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REVIEW 3 major objections 4 minor 64 references

Hadronic vacuum polarization contribution to the muon g-2 on Euclidean windows from tau data

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Tau decay data, converted to e+e- cross sections with isospin-breaking corrections, produce Euclidean-window contributions to the muon g-2 that agree with lattice QCD, while e+e- data-driven results are in tension.

desk verdict Tau-lattice window compatibility is plausible but rests on older isospin-breaking corrections; the proceedings adds no new numbers. read the letter →

arxiv 2411.09811 v1 pith:76WWJRFB submitted 2024-11-14 hep-ph

Pere Masjuan , Alejandro Miranda , Pablo Roig This is my paper · ORCID
classification hep-ph
keywords hadronicvacuumpolarizationmuong-2taudecaysEuclideanwindowsisospinbreakinge+e-annihilationlatticeQCDpionformfactor
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper computes, for the first time, the Euclidean-window contributions to the hadronic vacuum polarization part of the muon g-2 using tau decay data instead of e+e- annihilation data. It finds that tau-based results agree with lattice QCD window evaluations in the short, intermediate, and long windows, and with the full lattice result. In the intermediate window, where lattice evaluations are precise and mutually consistent, tau data are compatible with lattice, whereas e+e- data-driven values are in tension with both. A sympathetic reader would care because the e+e- vs lattice discrepancy in this window is the main driver of the current g-2 interpretive puzzle, and tau data offer an independent probe of whether the problem lies in the e+e- data.

What carries the argument

The central objects are the Euclidean window observables, which split the hadronic vacuum polarization contribution into short-distance, intermediate, and long-distance pieces (with sizes scaling roughly as $1:10:25$), and the conversion factor that turns tau decay spectra into $e^+e^-$ cross sections. That conversion is built from the long-distance radiative correction $G_{\rm EM}(s)$ and the form-factor ratio $|F_V(s)/f_+(s)|^2$, packaged into the isospin-breaking factor $R_{\rm IB}(s)$ that appears in the relation between the $\tau$ spectral function and the $\pi^+\pi^-$ cross section. The argument works by applying this conversion to the measured tau spectra, computing the three window integrals, and comparing them with lattice QCD and $e^+e^-$ data-driven window evaluations.

What would settle it

Measure the pion form factor ratio $|F_V(s)/f_+(s)|^2$ and the radiative factor $G_{\rm EM}(s)$ independently over the window energies (for example from a high-statistics study of $\tau^-\to\pi^-\pi^0\nu_\tau\gamma$ together with a precise $\pi^+\pi^-$ cross section measurement); if the resulting $R_{\rm IB}$ differs from the value used here by more than its uncertainty, the tau-based window moves and the claimed lattice agreement would be directly tested.

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Extended reading notes

Core claim

The paper's central claim is that, once isospin-breaking corrections are applied through the relation between the $\tau^-\to\pi^-\pi^0\nu_\tau$ spectral function and the $e^+e^-\to\pi^+\pi^-(\gamma)$ cross section, tau decay data produce Euclidean window integrals that sit on top of lattice QCD values. In the intermediate window the tau-based result agrees with the lattice average, while the $e^+e^-$-based result disagrees with both, and the discrepancy is almost entirely in the light-quark connected contribution dominated by the $\pi\pi$ channel. This supports the conclusion that the $e^+e^-$ data, not new physics, are the main source of the intermediate-window discrepancy affecting the interpretation of the measured $a_\mu$.

Load-bearing premise

The argument depends on the accuracy of the isospin-breaking conversion that turns tau decay spectra into $e^+e^-$ cross sections: the long-distance radiative factor $G_{\rm EM}(s)$ and the form-factor ratio $|F_V(s)/f_+(s)|^2$ must be correct over the window energies, or the tau-based windows shift and the agreement with lattice can disappear.

Editorial extensions

If this is right

  • If the tau-based windows are correct, the intermediate-window tension is attributed to $e^+e^-$ data rather than to new physics in muon g-2.
  • The light-quark connected $\pi\pi$ channel, which dominates the discrepancy, becomes the target for new precise $e^+e^-$ measurements around the $\rho$ peak.
  • Tau-based full HVP evaluations agree with lattice QCD, providing a cross-check that does not rely on $e^+e^-$ annihilation data.
  • The method can be applied to the short- and long-distance windows as lattice calculations reach comparable precision, giving a data-driven benchmark for each window.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • An implication the authors leave implicit: taking the tau-based windows at face value raises the Standard Model prediction for $a_\mu$ in the $\pi\pi$ channel by roughly $10\times10^{-10}$, shrinking the gap with the BNL/FNAL average and reducing the inferred new-physics signal.
  • A testable extension is to apply the same isospin-breaking machinery to the $\tau\to K\pi\nu_\tau$ channels, producing a tau-based prediction for the strange-quark contribution to the intermediate window as an independent cross-check of lattice and $e^+e^-$ results.
  • If future tau data reach percent-level precision on the $\pi\pi$ spectral function, the tau-based window method could become competitive with $e^+e^-$ data and help settle whether the CMD-3 cross section is the correct one.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. This proceedings contribution reports tau-data-driven evaluations of the short-distance (SD), intermediate (int), and long-distance (LD) Euclidean windows for the hadronic vacuum polarization (HVP) contribution to the muon g-2. The authors convert tau spectral functions into e+e- cross sections using the conserved-vector-current relation with isospin-breaking corrections from Resonance Chiral Theory, and they compare the resulting window quantities with lattice QCD and e+e- data-driven results. Their central claim is that tau-based results are compatible with the lattice evaluations in the intermediate window, whereas e+e- based results are in tension with both, suggesting that the e+e- data are the main source of the intermediate-window discrepancy rather than new physics. The results are presented graphically, with details referred to Refs. [50] and [61].

