Electromagnetic nucleon form factors in the extended vector meson dominance model

K.S. Kuzmin; M.I. Krivoruchenko; N.M. Levashko

REVIEW 4 major objections 5 minor 1 cited by

Electromagnetic nucleon form factors in the extended vector meson dominance model

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A six-parameter model built from four rho and four omega meson poles describes all four nucleon electromagnetic form factors across spacelike and timelike momentum transfers while respecting quark counting, OZI, and threshold identities.

desk verdict A careful, honest eVMD update—but the radii and couplings are fit outputs, not predictions, and the missing two-pion continuum needs quantification. read the letter →

arxiv 2412.13150 v2 pith:QZKJBUBK submitted 2024-12-17 hep-ph

K.S. Kuzmin , N.M. Levashko , M.I. Krivoruchenko This is my paper · ORCID

classification hep-ph PACS 14.20.Dh25.75.Dw13.30.Ce12.40.Yx

keywords nucleonformfactorsextendedvectormesondominanceradialexcitationsSachsquarkcountingrulesOkubo-Zweig-IizukaruletimelikemomentumtransferZemachradii

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 claims that the electromagnetic structure of the nucleon—the distribution of its electric charge and magnetization encoded in four Sachs form factors (electric and magnetic, proton and neutron)—can be described by one compact ansatz: the photon couples to nucleons through the ground-state $\rho$ and $\omega$ mesons plus their first three radial excitations. Six fitted parameters set the residues of the meson poles, while masses and widths are taken from experiment and theoretical constraints are imposed: quark counting rules (the QCD power-law falloff at large momentum transfer), OZI suppression of strange-meson contributions, approximate scaling of Sachs form factors at moderate transfers, and the threshold identity $G_E(4m_N^2)=G_M(4m_N^2)$. Fitting those six parameters to 395 spacelike and timelike data points yields $\chi^2/\mathrm{ndf}\approx 11.4$, and the extracted ground-state $\rho$ and $\omega$ couplings agree with Frazer-Fulco dispersion relations and the Bonn nucleon-nucleon potential. If correct, this gives applications such as neutrino-event generators, radiative-correction codes, and radius extractions a small, physically motivated parameterization over the whole measured range.

What carries the argument

The load-bearing object is the pole ansatz: each form factor is a rational function whose denominator is a product over the four mesons in an isospin channel, $G_{TN}(t)=P^{TN}_{n-2}(t)\prod_V m_V^2/(m_V^2-t)$, generalized to unstable mesons by replacing each pole with $m_V^2-i\sqrt{t}\,\Gamma_V(t)$. The polynomial $P^{TN}_{n-2}(t)$ (quadratic for the $F_1$ channels, linear for the $F_2$ channels) carries the six fitted parameters; the product of poles enforces the quark-counting falloff at large $|t|$, normalization at $t=0$ fixes charges and magnetic moments, and the threshold identity follows automatically because the electric and magnetic form factors share the same pole denominators. Energy-dependent widths—zero below the two- or three-pion thresholds and matched to empirical on-shell values—make the form factors continuous and give the pole residues their interpretation as vector-meson–nucleon couplings.

What would settle it

Evaluate the two-pion continuum contribution to the isovector spectral function using the measured pion form factor and pion-nucleon partial waves; if that contribution changes the isovector charge radius by more than a few percent or moves the fitted rho-nucleon residue outside the quoted unitarity band, the pole-saturation premise is falsified. A second decisive check would be a high-precision measurement of the proton electric-to-magnetic ratio at $Q^2 \approx 5$ to $8.5$ GeV$^2$, where the model already deviates from the JLab points.

