Pith. sign in

REVIEW 4 major objections 5 minor 12 references

The isospin-violating part of the hadronic vacuum polarisation

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Lattice QCD determines the isospin-violating hadronic vacuum polarisation contribution to the muon's anomalous magnetic moment as $7.3(2.1) \times 10^{-11}$ at the $SU(3)_\mathrm{f}$ symmetric point.

desk verdict A credible methods paper for the isospin-violating HVP at the symmetric point, but the headline number leans on an external TMR derivative that is never crosschecked against the CCS calculation. read the letter →

arxiv 2412.14760 v2 pith:NSGHZNVU submitted 2024-12-19 hep-lat

Dominik Erb , Antoine Gerardin , Harvey B. Meyer , Julian Parrino , Vladimir Pascalutsa , Volodymyr Biloshytskyi This is my paper · ORCID
classification hep-lat PACS 12.38.Gc14.60.Ef13.40.Em
keywords latticeQCDhadronicvacuumpolarisationmuong-2isospinbreakingPauli-Villarsregularisationcovariantcoordinate-spacemethodkaonmasssplittingSU(3)flavoursymmetricpoint
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 computes the isospin-violating part of the hadronic vacuum polarisation (HVP) contribution to the muon's anomalous magnetic moment $(g-2)_\mu$ in lattice QCD, working at the $SU(3)_\mathrm{f}$ flavour symmetric point. The central result is the value $7.3(2.1) \times 10^{-11}$ for this contribution, obtained after averaging continuum extrapolations at the two largest Pauli-Villars regulator masses. The calculation matters because isospin breaking is one of the corrections that must be controlled to reach sub-percent precision in the Standard Model prediction of $a_\mu$ from first principles. The paper also demonstrates that, with a Pauli-Villars regularised photon propagator and covariant coordinate-space methods, the continuum extrapolated result is essentially independent of the cutoff, so the $\Lambda \to \infty$ limit is taken cleanly.

What carries the argument

The calculation is carried through the covariant coordinate-space (CCS) representation of the HVP, using the traceless kernel $H^{\mathrm{TL}}_{\lambda\sigma}(z)$ from [6] and a doubly Pauli-Villars regulated photon propagator $[G(x)]_\Lambda$ given by a modified Bessel function $K_1$ with a cutoff scale $\Lambda$ kept well below the lattice cutoff. The photon regulator makes the otherwise UV-divergent electromagnetic correction finite and permits direct comparisons with continuum calculations on gluonless ensembles. The central object that fixes the numerical outcome is the counterterm, $CT$, which is proportional to the difference between the lattice-computed electromagnetic kaon mass splitting $\Delta m_K^{\mathrm{em}}(\Lambda)$ and its physical value $\Delta m_K^{\mathrm{phys}}$, divided by the matrix element $\langle K^+ | \bar u u - \bar d d | K^+\rangle$, and multiplied by the derivative of the leading-order HVP with respect to the light-quark mass difference. That last derivative is taken from the time-momentum-representation calculation of Ref. [9].

What would settle it

Compute the light-quark mass derivative of the HVP directly in the covariant coordinate-space representation on the same CLS ensembles and with the same renormalisation; if the resulting derivative differs from the TMR value used in the counterterm by more than its quoted uncertainty, the counterterm and the final $7.3(2.1) \times 10^{-11}$ would need to be revised.

Watch

Extended reading notes

Core claim

The authors establish that, at the $SU(3)_\mathrm{f}$ symmetric point, the leading isospin-violating correction to the hadronic vacuum polarisation contribution to the muon $g-2$ is small but nonzero: $a_{\mu}^{\mathrm{HVP,NLO,38}} = 7.3(2.1) \times 10^{-11}$. The value is dominated by a mass counterterm fixed through the charged-neutral kaon mass splitting, while the fully connected diagrams (self-energy and 2-loop) individually show large cancellations and contribute only at the level of $-0.25(33) \times 10^{-11}$ (for $\Lambda = 16 m_\mu$) before the disconnected part is added. Crucially, after separate continuum extrapolations at several Pauli-Villars masses, the combined result shows little to no dependence on $\Lambda$; averaging the two largest PV masses with a conservative correlation coefficient gives the final number. The use of a Pauli-Villars photon propagator below the lattice cutoff also allows the lattice results to be crosschecked against continuum perturbation theory in the quark-gluon-less (gluonless) limit, validating the coordinate-space methodology before applying it to full QCD ensembles.

