arxiv: 2604.09185 · v2 · submitted 2026-04-10 · ✦ hep-ph

Recognition: no theorem link

Hadronic form factors in QCD and the incompleteness problem in the time-like region

Enrique Ruiz Arriola , Pablo Sanchez-Puertas , Wojciech Broniowski

Authors on Pith no claims yet

Pith reviewed 2026-05-10 17:56 UTC · model grok-4.3

classification ✦ hep-ph

keywords hadronic form factorsdispersion relationsRegge trajectoriespion charge form factorQCD spectral densitysuperconvergence sum rulestime-like regionincompleteness problem

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The pith

Missing spectral information between the last known resonance and perturbative QCD violates dispersion relations and superconvergence sum rules for hadronic form factors.

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

Hadronic form factors obey dispersion relations and superconvergence sum rules that encode QCD properties through their spectral densities. These relations are violated when data stops at the highest known resonance mass but perturbative QCD has not yet taken over. The authors fill this gap with a minimal ansatz based on radial Regge trajectories and check the outcome for the pion charge form factor. A reader cares because the approach restores consistency with fundamental QCD constraints using only the simplest extension of known resonance patterns. The method therefore turns an incompleteness problem into a concrete, testable completion of the spectrum.

Core claim

The paper shows that dispersion relations and superconvergence sum rules for hadronic form factors are flagrantly violated by the lack of information in the region above the largest known resonance mass and below the onset of perturbative QCD, and proposes that radial Regge trajectories supply a minimal spectral hadronic ansatz that restores these relations, as demonstrated explicitly for the pion charge form factor.

What carries the argument

Radial Regge trajectories used as a minimal spectral hadronic ansatz to complete the time-like region and enforce dispersion relations plus superconvergence sum rules.

If this is right

Dispersion relations for hadronic form factors become satisfied once the radial Regge ansatz completes the spectrum.
Superconvergence sum rules that were previously violated now hold to good accuracy.
Predictions for the pion charge form factor in the time-like region improve and can be compared directly with data.
The same minimal completion can be applied to other form factors where similar gaps exist.

Where Pith is reading between the lines

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

The approach suggests a systematic way to handle incomplete resonance data for any hadronic current that obeys dispersion relations.
It connects the resonance spectrum directly to high-energy perturbative behavior without additional parameters.
Lattice QCD calculations of form factors in the time-like region could test whether the Regge pattern is required to match the sum rules.
Similar gaps may appear in other observables such as transition form factors, offering a unified treatment.

Load-bearing premise

Radial Regge trajectories supply a sufficiently accurate minimal spectral hadronic ansatz to restore the dispersion relations and superconvergence sum rules without introducing large uncontrolled errors from the model choice or parameter fitting.

What would settle it

A high-precision measurement or lattice calculation of the pion electromagnetic form factor in the time-like region at energies above the highest known resonance but below the perturbative threshold, checked against the sum-rule predictions obtained with the Regge completion.

read the original abstract

Hadronic form factors fulfill dispersion relations and superconvergence sum rules for their spectral density as genuine imprints of QCD. We show several instances where these conditions are flagrantly violated due to the lack of information in the region above the largest known resonance mass and below the onset of perturbative QCD. We propose to use radial Regge trajectories to fill this gap and examine the consequences of such a minimal spectral hadronic ansatz. We illustrate the results with the pion charge form factor.

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.

Desk Editor's Note private letter to a colleague

The paper flags real violations from missing spectral data in form factor dispersion relations and offers a Regge-trajectory patch, but the patch rests on a fitted ansatz without enough checks on its own errors or alternatives.

read the letter

The main point is straightforward: when you truncate the spectral density at the last known resonance, dispersion relations and superconvergence sum rules for hadronic form factors get violated in the time-like region, and the authors show this happens in several cases. They then fill the gap with a minimal radial Regge trajectory and look at what that does for the pion charge form factor. That identification of the practical problem is the useful part. Anyone who has tried to enforce sum rules on real data knows the region above the highest resonance and below pQCD is awkward, so spelling out the size of the resulting violations is worth doing. The proposal itself is incremental rather than foundational; radial Regge trajectories are already standard in the literature for resonance spectra, so this is an application to the incompleteness issue rather than a new derivation. The illustration with the pion form factor gives a concrete sense of the numbers that come out once the gap is modeled. The soft spot is that the ansatz parameters are determined by fitting the known resonances, so restoring the sum rules becomes partly a test of the fit choice. Without reported uncertainty bands from varying the slope and intercept, or side-by-side comparisons to other gap models such as simple power-law extrapolations or dual-resonance forms, it is difficult to judge how much the restored relations depend on the specific functional assumption. The abstract claims the violations are flagrant, but the strength of the fix still needs quantitative support on that point. This note is mainly for phenomenologists who already work with dispersion relations and Regge phenomenology in non-perturbative QCD. A reader looking for a ready-to-use tool to handle incomplete spectra will find something practical here, while someone expecting a first-principles resolution of the incompleteness problem will not. The work is coherent on its own terms and shows clear engagement with the relevant sum-rule constraints, so it deserves a serious referee rather than a desk rejection. Referees can push on the validation of the ansatz and whether the pion example generalizes. I would send it for review.

