Universal Mass Equation for Equal-Quantum Excited-States Sets II

Igor Strakovsky (GWU); L. David Roper (VTech)

REVIEW 5 major objections 5 minor 10 references

Universal Mass Equation for Equal-Quantum Excited-States Sets II

T0 review · 5 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Two measured masses can predict four higher hadron states

desk verdict A modest empirical pattern paper that overstates its 'two masses predict the rest' claim; the stress-test is right. read the letter →

arxiv 2506.06496 v1 pith:UBFQIUGJ submitted 2025-06-06 hep-ph hep-exnucl-exnucl-th

L. David Roper (VTech) , Igor Strakovsky (GWU) This is my paper

classification hep-phhep-exnucl-exnucl-th

keywords universalmassequationhadronradialexcitationslogarithmicformulabaryonresonancesmesonbottomoniumspectrumBreit-Wignermassestwo-stateprediction

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

The paper extends a proposed universal relation for hadron radial excitations: within a set of states having the same quantum numbers, masses follow $M_n = \alpha \ln(n) + \beta$, with $n$ the excitation level. The new claim is that only the two lowest accurately measured masses are needed, because $\beta = M_1$ and $\alpha = (M_2 - M_1)/\ln 2$, and these two constants then yield the next four masses with propagated uncertainties. The relation is checked on the six-state $N(1/2^+)$ and four-state $a_0$ families, then applied to twelve baryon and sixteen meson two-state sets from the current particle-data listings. The paper's additional claim is that the slope $\alpha$ for $b\bar{b}$ sets lies accurately on a line against the ground-state mass, so a single measured ground state determines the whole predicted spectrum.

What carries the argument

The load-bearing object is the one-parameter logarithmic spacing rule $M_n = \alpha \ln(n) + \beta$, where $n$ is the radial excitation level and the intercept $\beta$ is set equal to the lowest mass $M_1$. With only the first two masses, the slope is fixed by the identity $\alpha = (M_2 - M_1)/\ln 2$, and uncertainty propagation is carried by $\delta M_n = \sqrt{[\ln(n)\,\delta\alpha]^2 + (\delta\beta)^2}$. For bottomonium sets, the slope is further constrained by the linear relation $\alpha(M_1) = \gamma M_1 + \mu$, reducing the whole spectrum to a function of $M_1$. This machinery turns a two-point fit into a four-state prediction, and for $b\bar{b}$ sets into a one-input prediction.

What would settle it

Find a duo set where an intermediate state is later reported between the two input masses; for that set the predicted higher masses would have to move, and the calculation would need redoing. Alternatively, measure the predicted third state of a duo set and require the logarithmic prediction to agree with it within the quoted uncertainty; a clear miss at that point would falsify the one-parameter law for that family.

Watch

Extended reading notes

Core claim

The central discovery, stated as the authors would state it, is that the logarithmic mass formula is predictive rather than merely descriptive. Two accurate equal-quantum states determine the curve; four higher states follow, and the prediction carries a calculable uncertainty derived from the two input mass uncertainties. For bottomonium the predictive power sharpens: the fitted $\alpha$ values for the seven $b\bar{b}$ sets obey $\alpha(M_1) = (-0.6359 \pm 0.0028)M_1 + (6809.3 \pm 27.4)$ MeV, so $\alpha$ can be estimated from the ground-state mass alone. Substitution gives $M_n = [\gamma \ln(n) + 1]M_1 + \mu \ln(n)$, which the paper calls effectively a zero-parameter equation: knowing the ground state fixes all higher masses. The authors also use the relation to predict a maximum possible bottomonium ground-state mass near 10708 MeV and a common crossing point for all bottomonium curves.

Load-bearing premise

The paper's load-bearing premise is that when only two states of a set are known, they are the first two excitation levels; if an unseen state sits between them, $\alpha$ is overestimated and every higher predicted mass shifts upward.

