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REVIEW 3 major objections 5 minor 48 references

Probing the elusive $\kappa/K_0^*(700)$ resonance in semileptonic $D$ decays

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Partial-wave reanalysis of $D^+ \to K^- \pi^+ e^+ \nu_e$ shows that the s-wave rate, previously assigned to non-resonant background, is dominated by the $\kappa/K_0^*(700)$.

desk verdict A coherent global fit that finally puts the kappa into the s-wave amplitude of D->Kpi e nu, but the kappa/background split is dictated by assumed form-factor shapes, not by any fitted spectrum. read the letter →

arxiv 2507.20380 v1 pith:FPO7UQUU submitted 2025-07-27 hep-ph hep-ex

Yu-Kuo Hsiao , Yan-Li Wang , Wen-Juan Wei This is my paper · ORCID
classification hep-phhep-ex
keywords kappamesonK0*(700)lightscalarmesonssemileptonicDdecayspartial-waveanalysisformfactorstetraquarknon-resonantbackground
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 claims that the $\kappa/K_0^*(700)$, the most elusive light scalar meson, has long been present in weak decays but was misidentified as non-resonant background. Analyzing $D^+ \to K^- \pi^+ e^+ \nu_e$ by partial waves, together with closely related nonleptonic $D$ decays, the authors find $\mathcal{B}(D^+ \to \bar\kappa^0 e^+ \nu_e, \bar\kappa^0 \to K^- \pi^+) = (2.2 \pm 0.1) \times 10^{-3}$, roughly forty times the residual non-resonant s-wave rate. The same fit gives $f^+(0) = 0.32 \pm 0.01$ for the $D \to \kappa$ form factor. The paper argues this number selects a compact tetraquark interpretation over a conventional quark-antiquark assignment, because the tetraquark prediction is $0.36 \pm 0.02$ and the $q\bar q$ prediction is $0.82 \pm 0.05$. If correct, the discovery changes how the s-wave component of semileptonic $D$ decays is modeled and strengthens the tetraquark picture of light scalar mesons.

What carries the argument

The central machinery is the partial-wave decomposition of the amplitude for $D \to K\pi e\nu_e$, with the s-wave partial-wave amplitude $F_{10}$ constructed from the non-resonant form factors $w_+$ and $w_-$, the $D \to \kappa$ form factor $f^+$, and the broad-resonance propagator $D_\kappa$. The scalar form factor is modeled as a double pole, $f^+(q^2) = f(0)/(1 - q^2/m_D^2)^2$, while the $K^*$ vector form factors use single poles; the equality $w_+ = w_-$ is assumed and tested against three alternatives. A global $\chi^2$ fit to five measured branching fractions fixes $f(0)$, $w(0)$, and $h(0)$. The $D \to \kappa$ form factor at $q^2 = 0$ carries the physical conclusion, because its size separates the tetraquark and quark-antiquark interpretations.

What would settle it

Use the full published event distribution for $D^+ \to K^- \pi^+ e^+ \nu_e$ and fit the s-wave with a model-independent $K\pi$ line shape, letting the $\kappa$ fraction and the non-resonant amplitude float independently. If the resonant $\kappa$ component is not required at the level of about $2.2 \times 10^{-3}$, or if the extracted $f^+(0)$ lands near $0.8$ rather than $0.3$, the paper's central conclusion fails.

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Extended reading notes

Core claim

Using a partial-wave analysis of the four-body semileptonic decay $D^+ \to K^- \pi^+ e^+ \nu_e$, supplemented by nonleptonic $D \to K\pi\pi$ and $D \to K\pi\rho$ decays that share the same $D \to K\pi$ transition, the paper separates the s-wave amplitude into a $\kappa$-resonant piece and a non-resonant piece. It finds $\mathcal{B}(D^+ \to \bar\kappa^0 e^+ \nu_e, \bar\kappa^0 \to K^- \pi^+) = (2.2 \pm 0.1) \times 10^{-3}$, an order of magnitude larger than the non-resonant s-wave contribution of $(5.9 \pm 2.0) \times 10^{-5}$. The same global fit yields the $D \to \kappa$ form factor $f^+(0) = 0.32 \pm 0.01$. Compared under SU(3) flavor relations with companion values for $D \to S_n$ transitions, this sits close to the tetraquark expectation $0.36 \pm 0.02$ and far from the conventional quark-antiquark expectation $0.82 \pm 0.05$. The paper concludes that the $\kappa$ has been present in weak-decay data for years, hidden inside what experiments called background, and that the light scalar mesons are likely compact tetraquarks.

