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Likelihood of a zero in the proton elastic electric form factor

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Pith's one-line read A model-free analysis of 29 proton form factor ratios predicts a zero in the electric form factor at Q^2 = 10.37 GeV^2, with data inconsistent with no zero at the one-in-a-million level.

desk verdict A plausible data-driven estimate of the proton form factor zero, but the headline confidence claims rest on shape priors and on treating correlated results as independent. read the letter →

arxiv 2412.10598 v1 pith:WCZOJBLL submitted 2024-12-13 hep-ph hep-exhep-latnucl-exnucl-th

Peng Cheng , Zhao-Qian Yao , Daniele Binosi , Craig D. Roberts This is my paper · ORCID
classification hep-phhep-exhep-latnucl-exnucl-th PACS 13.40.Gp14.20.Dh
keywords Schlessingerpointmethodprotonelectricformfactorratiozerocrossingpolarizationtransfercontinued-fractioninterpolationemergenthadronmassQCD
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 asks a concrete question about the proton: does the ratio of its electric to magnetic form factor, mu_p G_E^p/G_M^p, actually cross zero at some momentum transfer? Using only 29 polarization-transfer data points and no model of proton structure, the authors build thousands of continued-fraction interpolations of the data and extrapolate each one. The ensemble predicts a zero at $Q^{2}$ = 10.$37^{{+0.87}}$_{-0.68} $GeV^{2}$, puts 99.9% confidence on the zero lying below 13.06 $GeV^{2}$, and estimates a one-in-a-million likelihood that the ratio stays positive through 14.49 $GeV^{2}$. If correct, the proton's electric form factor changes sign, a distinctive and testable feature of how charge is distributed inside the proton.

What carries the argument

The Schlessinger point method (SPM), a continued-fraction interpolation also known as a multipoint Pade approximant, is the engine of the analysis. For each of 15,000 Gaussian replicas of the 29 data points, random M-point subsets are used to build interpolating rational functions, which are accepted only if they are $C^{1}$ and monotonically decreasing on 0 < $Q^{2}$/$GeV^{2}$ < 20; 5,000 replicas with 25 accepted interpolators each define the ensemble, and the zero location is read from each curve. The paper excludes M = 4k+2 interpolator orders because those cannot be monotone-decreasing and cross zero at real $Q^{2}$, a hidden constraint the authors identify and remove.

What would settle it

Take polarization-transfer data on mu_p G_E^p/G_M^p at $Q^{2}$ = 10, 11, 12, and 13 $GeV^{2}$; if any measured value at or above 13 $GeV^{2}$ is positive and away from zero by more than the quoted uncertainty, the predicted zero below 13.06 $GeV^{2}$ is falsified.

Watch

Extended reading notes

Core claim

The paper's central claim is that existing data, extrapolated without any hadron model, already imply a zero in the proton electric form factor ratio. The Schlessinger point method, applied to 29 measured values of mu_p G_E^p/G_M^p, yields a predicted zero at $Q_z^{2}$ = 10.$37^{{+0.87}}$_{-0.68} $GeV^{2}$ (Eq. 8). With 99.9% confidence the data are consistent with a zero on $Q^{2}$ <= 13.06 $GeV^{2}$, and the likelihood that the data are consistent with a positive-definite ratio on $Q^{2}$ <= 14.49 $GeV^{2}$ is 1/1-million. The authors present this as an objective, function-form unbiased statement about what the data themselves imply.

Load-bearing premise

The analysis keeps only interpolating curves that are smooth and steadily decreasing, and it discards a whole family of curves that could not cross zero while staying decreasing; if the true ratio is positive but wiggles, this filter alone could manufacture the predicted zero.

