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arxiv: 2606.07776 · v1 · pith:I7MKRZBHnew · submitted 2026-06-05 · ✦ hep-ph

B Meson Semi-Invisible Decays via Perturbative QCD

Han-Bing Liu , Ye Xing , Bin Luo This is my paper

Pith reviewed 2026-06-27 21:24 UTC · model grok-4.3

classification ✦ hep-ph
keywords B meson decaysdark sectorperturbative QCDbranching ratiosB-Mesogenesissemi-invisible decaysflavor symmetryform factors
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The pith

B meson semi-invisible decays to light baryons plus dark particles reach branching ratios of order 10^{-5}.

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

The paper calculates branching ratios for B meson decays into a light baryon and an invisible dark baryon using perturbative QCD factorization together with flavor symmetry relations. It applies the method to two B-Mesogenesis scenarios and obtains numerical results for the form factors B to octet baryon. The largest values appear for B^0 to Lambda psi and Bs^0 to Xi^0 psi in the Type-I scenario, at the level of 10^{-5}. These rates are presented as potentially observable at hadron colliders and B factories, thereby providing a channel to search for dark matter.

Core claim

Using the perturbative QCD approach combined with flavor symmetry analysis, the branching ratios for B meson decays to light baryons and dark baryons are computed within two B-Mesogenesis scenarios. A detailed discussion of the B to B8 form factors supports the evaluation of effective couplings. The numerical analysis yields branching ratios on the order of 10^{-5} for the channels B^0 to Lambda psi and Bs^0 to Xi^0 psi in the Type-I model.

What carries the argument

Perturbative QCD factorization of B to octet-baryon form factors combined with flavor-symmetry relations to fix the effective couplings for dark-sector final states.

If this is right

  • Branching ratios reach O(10^{-5}) for B^0 to Lambda psi and Bs^0 to Xi^0 psi in the Type-I model.
  • The same framework produces smaller but still sizable rates for other channels and for the Type-II scenario.
  • The derived B to B8 form factors enter directly into the numerical branching-ratio predictions.
  • Such semi-invisible modes are expected to be searchable at hadron colliders and B factories.

Where Pith is reading between the lines

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

  • Observation of these modes at the predicted level would constrain the coupling strengths between visible and dark baryons in B-Mesogenesis models.
  • Non-observation at current sensitivity would tighten the allowed parameter space for the dark baryon masses and interactions.
  • The form-factor treatment could be reused for analogous semi-invisible decays of other heavy mesons.
  • Relative rates between the two model types could serve as a diagnostic once data become available.

Load-bearing premise

The perturbative QCD factorization and flavor symmetry relations remain valid when applied to the dark-sector final states.

What would settle it

An experimental upper limit on the branching fraction of B^0 to Lambda plus missing energy that lies well below 10^{-6} would contradict the calculated rate.

Figures

Figures reproduced from arXiv: 2606.07776 by Bin Luo, Han-Bing Liu, Ye Xing.

Figure 1
Figure 1. Figure 1: FIG. 1: Typical leading-order Feynman diagrams for the transition [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: The [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: The [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: The form factors [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: The dark baryon mass [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: The dark baryon mass [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
read the original abstract

This paper focuses on the dark sector decay processes of $B$ mesons ($B\to \mathcal{B}_8\ +$ invisible). Using the perturbative QCD (pQCD) approach combined with flavor symmetry analysis, we calculate the branching ratios for decays from $B$ mesons into light baryons and dark baryons within two distinct $B$-Mesogenesis scenarios. A detailed discussions of the form factor $B\to \mathcal{B}_8$ are presented. Based on the derived form factors and effective couplings, we then reach the final numerical analysis. The results show that the branching ratios are sizable, especially for $B^0\to \Lambda\psi$ and $B_s^0\to\Xi^0\psi$ in Type-I model, with values on the order of $\mathcal{O}(10^{-5})$ . Such processes are expected to facilitate the search for dark matter at hadron colliders and B factories.

