The role of the $f_0(1710)$ and $a_0(1710)$ resonances in the $D^0\to {\rho }^0\phi$, $\omega \phi$ decays

We study the $D^0\to {\rho }^0\phi$, $\omega \phi$ decays which proceed in a direct mode via internal emission with equal rates. Yet, the experimental branching ratio for the ${\rho }^0\phi$ mode is twice as big as that for the $\omega \phi$ mode. We find a natural explanation based on the extra indirect mechanism where $K^{\ast +}{K}^{\ast -}$ is produced via external emission and that channel undergoes final state interaction with other vector–vector channels to lead to the ${\rho }^0\phi$, $\omega \phi$ final states, with transition amplitudes dominated by the $a_0(1710)$ resonance, recently discovered, and $f_0(1710)$ respectively. The large coupling of the $a_0(1710)$ to the ${\rho }^0\phi$ channel is mostly responsible for this large ratio of the production rates.