Spiral structure and logarithmic evolution of deuteron form factors: evidence for a transitional regime in QCD

Iryna Myroshnykova; Yaroslav D. Krivenko-Emetov

arxiv: 2605.08225 · v1 · submitted 2026-05-06 · ✦ hep-ph · nucl-th

Spiral structure and logarithmic evolution of deuteron form factors: evidence for a transitional regime in QCD

Yaroslav D. Krivenko-Emetov , Iryna Myroshnykova This is my paper

Pith reviewed 2026-05-12 01:03 UTC · model grok-4.3

classification ✦ hep-ph nucl-th

keywords deuteron form factorselastic electron scatteringQCD transitional regimelogarithmic evolutionhelicity amplitudesvalence quark dynamicstwo-photon exchangetensor polarization

0 comments

The pith

Deuteron form factors fit best to a pre-asymptotic QCD parameterization with logarithmic corrections and valence-quark correlations.

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

The paper analyzes all available elastic electron-deuteron scattering data using perturbative QCD scaling, helicity amplitudes, and three families of phenomenological forms for the structure functions. It performs global fits that include two-photon exchange and logarithmic evolution tied to six-quark operator anomalous dimensions. The parameterization that adds pre-asymptotic logarithmic behavior together with correlated valence-quark dynamics yields the lowest chi-squared per degree of freedom. This outcome implies that the momentum transfers reached so far lie in a transitional window between purely hadronic and fully asymptotic quark-gluon regimes, where helicity-conserving amplitudes still dominate and hidden-color configurations may matter.

Core claim

A global fit to world data on A(Q²), B(Q²), differential cross sections, and tensor polarization shows that the f1 parameterization, which incorporates pre-asymptotic logarithmic behavior and correlated valence-quark dynamics, provides the best description with the smallest χ²/dof. The results indicate that the accessible Q² region corresponds to a transitional regime between hadronic and quark-gluon descriptions, with helicity-conserving amplitudes dominating while the asymptotic pQCD regime has not yet been realized; this points to possible nontrivial multiquark correlations in the short-distance deuteron structure.

What carries the argument

The f1 parameterization, which adds pre-asymptotic logarithmic evolution governed by anomalous dimensions of six-quark operators to correlated valence-quark dynamics.

If this is right

Helicity-conserving amplitudes continue to dominate in the present momentum-transfer window.
The fully asymptotic perturbative QCD regime for deuteron form factors has not yet been reached.
Nontrivial multiquark correlations, possibly linked to hidden-color configurations, remain relevant at short distances.
The observable t21 is especially sensitive to the onset of asymptotic helicity behavior.

Where Pith is reading between the lines

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

Similar transitional signatures may appear in other light nuclei once comparable data sets become available.
Improved two-photon-exchange calculations could further tighten the distinction among the three parameterization families.
If the transitional picture holds, lattice QCD calculations of six-quark operators should reproduce the observed logarithmic slope before full asymptotic scaling sets in.

Load-bearing premise

The three chosen classes of parameterizations are enough to separate dynamical regimes and that unaccounted systematic uncertainties in the world data set do not drive the global fit.

What would settle it

Precision measurements of the tensor polarization t21 at significantly larger Q² that either follow or clearly deviate from the helicity-amplitude behavior predicted by the f1 form.

Figures

Figures reproduced from arXiv: 2605.08225 by Iryna Myroshnykova, Yaroslav D. Krivenko-Emetov.

**Figure 1.** Figure 1: Comparison of the structure function A(Q2 ) with the predictions of the phenomenological parameterizations f1, f2, and f3. All models provide a qualitatively similar description of the experimental data in the accessible momentum-transfer region [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗

**Figure 2.** Figure 2: Comparison of the structure function B(Q2 ) with the predictions of the models f1, f2, and f3. The observable B(Q2 ) demonstrates enhanced sensitivity to helicity-flip amplitudes and subleading contributions. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

**Figure 3.** Figure 3: Comparison of the reduced differential cross section [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗

