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REVIEW 3 major objections 4 minor 40 references

Isospin-breaking corrections to the muon magnetic anomaly in Lattice QCD

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper reports a lattice QCD calculation of the leading electromagnetic and strong isospin-breaking corrections to the hadronic-vacuum-polarization contribution to the muon anomalous magnetic moment, finding $\delta a_\mu^{\rm…

desk verdict A solid, transparent proceedings summary of an already-published lattice result; the 'most accurate' claim rests on an estimated qQED+disconnected uncertainty, but the paper is honest about it. read the letter →

arxiv 1909.01962 v1 pith:QJWKWUNG submitted 2019-09-04 hep-lat hep-exhep-ph

Davide Giusti , Vittorio Lubicz , Guido Martinelli , Francesco Sanfilippo , Silvano Simula This is my paper · ORCID
classification hep-lathep-exhep-ph
keywords muonanomalousmagneticmomenthadronicvacuumpolarizationisospinbreakinglatticeQCDRM123methodquenchedQEDapproximationg-2strong
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 calculation of the leading electromagnetic and strong isospin-breaking corrections to the quark-connected hadronic-vacuum-polarization (HVP) contribution to the muon’s anomalous magnetic moment. Working in the RM123 expansion in powers of $\alpha_{\rm em}$ and $(m_d-m_u)/\Lambda_{\rm QCD}$, in the quenched-QED approximation, and using gauge ensembles at three lattice spacings, several volumes, and pion masses between about 220 and 490 MeV, the authors obtain $\delta a_\mu^{\rm HVP}(udsc)=7.1\,(2.9)\times10^{-10}$ after continuum, infinite-volume, and physical-mass extrapolations. The light-quark correction dominates, $\delta a_\mu^{\rm HVP}(ud)=7.1\,(2.5)\times10^{-10}$, about 85% of it from strong isospin breaking rather than QED; the strange and charm corrections are tiny. The paper states this is currently the most accurate determination of the isospin-breaking corrections to $a_\mu^{\rm HVP}$, and when added to leading-order HVP estimates it gives $a_\mu^{\rm HVP}=682\,(19)\times10^{-10}$, consistent with dispersive determinations. If the result holds, it tightens the first-principles HVP prediction at the precision target of the new muon $g-2$ experiments.

What carries the argument

The argument is carried by the RM123 method: an expansion of the QCD path integral in powers of the quark mass difference and of $\alpha_{\rm em}$ around an isosymmetric QCD baseline, with a mass and coupling matching prescription in the $\overline{\rm MS}$ scheme at 2 GeV. The vector-current correlator $V(t)$ is expanded as $V^{(0)}(t)+\delta V(t)$, and $\delta V(t)$ is built from self-energy, exchange, tadpole, pseudoscalar, and scalar insertion diagrams; the photon zero mode is removed with the QEDL prescription. The quenched-QED approximation treats sea quarks as electrically neutral, and quark-disconnected diagrams are neglected. The HVP integral uses the time-momentum representation, and for the light quark the ratio $\delta a_\mu^{\rm HVP}(ud)/a_\mu^{{\rm HVP},(0)}(ud)$ is extrapolated with a fit that includes chiral logarithms or quadratic terms, an $a^2$ discretization term, and exponential or power-law finite-volume corrections; strange and charm use linear fits with $1/L^3$ terms. Non-perturbative QCD+QED renormalization constants for the vector current and quark masses are the input that improves the strange and charm errors by roughly a factor of three relative to the earlier calculation.

What would settle it

Computing the quark-disconnected QED diagram for the light-quark contribution on the same gauge ensembles and comparing its size with the connected QED piece $1.1\,(1.0)\times10^{-10}$ would test the central systematic assumption; a disconnected contribution substantially larger than $1\times10^{-10}$ would move $\delta a_\mu^{\rm HVP}(udsc)$ outside the quoted total uncertainty of $2.9\times10^{-10}$.

