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REVIEW 3 major objections 4 minor 117 references

Improved global determination of two-meson distribution amplitudes from multi-body $B$ decays

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper shows that adding one free constant per meson pair makes the two-meson distribution amplitudes in B decays convergent and universal, and the updated fit reproduces the measured B_s->K*0 Kbar*0 polarization fraction.

desk verdict First transverse Kπ moments and an improved PQCD global fit are real, but the headline f0(Bs→K*Kbar*) 'prediction' is a fit output and the N_P parametrization's constancy remains untested. read the letter →

arxiv 2501.15150 v2 pith:JPTKCOER submitted 2025-01-25 hep-ph

Da-Cheng Yan , Hsiang-nan Li , Zhou Rui , Zhen-Jun Xiao , Ya Li This is my paper · ORCID
classification hep-ph
keywords two-mesondistributionamplitudesperturbativeQCDBmesondecaysGegenbauermomentspolarizationfractionsCPasymmetryglobalfit
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

Multi-body B meson decays, such as B -> V P3 -> P1 P2 P3, are described in perturbative QCD by factoring pair production into two-meson distribution amplitudes (DAs) whose shapes are controlled by Gegenbauer moments. This paper claims that the earlier disagreement between two ways of parametrizing the P-wave resonance in these DAs, time-like form factors versus Breit-Wigner propagators, can be cured by one momentum-independent constant N_P per meson pair (pi pi, K pi, KK). With those constants included in a global fit to measured branching ratios and polarization fractions, the extracted Gegenbauer moments become smaller and the expansion converges, supporting the universality of the two-meson DAs across different decay channels. The flagship consequence is the prediction f0($B_s^{0}$ -> K*0 Kbar*0) = 28.$2^{{+8.8}}$_{-9.5}%, which now matches the measured value, whereas earlier PQCD analyses disagreed. A reader should care because this is a step toward a predictive, data-driven description of multi-body B decays, a major LHCb and Belle-II program.

What carries the argument

The central objects are the two-meson distribution amplitudes (DAs) for the pairs pi pi, K pi, and KK, expanded in Gegenbauer polynomials with moments a_2^rho, a_{1,2}^{K*}, a_2^phi, and related twist-3 coefficients. They are normalized by time-like form factors, which for the narrow resonances K* and phi use relativistic Breit-Wigner line shapes and for the broad rho use the Gounaris-Sakurai model with rho-omega mixing and excited states. The new ingredient is the factor N_P in Eq. (48), a momentum-independent constant for each pair that absorbs the mismatch between the form-factor and Breit-Wigner descriptions of the P-wave resonance. The argument is carried by a leading-order PQCD factorization formula whose squared amplitudes are expanded as polynomials in the Gegenbauer moments; the moments and N_P are then determined by a standard nonlinear least-$chi^{2}$ fit to the measured branching ratios and polarization fractions.

What would settle it

Measure the branching ratio and polarization of B_s -> K*+ K*- (or another channel predicted in Table III) with enough precision to test the prediction; if the updated moments fail there, the universality claim collapses. More directly, fit the data in separate invariant-mass bins: if the best-fit N_P varies with omega, the parametrization is wrong and the moments are artifacts.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central claim is that the nonperturbative dynamics of a P-wave meson pair in multi-body B decays is captured by two-meson distribution amplitudes whose Gegenbauer moments are universal, and that the apparent earlier failure, moments above unity and a wrong prediction for the longitudinal polarization of $B_s^{0}$ -> K*0 Kbar*0, came from an inconsistency in how the intermediate vector resonance was parametrized. The resolution is to introduce a single factor N_P for each pair, relating the time-like form-factor normalization to the Breit-Wigner resonance amplitude. Fitting the moments and the N_P together to branching ratios and polarization fractions of three- and four-body decays yields a good global fit ($chi^{2}$/d.o.f. about 1.5 to 1.6), convergent Gegenbauer expansions, and predictions that track the data, prominently f0($B_s^{0}$ -> K*0 Kbar*0) = 28.$2^{{+8.8}}$_{-9.5}% and L_{K*0 Kbar*0} = 7.$7^{{+4.9}}$_{-3.8}. The paper takes the fit quality as evidence for the consistency of the LO PQCD framework and for the universality of the two-meson DAs.

Load-bearing premise

The load-bearing assumption is that a single momentum-independent constant N_P per meson pair fully absorbs the discrepancy between the time-like form-factor and Breit-Wigner descriptions; if N_P actually depends on invariant mass or on the decay channel, the extracted Gegenbauer moments are not universal.

Editorial extensions

If this is right

  • The two-meson DAs for pi pi, K pi, and KK become universal inputs: the same moments describe three-body and four-body decays and can be used for semileptonic B -> P1 P2 l+ l- form factors, which the paper computes and finds consistent with light-cone sum-rule results.
  • The longitudinal polarization fraction f0(B_s^0 -> K*0 Kbar*0) is no longer an anomaly; its agreement with data removes a standard-model tension and weakens the case for new physics in this channel.
  • The observable L_{K*0 Kbar*0} = 7.7^{+4.9}_{-3.8} is closer to the measured 4.43 +/- 0.92 than earlier QCDF and PQCD values, so potential new-physics signals inferred from the earlier discrepancy shrink.
  • The transverse K pi Gegenbauer moments a_perp_{1K*} and a_perp_{2K*} are determined for the first time, giving new constraints on SU(3) breaking in two-meson DAs.
  • Direct CP asymmetries in individual helicity states of four-body decays can be large even when the integrated asymmetry is small, so angular analyses are the right place to look for CP violation.

Reading between the lines

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

  • If N_P is actually a function of the invariant mass, the extracted moments are contaminated; this can be tested by fitting in bins of omega and checking whether the best-fit N_P stays constant.
  • The same method could be applied to S-wave and D-wave meson pairs, or to other heavy-quark decays, to test universality beyond the three pairs considered.
  • The claim of universality is only as strong as the leading-order approximation; next-to-leading-order corrections could shift the moments, so the moments should be re-fit once NLO kernels are available.
  • Because N_P and the moments are fitted to the same data, the good chi^2/d.o.f. is partly guaranteed; a sharper test is out-of-sample prediction, such as the B_s -> K* K* branching ratios that have not yet been measured.
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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

3 major / 4 minor

Summary. The manuscript revisits the PQCD treatment of charmless three- and four-body B decays with intermediate vector resonances. It introduces three fitted normalization constants N_P (P = pi pi, K pi, KK) in Eq. (48) to absorb the mismatch between time-like meson-pair form factors and Breit-Wigner resonance propagators, and re-fits the Gegenbauer moments of the two-meson DAs to branching ratios and polarization fractions of three- and four-body B decays. The authors report chi2/d.o.f. around 1.6, improved convergence of the Gegenbauer expansion, a first extraction of the transverse K pi moments, and predictions for B_s^0 -> K*0 Kbar*0 observables, the U-spin ratio L, and polarization-dependent CP asymmetries.

