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REVIEW 2 major objections 4 minor 43 references

Novel method for determining the light quark mass ratio using $\eta'\to\eta \pi\pi$ decays

T0 review · 2 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read A Dalitz-plot-to-unit-disk mapping turns the difference of two η′ decay distributions into a clean probe of isospin breaking, yielding Q = 22.3 ± 0.7.

desk verdict A clever new method for isolating isospin breaking in three-body decays, but the central Q extraction is not yet reliable at the claimed precision. read the letter →

arxiv 2502.02837 v3 pith:D4EELH5W submitted 2025-02-05 hep-ph hep-ex

Adolfo Guevara , Feng-Kun Guo , Hao-Jie Jing This is my paper · ORCID
classification hep-phhep-ex
keywords lightquarkmassesisospinbreakingDalitzplotunitdiskmappingetaprimedecaychiralperturbationtheorylargeNcN/Dunitarization
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 proposes a way to extract symmetry-breaking effects from three-body decays that are otherwise allowed by the symmetry. The idea is to map the Dalitz plot of each decay onto a unit disk and subtract the two disk distributions bin by bin; the dominant isospin-conserving amplitude cancels, leaving a difference that is purely isospin breaking. Applying this to the pair η′→ηπ⁺π⁻ and η′→ηπ⁰π⁰, using the published data, the authors fit the theoretical isospin-breaking amplitude and obtain Q = 22.3 ± 0.7, where Q²=(m_s²−m̂²)/(m_d²−m_u²). This is consistent with earlier determinations and shows the method can extract a fundamental Standard Model parameter from the full Dalitz-plot shape rather than from branching fractions alone. The same disk-difference technique is meant to generalize to other three-body decays.

What carries the argument

The load-bearing construction is a one-to-one mapping from the ordinary Dalitz plot to a unit disk, built from a boundary map f(θ)=(L(θ),R(θ)) that sends the circle to the boundary of the physical region. A point d inside the plot maps to radius r=|d−c₊|/|f(θ)−c₊|, so the entire phase space can be binned identically for two decay channels. Once both channels are placed on the same disk, the bin-by-bin difference dΓ_diff = d²Γ(π0)/dm²₁₂dm²₂₃ − d²Γ(π±)/dm²₁₂dm²₂₃ equals 2 Re(M_IC^* M_IB)+O(Q⁻⁸), so the known isospin-conserving amplitude M_IC drops out and the remaining signal is the isospin-breaking amplitude M_IB. That amplitude is computed in large-Nc chiral perturbation theory with the ππ final-state interaction included through N/D unitarization, following Ref. [20]; the fit to the difference disk then determines Q.

What would settle it

Apply the same unit-disk difference fit to the full BESIII η′ data set (eight times larger); if the resulting Q differs from 22.3 by more than the combined uncertainty, the assumed isospin-breaking amplitude is incomplete.

Watch

Extended reading notes

Core claim

The central claim is that the bin-by-bin difference of two Dalitz plots mapped onto a unit disk isolates pure isospin-breaking effects, because the leading isospin-conserving amplitude is identical for the two decays and cancels in the subtraction. For η′→ηπ⁺π⁻ and η′→ηπ⁰π⁰, the difference distribution is governed by 2 Re(M_IC^* M_IB), where M_IB itself starts at two isospin-breaking insertions and is proportional to (m_d−m_u)². Fitting the published difference disk with the large-Nc chiral perturbation theory amplitude unitarized by N/D for ππ rescattering, with L₂, L₃, Q and the two disk normalizations as free parameters, the authors obtain Q=22.3±0.7. They take this as a demonstration that the method works and can be extended to the full BESIII sample and to other three-body decays.

Load-bearing premise

The fit assumes the theoretical isospin-breaking amplitude taken from unitarized large-Nc chiral perturbation theory is accurate over the entire Dalitz plot; if that amplitude is incomplete, the extracted Q = 22.3 is biased.