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

0 major / 3 minor

Summary. The manuscript calculates branching ratios for B meson semi-invisible decays (B → ℬ₈ + invisible) in two B-Mesogenesis scenarios by combining perturbative QCD factorization with SU(3) flavor symmetry to obtain B → ℬ₈ form factors and effective couplings to the dark baryon ψ. It reports that the branching ratios are sizable, reaching O(10^{-5}) especially for B⁰ → Λψ and B_s⁰ → Ξ⁰ψ in the Type-I model.

Significance. If the pQCD factorization and flavor-symmetry relations remain valid for the dark-sector final states, the O(10^{-5}) predictions would supply concrete targets for dark-matter searches at B factories and hadron colliders.

minor comments (3)
  1. [Abstract] Abstract: the claim that branching ratios reach O(10^{-5}) is stated without any quoted central values, uncertainties, or input parameters, making it impossible to judge the numerical robustness from the abstract alone.
  2. The manuscript should supply at least one explicit form-factor expression (e.g., the leading-twist expression used for B → Λ or B_s → Ξ) together with the numerical inputs adopted for the effective couplings.
  3. No error estimates or variation ranges are mentioned for the final branching-ratio numbers; these should be added to the numerical-analysis section.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work on B meson semi-invisible decays using pQCD and flavor symmetry, and for recommending minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation applies standard pQCD factorization to B→B8 transitions and flavor symmetry to obtain form factors and couplings for the dark baryon final states. Branching ratios are computed numerically from these derived quantities in the two B-Mesogenesis scenarios. No step reduces by construction to a fitted input renamed as prediction, no self-citation is load-bearing for the central result, and no ansatz or uniqueness claim is smuggled via prior author work. The calculation chain is self-contained against external benchmarks and does not exhibit any of the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies insufficient detail to enumerate free parameters, axioms, or invented entities.

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Reference graph

Works this paper leans on

64 extracted references · 51 canonical work pages · 14 internal anchors

  1. [1]

    For the non-perturbative input wave functions, we adopt the light b aryon light cone distribution amplitude (LCDA) up to twist-6 and the leading-order B-meson LCDA

    Following the standard pQCD procedure, we can perturbatively calculate the hard kernels correspon ding to these three diagrams. For the non-perturbative input wave functions, we adopt the light b aryon light cone distribution amplitude (LCDA) up to twist-6 and the leading-order B-meson LCDA. After performing the convolution of the hard kernels with the wa...

  2. [2]

    large” and “small

    Usually, they can be constrained by LHC experiments, which provide upper 7 TABLE II: Form factors for the decay channels in the type-I/-II model at q2 = 0 (in units of 10 − 2). The uncertainties arise from the variation of Λ QCD and the wave function parameters. this work LCSR[25] LCSR[23, 24, 51] twist(3) twist(3-6) a1 twist(3) twist(3-6) B+ → p F (d) B→...

  3. [3]

    + (ξ− 5 − φ − 5 +ψ 0 5 )x3 + (ξ+ 5 − φ + 5 − ψ 0 5 )(1 − 2x3)]. (A9) • Twist-6 LCDAs 14 V6(xi) =2[φ 0 6 +φ + 6 (1 − 3x3)], A6(xi) =2(x2 − x1)φ − 6, T6(xi) =2[φ 0 6 + 1 2 (φ − 6 − φ + 6 )(1 − 3x3)], (A10) For Λ baryon, • Twist-3 LCDAs V1(xi) =120x1x2x3(x1 − x2) ( 21 √ 6 4 φ 11 − 7 √ 6 4 φ 10 ) , A1(xi) =120x1x2x3 [ −fΛ + 7 √ 6 4 (φ 11 +φ 10)(x1 +x2) − 14 √...

  4. [4]

    + 20m2 Bφ 0 3x1x2x3x4η ) ˆPL +mBmΣ + ( − 8φ ′ 4x1x2x3 +x4(40φ ′ 3x1x2x3 +x1(− 1 +x2 − x3 +x4)ξ0 4 +(x3 +x4)(x2ξ0 4 − x3ξ0 4 +ξ′ 4 − x4ξ′ 4)η) ) ̸ ¯n ̸n ˆPL ])] , (B5) Mc′ = ΛQCDCFm2 BηφB 288 √ 3ηπ E(tc) ∫ 1 0 da2da4da5 ∫ 1/ Λ QCD 0 b3db3b4db4h(x1,x 3,x 4,b 1,b 3,b 4) [ 8mΣ + ( m2 Σ +(x3 +x4)(φ 0 6 − 12φ 0 4x2x3 − 3φ 0 5x4 + 60φ 0 3x2x3x4 + 6ψ 0 4x4(− 1 +x...

