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Complete dispersive evaluation of the hadronic light-by-light contribution to muon $g-2$

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Pith's one-line read A complete dispersive evaluation determines the hadronic light-by-light contribution to the muon's magnetic moment as $a_\mu^\text{HLbL} = 101.9(7.9)\times 10^{-11}$.

desk verdict First complete dispersive HLbL evaluation with a realistic error budget; the U(3) axial-vector assumption is the main thing to probe, but it is handled as a transparent systematic. read the letter →

arxiv 2412.00190 v2 pith:AOYSNPLJ submitted 2024-11-29 hep-ph hep-exhep-latnucl-th

Martin Hoferichter , Peter Stoffer , Maximilian Zillinger This is my paper · ORCID
classification hep-phhep-exhep-latnucl-th
keywords muong-2hadroniclight-by-lightdispersiveapproachtransitionformfactorsshort-distanceconstraintsoperatorproductexpansionaxial-vectormesons
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

The paper presents the first complete dispersive evaluation of the hadronic light-by-light (HLbL) contribution to the muon's anomalous magnetic moment. The central result is $a_\mu^\text{HLbL} = 101.9(7.9)\times 10^{-11}$, which improves on the previous data-driven estimate by more than a factor of two and reaches the precision required for the final result of the muon g-2 experiment. The evaluation reconstructs the HLbL tensor from its discontinuities, expressing it in terms of hadronic matrix elements measured in experiment, and matches the low-energy hadronic description to short-distance constraints from perturbative QCD and the operator product expansion. The piece of the calculation that remains most assumption-dependent is the axial-vector meson contribution, fixed through U(3) symmetry with a 30% assigned uncertainty.

What carries the argument

The central mechanism is a Bardeen–Tung–Tarrach decomposition of the four-photon HLbL tensor, which reduces the amplitude to 54 scalar coefficient functions (12 independent after the equations of motion). The master formula for $a_\mu$ integrates these functions against known kernels over the three photon virtualities. The functions are reconstructed dispersively from their discontinuities, which are saturated by a finite set of hadronic intermediate states: pseudoscalar poles ($\pi^0,\eta,\eta'$) with transition form factors from data, two-meson boxes and rescattering, and narrow-resonance approximations for axial-vector, scalar, and tensor mesons. In the deep-Euclidean region the same functions are matched to the perturbative quark loop with $\alpha_s$ corrections, and in the asymmetric region to the operator product expansion controlled by the vector-vector-axial-vector correlator $w_{L,T}$. A newly introduced effective-pole parametrization estimates the error from intermediate states that lie between the hadronic and asymptotic descriptions.

What would settle it

New data on the $f_1$ or $f_1'$ transition form factor that deviate from the U(3)-symmetric parameterization by more than the assigned 30% would move the axial-vector contribution beyond the quoted systematic error.

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Extended reading notes

Core claim

The authors establish a data-driven, dispersive value for the hadronic light-by-light contribution to the muon's $g-2$, $a_\mu^\text{HLbL} = 101.9(7.9)\times 10^{-11}$, by reconstructing the HLbL tensor from its unitarity cuts and matching the sum of exclusive hadronic states to short-distance constraints. This is the first such evaluation that includes axial-vector resonances ($a_1$, $f_1$, $f_1'$), scalar and tensor resonances, the full vector-vector-axial-vector correlator for the OPE region, and a dedicated estimate of the matching uncertainty via an effective-pole ansatz. The result agrees with the 2020 phenomenological consensus, $92(19)\times 10^{-11}$, but reduces the uncertainty by more than half, meeting the precision target set for the final result of the muon g-2 experiment.

Load-bearing premise

The calculation assumes that the axial-vector meson transition form factors follow U(3) symmetry with no more than 30% deviation; if the true symmetry breaking is larger, the central value shifts outside the quoted uncertainty.

Editorial extensions

If this is right

  • The uncertainty of the HLbL contribution drops from $19\times 10^{-11}$ to $7.9\times 10^{-11}$, so the hadronic light-by-light piece no longer dominates the theory error for the muon $g-2$ prediction.
  • The quoted precision meets the requirement set by the final result of the muon g-2 experiment, making the comparison between the Standard-Model prediction and experiment sensitive mainly to the hadronic vacuum polarization contribution.
  • The new value agrees with the 2020 phenomenological estimate but with an uncertainty reduced by more than a factor of two, strengthening the data-driven benchmark against which lattice QCD calculations are compared.
  • The axial-vector resonance contribution is now the largest subleading term with an assumption-based error, so new measurements of the $f_1$ and $f_1'$ transition form factors directly test the main systematic uncertainty.