Load-bearing premise

The central result relies on the derivative of the hadronic vacuum polarisation with respect to the light-quark mass difference, taken from a separate time-momentum-representation calculation, being directly applicable to the covariant coordinate-space calculation at the $SU(3)_\mathrm{f}$ symmetric point with no additional corrections or uncertainties.

Editorial extensions

If this is right

  • At the $SU(3)_\mathrm{f}$ symmetric point, the isospin-violating HVP contribution is $7.3(2.1) \times 10^{-11}$, a small but non-negligible input for the Standard Model prediction of $a_\mu$.
  • The continuum extrapolated result shows negligible dependence on the Pauli-Villars mass, so the $\Lambda \to \infty$ limit is taken without an additional extrapolation ansatz.
  • The mass counterterm dominates the total value; improving the precision of the light-quark mass derivative of the HVP will directly reduce the uncertainty of the final result.
  • The method is validated against continuum calculations in the gluonless limit, supporting its extension to the physical point where isospin breaking is stronger.

Reading between the lines

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

  • If the external time-momentum-representation derivative of the HVP with respect to the light-quark mass difference is not consistent with the coordinate-space calculation, the counterterm and hence the final value would shift; a direct CCS computation of this derivative on the same ensembles would settle the question.
  • The strong cancellation between the self-energy and 2-loop connected diagrams suggests that partial contributions could be individually large at other kinematics, so monitoring each diagram's continuum limit may be more informative than relying only on the sum.
  • The same PV-regulated coordinate-space machinery could be applied to other isospin-violating corrections, such as the electromagnetic pion mass splitting or isospin-breaking effects in HVP window observables.
  • The observed $\Lambda$-independence may be specific to the $SU(3)_\mathrm{f}$ symmetric point; at the physical point, where light-quark isospin breaking is more pronounced, the PV-mass dependence should be re-examined before the limit is taken.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The manuscript computes the isospin-violating part of the hadronic vacuum polarisation contribution to the muon anomalous magnetic moment at the SU(3) flavour symmetric point, using lattice QCD with a Pauli-Villars regulated photon propagator and covariant coordinate-space methods. The fully connected diagrams are computed on CLS ensembles and cross-checked on gluonless ensembles against continuum perturbation theory; the only mass counterterm is determined from the charged-neutral kaon mass splitting and an external light-quark mass derivative. After continuum extrapolation, the results show little dependence on the Pauli-Villars mass, and averaging the two largest PV masses gives a final value of 7.3(2.1) x 10^-11. The central number is dominated by the counterterm rather than by the connected contribution computed in this work.

Significance. The calculation is a useful step toward a first-principles determination of a subleading isospin-violating contribution to HVP, and the use of a Pauli-Villars regulator to enable continuum cross-checks of individual diagrams is a genuine strength. The gluonless-ensemble checks and the observed logarithmic large-PV behaviour of the kaon mass splitting are concrete, falsifiable tests that support the methodology. If the final result is confirmed after resolving the external-input issue, it would provide a valuable data point at the SU(3) symmetric point for the isospin-breaking corrections to g-2. However, the final value and its uncertainty are presently controlled by a counterterm that depends on a derivative taken from a separate time-momentum representation calculation, and that dependence is not validated within this manuscript.

