{"id":"d6a95a51-ec09-4427-8063-eb08b69bc528","arxiv_id":"2412.14760","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"The isospin-violating connected part and mass counterterm of the hadronic vacuum polarisation are computed on the lattice at the SU(3) symmetric point, giving 7.3(2.1) x 10^-11.","lead":"Lattice QCD calculation of the isospin-violating part of the hadronic vacuum polarisation contribution to the muon anomalous magnetic moment at the SU(3) flavour symmetric point, using a Pauli-Villars regulated photon propagator. The result, 7.3(2.1) x 10^-11, is an early step toward the physical-point calculation needed to compare the Standard Model with the Fermilab muon g-2 measurement.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The final 7.3(2.1) x 10^-11 is numerically set by the external TMR mass derivative used in the counterterm, yet no test shows that this derivative is consistent with the CCS representation of this work.","rationale":"The reader's weakest_assumption correctly identifies the external TMR mass derivative as the load-bearing input, and my independent reading of the paper reaches the same conclusion. The final result is numerically dominated by the counterterm, so the consistency of this derivative with the CCS calculation is not an auxiliary concern but the central condition for the quoted number. The gluonless crosschecks and the OPE-based fit for the kaon mass splitting are genuine independent support, but they do not constrain the derivative from Ref. [9]. Because the paper already carries a CONDITIONAL verdict from the reader, and my concern confirms that condition rather than introducing a new one, the appropriate verdict is unchanged. The proposed concrete test, a direct CCS computation of the same derivative, would settle whether the concern actually lands; until such a test or an equivalent crosscheck is provided, the numerical result should not be treated as a final prediction.","tokens_in":8010,"tokens_out":5487,"duration_ms":43729,"concrete_test":"Compute d a_HVP,38 / d(m_u - m_d) at the SU(3)_f symmetric point directly in the CCS representation on the same CLS ensembles, using exactly the local vector currents and Z_V factors from Section 3, and extrapolate to the continuum. Compare this CCS value with the TMR-based value from Ref. [9] used in Eq. (4). If the two continuum values agree within their combined errors, the counterterm is validated; if they disagree by more than the quoted uncertainty of the Ref. [9] derivative, the final 7.3(2.1) x 10^-11 value is biased and needs to be revised or re-derived with the CCS derivative.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central number 7.3(2.1) x 10^-11 is controlled by the counterterm in Eq. (4), not by the connected diagrams computed in this work. On N202 at Lambda = 16 m_mu, the counterterm contributes 9.54(1.24) x 10^-11 while the connected contribution is -0.148(191) x 10^-11 (Section 5). The counterterm multiplies d a_HVP,38 / d(m_u - m_d), taken from Ref. [9], which was obtained in the time-momentum representation (TMR). For the final result to be valid, this TMR derivative must be the same quantity required by the covariant coordinate-space (CCS) expression in Eq. (2), after the same renormalisation and continuum extrapolation, and its quoted uncertainty must cover all relevant systematics. The proceedings do not demonstrate representation-independence of this derivative, do not quantify correlations between the external derivative and the CCS connected contribution, and do not establish that the ensembles, quark masses, and renormalisation scheme in Ref. [9] match the present calculation. Because the connected signal is consistent with zero and the counterterm dominates both the central value and the error, a mismatch in this single external input would shift the final result directly. The paper itself implicitly flags this dependency by acknowledging that the derivative was provided by a separate TMR calculation, but it provides no numerical crosscheck of that input.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":8326,"tokens_out":3823,"duration_ms":32443,"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":[{"comment":"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.","section":"§4, Eq. (4)"},{"comment":"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.","section":"§5, Table 3"},{"comment":"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.","section":"§4, Eq. (11)"},{"comment":"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.","section":"§3, Eq. (8)"}],"minor_comments":[{"comment":"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.","section":"Table 1"},{"comment":"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'.","section":"Introduction and §3"},{"comment":"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.","section":"§4"},{"comment":"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.","section":"§4, Eq. (4)"},{"comment":"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.","section":"§5"}],"recommendation":"major_revision","confidential_remarks":"The proceedings format is appropriate and the gluonless crosschecks give genuine support to the connected calculation. The main risk is the reliance on an external TMR mass derivative for the counterterm that dominates both the central value and the error; this should be resolved before the result is quoted as a final number. The authors may be able to address this with a modest additional analysis, so major revision rather than rejection seems appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a solid proceedings paper that does something genuinely new—first lattice estimate of the isospin-violating HVP at the SU(3)f symmetric point using a Pauli-Villars regulated photon in the covariant coordinate-space method. The gluonless crosschecks are convincing: the continuum extrapolations of the connected diagrams match continuum perturbation theory for the good discretisations. The kaon mass splitting calculation follows the expected OPE log behavior, and the fit is reasonable.