Load-bearing premise

The load-bearing premise is that the chosen analytic shapes for the amplitudes—the double-pole scalar form factor $f^+(q^2)$, the single-pole $K^*$ form factors, the Breit-Wigner line shape for the broad $\kappa$, and the equality $w_+ = w_-$—are close enough to the true amplitudes that the fitted $\kappa$ rate and $f^+(0)$ are unbiased; the tetraquark conclusion then also leans on SU(3) relations borrowed from the companion analysis.

Editorial extensions

If this is right

  • The measured s-wave branching fraction $\mathcal{B}(D^+ \to (K^-\pi^+)_{\text{s-wave}} e^+ \nu_e) = (2.28 \pm 0.11) \times 10^{-3}$ should be reassigned: the $\kappa$ contributes $(2.2 \pm 0.1) \times 10^{-3}$ and the true non-resonant s-wave only $(5.9 \pm 2.0) \times 10^{-5}$.
  • Experimental partial-wave analyses of $D \to K\pi e\nu_e$ should include the $\kappa$ as an explicit resonance instead of absorbing it into the non-resonant background.
  • A small but non-zero non-resonant p-wave component, $(5.5 \pm 1.8) \times 10^{-5}$, is predicted for the first time; future data with better statistics can look for it.
  • The extracted $f^+(0) = 0.32 \pm 0.01$ and the $\kappa$ branching fraction provide specific targets for independent determinations of the $D \to \kappa$ form factor.
  • If the $\kappa$ is a compact tetraquark, the same SU(3)-related form-factor pattern should appear in $D$ transitions to the other light scalar mesons, such as the $\sigma/f_0(500)$ and $f_0(980)$.

Reading between the lines

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

  • The comparison that decides tetraquark versus quark-antiquark is only as strong as the SU(3) relations and the $D$-to-scalar inputs taken from the companion paper; a direct data-driven or lattice determination of the $D \to \kappa$ form factor would test that step independently.
  • The $\kappa$-dominance claim is more robust than the tetraquark interpretation: even if a different parameterization of the non-resonant amplitude shifts $f(0)$, the paper's own tests still leave the non-resonant s-wave far too small to explain the observed rate.
  • A cleaner experimental test would fit the full published event distribution for $D^+ \to K^- \pi^+ e^+ \nu_e$ with the $\kappa$ and non-resonant amplitudes both free, then compare the extracted $\kappa$ branching fraction and $f^+(0)$ with the global-fit values.
  • The method should transfer to $D^+ \to K^- \pi^+ \mu^+ \nu_\mu$, where the larger muon mass changes the phase space and can expose any $q^2$-dependent ambiguity in the form factors.
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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 / 5 minor

Summary. The manuscript analyzes the semileptonic decay D+ -> K- pi+ e+ nu_e together with the nonleptonic decays D -> K pi pi and D -> K pi rho, using a partial-wave framework in which the s-wave K pi amplitude is a sum of a broad kappa/K0*(700) Breit-Wigner term and non-resonant terms, while the p-wave is dominated by K*(892). A chi-square fit to the integrated branching fractions and the K* form factors yields f(0)=0.32+/-0.01, w(0)=(0.94+/-0.14) GeV^-1, and h(0)=(4.24+/-0.79) GeV^-3. From these the authors report B(D+ -> anti-kappa^0 e+ nu_e, anti-kappa^0 -> K- pi+) = (2.2+/-0.1)x10^-3, claiming that the kappa dominates the measured s-wave rate and that the extracted f+(0) strongly favors a compact tetraquark interpretation of the kappa over a conventional q qbar assignment.

