Abstract
Isospin-breaking (IB) effects are required for an evaluation of hadronic vacuum polarization at subpercent precision. While the dominant contributions arise from the e+e− → π+π− channel, also IB in the subleading channels can become relevant for a detailed understanding, e.g., of the comparison to lattice QCD. Here, we provide such an analysis for e+e− → 3π by extending our dispersive description of the process, including estimates of final-state radiation (FSR) and ρ–ω mixing. In particular, we develop a formalism to capture the leading infrared-enhanced effects in terms of a correction factor η3π that generalizes the analog treatment of virtual and final-state photons in the 2π case. The global fit to the e+e− → 3π data base, subject to constraints from analyticity, unitarity, and the chiral anomaly, gives \( {\left.{a}_{\mu}^{3\pi}\right|}_{\le 1.8\ \textrm{GeV}}=45.91(53)\times {10}^{-10} \) for the total 3π contribution to the anomalous magnetic moment of the muon, of which \( {a}_{\mu}^{\textrm{FSR}}\left[3\pi \right]=0.51(1)\times {10}^{-10} \) and \( {a}_{\mu}^{\rho -\omega}\left[3\pi \right]=-2.68(70)\times {10}^{-10} \) can be ascribed to IB. We argue that the resulting cancellation with ρ–ω mixing in e+e− → 2π can be understood from a narrow-resonance picture, and provide updated values for the vacuum-polarization-subtracted vector-meson parameters Mω = 782.70(3) MeV, Mϕ = 1019.21(2) MeV, Γω = 8.71(3) MeV, and Γϕ = 4.27(1) MeV.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Muon g – 2 collaboration, Measurement of the Positive Muon Anomalous Magnetic Moment to 0.46 ppm, Phys. Rev. Lett. 126 (2021) 141801 [arXiv:2104.03281] [INSPIRE].
Muon g – 2 collaboration, Magnetic-field measurement and analysis for the Muon g2 Experiment at Fermilab, Phys. Rev. A 103 (2021) 042208 [arXiv:2104.03201] [INSPIRE].
Muon g – 2 collaboration, Beam dynamics corrections to the Run-1 measurement of the muon anomalous magnetic moment at Fermilab, Phys. Rev. Accel. Beams 24 (2021) 044002 [arXiv:2104.03240] [INSPIRE].
Muon g – 2 collaboration, Measurement of the anomalous precession frequency of the muon in the Fermilab Muon g – 2 Experiment, Phys. Rev. D 103 (2021) 072002 [arXiv:2104.03247] [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 et al., The anomalous magnetic moment of the muon in the Standard Model, Phys. Rept. 887 (2020) 1 [arXiv:2006.04822] [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].
B.-L. Hoid, M. Hoferichter and B. Kubis, Hadronic vacuum polarization and vector-meson resonance parameters from e+e− → π0γ, Eur. Phys. J. C 80 (2020) 988 [arXiv:2007.12696] [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].
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].
G. Colangelo, M. Hoferichter, M. Procura and P. Stoffer, Dispersive approach to hadronic light-by-light scattering, JHEP 09 (2014) 091 [arXiv:1402.7081] [INSPIRE].
G. Colangelo et al., Towards a data-driven analysis of hadronic light-by-light scattering, Phys. Lett. B 738 (2014) 6 [arXiv:1408.2517] [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].
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 (2017) 232001 [arXiv:1701.06554] [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].
M. Hoferichter et al., Pion-pole contribution to hadronic light-by-light scattering in the anomalous magnetic moment of the muon, Phys. Rev. Lett. 121 (2018) 112002 [arXiv:1805.01471] [INSPIRE].
M. Hoferichter et al., 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].
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].
G. Colangelo et al., 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].
G. Colangelo et al., 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].
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].
G. Colangelo et al., Remarks on higher-order hadronic corrections to the muon g – 2, Phys. Lett. B 735 (2014) 90 [arXiv:1403.7512] [INSPIRE].
