Structure of proton based on the classical string model

L. P. Sun; Y. F. Mo; Z. G. Tan

REVIEW 4 major objections 4 minor 15 references

Structure of proton based on the classical string model

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper claims that a classical relativistic string model of the proton forces the string tension to scale as $a = 0.03025/R$, making the quark interaction inversely proportional to separation, opposite to QCD asymptotic freedom.

desk verdict The paper's central claim is an artifact of a sign error and a curve fit, not a physical result. read the letter →

arxiv 1908.01307 v1 pith:4GLBVZMC submitted 2019-08-04 hep-ph

Z. G. Tan , Y. F. Mo , L. P. Sun This is my paper · ORCID

classification hep-ph PACS 12.38.Mh24.10.Lx

keywords protonstructureclassicalstringmodeltensionquarkcircularmotionrelativistickinematicsasymptoticfreedomhadronradiusmass

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 tries to show that classical string pictures of the proton can reproduce the observed proton mass and that the relativistic version forces an interaction law opposite to QCD's asymptotic freedom. Three quark–string configurations are proposed; two are equivalent up to a constant factor. In the relativistic calculation, length contraction of the quark orbit changes the kinematics and the string energy takes the form of Eq. (11). Fixing the proton mass at roughly 0.94 GeV and the quark mass at 5 MeV, the authors fit the string tension to $a = 0.03025/R$. On that basis they conclude that the interaction between quarks in a proton is inversely proportional to distance.

What carries the argument

The load-bearing object is the relativistic string segment of length $R$ with tension $a$. Its energy is $E_0 = (a/\omega)[(\omega R/2)\sqrt{1+\omega^2 R^2} + \tfrac12 \ln(\omega R + \sqrt{1+\omega^2 R^2})]$, and the relativistic circular motion of a quark gives $\omega$ through $m_q^2 R^4 \omega^6 + m_q^2 R^2 \omega^4 - a^2 = 0$. Feeding these into the additive mass formula $M = 3E_0 + 3m_q\sqrt{1+\omega^2 R^2}$ and requiring $M \approx 0.94$ GeV produces the fitted inverse law $a = 0.03025/R$; that fitted law is what carries the paper's central conclusion.

What would settle it

Measure the string tension from quarkonium spectra at a known quark separation and compare it with $0.03025/R$. At $R = 0.84$ fm the formula gives $a \approx 0.036$ GeV$^2$, nearly five times smaller than the input value $a \approx 0.176$ GeV$^2$ quoted from quarkonium data, so a single such comparison at any radius would settle whether the inverse law holds.

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Extended reading notes

Core claim

The central claim is that, inside a proton described by a classical string with tension $a$, a quark in relativistic circular motion feels a string force whose energy, together with the quark kinetic energy, has to add up to the measured proton mass. Solving that condition for a range of radii gives the fitted relation $a = 0.03025/R$ in Eq. (13). Since $R$ is the proton radius and therefore the quark separation, the paper concludes that the quark interaction is inversely proportional to distance. This is presented as contrary to asymptotic freedom, where the strong force weakens at short distances.

Load-bearing premise

The whole inverse-distance result assumes that the proton's mass is exactly the sum of three string energies and three quark kinetic energies, with no extra contribution from binding energy, gluon fields, or vacuum energy; if that additive mass formula is wrong, the fitted $a \propto 1/R$ collapses.

Editorial extensions

If this is right

If $a = 0.03025/R$ is correct, the proton's internal force grows as quarks are pushed closer together, reversing the qualitative behavior expected from asymptotic freedom at hadronic scales.
The fitted curve fixes $a$ once the proton radius is known; radii near 0.8–1.0 fm imply tensions far below the quarkonium value $a \approx 0.176$ GeV$^2$ used elsewhere in the paper.
Because structures B and C differ from structure A only by a constant geometric factor, all three proposed configurations carry the same inverse-distance behavior.
Relativistic length contraction is essential to the result: the non-relativistic version gives constant tension and radii 1.17 fm (A) and 0.67 fm (B, C), not the fitted inverse law.

