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One-boson-exchange dynamics in the Bs0N–BΛ–B*Λ–BΣ–B*Σ system produce three near-threshold poles that support molecular antibottom-strange pentaquarks.

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2026-07-11 15:17 UTC pith:FLDJOVXX

load-bearing objection Solid five-channel OBE prediction of three narrow near-threshold P_bs-bar poles; one of them is soft under the free short-range parameter a, but the paper maps that honestly. the 2 major comments →

arxiv 2607.04679 v2 pith:FLDJOVXX submitted 2026-07-06 hep-ph hep-th

Possibility of the antibottom-strange molecular pentaquarks near BSigma and B^*Sigma thresholds

Jian-Kang Zhao , Nijiati Yalikun This is my paper
classification hep-ph hep-th
keywords molecular pentaquarksone-boson-exchangeantibottom-strangeBΣ thresholdcoupled-channel polesS–D wave mixingshort-range δ(r)
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper asks whether antibottom-strange pentaquarks can form as molecules of a heavy antimeson and a light baryon near the BΣ and B*Σ thresholds. Using a one-boson-exchange potential that includes S–D wave mixing and a tunable short-range contact term, the authors solve the coupled-channel Schrödinger equation and analytically continue the S-matrix into the complex energy plane. Single-channel BΣ and B*Σ systems with JP = 1/2− and 3/2− already bind for reasonable cutoffs; when the lower open channels are switched on, those bound states become three nearby poles whose masses sit between 6.44 and 6.52 GeV and whose widths are only a few to several MeV. The poles couple most strongly to BΣ and B*Σ, so their experimental fingerprints should appear mainly as enhancements or cusps in the Bs0N, BΛ and B*Λ invariant-mass spectra. The result supplies concrete mass windows and preferred final states for collider searches of the still-unobserved Pbs-bar states.

Core claim

In the full Bs0N–BΛ–B*Λ–BΣ–B*Σ coupled system the OBE interaction generates three near-threshold poles (one below BΣ with JP = 1/2− and two associated with B*Σ with JP = 1/2− and 3/2−) whose masses lie in the 6.44–6.52 GeV window and whose widths are a few to several MeV; these poles are generated mainly by the BΣ and B*Σ interactions and produce observable line-shape enhancements in the lower open channels, supporting the existence of molecular Pbs-bar pentaquarks.

What carries the argument

Analytic continuation of the multi-channel S-matrix obtained from the coordinate-space Schrödinger equation with OBE potentials that include a monopole form factor and a dimensionless parameter a controlling the short-range δ(r) contact term; poles on Riemann sheets connected to the physical axis are identified as the molecular candidates.

Load-bearing premise

The short-range contact piece of the potential is controlled by a single free number a that is varied by hand; one of the three poles changes from a resonance into a virtual state when that number is switched off.

What would settle it

A prompt-production amplitude analysis at a hadron collider of the Bs0N, BΛ and B*Λ invariant-mass spectra near 6.44–6.52 GeV that finds no corresponding narrow peaks or threshold cusps would rule out the predicted poles.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The manuscript studies the coupled-channel system Bs0N–BΛ–B*Λ–BΣ–B*Σ in the one-boson-exchange framework with S–D mixing and a tunable short-range δ(r) contact term controlled by a dimensionless parameter a. Effective Lagrangians constrained by heavy-quark spin symmetry and SU(3) flavor symmetry are used to derive the potentials; a monopole form factor with cutoff Λ regularizes them. Bound, virtual and resonant poles are located by analytic continuation of the multi-channel S-matrix. Single-channel analysis yields three attractive bound states (1/2−(BΣ), 1/2−(B*Σ), 3/2−(B*Σ)). In the full coupled-channel calculation these evolve into near-threshold poles in the 6.44–6.52 GeV window with widths of a few to several MeV. Residue and partial-width analyses show that the poles are generated mainly by the BΣ and B*Σ interactions and should appear as enhancements or cusps in the open channels Bs0N, BΛ and B*Λ. The authors themselves emphasize that the higher 1/2− pole near the B*Σ threshold is sensitive to a and can become a virtual state when the contact term is removed.

