Open Access

Final-state rescattering mechanism of the $Δ (1232)^{+ +}$ production in $Λ_{c}^{+} \to K^{-} π^{+} p$ decay

Zu-Xin Cai^1,2, Cheng Chen^2,3, Si-Wei Liu^2,3, Xiang Wei^2,3, Gang Li ^1,*, Xiao-Hai Liu ^4,†, and Ju-Jun Xie ^2,3,5,‡

¹College of Physics and Engineering, Qufu Normal University, Qufu 273165, China
²State Key Laboratory of Heavy Ion Science and Technology, Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China
³School of Nuclear Science and Technology, University of Chinese Academy of Sciences, Beijing 101408, China
⁴Center for Joint Quantum Studies and Department of Physics, School of Science, Tianjin University, Tianjin 300350, China
⁵Southern Center for Nuclear-Science Theory (SCNT), Institute of Modern Physics, Chinese Academy of Sciences, Huizhou 516000, China

^*Contact author: gli@qfnu.edu.cn
^†Contact author: xiaohai.liu@tju.edu.cn
^‡Contact author: xiejujun@impcas.ac.cn

Phys. Rev. D 114, 034051 – Published 25 August, 2026

DOI: https://doi.org/10.1103/1j49-ks9s

Abstract

We investigate the production of the $Δ (1232)^{+ +}$ resonance in the charmed baryon weak decay $Λ_{c}^{+} \to K^{-} π^{+} p$ , focusing on the $π^{+} p$ final-state rescattering mechanism. The direct $W^{+}$ exchange diagram is expected to be suppressed, hence we adopt the $W^{+}$ internal emission process $Λ_{c}^{+} \to p {\bar{K}}^{* 0} (892)$ followed by the subsequent decay ${\bar{K}}^{* 0} \to K^{-} π^{+}$ as the dominant source of the final state particles. The $Δ (1232)^{+ +}$ resonance is then generated via $π^{+} p$ rescattering within a triangle loop mechanism. Our calculations incorporate both the tree-level ${\bar{K}}^{* 0} (892)$ and the dynamically generated ${\bar{K}}_{0}^{*} (700)$ state arising from the $S$ -wave $K π$ final state interaction. We find that our theoretical results can reproduce the bump and peak structures in the $K^{-} π^{+}$ invariant mass distributions for the ${\bar{K}}_{0}^{*} (700)$ and ${\bar{K}}^{* 0} (892)$ , respectively. Meanwhile, the peak for the $Δ (1232)^{+ +}$ in the $π^{+} p$ invariant mass distributions is also well described. The $Δ (1232)^{+ +}$ signal naturally emerges from rescattering effects, and adopting the pole parameters of $Δ (1232)$ resonance yields an improved description of the experimental data. In addition, we obtain a branching fraction ratio $B [Λ_{c}^{+} \to Δ (1232)^{+ +} K^{-}] / B [Λ_{c}^{+} \to p {\bar{K}}^{* 0} (892)] \approx 0.5$ , which is lower than the experimentally measured value. This discrepancy suggests that interference effects are likely significant in this decay process. Future high-precision measurements will further verify the proposed rescattering mechanism.

Physics Subject Headings (PhySH)

Article Text

References (68)