  5. [5]

    + 40m2 Bφ 0 3x1x2x3x4η ) ˆPL + ( 2m2 Σ +(x3 +x4)(ψ 5(1 +x2 − x3 − x4) +x4(−φ 0 5 + 20φ 0 3x2x3 +4ψ 0 4(− 1 +x2 − x3 +x4))) − 40m2 Bφ 0 3x1x2x3x4η +mBmΣ +(x2 − x3)(8x4ξ0 4 − ξ0 5 +ξ′ 5)(x1 + (x3 +x4)η) ) ̸ ¯n ̸n ˆPL ])] . (B6) 19 The hard functions hi(xi, bi)s are obtained by performing the Fourier transformation fr om mo- mentum space to impact-parameter ...

  6. [6]

    − 2x2x3ψ 0 4 } +mN (x1 − x2)(4x3ξ0 4 +ξ0 5) ] + Λ QCDCFm2 BmN ΦB 72 √ 3πη E(tc)b3b4h(x1,x 3,x 4,b 1,b 3,b 4) × [ − 12η2m2 Bsx2x3ψ 0 4 − 3ηmBsmN (− 2 +ηx2) ( 4x2 3ξ′0 4 − 4x3(x1ξ0 4 − x2ξ0 4 +ξ′0 4 ) + 2x3φ ′0 5 − x1ξ0 5 +x2ξ0 5 +x1ξ′0 5 +x2ξ′0 5 ) + 2m2 N (− 1 +ηx2) [ 60x1x2x3φ 0 3 − 12x1x2φ 0 4 − 6x3ψ 0 4 + 6x1x3ψ 0 4 + 6x2 3ψ 0 4 − 6x2x3ψ 0 4 − 3x3φ 0 5...

  7. [7]

    − 2x2x3ψ 0 4 } +mN (x1 − x2)(4x3ξ0 4 +ξ0 5) ] + √ 3 Λ QCDCFm2 Bm2 Bsη 36π E(tc)b3b4h(x1,x 3,x 4,b 1,b 3,b 4)x3ΦB × [ 20ηmBsx1x2(− 2 +ηx2)φ ′0 3 + 2mNx2(− 1 +ηx2)ψ 0 4 ]} . (B14) F4 = ∫ 1 0 dx1dx3dx4 ∫ 1/ Λ QCD 0 db1db3db4 { √ 6 Λ QCDCFm2 BmBsΦB 72π E(ta)b1b4h(x1,x 4,b 1,b 4) 21 × [ 20η2m2 Bsx1x2x3φ 0 3 +ηmBsmN [ − 8x1x2(− 1 +ηx2)φ ′0 4 +x3 ( 40x1x2(− 1 +η...

  8. [8]

    +φ 0 5 ) − 2x2ψ 0 5 ]] + √ 6 Λ QCDCFm2 BmBsΦB 144π E(tb)b3b4h(x1,x 3,x 4,b 1,b 3,b 4) × [ − 20η2m2 Bsx1x2x3φ 0 3 + 4η2mBsmNx2x3(x1 − x2)ξ0 4 − m2 N (− 1 +ηx2) { x3 [ − 4(5x1x2φ 0 3 − 2x2ψ 0

  9. [9]

    +φ 0 5 ] − 2x2ψ 0 5 }] + √ 6 Λ QCDCFm2 BmBsΦB 144π E(tc)b3b4h(x1,x 3,x 4,b 1,b 3,b 4) × [ 20η2m2 Bsx1x2x3φ 0 3 − 2ηmBsmN (− 2 +ηx2) ( − 40x1x2x3φ ′0 3 + 8x1x2φ ′0 4 − (x1 +x2)x3(ξ0 4 +ξ′0 4 ) ) +m2 N (− 1 +ηx2) [ x3 ( − 4(5x1x2φ 0 3 − 2x2ψ 0

