Pith. sign in

REVIEW 4 major objections 5 minor 25 references

Prediction of Toponium Levels Using a Logarithmic Potential Modeel

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

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A logarithmic mass formula predicts toponium resonance levels at 347 GeV and above, reachable by a 270 GeV e+e- collider.

desk verdict A clean empirical fit to bottomonium is extrapolated to toponium, but the top width kills the predicted discrete levels before n=2. read the letter →

arxiv 2412.12574 v1 pith:NIIENA6S submitted 2024-12-17 hep-ph

Yasushi Muraki , Shoichi Shibata This is my paper · ORCID
classification hep-ph
keywords toponiumlogarithmicpotentialresonancemassformulabottomoniumUpsilonfamilyquark-antiquarkboundstatesOZIruleelectron-positroncollider
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 proposes that the masses of toponium, the bound state of a top quark and its antiparticle, follow the same logarithmic spacing law that fits the Upsilon family. The formula offered is $M(n) = 0.81\ln(n) + 347$ GeV, so the ground state sits at 347 GeV and excited states climb with the logarithm of the excitation index $n$. If the law is right, an electron-positron collider with 270 GeV beams should see these states with a production cross-section of order $10^{-9}$ mb, and the spectrum should become nearly continuous above $n \approx 80$ and end near the $W^+b\,W^-\bar{b}$ threshold at about 524.5 GeV. The paper also interprets the logarithmic level spacing as the signature of a $1/r$ force, realized in a two-dimensional plane where the principal quantum number and angular momentum act as the same kind of label. That reading is extended to give a geometric account of the OZI rule.

What carries the argument

The load-bearing object is the empirical logarithmic mass formula $M(n) = a\ln(n) + M_0$, obtained by plotting measured resonance masses on a semi-logarithmic scale and fitting a straight line; for the Upsilon family the fit yields $a = 0.81$ GeV and the paper applies the same $a$ to toponium with $M_0 = 347$ GeV. The accompanying mechanism is a two-dimensional Schrödinger equation with potential $V(r) = q^2\ln(r)$, whose solutions give the state ordering and motivate treating $n$ and $\ell$ as equivalent quantum numbers, and which is used to interpret the OZI rule as a continuity problem of resonance planes.

What would settle it

Measure the $e^+e^- \to$ hadrons cross-section from roughly 340 to 530 GeV with resolution finer than a few hundred MeV. The formula predicts a first peak at 347.0 GeV, a second at about 347.56 GeV, and a ladder of progressively closer peaks; seeing no such sequence, or a single broad threshold caused by the top quark's fast decay, would settle that the logarithmic toponium ladder is not realized.

Watch

Extended reading notes

Core claim

The central claim is that quark-antiquark resonance families are logarithmically spaced, not linearly spaced. Fitting the Upsilon family gives a slope of $0.81$ GeV per unit $\ln n$ with a correlation coefficient close to one, and the paper transfers that slope to the top quark pair, setting $M(1)=347.0$ GeV and obtaining $M(n)=0.81\ln(n)+347$ GeV. The authors predict the first toponium resonance near 347 GeV, a ladder of increasingly close excited states up to roughly 524.5 GeV, and an $e^+e^-$ collider with each beam at 270 GeV as the experimental test. They further argue that the level spacing reflects motion under a logarithmic potential in a two-dimensional plane, so that the quantum numbers $n$ and $\ell$ can be combined as $(n+\ell)$, and that the OZI suppression is the need to erase one resonance plane and create another.

Load-bearing premise

The whole prediction rests on assuming the Upsilon-family slope carries over to top quarks and that toponium survives long enough to form sharp resonance levels.