**Figure 4.** Figure 4: Tensor polarization observable t20(Q2 ) for the parameterizations f1, f2, and f3. Although the current experimental uncertainties remain large, the models demonstrate visibly different asymptotic behavior. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗

**Figure 5.** Figure 5: Tensor polarization observable t21(Q2 ). The asymptotic behavior predicted by the three parameterizations differs substantially at large momentum transfer. Future precision measurements of t21 may therefore provide a decisive test of the helicity structure of the deuteron [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗

**Figure 6.** Figure 6: Tensor polarization observable t22(Q2 ) for the parameterizations f1, f2, and f3. The observable exhibits moderate sensitivity to subleading helicity amplitudes and logarithmic evolution effects. • the structure function A(Q2 ) is reproduced reliably by all models; 14 [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗

read the original abstract

We present a consistent analysis of elastic electron--deuteron scattering combining perturbative quantum chromodynamics (pQCD) scaling, helicity amplitudes, and phenomenological parameterizations of deuteron form factors. Particular attention is paid to logarithmic corrections governed by anomalous dimensions of six-quark operators and to two-photon exchange (TPE). Three classes of parameterizations corresponding to different dynamical regimes are considered: pre-asymptotic valence-quark dominance, effective higher-twist contributions, and modified logarithmic evolution. A global fit to the world data for the structure functions $A(Q^2)$ and $B(Q^2)$, differential cross sections, and tensor polarization observables is performed using a combined strategy of global and local minimization. The best description of the complete data set is achieved within the $f_1$ parameterization, which incorporates pre-asymptotic logarithmic behavior and correlated valence-quark dynamics. Among the considered models, $f_1$ gives the smallest value of $\chi^2/\mathrm{dof}$. The obtained results indicate that the presently accessible momentum-transfer region corresponds to a transitional regime between hadronic and quark--gluon descriptions. Helicity-conserving amplitudes dominate, whereas the asymptotic pQCD regime has not yet been fully realized. This may indicate nontrivial multiquark correlations related to hidden-color configurations in the short-distance deuteron structure. The tensor polarization observable $t_{21}$ is especially sensitive to the asymptotic behavior of the helicity amplitudes. Future measurements at larger $Q^2$ may provide a decisive test for distinguishing between the considered dynamical scenarios.

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 fits deuteron form factor data to three phenomenological classes and finds its f1 logarithmic form wins on chi-squared, but the transitional-regime claim rests on that ranking alone.

read the letter

The paper takes existing pQCD helicity-amplitude ideas and logarithmic corrections from six-quark operators, applies them to deuteron elastic form factors, and runs a global fit to the world data on A(Q²), B(Q²), cross sections, and tensor observables while folding in two-photon exchange. It reports that the f1 parameterization, built around pre-asymptotic valence-quark dynamics, returns the lowest chi-squared per degree of freedom and therefore concludes that present Q² values sit in a transitional regime with helicity-conserving amplitudes still dominant. The inclusion of t21 as a discriminator for asymptotic behavior is a sensible choice, and the attempt to compare three distinct model classes in one consistent framework is the clearest piece of new work here. The fit itself is the main deliverable; no new observable or first-principles derivation appears. The soft spots are straightforward. The abstract states that f1 wins but supplies no numerical chi-squared values, no breakdown of data cuts, no explicit treatment of normalization or radiative uncertainties, and no test of how sensitive the ranking is to plausible systematic floors. Because the world data set combines measurements from different experiments, any residual inter-experiment tension or incomplete TPE correction can shift which parameterization looks best without changing the underlying physics. The circularity is real: the claim that the data indicate a transitional regime reduces to the statement that one fitted function matches the input points better than the alternatives. That is useful for phenomenology but does not independently establish the dynamical picture. This work is for nuclear-structure and hadron-physics groups that need practical forms for deuteron observables at intermediate Q² or that are designing next polarization runs. A reader who wants a side-by-side comparison of pre-asymptotic, higher-twist, and modified-log models will find something concrete to use or to test further. The paper is coherent enough on its own terms to deserve a serious referee, even though the central claim will need the missing fit details and a clearer systematics discussion before it can be taken as evidence for a regime change. I would send it to review rather than desk-reject.

