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Calculation of meson charge radii using model-independent method in the PACS10 configuration

T0 review · 3 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper establishes that pion and kaon charge radii can be extracted from lattice QCD without any fit ansatz, reporting $\langle r^2\rangle_{\pi^+}=0.423(10)\,\mathrm{fm}^2$ and $\langle r^2\rangle_{K^+}=0.373(4)\,\mathrm{fm}^2$ on the…

desk verdict A clean preliminary lattice report that earns its place in the proceedings, but the headline K+ precision rests on coefficients and systematics the paper itself does not yet control. read the letter →

arxiv 2412.18166 v1 pith:UOCO6AFQ submitted 2024-12-24 hep-lat

Kohei Sato , Hiromasa Watanabe , Takeshi Yamazaki for PACS Collaboration This is my paper · ORCID
classification hep-lat MSC 81V0581T25 PACS 12.38.Gc13.40.Gp
keywords latticeQCDchargeradiuspionkaonelectromagneticformfactormodel-independentmethodspatialmomentPACS10
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 preliminary lattice QCD charge radii for the pion and kaon using a model-independent spatial-moment method that computes the first derivative of the electromagnetic form factor at zero momentum transfer without assuming a fit function. On the coarsest PACS10 ensemble, a physical-point lattice with extent $(10.9\,\mathrm{fm})^4$ and spacing $0.085\,\mathrm{fm}$, the method gives $\langle r^2\rangle_{\pi^+}=0.423(10)\,\mathrm{fm}^2$ and $\langle r^2\rangle_{K^+}=0.373(4)\,\mathrm{fm}^2$. These values are consistent with PDG22 and with conventional fit-ansatz analyses on the same ensemble, but the kaon radius comes with an uncertainty about eight times smaller than the experimental value. The paper's message is that one of the four standard systematic errors in lattice charge radii, the fit ansatz, can be removed outright when the volume is large enough for the moment combination to suppress higher-order momentum contamination.

What carries the argument

The central object is the normalized second spatial moment of the three-point function, $C_{X,\mathrm{3pt}}^{(1)}(t)=\sum_x x^2 C_{X,\mathrm{3pt}}(t;x)/\sum_x C_{X,\mathrm{3pt}}(t;x)$, which in infinite volume equals half the derivative with respect to $p^2$ of the momentum-space three-point function at $p^2=0$. Because the lattice volume is finite, the paper forms the linear combination $R_X^{\mathrm{MI}}(t)=\alpha_1 C_{X,\mathrm{3pt}}^{(1)}(t)+\alpha_2 C_{X,\mathrm{3pt}}^{(2)}(t)+h$, choosing $\alpha_1$, $\alpha_2$, and $h$ so that the unwanted higher-order contamination from $Q^4$, $Q^6$, and beyond cancels. The flat plateau of $R_X^{\mathrm{MI}}(t)$ in the source-sink time window then gives the first derivative of the electromagnetic form factor at zero momentum transfer directly, bypassing any assumed functional form. This combination is the mechanism by which the fit-ansatz systematic error is eliminated, and the improved log-moment variant is evaluated only as a consistency check.

What would settle it

Repeat the extraction on the same ensemble at a different source-sink separation, such as $t_{\mathrm{sink}}=48$, and on the two finer PACS10 lattice spacings; if the plateau value of $R_X^{\mathrm{MI}}(t)$ shifts by more than the quoted errors, the radii are contaminated by excited states or by uncancelled higher-order momentum terms. A complementary check is to compare the model-independent slope with the numerical derivative of directly computed form factors at sufficiently small $Q^2$ that the Taylor remainder is negligible.

