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REVIEW 3 major objections 3 minor 13 references

Valence leading isospin breaking contributions to $a_{\mu}^{\mathrm{HVP-LO}}$

T0 review · 3 major / 3 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read This lattice calculation reports the valence-connected leading isospin-breaking correction to the muon HVP as $3.41(44)$ and $4.79(86)\times10^{-10}$ for the light quarks, with strange and charm contributions about one and three orders of…

desk verdict Honest, well-executed ETMC progress report on valence LIB to HVP; the light chiral extrapolation is the main soft spot, but the paper is transparent about it. read the letter →

arxiv 2501.19350 v1 pith:JJ2HZBOL submitted 2025-01-31 hep-lat

Simone Bacchio , Antonio Evangelista , Roberto Frezzotti , Giuseppe Gagliardi , Marco Garofalo , Nikolaos Kalntis , Simone Romiti , Francesco Sanfilippo
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Nazario Tantalo
This is my paper · ORCID
classification hep-lat
keywords latticeQCDhadronicvacuumpolarizationmuong-2isospinbreakingQEDcorrectionsRM123approachtwistedmassfermionschiralextrapolation
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 reports a lattice QCD+QED calculation of the leading isospin-breaking corrections to the light, strange, and charm quark-connected contributions to the leading-order hadronic vacuum polarization (HVP) part of the muon anomalous magnetic moment, $a_\mu^{\mathrm{HVP}}$. The calculation uses the RM123 expansion, treating electromagnetic effects and the up-down quark mass difference as small perturbations around isospin-symmetric QCD to first order in $\alpha_{\mathrm{em}}$ and $(\mu_d-\mu_u)/\Lambda_{\mathrm{QCD}}$. At the current, explicitly preliminary stage, the light-quark correction is the dominant one, $\Delta a_\mu^{\mathrm{HVP}}(\ell) = 3.41(44)\times 10^{-10}$ on the smaller volume and $4.79(86)\times 10^{-10}$ on the larger, while the strange and charm corrections are one and three orders of magnitude smaller, respectively. The results are obtained at a single lattice spacing ($a \sim 0.08$ fm) with statistical errors only, so the paper's contribution is a feasibility demonstration that this method can reach the sub-percent accuracy needed for the comparison with the Fermilab measurement of the muon $g-2$.

What carries the argument

The machinery is the RM123 expansion, a first-order perturbation theory in $\alpha_{\mathrm{em}}$ and $\mu_u-\mu_d$ around an isospin-symmetric QCD ensemble: all QED and strong-isospin effects are obtained by differentiating correlation functions with respect to $e^2$, bare quark masses, and critical masses. The HVP integral uses the time-momentum representation with the analytic kernel $K(m_\mu t)$, and the counterterms (the critical-mass shift $\Delta \bar m_{cr}$ and the bare quark-mass shifts $\Delta \bar\mu$, $\Delta \mu_{ud}$, $\Delta \mu_s$, $\Delta \mu_c$) are fixed by parity-restoration conditions and by matching the $\pi^+$, $K^+$, $K^0$, and $D_s$ masses, with QED finite-size effects removed by a known $1/L$ formula. The light-quark result requires the chiral extrapolation ansatz $\Delta a_\mu^{\mathrm{HVP}}(\ell; t_{\mathrm{cut}}, r_m) = \Delta a_\mu^{\mathrm{HVP}}(\ell; t_{\mathrm{cut}}) + c_1 r_m$, applied at each time cutoff $t_{\mathrm{cut}}$.

What would settle it

Compute $\Delta a_\mu^{\mathrm{HVP}}(\ell)$ directly at $r_m = 1$, the physical light-quark mass, with enough statistics to avoid the chiral extrapolation; if the direct value falls outside the linear-fit band reported in Table 3, the light correction is not what the paper quotes.

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Extended reading notes

Core claim

The central result is the set of valence, quark-connected leading isospin-breaking corrections computed in the electro-quenched approximation, in which sea quarks carry no electric charge and the lattice spacing is kept fixed: $\Delta a_\mu^{\mathrm{HVP}}(\ell) = 3.41(44)\times 10^{-10}$ (B48) and $4.79(86)\times 10^{-10}$ (B64), $\Delta a_\mu^{\mathrm{HVP}}(s) = 0.0049(10)\times 10^{-10}$ and $0.0059(7)\times 10^{-10}$, and $\Delta a_\mu^{\mathrm{HVP}}(c) = 0.1369(12)\times 10^{-10}$ and $0.1363(11)\times 10^{-10}$. The light correction is obtained by a linear chiral extrapolation in the quark-mass factor $r_m$ from $r_m = 3,5,7,9$ down to the physical point $r_m = 1$; the strange and charm corrections come from plateaux in $t_{\mathrm{cut}}$ with no significant signal-to-noise degradation. The two volumes agree within about two standard deviations, and the quoted uncertainties are statistical only. The paper concludes that the accuracy is in line with earlier results by other collaborations and by the same collaboration, while a full account of systematic errors is deferred.

