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REVIEW 2 major objections 4 minor 69 references

Lattice QCD plus perturbative QCD yields the hadronic running of the electromagnetic coupling at the Z pole to 0.17 percent, twice as precise as data-driven estimates, while disagreeing with e+e- data by up to 7 sigma at low virtuality.

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2026-07-12 02:57 UTC pith:VC3YXMU3

load-bearing objection Solid factor-of-two improvement on Δα_had(M_Z) from lattice, with a real multi-σ tension at low Q^{2}; IB is estimated but sub-dominant and the error budget is careful. the 2 major comments →

arxiv 2607.03370 v1 pith:VC3YXMU3 submitted 2026-07-03 hep-lat hep-ph

The running of the electroweak gauge couplings from first principles

Alessandro Conigli , Dalibor Djukanovic , Georg von Hippel , Simon Kuberski , Harvey B. Meyer , Kohtaroh Miura , Konstantin Ottnad , Andreas Risch
show 1 more author
Hartmut Wittig
This is my paper
classification hep-lat hep-ph
keywords hadronic vacuum polarizationrunning of alphaweak mixing anglelattice QCDEuclidean split techniqueelectroweak precisionTime-Momentum Representation
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The electromagnetic coupling and the weak mixing angle run with energy because of hadronic vacuum polarization. That running is a leading uncertainty in Standard Model predictions used at colliders and in low-energy parity-violation experiments. This paper computes the hadronic piece from first principles: lattice QCD supplies the non-perturbative contribution for space-like virtualities up to about 12 GeV squared at permille precision, then the Euclidean split technique hands the evolution over to continuum perturbation theory. The result at the Z pole is more than twice as precise as recent phenomenological determinations that rely on e+e- cross sections. At the same time the lattice values lie systematically above those data-driven estimates, reaching a 7-sigma tension near 1 GeV squared. The authors also map how further lattice and perturbative improvements, together with a higher matching scale, can reach the precision targets of next-generation electroweak facilities.

Core claim

A first-principles lattice calculation of the subtracted hadronic vacuum polarization, combined with perturbative QCD via the Euclidean split technique, produces Δα_had^(5)(M_Z²) = 0.027821(34)_lat(35)_pQCD with a total relative uncertainty of 0.17 percent—more than twice as precise as recent dispersive evaluations—while the same lattice results for the running at low space-like virtualities deviate by as much as 7σ from estimates based on e+e- data.

What carries the argument

The Time-Momentum Representation that expresses the subtracted HVP as an integral of the Euclidean vector correlator against a known kernel, together with a telescopic window decomposition that isolates high-, mid- and low-virtuality contributions so that each can be extrapolated and noise-reduced separately before matching to the Adler function in pQCD.

Load-bearing premise

Isospin-breaking and QED corrections are small enough that a hybrid phenomenological-plus-lattice estimate at the matching scale is adequate; a full lattice calculation with dynamical photons and non-degenerate light quarks is left for future work.

What would settle it

An independent high-precision lattice determination of Δα_had(-Q²) at Q² ≃ 1 GeV² that either confirms or removes the reported multi-sigma excess over the existing e+e- based evaluations, or a complete dynamical-QED lattice calculation that shows the isospin-breaking correction is several times larger than the present hybrid estimate.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The Z-pole value of α can now be known to roughly 1.7 permille from hadronic effects alone, tightening the dominant theory error that enters global electroweak fits.
  • The lattice–data tension at low virtuality supplies an independent cross-check on the R-ratio that currently dominates data-driven evaluations of both Δα_had and a_μ^HVP.
  • A continuous Padé representation of the space-like HVP is provided for phenomenological use at any intermediate scale.
  • Moderate further lattice improvements plus a matching scale near 20 GeV² are projected to reach the 3×10^{-5} precision target required by FCC-ee measurements of α(M_Z).
  • The same correlators yield a first-principles determination of the hadronic running of the weak mixing angle that is free of model-dependent flavour separation.

Where Pith is reading between the lines

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

  • If the low-virtuality lattice–data discrepancy survives, the long-standing tension between lattice and dispersive evaluations of a_μ^HVP is unlikely to be resolved by simply re-weighting existing e+e- channels.
  • Because the same HVP enters both electroweak running and the muon g-2, a confirmed lattice excess would force a simultaneous re-assessment of both observables rather than treating them as independent.
  • The improvement roadmap shows that lattice progress alone is insufficient at fixed matching scale; raising the continuum matching point is the more efficient lever, which in turn prioritizes better control of short-distance discretization effects.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The manuscript reports a lattice-QCD determination of the hadronic running of the electromagnetic coupling and the weak mixing angle. Using 27 CLS ensembles, a telescopic window decomposition of the Euclidean HVP, hybrid finite-volume corrections, low-mode averaging and spectral reconstruction, and AIC-weighted chiral-continuum extrapolations, the authors obtain permille-level results for space-like virtualities Q^{2} ≲ 12 GeV^{2}. At Q^{2} ≃ 1 GeV^{2} their Δα_had^(5)(-Q^{2}) lies up to 7σ above standard e^{+}e^{-} dispersive evaluations. Matching lattice data to the Adler function at Q_{0}^{2} = 9 GeV^{2} yields Δα_had^(5)(M_Z^{2}) = 0.027821(34)_lat(35)_pQCD (relative uncertainty 0.17 %), more than twice as precise as recent phenomenological averages. A parametric error model is used to map the lattice and pQCD improvements needed for FCC-ee targets.