Reading between the lines

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

  • A natural next step would be a direct dispersive calculation of the $a_1$ transition form factor from $\tau$-decay data, which could replace the U(3)-symmetry assumption with an experimental determination.
  • If the effective-pole matching error is as small as quoted, then the dominant remaining model dependence in the subleading contributions sits in the tensor-meson treatment; a triangle-kinematics dispersive calculation of the $f_2$ contribution would provide a sharp cross-check.
  • The small tension with two of the lattice QCD evaluations, if it persists, may point to a systematic effect in one of the methods; the present result offers a data-driven reference point for locating the difference.
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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 / 5 minor

Summary. The paper presents a data-driven dispersive evaluation of the hadronic light-by-light (HLbL) contribution to the muon anomalous magnetic moment. The HLbL tensor is reconstructed from its discontinuities using four-point dispersion relations, with contributions from pseudoscalar poles, meson boxes, rescattering, axial-vector and tensor resonances, heavy scalars, and a matching to pQCD and OPE short-distance constraints. The final result is a_mu^HLbL = 101.9(7.9) x 10^-11 (Eq. (8)), which the authors state meets the precision requirements of the final Fermilab result. The paper relies on a companion paper (Ref. [82]) for many technical details.

Significance. If correct, this would be the most precise data-driven determination of a_mu^HLbL to date, reducing the uncertainty of the 2020 White Paper evaluation by more than a factor of two and providing a sharp benchmark for comparisons with lattice QCD. The paper is notable for its detailed error budget, separating experimental, matching, systematic, and effective-pole uncertainties, and for demonstrating stability of the integral under variation of the matching scale Q0 and the OPE parameter r (Fig. 5). The result is also falsifiable in the sense that future BESIII and Belle II measurements of axial-vector transition form factors, as well as improved lattice calculations, will directly test the assumptions that underlie the axial-vector and tensor contributions.

major comments (3)
  1. [Exclusive hadronic states / Table II / Eq. (7)] The axial-vector sector is the most load-bearing input for the claimed precision, yet the a1(1260) transition form factors are not extracted from direct data. The Letter fixes the f1 and f1' TFF normalizations and mixing angle from e+e- data and then defines the complete set of axial-vector TFFs using U(3) symmetry, assigning a global 30% uncertainty 'to account for possible symmetry violations.' This 30% is not derived from a spread of models, fits, or dispersion relations; the supporting comparison with the dispersive VVA calculation of Ref. [81] constrains only the singly-virtual a1 form factor, not the doubly-virtual TFFs that dominate the HLbL integral. Since the axial-vector contribution is 12.2(2.3) x 10^-11 (Table II), the largest non-pseudoscalar subleading piece, a U(3) violation of 40-50% in the a1 TFF normalizations or virtuality dependence would shift the result by more than the quoted 7.9 x 10^-11 total uncertainty and would invalidate the precision claim of Eq. (8). Please calibrate the 30% by explicit sensitivity studies (e.g., 50% variations or an alternative model spread) or soften the claim accordingly.
  2. [Matching to short-distance constraints / Fig. 4 / Table II] The effective-pole model (P(2200), A(1700)) is introduced to estimate the impact of missing higher intermediate states in the matching between hadronic and pQCD/OPE regions. The Letter assigns an effective-pole uncertainty larger than the entire effective-pole contribution (3.9 x 10^-11 vs. contributions of 2.0 and 1.2 x 10^-11 in Table II), which is conservative in that specific sense. However, the effective-pole amplitude is an ad-hoc construct with masses fixed at 2.2 GeV and 1.7 GeV and a TFF scale varied only over the same range as Q0; no explicit functional form is given in the Letter. A different pole structure or asymptotic behavior could shift the central value beyond the quoted eff uncertainty. Please provide the explicit effective-pole parameterization in the Letter or in a form that can be checked against Ref. [82], and demonstrate robustness to the assumed pole masses and the number of poles.
  3. [Tensor contributions / Table II / Eq. (7)] The tensor contribution is estimated using a quark-model-inspired TFF with a single nonvanishing form factor, with the scale set to M_rho. The Letter adds a 100% uncertainty on the total tensor contribution 'to protect against the cancellation observed between a_mu[Pi1,2] and a_mu[Pi3-12],' but this protects against the sign cancellation rather than against individual normalization or shape errors. The two tensor components in Table II are -2.6(3) and -5.1(7) x 10^-11; a 50% error in the f2(1270) TFF alone would shift the sum by roughly 3.9 x 10^-11, exceeding the quoted total sys error. Please provide a more robust estimate of the tensor uncertainty, for example by varying the TFF scale and form independently or by including a dispersive D-wave treatment as the paper itself identifies as future work.
minor comments (5)
  1. [Title and abstract] The phrase 'complete dispersive evaluation' overstates the treatment of axial-vector and tensor states, which are evaluated in narrow-resonance and quark-model approximations rather than fully dispersively; consider qualifying the claim as 'complete within the stated hadronic approximations' or moving the stronger claim to the conclusions.
  2. [Figure 4 caption] The caption refers to the effective pole P(2200) without defining its amplitude; please either define it in the text or refer explicitly to the equation in Ref. [82] where it is introduced.
  3. [Table II] The 'Eff.' rows list central values without uncertainties, while Eq. (7) quotes an effective-pole uncertainty; please clarify whether the quoted eff uncertainty is meant to cover these entries or is an additional assignment.
  4. [Exclusive hadronic states] The sentence 'all contributions but a single TFF vanish' is ambiguous; please specify which of the tensor TFFs is kept and in which basis the vanishing occurs.
  5. [Table I] The K± box entry is quoted as -0.5(0); please provide the actual uncertainty (even if negligible) to avoid the appearance of an exactly vanishing error.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the final result is a sum of externally fitted inputs and explicitly estimated matching contributions, not a redefinition of its own input.

