Baryon and Meson Excited States

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

REVIEW 4 major objections 5 minor 5 references

Baryon and Meson Excited States

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper claims that every equal-quantum set of excited baryons and mesons follows one logarithmic mass equation, $M_n = \alpha \ln(n) + \beta$, and uses that equation to predict missing and higher states.

desk verdict A two-parameter log fit to many hadron spectra is worth a footnote, but the radial-purity assumption and absence of error analysis keep the universality claim unproven. read the letter →

arxiv 2506.10646 v1 pith:6PG5ZBVJ submitted 2025-06-12 hep-ph hep-exnucl-exnucl-th

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

classification hep-phhep-exnucl-exnucl-th

keywords universalmassequationhadronexcitedstatesequal-quantumsetsradialexcitationsBreit-WignermassescharmoniumbottomoniumCornellpotential

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 claims that baryon and meson excited states are not randomly spaced: when grouped by identical quantum numbers ($J^P$ for baryons, $J^{PC}$ for mesons), the masses $M_n$ of successive radial excitations $n=1,2,3,\dots$ follow the universal mass equation (UME) $M_n = \alpha \ln(n) + \beta$. The authors fit this two-parameter curve to fifteen baryon sets and twenty-four meson sets with three or more known states, using Breit-Wigner masses from the PDG 2024 listings, and to another twelve baryon and sixteen meson sets with only two known states. If the conjecture is correct, all equal-quantum hadron spectra collapse to logarithmic sequences determined by just two numbers per channel, at least over the currently known mass range. That would give experimentalists concrete predicted masses to search for, and would give theorists a simple constraint that any model of quark confinement must reproduce.

What carries the argument

The central object is the universal mass equation (UME), $M_n = \alpha \ln(n) + \beta$, which assigns each state in a set of fixed $J^P$ (baryons) or $J^{PC}$ (mesons) a radial quantum number $n = 1,2,3,\dots$ ordered by increasing mass, with $\beta$ fixed to the ground-state mass. The argument is carried by $\chi^2$ fits of this two-parameter function to Breit-Wigner masses from the PDG listings; the fitted slope $\alpha$ produces predicted masses for missing and higher states. Two supporting tools carry the extensions: a systematic plot of $\alpha$ versus $M_1$, whose linear trend for $c\bar c$ and $b\bar b$ mesons lets the authors estimate $\alpha$ for channels with one known state, and the Breit-Wigner resonance formula used to judge whether predicted states are measurable against overlapping neighbors. A final link to a Cornell potential $V(r)=-(4/3)A/r + Br + C$ is made through a de Broglie-based radius for each excited state.

What would settle it

Choose any equal-quantum set with at least three accurately measured states, fit $M_n = \alpha \ln(n) + \beta$, and then measure the next predicted state with high statistics: if its mass disagrees with the extrapolated curve by more than the combined uncertainties, the universal mass equation fails for that channel, and one such failure would refute universality.

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Extended reading notes

Core claim

The central discovery claimed is that a single two-parameter logarithmic function describes the mass ladder of every equal-quantum excited-state set, including exotic $P_{c\bar c}^+$ baryon states and heavy quarkonia such as $s\bar s$, $s\bar c$, $c\bar c$, $c\bar b$, and $b\bar b$. On the paper's own terms, the UME is $M_n = \alpha \ln(n) + \beta$, with $n$ the radial excitation level, $\beta$ the ground-state mass $M_1$, and $\alpha$ a logarithmic slope obtained from a $\chi^2$ fit to PDG2024 Breit-Wigner masses. Beyond the fits, the authors derive systematic relations: baryon $\alpha$ versus $M_1$ roughly follows a power law, while for $c\bar c$ and $b\bar b$ mesons the relation is strongly linear, allowing $\alpha$ to be estimated for channels such as $\Upsilon$ with only one known state. They also use the Breit-Wigner lineshape to assess whether adjacent predicted resonances are separable enough to measure, and they connect the mass ladder to a Cornell-potential form through a de Broglie-based radius.