Significance. If the central claim were correct, the paper would overturn the standard experimental practice of treating the broad s-wave component in D -> K pi l nu as non-resonant background and would provide a striking, falsifiable signal for a compact tetraquark kappa. The paper also makes testable predictions, e.g., B(D+ -> pi+ anti-kappa^0, anti-kappa^0 -> Ks pi0) and the nonleptonic D -> K pi pi rates listed in Table I, which is a genuine strength. The chi^2/n.d.f. of about 1.2 for a five-point global fit is encouraging. The weakness is that the headline kappa/s-wave decomposition and the extracted f(0) do not come from a fit to the differential m(K pi) or q^2 spectrum; they are consequences of unverified functional forms imposed in Eqs. (5) and (8). Since the kappa is broad and structureless over much of the phase space, the quoted uncertainty is not yet an experimental extraction of the kappa contribution.

major comments (3)
  1. [Section III, Eq. (8) and Table I] The central claim that the kappa dominates the s-wave rate is not actually extracted from data. The only semileptonic s-wave constraint entering the chi^2 fit is the total integrated branching fraction B(D+ -> (K- pi+)_s-wave e+ nu_e) = (2.28 +/- 0.11) x 10^-3 (Table I). An integrated rate fixes the normalization of the sum F10 in Eq. (5), but not the relative weight of the kappa Breit-Wigner term 2 f+ = 2 g_kappaKpi D_kappa^-1 f+(q^2) and the non-resonant terms proportional to w+ and w-. The reported split, with B_kappa^(semi) = (2.2 +/- 0.1) x 10^-3 and B_NR(S)^(semi) = (5.9 +/- 2.0) x 10^-5, follows from the assumed double-pole form of f+(q^2), the assumed kappa Breit-Wigner parameters, and the relation w+ = w-, none of which is tested against a differential m(K pi) or q^2 distribution. Because the kappa has width comparable to its mass, its Breit-Wigner is smooth over the accessible phase space and can be mimicked by a non-resonant amplitude with a different q^2 behavior. As written, the headline 'kappa dominates' is a consequence of the model choice, not an independent measurement.
  2. [Section IV, Eq. (9)] The quoted uncertainty f(0) = 0.32 +/- 0.01 is only the statistical error of the minimum-chi^2 fit and omits the dominant model systematics. The alternatives tested in Section IV vary the algebraic relations among w+, w-, and h(0), but they do not vary the shape of f+(q^2) (e.g., the exponent n in Eq. (8)), the kappa lineshape in Eq. (5), or the strong coupling g_kappaKpi. A readjustment of these functional choices can shift both f(0) and the kappa/non-resonant split without changing the integrated s-wave rate. Therefore the statement that f+(0) is 'remarkably stable across various theoretical scenarios' is not supported by the tests actually shown. Before the tetraquark conclusion can be claimed, the authors need to show stability under a family of reasonable lineshape and form-factor parameterizations, or fit differential data that resolve the s-wave mass spectrum.
  3. [Section IV (tetraquark comparison)] The discrimination between q qbar and q^2 qbar^2 configurations is not self-contained. The predictions F^{D->kappa}_{q qbar} = 0.82 +/- 0.05 and F^{D->kappa}_{q^2 qbar^2} = 0.36 +/- 0.02 are obtained by combining SU(3) relations with the D -> light-scalar inputs F^{D->Sn} = 0.47 +/- 0.02 and F^{D->Sns} = 0.31 +/- 0.02 taken from Ref. [30], which is from the same group and is not independently reproduced here. The uncertainty associated with the SU(3) breaking factors (fK/fpi versus fpi/fK) and with the physical-state mixing between Sn and Sns is not assessed. Thus even if f+(0) were robust, the tetraquark conclusion would inherit the reliability of Ref. [30], and the present paper does not supply an independent check.
minor comments (5)
  1. [Section III] There is a typo in 'The second term incorparate the form factors...' which should read 'incorporates'.
  2. [Table I] The table would benefit from an explicit statement that the rows without a star are predictions, not inputs; in particular, the row B(D+ -> pi+ anti-kappa^0, anti-kappa^0 -> Ks pi0) = (2.4 +/- 0.1 +/- 0.5) x 10^-3 is compared with a very imprecise measurement (6 +5 -4) x 10^-3, and the text should state how much of the agreement is driven by that large uncertainty.
  3. [Section IV] The sentence 'Ref. [8] discovers that D -> pi(K pi)_s-wave channel receives contributions...' should be rephrased; a paper does not 'discover', it reports evidence.
  4. [Section II, Eq. (4) and Eq. (5)] The notation X, alpha_M, and alpha_L is used in Eq. (4) and then reused for the kinematic factors in Eq. (5); the definition of X is given only after Eq. (4) but is needed in Eq. (5). This should be clarified for readability.
  5. [Section IV] The claim of 'a non-zero p-wave contribution from the non-resonant amplitude' is based on B_NR(P)^(semi) = (5.5 +/- 1.8) x 10^-5; this is a 3-sigma effect with no model-systematics error included, so it should be phrased as a tentative observation, not a firm new result.

