pith. sign in

arxiv: 2508.09763 · v3 · submitted 2025-08-13 · ✦ hep-ph

U-spin sum rules for two-body decays of bottom baryons

Si-Jia Wen , Wei-Chen Fu , Di Wang This is my paper

Pith reviewed 2026-05-18 23:11 UTC · model grok-4.3

classification ✦ hep-ph
keywords U-spin symmetrybottom baryonstwo-body decayssum ruleseffective Hamiltonianbranching fractionsCP asymmetry
0
0 comments X

The pith

The effective Hamiltonian for b decays vanishes under U-spin lowering operators, yielding sum rules for bottom baryon two-body decays.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper demonstrates that the effective Hamiltonian governing b quark decays is annihilated by the U-spin lowering operators U_-^n. This property directly produces sum rules that relate only b to d or only b to s transitions in bottom baryon decays. A newly defined operator S_b is shown to generate additional sum rules that mix the two transitions. Master formulas derived from these operators cover the main classes of two-body decays and lead to concrete relations among rates, decay parameters, and CP asymmetries.

Core claim

The effective Hamiltonian for b quark decay is zero under the U-spin lowering operators U_-^n, which permits derivation of U-spin sum rules involving only b to d or b to s transitions; a new operator S_b = U_+ + r U_3 - r^2 U_- generates sum rules involving both transitions. The proof that the Hamiltonian is zero under these operators is given, and master formulas are obtained for transitions such as b to c c-bar d/s, b to c u-bar d/s, b to u u-bar d/s, and b to u c-bar d/s.

What carries the argument

The annihilation of the effective Hamiltonian by U-spin lowering operators U_-^n together with the new mixing operator S_b, which together produce the master formulas for generating sum rules.

If this is right

  • Numerous explicit U-spin sum rules are obtained for the listed transition classes, providing relations among observable rates.
  • Some branching fractions are predicted from the sum rules under U-spin symmetry.
  • Rate and decay-parameter sum rules that hold beyond the strict U-spin limit are found and can be tested more precisely.
  • CP asymmetry relations for U-spin conjugate pairs are derived for the first time while retaining partial-wave amplitudes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The master formulas could be applied to guide experimental searches for unobserved decay modes at current and future flavor facilities.
  • Quantifying the size of U-spin breaking in these baryon modes might allow the sum rules to be used for improved extraction of CKM elements or strong phases.
  • The same operator annihilation technique may extend to other flavor symmetries or to three-body decays once suitable data become available.

Load-bearing premise

U-spin symmetry is a sufficiently good approximation for the two-body decays of bottom baryons considered here, with corrections small enough that the derived sum rules remain useful for phenomenology.

What would settle it

Precision measurements of branching fractions or CP asymmetries for a U-spin related pair of bottom baryon decay modes that deviate from the predicted sum rule by more than the expected size of U-spin breaking effects.