Significance. If the continuum-extrapolated lattice HVP and the Euclidean-split matching are reliable, the work supplies a first-principles input for electroweak precision tests that is competitive with, and more precise than, data-driven determinations. The reported 7σ tension at low virtuality is a falsifiable claim that can be tested by independent lattice groups and by forthcoming e^{+}e^{-} data. The quantitative improvement scenarios (Fig. 3) give a concrete roadmap for reaching the 3 × 10^{-5} precision required by FCC-ee. Strengths include the multi-ensemble continuum limit, separate fits per flavor and window, two independent pQCD codes, and an explicit (if hybrid) isospin-breaking estimate that remains sub-dominant.

major comments (2)
  1. Eq. (9) and the accompanying text: the isospin-breaking correction at the matching scale is obtained from a hybrid of connected lattice estimates and a phenomenological model rather than a full dynamical QED + non-degenerate calculation. While the quoted uncertainty is treated as conservative and sub-dominant, the central claim of 0.17 % total precision at M_Z rests on this estimate remaining adequate. A short quantitative sensitivity study (e.g., shifting the IB central value by its full uncertainty and re-matching) would make the robustness of Eq. (11) more transparent.
  2. Matching procedure (text around Eq. (11)): the final Z-pole result is quoted for a single matching scale Q_{0}^{2} = 9 GeV^{2}. Although the parametric model of Fig. 3 explores other scales, the manuscript does not show the explicit variation of the matched central value and its lattice/pQCD error split when Q_{0}^{2} is changed within the range already accessible on the lattice (e.g., 5–12 GeV^{2}). Such a table would confirm that residual matching-scale dependence lies inside the quoted (34)_lat(35)_pQCD errors.
minor comments (4)
  1. Table I caption: the final bracketed uncertainty is described as the quadrature sum; it would help the reader if the three individual components (stat, model-average, scale-setting) were also listed in the caption for quick reference.
  2. Fig. 1: the orange vertical band is said to include the bottom-quark contribution, yet the figure legend does not state whether isospin-breaking is included or omitted; a one-line clarification would avoid ambiguity when comparing to the dispersive points.
  3. Eq. (12): the asymptotic value of the octet-singlet mixing is given without an explicit reference to the continuum-extrapolated lattice data that underlie it; a pointer to the companion paper or Supplemental Material would improve traceability.
  4. References: the companion paper is cited as arXiv:2511.01623; once it is published, the journal citation should be updated for permanence.

Circularity Check

0 steps flagged

No significant circularity: lattice HVP from Euclidean correlators is matched to independent pQCD Adler function; self-citations supply methods/ensembles, not the target observable.

full rationale

The central result is a first-principles lattice evaluation of the subtracted HVP via the TMR integral of Euclidean vector correlators on CLS ensembles, followed by continuum/chiral extrapolation, window decomposition, and hybrid finite-volume corrections. The Z-pole value is obtained by the Euclidean split technique: lattice data at a space-like matching scale (Q^{2}=9 GeV^{2}) are evolved with the perturbative Adler function (AdlerPy / updated pQCDAdler), whose inputs (α_s, m_c, m_b) are external and whose truncation uncertainty is propagated separately. Isospin-breaking is estimated rather than fully computed, but the estimate is treated as a sub-dominant systematic, not as a fitted input that forces the central claim. Self-citations (Mainz 2022, companion paper, CLS analyses) document ensembles, noise-reduction techniques and prior methodology; they do not define or force the numerical value of Δα_had^(5)(M_Z^{2}) or the reported 7σ tension at low Q^{2}. The Padé representation is a post-hoc fit to the lattice points for phenomenological convenience, not a prediction of those points. External benchmarks (dispersive R-ratio evaluations, global EW fits) are independent and are used only for comparison. Consequently the derivation chain does not reduce by construction to its own inputs.

