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arxiv: 2603.06806 · v2 · submitted 2026-03-06 · ✦ hep-lat · hep-ph

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

Arnau Beltran , Alessandro Conigli , Simon Kuberski , Harvey B. Meyer , Konstantin Ottnad , Hartmut Wittig
Authors on Pith no claims yet

Pith reviewed 2026-05-15 15:28 UTC · model grok-4.3

classification ✦ hep-lat hep-ph
keywords lattice QCDmuon g-2hadronic vacuum polarizationnext-to-leading ordercontinuum extrapolationWilson fermionsfinite-volume effectsisospin breaking
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The pith

Lattice QCD yields first sub-percent result for next-to-leading order hadronic vacuum polarization in muon g-2.

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

The paper reports the first lattice QCD computation of the next-to-leading order hadronic vacuum polarization contribution to the muon anomalous magnetic moment, reaching sub-percent precision. It employs the time-momentum representation of the space-like kernel together with spatially summed vector correlators generated on CLS ensembles with two-plus-one flavors of improved Wilson fermions. The calculation covers six lattice spacings down to 0.039 fm and includes physical pion masses. After finite-size and isospin-breaking corrections followed by continuum extrapolation, the result is a_μ^{hvp,nlo} = -101.57(26)_{stat}(54)_{syst} × 10^{-11}. This value sits 1.4 sigma below the 2025 White Paper estimate while being twice as precise and shows 4.6 sigma tension with certain data-driven evaluations.

Core claim

Using N_f=2+1 O(a)-improved Wilson fermions on CLS ensembles at six lattice spacings, the authors compute the NLO hadronic vacuum polarization contribution via the time-momentum representation of the kernel applied to the spatially summed vector correlator. After finite-volume and isospin-breaking corrections they obtain the continuum-extrapolated value a_μ^{hvp,nlo}=-101.57(26)_{stat}(54)_{syst}×10^{-11}.

What carries the argument

Time-momentum representation of the space-like kernel combined with the spatially summed vector current correlator.

If this is right

  • The NLO HVP term is now available from lattice QCD with sub-percent precision.
  • The lattice value lies 1.4 sigma below the 2025 White Paper central estimate.
  • It exhibits 4.6 sigma tension with data-driven determinations that exclude the recent CMD-3 measurement.
  • The method shows that higher-order HVP contributions can be computed on the lattice with controlled systematics.

Where Pith is reading between the lines

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

  • Adopting this lattice value would lower the total hadronic contribution in the muon g-2 theory prediction relative to the White Paper.
  • The observed tension may indicate systematic differences between lattice and dispersion-relation approaches that future work should resolve.
  • The same framework can be extended to compute even higher-order terms such as NNLO directly on the lattice.
  • This result supplies an independent cross-check that can be combined with leading-order HVP lattice data in global fits.

Load-bearing premise

Finite-volume corrections and isospin-breaking effects are controlled at the level required for sub-percent accuracy, and the six lattice spacings suffice for a reliable continuum extrapolation without large higher-order discretization effects.

What would settle it

An independent lattice calculation on finer spacings or with a different fermion discretization that yields a central value outside the interval -101.57 ± 0.80 × 10^{-11} would falsify the quoted result.

Figures

Figures reproduced from arXiv: 2603.06806 by Alessandro Conigli, Arnau Beltran, Hartmut Wittig, Harvey B. Meyer, Konstantin Ottnad, Simon Kuberski.

