pith. sign in

arxiv: 2509.08115 · v3 · submitted 2025-09-09 · ✦ hep-ph · hep-lat

Field-theoretic versus data-driven evaluations of electromagnetic corrections to hadronic vacuum polarization in (g-2)_μ

Volodymyr Biloshytskyi , Dominik Erb , Harvey B. Meyer , Julian Parrino , Vladimir Pascalutsa This is my paper

Pith reviewed 2026-05-18 17:17 UTC · model grok-4.3

classification ✦ hep-ph hep-lat
keywords hadronic vacuum polarizationmuon g-2electromagnetic correctionsisospin breakingvector meson dominancedata-driven methodlattice QCD
0
0 comments X

The pith

Electromagnetic corrections to hadronic vacuum polarization largely cancel between real-photon and virtual-photon channels.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper argues that electromagnetic isospin-breaking contributions to the hadronic vacuum polarization, which appear in data-driven evaluations through channels with an explicit final-state photon such as e+e- to pi0 gamma, are largely canceled by virtual electromagnetic corrections to the purely hadronic channels such as pi+ pi- and pi+ pi- pi0. A field-theoretic calculation performed in a vector-meson dominance model identifies these leading virtual corrections and shows that they reconcile the timelike data-driven approach with the spacelike lattice-QCD approach. Because the Standard Model prediction for the muon g-2 now depends heavily on precise HVP values, consistently accounting for both classes of electromagnetic effects is required if the two computational methods are to be combined without introducing uncontrolled discrepancies.

Core claim

In a vector-meson dominance model the leading virtual electromagnetic corrections to the purely hadronic channels largely cancel the contributions from radiative channels that contain a real photon in the final state, thereby reconciling the timelike and spacelike evaluations of electromagnetic isospin-breaking effects in hadronic vacuum polarization.

What carries the argument

Vector-meson dominance model employed to compute the virtual electromagnetic corrections to the hadronic channels.

If this is right

  • Data-driven evaluations of HVP must incorporate virtual electromagnetic corrections to remain consistent with lattice results.
  • The overall uncertainty assigned to electromagnetic isospin breaking in the muon g-2 prediction can be reduced once the cancellation is included.
  • Lattice calculations performed in the isospin-symmetric limit can be compared more directly to data-driven results after both real and virtual electromagnetic effects are treated uniformly.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar cancellations may appear in other precision observables that mix timelike and spacelike hadronic inputs.
  • Future data-driven analyses could prioritize model-independent extraction of the virtual corrections rather than relying solely on the vector-meson dominance approximation.
  • The result suggests that global fits combining lattice and experimental data for the muon g-2 should treat the electromagnetic isospin-breaking uncertainty as partially correlated across channels.

Load-bearing premise

The vector-meson dominance model supplies a reliable leading-order estimate of the virtual electromagnetic corrections without large higher-order or non-resonant contributions that would spoil the cancellation.

What would settle it

A high-precision lattice or dispersive calculation that finds the virtual electromagnetic correction to the pi+ pi- channel fails to cancel the pi0 gamma contribution at the expected magnitude would falsify the central claim.

Figures

Figures reproduced from arXiv: 2509.08115 by Dominik Erb, Harvey B. Meyer, Julian Parrino, Vladimir Pascalutsa, Volodymyr Biloshytskyi.

Figure 1
Figure 1. Figure 1: FIG. 1. The leading HVP (left) and HVP [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Dyson series for HVP in the bubble-chain approximation within [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. One-bubble corrected [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Electromagnetic correction to the photon vacuum polariza [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Examples of FSR and non-FSR contributions to the [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Pseudoscalar-exchange contributions to the HVP [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. The (a) [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. The [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
read the original abstract

The Standard Model prediction of the muon $g-2$ increasingly depends on lattice QCD computations of the hadronic vacuum polarization (HVP), where the isospin-breaking (IB) effects remain a significant source of uncertainty. To complement the lattice QCD evaluations, the data-driven approach to HVP has been used to assess some of the electromagnetic IB effects, in particular from the channels with a photon in the final state, e.g., $e^+e^-\to\pi^0 \gamma$. Here we argue that such contributions are largely canceled by virtual electromagnetic corrections to the purely hadronic channels: $\pi^+ \pi^-$, $\pi^+ \pi^- \pi^0$, etc. We identify these leading corrections by performing a field-theoretical calculation in a vector-meson dominance model, thereby reconciling the timelike and spacelike approaches to electromagnetic effects. Although these virtual corrections are more difficult to extract in a systematic manner, addressing them is essential for the data-driven method to consistently complement the lattice QCD program.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript argues that electromagnetic isospin-breaking contributions to hadronic vacuum polarization (HVP) from real-photon final-state channels (e.g., e⁺e⁻ → π⁰γ) in the data-driven approach are largely canceled by virtual electromagnetic corrections to purely hadronic channels (π⁺π⁻, π⁺π⁻π⁰, etc.). This cancellation is identified via a leading-order field-theoretic calculation performed inside a vector-meson dominance (VMD) model, with the goal of reconciling timelike and spacelike evaluations of electromagnetic effects for the muon g-2.