Load-bearing premise

The load-bearing premise is that every set of states carrying the same spin-parity quantum numbers consists purely of successive radial excitations ordered by mass, with no states of different internal structure mixed in; if a set mixes structures, the logarithm curve just connects unrelated states and its predicted masses carry no meaning.

Editorial extensions

If this is right

If the UME is right, each fixed-$J^P$ or fixed-$J^{PC}$ channel is a logarithmic sequence, so future measurements of any new excited state in a known channel either confirm the predicted mass or expose a deviation.
The predicted missing and higher states listed in the fits, such as states in the $N1/2^+$, $\Delta3/2^+$, $\pi_2$, $\rho$, $f_0$, $a_0$, $K^*$, and $\Upsilon$ sets, become concrete search targets for hadron-spectroscopy experiments.
The linear $\alpha$-$M_1$ trends for charmonium and bottomonium allow the mass ladder of the $\Upsilon$ system, where only one excited state is known, to be estimated before the remaining states are measured.
The Breit-Wigner overlap analysis in the paper tells which predicted states should be experimentally resolvable, so experimentalists can prioritize channels with separated resonances over channels where they are buried.

Reading between the lines

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

Going beyond the paper, a sharper test would be to take the published $\alpha$ and $\beta$ for each channel and check every subsequently measured state against the log prediction; a single clear miss would show the UME is approximate rather than universal.
Going beyond the paper, the linear $\alpha$-$M_1$ systematics for heavy quarkonia suggest that $\alpha$ is set mainly by the ground-state mass; if so, one could try to derive the UME from scale invariance of the confinement energy, a step the paper does not take.
Going beyond the paper, the fits depend on treating $n$ as a purely radial label; an alternative assignment that intermixes orbital excitations would either destroy the fits or reveal that the UME is a statement about all states at fixed quantum numbers, not just radial ladders.

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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

Short version: this is a proceedings summary of a two-parameter log fit to about forty baryon and meson spectra. The empirical pattern is interesting enough that I wouldn't dismiss it out of hand, but the paper does not establish the universality it claims. The load-bearing assumption is that each fixed J^P/PC list is a pure radial tower; if orbital angular momentum mixes into these sets, the log fit is just connecting unrelated points with two free parameters. That assumption is asserted, not tested.

What the paper does well: it compiles a large set of PDG masses and shows that a single functional form, M_n = alpha ln(n) + beta, produces decent-looking fits across many different channels, including heavy quarkonia and even LHCb pentaquarks. The BW overlap plots are a nice practical touch for judging whether predicted states could be seen. The Cornell potential connection is speculative but at least shows a plausible physical picture.

The soft spots are significant. First, with two free parameters per set and no error bars on the fits, the visual agreement is weak evidence; many simple functions would fit four or five points. The predictions for missing states are just evaluations of the fitted curve, so they are not independent. The alpha-versus-M1 systematics in Figs. 4 and 8 are fits to fitted parameters, adding another layer of circularity. The paper also gives no derivation of the logarithm; it is an empirical coincidence until shown otherwise.

The most serious problem is the radial-purity assumption. PDG quantum numbers do not fix L and S. For example, 1-- mesons include both ^3S1 and ^3D1 states, and 1/2+ baryons can come from L=0 and L=2. The paper assigns n by increasing mass without showing that all states in a set share the same radial Hamiltonian. If they do not, the universal mass equation is an interpolation, not a law.

Bottom line: as a conference record, this is acceptable. As a scientific claim, it needs much more supporting evidence. I would send it to peer review, but the referee's main job should be to demand a test of radial purity, error analysis, and at least one out-of-sample prediction. The citation to the authors' earlier papers is appropriate; the earlier papers may contain more detail, but this summary should stand on its own.

