Triangle Singularities and Charmonium-like XYZ States

The spectrum of hadrons is important for understanding the confinement of quantum chromodynamics. Many new puzzles arose since 2003 due to the abundance of experimental discoveries with the $XYZ$ structures in the heavy quarkonium mass region being the outstanding examples. Hadronic resonances correspond to poles of the $S$-matrix, which has other types of singularities such as the triangle singularity due to the simultaneous on-shellness of three intermediate particles. Here we briefly discuss a few possible manifestations of triangle singularities in the $XYZ$ physics, paying particular attention to the formalism that can be used to analyze the data for charged $Z_c$ structures in the $\psi\pi$ distributions of the reaction $e^+e^-\to \psi\pi^+\pi^-$.