Open Access
Access by Xinjiang University

Comparison of the hadronic vacuum polarization between hadronic $τ$ -decay data and lattice QCD

Noah Allen¹, Diogo Boito ², Maarten Golterman ^1,3, Kim Maltman ^4,5, Lucas M. Mansur ⁶, and Santiago Peris ³

¹Department of Physics and Astronomy, San Francisco State University, San Francisco, California 94132, USA
²Instituto de Física de São Carlos, Universidade de São Paulo, 13566-590 São Carlos, SP, Brazil
³Department of Physics and IFAE-BIST, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Barcelona, Spain
⁴Department of Mathematics and Statistics, York University, Toronto, Ontario M3J 1P3, Canada
⁵CSSM, University of Adelaide, Adelaide, South Australia 5005, Australia
⁶Institut für Kernphysik, Johannes Gutenberg-Universität Mainz, D-55128 Mainz, Germany

Phys. Rev. D 114, 034021 – Published 11 August, 2026

DOI: https://doi.org/10.1103/5y9v-c2h6

Abstract

We compare the isospin-one, vector-current hadronic vacuum polarization (HVP) obtained from isospin-symmetric lattice QCD with that obtained from a dispersive representation employing inclusive hadronic $τ$ decay data corrected for isospin breaking. We consider the subtracted HVP evaluated at squared Euclidean momenta ranging from $0.5 {GeV}^{2}$ to $12 {GeV}^{2}$ , together with the light-quark-connected HVP contribution to the muon anomalous magnetic moment and the short-, intermediate- and long-distance RBC/UKQCD window components thereof. Dispersive contributions from the region of hadronic invariant masses above the $τ$ mass are evaluated using perturbative QCD. We also consider dispersive determinations using $τ$ data only for contributions from two-pion, or two-pion and four-pion, modes, and evaluating the remaining contributions using exclusive-mode $e^{+} e^{-} \to hadrons$ cross sections up to about 2 GeV, lessening the dependence on perturbation theory. We find generally good agreement between lattice and $τ$ -based results. However, a comparison of $τ$ -based window-quantity contributions for the two four-pion modes to expectations for those contributions based on the Pais relations and $e^{+} e^{-}$ four-pion cross sections, reveals significant differences for the $2 π^{-} π^{+} π^{0}$ mode.

Physics Subject Headings (PhySH)

Article Text

References (83)