E.-H. Chao et al., Hadronic light-by-light contribution to (g – 2)μ from lattice QCD: a complete calculation, Eur. Phys. J. C 81 (2021) 651 [arXiv:2104.02632] [INSPIRE].
E.-H. Chao et al., The charm-quark contribution to light-by-light scattering in the muon (g − 2) from lattice QCD, Eur. Phys. J. C 82 (2022) 664 [arXiv:2204.08844] [INSPIRE].
T. Blum et al., Hadronic light-by-light contribution to the muon anomaly from lattice QCD with infinite volume QED at physical pion mass, arXiv:2304.04423 [INSPIRE].
C. Alexandrou et al., The η → γ∗γ∗ transition form factor and the hadronic light-by-light η-pole contribution to the muon g − 2 from lattice QCD, arXiv:2212.06704 [INSPIRE].
A. Gérardin et al., Lattice calculation of the π0, η and η′ transition form factors and the hadronic light-by-light contribution to the muon g − 2, arXiv:2305.04570 [INSPIRE].
M. Hoferichter and P. Stoffer, Asymptotic behavior of meson transition form factors, JHEP 05 (2020) 159 [arXiv:2004.06127] [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].
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. Bijnens, N. Hermansson-Truedsson, L. Laub and A. Rodríguez-Sánchez, The two-loop perturbative correction to the (g – 2)μ HLbL at short distances, JHEP 04 (2021) 240 [arXiv:2101.09169] [INSPIRE].
M. Zanke, M. Hoferichter and B. Kubis, On the transition form factors of the axial-vector resonance f1(1285) and its decay into e+e−, JHEP 07 (2021) 106 [arXiv:2103.09829] [INSPIRE].
I. Danilkin, M. Hoferichter and P. Stoffer, A dispersive estimate of scalar contributions to hadronic light-by-light scattering, Phys. Lett. B 820 (2021) 136502 [arXiv:2105.01666] [INSPIRE].
G. Colangelo et al., Short-distance constraints for the longitudinal component of the hadronic light-by-light amplitude: an update, Eur. Phys. J. C 81 (2021) 702 [arXiv:2106.13222] [INSPIRE].
S. Holz, C. Hanhart, M. Hoferichter and B. Kubis, A dispersive analysis of η′ → π+π−γ and η′ → ℓ+ℓ−γ, Eur. Phys. J. C 82 (2022) 434 [Addendum ibid. 82 (2022) 1159] [arXiv:2202.05846] [INSPIRE].
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 (2023) 054021 [arXiv:2211.16562] [INSPIRE].
J. Bijnens, N. Hermansson-Truedsson and A. Rodríguez-Sánchez, Constraints on the hadronic light-by-light in the Melnikov-Vainshtein regime, JHEP 02 (2023) 167 [arXiv:2211.17183] [INSPIRE].
J. Lüdtke, M. Procura and P. Stoffer, Dispersion relations for hadronic light-by-light scattering in triangle kinematics, JHEP 04 (2023) 125 [arXiv:2302.12264] [INSPIRE].
M. Hoferichter, B. Kubis and M. Zanke, Axial-vector transition form factors and e+e− → f1π+π−, arXiv:2307.14413 [INSPIRE].
Muon g – 2 collaboration, Muon (g – 2) Technical Design Report, arXiv:1501.06858 [INSPIRE].
G. Colangelo et al., Prospects for precise predictions of aμ in the Standard Model, arXiv:2203.15810 [INSPIRE].
J. Calmet, S. Narison, M. Perrottet and E. de Rafael, Higher Order Hadronic Corrections to the Anomalous Magnetic Moment of the Muon, Phys. Lett. B 61 (1976) 283 [INSPIRE].
M. Hoferichter and T. Teubner, Mixed Leptonic and Hadronic Corrections to the Anomalous Magnetic Moment of the Muon, Phys. Rev. Lett. 128 (2022) 112002 [arXiv:2112.06929] [INSPIRE].
S. Borsanyi et al., Leading hadronic contribution to the muon magnetic moment from lattice QCD, Nature 593 (2021) 51 [arXiv:2002.12347] [INSPIRE].