Axiom & Free-Parameter Ledger

4 free parameters · 6 axioms · 0 invented entities

The central claim rests on standard lattice QCD plus a controlled pQCD matching. Free parameters are the usual fit coefficients in chiral-continuum extrapolations, the chosen matching scale, and Padé coefficients used only for continuous representation. Axioms are domain-standard (continuum limit of lattice QCD, validity of the Euclidean split, reliability of the Adler function above a few GeV). No new particles or forces are introduced. The largest non-first-principles pieces are the estimated (not fully computed) isospin-breaking corrections and the external b-quark and pQCD input parameters.

free parameters (4)
  • Matching scale Q0² = 9 GeV²
    Fixed at 9 GeV² for the final Z-pole result; choice balances lattice discretization against pQCD truncation and is varied only in the prospective error model, not in the central value.
  • Chiral-continuum fit coefficients (per channel and window)
    Quark-mass and a² (and higher) coefficients in the global fits that produce the continuum physical-point HVP; varied across ansätze and combined by AIC weighting.
  • Padé approximant coefficients a_j, b_k for Π̄(γ,γ) and Π̄(Z,γ)
    Fitted to the lattice HVP points to provide a continuous analytic representation; used for phenomenology, not for the primary Z-pole number which uses the Euclidean split at fixed Q0².
  • Isospin-breaking correction central value and uncertainty = (0.75±1.28)×10^{-5}
    Hybrid lattice-plus-phenomenological estimate Δ_IB Δα_had = (0.75±1.28)×10^{-5} at Q²=9 GeV²; uncertainty taken as the size of the strong IB piece rather than a full dynamical calculation.
axioms (6)
  • domain assumption Continuum, infinite-volume lattice QCD with N_f=2+1(+c) correctly reproduces the Euclidean vector correlators of QCD once chiral and continuum extrapolations are controlled.
    Foundational premise of the entire calculation; invoked throughout the Methodology section and the companion paper.
  • domain assumption The Euclidean split technique correctly joins non-perturbative HVP at space-like Q0² to the perturbative Adler function for the run to M_Z².
    Used to obtain Eq. (11); standard in the literature (Eidelman et al., Jegerlehner) but still an assumption about the matching region.
  • domain assumption Perturbative QCD for the Adler function is reliable for Q² ≳ 9 GeV² once α_s, m_c, m_b and truncation are assigned the stated uncertainties.
    Controls the (35)_pQCD error; cross-checked with AdlerPy and updated pQCDAdler.
  • ad hoc to paper Isospin-breaking and QED corrections to the HVP at the matching scale are at the level of the quoted hybrid estimate and do not shift the central value outside the total error.
    Paper estimates rather than fully computes IB/QED; future work is flagged. Enters the final Z-pole number and the comparison figures.
  • domain assumption Bottom-quark HVP contribution can be taken from HPQCD lowest moments without spoiling the quoted precision.
    Stated explicitly as a small missing piece estimated from Ref. [53].
  • domain assumption sin²θ_W(0)=0.23857(5) from the PDG average is an adequate external input for converting Π̄(Z,γ) into (Δ sin²θ_W)_had.
    Adopted from Ref. [23] in the Methodology section.

pith-pipeline@v1.1.0-grok45 · 18037 in / 3986 out tokens · 37743 ms · 2026-07-12T02:57:18.206023+00:00 · methodology

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read the original abstract

We present a high-precision calculation of the hadronic running of electroweak gauge couplings from first principles. Employing lattice QCD in the low-energy regime, we achieve permille precision for virtualities $Q^2 \lesssim 12\;\mathrm{GeV}^2$. At $Q^2 \simeq 1\;\mathrm{GeV}^2$, our determination deviates by up to $7\sigma$ from estimates based on $e^+e^-$ measurements. Combining lattice QCD with perturbative QCD via the Euclidean split technique, we obtain for the electromagnetic coupling $\Delta\alpha^{(5)}_{\mathrm{had}}(M_Z^2) = 0.027821(34)_{\mathrm{lat}}(35)_{\mathrm{pQCD}}$, which is more than twice as precise as recent phenomenological determinations. We assess improvement scenarios by which the precision target for next-generation electroweak measurements could be reached.

Figures

Figures reproduced from arXiv: 2607.03370 by Alessandro Conigli, Andreas Risch, Dalibor Djukanovic, Georg von Hippel, Hartmut Wittig, Harvey B. Meyer, Kohtaroh Miura, Konstantin Ottnad, Simon Kuberski.

Figure 1
Figure 1. Figure 1: ) is less pronounced at the Z−pole, due to the ad￾ditional uncertainty from perturbative running. We also compare to global EW fits (blue open lower triangles), including results from the Gfitter group [61], from [62] (obtained using the HEPfit code [63]), from [64, 65] (ob￾tained from the Gfitter library), and from [66]. These typically favor lower values with reduced precision. The largest deviation, wit… view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Summary of results for ∆ [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Projected total uncertainty [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

discussion (0)

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