Reading between the lines

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

If the inverse-distance law were read as a universal statement, it would conflict with the constant string tension inferred from hadronic Regge trajectories at larger separations; the fitted law is more plausibly an effective low-radius description than a fundamental one.
A natural extension is to add a constant or logarithmic binding-energy term to the mass formula and see whether the best-fit $a(R)$ stays $\propto 1/R$ or flattens to a constant, which would decide whether the claimed reversal is an artifact of the additive ansatz.
The model implies a direct experimental handle: an independent proton-radius measurement (for example from electron scattering or muonic hydrogen) fixes $a = 0.03025/R$, and that prediction can be checked against string tensions extracted from heavy-quark bound states.
If the relation is taken seriously, it predicts that larger hadrons should have weaker string tensions, which could be searched for in the level spacings of excited mesons as a function of hadron size.

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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

The punchline: this paper doesn't establish that quark interactions are inversely proportional to distance. The stress-test note is right, and it's worse than the reader's report suggests. Eq. (11) rewrites 1/sqrt(1-u^2) as sqrt(1+u^2), which is simply false. The closed form that follows is the antiderivative of sqrt(1+u^2), so Eq. (12) is not the energy of the rotating string the paper describes. That single error is load-bearing: Eq. (13), the inverse-distance law, is fitted to points generated by Eq. (12). Fix the sign error and the fitted relation changes or disappears. The contradiction with asymptotic freedom is unsupported.

To give credit where it's due: the non-relativistic part of the paper is a legitimate exercise. With a standard string tension a ≈ 1 GeV/fm and mq = 5 MeV, the model gives proton radii around 1.17 fm (structure A) and 0.67 fm (structure B). Those are within the right order of magnitude, and the paper is transparent about the assumptions. The relativistic section is also explicit about its equations, which is what allows the error to be caught. The paper cites PDG and quarkonia string-tension sources. There's no self-citation problem and no hiding of the method.

But the soft spots are not minor. Eq. (12) treats the proton mass as an additive sum of three string energies plus three relativistic quark kinetic energies, with no binding energy, gluon self-interaction, or vacuum energy. That's an ad hoc ansatz, and the paper doesn't justify it. Even if Eq. (11) were correct, solving Eq. (12) for a(R) with Mp fixed and then fitting a = 0.03025/R is curve-fitting to the model's own output, not a derivation or a test. The conclusion overstates the evidence. The paper's own abstract says "contrary to asymptotic freedom," but that claim rests entirely on the algebraic slip and the fitted coefficient.

The paper is not a serious contribution to proton structure. It shows some elementary modeling instincts but not the care needed to produce a credible result. A desk reject is appropriate; sending this to referees would waste their time. There's nothing here worth citing or bringing to a reading group. If the authors want to pursue this, they should fix Eq. (11), redo the calculation, and test the resulting relation against data rather than fitting it.

Referee Report

4 major / 4 minor

Summary. The manuscript proposes a classical string model for the proton in which quarks rotate in circular orbits bound by strings, and it derives mass-radius relations in both non-relativistic and relativistic treatments. The central claim, stated in the abstract and conclusion, is that the string tension (interpreted as the quark interaction strength) is inversely proportional to the proton radius, in contrast to asymptotic freedom. The relativistic analysis leads to a fitted relation a = 0.03025/R, which the authors present as evidence against the QCD expectation. The paper is exploratory and acknowledges its rough character, but it presents the inverse-distance behavior as a concrete calculational result.

Significance. If the claimed inverse-distance relation were derived from valid physics and tested against data, it would constitute a serious challenge to a core QCD prediction. However, as presented, the result is not supported by independent data or a controlled derivation: the key relation emerges from an algebraic error and from fitting a curve to points generated by the model's own mass formula. The paper's strength is its transparency: the assumptions and equations are stated explicitly, so the flaws are easy to isolate. There are no machine-checked proofs or reproducible numerical codes. The falsifiable prediction is explicit, but it is an artifact of the calculation rather than a genuine physical prediction.

major comments (4)

