Pith. sign in

REVIEW 4 major objections 4 minor 1 cited by

Flavor SU(3) analysis of the charmless semileptonic $B \to PV\ell^+\nu_\ell$ decays

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Using SU(3) flavor symmetry, the paper predicts that most charmless B semileptonic decays through axial-vector resonances have branching ratios of order 10^-4 to 10^-3, making several modes accessible to Belle II and LHCb.

desk verdict Honest SU(3) survey of charmless semileptonic B -> PV with resonance chains, but the quoted A-resonance rates are not observable totals because same-partial-wave interference is left undetermined. read the letter →

arxiv 2507.17537 v2 pith:KZ7ME5O4 submitted 2025-07-23 hep-ph

Yi Qiao , Jin-Huan Sheng , Yuan-Guo Xu , Ru-Min Wang This is my paper
classification hep-ph
keywords flavorSU(3)symmetrysemileptonicBdecaysaxial-vectormesonstensorexcitedvectorbranchingratiopredictionsBelleIILHCb
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

The paper tries to establish that flavor SU(3) symmetry—treating up, down, and strange quarks as interchangeable—can organize the hadronic amplitudes of charmless semileptonic B decays into pseudoscalar-plus-vector final states through intermediate axial-vector, tensor, and excited-vector resonances. It derives branching-ratio predictions for the three-body B to resonance decays and then for the four-body resonance-plus-decay chains. The central numerical claim is that most B to axial-vector modes and many of the four-body axial-vector modes sit at $10^{-4}$ to $10^{-3}$, while tensor and excited-vector contributions are small. Since none of these modes has been measured yet, the paper gives Belle II and LHCb specific rate targets and relative-ratio relations to look for.

What carries the argument

The engine is the SU(3) flavor decomposition of the hadronic amplitudes, written as H(B to M l nu) = c0^M B_i M^j_i H_j plus symmetry-breaking terms; setting all modes in a multiplet to the same coefficient C^M turns relative form factors into fixed ratios. Hadronic helicity amplitudes in the appendix then convert these form factors into $q^{2}$ spectra and branching ratios. For excited vector mesons, the strong decays V' to PV are set by one coupling g_{V'} through g_{V'} V'^i_j P^k_i V^j_k, with the normalization fixed by K*(1680) data and, for the lighter excited vectors, by a model-based ratio; four-body rates follow by the narrow-width approximation or a width-averaged integral.

What would settle it

A precise measurement of the K*(1410) branching fractions that respects both existing 95 percent CL limits, K*(1410) to K* pi above 40 percent and to rho K below 7 percent, would test the single-coupling scheme; likewise, measuring B(omega(1650) to rho pi) at any value below the predicted order of 100 percent would falsify the SU(3) strong-decay relation for that resonance.

Watch

Extended reading notes

Core claim

The central claim is that a single SU(3) flavor amplitude parametrization, with one nonperturbative coefficient per meson multiplet, suffices to relate the hadronic form factors of B to axial-vector, tensor, and excited-vector mesons. Using these relations, the paper obtains branching ratios for the three-body decays and then combines them with the strong decay rates of the resonances to predict four-body B to PV lepton-neutrino branching ratios, both by the narrow-width approximation and by a width-averaging prescription. Its main quantitative finding is that most B to axial-vector rates and many axial-vector-mediated four-body rates lie in the $10^{-4}$ to $10^{-3}$ range, whereas tensor and excited-vector contributions are generically much smaller. The paper deliberately does not quote total four-body branching ratios because interference between same-partial-wave resonances cannot be fixed by SU(3) alone.

Load-bearing premise

The load-bearing premise is that one flavor-SU(3) coupling, with flavor breaking ignored, describes both the B-to-excited-vector form factors and the excited-vector strong decays; the paper's own results strain this by predicting an impossible sum for omega(1650) and by failing to satisfy the K*(1410) limits at the same time.

Editorial extensions

If this is right

  • Axial-vector three-body modes such as B0 -> b1(1235)^- l nu, predicted near 5 x 10^-3, move from unobserved to plausible first-discovery channels for Belle II and LHCb.
  • Many four-body axial-vector channels, for example B0 -> omega pi^- l nu and B0_s -> rho^- K^0 l nu, are predicted in the 10^-3 range and are the best early targets.
  • Tensor and excited-vector contributions remain below about 10^-4, so in most channels they cannot mimic the axial-vector signal; the exceptions are B -> rho eta' l nu modes, where excited vectors dominate.
  • Width effects reduce several axial-vector resonant rates noticeably, so comparisons with data must use the same resonance-shape prescription as the prediction.
  • SU(3) fixes ratios between related modes, meaning a single measured channel can calibrate an entire multiplet of predicted rates.

