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arxiv: 2512.12253 · v1 · submitted 2025-12-13 · ✦ hep-ph

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Neutral and charged pion Form Factors in the intermediate-energy region from double-dilaton HQCD model

H\'ector Cancio , Pere Masjuan
Authors on Pith no claims yet

Pith reviewed 2026-05-16 23:04 UTC · model grok-4.3

classification ✦ hep-ph
keywords pion form factorsholographic QCDnon-perturbative alpha_sintermediate energypion distribution amplitudeisospin breaking
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The pith

Using a double-dilaton holographic QCD model, pion form factor calculations show non-perturbative strong-interaction effects persist into the intermediate-energy region.

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

The paper computes the form factors for neutral and charged pions in the intermediate-energy range by incorporating a non-perturbative running of the strong coupling from a double-dilaton holographic QCD model into the pion distribution amplitude formalism. This approach addresses the observed deviation of the neutral pion form factor from its expected high-energy asymptotic behavior and provides predictions for the less-explored charged pion form factor. The results indicate that non-perturbative contributions remain important at energies where perturbative QCD is typically assumed to apply, suggesting the onset of the perturbative regime occurs at higher scales. The method also enables examination of isospin-breaking effects arising from the quadratic difference in pion masses.

Core claim

By applying the non-perturbative running of α_s obtained from the double-dilaton Holographic QCD model within the pion distribution amplitude formalism, the form factors of neutral and charged pions are computed in the intermediate-energy region. These calculations show that non-perturbative physics of the strong interaction is relevant even at energy scales traditionally considered perturbative, implying that the perturbative regime could occur at higher energies than previously thought. The approach further allows the study of isospin-breaking effects through the quadratic pion mass difference.

What carries the argument

The double-dilaton Holographic QCD model providing a non-perturbative running of the strong coupling α_s, used together with the pion distribution amplitude formalism to compute the form factors.

Load-bearing premise

The double-dilaton holographic QCD model together with the chosen pion distribution amplitude accurately reproduces the non-perturbative running of the strong coupling in the intermediate-energy region.

What would settle it

High-precision experimental data on the neutral pion form factor in the intermediate-energy region that agrees closely with perturbative QCD predictions without additional non-perturbative corrections would challenge the results.

Figures

Figures reproduced from arXiv: 2512.12253 by H\'ector Cancio, Pere Masjuan.

Figure 1
Figure 1. Figure 1: Comparison between experimental data of Q2F0(Q2 ) from CELLO Col. [3], purple triangles, CLEO Col. [4], red stars, BABAR Col. [1], black tri￾angles, and BELLE Col. [2], green dots; and the model prediction either fitted to data using Eq.(13) (solid blue line) or without fitting to them using Eq.(9) with n = 1 (dotted blue line). Dashed blue vertical line indicates the matching Q2 0 point. satisfy [42]: QM(… view at source ↗
Figure 3
Figure 3. Figure 3: Normalized neutral (charged) pion distribu [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Difference between neutral and charged pion [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Isospin breaking form factor FIso(Q2 ) from Eq.(20). estimate of the uncertainty takes into account the un￾certainty associated to both pion DAs from the corre￾sponding fits. Adding them up linearly, a rough esti￾mate of about 10% is found. Assuming f 2 π0 ≈ f 2 π± in Eq.(24), we obtain the approximate relation: ∆m2 π = FIso(0)m2 π0m2 π± f 2 π , (25) which numerically, using the inputs FIso(0) = 0.055(5) f… view at source ↗
read the original abstract

We compute the Form Factors of both neutral and charged pion using a non-perturbative running of the strong coupling constant $\alpha_s$ obtained using a double-dilaton Holographic QCD model. These form factors remain poorly understood in the intermediate-energy region, which marks the transition between low- and high-energy physics. In particular, experimental data for the neutral pion Form Factor exhibits a deviation from the expected asymptotic behavior, and the charged pion form factor remains comparatively less explored. To address these issues, we employ the pion distribution amplitude formalism to investigate the Form Factor behavior in this intermediate regime. Our results suggests that non-perturbative physics of the strong interaction is relevant even at energy scales traditionally considered perturbative, implying that the perturbative regime could occur at higher energies than previously thought. Finally, our approach allows us to study isospin-breaking effects through the quadratic pion mass difference.

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

2 major / 1 minor

Summary. The paper computes the electromagnetic form factors of neutral and charged pions in the intermediate-energy region (roughly 1-10 GeV²) by folding a non-perturbative running of α_s extracted from a double-dilaton holographic QCD model into the standard pion distribution-amplitude convolution. It reports deviations from the asymptotic perturbative-QCD prediction, concludes that non-perturbative strong-interaction effects remain relevant at scales traditionally viewed as perturbative, and examines isospin-breaking contributions arising from the quadratic pion-mass difference.