[Sec. III, Eq. (11)] The central derivation contains an algebraic error. The string energy is written as E0 = ∫_0^R a dx / sqrt(1 - ω²x²), which is then replaced by ∫_0^R a sqrt(1 + ω²x²) dx. These integrands are not equal: 1/sqrt(1-u²) ≠ sqrt(1+u²). The closed form that follows is the antiderivative of sqrt(1 + u²), not of the Lorentz factor. Consequently Eq. (12) is not the energy of the described rotating string, and the fitted relation in Eq. (13) rests on the incorrect expression.
[Sec. III, Eqs. (12)-(13)] The inverse-distance relation a = 0.03025/R is constructed, not tested. For fixed proton mass M_p and quark mass m_q, Eq. (12) is solved for a at various R, and the resulting (a, R) pairs are fitted to a power law. The functional form is therefore encoded in the mass ansatz and in the fitting procedure; it does not provide an independent check of any physical law. The claim of a contradiction with asymptotic freedom is unsupported by this procedure.
[Sec. III, Eq. (12)] The proton mass is assumed to be an additive sum of three string energies and three relativistic quark kinetic energies, with no binding energy, gluon self-interaction, or vacuum energy term. This ansatz is introduced without justification or a limiting argument. Since the central result depends entirely on this mass formula, the omission of binding energy is a load-bearing issue, not a minor refinement.
[Sec. III, Eq. (13) and Fig. 6] The units in Eq. (13) are ambiguous. The string tension a has dimensions of energy per length (GeV/fm or GeV² in natural units), while R is a length in fm. The fitted coefficient 0.03025 must carry dimensions for the equation to be consistent, but none are specified. In addition, the text mixes GeV and fm without stating a conversion convention, which complicates any test of the relation.

minor comments (4)

[Abstract and Sec. IV] The abstract and conclusion contain grammatical and formatting errors, e.g., 'imagination scenarios' and missing spaces after commas; these should be corrected.
[Sec. II, Eq. (2)] The factor sqrt(3) for structures B and C is stated without derivation. The text should explain how the geometry of three rotating planes leads to this factor.
[Sec. III, Eq. (7)] The derivation from Eq. (6) to Eq. (8) is compressed. The reader must infer intermediate algebraic steps; a few lines of detail would improve clarity and verifiability.
[References] Some references are incomplete or have formatting issues, e.g., Ref. [7] omits the journal name and Ref. [12] uses an inconsistent volume/pages format. The references should be harmonized.

Circularity Check

1 steps flagged · score 8.0 of 10

The claimed inverse-distance law is a fit to points generated by the paper's own mass formula, not an independent prediction.

fitted input called prediction [Sec. III, Eq. (12), Fig. 6, Eq. (13), and Conclusion]
"Where the small circle in the figure represents the data we have calculated by Eq.(12), and the real line is the function relationship fitted. a = 0.03025/R ... our calculations show that the tension of the string is inversely proportional to the proton radius and therefore to the distance between quarks."

Equation (12), M = 3E0 + 3mq sqrt(1+omega^2 R^2), is the model's assumed mass formula. With mq and the proton mass fixed, it supplies one relation among M, a, and R. The points in Fig. 6 are not measurements or independent theory; they are calculated by evaluating Eq. (12) at selected radii and solving for a. Fitting those generated points to a = 0.03025/R and then announcing that the string tension is inversely proportional to radius is therefore a restatement of the mass ansatz plus the fixed-M constraint. The conclusion is not a test of the model against data; it is a fit to the model's own output.

full rationale

The circularity is concentrated in the single load-bearing step: the central claim is Eq. (13), and Eq. (13) is explicitly a fit to 'data we have calculated by Eq.(12)'. Nothing external to the model is used to validate the inverse-distance law, so the 'prediction' is not independent of the input mass formula. This warrants a high circularity score. The nonrelativistic use of the experimental string tension a = 0.176 GeV^2 is not circular, but it is not the basis of the asymptotic-freedom contradiction. Separately, and not as a circularity issue, Eq. (11) rewrites 1/sqrt(1-omega^2 x^2) as sqrt(1+omega^2 x^2); these are not equal, so Eq. (12) does not describe the listed relativistic string energy. That algebraic error further undermines the numerical basis of the fitted curve, but the core circularity finding stands on the Eq. (12)-to-Eq. (13) reduction.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The model relies on several ad hoc classical assumptions and one fitted coefficient, and it does not use measured data beyond the proton mass and an external string tension. No new entities are introduced.

