Hadrons in group expansion

Hua-Xing Chen

REVIEW 4 major objections 3 minor 120 references

Hadrons in group expansion

T0 review · 4 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Ground-state baryons are mixtures of SU(4) flavor multiplets, with charmed baryons carrying a measurable 20S admixture.

desk verdict Useful SU(4) transition-matrix tabulation wrapped around a mass-mixing parametrization whose headline mixing fractions are calibrated inputs, not derived predictions, with one internal contradiction to fix. read the letter →

arxiv 2506.01272 v3 pith:WDL2BDK6 submitted 2025-06-02 hep-ph hep-thmath-phmath.MP

Hua-Xing Chen This is my paper

classification hep-phhep-thmath-phmath.MP

keywords flavorSU(4)baryonspectrumrepresentationmixingtransitionmatricescharmedbaryonsapproximatesymmetrygroupexpansionbreaking

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 argues that the mass spectrum of ground-state baryons, viewed under the severely broken flavor SU(4) symmetry of up, down, strange, and charm quarks, forces physical states to be mixtures of several flavor representations rather than pure members of one. Using systematically computed transition matrices between SU(4) multiplets and flavor-singlet mass Lagrangians, it fits the experimental baryon masses and extracts mixing fractions, most notably that the charmed baryons $\Sigma_c$, $\Xi'_c$, and $\Omega_c$ contain about 28% of the symmetric $\mathbf{20_S}$ multiplet, and that $\Xi_c$ contains about 10% of the SU(3) sextet. The broader idea, called group expansion, is that each broken symmetry dictates which representation admixtures appear, with subleading components characterized by the next multiplet in the decomposition. A sympathetic reader would care because these admixtures change the couplings and decay patterns of charmed and bottom baryons far beyond what pure multiplet assignments predict.

What carries the argument

The machinery is a set of transition matrices, objects such as $[T_\Lambda]$, $[T_\Delta]$, $D$, $F$, and $F_\Delta$, that express how a meson's quark--antiquark field acts on a baryon's three-quark state and redistributes it among the SU(4) representations $\mathbf{\bar{4}}_A$, $\mathbf{20_M}$, and $\mathbf{20_S}$. Contracting these matrices with meson fields and taking nonzero condensates $\langle M_0\rangle$, $\langle M_{15}\rangle$, $\langle M_8\rangle$ builds flavor-singlet mass Lagrangians; diagonalizing the resulting mass matrices yields the physical states as representation mixtures and, ultimately, the quoted mixing percentages.

What would settle it

Measure or compute the actual masses of the $\mathbf{20_S}$-plet baryons that the mass matrix assigns 2000/3000/7000 MeV, put them into the mixing equations of Sec. IV, and check whether the fitted 28% and 10% admixtures survive; alternatively, observe a decay or transition of $\Sigma_c/\Xi'_c/\Omega_c$ whose rate is forbidden for a pure $\mathbf{20_M}$ state but allowed at the quoted admixture level.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the measured masses of ground-state baryons cannot be accounted for within a single SU(4) flavor multiplet: the $\mathbf{20_M}$-plet baryons $\Sigma_c/\Xi'_c/\Omega_c$ mix with the $\mathbf{20_S}$-plet at the level $S_{15}\approx 492$ MeV, with $\Sigma_c$ about 72% $\mathbf{20_M}$ and 28% $\mathbf{20_S}$, and the SU(3) anti-triplet $\Xi_c$ mixes with the sextet at about 10%. The same mechanism, working downward through the symmetry chain $\mathrm{SU}(4)\supset\mathrm{SU}(3)\supset\mathrm{SU}(2)$, produces $\Lambda^0$--$\Sigma^0$ mixing from isospin breaking and makes $\Xi_c$--$\Xi'_c$ mixing automatic rather than an extra input. The paper further claims a general differential relation: the deviation of an approximate decuplet from the exact decuplet is characterized by the exact octet, $D(\mathbf{10})/D(\mathbf{8})=\text{const}$, which becomes exact in the symmetric limit.

Load-bearing premise

The quoted 72/28 and 90/10 fractions depend on a simplified mass matrix in which the masses of the $\mathbf{20_S}$ partners are put in by hand (2000/3000/7000 MeV) and the $\mathbf{\bar{4}}_A$ mixing term is dropped; if those inputs are wrong, the percentages change.