G. W. Bennett et al. (Muon g-2 Collaboration), Final report of the muon E821 anomalous magnetic moment measurement at BNL, Phys. Rev. D 73, 072003 (2006).
D. P. Aguillard et al. (Muon g-2 Collaboration), Measurement of the positive muon anomalous magnetic moment to 127 ppb, Phys. Rev. Lett. 135, 101802 (2025).
R. Aliberti, T. Aoyama, E. Balzani, A. Bashir, G. Benton, J. Bijnens, V. Biloshytskyi, T. Blum, D. Boito, M. Bruno et al., The anomalous magnetic moment of the muon in the Standard Model: An update, Phys. Rep. 1143, 1 (2025).
F. V. Ignatov et al. (CMD-3 Collaboration), Measurement of the $e^{+} e^{-} \to π^{+} π^{-}$ cross section from threshold to 1.2 GeV with the CMD-3 detector, Phys. Rev. D 109, 112002 (2024).
M. Davier, A. Hoecker, A. M. Lutz, B. Malaescu, and Z. Zhang, Tensions in $e^{+} e^{-} \to π^{+} π^{-} (γ)$ measurements: The new landscape of data-driven hadronic vacuum polarization predictions for the muon $g - 2$ , Eur. Phys. J. C 84, 721 (2024).
M. Davier, A. Hoecker, G. Lopez Castro, B. Malaescu, X. H. Mo, G. Toledo Sanchez, P. Wang, C. Z. Yuan, and Z. Zhang, The discrepancy between tau and $e^{+} e^{-}$ spectral functions revisited and the consequences for the muon magnetic anomaly, Eur. Phys. J. C 66, 127 (2010).
J. A. Miranda and P. Roig, New $τ$ -based evaluation of the hadronic contribution to the vacuum polarization piece of the muon anomalous magnetic moment, Phys. Rev. D 102, 114017 (2020).
P. Masjuan, A. Miranda, and P. Roig, $τ$ data-driven evaluation of Euclidean windows for the hadronic vacuum polarization, Phys. Lett. B 850, 138492 (2024).
G. L. Castro, A. Miranda, and P. Roig, Isospin breaking corrections in $2 π$ production in tau decays and $e^{+} e^{-}$ annihilation: Consequences for the muon g-2 and conserved vector current tests, Phys. Rev. D 111, 073004 (2025).
G. Colangelo, M. Hoferichter, B. Kubis, and P. Stoffer, Isospin-breaking effects in the two-pion contribution to hadronic vacuum polarization, J. High Energy Phys. 10 (2022) 032.
M. Hoferichter, G. Colangelo, B. L. Hoid, B. Kubis, J. R. de Elvira, D. Schuh, D. Stamen, and P. Stoffer, Phenomenological estimate of isospin breaking in hadronic vacuum polarization, Phys. Rev. Lett. 131, 161905 (2023).
M. Davier, B. Malaescu, and Z. Zhang, Data-based form factor corrections between the two-pion $τ$ and $e^{+} e^{-}$ spectral functions, arXiv:2504.13789.
G. Colangelo, M. Cottini, M. Hoferichter, and S. Holz, Improved calculation of radiative corrections to $τ \to π π ν_{τ}$ decays, Phys. Rev. Lett. 136, 101903 (2026) arXiv:2510.26871; Radiative corrections to $τ \to π π ν_{τ}$ , J. High Energy Phys. 02 (2026) 181.
T. Blum, P. A. Boyle, V. Gülpers, T. Izubuchi, L. Jin, C. Jung, A. Jüttner, C. Lehner, A. Portelli, and J. T. Tsang (RBC Collaboration and UKQCD Collaboration), Calculation of the hadronic vacuum polarization contribution to the muon anomalous magnetic moment, Phys. Rev. Lett. 121, 022003 (2018).
A. Conigli, D. Djukanovic, G. von Hippel, S. Kuberski, H. B. Meyer, K. Miura, K. Ottnad, A. Risch, and H. Wittig, Precision lattice calculation of the hadronic contribution to the running of the electroweak gauge couplings, arXiv:2511.01623.
S. Borsanyi, Z. Fodor, J. N. Guenther, C. Hoelbling, S. D. Katz, L. Lellouch, T. Lippert, K. Miura, L. Parato, K. K. Szabo et al., Leading hadronic contribution to the muon magnetic moment from lattice QCD, Nature (London) 593, 51 (2021).
G. Benton, D. Boito, M. Golterman, A. Keshavarzi, K. Maltman, and S. Peris, Data-driven determination of the light-quark connected component of the intermediate-window contribution to the muon $g - 2$ , Phys. Rev. Lett. 131, 251803 (2023).
D. Bernecker and H. B. Meyer, Vector correlators in lattice QCD: Methods and applications, Eur. Phys. J. A 47, 148 (2011).