free parameters (1)

coefficient C in fitted relation a = C/R (Eq. 13) = 0.03025
Fitted to the (a,R) curve generated by Eq. (12) with Mp fixed at 0.94 GeV. Not derived from external data.

assumptions (5)

ad hoc to paper Quarks perform steady circular motion inside hadrons, with centripetal force supplied by string tension (Sec. II, assumptions 1 and 2).
Classical mechanics assumption not derived from QCD or string theory.
ad hoc to paper All three quarks in a proton are treated as identical (taste symmetry), ignoring u/d distinctions (Sec. II, assumption 3).
Simplifies the picture but is not justified by QCD.
ad hoc to paper Proton mass is the additive sum of three string energies and three relativistic quark masses (Eq. 12).
Omits binding energy, gluon self-interaction, and vacuum contributions.
ad hoc to paper Special-relativistic length contraction applies to a rotating string's circumference (Eqs. 5-6).
Applying flat-space length contraction to a curved rotating string is not justified.
standard math Cardano's formula provides the real solution of the cubic for W (Eq. 10).
Standard algebraic result, but the branch choice for the physical root is not discussed.

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Cite this review

Pith. "Pith review of Structure of proton based on the classical string model." pith.science (2026). https://pith.science/paper/4GLBVZMC

@misc{pith2026190801307,
  author       = {Pith},
  title        = {Pith review of: Structure of proton based on the classical string model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4GLBVZMC}},
  note         = {Machine review of arXiv:1908.01307}
}

read the original abstract

In this paper, we propose several proton structure imagination scenarios based on classical string model. The radius and mass properties of protons in relativistic and non-relativistic cases are discussed, Contrary to asymptotic freedom, we find that the interaction between quarks in protons is inversely proportional to distance.

Figures

Figures reproduced from arXiv: 1908.01307 by the authors.

**Figure 1.** FIG. 1: Three possible pictures of proton structure do [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 3.** FIG. 3: Circumferential Motion Shape Considering [PITH_FULL_IMAGE:figures/full_fig_p002_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 5.** Figure 5: FIG. 5: The relation between the mass of proton and [PITH_FULL_IMAGE:figures/full_fig_p003_5.png]

**Figure 4.** Figure 4: FIG. 4: The relation between the angular velocity of [PITH_FULL_IMAGE:figures/full_fig_p003_4.png]

**Figure 6.** Figure 6: FIG. 6: The the relationship between string tension and [PITH_FULL_IMAGE:figures/full_fig_p004_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

[1]

The interaction between quarks in hadrons is ex- pressed by strings which strength parameter a, called string tension, can be used as ”rest mass den- sity” of the string

work page
[2]

Quarks must make steady circular motion inside hadrons, and their centripetal force comes from the tension of strings

work page
[3]

For protons, if you don’t distinguish between u quark and d quark, the possible internal picture is shown in FIG.1

Since quarks do not distinguish between tastes, quarks in protons should have the same status, that is, they have taste symmetry. For protons, if you don’t distinguish between u quark and d quark, the possible internal picture is shown in FIG.1. In the ﬁrst two pictures, three quarks that make up the proton are pulled by strings to rotate in a plane deter...

work page
[4]

Matteo Rinaldi, Federico Alberto Ceccopieri,Hadronic structure from double parton scattering,Phys. Rev. D 97, 071501 (2018)

work page 2018
[5]

Yuki Kamiya, Tetsuo Hyodo, Structure of hadron reso- nances with a nearby zero of the amplitude, Phys. Rev. D 97, 054019 (2018)

work page 2018
[6]

Yusuke Namekawa, Takeshi Yamazaki, Quark mass de- pendence of on-shell and half oﬀ-shell scattering ampli- tudes from Bethe-Salpeter wave function inside the in- teraction range, Phys. Rev. D 99, 114508 (2019)

work page 2019
[7]