major comments (4)
  1. [§4, Eq. (4)] The counterterm in Eq. (4) dominates the final result: in §5, at Λ=16m_mu on N202, the counterterm contributes 9.54(1.24) x 10^-11 while the connected contribution is -0.148(191) x 10^-11. The derivative ∂a_HVP,38/∂(m_u-m_d) is taken from Ref. [9], a time-momentum representation calculation, and the paper does not demonstrate that this derivative is the same quantity required by the covariant coordinate-space expression in Eq. (2) after the same renormalisation and continuum extrapolation, nor does it quantify correlations with the CCS connected contribution. Because any mismatch in this external input shifts the central value directly, the authors should either compute the derivative in the same CCS framework or provide a quantitative numerical crosscheck, for example by evaluating the derivative from the CCS correlators on at least one ensemble and comparing the result with the TMR input.
  2. [§5, Table 3] The Λ to infinity limit is based on only four PV masses (3, 5, 10 and 16 m_mu) and an average of the two largest values with a correlation coefficient taken to be unity. The data are consistent with no Λ dependence, but the errors are large and the counterterm dominates the total; the paper does not show the Λ dependence of the connected and counterterm contributions separately after continuum extrapolation, so the flatness of the total could mask opposite trends. Reporting the two components separately as a function of Λ would strengthen the claim that the limit is straightforward and would clarify where the 2.1 x 10^-11 uncertainty actually originates.
  3. [§4, Eq. (11)] The fit of Δm_K^em(Λ) is a load-bearing input to the counterterm, but the paper does not report the fit parameters a, b, d and C, the covariance matrix, or the justification for the correlation coefficient rho=0.9 between input parameters. In addition, the H200 ensemble is excluded from the continuum extrapolation of the kaon mass splitting because of 'large volume effects', but the criterion for this exclusion is not quantified. The authors should document the fit systematics and the exclusion criterion, or at least show that the final counterterm is stable under reasonable variations of these choices.
  4. [§3, Eq. (8)] The connected contribution is consistent with zero after a large cancellation between the self-energy and 2-loop diagrams, and the paper does not report the correlation between these two components or the covariance matrix of the continuum extrapolation in Eq. (9). Given that the final connected signal is an order of magnitude smaller than each individual diagram, a verification that the error on the sum is not underestimated, for example by comparing the fit result with a direct jackknife combination, would increase confidence in the quoted connected error.
minor comments (5)
  1. [Table 1] The row for the N202 ensemble is missing its beta value; the table header includes a beta column and all other rows provide this value.
  2. [Introduction and §3] There are several typos, including 'secound' in the introduction and 'renomalization' in §3; the text in §4 also says 'comes form' instead of 'comes from'.
  3. [§4] The disconnected (2+2)a diagram is added to the total in §5, but its calculation is described only as 'calculated separately without a PV regulator'; a short description or a reference for this continuum value should be added so the reader can assess its systematic error.
  4. [§4, Eq. (4)] The text accompanying Eq. (4) describes the 'first factor' as coming from the kaon mass splitting, but the notation is slightly confusing because the factor is a ratio involving the matrix element ⟨K+|u_bar u - d_bar d|K+⟩; rewriting this sentence would improve readability.
  5. [§5] The averaging procedure for the final result is described in words only; stating the explicit weighted average, including the treatment of correlations between the two PV-mass points, would make the final 7.3(2.1) x 10^-11 reproducible from Table 3 alone.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the connected diagrams are computed from first principles, the counterterm is fixed by a standard renormalisation condition, and the dominant external mass derivative is an independent published lattice input rather than a fitted version of the target result.