Editorial extensions

If this is right

If the 72/28 split is right, every coupling of $\Sigma_c$, $\Xi'_c$, and $\Omega_c$ to pions, kaons, and other baryons carries a $\mathbf{20_S}$ component of order 28%, so rates computed with pure $\mathbf{20_M}$ wave functions are off at the few-to-ten percent level.
The $\Xi_c$--$\Xi'_c$ mixing, equivalently the 90/10 $\mathbf{\bar{3}}_A/\mathbf{6_S}$ split, turns nominally suppressed $\Xi_c$ weak decays into accessible channels and shifts the corresponding semileptonic rates.
The fitted ratios $F_{15}/F_8\approx 7$--$10$ for both the $\mathbf{20_M}$ and $\mathbf{20_S}$ multiplets quantify how much stronger SU(4) breaking is than SU(3) breaking, giving a concrete hierarchy to compare with quark-mass ratios.
Mixing with the $\mathbf{20_S}$-plet shifts the masses of the charmed $\mathbf{20_M}$ baryons downward, so single-multiplet fits differ from the mixed fits; predictions for unobserved doubly charmed and bottom baryons inherit these shifts.
The group-expansion relation ties the breaking of one approximate symmetry to the next multiplet down the chain, so the same matrix construction can be repeated for other nearly degenerate multiplets.

Reading between the lines

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

A direct test would be to replace the hand-set 2000/3000/7000 MeV partner masses with measured or lattice masses and recompute the fit; the 28% and 10% fractions are input-dependent until that is done.
If the neglected $\mathbf{\bar{4}}_A$ mixing term from Eq. (26) is included, it will compete with the $\mathbf{20_S}$ admixture; the paper's own numbers therefore set an upper bound on how much of the $\Sigma_c$ mass shift can be attributed to the $\mathbf{20_S}$ component alone.
The group-expansion differential relation suggests a quantitative measure of symmetry-breaking distance between adjacent multiplets that could be carried over to other approximate symmetries, such as heavy-quark spin symmetry, by the same representation-mixing logic.
Because the mixing modifies meson-baryon couplings, precision data on charmed-baryon strong decays from ongoing facilities could extract the $\mathbf{20_S}$ fraction independently of the mass fit; agreement would convert the fitted fraction into a prediction.

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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

Useful tabulation, shaky headline. The genuinely new part is Sec. II and the supplemental matrix.nb: explicit SU(4) transition matrices (Eqs. 15–18) and the flavor-singlet Lagrangians (19–27). Those are checkable algebra and a reasonable reference for anyone doing SU(4) baryon wavefunctions. Credit where due.

The soft spots are in Sec. IV, and they are load-bearing. The 72% 20M / 28% 20S and 90% anti-3A / 10% 6S fractions come from a mass matrix whose 20S diagonal masses are fixed by hand to 2000/3000/7000 MeV, even though Sec. III fit the same charmed 20S states to about 2522/2646/2770 MeV. With off-diagonal S15 ~492 MeV, the mixing angle is sensitive to those assumed diagonal masses, so the fractions are calibrated inputs, not predictions. The matrix is truncated: Eq. (26), the 20M–anti-4A term that appears in the abstract, is dropped and Sec. VI concedes it is unexplored. And there is an internal contradiction: Eq. (51) sets S8 = 0, yet the fine-tuned solutions report S8 ≈ 53–70 MeV and the 28% fraction depends on it. That needs to be fixed or explained.

The mass formulas themselves are the usual Gell-Mann–Okubo style parametrization; they fit known masses because the parameters are solved from those same masses. That is not a crime, but it means the agreement in Eqs. (53)–(58) carries little weight. What is missing is an external observable—say a decay width or transition sensitive to the 20S admixture—that would discriminate 72/28 against a pure 20M assignment.

Bottom line: this paper is for readers who want the explicit SU(4) transition matrix tabulation and the Lagrangian building blocks. The numerical mixing percentages should not be quoted without a sensitivity analysis or an independent prediction. The paper deserves a serious referee: the formal core is reproducible, and the flaws are fixable in revision. I would accept it to peer review but come back with the S8 inconsistency and the hand-set masses as major points.

