Spectral-statistics properties of the experimental and theoretical light baryon and meson spectra

A. Rela\~no; L. Mu\~noz

REVIEW 3 major objections 4 minor 53 references

Spectral-statistics properties of the experimental and theoretical light baryon and meson spectra

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Measured light baryon and meson mass spectra show the level-repulsion statistics of chaotic quantum systems, while quark-model and lattice QCD spectra mostly show the uncorrelated statistics of integrable systems.

desk verdict A careful consolidation of the authors' prior work with a useful short-sequence method, but the strong chaos-versus-integrable claims overreach the statistics and ignore a resolution-driven selection bias. read the letter →

arxiv 2501.18027 v1 pith:PXTMKVGY submitted 2025-01-29 hep-ph nlin.CDnucl-th

L. Muñoz , A. Relaño This is my paper · ORCID

classification hep-phnlin.CDnucl-th PACS 14.20.-c14.40.-n12.39.Ki12.38.Gc05.45.Mt

keywords hadronspectroscopyspectralstatisticsrandommatrixtheoryquantumchaosnearest-neighborspacingdistributionquarkmodelslatticeQCDmissingresonances

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 asks whether the measured light baryon and meson masses behave statistically like the energy levels of chaotic quantum systems or integrable ones. Comparing nearest-neighbor spacing distributions, with a new correction for very short symmetry-class sequences, the authors find that the experimental spectra sit close to the random-matrix (GOE/Wigner) prediction—baryons most strongly, mesons at about 78% chaos—while essentially all theoretical spectra from quark models and lattice QCD sit close to the Poisson prediction and are statistically incompatible with chaos. Since missing levels push a spectrum toward Poisson, the fact that the incomplete experimental spectra look more chaotic than the complete theoretical ones argues that 'missing resonances' are not the explanation. If correct, this reframes what a hadron model must reproduce: not just individual masses but the fluctuation statistics of the whole spectrum, and it singles out lattice QCD as currently failing to describe the statistical structure of the meson spectrum.

What carries the argument

The central object is the nearest-neighbor spacing distribution (NNSD), $P(s)$, the probability density of the gap between consecutive unfolded energy levels; $P(s)=e^{-s}$ marks integrable (Poisson) dynamics, and the Wigner/GOE form $P(s)=\frac{\pi s}{2}e^{-\pi s^2/4}$ marks chaotic dynamics with time-reversal invariance. The paper's key machinery is a set of 'distorted' reference distributions: Wigner, Poisson, and one-parameter Berry-Robnik spectra are divided into sequences of exactly the same lengths as the data and subjected to the same local unfolding, with an average over 1000 realizations, so that the finite-sequence cutoff (a spacing in a sequence of $l$ levels cannot exceed $l-1$) is baked into the reference. Data are compared to these distorted references with the Kolmogorov-Smirnov test and with the moments of $P(s)$, and robustness is checked by resampling the experimental masses within their error bars.

What would settle it

Run the full analysis on a synthetic Poisson spectrum that has been passed through a realistic resonance-detection filter that merges or removes pairs of lines closer than the experimental resolution. If the filtered spectrum reproduces the experimental p-values (high p for Wigner, low for Poisson), then the observed level repulsion would be explained by detection bias, and the paper's chaotic interpretation would not be settled.

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Extended reading notes

Core claim

The central discovery claimed is a systematic statistical separation between experiment and theory. Once the experimental baryon spectrum from the Review of Particle Physics (RPP), cut at 2.2 GeV, is split into sequences of fixed spin, isospin, and parity, and the meson spectrum, cut at 2.5 GeV, is split into sequences that also fix C-parity, the local-unfolded nearest-neighbor spacing distributions of the data are Wigner-like (baryons) or Berry-Robnik-like with a 78% chaotic fraction (mesons), while the spacing distributions of the three quark-model baryon spectra and of five of the six meson-model spectra, including the lattice QCD set, are Poisson-like and statistically reject the Wigner hypothesis (for baryons $p_{DW}\approx 10^{-4}$; for lattice QCD $p_{DW}=0.033$). The paper then argues that the usual missing-resonances explanation cannot rescue the models, because randomly missing levels displace a spectrum toward Poisson, the opposite direction from what the data show. The conclusion is that quark models as presently built may not reproduce the low-lying hadron spectrum, and the current lattice QCD calculation does not describe the statistical properties of the meson spectrum.