Editorial extensions

If this is right

  • A zero near Q^2 = 10.4 GeV^2 means the proton electric form factor is not positive definite, implying a diffraction-like sign change in the electric charge distribution.
  • The prediction is directly testable with planned polarization-transfer measurements reaching Q^2 near 12 GeV^2; the 90% confidence band already extends to 11.49 GeV^2.
  • The SPM result agrees with parameter-free Faddeev equation predictions of the zero location, strengthening the case that emergent hadron mass controls the falloff of the ratio.
  • Phenomenological fits that force the proton electric form factor to remain positive are inconsistent with the data at the one-in-a-million likelihood level over the range up to 14.49 GeV^2.

Reading between the lines

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

  • The same bootstrap-plus-continued-fraction recipe could be applied to neutron or hyperon form factor ratios; a zero there would indicate whether the sign change is a generic feature of baryons or specific to the proton.
  • The 1/1-million figure is a likelihood within the SPM ensemble, not a classical p-value against a particular alternative model, so a single high-precision measurement above 12 GeV^2 could carry more evidential weight than the ensemble tail.
  • If future data confirm the zero, the crossing location and its asymmetric uncertainty become a precise target for lattice QCD and for relativistic quark models of the nucleon.
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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 / 6 minor

Summary. The paper applies the Schlessinger point method (SPM) to the 29 available polarization-transfer data points for the proton form factor ratio Rp(Q^2) = mu_p G_E^p(Q^2)/G_M^p(Q^2). Gaussian replicas of the data are generated, and for each replica the authors select 25 SPM interpolators that are C^1 and monotonically decreasing on 0 < Q^2/GeV^2 < 20; replicas that cannot supply 25 such interpolators are discarded. The analysis is repeated for M = 9, 8, 7, 5, 4, and the resulting zero locations are combined using an asymmetric Gaussian likelihood. The paper's central claims are a zero at Q_z^2 = 10.37^{+0.87}_{-0.68} GeV^2, a 99.9% confidence statement that the data are consistent with a zero on Q^2 <= 13.06 GeV^2, and a 1/1-million likelihood that the data are consistent with no zero on Q^2 <= 14.49 GeV^2.

Significance. If the central result is correct, the paper resolves a long-standing question about proton structure and provides a benchmark for QCD-based calculations. The analysis has several genuine strengths: it uses public data, it is transparent about the interpolation machinery, it checks stability across several M values and independent replica sets, and its central zero location agrees with the authors' Faddeev-equation prediction. However, the headline confidence statements are not supported by the present evidence because they are conditioned on a shape filter and on Gaussian tail extrapolations that are neither derived from the data nor validated by closure tests. The paper would be significant if the authors can show that the existence and location of the zero are robust to relaxing those assumptions, but as written the 99.9% and 1/1-million claims outrun the statistical methodology.