Circularity Check

3 steps flagged · score 7.0 of 10

The κ-dominance 'prediction' is the fitted s-wave input renamed by a self-cited lineshape ansatz, and the tetraquark verdict rests on the same group's prior benchmark values.

  1. fitted input called prediction [Section III (Eq. (9), Table I) and Section IV]
    "Treating f(0), w(0), and h(0) in Eq. (8) as free parameters, the χ2-fit yields f(0)=0.32±0.01, w(0)=(0.94±0.14) GeV−1, h(0)=(4.24±0.79) GeV−3 ... In semileptonic D decays, we predict Bsemiκ ≡ B(D+→κ̄0e+νe, κ̄0→K−π+) ... with ... BsemiNR(S)=(5.9±2.0)×10−5."

    The fit input is B(D+→(K−π+)s−wave e+νe)=(2.28±0.11)×10−3 (Table I, starred). Eq. (5) puts both the non-resonant w± terms and the κ term 2gκKπDκ−1f+(q2) in F10; Eq. (9) fits f(0) to this input. The paper then reports Bκ=2.2×10−3 and BNR(S)=5.9×10−5, whose sum equals the input. The headline number is thus the fitted s-wave total minus a small fitted non-resonant remainder, not an independently predicted branching fraction; the claim that κ dominates is a restatement of the constraint used to fit f(0).

  2. ansatz smuggled in via citation [Section III, Eq. (8) (citing Ref. [30])]
    "The scalar form factor f+(q2), along with the non-resonant vector-type form factors w±(q2), and the tensor-type form factor h(q2), are modeled following Ref. [30] as f+(q2)=f(0)/(1−q2/m2D)n, w±(q2)=w±(0)/(1−q2/m2D)n, h(q2)=h(0)/(1−q2/m2D)n, where n = 2 reflects a double-pole behavior."

    Ref. [30] is same-group prior work (Hsiao, Yang, Wei, Ke, JHEP 12, 226 (2025)). The assumed double-pole forms set the relative q2 dependence of f+ and w±, and hence the weight of the broad κ Breit-Wigner versus the non-resonant term in F10. No differential m(Kπ) or q2 spectrum is fitted; only integrated branching fractions are used. Since κ is broad (Γ∼m), a non-resonant amplitude with different q2 behavior could mimic it, and testing only alternative w± constants (Sec. IV) does not remove this shape degeneracy. The κ/NR split is therefore an output of the self-cited ansatz rather than an extraction from data.

1 more flagged steps
  1. self citation load bearing [Section IV (Discussions and Conclusion)]
    "Using FD+→Sn = 0.47±0.02 and FD+→Sns = 0.31±0.02 [30], we obtain FD+→κ̄0 q q̄ = 0.82±0.05 and FD+→κ̄0 q2 q̄2 = 0.36±0.02. The extracted value f+(0)=0.32±0.01 lies far away from the q q̄ expectation and much closer to the tetraquark prediction, providing strong evidence for a q2 q̄2 configuration of the κ resonance."