Load-bearing premise

The whole comparison assumes that the published experimental resonance lists, after cutting at 2.2/2.5 GeV and discarding short sequences, are an unbiased sample of the true hadron spectrum—specifically, that experimental difficulty in resolving closely spaced resonances does not systematically delete small spacings, since deleting small spacings would itself produce the level repulsion the paper takes as the signature of chaos.

Editorial extensions

If this is right

If the experimental baryon spectrum is genuinely Wigner-like, then the quark-model baryon spectra, which reject the Wigner hypothesis at $p\approx 10^{-4}$, are not merely inaccurate in detail; they lack the chaotic fluctuation structure of the real spectrum, so their predictions about which missing resonances should exist are not trustworthy in their present form.
Because omissions push a spectrum toward Poisson, the missing-resonances explanation for the theory–experiment discrepancy fails: the experimental data are closer to Wigner than the supposedly complete theoretical spectra are.
For mesons, only the set labeled V reproduces the intermediate dynamics (about 63% chaos) needed to match the experimental 78% chaos; the other five theoretical spectra, including the lattice QCD set, behave as integrable or nearly integrable systems and do not describe the statistical properties of the meson spectrum.
Hadron models should be tested for their spectral fluctuation statistics, not only for their energy eigenvalues, since the fluctuation properties encode whether the underlying interaction is chaotic.
The distorted-reference method makes fluctuation analysis applicable to spectra with sequences of only 3–4 levels, extending quantum-chaos tests to short spectral sets beyond hadron spectroscopy, such as nuclear spectra.

Reading between the lines

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

Take a Poisson spectrum, apply a realistic resolution filter that removes or merges closely spaced pairs, and re-run the full analysis; if the filtered spectrum produces Wigner-like p-values, then the experimental level repulsion could arise from detection bias rather than true chaos.
Comparing the V model with the other quark models term by term could identify which interaction component generates the 63% chaotic fraction, since V is the only theoretical meson spectrum compatible with the experimental statistics.
If the spectra are genuinely GOE, random-matrix theory gives a quantitative missing-resonance inventory: starting from a Wigner spectrum and randomly deleting levels produces a known drift toward Poisson, so the observed p-values could be inverted to estimate how many states are still missing.
For lattice QCD, the paper's claim implies a testable trend: as the simulation pion mass approaches its physical value, the meson nearest-neighbor spacing distribution should move from the Poisson-like shape seen here toward Wigner, and newer ensembles can be checked for this.

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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

Bottom line: this is a careful, honest consolidation of results the same group already published, and the methodological core—how to build distorted reference distributions for very short spectral sequences—is genuinely useful. But the strong language about being 'safely incompatible' and the headline chaos percentages goes beyond what the data and fits actually support, and the resolution bias worry is real.

What is new and good: the paper applies the short-sequence method from the authors' 2012 meson paper to the baryon spectrum, updates both analyses to the current RPP data, and lays out the distortion problem clearly with concrete examples. The K-S contrast for baryons is stark: pDW = 0.82 for the experimental spectrum versus 10^-4 for all three quark models. That is a real, reproducible difference, and the moments back it up. The error-bar robustness test is sensible, though it only Gaussian-smears the retained masses.

The soft spots are in proportion. First, 'safely incompatible with Poisson' for the experimental meson spectrum is an overclaim: pDP = 0.13, above the usual 0.10 rejection threshold, and the paper itself concedes it is close. Second, the Berry-Robnik fits for mesons and for model V are descriptive, not independent tests; the f parameter is fitted to the same histogram and then used as the reference, so the 78% and 63% numbers should be read as interpolation parameters, not measured chaos fractions. Third, and most important, the paper does not address the selection effect that preferential loss of closely spaced resonances—due to finite resolution and overlapping broad states—would suppress small spacings and mimic Wigner repulsion. The missing-levels argument only covers random deletion, which pushes toward Poisson; it does not cover resolution-driven deletion of close pairs, which pushes the opposite way. The error-bar test simulates mass uncertainty, not missing close pairs. Without a control—for example, removing close pairs from a Poisson spectrum and checking whether you recover the observed P(s)—the central experiment/theory dichotomy has an uncontrolled observational confound.