full rationale

The paper derives the isospin-violating HVP contribution from a lattice calculation of connected diagrams plus a mass counterterm. The connected contribution is not defined to be the final answer; it is computed from quark propagators on CLS ensembles and is checked against continuum perturbation theory on gluonless ensembles (Table 2a), which is an external benchmark. The counterterm in Eq. (4) is a renormalisation condition: it uses the physical charged-neutral kaon mass splitting from the PDG to fix the residual mass dependence introduced by the Pauli-Villars regulated photon propagator. The only quantity taken from outside the present calculation is the derivative of the leading-order HVP with respect to the light-quark mass difference, taken from Ref. [9] in the TMR representation. This is a separate published lattice-QCD result, not a parameter fitted to the target quantity in this paper, and it is not defined in terms of the final 7.3(2.1) x 10^-11. The paper also demonstrates that the continuum extrapolated results are nearly independent of the PV mass (Table 3 and Figure 6b), and the final value is obtained by averaging the two largest PV masses. While one could raise a correctness concern about whether the TMR derivative is fully consistent with the CCS framework and with the ensembles and renormalisation used here, that concern concerns external-input consistency, not circularity: the derivation does not reduce to its own assumptions by construction. Self-citation is present (Refs. [4,5,6,9,10] share authors), but the cited results are either methodology papers, an independent continuum QED calculation, or a published lattice computation of a different observable; none of them asserts the target result. The central claim therefore retains independent computational content.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the completeness of the mass counterterm and on external inputs (the HVP mass derivative from Ref. [9] and the physical kaon mass splitting). The analysis also uses fit functions with coefficients fitted to data and one ad hoc correlation coefficient.

free parameters (4)
  • b, c, d in the continuum extrapolation fit f_fit(a, m_pi L) = b + c a^2 + d exp(-m_pi L/2) (eq. 9) = not reported
    Fitted to the lattice data for each quantity (connected, kaon mass splitting, total HVP) to extract the continuum limit. The central result depends on these fitted coefficients.
  • a, b, d in the PV-mass fit of Delta m_K^em (eq. 11) = not reported
    Fitted to the nine PV-mass values of Delta m_K^em to check the OPE prediction; not directly used for the final total, but used as a plausibility check.
  • correlation coefficient rho = 0.9 between input parameters in the Delta m_K^em fit = 0.9
    Chosen by hand 'based on calculations for each ensemble' (Sec. 4). This ad hoc correlation affects the fit uncertainty of Delta m_K^em and the resulting systematic error.
  • selection of PV masses and averaging over the two largest values = Lambda = 10 and 16 m_mu (largest two in table 3)
    The final result is obtained by averaging the results with the largest two PV-masses rather than a fit to all Lambda values. This choice assumes the Lambda-dependence is negligible beyond 16 m_mu.
assumptions (5)
  • domain assumption The only UV counterterm needed at this order is the mass counterterm fixed by the kaon mass splitting.
    Stated in the abstract: 'the only (mass) counterterm'. If additional operator counterterms exist, the renormalisation is incomplete. No proof is provided.
  • domain assumption The mass derivative of the HVP from Ref. [9], computed in the TMR representation, can be used in the CCS calculation.
    Used in the counterterm (Sec. 4, 'Here, we use the mass derivatives from the calculation in reference [9] obtained in the TMR representation.'). The two representations are expected to agree in the continuum, but this is not demonstrated.
  • ad hoc to paper The OPE-based fit function (eq. 11) describes the Lambda-dependence of Delta m_K^em.
    The fit function is a modified OPE prediction with extra parameters to make it vanish at Lambda=0. It is used to corroborate the kaon mass splitting values.
  • domain assumption The exponential volume term in eq. (9) captures the finite-size effects.
    The continuum extrapolation assumes FVEs fall as exp(-m_pi L/2). Used for all continuum extrapolations.
  • domain assumption The (2+2)a disconnected diagram, computed without a PV regulator, can be combined with the PV-regulated connected and counterterm parts.
    Added in Sec. 5 from a separate calculation; the PV regularisation is claimed to be unnecessary for this finite diagram.

how reviews work

0 comments
Cite this review

Pith. "Pith review of The isospin-violating part of the hadronic vacuum polarisation." pith.science (2026). https://pith.science/paper/NSGHZNVU

@misc{pith2026241214760,
  author       = {Pith},
  title        = {Pith review of: The isospin-violating part of the hadronic vacuum polarisation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NSGHZNVU}},
  note         = {Machine review of arXiv:2412.14760}
}
abstract

We present our calculation of the isospin-violating part of the hadronic vacuum polarisation (HVP) contribution to muon $(g-2)$ in lattice QCD at the $SU(3)_{\mathrm{f}}$ symmetric point. The computation of the contributing fully connected diagrams with one internal photon as well as the computation of the only (mass) counterterm are shown. The latter is determined from the charged-neutral kaon mass splitting. We employ coordinate-space methods and a photon propagator which is regulated \`a la Pauli-Villars with a cutoff scale $\Lambda$ well below the lattice cutoff. This regularization makes it possible for us to do crosschecks of individual contributions with calculations in the continuum. Our continuum extrapolated results show little to no dependence on $\Lambda$. This makes our final limit $\Lambda \rightarrow \infty$ straightforward.