major comments (3)
  1. [Sec. 2, Step 2(i)-(ii) and the M=4k+2 discussion] The acceptance criterion that interpolators must be C^1 and monotonically decreasing on 0 < Q^2/GeV^2 < 20 is a shape prior, not a consequence of analyticity, QCD, or the data. The paper's claim that the SPM is 'blind to any and all prejudice' is therefore not supported by the procedure actually implemented. The exclusion of all M=4k+2 orders, because those interpolators cannot be monotone and have a single real zero, removes an entire family that could describe a positive-definite monotone ratio; the reported fractions of zero-crossing interpolators (e.g., 92% for M=9) are conditional on this filter. To support the existence claim, the authors should report the fraction of initial 15,000 replicas and of generated interpolators rejected by the monotonicity condition, and should repeat the analysis with relaxed filters that include non-monotone C^1 interpolators and the M=4k+2 family without the monotonicity requirement.
  2. [Sec. 3, Eqs. (4)-(6) and Fig. 6] The 99.9% and 1/1-million statements are Gaussian tail probabilities of the bootstrap distribution of the zero location, not likelihoods of the data under H1 or H2. The H2 tail is evaluated at 14.49 GeV^2 as (14.49 - 10.37)/0.87, which is about 4.7 sigma, and the quoted probability is read off a Gaussian tail. This assumes that the zero-location estimator is Gaussian with the fitted sigma and that the accepted SPM ensemble is an unbiased likelihood for the data; neither assumption is tested. Under H2, the acceptance filter would be applied to positive-definite curves, and the composition of the accepted ensemble, and hence the tail probability, would be different. The authors should provide a closure test: generate synthetic data from positive-definite functions (e.g., dipole or no-zero Faddeev-like forms) with the same noise, run Steps 1-4, and report how often a zero is claimed and at what confidence. Without this calibration, the '1/1-million' claim is not statistically meaningful.
  3. [Sec. 2, Steps 3-4] The procedure discards any replica for which 25 acceptable interpolators cannot be obtained, but the manuscript does not report how many of the 15,000 initial replicas are discarded. If the acceptance rate is low, the 5,000 retained replicas are a strongly selected subsample, and the uncertainty estimates derived from their zero locations will understate the variability implied by the data. The authors already check the effect of using 10 versus 50 interpolators per replica, but the selection rate itself is an essential diagnostic and should be reported for each M value. The absence of this information makes it difficult to assess whether the quoted uncertainties, and the subsequent confidence statements, reflect the data or the selection procedure.
minor comments (6)
  1. [Sec. 2, Step 1] The statement that 'systematic errors are small' should be justified with the systematic uncertainty estimates from Refs. [21-25], and a sensitivity test with inflated uncertainties would be useful given that the final confidence claims are at the 1e-6 level.
  2. [Sec. 2, after Step 4] The text says 'the 5,000 M = 9 interpolators that represent the N = 29 available data', but Step 2 describes 25 interpolators per replica; please clarify whether the plotted purple curves are the 25 individual interpolators or the replica-averaged representatives.
  3. [Eq. (3)] The continued-fraction notation with an ellipsis is ambiguous; please display the nested-fraction form or define the recursion for the coefficients a_i.
  4. [Fig. 6] The y-axis label 'Likelihood H1' is misleading because the plotted quantity is a tail probability or posterior probability under the SPM distribution, not a likelihood of the data; please relabel accordingly.
  5. [References] Ref. [31] is cited as an arXiv preprint; if a journal version now exists, please update the reference.
  6. [Notation] The phrase 'C1 function' should be written as 'C^1 function' for consistency with standard notation.

Circularity Check

0 steps flagged · score 2.0 of 10

No equation-level circularity: the zero location is an SPM extrapolation output, not a fitted input; the self-citations are not load-bearing.

full rationale

The central numerical claim, Q_z^2 = 10.37 +0.87/-0.68 GeV^2 (Eq. (8)), is produced by Schlessinger-point-method interpolation of the 29 measured ratio points. The zero crossing is not a fitted parameter; it is a derived feature of the accepted interpolators, so the main prediction does not reduce to its inputs by construction. The bootstrap replicas and the averaging over interpolators use only the data and their uncertainties, and the stability checks with different replica sets are genuine internal consistency checks. The acceptance filter (C^1, monotonically decreasing on 0 < Q^2/GeV^2 < 20) and the exclusion of M = 4k+2 interpolators are stated openly; these are shape priors rather than consequences of QCD or of the data, and they may bias the ensemble toward zero-crossing curves. This is a legitimate statistical and modeling concern, but it is not circular: the zero is not inserted as an input, and the paper does not hide the filter behind an equation. Similarly, the 1/1-million statement for H2 (positive definite on Q^2 <= 14.49 GeV^2) is obtained from a Gaussian tail model for the zero-location distribution, not from an actual likelihood of the data under H2; that is a calibration/risk issue, not a self-referential derivation. Several cited validation and comparison items (Refs. [30,31,44-48]) share authors with this paper, but the SPM method itself rests on the independent, older references [41-43], and the self-citations are used for context and comparison rather than to justify the central prediction. No specific equation or construction was found in which an input is renamed as the prediction.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

