Singly heavy Omega Baryon ($\Omega_c^0$ \& $\Omega_b^-$) Spectroscopy in the Relativistic Framework of Independent Quark Model

Bhoomika Pandya; Manan Shah; Rameshri V. Patel; Smruti Patel

REVIEW 3 major objections 7 minor 2 cited by

Singly heavy Omega Baryon ($\Omega_c^0$ \& $\Omega_b^-$) Spectroscopy in the Relativistic Framework of Independent Quark Model

T0 review · 3 major / 7 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper assigns concrete spin-parities to the newly seen excited states of Omega_c^0 and Omega_b^- using a single relativistic quark model fitted only to ground-state data.

desk verdict Solid IQM extension to Omega_c/b with useful tables, but the excited-state JP assignments are picked by eye from degenerate candidates, not derived by a quantified criterion. read the letter →

arxiv 2506.23594 v1 pith:GRYWA2RU submitted 2025-06-30 hep-ph

Rameshri V. Patel , Manan Shah , Smruti Patel , Bhoomika Pandya This is my paper · ORCID

classification hep-ph PACS 12.39.Ki14.20.Lq14.20.Mr

keywords independentquarkmodelsinglyheavybaryonsOmega_cbaryonOmega_bspectroscopyspin-parityassignmentsmagneticmomentsnon-leptonicweakdecays

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 asks whether one relativistic quark model can organize the spectra of the two singly heavy baryons $\Omega_c^0$ and $\Omega_b^-$, and answers yes: a parameter set fitted only to the spin-averaged ground-state masses generates the excited-state ladder, including the states seen by LHCb since 2017. On that basis it assigns spin-parities $\frac{5}{2}^-$, $\frac{3}{2}^+$, $\frac{3}{2}^+$, and $\frac{1}{2}^-$ to $\Omega_c(3050)$, $\Omega_c(3065)$, $\Omega_c(3120)$, and $\Omega_c(3185)$, leaves $\Omega_c(3000)$ as $\frac{1}{2}^-$ or $\frac{3}{2}^-$, and identifies the four new $\Omega_b$ states at 6316-6350 MeV as P-wave excitations. The same machinery produces magnetic moments, radiative widths, and non-leptonic weak decay branching ratios, turning the mass picture into several independently checkable predictions. The assignments matter because each spin-parity labels the orbital and spin pattern inside the baryon, which is a direct probe of the confining potential.

What carries the argument

The machinery is the Independent Quark Model with a Martin-like confining potential $V(r) = \frac{1+\gamma^0}{2}(\Lambda r^{0.1} + V_0)$, an equal mixture of scalar and vector Lorentz structures. Each quark obeys a Dirac equation in the baryon rest frame, and because of the special power $r^{0.1}$ the radial components reduce, via the dimensionless coordinate $\rho$, to the same ordinary differential equation as a Schr\"odinger equation; the Dirac energies are obtained by numerically solving that single equation. The spin-averaged mass is the sum of the three quark Dirac energies minus a center-of-mass correction. The degeneracy is then lifted by inter-quark interactions - the spin-spin term with fitted coupling $\sigma$, plus spin-orbit and tensor forces built from the confined gluon propagator (COGEP) - which together convert each spin-averaged level into the $J^P$ multiplet pattern the paper compares with experiment.

What would settle it

An independent spin-parity measurement of $\Omega_c(3050)$ would settle the central assignment: the model places it at $\frac{5}{2}^-$ (predicted $3.045 \pm 0.059$ GeV), and LHCb has already excluded $J = \frac{1}{2}$ at $2.2\sigma$; establishing $\frac{3}{2}^+$ would break the scheme. Likewise, measuring one of $\Omega_b(6316)$-$\Omega_b(6350)$ to have positive parity, or to be a 2S radial excitation rather than P-wave, would falsify the paper's identification of the four states.

