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Strange and charm quark contributions to the muon anomalous magnetic moment in lattice QCD with twisted-mass fermions

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Lattice QCD sets the strange- and charm-quark connected HVP contributions to the muon's anomalous magnetic moment at (53.57 ± 0.63) and (14.56 ± 0.13) × 10^-10.

desk verdict A careful, transparent incremental update of the ETMC strange/charm HVP results; the new finer lattice spacing and explicit mistuning corrections are real improvements, and it deserves a serious referee. read the letter →

arxiv 2411.08852 v1 pith:WWC6GEUE submitted 2024-11-13 hep-lat hep-ph

C. Alexandrou , S. Bacchio , A. De Santis , A. Evangelista , J. Finkenrath , R. Frezzotti , G. Gagliardi , M. Garofalo
show 10 more authors
N. Kalntis B. Kostrzewa V. Lubicz F. Pittler S. Romiti F. Sanfilippo S. Simula N. Tantalo C. Urbach U. Wenger
This is my paper · ORCID
classification hep-lathep-ph PACS 12.38.Gc14.60.Ef
keywords muong-2hadronicvacuumpolarizationlatticeQCDtwisted-massfermionsstrangequarkcontributioncharmcontinuumextrapolationAICmodelaveraging
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

This paper aims to establish first-principles lattice QCD values for the strange- and charm-quark connected contributions to the hadronic vacuum polarization (HVP), the largest theoretical uncertainty in the muon anomalous magnetic moment. Using gauge ensembles with Nf = 2+1+1 flavors of Wilson-clover twisted-mass quarks at four lattice spacings, the authors correct small sea-quark and critical-mass mistunings to land on the isospin-symmetric physical point and then extrapolate to the continuum with an information-criterion model average. The claimed results are a_mu^HVP(s) = (53.57 ± 0.63) × $10^{-10}$ and a_mu^HVP(c) = (14.56 ± 0.13) × $10^{-10}$. These values matter because independent lattice determinations of the flavor-by-flavor HVP contributions are needed to test the tension between e+e- data-driven estimates and the experimental g-2; if correct, they show the strange and charm channels agree with other lattice groups and are not the origin of that tension.

What carries the argument

The central object is the time-momentum representation of the HVP, which writes a_mu^HVP = 2 $alpha_em^{2}$ ∫_0^∞ dt $t^{2}$ K(m_mu t) V(t), where K is a known leptonic kernel and V(t) is the Euclidean vector-current correlator. On the lattice, V(t) is computed with the TM and OS regularizations, two discretizations that differ only by O($a^{2}$) artifacts, so forcing a common continuum limit in joint fits exposes the cutoff effects. The continuum extrapolation uses the ansatz P0 + P1 $a^{2}$ + P2 $a^{4}$ and variants with $a^{2}$/[log($a^{2}$/$lambda_0^{2}$)]^n terms, averaged with AIC weights; separately, leading-order reweighting converts the simulated ensembles to the isospin-symmetric point defined by the hadronic inputs M_pi = 135.0 MeV, M_K = 494.6 MeV, M_Ds = 1967 MeV, F_pi = 130.5 MeV.

What would settle it

A concrete test: compute the same strange and charm HVP contributions on an additional lattice spacing below 0.04 fm, or with an independent fermion action whose cutoff artifacts are known to differ, and see whether the continuum-extrapolated values remain within 53.57 ± 0.63 and 14.56 ± 0.13 (× $10^{-10}$); a shift exceeding the combined errors would show the model family is too narrow.

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Extended reading notes

Core claim

The central claim, stated as Eq. (1) and Eqs. (18)-(19), is that in isospin-symmetric QCD the quark-connected strange and charm HVP contributions are a_mu^HVP(s) = 53.57(63) × $10^{-10}$ and a_mu^HVP(c) = 14.56(13) × $10^{-10}$, where the error combines statistics, continuum extrapolation, finite-size, and tuning uncertainties in quadrature. The values come from vector correlators evaluated in two lattice regularizations that share the same continuum limit, corrected by leading-order reweighting for the small mistunings of the simulated bare parameters, and extrapolated with an AIC-weighted average over fit families of the form P0 + P1 $a^{2}$ + P2 $a^{4}$ (with logarithmic variants) using data at a = 0.049, 0.057, 0.068, and 0.080 fm. The same analysis yields short-distance, intermediate-window, and long-distance window contributions; the short-distance strange and intermediate-window charm determinations are the most precise lattice results in those windows.

