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Calibrated correlation between heavy-quark masses and Hadronic Vacuum Polarization observables at the precision frontier

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The charm and bottom quark masses and their hadronic vacuum polarization contributions are two views of one spectral integral, and the paper determines both at once from a single calibrated sum rule.

desk verdict A genuine new method—promoting the HVP kernel to the weight of a moment sum rule—presented with unusual honesty, but the one-parameter continuum ansatz carries the entire error budget and is calibrated only through the zeroth moment. read the letter →

arxiv 2608.10112 v1 pith:PI55U3JH submitted 2026-08-10 hep-ph hep-lat

Arnau Beltran , Pere Masjuan , Antonio Rivera This is my paper · ORCID
classification hep-phhep-lat
keywords heavy-quarkmasseshadronicvacuumpolarizationmuonanomalousmagneticmomentQCDsumrulesgeneralizedquark-hadrondualitycharmquarkbottom
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

The paper argues that the charm and bottom quark masses and their contributions to the muon anomalous magnetic moment are not independent quantities: both are weighted integrals of the same hadronic spectral function, with only the integration kernel differing. The authors promote the kernel to the weight of a generalized moment sum rule, so one self-consistency condition fixes the mass and the HVP contribution together, with their anticorrelation built into the uncertainty budget. They obtain $\hat m_c(\hat m_c)=1267.1(6.8)$ MeV, $a_\mu^{c,\mathrm{LO}}=14.46(13)\times 10^{-10}$, $\hat m_b(\hat m_b)=4182.3(7.2)$ MeV, and $a_\mu^{b,\mathrm{LO}}=0.3009(17)\times 10^{-10}$, with NLO charm pieces that agree with the first lattice determination. If the paper is right, at least part of the apparent spread among HVP determinations comes from treating correlated quantities as independent, and the construction is a template for any kernel-weighted dispersive observable.

What carries the argument

The carrier of the argument is the generalized kernel-weighted moment $A_n[K]$ of Eq. (23), which reduces to the ordinary moment $M_n$ when $K=1$ and to the HVP integral itself at $n=1$ when $K=\hat K^{(2)}$. The zeroth moment, defined through the ultraviolet limit of the dispersion relation, is the piece most sensitive to the continuum and is what breaks the near-degeneracy between the mass and the continuum shape parameter $\lambda_3^q$; imposing $A_n^{\rm th}[K]=A_n^{\rm had}[K]$ for a pair that includes it fixes $m_q$ and $\lambda_3^q$ simultaneously.

What would settle it

A sub-percent measurement of $R(s)$ for $e^+e^-\to$ hadrons across the open-charm window from about 3.7 to 4.8 GeV would settle it: computing the kernel-weighted zeroth and second moments directly from those data must give a continuum shape parameter compatible with the sum-rule value. A lattice computation of $a_\mu^{c,\mathrm{LO}}$ with uncertainty below $0.05\times 10^{-10}$ that disagrees with $14.46(13)\times 10^{-10}$ would likewise falsify the extraction.

Watch

Extended reading notes

Core claim

The central claim is that the heavy-quark mass $\hat m_q$ and its HVP contribution $a_\mu^{q}$ are determined by the same spectral function $R_q(s)$ through different kernels, so a consistent sum-rule determination must fix them jointly. The paper defines generalized moments $A_n[K]=\int ds\, K(s)R_q(s)/s^{n+1}$ and imposes equality between the perturbative and hadronic evaluations, Eq. (33), for a pair of moments that always includes the zeroth moment. With $K=\hat K^{(2)}$ as the weight, the pair $(M_0,M_2)$ for charm and $(M_0,M_6)$ for bottom yields the quoted masses and $a_\mu$ values together with their correlation matrices. Because the two quantities are anticorrelated, quoting them as a pair reduces the uncertainty on $a_\mu$ compared with treating the mass as an external input, and the residual mismatch of the two evaluations at other moment pairs becomes a direct measure of duality violation and continuum-model systematics.