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Extended reading notes

Core claim

The collaboration applies the model-independent spatial-moment method to the coarsest PACS10 ensemble, a $128^4$ lattice at $a=0.085\,\mathrm{fm}$ with physical pion and kaon masses in a $(10.9\,\mathrm{fm})^4$ box. The central claim is that the combination $R_X^{\mathrm{MI}}(t)=\alpha_1 C_{X,\mathrm{3pt}}^{(1)}(t)+\alpha_2 C_{X,\mathrm{3pt}}^{(2)}(t)+h$ has a time-independent plateau whose value directly equals $-\langle r^2\rangle_X/6$, with the coefficients chosen to cancel the $Q^4$ and higher terms in the form-factor Taylor expansion. From this plateau they obtain $\langle r^2\rangle_{\pi^+}=0.423(10)\,\mathrm{fm}^2$ and $\langle r^2\rangle_{K^+}=0.373(4)\,\mathrm{fm}^2$. The same ensemble analyzed with monopole, polynomial, z-expansion, and NLO chiral perturbation theory fits gives mutually consistent results with visible ansatz-to-ansatz spread; the model-independent values lie within that spread with smaller errors. The paper concludes that the fit-ansatz error is removed rather than estimated, and that the resulting $K^+$ radius is about eight times more precise than the PDG22 experimental value.

Load-bearing premise

The load-bearing premise is that the coefficients $\alpha_1$, $\alpha_2$, and $h$ in Eq. (3) cancel the higher-order $Q^2$ contamination of the form-factor Taylor expansion on the PACS10 volume to well below the quoted statistical errors, with the flat region of $R_X^{\mathrm{MI}}(t)$ in Fig. 1 dominated by the ground state at $t_{\mathrm{sink}}=36$.

Editorial extensions

If this is right

  • Fit-ansatz error can be dropped from the systematic-error budget for pion and kaon charge radii on physical-point, large-volume lattices.
  • The $K^+$ charge radius, $0.373(4)\,\mathrm{fm}^2$, is determined from this lattice calculation more precisely than from experiment.
  • The model-independent and traditional analyses agree on the same ensemble, so the ansatz-to-ansatz spread in the traditional method is confirmed as a genuine systematic rather than statistical noise.
  • The remaining work on this ensemble is to evaluate the source-sink time separation and continuum extrapolation, which the paper explicitly leaves for future study.

Reading between the lines

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

  • The same moment construction should transfer to other meson and baryon form-factor slopes, such as the neutron electric radius, since nothing in the derivation depends on the pion's quantum numbers.
  • A direct test of the ground-state assumption would be to repeat the extraction at a longer source-sink separation, such as $t_{\mathrm{sink}}=48$, on the same ensemble; the paper does not report such a variation.
  • The method fixes only the slope at $Q^2=0$, not the shape of $F(Q^2)$; combining the plateau value with a few low-$Q^2$ form-factor points would give a parameter-free curvature check that could be compared with z-expansion fits.
  • If the same precision survives the continuum extrapolation at the two finer PACS10 spacings, the combined result could reduce the pion charge radius uncertainty below the current lattice average and sharpen the comparison with electron-scattering measurements.
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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. The paper reports preliminary lattice QCD results for the pi+ and K+ charge radii on the coarsest PACS10 ensemble (a = 0.085 fm, spatial extent 10.9 fm, physical light and strange quark masses). The authors use a 'model-independent' spatial-moment method (Eq. 3) and compare it with a traditional form-factor analysis using six fit ansaetze. They quote <r^2>_{pi+} = 0.423(10) fm^2 and <r^2>_{K+} = 0.373(4) fm^2 in Eq. (9), state that both are consistent with PDG22 and previous lattice calculations, and note that the K+ result has a smaller quoted error than the experimental value. The paper is a proceedings contribution and explicitly acknowledges that source-sink separation and continuum-extrapolation systematics are not yet included.