Referee Report

3 major / 2 minor

Summary. The paper conducts a consistent analysis of elastic electron-deuteron scattering data by integrating pQCD scaling, helicity amplitudes, and phenomenological parameterizations of deuteron form factors, with emphasis on logarithmic corrections from six-quark operators and two-photon exchange effects. Three classes of parameterizations are examined: pre-asymptotic valence-quark dominance, effective higher-twist contributions, and modified logarithmic evolution. A global fit to world data on A(Q²), B(Q²), differential cross sections, and tensor polarization observables is performed, identifying the f1 parameterization as providing the best description with the smallest χ²/dof. The results suggest that the accessible Q² range represents a transitional regime in QCD, dominated by helicity-conserving amplitudes, with the asymptotic pQCD regime not yet fully realized, potentially pointing to multiquark correlations.

Significance. Should the findings hold, this analysis offers valuable insight into the transition from hadronic to quark-gluon descriptions of the deuteron at intermediate momentum transfers. By comparing different dynamical models through global fits, it underscores the sensitivity of tensor observables to asymptotic behavior and motivates future high-Q² experiments. The incorporation of logarithmic evolution and TPE corrections strengthens the phenomenological framework for nuclear form factors.

major comments (3)

[Global fit and results] The central claim that the data indicate a transitional regime is based on f1 yielding the lowest χ²/dof. However, without explicit numerical χ² values, error budgets, or discussion of data selection cuts and TPE corrections in the fit, it is difficult to evaluate whether the preference is robust or influenced by unaccounted systematics in the combined world data set.
[Model classes] The three parameterization classes are presented as corresponding to different dynamical regimes, but the manuscript does not demonstrate that these classes are exhaustive or that other possible functional forms would not produce similar or better fits, potentially affecting the interpretation of a 'transitional regime'.
[Discussion of implications] The conclusion regarding helicity-conserving amplitudes dominating and the absence of full asymptotic pQCD relies on the fit results; a more quantitative comparison of the helicity amplitudes across models would strengthen this point.

minor comments (2)

[Abstract] The abstract mentions the smallest value of χ²/dof but does not quote the actual values or dof for the models, which would aid quick assessment.
[Notation] The specific definitions of the f1 parameterization and the other classes should be introduced with equations early in the text for clarity.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. We address each of the major comments point by point below, providing clarifications and indicating where revisions will be made to improve the presentation and robustness of the analysis.

read point-by-point responses

Referee: The central claim that the data indicate a transitional regime is based on f1 yielding the lowest χ²/dof. However, without explicit numerical χ² values, error budgets, or discussion of data selection cuts and TPE corrections in the fit, it is difficult to evaluate whether the preference is robust or influenced by unaccounted systematics in the combined world data set.

Authors: We agree that explicit numerical values and additional details are necessary for a full assessment. The manuscript states that f1 yields the smallest χ²/dof but does not tabulate the values for all models or detail the precise data selection and TPE implementation. In the revised version we will add a table with χ²/dof for each parameterization class, describe the world-data cuts applied, and expand the discussion of how TPE corrections are incorporated into the global fit. This will allow readers to judge the robustness directly. revision: yes
Referee: The three parameterization classes are presented as corresponding to different dynamical regimes, but the manuscript does not demonstrate that these classes are exhaustive or that other possible functional forms would not produce similar or better fits, potentially affecting the interpretation of a 'transitional regime'.

Authors: The three classes were selected on the basis of established theoretical expectations for the hadronic-to-partonic transition (pre-asymptotic valence dominance, higher-twist corrections, and modified logarithmic evolution from six-quark operators). While we do not claim they exhaust all conceivable functional forms, they represent the principal dynamical scenarios discussed in the literature for deuteron form factors at intermediate Q². In the revision we will add a paragraph explaining this theoretical motivation and noting that a completely exhaustive survey lies beyond the scope of the present work, while the chosen classes suffice to demonstrate the sensitivity of the data to the transitional regime. revision: partial
Referee: The conclusion regarding helicity-conserving amplitudes dominating and the absence of full asymptotic pQCD relies on the fit results; a more quantitative comparison of the helicity amplitudes across models would strengthen this point.