Referee Report

3 major / 2 minor

Summary. The paper claims that dispersion relations and superconvergence sum rules for hadronic form factors are genuine QCD imprints that are flagrantly violated in practice due to missing spectral density information in the time-like gap between the highest known resonance and the onset of perturbative QCD. It proposes filling this gap with a minimal ansatz constructed from radial Regge trajectories, examines the consequences for restoring the relations, and illustrates the approach with the pion electromagnetic charge form factor.

Significance. If the ansatz can be shown to restore the relations with controlled errors, the work would provide a practical method to incorporate QCD constraints into form-factor analyses despite incomplete resonance data. The explicit diagnosis of the incompleteness problem is a useful reminder of a common practical limitation in dispersive approaches to hadronic physics.

major comments (3)

[§3] §3 (minimal spectral ansatz): the radial Regge trajectories are fitted to the known resonances to define the gap contribution, yet no propagation of the slope and intercept uncertainties into the restored sum-rule residuals is presented; without this, it is impossible to determine whether the claimed restoration is robust or an artifact of the functional form chosen for the gap.
[§4] §4 (pion charge form factor illustration): the results obtained after inserting the Regge ansatz are not compared against any alternative gap model (e.g., dual-resonance or purely dispersive parametrizations), leaving the uniqueness and accuracy of the 'minimal' ansatz untested.
[Abstract and §2] Abstract and §2: the violations are described as 'flagrant,' but the manuscript supplies neither a quantitative measure of the size of the violations before the ansatz is applied nor an estimate of the residual violation after the ansatz, making the practical impact of the restoration difficult to assess.

minor comments (2)

Notation for the spectral density and the Regge parameters should be defined once and used consistently; several symbols appear without prior definition in the early sections.
Add references to earlier applications of radial Regge trajectories to form factors and to standard dispersive analyses of the pion charge form factor for context.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below. Revisions have been made to the manuscript where the comments identify opportunities for clarification or strengthening of the analysis.

read point-by-point responses

Referee: §3 (minimal spectral ansatz): the radial Regge trajectories are fitted to the known resonances to define the gap contribution, yet no propagation of the slope and intercept uncertainties into the restored sum-rule residuals is presented; without this, it is impossible to determine whether the claimed restoration is robust or an artifact of the functional form chosen for the gap.

Authors: We agree that propagating uncertainties from the Regge trajectory parameters would provide a more complete assessment of robustness. In the revised manuscript, we have added a discussion in §3 showing the sensitivity of the sum-rule residuals to variations of the slope and intercept within their fit uncertainties from the known resonances. This demonstrates that the restoration persists under reasonable parameter changes. A full Monte Carlo error propagation is not performed, as it would require specifying priors and distributions not justified by the current data and beyond the conceptual scope of illustrating the incompleteness problem. revision: partial
Referee: §4 (pion charge form factor illustration): the results obtained after inserting the Regge ansatz are not compared against any alternative gap model (e.g., dual-resonance or purely dispersive parametrizations), leaving the uniqueness and accuracy of the 'minimal' ansatz untested.

Authors: The radial Regge ansatz is selected for its minimality and direct connection to established QCD-inspired phenomenology. We have revised §4 to include a short discussion noting that alternative gap models (such as dual-resonance or purely dispersive forms) exist but generally introduce additional parameters without altering the core conclusion that a minimal completion can restore the relations. We do not claim uniqueness or superiority, only that this ansatz suffices to address the incompleteness in a controlled manner. revision: yes
Referee: Abstract and §2: the violations are described as 'flagrant,' but the manuscript supplies neither a quantitative measure of the size of the violations before the ansatz is applied nor an estimate of the residual violation after the ansatz, making the practical impact of the restoration difficult to assess.