Significance. If the predicted poles survive a more constrained short-range treatment, the work supplies concrete, falsifiable guidance for experimental searches of antibottom-strange molecular pentaquarks at the LHC: masses, preferred final states, and the distinction between Breit–Wigner peaks and threshold cusps. The calculation is a pure prediction (no data are fitted), the free parameters Λ and a are varied openly, and the analytic continuation, residue extraction and partial-width formulae are standard and consistently applied. The explicit mapping of the short-range sensitivity (especially Fig. 5 and Tables V–VI) is a methodological strength that allows the reader to judge robustness. The results therefore constitute a useful, if still model-dependent, extension of the molecular picture from the charm to the bottom-strange sector.

major comments (2)
  1. Sec. II B, Eqs. (24)–(25) and Tables V–VI / Fig. 5: the higher JP=1/2− pole near the B*Σ threshold moves from a resonance on the physical sheet to a virtual state once a exceeds ~0.17. Because a is an essentially free phenomenological parameter, the resonant character of this particular state is not robust. The abstract and summary correctly flag the sensitivity, yet the claim of “three” near-threshold molecular states should be qualified more sharply: only the 1/2−(BΣ) and 3/2−(B*Σ) poles remain stable across the explored range of a. A short additional paragraph that isolates the two robust poles and treats the third as a possible virtual-state/cusp structure would bring the central claim into line with the paper’s own numerical findings.
  2. Sec. III B and Table VIII: partial widths are reported only for the extreme case a=0. Given that the higher 1/2− pole’s existence depends on a, the partial-width analysis should be repeated for at least one intermediate value (e.g. a=0.5) so that the experimental guidance for the robust poles is not tied exclusively to the most attractive short-range prescription.
minor comments (5)
  1. Throughout: inconsistent notation for the heavy mesons (B vs ¯B, Bs0 vs B0s). A single convention should be fixed in Sec. II and used uniformly.
  2. Fig. 1 caption and text: “cutoffset” appears to be a typographical error for “cutoff set”.
  3. Table III: the threshold values are quoted to 0.1 MeV while the Particle Data Group averages carry larger uncertainties; a brief remark on the adopted mass values would be helpful.
  4. Sec. IV: “indecate” → “indicate”; a few other minor spelling slips (e.g. “deu to”) remain.
  5. The experimental-search discussion correctly notes that weak decays of ground-state bottom baryons are kinematically closed; a short quantitative estimate of the expected prompt-production cross-section scale (even order-of-magnitude) would strengthen the guidance for LHCb.

Circularity Check

0 steps flagged

No significant circularity: pure OBE model prediction with literature couplings and openly varied free parameters Λ and a; mild self-citation of prior method only.

full rationale

The derivation chain is self-contained and non-circular. Effective Lagrangians (Eqs. 1–8) and couplings (Table I, values from Refs. [44,56,57,61,65–67]) are taken from external literature constrained by HQSS/SU(3); potentials (Eqs. 18, Table II) follow standard Breit approximation and Fourier transform (Eqs. 19–25) with monopole form factor. The free parameters Λ (cutoff) and a (δ(r) strength, a=0 fully includes, a=1 removes) are varied over ranges and their effects on single-channel bindings (Table IV) and coupled-channel poles (Tables V–VII, Figs. 2–6) are reported explicitly, including the transition of the JP=1/2− B*Σ pole to a virtual state for a≳0.17. No data are fitted; poles, couplings gi, and partial widths Γi are obtained by solving the coupled Schrödinger equation (Eq. 26) and analytic continuation of S/T matrices (Eqs. 28–32). The only minor self-reference is adoption of the a-parameter strategy from the authors’ prior work [36], which is not load-bearing (the paper maps the full a-dependence and claims only “possibility” plus experimental guidance). Central results (near-threshold poles at 6.44–6.52 GeV dominated by B(∗)Σ) are independent model outputs, not forced by construction or self-citation chain.

