–683. https://doi.org/10.1007/bf02787236Laporta, S. The analytical contribution of some eighth order graphs containing vacuum polarization insertions to the muon (g−2) in QED / S. Laporta // Phys. Lett. B. – 1993. – Vol. 312, № 4. – P. 495–500. https://doi.org/10.1016/03702693(93)90988-TLaporta S. The analytical contribution of some eighth order graphs containing vacuum polarization insertions to the muon (g−2) in QED. Physics Letters B, 1993, vol. 312, no. 4, pp. 495–500. https://doi.org/10.1016/0370-2693(93)90988-TFriot, S. Asymptotics of Feynman diagrams and the Mellin-Barnes representation / S. Friot, D. Greynat, E. de Rafael // Phys. Lett. B. – 2005. – Vol. 628, № 1–2. – P. 73–84. https://doi.org/10.1016/j.physletb.2005.08.126Friot S., Greynat D., Rafael E. Asymptotics of Feynman diagrams and the Mellin-Barnes representation. Physics Letters B, 2005, vol. 628, no. 1–2, pp. 73–84. https://doi.org/10.1016/j.physletb.2005.08.126Aguilar, J. P. Muon anomaly from lepton vacuum polarization and the Mellin-Barnes representation. / J. P. Aguilar, D. Greynat, E. Rafael // Phys. Rev. D. – 2008. – Vol. 77, № 9. – Art. ID 093010. https://doi.org/10.1103/physrevd.77.093010. Aguilar J. P., Greynat D., Rafael E. Muon anomaly from lepton vacuum polarization and the Mellin-Barnes representation. Physical Review D, 2008, vol. 77, no. 9, art. ID 093010. https://doi.org/10.1103/physrevd.77.093010Solovtsova, O. P. Lepton anomaly from QED diagrams with vacuum polarization insertions within the Mellin–Barnes representation / O. P. Solovtsova, V. I. Lashkevich, L. P. Kaptari // Eur. Phys. J. Plus. – 2023. – Vol. 138. – Art. ID 212. https://doi.org/10.1140/epjp/s13360-023-03834-4Solovtsova O. P., Lashkevich V. I., Kaptari L. P. Lepton anomaly from QED diagrams with vacuum polarization insertions within the Mellin–Barnes representation. European Physical Journal Plus, 2023, vol. 138, art. ID 212. https://doi. org/10.1140/epjp/s13360-023-03834-4Lautrup, B. On high order estimates in QED / B. Lautrup // Phys. Lett. B. – 1977. – Vol. 69, № 1. – P. 109–111. https://doi.org/10.1016/0370-2693(77)90145-9Lautrup B. On high order estimates in QED. Physics Letters B, 1977, vol. 69, no. 1, pp. 109–111. https://doi. org/10.1016/0370-2693(77)90145-9Berestetskii, V. B. Concerning the radiative corrections to the mu-meson magnetic moment / V. B. Berestetskii, O. N. Krohnin, A. K. Khlebnikov // J. Exp. Theor. Phys. – 1956. – Vol. 30, № 5. – P. 761–762.Berestetskii V. B., Krohnin O. N., Khlebnikov A. K. Concerning the radiative corrections to the mu-meson magnetic moment. Journal of Experimental and Theoretical Physics, 1956, vol. 30, no. 5, pp. 761–762.Brodsky, S. J. Suggested boson-lepton pair coupling and the anamalous magnetic moment of the muon / S. J. Brodsky, E. de Rafael, // Phys. Rev. – 1968. – Vol. 168. – P. 1620–1622. https://doi.org/10.1103/PhysRev.168.1620Brodsky S. J., Rafael E. Suggested boson-lepton pair coupling and the anamalous magnetic moment of the muon. Physical Review, 1968, vol. 168, pp. 1620–1622. https://doi.org/10.1103/PhysRev.168.1620Dubovyk, I. Mellin-Barnes Integrals: A Primer on Particle Physics Applications / I. Dubovyk, J. Gluza, G. Somogyi. – Springer Cham, 2022. – 266 p. – (Springer Nature Switzerland AG). https://doi.org/10.1007/978-3-031-14272-7Dubovyk I., Gluza J., Somogyi G. Mellin-Barnes Integrals: A Primer on Particle Physics Applications. Lecture Notes in Physics. Springer Cham, 2022. 