Figures

Figures reproduced from arXiv: 2412.14760 by the authors.

Figure 1
Figure 1. Diagrams that contribute to ⟨𝑗 3 𝜆 (𝑧) 𝑗 𝑒𝑚 𝜇 (𝑥) 𝑗 𝑒𝑚 𝜈 (𝑦) 𝑗 8 𝜎 (0)⟩ at the 𝑆𝑈(3)f symmetric point. four-point function ⟨𝑗 3 𝜆 (𝑧) 𝑗 𝑒𝑚 𝜇 (𝑥) 𝑗 𝑒𝑚 𝜈 (𝑦) 𝑗 8 𝜎 (0)⟩. These diagrams can be seen in figure 1. In the following, we will discuss the calculation of the fully connected diagrams. The result of the (2+2)a disconnected diagram will be added later on in section 5, already in the continuum. The fully connected… view at source ↗
Figure 2
Figure 2. Continuum extrapolation of the self-energy and 2-loop parts on the left side as well as the total value on the right side for the QCD ensembles. The Pauli-Villars mass is set to Λ = 16 𝑚𝜇. Equation (9) is used as an ansatz for the fit. The dots are the data from the lattices, while the crosses are the same data, but with the volume term of the fit function subtracted without adjusting the error bars. The straight li… view at source ↗
Figure 3
Figure 3. The Feynman diagrams of the leading order contributions to the electromagnetic mass splitting of the kaon. At the 𝑆𝑈(3)f symmetric point only these two diagrams contribute to the mass splitting [11]. to check the plausibility of these values, we can examine the large PV-mass behavior. Using an Operator Product Expansion (OPE), similar to the one used in [4] we get the following prediction for Λ → ∞: Δ𝑚 𝑒𝑚 𝐾 (Λ) ≈ 3𝛼… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: The methodology discussed in section 4 for extracting the short distance contribution of the kaon mass splitting. The calculations shown here were done on the H101 ensemble. 0.000 0.002 0.004 0.006 0.008 a 2 in fm2 0.5 1.0 1.5 2.0 2.5 ∆ m K in MeV Λ =3 mµ Λ =5 mµ Λ =8 …
Figure 5
Figure 5. Figure 5: Continuum extrapolation of the kaon mass splitting and PV-mass dependence of the resulting continuum values together with a fit using the fit function in equation 11. The H200 ensemble was excluded from the extrapolation since it showed large volume effects. value of −…
Figure 6
Figure 6. Figure 6: Continuum and PV-mass extrapolation of 𝑎 HVP,NLO,38 𝜇 . Figure (a) is analogous to the plots of figure 2 with the same fit function given by equation 9 of section 3. Λ in 𝑚𝜇 3 5 10 16 1/Λ 2 in 1/𝑚 2 𝜇 0.11 0.04 0.01 0.004 1011 × 𝑎 HVP,NLO,38 𝜇 (Λ) 6.63 ± 2.04 6.90 ± 2.…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

12 extracted references · 1 canonical work pages

  1. [9]

    Bruno, T

    M. Bruno, T. Korzec and S. Schaefer, Setting the scale for the CLS 2 + 1 flavor ensembles , https://doi.org/10.1103/PhysRevD.95.074504 Phys. Rev. D 95 (2017) 074504 [ https://arxiv.org/abs/1608.08900 1608.08900 ]

    arXiv 2017

  2. [1]

    write newline

    " write newline "" before.all 'output.state := FUNCTION blank.sep after.quote 'output.state := FUNCTION fin.entry output.state after.quoted.block = 'skip 'add.period if write newline FUNCTION new.block output.state before.all = 'skip output.state after.quote = after.quoted.block 'output.state := after.block 'output.state := if if FUNCTION new.sentence out...