Axiom & Free-Parameter Ledger

2 free parameters · 4 axioms · 1 invented entities

The central claim rests on the standard OBE machinery plus two free short-distance parameters and the assumption that the same light-meson exchanges that work for Pc states remain valid for antibottom-strange systems. No new dynamical entities beyond the predicted poles themselves are introduced.

free parameters (2)
  • cutoff Λ
    Monopole form-factor cutoff that regularizes the OBE potential; varied by hand in the range ~1.0–1.8 GeV; binding energies and pole locations depend strongly on its value (Tables IV–VII).
  • short-range strength a
    Dimensionless coefficient that multiplies the δ(r) contact term arising from the Fourier transform of the momentum-space potential; a=0 fully includes the term, a=1 removes it; one of the three poles changes character when a is varied (Eqs. (24)–(25), Fig. 5).
axioms (4)
  • domain assumption Heavy-quark spin symmetry and SU(3) flavor symmetry fix the relative strengths of the meson-baryon vertices.
    Used throughout Sec. II A to write the effective Lagrangians (1)–(8); standard but not exact for finite heavy-quark mass and SU(3) breaking.
  • domain assumption The interaction is adequately described by one-boson exchange of the lightest scalar, pseudoscalar and vector mesons plus a contact term.
    Core modeling choice of the entire paper; heavier exchanges and multi-meson forces are absorbed into a and Λ.
  • domain assumption The non-relativistic Schrödinger equation with Breit-approximated potentials and monopole form factors yields reliable near-threshold poles.
    Sec. II B–C; common in the molecular literature but introduces uncontrolled relativistic and off-shell uncertainties.
  • standard math Analytic continuation of the multi-channel S-matrix correctly classifies bound, virtual and resonant poles.
    Standard scattering theory (Taylor, RPP); applied without modification.
invented entities (1)
  • P_bs-bar molecular pentaquarks (three poles) no independent evidence
    purpose: The predicted near-threshold states whose existence is the paper’s central claim.
    They are model outputs, not input postulates; independent experimental confirmation would constitute external evidence.

pith-pipeline@v1.1.0-grok45 · 27151 in / 2982 out tokens · 28215 ms · 2026-07-11T15:17:56.806110+00:00 · methodology

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read the original abstract

The molecular states in coupled channel system of $B_s^0N- B\Lambda- B^*\Lambda- B\Sigma- B^*\Sigma$ are investigated within one-boson-exchange model that includes $S$-$D$ wave mixing and a tunable short-range $\delta(r)$ term. Bound, resonant, and virtual states are searched by analytically continuing the $S$ matrix in the complex energy plane. In the single-channel analysis, the $ B\Sigma$ and $ B^*\Sigma$ systems with $J^P=1/2^-$ and $3/2^-$ are found to be attractive and form three bound states with reasonable cutoff region, corresponding to $1/2^-( B\Sigma)$, $1/2^-( B^*\Sigma)$, and $3/2^-( B^*\Sigma)$ . When coupled-channel dynamics is included, these states evolve into near-threshold poles that produce enhancements in the $B_s^0N$, $ B\Lambda$, and $ B^*\Lambda$ invariant mass spectra. The pole below the $ B^*\Sigma$ threshold in the $J^P=1/2^-$ system depends strongly on the treatment of the short-range $\delta(r)$ term and becomes a virtual state when this contribution is removed, while the other two resonant poles and their line shapes are only moderately affected. For representative parameter region, the predicted masses of these states lie in the $6.44-6.52$ GeV region and have widths of a few to several MeV. The pole coupling and partial-width analyses indicate that the poles are generated mainly by the $ B\Sigma$ and $ B^*\Sigma$ channels interaction, and their observable signals are expected mainly in the lower open channels $B_s^0N$, $ B\Lambda$, and $ B^*\Lambda$. These results support the existence of near-threshold molecular pentaquark $P_{ \bar b s}$ and provide useful guidance in future experimental searches.