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Extended reading notes

Core claim

The central claim is that a single three-body relativistic quark model describes both $\Omega_c^0$ and $\Omega_b^-$ with four free numbers - the potential depth $V_0$, strength $\Lambda$, center-of-mass correction $E_{CM}$, and the $j$-$j$ spin coupling $\sigma$ - fixed once each by matching the spin-averaged S-wave ground-state masses. With those inputs the model places the first orbital excitations at 2.944-3.045 GeV for $\Omega_c^0$, where the observed $\Omega_c(3000)$, $\Omega_c(3050)$, $\Omega_c(3065)$, $\Omega_c(3090)$, and $\Omega_c(3120)$ sit, and at 6.323-6.379 GeV for $\Omega_b^-$, bracketing the four narrow peaks $\Omega_b(6316)$, $\Omega_b(6330)$, $\Omega_b(6340)$, and $\Omega_b(6350)$. The proposed assignments $J^P = \frac{5}{2}^-$, $\frac{3}{2}^+$, $\frac{3}{2}^+$, $\frac{1}{2}^-$ for $\Omega_c(3050)$, $\Omega_c(3065)$, $\Omega_c(3120)$, and $\Omega_c(3185)$ respect the LHCb constraints that rule out $J = \frac{1}{2}$ for the first two states. The claim that the model is predictive rather than merely descriptive is tested on decays: the computed relative branching ratio for $\Omega_c^0 \to \Xi^0 \bar{K}^0$ is 1.62 against the measured $1.64 \pm 0.29$. The $\Omega_b$ states are presented as $L = 1$ P-wave excitations without individual spin assignments.

Load-bearing premise

The whole prediction rests on the assumption that the same confining potential, with the same four numbers fitted to the known ground states, continues to work for excited P-, D-, and F-wave states; the ground-state matches come from the fit, so they cannot be counted as evidence.

Editorial extensions

If this is right

The four $\Omega_c$ assignments - $\frac{5}{2}^-$, $\frac{3}{2}^+$, $\frac{3}{2}^+$, $\frac{1}{2}^-$ - are directly testable through angular analysis of $\Omega_c^0 \to \Xi_c^+ K^-$ decays at LHCb.
If the $\Omega_b$ quartet is the predicted P-wave multiplet, the states should carry negative parity, with the $\frac{5}{2}^-$ member near 6.38 GeV.
The radial excitation ladder ($\Omega_c$ 2S near 3.05 GeV, $\Omega_b$ 2S near 6.44 GeV) gives concrete mass targets for searches of higher $\Omega_c$ and $\Omega_b$ states.
The radiative widths of the $\frac{3}{2}^+ \to \frac{1}{2}^+$ transitions - about 1.17 keV for $\Omega_c^0$ and 0.0057 keV for $\Omega_b^-$ - set the scale for future electromagnetic measurements.
For $\Omega_b^-$ non-leptonic decays, $\Omega_b^- \to \Omega_c^0 D_s^-$ is predicted as the dominant channel with branching fraction near $12 \times 10^{-3}$, about five times the pion mode.

Reading between the lines

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

Should the assignments survive measurement, the broader upshot would be that a flavor-independent confining potential with one parameter set per baryon reliably orders the singly heavy spectrum - a result that would constrain the quark-diquark picture used for tetraquarks and pentaquarks.
The paper leaves the $\Omega_b$ quartet unassigned; a natural next step is to map the four predicted P-wave members onto the observed peaks by decay width or production rate, since the predicted ordering runs from $\frac{1}{2}^-$ at 6.323 GeV to $\frac{5}{2}^-$ at 6.379 GeV.
A direct extension of the same fitting protocol would apply it to $\Xi_c$ and $\Xi_b$ baryons using only their ground states, testing whether the same potential reproduces their excited spectra and whether the fitted parameters scale smoothly with the heavy-quark mass.
Because the paper's own 5% parameter variation yields mass uncertainties of $\pm$50-110 MeV - larger than the 10-20 MeV gaps between nearby predicted states - the framework's most reliable output is likely the ordering and spin-parity pattern of a multiplet rather than masses accurate enough to resolve each observed peak.