Authors: We accept that a more explicit quantitative comparison would strengthen the argument. The present text infers dominance from the best-fit parameters and the resulting form-factor behavior. In the revised manuscript we will include an additional figure (or table) that displays the relative magnitudes of the helicity amplitudes for each parameterization at representative Q² values. This will make the statement that helicity-conserving amplitudes dominate in the accessible range more quantitative and transparent. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper defines three distinct classes of phenomenological parameterizations motivated by different dynamical regimes (pre-asymptotic valence-quark dominance, effective higher-twist, modified logarithmic evolution), performs a global fit of each to the aggregated world data on A(Q²), B(Q²), cross sections and tensor observables, and reports that the f1 form yields the lowest χ²/dof. This is a standard comparative model-selection procedure whose output is an interpretive statement that the data favor a transitional regime; it does not reduce by construction to the input data via self-definition, fitted-parameter renaming, or any self-citation chain. No equations are shown that equate a claimed prediction to a fitted quantity, and no load-bearing uniqueness theorem or ansatz is imported from prior work by the same authors. The analysis therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The claim rests on (1) the assumption that pQCD scaling plus anomalous-dimension logarithms from six-quark operators correctly describe the approach to asymptopia, (2) the validity of the helicity-amplitude decomposition, and (3) the completeness of the three chosen parameterization classes. No new entities are introduced; the 'hidden-color configurations' are mentioned only as a possible interpretation.

free parameters (1)

parameters of the f1 parameterization
Multiple numerical coefficients and scale parameters in the pre-asymptotic logarithmic form are adjusted to minimize χ² against the world data set.

axioms (2)

domain assumption Helicity-conserving amplitudes dominate in the transitional regime
Invoked to interpret why t21 is especially sensitive to asymptotic behavior.
domain assumption Logarithmic corrections are governed by anomalous dimensions of six-quark operators
Standard pQCD input used to motivate the modified logarithmic evolution class.

pith-pipeline@v0.9.0 · 5601 in / 1616 out tokens · 53268 ms · 2026-05-12T01:03:33.230345+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

[1]

J., Farrar G

Brodsky S. J., Farrar G. R. Scaling Laws at Large Transverse Momentum // Phys. Rev. Lett. — 1973. — Vol. 31. — P. 1153. — DOI: 10.1103/PhysRevLett.31.1153

work page doi:10.1103/physrevlett.31.1153 1973
[2]

A., Muradian R

Matveev V. A., Muradian R. M., Tavkhelidze A. N. Automodellism in the large-angle elastic scattering and structure of hadrons // Nuovo Cim. Lett. — 1973. — Vol. 7. — P. 719–723. — DOI: 10.1007/BF02728133

work page doi:10.1007/bf02728133 1973
[5]

Ji C.-R., Brodsky S. J. Perturbative QCD and deuteron form factors // Phys. Rev. D — 1987. — Vol. 34. — P. 1460–1463. — DOI: 10.1103/PhysRevD.34.1460

work page doi:10.1103/physrevd.34.1460 1987
[6]

Abbott D.et al.Phenomenology of the deuteron electromagnetic form factors // Eur. Phys. J. A — 2000. — Vol. 7. — P. 421–427. — DOI: 10.1007/s100500050489

work page doi:10.1007/s100500050489 2000
[7]

P., Syamtomov A

Kobushkin A. P., Syamtomov A. I. Polarization phenomena in elastic electron– deuteron scattering // Phys. Atom. Nucl. — 1994. — Vol. 57. — P. 1477–1485

work page 1994
[8]

D., Shevchuk O

Krivenko-Emetov Ya. D., Shevchuk O. S. Investigation of the internal structure of the deuteron against the background of two-photon exchange effects in elastic electron– deuteron scattering // Nucl. Phys. At. Energy — 2024. — Vol. 25, No. 4. — P. 309–

work page 2024
[9]