read the original abstract

$U$-spin symmetry, which reflects the symmetry between the down-type $d$ and $s$ quarks, is a powerful tool for analyzing heavy hadron weak decays. Motivated by recent experimental achievements in the bottom baryon sector, we study the $U$-spin sum rules for bottom baryon decays. The effective Hamiltonian for $b$ quark decay is zero under the $U$-spin lowering operators $U_-^n$, permitting us to derive $U$-spin sum rules involving only the $b\to d$ transition or $b\to s$ transition. Moreover, a new operator, $S_b=U_++rU_3-r^2U_-$, is proposed to generate $U$-spin sum rules involving both the $b\to d$ and $b\to s$ transitions. The proof that the effective Hamiltonian for $b$ quark decay is zero under $U_-^n$ and $S_b$ is presented. The master formulas for generating $U$-spin sum rules for the two-body decays of bottom baryons involving $b\to c\overline cd/s$, $b\to c\overline ud/s$, $b\to u\overline ud/s$, and $b\to u\overline cd/s$ transitions are derived. Numerous $U$-spin sum rules for the two-body decays of bottom baryons are obtained through these master formulas, which provide hints for new decay modes and enable the extraction of dynamical information. As a phenomenological analysis, some branching fractions are predicted according to $U$-spin symmetry. Several rate and decay parameter sum rules beyond the $U$-spin limit are found, providing a more precise test of flavor symmetry in the bottom baryon sector. Moreover, some $CP$ asymmetry relations for $U$-spin conjugate pairs in heavy baryon decays are derived for the first time by taking partial-wave amplitudes into account.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript claims that the effective Hamiltonian for b quark decay vanishes under the U-spin lowering operators U_-^n, allowing derivation of U-spin sum rules for b to d or b to s transitions in bottom baryon two-body decays. A new operator S_b = U_+ + r U_3 - r^2 U_- is proposed to generate sum rules involving both transitions. Master formulas are derived for transitions b to c bar c d/s, b to c bar u d/s, b to u bar u d/s, and b to u bar c d/s. These yield numerous sum rules, branching fraction predictions, rate and decay parameter sum rules beyond the U-spin limit, and CP asymmetry relations for U-spin conjugate pairs using partial-wave amplitudes.

Significance. This work provides a systematic framework for U-spin sum rules in bottom baryon decays, which is significant given recent experimental advances. The master formulas and explicit relations can aid in interpreting data, predicting new modes, and extracting dynamical information. The inclusion of partial-wave effects in CP relations is a valuable addition. The paper delivers concrete, testable predictions and relations that go beyond the symmetry limit.

major comments (2)
  1. [§2] The proof of the vanishing of the effective Hamiltonian under U_-^n and S_b is load-bearing for all subsequent results; the manuscript should include the explicit form of the Hamiltonian or the commutation relations used to establish this property to allow independent verification.
  2. [§4] In the phenomenological analysis, the predicted branching fractions (e.g., for specific modes listed in Table 1) are given without an accompanying estimate or discussion of U-spin breaking effects, which undermines the claim that these provide reliable hints for new decay modes.
minor comments (3)
  1. The notation for the U-spin operators and the parameter r in S_b should be introduced with more context in the introduction for readers unfamiliar with the formalism.
  2. A summary table collecting all the derived sum rules would improve the readability and utility of the results section.
  3. [CP asymmetry section] The definition of the partial-wave amplitudes A and B should be clarified to avoid ambiguity in the CP relations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the recommendation of minor revision. The comments are constructive, and we address them point by point below.

read point-by-point responses

  1. Referee: [§2] The proof of the vanishing of the effective Hamiltonian under U_-^n and S_b is load-bearing for all subsequent results; the manuscript should include the explicit form of the Hamiltonian or the commutation relations used to establish this property to allow independent verification.

    Authors: We agree that greater explicitness will aid independent verification. Although the manuscript states that the proof is presented, we will expand the relevant section to display the explicit form of the effective Hamiltonian for b-quark decays together with the commutation relations that establish its vanishing under U_-^n and S_b. This material will be added to §2 in the revised version. revision: yes

  2. Referee: [§4] In the phenomenological analysis, the predicted branching fractions (e.g., for specific modes listed in Table 1) are given without an accompanying estimate or discussion of U-spin breaking effects, which undermines the claim that these provide reliable hints for new decay modes.