H.-Y. Cheng, Charmed baryons circa 2015, Front. Phys. (Beijing) 10, 101406 (2015).
E. Oset et al., Weak decays of heavy hadrons into dynamically generated resonances, Int. J. Mod. Phys. E 25, 1630001 (2016).
H.-Y. Cheng, X.-W. Kang, and F. Xu, Singly Cabibbo-suppressed hadronic decays of $Λ_{c}^{+}$ , Phys. Rev. D 97, 074028 (2018).
K. Miyahara, T. Hyodo, and E. Oset, Weak decay of $Λ_{c}^{+}$ for the study of $Λ (1405)$ and $Λ (1670)$ , Phys. Rev. C 92, 055204 (2015).
J.-J. Xie and L.-S. Geng, The $a_{0} (980)$ and $Λ (1670)$ in the $Λ_{c}^{+} \to π^{+} η Λ$ decay, Eur. Phys. J. C 76, 496 (2016).
G.-Y. Wang, N.-C. Wei, H.-M. Yang, E. Wang, L.-S. Geng, and J.-J. Xie, Roles of $a_{0} (980)$ , $Λ (1670)$ , and $Σ (1385)$ in the $Λ_{c}^{+} \to η Λ π^{+}$ decay, Phys. Rev. D 106, 056001 (2022).
S.-C. Zhang, M.-Y. Duan, W.-T. Lyu, G.-Y. Wang, J.-Y. Zhu, and E. Wang, Explore the properties of $Λ (1670)$ in the Cabibbo-favored process $Λ_{c}^{+} \to p K^{-} π^{+}$ , Eur. Phys. J. C 84, 1253 (2024).
M.-Y. Duan, M. Bayar, and E. Oset, Precise determination of the $η Λ$ scattering length and effective range and relationship to the $Λ (1670)$ resonance, Phys. Lett. B 857, 139003 (2024).
J.-J. Xie and L.-S. Geng, $Σ_{1 / 2^{-}}^{*} (1380)$ in the $Λ_{c}^{+} \to η π^{+} Λ$ decay, Phys. Rev. D 95, 074024 (2017).
J.-J. Xie and L.-S. Geng, $Λ_{c}^{+} \to η π^{+} Λ$ decay and the possible $Σ_{1 / 2^{-}}^{*} (1380)$ , J. Phys. Soc. Jpn. Conf. Proc. 26, 031014 (2019).
W.-T. Lyu, S.-C. Zhang, G.-Y. Wang, J.-J. Wu, E. Wang, L.-S. Geng, and J.-J. Xie, Evidence of the low-lying baryon $Σ^{*} (1 / 2^{-})$ in the process $Λ_{c}^{+} \to η π^{+} Λ$ , Phys. Rev. D 110, 054020 (2024).
J.-J. Xie and L.-S. Geng, Role of the $N^{*} (1535)$ in the $Λ_{c}^{+} \to {\bar{K}}^{0} η p$ decay, Phys. Rev. D 96, 054009 (2017).
J.-J. Xie, L.-S. Geng, and F.-K. Guo, Role of $N^{*} (1535)$ in the $Λ_{c}^{+} \to {\bar{K}}^{0} η p$ decay and the possible $ϕ p$ state in the $Λ_{c}^{+} \to π^{0} ϕ p$ decay, Eur. Phys. J. Web Conf. 241, 02010 (2020).
Y. Li, S.-W. Liu, E. Wang, D.-M. Li, L.-S. Geng, and J.-J. Xie, Theoretical study of $N (1535)$ and $Σ^{*} (1 / 2^{-})$ in the Cabibbo-favored process $Λ_{c}^{+} \to p {\bar{K}}^{0} η$ , Phys. Rev. D 110, 074010 (2024).
Y. Li, E. Wang, L.-S. Geng, and J.-J. Xie, Dynamically generated $N (1535)$ state in the $Λ_{c}^{+} \to p {\bar{K}}^{0} π^{0}$ decay, Phys. Rev. D 113, 054039 (2026).
Z. Wang, Y.-Y. Wang, E. Wang, D.-M. Li, and J.-J. Xie, The scalar $f_{0} (500)$ and $f_{0} (980)$ resonances and vector mesons in the single Cabibbo-suppressed decays $Λ_{c} \to p K^{+} K^{-}$ and $p π^{+} π^{-}$ , Eur. Phys. J. C 80, 842 (2020).
X.-C. Feng, L.-L. Wei, M.-Y. Duan, E. Wang, and D.-M. Li, The $a_{0} (980)$ in the single Cabibbo-suppressed process $Λ_{c} \to π^{0} η p$ , Phys. Lett. B 846, 138185 (2023).
S.-W. Liu, Q.-H. Shen, and J.-J. Xie, Theoretical study on $Λ_{c}^{+} \to Λ K^{+} {\bar{K}}^{0}$ decay and $Ξ^{*} (1690)$ resonance, Phys. Rev. D 108, 114006 (2023).