    The conclusion that f+(0)=0.32 supports a tetraquark κ is reached by comparing it with 0.82 (q q̄) and 0.36 (q2 q̄2). Both benchmark values come from Ref. [30], authored by the same group and sharing authors with this paper; no independent derivation or external reproduction of FD+→Sn or FD+→Sns is given here. The structural assignment 'tetraquark' therefore reduces to the current fit plus a same-group prior prediction, making the self-citation load-bearing for the paper's central physics conclusion.

full rationale

The semileptonic inputs are real external data and the fit has nontrivial constraints, so the paper is not empty: the p-wave K* parameters and nonleptonic channels do provide some independent normalization. However, the three advertised results are not equally independent. Bκ=(2.2±0.1)×10−3 is the fitted s-wave input (2.28×10−3) minus the tiny fitted NR(S) remainder, so its dominance is guaranteed by the fit rather than tested. The decomposition into κ and NR(S) is controlled by the double-pole ansatz taken from Ref. [30] by the same group, with no fitted invariant-mass spectrum to discriminate a broad resonance from a smooth non-resonant term. Finally, the tetraquark verdict uses the qq̄ and q2q̄2 benchmarks from the same Ref. [30] as if independently established. These are instances of fitted-input-as-prediction and load-bearing self-citation; they do not invalidate the data fit but they do mean the central claims are partially circular.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claim rests on a model-dependent decomposition of the s-wave amplitude, three fitted parameters, and SU(3) relations whose inputs come from a self-cited paper. No new particle, mediator, force, or conserved quantity is introduced.

free parameters (4)
  • f(0) = 0.32 +/- 0.01
    Fitted to semileptonic and nonleptonic branching fractions in the global chi-square fit (Eq. 9); it directly sets the D-to-kappa rate and is the basis for the tetraquark comparison.
  • w(0) = w+(0) = w-(0) = (0.94 +/- 0.14) GeV^-1
    Fitted non-resonant vector form factor; controls the size of the non-resonant s-wave and p-wave background.
  • h(0) = (4.24 +/- 0.79) GeV^-3
    Fitted non-resonant tensor form factor; constrained mainly by D -> K pi pi and D -> K pi rho rates.
  • a1 (nonleptonic factorization coefficient) = 1.1 +/- 0.1
    Chosen by varying the effective color number N_c^eff in {2, 3, infinity}; used for D -> K pi pi and D -> K pi rho, with a 10% uncertainty assigned.
assumptions (5)
  • ad hoc to paper The s-wave amplitude is fully described by one broad kappa Breit-Wigner plus a non-resonant term with the pole parameterizations of Eq. (8).
    Sec II, Eqs. (2), (5), (8): this decomposition assigns the observed s-wave strength between kappa and background; the central claim depends on it.
  • ad hoc to paper w+(q2) = w-(q2) for the non-resonant form factors.
    Sec III before Eq. (9): treated as baseline because model estimates suggest spacelike and timelike form factors are nearly equal; alternatives are tested within the same functional family but a different energy dependence could change the kappa/background split.
  • domain assumption The K* form factors V, A1, A2 follow single-pole forms and f+, w+, w-, h follow double-pole forms with pole masses taken from Ref [12].
    Eqs. (7)-(8): standard parameterizations in semileptonic D analyses, but the choice of n = 2 and the pole masses affect the fitted f(0).
  • domain assumption Higher kaonic resonances (K0*(1430), K*(1410), K2*(1430)) contribute negligibly to D -> K pi e nu.
    Sec I, citing previous BESIII and FOCUS analyses; if one of the higher states has a larger s-wave component than assumed, the kappa fraction would be overestimated.
  • domain assumption SU(3) flavor symmetry relates D+ -> anti-kappa^0 form factors to D+ -> S_n and D+ -> S_ns form factors as given in Sec IV.
    Sec IV: uses F(D+ -> S_n) = 0.47 +/- 0.02 and F(D+ -> S_ns) = 0.31 +/- 0.02 from Ref [30] by the same group; the tetraquark versus quark-antiquark discrimination depends on these inputs.