RBC and UKQCD collaborations, Calculation of the hadronic vacuum polarization contribution to the muon anomalous magnetic moment, Phys. Rev. Lett. 121 (2018) 022003 [arXiv:1801.07224] [INSPIRE].
M. Cè et al., Window observable for the hadronic vacuum polarization contribution to the muon g – 2 from lattice QCD, Phys. Rev. D 106 (2022) 114502 [arXiv:2206.06582] [INSPIRE].
Extended Twisted Mass 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 (2023) 074506 [arXiv:2206.15084] [INSPIRE].
Fermilab Lattice et al. collaborations, Light-quark connected intermediate-window contributions to the muon g – 2 hadronic vacuum polarization from lattice QCD, Phys. Rev. D 107 (2023) 114514 [arXiv:2301.08274] [INSPIRE].
T. Blum et al., An update of Euclidean windows of the hadronic vacuum polarization, arXiv:2301.08696 [INSPIRE].
G. Colangelo et al., Data-driven evaluations of Euclidean windows to scrutinize hadronic vacuum polarization, Phys. Lett. B 833 (2022) 137313 [arXiv:2205.12963] [INSPIRE].
SND collaboration, Measurement of the e+e− → π+π− process cross section with the SND detector at the VEPP-2000 collider in the energy region 0.525 < \( \sqrt{s} \) < 0.883 GeV, JHEP 01 (2021) 113 [arXiv:2004.00263] [INSPIRE].
CMD-3 collaboration, Measurement of the e+e− → π+π− cross section from threshold to 1.2 GeV with the CMD-3 detector, arXiv:2302.08834 [INSPIRE].
CMD-2 collaboration, High-statistics measurement of the pion form factor in the ρ-meson energy range with the CMD-2 detector, Phys. Lett. B 648 (2007) 28 [hep-ex/0610021] [INSPIRE].
M.N. Achasov et al., Update of the e+e− → π+π− cross-section measured by SND detector in the energy region 400 < \( \sqrt{s} \) < 1000 MeV, J. Exp. Theor. Phys. 103 (2006) 380 [hep-ex/0605013] [INSPIRE].
BaBar collaboration, Precise Measurement of the e+e− → π+π−(γ) Cross Section with the Initial-State Radiation Method at BABAR, Phys. Rev. D 86 (2012) 032013 [arXiv:1205.2228] [INSPIRE].
KLOE-2 collaboration, Combination of KLOE σ (e+e− → π+π−γ(γ)) measurements and determination of \( {a}_{\mu}^{\pi^{+}{\pi}^{-}} \) in the energy range 0.10 < s < 0.95 GeV2, JHEP 03 (2018) 173 [arXiv:1711.03085] [INSPIRE].
BESIII collaboration, Measurement of the e+e− → π+π− cross section between 600 and 900 MeV using initial state radiation, Phys. Lett. B 753 (2016) 629 [Erratum ibid. 812 (2021) 135982] [arXiv:1507.08188] [INSPIRE].
L. Di Luzio, A. Masiero, P. Paradisi and M. Passera, New physics behind the new muon g – 2 puzzle?, Phys. Lett. B 829 (2022) 137037 [arXiv:2112.08312] [INSPIRE].
L. Darmé, G. Grilli di Cortona and E. Nardi, The muon g – 2 anomaly confronts new physics in e± and μ± final states scattering, JHEP 06 (2022) 122 [arXiv:2112.09139] [INSPIRE].
A. Crivellin and M. Hoferichter, Width effects of broad new resonances in loop observables and application to (g − 2)μ, Phys. Rev. D 108 (2023) 013005 [arXiv:2211.12516] [INSPIRE].
N.M. Coyle and C.E.M. Wagner, Resolving the muon g – 2 tension through Z′-induced modifications to σhad, arXiv:2305.02354 [INSPIRE].