\n\nThe soft spot is exactly where the stress-test puts it. The final 7.3(2.1) x 10^-11 is not set by the connected diagrams (which are consistent with zero at the symmetric point) but by the counterterm. That counterterm multiplies d a_HVP/d(m_u - m_d) taken from a TMR calculation in Ref. [9]. Nothing in this paper shows that this derivative is representation-independent, or that the renormalisation and discretisation systematics in the TMR calculation carry over to the CCS framework. If that derivative is not the same quantity, the central value shifts directly. The authors do acknowledge the derivative comes from a separate calculation, and thank the provider, but they give no numerical crosscheck. For a proceedings this might be acceptable as work in progress; as a final number it would not be.\n\nThere are smaller issues. The 'little to no dependence on Lambda' claim is stronger than the data: the four points in Table 3 are all within errors, but with errors of ~2 x 10^-11 they would cover a real slope. The H200 ensemble is excluded from the kaon mass splitting extrapolation because of large volume effects; that is a post-hoc selection, and the paper does not quantify the impact. The correlation coefficient of 0.9 in the Lambda fit is an estimate, not computed from data.\n\nWhat is good: the paper is honest about what dominates, the method is clearly described, and the crosschecks are a real virtue. This is not a claim about the physical point, so it should be read as a methods milestone. I would take the central number with a grain of salt until the external derivative is crosschecked in CCS, but the approach deserves serious engagement.","headline":"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.","tokens_in":8932,"tokens_out":1704,"would_cite":false,"duration_ms":13564,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["12.38.Gc","14.60.Ef","13.40.Em"],"model":"deepseek-v4-flash","headline":"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.","keywords":["lattice QCD","hadronic vacuum polarisation","muon g-2","isospin breaking","Pauli-Villars regularisation","covariant coordinate-space method","kaon mass splitting","SU(3) flavour symmetric point"],"falsifier":"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.","tokens_in":7750,"feed_emoji":"🧲","tokens_out":10329,"duration_ms":67941,"temperature":0.7,"pith_summary":"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.","feed_headline":"Lattice QCD finds isospin-breaking muon g-2 shift: 7.3(2.1) x 10^-11","feed_subtitle":"This correction is a required ingredient for a sub-percent Standard Model prediction of the muon's magnetic moment.","key_machinery":"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].","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the doubly Pauli-Villars regulated photon propagator and the continuum dispersion-relation calculation used for crosschecking the connected diagrams.","marker":"[4]"},{"why":"Provides the covariant coordinate-space representation of the HVP and its integration kernel.","marker":"[5]"},{"why":"Introduces the traceless kernel variant employed here and demonstrates the coordinate-space method as an alternative to the time-momentum representation.","marker":"[6]"},{"why":"Gives the physical charged-neutral kaon mass splitting entering the counterterm.","marker":"[7]"},{"why":"Supplies the derivative of the leading-order HVP with respect to the light-quark mass difference, the dominant ingredient of the counterterm.","marker":"[9]"},{"why":"Provides the nonperturbative renormalisation factor $\\hat Z_V$ for the local vector currents.","marker":"[10]"},{"why":"Establishes that only two Feynman diagrams contribute to the electromagnetic kaon mass splitting at the $SU(3)_\\mathrm{f}$ symmetric point.","marker":"[11]"}],"fun_headline_variants":["Isospin violation in muon g-2: lattice QCD gets 7.3(2.1)x10^-11","Lattice QCD: isospin shift in muon g-2 is 7.3(2.1)x10^-11","Muon g-2: lattice QCD quantifies isospin violation at 7.3(2.1)x10^-11","Small nonzero isospin effect on muon g-2: 7.3(2.1)x10^-11 from lattice"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Isospin violation in muon g-2: lattice QCD gets 7.3(2.1)x10^-11","Lattice QCD: isospin shift in muon g-2 is 7.3(2.1)x10^-11","Muon g-2: lattice QCD quantifies isospin violation at 7.3(2.1)x10^-11","Small nonzero isospin effect on muon g-2: 7.3(2.1)x10^-11 from lattice"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000953,"raw_usage":{"total_tokens":4074,"prompt_tokens":964,"completion_tokens":3110,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":580,"completion_tokens_details":{"reasoning_tokens":2977}},"tokens_in":580,"tokens_out":3110,"duration_ms":19440,"temperature":1.0,"reasoning_tokens":2977,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T11:56:42.209025+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"G\\'erardin, T","cited_arxiv_id":null,"evidence_quote":"Establishes that only two Feynman diagrams contribute to the electromagnetic kaon mass splitting at the $SU(3)_\\mathrm{f}$ symmetric point."}],"review_version":1}