Analysis of the $B^+\to J/\psi \phi K^+$ data at low $J/\psi \phi$ invariant masses and the $X(4140)$ and $X(4160)$ resonances

We have studied the $J/\psi \phi$ mass distribution of the $B^+\to J/\psi\phi K^+$ reaction from threshold to about 4250 MeV, and find that one needs the contribution of the $X(4140)$ with a narrow width, together with the $X(4160)$ which accounts for most of the strength of the distribution in that region. The existence of a clear cusp at the $D_s^* \bar{D}_s^*$ threshold indicates that the $X(4160)$ resonance is strongly tied to the $D_s^* \bar{D}_s^*$ channel, which finds a natural interpretation in the molecular picture of this resonance.