  10. [10]

    +φ 0 5 ) − 2x2ψ 0 5 ]]} . (B15)

  11. [11]

    G. Elor, M. Escudero and A. Nelson, Phys. Rev. D 99 (2019) no.3, 035031 doi:10.1103/PhysRevD.99.035031 [arXiv:1810.00880 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.99.035031 2019

  12. [13]

    Elahi, G

    F. Elahi, G. Elor and R. McGehee, Phys. Rev. D 105 (2022) no.5, 055024 doi:10.1103/PhysRevD.105.055024 [arXiv:2109.09751 [hep-ph]]

    work page doi:10.1103/physrevd.105.055024 2022

  13. [14]

    A. E. Nelson and H. Xiao, Phys. Rev. D 100 (2019) no.7, 075002 doi:10.1103/PhysRevD.100.075002 [arXiv:1901.08141 [hep-ph]]

    work page doi:10.1103/physrevd.100.075002 2019

  14. [15]

    Mir´ o, M

    C. Mir´ o, M. Escudero and M. Nebot, Phys. Rev. D 110 (2024) no.11, 115033 doi:10.1103/PhysRevD.110.115033 [arXiv:2410.13936 [hep-ph]]

    work page doi:10.1103/physrevd.110.115033 2024

  15. [16]

    Zheng, J

    Y. Zheng, J. N. Ding, D. H. Li, L. Y. Li, C. D. L¨ u and F. S. Yu, Chin. Phys. C 48 (2024) no.8, 083109 doi:10.1088/1674-1137/ad4afa [arXiv:2404.04337 [hep-ph]]

    work page doi:10.1088/1674-1137/ad4afa 2024

  16. [17]

    Khodjamirian, B

    A. Khodjamirian, B. Meli´ c and Y. M. Wang, Eur. Phys. J. ST 233 (2024) no.2, 271-298 doi:10.1140/epjs/s11734-023-01046-6 [arXiv:2311.08700 [hep-ph ]]

    work page doi:10.1140/epjs/s11734-023-01046-6 2024

  17. [18]

    A. Lenz, M. L. Piscopo and A. V. Rusov, JHEP 01 (2023), 004 doi:10.1007/JHEP01(2023)004 [arXiv:2208.02643 [hep-ph]]

    work page doi:10.1007/jhep01(2023)004 2023

  18. [19]

    Azatov, M

    A. Azatov, M. Vanvlasselaer and W. Yin, JHEP 10 (2021), 043 doi:10.1007/JHEP10(2021)043 [arXiv:2106.14913 [hep-ph]]

    work page doi:10.1007/jhep10(2021)043 2021

  19. [20]

    Bodeker and W

    D. Bodeker and W. Buchmuller, Rev. Mod. Phys. 93 (2021) no.3, 3 doi:10.1103/RevModPhys.93.035004 [arXiv:2009.07294 [hep-ph]]

    work page doi:10.1103/revmodphys.93.035004 2021

  20. [21]

    Alonso- ´Alvarez, G

    G. Alonso- ´Alvarez, G. Elor, A. E. Nelson and H. Xiao, JHEP 03 (2020), 046 doi:10.1007/JHEP03(2020)046 [arXiv:1907.10612 [hep-ph]]

    work page doi:10.1007/jhep03(2020)046 2020

  21. [22]

    Baryogenesis from neutron-dark matter oscillations

    T. Bringmann, J. M. Cline and J. M. Cornell, Phys. Rev. D 99 (2019) no.3, 035024 22 doi:10.1103/PhysRevD.99.035024 [arXiv:1810.08215 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.99.035024 2019

  22. [23]

    P. H. Gu, M. Lindner, U. Sarkar and X. Zhang, Phys. Rev. D 83 (2011), 055008 doi:10.1103/PhysRevD.83.055008 [arXiv:1009.2690 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.83.055008 2011

  23. [24]