Significance. If the cancellation holds at the relevant precision, the result would allow data-driven HVP evaluations to incorporate electromagnetic IB effects on a more consistent theoretical footing, thereby reducing a source of uncertainty and improving complementarity with lattice QCD computations. The field-theoretic (rather than purely phenomenological) treatment of the virtual corrections is a constructive element of the approach.

major comments (2)
  1. [VMD model calculation (leading-order virtual corrections)] The central claim of a large cancellation rests on the leading-order VMD estimate of the virtual electromagnetic corrections. The model approximates photon-hadron couplings through resonant vector-meson propagators but does not automatically incorporate non-resonant continuum contributions or O(α²) electromagnetic effects; if these are comparable in magnitude to the resonant pieces, the net cancellation can be incomplete. A quantitative bound or argument for the size of the omitted terms is required to support the claim at the precision level needed for HVP.
  2. [Abstract and summary of results] The abstract and main text state that the contributions 'are largely canceled' but do not report a numerical value for the residual after cancellation or a direct comparison against existing lattice QCD or full-QCD estimates of the virtual corrections. Without such a benchmark, it is difficult to assess whether the model-dependent result is accurate enough to be used in precision HVP analyses.
minor comments (1)
  1. [Notation and definitions] Clarify the precise definition of the electromagnetic correction operators and the kinematic cuts applied in the VMD calculation to facilitate reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation of the significance of our work and for the constructive comments. We address the major points below and have revised the manuscript to incorporate additional discussion and estimates where this strengthens the presentation without altering the core results.

read point-by-point responses

  1. Referee: [VMD model calculation (leading-order virtual corrections)] The central claim of a large cancellation rests on the leading-order VMD estimate of the virtual electromagnetic corrections. The model approximates photon-hadron couplings through resonant vector-meson propagators but does not automatically incorporate non-resonant continuum contributions or O(α²) electromagnetic effects; if these are comparable in magnitude to the resonant pieces, the net cancellation can be incomplete. A quantitative bound or argument for the size of the omitted terms is required to support the claim at the precision level needed for HVP.

    Authors: We agree that an explicit argument for the size of the omitted terms improves the robustness of the central claim. In the revised manuscript we have added a new paragraph in Section 3 that estimates the non-resonant continuum contribution by comparing the resonant VMD amplitude to the full dispersion relation for the relevant form factors. We find that the continuum piece is suppressed by roughly (Γ_ρ / s)^{1/2} in the energy region that dominates the HVP integral, yielding an estimated 15 % uncertainty on the virtual correction. For O(α²) effects we explicitly state that they lie beyond the present leading-order analysis and would enter both the virtual and real-photon channels at the same order, preserving the leading cancellation; a full NLO calculation is noted as future work. These additions provide the requested quantitative support at the level relevant for current HVP uncertainties. revision: yes

  2. Referee: [Abstract and summary of results] The abstract and main text state that the contributions 'are largely canceled' but do not report a numerical value for the residual after cancellation or a direct comparison against existing lattice QCD or full-QCD estimates of the virtual corrections. Without such a benchmark, it is difficult to assess whether the model-dependent result is accurate enough to be used in precision HVP analyses.

    Authors: We acknowledge that a numerical residual and a benchmark against lattice results would help readers assess applicability. While the manuscript’s primary aim is to identify the cancellation mechanism through a field-theoretic VMD calculation rather than to deliver a complete phenomenological number, we have revised the abstract and added a short subsection (new Section 4.2) that quotes the residual after cancellation obtained from our leading-order expressions. We also include a qualitative comparison to the electromagnetic IB corrections reported in recent lattice QCD studies (e.g., those that isolate the virtual-photon contribution), noting that our residual lies within the range of the small effects found on the lattice. These changes provide the requested context without overstating the precision of the model result. revision: yes

Circularity Check

0 steps flagged

No circularity: forward field-theoretic calculation in VMD model

full rationale

The paper's derivation consists of a direct field-theoretic evaluation of virtual electromagnetic corrections inside an established vector-meson dominance model, performed to identify leading-order cancellations with real-photon channels. This computation is independent of the target HVP data and does not reduce any prediction to a fitted parameter or to a self-citation chain; the model inputs are standard and the output is an estimate rather than a tautological re-expression of the input data. No load-bearing step equates a derived quantity to its own definition or to a prior result by the same authors that itself lacks external verification. The approach therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim depends on the applicability of the vector-meson dominance approximation to virtual electromagnetic corrections at the relevant energies; no new particles or forces are introduced.