Figures

Figures reproduced from arXiv: 2607.04679 by Jian-Kang Zhao, Nijiati Yalikun.

Figure 1
Figure 1. Figure 1: FIG. 1: Potentials of the [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: The [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: The absolute [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: For [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Trajectory of the pole below the [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: For [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗

discussion (0)

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Reference graph

Works this paper leans on

79 extracted references · 59 linked inside Pith

  1. [1]

    The scaled anti-bottomed meson field ˜P(∗) is defined in fla- vor/isospin space as (B +,B 0,B 0 s) and (B ∗+,B 0∗,B 0∗ s ) [58–60]

    withg π =3.73 [56, 57]. The scaled anti-bottomed meson field ˜P(∗) is defined in fla- vor/isospin space as (B +,B 0,B 0 s) and (B ∗+,B 0∗,B 0∗ s ) [58–60]. The pion decay constant isf π =132 MeV , and the vector cou- plings areg V =5.9,β=0.9,λ=0.56 GeV −1 [61]. Meanwhile,the SU(3) singlet terms of the 8⊗8⊗8 in- teracting vertices embedded into the effecti...

  2. [2]

    The potential in the momentum space in eq

    With the potentials in eq (18), the potentials forB 0 s N−B (∗)Λ−B (∗)Σsystem can be explicitly shown in Table II. The potential in the momentum space in eq. (18) is trans- formed into the coordinate space potential by Fourier trans- form [6, 8, 46]: Vex i (r,Λ, µ ex)= Z d3q (2π)3 Vex i (q,Q)F 2(q,Λ, µ ex)eiq·r,(19) whereF 2(q,Λ, µ ex) is the form factor,...

  3. [3]

    Resonance

    and has the asymptotic form uk j(r) r→∞ − →δ jkh− l j(q jr)−S jk(E)h+ l j(q jr),(28) 7 0 0.5 1.0 1.5 40 20 0 20 40 V(r) [MeV] J P = 1/2 − (B0 s N) 0.0 0.5 1.0 1.5 J P = 1/2 − ( ¯B ∗ Λ) 0.0 0.5 1.0 1.5 J P = 1/2 − ( ¯B ∗ Σ) 0.0 0.5 1.0 1.5 2.0 r [fm] J P = 3/2 − ( ¯B ∗ Σ) 0.0 0.5 1.0 1.5 r [fm] 40 20 0 20 40 V(r) [MeV] J P = 1/2 − ( ¯BΛ) 0.0 0.5 1.0 1.5 r ...

  4. [4]

    Gell-Mann, Phys

    M. Gell-Mann, Phys. Lett.8, 214 (1964)

    1964

  5. [5]

    Zweig, An SU(3) model for strong interaction symmetry and its breaking

    G. Zweig, An SU(3) model for strong interaction symmetry and its breaking. Version 2, inDEVELOPMENTS IN THE QUARK THEORY OF HADRONS. VOL. 1. 1964 - 1978, edited by D. B. Lichtenberg and S. P. Rosen (1964) pp. 22–101

    1964

  6. [6]

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

    Pith/arXiv arXiv 2003

  7. [7]

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

    Pith/arXiv arXiv 2017

  8. [8]

    Esposito, A

    A. Esposito, A. Pilloni, and A. D. Polosa, Phys. Rept.668, 1 (2017), arXiv:1611.07920 [hep-ph]

    Pith/arXiv arXiv 2017

  9. [9]

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

    Pith/arXiv arXiv 2016

  10. [10]

    F.-K. Guo, C. Hanhart, U.-G. Meißner, Q. Wang, Q. Zhao, and B.-S. Zou, Rev. Mod. Phys.90, 015004 (2018), [Erratum: Rev.Mod.Phys. 94, 029901 (2022)], arXiv:1705.00141 [hep- ph]

    Pith/arXiv arXiv 2018

  11. [11]

    Liu, H.-X

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

    Pith/arXiv arXiv 2019

  12. [12]