Referee Report

4 major / 3 minor

Summary. The paper develops a flavor-SU(4) group-theoretic framework for ground-state baryons. It lists transition matrices for the 15, 20_M, 20_S, and \bar{4}_A representations, builds flavor-singlet effective Lagrangians from meson-baryon-antibaryon combinations, and uses these to fit masses of the 20_S and 20_M baryon multiplets. The central claims are that the ground-state baryons are mixtures of different flavor representations, with the specific numerical estimates that \Sigma_c is approximately 72% 20_M and 28% 20_S, and that \Xi_c is approximately 90% of the SU(3) \bar{3}_A and 10% of the 6_S. The abstract additionally states that \Sigma_c/\Xi'_c/\Omega_c mix with the SU(4) \bar{4}_Aplet, but this mixing is not included in the body's mass-matrix analysis.

Significance. If the mixing percentages were robust, the paper would be a useful addition to the phenomenology of charmed baryons, since the admixtures would affect meson-baryon couplings and transition amplitudes. The group-theoretic identities in Eqs. (15)-(18) are a genuine and checkable contribution: the basis decomposition is explicit, and the accompanying Mathematica notebook supports reproducibility. The mass formulas for the 20_S multiplet in Sec. III are also useful and give reasonable fits to the known decuplet and singly charmed baryon masses. However, the numerical admixture claims are the headline result, and they depend on hand-set 20_S masses and on a truncated mass matrix; as presented, they are calibrated outputs rather than independent predictions. The stress-test concern in the reader's assessment lands directly on this point, and the manuscript itself concedes in Sec. VI that the \bar{4}_A contribution is unexplored and that the 20_S masses were fixed by hand.

major comments (4)

[Sec. IV, after Eq. (52)] The 20_S-plet diagonal masses used in the mixing matrix are set by hand to 2000 MeV for light baryons and 3000 MeV for charmed baryons (and 7000 MeV for bottom baryons), even though Sec. III's own fit gives charmed 20_S masses of approximately 2522, 2646, and 2770 MeV (Eqs. (30)-(34)). Because the off-diagonal coupling S15 is comparable to the resulting diagonal splittings, the mixing angle and therefore the quoted 72%/28% and 90%/10% fractions are highly sensitive to this arbitrary calibration. The paper should either use the internally consistent 20_S masses from Sec. III or provide a sensitivity study showing how the fractions change when the assumed 20_S masses are varied.
[Sec. IV, Eq. (51)] Equation (51) defines S8 as zero, but the subsequent fits in Eqs. (53), (55), and (58) report S8 values of approximately 70, 53, and 54 MeV. Since the mixing terms in Eq. (52) explicitly contain S8, the matrix actually diagonalized is not the one defined in Eq. (51). This internal inconsistency should be resolved by either treating S8 as a free parameter from the start or removing it from the mixing terms.
[Abstract and Sec. VI] The abstract states that \Sigma_c/\Xi'_c/\Omega_c are described as 20_M \oplus 20_S \oplus \bar{4}_A mixtures in SU(4), but Sec. IV explicitly neglects the 20_M-\bar{4}_A mixing term in Eq. (26) ("we take into account the latter and neglect the former"), and Sec. VI states that the \bar{4}_A contribution is "not explored". The abstract therefore attributes a derived result to the paper that the analysis does not support; it should be softened to describe only the 20_M-20_S mixing, or the \bar{4}_A mixing should be included and analyzed.
[Sec. IV, Eq. (55)] The charmed-baryon fit uses five free parameters (F8^M, D15^M, D8^M, S15, and S8) to reproduce the five masses in Eq. (55), so the agreement listed there is obtained by construction and is not an independent test of the model. The 72%/28% and 90%/10% percentages are outputs of this calibrated fit, not predictions. To make the mixing percentages credible, the paper should quantify their sensitivity to the assumed 20_S masses and, if possible, compare them with an independent observable such as a transition amplitude or decay width.

minor comments (3)

[Sec. VI, 'Group Expansion'] Several equations in this subsection (e.g., Eqs. (82)-(93)) are displayed with missing or undefined symbols, so the claimed relation between the approximate decuplet and the octet cannot be checked; please provide the actual Young diagram matrices or replace the symbolic placeholders with concrete expressions.
[Secs. III and IV, Eqs. (38) and (51)] The mixing parameters S'_{15}/S'_8 in Sec. III and S_{15}/S_8 in Sec. IV are introduced with very similar notation but are different objects; the paper should explicitly state their relationship (or lack thereof) to avoid confusion.
[Supplemental material] The Mathematica file 'matrix.nb' is referenced for the explicit transition matrices, but the paper does not describe the file's structure or how to verify the displayed identities; a short readme listing the main outputs would improve reproducibility.