Editorial extensions

If this is right

  • A 270 GeV × 270 GeV $e^+e^-$ collider can directly test the prediction; the lowest toponium state should appear near 347 GeV with a cross-section on the order of $10^{-9}$ mb.
  • The excited toponium states become denser as $n$ grows, with spacing below 10 MeV above $n \approx 80$, so experimenters should expect a continuous-looking rise toward 524.5 GeV rather than resolvable peaks at high $n$.
  • If the same logarithmic slope describes the $\rho$, charmonium, and bottomonium families, then the logarithmic spacing law is a general property of quark-antiquark confinement rather than an accident of one quark mass.
  • Baryon resonances also fall on logarithmic lines with slopes close to those of mesons, though the paper reports a roughly $3\sigma$ difference between the $N^*$ and Upsilon slopes that constrains how quarks pair inside baryons.
  • The plane-based interpretation predicts that resonances requiring the destruction of a quark pair, such as $J/\psi$, have narrow widths because the original resonance plane must be annihilated, while same-plane decays such as $\psi(3770)$ are broad.

Reading between the lines

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

  • The paper's own numbers imply the first toponium spacing is only $0.81\ln 2 \approx 0.56$ GeV, which is comparable to the top quark's natural decay width; a realistic observation may therefore look like a broad bump rather than sharp peaks, a consequence the paper only partially acknowledges.
  • Because the same logarithmic slope is used for charm and bottom quarks, the model makes a testable universality prediction: any other newly discovered quark-antiquark system should show the same slope in a $\ln n$ mass plot, a check the paper does not carry out.
  • The treatment of $n$ and $\ell$ as equivalent quantum numbers suggests near-degeneracies between states with the same $n+\ell$ in the Upsilon spectrum; searching for such pairs in existing data would provide a direct test of the two-dimensional-plane picture.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper proposes an empirical logarithmic mass formula M(n) = a ln n + b for hadron resonance families, fits it to the rho, J/psi, Upsilon, and baryon spectra, and then extrapolates the Upsilon slope to toponium, predicting M(n) = 0.81 ln n + 347 GeV. It further claims that toponium should be observable in e+e- collisions with a cross-section on the order of 10^-9 mb and that a collider with 270 GeV x 270 GeV beams would be needed. The paper also offers a two-dimensional-plane interpretation of the OZI rule.

Significance. If the toponium prediction were reliable, it would be a clean, falsifiable prediction for future e+e- colliders. The paper's empirical bottomonium fit is genuinely impressive (R = 0.99997 in Table 2), and the predicted Upsilon(5,6,7) masses agree with data. The interpretation of the OZI rule via two-dimensional resonance planes is speculative but does not affect the main numerical claims. However, the central toponium prediction is not supported by any error analysis or microscopic derivation, and it conflicts with standard-model facts about the top-quark width and the t-tbar threshold.