    Authors: The referee correctly notes the absence of a breaking discussion. The predictions are obtained in the exact U-spin limit. In the revision we will insert a short paragraph that estimates the size of U-spin breaking from the known d–s mass difference and from analogous meson-decay studies (typically 10–20 %), and we will qualify the branching-fraction numbers as approximate hints rather than precise values. This addition will be placed in the phenomenological section. revision: yes

Circularity Check

0 steps flagged

Symmetry algebra derivation is self-contained with no reduction to inputs

full rationale

The paper derives U-spin sum rules from the algebraic property that the effective Hamiltonian for b decays satisfies U_-^n H_eff = 0 (and likewise for the constructed operator S_b). It explicitly states that the proof of this annihilation property is presented internally, based on the U-spin singlet nature of the b quark and the quark content of the transition operators. Master formulas for the listed transitions (b to c cbar d/s, etc.) follow directly as consequences in the symmetry limit. No data fitting occurs in the derivation, no self-citations load-bear the central algebraic steps, and no ansatz is smuggled in; the phenomenological branching-fraction predictions are separate applications that invoke external normalizations but leave the sum-rule relations themselves independent. The chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The derivation relies on the effective weak Hamiltonian for b decays and the approximate validity of U-spin symmetry in the bottom baryon sector; no new particles or forces are introduced.

axioms (2)
  • domain assumption The effective Hamiltonian for b quark decay vanishes under U-spin lowering operators U_-^n
    Stated in the abstract as the basis for deriving sum rules involving only b to d or b to s transitions.
  • domain assumption U-spin symmetry is a good approximation for two-body bottom baryon decays
    Underlying assumption allowing phenomenological predictions and sum rules beyond the exact limit.

pith-pipeline@v0.9.0 · 5876 in / 1460 out tokens · 27611 ms · 2026-05-18T23:11:46.698444+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

73 extracted references · 73 canonical work pages

  1. [1]

    U-spin sum rules generated by Sb 19

    b → ucd/s modes 18 D. U-spin sum rules generated by Sb 19

    work page

  2. [2]

    INTRODUCTION Nonleptonic decays of bottom baryons provide laboratories for studying strong and weak interactions in heavy- to-light baryonic transitions

    b → ucd/s modes 27 References 30 I. INTRODUCTION Nonleptonic decays of bottom baryons provide laboratories for studying strong and weak interactions in heavy- to-light baryonic transitions. In recent years, numerous measurements of bottom baryon nonleptonic decays have been performed at the Large Hadron Collider (LHC) [1–25], permitting us to study the dy...

    work page

  3. [3]

    VubV ∗ uq 2X i=1 C(u) i (µ)O(u) i (µ) ! + VcbV ∗ cq 2X i=1 C(c) i (µ)O(c) i (µ) !# + GF√ 2 X q=d,s

    · U− = − (BΣ+ 8 ). (8) It indicates that ⟨DαΣ+|U− Heff|Bγ b ⟩ = X ω ⟨Dn[B8]l m|Oj k|[Bb]i⟩ × [... − (Dα)n(BΣ+ 8 )l mH(cu)j k (Bγ b )i + ...] = ... − A (Bγ b → DαΣ+) ... . (9) Summing over all the contributions arising from T (Dα), T (Bβ 8 ), and T (Bγ b ), the sum of decay amplitudes generated by T for the Bγ b → DαBβ 8 mode is derived to be SumT [γ, α, β...

    work page

  4. [4]

    (C14) 16

    b → ccd/s modes SumU−[Ξ0 b, Λ+ c , D−] = −A(Λ0 b → Λ+ c D−) + A(Ξ0 b → Λ+ c D− s ) + A(Ξ0 b → Ξ+ c D−) = 0, (C1) SumU 2 −[Ξ0 b, Λ+ c , D−] = −2 A(Λ0 b → Λ+ c D− s ) + A(Λ0 b → Ξ+ c D−) − A(Ξ0 b → Ξ+ c D− s ) = 0, (C2) SumU−[Ξ− b , Σ0 c, D−] = √ 2A(Ξ− b → Ξ0 cD−) + A(Ξ− b → Σ0 cD− s ) = 0, (C3) SumU 2 −[Ξ− b , Σ0 c, D−] = 2 √ 2A(Ξ− b → Ξ0 cD− s ) + A(Ξ− b ...

    work page

  5. [5]