No new physical entities are introduced; the zero is a property of the form factor ratio, not a new object. The main ledger items are methodological assumptions and the interpolator order choice, all of which affect the central prediction and its confidence levels.

free parameters (1)
  • SPM interpolator order M = 4, 5, 7, 8, 9; M=6,10 excluded
    The number of data points used in each continued-fraction interpolator is chosen by the authors. M=4k+2 values are excluded because they cannot represent a monotone-decreasing function with a real zero, a choice that may bias the ensemble toward finding a zero.
assumptions (5)
  • domain assumption The proton form factor ratio Rp(Q^2) is analytic in the interval up to Q^2 ~ 20 GeV^2 and is representable by the continued-fraction interpolator within its radius of convergence.
    Section 2 states SPM reliably reconstructs analytic functions within a radius set by the nearest branch point; this assumes the data lie on such an analytic curve beyond the measured range.
  • ad hoc to paper Rp(Q^2) is C^1 and monotonically decreasing on 0 < Q^2/GeV^2 < 20.
    Step 2(ii) accepts only monotone-decreasing interpolators; this shape prior is not derived from data or QCD and excludes non-monotone positive-definite possibilities.
  • domain assumption Experimental uncertainties are Gaussian, independent, and uncorrelated; systematic errors are negligible.
    Step 1 generates replicas by independent Gaussian smearing with the quoted experimental uncertainty; no systematic covariance is propagated.
  • domain assumption The zero-location distribution is well described by the variable-width Gaussian ln-likelihood of Eqs. (4)-(6).
    Section 3 assumes a linear variation of sigma and combines independent results with Eq. (7); the '1/1-million' tail follows from this assumed form.
  • domain assumption Results from different M values are statistically independent and can be combined by averaging their ln-likelihoods.
    Section 3 says all results are statistically independent and compatible; this permits the combination in Eq. (7).

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Cite this review

Pith. "Pith review of Likelihood of a zero in the proton elastic electric form factor." pith.science (2026). https://pith.science/paper/WCZOJBLL

@misc{pith2026241210598,
  author       = {Pith},
  title        = {Pith review of: Likelihood of a zero in the proton elastic electric form factor},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WCZOJBLL}},
  note         = {Machine review of arXiv:2412.10598}
}
abstract

Working with the $29$ available data on the ratio of proton electric and magnetic form factors, $\mu_p G_E^p(Q^2)/ G_M^p(Q^2)$, and independent of any model or theory of strong interactions, we use the Schlessinger point method to objectively address the question of whether the ratio possesses a zero and, if so, its location. Our analysis predicts that, with 50% confidence, the data are consistent with the existence of a zero in the ratio on $Q^2 \leq 10.37\,$GeV$^2$. The level of confidence increases to $99.9$\% on $Q^2 \leq 13.06\,$GeV$^2$. Significantly, the likelihood that existing data are consistent with the absence of a zero in the ratio on $Q^2 \leq 14.49\,$GeV$^2$ is $1/1$-million.

Figures

Figures reproduced from arXiv: 2412.10598 by the authors.

Figure 1
Figure 1. µpG p E (Q 2 )/G p M (Q 2 ). Available data obtained using the polarisation transfer reaction [21–25]; and M = 9 analysis of this data, showing the 5 000 data-replica interpolating functions (purple curves) and their mean (thicker pur￾ple curve) obtained as discussed in Steps 1 – 4. points, reaching to Q 2 = 8.5 GeV2 [21–25]. They are displayed in [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Each row shows the same 4 distinct, randomly selected replicas from our 5 000 member set. Then, as labelled from top to bottom, the images display the [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. ln-likelihood functions for the M-value SPM data replica analyses listed in [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figures from the paper (1 more)
Figure 6
Figure 6. Figure 6: SPM prediction for likelihood that available data on [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]

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