Pith.

High Energy Physics - Phenomenology · arXiv:hep-ph/0501217 v2 · submitted 24 Jan 2005 · 296 references

Quark-hadron duality in electron scattering

W. Melnitchouk, R. Ent, C. Keppel

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The paper says

“The duality between partonic and hadronic descriptions of physical phenomena is one of the most remarkable features of strong interaction physics.”

more of the abstract

A classic example of this is in electron-nucleon scattering, in which low-energy cross sections, when averaged over appropriate energy intervals, are found to exhibit the scaling behavior expected from perturbative QCD. We present a comprehensive review of data on structure functions in the resonance region, from which the global and local aspects of duality are quantified, including its flavor, spin and nuclear medium dependence. To interpret the experimental findings, we discuss various theoretical approaches which have been developed to understand the microscopic origins of quark-hadron duality in QCD. Examples from other reactions are used to place duality in a broader context, and future experimental and theoretical challenges are identified.

from the abstract

Pith says

Machine review · deepseek-v4.1-flash · 13 Nov 2026 · what was read was not recorded · not checked by a person

Reviewed in 2026 with later literature in view.

It can be wrong. Challenge this review or reply to it.

Holds up. The standard synthesis of quark-hadron duality in electron scattering, careful and honest, with one precision figure quoted tighter than its own systematic spread supports. This line is wrong

3 major objections · 12 minor

How often objections from reviews like this one are right: not yet measured

Are you an author? What you can do.

Not checked: figures, code, what the 296 references say.

The paper says, at Sec. IV A 1, Fig. 16

The review did not quote this passage.

Pith says: Objection 1 of 3 · major

The sharpest quantitative claim in the experimental section is that the ratio of resonance-region to perturbative integrals agrees to better than 5% above Q2 = 1 GeV2 (Fig. 16). The denominator of that ratio is a model-built scaling curve, and the manuscript itself supplies two reasons to think the denominator moves by more than 5%.

more of objection 1

First, Sec. IV A 1 states that evaluating the DIS parameterization at the (xi, Q2) of the resonance data 'causes an effective target mass correction to the scaling curve, which can increase the structure function strength by tens of percent'; a tens-of-percent lever on the reference curve is not obviously commensurate with a 5% agreement claim, and no target-mass-prescription variation band appears in Fig. 16. Second, the authors report that at the highest Q2 the MRST ratio rises above unity and attribute this to underestimated large-x strength in the pQCD parameterizations, so the reference curve is known to be biased low precisely where the resonance data sit. In that light, the sentence in Sec. IV A 1 that 'It is not expected that this choice affects any of the results presented here', referring to the choice among CTEQ, MRST, GRV and Bluemlein-Boettcher, is an expectation stated without demonstration and in some tension with the observed MRST overshoot. I recommend either adding a band in Fig. 16 that reflects the spread across PDF sets and target-mass prescriptions, or softening the 5% figure to a level the quoted systematics actually support, and qualifying the PDF-independence sentence accordingly.

The paper says, at Abstract; Sec. IV A 1 and Sec. IV D

The review did not quote this passage.

Pith says: Objection 2 of 3 · major

The abstract (sentence s2) claims that duality is quantified including its 'flavor, spin and nuclear medium dependence'. The spin and nuclear-medium parts are indeed developed at length in Secs. IV C and IV A 3. The flavor part is the thinnest of the three: it rests on the qualitative comparison of the duality-averaged deuteron F2 curve with neutrino xF3 data …

more of objection 2

… in Fig. 17, which the text itself describes as 'not perfect' though 'compelling', and on the low-energy dv/uv extraction in Fig. 42, which assumes factorization at W'^2 of only 2.0-3.3 GeV2. Neither comparison isolates flavor dependence of duality in the sense that the spin comparisons isolate spin dependence. I would ask the authors either to expand the flavor discussion into a dedicated subsection with an explicit statement of what is and is not established, or to soften the abstract so that flavor dependence is presented as an implication rather than a quantified result.

The paper says, at Sec. IV C 1, Fig. 33

The review did not quote this passage.