Editorial extensions

If this is right

Experimentalists with two accurately measured excited states in a new hadron family can immediately list the expected masses and uncertainties of the next four states, guiding searches.
Because the input masses for heavy quarkonium families are extremely precise, those families yield predictions with sub-MeV uncertainties and many more than four states.
The near-linear $\alpha$ versus $M_1$ trend for bottomonium implies that a single ground-state mass, as for the $\Upsilon_2$ example, is enough to generate a full predicted spectrum through the one-parameter equation.
Discrepant states such as $\psi(3770)$ and $\psi(4641)$ are identified as poorly determined relative to the log law, giving a criterion for questioning particular entries in the standard particle listings.
Predicted pentaquark-like states fall close to masses seen in a recent hadron-collider measurement, indicating where future data could confirm or rule out the extra states.

Reading between the lines

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

A direct test of the paper's weakest point would be to use a third independently known state as a check: if the log law is right, the third state should fall at an integer $n$; otherwise the two-state level assignment is suspect.
The bottomonium linearity suggests a dynamical relation between the confinement slope and the ground-state mass that could be compared with potential-model or lattice spectra, where level assignments are known by construction and missing states are not an issue.
Assigning predicted states the average width of known states is an ad hoc choice; a quantitative line-shape analysis of how many predicted bumps could be resolved in existing data would turn the measurability discussion into a concrete experimental proposal.

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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

Honestly, this is a better paper than the reader's score suggests, but the stress-test lands. The duo-set extension of Part I is a natural step, and the b-bbar linear trend (Eq. 5) is a real empirical observation. The arithmetic is transparent, and the two test cases (N(1/2+) and a0) do show that the logarithmic form tracks measured higher states when you start from the two lowest masses. If the paper stopped there, it would be a modest empirical note. The soft spots are not minor. Three of the twelve baryon sets (Delta7/2+, Delta5/2-, Lambda_c 3/2-) are not two-mass predictions: the body text computes alpha from the two PDG masses, then Section VIII replaces those alphas with lower values obtained by assuming an unobserved M2 derived from a global power-law fit. That turns a prediction into a post-hoc curve. The psi section does the same thing more aggressively: five missing states inserted, two measured states discarded, and uncertainties inflated by a factor of five to make the chi-square look acceptable. The 'zero-parameter' label for Eq. (6) is an overclaim; gamma and mu are fitted parameters, so the equation has three fitted inputs (gamma, mu, plus M1 as a measured anchor). And the universality claim rests on only two in-sample tests, not independent confirmation. One softer concern: on the b-bbar line, the chi_b1 and chi_b2 points scatter well above the fit, yet the reported gamma uncertainty is tiny. That looks like an underestimate, possibly from ignoring the scatter in the alpha input values. The citation pattern is fine: Part I, PDG, and a few foundational QCD references. No issue there. My verdict is close to the reader's: reject in current form, but with an invitation to resubmit a more modest version. The logarithmic formula might be a useful empirical guide for hadron spectroscopy, but the paper needs to stop calling it 'accurate' for sets where it has to invent missing states, and it needs an out-of-sample test before claiming universality. This is fixable with honest caveats and a clearer separation between measured-input predictions and model-assisted estimates. It deserves a serious referee, not a desk rejection.

Referee Report

5 major / 5 minor

Summary. The paper extends the Roper-Strakovsky 'universal mass equation' M_n = α ln(n) + M_1 to hadron families in which only the two lowest equal-quantum excited states are known. The authors fit α and β = M_1 from two masses, propagate uncertainties through Eqs. (2)-(4), and claim that the next four higher states can be predicted accurately. Twelve baryon and sixteen meson 'duo sets' are analyzed, with two test cases (N(1/2+) and a0) checked against measured higher states. The paper also argues that α for b-bbar mesons lies on a straight line, Eq. (5), allowing predictions for sets with only one known member, and it presents a revised analysis of ψ(1--) states in which five 'missing' states are inserted.