    (C34) 17

    b → cud/s modes SumU−[Ξ0 b, Ξ0 c, K0] = −A(Λ0 b → Ξ0 cK0) + 1√ 2 A(Ξ0 b → Ξ0 cπ0) − √ 3√ 2 A(Ξ0 b → Ξ0 cη8) = 0, (C15) SumU 2 −[Ξ0 b, Ξ0 c, K0] = − √ 2A(Λ0 b → Ξ0 cπ0) + √ 6A(Λ0 b → Ξ0 cη8) − 2A(Ξ0 b → Ξ0 cK 0 ) = 0, (C16) SumU−[Ξ0 b, Λ+ c , π−] = −A(Λ0 b → Λ+ c π−) + A(Ξ0 b → Λ+ c K −) + A(Ξ0 b → Ξ+ c π−) = 0, (C17) SumU 2 −[Ξ0 b, Λ+ c , π−] = −2A(Λ0 b →...

    work page

  6. [6]

    b → uud/s modes SumU−[Ξ− b , n, π−] = A(Ξ− b → nK −) + 1√ 2 − √ 3A(Ξ− b → Λ0π−) + A(Ξ− b → Σ0π−) = 0, (C35) SumU 2 −[Ξ− b , n, π−] = − √ 6A(Ξ− b → Λ0K −) − 2A(Ξ− b → Ξ0π−) + √ 2A(Ξ− b → Σ0K −) = 0, (C36) SumU 2 −[Ξ− b , Σ−, K0] = A(Ξ− b → Ξ−K0) + 1√ 2 A(Ξ− b → Σ−π0) − √ 3A(Ξ− b → Σ−η8) = 0, (C37) SumU 3 −[Ξ− b , Σ−, K0] = √ 2A(Ξ− b → Ξ−π0) − √ 6A(Ξ− b → Ξ...

    work page

  7. [7]

    b → ucd/s modes SumU−[Ξ− b , n, D−] = A(Ξ− b → nD− s ) + 1√ 2 − √ 3A(Ξ− b → Λ0D−) + A(Ξ− b → Σ0D−) = 0, (C43) SumU 2 −[Ξ− b , n, D−] = − √ 6A(Ξ− b → Λ0D− s ) − 2A(Ξ− b → Ξ0D−) + √ 2A(Ξ− b → Σ0D− s ) = 0, (C44) SumU−[Ξ0 b, P, D−] = −A(Λ0 b → P D−) + A(Ξ0 b → P D− s ) − A(Ξ0 b → Σ+D−) = 0, (C45) SumU 2 −[Ξ0 b, P, D−] = −2 A(Λ0 b → P D− s ) − A(Λ0 b → Σ+D−) ...

    work page

  8. [8]

    (D52) 21

    b → ccd/s modes SumS−[Ξ− b , Ξ0 c, D−] = r A(Ξ− b → Ξ0 cD−) − A(Ξ− b → Ξ0 cD− s ) = 0, (D1) SumS−[Ξ− b , Ξ0 c, D− s ] = A(Ξ− b → Ξ0 cD−) − A(Ξ− b → Ξ0 cD− s )r = 0, (D2) SumS−[Ξ0 b, Ξ0 c, D 0 ] = r A(Ξ0 b → Ξ0 cD 0 ) + A(Λ0 b → Ξ0 cD 0 )r = 0, (D3) SumS−[Λ0 b, Ξ0 c, D 0 ] = −A(Ξ0 b → Ξ0 cD 0 ) − A(Λ0 b → Ξ0 cD 0 )r = 0, (D4) SumS−[Ξ0 b, Ξ+ c , D−] = r A(Ξ...