Load-bearing premise

The load-bearing premise is that the continuum-limit fit family — smooth $a^{2}$ and $a^{4}$ terms with optional logarithmic variants — spans the true discretization effects at the four lattice spacings used.

Editorial extensions

If this is right

  • The strange and charm connected HVP contributions are now established at sub-percent precision, so future full lattice determinations of a_mu^HVP can treat these channels as fixed anchors rather than as free sources of error.
  • The short-distance strange value, 9.063(27) × 10^-10, and intermediate-window charm value, 2.920(64) × 10^-10, are the most precise lattice results in their respective windows, sharpening window-by-window comparisons with e+e- data.
  • Agreement with the other lattice determinations shown in the paper indicates that the longstanding g-2 tension is not driven by the strange or charm connected contributions.
  • The new fine lattice spacing (a ≈ 0.049 fm) and the explicit mistuning corrections reduce the total uncertainty relative to the earlier determination of Ref. [22], making these the reference values for those channels until the light-quark and disconnected contributions are added.

Reading between the lines

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

  • Beyond the paper: if the light-quark connected and disconnected contributions from the same ensembles reach comparable precision, the combined lattice HVP prediction will provide a sharp, independent test of whether e+e- data or lattice QCD is closer to the experimental g-2 value.
  • Beyond the paper: the AIC error is a conditional estimate — it measures spread within the assumed a^2/a^4 ansatz family. A future lattice spacing below 0.04 fm, or a different action with different cutoff effects, is the natural check; a shift larger than the quoted error would indicate the fit family is incomplete.
  • Beyond the paper: the charm sea-quark derivative was approximated by a mass-scaling assumption in Appendix D; computing it directly would test whether the strange-quark result carries a small unquantified bias from that approximation.
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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

2 major / 5 minor

Summary. This paper presents a lattice QCD calculation of the quark-connected strange and charm hadronic-vacuum-polarization contributions to the muon anomalous magnetic moment in isospin-symmetric QCD. The analysis uses ETMC Nf=2+1+1 twisted-mass Wilson-clover ensembles at four lattice spacings (a≈0.049–0.080 fm) with volumes up to 7.6 fm, corrects for small sea- and valence-quark mass mistunings and critical-mass mistuning using reweighting and hadronic inputs from the Edinburgh/FLAG consensus, and extrapolates to the continuum using an AIC model average over a^2/a^4 and logarithmic variants. The final results are a_mu^HVP(s)=53.57(63)×10^-10 and a_mu^HVP(c)=14.56(13)×10^-10, together with SD, W, and LD window contributions, and the paper compares these with other lattice determinations.

Significance. If the quoted results are correct, they are among the most precise lattice determinations of the strange and charm HVP contributions and provide useful independent cross-checks for the Standard Model prediction of the muon anomalous magnetic moment. The paper is unusually transparent in its error decomposition: Eqs. (18)-(19) separate statistical, continuum, and FSE errors, the continuum extrapolation uses two current regularizations (TM/OS) and a defined AIC averaging procedure, and the Z_V and Z_A renormalization constants are determined from Ward identities with high precision. The inclusion of the new fine E112 ensemble and explicit reweighting corrections for mistunings are concrete improvements over the authors' previous work. The main residual risks are the completeness of the continuum-extrapolation model family and the scaling assumption used for the charm sea-quark mass derivative; both are acknowledged in the paper but deserve quantitative robustness checks.