Load-bearing premise

The load-bearing premise is that one free shape parameter, $\lambda_3^q$, accurately describes the smooth part of the production cross-section above the charm and bottom thresholds for every weighting kernel and every moment order used; if the true spectrum bends differently from that single-parameter curve, the extracted masses and the muon g-2 pieces shift together.

Editorial extensions

If this is right

  • The extracted $\hat m_q$ and $a_\mu^q$ must be reported as a correlated pair; changing one while holding the other fixed is inconsistent with the sum-rule framework.
  • Compared with a naive data-driven estimate that ignores the correlation, the quoted uncertainty on $a_\mu^{c,\mathrm{LO}}$ drops by about 45% and that on $a_\mu^{b,\mathrm{LO}}$ by about 90%.
  • The residual spread between the theory-side and hadronic-side evaluations at moment pairs other than the calibration pair becomes an observable-specific diagnostic of quark-hadron duality violations and continuum-model ambiguity.
  • The same generalized construction applies to any observable expressible as a kernel-weighted dispersive integral over the same spectral function, provided the kernel admits the required analytic subtractions.

Reading between the lines

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

  • If the anticorrelation is as strong as reported, several existing heavy-quark $a_\mu$ values that treat the mass as external input may be quoting inflated uncertainties; re-running those analyses with the correlation could sharpen them without new data.
  • The likely next test is the light-quark sector, where the data-versus-lattice HVP tension lives; the $\hat K^{(4a)}$ example in this paper indicates the main obstacle there will be finding kernels with the right subtraction properties, not the sum-rule matching itself.
  • A direct check of the method's reach would be to apply the generalized condition to the NLO kernels $\hat K^{(4a)}_{\rm sub}$ and $\hat K^{(4b)}$ at several moment pairs and see whether the moment-pair independence that holds at LO survives, exposing where the one-parameter continuum ansatz breaks.
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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 / 5 minor

Summary. The paper proposes a generalized moment-sum-rule framework in which the HVP kernel K(s) is promoted to the weight of the moment, so that the heavy-quark mass m_q and the heavy-quark contribution to a_mu are extracted simultaneously from the same hadronic spectral function. Specializing to K = K^(2), the authors obtain m_c(m_c) = 1267.1(6.8) MeV with a_c(LO) = 14.46(13) x 10^-10, and m_b(m_b) = 4182.3(7.2) MeV with a_b(LO) = 0.3009(17) x 10^-10, together with NLO results for the 4a and 4b kernels. The paper includes detailed uncertainty breakdowns, correlation matrices, and a data-driven calibration of the continuum-shape parameter lambda_3^q. The central claim is that the mass and the HVP contribution are not independent observables, and that the anticorrelation between them can be exploited to reduce the final uncertainty, with the residual spread between theory-side and hadronic-side evaluations serving as a diagnostic of duality/model systematics.

Significance. If the framework is correct, it provides a genuinely new way to organize heavy-quark sum rules: instead of treating a_mu as a derived quantity after a mass determination, it determines both from one self-consistent condition, with an explicit correlation built in. The paper is unusually transparent about the point where the construction becomes circular, namely the 0th+1st moment pair with K = K^(2), and deliberately adopts other pairs for the final numbers. The explicit input tables, correlated uncertainty propagation, and correlation matrices are valuable and make the analysis reproducible in principle. The claimed uncertainty reductions relative to previous dispersive evaluations, roughly 45% for charm and 90% for bottom, are striking and would be important if the error budget can be defended. The main weakness, however, is that the entire extraction relies on a one-parameter continuum ansatz whose uncertainty is calibrated only through a partial zeroth moment in a limited energy window; this is the load-bearing assumption that needs further scrutiny before the quoted precision can be accepted.