Editorial extensions

If this is right

  • The full BESIII η′ sample, eight times larger than the one used here, should reduce the statistical uncertainty on Q substantially.
  • The method is not limited to η′ decays: it can be applied to any pair of three-body decays related by isospin, as the paper illustrates with heavy-quark decays into J/ψππ.
  • Because the whole Dalitz plot is used, the precision on Q does not have to rely on the ratio of branching fractions alone.
  • Including the Y X² and X⁴ terms in the Dalitz-plot parametrization, once available, could improve the extracted Q.
  • A dispersive treatment of the final-state interactions, as used in η→3π analyses, could replace the N/D unitarization and test the model dependence.

Reading between the lines

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

  • One can transfer the disk-difference idea to other isospin-related decay pairs, such as η→3π or charmonium decays, as long as both channels have precisely measured Dalitz plots and a reliable isospin-breaking amplitude; the paper itself mentions heavy-quark decays as candidates.
  • Because the normalizations of the two disks are fitted separately, the extracted Q should be insensitive to the absolute branching fractions; the information comes from the shape of the difference distribution. This follows from the fitting procedure described, though the paper does not state it as a separate claim.
  • At the eight-times-larger full BESIII statistics, the statistical error on Q is likely to fall well below 0.7, so the model dependence of the isospin-breaking amplitude (reflected in the fit quality) will become the limiting uncertainty; the paper does not quantify this model uncertainty.
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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

2 major / 4 minor

Summary. The paper proposes a new method for extracting isospin-breaking parameters from three-body decays by mapping each Dalitz plot onto a unit disk and taking a bin-by-bin difference of the two normalized disk distributions. As a first application, the method is applied to the BESIII Dalitz-plot measurements of eta' -> eta pi+ pi- and eta' -> eta pi0 pi0 from Ref. [6]. Working in large-Nc ChPT with N/D-unitarized pi pi rescattering (following Ref. [20]), the authors fit the difference disk with L2, L3, Q, and two disk normalizations as free parameters, obtaining Q = 22.3(7) with chi2/dof = 18549/7832. The abstract quotes Q = 22.5 +/- 1.0. The authors argue that the method uses the full Dalitz-plot information and will improve once the full BESIII data set becomes available.

Significance. If the method is sound, it offers a genuinely new way to extract symmetry-breaking quantities from three-body decays, going beyond rate-based analyses and using the entire Dalitz-plot shape. The paper is transparent in using published BESIII data and in listing the input parameters and correlations, which makes the numerical part reproducible. The central claim, however, is not yet secured: the reported fit is poor, the model-systematic uncertainty is not quantified, and the cancellation of the isospin-conserving contribution at equal disk coordinates is not demonstrated. These issues must be resolved before the result can be regarded as a competitive determination of Q.

major comments (2)
  1. [Section IV, Eq. (16)] The fit reported in Eq. (16) yields chi2/dof = 18549/7832 ≈ 2.37 for 7837 bins, which indicates that the model does not describe the difference disk within statistical uncertainties. The paper does not assign any systematic uncertainty from the truncation of the BESIII parameterization at the a, b, d terms; the YX^2 and X^4 terms are mentioned only as an outlook in the Conclusions. Consequently, the quoted Q = 22.3(7) is only a statistical error propagation and does not account for the model mismatch visible in the fit itself. A quantitative model-systematic estimate is required before the claimed competitive precision is supported.
  2. [Section III, Eqs. (12)-(13)] The cancellation of the isospin-conserving amplitude in the bin-by-bin difference is not established. The unit-disk mapping is mass-dependent, so the same disk coordinates (r, theta) correspond to different physical values of (m^2_12, m^2_23) for the charged and neutral channels. The statement after Eq. (13) that 'the former is the same for both decays' is therefore insufficient: the densities are evaluated at different kinematic points, so the |M_IC|^2 terms do not cancel exactly. The residual is of first order in m_pi+ - m_pi0, while the signal term 2 Re(M*_IC M_IB) is of second order, implying that this residual could dominate the difference disk. The authors should either prove an exact alignment of the isospin-related phase-space points in the mapping or quantify this background explicitly.
minor comments (4)
  1. [Abstract and Section IV] The abstract quotes Q = 22.5 ± 1.0, while Section IV, Eq. (16) and Table I quote Q = 22.3(7). This inconsistency must be corrected.
  2. [Section IV, Eq. (14) and the mapping] The conversion from the BESIII X,Y parameterization in Eq. (14) to the invariant-mass density used in the unit-disk mapping involves a Jacobian that is not discussed. The paper should state explicitly how the density was transformed.
  3. [Section IV, text after Eq. (14)] The description of the Monte Carlo procedure states that 7000 points per bin are used and that the parameters a, b, d are generated according to their BESIII means, errors, and correlations, but it does not specify how the normalization factors N in Eq. (14) are combined with the fitted disk normalizations N_pi+/- and N_pi0; this should be clarified.
  4. [Appendix C] In Eqs. (C2)-(C3) the correlation matrices are written with rows labeled in the order b, d, a; for readability, the label order should be made explicit or the matrices should be written with a uniform ordering.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity: Q is a fitted output from independent BESIII Dalitz data; the only self-citation (Ref. [29] for eta-eta' mixing inputs) is non-load-bearing.

