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Longitudinal Short-Distance Constraints on Hadronic Light-by-Light Scattering and Tensor-Meson Contributions to the Muon g−2

Jonas Mager1, Luigi Cappiello2,3, Josef Leutgeb1, and Anton Rebhan1

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Phys. Rev. Lett. 135, 091901 – Published 28 August, 2025

DOI: https://doi.org/10.1103/dxwr-gpsl

Abstract

Short-distance constraints from the operator product expansion in QCD play an important role in the evaluation of the hadronic light-by-light scattering contribution to the anomalous magnetic moment of the muon. Holographic QCD has been shown to naturally incorporate the Melnikov-Vainshtein constraint on the longitudinal amplitude following from the triangle anomaly in the asymmetric limit, where one photon virtuality remains small compared to the others. This is saturated by an infinite tower of axial vector mesons, and their numerical contribution to the muon g−2 in AdS/QCD models agrees rather well with a recent dispersive analysis. However, in AdS/QCD, the longitudinal short-distance constraint where all virtualities are large turns out to be matched only at the level of 81%. In this Letter we show that tensor mesons, whose contribution to the muon g−2 has recently been found to be underestimated, can fill this gap because in holographic QCD their infinite tower of excited mesons only contributes to the symmetric longitudinal short-distance constraint. Numerically, they give rise to a sizeable positive contribution from the low-energy region below 1.5 GeV, a small one from the mixed region, and a negligible one from the high-energy region, which could explain the remaining gap between the most recent dispersive and lattice results for the complete hadronic light-by-light contribution.

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References (69)

