Pith. sign in

REVIEW 4 major objections 5 minor 1 cited by

Roles of $f_{0}(500)$ and $f_{0}(980)$ in the $D_{(s)}^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}$ decays

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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

Pith's one-line read In $D_s^+\to\pi^+\pi^+\pi^-$ decays the $f_0(500)$ is absent because the weak decay hadronizes only into $K\bar K$ pairs, while in $D^+\to\pi^+\pi^+\pi^-$ all five coupled channels are produced and both $f_0(500)$ and $f_0(980)$ appear.

desk verdict The structural hadronization argument distinguishing D_s and D+ is clean and new, but the exact cancellation that removes f0(500) from D_s rests on an untested equal-weight vacuum ansatz. read the letter →

arxiv 2411.18134 v1 pith:RPJCKW7S submitted 2024-11-27 hep-ph

Zhong-Yu Wang , Wei Liang This is my paper · ORCID
classification hep-ph
keywords f0(500)mesonf0(980)chiralunitaryapproachfinal-stateinteractionshadronizationD_sthree-bodydecaysscalarmolecularstatesBethe-Salpeterequation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper sets out to explain a sharp experimental puzzle: in the Cabibbo-favored decay $D_s^+\to\pi^+\pi^+\pi^-$ only the $f_0(980)$ resonance shows up in the $\pi^+\pi^-$ S-wave amplitude, while the $D^+\to\pi^+\pi^+\pi^-$ decay shows both $f_0(500)$ and $f_0(980)$. The authors show that the difference follows from the dominant external and internal $W$-emission mechanisms at the quark level together with the final-state rescattering of the hadronized meson pairs. Because the $D_s^+$ weak decay hadronizes only into $K^+K^-$ and $K^0\bar K^0$ pairs, the $\pi\pi$ channel that would seed $f_0(500)$ is never produced; the $D^+$ decay, in contrast, produces all five coupled channels $(\pi^+\pi^-,\pi^0\pi^0,K^+K^-,K^0\bar K^0,\eta\eta)$, so both resonances are generated. If the calculation is right, it supports the picture of $f_0(500)$ as a $\pi\pi$ resonance and $f_0(980)$ as a $K\bar K$ molecular state, both dynamically generated by the chiral unitary approach.

What carries the argument

The load-bearing objects are the hadronization identities of Eqs. (1)–(7) and the coupled-channel amplitudes of the chiral unitary approach. The hadronization step replaces each quark–antiquark pair with $(M\cdot M)_{ij}$ using the pseudoscalar matrix $P$ of Eq. (4), with the vacuum creating $\bar u u + \bar d d + \bar s s$ pairs; this is what collapses the $D_s^+$ final state to the $K\bar K$ channels. The scattering of these pairs is then resummed through the Bethe–Salpeter equation $T=[1-VG]^{-1}V$, with the $S$-wave potentials $V_{ij}$ from the lowest-order chiral Lagrangian and the two-meson loop function $G$, producing the dynamically generated resonances that are identified with $f_0(500)$ and $f_0(980)$.

What would settle it

A decisive test would be a precision measurement of the $D_s^+\to\pi^+\pi^+\pi^-$ $\pi^+\pi^-$ S-wave amplitude in the $0.4$–$0.8$ GeV region, where the model undershoots the data: if the phase shows the rapid motion characteristic of the $f_0(500)$ pole, the cancellation in Eq. (7) is incomplete and the central explanation fails. A lattice or QCD-sum-rule evaluation of the hadronization matrix elements that finds the strange-pair weight differing from the up/down weight would also break the derivation.

Watch

Extended reading notes

Core claim

The central claim is that the absence of $f_0(500)$ in $D_s^+\to\pi^+\pi^+\pi^-$ is not accidental but follows from the hadronization of the weak decay. Using the quark-pair-creation ansatz $\bar u u + \bar d d + \bar s s$, the authors derive the hadronic amplitudes of the external and internal $W$-emission diagrams; after the $\eta\eta$ pieces cancel, the $D_s^+$ transition leaves only $\pi^+ K^+ K^-$ and $\pi^+ K^0 \bar K^0$ in S-wave (Eq. (7)). The final $\pi^+\pi^-$ pair then arises purely from rescattering $K\bar K\to\pi^+\pi^-$, which the chiral unitary approach produces through the $f_0(980)$. For $D^+$, the corresponding derivation keeps all five coupled channels and a tree-level $\pi^+\pi^+\pi^-$ term (Eq. (20)), so both $f_0(500)$ and $f_0(980)$ appear. The authors fit the measured magnitudes and phases and find the $D^+$ data are described well, with the $D_s^+$ data described reasonably above the $\pi\pi$ threshold region.