full rationale

The target quantity Q is not an input anywhere in the analysis: it is one of the free parameters fitted to the BESIII Dalitz plot parameters of Refs. [6,7] through the unit-disk difference distribution. The measured inputs (a,b,d and correlations for both decay modes) are external experimental results, and the theoretical amplitude, including the N/D unitarized M_IB, is taken from the independent prior work Ref. [20]; the paper's central result therefore does not reduce to its inputs by construction. The cancellation of the isospin-conserving amplitude in Eq. (13) is an algebraic consequence of the assumed amplitude decomposition, not a definition of the experimental difference, so the claim that the disk difference 'isolates' symmetry breaking is a model-dependent extraction rather than a circular one. The only self-citation is Ref. [29] by one of the authors used for the numerical values of the eta-eta' mixing parameters f8/0 and theta8/0; these are standard external inputs, and the cited values do not contain or presuppose the fitted Q. The large chi2/dof of 18549/7832 and the missing Y X^2 and X^4 terms are serious goodness-of-fit and model-systematic concerns, but they are correctness risks, not circularity. The derivation is therefore essentially self-contained against external data and independent theory inputs.

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

The paper introduces no new physical entities. It relies on standard ChPT/N/D input from the literature and fits five parameters (L2, L3, Q, two normalizations) to the experimentally derived difference disk. The main unverified input is the theoretical amplitude from Ref [20] and the assumed smallness of electromagnetic and ηπ-rescattering effects.

free parameters (5)
  • L2 = 1.01(3)×10^-3
    Low-energy constant fitted to the difference disk.
  • L3 = -4.40(10)×10^-3
    Low-energy constant fitted to the difference disk.
  • Q = 22.3(7)
    Target parameter; ratio of light quark masses.
  • N_pi± = 108.7(4)
    Overall normalization of the charged-pion decay disk.
  • N_pi0 = 61.8(4)
    Overall normalization of the neutral-pion decay disk.
assumptions (4)
  • domain assumption Electromagnetic corrections to η′→ηππ decays are negligible relative to quark-mass isospin breaking.
    Stated in Section II, argued by analogy with η→3π (Ref [13]); no dedicated estimate for η′ decays is given.
  • domain assumption The large-Nc ChPT amplitude with N/D unitarization from Ref [20] accurately describes both the isospin-conserving and isospin-breaking amplitudes.
    Adopted in Section II; the paper relies on this amplitude for the fit.
  • domain assumption ηπ rescattering (t- and u-channel) is negligible in these decays.
    Invoked in Sections II and IV, citing Refs [30,31].
  • ad hoc to paper The BESIII Dalitz plot parameterization truncated at a, b, d terms (without YX² and X⁴) is sufficient for the difference-disk extraction.
    The full BESIII analysis lacks these terms; the paper acknowledges this in Section V but does not assess its impact on Q.

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Pith. "Pith review of Novel method for determining the light quark mass ratio using $\eta'\to\eta \pi\pi$ decays." pith.science (2026). https://pith.science/paper/D4EELH5W

@misc{pith2026250202837,
  author       = {Pith},
  title        = {Pith review of: Novel method for determining the light quark mass ratio using $\eta'\to\eta \pi\pi$ decays},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/D4EELH5W}},
  note         = {Machine review of arXiv:2502.02837}
}
abstract

