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REVIEW 2 major objections 4 minor 73 references

Topological diagrams of $\Omega^0_c$ decays in the $SU(3)_F$ limit

T0 review · 2 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Omega_c^0 decays reduce to 30 independent SU(3) amplitudes

desk verdict The most complete SU(3) amplitude dictionary for Omega_c^0 decays to date, but the B8 basis count is asserted rather than proven and the paper's own reductions appear to give 17 independent amplitudes, not 20. read the letter →

arxiv 2608.12742 v1 pith:YS4TNONY submitted 2026-08-13 hep-ph

Ying-Xin Lai , Di Wang This is my paper · ORCID
classification hep-ph
keywords Omega_c^0baryoncharmedweakdecaysSU(3)flavorsymmetrytopologicalamplitudespenguindiagramsirreducibleisospinsumrulesKörner-Pati-Wootheorem
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 works out the complete, model-independent description of the weak decays of the charmed baryon $\Omega_c^0$ into a decuplet or octet baryon plus a pseudoscalar meson, assuming exact $SU(3)_F$ flavor symmetry. It lists every tree- and penguin-induced topological diagram and derives the linear relations that tie these diagrams to the $SU(3)$ irreducible amplitudes. The central result is that penguin-induced amplitudes never appear independently: each one is locked into a fixed combination with a tree amplitude, leaving 10 independent amplitudes for the decuplet channels and 20 for the octet channels. With a complete basis of this kind, measured branching fractions and CP asymmetries can be converted directly into dynamical parameters without quark-model assumptions.

What carries the argument

The central machinery is the tensor construction of decay amplitudes: every topological diagram is a contraction of the charmed-sextet tensor $(B_{c6})^{ij}$, the weak Hamiltonian tensor $(H^{(p)})^k_{ij}$, the meson tensor $M^i_j$, and the final baryon tensor $B_{10}^{ijk}$ or $(B_8)^i_j$ with Levi-Civita symbols. Completeness rests on the index-counting argument that all $5! = 120$ permutations of the five free indices in Eq. (12) are captured by 13 contractions, with analogous octet constructions in Eqs. (22)-(23). The $SU(3)$ irreducible amplitude basis of Eqs. (16) and (29) is the comparison object; the linear maps in Eqs. (17) and (30) connect the two bases, and CKM unitarity locks the penguin amplitudes into tree combinations through Eqs. (18)-(19) and (33)-(34).

What would settle it

Measure the ratio $\mathrm{Br}(\Omega_c^0\to\Sigma^{*+}K^-)/\mathrm{Br}(\Omega_c^0\to\Sigma^{*0}K^0_S)$ and check whether it equals 4, as predicted when the Körner-Pati-Woo theorem holds; a clear violation would show the basis or the theorem misses dynamics. Alternatively, exhibit an explicit SU(3)-invariant tensor contraction for these decays that is not among the listed ones, which would enlarge the amplitude basis.

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Extended reading notes

Core claim

Within the $SU(3)_F$ limit, this paper gives the full topological amplitude basis for $\Omega_c^0$ decays to a decuplet baryon plus a pseudoscalar meson and to an octet baryon plus a pseudoscalar meson. For the decuplet case, the amplitude is written as 13 tensor contractions of the charmed-sextet, weak Hamiltonian, meson, and decuplet tensors; the completeness argument is that these contractions account for all $5! = 120$ index permutations allowed by the symmetries of the sextet and decuplet. For octet final states, the required antisymmetry of the three-quark wave function forces a split into a symmetric octet $B_8^S$ and an antisymmetric octet $B_8^A$, each with its own set of diagrams. Comparing the topological basis with the $SU(3)$ irreducible basis, the paper derives linear maps and shows, through CKM unitarity, that the penguin-induced $3$ and $3'$ amplitudes enter only in fixed combinations with tree amplitudes, so the penguin diagrams collapse into the three combinations $A_8^{T+P}$, $A_9^{T+P}$, $A_{10}^{T+P}$ (and the five $B_{18}^{T+P},\ldots,B_{22}^{T+P}$ in the octet case). The result is a basis of 10 independent amplitudes for decuplet final states and 20 for octet final states, together with three isospin sum rules and branching-fraction and CP-asymmetry relations that test the Körner-Pati-Woo theorem.