Pith says: Objection 3 of 3 · major

The conclusion that the onset of duality in the proton g1 structure function occurs 'somewhere in the region of 1 < Q2 < 2 GeV2' is supported by ratios of integrated resonance strength to a global spin structure function parameterization (Ref.

more of objection 3

[116]), quoted as roughly 60%, 75% and 80% depending on the W2 interval and on whether the elastic contribution is included. These percentages are stated without an uncertainty on the reference parameterization itself, and the ratio is sensitive to the elastic-subtraction choice, which the text acknowledges is in principle arbitrary (Sec. IV C 3). Since the spin channel is one of the three headline claims of the review, a short statement of the uncertainty on the denominator, or a demonstration that the quoted onset range is stable against the elastic-inclusion choice, would materially strengthen this section.

12 minor objections

The paper says, at Sec. II D, Eqs. (46), (48), (49)

Pith says: minor objection 1 of 12

The Nachtmann moment expressions as typeset contain suspicious repeated factors, for example '2(n + 2)(n + 2)(n + 3)' in Eq. (46) and the bracketed coefficients in Eqs. (48) and (49). These should be checked against Refs. [12,14,17] and corrected if the duplication is a typesetting error rather than intended.

The paper says, at Sec. V A 2, Eq. (94)

Pith says: minor objection 2 of 12

The error budget in Eq. (94) is garbled as printed ('0.0000.018' and '0.0070.011'); the systematic and alpha_s uncertainties should be separated by an explicit plus-minus sign.

show 10 more

The paper says, at Sec. IV A 1

Pith says: minor objection 3 of 12

The phrase 'vary rapidly with Q62' appears to be a corrupted 'Q2'; please correct.

The paper says, at Fig. 13 caption

Pith says: minor objection 4 of 12

'wher the F p2 structure function' should read 'where'; the same caption also refers to 'the first (P33(1232) or Delta) and second (S11(1535)) resonance regions', which is clear, but the typo should be fixed.

The paper says, at Sec. IV C 4

Pith says: minor objection 5 of 12

'the the Burkhardt-Cottingham (BC) sum rule' contains a duplicated article.

The paper says, at Fig. 23 caption

Pith says: minor objection 6 of 12

The quantity 'd lnF 22/d lnQ2' appears to be a corrupted 'd ln F2 / d ln Q2'; please correct.

The paper says, at Fig. 50

Pith says: minor objection 7 of 12

The vertical axis label is printed as '1 100' and the horizontal axis as '2 (GeV )2Q'; both should be cleaned up so that the figure is readable independently of the caption.

The paper says, at Sec. IV C 1, Fig. 31

Pith says: minor objection 8 of 12

The power-law fit is printed as 'Ap1 = x0.7'; this should be A1 = x^0.7 with the exponent properly set.

The paper says, at Sec. IV B 1, Fig. 16

Pith says: minor objection 9 of 12

It would help to state explicitly what the integrated ratio for R (the last panel of Fig. 16) is testing, since R is itself a ratio of cross sections and the physical meaning of the ratio of its x-integrals is not immediate.

The paper says, at Sec. IV C 3, Eq. (71)

Pith says: minor objection 10 of 12

The reference to 'Eq. (16) in Sec. II' for the virtual photon flux K should be checked, since Eq. (16) defines K in the Hand convention and the flux entering Eq. (71) may differ by a factor; a one-line clarification would prevent confusion.

The paper says, at Sec. III A 1, Eq. (53)

Pith says: minor objection 11 of 12

The FESR is written with the same symbol for the integration limit and the asymptotic amplitude; a brief notational clarification (for example distinguishing the matching point from the Regge amplitude) would improve readability for readers outside Regge phenomenology.

The paper says, at Sec. VI

Pith says: minor objection 12 of 12

The cross-field section is valuable but long relative to its role in the argument; the authors might consider flagging at its start which of the analogies (QCD sum rules, e+e-, heavy meson decays, ppbar) carry the most direct lesson for electron scattering, so that readers can prioritize.