Reading between the lines

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

  • If one axial-vector channel is measured, the same symmetry coefficient calibrates every B to axial-vector rate, turning one experimental point into a global test of the scheme.
  • The excited-vector sector is the fragile part: the paper itself notes that the K*(1410) limits cannot be satisfied simultaneously and that the omega(1650) strong-decay sum exceeds 100 percent, so better data on these resonances would sharply constrain that sector.
  • The same SU(3) decomposition should apply to the corresponding tau-neutrino modes, giving a cross-check of the predicted axial-vector rates at future tau-capable facilities.
  • Because same-partial-wave interference is left uncalculated, the quoted 10^-3 estimates are order-of-magnitude guides rather than precise line-shape predictions.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper presents a flavor-SU(3) analysis of charmless semileptonic B -> PV l nu decays mediated by axial-vector (A), tensor (T), and excited-vector (V_E, V_D) resonances. It decomposes the B -> M l nu hadronic amplitudes into SU(3) invariant coefficients, uses external form-factor inputs and measured constraints (B+ -> pi+ pi- l nu for the tensor form factor A_T1(0), and K*(1680) -> K*pi, rho K decays for the strong coupling g_VD), and then combines B -> R l nu with R -> PV branching ratios through the narrow-width approximation (Eq. (10)) or a width-integrated formula (Eq. (11)) to predict four-body resonant rates. The main numerical conclusions are that many A-mediated modes have branching ratios of order 10^-4 to 10^-3, while T- and excited-vector-mediated modes are small, and that some of these modes could be tested at Belle II and LHCb.

Significance. If the predictions were physical observables, the paper would provide a useful survey of unmeasured charmless semileptonic channels and a consistency test of SU(3) for excited mesons. The strengths of the manuscript are its transparent amplitude decomposition, the explicit statements of caveats about interference and SU(3) breaking, and the use of experimental constraints to fix the tensor and excited-vector couplings. However, the quoted four-body numbers are incoherent resonance products rather than the physical coherent-sum rates, and the excited-vector sector contains internal inconsistencies: the SU(3) scheme predicts B(omega(1650) -> PV) above 100%, and the two 95% CL limits on K*(1410) decays cannot be simultaneously satisfied. These issues do not invalidate the SU(3) relations themselves, but they require the phenomenological claims to be reframed and the excited-vector part to be reworked.