Significance. If the claimed regularity were established, it would provide a strikingly simple empirical tool for estimating unobserved hadron masses, and the error propagation in Eqs. (2)-(4) is correct and clearly documented. The two test cases do reproduce the measured N(1/2+) and a0 higher states within uncertainties, and the paper makes concrete, falsifiable predictions. However, the universal claim is substantially weakened by the manuscript's own post-hoc modifications: three baryon sets are recalculated using model-inferred missing states rather than two measured masses, and the ψ analysis inserts five unobserved states after inflating the input uncertainties. As presented, the paper is a catalog of two-parameter fits rather than a validated universal predictive law.

major comments (5)

[§VIII.A, Table IV, Figs. 6, 7, 12] The abstract's central claim that 'two accurate masses that start a set are used to calculate four higher masses' is contradicted for three of the twelve baryon sets. For Δ7/2+, the two PDG masses 1930 and 2387 MeV give α=(2387−1930)/ln2=659.3 MeV, as stated in the Fig. 7 caption; Table IV instead reports α=415.9 MeV, obtained only after inserting a 'missing' state at 2218 MeV. Similarly, Δ5/2− is recalculated with a missing state at 2204 MeV (α=365.9 MeV rather than the two-mass value of about 580 MeV), and Λ_c 3/2− is recalculated with a missing state at 2824.6 MeV (α=283.6 MeV rather than the two-mass value 449.5 MeV). These missing states are inferred from the same global power-law fit (Fig. 40) whose validity is being tested, so the predictions for these sets are circular and do not use two measured masses.
[Introduction, p. 2] The assumption that two known states are the n=1 and n=2 states is explicitly acknowledged as uncertain ('There may be missing states... more likely M2 than M1'), yet every duo-set prediction depends on this assignment. The paper itself demonstrates that the assumption is not reliably satisfied by inserting missing states into several sets. No independent information (e.g., decay widths, quantum numbers, production amplitudes, or model calculations) is used to fix n, and no sensitivity analysis is provided. If an unseen state lies between the two known masses, α is overestimated and all predicted higher masses shift upward; the uncertainties from Eqs. (2)-(4) do not include this systematic error.
[§III and §§IV-V] The predictive claim is validated only on two test cases, N(1/2+) and a0, which already had measured higher states. For the other 26 duo sets, Eq. (4) is simply the two-parameter curve evaluated at n>2, with no measured masses to compare against. Agreement on two examples does not establish a universal law, especially because these same examples were used in Part I to motivate the functional form. A proper out-of-sample test, such as fitting a subset of states and predicting the remaining measured states, or a statistical statement about how many of the 28 sets would be expected to agree by chance given the resonance widths and uncertainties, is needed before the universality claim can be supported.
[§VIII.B, Eq. (5), Eq. (6)] The characterization of Eq. (6) as a 'zero-parameter' equation is overstated. The parameters γ and μ in Eq. (5) are free parameters fitted to the seven α values in Table III, and those α values themselves come from fits of Eq. (1) to measured masses. Using Eq. (5) to predict α for a single-member b-bbar set is therefore a two-parameter prediction, not a zero-parameter one. In addition, the uncertainty δM_n quoted after Eq. (6) does not propagate the full covariance of γ and μ with M_1; the claim that 'once the ground-state mass is known, all higher-state masses are known' is not yet supported by a proper error analysis.
[§VIII.C, Fig. 42] The ψ(1--) analysis is not a prediction from two measured masses. Five 'missing' states (3468.6, 3840.4, 3960.1, 4275.3, 4331.8 MeV) are inserted, the input uncertainties are arbitrarily increased by a factor of 5 to force χ2/DoF=1.2, and ψ(4641) is dismissed as 'not well determined' solely because it does not fit Eq. (1). This is post-hoc fitting with additional free parameters, and it cannot be used as evidence that the universal equation describes the ψ spectrum. Independent evidence for these missing states, or a genuine prediction from measured masses without uncertainty inflation, is required.

minor comments (5)