Watch

Extended reading notes

Core claim

At leading order in $\alpha_{\rm em}$ and $(m_d-m_u)/\Lambda_{\rm QCD}$, the paper’s central claim is that the total quark-connected isospin-breaking correction to the HVP contribution is $\delta a_\mu^{\rm HVP}(udsc)=7.1\,(2.9)\times10^{-10}$, the most accurate determination available. The individual flavor contributions after extrapolation are $\delta a_\mu^{\rm HVP}(ud)=7.1\,(2.5)\times10^{-10}$, $\delta a_\mu^{\rm HVP}(s)=-0.0053\,(33)\times10^{-10}$, and $\delta a_\mu^{\rm HVP}(c)=0.0182\,(36)\times10^{-10}$. The light-quark term splits into a QED part $1.1\,(1.0)\times10^{-10}$ and a strong-isospin part $6.0\,(2.3)\times10^{-10}$, so the correction is dominated by the up-down mass difference. Adding an estimated uncertainty for the quenched-QED and disconnected-diagram approximation, the authors quote $\delta a_\mu^{\rm HVP}(udsc)=7.1\,(2.6)\,(1.2)\times10^{-10}$ and combine it with leading-order connected and disconnected HVP results to obtain $a_\mu^{\rm HVP}=682\,(19)\times10^{-10}$, which agrees with dispersive determinations. The result agrees within errors with earlier dispersive and lattice estimates and is more precise than them.

Load-bearing premise

The load-bearing assumption is that the uncertainty from the quenched-QED approximation together with the neglect of quark-disconnected diagrams is about $1.0\times10^{-10}$, the size of the connected QED term; that estimate is not derived, so if either effect is larger the central value shifts beyond the quoted error.

Editorial extensions

If this is right

  • At the precision of the new muon $g-2$ experiments, the isospin-breaking correction of about $7\times10^{-10}$ is not negligible and must be included in a first-principles HVP prediction.
  • Because the light-quark correction is dominated by the strong up-down mass difference term $6.0\,(2.3)\times10^{-10}$, improving the input value of $m_d-m_u$ directly shrinks the total error.
  • The strange and charm corrections, both below $10^{-10}$, can be neglected at current precision without materially changing the result.
  • Adding the IB correction to leading-order HVP estimates gives $a_\mu^{\rm HVP}=682\,(19)\times10^{-10}$, compatible with dispersive determinations, so the IB corrections do not by themselves alter the muon $g-2$ discrepancy.

Reading between the lines

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

  • The assigned qQED-plus-disconnected uncertainty is about one-third of the total error; if the disconnected QED light-quark diagram is as large as one lattice group’s estimate, that uncertainty is a floor, and a future unquenched-QED simulation could shift the central value by roughly $1\times10^{-10}$ or more.
  • Because the paper extrapolates the ratio $\delta a_\mu^{\rm HVP}(ud)/a_\mu^{{\rm HVP},(0)}(ud)$, its absolute IB correction is tied to the leading-order HVP normalization; another group with a more precise $a_\mu^{{\rm HVP},(0)}$ could rescale the ratio to obtain an updated absolute value immediately.
  • The same RM123 machinery could be applied to the hadronic light-by-light contribution to $g-2$, whose leading isospin-breaking corrections are so far unknown; a disconnected diagram issue there may be more severe than in HVP.
  • The published strange-quark value differs from a separate lattice result by about two standard deviations; a combined reanalysis of the two raw datasets would clarify whether this is a statistical fluctuation or a systematic 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 / 4 minor

Summary. This proceedings contribution reports a lattice QCD calculation of the leading electromagnetic and strong isospin-breaking (IB) corrections to the quark-connected hadronic-vacuum-polarization (HVP) contribution to the muon magnetic anomaly. The calculation uses the RM123 expansion in the quenched-QED (qQED) approximation, ETMC N_f=2+1+1 configurations at three lattice spacings, several volumes, and pion masses between about 210 and 450 MeV, with QED_L zero-mode removal and nonperturbative renormalization constants. After chiral, continuum, and infinite-volume extrapolations the authors obtain delta a_mu^HVP(ud)=7.1(2.5) x 10^-10, delta a_mu^HVP(s)=-0.0053(33) x 10^-10, and delta a_mu^HVP(c)=0.0182(36) x 10^-10. Adding an estimated 1.2 x 10^-10 uncertainty for the qQED approximation and neglected quark-disconnected diagrams, they quote delta a_mu^HVP(udsc)=7.1(2.9) x 10^-10 and claim this is the most accurate determination of the IB corrections to a_mu^HVP to date.