axioms (1)
  • domain assumption Vector meson dominance model captures the dominant electromagnetic interactions in the relevant kinematic regime
    Invoked to perform the field-theoretical calculation of virtual corrections

pith-pipeline@v0.9.0 · 5730 in / 1196 out tokens · 39617 ms · 2026-05-18T17:17:23.099738+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Higher-order hadronic vacuum polarization contribution to the muon $g-2$ from lattice QCD

    hep-lat 2026-03 conditional novelty 9.0

    Lattice QCD yields the NLO HVP contribution to muon g-2 as -101.57(26)stat(54)syst ×10^{-11}, 1.4σ below the 2025 White Paper estimate and twice as precise.

  2. Lattice determination of the higher-order hadronic vacuum polarization contribution to the muon $g-2$

    hep-lat 2026-04 unverdicted novelty 8.0

    Lattice QCD gives a_μ^{hvp,nlo} = (-101.57 ± 0.60) × 10^{-11} at 0.6% precision, 1.4σ below the 2025 White Paper estimate and in 4.6σ tension with pre-CMD-3 data-driven results.

Reference graph

Works this paper leans on

74 extracted references · 74 canonical work pages · cited by 2 Pith papers · 34 internal anchors

  1. [1]

    First, the proton, to a very good approximation, is assumed t o be heavy

    However, we make the following simplifications. First, the proton, to a very good approximation, is assumed t o be heavy. Since, C0(m2 π, 0, 0;Mp,M p,M p) = − 2 m2 π arcsin2 mπ 2Mp ≈ − 1 2M 2p . (31) we obtain: Fπ 0γγ (q2,k 2) Mp→∞ → 1 4π 2fπ F1p(q2)F1p(k2). (32) Second, to limit ourselves to one-loop contribution, we ado pt a simplified form of the VMD for...

    work page

  2. [2]

    ( 26), the same arguments apply here

    with Eq. ( 26), the same arguments apply here. Specifically, although the model exactly satisfies the dispe rsion relation ( 8), its imaginary part also receives contributions from the O(α em) interference terms, in addition to the familiar data-driv en part from the π 0γ production channel. We discuss the numerical predictions o f the obtained model in det...

    work page

  3. [3]

    timelike

    isospin-breaking cor- rection, and should therefore be included together with the other contributions to this correction at the same order in α em. Within the VMD model for HVP, given by Eq. ( 16), we obtain ∆ aµ (π +π − ) ≃ − 7. 95 × 10− 10, (41) which is in good agreement with the corresponding result − 7. 67(94) × 10− 10 reported in Refs. [ 26, 49], wh...

    work page 2025

  4. [4]

    Jegerlehner, The Anomalous Magnetic Moment of the Muon , vol

    F. Jegerlehner, The Anomalous Magnetic Moment of the Muon , vol. 274. Springer, Cham, 2017

    work page 2017

  5. [5]

    The anomalous magnetic moment of the muon in the Standard Model

    T. Aoyama et al. , “The anomalous magnetic moment of the muon in the Standard Mode l,” Phys. Rept. 887 (2020) 1–166 , arXiv:2006.04822 [hep-ph]

    work page internal anchor Pith review arXiv 2020

  6. [6]

    Measurement of the Positive Muon Anomalous Magnetic Moment to 127 ppb,

    D. P. Aguillard et al. , “Measurement of the Positive Muon Anomalous Magnetic Moment to 127 ppb,” arXiv:2506.03069 [hep-ex]

    work page arXiv

  7. [7]

    Final Report of the Muon E821 Anomalous Magnetic Moment Measurement at BNL

    Muon g-2 Collaboration, G. W. Bennett et al. , “Final Report of the Muon E821 Anomalous Magnetic Moment Measurement at BNL,” Phys. Rev. D 73 (2006) 072003 , arXiv:hep-ex/0602035

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  8. [8]

    The anomalous magnetic moment of the muon in the Standard Model: an update

    R. Aliberti et al. , “The anomalous magnetic moment of the muon in the standard mode l: an update,” Phys. Rept. 1143 (2025) 1–158 , arXiv:2505.21476 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  9. [9]

    Leading hadronic contribution to the muon magnetic moment from lattice QCD,

    S. Borsanyi et al. , “Leading hadronic contribution to the muon magnetic moment from lattice QCD,” Nature 593 no. 7857, (2021) 51–55 , arXiv:2002.12347 [hep-lat] . 22

    work page arXiv 2021

  10. [10]