major comments (4)
  1. [Section 4] The slope transfer from bottomonium to toponium is asserted, not derived. The sentence "the same slope of 0.81 GeV as for the Upsilon family is applied" is the entire basis for the toponium mass formula, but no justification is given for why the logarithmic slope should be flavor-independent. The fitted slope has an error (Table 2 gives 809.2 ± 4.2 MeV or, in the weighted fit, 810.50 ± 0.39 MeV), and this uncertainty is not propagated to M(n). A prediction with no error bar and no theoretical argument for the extrapolation is not a quantitative prediction.
  2. [Section 4] The predicted discrete toponium levels are not consistent with standard-model top-quark physics. With m_t ≈ 172.7 GeV, 2m_t ≈ 345.4 GeV, so T(1) = 347 GeV has essentially zero binding and every state with n > 1 lies above the t-tbar threshold, where the system is a scattering continuum rather than a discrete bound spectrum. The top-quark width Γ_t ≈ 1.3–1.4 GeV implies a resonance width of order 2Γ_t, while the level spacing ΔM(n) = 0.81 ln(1 + 1/n) is 0.56 GeV at n = 2 and falls below 2Γ_t already there. The paper's caveat that intervals become narrower than 10 MeV above n = 80 sets the resolution problem roughly two orders of magnitude too high. No mechanism is proposed to suppress t → Wb, so the predicted many sharp resonances are not observable under the standard model.
  3. [Abstract and Section 4] The production cross-section is quoted as 3 × 10^-9 mb in the abstract and as 7 × 10^-9 mb in Section 4, and no derivation is given for either value. Because the observability claim depends on this cross-section, the discrepancy and the absent calculation are load-bearing issues.
  4. [Section 4] The collider energy requirement is internally inconsistent. A symmetric e+e- collider with 270 GeV beams has √s = 540 GeV, not 347 GeV; to produce the claimed T(1) = 347 GeV state one needs beams of about 173.5 GeV. The text does not distinguish beam energy from center-of-mass energy. In addition, the stated maximum mass 524.5 GeV = 177.5 GeV + 347 GeV appears to use a top mass of 177.5 GeV that differs from the 173 GeV quoted earlier in the same section; this inconsistency needs clarification.
minor comments (5)
  1. [Abstract] The formula is typeset as "0.81ln}(n)" with a stray closing brace.
  2. [Introduction and References] Several names and terms are misspelled: "Tang and Northbury" should be Tan and Norbury, "Okubo-Zwig-lizuka" should be Okubo-Zweig-Iizuka, "Zwing" should be Zweig, "uark" should be quark, and "botomnium" / "botomonium" are inconsistent spellings of bottomonium.
  3. [Figure 9] Figure 9 is described as showing the expected cross-section, but the manuscript contains no calculation, axis normalization, or tabulated values to support the plotted curve.
  4. [Section 4] The text quotes the top-quark mass as 173 GeV and also uses 177.5 GeV in the maximum-mass formula; a single consistent value with a PDG reference should be used.
  5. [Throughout] The mass formula in Section 4 is not numbered, which makes precise citation difficult.

Circularity Check

1 steps flagged · score 4.0 of 10

One fitted-input-called-prediction in the Upsilon validation; the toponium extrapolation itself is not circular.

  1. fitted input called prediction [Section 2-2, 'Mass plot on the logarithmic scale for mesons' (paragraph after Table 2)]
    "For Υ 5 (10753), Υ 6 (10885), and Υ 7 (11020), our empirical curve predicts correct resonance masses as Υ5(10754±4), Υ6(10900±4), and Υ7(11023±4MeV) respectively."

    Table 2 reports the Upsilon trajectory as a least-squares fit with 7 data points (slope 809.2±4.2 MeV, R=0.99997(4), '7 points'). The same fitted line is then used to 'predict' Υ5, Υ6, and Υ7. Since these states are part of the data set on which the fit was performed, the agreement is guaranteed by construction and is not an independent confirmation of the logarithmic law. This is the fitted-input-called-prediction pattern: the fit is presented as if it were a successful out-of-sample prediction. It does not, by itself, make the toponium extrapolation circular, because no toponium resonance data enter the fit, but it does inflate the empirical evidence that the toponium prediction inherits.

full rationale

The paper's central claim, M(n)=0.81 ln(n)+347 GeV for toponium, is obtained by transferring the logarithmic slope fitted to the Upsilon family and fixing the intercept from the measured top-quark mass (2m_t about 347 GeV). No toponium resonance data are used in the fit, so the n-dependence is an out-of-sample extrapolation rather than a tautology. The clearest circular step is in Section 2-2, where the paper calls the values of its own Upsilon fit 'predictions' for Υ5, Υ6, and Υ7; those agree because they lie on the fitted line, making that particular validation circular by construction. The application of the same 0.81 GeV slope to toponium is an explicit ansatz rather than a derived result, and the potential washout of discrete states by the top-quark width is a physical correctness concern, not a circularity. Overall score 4: one construction-reducing 'prediction' appears in the supporting empirical law, while the central toponium claim remains an extrapolation with independent content.

