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arxiv: 2604.12921 · v1 · submitted 2026-04-14 · ✦ hep-lat

Recognition: unknown

An improvement of model-independent method for meson charge radius calculation

Kohei Sato , Hiromasa Watanabe , Takeshi Yamazaki
Authors on Pith no claims yet

Pith reviewed 2026-05-10 13:42 UTC · model grok-4.3

classification ✦ hep-lat
keywords lattice QCDmeson charge radiuselectromagnetic form factormodel-independent methodfinite-volume effectsspatial momentsauxiliary function
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The pith

Introducing an auxiliary function of momentum transfer squared further suppresses higher-order terms in model-independent lattice calculations of meson charge radii.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a variant of the model-independent approach to meson charge radii that avoids assuming any specific functional form for the electromagnetic form factor. Earlier spatial-moment techniques already remove fit-related systematics but retain some finite-volume contamination from higher-order terms in the expansion around zero momentum transfer. The improvement comes from defining an auxiliary function of the squared momentum transfer and recasting the entire extraction around the product of this function and the form factor. Quadratic and logarithmic choices for the auxiliary function are tested and shown to damp the unwanted higher-order pieces more strongly than before. This damping is most pronounced on small volumes or for mesons with large radii, as verified both on monopole mock data and on Nf=2+1 ensembles at pion masses near 0.5 and 0.3 GeV.

Core claim

By introducing an auxiliary function of the momentum transfer squared and reformulating the method in terms of its product with the form factor rather than the form factor itself, higher-order contributions are further suppressed, notably in cases of small volume and large radius. Applying this method to mock data based on a monopole form factor as well as to actual lattice QCD data reduces residual finite-volume effects and provides an effective framework for meson charge radius determinations.

What carries the argument

An auxiliary function of the squared momentum transfer (implemented as quadratic or logarithmic) whose product with the form factor is inserted into the spatial-moment formulas, thereby shifting the leading truncation error to higher order.

If this is right

  • Residual finite-volume effects in the charge radius are smaller than in the earlier model-independent approach for the same lattice size.
  • The method remains free of any assumed functional form for the form factor.
  • Quadratic and logarithmic auxiliary functions both work on monopole mock data and on the tested Nf=2+1 ensembles.
  • The improvement is largest precisely when the physical radius is large relative to the lattice size.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Smaller lattices could become usable for radius determinations, lowering the computational cost of reaching a given precision.
  • The same auxiliary-function reformulation might be portable to other lattice quantities extracted from spatial moments, such as higher derivatives or transition form factors.
  • If the suppression remains effective across a broader class of form-factor shapes, the technique could become the default model-independent route for electromagnetic radii.

Load-bearing premise

That quadratic and logarithmic auxiliary functions will suppress higher-order contributions for any unknown true shape of the form factor without introducing new biases.

What would settle it

A comparison on a very large-volume ensemble showing whether radii extracted with the new auxiliary-function method agree with those from a high-precision multi-point fit while the older moment method still deviates.

Figures

Figures reproduced from arXiv: 2604.12921 by Hiromasa Watanabe, Kohei Sato, Takeshi Yamazaki.