Load-bearing premise

The load-bearing premise is that the light-quark correction follows a straight line in the quark-mass factor $r_m$ over the range from $r_m = 3$ down to $r_m = 1$; if the true curve bends in that interval, the quoted central value shifts by more than its statistical error.

Editorial extensions

If this is right

  • If these numbers hold, the light-quark connected correction shifts $a_\mu^{\mathrm{HVP}}$ by roughly $4\times 10^{-10}$, a few permille effect that must be included in the theory prediction.
  • The strange correction is negligible at current precision (about $0.005\times 10^{-10}$), while the charm correction is about $0.137\times 10^{-10}$ and becomes relevant only at sub-permille total accuracy.
  • The two-volume agreement within about two standard deviations means residual finite-size effects are not yet controlled at the quoted statistical precision.
  • Because the calculation is electro-quenched and uses one lattice spacing, the present numbers are not final LIB corrections; sea-quark QED, disconnected, and continuum-extrapolated contributions are all still missing.

Reading between the lines

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

  • A natural test of the extrapolation is to add a curvature term, e.g. $c_2 r_m^2$, to Eq. (18) or to include data at $r_m = 2$; even with existing points, a quadratic fit would show whether the linear ansatz is safe.
  • The near-vanishing of the strange correction suggests that the future struggle for the LIB part of $a_\mu^{\mathrm{HVP}}$ will concentrate on the light connected contribution and on the disconnected and sea-quark terms, not on heavy quarks.
  • The time dependence of the light integrand shown in the paper could be used to form short-distance and long-distance window quantities, which would localize the B48-B64 discrepancy in Euclidean time and sharpen the comparison with the BMW result.
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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 / 3 minor

Summary. This Lattice 2024 proceedings paper applies the RM123 expansion to compute valence, quark-connected, leading isospin-breaking corrections to the light, strange and charm contributions to a_mu^HVP-LO on two ETMC ensembles (B48 and B64) with lattice spacing a ~ 0.08 fm and linear sizes L ~ 3.8 fm and L ~ 5.1 fm. The counterterms are fixed by parity restoration for the critical mass shift and by the pi+, K+, K0 and D_s meson masses for the quark mass shifts, with QED finite-size effects on meson masses corrected via Eq. (10); the HVP correction is obtained from the time-momentum representation. For the light quark, the correction is computed at r_m = 3, 5, 7, 9 times the physical light mass and extrapolated linearly (Eq. (18)) to r_m = 1. The main results are Table 3: Delta a_mu^HVP(l) = 3.41(44) x 10^-10 (B48) and 4.79(86) x 10^-10 (B64); Delta a_mu^HVP(s) = 0.0049(10) and 0.0059(7) x 10^-10; Delta a_mu^HVP(c) = 0.1369(12) and 0.1363(11) x 10^-10. The paper explicitly states that all quoted errors are statistical only and that the results are preliminary.

Significance. The manuscript is a clear status report from an ongoing ETMC calculation. Its strengths are the use of the standard RM123 framework, an internally consistent counterterm setup that does not use a_mu itself as an input (so there is no circularity), two volumes for a first finite-size check, and precise and t_cut-stable strange and charm results. The authors are appropriately cautious in marking the results as preliminary. The main limitation is the light-quark chiral extrapolation: the central light values in Table 3 rest on a linear fit over a factor of three in quark mass with no data below r_m = 3, so the quoted statistical errors do not include the leading systematic risk. If the result holds up, the strange and charm numbers are useful intermediate checks, but the light values are not yet competitive with the final precision targets until the chiral form, continuum limit, and finite-size effects are addressed.