full rationale

The derivation chain is self-contained. Equation (8) is obtained by adding the dispersive pseudoscalar/box/rescattering contributions from Table I, the subleading hadronic and short-distance-matched contributions from Table II, and the independent charm-loop estimate. None of these inputs use a_mu^HLbL as a fit target or as a defining condition. The axial-vector TFFs are parameterized from fits to f1/f1' data in Refs. [73,74] and extended to a1 via an explicitly stated U(3) symmetry assumption with a 30% systematic uncertainty; this is a modeling input, not a quantity derived from the final result. The VVA correlator input [81] is a separate dispersive calculation used for the OPE region and as a cross-check, and the effective-pole and tensor contributions are estimates with uncertainties assigned at or above their full size. No equation in the paper defines the output in terms of its own fitted values, and no load-bearing conclusion is justified solely by a self-citation chain. The reliance on prior work by the same authors is reliance on data-driven analyses whose inputs are external experiments and dispersion relations, which does not constitute circularity.

Assumptions & free parameters 6 free parameters · 7 assumptions · 2 invented entities

The central result depends on a number of scales and models introduced or inherited from fits. The effective poles are ad hoc constructs with no independent evidence; the matching and TFF scales are varied or assigned uncertainties. The axial-vector TFF parameters are constrained by external data, so they are not ad hoc, but they carry fit uncertainties.