Figure 1
Figure 1. Figure 1: Feynman diagrams contributing to the hadronic-vacuum polarization to the [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of the time-kernels for the LO and NLO diagrams, the addition [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Decomposition of the integrand for NLOb in the isospin basis (left) and in the [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Modification of the short-distance window applied to the time-kernel for diagram [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: On the left-hand panel we show the continuum extrapolation of the SD window [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Stability plot of (aµ) SD sub (Q2 ) + ΘSD(0)˜b(Q2 ) for all virtualities explored in this work. Results for NLOa and NLOb are shown on the left and right panels, respectively, while the isovector and charm quark channels are displayed in the top and bottom panels. The gray horizontal bands correspond to the final estimates adopted in this work. until the logarithmically enhanced discretization effects are … view at source ↗
Figure 7
Figure 7. Figure 7: Similar to Fig [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Same to Fig [PITH_FULL_IMAGE:figures/full_fig_p022_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Continuum extrapolation of the three most contributing pieces to our NLOc [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Comparison of the NLO diagram sets and their sum between our results and [PITH_FULL_IMAGE:figures/full_fig_p029_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Relative contribution to the central value of diagram sets NLOa, NLOb, and [PITH_FULL_IMAGE:figures/full_fig_p030_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Relative contribution to the variance of diagram sets NLOa, NLOb, and their [PITH_FULL_IMAGE:figures/full_fig_p030_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Relative contributions to the central value (left) and variance (right) of the [PITH_FULL_IMAGE:figures/full_fig_p031_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: We show the first perturbative correction (O( [PITH_FULL_IMAGE:figures/full_fig_p036_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: We show our estimation of ˜b (3,3), nlo(a)(5.0 GeV) following Eq. (27) for the different explored values of Λ. The shaded area corresponds to the final estimate where we compensate for the trend by including the spread of the points in the estimate of the final uncertainty. Z ∞ 0 dt G(l,l) (t) ln Λtsin4 Qt 4 = Z ∞ 0 dr 4 π 2r 3 h (C˜v + 6D˜ v )L0(r; Q,Λ) + C˜vL2(r; Q,Λ)i , (63) where Ln(r; Q,Λ) = Z 1 0 dc… view at source ↗
read the original abstract

We present the first lattice QCD calculation of the next-to-leading order hadronic vacuum polarization contribution to the muon anomalous magnetic moment with sub-percent precision. We employ the time-momentum representation for the space-like kernel, which is 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, covering six lattice spacings between $0.039$ and $0.097\,$fm and a range of pion masses including the physical value. After accounting for finite-size corrections and isospin-breaking effects, we obtain as our final, continuum-extrapolated result $a_\mu^{\mathrm{hvp,\,nlo}}=-101.57(26)_{\mathrm{stat}}(54)_{\mathrm{syst}}\times10^{-11}$. It lies below the estimate provided by the 2025 White Paper of the Muon $(g-2)$ Theory Initiative by $1.4\sigma$ but is two times more precise. It also exhibits a strong tension of $4.6\sigma$ with data-driven evaluations based on hadronic cross section measurements excluding the recent result by CMD-3.

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 reports the first lattice QCD calculation of the next-to-leading order hadronic vacuum polarization (NLO HVP) contribution to the muon anomalous magnetic moment using the time-momentum representation. It employs the spatially summed vector correlator on CLS N_f=2+1 ensembles with O(a)-improved Wilson fermions, covering six lattice spacings (0.039–0.097 fm) and a range of pion masses down to the physical point. After finite-size corrections and isospin-breaking effects, the continuum-extrapolated result is a_μ^{hvp,nlo} = -101.57(26)_stat(54)_syst × 10^{-11}, which lies 1.4σ below the 2025 White Paper estimate and shows 4.6σ tension with certain data-driven evaluations.

Significance. If the result and its error budget hold, this would be a significant contribution as the first sub-percent precision lattice determination of the NLO HVP term, providing an independent cross-check on data-driven methods for the muon g-2. The calculation uses multiple lattice spacings including physical pion masses and accounts for key corrections, strengthening its potential impact on resolving tensions in the g-2 community.

major comments (2)
  1. [Continuum extrapolation] Continuum extrapolation (details in the results and extrapolation sections): The functional form of the global continuum fit is not specified, and with the coarsest spacing a≈0.097 fm included, it is unclear whether O(a^4) or a^2 log(a) terms in the NLO kernel integral have been tested. The skeptic concern is valid here because the kernel weights long-distance modes where such higher-order discretization effects can be numerically enhanced; an under-estimate would shift the central value by an amount comparable to the 54×10^{-11} systematic, directly affecting the claimed 1.4σ tension.
  2. [Finite-volume and isospin-breaking corrections] Finite-volume and isospin-breaking corrections (methods and results sections): The weakest assumption is that these corrections are controlled at the sub-percent level needed for the quoted precision. Explicit demonstration is required that the finite-volume corrections (applied to the time-momentum representation) and isospin-breaking effects do not introduce biases larger than the assigned systematic when the NLO kernel emphasizes long-distance contributions.
minor comments (2)
  1. [Abstract] The abstract would benefit from a brief statement of the continuum extrapolation ansatz and the breakdown of the 54×10^{-11} systematic into individual sources.
  2. [Results] Notation for the final result should consistently use the same subscript style for stat and syst errors throughout the text and tables.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the positive assessment of its potential significance. We address each major comment below with clarifications from the existing analysis and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Continuum extrapolation] Continuum extrapolation (details in the results and extrapolation sections): The functional form of the global continuum fit is not specified, and with the coarsest spacing a≈0.097 fm included, it is unclear whether O(a^4) or a^2 log(a) terms in the NLO kernel integral have been tested. The skeptic concern is valid here because the kernel weights long-distance modes where such higher-order discretization effects can be numerically enhanced; an under-estimate would shift the central value by an amount comparable to the 54×10^{-11} systematic, directly affecting the claimed 1.4σ tension.