Load-bearing premise

The argument depends on the assumption that the vacuum creates up, down, and strange quark–antiquark pairs with equal weight during hadronization, since that precise equality is what cancels the direct $\pi\pi$ content in the $D_s^+$ decay and leaves only $K\bar K$ channels.

Editorial extensions

If this is right

  • The $D_s^+\to\pi^+\pi^+\pi^-$ amplitude should show no $f_0(500)$ signal, exactly as observed in the current measurements.
  • The $D^+\to\pi^+\pi^+\pi^-$ amplitude contains both the broad $f_0(500)$ bump below 1 GeV and the narrow $f_0(980)$ cusp at the $K\bar K$ threshold.
  • The phase of the $\pi^+\pi^-$ amplitude in the $D_s^+$ decay is parameter-free once the magnitude is fitted, so an improved phase measurement in the low-mass region provides a direct check of the calculation.
  • Dropping the $\eta\eta$ channel widens the $f_0(980)$ and improves the description of the $D_s^+$ magnitude, which the model ascribes to the small width of $f_0(980)$ in the five-channel calculation.
  • The framework identifies the produced isoscalar channels, so $a_0(980)$ and other $I=1$ contributions are predicted to be strongly suppressed in these decays.

Reading between the lines

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

  • If the equal-weight quark-pair creation is replaced by a suppression of strange pairs, the $D_s^+$ decay would acquire a direct $\pi\pi$ component; the size of the $f_0(500)$ signal would then be a measure of the $SU(3)$ breaking in the hadronization vacuum, a quantity that could be extracted from data or from lattice calculations.
  • The same mechanism should govern other $D_s^+$ decays into three pseudoscalars, predicting, for example, which scalar resonances appear in $D_s^+\to\pi^+K^+K^-$ versus $D^+\to\pi^+K^+K^-$; those channels could be compared with existing experimental results.
  • The absence of $f_0(500)$ in $D_s^+$ decays and its presence in $D^+$ decays could serve as a production-side filter to separate $\pi\pi$-driven from $K\bar K$-driven scalar mesons in other final states.
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 / 5 minor

Summary. The paper studies the three-pion decays D_s+ -> pi+ pi+ pi- and D+ -> pi+ pi+ pi- in a model that combines quark-level weak hadronization with final-state rescattering in the chiral unitary approach (ChUA). Starting from external and internal W-emission mechanisms, the authors derive the primary two-meson channels produced in each decay. For D_s+ they find in Eq. (7) that only K+K- and K0Kbar0 channels survive in S-wave, because an exact cancellation removes the pi+ eta eta term; consequently only f0(980), which couples to K Kbar, is generated. For D+ the five coupled channels pi+ pi-, pi0 pi0, K+K-, K0Kbar0, and eta eta are produced, so both f0(500) and f0(980) are generated. The model is fitted to BaBar/BESIII/LHCb measurements of the pi+ pi- S-wave magnitude and phase, with two renormalization schemes and an alternative fit without the eta eta channel. The authors conclude that the observed pattern supports the molecular/dynamically-generated interpretation of these scalar mesons.

Significance. If correct, the paper gives a very simple selection mechanism for a striking experimental pattern: the same three-pion final state is f0(980)-dominated when produced from D_s+, but contains both f0(500) and f0(980) when produced from D+. The derivation leading to Eq. (7) is transparent, and the phase prediction in Fit I for D_s+ is parameter-free. The paper is also commendably explicit about the low-energy discrepancies and about the alternative eta-eta-excluded fit. The significance is moderate rather than high: the conclusion supports a picture that is already widely explored in ChUA, and the robustness of the central rule depends on the equal-weight vacuum-pair assumption discussed below, which needs to be quantified before the selection rule can be regarded as a firm prediction.