Significance. If the extracted moments and the N_P constants are genuinely universal, this is a useful step forward: the analysis adds four-body decay data, provides a covariance matrix for the fitted parameters, makes the Gegenbauer expansion more convergent, and gives form-factor predictions that are broadly consistent with LCSR results. The paper also makes a concrete, testable statement that per-helicity CP asymmetries in four-body decays can be large while the net CP asymmetry remains small. However, the headline phenomenological results are not independent predictions, and the universality claim rests on an untested constancy assumption for N_P. These issues materially reduce the strength of the conclusions as currently stated, although they do not invalidate the underlying analysis framework.

major comments (3)
  1. [Sec. II.C, Eq. (48)] The central universality claim depends on Eq. (48), where a single momentum-independent constant N_P per meson pair is introduced to absorb the mismatch between the time-like form-factor parametrization and the Breit-Wigner description of the intermediate vector resonance. This assumption is not tested: for K pi and KK the form factors in Eq. (37) contain only the lowest K*(892)/phi resonance, whereas the pi pi form factor in Eq. (40) includes rho-omega interference and excited rho states, so any omega^2-dependent mismatch from omitted states cannot be represented by a constant. Because N_P and the Gegenbauer moments are fitted simultaneously to the same data (Tables I-III), the agreement of the fitted N_Kpi = 1.48 +/- 0.03 and N_KK = 1.22 +/- 0.03 with the rough estimates in Eq. (49) does not validate the assumed functional form. I recommend adding a stability test with an omega^2-dependent or process-dependent N_P, or at least a clear statement that the extracted moments are conditional on this ansatz.
  2. [Sec. III.E, Table III] The abstract and Sec. III.E present f0(B_s^0 -> K*0 Kbar*0) = 28.2^{+8.8}_{-9.5}% and L = 7.7^{+4.9}_{-3.8} as predictions that match data. However, the data for B_s^0 -> K*0 Kbar*0 (branching ratio, f0, and f_perp) and for B^0 -> K*0 Kbar*0 are marked with a dagger in Table III, meaning they were included in the fit that determines the Gegenbauer moments and N_P used to compute these observables. The agreement is therefore a post-fit consistency check rather than an independent prediction; this should be stated explicitly and the wording in the abstract adjusted.
  3. [Sec. III.C, Tables II and VIII] The claim that the satisfactory fit quality (chi2/d.o.f. around 1.6) implies consistency of the PQCD framework is weakened by the selection of fitted data and by large discrepancies in related observables. The modes B^0 -> pi0 rho0 and B^0 -> rho0 rho0 are excluded from the fit because of subleading contributions, yet B(B^0 -> pi0 rho0) is predicted at 0.04 x 10^-6 against 2.0 +/- 0.5 x 10^-6 and B(B^0 -> rho0 rho0) at 0.35^{+0.12}_{-0.07} x 10^-6 against 0.96 +/- 0.15 x 10^-6. In addition, Table VIII gives A_CP(B^+ -> K^+ rho0) = 62.4^{+11.6}_{-13.5}%, while the LHCb value is 16 +/- 2%; the paper attributes this to NLO corrections, but the size of the discrepancy should be factored into the consistency claim. Please either include these channels in the fit with an assessment of the subleading uncertainties, or explicitly qualify the claimed consistency as applying only to the fitted set.
minor comments (4)
  1. [Abstract] The phrase 'destruction among them' should read 'cancellation among them', and the subject-verb agreement should be fixed ('results').
  2. [Sec. III.E, Eq. (65)] The notation for the longitudinal polarization fraction is inconsistent: Eq. (65) uses f_L while the rest of the text and Table III use f_0; please unify.
  3. [Sec. III.A] It would be useful to specify the number of fitted data points and the resulting number of degrees of freedom for each fit (Tables I, IV, V), since only chi2/d.o.f. is quoted.
  4. [Sec. III.D, Table VII] The comparison with LCSR form factors would be more informative if the theoretical errors in Table VII were broken into the same sources (omega_B, Gegenbauer moments, hard scale) as in Tables II and III.

Circularity Check

2 steps flagged · score 6.0 of 10

Headline f0(Bs→K*0Kbar*0) and U-spin ratio L are fitted inputs; agreement with data is fit quality, not prediction.

  1. fitted input called prediction [Sec. III.E; Table III; Abstract]
    "It is worth mentioning that the predicted longitudinal polarization fraction of the pure-penguin decay f0(B0s → K∗0K̄∗0) = (28.2+8.8−9.5)%, far below our previous result [23], is now in good agreement with the data f0(B0s → K∗0K̄∗0)exp = (24 ± 4)% [36]."

    The observable f0(B0s → K∗0K̄∗0) is itself an input to the global fit: Table III lists this channel with f0 = 24 ± 4 and marks it with †, and the table caption states 'Those data marked by † are included in the fit.' The Gegenbauer moments and N_P are determined by minimizing χ² to this very datum. The subsequently quoted 'prediction' 28.2% is therefore the fitted value of the same quantity, and its agreement with the measurement is a restatement of fit quality, not an independent prediction.

  2. fitted input called prediction [Sec. III.E, Eq. (65), Eq. (68)]
    "We update the ratio LK∗0K̄∗0 based on Table III, LPQCD K∗0K̄∗0 = 7.7+4.9−3.8, which turns closer to Eq. (66)."

