Onset of the asymptotic regime and logarithmic scaling-violation corrections for nucleon elastic form factors via Bloom-Gilman duality in asymptotically free theories

In 1974 Gross and Treiman combined the renormalization-group extrapolation of deep-inelastic structure functions with Bloom-Gilman local duality to predict the large-Q^2 behavior of nucleon elastic electromagnetic form factors: with the threshold exponent d=3 the leading power reproduces the empirical dipole formula, modified by calculable logarithmic corrections, and asymptotically the form factors fall faster than any inverse power of Q^2. The paper itself flagged what remains to be settled: whether the threshold limit is uniform near omega = 1 + O(1/Q^2) once the double-logarithmic corrections of relative order (ln ln omega)^2/ln(Q^2/mu^2) are resummed to all orders; whether experiments have reached the asymptotic regime where the analysis applies; and whether the predicted logarithmic deviations from the dipole can be extracted unambiguously from data rather than being masked by higher-twist corrections (power-suppressed contributions of relative order $1/Q^2$), duality violations, and the unknown normalization constants. In the subsequent literature the inclusive-exclusive bridge was formalized through moment relations in the Nachtmann variable, a target-mass-corrected scaling variable (De Rújula, Georgi, Politzer 1977) and validated quantitatively at low Q^2 (Ent, Keppel, Niculescu 2000), large-x double-log resummation became standard technology for inclusive DIS (Almasy, Soar, Vogt 2011), but the elastic data frontier (about 31 GeV^2 for the magnetic form factor) still does not demonstrate the asymptotic regime and the logarithmic corrections have not been unambiguously extracted. The open core survives in modern form as the question of the onset of the perturbative regime in exclusive nucleon reactions and of a controlled threshold resummation in the duality channel.

Background

In inclusive electron-nucleon scattering (deep inelastic scattering, DIS), a lepton transfers four-momentum $q$ (with $Q^2 = -q^2 > 0$) and breaks up the target; the cross section is encoded in structure functions such as $F_2(\nu, q^2)$, where $\nu$ is the energy transfer. Bjorken scaling is the observation that at large $Q^2$ these functions depend only on the dimensionless ratio $x = Q^2/(2M\nu)$ ($M$ being the nucleon mass) rather than on $Q^2$ and $\nu$ separately. In an asymptotically free field theory - one whose effective coupling vanishes at short distances, of which quantum chromodynamics (QCD) is the established example - Bjorken scaling holds only approximately: the moments $\int dx\, x^{N-2} F_2$ acquire calculable logarithmic $Q^2$ dependence, powers of $\ln(Q^2/\mu^2)$ multiplied by anomalous dimensions (numbers computable from the theory's operators; $\mu$ is an arbitrary reference scale). Gross showed how the moments could be extrapolated from one value of $Q^2$ to another ('How to Test Scaling in Asymptotically Free Theories', Phys. Rev. Lett. 32, 1071 (1974)); near the threshold $x \to 1$ the structure function behaves as $F_2 \sim (1-x)^d$ with a threshold exponent $d$, and the extrapolation is not uniform there.

In the resonance region, Bloom and Gilman ('Scaling and the Behavior of Nucleon Resonances in Inelastic Electron-Nucleon Scattering', Phys. Rev. D 4, 2901 (1971)) had found that resonances, when averaged, track the same scaling curve that describes truly deep inelastic scattering, and conjectured a local duality: the strength under an individual resonance - governed by that resonance's electromagnetic transition form factor - follows the scaling structure function in the same kinematic interval. Gross and Treiman combined the renormalization-group extrapolation (the renormalization group governs how quantities change with the momentum scale; here it extrapolates the structure function from a reference momentum $q'^2$ to the momentum of interest) with this local duality to connect the threshold behavior of $F_2$ to the nucleon's elastic electromagnetic form factors $G_E(Q^2)$ and $G_M(Q^2)$ (the electric and magnetic distributions measured in elastic electron-nucleon scattering), which at the time were well described by the dipole formula $(1+Q^2/0.71\,\mathrm{GeV}^2)^{-2}$. They predicted that the form factors fall faster than any inverse power of $Q^2$ up to calculable powers of $\ln(Q^2/\mu^2)$; with the experimentally inferred exponent $d = 3$ the leading power reproduces the dipole behavior, with logarithmic deviations given by their Eq. (12). The same power correspondence, $F_2 \sim (1-x)^3$ against $G_M \sim 1/Q^4$, is the Drell-Yan-West relation between the large-$x$ behavior of a structure function and the falloff of the corresponding form factor; the same power is implied by the dimensional counting rules of Brodsky and Farrar ('Scaling Laws at Large Transverse Momentum', Phys. Rev. Lett. 31, 1153 (1973)), according to which the elastic form factors of a bound state of $n$ constituents fall as $1/Q^{2(n-1)}$ up to logarithms.