    J. R. Ellis, M. K. Gaillard and D. V. Nanopoulos, Phys. Lett. B 80 (1979), 360 [erratum: Phys. Lett. B 82 (1979), 464] doi:10.1016/0370-2693(79)91190-0

    work page doi:10.1016/0370-2693(79)91190-0 1979

  24. [25]

    Recent Progress in Baryogenesis

    A. Riotto and M. Trodden, Ann. Rev. Nucl. Part. Sci. 49 (1999), 35-75 doi:10.1146/annurev.nucl.49.1.35 [arXiv:hep-ph/9901362 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1146/annurev.nucl.49.1.35 1999

  25. [26]

    Ablikim et al

    M. Ablikim et al. [BESIII], [arXiv:2505.22140 [hep-ex]]

    arXiv

  26. [27]

    X. Ai, W. Altmannshofer, P. Athron, X. Bai, L. Calibbi, L. Cao, Y. Che, C. Chen, J. Y. Chen and L. Chen, et al. Chin. Phys. C 49 (2025) no.10, 103003 doi:10.1088/1674-1137/adf1f0 [arXiv:2412.1 9743 [hep-ex]]

    work page doi:10.1088/1674-1137/adf1f0 2025

  27. [28]

    J. P. Lees et al. [BaBar], Phys. Rev. D 111 (2025) no.3, L031101 doi:10.1103/PhysRevD.111.L031101 [arXiv:2412.06950 [hep-ex]]

    work page doi:10.1103/physrevd.111.l031101 2025

  28. [29]

    J. P. Lees et al. [BaBar], Phys. Rev. Lett. 131 (2023) no.20, 201801 doi:10.1103/PhysRevLett.131.201801 [arXiv:2306.08490 [hep-ex]]

    work page doi:10.1103/physrevlett.131.201801 2023

  29. [30]

    J. P. Lees et al. [BaBar], Phys. Rev. D 107 (2023) no.9, 092001 doi:10.1103/PhysRevD.107.092001 [arXiv:2302.00208 [hep-ex]]

    work page doi:10.1103/physrevd.107.092001 2023

  30. [31]

    Shi [BESIII], PoS EPS-HEP2021 (2022), 663 doi:10.22323/1.398.0663

    X. Shi [BESIII], PoS EPS-HEP2021 (2022), 663 doi:10.22323/1.398.0663

    work page doi:10.22323/1.398.0663 2022

  31. [32]

    Hadjivasiliou et al

    C. Hadjivasiliou et al. [Belle], Phys. Rev. D 105 (2022) no.5, L051101 doi:10.1103/PhysRevD.105.L051101 [arXiv:2110.14086 [hep-ex]]

    work page doi:10.1103/physrevd.105.l051101 2022

  32. [33]

    Khodjamirian and M

    A. Khodjamirian and M. Wald, Phys. Lett. B 834 (2022), 137434 doi:10.1016/j.physletb.2022.137434 [arXiv:2206.11601 [hep-ph]]

    work page doi:10.1016/j.physletb.2022.137434 2022

  33. [34]

    Boushmelev and M

    A. Boushmelev and M. Wald, Phys. Rev. D 109 (2024) no.5, 055049 doi:10.1103/PhysRevD.109.055049 [arXiv:2311.13482 [hep-ph]]

    work page doi:10.1103/physrevd.109.055049 2024

  34. [35]

    Elor and A

    G. Elor and A. W. M. Guerrera, JHEP 02 (2023), 100 doi:10.1007/JHEP02(2023)100 [arXiv:2211.10553 [hep-ph]]

    work page doi:10.1007/jhep02(2023)100 2023

  35. [36]

    Biswas, A

    A. Biswas, A. Khodjamirian and A. Mohamed, [arXiv:2603.01723 [he p-ph]]

    Pith/arXiv arXiv

  36. [38]

    Y. J. Shi, Y. Xing and Z. P. Xing, Eur. Phys. J. C 84 (2024) no.3, 306 doi:10.1140/epjc/s10052-024- 12663-3 [arXiv:2401.14120 [hep-ph]]

    work page doi:10.1140/epjc/s10052-024- 2024

  37. [39]