major comments (4)
  1. [III, final paragraphs; Tables V and VI] The paper's central quantitative claim is that B -> A(A -> PV) l nu rates of order 10^-4 to 10^-3 are testable at Belle II and LHCb. But Tables V and VI list only incoherent products B(B -> R l nu) x B(R -> PV), while the physical B -> PV l nu rate is |sum_R A_R + A_nonres|^2. The text itself states in Section III that both a1(1260)^0 and h1(1170)^0 give large contributions to B+ -> rho- pi+ l nu and that their interference 'might be large', and that total branching ratios are not given because the interference cannot be determined by SU(3) flavor symmetry. With unknown relative strong phases, the sum of the two quoted contributions, near 1.3e-3 for B+ -> rho- pi+ l nu in the S1 case, can be largely cancelled or doubled. Since the abstract and conclusions nevertheless present these as 'branching ratio predictions' testable in experiments, the paper should either quote constructive/destructive interference envelopes for the affected channels or explicitly restrict the claim to resonant amplitudes that would be extracted from a Dalitz-plot analysis, not to total branching fractions.
  2. [III.C, Table VIII] Table VIII predicts B(omega(1650) -> rho pi) = 108.97 +/- 8.90%, and adding the predicted K*K and omega eta modes gives a total B(omega(1650) -> PV) above 130%, which is why the authors are forced to impose an ad hoc ceiling of 130%. A physical branching ratio cannot exceed 100%, so this is an internal inconsistency of the single-coupling SU(3) scheme for VD -> PV. Because the same g_VD is used for all VD modes, the Table X predictions for B -> VD(VD -> PV) l nu inherit this problem and cannot be regarded as quantitative until unitarity is imposed or SU(3) breaking is introduced in the strong couplings. This issue directly affects the VD-sector predictions that the paper claims are small but still part of the overall analysis.
  3. [III.C, Eq. (16) and Table IX] The two 95% CL limits on K*(1410) decays in Eq. (16), namely B(K*(1410) -> K*pi) > 40% and B(K*(1410) -> rho K) < 7%, are mutually inconsistent with the SU(3) relations, so no data-based determination of g_VE is possible. The authors therefore borrow g_VE from Ref. [48], which is a model calculation. This means that the VE branching ratios in Table IX and the VE contributions in Table X are model-dependent inputs rather than predictions obtained from SU(3) plus experimental data. The paper should state this limitation explicitly in the abstract and conclusions and should propagate the model uncertainty into the quoted VE-mediated four-body rates.
  4. [II.B/III.C, Eq. (14) and Table VIII] The g_VD fit does not provide a simultaneous description of the K*(1680) data. The paper fits g_VD to B(K*(1680) -> K*pi) = (29.9 +2.2 -5.0)% and B(K*(1680) -> rho K) = (31.4 +5.0 -2.1)%, but then predicts B(K*(1680) -> K*pi) = 22.5 +/- 5.4% and B(K*(1680) -> rho K) = 17.6 +/- 4.3%. The rho K prediction is more than 1 sigma below the measured value, and the authors state that both data points can be satisfied only when the experimental errors are extended to 2 sigma. The paper should report the fit quality and treat the K*(1680) -> K*pi row of Table VIII as a postdiction that partly recycles the fitted input rather than as an independent prediction.
minor comments (4)
  1. [III.C, after Eq. (14)] The sentence defining g_VD says it is obtained from B(K*(1680) -> K*pi)_Exp twice; the second instance should refer to B(K*(1680) -> rho K)_Exp.
  2. [Table XI and Ref. [44]] Ref. [44] is listed as an unpublished manuscript ('in preparing'), yet Table XI reports numerical form factors from it. The authors should either cite a published version or clearly mark the numbers as private communication with the authors' permission.
  3. [Table VI, B+ -> K*-K+ tau nu row] The entry '0.89 +/- 89^a' in the f1(1420) contribution appears to be a typo for '0.89 +/- 0.89^a'.
  4. [III, final paragraph] The notation ell vs ell' is used inconsistently in the bracketed lists of modes (for example, B(B+ -> K*∓K± ell'+nu_ell') is followed by ell+nu_ell without primes elsewhere); this makes the summary hard to parse and should be made uniform.

Circularity Check

2 steps flagged · score 4.0 of 10

Partial circularity: the K*(1680) strong-decay 'predictions' recycle the g_VD fit inputs, and the A/T four-body rates rest on the authors' prior Ref. [31]; the central B→Aℓν predictions themselves use external form factors.

  1. fitted input called prediction [Section III.C, Eqs. (12)-(15) and Tables VIII-X]
    "For the VD → PV strong decays, only K∗(1680) → ρK, K∗π decays have been measured [39], as follows: B(K ∗(1680) → ρK)Exp = (31.4+5.0−2.1)%, B(K ∗(1680) → K ∗π)Exp = (29.9+2.2−5.0)%, (14) and they can be used to constrain the nonperturbative parameter gVD . We obtain gVD = 6.48 ± 1.59 from B(K ∗(1680) → K ∗π)Exp and gVD = 5.33 ± 1.33 from B(K ∗(1680) → K ∗π)Exp. The communication between them, gVD ∈ [4.88, 6.66], will be used to obtained the branching ratios and decay widths of the VD → PV strong decays."

    The parameter gVD is fixed by the two measured K*(1680) branching ratios in Eq. (14), and the same two branching ratios are then recomputed from gVD in Table VIII (e.g., K*(1680)→K*π = 22.50±5.41% and K*(1680)→ρK = 17.61±4.27%) and presented as predictions that feed Table X. In the 2σ re-fit quoted in Eq. (15), the 'predicted' values (34.12±0.18% and 27.34±0.14%) are essentially the experimental inputs of Eq. (14) rewritten with reduced errors. These entries therefore reduce by construction to their own fit inputs rather than being independent predictions. The affected VD-mediated four-body rates are subdominant, so this is a partial, not global, circularity.