major comments (4)
  1. [Sec. II A, Eqs. (1), (5), (7)] The central prediction that only K+K- and K0Kbar0 are produced in the D_s+ decay follows from the exact cancellation of the pi+ eta eta term in Eq. (5). This cancellation uses the specific vacuum insertion written as ubar-u + dbar-d + sbar-s in Eq. (1) with equal weights for all three flavors. If the strange-pair insertion is weighted by lambda, the residual source is (2/3)(lambda-1) pi+ eta eta, which, through the nonzero ChUA amplitude T_{eta eta -> pi+ pi-}, feeds the pi+ pi- S-wave at low energies and can mimic f0(500). The paper neither justifies lambda = 1 independently nor studies the sensitivity to lambda. Since the D_s+ fit already underestimates the data below 0.8 GeV and describes the phase poorly in the 0.4-0.8 GeV range (Sec. III, Fig. 6), exactly the f0(500) region, this is a load-bearing gap for the qualitative D_s+ versus D+ difference.
  2. [Sec. III, Fig. 6, Tables I and II] The D_s+ fit combines data from BaBar, BESIII, and LHCb with a single global normalization C. These experiments use different amplitude conventions, with LHCb amplitude analyses typically reporting arbitrary-normalized amplitudes, so the authors must specify how the data sets are normalized to a common scale. Without this information, the combined fit quality and the claimed agreement are not well defined. In addition, no chi-squared per degree of freedom is quoted for any of the fits, which makes it difficult to assess the quantitative support for the model.
  3. [Sec. III, Tables I and II, Figs. 6 and 7] For D+, the quantitative agreement is obtained with two fitted normalizations D1 and D2, and in scheme II also with a fitted mu. In particular, the relative weight D2/D1 between the K Kbar contribution and the direct/eta eta contribution, which controls the relative strength of f0(980) and f0(500), is a free parameter rather than a prediction from the CKM and color input. The authors should report the fitted ratio D2/D1 and compare it with the naive expectation from V'_P V_cs V_us / (V_P V_cd V_ud), or explicitly state that this ratio is not predicted by the model.
  4. [Sec. III, Tables III and IV, Fig. 9] The improved description of the D_s+ magnitude is obtained only after removing the eta eta channel from the coupled-channel calculation. This removal is not derived within the formalism and changes the unitarity structure of the T-matrix; the paper justifies it only by citing Ref. [56] for the resulting larger f0(980) width. As presented, this is an ad hoc model alteration rather than a controlled check. The central Eq. (7) argument is unaffected, but the quantitative support for the claim that the model naturally explains the D_s+ data rests in part on this altered model; the authors should either justify the removal as a separate scheme or soften the corresponding claim.
minor comments (5)
  1. [Sec. II B, Eqs. (11), (13), and (18)] The vertex factor V'_P is introduced without definition; please clarify whether it denotes the same weak vertex factor as V_P or a different one, since the two enter the expressions for D1 and D2.
  2. [Sec. II C] There is a typo in the sentence 'In the present work. we use the dimensional regularization method'; the manuscript would benefit from a careful proofread of punctuation and spacing throughout.
  3. [Sec. III] The restriction of the comparison to sqrt(s) < 1.2 GeV is stated only in the numerical section; since f0(500) is a broad low-energy structure, the authors should state this restriction earlier and discuss the possible impact of the excluded higher-energy region on the conclusions.
  4. [Eq. (20)] The sentence regarding the factor of 2 cancelled by the 1/2 factor in the pi0 pi0 and eta eta propagators is hard to follow; please rephrase to state explicitly which identical-particle symmetry factor is used.
  5. [Eq. (4) and Figs. 6-9] The assumption eta = eta_8 ignores eta-eta' mixing; because the cancellation in Eq. (5) depends on the exact flavor coefficient of eta, the authors should at least acknowledge this approximation. Also, the figures and tables in the submitted version contain apparent encoding artifacts (e.g., repeated 'uni00000013' sequences) that must be cleaned in the final version.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the D_s vs D+ channel selection is parameter-free from Eqs. (5)–(7), and fitted constants only set normalization/phase.

full rationale

The central claim of the paper—that D_s^+ produces only K+K^- and K0Kbar0 at the primary vertex, so f0(980) appears while f0(500) does not, whereas D^+ produces all five coupled channels and both resonances—follows directly from the hadronization equations (5)–(7) and (18)–(20). It is not equivalent to the experimental observation by construction: it depends on the stated SU(3)-symmetric vacuum-creation ansatz (bar uu + bar dd + bar ss), and a different weighting for strange pairs would alter the cancellation in Eq. (5). That is a physical assumption with testable consequences, not a redefinition of the target result. The fitted constants C, D1, D2, and the regularization scale mu only set overall normalization and phase; they do not determine which channels enter the amplitudes. The paper explicitly labels the determinations as fits (“we fit the experimental data”) rather than presenting fitted quantities as predictions. The acknowledged shortcomings below 0.8 GeV in the D_s amplitude and phase (Fig. 6) are correctness or model-adequacy issues, not circularity. Self-citations, such as the use of Ref. [56] for the effect of the eta-eta channel on the f0(980) width and Refs. [22,52,58] for the choice mu = 0.6 GeV, are ancillary and not load-bearing; the main channel-selection argument rests on the standard ChUA equations (21)–(25) with established external references. No specific reduction of a prediction to its own fitted input or to a self-citation chain can be exhibited.

