Abstract
We address the contribution of the 3π channel to hadronic vacuum polarization (HVP) using a dispersive representation of the e+e− → 3π amplitude. This channel gives the second-largest individual contribution to the total HVP integral in the anomalous magnetic moment of the muon (g − 2)μ, both to its absolute value and uncertainty. It is largely dominated by the narrow resonances ω and ϕ, but not to the extent that the off-peak regions were negligible, so that at the level of accuracy relevant for (g − 2)μ an analysis of the available data as model independent as possible becomes critical. Here, we provide such an analysis based on a global fit function using analyticity and unitarity of the underlying γ∗ → 3π amplitude and its normalization from a chiral low-energy theorem, which, in particular, allows us to check the internal consistency of the various e+e− → 3π data sets. Overall, we obtain \( {a}_{\mu}^{3\pi } \)|≤1.8 GeV = 46.2(6)(6) × 10−10 as our best estimate for the total 3π contribution consistent with all (low-energy) constraints from QCD. In combination with a recent dispersive analysis imposing the same constraints on the 2π channel below 1 GeV, this covers nearly 80% of the total HVP contribution, leading to \( {a}_{\mu}^{\mathrm{HVP}} \) = 692.3(3.3) × 10−10 when the remainder is taken from the literature, and thus reaffirming the (g−2)μ anomaly at the level of at least 3.4σ. As side products, we find for the vacuum-polarization-subtracted masses Mω = 782.63(3)(1) MeV and Mϕ = 1019.20(2)(1) MeV, confirming the tension to the ω mass as extracted from the 2π channel.
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References
N.N. Khuri and S.B. Treiman, Pion-Pion Scattering and K ± → 3π Decay, Phys. Rev.119 (1960) 1115 [INSPIRE].
I.J.R. Aitchison and R.J.A. Golding, Relativistic Three Pion Dynamics in the ω 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].
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].
R. Garc ıa-Martín, R. Kaminski, J.R. Peláez, J. Ruiz de Elvira and F.J. Ynduráin, The Pion-pion scattering amplitude. IV: Improved analysis with once subtracted Roy-like equations up to 1100 MeV, Phys. Rev.D 83 (2011) 074004 [arXiv:1102.2183] [INSPIRE].
I. Caprini, G. Colangelo and H. Leutwyler, Regge analysis of the ππ scattering amplitude, Eur. Phys. J.C 72 (2012) 1860 [arXiv:1111.7160] [INSPIRE].
Muong − 2 collaboration, Final Report of the Muon E821 Anomalous Magnetic Moment Measurement at BNL, Phys. Rev.D 73 (2006) 072003 [hep-ex/0602035] [INSPIRE].
Muong − 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, Prog. Theor. Exp. Phys.2019 (2019) 053C02 [arXiv:1901.03047] [INSPIRE].
M. Hoferichter, G. Colangelo, M. Procura and P. Stoffer, Virtual photon-photon scattering, Int. J. Mod. Phys. Conf. Ser.35 (2014) 1460400 [arXiv:1309.6877] [INSPIRE].
G. Colangelo, M. Hoferichter, M. Procura and P. Stoffer, Dispersive approach to hadronic light-by-light scattering, JHEP09 (2014) 091 [arXiv:1402.7081] [INSPIRE].
G. Colangelo, M. Hoferichter, B. Kubis, M. Procura and P. Stoffer, 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, JHEP09 (2015) 074 [arXiv:1506.01386] [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, JHEP04 (2017) 161 [arXiv:1702.07347] [INSPIRE].
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 (2018) 112002 [arXiv:1805.01471] [INSPIRE].
M. Hoferichter, B.-L. Hoid, B. Kubis, S. Leupold and S.P. Schneider, Dispersion relation for hadronic light-by-light scattering: pion pole, JHEP10 (2018) 141 [arXiv:1808.04823] [INSPIRE].
T. Blum et al., Connected and Leading Disconnected Hadronic Light-by-Light Contribution to the Muon Anomalous Magnetic Moment with a Physical Pion Mass, Phys. Rev. Lett.118 (2017) 022005 [arXiv:1610.04603] [INSPIRE].
T. Blum et al., Using infinite volume, continuum QED and lattice QCD for the hadronic light-by-light contribution to the muon anomalous magnetic moment, Phys. Rev.D 96 (2017) 034515 [arXiv:1705.01067] [INSPIRE].
A. Gérardin, H.B. Meyer and A. Nyffeler, Lattice calculation of the pion transition form factor with N f = 2 + 1 Wilson quarks, arXiv:1903.09471 [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. Radium22 (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].