  2. self citation load bearing [Section III.A, Eq. (10) and Tables V-VI (A/T→PV inputs from Ref. [31])]
    "The branching ratios of the two-body non-leptonic decays B(A → P V) and B(T → P V) have been studied in our previous work [31]. Using the expressions of B(B → A/T ℓ+νℓ) given in Sec. II and B(A/T → P V) given in Ref. [31], we may predict the branching ratios of the B → A(A → P V)ℓ+νℓ and B → T (T → P V)ℓ+νℓ decays, which are listed in the second and third columns of Tabs. V and VI, respectively, named as the results in S1 case."

    The four-body A/T rates are assembled by Eq. (10) as products whose second factor is imported wholesale from the authors' own Ref. [31]; the paper performs no independent derivation, numerical check, or external benchmark of B(A→PV) or B(T→PV) in this work. Since those strong-decay amplitudes are load-bearing for the headline O(10^-4–10^-3) A-mediated branching ratios, the central four-body output rests on a same-author citation. This is not a definitional identity: Ref. [31] is a separate published SU(3) analysis, so the self-citation is a support-structure caveat rather than a proof of equivalence.

full rationale

The core three-body predictions are not globally circular. The B→Aℓν branching ratios use external form factors from Ref. [40] (with mixing angles from Refs. [31,41]); the B→Tℓν predictions use the measured π+π−ℓν rate to fix A_T1(0) plus external form-factor ratios from Ref. [42]; and the B→VE,Dℓν predictions use external form factors from Refs. [43,44] with SU(3) relating the modes. The four-body product formula Eq. (10) is a standard narrow-width factorization, and the paper explicitly warns that interference among same-partial-wave resonances is not determined by SU(3), so the quoted entries are resonant products rather than measurable coherent sums. The main circularity found is confined to the VD strong-decay sector: gVD is fitted to Eqs. (14), and Table VIII then 'predicts' those same K*(1680) branching ratios, with Eq. (15) reproducing the inputs after a 2σ refit. The A/T→PV factors are taken from the authors' own prior Ref. [31], which is a load-bearing self-citation but not a definitional reduction, since that prior work is a separate published analysis. The f2 entry in Table III is also an experimental input relabeled in an 'Our predictions' column, but it is explicitly daggered and therefore transparent. Overall the central B→Aℓν results retain independent external content, while a subset of the VD-mediated predictions is partially circular.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new particles or interactions. Its results rest instead on fitted form-factor normalizations, strong-decay couplings, mixing angles from the literature, and three symmetry and factorization assumptions, several of which are explicitly acknowledged to be imperfect.

free parameters (4)
  • A_T1(0) tensor form factor at q2=0 = 0.14 ± 0.06
    Determined from the measured B(B+ to f2(1270) l nu) extracted from B(B+ to pi+ pi- l nu) via Eq. (9); used as the normalization for all B to T l nu branching ratio predictions.
  • g_VD (SU(3) strong coupling for VD to PV) = interval 4.88 to 6.66
    Obtained from the two measured K*(1680) branching fractions (K* pi and rho K) in Eq. (14); sets the scale of all VD to PV strong decay rates.
  • g_VE (SU(3) strong coupling for VE to PV) = 5.78 ± 2.07 (derived from g_VD with a ratio from Ref. [48])
    Because the K*(1410) limits cannot be satisfied simultaneously within SU(3), g_VE is taken from the model calculation of Piotrowska et al., so it is effectively a model input, not fitted to data.
  • Mixing angles (theta_K1, theta_3P1, theta_1P1, alpha_f1, alpha_h1, theta_f2, theta_P) = ranges from PDG and Refs. [31,41,64] (e.g., theta_K1: 52 to 65 degrees or 33 ± 4 degrees; theta_3P1: 56 to 125…
    Chosen from the literature; different choices for K1 and f1/h1 mixing produce order-of-magnitude variations in several branching ratio predictions (case a vs b, natural vs sharp).
assumptions (6)
  • domain assumption The b to u l nu transition is described by the SM effective Hamiltonian H_eff = (G_F/√2) V_ub b-bar gamma_mu (1 - gamma_5) u nu-bar_l gamma^mu (1 - gamma_5) l (Eq. 1).
    Standard electroweak input; no new physics is considered.
  • domain assumption Flavor SU(3) symmetry relates hadronic amplitudes and form factors across the B to A/T/VE/D multiplets; SU(3) breaking terms c_1,2 are set to zero (Eq. 8, Table I, Sec. II A).
    This is the load-bearing symmetry assumption. The authors note breaking is ignored and may cause roughly 40 percent errors in some branching ratios.
  • domain assumption The narrow width approximation B(B to PV l nu) = B(B to R l nu) B(R to PV) and the width-convoluted formula Eq. (11) correctly factorize resonant contributions.
    Used throughout Sec. III to combine three-body and two-body rates; interference between resonances is neglected.
  • domain assumption The A/T to PV branching ratios taken from Ref. [31] (same author group) are correct.
    These are inputs to all B to A, A to PV, l nu and B to T, T to PV, l nu predictions; no independent measurement is cited.
  • domain assumption External form factors from Refs. [40], [42], [43], [44] describe B to A/T/VE,D transitions.
    The central B to R l nu rates are computed from these form factors; Ref. [44] is unpublished, and different models (LEET vs light-front) give different predictions.
  • domain assumption The strong V prime to PV couplings g_{V prime to PV} are parametrized by a single SU(3) coupling g_{V prime} per multiplet (Eq. 13).
    This parametrization produces B(omega(1650) to PV) above 100 percent and cannot satisfy the K*(1410) limits simultaneously, indicating the assumption is strained.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Flavor SU(3) analysis of the charmless semileptonic $B \to PV\ell^+\nu_\ell$ decays." pith.science (2026). https://pith.science/paper/KZ7ME5O4