266 p. https://doi.org/10.1007/978-3-031-14272-7Smirnov, V. A. Analytic Tools for Feynman Integrals / V. A. Smirnov. – Berlin; Heidelberg: Springer, 2012. – 298 p. – (Springer Tracts Mod. Phys.). https://doi.org/10.1007/978-3-642-34886-0Smirnov V. A. Analytic Tools for Feynman Integrals. Springer Tracts in Modern Physics. Berlin, Heidelberg, Springer, 2012. 298 p. https://doi.org/10.1007/978-3-642-34886-0Boos, E. E. A method of evaluation massive Feynman diagrams / E. E. Boos, A. I. Davydychev // Theor. Math. Phys. – 1991. – Vol. 89. – P. 1052–1064. https://doi.org/10.1007/bf01016805Boos E. E., Davydychev A. I. A method of evaluation massive Feynman diagrams. Theoretical and Mathematical Physics, 1991, vol. 89, pp. 1052–1064. https://doi.org/10.1007/bf01016805Lautrup, B. E. Calculation of the sixth-order contribution from the fourth-order vacuum polarization to the difference of the anomalous magnetic moments of muon and electron / B. E. Lautrup, E. de Rafael // Phys. Rev. – 1968. – Vol. 174. – P. 1835–1842. https://doi.org/10.1103/PhysRev.174.1835Lautrup B. E., Rafael E. Calculation of the sixth-order contribution from the fourth-order vacuum polarization to the difference of the anomalous magnetic moments of muon and electron. Physical Review, 1968, vol. 174, pp. 1835–1842. https:// doi.org/10.1103/PhysRev.174.1835Laursen, M. L. The n-bubble diagram contribution to g-2 / M. L. Laursen, M. A. Samuel // J. Math. Phys. – 1981. – Vol. 22, № 5. – P. 1114–1126. https://doi.org/10.1063/1.524995Laursen M. L., Samuel M. A. The n-bubble diagram contribution to g-2. Journal of Mathematical Physics, 1981, vol. 22, no. 5, pp. 1114–1126. https://doi.org/10.1063/1.524995Лашкевич, В. И. О вкладах высших порядков в аномальные магнитные моменты лептонов от поляризации вакуума лептонными петлями / В. И. Лашкевич, О. П. Соловцова, О. В. Теряев // Вес. Нац. акад. навук Беларусі. Сер. фіз.-мат. навук. – 2022. – Т. 58, № 4. – С. 412–423. https://doi.org/10.29235/1561-2430-2022-58-4-412-423Lashkevich V. I., Solovtsova O. P., Tetyaev O. V. On high order contributions to the anomalous magnetic moments of leptons due to the vacuum polarization by lepton loops. Vestsі Natsyianal’nai akademіі navuk Belarusі. Seryia fіzіkamatematychnykh navuk = Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics series, 2022, vol. 58, no. 4, pp. 412–423 (in Russian). https://doi.org/10.29235/1561-2430-2022-58-4-412-423. CODATA recommended values of the fundamental physical constants: 2018 / E. Tiesinga [et al.] // Rev. Mod. Phys. – 2021. – Vol. 93, № 2. – Art. ID 025010. https://doi.org/10.1103/RevModPhys.93.025010Tiesinga E., Mohr P. J., Newell D. B., Taylor B. N. CODATA recommended values of the fundamental physical constants: 2018. Reviews of Modern Physics, 2021, vol. 93, no. 2, art. ID 025010. https://doi.org/10.1103/RevModPhys.93.025010Laporta, S. Analytical and numerical contributions of some tenth-order graphs containing vacuum polarization insertions to the muon (g−2) in QED / S. Laporta // Phys. Lett. B. – 1994. – Vol. 328, № 3–4. – P. 522–527. https://doi.org/10.1016/0370-2693(94)91513-xLaporta S. Analytical and numerical contributions of some tenth-order graphs containing vacuum polarization insertions to the muon (g−2) in QED. Physics Letters B, 1994, vol. 328, no. 3–4, pp. 522–527. https://doi.org/10.1016/03702693(94)91513-x

The authors declare that there are no conflicts of interest present.