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

The free parameters are normalization constants and the regularization scale mu used in the loop functions. The structural claim about which resonances appear is independent of these, but the quantitative agreement is not. No new particles or forces are introduced.

free parameters (4)
  • C (global normalization for D_s amplitude) = 23.89 +/- 0.14; 28.75 +/- 0.22; 13.15 +/- 0.07; 13.87 +/- 0.13
    Normalizes the D_s amplitude to the experimental magnitude. It absorbs the weak vertex strength, (1+beta), CKM factors, and the data normalization. Fitted in all schemes (Tables I and III).
  • D1 (normalization for D+ direct channels) = 12.22 +/- 0.14; 12.05 +/- 0.15; 9.89 +/- 0.13; 10.03 +/- 0.13
    Controls the weight of the pi+pi-, pi0pi0, KKbar, and eta eta terms in the D+ amplitude. Fitted to the LHCb D+ data (Tables II and IV).
  • D2 (normalization for D+ KKbar contribution) = -12.05 +/- 0.15; -16.74 +/- 0.43; -8.01 +/- 0.11; -7.50 +/- 0.16
    Controls the weight of the K+K- and K0Kbar0 rescattering terms in the D+ amplitude. Fitted jointly with D1 (Tables II and IV).
  • mu (regularization scale in the loop function) = 0.561, 0.525, 0.894, 0.980 GeV (Fit II schemes)
    Sets the scale of the two-meson loop function and strongly affects the phase. Held fixed in Fit I (0.6 or 0.931 GeV), but treated as a free parameter in Fit II and fitted to the same data.
assumptions (6)
  • domain assumption Equal-weight vacuum pair creation for anti-u u, anti-d d, anti-s s in hadronization
    Introduced in Eq. (1) to build (M.M) products; the cancellation that leaves only KKbar in D_s depends on equal weights.
  • domain assumption Dominant weak decay topologies are external and internal W-emission only
    Sec. II A/B builds the full amplitude from these diagrams; W-annihilation, penguins, and other topologies are neglected.
  • domain assumption Chiral unitary approach with on-shell factorized Bethe-Salpeter equation describes the S-wave coupled channels
    Eqs. (21)-(22) use the lowest-order chiral Lagrangian and unitarization to generate f0(500) and f0(980); the interpretation relies on this framework.
  • domain assumption Only the S-wave part of the measured pi-pi amplitude is compared
    Amplitudes in Eqs. (8) and (19) include only S-wave rescattering; higher partial waves are omitted.
  • ad hoc to paper ChUA is valid only for sqrt(s) < 1.2 GeV
    Data above 1.2 GeV are excluded from the fits because the model is assumed not to apply there, stated before Fig. 6.
  • ad hoc to paper The eta-eta channel can be excluded in an alternative fit to widen f0(980)
    Tables III and IV and Fig. 9 drop eta-eta to improve the D_s magnitude, a post hoc model choice.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Roles of $f_{0}(500)$ and $f_{0}(980)$ in the $D_{(s)}^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}$ decays." pith.science (2026). https://pith.science/paper/RPJCKW7S

@misc{pith2026241118134,
  author       = {Pith},
  title        = {Pith review of: Roles of $f_0(500)$ and $f_0(980)$ in the $D_(s)^+\rightarrow\pi^+\pi^+\pi^-$ decays},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RPJCKW7S}},
  note         = {Machine review of arXiv:2411.18134}
}
abstract

With the recent measurements of the $D_{s}^{+}\rightarrow \pi^{+}\pi^{+}\pi^{-}$ and $D^{+}\rightarrow \pi^{+}\pi^{+}\pi^{-}$ decays by the LHCb Collaboration, we study these two decay processes by considering the final state interaction formalism. Taking into account the external and internal $W$-emission dominant mechanisms at the quark level, our model can naturally explain why the $D_{s}^{+}$ decay only has a contribution from the $f_{0}(980)$ resonance, while the $D^{+}$ decay has contributions from both the $f_{0}(500)$ and $f_{0}(980)$ states. The magnitudes and phases of $\pi^{+}\pi^{-}$ for $S$-wave amplitudes in our model are in agreement with the experimental measurements. These results support the interpretations of $f_{0}(500)$ and $f_{0}(980)$ as the $\pi\pi$ and $K\bar{K}$ resonances, respectively, where they are dynamically generated from the $S$-wave pseudoscalar-pseudoscalar meson interactions within the chiral unitary approach.