major comments (3)
  1. [Sec. IV, Eq. (35)] The continuum model in Eq. (9) has a single free shape parameter lambda_3^q, which is assumed to describe R_cont(s) above the open-heavy-flavor threshold for every kernel and every moment order used in Eq. (33). The data calibration in Sec. V.B, however, constrains only the partial zeroth moment over finite windows (up to 4.8 GeV for charm and 11.2 GeV for bottom), as summarized in Table IV. This does not independently validate the kernel-weighted higher moments that carry the bulk of the a_mu information. Because the spread across moment pairs shown in Figs. 3b and 4b is generated inside the same ansatz, it cannot expose a common model error. I therefore request a direct sensitivity test, for example allowing a second shape parameter for the O(m^4/s^2) term or an n-dependent lambda_3, and an estimate of the shift in both m_q and a_mu when the calibrated window and the continuum shape are varied; the resulting shift should be included in the error budget.
  2. [Sec. V.A, Tables II and III] The extra Euclidean term in A_0^{pQCD}[K^(i)] is dropped with the statement that it is numerically suppressed at the per-mil level, but no numerical estimate or bound is provided. Since the zeroth moment is one of the two constraints used in Eq. (33) to fix m_q and lambda_3^q, even a per-mil shift in A_0 can propagate into a larger shift in the extracted mass and in a_mu. The authors should quantify this term explicitly, or include it in the analysis, before the quoted uncertainties can be considered complete.
  3. [Sec. V.A, Tables II and III] The final a_mu values are described as the average of the central values and uncertainties of the theory-side and hadronic-side evaluations at the default moment pair. These two evaluations are strongly correlated (rho = 0.90 for charm and rho = 0.81 for bottom in Tables VI and VII), so a simple arithmetic average of their uncertainties is not a statistically defined combination. The paper should state the exact prescription used to form the blue band, including whether the covariance is used, and should report the resulting covariance or correlation. This is directly relevant to the central precision claim, since the quoted errors of 0.13 x 10^-10 and 0.0017 x 10^-10 are the headline results.
minor comments (5)
  1. [Sec. V.B] There is a typo in the sentence 'due to the extensions versos these previous works'; 'versos' should be 'versus' or 'with respect to'.
  2. [Fig. 10] The label 'HPQCD-14' appears twice in the same figure; the two points should be distinguished, for example by citing the two different HPQCD determinations explicitly.
  3. [Eq. (35)] The phrase 'scope at the per-mil level' is unclear; it should be rephrased as, for example, 'accurate at the per-mil level'.
  4. [Sec. V.A] The description of the blue band as 'generously covers' the deviations should be replaced by a numerical prescription for how the band width is obtained from the two evaluations.
  5. [Abstract] The abstract and main text use inconsistent formatting for the muon symbol and for the units of a_mu; a uniform notation would improve readability.

Circularity Check

1 steps flagged · score 4.0 of 10

Final central values avoid the explicit self-pinning of the (A0,A1) pair; one acknowledged by-construction identity is confined to an illustrative point, while the continuum model is a correctness risk rather than circularity.

  1. self definitional [Sec. IV (after Eq. 33) and Sec. V A, Fig. 3b, point '0th+1st']
    "Using a pair including the n=1 generalized moment, A_1[\hat K^{(i)}], together with the zeroth moment, to fix m_q and \lambda_3 makes the two descriptions of a^{(i)}_\mu coincide exactly by construction."

    A_1[\hat K] is defined in Eq. (23) as the same integral over the hadronic spectral function that enters a_\mu in Eq. (1), up to a known prefactor. Imposing Eq. (33) for the pair (A_0[\hat K], A_1[\hat K]) therefore uses the equality of the two descriptions of a_\mu as one of the two equations that fix (m_q, \lambda_3), so the 'agreement' of the blue solid and empty points at 0th+1st is an identity, not a test. The paper openly acknowledges this and does not use that pair for the final quotes: the adopted pairs are (A_0[\hat K^{(2)}], A_2[\hat K^{(2)}]) for charm and (A_0[\hat K^{(2)}], A_6[\hat K^{(2)}]) for bottom. The quoted a_\mu values are therefore not literally fitted to themselves; the circularity is confined to the illustrative self-consistency point.