    H.-X. Chen, W. Chen, X. Liu, Y .-R. Liu, and S.-L. Zhu, Rept. Prog. Phys.86, 026201 (2023), arXiv:2204.02649 [hep-ph]

    Pith/arXiv arXiv 2023

  13. [13]

    Wang, Front

    Z.-G. Wang, Front. Phys. (Beijing)21, 016300 (2026), arXiv:2502.11351 [hep-ph]

    Pith/arXiv arXiv 2026

  14. [14]

    Aaijet al.(LHCb), Phys

    R. Aaijet al.(LHCb), Phys. Rev. Lett.115, 072001 (2015), arXiv:1507.03414 [hep-ex]

    Pith/arXiv arXiv 2015

  15. [15]

    Aaijet al.(LHCb), Phys

    R. Aaijet al.(LHCb), Phys. Rev. Lett.122, 222001 (2019), arXiv:1904.03947 [hep-ex]

    Pith/arXiv arXiv 2019

  16. [16]

    Aaijet al.(LHCb), Sci

    R. Aaijet al.(LHCb), Sci. Bull.66, 1278 (2021), arXiv:2012.10380 [hep-ex]

    arXiv 2021

  17. [17]

    Aaijet al.(LHCb), Phys

    R. Aaijet al.(LHCb), Phys. Rev. Lett.131, 031901 (2023), arXiv:2210.10346 [hep-ex]

    Pith/arXiv arXiv 2023

  18. [18]

    Dong, F.-K

    X.-K. Dong, F.-K. Guo, and B.-S. Zou, Phys. Rev. Lett.126, 152001 (2021), arXiv:2011.14517 [hep-ph]

    Pith/arXiv arXiv 2021

  19. [19]

    Mutuk, Chin

    H. Mutuk, Chin. J. Phys.97, 1406 (2025), arXiv:2411.16486 [hep-ph]

    Pith/arXiv arXiv 2025

  20. [20]

    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 (2020), arXiv:1907.07583 [hep-ex]

    Pith/arXiv arXiv 2020

  21. [21]

    Yamaguchi, A

    Y . Yamaguchi, A. Hosaka, S. Takeuchi, and M. Takizawa, J. Phys. G47, 053001 (2020), arXiv:1908.08790 [hep-ph]

    Pith/arXiv arXiv 2020

  22. [22]

    Dong, F.-K

    X.-K. Dong, F.-K. Guo, and B.-S. Zou, Progr. Phys.41, 65 (2021), arXiv:2101.01021 [hep-ph]

    Pith/arXiv arXiv 2021

  23. [23]

    Dong, F.-K

    X.-K. Dong, F.-K. Guo, and B.-S. Zou, Commun. Theor. Phys. 73, 125201 (2021), arXiv:2108.02673 [hep-ph]

    Pith/arXiv arXiv 2021

  24. [24]

    Mai, U.-G

    M. Mai, U.-G. Meißner, and C. Urbach, Phys. Rept.1001, 1 (2023), arXiv:2206.01477 [hep-ph]

    Pith/arXiv arXiv 2023

  25. [25]

    Zou, Sci

    B.-S. Zou, Sci. Bull.66, 1258 (2021), arXiv:2103.15273 [hep- ph] . 14

    Pith/arXiv arXiv 2021

  26. [26]

    Q. X. Yu, R. Pavao, V . R. Debastiani, and E. Oset, Eur. Phys. J. C79, 167 (2019), arXiv:1811.11738 [hep-ph]

    Pith/arXiv arXiv 2019

  27. [27]

    Huang, C.-j

    Y . Huang, C.-j. Xiao, L.-S. Geng, and J. He, Phys. Rev. D99, 014008 (2019), arXiv:1811.10769 [hep-ph]

    Pith/arXiv arXiv 2019

  28. [28]

    Song, M.-Y

    J. Song, M.-Y . Duan, L. Roca, and E. Oset, Eur. Phys. J. C84, 1055 (2024), arXiv:2406.14895 [hep-ph]

    Pith/arXiv arXiv 2024

  29. [29]