Watch

Extended reading notes

Core claim

The central claim is that the nucleon form factors are saturated by four isoscalar and four isovector vector mesons—$\rho(770)$, $\rho(1250)$, $\rho(1450)$, $\rho(1700)$ and their $\omega$ partners—with masses and widths fixed to empirical values, so that only six parameters remain free. In the zero-width limit the form factors take the multiplicative form $G_{TN}(t)=P^{TN}_{n-2}(t)\prod_V m_V^2/(m_V^2-t)$, which satisfies quark counting rules identically and makes the threshold identity $G_E(4m_N^2)=G_M(4m_N^2)$ automatic; the model is then converted to a sum of Breit-Wigner poles with energy-dependent widths. With those six parameters the model reproduces the smooth part of the proton and neutron electric and magnetic form factors in both spacelike and timelike regions, gives ground-state couplings consistent with Frazer-Fulco unitarity and the Bonn potential, and returns nucleon and Zemach radii close to the measured values.

Load-bearing premise

The entire description rests on four rho and four omega poles with fixed masses and widths saturating the form factors; the continuous two-pion background and other non-resonant contributions are neglected, and if those continua are significant the extracted radii and couplings shift.

Editorial extensions

If this is right

If the model is right, a six-parameter physically constrained parameterization suffices for applications that currently use fits with many more free parameters, including neutrino-event generators and radiative-correction codes.
The extracted $\rho$- and $\omega$-nucleon couplings can be used as input to meson-exchange models of the nucleon-nucleon interaction, since they agree with the Bonn potential.
The predicted radii—proton charge radius 0.815 fm, neutron charge radius squared $-0.067$ fm$^2$, magnetic radii near 0.79 fm, and Zemach moments 1.034 fm and 1.015 fm with third Zemach moment 2.036 fm$^3$—provide definite targets for atomic-physics and electron-scattering experiments.
In the timelike region the model fixes the smooth background; the observed sinusoidal oscillations would then have to come from vector mesons heavier than 2 GeV, motivating their spectroscopy.
The compatibility of quark counting, OZI, scaling, and threshold identities with the full dataset argues that these constraints should be imposed in any future form-factor parameterization.

Reading between the lines

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

Because the model omits the two-pion continuum, its isovector charge radius is probably somewhat low; adding that continuum through dispersion relations could move the neutron charge radius toward the measured $-0.1155$ fm$^2$ without changing the high-$Q^2$ behavior.
The same pole ansatz could be applied to $\Sigma$ and $\Xi$ hyperon form factors, where timelike data are sparser; a comparable fit there would show whether radial-excitation saturation is a general hadron property rather than a nucleon-specific arrangement.
The global $\chi^2/\mathrm{ndf}\approx 11.4$ points to internal tensions among datasets; refitting with floating per-experiment normalizations would show whether the tension is concentrated in a few discrepant sets or reflects a genuine shape disagreement, and would make the parameter errors more honest.
Should future $e^+e^-$ data resolve the timelike oscillations as interference from a meson just above the $N\bar{N}$ threshold, the residue extracted for that meson would provide a sharp test of the same unitarity-based coupling pattern the model uses for the ground states.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This paper extends the MFK eVMD model from two to six parameters, adds a fourth rho/omega pair at 1.70 GeV, and fits 395 nucleon form factor points spanning spacelike and timelike regions. The model construction is transparent: it imposes QCR, OZI, scaling, threshold identities, and uses empirical masses and widths. The appendix derivation of the radius formulas is sound, and the authors deserve credit for reporting the bad news themselves—the global chi2/ndf is 11.4, and separate fits to spacelike and timelike data give incompatible parameter sets (12.3 and 5.2).

What is genuinely new is the updated fit with the BESIII 2023 neutron timelike data and the six-parameter parameterization. For neutrino event generators that need a compact analytic form, the eVMD-VI curves could be useful despite the imperfect chi-square.

The soft spots are real but not hidden. The radii and Zemach moments in Table 1 are computed from the same fitted c, d, h parameters via Eqs. (2.23)-(2.24), so they are not independent predictions; the agreement with PDG values is a consistency check, not a validation. The stress-test concern about the isovector two-pion continuum is on target. Section III.C admits that the logarithmic singularity in the isovector spectral function below the two-pion threshold contributes positively to the isovector charge radius and is outside the model. Since the six parameters are fit globally, the omitted continuum is absorbed into the fitted coefficients; it biases both the extracted radii and the ground-state residues that are then compared with Frazer-Fulco and the Bonn model. The paper does not quantify this bias. That is a genuine gap, and it is the main reason the 'predictions' should be treated with caution.