Figures

Figures reproduced from arXiv: 2411.18134 by the authors.

Figure 1
Figure 1. FIG. 1: Diagrams for the [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Procedure for the hadronization [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Mechanisms of the [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Diagrams for the [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Mechanisms of the [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: The magnitude (above) and phase (below) of the [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: The magnitude (above) and phase (below) of the [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Modulus square of the amplitudes in coupled channels. [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: The magnitudes (left) and phases (right) of the [PITH_FULL_IMAGE:figures/full_fig_p008_9.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

  1. Scalar resonance contributions in the $D_{s1}(2460)^{+} \rightarrow D_{s}^{+}\pi^{+}\pi^{-}$ reaction

    hep-ph 2024-12 conditional novelty 6.0 of 10

    In D_s1(2460)+ -> D_s+ pi+ pi-, a chiral unitary coupled-channel model reproduces the LHCb mass distributions and generates T_cbar-s and f0(500) as hadronic molecules.

Reference graph

Works this paper leans on

73 extracted references · 34 canonical work pages · cited by 1 Pith paper

  1. [52]

    J. M. Dias, F. S. Navarra, M. Nielsen and E. Oset, Phys. Rev. D 94, no.9, 096002 (2016) [arXiv:1601.04635 [hep-ph]]

    arXiv 2016

  2. [56]

    H. A. Ahmed, Z. Y . Wang, Z. F. Sun and C. W. Xiao, Eur. Phys. J. C 81, no.8, 695 (2021) [arXiv:2011.08758 [hep-ph]]

    work page Pith review arXiv 2021

  3. [1]

    S. K. Choi et al. [Belle], Phys. Rev. Lett. 91, 262001 (2003) [arXiv:hep-ex/0309032 [hep-ex]]

    arXiv 2003

  4. [2]

    Liu, Chin

    X. Liu, Chin. Sci. Bull. 59, 3815-3830 (2014) [arXiv:1312.7408 [hep-ph]]

    arXiv 2014

  5. [3]

    Hosaka, T

    A. Hosaka, T. Iijima, K. Miyabayashi, Y . Sakai and S. Yasui, PTEP 2016, no.6, 062C01 (2016) [arXiv:1603.09229 [hep-ph]]

    arXiv 2016

  6. [4]

    H. X. Chen, W. Chen, X. Liu and S. L. Zhu, Phys. Rept. 639, 1-121 (2016) [arXiv:1601.02092 [hep-ph]]

    arXiv 2016

  7. [5]

    J. M. Richard, Few Body Syst. 57, no.12, 1185-1212 (2016) [arXiv:1606.08593 [hep-ph]]

    arXiv 2016

  8. [6]

    R. F. Lebed, R. E. Mitchell and E. S. Swanson, Prog. Part. Nucl. Phys. 93, 143-194 (2017) [arXiv:1610.04528 [hep-ph]]

    arXiv 2017

Show all 73 references
  1. [7]

    S. L. Olsen, T. Skwarnicki and D. Zieminska, Rev. Mod. Phys. 90, no.1, 015003 (2018) [arXiv:1708.04012 [hep-ph]]

    2018 arXiv

  2. [8]

    F. K. Guo, C. Hanhart, U. G. Meißner, Q. Wang, Q. Zhao and B. S. Zou, Rev. Mod. Phys. 90, no.1, 015004 (2018) [erratum: Rev. Mod. Phys. 94, no.2, 029901 (2022)] [arXiv:1705.00141 [hep-ph]]

    2018 arXiv

  3. [9]

    Y . R. Liu, H. X. Chen, W. Chen, X. Liu and S. L. Zhu, Prog. Part. Nucl. Phys. 107, 237-320 (2019) [arXiv:1903.11976 [hep- ph]]

    2019 arXiv

  4. [10]