We propose a novel approach for extracting symmetry breaking effects from symmetry conserving three-body decays. The method is based on mapping the Dalitz plot to a unit disk, and the difference of the disk distributions of two related decays isolates purely symmetry breaking effects. We demonstrate this method by extracting the fundamental parameter $Q$, an isospin breaking ratio of light quark masses defined as $Q^2\equiv (m_s^2-\hat m^2)/(m_d^2-m_u^2)$ with $\hat m$ the average of up and down quark masses, from the decays $\eta'\to\eta\pi^+\pi^-$ and $\eta'\to\eta\pi^0\pi^0$. With the Dalitz plot distributions for these two decays reported by BESIII, we illustrate the method and obtain $Q=22.5\pm1.0$, which is consistent with previous determinations and has a comparable uncertainty. With the full BESIII data set, which is eight times larger than the one used here, a more precise determination of $Q$ should become possible. This promising and novel method can be generalized to other three-body decays to extract symmetry breaking effects.

Figures

Figures reproduced from arXiv: 2502.02837 by the authors.

Figure 1
Figure 1. FIG. 1. Feynman diagrams of the contributions with two [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Illustration of the mapping from a unit disk [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Unnormalized Dalitz disk of the [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (2 more)
Figure 5
Figure 5. Figure 5: FIG. 5. Left: the unit disk with its boundary divided into two [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Mapping from a Dalitz plot to the unit disk, where [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

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Reference graph

Works this paper leans on

43 extracted references · 19 canonical work pages

  1. [20]

    Escribano, P

    R. Escribano, P. Masjuan, and J. J. Sanz-Cillero, Chi- ral dynamics predictions for η′ → ηππ , JHEP 05, 094, arXiv:1011.5884 [hep-ph]

    arXiv

  2. [6]

    Measurement of the matrix elements for the decays $\eta^{\prime}\rightarrow\eta\pi^+\pi^-$ and $\eta^{\prime}\rightarrow\eta\pi^0\pi^0$

    M. Ablikim et al. (BESIII), Measurement of the matrix elements for the decays η′ → ηπ +π− and η′ → ηπ 0π0, Phys. Rev. D 97, 012003 (2018), arXiv:1709.04627 [hep- ex]

    work page Pith review arXiv 2018

  3. [1]

    R. H. Dalitz, On the analysis of tau-meson data and the nature of the tau-meson, Phil. Mag. Ser. 7 44, 1068 (1953)

    work page 1953

  4. [2]

    Fabri, A study of tau-meson decay, Nuovo Cim

    E. Fabri, A study of tau-meson decay, Nuovo Cim. 11, 479 (1954)

    work page 1954

  5. [3]

    Leutwyler, Bounds on the light quark masses, Phys

    H. Leutwyler, Bounds on the light quark masses, Phys. Lett. B 374, 163 (1996), arXiv:hep-ph/9601234

    arXiv 1996

  6. [4]

    Gasser and H

    J. Gasser and H. Leutwyler, η → 3π to One Loop, Nucl. Phys. B 250, 539 (1985)

    work page 1985

  7. [5]

    R. L. Workman et al. (Particle Data Group), Review of Particle Physics, PTEP 2022, 083C01 (2022)

    2022

  8. [7]

    Ablikim et al

    M. Ablikim et al. (BESIII), Evidence for the Cusp Effect in η′ Decays into ηπ 0π0, Phys. Rev. Lett. 130, 081901 (2023), arXiv:2207.01004 [hep-ex]

    work page arXiv 2023

Show all 43 references
  1. [8]

    L. Gan, B. Kubis, E. Passemar, and S. Tulin, Precision tests of fundamental physics with η and η′ mesons, Phys. Rept. 945, 1 (2022), arXiv:2007.00664 [hep-ph]

    2022 arXiv

  2. [9]

    Weinberg, Phenomenological Lagrangians, Physica A 96, 327 (1979)

    S. Weinberg, Phenomenological Lagrangians, Physica A 96, 327 (1979)

    1979

  3. [10]

    Gasser and H

    J. Gasser and H. Leutwyler, Chiral Perturbation Theory to One Loop, Annals Phys. 158, 142 (1984)

    1984

  4. [11]

    Gasser and H

    J. Gasser and H. Leutwyler, Chiral Perturbation The- ory: Expansions in the Mass of the Strange Quark, Nucl. Phys. B 250, 465 (1985)