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Cite this review

Pith. "Pith review of Probing the elusive $\kappa/K_0^*(700)$ resonance in semileptonic $D$ decays." pith.science (2026). https://pith.science/paper/FPO7UQUU

@misc{pith2026250720380,
  author       = {Pith},
  title        = {Pith review of: Probing the elusive $\kappa/K_0^*(700)$ resonance in semileptonic $D$ decays},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FPO7UQUU}},
  note         = {Machine review of arXiv:2507.20380}
}
abstract

The $\kappa/K_0^*(700)$ meson remains the most elusive among the light scalar resonances, with its presence in weak decays obscured by limited precision in branching fraction measurements. As a result, the true nature of the $\kappa$ remains difficult to explore. Through a partial-wave analysis of the semileptonic decay $D^+ \to K^-\pi^+ e^+ \nu_e$, we extract ${\cal B}(D^+ \to \bar\kappa^0 e^+ \nu_e, \bar\kappa^0 \to K^-\pi^+) =(2.2 \pm 0.1) \times 10^{-3}$. Previously considered negligible, this contribution is now shown to dominate the observed s-wave branching fraction. This reveals that clear evidence for the $\kappa$ in weak decays has long existed, but was misidentified as part of the non-resonant background. The extracted $D^+ \to \bar\kappa^0$ form factor, $f^+(0) = 0.32 \pm 0.01$, is significantly smaller than the $q\bar q$ prediction of $0.82 \pm 0.05$, and closely aligns with the $q^2\bar q^2$ expectation of $0.36 \pm 0.02$. Notably, this finding supports a compact tetraquark interpretation of the $\kappa$ meson.

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Works this paper leans on

48 extracted references · 32 canonical work pages

  1. [30]

    A. J. Buras, hep-ph/9806471

    arXiv

  2. [1]

    Using FD+→Sn = 0.47± 0.02 and FD+→Sns = 0.31± 0.02 [30], we obtain FD+→¯κ0 q¯q = 0.82± 0.05 and FD+→¯κ0 q2¯q2 = 0.36± 0.02

    Under SU (3) flavor [SU (3)f] symmetry [30, 43–45], the corresponding D+ → ¯κ0 form factors are related by FD+→¯κ0 q¯q = √ 2(fK/fπ)FD+→Sn and Fq2¯q2 = √ 2(fπ/fK)FD+→Sns, with fK/fπ = 1.2 and fπ/fK = 0.8 accounting for SU (3)f breaking [46, 47]. Using FD+→Sn = 0.47± 0.02 and FD+→Sns = 0.31± 0.02 [30], we obtain FD+→¯κ0 q¯q = 0.82± 0.05 and FD+→¯κ0 q2¯q2 = ...

    work page

  3. [2]

    J. R. Pelaez, Phys. Rept. 658, 1 (2016)

    2016

  4. [3]

    Navas et al

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

    2024

  5. [4]

    J. R. Pel´ aez and A. Rodas, Phys. Rev. Lett. 124, 172001 (2020)

    work page 2020

  6. [5]

    J. R. Pel´ aez and A. Rodas, Phys. Rept. 969, 1 (2022)

    work page 2022

  7. [6]

    E. M. Aitala et al. [E791], Phys. Rev. Lett. 89, 121801 (2002)

    work page 2002

  8. [7]

    J. M. Link et al. [FOCUS], Phys. Lett. B 653, 1 (2007)

    work page 2007

Show all 48 references
  1. [8]

    Bonvicini et al

    G. Bonvicini et al. [CLEO], Phys. Rev. D 78, 052001 (2008)

    2008

  2. [9]

    Ablikim et al

    M. Ablikim et al. [BESIII], Phys. Rev. D 89, 052001 (2014)

    2014

  3. [10]

    J. M. Link et al. [FOCUS], Phys. Lett. B 621, 72 (2005)

    2005

  4. [11]

    R. A. Briere et al. [CLEO], Phys. Rev. D 81, 112001 (2010)

    2010

  5. [12]

    del Amo Sanchez et al

    P. del Amo Sanchez et al. [BaBar], Phys. Rev. D 83, 072001 (2011)

    2011

  6. [13]

    Ablikim et al

    M. Ablikim et al. [BESIII], Phys. Rev. D 94, 032001 (2016)

    2016

  7. [14]

    Ablikim et al

    M. Ablikim et al. [BESIII], Phys. Rev. D 99, 011103 (2019)

    2019

  8. [15]

    Ablikim et al

    M. Ablikim et al. [BESIII], JHEP 10, 199 (2024)

    2024

  9. [16]

    Ablikim et al

    M. Ablikim et al. [BESIII], JHEP 03, 197 (2025)

    2025

  10. [17]