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 α(\( {m}_Z^2 \)) 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 α(\( {M}_Z^2 \)): a new data-based analysis, Phys. Rev.D 97 (2018) 114025 [arXiv:1802.02995] [INSPIRE].
F. Jegerlehner, The role of mesons in muon g − 2, EPJ Web Conf.199 (2019) 01010 [arXiv:1809.07413] [INSPIRE].
M. Benayoun, L. Delbuono and F. Jegerlehner, BHLS 2, a New Breaking of the HLS Model and its Phenomenology, arXiv:1903.11034 [INSPIRE].
Budapest-M-Wuppertal collaboration, Hadronic vacuum polarization contribution to the anomalous magnetic moments of leptons from first principles, Phys. Rev. Lett.121 (2018) 022002 [arXiv:1711.04980] [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].
D. Giusti, F. Sanfilippo and S. Simula, Light-quark contribution to the leading hadronic vacuum polarization term of the muon g − 2 from twisted-mass fermions, Phys. Rev.D 98 (2018) 114504 [arXiv:1808.00887] [INSPIRE].
E. Shintani and Y. Kuramashi, Study of systematic uncertainties in hadronic vacuum polarization contribution to muon g − 2 with 2 + 1 flavor lattice QCD, arXiv:1902.00885 [INSPIRE].
Fermilab Lattice, LATTICE-HPQCD and MILC collaborations, Hadronic-Vacuum-Polarization Contribution to the Muon’s Anomalous Magnetic Moment from Four-Flavor Lattice QCD, arXiv:1902.04223 [INSPIRE].
A. Gérardin et al., The leading hadronic contribution to (g − 2)μfrom lattice QCD with N f = 2 + 1 flavours of O(a) improved Wilson quarks, Phys. Rev.D 100 (2019) 014510 [arXiv:1904.03120] [INSPIRE].
C. Aubin, T. Blum, C. Tu, M. Golterman, C. Jung and S. Peris, Light quark vacuum polarization at the physical point and contribution to the muon g − 2, arXiv:1905.09307 [INSPIRE].
G. Abbiendi et al., Measuring the leading hadronic contribution to the muon g − 2 via μe scattering, Eur. Phys. J.C 77 (2017) 139 [arXiv:1609.08987] [INSPIRE].
G. Colangelo, M. Hoferichter and P. Stoffer, Two-pion contribution to hadronic vacuum polarization, JHEP02 (2019) 006 [arXiv:1810.00007] [INSPIRE].
J.F. de Trocóniz and F.J. Ynduráin, Precision determination of the pion form-factor and calculation of the muon g − 2, Phys. Rev.D 65 (2002) 093001 [hep-ph/0106025] [INSPIRE].
H. Leutwyler, Electromagnetic form-factor of the pion, in proceedings of the Continuous Advances in QCD 2002/ARKADYFEST (honoring the 60th birthday of Prof. Arkady Vainshtein), Minneapolis, Minnesota, U.S.A., 17-23 May 2002, pp. 23-40 [https://doi.org/10.1142/97898127763100002] [hep-ph/0212324] [INSPIRE].
G. Colangelo, Hadronic contributions to a μbelow one GeV, Nucl. Phys. Proc. Suppl.131 (2004) 185 [hep-ph/0312017] [INSPIRE].
J.F. de Trocóniz and F.J. Ynduráin, The Hadronic contributions to the anomalous magnetic moment of the muon, Phys. Rev.D 71 (2005) 073008 [hep-ph/0402285] [INSPIRE].
B. Ananthanarayan, I. Caprini and D. Das, Pion electromagnetic form factor at high precision with implications to \( {a}_{\mu}^{\pi \pi} \)and the onset of perturbative QCD, Phys. Rev.D 98 (2018) 114015 [arXiv:1810.09265] [INSPIRE].
BaBar collaboration, Precise measurement of the e +e − → π +π −(γ) cross section with the Initial State Radiation method at BABAR, Phys. Rev. Lett.103 (2009) 231801 [arXiv:0908.3589] [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 collaboration, Measurement of σ(e +e − → π +π −γ (γ)) and the dipion contribution to the muon anomaly with the KLOE detector, Phys. Lett.B 670 (2009) 285 [arXiv:0809.3950] [INSPIRE].
KLOE collaboration, Measurement of σ(e +e − → π +π −) from threshold to 0.85 GeV 2using Initial State Radiation with the KLOE detector, Phys. Lett.B 700 (2011) 102 [arXiv:1006.5313] [INSPIRE].