— DOI: 10.15407/jnpae2024.04.309

work page doi:10.15407/jnpae2024.04.309
[10]

M., Derkachov S

Braun V. M., Derkachov S. E., Korchemsky G. P., Manashov A. N. Baryon distri- bution amplitudes in QCD // Nucl. Phys. B — 1999. — Vol. 553. — P. 355–426. — DOI: 10.1016/S0550-3213(99)00222-0

work page doi:10.1016/s0550-3213(99)00222-0 1999
[11]

Garcon M., Van Orden J. W. The deuteron: structure and form factors // Adv. Nucl. Phys. — 2001. — Vol. 26. — P. 293–378. — DOI: 10.1007/0-306-47915-X_6

work page doi:10.1007/0-306-47915-x_6 2001
[12]

P., Krivenko-Emetov Y

Kobushkin A. P., Krivenko-Emetov Y. D., Dubnicka S., Dubnickova A. Elastic electron–deuteron scattering beyond one-photon exchange // Phys. Rev. C — 2010. — Vol. 81. — P. 054001. — DOI: 10.1103/PhysRevC.81.054001. 17

work page doi:10.1103/physrevc.81.054001 2010
[13]

Wang, Physical Review B94, 10.1103/phys- revb.94.195105 (2016)

Kobushkin A. P., Krivenko-Emetov Y. D., Dubnicka S., Dubnickova A. Two-photon exchange and elastic scattering of longitudinally polarized electrons on polarized deuterons // Phys. Rev. C — 2011. — Vol. 84. — P. 054007. — DOI: 10.1103/Phys- RevC.84.054007

work page doi:10.1103/phys- 2011
[14]

J., Ji C.-R., Lepage G

Brodsky S. J., Ji C.-R., Lepage G. P. Quantum chromodynamic predictions for the deuteron form factor // Phys. Rev. Lett. — 1983. — Vol. 51. — P. 83–86. — DOI: 10.1103/PhysRevLett.51.83

work page doi:10.1103/physrevlett.51.83 1983
[15]

P., Krivenko-Emetov Ya

Kobushkin A. P., Krivenko-Emetov Ya. D. Perturbative QCD phenomenology of elastic ed scattering // Nucl. Phys. At. Energy — 2003. — Vol. 4, No. 3. — P. 49–59. — DOI: 10.15407/jnpae2003.03.049

work page doi:10.15407/jnpae2003.03.049 2003
[16]

J., Ji C.-R

Brodsky S. J., Ji C.-R. Perturbative QCD and nuclear physics // Phys. Lett. B —

work page
[17]

— Vol. 727. — P. 438–446. — DOI: 10.1016/j.physletb.2013.10.055

work page doi:10.1016/j.physletb.2013.10.055 2013
[18]

J., Clement H

Bashkanov M., Brodsky S. J., Clement H. Novel six-quark hidden-color dibaryon states in QCD // Phys. Lett. B — 2013. — Vol. 727. — P. 438–442. — DOI: 10.1016/j.physletb.2013.10.059. 18

work page doi:10.1016/j.physletb.2013.10.059 2013

[1] [1]

J., Farrar G

Brodsky S. J., Farrar G. R. Scaling Laws at Large Transverse Momentum // Phys. Rev. Lett. — 1973. — Vol. 31. — P. 1153. — DOI: 10.1103/PhysRevLett.31.1153

work page doi:10.1103/physrevlett.31.1153 1973

[2] [2]

A., Muradian R

Matveev V. A., Muradian R. M., Tavkhelidze A. N. Automodellism in the large-angle elastic scattering and structure of hadrons // Nuovo Cim. Lett. — 1973. — Vol. 7. — P. 719–723. — DOI: 10.1007/BF02728133

work page doi:10.1007/bf02728133 1973

[3] [5]

Ji C.-R., Brodsky S. J. Perturbative QCD and deuteron form factors // Phys. Rev. D — 1987. — Vol. 34. — P. 1460–1463. — DOI: 10.1103/PhysRevD.34.1460

work page doi:10.1103/physrevd.34.1460 1987

[4] [6]