P. A. Baikov, K. G. Chetyrkin, and J. H. Kühn, Order $α_{s}^{4}$ QCD corrections to $Z$ and $τ$ decays, Phys. Rev. Lett. 101, 012002 (2008).
D. Boito, A. Eiben, M. Golterman, K. Maltman, L. M. Mansur, and S. Peris, Strong coupling from hadronic $τ \to π^{-} π^{0} ν_{τ}$ from Belle, Phys. Rev. D 111, 074010 (2025).
R. Barate et al. (ALEPH Collaboration), Measurement of the spectral functions of axial-vector hadronic tau decays and determination of $α_{s} (M_{τ}^{2})$ , Eur. Phys. J. C 4, 409 (1998).
S. Schael et al. (ALEPH Collaboration), Branching ratios and spectral functions of tau decays: Final ALEPH measurements and physics implications, Phys. Rep. 421, 191 (2005).
M. Davier, A. Hoecker, B. Malaescu, C. Z. Yuan, and Z. Zhang, Update of the ALEPH non-strange spectral functions from hadronic $τ$ decays, Eur. Phys. J. C 74, 2803 (2014).
K. Ackerstaff et al. (OPAL Collaboration), Measurement of the strong coupling constant alpha(s) and the vector and axial-vector spectral functions in hadronic tau decays, Eur. Phys. J. C 7, 571 (1999).
M. Fujikawa et al. (Belle Collaboration), High-statistics study of the $τ^{-} \to π^{-} π^{0} ν (τ)$ decay, Phys. Rev. D 78, 072006 (2008).
D. Boito, M. Golterman, K. Maltman, S. Peris, M. V. Rodrigues, and W. Schaaf, Strong coupling from an improved $τ$ vector isovector spectral function, Phys. Rev. D 103, 034028 (2021).
J. P. Lees et al. (BABAR Collaboration), Measurement of the spectra function for the $τ^{-} \to K^{-} K_{S} ν_{τ}$ decay, Phys. Rev. D 98, 032010 (2018).
A. Keshavarzi, D. Nomura, and T. Teubner, $g - 2$ of charged leptons, $α (M_{Z}^{2})$ and the hyperfine splitting of muonium: A new data-based analysis, Phys. Rev. D 101, 014029 (2020).
Y. Aoki et al. (Flavour Lattice Averaging Group (FLAG), FLAG review 2024, Phys. Rev. D 113, 014508 (2026).
J. Erler, Electroweak radiative corrections to semileptonic tau decays, Rev. Mex. Fis. 50, 200 (2004).
A. Pais, The many pi-meson problem, Ann. Phys. (Amsterdam) 9, 548 (1960).
P. A. Baikov, K. G. Chetyrkin, and J. H. Kühn, Five-loop running of the QCD coupling constant, Phys. Rev. Lett. 118, 082002 (2017).
F. Herzog, B. Ruijl, T. Ueda, J. A. M. Vermaseren, and A. Vogt, The five-loop beta function of Yang-Mills theory with fermions, J. High Energy Phys. 02 (2017) 090.
M. Jamin, Contour-improved versus fixed-order perturbation theory in hadronic $τ$ decays, J. High Energy Phys. 09 (2005) 058.
M. Beneke and M. Jamin, $α_{s}$ and the $τ$ hadronic width: Fixed-order, contour-improved and higher-order perturbation theory, J. High Energy Phys. 09 (2008) 044.
D. Boito, P. Masjuan, and F. Oliani, Higher-order QCD corrections to hadronic $τ$ decays from Padé approximants, J. High Energy Phys. 08 (2018) 075.
M. Cè, A. Gérardin, G. von Hippel, H. B. Meyer, K. Miura, K. Ottnad, A. Risch, T. San José, J. Wilhelm, and H. Wittig, The hadronic running of the electromagnetic coupling and the electroweak mixing angle from lattice QCD, J. High Energy Phys. 08 (2022) 220.
S. Kuberski, M. Cè, G. von Hippel, H. B. Meyer, K. Ottnad, A. Risch, and H. Wittig, Hadronic vacuum polarization in the muon $g - 2$ : The short-distance contribution from lattice QCD, J. High Energy Phys. 03 (2024) 172.
D. Boito, O. Catà, M. Golterman, M. Jamin, K. Maltman, J. Osborne, and S. Peris, A new determination of $α_{s}$ from hadronic $τ$ decays, Phys. Rev. D 84, 113006 (2011).
D. Boito, I. Caprini, M. Golterman, K. Maltman, and S. Peris, Hyperasymptotics and quark-hadron duality violations in QCD, Phys. Rev. D 97, 054007 (2018).
E. C. Poggio, H. R. Quinn, and S. Weinberg, Smearing the quark model, Phys. Rev. D 13, 1958 (1976).
O. Catà, M. Golterman, and S. Peris, Possible duality violations in $τ$ decay and their impact on the determination of $α_{s}$ , Phys. Rev. D 79, 053002 (2009).