Significance. If the quoted uncertainty is reliable, this is a valuable first-principles cross-check of both the dispersive estimates and other lattice determinations, improving on the BMW value 7.8(5.1) x 10^-10 and the RBC/UKQCD value 9.5(10.2) x 10^-10. The paper is methodologically transparent in several respects: it uses the time-momentum representation, an O(a)-improved twisted-mass setup, dedicated finite-volume ensembles, and nonperturbative renormalization constants from Ref. [27]. It also gives a detailed error budget separating statistical, input, chiral, finite-volume, and discretization effects. The main caveat is that the 'most accurate' claim depends on an estimated, rather than computed, uncertainty for the qQED approximation and disconnected diagrams; this is a load-bearing point that needs to be addressed before the headline claim can be accepted.

major comments (3)
  1. [Section 3, Eq. (3.15)] The central quantitative claim rests on an estimate rather than a calculation. The text states that the uncertainty from the qQED approximation and the neglect of quark-disconnected diagrams is 'approximately equal to our QED contribution (3.8)', i.e. about 1.1 x 10^-10, quoted as 1.2 x 10^-10 in Eq. (3.15). This estimate is not derived: the qQED sea-quark effect is not evaluated, and the cited RBC/UKQCD observation concerns a single disconnected diagram, not the full set of disconnected contractions nor the qQED sea effect. The 'most accurate' claim fails if the neglected effect is larger than estimated; for example, if the combined qQED/disconnected systematic were about 4.4 x 10^-10 (a factor of roughly four larger), the total uncertainty in Eq. (3.15) would become comparable to the BMW error of 5.1 x 10^-10, and the claimed superiority over BMW would disappear. I do not see an internal inconsistency, but the precision headline needs either a computed qQED/disconnected contribution, a conservative upper bound, or a revised wording that restricts the claim to the connected qQED contribution.
  2. [Section 3, Eq. (3.13)] The paper reports a roughly 2-sigma deviation from RBC/UKQCD for the strange-quark contribution, delta a_mu^HVP(s). Although the strange term is numerically small and does not by itself change the total central value, this tension between two lattice determinations is exactly the kind of signal that could indicate a missing systematic in one of the calculations, and it is directly relevant to the credibility of the estimated qQED/disconnected systematics. The revised manuscript should discuss possible sources of this discrepancy, such as renormalization constants, qQED sea effects, finite-volume effects, or the different treatment of disconnected diagrams.
  3. [Section 3, Eqs. (3.15)-(3.16)] The comparison with BMW and RBC/UKQCD is not fully apples-to-apples. Eq. (3.15) is a connected-only result within the qQED approximation plus an estimated uncertainty, whereas the quoted BMW estimate is a full dispersive QCD+QED estimate and the RBC/UKQCD result includes at least one disconnected diagram. If the qQED/disconnected uncertainty remains an estimate, the phrase 'most accurate determination of the IB contribution to a_mu^HVP' should be either qualified to 'most accurate connected qQED determination' or backed by a calculation that places the neglected effects on the same footing as the other determinations.
minor comments (4)
  1. [Section 2 (near Figure 1)] The text between Section 2 and Section 3 contains a large duplicated passage from Ref. [14], beginning with 'where mud 1/4...' and repeated several times. This appears to be a copy-paste error and should be removed from the submitted version.
  2. [Section 3, Eq. (3.4)] The fit coefficients delta A_l1l, delta A_l2, delta D_l, and delta F_l in Eq. (3.4) are introduced only in prose; they should be defined explicitly in the equation or immediately after it so the fit ansatz is self-contained.
  3. [Section 3, Eq. (3.15)] The symbol 'qQED+disc' is used for an uncertainty, but the text says only that it is 'approximately equal to our QED contribution (3.8)'. Please state explicitly whether this is meant as a 1-sigma systematic uncertainty and justify the rounding from 1.1 x 10^-10 to 1.2 x 10^-10.
  4. [Figure 3] The figure caption states that the errors are statistical only, but the final result in Eq. (3.6) includes several systematic errors. It would be useful to show the fully combined uncertainty on the physical-point marker, or to explain in the caption why the systematic errors are not displayed there.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the IB corrections are computed from independent lattice inputs, and the qQED/disconnected uncertainty is a stated estimate, not a fitted quantity.