The paper also ships no code or data files, which would make the fit reproducible. The per-set chi-squares in Sec. III are useful, but I would like to see the continuum version.

Who is this for? Researchers needing a compact, physically motivated parameterization with known constraints, and anyone working on the eVMD program. The paper deserves a serious referee: the constraints and the data handling are careful enough, and the continuum question is worth forcing. I would ask the authors to (i) add the explicit two-pion continuum and show how the radii and residues shift, and (ii) soften the abstract and conclusion claims given chi2/ndf ~ 11.4 and the region-incompatibility. With those changes the paper would be more honest about what it delivers.

Referee Report

4 major / 5 minor

Summary. The paper proposes an extended vector meson dominance (eVMD-VI) model with four rho/omega radial excitations and six free parameters to describe all four nucleon electromagnetic form factors in both spacelike and timelike regions. The model implements threshold identities, quark counting rules, OZI suppression, and Sachs scaling. A global fit to 395 data points yields chi2/ndf about 11.4. From the fitted parameters the authors compute nucleon radii, Zemach moments, and ground-state rho/omega nucleon coupling constants, and compare these with experimental and dispersion-theory values. An appendix derives closed-form expressions for radius moments in terms of pole residues.

Significance. If validated, the model would provide a compact, physically motivated parameterization of nucleon form factors with few free parameters, and its pole-residue framework would connect electromagnetic form factors to vector-meson nucleon couplings. The paper's strengths include explicit analytic derivations of the radius and Zemach moment formulas (Appendix A and Sec. II.D), careful compilation of a large modern dataset, bootstrap and covariance-matrix error estimates, and consistent enforcement of several theoretical constraints. However, the statistical quality of the global fit is poor, the radius and coupling 'predictions' are algebraically dependent on the fitted parameters rather than independent, and the model's own text concedes that the two-pion continuum is omitted. These issues substantially temper the significance of the quantitative claims as they now stand.

major comments (4)

[Sec. III.B] The reported global fit quality, chi2/ndf = 4443.3/389 ≈ 11.4, is far above unity, and the paper itself states that separate fits to spacelike and timelike data give chi2/ndf = 12.3 and 5.2 with inconsistent parameter sets. This means the model does not simultaneously describe both regions with one parameter set, contrary to the central claim. The authors should either treat the model as a qualitative 'reasonable description' with an explicit caveat about the reduced chi-square, or provide a quantitative account of the discrepancy (e.g., underestimated systematic uncertainties, missing contributions above 1.7 GeV, or deficiencies in the width parameterization). Without such an account, the claim of a simultaneous description is not statistically supported.
[Sec. II.D, Eqs. (2.23)-(2.24)] The radius 'predictions' in Table 1 are not independent of the fit: Eqs. (2.23) and (2.24) express <r^2>_EN and <r^2>_MN as linear combinations of the fitted parameters c_N and h_N plus fixed meson masses. The agreement with PDG/experimental radii is therefore a consistency check of the fitted low-Q^2 behavior, not a model prediction. The text should state this explicitly and avoid implying that the radii are an output with predictive power beyond the data already included in the fit.
[Sec. III.C] The paper concedes that the logarithmic singularity in the isovector spectral function below the two-pion threshold contributes positively to the nucleon isovector charge radius and lies outside the eVMD model. Since Eqs. (2.20)-(2.24) compute all radius moments from the pole residues of the fitted form factors, the omission of the two-pion cut biases the extracted radii and also the ground-state rho/omega coupling constants that are claimed to be Frazer-Fulco/Bonn-consistent. Please quantify this systematic effect, for example by adding a spectral function based on two-pion unitarity (Hohler/Hammer-Meissner input) and showing how the extracted radii and residues shift, or by demonstrating that the pole approximation is numerically adequate for the low-Q^2 derivatives. This is load-bearing because the same six parameters absorb the missing continuum in the global fit.
[Sec. III.C] The text reports that 12 experimental data sets with 101 points (27% of the total) have chi2/np values higher than 10, including most proton spacelike data. This quantitative breakdown, combined with the global chi2/ndf ≈ 11.4, shows that the model's description is not uniform across the dataset. The authors should address whether this indicates a systematic deficiency of the model rather than random scatter, and discuss the implications for the extracted parameter values and their quoted 1σ uncertainties.

