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arxiv: 2605.12205 · v1 · submitted 2026-05-12 · ✦ hep-ph · hep-lat

Recognition: 1 theorem link

· Lean Theorem

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

Noah Allen , Diogo Boito , Maarten Golterman , Kim Maltman , Lucas M. Mansur , Santiago Peris
Authors on Pith no claims yet

Pith reviewed 2026-05-13 04:45 UTC · model grok-4.3

classification ✦ hep-ph hep-lat
keywords hadronic vacuum polarizationtau decayslattice QCDmuon g-2isospin breakingfour-pion modesdispersive integrals
0
0 comments X

The pith

Lattice QCD and tau-decay data agree on hadronic vacuum polarization except for one four-pion mode.

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

The paper compares the isospin-one vector-current hadronic vacuum polarization computed in isospin-symmetric lattice QCD with results from a dispersive representation of inclusive hadronic tau decay data after isospin-breaking corrections. It evaluates the subtracted HVP at Euclidean momenta from 0.5 to 12 GeV squared, the light-quark-connected contribution to the muon magnetic moment, and the short, intermediate, and long-distance RBC/UKQCD window quantities. Dispersive contributions above the tau mass are handled with perturbative QCD, while alternative determinations restrict tau data to two-pion or two-plus-four-pion modes and use e+e- cross sections for the rest. The authors report generally good agreement between the lattice and tau-based determinations. A specific comparison for the four-pion modes, however, reveals significant differences for the 2 pi minus pi plus pi zero channel relative to expectations from Pais relations and e+e- data.

Core claim

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 2π−π+π0 mode.

What carries the argument

The subtracted isospin-one vector-current hadronic vacuum polarization evaluated at Euclidean momenta and in RBC/UKQCD windows, obtained via dispersive integrals from inclusive τ-decay spectra corrected for isospin breaking and compared directly to lattice QCD.

If this is right

  • The two independent approaches can be used interchangeably for HVP contributions to the muon anomalous magnetic moment.
  • Perturbative QCD can be applied reliably above the τ mass for the remaining dispersive tail.
  • Restricting τ data to two- and four-pion modes and supplementing with e+e− cross sections reduces reliance on perturbation theory while preserving overall consistency.
  • The observed discrepancy isolates to the 2π−π+π0 channel and calls for targeted checks on that mode's spectral function.

Where Pith is reading between the lines

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

  • The mismatch in the 2π−π+π0 channel suggests that current isospin-breaking corrections or branching-fraction inputs for that mode may require re-examination.
  • Good global agreement implies that future combined lattice-plus-τ analyses could tighten the HVP uncertainty for muon g-2 without introducing new tensions.
  • The result motivates direct experimental re-measurement of the 2π−π+π0 spectral function in τ decays to test the Pais-relation expectation.

Load-bearing premise

The isospin-breaking corrections applied to the inclusive hadronic τ decay data are accurate and the perturbative QCD evaluation for contributions above the τ mass is reliable without introducing significant bias.

What would settle it

A new lattice simulation or refined τ-decay analysis that produces a total subtracted HVP differing by more than the quoted uncertainties at 1 GeV squared, or a 2π−π+π0 window contribution that matches the e+e− expectation, would falsify the reported agreement.

Figures

Figures reproduced from arXiv: 2605.12205 by Diogo Boito, Kim Maltman, Lucas M. Mansur, Maarten Golterman, Noah Allen, Santiago Peris.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: 2 [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗
read the original 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 $\tau$ 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 $\tau$ mass are evaluated using perturbative QCD. We also consider dispersive determinations using $\tau$ data only for contributions from two-pion, or two-pion and four-pion, modes, and evaluating the remaining contributions using exclusive-mode $e^+e^-\to\mbox{hadrons}$ cross sections up to about 2 GeV, lessening the dependence on perturbation theory. We find generally good agreement between lattice and $\tau$-based results. However, a comparison of $\tau$-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 $2\pi^-\pi^+\pi^0$ mode.

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 / 2 minor

Summary. The manuscript compares the isospin-one vector-current hadronic vacuum polarization (HVP) from isospin-symmetric lattice QCD with dispersive representations based on inclusive hadronic τ-decay data after isospin-breaking corrections. It evaluates the subtracted HVP at Euclidean momenta from 0.5 to 12 GeV², the light-quark-connected HVP contribution to a_μ, and the RBC/UKQCD short-, intermediate-, and long-distance window components. Contributions above the τ mass use perturbative QCD, with additional hybrid determinations restricting τ data to two-pion or two- plus four-pion modes and using e⁺e⁻ cross sections elsewhere. The authors report generally good agreement between lattice and τ-based results but identify significant differences for the 2π⁻π⁺π⁰ four-pion mode in τ-based window contributions relative to Pais-relation expectations from e⁺e⁻ four-pion cross sections.