B. Blok, M. A. Shifman, and D. X. Zhang, An illustrative example of how quark-hadron duality might work, Phys. Rev. D 57, 2691 (1998); 59, 019901(E) (1999).
I. I. Y. Bigi, M. A. Shifman, N. Uraltsev, and A. I. Vainshtein, Heavy flavor decays, OPE and duality in two-dimensional ’t Hooft model, Phys. Rev. D 59, 054011 (1999).
M. A. Shifman, Quark Hadron Duality (2021), pp. 1447–1494.
M. Golterman, S. Peris, B. Phily, and E. de Rafael, Testing an approximation to large- $N_{c}$ QCD with a toy model, J. High Energy Phys. 01 (2002) 024.
O. Catà, M. Golterman, and S. Peris, Duality violations and spectral sum rules, J. High Energy Phys. 08 (2005) 076.
O. Catà, M. Golterman, and S. Peris, Unraveling duality violations in hadronic tau decays, Phys. Rev. D 77, 093006 (2008).
M. Della Morte, A. Francis, V. Gülpers, G. Herdoíza, G. von Hippel, H. Horch, B. Jäger, H. B. Meyer, A. Nyffeler, and H. Wittig, The hadronic vacuum polarization contribution to the muon $g - 2$ from lattice QCD, J. High Energy Phys. 10 (2017) 020.
V. Cirigliano, G. Ecker, and H. Neufeld, Isospin violation and the magnetic moment of the muon, Phys. Lett. B 513, 361 (2001).
C. Alexandrou et al. (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, 074506 (2023).
T. Blum et al. (RBC Collaboration and UKQCD Collaboration), Update of Euclidean windows of the hadronic vacuum polarization, Phys. Rev. D 108, 054507 (2023).
A. Bazavov et al. (MILC Collaboration, Fermilab Lattice Collaboration and HPQCD Collaboration), Hadronic vacuum polarization for the muon $g - 2$ from lattice QCD: Complete short and intermediate windows, Phys. Rev. D 111, 094508 (2025).
V. Cirigliano, M. Hoferichter, and N. Valori, Pion $β$ decay and $τ \to π π ν_{τ}$ beyond leading logarithms, arXiv:2602.11253.
F. Guerrero and A. Pich, Effective field theory description of the pion form-factor, Phys. Lett. B 412, 382 (1997).
M. Davier, A. Hoecker, G. Lopez Castro, B. Malaescu, X. H. Mo, G. Toledo Sanchez, P. Wang, C. Z. Yuan, and Z. Zhang, The discrepancy between tau and $e^{+} e^{-}$ spectral functions revisited and the consequences for the muon magnetic anomaly, Eur. Phys. J. C 66, 127 (2010).
S. Navas et al. (Particle Data Group), Review of particle physics, Phys. Rev. D 110, 030001 (2024).
Y. Aoki et al. (Flavour Lattice Averaging Group (FLAG), FLAG review 2024, Phys. Rev. D 113, 014508 (2026).
T. Coan et al. (CLEO Collaboration), Measurement of $α_{s}$ from tau decays, Phys. Lett. B 356, 580 (1995).
S. Anderson et al. (CLEO Collaboration), Hadronic structure in the decay $τ^{-} \to π^{-} π^{0} ν_{τ}$ , Phys. Rev. D 61, 112002 (2000).
M. Davier, A. Hoecker, A. M. Lutz, B. Malaescu, and Z. Zhang, Tensions in $e^{+} e^{-} \to π^{+} π^{-} (γ)$ measurements: The new landscape of data-driven hadronic vacuum polarization predictions for the muon $g - 2$ , Eur. Phys. J. C 84, 721 (2024).
R. Alemany, M. Davier, and A. Hoecker, Improved determination of the hadronic contribution to the muon ( $g - 2$ ) and to $α (M_{Z}^{2})$ using new data from hadronic tau decays, Eur. Phys. J. C 2, 123 (1998).
A. del Pino, D. A. Clarke, C. DeTar, A. X. El-Khadra, E. Gámiz, S. Gottlieb, A. V. Grebe, L. Hostetler, W. I. Jay, A. S. Kronfeld et al., Hadronic contributions to $α (Q^{2})$ and $\sin_{W}^{2} (Q^{2})$ from spectral reconstruction of lattice-QCD data, Proc. Sci. LATTICE2025 (2025) 518 [arXiv:2604.22349].
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, 114504 (2018).
E. Shintani and Y. Kuramashi (PACS Collaboration), Hadronic vacuum polarization contribution to the muon $g - 2$ with $2 + 1$ flavor lattice QCD on a larger than ${(10 fm)}^{4}$ lattice at the physical point, Phys. Rev. D 100, 034517 (2019).
D. Giusti and S. Simula, Lepton anomalous magnetic moments in Lattice $QCD + QED$ , Proc. Sci. LATTICE2019 (2019) 104.