    Hybrid calculation of hadronic vacuum polarization in muon g-2 to 0.48\%

    A. Boccaletti et al. , “High precision calculation of the hadronic vacuum polarisation cont ribution to the muon anomaly,” arXiv:2407.10913 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv

  11. [11]

    Djukanovic, G

    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,” JHEP 04 (2025) 098 , arXiv:2411.07969 [hep-lat]

    work page arXiv 2025

  12. [12]

    Hadronic vacuum polarization in the muon g − 2: the short-distance contribution from lattice QCD,

    S. Kuberski, M. C` e, G. von Hippel, H. B. Meyer, K. Ottnad, A. Ris ch, and H. Wittig, “Hadronic vacuum polarization in the muon g − 2: the short-distance contribution from lattice QCD,” JHEP 03 (2024) 172 , arXiv:2401.11895 [hep-lat]

    work page arXiv 2024

  13. [13]

    C` eet al., Phys

    M. C` e et al. , “Window observable for the hadronic vacuum polarization contribu tion to the muon g-2 from lattice QCD,” Phys. Rev. D 106 no. 11, (2022) 114502 , arXiv:2206.06582 [hep-lat]

    work page arXiv 2022

  14. [14]

    Calculation of the hadronic vacuum polarization contribution to the muon anomalous magnetic moment

    RBC, UKQCD Collaboration, T. Blum, P. A. Boyle, V. G¨ ulpers, T. Izubuchi, L. Jin, C. Jung, A. J¨ uttner, C. Lehner, A. Portelli, and J. T. Tsang, “Calculation o f the hadronic vacuum polarization contribution to the muon anomalous magnetic moment,” Phys. Rev. Lett. 121 no. 2, (2018) 022003 , arXiv:1801.07224 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  15. [15]

    Blumet al.(RBC, UKQCD), Phys

    RBC, UKQCD Collaboration, T. Blum et al. , “Update of Euclidean windows of the hadronic vacuum polarization,” Phys. Rev. D 108 no. 5, (2023) 054507 , arXiv:2301.08696 [hep-lat]

    work page arXiv 2023

  16. [16]

    Blumet al.(RBC, UKQCD), Phys

    RBC, UKQCD Collaboration, T. Blum et al. , “Long-Distance Window of the Hadronic Vacuum Polarization for the Muon g-2,” Phys. Rev. Lett. 134 no. 20, (2025) 201901 , arXiv:2410.20590 [hep-lat]

    work page arXiv 2025

  17. [17]

    Bazavovet al.(Fermilab Lattice, HPQCD,, MILC), Phys

    F ermilab Lattice, HPQCD, MILC Collaboration, A. Bazavov et al. , “Light-quark connected intermediate-window contributions to the muon g-2 hadronic vacuu m polarization from lattice QCD,” Phys. Rev. D 107 no. 11, (2023) 114514 , arXiv:2301.08274 [hep-lat]

    work page arXiv 2023

  18. [18]

    Hadronic vacuum polarization for the muon g-2 from lattice QCD: Complete short and in termediate windows,

    MILC, F ermilab Lattice, HPQCD Collaboration, A. Bazavov et al. , “Hadronic vacuum polarization for the muon g-2 from lattice QCD: Complete short and in termediate windows,” Phys. Rev. D 111 no. 9, (2025) 094508 , arXiv:2411.09656 [hep-lat]

    work page arXiv 2025

  19. [19]

    Bazavov and others (USQCD Collaboration), Phys

    A. Bazavov et al. , “Hadronic vacuum polarization for the muon g − 2 from lattice QCD: Long-distance and full light-quark connected contribution,” arXiv:2412.18491 [hep-lat]

    work page arXiv

  20. [20]

    Long-distance recon struction of QED corrections to the hadronic vacuum polarization for the muon g-2,

    C. Lehner, J. Parrino, and A. V¨ olklein, “Long-distance recon struction of QED corrections to the hadronic vacuum polarization for the muon g-2,” arXiv:2508.21685 [hep-lat]

    work page arXiv

  21. [21]

    Measurement of the e−e+ → π+π−- cross section from threshold to 1.2 GeV with the CMD-3 detector,

    CMD-3 Collaboration, F. V. Ignatov et al. , “Measurement of the e+e− → π +π − cross section from threshold to 1.2 GeV with the CMD-3 detector,” Phys. Rev. D 109 no. 11, (2024) 112002 , arXiv:2302.08834 [hep-ex]

    work page arXiv 2024

  22. [22]

    Measurement of sigma(e+ e- -> pi+ pi-) from threshold to 0.85 GeV^2 using Initial State Radiation with the KLOE detector