    Brambilla, S

    N. Brambilla, S. Eidelman, C. Hanhart, A. Nefediev, C. P. Shen, C. E. Thomas, A. Vairo and C. Z. Yuan, Phys. Rept.873, 1-154 (2020) [arXiv:1907.07583 [hep-ex]]

    2020 arXiv

  5. [11]

    L. Meng, B. Wang, G. J. Wang and S. L. Zhu, Phys. Rept.1019, 1-149 (2023) [arXiv:2204.08716 [hep-ph]]

    2023 arXiv

  6. [12]

    H. X. Chen, W. Chen, X. Liu, Y . R. Liu and S. L. Zhu, Rept. Prog. Phys. 86, no.2, 026201 (2023) [arXiv:2204.02649 [hep- ph]]

    2023 arXiv

  7. [13]

    M. Z. Liu, Y . W. Pan, Z. W. Liu, T. W. Wu, J. X. Lu and L. S. Geng, [arXiv:2404.06399 [hep-ph]]

    arXiv

  8. [14]

    Aaij et al

    R. Aaij et al. [LHCb], JHEP 04, 063 (2019) [arXiv:1902.05884 [hep-ex]]

    2019 arXiv

  9. [15]

    Ablikim et al

    M. Ablikim et al. [BESIII], Phys. Rev. D 101, no.5, 052009 (2020) [arXiv:1912.12411 [hep-ex]]

    2020

  10. [16]

    Ablikim et al

    M. Ablikim et al. [BESIII], [arXiv:2404.09219 [hep-ex]]

  11. [17]

    Ablikim et al

    M. Ablikim et al. [BESIII], Phys. Lett. B 781, 368-375 (2018) [arXiv:1803.05769 [hep-ex]]

    2018 arXiv

  12. [18]

    Ablikim et al

    M. Ablikim et al. [BESIII], Phys. Rev. D 104, no.1, 012016 (2021) [arXiv:2011.08041 [hep-ex]]

    2021

  13. [19]

    Ablikim et al

    M. Ablikim et al. [BESIII], JHEP 08, 196 (2022) [arXiv:2205.08844 [hep-ex]]

    2022

  14. [20]

    Ablikim et al

    M. Ablikim et al. [BESIII], Phys. Rev. Lett.123, no.11, 112001 (2019) [arXiv:1903.04118 [hep-ex]]

    2019

  15. [21]

    Roca and E

    L. Roca and E. Oset, Phys. Rev. D 103, no.3, 034020 (2021) [arXiv:2011.05185 [hep-ph]]

    2021 arXiv

  16. [22]

    M. Y . Duan, J. Y . Wang, G. Y . Wang, E. Wang and D. M. Li, Eur. Phys. J. C80, no.11, 1041 (2020) [arXiv:2008.10139 [hep- ph]]

    2020 arXiv

  17. [23]

    Ikeno, M

    N. Ikeno, M. Bayar and E. Oset, Eur. Phys. J. C 81, no.4, 377 (2021) [arXiv:2102.01650 [hep-ph]]

    2021 arXiv

  18. [24]

    Z. Y . Wang, H. A. Ahmed and C. W. Xiao, Phys. Rev. D 105, no.1, 016030 (2022) [arXiv:2110.05359 [hep-ph]]

    2022 arXiv

  19. [25]

    J. Y . Wang, M. Y . Duan, G. Y . Wang, D. M. Li, L. J. Liu and E. Wang, Phys. Lett. B 821, 136617 (2021) [arXiv:2105.04907 [hep-ph]]

    2021 arXiv

  20. [26]

    Z. Y . Wang, J. Y . Yi, Z. F. Sun and C. W. Xiao, Phys. Rev. D 105, no.1, 016025 (2022) [arXiv:2109.00153 [hep-ph]]

    2022 arXiv

  21. [27]

    L. R. Dai and E. Oset, Phys. Rev. D 109, no.5, 054008 (2024) [arXiv:2307.03014 [hep-ph]]

    2024 arXiv

  22. [28]

    Liang, J

    W. Liang, J. Y . Yi, C. W. Xiao and S. Q. Zhou, Phys. Rev. D 108, no.11, 116001 (2023)

    2023

  23. [29]

    Molina, J

    R. Molina, J. J. Xie, W. H. Liang, L. S. Geng and E. Oset, Phys. Lett. B 803, 135279 (2020) [arXiv:1908.11557 [hep-ph]]

    2020 arXiv

  24. [30]

    X. Z. Ling, M. Z. Liu, J. X. Lu, L. S. Geng and J. J. Xie, Phys. Rev. D 103, no.11, 116016 (2021) [arXiv:2102.05349 [hep- ph]]