    The observable LK∗0K̄∗0 is defined through the branching fractions and longitudinal polarization fractions of B0s → K∗0K̄∗0 and B0 → K∗0K̄∗0. Both channels are fitted inputs: Table III marks B0s → K∗0K̄∗0 and B0 → K∗0K̄∗0 with † for their branching ratios and/or polarization fractions. Thus the 'prediction' L = 7.7 is a function of fitted central values; its movement toward the experimental value reflects the fit absorbing the data, not a parameter-free derivation. The NP-signal comparison in Eqs. (66)-(68) is consequently weaker than presented.

full rationale

The paper's central numerical showcase, f0(B0s → K∗0K̄∗0), is marked with † in Table III and therefore included in the fit that determines the Gegenbauer moments and the normalization constants N_P. Calling the resulting number a 'prediction' that 'matches well the measurement' is a fitted-input-called-prediction reduction: the χ² minimization was performed against that exact datum. The U-spin ratio LK∗0K̄∗0 inherits the same issue because it is built from two †-marked fitted channels. The paper does contain genuinely independent elements: predictions for modes excluded from the fit (e.g., B0 → π0ρ0, B0 → ρ0ρ0) and the B(s) → P1P2 transition form factors compared with LCSR results in Table VII. Those checks give the work independent content and prevent a higher score. The constancy of N_P in Eq. (48) is an untested modeling assumption, but it is not itself circular; the closer circularity is that N_P and the moments are fit together to the same †-marked data and then used to validate the framework and to 'confirm' Eq. (49). Overall, the claim of universality is partially supported by external comparisons, but the headline agreement for f0 and L is enforced by construction.

Assumptions & free parameters 15 free parameters · 6 assumptions · 0 invented entities

The central results rest on 12 fitted parameters (9 Gegenbauer moments and 3 N_P coefficients) plus several adopted input parameters from prior fits. No new particles or forces are introduced.

free parameters (15)
  • a^0_{2ρ} = 0.16 ± 0.10
    Fitted to B→ρππ, ρρ, ρK* data in the global χ2 fit (Table I).
  • a^s_{2ρ} = -0.11 ± 0.14
    Twist-3 ππ DA moment, fitted in the global χ2 fit (Table I).
  • a^t_{2ρ} = -0.21 ± 0.04
    Twist-3 ππ DA moment, fitted in the global χ2 fit (Table I).
  • a^||_{1K*} = 0.45 ± 0.11
    Twist-2 longitudinal Kπ DA moment, fitted to B(s)→Kπ and K*ρ, K*K*, K*φ data (Table I).
  • a^||_{2K*} = -0.75 ± 0.08
    Twist-2 longitudinal Kπ DA moment, fitted in the global χ2 fit (Table I).
  • a^⊥_{1K*} = 0.61 ± 0.21
    Twist-2 transverse Kπ DA moment, extracted for the first time from four-body data (Table I).
  • a^⊥_{2K*} = 0.45 ± 0.06
    Twist-2 transverse Kπ DA moment, extracted for the first time (Table I).
  • a^0_{2φ} = -0.54 ± 0.14
    Twist-2 longitudinal KK DA moment, fitted to B→Kφ and φK*, φφ data (Table I).
  • a^T_{2φ} = 0.77 ± 0.04
    Twist-2 transverse KK DA moment, fitted in the global χ2 fit (Table I).
  • N_{ππ} = 1.05 ± 0.04
    Introduced in Eq. (48) to reconcile form-factor and BW parametrizations; fitted to data (Table I).
  • N_{Kπ} = 1.48 ± 0.03
    Introduced in Eq. (48); fitted to data (Table I).
  • N_{KK} = 1.22 ± 0.03
    Introduced in Eq. (48); fitted to data (Table I).
  • a^T_{2ρ} = 0.5 ± 0.5
    Not fitted in this work; adopted from Ref [50] as an input for transverse ππ DA.
  • a^a_{2ρ} = 0.4 ± 0.4
    Not fitted; adopted from Ref [50].
  • a^v_{2ρ} = -0.5 ± 0.5
    Not fitted; adopted from Ref [50].
assumptions (6)
  • domain assumption PQCD factorization for multi-body B decays (Eq. (7)): M = Φ_B ⊗ H ⊗ Φ_{P1P2} ⊗ Φ_{P3}
    Assumes quasi-two-body factorization when the meson pair is collimated; invoked throughout Sec. II.
  • domain assumption Watson theorem applies, absorbing elastic rescattering in the meson pair into time-like form factors
    Provides the basis for the BW/GS parametrization of F^{||,⊥}(ω^2); cited in Sec. II.B.
  • ad hoc to paper A single momentum-independent constant N_P per pair fixes the mismatch between form-factor and BW descriptions (Eq. (48))
    No derivation of the ω-independence; N_P is fitted to data and the fitted values are used as confirmation.
  • ad hoc to paper Twist-3 Kπ and KK DAs are set to asymptotic forms
    Limited data; φ^s,φ^t for Kπ and KK are fixed to simple t or t^2 forms (Eqs. (26)-(27), (32)-(33)).
  • domain assumption Leading-order hard kernels with hard-scale variation to estimate NLO corrections
    All amplitudes are LO in αs; NLO effects are approximated by varying t from 0.75t to 1.25t (Sec. III.A).
  • domain assumption B meson DA shape parameter ωB = 0.40 GeV (ωBs = 0.48 GeV)
    Taken from prior PQCD analyses and lattice inputs; varied by 10% for uncertainty (Sec. III.A).

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Cite this review

Pith. "Pith review of Improved global determination of two-meson distribution amplitudes from multi-body $B$ decays." pith.science (2026). https://pith.science/paper/JPTKCOER

@misc{pith2026250115150,
  author       = {Pith},
  title        = {Pith review of: Improved global determination of two-meson distribution amplitudes from multi-body $B$ decays},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JPTKCOER}},
  note         = {Machine review of arXiv:2501.15150}
}
abstract

We improve the perturbative QCD (PQCD) formalism for multi-body charmless hadronic $B$ meson decays, such as $B\to VP_3\to P_1P_2P_3$, by resolving the possible discrepancy in parametrizing the contribution of the $P$-wave resonance $V$ to the two-meson distribution amplitudes (DAs) associated with the pairs $P_1P_2=\pi\pi, K\pi, KK$. The determination of the Gegenbauer moments in the two-meson DAs is then updated in the global fit of the improved PQCD factorization formulas at leading order in the strong coupling $\alpha_s$ to available data for branching ratios and polarization fractions of three- and four-body $B$ decays. The convergence of the Gegenbauer expansion of the resultant two-meson DAs is manifest. The satisfactory quality of the fit implies the consistency of the PQCD framework for multi-body $B$ decays and the universality of the nonperturbative two-meson DAs. In particular, the predicted longitudinal polarization fraction $f_0(B_s^0\to K^{*0} {\bar K}^{*0})=28.2^{+8.8}_{-9.5} \%$ with the updated Gegenbauer moments matches well the measurement. The observable $L_{K^{*0}{\bar K}^{*0}}=7.7^{+4.9}_{-3.8}$, defined as the ratio of the longitudinal amplitudes of the two $U$-spin related channels $B_s^0\to K^{*0} {\bar K}^{*0}$ and $B^0\to K^{*0} {\bar K}^{*0}$, accommodates the current data within errors. It is found that the direct $CP$ asymmetries ${\cal A}^{0,||,\bot}_{\rm CP}$ in the polarization states of some four-body decays $B\to V_1V_2\to (P_1P_2)(P_3P_4)$ might be large, but the destruction among them result in small net $CP$ violation. Our predictions can be confronted with LHCb and Belle-II data in the future.