minor comments (5)

[Sec. II.C] There is a typo: 'form actors' should read 'form factors' in the sentence describing parameterizations and phenomenological models.
[Sec. III.D] The text refers to 'BIND' experiments; this should be 'BINP' (the Budker Institute of Nuclear Physics), matching the experimental collaborations cited.
[Eq. (3.1)] The variable eta in the definition of the effective form factor is used but not defined in the text; it is the standard eta = t/(2 m_N^2), but this should be stated explicitly for clarity.
[Reference [38]] The publisher location 'Gernamy' is a typo for 'Germany'.
[Sec. II.C, Eq. (2.14)] The notation P^1I_2(t) and P^2I_1(t) is not immediately transparent; a brief statement that these are the polynomial numerators for the Dirac form factors F_1I and F_2I with the specified degrees would improve readability.

Circularity Check

1 steps flagged · score 4.0 of 10

Radii and Zemach moments are presented as predictions but reduce by Eqs. (2.23)-(2.24) to the six fitted parameters; the central pole fit itself is anchored by external Frazer-Fulco/Bonn coupling comparisons.

fitted input called prediction [Sec. II.D, Eqs. (2.23)-(2.24); Sec. III.C, Tab. 1]
"The important case of the second moments allows the result to be presented in terms of the model parameters in a simple way: 1/6 ⟨r²⟩_EN = c_N + (µ_N − e_N)/(4m_N²) + e_N Σ_V 1/m_V², 1/6 ⟨r²⟩_MN = c_N + h_N/µ_N + Σ_V 1/m_V². ... The numerical values of the second moments obtained in the model are displayed in Tab. 1 together with the experimental values."

The c_N and h_N in Eqs. (2.23)-(2.24) are not external inputs: Sec. III.B states 'The model parameters c_I, d_I, and h_I are determined by fitting the experimental data.' The quoted radii, e.g. ⟨r²⟩^{1/2}_{Ep}=0.815 fm, are thus Taylor coefficients of the same fitted form factors, not independent predictions. The low-Q² spacelike data that most directly constrain these coefficients are part of the 395-point global fit, so the apparent agreement with PDG radii is largely a restatement of the fitted low-t behavior. The Zemach moments, being convolutions of the same fitted G_E and G_M curves, inherit the same status. This does not affect the separate, genuinely external comparison of ground-state residues with Frazer-Fulco and Bonn values.

full rationale

The central global fit is not circular: six parameters are fitted directly to 395 form-factor data points, and the ground-state ρ/ω residues are compared with Frazer-Fulco unitarity and Bonn-potential values that are not part of the fit, providing independent grounding for the pole content. The quantitative circularity is confined to the radii and Zemach moments: Eqs. (2.23)-(2.24) express them as explicit functions of the fitted c_N and h_N, and the paper's own Sec. III.C admits that the two-pion continuum below threshold contributes to the isovector charge radius but is outside the eVMD model. That admitted omission is a robustness limitation rather than a logical circle, and should be weighed as a correctness risk: the fitted parameters can absorb the missing continuum and shift both the quoted radii and extracted residues. Self-citations appear (e.g., Refs. [14], [48], [128]) but are not load-bearing, since external comparisons and standard unitarity relations carry the weight. Overall, the derivation is partially circular in its secondary 'predictions' while the main fit-based claim retains independent content.