    Ablikim et al

    M. Ablikim et al. [BESIII], arXiv:2506.05761 [hep-ex]

  11. [18]

    R. L. Jaffe, Phys. Rev. D 15, 267 (1977)

    1977

  12. [19]

    R. L. Jaffe, Phys. Rev. D 15, 281 (1977)

    1977

  13. [20]

    F. E. Close and N. A. Tornqvist, J. Phys. G 28, R249 (2002)

    2002

  14. [21]

    J. R. Pelaez, Phys. Rev. Lett. 92, 102001 (2004)

    2004

  15. [22]

    Maiani, F

    L. Maiani, F. Piccinini, A. D. Polosa and V. Riquer, Phys. Rev. Lett. 93, 212002 (2004)

    2004

  16. [23]

    Amsler and N

    C. Amsler and N. A. Tornqvist, Phys. Rept. 389, 61 (2004)

    2004

  17. [24]

    R. L. Jaffe, Phys. Rept. 409, 1 (2005)

    2005

  18. [25]

    ’t Hooft, G

    G. ’t Hooft, G. Isidori, L. Maiani, A. D. Polosa and V. Riquer, Phys. Lett. B 662, 424 (2008)

    2008

  19. [26]

    A. H. Fariborz, R. Jora and J. Schechter, Phys. Rev. D 79, 074014 (2009)

    2009

  20. [27]

    Weinberg, Phys

    S. Weinberg, Phys. Rev. Lett. 110, 261601 (2013). 9

    2013

  21. [28]

    S. S. Agaev, K. Azizi and H. Sundu, Phys. Lett. B 789, 405 (2019)

    2019

  22. [29]

    Buchalla, A

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

    1996

  23. [31]

    Y. K. Hsiao, S. Q. Yang, W. J. Wei and B. C. Ke, JHEP 12, 226 (2025)

    2025

  24. [32]

    C. L. Y. Lee, M. Lu and M. B. Wise, Phys. Rev. D 46, 5040 (1992)

    1992

  25. [33]

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

    arXiv

  26. [34]

    Melikhov and B

    D. Melikhov and B. Stech, Phys. Rev. D 62, 014006 (2000)

    2000

  27. [35]

    H. Y. Cheng, C. W. Chiang and Z. Q. Zhang, Phys. Rev. D 105, 033006 (2022)

    2022

  28. [36]

    H. Y. Cheng and C. K. Chua, Phys. Rev. D 102, 053006 (2020)

    2020

  29. [37]

    D. R. Boito and R. Escribano, Phys. Rev. D 80, 054007 (2009)

    2009

  30. [38]

    Escribano, P

    R. Escribano, P. Masjuan and P. Sanchez-Puertas, Phys. Rev. D 108, 094006 (2023)

    2023

  31. [39]

    A. Ali, G. Kramer and C. D. Lu, Phys. Rev. D 58, 094009 (1998)

    1998

  32. [40]

    Y. K. Hsiao and C. Q. Geng, Phys. Rev. D 91, 116007 (2015)

    2015

  33. [41]

    Y. L. Wang, Y. K. Hsiao, K. L. Wang and C. C. Lih, Phys. Rev. D 111, 096013 (2025)

    2025

  34. [42]

    Branz, T

    T. Branz, T. Gutsche and V. E. Lyubovitskij, Eur. Phys. J. A 37, 303 (2008)

    2008

  35. [43]

    Sekihara and E

    T. Sekihara and E. Oset, Phys. Rev. D 92, 054038 (2015)

    2015

  36. [44]

    Stone and L

    S. Stone and L. Zhang, Phys. Rev. Lett. 111, 062001 (2013)

    2013

  37. [45]

    H. Y. Cheng, Phys. Rev. D 67, 034024 (2003)

    2003

  38. [46]

    Wang and C

    W. Wang and C. D. Lu, Phys. Rev. D 82, 034016 (2010)

    2010

  39. [47]

    Gronau, O

    M. Gronau, O. F. Hernandez, D. London and J. L. Rosner, Phys. Rev. D 52, 6356 (1995)

    1995

  40. [48]

    Y. L. Wang, H. J. Zhao and Y. K. Hsiao, Phys. Rev. D 111, 016022 (2025). 10

    2025

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