M. Passera, W.J. Marciano and A. Sirlin, The Muon g – 2 and the bounds on the Higgs boson mass, Phys. Rev. D 78 (2008) 013009 [arXiv:0804.1142] [INSPIRE].
A. Crivellin, M. Hoferichter, C.A. Manzari and M. Montull, Hadronic Vacuum Polarization: (g – 2)μ versus Global Electroweak Fits, Phys. Rev. Lett. 125 (2020) 091801 [arXiv:2003.04886] [INSPIRE].
A. Keshavarzi, W.J. Marciano, M. Passera and A. Sirlin, Muon g – 2 and ∆α connection, Phys. Rev. D 102 (2020) 033002 [arXiv:2006.12666] [INSPIRE].
B. Malaescu and M. Schott, Impact of correlations between aμ and αQED on the EW fit, Eur. Phys. J. C 81 (2021) 46 [arXiv:2008.08107] [INSPIRE].
G. Colangelo, M. Hoferichter and P. Stoffer, Constraints on the two-pion contribution to hadronic vacuum polarization, Phys. Lett. B 814 (2021) 136073 [arXiv:2010.07943] [INSPIRE].
M. Cè et al., The hadronic running of the electromagnetic coupling and the electroweak mixing angle from lattice QCD, JHEP 08 (2022) 220 [arXiv:2203.08676] [INSPIRE].
G. Colangelo, M. Hoferichter, J. Monnard and J. Ruiz de Elvira, Radiative corrections to the forward-backward asymmetry in e+e− → π+π−, JHEP 08 (2022) 295 [arXiv:2207.03495] [INSPIRE].
G. Chanturia, A two-potential formalism for the pion vector form factor, PoS Regio2021 (2022) 030 [INSPIRE].
G. Colangelo, M. Hoferichter, B. Kubis and P. Stoffer, Isospin-breaking effects in the two-pion contribution to hadronic vacuum polarization, JHEP 10 (2022) 032 [arXiv:2208.08993] [INSPIRE].
D. Stamen et al., Kaon electromagnetic form factors in dispersion theory, Eur. Phys. J. C 82 (2022) 432 [arXiv:2202.11106] [INSPIRE].
ETMC collaboration, Probing the Energy-Smeared R Ratio Using Lattice QCD, Phys. Rev. Lett. 130 (2023) 241901 [arXiv:2212.08467] [INSPIRE].
M. Hoferichter et al., Chiral extrapolation of hadronic vacuum polarization and isospin-breaking corrections, PoS LATTICE2022 (2022) 316 [arXiv:2210.11904] [INSPIRE].
C.L. James, R. Lewis and K. Maltman, ChPT estimate of the strong-isospin-breaking contribution to the anomalous magnetic moment of the muon, Phys. Rev. D 105 (2022) 053010 [arXiv:2109.13729] [INSPIRE].
BABAR collaboration, Study of the process e+e− → π+π−π0 using initial state radiation with BABAR, Phys. Rev. D 104 (2021) 112003 [arXiv:2110.00520] [INSPIRE].
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 (2022) 093003 [arXiv:2203.05070] [INSPIRE].
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 (2023) 074001 [arXiv:2211.11055] [INSPIRE].
G. Benton et al., Data-driven determination of the light-quark connected component of the intermediate-window contribution to the muon g − 2, arXiv:2306.16808 [INSPIRE].
A. Hoefer, J. Gluza and F. Jegerlehner, Pion pair production with higher order radiative corrections in low energy e+e− collisions, Eur. Phys. J. C 24 (2002) 51 [hep-ph/0107154] [INSPIRE].
H. Czyż, A. Grzelińska, J.H. Kühn and G. Rodrigo, The Radiative return at Φ and B factories: FSR for muon pair production at next-to-leading order, Eur. Phys. J. C 39 (2005) 411 [hep-ph/0404078] [INSPIRE].
J. Gluza, A. Hoefer, S. Jadach and F. Jegerlehner, Measuring the FSR inclusive π+π− cross-section, Eur. Phys. J. C 28 (2003) 261 [hep-ph/0212386] [INSPIRE].