Assumptions & free parameters 7 free parameters · 8 assumptions · 0 invented entities

The model rests on a VMD saturation assumption plus constraints (QCR, OZI, SR, TI). The six polynomial coefficients are fitted, and the masses/widths of the two highest radial excitations are fixed by hand. No new particles are invented.

free parameters (7)

cp = -0.812 +/- 0.008 GeV^-2
Linear coefficient of the proton polynomial P^1I_2(t), fitted to 395 data points.
dp = 0.221 +/- 0.003 GeV^-4
Quadratic coefficient of the proton polynomial P^1I_2(t), fitted.
hp = -0.552 +/- 0.009 GeV^-2
Linear coefficient of the proton polynomial P^2I_1(t), fitted.
cn = 0.256 +/- 0.016 GeV^-2
Linear coefficient of the neutron polynomial P^1I_2(t), fitted.
dn = -0.110 +/- 0.010 GeV^-4
Quadratic coefficient of the neutron polynomial P^1I_2(t), fitted.
hn = 0.650 +/- 0.039 GeV^-2
Linear coefficient of the neutron polynomial P^2I_1(t), fitted.
Masses and widths of the 1.25 GeV and 1.70 GeV rho/omega states = m=1.25, 1.70 GeV; widths 0.3, 0.13, 0.25, 0.315 GeV
Fixed by hand from PDG and other evidence; the 1.25 GeV states are not reliably established, and the rho/omega mass degeneracy is imposed.

assumptions (8)

domain assumption Spectral functions of nucleon form factors are saturated by four rho and four omega radial excitations
Central VMD ansatz, Sec. II.B-II.C; neglects non-resonant continua.
domain assumption Quark counting rules (1.1)-(1.2) hold for the form factors
Imposed via polynomial structure, Sec. II.A.2.
domain assumption OZI rule suppresses strange vector meson contributions
Sec. II.A.4; phi and radial excitations are excluded.
domain assumption Sachs scaling relations (1.3) hold at moderate Q^2
Used to relate proton/neutron polynomials, Sec. II.A.3.
standard math Analyticity with cuts starting at 4m_pi^2 (I=1) and 9m_pi^2 (I=0)
Standard dispersion theory background, Sec. II.A.5.
standard math Threshold identity G_E(4m_N^2)=G_M(4m_N^2)
Follows from definitions of Sachs form factors, Sec. II.A.1.
ad hoc to paper Mass degeneracy of rho and omega families
Needed to satisfy TI, QCR and SR simultaneously in the no-width limit, Sec. II.B; used for all four pairs.
domain assumption Neglect of two-pion and multi-pion continuum below the vector meson poles
Acknowledged in Sec. III.C as beyond the eVMD scope; affects the isovector charge radius.

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Cite this review

Pith. "Pith review of Electromagnetic nucleon form factors in the extended vector meson dominance model." pith.science (2026). https://pith.science/paper/QZKJBUBK

@misc{pith2026241213150,
  author       = {Pith},
  title        = {Pith review of: Electromagnetic nucleon form factors in the extended vector meson dominance model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QZKJBUBK}},
  note         = {Machine review of arXiv:2412.13150}
}

abstract

An extended vector meson dominance model is developed to describe electromagnetic nucleon form factors. The model includes families of the $\rho$- and $\omega$-mesons with the associated radial excitations. The free parameters of the model are determined using a global statistical analysis of experimental data on the electromagnetic nucleon form factors in space- and timelike regions of transferred momenta. The vector meson masses and widths are equal to their empirical values, while the residues of form factors at the poles corresponding to the ground states of the $\rho$- and $\omega$-mesons are consistent with the findings of both the Frazer-Fulco unitarity relations and the Bonn potential for coupling constants of the $\rho$- and $\omega$-mesons with nucleons. Theoretical constraints imposed on the model include the quark counting rules, the Okubo-Zweig-Iizuka rule, the scaling law of Sachs form factors at moderate momentum transfers, and the suppression of Sachs form factors near the nucleon-antinucleon threshold. A reasonable description of the nucleon form factors in the experimentally accessible range of transferred momenta, as well as the electric and magnetic nucleon radii and Zemach radii, is obtained.

Figures

Figures reproduced from arXiv: 2412.13150 by the authors.