Load-bearing premise

The whole construction stands on the claim that the listed diagrams are all there are: the paper counts 5! index contractions and takes that as exhaustive, but it does not prove that no other SU(3)-preserving contribution exists. If another term were possible, the 10- and 20-amplitude bases and every derived sum rule would be incomplete.

Editorial extensions

If this is right

  • Only 10 independent parameters describe all $\Omega_c^0\to B_{10}M$ decays and 20 describe all $\Omega_c^0\to B_8 M$ decays in the $SU(3)_F$ limit, so once enough channels are measured these parameters can be extracted without model input.
  • No penguin amplitude is separately measurable: any CP asymmetry must arise from the fixed tree-plus-penguin combinations $A_8^{T+P}$, $A_9^{T+P}$, $A_{10}^{T+P}$ and their octet analogues, so the number of strong phases controlling CP violation is small.
  • The isospin sum rules (35)-(37) are exact in the isospin limit and give cross-checks for future data on $\Omega_c^0$ decays.
  • The branching-fraction equalities (51)-(53) and (55)-(58), together with the decay-parameter equalities (59)-(63), give direct experimental tests of the Körner-Pati-Woo theorem.
  • The predicted Cabibbo ratio for $\mathrm{Br}(\Omega_c^0\to\Omega^-K^+)/\mathrm{Br}(\Omega_c^0\to\Omega^-\pi^+)$ is consistent with the currently measured value, supporting the dominance of the color-favored emitted diagram.

Reading between the lines

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

  • The same index-counting construction should apply to the other charmed sextet baryons, giving complete topological bases for their weak decays as well; the paper does not state this extension.
  • If a measured CP asymmetry in a singly Cabibbo-suppressed channel cannot be reproduced with the fixed tree-plus-penguin combinations, that would signal new physics or substantial $SU(3)_F$ breaking beyond the basis.
  • Because the paper shows rescattering triangle and bubble graphs feed exactly the amplitudes the Körner-Pati-Woo theorem sets to zero, failure of the branching relations (51)-(58) would provide a quantitative diagnostic of long-distance dynamics.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper constructs topological-diagram decompositions for Omega_c^0 -> B_10 M and Omega_c^0 -> B_8 M weak decays in the SU(3)_F limit. For the decuplet sector it lists 13 contraction structures, argues completeness by a 5! = 120 permutation count, maps them onto SU(3) irreducible amplitudes, and shows that all penguin-induced amplitudes are locked with three tree amplitudes through Eqs. (18)-(19). For the octet sector it constructs 33 symmetric and 27 antisymmetric three-index contractions, maps them onto 27 irreducible amplitudes, reduces the ten b_6 structures to five b'^6 structures in Eqs. (31)-(32), and asserts that 20 independent amplitudes remain. The paper then derives isospin sum rules, uses the framework to test the Koerner-Pati-Woo theorem, and compares the predicted ratio Br(Omega_c^0 -> Omega^- K^+)/Br(Omega_c^0 -> Omega^- pi^+) with LHCb data.

Significance. If the proposed basis is complete, the paper provides a useful model-independent SU(3) parameterization of Omega_c^0 decays, with explicit linear relations between topological amplitudes and SU(3) irreducible amplitudes that would be directly applicable to global fits. The decuplet sector is well supported: the contraction count is explicit, the mapping in Eq. (17) is one-to-one, and the penguin-locking relations in Eqs. (18)-(19) are transparent. The prediction Br(Omega_c^0 -> Omega^- K^+)/Br(Omega_c^0 -> Omega^- pi^+) ~ |V_us|^2/|V_ud|^2 is parameter-free and agrees with the LHCb measurement, and the isospin sum rules in Eqs. (35)-(37) are concrete and check out from the tables. The central weakness is the octet sector: the claimed number of 20 independent amplitudes is neither derived nor consistent with the reductions already present in the paper, and the completeness of the 33-plus-27 contraction list in Eqs. (22)-(23) is asserted without a counting argument. Because the completeness claim is the main advertised result, this issue is load-bearing.