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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

Two things to know. This is a competent, readable application of the authors' Independent Quark Model to Omega_c and Omega_b spectroscopy, with a lot of numbers and extensive comparisons to other calculations. The soft spot is the headline: the spin-parity assignments for the excited Omega_c states are not forced by the model. They are selected by hand among several candidate rows in the mass tables.

What the paper does well: it extends the same group's earlier IQM baryon work to these two baryons, gives a clear formalism, and provides complete mass tables for S, P, D, F states plus magnetic moments, radiative widths, and weak-decay branching ratios. The Omega_b weak-decay predictions are straightforward and clearly labeled as model-dependent, and they are honest that the topological amplitude parameters come from ref [88] and form factors from ref [90]. The comparison with other quark models is thorough. That is real work, useful for people cataloging predictions for these states.

Where it is soft: the ground-state agreement is built in by fitting V0, Lambda, ECM, and sigma to the two ground-state masses. Four parameters, two constraints; the 5% variations in Tables I and II are variations around one arbitrary point, not a scan of the full solution manifold. When I look at Tables IV and V, the quoted JP assignments are not unique. For example, 3.073, 3.102, and 3.118 are all within about one sigma of the Omega_c(3067)-class masses; 3.118 and 3.113 both match Omega_c(3120); 3.182 and 3.179 both match Omega_c(3185). No chi-squared, likelihood, width estimate, or LHCb spin-preference input is used to choose one row over another. The claim that Omega_c(3050) is 5/2- is plausible but is basically proximity matching. The Omega_b section is more careful: they say the four states are likely P-wave and refrain from specific JP assignments. Good.

Is the paper worth engaging with? Yes, but with the right expectations. It is not a determination of quantum numbers; it is a model application that gives a plausible organizing scheme. The citations are extensive and the authors are transparent about what is fit and what is predicted. There is no fundamental incoherence, and the calculation is reproducible in principle from the equations given.

Recommendation: send to peer review. A serious referee should ask for a quantified assignment procedure, such as a chi-squared comparison over all candidate states, proper treatment of the parameter degeneracy, and explicit use of LHCb spin information where available. With that revision the paper would be a respectable contribution to heavy-baryon phenomenology. As written, I would not cite it as a source for the JP assignments, but I might cite the mass table in a comparison plot. I would not bring it to reading group unless someone there is actively working on Omega baryon assignments.

Referee Report

3 major / 7 minor

Summary. The paper applies the relativistic Independent Quark Model with a Martin-like potential (an equal admixture of scalar and vector confinement) to the singly heavy baryons Ω_c^0 and Ω_b^-. After fitting the model parameters to the two ground-state masses of each baryon, the authors compute S-, P-, D-, and F-wave spectra, propose spin-parity assignments for four recently observed Ω_c states, and identify the four new Ω_b states as P-wave excitations. They also compute magnetic moments, radiative decay widths, and non-leptonic weak decay branching ratios for both baryons. The central claims are the excited-state spectra and the specific J^P assignments for Ω_c(3050), Ω_c(3065), Ω_c(3120), and Ω_c(3185).

Significance. If the spectral predictions were robust, this would be a useful single-model description of two heavy-baryon families, generating concrete assignments for states whose quantum numbers remain experimentally unsettled, plus falsifiable predictions for higher excitations and decay channels. The paper's strengths are the breadth of computed observables and the comparison with many alternative approaches; the decay calculations follow established frameworks. However, the significance is limited because the ground-state masses are fit inputs, the parameter determination is underdetermined, and the excited-state assignments are not uniquely selected by the calculation. The model's independent predictive content therefore rests on the excited spectrum, moments, and decay rates, and on demonstrating that the assignments are stable once the fitting ambiguity is resolved.

major comments (3)