[Table V and Fig. 29] The ηb ground-state mass is listed as 9299 MeV in Table V but as 9399 MeV in Fig. 29 and Table III; one of these is a typo and should be corrected.
[§V.A.3 and Table V] The η2 quantum numbers appear as 2−+ in the text and figure captions but as 2++ in Table V; please reconcile the assignment.
[Fig. 15 caption] The caption lists Ξ_b 3/2− in both 'Top Right' and 'Bottom', and says 'twelfth baryon data sets' while only eleven sets are shown; please correct the caption and remove the duplicate.
[Throughout] The notation 'Ln' should be the standard 'ln', and there are several typographical errors in section headings ('F our', 'T est Case') that should be cleaned up.
[§V.H] The claimed common crossing point n_c≈4.82, r_c≈0.13 fm, and M_c≈10708 MeV is introduced without derivation or uncertainty; if retained, it should be derived more carefully.

Circularity Check

2 steps flagged · score 6.0 of 10

Partial circularity: three baryon sets and the ψ fit insert model-calculated 'missing' states as inputs, so the claimed two-mass predictions reduce to the global fit for these cases.

fitted input called prediction [Sections IV.A.4, IV.A.5, IV.D.1 and VIII.A (Baryon Power Equation), Table IV]
"The recalculation for those three sets, with lower α values due to an assumed missing M2, is included here rather than the original calculations. The power-equation [(α= 1.688×10 7)M −−1.446 1 MeV] fit with the three lower α values is shown in Fig. 40."

The three outlier sets are re-fit using an 'assumed missing M2' so that α falls on the baryon power-law curve; the missing states (e.g., 2218, 2204, 2824.6 MeV) are not PDG measurements but are 'calculated' in the text. Table IV reports these recalculated α values (415.9, 365.9, 283.6 MeV) rather than the α values from the two measured masses (659.3, 580.1, 450.2 MeV). The power-law curve is itself fitted using these three lower α values, so the input M2 is defined by the very regularity the paper claims to discover. The 'two accurate masses' prediction therefore reduces, for these sets, to one measured mass plus a model-inferred state.
fitted input called prediction [Section VIII.C (ψ(1−−) Excited-States)]
"ψ has two states (instead of three in our previous analysis) reported in the PDG with masses ψ(3770) and ψ(4641) that do not fit Eq. (1). ... In addition, the five missing states (green triangles) are shown as calculated."

The fit discards measured states that disagree with Eq. (1) and adds five unobserved 'missing' states calculated to lie on the logarithmic curve; the resulting fit is then presented as evidence for the equation. The missing states are generated by the model, so the agreement is by construction, not independent confirmation.

full rationale

The paper's basic two-point logarithmic extrapolation M_n = α ln n + M1 is not circular: for a given functional form, M1 and M2 determine α and the higher masses are genuine extrapolations, and the N(1/2+) and a0 test cases compare those extrapolations to measured data. The b-bbar line (Eq. 5) is a calibration, not a circularity, although calling Eq. (6) 'zero-parameter' overstates the fact that γ and μ are fitted. However, two places make the central claim circular for subsets of the data. First, three baryon sets (Δ7/2+, Δ5/2−, Λc3/2−) are re-fit in Sec. VIII.A using an 'assumed missing M2' that is calculated in the figures; the resulting α values replace the two-measured-mass values in Table IV. Since the missing M2 is chosen so that α lies on the baryon power-law curve, and that curve is then fitted using these same α values, the 'predictions' for these sets reduce to the global fit rather than to two accurate measured masses. Second, the ψ fit in Sec. VIII.C discards two measured states that do not fit Eq. (1) and inserts five calculated missing states that do, so the resulting fit is partly self-consistent by construction. These are partial circularities: the meson duo sets and the other nine baryon duo sets are unaffected, and the test cases provide independent support for the logarithmic form. Score 6.