Y.M. Bystritskiy, E.A. Kuraev, G.V. Fedotovich and F.V. Ignatov, The Cross sections of the muons and charged pions pairs production at electron-positron annihilation near the threshold, Phys. Rev. D 72 (2005) 114019 [hep-ph/0505236] [INSPIRE].
J. Wess and B. Zumino, Consequences of anomalous Ward identities, Phys. Lett. B 37 (1971) 95 [INSPIRE].
E. Witten, Global Aspects of Current Algebra, Nucl. Phys. B 223 (1983) 422 [INSPIRE].
L. Ametller, M. Knecht and P. Talavera, Electromagnetic corrections to γπ± → π0π±, Phys. Rev. D 64 (2001) 094009 [hep-ph/0107127] [INSPIRE].
A.I. Ahmedov, G.V. Fedotovich, E.A. Kuraev and Z.K. Silagadze, Near threshold radiative 3π production in e+e− annihilation, JHEP 09 (2002) 008 [hep-ph/0201157] [INSPIRE].
S. Bakmaev, Y.M. Bystritskiy and E.A. Kuraev, Process e+e− → 3π(γ) with final state radiative corrections, Phys. Rev. D 73 (2006) 034010 [hep-ph/0507219] [INSPIRE].
B. Moussallam, Unified dispersive approach to real and virtual photon-photon scattering at low energy, Eur. Phys. J. C 73 (2013) 2539 [arXiv:1305.3143] [INSPIRE].
N.N. Khuri and S.B. Treiman, Pion-Pion Scattering and K± → 3π Decay, Phys. Rev. 119 (1960) 1115 [INSPIRE].
M. Hoferichter et al., Dispersive analysis of the pion transition form factor, Eur. Phys. J. C 74 (2014) 3180 [arXiv:1410.4691] [INSPIRE].
S.L. Adler, B.W. Lee, S.B. Treiman and A. Zee, Low Energy Theorem for γ + γ → π + π + π, Phys. Rev. D 4 (1971) 3497 [INSPIRE].
M.V. Terent’ev, Process π± → π0π± in Coulomb field and anomalous divergence of neutral axial vector current, Phys. Lett. B 38 (1972) 419 [INSPIRE].
R. Aviv and A. Zee, Low-energy theorem for γ → 3π, Phys. Rev. D 5 (1972) 2372 [INSPIRE].
I.J.R. Aitchison and R.J.A. Golding, Relativistic Three Pion Dynamics in the omega Channel, J. Phys. G 4 (1978) 43 [INSPIRE].
F. Niecknig, B. Kubis and S.P. Schneider, Dispersive analysis of ω → 3π and ϕ → 3π decays, Eur. Phys. J. C 72 (2012) 2014 [arXiv:1203.2501] [INSPIRE].
S.P. Schneider, B. Kubis and F. Niecknig, The ω → π0γ∗ and ϕ → π0γ∗ transition form factors in dispersion theory, Phys. Rev. D 86 (2012) 054013 [arXiv:1206.3098] [INSPIRE].
M. Hoferichter, B. Kubis and D. Sakkas, Extracting the chiral anomaly from γπ → ππ, Phys. Rev. D 86 (2012) 116009 [arXiv:1210.6793] [INSPIRE].
I.V. Danilkin et al., Dispersive analysis of ω/ϕ → 3π, πγ∗, Phys. Rev. D 91 (2015) 094029 [arXiv:1409.7708] [INSPIRE].
M. Dax, T. Isken and B. Kubis, Quark-mass dependence in ω → 3π decays, Eur. Phys. J. C 78 (2018) 859 [arXiv:1808.08957] [INSPIRE].
M. Jacob and G.C. Wick, On the General Theory of Collisions for Particles with Spin, Annals Phys. 7 (1959) 404 [INSPIRE].
M. Hoferichter, B. Kubis and M. Zanke, Radiative resonance couplings in γπ → ππ, Phys. Rev. D 96 (2017) 114016 [arXiv:1710.00824] [INSPIRE].