    work page

  9. [9]

    b → cud/s modes SumS−[Λ0 b, Λ+ c , π−] = r A(Λ0 b → Λ+ c π−) − rA(Λ0 b → Λ+ c K −) − rA(Λ0 b → Ξ+ c π−) = 0, (D53) SumS−[Λ0 b, Λ+ c , K−] = A(Λ0 b → Λ+ c π−) − A(Ξ0 b → Λ+ c K −) − rA(Λ0 b → Λ+ c K −) = 0, (D54) SumS−[Λ0 b, Ξ+ c , K−] = A(Λ0 b → Λ+ c K −) + A(Λ0 b → Ξ+ c π−) − A(Ξ0 b → Ξ+ c K −) = 0, (D55) SumS−[Λ0 b, Ξ+ c , π−] = A(Λ0 b → Λ+ c π−) − A(Ξ0...

    work page

  10. [10]

    b → ucd/s modes SumS−[Ξ− b , Σ0, D−] = 1 2 r 2A(Ξ− b → Σ0D−) + √ 2rA(Ξ− b → Ξ0D−) − 2rA(Ξ− b → Σ0D− s ) = 0, (D147) SumS−[Ξ− b , Σ0, D− s ] = 1√ 2 A(Ξ− b → nD− s ) + A(Ξ− b → Σ0D−) − rA(Ξ− b → Σ0D− s ) = 0, (D148) SumS−[Ξ− b , n, D−] = −1 2 r2 2A(Ξ− b → nD− s ) − √ 6A(Ξ− b → Λ0D−) + √ 2A(Ξ− b → Σ0D−) = 0, (D149) SumS−[Ξ− b , n, D− s ] = rA(Ξ− b → nD− s ) ...

    work page

  11. [11]

    Aaij et al

    R. Aaij et al. [LHCb], JHEP 08, 039 (2018)

    work page 2018

  12. [12]

    Aaij et al

    R. Aaij et al. [LHCb], Phys. Lett. B 787, 124-133 (2018)

    work page 2018

  13. [13]

    Aaij et al

    R. Aaij et al. [LHCb], JHEP 03, 126 (2019)

    work page 2019

  14. [14]

    Aaij et al

    R. Aaij et al. [LHCb], JHEP 09, 028 (2019)

    work page 2019

  15. [15]

    Aaij et al

    R. Aaij et al. [LHCb], Phys. Rev. Lett. 124, no.11, 111802 (2020)

    work page 2020

  16. [16]

    Aaij et al

    R. Aaij et al. [LHCb], Phys. Rev. D 102, no.5, 051101 (2020)

    work page 2020

  17. [17]

    Aaij et al

    R. Aaij et al. [LHCb], JHEP 06, 110 (2020)

    work page 2020

  18. [18]

    Aaij et al

    R. Aaij et al. [LHCb], Phys. Rev. D 102, no.11, 112012 (2020)

    work page 2020

  19. [19]

    Aaij et al

    R. Aaij et al. [LHCb], Phys. Lett. B 815, 136172 (2021)

    work page 2021

  20. [20]

    Aaij et al

    R. Aaij et al. [LHCb], JHEP 05, 095 (2021)

    work page 2021

  21. [21]

    Aaij et al

    R. Aaij et al. [LHCb], Phys. Rev. D 104, no.5, 052010 (2021)

    work page 2021

  22. [22]

    Aaij et al

    R. Aaij et al. [LHCb], Phys. Rev. D 104, no.11, 112008 (2021)

    work page 2021

  23. [23]

    Aaij et al

    R. Aaij et al. [LHCb], JHEP 03, 153 (2022)

    work page 2022

  24. [24]

    Aaij et al

    R. Aaij et al. [LHCb], JHEP 07, 075 (2023)

    work page 2023

  25. [25]

    Aaij et al

    R. Aaij et al. [LHCb], Phys. Rev. D 108, no.7, 072002 (2023)

    work page 2023

  26. [26]

    Aaij et al

    R. Aaij et al. [LHCb], Eur. Phys. J. C 84, no.3, 237 (2024)

    work page 2024

  27. [27]