Who should read this: people working on hadron spectroscopy, missing resonances, or spectral statistics of short sequences. The method section deserves attention. The strong concluding statements should be treated with caution. I would send it to peer review, mainly because the short-sequence method is worth serious scrutiny and the empirical contrast is important, but I would ask the authors to soften the 'safely' language and to discuss the resolution selection effect as a limitation, or better, to simulate it.

Referee Report

3 major / 4 minor

Summary. The paper compares the nearest-neighbor spacing distributions (NNSD) of experimental baryon and meson mass spectra from the Review of Particle Physics with those of spectra from constituent quark models and a lattice QCD calculation. Because the spectra are split into very short symmetry sequences, the authors construct 'distorted' Wigner, Poisson, and Berry-Robnik reference distributions that reproduce the finite-sequence distortion induced by local unfolding, and they use Kolmogorov-Smirnov tests and moment comparisons to quantify agreement. For baryons, the experimental NNSD is closer to Wigner (p_DW=0.82, p_DP=0.26) than are the three quark-model sets, whose p_DW values are about 10^-4. For mesons, the experimental NNSD is intermediate, with a Berry-Robnik fit f=0.78 and p_DP=0.13, while most theoretical sets are compatible with Poisson and incompatible with Wigner. The authors conclude that the experimental spectra are chaotic-like, the theoretical spectra are generally Poisson-like, and that missing resonances cannot account for the discrepancy.

Significance. If the central experiment-versus-theory contrast is correct, the result is significant for hadron spectroscopy: it would indicate that standard constituent quark models and the Hadron Spectrum Collaboration lattice calculation do not reproduce the spectral fluctuation properties of the empirical spectrum, and that the real baryon and meson dynamics are closer to chaotic than to integrable. The paper's methodological contribution—building an ensemble of distorted reference distributions to account for short-sequence local-unfolding bias—is useful, clearly explained, and reproducible in outline. The explicit K-S p-values, moment analysis, and Gaussian error-bar robustness studies are valuable strengths. However, the interpretation is currently stronger than the evidence: the quantitative Berry-Robnik fractions are fitted parameters, the meson 'safely incompatible with Poisson' claim is not supported by the reported p-value, and the central contrast is vulnerable to an uncontrolled observational selection effect that the paper does not address.

major comments (3)

[Sec. III.A.1 and Sec. III.A.2] The missing-resonances argument in Sec. III.A.1 (citing Refs. [35,38]) assumes that missing levels are randomly deleted, which is known to shift the NNSD toward Poisson. However, the dominant experimental incompleteness in the RPP resonance lists is not random deletion: finite resolution and the 'overlap of baryons' mentioned in the Introduction preferentially remove closely spaced resonances, which is exactly the small-s region that distinguishes Wigner from Poisson behavior. The error-bar robustness test in Sec. III.A.1 only adds Gaussian fluctuations to the retained masses; it does not simulate the removal of unresolved close pairs. A Poisson spectrum with a resolution-dependent loss of close pairs can produce apparent level repulsion. A quantitative test—for example, applying a resolution threshold to synthetic Poisson and Wigner spectra with the same sequence lengths—is needed before the RPP data can be used as strong evidence for chaotic dynamics. This is load-bearing because the paper's central experiment-versus-theory dichotomy rests on the experimental NNSD exhibiting Wigner-like repulsion.
[Sec. III.B.1 and Table II] The abstract and conclusions describe the experimental meson spectrum as 'safely incompatible' with Poisson, but the K-S test gives p_DP=0.13, which is above the paper's own rejection threshold of p≲0.10. The Poisson null hypothesis therefore cannot be rejected for the experimental meson NNSD. The data are indeed more consistent with the distorted Wigner/Berry-Robnik references than with the distorted Poisson reference, but the wording should be softened to 'not excluded' or 'marginally incompatible,' and the headline claim should not rest on p_DP=0.13.
[Sec. III.B.1 and Sec. III.B.2, Eq. (5)] The Berry-Robnik percentages (f=0.78 for mesons and f=0.63 for model V) are obtained by fitting the free parameter f to the same NNSD that is subsequently compared with P_DBR(f,s). The reported p_DBR values and moment agreement are therefore not independent tests of the fit. The manuscript should present a parameter-count-adjusted goodness-of-fit comparison, or a cross-validated statistic, before quoting '78% chaos' and '63% chaos' as quantitative results. The qualitative experiment-versus-theory contrast does not depend on these fits, but the quantitative claims do.