@misc{pith2026250717537,
  author       = {Pith},
  title        = {Pith review of: Flavor SU(3) analysis of the charmless semileptonic $B \to PV\ell^+\nu_\ell$ decays},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KZ7ME5O4}},
  note         = {Machine review of arXiv:2507.17537}
}
abstract

All charmless $B \to PV\ell^+\nu_\ell$ ($P$ denotes the light pseudoscalar meson and $V$ denotes the vector meson) decays have not been experimentally observed to date, but some of them might be measured in the near future. The charmless $B \to PV\ell^+\nu_\ell$ decays with the axial-vector ($A$) resonance states, the tensor ($T$) resonance states and the excited vector ($V_{E,D}$) resonance states are studied based on flavor SU(3) analysis in this work, where $V_E$ contains $\{\rho(1450), K^*(1410), \omega(1420), \phi(1680)\}$ with quantum numbers $n^{2S+1}L_J=2^3S_1$ and $V_D$ contains $\{\rho(1700), K^*(1680), \omega(1650), \phi(2170)\}$ with quantum numbers $n^{2S+1}L_J=1^3D_1$. The hadronic amplitudes of the $B \to A/T \ell^+\nu_{\ell}$ decays are related by the nonperturbative parameters, and the branching ratio predictions of $B \to A/T/V_{E,D} \ell^+\nu_{\ell}$ are obtained, and then the branching ratios of the $B\to A(A\to PV)\ell^+\nu_\ell$, $B\to T(T\to PV)\ell^+\nu_\ell$, $B\to V_E(V_E\to PV)\ell^+\nu_\ell$ and $B\to V_D(V_D\to PV)\ell^+\nu_\ell$ are obtained by the narrow width approximation or further considering the width effects. We find that most branching ratios of the $B \to A \ell^+\nu_{\ell}$ decays and many branching ratios of the $B\to A(A\to PV)\ell^+\nu_\ell$ decays are on the order of $\mathcal{O}(10^{-4}-10^{-3})$, and all branching ratios with the tensor resonance states and the excited vector resonance states are small. Our results could be tested in the future BelleII and LHCb experiments.

Discussion (0). Sign in to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Studying the tensor resonance contributions in $B \to PP\ell^+\ell^-$ and $B \to PV\ell^+\ell^-$ decays

    hep-ph 2026-07 conditional novelty 4.0 of 10

    Tensor-resonance contributions to B → PPℓ⁺ℓ⁻ and B → PVℓ⁺ℓ⁻ are small compared with measured totals once SU(3)-related B → Tℓ⁺ℓ⁻ rates are fixed to Bs → f′₂(1525)μ⁺μ⁻.