Figures

Figures reproduced from arXiv: 2501.15150 by the authors.

Figure 1
Figure 1. FIG. 1: Covariance matrix of the fitted Gegenbauer moments in [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Dependence of the [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Dependence of [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Longitudinal polarization fraction [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]

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Works this paper leans on

117 extracted references · 23 canonical work pages

  1. [1]

    R. H. Dalitz, Phil. Mag. Ser. 7 44 (1953), 1068-1080. doi:10.1080/14786441008520365

    doi:10.1080/14786441008520365 1953

  2. [2]

    R. H. Dalitz, Phys. Rev. 94 (1954), 1046-1051 doi:10.1103/PhysRev.94.1046

    doi:10.1103/physrev.94.1046 1954

  3. [3]

    Aaij et al

    R. Aaij et al. [LHCb], Phys. Rev. Lett. 111 (2013), 101801 doi:10.1103/PhysRevLett.111.101801 [ arXiv:1306.1246 [hep-ex]]

    arXiv 2013

  4. [4]

    Aaij et al

    R. Aaij et al. [LHCb], Phys. Rev. Lett. 112 (2014) no.1, 011801 doi:10.1103/PhysRevLett.112.011801 [arXiv:1310.4740 [hep-ex]]

    arXiv 2014

  5. [5]

    Aaij et al

    R. Aaij et al. [LHCb], Phys. Rev. D 108 (2023) no.1, 012008 doi:10.1103/PhysRevD.108.012008 [ arXiv:2206.07622 [hep-ex]]

    arXiv 2023

  6. [6]

    Aaij et al

    R. Aaij et al. [LHCb], Phys. Rev. Lett. 124 (2020) no.3, 031801 doi:10.1103/PhysRevLett.124.031801 [arXiv:1909.05211 [hep-ex]]

    arXiv 2020

  7. [7]

    Bediaga, T

    I. Bediaga, T. Frederico and O. Lourenc ¸o, Phys. Rev. D 89 (2014) no.9, 094013 doi:10.1103/PhysRevD.89.094013 [ arXiv:1307.8164 [hep-ph]]

    arXiv 2014

  8. [8]

    Bediaga and P

    I. Bediaga and P . C. Magalh˜ aes, [arXiv:1512.09284 [hep-ph]]

    arXiv

Show all 117 references
  1. [9]

    X. W. Kang, B. Kubis, C. Hanhart and U. G. Meißner, Phys. Re v. D 89 (2014), 053015 doi:10.1103/PhysRevD.89.053015 [arXiv:1312.1193 [hep-ph]]

    2014 arXiv

  2. [10]

    C. H. Chen and H. n. Li, Phys. Lett. B 561 (2003), 258-265 doi:10.1016/S0370-2693(03)00486-6 [ arXiv:hep-ph/0209043 [hep-ph]]

    2003 arXiv

  3. [11]

    El-Bennich, A

    B. El-Bennich, A. Furman, R. Kaminski, L. Lesniak, B. Lo iseau and B. Moussallam, Phys. Rev. D 79 (2009), 094005 [erratum: Phys. Rev. D 83 (2011), 039903] doi:10.1103/PhysRevD.83.039903 [ arXiv:0902.3645 [hep-ph]]

    2009 arXiv

  4. [12]

    Virto, PoS FPCP2016 (2017), 007 doi:10.22323/1.280.0007 [ arXiv:1609.07430 [hep-ph]]

    J. Virto, PoS FPCP2016 (2017), 007 doi:10.22323/1.280.0007 [ arXiv:1609.07430 [hep-ph]]

    2017 arXiv

  5. [13]

    Kr¨ ankl, T

    S. Kr¨ ankl, T. Mannel and J. Virto, Nucl. Phys. B 899 (2015), 247-264 doi:10.1016/j.nuclphysb.2015.08.004 [ arXiv:1505.04111 [hep- ph]]

    2015 arXiv

  6. [14]

    A. G. Grozin, Sov. J. Nucl. Phys. 38 (1983), 289-292

    1983

  7. [15]

    A. G. Grozin, Theor. Math. Phys. 69 (1986), 1109-1121 doi:10.1007/BF01037870

    1986 doi

  8. [16]

    M¨ uller, D

    D. M¨ uller, D. Robaschik, B. Geyer, F. M. Dittes and J. Ho ˇ rejˇ si, Fortsch. Phys.42 (1994), 101-141 doi:10.1002/prop.2190420202 [arXiv:hep-ph/9812448 [hep-ph]]

    1994 arXiv

  9. [17]

    Diehl, T

    M. Diehl, T. Gousset, B. Pire and O. Teryaev, Phys. Rev. L ett. 81 (1998), 1782-1785 doi:10.1103/PhysRevLett.81.1782 [arXiv:hep-ph/9805380 [hep-ph]]

    1998 arXiv

  10. [18]

    Diehl, T

    M. Diehl, T. Gousset and B. Pire, Phys. Rev. D 62 (2000), 073014 doi:10.1103/PhysRevD.62.073014 [arXiv:hep-ph/0003233 [hep-ph]]

    2000 arXiv

  11. [19]

    Pire and L

    B. Pire and L. Szymanowski, Phys. Lett. B 556 (2003), 129-134 doi:10.1016/S0370-2693(03)00134-5 [ arXiv:hep-ph/0212296 [hep- ph]]

    2003 arXiv

  12. [20]

    M. V . Polyakov, Nucl. Phys. B 555 (1999), 231 doi:10.1016/S0550-3213(99)00314-4 [ arXiv:hep-ph/9809483 [hep-ph]]

    1999 arXiv

  13. [21]