Assumptions & free parameters 3 free parameters · 4 assumptions · 4 invented entities

The central claim rests on the unproven logarithmic form, the assumption that the two known states are the first two radial excitations, and the ad hoc insertion of missing states in several sets. The b-bbar line adds two fitted constants, gamma and mu, while the baryon power equation adds two more.

free parameters (3)

gamma (b-bbar line slope) = -0.6359 +/- 0.0028
Fitted to the seven alpha values of b-bbar sets in Table III; used in Eq. (5) to compute alpha from M1.
mu (b-bbar line intercept) = 6809.3 +/- 27.4 MeV
Fitted together with gamma to the same seven b-bbar sets.
constants in baryon power equation = 1.688e7 and exponent -1.446
Fit to the alpha vs M1 scatter for baryon sets (Fig. 40).

assumptions (4)

domain assumption The logarithmic mass formula M_n = alpha Ln(n) + M_1 holds for every equal-quantum excited-state set.
Adopted from Part I; no derivation given. Eq. (1).
domain assumption For a duo set, the two known states are the n=1 and n=2 states.
Stated in the Introduction: 'It seems highly likely that, when only two excited states in a set have been measured, they are the first two states in a set.'
domain assumption Radial excitation level n is assigned by ordering the known masses in a J^P / J^PC set, with no missing low-lying states.
This is required for Eq. (3) to give alpha; if states are missing, the fitted alpha and all predictions shift.
domain assumption Breit-Wigner masses from PDG2024 represent the physical masses of the states.
Used for all inputs; standard practice in hadron spectroscopy.

invented entities (4)

Missing state in Delta5/2- set (labeled N(2204))
purpose: Inserted as a green triangle to lower the fitted alpha so the set conforms to the baryon power equation.
No experimental observation is cited; the mass is computed, not measured.
Missing state in Delta7/2+ set (labeled N(2218))
purpose: Same as above, to force the alpha value onto the power curve.
No experimental evidence; added post hoc.
Missing state in Lambda_c 3/2- set (labeled Lambda_c(2825))
purpose: Same as above.
No experimental evidence; added post hoc.
Five missing psi states (3469, 3840, 3960, 4275, 4332)
purpose: Included in the psi(1--) analysis to make the logarithmic fit pass with chi^2/DoF=1.2; these are calculated, not observed.
The paper shows them as green triangles; they are not in PDG as established states.

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Cite this review

Pith. "Pith review of Universal Mass Equation for Equal-Quantum Excited-States Sets II." pith.science (2026). https://pith.science/paper/UBFQIUGJ

@misc{pith2026250606496,
  author       = {Pith},
  title        = {Pith review of: Universal Mass Equation for Equal-Quantum Excited-States Sets II},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UBFQIUGJ}},
  note         = {Machine review of arXiv:2506.06496}
}

abstract

We extend our recent study of the universal mass equation for equal-quantum excited-states sets reported by Roper and Strakovsky~\cite{Roper:2024ovj}. The masses of twelve baryon sets and sixteen meson sets, with only two equal-quantum excited states in each set, using Breit-Wigner PDG2024 masses and their uncertainties at fixed $J^P$ for baryons and $J^{PC}$ for mesons, are fitted by a simple one-parameter logarithmic function, $M_n = \alpha Ln(n) + M_1$, where $n$ is the level of radial excitation. Two accurate masses that start a set are used to calculate four higher masses in the set accurately. It is noted that $\alpha$ values for $b\bar{b}$ equal-quantum excited-states sets accurately lie on a straight line, whose line parameters can be used to calculate $\alpha$ and predict higher mass states for $b\bar{b}$ sets that have only one known member.

Figures

Figures reproduced from arXiv: 2506.06496 by the authors.