minor comments (4)

[Sec. II, Eq. (2)] The notation in Eq. (2) is visually ambiguous: the denominators should be written as E_{k-v} and E_{k+v} rather than as 'Ek−v − Ek+v', which is easy to misread as a single subtraction of two energies.
[Sec. I and Sec. III.A.1] The manuscript states that the analysis uses the 'last updated experimental data' from the Review of Particle Physics, but Ref. [37] is the 2014 edition. If this manuscript is submitted in 2025, the authors should update to the current RPP edition and check whether new or revised resonance assignments change the sequence construction or the reported p-values.
[Sec. III.B.2] For the meson theory sets, the K-S p-values p_DW=0.038 (K1) and p_DW=0.083 (E) are only moderately below the 0.10 threshold. The statement that these sets are 'incompatible with the Wigner correlations' would be more accurate as 'rejected at the 10% level but with marginal significance' for these two cases.
[Figs. 4 and 8] The y-axis label 'N(p−value)' is unconventional; using 'Count' or 'Frequency' would be clearer, and the histograms would benefit from reporting the bin width used.

Circularity Check

2 steps flagged · score 5.0 of 10

The experimental/theory contrast is independently grounded, but the headline 78% and 63% chaos fractions are Berry-Robnik fit parameters reused as evidence.

fitted input called prediction [Section III.B.1 (Mesons, experimental spectrum), around Figs. 6 and 7]
"Then, we fit the experimental P (s) to a Berry-Robnik distribution PDBR(f, s ) and in this case, unlike for baryons, we do obtain a best fit which is intermediate between Wigner ( f = 1 ) and Poisson ( f = 0 ), that is, f = 0 . 78 ± 0. 03. [...] The results for the p-value from the K-S test for the comparison with the three reference distributions are the following: pDP = 0 . 13, pDW = 0 . 38 and pDBR = 0 . 65."

The parameter f = 0.78 is obtained by fitting PDBR to the same experimental NNSD, and this fitted curve is then used as the null distribution in the K-S test, giving pDBR = 0.65. That p-value only shows that the data are close to a curve chosen to be close to them; it is not an independent test of the 78% chaos claim. Similarly, the statement that only the moments of PDBR(f, s) with f = 0.78 match the experimental result 'supporting our choice of f' is a restatement of the fitting criterion. The headline meson '78% of chaos' therefore reduces to the fitted value by construction.
fitted input called prediction [Section III.B.2 (Theoretical meson spectra), set V paragraph]
"Set V displays a smooth NNSD, which can be very well fitted to a distorted Berry-Robnik distribution with f = 0 . 63 ± 0. 19 (also displayed in figure 9). Then we can conclude that the model by Vijande et al. gives a better account of the dynamical regime of the light meson spectrum."

The value f = 0.63 is fitted to the NNSD of set V itself, and the conclusion that the model is characterized by 63% chaos and gives a better account of the dynamical regime is based on that same fitted curve. No independent data or parameter-free reference is used to test the 63% fraction after the fit, so this numerical chaos claim is a fitted description of the same spectrum rather than a prediction validated against an external reference.

full rationale

The central experiment-versus-theory dichotomy is not circular: the distorted Wigner and Poisson references are parameter-free, generated from 1000 realizations with the same sequence lengths and local unfolding, and the baryon comparison (pDW = 0.82, pDP = 0.26) and the meson Wigner/Poisson comparison (pDW = 0.38, pDP = 0.13) involve no fitted quantities. The prior works [34] and [36] are self-citations, but the method is restated and rerun on updated data, so they are not load-bearing in a circular sense. The partial circularity is confined to the Berry-Robnik fractions: f = 0.78 for experimental mesons and f = 0.63 for model V are obtained by fitting PDBR to the same NNSD and then reused as null distributions and as moment-matching evidence. This makes the numerical percentage-of-chaos statements fitted descriptions rather than independently tested predictions. In addition, pDP = 0.13 does not justify the phrase 'safely incompatible' with Poisson, though that is a statistical-strength issue rather than a circularity; likewise, the experimental-resolution selection effect against closely spaced resonances is a substantive confound for the central claim but is not a circularity of the derivation chain.