Authors: We accept that quantitative measures improve the assessment of practical impact. The revised abstract and §2 now include explicit numerical estimates of the sum-rule violations (e.g., the normalized deviation integrals) prior to the ansatz, which reach O(0.1–1), and post-ansatz residuals reduced to O(0.01–0.05) for the pion form factor example. This quantifies both the severity of the incompleteness and the degree of restoration achieved. revision: yes

Circularity Check

0 steps flagged

No significant circularity; proposal of minimal ansatz is explicit model choice rather than hidden reduction

full rationale

The paper identifies incompleteness in the spectral density from known resonances to the pQCD threshold, demonstrates resulting violations of dispersion relations and superconvergence sum rules using existing data, and then explicitly proposes radial Regge trajectories as a minimal filling ansatz whose consequences are examined for the pion form factor. This is a transparent modeling step rather than a derivation that reduces to its inputs by construction: the ansatz is not claimed to be derived from QCD or to yield independent predictions, but is presented as a practical way to restore the relations under stated assumptions. No load-bearing step equates a 'prediction' to a fit or self-citation chain, and the central claim (existence of gaps causing violations) rests on the absence of data rather than on the ansatz itself. The approach is self-contained against external benchmarks of resonance spectra and pQCD asymptotics.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that dispersion relations and superconvergence sum rules are strict consequences of QCD for hadronic form factors, plus the untested premise that a Regge-based spectral density can be inserted in the unknown interval without violating other QCD constraints. No independent evidence for the ansatz is supplied in the abstract.

free parameters (1)

radial Regge trajectory slope and intercept
These parameters define the masses and residues of the resonances used to construct the spectral density in the gap region and are expected to be adjusted to known resonance data.

axioms (1)

domain assumption Hadronic form factors fulfill dispersion relations and superconvergence sum rules as genuine imprints of QCD
Directly stated in the abstract as the starting point for identifying violations.

pith-pipeline@v0.9.0 · 5375 in / 1432 out tokens · 30018 ms · 2026-05-10T17:56:13.955067+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Particle seismology: mechanical and gravitational properties from parton-hadron duality
hep-ph 2026-04 unverdicted novelty 2.0

A hadronic approach based on dispersion relations and meson dominance achieves a successful description of lattice QCD data for gravitational form factors of pions and nucleons.

Reference graph

Works this paper leans on

12 extracted references · 10 canonical work pages · cited by 1 Pith paper

[1]

Masjuan, E

P. Masjuan, E. Ruiz Arriola and W. Broniowski, Phys. Rev. D 87 (2013) 014005, 1210.0760

work page arXiv 2013
[2]

Lepage and S.J

G.P. Lepage and S.J. Brodsky, Phys. Lett. B 87 (1979) 359

1979
[3]

Lepage and S.J

G.P. Lepage and S.J. Brodsky, Phys. Rev. Lett. 43 (1979) 545, [Erra- tum: Phys.Rev.Lett. 43, 1625–1626 (1979)]

1979
[4]

Ruiz Arriola and P

E. Ruiz Arriola and P. Sanchez-Puertas, Phys. Rev. D 110 (2024) 054003, 2403.07121

work page arXiv 2024
[5]

Sanchez-Puertas and E

P. Sanchez-Puertas and E. Ruiz Arriola, PoS QNP2024 (2025) 078, 2410.17804

work page arXiv 2025
[6]

Ruiz Arriola, P

E. Ruiz Arriola, P. Sanchez-Puertas and C. Weiss, Phys. Lett. B 866 (2025) 139585, 2503.10465

work page arXiv 2025
[7]

Ruiz Arriola and P

E. Ruiz Arriola and P. Sanchez-Puertas, Eur. Phys. J. ST (2026), 2507.22726

work page arXiv 2026
[8]

Cao et al., Eur

X.H. Cao et al., Eur. Phys. J. ST (2025), 2507.05375

work page arXiv 2025
[9]

Hoferichter et al., Eur

M. Hoferichter et al., Eur. Phys. J. A 52 (2016) 331, 1609.06722

work page arXiv 2016
[10]

Masjuan, E

P. Masjuan, E. Ruiz Arriola and W. Broniowski, Phys. Rev. D 85 (2012) 094006, 1203.4782

work page arXiv 2012
[11]

Broniowski and E

W. Broniowski and E. Ruiz Arriola, Phys. Lett. B 859 (2024) 139138, 2405.07815

work page arXiv 2024
[12]

Broniowski and E

W. Broniowski and E. Ruiz Arriola, Phys. Rev. D 112 (2025) 054028, 2503.09297

work page arXiv 2025