major comments (2)
  1. [Section III, Eq. (15) and Fig. 3] The dominant error in the final results is the continuum-extrapolation uncertainty (0.48 vs 0.41 in units of 10^-10 for strange; 0.09 vs 0.10 for charm), yet this error is defined as the AIC spread within a single parametric family P0+P1^reg a^2+P2^reg a^4 and its logarithmic variants. With only four lattice spacings spanning a factor of about 2.7 in a^2, the fitted data have limited power to discriminate between these smooth forms and, for example, a residual O(a) contribution or a spacing-dependent artifact that affects mainly the finest ensemble E112. Since the model family is assumed complete, the quoted σ_cont does not include this risk. I request a leave-one-out analysis that omits the finest spacing E112 and, where possible, a fit including an explicit O(a) term, together with a discussion of how the central values in Eqs. (18)-(19) shift under these variations.
  2. [Appendix D, Eq. (D2)] The charm sea-quark mass derivative is estimated from the strange one by the scaling ∂_c^sea O ≃ (m_s/m_c) ∂_s^sea O because the direct derivative is too noisy. The text states that including the charm sea mistuning correction increased the errors on a_HVP(s) and on the intermediate window, so this approximation contributes to the final error budget, but no uncertainty is assigned to the scaling assumption itself. Please quantify the sensitivity of Eqs. (18)-(19) and Eqs. (22)-(23) to this assumption, for example by varying the proportionality factor within a conservative range or by computing ∂_c^sea on one ensemble with increased statistics, and add the corresponding systematic if it is not already covered.
minor comments (5)
  1. [Footnote 1] The strange and charm analysis is stated to be unblinded; please include a brief statement of the analysis choices fixed in advance (e.g., model set, tmin and tcut criteria) so that the AIC-based error estimate can be assessed as a pre-specified procedure.
  2. [Section III and Eq. (13)] There are several rendering or typographical artifacts, such as 'tcut7→∞' in Eq. (13) and 'Appedices' in the text of Section III; these should be corrected.
  3. [Figs. 3, 4, and 6] The many grey fit lines in the continuum-extrapolation plots are not individually labeled; adding a legend that maps each line to the corresponding AIC model variant would make the spread of the extrapolations easier to audit.
  4. [Footnote 5 and Figs. 7-8] The comparison with other lattice groups neglects possible differences in the definition of isoQCD; a quantitative estimate of the expected size of these differences would help interpret the good agreement shown in Figs. 7 and 8.
  5. [Eq. (17)] The AIC weight contains the term -2Ndata, which is nonstandard; since the authors state that the alternative definition from Ref. [38] gives similar results, reporting the shift in σ_cont between the two definitions would strengthen the robustness statement.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the strange/charm HVP values are continuum limits of first-principles lattice correlators, with external hadronic inputs fixing the isoQCD scheme rather than the target a_mu.

full rationale

The central results in Eqs. (18)-(19) are obtained from lattice-evaluated vector correlators through Eq. (13), followed by a continuum extrapolation of the finite-spacing data. Nothing in the chain is tuned to reproduce a_mu^HVP(s) or a_mu^HVP(c). The isoQCD point is defined by external Edinburgh/FLAG hadronic inputs in Eq. (12) (M_pi, M_K, M_Ds, F_pi), which are used to set quark masses and the lattice spacing; those inputs do not include the target HVP values. The continuum extrapolation uses the model family of Eq. (15) with an AIC average (Eqs. (16)-(17)); the 'cont' error is the AIC spread. This is a model-uncertainty risk, not circularity, because the finite-spacing data are independent lattice observables and the target value is not an input to the fit. The paper's self-citations (e.g., Ref. [22] for the hadronic method of ZV/ZA, Refs. [23-26] for ensemble generation, and Ref. [16] for the FSE estimate) provide technical methods or earlier results for comparison; the ZV/ZA values used here are computed in this work from Ward identities, and the previous ETMC-22 values are not used as inputs to the present central determinations. Acknowledged limitations, such as the non-blinded analysis noted in footnote 1 and the estimate of the charm sea-quark derivative via the approximate scaling of Eq. (D2), affect systematics but do not make the derivation self-referential. The dominant concern is the completeness of the continuum-extrapolation ansatz at four lattice spacings, which is a correctness risk outside the category of circularity.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The calculation rests on the standard lattice QCD framework of the ETMC collaboration: the mixed-action theory, the reweighting-based fine-tuning, and the continuum extrapolation. The only genuinely ad hoc element is the scaling estimate of the charm sea-quark derivative (Eq. D2). No parameters are fitted to the target a_mu^HVP(s,c) values themselves, so the result is not circular, but its error budget depends on the stated assumptions.