Reference graph

Works this paper leans on

64 extracted references · 37 canonical work pages · cited by 1 Pith paper

  1. [44]

    Liting Wang, Qin Chang, in preparing

    work page

  2. [48]

    Strong and radiative decays of excited vector mesons and predictions for a new $\phi(1930)$ resonance

    M. Piotrowska, C. Reisinger and F. Giacosa, Phys. Rev. D 96, 054033 (2017) [arXiv:1708.02593 [hep-ph]]

    work page Pith review arXiv 2017

  3. [1]

    X. G. He, Eur. Phys. J. C 9, 443 (1999) [hep-ph/9810397]

    work page Pith review arXiv 1999

  4. [2]

    X. G. He, Y. K. Hsiao, J. Q. Shi, Y. L. Wu and Y. F. Zhou, Phys. Rev. D 64, 034002 (2001) [hep-ph/0011337]

    work page Pith review arXiv 2001

  5. [3]

    H. K. Fu, X. G. He and Y. K. Hsiao, Phys. Rev. D 69, 074002 (2004) [hep-ph/0304242]

    work page Pith review arXiv 2004

  6. [4]

    Y. K. Hsiao, C. F. Chang and X. G. He, Phys. Rev. D 93, 114002 (2016) [arXiv:1512.09223 [hep-ph]]

    arXiv 2016

  7. [5]

    X. G. He and G. N. Li, Phys. Lett. B 750, 82 (2015) [arXiv:1501.00646 [hep-ph]]

    arXiv 2015

  8. [6]

    Gronau, O

    M. Gronau, O. F. Hernandez, D. London and J. L. Rosner, Phys. Rev. D 50, 4529 (1994) [hep-ph/9404283]

    arXiv 1994

Show all 64 references
  1. [7]

    Gronau, O

    M. Gronau, O. F. Hernandez, D. London and J. L. Rosner, Phys. Rev. D 52, 6356 (1995) [hep-ph/9504326]

    1995 arXiv

  2. [8]

    S. H. Zhou, Q. A. Zhang, W. R. Lyu and C. D. L, Eur. Phys. J. C 77, 125 (2017) [arXiv:1608.02819 [hep-ph]]. 18

    2017 arXiv

  3. [9]

    H. Y. Cheng, C. W. Chiang and A. L. Kuo, Phys. Rev. D 91, 014011 (2015) [arXiv:1409.5026 [hep-ph]]

    2015 arXiv

  4. [10]

    M. He, X. G. He and G. N. Li, Phys. Rev. D 92, 036010 (2015) [arXiv:1507.07990 [hep-ph]]

    2015 arXiv

  5. [11]

    N. G. Deshpande and X. G. He, Phys. Rev. Lett. 75, 1703 (1995) [hep-ph/9412393]

    1995 arXiv

  6. [12]

    Shivashankara, W

    S. Shivashankara, W. Wu and A. Datta, Phys. Rev. D 91, 115003 (2015) [arXiv:1502.07230 [hep-ph]]

    2015 arXiv

  7. [13]

    R. M. Wang, Y. G. Xu, C. Hua and X. D. Cheng, Phys. Rev. D 103, 013007 (2021) [arXiv:2101.02421 [hep-ph]]

    2021 arXiv

  8. [14]

    R. M. Wang, X. D. Cheng, Y. Y. Fan, J. L. Zhang and Y. G. Xu, J. Phys. G 48, 085001 (2021) [arXiv:2008.06624 [hep-ph]]

    2021 arXiv

  9. [15]

    Grossman and D

    Y. Grossman and D. J. Robinson, JHEP 1304, 067 (2013) [arXiv:1211.3361 [hep-ph]]

    2013 arXiv

  10. [16]

    Pirtskhalava and P

    D. Pirtskhalava and P. Uttayarat, Phys. Lett. B 712, 81 (2012) [arXiv:1112.5451 [hep-ph]]

    2012 arXiv

  11. [17]

    H. Y. Cheng and C. W. Chiang, Phys. Rev. D 86, 014014 (2012) [arXiv:1205.0580 [hep-ph]]

    2012 arXiv

  12. [18]

    M. J. Savage and R. P. Springer, Phys. Rev. D 42, 1527 (1990)

    1990

  13. [19]

    M. J. Savage, Phys. Lett. B 257, 414 (1991)

    1991

  14. [20]

    Altarelli, N

    G. Altarelli, N. Cabibbo, and L. Maiani, Phys. Lett. B 57, 277 (1975)

    1975

  15. [21]

    C. D. L¨ u, W. Wang and F. S. Yu, Phys. Rev. D 93, 056008 (2016) [arXiv:1601.04241 [hep-ph]]

    2016 arXiv

  16. [22]