    1985

  5. [12]

    Urech, Virtual photons in chiral perturbation theory, Nucl

    R. Urech, Virtual photons in chiral perturbation theory, Nucl. Phys. B 433, 234 (1995), arXiv:hep-ph/9405341

    1995 arXiv

  6. [13]

    Ditsche, B

    C. Ditsche, B. Kubis, and U.-G. Meißner, Electromag- netic corrections in η → 3π decays, Eur. Phys. J. C 60, 83 (2009), arXiv:0812.0344 [hep-ph]

    2009 arXiv

  7. [14]

    Kubis and S

    B. Kubis and S. P. Schneider, The Cusp effect in η′ → ηππ decays, Eur. Phys. J. C 62, 511 (2009), arXiv:0904.1320 [hep-ph]. 6

    2009 arXiv

  8. [15]

    Kaiser and H

    R. Kaiser and H. Leutwyler, Large Nc in chiral perturba- tion theory, Eur. Phys. J. C 17, 623 (2000), arXiv:hep- ph/0007101

    2000

  9. [16]

    A. H. Fariborz and J. Schechter, η′ → ηππ decay as a probe of a possible lowest lying scalar nonet, Phys. Rev. D 60, 034002 (1999), arXiv:hep-ph/9902238

    1999 arXiv

  10. [17]

    A. V. Anisovich and H. Leutwyler, Dispersive analysis of the decay η → 3π, Phys. Lett. B 375, 335 (1996), arXiv:hep-ph/9601237

    1996 arXiv

  11. [18]

    Borasoy and R

    B. Borasoy and R. Nißler, Hadronic η and η′ decays, Eur. Phys. J. A 26, 383 (2005), arXiv:hep-ph/0510384

    2005 arXiv

  12. [19]

    Borasoy, U.-G

    B. Borasoy, U.-G. Meißner, and R. Nißler, On the ex- traction of the quark mass ratio ( md − mu)/ms from Γ(η′ → π0π+π−)/Γ(η′ → ηπ +π−), Phys. Lett. B 643, 41 (2006), arXiv:hep-ph/0609010

    2006 arXiv

  13. [21]

    Gonz` alez-Sol ´ ıs and E

    S. Gonz` alez-Sol ´ ıs and E. Passemar,η′ → ηππ decays in unitarized resonance chiral theory, Eur. Phys. J. C 78, 758 (2018), arXiv:1807.04313 [hep-ph]

    2018 arXiv

  14. [22]

    Isken, B

    T. Isken, B. Kubis, S. P. Schneider, and P. Stoffer, Dis- persion relations for η′ → ηππ , Eur. Phys. J. C 77, 489 (2017), arXiv:1705.04339 [hep-ph]

    2017 arXiv

  15. [23]

    The decay amplitude for each decay contains both the IC and IB contributions

    will be included as a factor of each differential decay width obtained from experimental data. The decay amplitude for each decay contains both the IC and IB contributions. Since the former is the same for both decays, we have dΓdiff(r, θ) = 2 Re(M∗ ICMIB) + O Q−8 , (13) where...

  16. [24]

    Akdag, T

    H. Akdag, T. Isken, and B. Kubis, Patterns of C- and CP-violation in hadronic η and η′ three-body de- cays, JHEP 02, 137, [Erratum: JHEP 12, 156 (2022)], arXiv:2111.02417 [hep-ph]

    2022 arXiv

  17. [25]

    G. F. Chew and S. Mandelstam, Theory of low-energy pion pion interactions, Phys. Rev. 119, 467 (1960)

    1960

  18. [26]

    J. A. Oller and E. Oset, N/D description of two me- son amplitudes and chiral symmetry, Phys. Rev. D 60, 074023 (1999), arXiv:hep-ph/9809337

    1999 arXiv

  19. [27]

    Osborn and D

    H. Osborn and D. J. Wallace, η-X mixing, η → 3π and chiral lagrangians, Nucl. Phys. B 20, 23 (1970)

    1970

  20. [28]

    Feldmann, P

    T. Feldmann, P. Kroll, and B. Stech, Mixing and de- cay constants of pseudoscalar mesons, Phys. Rev. D 58, 114006 (1998), arXiv:hep-ph/9802409

    1998 arXiv

  21. [29]

