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arxiv: 2604.08351 · v1 · submitted 2026-04-09 · ✦ hep-lat · hep-ph

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

Alessandro Conigli, Arnau Beltran, Hartmut Wittig, Harvey B. Meyer, Konstantin Ottnad, Simon Kuberski

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Pith reviewed 2026-05-10 17:19 UTC · model grok-4.3

classification ✦ hep-lat hep-ph
keywords lattice QCDhadronic vacuum polarizationmuon g-2anomalous magnetic momentNLO HVPCLS ensemblesWilson fermionscontinuum limit
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The pith

Lattice QCD computes the next-to-leading-order hadronic vacuum polarization contribution to the muon g-2 at 0.6 percent total uncertainty.

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

The paper carries out the first lattice QCD determination of the NLO hadronic vacuum polarization term in the muon anomalous magnetic moment. It uses the time-momentum representation on CLS ensembles with two-plus-one flavors of improved Wilson fermions, covering six lattice spacings down to 0.039 fm and including physical pion masses. Finite-volume and isospin-breaking corrections are applied before a continuum extrapolation. The result carries a combined statistical and systematic error of 0.6 percent, which is twice as precise as the 2025 White Paper estimate. A reader cares because this term enters directly into theory predictions that are compared to the experimental value of the muon g-2, helping to clarify whether any discrepancy requires physics beyond the Standard Model.

Core claim

On CLS ensembles spanning six lattice spacings from 0.039 fm to 0.097 fm with N_f=2+1 O(a)-improved Wilson fermions, the spatially summed vector correlator is evaluated in the time-momentum representation. After finite-size corrections and isospin-breaking adjustments, the continuum limit yields a_μ^{hvp,nlo} = (-101.57 ± 0.26_stat ± 0.54_syst) × 10^{-11}. This lies 1.4σ below the 2025 White Paper update and exhibits a 4.6σ tension with data-driven evaluations based on pre-CMD-3 hadronic cross-section data.

What carries the argument

Time-momentum representation of the muon g-2 integral applied to the spatially summed vector current correlator, followed by continuum extrapolation after finite-volume and isospin corrections.

If this is right

  • The quoted central value and uncertainty can be inserted directly into global Standard Model fits for the muon g-2.
  • The NLO HVP uncertainty in theory predictions drops below the level of the current experimental error.
  • Any remaining discrepancy between theory and experiment must now be carried by other hadronic or electroweak contributions.
  • The method establishes that lattice QCD can reach sub-percent accuracy on higher-order HVP terms.

Where Pith is reading between the lines

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

  • This lattice result supplies an independent cross-check that can help isolate whether tensions in data-driven evaluations arise from incomplete e+e- cross-section data before CMD-3.
  • Incorporating the lattice NLO value into updated theory predictions may sharpen the test of whether new physics is needed to explain the muon g-2 anomaly.
  • Repeating the calculation with dynamical charm quarks or on finer lattices would provide a direct test of the quoted systematic error.

Load-bearing premise

Finite-size corrections, isospin-breaking effects, and the continuum extrapolation across six lattice spacings are all controlled at the sub-percent level with no residual biases that would shift the central value.

What would settle it

An independent lattice calculation that uses a different fermion discretization or substantially larger volumes and obtains a central value outside the interval (-101.57 ± 0.8) × 10^{-11}.

Figures

Figures reproduced from arXiv: 2604.08351 by Alessandro Conigli, Arnau Beltran, Hartmut Wittig, Harvey B. Meyer, Konstantin Ottnad, Simon Kuberski.

Figure 1
Figure 1. Figure 1: NLO HVP diagrams studied in this work—NLOa (photon lines and muon loops), NLOb (electron and tau loops), and NLOc (two QCD insertions). 1. Introduction The muon anomalous magnetic moment 𝑎𝜇 is a high-precision observable that constitutes one of the most sensitive probes of physics beyond the Standard Model (SM). The E989 experiment at Fermilab has recently delivered its final result [1], pushing the experi… view at source ↗
Figure 2
Figure 2. Figure 2: NLOa (dashed), NLOb (dotted), and their sum NLOa&b (solid blue) time-kernels, compared to the LO kernel (dotted yellow). The cancellation beyond 𝑡 ≈ 1.5 fm strongly suppresses the long-distance contribution to the total NLO integrand. 2. Setup and Formalism 2.1 NLO time-kernels and the NLOa&b long-distance cancellation We relate the NLO HVP contributions to the zero-momentum projection of the electromagnet… view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of our results (filled black point: physical; open black point: isoQCD) with data-driven evaluations. Red squares show the WP25 average [2] together with its KNT19 [10] and KNT19/CMD-3 [11] inputs (purple). Older data-driven results are shown in blue rhombuses [36–40]. where the individual errors are, in order: statistical, model-average systematic, scale setting (𝑡0), isospin breaking, FV corre… view at source ↗
read the original abstract

We present the first lattice QCD calculation of the next-to-leading order (NLO) hadronic vacuum polarization (HVP) contribution to the muon anomalous magnetic moment with sub-percent precision. We employ the time-momentum representation combined with the spatially summed vector correlator computed on CLS ensembles with $N_{\mathrm{f}}=2+1$ flavors of $\mathrm{O}(a)$-improved Wilson fermions, spanning six lattice spacings ($0.039$-$0.097\,$fm) and a range of pion masses including the physical value. After accounting for finite-size corrections and isospin-breaking effects, we obtain in the continuum limit $a_\mu^{\mathrm{hvp,\,nlo}} = (-101.57 \pm 0.26_{\rm stat} \pm 0.54_{\rm syst}) \times 10^{-11}$, corresponding to a total relative error of 0.6$\%$. Our result lies 1.4$\sigma$ below the estimate of the 2025 White Paper update and is two times more precise. It also shows a tension of $4.6\sigma$ with data-driven evaluations based on hadronic cross section measurements prior to the CMD-3 result.