Significance. If the double-dilaton model's α_s(Q²) is shown to be consistent with independent non-perturbative determinations, the calculation would supply a concrete, model-based estimate of the scale at which perturbative QCD becomes reliable for exclusive processes and would help interpret existing neutral-pion form-factor data. The simultaneous treatment of charged and neutral channels plus explicit isospin violation is a constructive feature. The result's broader impact, however, rests entirely on the unvalidated extrapolation of the holographic coupling into the transition region.

major comments (2)
  1. [Model description and results] The central claim that non-perturbative physics remains relevant up to higher scales is carried solely by the specific α_s(Q²) running obtained from the double-dilaton model. No section supplies a direct, quantitative comparison of this running against lattice-QCD or other holographic determinations in the 1-10 GeV² window; without such a benchmark the reported deviation from asymptotic behavior cannot be distinguished from a model artifact.
  2. [Methodology] The double-dilaton parameters are fitted to reproduce low-energy QCD features; using the same parameters to generate α_s at intermediate scales and then interpreting the output form factors as evidence for delayed perturbative onset introduces a circularity that must be addressed explicitly (e.g., by showing that the form-factor predictions are stable under reasonable variations of the fit parameters or by cross-checking against an independent non-perturbative α_s).
minor comments (1)
  1. [Abstract] The abstract contains no numerical results, error estimates, or explicit energy range; adding a short statement of the computed Q² interval and the size of the deviation from the asymptotic limit would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The points raised regarding model validation and potential circularity are important, and we address them point by point below. We will incorporate revisions to strengthen the presentation and support for our conclusions.

read point-by-point responses

  1. Referee: The central claim that non-perturbative physics remains relevant up to higher scales is carried solely by the specific α_s(Q²) running obtained from the double-dilaton model. No section supplies a direct, quantitative comparison of this running against lattice-QCD or other holographic determinations in the 1-10 GeV² window; without such a benchmark the reported deviation from asymptotic behavior cannot be distinguished from a model artifact.

    Authors: We agree that a direct benchmark of our α_s(Q²) in the intermediate-energy window would strengthen the manuscript and help distinguish our results from model-specific features. In the revised version we will add a new subsection (with accompanying figure) that quantitatively compares the double-dilaton running coupling to available lattice-QCD determinations and to other holographic QCD extractions over the 1–10 GeV² range. This addition will provide the requested validation and allow readers to assess the robustness of the claimed deviation from asymptotic perturbative behavior. revision: yes

  2. Referee: The double-dilaton parameters are fitted to reproduce low-energy QCD features; using the same parameters to generate α_s at intermediate scales and then interpreting the output form factors as evidence for delayed perturbative onset introduces a circularity that must be addressed explicitly (e.g., by showing that the form-factor predictions are stable under reasonable variations of the fit parameters or by cross-checking against an independent non-perturbative α_s).

    Authors: We acknowledge the concern about circularity. To address it explicitly, the revised manuscript will include a parameter-variation study: we will vary the double-dilaton fit parameters within the uncertainties fixed by the low-energy observables and demonstrate that the qualitative deviation of the pion form factors from the asymptotic pQCD prediction remains stable. In addition, we will discuss and reference independent non-perturbative determinations of α_s(Q²) in the same energy window to provide an external cross-check. These additions will clarify that the reported behavior is not an artifact of the specific low-energy fit. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper obtains a non-perturbative running of alpha_s from the double-dilaton HQCD model and convolves it with the pion distribution amplitude to compute the neutral and charged pion form factors. This is a standard perturbative QCD calculation in the intermediate regime with the holographic model supplying an external input for the coupling. No equation or step reduces the output form factors to the model parameters by construction, nor does any load-bearing claim rely on a self-citation whose content is unverified or tautological. The interpretation that non-perturbative effects persist at higher scales follows from the numerical results rather than being forced by the inputs themselves.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Based solely on abstract; model parameters for the two dilatons are expected to be fitted, and the holographic mapping itself is an assumption whose validity is not independently verified here.

free parameters (1)
  • double-dilaton parameters
    Typical in holographic QCD models; values are chosen or fitted to match low-energy QCD observables and then used to generate the running coupling.
axioms (1)
  • domain assumption Holographic QCD with double dilaton accurately captures non-perturbative running of alpha_s in the intermediate-energy domain
    Invoked when applying the model to pion distribution amplitudes and form factors.

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discussion (0)

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Reference graph

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