    Feldmann, P

    T. Feldmann, P. Kroll, and B. Stech, Mixing and de- cay constants of pseudoscalar mesons: The Sequel, Phys. Lett. B 449, 339 (1999), arXiv:hep-ph/9812269

    1999 arXiv

  22. [30]

    Guevara, P

    A. Guevara, P. Roig, and J. J. Sanz-Cillero, Pseu- doscalar pole light-by-light contributions to the muon (g − 2) in Resonance Chiral Theory, JHEP 06, 160, arXiv:1803.08099 [hep-ph]

    arXiv

  23. [31]

    S. P. Schneider and B. Kubis, Cusps in η′ → ηππ decays, PoS CD09, 120 (2009), arXiv:0910.0200 [hep-ph]

    2009 arXiv

  24. [32]

    Kubis, Cusp effects in meson decays, EPJ Web Conf

    B. Kubis, Cusp effects in meson decays, EPJ Web Conf. 3, 01008 (2010), arXiv:0912.3440 [hep-ph]

    2010 arXiv

  25. [33]

    Navas and Others (Particle Data Group), Review of Particle Physics, Phys

    S. Navas and Others (Particle Data Group), Review of Particle Physics, Phys. Rev. D 110, 030001 (2024)

    2024

  26. [34]

    Kambor, C

    J. Kambor, C. Wiesendanger, and D. Wyler, Final state interactions and Khuri-Treiman equations in η → 3π decays, Nucl. Phys. B 465, 215 (1996), arXiv:hep- ph/9509374

    1996

  27. [35]

    Bijnens and K

    J. Bijnens and K. Ghorbani, η → 3π at Two Loops In Chiral Perturbation Theory, JHEP 11, 030, arXiv:0709.0230 [hep-ph]

    arXiv

  28. [36]

    Kampf, M

    K. Kampf, M. Knecht, J. Novotny, and M. Zdrahal, An- alytical dispersive construction of η → 3π amplitude: first order in isospin breaking, Phys. Rev. D 84, 114015 (2011), arXiv:1103.0982 [hep-ph]

    2011 arXiv

  29. [37]

    Colangelo et al

    G. Colangelo et al. , Review of lattice results concerning low energy particle physics, Eur. Phys. J. C 71, 1695 (2011), arXiv:1011.4408 [hep-lat]

    2011 arXiv

  30. [38]

    Colangelo, S

    G. Colangelo, S. Lanz, H. Leutwyler, and E. Passemar, η → 3π: Study of the Dalitz plot and extraction of the quark mass ratio Q, Phys. Rev. Lett. 118, 022001 (2017), arXiv:1610.03494 [hep-ph]

    2017 arXiv

  31. [39]

    Colangelo, S

    G. Colangelo, S. Lanz, H. Leutwyler, and E. Passemar, Dispersive analysis of η → 3π, Eur. Phys. J. C 78, 947 (2018), arXiv:1807.11937 [hep-ph]

    2018 arXiv

  32. [40]

    Albaladejo and B

    M. Albaladejo and B. Moussallam, Extended chiral Khuri-Treiman formalism for η → 3π and the role of the a0(980), f0(980) resonances, Eur. Phys. J. C 77, 508 (2017), arXiv:1702.04931 [hep-ph]

    2017 arXiv

  33. [41]

    Aoki et al

    Y. Aoki et al. (Flavour Lattice Averaging Group (FLAG)), FLAG Review 2024, (2024), arXiv:2411.04268 [hep-lat]

    2024 arXiv

  34. [42]

    R. F. Dashen, Chiral SU(3) × SU(3) as a symmetry of the strong interactions, Phys. Rev. 183, 1245 (1969). Appendix A: Isospin conserving contribution The ChPT Lagrangian density gives the dynamics of the pseudo-Nambu-Goldstone bosons in a nonlinear re- alization of the symmetr...

    1969

  35. [43]

    + u(m2 3m2 1 +p2m2 2), where s = m2 12, t = m2 23, u = m2 13 = m2 + m2 1 + m2 2 + m2 3 − m2 12 − m2 23 and m is the mass of the initial particle. It is easy to obtain the coordinates in the conventional Dalitz plot where m12 has its maximum and minimum values, which we call, r...

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