    Aaij et al

    R. Aaij et al. [LHCb], Eur. Phys. J. C 84, no.6, 575 (2024)

    work page 2024

  28. [28]

    Aaij et al

    R. Aaij et al. [LHCb], JHEP 07, 140 (2024)

    work page 2024

  29. [29]

    Aaij et al

    R. Aaij et al. [LHCb], Phys. Rev. D 110, no.3, L031104 (2024)

    work page 2024

  30. [30]

    Aaij et al

    R. Aaij et al. [LHCb], JHEP 08, 132 (2024)

    work page 2024

  31. [31]

    Aaij et al

    R. Aaij et al. [LHCb], Phys. Rev. Lett. 133, no.26, 261804 (2024)

    work page 2024

  32. [32]

    Aaij et al

    R. Aaij et al. [LHCb], Phys. Rev. Lett. 134, no.10, 101802 (2025)

    work page 2025

  33. [33]

    Aaij et al

    R. Aaij et al. [LHCb], Phys. Rev. D 111, no.9, 092004 (2025)

    work page 2025

  34. [34]

    Aaij et al

    R. Aaij et al. [LHCb], [arXiv:2501.12779 [hep-ex]]

    work page arXiv

  35. [35]

    Aaij et al

    R. Aaij et al. [LHCb], [arXiv:2507.10713 [hep-ex]]

    work page arXiv

  36. [36]

    Aaij et al

    R. Aaij et al. [LHCb], Nature (2025) [arXiv:2503.16954 [hep-ex]]

    work page arXiv 2025

  37. [37]

    J. H. Christenson, J. W. Cronin, V. L. Fitch and R. Turlay, Phys. Rev. Lett. 13, 138-140 (1964)

    work page 1964

  38. [38]

    Aubert et al

    B. Aubert et al. [BaBar], Phys. Rev. Lett. 86, 2515-2522 (2001)

    work page 2001

  39. [39]

    Abe et al

    K. Abe et al. [Belle], Phys. Rev. Lett. 87, 091802 (2001)

    work page 2001

  40. [40]

    Aaij et al

    R. Aaij et al. [LHCb], Phys. Rev. Lett. 122, no.21, 211803 (2019)

    work page 2019

  41. [41]

    J. J. Han, Y. Li, H. n. Li, Y. L. Shen, Z. J. Xiao and F. S. Yu, Eur. Phys. J. C 82, no.8, 686 (2022)

    work page 2022

  42. [42]

    J. J. Han, J. X. Yu, Y. Li, H. n. Li, J. P. Wang, Z. J. Xiao and F. S. Yu, Phys. Rev. Lett. 134, no.22, 221801 (2025)

    work page 2025

  43. [43]

    J. P. Wang and F. S. Yu, Chin. Phys. C 48, no.10, 101002 (2024)

    work page 2024

  44. [44]

    Z. D. Duan, J. P. Wang, R. H. Li, C. D. Lv and F. S. Yu, [arXiv:2412.20458 [hep-ph]]

    work page arXiv

  45. [45]

    J. J. Han, J. X. Yu, Y. Li, H. n. Li, J. P. Wang, Z. J. Xiao and F. S. Yu, [arXiv:2506.07197 [hep-ph]]

    work page arXiv

  46. [46]

    X. G. He and G. N. Li, Phys. Lett. B 750, 82-88 (2015)

    work page 2015

  47. [47]

    M. He, X. G. He and G. N. Li, Phys. Rev. D 92, no.3, 036010 (2015)

    work page 2015

  48. [48]

    X. G. He, Y. J. Shi and W. Wang, Eur. Phys. J. C 80, no.5, 359 (2020)

    work page 2020

  49. [49]

    S. Roy, R. Sinha and N. G. Deshpande, Phys. Rev. D 101, no.3, 036018 (2020)

    work page 2020

  50. [50]