    L. Y. Li, C. D. L¨ u, J. Wang and Y. B. Wei, Phys. Rev. D 109 (2024) no.11, 116012 doi:10.1103/PhysRevD.109.116012 [arXiv:2401.11978 [hep-ph]]

    work page doi:10.1103/physrevd.109.116012 2024

  38. [40]

    Y. Xing, Y. J. Shi and X. H. Hu, Phys. Rev. D 112 (2025) no.11, 116018 doi:10.1103/xr47-p5n2 [arXiv:2508.05181 [hep-ph]]

    work page doi:10.1103/xr47-p5n2 2025

  39. [41]

    Alonso- ´Alvarez, G

    G. Alonso- ´Alvarez, G. Elor and M. Escudero, Phys. Rev. D 104 (2021) no.3, 035028 doi:10.1103/PhysRevD.104.035028 [arXiv:2101.02706 [hep-ph]]

    work page doi:10.1103/physrevd.104.035028 2021

  40. [42]

    Xing and Z

    Y. Xing and Z. P. Xing, Chin. Phys. C 43 (2019) no.7, 073103 doi:10.1088/1674-1137/43/7/073103 [arXiv:1903.04255 [hep-ph]]

    work page doi:10.1088/1674-1137/43/7/073103 2019

  41. [43]

    J. J. Han, Y. Li, H. n. Li, Y. L. Shen, Z. J. Xiao and F. S. Yu, Eur. Phys. J. C 82 (2022) no.8, 686 doi:10.1140/epjc/s10052-022-10642-0 [arXiv:2202.04804 [hep-ph ]]

    work page doi:10.1140/epjc/s10052-022-10642-0 2022

  42. [44]

    X. G. He, T. Li, X. Q. Li and Y. M. Wang, Phys. Rev. D 74 (2006), 034026 doi:10.1103/PhysRevD.74.034026 [arXiv:hep-ph/0606025 [hep-ph]]. 23

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.74.034026 2006

  43. [45]

    R. H. Li, C. D. Lu, W. Wang and X. X. Wang, Phys. Rev. D 79 (2009), 014013 doi:10.1103/PhysRevD.79.014013 [arXiv:0811.2648 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.79.014013 2009

  44. [46]

    Z. Rui, C. Q. Zhang, J. M. Li and M. K. Jia, Phys. Rev. D 106 (2022) no.5, 053005 doi:10.1103/PhysRevD.106.053005 [arXiv:2206.04501 [hep-ph]]

    work page doi:10.1103/physrevd.106.053005 2022

  45. [47]

    Ou-Yang, R

    J. Ou-Yang, R. H. Li and S. H. Zhou, Phys. Rev. D 112 (2025) no.5, 056005 doi:10.1103/vl97-y4ql [arXiv:2506.14675 [hep-ph]]

    work page doi:10.1103/vl97-y4ql 2025

  46. [48]

    Chang, D

    Q. Chang, D. H. Yao and X. Liu, Eur. Phys. J. C 85 (2025) no.3, 292 doi:10.1140/epjc/s10052-025- 13939-y [arXiv:2501.01075 [hep-ph]]

    work page doi:10.1140/epjc/s10052-025- 2025

  47. [49]

    C. Q. Zhang, J. Sun, Z. P. Xing and R. L. Zhu, Phys. Rev. D 111 (2025) no.11, 113003 doi:10.1103/6g56- v496 [arXiv:2501.00512 [hep-ph]]

    work page doi:10.1103/6g56- 2025

  48. [50]

    J. L. Ren, M. Q. Li, X. Liu, Z. T. Zou, Y. Li and Z. J. Xiao, Eur. Ph ys. J. C 84 (2024) no.4, 358 doi:10.1140/epjc/s10052-024-12702-z [arXiv:2311.16824 [hep-ph ]]

    work page doi:10.1140/epjc/s10052-024-12702-z 2024

  49. [51]

    Chai and S

    J. Chai and S. Cheng, JHEP 06 (2025), 229 doi:10.1007/JHEP06(2025)229 [arXiv:2501.08783 [hep- ph]]

    work page doi:10.1007/jhep06(2025)229 2025

  50. [52]