  1. Muon g−2 Collaboration, Measurement of the positive muon anomalous magnetic moment to 0.20 ppm, Phys. Rev. Lett. 131, 161802 (2023).
  2. Muon g−2 Collaboration, Detailed report on the measurement of the positive muon anomalous magnetic moment to 0.20 ppm, Phys. Rev. D 110, 032009 (2024).
  3. T. Aoyama et al., The anomalous magnetic moment of the muon in the standard model, Phys. Rep. 887, 1 (2020).
  4. G. Colangelo et al., Prospects for precise predictions of aμ in the standard model, arXiv:2203.15810.
  5. S. Borsányi et al., Leading hadronic contribution to the muon magnetic moment from lattice QCD, Nature (London) 593, 51 (2021).
  6. 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).
  7. 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).
  8. Fermilab Lattice, HPQCD, and MILC Collaborations, Light-quark connected intermediate-window contributions to the muon g-2 hadronic vacuum polarization from lattice QCD, Phys. Rev. D 107, 114514 (2023).
  9. RBC and UKQCD Collaborations, Update of Euclidean windows of the hadronic vacuum polarization, Phys. Rev. D 108, 054507 (2023).
  10. A. Boccaletti et al., High precision calculation of the hadronic vacuum polarisation contribution to the muon anomaly, arXiv:2407.10913.
  11. RBC and UKQCD Collaborations, The long-distance window of the hadronic vacuum polarization for the muon g-2, Phys. Rev. Lett. 134, 201901 (2025).
  12. D. Djukanovic, G. von Hippel, S. Kuberski, H. B. Meyer, N. Miller, K. Ottnad, J. Parrino, A. Risch, and H. Wittig, The hadronic vacuum polarization contribution to the muon g−2 at long distances, J. High Energy Phys. 04 (2025) 098.
  13. K. Melnikov and A. Vainshtein, Hadronic light-by-light scattering contribution to the muon anomalous magnetic moment revisited, Phys. Rev. D 70, 113006 (2004).
  14. P. Masjuan and P. Sánchez-Puertas, Pseudoscalar-pole contribution to the (gμ−2): A rational approach, Phys. Rev. D 95, 054026 (2017).
  15. G. Colangelo, M. Hoferichter, M. Procura, and P. Stoffer, Rescattering effects in the hadronic-light-by-light contribution to the anomalous magnetic moment of the muon, Phys. Rev. Lett. 118, 232001 (2017).
  16. 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.
  17. M. Hoferichter, B.-L. Hoid, B. Kubis, S. Leupold, and S. P. Schneider, Pion-pole contribution to hadronic light-by-light scattering in the anomalous magnetic moment of the muon, Phys. Rev. Lett. 121, 112002 (2018).
  18. 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.
  19. 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, 034520 (2019).
  20. 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).
  21. 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, 051501(R) (2020).
  22. 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, J. High Energy Phys. 03 (2020) 101.
  23. V. Pauk and M. Vanderhaeghen, Single meson contributions to the muon‘s anomalous magnetic moment, Eur. Phys. J. C 74, 3008 (2014).
  24. I. Danilkin and M. Vanderhaeghen, Light-by-light scattering sum rules in light of new data, Phys. Rev. D 95, 014019 (2017).
  25. F. Jegerlehner, The anomalous magnetic moment of the muon, Springer Tracts Mod. Phys. 274, 1 (2017).
  26. M. Knecht, S. Narison, A. Rabemananjara, and D. Rabetiarivony, Scalar meson contributions to aμ from hadronic light-by-light scattering, Phys. Lett. B 787, 111 (2018).
  27. G. Eichmann, C. S. Fischer, and R. Williams, Kaon-box contribution to the anomalous magnetic moment of the muon, Phys. Rev. D 101, 054015 (2020).
  28. M. Hoferichter, P. Stoffer, and M. Zillinger, Dispersion relation for hadronic light-by-light scattering: Subleading contributions, J. High Energy Phys. 02 (2025) 121.
  29. M. Hoferichter, P. Stoffer, and M. Zillinger, Complete dispersive evaluation of the hadronic light-by-light contribution to muon g-2, Phys. Rev. Lett. 134, 061902 (2025).
  30. 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).
  31. 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).
  32. J. Leutgeb and A. Rebhan, Hadronic light-by-light contribution to the muon g−2 from holographic QCD with massive pions, Phys. Rev. D 104, 094017 (2021).
  33. J. Leutgeb, J. Mager, and A. Rebhan, Holographic QCD and the muon anomalous magnetic moment, Eur. Phys. J. C 81, 1008 (2021).
  34. 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).
  35. P. Colangelo, F. Giannuzzi, and S. Nicotri, Hadronic light-by-light scattering contributions to (g−2)μ from axial-vector and tensor mesons in the holographic soft-wall model, Phys. Rev. D 109, 094036 (2024).
  36. J. Leutgeb, J. Mager, and A. Rebhan, Superconnections in AdS/QCD and the hadronic light-by-light contribution to the muon g-2, Phys. Rev. D 111, 114001 (2025).
  37. 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).
  38. 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).
  39. M. Hoferichter, P. Stoffer, and M. Zillinger, An optimized basis for hadronic light-by-light scattering, J. High Energy Phys. 04 (2024) 092.
  40. Belle Collaboration, Study of π0 pair production in single-tag two-photon collisions, Phys. Rev. D 93, 032003 (2016).
  41. 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).
  42. 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).
  43. RBC and UKQCD Collaborations, 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).
  44. Z. Fodor, A. Gerardin, L. Lellouch, K. K. Szabo, 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).
  45. W. A. Bardeen and W. K. Tung, Invariant amplitudes for photon processes, Phys. Rev. 173, 1423 (1968).
  46. R. Tarrach, Invariant amplitudes for virtual compton scattering off polarized nucleons free from kinematical singularities, zeros and constraints, Nuovo Cimento Soc. Ital. Fis. 28A, 409 (1975).
  47. 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.
  48. J. Lüdtke, M. Procura, and P. Stoffer, Dispersion relations for hadronic light-by-light scattering in triangle kinematics, J. High Energy Phys. 04 (2023) 125.
  49. J. Lüdtke, M. Procura, and P. Stoffer, Dispersion relations for the hadronic VVA correlator, J. High Energy Phys. 04 (2025) 130.
  50. M. Hoferichter and P. Stoffer, Asymptotic behavior of meson transition form factors, J. High Energy Phys. 05 (2020) 159.
  51. 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.
  52. 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.
  53. T. Sakai and S. Sugimoto, Low energy hadron physics in holographic QCD, Prog. Theor. Phys. 113, 843 (2005).
  54. J. Erlich, E. Katz, D. T. Son, and M. A. Stephanov, QCD and a holographic model of hadrons, Phys. Rev. Lett. 95, 261602 (2005).
  55. L. Da Rold and A. Pomarol, Chiral symmetry breaking from five-dimensional spaces, Nucl. Phys. B721, 79 (2005).
  56. J. Hirn and V. Sanz, Interpolating between low and high energy QCD via a 5-D Yang-Mills model, J. High Energy Phys. 12 (2005) 030.
  57. A. Karch, E. Katz, D. T. Son, and M. A. Stephanov, Linear confinement and AdS/QCD, Phys. Rev. D 74, 015005 (2006).
  58. J. M. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38, 1113 (1999).
  59. S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428, 105 (1998).
  60. E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2, 253 (1998).
  61. L. D. Landau, On the angular momentum of a system of two photons, Dokl. Akad. Nauk Ser. Fiz. 60, 207 (1948).
  62. C.-N. Yang, Selection rules for the dematerialization of a particle into two photons, Phys. Rev. 77, 242 (1950).
  63. E. Katz, A. Lewandowski, and M. D. Schwartz, Tensor mesons in AdS/QCD, Phys. Rev. D 74, 086004 (2006).
  64. S. Mamedov, Z. Hashimli, and S. Jafarzade, Tensor meson couplings in AdS/QCD, Phys. Rev. D 108, 114032 (2023).
  65. G. A. Schuler, F. A. Berends, and R. van Gulik, Meson photon transition form-factors and resonance cross-sections in e+e− collisions, Nucl. Phys. B523, 423 (1998).
  66. 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.

  67. References [28, 29] obtained −2.5×10−11 for the pole contribution of the ground-state tensor multiplet in the IR region, which in the hQCD case with Fρ fit reads +2.9×10−11 (see Table 1). References [28, 29] also include additional effective poles to account for other higher intermediate states. The amount of their contribution (+2.0×10−11) is comparable to the contributions of excited pseudoscalar and axial-vector states in the hQCD model of [34] (+2.3×10−11 for Fρ fit). The hQCD (Fρ-fit) result for the contribution of excited tensors to the IR region amounts to an extra 8.5−2.9=5.6×10−11 in this category.

  68. G. P. Lepage and S. J. Brodsky, Exclusive processes in quantum chromodynamics: Evolution equations for hadronic wave functions and the form-factors of mesons, Phys. Lett. B 87, 359 (1979).
  69. S. J. Brodsky and G. P. Lepage, Large angle two photon exclusive channels in quantum chromodynamics, Phys. Rev. D 24, 1808 (1981).

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Phys. Rev. Lett. 135, 091901– Published 28 August, 2025
  • Received 21 March 2025
  • Revised 23 May 2025
  • Accepted 21 July 2025
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