    C. Q. Geng, Y. K. Hsiao, Y. H. Lin and L. L. Liu, Phys. Lett. B 776, 265 (2018) [arXiv:1708.02460 [hep-ph]]

    2018 arXiv

  17. [23]

    C. Q. Geng, Y. K. Hsiao, C. W. Liu and T. H. Tsai, Phys. Rev. D 97, 073006 (2018) [arXiv:1801.03276 [hep-ph]]

    2018 arXiv

  18. [24]

    C. Q. Geng, Y. K. Hsiao, C. W. Liu and T. H. Tsai, JHEP 1711, 147 (2017) [arXiv:1709.00808 [hep-ph]]

    2017 arXiv

  19. [25]

    C. Q. Geng, C. W. Liu, T. H. Tsai and S. W. Yeh, Phys. Lett. B 792, 214 (2019) [arXiv:1901.05610 [hep-ph]]

    2019 arXiv

  20. [26]

    W. Wang, Z. P. Xing and J. Xu, Eur. Phys. J. C 77, 800 (2017) [arXiv:1707.06570 [hep-ph]]

    2017 arXiv

  21. [27]

    Wang, Eur

    D. Wang, Eur. Phys. J. C 79, 429 (2019) [arXiv:1901.01776 [hep-ph]]

    2019 arXiv

  22. [28]

    D. Wang, P. F. Guo, W. H. Long and F. S. Yu, JHEP 1803, 066 (2018) [arXiv:1709.09873 [hep-ph]]

    2018 arXiv

  23. [29]

    M¨ uller, U

    S. M¨ uller, U. Nierste and S. Schacht, Phys. Rev. D 92, 014004 (2015) [arXiv:1503.06759 [hep-ph]]

    2015 arXiv

  24. [30]

    Y. Qiao, Y. X. Liu, Y. G. Xu and R. M. Wang, Eur. Phys. J. C 84, 1110 (2024) [arXiv:2404.03857 [hep-ph]]

    2024 arXiv

  25. [31]

    R. M. Wang, Y. Qiao, Y. J. Zhang, X. D. Cheng and Y. G. Xu, Phys. Rev. D 107, 056022 (2023) [arXiv:2301.00090 [hep-ph]]

    2023 arXiv

  26. [32]

    Feldmann, D

    T. Feldmann, D. Van Dyk and K. K. Vos, JHEP 10, 030 (2018) [arXiv:1807.01924 [hep-ph]]

    2018 arXiv

  27. [33]

    Faller, T

    S. Faller, T. Feldmann, A. Khodjamirian, T. Mannel and D. van Dyk, Phys. Rev. D 89, 014015 (2014) [arXiv:1310.6660 [hep-ph]]

    2014 arXiv

  28. [34]

    C. S. Kim, G. L. Castro and S. L. Tostado, Phys. Rev. D 95, 073003 (2017) [arXiv:1702.01704 [hep-ph]]

    2017 arXiv

  29. [35]

    Ananthanarayan and K

    B. Ananthanarayan and K. Shivaraj, Phys. Lett. B 628, 223 (2005) [arXiv:hep-ph/0508116 [hep-ph]]

    2005 arXiv

  30. [36]

    M. A. Ivanov, J. G. K¨ orner, J. N. Pandya, P. Santorelli, N. R. Soni and C. T. Tran, Front. Phys. (Beijing) 14, 64401 (2019) [arXiv:1904.07740 [hep-ph]]

    2019 arXiv

  31. [37]

    D. Xu, G. N. Li and X. G. He, Int. J. Mod. Phys. A 29, 1450011 (2014) [arXiv:1307.7186 [hep-ph]]

    2014 arXiv

  32. [38]

    X. G. He, G. N. Li and D. Xu, Phys. Rev. D 91, 014029 (2015) [arXiv:1410.0476 [hep-ph]]

    2015 arXiv

  33. [39]

    Navas et al

    S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)

    2024

  34. [40]

    R. C. Verma, J. Phys. G 39, 025005 (2012) [arXiv:1103.2973 [hep-ph]]

    2012 arXiv

  35. [41]

    X. W. Kang, T. Luo, Y. Zhang, L. Y. Dai and C. Wang, Eur. Phys. J. C 78, 909 (2018) [arXiv:1808.02432 [hep-ph]]

    2018 arXiv

  36. [42]