    H. n. Li and G. F. Sterman, Nucl. Phys. B 381 (1992), 129-140 doi:10.1016/0550-3213(92)90643-P

    work page doi:10.1016/0550-3213(92)90643-p 1992

  51. [53]

    H. n. Li and H. L. Yu, Phys. Rev. Lett. 74 (1995), 4388-4391 doi:10.1103/PhysRevLett.74.4388 [arXiv:hep-ph/9409313 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.74.4388 1995

  52. [54]

    H. n. Li, C. D. Lu and F. S. Yu, Phys. Rev. D 86 (2012), 036012 doi:10.1103/PhysRevD.86.036012 [arXiv:1203.3120 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.86.036012 2012

  53. [55]

    M. J. Savage and M. B. Wise, Phys. Rev. D 39 (1989), 3346 [erratum: Phys. Rev. D 40 (1989), 3127] doi:10.1103/PhysRevD.39.3346

    work page doi:10.1103/physrevd.39.3346 1989

  54. [56]

    C. W. Chiang, M. Gronau, J. L. Rosner and D. A. Suprun, Phys. Rev. D 70 (2004), 034020 doi:10.1103/PhysRevD.70.034020 [arXiv:hep-ph/0404073 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.70.034020 2004

  55. [57]

    N. Li, Y. Xing and X. H. Hu, Eur. Phys. J. C 83 (2023) no.11, 1013 doi:10.1140/epjc/s10052-023- 12188-1 [arXiv:2303.08008 [hep-ph]]

    work page doi:10.1140/epjc/s10052-023- 2023

  56. [58]

    Xing, Eur

    Y. Xing, Eur. Phys. J. C 80 (2020) no.1, 57 doi:10.1140/epjc/s10052-020-7625-3 [arXiv:1910 .11593 [hep-ph]]

    work page doi:10.1140/epjc/s10052-020-7625-3 2020

  57. [59]

    Y. J. Shi, W. Wang, Y. Xing and J. Xu, Eur. Phys. J. C 78 (2018) no.1, 56 doi:10.1140/epjc/s10052- 018-5532-7 [arXiv:1712.03830 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1140/epjc/s10052- 2018

  58. [60]

    Model-independent description of $B\to \pi l\nu$ decays and a determination of $|V_{ub}|$

    C. Bourrely, I. Caprini and L. Lellouch, Phys. Rev. D 79 (2009), 013008 [erratum: Phys. Rev. D 82 (2010), 099902] doi:10.1103/PhysRevD.82.099902 [arXiv:0807.2722 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.82.099902 2009

  59. [61]

    M. A. Abri, N. Hajirasouliha and K. Azizi, [arXiv:2605.13701 [hep-ph ]]

    Pith/arXiv arXiv

  60. [62]

    I. D. King and C. T. Sachrajda, Nucl. Phys. B 279 (1987), 785-803 doi:10.1016/0550-3213(87)90019-8

    work page doi:10.1016/0550-3213(87)90019-8 1987

  61. [63]

    Braun, R

    V. Braun, R. J. Fries, N. Mahnke and E. Stein, Nucl. Phys. B 589 (2000), 381-409 [erratum: Nucl. Phys. B 607 (2001), 433-433] doi:10.1016/S0550-3213(00)00516-2 [arXiv:he p-ph/0007279 [hep-ph]]

    work page doi:10.1016/s0550-3213(00)00516-2 2000

  62. [64]

    G. S. Bali et al. [RQCD], Eur. Phys. J. A 55 (2019) no.7, 116 doi:10.1140/epja/i2019-12803-6 [arXiv:1903.12590 [hep-lat]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1140/epja/i2019-12803-6 2019

  63. [65]

    Y. L. Liu, C. Y. Cui and M. Q. Huang, Phys. Rev. D 89 (2014) no.3, 035005 doi:10.1103/PhysRevD.89.035005 [arXiv:1311.5960 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.89.035005 2014

  64. [66]

    A. Ali, G. Kramer, Y. Li, C. D. Lu, Y. L. Shen, W. Wang and Y. M. Wa ng, Phys. Rev. D 76 (2007), 074018 doi:10.1103/PhysRevD.76.074018 [arXiv:hep-ph/0703162 [he p-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.76.074018 2007