Abbott D.et al.Phenomenology of the deuteron electromagnetic form factors // Eur. Phys. J. A — 2000. — Vol. 7. — P. 421–427. — DOI: 10.1007/s100500050489

work page doi:10.1007/s100500050489 2000

[5] [7]

P., Syamtomov A

Kobushkin A. P., Syamtomov A. I. Polarization phenomena in elastic electron– deuteron scattering // Phys. Atom. Nucl. — 1994. — Vol. 57. — P. 1477–1485

work page 1994

[6] [8]

D., Shevchuk O

Krivenko-Emetov Ya. D., Shevchuk O. S. Investigation of the internal structure of the deuteron against the background of two-photon exchange effects in elastic electron– deuteron scattering // Nucl. Phys. At. Energy — 2024. — Vol. 25, No. 4. — P. 309–

work page 2024

[7] [9]

— DOI: 10.15407/jnpae2024.04.309

work page doi:10.15407/jnpae2024.04.309

[8] [10]

M., Derkachov S

Braun V. M., Derkachov S. E., Korchemsky G. P., Manashov A. N. Baryon distri- bution amplitudes in QCD // Nucl. Phys. B — 1999. — Vol. 553. — P. 355–426. — DOI: 10.1016/S0550-3213(99)00222-0

work page doi:10.1016/s0550-3213(99)00222-0 1999

[9] [11]

Garcon M., Van Orden J. W. The deuteron: structure and form factors // Adv. Nucl. Phys. — 2001. — Vol. 26. — P. 293–378. — DOI: 10.1007/0-306-47915-X_6

work page doi:10.1007/0-306-47915-x_6 2001

[10] [12]

P., Krivenko-Emetov Y

Kobushkin A. P., Krivenko-Emetov Y. D., Dubnicka S., Dubnickova A. Elastic electron–deuteron scattering beyond one-photon exchange // Phys. Rev. C — 2010. — Vol. 81. — P. 054001. — DOI: 10.1103/PhysRevC.81.054001. 17

work page doi:10.1103/physrevc.81.054001 2010

[11] [13]

Wang, Physical Review B94, 10.1103/phys- revb.94.195105 (2016)

Kobushkin A. P., Krivenko-Emetov Y. D., Dubnicka S., Dubnickova A. Two-photon exchange and elastic scattering of longitudinally polarized electrons on polarized deuterons // Phys. Rev. C — 2011. — Vol. 84. — P. 054007. — DOI: 10.1103/Phys- RevC.84.054007

work page doi:10.1103/phys- 2011

[12] [14]

J., Ji C.-R., Lepage G

Brodsky S. J., Ji C.-R., Lepage G. P. Quantum chromodynamic predictions for the deuteron form factor // Phys. Rev. Lett. — 1983. — Vol. 51. — P. 83–86. — DOI: 10.1103/PhysRevLett.51.83

work page doi:10.1103/physrevlett.51.83 1983

[13] [15]

P., Krivenko-Emetov Ya

Kobushkin A. P., Krivenko-Emetov Ya. D. Perturbative QCD phenomenology of elastic ed scattering // Nucl. Phys. At. Energy — 2003. — Vol. 4, No. 3. — P. 49–59. — DOI: 10.15407/jnpae2003.03.049

work page doi:10.15407/jnpae2003.03.049 2003

[14] [16]

J., Ji C.-R

Brodsky S. J., Ji C.-R. Perturbative QCD and nuclear physics // Phys. Lett. B —

work page

[15] [17]

— Vol. 727. — P. 438–446. — DOI: 10.1016/j.physletb.2013.10.055

work page doi:10.1016/j.physletb.2013.10.055 2013

[16] [18]

J., Clement H

Bashkanov M., Brodsky S. J., Clement H. Novel six-quark hidden-color dibaryon states in QCD // Phys. Lett. B — 2013. — Vol. 727. — P. 438–442. — DOI: 10.1016/j.physletb.2013.10.059. 18

work page doi:10.1016/j.physletb.2013.10.059 2013