Heikki M¨antysaari, Bj¨orn Schenke, Chun Shen, Prithwish Tribedy ,Proton structure ﬂuctuations: constraints from HERA and applications to p+A collisions, Nucl. Phys. A 967, 317-320 (2017)

work page 2017
[8]

Tanmay Maji, Dipankar Chakrabarti ,Transverse struc- ture of proton in a light-front quark-diquark model,Phys. Rev. D 95, 074009 (2017)

work page 2017

Show all 15 references

[9]

Stephen Farry, Olli Lupton, Martina Pili, Mika Vesteri- nen ,Understanding and constraining the PDF uncertain- ties in a W boson mass measurement with forward muons at the LHC,Eur. Phys. J. C (2019) 79: 497

2019
[10]

Begun, M.I

V.V. Begun, M.I. Gorenstein, O.A. Mogilevsky,Modiﬁed Bag Models for the Quark Gluon Plasma Equation of State,nt.J.Mod.Phys.E20:1805-1815,2011

2011
[11]

Kaidalov, M.G

A.B. Kaidalov, M.G. Poghosyan,Predictions of Quark- Gluon String Model for pp at LHC ,Eur.Phys.J.C67:397- 404,2010

2010
[12]

Glazek, Patryk Kubiczek ,Proton structure in high-energy high-multiplicity p-p collisions,Few-Bod y Syst (2016) 57:425-430

Stanislaw D. Glazek, Patryk Kubiczek ,Proton structure in high-energy high-multiplicity p-p collisions,Few-Bod y Syst (2016) 57:425-430

2016
[13]

L. D. Soloviev ,Masses and Internal Structure of Mesons in the String Quark Model ,Phys.Rev. D61 (2000) 015009

2000
[14]

Simeon Hellerman, Ian Swanson,String Theory of the Regge Intercept,Phys. Rev. Lett. 114, 111601 (2015)

2015
[15]

Patrignani et al (Particle Data Group) ,Review of Par - ticle Physics,Chinese Physics C,2016, Vol

C. Patrignani et al (Particle Data Group) ,Review of Par - ticle Physics,Chinese Physics C,2016, Vol. 40(10)

2016

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

The interaction between quarks in hadrons is ex- pressed by strings which strength parameter a, called string tension, can be used as ”rest mass den- sity” of the string

work page

[2] [2]

Quarks must make steady circular motion inside hadrons, and their centripetal force comes from the tension of strings

work page

[3] [3]

For protons, if you don’t distinguish between u quark and d quark, the possible internal picture is shown in FIG.1

Since quarks do not distinguish between tastes, quarks in protons should have the same status, that is, they have taste symmetry. For protons, if you don’t distinguish between u quark and d quark, the possible internal picture is shown in FIG.1. In the ﬁrst two pictures, three quarks that make up the proton are pulled by strings to rotate in a plane deter...

work page

[4] [4]

Matteo Rinaldi, Federico Alberto Ceccopieri,Hadronic structure from double parton scattering,Phys. Rev. D 97, 071501 (2018)

work page 2018

[5] [5]

Yuki Kamiya, Tetsuo Hyodo, Structure of hadron reso- nances with a nearby zero of the amplitude, Phys. Rev. D 97, 054019 (2018)

work page 2018

[6] [6]

Yusuke Namekawa, Takeshi Yamazaki, Quark mass de- pendence of on-shell and half oﬀ-shell scattering ampli- tudes from Bethe-Salpeter wave function inside the in- teraction range, Phys. Rev. D 99, 114508 (2019)

work page 2019

[7] [7]

Heikki M¨antysaari, Bj¨orn Schenke, Chun Shen, Prithwish Tribedy ,Proton structure ﬂuctuations: constraints from HERA and applications to p+A collisions, Nucl. Phys. A 967, 317-320 (2017)

work page 2017

[8] [8]

Tanmay Maji, Dipankar Chakrabarti ,Transverse struc- ture of proton in a light-front quark-diquark model,Phys. Rev. D 95, 074009 (2017)

work page 2017