full rationale

The central derivation is otherwise self-contained: masses and HVP contributions are determined by solving Eq. (33) for a pair of generalized moments and then evaluating a_\mu from the same spectral function with the fitted parameters, with the default pairs avoiding the n=1 HVP moment whose agreement is definitional. The paper explicitly flags the by-construction case and uses other moment pairs for its quoted results, so the final numbers are not statistically forced to reproduce a_\mu. The calibration of \lambda_3^{q,exp} against experimental moments is used only to assign an uncertainty, not to set the central value, as stated in Sec. V B. The main residual concern, the one-parameter continuum ansatz of Eq. (9) controlling both the mass and a_\mu, is a modeling and correctness risk (the data calibration covers only partial zeroth moments and limited energy windows), not a circular reduction. Self-citations to Refs. [7-9] and [51] supply the adopted formalism and a lattice comparison, but the framework extends the ansatz and is checked against independent experimental data; they do not by themselves force the claimed result. Score 4 reflects the explicit self-definitional identity at (A_0, A_1), which is acknowledged and avoided in the final extraction.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new physical entities. Its central claim rests on a model of the hadronic continuum (with a fitted shape parameter), global quark-hadron duality, a truncated high-energy kernel expansion, the standard O(α_s^3) pQCD series, and reconstructed logarithmic-moment coefficients. The neglected term in Eq. (35) is an assumption specific to this paper. These are all plausible, but they constitute the main burden a reader must accept.

free parameters (2)
  • λ3^q (continuum shape parameter, charm and bottom) = not stated explicitly for the sum-rule solution; λ3^exp = 0.673(86) (charm, K=Khat) and 0.696(45) (bottom, K=Khat) for…
    Free parameter of the continuum ansatz Eq. (9), fixed by the sum-rule self-consistency condition Eq. (33). It absorbs the unknown shape of the non-resonant hadronic spectral function above threshold. The experimental calibration values in Table IV differ slightly between kernels.
  • κ (global normalization of sub-threshold data) = 0.998 ± 0.011
    Global rescaling of the light-quark background subtraction used only in the data calibration of Sec. V B, not in the central extraction. It is fitted to the combined sub-threshold data.
assumptions (6)
  • domain assumption The continuum ansatz Eq. (9), with a single shape parameter λ3^q, correctly describes the hadronic spectral function above the open-heavy-flavor threshold for all kernels and moment orders used.
    Entering in Sec. III B and used throughout Sec. V; the entire extraction of both m_q and a_mu depends on this model.
  • domain assumption Global quark-hadron duality holds for the integrated moments, meaning the pQCD expression can be equated to the hadronic integral on average.
    Invoked in Eq. (22) and Eq. (33) as the basis of all moment sum rules; a weaker assumption than the local duality discussed in Sec. II.
  • standard math The high-energy expansion of the HVP kernels, Eq. (26), truncated at j=3, is accurate at the quoted precision.
    Used in Eq. (29); m_μ^2/(4 m_q^2) is ~10^-3 for charm and ~10^-4 for bottom, so truncation is expected to be safe, but is an approximation.
  • domain assumption The perturbative QCD series for the vector correlator through O(α_s^3), with the truncation estimate of Eq. (19), is valid for the moments used.
    Theoretical side of the sum rule; the truncation uncertainty is included but the functional form is assumed.
  • domain assumption The logarithmic-moment coefficients C^(i)_{n,r} reconstructed following Refs. [21,33] are correct.
    Table V provides these coefficients; a reconstruction error would feed directly into Eq. (29).
  • ad hoc to paper The additional term in the generalized zeroth moment, Eq. (35), is numerically suppressed and can be dropped at the per-mil level.
    The paper states it is not considered further for present purposes; no quantitative bound is given.