[Sec. II, Eqs. (1), (15)-(16); Tables I-II] The four parameters V0, Lambda, ECM, and sigma are fitted to two ground-state observables per baryon: the spin-averaged mass (Eq. (15)) and the 1/2^+ – 3/2^+ splitting (via Eq. (16)). This leaves a continuum of parameter sets that satisfy the fit, and the manuscript does not state any additional constraint (e.g., transfer from meson fits, a chi-square over more states, or a regularization). All excited-state masses in Tables III–VIII therefore depend on an arbitrary point in the solution manifold, and the '5% variation' shown in Tables I–II samples only that point, not the degeneracy. The authors must either specify how a unique fit is obtained or demonstrate that the qualitative spectral predictions and assignments are unchanged over the full acceptable parameter region.
[Sec. V (Conclusion) and Tables IV–V] The proposed spin-parity assignments are not uniquely determined by the mass tables. For example, Ω_c(3065) is within the quoted uncertainties of 12D3/2 (3.073 GeV), 2S3/2 (3.102 GeV), and 14D3/2 (3.118 GeV); Ω_c(3120) is consistent with both 14D3/2 (3.118 GeV) and 14D1/2 (3.113 GeV); and Ω_c(3185) could correspond to either 24P1/2 (3.182 GeV) or 24P3/2 (3.179 GeV). No explicit criterion (chi-square, LHCb spin-parity likelihoods, widths, production rates) is given to select the claimed 5/2^-, 3/2^+, 3/2^+, 1/2^- assignment. The authors should either adopt a well-defined selection rule or present these as allowed, rather than unique, assignments.
[Sec. II and Abstract] The agreement of the ground-state masses with experiment is presented as validation, but these masses are the inputs used to fix the model parameters: Eq. (1) defines the spin-averaged mass from the fitted Dirac energies and ECM, and Eq. (16) with fitted sigma reproduces the S-wave splitting. The ground-state agreement is therefore a consistency check by construction, not a prediction. The paper should state this explicitly and base its claims of predictive success on the excited-state masses, magnetic moments, and decay widths, which are not used in the fit.

minor comments (7)

[Eq. (30)] The transition magnetic moment is written as (2√2/3)(μ_u − μ_s), but Ω_c^0 is (css) and Ω_b^- is (bss); no u quark is present. The formula should involve μ_s and μ_c (or μ_b). Please correct and re-evaluate Table X if the numerical results used the displayed formula.
[Table IV, row 14P5/2] The experimental mass is listed as '3.050 ± 0.0001', which appears to be a typo; it should presumably be '3.050 ± 0.1'.
[Table VII, rows 32P1/2 and 34P1/2] The uncertainties for these rows are given as ±0.010, while neighboring rows use uncertainties of order ±0.099; this inconsistency should be corrected.
[Abstract/Introduction/Conclusion] The state Ω_c(3065) in the conclusion and abstract is called Ω_c(3067) in the introduction; please use a consistent naming convention.
[Eq. (28)] The formula for the effective quark mass is typeset ambiguously (the parentheses in the denominator are unclear). Please rewrite it so that the intended operation is unambiguous.
[Reference [90]] The reference title 'Restudy of the color- spectroscopy' is incomplete or garbled; please provide the full reference.
[Sec. III.B and Sec. IV] The non-leptonic decay calculations for Ω_c use the TDA parameters of Ref. [88] and the Ω_b decays use form factors from Ref. [90]; these are not independent tests of the IQM but rather applications of the model masses in established frameworks. This should be made clear when comparing with experiment.

Circularity Check

1 steps flagged · score 2.0 of 10

Minor circularity: the ground-state "prediction" is a fitted consistency check, while the excited-state masses and decay properties remain genuinely predictive model output.

fitted input called prediction [Sec. II (parameter fitting) and Sec. V (conclusion)]
"The parameters of the potential are determined by matching the theoretical spin-averaged mass, as given by Eq. (1), to the experimental spin-averaged mass of the S wave. ... Our predictions for the ground and first excited states of Ω0c show good agreement with experimental observations. ... Our approach successfully predicts experimental data by fixing model parameters (λ, V0, and σ)."