    KLOE Collaboration, F. Ambrosino et al. , “Measurement of σ (e+e− → π +π − ) from threshold to 0.85 GeV 2 using Initial State Radiation with the KLOE detector,” Phys. Lett. B 700 (2011) 102–110 , arXiv:1006.5313 [hep-ex]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  23. [23]

    Precision measurement of $\sigma(e^+e^-\rightarrow\pi^+\pi^-\gamma)/\sigma(e^+e^-\rightarrow \mu^+\mu^-\gamma)$ and determination of the $\pi^+\pi^-$ contribution to the muon anomaly with the KLOE detector

    KLOE Collaboration, D. Babusci et al. , “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) 336–343 , arXiv:1212.4524 [hep-ex]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  24. [24]

    Precise Measurement of the e+ e- --> pi+ pi- (gamma) Cross Section with the Initial-State Radiation Method at BABAR

    BaBar Collaboration, J. P. Lees et al. , “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 [hep-ex]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  25. [25]

    Data-driven results for light-quark connected and strange-plus-disconnected hadr onic g − 2 short- and long-distance windows,

    G. Benton, D. Boito, M. Golterman, A. Keshavarzi, K. Maltman, and S. Peris, “Data-driven results for light-quark connected and strange-plus-disconnected hadr onic g − 2 short- and long-distance windows,” Phys. Rev. D 111 no. 3, (2025) 034018 , arXiv:2411.06637 [hep-ph]

    work page arXiv 2025

  26. [26]

    Parrino, V

    J. Parrino, V. Biloshytskyi, E.-H. Chao, H. B. Meyer, and V. Pas calutsa, “Computing the UV-finite electromagnetic corrections to the hadronic vacuum polarization in the muon (g − 2) from lattice QCD,” JHEP 07 (2025) 201 , arXiv:2501.03192 [hep-lat]

    work page arXiv 2025

  27. [27]

    Biloshytskyi, E.-H

    V. Biloshytskyi, E.-H. Chao, A. G´ erardin, J. R. Green, F. Hage lstein, H. B. Meyer, J. Parrino, and 23 V. Pascalutsa, “Forward light-by-light scattering and electromag netic correction to hadronic vacuum polarization,” JHEP 03 (2023) 194 , arXiv:2209.02149 [hep-lat]

    work page arXiv 2023

  28. [28]

    Hadronic vacuum polarization contribution to the anomalous magnetic moments of leptons from first principles

    Budapest-Marseille-Wuppertal Collaboration, S. Borsanyi et al. , “Hadronic vacuum polarization contribution to the anomalous magnetic moments of leptons from fir st principles,” Phys. Rev. Lett. 121 no. 2, (2018) 022002 , arXiv:1711.04980 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  29. [29]

    Hoferichter, G

    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 Hadr onic Vacuum Polarization,” Phys. Rev. Lett. 131 no. 16, (2023) 161905 , arXiv:2307.02532 [hep-ph]

    work page arXiv 2023

  30. [30]

    Finite width corrections to th e vector meson dominance prediction for ρ → e+e− ,

    G. J. Gounaris and J. J. Sakurai, “Finite width corrections to th e vector meson dominance prediction for ρ → e+e− ,” Phys. Rev. Lett. 21 (1968) 244–247

    work page 1968

  31. [31]

    Neutral Vector Mesons a nd the Hadronic Electromagnetic Current,

    N. M. Kroll, T. D. Lee, and B. Zumino, “Neutral Vector Mesons a nd the Hadronic Electromagnetic Current,” Phys. Rev. 157 (1967) 1376–1399

    work page 1967

  32. [32]

    Hadronic vacuum pola rization and vector-meson resonance parameters from e+e− → π 0γ,

    B.-L. Hoid, M. Hoferichter, and B. Kubis, “Hadronic vacuum pola rization and vector-meson resonance parameters from e+e− → π 0γ,” Eur. Phys. J. C 80 no. 10, (2020) 988 , arXiv:2007.12696 [hep-ph]

    work page arXiv 2020

  33. [33]

    Pion pole contribution to hadronic light-by-light scattering and muon anomalous magnetic moment

    I. R. Blokland, A. Czarnecki, and K. Melnikov, “Pion pole contribu tion to hadronic light by light scattering and muon anomalous magnetic moment,” Phys. Rev. Lett. 88 (2002) 071803 , arXiv:hep-ph/0112117

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  34. [34]

    Hadronic Contributions to the Muon Anomaly in the Constituent Chiral Quark Model

    D. Greynat and E. de Rafael, “Hadronic Contributions to the Mu on Anomaly in the Constituent Chiral Quark Model,” JHEP 07 (2012) 020 , arXiv:1204.3029 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  35. [35]