    2021 arXiv

  25. [31]

    Bayar, R

    M. Bayar, R. Molina, E. Oset, M. Z. Liu and L. S. Geng, Phys. Rev. D 109, no.7, 076027 (2024) [arXiv:2312.12004 [hep-ph]]

    2024 arXiv

  26. [32]

    N. N. Achasov and G. N. Shestakov, Phys. Rev. D 96, no.3, 036013 (2017) [arXiv:1704.01763 [hep-ph]]

    2017 arXiv

  27. [33]

    H. Y . Cheng, C. W. Chiang and Z. Q. Zhang, Phys. Rev. D105, no.3, 033006 (2022) [arXiv:2201.00460 [hep-ph]]

    2022 arXiv

  28. [34]

    N. N. Achasov and G. N. Shestakov, Phys. Rev. D 110, no.1, 016025 (2024) [arXiv:2405.13332 [hep-ph]]

    2024 arXiv

  29. [35]

    H. Y . Cheng, C. W. Chiang and F. Xu, [arXiv:2408.13942 [hep- ph]]

    arXiv

  30. [36]

    Ablikim et al

    M. Ablikim et al. [BESIII], Phys. Rev. D 105, no.5, L051103 (2022) [arXiv:2110.07650 [hep-ex]]

    2022

  31. [37]

    Ablikim et al

    M. Ablikim et al. [BESIII], Phys. Rev. Lett. 129, no.18, 18 (2022) [arXiv:2204.09614 [hep-ex]]

    2022

  32. [38]

    D. Guo, W. Chen, H. X. Chen, X. Liu and S. L. Zhu, Phys. Rev. D 105, no.11, 114014 (2022) [arXiv:2204.13092 [hep-ph]]

    2022 arXiv

  33. [39]

    L. R. Dai, E. Oset and L. S. Geng, Eur. Phys. J. C 82, no.3, 225 (2022) [arXiv:2111.10230 [hep-ph]]

    2022 arXiv

  34. [40]

    X. Zhu, D. M. Li, E. Wang, L. S. Geng and J. J. Xie, Phys. Rev. D 105, no.11, 116010 (2022) [arXiv:2204.09384 [hep-ph]]

    2022 arXiv

  35. [41]

    Y . W. Peng, W. Liang, X. Xiong and C. W. Xiao, Phys. Rev. D 110, no.3, 036013 (2024) [arXiv:2402.02539 [hep-ph]]

    2024 arXiv

  36. [42]

    X. Zhu, H. N. Wang, D. M. Li, E. Wang, L. S. Geng and J. J. Xie, Phys. Rev. D 107, no.3, 034001 (2023) [arXiv:2210.12992 [hep-ph]]. 10

    2023 arXiv

  37. [43]

    E. Oset, L. R. Dai and L. S. Geng, Sci. Bull.68, 243-246 (2023) [arXiv:2301.08532 [hep-ph]]

    2023 arXiv

  38. [44]

    Z. Y . Wang, Y . W. Peng, J. Y . Yi, W. C. Luo and C. W. Xiao, Phys. Rev. D 107, no.11, 116018 (2023) [arXiv:2306.06395 [hep-ph]]

    2023 arXiv

  39. [45]

    N. N. Achasov and G. N. Shestakov, Phys. Rev. D 108, no.3, 036018 (2023) [arXiv:2306.04478 [hep-ph]]

    2023 arXiv

  40. [46]

    Kim and K

    H. Kim and K. S. Kim, [arXiv:2410.18589 [hep-ph]]

    arXiv

  41. [47]

    Aaij et al

    R. Aaij et al. [LHCb], JHEP 06, 044 (2023) [arXiv:2208.03300 [hep-ex]]

    2023 arXiv

  42. [48]

    Aubert et al

    B. Aubert et al. [BaBar], Phys. Rev. D 79, 032003 (2009) [arXiv:0808.0971 [hep-ex]]

    2009 arXiv

  43. [49]

    Ablikim et al

    M. Ablikim et al. [BESIII], Phys. Rev. D 106, no.11, 112006 (2022) [arXiv:2108.10050 [hep-ex]]

    2022

  44. [50]

    Aaij et al

    R. Aaij et al. [LHCb], JHEP 07, 204 (2023) [arXiv:2209.09840 [hep-ex]]

    2023 arXiv

  45. [51]