    S. Roy, R. Sinha and N. G. Deshpande, Phys. Rev. D 102, no.5, 053007 (2020)

    work page 2020

  51. [51]

    Huang, Y

    F. Huang, Y. Xing and J. Xu, Eur. Phys. J. C 82, no.11, 1075 (2022)

    work page 2022

  52. [52]

    Gavrilova, Y

    M. Gavrilova, Y. Grossman and S. Schacht, JHEP 08, 278 (2022)

    work page 2022

  53. [53]

    W. H. Han, Y. Xing and J. Xu, Chin. Phys. C 48, no.5, 053106 (2024). 30

    work page 2024

  54. [54]

    W. Wang, Z. P. Xing and Z. X. Zhao, Phys. Rev. D 111, no.5, 053006 (2025)

    work page 2025

  55. [55]

    B. n. Zhang and D. Wang, Phys. Lett. B 868, 139674 (2025)

    work page 2025

  56. [56]

    Wang and W

    D. Wang and W. C. Fu, [arXiv:2503.22170 [hep-ph]]

    work page arXiv

  57. [57]

    X. G. He, C. W. Liu and J. Tandean, [arXiv:2503.24350 [hep-ph]]

    work page arXiv

  58. [58]

    Q. Chen, X. Wu, Z. P. Xing and R. Zhu, [arXiv:2504.01853 [hep-ph]]

    work page arXiv

  59. [59]

    Roy, [arXiv:2504.10891 [hep-ph]]

    S. Roy, [arXiv:2504.10891 [hep-ph]]

    work page arXiv

  60. [60]

    Wang, JHEP 05, 064 (2023)

    D. Wang, JHEP 05, 064 (2023)

    work page 2023

  61. [61]

    J. F. Luo and D. Wang, Eur. Phys. J. C 84, no.10, 1010 (2024)

    work page 2024

  62. [62]

    Wang, Eur

    D. Wang, Eur. Phys. J. C 84, no.9, 949 (2024)

    work page 2024

  63. [63]

    W. C. Fu, S. J. Wen and D. Wang, Eur. Phys. J. C 85, no.6, 610 (2025)

    work page 2025

  64. [64]

    Eckart, Rev

    C. Eckart, Rev. Mod. Phys. 2, 305 (1930). Google ScholarCrossref, CAS

    work page 1930

  65. [65]

    Wigner, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra (Academic, New York, 1959)

    E. Wigner, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra (Academic, New York, 1959)

    work page 1959

  66. [66]

    Grossman and D

    Y. Grossman and D. J. Robinson, JHEP 04, 067 (2013)

    work page 2013

  67. [67]

    D. A. Varshalovich, A. N. Moskalev and V. K. Khersonskii, ”Quantum Theory of Angular Momentum: Irreducible Tensors, Spherical Harmonics, Vector Coupling Coefficients, 3nj Symbols ,” World Scientific Publishing Company, 1988

    work page 1988

  68. [68]

    Navas et al

    S. Navas et al. [Particle Data Group], Phys. Rev. D 110, no.3, 030001 (2024)

    work page 2024

  69. [69]

    Buchalla, A

    G. Buchalla, A. J. Buras and M. E. Lautenbacher, Rev. Mod. Phys. 68, 1125 (1996)

    work page 1996

  70. [70]

    Beneke, G

    M. Beneke, G. Buchalla, M. Neubert and C. T. Sachrajda, Phys. Rev. Lett. 83, 1914 (1999)

    work page 1914

  71. [71]

    Beneke, G

    M. Beneke, G. Buchalla, M. Neubert and C. T. Sachrajda, Nucl. Phys. B 591, 313 (2000)

    work page 2000

  72. [72]

    Beneke and M

    M. Beneke and M. Neubert, Nucl. Phys. B 675, 333 (2003)

    work page 2003

  73. [73]

    D. Wang, C. P. Jia and F. S. Yu, JHEP 09, 126 (2021). 31

    work page 2021