The four parameters V0, Λ, ECM, and σ are fitted to the two S-wave ground-state masses: the spin-averaged combination (M1/2 + 2M3/2)/3 fixes V0, Λ, and ECM jointly, and the 1/2+–3/2+ splitting fixes σ through Eq. (16). The subsequent agreement of the 1S 1/2+ and 1S 3/2+ masses with experiment is therefore enforced by construction; quoting it as a "prediction" and as evidence that the model "successfully predicts experimental data" is a fitted-input-called-prediction step. This is a labeling/validation issue rather than a collapse of the central claim: the excited S, P, D, and F masses shown in Tables III–VIII are computed from the same potential without being fitted to the excited-state data, so the central spectroscopy claim retains independent content.

full rationale

The paper's central claim is the assignment of spin-parities to the observed excited Ωc0 and Ωb− states, based on computed P-, D-, and F-wave masses. After the four potential and coupling parameters are fixed to the two ground-state masses, the excited-state masses follow from solving the Dirac/Schrödinger system with no additional fit; no equation in the manuscript reduces an excited-state mass to a ground-state input by construction, and no uniqueness theorem or load-bearing self-citation is invoked. The one clear circular element is the presentation of the ground-state agreement—which is exactly the fitting target—as a "prediction." The spin-parity assignments are chosen by matching computed masses to observed states without an explicit quantitative selection criterion among competing multiplets; this underdetermination is a model-selection weakness, not circularity. The magnetic moments, radiative widths, and non-leptonic decay branching ratios are computed from the model wavefunctions plus external inputs (topological parameters [88], form factors [90], CKM and decay constants); they are not defined to reproduce the experimental quantities they are compared with, so they do not create further circular steps. Self-citations [75–77] document methodological provenance rather than supply the derivation. Overall, the core spectroscopy is self-contained and predictive; the score is elevated only for the fitted ground state being relabeled as a prediction.

Assumptions & free parameters 11 free parameters · 6 assumptions · 0 invented entities

The ledger contains no new particles or forces; the model is phenomenological with 8 fitted parameters and several imported model assumptions (CGP, topological amplitudes, form factors).

free parameters (11)

V0: potential depth for Omega_c = -1.011 +/- 0.050 GeV
Fitted to reproduce the experimental spin-averaged ground-state mass of Omega_c (Eq. 15, Tables I-II).
Lambda: potential strength for Omega_c = 1.250 +/- 0.062 GeV^1.1
Fitted to ground-state S-wave mass of Omega_c.
ECM: center-of-mass correction for Omega_c = 0.236 +/- 0.012 GeV
Fitted parameter in Eq. (1); removes spurious c.m. motion, but its numerical value is chosen by fit to the ground state.
sigma: j-j coupling constant for Omega_c = 0.075 +/- 0.004 GeV^3
Determined by fitting to experimental data (Eq. 16).
V0: potential depth for Omega_b = -1.110 +/- 0.055 GeV
Fitted to the ground-state spin-averaged mass of Omega_b.
Lambda: potential strength for Omega_b = 1.450 +/- 0.072 GeV^1.1
Fitted to Omega_b ground state.
ECM: center-of-mass correction for Omega_b = 0.102 +/- 0.005 GeV
Fitted to Omega_b ground state.
sigma: j-j coupling constant for Omega_b = 0.069 +/- 0.003 GeV^3
Fitted to experimental data.
alpha_s(mu_0=1 GeV) = 0.6 = 0.6
Chosen coupling strength in Eq. (19) for spin-orbit and tensor interactions; affects P-, D-, F-state splittings.
CGP constants alpha1, alpha2, c0, c1, gamma = 0.036, 0.056, 0.1017 GeV, 0.1522 GeV, 0.0139
Confined gluon propagator parameters taken from ref. [81], not derived here.
MST_DA topological parameters from ref [88] = given in [88]
The Omega_c non-leptonic branching ratios use the topological parameters (c, c', sc, cd, etc.) from ref [88], not fixed by this paper.