A. Zupanc et al. (Belle Collaboration), Measurement of the branching fraction $B (Λ_{c}^{+} \to p K^{-} π^{+})$ , Phys. Rev. Lett. 113, 042002 (2014).
M. Ablikim et al. (BESIII Collaboration), Measurements of absolute hadronic branching fractions of $Λ_{c}^{+}$ baryon, Phys. Rev. Lett. 116, 052001 (2016).
R. Aaij et al. (LHCb Collaboration), Measurements of the branching fractions of $Λ_{c}^{+} \to p π^{-} π^{+}$ , $Λ_{c}^{+} \to p K^{-} K^{+}$ , and $Λ_{c}^{+} \to p π^{-} K^{+}$ , J. High Energy Phys. 03 (2018) 043.
M. Berger et al. (Belle Collaboration), Measurement of the decays $Λ_{c} \to Σ π π$ at Belle, Phys. Rev. D 98, 112006 (2018).
J. Y. Lee et al. (Belle Collaboration), Measurement of branching fractions of $Λ_{c}^{+} \to η Λ π^{+}$ , $η Σ^{0} π^{+}$ , $Λ (1670) π^{+}$ , and $η Σ (1385)^{+}$ , Phys. Rev. D 103, 052005 (2021).
M. Ablikim et al. (BESIII Collaboration), Measurement of the absolute branching fraction of $Λ_{c}^{+} \to p K_{S}^{0} η$ decays, Phys. Lett. B 817, 136327 (2021).
M. Ablikim et al. (BESIII Collaboration), Partial wave analysis of the charmed baryon hadronic decay $Λ_{c}^{+} \to Λ π^{+} π^{0}$ , J. High Energy Phys. 12 (2022) 033.
M. Ablikim et al. (BESIII Collaboration), Measurement of the branching fractions of the decays $Λ_{c}^{+} \to Λ K_{S}^{0} K^{+}$ , $Λ_{c}^{+} \to Λ K_{S}^{0} π^{+}$ , and $Λ_{c}^{+} \to Λ K^{* +}$ , Phys. Rev. D 111, 012014 (2025).
M. Ablikim et al. (BESIII Collaboration), Measurement of the branching fractions of the Cabibbo-favored decays $Λ_{c}^{+} \to Λ K_{S}^{0} K^{+}$ and $Λ_{c}^{+} \to Ξ^{0} K_{S}^{0} π^{+}$ and search for $Λ_{c}^{+} \to Σ^{0} K_{S}^{0} K^{+}$ , Phys. Rev. D 112, 032006 (2025).
M. Ablikim et al. (BESIII Collaboration), Observation of $Λ_{c}^{+} \to n π^{+} η$ and search for $Λ_{c}^{+} \to n a_{0} (980)^{+}$ , arXiv:2603.28232.
M. Ablikim et al. (BESIII Collaboration), Measurements of the absolute branching fractions of the $Λ_{c}^{+}$ hadronic decays, arXiv:2601.01503.
M. Ablikim et al. (BESIII Collaboration), Measurement of the absolute branching fractions of $Λ_{c}^{+} \to Λ η π^{+}$ and $Σ (1385)^{+} η$ , Phys. Rev. D 99, 032010 (2019).
M. Ablikim et al. (BESIII Collaboration), Observation of $Λ_{c}^{+} \to Λ a_{0} (980)^{+}$ and evidence for $Σ (1380)^{+}$ in $Λ_{c}^{+} \to Λ π^{+} η$ , Phys. Rev. Lett. 134, 021901 (2025).
L. Li et al. (Belle Collaboration), Measurement of branching fractions of $Λ_{c}^{+} \to p K_{S}^{0} K_{S}^{0}$ and $Λ_{c}^{+} \to p K_{S}^{0} η$ at Belle, Phys. Rev. D 107, 032004 (2023).
E. M. Aitala et al. (E791 Collaboration), Multidimensional resonance analysis of $Λ_{c}^{+} \to p K^{-} π^{+}$ , Phys. Lett. B 471, 449 (2000).
X.-H. Liu, G. Li, J.-J. Xie, and Q. Zhao, Visible narrow cusp structure in $Λ_{c}^{+} \to p K^{-} π^{+}$ enhanced by triangle singularity, Phys. Rev. D 100, 054006 (2019).
J. K. Ahn, S. Yang, and S.-I. Nam, Hyperon production in $Λ_{c}^{+} \to K^{-} p π^{+}$ and $Λ_{c}^{+} \to K_{S}^{0} p π^{0}$ , Phys. Rev. D 100, 034027 (2019).
R. Aaij et al. (LHCb Collaboration), Amplitude analysis of the $Λ_{c}^{+} \to p K^{-} π^{+}$ decay and $Λ_{c}^{+}$ baryon polarization measurement in semileptonic beauty hadron decays, Phys. Rev. D 108, 012023 (2023).