KLOE collaboration, Precision measurement of σ(e +e − → π +π −γ)/σ(e +e − → μ +μ −γ) and determination of the π +π −contribution to the muon anomaly with the KLOE detector, Phys. Lett.B 720 (2013) 336 [arXiv:1212.4524] [INSPIRE].
KLOE-2 collaboration, Combination of KLOE σ(e +e − → π +π −γ (γ)) measurements and determination of \( {a}_{\mu}^{\uppi +\uppi -} \)in the energy range 0.10 < s < 0.95 GeV 2, JHEP03 (2018) 173 [arXiv:1711.03085] [INSPIRE].
S.P. Schneider, B. Kubis and F. Niecknig, The ω → π 0γ ∗and ϕ → π 0γ ∗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 fromγ π → ππ, Phys. Rev.D 86 (2012) 116009 [arXiv:1210.6793] [INSPIRE].
M. Hoferichter, B. Kubis, S. Leupold, F. Niecknig and S.P. Schneider, Dispersive analysis of the pion transition form factor, Eur. Phys. J.C 74 (2014) 3180 [arXiv:1410.4691] [INSPIRE].
M. Hoferichter, B. Kubis and M. Zanke, Radiative resonance couplings in γπ → ππ, Phys. Rev.D 96 (2017) 114016 [arXiv:1710.00824] [INSPIRE].
R. Garcıa-Mart´ın and B. Moussallam, MO analysis of the high statistics Belle results on γγ → π +π −,π 0π 0with chiral constraints, Eur. Phys. J.C 70 (2010) 155 [arXiv:1006.5373] [INSPIRE].
M. Hoferichter, D.R. Phillips and C. Schat, Roy-Steiner equations for γγ → ππ, Eur. Phys. J.C 71 (2011) 1743 [arXiv:1106.4147] [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].
I. Danilkin and M. Vanderhaeghen, Dispersive analysis of the γγ ∗ → ππ process, Phys. Lett.B 789 (2019) 366 [arXiv:1810.03669] [INSPIRE].
M. Hoferichter and P. Stoffer, Dispersion relations for γ ∗γ ∗ → ππ: helicity amplitudes, subtractions and anomalous thresholds, JHEP07 (2019) 073 [arXiv:1905.13198] [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].
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 ± → π 0π ±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].
CMD-2 collaboration, Measurement of e +e − → π +π −cross-section with CMD-2 around ρ-meson, Phys. Lett.B 527 (2002) 161 [hep-ex/0112031] [INSPIRE].
CMD-2 collaboration, Reanalysis of hadronic cross-section measurements at CMD-2, Phys. Lett.B 578 (200 4) 285 [hep-ex/0308008] [INSPIRE].
M.N. Achasov et al., Study of the process e +e − → π +π −in the energy region 400 < \( \sqrt{s} \)< 1000 MeV, J. Exp. Theor. Phys.101 (2005) 1053 [Zh. Eksp. Teor. Fiz.128 (2005) 1201] [hep-ex/0506076] [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 [Zh. Eksp. Teor. Fiz.130 (2006) 437] [hep-ex/0605013] [INSPIRE].
V.M. Aul’chenko et al., Measurement of the e +e − → π +π −cross section with the CMD-2 detector in the 370-520 MeV energy range, JETP Lett.84 (2006) 413 [Pisma Zh. Eksp. Teor. Fiz.84 (2006) 491] [hep-ex/0610016] [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].
BaBar collaboration, Study of e +e − → π +π −π 0process 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 −, K SK Land π +π −π 0, Phys. Rev.D 63 (2001) 072002 [hep-ex/0009036] [INSPIRE].
M.N. Achasov et al., Study of the process e +e − → π +π −π 0in 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 −π 0in the energy region \( \sqrt{s} \)below 0.98 GeV, Phys. Rev.D 68 (2003) 052006 [hep-ex/0305049] [INSPIRE].
V.M. Aul’chenko et al., Study of the e +e − → π +π −π 0process in the energy range 1.05-2.00 GeV, J. Exp. Theor. Phys.121 (2015) 27 [Zh. Eksp. Teor. Fiz.148 (2015) 34] [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 −π 0decay with CMD-2 detector, Phys. Lett.B 434 (1998) 426 [INSPIRE].
R.R. Akhmetshin et al., Study of ϕ → π +π −π 0with CMD-2 detector, Phys. Lett.B 642 (2006) 203 [INSPIRE].