**Figure 2.** FIG. 2. Two [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 4.** [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

Figures from the paper (47 more)

**Figure 3.** Figure 3: FIG. 3. Data for ∆1 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png]

**Figure 4.** Figure 4: FIG. 4. Data for ∆3 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: FIG. 5. Data for ∆5 [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: FIG. 6. Data for ∆5 [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

**Figure 7.** Figure 7: FIG. 7. Data for ∆7 [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 8.** Figure 8: FIG. 8. Data for Λ3 [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]

**Figure 9.** Figure 9: FIG. 9. Data for Λ5 [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]

**Figure 10.** Figure 10: FIG. 10. Data for Λ5 [PITH_FULL_IMAGE:figures/full_fig_p011_10.png]

**Figure 11.** Figure 11: FIG. 11. Data for Σ5 [PITH_FULL_IMAGE:figures/full_fig_p011_11.png]

**Figure 12.** Figure 12: FIG. 12. Data for Λ [PITH_FULL_IMAGE:figures/full_fig_p012_12.png]

**Figure 13.** Figure 13: FIG. 13. Data for Ξ [PITH_FULL_IMAGE:figures/full_fig_p013_13.png]

**Figure 14.** Figure 14: FIG. 14. Data for [PITH_FULL_IMAGE:figures/full_fig_p014_14.png]

**Figure 15.** Figure 15: FIG. 15. Logarithmic curves for twelfth baryon data sets that only have two [PITH_FULL_IMAGE:figures/full_fig_p015_15.png]

**Figure 17.** Figure 17 [PITH_FULL_IMAGE:figures/full_fig_p015_17.png]

**Figure 16.** Figure 16: FIG. 16. Samples of spectral diagrams of baryon excited-states sets analyzed [PITH_FULL_IMAGE:figures/full_fig_p016_16.png]

**Figure 18.** Figure 18: 3. η2(2−+) Two Known Excited States: Predict Four More Two excited states η2(2−+) are recorded in the Particle Data Listings [2]. η2(2−+): I G (J P C)S = 0+(2−+)0. The logarithmic fit to the BW masses (MeV) of the two known excited states of η2(2−+) (blue circles) and…

**Figure 17.** Figure 17: FIG. 17. Data for [PITH_FULL_IMAGE:figures/full_fig_p017_17.png]

**Figure 18.** Figure 18: FIG. 18. Data for [PITH_FULL_IMAGE:figures/full_fig_p017_18.png]

**Figure 19.** Figure 19 [PITH_FULL_IMAGE:figures/full_fig_p017_19.png]

**Figure 19.** Figure 19: FIG. 19. Data for [PITH_FULL_IMAGE:figures/full_fig_p018_19.png]

**Figure 20.** Figure 20: B. Strange Mesons 1. K∗ 2 (2+) Two Known Excited States: Predict Four More Two excited states K∗ 2 (2+) are recorded in the Particle Data Listings [2]. K∗ 2 (2+): I(J P ) = 1/2(2+). The logarithmic fit to the BW masses (MeV) of the two known excited states of K∗ 2 (2+…