Figure 1
Figure 1. Figure 1: FIG. 1. Results of [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Results of [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Data of [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Dependence of [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The dependence of [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Results of [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Results of [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Results of [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. The same format as Fig. 3, but using the logarithmic function for several values of [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Dependence of [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. The same format as Fig. 6, but using the logarithmic function. [PITH_FULL_IMAGE:figures/full_fig_p018_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Result of [PITH_FULL_IMAGE:figures/full_fig_p018_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. The same format as Fig. 7, but using our method with the logarithmic function for [PITH_FULL_IMAGE:figures/full_fig_p019_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. The same format as Fig. 8, but using our method with the logarithmic function for [PITH_FULL_IMAGE:figures/full_fig_p019_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. Result of [PITH_FULL_IMAGE:figures/full_fig_p025_15.png] view at source ↗
Figure 17
Figure 17. Figure 17: The results for ⟨r 2 π ⟩ obtained from the three fits are summarized in Table VI. For the fit ranges on L = 64 lattice, we use all six Q2 data points for all fits. The results of ⟨r 2 π ⟩ are summarized in [PITH_FULL_IMAGE:figures/full_fig_p026_17.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. Results of [PITH_FULL_IMAGE:figures/full_fig_p027_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17. The same format as Fig. 15, but for [PITH_FULL_IMAGE:figures/full_fig_p027_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: shows the consistency of the model-independent analysis with the measured data determined from the ratio in Eq. (69). Since RZ(0) = 1 by definition, the de￾sired first Taylor coefficient of Fπ(Q2 ) is then obtained by subtracting Z1 from the slope extracted for Fπ(Q2 )RZ(Q2 ), i.e., R(t) − Z1 (or, equivalently, from that extracted for Fπ(Q2 )G(Q2 )RZ(Q2 )). 29 [PITH_FULL_IMAGE:figures/full_fig_p029_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19. Data of [PITH_FULL_IMAGE:figures/full_fig_p031_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20. The time dependence of [PITH_FULL_IMAGE:figures/full_fig_p031_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: FIG. 21. Dependence of the charge radius on the parameter [PITH_FULL_IMAGE:figures/full_fig_p032_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: FIG. 22. Dependence of the charge radius on the parameter [PITH_FULL_IMAGE:figures/full_fig_p033_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: FIG. 23. All the results of [PITH_FULL_IMAGE:figures/full_fig_p034_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: FIG. 24. The same format as Fig. 23, but for [PITH_FULL_IMAGE:figures/full_fig_p035_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: FIG. 25. Relative difference [PITH_FULL_IMAGE:figures/full_fig_p041_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: FIG. 26. Results of [PITH_FULL_IMAGE:figures/full_fig_p041_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: shows that δlin is roughly 1% and 0.1% at L = 6 and 10 fm, respectively, corresponding to the volumes for the calculations in Refs. [18] and [30]. The results for δquad and δlog are significantly smaller than δlin. However, it is expected that they would change significantly if the form factor has a correction from the monopole as discussed in the previous subsection. 0 5 10 15 20 L [fm] 10-4 10-3 10-2 10… view at source ↗
Figure 28
Figure 28. Figure 28: FIG. 28. Result of [PITH_FULL_IMAGE:figures/full_fig_p045_28.png] view at source ↗
Figure 29
Figure 29. Figure 29: FIG. 29. Result of [PITH_FULL_IMAGE:figures/full_fig_p046_29.png] view at source ↗
Figure 30
Figure 30. Figure 30: FIG. 30. Results of [PITH_FULL_IMAGE:figures/full_fig_p047_30.png] view at source ↗
read the original abstract

We propose a variant of the model-independent method for determining meson charge radii from spatial moments of correlation functions on the lattice. Traditional determinations based on fits to the momentum transfer squared dependence of form factors are subject to systematic uncertainties arising from the choice of fit ansatz. By contrast, model-independent methods based on spatial moments provide a useful framework for determining the slope of the form factor without assuming its functional form. Recently, Feng et al. proposed a model-independent method, which drastically suppresses the finite-volume effect in the charge radius coming from higher-order contributions of the expansion of the form factor with respect to the momentum transfer squared. In this work, we introduce an auxiliary function of the momentum transfer squared and reformulate the method in terms of its product with the form factor, rather than the form factor itself, thereby further suppressing higher-order contributions, notably in cases of small volume and large radius. In particular, we investigate quadratic and logarithmic forms as practical choices for this auxiliary function. Applying this method to mock data based on a monopole form factor, as well as to actual lattice QCD data using $N_f=2+1$ gauge ensembles at $m_\pi \simeq 0.5$ and $0.3$ GeV, we find that it reduces residual finite-volume effects and provides an effective framework for meson charge radius determinations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript proposes an improvement to the model-independent method of Feng et al. for extracting meson charge radii from lattice QCD spatial moments of correlation functions. The authors introduce an auxiliary function f(Q²) of the squared momentum transfer (considering quadratic and logarithmic choices) and reformulate the extraction procedure in terms of the product f(Q²) × F(Q²) rather than F(Q²) alone. This is claimed to further suppress higher-order terms in the small-Q² expansion, thereby reducing residual finite-volume effects in the charge radius, particularly for small volumes and large radii. The method is tested on mock data generated from a monopole form factor and applied to two N_f=2+1 gauge ensembles at m_π ≈ 0.5 and 0.3 GeV.

Significance. If the central claim holds, the approach would offer a practical enhancement to existing model-independent techniques for lattice determinations of hadron charge radii, potentially allowing more reliable extractions on smaller volumes without sacrificing the avoidance of fit ansatz biases. The direct mathematical re-expression preserves the model-independent character in principle, which is a methodological strength.

