Abstract
The short-distance behaviour of the hadronic light-by-light (HLbL) contribution to (g − 2)μ has recently been studied by means of an operator product expansion in a background electromagnetic field. The leading term in this expansion has been shown to be given by the massless quark loop, and the non-perturbative corrections are numerically very suppressed. Here, we calculate the perturbative QCD correction to the massless quark loop. The correction is found to be fairly small compared to the quark loop as far as we study energy scales where the perturbative running for the QCD coupling is well-defined, i.e. for scales μ ≳ 1 GeV. This should allow to reduce the large systematic uncertainty associated to high-multiplicity hadronic states.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
T. Aoyama et al., The anomalous magnetic moment of the muon in the Standard Model, Phys. Rept. 887 (2020) 1 [arXiv:2006.04822] [INSPIRE].
Muon g-2 collaboration, Muon (g − 2) technical design report, arXiv:1501.06858 [INSPIRE].
M. Abe et al., A new approach for measuring the muon anomalous magnetic moment and electric dipole moment, PTEP 2019 (2019) 053C02 [arXiv:1901.03047] [INSPIRE].
Muon g-2 collaboration, Final report of the muon E821 anomalous magnetic moment measurement at BNL, Phys. Rev. D 73 (2006) 072003 [hep-ex/0602035] [INSPIRE].
T. Aoyama, M. Hayakawa, T. Kinoshita and M. Nio, Complete tenth-order QED contribution to the muon g − 2, Phys. Rev. Lett. 109 (2012) 111808 [arXiv:1205.5370] [INSPIRE].
T. Aoyama, T. Kinoshita and M. Nio, Theory of the anomalous magnetic moment of the electron, Atoms 7 (2019) 28 [INSPIRE].
A. Czarnecki, W. J. Marciano and A. Vainshtein, Refinements in electroweak contributions to the muon anomalous magnetic moment, Phys. Rev. D 67 (2003) 073006 [Erratum ibid. 73 (2006) 119901] [hep-ph/0212229] [INSPIRE].
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 (2013) 053005 [arXiv:1306.5546] [INSPIRE].
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 \( \alpha \left({m}_Z^2\right) \) using newest hadronic cross-section data, Eur. Phys. J. C 77 (2017) 827 [arXiv:1706.09436] [INSPIRE].
A. Keshavarzi, D. Nomura and T. Teubner, Muon g − 2 and \( \alpha \left({M}_Z^2\right) \): a new data-based analysis, Phys. Rev. D 97 (2018) 114025 [arXiv:1802.02995] [INSPIRE].
G. Colangelo, M. Hoferichter and P. Stoffer, Two-pion contribution to hadronic vacuum polarization, JHEP 02 (2019) 006 [arXiv:1810.00007] [INSPIRE].
M. Hoferichter, B.-L. Hoid and B. Kubis, Three-pion contribution to hadronic vacuum polarization, JHEP 08 (2019) 137 [arXiv:1907.01556] [INSPIRE].
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 \( \alpha \left({m}_Z^2\right) \), Eur. Phys. J. C 80 (2020) 241 [Erratum ibid. 80 (2020) 410] [arXiv:1908.00921] [INSPIRE].
A. Keshavarzi, D. Nomura and T. Teubner, g − 2 of charged leptons, \( \alpha \left({M}_Z^2\right) \), and the hyperfine splitting of muonium, Phys. Rev. D 101 (2020) 014029 [arXiv:1911.00367] [INSPIRE].
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 (2014) 144 [arXiv:1403.6400] [INSPIRE].
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 (2014) 90 [arXiv:1403.7512] [INSPIRE].
T. Blum et al., Hadronic light-by-light scattering contribution to the muon anomalous magnetic moment from lattice QCD, Phys. Rev. Lett. 124 (2020) 132002 [arXiv:1911.08123] [INSPIRE].
J. Bijnens, E. Pallante and J. Prades, Hadronic light by light contributions to the muon g – 2 in the large Nc limit, Phys. Rev. Lett. 75 (1995) 1447 [Erratum ibid. 75 (1995) 3781] [hep-ph/9505251] [INSPIRE].
J. Bijnens, E. Pallante and J. Prades, Analysis of the hadronic light by light contributions to the muon g − 2, Nucl. Phys. B 474 (1996) 379 [hep-ph/9511388] [INSPIRE].
M. Hayakawa and T. Kinoshita, Pseudoscalar pole terms in the hadronic light by light scattering contribution to muon g − 2, Phys. Rev. D 57 (1998) 465 [Erratum ibid. 66 (2002) 019902] [hep-ph/9708227] [INSPIRE].
J. Bijnens, E. Pallante and J. Prades, Comment on the pion pole part of the light by light contribution to the muon g − 2, Nucl. Phys. B 626 (2002) 410 [hep-ph/0112255] [INSPIRE].
M. Hayakawa and T. Kinoshita, Comment on the sign of the pseudoscalar pole contribution to the muon g − 2, hep-ph/0112102 [INSPIRE].