major comments (3)
  1. [§4.1, Eq. (18), Fig. 3] The central light-quark values in Table 3 are obtained by the linear ansatz Delta a_mu^HVP(l; t_cut, r_m) = Delta a_mu^HVP(l; t_cut) + c1 r_m, fitted to data at r_m = 3, 5, 7, 9 and evaluated at r_m = 1. No simulated point lies below three times the physical light mass, and the alternative fits with r_m in [3,7] and [5,9] only probe the slope inside the fitted window; they cannot detect curvature setting in below r_m = 3. Because the QED and SIB components of the integrand both vary steeply with r_m (Fig. 3, top panels), the linear form is a nontrivial assumption. Please add a curvature test (for example, a c2 r_m^2 term or a data point at r_m close to 1) or explicitly state that the Table 3 light values are model-dependent estimates pending such a check.
  2. [§5 and Table 3] The errors quoted in Table 3 are statistical only, yet the spread among the three chiral fits shown in Fig. 3 (bottom-left: 4.34(76), 4.84(95), 4.75(74) at t_cut = 2.54 on B64) is described as the estimated systematic error of the extrapolation but is not propagated into the final results. This makes it difficult to interpret the comparison in Sec. 5 of the paper's accuracy with that of other collaborations. Please either include the extrapolation systematic in the quoted uncertainties or clearly identify the Table 3 light values as central values of a preliminary analysis with an additional unquantified systematic.
  3. [§4.1 and Table 3] The two light results, 3.41(44) (B48) and 4.79(86) (B64), differ by 1.38 x 10^-10 against a combined statistical error of about 0.97 x 10^-10, i.e., only 1.4 sigma, not 'about two standard deviations' as stated. Since no finite-volume correction is applied to the light HVP correlator (Eq. (10) is used only for the meson masses in the counterterm determination), the volume dependence of the light value is unresolved. Please report the actual significance and comment on the implications for the final error budget.
minor comments (3)
  1. [§4.1] The heading 'The LIB correntions' contains a typo; it should be 'corrections', and later in the same section 'respecively' should be 'respectively'.
  2. [Acknowledgments] The heading 'Acknowlogments' is misspelled; it should be 'Acknowledgments'.
  3. [§2] The electro-quenched approximation is never explicitly defined in the text; please add a sentence clarifying that it neglects QED effects on sea quarks and the associated determinant expansion.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the target a_mu HVP correction is not used to fix any input parameter; the light r_m=1 value is the extrapolated intercept of Eq. (18), not an input.

full rationale

The derivation chain is self-contained. Counterterms are fixed by parity restoration (Eqs. 6-8) and by matching FSE-corrected experimental meson masses against the isoQCD inputs (Eqs. 9-11); the target a_mu HVP value never appears in these conditions. The vector-current renormalization correction Delta Z_V is determined from the correlator ratio in Eq. (15), and the RM123 expansion (Eqs. 5, 14) is then used to compute Delta a_mu^HVP(f). The light-quark result is obtained by the linear chiral extrapolation of Eq. (18), fitted to data at r_m = 3, 5, 7, 9 and evaluated at r_m = 1; the r_m = 1 target is an extrapolated intercept, not a data point fed into the fit, so this is an ordinary model-dependent extrapolation rather than a circular reduction. The self-citations (e.g., [4], [5], [7], [10]) are to published method papers whose relevant conditions are stated explicitly in the present text; they are not invoked to forbid alternatives or to import the final numerical result. Thus no circular step is identified; the quoted 2 reflects only the presence of non-load-bearing methodological self-citations.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The computation rests on standard lattice QCD+QED machinery. The free parameters are the quark mass counterterms (fixed by external meson masses), the critical mass shift (fixed by parity conditions), and the chiral fit slope. The target a_mu correction is not used to set any of these. The axioms are mostly domain assumptions about the RM123 expansion, the electro-quenched approximation, the mixed action improvement, and the FLAG isoQCD inputs; the only paper-specific ad hoc choice is the linear chiral extrapolation ansatz. No new physical entities are introduced.

free parameters (3)
  • Bare quark mass counterterms a Delta mu_f = B48: aDelta mu_ud = 2.65(3)e-4, aDelta mu_s = -0.436(6)e-4, aDelta mu_c = -23.72(6)e-4
    Fixed by Eq. (9) so that lattice meson masses match FLAG/Edinburgh inputs after QED finite-size correction; the target a_mu HVP correction is not among the inputs.
  • Critical mass shift a Delta mbar_cr = B48: -64.63(3)e-4; B64: -64.71(2)e-4
    Determined by imposing parity restoration conditions (Eqs. 6 and 7) and fitting a plateau from 1.0 to 2.5 fm.
  • Chiral extrapolation slope c1(t_cut) = Not tabulated; fit over r_m in [3,9], [3,7], and [5,9]
    In Eq. (18), a linear ansatz in r_m is fit to Delta a_mu(l;t_cut,r_m) to extrapolate to the physical light mass r_m=1; the spread among fit ranges is used as an estimate of the extrapolation systematic.
assumptions (5)
  • domain assumption The RM123 expansion to first order in alpha_em and delta_ud=(mu_u-mu_d)/Lambda_QCD is valid, with higher-order terms negligible.
    Used as the basis of Eq. (4) and (5); the electro-quenched approximation drops sea-quark QED effects, which are deferred to future work.
  • domain assumption The mixed action S_mix preserves automatic O(a) improvement, and the critical mass shifts satisfy m_c^cr=m_u^cr and m_s^cr=m_d^cr because the shifts arise only from QED.
    Stated in Sec. 2 after Eq. (1); needed to define the counterterm structure and to use the parity-restoration conditions.
  • domain assumption The isoQCD reference theory is defined by f_pi=130.5 MeV, M_pi=135.0 MeV, M_K=494.6 MeV, M_Ds=1967 MeV (Eq. 11).
    These values enter Eq. (9) to fix the physical quark mass counterterms; different inputs would shift the central values.
  • domain assumption The QED finite-size correction to meson masses is given by Eq. (10), with the 1/L and 1/L^2 coefficients from BMW.
    Used in Eq. (9) to correct the input meson masses for finite-volume effects; no independent verification in this paper.
  • ad hoc to paper The light-quark HVP correction depends linearly on the mass factor r_m at each t_cut (Eq. 18).
    The ansatz is fit to r_m=3,5,7,9 and evaluated at r_m=1; no curvature or higher-order term is tested, making this the weakest structural assumption.