J. Bijnens, A. Bramon and F. Cornet, Three Pseudoscalar Photon Interactions in Chiral Perturbation Theory, Phys. Lett. B 237 (1990) 488 [INSPIRE].
R.A. Briceño et al., The ππ → πγ⋆ amplitude and the resonant ρ → πγ⋆ transition from lattice QCD, Phys. Rev. D 93 (2016) 114508 [Erratum ibid. 105 (2022) 079902] [arXiv:1604.03530] [INSPIRE].
C. Alexandrou et al., πγ → ππ transition and the ρ radiative decay width from lattice QCD, Phys. Rev. D 98 (2018) 074502 [Erratum ibid. 105 (2022) 019902] [arXiv:1807.08357] [INSPIRE].
M. Niehus, M. Hoferichter and B. Kubis, The γπ → ππ anomaly from lattice QCD and dispersion relations, JHEP 12 (2021) 038 [arXiv:2110.11372] [INSPIRE].
Particle Data Group collaboration, Review of Particle Physics, PTEP 2022 (2022) 083C01 [INSPIRE].
F. Campanario et al., Standard model radiative corrections in the pion form factor measurements do not explain the aμ anomaly, Phys. Rev. D 100 (2019) 076004 [arXiv:1903.10197] [INSPIRE].
F. Ignatov and R.N. Lee, Charge asymmetry in e+e− → π+π− process, Phys. Lett. B 833 (2022) 137283 [arXiv:2204.12235] [INSPIRE].
J. Monnard, Radiative corrections for the two-pion contribution to the hadronic vacuum polarization contribution to the muon g – 2, Ph.D. Thesis, Bern University (2020) [https://boristheses.unibe.ch/2825/].
G. Abbiendi et al., Mini-Proceedings of the STRONG2020 Virtual Workshop on “Space-like and Time-like determination of the Hadronic Leading Order contribution to the Muon g – 2”, (2022) [arXiv:2201.12102] [INSPIRE].
D. Stamen et al., Analysis of rescattering effects in 3π final states, Eur. Phys. J. C 83 (2023) 510 [Erratum ibid. 83 (2023) 586] [arXiv:2212.11767] [INSPIRE].
C. Hanhart, A New Parameterization for the Pion Vector Form Factor, Phys. Lett. B 715 (2012) 170 [arXiv:1203.6839] [INSPIRE].
S. Ropertz, C. Hanhart and B. Kubis, A new parametrization for the scalar pion form factors, Eur. Phys. J. C 78 (2018) 1000 [arXiv:1809.06867] [INSPIRE].
L. von Detten et al., On the scalar πK form factor beyond the elastic region, Eur. Phys. J. C 81 (2021) 420 [arXiv:2103.01966] [INSPIRE].
R. Omnès, On the Solution of certain singular integral equations of quantum field theory, Nuovo Cim. 8 (1958) 316 [INSPIRE].
J.J. Sakurai, Currents and Mesons, University of Chicago Press (1969).
F. Klingl, N. Kaiser and W. Weise, Effective Lagrangian approach to vector mesons, their structure and decays, Z. Phys. A 356 (1996) 193 [hep-ph/9607431] [INSPIRE].
C. Bouchiat and L. Michel, La résonance dans la diffusion méson π-méson π et le moment magnétique anormal du méson μ, J. Phys. Radium 22 (1961) 121 [INSPIRE].
S.J. Brodsky and E. de Rafael, Suggested boson-lepton pair couplings and the anomalous magnetic moment of the muon, Phys. Rev. 168 (1968) 1620 [INSPIRE].
V.M. Aul’chenko et al., Study of the e+e− → π+π−π0 process in the energy range 1.05–2.00 GeV, J. Exp. Theor. Phys. 121 (2015) 27 [INSPIRE].
SND collaboration, Study of dynamics of the process e+e− → π+π−π0 in the energy range 1.15–2.00 GeV, Eur. Phys. J. C 80 (2020) 993 [arXiv:2007.14595] [INSPIRE].