Significance. If the quoted numbers survive a full systematic analysis, the work would demonstrate a useful cross-check of charge-radius determinations that avoids fit-ansatz bias, and the K+ radius could become one of the most precise available values. The strengths are the physical-point, large-volume ensemble, the comparison of six form-factor ansaetze, the agreement between the original and improved spatial-moment variants, and the transparency about missing systematics. However, the current version does not provide the quantitative residual bounds needed to turn Eq. (9) into a precision statement; at present the quoted errors are statistical only.

major comments (3)
  1. [Section 2, Eq. (3) and Section 3.3, Eq. (9)] The coefficients alpha1, alpha2 and h that enter R_MI(t) are not specified in this manuscript, and no numerical estimate is given for the residual higher-order Taylor contamination that they are meant to cancel. Since the central values in Eq. (9) come from a constant fit to R_MI(t), the extraction is only as model-independent as this cancellation. The authors should either give the explicit coefficients and the derivation of the cancellation in the text, or provide a quantitative bound from the next uncancelled term at L = 10.9 fm; agreement between the original and improved variants does not by itself bound the absolute residual.
  2. [Section 3.2 and Fig. 1] All results are obtained at a single source-sink separation t_sink = 36, and the 'time-independent region' shown in Fig. 1 is not sufficient to exclude a common excited-state offset. The manuscript should show at least one additional t_sink value or an equivalent excited-state analysis to demonstrate that the flat plateau in R_MI(t) corresponds to the ground-state matrix element; otherwise the quoted error bars on Eq. (9) omit a potentially dominant systematic effect.
  3. [Section 3.1 and Section 4] The calculation is performed on a single lattice spacing a = 0.085 fm, and no estimate of discretization effects is given, even though the collaboration has finer PACS10 ensembles. Consequently, the agreement with PDG22 shown in Fig. 3 is only a statistical comparison, and the statement that the K+ radius is 'more accurate than the experimental value' should be restricted to statistical precision. The text should either include a continuum-extrapolation estimate or explicitly frame Eq. (9) as a single-lattice-spacing check.
minor comments (3)
  1. [Eq. (2) and Eq. (4)] The derivative identity in Eq. (2) is written for the unnormalized momentum-space correlation function, while the moments in Eq. (4) are normalized by the sum over x of C_3pt(t;x); please clarify the normalization so that the relation to R_MI(t) in Eq. (3) is unambiguous.
  2. [Section 3.3 and Fig. 1] Please state the exact fitting window and the criterion used to select the flat region for the constant fits to R_MI(t), since the final values in Eq. (9) depend on that choice.
  3. [Section 2 and Fig. 2] The improved model-independent method 'with log function' is not defined in the text; please include the defining equation or indicate precisely which equations of Refs. [3,4] are being used.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Eq. (9) is obtained from a direct spatial-moment derivative extraction, with the improved-method self-citations used only as a cross-check, so the central result is not reduced to its inputs.

full rationale

The paper's central quantities, the pion and kaon charge radii in Eq. (9), are extracted as the intercept of the flat region of R_MI(t) in Eq. (3), which is built from spatial moments of the three-point function via the Fourier identity in Eq. (2). This is not a fit of the radius to an assumed form factor; no fitted parameter is renamed as a prediction. The coefficients alpha1, alpha2 and h in Eq. (3) are stated to cancel higher-order contamination, but the manuscript does not determine the charge radius from those coefficients themselves; the extraction is a constant fit to a time-independent region. Even if the coefficients are underdocumented, that is a systematic uncertainty rather than a circular reduction. The renormalization factor Z_V is fixed by the zero-momentum form factor being the electric charge, which constrains normalization at Q^2=0 and does not determine the derivative at Q^2=0. Self-citations [3,4] appear only when the authors' improved model-independent method is used as a consistency check, while the main results use the original model-independent method [2], which is an external reference. The results are additionally compared with PDG22 and previous independent lattice calculations, providing external benchmarks. The concerns about unquantified higher-order cancellation and fixed t_sink are genuine systematic-error risks, but they are not instances of the derivation being equivalent to its inputs by construction, so no circular step can be exhibited.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The calculation uses standard lattice QCD machinery and the moment method from Refs. [2-4]. The main unexamined ingredient is the set of coefficients in Eq. (3) and the assumption that residual O(Q^4) contamination and excited states are below the statistical errors. No new entities are introduced. The reported errors are statistical only.