Circularity Check

2 steps flagged · score 6.0 of 10

The headline mixing fractions (72/28 and 90/10) are outputs of a mass-matrix fit with hand-set 20S masses, and the Eq. (93) group-expansion relation is definitional, so the paper's central numerical claims reduce to calibrated inputs rather than independent predictions.

fitted input called prediction [Sec. IV, paragraph after Eq. (56), mass-matrix fit with 20M-20S mixing (Eqs. 52-56)]
"After appropriate fine-tuning, we obtain the following results: ... we can estimate the five parameters F^M_8 ≈ −190 MeV, D^M_15 ≈ 325 MeV, D^M_8 ≈ 44 MeV, S15 ≈ 492 MeV, and S8 ≈ 53 MeV, which yield the fitted mass values ... Specifically, we estimate that the Σc baryon contains approximately 72% of the 20M component and 28% of the 20S component."

The five parameters entering the mixing matrix are solved from the same baryon masses (Λc, Ξc, Σc, Ξ′c, Ωc) that the matrix then reproduces; the 72%/28% (and the analogous 90%/10% for Ξc) are the eigenvector components of this fitted matrix. They are therefore not predicted from the SU(4) group structure or from independent data; they are re-expressions of fitted parameters S15, D^M_8, and S8. The fit is further calibrated by hand-setting the 20S diagonal masses to 2000/3000/7000 MeV and dropping the anti-4A mixing term of Eq. (26), as Sec. VI later concedes, so the quoted fractions are dependent on these choices and are not a derived prediction.
self definitional [Sec. VI, 'Group Expansion' subsection, Eqs. (87)-(93)]
"we use the two reversed symbols and to denote the proportion of the symmetric and antisymmetric components in the total wave function, respectively: = cos θ and = sin θ ... This relation may no longer be merely approximate; rather, it becomes exact in the limit of exact isospin SU(2) symmetry, where its “differential” form can be written as D /D | = . (93)"

Equation (93) is presented as a result ('This relation ... becomes exact'), but the pictorial symbols appearing in D /D are introduced in Eq. (88) as the proportions cos θ and sin θ of the two components in the expansion |Δ+⟩ = ⊕. The quotient D /D therefore equals by construction the ratio of the antisymmetric to symmetric components; no independent content is added. The relation is a restatement of the definition of the pictorial symbols, not a derivation from symmetry breaking.

full rationale

The group-theoretic transition matrices of Sec. II and Appendix A are explicit tensor decompositions supplied with a Mathematica file, so that part is self-contained and not circular. The mass formulas in Sec. III are fits to experimental masses and are labeled as fitted values. Circularity is concentrated in the paper's headline quantitative claims. First, the vector of claims that Σc ~ 20M ⊕ 20S with 72%/28% and Ξc ~ anti-3A ⊕ 6S with 90%/10% is obtained by diagonalizing a mass matrix whose parameters are fine-tuned to the same spectrum, with the 20S diagonal masses set to 2000/3000/7000 MeV by hand and the anti-4A mixing term of Eq. (26) excluded. Sec. VI itself states that the 20S masses 'should in principle follow the expressions given in Eqs. (30)' but were 'fixed to 2000/3000 MeV to simplify the analysis,' and that the anti-4A contribution 'is not explored.' Thus the percentages are calibrated outputs of a truncated, hand-calibrated matrix, not first-principles predictions. Second, the 'differential' decuplet/octet relation in the Group Expansion (Eq. 93) is true by definition of the pictorial symbols as cos θ and sin θ, making it a notational restatement rather than a derived physical law. The paper also defines S8 = 0 in Eq. (51) but later fits S8 ≈ 53-70 MeV, and the abstract's anti-4A component is never computed; these are support and consistency gaps that reinforce the calibration nature of the numerical claims. No load-bearing self-citation or imported uniqueness theorem is present, so the score reflects definitional and fitted-input circularity rather than citation circularity.