assumptions (6)

domain assumption The Dirac equation for a single quark in a baryon with an average potential is a valid description of quark dynamics.
The whole methodology is built on this idea (Eq. 2), following Kobushkin and Ferreira (refs. [63, 64]).
ad hoc to paper The potential has the Martin-like form V(r) = (1+gamma^0)/2 (Lambda r^0.1 + V0) with an equal mixture of scalar and vector components.
The 0.1 power is a phenomenological choice from meson studies [68-74], not derived from QCD.
domain assumption The three-body system can be described by three independent single-quark Dirac equations with a single center-of-mass subtraction.
This neglects genuine three-body correlations and is only approximate for baryons.
domain assumption The spin-averaged mass equals the sum of quark Dirac energies minus ECM (Eq. 1).
Assumed rather than derived.
domain assumption Spin-spin, spin-orbit, and tensor interactions generated by a confined gluon propagator (COGEP) from ref [81] are valid for excited states.
The paper imports the CGP model and its constants from ref [81] without independent derivation.
standard math The standard Dirac-Schrodinger equivalence of Eqs. (6)-(12) holds for the Martin-like potential.
This is a standard transformation for the Dirac equation with scalar-vector potentials.

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Cite this review

Pith. "Pith review of Singly heavy Omega Baryon ($\Omega_c^0$ \& $\Omega_b^-$) Spectroscopy in the Relativistic Framework of Independent Quark Model." pith.science (2026). https://pith.science/paper/GRYWA2RU

@misc{pith2026250623594,
  author       = {Pith},
  title        = {Pith review of: Singly heavy Omega Baryon ($\Omega_c^0$ \& $\Omega_b^-$) Spectroscopy in the Relativistic Framework of Independent Quark Model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GRYWA2RU}},
  note         = {Machine review of arXiv:2506.23594}
}

abstract

The Independent Quark Model, formulated for a three-body system within a relativistic framework, is applied to singly heavy baryons $ \Omega_c^0 $ and $ \Omega_b^- $ to investigate their spectroscopic properties. A Martin-like potential with an equal mixture of scalar and vector components is employed, with potential parameters fitted using ground-state experimental data. The resulting mass spectra include both radial and orbital excitations. The spin-parity for the recently observed states $ \Omega_c^0(3000) $, $ \Omega_c^0(3050) $, $ \Omega_c^0(3067) $, $ \Omega_c^0(3120) $, and $ \Omega_c^0(3185) $ as well as possible spin assignments for the four newly observed excited states of $ \Omega_b^- $ ($\Omega_b^-(6316)$, $\Omega_b^-(6330)$, $\Omega_b^-(6340)$, and $\Omega_b^-(6350)$) are proposed. The magnetic moments of the ground and first excited states are also calculated, along with radiative decay widths and transition magnetic moments. The non-leptonic weak decays of $ \Omega_c^0 $ are analyzed, with decay widths and branching ratios computed and compared with experimental data to validate the predictive power of the model. The branching ratios for the non-leptonic decays of $\Omega_b^-$ are also predicted for future observations.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Phenomenology of Hypothetical Single-Top Hadronic States
hep-ph 2026-05 unverdicted novelty 5.0 of 10

QCD sum-rule calculations yield single-top baryon and meson masses near the top-quark mass, with a few channels slightly below the naive quark-sum threshold.
Semileptonic $\Omega_{b}^{*}\rightarrow\Omega_{c}^{*} \ell \bar{\nu}_{\ell}$ transition in QCD
hep-ph 2025-09 conditional novelty 5.0 of 10

The Omega_b* -> Omega_c* l nu semileptonic decay widths are predicted with QCD sum rules: about 2.5e-12 GeV for electron/muon channels and 0.71e-12 GeV for the tau channel, with R = 0.29.

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