Referee Report

4 major / 5 minor

Summary. The paper proposes a universal mass equation (UME) for hadron excited states: M_n = α ln(n) + β, where n is the radial excitation level, and applies it to baryon sets at fixed J^P and meson sets at fixed J^{PC} using PDG 2024 Breit-Wigner masses. It reports fits to fifteen baryon and twenty-four meson sets with at least three states, and to additional two-state sets. The paper predicts missing or higher-mass states, discusses their measurability via Breit-Wigner overlap, and attempts to relate the resulting masses to a Cornell potential using an assumed wavefunction radius. The central claim is that all equal-quantum excited-state hadron masses follow this logarithmic sequence.

Significance. If the UME were established, it would provide a remarkably simple empirical regularity for hadron spectroscopy, useful for guiding experimental searches. The paper merits credit for working directly with PDG data and for explicitly listing the fitted functional form and the data sources. However, the claim is supported only by in-sample fits with two free parameters per set, no statistical diagnostics, no out-of-sample validation, and no theoretical derivation. The proposed universality therefore remains a conjecture of limited evidential weight in its current form.

major comments (4)

[Sec. 2, Eq. (1)] The logarithmic form M_n = α ln(n) + β is assumed without derivation, and the predictive content of the paper is in-sample. For a set with known masses M_1,...,M_N, the parameters α and β are determined by those same masses, so the 'predicted' masses for n>N (or for a missing intermediate state) are extrapolations/interpolations of the fitted curve with no uncertainty propagation. The manuscript reports no χ²/dof, parameter uncertainties, or cross-validation for the fits to individual sets, so the claimed universality is not statistically supported.
[Secs. 2 and 3] The identification of n as the radial excitation number and the ordering n=1,2,3,... by increasing mass is unjustified. Quantum numbers J^P (baryons) or J^{PC} (mesons) do not uniquely specify the orbital angular momentum L; for example, J^{PC}=1^{--} mesons include both ^3S_1 and ^3D_1 states, and 1/2^+ baryons can arise from L=0 or L=2. If a fitted set contains states of different L, the log fit connects unrelated states and the predicted masses have no physical basis. The manuscript provides no evidence that each set is a pure radial tower; the systematics in Fig. 4 and Fig. 8, which correlate fitted α with M_1, do not test this purity.
[Sec. 6] The radius ansatz r_n = n λ_n / 4 is introduced without physical justification, and the Cornell potential fit of Eq. (3) introduces three additional free parameters (A, B, C) for what appears to be a small number of data points. The 'good fit' in Fig. 11 is therefore not an independent check of the UME; it is a fit to a derived quantity using an ad hoc scaling of the radius. Without a derivation of this radius assumption, the potential-energy connection does not lend support to the mass equation.
[Secs. 3.1 and 4.1] The α versus M_1 systematics are fits to fitted parameters, introducing another layer of circular reasoning. The power-law fit for baryons (Fig. 4) and the linear fits for charmonium/bottomonium (Fig. 8) are not derived from any underlying theory, yet they are used to estimate α for Υ(2) (Fig. 9) and then to predict its higher masses. These predictions thus depend on correlations among best-fit parameters rather than on a physical law, and the manuscript gives no uncertainty estimates for the extrapolated masses.

minor comments (5)

[Abstract] There are typographical errors: 'Breat-Wigner' should be 'Breit-Wigner', and 'thr' should be 'the'.
[Reference [3]] The arXiv identifier for reference [3] appears to be mistyped ('2506.006496'); please verify and correct it.
[Sec. 2] The sentence 'An excited state is a quantum state of a system that has a higher energy than the ground state M1. In this work, the authors often label a ground state as an excited state of the vacuum.' is confusing and should be clarified or removed.
[Notation] The notation 'Ln(n)' is nonstandard; use '\ln(n)' throughout.
[Sec. 5, Fig. 10] The measurability discussion is qualitative and the figure does not show error bands or overlap integrals; please quantify the overlap criterion or label the figure as an illustration only.

Circularity Check

1 steps flagged · score 2.0 of 10

Minimal circularity: duo-set 'fits' are two-point identities; the core three-or-more-state log fits are ordinary empirical fits to PDG masses and are not circular.

self definitional [Abstract; Section 2, Eq. (1)]
"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, are fitted by thr universal mass equation."