What would settle it

Recompute the Fig. 16 ratios for F2, 2xF1 and FL at Q2 = 1.5, 2.5 and 3.5 GeV^2 using at least three reference curves (MRST, CTEQ, Alekhin) each with and without the Barbieri et al. target-mass correction, and plot the envelope rather than a single ratio. If the envelope at Q2 > 2 GeV^2 exceeds the quoted 5% (the authors' own text implies a tens-of-percent TMC lever and a visible MRST overshoot), the 'better than 5%' statement should be restated as a central value with that band attached.

from the skeptic's reading

How the review went

4 readings by deepseek-v4.1-flash · 13 Nov 2026

  1. Reader holds up confidence high
  2. Skeptic partly agrees 1 min 6 s · 57,110 tokens read, 3,205 written
  3. Editor wrote the verdict above 24 s · 58,116 tokens read, 1,352 written
  4. Referee minor revision: 3 major objections, 12 minor 1 min 40 s · 57,521 tokens read, 4,638 written

Not run on this paper: the simulated authors' defence.

Pith did not check this review's objections against the paper.

The review in full

The review in brief, machine-written

This is the review you hand someone who needs the resonance-to-scaling story in one place, and it holds up. Melnitchouk, Ent and Keppel do the unglamorous work: they pull together the JLab, SLAC, NMC, HERMES and Cornell data, separate global duality (moments) from local duality (resonance by resonance), and extend the discussion to spin, flavor and nuclear targets. The theory half is a genuine service — the OPE and higher-twist section, the model survey, and the cross-field material on e+e- annihilation, heavy meson decays and QCD sum rules are the parts people actually cite. It is a review and says so; nobody should read it looking for a new measurement or derivation, and the authors don't pretend otherwise. They are also candid about where the evidence thins: the assumed R in the historical Bloom-Gilman plots, the d/u ambiguity at large x, the preliminary status of the moment analyses and of E00-108. That candor is worth something and I'd weigh it in the paper's favor rather than against it. The soft spot is the 5% agreement figure in Fig. 16. The denominator there is a model-built scaling curve, and the paper's own text says the target-mass prescription can move that curve by tens of percent, yet no TMC-variation band appears in the ratio. The line that the choice among CTEQ, MRST, GRV and Bluemlein-Boettcher 'is not expected' to affect the results is an expectation, not a demonstration, and it sits awkwardly next to the MRST overshoot at high Q2 that the authors themselves attribute to missing large-x strength. That is a bookkeeping complaint about a precision claim, not a challenge to duality itself — the phenomenon survives any reasonable variation of the reference curve. The reader's R worry is largely retired for the proton by the E94-110 separations, but the nuclear panels still inherit it, and the review could have said so more sharply. Who should read it: anyone working on nucleon structure, lattice moments, or neutrino oscillation flux modeling at low Q2. It deserves a serious referee, though for a review the questions are accuracy and completeness rather than correctness of a new claim.

The strongest objection

The review's sharpest quantitative claim is that the ratio of resonance-region to perturbative integrals agrees to better than 5% above Q2 = 1 GeV^2 (Fig. 16, Sec. IV A 1). That number is a ratio whose denominator is a model-built scaling curve, and the paper supplies two reasons to think the denominator moves by more than 5%. First, Sec. IV A 1 states that evaluating the DIS parameterization at the (xi, Q2) of the resonance data 'causes an effective target mass correction to the scaling curve, which can increase the structure function strength by tens of percent' (Barbieri et al. prescription). A tens-of-percent lever on the reference curve is not commensurate with a 5% agreement claim; the ratio is only as good as the TMC prescription, and no TMC-variation band appears in Fig. 16. Second, the authors report that at the highest Q2 the MRST ratio rises above unity and attribute it to 'an underestimation of large-x strength in the pQCD parameterizations', noting the d/u ambiguity beyond x ~ 0.5. So the reference curve is known to be biased low exactly where the resonance data sit. Yet Sec. IV A 1 also asserts that the choice among CTEQ, MRST, GRV and Bluemlein-Boettcher 'is not expected that this choice affects any of the results presented here' — an expectation stated without a demonstration, in tension with the observed MRST overshoot. The phenomenon of duality is not in doubt; what is under-supported is the specific precision figure, which is quoted more tightly than the systematic spread on the comparison allows. This is distinct from the reader's R concern, which the LT-separated E94-110 data (Figs. 25, 26, 16) largely retire for the proton.

