The spectrum of hadrons is the manifestation of color confinement of quantum chromodynamics. Hadronic resonances correspond to poles of the S-matrix. Since 2003, lots of new hadron resonant structures were discovered in the mass regions from light mesons to hadrons containing a pair of a heavy quark and an antiquark. Many of them are candidates of exotic hadrons, and they are usually observed as peaks in invariant mass distributions. However, the S-matrix also has kinematical singularities due to the on-shellness of intermediate particles for a process, such as two-body thresholds and triangle singularities (TSs), and they can produce peaks as well. On the one hand, such singularities may be misidentified as resonances; on the other hand, they can be used as tools for precision measurements. In this paper, we review the threshold cusps and various triangle singularities in hadronic reactions, paying attention to their manifestations in phenomena related to exotic hadron candidates. * fkguo@itp.ac.cn † xiaohai.liu@tju.edu.cn ‡ shsakai@itp.ac.cn 1 The X(3872) is named χ c1 (3872) according to its quantum numbers I G (J P C ) = 0 + (1 ++ ) by the PDG [10]. Similarly, the vector charmonium-like states Y (4260) and Y (4660) mentioned below are called ψ(4260) and ψ(4660), respectively. This naming scheme does not mean that the PDG assumes them to be normal cc charmonium states. Here we follow the XY Z naming scheme that is still used in most of the relevant publications.2 A conventional explanation of the observed peaks was given in Refs. [26,27].3 region Z c (4430) [30], Z c (3900) [31,32] and Z c (4020) [33], the charged bottomonium-like structures Z b (10610) and Z b (10650) [34], and the pentaquark candidates with hidden charm P c (4312), P c (4440) and P c (4457) [35,36]. Most of these new structures were observed in the heavy-flavor sector. In particular, the heavy quarkonium-like ones are often called XY Z states in the literature due to the undetermined internal structure. On the one hand, these discoveries enlarged the known QCD spectrum to a large extent; on the other hand, they became a nice showcase of the intricate nonperturbative nature of QCD at low energies 3 : most of them fall off the expectations from quark model, which despite being just a model had provided useful guidance in classifying a large amount of hadrons into various multiplets. Therefore, they are regarded as prominent candidates of exotic hadrons. However, how the spectrum of exotic hadrons should be organized and even what types of exotic hadrons can be well defined are still unclear. Partly because of this, the observation of each of these new structures leads to different models such as compact tetraquarks (or pentaquarks), hadronic molecules, hybrid states, hadro-charmonia, and kinematic effects, etc. Nevertheless, a deeper understanding of how the hadron spectrum, in particular that of the excited hadrons above (or at least close to) strong decay thresholds, is organized can shed light on the color confinement problem of QCD. For th...

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The LHCb Collaboration announced the observation of three narrow structures consistent with hidden-charm pentaquark states. They are candidates of hadronic molecules formed of a pair of a charmed baryon and an anticharmed meson. Among them, the P c (4457) mass is consistent with earlier predictions of a Σ cD * molecule with I = 1/2. We point out that if such a picture were true, one would have B(P c (4457) → J/ψ∆ + )/B(P c (4457) → J/ψp) at the level ranging from a few percent to about 30%. Such a large isospin breaking decay ratio is two to three orders of magnitude larger than that for normal hadron resonances. It is a unique feature of the Σ cD * molecular model, and can be checked by LHCb. * Four years after the discovery of the hidden-charm pentaquark-like states P c (4380) and P c (4450) [1], with the full set of Run-1 and Run-2 data the LHCb Collaboration announced the observation of more structures consistent with hidden-charm pentaquark states with masses and widths given by [2, 3] M Pc(4312) + = 4311.9 ± 0.7 +6.8 −0.6 MeV, Γ Pc(4312) + = 9.8 ± 2.7 +3.7 −4.5 MeV, M Pc(4440) + = 4440.3 ± 1.3 +4.1 −4.7 MeV, Γ Pc(4440) + = 20.6 ± 4.9 +8.7 −10.1 MeV,

We investigate the consequences of heavy quark spin symmetry (HQSS) on hidden-charm pentaquark P c states. As has been proposed before, assuming the P c (4440) and the P c (4457) as S-wavē D * Σ c molecules, seven hadronic molecular states composed ofDΣ c ,DΣ * c ,D * Σ c , andD * Σ * c can be obtained, with theDΣ c molecule corresponding to the P c (4312). These seven states can decay into J/ψN and η c N , and we use HQSS to predict ratios of partial widths of the S-wave decays.For the decays into J/ψN , it is found that among all six P c molecules with spin 1/2 or 3/2, at least four states decay much more easily into the J/ψN than the P c (4312), and two of them couple dominantly to theD * Σ * c . While no significant peak around theD * Σ * c threshold is found in the J/ψp distribution, these higher P c states either are produced with lower rates or some special production mechanism for the observed P c states might play an important role, such as an intricate interplay between the production of pentaquarks and triangle singularities.

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The decay of Λ þ c into π þ π 0 Λð1405Þ with the Λð1405Þ decay into π 0 Σ 0 through a triangle diagram is studied. This process is initiated by Λ þ c → π þKÃ N, and then theK Ã decays intoKπ andKN produce the Λð1405Þ through a triangle loop containingK Ã NK which develops a singularity around 1890 MeV. This process is prohibited by the isospin symmetry, but the decay into this channel is enhanced by the contribution of the triangle diagram, which is sensitive to the mass of the internal particles. We find a narrow peak in the π 0 Σ 0 invariant mass distribution, which originates from the Λð1405Þ amplitude, but is tied to the mass differences between the charged and neutralK or N states. The observation of the unavoidable peak of the triangle singularity in the isospin-violating Λð1405Þ production would provide further support for the hadronic molecular picture of the Λð1405Þ and further information on theKN interaction.