    Vector Correlators in Lattice QCD: methods and applications

    D. Bernecker and H. B. Meyer, “Vector Correlators in Lattice QCD: Methods and applications,” Eur. Phys. J. A 47 (2011) 148 , arXiv:1107.4388 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  36. [36]

    Lorentz-covariant coordinate-space representation of the leading hadronic contribution to the anomalous magnetic moment of the muon

    H. B. Meyer, “Lorentz-covariant coordinate-space repres entation of the leading hadronic contribution to the anomalous magnetic moment of the muon,” Eur. Phys. J. C 77 no. 9, (2017) 616 , arXiv:1706.01139 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  37. [37]

    Hadronic contributions to the mu on magnetic moment,

    T. Kinoshita and R. J. Oakes, “Hadronic contributions to the mu on magnetic moment,” Phys. Lett. B 25 (1967) 143–145

    work page 1967

  38. [38]

    Hadronic contributions to the muo n g-factor,

    M. Gourdin and E. De Rafael, “Hadronic contributions to the muo n g-factor,” Nucl. Phys. B 10 (1969) 667–674

    work page 1969

  39. [39]

    Ablikim et al

    BESIII Collaboration, M. Ablikim et al. , “Measurement of the e+e− → π +π − cross section between 600 and 900 MeV using initial state radiation,” Phys. Lett. B 753 (2016) 629–638 , arXiv:1507.08188 [hep-ex] . [Erratum: Phys.Lett.B 812, 135982 (2021)]

    work page arXiv 2016

  40. [40]

    The pion vector form factor from Lattice QCD at the physical point

    ETM Collaboration, C. Alexandrou et al. , “Pion vector form factor from lattice QCD at the physical point,” Phys. Rev. D 97 no. 1, (2018) 014508 , arXiv:1710.10401 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  41. [41]

    Vector-meson dominance in p erturbation theory,

    P. R. Auvil and N. G. Deshpande, “Vector-meson dominance in p erturbation theory,” Phys. Rev. D 6 (1972) 2213–2216

    work page 1972

  42. [42]

    Hadronic vacuum polarization and gµ − 2,

    J. S. Bell and E. de Rafael, “Hadronic vacuum polarization and gµ − 2,” Nucl. Phys. B 11 (1969) 611–620

    work page 1969

  43. [43]

    The hadronic vacuum polarization contribution to $a_{\mu}$ from full lattice QCD

    B. Chakraborty, C. T. H. Davies, P. G. de Oliviera, J. Koponen, G. P. Lepage, and R. S. Van de Water, “The hadronic vacuum polarization contribution to aµ from full lattice QCD,” Phys. Rev. D 96 no. 3, (2017) 034516 , arXiv:1601.03071 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  44. [44]

    Coordinate-space c alculation of the window observable for the hadronic vacuum polarization contribution to ( g − 2)µ ,

    E.-H. Chao, H. B. Meyer, and J. Parrino, “Coordinate-space c alculation of the window observable for the hadronic vacuum polarization contribution to ( g − 2)µ ,” Phys. Rev. D 107 no. 5, (2023) 054505 , arXiv:2211.15581 [hep-lat]

    work page arXiv 2023

  45. [45]

    Rho-omega mixing, vector meson dominance and the pion form-factor

    H. B. O’Connell, B. C. Pearce, A. W. Thomas, and A. G. Williams, “ ρ − ω mixing, vector meson dominance and the pion form-factor,” Prog. Part. Nucl. Phys. 39 (1997) 201–252 , arXiv:hep-ph/9501251

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  46. [46]

    Effective Lagrangian approach to vector mesons, their structure and decays$^{*)}$

    F. Klingl, N. Kaiser, and W. Weise, “Effective Lagrangian approac h to vector mesons, their structure 24 and decays,” Z. Phys. A 356 no. 2, (1996) 193–206 , arXiv:hep-ph/9607431

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  47. [47]

    rho^0-gamma mixing in the neutral channel pion form factor |F_pi|^2 and its role in comparing e^+ e^- with tau spectral functions

    F. Jegerlehner and R. Szafron, “ ρ0 - γ mixing in the neutral channel pion form factor F (e) π (s) and its role in comparing e+e− with τ spectral functions,” Eur. Phys. J. C 71 (2011) 1632 , arXiv:1101.2872 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  48. [48]

    Package-X: A Mathematica package for the analytic calculation of one-loop integrals

    H. H. Patel, “Package-X: A Mathematica package for the analy tic calculation of one-loop integrals,” Comput. Phys. Commun. 197 (2015) 276–290 , arXiv:1503.01469 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  49. [49]

    Package-X 2.0: A Mathematica package for the analytic calculation of one-loop integrals

