Exploring the threshold behavior and implications on the nature of Y(4260) and $Z_c(3900)$

By assuming that Y(4260) either is a $D_1\bar{D}$ molecular state, or has sizeable couplings with $D_0 \bar{D}^*$ and $D_1^\prime \bar{D}$, we investigate several decay modes of Y(4260). Under the special kinematic configurations, triangle singularity (TS) may occur in the rescattering amplitude, which will significantly change the threshold behavior. Obvious threshold enhancements or narrow cusp structures appear quite naturally without introducing a genuine resonance. We emphasize that the radiative decay modes may be useful for studying $D^{(*)}\bar{D}$ $S$-wave scattering.