Studying triangle singularity through spin observables

In this work, we study the spin density matrix element $\rho_{00}$ of the $\phi$ in the decay $J/\psi \rightarrow \eta \pi \phi$. In previous studies, a band around 1.4 GeV on the $\pi^0\phi$ distribution in Dalitz plot was reported by the BESIII Collaboration. This structure may be caused by the production of a resonance or the triangle singularity mechanism. We find that the predictions of the spin density matrix elements of the final $\phi$ based on these mechanisms show distinct features. Thus the measurement of the spin density matrix elements of the $\phi$ in this reaction may offer an alternative way to study the triangle singularity and to clarify the reaction mechanisms, i.e. resonance production or kinematic effects. This work also shows the potential of spin observables in studying kinematic singularities.