The paper is a review, and the central claim — that averaged resonance structure functions reproduce the perturbative scaling curves down to Q2 ~ 1 GeV^2, globally in moments and semi-locally resonance by resonance, for spin-averaged, spin-dependent and nuclear targets — is well supported by the compiled data and is argued with appropriate care. The authors are candid about the weak joints: the ambiguity of comparing low-Q2 resonance data to higher-Q2 scaling curves (Sec. IV A 1), the necessity of averaging for local duality ('local duality has a limit'), the later onset of duality in g1 where the Delta region stays negative to Q2 ~ 2 GeV^2, the preliminary status of the moment analyses (Figs. 27-29), the limited nuclear coverage, and the 20% uncertainties in the semi-inclusive pion data. I therefore do not find a soundness or circularity failure. The reader's weakest_assumption (the assumed R entering F2 extraction) is a real historical dependence, but the review itself retires most of it for the proton via the E94-110 longitudinal/transverse separations, and Fig. 16 treats F1, FL and R separately, so it is no longer the load-bearing soft spot. The residual concern is narrower and quantitative: the headline '5%' agreement is a ratio against a constructed reference curve, and the paper's own text reports two effects that move that curve by more than 5% — target-mass corrections of 'tens of percent' and a large-x PDF deficit visible as the MRST ratio rising above unity. The accompanying assertion that the PDF-set choice does not affect the results is an expectation, not a demonstrated invariance. This does not undermine the existence or the qualitative robustness of duality, which is corroborated by independent handles (the neutrino xF3 comparison in Fig. 17, the agreement of separated F1 and FL with NNLO fits in Figs. 25-26, the near Q2-independence of elastic-subtracted moments in Figs. 27-29). It only means the precision figure should carry the systematic band of its denominator. That is a presentational tightening, not a reason to change the verdict, so I recommend UNCHANGED. The reader's rationale already gestures at the large-x PDF issue as 'the one red flag'; my concern sharpens that into the specific claim that the 5% number is quoted below its own systematic floor, which is why I record partial rather than full agreement with the reader's stated weakest assumption.

The referee's report

If the central claims hold, this is a valuable and timely synthesis. The review assembles world data spanning several orders of magnitude in x and Q2 and quantifies duality in a way that individual experimental papers cannot, and it does so with an unusual degree of self-scrutiny: it states the systematic uncertainty on the Jefferson Lab data (below 4%), propagates low-x extrapolation and alpha_s uncertainties into the extracted higher-twist matrix elements (Eqs. (94)-(95)), and explicitly flags where duality does not work locally, notably the negative Delta-region contribution to the proton g1 (Sec. IV C 1) and the apparent proton Burkhardt-Cottingham violation (Sec. IV C 4). The review also makes its consequences falsifiable and operational: duality as a route to large-x parton distributions, as a bridge to lattice QCD moment calculations in the 1-10 GeV2 range, and as a tool for neutrino-oscillation and hadronization physics. The cross-field comparisons in Sec. VI and the explicit list of open problems in Sec. VII are genuine added value for the community.

Pith's reading of the paper (a machine's summary, not the authors')

This review asks whether the parton description of electron-nucleon scattering survives in the region where individual nucleon resonances dominate. The authors assemble the modern structure-function data and argue that, once averaged over suitable energy intervals, the resonance cross sections follow the same scaling curves and the same perturbative evolution as deep inelastic data. They extend the claim from global moments to local resonance regions, and from spin-averaged to spin-dependent structure functions and nuclear targets. If the case holds, the resonance region becomes a usable window on parton distributions, higher-twist matrix elements, and the transition between confinement and asymptotic freedom.