Significance. If the central claim holds, it sharpens the interpretation of the muon g-2 discrepancy: it would support the view that the intermediate-window tension originates in e+e- data rather than in new physics. The paper has the strength of being based on external tau spectra and compared with external lattice benchmarks, so the central claim is not defined into existence; it is also independently mirrored by the corroborating study of Ref. [61]. The main technical risk is the reliance on the isospin-breaking conversion factor R_IB(s), whose s-dependence could shift the window values if the more recent treatment of Ref. [49] is not propagated. Because the proceedings does not provide numeric values, uncertainties, or integration details, the quantitative strength of the compatibility claim cannot be assessed from the manuscript alone.

major comments (3)
  1. [Section 2, Eqs. (2)-(3)] The tau-to-e+e- conversion factor R_IB(s) is the backbone of the analysis, and the text states that the isospin-breaking correction to a_mu^HVP|pi pi lies in the range [-20.52,-6.96] x 10^-10 at 68% CL, with the parenthetical 'we are more precise in the recent [49]'. The window results shown in Figs. 2 and 3, however, are taken from Ref. [50], which uses the earlier treatment of Ref. [34]. Since R_IB(s) is s-dependent, an updated G_EM(s) or form-factor ratio can shift the window integrals by more than the quoted window uncertainties. The authors should either fold the Ref. [49] corrections into the window values or explicitly quantify that the shift is negligible; as written, the claimed compatibility with lattice in the intermediate window is not yet established at the precision implied by the figures.
  2. [Section 3, Fig. 2] No numerical central values or uncertainties are given for the tau-based SD, int, and LD window contributions; the comparison with lattice and e+e- results is purely graphical. Since the central claim is a quantitative compatibility statement, the reader cannot verify it (e.g., by a chi-square or pull test) from this manuscript alone. A table with the window values, statistical and systematic uncertainties, and a definition of the 'two approaches' used to assign the associated error should be added.
  3. [Section 3, Fig. 3] The plot mixes total intermediate-window evaluations obtained with different procedures (various tau experiments, e+e- datasets, and lattice groups), and the text says the difference between two approaches gave the associated error without specifying what those approaches are. This definition is needed to judge whether the error bars and the resulting compatibility with lattice are robust. The abstract's statement that tau-based results agree 'with the full result' is also not supported by any explicitly quoted full tau-based a_mu value in the text.
minor comments (4)
  1. [Fig. 1] Axis labels show 'x 10 10' instead of a formatted x10^10; please fix for readability.
  2. [References] References [35], [39], and [40] do not appear to be cited in the text; either cite them where S_EW and G_EM are introduced or remove them from the bibliography.
  3. [Eq. (4)] The Wilson coefficients epsilon_i are introduced without specifying the operator basis or renormalization scale; a sentence pointing to Ref. [11] would suffice.
  4. [Notation] The notation for the HVP contribution is inconsistent (e.g., 'aHVP,LO mu', 'a_mu^HVP, LO'); unify the notation throughout.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the tau-window values are derived from external tau spectra and compared with external lattice benchmarks; the only caveat is that the figures use the same group's older IB correction rather than their newer one.

full rationale

The central comparison is not circular. The tau-based Euclidean-window values are built from external tau spectral functions (ALEPH, Belle, CLEO, OPAL) through Eqs. (2)-(3), with the isospin-breaking factor RIB(s) computed in Resonance Chiral Theory in Refs. [9,10,34], and are then compared with lattice window values [15,57-60] and with the e+e- window evaluation of Ref. [52]. Nothing in this chain defines the tau windows in terms of the lattice values they are claimed to match: the tau spectra are experimental inputs and RIB(s) is an independent theoretical correction, not a parameter fitted to the lattice or to a_mu. The window definitions themselves come from Ref. [51]. The proceedings relies on the authors' previous work - Ref. [34] for RIB(s) and Ref. [50] for the window integrals - but this self-citation is not circular: Ref. [50] is a separate published calculation whose results the proceedings states were 'corroborated by ref. [61]', and the IB range is cross-checked against other tau-based determinations [41-48]. The relevant flagged limitation is this sentence in Sec. 2: 'with our IB contributions to a_mu^HVP|pi pi in the range [-20.52,-6.96] x 10^-10 at 68% confidence level (we are more precise in the recent [49])', together with 'In ref. [50] we applied these results to the window quantities introduced in Ref. [51]'. This means the figures shown use the older IB treatment of Ref. [34] and do not propagate the improved correction of Ref. [49]; if the s-dependent RIB(s) changes, the claimed tau-lattice compatibility in the intermediate window could shift. That is a precision/update risk, not a circularity, because Ref. [49]'s correction is not fitted to the lattice values being compared. No prediction is forced by construction, so no circular step is identified.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The tau window values are built from external experimental tau spectra and compared with external lattice results; the model content sits in the isospin-breaking and radiative corrections (R_IB(s), G_EM(s)), computed in the authors' RChT framework rather than fitted to the g-2 target. No new particles, forces, or dimensions are introduced. The main uncharged inputs are the reliability of the RChT corrections and of the lattice benchmarks.