full rationale

The derivation chain is self-contained: the final IB correction in Eq. (3.15) is the sum of lattice-determined connected contributions, each obtained from measured δV_f(t) correlators and extrapolated with Eqs. (3.4) and (3.10). No parameter is fitted to the target quantity δa_mu^HVP(udsc). The external inputs—md-mu from kaon masses [17], renormalization constants from [24,27], leading-order HVP values from [23,28], and finite-volume corrections from [28]—do not contain the target result, and the cited works are independent lattice or experimental determinations rather than definitions of the IB correction. The ansätze in Eqs. (3.4) and (3.10) are extrapolation models whose systematic effects are estimated by comparing variants; they do not build the final number into the fit. The qQED and quark-disconnected systematic is admittedly an estimate ('we estimate that the uncertainty related to the qQED approximation and to the neglect of quark-disconnected diagrams is approximately equal to our QED contribution (3.8)', Sec. 3), which is a limitation in rigor but not a circular reduction: the quoted central value is not constructed to equal that estimate. Self-citations are numerous, but the load-bearing inputs trace to separately published, externally falsifiable lattice calculations, which under the stated rules constitute independent evidence rather than circularity.

Assumptions & free parameters 2 free parameters · 7 assumptions · 0 invented entities

The calculation relies on standard lattice QCD machinery and on prior determinations of renormalization constants, md-mu, and lower-order HVP. No new particles or forces are introduced. The principal extra assumptions are qQED, the neglect of disconnected diagrams, and an order-of-magnitude systematic for both effects.

free parameters (2)
  • Light ratio chiral, continuum, and finite-volume fit coefficients (delta A_l0, delta A_l1, delta A_l1l, delta A_l2…
    Fit parameters in Eq. (3.4) fitted to the lattice data for delta a(ud)/a0(ud); they drive the chiral, continuum, and finite-volume extrapolations.
  • Strange and charm ratio fit coefficients (delta A_s,c0, delta A_s,c1, delta D_s,c, delta F_s,c)
    Fit parameters in Eq. (3.10) used for the strange and charm extrapolations.
assumptions (7)
  • domain assumption RM123 expansion truncated at first order in alpha_em and (md-mu)/Lambda_QCD.
    Section 2 states higher-order terms are neglected; the central result relies on this truncation.
  • domain assumption Quenched-QED approximation: dynamical quarks are electrically neutral and QED sea effects are omitted.
    Section 2 explicitly adopts qQED; this is not exact QCD+QED.
  • domain assumption Quark-disconnected contractions are omitted, except that one QED disconnected diagram from Ref. [38] is used only to estimate the uncertainty.
    Section 3 notes the neglect and approximates its effect.
  • ad hoc to paper The systematic uncertainty for qQED plus disconnected diagrams is estimated as equal to the QED connected contribution of Eq. (3.8).
    Eq. (3.15) adds a 1.2 x 10^-10 uncertainty based on an estimate, not a calculation.
  • domain assumption GRS prescription: renormalized masses and couplings in the full theory match isosymmetric QCD at MS scale 2 GeV.
    Section 2 imposes this matching condition; the final result is expected to be prescription dependent only beyond leading order.
  • domain assumption The value md-mu = 2.38 (18) MeV from Ref. [17] is used for all ensembles.
    Section 2 uses this external input to set the strong isospin-breaking term; an error here shifts the SIB contribution proportionally.
  • domain assumption Finite-volume corrections are modeled by the ChPT-inspired forms in Eq. (3.5), with either exponential or power-law behavior.
    Section 3 chooses these functional forms for the extrapolation; the systematic is estimated by comparing variants.