major comments (2)
  1. [II.B, after Eq. (34)] The statement that 'there are 20 independent amplitudes' is not derived and is inconsistent with the paper's own equations. Equation (29) contains 7 b_15, 10 b_6, 5 b_3, and 5 b_3' coefficients. Equations (31)-(32) reduce the ten b_6 coefficients to five b'^6 coefficients. Since H(0)(3') = 0 in Eq. (14), the five b_3' coefficients associated with H(0)(3') do not contribute to physical amplitudes, and Eq. (33) combines the remaining five b_3 coefficients with the penguin b_3^P and b_3'^P coefficients into five fixed combinations. This gives 7 + 5 + 5 = 17 independent physical amplitudes, not 20. Please provide a complete rank computation of the physical amplitude space; if the true count is 20, identify explicitly which three additional combinations survive, and if the true count is 17, correct the paper and state the implications for global parameter extraction, since three of the quoted parameters would be redundant.
  2. [II.B, Eqs. (22)-(23)] Completeness of the octet contraction list is asserted but not demonstrated. Unlike the 5! = 120 counting argument given for Eq. (12), no analogous enumeration is provided for the 33 terms in Eq. (22) or the 27 terms in Eq. (23); the paper only lists the contractions and later states the final number of independent amplitudes. Because the central claim of the paper is that this is the complete topological amplitude basis for Omega_c^0 -> B_8 M decays, an explicit index-contraction count using the symmetries of (B^S_8)_{ijk}, (B^A_8)_{ijk}, (B_c6)_{ij}, H^k_{lm}, and M^i_m, or an explicit independent-set check, must be supplied. Please also state which parts of the third-rank octet construction are quoted from Refs. [41-43] and which are new in this paper.
minor comments (4)
  1. [Eq. (11)] The symmetrization statement contains a repeated term: the list should give the distinct permutations of the three decuplet indices, not repeat 'B^{jik}_{10}'.
  2. [Tables I-III and Sec. III] The symbols lambda_1, lambda_2, lambda_d, lambda_s, and lambda_b are used throughout the tables and in the text without definitions; please define them before their first use.
  3. [Eq. (60)] The definition of the partial-wave CP asymmetry, 'A^alpha_CP = alpha+ alpha /2', is ambiguous; it should read (alpha + bar{alpha})/2 with alpha and bar{alpha} the CP-conjugate decay parameters, and analogously for beta and gamma.
  4. [Sec. III, Eqs. (55)-(58)] The tables provide amplitudes for eta_8 and eta_1, while the results are quoted for the physical eta and eta'; the eta-eta' mixing convention from Eq. (10) should be used explicitly to show how the quoted relations for eta and eta' follow.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the amplitude decompositions are index-contraction ansätze with basis-change dictionaries, penguin locking follows from CKM unitarity, and the LHCb ratio is only a post-hoc consistency check.

full rationale

The derivation is self-contained and does not reduce to its inputs. The B10 amplitude Eq. (12) is an explicit list of all 5! index contractions (counted in Sec. II.A), and Eq. (16) is an independent SU(3)-irrep basis; Eq. (17) is a change of basis, not a prediction. The locking of penguin amplitudes in Eqs. (18)-(19) and (33)-(34) follows from the CKM relation H(0)(3):H(1)(3):H(1)(3') = -V_cb*V_ub : -V_cb*V_ub : -3V_cb*V_ub stated after Eq. (15), so those combinations are derived, not fitted. The B8 sector uses the spin-flavor split into B_S8/B_A8 and Eq. (26) from Refs. [41-43]; this is a parameter-free group-theoretic input whose stated assumptions (SU(3)_F, antisymmetry of the baryon wavefunction) do not include the Omega_c amplitudes, so under the review rules it is real evidence and does not create circularity. The phenomenological ratio Br(Omega_c->Omega^-K+)/Br(Omega_c->Omega^-pi+) ~ |V_us|^2/|V_ud|^2 (Eq. (40)) is compared with the LHCb value (Eq. (41)) after the fact; no parameter is fitted to that datum, so this is not a fitted input called a prediction. One non-circular completeness risk should be flagged: the paper asserts 'Ultimately, there are 20 independent amplitudes contributing to B_c6->B_8M decays' at the end of Sec. II.B without exhibiting the linear-algebra rank count, and Eqs. (22)-(23) have no 5!=120-style exhaustiveness check analogous to the B10 sector. That is a missing proof/correctness concern, not a circular reduction.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The paper introduces no fitted constants or new dynamical entities. It does rely on the topological-diagram framework developed in the authors' previous papers (Refs. [41-43]) and on the standard weak effective Hamiltonian; the central result is a dictionary between topological and irreducible amplitudes, so the ledger of assumptions is short.