A. Cordier et al., Cross-section of the Reaction e +e − → π +π −π 0for 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 −π 0and e +e − → ω π +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].
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].
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].
F. Campanario et al., Standard Model radiative corrections in the pion form factor measurements do not explain the a μanomaly, arXiv:1903.10197 [INSPIRE].
M. Jacob and G.C. Wick, On the general theory of collisions for particles with spin, Annals Phys.7 (1959) 404 [Annals Phys.281 (2000) 774] [INSPIRE].
R. Omnès, On the Solution of certain singular integral equations of quantum field theory, Nuovo Cim.8 (1958) 316 [INSPIRE].
J. Bijnens, A. Bramon and F. Cornet, Three Pseudoscalar Photon Interactions in Chiral Perturbation Theory, Phys. Lett.B 237 (1990) 488 [INSPIRE].
Particle Data Group collaboration, Review of Particle Physics, Phys. Rev.D 98 (2018) 030001 [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, JHEP05 (2010) 075 [arXiv:0912.2276] [INSPIRE].
CMD-2 collaboration, Study of the processes e +e − → ηγ, π 0γ → 3γ in the c.m. energy range 600-1380 MeV at CMD-2, Phys. Lett.B 605 (2005) 26 [hep-ex/0409030] [INSPIRE].
Crystal Barrel collaboration, Anti-proton-proton annihilation at rest into ωπ 0π 0, Phys. Lett.B 311 (1993) 362 [INSPIRE].
F. Jegerlehner, The Anomalous Magnetic Moment of the Muon, Springer Tracts Mod. Phys.274 (2017) 1 [INSPIRE].
K. Hagiwara, A.D. Martin, D. Nomura and T. Teubner, Predictions for g − 2 of the muon and α QED (\( {M}_2^Z \)), Phys. Rev.D 69 (2004) 093003 [hep-ph/0312250] [INSPIRE].
E.A. Kuraev and Z.K. Silagadze, Once more about the ω → 3π contact term, Phys. Atom. Nucl.58 (1995) 1589 [Yad. Fiz.58N9 (1995) 1687] [hep-ph/9502406] [INSPIRE].
A.I. Ahmedov, G.V. Fedotovich, E.A. Kuraev and Z.K. Silagadze, Near threshold radiative 3π production in e +e −annihilation, JHEP09 (2002) 008 [hep-ph/0201157] [INSPIRE].
T. Aoyama, T. Kinoshita and M. Nio, Revised and Improved Value of the QED Tenth-Order Electron Anomalous Magnetic Moment, Phys. Rev.D 97 (2018) 036001 [arXiv:1712.06060] [INSPIRE].
C. Gnendiger, D. Stöckinger and H. Stöckinger-Kim, The electroweak contributions to(g − 2)μafter the Higgs boson mass measurement, Phys. Rev.D 88 (2013) 053005 [arXiv:1306.5546] [INSPIRE].
D. Hanneke, S. Fogwell and G. Gabrielse, New Measurement of the Electron Magnetic Moment and the Fine Structure Constant, Phys. Rev. Lett.100 (2008) 120801 [arXiv:0801.1134] [INSPIRE].
R.H. Parker, C. Yu, W. Zhong, B. Estey and H. Müller, Measurement of the fine-structure constant as a test of the Standard Model, Science360 (2018) 191 [arXiv:1812.04130] [INSPIRE].
H. Davoudiasl and W.J. Marciano, Tale of two anomalies, Phys. Rev.D 98 (2018) 075011 [arXiv:1806.10252] [INSPIRE].
A. Crivellin, M. Hoferichter and P. Schmidt-Wellenburg, Combined explanations of (g − 2)μ,eand implications for a large muon EDM, Phys. Rev.D 98 (2018) 113002 [arXiv:1807.11484] [INSPIRE].
C. Hanhart, A New Parameterization for the Pion Vector Form Factor, Phys. Lett.B 715 (2012) 170 [arXiv:1203.6839] [INSPIRE].
C. Hanhart, M. Hoferichter, S. Holz and B. Kubis, in preparation.
J. Seyfried, Determination of the Chiral Anomaly and Studies on the Pion Polarizability in Pion-Nickel Reactions from COMPASS at CERN, Master’s Thesis, Technical University of Munich, Munich Germany (2017).
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Hoferichter, M., Hoid, BL. & Kubis, B. Three-pion contribution to hadronic vacuum polarization. J. High Energ. Phys. 2019, 137 (2019). https://doi.org/10.1007/JHEP08(2019)137
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DOI: https://doi.org/10.1007/JHEP08(2019)137