Significance. If the agreement holds after accounting for the identified discrepancy, the work provides a useful cross-validation of lattice HVP results against τ-decay data, helping constrain uncertainties relevant to the muon anomalous magnetic moment. The explicit flagging of a channel-specific discrepancy supplies a concrete diagnostic for refining isospin-breaking corrections.

major comments (2)
  1. [four-pion modes comparison] The significant differences reported for the 2π⁻π⁺π⁰ mode in the τ-based window-quantity contributions (abstract and four-pion comparison) directly test the internal consistency of the isospin-breaking corrections applied to inclusive τ data. This discrepancy, when set against Pais-relation expectations from e⁺e⁻ data, is load-bearing for the central claim of generally good agreement; the manuscript must quantify its propagation into the total subtracted HVP, a_μ, and window sums.
  2. [hybrid determinations] The hybrid τ + e⁺e⁻ determinations still employ the same IB-corrected τ inputs for the two- and four-pion sectors; the impact of the 2π⁻π⁺π⁰ discrepancy on these hybrid results (and on the claimed reduction in pQCD dependence) is not shown explicitly.
minor comments (2)
  1. The Pais relations invoked for the four-pion modes should be stated or referenced explicitly to aid readers unfamiliar with the isospin decomposition.
  2. Error budgets for the τ-based results, especially the four-pion channels, require fuller tabulation or supplementary material to allow assessment of whether the reported agreement lies within combined uncertainties.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below and will revise the manuscript to incorporate the suggested clarifications and quantifications.

read point-by-point responses

  1. Referee: [four-pion modes comparison] The significant differences reported for the 2π⁻π⁺π⁰ mode in the τ-based window-quantity contributions (abstract and four-pion comparison) directly test the internal consistency of the isospin-breaking corrections applied to inclusive τ data. This discrepancy, when set against Pais-relation expectations from e⁺e⁻ data, is load-bearing for the central claim of generally good agreement; the manuscript must quantify its propagation into the total subtracted HVP, a_μ, and window sums.

    Authors: We agree that an explicit quantification of the impact is necessary to fully assess the robustness of the comparison. In the revised manuscript we will add a new paragraph (or short subsection) that propagates the observed difference in the 2π⁻π⁺π⁰ channel through to the total subtracted HVP at the quoted Euclidean momenta, to the light-quark-connected contribution to a_μ, and to each of the RBC/UKQCD window quantities. This will be done by isolating the four-pion contribution and rescaling it according to the size of the discrepancy relative to the Pais-relation expectation, thereby showing the numerical effect on the overall results. We continue to view the agreement as generally good, with the noted channel-specific difference serving as a diagnostic for future refinements of isospin-breaking corrections. revision: yes

  2. Referee: [hybrid determinations] The hybrid τ + e⁺e⁻ determinations still employ the same IB-corrected τ inputs for the two- and four-pion sectors; the impact of the 2π⁻π⁺π⁰ discrepancy on these hybrid results (and on the claimed reduction in pQCD dependence) is not shown explicitly.

    Authors: We acknowledge that the hybrid determinations retain the IB-corrected τ data for the two- and four-pion modes. In the revision we will add an explicit statement (and, if space permits, a supplementary table) that isolates the effect of the 2π⁻π⁺π⁰ discrepancy on the hybrid results. This will include a brief comparison of the hybrid a_μ and window values obtained with the current τ four-pion input versus a version in which the 2π⁻π⁺π⁰ contribution is replaced by the Pais-relation expectation from e⁺e⁻ data. We will also clarify that, even with this sensitivity, the hybrid construction still reduces the pQCD contribution relative to the pure τ-based determination by replacing the high-mass region with exclusive e⁺e⁻ cross sections up to ~2 GeV. revision: yes

Circularity Check

0 steps flagged

No circularity: direct comparison of independent lattice and tau-decay HVP inputs

full rationale

The paper's central results consist of a numerical comparison between subtracted HVP and window quantities computed from lattice QCD simulations on one side and from a dispersive integral over inclusive hadronic tau-decay spectra (with standard isospin-breaking corrections plus pQCD above the tau mass) on the other. No equation in the derivation chain equates a claimed prediction or result to a parameter fitted from the same data or to a self-referential definition. The additional consistency check against Pais relations and e+e- four-pion cross sections is an external cross-validation, not a reduction of the main agreement to the input data. The analysis therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The comparison rests on standard QCD assumptions and established mathematical tools without new free parameters or postulated entities introduced in the abstract.

axioms (3)
  • domain assumption Isospin symmetry holds for the lattice QCD calculations
    Explicitly stated as isospin-symmetric lattice QCD.
  • standard math Dispersive representations accurately reconstruct the HVP from tau decay data
    Core method employed for the tau-based side.
  • domain assumption Perturbative QCD is reliable for hadronic masses above the tau mass
    Used to evaluate high-mass contributions.

pith-pipeline@v0.9.0 · 5574 in / 1611 out tokens · 64172 ms · 2026-05-13T04:45:05.762552+00:00 · methodology

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Lean theorems connected to this paper

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

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    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.

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.

Reference graph

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