Assumptions & free parameters 2 free parameters · 4 assumptions · 1 invented entities

The central toponium prediction rests on the empirical log-spacing law, a slope transferred from bottomonium, the assumed T(1)=347 GeV, the unmentioned top width, and the 2D-plane interpretation. The only numerically fitted free parameter carried into the prediction is the Upsilon slope; the intercept is externally set by the top quark mass.

free parameters (2)
  • Logarithmic slope s (bottomonium fit) = 0.81 GeV (810.50 +/- 0.39 MeV, Table 2)
    Least-squares slope of the Upsilon mass-vs-ln(n) line; carried over unchanged to toponium in Section 4.
  • T(1) mass = 347.0 GeV
    Set to roughly twice the top quark mass and used as the lowest toponium level; not measured, chosen as an input.
assumptions (4)
  • domain assumption Hadron resonance masses satisfy M(n) = m0 + s ln(n) for each quarkonium family.
    Empirical law stated in Sections 2-2 and 2-3 and used to fit rho, charmonium, bottomonium, and baryons; not derived from QCD in this paper.
  • ad hoc to paper The logarithmic slope s is universal across quark flavors (0.81 GeV for bottomonium applies to toponium).
    Section 4: 'the same slope of 0.81 GeV as for the Upsilon family is applied' with no derivation or error propagation.
  • domain assumption Toponium bound states are narrow enough to display discrete levels despite top quark weak decay.
    The paper never mentions the top quark width (about 1.4 GeV); its prediction of many sharp resonances requires lifetimes longer than the inverse level spacing.
  • domain assumption Quark-antiquark motion is confined to a two-dimensional plane, so n and l are equivalent and the argument is (n+l).
    Section 3.1 (iii)-(iv) and Appendix I; this yields the logarithmic spacing and the OZI-plane interpretation but is not independently evidenced.
invented entities (1)
  • Two-dimensional resonance plane
    purpose: To interpret quark-antiquark resonance states and the OZI rule as continuity/discontinuity of a 2D plane.
    Introduced in Section 3.2 as a geometric model; no falsifiable handle outside the paper beyond the mass predictions.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Prediction of Toponium Levels Using a Logarithmic Potential Modeel." pith.science (2026). https://pith.science/paper/NIIENA6S

@misc{pith2026241212574,
  author       = {Pith},
  title        = {Pith review of: Prediction of Toponium Levels Using a Logarithmic Potential Modeel},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NIIENA6S}},
  note         = {Machine review of arXiv:2412.12574}
}
read the original abstract

In this paper, the energy levels of the resonant states of toponium, composed of top quark and anti-top quark, are given on the basis of an empirical law. We predict that the mass of the n-th resonant state of toponium is given by Mass(n)=0.81ln}(n) + 347GeV from the empirical law on the resonance level of the bottomonium. The cross-section produced by electron-positron collisions is 3X10^{-9}mb and an electron-positron collider would need an energy of 270GeV X 270 GeV to find out the resonance state of toponium. This prediction is based on the empirical law that the energy levels of hadron resonance states are expressed in logarithms. An interpretation of the appearance of quark resonance states in logarithmic intervals is also given in the paper. An application of this model, we present that the Okubo-Zwig-lizuka law can be viewed as a creation 11and annihilation problem of the two-dimensional resonance planes.

Figures

Figures reproduced from arXiv: 2412.12574 by the authors.