J. Prades, E. de Rafael and A. Vainshtein, The hadronic light-by-light scattering contribution to the muon and electron anomalous magnetic moments, Adv. Ser. Direct. High Energy Phys. 20 (2009) 303 [arXiv:0901.0306] [INSPIRE].
G. Colangelo, M. Hoferichter, M. Procura and P. Stoffer, Dispersion relation for hadronic light-by-light scattering: theoretical foundations, JHEP 09 (2015) 074 [arXiv:1506.01386] [INSPIRE].
P. Masjuan and P. Sánchez-Puertas, Pseudoscalar-pole contribution to the (gμ − 2): a rational approach, Phys. Rev. D 95 (2017) 054026 [arXiv:1701.05829] [INSPIRE].
M. Hoferichter, B.-L. Hoid, B. Kubis, S. Leupold and S. P. Schneider, Dispersion relation for hadronic light-by-light scattering: pion pole, JHEP 10 (2018) 141 [arXiv:1808.04823] [INSPIRE].
A. Gérardin, H. B. Meyer and A. Nyffeler, Lattice calculation of the pion transition form factor with Nf = 2 + 1 Wilson quarks, Phys. Rev. D 100 (2019) 034520 [arXiv:1903.09471] [INSPIRE].
G. Colangelo, M. Hoferichter, M. Procura and P. Stoffer, Dispersion relation for hadronic light-by-light scattering: two-pion contributions, JHEP 04 (2017) 161 [arXiv:1702.07347] [INSPIRE].
V. Pauk and M. Vanderhaeghen, Single meson contributions to the muon’s anomalous magnetic moment, Eur. Phys. J. C 74 (2014) 3008 [arXiv:1401.0832] [INSPIRE].
I. Danilkin and M. Vanderhaeghen, Light-by-light scattering sum rules in light of new data, Phys. Rev. D 95 (2017) 014019 [arXiv:1611.04646] [INSPIRE].
F. Jegerlehner, The anomalous magnetic moment of the muon, Springer, Cham, Switzerland (2017) [INSPIRE].
M. Knecht, S. Narison, A. Rabemananjara and D. Rabetiarivony, Scalar meson contributions to aμ from hadronic light-by-light scattering, Phys. Lett. B 787 (2018) 111 [arXiv:1808.03848] [INSPIRE].
G. Eichmann, C. S. Fischer and R. Williams, Kaon-box contribution to the anomalous magnetic moment of the muon, Phys. Rev. D 101 (2020) 054015 [arXiv:1910.06795] [INSPIRE].
P. Roig and P. Sanchez-Puertas, Axial-vector exchange contribution to the hadronic light-by-light piece of the muon anomalous magnetic moment, Phys. Rev. D 101 (2020) 074019 [arXiv:1910.02881] [INSPIRE].
J. H. Kühn, A. I. Onishchenko, A. A. Pivovarov and O. L. Veretin, Heavy mass expansion, light by light scattering and the anomalous magnetic moment of the muon, Phys. Rev. D 68 (2003) 033018 [hep-ph/0301151] [INSPIRE].
G. Colangelo, F. Hagelstein, M. Hoferichter, L. Laub and P. Stoffer, Longitudinal short-distance constraints for the hadronic light-by-light contribution to (g − 2)μ with large-Nc Regge models, JHEP 03 (2020) 101 [arXiv:1910.13432] [INSPIRE].
G. Colangelo, F. Hagelstein, M. Hoferichter, L. Laub and P. Stoffer, Short-distance constraints on hadronic light-by-light scattering in the anomalous magnetic moment of the muon, Phys. Rev. D 101 (2020) 051501 [arXiv:1910.11881] [INSPIRE].
M. Knecht and A. Nyffeler, Hadronic light by light corrections to the muon g − 2: the pion pole contribution, Phys. Rev. D 65 (2002) 073034 [hep-ph/0111058] [INSPIRE].
T. Kinoshita, B. Nizic and Y. Okamoto, Hadronic contributions to the anomalous magnetic moment of the muon, Phys. Rev. D 31 (1985) 2108 [INSPIRE].
T. Goecke, C. S. Fischer and R. Williams, Hadronic light-by-light scattering in the muon g − 2: a Dyson-Schwinger equation approach, Phys. Rev. D 83 (2011) 094006 [Erratum ibid. 86 (2012) 099901] [arXiv:1012.3886] [INSPIRE].
R. Boughezal and K. Melnikov, Hadronic light-by-light scattering contribution to the muon magnetic anomaly: constituent quark loops and QCD effects, Phys. Lett. B 704 (2011) 193 [arXiv:1104.4510] [INSPIRE].
D. Greynat and E. de Rafael, Hadronic contributions to the muon anomaly in the constituent chiral quark model, JHEP 07 (2012) 020 [arXiv:1204.3029] [INSPIRE].
P. Masjuan and M. Vanderhaeghen, Ballpark prediction for the hadronic light-by-light contribution to the muon (g − 2)μ, J. Phys. G 42 (2015) 125004 [arXiv:1212.0357] [INSPIRE].