    Shen and U.-G

    C.-W. Shen and U.-G. Meißner, Phys. Lett. B831, 137197 (2022), arXiv:2203.09804 [hep-ph]

    Pith/arXiv arXiv 2022

  30. [30]

    Wang and X

    F.-L. Wang and X. Liu, Phys. Rev. D113, 094037 (2026), arXiv:2603.23287 [hep-ph]

    Pith/arXiv arXiv 2026

  31. [31]

    Suntharawirat, N

    R. Suntharawirat, N. Ponkhuha, and D. Samart, (2026), arXiv:2606.09202 [hep-ph]

    Pith/arXiv arXiv 2026

  32. [32]

    W. H. Liang, C. W. Xiao, and E. Oset, Phys. Rev. D89, 054023 (2014), arXiv:1401.1441 [hep-ph]

    Pith/arXiv arXiv 2014

  33. [33]

    Z.-Y . Jian, H. Q. Zhu, F. Yang, Q.-H. Chen, Y . Huang, and J. He, Eur. Phys. J. A59, 262 (2023), arXiv:2204.01961 [hep-ph]

    Pith/arXiv arXiv 2023

  34. [34]

    Song, Y .-Y

    J. Song, Y .-Y . Li, and E. Oset, Eur. Phys. J. C85, 1101 (2025), arXiv:2507.00840 [hep-ph]

    Pith/arXiv arXiv 2025

  35. [35]

    He and D.-Y

    J. He and D.-Y . Chen, Eur. Phys. J. C79, 887 (2019), arXiv:1909.05681 [hep-ph]

    Pith/arXiv arXiv 2019

  36. [36]

    Chen, Z.-F

    R. Chen, Z.-F. Sun, X. Liu, and S.-L. Zhu, Phys. Rev. D100, 011502 (2019), arXiv:1903.11013 [hep-ph]

    Pith/arXiv arXiv 2019

  37. [37]

    Liu, T.-W

    M.-Z. Liu, T.-W. Wu, M. S ´anchez S´anchez, M. P. Valderrama, L.-S. Geng, and J.-J. Xie, Phys. Rev. D103, 054004 (2021), arXiv:1907.06093 [hep-ph]

    Pith/arXiv arXiv 2021

  38. [38]

    M.-L. Du, V . Baru, F.-K. Guo, C. Hanhart, U.-G. Meißner, J. A. Oller, and Q. Wang, JHEP08, 157, arXiv:2102.07159 [hep-ph]

    Pith/arXiv arXiv

  39. [39]

    Yalikun, Y .-H

    N. Yalikun, Y .-H. Lin, F.-K. Guo, Y . Kamiya, and B.-S. Zou, Phys. Rev. D104, 094039 (2021), arXiv:2109.03504 [hep-ph]

    Pith/arXiv arXiv 2021

  40. [40]

    Yalikun, X.-K

    N. Yalikun, X.-K. Dong, and B.-S. Zou, Chin. Phys. C47, 123101 (2023), arXiv:2303.03629 [hep-ph]

    Pith/arXiv arXiv 2023

  41. [41]

    Z. Wang, N. Yalikun, and Y . Reyimuaji, Phys. Rev. C112, 045205 (2025), arXiv:2505.02601 [hep-ph]

    arXiv 2025

  42. [42]

    Yan, H.-Y

    T.-M. Yan, H.-Y . Cheng, C.-Y . Cheung, G.-L. Lin, Y . C. Lin, and H.-L. Yu, Phys. Rev. D46, 1148 (1992), [Erratum: Phys.Rev.D 55, 5851 (1997)]

    1992

  43. [43]

    M. B. Wise, Phys. Rev. D45, R2188 (1992)

    1992

  44. [44]

    P. L. Cho, Phys. Rev. D50, 3295 (1994), arXiv:hep-ph/9401276

    Pith/arXiv arXiv 1994

  45. [45]

