$J/\psi$ decays into $\omega (\phi) f_1(1285)$ and $\omega (\phi) \,''f_1(1420)''$

We perform a theoretical study of the $J/\psi \to \omega (\phi) K^* \bar{K} + c.c. \to \omega (\phi) K^0 \pi^+ K^-$ reactions with the assumption that the $f_1(1285)$ is dynamically generated from a single channel $K^* \bar{K} + c.c$ interaction in the chiral unitary approach. Two peaks in the $K^0 \pi^+ K^-$ invariant mass distribution are observed, one clear peak locates at the $f_1(1285)$ nominal mass, the other peak locates at around $1420\; \rm MeV$ with about $70 \;\rm MeV$ width. We conclude that the former peak is associated with the $f_1(1285)$ and the latter peak is not a genuine resonance but a manifestation of the kinematic effect in the higher energy region caused by the $K^* \bar{K} + c.c.$ decay mode of the $f_1(1285)$.