Figure 1
Figure 1. The rho-meson trajectory, from top to bottom, (a) charmonium (b) and botomonium [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. The three meson resonance families are plotted on the semi-log diagram. All three [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Top figure shows the logarithmic plot for the nucleon resonances of [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: (rose model)shows the pictorial pre￾sentation of the resonance wave state inside very small area. The color of each wave corre￾sponds to the number of waves n from 1 (red), 2 (orange), 3 (green), 4 (light blue), 5 (blue), and 6 (violet) respectively [PITH_FULL_IMAGE:f…
Figure 6
Figure 6. Figure 6: The slope of each resonance trajectory is compared. The error bars of the baryon [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]
Figure 7
Figure 7. Figure 7: The decay of charmonium J/ψ into ρ and π mesons (right) and ψ (3770) (left) are shown on the plane with the quark line. The decay rate of the right side is limited due to the OZI rule. In our model, the OZI rule may be interpreted as the connection problem between the …
Figure 8
Figure 8. Figure 8: The resonance states of ρ meson series (green), charmonium (red) and botomonium (blue) are plotted on the semi-logarithmic scale diagram (left). The open circles correspond to the states that peak energy is not well determined with narrow width. The right side picture …
Figure 9
Figure 9. Figure 9: The total cross section of e +e − →hadrons as a function of collision energy √ s [GeV]. Possible production of toponium resonance states are also plotted over 347 GeV to 525 GeV by the red bars. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_9.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

25 extracted references · 12 canonical work pages

  1. [1]

    G. F. Chew and S.C. Frautschi, Physical Review Letters, 5 (1960) 580. Unified approach to high-and low-energy strong interactions on the basis of the Mandelstam representation. DOI:https://doi.org/10.1103/PhysRevLett.5.580 G. F. Chew and S.C. Frautschi, Physical Review, 123 (1961) 1478. Dynamical theory for strong interactions at low momentum transfers bu...

    work page doi:10.1103/physrevlett.5.580 1960

  2. [2]

    G. F. Chew and S.C. Frautschi, Physical Review Letters 7 (1961) 394. Principle of equiva- lence for all strongly interacting particles within the S-matrix framework DOI: https://doi.org/10.1103/PhysRevLett.7.394

    doi:10.1103/physrevlett.7.394 1961

  3. [3]

    G. F. Chew and S.C. Frautschi, Physical Review Letters, 8 (1961) 41, Regge Trajectories and the principle of maximum strength for strong interactions. DOI: https://doi.org/10.1103/PhysRevLett.8.41

    doi:10.1103/physrevlett.8.41 1961

  4. [4]

    Regge, Nuovo Cimento 14 (1959) 951, 18 (1960) 947

    T. Regge, Nuovo Cimento 14 (1959) 951, 18 (1960) 947. https://doi.org/10.1007/BF02728177

    doi:10.1007/bf02728177 1959

  5. [5]

    Tang and J

    A. Tang and J. W. Norbury, Physical Review D62 (2000), 016006. https://doi.org/10.1103/PhysRevD.62.016006

    doi:10.1103/physrevd.62.016006 2000

  6. [6]

    Muraki, Prog

    Y. Muraki, Prog. Theo. Phys. 41 (1969) 473. A new mass formula of elementary particles, DOI: 10.1143/PTP.41.473. Y. Muraki, Soryusiron Kenkyuu Electronnics Vol.38-1 (1968) 34-42 in English. https://doi.org/10.24532/soken.38.1.34

    work page doi:10.1143/ptp.41.473 1969

  7. [7]

    Iida and Y

    S. Iida and Y. Muraki, Proceed. 11 th of Int. Conf. on Cosmic Rays (Budapest) (1969). Acta Physica Hungarica Supplement 29 pp 271-277 (1970)

    work page 1969

  8. [8]

    Jackson, C

    J.D. Jackson, C. Quigg, and L.L. Rosner, New Particles, Theoretical Proceeding of the 19 th International Conference on High Energy Physics (Tokyo) 1978 B8. (ICHEP78), CERN Document Server p391, New Particles, Theoretical, by Jakson, David

    work page 1978

Show all 25 references
  1. [9]

    Machacek and Y

    M. Machacek and Y. Tomozawa, Psi Phenomenology and nature of Quark Confinment, Annals of Physics, 110 (1978) 407-420. 9 http://dx.dvi.org/10.1016/0003-4916(78)90037-4

    1978 doi

  2. [10]