What carries the argument. The carrying object is the comparison of structure functions and their moments in the resonance region with perturbative scaling curves at the same kinematics. Global duality is the statement that moments of the structure function, integrated over all $x$, match the leading-twist prediction even when resonances dominate the integral; local duality is the stronger statement that the average over a single resonance region already matches the scaling curve. The review uses the Nachtmann variable $xi$ to absorb target-mass effects, and the operator product expansion to organize the comparison as a competition between leading-twist and higher-twist terms.

Core claim. The paper's central claim is that quark-hadron duality in inclusive electron scattering is not a loose historical curiosity but a quantified feature of the data. The review reports that resonance-region averages reproduce perturbative scaling curves to better than about 10% locally, and that integrated resonance strengths for $F_2$, $F_1$, $F_L$, and $R$ agree with perturbative estimates to better than about 5% above $Q^2 = 1$ GeV$^2$. It further argues that the resonance data track the same $Q^2$ evolution as parton-distribution fits, that local duality holds resonance by resonance once each region is averaged, and that the pattern extends to spin-dependent structure functions, where the onset is later, and to nuclei, where Fermi motion washes out the resonance structure.

The premise it rests on. The load-bearing premise is that measured resonance-region cross sections can be converted into structure functions and compared with perturbative curves at the same kinematics; where the longitudinal-to-transverse ratio $R$ was not measured, the extraction assumes a value for $R$.

If the paper is right (Pith's inferences, not the authors')

Resonance-region data can be used to constrain parton distributions at large $x$, where direct deep inelastic measurements are sparse.

The small higher-twist corrections inferred from duality give experimental access to quark-gluon correlation matrix elements and color polarizabilities.

Lattice QCD calculations of structure-function moments at $Q^2 \approx 1$–10 GeV$^2$ can be checked against experimental moments that include resonance strength.

The same duality logic can be tested in semi-inclusive pion production and in exclusive reactions, where the onset of factorization and scaling is less established.

If the review's case holds, resonance-region data become an independent cross-check on large-$x$ parton distributions, where the $d/u$ ratio and nuclear corrections remain uncertain.

The argument suggests a low-energy test of semi-inclusive factorization: pion yields in the resonance region should already yield fragmentation functions consistent with those extracted at much higher energies.

A natural extension is neutrino scattering, where the axial structure function is not suppressed by the same gauge-invariance constraints at low $Q^2$ and may show a different onset of duality.

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The paper

Abstract

The duality between partonic and hadronic descriptions of physical phenomena is one of the most remarkable features of strong interaction physics. A classic example of this is in electron-nucleon scattering, in which low-energy cross sections, when averaged over appropriate energy intervals, are found to exhibit the scaling behavior expected from perturbative QCD. We present a comprehensive review of data on structure functions in the resonance region, from which the global and local aspects of duality are quantified, including its flavor, spin and nuclear medium dependence. To interpret the experimental findings, we discuss various theoretical approaches which have been developed to understand the microscopic origins of quark-hadron duality in QCD. Examples from other reactions are used to place duality in a broader context, and future experimental and theoretical challenges are identified.

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. No sentence in this record was written by a model.

abstract compiled · 2026-09-23

The authors' main claim, as the classifier picked it out 0.76

“We present a comprehensive review of data on structure functions in the resonance region, from which the global and local aspects of duality are quantified, including its flavor, spin and nuclear medium dependence.”

How the main claim is stated: asserted 0.99

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.07 · human participants 0.02 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.17 · mentions code or data availability 0.02

The abstract, sentence by sentence

This table labels the abstract only (arXiv listing abstract v2). The number is the probability the classifier gave the label.