free parameters (1)
  • Topological amplitude parameters A_i, B_i (tree and penguin combinations)
    The framework defines 10 claimed independent amplitudes for B10M and 20 claimed for B8M, but their magnitudes and strong phases are not determined in this paper. They are structural unknowns of the decomposition, not fitted constants.
assumptions (6)
  • domain assumption SU(3)_F symmetry is exact for the two-body charmed baryon decays considered.
    The title and every amplitude table are constructed in the SU(3)_F limit; SU(3) breaking and electromagnetic corrections are not estimated.
  • domain assumption The weak effective Hamiltonian in Eq. (1), with operators up to dimension six plus the chromomagnetic penguin, describes Omega_c^0 decays.
    Taken from Ref. [44]; the CKM unitarity argument equating H(0)(3), H(1)(3), and H(1)(3') depends on this operator basis.
  • domain assumption Every allowed topological diagram corresponds to one of the index contractions counted after Eq. (12).
    The completeness argument is the 5! = 120 contraction count; the symmetries of Bc6 and B10 reduce this to the listed 13 terms. This is a model assumption about the diagrammatic basis, not a theorem about QCD.
  • domain assumption The spin-flavor decomposition of the final octet into symmetric (BS8) and antisymmetric (BA8) tensor sets, taken from Refs. [41-43], exhausts the possible topological structures.
    Used to write Eqs. (22)-(24); if this decomposition misses structures, the B8M completeness claim fails. This is the load-bearing framework assumption.
  • domain assumption The Korner-Pati-Woo theorem applies to the two quarks emitted by the weak vertex when they enter the same final-state baryon.
    Invoked in Sec. III to derive Eqs. (50) and (54); the paper treats it as a testable assumption, and Ref. [47] argues it is violated by rescattering.
  • domain assumption The chiral Lagrangians in Eq. (43) and the closure relations Eq. (49) capture the u-, t-, and s-channel rescattering contributions.
    Used only for the qualitative rescattering discussion in Sec. III and Table IV, not for the central algebraic decomposition.

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Pith. "Pith review of Topological diagrams of $\Omega^0_c$ decays in the $SU(3)_F$ limit." pith.science (2026). https://pith.science/paper/YS4TNONY

@misc{pith2026260812742,
  author       = {Pith},
  title        = {Pith review of: Topological diagrams of $\Omega^0_c$ decays in the $SU(3)_F$ limit},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YS4TNONY}},
  note         = {Machine review of arXiv:2608.12742}
}
abstract

The $\Omega^0_c$ baryon is a unique charmed sextet baryon as it decays through weak interaction. In this work, we investigate the topological amplitudes of $\Omega_c^0$ decays in the $SU(3)_F$ limit. The tree- and penguin-induced diagrams contributing to $\Omega_c^0$ decays into decuplet and octet baryons are presented completely. The linear relations between the topological amplitudes and the $SU(3)$ irreducible amplitudes are derived via tensor analysis. Several isospin relations are obtained, and some relations are derived to test the K\"orner-Pati-Woo theorem.

Figures

Figures reproduced from arXiv: 2608.12742 by the authors.

Figure 1
Figure 1. FIG. 1: Topological diagrams contributing to charmed sextet baryon decays into a decuplet baryon and a pseudoscalar meson. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Topological diagrams contributing to [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: The [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗

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