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Cite this review

Pith. "Pith review of Valence leading isospin breaking contributions to $a_{\mu}^{\mathrm{HVP-LO}}$." pith.science (2026). https://pith.science/paper/JJ2HZBOL

@misc{pith2026250119350,
  author       = {Pith},
  title        = {Pith review of: Valence leading isospin breaking contributions to $a_\mu^\mathrmHVP-LO$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JJ2HZBOL}},
  note         = {Machine review of arXiv:2501.19350}
}
abstract

By employing the RM123 approach to QCD+QED, we computed the valence quark-connected isospin-breaking corrections to the light, strange and charm contributions at leading order in $\alpha_{\mathrm{em}}$ and $\left(\mu_d-\mu_u\right) / \Lambda_{\mathrm{QCD}}$. Here we report the preliminary results on two different volumes ($L \sim 3.8$ fm and $L \sim 5.1$ fm) and a fixed lattice spacing (corresponding to $a_{\text {isoQCD }} \sim 0.07951(4)$ fm), obtained in the framework of the ongoing computation by ETMC of the leading-order hadronic vacuum polarization (HVP) contribution to the muon anomalous magnetic moment $a_\mu^{\mathrm{HVP}}$ in QCD+QED.

Figures

Figures reproduced from arXiv: 2501.19350 by the authors.

Figure 1
Figure 1. Critical mass shift Δ𝑚¯ 𝑐𝑟 determinations on B48 (left panel) and B64 (right panel). To determine 𝑚 𝑢 cr − 𝑚 0 cr and 𝑚 𝑑 cr − 𝑚 0 cr in general we have to exploit two independent conditions, which here we choose by requiring restoration of parity invariance [7]. 𝐶¯(𝑡) ≡ 𝜕0 ∑︁ 𝑥® [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. The time behaviour of the estimator of Δ𝑍𝑉 /𝑍𝑉 and the constant fit results are shown for B48 (left panel) and B64 (right panel). Before computing the correction to the light, strange and charm contribution to the HVP, we discuss the determination of the correction to the vector current renormalization constant. Following the discussion of Appendix B of [10], and noticing that the relation used to determine 𝑍𝑉, 𝑓 no… view at source ↗
Figure 3
Figure 3. In the case of ensemble B64, we show the integrand of Eq. (17) for 𝑟𝑚 = 3, 5, 7, 9 and the SIB and QED correction to 𝑉 𝑢 (𝑡) + 𝑉 𝑑 (𝑡) with 𝑟𝑚 = 5 (left and right top panel, respectively). The left and right bottom panels show the chiral extrapolation and the time dependence of the simulated and extrapolated Δ𝑎 HVP 𝜇 (ℓ; 𝑡cut). 1.0 1.5 2.0 2.5 3.0 3.5 4.0 tcut [fm] 2 0 2 4 6 ¢ a H V P ¹ (`;tcut) £ 10 10 B48 B64 [PI… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: The time dependence of the extrapolated Δ𝑎 HVP 𝜇 (ℓ; 𝑡cut) on the two volumes 𝐿 ∼ 3.8 fm (B48 in blue) and 𝐿 ∼ 5.1 fm (B64 in orange). values, while we estimated the systematic errors on the extrapolation from the spread between the second and third kind of fit. Finall…
Figure 5
Figure 5. Figure 5: In the left panel, we show the SIB and QED correction to 𝐶 𝑠 𝐽 𝐽 (𝑡) and 𝐶 𝑐 𝐽 𝐽 (𝑡) for the B64 ensemble. The plateaux analysis of 𝑎 HVP 𝜇 (𝑠) for both the ensembles (red point for B48 and blue points for B64) is shown in the right panel. Δ𝑎 HVP 𝜇 (ℓ) × 1010 Δ𝑎 HVP 𝜇 …

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