free parameters (2)
  • alpha1, alpha2, h (coefficients in Eq. (3)) = not reported
    Parameters chosen to cancel higher-order contamination in the model-independent combination. They are not fitted to the form-factor data in this paper, but their values and derivation are only given by reference to Refs. [2-4].
  • Traditional fit ansatz parameters (monopole mass, polynomial and z-expansion coefficients, ChPT LECs) = not tabulated
    Used only for the traditional-method comparison in Section 3.3; they do not enter the model-independent result.
assumptions (5)
  • domain assumption The Iwasaki gauge action with Nf=2+1 stout-smeared nonperturbatively O(a)-improved Wilson quarks at a=0.085 fm reproduces continuum QCD to within the quoted precision, despite being a single lattice spacing.
    Invoked throughout Sections 3.1 and 4; no continuum extrapolation is performed, so this assumption is load-bearing.
  • standard math The Fourier derivative identity d C3pt/dp^2 at p^2=0 equals (1/2) sum_x x^2 C3pt(x) in the infinite-volume limit (Eq. 2).
    Used in Section 2 to justify the moment method; it is exact in infinite volume but relies on the infinite-volume limit.
  • domain assumption Finite-volume corrections to the moment method are negligible on the 10.9 fm PACS10 lattice.
    The paper argues contamination is suppressed at large volume but does not quantify the residual finite-volume effect.
  • domain assumption The coefficients alpha1, alpha2, h in Eq. (3) cancel the higher-order Q^2 contamination to below the statistical precision, as derived in Refs. [2-4].
    This is the core of the model-independent method; the present paper does not reproduce the derivation or demonstrate the cancellation on this ensemble.
  • domain assumption The flat region of R_MI(t) between source and sink is ground-state dominated at t_sink=36, so constant fits extract the true derivative.
    Section 3.3 uses a constant fit of a flat region; no source-sink variation is performed to check excited-state contamination.

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Cite this review

Pith. "Pith review of Calculation of meson charge radii using model-independent method in the PACS10 configuration." pith.science (2026). https://pith.science/paper/UOCO6AFQ

@misc{pith2026241218166,
  author       = {Pith},
  title        = {Pith review of: Calculation of meson charge radii using model-independent method in the PACS10 configuration},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UOCO6AFQ}},
  note         = {Machine review of arXiv:2412.18166}
}
abstract

We report our preliminary results for the charge radii of $\pi^{+}$ and $K^{+}$ mesons with the PACS10 configuration generated at the physical point using the Iwasaki gauge action and $N_{f}=2+1$ stout-smeared nonperturbatively $\mathcal{O}(a)$ improved Wilson quark action, especially at $0.085$ fm corresponding lattice size $128^4$. The charge radii are obtained from a model-independent method that directly calculates the first-order differential coefficient of the electromagnetic form factor and also from a traditional method that analyzes the form factor using a fit ansatz. We compare our preliminary results obtained by these methods with previous lattice calculations and experiments.

Figures

Figures reproduced from arXiv: 2412.18166 by the authors.

Figure 1
Figure 1. Results of 𝑅 𝑋 MI(𝑡) in Eq. (3) for 𝜋 + meson (left) and 𝐾 + meson (right). The solid red lines express the first derivative of the form factor obtained from a constant fit of a flat region. Traditional Model-independent Our improved Original Traditional Model-independent Our improved Original [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Preliminary results for 𝜋 + meson (left) and 𝐾 + meson (right) charge radii obtained from traditional and model-independent methods. The black and blue symbols are the results of the traditional method. The black circle, inverted triangle, triangle, diamond, square, and crosse symbols are the results of using monopole, quadratic polynomial, cubic polynomial, quadratic z-expansion, cubic z-expansion, and ChPT NLO, re… view at source ↗
Figure 3
Figure 3. The comparison of our preliminary result of the charge radius between those for previous lattice QCD calculations and PDG value. The left graph shows the 𝜋 + charge radius and right graph is 𝐾 + one. The colored symbols are the lattice calculations[2, 20–23], especially the red symbol is our preliminary result. The black line and gray band represent the central value and error of PDG22 [19]. We compare our results, … view at source ↗

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Reference graph

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