C. Lehner and A. S. Meyer, Consistency of hadronic vacuum polarization between lattice QCD and the R-ratio, Phys. Rev. D 101, 074515 (2020).
G. Wang , T. Draper, K.-F. Liu, and Y.-B. Yang (chiQCD Collaboration), Muon $g - 2$ with overlap valence fermions, Phys. Rev. D 107, 034513 (2023).
C. Aubin, T. Blum, M. Golterman, and S. Peris, Muon anomalous magnetic moment with staggered fermions: Is the lattice spacing small enough?, Phys. Rev. D 106, 054503 (2022).
M. Cè, A. Gérardin, G. von Hippel, R. J. Hudspith, S. Kuberski, H. B. Meyer, K. Miura, D. Mohler, K. Ottnad, P. Srijit et al., Window observable for the hadronic vacuum polarization contribution to the muon $g - 2$ from lattice QCD, Phys. Rev. D 106, 114502 (2022).
A. Boccaletti, S. Borsanyi, M. Davier, Z. Fodor, F. Frech, A. Gerardin, D. Giusti, A. Y. Kotov, L. Lellouch, T. Lippert et al., High precision calculation of the hadronic vacuum polarisation contribution to the muon anomaly, Nature (London) 653, 373 (2026).
S. Spiegel and C. Lehner, High-precision continuum limit study of the HVP short-distance window, Phys. Rev. D 111, 114517 (2025).
T. Blum et al. (RBC Collaboration and UKQCD Collaboration), Long-distance window of the hadronic vacuum polarization for the muon $g - 2$ , Phys. Rev. Lett. 134, 201901 (2025).
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.
C. Alexandrou et al. (Extended Twisted Mass Collaboration), Strange and charm quark contributions to the muon anomalous magnetic moment in lattice QCD with twisted-mass fermions, Phys. Rev. D 111, 054502 (2025).
A. Bazavov et al. (Fermilab Lattice Collaboration, HPQCD Collaboration, and MILC Collaboration), Hadronic vacuum polarization for the muon $g - 2$ from lattice QCD: Long-distance and full light-quark connected contribution, Phys. Rev. Lett. 135, 011901 (2025).
K. G. Chetyrkin and A. Maier, Massless correlators of vector, scalar and tensor currents in position space at orders $α_{s}^{3}$ and $α_{s}^{4}$ : Explicit analytical results, Nucl. Phys. B844, 266 (2011).
G. Benton, D. Boito, M. Golterman, A. Keshavarzi, K. Maltman, and S. Peris, Data-driven results for light-quark connected and strange-plus-disconnected hadronic $g - 2$ short- and long-distance windows, Phys. Rev. D 111, 034018 (2025).
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, 074001 (2023).
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, 093003 (2022).
G. Benton, D. Boito, M. Golterman, A. Keshavarzi, K. Maltman, and S. Peris, Data-driven estimates for light-quark-connected and strange-plus-disconnected hadronic $g - 2$ window quantities, Phys. Rev. D 109, 036010 (2024).
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 $α_{s} (m_{Z}^{2})$ , Eur. Phys. J. C 80, 241 (2020); 80, 410(E) (2020).
B. Aubert et al., BABAR Collaboration), Measurements of $e^{+} e^{-} \to K^{+} K^{-} η$ , $K^{+} K^{-} π^{0}$ and $K_{S}^{0} K^{\pm} π^{\mp}$ cross sections using initial state radiation events, Phys. Rev. D 77, 092002 (2008).

Phys. Rev. D 114, 034021– Published 11 August, 2026

Vol. 114, Iss. 3 — 1 August 2026

Received 3 June 2026
Accepted 9 July 2026

DOI: https://doi.org/10.1103/5y9v-c2h6

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

Published by the American Physical Society

Physical Review D

Reuse & Permissions

Export citation

Export citation

Comparison of the hadronic vacuum polarization between hadronic $τ$ -decay data and lattice QCD

Abstract

Physics Subject Headings (PhySH)

Article Text

References (83)

Outline

Information

Filter

Filter

Article Lookup

Enter a citation

Comparison of the hadronic vacuum polarization between hadronic τ-decay data and lattice QCD

Abstract

Physics Subject Headings (PhySH)

Article Text

References (83)

Comparison of the hadronic vacuum polarization between hadronic $τ$ -decay data and lattice QCD