free parameters (2)
  • Continuum extrapolation coefficients P1^reg, P2^reg (Eq. 15) = not quoted
    Fitted to lattice data at four lattice spacings to remove O(a^2) and O(a^4) discretization effects; the AIC average over the model family sets the continuum error.
  • Valence mass interpolation slopes A_w^s, A_w^c, B_w^c (Eq. C1) = not quoted
    Fitted to correlator data at two or three valence masses per ensemble to interpolate to the tuned isoQCD strange and charm masses.
assumptions (4)
  • domain assumption The mixed-action lattice theory (TM sea, OS valence, ghosts) is unitary and equivalent to continuum QCD up to O(a^2) corrections; the TM/OS determinant ratio equals 1 + O(a^2) (Eq. A12).
    Invoked in Appendix A to justify using OS valence quarks on TM sea configurations; standard for twisted-mass lattice QCD but not proven nonperturbatively at finite a.
  • domain assumption Leading-order linear reweighting in the mass mismatches (Eq. B6) is accurate; O(Delta m^2) corrections are negligible.
    Used throughout Appendix B and D to correct hadronic inputs and HVP values for small mistunings of sea-quark masses and critical mass; the paper cross-checks with global fits for light quantities.
  • ad hoc to paper The charm sea-quark mass derivative is estimated from the strange one by the scaling d_c^sea O ~ (m_s/m_c) d_s^sea O (Eq. D2).
    The direct charm derivative is too noisy; the 1/m_c scaling is a physically motivated approximation introduced in this work and not directly validated for the HVP observables.
  • domain assumption Perturbative QCD (RHAD at NNLO) is used to evaluate the short-time region [0, tmin] in the second analysis branch (Section III A, Fig. 5).
    The tmin dependence of the SD window is checked by adding the continuum NNLO contribution to the short-distance tail; standard assumption of perturbation theory for small t.

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Pith. "Pith review of Strange and charm quark contributions to the muon anomalous magnetic moment in lattice QCD with twisted-mass fermions." pith.science (2026). https://pith.science/paper/WWC6GEUE

@misc{pith2026241108852,
  author       = {Pith},
  title        = {Pith review of: Strange and charm quark contributions to the muon anomalous magnetic moment in lattice QCD with twisted-mass fermions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WWC6GEUE}},
  note         = {Machine review of arXiv:2411.08852}
}
abstract

We present a lattice calculation of the Hadronic Vacuum Polarization (HVP) contribution of the strange and charm quarks to the anomalous magnetic moment of the muon in isospin symmetric QCD. We employ the gauge configurations generated by the Extended Twisted Mass Collaboration (ETMC) with $N_f = 2 + 1 + 1$ flavors of Wilson-clover twisted-mass quarks at five lattice spacings and at values of the quark mass parameters that are close and/or include the isospin symmetric QCD point of interest. After computing the small corrections necessary to precisely match this point, and carrying out an extrapolation to the continuum limit based on the data at lattice spacings $a \simeq 0.049, 0.057, 0.068, 0.080$~fm and spatial lattice sizes up to $L \simeq 7.6$~fm, we obtain $a_\mu^{\rm HVP}(s) = (53.57 \pm 0.63) \times 10^{-10}$ and $a_\mu^{\rm HVP}(c) = (14.56 \pm 0.13) \times 10^{-10}$, for the quark-connected strange and charm contributions, respectively. Our findings agree well with the corresponding results by other lattice groups.

Figures

Figures reproduced from arXiv: 2411.08852 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
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Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (15 more)
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
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Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
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Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]
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Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]
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Figure 10. Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p020_10.png]
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Figure 11. Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p023_11.png]
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Figure 12. Figure 12: FIG. 12 [PITH_FULL_IMAGE:figures/full_fig_p024_12.png]
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Figure 13. Figure 13: FIG. 13 [PITH_FULL_IMAGE:figures/full_fig_p025_13.png]
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Figure 14. Figure 14: FIG. 14 [PITH_FULL_IMAGE:figures/full_fig_p028_14.png]
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Figure 15. Figure 15: FIG. 15 [PITH_FULL_IMAGE:figures/full_fig_p028_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16 [PITH_FULL_IMAGE:figures/full_fig_p029_16.png]
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Figure 17. Figure 17: FIG. 17 [PITH_FULL_IMAGE:figures/full_fig_p029_17.png]
Figure 18
Figure 18. Figure 18: FIG. 18 [PITH_FULL_IMAGE:figures/full_fig_p030_18.png]

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Reviewed August 12, 2026 · model on record in the stance chip above.