SentenceRoleThe authors' own workStates the main conclusionStrength
The duality between partonic and hadronic descriptions of physical phenomena is one of the most remarkable features of strong interaction physics. background 1.00 0.03 0.05 asserted 1.00
A classic example of this is in electron-nucleon scattering, in which low-energy cross sections, when averaged over appropriate energy intervals, are found to exhibit the scaling behavior expected from perturbative QCD. background 1.00 0.07 0.07 asserted 1.00
We present a comprehensive review of data on structure functions in the resonance region, from which the global and local aspects of duality are quantified, including its flavor, spin and nuclear medium dependence. contribution 1.00 0.87 0.18 asserted 0.99
To interpret the experimental findings, we discuss various theoretical approaches which have been developed to understand the microscopic origins of quark-hadron duality in QCD. interpretation 0.61 0.21 0.04 asserted 0.65
Examples from other reactions are used to place duality in a broader context, and future experimental and theoretical challenges are identified. future work 0.90 0.47 0.06 asserted 0.87

What kind of paper the classifier reports this is: review 1.00 · evidence from analysis of existing data 0.98

What was examined, and what was not

Sources read

  • arXiv listing abstract v2

Not examined

  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers

Levels run

  • P abstract

Not run

  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

How the labels were made

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,291 input tokens, $0.0002 reported.