D.-H. Wei, Y.-X. Yang, and R.-G. Ping, Using $Λ_{c}^{+} \to p K^{-} π^{+}$ as a spin polarimeter, Chin. Phys. C 46, 074002 (2022).
S. Navas et al. (Particle Data Group), Review of particle physics, Phys. Rev. D 110, 030001 (2024).
D. Marangotto, Extracting maximum information from polarised baryon decays via amplitude analysis: The $Λ_{c}^{+} \to p K^{-} π^{+}$ case, Adv. High Energy Phys. 2020, 7463073 (2020).
H. L. Anderson, E. Fermi, E. A. Long, and D. E. Nagle, Total cross-sections of positive pions in hydrogen, Phys. Rev. 85, 936 (1952).
F. Huang, M. Doring, H. Haberzettl, J. Haidenbauer, C. Hanhart, S. Krewald, U. G. Meissner, and K. Nakayama, Pion photoproduction in a dynamical coupled-channels model, Phys. Rev. C 85, 054003 (2012).
M.-Y. Dai, S.-W. Liu, C. Chen, D.-M. Li, E. Wang, and J.-J. Xie, Roles of $Δ (1232)$ , $N^{*} (1520)$ , and $N^{*} (1650)$ resonances in $γ p \to π^{0} π^{0} p$ reaction within an effective Lagrangian approach, Chin. Phys. C 49, 063102 (2025).
N. Muramatsu et al. (LEPS2/BGOegg Collaboration), First measurement of differential cross sections and photon beam asymmetries for photoproduction of the $f_{0} (980)$ meson decaying into $π^{0} π^{0}$ at $E_{γ} < 2.4 GeV$ , Phys. Rev. C 107, L042201 (2023).
F. Aceti, L. R. Dai, L. S. Geng, E. Oset, and Y. Zhang, Meson-baryon components in the states of the baryon decuplet, Eur. Phys. J. A 50, 57 (2014).
T. Sekihara, T. Arai, J. Yamagata-Sekihara, and S. Yasui, Compositeness of the $Δ (1232)$ resonance in $π N$ scattering, J. Phys. Soc. Jpn. Conf. Proc. 10, 022005 (2016).
T. Sekihara, T. Arai, J. Yamagata-Sekihara, and S. Yasui, Compositeness of baryonic resonances: Application to the $Δ (1232)$ , $N (1535)$ , and $N (1650)$ resonances, Phys. Rev. C 93, 035204 (2016).
H.-Y. Cheng and C.-W. Chiang, Hadronic $D$ decays involving even-parity light mesons, Phys. Rev. D 81, 074031 (2010).
K. Miyahara, T. Hyodo, M. Oka, J. Nieves, and E. Oset, Theoretical study of the $Ξ (1620)$ and $Ξ (1690)$ resonances in $Ξ_{c} \to π^{+} M B$ decays, Phys. Rev. C 95, 035212 (2017).
J.-J. Xie, L.-R. Dai, and E. Oset, The low lying scalar resonances in the $D^{0}$ decays into $K_{s}^{0}$ and $f_{0} (500)$ , $f_{0} (980)$ , $a_{0} (980)$ , Phys. Lett. B 742, 363 (2015).
J. R. Peláez and A. Rodas, Dispersive $π K \to π K$ and $π π \to K \bar{K}$ amplitudes from scattering data, threshold parameters, and the lightest strange resonance $κ$ or $K_{0}^{*} (700)$ , Phys. Rep. 969, 1 (2022).
M. Albaladejo, A. Canoa, J. Nieves, J. R. Peláez, E. Ruiz-Arriola, and J. R. de Elvira, The role of chiral symmetry and the non-ordinary $κ / K_{0}^{*} (700)$ nature in $π^{\pm} K_{S}$ femtoscopic correlations, Phys. Lett. B 866, 139552 (2025).
K. P. Khemchandani, A. Martínez Torres, A. Hosaka, H. Nagahiro, F. S. Navarra, and M. Nielsen, Why $Ξ (1690)$ and $Ξ (2120)$ are so narrow?, Phys. Rev. D 97, 034005 (2018).
G. Toledo, N. Ikeno, and E. Oset, Theoretical study of the $D^{0} \to K^{-} π^{+} η$ reaction, Eur. Phys. J. C 81, 268 (2021).