    Y . Li, D. C. Yan, J. Hua, Z. Rui and H. n. Li, Phys. Rev. D 104 (2021) no.9, 096014 doi:10.1103/PhysRevD.104.096014 [arXiv:2105.03899 [hep-ph]]

    2021 arXiv

  14. [22]

    J. Hua, H. n. Li, C. D. Lu, W. Wang and Z. P . Xing, Phys. Rev. D 104 (2021) no.1, 016025 doi:10.1103/PhysRevD.104.016025 [arXiv:2012.15074 [hep-ph]]

    2021 arXiv

  15. [23]

    Z. Rui, Y . Li and H. n. Li, JHEP 05, 082 (2021) doi:10.1007/JHEP05(2021)082 [ arXiv:2103.00642 [hep-ph]]. 20

    2021 arXiv

  16. [24]

    Y . Li, D. C. Yan, Z. Rui and Z. J. Xiao, Eur. Phys. J. C 81, no.9, 806 (2021) doi:10.1140/epjc/s10052-021-09608-5 [ arXiv:2107.10684 [hep-ph]]

    2021 arXiv

  17. [25]

    C. Q. Zhang, J. M. Li, M. K. Jia, Y . Li and Z. Rui, Phys. Rev. D 105, no.5, 053002 (2022) doi:10.1103/PhysRevD.105.053002 [arXiv:2112.10939 [hep-ph]]

    2022 arXiv

  18. [26]

    D. C. Yan, Z. Rui, Z. J. Xiao and Y . Li, Phys. Rev. D 105, no.9, 093001 (2022) doi:10.1103/PhysRevD.105.093001 [ arXiv:2204.01092 [hep-ph]]

    2022 arXiv

  19. [27]

    Z. Q. Zhang, Y . C. Zhao, Z. L. Guan, Z. J. Sun, Z. Y . Zhang an d K. Y . He, Chin. Phys. C 46, no.12, 123105 (2022) doi:10.1088/1674- 1137/ac89d1 [arXiv:2207.02043 [hep-ph]]

    2022 arXiv

  20. [28]

    will be derived for the first time. We specify the inputted ma sses and widths (in units of GeV) [ 36] in the numerical analysis, mB = 5 .280, m Bs = 5.367, m b = 4.8, m K ± = 0.494, mK 0 = 0 .498, m π ± = 0.140, m π 0 = 0.135, Γ ρ = 0.1496 Γ K ∗ = 0 .0473, Γ φ = 0.00425. (50) ...

  21. [29]

    Z. T. Zou, W. S. Fang, X. Liu and Y . Li, Eur. Phys. J. C82, no.11, 1076 (2022) doi:10.1140/epjc/s10052-022-11060-y [arXiv:2210.08522 [hep-ph]]

    2022 arXiv

  22. [30]

    D. C. Yan, Z. Rui, Y . Yan and Y . Li, Eur. Phys. J. C 83, no.10, 974 (2023) doi:10.1140/epjc/s10052-023-12152-z [arXiv:2308.12543 [hep-ph]]

    2023 arXiv

  23. [31]

    D. C. Yan, Y . Yan and Z. Rui, Eur. Phys. J. C 84, no.7, 754 (2024) doi:10.1140/epjc/s10052-024-13087-9 [ arXiv:2404.19198 [hep-ph]]

    2024 arXiv

  24. [32]

    J. P . Dedonder, A. Furman, R. Kaminski, L. Lesniak and B. Loiseau, Acta Phys. Polon. B 42 (2011), 2013 doi:10.5506/APhysPolB.42.2013 [arXiv:1011.0960 [hep-ph]]

    2011 arXiv

  25. [33]

    Alguer´ o, A

    M. Alguer´ o, A. Crivellin, S. Descotes-Genon, J. Matia s and M. Novoa-Brunet, JHEP 04, 066 (2021) doi:10.1007/JHEP04(2021)066 [arXiv:2011.07867 [hep-ph]]

    2021 arXiv

  26. [34]

    Aaij et al

    R. Aaij et al. [LHCb], Eur. Phys. J. C 78 (2018) no.12, 1019 doi:10.1140/epjc/s10052-018-6447-z [ arXiv:1809.07416 [hep-ex]]

    2018 arXiv

  27. [35]

    G. J. Gounaris and J. J. Sakurai, Phys. Rev. Lett. 21 (1968), 244-247 doi:10.1103/PhysRevLett.21.244

    1968 doi

  28. [36]

    K. M. Watson, Phys. Rev. 88 (1952), 1163-1171 doi:10.1103/PhysRev.88.1163

    1952 doi

  29. [37]

    Navas et al

    S. Navas et al. [Particle Data Group], Phys. Rev. D 110 (2024) no.3, 030001 doi:10.1103/PhysRevD.110.030001

    2024 doi

  30. [38]

    Y . Y . Keum, H. N. Li and A. I. Sanda, Phys. Rev. D 63 (2001), 054008 doi:10.1103/PhysRevD.63.054008 [ arXiv:hep-ph/0004173 [hep-ph]]

    2001 arXiv

  31. [39]

    Kurimoto, H

    T. Kurimoto, H. n. Li and A. I. Sanda, Phys. Rev. D 65 (2002), 014007 doi:10.1103/PhysRevD.65.014007 [ arXiv:hep-ph/0105003 [hep-ph]]

    2002 arXiv

  32. [40]

    C. D. Lu and M. Z. Yang, Eur. Phys. J. C 28 (2003), 515-523 doi:10.1140/epjc/s2003-01199-y [ arXiv:hep-ph/0212373 [hep-ph]]

    2003 arXiv

  33. [41]

    H. n. Li, Prog. Part. Nucl. Phys. 51 (2003), 85-171 doi:10.1016/S0146-6410(03)90013-5 [ arXiv:hep-ph/0303116 [hep-ph]]

    2003 arXiv

  34. [42]

    Z. J. Xiao, W. F. Wang and Y . y. Fan, Phys. Rev. D 85 (2012), 094003 doi:10.1103/PhysRevD.85.094003 [ arXiv:1111.6264 [hep-ph]]

    2012 arXiv

  35. [43]

    H. N. Li, Y . L. Shen and Y . M. Wang, JHEP 02 (2013), 008 doi:10.1007/JHEP02(2013)008 [ arXiv:1210.2978 [hep-ph]]

    2013 arXiv

  36. [44]