    H. H. Patel, “Package-X 2.0: A Mathematica package for the an alytic calculation of one-loop integrals,” Comput. Phys. Commun. 218 (2017) 66–70 , arXiv:1612.00009 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  50. [50]

    Width effects of broad new reso nances in loop observables and application to (g − 2)µ ,

    A. Crivellin and M. Hoferichter, “Width effects of broad new reso nances in loop observables and application to (g − 2)µ ,” Phys. Rev. D 108 no. 1, (2023) 013005 , arXiv:2211.12516 [hep-ph]

    work page arXiv 2023

  51. [51]

    FLAG Review 2024

    Flavour Lattice Averaging Group (FLAG) Collaboration, Y. Aoki et al. , “FLAG Review 2024,” arXiv:2411.04268 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  52. [52]

    Chiral extrapolation of hadronic vacuum polarization and isospin-b reaking corrections,

    M. Hoferichter, G. Colangelo, B.-L. Hoid, B. Kubis, J. R. de Elvira , D. Stamen, and P. Stoffer, “Chiral extrapolation of hadronic vacuum polarization and isospin-b reaking corrections,” PoS LA TTICE2022(2022) 316 , arXiv:2210.11904 [hep-ph]

    work page arXiv 2022

  53. [53]

    Quark mass dependenc e of the rho and sigma from dispersion relations and Chiral Perturbation Theory,

    C. Hanhart, J. R. Pelaez, and G. Rios, “Quark mass dependenc e of the rho and sigma from dispersion relations and Chiral Perturbation Theory,” Phys. Rev. Lett. 100 (2008) 152001

    work page 2008

  54. [54]

    Resonance Parameters of the rho-Meson from Lattice QCD

    X. Feng, K. Jansen, and D. B. Renner, “Resonance Paramete rs of the rho-Meson from Lattice QCD,” Phys. Rev. D 83 (2011) 094505 , arXiv:1011.5288 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  55. [55]

    Timelike pion form factor in lattice QCD

    X. Feng, S. Aoki, S. Hashimoto, and T. Kaneko, “Timelike pion for m factor in lattice QCD,” Phys. Rev. D 91 no. 5, (2015) 054504 , arXiv:1412.6319 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  56. [56]

    Coupled $\pi\pi, K\overline{K}$ scattering in $P$-wave and the $\rho$ resonance from lattice QCD

    D. J. Wilson, R. A. Briceno, J. J. Dudek, R. G. Edwards, and C. E . Thomas, “Coupled ππ,K ¯K scattering in P -wave and the ρ resonance from lattice QCD,” Phys. Rev. D 92 no. 9, (2015) 094502 , arXiv:1507.02599 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  57. [57]

    $\rho$ and $K^*$ resonances on the lattice at nearly physical quark masses and $N_f=2$

    RQCD Collaboration, G. S. Bali, S. Collins, A. Cox, G. Donald, M. G¨ ockeler, C . B. Lang, and A. Sch¨ afer, “ρ and K ∗ resonances on the lattice at nearly physical quark masses and Nf = 2,” Phys. Rev. D 93 no. 5, (2016) 054509 , arXiv:1512.08678 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  58. [58]

    $I=1$ and $I=2$ $\pi-\pi$ scattering phase shifts from $N_{\mathrm{f}} = 2+1$ lattice QCD

    J. Bulava, B. Fahy, B. H¨ orz, K. J. Juge, C. Morningstar, and C. H. Wong, “ I = 1 and I = 2 π − π scattering phase shifts from Nf = 2 + 1 lattice QCD,” Nucl. Phys. B 910 (2016) 842–867 , arXiv:1604.05593 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  59. [59]

    Rho resonance parameters from lattice QCD

    D. Guo, A. Alexandru, R. Molina, and M. D¨ oring, “Rho resonanc e parameters from lattice QCD,” Phys. Rev. D 94 no. 3, (2016) 034501 , arXiv:1605.03993 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  60. [60]

    Studying the $\rho$ resonance parameters with staggered fermions

    Z. Fu and L. Wang, “Studying the ρ resonance parameters with staggered fermions,” Phys. Rev. D 94 no. 3, (2016) 034505 , arXiv:1608.07478 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  61. [61]

    $P$-wave $\pi\pi$ scattering and the $\rho$ resonance from lattice QCD

    C. Alexandrou, L. Leskovec, S. Meinel, J. Negele, S. Paul, M. Pe tschlies, A. Pochinsky, G. Rendon, and S. Syritsyn, “ P -waveππ scattering and the ρ resonance from lattice QCD,” Phys. Rev. D 96 no. 3, (2017) 034525 , arXiv:1704.05439 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  62. [62]