major comments (3)
  1. [§4.1] §4.1 (mock data tests): The suppression of finite-volume effects and absence of bias from the auxiliary function are demonstrated exclusively on mock data generated assuming a monopole form factor. This does not test whether quadratic or logarithmic choices alter the extracted radius when the true form factor contains different higher-order structure (e.g., dipole or resonance contributions), which is load-bearing for the claim of improved model independence.
  2. [§3] §3 (reformulation): The derivation of the product f(Q²) × F(Q²) and the corresponding spatial-moment expression does not explicitly quantify how the auxiliary-function coefficients enter the final radius extraction or demonstrate that they remain free of new systematic bias when the true form factor deviates from the test case; this step is central to asserting further suppression without compromising the result.
  3. [§4.2] §4.2 (lattice ensembles): The application to the two N_f=2+1 ensembles reports reduced residual finite-volume effects but provides no tabulated quantitative comparison (e.g., radius values and uncertainties with/without the auxiliary function, or explicit FV correction magnitudes), leaving the magnitude of the claimed improvement unverifiable.
minor comments (2)
  1. [§3] The notation for the auxiliary function and its product with the form factor should be introduced with an explicit example expansion up to O(Q⁴) to clarify the suppression mechanism.
  2. [§4] Figure captions for the mock-data and ensemble results should include the specific auxiliary-function parameters used and the volume sizes to aid reproducibility.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major point below and indicate the revisions we will make to improve clarity and verifiability.

read point-by-point responses

  1. Referee: [§4.1] §4.1 (mock data tests): The suppression of finite-volume effects and absence of bias from the auxiliary function are demonstrated exclusively on mock data generated assuming a monopole form factor. This does not test whether quadratic or logarithmic choices alter the extracted radius when the true form factor contains different higher-order structure (e.g., dipole or resonance contributions), which is load-bearing for the claim of improved model independence.

    Authors: The reformulation is derived generally, without assuming any specific form for F(Q²). The auxiliary function f(Q²) is selected to reduce the size of Q⁴ and higher coefficients in the expansion of the product around Q²=0, independent of the coefficients present in F. The charge radius is recovered from the slope of the product after a known correction from f, preserving model independence. To strengthen the demonstration, we will add mock-data tests with dipole and resonance-inspired form factors in the revised §4.1. revision: yes

  2. Referee: [§3] §3 (reformulation): The derivation of the product f(Q²) × F(Q²) and the corresponding spatial-moment expression does not explicitly quantify how the auxiliary-function coefficients enter the final radius extraction or demonstrate that they remain free of new systematic bias when the true form factor deviates from the test case; this step is central to asserting further suppression without compromising the result.

    Authors: We will expand the derivation in the revised §3 to show the explicit Taylor expansion of the product, demonstrating that the auxiliary coefficients affect only the Q⁴ and higher terms while the linear term (which determines the radius) receives a known, exact adjustment from f. Because the spatial-moment relation is applied to the product and the procedure remains exact in infinite volume, no new systematic bias is introduced when F deviates from the monopole case. revision: partial

  3. Referee: [§4.2] §4.2 (lattice ensembles): The application to the two N_f=2+1 ensembles reports reduced residual finite-volume effects but provides no tabulated quantitative comparison (e.g., radius values and uncertainties with/without the auxiliary function, or explicit FV correction magnitudes), leaving the magnitude of the claimed improvement unverifiable.

    Authors: We agree that tabulated comparisons are required for verifiability. In the revised §4.2 we will insert a table listing the extracted radii (with uncertainties) obtained both with and without the auxiliary function for each ensemble, together with the estimated residual finite-volume corrections. This will make the size of the improvement quantitative and directly comparable. revision: yes

Circularity Check

0 steps flagged

No circularity: reformulation is an independent mathematical re-expression with external validation.

full rationale

The paper's central step is a direct algebraic reformulation of the Feng et al. spatial-moment method by multiplying the form factor by a chosen auxiliary function of Q^2 (quadratic or logarithmic) to suppress higher-order terms in the small-volume expansion. This re-expression does not define the radius in terms of itself, nor rename a fitted parameter as a prediction, nor invoke any self-citation chain for uniqueness or ansatz justification. The auxiliary forms are presented as practical choices and are validated on independent monopole mock data plus two external Nf=2+1 ensembles; the derivation chain therefore remains self-contained against external benchmarks and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The approach rests on standard lattice QCD assumptions about form factor expansions and finite-volume corrections, with the auxiliary function as the primary addition; no new physical entities are postulated.

free parameters (1)
  • auxiliary function coefficients
    Choice of specific quadratic or logarithmic forms may implicitly involve scale or coefficient selections, though not explicitly fitted in the abstract description.
axioms (2)
  • domain assumption The electromagnetic form factor of the meson admits a Taylor expansion in powers of Q^2 around zero momentum transfer.
    This underpins extraction of the charge radius as the linear coefficient and is invoked throughout the model-independent framework.
  • domain assumption Finite-volume corrections to the form factor can be organized as an expansion in momentum transfer whose higher terms are suppressible by weighting with an auxiliary function.
    This is the key premise enabling the reformulation to reduce residual volume effects.

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discussion (0)

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Reference graph

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