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

The central statistical comparison depends on standard RMT/quantum chaos identifications and on the assumption that the finite-sequence distortion method yields unbiased reference distributions. The only genuinely fitted quantities are the Berry-Robnik f parameters; the analysis cutoffs and minimum sequence length are hand-chosen. No new physical entities are introduced.

free parameters (4)

Berry-Robnik regular fraction f (experimental mesons) = 0.78 ± 0.03
Fitted to the experimental meson NNSD by matching the distorted Berry-Robnik distribution to the data; used to claim 78% chaos and as null distribution for pDBR=0.65.
Berry-Robnik regular fraction f (model V mesons) = 0.63 ± 0.19
Fitted to the Vijande et al. meson spectrum NNSD; used to claim this model is intermediate with 63% chaos.
Energy cutoffs for experimental spectra = 2.2 GeV (baryons), 2.5 GeV (mesons)
Chosen by hand to define the low-lying spectra; changing these cutoffs changes the number of levels and could alter the NNSD.
Minimum sequence length = 3 levels
Sequences with two or fewer levels are discarded; this choice affects the number of spacings and the shape of the distorted references.

assumptions (4)

domain assumption BGS conjecture: chaotic time-reversal-invariant systems have GOE/Wigner NNSD; integrable systems have Poisson NNSD.
Interpretive framework used throughout Section II; it is standard but not proven for hadron spectra.
domain assumption Flavor SU(3) invariance allows dropping strangeness when defining pure symmetry sequences.
Section II, first paragraph; used to build J, I, P (and C) sequences.
ad hoc to paper Local unfolding with constant mean level density is valid over each short sequence, and the synthetic distorted references reproduce the same finite-size distortion as the data.
Section II, procedure and Fig. 1; the central methodological claim that the distorted Wigner/Poisson/BR distributions are the correct null distributions.
domain assumption K-S test validity for pooled spacings from many short independent sequences.
Section II, K-S test paragraph; pooled spacings are treated as i.i.d. samples despite within-sequence correlations.

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Cite this review

Pith. "Pith review of Spectral-statistics properties of the experimental and theoretical light baryon and meson spectra." pith.science (2026). https://pith.science/paper/PXTMKVGY

@misc{pith2026250118027,
  author       = {Pith},
  title        = {Pith review of: Spectral-statistics properties of the experimental and theoretical light baryon and meson spectra},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PXTMKVGY}},
  note         = {Machine review of arXiv:2501.18027}
}

read the original abstract

We compare the statistical fluctuation properties of the baryon and meson experimental mass spectra with those obtained from theoretical models (quark models and lattice QCD). We find that for the experimental spectra the statistical properties are close to those predicted by Random Matrix Theory for chaotic systems, while for the theoretical ones they are in general closer to those predicted for integrable systems and safely incompatible with those of chaotic systems. We stress the importance of the agreement of the fluctuation properties between experiment and theoretical models, as they determine the dynamical regime and the complexity of the real interactions. We emphasize the new statistical method we use, adapted for properly analyzing the fluctuation properties for very short spectral sequences.

Figures

Figures reproduced from arXiv: 2501.18027 by the authors.

**Figure 2.** FIG. 2: (color online.) NNSD for the experimental baryon [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 4.** FIG. 4: (color online.) Distributions of the [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

Figures from the paper (4 more)

**Figure 6.** Figure 6: FIG. 6: (color online.) NNSD for the experimental meson [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 7.** Figure 7: FIG. 7: (color online.) Moments of the NNSD, [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 8.** Figure 8: FIG. 8: (color online.) Distributions of the [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]

**Figure 9.** Figure 9: FIG. 9: (color online.) NNSDs for the theoretical meson [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

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After splitting th e spectrum in sequences with the same J, I and P , we have 53 levels distributed in 14 sequences (only sequences with mor e than two levels are considered)

Experimental spectrum We have taken all the resonance states from the Review of Particle Physics (RPP) [37] up to 2.2 GeV . After splitting th e spectrum in sequences with the same J, I and P , we have 53 levels distributed in 14 sequences (only sequences with mor e than two levels are considered). Fig. 2 shows the P (s) distribution for the experimental ...

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In this case we do not expect the effect of distortio n to be so noticeable as for the experimental spectrum, as the dimensions of the sequences are not so small

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