# Exploring Possible Triangle Singularities in the Ξ b − → K − J / ψ Λ Decay

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Framework and Analysis Results of All Possible Triangle Singularities

## 3. Detailed Analysis of the Amplitudes for the Diagram with ${\mathbf{\chi}}_{\mathbf{c}\mathbf{1}}\Xi \left(\mathbf{2120}\right)\Lambda $ Loop

## 4. Summary

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Four possible kinds of triangle diagrams for the ${\Xi}_{b}^{-}\to {K}^{-}\Lambda J/\psi $ decay that may have a triangle singularity of interest. The momentum of each particle and the numbering of the exchanged particles used here are given in the brackets in the first diagram.

**Figure 2.**The value of ${\left|I\right|}^{2}$ in the ${m}_{J/\psi \Lambda}$ invariant mass distribution. The red solid (orange dashed), green dotted and blue dotdashed lines correspond to the Nos. 1, 4 and 11 cases in Table 1, respectively. For the case of No.1, the red solid and orange dashed lines represent the different choices of the $\Xi \left(2500\right)$ width: the narrower and wider peaks correspond to taking 59 and 150 MeV for the width, respectively.

**Figure 3.**The ${m}_{J/\psi \Lambda}$ invariant mass distribution for the ${\Xi}_{b}^{-}\to {K}^{-}J/\psi \Lambda $ process via the ${\chi}_{c1}\Xi \left(2120\right)\Lambda $ loop with the quantum numbers of the $J/\psi \Lambda $ system being (

**a**) $1/{2}^{+}$, and (

**b**) $1/{2}^{-}$, respectively.

**Table 1.**The charmonium and ${\Xi}^{*-}$ states in Figure 1a that could generate triangle singularities and the positions of the corresponding singularities. Among the listed hyperons, $\Xi \left(2030\right)$ is a three-star state; $\Xi \left(2250\right)$ and $\Xi \left(2370\right)$ are two-star states; $\Xi \left(2120\right)$ and $\Xi \left(2500\right)$ are one-star states [69].

No. | $\mathit{c}\overline{\mathit{c}}$ | ${\mathbf{\Xi}}^{*-}$ | Position of Triangle Singularity (MeV) |
---|---|---|---|

1 | $J/\psi $ | $\Xi \left(2500\right)$ | 4232 |

2 | ${\chi}_{c0}$ | $\Xi \left(2250\right)$ | 4546 |

3 | ${\chi}_{c0}$ | $\Xi \left(2370\right)$ | 4665 |

4 | ${\chi}_{c1}$ | $\Xi \left(2120\right)$ | 4628 |

5 | ${\chi}_{c1}$ | $\Xi \left(2250\right)$ | 4696 |

6 | ${\chi}_{c2}$ | $\Xi \left(2120\right)$ | 4680 |

7 | ${h}_{c}$ | $\Xi \left(2120\right)$ | 4644 |

8 | ${h}_{c}$ | $\Xi \left(2250\right)$ | 4730 |

9 | ${\eta}_{c}\left(2S\right)$ | $\Xi \left(2030\right)$ | 4754 |

10 | ${\eta}_{c}\left(2S\right)$ | $\Xi \left(2120\right)$ | 4797 |

11 | $\psi \left(2S\right)$ | $\Xi \left(2030\right)$ | 4810 |

**Table 2.**The ${D}_{s}^{*-}$, ${\Xi}_{c}^{*0}$ and intermediate particle 3 in Figure 1b,c that could generate triangle singularities and the positions of the corresponding singularities.