    C. D. Lu, K. Ukai and M. Z. Yang, Phys. Rev. D 63 (2001), 074009 doi:10.1103/PhysRevD.63.074009 [ arXiv:hep-ph/0004213 [hep- ph]]

    2001 arXiv

  37. [45]

    Y . Y . Keum, H. n. Li and A. I. Sanda, Phys. Lett. B 504 (2001), 6-14 doi:10.1016/S0370-2693(01)00247-7 [ arXiv:hep-ph/0004004 [hep-ph]]

    2001 arXiv

  38. [46]

    A. G. Grozin and M. Neubert, Phys. Rev. D 55 (1997), 272-290 doi:10.1103/PhysRevD.55.272 [ arXiv:hep-ph/9607366 [hep-ph]]

    1997 arXiv

  39. [47]

    Y . Yang, L. Lang, X. Zhao, J. Huang and J. Sun, Phys. Rev. D 103 (2021) no.5, 056006 doi:10.1103/PhysRevD.103.056006 [arXiv:2012.10581 [hep-ph]]

    2021 arXiv

  40. [48]

    V . M. Braun, Y . Ji and A. N. Manashov, JHEP 05 (2017), 022 doi:10.1007/JHEP05(2017)022 [ arXiv:1703.02446 [hep-ph]]

    2017 arXiv

  41. [49]

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

    2007 arXiv

  42. [50]

    W. F. Wang and H. n. Li, Phys. Lett. B 763 (2016), 29-39 doi:10.1016/j.physletb.2016.10.026 [ arXiv:1609.04614 [hep-ph]]

    2016 arXiv

  43. [51]

    Z. Rui, Y . Li and H. N. Li, Phys. Rev. D 98 (2018) no.11, 113003 doi:10.1103/PhysRevD.98.113003 [ arXiv:1809.04754 [hep-ph]]

    2018 arXiv

  44. [52]

    Aaij et al

    R. Aaij et al. [LHCb], JHEP 05, 026 (2019) doi:10.1007/JHEP05(2019)026 [ arXiv:1812.07008 [hep-ex]]

    2019 arXiv

  45. [53]

    Breit and E

    G. Breit and E. Wigner, Phys. Rev. 49 (1936), 519-531 doi:10.1103/PhysRev.49.519

    1936 doi

  46. [54]

    Y . Li, A. J. Ma, W. F. Wang and Z. J. Xiao, Phys. Rev. D 95 (2017) no.5, 056008 doi:10.1103/PhysRevD.95.056008 [ arXiv:1612.05934 [hep-ph]]

    2017 arXiv

  47. [55]

    J. P . Lees et al. [BaBar], Phys. Rev. D 86 (2012), 032013 doi:10.1103/PhysRevD.86.032013 [ arXiv:1205.2228 [hep-ex]]

    2012 arXiv

  48. [56]

    A. Ali, G. Kramer and C. D. Lu, Phys. Rev. D 58 (1998), 094009 doi:10.1103/PhysRevD.58.094009 [ arXiv:hep-ph/9804363 [hep-ph]]

    1998 arXiv

  49. [57]

    H. Y . Cheng, Phys. Rev. D 106 (2022) no.11, 113004 doi:10.1103/PhysRevD.106.113004 [ arXiv:2211.03965 [hep-ph]]

    2022 arXiv

  50. [58]

    Lepage and C

    P . Lepage and C. Gohlke, gplepage/lsqfit: lsqfit version 11.7, Zenodo. http://doi.org/10.5281/zenodo.4037174

    doi

  51. [59]

    H. n. Li and S. Mishima, Phys. Rev. D 74 (2006), 094020 doi:10.1103/PhysRevD.74.094020 [ arXiv:hep-ph/0608277 [hep-ph]]

    2006 arXiv

  52. [60]

    Z. Rui, X. Gao and C. D. Lu, Eur. Phys. J. C 72 (2012), 1923 doi:10.1140/epjc/s10052-012-1923-3 [ arXiv:1111.0181 [hep-ph]]

    2012 arXiv

  53. [61]

    P . Ball, V . M. Braun, Y . Koike and K. Tanaka, Nucl. Phys. B 529 (1998), 323-382 doi:10.1016/S0550-3213(98)00356-3 [arXiv:hep-ph/9802299 [hep-ph]]

    1998 arXiv

  54. [62]

    X. Liu, H. n. Li and Z. J. Xiao, Phys. Rev. D 91, no.11, 114019 (2015) doi:10.1103/PhysRevD.91.114019 [ arXiv:1502.04162 [hep-ph]]

    2015 arXiv

  55. [63]

    Y . M. Wang and Y . L. Shen, Nucl. Phys. B 898 (2015), 563-604 doi:10.1016/j.nuclphysb.2015.07.016 [ arXiv:1506.00667 [hep-ph]]

    2015 arXiv

  56. [64]

    Y . M. Wang, JHEP 09 (2016), 159 doi:10.1007/JHEP09(2016)159 [ arXiv:1606.03080 [hep-ph]]

    2016 arXiv

  57. [65]

    Y . M. Wang and Y . L. Shen, JHEP 05 (2018), 184 doi:10.1007/JHEP05(2018)184 [ arXiv:1803.06667 [hep-ph]]

    2018 arXiv

  58. [66]

    W. Wang, Y . M. Wang, J. Xu and S. Zhao, Phys. Rev. D102 (2020) no.1, 011502 doi:10.1103/PhysRevD.102.011502 [arXiv:1908.09933 [hep-ph]]

    2020 arXiv

  59. [67]

    X. Y . Han, J. Hua, X. Ji, C. D. L¨ u, W. Wang, J. Xu, Q. A. Zhang and S. Zhao, [ arXiv:2403.17492 [hep-ph]]

    arXiv

  60. [68]

    X. Y . Han et al. [Lattice Parton], Phys. Rev. D 111 (2025) no.3, 034503 doi:10.1103/PhysRevD.111.034503 [ arXiv:2410.18654 [hep- lat]]

    2025 arXiv

  61. [69]

    D. C. Yan, P . Yang, X. Liu and Z. J. Xiao, Nucl. Phys. B 931, 79-104 (2018) doi:10.1016/j.nuclphysb.2018.04.007 [ arXiv:1707.06043 [hep-ph]]. 21

    2018 arXiv

  62. [70]

    Gubernari, A

    N. Gubernari, A. Kokulu and D. van Dyk, JHEP 01, 150 (2019) doi:10.1007/JHEP01(2019)150 [ arXiv:1811.00983 [hep-ph]]

    2019 arXiv

  63. [71]