BaBar collaboration, Study of e+e− → π+π−π0 process using initial state radiation with BaBar, Phys. Rev. D 70 (2004) 072004 [hep-ex/0408078] [INSPIRE].
M.N. Achasov et al., Measurements of the parameters of the ϕ(1020) resonance through studies of the processes e+e− → K+K−, KSKL, and π+π−π0, Phys. Rev. D 63 (2001) 072002 [hep-ex/0009036] [INSPIRE].
M.N. Achasov et al., Study of the process e+e− → π+π−π0 in the energy region \( \sqrt{s} \) from 0.98 to 1.38 GeV, Phys. Rev. D 66 (2002) 032001 [hep-ex/0201040] [INSPIRE].
M.N. Achasov et al., Study of the process e+e− → π+π−π0 in the energy region \( \sqrt{s} \) below 0.98 GeV, Phys. Rev. D 68 (2003) 052006 [hep-ex/0305049] [INSPIRE].
R.R. Akhmetshin et al., Measurement of ϕ meson parameters with CMD-2 detector at VEPP-2M collider, Phys. Lett. B 364 (1995) 199 [INSPIRE].
R.R. Akhmetshin et al., Study of dynamics of ϕ → π+π−π0 decay with CMD-2 detector, Phys. Lett. B 434 (1998) 426 [INSPIRE].
CMD-2 collaboration, Reanalysis of hadronic cross-section measurements at CMD-2, Phys. Lett. B 578 (2004) 285 [hep-ex/0308008] [INSPIRE].
R.R. Akhmetshin et al., Study of ϕ → π+π−π0 with CMD-2 detector, Phys. Lett. B 642 (2006) 203 [INSPIRE].
A. Cordier et al., Cross-section of the Reaction e+e− → π+π−π0 for Center-of-mass Energies From 750 to 1100 MeV, Nucl. Phys. B 172 (1980) 13 [INSPIRE].
DM2 collaboration, Measurement of the e+e− → π+π−π0 and e+e− → ωπ+π− reactions in the energy interval 1350–2400 MeV, Z. Phys. C 56 (1992) 15 [INSPIRE].
S.I. Dolinsky et al., Summary of experiments with the neutral detector at the e+e− storage ring VEPP-2M, Phys. Rept. 202 (1991) 99 [INSPIRE].
G. D’Agostini, On the use of the covariance matrix to fit correlated data, Nucl. Instrum. Meth. A 346 (1994) 306 [INSPIRE].
NNPDF collaboration, Fitting Parton Distribution Data with Multiplicative Normalization Uncertainties, JHEP 05 (2010) 075 [arXiv:0912.2276] [INSPIRE].
Crystal Barrel collaboration, Antiproton-proton annihilation at rest into ωπ0π0, Phys. Lett. B 311 (1993) 362 [INSPIRE].
M. Hoferichter et al., A phenomenological estimate of isospin breaking in hadronic vacuum polarization, arXiv:2307.02532 [INSPIRE].
Acknowledgments
We thank M. Davier and V. Druzhinin for helpful communication on ref. [86], Dominik Stamen for providing 3π KT basis functions, and Janak Prabhu for collaboration on the electromagnetic corrections in an early stage of this project. Financial support by the DFG through the funds provided to the Sino-German Collaborative Research Center TRR110 “Symmetries and the Emergence of Structure in QCD” (DFG Project-ID 196253076 — TRR 110) and the SNSF (Project No. PCEFP2_181117) is gratefully acknowledged. MH thanks the INT at the University of Washington for its hospitality and the DOE for partial support (grant No. DE-FG02-00ER41132) during a visit when part of this work was performed.
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: 2307.02546
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
Hoferichter, M., Hoid, BL., Kubis, B. et al. Isospin-breaking effects in the three-pion contribution to hadronic vacuum polarization. J. High Energ. Phys. 2023, 208 (2023). https://doi.org/10.1007/JHEP08(2023)208
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2023)208