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Cite this review

Pith. "Pith review of Isospin-breaking corrections to the muon magnetic anomaly in Lattice QCD." pith.science (2026). https://pith.science/paper/QJWKWUNG

@misc{pith2026190901962,
  author       = {Pith},
  title        = {Pith review of: Isospin-breaking corrections to the muon magnetic anomaly in Lattice QCD},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QJWKWUNG}},
  note         = {Machine review of arXiv:1909.01962}
}
abstract

In this contribution we present a lattice calculation of the leading-order electromagnetic and strong isospin-breaking (IB) corrections to the quark-connected hadronic-vacuum-polarization (HVP) contribution to the anomalous magnetic moment of the muon. The results are obtained adopting the RM123 approach in the quenched-QED approximation and using the QCD gauge configurations generated by the ETM Collaboration with $N_f = 2+1+1$ dynamical quarks, at three values of the lattice spacing ($a \simeq 0.062, 0.082, 0.089$ fm), at several lattice volumes and with pion masses between $\simeq 210$ and $\simeq 450$ MeV. After the extrapolations to the physical pion mass and to the continuum and infinite-volume limits the contributions of the light, strange and charm quarks are respectively equal to $\delta a_\mu^{\rm HVP}(ud) = 7.1 ~ (2.5) \cdot 10^{-10}$, $\delta a_\mu^{\rm HVP}(s) = -0.0053 ~ (33) \cdot 10^{-10}$ and $\delta a_\mu^{\rm HVP}(c) = 0.0182 ~ (36) \cdot 10^{-10}$. At leading order in $\alpha_{em}$ and $(m_d - m_u) / \Lambda_{QCD}$ we obtain $\delta a_\mu^{\rm HVP}(udsc) = 7.1 ~ (2.9) \cdot 10^{-10}$, which is currently the most accurate determination of the IB corrections to $a_\mu^{\rm HVP}$.

Figures

Figures reproduced from arXiv: 1909.01962 by the authors.

Figure 1
Figure 1. Fermionic connected diagrams contributing to the IB corrections δa HVP µ (f): self-energy (a), exchange (b), tadpole (c), pseudoscalar (d) and scalar (e) insertions. Solid lines represent the propagators of the quark with flavor f in isosymmetric QCD. corrections δVf(t) consists of two (prescription-dependent) contributions: the em, δV QED f (t), and the strong IB (SIB), δV SIB f (t), one. Diagrams (1a)-(1d) contrib… view at source ↗
Figure 2
Figure 2. Time dependence of the integrand functions Kµ (t)δV SIB ud (t) (top-right panel) and Kµ (t)δV QED f (t) for the light- (top-left panel), strange- (bottom-left panel) and charm-quark (bottom-right panel) contributions to the IB correc￾tions δa HVP µ (f) [see Eq. (3.2)] in the cases of the ETMC gauge ensembles B55.32 (Mπ ' 375 MeV, a ' 0.082 fm) and D20.48 (Mπ ' 260 MeV, a ' 0.062 fm). In the panels the labels “self”,… view at source ↗
Figure 3
Figure 3. Results for the ratio δa HVP µ (ud)/a HVP,(0) µ (ud) versus the renormalized average u/d mass mud in the MS(2 GeV) scheme. The empty markers correspond to the raw data, while the full ones represent the lattice data corrected by the FVEs obtained in the fitting procedure (3.4) with δA ` 1` = 0 and δA ` 2 6= 0. The solid lines correspond to the results of the combined fit (3.4) obtained in the infinite-volume limit a… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Results for the strange (left panel) and charm (right panel) contributions to δa HVP µ /a HVP,(0) µ versus the renormalized average u/d mass mud. The solid lines correspond to the linear fit (3.10) including the discretization term in the infinite-volume limit. The bla…

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