**Figure 21.** Figure 21 [PITH_FULL_IMAGE:figures/full_fig_p018_21.png]

**Figure 20.** Figure 20: FIG. 20. Data for [PITH_FULL_IMAGE:figures/full_fig_p019_20.png]

**Figure 21.** Figure 21: FIG. 21. Data for [PITH_FULL_IMAGE:figures/full_fig_p019_21.png]

**Figure 22.** Figure 22: FIG. 22. Data for [PITH_FULL_IMAGE:figures/full_fig_p020_22.png]

**Figure 23.** Figure 23: FIG. 23. Data for [PITH_FULL_IMAGE:figures/full_fig_p021_23.png]

**Figure 24.** Figure 24: FIG. 24. Data for [PITH_FULL_IMAGE:figures/full_fig_p022_24.png]

**Figure 25.** Figure 25: FIG. 25. Data for [PITH_FULL_IMAGE:figures/full_fig_p022_25.png]

**Figure 26.** Figure 26: FIG. 26. Data for [PITH_FULL_IMAGE:figures/full_fig_p023_26.png]

**Figure 27.** Figure 27: FIG. 27. Data for [PITH_FULL_IMAGE:figures/full_fig_p024_27.png]

**Figure 28.** Figure 28: FIG. 28. Data for [PITH_FULL_IMAGE:figures/full_fig_p025_28.png]

**Figure 29.** Figure 29: FIG. 29. Data for [PITH_FULL_IMAGE:figures/full_fig_p025_29.png]

**Figure 30.** Figure 30: FIG. 30. Data for [PITH_FULL_IMAGE:figures/full_fig_p026_30.png]

**Figure 31.** Figure 31: FIG. 31. Data for [PITH_FULL_IMAGE:figures/full_fig_p027_31.png]

**Figure 32.** Figure 32: FIG. 32. Data for [PITH_FULL_IMAGE:figures/full_fig_p028_32.png]

**Figure 33.** Figure 33: FIG. 33. Logarithmic curves for sixteen meson data sets that only have two [PITH_FULL_IMAGE:figures/full_fig_p029_33.png]

**Figure 34.** Figure 34: FIG. 34. Eight [PITH_FULL_IMAGE:figures/full_fig_p030_34.png]

**Figure 36.** Figure 36: VI. LHCB EXOTICS QCD gives rise to Hadron Spectrum [4, 5]. PDG2024 [2] reports that many qq¯ and qqq states have been observed - more than 200 and 100 states, respectively. In addition, qqq¯ q¯ and qqqqq¯ are not forbidden, or we do not know it yet. Recently, LHCb Col…

**Figure 35.** Figure 35: FIG. 35. Fit parameter [PITH_FULL_IMAGE:figures/full_fig_p031_35.png]

**Figure 36.** Figure 36: FIG. 36. Samples of spectral diagrams of meson excited-states sets analyzed [PITH_FULL_IMAGE:figures/full_fig_p032_36.png]

**Figure 37.** Figure 37: FIG. 37. Left [PITH_FULL_IMAGE:figures/full_fig_p033_37.png]

**Figure 38.** Figure 38: FIG. 38. Samples for the “bump hunting” in the baryon ( [PITH_FULL_IMAGE:figures/full_fig_p034_38.png]

**Figure 39.** Figure 39: FIG. 39. Sample for the “bump hunting” in the heavy vector meson (Υ) case. Blue BW curves [PITH_FULL_IMAGE:figures/full_fig_p035_39.png]

**Figure 40.** Figure 40: FIG. 40. Baryon [PITH_FULL_IMAGE:figures/full_fig_p035_40.png]

**Figure 41.** Figure 41: FIG. 41. Data for Υ2(2− −) (blue circle): Υ2(10164) [2]. Predicted states generated by Eq. (6) (magenta diamonds): Υ2(10404), Υ2(10544), Υ2(10644), and Υ2(10721), with masses of 10404 ± 27 MeV, 10544 ± 43 MeV, 10644 ± 55 MeV, and 10721 ± 64 MeV, respectively. The soli…

**Figure 42.** Figure 42: ψ has two states (instead of three in our previous analysis) reported in the PDG with masses ψ(3770) and ψ(4641) that do not fit Eq. (1). We conclude that this mass is not well determined. PDG gives very small uncertainties for heavy-mass mesons of approximately 0.2% …

**Figure 42.** Figure 42: FIG. 42. Data for [PITH_FULL_IMAGE:figures/full_fig_p037_42.png]

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Reference graph

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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Produce the linear relationship betweenα(M 1) andM 1 (Eq. (5) forb ¯bequal-quantum excited states sets. ACKNOWLEDGMENTS This work was supported in part by the U. S. Department of Energy, Office of Science, Office of Nuclear Physics, under Award No. DE–SC0016583. Both authors dedicate this document to the memory of their excellent colleague and friend 40 R...

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Observation ofJ/ψpresonances consistent with pentaquark states in Λ0 b →J/ψK −pdecays,

R. Aaijet al.[LHCb], “Observation ofJ/ψpresonances consistent with pentaquark states in Λ0 b →J/ψK −pdecays,” Phys. Rev. Lett.115, 072001 (2015)

work page 2015