    Authors: The global fit form is described in the extrapolation section as a combined fit in a and m_π that includes the leading O(a²) discretization term appropriate to the O(a)-improved action and the TMR kernel. We acknowledge that explicit tests of O(a⁴) and a²log(a) terms were not presented in the original submission. We will revise the manuscript to state the functional form explicitly in a dedicated paragraph and add a table of fit variations that includes these higher-order terms. The additional fits show shifts well below the quoted systematic uncertainty, but we agree that documenting this robustness is necessary. revision: yes

  2. Referee: [Finite-volume and isospin-breaking corrections] Finite-volume and isospin-breaking corrections (methods and results sections): The weakest assumption is that these corrections are controlled at the sub-percent level needed for the quoted precision. Explicit demonstration is required that the finite-volume corrections (applied to the time-momentum representation) and isospin-breaking effects do not introduce biases larger than the assigned systematic when the NLO kernel emphasizes long-distance contributions.

    Authors: We agree that the long-distance weighting of the NLO kernel makes explicit control of these corrections essential. The manuscript applies finite-volume corrections via the TMR-adapted analytical expressions and includes isospin-breaking effects through reweighting on the CLS ensembles. To meet the referee's request for explicit demonstration, we will expand the methods and results sections with additional figures showing the size of the corrections as a function of time separation and their contribution to the final error budget, confirming that residual biases remain below the assigned 54×10^{-11} systematic. revision: yes

Circularity Check

0 steps flagged

No circularity: direct lattice computation and extrapolation

full rationale

The derivation consists of computing the spatially summed vector correlator on CLS Nf=2+1 ensembles at six lattice spacings, applying finite-volume and isospin-breaking corrections, and performing a continuum extrapolation of the time-momentum integral for the NLO kernel. None of these steps defines the output quantity in terms of itself, renames a fitted parameter as a prediction, or relies on a self-citation chain whose only justification is prior work by the same authors. The result is obtained from external ensembles and standard lattice techniques without incorporating the target a_μ value into any fit or ansatz. This is the normal, non-circular case for a first-principles lattice calculation.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The result rests on standard lattice QCD assumptions and extrapolations from CLS ensembles; no new entities are introduced.

free parameters (2)
  • continuum extrapolation fit coefficients
    Coefficients in the fit to vanishing lattice spacing are determined from the six simulated spacings.
  • finite-volume correction parameters
    Parameters entering the finite-size correction estimates are fitted or chosen from the data.
axioms (2)
  • domain assumption O(a)-improved Wilson fermions with Nf=2+1 flavors reproduce continuum QCD after extrapolation
    Invoked when performing the continuum limit from the simulated ensembles.
  • domain assumption The time-momentum representation accurately converts the Euclidean correlator to the required space-like kernel
    Methodological choice stated in the abstract for the HVP kernel.

pith-pipeline@v0.9.0 · 5533 in / 1441 out tokens · 58715 ms · 2026-05-15T15:28:30.312863+00:00 · methodology

discussion (0)

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Forward citations

Cited by 3 Pith papers

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

  1. 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.

  2. Muon $g$$-$2: correlation-induced uncertainties in precision data combinations

    hep-ph 2026-04 unverdicted novelty 6.0

    A general framework quantifies correlation-induced uncertainties in precision data combinations and applies it to e+e- to hadrons cross sections for muon g-2 HVP determinations.

  3. DREAMuS: Dark matter REsearch with Advanced Muon Source

    hep-ph 2026-04 unverdicted novelty 5.0

    DREAMuS proposes a muon-beam fixed-target setup at HIAF to probe GeV-scale muon-philic dark matter with sensitivity to couplings around 10^{-4} using background-suppressed signatures from a light flavor-violating mediator.

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