    L. Chen, Y. W. Ren, L. T. Wang and Q. Chang, Eur. Phys. J. C 82 (2022), 451 [arXiv:2112.08016 [hep-ph]]

    2022 arXiv

  37. [43]

    Hatanaka and K

    H. Hatanaka and K. C. Yang, Eur. Phys. J. C 67, 149 (2010) [arXiv:0907.1496 [hep-ph]]

    2010 arXiv

  38. [45]

    H. Y. Cheng, C. Y. Cheung, W. Dimm, G. L. Lin, Y. C. Lin, T. M. Yan and H. L. Yu, Phys. Rev. D 48, 3204 (1993) [arXiv:hep-ph/9305340 [hep-ph]]. 19

    1993 arXiv

  39. [46]

    S. Y. Tsai and Y. K. Hsiao, [arXiv:2107.03634 [hep-ph]]

    arXiv

  40. [47]

    Liang, S

    J. Liang, S. Chen, Y. Chen, C. Shi and W. Sun, Sci. China Phys. Mech. Astron. 68, 251011 (2025) [arXiv:2409.14410 [hep-lat]]

    2025 arXiv

  41. [49]

    X. W. Kang, B. Kubis, C. Hanhart and U. G. Meißner, Phys. Rev. D 89, 053015 (2014) [arXiv:1312.1193 [hep-ph]]

    2014 arXiv

  42. [50]

    G. S. Yang and H. C. Kim, Phys. Rev. C 92, 035206 (2015) [arXiv:1504.04453 [hep-ph]]

    2015 arXiv

  43. [51]

    M. E. Carrillo-Serrano, I. C. Clo¨ et and A. W. Thomas, Phys. Rev. C 90, 064316 (2014) [arXiv:1409.1653 [nucl-th]]

    2014 arXiv

  44. [52]

    C. Q. Geng, Y. K. Hsiao, C. W. Liu and T. H. Tsai, Eur. Phys. J. C 78, 593 (2018) [arXiv:1804.01666 [hep-ph]]

    2018 arXiv

  45. [53]

    Flores-Mendieta, E

    R. Flores-Mendieta, E. E. Jenkins and A. V. Manohar, Phys. Rev. D 58, 094028 (1998) [arXiv:hep-ph/9805416 [hep-ph]]

    1998 arXiv

  46. [54]

    Y. J. Sun, Z. G. Wang and T. Huang, Chin. Phys. C 36, 1046 (2012) [arXiv:1106.4915 [hep-ph]]

    2012 arXiv

  47. [55]

    R. H. Li, C. D. Lu and W. Wang, Phys. Rev. D 79, 034014 (2009) [arXiv:0901.0307 [hep-ph]]

    2009 arXiv

  48. [56]

    Colangelo, F

    P. Colangelo, F. De Fazio and F. Loparco, Phys. Rev. D 100, 075037 (2019) [arXiv:1906.07068 [hep-ph]]

    2019 arXiv

  49. [57]

    H. Y. Cheng and X. W. Kang, Eur. Phys. J. C 77, 587 (2017) [erratum: Eur. Phys. J. C 77, 863 (2017)] [arXiv:1707.02851 [hep-ph]]

    2017 arXiv

  50. [58]

    H. Y. Cheng, C. K. Chua and C. W. Hwang, Phys. Rev. D 69, 074025 (2004) [arXiv:hep-ph/0310359 [hep-ph]]

    2004 arXiv

  51. [59]

    L. Roca, J. E. Palomar and E. Oset, Phys. Rev. D 70, 094006 (2004) [arXiv:hep-ph/0306188 [hep-ph]]

    2004 arXiv

  52. [60]

    X. G. He, Y. J. Shi and W. Wang, Eur. Phys. J. C 80, 359 (2020) [arXiv:1811.03480 [hep-ph]]

    2020 arXiv

  53. [61]

    Bramon, A

    A. Bramon, A. Grau and G. Pancheri, Phys. Lett. B 283, 416 (1992)

    1992

  54. [62]

    C. Chen, N. Q. Cheng, L. W. Yan, C. G. Duan and Z. H. Guo, Phys. Rev. D 108, 014002 (2023) [arXiv:2302.11316 [hep-ph]]

    2023 arXiv

  55. [63]

    Ecker and C

    G. Ecker and C. Zauner, Eur. Phys. J. C 52, 315 (2007) [arXiv:0705.0624 [hep-ph]]

    2007 arXiv

  56. [64]

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

    2011 arXiv

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.