    N. N. Achasov and G. N. Shestakov, Phys. Rev. D 107, no.5, 056009 (2023) [arXiv:2211.05526 [hep-ph]]

    2023 arXiv

  46. [53]

    L. L. Chau, Phys. Rept. 95, 1-94 (1983)

    1983

  47. [54]

    L. L. Chau and H. Y . Cheng, Phys. Rev. D36, 137 (1987)

    1987

  48. [55]

    W. H. Liang, J. J. Xie and E. Oset, Eur. Phys. J. C 75, no.12, 609 (2015) [arXiv:1510.03175 [hep-ph]]

    2015 arXiv

  49. [57]

    L. L. Wei, H. S. Li, E. Wang, J. J. Xie, D. M. Li and Y . X. Li, Phys. Rev. D103, 114013 (2021) [arXiv:2102.03704 [hep-ph]]

    2021 arXiv

  50. [58]

    W. H. Liang and E. Oset, Phys. Lett. B 737, 70-74 (2014) [arXiv:1406.7228 [hep-ph]]

    2014 arXiv

  51. [59]

    H. A. Ahmed and C. W. Xiao, Phys. Rev. D 101, no.9, 094034 (2020) [arXiv:2001.08141 [hep-ph]]

    2020 arXiv

  52. [60]

    J. A. Oller and E. Oset, Nucl. Phys. A 620, 438-456 (1997) [erratum: Nucl. Phys. A 652, 407-409 (1999)] [arXiv:hep- ph/9702314 [hep-ph]]

    1997

  53. [61]

    Oset and A

    E. Oset and A. Ramos, Nucl. Phys. A 635, 99-120 (1998) [arXiv:nucl-th/9711022 [nucl-th]]

    1998 arXiv

  54. [62]

    J. A. Oller, E. Oset and J. R. Pelaez, Phys. Rev. Lett. 80, 3452- 3455 (1998) [arXiv:hep-ph/9803242 [hep-ph]]

    1998 arXiv

  55. [63]

    J. A. Oller and E. Oset, Phys. Rev. D 60, 074023 (1999) [arXiv:hep-ph/9809337 [hep-ph]]

    1999 arXiv

  56. [64]

    J. A. Oller and U. G. Meissner, Phys. Lett. B 500, 263-272 (2001) [arXiv:hep-ph/0011146 [hep-ph]]

    2001 arXiv

  57. [65]

    Gamermann, E

    D. Gamermann, E. Oset, D. Strottman and M. J. Vicente Vacas, Phys. Rev. D 76, 074016 (2007) [arXiv:hep-ph/0612179 [hep- ph]]

    2007 arXiv

  58. [66]

    Alvarez-Ruso, J

    L. Alvarez-Ruso, J. A. Oller and J. M. Alarcon, Phys. Rev. D 82, 094028 (2010) [arXiv:1007.4512 [hep-ph]]

    2010 arXiv

  59. [67]

    Z. H. Guo, L. Liu, U. G. Meißner, J. A. Oller and A. Rusetsky, Phys. Rev. D 95, no.5, 054004 (2017) [arXiv:1609.08096 [hep- ph]]

    2017 arXiv

  60. [68]

    Z. H. Guo, L. Liu, U. G. Meißner, J. A. Oller and A. Rusetsky, Eur. Phys. J. C 79, no.1, 13 (2019) [arXiv:1811.05585 [hep- ph]]

    2019 arXiv

  61. [69]

    E. Oset, A. Ramos and C. Bennhold, Phys. Lett. B 527, 99- 105 (2002) [erratum: Phys. Lett. B 530, 260-260 (2002)] [arXiv:nucl-th/0109006 [nucl-th]]

    2002 arXiv

  62. [70]

    Monta ˜na, A

    G. Monta ˜na, A. Ramos, L. Tolos and J. M. Torres-Rincon, Phys. Rev. D 107, no.5, 054014 (2023) [arXiv:2211.01896 [hep-ph]]

    2023 arXiv

  63. [71]

    Navas et al

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

    2024

  64. [72]

    V . R. Debastiani, W. H. Liang, J. J. Xie and E. Oset, Phys. Lett. B 766, 59-64 (2017) [arXiv:1609.09201 [hep-ph]]

    2017 arXiv

  65. [73]

    C. W. Xiao, U. G. Meißner and J. A. Oller, Eur. Phys. J. A 56, no.1, 23 (2020) [arXiv:1907.09072 [hep-ph]]

    2020 arXiv

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.