    Quigg and J

    C. Quigg and J. L. Rosner, Quarkonium Level Spacings, Physics Letters B71 (1977) 153. DOI: 10.1016/0370-2693(77)90765-1

    1977 doi

  3. [11]

    Quigg and J

    C. Quigg and J. L. Rosner, Quantum Mechanics with Applications to Quarkonium, Physics Reports, 56 No. 4 (1979) 167-235. https://doi.org/10.1016/0370-1573(79)90095-4

    1979 doi

  4. [12]

    Martin, Potential Models of Hadrons, Nuclear Physics B (Proc

    A. Martin, Potential Models of Hadrons, Nuclear Physics B (Proc. Suppl.) 23B (1991) 316-327. https://doi.org/10.1016/0920-5632(91)90698-E

    1991 doi

  5. [13]

    Jean-Marc Richard, An introduction to the quark model, arXiv:1205.4326v2, [hep-ph] 24 May 2012

    2012 arXiv

  6. [14]

    Okubo, Physics Leters 5(2) (1963) 165-168

    S. Okubo, Physics Leters 5(2) (1963) 165-168. doi:10.1016/s0375-9601(63)925-48-9

    1963 doi

  7. [15]

    Zwing (1964) CERN report No.8419/TH412(Report)

    G. Zwing (1964) CERN report No.8419/TH412(Report)

    1964

  8. [16]

    Iizuka, Progr.Theor.Phys

    J. Iizuka, Progr.Theor.Phys. Suppl. 37-38(1966) 21-34. https://doi.org/10.1143/PTPS.37.21

    1966 doi

  9. [17]

    Eichten and K

    E. Eichten and K. Gottfried, Physics Letters B66 (1977) 286. https://doi.org/10.1016/0370-2693(77)90882-6 18)A. Tang and J.W. Norbury, Physical Review D62 (20009 016006. https://doi/10.1103/PhysRev.D.62.016006

    1977 doi

  10. [19]

    Klempt and B

    E. Klempt and B. Ch. Metsch, Multiplet classification of light-uark baryons, Eur. Phys. J. A 48(2012) 127 DOI:10.1140/epja/i2012-12127-1

    2012 doi

  11. [20]

    Atsbek, C

    O. Atsbek, C. Deutsch, and M. Lavaud, Physical Review A9 (1974) 2617. Schr¨ odinger equation for the two-dimensional Coulomb potential

    1974

  12. [21]

    Abe et al

    F. Abe et al. (CDF collaboration), Phys. Rev. Lett., 74 (1996) 2626. https://doi.org/10.1103/PhysRevLet.742626

    1996 doi

  13. [22]

    Abachi et al

    S. Abachi et al. (D0 collaboration), Phys. Rev. Lett. 74 (1996) 2632-2637. DOI:10.1103/PhysRevLett. 742632

    1996 doi

  14. [23]

    Aabound et al

    M. Aabound et al. (ATLAS collaboration) Eur. Phys. J. C79 (2019) 290. arXiv:1810.01772 [hep ex]

    2019 arXiv

  15. [24]

    Hachatryan et al., Phys.Rev

    V.K. Hachatryan et al., Phys.Rev. D93 7 (2016) 072004. arXiv:1509.04044

    2016 arXiv

  16. [25]

    Brunner et al., The CLIC project, ar Xiv:2203.09186v2 [physics.acc-ph] 18 Apr 2022

    O. Brunner et al., The CLIC project, ar Xiv:2203.09186v2 [physics.acc-ph] 18 Apr 2022

    2022 arXiv

  17. [26]

    Particle Data Group, Progr. Theor. Exp. Phys. 2020 083c01 (2020) 741-761. DOI:10.1093/ptep/ptaa104 Appendix I In the paper by Atabek, Deutsch, and lavaud, the Schr¨ odinger equation in the two-dimensional space is discussed under the logarithmic potential; ∆ ψ + [λ - V(r)] ψ= ...

    2020 doi

Pith tools

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