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. This record as JSON

Figures (80)
Figure 1 of the paper
Figure 1. FIG. 1. Summary of measurements of
Figure 2 of the paper
Figure 2. FIG. 2. Inclusive lepton–nucleon scattering in the one-pho
Figure 3 of the paper
Figure 3. FIG. 3. Dual descriptions of the scattering process, in term
Figure 4 of the paper
Figure 4. FIG. 4. Chew-Frautschi plot for several degenerate meson fa
Figure 5 of the paper
Figure 5. FIG. 5. Isovector
Figure 6 of the paper
Figure 6. FIG. 6. Total
Figure 7 of the paper
Figure 7. FIG. 7. Total
Figure 8 of the paper
Figure 8. FIG. 8. Illustration of two-component duality in
Figure 9 of the paper
Figure 9. FIG. 9. Early proton
Figure 10 of the paper
Figure 10. FIG. 10. Proton
Figure 11 of the paper
Figure 11. FIG. 11. Proton
Figure 12 of the paper
Figure 12. FIG. 12. Proton
Figure 13 of the paper
Figure 13. FIG. 13. Proton
Figure 14 of the paper
Figure 14. FIG. 14. Proton
Figure 15 of the paper
Figure 15. FIG. 15. The difference ∆
Figure 16 of the paper
Figure 16. FIG. 16. Ratios of the resonance to scaling integrals of the p
Figure 17 of the paper
Figure 17. FIG. 17. A comparison of the duality-averaged deuteron
Figure 18 of the paper
Figure 18. FIG. 18. Global data on the proton
Figure 19 of the paper
Figure 19. FIG. 19. Moments of the proton
Figure 20 of the paper
Figure 20. FIG. 20. Cornwall-Norton moments of the
Figure 21 of the paper
Figure 21. FIG. 21. The
Figure 24 of the paper
Figure 24. FIG. 24. Ratio of nuclear to deuterium cross sections per nuc
Figure 25 of the paper
Figure 25. FIG. 25. The purely transverse proton structure function 2
Figure 26 of the paper
Figure 26. FIG. 26. As in Fig. 25, but for the longitudinal structure fun
Figure 27 of the paper
Figure 27. FIG. 27. Second (
Figure 28 of the paper
Figure 28. FIG. 28. As in Fig. 27 but for the
Figure 29 of the paper
Figure 29. FIG. 29. Second (
Figure 30 of the paper
Figure 30. FIG. 30. Proton
Figure 31 of the paper
Figure 31. FIG. 31. Proton spin asymmetry
Figure 32 of the paper
Figure 32. FIG. 32. Proton spin structure function
Figure 33 of the paper
Figure 33. FIG. 33. Ratio of the integrated strength of the
Figure 34 of the paper
Figure 34. FIG. 34. Neutron
Figure 35 of the paper
Figure 35. FIG. 35. Inelastic contribution to the moment Γ
Figure 36 of the paper
Figure 36. FIG. 36. Inelastic contribution to the moment Γ
Figure 37 of the paper
Figure 37. FIG. 37. Difference of the proton and neutron moments of the
Figure 38 of the paper
Figure 38. FIG. 38. The structure function
Figure 39 of the paper
Figure 39. FIG. 39. Contributions to the moment Γ
Figure 40 of the paper
Figure 40. FIG. 40. A representation of semi-inclusive electroproduc
Figure 41 of the paper
Figure 41. FIG. 41. Relation between elasticity
Figure 42 of the paper
Figure 42. FIG. 42. The ratio
Figure 43 of the paper
Figure 43. FIG. 43. The distribution (
Figure 44 of the paper
Figure 44. FIG. 44. The ratio of
Figure 45 of the paper
Figure 45. FIG. 45. The ratio of unfavored to favored fragmentation fun
Figure 46 of the paper
Figure 46. FIG. 46. (a) Leading-twist (“handbag diagram”) contributi
Figure 47 of the paper
Figure 47. FIG. 47. Ratio of the
Figure 48 of the paper
Figure 48. The Q2 dependence of the higher-twist operators is governed by the appropriate anomalous dimensions of the twist-4 operators [180], which results in a (logarithmic) Q2 dependence of the matrix element Ae(2) 4 (Q2 ). For all n <∼ 10, Ae(n) 4 remains approximately constant, in contrast to the rapidly decreasing size of the leading-twist moment, A (n) 2 , as shown in
Figure 49 of the paper
Figure 49. FIG. 49. Contribution to the moments Γ
Figure 50 of the paper
Figure 50. FIG. 50. (Left panel) Lowest moment of the proton
Figure 51 of the paper
Figure 51. FIG. 51. Total inclusive photoproduction cross section dat
Figure 54 of the paper
Figure 54. FIG. 54. Comparison of the calculated
Figure 55 of the paper
Figure 55. FIG. 55. Schematic illustration of quark-hadron duality in
Figure 56 of the paper
Figure 56. FIG. 56. Resonance contributions to the proton
Figure 57 of the paper
Figure 57. FIG. 57. Onset of scaling for the structure function
Figure 58 of the paper
Figure 58. FIG. 58. Structure function ˜q
Figure 59 of the paper
Figure 59. FIG. 59. Proton magnetic form factor
Figure 60 of the paper
Figure 60. FIG. 60. Neutron to proton ratio for
Figure 61 of the paper
Figure 61. FIG. 61. Ratio
Figure 62 of the paper
Figure 62. FIG. 62. Proton (left) and neutron (right) polarization asy
Figure 63 of the paper
Figure 63. FIG. 63. (Left panel) Local duality prediction [267] for the
Figure 64 of the paper
Figure 64. FIG. 64. Lowest (
Figure 65 of the paper
Figure 65. FIG. 65. Duality between descriptions of semi-inclusive me
Figure 66 of the paper
Figure 66. FIG. 66. Preliminary data from Jefferson Lab experiment E00-
Figure 67 of the paper
Figure 67. FIG. 67. Schematic illustration of hadron production withi
Figure 68 of the paper
Figure 68. FIG. 68. Average jet multiplicity
Figure 69 of the paper
Figure 69. FIG. 69. Factorial moment
Figure 70 of the paper
Figure 70. FIG. 70. Momentum spectrum of produced hadrons in the inclus
Figure 71 of the paper
Figure 71. FIG. 71. Cross section for wide-angle Compton scattering us
Figure 72 of the paper
Figure 72. FIG. 72. Schematic diagram for quark-hadron duality in non-
Figure 73 of the paper
Figure 73. FIG. 73. Energy dependence of the
Figure 74 of the paper
Figure 74. FIG. 74. Spectral density
Figure 75 of the paper
Figure 75. FIG. 75. Simple model of the spectral density for the ratio
Figure 76 of the paper
Figure 76. FIG. 76. Quark-hadron duality in
Figure 77 of the paper
Figure 77. FIG. 77. (a) Ratio
Figure 78 of the paper
Figure 78. FIG. 78. A sketch of the
Figure 79 of the paper
Figure 79. FIG. 79. Sketch of the radial time projection chamber for the
Figure 80 of the paper
Figure 80. FIG. 80. Schematic layout of the electron-Relativistic Heav

Showing 76 of the paper's 80 figures. Figures are the authors', reproduced from the paper.

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