A. E. Dorokhov, A. E. Radzhabov and A. S. Zhevlakov, Dynamical quark loop light-by-light contribution to muon g − 2 within the nonlocal chiral quark model, Eur. Phys. J. C 75 (2015) 417 [arXiv:1502.04487] [INSPIRE].
K. Melnikov and A. Vainshtein, Hadronic light-by-light scattering contribution to the muon anomalous magnetic moment revisited, Phys. Rev. D 70 (2004) 113006 [hep-ph/0312226] [INSPIRE].
K. Melnikov and A. Vainshtein, On dispersion relations and hadronic light-by-light scattering contribution to the muon anomalous magnetic moment, arXiv:1911.05874 [INSPIRE].
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 (2020) 114015 [arXiv:1912.01596] [INSPIRE].
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 (2020) 016009 [arXiv:1912.02779] [INSPIRE].
M. Knecht, On some short-distance properties of the fourth-rank hadronic vacuum polarization tensor and the anomalous magnetic moment of the muon, JHEP 08 (2020) 056 [arXiv:2005.09929] [INSPIRE].
P. Masjuan, P. Roig and P. Sanchez-Puertas, The interplay of transverse degrees of freedom and axial-vector mesons with short-distance constraints in g − 2, arXiv:2005.11761 [INSPIRE].
J. Lüdtke and M. Procura, Effects of longitudinal short-distance constraints on the hadronic light-by-light contribution to the muon g − 2, Eur. Phys. J. C 80 (2020) 1108 [arXiv:2006.00007] [INSPIRE].
M. Hoferichter and P. Stoffer, Asymptotic behavior of meson transition form factors, JHEP 05 (2020) 159 [arXiv:2004.06127] [INSPIRE].
M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, QCD and resonance physics. Theoretical foundations, Nucl. Phys. B 147 (1979) 385 [INSPIRE].
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 (2019) 134994 [arXiv:1908.03331] [INSPIRE].
I. I. Balitsky and A. V. Yung, Proton and neutron magnetic moments from QCD sum rules, Phys. Lett. B 129 (1983) 328 [INSPIRE].
B. L. Ioffe and A. V. Smilga, Nucleon magnetic moments and magnetic properties of vacuum in QCD, Nucl. Phys. B 232 (1984) 109 [INSPIRE].
J. Bijnens, N. Hermansson-Truedsson, L. Laub and A. Rodríguez-Sánchez, Short-distance HLbL contributions to the muon anomalous magnetic moment beyond perturbation theory, JHEP 10 (2020) 203 [arXiv:2008.13487] [INSPIRE].
J. Aldins, T. Kinoshita, S. J. Brodsky and A. J. Dufner, Photon-photon scattering contribution to the sixth order magnetic moments of the muon and electron, Phys. Rev. D 1 (1970) 2378 [INSPIRE].
J. A. M. Vermaseren, New features of FORM, math-ph/0010025 [INSPIRE].
P. Maierhöfer, J. Usovitsch and P. Uwer, Kira — a Feynman integral reduction program, Comput. Phys. Commun. 230 (2018) 99 [arXiv:1705.05610] [INSPIRE].
T. G. Birthwright, E. W. N. Glover and P. Marquard, Master integrals for massless two-loop vertex diagrams with three offshell legs, JHEP 09 (2004) 042 [hep-ph/0407343] [INSPIRE].
F. Chavez and C. Duhr, Three-mass triangle integrals and single-valued polylogarithms, JHEP 11 (2012) 114 [arXiv:1209.2722] [INSPIRE].
J. Kodaira, S. Matsuda, T. Muta, K. Sasaki and T. Uematsu, QCD effects in polarized electroproduction, Phys. Rev. D 20 (1979) 627 [INSPIRE].
J. Kodaira, S. Matsuda, K. Sasaki and T. Uematsu, QCD higher order effects in spin dependent deep inelastic electroproduction, Nucl. Phys. B 159 (1979) 99 [INSPIRE].
J. Kodaira, QCD higher order effects in polarized electroproduction: flavor singlet coefficient functions, Nucl. Phys. B 165 (1980) 129 [INSPIRE].
Flavour Lattice Averaging Group collaboration, FLAG review 2019: Flavour Lattice Averaging Group (FLAG), Eur. Phys. J. C 80 (2020) 113 [arXiv:1902.08191] [INSPIRE].
F. Herren and M. Steinhauser, Version 3 of RunDec and CRunDec, Comput. Phys. Commun. 224 (2018) 333 [arXiv:1703.03751] [INSPIRE].
N. I. Usyukina and A. I. Davydychev, New results for two loop off-shell three point diagrams, Phys. Lett. B 332 (1994) 159 [hep-ph/9402223] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2101.09169
Supplementary Information
ESM 1
(ZIP 123 kb)
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Bijnens, J., Hermansson-Truedsson, N., Laub, L. et al. The two-loop perturbative correction to the (g − 2)μ HLbL at short distances. J. High Energ. Phys. 2021, 240 (2021). https://doi.org/10.1007/JHEP04(2021)240
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2021)240