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Pith. "Pith review of Calibrated correlation between heavy-quark masses and Hadronic Vacuum Polarization observables at the precision frontier." pith.science (2026). https://pith.science/paper/PI55U3JH

@misc{pith2026260810112,
  author       = {Pith},
  title        = {Pith review of: Calibrated correlation between heavy-quark masses and Hadronic Vacuum Polarization observables at the precision frontier},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PI55U3JH}},
  note         = {Machine review of arXiv:2608.10112}
}
abstract

The theoretical prediction of the muon anomalous magnetic moment $a_\mu$ depends crucially on the Hadronic Vacuum Polarization (HVP), and the tension between its dispersive and lattice-QCD determinations remains unresolved. We show that part of this puzzle can be addressed in the heavy-quark sector, where both descriptions are theoretically clean, by recognizing that the heavy-quark mass and its contribution to $a_\mu$ are not independent quantities: both follow from integrals of the same hadronic spectral function, differing only in their integration kernel. Promoting this kernel to a free choice within the relativistic QCD Sum Rules used to determine heavy-quark masses, we break with the conventional notion of a single valid sum rule and instead determine the mass and its HVP contribution simultaneously, from a common, self-consistent framework. This intrinsic construction exploits the anticorrelation between the two quantities to sharpen the final uncertainty, and turns the residual disagreement between the perturbative and hadronic descriptions of the observable into a direct observable-specific diagnostic of residual theory/model dependence, including duality-violation and continuum-modeling effects, unavailable to a determination of the mass alone. We obtain $a_\mu^{\rm HVP_{c+b},LO} =(14.46(13)+0.3009(17))\times 10^{-10}$ at leading and $a_\mu^{\rm HVP_{c+b}, NLO_{a,b}} = ( -0.5738(95) - 0.01822(13) )\times 10^{-10}$ at next-to-leading order, for charm and bottom contributions, respectively. We compare our next-to-leading-order results with its first available lattice determination, finding good agreement in the charm sector. As a byproduct, we obtain $\hat m_c(\hat m_c)=1267.1(6.8)$ MeV and $\hat m_b(\hat m_b) = 4182.3(7.2)$ MeV, with unprecedented phenomenological precision.

Figures

Figures reproduced from arXiv: 2608.10112 by the authors.

Figure 1
Figure 1. FIG. 1: Charm-quark mass ˆm [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Bottom-quark mass ˆm [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. b is given in Table II. As final result at a given moment pair, we adopt the combination of the theory￾side and hadronic-side evaluations of a q,(i) µ , since this pro￾vides the most conservative determination, incorporating not only the sources of uncertainty discussed in Sec. III but also the residual duality-violation and continuum￾modeling systematic exposed by their mutual spread. We stress that the results sho… view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: FIG. 4: (a) ˆm [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Sub-threshold data ( [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 8
Figure 8. Figure 8: FIG. 8: ( [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 9
Figure 9. Figure 9: compares our determination of a c µ (LO) with other dispersive evaluations (Bodenstain-12 from Ref. [44], Keshavarzi-18 from [36], Erler-20 from [45]) and with the recent reanalysis Kennedy-21 of Ref. [46], together with lattice results Refs. [47–50]. The result Bodens…
Figure 10
Figure 10. Figure 10: similarly compares our determination of a b µ (LO) with the same dispersive evaluations, together with HPQCD lattice estimates from Refs. [42, 43]. Figures 11 and 12 show the analogous comparison for a c µ (NLOa) and a c µ (NLOb), respectively, together with the latti…
Figure 11
Figure 11. Figure 11: FIG. 11: Comparison of [PITH_FULL_IMAGE:figures/full_fig_p014_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12: Comparison of [PITH_FULL_IMAGE:figures/full_fig_p014_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13: Bands in the ( ˆm [PITH_FULL_IMAGE:figures/full_fig_p018_13.png]

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