    Casalbuoni, A

    R. Casalbuoni, A. Deandrea, N. Di Bartolomeo, R. Gatto, F. Feruglio, and G. Nardulli, Phys. Rept.281, 145 (1997), arXiv:hep-ph/9605342

    Pith/arXiv arXiv 1997

  46. [46]

    Doring, C

    M. Doring, C. Hanhart, F. Huang, S. Krewald, U. G. Meissner, and D. Ronchen, Nucl. Phys. A851, 58 (2011), arXiv:1009.3781 [nucl-th]

    Pith/arXiv arXiv 2011

  47. [47]

    J. J. de Swart, Rev. Mod. Phys.35, 916 (1963), [Erratum: Rev.Mod.Phys. 37, 326–326 (1965)]

    1963

  48. [48]

    Liu, Z.-G

    X. Liu, Z.-G. Luo, Y .-R. Liu, and S.-L. Zhu, Eur. Phys. J. C61, 411 (2009), arXiv:0808.0073 [hep-ph]

    Pith/arXiv arXiv 2009

  49. [49]

    Wang and X

    F.-L. Wang and X. Liu, Phys. Rev. D102, 094006 (2020), arXiv:2008.13484 [hep-ph]

    Pith/arXiv arXiv 2020

  50. [50]

    Chen, Eur

    R. Chen, Eur. Phys. J. C81, 122 (2021), arXiv:2101.10614 [hep-ph]

    Pith/arXiv arXiv 2021

  51. [51]

    Wang and X

    F.-L. Wang and X. Liu, Phys. Lett. B835, 137583 (2022), arXiv:2207.10493 [hep-ph]

    Pith/arXiv arXiv 2022

  52. [52]

    Wang, Z.-Y

    J.-Z. Wang, Z.-Y . Lin, B. Wang, L. Meng, and S.-L. Zhu, Phys. Rev. D110, 114003 (2024), arXiv:2408.08965 [hep-ph]

    Pith/arXiv arXiv 2024

  53. [53]

    C. E. Thomas and F. E. Close, Phys. Rev. D78, 034007 (2008), arXiv:0805.3653 [hep-ph]

    Pith/arXiv arXiv 2008

  54. [54]

    Ling, M.-Z

    X.-Z. Ling, M.-Z. Liu, and L.-S. Geng, Eur. Phys. J. C81, 1090 (2021), arXiv:2110.13792 [hep-ph]

    Pith/arXiv arXiv 2021

  55. [55]

    R. Xu, L. Meng, H.-X. Zhu, N. Li, and W. Chen, Phys. Rev. D 111, 094015 (2025), arXiv:2503.04264 [hep-ph]

    Pith/arXiv arXiv 2025

  56. [56]

    Cheng, C.-Y

    H.-Y . Cheng, C.-Y . Cheung, G.-L. Lin, Y . C. Lin, T.-M. Yan, and H.-L. Yu, Phys. Rev. D47, 1030 (1993), arXiv:hep- ph/9209262

    arXiv 1993

  57. [57]

    Pirjol and T.-M

    D. Pirjol and T.-M. Yan, Phys. Rev. D56, 5483 (1997), arXiv:hep-ph/9701291

    Pith/arXiv arXiv 1997

  58. [58]

    Liu and M

    Y .-R. Liu and M. Oka, Phys. Rev. D85, 014015 (2012), arXiv:1103.4624 [hep-ph]

    Pith/arXiv arXiv 2012

  59. [59]

    W. A. Bardeen, E. J. Eichten, and C. T. Hill, Phys. Rev. D68, 054024 (2003), arXiv:hep-ph/0305049

    Pith/arXiv arXiv 2003

  60. [60]

    Yalikun and B.-S

    N. Yalikun and B.-S. Zou, Phys. Rev. D105, 094026 (2022), arXiv:2112.06426 [hep-ph]

    Pith/arXiv arXiv 2022

  61. [61]