No. | ${\mathit{D}}_{\mathit{s}}^{*-}$ | ${\mathbf{\Xi}}_{\mathit{c}}^{*0}$ | Intermediate Particle 3 | Position of Triangle Singularity (MeV) |
---|---|---|---|---|

1 | ${\overline{D}}_{s1/s3}^{*}\left(2860\right)$ | ${\Xi}_{c}\left(2930\right)$ | ${\overline{D}}^{0}$ | 4838 |

2 | ${\overline{D}}_{s1}^{*}\left(2700\right)$ | ${\Xi}_{c}\left(3055\right)$ | ${\overline{D}}^{0}$ | 4922 |

3 | ${\overline{D}}_{s1}^{*}\left(2700\right)$ | ${\Xi}_{c}\left(3080\right)$ | ${\overline{D}}^{0}$ | 4957 |

4 | ${\overline{D}}_{s1/s3}^{*}\left(2860\right)$ | ${\Xi}_{c}\left(2930\right)$ | ${\overline{D}}^{*0}$ | 4959 |

5 | ${\overline{D}}_{s1}^{*}\left(2700\right)$ | ${\Xi}_{c}\left(3080\right)$ | ${\overline{D}}^{*0}$ | 5089 |

6 | ${\overline{D}}_{s1}^{*}\left(2700\right)$ | ${\Xi}_{c}\left(3080\right)$ | ${\Lambda}_{c}^{+}$ | 4990 |

7 | ${\overline{D}}_{s1/s3}^{*}\left(2860\right)$ | ${\Xi}_{c}\left(2930\right)$ | ${\Lambda}_{c}^{+}$ | 5149 |

$\mathit{c}\overline{\mathit{c}}$ | $\mathit{BR}({\mathit{B}}^{0}\to \mathit{c}\overline{\mathit{c}}{\mathit{K}}^{0})$ | $\mathit{BR}({\mathit{B}}^{+}\to \mathit{c}\overline{\mathit{c}}{\mathit{K}}^{+})$ |
---|---|---|

$J/\psi $ | $8.73\times {10}^{-4}$ | $1.01\times {10}^{-3}$ |

${\chi}_{c0}$ | $1.11\times {10}^{-6}$ | $1.49\times {10}^{-4}$ |

${\chi}_{c1}$ | $3.93\times {10}^{-4}$ | $4.84\times {10}^{-4}$ |

${\chi}_{c2}$ | $<1.5\times {10}^{-5}$ | $1.1\times {10}^{-5}$ |

${h}_{c}$ | $<3.8\times {10}^{-5}$ | |

${\eta}_{c}\left(2S\right)$ | $4.4\times {10}^{-4}$ | |

$\psi \left(2S\right)$ | $5.8\times {10}^{-4}$ | $6.21\times {10}^{-4}$ |

**Table 4.**The interactions of the ${\Xi}_{b}\Xi \left(2120\right){\chi}_{c1}$ and $\Xi \left(2120\right)\Lambda \overline{K}$ vertices with different spin-parities of $\Xi \left(2120\right)$.

${\mathit{J}}^{\mathit{P}}$ | ${\mathbf{\Xi}}_{\mathit{b}}\mathbf{\Xi}\left(2120\right){\mathit{\chi}}_{\mathit{c}1}$ | $\mathbf{\Xi}\left(2120\right)\mathbf{\Lambda}\overline{\mathit{K}}$ |
---|---|---|

$1/{2}^{+}$ | S-wave | P-wave |

$1/{2}^{-}$ | S-wave | S-wave |

$3/{2}^{+}$ | S-wave | P-wave |

$3/{2}^{-}$ | S-wave | D-wave |

$5/{2}^{+}$ | P-wave | F-wave |

$5/{2}^{-}$ | P-wave | D-wave |

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Shen, C.-W.; Jing, H.-J.; Guo, F.-K.; Wu, J.-J.
Exploring Possible Triangle Singularities in the Ξ b − → K − J / ψ Λ Decay. *Symmetry* **2020**, *12*, 1611.
https://doi.org/10.3390/sym12101611

**AMA Style**

Shen C-W, Jing H-J, Guo F-K, Wu J-J.
Exploring Possible Triangle Singularities in the Ξ b − → K − J / ψ Λ Decay. *Symmetry*. 2020; 12(10):1611.
https://doi.org/10.3390/sym12101611

**Chicago/Turabian Style**

Shen, Chao-Wei, Hao-Jie Jing, Feng-Kun Guo, and Jia-Jun Wu.
2020. "Exploring Possible Triangle Singularities in the Ξ b − → K − J / ψ Λ Decay" *Symmetry* 12, no. 10: 1611.
https://doi.org/10.3390/sym12101611