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We analyze possible singularities in the $J/\psi \Lambda$ invariant massdistribution of the $\Xi^-_{b}~\to~K^- J/\psi \Lambda$ process via triangleloop diagrams. Triangle singularities in the physical region are found in 18different triangle loop diagrams. Among those with $\Xi^*$-charmonium-$\Lambda$intermediate states, the one from the $\chi_{c1} \Xi(2120) \Lambda$ loop, whichis located around 4628 MeV, is found the most likely to cause observableeffects. One needs $S$- and $P$-waves in $\chi_{c1} \Lambda$ and $J/\psi\Lambda$ systems, respectively, when the quantum numbers of these systems are$1/2^+$ or $3/2^+$. When the quantum numbers of the $\Xi(2120)$ are$J^P=1/2^+$, $1/2^-$ or $3/2^+$, the peak structure should be sharper than theother $J^P$ choices. This suggests that although the whole strength is unknown,we should pay attention to the contributions from the$\Xi^*$-charmonium-$\Lambda$ triangle diagram if structures are observed in the$J/\psi \Lambda$ invariant mass spectrum experimentally. In addition, a fewtriangle diagrams with the $D_{s1}^*(2700)$ as one of the intermediateparticles can also produce singularities in the $J/\psi\Lambda$ distribution,but at higher energies above 4.9 GeV.

Exploring possible triangle singularities in the $\Xi^-_{b} \to K^- J/\psi \Lambda$ decay