    Harrison and C

    J. Harrison and C. T. H. Davies (HPQCD, (HPQCD Collabo- ration)‡), Phys. Rev. D109, 094515 (2024), arXiv:2304.03137 [hep-lat]

    Pith/arXiv arXiv 2024

  62. [62]

    Aaijet al.(LHCb), JHEP03, 047, arXiv:2411.10219 [hep- ex]

    R. Aaijet al.(LHCb), JHEP03, 047, arXiv:2411.10219 [hep- ex]

    arXiv

  63. [63]

    Machleidt, K

    R. Machleidt, K. Holinde, and C. Elster, Phys. Rept.149, 1 (1987)

    1987

  64. [64]

    Bando, T

    M. Bando, T. Kugo, and K. Yamawaki, Phys. Rept.164, 217 (1988)

    1988

  65. [65]

    Pich, Rept

    A. Pich, Rept. Prog. Phys.58, 563 (1995), arXiv:hep- ph/9502366

    arXiv 1995

  66. [66]

    Bernard, N

    V . Bernard, N. Kaiser, and U.-G. Meissner, Int. J. Mod. Phys. E 4, 193 (1995), arXiv:hep-ph/9501384

    Pith/arXiv arXiv 1995

  67. [67]

    Takahashiet al.(Particle Data Group), Int

    F. Takahashiet al.(Particle Data Group), Int. J. Mod. Phys. A 41, 2630011 (2026)

    2026

  68. [68]

    Ronchen, M

    D. Ronchen, M. Doring, F. Huang, H. Haberzettl, J. Haiden- bauer, C. Hanhart, S. Krewald, U. G. Meissner, and K. Nakayama, Eur. Phys. J. A49, 44 (2013), arXiv:1211.6998 [nucl-th]

    Pith/arXiv arXiv 2013

  69. [69]

    Doi, H.-c

    T. Doi, H.-c. Kim, Y . Kondo, and M. Oka, Nucl. Phys. A721, 755 (2003), arXiv:hep-ph/0212043

    Pith/arXiv arXiv 2003

  70. [70]

    Adamu ˇsˇcin, E

    C. Adamu ˇsˇcin, E. Bartoˇs, S. Dubniˇcka, and A. Z. Dubni ˇckov´a, Phys. Rev. C93, 055208 (2016), arXiv:1601.06190 [hep-ph]

    Pith/arXiv arXiv 2016

  71. [71]

    Breit, Phys

    G. Breit, Phys. Rev.34, 553 (1929)

    1929

  72. [72]

    Breit, Phys

    G. Breit, Phys. Rev.36, 383 (1930)

    1930

  73. [73]

    Lu, L.-S

    J.-X. Lu, L.-S. Geng, and M. P. Valderrama, Phys. Rev. D99, 074026 (2019), arXiv:1706.02588 [hep-ph]

    Pith/arXiv arXiv 2019

  74. [74]

    N. A. Tornqvist, Z. Phys. C61, 525 (1994), arXiv:hep- ph/9310247

    arXiv 1994

  75. [75]

    R. Chen, A. Hosaka, and X. Liu, Phys. Rev. D96, 116012 (2017), arXiv:1707.08306 [hep-ph] . 15

    Pith/arXiv arXiv 2017

  76. [76]

    J. R. Taylor,Scattering Theory: The Quantum Theory of Non- relativistic Collisions(John Wiley & Sons, Inc., New York, 1972)

    1972

  77. [77]

    Oset and A

    E. Oset and A. Ramos, Eur. Phys. J. A44, 445 (2010), arXiv:0905.0973 [hep-ph]

    Pith/arXiv arXiv 2010

  78. [78]

    Sakai, H.-J

    S. Sakai, H.-J. Jing, and F.-K. Guo, Phys. Rev. D100, 074007 (2019), arXiv:1907.03414 [hep-ph]

    Pith/arXiv arXiv 2019

  79. [79]

    E. J. Garzon and E. Oset, Eur. Phys. J. A48, 5 (2012), arXiv:1201.3756 [hep-ph]

    Pith/arXiv arXiv 2012