**Figure 2.** Normalized electric form factor of the proton mea [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Normalized magnetic form factor of the proton (a) and the ratio of normalized electric to magnetic form factors of [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

Figures from the paper (9 more)

**Figure 4.** Figure 4: Module of electric form factor of the neutron (a) measured for the timelike region with BESIII 2023 [123]; normalized [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

**Figure 5.** Figure 5: Proton effective form factor (3.1) measured for the timelike region with BESIII 2021 [35], 2020 [121], 2019 [120], [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

**Figure 6.** Figure 6: Neutron effective form factor (3.1) measured for the timelike region with BESIII 2021 [122], BINP VEPP-2000 SND [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]

**Figure 7.** Figure 7: Modules of electric (a) and magnetic (b) form factors of the proton measured for the timelike region with BESIII [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]

**Figure 8.** Figure 8: Modules of electric (a) and magnetic (b) form factor of the neutron measured for the timelike region with BESIII [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]

**Figure 9.** Figure 9: Modules of the ratios of electric to magnetic form factors of the proton (a) and neutron (b) measured for the timelike [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]

**Figure 10.** Figure 10: The directed closed curve, C, of the contour integral (A.1) in the complex Q-plane. The crosses show simple poles of the form factor GT N −Q2 corresponding to the vector meson masses and vector meson widths. Now we change the order of integrals and obtain Z dr r 2…

**Figure 12.** Figure 12: The contour of integration over the complex vari [PITH_FULL_IMAGE:figures/full_fig_p018_12.png]

**Figure 11.** Figure 11: The contour for integration over the complex vari [PITH_FULL_IMAGE:figures/full_fig_p018_11.png]

Discussion (0). Continue with ORCID to comment.

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Reference graph

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

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(2.4) The identities are a direct consequence of definitions (2.2) and (2.3), and they are obtained automatically when the parameterization is carried out using FT N(t)

Threshold identities At the nucleon–antinucleon threshold ( N N ), the elec- tric ( E) and magnetic ( M ) form factors satisfy the threshold identities (TI’s): GEN 4m2 N = GM N 4m2 N . (2.4) The identities are a direct consequence of definitions (2.2) and (2.3), and they are obtained automatically when the parameterization is carried out using FT N(t). If...

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Taking into account additional suppression of amplitudes, generated by spin flip of quarks [8], QCR’s take the form of Eqs

Quark counting rules Hadron form factors obey QCR’s [5–7]. Taking into account additional suppression of amplitudes, generated by spin flip of quarks [8], QCR’s take the form of Eqs. (1.1) and (1.2) and give rise to GEN (t) ∼ GM N(t) = O 1/t2 as t → −∞. (2.5) The spin flip has a particularly substantial impact on the electromagnetic transitions of high-sp...

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Scaling relations In the spacelike region for low momentum transfers Sachs form factors fulfill SR’s (1.3). With acceptable ac- curacy GEp(t), GM p(t)/µp and GM n(t)/µn are described 3 10 -3 10 -2 10 -1 1 10 10 2 10 -3 10 -2 10 -1 1 10 10 2 10 -3 10 -2 10 -1 1 10 10 2 10 -3 10 -2 10 -1 1 10 10 2 | GEp | (a) | GMp | (b) | GEn | (c) | GMn | (GeV2) −10 −9 −8...

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Okubo-Zweig-Iizuka rule According to the Okubo-Zweig-Iizuka (OZI) rule, me- sons with valence strange quark–antiquark pairs are only weakly coupled to non-strange hadrons [38]. The theo- retical explanation of this rule is based on the fact that annihilation of a strange quark–antiquark pair causes the appearance of a large momentum in the QCD coupling co...

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Form factors are functions with the established analytical properties

Analyticity Analyticity is a universal attribute of quantum field theory, arising from causality and Lorentz covariance. Form factors are functions with the established analytical properties. The functions are analytical in the complex t-plane with a cut 4m2 π, +∞ in the isovector channel and a cut 9m2 π, +∞ in the isoscalar channel, where mπ is the pion ...

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bootstrap

Frazer-Fulco unitarity The formulation of two-body unitarity relations for the electromagnetic pion form factor [1, 4] and nucleon form factors [1–3] became a noteworthy achievement. The for- malism used provides form factors with the correct ana- lytical properties. The two-body unitarity is justified for spectral functions with the pion energy t <1 GeV2...

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