The prediction rests on assumptions the 1974 paper explicitly could not justify. The derivation assumed the $Q^2 \to \infty$ limit is uniform in the scaling variable $\omega$ even near threshold, $\omega = 1 + O(1/Q^2)$ (here $\omega$ is the Bloom-Gilman variable $\omega = 1 + W^2/Q^2$, with $W$ the invariant mass of the produced hadrons, so that near threshold $\omega - 1$ is proportional to $1 - x$), while the paper's own footnote warned that corrections of relative order $(\ln\ln\omega)^2/\ln(Q^2/\mu^2)$ grow in that region and would require summing all orders in the coupling - 'a difficult task'. Whether the reference structure function at the then-available momenta ($q'^2$ of a few to ten $\mathrm{GeV}^2$) had 'penetrated this asymptotic region' was stated to be an open question, with evidence 'too sparse to determine' whether the exponent $d$ varies with $Q^2$. The detailed comparison with experiment was called premature because of sensitivity to the reference choices $\mu^2$ and $q'^2$; the observed deviation from the dipole formula could reflect either scaling violation or a failure of the duality arguments; and the absolute normalization is not predicted because the relevant operator matrix elements (the constants $C_n$ in the moment formula) are unknown - the companion letter by De Rújula ('Proton Magnetic Form Factor in Asymptotically Free Field Theories', Phys. Rev. Lett. 32, 1143 (1974)) phrased this as the inverse-moment problem, through which asymptotic freedom 'loses a lot of predictive power' for the absolute scale. The question is whether these gaps have since been closed.

Problem Statement

Determine whether the asymptotic-freedom prediction for the large-$Q^2$ behavior of nucleon elastic form factors - dipole-like power fall-off modified by calculable logarithmic scaling-violation corrections, derived through local Bloom-Gilman duality from the threshold behavior of the deep-inelastic structure functions - is theoretically sound and experimentally realizable. All parts of the question serve this single objective of establishing the validity domain of the prediction. Specifically: (a) Uniformity and resummation: is the $Q^2 \to \infty$ limit of the renormalization-group extrapolation uniform near threshold, $\omega = 1 + O(1/Q^2)$, once the double-logarithmic corrections of relative order $(\ln\ln\omega)^2/\ln(Q^2/\mu^2)$ are resummed to all orders in the coupling, and does the duality-based form-factor prediction survive in a definite corrected form? (b) Onset and threshold exponent: at what $Q^2$ does the asymptotic regime, in which the perturbative renormalization-group analysis of the structure functions actually applies, begin - and does the threshold exponent $d$ vary with $Q^2$ there, i.e., how large are the higher-twist corrections - contributions suppressed by additional powers of $1/Q^2$, associated with multi-parton correlations rather than single-parton scattering - at large $x$? (c) Extraction and interpretation: can the predicted logarithmic deviations from the dipole behavior be extracted unambiguously from experimental data, given the sensitivity to the reference choices $\mu^2$ and $q'^2$, the alternative explanation of the observed dipole deviation as a failure of local duality rather than scaling violation, and the obstruction of absolute-scale predictions by the unknown operator matrix elements (the inverse-moment problem)? The 1974 prediction stands or falls with both the resummed threshold limit in the elastic duality channel and the onset scale and the extractability of the logarithmic corrections.