Assumptions & free parameters 8 free parameters · 6 assumptions · 0 invented entities

The central mass formulas depend on a large set of fitted coupling and condensate parameters, plus two ad hoc modeling choices (fixed 20S masses and neglected anti-4A mixing). No new particles, forces, or conserved quantities are introduced. The group-theoretic decomposition itself is standard.

free parameters (8)

F^S_8 (20S-plet SU(3)-breaking parameter) = -253 MeV (light), -215 MeV (charmed), -210 MeV (bottom)
Fitted to mass splittings within the SU(4) 20S-plet; used in the 20S mass formulas Eq. (30).
F^S_15 (20S-plet SU(4)-breaking parameter) = approximately -2241 MeV
Estimated from light and charmed 20S masses; enters Eq. (30) and the ratio F15/F8 about 10.
F^M_8 (20M-plet SU(3)-breaking parameter) = -219 MeV (light), -146 or -190 MeV (charmed), -139 or -187 MeV (bottom)
Fitted to octet and charmed baryon masses in Eqs. (42), (53), (55), and (58).
D^M_8 (20M-plet D-type parameter) = 67 MeV (light), 80 or 44 MeV (charmed), 91 or 45 MeV (bottom)
Fitted to Lambda-Sigma and Xi_c-Xi'_c splittings; induces Xi_c-Xi'_c mixing via Eq. (43).
D^M_15 (20M-plet charm-breaking D parameter) = 108 or 325 MeV (charmed), 133 or 795 MeV (bottom)
Fitted to charmed baryon masses in Eqs. (46) and (55).
F^M_15 (20M-plet SU(4)-breaking parameter) = approximately -1585 MeV or -1368 MeV
Estimated from the 20M mass formulas; used for ratios in Eqs. (50) and (59).
S_8 and S_15 (20M-20S mixing parameters) = S8 about 70 MeV (light), 53 MeV (charmed), 54 MeV (bottom); S15 about 492 MeV (charmed), 1339 MeV (bottom)
Fitted when mixing with the 20S-plet is included; controls the 72%/28% composition claim.
Fixed 20S-plet masses in the mixing analysis = 2000 MeV (light), 3000 MeV (charmed), 7000 MeV (bottom)
Chosen by hand in Sec. IV to reduce parameters; the extracted mixing fractions depend on these values.

assumptions (6)

standard math Flavor SU(4) and SU(3) tensor decompositions and Young diagram rules are standard group theory.
Used in Sec. II and Appendix A to construct baryon fields and transition matrices.
domain assumption Physical baryon masses are dominated by flavor-singlet couplings of one meson condensate to baryon-antibaryon pairs; the chiral condensates M0, M15, M8, M3 acquire nonzero vacuum expectation values.
Eqs. (19)-(27) and Sec. II.C; this replaces an explicit QCD calculation.
domain assumption Ground-state baryon wavefunctions can be linearly decomposed into exact flavor-SU(4) representation components with coefficients determined by mass mixing.
Stated explicitly at the end of Sec. VI as the validity condition for group expansion, and used throughout Secs. III-V.
ad hoc to paper The anti-4A-plet contribution to 20M baryons is neglected in the fits.
Sec. IV: 'we take into account the latter and neglect the former'; this contradicts the abstract's inclusion of anti-4A in the headline formula.
ad hoc to paper The 20S-plet masses entering the 20M mixing calculation are set to 2000, 3000, and 7000 MeV by hand.
After Eq. (52), 'To simplify the model and reduce the number of free parameters, we set...' This choice changes the extracted S15 and S8 and hence the 72% fraction.
domain assumption Isospin SU(2) breaking is neglected in the main analysis, with M3 set to zero.
Eq. (28) and Sec. VI; Appendix B shows the SU(2)-breaking terms would introduce Lambda-Sigma mixing, but they are not included in the fits.

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Cite this review

Pith. "Pith review of Hadrons in group expansion." pith.science (2026). https://pith.science/paper/WDL2BDK6

@misc{pith2026250601272,
  author       = {Pith},
  title        = {Pith review of: Hadrons in group expansion},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WDL2BDK6}},
  note         = {Machine review of arXiv:2506.01272}
}