free parameters (2)
  • Resonance Chiral Theory couplings and low-energy constants entering G_EM(s)
    The long-distance radiative correction factor G_EM(s) in Eq. (3) is computed in RChT up to O(p^6) in the authors' prior work [34,49]; the parameters are not given in this proceedings but affect every window value through the tau-to-e+e- conversion.
  • Isospin-breaking form-factor ratio |F_V(s)/f_+(s)|^2
    This ratio enters R_IB(s) in Eq. (3) and is model-dependent. Its uncertainty drives the quoted [-20.52,-6.96] x 10^-10 isospin-breaking range for the pi-pi HVP contribution.
assumptions (4)
  • domain assumption CVC and isospin rotation relate tau spectral functions to the isovector e+e- cross section (Eq. 2)
    Central to mapping tau decay spectra to R(s); the corrections are parameterized by R_IB(s).
  • standard math The dispersion relation Eq. (1) with the QED kernel K(s) gives a_mu^HVP,LO from R(s)
    Standard analyticity and unitarity result from Gourdin-de Rafael and Brodsky-de Rafael, accepted in the field.
  • ad hoc to paper G_EM(s) from RChT at O(p^6) with short-distance constraints is accurate over window energies
    The paper relies on its own RChT computation [34,49] for long-distance electromagnetic corrections; the quoted wide IB range indicates sensitivity to this assumption.
  • domain assumption Lattice QCD window results from Refs. [57-60] are correct benchmarks
    The agreement claim is only as strong as the lattice values used for comparison.

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Cite this review

Pith. "Pith review of Hadronic vacuum polarization contribution to the muon g-2 on Euclidean windows from tau data." pith.science (2026). https://pith.science/paper/76WWJRFB

@misc{pith2026241109811,
  author       = {Pith},
  title        = {Pith review of: Hadronic vacuum polarization contribution to the muon g-2 on Euclidean windows from tau data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/76WWJRFB}},
  note         = {Machine review of arXiv:2411.09811}
}
abstract

We computed for the first time the $\tau$ data-driven Euclidean windows for the hadronic vacuum polarization contribution to the muon g-2. We showed that $\tau$-based results agree with the available lattice window evaluations and with the full result. On the intermediate window, where all lattice evaluations are rather precise and agree, $\tau$-based results are compatible with them. This is particularly interesting, given that the disagreement of the $e^+e^-$ data-driven result with the lattice values in this window is the main cause for their discrepancy, affecting the interpretation of the $a_\mu$ measurement in terms of possible new physics.

Figures

Figures reproduced from arXiv: 2411.09811 by the authors.

Figure 1
Figure 1. The 𝜋𝜋(𝛾) contribution to 𝑎 HVP, LO 𝜇 around the 𝜌 peak, obtained from the 𝑒 + 𝑒 − → 𝜋 +𝜋 − (𝛾) cross section (top) and di-pion 𝜏 decays (bottom). the results displayed in [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Window quantities (𝑆𝐷 top, 𝑖𝑛𝑡 medium, and 𝐿𝐷 bottom) for the 2𝜋 contribution below 1.0 GeV to 𝑎 HVP 𝜇 , corresponding to our reference results. The 𝜏 data mean is shown in blue, with the 𝑒 + 𝑒 − result from [52]. References [1] T. Aoyama, et al. “The anomalous magnetic moment of the muon in the Standard Model,” Phys. Rept. 887 (2020), 1-166, doi:10.1016/j.physrep.2020.07.006, [arXiv:2006.04822 [hep-ph]]. 5 [PITH_F… view at source ↗
Figure 3
Figure 3. Comparison of the total intermediate window contribution to 𝑎 HVP, LO 𝜇 according to lattice QCD, 𝑒 + 𝑒 − and 𝜏 data-driven evaluations. The blue band is the weighted average of the lattice results excluding those superseded, RBC/UKQCD 2018 [51] and ETMC 2021 [57] (by refs. [58] and [60], respectively). [2] G. W. Bennett et al. [Muon g-2], “Final Report of the Muon E821 Anoma￾lous Magnetic Moment Measurement at BNL,… view at source ↗

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Pith tools

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