    Khodjamirian, T

    A. Khodjamirian, T. Mannel and N. Offen, Phys. Rev. D 75, 054013 (2007) doi:10.1103/PhysRevD.75.054013 [ arXiv:hep-ph/0611193 [hep-ph]]

    2007 arXiv

  64. [72]

    Bharucha, D

    A. Bharucha, D. M. Straub and R. Zwicky, JHEP 08, 098 (2016) doi:10.1007/JHEP08(2016)098 [ arXiv:1503.05534 [hep-ph]]

    2016 arXiv

  65. [73]

    Cheng, A

    S. Cheng, A. Khodjamirian and J. Virto, JHEP 05, 157 (2017) doi:10.1007/JHEP05(2017)157 [ arXiv:1701.01633 [hep-ph]]

    2017 arXiv

  66. [74]

    Ball and R

    P . Ball and R. Zwicky, Phys. Rev. D 71, 014029 (2005) doi:10.1103/PhysRevD.71.014029 [ arXiv:hep-ph/0412079 [hep-ph]]

    2005 arXiv

  67. [75]

    Khodjamirian, T

    A. Khodjamirian, T. Mannel, A. A. Pivovarov and Y . M. Wan g, JHEP 09, 089 (2010) doi:10.1007/JHEP09(2010)089 [ arXiv:1006.4945 [hep-ph]]

    2010 arXiv

  68. [76]

    Descotes-Genon, A

    S. Descotes-Genon, A. Khodjamirian and J. Virto, JHEP 12, 083 (2019) doi:10.1007/JHEP12(2019)083 [ arXiv:1908.02267 [hep-ph]]

    2019 arXiv

  69. [77]

    H. Y . Cheng and C. K. Chua, Phys. Rev. D 80 (2009), 114026 doi:10.1103/PhysRevD.80.114026 [ arXiv:0910.5237 [hep-ph]]

    2009 arXiv

  70. [78]

    C. Wang, S. H. Zhou, Y . Li and C. D. Lu, Phys. Rev. D 96 (2017) no.7, 073004 doi:10.1103/PhysRevD.96.073004 [ arXiv:1708.04861 [hep-ph]]

    2017 arXiv

  71. [79]

    C. Wang, Q. A. Zhang, Y . Li and C. D. Lu, Eur. Phys. J. C 77 (2017) no.5, 333 doi:10.1140/epjc/s10052-017-4889-3 [ arXiv:1701.01300 [hep-ph]]

    2017 arXiv

  72. [80]

    Z. T. Zou, A. Ali, C. D. Lu, X. Liu and Y . Li, Phys. Rev. D 91 (2015), 054033 doi:10.1103/PhysRevD.91.054033 [ arXiv:1501.00784 [hep-ph]]

    2015 arXiv

  73. [81]

    H. n. Li, Phys. Lett. B 622 (2005), 63-68 doi:10.1016/j.physletb.2005.06.077 [ arXiv:hep-ph/0411305 [hep-ph]]

    2005 arXiv

  74. [82]

    Aaltonen et al

    T. Aaltonen et al. [CDF], Phys. Rev. Lett. 107 (2011), 261802 doi:10.1103/PhysRevLett.107.261802 [ arXiv:1107.4999 [hep-ex]]

    2011 arXiv

  75. [83]

    Aaij et al

    R. Aaij et al. [LHCb], Phys. Lett. B 713 (2012), 369-377 doi:10.1016/j.physletb.2012.06.012 [ arXiv:1204.2813 [hep-ex]]

    2012 arXiv

  76. [84]

    K. F. Chen et al. [Belle], Phys. Rev. Lett. 91 (2003), 201801 doi:10.1103/PhysRevLett.91.201801 [ arXiv:hep-ex/0307014 [hep-ex]]

    2003 arXiv

  77. [85]

    Aubert et al

    B. Aubert et al. [BaBar], Phys. Rev. D 78 (2008), 092008 doi:10.1103/PhysRevD.78.092008 [ arXiv:0808.3586 [hep-ex]]

    2008 arXiv

  78. [86]

    J. P . Lees et al. [BaBar], Phys. Rev. D 85 (2012), 072005 doi:10.1103/PhysRevD.85.072005 [ arXiv:1112.3896 [hep-ex]]

    2012 arXiv

  79. [87]

    Beneke, J

    M. Beneke, J. Rohrer and D. Yang, Nucl. Phys. B 774 (2007), 64-101 doi:10.1016/j.nuclphysb.2007.03.020 [ arXiv:hep-ph/0612290 [hep-ph]]

    2007 arXiv

  80. [88]

    C. W. Bauer, D. Pirjol, I. Z. Rothstein and I. W. Stewart, Phys. Rev. D 70 (2004), 054015 doi:10.1103/PhysRevD.70.054015 [arXiv:hep-ph/0401188 [hep-ph]]

    2004 arXiv

  81. [89]

    H. n. Li and S. Mishima, Phys. Rev. D 71 (2005), 054025 doi:10.1103/PhysRevD.71.054025 [ arXiv:hep-ph/0411146 [hep-ph]]

    2005 arXiv

  82. [90]

    H. Y . Cheng and K. C. Yang, Phys. Rev. D 78 (2008), 094001 [erratum: Phys. Rev. D 79 (2009), 039903] doi:10.1103/PhysRevD.79.039903 [arXiv:0805.0329 [hep-ph]]

    2008 arXiv

  83. [91]

    Grossman, Int

    Y . Grossman, Int. J. Mod. Phys. A 19 (2004), 907-917 doi:10.1142/S0217751X04018865 [ arXiv:hep-ph/0310229 [hep-ph]]

    2004 arXiv

  84. [92]

    P . K. Das and K. C. Yang, Phys. Rev. D 71 (2005), 094002 doi:10.1103/PhysRevD.71.094002 [ arXiv:hep-ph/0412313 [hep-ph]]

    2005 arXiv

  85. [93]

    C. H. Chen and C. Q. Geng, Phys. Rev. D 71 (2005), 115004 doi:10.1103/PhysRevD.71.115004 [ arXiv:hep-ph/0504145 [hep-ph]]

    2005 arXiv

  86. [94]

    Y . D. Yang, R. M. Wang and G. R. Lu, Phys. Rev. D 72 (2005), 015009 doi:10.1103/PhysRevD.72.015009 [ arXiv:hep-ph/0411211 [hep-ph]]