J. A. Oller and E. Oset, Chiral symmetry amplitudes in the $S$ -wave isoscalar and isovector channels and the $σ$ , $f_{0} (980)$ , $a_{0} (980)$ scalar mesons, Nucl. Phys. A620, 438 (1997); A652, 407(E) (1999).
J. A. Oller, E. Oset, and J. R. Pelaez, Meson meson interaction in a nonperturbative chiral approach, Phys. Rev. D 59, 074001 (1999); 60, 099906(E) (1999); 75, 099903(E) (2007).
J. A. Oller, E. Oset, and J. R. Pelaez, Nonperturbative approach to effective chiral Lagrangians and meson interactions, Phys. Rev. Lett. 80, 3452 (1998).
D. Gamermann, E. Oset, D. Strottman, and M. J. Vicente Vacas, Dynamically generated open and hidden charm meson systems, Phys. Rev. D 76, 074016 (2007).
D. Lee and S.-i. Nam, Strangeness plus-one ( $S = + 1$ ) resonance-state $P_{0}^{+ *}$ via $K^{+} n \to K^{* 0} p$ , Phys. Rev. C 109, 045207 (2024).
Y.-K. Chen, Y. Pan, and B.-C. Liu, $Δ (1232)$ production in the $γ p \to ϕ π p$ reaction, Phys. Rev. D 112, 074013 (2025).
O. Gortchakov, M. P. Locher, V. E. Markushin, and S. von Rotz, Two meson doorway calculation for $\bar{p} p \to ϕ π$ including off-shell effects and the OZI rule, Z. Phys. A 353, 447 (1996).
N. A. Tornqvist, From the deuteron to deusons, an analysis of deuteronlike meson meson bound states, Z. Phys. C 61, 525 (1994).
H.-Y. Cheng, C.-K. Chua, and A. Soni, Final state interactions in hadronic $B$ decays, Phys. Rev. D 71, 014030 (2005).
X.-Q. Li, D. V. Bugg, and B.-S. Zou, Possible explanation of the “ $ρ π$ puzzle” in $J / ψ$ , $ψ^{'}$ decays, Phys. Rev. D 55, 1421 (1997).
M. P. Locher, Y. Lu, and B. S. Zou, Rates for the reactions $\bar{p} p \to π ϕ$ and $γ ϕ$ , Z. Phys. A 347, 281 (1994).
V. Shklyar, H. Lenske, and U. Mosel, A coupled-channel analysis of $K Λ$ production in the nucleon resonance region, Phys. Rev. C 72, 015210 (2005).
T. Feuster and U. Mosel, A unitary model for meson nucleon scattering, Phys. Rev. C 58, 457 (1998).
R. Brockmann and R. Machleidt, Relativistic nuclear structure. I. Nuclear matter, Phys. Rev. C 42, 1965 (1990).
R. Machleidt, K. Holinde, and C. Elster, The Bonn meson-exchange model for the nucleon-nucleon interaction, Phys. Rep. 149, 1 (1987).

Phys. Rev. D 114, 034051– Published 25 August, 2026

Vol. 114, Iss. 3 — 1 August 2026

Received 7 June 2026
Accepted 4 August 2026

DOI: https://doi.org/10.1103/1j49-ks9s

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

Published by the American Physical Society

Physical Review D

Reuse & Permissions

Export citation

Export citation

Final-state rescattering mechanism of the $Δ (1232)^{+ +}$ production in $Λ_{c}^{+} \to K^{-} π^{+} p$ decay

Abstract

Physics Subject Headings (PhySH)

Article Text

References (68)

Outline

Information

Filter

Filter

Article Lookup

Enter a citation

Final-state rescattering mechanism of the Δ(1232)++ production in Λc+→K−π+p decay

Abstract

Physics Subject Headings (PhySH)

Article Text

References (68)

Final-state rescattering mechanism of the $Δ (1232)^{+ +}$ production in $Λ_{c}^{+} \to K^{-} π^{+} p$ decay