free parameters (6)
  • Matching scale Q0 = 1.5 GeV (central), varied 1.2-2.0 GeV
    Splits the integration into hadronic (Qi<Q0) and pQCD/OPE (Qi>Q0) regions; the final result's stability under its variation is used to estimate the matching uncertainty.
  • OPE matching parameter r = 1/4 (central), varied 1/8-1/2
    Defines the mixed region where one virtuality is smaller than the other two; scanned to estimate the matching uncertainty.
  • Effective pole masses M_P^eff, M_A^eff = 2.2 GeV and 1.7 GeV
    Mass parameters for the fictitious poles introduced to model the intermediate-energy asymptotic matching, determined from the symmetric asymptotic limit rather than from data.
  • Effective pole TFF scale = varied with Q0 in [1.2,2.0] GeV
    Scale parameter of the effective-pole transition form factors, one of the dominant systematic uncertainties.
  • Tensor TFF scale = M_rho = 0.775 GeV
    Quark-model-inspired scale for the f2/a2/f2' form factors, assumed equal to the rho mass based on f0(980)/a0(980) studies.
  • Axial-vector TFF normalizations and f1-f1' mixing angle
    Input from global fits to e+e- and radiative decay data in Refs [73,74]; not fitted in this paper, but the central result depends on them.
assumptions (7)
  • domain assumption The HLbL tensor can be reconstructed from its discontinuities via dispersion relations in four-point kinematics, and the infinite sum of intermediate states can be truncated to the listed exclusive hadronic states plus pQCD/OPE matching.
    Invoked in 'Dispersive formalism' and 'Exclusive hadronic states'; if higher multi-meson states contribute significantly beyond the effective-pole estimate, the central value shifts.
  • domain assumption The narrow-resonance approximation is valid for the axial-vector states a1, f1, f1' and heavy scalars/tensors.
    Explicitly verified only for f0(980) and a0(980) in Refs [107,108]; assumed for the other resonances by mass suppression.
  • domain assumption U(3) symmetry determines the full set of axial-vector TFFs from the f1 data.
    Section 'Exclusive hadronic states'; the paper adds a global 30% uncertainty for symmetry violations, indicating the assumption is not exact.
  • ad hoc to paper The quark-model-inspired TFF with a single nonzero form and scale M_rho describes the tensor and heavy-scalar contributions.
    Section 'Exclusive hadronic states'; this is a simplified set-up, with a 100% uncertainty added on the total tensor contribution.
  • ad hoc to paper The effective-pole model with parameters fixed in the symmetric asymptotic limit captures the intermediate-energy matching between hadronic and pQCD/OPE regions.
    Section 'Matching to short-distance constraints'; the assigned uncertainty is larger than the entire effective-pole contribution.
  • domain assumption The OPE limit relates the HLbL tensor to the VVA correlator, and the cancellation found in Ref [80] holds beyond leading order.
    Section 'Matching to short-distance constraints'; relies on independent Bijnens et al. analysis.
  • standard math Standard QCD factorization and alpha_s evolution from Refs [124,125] apply in the pQCD quark-loop region.
    Section 'Matching to short-distance constraints'.
invented entities (2)
  • Effective pole P(2200)
    purpose: Models the asymptotic matching for the II1,2 scalar functions in the mixed region, representing missing higher intermediate states.
    Not a physical resonance; mass 2.2 GeV chosen to reproduce the asymptotic coefficient. No independent observable.
  • Effective pole A(1700)
    purpose: Models the asymptotic matching for II3-12 scalar functions.
    Not a physical resonance; mass 1.7 GeV chosen for matching. No independent observable.

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Cite this review

Pith. "Pith review of Complete dispersive evaluation of the hadronic light-by-light contribution to muon $g-2$." pith.science (2026). https://pith.science/paper/AOYSNPLJ

@misc{pith2026241200190,
  author       = {Pith},
  title        = {Pith review of: Complete dispersive evaluation of the hadronic light-by-light contribution to muon $g-2$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AOYSNPLJ}},
  note         = {Machine review of arXiv:2412.00190}
}
abstract

Hadronic light-by-light (HLbL) scattering defines one of the critical contributions in the Standard-Model prediction of the anomalous magnetic moment of the muon. In this Letter, we present a complete evaluation using a dispersive formalism, in which the HLbL tensor is reconstructed from its discontinuities, expressed in terms of simpler hadronic matrix elements that can be extracted from experiment. Profiting from recent developments in the determination of axial-vector transition form factors, short-distance constraints for the HLbL tensor, and the vector-vector-axial-vector correlator, we obtain $a_\mu^\text{HLbL}=101.9(7.9)\times 10^{-11}$, which meets the precision requirements set by the final result of the Fermilab experiment.

Figures

Figures reproduced from arXiv: 2412.00190 by the authors.

Figure 1
Figure 1. FIG. 1: HVP (left) and HLbL (right) contributions to [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Unitarity diagrams in a dispersive approach to HLbL [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Diagrams relevant for SDCs: ( [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Matching between the sum of hadronic states, [PITH_FULL_IMAGE:figures/full_fig_p003_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Stability of the [PITH_FULL_IMAGE:figures/full_fig_p004_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Comparison of our result for [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 6 Pith papers

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