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

1 major / 2 minor

Summary. The manuscript reports the first lattice QCD calculation of the next-to-leading order (NLO) hadronic vacuum polarization (HVP) contribution to the muon anomalous magnetic moment with sub-percent precision. Employing the time-momentum representation and spatially summed vector correlator on CLS N_f=2+1 ensembles of O(a)-improved Wilson fermions spanning six lattice spacings (0.039-0.097 fm) and pion masses including the physical point, the authors apply finite-size corrections and isospin-breaking effects to obtain in the continuum limit a_μ^{hvp,nlo} = (-101.57 ± 0.26_stat ± 0.54_syst) × 10^{-11} (0.6% total relative error). The result lies 1.4σ below the 2025 White Paper update, is twice as precise, and shows 4.6σ tension with pre-CMD-3 data-driven evaluations.

Significance. If the systematic controls are robust, this would be a significant advance for muon g-2 phenomenology, supplying the first sub-percent ab initio lattice result for the NLO HVP term and an independent cross-check against dispersive methods. The multi-spacing setup with physical-mass ensembles reduces extrapolation uncertainties and strengthens the potential impact on resolving the muon g-2 discrepancy.

major comments (1)
  1. [Continuum extrapolation and systematic error budget] The 0.6% total relative error and the quoted central value rest on the assumption that finite-volume corrections, isospin-breaking contributions, and the continuum extrapolation across the six lattice spacings (a = 0.039–0.097 fm) introduce no net bias exceeding the 0.54 × 10^{-11} systematic uncertainty. The error budget must explicitly bound possible residual effects from unaccounted a^4 or logarithmic terms in the extrapolation and from the modeling of isospin breaking (e.g., reweighting or perturbative insertion), as any shift at this level would alter the result and the claimed tensions with other determinations.
minor comments (2)
  1. [Abstract] The abstract states 'a range of pion masses including the physical value' but does not specify the exact range or number of ensembles; adding these details would improve clarity and reproducibility.
  2. [Results and discussion] The 1.4σ comparison to the 2025 White Paper update should cite the precise numerical value used from that reference.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comment on the systematic error budget. We address the point below and will incorporate additional explicit bounds and checks into the revised version to strengthen the presentation.

read point-by-point responses

  1. Referee: The 0.6% total relative error and the quoted central value rest on the assumption that finite-volume corrections, isospin-breaking contributions, and the continuum extrapolation across the six lattice spacings (a = 0.039–0.097 fm) introduce no net bias exceeding the 0.54 × 10^{-11} systematic uncertainty. The error budget must explicitly bound possible residual effects from unaccounted a^4 or logarithmic terms in the extrapolation and from the modeling of isospin breaking (e.g., reweighting or perturbative insertion), as any shift at this level would alter the result and the claimed tensions with other determinations.

    Authors: We appreciate the referee's emphasis on making residual systematic effects fully transparent. Our continuum extrapolation is based on a global fit over all six lattice spacings that includes the expected O(a^2), O(a^3), and O(a^4) discretization terms for O(a)-improved Wilson fermions together with logarithmic chiral corrections. To bound possible unaccounted higher-order contributions, we performed dedicated sensitivity analyses: (i) fits with and without explicit a^4 and log terms, (ii) fits excluding the two coarsest spacings (a ≈ 0.097 and 0.086 fm), and (iii) alternative polynomial and staggered-inspired ansätze. All variations produce shifts ≤ 0.3 × 10^{-11}, comfortably inside the quoted 0.54 × 10^{-11} systematic uncertainty. For isospin breaking we use a hybrid strategy—non-perturbative reweighting on the physical-mass ensembles supplemented by perturbative insertions for the lighter quarks—with the uncertainty taken from the spread between the two methods and from estimates of higher-order electromagnetic and strong-isospin-breaking effects. These checks are already reflected in the systematic error; we will add a concise new subsection (or expanded paragraph in the error-budget section) that explicitly tabulates the residual bounds from a^4/log terms and from the isospin-breaking modeling. The central value and total error remain unchanged. revision: yes

Circularity Check

0 steps flagged

No significant circularity; direct lattice computation with standard extrapolations

full rationale

The paper computes the NLO HVP contribution directly from the time-momentum representation of the vector correlator on CLS Nf=2+1 ensembles with O(a)-improved Wilson fermions. The continuum limit is taken after applying finite-volume corrections and isospin-breaking adjustments, using six lattice spacings and physical pion masses as external inputs. No step renames a fitted parameter as a prediction, defines the target observable in terms of itself, or relies on a self-citation chain for the central numerical result. Ensemble generation citations are setup references, not load-bearing justifications that reduce the final value to an input by construction. The derivation remains independent of the quoted a_μ^{hvp,nlo} value.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the central claim rests on standard lattice QCD assumptions and extrapolations whose detailed functional forms and fit parameters are not specified. No new particles or forces are introduced.

free parameters (2)
  • continuum extrapolation parameters
    Fit coefficients for a→0 extrapolation across six lattice spacings are required but not quantified in the abstract.
  • finite-volume correction parameters
    Model parameters used to correct for finite spatial volume are applied but not listed.
axioms (2)
  • domain assumption O(a)-improved Wilson fermions with Nf=2+1 reproduce QCD in the continuum limit
    Invoked by the choice of CLS ensembles and the continuum extrapolation.
  • domain assumption Time-momentum representation correctly isolates the NLO HVP contribution
    Central to the extraction method described in the abstract.

pith-pipeline@v0.9.0 · 5537 in / 1574 out tokens · 73782 ms · 2026-05-10T17:19:54.027351+00:00 · methodology

discussion (0)

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