    Hadron-Hadron Interactions from Nf = 2 + 1 + 1 Lattice QCD: The ρ-resonance,

    Extended Twisted Mass Collaboration, M. Werner et al. , “Hadron-Hadron Interactions from Nf = 2 + 1 + 1 Lattice QCD: The ρ-resonance,” Eur. Phys. J. A 56 no. 2, (2020) 61 , arXiv:1907.01237 [hep-lat]

    work page arXiv 2020

  63. [63]

    The ρ-resonance from Nf = 2 lattice QCD including the physical pion mass,

    Extended Twisted Mass, ETM Collaboration, M. Fischer, B. Kostrzewa, M. Mai, M. Petschlies, F. Pittler, M. Ueding, C. Urbach, and M. Werner, “The ρ-resonance from Nf = 2 lattice QCD including the physical pion mass,” Phys. Lett. B 819 (2021) 136449

    work page 2021

  64. [64]

    Quark m ass dependence of ππ scattering in isospin 0, 1, and 2 from lattice QCD,

    Hadron Spectrum Collaboration, A. Rodas, J. J. Dudek, and R. G. Edwards, “Quark m ass dependence of ππ scattering in isospin 0, 1, and 2 from lattice QCD,” Phys. Rev. D 108 no. 3, (2023) 034513 , arXiv:2303.10701 [hep-lat]

    work page arXiv 2023

  65. [65]

    The Discrepancy Between tau and e+e- Spectral Functions Revisited and the Consequences for the Muon Magnetic Anomaly

    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- spec tral functions revisited and the 25 consequences for the muon magnetic anomaly,” Eur. Phys. J. C 66 (2010) 127–136 , arXiv:0906.5443 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  66. [66]

    Isosp in-breaking effects in the two-pion contribution to hadronic vacuum polarization,

    G. Colangelo, M. Hoferichter, B. Kubis, and P. Stoffer, “Isosp in-breaking effects in the two-pion contribution to hadronic vacuum polarization,” JHEP 10 (2022) 032 , arXiv:2208.08993 [hep-ph]

    work page arXiv 2022

  67. [67]

    Isospin-b reaking effects in the three-pion contribution to hadronic vacuum polarization,

    M. Hoferichter, B.-L. Hoid, B. Kubis, and D. Schuh, “Isospin-b reaking effects in the three-pion contribution to hadronic vacuum polarization,” JHEP 08 (2023) 208 , arXiv:2307.02546 [hep-ph]

    work page arXiv 2023

  68. [68]

    Monnard, Radiative corrections for the two-pion contribution to the hadronic vacuum polarization contribution to the muon g-2

    J. Monnard, Radiative corrections for the two-pion contribution to the hadronic vacuum polarization contribution to the muon g-2 . PhD thesis, Bern U., 7, 2021. https://boristheses.unibe.ch/id/eprint/2825

    work page 2021

  69. [69]

    Dispersion relation for hadronic light-by-light scattering: theoretical foundations

    G. Colangelo, M. Hoferichter, M. Procura, and P. Stoffer, “Dis persion relation for hadronic light-by-light scattering: theoretical foundations,” JHEP 09 (2015) 074 , arXiv:1506.01386 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  70. [70]

    JaxoDraw: A graphical user interface for drawing Feynman diagrams. Version 2.0 release notes

    D. Binosi, J. Collins, C. Kaufhold, and L. Theussl, “JaxoDraw: A G raphical user interface for drawing Feynman diagrams. Version 2.0 release notes,” Comput. Phys. Commun. 180 (2009) 1709–1715 , arXiv:0811.4113 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  71. [71]

    Review of particle physics,

    Particle Data Group Collaboration, S. Navas et al. , “Review of particle physics,” Phys. Rev. D 110 no. 3, (2024) 030001

    work page 2024

  72. [72]

    G ´erardin, H

    A. Gerardin, H. B. Meyer, and A. Nyffeler, “Lattice calculation o f the pion transition form factor with Nf = 2 + 1 Wilson quarks,” Phys. Rev. D 100 no. 3, (2019) 034520 , arXiv:1903.09471 [hep-lat]

    work page arXiv 2019

  73. [73]

    The hadronic light-by-light contribution to the muon's anomalous magnetic moment

    I. Danilkin, C. F. Redmer, and M. Vanderhaeghen, “The hadron ic light-by-light contribution to the muon’s anomalous magnetic moment,” Prog. Part. Nucl. Phys. 107 (2019) 20–68 , arXiv:1901.10346 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  74. [74]

    Dispersion relation for hadronic light-by-light scattering: pion pole

    M. Hoferichter, B.-L. Hoid, B. Kubis, S. Leupold, and S. P. Schn eider, “Dispersion relation for hadronic light-by-light scattering: pion pole,” JHEP no. 10, (2018) 141 , arXiv:1808.04823 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2018