abstract

Various approximate symmetries exist in nature. For example, the flavor $SU(4)$ symmetry involving the $up/down/strange/charm$ quarks is severely broken, the flavor $SU(3)$ symmetry involving the $up/down/strange$ quarks is moderately broken, and the isospin $SU(2)$ symmetry involving the $up/down$ quarks is slightly broken. These broken symmetries are primarily governed by the strong interaction, making them an ideal platform for investigating the general behavior of approximate symmetries. To explore the application of the flavor $SU(4)$ group to ground-state baryons, we systematically calculate the transition matrices associated with various flavor $SU(4)$ representations as well as the matrices that describe their connections. These matrices are then employed to analyze the mass spectrum of ground-state baryons. Our results indicate that these states can be described as mixtures of various flavor representations, such as $\Sigma_c/\Xi_c^\prime/\Omega_c \sim \mathbf{20_M} \oplus \mathbf{20_S}\oplus \mathbf{\bar{4}_A}~[SU(4)]$, $\Xi_c/\Xi_c^\prime \sim \mathbf{\bar 3_A} \oplus \mathbf{6_S}~[SU(3)]$, $\Lambda^0/\Sigma^0 \sim \mathbf{1_A} \oplus \mathbf{3_S}~[SU(2)]$, where the subscripts $\mathbf{S}$, $\mathbf{A}$, and $\mathbf{M}$ denote the symmetric, antisymmetric, and mixed flavor wave functions, respectively. Our results also indicate that the flavor symmetries, as they break, necessitate the mixing of these flavor representations according to specific rules. For example, the approximate $SU(3)$ flavor decuplet, with one of its flavor components slightly differing from the other two, deviates from the exact $SU(3)$ flavor decuplet, and this deviation is characterized by the exact $SU(3)$ flavor octet.

Figures

Figures reproduced from arXiv: 2506.01272 by the authors.

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Works this paper leans on

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We explicitly define the matrices λN AB, MI ABC , and SP ABC , which serve as the building blocks for the decomposition
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We contract the four matrices ϵABCF , SP ABC , MI ABC , and MI BCA with three quarks qA, qB, and qC to generate the relevant tensor combinations
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This equation corresponds to the transformation involving three up quarks

As an illustrative example, we set A = B = C = E = N = 1, leading to the specific equation: ϵ11D1 × λ1 D1 = A1 1F × ϵ111F + B1 1P × SP 111 (A3) + C1 1I × MI 111 + D1 1I × MI 111 . This equation corresponds to the transformation involving three up quarks
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By solving this system of equations, we determine the four transition matrices: AN EF = −λN F E/3 , (A4) BN EP = 0 , (A5) CN EI = [ TΛ]N EI , (A6) DN EI = 0

We then generate a large number of such equations by randomly assigning values to the indices A, B, C, E, and N . By solving this system of equations, we determine the four transition matrices: AN EF = −λN F E/3 , (A4) BN EP = 0 , (A5) CN EI = [ TΛ]N EI , (A6) DN EI = 0 . (A7) Appendix B: Isospin SU (2) symmetry In this appendix we list the mass terms con...
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(21) and Eq

The 20 S-plet We use the flavor-singlet combinations given in Eq. (21) and Eq. (25) to derive the mass terms for the baryons belonging to the SU (4) flavor 20S-plet. These terms are purely diagonal: m∆++ = m∆ + F S 3 , m∆+ = m∆ + F S 3 /3 , m∆0 = m∆ − F S 3 /3 , m∆− = m∆ − F S 3 , mΣ∗+ = mΣ∗ + 2F S 3 /3 , mΣ∗0 = mΣ∗ , mΣ∗− = mΣ∗ − 2F S 3 /3 , mΞ∗0 = mΞ∗ +...
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(20), Eq

The 20 M-plet We use the flavor-singlet combinations given in Eq. (20), Eq. (23), and Eq. (24) to derive the mass terms for the baryons belonging to the SU (4) flavor 20M-plet. These terms include both diagonal and off-diagonal com- ponents. The diagonal terms are mN + = mN + F M 3 /2 + DM 3 /2 , mN 0 = mN − F M 3 /2 − DM 3 /2 , mΣ+ = mΣ + F M 3 , mΣ0 = m...
[7]

(19) and Eq

The ¯4A-plet We use the flavor-singlet combinations given in Eq. (19) and Eq. (22) to derive the mass terms for the baryons belonging to the SU (4) flavor ¯4A-plet. These terms are purely diagonal: mΛ0 A = mA − r 3 2 F A 15 , mΛ+ cA = mA + r 1 6 F A 15 − r 4 3 F A 8 , mΞ+ cA = mA + r 1 6 F A 15 + r 1 3 F A 8 − F A 3 , mΞ0 cA = mA + r 1 6 F A 15 + r 1 3 F ...
[8]

(26) cor- responds to the mixing between the 20M-plet and the 13 ¯4A-plet

Mixing terms The flavor-singlet combination given in Eq. (26) cor- responds to the mixing between the 20M-plet and the 13 ¯4A-plet. This combination gives rise to the following mixing terms, with the charge-conjugated terms omitted for simplicity: mΛAΣ = 2 A3 × ¯Λ0 AΣ0 , mΛAΛ = 2 A8 × ¯Λ0 AΛ0 , mΛcAΛc = r 32 9 A15 × ¯Λ+ cAΛ+ c + 2 3 A8 × ¯Λ+ cAΛ+ c , mΞcA...