For a duo set, Eq. (1) has two free parameters, and Section 2 fixes beta = M1 because Ln(1)=0. Then alpha = (M2 - M1)/Ln(2) is fully determined by the two input masses, so the 'fit' passes through both points exactly for any pair of masses. This is an identity, not a test: the statement that a two-state set is 'fitted by the UME' is true by construction and provides no support for the logarithmic law. The sets with three or more states do not have this defect, since after anchoring M1 the remaining masses must independently follow Ln(n), so the central claim retains independent content.

full rationale

The paper's central object is an empirical fit, not a first-principles derivation: M_n = alpha Ln(n) + beta is fitted to PDG2024 masses and then used to interpolate or extrapolate missing or higher states. Using a fitted curve to predict a mass not included in the fit is standard model-based extrapolation, not circularity. The predicted masses are not fitted inputs; they are outputs of the fitted formula. The self-citations [2] and [3] are the authors' own earlier papers containing the detailed fits, but no uniqueness theorem or external authority is imported from them, so this is not load-bearing self-citation. The alpha-versus-M1 systematics (Figs. 4 and 8) are second-order fits of fitted parameters against an input mass; for duo sets alpha = (M2-M1)/Ln(2) makes the correlation partially built-in, and the paper overstates the predictive power of these trends, but this is a statistical fragility rather than a definitional circularity. The one genuine reduction is the abstract's treatment of two-state sets: with beta = M1 and only M1 and M2 known, the UME is forced through both points with zero degrees of freedom, so those 'fits' are tautological. Because the main three-or-more-state analysis is a real, falsifiable empirical fit against PDG masses, the overall circularity is minor.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The framework adds many free parameters: α and β for every set, power-law and linear systematics relating α to M1, and Cornell potential parameters. There are no new physical entities. The main axioms are that the mass-ordered states at fixed J^P/J^PC are radial excitations and that the logarithmic form is valid without derivation.

free parameters (4)

α and β for each hadronic set = Not tabulated in this paper; fitted to PDG masses
Each of the 39 three-or-more-state sets and 28 two-state sets is fitted independently, giving two free parameters per set. The fit is therefore trivially flexible; no global fit or cross-validation is reported.
Power-law parameters for α vs M1 (baryons) = 1.688e7 and -1.446
A power equation α = 1.688e7 * M1^(-1.446) is fitted to the set-dependent α values (Sec. 3.1), adding two tuned parameters.
Slope and intercept for charmonium and bottomonium α vs M1 = c-cbar: slope -0.7539, intercept 3190.7; b-bbar: slope -0.6359, intercept 6809.3
Linear fits in Fig. 8 use two parameters each to describe how α varies with ground-state mass.
Cornell potential parameters A, B, C = Not given in text
In Eq. (3), A, B, C are fitted to the excited-state potential energy curves (Sec. 6), adding further free parameters to the overall framework.

assumptions (4)

domain assumption The states in each set are radial excitations at fixed J^P or J^PC
The entire analysis groups resonances that share J^P or J^PC but does not verify that they differ only in radial quantum number; mixing with other excitation modes would invalidate the fit.
ad hoc to paper The logarithmic functional form M_n = α ln(n) + β is assumed without derivation
Eq. (1) is posited as the universal mass equation; no QCD or string-theory derivation is provided.
ad hoc to paper Wavefunction radius r_n = n λ_n / 4
In Sec. 6, the radius is assumed to scale as n times a quarter of the de Broglie wavelength; this relation is introduced for this work and not derived or validated.
domain assumption PDG Breit-Wigner masses are accurate
The fits rely on the masses reported in PDG2024 without accounting for extraction model dependence beyond the quoted uncertainties.