    2005 arXiv

  87. [95]

    A. L. Kagan, Phys. Lett. B 601 (2004), 151-163 doi:10.1016/j.physletb.2004.09.030 [ arXiv:hep-ph/0405134 [hep-ph]]

    2004 arXiv

  88. [96]

    Beneke, J

    M. Beneke, J. Rohrer and D. Yang, Phys. Rev. Lett. 96 (2006), 141801 doi:10.1103/PhysRevLett.96.141801 [ arXiv:hep-ph/0512258 [hep-ph]]

    2006 arXiv

  89. [97]

    Datta, A

    A. Datta, A. V . Gritsan, D. London, M. Nagashima and A. Sz ynkman, Phys. Rev. D 76 (2007), 034015 doi:10.1103/PhysRevD.76.034015 [arXiv:0705.3915 [hep-ph]]

    2007 arXiv

  90. [98]

    Colangelo, F

    P . Colangelo, F. De Fazio and T. N. Pham, Phys. Lett. B597 (2004), 291-298 doi:10.1016/j.physletb.2004.07.024 [arXiv:hep-ph/0406162 [hep-ph]]

    2004 arXiv

  91. [99]

    Ladisa, V

    M. Ladisa, V . Laporta, G. Nardulli and P . Santorelli, Ph ys. Rev. D 70 (2004), 114025 doi:10.1103/PhysRevD.70.114025 [arXiv:hep-ph/0409286 [hep-ph]]

    2004 arXiv

  92. [100]

    H. Y . Cheng, C. K. Chua and A. Soni, Phys. Rev. D 71 (2005), 014030 doi:10.1103/PhysRevD.71.014030 [ arXiv:hep-ph/0409317 [hep-ph]]

    2005 arXiv

  93. [101]

    Alvarez, L

    E. Alvarez, L. N. Epele, D. Gomez Dumm and A. Szynkman, P hys. Rev. D 70 (2004), 115014 doi:10.1103/PhysRevD.70.115014 [arXiv:hep-ph/0410096 [hep-ph]]

    2004 arXiv

  94. [102]

    K. C. Yang, Phys. Rev. D 72 (2005), 034009 [erratum: Phys. Rev. D 72 (2005), 059901] doi:10.1103/PhysRevD.72.034009 [arXiv:hep-ph/0506040 [hep-ph]]

    2005 arXiv

  95. [103]

    S. Baek, A. Datta, P . Hamel, O. F. Hernandez and D. Londo n, Phys. Rev. D 72 (2005), 094008 doi:10.1103/PhysRevD.72.094008 [arXiv:hep-ph/0508149 [hep-ph]]

    2005 arXiv

  96. [104]

    C. S. Huang, P . Ko, X. H. Wu and Y . D. Yang, Phys. Rev. D 73 (2006), 034026 doi:10.1103/PhysRevD.73.034026 [arXiv:hep-ph/0511129 [hep-ph]]

    2006 arXiv

  97. [105]

    C. H. Chen and H. Hatanaka, Phys. Rev. D 73 (2006), 075003 doi:10.1103/PhysRevD.73.075003 [ arXiv:hep-ph/0602140 [hep-ph]]

    2006 arXiv

  98. [106]

    Faessler, T

    A. Faessler, T. Gutsche, J. C. Helo, S. Kovalenko and V . E. Lyubovitskij, Phys. Rev. D 75 (2007), 074029 doi:10.1103/PhysRevD.75.074029 [arXiv:hep-ph/0702020 [hep-ph]]

    2007 arXiv

  99. [107]

    C. H. Chen, C. Q. Geng, Y . K. Hsiao and Z. T. Wei, Phys. Rev . D 72 (2005), 054011 doi:10.1103/PhysRevD.72.054011 [arXiv:hep-ph/0507012 [hep-ph]]

    2005 arXiv

  100. [108]

    C. H. Chen and C. Q. Geng, Phys. Rev. D 75 (2007), 054010 doi:10.1103/PhysRevD.75.054010 [ arXiv:hep-ph/0701023 [hep-ph]]

    2007 arXiv

  101. [109]

    H. Y . Cheng and K. C. Yang, Phys. Rev. D 83 (2011), 034001 doi:10.1103/PhysRevD.83.034001 [ arXiv:1010.3309 [hep-ph]]

    2011 arXiv

  102. [110]

    Bobeth, M

    C. Bobeth, M. Gorbahn and S. Vickers, Eur. Phys. J. C 75 (2015) no.7, 340 doi:10.1140/epjc/s10052-015-3535-1 [ arXiv:1409.3252 [hep-ph]]

    2015 arXiv

  103. [111]

    H. n. Li and S. Mishima, Phys. Rev. D 73, 114014 (2006) doi:10.1103/PhysRevD.73.114014 [ arXiv:hep-ph/0602214 [hep-ph]]

    2006 arXiv

  104. [112]

    Y . Li, G. H. Zhao, Y . J. Sun and Z. T. Zou, Phys. Rev. D 106, no.9, 093009 (2022) doi:10.1103/PhysRevD.106.093009 22 [arXiv:2209.13389 [hep-ph]]

    2022 arXiv

  105. [113]

    Aaij et al

    R. Aaij et al. [LHCb], Phys. Rev. D 108, no.1, 012013 (2023) doi:10.1103/PhysRevD.108.012013 [ arXiv:2206.02038 [hep-ex]]

    2023 arXiv

  106. [114]

    H. Y . Cheng, [arXiv:2005.06080 [hep-ph]]

    2005 arXiv

  107. [115]

    J. Chai, S. Cheng, Y . h. Ju, D. C. Yan, C. D. L¨ u and Z. J. Xiao, Chin. Phys. C 46, no.12, 123103 (2022) doi:10.1088/1674-1137/ac88bd [arXiv:2207.04190 [hep-ph]]

    2022 arXiv

  108. [116]

    Cheng, Z

    S. Cheng, Z. J. Xiao and Y . L. Zhang, Nucl. Phys. B 896, 255-280 (2015) doi:10.1016/j.nuclphysb.2015.04.021 [ arXiv:1409.5947 [hep-ph]]

    2015 arXiv

  109. [117]

    J. X. Y u, J. J. Han, Y . Li, H. n. Li, Z. J. Xiao and F. S. Y u, [ arXiv:2409.02821 [hep-ph]]

    arXiv

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