  3. [2]

    Muon g-2 collaboration, Measurement of the Positive Muon Anomalous Magnetic Moment to 0.46 ppm , https://doi.org/10.1103/PhysRevLett.126.141801 Phys. Rev. Lett. 126 (2021) 141801 [ https://arxiv.org/abs/2104.03281 2104.03281 ]

    arXiv 2021

  4. [3]

    Muon g-2 collaboration, Measurement of the Positive Muon Anomalous Magnetic Moment to 0.20 ppm , https://arxiv.org/abs/2308.06230 2308.06230

    arXiv

  5. [4]

    Aoyama et al., The anomalous magnetic moment of the muon in the Standard Model , https://doi.org/10.1016/j.physrep.2020.07.006 Phys

    T. Aoyama et al., The anomalous magnetic moment of the muon in the Standard Model , https://doi.org/10.1016/j.physrep.2020.07.006 Phys. Rept. 887 (2020) 1 [ https://arxiv.org/abs/2006.04822 2006.04822 ]

    arXiv 2020

  6. [5]

    Biloshytskyi, E.-H

    V. Biloshytskyi, E.-H. Chao, A. G\'erardin, J.R. Green, F. Hagelstein, H.B. Meyer et al., Forward light-by-light scattering and electromagnetic correction to hadronic vacuum polarization , https://doi.org/10.1007/JHEP03(2023)194 JHEP 03 (2023) 194 [ https://arxiv.org/abs/2209.02149 2209.02149 ]

    arXiv 2023

  7. [6]

    H.B. Meyer, Lorentz-covariant coordinate-space representation of the leading hadronic contribution to the anomalous magnetic moment of the muon , https://doi.org/10.1140/epjc/s10052-017-5200-3 Eur. Phys. J. C 77 (2017) 616 [ https://arxiv.org/abs/1706.01139 1706.01139 ]

    arXiv 2017

  8. [7]

    Chao, H.B

    E.-H. Chao, H.B. Meyer and J. Parrino, Coordinate-space calculation of the window observable for the hadronic vacuum polarization contribution to (g-2)_ , https://doi.org/10.1103/PhysRevD.107.054505 Phys. Rev. D 107 (2023) 054505 [ https://arxiv.org/abs/2211.15581 2211.15581 ]

    arXiv 2023

Show all 12 references
  1. [8]

    S.N. et al. (Particle Data Group) https://doi.org/10.1103/PhysRevD.110.030001 , Phys. Rev. D 110 (2024) 030001

    2024 doi

  2. [10]

    C\`e et al., Window observable for the hadronic vacuum polarization contribution to the muon g-2 from lattice QCD , https://doi.org/10.1103/PhysRevD.106.114502 Phys

    M. C\`e et al., Window observable for the hadronic vacuum polarization contribution to the muon g-2 from lattice QCD , https://doi.org/10.1103/PhysRevD.106.114502 Phys. Rev. D 106 (2022) 114502 [ https://arxiv.org/abs/2206.06582 2206.06582 ]

    2022 arXiv

  3. [11]

    G\'erardin, T

    A. G\'erardin, T. Harris and H.B. Meyer, Nonperturbative renormalization and O(a) -improvement of the nonsinglet vector current with N _ f =2+1 wilson fermions and tree-level symanzik improved gauge action , https://doi.org/10.1103/PhysRevD.99.014519 Phys. Rev. D 99 (2019) 014519

    2019 doi

  4. [12]

    RM123 collaboration, Leading isospin breaking effects on the lattice , https://doi.org/10.1103/PhysRevD.87.114505 Phys. Rev. D 87 (2013) 114505 [ https://arxiv.org/abs/1303.4896 1303.4896 ]

    2013 arXiv

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.