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Cite this review

Pith. "Pith review of Baryon and Meson Excited States." pith.science (2026). https://pith.science/paper/6PG5ZBVJ

@misc{pith2026250610646,
  author       = {Pith},
  title        = {Pith review of: Baryon and Meson Excited States},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6PG5ZBVJ}},
  note         = {Machine review of arXiv:2506.10646}
}

abstract

The masses of fifteen baryon sets and twenty-four meson sets of three or more equal-quantum excited states are fitted by a simple two-parameter logarithm function, $M_n = \alpha Ln(n) + \beta$, where $n$ is the level of radial excitation. The conjecture is made that accurately measured masses using Breat-Wigner PDG2024 data at fixed $J^P$ for baryons and $J^{PC}$ for mesons of all equal-quantum baryons (including LHCb exotic $P_{c\bar{c}}^+$s) and meson (including $s\bar{s}$, $s\bar{c}$, $c\bar{c}$, $c\bar{b}$, and $b\bar{b}$) excited states are related by the logarithm function used here; at least for the mass range of currently known excited states. Thus, a universal mass equation for equal-quantum excited-states sets is presented. 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, are fitted by thr universal mass equation.

Figures

Figures reproduced from arXiv: 2506.10646 by the authors.

**Figure 1.** 𝑁1/2 + set (left) and Δ3/2 + set (right). PDG2024 data (blue circles) [1]. The green triangle is the calculated mass of the missing states. Predicted states (magenta diamonds). The solid red curve presents the best-fit. Part I [2]. 2 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** Λ1/2 + set (left) and Σ3/2 + set (right). The notation is the same as in [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Λ𝑐3/2 − duo set (left) and Ξ𝑏3/2 − duo set (right). The notation is the same as in [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: Baryon 𝛼 vs 𝑀1 power-equation fit [(𝛼 = 1.688 × 107 )𝑀−1.446 1 MeV]. Blue filled circles are from Part I baryon sets [2], green filled triangles are from this Part II baryon sets [3], and red curve is Parts I and II fit to the power equation. 3 [PITH_FULL_IMAGE:figure…

**Figure 5.** Figure 5: 𝜋2 set (left) and 𝜌 set (right). The notation is the same as in [PITH_FULL_IMAGE:figures/full_fig_p004_5.png]

**Figure 6.** Figure 6: 𝑓0 set (left) and 𝑎0 set (right). The notation is the same as in [PITH_FULL_IMAGE:figures/full_fig_p004_6.png]

**Figure 7.** Figure 7: 𝐾 ∗ set (left) and Υ set (right). The notation is the same as in [PITH_FULL_IMAGE:figures/full_fig_p004_7.png]

**Figure 8.** Figure 8: 𝑐𝑐¯: 𝛼 = −(0.7539 ± 0.0045)𝑀1 + (3190.7 ± 14.1) MeV (magenta filled squares). 𝑏𝑏¯: 𝛼 = −(0.6359±0.0028)𝑀1 + (6809.3±27.4) MeV (blue filled circles). Line fits for 𝑞𝑞¯ and 𝑠𝑠¯ are not good. The red open triangle shows Υ2 ( [PITH_FULL_IMAGE:figures/full_fig_p005_8.png]

**Figure 9.** Figure 9: Υ2, a 𝑏𝑏¯ meson, has only one excited state known [1]. The 𝑏𝑏¯-line fit ( [PITH_FULL_IMAGE:figures/full_fig_p005_9.png]

**Figure 10.** Figure 10: The final black curves are for the known, missed, and predicted states with average width of the preceding known states. 𝑁1/2 + BW curves (top left) - possibly not measurable. 𝑎0 (top right) BW curves - possibly measurable. Υ (bottom) - measurable. The notation is the…

**Figure 11.** Figure 11: 𝑁1/2 + baryon set (left) and 𝑎0 meson set (right). The blue curves are the Cornell potential-energy function (Eq. (3)) fits to the excited-states set’s potential energy. Part I [2]. 7. Conclusion We have identified a UME that fits known equal-quantum excited-state set…

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5 extracted references · 5 canonical work pages

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Pith tools

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

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