References

1. D. J. Gross and S. B. Treiman, 'Hadronic Form Factors in Asymptotically Free Field Theories', Physical Review Letters 32, 1145-1148 (1974). DOI: 10.1103/PhysRevLett.32.1145, https://doi.org/10.1103/PhysRevLett.32.1145 (primary source)
2. D. J. Gross, 'How to Test Scaling in Asymptotically Free Theories', Physical Review Letters 32, 1071-1073 (1974). DOI: 10.1103/PhysRevLett.32.1071, https://doi.org/10.1103/PhysRevLett.32.1071 (source of the extrapolation formula used by the primary paper)
3. A. De Rújula, 'Proton Magnetic Form Factor in Asymptotically Free Field Theories', Physical Review Letters 32, 1143-1145 (1974). DOI: 10.1103/PhysRevLett.32.1143, https://doi.org/10.1103/PhysRevLett.32.1143 (companion letter; source of the inverse-moment-problem phrasing)
4. A. De Rújula, H. Georgi, and H. D. Politzer, 'Demythification of Electroproduction, Local Duality and Precocious Scaling', Annals of Physics 103, 315-353 (1977). DOI: 10.1016/S0003-4916(97)90003-8, https://doi.org/10.1016/S0003-4916(97)90003-8
5. E. D. Bloom and F. J. Gilman, 'Scaling and the Behavior of Nucleon Resonances in Inelastic Electron-Nucleon Scattering', Physical Review D 4, 2901-2916 (1971). DOI: 10.1103/PhysRevD.4.2901, https://doi.org/10.1103/PhysRevD.4.2901 (the duality hypothesis source)
6. S. J. Brodsky and G. R. Farrar, 'Scaling Laws at Large Transverse Momentum', Physical Review Letters 31, 1153-1156 (1973). DOI: 10.1103/PhysRevLett.31.1153, https://doi.org/10.1103/PhysRevLett.31.1153 (dimensional counting rules)
7. R. Ent, C. E. Keppel, and I. Niculescu, 'Nucleon elastic form factors and local duality', Physical Review D 62, 073008 (2000). DOI: 10.1103/PhysRevD.62.073008, https://doi.org/10.1103/PhysRevD.62.073008 (local-duality extraction of G_M^p)
8. S. Simula, 'Comment on "Nucleon elastic form factors and local duality"', Physical Review D 64, 038301 (2001). DOI: 10.1103/PhysRevD.64.038301, https://arxiv.org/abs/hep-ph/0012193
9. W. Melnitchouk, R. Ent, and C. Keppel, 'Quark-hadron duality in electron scattering', Physics Reports 406, 127-301 (2005). DOI: 10.1016/j.physrep.2004.10.004, https://arxiv.org/abs/hep-ph/0501217 (local elastic duality section and outlook)
10. A. A. Almasy, G. Soar, and A. Vogt, 'Generalized double-logarithmic large-x resummation in inclusive deep-inelastic scattering', Journal of High Energy Physics 2011, 030 (2011). DOI: 10.1007/JHEP03(2011)030, https://arxiv.org/abs/1012.3352
11. V. Punjabi, C. F. Perdrisat, M. K. Jones, E. J. Brash, and C. E. Carlson, 'The structure of the nucleon: Elastic electromagnetic form factors', European Physical Journal A 51, 79 (2015). DOI: 10.1140/epja/i2015-15079-x, https://arxiv.org/abs/1503.01452
12. Y. Wunderlich, A. Bulgakov, K. Joo, T.-S. H. Lee, and V. I. Mokeev, 'Quark-hadron duality in inclusive electron-proton scattering at high Q^2: structure functions and truncated moments from CLAS12', arXiv:2606.29690 (2026), https://arxiv.org/abs/2606.29690 (preprint; extends quantitative local-duality evidence in the inclusive channel to Q^2 of about 10 GeV^2)