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arXiv:hep-ph/9706531, doi:10.1103/PhysRevD. 57.2859

arXiv

Pith tools

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

[1] [1]

We explicitly define the matrices λN AB, MI ABC , and SP ABC , which serve as the building blocks for the decomposition

[2] [2]

We contract the four matrices ϵABCF , SP ABC , MI ABC , and MI BCA with three quarks qA, qB, and qC to generate the relevant tensor combinations

[3] [3]

This equation corresponds to the transformation involving three up quarks

As an illustrative example, we set A = B = C = E = N = 1, leading to the specific equation: ϵ11D1 × λ1 D1 = A1 1F × ϵ111F + B1 1P × SP 111 (A3) + C1 1I × MI 111 + D1 1I × MI 111 . This equation corresponds to the transformation involving three up quarks

[4] [4]

By solving this system of equations, we determine the four transition matrices: AN EF = −λN F E/3 , (A4) BN EP = 0 , (A5) CN EI = [ TΛ]N EI , (A6) DN EI = 0

We then generate a large number of such equations by randomly assigning values to the indices A, B, C, E, and N . By solving this system of equations, we determine the four transition matrices: AN EF = −λN F E/3 , (A4) BN EP = 0 , (A5) CN EI = [ TΛ]N EI , (A6) DN EI = 0 . (A7) Appendix B: Isospin SU (2) symmetry In this appendix we list the mass terms con...

[5] [5]

(21) and Eq

The 20 S-plet We use the flavor-singlet combinations given in Eq. (21) and Eq. (25) to derive the mass terms for the baryons belonging to the SU (4) flavor 20S-plet. These terms are purely diagonal: m∆++ = m∆ + F S 3 , m∆+ = m∆ + F S 3 /3 , m∆0 = m∆ − F S 3 /3 , m∆− = m∆ − F S 3 , mΣ∗+ = mΣ∗ + 2F S 3 /3 , mΣ∗0 = mΣ∗ , mΣ∗− = mΣ∗ − 2F S 3 /3 , mΞ∗0 = mΞ∗ +...

[6] [6]

(20), Eq

The 20 M-plet We use the flavor-singlet combinations given in Eq. (20), Eq. (23), and Eq. (24) to derive the mass terms for the baryons belonging to the SU (4) flavor 20M-plet. These terms include both diagonal and off-diagonal com- ponents. The diagonal terms are mN + = mN + F M 3 /2 + DM 3 /2 , mN 0 = mN − F M 3 /2 − DM 3 /2 , mΣ+ = mΣ + F M 3 , mΣ0 = m...

[7] [7]

(19) and Eq

The ¯4A-plet We use the flavor-singlet combinations given in Eq. (19) and Eq. (22) to derive the mass terms for the baryons belonging to the SU (4) flavor ¯4A-plet. These terms are purely diagonal: mΛ0 A = mA − r 3 2 F A 15 , mΛ+ cA = mA + r 1 6 F A 15 − r 4 3 F A 8 , mΞ+ cA = mA + r 1 6 F A 15 + r 1 3 F A 8 − F A 3 , mΞ0 cA = mA + r 1 6 F A 15 + r 1 3 F ...

[8] [8]

(26) cor- responds to the mixing between the 20M-plet and the 13 ¯4A-plet

Mixing terms The flavor-singlet combination given in Eq. (26) cor- responds to the mixing between the 20M-plet and the 13 ¯4A-plet. This combination gives rise to the following mixing terms, with the charge-conjugated terms omitted for simplicity: mΛAΣ = 2 A3 × ¯Λ0 AΣ0 , mΛAΛ = 2 A8 × ¯Λ0 AΛ0 , mΛcAΛc = r 32 9 A15 × ¯Λ+ cAΛ+ c + 2 3 A8 × ¯Λ+ cAΛ+ c , mΞcA...